Interior Multipeaked Solutions for Slightly Subcritical Elliptic Problems

Abstract

In this paper, we consider the nonlinear elliptic equation $-\Delta u+V\left(x\right)u=g\left(x\right)u^{{\displaystyle \frac{n+2}{n-2}}-\epsilon }$ in a bounded, smooth domain $\Omega$ in ${\mathbb{R}}^n$, under zero Neumann boundary conditions, where $n\ge 4$, $\epsilon$ is a small positive parameter, and V and g are non-constant smooth positive functions on $\overline{\Omega }$. Under certain flatness conditions on the function g, we provide a complete description of the single interior blow-up scenario for solutions that weakly converge to zero. We also construct interior multipeaked solutions, both with isolated and clustered bubbles. The proofs of our results rely on a refined asymptotic expansion of the gradient of the corresponding functional. Furthermore, no assumption regarding the symmetry of the domain is required.

1. Introduction and Main Results

Over the past few decades, considerable attention has been given to studying the following elliptic problem:

$$\left({\mathbf{P}}_{\mu ,q}\right)\left\{\begin{array}{cc}-\Delta u+\mu u=u^q,u>0\hfill & \mathrm{in}\Omega ,\hfill \\ {\displaystyle \frac{\partial u}{\partial \nu }}=0,\hfill & \mathrm{on}\partial \Omega ,\hfill \end{array}\right.$$

where $\Omega$ is a smooth and bounded open set of ${\mathbb{R}}^n$ with $n\ge 3$, $\mu$ is a positive real number, and $q>1$.

Problem $\left({\mathbf{P}}_{\mu ,q}\right)$ is a widely recognized example found in numerous applied scientific fields. For example, it can be interpreted as the stationary problem within a chemotaxis model [1,2], or as a shadow system derived from a reaction–diffusion framework in morphogenesis [3]. Additionally, from a mathematical perspective, problem $\left({\mathbf{P}}_{\mu ,q}\right)$ is of particular interest because its solutions frequently exhibit the bubbling phenomenon. This refers to the emergence of concentration peaks around one or more points within the domain or on its boundary, with the solutions remaining negligibly small in other regions.

A substantial body of research has investigated the case where the exponent q is fixed, and $\mu$ is treated as a parameter. In some studies, $\mu$ is assumed to converge to zero, while in others, it is considered in the limit as it goes to infinity. When q is fixed and subcritical (i.e., $1<q<{\displaystyle \frac{n+2}{n-2}}$), the only solution to problem $\left({\mathbf{P}}_{\mu ,q}\right)$ for small $\mu$ is the constant solution. However, as $\mu$ increases, non-constant solutions emerge, which exhibit blow-up at one or more points as $\mu \to \infty$ [4]. For large $\mu$, the least-energy solution blows up at a boundary point where the mean curvature is maximized [4,5,6,7]. Several studies, including [4,8,9,10,11], have examined higher-energy solutions of $\left({\mathbf{P}}_{\mu ,q}\right)$ that exhibit this asymptotic profile, with blow-up occurring at either boundary or interior points as $\mu \to \infty$. When q is critical (i.e., $q={\displaystyle \frac{n+2}{n-2}}$), the situation is significantly different. For $n\in \{4,5,6\}$ and small $\mu$, problem $\left({\mathbf{P}}_{\mu ,q}\right)$ admits non-constant solutions [12,13,14]. However, the limiting equation of problem $\left({\mathbf{P}}_{\mu ,q}\right)$, arising when studying the asymptotic behavior of the least-energy solution as $\mu \to \infty$, has no solutions. Nevertheless, least-energy solutions $u_{\mu }$ still exist for large $\mu$, and concentration phenomena appear in the following form [15,16]:

$$u_{\mu }\left(x\right)\sim {\delta }_{a_{\mu },{\lambda }_{\mu }}\left(x\right),$$

where, for $a\in {\mathbb{R}}^n$ and $\lambda >0$, ${\delta }_{a,\lambda }$ represents the standard bubble defined by

$${\delta }_{a,\lambda }\left(x\right)=c_0{\displaystyle \frac{{\lambda }^{\frac{n-2}{2}}}{{(1+{\lambda }^2\mid x-a{\mid }^2)}^{\frac{n-2}{2}}}},x\in {\mathbb{R}}^n,c_0={\left(n(n-2)\right)}^{{\displaystyle \frac{n-2}{4}}}$$

(1)

and these are the only solutions [17] to the following problem:

$$-\Delta u=u^{{\displaystyle \frac{n+2}{n-2}}},u>0\mathrm{in}{\mathbb{R}}^n.$$

Several studies, including [15,16,18,19,20,21,22,23,24,25] and the references therein have constructed higher-energy solutions of $\left({\mathbf{P}}_{\mu ,q}\right)$ with concentration at the boundary as $\mu \to \infty$. In contrast to the subcritical case, these solutions require at least one blow-up point to be located on the boundary [26].

Another interesting direction of research for problem $\left({\mathbf{P}}_{\mu ,q}\right)$ involves investigating blow-up phenomena by fixing $\mu$ while allowing the exponent q to approach the critical exponent, i.e., $q={\displaystyle \frac{n+2}{n-2}}\pm \epsilon$, where $\epsilon$ is a small positive parameter. This was initially explored by Rey and Wei [27,28]. For $n\ge 4$ and $q={\displaystyle \frac{n+2}{n-2}}+\epsilon$, they demonstrated the existence of a solution that blows up at a boundary point where the mean curvature is maximized [27]. They also established the existence of a solution that blows up at a boundary point where the mean curvature is minimized when $q={\displaystyle \frac{n+2}{n-2}}-\epsilon$ and $\Omega$ is not convex [27]. In dimension 3, they identified a solution with a single interior blow-up point [28]. More recently, it was shown that for $n\ge 4$ and $q={\displaystyle \frac{n+2}{n-2}}+\epsilon$, no solutions exist that exhibit blow-up solely at interior points when $\epsilon$ is a small positive number [29]. Additionally, in [30], the authors extended the problem by replacing the constant $\mu$ with a function V and studied the scenario when $q={\displaystyle \frac{n+2}{n-2}}-\epsilon$, constructing interior bubbling solutions. In this case, the interior blow-up points of these solutions converge, as $\epsilon \to 0$, to the critical points of the function V. More recently, in [31], the authors examined the case where a function g is introduced in front of the nonlinear term, considering the following problem

$$\left({\mathbf{P}}_{\epsilon }\right):\left\{\begin{array}{c}-\Delta u+V\left(x\right)u=g\left(x\right)u^{p-\epsilon },u>0\mathrm{in}\Omega \hfill \\ {\displaystyle \frac{\partial u}{\partial \nu }}=0\mathrm{on}\partial \Omega ,\hfill \end{array}\right.$$

where $\Omega$ is a smooth bounded domain in ${\mathbb{R}}^n$, $n\ge 4$, $p+1=\left(2n\right)/(n-2)$ is the critical Sobolev exponent for the embedding $H^1(\Omega )\hookrightarrow L^q(\Omega )$, $\epsilon$ is a small positive parameter, and g and V are $C^3$ positive functions defined on $\overline{\Omega }$.

Assuming that the critical points of g are non-degenerate, the authors demonstrated that, in contrast to the case where $g\equiv 1$ studied in [30], problem $\left({\mathbf{P}}_{\epsilon }\right)$ does not admit interior bubbling solutions with clustered bubbles. However, they were able to construct solutions $u_{\epsilon }$ to $\left({\mathbf{P}}_{\epsilon }\right)$ with isolated interior multiple blow-up points. By “clustered bubbles”, we refer to a sum of bubbles that have rates of the same order, with their concentration points converging to a single point y. In this scenario, it holds that

$$\underset{r\to 0}{lim}\underset{\epsilon \to 0}{lim}{\int }_{B(y,r)}g\left(x\right)u_{\epsilon }^{2n/(n-2)}\left(x\right)dx=kg\left(y\right)S_n\mathrm{with} k\ge 2 \mathrm{and} S_n:={\int }_{{\mathbb{R}}^n}{\delta }_{0,1}^{2n/(n-2)}.$$

On the other hand, by “isolated bubbles”, we refer to a sum of bubbles with concentration points $a_i$ satisfying $|a_i-a_j|\ge c>0$ for each $j\ne i$.

The results mentioned above raise a natural question: what happens when the critical points of g are degenerate, particularly when g satisfies a specific ’flatness’ condition? The aim of this paper is to address this question. Throughout the paper, we assume that g satisfies a certain flatness condition (see Equation (2)) for the precise statement). First, we provide a complete description of the asymptotic behavior of interior single-peaked solutions. Second, we show that, unlike the case where the non-degeneracy assumption is made, as studied by the authors in [31], interior multipeaked solutions with clustered bubbles do indeed exist. We will also prove that solutions with isolated bubbles continue to exist. These existence results hold in both cases, whether the concentration points are critical points of the function V or not.

To present our results, we need to establish some definitions. We say that a function f satisfies a flatness condition near a critical point y if it can be expressed as

$$f\left(x\right)={f\left(y\right)+h(x-y)+o\left(\right|x-y|}^{\gamma }) \mathrm{with} \gamma :={\gamma }_y>2,$$

(2)

where h satisfies

$$h\left(tx\right)={\left|t\right|}^{\gamma }h\left(x\right)\mathrm{for} \mathrm{each}t\in \mathbb{R} \mathrm{and} x\in {\mathbb{R}}^n.$$

Additionally, we say that f satisfies assumption $\left(H1\right)$ if the following holds:

Hypothesis 1.

For $\omega \in {\mathbb{R}}^n$, the equation

$$\nabla h\left(\xi \right)=admits a solution \xi with D^{2}h\left(\xi \right) is invertible.$$

(3)

Note that if y is a non-degenerate critical point of f, then Equation (2) holds immediately with $\gamma =2$ and the assumptions given in Equation (3) are satisfied. Furthermore, as an example of functions h that satisfy these assumptions, we can take $h\left(x\right)={\left|x\right|}^{\gamma }$ for any $\gamma \ge 2$.

For $a\in \Omega ,$ and $\lambda >0$, we define the following projection ${\phi }_{a,\lambda }$:

$$\left\{\begin{array}{cc}-\Delta {\phi }_{a,\lambda }+V\left(x\right){\phi }_{a,\lambda }={\delta }_{a,\lambda }^p\hfill & \mathrm{in}\Omega \hfill \\ {\displaystyle \frac{\partial {\phi }_{a,\lambda }}{\partial \nu }}=0\hfill & \mathrm{on}\partial \Omega .\hfill \end{array}\right.$$

(4)

The functions ${\phi }_{a,\lambda }$ serve as the appropriate approximate solutions, and in their neighborhood, we find the true solution to the problem.

Now, we begin by analyzing the asymptotic behavior of solutions to $\left({\mathbf{P}}_{\epsilon }\right)$ that blow up at a single interior point as $\epsilon \to 0$. We provide a complete description of the single interior blow-up scenario for solutions that weakly converge to zero. Specifically, we prove the following.

Theorem 1.

Let $n\ge 4$ and let $\left(u_{\epsilon }\right)$ be a family of solutions of $\left({\mathbf{P}}_{\epsilon }\right)$ having the form

$$\begin{array}{cc}\hfill & u_{\epsilon }={\alpha }_{\epsilon }{\phi }_{a_{\epsilon },{\lambda }_{\epsilon }}+v_{\epsilon }with\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & {\parallel v_{\epsilon }\parallel }_{H^1(\Omega )}\to 0,{\alpha }_{\epsilon }^{4/(n-2)}g\left(a_{\epsilon }\right)\to 1,{\lambda }_{\epsilon }\to \infty andd(a_{\epsilon },\partial )\ge c>0.\hfill \end{array}$$

(6)

Then, the concentration point $a_{\epsilon }$ converges to a critical point y of g.

Furthermore, if Equation (2) is satisfied in the sense of $C^2$, then the concentration rate ${\lambda }_{\epsilon }$ satisfies

$$E_{n}V\left(y\right){\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_{\epsilon }}{{\lambda }_{\epsilon }^2}}=c_3\epsilon (1+o\left(1\right)),$$

(7)

where ${\sigma }_4=1$, ${\sigma }_n=0$ for $n\ge 5$,

$$\begin{array}{cc}\hfill & c_3:={\displaystyle \frac{{(n-2)}^2}{4}}{c}_0^{p+1}{\int }_{R^n}{\displaystyle \frac{{\left(\right|x|}^2-1)ln(1+{\left|x\right|}^2)}{{(1+|x|}^2{)}^{n+1}}}dx>0,\hfill \\ \hfill & E_4=c_0^2\left|{\mathbb{S}}^3\right|,E_n:={\displaystyle \frac{n-2}{2}}{c}_0^2{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{{\left|x\right|}^2-1}{{(1+|x|}^2{)}^{n-1}}}dx>0forn\ge 5.\hfill \end{array}$$

Additionally, if there exists a positive constant ${\beta }_1$ such that $|\nabla g\left(x\right)|\ge {\beta }_1{|x-y|}^{\gamma -1}$ in a neighborhood of y, then the concentration point $a_{\epsilon }$ satisfies the following: ${\epsilon }^{-1}{|a_{\epsilon }-y|}^{\gamma -1}$ is bounded and two cases may arise:

• If there exists a positive constant β such that ${\epsilon }^{-1}{|a_{\epsilon }-y|}^{\gamma -1}\ge \beta$, then $\nabla V\left(y\right)\ne 0$.
• If $lim{\epsilon }^{-1}{|a_{\epsilon }-y|}^{\gamma -1}=0$, then y has to be a critical point of V.

Before presenting the rest of our results, we note that the dimension $n=3$ is excluded from our work, due to the following reasons:

• For $n\ge 5$, the function ${\delta }_{a,\lambda }\in L^2\left({\mathbb{R}}^n\right)$ and satisfies

  $${\int }_{{\mathbb{R}}^n}{\delta }_{a,\lambda }^2={\displaystyle \frac{c^{\prime }}{{\lambda }^2}}\mathrm{and}{\int }_{\Omega }{\delta }_{a,\lambda }^2={\displaystyle \frac{c^{\prime }}{{\lambda }^2}}+O\left({\displaystyle \frac{1}{{\lambda }^n}}\right).$$
• For $n\in \{3,4\}$, the previous integral diverges. But, since $\Omega$ is bounded, for $n=4$, we have

  $${\int }_{\Omega }{\delta }_{a,\lambda }^2=c{\displaystyle \frac{ln\lambda }{{\lambda }^2}}+O\left({\displaystyle \frac{1}{{\lambda }^2}}\right).$$
• For $n=3$, this integral depends on the geometry of $\Omega$, and precise computations are required to determine the principal part. In fact, we have

  $${\int }_{\Omega }{\delta }_{a,\lambda }^2={\displaystyle \frac{c(a,\Omega )}{\lambda }}+O\left({\displaystyle \frac{1}{{\lambda }^2}}\right)$$

  where $c(a,\Omega )$ depends on a and $\Omega$. Hence, this case must be considered separately.

Next, our objective is to establish the converse of Theorem 1. More precisely, we aim to construct interior multipeaked solutions for problem $\left({\mathbf{P}}_{\epsilon }\right)$. According to Theorem 1, we see that such a construction must be centered around a critical point y of g. Furthermore, when the flatness condition Equation (2) on the function g holds in the $C^2$ sense, there are two distinct cases to consider: one where the gradient of V at y is non-zero, and the other where it is zero. Roughly speaking, in the first case, the rate of convergence is on the order of ${\epsilon }^{1/(\gamma -1)}$, while in the second case, it is negligible in comparison to ${\epsilon }^{1/(\gamma -1)}$. We begin by constructing interior multipeaked solutions in the first case, and specifically, we prove the following.

Theorem 2.

Let $n\ge 4$ and let $y_1,\cdots ,y_l$ be l critical points of g such that, near each $y_i$, the function g satisfies Equation (2) in the sense of $C^3$ with ${\gamma }_i>3$. Assume that $\nabla V\left(y_i\right)\ne 0$ for $i\in \{1,\cdots ,l\}$ and that condition Equation (3) is satisfied. Then, for any integer $N\le l$, there exists a small positive real number ${\epsilon }_0$ such that for every $\epsilon \in (0,{\epsilon }_0)$, the problem $\left({\mathbf{P}}_{\epsilon }\right)$ admits a solution $u_{\epsilon }$ which develops exactly one bubble at each point $y_{i_j}$ for $j\in \{1,\cdots ,N\}$ and weakly converges to zero in $H^1(\Omega )$. More precisely, there exist values ${\lambda }_{1,\epsilon }$, …, ${\lambda }_{N,\epsilon }$ satisfying Equation (7) and points $a_{j,\epsilon }\to y_{i_j}$ as $\epsilon \to 0$ for all j such that

$$\left|\right|u_{\epsilon }-\sum _{j=1}^N{\phi }_{a_{j,\epsilon },{\lambda }_{j,\epsilon }}|{|}_{H^1(\Omega )}\to 0,as\epsilon \to 0.$$

Additionally, for all j, ${\epsilon }^{1/({\gamma }_{i_j}-1)}(a_{j,\epsilon }-y_{i_j})$ converges to a solution of the equation

$$\nabla h\left({\xi }_j\right)={\displaystyle \frac{g\left(y_{i_j}\right)}{{\overline{c}}_5}}{\displaystyle \frac{c_3{D}_n}{E_{n}V\left(y_{i_j}\right)}}\nabla V\left(y_{i_j}\right),$$

where $c_3$ and $E_n$ are the constants defined in Theorem 1, ${\overline{c}}_5={\displaystyle \frac{n-2}{n}}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{c_0^{p+1}{\left|x\right|}^2}{{(1+|x|}^2{)}^{n+1}}}dx$, and

$$D_4={\displaystyle \frac{c_0^2}{2}}\left|{\mathbb{S}}^3\right|,D_n={\displaystyle \frac{(n-2)}{n}}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{c_0^2{\left|x\right|}^2}{{(1+|x|}^2{)}^{n-1}}}dxfor n\ge 5.$$

Theorem 2 provides the following multiplicity result concerning the number of critical points of g that are not critical points of V.

Theorem 3.

Under the assumptions of Theorem 2, there exists an ${\epsilon }_0>0$ such that for $\epsilon \in (0,{\epsilon }_0]$, problem $\left({\mathbf{P}}_{\epsilon }\right)$ admits at least $2^l-1$ solutions, where l denotes the number of critical points of g that are not critical points of V.

Next, we consider the case where the functions g and V share common critical points. Specifically, we have the following.

Theorem 4.

Let $n\ge 5$ and let $y_1,\cdots ,y_m$ be m common critical points of g and V. We assume that these critical points are non-degenerate for V, and that near each such point, g takes the form.

$$g\left(x\right)=g\left(y_i\right)+O\left(\right|x-y_i{|}^{{\gamma }_i})with{\gamma }_i>4(in the C^3 sense).$$

(8)

Then, for any integer $N\le m$, there exists a small ${\epsilon }_0>0$ such that for every $\epsilon \in (0,{\epsilon }_0]$, the problem $\left({\mathbf{P}}_{\epsilon }\right)$ admits a solution $u_{\epsilon }$ satisfying the following: $u_{\epsilon }$ develops exactly one bubble at each point $y_{i_j}$ for $j\in \{1,\cdots ,N\}$ and weakly converges to zero in $H^1(\Omega )$ as $\epsilon \to 0$. More precisely there exist values ${\lambda }_{1,\epsilon }$,…, ${\lambda }_{N,\epsilon }$ of order ${\epsilon }^{-1/2}$ and points $a_{j,\epsilon }\to y_{i_j}$ as $\epsilon \to 0$ for all j such that

$$\left|\right|u_{\epsilon }-\sum _{j=1}^N{\phi }_{a_{j,\epsilon },{\lambda }_{j,\epsilon }}|{|}_{H^1(\Omega )}\to 0,as\epsilon \to 0.$$

In addition, we have ${\epsilon }^{1/({\gamma }_{i_j}-2)}|a_{j,\epsilon }-y_{i_j}|\to 0$ as $\epsilon \to 0$ for all $j\in \{1,\cdots ,N\}$.

Theorem 4 provides the following multiplicity result related to the number of common critical points of g and V.

Theorem 5.

Under the assumptions of Theorem 4, there exists an ${\epsilon }_0>0$ such that for $\epsilon \in (0,{\epsilon }_0]$, problem $\left({\mathbf{P}}_{\epsilon }\right)$ admits at least $2^m-1$ solutions, where m denotes the number of common critical points of g and V.

The constructions of interior multipeaked solutions we presented in Theorems 2 and 4 can be combined to yield solutions that concentrate at interior points, which divide into two blocks: one consisting of common critical points of g and V, and the other consisting of critical points of g that are not critical points of V. This leads to the following result.

Theorem 6.

Let $n\ge 5$ and let $y_1,\cdots ,y_l,z_l,\cdots ,z_m$ be $l+m$ critical points of g such that, near each points $y_i$ for $1\le i\le l$, g takes the form Equation (2) in the sense of $C^3$ with ${\gamma }_i>3$. Assume that $\nabla V\left(y_i\right)\ne 0$ for $1\le i\le l$ and that Equation (3) holds. Assume further that $z_1,\cdots ,z_m$ are non-degenerate critical points V, and that near each point $z_i$ for $1\le i\le m$, g takes the form Equation (8). Then, for any $N_1\le l$ and $N_2\le m$, there exists a small ${\epsilon }_0>0$ such that for every $\epsilon \in (0,{\epsilon }_0]$, the problem $\left({\mathbf{P}}_{\epsilon }\right)$ admits a solution $u_{\epsilon }$ satisfying the following:

$$\left|\right|u_{\epsilon }-\sum _{j=1}^{N_1}{\phi }_{a_{1,j,\epsilon },{\lambda }_{1,j,\epsilon }}-\sum _{j=1}^{N_2}{\phi }_{a_{2,j,\epsilon },{\lambda }_{2,j,\epsilon }}|{|}_{H^1(\Omega )}\to 0,as\epsilon \to 0.$$

with the values ${\lambda }_{1,j,\epsilon }$’s and ${\lambda }_{2,j,\epsilon }$’s, which are of order ${\epsilon }^{-1/2}$ and the points $a_{1,j,\epsilon }\to y_{i_j},a_{2,j,\epsilon }\to z_{i_j}$ as $\epsilon \to 0$ for all j. In addition, for all j, ${\epsilon }^{1/({\gamma }_{i_j}-2)}|a_{2,j,\epsilon }-z_{i_j}|\to 0$, and ${\epsilon }^{1/({\gamma }_{i_j}-1)}(a_{1,j,\epsilon }-y_{i_j})$ converges to a solution of the equation

$$\nabla h\left({\xi }_j\right)={\displaystyle \frac{g\left(y_{i_j}\right)}{{\overline{c}}_5}}{\displaystyle \frac{c_3{D}_n}{E_{n}V\left(y_{i_j}\right)}}\nabla V\left(y_{i_j}\right),$$

where $c_3,{\overline{c}}_5,D_n$, and $E_n$ are the constants defined in Theorem 2.

Note that, up to this point, all of our existence theorems have dealt with interior blowing-up solutions featuring isolated bubbles. The goal of the next results is to construct interior bubbling solutions $u_{\epsilon }$ with clustered bubbles. As mentioned, this is the key difference between the case studied in [31], where the critical points of g are non-degenerate, and our case, where the critical points are degenerate. In [31], the authors demonstrated that there are no interior blowing-up solutions with clustered bubbles. In contrast, we will prove here that such solutions do exist.

To achieve this, we introduce the following notation. For $N\in \mathbb{N}$ and y, a common critical point of both g and V, we define the function

$$F_{N,y}(z_1,\cdots ,z_N)=\sum _{j=1}^N{D}^{2}V\left(y\right)(z_j,z_j)-\sum _{k\ne r}{\displaystyle \frac{1}{|z_k-z_r{|}^{n-2}}},$$

(9)

where $(z_1,\cdots ,z_N)\in {\left({\mathbb{R}}^n\right)}^N$ such that $z_i\ne z_j$ if $i\ne j$.

Our result is stated as follows.

Theorem 7.

Let $n\ge 5$ and let y be a common critical point of g and V. We assume that y is non-degenerate for V and that g satisfies Equation (8) with $\gamma >4+8/(n-4)$. Let $N\in {\mathbb{N}}^{*}$ with $N\ge 2$ and assume that the function $F_{N,y}$ has a non-degenerate critical point $({\overline{z}}_1,\cdots ,{\overline{z}}_N)$. Then, for any integer $N\ge 2$, there exists a small ${\epsilon }_0>0$ such that for every $\epsilon \in (0,{\epsilon }_0]$, the problem $\left({\mathbf{P}}_{\epsilon }\right)$ admits a solution $u_{\epsilon }$ satisfying the following

$$\left|\right|u_{\epsilon }-\sum _{i=1}^N{\phi }_{a_{i,\epsilon },{\lambda }_{i,\epsilon }}|{|}_{H^1(\Omega )}\to 0,as\epsilon \to 0$$

with the values ${\lambda }_{1,\epsilon }$, …, ${\lambda }_{N,\epsilon }$ which are of the order ${\epsilon }^{-1/2}$, and the concentration points $a_{i,\epsilon }^{\prime }s$ satisfy

$$|a_{i,\epsilon }-y+\sigma {\epsilon }^{{\displaystyle \frac{n-4}{2n}}}{\overline{z}}_i|\le {\epsilon }^{{\displaystyle \frac{n-4}{2n}}}{\eta }_0\forall i,$$

(10)

where ${\eta }_0$ is a small positive real, and σ is the constant defined in Equation (142).

Moreover, if for each N, $F_{N,y}$ has a non-degenerate critical point, then the problem $\left({\mathcal{P}}_{\epsilon }\right)$ admits an arbitrary number of non-constant distinct solutions provided that ε is sufficiently small.

Theorem 8.

Let $n\ge 5$, $2\le m\in \mathbb{N}$, $N_1,\cdots ,N_m\in \mathbb{N}$ and let $y_1,\cdots ,y_m$ satisfy the assumptions stated in Theorem 7. For $k\le m$, if $N_k\ge 2$, we assume that the function $F_{N_k,y_k}$ has a non-degenerate critical point $({\overline{z}}_{k,1},\cdots ,{\overline{z}}_{k,N_k})$. Then, there exists a small ${\epsilon }_0>0$ such that for every $\epsilon \in (0,{\epsilon }_0]$, the problem $\left({\mathbf{P}}_{\epsilon }\right)$ admits a solution $u_{\epsilon ,}$ satisfying the following

$$\left|\right|u_{\epsilon }-\sum _{i=1}^{N_1}{\phi }_{a_{1,i,\epsilon },{\lambda }_{1,i,\epsilon }}-\cdots -\sum _{i=1}^{N_m}{\phi }_{a_{m,i,\epsilon },{\lambda }_{m,i,\epsilon }}|{|}_{H^1(\Omega )}\to 0,as\epsilon \to 0$$

with the values ${\lambda }_{1,i,\epsilon }^{\prime }s,\cdots ,{\lambda }_{m,i,\epsilon }^{\prime }s$ which are of the order ${\epsilon }^{-1/2}$, and for each $k\le m$, the concentration points $a_{k,i,\epsilon }^{\prime }s$ satisfy Equation (10) when $N_k\ge 2$, and the following property when $N_k=1$:

$${\epsilon }^{1/({\gamma }_{i_k}-2)}|a_{k,i,\epsilon }-y_{i_k}|\to 0as\epsilon \to 0.$$

To prove our results, we perform a refined asymptotic expansion of the gradient of the associated functional and subsequently test the equation using vector fields. This allows us to derive relationships between the concentration parameters. By carefully analyzing these relationships, we obtain our results. Furthermore, no assumption regarding the symmetry of the domain is required. Note that traditional blow-up analysis methods typically depend on detailed pointwise $C^0$-estimates and the frequent utilization of Pohozaev identities. In contrast, the approach outlined in this paper deviates from these conventional techniques. We argue that our method, which bypasses the need for pointwise estimates and Pohozaev identities, holds significant promise for addressing non-compact variational problems that involve more intricate blow-up behaviors. Indeed, the occurrence of non-simple blow-up points adds complexity to the derivation of pointwise $C^0$-estimates, rendering this process particularly challenging.

The remainder of this paper is organized as follows: In Section 2, we provide precise estimates for the infinite-dimensional part of $u_{\epsilon }$. Section 3 is dedicated to the expansion of the gradient of the functional associated with problem $\left({\mathbf{P}}_{\epsilon }\right)$. In Section 4, we study the asymptotic behavior of solutions to $\left({\mathbf{P}}_{\epsilon }\right)$ that blow up at a single interior point as $\epsilon \to 0$, leading to the proof of Theorem 1. Section 5 focuses on the construction of interior blowing-up solutions with isolated bubbles, which are used to prove Theorems 2, 3, 4, 5, and 6. In Section 6, we construct interior blowing-up solutions with clustered bubbles, thereby proving Theorems 7 and 8. Section 7 explores possible avenues for future research. Finally, in Appendix A, we collect several estimates used throughout the paper.

2. The Infinite-Dimensional Part

The objective of this section is to estimate the infinite-dimensional component of the solutions $u_{\epsilon }$ to the problem $\left({\mathbf{P}}_{\epsilon }\right)$. It is important to note that every solution $u_{\epsilon }$ of $\left({\mathbf{P}}_{\epsilon }\right)$ satisfies the condition $\parallel u_{\epsilon }\parallel \ge C$ with C as a positive constant independent of $\epsilon$. Consequently, the concentration compactness principle (see [32,33]) implies that if $u_{\epsilon }$ is an energy-bounded solution of $\left({\mathbf{P}}_{\epsilon }\right)$ that weakly converges to zero, then $u_{\epsilon }$ must blow up at finite number k of points within $\overline{\Omega }$. More specifically, $u_{\epsilon }$ can be expressed as

$$\begin{array}{cc}\hfill & u_{\epsilon }=\sum _{i=1}^{k}g{\left(a_{i,\epsilon }\right)}^{(2-n)/4}{\delta }_{a_{i,\epsilon },{\lambda }_{i,\epsilon }}+v_{\epsilon }\mathrm{where}\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & {∥v_{\epsilon }∥}_{H^1}\to 0,{\lambda }_{i,\epsilon }\to \infty ,a_{i,\epsilon }\to {\overline{a}}_i\in \overline{\Omega }\forall i\mathrm{as}\epsilon \to 0,\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & {\epsilon }_{ij}:={\left({\displaystyle \frac{{\lambda }_{i,\epsilon }}{{\lambda }_{j,\epsilon }}}+{\displaystyle \frac{{\lambda }_{j,\epsilon }}{{\lambda }_{i,\epsilon }}}+{\lambda }_{i,\epsilon }{\lambda }_{j,\epsilon }{\left|a_{i,\epsilon }-a_{j,\epsilon }\right|}^2\right)}^{{\displaystyle \frac{2-n}{2}}}\to 0\mathrm{as}\epsilon \to 0\forall i\ne j.\hfill \end{array}$$

(13)

In this paper, we focus on examining the qualitative characteristics and the existence of interior blowing up solutions for problem $\left({\mathbf{P}}_{\epsilon }\right)$. Specifically, we investigate the scenario where

$$d(a_{i,\epsilon },\partial \Omega )\ge d_0>0\forall i\in \{1,\cdots ,N\}.$$

(14)

Following the proof of Proposition 7 of [34] and Proposition $2.1$ of [25], we see that, for $k\in \mathbb{N}$ and $\left(u_{\epsilon }\right)$ to be a family of functions having the formof Equation (11) and satisfying the properties Equations (12)–(14), there is a unique way to choose ${\lambda }_{i,\epsilon }, a_{i,\epsilon }$ and $v_{\epsilon }$ such that

$$u_{\epsilon }=\sum _{i=1}^k{\alpha }_{i,\epsilon }{\phi }_{a_{i,\epsilon }{\lambda }_{i,\epsilon }}+v_{\epsilon }$$

(15)

with $(a_{\epsilon },{\lambda }_{\epsilon }):=((a_{1,\epsilon },\cdots ,a_{k,\epsilon }),({\lambda }_{1,\epsilon },\cdots ,{\lambda }_{k,\epsilon }))\in {\Omega }^k\times {(0,\infty )}^k$ and

$$\left\{\begin{array}{c}|{\alpha }_{i,\epsilon }^{{\displaystyle \frac{4}{n-2}}}g\left(a_{i,\epsilon }\right)-1|\to 0\mathrm{as}\epsilon \to 0\forall i\in \{1,\cdots ,k\},\hfill \\ a_{i,\epsilon }\to {\overline{a}}_i\in \Omega ,{\lambda }_{i,\epsilon }\to \infty \mathrm{as}\epsilon \to 0\forall i\in \{1,\cdots ,k\},\hfill \\ {\epsilon }_{ij}\to 0\mathrm{as}\epsilon \to 0\forall i\ne j,\parallel v_{\epsilon }\parallel \to 0,v_{\epsilon }\in E_{a_{\epsilon }{\lambda }_{\epsilon }},\hfill \end{array}\right.$$

(16)

where $E_{a_{\epsilon },{\lambda }_{\epsilon }}$ is defined by

$$\begin{array}{cc}\hfill E_{a_{\epsilon },{\lambda }_{\epsilon }}=\{v\in H^1(\Omega ):(v,{\phi }_{a_{i,\epsilon },{\lambda }_{i,\epsilon }})& =(v,{\displaystyle \frac{\partial {\phi }_{a_{i,\epsilon },{\lambda }_{i,\epsilon }}}{\partial {\lambda }_{i,\epsilon }}})\hfill \\ \hfill & =(v,{\displaystyle \frac{\partial {\phi }_{a_{i,\epsilon },{\lambda }_{i,\epsilon }}}{\partial {\left(a_{i,\epsilon }\right)}_j}})=0\forall i⩽k,\forall j⩽n\}.\hfill \end{array}$$

(17)

Here and in the sequel, $H^1(\Omega )$ is equipped with the norm $\parallel .\parallel$ and the corresponding inner product given by

$${\parallel u\parallel }^2={\int }_{\Omega }\left(\mid \nabla u{\mid }^2+V\left(x\right)u^2\right)dx,(u,v)={\int }_{\Omega }\left(\nabla u\nabla v+V\left(x\right)uv\right)dx.$$

Throughout the sequel, we write ${\alpha }_i, {\lambda }_i, a_i, {\phi }_i$ and $E_{a,\lambda }$ instead of ${\alpha }_{i,\epsilon }, {\lambda }_{i,\epsilon }, a_{i,\epsilon }, {\phi }_{a_{i,\epsilon },{\lambda }_{i,\epsilon }}$ and $E_{a_{\epsilon }{\lambda }_{\epsilon }}$, respectively, and we assume that $u_{\epsilon }$ in written as in Equation (15) with the properties listed in Equation (16).

To study interior bubbling solutions, we introduce the following sets:

$$\begin{array}{cc}\hfill & \mathcal{O}\left(k,{\nu }_0\right)=\{(\alpha ,\lambda ,a,v)\in {\left({\mathbb{R}}_{+}\right)}^k\times {\left({\mathbb{R}}_{+}\right)}^k\times {\Omega }^k\times H^1(\Omega ):|{\alpha }_i^{4/n-2}g\left(a_i\right)-1|<{\nu }_0,\hfill \\ \hfill & {\lambda }_i>{\nu }_0^{-1},\epsilon ln{\lambda }_i<{\nu }_0,d(a_i,\partial \Omega )\ge c>0,{\epsilon }_{ij}<{\nu }_0,v\in E_{a,\lambda },\parallel v\parallel <{\nu }_0\},\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & \beta \left(k,{\nu }_0\right)=\left\{(\alpha ,\lambda ,a)\in {\left({\mathbb{R}}_{+}\right)}^k\times {\left({\mathbb{R}}_{+}\right)}^k\times {\Omega }^k:\left(\alpha ,\lambda ,a,0\right)\in \mathcal{O}\left(k,{\nu }_0\right)\right\},\hfill \end{array}$$

(19)

where ${\nu }_0$ is a positive small real.

Problem $\left({\mathbf{P}}_{\epsilon }\right)$ has a variational structure. Solutions of $\left({\mathbf{P}}_{\epsilon }\right)$ are the positive critical points of the functional $I_{\epsilon }$ defined on $H^1(\Omega )$ by:

$$I_{\epsilon }\left(u\right)={\displaystyle \frac{1}{2}}{\int }_{\Omega }\left({|\nabla u|}^2+V\left(x\right)u^2\right)-{\displaystyle \frac{1}{p+1-\epsilon }}{\int }_{\Omega }g\left(x\right){\left|u\right|}^{p+1-\epsilon }.$$

(20)

Now, for $\left(\alpha ,\lambda ,a\right)\in \beta (k, {\nu }_0)$, we denote by $\phi :={\sum }_{i=1}^k{\alpha }_i{\phi }_i$. Letting $v\in E_{a,\lambda }$ with $\parallel v\parallel <{\nu }_0,$ we see that

$$\begin{array}{cc}\hfill I_{\epsilon }\left(\sum _{i=1}^k{\alpha }_i{\phi }_i+v\right)& ={\displaystyle \frac{1}{2}}{\int }_{\Omega }|\nabla \left(\sum _{i=1}^k{\alpha }_i{\phi }_i\right){|}^2+{\displaystyle \frac{1}{2}}{\int }_{\Omega }{|\nabla v|}^2+{\displaystyle \frac{1}{2}}{\int }_{\Omega }V\left(x\right){\left(\sum _{i=1}^k{\alpha }_i{\phi }_i\right)}^2\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\int }_{\Omega }V\left(x\right)v^2-{\displaystyle \frac{1}{p+1-\epsilon }}{\int }_{\Omega }g\left(x\right){|\sum _{i=1}^k{\alpha }_i{\phi }_i+v|}^{p+1-\epsilon }.\hfill \end{array}$$

(21)

But we have

$$\begin{array}{cc}\hfill {\int }_{\Omega }& g\left(x\right)|\sum _{i=1}^k{\alpha }_i{\phi }_i+v{|}^{p+1-\epsilon }\hfill \\ \hfill & ={\int }_{\Omega }g\left(x\right)|\sum _{i=1}^k{\alpha }_i{\phi }_i{|}^{p+1-\epsilon }+(p-\epsilon ){\int }_{\Omega }g\left(x\right)|\sum _{i=1}^k{\alpha }_i{\phi }_i{|}^{p-1-\epsilon }\sum _{i=1}^k{\alpha }_i{\phi }_{i}v\hfill \\ \hfill & +{\displaystyle \frac{(p+1-\epsilon )(p-\epsilon )}{2}}{\int }_{\Omega }g\left(x\right)|\sum _{i=1}^k{\alpha }_i{\phi }_i{|}^{p-1-\epsilon }{v}^2+O\left({\int }_{\Omega }{\left|v\right|}^{p+1-e}\right)\hfill \\ \hfill & +\left(\mathrm{if}n⩽5\right)O\left({\int }_{\Omega }|\sum _{i=1}^k{\alpha }_i{\phi }_i{|}^{p-2-\epsilon }{\left|v\right|}^3\right).\hfill \end{array}$$

(22)

This implies that

$$I_{\epsilon }\left(\sum _{i=1}^k{\alpha }_i{\phi }_i+v\right)=I_{\epsilon }\left(\sum _{i=1}^k{\alpha }_i{\phi }_i\right)-⟨L_{\epsilon },v⟩+{\displaystyle \frac{1}{2}}{Q}_{\epsilon }\left(v\right)+R_{\epsilon }\left(v\right),$$

where

$$\begin{array}{cc}\hfill & ⟨L_{\epsilon },v⟩={\int }_{\Omega }g\left(x\right)|\sum _{i=1}^k{\alpha }_i{\phi }_i{|}^{p-1-\epsilon }\sum _{i=1}^k{\alpha }_i{\phi }_{i}v,\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill & Q_{\epsilon }\left(v\right)={\int }_{\Omega }{\left|\nabla v\right|}^2+{\int }_{\Omega }V\left(x\right)v^2-(p-\epsilon ){\int }_{\Omega }g\left(x\right)|\sum _{i=1}^k{\alpha }_i{\phi }_i{|}^{p-1-\epsilon }{v}^2\hfill \end{array}$$

(24)

and $R_{\epsilon }$ satisfies $R_{\epsilon }\left(v\right)=O\left({\parallel v\parallel }^{min(3,p+1-\epsilon )}\right)$, and

$$R_{\epsilon }^{\prime }\left(v\right)=O\left({\parallel v\parallel }^{min(2,p-\epsilon )}\right),R_{\epsilon }^{″}\left(v\right)=O\left({\parallel v\parallel }^{min(1,p-1-\epsilon )}\right).$$

(25)

To proceed further, we need to prove the uniform coercivity of the quadratic form $Q_{\epsilon }$.

Proposition 1.

Let $n\ge 3$ and $(\alpha ,\lambda ,a)\in \beta (k,{\nu }_0)$ with ${\nu }_0$ small. Then, there exist ${\epsilon }_0>0$ and $C>0$ such that for $\epsilon \in \left(0,{\epsilon }_0\right)$, the following holds:

$$Q_{\epsilon }\left(v\right)\ge C{\parallel v\parallel }^2\forall v\in E_{a_{\epsilon },{\lambda }_{\epsilon }}.$$

Proof. 

On one hand, since $\epsilon ln{\lambda }_i$ is small and $\Omega$ is bounded, Taylor’s expansion implies that

$${\delta }_i^{-\epsilon }=c_0^{-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}\left(1+O\left(\epsilon ln\left(1+{\lambda }_i^2{\left|x-a_i\right|}^2\right)\right)\right)=1+o\left(1\right).$$

(26)

On the other hand, let $v\in E_{a_{\epsilon },{\lambda }_{\epsilon }}$, we have

$$\begin{array}{cc}\hfill & \epsilon {\int }_{\Omega }g{\overline{u}}^{p-1-\epsilon }{v}^2⩽c\epsilon \sum _{i=1}^N{\int }_{\Omega }{\delta }_i^{p-1-\epsilon }{v}^2⩽c\epsilon {\parallel v\parallel }^2,\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill & {\int }_{\Omega }g{\overline{u}}^{p-1-\epsilon }{v}^2=\sum _{i=1}^N{\int }_{\Omega }g{\left({\alpha }_i{\delta }_i\right)}^{p-1-\epsilon }{v}^2+\left\{\begin{array}{cc}\sum _{j\ne i}O\left({\int }_{\Omega }{\left({\delta }_i{\delta }_j\right)}^{(p-1)/2}{v}^2\right)\hfill & \mathrm{if}n\ge 4,\hfill \\ \sum _{j\ne i}O\left({\int }_{\Omega }{\delta }_i^3{\delta }_j{v}^2\right)\hfill & \mathrm{if}n=3.\hfill \end{array}\right.\hfill \end{array}$$

(28)

But, using Holder’s inequality and estimate $E_2$ of [35], we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }& ({\delta }_i{\delta }_j{)}^{(p-1)/2}{v}^2\hfill \\ \hfill & ⩽c{\left({\epsilon }_{ij}^{n/(n-2)}ln{\epsilon }_{ij}^{-1}\right)}^{2/n}{\parallel v\parallel }^2=c{\epsilon }_{ij}^{2/(n-2)}{(ln{\epsilon }_{ij}^{-1})}^{2/n}{\parallel v\parallel }^2=o\left({\parallel v\parallel }^2\right).\hfill \end{array}$$

(29)

Thus, combining Equations (26)–(29), we deduce that

$$Q_{\epsilon }\left(v\right)={\int }_{\Omega }{|\nabla v|}^2+{\int }_{\Omega }V{v}^2-p\sum _{i=1}^N{\alpha }_i^{p-1}{\int }_{\Omega }g{\delta }_i^{p-1}{v}^2+o\left({\parallel v\parallel }^2\right).$$

(30)

Observe that

$${\alpha }_i^{p-1}{\int }_{\Omega }g{\delta }_i^{p-1}{v}^2={\alpha }_i^{p-1}g\left(a_i\right){\int }_{\Omega }{\delta }_i^{p-1}{v}^2+O\left({\int }_{\Omega }|g\left(x\right)-g\left(a_i\right)|{\delta }_i^{p-1}{v}^2\right)$$

and

$${\int }_{\Omega }|g\left(x\right)-g(a_i\left)\right|{\delta }_i^{p-1}{v}^2\le c{\int }_{\Omega }|x-a_i|{\delta }_i^{p-1}{v}^2=o\left({\parallel v\parallel }^2\right).$$

Using the fact that ${\alpha }_i^{p-1}g\left(a_i\right)=1+o\left(1\right),$ we obtain

$$Q_{\epsilon }\left(v\right)={\int }_{\Omega }{\left|\nabla v\right|}^2+{\int }_{\Omega }V{v}^2-p\sum _{i=1}^k{\int }_{\Omega }{\delta }_i^{p-1}{v}^2+o\left({\parallel v\parallel }^2\right).$$

(31)

However, as shown in the proof of Proposition 1 in [30], there exists a positive constant c such that

$${\int }_{\Omega }{\left|\nabla v\right|}^2+{\int }_{\Omega }V{v}^2-p\sum _{i=1}^N{\int }_{\Omega }{\delta }_i^{p-1}{v}^2\ge c{\parallel v\parallel }^2.$$

This leads to the desired conclusion, thus completing the proof of the proposition. □

Next, we are going to estimate the infinite dimensional part $v_{\epsilon }$. Namely, we prove

Proposition 2.

Let $n\ge 4$ and $(\alpha ,\lambda ,a)\in \beta (k,{\nu }_0)$. Then, for ${\nu }_0$ positive small, there exists a unique ${\overline{v}}_{\epsilon }\in E_{a,\lambda }$ which minimizes $I_{\epsilon }\left({\sum }_{i=1}^k{\alpha }_i{\phi }_{a_i{\lambda }_i}+v\right)$ with respect to $v\in E_{a,\lambda }$ and $\parallel v\parallel$ small. In particular, we obtain

$$⟨I_{\epsilon }^{\prime }\left(\sum _{i=1}^k{\alpha }_i{\phi }_{a_i{\lambda }_i}+{\overline{v}}_{\epsilon }\right),h⟩=0\forall h\in E_{a,\lambda }.$$

(32)

Moreover, the following estimate holds:

$$\parallel {\overline{v}}_{\epsilon }\parallel \le c{T}_{\epsilon ,a,\lambda },where$$

(33)

$$T_{\epsilon ,a,\lambda }=\epsilon +\sum _{i=1}^k{\displaystyle \frac{|\nabla g(a_i\left)\right|}{{\lambda }_i}}$$

$$\begin{array}{cc}\hfill & +\left\{\begin{array}{cc}\sum _{j\ne i}{\epsilon }_{ij}+\sum _{i=1}^k{\displaystyle \frac{ln{\lambda }_i}{{\lambda }_i^2}}\hfill & ifn=4,\hfill \\ \sum _{j\ne i}{\epsilon }_{ij}+\sum _{i=1}^k{\displaystyle \frac{1}{{\lambda }_i^2}}\hfill & ifn=5,\hfill \\ \sum _{j\ne i}{\epsilon }_{ij}^{(n+2)/\left(2\right(n-2\left)\right)}{(ln{\epsilon }_{ij}^{-1})}^{(n+2)/\left(2n\right)}+\sum _{i=1}^k{\displaystyle \frac{1}{{\lambda }_i^2}}\hfill & ifn\ge 6.\hfill \end{array}\right.\hfill \end{array}$$

(34)

Proof. 

Using the implicit function theorem, estimate Equation (25), and Proposition 1, we see that there exists ${\overline{v}}_{\epsilon }\in E_{a,\lambda }$ satisfying $\parallel {\overline{v}}_{\epsilon }\parallel = O\left(\parallel L_{\epsilon }\parallel \right)$, where $L_{\epsilon }$ is defined by Equation (23). Next, we are going to estimate $\parallel L_{\epsilon }\parallel$. Taking $v\in E_{a,\lambda }$, we have

$$\begin{array}{cc}\hfill L_{\epsilon }\left(v\right)& ={\int }_{\Omega }g\left(x\right){\left|\phi \left(x\right)\right|}^{p-1-\epsilon }\phi \left(x\right)v\hfill \\ \hfill & =\sum _{i=1}^k{\int }_{\Omega }g\left(x\right){\alpha }_i^{p-\epsilon }{\left|{\phi }_i\left(x\right)\right|}^{p-1-\epsilon }{\phi }_{i}v+(ifn⩾6)O\left(\sum _{i\ne j}{\int }_{\Omega }|{\phi }_i{\phi }_j{|}^{{\displaystyle \frac{p-\epsilon }{2}}}\left|v\right|\right)\hfill \\ \hfill & +(ifn⩽5)O\left(\sum _{i\ne j}{\int }_{\Omega }|{\phi }_i{|}^{p-1-\epsilon }|{\phi }_j|\left|v\right|\right).\hfill \end{array}$$

(35)

Using estimate E2 of [35], Proposition A1 and the fact that

$${\delta }_i^{-\epsilon }=c_0^{-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}\left(1+O\left(\epsilon ln\left(1+{\lambda }_i^2{|x-a_i|}^2\right)\right)\right)=1+o\left(1\right),$$

(36)

we obtain, for $n⩾6$,

$${\int }_{\Omega }|{\phi }_i{\phi }_j{|}^{{\displaystyle \frac{p-\epsilon }{2}}}\left|v\right|⩽c{\int }_{\Omega }{\left({\delta }_i{\delta }_j\right)}^{{\displaystyle \frac{p-\epsilon }{2}}}\left|v\right|⩽c\parallel v\parallel {\epsilon }_{ij}^{{\displaystyle \frac{n+2}{2\left(n-2\right)}}}{(ln{\epsilon }_{ij}^{-1})}^{{\displaystyle \frac{n+2}{2n}}}$$

(37)

and, for $n⩽5$, using Lemma $6.6$ of [29], we obtain

$${\int }_{\Omega }|{\phi }_i{|}^{p-1-\epsilon }|{\phi }_j||v|⩽c{\int }_{\Omega }{\delta }_i^{p-1}{\delta }_j\left|v\right|⩽c\parallel v\parallel {\left({\int }_{\Omega }{\delta }_i^{{\displaystyle \frac{8n}{n^2-4}}}{\delta }_j^{{\displaystyle \frac{2n}{n+2}}}\right)}^{{\displaystyle \frac{n+2}{2n}}}⩽c\parallel v\parallel {\epsilon }_{ij}.$$

(38)

For the other term in Equation (35), using Equation (36) and Proposition A1, we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }& g\left(x\right)|{\phi }_i{|}^{p-1-\epsilon }{\phi }_{i}v\hfill \\ \hfill & ={\int }_{\Omega }g\left(x\right){|{\delta }_i-{\psi }_i|}^{p-1-\epsilon }\left({\delta }_i-{\psi }_i\right)v\hfill \\ \hfill & ={\int }_{\Omega }g\left(x\right){\delta }_i^{p-\epsilon }v+O\left({\int }_{\Omega }{\delta }_i^{p-1}|{\psi }_i|\left|v\right|\right)\hfill \\ \hfill & ={\int }_{\Omega }g\left(x\right){\delta }_i^{p-\epsilon }v+\left\{\begin{array}{cc}O\left({\displaystyle \frac{ln{\lambda }_i}{{\lambda }_i^2}}{\int }_{\Omega }{\delta }_i^3\left|v\right|+{\int }_{\Omega }{|x-a_i|}^2|ln|x-a_i\left|\right|{\delta }_i^3\left|v\right|\right)\hfill & \mathrm{if}n=4,\hfill \\ O\left({\displaystyle \frac{1}{{\lambda }_i^2}}{\int }_{\Omega }{\delta }_i^p\left|v\right|+{\int }_{\Omega }{|x-a_i|}^2{\delta }_i^p\left|v\right|\right)\hfill & \mathrm{if}n\ge 5,\hfill \end{array}\right.\hfill \end{array}$$

(39)

where

$${\psi }_{a,\lambda }={\delta }_{a,\lambda }-{\phi }_{a,\lambda }.$$

(40)

Notice that, for $n⩾5$, we have

$${\displaystyle \frac{1}{{\lambda }_i^2}}{\int }_{\Omega }{\delta }_i^p|v|⩽{\displaystyle \frac{c}{{\lambda }_i^2}}\parallel v\parallel \mathrm{and}{\int }_{\Omega }|x-a_i{|}^2{\delta }_i^p|v|⩽c\parallel v\parallel {\left({\int }_{\Omega }{|x-a_i|}^{{\displaystyle \frac{4n}{n+2}}}{\delta }_i^{{\displaystyle \frac{2n}{n-2}}}\right)}^{{\displaystyle \frac{n+2}{2n}}}.$$

(41)

But we have

$${\int }_{\Omega }{|x-a_i|}^{{\displaystyle \frac{4n}{n+2}}}{\delta }_i^{{\displaystyle \frac{2n}{n-2}}}⩽{\displaystyle \frac{c}{{\lambda }_i^{4n/(n+2)}}}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{{|x|}^{\frac{4n}{n+2}}}{{(1+|x|}^2{)}^n}}⩽{\displaystyle \frac{c}{{\lambda }_i^{4n/(n+2)}}}.$$

Thus

$${\int }_{\Omega }|x-a_i{|}^2{\delta }_i^p|v|⩽{\displaystyle \frac{c}{{\lambda }_i^2}}\parallel v\parallel .$$

(42)

For $n=4$, we have ${\int }_{\Omega }{\delta }_i^3|v|⩽c\parallel v\parallel$, and

$${\int }_{\Omega }|x-a_i{|}^2|ln|x-a_i||{\delta }_i^3|v|⩽\parallel v\parallel {({\int }_{\Omega }|x-a_i{|}^{{\displaystyle \frac{8}{3}}}|ln|x-a_i{||}^{{\displaystyle \frac{4}{3}}}{\delta }_i^4)}^{{\displaystyle \frac{3}{4}}}.$$

But we have

$${\int }_{\Omega }|x-a_i{|}^{8/3}|ln|x-a_i{||}^{4/3}{\delta }_i^4⩽{\displaystyle \frac{c}{{\lambda }_i^{8/3}}}{\int }_{{\mathbb{R}}^4}{\displaystyle \frac{{|x|}^{8/3}{|ln\frac{|x|}{{\lambda }_i}|}^{4/3}}{{(1+|x|}^2{)}^4}}dx⩽c{\displaystyle \frac{{(ln{\lambda }_i)}^{4/3}}{{\lambda }_i^{8/3}}}.$$

(43)

Combining Equations (39), (41)–(43), we obtain

$${\int }_{\Omega }g\left(x\right){|{\phi }_i\left(x\right)|}^{p-1-\epsilon }{\phi }_i\left(x\right)v\left(x\right)={\int }_{\Omega }g{\delta }_i^{p-\epsilon }v+\left\{\begin{array}{cc}O\left({\displaystyle \frac{ln{\lambda }_i}{{\lambda }_i^2}}\parallel v\parallel \right)\hfill & \mathrm{if}n=4,\hfill \\ O\left({\displaystyle \frac{1}{{\lambda }_i^2}}\parallel v\parallel \right)\hfill & \mathrm{if}n⩾5.\hfill \end{array}\right.$$

(44)

Now, we deal with the first term on the right-hand side of Equation (44). Using Equation (36), we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }g\left(x\right){\delta }_i^{p-\epsilon }v& =c_0^{-\epsilon }{\lambda }_{\epsilon }^{{\displaystyle \frac{-\epsilon (n-2)}{2}}}{\int }_{\Omega }g{\delta }_i^{p}v+O(\epsilon {\int }_{\Omega }{\delta }_i^p|v|ln(1+{\lambda }_i^2|x-a_i{|}^2)\hfill \\ \hfill & =c_0^{-\epsilon }{\lambda }_{\epsilon }^{{\displaystyle \frac{-\epsilon (n-2)}{2}}}g\left(a_i\right){\int }_{\Omega }{\delta }_i^{p}v+O\left(|\nabla g\left(a_i\right)|{\int }_{\Omega }|x-a_i|{\delta }_i^p|v|\right)\hfill \\ & +O({\int }_{\Omega }|x-a_i{|}^2{\delta }_i^p|v|+\epsilon \parallel v\parallel ).\hfill \end{array}$$

(45)

Notice that

$${\int }_{\Omega }|x-a_i|{\delta }_i^p|v|=O\left({\displaystyle \frac{\parallel v\parallel }{{\lambda }_i}}\right)\mathrm{and}{\int }_{\Omega }|x-a_i{|}^2{\delta }_i|v|=O\left({\displaystyle \frac{\parallel v\parallel }{{\lambda }_i^2}}\right).$$

(46)

We also note that

$${\int }_{\Omega }{\delta }_i^{p}v={\int }_{\Omega }(-\Delta +V\left(x\right)){\phi }_{i}v={\int }_{\Omega }(\nabla {\phi }_i\nabla v+V{\phi }_{i}v)=({\phi }_i,v)=0.$$

(47)

Combining Equations (45) and (47), we obtain

$${\int }_{\Omega }g{\delta }_i^{p-\epsilon }v=O\left(\parallel v\parallel \left({\displaystyle \frac{|\nabla g(a_i)|}{{\lambda }_i}}+\epsilon \right)\right).$$

(48)

Now, using Equations (35), (37)–(39), (42), (44) and (48), we easily derive the desired estimate. □

3. Expansion of the Gradient in a Neighborhood of Bubbles

In this section, our aim is to give the expansion of the gradient of the functional $I_{\epsilon }$ defined by Equation (20) in a neighborhood of bubbles. Observe that, for $u,h\in H^1(\Omega )$, we have

$$⟨I_{\epsilon }^{\prime }\left(u\right),h⟩={\int }_{\Omega }\nabla u·\nabla h+{\int }_{\Omega }V\left(x\right)uh-{\int }_{\Omega }g{|u|}^{p-1-\epsilon }uh.$$

(49)

In Equation (49), we will take $u={\sum }_{i=1}^k{\alpha }_i{\phi }_i+v_{\epsilon }:=\phi +v_{\epsilon }$ and $h=h_i\in \{{\phi }_i, {\lambda }_i\partial {\phi }_i/\partial {\lambda }_i,$${\lambda }_i^{-1}\partial {\phi }_i/\partial a_i\}$ with $1\le i\le k$. We are going to estimate each integral in Equation (49). We begin by some integrals involving $v_{\epsilon }$.

Lemma 1.

Let $n\ge 4$, $(\alpha ,\lambda ,a,v)\in \mathcal{O}(k,{\nu }_0)$ and $u:=\sum {\alpha }_i{\phi }_i+v:=\phi +v$. Then, for $h_i\in \{{\phi }_i,{\lambda }_i\partial {\phi }_i/\partial {\lambda }_i,{\lambda }_i^{-1}\partial {\phi }_i/\partial a_i\}$ with $1\le i\le k$, we have

$${\int }_{{\Omega }_1}{v}^2{|\phi |}^{p-2-\epsilon }|h_i|,{\int }_{{\Omega }_2}{|v|}^{p-\epsilon }|h_i|\le \left\{\begin{array}{cc}c{\parallel v\parallel }^2\hfill & ifn\le 5\hfill \\ c{A}_{\epsilon }\left(v\right)\hfill & ifn\ge 6\hfill \end{array}\right.,$$

where ${\Omega }_1:=\{x\in \Omega :|v|\le |\phi |\}$, ${\Omega }_2:=\Omega \setminus {\Omega }_1$ and

$$A_{\epsilon }\left(v\right):={\parallel v\parallel }^{p-\epsilon }{\lambda }_i^{-{\displaystyle \frac{n-2}{2}}}+{\parallel v\parallel }^2+{\parallel v\parallel }^{p-\epsilon }\sum _{j\ne i}{\epsilon }_{ij}^{{\displaystyle \frac{1}{2}}}{\left(ln{\epsilon }_{ij}^{-1}\right)}^{{\displaystyle \frac{n-2}{2n}}}\le {\parallel v\parallel }^2+{\displaystyle \frac{1}{{\lambda }_i^4}}+\sum _{j\ne i}{\epsilon }_{ij}^{n/(n-2)}.$$

Proof. 

Let us start with the case where $n\le 5$. In this case, note that if $n\le 5$, we have $p>2$ and therefore, using Proposition A1 and estimate Equation (36), we obtain

$${\int }_{{\Omega }_1}{v}^2{|\phi |}^{p-2-\epsilon }|h_i|\le c\sum _{j=1}^k{\int }_{{\Omega }_1}{v}^2{\delta }_j^{p-2}{\delta }_i\le c{\parallel v\parallel }^2$$

(50)

We also have for $n\le 5$

$${\int }_{{\Omega }_2}{|v|}^{p-\epsilon }|h_i|\le {\int }_{{\Omega }_2}{|v|}^{p-\epsilon }{\delta }_i\le c{\parallel v\parallel }^{p-\epsilon }\le c{\parallel v\parallel }^2.$$

(51)

Next, we consider the case where $n\ge 6$. Notice that Proposition A1 implies the existence of a small positive real $\eta$ such that

$$|{\psi }_{a,\lambda }|\le {\displaystyle \frac{1}{2}}{\delta }_{a,\lambda }\mathrm{and}{\phi }_{a,\lambda }\ge {\displaystyle \frac{1}{2}}{\delta }_{a,\lambda }\mathrm{in}B(a,\eta ).$$

(52)

Letting $B_i=B(a_i,\eta )$, we see that

$${\int }_{{\Omega }_1\setminus B_i}{v}^2{|\phi |}^{p-2-\epsilon }|h_i|\le {\int }_{{\Omega }_1\setminus B_i}{|v|}^{p-\epsilon }|h_i|\le c{\int }_{\Omega \setminus B_i}{|v|}^{p-\epsilon }{\delta }_i\le {\displaystyle \frac{{c\parallel v\parallel }^{p-\epsilon }}{{\lambda }_i^{(n-2)/2}}},$$

(53)

$${\int }_{{\Omega }_2\setminus B_i}{|v|}^{p-\epsilon }|h_i|\le c{\int }_{\Omega \setminus B_i}{|v|}^{p-\epsilon }{\delta }_i\le {\displaystyle \frac{{c\parallel v\parallel }^{p-\epsilon }}{{\lambda }_i^{(n-2)/2}}}.$$

(54)

It remains to estimate the integral in $B_i$. Let

$${\Omega }_3=\{x\in \Omega :|{\alpha }_i{\phi }_i|\le 2|\phi |\}.$$

Notice that, on one hand, we have

$$|{\alpha }_i{\phi }_i\left(x\right)|\ge 2|\phi \left(x\right)|\mathrm{and}|\phi \left(x\right)|\ge |{\alpha }_i{\phi }_i\left(x\right)|-|\sum _{j\ne i}{\alpha }_j{\phi }_j\left(x\right)|\forall x\in \setminus {\Omega }_3.$$

This implies that

$$|{\alpha }_i{\phi }_i\left(x\right)|\le 2\left|\sum _{j\ne i}{\alpha }_j{\phi }_j\left(x\right)\right|\forall x\in \Omega \setminus {\Omega }_3.$$

(55)

On the other hand, we have

$$|h_i\left(x\right)|\le c{\delta }_i\left(x\right)\le c{\phi }_i\left(x\right)\le c|\phi \left(x\right)|\forall x\in {\Omega }_3\cap B_i.$$

Thus, we have

$${\int }_{{\Omega }_1\cap B_i\cap {\Omega }_3}{v}^2{|\phi |}^{p-2-\epsilon }|h_i|\le c{\int }_{\Omega }{v}^2|h_i{|}^{p-1-\epsilon }\le c{\parallel v\parallel }^2,$$

(56)

$${\int }_{{\Omega }_2\cap B_i\cap {\Omega }_3}{|v|}^{p-\epsilon }|h_i|\le {\int }_{\Omega }{|v|}^{p+1-\epsilon }\le c{\parallel v\parallel }^{p+1-\epsilon }\le c{\parallel v\parallel }^2,$$

(57)

$$\begin{array}{cc}\hfill {\int }_{({\Omega }_1\cap B_i)\setminus {\Omega }_3}{v}^2{|\phi |}^{p-2-\epsilon }|h_i|& \le c{\int }_{({\Omega }_1\cap B_i)\setminus {\Omega }_3}{|v|}^{p-\epsilon }|h_i|\le c{\int }_{({\Omega }_1\cap B_i)\setminus {\Omega }_3}{|v|}^{p-\epsilon }\sqrt{{\delta }_i}\sqrt{{\phi }_i}\hfill \\ \hfill & \le c\sum _{j\ne i}{\int }_{\Omega }{|v|}^{p-\epsilon }\sqrt{{\delta }_i{\delta }_j}\le c{\parallel v\parallel }^{p-\epsilon }\sum _{j\ne i}{\left({\epsilon }_{ij}^{{\displaystyle \frac{n}{n-2}}}ln{\epsilon }_{ij}^{-1}\right)}^{{\displaystyle \frac{n-2}{2n}}},\hfill \end{array}$$

(58)

$${\int }_{({\Omega }_2\cap B_i)\setminus {\Omega }_3}{|v|}^{p-\epsilon }|h_i|\le c{\int }_{(\Omega \cap B_i)\setminus {\Omega }_3}{|v|}^{p-\epsilon }\sqrt{{\delta }_i}\sqrt{{\phi }_i}\le c{\parallel v\parallel }^{p-\epsilon }\sum _{j\ne i}{\epsilon }_{ij}^{{\displaystyle \frac{1}{2}}}{\left(ln{\epsilon }_{ij}^{-1}\right)}^{{\displaystyle \frac{n-2}{2n}}},$$

(59)

where we use Equations (52) and (55) and Proposition A1.

Combining Equations (50), (51), (53), (54), (56)–(59), we derive our lemma. □

Next, we estimate the nonlinear term in Equation (49).

Proposition 3.

Let $n\ge 4$, $(\alpha ,\lambda ,a,v)\in \mathcal{O}(k,{\nu }_0)$ and $u_{\epsilon }:=\sum {\alpha }_i{\phi }_i+v:=\phi +v$. Then, for $h_i\in \{{\phi }_i,{\lambda }_i\partial {\phi }_i/\partial {\lambda }_i,{\lambda }_i^{-1}\partial {\phi }_i/\partial a_i\}$ with $1\le i\le k$, we have

$$\begin{array}{cc}\hfill {\int }_{\Omega }& g|u_{\epsilon }{|}^{p-1-\epsilon }{u}_{\epsilon }{h}_i\hfill \\ \hfill & =\sum _{j=1}^k{\int }_{\Omega }g|{\alpha }_j{\phi }_j{|}^{p-1-\epsilon }{\alpha }_j{\phi }_j{h}_i+(p-\epsilon )\sum _{j\ne i}{\int }_{\Omega }g{|{\alpha }_i{\phi }_i|}^{p-1-\epsilon }\left({\alpha }_j{\phi }_j\right)h_i\hfill \\ & +(p-\epsilon ){\int }_{\Omega }g{|{\alpha }_i{\phi }_i|}^{p-1-\epsilon }v{h}_i+O\left(R_2\left(i\right)\right),\hfill \end{array}$$

(60)

where

$$R_2\left(i\right):={\parallel v\parallel }^2+{\displaystyle \frac{1}{{\lambda }_i^4}}+\sum _{j\ne \mathsf{\ell }}{\epsilon }_{j\mathsf{\ell }}^{{\displaystyle \frac{n}{n-2}}}(ln{\epsilon }_{j\mathsf{\ell }}^{-1})+\sum _{j=1}^k{\displaystyle \frac{T_1^2\left({\lambda }_j\right)}{{\lambda }_i^{n-2}}}+\sum _{j\ne i}{T}_2^2\left({\epsilon }_{ij}\right).$$

Here, $T_1$ and $T_2$ are defined by

$$\begin{array}{cc}\hfill & T_1\left(t\right):=\left(t^{(2-n)/2)} if n\le 5;t^{-2}lnt if n=6;t^{-2} if n\ge 7\right),\hfill \\ \hfill & T_2\left(t\right):=\left(t^{{\displaystyle \frac{n+2}{2(n-2)}}}{(ln{t}^{-1})}^{{\displaystyle \frac{n+2}{2n}}}if n\ge 6;tif n\le 5\right).\hfill \end{array}$$

Proof. 

Observe that, for any $t,s\in \mathbb{R}$ and $\gamma >0$, we have

$${|s+t|}^{\gamma }(s+t)={|s|}^{\gamma }s+(\gamma +1){|s|}^{\gamma }t+\left\{\begin{array}{cc}O(|s{|}^{\gamma -1}{t}^2)\hfill & \mathrm{if} |t|\le |s|,\hfill \\ O(|t{|}^{\gamma +1})\hfill & \mathrm{if} |s|\le |t|.\hfill \end{array}\right.$$

Thus, writing $u_{\epsilon }=\phi +v$, we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }g{|u_{\epsilon }|}^{p-1-\epsilon }{u}_{\epsilon }{h}_i& ={\int }_{\Omega }{g|\phi |}^{p-1-\epsilon }\phi h_i+(p-\epsilon ){\int }_{\Omega }g{|\phi |}^{p-1-\epsilon }v{h}_i\hfill \\ \hfill & +O\left({\int }_{|v|\le |\phi |}{|\phi |}^{p-2-\epsilon }{v}^2|h_i|+{\int }_{|\phi |\le |v|}{|v|}^{p-\epsilon }|h_i|\right).\hfill \end{array}$$

(61)

We remark that the two last integrals are estimated in Lemma 1. To study the second integral on the right hand side of Equation (61), let $B_i=B(a_i,\eta )$ with the properties Equation (52). It is easy to deduce that

$${\int }_{\Omega \setminus B_i}{g|\phi |}^{p-1-\epsilon }|v||h_i|\le {\displaystyle \frac{c}{{\lambda }_i^{(n-2)/2}}}\sum _j{\int }_{\Omega }|v|{\delta }_j^{p-1}\le {\displaystyle \frac{c\parallel v\parallel }{{\lambda }_i^{(n-2)/2}}}\sum _j{\left({\int }_{\Omega }{\delta }_j^{{\displaystyle \frac{8n}{n^2-4}}}\right)}^{(n+2)/2n},$$

(62)

and, by easy computations,

$${\left({\int }_{\Omega }{\delta }_j^{{\displaystyle \frac{8n}{n^2-4}}}\right)}^{(n+2)/2n}\le c{T}_1\left({\lambda }_j\right).$$

To estimate the integral over $\Omega \cap B_i$, we notice that, for $t_j\in \mathbb{R}$ with $j\in \{1,\cdots ,k\}$ and $|{\zeta }_i|\le c|t_i|$, we have

$$||\sum _{j\ne i}{t}_j{|}^{\gamma }{\zeta }_i-{|t_i|}^{\gamma }{\zeta }_i|\le c\left\{\begin{array}{cc}\sum _{j\ne i}{\left(t_i{t}_j\right)}^{{\displaystyle \frac{\gamma +1}{2}}}\hfill & \mathrm{if}\gamma \le 1,\hfill \\ \sum _{j\ne i}(t_i^{\gamma }{t}_j+t_i{t}_j^{\gamma })\hfill & \mathrm{if}\gamma >1.\hfill \end{array}\right.$$

Thus, using Proposition A1, estimate $E_2$ of [35] and estimate Equation (36), we have

$$\begin{array}{cc}\hfill {\int }_{\Omega \cap B_i}g& {|\phi |}^{p-1-\epsilon }v{h}_i\hfill \\ \hfill & ={\int }_{\cap B_i}g{|{\alpha }_i{\phi }_i|}^{p-1-\epsilon }v{h}_i+(ifn\ge 6)O\left(\sum _{j\ne i}{\int }_{\Omega }|v|{\left({\delta }_i{\delta }_j\right)}^{{\displaystyle \frac{p}{2}}}\right)\hfill \\ \hfill & +(ifn\le 5)O\left(\sum _{j\ne i}\left({\int }_{\Omega }|v|({\delta }_i^{p-1}{\delta }_j+{\delta }_i{\delta }_j^{p-1})\right)\right)\hfill \\ \hfill & ={\int }_{\Omega }g|{\alpha }_i{\phi }_i{|}^{p-1-\epsilon }v{h}_i-{\int }_{\Omega \setminus B_i}g|{\alpha }_i{\phi }_i{|}^{p-1-\epsilon }v{h}_i+\sum _{j\ne i}O\left(\parallel v\parallel T_2\left({\epsilon }_{ij}\right)\right).\hfill \end{array}$$

(63)

Furthermore, we have

$$\left|{\int }_{\Omega \setminus B_i}g|{\alpha }_i{\phi }_i{|}^{p-1-\epsilon }v{h}_i\right|\le c{\int }_{\setminus B_i}|v|{\delta }_i^p\le c{\displaystyle \frac{\parallel v\parallel }{{\lambda }_i^{(n+2)/2}}}.$$

(64)

This completes the estimate of the second integral of the right hand side of Equation (61) and we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }& {g|\phi |}^{p-1-\epsilon }v{h}_i\hfill \\ \hfill & ={\alpha }_i^{p-1-\epsilon }{\int }_{\Omega }g{|{\phi }_i|}^{p-1-\epsilon }v{h}_i+O\left(\parallel v\parallel \left[{\displaystyle \frac{1}{{\lambda }_i^{(n+2)/2}}}+\sum _j{\displaystyle \frac{T_1\left({\lambda }_j\right)}{{\lambda }_i^{(n-2)/2}}}+\sum _{j\ne i}{T}_2\left({\epsilon }_{ij}\right)\right]\right).\hfill \end{array}$$

(65)

To deal with the first integral, we note that, for $a,b\in \mathbb{R}$, it holds

$$|{|a+b|}^{p-1-\epsilon }(a+b)-{|a|}^{p-1-\epsilon }a-{(p-\epsilon )|a|}^{p-1-\epsilon }b-{|b|}^{p-1-\epsilon }b|\le {c|a|}^{{\displaystyle \frac{2}{n-2}}-\epsilon }{|b|}^{{\displaystyle \frac{n}{n-2}}}.$$

Thus using estimate $E_2$ of [35] and Proposition A1, we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }g& {|\phi |}^{p-1-\epsilon }\phi h_i\hfill \\ \hfill & ={\int }_{\Omega }g|{\alpha }_i{\phi }_i{|}^{p-1-\epsilon }\left({\alpha }_i{\phi }_i\right)h_i+(p-\epsilon ){\int }_{\Omega }g{|{\alpha }_i{\phi }_i|}^{p-1-\epsilon }\left(\sum _{j\ne i}{\alpha }_j{\phi }_j\right)h_i\hfill \\ \hfill & +{\int }_{\Omega }g|\sum _{j\ne i}{\alpha }_j{\phi }_j{|}^{p-1-\epsilon }\left(\sum _{j\ne i}{\alpha }_j{\phi }_j\right)h_i+O\left(\sum _{j\ne i}{\epsilon }_{ij}^{{\displaystyle \frac{n}{n-2}}}ln{\epsilon }_{ij}^{-1}\right).\hfill \end{array}$$

(66)

Using the fact that

$$||\sum _j{t}_j{|}^{\gamma -1}(\sum t_j)-\sum _j|t_j{|}^{\gamma -1}{t}_j|\le c\sum _{k\ne j}|t_k{|}^{\gamma -1}|t_j|\forall \gamma >0$$

and Proposition A1, we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }g{\left|\sum _{j\ne i}{\alpha }_j{\phi }_j\right|}^{p-1-\epsilon }& (\sum _{j\ne i}{\alpha }_j{\phi }_j)h_i\hfill \\ \hfill & =\sum _{j\ne i}{\int }_{\Omega }g{|{\alpha }_j{\phi }_j|}^{p-1-\epsilon }{\alpha }_j{\phi }_j{h}_i+O(\sum _{\begin{array}{c}k\ne j\\ k,j\ne i\end{array}}{\int }_{\Omega }{\delta }_j^{p-1}{\delta }_k{\delta }_i).\hfill \end{array}$$

(67)

Now, using estimate $E_2$ of [35], we have

$$\sum _{\begin{array}{c}k\ne j\\ k,j\ne i\end{array}}{\int }_{\Omega }{\delta }_j^{p-1}{\delta }_k{\delta }_j\le \sum _{k\ne j}{\int }_{\Omega }{\left({\delta }_j{\delta }_k\right)}^{{\displaystyle \frac{n}{n-2}}}\le c\sum _{k\ne j}{\epsilon }_{kj}^{{\displaystyle \frac{n}{n-2}}}ln{\epsilon }_{kj}^{-1}.$$

(68)

Combining Lemma 1 and estimate Equations (61)–(68), we easily derive our result. □

Next, we deal with the first integral on the right-hand side of Equation (60). More precisely, we prove

Lemma 2.

Let $n\ge 4$, $(\alpha ,\lambda ,a)\in \beta (k,{\nu }_0)$ with ${\nu }_0$ positive small. Then, for $i,j\in \{1,\cdots ,k\}$ with $j\ne i$ and $h_i\in \{{\phi }_i,{\lambda }_i\partial {\phi }_i/\partial {\lambda }_i,{\lambda }_i^{-1}\partial {\phi }_i/\partial a_i\}$, we have

$$\begin{array}{cc}\hfill {\int }_{\Omega }& g|{\phi }_j{|}^{p-1-\epsilon }{\phi }_j{h}_i\hfill \\ \hfill & =c_0^{-\epsilon }{\lambda }_j^{-\epsilon (n-2)/2}g\left(a_j\right){\int }_{\Omega }{\delta }_j^p{h}_i+O\left({\displaystyle \frac{|\nabla g(a_j)|}{{\lambda }_j}}{\epsilon }_{ij}+\epsilon {\epsilon }_{ij}+{\displaystyle \frac{{ln}^{1+{\sigma }_n/2}{\lambda }_j}{{\lambda }_j^2}}{\epsilon }_{ij}{(ln{\epsilon }_{ij}^{-1})}^{{\displaystyle \frac{n-2}{n}}}\right),\hfill \end{array}$$

where ${\sigma }_n=0$ for $n\ge 5$ and ${\sigma }_4=1$.

Proof. 

Using Proposition A1 and estimate Equation (36), we have

$${\int }_{\Omega }g{|{\phi }_j|}^{p-1-\epsilon }{\phi }_j{h}_i={\int }_{\Omega }g{\delta }_j^{p-\epsilon }{h}_i+O\left({\int }_{\Omega }{\delta }_j^{p-1}|{\psi }_j|{\delta }_i\right).$$

(69)

But, using estimate $E_2$ of [35], Lemma $2.2$ of [36] and Proposition A1, we obtain for $n\ge 5$

$$\begin{array}{cc}\hfill {\int }_{\Omega }& {\delta }_j^{p-1}|{\psi }_j|{\delta }_i\hfill \\ \hfill & \le {\displaystyle \frac{c}{{\lambda }_j^2}}{\int }_{\Omega }{\delta }_j^p{\delta }_i+c{\int }_{\Omega }{|x-a_j|}^2{\delta }_j^p{\delta }_i\hfill \\ \hfill & \le {\displaystyle \frac{c{\epsilon }_{ij}}{{\lambda }_j^2}}+c{\left({\int }_{\Omega }{\left({\delta }_i{\delta }_j\right)}^{{\displaystyle \frac{n}{n-2}}}\right)}^{{\displaystyle \frac{n-2}{n}}}{\left(\int |x-a_j{|}^n{\delta }_j^{{\displaystyle \frac{2n}{n-2}}}\right)}^{{\displaystyle \frac{2}{n}}}\le c{\displaystyle \frac{ln{\lambda }_j}{{\lambda }_j^2}}{\epsilon }_{ij}{(ln{\epsilon }_{ij}^{-1})}^{{\displaystyle \frac{n-2}{n}}}\hfill \end{array}$$

(70)

and for $n=4$

$$\begin{array}{cc}\hfill {\int }_{\Omega }{\delta }_j^{p-1}|{\psi }_j|{\delta }_i& \le {\displaystyle \frac{cln{\lambda }_j}{{\lambda }_j^2}}{\int }_{\Omega }{\delta }_j^3{\delta }_i+c{\int }_{\Omega }{|x-a|}^2|ln|x-a||{\delta }_j^3{\delta }_i\hfill \\ \hfill & \le {\displaystyle \frac{cln{\lambda }_j}{{\lambda }_j^2}}{\epsilon }_{ij}+c{\left({\int }_{\Omega }{\left({\delta }_i{\delta }_j\right)}^2\right)}^{{\displaystyle \frac{1}{2}}}{\left({\int }_{\Omega }|x-a_j{|}^4{ln}^2|x-a_j|{\delta }_j^4\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \le C{\displaystyle \frac{{ln}^{\frac{3}{2}}{\lambda }_j}{{\lambda }_j^2}}{\epsilon }_{ij}{(ln{\epsilon }_{ij}^{-1})}^{{\displaystyle \frac{1}{2}}}.\hfill \end{array}$$

(71)

For the first term on the right-hand side of Equation (69), using estimate Equation (36) and Proposition A1, we write

$${\int }_{\Omega }g{\delta }_j^{p-\epsilon }{h}_i=c_0^{-\epsilon }{\lambda }_j^{-\epsilon (n-2)/2}{\int }_{\Omega }g{\delta }_j^p{h}_i+O\left(\epsilon {\int }_{\Omega }{\delta }_j^{p}ln(1+{\lambda }_j^2|x-a_j{|}^2){\delta }_i\right).$$

(72)

Note that, using Lemma $2.2$ of [36], we have

$$\begin{array}{cc}\hfill {\int }_{\Omega }& {\delta }_j^{p}ln(1+{\lambda }_j^2|x-a_j{|}^2){\delta }_i\hfill \\ \hfill & ={\int }_{\Omega }\left({\delta }_i{\delta }_j^{{\displaystyle \frac{n}{n-2}}}\right){\delta }_j^{{\displaystyle \frac{2}{n-2}}}ln(1+{\lambda }_j^2|x-a_j{|}^2)\hfill \\ \hfill & \le {\left({\int }_{\Omega }{\delta }_i^{{\displaystyle \frac{2n}{2n-2}}}{\delta }_j^{{\displaystyle \frac{n}{n-2}}.{\displaystyle \frac{2n}{2n-2}}}\right)}^{{\displaystyle \frac{2n-2}{2n}}}{\left({\int }_{\Omega }{\delta }_j^{{\displaystyle \frac{2n}{n-2}}}{ln}^n(1+{\lambda }_j^2{|x-a_j|}^2\right)}^{{\displaystyle \frac{1}{n}}}\le c{\epsilon }_{ij}.\hfill \end{array}$$

(73)

We also note that

$${\int }_{\Omega }g{\delta }_j^p{h}_i=g\left(a_i\right){\int }_{\Omega }{\delta }_j^p{h}_i+O\left(|\nabla g\left(a_i\right)|{\int }_{\Omega }|x-a_j|{\delta }_j^p{\delta }_i\right)+O\left({\int }_{\Omega }{|x-a_j|}^2{\delta }_j^p{\delta }_i\right).$$

But, using Lemma $2.2$ of [36] and estimate $E_2$ of [35], we have

$$\begin{array}{cc}{\int }_{\Omega }|x-a_j|{\delta }_j^p{\delta }_i\hfill & ={\int }_{\Omega }|x-a_j|{\delta }_j^{{\displaystyle \frac{3}{n-2}}}\left({\delta }_j^{{\displaystyle \frac{n-1}{n-2}}}{\delta }_i\right)\hfill \\ & \le {\left({\int }_{\Omega }{\delta }_j^{{\displaystyle \frac{(n-1)}{n-2}}{\displaystyle \frac{2n}{n-3}}}{\delta }_i^{{\displaystyle \frac{2n}{n-3}}}\right)}^{{\displaystyle \frac{2n-3}{2n}}}{\left({\int }_{\Omega }{|x-a_j|}^{{\displaystyle \frac{2n}{3}}}{\delta }_j^{{\displaystyle \frac{2n}{n-2}}}\right)}^{{\displaystyle \frac{3}{2n}}}\le c{\epsilon }_{ij}{\displaystyle \frac{1}{{\lambda }_j}}\hfill \end{array}$$

(74)

$$\begin{array}{cc}{\int }_{\Omega }|x-a_j{|}^2{\delta }_j^p{\delta }_i\hfill & ={\int }_{\Omega }{|x-a_j|}^2{\delta }_j^{{\displaystyle \frac{4}{n-2}}}\left({\delta }_j{\delta }_i\right)\le {\left({\int }_{\Omega }{\left({\delta }_j{\delta }_i\right)}^{{\displaystyle \frac{n}{n-2}}}\right)}^{{\displaystyle \frac{n-2}{n}}}{\left({\int }_{\Omega }{|x-a_j|}^n{\delta }_j^{{\displaystyle \frac{2n}{n-2}}}\right)}^{{\displaystyle \frac{2}{n}}}\hfill \\ & \le c{\displaystyle \frac{ln{\lambda }_j}{{\lambda }_j^2}}{\epsilon }_{ij}{ln}^{{\displaystyle \frac{n-2}{n}}}{\epsilon }_{ij}^{-1}.\hfill \end{array}$$

(75)

Thus

$${\int }_{\Omega }g{\delta }_j^p{h}_i=g\left(a_i\right){\int }_{\Omega }{\delta }_j^p{h}_i+O\left({\displaystyle \frac{|\nabla g(a_j)|}{{\lambda }_j}}{\epsilon }_{ij}+{\displaystyle \frac{ln{\lambda }_j}{{\lambda }_j^2}}{\epsilon }_{ij}{ln}^{{\displaystyle \frac{n-2}{n}}}{\epsilon }_{ij}^{-1}\right).$$

(76)

Combining Equations (69)–(76), we easily obtain the lemma. □

Next, we are going to deal with the linear term in the expansion Equation (60) with respect to $v\in E_{a,\lambda }$. More precisely, we have the following.

Lemma 3.

Let $n\ge 4$ and $(\alpha ,\lambda ,a,v)\in \mathcal{O}(k,{\nu }_0)$. Then, for $h_i\in \{{\phi }_i,{\lambda }_i{\displaystyle \frac{\partial {\phi }_i}{\partial {\lambda }_i}}{\lambda }_i^{-1}{\displaystyle \frac{\partial {\phi }_i}{\partial a_i}}\}$ with $1\le i\le k$, we have

$$\left|{\int }_{\Omega }g|{\phi }_i{|}^{p-1-\epsilon }v{h}_i\right|\le c\parallel v\parallel \left(\epsilon +{\displaystyle \frac{|\nabla g(a_i)|}{{\lambda }_i}}+{\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^2}}\right),$$

where ${\sigma }_4=1$ and ${\sigma }_n=0$ for $n\ge 5$.

Proof. 

First, combining Equations (44) and (48), we obtain the desired result for $h_i={\phi }_i$. Second, following the proof of Equations (44) and (48), we obtain the desired result for $h_i={\lambda }_i{\displaystyle \frac{\partial {\phi }_i}{\partial {\lambda }_i}}$ and $h_i={\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\phi }_i}{\partial a_i}}.$ This completes the proof of the lemma. □

The aim of the following three propositions is to specify the statement of Proposition 3.

Proposition 4.

Let $n\ge 4$, $(\alpha ,\lambda ,a,v)\in \mathcal{O}(k,{\nu }_0)$ with ${\nu }_0$ positive small and $u_{\epsilon }={\sum }_{i=1}^k{\alpha }_i{\phi }_i+v$. Then, for $1\le i\le k$, the following statement holds

$${\int }_{\Omega }g{|u_{\epsilon }|}^{p-1-\epsilon }{u}_{\epsilon }{\phi }_i=g\left(a_i\right){\alpha }_i^{p-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}{S}_n+O\left(R_3\left(i\right)\right),$$

where

$$S_n=c_0^{p+1}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{dx}{{(1+|x|}^2{)}^n}},R_3\left(i\right)=\epsilon +{\parallel v_{\epsilon }\parallel }^2+{\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^2}}+\sum _{j\ne i}{\epsilon }_{ij},$$

(77)

with ${\sigma }_4=1$ and ${\sigma }_n=0$ for $n\ge 5$.

Proof. 

Combining Equation (61), Lemma 1 and Equation (65), we derive that

$$\begin{array}{cc}\hfill {\int }_{\Omega }g& |u_{\epsilon }{|}^{p-1-\epsilon }{u}_{\epsilon }{\phi }_i={\int }_{\Omega }g{|\phi |}^{p-1-\epsilon }\phi {\phi }_i+O\left({\parallel v\parallel }^2+{\epsilon }^2+{\displaystyle \frac{1}{{\lambda }_i^2}}+\sum {\epsilon }_{ij}^{n/(n-2)}\right).\hfill \end{array}$$

(78)

Furthermore, using Lemma A1 and the fact that

$${|s+t|}^{\gamma }(s+t)={|s|}^{\gamma }s+{O(|s|}^{\gamma }|t|+{|s||t|}^{\gamma })\forall s,t\in \mathbb{R} \mathrm{and} \gamma >0,$$

we obtain

$${\int }_{\Omega }{g|\phi |}^{p-1-\epsilon }\phi {\phi }_i={\alpha }_i^{p-\epsilon }{\int }_{\Omega }g{|{\phi }_i|}^{p+1-\epsilon }+O\left({\int }_{\Omega }{\delta }_i^p{\delta }_j+{\int }_{\Omega }{\delta }_i{\delta }_j^p\right).$$

(79)

Note that, by Estimate $E1$ of [35], we have

$${\int }_{\Omega }{\delta }_j^p{\delta }_i=O\left({\epsilon }_{ij}\right).$$

(80)

In addition, using again Proposition A1 and estimate Equation (36), for $r>0$ small, by oddness, we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }& g|{\phi }_i{|}^{p+1-\epsilon }\hfill \\ \hfill & ={\int }_{\Omega }g\left({\delta }_i^{p+1-\epsilon }+O({\delta }_i^p|{\psi }_i|)\right)\hfill \\ \hfill & =g\left(a_i\right){\int }_{B(a_i,r)}{\delta }_i^{p+1-\epsilon }+O\left({\int }_{B(a_i,r)}{|x-a_i|}^2{\delta }_i^{p+1}+{\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^2}}+{\int }_{\Omega \setminus B(a_i,r)}{\delta }_i^{p+1}\right)\hfill \\ \hfill & =g\left(a_i\right)c_0^{-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}{\int }_{B(a_i,r)}{\delta }_i^{p+1}+O\left(\epsilon {\int }_{B(a_i,r)}{\delta }_i^{p+1}ln(1+{\lambda }_i^2{|x-a_i|}^2+{\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^2}}\right)\hfill \\ \hfill & =g\left(a_i\right)c_0^{-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}{S}_n+O\left(\epsilon +{\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^2}}\right).\hfill \end{array}$$

(81)

Combining Equations (78)–(81) and the fact that $c_0^{-\epsilon }=1+O\left(\epsilon \right)$, we derive the desired result. □

Proposition 5.

Let $n\ge 4$, $(\alpha ,\lambda ,a,v)\in \mathcal{O}(k,{\nu }_0)$ with ${\nu }_0$ positive small and $u_{\epsilon }={\sum }_{i=1}^k{\alpha }_i{\phi }_i+v$. Then, for $1\le i\le k$, the following statement holds:

$$\begin{array}{cc}\hfill {\int }_{\Omega }& g|u_{\epsilon }{|}^{p-1-\epsilon }{u}_{\epsilon }{\lambda }_i{\displaystyle \frac{\partial {\phi }_i}{\partial {\lambda }_i}}\hfill \\ \hfill & ={\alpha }_i^{p-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}g\left(a_i\right)\left[-c_3\epsilon +2E_n{\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^2}}V\left(a_i\right)\right]\hfill \\ \hfill & +c_1\sum _{j\ne i}{\lambda }_i{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial {\lambda }_i}}{\alpha }_j\left({\alpha }_i^{p-1-\epsilon }g\left(a_i\right){\lambda }_i^{-\epsilon (n-2)/2}+{\alpha }_j^{p-1-\epsilon }g\left(a_j\right){\lambda }_j^{-\epsilon (n-2)/2}\right)+O\left(R_4\left(i\right)\right),\hfill \end{array}$$

where ${\sigma }_4=1$, ${\sigma }_n=0$ for $n\ge 5$, $c_3$ and $E_n$ are defined in Theorem 1 and

$$\begin{array}{cc}\hfill & R_4\left(i\right):={\parallel v\parallel }^2+{\epsilon }^2+{\displaystyle \frac{|\nabla g\left(a_i\right){|}^2}{{\lambda }_i^2}}+{\displaystyle \frac{|D^{2}g\left(a_i\right)|}{{\lambda }_i^2}}+\sum _{j\ne \mathsf{\ell }}{\epsilon }_{j\mathsf{\ell }}^{{\displaystyle \frac{n}{n-2}}}ln{\epsilon }_{j\mathsf{\ell }}^{-1}+{\displaystyle \frac{1}{{\lambda }_i^3}}+\sum _{j=1}^k{\displaystyle \frac{1}{{\left({\lambda }_i{\lambda }_j\right)}^{(n-2)/2}}}\hfill \\ \hfill & +\sum _{j\ne i}{\epsilon }_{ij}\left({\displaystyle \frac{|\nabla g(a_j)|}{{\lambda }_j}}+|a_i-a_j{|}^2|ln|a_i-a_j{||}^{{\sigma }_n}\right)+\sum _{j\ne i}{\epsilon }_{ij}{ln}^{{\displaystyle \frac{n-2}{n}}}{\epsilon }_{ij}^{-1}{\displaystyle \frac{{ln}^{1+{\sigma }_n/2}{\lambda }_j}{{\lambda }_j^2}},\hfill \\ \hfill & c_1:=c_0^{2n/(n-2)}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{1}{{(1+|x|}^2{)}^{(n+2)/2}}}dx.\hfill \end{array}$$

Proof. 

Applying Proposition 3, Lemmas 2 and 3, we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }& g|u_{\epsilon }{|}^{p-1-\epsilon }{u}_{\epsilon }{\lambda }_i{\displaystyle \frac{\partial {\phi }_i}{\partial {\lambda }_i}}\hfill \\ \hfill & ={\alpha }_i^{p-\epsilon }{\int }_{\Omega }g{|{\phi }_i|}^{p-1-\epsilon }{\phi }_i{\lambda }_i{\displaystyle \frac{\partial {\phi }_i}{\partial {\lambda }_i}}+\sum _{j\ne i}{c}_0^{-\epsilon }{\lambda }_j^{-\epsilon (n-2)/2}{\alpha }_j^{p-\epsilon }g\left(a_j\right)({\phi }_j,{\lambda }_i{\displaystyle \frac{\partial {\phi }_i}{\partial {\lambda }_i}})\hfill \\ \hfill & +\sum _{j\ne i}O\left({\displaystyle \frac{|\nabla g(a_j)|}{{\lambda }_j}}{\epsilon }_{ij}+\epsilon {\epsilon }_{ij}+{\displaystyle \frac{{ln}^{1+{\sigma }_n/2}{\lambda }_j}{{\lambda }_j^2}}{\epsilon }_{ij}{ln}^{{\displaystyle \frac{n-2}{n}}}{\epsilon }_{ij}^{-1}\right)+O\left(R_2\left(i\right)\right)\hfill \\ \hfill & +(p-\epsilon )\sum _{j\ne i}{\int }_{\Omega }g{|{\alpha }_i{\phi }_i|}^{p-1-\epsilon }{\alpha }_j{\phi }_j{\lambda }_i{\displaystyle \frac{\partial {\phi }_i}{\partial {\lambda }_i}}+O\left(\parallel v_{\epsilon }\parallel \left[\epsilon +{\displaystyle \frac{|\nabla g(a_i)|}{{\lambda }_i}}+{\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^2}}\right]\right).\hfill \end{array}$$

(82)

First, we focus on studying the first integral on the right-hand side of Equation (82). We have

$$\begin{array}{cc}\hfill {\int }_{\Omega }& g|{\phi }_i{|}^{p-1-\epsilon }{\phi }_i{\lambda }_i{\displaystyle \frac{\partial {\phi }_i}{\partial {\lambda }_i}}\hfill \\ \hfill & ={\int }_{\Omega }g\left({\delta }_i^{p-\epsilon }-(p-\epsilon ){\delta }_i^{p-1-\epsilon }{\psi }_i+O\left({\delta }_i^{p-2}{\psi }_i^2\right)\right)\left({\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}-{\lambda }_i{\displaystyle \frac{\partial {\psi }_i}{\partial {\lambda }_i}}\right)\hfill \\ \hfill & ={\int }_{\Omega }g{\delta }_i^{p-\epsilon }{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}-{\int }_{\Omega }g{\delta }_i^{p-\epsilon }{\lambda }_i{\displaystyle \frac{\partial {\psi }_i}{\partial {\lambda }_i}}-(p-\epsilon ){\int }_{\Omega }g{\delta }_i^{p-1-\epsilon }{\psi }_i{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}\hfill \\ \hfill & +O\left({\int }_{\Omega }{\delta }_i^{p-1}|{\psi }_i|\left|{\lambda }_i{\displaystyle \frac{\partial {\psi }_i}{\partial {\lambda }_i}}\right|+{\int }_{\Omega }{\delta }_i^{p-1}{\psi }_i^2\right).\hfill \end{array}$$

(83)

But, using Proposition A1, easy computations imply that

$${\int }_{\Omega }{\delta }_i^{p-1}|{\psi }_i|\left|{\lambda }_i{\displaystyle \frac{\partial {\psi }_i}{\partial {\lambda }_i}}\right|+{\int }_{\Omega }{\delta }_i^{p-1}{\psi }_i^2\le C{\int }_{\Omega }{\chi }_1^2{\delta }_i^{p+1}\le C{\displaystyle \frac{{ln}^{3{\sigma }_n}{\lambda }_i}{{\lambda }_i^4}}.$$

(84)

Now, expanding g around $a_i$ in $B_i:=B(a_i,\rho )$ with $\rho$ positive small, by oddness, we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }& g{\delta }_i^{p-\epsilon }{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}\hfill \\ \hfill & =g\left(a_i\right){\int }_{B_i}{\delta }_i^{p-\epsilon }{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}+O\left(|D^{2}g\left(a_i\right)|{\int }_{B_i}|x-a_i{|}^2{\delta }_i^{p+1}+{\int }_{B_i}{|x-a_i|}^4{\delta }_i^{p+1}+{\displaystyle \frac{1}{{\lambda }_i^n}}\right)\hfill \\ \hfill & =-g\left(a_i\right)\epsilon c_3{c}_0^{-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}+O\left({\epsilon }^2+{\displaystyle \frac{|D^{2}g\left(a_i\right)|}{{\lambda }_i^2}}+{\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^4}}\right),\hfill \end{array}$$

(85)

where we use estimate $\left(91\right)$ of [30] and Equation (36).

For the other integrals on the right-hand side of Equation (83), using Lemma A1, we write

$$\begin{array}{cc}\hfill -{\int }_{\Omega }& g{\delta }_i^{p-\epsilon }{\lambda }_i{\displaystyle \frac{\partial {\psi }_i}{\partial {\lambda }_i}}\hfill \\ \hfill & =-g\left(a_i\right){\int }_{\Omega }{\delta }_i^{p-\epsilon }{\lambda }_i{\displaystyle \frac{\partial {\psi }_i}{\partial {\lambda }_i}}+O\left({\int }_{\Omega }|x-a_i|{\delta }_i^{p+1}{\chi }_1\left(x\right)\right)\hfill \\ \hfill & =c_0^{-\epsilon }g\left(a_i\right){\lambda }_i^{-\epsilon (n-2)/2}\left({\phi }_i,{\lambda }_i{\displaystyle \frac{\partial {\phi }_i}{\partial {\lambda }_i}}\right)+O\left(\epsilon {\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^2}}+{\displaystyle \frac{1}{{\lambda }_i^n}}\right)+O\left({\displaystyle \frac{{ln}^{{\sigma }_n}\left({\lambda }_i\right)}{{\lambda }_i^3}}\right),\hfill \end{array}$$

(86)

$$\begin{array}{cc}\hfill -p{\int }_{\Omega }& g{\delta }_i^{p-1-\epsilon }{\psi }_i{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}\hfill \\ \hfill & =-pg\left(a_i\right){\int }_{\Omega }{\delta }_i^{p-\epsilon }{\psi }_i{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}+O\left({\int }_{\Omega }|x-a_i|{\delta }_i^{p+1}{\chi }_1\left(x\right)\right)\hfill \\ \hfill & =c_0^{-\epsilon }g\left(a_i\right){\lambda }_i^{-\epsilon (n-2)/2}\left({\phi }_i,{\lambda }_i{\displaystyle \frac{\partial {\phi }_i}{\partial {\lambda }_i}}\right)+O\left(\epsilon {\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^2}}+{\displaystyle \frac{1}{{\lambda }_i^n}}\right)+O\left({\displaystyle \frac{{ln}^{{\sigma }_n}\left({\lambda }_i\right)}{{\lambda }_i^3}}\right).\hfill \end{array}$$

(87)

Combining Equations (83)–(86) and using Lemma A2, we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }& g|{\phi }_i{|}^{p-1-\epsilon }{\phi }_i{\lambda }_i{\displaystyle \frac{\partial {\phi }_i}{\partial {\lambda }_i}}\hfill \\ \hfill =& c_0^{-\epsilon }{\lambda }_i^{-\epsilon {\displaystyle \frac{n-2}{2}}}g\left(a_i\right)\left[-c_3\epsilon +2E_n{\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^2}}V\left(a_i\right)\right]+O\left({\epsilon }^2+{\displaystyle \frac{|D^{2}g\left(a_i\right)|}{{\lambda }_i^2}}+{\displaystyle \frac{1}{{\lambda }_i^3}}\right).\hfill \end{array}$$

(88)

It remains to deal with the last integral on the right-hand side of Equation (82). To this aim, using estimate Equation (36), we have

$$\begin{array}{cc}\hfill {\int }_{\Omega }& g|{\phi }_i{|}^{p-1-\epsilon }{\phi }_j{\lambda }_i{\displaystyle \frac{\partial {\phi }_i}{\partial {\lambda }_i}}\hfill \\ \hfill & ={\int }_{\Omega }g\left({\delta }_i^{p-1-\epsilon }+O\left({\delta }_i^{p-2-\epsilon }|{\psi }_i|\right)\right){\phi }_j\left({\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}-{\lambda }_i{\displaystyle \frac{\partial {\psi }_i}{\partial {\lambda }_i}}\right)\hfill \\ \hfill & ={\int }_{\Omega }g{\delta }_i^{p-1-\epsilon }{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}{\phi }_j+O\left({\int }_{\Omega }{\delta }_i^{p-1}|{\phi }_j||{\lambda }_i{\displaystyle \frac{\partial {\psi }_i}{\partial {\lambda }_i}}|+{\int }_{\Omega }{\delta }_i^{p-1}|{\psi }_i||{\phi }_j|\right).\hfill \end{array}$$

(89)

For the remainder term, using Proposition A1, Equations (70) and (71), we deduce that

$${\int }_{\Omega }{\delta }_i^{p-1}|{\phi }_j||{\lambda }_i{\displaystyle \frac{\partial {\psi }_i}{\partial {\lambda }_i}}|+{\int }_{\Omega }{\delta }_i^{p-1}|{\psi }_i||{\phi }_j|\le c{\displaystyle \frac{{(ln{\lambda }_i)}^{1+{\sigma }_n/2}}{{\lambda }_i^2}}{\epsilon }_{ij}{(ln{\epsilon }_{ij}^{-1})}^{(n-2)/n}.$$

For the first integral on the right-hand side of Equation (89), using Proposition A1, we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }g& {\delta }_i^{p-1-\epsilon }{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}{\phi }_j\hfill \\ \hfill & =c_0^{-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}{\int }_{\Omega }g{\delta }_i^{p-1}{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}{\phi }_j+O\left(\epsilon {\int }_{\Omega }{\delta }_i^{p}ln(1+{ł}_i^2|x-a_i{|}^2){\delta }_j\right).\hfill \end{array}$$

(90)

Expanding g around $a_i$ and following the argument in the proof of Equation (76), we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }g{\delta }_i^{p-1}{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}{\phi }_j& =g\left(a_i\right){\int }_{\Omega }{\delta }_i^{p-1}{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}{\phi }_j+O\left({\displaystyle \frac{|\nabla g(a_i)|}{{\lambda }_i}}{\epsilon }_{ij}+{\displaystyle \frac{ln{\lambda }_i}{{\lambda }_i^2}}{\epsilon }_{ij}{(ln{\epsilon }_{ij}^{-1})}^{(n-2)/n}\right)\hfill \\ \hfill & =g\left(a_i\right){\displaystyle \frac{1}{p}}\left({\phi }_j,{\lambda }_i{\displaystyle \frac{\partial {\phi }_i}{\partial {\lambda }_i}}\right)+O\left({\displaystyle \frac{|\nabla g(a_i)|}{{\lambda }_i}}{\epsilon }_{ij}+{\displaystyle \frac{ln{\lambda }_i}{{\lambda }_i^2}}{\epsilon }_{ij}{(ln{\epsilon }_{ij}^{-1})}^{(n-2)/n}\right).\hfill \end{array}$$

Thus, using Equation (73), Equation (89) becomes

$$\begin{array}{cc}\hfill (p-\epsilon ){\int }_{\Omega }g{|{\phi }_i|}^{p-1-\epsilon }{\phi }_j{\lambda }_i{\displaystyle \frac{\partial {\phi }_i}{\partial {\lambda }_i}}=& \left(1-{\displaystyle \frac{\epsilon }{p}}\right)g\left(a_i\right)c_0^{-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}\left({\phi }_j,{\lambda }_i{\displaystyle \frac{\partial {\phi }_i}{\partial {\lambda }_i}}\right)\hfill \\ \hfill & +O\left({\displaystyle \frac{|\nabla g(a_i)|}{{\lambda }_i}}{\epsilon }_{ij}+{\displaystyle \frac{ln{\lambda }_i}{{\lambda }_i^2}}{\epsilon }_{ij}{(ln{\epsilon }_{ij}^{-1})}^{(n-2)/n}\right).\hfill \end{array}$$

(91)

Thus, combining estimate Equations (82)–(88), (91) and Lemma A4, we obtain the desired result. □

Proposition 6.

Let $n\ge 4,(\alpha ,\lambda ,a,v)\in \mathcal{O}(k,{\nu }_0)$ with ${\nu }_0$ positive small and $u_{\epsilon }={\sum }_{i=1}^k{\alpha }_i{\phi }_i+v$. Then, for $1\le i\le k$, the following statement holds:

$$\begin{array}{cc}\hfill {\int }_{\Omega }& g|u_{\epsilon }{|}^{p-1-\epsilon }{u}_{\epsilon }{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\phi }_i}{\partial a_i}}\hfill \\ \hfill & ={\alpha }_i^{p-\epsilon }{c}_0^{-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}{\overline{c}}_5{\displaystyle \frac{\nabla g\left(a_i\right)}{{\lambda }_i}}+2c_0^{-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}{\alpha }_i^{p-\epsilon }g\left(a_i\right)D_n{\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^3}}\nabla V\left(a_i\right)\hfill \\ \hfill & +\sum _{j\ne i}{c}_0^{-\epsilon }{c}_1{\alpha }_j{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial a_i}}\left({\lambda }_i^{-\epsilon (n-2)/2}{\alpha }_i^{p-1-\epsilon }g\left(a_i\right)+{\lambda }_j^{-\epsilon (n-2)/2}{\alpha }_j^{p-1-\epsilon }g\left(a_j\right)\right)+O\left(R_5\left(i\right)\right),\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill R_5\left(i\right)=& {\parallel v\parallel }^2+{\epsilon }^2+{\displaystyle \frac{|\nabla g\left(a_i\right){|}^2}{{\lambda }_i^2}}+{\displaystyle \frac{|D^{3}g\left(a_i\right)|}{{\lambda }_i^3}}+{\displaystyle \frac{1}{{\lambda }_i^4}}+(ifn=4){\displaystyle \frac{1}{{\lambda }_i^3}}+(ifn=5){\displaystyle \frac{ln{\lambda }_i}{{\lambda }_i^4}}\hfill \\ \hfill & +\sum _{j\ne \mathsf{\ell }}{\epsilon }_{j\mathsf{\ell }}^{n/(n-2)}(ln{\epsilon }_{j\mathsf{\ell }}^{-1})+\sum _{j\ne i}{\lambda }_j|a_i-a_j|{\epsilon }_{ij}^{{\displaystyle \frac{n+1}{n-2}}}+\sum _{j\ne i}{\displaystyle \frac{|\nabla g(a_j)|}{{\lambda }_j}}{\epsilon }_{ij}\hfill \\ \hfill & +\sum _{j\ne i}\left({\displaystyle \frac{{ln}^{1+{\sigma }_n/2}{\lambda }_j}{{\lambda }_j^2}}{\epsilon }_{ij}{(ln{\epsilon }_{ij}^{-1})}^{(n-2)/n}+{\displaystyle \frac{{\epsilon }_{ij}}{{\lambda }_i}}+{\displaystyle \frac{1}{{\lambda }_i^{(n-2)/2}{\lambda }_j^{(n+2)/2}}}\right)\hfill \end{array}$$

where $c_1$ is defined in Proposition 5, ${\overline{c}}_5$ and $D_n$ are defined in Theorem 2.

Proof. 

Applying Proposition 3, and Lemmas 2 and 3, we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }& g|u_{\epsilon }{|}^{p-1-\epsilon }{u}_{\epsilon }{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\phi }_i}{\partial a_i}}\hfill \\ \hfill & ={\alpha }_i^{p-\epsilon }{\int }_{\Omega }g{|{\phi }_i|}^{p-1-\epsilon }{\phi }_i{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\phi }_i}{\partial a_i}}\hfill \\ \hfill & +\sum _{j\ne i}{\alpha }_j^{p-\epsilon }{C}_0^{-\epsilon }{\lambda }_j^{{\displaystyle \frac{-\epsilon (n-2)}{2}}}g\left(a_j\right){\int }_{\Omega }{\delta }_j^p{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\phi }_i}{\partial a_i}}+O(\sum _{j\ne i}({\displaystyle \frac{|\nabla g(a_i)|}{{\lambda }_j}}{\epsilon }_{ij}+\epsilon {\epsilon }_{ij}))\hfill \\ \hfill & +O(\sum _{j\ne i}{\displaystyle \frac{{ln}^{1+{\sigma }_n/2}{\lambda }_j}{{\lambda }_j^2}}{\epsilon }_{ij}{ln}^{{\displaystyle \frac{n-2}{n}}}{\epsilon }_{ij}^{-1})+(p-\epsilon )\sum _{j\ne i}{\int }_{\Omega }g{|{\alpha }_i{\phi }_i|}^{p-1-\epsilon }{\alpha }_j{\phi }_j{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\phi }_i}{\partial a_i}}\hfill \\ \hfill & +O({\parallel v_{\epsilon }\parallel }^2+{\epsilon }^2+{\displaystyle \frac{|\nabla g\left(a_i\right){|}^2}{{\lambda }_i^2}}+{\displaystyle \frac{{ln}^{2{\sigma }_n}{\lambda }_i}{{\lambda }_i^4}})+R_2\left(i\right).\hfill \end{array}$$

(92)

We start by dealing with the first integral on the right-hand side of Equation (92). Using the fact that

$${|s-t|}^{\gamma }(s-t)={|s|}^{\gamma }s-{(\gamma +1)|s|}^{\gamma }t+O\left({|s|}^{\gamma -1}{t}^2\right)\mathrm{for}t,s\in \mathbb{R}\mathrm{with}|t|\le c|s|,$$

we derive that

$$\begin{array}{cc}\hfill {\int }_{\Omega }& g|{\phi }_i{|}^{p-1-\epsilon }{\phi }_i{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\phi }_i}{\partial a_i}}\hfill \\ \hfill & ={\int }_{\Omega }g\left({\delta }_i^{p-\epsilon }-(p-\epsilon ){\delta }_i^{p-1-\epsilon }{\psi }_i+O\left({\delta }_i^{p-2}{\psi }_i^2\right)\right)({\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}-{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\psi }_i}{\partial a_i}})\hfill \\ & ={\int }_{\Omega }g{\delta }_i^{p-\epsilon }{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}-{\int }_{\Omega }g{\delta }_i^{p-\epsilon }{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\psi }_i}{\partial a_i}}-(p-\epsilon ){\int }_{\Omega }g{\delta }_i^{p-1-\epsilon }{\psi }_i{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}\hfill \\ \hfill & +O\left({\int }_{\Omega }{\delta }_i^{p-1}|{\psi }_i||{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\psi }_i}{\partial a_i}}|+{\int }_{\Omega }{\delta }_i^{p-1}{\psi }_i^2\right).\hfill \end{array}$$

(93)

Using Proposition A1 and Lemma A6, we see that

$$\begin{array}{cc}\hfill {\int }_{\Omega }& g{\delta }_i^{p-\epsilon }{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\psi }_i}{\partial a_i}}\hfill \\ \hfill & =g\left(a_i\right){\int }_{\Omega }{\delta }_i^{p-\epsilon }{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\psi }_i}{\partial a_i}}+O\left(|\nabla g\left(a_i\right)|{\int }_{\Omega }|x-a_i|{\delta }_i^{p+1}{\chi }_2+{\int }_{\Omega }{|x-a_i|}^2{\delta }_i^{p+1}{\chi }_2\right)\hfill \\ \hfill & =-c_0^{-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}g\left(a_i\right)\left({\phi }_i,{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\phi }_i}{\partial a_i}}\right)+O\left({\displaystyle \frac{\epsilon }{{\lambda }_i^2}}+{\displaystyle \frac{|\nabla g(a_i)|}{{\lambda }_i^3}}+{\displaystyle \frac{1}{{\lambda }_i^4}}\right),\hfill \end{array}$$

(94)

$$\begin{array}{cc}\hfill & {\int }_{\Omega }g{\delta }_i^{p-1-\epsilon }{\psi }_i{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}\hfill \\ \hfill & =g\left(a_i\right){\int }_{\Omega }{\delta }_i^{p-1-\epsilon }{\psi }_i{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}+O\left(|\nabla g\left(a_i\right)|{\int }_{\Omega }|x-a_i|{\delta }_i^{p+1}{\chi }_1\right)+O\left({\int }_{\Omega }{|x-a_i|}^2{\chi }_1{\delta }_i^{p+1}\right)\hfill \\ \hfill & =-{\displaystyle \frac{1}{p}}{c}_0^{-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}g\left(a_i\right)\left({\phi }_i,{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\phi }_i}{\partial a_i}}\right)+O\left(\epsilon {\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^2}}+|\nabla g\left(a_i\right)|{\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^3}}+{\displaystyle \frac{{ln}^{1+{\sigma }_n}{\lambda }_i}{{\lambda }_i^4}}\right).\hfill \end{array}$$

(95)

For the first integral on the right-hand side of Equation (93), we write for r positive small and $B_i:=B(a_i,r)$,

$$\begin{array}{cc}\hfill {\int }_{\Omega }g{\delta }_i^{p-\epsilon }{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}& ={\int }_{B_i}g{\delta }_i^{p-\epsilon }{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}+O\left({\int }_{{\mathbb{R}}^n\setminus B_i}{\delta }_i^p{\displaystyle \frac{1}{{\lambda }_i}}|{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}|\right)\hfill \\ \hfill & ={\int }_{B_i}\nabla g\left(a_i\right)(x-a_i){\delta }_i^{p-\epsilon }{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}+O\left(|D^{3}g\left(a_i\right)|{\int }_{B_i}{|x-a_i|}^3{\delta }_i^{p+1}\right)\hfill \\ \hfill & +O\left({\int }_{B_i}{|x-a_i|}^4{\displaystyle \frac{{\delta }_i^{p+1}}{{\lambda }_i|x-a_i|}}+{\int }_{{\mathbb{R}}^n\setminus B_i}{\displaystyle \frac{{\delta }_i^{p+1}}{{\lambda }_i|x-a_i|}}\right)\hfill \\ \hfill & =c_0^{-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}{\overline{c}}_5{\displaystyle \frac{\nabla g\left(a_i\right)}{{\lambda }_i}}+O\left(\epsilon {\displaystyle \frac{|\nabla g(a_i)|}{{\lambda }_i}}+{\displaystyle \frac{|D^{3}g\left(a_i\right)|}{{\lambda }_i^3}}+{\displaystyle \frac{1}{{\lambda }_i^4}}\right),\hfill \end{array}$$

(96)

where we use Lemma $6.3$ of [29], estimate Equation (36) and the fact that for $1\le j\le n$, we have

$$\begin{array}{cc}\hfill {\int }_{B_i}\nabla g\left(a_i\right)(x-a_i){\delta }_i^p{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial {\left(a_i\right)}_j}}& =\sum _{l=1}^n{\displaystyle \frac{\partial g\left(a_i\right)}{\partial x_l}}{\int }_{B_i}{(x-a_i)}_l{\delta }_i^p{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial {\left(a_i\right)}_j}}\hfill \\ \hfill & ={\displaystyle \frac{\partial g\left(a_i\right)}{\partial x_j}}{\int }_{B_i}{(x-a_i)}_j{\delta }_i^p{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial {\left(a_i\right)}_j}}\hfill \\ \hfill & =(n-2)c_0^{p+1}{\displaystyle \frac{\partial g\left(a_i\right)}{\partial x_j}}{\int }_{B_i}{\displaystyle \frac{{\lambda }_i^{n+1}{(x-a_i)}_j^2}{(1+{\lambda }_i^2|x-a_i{{|}^2)}^{n+1}}}dx\hfill \\ \hfill & ={\displaystyle \frac{(n-2)}{n}}{c}_0^{p+1}{\displaystyle \frac{\partial g\left(a_i\right)}{\partial x_j}}{\int }_{B_i}{\displaystyle \frac{{\lambda }_i^{n+1}{|x-a_i|}^2}{(1+{\lambda }_i^2|x-a_i{{|}^2)}^{n+1}}}dx\hfill \\ & ={\displaystyle \frac{\partial g\left(a_i\right)}{\partial x_j}}{\displaystyle \frac{{\overline{c}}_5}{{\lambda }_i}}+O\left({\displaystyle \frac{1}{{\lambda }_i^{n+1}}}\right).\hfill \end{array}$$

Combining Equations (93)–(96) and Lemma A7, we derive that

$$\begin{array}{cc}\hfill {\int }_{\Omega }& g|{\phi }_i{|}^{p-1-\epsilon }{\phi }_i{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\phi }_i}{\partial a_i}}\hfill \\ \hfill & =c_0^{-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}{\overline{c}}_5{\displaystyle \frac{\nabla g\left(a_i\right)}{{\lambda }_i}}+2c_0^{-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}g\left(a_i\right)D_n{\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^3}}\nabla V\left(a_i\right)\hfill \\ \hfill & +O\left(\epsilon {\displaystyle \frac{|\nabla g(a_i)|}{{\lambda }_i}}+{\displaystyle \frac{|D^{3}g\left(a_i\right)|}{{\lambda }_i^3}}+{\displaystyle \frac{|\nabla g\left(a_i\right)|{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^3}}+{\displaystyle \frac{\epsilon {ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^2}}\right)\hfill \\ \hfill & +(ifn=4)O({\displaystyle \frac{1}{{\lambda }_i^3}})+(ifn=5)O({\displaystyle \frac{ln{\lambda }_i}{{\lambda }_i^4}})+(ifn\ge 6)O({\displaystyle \frac{1}{{\lambda }_i^4}}).\hfill \end{array}$$

(97)

Now, we deal with the third integral on the right-hand side of Equation (92). Using estimate Equation (36) and Proposition A1, we write

$$\begin{array}{cc}\hfill {\int }_{\Omega }& g|{\phi }_i{|}^{p-1-\epsilon }{\phi }_j{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\phi }_i}{\partial a_i}}\hfill \\ \hfill & ={\int }_{\Omega }g\left({\delta }_i^{p-1-\epsilon }+O\left({\delta }_i^{p-2}|{\psi }_i|\right)\right){\phi }_j\left({\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}-{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\psi }_i}{\partial a_i}}\right)\hfill \\ \hfill & ={\int }_{\Omega }g{\delta }_i^{p-1-\epsilon }{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}{\phi }_j+O\left({\int }_{\Omega }{\delta }_i^{p-1}|{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\psi }_i}{\partial a_i}}|{\delta }_j+{\int }_{\Omega }{\delta }_i^{p-2}{\displaystyle \frac{1}{{\lambda }_i}}|{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}||{\psi }_i|{\delta }_j\right)\hfill \\ & +O\left({\int }_{\Omega }{\delta }_i^{p-2}|{\psi }_i|{\displaystyle \frac{1}{{\lambda }_i}}|{\displaystyle \frac{\partial {\psi }_i}{\partial a_i}}|{\delta }_j\right).\hfill \end{array}$$

(98)

But, Lemma $2.2$ of [36], Proposition A1, and estimate Equation (74) imply that

$$\begin{array}{c}\hfill {\int }_{\Omega }{\delta }_i^{p-1}|{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\psi }_i}{\partial a_i}}|{\delta }_j\le {\displaystyle \frac{c}{{\lambda }_i^2}}{\int }_{\Omega }{\delta }_i^p{\delta }_j+{\displaystyle \frac{1}{{\lambda }_i}}{\int }_{\Omega }|x-a_i|{\delta }_i^p{\delta }_j\le {\displaystyle \frac{c}{{\lambda }_i^2}}{\epsilon }_{ij}.\end{array}$$

(99)

Now, for $n\ge 5$, using again Lemma $2.2$ of [36], Proposition A1 and the fact that ${\displaystyle \frac{1}{{\lambda }_i}}|{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}|\le {\displaystyle \frac{1}{{\lambda }_i|x-a_i|}}{\delta }_i$, we see that

$$\begin{array}{c}\hfill {\int }_{\Omega }{\delta }_i^{p-2}{\displaystyle \frac{1}{{\lambda }_i}}|{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}||{\psi }_i|{\delta }_j\le {\displaystyle \frac{c}{{\lambda }_i^2}}{\int }_{\Omega }{\delta }_i^p{\delta }_j+c{\int }_{\Omega }{|x-a_i|}^2{\displaystyle \frac{{\delta }_i^p}{{\lambda }_i|x-a_i|}}{\delta }_j\le {\displaystyle \frac{c}{{\lambda }_i^2}}{\epsilon }_{ij}.\end{array}$$

(100)

In the same way, for $n=4$, we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }{\delta }_i^{p-2}{\displaystyle \frac{1}{{\lambda }_i}}|{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}||{\psi }_i|{\delta }_j& \le C{\displaystyle \frac{ln{\lambda }_i}{{\lambda }_i^2}}{\int }_{\Omega }{\delta }_i^p{\delta }_j+{\displaystyle \frac{c}{{\lambda }_i}}{\int }_{\Omega }|x-a_i||ln|x-a_i||{\delta }_i^p{\delta }_j\le {\displaystyle \frac{cln{\lambda }_i}{{\lambda }_i^2}}{\epsilon }_{ij},\hfill \end{array}$$

(101)

$${\int }_{\Omega }{\delta }_i^{p-2}|{\psi }_i|{\displaystyle \frac{1}{{\lambda }_i}}|{\displaystyle \frac{\partial {\psi }_i}{\partial a_i}}|{\delta }_j\le C{\int }_{\Omega }{\delta }_i^{p-1}{\displaystyle \frac{1}{{\lambda }_i}}|{\displaystyle \frac{\partial {\psi }_i}{\partial a_i}}|{\delta }_j\le {\displaystyle \frac{c}{{\lambda }_i^2}}{\epsilon }_{ij}.$$

(102)

For the first integral on the right-hand side of Equation (98), using Equations (36), (74), and (102), we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }& g{\delta }_i^{p-1-\epsilon }{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}{\phi }_j\hfill \\ \hfill & =c_0^{-\epsilon }{\lambda }_i^{{\displaystyle \frac{-\epsilon (n-2)}{2}}}{\int }_{\Omega }g{\delta }_i^{p-1}{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}{\phi }_j+O\left(\epsilon {\int }_{\Omega }{\delta }_i^{p}ln(1+{\lambda }_i^2|x-a_i{|}^2){\delta }_j\right)\hfill \\ \hfill & =c_0^{-\epsilon }{\lambda }_i^{{\displaystyle \frac{-\epsilon (n-2)}{2}}}g\left(a_i\right){\int }_{\Omega }g{\delta }_i^{p-1}{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}{\phi }_j+O\left(|\nabla g\left(a_i\right)|{\int }_{\Omega }|x-a_i|{\delta }_i^p{\delta }_j\right)\hfill \\ & +O\left({\int }_{\Omega }{|x-a_i|}^2{\displaystyle \frac{{\delta }_i^p}{{\lambda }_i|x-a_i|}}{\delta }_j\right)+\left(\epsilon {\epsilon }_{ij}\right)\hfill \\ \hfill & =c_0^{-\epsilon }{\lambda }_i^{{\displaystyle \frac{-\epsilon (n-2)}{2}}}g\left(a_i\right){\int }_{\Omega }g{\delta }_i^{p-1}{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}{\phi }_j+O\left(|\nabla g\left(a_i\right)|{\displaystyle \frac{{\epsilon }_{ij}}{{\lambda }_i}}+{\displaystyle \frac{{\epsilon }_{ij}}{{\lambda }_i^2}}+\epsilon {\epsilon }_{ij}\right)\hfill \\ \hfill & =c_0^{-\epsilon }{\lambda }_i^{{\displaystyle \frac{-\epsilon (n-2)}{2}}}g\left(a_i\right){\displaystyle \frac{1}{p}}\left({\phi }_j,{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\phi }_i}{\partial a_i}}\right)+O\left(|\nabla g\left(a_i\right)|{\displaystyle \frac{{\epsilon }_{ij}}{{\lambda }_i}}+{\displaystyle \frac{{\epsilon }_{ij}}{{\lambda }_i^2}}+\epsilon {\epsilon }_{ij}\right).\hfill \end{array}$$

(103)

Combining Lemma A4, estimates Equations (92)–(103), we derive our proposition. □

Next, we give the expansion of the gradient of the function $I_{\epsilon }$ defined by Equation (20) in $\beta (k, {\nu }_0)$ with ${\nu }_0$ being a positive small real. We start by giving the expansion of $I_{\epsilon }$ with respect to the gluing parameter ${\alpha }_i$’s.

Proposition 7.

Let $n\ge 4,(\alpha ,\lambda ,a,v)\in \mathcal{O}(k,{\nu }_0)$ with ${\nu }_0$ positive small and $u_{\epsilon }={\sum }_{i=1}^k{\alpha }_i{\phi }_i+v$. Then, for $1\le i\le k$, the following statement holds:

$$\begin{array}{cc}\hfill & ⟨I_{\epsilon }^{\prime }\left(u_{\epsilon }\right),{\phi }_i⟩={\alpha }_i{S}_n(1-g\left(a_i\right){\alpha }_i^{p-1-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2})+O\left(R_6\left(i\right)\right),with\hfill \\ \hfill & R_6\left(i\right):={\parallel v\parallel }^2+\epsilon +{\displaystyle \frac{|\nabla g(a_i)|}{{\lambda }_i}}+\sum _{j\ne i}{\epsilon }_{ij}+{\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^2}}+\sum _{j=1}^k{\displaystyle \frac{1}{{\lambda }_j^{n-2}}},\hfill \end{array}$$

where $S_n$ is defined in Proposition 4.

Proof. 

Applying Equation (49), Lemmas A1 and A4, Proposition 4, and the fact that $v\in E_{a,\lambda }$, we easily obtain the desired result. □

Next, we provide a balancing condition involving the mutual interaction of bubble ${\epsilon }_{ij}$ and the rate of the concentration ${\lambda }_i$.

Proposition 8.

Let $n\ge 4,(\alpha ,\lambda ,a,v)\in \mathcal{O}(k,{\nu }_0)$ with ${\nu }_0$ being positive small and $u_{\epsilon }={\sum }_{i=1}^k{\alpha }_i{\phi }_i+v$. Then, for $1\le i\le k$, it holds that

$$\begin{array}{c}⟨I_{\epsilon }^{\prime }\left(u\right),{\lambda }_i{\displaystyle \frac{\partial {\phi }_i}{\partial {\lambda }_i}}⟩\hfill \\ ={\alpha }_i^{p-\epsilon }{c}_3{\lambda }_i^{-\epsilon (n-2)/2}\epsilon g\left(a_i\right)-{\alpha }_i{E}_{n}V\left(a_i\right){\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^2}}\left(-1+2{\alpha }_i^{p-1-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}g\left(a_i\right)\right)\hfill \\ -c_1\sum _{j\ne i}{\alpha }_j{\lambda }_i{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial {\lambda }_i}}\left(-1+{\alpha }_i^{p-1-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}g\left(a_i\right)+{\alpha }_j^{p-1-\epsilon }{\lambda }_j^{-\epsilon (n-2)/2}g\left(a_j\right)\right)+O\left(R_4\left(i\right)\right),\hfill \end{array}$$

where $R_4\left(i\right)$ and $c_1$ are defined in Proposition 5, $E_n$ and $c_3$ are defined in Theorem 1, ${\sigma }_4=1$, ${\sigma }_n=0$ for $n\ge 5$.

Proof. 

Applying Equation (49), and Lemmas 5, A2, and A4, and the fact that $v\in E_{a,ł}$, we easily obtain the proof of the proposition. □

Finally, we give the following balancing condition involving the concentration point $a_i$.

Proposition 9.

Let $n\ge 4,(\alpha ,\lambda ,a,v)\in \mathcal{O}(k,{\nu }_0)$ with ${\nu }_0$ positive small and $u_{\epsilon }={\sum }_{i=1}^k{\alpha }_i{\phi }_i+v$. Then, for $1\le i\le k$, the following statement holds:

$$\begin{array}{c}⟨I_{\epsilon }^{\prime }\left(u\right),{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\phi }_i}{\partial a_i}}⟩\hfill \\ =-{\alpha }_i^{p-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}{\overline{c}}_5{\displaystyle \frac{\nabla g\left(a_i\right)}{{\lambda }_i}}+{\alpha }_i{D}_n{\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^3}}\nabla V\left(a_i\right)\left(-1+2{\alpha }_i^{p-1-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}g\left(a_i\right)\right)\hfill \\ -c_1\sum _{j\ne i}{\alpha }_j{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial a_i}}\left[-1+{\alpha }_i^{p-1-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}g\left(a_i\right)+{\alpha }_j^{p-1-\epsilon }{\lambda }_j^{-\epsilon (n-2)/2}g\left(a_j\right)\right]+O\left(R_5\left(i\right)\right),\hfill \end{array}$$

where ${\overline{c}}_5$, $D_n$ are defined in Proposition 6, $c_1$ is defined in Proposition 5, and $R_5\left(i\right)$ is defined in Proposition 6.

Proof. 

Applying Equation (49), Lemmas A3, A4, Proposition 6, and the fact that $v\in E_{a,\lambda }$, we easily derive the proof of the proposition. □

4. Asymptotic Behavior of Interior Single Peaked Solutions

Our aim in this section is to study the asymptotic behavior of solutions of $\left({\mathbf{P}}_{\epsilon }\right)$ which blow up at one interior point as $\epsilon$ goes to zero.

First, we consider a general situation, that is, let $\left(u_{\epsilon }\right)$ be a family of solutions of $\left({\mathbf{P}}_{\epsilon }\right)$ having the form Equation (15) with $k\ge 1$, and the properties introduced in Equation (16) are satisfied. We begin by proving the following crucial fact:

Lemma 4.

Let $n\ge 4$, then, for all $i\in \{1,\dots ,k\}$, the following fact holds:

$$\epsilon ln{\lambda }_i⟶0\mathit{as}\epsilon ⟶0.$$

Proof. 

Multiplying $\left({\mathbf{P}}_{\epsilon }\right)$ by ${\phi }_i$ and integrating over Ω, we obtain

$$\sum _{j=1}^k{\alpha }_j({\phi }_j,{\phi }_i)={\int }_{\Omega }g{\left(\sum _{j=1}^k{\alpha }_j{\phi }_j+v_{\epsilon }\right)}^{p-\epsilon }{\phi }_i.$$

(104)

First, using Lemmas A1 and A4, we deduce that

$$\sum _{j=1}^k{\alpha }_j({\phi }_j,{\phi }_i)={\alpha }_i{S}_n+o\left(1\right).$$

Next, we are going to estimate the right-hand side of Equation (104). To this aim, using Proposition A1, we write

$$\begin{array}{cc}\hfill {\int }_{\Omega }g& {\left(\sum _{j=1}^N{\alpha }_j{\phi }_j+v_{\epsilon }\right)}^{p-\epsilon }{\phi }_i={\int }_{\Omega }g{\left({\alpha }_i{\phi }_i\right)}^{p-\epsilon }{\phi }_i\hfill \\ \hfill & +O\left(\sum _{j\ne i}{\int }_{\Omega }\left({\delta }_j^{p-\epsilon }{\delta }_i+{\delta }_i^{p-\epsilon }{\delta }_j\right)\right)+O\left({\int }_{\Omega }\left(|v_{\epsilon }{|}^{p-\epsilon }{\delta }_i+{\delta }_i^{p-\epsilon }|v_{\epsilon }|\right)\right).\hfill \end{array}$$

(105)

But, using Estimate $E2$ of [35], we have

$$\begin{array}{cc}\hfill {\int }_{\Omega }{\delta }_r^{p-\epsilon }{\delta }_{\mathsf{\ell }}& ={\int }_{\Omega }\left({\delta }_{\mathsf{\ell }}{\delta }_r\right){\delta }_r^{p-1-\epsilon }\hfill \\ \hfill & \le {\left({\int }_{\Omega }{\left({\delta }_{\mathsf{\ell }}{\delta }_r\right)}^{{\displaystyle \frac{n}{n-2}}}\right)}^{{\displaystyle \frac{n-2}{n}}}{\left({\int }_{\Omega }{\delta }_r^{(p-1-\epsilon ){\displaystyle \frac{n}{2}}}\right)}^{{\displaystyle \frac{2}{n}}}\le c{\epsilon }_{\mathsf{\ell }r}{(ln{\epsilon }_{\mathsf{\ell }r}^{-1})}^{(n-2)/n}=o\left(1\right).\hfill \end{array}$$

(106)

We also have

$$\begin{array}{cc}\hfill {\int }_{\Omega }& \left({\left|v_{\epsilon }\right|}^{p-\epsilon }{\delta }_i+{\delta }_i^{p-\epsilon }\left|v_{\epsilon }\right|\right)\hfill \\ \hfill & \le c{\parallel v_{\epsilon }\parallel }^{p-\epsilon }{\left({\int }_{\Omega }{\delta }_i^{{\displaystyle \frac{2n}{n-2}}}\right)}^{{\displaystyle \frac{n-2}{2n}}}+\parallel v_{\epsilon }\parallel {\left({\int }_{\Omega }{\delta }_i^{(p-\epsilon ){\displaystyle \frac{2n}{n+2}}}\right)}^{{\displaystyle \frac{n+2}{2n}}}=o\left(1\right).\hfill \end{array}$$

(107)

Concerning the first integral on the right-hand side of Equation (105), it holds that

$$\begin{array}{cc}\hfill {\int }_{\Omega }& g{\phi }_i^{p+1-\epsilon }={\int }_{\Omega }g{\delta }_i^{p+1-\epsilon }+O\left({\int }_{\Omega }g{\delta }_i^{p-\epsilon }|{\psi }_i|\right)\hfill \\ \hfill & =g\left(a_i\right){\int }_{{\mathbb{R}}^n}{\delta }_i^{p+1-\epsilon }+O\left({\int }_{\Omega }|x-a_i|{\delta }_i^{p+1-\epsilon }+{\int }_{{\mathbb{R}}^n\setminus \Omega }{\delta }_i^{p+1-\epsilon }+{\int }_{\Omega }{\delta }_i^{p-\epsilon }|{\psi }_i|\right).\hfill \end{array}$$

(108)

But, we observe that

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^n}{\delta }_i^{p+1-\epsilon }={\int }_{{\mathbb{R}}^n}{\displaystyle \frac{c_0^{p+1}{c}_0^{-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}}{{\left(1+{|x|}^2\right)}^{n-\epsilon (n-2)/2}}}={\lambda }_i^{-\epsilon (n-2)/2}\left(S_n+O\left(\epsilon \right)\right),\hfill \\ \hfill & {\int }_{{\mathbb{R}}^n\setminus \Omega }{\displaystyle \frac{{\lambda }_i^{n-\epsilon (n-2)/2}}{{\left(1+{\lambda }_i^2{\left|x-a_i\right|}^2\right)}^{n-\epsilon (n-2)/2}}}\le {\displaystyle \frac{c}{{\lambda }_i^{n-\epsilon (n-2)/2}}}{\int }_{{\mathbb{R}}^n\setminus B(a_i,r)}{\displaystyle \frac{dx}{|x-a_i{|}^{2n-\epsilon (n-2)}}}=o\left(1\right).\hfill \end{array}$$

where r is a fixed positive constant satisfying $B(a_i,r)\subset \Omega$.

For the other integrals in Equation (108), since $\Omega$ is bounded, observe that, for $a,x\in \Omega$, we have

$${\delta }_{a,\lambda }^{-\epsilon }\left(x\right)={\displaystyle \frac{c_0^{-\epsilon }}{{\lambda }^{\epsilon (n-2)/2}}}(1+{\lambda }^2{|x-a|}^2{)}^{\epsilon (n-2)/2}\le {\displaystyle \frac{c}{{\lambda }^{\epsilon (n-2)/2}}}+c{\lambda }^{\epsilon (n-2)/2}\le c{\lambda }^{\epsilon (n-2)/2}.$$

Thus, we obtain

$${\int }_{\Omega }|x-a_i|{\delta }_i^{p+1-\epsilon }\le c{\lambda }^{\epsilon (n-2)/2}{\int }_{{\mathbb{R}}^n}|x-a_i|{\delta }_i^{p+1}\le c{\displaystyle \frac{{\lambda }^{\epsilon (n-2)/2}}{{\lambda }_i}}=o\left(1\right),$$

and using Proposition A1, it holds that

$$\begin{array}{cc}\hfill {\int }_{\Omega }{\delta }_i^{p-\epsilon }|{\psi }_i|& \le c{\lambda }_i^{\epsilon {\displaystyle \frac{n-2}{2}}}\left({\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^2}}{\int }_{\Omega }{\delta }_i^{p+1}+{\int }_{\Omega }{|x-a_i|}^2{|ln|x-a_i||}^{{\sigma }_n}{\delta }_i^{p+1}\right)\hfill \\ \hfill & \le c{\lambda }_i^{\epsilon {\displaystyle \frac{n-2}{2}}}{\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^2}}=o\left(1\right).\hfill \end{array}$$

The above estimates imply that

$${\alpha }_i^{p-\epsilon }{\int }_{\Omega }g{\phi }_i^{p+1-\epsilon }={\alpha }_i^{p-\epsilon }g\left(a_i\right){\lambda }_i^{-\epsilon (n-2)/2}\left(S_n+O\left(\epsilon \right)\right)+o\left(1\right).$$

(109)

Combining the estimates of Equations (104)–(109) and using the fact that ${\alpha }_i^{p-1}g\left(a_i\right)=1+o\left(1\right)$, we obtain

$$S_n+o\left(1\right)={\lambda }_i^{-\epsilon (n-2)/2}(S_n+o\left(1\right))$$

which implies that ${\lambda }_i^{-\epsilon (n-2)/2}=1+o\left(1\right)$. The proof of the lemma is thereby complete. □

Next, we consider $\left(u_{\epsilon }\right)$ to be a family of solutions of $\left({\mathbf{P}}_{\epsilon }\right)$ having the form Equation (11) with $k=1$ and satisfying Equations (12)–(14). We know that $u_{\epsilon }$ can be written in the form Equation (15) with $k=1$, that is

$$u_{\epsilon }={\alpha }_{\epsilon }{\phi }_{a_{\epsilon }{\lambda }_{\epsilon }}+v_{\epsilon }$$

(110)

with $(a_{\epsilon },{\lambda }_{\epsilon })\in \Omega \times (0,\infty )$ and

$$\begin{array}{cc}\hfill & |{\alpha }_{\epsilon }^{{\displaystyle \frac{4}{n-2}}}g\left(a_{\epsilon }\right)-1|\to 0,a_{\epsilon }\to \overline{a}\in \Omega ,{\lambda }_{\epsilon }\to \infty ,\parallel v_{\epsilon }\parallel \to 0\mathrm{as}\epsilon \to 0,v_{\epsilon }\in E_{a_{\epsilon }{\lambda }_{\epsilon }},\hfill \end{array}$$

(111)

$$\begin{array}{cc}\hfill & E_{a_{\epsilon },{\lambda }_{\epsilon }}:=\left\{v\in H^1(\Omega ):(v,{\phi }_{a_{\epsilon },{\lambda }_{\epsilon }})=(v,{\displaystyle \frac{\partial {\phi }_{a_{\epsilon },{\lambda }_{\epsilon }}}{\partial {\lambda }_{\epsilon }}})=(v,{\displaystyle \frac{\partial {\phi }_{a_{\epsilon },{\lambda }_{\epsilon }}}{\partial {\left(a_{\epsilon }\right)}_j}})=0\forall j⩽n\right\}.\hfill \end{array}$$

(112)

Using Lemma 4, we see that $({\alpha }_{\epsilon },{\lambda }_{\epsilon },a_{\epsilon },v_{\epsilon })\in \mathcal{O}(1,{\nu }_0)$ which is defined in Equation (18). Since $u_{\epsilon }$ is a solution of $\left({\mathbf{P}}_{\epsilon }\right)$, we see that Equation (32) is satisfied with $v_{\epsilon }$. Thus, by the uniqueness, we obtain $v_{\epsilon }={\overline{v}}_{\epsilon }$, where ${\overline{v}}_{\epsilon }$ is defined in Proposition 2. Therefore $v_{\epsilon }$ satisfies estimate Equation (33). We start by proving Theorem 1.

Proof of Theorem 1. 

In the case of a single interior blow-up point, that is $k=1$ and $u_{\epsilon }=\alpha {\phi }_{a,\lambda }+v_{\epsilon }$, the estimate Equation (33) becomes

$$\parallel v_{\epsilon }\parallel \le c\left(\epsilon +{\displaystyle \frac{|\nabla g(a)|}{\lambda }}\right)+c\left\{\begin{array}{cc}{\lambda }^{-2}\hfill & \mathrm{if}n\ge 5,\hfill \\ {\lambda }^{-2}ln\lambda \hfill & \mathrm{if}n=4.\hfill \end{array}\right.$$

(113)

Combining Equation (113) and Propositions 7–9, we obtain

$$\begin{array}{cc}\hfill & {\alpha }^{p-1-\epsilon }{\lambda }^{-\epsilon (n-2)/2}g\left(a\right)=1+O\left(\epsilon +{\displaystyle \frac{|\nabla g(a)|}{\lambda }}+{\displaystyle \frac{{ln}^{{\sigma }_n}\lambda }{{\lambda }^2}}\right),\hfill \end{array}$$

(114)

$$\begin{array}{cc}\hfill & -E_{n}V\left(a\right){\displaystyle \frac{{ln}^{{\sigma }_n}\lambda }{{\lambda }^2}}\left(-1+2{\alpha }^{p-1-\epsilon }{\lambda }^{-\epsilon (n-2)/2}g\left(a\right)\right)+{\alpha }^{p-1-\epsilon }{\lambda }^{-\epsilon (n-2)/2}g\left(a\right)c_3\epsilon \hfill \\ \hfill & =O\left({\epsilon }^2+{\displaystyle \frac{1}{{\lambda }^{n-2}}}+{\displaystyle \frac{1}{{\lambda }^3}}+{\displaystyle \frac{|D^{2}g\left(a\right)|}{{\lambda }^2}}+{\displaystyle \frac{{|\nabla g\left(a\right)|}^2}{{\lambda }^2}}\right),\hfill \end{array}$$

(115)

$$\begin{array}{cc}\hfill & -{\alpha }^{p-1-\epsilon }{\lambda }^{-\epsilon (n-2)/2}{\overline{c}}_5{\displaystyle \frac{\nabla g\left(a\right)}{\lambda }}+D_n{\displaystyle \frac{{ln}^{{\sigma }_n}\lambda }{{\lambda }^3}}\nabla V\left(a\right)\left(-1+2{\alpha }^{p-1-\epsilon }{\lambda }^{-\epsilon (n-2)/2}g\left(a\right)\right)\hfill \\ \hfill & =O\left({\epsilon }^2+{\displaystyle \frac{1}{{\lambda }^{n-1}}}+{\displaystyle \frac{1}{{\lambda }^4}}+{\displaystyle \frac{|D^{3}g\left(a\right)|}{{\lambda }^3}}+{\displaystyle \frac{{|\nabla g\left(a\right)|}^2}{{\lambda }^2}}\right)+(\mathrm{if}n=5)O\left({\displaystyle \frac{ln\lambda }{{\lambda }^4}}\right).\hfill \end{array}$$

(116)

Putting Equation (114) in Equations (115) and (116), we obtain

$$\begin{array}{cc}\hfill & -E_{n}V\left(a\right){\displaystyle \frac{{ln}^{{\sigma }_n}\lambda }{{\lambda }^2}}+c_3\epsilon =O\left({\epsilon }^2+{\displaystyle \frac{1}{{\lambda }^{n-2}}}+{\displaystyle \frac{1}{{\lambda }^3}}+{\displaystyle \frac{|D^{2}g\left(a\right)|}{{\lambda }^2}}+{\displaystyle \frac{{|\nabla g\left(a\right)|}^2}{{\lambda }^2}}\right),\hfill \end{array}$$

(117)

$$\begin{array}{cc}\hfill & -{\displaystyle \frac{{\overline{c}}_5}{g\left(a\right)}}{\displaystyle \frac{\nabla g\left(a\right)}{\lambda }}+D_n{\displaystyle \frac{{ln}^{{\sigma }_n}\lambda }{{\lambda }^3}}\nabla V\left(a\right)\hfill \\ \hfill & =O\left({\epsilon }^2+{\displaystyle \frac{1}{{\lambda }^{n-1}}}+{\displaystyle \frac{1}{{\lambda }^4}}+{\displaystyle \frac{|D^{3}g\left(a\right)|}{{\lambda }^3}}+{\displaystyle \frac{{|\nabla g\left(a\right)|}^2}{{\lambda }^2}}\right)+(ifn=5)O\left({\displaystyle \frac{ln\lambda }{{\lambda }^4}}\right).\hfill \end{array}$$

(118)

Using Equation (117), we obtain

$$\epsilon \le c{\displaystyle \frac{{ln}^{{\sigma }_n}\lambda }{{\lambda }^2}}.$$

(119)

Putting Equation (119) in Equation (118), we obtain

$${\displaystyle \frac{\nabla g\left(a\right)}{\lambda }}=O\left({\displaystyle \frac{{ln}^{{\sigma }_n}\lambda }{{\lambda }^3}}\right)$$

(120)

which implies that a converges to a critical point of g.

Now, using the assumption Equation (2) on the critical points on g, we deduce that

$${\displaystyle \frac{|D^{2}g\left(a\right)|}{{\lambda }^2}}+{\displaystyle \frac{{|\nabla g\left(a\right)|}^2}{{\lambda }^2}}=o\left({\displaystyle \frac{1}{{\lambda }^2}}\right)$$

and therefore, from Equation (117), we derive that

$$E_{n}V\left(a\right){\displaystyle \frac{{ln}^{{\sigma }_n}\lambda }{{\lambda }^2}}=c_3\epsilon (1+o\left(1\right)).$$

(121)

This completes the proof of the first part of Theorem 1.

Concerning the second part, recall that a converges to a critical point y of g and, by assumption, we know that

$${\beta }_1{|a-y|}^{\gamma -1}\le |\nabla g\left(a\right)|\le {\beta }_2{|a-y|}^{\gamma -1},$$

(122)

where ${\beta }_1$ and ${\beta }_2$ are two positive constants. Thus, combining Equations (120)–(122), we deduce that

$${\epsilon }^{-1}{|a-y|}^{\gamma -1}\le c.$$

Furthermore, using Equations (120) and (121), Equation (118) implies that

$${\displaystyle \frac{D_n{c}_3}{E_{n}V\left(a\right)}}\epsilon (1+o\left(1\right))\nabla V\left(a\right)={\displaystyle \frac{{\overline{c}}_5}{g\left(a\right)}}\nabla g\left(a\right)+o\left(\epsilon \right).$$

(123)

Observe the following:

• If ${\epsilon }^{-1}{|a-y|}^{\gamma -1}$ goes to zero, then Equation (122) implies that $|\nabla g(a)|=o\left(\epsilon \right)$ and therefore, using Equation (123), we deduce that $|\nabla V(a)|$ goes to zero which implies that y is a critical point of V.
• If ${\epsilon }^{-1}{|a-y|}^{\gamma -1}\ge \beta$, Equation (122) implies that $|\nabla g(a)|\ge c\epsilon$, and therefore, using Equation (123), we deduce that $|\nabla V(a)|\ge c>0$.

This completes the proof of Theorem 1. □

5. Construction of Interior Blowing up Solutions with Isolated Bubbles

In this section, we focus on the construction of interior blowing-up solutions with isolated bubbles, thereby proving Theorems 2–6. We begin with the case where the solutions concentrate around critical points of g that are not critical points of V.

5.1. Around Critical Points of g That Are Not Critical Points of V

This subsection is dedicated to the proof of Theorems 2 and 3. We begin by proving the first theorem. To this aim, let $y_1,\cdots ,y_N$ be critical points of g such that $\nabla V\left(y_i\right)\ne 0$ and, near each one, the function g satisfies Equation (2) with ${\gamma }_i>3$. Assume further that, near each one, Equation (3) holds. Let ${\xi }_i$ be a solution of the equation

$$\nabla h\left({\xi }_i\right)={\displaystyle \frac{g\left(y_i\right)}{{\overline{c}}_5}}{\displaystyle \frac{c_3{D}_n}{E_{n}V\left(y_i\right)}}\nabla V\left(y_i\right).$$

(124)

We adopt the proof strategy from [37]. To proceed, let us define the following set:

$$\begin{array}{cc}\hfill {\mathbb{M}}_1\left(\epsilon \right)=& \left\{\right(\alpha ,\lambda ,a,v)\in {\left({\mathbb{R}}_{+}\right)}^N\times {\left({\mathbb{R}}_{+}\right)}^N\times {\Omega }^N\times H^1(\Omega ):|{\alpha }_i^{{\displaystyle \frac{4}{n-2}}}g\left(a_i\right)-1|<\epsilon {ln}^2\epsilon ,\hfill \\ \hfill & {\displaystyle \frac{1}{c}}<{\displaystyle \frac{{\lambda }_i^2\epsilon }{{ln}^{{\sigma }_n}{\lambda }_i}}<c,|a_i-y_i-{\epsilon }^{1/({\gamma }_i-1)}{\xi }_i|<{\mu }_0{\epsilon }^{1/({\gamma }_i-1)}\forall i\in \{1,\cdots ,N\}\hfill \\ \hfill & ,v\in E_{a,\lambda }\mathrm{and}\parallel v\parallel <\sqrt{\epsilon }\},\hfill \end{array}$$

(125)

where ${\mu }_0$ is small constant, c is a positive constant, $E_{a,\lambda }$ is defined by Equation (17), ${\sigma }_n=0$ if $n\ge 5$, and ${\sigma }_4=1$. We also define the following function:

$${\theta }_{\epsilon }:{\mathbb{M}}_1\left(\epsilon \right)\to \mathbb{R},(\alpha ,\lambda ,a,v)\mapsto {\theta }_{\epsilon }(\alpha ,\lambda ,a,v)=I_{\epsilon }\left(\sum _{i=1}^N{\alpha }_i{\phi }_{a_i,{\lambda }_i}+v\right).$$

Since the variable $v\in E_{a,\lambda }$, the Euler–Lagrange multiplier theorem implies that the following proposition holds:

Proposition 10. 

$(\alpha ,\lambda ,a,v)\in {\mathbb{M}}_1\left(\epsilon \right)$ is a critical point of ${\theta }_{\epsilon }$ if and only if $u={\sum }_{i=1}^N{\alpha }_i{\phi }_{a_i,{\lambda }_i}+v$ is a critical point of $I_{\epsilon }$, that is, if and only if there exists $(A,B,C)\in {\mathbb{R}}^N\times {\mathbb{R}}^N\times {\left({\mathbb{R}}^n\right)}^N$ such that the following system holds

$$\begin{array}{ccc}\hfill & {\displaystyle \frac{\partial {\theta }_{\epsilon }}{\partial {\alpha }_i}}(\alpha ,\lambda ,a,v)=0\hfill & \hfill \forall i\in \{1,\cdots ,N\},\end{array}$$

(126)

$$\begin{array}{ccc}\hfill & {\displaystyle \frac{\partial {\theta }_{\epsilon }}{\partial {\lambda }_i}}(\alpha ,\lambda ,a,v)=B_i{\int }_{\Omega }\nabla v{\lambda }_i{\displaystyle \frac{{\partial }^2{\phi }_{a_i,{\lambda }_i}}{\partial {\lambda }_i^2}}+\sum _{j=1}^n{C}_{ij}{\int }_{\Omega }\nabla v{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{{\partial }^2{\phi }_{a_i,{\lambda }_i}}{\partial {\lambda }_i\partial a_i^j}}\hfill & \hfill \forall i\in \{1,\cdots ,N\},\end{array}$$

(127)

$$\begin{array}{ccc}\hfill & {\displaystyle \frac{\partial {\theta }_{\epsilon }}{\partial a_i}}(\alpha ,\lambda ,a,v)=B_i{\int }_{\Omega }\nabla v{\lambda }_i{\displaystyle \frac{{\partial }^2{\phi }_{a_i,{\lambda }_i}}{\partial {\lambda }_i\partial a_i}}+\sum _{j=1}^n{C}_{ij}{\int }_{\Omega }\nabla v{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{{\partial }^2{\phi }_{a_i,{\lambda }_i}}{\partial a_i^j\partial a_i}}\hfill & \hfill \forall i\in \{1,\cdots ,N\},\end{array}$$

(128)

$$\begin{array}{ccc}\hfill & {\displaystyle \frac{\partial {\theta }_{\epsilon }}{\partial v}}(\alpha ,\lambda ,a,v)=\sum _{i=1}^N\left(A_i{\displaystyle \frac{\partial f_{i,1}}{\partial v}}+B_i{\displaystyle \frac{\partial f_{i,2}}{\partial v}}+\sum _{j=1}^n{C}_{ij}{\displaystyle \frac{\partial f_{i,j+2}}{\partial v}}\right),\hfill & \hfill \end{array}$$

(129)

where the image of $(\alpha ,\lambda ,a,v)$ by the functions appearing in Equation (129) is defined by

$$f_{i,1}={\int }_{\Omega }\nabla v\nabla {\phi }_{a_i,{\lambda }_i},f_{i,2}={\lambda }_i{\int }_{\Omega }\nabla v\nabla {\displaystyle \frac{\partial {\phi }_{a_i,{\lambda }_i}}{\partial {\lambda }_i}},f_{i,j+2}={\displaystyle \frac{1}{{\lambda }_i}}{\int }_{\Omega }\nabla v\nabla {\displaystyle \frac{\partial {\phi }_{a_i,{\lambda }_i}}{\partial a_i^j}}.$$

The proof of Theorem 2 will be performed through a careful analysis of the previous system on ${\mathbb{M}}_1\left(\epsilon \right)$. Observe that ${\overline{v}}_{\epsilon }$, defined in Proposition 2, satisfies Equation (129). In the sequel, we will write $v_{\epsilon }$ instead of ${\overline{v}}_{\epsilon }$. Taking $(\alpha ,\lambda ,a,0)\in {\mathbb{M}}_1\left(\epsilon \right)$, we see that $u_{\epsilon }={\sum }_{i=1}^N{\alpha }_i{\phi }_{a_i,{\lambda }_i}+v_{\epsilon }$ is a critical point of $I_{\epsilon }$ if and only if $(\alpha ,\lambda ,a)$ satisfies the following system (for $1\le i\le N$):

$$\begin{array}{cc}\hfill & \left(E_{{\alpha }_i}\right)⟨I_{\epsilon }^{\prime }\left(u_{\epsilon }\right),{\phi }_{a_i,{\lambda }_i}⟩=0,\hfill \end{array}$$

(130)

$$\begin{array}{cc}\hfill & \left(E_{{\lambda }_i}\right)⟨I_{\epsilon }^{\prime }\left(u_{\epsilon }\right),{\alpha }_i{\displaystyle \frac{\partial {\phi }_{a_i,{\lambda }_i}}{\partial {\lambda }_i}}⟩=B_i{\int }_{\Omega }\nabla v{\lambda }_i{\displaystyle \frac{{\partial }^2{\phi }_{a_i,{\lambda }_i}}{\partial {\lambda }_i^2}}+\sum _{j=1}^n{C}_{ij}{\int }_{\Omega }\nabla v{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{{\partial }^2{\phi }_{a_i,{\lambda }_i}}{\partial {\lambda }_i\partial a_i^j}}\hfill \end{array}$$

(131)

$$\begin{array}{cc}\hfill & \left(E_{a_i}\right)⟨I_{\epsilon }^{\prime }\left(u_{\epsilon }\right),{\alpha }_i{\displaystyle \frac{\partial {\phi }_{a_i,{\lambda }_i}}{\partial a_i}}⟩=B_i{\int }_{\Omega }\nabla v{ł}_i{\displaystyle \frac{{\partial }^2{\phi }_{a_i,{\lambda }_i}}{\partial {\lambda }_i\partial a_i}}+\sum _{j=1}^n{C}_{ij}{\int }_{\Omega }\nabla v{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{{\partial }^2{\phi }_{a_i,{\lambda }_i}}{\partial a_i^j\partial a_i}}\hfill \end{array}$$

(132)

Next, we present the following estimates, which directly follow from Propositions 2, 7–9.

Lemma 5.

The function $v_{\epsilon }$ found in Proposition 2 and the quantities $R_6$, $R_4$ and $R_5$ defined in Propositions 7–9 respectively satisfy

$$\begin{array}{cc}& \parallel v_{\epsilon }\parallel \le c\epsilon ,R_6\le c\epsilon ,R_4\le {\epsilon }^{1+(\gamma -2)/(\gamma -1)}+\left(\epsilon {|ln\epsilon |}^{-1}ifn=4;{\epsilon }^{3/2}ifn\ge 5\right),\hfill \\ \hfill & R_5\le c({(\epsilon {|ln\epsilon |}^{-1})}^{3/2}(ifn=4);{\epsilon }^2|ln\epsilon |+{\epsilon }^{{\displaystyle \frac{3}{2}}+{\displaystyle \frac{\gamma -3}{\gamma -1}}}(ifn=5);{\epsilon }^2+{\epsilon }^{{\displaystyle \frac{3}{2}}+{\displaystyle \frac{\gamma -3}{\gamma -1}}}(ifn\ge 6)).\hfill \end{array}$$

Lemma 6.

Let $(\alpha ,\lambda ,a,0)\in {\mathbb{M}}_1\left(\epsilon \right)$. Then, for ε small, the following estimates hold:

$$A_i=O\left(\epsilon {ln}^2\epsilon \right),B_i=O\left(\epsilon \right)and|C_{ij}|\le c{\epsilon }^{3/2}{|ln\epsilon |}^{-{\sigma }_n/2}\forall 1\le i\le N,\forall 1\le j\le n.$$

Next, we are going to study equations $\left(E_{{\alpha }_i}\right)$, $\left(E_{{\lambda }_i}\right)$, $\left(E_{a_i}\right)$. To obtain an easy system to solve, we perform the following change of variables

$${\beta }_i=1-{\alpha }_i^{p-1}g\left(y_i\right),{\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^2}}={\displaystyle \frac{c_3}{E_{n}V\left(y_i\right)}}\epsilon (1+{\wedge }_i),{\zeta }_i={\epsilon }^{-1/({\gamma }_i-1)}(a_i-y_i)-{\xi }_i.$$

(133)

Using this change of variables, we rewrite our system in the following simple form:

Lemma 7.

For ε small, the system of Equations (130)–(132) is equivalent to the following system (for $1\le i\le N$):

$$\left(\mathcal{S}\right)\left\{\begin{array}{c}{\beta }_i=O(\epsilon |ln\epsilon |),\hfill \\ {\wedge }_i=O({\epsilon }^{1/({\gamma }_i-1)}+{\epsilon }^{({\gamma }_i-2)/({\gamma }_i-1)})+O({|ln\epsilon |}^{-1}ifn=4;{\epsilon }^{1/2}ifn\ge 5),\hfill \\ D^{2}h\left({\xi }_i\right){\zeta }_i=O(|{\zeta }_i{|}^2)+o\left(1\right),\hfill \end{array}\right.$$

where the function h is introduced in Equation (2).

Proof. 

Using the fact that

$${\alpha }_i^{p-1-\epsilon }={\alpha }_i^{p-1}+O\left(\epsilon \right)\mathrm{and}{ł}_i^{-\epsilon (n-2)/2}=1+O(\epsilon ln{ł}_i),$$

we see that Equation (130) is equivalent to the first equation of the system $\left(S\right)$.

For the second equation of the system $\left(S\right)$, using Equation (131), Proposition 8, Lemmas 5 and 6, we obtain

$$\begin{array}{cc}\hfill c_3\epsilon (1+O\left({\beta }_i\right))& -E_{n}V\left(a_i\right){\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^2}}(1+O\left({\beta }_i\right))\hfill \\ \hfill & =O\left({\epsilon }^{1+({\gamma }_i-2)/({\gamma }_i-1)}\right)+\left\{\begin{array}{cc}O(\epsilon |ln\epsilon {|}^{-1})\hfill & \mathrm{if}n=4,\hfill \\ O\left({\epsilon }^{3/2}\right)\hfill & \mathrm{if}n\ge 5.\hfill \end{array}\right.\hfill \end{array}$$

Observe that

$$\begin{array}{c}\hfill \left\{\begin{array}{c}V\left(a_i\right)=V\left(y_i\right)+O(|a_i-y_i|)=V\left(y_i\right)+O({\epsilon }^{1/({\gamma }_i-1)},\hfill \\ \nabla V\left(a_i\right)=\nabla V\left(y_i\right)+O\left({\epsilon }^{1/({\gamma }_i-1)}\right).\hfill \end{array}\right.\end{array}$$

(134)

Hence, using the first equation of $\left(S\right)$ and Equation (133), we derive the second equation of the system $\left(S\right)$.

Finally, using Equation (132), Proposition 9, Lemmas 5 and 6, we obtain

$$-{\displaystyle \frac{1}{g\left(a_i\right)}}(1+O\left({\beta }_i\right)){\overline{c}}_5\nabla g\left(a_i\right)+D_n{\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^2}}\nabla V\left(a_i\right)(1+O\left({\beta }_i\right))=o\left(\epsilon \right).$$

Note that, using Equations (2), (133) and (134), and the first equation of $\left(S\right)$, the previous equation becomes

$$-\nabla h\left({\epsilon }^{-1/({\gamma }_i-1)}[a_i-y_i]\right)+{\displaystyle \frac{g\left(y_i\right)}{{\overline{c}}_5}}{\displaystyle \frac{c_3{D}_n}{E_{n}V\left(y_i\right)}}\nabla V\left(y_i\right)=o\left(1\right).$$

Now, using Equations (124) and (133) and expanding $\nabla h$ near ${\xi }_i$, we derive that

$$-D^{2}h\left({\xi }_i\right){\zeta }_i=O(|{\zeta }_i{|}^2)+o\left(1\right).$$

This ends the proof of Lemma 7. □

Now, we are ready to present the proof of Theorem 2.

Proof of Theorem 2. 

The system $\left(S\right)$, as stated in Lemma 7, can be expressed in an equivalent form:

$$\left\{\begin{array}{cc}{\beta }_i=U_{1,i}(\epsilon ,\beta ,\wedge ,\zeta )\hfill & \forall 1\le i\le N,\hfill \\ {\wedge }_i=U_{2,i}(\epsilon ,\beta ,\wedge ,\zeta )\hfill & \forall 1\le i\le N,\hfill \\ D^{2}h\left({\xi }_i\right){\zeta }_i=U_{3,i}(\epsilon ,\beta ,\wedge ,{\zeta }_i)\hfill & \forall 1\le i\le N,\hfill \end{array}\right.$$

with

$$\left\{\begin{array}{c}{U}_{1,i}(\epsilon ,\beta ,\wedge ,\zeta )=O(\epsilon |ln\epsilon |)\hfill \\ U_{2,i}(\epsilon ,\beta ,\wedge ,\zeta )=O({\epsilon }^{1/({\gamma }_i-1)}+{\epsilon }^{({\gamma }_i-2)/({\gamma }_i-1)})+{O(|ln\epsilon |}^{-1}\mathrm{if}n=4;{\epsilon }^{1/2}\mathrm{if}n\ge 5)\hfill \\ U_{3,i}(\epsilon ,\beta ,\wedge ,\zeta )=O(|{\zeta }_i{|}^2)+o\left(1\right),\hfill \end{array}\right.$$

where $\beta =({\beta }_1,\cdots ,{\beta }_N)$, $\wedge =({\wedge }_1,\cdots ,{\wedge }_N)$ and $\zeta =({\zeta }_1,\cdots ,{\zeta }_N)$.

By defining the linear map

$$\begin{array}{cc}\hfill & l:{\mathbb{R}}^N\times {\mathbb{R}}^N\times {\left({\mathbb{R}}^n\right)}^N\to {\mathbb{R}}^N\times {\mathbb{R}}^N\times {\left({\mathbb{R}}^n\right)}^N\hfill \\ \hfill & (\beta ,\wedge ,\zeta )\mapsto (\beta ,\wedge ,D^{2}h\left({\xi }_1\right)({\zeta }_1,.),\cdots ,D^{2}h\left({\xi }_N\right)({\zeta }_N,.)),\hfill \end{array}$$

we observe that the system is equivalent to

$$l(\beta ,\wedge ,\zeta )=(U_1(\epsilon ,\beta ,\wedge ,\zeta ),U_2(\epsilon ,\beta ,\wedge ,\zeta ),U_3(\epsilon ,\beta ,\wedge ,\zeta )),$$

(135)

where

$$\begin{array}{cc}\hfill & U_1(\epsilon ,\beta ,\wedge ,\zeta )=(U_{1,1},\cdots ,U_{1,N}),U_2(\epsilon ,\beta ,\wedge ,\zeta )=(U_{2,1},\cdots ,U_{2,N}),\hfill \\ \hfill & \mathrm{and}{U}_3(\epsilon ,\beta ,\wedge ,\zeta )=(U_{3,1},\cdots ,U_{3,N}).\hfill \end{array}$$

Furthermore, given that ${\xi }_i$ is a non-degenerate critical point of h, we conclude that l is invertible. As a result, Equation (135) is equivalent to.

$$(\beta ,\wedge ,\zeta )=l^{-1}(U_1(\epsilon ,\beta ,\wedge ,\zeta ),U_2(\epsilon ,\beta ,\wedge ,\zeta ),U_3(\epsilon ,\beta ,\wedge ,\zeta )).$$

(136)

By selecting a small positive value for r and letting $(\beta ,\wedge ,\zeta )\in \overline{B(0,r)}$, we obtain

$$\begin{array}{c}\parallel l^{-1}(U_1(\epsilon ,\beta ,\wedge ,\zeta ),U_2(\epsilon ,\beta ,\wedge ,\zeta ),U_3(\epsilon ,\beta ,\wedge ,\zeta ))\parallel \hfill \\ \le \parallel l^{-1}\parallel \parallel U_1(\epsilon ,\beta ,\wedge ,\zeta ),U_2(\epsilon ,\beta ,\wedge ,\zeta ),U_3(\epsilon ,\beta ,\wedge ,\zeta ))\parallel \hfill \\ \le C\parallel l^{-1}\parallel ({|\zeta |}^{\gamma -1}+o\left(1\right))\hfill \\ \le C\parallel l^{-1}\parallel 2r^{\gamma -1}\hfill \end{array}$$

and hence if we choose $r\le {[1/(2C\parallel l^{-1}\parallel \left)\right]}^{1/(\gamma -2)}$, we note that the function

$$f:\overline{B(0,r)}\to \overline{B(0,r)},f(\beta ,\wedge ,\zeta )=l^{-1}(U_1(\epsilon ,\beta ,\wedge ,\zeta ),U_2(\epsilon ,\beta ,\wedge ,\zeta ),U_3(\epsilon ,\beta ,\wedge ,\zeta ))$$

(137)

is well defined and continuous. Therefore, by applying Brouwer’s fixed point theorem, we conclude that f has a fixed point. This implies that the system $\left(S\right)$ has at least one solution $({\beta }_{\epsilon },{\wedge }_{\epsilon },{\zeta }_{\epsilon })$ for sufficiently small $\epsilon$. To complete the proof of the theorem, it remains to show that the constructed function $u_{\epsilon }={\sum }_{i=1}^N{\alpha }_{i,\epsilon }{\phi }_{a_{i,\epsilon },{\lambda }_{i,\epsilon }}+v_{\epsilon }$ is positive. To this end, we first observe that since $|v_{\epsilon }|\to 0$, it follows that $u_{\epsilon }\not\equiv 0$ for small $\epsilon$. By the construction of $u_{\epsilon }$, it is evident that it satisfies the problem defined by

$$\left({\mathcal{P}}_{\epsilon }^{\prime }\right):\left\{\begin{array}{cc}-\Delta u_{\epsilon }+V\left(x\right)u_{\epsilon }=g{|u_{\epsilon }|}^{p-1-\epsilon }{u}_{\epsilon }\hfill & \mathrm{in}\Omega ,\hfill \\ {\displaystyle \frac{\partial u_{\epsilon }}{\partial \nu }}=0\hfill & \mathrm{on}\partial \Omega .\hfill \end{array}\right.$$

Multiplying $\left({\mathcal{P}}_{\epsilon }^{\prime }\right)$ by $u_{\epsilon }^{-}:=max(0,-u_{\epsilon })$ and integrating on Ω, we obtain

$${\int }_{\Omega }{|\nabla u_{\epsilon }^{-}|}^2+{\int }_{\Omega }V{\left(u_{\epsilon }^{-}\right)}^2={\int }_{\Omega }g{\left(u_{\epsilon }^{-}\right)}^{p+1-\epsilon }.$$

But we have

$${\int }_{\Omega }g{\left(u_{\epsilon }^{-}\right)}^{p+1-\epsilon }\le c{\left({\int }_{\Omega }{\left(u_{\epsilon }^{-}\right)}^{p+1}\right)}^{{\displaystyle \frac{p+1-\epsilon }{p+1}}}\le C{\left({\int }_{\Omega }{|\nabla u_{\epsilon }^{-}|}^2+{\int }_{\Omega }V{\left(u_{\epsilon }^{-}\right)}^2\right)}^{{\displaystyle \frac{p+1-\epsilon }{2}}}$$

which implies that

$$\mathrm{either}{u}_{\epsilon }^{-}\equiv 0\mathrm{or}{\int }_{\Omega }{\left(u_{\epsilon }^{-}\right)}^{p+1}\ge C^{-2(n-2)/(4-\epsilon (n-2\left)\right)}.$$

However, since $u_{\epsilon }^{-}\le |v_{\epsilon }|$ and $|v_{\epsilon }|L^{p+1}\to 0$ as $\epsilon \to 0$, we deduce that $u_{\epsilon }^{-}\equiv 0$ and $u_{\epsilon }\ge 0$. Therefore, by the maximum principle, $u_{\epsilon }$ must be positive. This completes the proof of the theorem. □

Proof of Theorem 3. 

Notice that, from Theorem 2, for each collection of points $(y_{i_1},\cdots ,y_{i_k})$, there exists a solution $u_{\epsilon }$ which blows up at the points $y_{i_j}$’s, for $j\in \{1,\cdots ,k\}$. Thus the number of solutions is at least equal to

$$\sum _{k=1}^{\mathsf{\ell }}{C}_{\mathsf{\ell }}^k=2^{\mathsf{\ell }}-1.$$

This completes the proof of Theorem 3. □

5.2. Around the Common Critical Points of g and V

In this subsection, we assume that $n\ge 5$, and we take $N\le m$, where m is the number of common critical points of V and g, and let $y_1$, …, $y_N$ be some distinct critical points of V and g. We assume that these points are non-degenerate for V and that g satisfies Equation (8) near each one. The proofs of Theorems 4, 6–8 closely follow the proof of Theorem 2 presented in the previous subsection. We will only outline the necessary changes in the remainder of the paper to prove these results. To proceed, let

$$\begin{array}{c}{\mathbb{M}}_2\left(\epsilon \right)=\{(\alpha ,\lambda ,a,v)\in {\left({\mathbb{R}}_{+}\right)}^N\times {\left({\mathbb{R}}_{+}\right)}^N\times {\Omega }^N\times H^1(\Omega ):|{\alpha }_i^{4/(n-2)}g\left(a_i\right)-1|<\epsilon {ln}^2\epsilon ,\hfill \\ {\displaystyle \frac{1}{M}}<{\lambda }_i^2\epsilon <M,|a_i-y_i|<M\eta \left(\epsilon \right)\forall 1\le i\le N,v\in E_{a,\lambda }\mathrm{and}\parallel v\parallel <\sqrt{\epsilon }\},\hfill \end{array}$$

(138)

where M is a large positive constant, $E_{a,\lambda }$ is defined by Equation (17) and

$$\eta \left(\epsilon \right):=\left(\sqrt{\epsilon }\mathrm{if}n\ge 6;\sqrt{\epsilon }|ln\epsilon |\mathrm{if}n=5\right).$$

First, we note that Proposition 10 also holds in this case. Second, as in the previous subsection, we need to find $(\alpha ,\lambda ,a,0)\in {\mathbb{M}}_2\left(\epsilon \right)$ satisfying Equations (130)–(132). Using Propositions 2, 7–9, we easily derive the following estimates.

Lemma 8.

For ε small, the following statements hold:

$$\parallel v_{\epsilon }\parallel \le c\epsilon ,R_6\left(i\right)\le c\epsilon ,R_4\left(i\right)\le c{\epsilon }^{3/2},R_5\left(i\right)\le c\left\{\begin{array}{c}{\epsilon }^{2}ifn\ge 6,\hfill \\ {\epsilon }^2|ln\epsilon |ifn=5,\hfill \end{array}\right.\forall i,$$

where $R_6\left(i\right)$, $R_4\left(i\right)$, and $R_5\left(i\right)$ are defined in Propositions 7, 8, and 9, respectively.

Now, by combining Propositions 7–9 and Lemmas A1, …, A4, we observe that the constants $A_i^{\prime }s$, $B_i^{\prime }s$, and $C_{ij}^{\prime }s$ appearing in equations $\left(E_{{\alpha }_i}\right)$, $\left(E_{{\lambda }_i}\right)$, and $\left(E_{a_i}\right)$ satisfy the following.

Lemma 9.

Let $(\alpha ,\lambda ,a,0)\in {\mathbb{M}}_2\left(\epsilon \right)$. Then, for ε small, the following statements hold:

$$A_i=O\left(\epsilon {ln}^2\epsilon \right),B_i=O\left(\epsilon \right)and{C}_{ij}=O\left({\epsilon }^{3/2}\right)\forall 1\le i\le N,\forall 1\le j\le n.$$

Next, for $1\le i\le N$, we consider the following change of variables

$${\beta }_i=1-{\alpha }_i^{p-1}g\left(y_i\right),{\displaystyle \frac{1}{{\lambda }_i^2}}={\displaystyle \frac{c_3}{E_n}}{\displaystyle \frac{1}{V\left(y_i\right)}}\epsilon (1+{\wedge }_i),{\zeta }_i={\epsilon }^{1/({\gamma }_i-2)}(a_i-y_i),$$

(139)

where $c_1$ and $c_2$ are defined in Proposition 8.

This change of variables enables us to express the system in the following simplified form.

Lemma 10.

For ε small, equations $\left(E_{{\alpha }_i}\right)$, $\left(E_{{\lambda }_i}\right)$, and $\left(E_{a_i}\right)$ are equivalent to the following system:

$$\left(S_1\right)\left\{\begin{array}{c}{\beta }_i=O(\epsilon |ln\epsilon |+{\epsilon }^{({\gamma }_i-1)/({\gamma }_i-2)}|{\zeta }_i{|}^{{\gamma }_i-1})\forall 1\le i\le N,\hfill \\ {\wedge }_i=O({\epsilon }^{2/({\gamma }_i-2)}|{\zeta }_i{|}^2+{\epsilon }^{1/2})\forall 1\le i\le N,\hfill \\ D^{2}V\left(y_i\right)({\zeta }_i,.)=O(|{\zeta }_i{|}^2{\epsilon }^{1/({\gamma }_i-2)}+|{\zeta }_i{|}^{{\gamma }_i-1}+R_7\left(\epsilon \right))\forall 1\le i\le N,\hfill \end{array}\right.$$

where

$$R_7\left(\epsilon \right):=\left\{\begin{array}{cc}{\epsilon }^{(\gamma -4)/\left[2\right(\gamma -2\left)\right]}\hfill & ifn\ge 6,\hfill \\ {\epsilon }^{(\gamma -4)/\left[2\right(\gamma -2\left)\right]}|ln\epsilon |\hfill & ifn=5,\hfill \end{array}\right.with\gamma :=min{\gamma }_i.$$

Proof. 

Using the fact that

$${\alpha }_i^{p-1-\epsilon }={\alpha }_i^{p-1}+O\left(\epsilon \right),g\left(a_i\right)=g\left(y_i\right)+O(|{\zeta }_i{|}^2)\mathrm{and}{\lambda }_i^{-\epsilon (n-2)/2}=1+O(\epsilon ln{\lambda }_i),$$

(140)

we see that $\left(E_{{\alpha }_i}\right)$ is equivalent to the first equation of the system $\left(S_1\right)$.

For the second equation $\left(E_{{\lambda }_i}\right)$, Proposition 8, Lemmas 8 and 9 imply that

$$(1-{\beta }_i)c_3\epsilon -{\displaystyle \frac{E_{n}V\left(a_i\right)}{{\lambda }_i^2}}(1-2{\beta }_i)=O\left({\epsilon }^{3/2}\right).$$

Writing

$$V\left(a_i\right)=V\left(y_i\right)+O({\epsilon }^{2/({\gamma }_i-2)}|{\zeta }_i{|}^2),$$

we obtain the second equation in the system $\left(S_1\right)$.

Lastly, writing

$$\left\{\begin{array}{c}\nabla V\left(a_i\right)=D^{2}V\left(y_i\right)({\epsilon }^{{\displaystyle \frac{1}{{\gamma }_i-2}}}{\zeta }_i,.)+O({\epsilon }^{{\displaystyle \frac{2}{{\gamma }_i-2}}}|{\zeta }_i{|}^2),\hfill \\ |\nabla g\left(a_i\right)|=O(|a_i-y_i{|}^{\gamma -1})=O({\epsilon }^{{\displaystyle \frac{{\gamma }_i-1}{{\gamma }_i-2}}}|{\zeta }_i{|}^{{\gamma }_i-1})\hfill \end{array}\right.$$

and using Proposition 9, Lemmas 8 and 9, we see that equation $\left(E_{a_i}\right)$ is equivalent to the third equation in the system $\left(S_1\right)$ which completes the proof of Lemma 10. □

Now, we are ready to present the proof of Theorems 4 and 5.

Proof of Theorem 4. 

Note that, as mentioned earlier, Proposition 10 holds in this case. Thus, the existence of a solution $u_{\epsilon }$ of the form of Equation (11) with the properties of Equations (12) and (13) is equivalent to solving the system Equations (126)–(129). Furthermore, by Proposition 2, taking $(\alpha ,\lambda ,a,0)\in {\mathbb{M}}_2\left(\epsilon \right)$, we see that $u_{\epsilon }={\sum }_{i=1}^N{\alpha }_i{\phi }_{a_i,{\lambda }_i}+v_{\epsilon }$ is a critical point of $I_{\epsilon }$ if and only if $(\alpha ,\lambda ,a)$ satisfies the system $\left(E_{{\alpha }_i}\right),\left(E_{{\lambda }_i}\right),\left(E_{a_i}\right)$ with $i\in \{1,\cdots ,N\}$ which is equivalent, by using Lemma 10, to solving the system $\left(S_1\right)$. Notice that the system $\left(S_1\right)$ and the system $\left(S\right)$ (introduced in the proof of Theorem 2) have the same form. At this stage, since the $y_i$’s are non-degenerate critical points of V, we can replicate the final part of the proof of Theorem 2 using V in place of h and $y_i$ in place of ${\xi }_i$. Thus, we conclude the existence of a positive critical point $u_{\epsilon }$ of the functional $I_{\epsilon }$, which completes the proof of Theorem 4. □

Proof of Theorem 5. 

The proof is identical to that of Theorem 3. □

5.3. Combination of the Two Previous Cases: Proof of Theorem 6

In this subsection, our goal is to construct interior blowing-up solutions that concentrate around two groups. One group consists of critical points of g that are not critical for V, while the other consists of common critical points of both g and V. Since the proof of the desired result is simply a straightforward combination of the two proofs presented in the previous two subsections, we will be brief in our presentation. To proceed, let

$$M\left(\epsilon \right)=\{({\alpha }^1,{\alpha }^2,{\lambda }^1,{\lambda }^2,a^1,a^2,v):({\alpha }^1,{\lambda }^1,a^1,v)\in {\mathbb{M}}_1\left(\epsilon \right)\mathrm{and}({\alpha }^2,{\lambda }^2,a^2,v)\in {\mathbb{M}}_2\left(\epsilon \right)\}.$$

Following the proof of Theorems 2 and 4 and applying the change of variables for $({\alpha }^1,{\lambda }^1,a^1)$, given in Equation (133) and for $({\alpha }^2,{\lambda }^2,a^2)$, given in Equation (139), we deduce the proof of this theorem in the same manner as before.

6. Construction of Interior Blowing up Solutions with Clustered Bubbles

This section is devoted to the proof of Theorems 7 and 8. We begin by proving Theorem 7. Let $n\ge 5$, and y be a common critical point of g and V. We assume that y is non-degenerate for V and that g satisfies Equation (8) with $\gamma >4+8/(n-4)$. Let $N\in \mathbb{N}$ with $N\ge 2$ and assume that the function $F_{N,y}$ has a non-degenerate critical point $({\overline{z}}_1,\cdots , {\overline{z}}_N)$ where $F_{N,y}$ is defined by Equation (9). The proof strategy for Theorem 7 follows the same approach as that of Theorem 4. We begin by introducing a neighborhood of the desired constructed solutions. Let

$$\begin{array}{c}\mathcal{V}(N,y,\epsilon )=\{(\alpha ,\lambda ,a,v)\in {\mathbb{R}}_{+}^N\times {\mathbb{R}}_{+}^N\times {\Omega }^N\times H^1(\Omega ):|{\alpha }_i^{4/(n-2)}g\left(a_i\right)-1|<\epsilon {ln}^2\epsilon ,\hfill \\ {\displaystyle \frac{1}{c}}<{\lambda }_i^2\epsilon <c,|a_i-y-{\epsilon }^{\mu }\sigma {\overline{z}}_i|\le {\epsilon }^{\mu }{\eta }_0\forall 1\le i\le N,v\in E_{a,\lambda }\parallel v\parallel <\sqrt{\epsilon }\},\hfill \end{array}$$

(141)

where $E_{a,\lambda }$ is defined by Equation (17),

$$\mu ={\displaystyle \frac{n-4}{2n}}\mathrm{and}\sigma ={\left({\displaystyle \frac{{\overline{c}}_1}{D_n}}\right)}^{1/n}{\left({\displaystyle \frac{c_3}{E_{n}V\left(y\right)}}\right)}^{\mu }.$$

(142)

Here, $c_1$, $D_n$, $c_3$ and $E_n$ denote the constants defined in Proposition 8 and 9.

Following the approach in the proof of Theorem 4, we reduce the problem to a finite-dimensional system. Using Proposition 2, we can achieve this reduction by identifying $\overline{v}$ that satisfies Equation (129). Consequently, we seek a solution $(\alpha ,\lambda ,a)\in V_{N,y,\epsilon }$ that satisfies the system defined by Equations (130)–(132), where $V_{N,y,\epsilon }$ is defined by

$$V_{N,y,\epsilon }=\{(\alpha ,\lambda ,a)\in {\left({\mathbb{R}}_{+}\right)}^N\times {\left({\mathbb{R}}_{+}\right)}^N\times {\Omega }^N:(\alpha ,\lambda ,a,0)\in \mathcal{V}(N,y,\epsilon )\}.$$

(143)

Following a similar approach as in the proof of Theorem 4, we introduce the change of variables as follows:

$${\beta }_i=1-{\alpha }_i^{p-1}g\left(y\right),{\displaystyle \frac{1}{{\lambda }_i^2}}={\displaystyle \frac{c_3}{E_n}}{\displaystyle \frac{1}{V\left(y\right)}}\epsilon (1+{\wedge }_i),(a_i-y)={\epsilon }^{\mu }\sigma ({\zeta }_i+{\overline{z}}_i),$$

(144)

where $1\le i\le N$, $c_3$, and $E_n$ are defined in Proposition 8. With these variable changes, it becomes straightforward to observe that

$$\begin{array}{cc}\hfill & {\epsilon }_{ij}={\displaystyle \frac{1}{({\lambda }_i{\lambda }_j|a_i-a_j{{|}^2)}^{\frac{n-2}{2}}}}(1+O({\epsilon }^{4/n}))=O\left({\epsilon }^{2\mu +1}\right)\hfill \end{array}$$

(145)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial a_i}}={\displaystyle \frac{(n-2){\epsilon }^{\mu +1}}{{\sigma }^{n-1}}}{\left({\displaystyle \frac{c_3}{E_{n}V(y)}}\right)}^{(n-2)/2}{\displaystyle \frac{({\zeta }_j+{\overline{z}}_j-{\zeta }_i-{\overline{z}}_i)}{{\left|{\zeta }_i+{\overline{z}}_i-{\zeta }_j-{\overline{z}}_j\right|}^n}}\left(1+R({\wedge }_i,{\wedge }_j\right)(1+O({\epsilon }^{{\displaystyle \frac{4}{n}}}))\hfill \\ \hfill & =O\left({\epsilon }^{\mu +1}\right),\hfill \end{array}$$

(146)

where

$$R({\wedge }_i,{\wedge }_j)={\displaystyle \frac{n-2}{4}}{\wedge }_i+{\displaystyle \frac{n-2}{4}}{\wedge }_j+O\left({\wedge }_i^2\right)+O\left({\wedge }_j^2\right).$$

(147)

Subsequently, by applying Propositions 2, 7–9, we can conclude that the following estimates are valid.

Lemma 11.

For ε small, the following statements hold:

$$\begin{array}{cc}\hfill & \parallel {\overline{v}}_{\epsilon }\parallel \le c\epsilon ,R_6\left(i\right)=O\left(\epsilon \right),R_4\left(i\right)=O\left({\epsilon }^{3/2}+{\epsilon }^{4\mu +1}\right),R_5\left(i\right)\le c{\epsilon }^2|ln\epsilon |+{\epsilon }^{2\mu +3/2},\hfill \end{array}$$

where $R_6$, $R_4$, and $R_5$ are defined in Propositions 7, 8 and 9, respectively.

At this point, following the argument used in the proof of Lemma 9, we deduce that the constants $A_i^{\prime }s$, $B_i^{\prime }s$, and $C_{ij}^{\prime }s$ that appear in equations $\left(E_{{\alpha }_i}\right)$, $\left(E_{{\lambda }_i}\right)$, and $\left(E_{a_i}\right)$ satisfy the following estimates:

Lemma 12.

Let $(\alpha ,\lambda ,a,0)\in \mathcal{V}(N,y,\epsilon )$. Then, for ε small, the following statements hold:

$$A_i=O\left(\epsilon {ln}^2\epsilon \right),B_i=O\left(\epsilon \right)and{C}_{ij}=O\left({\epsilon }^{\mu +3/2}\right)\forall 1\le i\le N,\forall 1\le j\le n.$$

Next, we seek to express the equations $\left(E_{{\alpha }_i}\right)$, $\left(E_{{\lambda }_i}\right)$, $\left(E_{a_i}\right)$ in a more simplified form.

Lemma 13.

For ε small, equations $\left(E_{{\alpha }_i}\right)$, $\left(E_{{\lambda }_i}\right)$, $\left(E_{a_i}\right)$ are equivalent to the following system:

$$\left(S^{\prime }\right)\left\{\begin{array}{c}{\beta }_i=O(\epsilon |ln\epsilon |)\forall 1\le i\le N\hfill \\ {\wedge }_i=O\left({\epsilon }^{1/2}+{\epsilon }^{2\mu }\right)\forall 1\le i\le N\hfill \\ D^{2}V\left(y\right)({\zeta }_i,.)-(n-2)\sum _{j\ne i}{\displaystyle \frac{{\zeta }_j-{\zeta }_i}{|{\overline{z}}_j-{\overline{z}}_i{|}^n}}+n(n-2)\sum _{j\ne i}⟨{\displaystyle \frac{{\overline{z}}_j-{\overline{z}}_i}{|{\overline{z}}_j-{\overline{z}}_i{|}^2}},{\zeta }_j-{\zeta }_i⟩{\displaystyle \frac{{\overline{z}}_j-{\overline{z}}_i}{|{\overline{z}}_j-{\overline{z}}_i{|}^n}}\hfill \\ -(n-2)\sum _{j\ne i}{\displaystyle \frac{{\overline{z}}_j-{\overline{z}}_i}{|{\overline{z}}_j-{\overline{z}}_i{|}^n}}\left({\displaystyle \frac{n-6}{4}}{\wedge }_i+{\displaystyle \frac{n-2}{4}}{\wedge }_j\right)=O\left(R_8\right)\forall 1\le i\le N,\hfill \end{array}\right.$$

where

$$R_8=\sum _{j=1}^N{\wedge }_j^2+\sum _{j=1}^N|{\zeta }_j{|}^2+{\epsilon }^{\mu }+{\epsilon }^{2/n}|ln\epsilon |+{\epsilon }^{\left[\gamma \right(n-4)-(4n-8\left)\right]/\left(2n\right)}.$$

Proof. 

First, using Proposition 7, Lemma 11, Equation (140) and the fact that $(\alpha ,\lambda ,a)\in V_{N,y,\epsilon }$, we see that $\left(E_{{\alpha }_i}\right)$ is equivalent to the first equation of the system $\left(S^{\prime }\right)$.

Second, using Lemma 12, we write

$$\begin{array}{cc}\hfill ⟨I_{\epsilon }^{\prime }\left(u\right),{\lambda }_i{\displaystyle \frac{\partial {\phi }_i}{\partial {\lambda }_i}}⟩& =O\left(|B_i|{\parallel \nabla v\parallel }_{L^2}{\parallel {\lambda }_i{\displaystyle \frac{{\partial }^2{\phi }_i}{\partial {\lambda }_i^2}}\parallel }_{L^2}+\sum _{j=1}^N|C_{ij}|{\parallel \nabla v\parallel }_{L^2}{\parallel {\displaystyle \frac{{\partial }^2{\phi }_i}{\partial {\lambda }_i\partial a_i^j}}\parallel }_{L^2}\right)\hfill \\ \hfill & =O\left({\displaystyle \frac{\epsilon \parallel v\parallel }{{\lambda }_i}}+{\epsilon }^{\mu +3/2}\parallel v\parallel \right)\hfill \\ \hfill & =O\left({\epsilon }^{3/2}\parallel v\parallel \right).\hfill \end{array}$$

Using Proposition 8 and Lemma 11, we obtain

$$c_3\epsilon -E_n{\displaystyle \frac{V\left(a_i\right)}{{\lambda }_i^2}}=O\left(\epsilon |{\beta }_i|+{\epsilon }^{3/2}+{\epsilon }^{2\mu +1}\right).$$

But we have

$$V\left(a_i\right)=V\left(y\right)+O\left(|a_i{-y|}^2\right)=V\left(y\right)+O\left({\epsilon }^{2\mu }\right).$$

This implies that $\left(E_{{\lambda }_i}\right)$ is equivalent to the second equation of the system $\left(S^{\prime }\right)$.

To deal with the third equation $\left(E_{a_i}\right)$, we write

$$\begin{array}{cc}\hfill ⟨I_{\epsilon }^{\prime }\left(u\right),{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\phi }_i}{\partial a_i}}⟩& =O\left(|B_i|{\parallel \nabla v\parallel }_{L^2}{\left|\right|{\displaystyle \frac{{\partial }^2{\phi }_i}{\partial {\lambda }_i\partial a_i}}\left|\right|}_{L^2}+\sum _{j=1}^N|C_{ij}|{\parallel \nabla v\parallel }_{L^2}{\left|\right|{\displaystyle \frac{1}{{\lambda }_i^2}}{\displaystyle \frac{{\partial }^2{\phi }_i}{\partial a_i\partial a_i^j}}\left|\right|}_{L^2}\right)\hfill \\ \hfill & =O\left(\epsilon \parallel v\parallel +{\epsilon }^{\mu +3/2}\parallel v\parallel \right)=O\left(\epsilon \parallel v\parallel \right).\hfill \end{array}$$

(148)

But, combining Equation (146), Proposition 9 and Lemma 11, Equation (148) becomes

$$\begin{array}{cc}\hfill {\displaystyle \frac{D_n}{{\lambda }_i^3}}\nabla V\left(a_i\right)& -c_5{\displaystyle \frac{{\epsilon }^{1+\mu }}{{\lambda }_i}}\sum _{j\ne i}{\displaystyle \frac{({\zeta }_j+{\overline{z}}_j-{\zeta }_i-{\overline{z}}_i)}{|{\zeta }_j+{\overline{z}}_j-{\zeta }_i-{\overline{z}}_i{|}^n}}\left(1+{\displaystyle \frac{n-2}{4}}{\wedge }_i+{\displaystyle \frac{n-2}{4}}{\wedge }_j+O\left({\wedge }_i^2+{\wedge }_j^2\right)\right)\hfill \\ \hfill & =O\left({\epsilon }^{\mu (\gamma -1)+1/2}+{\epsilon }^2|ln\epsilon |+{\epsilon }^{2\mu +3/2}\right),\hfill \end{array}$$

(149)

where

$$c_5={\displaystyle \frac{c_1(n-2)}{{\sigma }^{n-1}}}{\left({\displaystyle \frac{c_3}{E_{n}V\left(y\right)}}\right)}^{(n-2)/2}.$$

Observe that

$$\begin{array}{cc}\hfill \nabla V\left(a_i\right)& =D^{2}V\left(y\right)(a_i-y,.)+O\left(|a_i{-y|}^2\right)\hfill \\ \hfill & ={\epsilon }^{\mu }\sigma D^{2}V\left(y\right)({\zeta }_i+{\overline{z}}_i,.)+O\left({\epsilon }^{2\mu }\right).\hfill \end{array}$$

(150)

Combining Equation (149) and Equation (150), we obtain

$$\begin{array}{cc}\hfill D^{2}V\left(y\right)({\zeta }_i+{\overline{z}}_i,.)& -(n-2)\sum _{j\ne i}{\displaystyle \frac{({\zeta }_j+{\overline{z}}_j-{\zeta }_i-{\overline{z}}_i)}{|{\zeta }_j+{\overline{z}}_j-{\zeta }_i-{\overline{z}}_i{|}^n}}(1+{\displaystyle \frac{n-6}{4}}{\wedge }_i\hfill \\ \hfill & +{\displaystyle \frac{n-2}{4}}{\wedge }_j+O\left({\wedge }_i^2+{\wedge }_j^2\right))\hfill \\ \hfill & =O\left({\epsilon }^{\mu }+{\epsilon }^{\mu (\gamma -2)-1}+{\epsilon }^{{\displaystyle \frac{1}{2}}-\mu }|ln\epsilon |\right).\hfill \end{array}$$

(151)

But we have

$$D^{2}V\left(y\right)({\zeta }_i+{\overline{z}}_i,.)=D^{2}V\left(y\right)({\zeta }_i,.)+D^{2}V\left(y\right)({\overline{z}}_i,.)$$

(152)

and

$$\begin{array}{cc}\hfill & {\displaystyle \frac{({\zeta }_j+{\overline{z}}_j-{\zeta }_i-{\overline{z}}_i)}{|{\zeta }_j+{\overline{z}}_j-{\zeta }_i-{\overline{z}}_i{|}^n}}=\left({\displaystyle \frac{({\overline{z}}_j-{\overline{z}}_i)}{|{\overline{z}}_j-{\overline{z}}_i{|}^n}}+{\displaystyle \frac{({\zeta }_j-{\zeta }_i)}{|{\overline{z}}_j-{\overline{z}}_i{|}^n}}\right)\left[1-n⟨{\displaystyle \frac{({\overline{z}}_j-{\overline{z}}_i)}{|{\overline{z}}_j-{\overline{z}}_i{|}^2}},{\zeta }_j-{\zeta }_i⟩+O\left(\sum _{j=1}^N{|{\zeta }_j|}^2\right)\right]\hfill \\ \hfill & ={\displaystyle \frac{({\overline{z}}_j-{\overline{z}}_i)}{|{\overline{z}}_j-{\overline{z}}_i{|}^n}}-n⟨{\displaystyle \frac{({\overline{z}}_j-{\overline{z}}_i)}{|{\overline{z}}_j-{\overline{z}}_i{|}^2}},{\zeta }_j-{\zeta }_i⟩{\displaystyle \frac{({\overline{z}}_j-{\overline{z}}_i)}{|{\overline{z}}_j-{\overline{z}}_i{|}^n}}+{\displaystyle \frac{({\zeta }_j-{\zeta }_i)}{|{\overline{z}}_j-{\overline{z}}_i{|}^n}}+O\left(\sum _{j=1}^N{|{\zeta }_j|}^2\right).\hfill \end{array}$$

(153)

Combining Equation (151)–(153) and the fact that $({\overline{z}}_1,$…$,{\overline{z}}_N)$ is a critical point of $F_{N,y}$, we see that equation $\left(E_{a_i}\right)$ is equivalent to

$$\begin{array}{cc}\hfill \left(F_{a_i}^{\prime }\right)D^{2}V\left(y\right)({\zeta }_i,.)& -(n-2)\sum _{j\ne i}{\displaystyle \frac{{\zeta }_j-{\zeta }_i}{|{\overline{z}}_j-{\overline{z}}_i{|}^n}}+n(n-2)\sum _{j\ne i}⟨{\displaystyle \frac{{\overline{z}}_j-{\overline{z}}_i}{|{\overline{z}}_j-{\overline{z}}_i{|}^2}},{\zeta }_j-{\zeta }_i⟩{\displaystyle \frac{{\overline{z}}_j-{\overline{z}}_i}{|{\overline{z}}_j-{\overline{z}}_i{|}^n}}\hfill \\ \hfill & -(n-2)\sum _{j\ne i}{\displaystyle \frac{{\overline{z}}_j-{\overline{z}}_i}{|{\overline{z}}_j-{\overline{z}}_i{|}^n}}\left({\displaystyle \frac{n-6}{4}}{\wedge }_i+{\displaystyle \frac{n-2}{4}}{\wedge }_j\right)=O\left(R_8\right),\hfill \end{array}$$

which completes the proof of Lemma 13. □

At this point, we are prepared to prove the results concerning the construction of clustered bubbling solutions.

Proof of Theorem 7. 

Note that the system $\left(F_{a_1}^{\prime }\right)$, …, $\left(F_{a_N}^{\prime }\right)$ is equivalent to

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{D}^2{F}_{N,y}({\overline{z}}_1,...,{\overline{z}}_N)& ({\zeta }_1,...,{\zeta }_N)-\Gamma \hfill \\ \hfill & =O\left(\sum _{j=1}^N{\wedge }_j^2+\sum _{j=1}^N|{\zeta }_j{|}^2+{\epsilon }^{\mu }+{\epsilon }^{2/n}|ln\epsilon |+{\epsilon }^{\left[\gamma \right(n-4)-(4n-8\left)\right]/\left(2n\right)}\right),\hfill \end{array}$$

where $\Gamma :=({\Gamma }_1,\cdots ,{\Gamma }_N)$ defined by

$${\Gamma }_i:=(n-2)\sum _{j\ne i}{\displaystyle \frac{{\overline{z}}_j-{\overline{z}}_i}{|{\overline{z}}_j-{\overline{z}}_i{|}^n}}\left({\displaystyle \frac{n-6}{4}}{\wedge }_i+{\displaystyle \frac{n-2}{4}}{\wedge }_j\right).$$

As in the proof of Theorem 4, we define a linear map by taking the left-hand side of the system defined by $\left(F_{{\alpha }_i}^{\prime }\right)$, $\left(F_{{\lambda }_i}^{\prime }\right)$ and $\left(F_{a_i}^{\prime }\right)$. Since $({\overline{z}}_1,...,{\overline{z}}_N)$ is a non-degenerate critical point of $F_{N,y}$, we deduce that such a linear map is invertible and arguing as in the proof of Theorem 4, we derive that the system $\left(S^{\prime }\right)$ has a solution $({\beta }_{\epsilon },{\wedge }_{\epsilon },{\zeta }_{\epsilon })$ for $\epsilon$ small. This implies that $\left({\mathbf{P}}_{\epsilon }\right)$ admits a solution $u_{\epsilon ,y}={\sum }_{i=1}^N{\alpha }_{i,\epsilon }{\phi }_{a_{i,\epsilon },{\lambda }_{i,\epsilon }}+v_{\epsilon }.$ and by construction, Equation (10) is satisfied. The proof of Theorem 7 is thereby completed. □

Proof of Theorem 8. 

Let $n\ge 5$, $2\le m\in \mathbb{N}$, $N_1,\cdots ,N_m\in \mathbb{N}$ and let $y_1,\cdots ,y_m$ satisfy the assumptions stated in Theorem 7. For $k\le m$, we assume that $N_k\ge 2$ and the function $F_{N_k,y_k}$ has a non-degenerate critical point $({\overline{z}}_{k,1},\cdots ,{\overline{z}}_{k,N_k})$. We introduce the following set

$$\mathcal{V}(y_1,\cdots ,y_m,\epsilon )=\mathcal{V}(N_1,y_1,\epsilon )\times \cdots \times \mathcal{V}(N_m,y_m,\epsilon ).$$

We need to construct solution in the form

$$u_{\epsilon }=\sum _{k=1}^m\sum _{i=1}^{N_k}{\alpha }_{i,k}{\phi }_{a_{i,k},{\lambda }_{i,k}}+{\overline{v}}_{\epsilon }:=\sum _{k=1}^m\sum _{i=1}^{N_k}{\alpha }_{i,k}{\phi }_{i,k}+{\overline{v}}_{\epsilon }.$$

We remark that, for $a_i$, $a_j\in$ such that $|a_i-y_i|<<1$ and $|a_j-y_j|<<1$, we have $|a_i-a_j|\ge c>0$ and thus

$$\begin{array}{cc}\hfill & {\epsilon }_{ij}\le {\displaystyle \frac{c}{{\left({\lambda }_{i,\epsilon }{\lambda }_{j,\epsilon }\right)}^{(n-2)/2}}}=O\left({\epsilon }^{(n-2))/2}\right)=o(\epsilon |ln\epsilon |),\hfill \\ \hfill & {\lambda }_i\left|{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial {\lambda }_i}}\right|\le {\displaystyle \frac{c}{{\left({\lambda }_{i,\epsilon }{\lambda }_{j,\epsilon }\right)}^{(n-2)/2}}}\le c{\epsilon }^{(n-2))/2}\le c{\epsilon }^{3/2},\hfill \\ \hfill & {\displaystyle \frac{1}{{\lambda }_i}}\left|{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial a_i}}\right|\le {\displaystyle \frac{c}{{\lambda }_i^{n/2}{\lambda }_j^{(n-2)/2}}}\le c{\epsilon }^{(n-1)/2}\le c{\epsilon }^2.\hfill \end{array}$$

Therefore, Propositions 7–9 can be rewritten as

Proposition 11. 

Let $1\le k\le m$ and $1\le i\le N_k$. It holds that

$$\begin{array}{cc}\hfill & ⟨I_{\epsilon }^{\prime }\left(u_{\epsilon }\right),{\phi }_{k,i}⟩={\alpha }_{k,i}{S}_n(1-g\left(a_{k,i}\right){\alpha }_{k,i}^{p-1-\epsilon }{\lambda }_{k,i}^{-\epsilon (n-2)/2})+O\left(\epsilon \right),\hfill \\ \hfill & ⟨I_{\epsilon }^{\prime }\left(u\right),{\lambda }_{k,i}{\displaystyle \frac{\partial {\phi }_{k,i}}{\partial {\lambda }_{k,i}}}⟩={\alpha }_{k,i}^{p-\epsilon }{\lambda }_{k,i}^{{\displaystyle \frac{-\epsilon (n-2)}{2}}}{c}_3\epsilon g\left(a_{k,i}\right)-{\alpha }_{k,i}{E}_{n}V\left(a_{k,i}\right){\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_{k,i}}{{\lambda }_{k,i}^2}}(-1+2{\alpha }_{k,i}^{p-1-\epsilon }{\lambda }_{k,i}^{{\displaystyle \frac{-\epsilon (n-2)}{2}}}\hfill \\ \hfill & \times g\left(a_{k,i}\right))-c_1\sum _{j\ne i}{\alpha }_{k,j}{\lambda }_{k,i}{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial {\lambda }_{k,i}}}(-1+{\alpha }_{k,i}^{p-1-\epsilon }{\lambda }_{k,i}^{{\displaystyle \frac{-\epsilon (n-2)}{2}}}g\left(a_{k,i}\right)+{\alpha }_{k,j}^{p-1-\epsilon }{\lambda }_{k,j}^{{\displaystyle \frac{-\epsilon (n-2)}{2}}}g\left(a_{k,j}\right))\hfill \\ \hfill & +O({\epsilon }^{{\displaystyle \frac{3}{2}}}+{\epsilon }^{4\mu +1}),\hfill \\ \hfill & ⟨I_{\epsilon }^{\prime }\left(u\right),{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\phi }_i}{\partial a_{k,i}}}⟩=-{\alpha }_{k,i}^{p-\epsilon }{\lambda }_{k,i}^{{\displaystyle \frac{-\epsilon (n-2)}{2}}}{\overline{c}}_5{\displaystyle \frac{\nabla g\left(a_{k,i}\right)}{{\lambda }_{k,i}}}+{\alpha }_{k,i}{D}_n{\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_{k,i}}{{\lambda }_{k,i}^3}}\nabla V\left(a_{k,i}\right)(-1+2{\alpha }_{k,i}^{p-1-\epsilon }\hfill \\ \hfill & \times {\lambda }_{k,i}^{{\displaystyle \frac{-\epsilon (n-2)}{2}}}g\left(a_{k,i}\right))-c_1\sum _{j\ne i}{\alpha }_{k,j}{\displaystyle \frac{1}{{\lambda }_{k,i}}}{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial a_{k,i}}}[-1+{\alpha }_{k,i}^{p-1-\epsilon }{\lambda }_{k,i}^{-\epsilon (n-2)/2}g\left(a_{k,i}\right)\hfill \\ \hfill & +{\alpha }_{k,j}^{p-1-\epsilon }{\lambda }_{k,j}^{-\epsilon (n-2)/2}g\left(a_{k,j}\right)]+O({\epsilon }^2|ln\epsilon |+{\epsilon }^{2\mu +3/2}).\hfill \end{array}$$

This means that only the indices in the same bloc are involved in the expansions. That is the contribution of the other blocs are taken in the remainder term.

As in the proof of Theorem 7, we derive the following.

Lemma 14. 

For ε small, $u_{\epsilon }$ is a critical point of $I_{\epsilon }$ if and the only if the following system $(S_1^{\prime },\cdots ,S_m^{\prime })$ is satisfied where, for $1\le k\le m$ and $1\le i\le N$,

$$\left(S_k^{\prime }\right)\left\{\begin{array}{c}{\beta }_{k,i}=O(\epsilon |ln\epsilon |)\hfill \\ {\wedge }_{k,i}=O\left({\epsilon }^{1/2}+{\epsilon }^{2\mu }\right)\hfill \\ D^{2}V\left(y_i\right)({\zeta }_{k,i},.)-(n-2)\sum _{j\ne i}{\displaystyle \frac{{\zeta }_{k,j}-{\zeta }_{k,i}}{|{\overline{z}}_{k,j}-{\overline{z}}_{k,i}{|}^n}}+n(n-2)\sum _{j\ne i}⟨{\displaystyle \frac{{\overline{z}}_{k,j}-{\overline{z}}_{k,i}}{|{\overline{z}}_{k,j}-{\overline{z}}_{k,i}{|}^2}},{\zeta }_{k,j}-{\zeta }_{k,i}⟩\hfill \\ \times {\displaystyle \frac{{\overline{z}}_{k,j}-{\overline{z}}_{k,i}}{|{\overline{z}}_{k,j}-{\overline{z}}_{k,i}{|}^n}}-(n-2)\sum _{j\ne i}{\displaystyle \frac{{\overline{z}}_{k,j}-{\overline{z}}_{k,i}}{|{\overline{z}}_{k,j}-{\overline{z}}_{k,i}{|}^n}}\left({\displaystyle \frac{n-6}{4}}{\wedge }_{k,i}+{\displaystyle \frac{n-2}{4}}{\wedge }_{k,j}\right)=O\left(R_8\right),\hfill \end{array}\right.$$

where

$$R_8=\sum _{j=1}^N{\wedge }_{k,j}^2+\sum _{j=1}^N|{\zeta }_{k,j}{|}^2+{\epsilon }^{\mu }+{\epsilon }^{2/n}|ln\epsilon |+{\epsilon }^{\left[\gamma \right(n-4)-(4n-8\left)\right]/\left(2n\right)},$$

$$\left\{\begin{array}{c}{\beta }_{k,i}=1-{\alpha }_{k,i}^{p-1}g\left(y_k\right),{\displaystyle \frac{1}{{\lambda }_{k,i}^2}}={\displaystyle \frac{c_3}{E_n}}{\displaystyle \frac{1}{V\left(y_k\right)}}\epsilon (1+{\wedge }_{k,i}),1\le i\le N_k,\hfill \\ (a_{k,i}-y_k)={\epsilon }^{\mu }{\sigma }_k({\zeta }_{k,i}+{\overline{z}}_{k,i})1\le i\le N_k,\hfill \end{array}\right.$$

(154)

${\sigma }_k$ is defined by Equation (142) with $y_k$ instead of y and μ is defined in Equation (142).

The existence of a solution for the system $(S_1^{\prime },\cdots ,S_m^{\prime })$ follows as in the previous proofs by using the fact that for each $1\le k\le m$, the point $({\overline{z}}_{k,1},\cdots ,{\overline{z}}_{k,N_k})$ is a non-degenerate critical point of $F_{N_k,y_k}$. Following the end of the proof of Theorem 2, the constructed solution is positive. Hence the proof of Theorem 8 is completed. □

7. Conclusions

In this paper, we studied a nonlinear elliptic equation with zero Neumann boundary conditions. Under a certain flatness condition, we provided a detailed characterization of the single interior blow-up scenario for solutions that weakly converge to zero. Additionally, we constructed interior multipeaked solutions, featuring both isolated and clustered bubbles. In particular, we have highlighted that the key difference between the case where the critical points of the function appearing in front of the nonlinear term are non-degenerate, and our case, where the critical points are degenerate, is that interior blowing-up solutions with clustered bubbles do, in fact, exist. While this work focused on the case where the exponent is slightly subcritical and on solutions that weakly converge to zero, it also opens several promising avenues for future research and unresolved questions. The following are examples:

(i) 

Impact of the Nonlinear Exponent: This work examines the case of a slightly subcritical exponent in the context of Sobolev embedding. Future research could explore the problem for exponents that are slightly supercritical, that is, when $\epsilon <0$ but close to zero.

(ii) 

Limit of the solutions: This paper focuses on solutions that weakly converge to zero. What occurs in the case where the solutions of the problem have a non-zero weak limit?