Interior Bubbling Solutions with Residual Mass for a Neumann Problem with Slightly Subcritical Exponent

Abstract

In this paper, we consider the Neumann elliptic problem $\left({\mathcal{I}}_{\epsilon }\right)$: $-\Delta u+V\left(x\right)u=u^{p-\epsilon }$, $u>0$ in $\mathsf{\Omega }$, $\partial u/\partial \nu =0$ on $\partial \mathsf{\Omega }$, where $\mathsf{\Omega }$ is a bounded smooth domain in ${\mathbb{R}}^n$, $n\ge 3$, $p+1=2n/(n-2)$ is the critical Sobolev exponent, $\epsilon$ is a small positive parameter and V is a smooth positive function on $\overline{\mathsf{\Omega }}$. First, in contrast to the case where solutions converge weakly to zero, we rule out the existence of interior bubbling solutions with a non-zero weak limit in small-dimensional domains. Second, for large-dimensional domains, we construct both simple and non-simple interior bubbling solutions with residual mass. This construction allows us to establish the multiplicity results. The proofs of these results are based on refined asymptotic expansions of the gradient of the associated Euler–Lagrange functional.

1. Introduction

In this paper, we are interested in the following Neumann problem:

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

where $\mathsf{\Omega }$ is an open bounded smooth subset of ${\mathbb{R}}^n$, $n\ge 3$, $1<q<\infty$ and V is a smooth positive function.

Such a problem arises in various areas of applied sciences, for example, stationary waves for nonlinear Schrödinger equations, see [1,2,3], the Keller–Segel model in chemotaxis [4,5] and the Gierer–Meinhardt model for biological pattern formation [6].

A difficult feature of solutions to $\left({\mathcal{P}}_{V,q}\right)$ is that they present concentration phenomena as the real q approaches some critical exponent. In the special case where $V\left(x\right)=\mu >0$, that is, problem $\left({\mathcal{P}}_{\mu ,q}\right)$, a considerable amount of works have been conducted. In the subcritical case, that is, $q<{\displaystyle \frac{n+2}{n-2}}$, $\left({\mathcal{P}}_{\mu ,q}\right)$ has no non-constant solutions for $\mu$ small, while non-constant solutions exist for $\mu$ large, which blow up when $\mu$ tends to infinity, at one or more points [7] (see also the review in [8]). When q is critical, that is, $q={\displaystyle \frac{n+2}{n-2}}$, the non-compactness of the embedding $H^1\left(\mathsf{\Omega }\right)\hookrightarrow L^{q+1}\left(\mathsf{\Omega }\right)$ makes the problem more difficult. $\left({\mathcal{P}}_{\mu ,q}\right)$ has only constant solutions for $\mu$ small, $n=3$, and $\mathsf{\Omega }$ is convex [9]. On the contrary, it admits non-constant solutions for $\mu$ small and $n\in \{4,5,6\}$ [10,11,12]. When $\mu$ is large, $\left({\mathcal{P}}_{\mu ,q}\right)$ also admits solutions [13,14,15,16]. However, in contrast with the subcritical situation, $\left({\mathcal{P}}_{\mu ,q}\right)$ has no solutions which blow up only in interior points [17]. In [18,19], the authors are interested in problem $\left({\mathcal{P}}_{\mu ,q}\right)$ when $\mu$ is fixed and q is almost critical, $q=(n+2)/(n-2)\pm \epsilon$ and $\epsilon$ is a small positive real. They constructed solutions which blow up, as $\epsilon$ goes to zero, at a single interior point when $n=3$. They also construct single-boundary blow-up solutions for $n\ge 4$. It is worth observing that unlike dimension 3, problem $\left({\mathcal{P}}_{\mu ,(n+2)/(n-2)+\epsilon }\right)$ does not admit solutions which blow up at only interior points when $n\ge 4$ [20]. Recently, the authors in [21] studied the case where the function V is a smooth non-constant positive function and q is a slightly subcritical exponent, that is, problem $\left({\mathcal{P}}_{V,(n+2)/(n-2)-\epsilon }\right)$ with $\epsilon$ is a small positive parameter. They showed, for $n\ge 6$, the existence of interior bubbling solutions (with isolated or clustered bubbles), which converge weakly to zero. Their results were generalized to small dimensions [22]. By interior bubbling solutions, we refer to solutions whose local maxima occur at points that converge to interior points. In other words, these solutions remain bounded in a neighborhood of the boundary. On the other hand, clustered bubbles refer to a sum of bubbles with comparable concentration rates, whose centers converge to a common point y. In this setting, it holds that

$$\underset{r\to 0}{lim}\underset{\epsilon \to 0}{lim}{\int }_{B(y,r)}{u}_{\epsilon }^{2n/(n-2)}\left(x\right)dx=k{S}_{n}withk\ge 2and{S}_n:={\int }_{{\mathbb{R}}^n}{\delta }_{0,1}^{2n/(n-2)}.$$

(1)

Here and in the sequel, ${\delta }_{(a,\lambda )}$ denotes the so-called bubbles, defined by

$${\delta }_{(a,\lambda )}\left(x\right)=c_0{\displaystyle \frac{{\lambda }^{(n-2)/2}}{(1+{\lambda }^2{|x-a|}^2{)}^{(n-2)/2}}},\lambda >0,a,x\in {\mathbb{R}}^n,c_0={\left(n(n-2)\right)}^{{\displaystyle \frac{n-2}{4}}},$$

(2)

which are the only solutions to problem [23]

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

By isolated bubbles, we refer to a finite sum of bubbles whose concentration points $a_i^{\prime }s$ are uniformly separated, i.e., there exists a constant $c>0$ such that the $a_i^{\prime }s$ satisfy $|a_i-a_j|\ge c$ for all $j\ne i$. In this case, we have

$$\underset{r\to 0}{lim}\underset{\epsilon \to 0}{lim}{\int }_{B(y,r)}{u}_{\epsilon }^{2n/(n-2)}\left(x\right)dx=S_n.$$

(3)

Considering the above results, it is natural to ask: what happens in the scenario where solutions of the problem $\left({\mathcal{P}}_{V,(n+2)/(n-2)-\epsilon }\right)$ have a non-zero weak limit? In particular, do interior blowing-up solutions (with isolated or clustered bubbles) still exist? Notice that the case of bubbling solutions with a non-zero weak limit presents a challenging and intriguing mathematical problem. It corresponds to situations in which the solution, as it blows up, splits into two distinct components: one remains within a compact set and eventually converges to a genuine solution of the original problem, while the other escapes to infinity. This type of phenomenon arises in various geometric contexts—for example, in the well-known harmonic map problem [24], as well as in relation to the Yang–Mills equations in gauge theory [25]. A similar behavior is also observed in the Toda system, where the converging component satisfies what is known as the shadow system [26].

Motivated by the above intricate questions, we undertake in this paper the problem of the existence and non-existence of interior blowing-up solutions of $\left({\mathcal{P}}_{V,(n+2)/(n-2)-\epsilon }\right)$, which converge weakly to a non-zero function as $\epsilon$ goes to zero. First, contrary to the results of [22], i.e., in the case where the solutions of $\left({\mathcal{P}}_{V,(n+2)/(n-2)-\epsilon }\right)$ converge weakly to zero, we exclude the existence of interior blowing-up solutions having a non-zero weak limit in small-dimensional domains. Second, in the large-dimensional cases, we prove that $\left({\mathcal{P}}_{V,(n+2)/(n-2)-\epsilon }\right)$ admits interior blowing-up solutions (with isolated or clustered bubbles) having a non-zero weak limit.

The remainder of the paper is organized as follows: Section 2 presents the statement of our main results. In Section 3, we analyze the infinite-dimensional component of the solutions. Section 4 is dedicated to the expansion of the gradient of the functional associated with the problem. Section 5 proves the non-existence result in small dimensions (Theorem 1 below). In Section 6, we construct interior blowing-up solutions with isolated bubbles (proofs of Theorems 2 and 3 below), while Section 7 focuses on the construction of interior blowing-up solutions with clustered bubbles (proofs of Theorems 4 and 5 below). Finally, Section 8 presents the paper’s conclusion and outlines potential directions for future research.

2. Main Results

To state our results, we need to introduce some notation. Throughout this paper, we consider the following nonlinear elliptic equation with subcritical nonlinearity:

$$\left({\mathcal{I}}_{\epsilon }\right):\left\{\begin{array}{cc}-\Delta u+Vu=u^{p-\epsilon },u>0\hfill & \mathrm{in}\mathsf{\Omega },\hfill \\ {\displaystyle \frac{\partial u}{\partial \nu }}=0\hfill & \mathrm{on}\partial \mathsf{\Omega },\hfill \end{array}\right.$$

where $\mathsf{\Omega }$ is an open-bounded smooth subset of ${\mathbb{R}}^n$, $n\ge 3$, $\epsilon >0$ small, $p+1=\left(2n\right)/(n-2)$ is the limiting Sobolev exponent for the embedding $H^1\left(\mathsf{\Omega }\right)\hookrightarrow L^q\left(\mathsf{\Omega }\right)$, and V is a smooth positive function.

Problem $\left({\mathcal{I}}_{\epsilon }\right)$ has a variational structure. Solutions of $\left({\mathcal{I}}_{\epsilon }\right)$ are the positive critical points of the functional

$$J_{\epsilon }\left(u\right):={\displaystyle \frac{1}{2}}{\int }_{\mathsf{\Omega }}{\left(\right|\nabla u|}^2+V\left(x\right)u^2)-{\displaystyle \frac{1}{p+1-\epsilon }}{\int }_{\mathsf{\Omega }}{\left|u\right|}^{p+1-\epsilon },$$

(4)

defined on $H^1\left(\mathsf{\Omega }\right)$ equipped with the norm $\parallel .\parallel$, and its corresponding inner product is given by

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

(5)

For $\epsilon =0$, the functional $J_{\epsilon }$ fails to satisfy the Palais–Smale condition, and the reason for such a lack of compactness is the existence of almost solutions ${\delta }_{(a,\lambda )}$ of the equation $\left({\mathcal{I}}_{\epsilon }\right)$. Our first result deals with the case of small dimensions. It is stated as follows.

Theorem 1. 

Let $3\le n\le 5$, ω be a non-degenerate solution of Problem $\left({\mathcal{I}}_0\right)$ and $k\in \mathbb{N}$. Then, $\left({\mathcal{I}}_{\epsilon }\right)$ has no interior blowing-up solution $u_{\epsilon }$ converging weakly to ω. More precisely, there is no solution $u_{\epsilon }$ such that

$$\begin{array}{cc}\hfill & u_{\epsilon }=\sum _{i=1}^k{\delta }_{(a_{i,\epsilon },{\lambda }_{i,\epsilon })}+\omega +v_{\epsilon }with\hfill \\ \hfill & v_{\epsilon }\to 0in{H}^1\left(\mathsf{\Omega }\right),{\lambda }_{i,\epsilon }\to \infty ,a_{i,\epsilon }\to {\overline{a}}_i\in \mathsf{\Omega }as\epsilon \to 0,\hfill \\ \hfill & \left({\displaystyle \frac{{\lambda }_{i,\epsilon }}{{\lambda }_{j,\epsilon }}}+{\displaystyle \frac{{\lambda }_{j,\epsilon }}{{\lambda }_{i,\epsilon }}}+{\lambda }_{i,\epsilon }{\lambda }_{j,\epsilon }{|a_{i,\epsilon }-a_{j,\epsilon }|}^2\right)\to \infty \forall i\ne jas\epsilon \to 0.\hfill \end{array}$$

The aim of our second result is to construct interior blowing-up solutions with isolated bubbles. More precisely, we have the following.

Theorem 2. 

Let $n\ge 7$, $V:\overline{\mathsf{\Omega }}\to \mathbb{R}$ be a $C^3$ positive function having m non-degenerate critical points $y_1$, …, $y_m$, and ω be a non-degenerate solution of Problem $\left({\mathcal{I}}_0\right)$. Then, for any $k\le m$, there exists ${\epsilon }_0>0$ small such that for any $\epsilon \in (0,{\epsilon }_0]$, Problem $\left({\mathcal{I}}_{\epsilon }\right)$ admits a solution $u_{\epsilon ,y_{i_1},\dots ,y_{i_k}}$ satisfying $u_{\epsilon ,y_{i_1},\dots ,y_{i_k}}$, which develops exactly one bubble at each point $y_{i_j}$ and converges weakly to ω in $H^1\left(\mathsf{\Omega }\right)$ as $\epsilon \to 0$. More precisely, there exist ${\lambda }_{j,\epsilon }>0$ and $a_{j,\epsilon }\to y_{i_j}$ with $j\in \{1,\dots ,k\}$ such that

$$\left|\right|u_{\epsilon ,y_{i_1},\dots ,y_{i_k}}-\sum _{j=1}^k{\delta }_{(a_{j,\epsilon },{\lambda }_{j,\epsilon })}-\omega \left|\right|\to 0,as\epsilon \to 0.$$

In addition, we have

$$\underset{\epsilon \to 0}{lim}{\lambda }_{j,\epsilon }\sqrt{\epsilon }=C>0and|a_{j,\epsilon }-y_{i_j}|\le c\left({\epsilon }^{(n-6)/2}|ln\epsilon |ifn\le 8,{\epsilon }^{1/2}ifn\ge 9\right).$$

Theorem 2 leads to the following multiplicity result in conjunction with the number of critical points of V.

Theorem 3. 

Let $n\ge 7$, $V:\overline{\mathsf{\Omega }}\to \mathbb{R}$ be a $C^3$ positive function having m non-degenerate critical points $y_1$, …, $y_m$, and ω be a non-degenerate solution of Problem $\left({\mathcal{I}}_0\right)$. Then, for $\epsilon >0$ small, $\left({\mathcal{I}}_{\epsilon }\right)$ admits at least $2^m-1$ interior blowing-up solutions which converge weakly to ω.

We note that a result similar to Theorem 2 was obtained in [27] in the context of the scalar curvature problem on high-dimensional spheres. However, the presence of a non-constant Morse function multiplying the nonlinear term in [27] prevents the formation of clustered bubbles, thereby simplifying the analysis. In contrast, in our setting, clustered bubbling does occur, requiring a more delicate and detailed analysis. The following results address this situation: specifically, we construct interior bubbling solutions featuring clustered bubbles centered at a critical point of V. To this aim, we fix some notation. For $k\in \mathbb{N}$ and y a critical point of V, we define the following function:

$$F_{k,y}(b_1,\dots ,b_k)=\sum _{j=1}^k{D}^{2}V\left(y\right)(b_j,b_j)-\sum _{l\ne r}{\displaystyle \frac{1}{|b_l-b_r{|}^{n-2}}},$$

(6)

where $(b_1,\dots ,b_k)\in {\left({\mathbb{R}}^n\right)}^k$ such that $b_i\ne b_j$ if $i\ne j$.

Our result is stated as follows.

Theorem 4. 

Let $n\ge 7$, y be a non-degenerate critical point of V, ω be a non-degenerate solution of Problem $\left({\mathcal{I}}_0\right)$, and $k\in \mathbb{N}$ with $k\ge 2$. Assume that the function $F_{k,y}$ has a non-degenerate critical point $({\overline{b}}_1,\dots ,{\overline{b}}_k)$. Then, there exists ${\epsilon }_0>0$ small such that for any $\epsilon \in (0,{\epsilon }_0]$, problem $\left({\mathcal{I}}_{\epsilon }\right)$ admits a solution $u_{\epsilon ,y}$ with the following properties:

$$\begin{array}{cc}\hfill & \parallel u_{\epsilon ,y}-\sum _{i=1}^k{\delta }_{(a_{i,\epsilon },{\lambda }_{i,\epsilon })}-\omega \parallel \to 0as\epsilon \to 0with\hfill \\ \hfill & \underset{\epsilon \to 0}{lim}{\lambda }_{i,\epsilon }\sqrt{\epsilon }=c>0and\underset{\epsilon \to 0}{lim}{\epsilon }^{{\displaystyle \frac{4-n}{2n}}}(a_{i,\epsilon }-y)-\sigma {\overline{b}}_i=0\forall i\in \{1,\dots ,k\}\hfill \end{array}$$

(7)

where σ is the constant defined by (121).

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

The aim of our last result is to study the case of many interior blow-up points. Namely, we have the following.

Theorem 5. 

Let $n\ge 7$, $y_1,\dots ,y_m$ be non-degenerate critical points of V and ω be a non-degenerate solution of Problem $\left({\mathcal{I}}_0\right)$. Let $\ell \le m$ and $k_1,\dots ,k_{\ell }\in \mathbb{N}$. For $k_j\ge 2$, we assume that the function $F_{k_j,y_{i_j}}$ has a non-degenerate critical point $({\overline{b}}_{j,1},\dots ,{\overline{b}}_{j,k_j})$. Then, there exists ${\epsilon }_0>0$ small such that for any $\epsilon \in (0,{\epsilon }_0]$, problem $\left({\mathcal{I}}_{\epsilon }\right)$ admits a solution $u_{\epsilon ,y_1,\dots ,y_m}$, satisfying

$$\parallel u_{\epsilon ,y_1,\dots ,y_N}-\sum _{i=1}^{k_1}{\delta }_{(a_{1,i,\epsilon },{\lambda }_{1,i,\epsilon })}-\dots -\sum _{i=1}^{k_{\ell }}{\delta }_{(a_{\ell ,i,\epsilon },{\lambda }_{\ell ,i,\epsilon })}-\omega \parallel \to 0as\epsilon \to 0$$

with the rates ${\lambda }_{j,r,\epsilon }$ and the concentration points $a_{j,r,\epsilon }$ satisfying the properties introduced in (7) (with $y_{i_j}$ instead of y).

Remark 1. 

1. 

The solutions constructed in Theorems 2–5 are non-degenerate critical points of the functional $J_{\epsilon }$.

2. 

The $L^{\infty }\left(\mathsf{\Omega }\right)$-norm of the solutions constructed in Theorems 2–5 diverges as $\epsilon \to 0$.

3. 

The $L^{2n/(n-2)}\left(\mathsf{\Omega }\right)$-norm of the solutions constructed in Theorems 2–5 remains uniformly bounded as $\epsilon \to 0$.

4. 

The energy of the solutions constructed in Theorems 2 and 3 is uniformly distributed among the concentration points (see (3)). In contrast, this uniform distribution does not necessarily hold for the solutions in Theorem 5, where a significant portion of the energy may concentrate around a single point (see (1)).

5. 

The results obtained in this paper should be extended to similar boundary value problems with Dirichlet or Robin conditions. We will return to these problems in future work.

The strategy of the proof of our results is based on refined asymptotic expansions of the gradient of the associated Euler–Lagrange functional. The goal is to determine the equilibrium conditions that the concentration parameters must satisfy. These conditions are derived by evaluating the equation using vector fields, which represent the dominant terms in the gradient with respect to the concentration parameters. By analyzing these balancing conditions, we can extract the necessary information to establish our results. It is worth noting that traditional blow-up analysis methods often rely heavily on precise pointwise $C^0$ estimates and the frequent application of Pohozaev identities. In contrast, the approach adopted in this paper deliberately avoids these classical tools. We contend that our method—free from the dependence on pointwise estimates and Pohozaev identities—offers significant potential for tackling non-compact variational problems characterized by more complex blow-up behavior. In particular, the presence of non-simple blow-up points substantially complicates the derivation of pointwise $C^0$-estimates, making such techniques especially difficult to implement in these settings.

To better illustrate the distinction between the low-dimensional and high-dimensional cases, we now outline the key elements of our proofs. In the gradient expansion (see Proposition 9), there are two principal terms: ${\displaystyle \frac{\omega \left(a\right)}{{\lambda }^{(n-2)/2}}}$ and ${\displaystyle \frac{V\left(a\right)}{{\lambda }^2}}$. For dimensions $n\le 5$, the second term is negligible relative to the first, meaning the presence of the first term ensures the non-existence result in Theorem 1. In contrast, when $n\ge 7$, the situation reverses: the first term becomes negligible compared to the second term. This shift enables the construction of interior blowing-up solutions, as stated in Theorems 2–5, by exploiting the dominance of the function V. However, for $n=6$, the two terms are of the same order, leading to a balance phenomenon. As a result, we cannot draw a definitive conclusion about the existence or non-existence of interior blowing-up solutions with residual mass in this case. This remains an open problem. We conjecture that Theorem 1 holds in dimension 6 as well.

3. Study of the Infinite Dimensional Part

Let $\omega$ be a non-degenerate solution of Problem $\left({\mathcal{I}}_0\right)$. Notice that, by the concentration compactness principle [28,29], we know that if $u_{\epsilon }$ is an energy-bounded solution of (${\mathcal{I}}_{\epsilon }$) which converges weakly to $\omega$, then there exists $k\in \mathbb{N}$ such that $u_{\epsilon }$ can be written as

$$\begin{array}{cc}\hfill & u_{\epsilon }=\omega +\sum _{i=1}^k{\delta }_{(a_{i,\epsilon },{\lambda }_{i,\epsilon })}+v_{\epsilon }\mathrm{with}\hfill \end{array}$$

(8)

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

(9)

$$\begin{array}{cc}\hfill & \parallel v_{\epsilon }\parallel ⟶0,{\lambda }_{i,\epsilon }⟶\infty ,a_{i,\epsilon }⟶a_i\in \overline{\mathsf{\Omega }},\forall i\in \{1,\dots ,k\}.\hfill \end{array}$$

(10)

As we are dealing with the interior concentration solution, we assume that

$$d(a_{i,\epsilon },\partial \mathsf{\Omega })\ge c>0.$$

(11)

It is well known that, see [30], if $\left(u_{\epsilon }\right)$ is a sequence having the form (8) and satisfying (9)–(11), then there is a unique way to choose ${\lambda }_{i,\epsilon }$, $a_{i,\epsilon }$ and $v_{\epsilon }$ such that

$$\begin{array}{cc}\hfill & u_{\epsilon }={\alpha }_{0,\epsilon }\omega +\sum _{i=1}^k{\alpha }_{i,\epsilon }{\delta }_{(a_{i,\epsilon },{\lambda }_{i,\epsilon })}+v_{\epsilon },\mathrm{with}{\alpha }_{0,\epsilon }⟶0,\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & \left\{\begin{array}{c}{\alpha }_{i,\epsilon }⟶1,a_{i,\epsilon }⟶{\overline{a}}_i\in \mathsf{\Omega },{\lambda }_{i,\epsilon }⟶\infty as\epsilon ⟶0\forall i\in \{1,\dots ,k\},\hfill \\ {\epsilon }_{ij}⟶0\forall i\ne j,\parallel v_{\epsilon }\parallel ⟶0as\epsilon ⟶0\mathrm{and}{v}_{\epsilon }\in F_{a_{\epsilon },{\lambda }_{\epsilon },\omega },\hfill \end{array}\right.\hfill \end{array}$$

(13)

where $a_{\epsilon }=(a_{1,\epsilon },\dots ,a_{k,\epsilon })\in {\mathsf{\Omega }}^k,{\lambda }_{\epsilon }=({\lambda }_{1,\epsilon },\dots ,{\lambda }_{k,\epsilon })\in {(0,\infty )}^k$ and $F_{a_{\epsilon },{\lambda }_{\epsilon },\omega }$ denotes

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

(14)

In order to simplify the notation, we write in the sequel ${\alpha }_i$, $a_i$, ${\lambda }_i$, $F_{a,\lambda ,\omega }$ instead of ${\alpha }_{i,\epsilon }$, $a_{i,\epsilon }$, ${\lambda }_{i,\epsilon }$ and $F_{a_{\epsilon },{\lambda }_{\epsilon },\omega }$, respectively. For ${\mu }_0$ that is positive small, we introduce the following sets:

$$\begin{array}{cc}\hfill \mathcal{O}(k,{\mu }_0)=& \{(\alpha ,\lambda ,a,v)\in {\mathbb{R}}_{+}^k\times {\mathbb{R}}_{+}^k\times {\mathsf{\Omega }}^k\times H^1\left(\mathsf{\Omega }\right):|{\alpha }_i-1|<{\mu }_0,{\lambda }_i>{\mu }_0^{-1},\epsilon ln{\lambda }_i<{\mu }_0,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & d(a_i,\partial \mathsf{\Omega })>c>0,{\epsilon }_{ij}<{\mu }_0,v\in F_{a,\lambda ,\omega },\parallel v\parallel <{\mu }_0\},\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill \mathcal{B}(k,{\mu }_0)=& \{(\alpha ,\lambda ,a)\in {\mathbb{R}}_{+}^k\times {\mathbb{R}}_{+}^k\times {\mathsf{\Omega }}^k:(\alpha ,\lambda ,a,0)\in \mathcal{O}(k,{\mu }_0)\}.\hfill \end{array}$$

(16)

As usual, in these kinds of problems, we start by dealing with the v-part of $u_{\epsilon }$, which is the infinite dimensional variable. To this aim, we are going to perform an expansion of the functional $J_{\epsilon }$ defined by (4) with respect to $v\in F_{a,\lambda ,\omega }$ satisfying $\parallel v\parallel <{\mu }_0$. Let $(\alpha ,\lambda ,a)$$\in \mathcal{B}(k,{\mu }_0)$, taking

$$\underline{u}=\sum _{i=1}^k{\alpha }_i{\delta }_{(a_i,{\lambda }_i)}+{\alpha }_0\omega$$

and $v\in F_{\alpha ,\lambda ,\omega }$ with $\parallel v\parallel <{\mu }_0$, as in [31], we have

$$\begin{array}{cc}\hfill & J_{\epsilon }(\underline{u}+v)=J_{\epsilon }\left(\underline{u}\right)-〈L_{\epsilon },v〉+{\displaystyle \frac{1}{2}}{Q}_{\epsilon }\left(v\right)+T_{\epsilon }\left(v\right),\mathrm{where}\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & 〈L_{\epsilon },v〉={\int }_{\mathsf{\Omega }}{\underline{u}}^{p-\epsilon }v,Q_{\epsilon }\left(v\right)={\int }_{\mathsf{\Omega }}{\left|\nabla v\right|}^2+{\int }_{\mathsf{\Omega }}V\left(x\right)v^2-(p-\epsilon ){\int }_{\mathsf{\Omega }}{\underline{u}}^{p-1-\epsilon }{v}^2,\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & \left\{\begin{array}{c}{T}_{\epsilon }\left(v\right)=O\left({\parallel v\parallel }^{min(3,p+1-\epsilon )}\right),T_{\epsilon }^{\prime }\left(v\right)=O\left({\parallel v\parallel }^{min(2,p-\epsilon )}\right)\hfill \\ T_{\epsilon }^{\prime \prime }\left(v\right)=O\left({\parallel v\parallel }^{\min (1,p-1-\epsilon )}\right).\hfill \end{array}\right.\hfill \end{array}$$

(19)

To proceed further, we need to prove that $Q_{\epsilon }$ is non-degenerate.

To this aim, we introduce the following quadratic forms:

$$\begin{array}{cc}\hfill & Q_{\omega }\left(v\right):={\parallel v\parallel }^2-p{\int }_{\mathsf{\Omega }}{\omega }^{p-1}{v}^2,Q_0\left(v\right):={\parallel v\parallel }^2-p\sum _{i=1}^k{\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^{p-1}{v}^2,\hfill \\ \hfill & Q\left(v\right):={\parallel v\parallel }^2-p\sum _{i=1}^k{\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^{p-1}{v}^2-p{\int }_{\mathsf{\Omega }}{\omega }^{p-1}{v}^2,\hfill \end{array}$$

(20)

Our aim is to prove that Q is non-degenerate on the space $F_{a,\lambda ,\omega }$. First, recall that, in [21], it is proved that $Q_0$ is coercive on the space $E_{a,\lambda }:=F_{a,\lambda ,0}$, which means there exists ${\beta }_0>0$ (independent of $(a,\lambda )$) such that

$$\begin{array}{c}\hfill Q_0\left(v\right)\ge {\beta }_0{\parallel v\parallel }^2\forall v\in E_{a,\lambda }.\end{array}$$

(21)

Second, since $\omega$ is non-degenerate, we can decompose $H^1\left(\mathsf{\Omega }\right)$ as follows:

$$\begin{array}{c}\hfill H^1\left(\mathsf{\Omega }\right)=N_{-}\left(\omega \right)\oplus <\omega >\oplus N_{+}\left(\omega \right),\end{array}$$

(22)

where $N_{+}\left(\omega \right)$ and $N_{-}\left(\omega \right)$ are the space of positivity and the space of negativity (respectively) of $Q_{\omega }$ in $<\omega {>}^{\perp }$. We remark that the spaces $N_{+}\left(\omega \right)$, $N_{-}\left(\omega \right)$ and $<\omega >$ are orthogonal spaces with respect to the scalar product $(·,·)$ and the bilinear form $(h,k)\mapsto {\int }_{\mathsf{\Omega }}{\omega }^{p-1}hk$.

In addition, let $\left({\sigma }_k\right)$ be the eigenvalues corresponding to $Q_{\omega }$. It is known that ${\sigma }_k\nearrow 1$, and therefore there exists $c>0$ such that

$$\begin{array}{cc}\hfill & Q_{\omega }\left(v\right)⩽-{c\parallel v\parallel }^2\forall v\in N_{-}\left(\omega \right),\mathrm{and}{Q}_{\omega }\left(v\right)⩾c{\parallel v\parallel }^2\forall v\in N_{+}\left(\omega \right).\hfill \end{array}$$

(23)

Furthermore, let $V_k$ be the eigenspace associated to the eigenvalue ${\sigma }_k$. Then, there exists $k_0\in \mathbb{N}$ such that

$$\begin{array}{c}\hfill Q_{\omega }\left(v\right)⩾(1-{\displaystyle \frac{{\beta }_0}{10}}){\parallel v\parallel }^2\forall v\in \bigcup _{k⩾k_0}{V}_k,\end{array}$$

(24)

where ${\beta }_0$ is defined in (21). Notice that (24) implies that

$$\begin{array}{c}\hfill p{\int }_{\mathsf{\Omega }}{\omega }^{p-1}{v}^2⩽({\beta }_0/10){\parallel v\parallel }^2\forall v\in \bigcup _{k⩾k_0}{V}_k.\end{array}$$

(25)

Now, we are able to state the following crucial result.

Proposition 1. 

Let ω be a non-degenerate critical point of $J_0$, and let $(a,\lambda )\in {\mathsf{\Omega }}^k\times {(0,\infty )}^k$ satisfy

$$\begin{array}{c}\hfill d(a_i,\partial \mathsf{\Omega })\ge c>0,{\lambda }_i\ge {\mu }_0^{-1}\forall i,{\epsilon }_{ij}\le {\mu }_0\forall i\ne j.\end{array}$$

(26)

Then, the quadratic form Q (defined by (20)) is non-degenerate on the space’ $F_{a,\lambda ,\omega }$.

More precisely, let $v\in F_{a,\lambda ,\omega }$, writing v as

$$\begin{array}{c}\hfill v:=v_{+}+v_{-}with{v}_{+}\in N_{+}\left(\omega \right)and{v}_{-}\in N_{-}\left(\omega \right),\end{array}$$

(27)

and there exists a constant ${\beta }_1>0$ such that

$$\begin{array}{cc}\hfill & Q\left(v_{-}\right)\le -{\beta }_1{\parallel v_{-}\parallel }^2,\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill & Q\left(v_{+}\right)\ge {\beta }_1\parallel v_{+}{\parallel }^2+{o(\parallel v\parallel }^2).\hfill \end{array}$$

(29)

Proof. 

First, since $v\in F_{a,\lambda ,\omega }$, it follows that $(v,\omega )=0$ which justifies (27) (by using (22) and the fact that $N_{+}\left(\omega \right)$, $N_{-}\left(\omega \right)$ and $<\omega >$ are orthogonal spaces).

Second, we remark that (28) and (29) imply that Q is non-degenerate on the space $F_{a,\lambda ,\omega }$. Hence, we will focus on proving Equations (28) and (29).

Notice that $N_{-}\left(\omega \right)$ is finite dimensional space, and therefore ${\parallel g\parallel }_{\infty }\le c\parallel g\parallel$ for each $g\in N_{-}\left(\omega \right).$

Thus, we obtain

$$\begin{array}{c}\hfill {\int }_{\mathsf{\Omega }}{\delta }_{a_i,{\lambda }_i}^{p-1}{\left(v_{-}\right)}^2\le {\parallel v_{-}\parallel }_{\infty }^2{\int }_{\mathsf{\Omega }}{\delta }_{a_i,{\lambda }_i}^{p-1}=o\left(\parallel v_{-}{\parallel }^2\right)\forall i\end{array}$$

which implies (28) (by using (23)).

To prove (29), we need to decompose $N_{+}\left(\omega \right)$ into two subspaces. Let $k_0$ be the integer introduced in (24), and we have

$$N_{+}\left(\omega \right):=\left(\bigcup _{k⩾k_0}{V}_k\right)\oplus \left(N_{+}\left(\omega \right)\cap \left[\bigcup _{k<k_0}{V}_k\right]\right):=N_{+,1}\left(\omega \right)\oplus N_{+,2}\left(\omega \right).$$

Therefore, we can write $v_{+}$ as

$$v_{+}=v_{+,1}+v_{+,2}with{v}_{+,1}\in N_{+,1}\left(\omega \right)and{v}_{+,2}\in N_{+,2}\left(\omega \right).$$

Thus, since $N_{+,1}\left(\omega \right)$ and $N_{+,2}\left(\omega \right)$ are orthogonal spaces with respect to $(·,·)$ and the bilinear form $(h,k)\mapsto {\int }_{\mathsf{\Omega }}{\omega }^{p-1}hk$, we deduce that

$$\begin{array}{cc}\hfill Q\left(v_{+}\right)=& \left(\parallel v_{+,2}{\parallel }^2-p{\int }_{\mathsf{\Omega }}{\omega }^{p-1}{v}_{+,2}^2\right)+\left(\parallel v_{+,1}{\parallel }^2-p\sum _{i=1}^k{\int }_{\mathsf{\Omega }}{\delta }_i^{p-1}{v}_{+,1}^2\right)\hfill \\ \hfill & -p{\int }_{\mathsf{\Omega }}{\omega }^{p-1}{v}_{+,1}^2-p\sum _{i=1}^k{\int }_{\mathsf{\Omega }}{\delta }_i^{p-1}{v}_{+,2}^2-2p\sum _{i=1}^k{\int }_{\mathsf{\Omega }}{\delta }_i^{p-1}{v}_{+,1}{v}_{+,2}.\hfill \end{array}$$

(30)

In addition, since the dimension of $N_{+,2}$ is finite, we obtain

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}{\delta }_i^{p-1}{v}_{+,2}^2& \le \parallel v_{+,2}{\parallel }_{\infty }^2{\int }_{\mathsf{\Omega }}{\delta }_i^{p-1}=o\left(\parallel v_{+,2}{\parallel }^2\right),\hfill \\ \hfill \left|{\int }_{\mathsf{\Omega }}{\delta }_i^{p-1}{v}_{+,2}{v}_{+,1}\right|& \le \parallel v_{+,2}{\parallel }_{\infty }{\int }_{\mathsf{\Omega }}{\delta }_i^{p-1}|v_{+,1}|\le c\parallel v_{+,2}\parallel \parallel v_{+,1}\parallel {\left({\int }_{\mathsf{\Omega }}{\delta }_i^{{\displaystyle \frac{8n}{n^2-4}}}\right)}^{{\displaystyle \frac{n+2}{2n}}}=o\left(\parallel v_{+}{\parallel }^2\right).\hfill \end{array}$$

Thus, (30) becomes

$$\begin{array}{c}\hfill Q\left(v_{+}\right)=Q_{\omega }\left(v_{+,2}\right)+Q_0\left(v_{+,1}\right)-p{\int }_{\mathsf{\Omega }}{\omega }^{p-1}{v}_{+,1}^2+o\left(\parallel v_{+}{\parallel }^2\right).\end{array}$$

(31)

The first term and the third one on the right hand side of (31) are estimated in (23) and (25), respectively. However, for the second term, the function $v_{+,1}$ does not belong to $E_{a,\lambda }$ necessarily.

For this reason, we decompose $v_{+,1}$ as

$$\begin{array}{c}\hfill v_{+,1}:=\sum _{i=1}^k\left(t_i{\delta }_{a_i,{\lambda }_i}+{\gamma }_i{\lambda }_i{\displaystyle \frac{\partial {\delta }_{a_i,{\lambda }_i}}{\partial {\lambda }_i}}+\sum _{j=1}^n{\eta }_{ij}{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_{a_i,{\lambda }_i}}{\partial a_{i,j}}}\right)+v_{+,1}^{\perp },\end{array}$$

(32)

where $a_{ij}$ is the $j^{th}$ component of $a_i$ and $v_{+,1}^{\perp }\in E_{a,\lambda }.$ Now, we need to estimate the parameters $t_i^{\prime }s,{\gamma }_i^{\prime }s$ and ${\eta }_{ij}^{\prime }s$. To this aim, taking the scalar product of $v_{+,1}$ with ${\delta }_{a_i,{\lambda }_i}$, on one hand, we have

$$\begin{array}{c}\hfill (v_{+,1},{\delta }_{a_i,{\lambda }_i})=c{t}_i+\sum o\left(|t_l|+|{\gamma }_l|+\sum |{\eta }_{l,j}|\right).\end{array}$$

On the other hand, we have

$$(v_{+,1},{\delta }_{a_i,{\lambda }_i})=(v,{\delta }_{a_i,{\lambda }_i})-(v_{+,2},{\delta }_{a_i,{\lambda }_i})-(v_{-},{\delta }_{a_i,{\lambda }_i})=o(\parallel v_{+,2}\parallel +\parallel v_{-}\parallel ),$$

since $v\in F_{a,\lambda ,\omega }$, $v_{+,2}$ and $v_{-}$ are in finite dimensional spaces. Hence, we deduce that

$$|t_i|=\sum o\left(|t_l|+|{\gamma }_l|+\sum |{\eta }_{l,j}|\right)+o(\parallel v\parallel )\forall i.$$

In the same way, we prove that ${\gamma }_i$ and ${\eta }_{i,r}$ satisfy

$$|{\gamma }_i|;|{\eta }_{ir}|=\sum o\left(|t_l|+|{\gamma }_l|+\sum _{j=1}^n|{\eta }_{l,j}|\right)+o(\parallel v\parallel )\forall i,\forall r.$$

Thus, we deduce that

$$\begin{array}{c}\hfill v_{+,1}-v_{+,1}^{\perp }=o(\parallel v\parallel \sum {\delta }_{a_i,{\lambda }_i})and\parallel v_{+,1}\parallel =\parallel v_{+,1}^{\perp }\parallel +o(\parallel v\parallel ).\end{array}$$

(33)

Therefore, we obtain

$$Q_0\left(v_{+,1}\right)=\parallel v_{+,1}^{\perp }{\parallel }^2-p\sum _{i=1}^k{\int }_{\mathsf{\Omega }}{\delta }_{a_i,{\lambda }_i}^{p-1}{\left(v_{+,1}^{\perp }\right)}^2+{o(\parallel v\parallel }^2)=Q_0\left(v_{+,1}^{\perp }\right)+{o(\parallel v\parallel }^2).$$

(34)

Finally, using (21), (23), (25), (33) and (34), Equation (31) becomes

$$Q\left(v_{+}\right)\ge c{\parallel v_{+,2}\parallel }^2+{\beta }_0\parallel v_{+,1}^{\perp }{\parallel }^2+{o(\parallel v\parallel }^2)-({\beta }_0/10)\parallel v_{+,1}{\parallel }^2+o({\parallel v_{+}\parallel }^2)\ge c{\parallel v_{+}\parallel }^2+{o(\parallel v\parallel }^2),$$

which completes the proof of Equation (29). Hence, the proof of Proposition 1 is completed. □

Next, we come back to the question of the non-degeneracy of $Q_{\epsilon }$. Namely, we have the following.

Proposition 2. 

Let $n\ge 3$ and $(\alpha ,\lambda ,a)\in \mathcal{B}(k,{\mu }_0)$ with ${\mu }_0$ positive small. Then, for ε small, the quadratic form $Q_{\epsilon }$ is non-degenerate on the space $F_{a,\lambda ,\omega }$.

Proof. 

First, since $\epsilon ln{\lambda }_i\to 0$, and $\mathsf{\Omega }$ is bounded, Taylor’s expansion implies that

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

(35)

Second, using (35), we deduce that ${\underline{u}}^{-\epsilon }\le c$, and therefore we obtain

$$\begin{array}{cc}\hfill \epsilon {\int }_{\mathsf{\Omega }}{\underline{u}}^{p-1-\epsilon }{v}^2& \le {c\epsilon \parallel v\parallel }^2{\left(\sum {\int }_{\mathsf{\Omega }}{\delta }_i^{p+1}+{\int }_{\mathsf{\Omega }}{\omega }^{p+1}\right)}^{2/n}={o(\parallel v\parallel }^2),\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}{\underline{u}}^{p-1-\epsilon }{v}^2& =\sum _{i=1}^k{\alpha }_i^{p-1-\epsilon }{\int }_{\mathsf{\Omega }}{\delta }_i^{p-1-\epsilon }{v}^2+{\alpha }_0^{p-1-\epsilon }{\int }_{\mathsf{\Omega }}{\omega }^{p-1-\epsilon }{v}^2+{o(\parallel v\parallel }^2).\hfill \end{array}$$

(37)

Observe that (35), and the fact that the ${\alpha }_i$’s are close to 1 imply that

$$\begin{array}{cc}\hfill & {\alpha }_i^{p-1-\epsilon }{\int }_{\mathsf{\Omega }}{\delta }_i^{p-1-\epsilon }{v}^2=(1+o\left(1\right)){\int }_{\mathsf{\Omega }}{\delta }_i^{p-1}{v}^2={\int }_{\mathsf{\Omega }}{\delta }_i^{p-1}{v}^2+{o(\parallel v\parallel }^2),\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill & {\alpha }_0^{p-1-\epsilon }{\int }_{\mathsf{\Omega }}{\omega }^{p-1-\epsilon }{v}^2=(1+o\left(1\right)){\int }_{\mathsf{\Omega }}{\omega }^{p-1}{v}^2={\int }_{\mathsf{\Omega }}{\omega }^{p-1}{v}^2+{o(\parallel v\parallel }^2).\hfill \end{array}$$

(39)

Combining (36)–(39), we obtain $Q_{\epsilon }\left(v\right)=Q\left(v\right)+{o(\parallel v\parallel }^2)$, which implies the proof of the proposition by using Proposition 1. □

Now, we are ready to deal with the infinite dimensional variable v. Namely, we prove the following.

Proposition 3. 

Let $n\ge 3$, and $(\alpha ,\lambda ,a)\in \mathcal{B}(k,{\mu }_0)$. Then, for ${\mu }_0$ positive small, there exists a unique ${\overline{v}}_{\epsilon }\in F_{a,\lambda ,\omega }$ which minimizes $J_{\epsilon }({\sum }_{i=1}^k{\alpha }_i{\delta }_{(a_i,{\lambda }_i)}+{\alpha }_0\omega +v)$ with respect to $v\in F_{a,\lambda ,\omega }$ and $\parallel v\parallel$ small. In particular, we obtain

$$\langle J_{\epsilon }^{\prime }(\sum _{i=1}^k{\alpha }_i{\delta }_{(a_i,{\lambda }_i)}+{\alpha }_0\omega +{\overline{v}}_{\epsilon }),h\rangle =0\forall h\in F_{a,\lambda ,\omega }.$$

In addition, the following estimate holds true: $\parallel {\overline{v}}_{\epsilon }\parallel \le c{T}_{(\epsilon ,a,\lambda )}$, where

$$\begin{array}{cc}\hfill & T_{(\epsilon ,a,\lambda )}=\epsilon +\sum _{i=1}^k{R}_1\left(i\right)+\sum _{j\ne i}{\epsilon }_{ij}+\sum _{j\ne i}{\epsilon }_{ij}^{{\displaystyle \frac{n+2}{2(n-2)}}}{ln}^{{\displaystyle \frac{n+2}{2n}}}{\epsilon }_{ij}^{-1}\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill & with{R}_1\left(i\right)=\left({\displaystyle \frac{1}{{\lambda }_i^{(n-2)/2}}}(ifn\le 5);{\displaystyle \frac{{ln}^{2/3}\left({\lambda }_i\right)}{{\lambda }_i^2}}(ifn=6);{\displaystyle \frac{1}{{\lambda }_i^2}}(ifn\ge 7)\right).\hfill \end{array}$$

(41)

Proof. 

Combining the implicit theorem, Proposition 2, and estimate (17), we see that, for $\epsilon$ small, there exists ${\overline{v}}_{\epsilon }\in F_{a,\lambda ,\omega }$ such that $\parallel {\overline{v}}_{\epsilon }\parallel =O(\parallel L_{\epsilon }\parallel )$, where $L_{\epsilon }$ is defined in (16). Next, we are going to estimate $\parallel L_{\epsilon }\parallel$. For $\underline{u}:=\overline{u}+{\alpha }_0\omega$ with $\overline{u}={\sum }_{i=1}^k{\alpha }_i{\delta }_{(a_i,{\lambda }_i)}$, we write

$$\begin{array}{cc}\hfill L_{\epsilon }\left(v\right)=& \sum _{i=1}^k{\alpha }_i^{p-\epsilon }{\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^{p-\epsilon }v+{\alpha }_0^{p-\epsilon }{\int }_{\mathsf{\Omega }}{\omega }^{p-\epsilon }v\hfill \\ \hfill & +(ifn\le 5)O\left[\sum _{j\ne i}{\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^{p-1-\epsilon }{\delta }_{(a_i,{\lambda }_i)}\left|v\right|+\sum _{i=1}^k{\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^{p-1-\epsilon }\omega \left|v\right|+\sum _{i=1}^k{\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}{\omega }^{p-1-\epsilon }\left|v\right|\right]\hfill \\ \hfill & +(ifn\ge 6)O\left[\sum _{j\ne i}{\int }_{\mathsf{\Omega }}{\left({\delta }_{(a_i,{\lambda }_i)}{\delta }_{(a_j,{\lambda }_j)}\right)}^{{\displaystyle \frac{p-\epsilon }{2}}}\left|v\right|+\sum _{i=1}^k{\int }_{\mathsf{\Omega }}{\left({\delta }_{(a_i,{\lambda }_i)}\omega \right)}^{{\displaystyle \frac{p-\epsilon }{2}}}\left|v\right|\right].\hfill \end{array}$$

(42)

Using estimate E2 of [32] and (35), we obtain

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

(43)

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^{{\displaystyle \frac{p-\epsilon }{2}}}{\omega }^{{\displaystyle \frac{p-\epsilon }{2}}}\left|v\right|\le c{\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^{p/2}\left|v\right|\le {\displaystyle \frac{c\parallel v\parallel }{{\lambda }_i^{(n+2)/4}}}{(ln{\lambda }_i)}^{(n+2)/\left(2n\right)}.\hfill \end{array}$$

(44)

To estimate the rest in (42) for $n\le 5$, we observe that $1<{\displaystyle \frac{2n}{n+2}}<{\displaystyle \frac{8n}{n^2-4}}$. Thus, using Lemma 6.6 of [20], we obtain

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^{p-1-\epsilon }{\delta }_{(a_j,{\lambda }_j)}\left|v\right|\le c\parallel v\parallel {\left({\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^{8n/(n^2-4)}{\delta }_{(a_j,{\lambda }_j)}^{2n/(n+2)}\right)}^{(n+2)/\left(2n\right)}\le c\parallel v\parallel {\epsilon }_{ij},\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^{p-1-\epsilon }\omega \left|v\right|\le c\parallel v\parallel {\left({\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^{8n/(n^2-4)}\right)}^{(n+2)/2n}\le {\displaystyle \frac{c\parallel v\parallel }{{\lambda }_i^{(n-2)/2}}},\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}{\omega }^{p-1-\epsilon }\left|v\right|\le c\parallel v\parallel {\left({\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^{2n/(n+2)}\right)}^{(n+2)/2n}\le {\displaystyle \frac{c\parallel v\parallel }{{\lambda }_i^{(n-2)/2}}}.\hfill \end{array}$$

(47)

For the second integral in the right hand side of (42), we write

$${\int }_{\mathsf{\Omega }}{\omega }^{p-\epsilon }v={\int }_{\mathsf{\Omega }}{\omega }^{p}v+O({\int }_{\mathsf{\Omega }}\epsilon \left|v\right|)=(\omega ,v)+O(\epsilon \parallel v\parallel )=O(\epsilon \parallel v\parallel ).$$

(48)

Lastly, for the first integral in the right hand side of (42), using (35), we have

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

(49)

But, since $v\in F_{a,\lambda ,\omega }$ we have

$${\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^{p}v={\int }_{\mathsf{\Omega }}\nabla {\delta }_{(a_i,{\lambda }_i)}\nabla v-{\int }_{\partial \mathsf{\Omega }}{\displaystyle \frac{\partial {\delta }_{(a_i,{\lambda }_i)}}{\partial \nu }}v=-{\int }_{\mathsf{\Omega }}V{\delta }_{(a_i,{\lambda }_i)}v-{\int }_{\partial \mathsf{\Omega }}{\displaystyle \frac{\partial {\delta }_{(a_i,{\lambda }_i)}}{\partial \nu }}v.$$

Note that

$$\left|{\int }_{\mathsf{\Omega }}V{\delta }_{(a_i,{\lambda }_i)}v\right|\le c{\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}\left|v\right|\le c\parallel v\parallel {\left({\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^{2n/(n+2)}\right)}^{(n+2)/\left(2n\right)}\le c{R}_1\left(i\right)\parallel v\parallel ,$$

(50)

where $R_1\left(i\right)$ is defined in (41).

For the other integral on the boundary, since $d(a_i,\partial \mathsf{\Omega })\ge c>0$, we deuce that

$$\left|{\int }_{\partial \mathsf{\Omega }}{\displaystyle \frac{\partial {\delta }_{(a_i,{\lambda }_i)}}{\partial \nu }}v\right|\le {\displaystyle \frac{c}{{\lambda }_i^{(n-2)/2}}}{\int }_{\partial \mathsf{\Omega }}\left|v\right|\le {\displaystyle \frac{c\parallel v\parallel }{{\lambda }_i^{(n-2)/2}}}.$$

Thus

$$\left|{\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^{p}v\right|\le c\parallel v\parallel \left({\displaystyle \frac{1}{{\lambda }_i^{(n-2)/2}}}+R_1\left(i\right)\right).$$

(51)

Combining (42), (43)–(49) and (51), we easily derive the desired result. □

4. Estimate of the Gradient of the Associated Functional

In this section, we are going to give careful estimates of the gradient of the associated functional $J_{\epsilon }$ defined by (4). To this aim, we note that for $u,h\in H^1\left(\mathsf{\Omega }\right)$, we have

$$\begin{array}{c}\hfill 〈J_{\epsilon }^{\prime }\left(u\right),h〉={\int }_{\mathsf{\Omega }}\nabla u.\nabla h+{\int }_{\mathsf{\Omega }}Vuh-{\int }_{\mathsf{\Omega }}{\left|u\right|}^{p-1-\epsilon }uh.\end{array}$$

(52)

In (52), we will take $u={\sum }_{i=1}^k{\alpha }_i{\delta }_{(a_i,{\lambda }_i)}+{\alpha }_0\omega +{\overline{v}}_{\epsilon }:=\overline{u}+{\alpha }_0\omega +{\overline{v}}_{\epsilon }:=\underline{u}+{\overline{v}}_{\epsilon }$ where ${\overline{v}}_{\epsilon }$ is defined in Proposition 3, and $h={\phi }_i\in \{{\delta }_{(a_i,{\lambda }_i)},{\lambda }_i\partial {\delta }_{(a_i,{\lambda }_i)}/\partial {\lambda }_i,{\lambda }_i^{-1}\partial {\delta }_{(a_i,{\lambda }_i)}/\partial a_i\}$ with $1\le i\le k$ and our goal is to perform careful estimates of $〈J_{\epsilon }^{\prime }\left(u\right),{\phi }_i〉$ for $1\le i\le k$. We start by estimating the nonlinear term in (52).

Proposition 4. 

Let $n\ge 3$ and $(\alpha ,\lambda ,a)$$\in \mathcal{B}(k,{\mu }_0)$ with ${\mu }_0$ positive small. Then, for ${\phi }_i\in \{{\delta }_{(a_i,{\lambda }_i)},{\lambda }_i\partial {\delta }_{(a_i,{\lambda }_i)}/\partial {\lambda }_i,{\lambda }_i^{-1}\partial {\delta }_{(a_i,{\lambda }_i)}/\partial a_i\}$ with $1\le i\le k$ and $u={\sum }_{i=1}^k{\alpha }_i{\delta }_{(a_i,{\lambda }_i)}+{\alpha }_0\omega +v$, where $v\in F_{a,\lambda ,\omega }$, the following expansion holds

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}{\left|u\right|}^{p-1-\epsilon }u{\phi }_i& ={\alpha }_i^{p-\epsilon }{\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^{p-\epsilon }{\phi }_i+(p-\epsilon )\sum _{j\ne i}{\int }_{\mathsf{\Omega }}{\left({\alpha }_i{\delta }_{\left(a_i{\lambda }_i\right)}\right)}^{p-1-\epsilon }\left({\alpha }_j{\delta }_{(a_j,{\lambda }_j)}\right){\phi }_i\hfill \\ \hfill & +\sum _{j\ne i}{\int }_{\mathsf{\Omega }}{\left({\alpha }_j{\delta }_{(a_j,{\lambda }_j)}\right)}^{p-\epsilon }{\phi }_i+{\alpha }_0^{p-\epsilon }{\int }_{\mathsf{\Omega }}{\omega }^p{\phi }_i+(p-\epsilon ){\alpha }_i^{p-1-\epsilon }{\alpha }_0{\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^{p-1-\epsilon }\omega {\phi }_i\hfill \\ \hfill & +(p-\epsilon ){\alpha }_i^{p-1-\epsilon }{\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^{p-1-\epsilon }{\phi }_{i}v+O\left(R_2\left(i\right)\right),\hfill \end{array}$$

(53)

where

$$\begin{array}{cc}\hfill R_2\left(i\right)=& {\parallel v\parallel }^2+{\displaystyle \frac{\epsilon }{{\lambda }_i^{\frac{n-2}{2}}}}+\sum _{j=1}^k\left({\displaystyle \frac{ln{\lambda }_j}{{\lambda }_j^{n/2}}}+{\displaystyle \frac{1}{{\lambda }_j^4}}\right)+\sum _{j\ne r}{\epsilon }_{jr}^{{\displaystyle \frac{n}{n-2}}}ln{\epsilon }_{jr}^{-1}+(ifn=3)\sum _{j\ne i}\left[{\displaystyle \frac{1}{{\lambda }_j}}+{\epsilon }_{ij}^2{ln}^{{\displaystyle \frac{2}{3}}}{\epsilon }_{ij}^{-1}\right].\hfill \end{array}$$

Proof. 

Observe that

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

(54)

But, since $|{\phi }_i|\le c\underline{u}$, we have

$${\int }_{\left[\right|\underline{u}|\le |v\left|\right]}{\left|v\right|}^{p-\epsilon }|{\phi }_i|+{\int }_{\mathsf{\Omega }}{\underline{u}}^{p-2-\epsilon }{v}^2|{\phi }_i|\le {\int }_{\left[\right|\underline{u}|\le |v\left|\right]}{\left|v\right|}^{p+1-\epsilon }+{\int }_{\mathsf{\Omega }}{\underline{u}}^{p-1-\epsilon }{v}^2\le c{\parallel v\parallel }^2.$$

(55)

Next, we are going to estimate the second integral in the right hand side of (54). Using (35) and the fact that $|{\phi }_i|\le c{\delta }_{(a_i,{\lambda }_i)}$, we see that

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}{\underline{u}}^{p-1-\epsilon }v{\phi }_i& ={\int }_{\mathsf{\Omega }}{\overline{u}}^{p-1-\epsilon }v{\phi }_i+O\left({\int }_{\mathsf{\Omega }}{\overline{u}}^{p-2-\epsilon }\omega \left|v\right||{\phi }_i|+{\int }_{\mathsf{\Omega }}{\omega }^{p-1-\epsilon }\left|v\right||{\phi }_i|\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & ={\int }_{\mathsf{\Omega }}{\overline{u}}^{p-1-\epsilon }v{\phi }_i+O\left(\sum _{j=1}^k{\int }_{\mathsf{\Omega }}{\delta }_{(a_j,{\lambda }_j)}^{p-1}\left|v\right|+{\int }_{\mathsf{\Omega }}\left|v\right|{\delta }_{(a_i,{\lambda }_i)}\right)\hfill \end{array}$$

(56)

$$\begin{array}{cc}\hfill & ={\int }_{\mathsf{\Omega }}{\overline{u}}^{p-1-\epsilon }v{\phi }_i+O\left(\parallel v\parallel \sum _{j=1}^k{R}_1\left(j\right)\right),\hfill \end{array}$$

(57)

where $R_1\left(j\right)$ is defined in (41) and where we have used (50) for the last integral of (56) and similar computations for the second one. Combining estimate (43) of [31] and (57), we obtain

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}{\underline{u}}^{p-1-\epsilon }v{\phi }_i=& {\alpha }_i^{p-1-\epsilon }{\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^{p-1-\epsilon }v{\phi }_i+O\left({\parallel v\parallel }^2+\sum _{j\ne i}{({\epsilon }_{ij}^{{\displaystyle \frac{n}{n-2}}}ln{\epsilon }_{ij}^{-1})}^{inf({\displaystyle \frac{2\dot{(n-2)}}{n}},{\displaystyle \frac{n+2}{n}})}\right)\hfill \\ \hfill & +O\left(\sum _{j=1}^k{R}_1^2\left(j\right)\right).\hfill \end{array}$$

(58)

It remains to estimate the first integral in the right hand side of (54). First, observe that, for $t,s\in {\mathbb{R}}_{+}$ and $\beta >1$, it holds

$${(t+s)}^{\beta }=t^{\beta }+\beta t^{\beta -1}s+s^{\beta }+\left\{\begin{array}{cc}O\left(t^{(\beta -1)/2}{s}^{(\beta +1)/2}\right)\hfill & if\beta \le 3,\hfill \\ O(t^{\beta -2}{s}^2+t{s}^{\beta -1})\hfill & if\beta >3.\hfill \end{array}\right.$$

Thus, we deduce that

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}& {\underline{u}}^{p-\epsilon }{\phi }_i={\int }_{\mathsf{\Omega }}{\overline{u}}^{p-\epsilon }{\phi }_i+(p-\epsilon ){\int }_{\mathsf{\Omega }}{\overline{u}}^{p-\epsilon -1}\left({\alpha }_0\omega \right){\phi }_i+{\int }_{\mathsf{\Omega }}{\left({\alpha }_0\omega \right)}^{p-\epsilon }{\phi }_i\hfill \\ \hfill & +O\left({\int }_{\mathsf{\Omega }}{\overline{u}}^{{\displaystyle \frac{p-\epsilon -1}{2}}}{\omega }^{{\displaystyle \frac{p-\epsilon +1}{2}}}{\delta }_{(a_i,{\lambda }_i)}\right)+(ifn=3)O\left({\int }_{\mathsf{\Omega }}\overline{u}{\omega }^4{\delta }_{(a_i,{\lambda }_i)}+{\int }_{\mathsf{\Omega }}{\overline{u}}^3{\omega }^2{\delta }_{(a_i,{\lambda }_i)}\right).\hfill \end{array}$$

(59)

Notice that

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}{\left({\alpha }_0\omega \right)}^{p-\epsilon }{\phi }_i={\alpha }_0^{p-\epsilon }{\int }_{\mathsf{\Omega }}{\omega }^p{\phi }_i+O\left(\epsilon {\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}\right)={\alpha }_0^{p-\epsilon }{\int }_{\mathsf{\Omega }}{\omega }^p{\phi }_i+O\left({\displaystyle \frac{\epsilon }{{\lambda }_i^{(n-2)/2}}}\right),\hfill \end{array}$$

(60)

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}{\overline{u}}^{(p-\epsilon -1)/2}{\omega }^{(p-\epsilon +1)/2}{\delta }_{(a_i,{\lambda }_i)}\le c\sum _{1\le j\le k}{\int }_{\mathsf{\Omega }}{\delta }_{(a_j,{\lambda }_j)}^{n/(n-2)}\le c\sum _{1\le j\le k}{\displaystyle \frac{ln\left({\lambda }_j\right)}{{\lambda }_j^{n/2}}},\hfill \end{array}$$

(61)

and for $n=3$,

$${\int }_{\mathsf{\Omega }}\overline{u}{\omega }^4{\delta }_{(a_i,{\lambda }_i)}+{\int }_{\mathsf{\Omega }}{\overline{u}}^3{\omega }^2{\delta }_{(a_i,{\lambda }_i)}\le c\sum _{1\le j\le k}{\int }_{\mathsf{\Omega }}{\delta }_{(a_j,{\lambda }_j)}^2+{\delta }_{(a_j,{\lambda }_j)}^4\le c\sum _{1\le j\le k}{\displaystyle \frac{1}{{\lambda }_j}}.$$

(62)

Concerning the first integral in the right hand side of (59), using (44)–(46) of [31] with $K\equiv 1$, we obtain

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}{\overline{u}}^{p-\epsilon }{\phi }_i=& {\int }_{\mathsf{\Omega }}{\left({\alpha }_i{\delta }_{(a_i,{\lambda }_i)}\right)}^{p-\epsilon }{\phi }_i+(p-\epsilon )\sum _{j\ne i}{\int }_{\mathsf{\Omega }}{\left({\alpha }_i{\delta }_{(a_i,{\lambda }_i)}\right)}^{p-1-\epsilon }\left({\alpha }_j{\delta }_{(a_j,{\lambda }_j)}\right){\phi }_i\hfill \\ \hfill & +\sum _{j\ne i}{\int }_{\mathsf{\Omega }}{\left({\alpha }_j{\delta }_{(a_j,{\lambda }_j)}\right)}^{p-\epsilon }{\phi }_i+O\left(\sum _{j\ne r}{\epsilon }_{jr}^{{\displaystyle \frac{n}{n-2}}}ln{\epsilon }_{jr}^{-1}+\sum _{j\ne i}{\epsilon }_{ij}^2{ln}^{{\displaystyle \frac{2}{3}}}{\epsilon }_{ij}^{-1}\right).\hfill \end{array}$$

(63)

Now, we will focus on estimating the second integral of the right hand side of (59). Using the fact that

$${(t+s)}^{\beta }=t^{\beta }+O\left(t^{\beta -1}s+s^{\beta }\right)\forall t,s,\beta \in (0,\infty ),$$

(64)

we deduce that

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}& {\overline{u}}^{p-\epsilon -1}\omega {\phi }_i\hfill \\ \hfill & ={\int }_{\mathsf{\Omega }}{\left({\alpha }_i{\delta }_{(a_i,{\lambda }_i)}\right)}^{p-\epsilon -1}\omega {\phi }_i+\sum _{j\ne i}O\left({\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^{p-1}{\delta }_{(a_j,{\lambda }_j)}\omega +{\int }_{\mathsf{\Omega }}{\delta }_{(a_j,{\lambda }_j)}^{p-1}{\delta }_{(a_i,{\lambda }_i)}\omega \right).\hfill \end{array}$$

(65)

Notice that, for $n\ge 4$, we have $p-1:=4/(n-2)\le n/(n-2)$ and therefore, it follows that

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

But for $n=3$, it holds

$${\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^{p-1}{\delta }_{(a_j,{\lambda }_j)}\omega +{\int }_{\mathsf{\Omega }}{\delta }_{(a_j,{\lambda }_j)}^{p-1}{\delta }_{(a_i,{\lambda }_i)}\omega \le c{\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^4{\delta }_{(a_j,{\lambda }_j)}+c{\int }_{\mathsf{\Omega }}{\delta }_{(a_j,{\lambda }_j)}^4{\delta }_{(a_i,{\lambda }_i)}$$

and

$${\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^4{\delta }_{(a_j,{\lambda }_j)}\le {\left({\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^3{\delta }_{(a_j,{\lambda }_j)}^3\right)}^{{\displaystyle \frac{1}{3}}}{\left({\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^{9/2}\right)}^{{\displaystyle \frac{2}{3}}}\le {\displaystyle \frac{c{\epsilon }_{ij}}{\sqrt{{\lambda }_i}}}{ln}^{{\displaystyle \frac{1}{3}}}\left({\epsilon }_{ij}^{-1}\right)\le c{\epsilon }_{ij}^{2}ln{\left({\epsilon }_{ij}^{-1}\right)}^{{\displaystyle \frac{2}{3}}}+{\displaystyle \frac{c}{{\lambda }_i}}.$$

Thus, (65) becomes

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}& {\overline{u}}^{p-\epsilon -1}\omega {\phi }_i\hfill \\ \hfill & ={\int }_{\mathsf{\Omega }}{\left({\alpha }_i{\delta }_{(a_i,{\lambda }_i)}\right)}^{p-\epsilon -1}\omega {\phi }_i+\left\{\begin{array}{cc}O\left(\sum _{j\ne i}{\epsilon }_{ij}^{{\displaystyle \frac{n}{n-2}}}ln\left({\epsilon }_{ij}^{-1}\right)+\sum _{j\le k}{\displaystyle \frac{ln{\lambda }_j}{{\lambda }_j^{n/2}}}\right)\hfill & \mathrm{if}n\ge 4,\hfill \\ O\left(\sum _{j\ne i}{\epsilon }_{ij}^{2}ln{\left({\epsilon }_{ij}^{-1}\right)}^{2/3}+\sum _{j\le k}{\displaystyle \frac{1}{{\lambda }_j}}\right)\hfill & \mathrm{if}n=3.\hfill \end{array}\right.\hfill \end{array}$$

(66)

Combining (54), (58)–(63) and (66), the proof of Proposition 4 follows. □

Next, we are going to estimate the linear term in Proposition 4 with respect to ${\overline{v}}_{\epsilon }$.Our result reads as follows.

Lemma 1. 

Let $n\ge 3$, $(\alpha ,\lambda ,a)\in \mathcal{B}(k,{\mu }_0)$ and $v\in F_{a,\lambda ,\omega }$. Then, for $i\in \{1,\dots ,k\}$, we have

$$\begin{array}{c}\hfill \left|{\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^{p-1-\epsilon }v{\phi }_i\right|\le c\parallel v\parallel \left(\epsilon +R_1\left(i\right)\right),\end{array}$$

where $R_1\left(i\right)$ is defined in (41).

Proof. 

Using (35) and the fact that $|{\phi }_i|\le c{\delta }_{(a_i,{\lambda }_i)}$, we obtain

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

(67)

If ${\phi }_i={\delta }_{(a_i,{\lambda }_i)}$, the estimates (51) and (67) clearly give the desired result.

If ${\phi }_i={\lambda }_i\partial {\delta }_{(a_i,{\lambda }_i)}/\partial {\lambda }_i$, using the fact that $(v,\partial {\delta }_{(a_i,{\lambda }_i)}/\partial {\lambda }_i)=0$, $|{\phi }_i|\le c{\delta }_{(a_i,{\lambda }_i)}$, $d(a_i,\partial \mathsf{\Omega })\ge c>0$ and (50), we obtain

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^{p-1}v{\lambda }_i{\displaystyle \frac{\partial {\delta }_{(a_i,{\lambda }_i)}}{\partial {\lambda }_i}}& ={\displaystyle \frac{1}{p}}{\int }_{\mathsf{\Omega }}-\Delta \left({\lambda }_i{\displaystyle \frac{\partial {\delta }_{(a_i,{\lambda }_i)}}{\partial {\lambda }_i}}\right)v\hfill \\ \hfill & =-{\displaystyle \frac{1}{p}}{\int }_{\mathsf{\Omega }}V{\lambda }_i{\displaystyle \frac{\partial {\delta }_{(a_i,{\lambda }_i)}}{\partial {\lambda }_i}}v-{\displaystyle \frac{1}{p}}{\int }_{\partial \mathsf{\Omega }}{\displaystyle \frac{\partial }{\partial \nu }}\left({\lambda }_i{\displaystyle \frac{\partial {\delta }_{(a_i,{\lambda }_i)}}{\partial {\lambda }_i}}\right)v\hfill \\ \hfill & =O\left({\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}\left|v\right|+{\displaystyle \frac{1}{{\lambda }_i^{(n-2)/2}}}{\int }_{\partial \mathsf{\Omega }}\left|v\right|\right)=O\left(\parallel v\parallel R_1\left({\lambda }_i\right)+{\displaystyle \frac{\parallel v\parallel }{{\lambda }_i^{(n-2)/2}}}\right)\hfill \end{array}$$

which also gives the desired result in this case.

Lastly if ${\phi }_i={\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_{(a_i,{\lambda }_i)}}{\partial a_i}}$, using again (50), we write in the same way

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^{p-1}v{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_{(a_i,{\lambda }_i)}}{\partial a_i}}& =-{\displaystyle \frac{1}{p}}{\int }_{\mathsf{\Omega }}V{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_{(a_i,{\lambda }_i)}}{\partial a_i}}v+O\left({\int }_{\partial \mathsf{\Omega }}{\displaystyle \frac{1}{{\lambda }_i}}\left|{\displaystyle \frac{\partial {\delta }_{(a_i,{\lambda }_i)}}{\partial a_i}}\right|\left|v\right|\right)\hfill \\ \hfill & =O\left({\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}\left|v\right|+{\displaystyle \frac{1}{{\lambda }_i^{n/2}}}{\int }_{\partial \mathsf{\Omega }}\left|v\right|\right)=O\left(\parallel v\parallel R_1\left(i\right)+{\displaystyle \frac{\parallel v\parallel }{{\lambda }^{n/2}}}\right).\hfill \end{array}$$

This completes the proof of Lemma 1. □

Our aim in the next three propositions is to precise the statement of Proposition 4 for the three possible values of ${\phi }_i$. We begin by the case where ${\phi }_i={\delta }_{(a_i,{\lambda }_i)}$.

Proposition 5. 

Let $n\ge 3$, $(\alpha ,\lambda ,a)\in \mathcal{B}(k,{\mu }_0)$ with ${\mu }_0$ positive small and $v\in F_{a,\lambda ,\omega }$. Then, for $1\le i\le k$ and $u={\sum }_{i=1}^k{\alpha }_i{\delta }_{(a_i,{\lambda }_i)}+{\alpha }_0\omega +v$, the following expansion holds.

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}{\left|u\right|}^{p-1-\epsilon }u{\delta }_{(a_i,{\lambda }_i)}=& {\alpha }_i^{p-\epsilon }{\lambda }_i^{-\epsilon {\displaystyle \frac{n-2}{2}}}{S}_n+O\left({\parallel v\parallel }^2+\epsilon +{\displaystyle \frac{1}{{\lambda }_i^{(n-2)/2}}}\right)\hfill \\ \hfill & +O\left(\sum _{j=1}^k\left({\displaystyle \frac{ln{\lambda }_j}{{\lambda }_j^{n/2}}}+{\displaystyle \frac{1}{{\lambda }_j^4}}\right)+\sum _{j\ne r}{\epsilon }_{jr}\right)+(\mathrm{if}n=3)O\left(\sum _{j=1}^k{\displaystyle \frac{1}{{\lambda }_j}}\right),\hfill \end{array}$$

where $S_n=c_0^{2n/(n-2)}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{dx}{{(1+|x|}^2{)}^n}}$.

Proof. 

We will apply Proposition 4 with ${\phi }_i={\delta }_{(a_i,{\lambda }_i)}$. Thus, we need to estimate each integral in (53). To this aim, we observe that

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}{\omega }^p{\delta }_{(a_i,{\lambda }_i)}\le c{\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}\le {\displaystyle \frac{c}{{\lambda }_i^{(n-2)/2}}}{\int }_{\mathsf{\Omega }}{\displaystyle \frac{dx}{{|x-a|}^{n-2}}}\le {\displaystyle \frac{c}{{\lambda }_i^{(n-2)/2}}},\hfill \end{array}$$

(68)

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^{p-\epsilon }\omega \le c{\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^p=O\left({\displaystyle \frac{1}{{\lambda }_i^{(n-2)/2}}}\right),\hfill \end{array}$$

(69)

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^{p-\epsilon }{\delta }_{(a_j,{\lambda }_j)}+{\int }_{\mathsf{\Omega }}{\delta }_{(a_j,{\lambda }_j)}^{p-\epsilon }{\delta }_{(a_i,{\lambda }_i)}=O\left({\epsilon }_{ij}\right),\hfill \end{array}$$

(70)

where we used in (70) Lemma 6.6 of [20]. Furthermore, using (35), we deduce that

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

(71)

Combining (68)–(71), Lemma 1 and Proposition 4, we easily obtain the desired result. □

Now, we deal with the case where ${\phi }_i={\lambda }_i\partial {\delta }_{(a_i,{\lambda }_i)}/\partial {\lambda }_i$ in Proposition 4. Precisely, we prove

Proposition 6. 

Let $n\ge 3$, $(\alpha ,\lambda ,a)\in \mathcal{B}(k,{\mu }_0)$ with ${\mu }_0$ positive small and $v\in F_{a,\lambda ,\omega }$. Then, for $1\le i\le k$ and $u={\sum }_{i=1}^k{\alpha }_i{\delta }_{(a_i,{\lambda }_i)}+{\alpha }_0\omega +v$, the following expansion holds.

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}{\left|u\right|}^{p-1-\epsilon }u{\lambda }_i{\displaystyle \frac{\partial {\delta }_{(a_i,{\lambda }_i)}}{\partial {ł}_i}}={\alpha }_0^{p-\epsilon }(\omega ,{\lambda }_i{\displaystyle \frac{\partial {\delta }_{(a_i,{\lambda }_i)}}{\partial {\lambda }_i}})-{\alpha }_i^{p-1-\epsilon }{\alpha }_0{\lambda }_i^{{\displaystyle \frac{-\epsilon (n-2)}{2}}}{\displaystyle \frac{c_1}{{\lambda }_i^{\frac{(n-2)}{2}}}}\omega \left(a_i\right)\hfill \\ \hfill & -c_3\epsilon {\alpha }_i^{p-\epsilon }{\lambda }_i^{{\displaystyle \frac{-\epsilon (n-2)}{2}}}+c_2\sum _{j\ne i}{\alpha }_j{\lambda }_i{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial {\lambda }_i}}\left({\alpha }_i^{p-1-\epsilon }{\lambda }_i^{{\displaystyle \frac{-\epsilon (n-2)}{2}}}+{\alpha }_j^{p-1-\epsilon }{\lambda }_j^{{\displaystyle \frac{-\epsilon (n-2)}{2}}}\right)+O\left(R_3\left(i\right)\right),\hfill \end{array}$$

where $c_1={\displaystyle \frac{(n+2)}{2}}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{c_0^p{\left(\right|x|}^2-1)}{{(1+|x|}^2{)}^{\frac{n+4}{2}}}}dx$, $c_2={\int }_{{\mathbb{R}}^n}{\displaystyle \frac{c_0^{p+1}}{{(1+|x|}^2{)}^{\frac{n+2}{2}}}}dx$, $c_3={\displaystyle \frac{{(n-2)}^2}{4}}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{c_0^{p+1}{\left(\right|x|}^2-1\left)\right(1+{\left|x\right|}^2)}{{(1+|x|}^2{)}^{n+1}}}dx$ and where $R_3\left(i\right)={\epsilon }^2+R_2\left(i\right)$ and $R_2\left(i\right)$ is defined in Proposition 4.

Proof. 

The proof is based on Proposition 4 and Lemma 1. Thus, we need to estimate each integral that appears in (53). For the first integral in (53), we recall that (see (91) of [21])

$$\begin{array}{c}\hfill {\int }_{\mathsf{\Omega }}{\delta }_{(a_i,{\lambda }_i)}^{p-\epsilon }{\lambda }_i{\displaystyle \frac{\partial {\delta }_{(a_i,{\lambda }_i)}}{\partial {\lambda }_i}}=-c_3{c}_0^{-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}\epsilon +O\left({\epsilon }^2+{\displaystyle \frac{ln{\lambda }_i}{{\lambda }_i^n}}\right).\end{array}$$

(72)

Concerning the second integral in (53), using (35), it holds

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

But, since $d(a_{\ell },\partial \mathsf{\Omega })\ge c>0$ for $\ell \in \{i,j\}$, we deduce that

$${\int }_{{\mathbb{R}}^n\setminus \mathsf{\Omega }}{\delta }_i^p{\delta }_j\le {\displaystyle \frac{c}{{\lambda }_i^{(n+2)/2}{\lambda }_j^{(n-2)/2}}}{\int }_{{\mathbb{R}}^n\setminus \mathsf{\Omega }}{\displaystyle \frac{1}{|x-a_i{|}^{n+2}{|x-a_j|}^{n-2}}}\le {\displaystyle \frac{c}{{\lambda }_i^{(n+2)/2}{\lambda }_j^{(n-2)/2}}},$$

and, using Lemmas 6.4 and 6.6 of [20], we have

$$\begin{array}{cc}\hfill & p{\int }_{{\mathbb{R}}^n}{\delta }_i^{p-1}{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}{\delta }_j=c_2{\lambda }_i{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial {\lambda }_i}}+O\left({\epsilon }_{ij}^{{\displaystyle \frac{n}{n-2}}}ln{\epsilon }_{ij}^{-1}\right),\hfill \\ \hfill & {\int }_{\mathsf{\Omega }}{\delta }_i^p{\delta }_{j}ln(1+{\lambda }_i^2|x-a_i{|}^2)\le {\left({\int }_{\mathsf{\Omega }}{\delta }_j^{{\displaystyle \frac{n}{n-1}}}{\delta }_i^{{\displaystyle \frac{n^2}{(n-2)(n-1)}}}\right)}^{{\displaystyle \frac{n-1}{n}}}{\left({\int }_{\mathsf{\Omega }}{\delta }_i^{{\displaystyle \frac{2n}{n-2}}}{ln}^n(1+{\lambda }_i^2|x-a_i{|}^2)\right)}^{{\displaystyle \frac{1}{n}}}\hfill \\ \hfill & \le c{\epsilon }_{ij}.\hfill \end{array}$$

Thus we deduce that

$$\begin{array}{cc}\hfill p{\int }_{\mathsf{\Omega }}{\delta }_i^{p-1-\epsilon }& {\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}{\delta }_j\hfill \\ \hfill & =c_0^{-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}{c}_2{\lambda }_i{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial {\lambda }_i}}+O\left({\epsilon }_{ij}^{{\displaystyle \frac{n}{n-2}}}ln{\epsilon }_{ij}^{-1}+{\displaystyle \frac{1}{{\lambda }_i^{(n+2)/2}{\lambda }_j^{(n-2)/2}}}+\epsilon {\epsilon }_{ij}\right).\hfill \end{array}$$

(73)

In the same way, using Lemmas 6.4 and 6.6 of [20], we estimate the third integral of (53) as

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}{\delta }_j^{p-\epsilon }{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}& =c_0^{-\epsilon }{\lambda }_j^{-\epsilon (n-2)/2}{c}_2{\lambda }_i{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial {\lambda }_i}}+O\left({\epsilon }_{ij}^{{\displaystyle \frac{n}{n-2}}}ln{\epsilon }_{ij}^{-1}+{\displaystyle \frac{1}{{\lambda }_j^{(n+2)/2}{\lambda }_i^{(n-2)/2}}}+\epsilon {\epsilon }_{ij}\right).\hfill \end{array}$$

(74)

Concerning the fourth integral of (53), it holds

$$\begin{array}{c}\hfill {\int }_{\mathsf{\Omega }}{\omega }^p{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}={\int }_{\mathsf{\Omega }}(-\Delta +V)\omega {\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}=\left(\omega ,{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}\right).\end{array}$$

(75)

Now, we focus on the fifth integral of (53). Using (35), we deduce that

$${\int }_{\mathsf{\Omega }}{\delta }_i^{p-1-\epsilon }{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}\omega =c_0^{-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}{\int }_{\mathsf{\Omega }}{\delta }_i^{p-1}{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}\omega +O\left(\epsilon {\int }_{\mathsf{\Omega }}{\delta }_i^{p}ln(1+{\lambda }_i^2|x-a_i{|}^2)\omega \right).$$

Observe that

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}{\delta }_{(a,\lambda )}^{p}ln(1+{\lambda }^2{|x-a|}^2)\omega \le {\displaystyle \frac{c}{{\lambda }^{(n-2)/2}}}{\int }_0^{\infty }{\displaystyle \frac{ln(1+r^2)r^{n-1}}{{(1+r^2)}^{(n+2)/2}}}\le {\displaystyle \frac{c}{{\lambda }^{(n-2)/2}}},\hfill \\ \hfill & {\int }_{\mathsf{\Omega }}{\delta }_i^{p-1}{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}\omega =\omega \left(a_i\right){\int }_{B(a_i,\rho )}{\delta }_i^{p-1}{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}+O\left({\int }_{B(a_i,\rho )}{|x-a_i|}^2{\delta }_i^p+{\int }_{\mathsf{\Omega }\setminus B(a_i,\rho )}{\delta }_i^p\right),\hfill \end{array}$$

where $\rho$ is a fixed radius satisfying $B(a_i,\rho )\subset \mathsf{\Omega }$. This radius can be chosen independent of $a_i$ since $d(a_i,\partial \mathsf{\Omega })\ge c>0$. But we have

$${\int }_{\mathsf{\Omega }\setminus B(a_i,\rho )}{\delta }_i^p\le {\int }_{{\mathbb{R}}^n\setminus B(a_i,\rho )}{\delta }_i^p\le {\displaystyle \frac{c}{{\lambda }_i^{n+2)/2}}};{\int }_{B(a_i,\rho )}{|x-a_i|}^2{\delta }_i^p\le c{\displaystyle \frac{ln\left({\lambda }_i\right)}{{\lambda }_i^{(n+2)/2}}}$$

$$\begin{array}{c}\hfill p{\int }_{{\mathbb{R}}^n}{\delta }_i^{p-1}{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}=c_0^p{\lambda }_i^{(n+2)/2}\left({\displaystyle \frac{n+2}{2}}\right){\int }_{{\mathbb{R}}^n}{\displaystyle \frac{(1-{\lambda }_i|x-a_i{|}^2)}{(1+{\lambda }_i^2|x-a_i{{|}^2)}^{(n+4)/2}}}dx=-{\displaystyle \frac{c_1}{{\lambda }_i^{(n-2)/2}}}.\end{array}$$

Thus, combining the previous estimates, we deduce that

$$\begin{array}{c}\hfill p{\int }_{\mathsf{\Omega }}{\delta }_i^{p-1-\epsilon }{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}\omega =-c_0^{-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}{\displaystyle \frac{c_1}{{\lambda }_i^{(n-2)/2}}}\omega \left(a_i\right)+O\left({\displaystyle \frac{\epsilon }{{\lambda }_i^{(n-2)/2}}}+{\displaystyle \frac{ln{\lambda }_i}{{\lambda }_i^{(n+2)/2}}}\right).\end{array}$$

(76)

Notice that the last integral of (53) is computed in Lemma 1.

Combining (75)–(72), Lemma 1 and Proposition 4, we easily derive the desired result. □

Lastly, we take ${\phi }_i={\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}$ in Proposition 4 and our goal is to prove the following result.

Proposition 7. 

Let $n\ge 7$, $(\alpha ,\lambda ,a)\in \mathcal{B}(k,{\mu }_0)$ with ${\mu }_0$ positive small and $v\in F_{a,\lambda ,\omega }$. Then, for $1\le i\le k$ and $u={\sum }_{i=1}^k{\alpha }_i{\delta }_{(a_i,{\lambda }_i)}+{\alpha }_0\omega +v$, the following expansion holds:

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}{\left|u\right|}^{p-1-\epsilon }u{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}=c_2\sum _{j\ne i}{\alpha }_j{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial a_i}}\left({\alpha }_i^{p-1-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}+{\alpha }_j^{p-1-\epsilon }{\lambda }_j^{-\epsilon (n-2)/2}\right)+O\left(R_4\left(i\right)\right),\hfill \\ \hfill & with{R}_4\left(i\right):={\parallel v\parallel }^2+{\epsilon }^2+\sum _{1\le j\le k}\left({\displaystyle \frac{ln{\lambda }_j}{{\lambda }_j^{n/2}}}+{\displaystyle \frac{1}{{\lambda }_j^4}}\right)+\sum _{j\ne i}{\lambda }_j|a_i-a_j|{\epsilon }_{ij}^{{\displaystyle \frac{n+1}{n-2}}}+\sum _{j\ne \ell }{\epsilon }_{\ell j}^{{\displaystyle \frac{n}{n-2}}}ln{\epsilon }_{\ell j}^{-1},\hfill \end{array}$$

when $c_2$ is defined in Proposition 6.

Proof. 

As in the proof of Proposition 6, the proof is based on Proposition 4 (by taking ${\phi }_i={{\lambda }_i}^{-1}\partial {\delta }_i/\partial a_i$) and Lemma 1. Thus, we need to estimate the integrals that appear in (53).

First, by oddness, we see that the first integral in (53) is estimated as

$${\int }_{\mathsf{\Omega }}{\delta }_i^{p-\epsilon }{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}=O\left({\displaystyle \frac{1}{{\lambda }_i^n}}\right).$$

For the second and the third integrals of (53), using (108)–(111) of [21], we obtain

$$\begin{array}{cc}\hfill (p-\epsilon )\sum _{j\ne i}& {\alpha }_i^{p-1-\epsilon }{\alpha }_j{\int }_{\mathsf{\Omega }}{\delta }_i^{p-1-\epsilon }{\delta }_j{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}+\sum _{j\ne i}{\alpha }_j^{p-\epsilon }{\int }_{\mathsf{\Omega }}{\delta }_j^{p-\epsilon }{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}\hfill \\ \hfill & =c_2\sum _{j\ne i}{\alpha }_j{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial a_i}}\left({\alpha }_i^{p-1-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}+{\alpha }_j^{p-1-\epsilon }{\lambda }_j^{-\epsilon (n-2)/2}\right)\hfill \\ \hfill & +O\left(\sum _{j\ne i}{\lambda }_j|a_i-a_j|{\epsilon }_{ij}^{{\displaystyle \frac{n+1}{n-2}}}\right)+O\left(\sum _{j=1}^k{\displaystyle \frac{1}{{\lambda }_j^{n-2}}}+{\epsilon }^2+\sum _{j\ne i}{\epsilon }_{ij}^2{ln}^{{\displaystyle \frac{2(n-2)}{n}}}{\epsilon }_{ij}^{-1}\right).\hfill \end{array}$$

Concerning the fourth integral of (53), we have

$$\begin{array}{c}\hfill {\int }_{\mathsf{\Omega }}{\omega }^p{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}\le c{\int }_{\mathsf{\Omega }}{\displaystyle \frac{1}{{\lambda }_i|x-a_i|}}{\delta }_i\le {\displaystyle \frac{c}{{\lambda }_i^{n/2}}}{\int }_{\mathsf{\Omega }}{\displaystyle \frac{dx}{|x-a_i{|}^{n-1}}}\le {\displaystyle \frac{c}{{\lambda }_i^{n/2}}},\end{array}$$

Now, for the fifth integral of (53), let $\rho$ be a fixed radius such that $B(a_i,\rho )\subset \mathsf{\Omega }$, expanding $\omega$ around $a_i$, by oddness, we deduce that

$${\int }_{\mathsf{\Omega }}{\delta }_i^{p-1-\epsilon }\omega {\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}=O\left({\int }_{B(a_i,\rho )}|x-a_i|{\delta }_i^p+{\int }_{\mathsf{\Omega }\setminus B(a_i,\rho )}{\delta }_i^p\right)=O\left({\displaystyle \frac{1}{{\lambda }_i^{n/2}}}\right).$$

Lastly, we notice that the last integral of (53) can be deduced from Lemma 1. Combining the previous estimates with Lemma 1 and Proposition 4, we obtain our proposition. □

Now, we are ready to perform careful estimate of the gradient of the associated functional $J_{\epsilon }$ in the set $\mathcal{B}(k,{\mu }_0)$ with ${\mu }_0$ is a small positive real. We begin by the expansion with respect to the parameter ${\alpha }_i$. More precisely, we prove

Proposition 8. 

Let $n\ge 3$, $(\alpha ,\lambda ,a)\in \mathcal{B}(k,{\mu }_0)$ with ${\mu }_0$ positive small, $v\in F_{a,\lambda ,\omega }$ and $u={\sum }_{i=1}^k{\alpha }_i{\delta }_{(a_i,{\lambda }_i)}+{\alpha }_0\omega +v$. Then, for $1\le i\le k$, the following expansion holds.

$$\begin{array}{c}\hfill 〈J_{\epsilon }^{\prime }\left(u\right),{\delta }_{(a_i,{\lambda }_i)}〉={\alpha }_i{S}_n\left(1-{\alpha }_i^{p-1-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}\right)+O\left(R_5\left(i\right)\right),\end{array}$$

where $S_n$ is defined in Proposition 5 and where

$$R_5\left(i\right)={\parallel v_{\epsilon }\parallel }^2+\epsilon +{\displaystyle \frac{1}{{\lambda }_i^{(n-2)/2}}}+\sum _{j\ne \ell }{\epsilon }_{\ell j}+\sum _{1\le j\le k}\left({\displaystyle \frac{ln{\lambda }_j}{{\lambda }_j^{n/2}}}+{\displaystyle \frac{1}{{\lambda }_j^4}}\right)+(ifn\ge 7){\displaystyle \frac{1}{{\lambda }_i^2}}.$$

Proof. 

It follows from (52) and the fact that ${\overline{v}}_{\epsilon }\in F_{a,\lambda ,\omega }$

$$\begin{array}{c}\hfill 〈J_{\epsilon }^{\prime }\left(u\right),{\delta }_{(a_i,{\lambda }_i)}〉=\sum _{j=1}^k{\alpha }_j({\delta }_j,{\delta }_i)+{\alpha }_0(\omega ,{\delta }_i)-{\int }_{\mathsf{\Omega }}{\left|u\right|}^{p-1-\epsilon }u{\delta }_i.\end{array}$$

(77)

Observe that

$$\begin{array}{cc}\hfill & (\omega ,{\delta }_i)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & ={\int }_{\mathsf{\Omega }}\nabla \omega \nabla {\delta }_i+{\int }_{\mathsf{\Omega }}V\omega {\delta }_i={\int }_{\mathsf{\Omega }}(-\Delta +V)\omega {\delta }_i={\int }_{\mathsf{\Omega }}{\omega }^p{\delta }_i\le c{\int }_{\mathsf{\Omega }}{\delta }_i\le {\displaystyle \frac{c}{{\lambda }_i^{(n-2)/2}}},\hfill \\ \hfill & ({\delta }_i,{\delta }_i)\hfill \end{array}$$

(78)

$$\begin{array}{cc}\hfill & ={\int }_{\mathsf{\Omega }}{|\nabla {\delta }_i|}^2+{\int }_{\mathsf{\Omega }}V{\delta }_i^2=S_n+O\left((ifn=3){\displaystyle \frac{1}{{\lambda }_i}}+(ifn=4){\displaystyle \frac{ln{\lambda }_i}{{\lambda }_i^2}}+(ifn\ge 5){\displaystyle \frac{1}{{\lambda }_i^2}}\right),\hfill \end{array}$$

(79)

and, for $j\ne i$, using Lemma 6.6 of [20], we deduce that

$$\begin{array}{cc}\hfill ({\delta }_j,{\delta }_i)={\int }_{\mathsf{\Omega }}\nabla {\delta }_j\nabla {\delta }_i+{\int }_{\mathsf{\Omega }}V{\delta }_j{\delta }_i& ={\int }_{\mathsf{\Omega }}-\Delta {\delta }_i{\delta }_j+{\int }_{\partial \mathsf{\Omega }}{\delta }_j{\displaystyle \frac{\partial {\delta }_i}{\partial \nu }}+{\int }_{\mathsf{\Omega }}V{\delta }_j{\delta }_i\hfill \\ \hfill & =O\left({\epsilon }_{ij}+{\displaystyle \frac{1}{{\left({\lambda }_i{\lambda }_j\right)}^{(n-2)/2}}}\right).\hfill \end{array}$$

(80)

Combining (77)–(80) and Proposition 5, we derive Proposition 8. □

Next, we provide the expansion of the gradient of $J_{\epsilon }$ with respect to the rate of concentration ${\lambda }_i$. Namely, we have

Proposition 9. 

Let $n\ge 3$, $(\alpha ,\lambda ,a)\in \mathcal{B}(k,{\mu }_0)$ with ${\mu }_0$ positive small, $v\in F_{a,\lambda ,\omega }$ and $u={\sum }_{i=1}^k{\alpha }_i{\delta }_{(a_i,{\lambda }_i)}+{\alpha }_0\omega +v$. Then, for $1\le i\le k$, the following expansion holds.

$$\begin{array}{cc}\hfill \langle & J_{\epsilon }^{\prime }\left(u\right),{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}\rangle \hfill \\ \hfill & =-{\alpha }_{i}V\left(a_i\right)d_n{\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^2}}+\sum _{j\ne i}{\alpha }_j{c}_2{\lambda }_i{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial {\lambda }_i}}\left(1-{\alpha }_i^{p-1-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}-{\alpha }_j^{p-1-\epsilon }{\lambda }_j^{-\epsilon (n-2)/2}\right)\hfill \\ \hfill & +{\alpha }_0(\omega ,{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}})(1-{\alpha }_0^{p-1-\epsilon })+c_3\epsilon {\alpha }_i^{p-\epsilon }{\lambda }_i^{{\displaystyle \frac{-\epsilon (n-2)}{2}}}+{\alpha }_i^{p-1-\epsilon }{\lambda }_i^{{\displaystyle \frac{-\epsilon (n-2)}{2}}}{\alpha }_0{c}_1{\displaystyle \frac{\omega \left(a_i\right)}{{\lambda }_i^{(n-2)/2}}}+O\left(R_6\left(i\right)\right),\hfill \end{array}$$

with

$$R_6\left(i\right)=R_3\left(i\right)+(ifn\ge 4)\sum _{j\ne i}{\epsilon }_{ij}\left(|a_i-a_j{|}^2|ln|a_i-a_j{\left|\right|}^{{\sigma }_n}\right),$$

where $R_3\left(i\right)$, $c_2$ and $c_3$ are defined in Proposition 6,

$$d_4=c_0^2\left|{\mathbb{S}}^3\right|,d_n={\displaystyle \frac{(n-2)}{2}}{c}_0^2{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{{\left|x\right|}^2-1}{{(1+|x|}^2{)}^{n-1}}}dxifn\ge 5,{\sigma }_4=1,{\sigma }_n=0ifn\ge 5.$$

Proof. 

Using (52) and the fact that ${\overline{v}}_{\epsilon }\in F_{a,\lambda ,\omega }$, we obtain

$$\begin{array}{cc}\hfill 〈J_{\epsilon }^{\prime }\left(u\right),{\lambda }_i{\displaystyle \frac{\partial {\delta }_{(a_i,{\lambda }_i)}}{\partial {\lambda }_i}}〉& =\sum _{j=1}^k{\alpha }_j({\delta }_j,{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}})+{\alpha }_0(\omega ,{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}})-{\int }_{\mathsf{\Omega }}{\left|u\right|}^{p-1-\epsilon }u{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}\hfill \\ \hfill & ={\alpha }_i{\int }_{\mathsf{\Omega }}V{\delta }_i{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}+\sum _{j\ne i}{\int }_{\mathsf{\Omega }}{\alpha }_{j}V{\delta }_j{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}+{\alpha }_i{\int }_{\mathsf{\Omega }}\nabla {\delta }_i\nabla \left({\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}\right)\hfill \\ \hfill & +\sum _{j\ne i}{\alpha }_j{\int }_{\mathsf{\Omega }}\nabla {\delta }_j\nabla \left({\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}\right)+{\alpha }_0(\omega ,{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}})-{\int }_{\mathsf{\Omega }}{\left|u\right|}^{p-1-\epsilon }u{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}.\hfill \end{array}$$

(81)

For the first term on the right hand side of (81), observe that, for $n=3$,

$$\begin{array}{c}\hfill {\int }_{\mathsf{\Omega }}V{\delta }_i{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}=O\left({\int }_{\mathsf{\Omega }}{\delta }_i^2\right)=O\left({\displaystyle \frac{1}{{\lambda }_i}}\right)\end{array}$$

(82)

and, for $n\ge 4$, easy computations show that, for $r>0$, in $B(a_i,r)$, expanding V around $a_i$, we obtain

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}V{\delta }_i& {\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}=V\left(a_i\right){\int }_{B(a_i,r)}{\delta }_i{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}+O\left({\int }_{B(a_i,r)}{|x-a_i|}^2{\delta }_i^2\right)+O\left({\int }_{\mathsf{\Omega }\setminus B(a_i,r)}{\delta }_i^2\right)\hfill \\ \hfill & =-V\left(a_i\right)d_n{\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^2}}+O\left({\displaystyle \frac{1}{{\lambda }_i^{n-2}}}\right)+(ifn\ge 7)O\left({\displaystyle \frac{1}{{\lambda }_i^4}}\right)+(ifn=6)O\left({\displaystyle \frac{ln{\lambda }_i}{{\lambda }_i^4}}\right).\hfill \end{array}$$

(83)

Concerning the second integral on the right hand side of (81), using Lemma 6.6 of [20], we have

$$\begin{array}{c}\hfill |{\int }_{\mathsf{\Omega }}V{\delta }_j{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}|\le c{\int }_{\mathsf{\Omega }}{\delta }_j{\delta }_i\le c\left\{\begin{array}{cc}{\epsilon }_{ij}{|a_i-a_j|}^2+{\displaystyle \frac{{\epsilon }_{ij}}{min{({\lambda }_i,{\lambda }_j)}^2}}\hfill & \mathrm{if}n\ge 5,\hfill \\ {\epsilon }_{ij}|a_i-a_j{|}^2|ln|a_i-a_j\left|\right|+{\epsilon }_{ij}{\displaystyle \frac{lnmin({\lambda }_i,{\lambda }_j)}{{\left(min({\lambda }_i,{\lambda }_j)\right)}^2}}\hfill & \mathrm{if}n=4,\hfill \\ {\displaystyle \frac{1}{\sqrt{{\lambda }_i{\lambda }_j}}}\hfill & \mathrm{if}n=3.\hfill \end{array}\right.\end{array}$$

(84)

Furthermore, for the third term on the right hand side of (81), we have

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}& \nabla {\delta }_i\nabla \left({\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}\right)\hfill \\ \hfill & ={\int }_{\mathsf{\Omega }}{\delta }_i^p{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}+{\int }_{\partial \mathsf{\Omega }}{\displaystyle \frac{\partial {\delta }_i}{\partial \nu }}{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}=-{\int }_{{\mathbb{R}}^n\setminus \mathsf{\Omega }}{\delta }_i^p{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}+O\left({\displaystyle \frac{1}{{\lambda }_i^{n-2}}}\right)=O\left({\displaystyle \frac{1}{{\lambda }_i^{n-2}}}\right).\hfill \end{array}$$

(85)

Concerning the fourth integral on the right hand side of (81), using Lemma 6.4 of [20], we obtain

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}\nabla {\delta }_j\nabla \left({\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}\right)& ={\int }_{\mathsf{\Omega }}{\delta }_j^p{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}+{\int }_{\partial \mathsf{\Omega }}{\displaystyle \frac{\partial {\delta }_j}{\partial \nu }}{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}\hfill \\ \hfill & =c_2{\lambda }_i{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial {\lambda }_i}}+O\left({\epsilon }_{ij}^{{\displaystyle \frac{n}{n-2}}}ln{\epsilon }_{ij}^{-1}\right)+O\left({\displaystyle \frac{1}{{\left({\lambda }_i{\lambda }_j\right)}^{(n-2)/2}}}\right).\hfill \end{array}$$

(86)

Thus, combining (81)–(86) and Proposition 6, we easily obain the desired result. □

Now, we will give the expansion of the gradient of the functional $J_{\epsilon }$ with respect to the concentration point $a_i$. Namely, we prove the following.

Proposition 10. 

Let $n\ge 7$, $(\alpha ,\lambda ,a)\in \mathcal{B}(k,{\mu }_0)$ with ${\mu }_0$ positive small, $v\in F_{a,\lambda ,\omega }$ and $u={\sum }_{i=1}^k{\alpha }_i{\delta }_{(a_i,{\lambda }_i)}+{\alpha }_0\omega +v$. Then, for $1\le i\le k$, the following expansion holds:

$$\begin{array}{cc}\hfill 〈J_{\epsilon }^{\prime }\left(u\right),{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}〉=& {\alpha }_i{c}_4{\displaystyle \frac{\nabla V\left(a_i\right)}{{\lambda }_i^3}}+c_2\sum _{j\ne i}{\alpha }_j{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial a_i}}\left(1-{\alpha }_i^{p-1-\epsilon }{\lambda }_i^{{\displaystyle \frac{-\epsilon (n-2)}{2}}}-{\alpha }_j^{p-1-\epsilon }{\lambda }_j^{{\displaystyle \frac{-\epsilon (n-2)}{2}}}\right)\hfill \\ \hfill & +{\alpha }_0(\omega ,{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}})+O\left(R_7\left(i\right)\right)with{R}_7\left(i\right):=R_4\left(i\right)+{\displaystyle \frac{1}{{\lambda }_i}}\sum _{j\ne i}{\epsilon }_{ij},\hfill \end{array}$$

where $c_2$ and $R_4\left(i\right)$ are defined in Propositions 6, 7 and $c_4:={\displaystyle \frac{n-2}{n}}{c}_0^2{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{{\left|x\right|}^2}{{(1+|x|}^2{)}^{n-1}}}dx$.

Proof. 

Using (52) and the fact that ${\overline{v}}_{\epsilon }\in F_{a,\lambda ,\omega }$, we obtain

$$\begin{array}{cc}\hfill 〈J_{\epsilon }^{\prime }\left(u\right),{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}〉& ={\alpha }_i{\int }_{\mathsf{\Omega }}V{\delta }_i{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}+\sum _{j\ne i}{\alpha }_j{\int }_{\mathsf{\Omega }}V{\delta }_j{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}+{\alpha }_i{\int }_{\mathsf{\Omega }}\nabla {\delta }_i\nabla \left({\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}\right)\hfill \\ \hfill & +\sum _{j\ne i}{\alpha }_j{\int }_{\mathsf{\Omega }}\nabla {\delta }_j\nabla \left({\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}\right)+{\alpha }_0\left(\omega ,{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}\right)-{\int }_{\mathsf{\Omega }}{\left|u\right|}^{p-1-\epsilon }u{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}.\hfill \end{array}$$

(87)

We notice that, for $r>0$ small, let $B_i:=B(a_i,r)$, we have

$$\begin{array}{c}\hfill {\int }_{\mathsf{\Omega }}V{\delta }_i{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}={\int }_{B_i}V{\delta }_i{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}+O\left({\int }_{\mathsf{\Omega }\setminus B_i}{\delta }_i{\displaystyle \frac{1}{{\lambda }_i}}\left|{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}\right|\right)\end{array}$$

and thus by oddness, we obtain, by using Lemma 6.3 of [20],

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}V{\delta }_i{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}& ={\int }_{B_i}\nabla V\left(a_i\right){\delta }_i{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}(x-a_i)+O\left({\int }_{B_i}{\delta }_i{\displaystyle \frac{1}{{\lambda }_i}}\left|{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}\right|{|x-a_i|}^3\right)+O\left({\displaystyle \frac{1}{{\lambda }_i^{n-1}}}\right)\hfill \\ \hfill & =c_4{\displaystyle \frac{\nabla V\left(a_i\right)}{{\lambda }_i^3}}+O\left({\displaystyle \frac{1}{{\lambda }_i^5}}\right).\hfill \end{array}$$

(88)

In addition, for $j\ne i$, we have by using Lemma 6.6 of [20]

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

(89)

We also have

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}\nabla {\delta }_i\nabla \left({\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}\right)& ={\int }_{{\mathbb{R}}^n}\nabla {\delta }_i\nabla \left({\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}\right)-{\int }_{{\mathbb{R}}^n\setminus \mathsf{\Omega }}\nabla {\delta }_i\nabla \left({\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}\right)=O\left({\displaystyle \frac{1}{{\lambda }_i^{n-1}}}\right).\hfill \end{array}$$

(90)

Now, for $j\ne i$, using Lemma 6.4 of [20], we obtain

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}\nabla {\delta }_j\nabla \left({\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}\right)& =-{\int }_{\mathsf{\Omega }}\Delta {\delta }_j{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}+{\int }_{\partial \mathsf{\Omega }}{\displaystyle \frac{\partial {\delta }_j}{\partial \nu }}{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}\hfill \\ \hfill & ={\int }_{\mathsf{\Omega }}{\delta }_j^p{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}+O\left({\displaystyle \frac{1}{{\lambda }_j^{(n-2)/2}}}{\displaystyle \frac{1}{{\lambda }_i^{n/2}}}\right)\hfill \\ \hfill & =c_2{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial a_i}}+O\left({\lambda }_j|a_i-a_j|{\epsilon }_{ij}^{{\displaystyle \frac{n+1}{n-2}}}+{\displaystyle \frac{1}{{\lambda }_j^{(n-2)/2}}}{\displaystyle \frac{1}{{\lambda }_i^{n/2}}}\right).\hfill \end{array}$$

(91)

Combining (87)–(91) and Proposition 7, we easily obtain the desired result. □

Before ending this section, we perform the expansion of the gradient of the functional $J_{\epsilon }$ with respect to the parameter ${\alpha }_0$. More precisely, we have

Proposition 11. 

Let $n\ge 3$, $(\alpha ,\lambda ,a)\in \mathcal{B}(k,{\mu }_0)$ with ${\mu }_0$ positive small, $v\in F_{a,\lambda ,\omega }$ and $u={\sum }_{i=1}^k{\alpha }_i{\delta }_{(a_i,{\lambda }_i)}+{\alpha }_0\omega +v$. Then, the following expansion holds:

$$\begin{array}{cc}\hfill & 〈J_{\epsilon }^{\prime }\left(u\right),\omega 〉={\alpha }_0\left(1-{\alpha }_0^{p-1-\epsilon }\right)A+O\left(R_8\left(i\right)\right),where\hfill \\ \hfill & A:={\int }_{\mathsf{\Omega }}{\omega }^{p+1}\left(x\right)dxand{R}_8\left(i\right)=\epsilon +\sum _{i=1}^k\left({\displaystyle \frac{1}{{\lambda }_i^{(n-2)/2}}}+{\displaystyle \frac{1}{{\lambda }_i^4}}\right)+{\parallel v\parallel }^2.\hfill \end{array}$$

Proof. 

Using (52) and the fact that $v\in F_{a,\lambda ,\omega }$, we see that

$$\begin{array}{c}\hfill 〈J_{\epsilon }^{\prime }\left(u\right),\omega 〉=\sum _{i=1}^k{\alpha }_i({\delta }_i,\omega )+{\alpha }_0{\parallel \omega \parallel }^2-{\int }_{\mathsf{\Omega }}{\left|u\right|}^{p-1-\epsilon }u\omega .\end{array}$$

(92)

Note that ${\parallel \omega \parallel }^2=A$ and

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}& {\left|u\right|}^{p-1-\epsilon }u\omega \hfill \\ \hfill & ={\int }_{\mathsf{\Omega }}{\underline{u}}^{p-\epsilon }\omega +(p-\epsilon ){\int }_{\mathsf{\Omega }}{\underline{u}}^{p-1-\epsilon }v\omega +O\left({\int }_{[\underline{u}\le |v\left|\right]}{\left|v\right|}^{p-\epsilon }\omega +{\int }_{\left[\right|v|\le \underline{u}]}{\underline{u}}^{p-2-\epsilon }{v}^2\omega \right).\hfill \end{array}$$

Observe that

$${\int }_{[\underline{u}\le |v\left|\right]}{\left|v\right|}^{p-\epsilon }\omega +{\int }_{\left[\right|v|\le \underline{u}]}{\underline{u}}^{p-2-\epsilon }{v}^2\omega \le c{\int }_{\mathsf{\Omega }}{\left|v\right|}^{p+1-\epsilon }+c{\int }_{\mathsf{\Omega }}{\underline{u}}^{p-1-\epsilon }{v}^2\le c{\parallel v\parallel }^2.$$

Now, we write

$${\int }_{\mathsf{\Omega }}{\underline{u}}^{p-1-\epsilon }v\omega ={\alpha }_0^{p-1-\epsilon }{\int }_{\mathsf{\Omega }}{\omega }^{p-\epsilon }v+O\left({\int }_{\mathsf{\Omega }}{\omega }^{p-2-\epsilon }\overline{u}|v_{\epsilon }|+{\int }_{\mathsf{\Omega }}{\overline{u}}^{p-1-\epsilon }\left|v\right|\right).$$

(93)

But, we have

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}{\omega }^{p-\epsilon }v& ={\int }_{\mathsf{\Omega }}{\omega }^{p}v+O\left(\epsilon {\int }_{\mathsf{\Omega }}\left|v\right|\right)=\left(\omega ,v\right)+O\left(\epsilon \parallel v\parallel \right)=O\left(\epsilon \parallel v\parallel \right),\hfill \end{array}$$

(94)

where we use the fact that $v\in F_{a,\lambda ,\omega }$. Using (50), we deduce that

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}{\omega }^{p-2-\epsilon }\overline{u}\left|v\right|\le c{\int }_{\mathsf{\Omega }}\overline{u}\left|v\right|\le c\sum {\int }_{\mathsf{\Omega }}{\delta }_i\left|v\right|\le c\parallel v\parallel \sum R_1\left(i\right),\hfill \end{array}$$

(95)

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}|\overline{u}{|}^{p-1-\epsilon }\left|v\right|\le c\sum {\int }_{\mathsf{\Omega }}{\delta }_i^{p-1}\left|v\right|\le c\sum \parallel v\parallel {\left({\int }_{\mathsf{\Omega }}{\delta }_i^{{\displaystyle \frac{8n}{n^2-4}}}\right)}^{{\displaystyle \frac{n+2}{2n}}}\le c\parallel v\parallel \sum R_1\left(i\right)\hfill \end{array}$$

(96)

where $R_1\left(i\right)$ is defined in (41). Hence, (94), (95) and (96) imply the estimate of (93).

Now, we observe that

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}{\underline{u}}^{p-\epsilon }\omega & ={\alpha }_0^{p-\epsilon }{\int }_{\mathsf{\Omega }}{\omega }^{p+1-\epsilon }+O\left({\int }_{\mathsf{\Omega }}{\omega }^{p-\epsilon }\overline{u}+{\int }_{\mathsf{\Omega }}{\overline{u}}^{p-\epsilon }\omega \right)\hfill \\ \hfill & ={\alpha }_0^{p-\epsilon }{\int }_{\mathsf{\Omega }}{\omega }^{p+1}+O\left(\epsilon \right)+\sum _{i=1}^{k}O\left({\int }_{\mathsf{\Omega }}{\delta }_i+{\int }_{\mathsf{\Omega }}{\delta }_i^p\right)={\alpha }_0^{p-\epsilon }A+O\left(\epsilon +\sum _{i=1}^k{\displaystyle \frac{1}{{\lambda }_i^{(n-2)/2}}}\right).\hfill \end{array}$$

Combining the previous estimates, we easily obtain the result. □

5. The Non-Existence Result for Small Dimensions

The aim of this section is to prove the nonexistence of interior bubbling solutions with residual mass for small dimensions. Namely, we prove Theorem 1. We start by proving the following crucial lemma.

Lemma 2. 

Let $n\ge 3$ and $\left(u_{\epsilon }\right)$ be a sequence of solutions of $\left({\mathcal{I}}_{\epsilon }\right)$ having the form (12) and satisfying (13). Then, for all $i\in \{1,\dots ,k\}$, the following limit holds: $\epsilon ln{\lambda }_{i,\epsilon }⟶0$ as $\epsilon ⟶0$.

Proof. 

Let $u_{\epsilon }={\sum }_{j=1}^k{\alpha }_{j,\epsilon }{\delta }_{(a_{j,\epsilon },{\lambda }_{j,\epsilon })}+{\alpha }_{0,\epsilon }\omega +v_{\epsilon }$ with the properties introduced in (13). For sake of simplicity of presentation, we omit the index $\epsilon$ from the indices of the parameters. Multiplying $\left({\mathcal{I}}_{\epsilon }\right)$ by ${\delta }_i:={\delta }_{(a_i,{\lambda }_i)}$ we obtain by integrating on $\mathsf{\Omega }$

$$(u_{\epsilon },{\delta }_i):={\int }_{\mathsf{\Omega }}\nabla u_{\epsilon }·\nabla {\delta }_i+{\int }_{\mathsf{\Omega }}V{u}_{\epsilon }{\delta }_i={\int }_{\mathsf{\Omega }}(-\Delta +V)u_{\epsilon }{\delta }_i={\int }_{\mathsf{\Omega }}{u}_{\epsilon }^{p-\epsilon }{\delta }_i.$$

(97)

Using (78)–(80) and the fact that $v_{\epsilon }\in F_{a,\lambda ,\omega }$, we obtain

$$(u_{\epsilon },{\delta }_i)={\alpha }_i{S}_n+o\left(1\right).$$

(98)

Concerning the right hand side of (97), using (64), we obtain

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}{\left(\sum _{j=1}^k{\alpha }_j{\delta }_j+{\alpha }_0\omega +v_{\epsilon }\right)}^{p-\epsilon }{\delta }_i=& {\alpha }_i^{p-\epsilon }{\int }_{\mathsf{\Omega }}{\delta }_i^{p+1-\epsilon }+O\left(\sum _{j\ne i}{\int }_{\mathsf{\Omega }}\left({\delta }_i^{p-\epsilon }{\delta }_j+{\delta }_j^{p-\epsilon }{\delta }_i\right)\right)\hfill \\ \hfill & +O\left({\int }_{\mathsf{\Omega }}\left({\delta }_i^{p-\epsilon }\omega +{\delta }_i^{p-\epsilon }|v_{\epsilon }|+{\omega }^{p-\epsilon }{\delta }_i+{\left|v_{\epsilon }\right|}^{p-\epsilon }{\delta }_i\right)\right)\hfill \\ \hfill & ={\alpha }_i^{p-\epsilon }{\int }_{\mathsf{\Omega }}{\delta }_i^{p+1-\epsilon }+o\left(1\right).\hfill \end{array}$$

(99)

Finally, by standard computations, we have

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

(100)

Combining (97)–(100), the proof of the lemma follows. □

Next, we are going to prove Theorem 1. Arguing by contradiction, we assume that such a solution $u_{\epsilon }$ exists, that is $u_{\epsilon }=\sum {\alpha }_{i,\epsilon }{\delta }_{(a_{i,\epsilon },{\lambda }_{i,\epsilon })}+{\alpha }_{0,\epsilon }\omega +v_{\epsilon }$ is a solution of $\left({\mathcal{I}}_{\epsilon }\right)$ satisfying the properties introduced in (13).

Notice that, using Lemma 2, we deduce that Propositions 3, 8–11 hold true. We also notice that the left hand sides in Propositions 8–10 are equal to zero. In addition, to simplify the notation, we omit, in the sequel, the index $\epsilon$ of the variables. Without loss of generality, we can assume that ${\lambda }_1\le {\lambda }_2\le \dots \le {\lambda }_k$. Observe that

$$\begin{array}{cc}\hfill 1& -{\alpha }_0^{p-1-\epsilon }=o\left(1\right);{\alpha }_i^{p-1-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}=1+o\left(1\right)\forall i,{\lambda }_i{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial {\lambda }_i}}=O\left({\epsilon }_{ij}\right),\hfill \\ \hfill (& \omega ,{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}})\hfill \end{array}$$

(101)

$$\begin{array}{cc}\hfill & ={\int }_{\mathsf{\Omega }}\nabla \omega \nabla \left({\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}\right)+{\int }_{\mathsf{\Omega }}V\omega {\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}={\int }_{\mathsf{\Omega }}{\omega }^p{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {ł}_i}}=O\left({\int }_{\mathsf{\Omega }}{\delta }_i\right)=O\left({\displaystyle \frac{1}{{\lambda }_i^{\frac{n-2}{2}}}}\right).\hfill \end{array}$$

(102)

Thus, using (101) and (102), Proposition 9 implies that

$$\begin{array}{cc}\hfill 0=& -c_2\sum _{j\ne i}{\lambda }_i{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial {\lambda }_i}}+c_3\epsilon +{\displaystyle \frac{c_1}{{\lambda }_i^{(n-2)/2}}}\omega \left(a_i\right)\hfill \\ \hfill & +o\left(\epsilon +\sum {\displaystyle \frac{1}{{\lambda }_j^{(n-2)/2}}}+\sum _{j\ne r}{\epsilon }_{jr}\right)+(ifn\ge 4)O\left(\sum _{j\ne i}|a_i-a_j{|}^2|ln|a_i-a_j{\left|\right|}^{{\sigma }_n}{\epsilon }_{ij}\right).\hfill \end{array}$$

(103)

Note that

$$\begin{array}{c}\hfill |a_i-a_j{|}^2|ln|a_i-a_j{\left|\right|}^{{\sigma }_n}{\epsilon }_{ij}=\left\{\begin{array}{c}o\left({\epsilon }_{ij}\right)if|a_i-a_j|\to 0,\hfill \\ O\left({\displaystyle \frac{1}{{\lambda }_i^{(n-2)/2}{\lambda }_j^{(n-2)/2}}}\right)=o\left({\displaystyle \frac{1}{{\lambda }_i^{(n-2)/2}}}\right)if|a_i-a_j|\nrightarrow 0.\hfill \end{array}\right.\end{array}$$

Thus, (103) becomes

$$\begin{array}{c}\hfill -c_2\sum _{j\ne i}{\lambda }_i{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial {\lambda }_i}}+c_3\epsilon +{\displaystyle \frac{c_1}{{\lambda }_i^{(n-2)/2}}}\omega \left(a_i\right)=o\left(\sum _{j\ne r}{\epsilon }_{jr}+\sum _{j=1}^k{\displaystyle \frac{1}{{\lambda }_j^{(n-2)/2}}}+\epsilon \right).\end{array}$$

(104)

Notice that Lemma $6.5$ of [20] implies that

$$\begin{array}{c}\hfill -{\gamma }_i{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial {\lambda }_i}}-\gamma {\lambda }_j{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial {\lambda }_j}}⩾c{\epsilon }_{ij}if{\lambda }_i⩽{\lambda }_j\forall \gamma >3/2.\end{array}$$

(105)

Multiplying (104) by $2^{i-1}$ and summing over $i=1,\dots ,k$, we obtain

$$\begin{array}{c}\hfill \sum _{j\ne r}{\epsilon }_{jr}+\sum _{j=1}^k{\displaystyle \frac{1}{{\lambda }_j^{(n-2)/2}}}+\epsilon =o\left(\sum _{j\ne r}{\epsilon }_{jr}+\sum _{j=1}^k{\displaystyle \frac{1}{{\lambda }_j^{(n-2)/2}}}+\epsilon \right),\end{array}$$

(106)

where we use (105).

Clearly, (106) gives a contradiction, which completes the proof of the theorem.

6. Construction of Interior Blowing-Up Solutions with Isolated Bubbles

This section is devoted to the proof of Theorems 2 and 3. Namely, our aim is to construct solutions of $\left({\mathcal{I}}_{\epsilon }\right)$, which concentrate at k interior points with residual mass as $\epsilon$ tends to zero, with $1\le k\le N$, where N is the number of critical points of V. Let $n\ge 7$ and $y_1,\dots ,y_k$ be non-degenerate distinct critical points of V and $\omega$ be a non-degenerate solution of the limit problem $\left({\mathcal{I}}_0\right)$. To construct the desired solution, we follow [21,31,33]. To this aim, we introduce the following set:

$$\begin{array}{cc}\hfill \mathbb{A}(k,\epsilon ,\omega )=& \{(\alpha ,\lambda ,a,v)\in {\left({\mathbb{R}}_{+}\right)}^{k+1}\times {\left({\mathbb{R}}_{+}\right)}^k\times {\mathsf{\Omega }}^k\times H^1\left(\mathsf{\Omega }\right):|{\alpha }_i-1|<c\epsilon {ln}^2\epsilon ,1\le i\le k,\hfill \\ \hfill & {\displaystyle \frac{1}{c}}<{\lambda }_i^2\epsilon <c,|a_i-y_i|<c{\epsilon }^{1/5}\forall 1\le i\le k,v\in F_{a,\lambda ,\omega }and\parallel v\parallel <c\sqrt{\epsilon }\},\hfill \end{array}$$

(107)

where $F_{a,\lambda ,\omega }$ is defined by (14), and c is a fixed positive constant.

We also introduce the following map:

$${\phi }_{\epsilon }:\mathbb{A}(k,\epsilon ,\omega )\to \mathbb{R},(\alpha ,\lambda ,a,v)\to {\phi }_{\epsilon }(\alpha ,\lambda ,a,v)=J_{\epsilon }\left(\sum _{i=1}^k{\alpha }_i{\delta }_{(a_i,{\lambda }_i)}+{\alpha }_0\omega +v\right).$$

Clearly, $(\alpha ,\lambda ,a,v)$ is a critical point of ${\phi }_{\epsilon }$ if and only if $u={\sum }_{i=1}^k{\alpha }_i{\delta }_{(a_i,{\lambda }_i)}+{\alpha }_0\omega +v$ is a critical point $J_{\epsilon }$. Since the variable $v\in F_{a,\lambda ,\omega }$, the Lagrange multiplier theorem implies that the following fact holds: $(\alpha ,\lambda ,a,v)\in \mathbb{A}(k,\epsilon ,\omega )$ is a critical point of ${\phi }_{\epsilon }$ if and only if there exists $(A,B,C,D)\in {\mathbb{R}}^k\times {\mathbb{R}}^k\times {\left({\mathbb{R}}^n\right)}^k\times \mathbb{R}$ such that

$$\begin{array}{cc}\hfill & {\phi }_{\epsilon }^{\prime }=\sum _{i=1}^k{A}_i{\Psi }_i^{\prime }+\sum _{i=1}^k{B}_i{\varphi }_i^{\prime }+\sum _{i=1}^k\sum _{j=1}^n{C}_{ij}{\xi }_{ij}^{\prime }+D{\psi }_{\omega }^{\prime },\mathrm{where}\hfill \\ \hfill & {\Psi }_i(\alpha ,\lambda ,a,v)=(v,{\delta }_i),{\varphi }_i(\alpha ,\lambda ,a,v)=(v,{\lambda }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}),\hfill \\ \hfill & {\xi }_{ij}(\alpha ,\lambda ,a,v)=(v,{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_{i,j}}}),{\psi }_{\omega }(\alpha ,\lambda ,a,v)=(v,\omega ).\hfill \end{array}$$

This means that $(\alpha ,\lambda ,a,v)$ is a critical point of ${\phi }_{\epsilon }$ if and only if the following system holds:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial {\phi }_{\epsilon }}{\partial {\alpha }_i}}(\alpha ,\lambda ,a,v)=0\forall i\in \{0,1,\dots ,k\},\hfill \end{array}$$

(108)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial {\phi }_{\epsilon }}{\partial {\lambda }_i}}(\alpha ,\lambda ,a,v)=B_i(v,{\lambda }_i{\displaystyle \frac{{\partial }^2{\delta }_i}{\partial {\lambda }_i^2}})+\sum _{j=1}^n{C}_{ij}(v,{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{{\partial }^2{\delta }_i}{\partial {\lambda }_i\partial a_{ij}}})\forall i\in \{1,\dots ,k\},\hfill \end{array}$$

(109)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial {\phi }_{\epsilon }}{\partial a_i}}(\alpha ,\lambda ,a,v)=B_i(v,{\lambda }_i{\displaystyle \frac{{\partial }^2{\delta }_i}{\partial a_i\partial {\lambda }_i}})+\sum _{j=1}^n{C}_{ij}(v,{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{{\partial }^2{\delta }_i}{\partial a_i\partial a_{ij}}}),\hfill \end{array}$$

(110)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial {\phi }_{\epsilon }}{\partial v}}(\alpha ,\lambda ,a,v)=\sum _{j=1}^k{A}_j{\delta }_j+\sum _{j=1}^k{B}_j{\lambda }_j{\displaystyle \frac{\partial {\delta }_j}{\partial {\lambda }_i}}+\sum _{i=1}^k\sum _{j=1}^n{C}_{ij}{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_i}{\partial a_{i,j}}}+D\omega .\hfill \end{array}$$

(111)

To construct the desired solution, we perform a careful study of the system (108)–(111). Observe that, for $\tau \in \{{\alpha }_j,{\lambda }_i,a_i\}$, it holds that

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial {\phi }_{\epsilon }}{\partial \tau }}(\alpha ,\lambda ,a,v)=\langle J_{\epsilon }^{\prime }(\sum _{i=1}^k{\alpha }_i{\delta }_i+{\alpha }_0\omega +v),{\displaystyle \frac{\partial }{\partial \tau }}(\sum _{i=1}^k{\alpha }_i{\delta }_i+{\alpha }_0\omega +v)\rangle ,\hfill \end{array}$$

(112)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial {\phi }_{\epsilon }}{\partial v}}(\alpha ,\lambda ,a,v)=J_{\epsilon }^{\prime }(\sum _{i=1}^k{\alpha }_i{\delta }_i+{\alpha }_0\omega +v).\hfill \end{array}$$

(113)

First, we notice that the element ${\overline{v}}_{\epsilon }$, defined in Proposition 3, satisfies Equation (111). In the sequel, we write v instead of ${\overline{v}}_{\epsilon }$.

Second, Equations (108)–(113) imply that $u={\sum }_{i=1}^k{\alpha }_i{\delta }_i+{\alpha }_0\omega +v$ is a critical point of $J_{\epsilon }$ if and only if $(\alpha ,\lambda ,a)$ solves the following system:

$$\begin{array}{cc}\hfill & \langle J_{\epsilon }^{\prime }\left(u\right),\omega \rangle =0,\hfill \end{array}$$

(114)

$$\begin{array}{cc}\hfill & \langle J_{\epsilon }^{\prime }\left(u\right),{\delta }_i\rangle =0\forall i\in \{1,\dots ,k\},\hfill \end{array}$$

(115)

$$\begin{array}{cc}\hfill & \langle J_{\epsilon }^{\prime }\left(u\right),{\alpha }_i{\displaystyle \frac{\partial {\delta }_i}{\partial {\lambda }_i}}\rangle =B_i(v,{\lambda }_i{\displaystyle \frac{{\partial }^2{\delta }_i}{\partial {\lambda }_i^2}})+\sum _{j=1}^n{C}_{ij}(v,{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{{\partial }^2{\delta }_i}{\partial {\lambda }_i\partial a_{ij}}})\forall i\in \{1,\dots ,k\},\hfill \end{array}$$

(116)

$$\begin{array}{cc}\hfill & \langle J_{\epsilon }^{\prime }\left(u\right),{\alpha }_i{\displaystyle \frac{\partial {\delta }_i}{\partial a_i}}\rangle =B_i(v,{\lambda }_i{\displaystyle \frac{{\partial }^2{\delta }_i}{\partial a_i\partial {\lambda }_i}})+\sum _{j=1}^n{C}_{ij}(v,{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{{\partial }^2{\delta }_i}{\partial a_i\partial a_{ij}}})\forall i\in \{1,\dots ,k\}.\hfill \end{array}$$

(117)

Notice that, since $a_i$ is close to the critical point $y_i$ of V and ${\displaystyle \frac{1}{c}}<{\lambda }_i^2\epsilon <c$, we see that

$$\begin{array}{c}\hfill |a_i-a_j|\ge c>0and{\epsilon }_{ij}=O\left({\epsilon }^{(n-2)/2}\right).\end{array}$$

(118)

Using (118), Propositions 3, 8, 9, 10 and 11, we obtain

$$\begin{array}{c}\hfill \parallel v_{\epsilon }\parallel ;R_8\left(i\right);R_5\left(i\right)=O\left(\epsilon \right),R_6\left(i\right);R_7\left(i\right)\le c\left\{\begin{array}{cc}{\epsilon }^2\hfill & \mathrm{if}n\ge 9,\hfill \\ {\epsilon }^{n/4}|ln\epsilon |\hfill & \mathrm{if}n\in \{7,8\}.\hfill \end{array}\right.\end{array}$$

(119)

Next, we estimate the constants $A_i$, $B_i$, $C_{ij}$, and D, which appear in Equations (108)–(111). Namely, we prove the following.

Lemma 3. 

Let $(\alpha ,\lambda ,a,0)\in \mathbb{A}(k,\epsilon ,\omega )$. Then, for ε small, we have

$$\begin{array}{c}\hfill D=O\left(\epsilon {ln}^2\epsilon \right),A_i=O\left(\epsilon {ln}^2\epsilon \right),B_i=O\left(\epsilon \right)and{c}_{ij}=O\left({\epsilon }^{3/2}\right)\forall 1\le i\le k\forall 1\le j\le n.\end{array}$$

Proof. 

The proof can be conducted in the same way as the proof of Lemma 3 of [21]. Hence, we omit it here. □

Now, to simplify the study of Equations (114)–(117), we take the following change of variables:

$${\beta }_i=1-{\alpha }_i^{p-1},0\le i\le k;{\displaystyle \frac{1}{{\lambda }_i^2}}={\displaystyle \frac{c_3}{d_{n}V\left(y_i\right)}}\epsilon (1+{\gamma }_i),1\le i\le k;{\zeta }_i:=a_i-y_{i}1\le i\le k,$$

where $c_3$ and $d_n$ are defined in Propositions 6 and 9, respectively. Combining Proposition 11 and estimate (119), we obtain ${\beta }_0=O\left(\epsilon \right)$. Furthermore, Proposition 8 and estimate (119) imply that ${\beta }_i=O\left(\epsilon \right|ln\epsilon \left|\right)$ for $1\le i\le k$.

Now, using Proposition 9, Lemma 3, and estimates (116), (119), we obtain

$$c_3\epsilon -{\displaystyle \frac{d_n}{{\lambda }_i^2}}V\left(a_i\right)=O\left({\epsilon }^2(\mathrm{if}n\ge 10);{\epsilon }^{{\displaystyle \frac{n-2}{4}}}(\mathrm{if}n\in \{7,8,9\})\right).$$

Writing

$$V\left(a_i\right)=V\left(y_i\right)+O\left(\right|y_i-a_i{|}^2),$$

we obtain

$${\gamma }_i=O\left(\right|{\zeta }_i{|}^2)+O\left(\epsilon (\mathrm{if}n\ge 10);{\epsilon }^{{\displaystyle \frac{n-6}{4}}}(\mathrm{if}n\in \{7,8,9\})\right)\forall i\in \{1,\dots ,k\}.$$

Lastly, writing

$$\nabla V\left(a_i\right)=D^{2}V\left(y_i\right)({\zeta }_i,·)+O\left(\right|{\zeta }_i{|}^2)$$

and using Proposition 10, Lemma 3 and estimates (117)–(119), we obtain

$$D^{2}V\left(y_i\right)({\zeta }_i,·)=O\left(\right|{\zeta }_i{|}^2)+O\left({\epsilon }^{1/2}(\mathrm{if}n\ge 9);{\epsilon }^{{\displaystyle \frac{n-6}{4}}}|ln\epsilon |(\mathrm{if}n\in \{7,8\})\right).$$

In conclusion, we have proved that Equations (114)–(117) are equivalent to the system

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\beta }_0=O\left(\epsilon \right)\hfill \\ {\beta }_i=O\left(\epsilon \right|ln\epsilon \left|\right)1\le i\le k\hfill \\ {\gamma }_i=O\left(\right|{\zeta }_i{|}^2)+(ifn\ge 10)O\left(\epsilon \right)+(if7\le n\le 9)O\left({\epsilon }^{{\displaystyle \frac{n-6}{4}}}\right)1\le i\le k\hfill \\ D^{2}V\left(y_i\right)({\zeta }_i,·)=O\left(\right|{\zeta }_i{|}^2)+(ifn\ge 9)O\left({\epsilon }^{{\displaystyle \frac{1}{2}}}\right)+(if7\le n\le 8)O({\epsilon }^{{\displaystyle \frac{n-6}{4}}}|ln\epsilon |)1\le i\le k.\hfill \end{array}\right.\end{array}$$

Thus, applying Brouwer’s fixed point theorem, we see that the previous system has at last one solution $({\beta }_{\epsilon },{\gamma }_{\epsilon },{\zeta }_{\epsilon })$ for $\epsilon$ small, where ${\beta }_{\epsilon }=({\beta }_0,{\beta }_1,\dots ,{\beta }_k)$,${\gamma }_{\epsilon }=({\gamma }_1,\dots ,{\gamma }_k)$ and ${\zeta }_{\epsilon }=({\zeta }_1,\dots ,{\zeta }_k).$

Clearly, $u_{\epsilon }={\sum }_{i=1}^k{\alpha }_i{\delta }_i+{\alpha }_0\omega +v$ satisfies

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

Multiplying $\left({\mathcal{N}}_{\epsilon }\right)$ by ${\left(u_{\epsilon }\right)}^{-}:=max(0,-u_{\epsilon })$, and using the fact that ${\left(u_{\epsilon }\right)}^{-}\le \left|v_{\epsilon }\right|,$ we see that ${\left(u_{\epsilon }\right)}^{-}\equiv 0$ and $u_{\epsilon }\ge 0.$ Applying the maximum principal, we derive that $u_{\epsilon }$ has to be positive.

This achieves the proof of Theorem 2. Theorem 3 is a straightforward consequence of Theorem 2.

7. Construction of Clustered Blowing-Up Solutions

In this section, we prove Theorems 4 and 5. That is, our aim is to construct interior blowing-up solutions with clustered bubbles at a critical point of V. We assume that $n\ge 7$ and we begin by proving Theorem 5. To this aim, we consider $y\in \mathsf{\Omega }$ to be a critical point of V, and $(b_1,\dots ,b_k)$ to be a non-degenerate critical point of the function $F_{k,y}$ defined by (5). We follow the proof of Theorem 4. We introduce the following set:

$$\begin{array}{cc}\hfill \mathcal{u}(k,y,\epsilon ,\omega )=& \{(\alpha ,\lambda ,a,v)\in {\left({\mathbb{R}}_{+}\right)}^{k+1}\times {\left({\mathbb{R}}_{+}\right)}^k\times {\mathsf{\Omega }}^k\times H^1\left(\mathsf{\Omega }\right):|{\alpha }_i-1|<\epsilon {ln}^2\epsilon ,\hfill \\ \hfill & {\displaystyle \frac{1}{c}}<{\lambda }_i^2\epsilon <c,|a_i-y-{\epsilon }^{\gamma }\sigma b_i|<\epsilon \forall 1\le i\le k,v\in F_{a,\lambda ,\omega }and\parallel v\parallel <\sqrt{\epsilon }\},\hfill \end{array}$$

(120)

where $F_{a,\lambda ,\omega }$ is defined by (14),

$$\gamma ={\displaystyle \frac{n-4}{2n}}and\sigma ={\left({\displaystyle \frac{c_2}{c_4}}\right)}^{1/n}{\left({\displaystyle \frac{c_3}{d_{n}V\left(y\right)}}\right)}^{\gamma }$$

(121)

with $c_2$, $c_3$, $c_4$ and $d_n$ are the constants defined in Propositions 6, 9 and 10. As in the proof of Theorem 2, we see that Equations (108)–(111) and (114)–(117) are satisfied. To deal with these equations, we consider the following change of variables:

$$\begin{array}{cc}\hfill {\beta }_i=& 1-{\alpha }_i^{p-1},0\le i\le k;{\displaystyle \frac{1}{{\lambda }_i^2}}={\displaystyle \frac{c_3}{d_{n}V\left(y\right)}}\epsilon (1+{\Gamma }_i);a_i-y={\epsilon }^{\gamma }\sigma ({\tau }_i+b_i),1\le i\le k,\hfill \end{array}$$

(122)

It follows that

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

(123)

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{(n-2){\epsilon }^{\gamma +1}{e}_3^{\frac{n-2}{2}}}{{\sigma }^{n-1}{\left(d_{n}V\left(y\right)\right)}^{\frac{n-2}{2}}}}{\displaystyle \frac{({\tau }_j+b_j-{\tau }_i-b_i)}{|{\tau }_i+b_i-{\tau }_j-b_j{|}^n}}(1+R({\Gamma }_i,{\Gamma }_j))(1+O\left({\epsilon }^{1-2\gamma }\right))=O\left({\epsilon }^{\gamma +1}\right),\hfill \end{array}$$

(124)

where

$$\begin{array}{c}\hfill R({\Gamma }_i,{\Gamma }_j)={\displaystyle \frac{n-2}{4}}{\Gamma }_i+{\displaystyle \frac{n-2}{4}}{\Gamma }_j+O\left({\Gamma }_i^2\right)+O\left({\Gamma }_j^2\right).\end{array}$$

(125)

Using estimate (123), Propositions 3 and 8–11, we obtain

$$\parallel v_{\epsilon }\parallel =O\left(\epsilon \right);R_8\left(i\right)=O\left(\epsilon \right);R_5\left(i\right)=O\left(\epsilon \right);R_6\left(i\right),R_7\left(i\right)\le c\left\{\begin{array}{cc}{\epsilon }^2|ln\epsilon |\hfill & \mathrm{if}n\ge 8,\hfill \\ {\epsilon }^{7/4}|ln\epsilon |\hfill & \mathrm{if}n=7.\hfill \end{array}\right.$$

(126)

Arguing as in the proof of Lemma 3, we see that the constants $A_i^{\prime }s$,$B_i^{\prime }s$,$c_{ij}^{\prime }s$ and $D^{\prime }s$ satisfy

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

(127)

Next, we need to rewrite Equations (114)–(117) using the change of variables introduced in (122).

First, estimate (126) and Propositions 6 and 8 imply that

$${\beta }_0=O\left(\epsilon \right);{\beta }_i=O\left(\epsilon \right|ln\epsilon \left|\right)1\le i\le k.$$

(128)

Second, estimates (123), (126)–(128) and Proposition 9 imply that

$$c_3\epsilon -{\displaystyle \frac{d_{n}V\left(a_i\right)}{{\lambda }_i^2}}=O\left({\epsilon }^{2\gamma +1}\mathrm{if}n\ge 8,{\epsilon }^{5/4}\mathrm{if}n=7\right).$$

Writing

$$\begin{array}{cc}\hfill V\left(a_i\right)& =V\left(y\right)+O\left(\right|a_i{-y|}^2)=V\left(y\right)+O({\epsilon }^{2\gamma }|{\tau }_i+b_i{|}^2),\hfill \end{array}$$

we see that

$${\Gamma }_i=O\left({\epsilon }^{2\gamma }\mathrm{if}n\ge 8,{\epsilon }^{1/4}ifn=7\right)\forall 1\le i\le k$$

(129)

Lastly, using Proposition 10, and estimates (124), (126)–(128), we obtain

$$\begin{array}{cc}\hfill c_4& {\displaystyle \frac{\nabla V\left(a_i\right)}{{\lambda }_i^3}}-{\displaystyle \frac{c_2(n-2)}{{\sigma }^{n-1}}}{\left({\displaystyle \frac{c_3}{d_{n}V\left(y\right)}}\right)}^{{\displaystyle \frac{n-2}{2}}}{\displaystyle \frac{{\epsilon }^{\gamma +1}}{{\lambda }_i}}\sum _{j\ne i}{\displaystyle \frac{({\tau }_j+b_j-{\tau }_i-b_i)}{|{\tau }_j+b_j-{\tau }_i-b_i{|}^n}}\hfill \\ \hfill & \times \left(1+{\displaystyle \frac{(n-2)}{4}}{\Gamma }_i{\displaystyle \frac{(n-2)}{4}}{\Gamma }_j\right)\hfill \\ \hfill & =O\left({|\Gamma |}^2{\epsilon }^{\gamma +3/2}\right)+O\left({\epsilon }^2|ln\epsilon |\mathrm{if}n\ge 8,{\epsilon }^{7/4}|ln\epsilon |\mathrm{if}n=7\right).\hfill \end{array}$$

Writing

$$\begin{array}{cc}\hfill \nabla V\left(a_i\right)& =D^{2}V\left(y\right)({\epsilon }^{\gamma }\sigma ({\tau }_i+b_i),·)+O({\epsilon }^{2\gamma }|{\tau }_i+b_i{|}^2),\hfill \end{array}$$

we obtain

$$\begin{array}{cc}\hfill D^{2}V\left(y\right)& ({\tau }_i+b_i,·)-(n-2)\sum _{j\ne i}{\displaystyle \frac{({\tau }_j+b_j-{\tau }_i-b_i)}{|{\tau }_j+b_j-{\tau }_i-b_i{|}^n}}(1+{\displaystyle \frac{(n-6)}{4}}{\Gamma }_i\hfill \\ \hfill & +{\displaystyle \frac{(n-2)}{4}}{\Gamma }_j+O({\Gamma }_i^2+{\Gamma }_j^2))\hfill \\ \hfill & =O({\epsilon }^{\gamma }|{\tau }_i+b_i{|}^2)+(ifn\ge 8)O({\epsilon }^{1/2-\gamma }|ln\epsilon |)+(ifn=7)O({\epsilon }^{1/4-\gamma }|ln\epsilon |)\hfill \\ \hfill & =O\left({\epsilon }^{1/2-\gamma }|ln\epsilon |\mathrm{if}n\ge 8,{\epsilon }^{1/4-\gamma }|ln\epsilon |\mathrm{if}n=7\right).\hfill \end{array}$$

Note that

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\tau }_j+b_j-{\tau }_i-b_i}{|{\tau }_j+b_j-{\tau }_i-b_i{|}^n}}\hfill \\ \hfill & ={\displaystyle \frac{b_j-b_i}{|b_j-b_i{|}^n}}-n<{\displaystyle \frac{b_j-b_i}{|b_j-b_i{|}^2}},{\tau }_j-{\tau }_i>{\displaystyle \frac{(b_j-b_i)}{|b_j-b_i{|}^n}}+{\displaystyle \frac{{\tau }_j-{\tau }_i}{|b_j-b_i{|}^n}}+O(\sum |{\tau }_{\epsilon }{|}^2).\hfill \end{array}$$

Thus, we obtain

$$\begin{array}{cc}\hfill D^{2}V\left(y\right)({\tau }_i,·)-& (n-2)\sum _{j\ne i}{\displaystyle \frac{{\tau }_j-{\tau }_i}{|b_j-b_i{|}^n}}-(n-2)\sum _{j\ne i}{\displaystyle \frac{b_j-b_i}{|b_j-b_i{|}^n}}\left({\displaystyle \frac{(n-6)}{4}}{\Gamma }_i+{\displaystyle \frac{(n-2)}{4}}{\Gamma }_j\right)\hfill \\ \hfill & +n(n-2)\sum _{j\ne i}<{\displaystyle \frac{b_j-b_i}{|b_j-b_i{|}^2}},{\tau }_j-{\tau }_i>{\displaystyle \frac{(b_j-b_i)}{|b_j-b_i{|}^n}}+O\left(\sum _l|{\tau }_l{|}^2+\sum _l{\left|{\Gamma }_l\right|}^2\right)\hfill \\ \hfill & =O\left({\epsilon }^{1/2-\gamma }|ln\epsilon |\mathrm{if}n\ge 8,{\epsilon }^{1/4-\gamma }|ln\epsilon |\mathrm{if}n=7\right).\hfill \end{array}$$

(130)

Thus, we have proved that, for $\epsilon$ small, Equations (114)–(117) are equivalent to the system whose equations are defined by (128)–(130). In addition, it is easy to see that Equation (130) for $1\le i\le k$ is equivalent to

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2}}{D}^2{F}_{k,y}(b_1,\dots ,b_k)({\tau }_1,\dots ,{\tau }_k)-(n-2)\wedge \hfill \\ \hfill & =O\left(\sum _{l=1}^k|{\tau }_l{|}^2+\sum _{l=1}^k{\left|{\Gamma }_l\right|}^2\right)+O\left({\epsilon }^{1/2-\gamma }|ln\epsilon |\mathrm{if}n\ge 8,{\epsilon }^{1/4-\gamma }|ln\epsilon |\mathrm{if}n=7\right),\hfill \end{array}$$

where $\wedge :=({\wedge }_1,\dots ,{\wedge }_k)$ with

$${\wedge }_i=\sum _{j\ne i}{\displaystyle \frac{(b_j-b_i)}{|b_j-b_i{|}^n}}\left({\displaystyle \frac{(n-6)}{4}}{\Gamma }_i+{\displaystyle \frac{(n-2)}{4}}{\Gamma }_j\right)\mathrm{for}1\le i\le k.$$

Using the fact that $(b_1,\dots ,b_k)$ is a non-degenerate critical point of $F_{k,y}$, we see that our system (123)–(130) admits a solution $({\beta }_{\epsilon },{\Gamma }_{\epsilon },{\tau }_{\epsilon })$ for $\epsilon$ small. This shows that $\left({\mathcal{N}}_{\epsilon }\right)$ admits a solution $u_{\epsilon ,y,\omega }={\sum }_{i=1}^k{\alpha }_{i,\epsilon }{\delta }_{(a_{i,\epsilon },{\lambda }_{i,\epsilon })}+v_{\epsilon }+{\alpha }_{0,\epsilon }\omega$, which satisfies the desired properties. Furthermore, arguing as before, we prove that this solution is positive. This completes the proof of Theorem 4.

To prove Theorem 5, let $y_1,y_2$ be two non-degenerate critical points of V and $a_{1,\epsilon },a_{2,\epsilon }\in \mathsf{\Omega }$ satisfying $|a_{1,\epsilon }-y_1|=o\left(1\right)$ and $|a_{2,\epsilon }-y_2|=o\left(1\right)$. It follows that

$$\begin{array}{c}\hfill ({\delta }_{(a_{1,\epsilon },{\lambda }_{1,\epsilon })},{\delta }_{(a_{2,\epsilon },{\lambda }_{2,\epsilon })})=O\left({\epsilon }_{ij}\right)=O\left({\displaystyle \frac{1}{{\left({\lambda }_{1,\epsilon }{\lambda }_{2,\epsilon }\right)}^{(n-2)/2}}}\right)=O\left({\epsilon }^{(n-2)/2}\right).\end{array}$$

This shows that the interaction between the bubbles in two different blocks is negligible for $n\ge 7$ and enters in the remainder of Equations (128)–(130). This implies that we may separate each block alone. Hence, arguing as in the proof of Theorem 4, we obtain the desired results.

8. Conclusions

By employing refined asymptotic estimates of the gradient of the associated Euler–Lagrange functional, we were able to prove the non-existence of interior bubbling solutions with residual mass for small dimensions. In contrast, we constructed both simple and non-simple interior bubbling solutions with residual mass for large dimensions. This construction enabled us to establish the multiplicity results for the problem $\left({\mathcal{P}}_{\epsilon }\right)$. However, several promising avenues for further research and open questions remain:

(i)

Location of the Concentration Points and the Rate of Concentration: This thesis focuses on the construction of interior bubbling solutions with residual mass. A natural extension of this work would be to examine the asymptotic profile of these solutions to fully understand their behavior as the concentration points and their rates of concentration.

(ii)

Impact of the Nature of Critical Points: The solutions presented in this thesis rely on the assumption that the critical point y of the potential V is non-degenerate, and that the corresponding function $F_{k,y}$ has a non-degenerate critical point, where $F_{k,y}$ is defined by Equation (6), and k represents the number of bubbles clustered at point y. A natural question arises: what happens if y is degenerate, or if $F_{k,y}$ lacks non-degenerate critical points?

(iii)

Impact of the Subcritical Exponent: This work addresses a slightly subcritical exponent in the context of Sobolev embedding. Future studies could extend the analysis to slightly supercritical exponents, that is, when $\epsilon <0$ but is close to zero.