Boundary Concentrated Solutions for an Elliptic Equation with Subcritical Nonlinearity

Abstract

In this paper, we consider the nonlinear Neumann problem $\left(Q_{\epsilon }\right):-\Delta u+V\left(x\right)u=u^{{\displaystyle \frac{n+2}{n-2}}-\epsilon }$, with $u>0$ in $\Omega$ and $\partial u/\partial \nu =0$ on $\partial \Omega$, where $\Omega$ is a bounded regular domain in ${\mathbb{R}}^n$, with $n\ge 4$, $\epsilon$ is a small positive parameter, and V is a non-constant smooth positive function on $\overline{\Omega }$. Assuming the flatness of the boundary near the critical points of the restriction of the function V on the boundary, we construct boundary peak solutions with isolated bubbles, leading to a multiplicity result for $\left(Q_{\epsilon }\right)$. The proof of our results relies on expanding the gradient of the associated functional and testing the equation with the appropriate vector fields, which yields constraints for the concentration points and blow-up rates. A thorough analysis of these constraints leads to our results.

1. Introduction and Main Results

Let us consider the following boundary value Neumann problem

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

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

Problems of this type arise in various areas of applied sciences, such as the Gierer–Meinhardt model for biological pattern formation [1] and the Keller–Segel model in chemotaxis [2].

A substantial body of literature has focused on this problem when the exponent q is a fixed real and $\mu$ is a parameter. An interesting feature of problem $\left(P_{\mu }\right)$ is the existence of families of solutions, $u_{\mu }$, that display point concentration phenomena as the parameter $\mu$ varies. This means that solutions display concentration peaks around one or more points of or ∂, while remaining negligibly small elsewhere.

For subcritical q, that is $1<q<{\displaystyle \frac{n+2}{n-2}}$, the only solution of $\left(P_{\mu }\right)$ for small $\mu$ is the constant solution. However, non-constant solutions emerge for large $\mu$, which blow up at one or more points as $\mu \to \infty$ [3]. The least energy solution $u_{\mu }$ of $\left(P_{\mu }\right)$ for $\mu \to \infty$ must look like

$${\mu }^{1/(q-1)}U\left(\sqrt{\mu }(x-a_{\mu })\right),$$

where U is the unique radial solution of

$$-\Delta U+U=U^q,U>0\mathrm{in}{\mathbb{R}}^n,\underset{\left|y\right|\to \infty }{lim}U\left(y\right)=0$$

(1)

and $a_{\mu }\in \partial \Omega$ tends to a point that maximizes the mean curvature of the boundary [3,4,5,6]. Higher energy solutions of $\left(P_{\mu }\right)$ exhibiting this asymptotic profile near one or more points on the boundary or in the interior of have been constructed and analyzed in numerous works, such as [3,7,8,9,10] and their references. In particular, solutions with any specified number of concentration points, both interior and boundary, are known to exist as $\mu \to \infty$.

The case of the critical exponent, i.e., $q={\displaystyle \frac{n+2}{n-2}}$, is quite distinct. On the one hand, if $n\in \{4,5,6\}$ and $\mu$ is small, $\left(P_{\mu }\right)$ admits non-constant solutions [11,12,13]. On the other hand, the limiting Equation (1), which arises when examining the asymptotic profile of the least energy solution as $\mu \to \infty$, has no solutions. However, in this case, least energy solutions $u_{\mu }$ to $\left(P_{\mu }\right)$ do exist for sufficiently large values of $\mu$, with the following properties: there exist $a_{\mu }\in \partial \Omega$ (as in the subcritical cases) which converges to a point that maximizes the mean curvature of the boundary, and ${\lambda }_{\mu }$ (with ${\lambda }_{\mu }^{-1}=o\left({\mu }^{-1}\right)$ as $\mu \to \infty$) such that

$$\parallel u_{\mu }-{\delta }_{a_{\mu },{\lambda }_{\mu }}{\parallel }_{H^1(\Omega )}\to 0\mathrm{as}\mu \to \infty ,$$

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

$${\delta }_{a,\lambda }\left(x\right)=c_0{\displaystyle \frac{{\lambda }^{(n-2)/2}}{(1+{\lambda }^2{|x-a|}^2{)}^{(n-2)/2}}}\mathrm{with}{c}_0={\left[n(n-2)\right]}^{(n-2)/4}$$

(2)

which are the only solutions [14] of

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

Higher energy solutions to $\left(P_{\mu }\right)$ with boundary concentration, such as $\mu \to \infty$, have been constructed, with their dimension-dependent concentration rates ${\lambda }_{\mu }$ analyzed in various works, such as [15,16,17,18,19,20,21,22,23,24] and the references therein. Unlike the subcritical case, it is important to note that at least one concentration point must lie on the boundary [25].

Another research question related to problem $\left(P_{\mu }\right)$ is the study of the concentration phenomenon by fixing $\mu$ and allowing the exponent q to approach the critical exponent, that is, $q={\displaystyle \frac{n+2}{n-2}}\pm \epsilon$, where $\epsilon$ is a small positive parameter. This question was first addressed by Rey and Wei. For $n\ge 4$ and $q={\displaystyle \frac{n+2}{n-2}}+\epsilon$, they demonstrated the existence of a solution that blows up at a boundary point maximizing the mean curvature of the boundary [26]. They also constructed a solution that blows up at a boundary point minimizing the mean curvature when $n\ge 4$ and $q={\displaystyle \frac{n+2}{n-2}}-\epsilon$ [26]. Additionally, they showed the existence of single interior blow-up solutions when $n=3$ [27]. Recently, it was shown that, unlike in dimension 3, problem $\left(P_{\mu }\right)$ has no solution exhibiting blow-up only at interior points when $n\ge 4$ and $q={\displaystyle \frac{n+2}{n-2}}+\epsilon$, with $\epsilon$ being a small positive real number [28]. Recently, in [29], the authors replaced the constant $\mu$ with a function V and studied the problem.

$$\left(Q_{\epsilon }\right)\left\{\begin{array}{cc}-\Delta u+Vu=u^{{\displaystyle \frac{n+2}{n-2}}-\epsilon },u>0\hfill & \mathrm{in}\Omega ,\hfill \\ {\displaystyle \frac{\partial u}{\partial \nu }}=0,\hfill & \mathrm{on}\partial \Omega ,\hfill \end{array}\right.$$

where $\Omega$ is a bounded and smooth open set of ${\mathbb{R}}^n$ with $n\ge 3$, V is a positive $C^2$-function on $\overline{\Omega }$ and $\epsilon$ is a small positive parameter.

They constructed simple interior bubbling solutions and also demonstrated the existence of interior bubbling solutions with clustered bubbles. The concentration points of both simple and clustered interior bubbling solutions converge to the critical points of the function V as $\epsilon \to 0$.

In all results concerning the construction of solutions that blow up on the boundary, it is evident that the mean curvature $\mathcal{H}$ of the boundary plays a crucial role. This raises the following natural question: what happens when this function vanishes? The aim of this paper is to provide an answer to this question. Specifically, we assume that the boundary near the critical points of the restriction of the function V on the boundary is flat, and our goal is to construct solutions that concentrate at these points. To clarify this approach, we first introduce the following definition: we will say that $\Omega$ satisfies the condition $\left(\mathcal{F}\right)$ at a point $y\in \partial \Omega$, if there exists a radius $\rho >0$ such that

$$\Omega \cap B(y,2\rho )=B^{+}(y,2\rho ),$$

(3)

where $B^{+}(y,2\rho )$ denotes a half ball.

We note that, near such a point y, the function $\mathcal{H}$ vanishes. Under this assumption, our result is expressed as follows:

Theorem 1. 

Let $n\ge 4$, V be a positive $C^2$-function on $\overline{\Omega }$ and ${\xi }_1,\cdots ,{\xi }_q$ be q non-degenerate critical points of $V_b:=V_{|\partial \Omega }$. We assume that Ω satisfies the condition $\left(\mathcal{F}\right)$ at each point ${\xi }_i$. Then, there exists a positive real ${\epsilon }_0$ such that, for each $0<\epsilon \le {\epsilon }_0$, problem $\left(Q_{\epsilon }\right)$ has a solution $u_{\epsilon }\rightharpoonup 0$ (converging weakly to zero) blowing up at the points ${\xi }_i$’s and satisfying,

$$\underset{\eta \to 0}{lim}\underset{\epsilon \to 0}{lim}{\int }_{B({\xi }_i,\eta )\cap \Omega }{u}_{\epsilon }^{2n/(n-2)}={\displaystyle \frac{1}{2}}{S}_n\forall i\in \{1,\cdots ,q\},$$

where

$$S_n:={\left[n(n-2)\right]}^{n/2}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{1}{{(1+|x|}^2{)}^n}}dx.$$

(4)

Remark 1. 

In fact, the solution $u_{\epsilon }$ constructed in Theorem 1 is provided with the precise blow-up rate and the locations of the concentration points. More precisely, there exist ${\mu }_{1,\epsilon }$, …, ${\mu }_{q,\epsilon }$ having the same order as ${\epsilon }^{-1/2}$ for $n\ge 5$, and as ${\epsilon }^{-1/2}{|ln\epsilon |}^{1/2}$ for $n=4$ and q points $a_{j,\epsilon }\to {\xi }_{i_j}$ for all $j\in \{1,\cdots ,q\}$ such that

$$||u_{\epsilon }-\sum _{j=1}^q{\delta }_{a_{j,\epsilon },{\mu }_{j,\epsilon }}\left|\right|\to 0,as\epsilon \to 0.$$

Theorem 1 enables us to derive the following multiplicity result related to the number of non-degenerate critical points of the restriction of V to the boundary.

Theorem 2. 

Let $n\ge 4$ and $V:\overline{\Omega }\to \mathbb{R}$ be a positive $C^2$-function such that the restriction of V to the boundary, $V_{|\partial \Omega }$, has q non-degenerate critical points ${\xi }_1,\cdots ,{\xi }_q$. We assume that satisfies condition $\left(\mathcal{F}\right)$ at each ${\xi }_i$. Then, for a sufficiently small positive ε, the number of solutions to $\left(Q_{\epsilon }\right)$ that concentrate on the boundary is at least $2^q-1$.

Remark 2. 

The minimal regularity assumption on function V required to ensure the validity of our results is as follows: $V\in C^0\left(\overline{\Omega }\right)$ and V is a $C^2$-function near the boundary.

Note that the constructions of boundary concentration solutions in the literature rely on the fact that the mean curvature of the boundary is non-zero at the concentration points, which is not the case due to the assumption $\left(\mathcal{F}\right)$ (see Equation (3)). To overcome this difficulty, the proof of our results is based on the relationships governing the blow-up behavior of the solutions. These relationships are derived by performing an asymptotic expansion of the gradient of the associated functional and testing the equation with appropriate vector fields, which leads to constraints on the concentration points and the corresponding blow-up rates. A thorough analysis of these constraints leads to our results.

The remainder of the paper is organized as follows: In Section 2, we present a two-parameters family of approximate solutions to the problem $\left(Q_{\epsilon }\right)$. Section 3 is devoted to the optimization of the infinite-dimensional part of the solutions. In Section 4, we perform an asymptotic expansion of the gradient of the Euler–Lagrange functional associated with $\left(Q_{\epsilon }\right)$. Section 5 contains the proof of our main results. Section 6 explores possible avenues for future research. Finally, the proofs require some technical integral estimates, which, for the convenience of the reader, are deferred to the appendix in Appendix A.

2. The Approximate Solution

In this section, we introduce the appropriate approximate solutions around which we will identify a true solution to the problem. For $a\in \partial \Omega$ and $\mu >0$, we define the projection $U_{a,\mu }$ of the function ${\delta }_{a,\mu }$ by

$$\left\{\begin{array}{cc}{\displaystyle -\Delta U_{a,\mu }+V{U}_{a,\mu }={\delta }_{a,\mu }^{\frac{n+2}{n-2}}}\hfill & \mathrm{in} \Omega ,\hfill \\ {\displaystyle \partial U_{a,\mu }/\partial \nu =0}\hfill & \mathrm{on} \partial \Omega \hfill \end{array}\right.$$

(5)

and we set

$${\psi }_{a,\mu }:={\delta }_{a,\mu }-U_{a,\mu }.$$

(6)

Note that the advantage of this projection, $U_{a,\mu }$, is that it allows us to handle the small dimensions. The aim of the rest of this section is to estimate the functions $U_{a,\mu }$ and ${\psi }_{a,\mu }$. To do this, let $G_V$ be the Green function defined by, for $b\in \overline{\Omega }$,

$$-\Delta G_V(b,.)+V{G}_V(b,.)={\rho }_n{\delta }_b\mathrm{in}\Omega ,{\displaystyle \frac{\partial G_V(b,.)}{\partial \nu }}=0\mathrm{on}\partial \Omega ,$$

(7)

where ${\rho }_n=(n-2)\mathrm{meas}\left({\mathbb{S}}^{n-1}\right)$ and ${\delta }_b$ denotes the Dirac mass at point b.

Following the proof of Lemma $3,2$ of [26], we see that function $G_V$ satisfies

$$|G_V(b,x)|\le {\displaystyle \frac{c}{{|x-b|}^{n-2}}}\forall x\ne b.$$

(8)

Now, we recall the following inequalities which are extracted from [30].

$$\begin{array}{cc}\hfill & {\int }_{\Omega }{\displaystyle \frac{1}{{|\zeta -b|}^{n-2}}}{\delta }_{a,\mu }\left(\zeta \right)d\zeta \le c{\chi }_1(a,\mu ,b){\delta }_{a,\mu }\left(b\right)\forall b\in \overline{\Omega },\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & {\int }_{\Omega }{\displaystyle \frac{1}{{|\zeta -b|}^{n-2}}}{\displaystyle \frac{1}{\mu }}|{\displaystyle \frac{\partial {\delta }_{a,\mu }\left(\zeta \right)}{\partial a}}|d\zeta \le c({\mu }^{-2}+|a-b|{\mu }^{-1}){\delta }_{a,\mu }\left(b\right)\forall b\in \overline{\Omega },\hfill \end{array}$$

(10)

where

$${\chi }_1(a,\mu ,b)=\left\{\begin{array}{cc}{\mu }^{-2}+{|b-a|}^2\hfill & \mathrm{if}n\ge 5,\hfill \\ {\mu }^{-2}{ln\mu +|b-a|}^2|ln|b-a\left|\right|\hfill & \mathrm{if}n=4.\hfill \end{array}\right.$$

(11)

We are now in a position to provide the estimates for the function ${\psi }_{a,\mu }$, defined by (6). Indeed, we have the following:

Proposition 1. 

Let $a\in \partial \Omega$ and $\mu >0$ be a large real number. We assume that Ω satisfies the condition $\left(\mathcal{F}\right)$ at point a (see (3)). Then, the following estimates hold

(i) 

${\displaystyle \left|{\psi }_{a,\mu }\left(x\right)\right|\le C{\chi }_1(a,\mu ,x){\delta }_{a,\mu }\left(x\right)\forall x\in \overline{\Omega }}$;

(ii) 

${\displaystyle \mu \left|\frac{\partial {\psi }_{a,\mu }}{\partial \mu }\left(x\right)\right|\le C{\chi }_1(a,\mu ,x){\delta }_{a,\mu }\left(x\right)\forall x\in \overline{\Omega }}$;

(iii) 

${\displaystyle \frac{1}{\mu }\left|\frac{\partial {\psi }_{a,\mu }}{\partial a}\left(x\right)\right|\le C({\mu }^{-2}+|x-a|{\mu }^{-1}){\delta }_{a,\mu }\left(x\right)\forall x\in \overline{\Omega }}$.

Proof. 

First, we will focus on proving $\left(i\right)$. We observe that ${\psi }_{a,\mu }$ satisfies

$$(-\Delta +V){\psi }_{a,\mu }=V{\delta }_{a,\mu }\mathrm{in}\Omega ,{\displaystyle \frac{\partial {\psi }_{a,\mu }}{\partial \nu }}={\displaystyle \frac{\partial {\delta }_{a,\mu }}{\partial \nu }}\mathrm{on}\partial \Omega .$$

(12)

Furthermore, let ${\Gamma }_1:=\partial \Omega \cap B(a,\rho )$. By Assumption $\left(\mathcal{F}\right)$ (see (3)), ${\Gamma }_1$ is contained in the tangent space to ∂ at point a. Thus, it is easy to see that ${\nu }_x$ is a constant vector and $(x-a).{\nu }_x=0$ for each $x\in {\Gamma }_1$, and therefore, we obtain

$${\displaystyle \frac{\partial {\delta }_{a,\lambda }}{\partial \nu }}\left(x\right)=\left\{\begin{array}{cc}0\hfill & \mathrm{if}x\in {\Gamma }_1,\hfill \\ O\left({\mu }^{-(n-2)/2}\right)\hfill & \mathrm{if}x\in \partial \Omega \setminus {\Gamma }_1,\hfill \end{array}\right.$$

(13)

since, for $x\in \partial \Omega \setminus {\Gamma }_1$, we have $|x-a|\ge \rho$. Then, ${\psi }_{a,\mu }$ can be written as follows:

$${\rho }_n{\psi }_{a,\mu }\left(x\right)={\int }_{\Omega }{G}_V(x,\zeta )V\left(\zeta \right){\delta }_{a,\mu }\left(\zeta \right)d\zeta +O\left({\mu }^{-(n-2)/2}{\int }_{\partial \Omega \setminus {\Gamma }_1}\left|G_V(x,\zeta )\right|d\zeta \right).$$

The estimate of the first integral can be deduced from (9) (by using (8) and the fact that V is bounded). The other integral is $O\left({\mu }^{-(n-2)/2}\right)$ and it is easy to see that

$${\mu }^{-(n-2)/2}\le C({\mu }^{-2}{+|x-a|}^2){\delta }_{a,\mu }\left(x\right)\forall x\in \overline{\Omega }.$$

(14)

Hence, the proof of Claim $\left(i\right)$ is completed. The other claims follow in the same way. □

Remark 3. 

As an immediate consequence of Proposition 1, there exists a radius $\zeta >0$ (independent of μ and a) such that

$$|{\psi }_{a,\mu }|\le {\displaystyle \frac{1}{2}}{\delta }_{a,\mu }inB(a,\zeta )and{U}_{a,\mu }\ge {\displaystyle \frac{1}{2}}{\delta }_{a,\mu }inB(a,\zeta ).$$

(15)

Corollary 1. 

Under the assumptions of Proposition 1, we have

$$|{\psi }_{a,\mu }|;\mu \left|{\displaystyle \frac{\partial {\psi }_{a,\mu }}{\partial \mu }}\right|;\left|U_{a,\mu }\right|;\mu \left|{\displaystyle \frac{\partial U_{a,\mu }}{\partial \mu }}\right|\le C{\delta }_{a,\mu },$$

$${\displaystyle \frac{1}{\mu }}\left|{\displaystyle \frac{\partial {\psi }_{a,\mu }}{\partial a}}\right|\le {\displaystyle \frac{C}{\mu }}{\delta }_{a,\mu };{\displaystyle \frac{1}{\mu }}\left|{\displaystyle \frac{\partial U_{a,\mu }}{\partial a}}\right|\le {\displaystyle \frac{C}{\mu }}\left|{\displaystyle \frac{\partial {\delta }_{a,\mu }}{\partial a}}\right|+{\displaystyle \frac{C}{\mu }}{\delta }_{a,\mu }.$$

3. Optimization with Respect to the Infinite Dimensional Part

As is typical in this type of problem, we first focus on the infinite-dimensional part of the solutions to $\left(Q_{\epsilon }\right)$, showing that it is negligible in comparison to the concentration parameters. This is the main goal of this section. To this aim, notice that $\left(Q_{\epsilon }\right)$ is a variational problem. Indeed, the solutions of ($Q_{\epsilon }$) are the positive critical points of the functional

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

(16)

defined on $H^1(\Omega )$ equipped with the inner product $⟨.,.⟩$ and its corresponding norm $\parallel .\parallel$ defined by

$$⟨u,w⟩={\int }_{\Omega }\nabla u\nabla w+{\int }_{\Omega }{Vuw\mathrm{and}\parallel u\parallel }^2={\int }_{\Omega }{\left(\right|\nabla u|}^2+V{u}^2).$$

Since V is a positive $C^2$-function on $\overline{\Omega }$, the norm $\parallel .\parallel$ is equivalent to the norm ${\parallel .\parallel }_{H^1}$ of $H^1(\Omega )$.

Now, let $n\ge 4$ and ${\xi }_1,...,{\xi }_q$ be q distinct points in $\partial \Omega$. We assume that $\Omega$ satisfies the condition $\left(\mathcal{F}\right)$ (see (3)) at each point ${\xi }_i$. Let $\rho \in (0,\zeta ]$ (where $\zeta$ is introduced in Remark 3) be such that

$$B({\xi }_i,2\rho )\cap B({\xi }_j,2\rho )=\varnothing \forall i\ne j\mathrm{and}\Omega \cap B({\xi }_i,2\rho )=B^{+}({\xi }_i,2\rho )\forall i.$$

(17)

Let ${\rho }_0>0$ be a small real and $M_1$ be a large constant. We introduce the following set

$$\begin{array}{cc}\hfill {\aleph }_S(q,{\rho }_0):=\{(\alpha ,a,\mu )\in {(0,\infty )}^q\times {(\partial \Omega )}^q& \times {({\rho }_0^{-1},\infty )}^q:|a_i-{\xi }_i|<\rho ;|{\alpha }_i-1|<{\rho }_0;\hfill \\ \hfill & M_1^{-1}\le {\displaystyle \frac{{ln}^{\sigma \left(n\right)}\left({\mu }_i\right)}{\epsilon {\mu }_i^2}}\le M_1\forall i\in \{1,\cdots ,q\}\}\hfill \end{array}$$

(18)

where ${\sigma }_4=1$ and ${\sigma }_n=0$ for $n\ge 5$. First, we remark that, for $(\overline{\alpha },a,\mu )\in {\aleph }_S(q,{\rho }_0)$ with ${\overline{\alpha }}_i=1$ for each i, it follows that $\epsilon ln{\mu }_i$ is small for each $i\in \{1,\cdots ,q\}$. Therefore, by using Taylor expansion, we derive that, for each $x\in \Omega$,

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

(19)

In addition, for $(\overline{\alpha },a,\mu )\in {\aleph }_S(q,{\rho }_0)$, we denote

$$F_{a,\mu }:=span\{U_{a_i,{\mu }_i},{\mu }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\mu }_i}},{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\tau }_{i,j}}},i=1,...,q,j=1,...,n-1\}$$

(20)

where, for $i\in \{1,\cdots ,q\}$, the ${\tau }_{i,j}$’s build an orthonormal system of coordinates on the tangent space to $\partial \Omega$ at point $a_i$. In this paper, we say that $v\in F_{a,\mu }^{\perp }$ if

$$⟨v,w⟩:={\int }_{\Omega }\nabla v\nabla w+{\int }_{\Omega }Vvw=0\forall w\in F_{a,\mu }.$$

(21)

For $(\alpha ,a,\mu )\in {\aleph }_S(q,{\rho }_0)$ and $v\in F_{a,\mu }^{\perp }$ with $\parallel v\parallel <{\rho }_0$, we denote

$$U:=\sum _{i=1}^q{\alpha }_i{U}_{a_i,{\mu }_i}\mathrm{and}u:=\sum _{i=1}^q{\alpha }_i{U}_{a_i,{\mu }_i}+v=U+v.$$

(22)

Using the fact that, for $\gamma >2$ and $s,t\in \mathbb{R}$, we have

$${|s+t|}^{\gamma }={\left|s\right|}^{\gamma }+{\gamma \left|s\right|}^{\gamma -2}st+{\displaystyle \frac{\gamma (\gamma -1)}{2}}{\left|s\right|}^{\gamma -2}{t}^2+\left\{\begin{array}{cc}O\left(\right|s{|}^{\gamma -3}\left|t{|}^3\right)\hfill & \mathrm{if}2\left|t\right|<\left|s\right|,\hfill \\ O\left(\right|t{|}^{\gamma })\hfill & \mathrm{if}\left|s\right|\le 2\left|t\right|,\hfill \end{array}\right.$$

we derive that (with $\gamma =p+1-\epsilon :=(2n/(n-2\left)\right)-\epsilon >2$ for $\epsilon$ small)

$$\begin{array}{cc}\hfill I_{\epsilon }\left(u\right)& ={\displaystyle \frac{1}{2}}{\parallel U\parallel }^2+{\displaystyle \frac{1}{2}}{\parallel v\parallel }^2-{\displaystyle \frac{1}{p+1-\epsilon }}{\int }_{\Omega }{\left|U\right|}^{p+1-\epsilon }\hfill \\ \hfill & -{\int }_{\Omega }{\left|U\right|}^{p-1-\epsilon }Uv-{\displaystyle \frac{p-\epsilon }{2}}{\int }_{\Omega }{\left|U\right|}^{p-1-\epsilon }{v}^2+{o\left(\right|\left|v\right||}^2)\hfill \\ \hfill & =I_{\epsilon }\left(U\right)-f_{\epsilon }\left(v\right)+{\displaystyle \frac{1}{2}}{Q}_{\epsilon }\left(v\right)+R\left(v\right)\hfill \end{array}$$

(23)

where $f_{\epsilon }$ is a linear form, $Q_{\epsilon }$ is a quadratic form defined by

$$f_{\epsilon }\left(v\right):={\int }_{\Omega }{\left|U\right|}^{p-1-\epsilon }Uv\mathrm{and}{Q}_{\epsilon }\left(v\right):={\parallel v\parallel }^2-(p-\epsilon ){\int }_{\Omega }{\left|U\right|}^{p-1-\epsilon }{v}^2$$

(24)

and $R\left(v\right)$ satisfies

$$R\left(v\right)={o(\parallel v\parallel }^2);R^{\prime }\left(v\right)=o(\parallel v\parallel );R^{″}\left(v\right)=o\left(1\right).$$

(25)

We observe that, since $(\alpha ,a,\mu )\in {\aleph }_S(q,{\rho }_0)$, using (17), we derive that $|a_i-a_j|\ge 2\rho$ for each $i\ne j$. Let $B_i:=B(a_i,\rho )$. It follows that $B_i\cap B_j=\varnothing$ for each $i\ne j$, $B_i\subset B({\xi }_i,2\rho )$ and

$$\begin{array}{cc}\hfill & U_{a_k,{\mu }_k}\le {\displaystyle \frac{C}{{\mu }_k^{(n-2)/2}}}\mathrm{in}\Omega \setminus B_k\mathrm{and}\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill & \sum _{k=1}^q{\alpha }_k{U}_{a_k,{\mu }_k}={\alpha }_i{U}_{a_i,{\mu }_i}+\sum _{j\ne i}O\left({\displaystyle \frac{1}{{\mu }_j^{(n-2)/2}}}\right)\mathrm{in}\cap B_i.\hfill \end{array}$$

(27)

Now, we deal with the linear form $f_{\epsilon }$.

Lemma 1. 

Let $(\alpha ,a,\mu )\in {\aleph }_S(q,{\rho }_0)$ and $v\in F_{a,\mu }^{\perp }$ with $\parallel v\parallel$ small. It holds that

$$|f_{\epsilon }\left(v\right)|\le C\parallel v\parallel \left(\epsilon +\sum {\displaystyle \frac{1}{{\mu }_j^2}}{ln}^{{\sigma }_n}{\mu }_j\right),$$

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

Proof. 

Using (26), (27) and the fact that

$${|s+t|}^{\gamma }(s+t)={\left|s\right|}^{\gamma }s+{O\left(\right|s|}^{\gamma }\left|t\right|+{\left|t\right|}^{\gamma +1})for\gamma >0,s,t\in \mathbb{R},$$

(28)

we derive that

$$\begin{array}{cc}\hfill f_{\epsilon }\left(v\right)& =\sum _{i=1}^q\left[{\alpha }_i^{p-\epsilon }{\int }_{B_i}{\left|U_{a_i,{\mu }_i}\right|}^{p-1-\epsilon }{U}_{a_i,{\mu }_i}v+\sum _{j\ne i}O\left({\displaystyle \frac{1}{{\mu }_j^{\frac{n-2}{2}}}}{\int }_{B_i}|U_{a_i,{\mu }_i}{|}^{p-1-\epsilon }\left|v\right|\right)\right]\hfill \\ \hfill & +O\left(\sum _{j=1}^q{\displaystyle \frac{1}{{\mu }_j^{\frac{n+2}{2}}}}{\int }_{\Omega }\left|v\right|\right)\hfill \\ \hfill & =\sum _{i=1}^q{\alpha }_i^{p-\epsilon }{\int }_{B_i}{\left|U_{a_i,{\mu }_i}\right|}^{p-1-\epsilon }{U}_{a_i,{\mu }_i}v+O\left(\sum _{j=1}^q{\displaystyle \frac{\left|\right|v\left|\right|}{{\mu }_j^{\frac{(n+2)}{2}}}}+\sum _{j\ne i}{\displaystyle \frac{1}{{\mu }_j^{\frac{(n-2)}{2}}}}{\int }_{B_i}{\delta }_{a_i,{\mu }_i}^{p-1-\epsilon }\left|v\right|\right).\hfill \end{array}$$

(29)

Note that, using (19) and Holder’s inequality, we deduce that

$${\int }_{B_i}{\delta }_{a_i,{\mu }_i}^{p-1-\epsilon }\left|v\right|\le C\left|\right|v\left|\right|{\left({\int }_{B_i}{\delta }_{a_i,{\mu }_i}^{{\displaystyle \frac{8n}{n^2-4}}}\right)}^{{\displaystyle \frac{n+2}{2n}}}\le C\left|\right|v\left|\right|{\chi }_2\left({\mu }_i\right)$$

(30)

with

$${\chi }_2\left(\mu \right):=\left\{\begin{array}{cc}{\mu }^{(2-n)/2}\hfill & \mathrm{if}n\le 5,\hfill \\ {\mu }^{-2}{(ln\mu )}^{2/3}\hfill & \mathrm{if}n=6,\hfill \\ {\mu }^{-2}\hfill & \mathrm{if}n\ge 7.\hfill \end{array}\right.$$

(31)

It remains to estimate the first integral in (29). Using Proposition 1, Remark 3 and (28), we obtain

$${\int }_{B_i}{\left|U_{a_i,{\mu }_i}\right|}^{p-1-\epsilon }{U}_{a_i,{\mu }_i}v={\int }_{\Omega }{\delta }_{a_i,{\mu }_i}^{p-\epsilon }v+O\left({\displaystyle \frac{1}{{\mu }_i^{\frac{n+2}{2}}}}{\int }_{\Omega \setminus B_i}\left|v\right|+{\int }_{B_i}{\chi }_1(a_i,{\mu }_i,x){\delta }_{a_i,{\mu }_i}^p\left|v\right|\right).$$

(32)

For the last integral in (32), we have, for $n\ge 5$,

$${\int }_{\Omega }{\chi }_1(a,\mu ,x){\delta }_{a,\mu }^p\left(x\right)\left|v\left(x\right)\right|dx\le {\displaystyle \frac{C}{{\mu }^2}}{\int }_{\Omega }{\delta }_{a,\mu }^p\left|v\right|+C{\int }_{\Omega }{|x-a|}^2{\delta }_{a,\mu }^p\left|v\right|\le C{\displaystyle \frac{1}{{\mu }^2}}\left|\right|v\left|\right|$$

(33)

by using the Holder inequality. For $n=4$, we have

$$\begin{array}{cc}\hfill {\int }_{\Omega }{\chi }_1(a,\mu ,x){\delta }_{a,\mu }^p\left(x\right)\left|v\left(x\right)\right|dx& \le C{\displaystyle \frac{ln\mu }{{\mu }^2}}{\int }_{\Omega }{\delta }_{a,\mu }^p\left|v\right|+C{\int }_{\Omega }{|x-a|}^2|ln|x-a\left|\right|{\delta }_{a,\mu }^p\left|v\right|\hfill \\ \hfill & \le c{\displaystyle \frac{ln\mu }{{\mu }^2}}\left|\right|v\left|\right|.\hfill \end{array}$$

(34)

Finally, using (19), we deduce that

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

since $v\in F_{a,\mu }^{\perp }$ and $(\partial U_{a_i,{\mu }_i}/\partial \nu )=0$ on $\partial \Omega$. Thus, (32) becomes

$${\int }_{B_i}{\left|U_{a_i,{\mu }_i}\right|}^{p-1-\epsilon }{U}_{a_i,{\mu }_i}v=O\left(\parallel v\parallel \left(\epsilon +{\displaystyle \frac{1}{{\mu }_i^{(n+2)/2}}}+{\displaystyle \frac{{ln}^{{\sigma }_n}\left({\mu }_i\right)}{{\mu }_i^2}}\right)\right).$$

(35)

Combining the previous estimates, the proof follows. □

Now, we will focus on the quadratic form $Q_{\epsilon }$ defined in (24).

Proposition 2. 

Let $(\overline{\alpha },a,\mu )\in {\aleph }_S(q,{\rho }_0)$ with ${\overline{\alpha }}_i=1$ for each i. Then, there exists ${\beta }_1>0$ such that

$$Q_{\epsilon }\left(v\right)\ge {\beta }_1{\parallel v\parallel }^2\forall v\in F_{a,\mu }^{\perp }.$$

Proof. 

For $i\in \{1,...,q\}$, let $B_i:=B({\xi }_i,2\rho )$, where $\rho$ is defined in the assumption $\left(\mathcal{F}\right)$ (see (17)). Using (26), (17) and the fact that

$${|s+t|}^{\gamma }={\left|s\right|}^{\gamma }+{O\left(\right|s|}^{\gamma -1}\left|t\right|+{\left|t\right|}^{\gamma })\forall s,t\in \mathbb{R},\forall \gamma >0,$$

(36)

we deduce that

$$\begin{array}{cc}\hfill Q_{\epsilon }\left(v\right)=& \sum _{i=1}^q\left\{{\int }_{B_i\cap \Omega }{\left(\right|\nabla v|}^2+V{v}^2)-p{\int }_{B_i\cap \Omega }{\left|{\alpha }_i{U}_{a_i,{\mu }_i}\right|}^{p-1-\epsilon }{v}^2\right\}\hfill \\ \hfill & +{\int }_{\Omega \setminus (\cup B_j)}{\left(\right|\nabla v|}^2+V{v}^2{)+o(\left|\right|v\left|\right|}^2).\hfill \end{array}$$

(37)

First, using (6), (36) and (19), it follows that

$$\begin{array}{cc}\hfill {\int }_{B_i\cap \Omega }{\left|U_{a_i,{\mu }_i}\right|}^{p-1-\epsilon }{v}^2& ={\int }_{B_i\cap \Omega }{\delta }_{a_i,{\mu }_i}^{p-1}{v}^2\hfill \\ \hfill & +O\left({\int }_{B_i\cap \Omega }{\delta }_{a_i,{\mu }_i}^{p-2}|{\psi }_{a_i,{\mu }_i}|v^2+{\int }_{B_i\cap \Omega }{\left|{\psi }_{a_i,{\mu }_i}\right|}^{p-1-\epsilon }{v}^2\right)+{o\left(\right|\left|v\right||}^2).\hfill \end{array}$$

(38)

Note that, using Proposition 1, we deduce that

$${\int }_{B_i\cap \Omega }|{\psi }_{a_i,{\mu }_i}{|}^{p-1-\epsilon }{v}^2+{\int }_{B_i\cap \Omega }{\delta }_{a_i,{\mu }_i}^{p-2}|{\psi }_{a_i,{\mu }_i}|v^2={o\left(\right|\left|v\right||}^2).$$

Second, expanding V around ${\xi }_i$, we deduce that

$$\begin{array}{cc}\hfill {\int }_{B_i\cap \Omega }V{v}^2& =V\left({\xi }_i\right){\int }_{B_i\cap \Omega }{v}^2+O\left({\int }_{B_i\cap \Omega }|x-{\xi }_i|v^2\right)=V\left({\xi }_i\right){\int }_{B_i\cap \Omega }{v}^2+O\left(\rho {\int }_{B_i\cap \Omega }{v}^2\right).\hfill \end{array}$$

Thus, (37) becomes (by choosing $\rho$ small)

$$\begin{array}{cc}\hfill & Q_{\epsilon }\left(v\right)=\sum _{i=1}^q{Q}_i\left(v\right)+{\int }_{\Omega \setminus (\cup B_j)}{\left(\right|\nabla v|}^2+V{v}^2{)+o(\left|\right|v\left|\right|}^2)\mathrm{with}\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill & Q_i\left(v\right)={\int }_{B_i\cap \Omega }{|\nabla v|}^2+V\left({\zeta }_i\right){\int }_{B_i\cap \Omega }{v}^2-p{\int }_{B_i\cap \Omega }{\delta }_{a_i,{\mu }_i}^{p-1}{v}^2.\hfill \end{array}$$

(40)

Since $\Omega$ satisfies the assumption ($\mathcal{F}$) at ${\xi }_i$, it follows that $B_i\cap \Omega$ is a half ball. To simplify the presentation, we can assume that ${\xi }_i=0$ and ${\nu }_{{\xi }_i}=-e_n$, which imply that $\partial \Omega \cap B_i\subset {\mathbb{R}}^{n-1}\times \left\{0\right\}$. Hence, for $x=(x^{\prime },x_n)\in B_i:=B(0,2\rho )$, it follows that $x\in B_i\cap \Omega$ if $x_n>0$. Let

$$\mathrm{for}x=(x^{\prime },x_n)\in B_i,\tilde{v}\left(x\right)=\left\{\begin{array}{cc}v\left(x\right)\hfill & \mathrm{if}{x}_n>0,\hfill \\ v(x^{\prime },-x_n)\hfill & \mathrm{if}{x}_n<0.\hfill \end{array}\right.$$

(41)

It is easy to see that

$$\tilde{v}\in H^1\left(B_i\right),2Q_i\left(v\right)={\tilde{Q}}_i\left(\tilde{v}\right):={\int }_{B_i}{|\nabla \tilde{v}|}^2+V\left({\zeta }_i\right){\int }_{B_i}{\left(\tilde{v}\right)}^2-p{\int }_{B_i}{\delta }_{a_i,{\mu }_i}^{p-1}{\left(\tilde{v}\right)}^2.$$

(42)

At this step, we need to apply Proposition 1 of [29] with $N=1$ and $K=V\left({\zeta }_i\right)$. In fact, we observe that point $a_i$ satisfies $d(a_i,\partial B_i)\ge \rho$, but the function $\tilde{v}$ does not satisfy the orthogonality assumptions required in [29]. To this aim, we decompose $\tilde{v}$ as follows

$$\tilde{v}:={\gamma }_1{\delta }_{a_i,{\mu }_i}+{\gamma }_2{\mu }_i{\displaystyle \frac{\partial {\delta }_{a_i,{\mu }_i}}{\partial {\mu }_i}}+\sum _{j=1}^n{\sigma }_j^{\prime }{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial {\delta }_{a_i,{\mu }_i}}{\partial {\left(a_i\right)}_j}}+{\tilde{v}}_{\perp }\mathrm{with}{\tilde{v}}_{\perp }\in E_i$$

(43)

where

$$E_i=\{w\in H^1\left(B_i\right):{\int }_{B_i}\nabla w\nabla {\delta }_{a_i,{\mu }_i}={\int }_{B_i}\nabla w\nabla {\displaystyle \frac{\partial {\delta }_{a_i,{\mu }_i}}{\partial {\mu }_i}}={\int }_{B_i}\nabla w\nabla {\displaystyle \frac{\partial {\delta }_{a_i,{\mu }_i}}{\partial {\left(a_i\right)}_j}}=0,j\le n\}.$$

We notice that ${\tilde{v}}_{\perp }$ satisfies the assumptions of Proposition 1 of [29] with $N=1$ and $K=V\left({\zeta }_i\right)$. Thus, applying this proposition, there exists ${\beta }_0>0$ such that

$${\tilde{Q}}_i\left({\tilde{v}}_{\perp }\right)\ge {\beta }_0\left({\int }_{B_i}{|\nabla {\tilde{v}}_{\perp }|}^2+V\left({\zeta }_i\right){\int }_{B_i}{\left({\tilde{v}}_{\perp }\right)}^2\right).$$

(44)

In addition, on the one hand, using (43), it follows that

$${\int }_{B_i}\nabla \tilde{v}\nabla {\delta }_{a_i,{\mu }_i}=c_1{\gamma }_1+o\left(\right|{\gamma }_1|+|{\gamma }_2|+\sum |{\sigma }_j^{\prime }\left|\right).$$

(45)

On the other hand, using (41) and the fact that $v\in F_{a,\mu }^{\perp }$, we obtain

$$\begin{array}{cc}\hfill {\int }_{B_i}\nabla \tilde{v}\nabla {\delta }_{a_i,{\mu }_i}& ={\int }_{B_i}\tilde{v}{\delta }_i^p+O\left({\displaystyle \frac{1}{{\mu }_i^{(n-2)/2}}}{\int }_{\partial B_i}\left|\tilde{v}\right|\right)\hfill \\ \hfill & =2{\int }_{B_i\cap \Omega }v{\delta }_i^p+O\left({\displaystyle \frac{1}{{\mu }_i^{(n-2)/2}}}{\left[{\int }_{B_i}|\nabla \tilde{v}{|}^2+{\int }_{B_i}{\left|\tilde{v}\right|}^2\right]}^{1/2}\right)\hfill \\ \hfill & =2{\int }_{\Omega }v{\delta }_i^p+O\left({\displaystyle \frac{1}{{\mu }_i^{(n+2)/2}}}{\int }_{\Omega \setminus B_i}\left|v\right|\right)+o(\parallel v\parallel )=o(\parallel v\parallel ).\hfill \end{array}$$

Therefore, we obtain the following:

$${\gamma }_1=o\left(\right|{\gamma }_1|+|{\gamma }_2|+\sum |{\sigma }_j^{\prime }|+|\left|v\right|\left|\right).$$

(46)

In the same way, we derive the estimate of ${\gamma }_2$ and ${\sigma }_j^{\prime }$ for $1\le j\le n-1$ and we obtain

$${\gamma }_2;{\sigma }_j^{\prime }=o\left(\right|{\gamma }_1|+|{\gamma }_2|+\sum _{1\le k\le n}|{\sigma }_k^{\prime }|+|\left|v\right|\left|\right).$$

(47)

However, for ${\sigma }_n^{\prime }$, the argument is different. In fact, we remark that $\tilde{v}$ is an even function with respect to the variable $x_n$. But

$${\displaystyle \frac{\partial {\delta }_{a_i,{\mu }_i}}{\partial {\left(a_i\right)}_n}}=c_0(n-2){\displaystyle \frac{{\mu }_i^{(n+2)/2}{x}_n}{(1+{\mu }_i^2|x-a_i{{|}^2)}^{n/2}}}$$

(48)

which is an odd function with respect to variable $x_n$. Hence, we obtain

$$\begin{array}{cc}\hfill {\int }_{B_i}\nabla \tilde{v}\nabla {\displaystyle \frac{\partial {\delta }_{a_i,{\mu }_i}}{\partial {\left(a_i\right)}_n}}& ={\int }_{B_i}\tilde{v}(-\Delta ){\displaystyle \frac{\partial {\delta }_{a_i,{\mu }_i}}{\partial {\left(a_i\right)}_n}}+{\int }_{\partial B_i}\tilde{v}{\displaystyle \frac{\partial }{\partial \nu }}\left({\displaystyle \frac{\partial {\delta }_{a_i,{\mu }_i}}{\partial {\left(a_i\right)}_n}}\right)\hfill \\ \hfill & ={\displaystyle \frac{n+2}{n-2}}{\int }_{B_i}\tilde{v}{\delta }_{a_i,{\mu }_i}^{{\displaystyle \frac{4}{n-2}}}{\displaystyle \frac{\partial {\delta }_{a_i,{\mu }_i}}{\partial {\left(a_i\right)}_n}}+O\left({\displaystyle \frac{1}{{\mu }_i^{(n-2)/2}}}{\int }_{\partial B_i}\left|\tilde{v}\right|\right)\hfill \\ \hfill & =O\left({\displaystyle \frac{1}{{\mu }_i^{(n-2)/2}}}{\left[{\int }_{B_i}|\nabla \tilde{v}{|}^2+{\int }_{B_i}{\left|\tilde{v}\right|}^2\right]}^{1/2}\right)=o\left(\right|\left|v\right|\left|\right)\hfill \end{array}$$

by using Equation (41) in the last line. Furthermore, using (43), we obtain

$${\int }_{B_i}\nabla \tilde{v}\nabla {\displaystyle \frac{\partial {\delta }_{a_i,{\mu }_i}}{\partial {\left(a_i\right)}_n}}=C_3{\sigma }_n^{\prime }+o\left(\right|{\gamma }_1|+|{\gamma }_2|+\sum |{\sigma }_k^{\prime }\left|\right).$$

(49)

Thus, (47) is also true for ${\sigma }_n^{\prime }$, and therefore, we obtain

$$\begin{array}{cc}\hfill & {\int }_{B_i}|\nabla \tilde{v}{|}^2={\int }_{B_i}|\nabla {\tilde{v}}_{\perp }{|}^2+{o\left(\right|\left|v\right||}^2),\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill & {\int }_{B_i}|\tilde{v}{|}^2={\int }_{B_i}|{\tilde{v}}_{\perp }{|}^2+{o\left(\right|\left|v\right||}^2+\left|\right|{\tilde{v}}_{\perp }{\left|\right|}_{L^2\left(B_i\right)}^2).\hfill \end{array}$$

(51)

Finally, using (50), (51), (41) and (43), we derive that

$$\begin{array}{cc}\hfill {\tilde{Q}}_i\left(\tilde{v}\right)& ={\int }_{B_i}|\nabla {\tilde{v}}_{\perp }{|}^2+V\left({\zeta }_i\right){\int }_{B_i}|{\tilde{v}}_{\perp }{|}^2-p{\int }_{B_i}{\delta }_{a_i,{\mu }_i}^{p-1}{\tilde{v}}_{\perp }^2+{o\left(\right|\left|v\right||}^2+\parallel {\tilde{v}}_{\perp }{\parallel }_{H^1\left(B_i\right)}^2)\hfill \\ \hfill & ={\tilde{Q}}_i\left({\tilde{v}}_{\perp }\right)+{o\left(\right|\left|v\right||}^2+\left|\right|{\tilde{v}}_{\perp }{\left|\right|}_{H^1\left(B_i\right)}^2)\hfill \\ \hfill & \ge {\beta }_0({\int }_{B_i}|\nabla {\tilde{v}}_{\perp }{|}^2+V\left({\zeta }_i\right){\int }_{B_i}|{\tilde{v}}_{\perp }{|}^2{)+o(\parallel v\parallel }^2+\parallel {\tilde{v}}_{\perp }{\parallel }_{H^1\left(B_i\right)}^2)\hfill \\ \hfill & \ge {\displaystyle \frac{3}{2}}{\beta }_0{\int }_{B_i\cap \Omega }{|\nabla v|}^2+V\left({\zeta }_i\right){\int }_{B_i\cap \Omega }{v}^2+{o\left(\right|\left|v\right||}^2).\hfill \end{array}$$

(52)

Now, using (39), (42) and (52), we obtain

$$\begin{array}{cc}\hfill Q_i\left(v\right)& \ge \sum _{i=1}^q{\displaystyle \frac{1}{2}}{\displaystyle \frac{3}{2}}{\beta }_0\left({\int }_{B_i\cap \Omega }{|\nabla v|}^2+V\left({\zeta }_i\right){\int }_{B_i\cap \Omega }{v}^2\right)+{\int }_{\Omega \setminus (\cup B_i)}{|\nabla v|}^2+V{v}^2+{o\left(\right|\left|v\right||}^2)\hfill \\ \hfill & \ge {\beta }_1{\left|\right|v\left|\right|}^2\hfill \end{array}$$

for some positive constant ${\beta }_1$. This achieves the proof of the proposition. □

As a consequence of (23), (24), Proposition 2 and Lemma 1, we deduce the following result

Proposition 3. 

Let $\epsilon >0$ be small. For each $(\alpha ,a,\mu )\in {\aleph }_S(q,{\rho }_0)$, there exists $\overline{v}:={\overline{v}}_{(\epsilon ,\alpha ,a,\mu )}\in F_{a,\mu }^{\perp }$ satisfying

$$⟨\nabla I_{\epsilon }\left(\sum _{i=1}^q{\alpha }_i{U}_{a_i,{\mu }_i}+\overline{v}\right),h⟩=0\forall h\in F_{a,\mu }^{\perp }and\parallel \overline{v}\parallel \le C\epsilon +C\sum {\displaystyle \frac{1}{{\mu }_j^2}}{ln}^{{\sigma }_n}{\mu }_j.$$

4. Expansion of the Gradient of the Associated Functional

In this section, we are going to perform an asymptotic expansion of the gradient of the functional $I_{\epsilon }$. To this aim, we consider $(\alpha ,a,\mu )\in {\aleph }_S(q,{\rho }_0)$ and $v\in F_{a,\mu }^{\perp }$. For $i\in \{1,...,q\}$, let $B_i^{+}:=B(a_i,\rho )\cap \Omega$, where $\rho$ is the real defined in (3). We remark that $B_i^{+}$ is a half ball since $\Omega$ satisfies the assumption $\left(\mathcal{F}\right)$ at each point ${\xi }_i$. We start by the following expansion.

Lemma 2. 

For $i\in \{1,\cdots ,q\}$, let $\phi \in \{U_{a_i,{\mu }_i},{\mu }_i\partial U_{a_i,{\mu }_i}/\partial {\mu }_i,{{\mu }_i}^{-1}\partial U_{a_i,{\mu }_i}/\partial a_i\}$.

(i) 

In $\Omega \setminus B_i^{+}$, it holds that

$${|U+v|}^{p-\epsilon }\left|{\phi }_i\right|\le c\left(\sum _{j=1}^q{\delta }_{a_j,{\mu }_j}^p+{\left|v\right|}^{p-\epsilon }\right)\times \left\{\begin{array}{cc}{\displaystyle {\mu }_i^{-n/2}}\hfill & if{\phi }_i={\mu }_i^{-1}\partial U_{a_i,{\mu }_i}/\partial a_i,\hfill \\ {\displaystyle {\mu }_i^{(2-n)/2}}\hfill & otherwise.\hfill \end{array}\right.$$

(ii) 

In the set $B_i^{+}$, it holds that

$$\begin{array}{cc}\hfill {|U+v|}^{p-1-\epsilon }(U+v){\phi }_i=& {\alpha }_i^{p-\epsilon }{\left|U_{a_i,{\mu }_i}\right|}^{p-1-\epsilon }{U}_{a_i,{\mu }_i}{\phi }_i\hfill \\ \hfill & +(p-\epsilon )|{\alpha }_i{U}_{a_i,{\mu }_i}{|}^{p-1-\epsilon }\left(\sum _{j\ne i}{\alpha }_j{U}_{a_j,{\mu }_j}+v\right){\phi }_i\hfill \\ \hfill & +O\left({\delta }_{a_i,{\mu }_i}^{p-1}[v^2+\sum {\displaystyle \frac{1}{{\mu }_j^{n-2}}}]+{\left|v\right|}^{p+1-\epsilon }+\sum {\displaystyle \frac{1}{{\mu }_j^n}}\right).\hfill \end{array}$$

Proof. 

Using (26), (19), Corollary 1 and the fact that

$$|\sum t_j{|}^{\gamma }\le c\sum {\left|t_j\right|}^{\gamma }\forall t_j\in \mathbb{R},\forall \gamma >0,$$

(53)

the proof of assertion $\left(i\right)$ follows.

Concerning the second assertion, we notice that Remark 3 and Corollary 1 imply that

$$|{\phi }_i|\le c{U}_{a_i,{\mu }_i}\mathrm{in}{B}_i\mathrm{for}\rho \mathrm{small}.$$

(54)

Furthermore, for each $\gamma >0,s,t,\zeta \in \mathbb{R}$ with $\left|\zeta \right|\le c\left|s\right|$, we have

$${|s+t|}^{\gamma }(s+t)\zeta ={\left|s\right|}^{\gamma }s\zeta +{(\gamma +1)\left|s\right|}^{\gamma }t\zeta +{O\left(\right|s|}^{\gamma }{t}^2+{\left|t\right|}^{\gamma +2}).$$

(55)

Taking $s={\alpha }_i{U}_{a_i,{\mu }_i}$ and $\zeta ={\phi }_i$, the proof follows from (55) and (54) by using (53), (26), (19) and Corollary 1. □

We are now in a position to provide the expansion in terms of the gluing parameter ${\alpha }_i$. Namely, we prove

Proposition 4. 

Let $(\alpha ,a,\mu )\in {\aleph }_S(q,{\rho }_0)$ and $v\in F_{a,\mu }^{\perp }$. For each $i\in \{1,\cdots ,q\}$, it holds that

$$\begin{array}{cc}\hfill & ⟨\nabla I_{\epsilon }(U+v),U_{a_i,{\mu }_i}⟩={\displaystyle \frac{{\alpha }_i}{2}}{S}_n\left(1-c_0^{-\epsilon }{\mu }_i^{-\epsilon (n-2)/2}{\alpha }_i^{p-1-\epsilon }\right)+O\left(\epsilon +{\displaystyle \frac{{ln}^{{\sigma }_n}\left({\mu }_i\right)}{{\mu }_i^2}}+{\chi }_0\left(i\right)\right),\hfill \\ \hfill & where{\chi }_0\left(i\right):=\sum {\displaystyle \frac{1}{{\left({\mu }_i{\mu }_j\right)}^{(n-2)/2}}}+{\left|\right|v\left|\right|}^2+\sum {\displaystyle \frac{1}{{\mu }_j^n}},{\sigma }_4=1,{\sigma }_n=0forn\ge 5\hfill \end{array}$$

(56)

and where $S_n$ is defined in (4).

Proof. 

Notice that we have

$$⟨\nabla I_{\epsilon }\left(u\right),h⟩=⟨u,h⟩-{\int }_{\Omega }{\left|u\right|}^{p-1-\epsilon }uh\forall u\in H^1(\Omega ),\forall h\in H^1(\Omega ).$$

(57)

In this proof, we will take $u=U+v$ and $h=U_{a_i,{\mu }_i}$. Since $v\in F_{a,\mu }^{\perp }$, using Lemmas A4 and A5, we deduce that

$$\begin{array}{cc}\hfill ⟨U+v,U_{a_i,{\mu }_i}⟩& ={\alpha }_i\left|\right|U_{a_i,{\mu }_i}{\left|\right|}^2+\sum _{j\ne i}{\alpha }_j⟨U_{a_j,{\mu }_j},U_{a_i,{\mu }_i}⟩\hfill \\ \hfill & ={\displaystyle \frac{{\alpha }_i}{2}}{S}_n+O\left({\displaystyle \frac{{ln}^{{\sigma }_n}\left({\mu }_i\right)}{{\mu }_i^2}}\right)+\sum _{j\ne i}O\left({\displaystyle \frac{1}{{\left({\mu }_i{\mu }_j\right)}^{(n-2)/2}}}\right).\hfill \end{array}$$

(58)

This achieves the first part of (57). Concerning the second part, applying Lemma 2 and (26), we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }{|U+v|}^{p-1-\epsilon }(U+v)U_{a_i,{\mu }_i}=& {\alpha }_i^{p-\epsilon }{\int }_{B_i^{+}}{\left|U_{a_i,{\mu }_i}\right|}^{p+1-\epsilon }\hfill \\ \hfill & +(p-\epsilon ){\int }_{B_i^{+}}{\left|{\alpha }_i{U}_{a_i,{\mu }_i}\right|}^{p-1-\epsilon }v{U}_{a_i,{\mu }_i}+O\left({\chi }_0\left(i\right)\right).\hfill \end{array}$$

(59)

The last integral in (59) is computed in (35). For the other one, using (19) and Lemma A1, we obtain

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

(60)

Combining (57)–(60), the proof of the lemma follows. □

Next, we perform the expansion with respect to the rate of concentration.

Proposition 5. 

Let $(\alpha ,a,\mu )\in {\aleph }_S(q,{\rho }_0)$ and $v\in F_{a,\mu }^{\perp }$. For each $i\in \{1,\cdots ,q\}$, it holds that

$$\begin{array}{cc}\hfill ⟨\nabla I_{\epsilon }& (U+v),{\mu }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\mu }_i}}⟩\hfill \\ \hfill & =c_0^{-\epsilon }{\mu }_i^{-\epsilon (n-2)/2}{\alpha }_i^{p-\epsilon }{\displaystyle \frac{\overline{c}}{2}}\epsilon +{\displaystyle \frac{1}{2}}{\alpha }_i{d}_n{\displaystyle \frac{{ln}^{{\sigma }_n}\left({\mu }_i\right)}{{\mu }_i^2}}V\left(a_i\right)\left(1-2c_0^{-\epsilon }{\mu }_i^{-\epsilon (n-2)/2}{\alpha }_i^{p-1-\epsilon }\right)\hfill \\ \hfill & +O\left({\chi }_0\left(i\right)+{\epsilon }^2+{\displaystyle \frac{{ln}^{2{\sigma }_n}\left({\mu }_i\right)}{{\mu }_i^4}}\right)\hfill \end{array}$$

where ${\sigma }_4=1$, ${\sigma }_n=0$ for $n\ge 5$, $d_n$ is defined in Lemma A1, ${\chi }_0$ is defined in (56) and

$${\displaystyle \overline{c}=\frac{{(n-2)}^2}{4}{c}_0^{\frac{2n}{n-2}}{\int }_{R^n}\frac{{\left|x\right|}^2-1}{{(1+|x|}^2{)}^{n+1}}{ln(1+|x|}^2)dx>0.}$$

Proof. 

In this proof, we take $u=U+v$ and $h={\mu }_i\partial U_{a_i,{\mu }_i}/\partial {\mu }_i$ in (57). Since $v\in F_{a,\mu }^{\perp }$, it follows that

$$⟨U+v,{\mu }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\mu }_i}}⟩=\sum _{j=1}^q{\alpha }_j⟨U_{a_j,{\mu }_j},{\mu }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\mu }_i}}⟩.$$

(61)

For $j=i$, the scalar product is computed in Lemma A4. However for $j\ne i$, using (5), (26) and Corollary 1, it holds that

$$\begin{array}{cc}\hfill ⟨U_{a_j,{\mu }_j},{\mu }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\mu }_i}}⟩& ={\int }_{\Omega }(-\Delta +V)U_{a_j,{\mu }_j}{\mu }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\mu }_i}}\hfill \\ \hfill & =O\left({\int }_{\Omega }{\delta }_{a_j,{\mu }_j}^{{\displaystyle \frac{n+2}{n-2}}}{\delta }_{a_i,{\mu }_i}\right)\hfill \\ \hfill & =O\left({\displaystyle \frac{1}{{\mu }_i^{(n-2)/2}}}{\int }_{B_j^{+}}{\delta }_{a_j,{\mu }_j}^{{\displaystyle \frac{n+2}{n-2}}}+{\displaystyle \frac{1}{{\mu }_j^{(n+2)/2}}}{\int }_{\Omega \setminus B_j^{+}}{\delta }_{a_i,{\mu }_i}\right)\hfill \\ \hfill & =O\left({\displaystyle \frac{1}{{\left({\mu }_i{\mu }_j\right)}^{(n-2)/2}}}\right)\hfill \end{array}$$

where $B_j^{+}:=B(a_j,\rho )\cap \Omega$. Thus, (61) becomes

$$⟨U+v,{\mu }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\mu }_i}}⟩={\displaystyle \frac{1}{2}}{\alpha }_i{d}_n{\displaystyle \frac{{ln}^{{\sigma }_n}\left({\mu }_i\right)}{{\mu }_i^2}}V\left(a_i\right)+O\left({\chi }_3\left({\mu }_i\right))+\sum _{j\ne i}{\displaystyle \frac{1}{{\left({\mu }_i{\mu }_j\right)}^{(n-2)/2}}}\right).$$

(62)

Regarding the integral of (57), applying Lemma 2, we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }{|U+v|}^{p-1-\epsilon }(U+v){\mu }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\mu }_i}}& ={\alpha }_i^{p-\epsilon }{\int }_{B_i^{+}}{\left|U_{a_i,{\mu }_i}\right|}^{p-1-\epsilon }{U}_{a_i,{\mu }_i}{\mu }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\mu }_i}}\hfill \\ \hfill & +(p-\epsilon ){\int }_{B_i^{+}}{\left|{\alpha }_i{U}_{a_i,{\mu }_i}\right|}^{p-1-\epsilon }v{\mu }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\mu }_i}}+O\left({\chi }_0\left(i\right)\right).\hfill \end{array}$$

(63)

The last integral in (63) can be computed as

$$\begin{array}{cc}\hfill {\int }_{B_i^{+}}{\left|U_{a_i,{\mu }_i}\right|}^{p-1-\epsilon }v{\mu }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\mu }_i}}& ={\int }_{B_i^{+}}\left({\delta }_{a_i,{\mu }_i}^{p-1-\epsilon }+O({\delta }_{a_i,{\mu }_i}^{p-2}|{\psi }_{a_i,{\mu }_i}\left|\right)\right)\left({\mu }_i{\displaystyle \frac{\partial {\delta }_{a_i,{\mu }_i}}{\partial {\mu }_i}}-{\mu }_i{\displaystyle \frac{\partial {\psi }_{a_i,{\mu }_i}}{\partial {\mu }_i}}\right)v\hfill \\ \hfill & ={\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^{p-1-\epsilon }{\mu }_i{\displaystyle \frac{\partial {\delta }_{a_i,{\mu }_i}}{\partial {\mu }_i}}v+O\left(\int {\chi }_1(a_i,{\mu }_i,x){\delta }_{a_i,{\mu }_i}^{{\displaystyle \frac{n+2}{n-2}}}\left|v\right|\right).\hfill \end{array}$$

(64)

Notice that, the last integral is computed in (33) and (34). Concerning the first one, since $v\in F_{a,\mu }^{\perp }$ and using (19), we obtain

$$\begin{array}{cc}\hfill {\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^{p-1-\epsilon }{\mu }_i{\displaystyle \frac{\partial {\delta }_{a_i,{\mu }_i}}{\partial {\mu }_i}}v& =c_0^{-\epsilon }{\mu }_i^{-\epsilon (n-2)/2}{\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^{p-1}{\mu }_i{\displaystyle \frac{\partial {\delta }_{a_i,{\mu }_i}}{\partial {\mu }_i}}v\hfill \\ \hfill & +O\left({\int }_{\Omega \setminus B_i^{+}}{\delta }_{a_i,{\mu }_i}^p\left|v\right|+\epsilon {\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^{p}ln(1+{\mu }_i^2|x-a_i{|}^2\left)\right|v|\right)\hfill \\ \hfill & =c_0^{-\epsilon }{\mu }_i^{-\epsilon {\displaystyle \frac{n-2}{2}}}{\displaystyle \frac{1}{p}}{\int }_{\Omega }(-\Delta +V){\mu }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\mu }_i}}v+O\left({\displaystyle \frac{\left|\right|v\left|\right|}{{\mu }_i^{\frac{n+2}{2}}}}+\epsilon \left|\right|v\left|\right|\right)\hfill \\ \hfill & =O\left(||v||(\epsilon +{\displaystyle \frac{1}{{\mu }_i^{(n+2)/2}}})\right).\hfill \end{array}$$

(65)

This completes the estimate of (64) and we obtain

$${\int }_{B_i^{+}}{\left|U_{a_i,{\mu }_i}\right|}^{p-1-\epsilon }v{\mu }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\mu }_i}}=O\left(||v||(\epsilon +{\displaystyle \frac{{ln}^{{\sigma }_n}\left({\mu }_i\right)}{{\mu }_i^{(n+2)/2}}})\right).$$

(66)

Concerning the other integral of (63), using Corollary 1 and (55), it holds that

$$\begin{array}{cc}\hfill {\int }_{B_i^{+}}& |U_{a_i,{\mu }_i}{|}^{p-1-\epsilon }{U}_{a_i,{\mu }_i}{\mu }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\mu }_i}}\hfill \\ \hfill & ={\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^{p-\epsilon }{\mu }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\mu }_i}}-(p-\epsilon ){\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^{p-1-\epsilon }{\psi }_{a_i,{\mu }_i}{\mu }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\mu }_i}}+O\left({\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^{p-1}{\psi }_{a_i,{\mu }_i}^2\right).\hfill \end{array}$$

(67)

For the last integral in (67), using (11) and Proposition 1, easy computations imply that

$${\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^{p-1}{\psi }_{a_i,{\mu }_i}^2\le {\int }_{B_i^{+}}{\chi }_1^2(a_i,{\mu }_i,x){\delta }_{a_i,{\mu }_i}^{p+1}\le C{\displaystyle \frac{{ln}^{2{\sigma }_n}\left({\mu }_i\right)}{{\mu }_i^4}}.$$

(68)

Before giving the estimate of the two other integrals, we remark that, using Lemma A1, we have

$$\begin{array}{cc}\hfill ⟨U_{a_i,{\mu }_i},{\mu }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\mu }_i}}⟩& ={\int }_{\Omega }{\delta }_{a_i,{\mu }_i}^{{\displaystyle \frac{n+2}{n-2}}}{\mu }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\mu }_i}}=-{\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^{{\displaystyle \frac{n+2}{n-2}}}{\mu }_i{\displaystyle \frac{\partial {\psi }_{a_i,{\mu }_i}}{\partial {\mu }_i}}+O\left({\displaystyle \frac{1}{{\mu }_i^n}}\right),\hfill \end{array}$$

(69)

$$\begin{array}{cc}\hfill & ={\int }_{\Omega }{U}_{a_i,{\mu }_i}p{\delta }_{a_i,{\mu }_i}^{{\displaystyle \frac{4}{n-2}}}{\mu }_i{\displaystyle \frac{\partial {\delta }_{a_i,{\mu }_i}}{\partial {\mu }_i}}=-p{\int }_{B_i^{+}}{\psi }_{a_i,{\mu }_i}{\delta }_{a_i,{\mu }_i}^{{\displaystyle \frac{4}{n-2}}}{\mu }_i{\displaystyle \frac{\partial {\delta }_{a_i,{\mu }_i}}{\partial {\mu }_i}}+O\left({\displaystyle \frac{1}{{\mu }_i^n}}\right).\hfill \end{array}$$

(70)

Now, we focus on the first integral of (67). We have

$${\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^{p-\epsilon }{\mu }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\mu }_i}}={\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^{p-\epsilon }{\mu }_i{\displaystyle \frac{\partial {\delta }_{a_i,{\mu }_i}}{\partial {\mu }_i}}-{\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^{p-\epsilon }{\mu }_i{\displaystyle \frac{\partial {\psi }_{a_i,{\mu }_i}}{\partial {\mu }_i}}:=A_1-A_2.$$

(71)

We notice that, $B_i^{+}=B(a_i,\rho )\cap \Omega$ is a half ball and a similar integral to $A_1$ is computed in the equation $\left(91\right)$ of [29] in the ball $B(a_i,\rho )$. However, since the function is even, we obtain

$$A_1={\displaystyle \frac{1}{2}}{\int }_{B(a_i,\rho )}{\delta }_{a_i,{\mu }_i}^{p-\epsilon }{\mu }_i{\displaystyle \frac{\partial {\delta }_{a_i,{\mu }_i}}{\partial {\mu }_i}}=-{\displaystyle \frac{1}{2}}\overline{c}\epsilon c_0^{-\epsilon }{\mu }_i^{-\epsilon (n-2)/2}+O\left({\epsilon }^2+{\displaystyle \frac{ln\left({\mu }_i\right)}{{\mu }_i^n}}\right).$$

(72)

Concerning $A_2$, using (19) and (69), we obtain

$$\begin{array}{cc}\hfill A_2& =c_0^{-\epsilon }{\mu }_i^{-\epsilon (n-2)/2}{\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^p{\mu }_i{\displaystyle \frac{\partial {\psi }_{a_i,{\mu }_i}}{\partial {\mu }_i}}+O\left(\epsilon {\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^{p+1}ln(1+{\mu }_i^2|x-a_i{|}^2){\chi }_1(a_i,{\mu }_i,x)\right)\hfill \\ \hfill & =-c_0^{-\epsilon }{\mu }_i^{-\epsilon (n-2)/2}⟨U_{a_i,{\mu }_i},{\mu }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\mu }_i}}⟩+O\left({\displaystyle \frac{1}{{\mu }_i^n}}+\epsilon {\displaystyle \frac{{ln}^{{\sigma }_n}\left({\mu }_i\right)}{{\mu }_i^2}}\right).\hfill \end{array}$$

Thus, we obtain

$$\begin{array}{cc}\hfill {\int }_{B_i^{+}}& {\delta }_{a_i,{\mu }_i}^{p-\epsilon }{\mu }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\mu }_i}}\hfill \\ \hfill & =c_0^{-\epsilon }{\mu }_i^{-\epsilon {\displaystyle \frac{n-2}{2}}}\left(-{\displaystyle \frac{\overline{c}}{2}}\epsilon +⟨U_{a_i,{\mu }_i},{\mu }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\mu }_i}}⟩\right)+O\left({\displaystyle \frac{ln\left({\mu }_i\right)}{{\mu }_i^n}}+\epsilon {\displaystyle \frac{{ln}^{{\sigma }_n}\left({\mu }_i\right)}{{\mu }_i^2}}+{\epsilon }^2\right).\hfill \end{array}$$

(73)

It remains to estimate the second integral of (67). Using (19), (68) and Proposition 1, we obtain

$$\begin{array}{cc}\hfill {\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^{p-1-\epsilon }{\psi }_{a_i,{\mu }_i}{\mu }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\mu }_i}}& ={\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^{p-1-\epsilon }{\psi }_{a_i,{\mu }_i}{\mu }_i{\displaystyle \frac{\partial {\delta }_{a_i,{\mu }_i}}{\partial {\mu }_i}}+O\left({\int }_{B_i^{+}}{\chi }_1^2(a_i,{\mu }_i,x){\delta }_{a_i,{\mu }_i}^{p+1}\right)\hfill \\ \hfill & =c_0^{-\epsilon }{\mu }_i^{-\epsilon (n-2)/2}{\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^{p-1}{\mu }_i{\displaystyle \frac{\partial {\delta }_{a_i,{\mu }_i}}{\partial {\mu }_i}}{\psi }_{a_i,{\mu }_i}+O\left({\displaystyle \frac{{ln}^{2{\sigma }_n}\left({\mu }_i\right)}{{\mu }_i^4}}\right)\hfill \\ \hfill & +O\left(\epsilon \int {\chi }_1(a_i,{\mu }_i,x){\delta }_{a_i,{\mu }_i}^{p+1}ln(1+{\mu }_i^2|x-a_i{|}^2)\right)\hfill \\ \hfill & =-c_0^{-\epsilon }{\mu }_i^{-\epsilon (n-2)/2}{\displaystyle \frac{1}{p}}⟨U_{a_i,{\mu }_i},{\mu }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\mu }_i}}⟩\hfill \\ \hfill & +O\left({\displaystyle \frac{1}{{\mu }_i^n}}+{\displaystyle \frac{{ln}^{2{\sigma }_n}\left({\mu }_i\right)}{{\mu }_i^4}}+{\epsilon }^2\right)\hfill \end{array}$$

(74)

where we have used (69) and the fact that $\left|st\right|\le s^2+t^2$. Combining (63), (66)–(68), (73) and (74), we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }{|U+v|}^{p-1-\epsilon }(U+v){\mu }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\mu }_i}}& =c_0^{-\epsilon }{\mu }_i^{-\epsilon {\displaystyle \frac{n-2}{2}}}{\alpha }_i^{p-\epsilon }\left(-{\displaystyle \frac{\overline{c}}{2}}\epsilon +2⟨U_{a_i,{\mu }_i},{\mu }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\mu }_i}}⟩\right)\hfill \\ \hfill & +O\left({\left|\right|v\left|\right|}^2+{\epsilon }^2+\sum {\displaystyle \frac{1}{{\left({\mu }_i{\mu }_j\right)}^{(n-2)/2}}}+{\displaystyle \frac{{ln}^{2{\sigma }_n}\left({\mu }_i\right)}{{\mu }_i^4}}\right).\hfill \end{array}$$

(75)

Thus, the proof of Proposition 5 follows from (57), (62), (75) and Lemma A4. □

Next, we perform the expansion with respect to the points of concentration.

Proposition 6. 

Let $(\alpha ,a,\mu )\in {\aleph }_S(q,{\rho }_0)$ and $v\in F_{a,\mu }^{\perp }$. For $i\in \{1,\cdots ,q\}$, let ${\tau }_{i,j}$’s, with $j\in \{1,\cdots ,n-1\}$ be an orthonormal system of coordinates on the tangent space to $\partial \Omega$ at point $a_i$. It holds that

$$\begin{array}{cc}\hfill ⟨\nabla I_{\epsilon }& (U+v),{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\tau }_{i,j}}}⟩={\overline{c}}_9\left(n\right){\alpha }_i{\displaystyle \frac{{ln}^{{\sigma }_n}\left({\mu }_i\right)}{{\mu }_i^3}}{\displaystyle \frac{\partial V}{\partial {\tau }_{i,j}}}\left(a_i\right)\left(-1+2c_0^{-\epsilon }{\mu }_i^{-\epsilon (n-2)/2}{\alpha }_i^{p-1-\epsilon }\right)\hfill \\ \hfill & +O\left({\displaystyle \frac{1}{{\mu }_i}}{\chi }_3\left({\mu }_i\right)+\sum {\displaystyle \frac{1}{{\mu }_i^{n-1}}}+{\left|\right|v\left|\right|}^2+\epsilon \left|\right|v\left|\right|+\sum {\displaystyle \frac{\epsilon }{{\left({\mu }_i{\mu }_j\right)}^{(n-2)/2}}}+\epsilon {\displaystyle \frac{{ln}^{{\sigma }_n}\left({\mu }_i\right)}{{\mu }_i^2}}\right)\hfill \end{array}$$

where ${\sigma }_4=1$, ${\sigma }_n=0$ for $n\ge 5$ and ${\chi }_3$ is defined in (A3).

Proof. 

Without loss of generality, we can assume that $a_i=0$ and ${\nu }_{a_i}=-e_n$, where $(e_1,\cdots ,e_n)$ is the canonical basis of ${\mathbb{R}}^n$. Thus, the tangent space to $\partial \Omega$ at $a_i=0$ is ${\mathbb{R}}^{n-1}\times \left\{0\right\}$ and ${\tau }_{i,j}$’s becomes the classical coordinates.

In this proof, we will take $u=U+v$ and $h={\mu }_i^{-1}\partial U_{a_i,{\mu }_i}/\partial a_{i,j}$ (for $j\in \{1,\cdots ,n-1\}$) in (57). Since $v\in F_{a,\mu }^{\perp }$, it follows that

$$⟨U+v,{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial a_{i,j}}}⟩=\sum _{k=1}^q{\alpha }_k⟨U_{a_k,{\mu }_k},{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial a_{i,j}}}⟩.$$

(76)

For $k=i$, the scalar product is computed in Lemma A4. However, for $k\ne i$, using (26) and Corollary 1, it holds

$$\begin{array}{cc}\hfill ⟨U_{a_k,{\mu }_k},{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial a_{i,j}}}⟩& ={\int }_{\Omega }(-\Delta +V)U_{a_k,{\mu }_k}{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial a_{i,j}}}={\int }_{\Omega }{\delta }_{a_k,{\mu }_k}^{{\displaystyle \frac{n+2}{n-2}}}\left({\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial {\delta }_{a_i,{\mu }_i}}{\partial a_{i,j}}}-{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial {\psi }_{a_i,{\mu }_i}}{\partial a_{i,j}}}\right)\hfill \\ \hfill & =O\left({\displaystyle \frac{1}{{\mu }_i^{n/2}}}{\int }_{B_k^{+}}{\delta }_{a_k,{\mu }_k}^{{\displaystyle \frac{n+2}{n-2}}}+{\displaystyle \frac{1}{{\mu }_k^{(n+2)/2}}}{\int }_{\Omega \setminus B_k^{+}}{\delta }_{a_i,{\mu }_i}\right)\hfill \\ \hfill & =O\left({\displaystyle \frac{1}{{\mu }_i^{n/2}{\mu }_k^{(n-2)/2}}}+{\displaystyle \frac{1}{{\mu }_k^{(n+2)/2}{\mu }_i^{(n-2)/2}}}\right)=O\left({\displaystyle \frac{1}{{\mu }_i^{n-1}}}+{\displaystyle \frac{1}{{\mu }_k^{n-1}}}\right)\hfill \end{array}$$

by using $s^{\gamma }{t}^{\sigma }\le s^{\gamma +\sigma }+t^{\gamma +\sigma }$ for $s,t,\sigma ,\gamma \in (0,\infty )$. Thus, (76) becomes

$$⟨U+v,{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial a_{i,j}}}⟩=-{\alpha }_i{\overline{c}}_9\left(n\right){\displaystyle \frac{{ln}^{{\sigma }_n}\left({\mu }_i\right)}{{\mu }_i^3}}{\displaystyle \frac{\partial V}{\partial x_j}}\left(a_i\right)+O\left({\displaystyle \frac{1}{{\mu }_i}}{\chi }_3\left({\mu }_i\right)+\sum {\displaystyle \frac{1}{{\mu }_k^{n-1}}}\right).$$

(77)

Regarding the integral of (57), applying Lemma 2, we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }& {|U+v|}^{p-1-\epsilon }(U+v){\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial a_{i,j}}}\hfill \\ \hfill & ={\alpha }_i^{p-\epsilon }{\int }_{B_i^{+}}|U_{a_i,{\mu }_i}{|}^{p-1-\epsilon }{U}_{a_i,{\mu }_i}{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial a_{i,j}}}+(p-\epsilon ){\int }_{B_i^{+}}{\left|{\alpha }_i{U}_{a_i,{\mu }_i}\right|}^{p-1-\epsilon }v{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial a_{i,j}}}\hfill \\ \hfill & +(p-\epsilon ){\int }_{B_i^{+}}{\left|{\alpha }_i{U}_{a_i,{\mu }_i}\right|}^{p-1-\epsilon }{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial a_{i,j}}}\left(\sum _{k\ne i}{\alpha }_j{U}_{a_k,{\mu }_k}\right)\hfill \\ \hfill & +O\left({\left|\right|v\left|\right|}^{p+1-\epsilon }+{\left|\right|v\left|\right|}^2+{\displaystyle \frac{{\left|\right|v\left|\right|}^{p-\epsilon }}{{\mu }_i^{n/2}}}+\sum _k\left({\displaystyle \frac{1}{{\mu }_k^n}}+{\displaystyle \frac{1}{{\mu }_i^{\frac{n}{2}}{\mu }_k^{\frac{n-2}{2}}}}\right)\right).\hfill \end{array}$$

(78)

Following the same computation to prove (66), we deduce that

$${\int }_{B_i^{+}}{\left|U_{a_i,{\mu }_i}\right|}^{p-1-\epsilon }v{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial a_{i,j}}}=O\left(||v||(\epsilon +{\displaystyle \frac{1}{{\mu }_i^2}})\right).$$

(79)

Concerning the last integral in (78), using Remark 3, for a small $\rho$, we have ${\psi }_{a_i,{\mu }_i}\le (1/2){\delta }_{a_i,{\mu }_i}$ in $B_i^{+}$. Thus, we obtain (by using (19), Proposition 1 and Corollary 1)

$$\begin{array}{cc}\hfill {\int }_{B_i^{+}}& |U_{a_i,{\mu }_i}{|}^{p-1-\epsilon }{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial a_{i,j}}}{U}_{a_k,{\mu }_k}\hfill \\ \hfill & ={\int }_{B_i^{+}}({\delta }_{a_i,{\mu }_i}^{p-1-\epsilon }+O({\delta }_{a_i,{\mu }_i}^{p-2-\epsilon }|{\psi }_{a_i,{\mu }_i}\left|\right))\left({\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial {\delta }_{a_i,{\mu }_i}}{\partial a_{i,j}}}-{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial {\psi }_{a_i,{\mu }_i}}{\partial a_{i,j}}}\right)U_{a_k,{\mu }_k}\hfill \\ \hfill & ={\int }_{B_i^{+}}({\delta }_{a_i,{\mu }_i}^{p-1-\epsilon }{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial {\delta }_{a_i,{\mu }_i}}{\partial a_{i,j}}}{U}_{a_k,{\mu }_k}+O\left({\int }_{B_i^{+}}\left({\chi }_1(a_i,{\mu }_i,x)+{\displaystyle \frac{1}{{\mu }_i^2}}+{\displaystyle \frac{|x-a_i|}{{\mu }_i}}\right){\delta }_{a_i,{\mu }_i}^p{\delta }_{a_k,{\mu }_k}\right)\hfill \\ \hfill & =c_0^{-\epsilon }{\mu }_i^{-\epsilon {\displaystyle \frac{n-2}{2}}}{\displaystyle \frac{1}{p}}{\int }_{B_i^{+}}(-\Delta +V)\left({\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial a_{i,j}}}\right)U_{a_k,{\mu }_k}\hfill \\ \hfill & +O\left(\epsilon {\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^{p}ln(1+{\mu }_i^2|x-a_i{|}^2){\delta }_{a_k,{\mu }_k}\right)+O\left({\displaystyle \frac{{ln}^{2{\sigma }_n}\left({\mu }_i\right)}{{\mu }_i^2}}{\displaystyle \frac{1}{{\mu }_k^{(n-2)/2}{\mu }_i^{(n-2)/2}}}\right)\hfill \\ \hfill & =c_0^{-\epsilon }{\mu }_i^{-\epsilon {\displaystyle \frac{n-2}{2}}}{\displaystyle \frac{1}{p}}{\int }_{\Omega }(-\Delta +V)\left({\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial a_{i,j}}}\right)U_{a_k,{\mu }_k}+O\left({\displaystyle \frac{1}{{\mu }_i^{(n+2)/2}}}{\int }_{\Omega \setminus B_i^{+}}{\delta }_{a_k,{\mu }_k}\right)\hfill \\ \hfill & +O\left({\displaystyle \frac{\epsilon }{{\left({\mu }_i{\mu }_k\right)}^{(n-2)/2}}}+{\displaystyle \frac{1}{{\mu }_i^{n-1}}}+{\displaystyle \frac{1}{{\mu }_k^{n-1}}}\right).\hfill \end{array}$$

(80)

Notice that

$$\begin{array}{cc}\hfill {\int }_{\Omega }(-\Delta +V)& \left({\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial a_{i,j}}}\right)U_{a_k,{\mu }_k}\hfill \\ \hfill & ={\int }_{\Omega }{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial a_{i,j}}}(-\Delta +V)U_{a_k,{\mu }_k}\hfill \\ \hfill & ={\int }_{\Omega }{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial a_{i,j}}}{\delta }_{a_k,{\mu }_k}^p=O\left({\displaystyle \frac{1}{{\mu }_k^{(n+2)/2}}}{\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}+{\displaystyle \frac{1}{{\mu }_i^{n/2}}}{\int }_{\Omega \setminus B_i^{+}}{\delta }_{a_k,{\mu }_k}^p\right)\hfill \\ \hfill & =O\left({\displaystyle \frac{1}{{\mu }_k^{(n+2)/2}{\mu }_i^{(n-2)/2}}}+{\displaystyle \frac{1}{{\mu }_i^{n/2}{\mu }_k^{(n-2)/2}}}\right)=O\left({\displaystyle \frac{1}{{\mu }_i^{n-1}}}+{\displaystyle \frac{1}{{\mu }_k^{n-1}}}\right)\hfill \end{array}$$

(81)

by using (5) and Corollary 1. This completes the estimate of the last integral in (78) and we obtain

$${\int }_{B_i^{+}}{\left|U_{a_i,{\mu }_i}\right|}^{p-1-\epsilon }{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial a_{i,j}}}{U}_{a_k,{\mu }_k}=O\left({\displaystyle \frac{\epsilon }{{\left({\mu }_i{\mu }_k\right)}^{(n-2)/2}}}+{\displaystyle \frac{1}{{\mu }_i^{n-1}}}+{\displaystyle \frac{1}{{\mu }_k^{n-1}}}\right).$$

(82)

It remains to estimate the first integral of (78). Following the proof of (67), we have

$$\begin{array}{cc}\hfill {\int }_{B_i^{+}}& |U_{a_i,{\mu }_i}{|}^{p-1-\epsilon }{U}_{a_i,{\mu }_i}{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial a_{i,j}}}\hfill \\ \hfill & ={\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^{p-\epsilon }{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial a_{i,j}}}-(p-\epsilon ){\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^{p-1-\epsilon }{\psi }_{a_i,{\mu }_i}{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial a_{i,j}}}+O\left({\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^{p-1}{\psi }_{a_i,{\mu }_i}^2\right).\hfill \end{array}$$

(83)

The last integral in (83) is computed in (68). For the first one of (83), notice that $1\le j\le n-1$ and ${\displaystyle \frac{\partial {\delta }_{a_i,{\mu }_i}}{\partial a_{i,j}}}$ is odd with respect to $x_j$ (recall that, in the beginning of the proof, we assumed that $a_i=0$). Furthermore, $B_i^{+}:=\{(x^{\prime },x_n)\in B(0,\rho ):x_n>0\}$ is a half ball. Thus, we obtain

$$\begin{array}{cc}\hfill {\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^{p-\epsilon }{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial a_{i,j}}}& ={\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^{p-1}{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial {\delta }_{a_i,{\mu }_i}}{\partial a_{i,j}}}-{\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^{p-\epsilon }{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial {\psi }_{a_i,{\mu }_i}}{\partial a_{i,j}}}=-{\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^{p-\epsilon }{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial {\psi }_{a_i,{\mu }_i}}{\partial a_{i,j}}}.\hfill \end{array}$$

(84)

Using (19), Lemma A3 and Proposition 1, we obtain

$$\begin{array}{cc}\hfill {\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^{p-\epsilon }{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial {\psi }_{a_i,{\mu }_i}}{\partial a_{i,j}}}& =c_0^{-\epsilon }{\mu }_i^{-\epsilon (n-2)/2}{\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^p{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial {\psi }_{a_i,{\mu }_i}}{\partial a_{i,j}}}\hfill \\ \hfill & +O\left(\epsilon {\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^{p+1}(ln(1+{\mu }_i^2|x-a_i{|}^2\left)\right({\displaystyle \frac{1}{{\mu }_i^2}}+{\displaystyle \frac{|x-a_i|}{{\mu }_i}}\right)\hfill \\ \hfill & =c_0^{-\epsilon }{\mu }_i^{-\epsilon (n-2)/2}{\overline{c}}_9\left(n\right){\displaystyle \frac{{ln}^{{\sigma }_n}\left({\mu }_i\right)}{{\mu }_i^3}}{\displaystyle \frac{\partial V}{\partial x_j}}\left(a_i\right)+O\left({\displaystyle \frac{\epsilon }{{\mu }_i^2}}+{\displaystyle \frac{1}{{\mu }_i}}{\chi }_3\left({\mu }_i\right)\right).\hfill \end{array}$$

Hence, we deduce that

$${\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^{p-\epsilon }{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial {\psi }_{a_i,{\mu }_i}}{\partial a_{i,j}}}=c_0^{-\epsilon }{\mu }_i^{-\epsilon (n-2)/2}{\overline{c}}_9\left(n\right){\displaystyle \frac{{ln}^{{\sigma }_n}\left({\mu }_i\right)}{{\mu }_i^3}}{\displaystyle \frac{\partial V}{\partial x_j}}\left(a_i\right)+O\left({\displaystyle \frac{\epsilon }{{\mu }_i^2}}+{\displaystyle \frac{1}{{\mu }_i}}{\chi }_3\left({\mu }_i\right)\right)$$

(85)

which gives the estimate of the first integral of (83).

Regarding the second integral of (83), using (19) and Proposition 1, we obtain

$$\begin{array}{cc}\hfill {\int }_{B_i^{+}}& {\delta }_{a_i,{\mu }_i}^{p-1-\epsilon }{\psi }_{a_i,{\mu }_i}{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial a_{i,j}}}\hfill \\ \hfill & ={\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^{p-1-\epsilon }{\psi }_{a_i,{\mu }_i}{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial {\delta }_{a_i,{\mu }_i}}{\partial a_{i,j}}}+O\left({\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^{p+1}{\chi }_1(a_i,{\mu }_i,x)\left({\displaystyle \frac{1}{{\mu }_i^2}}+{\displaystyle \frac{|x-a_i|}{{\mu }_i}}\right)\right)\hfill \\ \hfill & =c_0^{-\epsilon }{\mu }_i^{-\epsilon (n-2)/2}{\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^{p-1}{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial {\delta }_{a_i,{\mu }_i}}{\partial a_{i,j}}}{\psi }_{a_i,{\mu }_i}\hfill \\ \hfill & +O\left(\epsilon {\int }_{B_i^{+}}{\delta }_{a_i,{\mu }_i}^{p+1}ln(1+{\mu }_i^2|x-a_i{|}^2){\chi }_1(a_i,{\mu }_i,x)+{\displaystyle \frac{{ln}^{{\sigma }_n}\left({\mu }_i\right)}{{\mu }_i^4}}\right)\hfill \\ \hfill & =-c_0^{-\epsilon }{\mu }_i^{-\epsilon (n-2)/2}{\displaystyle \frac{1}{p}}⟨U_{a_i,{\mu }_i},{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial a_{i,j}}}⟩+O\left(\epsilon {\displaystyle \frac{{ln}^{{\sigma }_n}\left({\mu }_i\right)}{{\mu }_i^2}}+{\displaystyle \frac{{ln}^{{\sigma }_n}\left({\mu }_i\right)}{{\mu }_i^4}}\right)\hfill \end{array}$$

(86)

(by following the computations performed in (70)). Combining (83), (68), (85), (86) and Lemma A4, we obtain

$$\begin{array}{cc}\hfill {\int }_{B_i^{+}}& |U_{a_i,{\mu }_i}{|}^{p-1-\epsilon }{U}_{a_i,{\mu }_i}{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial a_{i,j}}}\hfill \\ \hfill & =-2c_0^{-\epsilon }{\mu }_i^{-\epsilon {\displaystyle \frac{n-2}{2}}}{\overline{c}}_9\left(n\right){\displaystyle \frac{{ln}^{{\sigma }_n}\left({\mu }_i\right)}{{\mu }_i^3}}{\displaystyle \frac{\partial V}{\partial x_j}}\left(a_i\right)+O\left(\epsilon {\displaystyle \frac{{ln}^{{\sigma }_n}\left({\mu }_i\right)}{{\mu }_i^2}}+{\displaystyle \frac{1}{{\mu }_i}}{\chi }_3\left({\mu }_i\right)\right).\hfill \end{array}$$

(87)

Combining (78), (79), (82) and (87), we deduce that

$$\begin{array}{cc}\hfill {\int }_{B_i}& {|U+v|}^{p-1-\epsilon }(U+v){\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial a_{i,j}}}=-2c_0^{-\epsilon }{\mu }_i^{-\epsilon (n-2)/2}{\overline{c}}_9\left(n\right){\alpha }_i^{p-\epsilon }{\displaystyle \frac{{ln}^{{\sigma }_n}\left({\mu }_i\right)}{{\mu }_i^3}}{\displaystyle \frac{\partial V}{\partial x_j}}\left(a_i\right)\hfill \\ \hfill & +O\left({\left|\right|v\left|\right|}^2+\epsilon \left|\right|v\left|\right|+\sum {\displaystyle \frac{1}{{\mu }_k^{n-1}}}+\sum {\displaystyle \frac{\epsilon }{{\left({\mu }_i{\mu }_k\right)}^{(n-2)/2}}}+\epsilon {\displaystyle \frac{{ln}^{{\sigma }_n}\left({\mu }_i\right)}{{\mu }_i^2}}+{\displaystyle \frac{1}{{\mu }_i}}{\chi }_3\left({\mu }_i\right)\right).\hfill \end{array}$$

(88)

Thus, the proof of Proposition 6 follows from (57), (77) and (88). □

5. Proof of Theorems 1 and 2

Since Theorem 2 follows directly from Theorem 1, it is sufficient to prove the latter. To this aim, let ${\xi }_1,\dots ,{\xi }_q$ be non-degenerate critical points of $V_b:=V_{|\partial \Omega }$. We assume that $\Omega$ satisfies the assumption ($\mathcal{F}$) at each point ${\xi }_i$ (see (3)). For a small $\epsilon$, we define

$$\begin{array}{cc}\hfill {\mathcal{A}}_{\epsilon }^q=\{& m:=(\alpha ,a,\mu ,v)\in {(0,\infty )}^q\times {(\partial \Omega )}^q\times {({\epsilon }^{-1},\infty )}^q\times H^1(\Omega ):|{\alpha }_i-1|<\epsilon {ln}^2\epsilon ;\hfill \\ \hfill & |a_i-{\xi }_i{|<|ln\epsilon |}^{-1/2};M_1^{-1}\epsilon <{\displaystyle \frac{{ln}^{{\sigma }_n}\left({\mu }_i\right)}{{\mu }_i^2}}<M_1\epsilon ,v\in F_{a,\mu }^{\perp }with\left|\right|v\left|\right|<\sqrt{\epsilon }\}\hfill \end{array}$$

(89)

where ${\sigma }_4=1$, ${\sigma }_n=0$ for $n\ge 5$ and $M_1$ is a large constant. In ${\mathcal{A}}_{\epsilon }^q$, we define the function $g_{\epsilon }$ by

$$g_{\epsilon }:{\mathcal{A}}_{\epsilon }^q\to \mathbb{R};m:=(\alpha ,a,\mu ,v)\mapsto g_{\epsilon }\left(m\right):=I_{\epsilon }\left(\sum _{i=1}^q{\alpha }_i{U}_{a_i,{\mu }_i}+v\right).$$

(90)

Following [31], we have the following result.

Proposition 7. 

Let $m\in {\mathcal{A}}_{\epsilon }^q$. m is a critical point of $g_{\epsilon }$ iff $u:={\sum }_{i=1}^q{\alpha }_i{U}_{a_i,{\mu }_i}+v$ is a critical point of $I_{\epsilon }$. In other words, there exists $({\sigma }^{\prime },\gamma ,\zeta )\in {\mathbb{R}}^q\times {\mathbb{R}}^q\times {\left({\mathbb{R}}^{n-1}\right)}^q$ such that

($D_{{\alpha }_i}$)

${\displaystyle \frac{\partial g_{\epsilon }}{\partial {\alpha }_i}\left(m\right)=0\forall i}$

($D_{{\mu }_i}$)

${\displaystyle \frac{\partial g_{\epsilon }}{\partial {\mu }_i}\left(m\right)={\gamma }_i⟨{\mu }_i\frac{{\partial }^2{U}_{a_i,{\mu }_i}}{\partial {\mu }_i^2},v⟩+\sum _{j=1}^{n-1}{\zeta }_{i,j}⟨\frac{1}{{\mu }_i}\frac{{\partial }^2{U}_{a_i,{\mu }_i}}{\partial {\tau }_{i,j}\partial {\mu }_i},v⟩\forall i}$

($D_{a_i}$)

${\displaystyle \frac{\partial g_{\epsilon }}{\partial {\tau }_{i,j}}\left(m\right)={\gamma }_i⟨{\mu }_i\frac{{\partial }^2{U}_{a_i,{\mu }_i}}{\partial {\mu }_i\partial {\tau }_{i,j}},v⟩+\sum _{k=1}^{n-1}{\zeta }_{i,j}⟨\frac{1}{{\mu }_i}\frac{{\partial }^2{U}_{a_i,{\mu }_i}}{\partial {\tau }_{i,j}\partial {\tau }_{i,k}},v⟩\forall i,\forall j}$

($D_v$)

${\displaystyle \frac{\partial g_{\epsilon }}{\partial v}\left(m\right)=\sum _{j=1}^{n-1}\left({\sigma }_i^{\prime }{U}_{a_i,{\mu }_i}+{\gamma }_i{\mu }_i\frac{\partial U_{a_i,{\mu }_i}}{{\mu }_i}+\sum _{j=1}^{n-1}{\zeta }_{i,j}\frac{1}{{\mu }_i}\frac{\partial U_{a_i,{\mu }_i}}{\partial {\tau }_{i,j}}\right)}$.

Proof. 

Notice that, for $m:=(\alpha ,a,\mu ,v)\in {\mathcal{A}}_{\epsilon }^q$, some orthogonality constraints have to be satisfied which are

$$\begin{array}{cc}\hfill & {\psi }_i^1\left(m\right):=⟨U_{a_i,{\mu }_i},v⟩=0,{\psi }_i^2\left(m\right):=⟨{\mu }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\mu }_i}},v⟩=0,\hfill \end{array}$$

(91)

$$\begin{array}{cc}\hfill & {\psi }_{i,j}^3\left(m\right):=⟨{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial U_{a_{i,j}}}{\partial {\tau }_{i,j}}},v⟩=0,i=1,...,q,j=1,...,n-1.\hfill \end{array}$$

(92)

Thus, using the multiplier Lagrange theorem, we derive that $\nabla g_{\epsilon }\left(m\right)$ has to be a linear combination of $\nabla {\psi }_i^1$’s, $\nabla {\psi }_i^2$’s and the $\nabla {\psi }_{i,j}^3$’s. Hence, there exists $({\sigma }^{\prime },\gamma ,\zeta )\in {\mathbb{R}}^q\times {\mathbb{R}}^q\times {\left({\mathbb{R}}^{n-1}\right)}^q$ such that

$$\nabla g_{\epsilon }\left(m\right)=\sum _{i=1}^q\left({\sigma }_i^{\prime }\nabla {\psi }_i^1\left(m\right)+{\gamma }_i\nabla {\psi }_i^2\left(m\right)+\sum _{j=1}^{n-1}{\zeta }_{i,j}\nabla {\psi }_{i,j}^3\left(m\right)\right).$$

(93)

This completes the proof of Proposition 7. □

By Proposition 7, to prove Theorem 1, we have to solve the system $(D_{{\alpha }_i},D_{{\mu }_i},D_{a_i},D_v)$. Notice that, from the definition of $g_{\epsilon }$, we deduce that

$$\begin{array}{cc}& {\displaystyle \frac{\partial g_{\epsilon }}{\partial v}}\left(m\right)=\nabla I_{\epsilon }\left(u\right);{\displaystyle \frac{\partial g_{\epsilon }}{\partial {\mu }_i}}\left(m\right)=⟨\nabla I_{\epsilon }\left(u\right),{\alpha }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\mu }_i}}⟩,\hfill \\ & {\displaystyle \frac{\partial g_{\epsilon }}{\partial {\alpha }_i}}\left(m\right)=⟨\nabla I_{\epsilon }\left(u\right),U_{a_i,{\mu }_i}⟩;{\displaystyle \frac{\partial g_{\epsilon }}{\partial {\tau }_{i,j}}}\left(m\right)=⟨\nabla I_{\epsilon }\left(u\right),{\alpha }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\tau }_{i,j}}}⟩.\hfill \end{array}$$

(94)

Furthermore, Proposition 3 tells us that, for each $(\alpha ,a,\mu ,v)\in {\mathcal{A}}_{\epsilon }^q$, there exists a unique $\overline{v}$ satisfying

$$⟨\nabla I_{\epsilon }(\sum {\alpha }_i{U}_{a_i,{\mu }_i}+\overline{v}),h⟩=0,\forall h\in F_{a,\mu }^{\perp }\mathrm{and}\parallel \overline{v}\parallel \le C\epsilon .$$

(95)

Notice that (95) implies the existence of $(\sigma ,\gamma ,\zeta )$ such that

$$\nabla I_{\epsilon }(\sum {\alpha }_i{U}_{a_i,{\mu }_i}+\overline{v})=\sum _{i=1}^q\left({\sigma }_i^{\prime }{U}_{a_i,{\mu }_i}+{\gamma }_i{\mu }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\mu }_i}}+\sum _{j=1}^{n-1}{\zeta }_{i,j}{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\tau }_{i,j}}}\right).$$

(96)

Hence, $\left(D_v\right)$ is satisfied for $m=\overline{m}=(\alpha ,a,\mu ,\overline{v})$.

• Now, it remains to study the other equations. To this aim, we start by estimating the parameters $({\sigma }^{\prime },\gamma ,\zeta )$.

Lemma 3. 

Let $({\sigma }^{\prime },\gamma ,\zeta )$ be the parameters found in (96). It holds that

$$|{\sigma }_i^{\prime }|\le c\epsilon {ln}^2\epsilon ;|{\gamma }_i|\le c\epsilon ;|{\zeta }_i|\le c{\epsilon }^{3/2}\forall i.$$

(97)

Proof. 

Let ${\phi }_k;{\phi }_l\in \{U_{a_i,{\mu }_i},{\mu }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\mu }_i}},{\displaystyle \frac{1}{{\mu }_i}}{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\tau }_{i,j}}},1\le i\le q,1\le j\le n-1\}$. Easy computations imply that

$$⟨{\phi }_k,{\phi }_k⟩=C+O\left(\epsilon \right);⟨{\phi }_k,{\phi }_l⟩=O\left(\epsilon \right)\mathrm{for}k\ne l.$$

(98)

Now, taking the scalar product of (96) with the functions $U_{a_i,{\mu }_i}$, ${\mu }_i{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\mu }_i}}$ and ${\mu }_i^{-1}{\displaystyle \frac{\partial U_{a_i,{\mu }_i}}{\partial {\tau }_{i,j}}}$, respectively, and using Propositions 4–6 and the fact that $(\alpha ,a,\mu ,0)\in {\mathcal{A}}_{\epsilon }^q$, we obtain a quasi-diagonal system. The result follows easily by solving this system. □

In the next step, we will study the system $(D_{\alpha },D_{\mu },D_a)$ defined in Proposition 7. Using (94), Propositions 4–6 and the fact that $(\alpha ,a,\mu ,0)\in {\mathcal{A}}_{\epsilon }^q$, we deduce that the system $(D_{\alpha },D_{\mu },D_a)$ is equivalent to

$$\left(S_1\right)\begin{array}{c}1-c_0^{-\epsilon }{\mu }_i^{-\epsilon {\displaystyle \frac{n-2}{2}}}{\alpha }_i^{p-1-\epsilon }=O\left(\epsilon \right)\forall i,\hfill \\ \overline{c}\epsilon -d_n{\displaystyle \frac{{ln}^{{\sigma }_n}\left({\mu }_i\right)}{{\mu }_i^2}}V\left(a_i\right)=O\left({\epsilon }^2\right)\forall i,\hfill \\ {\displaystyle \frac{{ln}^{{\sigma }_n}\left({\mu }_i\right)}{{\mu }_i^3}}\nabla V_b\left(a_i\right)=O({\epsilon }^2+\sum {\displaystyle \frac{1}{{\mu }_k^{n-1}}}+{\mu }_i^{-1}{\chi }_3\left({\mu }_i\right))\forall i.\hfill \end{array}$$

To obtain an easy system to solve, we introduce the change of variables

$${\overline{a}}_i:=a_i-{\xi }_i,{\overline{\alpha }}_i:=1-{\alpha }_i^{p-1},{\displaystyle \frac{{ln}^{{\sigma }_n}\left({\mu }_i\right)}{{\mu }_i^2}}={\displaystyle \frac{\overline{c}}{d_{n}V\left({\xi }_i\right)}}\epsilon (1+{\overline{\gamma }}_i)\forall i$$

(99)

Notice that $V\left(a_i\right)=V_b\left(a_i\right)=V_b\left({\xi }_i\right)+O\left(\right|a_i-{\xi }_i{|}^2)$. Thus, the system $\left(S_1\right)$ becomes

$$\left(S_2\right)\begin{array}{c}{\overline{\alpha }}_i=O\left(\epsilon \right|ln\epsilon \left|\right)\forall i,\hfill \\ {\overline{\gamma }}_i=O(\epsilon +|{\overline{a}}_i{|}^2)\forall i,\hfill \\ D^2{V}_b\left({\xi }_i\right)({\overline{a}}_i,.)=O\left(\right|{\overline{a}}_i{|}^2+{\chi }_8\left(\epsilon \right))\forall i,\hfill \end{array}$$

where

$${\chi }_8\left(\epsilon \right)={\left(\right|ln\epsilon |}^{-1}\mathrm{if}n=4,\sqrt{\epsilon }|ln\epsilon |\mathrm{if}n=5,\sqrt{\epsilon }\mathrm{if}n\ge 6).$$

Finally, for a small $\epsilon$, since the ${\xi }_i$’s are non-degenerate critical points of $V_b$, Brouwer’s fixed point theorem implies that $\left(S_2\right)$ has a solution $({\overline{\alpha }}^{\epsilon },{\overline{a}}^{\epsilon },{\overline{\gamma }}^{\epsilon })$. In addition, we have

$${\overline{\alpha }}_i^{\epsilon }=O\left(\epsilon \right|ln\epsilon \left|\right),{\overline{\gamma }}_i^{\epsilon }=O\left(\epsilon \right),{\overline{a}}_i^{\epsilon }=O\left({\chi }_8\left(\epsilon \right)\right)\forall i.$$

(100)

Taking ${\alpha }_i^{\epsilon },a_i^{\epsilon }$ and ${\mu }_i^{\epsilon }$, using (99) and considering $u_{\epsilon }={\sum }_{i=1}^q{\alpha }_i^{\epsilon }{U}_{a_i^{\epsilon },{\mu }_i^{\epsilon }}+\overline{v},$ it follows from Proposition 7 that $u_{\epsilon }$ is a solution of Problem $\left(Q_{\epsilon }\right)$. This achieves the proof of the theorem.

6. Conclusions

In this paper, we studied the existence of solutions to a nonlinear problem with Neumann boundary conditions, featuring a slightly subcritical growth for Sobolev embedding $H_0^1(\Omega )\hookrightarrow L^q(\Omega )$. By assuming that the boundary is flat near the critical points of the restriction of a given potential V on the boundary, we constructed boundary peak solutions with isolated bubbles. This result allowed us to establish a multiplicity result for the problem. The approach adopted is specialized for variational problems. While this paper focuses on the existence of solutions and multiplicity results for the given problem, several promising directions for future research and unresolved questions remain as follows:

(i) 

Effect of the type of critical points of the potential: The solutions constructed in this paper are based on the assumption that the critical points of the restriction of the potential V on the boundary are non-degenerate. What happens if this assumption is not met, especially when V satisfies a flatness condition?

(ii) 

Type of blow-up points: This paper concentrates on constructing solutions that localize at isolated boundary points. An intriguing extension would be to explore the existence of solutions that concentrate at non-isolated boundary points.

(iii) 

Effect of the nonlinear exponent: This work focuses on a slightly subcritical exponent for Sobolev embedding. Future studies could investigate the problem with exponents that are slightly supercritical., i.e., when $\epsilon <0$ but close to zero.