Interior Peak Solutions for a Semilinear Dirichlet Problem

Abstract

In this paper, we consider the semilinear Dirichlet problem $\left({\mathcal{P}}_{\epsilon }\right):-\Delta u+V\left(x\right)u=u^{{\displaystyle \frac{n+2}{n-2}}-\epsilon }$, $u>0$ in $\Omega$, $u=0$ on ∂$\Omega$, where $\Omega$ is a bounded regular domain in ${\mathbb{R}}^n$, $n\ge 4$, $\epsilon$ is a small positive parameter, and V is a non-constant positive $C^2$-function on $\overline{\Omega }$. We construct interior peak solutions with isolated bubbles. This leads to a multiplicity result for $\left({\mathcal{P}}_{\epsilon }\right)$. The proof of our results relies on precise expansions of the gradient of the Euler–Lagrange functional associated with $\left({\mathcal{P}}_{\epsilon }\right)$, along with a suitable projection of the bubbles. This projection and its associated estimates are new and play a crucial role in tackling such types of problems.

1. Introduction and Main Results

Let us consider the following semilinear Dirichlet problem

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

where $\Omega$ is a bounded regular domain in ${\mathbb{R}}^n$, $n\ge 3$, $1<q<\infty$ and V is a smooth positive function on $\overline{\Omega }$.

The problem in the form of $\left({\mathcal{P}}_{V,q}\right)$ appears in various physical models, including quantum transport and non-relativistic Newtonian gravity, as discussed in [1,2,3] and the references cited therein. It is also related to the Yamabe problem in differential geometry; see, for example, [4] and the references therein.

In the subcritical case, that is, $1<q<{\displaystyle \frac{n+2}{n-2}}$, proving the existence of a solution to $\left({\mathcal{P}}_{V,q}\right)$ is relatively straightforward. This can be demonstrated by noting that the following infimum

$$inf\{{\int }_{\Omega }\left({|\nabla u|}^2+V{u}^2\right):u\in H_0^1(\Omega )and{\int }_{\Omega }|u{|}^{q+1}=1\}$$

is attained due to the compactness of the embedding of $H_0^1(\Omega )\hookrightarrow L^q(\Omega )$.

In the critical case, where $q={\displaystyle \frac{n+2}{n-2}}$, it is well known that the existence of solutions to $\left({\mathcal{P}}_{V,q}\right)$ depends on the geometry of the domain $\Omega$, the properties of V, and the dimension n. For example, if V is constant and $\Omega$ is starshaped, the problem $\left({\mathcal{P}}_{V,q}\right)$ has no solutions. Given the extensive literature on this topic, we will only highlight the pioneering papers by Brezis–Nirenberg [5] and Bahri–Coron [6].

The supercritical case, $q>{\displaystyle \frac{n+2}{n-2}}$, is significantly more challenging to address, as there is no embedding of $H_0^1(\Omega )\hookrightarrow L^q(\Omega )$. Notable existence and nonexistence results in specific domains are due to Passaseo [7,8]. We also refer to the paper [9], which addresses domains with small circular holes. For the case of the ball, we refer to the results in [10,11,12,13]. It is important to note that the general domain case does not appear to be fully understood.

Note that, for $q\le {\displaystyle \frac{n+2}{n-2}}$, the problem $\left({\mathcal{P}}_{V,q}\right)$ is a variational problem and its Euler–Lagrange function is defined on $H_0^1(\Omega )$ by

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

When q is critical, the functional $I_{V,q}$ does not satisfy the Palais–Smale condition; that is, there exist sequences along which $I_{V,q}$ is bounded, with a gradient that goes to zero, and which do not converge. This is violated for all levels $N{S}_n^{n/2}$, where

$$S_n=\underset{u\in H_0^1(\Omega )\setminus \left\{0\right\}}{inf}\left({\int }_{\Omega }{|\nabla u|}^2\right){\left({\int }_{\Omega }{\left|u\right|}^{p+1}\right)}^{-2/(p+1)}\mathrm{where}p={\displaystyle \frac{n+2}{n-2}}.$$

These compactness defects correspond to a concentration in N points $a_i$ (possibly confused) of $\overline{\Omega }$ sequences $\left(u_k\right)$ in $H_0^1(\Omega )$; that is,

$$|\nabla u_k{|}^2\rightharpoonup S_n^{n/2}\sum _{i=1}^N{\delta }_{a_i}inthesenseofmeasure,$$

where ${\delta }_{a_i}$ denotes the Dirac mass at $a_i$ (see [14,15]).

In this paper, we are interested in slightly subcritical problems that actually admit solutions concentrating, as $\epsilon$ goes to zero, at certain interior points of the domain $\Omega$. More precisely, we consider the problem

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

where $\Omega$ is a bounded regular domain of ${\mathbb{R}}^n$, $n\ge 4$, $p+1:={\displaystyle \frac{n+2}{n-2}}+1={\displaystyle \frac{2n}{n-2}}$ is the critical Sobolev exponent for the embedding $H_0^1(\Omega )\hookrightarrow L^q(\Omega )$, V is a $C^2$ positive function on $\overline{\Omega }$, and $\epsilon$ is a small positive parameter.

Our aim is to establish the existence and multiplicity of interior peak solutions of $\left({\mathcal{P}}_{\epsilon }\right)$ for any positive $C^2$ functions V and for any smooth bounded domain $\Omega$ provided that $\epsilon$ is positive and sufficiently small.

To state our result, we need to fix some notation. We denote by $I_{\epsilon }$ the Euler–Lagrange functional associated with $\left({\mathcal{P}}_{\epsilon }\right)$ defined on $H_0^1(\Omega )$ by

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

In this paper, we will use the following scalar product and its corresponding norm

$$〈v,w〉:={\int }_{\Omega }\nabla v·\nabla w+{\int }_{\Omega }Vvw;\parallel w\parallel :={\int }_{\Omega }{|\nabla w|}^2+{\int }_{\Omega }V{w}^2.$$

(1)

Notice that this norm is equivalent to the two classical norms of $H_0^1(\Omega )$ and $H^1(\Omega )$.

Let $a\in {\mathbb{R}}^n$ and $\lambda >0$; we define

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

These functions, for $a\in {\mathbb{R}}^n$ and $\lambda >0$, are the only positive solutions [16] of

$$-\Delta w={\left|w\right|}^{4/(n-2)}win{\mathbb{R}}^n.$$

The aim of our first result is to construct interior peak solutions. More precisely, we prove the following theorem.

Theorem 1.

Let $n\ge 4$ and $V:\overline{\Omega }\to \mathbb{R}$ be a positive $C^2$-function with m non- degenerate critical points $b_1,\cdots ,b_m$. Then, for any $N\le m$, there exists small ${\epsilon }_0>0$ such that for any $\epsilon \in (0,{\epsilon }_0]$, problem $\left({\mathcal{P}}_{\epsilon }\right)$ admits a solution $u_{\epsilon ,b_{i_1},\cdots ,b_{i_N}}$ satisfying that $u_{\epsilon ,b_{i_1},\cdots ,b_{i_N}}$ develops exactly one peak at each point $b_{i_j}$ and converges weakly to zero in $H_0^1(\Omega )$ as $\epsilon \to 0$. More precisely, there exist ${\lambda }_{1,\epsilon }$,..., ${\lambda }_{N,\epsilon }$ with the same order as ${\epsilon }^{-1/2}$ for $n\ge 5$ and as ${\epsilon }^{-1/2}{|ln\epsilon |}^{1/2}$ for $n=4$ and N points $a_{i_j,\epsilon }\to b_{i_j}$ for all j such that

$$\left|\right|u_{\epsilon ,b_{i_1},...,b_{i_N}}-\sum _{j=1}^N{\delta }_{a_{i_j,\epsilon },{\lambda }_{i_j,\epsilon }}\left|\right|\to 0,as\epsilon \to 0.$$

Theorem 1 allows us to obtain the following multiplicity result in connection with the number of critical points of V.

Theorem 2.

Let $n\ge 4$ and let $V:\overline{\Omega }\to \mathbb{R}$ be a positive $C^2$-function with m non-degenerate critical points. Then, for positive and sufficiently small ε, the number of solutions of $\left({\mathcal{P}}_{\epsilon }\right)$ is at least equal to $2^m-1$.

The structure of the proof follows a similar approach to that in [17,18,19]. It is based on fine expansions of the gradient of the functional $I_{\epsilon }$. However, we adopt a different framework—that is, an appropriate projection—to handle our case. This projection, along with its estimates, proves instrumental in addressing such types of problems. In the next section, we introduce a two-parameter family of approximate solutions to the problem $\left({\mathcal{P}}_{\epsilon }\right)$ and seek a true solution in a neighborhood of this set. We also introduce the neighborhood at infinity, provide precise estimates for the infinite-dimensional part, and perform the expansion of the gradient of the associated functional. Section 3 is devoted to the proof of our results. Section 4 discusses potential directions for future research. The proofs involve certain technical estimates, which are provided in the Appendix A for the reader’s convenience.

2. Analytical Frameworks

Let $a\in \Omega$ and $\lambda >0$; we notice that the function ${\delta }_{a,\lambda }$ does not belong to $H_0^1(\Omega )$; thus, we need to introduce its projection onto this space. Different projections have been introduced into the literature. In our case, we will use the following projection

$$\left\{\begin{array}{cc}(-\Delta +V)\left(\pi {\delta }_{a,\lambda }\right)={\delta }_{a,\lambda }^{{\displaystyle \frac{n+2}{n-2}}}\hfill & in\Omega ,\hfill \\ \pi {\delta }_{a,\lambda }=0\hfill & on\partial \Omega .\hfill \end{array}\right.$$

(2)

In the sequel, for $a\in \Omega$ and $\lambda >0$, we denote

$${\theta }_{a,\lambda }:={\delta }_{a,\lambda }-\pi {\delta }_{a,\lambda }.$$

(3)

To enhance readability, the estimates of $\pi {\delta }_{a,\lambda }$ and ${\theta }_{a,\lambda }$ are deferred to the Appendix A, allowing the main argument to be presented more clearly.

Next, we introduce the neighborhood at infinity, provide precise estimates for the infinite-dimensional part, and finally, perform the expansion of the gradient of the associated functional.

To this aim, first, let $N\in \mathbb{N}$, $a_1,\cdots ,a_N$ be N points in $\Omega$ satisfying $d(a_i,\partial \Omega )\ge c$ for some constant $c>0$, and let ${\lambda }_1,\cdots ,{\lambda }_N$ be large so that, for each $i\ne j$,

$${\epsilon }_{ij}:={\left({\displaystyle \frac{{\lambda }_i}{{\lambda }_j}}+{\displaystyle \frac{{\lambda }_j}{{\lambda }_i}}+{\lambda }_i{\lambda }_j{|a_i-a_j|}^2\right)}^{-(n-2)/2}$$

(4)

is small enough, and we denote

$$E_{a,\lambda }:=\mathrm{span}\left\{\pi {\delta }_{a_i,{\lambda }_i},{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial {\lambda }_i}},{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial a_{i,j}}},i\in \{1,\cdots ,N\},j\in \{1,\cdots ,n\}\right\},$$

(5)

where $a_{i,j}$ is the jth component of $a_i$.

Second, let $\mu >0$ be a small real and $d_0>0$ be a fixed small constant; we introduce the following set:

$$\begin{array}{cc}\hfill \mathcal{O}(N,\mu ,d_0):=\{& (\alpha ,a,\lambda )\in {(0,\infty )}^N\times {\Omega }^N\times {({\mu }^{-1},\infty )}^N:|{\alpha }_i-1|<\mu ;d(a_i,\partial \Omega )>2d_0;\hfill \\ \hfill & M^{-1}\sqrt{{|ln\epsilon |}^{{\sigma }_n}/\epsilon }\le {\lambda }_i\le M\sqrt{{|ln\epsilon |}^{{\sigma }_n}/\epsilon };\forall iand|a_i-a_j|>2d_0\forall i\ne j\}\hfill \end{array}$$

(6)

where M is a fixed large constant, ${\sigma }_4=1$ and ${\sigma }_n=0$ for $n\ge 5$.

Notice that, for $(\alpha ,a,\lambda )\in \mathcal{O}(N,\mu ,d_0)$, since we have $|a_i-a_j|>2d_0$ and $d(a_i,\partial \Omega )>2d_0$, we deduce that

$$B_i:=B(a_i,d_0)\subset {\Omega }_0:=\{x\in \Omega :d(x,\partial \Omega )\ge d_0\}\forall iand{B}_i\cap B_j=\varnothing \forall j\ne i.$$

(7)

For each $(\alpha ,a,\lambda )\in \mathcal{O}(N,\mu ,d_0)$ and $v\in E_{a,\lambda }^{\perp }$, we associate a function

$$u:=\sum _{i=1}^N{\alpha }_i\left(\pi {\delta }_{a_i,{\lambda }_i}\right)+v:=\underline{u}+v.$$

(8)

We notice that the orthogonality is taken with respect to the scalar product defined in (1).

Furthermore, for each $i\in \{1,\cdots ,N\}$, the following estimates hold

$$\left\{\begin{array}{cc}\underline{u}={\alpha }_i\pi {\delta }_{a_i,{\lambda }_i}+O\left(\gamma \left(\epsilon \right)\right)\hfill & in{B}_i,\hfill \\ \underline{u}=O\left(\gamma \left(\epsilon \right)\right)\hfill & in\Omega \setminus (\cup B_i)\hfill \end{array}\right.\mathrm{where}\gamma \left(\epsilon \right):=\left\{\begin{array}{cc}{\epsilon }^{1/2}/{|ln\epsilon |}^{1/2}\hfill & ifn=4,\hfill \\ {\epsilon }^{(n-2)/4}\hfill & ifn\ge 5.\hfill \end{array}\right.$$

(9)

Below, our aim is to find some positive critical points u of $I_{\epsilon }$ with the form (8). We start by studying the v-part of u.

2.1. Minimizing with Respect to the Infinite-Dimensional Part

Let $(\alpha ,a,\lambda )\in \mathcal{O}(N,\mu ,d_0)$ and u be defined in (8). To study the v-part of $u:={\sum }_{i=1}^N{\alpha }_i\left(\pi {\delta }_{a_i,{\lambda }_i}\right)+v:=\underline{u}+v$, we need to expand $I_{\epsilon }$ with respect to v.

Using Lemma A1 and the fact that $〈v,\underline{u}〉=0$, we obtain

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

(10)

where

$$f_{\epsilon }\left(v\right):={\int }_{\Omega }{\left(\underline{u}\right)}^{p-\epsilon }vand{Q}_{\epsilon }\left(v\right):={\left|\left|v\right|\right|}^2-(p-\epsilon ){\int }_{\Omega }{\left(\underline{u}\right)}^{p-\epsilon -1}{v}^2.$$

(11)

Proposition 1.

Let $N\in \mathbb{N}$, ${\alpha }_1,\cdots ,{\alpha }_n$ be N positive reals close to 1, let $a_1,\cdots ,a_N$ be N points in Ω satisfying $d(a_i,\partial \Omega )\ge c$ for some constant $c>0$, and let ${\lambda }_1,\cdots ,{\lambda }_N$ be large reals. Assume that the ${\epsilon }_{ij}$’s, for $i\ne j$, and the $\epsilon ln{\lambda }_i$’s, for $i\in \{1,\cdots ,N\}$, are small enough. Then, there exists a positive constant ${\rho }_0>0$ such that

$$\begin{array}{c}\hfill Q_{\epsilon }\left(v\right)\ge {\rho }_0{∥v∥}^2\forall v\in E_{a,\lambda }^{\perp }.\end{array}$$

Proof. 

Using Lemma A1 and the fact that ${\alpha }_i=1+o\left(1\right)$ for each i, we derive that

$${\int }_{\Omega }{\left(\underline{u}\right)}^{p-\epsilon -1}{v}^2=\sum {\int }_{\Omega }{\left(\pi {\delta }_{a_i,{\lambda }_i}\right)}^{p-\epsilon -1}{v}^2+{o(\parallel v\parallel }^2).$$

Observe that, since $\Omega$ is bounded, for each $b\in \Omega$ and $\lambda >0$ satisfying that $\epsilon ln\lambda$ is small, it follows that

$${\delta }_{b,\lambda }^{-\epsilon }=c_0^{-\epsilon }{\lambda }^{-\epsilon (n-2)/2}\left(1+{\displaystyle \frac{n-2}{2}}\epsilon ln(1+{\lambda }^2{|x-b|}^2)\right)+O\left({\epsilon }^2{ln}^2(1+{\lambda }^2{|x-b|}^2)\right)=1+o\left(1\right).$$

(12)

Now, using (12) and Proposition A1, we deduce that

$${\int }_{\Omega }{\left(\pi {\delta }_{a_i,{\lambda }_i}\right)}^{p-\epsilon -1}{v}^2={\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^{p-1}{v}^2+{o(\parallel v\parallel }^2).$$

Therefore, the quadratic form becomes

$$Q_{\epsilon }\left(v\right)={\parallel v\parallel }^2-p\sum {\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^{p-1}{v}^2+{o(\parallel v\parallel }^2)\ge \underset{:=\overline{Q}\left(v\right)}{\underbrace{{\int }_{\Omega }{|\nabla v|}^2-p\sum {\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^{p-1}{v}^2}}+{o(\parallel v\parallel }^2).$$

(13)

Observe that $\overline{Q}$ is studied in several works, and it is proved that it is coercive on the space (see Proposition 3.1 of [20])

$$F_{a,\lambda }:=\bigcap _{1\le i\le N}\left\{v\in H_0^1(\Omega ):{\int }_{\Omega }\nabla v\nabla P{\delta }_{a_i,{\lambda }_i}={\int }_{\Omega }\nabla v\nabla {\displaystyle \frac{\partial P{\delta }_{a_i,{\lambda }_i}}{\partial {\lambda }_i}}=0,{\int }_{\Omega }\nabla v\nabla {\displaystyle \frac{\partial P{\delta }_{a_i,{\lambda }_i}}{\partial a_i}}=0\right\},$$

where, for $b\in \Omega$ and $\mu >0$, the function $P{\delta }_{b,\mu }$ is defined by the following PDE:

$$-\Delta P{\delta }_{b,\mu }={\delta }_{b,\mu }^{(n+2)/(n-2)}\mathrm{in}\Omega ,P{\delta }_{b,\mu }=0\mathrm{on}\partial \Omega .$$

Note that the coercivity of $\overline{Q}$ implies the existence of ${\rho }_0>0$ such that

$$\overline{Q}\left(v\right)\ge {\rho }_0{\int }_{\Omega }{|\nabla v|}^2\forall v\in F_{a,\lambda }.$$

(14)

Now, for $b\in \Omega$ and $\mu >0$, observe that

$$\begin{array}{cc}\hfill \langle v,\pi {\delta }_{b,\mu }\rangle & ={\int }_{\Omega }\nabla v\nabla \left(\pi {\delta }_{b,\mu }\right)+{\int }_{\Omega }Vv\left(\pi {\delta }_{b,\mu }\right)={\int }_{\Omega }(-\Delta +V)\left(\pi {\delta }_{b,\mu }\right)v\hfill \\ \hfill & ={\int }_{\Omega }{\delta }_{b,\mu }^{{\displaystyle \frac{n+2}{n-2}}}v={\int }_{\Omega }(-\Delta \left(P{\delta }_{b,\mu }\right))v={\int }_{\Omega }\nabla v\nabla \left(P{\delta }_{b,\mu }\right),\hfill \end{array}$$

and for $\tau \in \{\mu ,b\}$,

$$\begin{array}{cc}\hfill \langle v,{\displaystyle \frac{\partial \left(\pi {\delta }_{b,\mu }\right)}{\partial \tau }}\rangle & ={\int }_{\Omega }\nabla v\nabla {\displaystyle \frac{\partial \left(\pi {\delta }_{b,\mu }\right)}{\partial \tau }}+{\int }_{\Omega }Vv{\displaystyle \frac{\partial \left(\pi {\delta }_{b,\mu }\right)}{\partial \tau }}={\int }_{\Omega }(-\Delta +V){\displaystyle \frac{\partial \left(\pi {\delta }_{b,\mu }\right)}{\partial \tau }}v\hfill \\ \hfill & ={\displaystyle \frac{n+2}{n-2}}{\int }_{\Omega }{\delta }_{b,\mu }^{{\displaystyle \frac{4}{n-2}}}{\displaystyle \frac{\partial {\delta }_{b,\mu }}{\partial \tau }}v={\int }_{\Omega }(-\Delta {\displaystyle \frac{\partial \left(P{\delta }_{b,\mu }\right)}{\partial \tau }})v={\int }_{\Omega }\nabla v\nabla {\displaystyle \frac{\partial \left(P{\delta }_{b,\mu }\right)}{\partial \tau }}.\hfill \end{array}$$

We deduce that $E_{a,\lambda }^{\perp }=F_{a,\lambda }.$ Finally, since the norm $\parallel .\parallel$ and the norm of $H_0^1(\Omega )$ are equivalent, the proof of the proposition follows from (13) and (14). □

Now, we will estimate the linear form on v defined in (11). More precisely, we have

Lemma 1.

Let $(\alpha ,a,\lambda )\in \mathcal{O}(N,\mu ,d_0)$. Then, for each $v\in E_{a,\lambda }^{\perp }$, we have

$$\left|f_{\epsilon }\left(v\right)\right|\le c\epsilon ∥v∥.$$

Proof. 

Using (9) and Lemma A1, we obtain

$$\begin{array}{cc}\hfill f_{\epsilon }\left(v\right)& =\sum _{i=1}^N{\int }_{B_i}{\left({\alpha }_i\pi {\delta }_{a_i,{\lambda }_i}+\gamma \left(\epsilon \right)\right)}^{p-\epsilon }v+O\left(\gamma {\left(\epsilon \right)}^{p-\epsilon }{\int }_{\Omega \setminus (\cup B_i)}\left|v\right|\right)\hfill \\ \hfill & =\sum _{i=1}^N{\int }_{B_i}{\left({\alpha }_i\pi {\delta }_{a_i,{\lambda }_i}\right)}^{p-\epsilon }v+O\left(\gamma \left(\epsilon \right){\int }_{B_i}{\delta }_{a_i,{\lambda }_i}^{p-\epsilon -1}\left|v\right|+\gamma {\left(\epsilon \right)}^{p-\epsilon }{\int }_{\Omega }\left|v\right|\right)\hfill \\ \hfill & =\sum _{i=1}^N{\alpha }_i^{p-\epsilon }{\int }_{\Omega }{\left(\pi {\delta }_{a_i,{\lambda }_i}\right)}^{p-\epsilon }v+O\left(\gamma {\left(\epsilon \right)}^{p-\epsilon }\parallel v\parallel +\gamma \left(\epsilon \right)\parallel v\parallel {\left({\int }_{B_i}{\delta }_{a_i,{\lambda }_i}^{{\displaystyle \frac{8n}{n^2-4}}}\right)}^{(n+2)/\left(2n\right)}\right)\hfill \end{array}$$

(15)

by using the embedding Theorem of $H_0^1(\Omega )\hookrightarrow L^q(\Omega )$ for each $q\in [1,2n/(n-2\left)\right]$.

Observe that easy computations imply that

$$\begin{array}{cc}\hfill {\left({\int }_{B_i}{\delta }_{a_i,{\lambda }_i}^{{\displaystyle \frac{8n}{n^2-4}}}\right)}^{(n+2)/\left(2n\right)}& \le c\left({\displaystyle \frac{1}{{\lambda }_i^{(n-2)/2}}}(ifn\le 5);{\displaystyle \frac{ln{\left({\lambda }_i\right)}^{2/3}}{{\lambda }_i^2}}(ifn=6);{\displaystyle \frac{1}{{\lambda }_i^2}}(ifn\ge 7)\right)\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & \le c\left(\gamma \left(\epsilon \right)(ifn\le 5);{|ln\epsilon |}^{2/3}\epsilon (ifn=6);\epsilon (ifn\ge 7)\right).\hfill \end{array}$$

(17)

It remains to estimate the first integral in (15). Using Lemma A1, we obtain

$${\int }_{\Omega }{\left(\pi {\delta }_{a_i,{\lambda }_i}\right)}^{p-\epsilon }v={\int }_{\Omega }{\left({\delta }_{a_i,{\lambda }_i}-{\theta }_{ai,\lambda i}\right)}^{p-\epsilon }v={\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^{p-\epsilon }v+O\left({\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^{{\displaystyle \frac{4}{n-2}}-\epsilon }{\theta }_{ai,\lambda i}\left|v\right|\right).$$

The last integral can be deduced from Lemma A5 by using (12).

For the other one, using (12) and the fact that $v\perp \pi {\delta }_{a_i,{\lambda }_i}$, we obtain

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

(18)

Thus, Lemma A5 and (18) imply that

$${\int }_{\Omega }{\left(\pi {\delta }_{a_i,{\lambda }_i}\right)}^{p-\epsilon }v=O\left(\epsilon \parallel v\parallel \right).$$

(19)

Putting (17) and (19) in (15), the result follows. □

Note that Proposition 1 implies that $Q_{\epsilon }$ is a positive definite quadratic form on $E_{a,\lambda }^{\perp }$. Hence, combining (10), Proposition 1, and Lemma 1, we deduce that

Proposition 2.

Let $(\alpha ,a,\lambda )\in \mathcal{O}(N,\mu ,d_0)$ and let $\underline{u}:={\sum }_{i=1}^N{\alpha }_i\pi {\delta }_{a_i,{\lambda }_i}$. Then, for ε small, there exists a unique $\overline{v}\in E_{a,\lambda }^{\perp }$ satisfying

$$\begin{array}{c}\hfill \langle \nabla I_{\epsilon }(\underline{u}+\overline{v}),h\rangle =0\forall h\in E_{a,\lambda }^{\perp }.\end{array}$$

(20)

Moreover, $\overline{v}$ satisfies $∥\overline{v}∥\le c\epsilon$.

Next, we perform the expansion of the gradient of the functional $I_{\epsilon }$.

2.2. Expansion of the Gradient

We begin with the expansion in terms of the gluing parameter ${\alpha }_i$. Specifically, we prove

Proposition 3.

Let $(\alpha ,a,\lambda )\in \mathcal{O}(N,\mu ,d_0)$, $v\in E_{a,\lambda }^{\perp }$ and $u={\sum }_{i=1}^N{\alpha }_i\pi {\delta }_{a_i,{\lambda }_i}+v$. Then, for ε small and $i\in \left\{1,\cdots ,N\right\}$, we have

$$〈\nabla I_{\epsilon }\left(u\right),\pi {\delta }_{a_i,{\lambda }_i}〉={\alpha }_i{S}_n\left(1-{\alpha }_i^{p-1-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}\right)+O\left(\epsilon \right)where{S}_n:={\int }_{{\mathbb{R}}^n}{\delta }_{0,1}^{2n/(n-2)}.$$

Proof. 

Observe that, since $v\in E_{a,\lambda }^{\perp }$, it holds that

$$\begin{array}{c}\hfill 〈\nabla I_{\epsilon }\left(u\right),\pi {\delta }_{a_i,{\lambda }_i}〉=\sum _{j=1}^N{\alpha }_j〈\pi {\delta }_{a_j,{\lambda }_j},\pi {\delta }_{a_i,{\lambda }_i}〉-{\int }_{\Omega }{\left|u\right|}^{p-1-\epsilon }u\left(\pi {\delta }_{a_i,{\lambda }_i}\right).\end{array}$$

(21)

First, for $j\ne i$, we have

$$\begin{array}{cc}\hfill 〈\pi {\delta }_{a_j,{\lambda }_j},\pi {\delta }_{a_i,{\lambda }_i}〉& ={\int }_{\Omega }\nabla \left(\pi {\delta }_{a_j,{\lambda }_j}\right)\nabla \left(\pi {\delta }_{a_i,{\lambda }_i}\right)+{\int }_{\Omega }V\left(\pi {\delta }_{a_j,{\lambda }_j}\right)\left(\pi {\delta }_{a_i,{\lambda }_i}\right)\hfill \\ \hfill & ={\int }_{\Omega }\left(-\Delta +V\right)\left(\pi {\delta }_{a_j,{\lambda }_j}\right)\left(\pi {\delta }_{a_i,{\lambda }_i}\right)\le {\int }_{\Omega }{\delta }_{a_j,{\lambda }_j}^{{\displaystyle \frac{n+2}{n-2}}}{\delta }_{a_i,{\lambda }_i}\hfill \end{array}$$

(22)

and we have

$${\int }_{\Omega }{\delta }_{a_j,{\lambda }_j}^{{\displaystyle \frac{n+2}{n-2}}}{\delta }_{a_i,{\lambda }_i}\le {\displaystyle \frac{c}{{\lambda }_j^{(n+2)/2}}}{\int }_{B_i}{\delta }_{a_i,{\lambda }_i}+{\displaystyle \frac{c}{{\lambda }_i^{(n-2)/2}}}{\int }_{\Omega \setminus B_i}{\delta }_{a_j,{\lambda }_j}^{{\displaystyle \frac{n+2}{n-2}}}\le {\displaystyle \frac{c}{{\left({\lambda }_i{\lambda }_j\right)}^{(n-2)/2}}}\le c\gamma {\left(\epsilon \right)}^2.$$

(23)

Second, for $j=i$, as in the previous computation, we have

$$\begin{array}{c}\hfill {∥\pi {\delta }_{a_i,{\lambda }_i}∥}^2={\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^{{\displaystyle \frac{n+2}{n-2}}}\left(\pi {\delta }_{a_i,{\lambda }_i}\right)={\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^{{\displaystyle \frac{2n}{n-2}}}-{\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^{{\displaystyle \frac{n+2}{n-2}}}{\theta }_{a_i,{\lambda }_i}.\end{array}$$

(24)

Estimate of the first integral is well known, and we have

$$\begin{array}{c}\hfill {\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^{{\displaystyle \frac{2n}{n-2}}}=S_n+O\left({\displaystyle \frac{1}{{\lambda }_i^n}}\right).\end{array}$$

(25)

For the second integral, using Proposition A1, easy computations imply that

$$\begin{array}{cc}\hfill {\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^{{\displaystyle \frac{n+2}{n-2}}}{\theta }_{a_i,{\lambda }_i}& \le c{\int }_{B_i}{R}_1(x,a_i,{\lambda }_i){\delta }_{a_i,{\lambda }_i}^{{\displaystyle \frac{2n}{n-2}}}+{\int }_{\Omega \setminus B_i}{\delta }_{a_i,{\lambda }_i}^{{\displaystyle \frac{2n}{n-2}}}\le c\epsilon +{\displaystyle \frac{c}{{\lambda }_i^n}}\le c\epsilon .\hfill \end{array}$$

(26)

Now, we focus on the last term in (21). Let $\underline{u}:={\sum }_{j=1}^N{\alpha }_j\left(\pi {\delta }_{a_j,{\lambda }_j}\right)$, using Lemma A1, we obtain

$$\begin{array}{c}\hfill {\int }_{\Omega }{\left|u\right|}^{p-\epsilon -1}u\left(\pi {\delta }_{a_i,{\lambda }_i}\right)={\int }_{\Omega }{\left(\underline{u}\right)}^{p-\epsilon }\left(\pi {\delta }_{a_i,{\lambda }_i}\right)+(p-\epsilon ){\int }_{\Omega }{\left(\underline{u}\right)}^{p-\epsilon -1}v\left(\pi {\delta }_{a_i,{\lambda }_i}\right)+O\left({∥v∥}^2\right).\end{array}$$

(27)

The first integral in the right hand side of (27) can be written as (using Lemma A1, Proposition A1, (12), (23) and (26))

$$\begin{array}{cc}\hfill {\int }_{\Omega }{\left(\underline{u}\right)}^{p-\epsilon }\left(\pi {\delta }_{a_i,{\lambda }_i}\right)& ={\alpha }_i^{p-\epsilon }{\int }_{\Omega }{\left(\pi {\delta }_{a_i,{\lambda }_i}\right)}^{p+1-\epsilon }+\sum _{j\ne i}O\left({\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^{p-\epsilon }{\delta }_{a_j,{\lambda }_j}+{\int }_{\Omega }{\delta }_{a_j,{\lambda }_j}^{p-\epsilon }{\delta }_{a_i,{\lambda }_i}\right)\hfill \\ \hfill & ={\alpha }_i^{p-\epsilon }{\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^{p+1-\epsilon }+O\left({\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^p{\theta }_{a_i,{\lambda }_i}+\sum _{j\ne i}{\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^p{\delta }_{a_j,{\lambda }_j}+{\int }_{\Omega }{\delta }_{a_j,{\lambda }_j}^p{\delta }_{a_i,{\lambda }_i}\right)\hfill \\ \hfill & ={\alpha }_i^{p-\epsilon }{c}_0^{-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}{S}_n+O\left(\epsilon +{\displaystyle \frac{1}{{\lambda }_i^n}}+\gamma {\left(\epsilon \right)}^2\right),\hfill \end{array}$$

(28)

where we have used (64) of [19]. Concerning the second integral in (27), it is computed in Lemma A6.

Combining the previous estimates, the proof follows. □

Next, we present a balancing formula that relates to the rate of concentration. Specifically, we prove

Proposition 4.

Let $n\ge 4$, $(\alpha ,a,\lambda )\in \mathcal{O}(N,\mu ,d_0)$, $v\in E_{a,\lambda }^{\perp }$ and $u={\sum }_{j=1}^N{\alpha }_j\pi {\delta }_{a_j,{\lambda }_j}+v$. For small ε and $i\in \left\{1,...,N\right\}$, we have

$$\begin{array}{cc}\hfill 〈\nabla I_{\epsilon }\left(u\right),{\lambda }_i{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial {\lambda }_i}}〉=& c_0^{-\epsilon }{\lambda }_i^{-\epsilon {\displaystyle \frac{n-2}{2}}}{c}_2{\alpha }_i^{p-\epsilon }\epsilon \hfill \\ \hfill & -c\left(n\right){\alpha }_i{\displaystyle \frac{{ln}^{{\sigma }_n}\left({\lambda }_i\right)}{{\lambda }_i^2}}V\left(a_i\right)\left(2{\alpha }_i^{p-\epsilon -1}{c}_0^{-\epsilon }{\lambda }_i^{-\epsilon {\displaystyle \frac{n-2}{2}}}-1\right)+O\left(R_1\left(\epsilon \right)\right),\hfill \end{array}$$

where $c\left(n\right)$ is defined in Lemma A3 and

$$\begin{array}{cc}\hfill & R_1\left(\epsilon \right):=\left(\epsilon /|ln\epsilon |(ifn=4);{\epsilon }^{3/2}(ifn=5);{\epsilon }^2|ln\epsilon |(ifn=6);{\epsilon }^2(ifn\ge 7)\right),\hfill \\ \hfill & c_2:={\left({\displaystyle \frac{n-2}{2}}\right)}^2{c}_0^{2n/(n-2)}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{({\left|x\right|}^2-1)}{{(1+{\left|x\right|}^2)}^{n+1}}}ln(1+{\left|x\right|}^2)dx>0.\hfill \end{array}$$

Proof. 

We will follow the proof of Proposition 3, and we need to be more precise in some integrals. Since $v\in E_{a,\lambda }^{\perp }$, we have

$$〈u,{\lambda }_i{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial {\lambda }_i}}〉=\sum {\alpha }_j〈\pi {\delta }_{a_j,{\lambda }_j},{\lambda }_i{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial {\lambda }_i}}〉.$$

First, for $j\ne i$, using Remark A1 and (23), we have

$$〈\pi {\delta }_{a_j,{\lambda }_j},{\lambda }_i{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial {\lambda }_i}}〉={\int }_{\Omega }{\delta }_{a_j,{\lambda }_j}^{{\displaystyle \frac{n+2}{n-2}}}{\lambda }_i{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial {\lambda }_i}}=O\left({\int }_{\Omega }{\delta }_{a_j,{\lambda }_j}^{{\displaystyle \frac{n+2}{n-2}}}{\delta }_{a_i,{\lambda }_i}\right)=O\left(\gamma {\left(\epsilon \right)}^2\right).$$

(29)

Second, for $j=i$, using Lemma A3, we obtain

$$\begin{array}{cc}\hfill 〈\pi {\delta }_{a_i,{\lambda }_i},{\lambda }_i{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial {\lambda }_i}}〉& ={\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^{{\displaystyle \frac{n+2}{n-2}}}\left({\lambda }_i{\displaystyle \frac{\partial {\delta }_{a_i,{\lambda }_i}}{\partial {\lambda }_i}}-{\lambda }_i{\displaystyle \frac{\partial {\theta }_{a_i,{\lambda }_i}}{\partial {\lambda }_i}}\right)\hfill \\ \hfill & =c\left(n\right){\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^2}}V\left(a_i\right)+O\left(\gamma {\left(\epsilon \right)}^2+{\epsilon }^2+(ifn=6){\epsilon }^2|ln\epsilon |\right).\hfill \end{array}$$

We derive that

$$〈u,{\lambda }_i{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial {\lambda }_i}}〉={\alpha }_{i}c\left(n\right){\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^2}}V\left(a_i\right)+O\left(\gamma {\left(\epsilon \right)}^2+{\epsilon }^2+(ifn=6){\epsilon }^2|ln\epsilon |\right).$$

(30)

Now, using Lemma A1, we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }& {\left|u\right|}^{p-\epsilon -1}u{\lambda }_i{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial {\lambda }_i}}\hfill \\ \hfill & ={\int }_{\Omega }{\left(\underline{u}\right)}^{p-\epsilon }{\lambda }_i{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial {\lambda }_i}}+(p-\epsilon ){\int }_{\Omega }{\left(\underline{u}\right)}^{p-\epsilon -1}v{\lambda }_i{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial {\lambda }_i}}+{O(\parallel v\parallel }^2).\hfill \end{array}$$

(31)

The last integral is computed in Lemma A6. For the first one, using Lemma A1 and Remark A1, we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }& {\left(\underline{u}\right)}^{p-\epsilon }{\lambda }_i{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial {\lambda }_i}}\hfill \\ \hfill & ={\alpha }_i^{p-\epsilon }{\int }_{\Omega }{\left(\pi {\delta }_{a_i,{\lambda }_i}\right)}^{p-\epsilon }{\lambda }_i{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial {\lambda }_i}}+\sum _{j\ne i}O\left({\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^p{\delta }_{a_j,{\lambda }_j}+{\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}{\delta }_{a_j,{\lambda }_j}^p\right).\hfill \end{array}$$

(32)

First, we remark that the remainder term is computed in (23). Second, we focus on estimating the first integral. Using (12) and Lemma A1, we obtain

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

(33)

To estimate the last term in (33), using Propositions A1 and A2, we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^{p-1}{\theta }_{a_i,{\lambda }_i}^2+{\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^{p-1}{\theta }_{a_i,{\lambda }_i}{\lambda }_i\left|{\displaystyle \frac{\partial {\theta }_{a_i,{\lambda }_i}}{\partial {\lambda }_i}}\right|\le & c{\int }_{\Omega }{R}_1^2(x,a_i,{\lambda }_i){\delta }_{a_i,{\lambda }_i}^{p+1}\hfill \\ \hfill \le & c\left({\displaystyle \frac{1}{{\lambda }_i^4}}(ifn\ge 5),{\displaystyle \frac{{ln}^3{\lambda }_i}{{\lambda }_i^4}}(ifn=4)\right).\hfill \end{array}$$

Concerning the other integrals of (33), using (12) and Propositions A1 and A2, we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^{p-\epsilon }{\lambda }_i{\displaystyle \frac{\partial {\theta }_{a_i,{\lambda }_i}}{\partial {\lambda }_i}}& =c_0^{-\epsilon }{\lambda }_i^{-\epsilon {\displaystyle \frac{n-2}{2}}}{\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^p{\lambda }_i{\displaystyle \frac{\partial {\theta }_{a_i,{\lambda }_i}}{\partial {\lambda }_i}}\hfill \\ \hfill & +O\left(\epsilon \int R_1(x,a_i,{\lambda }_i){\delta }_{a_i,{\lambda }_i}^{p+1}ln(1+{\lambda }_i^2|x-a_i{|}^2)\right),\hfill \\ \hfill {\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^{p-\epsilon -1}{\lambda }_i{\displaystyle \frac{\partial {\delta }_{a_i,{\lambda }_i}}{\partial {\lambda }_i}}{\theta }_{a_i,{\lambda }_i}& =c_0^{-\epsilon }{\lambda }_i^{-\epsilon {\displaystyle \frac{n-2}{2}}}{\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^{p-1}{\lambda }_i{\displaystyle \frac{\partial {\delta }_{a_i,{\lambda }_i}}{\partial {\lambda }_i}}{\theta }_{a_i,{\lambda }_i}\hfill \\ \hfill & +O\left(\epsilon \int R_1(x,a_i,{\lambda }_i){\delta }_{a_i,{\lambda }_i}^{p+1}ln(1+{\lambda }_i^2|x-a_i{|}^2)\right).\hfill \end{array}$$

Thus, using Lemma A3, we obtain

$$\begin{array}{cc}\hfill {-\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^{p-\epsilon }& {\lambda }_i{\displaystyle \frac{\partial {\theta }_{a_i,{\lambda }_i}}{\partial {\lambda }_i}}-(p-\epsilon ){\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^{p-\epsilon -1}{\lambda }_i{\displaystyle \frac{\partial {\delta }_{a_i,{\lambda }_i}}{\partial {\lambda }_i}}{\theta }_{a_i,{\lambda }_i}\hfill \\ \hfill =& c_0^{-\epsilon }{\lambda }_i^{-\epsilon {\displaystyle \frac{n-2}{2}}}\left(2c\left(n\right){\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^2}}V\left(a_i\right)\right)+O\left({\displaystyle \frac{1}{{\lambda }_i^{n-2}}}+{\displaystyle \frac{1}{{\lambda }_i^4}}+(ifn=6){\displaystyle \frac{ln{\lambda }_i}{{\lambda }_i^4}}+\epsilon {\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^2}}\right).\hfill \end{array}$$

It remains to study the first integral of (33). Using (91) of [19], we obtain

$${\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^{p-\epsilon }{\lambda }_i{\displaystyle \frac{\partial {\delta }_{a_i,{\lambda }_i}}{\partial {\lambda }_i}}=-c_2\epsilon c_0^{-\epsilon }{\lambda }_i^{-\epsilon {\displaystyle \frac{n-2}{2}}}+O\left({\epsilon }^2+{\displaystyle \frac{ln{\lambda }_i}{{\lambda }_i^n}}\right).$$

Combining the previous estimates, (33) becomes

$$\begin{array}{cc}\hfill {\int }_{\Omega }{\left(\pi {\delta }_{a_i,{\lambda }_i}\right)}^{p-\epsilon }{\lambda }_i{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial {\lambda }_i}}=& c_0^{-\epsilon }{\lambda }_i^{-\epsilon {\displaystyle \frac{n-2}{2}}}\left(-c_2\epsilon +2c\left(n\right){\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^2}}V\left(a_i\right)\right)\hfill \\ \hfill & +O\left({\epsilon }^2+{\displaystyle \frac{1}{{\lambda }_i^{n-2}}}+{\displaystyle \frac{1}{{\lambda }_i^4}}+(ifn=6){\displaystyle \frac{ln{\lambda }_i}{{\lambda }_i^4}}\right).\hfill \end{array}$$

(34)

This completes the estimate of the first integral of (32).

Finally, combining (30)–(32), (34) and Lemma A6, the result follows. □

We end this section by presenting a balancing condition related to the points of concentration.

Proposition 5.

Let $n\ge 4$, $(\alpha ,a,\lambda )\in \mathcal{O}(N,\mu ,d_0)$, $v\in E_{a,\lambda }^{\perp }$ and $u={\sum }_{j=1}^N{\alpha }_j\left(\pi {\delta }_{a_j,{\lambda }_j}\right)+v$. For small ε and $i\in \left\{1,...,N\right\}$, we have

$$〈\nabla I_{\epsilon }\left(u\right),{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial a_i}}〉=c_2\left(n\right){\alpha }_i{\displaystyle \frac{{ln}^{{\sigma }_n}\left({\lambda }_i\right)}{{\lambda }_i^3}}\nabla V\left(a_i\right)\left(2{\alpha }_i^{p-\epsilon -1}{c}_0^{-\epsilon }{\lambda }_i^{-\epsilon {\displaystyle \frac{n-2}{2}}}-1\right)+O\left(R_2\left(\epsilon \right)\right),$$

where $c_2\left(n\right)$ is defined in Lemma A4 and

$$R_2\left(\epsilon \right):=\left({[\epsilon /|ln\epsilon \left|\right]}^{3/2}ifn=4;{\epsilon }^2|ln\epsilon |ifn=5;{\epsilon }^{2}ifn\ge 6\right).$$

Proof. 

We will follow the proof of Proposition 4. Since $v\in E_{a,\lambda }^{\perp }$, we have

$$〈u,{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial a_i}}〉=\sum {\alpha }_j〈\pi {\delta }_{a_j,{\lambda }_j},{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial a_i}}〉.$$

First, for $j\ne i$, using Remark A1 and (7), we obtain

$$\begin{array}{cc}\hfill 〈\pi {\delta }_{a_j,{\lambda }_j},{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial a_i}}〉& ={\int }_{\Omega }{\delta }_{a_j,{\lambda }_j}^{{\displaystyle \frac{n+2}{n-2}}}\left({\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_{a_i,{\lambda }_i}}{\partial a_i}}-{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\theta }_{a_i,{\lambda }_i}}{\partial a_i}}\right)\hfill \\ \hfill & =O\left({\displaystyle \frac{1}{{\lambda }_i^{n/2}}}{\int }_{B_j}{\delta }_{a_j,{\lambda }_j}^{{\displaystyle \frac{n+2}{n-2}}}+{\displaystyle \frac{1}{{\lambda }_j^{(n+2)/2}}}{\int }_{\Omega \setminus B_j}{\delta }_{a_i,{\lambda }_i}\right)\hfill \\ \hfill & =O\left({\displaystyle \frac{1}{{\lambda }_i^{n/2}{\lambda }_j^{(n-2)/2}}}+{\displaystyle \frac{1}{{\lambda }_j^{(n+2)/2}{\lambda }_i^{(n-2)/2}}}\right)\hfill \\ \hfill & =O\left(\gamma {\left(\epsilon \right)}^{{\displaystyle \frac{2n-2}{n-2}}}\right)=O\left({[\epsilon /|ln\epsilon \left|\right]}^{3/2}(ifn=4);{\epsilon }^{(n-1)/2}(ifn\ge 5)\right).\hfill \end{array}$$

(35)

Second, for $j=i$, using Lemma A4, we obtain

$$\begin{array}{cc}\hfill 〈\pi {\delta }_{a_i,{\lambda }_i},{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial a_i}}〉& ={\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^{{\displaystyle \frac{n+2}{n-2}}}\left({\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_{a_i,{\lambda }_i}}{\partial a_i}}-{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\theta }_{a_i,{\lambda }_i}}{\partial a_i}}\right)\hfill \\ \hfill & =-c_2\left(n\right){\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^3}}\nabla V\left(a_i\right)+O\left({\displaystyle \frac{1}{{\lambda }_i^{n-1}}}+{\displaystyle \frac{1}{{\lambda }_i^4}}+(ifn=5){\displaystyle \frac{ln{\lambda }_i}{{\lambda }_i^4}}\right).\hfill \end{array}$$

We derive that

$$\begin{array}{cc}\hfill 〈u,{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial a_i}}〉& =-{\alpha }_i{c}_2\left(n\right){\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^3}}\nabla V\left(a_i\right)\hfill \\ \hfill & +O\left({[\epsilon /|ln\epsilon \left|\right]}^{3/2}(ifn=4);{\epsilon }^2|ln\epsilon |(ifn=5);{\epsilon }^2(ifn\ge 6)\right).\hfill \end{array}$$

(36)

Now, using Lemma A1, we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }& {\left|u\right|}^{p-\epsilon -1}u{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial a_i}}\hfill \\ \hfill & ={\int }_{\Omega }{\left(\underline{u}\right)}^{p-\epsilon }{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial a_i}}+(p-\epsilon ){\int }_{\Omega }{\left(\underline{u}\right)}^{p-\epsilon -1}v{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial a_i}}+{O(\parallel v\parallel }^2).\hfill \end{array}$$

(37)

The last integral is computed in Lemma A6. For the first one, using Lemma A1, we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }{\left(\underline{u}\right)}^{p-\epsilon }{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial a_i}}=& \sum _{j=1}^N{\alpha }_j^{p-\epsilon }{\int }_{\Omega }{\left(\pi {\delta }_{a_j,{\lambda }_j}\right)}^{p-\epsilon }{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial a_i}}\hfill \\ \hfill & +(p-\epsilon ){\int }_{\Omega }{\left({\alpha }_i\pi {\delta }_{a_i,{\lambda }_i}\right)}^{p-\epsilon -1}\left(\sum _{j\ne i}{\alpha }_j\pi {\delta }_{a_j,{\lambda }_j}\right){\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial a_i}}\hfill \\ \hfill & +\sum _{j\ne k}O\left({\int }_{\Omega }{\left({\delta }_{a_j,{\lambda }_j}{\delta }_{a_k,{\lambda }_k}\right)}^{(p+1-\epsilon )/2}\right).\hfill \end{array}$$

(38)

To estimate the last integral in (38), using (12) and (7), we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }{\left({\delta }_{a_j,{\lambda }_j}{\delta }_{a_k,{\lambda }_k}\right)}^{(p+1-\epsilon )/2}& \le {\int }_{\Omega }{\left({\delta }_{a_j,{\lambda }_j}{\delta }_{a_k,{\lambda }_k}\right)}^{n/(n-2)}\hfill \\ \hfill & \le {\displaystyle \frac{c}{{\lambda }_k^{n/2}}}{\int }_{B_j}{\delta }_{a_j,{\lambda }_j}^{n/(n-2)}+{\displaystyle \frac{c}{{\lambda }_j^{n/2}}}{\int }_{\Omega \setminus B_j}{\delta }_{a_k,{\lambda }_k}^{n/(n-2)}\le c{\displaystyle \frac{ln{\lambda }_k+ln{\lambda }_j}{{\left({\lambda }_k{\lambda }_j\right)}^{n/2}}}\hfill \\ \hfill & \le c\left({\epsilon }^2{|ln\epsilon |}^{-1}(ifn=4);|ln\epsilon |{\epsilon }^{n/2}(ifn\ge 5)\right).\hfill \end{array}$$

(39)

For the first integral in (38) with $j\ne i$, following the same computations as in (35), we obtain

$${\int }_{\Omega }{\left(\pi {\delta }_{a_j,{\lambda }_j}\right)}^{p-\epsilon }{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial a_i}}=O\left(\gamma {\left(\epsilon \right)}^{(2n-2)/(n-2)}\right).$$

(40)

Concerning the first integral in (38) with $j=i$, using Lemma A1 and (12), we obtain

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

(41)

For the last two terms in (41), using Propositions A1 and A3, we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^{p-1}{\theta }_{a_i,{\lambda }_i}^2+{\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^{p-1}{\theta }_{a_i,{\lambda }_i}{\displaystyle \frac{1}{{\lambda }_i}}\left|{\displaystyle \frac{\partial {\theta }_{a_i,{\lambda }_i}}{\partial a_i}}\right|\le & c{\int }_{\Omega }\left(R_1^2(x,a_i,{\lambda }_i)+{\displaystyle \frac{1}{{\lambda }_i^4}}+{\displaystyle \frac{|x-a_i{|}^2}{{\lambda }_i^2}}\right){\delta }_{a_i,{\lambda }_i}^{p+1}\hfill \\ \hfill \le & c\left({\displaystyle \frac{1}{{\lambda }_i^4}}(ifn\ge 5),{\displaystyle \frac{{ln}^3{\lambda }_i}{{\lambda }_i^4}}(ifn=4)\right).\hfill \end{array}$$

Concerning the other integrals of (41), using (12) and Propositions A1 and A3, by oddness, we obtain

$$\begin{array}{cc}\hfill & {\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^{p-\epsilon }{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_{a_i,{\lambda }_i}}{\partial a_i}}={\int }_{B(a_i,d)}\cdots +{\int }_{\Omega \setminus B(a_i,d)}\cdots =O\left({\displaystyle \frac{1}{{\lambda }_i^{n+1}}}\right),\hfill \\ \hfill & {\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^{p-\epsilon }{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\theta }_{a_i,{\lambda }_i}}{\partial a_i}}=c_0^{-\epsilon }{\lambda }_i^{-\epsilon {\displaystyle \frac{n-2}{2}}}{\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^p{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\theta }_{a_i,{\lambda }_i}}{\partial a_i}}\hfill \\ \hfill & +O\left(\epsilon \int R_2(x,a_i,{\lambda }_i){\delta }_{a_i,{\lambda }_i}^{p+1}ln(1+{\lambda }_i^2|x-a_i{|}^2)\right),\hfill \\ \hfill & {\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^{p-\epsilon -1}{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_{a_i,{\lambda }_i}}{\partial a_i}}{\theta }_{a_i,{\lambda }_i}=c_0^{-\epsilon }{\lambda }_i^{-\epsilon {\displaystyle \frac{n-2}{2}}}{\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^{p-1}{\lambda }_i{\displaystyle \frac{\partial {\delta }_{a_i,{\lambda }_i}}{\partial {\lambda }_i}}{\theta }_{a_i,{\lambda }_i}\hfill \\ \hfill & +O\left(\epsilon \int R_1(x,a_i,{\lambda }_i){\delta }_{a_i,{\lambda }_i}^{p+1}ln(1+{\lambda }_i^2|x-a_i{|}^2)\right).\hfill \end{array}$$

Using Lemma A4, we obtain

$$\begin{array}{cc}\hfill & {-\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^{p-\epsilon }{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\theta }_{a_i,{\lambda }_i}}{\partial a_i}}-(p-\epsilon ){\int }_{\Omega }{\delta }_{a_i,{\lambda }_i}^{p-\epsilon -1}{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_{a_i,{\lambda }_i}}{\partial a_i}}{\theta }_{a_i,{\lambda }_i}\hfill \\ \hfill & =-2c_2\left(n\right)c_0^{-\epsilon }{\lambda }_i^{-\epsilon {\displaystyle \frac{n-2}{2}}}{\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^3}}\nabla V\left(a_i\right)+O\left({\displaystyle \frac{1}{{\lambda }_i^{n-1}}}+{\displaystyle \frac{1}{{\lambda }_i^4}}+(ifn=5){\displaystyle \frac{ln{\lambda }_i}{{\lambda }_i^4}}+\epsilon {\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^2}}\right).\hfill \end{array}$$

Combining the previous estimates, (41) becomes

$$\begin{array}{cc}\hfill {\int }_{\Omega }{\left(\pi {\delta }_{a_i,{\lambda }_i}\right)}^{p-\epsilon }{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial a_i}}=& -2c_2\left(n\right)c_0^{-\epsilon }{\lambda }_i^{-\epsilon {\displaystyle \frac{n-2}{2}}}{\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^3}}\nabla V\left(a_i\right)\hfill \\ \hfill & +O\left({\displaystyle \frac{1}{{\lambda }_i^{n-1}}}+{\displaystyle \frac{1}{{\lambda }_i^4}}+(ifn=5){\displaystyle \frac{ln{\lambda }_i}{{\lambda }_i^4}}+\epsilon {\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^2}}\right).\hfill \end{array}$$

(42)

It remains to study the second integral in the right-hand side of (38). Using Propositions A1 and A3, we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }& {\left(\pi {\delta }_{a_i,{\lambda }_i}\right)}^{p-\epsilon -1}{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial a_i}}\left(\pi {\delta }_{a_j,{\lambda }_j}\right)\hfill \\ \hfill & ={\int }_{B_i}\left[{\delta }_{a_i,{\lambda }_i}^{p-\epsilon -1}+O\left({\delta }_{a_i,{\lambda }_i}^{p-2}{\theta }_{a_i{\lambda }_i}\right)\right]\left[{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_{a_i,{\lambda }_i}}{\partial a_i}}-{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\theta }_{a_i,{\lambda }_i}}{\partial a_i}}\right]\left(\pi {\delta }_{a_j,{\lambda }_j}\right)+O\left({\int }_{\Omega \setminus B_i}{\delta }_{a_i,{\lambda }_i}^p{\delta }_{a_j,{\lambda }_j}\right)\hfill \\ \hfill & ={\int }_{B_i}{\delta }_{a_i,{\lambda }_i}^{p-\epsilon -1}{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_{a_i,{\lambda }_i}}{\partial a_i}}\left(\pi {\delta }_{a_j,{\lambda }_j}\right)+O\left({\displaystyle \frac{1}{{\lambda }_j^{(n-2)/2}}}{\int }_{B_i}{\theta }_{a_i{\lambda }_i}{\delta }_{a_i,{\lambda }_i}^{p-1}+{\displaystyle \frac{1}{{\lambda }_i^{(n+2)/2}}}{\int }_{\Omega \setminus B_i}{\delta }_{a_j,{\lambda }_j}\right).\hfill \end{array}$$

Observe that, using Proposition A1, we have

$$\begin{array}{cc}\hfill {\int }_{B_i}{\theta }_{a_i{\lambda }_i}{\delta }_{a_i,{\lambda }_i}^{p-1}\le c{\int }_{B_i}{R}_1(x,a_i,{\lambda }_i){\delta }_{a_i,{\lambda }_i}^p& \le c{\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^2}}{\int }_{B_i}{\delta }_{a_i,{\lambda }_i}^p+{\int }_{B_i}|x-a_i{|}^2|ln|x-a_i{\left|\right|}^{{\sigma }_n}{\delta }_{a_i,{\lambda }_i}^p\hfill \\ \hfill & \le c{\displaystyle \frac{{ln}^{{\sigma }_n+1}{\lambda }_i}{{\lambda }_i^{(n+2)/2}}}\hfill \end{array}$$

and, expanding $\pi {\delta }_{a_j,{\lambda }_j}$ around $a_i$, by oddness, we obtain

$${\int }_{B_i}{\delta }_{a_i,{\lambda }_i}^{p-\epsilon -1}{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial {\delta }_{a_i,{\lambda }_i}}{\partial a_i}}\left(\pi {\delta }_{a_j,{\lambda }_j}\right)=O\left({\displaystyle \frac{1}{{\lambda }_j^{(n-2)/2}}}{\int }_{B_i}|x-a_i|{\delta }_{a_i,{\lambda }_i}^p\right)=O\left({\displaystyle \frac{1}{{\lambda }_j^{(n-2)/2}{\lambda }_i^{n/2}}}\right).$$

We derive that

$$\begin{array}{cc}\hfill {\int }_{\Omega }{\left(\pi {\delta }_{a_i,{\lambda }_i}\right)}^{p-\epsilon -1}{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial a_i}}\left(\pi {\delta }_{a_j,{\lambda }_j}\right)& =O\left({\displaystyle \frac{1}{({\lambda }_j^{(n-2)/2}{\lambda }_i^{n/2}}}\right)\hfill \\ \hfill & =O\left({[\epsilon /|ln\epsilon \left|\right]}^{3/2}(ifn=4);{\epsilon }^{(n-1)/2}(ifn\ge 5)\right).\hfill \end{array}$$

(43)

Finally, combining (36)–(43) and Lemma A6, the result follows. □

3. Construction of Interior Peak Solutions

Let $b_1,\cdots b_N$ be non-degenerate distinct critical points of V. As in [17] (see also [18,19]), the strategy of the proof of Theorem 1 is as follows. Let

$$\begin{array}{cc}\hfill {\mathcal{O}}_1(N,\epsilon )=\{& (\alpha ,\lambda ,a,v)\in {\left({\mathbb{R}}_{+}\right)}^N\times {\left({\mathbb{R}}_{+}\right)}^N\times {\Omega }^N\times H^1(\Omega ):|{\alpha }_i-1|<\epsilon {ln}^2\epsilon ,|a_i-b_i|<{\epsilon }^{1/5},\hfill \\ \hfill & M^{-1}\sqrt{{|ln\epsilon |}^{{\sigma }_n}/\epsilon }\le {\lambda }_i\le M\sqrt{{|ln\epsilon |}^{{\sigma }_n}/\epsilon },\forall 1\le i\le N,v\in E_{a,\lambda }^{\perp }and\parallel v\parallel <\sqrt{\epsilon }\}\hfill \end{array}$$

(44)

where M is a positive constant and $E_{a,\lambda }$ is defined by (5).

Note that, for each $(\alpha ,\lambda ,a,v)\in {\mathcal{O}}_1(N,\epsilon )$, it follows that $(\alpha ,\lambda ,a)\in \mathcal{O}(N,\mu ,d_0)$ by taking $\mu >\epsilon {ln}^2\epsilon$ and $d_0$ to satisfy $d_0\le min\left\{\right|b_i-b_j|/4:j\ne i\}$ and $d_0\le min\{d(b_i,\partial \Omega )/4:1\le i\le N\}$.

In addition, we consider the following function

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

We observe that $(\alpha ,\lambda ,a,v)$ is a critical point of ${\psi }_{\epsilon }$ if and only if $u={\sum }_{i=1}^N{\alpha }_i\pi {\delta }_i+v$ is a critical point of $I_{\epsilon }$. Therefore, our goal is to find the critical points of ${\psi }_{\epsilon }$. Since the variable v belongs to $E_{a,\lambda }^{\perp }$, the Lagrange multiplier theorem enables us to derive the following proposition.

Proposition 6.

$(\alpha ,\lambda ,a,v)\in {\mathcal{O}}_1(N,\epsilon )$ is a critical point of ${\psi }_{\epsilon }$ if and only if there exists $(A,B,C)\in {\mathbb{R}}^N\times {\mathbb{R}}^N\times {\left({\mathbb{R}}^n\right)}^N$ such that the following holds:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial {\psi }_{\epsilon }}{\partial {\alpha }_i}}(\alpha ,\lambda ,a,v)=0\forall i\in \left\{1,...,N\right\},\hfill \end{array}$$

(45)

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

(46)

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

(47)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial {\psi }_{\epsilon }}{\partial v}}(\alpha ,\lambda ,a,v)=\sum _{k=1}^N\left(A_k\pi {\delta }_{a_k,{\lambda }_k}+B_k{\lambda }_k{\displaystyle \frac{\partial \left(\pi {\delta }_{a_k,{\lambda }_k}\right)}{\partial {\lambda }_k}}+\sum _{j=1}^n{C}_{kj}{\displaystyle \frac{1}{{\lambda }_k}}{\displaystyle \frac{\partial \left(\pi {\delta }_{a_k,{\lambda }_k}\right)}{\partial a_k^j}}\right).\hfill \end{array}$$

(48)

To prove Theorem 1, we will conduct a detailed analysis of the preceding equations. Observe that

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial {\psi }_{\epsilon }}{\partial {\alpha }_i}}(\alpha ,\lambda ,a,v)=〈\nabla I_{\epsilon }\left(\sum _{i=1}^N{\alpha }_i\pi {\delta }_{a_i,{\lambda }_i}+v\right),\pi {\delta }_{a_i,{\lambda }_i}〉,\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial {\psi }_{\epsilon }}{\partial {\lambda }_i}}(\alpha ,\lambda ,a,v)=〈\nabla I_{\epsilon }\left(\sum _{i=1}^N{\alpha }_i\pi {\delta }_{a_i,{\lambda }_i}+v\right),{\alpha }_i{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial {\lambda }_i}}〉,\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial {\psi }_{\epsilon }}{\partial a_i}}(\alpha ,\lambda ,a,v)=〈\nabla I_{\epsilon }\left(\sum _{i=1}^N{\alpha }_i\pi {\delta }_{a_i,{\lambda }_i}+v\right),{\alpha }_i{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial a_i}}〉,\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial {\psi }_{\epsilon }}{\partial v}}(\alpha ,\lambda ,a,v)=\nabla I_{\epsilon }\left(\sum _{i=1}^N{\alpha }_i\pi {\delta }_{a_i,{\lambda }_i}+v\right).\hfill \end{array}$$

(52)

Let $(\alpha ,\lambda ,a,0)\in {\mathcal{O}}_1(N,\epsilon )$, which is defined by (44). Clearly, $\overline{v}$, as obtained in Proposition 2, satisfies Equation (48). Hence, we need to solve the system defined by Equations (45)–(47). By combining Equations (45)–(52), we observe that $u={\sum }_{i=1}^N{\alpha }_i\pi {\delta }_{a_i,{\lambda }_i}+\overline{v}$ is a critical point of $I_{\epsilon }$ if and only if $(\alpha ,\lambda ,a)$ solves the following system: for each $1\le i\le N$

$$\begin{array}{cc}\hfill & \left(E_{{\alpha }_i}\right)〈\nabla I_{\epsilon }\left(u\right),\pi {\delta }_{a_i,{\lambda }_i}〉=0,\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill & \left(E_{{\lambda }_i}\right)〈\nabla I_{\epsilon }\left(u\right),{\alpha }_i{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial {\lambda }_i}}〉=B_i〈\overline{v},{\lambda }_i{\displaystyle \frac{{\partial }^2\left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial {\lambda }_i^2}}〉+\sum _{j=1}^n{C}_{ij}〈\overline{v},{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{{\partial }^2\left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial {\lambda }_i\partial a_i^j}}〉,\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill & \left(E_{a_i}\right)〈\nabla I_{\epsilon }\left(u\right),{\alpha }_i{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial a_i}}〉=B_i〈\overline{v},{\lambda }_i{\displaystyle \frac{{\partial }^2\left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial {\lambda }_i\partial a_i}}〉+\sum _{j=1}^n{C}_{ij}〈\overline{v},{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{{\partial }^2\left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial a_i^j\partial a_i}}〉.\hfill \end{array}$$

(55)

To analyze the system $\left(E_{{\alpha }_i}\right)$, $\left(E_{{\lambda }_i}\right)$, $\left(E_{a_i}\right)$, we need to estimate the constants $A_i$s, $B_i$s and $C_{ij}$s that appear in the equations $\left(E_{{\lambda }_i}\right)$ and $\left(E_{a_i}\right)$. This is the aim of the following lemma

Lemma 2.

Let $(\alpha ,\lambda ,a,0)\in {\mathcal{O}}_1(N,\epsilon )$. Then, for ε small, the following statements hold:

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

Proof. 

Taking the scalar product of ${\displaystyle \frac{\partial {\psi }_{\epsilon }}{\partial v}}$ (see (48)) with the functions $\pi {\delta }_{a_i,{\lambda }_i}$, ${\lambda }_i\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)/\partial {\lambda }_i$ and ${{\lambda }_i}^{-1}\partial (\pi {\delta }_{a_i,{\lambda }_i})/\partial a_i^j$, respectively, we obtain the following quasi-diagonal system:

$$\begin{array}{cc}\hfill & c{A}_i+O\left(\epsilon \left(\sum _{k=1}^N(|A_k|+|B_k|+\sum _{j=1}^n|C_{kj}|)\right)\right)=〈{\displaystyle \frac{\partial {\psi }_{\epsilon }}{\partial v}},\pi {\delta }_{a_i,{\lambda }_i}〉\forall i\in \{1,\cdots ,N\},\hfill \\ \hfill & c^{\prime }{B}_i+O\left(\epsilon \left(\sum _{k=1}^N(|A_k|+|B_k|+\sum _{j=1}^n|C_{kj}|)\right)\right)=〈{\displaystyle \frac{\partial {\psi }_{\epsilon }}{\partial v}},{\lambda }_i{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial {\lambda }_i}}〉\forall i\in \{1,\cdots ,N\}\hfill \\ \hfill & c^{″}{C}_{il}+O\left(\epsilon \left(\sum _{k=1}^N(|A_k|+|B_k|+\sum _{j=1}^n|C_{kj}|)\right)\right)=〈{\displaystyle \frac{\partial {\psi }_{\epsilon }}{\partial v}},{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial \left(\pi {\delta }_{a_i,{\lambda }_i}\right)}{\partial a_i^l}}〉\forall 1\le i\le N,\forall 1\le \ell \le n\hfill \end{array}$$

where c, $c^{\prime }$ and $c^{″}$ are some positive universal constants.

Combining Propositions 3, 4 and 5 and the fact that $(\alpha ,\lambda ,a,0)\in {\mathcal{O}}_1(N,\epsilon )$, we see that, for all $i\in \{1,\cdots ,N\}$, we have

$$〈{\displaystyle \frac{\partial {\psi }_{\epsilon }}{\partial v}},\pi {\delta }_{a_i,{\lambda }_i}〉=O\left(\epsilon {ln}^2\epsilon \right),〈{\displaystyle \frac{\partial {\psi }_{\epsilon }}{\partial v}},{\lambda }_i{\displaystyle \frac{\partial \pi {\delta }_{a_i,{\lambda }_i}}{\partial {\lambda }_i}}〉=O\left(\epsilon \right),〈{\displaystyle \frac{\partial {\psi }_{\epsilon }}{\partial v}},{\displaystyle \frac{1}{{\lambda }_i}}{\displaystyle \frac{\partial \pi {\delta }_{a_i,{\lambda }_i}}{\partial a_i}}〉=O({\epsilon }^{{\displaystyle \frac{3}{2}}}{|ln\epsilon |}^{-{\displaystyle \frac{{\sigma }_n}{2}}}).$$

Thus Lemma 2 follows. □

Next, we will analyze the equations $\left(E_{{\alpha }_i}\right)$, $\left(E_{{\lambda }_i}\right)$ and $\left(E_{a_i}\right)$. To simplify the system for solving, we introduce the following change of variables

$${\beta }_i=1-{\alpha }_i^{p-1},{\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^2}}={\displaystyle \frac{c_2}{c\left(n\right)}}{\displaystyle \frac{1}{V\left(b_i\right)}}\epsilon (1+{\wedge }_i),z_i=a_i-b_{i}1\le i\le N,$$

(56)

where $c_2$ and $c\left(n\right)$ are defined in Proposition 4.

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

Lemma 3.

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

$$\left(S\right)\left\{\begin{array}{c}{\beta }_i=O\left(\epsilon \right|ln\epsilon \left|\right)\forall 1\le i\le N,\hfill \\ {\wedge }_i=O\left(\right|z_i{|}^2+R_1\left(\epsilon \right){\epsilon }^{-1})\forall 1\le i\le N,\hfill \\ D^{2}V\left(b_i\right)(z_i,.)=O\left(R_2\left(\epsilon \right){\epsilon }^{-3/2}{|ln\epsilon |}^{1/2}+{\left|z_i\right|}^2\right)\forall 1\le i\le N,\hfill \end{array}\right.$$

where $R_1\left(\epsilon \right)$ and $R_2\left(\epsilon \right)$ are defined in Propositions 4 and 5.

Proof. 

Using Proposition 3, it follows that

$$1-c_0^{-\epsilon }{\lambda }_i^{-\epsilon (n-2)/2}{\alpha }_i^{p-1-\epsilon }=O\left(\epsilon \right).$$

(57)

Now, observe that

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

Thus, $\left(E_{{\alpha }_i}\right)$ is equivalent to

$$\left(E_{{\alpha }_i}^{\prime }\right){\beta }_i=O\left(\epsilon \right|ln\epsilon \left|\right)\forall 1\le i\le N.$$

For the second equation $\left(E_{{\lambda }_i}\right)$, Proposition 4 and Lemma 2 imply that

$$(1+O\left(\epsilon \right))c_2\epsilon -c\left(n\right)(1+O\left(\epsilon \right))V\left(a_i\right){\displaystyle \frac{{ln}^{{\sigma }_n}{\lambda }_i}{{\lambda }_i^2}}=O\left(R_1\left(\epsilon \right)\right),$$

where $R_1\left(\epsilon \right)$ is defined in Proposition 4. Writing

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

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

Lastly, writing

$$\nabla V\left(a_i\right)=D^{2}V\left(b_i\right)(z_i,.)+O\left(\right|z_i{|}^2)$$

and using Proposition 5 and Lemma 2, we see that Equation $\left(E_{a_i}\right)$ is equivalent to the third equation in the system $\left(S\right)$ which completes the proof of Lemma 3. □

We are now prepared to prove our results regarding the existence of interior peak solutions. Since Theorem 2 follows directly from Theorem 1, it suffices to prove the latter.

Proof of Theorem 1.

The system $\left(S\right)$, as presented in Lemma 3, can be expressed in the following form:

$$\left(S^{\prime }\right)\left\{\begin{array}{c}{\beta }_i=U_{1,i}(\epsilon ,\beta ,\wedge ,z)=O\left(\epsilon \right|ln\epsilon \left|\right)\forall 1\le i\le N\hfill \\ {\wedge }_i=U_{2,i}(\epsilon ,\beta ,\wedge ,z)=O\left(\epsilon \right|ln\epsilon |+|z_i{|}^2+R_1\left(\epsilon \right){\epsilon }^{-1})\forall 1\le i\le N\hfill \\ D^{2}V\left(b_i\right)(z_i,.)=U_{3,i}(\epsilon ,\beta ,\wedge ,z)=O\left(R_2\left(\epsilon \right){\epsilon }^{-3/2}{|ln\epsilon |}^{1/2}+{\left|z_i\right|}^2\right)\forall 1\le i\le N\hfill \end{array}\right.$$

where $\beta =({\beta }_1,\cdots ,{\beta }_N)$, $\wedge =({\wedge }_1,\cdots ,{\wedge }_N)$ and $z=(z_1,\cdots ,z_N)$.

Thus, using Brouwer’s fixed point theorem, we derive that the system $\left(S^{\prime }\right)$ has at least one solution $({\beta }_{\epsilon },{\wedge }_{\epsilon },z_{\epsilon })$ for small $\epsilon$. To complete the proof of the theorem, it remains to prove that the constructed function $u_{\epsilon }={\sum }_{i=1}^N{\alpha }_{i,\epsilon }\pi {\delta }_{a_{i,\epsilon },{\lambda }_{i,\epsilon }}+{\overline{v}}_{\epsilon }$ is positive. To this aim, we first remark, since $\parallel v\parallel \to 0$, that $u_{\epsilon }\cancel{\equiv }0$ for small $\epsilon$. By construction, $u_{\epsilon }$ satisfies

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

Multiplying $\left(Q_{V,\epsilon }^{\prime }\right)$ by $u_{\epsilon }^{-}=max(0,-u_{\epsilon })$ and integrating on $\Omega$, we obtain

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

But we have

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

which implies that

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

Since $u_{\epsilon }^{-}\le \left|v_{\epsilon }\right|$ and $\parallel v_{\epsilon }{\parallel }_{L^{p+1}}\to 0$ as $\epsilon \to 0$, we derive that $u_{\epsilon }^{-}\equiv 0$ and $u_{\epsilon }\ge 0$. Thus, using the maximum principle, $u_{\epsilon }$ has to be positive. This completes the proof of the theorem. □

4. Conclusions

In this paper, we have explored the existence of solutions to a nonlinear elliptic problem with Dirichlet boundary conditions, featuring a slightly subcritical exponent for Sobolev embedding $H_0^1(\Omega )\hookrightarrow L^q(\Omega )$. By performing a detailed asymptotic analysis of the gradient of the corresponding Euler–Lagrange functional near the so-called “bubbles”, we were able to construct solutions that concentrate at isolated interior points. These techniques enabled us to establish a multiplicity result for the problem. The approach employed is specific to variational problems. While this paper addresses the existence of solutions and multiplicity results for the given problem, several natural directions for future research and open questions remain:

(i) 

Impact of the Nature of Critical Points of the Potential: The solutions we have constructed in this paper rely on the assumption that the critical points of the potential V are non-degenerate. What happens if this hypothesis is not satisfied, particularly when V satisfies some flatness condition?

(ii) 

Location of the Concentration Points: This paper focuses on constructing solutions that concentrate at isolated interior points. An interesting extension would be to study the existence of solutions that concentrate at non-isolated points, boundary points, or interior points that converge to the boundary.

(iii) 

Asymptotic Behavior of Solutions: Another important question is to fully characterize the asymptotic behavior of the solutions, providing a comprehensive understanding of their long-term behavior.

(iv) 

Impact of Subcritical Exponent: The current work focuses on a slightly subcritical exponent for Sobolev embedding. Future research could explore the problem with exponents that are slightly supercritical, i.e., when $\epsilon <0$ but close to zero.