Interior Bubbling Solutions for an Elliptic Equation with Slightly Subcritical Nonlinearity

Abstract

In this paper, we considered the Neumann elliptic equation $\left({\mathcal{P}}_{\epsilon }\right)$: $-\Delta u+K\left(x\right)u=u^{(n+2)/(n-2)-\epsilon }$, $u>0$ in $\Omega$, $\partial u/\partial \nu =0$ on $\partial \Omega$, where $\Omega$ is a smooth bounded domain in ${\mathbb{R}}^n$, $n\ge 6$, $\epsilon$ is a small positive real and K is a smooth positive function on $\overline{\Omega }$. Using refined asymptotic estimates of the gradient of the associated Euler–Lagrange functional, we constructed simple and non-simple interior bubbling solutions of $\left({\mathcal{P}}_{\epsilon }\right)$ which allowed us to prove multiplicity results for $\left({\mathcal{P}}_{\epsilon }\right)$ provided that $\epsilon$ is small. The existence of non-simple interior bubbling solutions is a new phenomenon for the positive solutions of subcritical problems.

1. Introduction

In this paper, we consider the following nonlinear elliptic equation

$$\left({\mathcal{P}}_q\right):\left\{\begin{array}{c}-\Delta u+\mu u=u^q,u>0\mathrm{in}\Omega \hfill \\ \frac{\partial u}{\partial \nu }=0\mathrm{on}\partial \Omega ,\hfill \end{array}\right.$$

where $1<q<\infty$, $\Omega$ is a smooth bounded domain in ${\mathbb{R}}^n$, $n\ge 3$, and $\mu$ is a positive real.

Equation $\left({\mathcal{P}}_q\right)$ arises in many areas of applied sciences, including the Gierer–Meinhardt model in biological pattern formation [1] and the Keller– Segel model in chemotaxis [2].

The problem $\left({\mathcal{P}}_q\right)$ has been studied extensively in the last two decades, and it is well known that the situation depends on the exponent q, the parameter $\mu$, and the dimension n.

For a subcritical case, that is $q<\frac{n+2}{n-2}$; Lin, Ni, and Takagi [3] showed that $\left({\mathcal{P}}_q\right)$ does not have any non constant solution for small $\mu$, whereas nonconstant solutions appear for large $\mu$ which blow up, as $\mu$ goes to infinity, at one or several points. The least energy solution blows up, for large $\mu$, at the maximum point of the mean curvature of the boundary [4,5]. This concentration phenomenon was shown by several authors (see [6] for good review).

Due to the lack of compactness of the associated variational Euler–Lagrange functional, the critical case, that is $q=\frac{n+2}{n-2}$, is much more difficult to handle. For $n=3$, Zhu [7] showed that when $\Omega$ is convex $\left({\mathcal{P}}_q\right)$ has only constant solution for small $\mu$ (see also [8]). However, for $\mu$, small and $n=4,5,6$, nonconstant solutions exist (see [9] when $\Omega$ is a ball and [8,10] for general domains). In [11,12], the authors proved the existence of solutions for large $\mu$. As in the subcritical case, solutions blow up, for large $\mu$, at one or several boundary points as $\mu$ goes to infinity (see [13,14,15,16,17,18,19,20,21,22]). In [23], Rey proved, in contrast with the subcritical case, that at least one blow-up point must be on the boundary. In [24,25], the authors studied the problem for fixed $\mu$, when the exponent q is close to the critical one, i.e., $q=\frac{n+2}{n-2}-\epsilon$ and $\epsilon$ is a small nonzero number. When the dimension $n=3$, they proved the existence of single interior peak solution and for $n\ge 4$, they also constructed a single boundary peak solution.

In this work, we consider the case where the constant $\mu$ in $\left({\mathcal{P}}_q\right)$ is replaced by a function K, and we are going to prove that multiple interior bubbling solutions exist provided that q is close enough to the critical exponent. More precisely, we considered the following nonlinear elliptic equation with subcritical nonlinearity

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

where $\Omega$ is a smooth bounded domain in ${\mathbb{R}}^n$, $n\ge 6$, $\epsilon$ is a small positive real, $p+1=\left(2n\right)/(n-2)$ is the critical Sobolev exponent for the embedding $H^1(\Omega )\to L^q(\Omega )$, and K is a $C^3$ positive function on $\overline{\Omega }$.

As we mentioned above, there have been many works on $\left({\mathcal{P}}_q\right)$ (i.e., when $\mu$ is constant). On the contrary, in the case of problem $\left({\mathcal{P}}_{\epsilon }\right)$ there is a very poor literature. To our knowledge, the only results are made on the whole space [26,27,28].

In the first part of this work, our aim is to construct simple interior bubbling solutions for problem $\left({\mathcal{P}}_{\epsilon }\right)$. This construction allows us to prove a multiplicity result for problem $\left({\mathcal{P}}_{\epsilon }\right)$ in connection with the number of critical points of K.

In the second part, we constructed interior bubbling solutions with clustered bubbles at a critical point of K. Such solutions give a new phenomenon for the positive solutions, that is, the existence of non simple blow up points in the interior for the subcritical problem. This phenomenon is known for changing-sign solution with Dirichlet boundary conditions on some symmetric domains (see [29]).

The remainder of the paper is organized as follows: in Section 2, we state our main results. In Section 3, we set up the analytical framework of the problems $\left({\mathcal{P}}_{\epsilon }\right)$, introduce the neighbourhood at infinity and its parametrization. We also give a precise estimates of the infinite dimensional part. Section 4 is devoted to the expansion of the gradient of the associated functional. In Section 5, we prove Theorems 1 and 2, while Section 6 is devoted to the proof Theorems 3 and 4.

2. Main Results

To state ours results, we need to introduce some notation. Problem $\left({\mathcal{P}}_{\epsilon }\right)$ has a variational structure. Indeed, its solutions correspond to the positive critical points of the functional

$$I_{\epsilon }\left(u\right):=\frac{1}{2}{\parallel u\parallel }^2-\frac{1}{p+1-\epsilon }{\int }_{\Omega }{\left|u\right|}^{p+1-\epsilon },\mathrm{with}p:=\frac{n+2}{n-2}.$$

(1)

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

$${\parallel u\parallel }^2={\int }_{\Omega }\left(\mid \nabla u{\mid }^2+K{u}^2\right),(u,v)={\int }_{\Omega }\left(\nabla u\nabla v+Kuv\right).$$

For $\epsilon =0$, the functional $I_{\epsilon }$ fails to satisfy the Palais Smale condition, and the reason for such a lack of compactness is the existence of almost solution of the equation $\left({\mathcal{P}}_{\epsilon }\right)$. These almost solutions, called bubbles, are defined as follows:

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

(2)

which are the only solutions to the problem [30]

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

The aim of our first result is to construct simple interior bubbling solutions. More precisely, we prove:

Theorem 1. 

Let $n\ge 6$ and let $K:\overline{\Omega }\to \mathbb{R}$ be a smooth positive function having m non- degenerate critical points $y_1$, ..., $y_m$. Then, for any $N\le m$, there exists ${\epsilon }_0>0$ small such that for any $\epsilon \in (0,{\epsilon }_0]$, problem $\left({\mathcal{P}}_{\epsilon }\right)$ admits a solution $u_{\epsilon ,y_{i_1},\dots ,y_{i_N}}$ satisfying: $u_{\epsilon ,y_{i_1},\dots ,y_{i_N}}$ develops exactly one bubble at each point $y_{i_j}$ and converges weakly to zero in $H^1(\Omega )$ as $\epsilon \to 0$. More precisely there exist ${\lambda }_{1,\epsilon }$, ..., ${\lambda }_{N,\epsilon }$ having the same order as ${\epsilon }^{-1/2}$ and points $a_{i_j,\epsilon }\to y_{i_j}$ for all j such that

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

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

Theorem 2. 

Let $n\ge 6$ and let $K:\overline{\Omega }\to \mathbb{R}$ be a smooth positive function having m non-degenerate critical points. Then, for $\epsilon >0$ small, $\left({\mathcal{P}}_{\epsilon }\right)$ admits at least $2^m-1$ solutions.

Our aim in the next result was to construct interior bubbling solutions with clustered bubbles at a critical point of K. To this aim, we introduced some notation. For $N\in \mathbb{N}$ and y a critical point of K, we defined the following function

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

(3)

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

Our result reads as follows:

Theorem 3. 

Let $n\ge 6$ and let y be a non-degenerate critical point of K. Let $N\in {\mathbb{N}}^{*}$ and assume that the function $F_{N,y}$ has a non-degenerate critical point $({\overline{z}}_1,\dots ,{\overline{z}}_N)$. Then, for any integer, there exists ${\epsilon }_0>0$ small such that for any $\epsilon \in (0,{\epsilon }_0]$, problem $\left({\mathcal{P}}_{\epsilon }\right)$ admits a solution $u_{\epsilon ,y}$ with the following properties

$$u_{\epsilon ,y}=\sum _{i=1}^N{\alpha }_{i,\epsilon }{\delta }_{a_{i,\epsilon },{\lambda }_{i,\epsilon }}+v_{\epsilon },$$

with

$$\begin{array}{cc}\hfill & \parallel v_{\epsilon }\parallel \to 0as\epsilon \to 0,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & |{\alpha }_{i,\epsilon }-1|\le c_1^{\prime }\epsilon {ln}^2\epsilon \forall i\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & \frac{\sqrt{\epsilon }}{c_2^{\prime }}\le \frac{1}{{\lambda }_{i,\epsilon }}\le c_2^{\prime }\sqrt{\epsilon }\forall i\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & |a_{i,\epsilon }-y+\sigma {\epsilon }^{\frac{n-4}{2n}}{\overline{z}}_i|\le {\epsilon }^{\frac{n-4}{2n}}{\eta }_0\forall i,\hfill \end{array}$$

(7)

where $c_1^{\prime }$, $c_2^{\prime }$ are positive constants, ${\eta }_0$ is a small positive real and σ is the constant defined by (135). Moreover, if for each N, $F_{N,y}$ has a non-degenerate critical point, then, problem $\left({\mathcal{P}}_{\epsilon }\right)$ has an arbitrary number of non-constant distinct solutions provided that ε is small.

Remark 1. 

If we assume that $D^{2}K\left(y\right)(·,·)$ is negative definite, we see that for each N, $F_{N,y}$ tends to $-\infty$ near the boundary of its definition domain and therefore $F_{N,y}$ achieves its maximum.

Our last result deals with the case of many interior blow up points. Namely, we prove:

Theorem 4. 

Let $n\ge 6$ and let $y_1,\dots ,y_m$ be non-degenerate critical points of K. For each $k\le m$ and $N_1,\dots ,N_m\in {\mathbb{N}}^{*}$, we assume that the function $F_{N_k,y_k}$ has a non-degenerate critical point $({\overline{z}}_{k,1},\dots ,{\overline{z}}_{k,N_k})$. Then, there exists ${\epsilon }_0>0$ small such that for any $\epsilon \in (0,{\epsilon }_0]$, problem $\left({\mathcal{P}}_{\epsilon }\right)$ admits a solution $u_{\epsilon ,y_1,\dots ,y_m}$ satisfying

$$u_{\epsilon ,y_1,\dots ,y_m}=\sum _{i=1}^{N_1}{\alpha }_{1,i,\epsilon }{\delta }_{a_{1,i,\epsilon },{\lambda }_{1,i,\epsilon }}+\dots +\sum _{i=1}^{N_m}{\alpha }_{m,i,\epsilon }{\delta }_{a_{m,i,\epsilon },{\lambda }_{m,i,\epsilon }}+v_{\epsilon },$$

with $\parallel v_{\epsilon }\parallel \to 0$ as $\epsilon \to 0$ and for each $k\le m$ the coefficients ${\alpha }_{k,i,\epsilon }$, the speeds ${\lambda }_{k,i,\epsilon }$ and the points $a_{k,i,\epsilon }$ satisfy the properties (5), (6), and (7), respectively.

Remark 2. 

We believe that our results should also be true in small dimensions, but we need more careful computations. We will come back to these dimensions in future work.

The strategy of the proof of our results is based on refined asymptotic estimates of the gradient of the associated Euler–Lagrange functional in the so-called neighborhood of critical points at infinity [31]. The aim is to find the equilibrium conditions satisfied by the concentration parameters. These balancing conditions are obtained by testing the equation by vector fields which are the dominant term of the gradient with respect to the parameters of concentration. The analysis of these balancing conditions allows us to obtain all the information we need to prove our results.

3. Analytical Framework

Let $n\ge 6$ and $y_1,\dots ,y_m$ be distinct critical points of K. The analytical framework is similar to the one considered in [32]. We are going to proceed to the suitable parametrization of the associated variational problem. For $N\le m$ and ${\nu }_0$ positive small, we introduce the following set

$$\begin{array}{c}\hfill \mathcal{O}(N,{\nu }_0)=\{(\alpha ,\lambda ,a)\in {\left({\mathbb{R}}_{+}\right)}^N\times {\left({\mathbb{R}}_{+}\right)}^N\times {\Omega }^N:|{\alpha }_i-1|<{\nu }_0,{\lambda }_i>{\nu }_0^{-1},\epsilon ln{\lambda }_i<{\nu }_0,\\ d(a_i,\partial )>c,\mathrm{and}{\epsilon }_{ij}<{\nu }_0\},\hfill \end{array}$$

(8)

where c is a positive constant and ${\epsilon }_{ij}:={\left(\frac{{\lambda }_i}{{\lambda }_j}+\frac{{\lambda }_j}{{\lambda }_i}+{\lambda }_i{\lambda }_j{|a_i-a_j|}^2\right)}^{(2-n)/2}$.

• For each $(\alpha ,\lambda ,a)\in \mathcal{O}(N,{\nu }_0)$, we consider $u={\sum }_{i=1}^N{\alpha }_i{\delta }_{a_i,{\lambda }_i}+v$, with $v\in E_{a,\lambda }$, where

  $$E_{a,\lambda }=\{v\in H^1(\Omega )/{\int }_{\Omega }\nabla v\nabla {\delta }_i={\int }_{\Omega }\nabla v\nabla \frac{\partial {\delta }_i}{\partial {\lambda }_i}={\int }_{\Omega }\nabla v\nabla \frac{\partial {\delta }_i}{\partial a_i^j}=0,1\le i\le N,1\le j\le n\},$$

  (9)

  with ${\delta }_i={\delta }_{a_i,{\lambda }_i}$.

As usual in these type of equations, we first studied the infinite dimensional variable v. To this aim, we need to perform and expansion of $I_{\epsilon }$ with respect to $v\in E_{a,\lambda }$. Writing $U={\sum }_{i=1}^N{\alpha }_i{\delta }_i$, we see that

$$\begin{array}{cc}\hfill I_{\epsilon }\left(U+v\right)& =\frac{1}{2}{\int }_{\Omega }{\left|\nabla (U+v)\right|}^2+\frac{1}{2}{\int }_{\Omega }K{(U+v)}^2-\frac{1}{p+1-\epsilon }{\int }_{\Omega }{\left|U+v\right|}^{p+1-\epsilon }\hfill \\ & =I_{\epsilon }\left(U\right)+〈f_{\epsilon },v〉+\frac{1}{2}{Q}_{\epsilon }\left(v\right)+R_{\epsilon }\left(v\right)\hfill \end{array}$$

with

$$\begin{array}{cc}\hfill & 〈f_{\epsilon },v〉={\int }_{\Omega }KUv-{\int }_{\Omega }{U}^{p-\epsilon }v,\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & Q_{\epsilon }\left(v\right)={\int }_{\Omega }{\left|\nabla v\right|}^2+{\int }_{\Omega }K{v}^2-(p-\epsilon ){\int }_{\Omega }{U}^{p-1-\epsilon }{v}^2,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & R_{\epsilon }\left(v\right)=O\left({∥v∥}^{min(3,p+1-\epsilon )}\right).\hfill \end{array}$$

(12)

In addition the derivatives of $R_{\epsilon }$ satisfy

$$R_{\epsilon }^{\prime }\left(v\right)=O\left({∥v∥}^{min(2,p-\epsilon )}\right)\mathrm{and}{R}_{\epsilon }^{″}\left(v\right)=O\left({∥v∥}^{min(1,p-1-\epsilon )}\right).$$

(13)

Now, to simplify the expression of $Q_{\epsilon }$, we prove the following lemma

Lemma 1. 

Let $(\alpha ,\lambda ,a)\in \mathcal{O}(N,{\nu }_0)$ and $v\in E_{a,\lambda }$. Then, for ε small, the following statement holds

$$Q_{\epsilon }\left(v\right)={\int }_{\Omega }{\left|\nabla v\right|}^2+{\int }_{\Omega }K{v}^2-p\sum _{i=1}^N{\int }_{\Omega }{\delta }_i^{p-1}{v}^2+o\left({∥v∥}^2\right),$$

where $Q_{\epsilon }$ is defined in (11).

Proof. 

First, since $\epsilon ln{\lambda }_i\to 0$, we have

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

(14)

Second, we observe that

$$\epsilon {\int }_{\Omega }{U}^{p-1-\epsilon }{v}^2\le c\epsilon \sum _{i=1}^N{\int }_{\Omega }{\delta }_i^{p-1-\epsilon }{v}^2\le c\sum _{i=1}^N{\left({\int }_{\Omega }{\delta }_i^{p+1}\right)}^{\frac{2}{n}}{∥v∥}^2\le c\epsilon {∥v∥}^2$$

(15)

and

$${\int }_{\Omega }{U}^{p-1-\epsilon }{v}^2=\sum _{i=1}^N{\int }_{\Omega }{\left({\alpha }_i{\delta }_i\right)}^{p-1-\epsilon }{v}^2+\sum _{i\ne j}O\left({\int }_{\Omega }{\left({\delta }_i{\delta }_j\right)}^{\frac{p-1}{2}}{v}^2\right).$$

(16)

However, using estimate $E2$ of [31], we see that

$${\int }_{\Omega }{\left({\delta }_i{\delta }_j\right)}^{\frac{p-1}{2}}{v}^2\le c{\left({\int }_{\Omega }{\left({\delta }_i{\delta }_j\right)}^{\frac{n}{n-2}}\right)}^{\frac{2}{n}}{\parallel v\parallel }^2\le c{({\epsilon }_{ij}^{\frac{n}{n-2}}ln{\epsilon }_{ij}^{-1})}^{\frac{2}{n}}{\parallel v\parallel }^2\le c{\epsilon }^{\frac{2}{n-2}}{(ln{\epsilon }_{ij}^{-1})}^{\frac{2}{n}}{\parallel v\parallel }^2.$$

(17)

Using (14) and the fact that $|{\alpha }_i-1|$ is small, we obtained

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

(18)

Combining (15)–(18), our lemma follows. □

To proceed further, we need to prove the uniform coercivity of the quadratic form $Q_{\epsilon }$. Notice that such a kind of coercivity has been proved by Bahri [31] in the case of Dirichlet boundary conditions. Such a result has been adapted in [33] for our case when the concentration points do not approach each other. Hpwever, here we need a more general result which holds even if the points are close from each other. More precisely, without any assumption on the distance between the points, we have:

Proposition 1. 

Let $(\alpha ,\lambda ,a)\in \mathcal{O}(N,{\nu }_0)$ and $v\in E_{a,\lambda }$. Then, for ε small, there exists a constant $C>0$ such that

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

Proof. 

By Lemma 1, we have

$$Q_{\epsilon }\left(v\right)=Q\left(v\right)+{o(\parallel v\parallel }^2),$$

(19)

where

$$Q\left(v\right)={\int }_{\Omega }{\left|\nabla v\right|}^2+{\int }_{\Omega }K{v}^2-p\sum _{i=1}^N{\int }_{\Omega }{\delta }_i^{p-1}{v}^2.$$

Now, let $Pv$ be the projection of v onto $H_0^1(\Omega )$ defined by

$$\Delta Pv=\Delta v\mathrm{in}\Omega ,Pv=0\mathrm{on}\partial \Omega$$

and define $w=v-Pv$. Clearly, we have

$${\int }_{\Omega }\nabla Pv\nabla w=-{\int }_{\Omega }Pv\Delta w+{\int }_{\partial \Omega }Pv\frac{\partial w}{\partial \nu }=0.$$

(20)

Thus

$${\int }_{\Omega }{|\nabla v|}^2={\int }_{\Omega }|\nabla Pv{{|}^2+{\int }_{\Omega }|\nabla w|}^2.$$

(21)

Now, let ${\Omega }_0$ be a compact subset of $\Omega$ such that $B(a_i,r_0)\subset {\Omega }_0$ for all i with a fixed positive real $r_0$. For $y\in {\Omega }_0$, since w is a harmonic function in Ω, we see that

$$\left|w\left(y\right)\right|\le c{\int }_{\partial \Omega }|\frac{\partial G_0}{\partial \nu }(x,y)\left|\right|w\left(x\right)|dx\le C\left({\Omega }_0\right){\partial }_{\Omega }|w\left(x\right){|dx\le C\parallel w\parallel }_{H^1},$$

(22)

where $G_0$ is the Green’s function of the Laplace operator with Dirichlet boundary conditions.

• In the same way, we have

$$|\nabla w\left(y\right)|\le C\left({\Omega }_0\right){\parallel w\parallel }_{H^1}\forall y\in {\Omega }_0.$$

(23)

Next, for $1\le i\le N$, taking

$${\psi }_i\in \left\{{\delta }_i,{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i},\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial {\left(a_i\right)}_j},1\le j\le n\right\},$$

(24)

we observe that

$$\begin{array}{cc}\hfill {\int }_{\Omega }\nabla Pv\nabla {\psi }_i& ={\int }_{\Omega }\nabla v\nabla {\psi }_i-{\int }_{\Omega }\nabla w\nabla {\psi }_i\hfill \\ \hfill & =-{\int }_{{\Omega }_0}\nabla w\nabla {\psi }_i+O\left(\frac{1}{{\lambda }_i^{(n-2)/2}}{\int }_{\Omega \backslash \Omega 0}|\nabla w|\right)\hfill \\ \hfill & =O\left({\parallel w\parallel }_{H^1}{\int }_{{\Omega }_0}|\nabla {\psi }_i|+\frac{{\parallel w\parallel }_{H^1}}{{\lambda }_i^{(n-2)/2}}\right)=O\left(\frac{{\parallel w\parallel }_{H^1}}{{\lambda }_i^{(n-2)/2}}\right),\hfill \end{array}$$

(25)

where we have used (23) and the fact that ${\int }_{{\Omega }_0}|\nabla {\psi }_i|=O\left(\frac{1}{{\lambda }_i^{(n-2)/2}}\right)$. Now, we write

$$Q\left(v\right)=Q_0\left(Pv\right)+{\int }_{\Omega }{|\nabla w|}^2+{\int }_{\Omega }{Kv}^2-p\sum _{i=1}^N{\int }_{\Omega }{\delta }_i^{p-1}(2Pv+w)w,$$

(26)

where

$$Q_0\left(Pv\right)={\int }_{\Omega }{|\nabla Pv|}^2-p\sum _{i=1}^N{\int }_{\Omega }{\delta }_i^{p-1}{\left(Pv\right)}^2.$$

However, using (22), we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }{\delta }_i^{p-1}\left(2\right|Pv|& +\left|w\right|\left)\right|w|={\int }_{{\Omega }_0}{\delta }_i^{p-1}\left(2\right|Pv|+\left|w\right|\left)\right|w|+{\int }_{\Omega \backslash \Omega 0}{\delta }_i^{p-1}\left(2\right|Pv|+\left|w\right|\left)\right|w|\hfill \\ \hfill & \le {C\parallel w\parallel }_{H^1}{(\parallel Pv\parallel }_{H^1}+{\parallel w\parallel }_{H^1})\left[{\left({\int }_{{\Omega }_0}{\delta }_i^{\frac{8n}{n^2-4}}\right)}^{\frac{n+2}{2n}}+{\left({\int }_{\Omega \backslash \Omega 0}{\delta }_i^{p+1}\right)}^{\frac{2}{n}}\right].\hfill \end{array}$$

(27)

By easy computations, we have

$$\left[{\left({\int }_{{\Omega }_0}{\delta }_i^{\frac{8n}{n^2-4}}\right)}^{\frac{n+2}{2n}}+{\left({\int }_{\Omega \backslash \Omega 0}{\delta }_i^{p+1}\right)}^{\frac{2}{n}}\right]=\left\{\begin{array}{c}\frac{{ln}^{2/3}{\lambda }_i}{{\lambda }_i^2}\mathrm{if}n=6\hfill \\ \frac{1}{{\lambda }_i^2}\mathrm{if}n\ne 6.\hfill \end{array}\right.$$

Thus

$${\int }_{\Omega }{\delta }_i^{p-1}\left(2\right|Pv|+\left|w\right|\left)\right|w|=o({\parallel v_1\parallel }_{H^1}^2+\parallel v_2{\parallel }_{H^1}^2).$$

(28)

To proceed further, let

$$v_1=Pv\mathrm{in}\Omega ,v_1=0\mathrm{in}{\mathbb{R}}^n\backslash \Omega .$$

Clearly $v_1\in {\mathcal{D}}^{1,2}\left({\mathbb{R}}^n\right)$. Now, we split

$$v_1=\sum _{i=1}^{N(n+2)}{\gamma }_i{\psi }_i+{\overline{v}}_1.$$

where ${\overline{v}}_1$ satisfies ${\int }_{{\mathbb{R}}^n}\nabla {\overline{v}}_1\nabla {\psi }_i=0$ for all i and ${\psi }_i$ goes through all the $N(n+2)$ functions defined by (24).

Note that, by easy computations, the following estimates can be verified

$$\begin{array}{cc}\hfill & {\int }_{\Omega }|\nabla {\delta }_i{|}^2=c+O\left(\frac{1}{{\lambda }_i^{n-2}}\right);{\int }_{\Omega }\nabla {\delta }_i\nabla {\delta }_k=O\left({\epsilon }_{ik}\right)\mathrm{for}k\ne i,\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & {\int }_{\Omega }\nabla {\delta }_i\nabla \left({\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}\right)=O\left(\frac{1}{{\lambda }_i^{n-2}}\right);{\int }_{\Omega }\nabla {\delta }_i\nabla \left({\lambda }_k\frac{\partial {\delta }_k}{\partial {\lambda }_k}\right)=O\left({\epsilon }_{ik}\right)\mathrm{for}k\ne i,\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill & {\int }_{\Omega }\nabla {\delta }_i\nabla \left(\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i^j}\right)=O\left(\frac{1}{{\lambda }_i^{n-1}}\right);{\int }_{\Omega }\nabla {\delta }_i\nabla \left(\frac{1}{{\lambda }_k}\frac{\partial {\delta }_k}{\partial a_k^j}\right)=O\left({\epsilon }_{ik}\right)\mathrm{for}k\ne i,\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill & {\int }_{\Omega }\nabla {\delta }_i\nabla \left({\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}\right)=O\left(\frac{1}{{\lambda }_i^{n-2}}\right);{\int }_{\Omega }\nabla \left({\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}\right)\nabla {\delta }_k=O\left({\epsilon }_{ik}\right)\mathrm{for}k\ne i,\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill & {\int }_{\Omega }\nabla \left({\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}\right)\nabla \left({\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}\right)=c+O\left(\frac{1}{{\lambda }_i^{n-2}}\right);{\int }_{\Omega }\nabla \left({\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}\right)\nabla \left({\lambda }_k\frac{\partial {\delta }_k}{\partial {\lambda }_k}\right)=O\left({\epsilon }_{ik}\right)\mathrm{for}k\ne i,\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill & {\int }_{\Omega }\nabla \left({\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}\right)\nabla \left(\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i^j}\right)=O\left(\frac{1}{{\lambda }_i^{n-1}}\right);{\int }_{\Omega }\nabla \left({\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}\right)\nabla \left(\frac{1}{{\lambda }_k}\frac{\partial {\delta }_k}{\partial a_k^j}\right)=O\left({\epsilon }_{ik}\right)\mathrm{for}k\ne i,\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill & {\int }_{\Omega }\nabla {\delta }_i\nabla \left(\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i^j}\right)=O\left(\frac{1}{{\lambda }_i^{n-1}}\right);{\int }_{\Omega }\nabla \left(\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i^j}\right)\nabla {\delta }_k=O\left({\epsilon }_{ik}\right)\mathrm{for}k\ne i,\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill & {\int }_{\Omega }\nabla \left(\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i^j}\right)\nabla \left({\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}\right)=O\left(\frac{1}{{\lambda }_i^{n-1}}\right);{\int }_{\Omega }\nabla \left(\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i^j}\right)\nabla \left({\lambda }_k\frac{\partial {\delta }_k}{\partial {\lambda }_k}\right)=O\left({\epsilon }_{ik}\right)\mathrm{for}k\ne i,\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill & {\int }_{\Omega }\nabla \left(\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i^j}\right)\nabla \left(\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i^l}\right)=c^{″}{\delta }_{jl}+O\left(\frac{1}{{\lambda }_i^{n-1}}\right);{\int }_{\Omega }\nabla \left(\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i^j}\right)\nabla \left(\frac{1}{{\lambda }_k}\frac{\partial {\delta }_k}{\partial a_k^j}\right)=O\left({\epsilon }_{ik}\right)\mathrm{for}k\ne i,\hfill \end{array}$$

(37)

where ${\delta }_{jl}$ denotes the Kronecker delta, i.e., ${\delta }_{jl}=1$ if $j=l$ and ${\delta }_{jl}=0$ if $j\ne l$.

Now, using (25), (29)–(37), we obtain

$${\int }_{{\mathbb{R}}^n}\nabla v_1\nabla {\psi }_i=c{\gamma }_i+o(\sum {\gamma }_j)={\int }_{\Omega }\nabla Pv\nabla {\psi }_i=O\left({\parallel w\parallel }_{H^1}\sum {\lambda }_l^{\frac{2-n}{2}}\right)=o\left({\parallel w\parallel }^{H1}\right).$$

(38)

This implies that

$${\int }_{{\mathbb{R}}^n}|\nabla {\overline{v}}_1{|}^2={\int }_{{\mathbb{R}}^n}|\nabla v_1{|}^2+{o(\parallel w\parallel }_{H^1}^2).$$

(39)

Using (39), we get

$$\begin{array}{cc}\hfill Q_0\left(Pv\right)& ={\int }_{{\mathbb{R}}^n}{|\nabla v_1|}^2-p\sum {\int }_{{\mathbb{R}}^n}{\delta }_i^{p-1}{v}_1^2\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^n}|\nabla {\overline{v}}_1{|}^2-p\sum {\int }_{{\mathbb{R}}^n}{\delta }_i^{p-1}{\overline{v}}_1^2+o(\parallel \nabla {\overline{v}}_1{\parallel }_{L^2}^2+{\parallel w\parallel }_{H^1}^2).\hfill \end{array}$$

(40)

Combining (38) and Proposition $3.1$ of [31], we obtain

$$\begin{array}{cc}\hfill Q_0\left(Pv\right)& \ge \rho {\int }_{{\mathbb{R}}^n}|\nabla {\overline{v}}_1{|}^2+o(\parallel \nabla {\overline{v}}_1{\parallel }_{L^2}^2+{\parallel w\parallel }_{H^1}^2)\hfill \\ \hfill & \ge \frac{\rho }{2}{\int }_{{\mathbb{R}}^n}|\nabla v_1{|}^2+{o(\parallel w\parallel }_{H^1}^2)=\frac{\rho }{2}{\int }_{\Omega }{|\nabla Pv|}^2+{o(\parallel w\parallel }_{H^1}^2).\hfill \end{array}$$

(41)

Therefore (41) and (26) imply that

$$\begin{array}{cc}\hfill Q\left(v\right)& \ge \frac{\rho }{2}{\int }_{\Omega }{|\nabla Pv|}^2+{\int }_{\Omega }{|\nabla w|}^2+\mu {\int }_{\Omega }v2+{o(\parallel Pv\parallel }^2+{\parallel w\parallel }_{H^1}^2{)\ge C\parallel v\parallel }^2\hfill \end{array}$$

which completes the proof of the proposition. □

Now, we are able to deal with the infinite dimensional variable v. More precisely, we prove the following:

Proposition 2. 

Let $(\alpha ,\lambda ,a)\in \mathcal{O}(N,{\nu }_0)$. Then, for ε small, there exists a unique $\overline{v}\in E_{a,\lambda }$ satisfying

$$〈I_{\epsilon }^{\prime }(\sum _{i=1}^N{\alpha }_i{\delta }_{a_i,{\lambda }_i}+\overline{v}),h〉=0\forall h\in E_{a,\lambda }.$$

Moreover, $\overline{v}$ satisfies $\parallel \overline{v}\parallel \le cT(\epsilon ,a\lambda )$ with

$$T(\epsilon ,a,\lambda ):=\epsilon +\sum _{i=1}^N\frac{1}{{\lambda }_i^{\frac{n-2}{2}}}+T_1(a,\lambda )+\sum _{i=1}^N{T}_2\left({\lambda }_i\right),$$

where

$$T_1(a,\lambda )=\sum _{i\ne j}{\epsilon }_{ij}^{\frac{n+2}{2(n-2)}}{ln}^{\frac{n+2}{2n}}{\epsilon }_{ij}^{-1}and{T}_2\left({\lambda }_i\right)=\left\{\begin{array}{c}{ln}^{\frac{2}{3}}{\lambda }_i/{\lambda }_i^{2}ifn=6\hfill \\ c{\lambda }_i^{-2}ifn\ge 7.\hfill \end{array}\right.$$

Proof. 

Applying Proposition 1 and estimate (13), we see that the implicit function theorem implies, for $\epsilon$ small, the existence of $\overline{v}\in E_{a,\lambda }$ satisfying the estimate $\parallel \overline{v}\parallel =O(\parallel f_{\epsilon }\parallel )$ where $f_{\epsilon }$ is defined by (10). However, we have

$${\int }_{\Omega }KU\left|v\right|\le c\sum {\int }_{\Omega }{\delta }_i\left|v\right|\le c\parallel v\parallel \sum _{i=1}^N{\left({\int }_{\Omega }{\delta }_i^{\frac{2n}{n+2}}\right)}^{\frac{n+2}{2n}}.$$

(42)

By easy computations, we obtain

$${\left({\int }_{\Omega }{\delta }_i^{\frac{2n}{n+2}}\right)}^{\frac{n+2}{2n}}=T_2\left({\lambda }_i\right).$$

(43)

Using the fact that, for $a,b>0$ and $\beta >0$, we have

$${(a+b)}^{\beta }=a^{\beta }+b^{\beta }+O({\left(ab\right)}^{\frac{\beta }{2}}),$$

we see that

$${\int }_{\Omega }{U}^{p-\epsilon }v=\sum _{i=1}^N{\alpha }_i^{p-\epsilon }{\int }_{\Omega }{\delta }_i^{p-\epsilon }v+\sum _{i\ne j}O\left({\int }_{\Omega }{\left({\delta }_i{\delta }_j\right)}^{\frac{p-\epsilon }{2}}\left|v\right|\right).$$

(44)

Using (14) and estimate $E2$ of [31], we obtain

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

(45)

For the first term in right hand side of (44), we use (14) to derive

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

(46)

However, since $v\in E_{a,\lambda }$, we have

$$\left|{\int }_{\Omega }{\delta }_i^{p}v\right|=\left|{\int }_{\Omega }-\Delta {\delta }_{i}v\right|=\left|{\int }_{\partial \Omega }\frac{\partial {\delta }_i}{\partial \nu }v\right|\le \frac{c}{{\lambda }_i^{\frac{n-2}{2}}}{\int }_{\partial \Omega }\left|v\right|\le \frac{c}{{\lambda }_i^{\frac{n-2}{2}}}\parallel v\parallel .$$

(47)

Combining (42)–(47), we easily derive our proposition. □

4. Expansion of the Gradient in the Neighbourhood at Infinity

In this section, we are going to give asymptotic expansions of the gradient of the Euler-Lagrange function $I_{\epsilon }$. We start by the expansion with respect to the gluing parameter ${\alpha }_i^{\prime }s$. Namely, we prove:

Proposition 3. 

Let $(\alpha ,\lambda ,a)\in \mathcal{O}(N,{\nu }_0)$, $v\in E_{a,\lambda }$ and $u={\sum }_{i=1}^N{\alpha }_i{\delta }_{a_i,{\lambda }_i}+v$. Then, for ε small, the following statement holds

$$〈I_{\epsilon }^{\prime }\left(u\right),{\delta }_i〉={\alpha }_i{S}_n\left(1-{\alpha }_i^{p-1-\epsilon }{\lambda }_i^{\frac{-\epsilon (n-2)}{2}}\right)+O\left(R_1\right),$$

where

$$S_n=c_0^{\frac{2n}{n-2}}{\int }_{{\mathbb{R}}^n}\frac{dx}{{(1+{\left|x\right|}^2)}^n}and{R}_1=\epsilon +\frac{1}{{\lambda }_i^2}+\sum _{j\ne i}\frac{1}{{\left({\lambda }_i{\lambda }_j\right)}^{\frac{n-2}{2}}}+\sum _{j\ne i}{\epsilon }_{ij}+{\parallel v\parallel }^2.$$

Proof. 

For $h\in H^1(\Omega )$, we notice that

$$〈I_{\epsilon }^{\prime }\left(u\right),h〉={\int }_{\Omega }\nabla u\nabla h+{\int }_{\Omega }Kuh-{\int }_{\Omega }{\left|u\right|}^{p-1-\epsilon }uh,$$

(48)

where

$$u=\sum _{i=1}^N{\alpha }_i{\delta }_i+v:=U+v.$$

(49)

Taking $h={\delta }_i$ in (48), we see that

$${\int }_{\Omega }\nabla u\nabla {\delta }_i={\alpha }_i{\int }_{\Omega }{\left|\nabla {\delta }_i\right|}^2+\sum _{j\ne i}{\alpha }_j{\int }_{\Omega }\nabla {\delta }_j\nabla {\delta }_i.$$

(50)

However, we have

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

(51)

and

$${\alpha }_i{\int }_{\Omega }{\left|\nabla {\delta }_i\right|}^2={\alpha }_i{S}_n+O\left(\frac{1}{{\lambda }_i^{n-2}}\right).$$

(52)

We also have

$${\int }_{\Omega }Ku{\delta }_i={\alpha }_i{\int }_{\Omega }K{\delta }_i^2+\sum _{j\ne i}{\alpha }_j{\int }_{\Omega }K{\delta }_j{\delta }_i+{\int }_{\Omega }Kv{\delta }_i.$$

(53)

For the last term in the right hand side of (53), we have

$$\left|{\int }_{\Omega }Kv{\delta }_i\right|\le c\int \left|v\right|{\delta }_i\le c\parallel v\parallel T_2\left({\lambda }_i\right),$$

(54)

where $T_2$ is defined in Proposition 2.

• For the other term, using Lemma 2.2 of [34], we observe that

$$\begin{array}{cc}\hfill {\int }_{\Omega }K{\delta }_j{\delta }_i& \le c{\int }_{\Omega }{\delta }_j{\delta }_i\le c{\epsilon }_{ij}\mathrm{for}i\ne j\hfill \end{array}$$

(55)

$$\begin{array}{cc}\hfill {\int }_{\Omega }K{\delta }_i^2& \le c{\int }_{\Omega }{\delta }_i^2=O\left(\frac{1}{{\lambda }_i^2}\right).\hfill \end{array}$$

(56)

It remains to deal with the last term in the right hand side of (48). To this aim, we write

$${\Omega }_1=\{x:\left|v\right|\le \frac{1}{2}U\},{\Omega }_2=\Omega \backslash {\Omega }_1,$$

(57)

where U is defined in (49).

Observe that

$$\begin{array}{cc}\hfill \left|{\int }_{{\Omega }_2}\mid u{\mid }^{p-1-\epsilon }u{\delta }_i\right|& \le {\int }_{{\Omega }_2}\mid v{\mid }^{p+1-\epsilon }\le c{\parallel v\parallel }^{p+1-\epsilon },\hfill \end{array}$$

(58)

$$\begin{array}{cc}\hfill {\int }_{{\Omega }_1}\mid u{\mid }^{p-1-\epsilon }u{\delta }_i& ={\int }_{{\Omega }_1}{U}^{p-\epsilon }{\delta }_i+(p-\epsilon ){\int }_{{\Omega }_1}{U}^{p-1-\epsilon }v{\delta }_i+O\left({\int }_{{\Omega }_1}{U}^{p-2-\epsilon }{v}^2{\delta }_i\right),\hfill \end{array}$$

(59)

$$\begin{array}{cc}\hfill {\int }_{{\Omega }_1}{U}^{p-2-\epsilon }{v}^2{\delta }_i& \le {\int }_{{\Omega }_1}{U}^{p-1-\epsilon }{v}^2={O(\parallel v\parallel }^2),\hfill \end{array}$$

(60)

$$\begin{array}{cc}\hfill {\int }_{{\Omega }_1}{U}^{p-\epsilon }{\delta }_i& ={\alpha }_i^{p-\epsilon }{\int }_{\Omega }{\delta }_i^{p+1-\epsilon }+O\left(\sum _{j\ne i}{\int }_{\Omega }{\delta }_j^{p-\epsilon }{\delta }_i+\sum _{j\ne i}{\int }_{\Omega }{\delta }_i^{p-\epsilon }{\delta }_j\right)+O\left({\int }_{{\Omega }_2}\mid v{\mid }^{p+1-\epsilon }\right)\hfill \\ \hfill & ={\alpha }_i^{p-\epsilon }{\int }_{\Omega }{\delta }_i^{p+1-\epsilon }+\sum _{j\ne i}O\left({\epsilon }_{ij}\right)+{O(\parallel v\parallel }^2),\hfill \end{array}$$

(61)

$$\begin{array}{cc}\hfill {\int }_{{\Omega }_1}{U}^{p-1-\epsilon }v{\delta }_i& ={\int }_{\Omega }{U}^{p-1-\epsilon }v{\delta }_i+O\left({\int }_{{\Omega }_2}\mid v{\mid }^{p+1-\epsilon }\right)\hfill \\ \hfill & ={\alpha }_i^{p-1-\epsilon }{\int }_{\Omega }{\delta }_i^{p-\epsilon }v+O\left(\sum _{j\ne i}{\int }_{{\Omega }^i}{\delta }_i^{p-1-\epsilon }\left|v\right|{\delta }_j+{\int }_{\Omega \backslash {\Omega }^i}{\delta }_j^{p-1}\left|v\right|{\delta }_i\right)+{O(\Vert v\Vert }^2),\hfill \end{array}$$

(62)

where

$${\Omega }^i=\{x\in \Omega :\sum _{j\ne i}{\alpha }_j{\delta }_j\left(x\right)\le {\alpha }_i{\delta }_i\left(x\right)\}\mathrm{and}{\left({\Omega }^i\right)}^c=\Omega \backslash {\Omega }^i.$$

(63)

Observe that

$$\begin{array}{cc}\hfill \sum _{j\ne i}{\int }_{{\Omega }^i}{\delta }_i^{p-1-\epsilon }\left|v\right|{\delta }_j+{\int }_{\Omega \backslash {\Omega }^i}{\delta }_j^{p-1}\left|v\right|{\delta }_i& \le c\parallel v\parallel \sum _{j\ne i}{\int }_{\Omega }({\delta }_i{\delta }_j{)}^{p/2}\left|v\right|\hfill \\ \hfill & \le \parallel v\parallel {\left({\int }_{\Omega }({\delta }_i{\delta }_j{)}^{\frac{n}{n-2}}\right)}^{\frac{n+2}{2n}}\hfill \\ \hfill & =O\left(\parallel v\parallel {\epsilon }_{ij}^{\frac{n+2}{2(n-2)}}{ln}^{\frac{n+2}{2n}}{\epsilon }_{ij}^{-1}\right).\hfill \end{array}$$

Lastly, we write

$$\begin{array}{cc}\hfill {\alpha }_i^{p-\epsilon }{\int }_{\Omega }{\delta }_i^{p+1-\epsilon }& =\frac{{\alpha }_i^{p-\epsilon }{c}_0^{-\epsilon }}{{\lambda }_i^{\frac{\epsilon (n-2)}{2}}}{\int }_{\Omega }{\delta }_i^{p+1}+O\left(\epsilon {\int }_{\Omega }{\delta }_i^{p+1}ln(1+{\lambda }_i^2|x-a_i{|}^2)\right)\hfill \\ \hfill & =\frac{{\alpha }_i^{p-\epsilon }{c}_0^{-\epsilon }}{{\lambda }_i^{\frac{\epsilon (n-2)}{2}}}{S}_n+O\left(\epsilon +\frac{1}{{\lambda }_i^n}\right)\hfill \end{array}$$

(64)

and, since $v\in E_{a,\lambda }$,

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

(65)

Combining (48)–(65), we easily derive our proposition. □

Now we are going to provide a balancing formula involving the rate of the concentration ${\lambda }_i$ and the self-interaction of bubbles ${\epsilon }_{ij}$. Namely, we prove:

Proposition 4. 

Let $(\alpha ,\lambda ,a)\in \mathcal{O}(N,{\nu }_0)$, $v\in E_{a,\lambda }$ and $u={\sum }_{i=1}^N{\alpha }_i{\delta }_{a_i,{\lambda }_i}+v$. Then, for ε small, the following statement holds for all $1\le i\le N$

$$〈I_{\epsilon }^{\prime }\left(u\right),{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}〉={\alpha }_i^p{c}_2\epsilon -{\alpha }_i{c}_1\frac{K\left(a_i\right)}{{\lambda }_i^2}+O\left(R_2\right),$$

where

$$R_2={\epsilon }^2+{\Vert v\Vert }^2+\sum _{j=1}^N\frac{1}{{\left({\lambda }_j{\lambda }_i\right)}^{\frac{n-2}{2}}}+\sum _{j\ne i}{\epsilon }_{ij}+\parallel v\parallel T_2\left({\lambda }_i\right)+T_3\left({\lambda }_i\right),$$

$$c_1=\frac{n-2}{2}{c}_0^2{\int }_{{\mathbb{R}}^{⋉}}\frac{(\mid x{\mid }^2-1)dx}{{(1+\mid x{\mid }^2)}^{n-1}}>0,c_2={\left(\frac{n-2}{2}\right)}^2{\int }_{{\mathbb{R}}^n}\frac{c_0^{\frac{2n}{n-2}}({\left|x\right|}^2-1)}{{(1+{\left|x\right|}^2)}^{n+1}}ln(1+{\left|x\right|}^2)dx$$

and

$$T_3\left({\lambda }_i\right)\left\{\begin{array}{c}{\lambda }_i^{-4}ifn\ge 7\hfill \\ {\lambda }_i^{-4}ln{\lambda }_{i}ifn=6.\hfill \end{array}\right.$$

Proof. 

We will take $h={\lambda }_i\frac{\partial {\delta }_i}{\partial \lambda }$ in (48) and estimate each term.

• Observe that:

$$\begin{array}{cc}\hfill & {\int }_{\Omega }Ku{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}=\sum _{j=1}^N{\alpha }_j{\int }_{\Omega }K{\delta }_j{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}+{\int }_{\Omega }Kv{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i},\hfill \end{array}$$

(66)

$$\begin{array}{cc}\hfill & \left|{\int }_{\Omega }Kv{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}\right|\le c{\int }_{\Omega }\left|v\right|{\delta }_i\le c\parallel v\parallel T_2\left({\lambda }_i\right),\hfill \end{array}$$

(67)

$$\begin{array}{cc}\hfill & |{\int }_{\Omega }K{\delta }_j{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}|\le c{\int }_{\Omega }{\delta }_j{\delta }_i\le c{\epsilon }_{ij}\forall j\ne i,\hfill \end{array}$$

(68)

where we have used in the last formula Lemma 2.2 of [34].

For $r>0$ small, we have

$$\begin{array}{cc}\hfill & {\int }_{\Omega }K{\delta }_i{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}={\int }_{B\left(a_i,r\right)}K{\delta }_i{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}+{\int }_{\Omega \backslash B\left(a_i,r\right)}K{\delta }_i{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}\hfill \end{array}$$

(69)

$$\begin{array}{cc}\hfill & {\int }_{\Omega \backslash B\left(a_i,r\right)}K{\delta }_i\left|{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}\right|\le C{\int }_{\Omega \backslash B\left(a_i,r\right)}{\delta }_i^2\le c{\int }_{\Omega \backslash B\left(a_i,r\right)}\frac{dx}{{\lambda }_i^{n-2}{|x-a_i|}^{2n-4}}\le \frac{C}{{\lambda }_i^{n-2}}.\hfill \end{array}$$

(70)

On $B\left(a_i,r\right):=B_i$, we write

$${\int }_{B_i}K{\delta }_i{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}=K\left(a_i\right){\int }_{B_i}{\delta }_i{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}+O\left({\int }_{B_i}{\left|x-a_i\right|}^2{\delta }_i^2\right).$$

(71)

Notice that

$${\int }_{B_i}{|x-a_i|}^2{\delta }_i^2\le c{\int }_{B_i}\frac{{\lambda }_i^{n-2}{|x-a_i|}^2}{{\left(1+{\lambda }_i^2{|x-a_i|}^2\right)}^{n-2}}dx\le \frac{c}{{\lambda }_i^4}{\int }_0^{{\lambda }_{i}r}\frac{t^{n+1}}{{\left(1+t^2\right)}^{n-2}}dt=O\left(T_3\left({\lambda }_i\right)\right),$$

(72)

$$\begin{array}{cc}\hfill {\int }_{B_i}{\delta }_i{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}& =\frac{(n-2)}{2}{c}_0^2{\int }_{B_i}\frac{{\lambda }_i^{n-2}\left(1-{\lambda }_i^2{\left|x-a_i\right|}^2\right)}{{\left(1+{\lambda }_i^2{\left|x-a_i\right|}^2\right)}^{n-1}}\hfill \\ \hfill & =\frac{(n-2)}{2}\frac{c_0^2}{{\lambda }_i^2}\left[{\int }_{{\mathbb{R}}^n}\frac{\left(1-{\left|x\right|}^2\right)}{{\left(1+{\left|x\right|}^2\right)}^{n-1}}dx+O\left(\frac{1}{{\lambda }_i^{n-4}}\right)\right]\hfill \\ \hfill & =-\frac{c_1}{{\lambda }_i^2}+O\left(\frac{1}{{\lambda }_i^{n-2}}\right).\hfill \end{array}$$

(73)

Combining (66)–(73), we obtain

$${\int }_{\Omega }Ku{\lambda }_i\frac{\partial \delta }{\partial {\lambda }_i}=-K\left(a_i\right)\frac{c_1{\alpha }_i}{{\lambda }_i^2}+O\left(\frac{1}{{\lambda }_i^{n-2}}+\sum _{j\ne i}{\epsilon }_{ij}+\parallel v\parallel T_2\left({\lambda }_i\right)+T_3\left({\lambda }_i\right)\right).$$

(74)

Next, we are going to estimate the first term in the right hand side of (48) with $h={\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}$. We observe that

$$\begin{array}{cc}\hfill {\int }_{\Omega }\nabla u\nabla \left({\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}\right)& =\sum _{j=1}^N{\alpha }_j{\int }_{\Omega }\nabla {\delta }_j\nabla \left({\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}\right)\hfill \\ \hfill & =\sum _{j=1}^N{\alpha }_j\left[{\int }_{\Omega }\left(-\Delta {\delta }_j\right){\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}+{\int }_{\partial \Omega }\frac{\partial {\delta }_j}{\partial \nu }{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}\right]\hfill \\ \hfill & =\sum _{j=1}^N{\alpha }_j\left[{\int }_{\Omega }{\delta }_j^p{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}+{\int }_{\partial \Omega }\frac{\partial {\delta }_j}{\partial \nu }{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}\right].\hfill \end{array}$$

(75)

However, using estimate $E1$ of [31], we have

$${\int }_{\Omega }{\delta }_j^p{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}\le c{\int }_{\Omega }{\delta }_j^p{\delta }_i\le c{\epsilon }_{ij}\forall j\ne i.$$

(76)

We also have

$$\begin{array}{cc}\hfill & {\int }_{\Omega }{\delta }_i^p{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}={\int }_{{\mathbb{R}}^n}{\delta }_i^p{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}-{\int }_{{\mathbb{R}}^n\backslash \Omega }{\delta }_i^p{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}=-{\int }_{{\mathbb{R}}^n\backslash \Omega }{\delta }_i^p{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}=O\left(\frac{1}{{\lambda }_i^n}\right),\hfill \end{array}$$

(77)

$$\begin{array}{cc}\hfill & \left|{\int }_{\partial \Omega }\frac{\partial {\delta }_j}{\partial \nu }{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}\right|\le \frac{c}{{\left({\lambda }_i{\lambda }_j\right)}^{\frac{n-2}{2}}}\forall i,j.\hfill \end{array}$$

(78)

Combining (75)–(78), we see that

$${\int }_{\Omega }\nabla u\nabla \left({\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}\right)=O\left(\sum _{j=1}^N\frac{1}{{\left({\lambda }_i{\lambda }_j\right)}^{(n-2)/2}}+\sum _{j\ne i}{\epsilon }_{ij}\right).$$

(79)

Now, we deal with the last term in the right hand side of (48) with $h={\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}$. To this aim, we write

$${\int }_{\Omega }{\left|u\right|}^{p-1-\epsilon }u{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}={\int }_{{\Omega }_2}{\left|u\right|}^{p-1-\epsilon }u{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}+{\int }_{{\Omega }_1}{\left|u\right|}^{p-1-\epsilon }u{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i},$$

(80)

where ${\Omega }_1$, ${\Omega }_2$ are defined in (57) and $u={\sum }_{i=1}^N{\alpha }_i{\delta }_i+v:=U+v.$

Observe that

$${\int }_{{\Omega }_2}{\left|u\right|}^{p-1-\epsilon }u{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}=O\left({\int }_{{\Omega }_2}{\left|v\right|}^{p+1-\epsilon }\right)=O\left({\parallel v\parallel }^{p+1-\epsilon }\right)=o\left({\parallel v\parallel }^2\right).$$

(81)

In addition, we write

$$\begin{array}{cc}\hfill {\int }_{{\Omega }_1}& {\left|u\right|}^{p-1-\epsilon }u{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}={\int }_{{\Omega }_1}{U}^{p-\epsilon }{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}+\left(p-\epsilon \right){\int }_{{\Omega }_1}{U}^{p-1-\epsilon }v{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}+O\left({\int }_{{\Omega }_1}{U}^{p-2-\epsilon }{v}^2{\delta }_i\right)\hfill \\ \hfill & ={\int }_{\Omega }{U}^{p-\epsilon }{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}+\left(p-\epsilon \right){\int }_{\Omega }{U}^{p-1-\epsilon }v{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}+O\left({\int }_{\Omega }{U}^{p-1-\epsilon }{v}^2+{\int }_{{\Omega }_2}\mid v{\mid }^{p+1-\epsilon }\right).\hfill \end{array}$$

(82)

For the remaining term in (82), we have

$${\int }_{\Omega }{U}^{p-1-\epsilon }{v}^2+{\int }_{{\Omega }_2}\mid v{\mid }^{p+1-\epsilon }\le c{\int }_{\Omega }{U}^{p-1}{v}^2+c{\int }_{{\Omega }_2}\mid v{\mid }^{p+1}\le c{\parallel v\parallel }^2.$$

(83)

For the other terms, from one hand, we write

$$\begin{array}{cc}\hfill {\int }_{\Omega }{U}^{p-1-\epsilon }v{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}=& {\int }_{\Omega }({\alpha }_i{\delta }_i{)}^{p-1-\epsilon }v{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}+O\left({\int }_{{\Omega }^i}{\left({\alpha }_i{\delta }_i\right)}^{p-2}(\sum _{j\ne i}{\alpha }_j{\delta }_j)\left|v\right||{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}|\right)\hfill \\ \hfill & +O\left({\int }_{{\left({\Omega }^i\right)}^c}{(\sum _{j\ne i}{\alpha }_j{\delta }_j)}^{p-1}\left|v\right||{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}|\right),\hfill \end{array}$$

(84)

where ${\Omega }^i$ is defined in (63).

Note that, since $n\ge 6$, we have $p-1:=4/(n-2)\le 1$. Thus, it follows that

$$\begin{array}{cc}\hfill {\int }_{{\Omega }^i}{\left({\alpha }_i{\delta }_i\right)}^{p-2}& \left(\sum _{j\ne i}{\alpha }_j{\delta }_j\right)\left|v\right||{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}|+{\int }_{{\left({\Omega }^i\right)}^c}{\left(\sum _{j\ne i}{\alpha }_j{\delta }_j\right)}^{p-1}\left|v\right||{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}|\hfill \\ \hfill & \le C{\int }_{{\Omega }^i}{\left({\alpha }_i{\delta }_i\right)}^{p-1}\left(\sum _{j\ne i}{\alpha }_j{\delta }_j\right)\left|v\right|+C{\int }_{{\left({\Omega }^i\right)}^c}{\left(\sum _{j\ne i}{\alpha }_j{\delta }_j\right)}^{p-1}\left|v\right|{\delta }_i\hfill \\ \hfill & \le C\sum _{j\ne i}{\int }_{\Omega }({\delta }_i{\delta }_j{)}^{p/2}\left|v\right|\le c\parallel v\parallel {\left({\int }_{\Omega }({\delta }_i{\delta }_j{)}^{n/(n-2)}\right)}^{(n+2)/\left(2n\right)}\hfill \\ \hfill & \le C\parallel v\parallel \sum {\epsilon }_{ij}^{(n+2)/\left(2\right(n-2\left)\right)}ln{\left({\epsilon }_{ij}^{-1}\right)}^{(n+2)/\left(2n\right)}.\hfill \end{array}$$

(85)

Now, as the computation of (65), we obtain

$${\int }_{\Omega }{\delta }_i^{p-1-\epsilon }{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}v=O\left(\epsilon \parallel v\parallel +\frac{\parallel v\parallel }{{\lambda }_i^{\frac{n-2}{2}}}\right).$$

(86)

On the other hand, we write

$${\int }_{\Omega }{U}^{p-\epsilon }{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}={\int }_{\Omega }({\alpha }_i{\delta }_i{)}^{p-\epsilon }{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}+O\left({\int }_{\Omega }{\delta }_i^{p-\epsilon }\left(\sum _{j\ne i}{\alpha }_j{\delta }_j\right)+\sum _{j\ne i}{\int }_{\Omega }{\delta }_j^{p-\epsilon }{\delta }_i\right).$$

(87)

For the remaining term in (87), using the fact that $\epsilon ln{\lambda }_i\to 0$, we see that

$${\int }_{\Omega }{\delta }_i^{p-\epsilon }\left(\sum _{j\ne i}{\alpha }_j{\delta }_j\right)+\sum _{j\ne i}{\int }_{\Omega }{\delta }_i^{p-\epsilon }{\delta }_i\le c\sum _{j\ne i}{\epsilon }_{ij}.$$

(88)

Now, since $\epsilon ln{\lambda }_i\to 0$, we notice that

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

(89)

Thus

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

However, we have

$${\int }_{\Omega }{\delta }_i^p\frac{\partial {\delta }_i}{\partial {\lambda }_i}={\int }_{{\mathbb{R}}^n}{\delta }_i^p{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}-{\int }_{{\mathbb{R}}^n\backslash \Omega }{\delta }_i^p{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}=O\left({\int }_{{\mathbb{R}}^n\backslash \Omega }{\delta }_i^{p+1}\right)=O\left(\frac{1}{{\lambda }_i^n}\right)$$

(90)

and

$${\int }_{\Omega }{\delta }_i^p{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}ln(1+{\lambda }_i^2{\left|x-a_i\right|}^2)=\frac{n-2}{2}{\int }_{{\mathbb{R}}^n}\frac{c_0^{\frac{2n}{n-2}}(1-{\left|x\right|}^2)}{{(1+{\left|x\right|}^2)}^{n+1}}ln(1+{\left|x\right|}^2)dx+O\left(\frac{ln{\lambda }_i}{{\lambda }_i^n}\right).$$

This implies that

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

(91)

Combining (80)–(91), we obtain

$${\int }_{\Omega }{\left|u\right|}^{p-1-\epsilon }u{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}=-{\alpha }_i^{p-\epsilon }{c}_2\frac{\epsilon c_0^{-\epsilon }}{{\lambda }_i^{\frac{\epsilon (n-2)}{2}}}+O\left({\epsilon }^2+{\parallel v\parallel }^2+\sum _{j\ne i}{\epsilon }_{ij}+\sum _{j=1}^N\frac{1}{{\left({\lambda }_i{\lambda }_j\right)}^{\frac{n-2}{2}}}\right).$$

(92)

Clearly (74), (79), and (82) give the desired result. □

Next, we provide a balancing condition involving the point of concentration.

Proposition 5. 

Let $(\alpha ,\lambda ,a)\in \mathcal{O}(N,{\nu }_0)$, $v\in E_{a,\lambda }$ and $u={\sum }_{i=1}^N{\alpha }_i{\delta }_{a_i,{\lambda }_i}+v$. Then for ε small and for each $i\le N$, the following statement holds

$$\begin{array}{cc}\hfill 〈I_{\epsilon }^{\prime }\left(u\right),\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}〉& ={\alpha }_i\frac{c_4}{{\lambda }_i^3}\nabla K\left(a_i\right)+c_3\sum _{j\ne i}\frac{{\alpha }_j}{{\lambda }_i}\frac{\partial {\epsilon }_{ij}}{\partial a_i}\left(1-c_0^{-\epsilon }{\alpha }_j^{p-1-\epsilon }{\lambda }_j^{\frac{-\epsilon (n-2)}{2}}-c_0^{-\epsilon }{\alpha }_i^{p-1-\epsilon }{\lambda }_i^{\frac{-\epsilon (n-2)}{2}}\right)\hfill \\ \hfill & +R_3,\hfill \end{array}$$

where

$$R_3=O\left({\epsilon }^2+{\parallel v\parallel }^2+\sum _{k\ne r}{\epsilon }_{kr}^{\frac{n}{n-2}}ln{\epsilon }_{kr}^{-1}+\sum _{j=1}^N\frac{1}{{\lambda }_j^{\frac{n-2}{2}}{\lambda }_i^{\frac{n}{2}}}+\sum _{j\ne i}{\lambda }_j|a_i-a_j|{\epsilon }_{ij}^{\frac{n+1}{n-2}}+T_2^2\left({\lambda }_i\right)+\frac{1}{{\lambda }_i}\sum _{j\ne i}{\epsilon }_{ij}{ln}^{\frac{n-2}{n}}\right),$$

$$c_3=c_0^{\frac{2n}{n-2}}{\int }_{{\mathbb{R}}^n}\frac{dx}{{(1+\mid x{\mid }^2)}^{\frac{n+2}{2}}},c_4=\frac{(n-2)}{n}{c}_0^2{\int }_{{\mathbb{R}}^n}\frac{{\left|x\right|}^2}{{(1+|x|}^2{)}^{n-1}}dx.$$

Proof. 

As in the proof of the previous proposition, we will take $h=\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}$ in (48) and estimate each term.

First, using estimate $F11$ of [31], we observe that for $i\ne j$, we have

$$\begin{array}{cc}\hfill {\int }_{\Omega }\nabla {\delta }_j\nabla \left(\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}\right)& =-{\int }_{\Omega }\Delta {\delta }_j\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}+{\int }_{\partial \Omega }\frac{\partial {\delta }_j}{\partial \nu }\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}\hfill \\ \hfill & ={\int }_{\Omega }{\delta }_j^p\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}+O\left(\frac{1}{{\lambda }_j^{\frac{n-2}{2}}}\frac{1}{{\lambda }_i^{\frac{n}{2}}}\right)\hfill \\ \hfill & =c_3\frac{1}{{\lambda }_i}\frac{\partial {\epsilon }_{ij}}{\partial a_i}+O\left({\lambda }_j|a_i-a_j|{\epsilon }_{ij}^{\frac{n+1}{n-2}}\right)+O\left(\frac{1}{{\lambda }_j^{\frac{n-2}{2}}}\frac{1}{{\lambda }_i^{\frac{n}{2}}}\right)\hfill \end{array}$$

(93)

and

$${\int }_{\Omega }\nabla {\delta }_i\nabla \left(\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}\right)={\int }_{{\mathbb{R}}^n\backslash \Omega }\nabla {\delta }_i\nabla \left(\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}\right)=O\left(\frac{1}{{\lambda }_i^{n-1}}\right).$$

(94)

Combining (93) and (94), we get

$${\int }_{\Omega }\nabla u\nabla \left(\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}\right)=c_3\sum _{j\ne i}{\alpha }_j\frac{1}{{\lambda }_i}\frac{\partial {\epsilon }_{ij}}{\partial a_i}+O\left(\sum _{j\ne i}{\lambda }_j|a_i-a_j|{\epsilon }_{ij}^{\frac{n+1}{n-2}}+\sum _{j=1}^N\frac{1}{{\lambda }_j^{\frac{n-2}{2}}}\frac{1}{{\lambda }_i^{\frac{n}{2}}}\right).$$

(95)

Second, we notice that

$$\begin{array}{cc}\hfill & \left|{\int }_{\Omega }Kv\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}\right|\le c{\int }_{\Omega }\left|v\right|{\delta }_i\le c\parallel v\parallel T_2\left({\lambda }_i\right),\hfill \\ \hfill & \left|{\int }_{\Omega }K{\delta }_j\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}\right|\le c{\int }_{\Omega }\frac{{\delta }_j{\delta }_i}{{\lambda }_i|x-a_i|}\le \frac{c}{{\lambda }_i}{\left({\int }_{\Omega }({\delta }_i{\delta }_j{)}^{\frac{n}{n-2}}\right)}^{\frac{n-2}{n}}{\left({\int }_{\Omega }\frac{dx}{{|x-a|}^{n/2}}\right)}^{\frac{2}{n}}\hfill \end{array}$$

(96)

$$\begin{array}{cc}\hfill & \le \frac{c}{{\lambda }_i}{\epsilon }_{ij}{ln}^{(n-2)/n}{\epsilon }_{ij}^{-1}\mathrm{for}\mathrm{i}\ne j,\hfill \end{array}$$

(97)

$$\begin{array}{cc}\hfill & \left|{\int }_{\Omega \backslash B_i}K{\delta }_i\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}\right|\le \frac{c}{{\lambda }_i^{n-1}}\hfill \end{array}$$

(98)

and

$${\int }_{B_i}K{\delta }_i\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}={\int }_{B_i}\nabla K\left(a_i\right)(x-a_i){\delta }_i\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}+O\left({\int }_{B_i}{\delta }_i\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}{(x-a_i)}^3\right),$$

(99)

where $B_i=B(a_i,r)$ with $r>0$ small.

Note that

$$\begin{array}{cc}\hfill {\int }_{B_i}\nabla K\left(x-a_i\right)\left(a_i\right){\delta }_i\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial {\left(a_i\right)}_j}& =\sum _{l=1}^n\frac{\partial K}{\partial x_l}\left(a_i\right){\int }_{B_i}{\delta }_i\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial {\left(a_i\right)}_j}{\left(x-a_i\right)}_l\hfill \\ & =\frac{\partial K}{\partial x_j}\left(a_i\right){\int }_{B_i}{\delta }_i\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial \left(a_i\right)j}{\left(x-a_i\right)}_j\hfill \\ & =\frac{\partial K}{\partial x_j}\left(a_i\right)(n-2)c_0^2{\int }_{B_i}\frac{{\lambda }_i^{n-1}{\left(x-a_i\right)}_j^2}{{\left(1+{\lambda }_i^2{\left|x-a_i\right|}^2\right)}^{n-1}}dx\hfill \\ & =\frac{\partial K}{\partial x_j}\left(a_i\right)(n-2)\frac{c_0^2}{n}{\int }_{B_i}\frac{{\lambda }_i^{n-1}{\left|x-a_i\right|}^2}{{\left(1+{\lambda }_i^2{\left|x-a_i\right|}^2\right)}^{n-1}}dx\hfill \\ & =\frac{\partial K}{\partial x_j}\left(a_i\right)\frac{(n-2)}{n}\frac{c_0^2}{{\lambda }_i^3}{\int }_{B(0,{\lambda }_{i}r)}\frac{{\left|x\right|}^2}{{\left(1+{\left|x\right|}^2\right)}^{n-1}}dx\hfill \\ & =\frac{\partial K}{\partial x_j}\left(a_i\right)\frac{c_4}{{\lambda }_i^3}+O\left(\frac{1}{{\lambda }_i^{n-1}}\right).\hfill \end{array}$$

This implies that

$${\int }_{B_i}\nabla K\left(a_i\right)\left(x-a_i\right){\delta }_i\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}=\frac{c_4}{{\lambda }_i^3}\nabla K\left(a_i\right)+O\left(\frac{1}{{\lambda }_i^{n-1}}\right).$$

(100)

We also note that, since $\frac{1}{{\lambda }_i}\left|\frac{\partial {\delta }_i}{\partial a_i}\right|\le \frac{c}{{\lambda }_i|x-a_i|}{\delta }_i,$ we have

$$\begin{array}{cc}\hfill {\int }_{B_i}{\delta }_i\frac{1}{{\lambda }_i}\left|\frac{\partial {\delta }_i}{\partial a_i}\right|{|x-a_i|}^{3}dx& \le \frac{c}{{\lambda }_i}{\int }_{B_i}{\delta }_i^2{|x-a_i|}^{2}dx\hfill \\ \hfill & \le \frac{c}{{\lambda }_i}{\int }_{B_i}\frac{{\lambda }_i^{n-2}{\left|x-a_i\right|}^2}{{\left(1+{\lambda }_i^2{\left|x-a_i\right|}^2\right)}^{n-2}}dx\hfill \\ \hfill & \le c\left\{\begin{array}{c}{\lambda }_i^{-5}\mathrm{if}n\ge 7\hfill \\ {\lambda }_i^{-5}ln{\lambda }_{i}ifn\ge 6.\hfill \end{array}\right.\hfill \end{array}$$

(101)

Combining (96)–(101), we obtain

$${\int }_{\Omega }Ku\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}=\frac{{\alpha }_i{c}_4}{{\lambda }_i^3}\nabla K\left(a_i\right)+O\left(\frac{1}{{\lambda }_i^{n-1}}+\frac{1}{{\lambda }_i}\sum _{j\ne i}{\epsilon }_{ij}{ln}^{\frac{n-2}{n}}{\epsilon }_{ij}^{-1}+{\parallel v\parallel }^2+T_2^2\left({\lambda }_i\right)\right).$$

(102)

Next, we are going to estimate the last term of the right hand side of (48) with $h=\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}.$

• To this aim, we observe that

$$\left|{\int }_{{\Omega }_2}{\left|u\right|}^{p-1-\epsilon }u\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}\right|\le {\int }_{{\Omega }_2}{\left|v\right|}^{p+1-\epsilon }=O\left({\parallel v\parallel }^{p+1-\epsilon }\right)$$

and

$$\begin{array}{cc}\hfill {\int }_{{\Omega }_1}& u^{p-\epsilon }\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}={\int }_{{\Omega }_1}{U}^{p-\epsilon }\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}+(p-\epsilon ){\int }_{{\Omega }_1}{U}^{p-1-\epsilon }v\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}+O\left({\int }_{{\Omega }_1}{U}^{p-2-\epsilon }{v}^2\left|\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}\right|\right)\hfill \\ \hfill & ={\int }_{\Omega }{U}^{p-\epsilon }\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}+(p-\epsilon ){\int }_{\Omega }{U}^{p-1-\epsilon }v\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}+O\left({\int }_{\Omega }{U}^{p-1-\epsilon }{v}^2+{\int }_{{\Omega }_2}{\left|v\right|}^{p+1-\epsilon }\right),\hfill \end{array}$$

(103)

where ${\Omega }_1$ and ${\Omega }_2$ are defined in (57).

However, we have

$${\int }_{\Omega }{U}^{p-1-\epsilon }{v}^2+{\int }_{{\Omega }_2}{\left|v\right|}^{p+1-\epsilon }={O(\parallel v\parallel }^2)$$

(104)

and

$$\begin{array}{cc}\hfill {\int }_{\Omega }& U^{p-\epsilon }\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}={\int }_{\Omega }({\alpha }_i{\delta }_i{{)}^{p-\epsilon }\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}+(p-\epsilon ){\int }_{\Omega }({\alpha }_i{\delta }_i)}^{p-\epsilon -1}\left(\sum _{j\ne i}{\alpha }_j{\delta }_j\right)\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}\hfill \\ \hfill & +{\int }_{\Omega }(\sum _{j\ne i}{\alpha }_j{\delta }_j{)}^{p-\epsilon }\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}+O\left({\int }_{{\Omega }^i}\left[{\left({\alpha }_i{\delta }_i\right)}^{p-2-\epsilon }{\left(\sum _{j\ne i}{\alpha }_j{\delta }_j\right)}^2+{\left(\sum _{j\ne i}{\alpha }_j{\delta }_j\right)}^{p-\epsilon }\right]\left|\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}\right|\right)\hfill \\ \hfill & +O\left({\int }_{\Omega \backslash {\Omega }^i}\left[{\alpha }_i{\delta }_i{\left(\sum _{j\ne i}{\alpha }_j{\delta }_j\right)}^{p-\epsilon -1}+{\left({\alpha }_i{\delta }_i\right)}^{p-1-\epsilon }\left(\sum _{j\ne i}{\alpha }_j{\delta }_j\right)\right]\left|\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}\right|\right),\hfill \end{array}$$

(105)

where ${\Omega }^i$ is defined in (63).

We need to estimate each term of the right hand side of (105). First, we observe that

$${\int }_{{\Omega }^i}[\dots ]\left|\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}\right|+{\int }_{\Omega \backslash {\Omega }^i}[\dots ]\left|\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}\right|\le C\sum _{j\ne i}{\int }_{\Omega }({\delta }_i{\delta }_j{)}^{\frac{n}{n-2}}\le C\sum _{j\ne i}{\epsilon }_{ij}^{\frac{n}{n-2}}ln{\epsilon }_{ij}^{-1}.$$

(106)

Second, by oddness we have

$${\int }_{\Omega }({\alpha }_i{\delta }_i{)}^{p-\epsilon }\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}=O\left({\int }_{\Omega \backslash B(a_i,r)}{\delta }_i^p\left|\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}\right|\right)=O\left({\int }_{\Omega \backslash B(a_i,r)}{\delta }_i^{p+1}\right)=O\left(\frac{1}{{\lambda }_i^n}\right),$$

(107)

where r is a fixed small positive constant.

Third, we write

$$\begin{array}{cc}\hfill {\int }_{\Omega }& {\left({\alpha }_i{\delta }_i\right)}^{p-\epsilon -1}\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}{\delta }_j=c_0^{-\epsilon }{\lambda }_i^{\frac{\epsilon (2-n)}{2}}{\int }_{\Omega }{\delta }_i^{p-1}\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}{\delta }_j+O\left(\epsilon {\int }_{\Omega }{\delta }_i^{p-\epsilon }{\delta }_{j}ln(1+{\lambda }_i^2{|x-a_i|}^2\right)\hfill \\ \hfill & =c_0^{-\epsilon }{\lambda }_i^{\frac{\epsilon (2-n)}{2}}{\int }_{{\mathbb{R}}^n}{\delta }_i^{p-1}\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}{\delta }_j+O\left({\int }_{{\mathbb{R}}^n\backslash \Omega }{\delta }_i^p{\delta }_j+\epsilon {\int }_{\Omega }({\delta }_i{\delta }_j){\delta }_i^{p-1}ln(1+{\lambda }_i^2|x-a_i{|}^2)\right)\hfill \\ \hfill & =c_0^{-\epsilon }{\lambda }_i^{\frac{\epsilon (2-n)}{2}}\frac{c_3}{p{\lambda }_i}\frac{\partial {\epsilon }_{ij}}{\partial a_i}+O\left({\lambda }_j|a_i-a_j|{\epsilon }_{ij}^{\frac{n+1}{n-2}}+{\lambda }_i^{-\frac{(n+2)}{2}}{\lambda }_j^{\frac{2-n}{2}}+\epsilon {\epsilon }_{ij}{ln}^{\frac{n-2}{n}}{\epsilon }_{ij}^{-1}\right).\hfill \end{array}$$

(108)

Lastly, we write

$$\begin{array}{cc}\hfill {\int }_{\Omega }(\sum _{j\ne i}{\alpha }_j{\delta }_j{)}^{p-\epsilon }& \frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}=\sum _{j\ne i}{\int }_{\Omega }({\alpha }_j{\delta }_j{)}^{p-\epsilon }\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}+O\left(\sum _{k\notin \{i,j\}}{\int }_{\Omega }\delta k{\delta }_j^{p-1}|\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}|\right)\hfill \\ \hfill & =\sum _{j\ne i}{\alpha }_j^{p-\epsilon }{c}_0^{-\epsilon }{\lambda }_i^{\frac{\epsilon (2-n)}{2}}{\int }_{\Omega }{\delta }_j^p\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}+O\left(\epsilon {\int }_{\Omega }{\delta }_j^p|\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}|ln(1+{\lambda }_j^2|x-a_j{|}^2)\right)\hfill \\ \hfill & +O\left(\sum _{k\notin \{i,j\}}{\int }_{\Omega }{\delta k}_{\delta j}^{p-1}\left|\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}\right|\right).\hfill \end{array}$$

(109)

However, we have

$$\begin{array}{cc}\hfill & \sum _{k\notin \{i,j\}}{\int }_{\Omega }{\delta }_k{\delta }_j^{p-1}|\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}{|\le \sum _{l\ne r}{\int }_{\Omega }({\delta }_l{\delta }_r)}^{n/(n-2)}\le C\sum _{l\ne r}{\epsilon }_{lr}^{n/(n-2)}ln{\epsilon }_{lr}^{-1},\hfill \\ \hfill & {\int }_{\Omega }{\delta }_j^p|\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}|ln(1+{\lambda }_j^2|x-a_j{|}^2)\le {\int }_{\Omega }{\delta }_j^{p-1}\left({\delta }_i{\delta }_j\right)ln(1+{\lambda }_j^2|x-a_j{|}^2)\hfill \end{array}$$

(110)

$$\begin{array}{cc}\hfill & \le C{\left({\int }_{\Omega }({\delta }_i{\delta }_j{)}^{n/(n-2)}\right)}^{\frac{n-2}{n}}\le C{\epsilon }_{ij}{ln}^{\frac{n-2}{n}}{\epsilon }_{ij}^{-1},\hfill \end{array}$$

(111)

$$\begin{array}{cc}\hfill & {\int }_{\Omega }{\delta }_j^p\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}={\int }_{{\mathbb{R}}^n}{\delta }_j^p\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}+O\left({\int }_{{\mathbb{R}}^n\backslash \Omega }{\delta }_j^p{\delta }_i\right)\hfill \\ \hfill & =\frac{c_3}{{\lambda }_i}\frac{\partial {\epsilon }_{ij}}{\partial a_i}+O\left({\lambda }_j|a_i-a_j|{\epsilon }_{ij}^{\frac{n+1}{n-2}}+{\lambda }_j^{-\frac{(n+2)}{2}}{\lambda }_i^{\frac{2-n}{2}}\right).\hfill \end{array}$$

(112)

To deal with the second integral in the right hand side of (103), we write

$$\begin{array}{cc}\hfill {\int }_{\Omega }{U}^{p-1-\epsilon }v\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}& ={\int }_{\Omega }({\alpha }_i{\delta }_i{)}^{p-1-\epsilon }v\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}+O\left({\int }_{{\Omega }^i}{\left({\alpha }_i{\delta }_i\right)}^{p-2-\epsilon }\left(\sum _{j\ne i}{\alpha }_j{\delta }_j\right)\left|v\right||\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}|\right)\hfill \\ \hfill & +O\left({\int }_{\Omega \backslash {\Omega }^i}{\left(\sum _{j\ne i}{\alpha }_j{\delta }_j\right)}^{p-1-\epsilon }\left|v\right||\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}|\right),\hfill \end{array}$$

(113)

where ${\Omega }^i$ is defined in (63).

However, since $n\ge 6$, we have

$$\begin{array}{cc}\hfill {\int }_{{\Omega }_i}{\left({\alpha }_i{\delta }_i\right)}^{p-2+\epsilon }& \left(\sum _{j\ne i}{\alpha }_j{\delta }_j\right)\left|v\right||\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}|+{\int }_{{\Omega }_i^c}{\left(\sum _{j\ne i}{\alpha }_j{\delta }_j\right)}^{p-1+\epsilon }\left|v\right||\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}|\hfill \\ \hfill & \le C{\int }_{{\Omega }_i}{\left({\alpha }_i{\delta }_i\right)}^{p-1}\left(\sum _{j\ne i}{\alpha }_j{\delta }_j\right)\left|v\right||+C{\int }_{{\Omega }_i^c}{\left(\sum _{j\ne i}{\alpha }_j{\delta }_j\right)}^{p-1}\left|v\right|{\delta }_i\hfill \\ \hfill & \le C\sum _{j\ne i}{\int }_{\Omega }({\delta }_i{\delta }_j{)}^{\frac{p}{2}}\left|v\right|\le c\parallel v\parallel {\left({\int }_{\Omega }({\delta }_i{\delta }_j{)}^{\frac{n}{n-2}}\right)}^{\frac{n+2}{2n}}\hfill \\ \hfill & \le C\parallel v\parallel \sum {\epsilon }_{ij}^{\frac{n+2}{2(n-2)}}{ln}^{\frac{n+2}{2n}}{\epsilon }_{ij}^{-1}.\hfill \end{array}$$

(114)

To complete the estimate (113), we write

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

(115)

Combining the above estimates, we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }& {\left|u\right|}^{p-\epsilon }u\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}=c_3{c}_0^{-\epsilon }\sum _{j\ne i}{\alpha }_j\frac{1}{{\lambda }_i}\frac{\partial {\epsilon }_{ij}}{\partial a_i}\left[\sum _{k=i,j}{\alpha }_k^{p-1-\epsilon }{\lambda }_k^{-\epsilon \frac{n-2}{2}}\right]+O\left(\sum _{k\ne r}{\epsilon }_{kr}^{\frac{n}{n-2}}ln{\epsilon }_{kr}^{-1}\right)\hfill \\ \hfill & +O\left(\sum _{j\ne i}\left(\frac{1}{{\lambda }_i^{\frac{n+2}{2}}{\lambda }_j^{\frac{n-2}{2}}}+\frac{1}{{\lambda }_i^{\frac{n-2}{2}}{\lambda }_j^{\frac{n+2}{2}}}\right)+\frac{\parallel v\parallel }{{\lambda }_i^{\frac{n}{2}}}+\epsilon \parallel v\parallel +\sum _{j\ne i}{\lambda }_j|a_i-a_j|{\epsilon }_{ij}^{\frac{n+1}{n-2}}+{\parallel v\parallel }^2\right)\hfill \\ \hfill & +O\left(\parallel v\parallel T_1(a,\lambda )+\frac{1}{{\lambda }_i^n}+\epsilon \sum _{j\ne i}{\epsilon }_{ij}{ln}^{(n-2)/n}{\epsilon }_{ij}^{-1}\right).\hfill \end{array}$$

(116)

Combining (95), (102) and (116), we easily obtain the desired result. □

5. Construction of Simple Interior Bubbling Solutions

In this section, we assume that $n\ge 6$ and we take $N\le m$, where m is the number of critical points of K, and let $y_1$, ..., $y_N$ be non-degenerate distinct critical points of K. As in [32], the strategy of the proof of Theorem 1 is the following: let

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

(117)

where c is a positive constant and $E_{a,\lambda }$ is defined by (9).

In addition, we consider the following function

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

We notice that $(\alpha ,\lambda ,a,v)$ is a critical point of ${\psi }_{\epsilon }$ if and only if $u={\sum }_{i=1}^N{\alpha }_i{\delta }_i+v$ is a critical point of $I_{\epsilon }$. Thus, we need to look for critical point of ${\psi }_{\epsilon }$. Since the variable v belongs to $E_{a,\lambda }$, the lagrange multiplier theorem allows us to get the following proposition.

Proposition 6. 

$(\alpha ,\lambda ,a,v)\in \mathcal{M}(N,\epsilon )$ 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:

$${\psi }_{\epsilon }^{\prime }=\sum _{k=1}^N(A_k{\phi }_k^{\prime }+B_k{\varphi }_k^{\prime }+\sum _{j=1}^n{c}_{kj}{\xi }_{kj}^{\prime }),$$

where

$${\phi }_k(\alpha ,\lambda ,a,v)={\int }_{\Omega }\nabla v\nabla {\delta }_k,{\varphi }_k(\alpha ,\lambda ,a,v)={\lambda }_k{\int }_{\Omega }\nabla v\nabla \frac{\partial {\delta }_k}{\partial {\lambda }_k},{\xi }_{kj}(\alpha ,\lambda ,a,v)=\frac{1}{{\lambda }_k}{\int }_{\Omega }\nabla v\nabla \frac{\partial {\delta }_k}{\partial a_k^j}.$$

In other words, $(\alpha ,\lambda ,a,v)$ is a critical point of ${\psi }_{\epsilon }$ is equivalent to the following system:

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

(118)

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

(119)

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

(120)

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

(121)

• To prove Theorem 1, we will make a careful study of the previous equations. Notice that

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

(122)

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

(123)

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

(124)

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

(125)

Let $(\alpha ,\lambda ,a,0)\in \mathcal{M}(N,\epsilon )$, where $\mathcal{M}(N,\epsilon )$ is defined by (117). We are going to solve the system defined by Equations (118)–(121). Clearly $\overline{v}$, obtained in Proposition 2, satisfies Equation (121). To simplify the notation, we will write in the sequel v instead of $\overline{v}$. Combining Equations (118)–(125), we see that $u={\sum }_{i=1}^N{\alpha }_i{\delta }_i+v$ is a critical point of $I_{\epsilon }$ if and only if $(\alpha ,\lambda ,a)$ solves the following system for each $1\le i\le N$

$$\left(E_{{\alpha }_i}\right)〈I_{\epsilon }^{\prime }\left(u\right),{\delta }_i〉)=0$$

(126)

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

(127)

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

(128)

Recall that $(\alpha ,\lambda ,a,0)\in \mathcal{M}(N,\epsilon )$, $a_i$ is close to the critical point $y_i$ of K and $0<1/c\le {\lambda }_i/{\lambda }_j\le c$ for some fixed positive constant. Therefore $|a_i-a_j|\ge c>0$ and ${\epsilon }_{ij}=O\left({\epsilon }^{(n-2)/2}\right)$.

Next, we state the following crucial estimates, which are a direct consequence of Propositions 2–5.

Lemma 2. 

For ε small, the following statements hold:

$$\parallel v\parallel \le c\left\{\begin{array}{c}{\epsilon }^{1}ifn\ge 7,\hfill \\ {\epsilon |ln\epsilon |}^{\frac{2}{3}}ifn=6,\hfill \end{array}\right.R_1=O\left(\epsilon \right),R_2;R_3\le c\left\{\begin{array}{c}{\epsilon }^{2}ifn\ne 6\hfill \\ {\epsilon }^2{|ln\epsilon |}^{4/3}ifn=6,\hfill \end{array}\right.$$

where $R_1$, $R_2$, and $R_3$ are defined in Propositions 3, 4, and 5, respectively.

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

Lemma 3. 

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

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

Proof. 

Applying $\frac{\partial {\psi }_{\epsilon }}{\partial v}$ (see (121)) to the functions ${\delta }_i$, ${\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}$ and $\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i^j}$, we obtained the following quasi-diagonal system

$$\begin{array}{cc}\hfill & c{A}_i+O\left({\epsilon }^{\frac{n-2}{2}}\left(\sum _{k=1}^N\left(\right|A_k|+|B_k|+\sum _{j=1}^n\left|C_{kj}\right)\right)\right)=〈\frac{\partial {\psi }_{\epsilon }}{\partial v},{\delta }_i〉\hfill \\ \hfill & c^{\prime }{B}_i+O\left({\epsilon }^{\frac{n-2}{2}}\left(\sum _{k=1}^N\left(\right|A_k|+|B_k|+\sum _{j=1}^n\left|C_{kj}\right)\right)\right)=〈\frac{\partial {\psi }_{\epsilon }}{\partial v},{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}〉\hfill \\ \hfill & c^{″}{C}_{il}+O\left({\epsilon }^{\frac{n-2}{2}}\left(\sum _{k=1}^N\left(\right|A_k|+|B_k|+\sum _{j=1}^n\left|C_{kj}\right)\right)\right)=〈\frac{\partial {\psi }_{\epsilon }}{\partial v},\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i^l}〉,\hfill \end{array}$$

where

$$c={\int }_{{\mathbb{R}}^n}{|\nabla {\delta }_i|}^2,c^{\prime }={\int }_{{\mathbb{R}}^n}{\left|{\lambda }_i\frac{\nabla \partial {\delta }_i}{\partial {\lambda }_i}\right|}^2\mathrm{and}{c}^{″}={\int }_{{\mathbb{R}}^n}{\left|\frac{1}{{\lambda }_i}\frac{\nabla \partial {\delta }_i}{\partial a_i^j}\right|}^2.$$

(129)

Combining Propositions 3–5, estimates (29)–(37) and the fact that $(\alpha ,\lambda ,a,0)\in \mathcal{M}(N,\epsilon )$, we see that for all $i\in \{1,\dots ,N\}$ we have

$$〈\frac{\partial {\psi }_{\epsilon }}{\partial v},{\delta }_i〉=O\left(\epsilon {ln}^2\epsilon \right),〈\frac{\partial {\psi }_{\epsilon }}{\partial v},{\lambda }_i\frac{\partial {\delta }_i}{\partial {\lambda }_i}〉=O\left(\epsilon \right)\mathrm{and}〈\frac{\partial {\psi }_{\epsilon }}{\partial v},\frac{1}{{\lambda }_i}\frac{\partial {\delta }_i}{\partial a_i}〉=O\left({\epsilon }^{\frac{3}{2}}\right).$$

This implies that

$$M{(A,B,C)}^T=(O\left(\epsilon {ln}^2\epsilon \right),O\left(\epsilon \right),O{\left({\epsilon }^{3/2}\right)}^T,$$

where M is the matrix defined by

$$m_{ij}=O\left({\epsilon }^{\frac{n-2}{2}}\right)\forall i\ne j\mathrm{and}{m}_{ii}=\left\{\begin{array}{c}c+O\left({\epsilon }^{\frac{n-2}{2}}\right)\mathrm{for}1\le i\le N\hfill \\ c^{\prime }+O\left({\epsilon }^{\frac{n-2}{2}}\right)\mathrm{for}N+1\le i\le 2N\hfill \\ c^{″}+O\left({\epsilon }^{\frac{n-2}{2}}\right)\mathrm{for}2N+1\le i\le N(n+2),\hfill \end{array}\right.$$

where c, $c^{\prime }$, and $c^{″}$ are defined in (129).

Clearly, M is an invertible matrix. We see that M and $M^{-1}$ have the same form. Thus, Lemma 3 follows. □

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

$${\beta }_i=1-{\alpha }_i^{p-1},\frac{1}{{\lambda }_i^2}=\frac{c_2}{c_1}\frac{1}{K\left(y_i\right)}\epsilon (1+{\wedge }_i),z_i=a_i-y_{i}1\le i\le N,$$

(130)

where $c_1$ and $c_2$ are defined in Proposition 4.

This change of variables allows us to rewrite the system in the following simple form:

Lemma 4. 

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

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

Proof. 

Using the fact that

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

we see that $\left(E_{{\alpha }_i}\right)$ 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, Lemmas 2 and 3 imply that

$$(1-{\beta }_i)c_2\epsilon -\frac{c_{1}K\left(a_i\right)}{{\lambda }_i^2}\le c\left\{\begin{array}{c}{\epsilon }^{min(2,(n-2)/2})\mathrm{if}n\ne 6\hfill \\ {\epsilon }^2{|ln\epsilon |}^{4/3}\mathrm{if}n=6.\hfill \end{array}\right.$$

Writing

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

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

Lastly, writing

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

and using Proposition 5, Lemmas 2 and 3, 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 4. □

Now, we are ready to prove our results related to the construction of simple interior bubbling solutions. We note that, since Theorem 2 is a straightforward consequence of Theorem 1, we only need to prove this result.

Proof of Theorem 1 

The system $\left(S\right)$, given in Lemma 4, can be rewritten

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

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

Defining the following linear map

$$\begin{array}{cc}\hfill & l:{\mathbb{R}}^N\times {\mathbb{R}}^N\times {\left({\mathbb{R}}^n\right)}^N\to {\mathbb{R}}^N\times {\mathbb{R}}^N\times {\left({\mathbb{R}}^n\right)}^N\hfill \\ \hfill & (\beta ,\wedge ,z)\mapsto (\beta ,\wedge ,D^{2}K\left(y_1\right)(z_1,.),\dots ,D^{2}K\left(y_N\right)(z_N,.)),\hfill \end{array}$$

we see that the system is equivalent to

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

(131)

where

$$\begin{array}{cc}\hfill & U_1(\epsilon ,\beta ,\wedge ,z)=(U_{1,1},\dots ,U_{1,N}),U_2(\epsilon ,\beta ,\wedge ,z)=(U_{2,1},\dots ,U_{2,N}),\hfill \\ \hfill & U_3(\epsilon ,\beta ,\wedge ,z)=(U_{3,1},\dots ,U_{3,N}).\hfill \end{array}$$

In addition, since $y_i$ is a non-degenerate critical point of K, we deduced that l is invertible. This implies that (131) is equivalent to

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

(132)

Choosing r positive small and $(\beta ,\wedge ,z)\in \overline{B(0,r)}$, we obtain

$$\begin{array}{cc}\hfill \parallel l^{-1}(U_1(\epsilon ,\beta ,\wedge ,z),U_2(\epsilon ,\beta ,\wedge ,z),& U_3(\epsilon ,\beta ,\wedge ,z))\parallel \hfill \\ \hfill & \le \parallel l^{-1}\parallel \parallel (\epsilon ,\beta ,\wedge ,z),U_2(\epsilon ,\beta ,\wedge ,z),U_3(\epsilon ,\beta ,\wedge ,z))\parallel \hfill \\ \hfill & \le C\parallel l^{-1}{\parallel \left(\right|z|}^2+\left|e^{1/4}\right)\hfill \\ \hfill & \le C\parallel l^{-1}\parallel 2r^2\hfill \end{array}$$

and hence if we choose $r\le 1/(2C\parallel l^{-1}\parallel$, we see that the function

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

(133)

is well-defined and continuous. Thus, applying Brouwer’s fixed point theorem, we derived that f has a fixed point. Therefore, the system $\left(S\right)$ has at least one solution $({\beta }_{\epsilon },{\wedge }_{\epsilon },z_{\epsilon })$ for $\epsilon$ small. To complete the proof of the theorem, it remains to be proven that the constructed function $u_{\epsilon }={\sum }_{i=1}^N{\alpha }_i{\delta }_i+v$ is positive. To this aim, we first remark, since $\parallel v\parallel \to 0$, that $u_{\epsilon }\not\equiv 0$ for $\epsilon$ small. By construction, $u_{\epsilon }$ satisfies

$$\left({\mathcal{P}}_{\epsilon }^{\prime }\right):\left\{\begin{array}{c}-\Delta u_{\epsilon }+K\left(x\right)u_{\epsilon }={\left|u_{\epsilon }\right|}^{p-1-\epsilon }{u}_{\epsilon }\mathrm{in}\Omega \hfill \\ \frac{\partial u_{\epsilon }}{\partial n}=0\mathrm{on}\partial \Omega .\hfill \end{array}\right.$$

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

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

However, we have

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

which implies that

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

However, 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 principal, $u_{\epsilon }$ has to be positive. This completes the proof of the theorem. □

6. Construction of Clustered Bubbling Solutions

This section is devoted to the proof of Theorems 3 and 4. We start by proving Theorem 3. Let $n\ge 6$, $y\in$ be a non-degenerate critical point of K and $({\overline{z}}_1,\dots ,{\overline{z}}_N)$ be a non-degenerate critical point of $F_{N,y}$, where $F_{N,y}$ is defined by (3). The strategy of the proof of Theorem 3 is the same as that of Theorem 1. We start by introducing a neighborhood of the desired constructed solutions. Let

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

(134)

where $E_{a,\lambda }$ is defined by (9),

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

(135)

with $c_3$, $c_4$, $c_2$ and $c_1$ are the constants defined in Propositions 4 and 5. As in the proof of Theorem 1, we reduced the problem to a finite dimensional system. Proposition 2 allows us to obtain such a reduction by finding $\overline{v}$ verifying the Equation (121). Thus, we are looking for $(\alpha ,\lambda ,a)\in V_{N,y,\epsilon }$ solution of the system defined by Equations (126)–(128), where $V_{N,y,\epsilon }$ is defined by

$$V_{N,y,\epsilon }=\{(\alpha ,\lambda ,a)\in {\left({\mathbb{R}}_{+}\right)}^N\times {\left({\mathbb{R}}_{+}\right)}^N\times {\Omega }^N:(\alpha ,\lambda ,a,0)\in \mathcal{V}(N,y,\epsilon )\forall 1\le i\le N\}.$$

(136)

As in the proof of Theorem 1 and in order to work with a simpler system, we made the following change of variables:

$${\beta }_i=1-{\alpha }_i^{p-1},\frac{1}{{\lambda }_i^2}=\frac{c_2}{c_1}\frac{1}{K\left(y\right)}\epsilon (1+{\wedge }_i),(a_i-y)={\epsilon }^{\gamma }\sigma ({\zeta }_i+{\overline{z}}_i)1\le i\le N,$$

(137)

where $c_1$ and $c_2$ are defined in Proposition 4.

Using these changes of variables, it is easy to see that

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

(138)

$$\begin{array}{cc}\hfill & \frac{\partial {\epsilon }_{ij}}{\partial a_i}=\frac{(n-2){\epsilon }^{\gamma +1}{c}_2^{\frac{n-2}{2}}}{{\sigma }^{n-1}{\left(c_{1}K\left(y\right)\right)}^{\frac{n-2}{2}}}\frac{({\zeta }_j+{\overline{z}}_j-{\zeta }_i-{\overline{z}}_i)}{{\left|{\zeta }_i+{\overline{z}}_i-{\zeta }_j-{\overline{z}}_j\right|}^n}\left(1+R({\wedge }_i,{\wedge }_j\right)(1+O\left({\epsilon }^{\frac{4}{n}}\right))=O\left({\epsilon }^{\gamma +1}\right),\hfill \end{array}$$

(139)

where

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

(140)

Next, using Propositions 2–5, we deduced that the following estimates hold:

Lemma 5. 

For ε small, the following statements hold:

$$\begin{array}{cc}\hfill & \parallel v\parallel \le c\left\{\begin{array}{c}\epsilon ifn\ge 7,\hfill \\ {\epsilon |ln\epsilon |}^{\frac{2}{3}}ifn=6,\hfill \end{array}\right.R_1=O\left(\epsilon \right),\hfill \\ \hfill & R_3\le c\left\{\begin{array}{c}{\epsilon }^2|ln\epsilon |ifn\ge 7\hfill \\ {\epsilon }^2{|ln\epsilon |}^{4/3}ifn=6,\hfill \end{array}\right.R_2=O\left({\epsilon }^{2\gamma +1}\right),\hfill \end{array}$$

where $R_1$, $R_2$, and $R_3$ are defined in Propositions 3, 4, and 5, respectively.

Now, arguing as in the proof of Lemma 3, we derive that the constants $A_i^{\prime }s$, $B_i^{\prime }s$, and $C_{ij}^{\prime }s$, which appear in equations $\left(E_{{\alpha }_i}\right)$, $\left(E_{{\lambda }_i}\right)$ and $\left(E_{a_i}\right)$, satisfy the following estimates:

Lemma 6. 

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

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

Next, our aim is to rewrite equations $\left(E_{{\alpha }_i}\right)$, $\left(E_{{\lambda }_i}\right)$, $\left(E_{a_i}\right)$ in a simple form.

Lemma 7. 

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

$$\left(S^{\prime }\right)\left\{\begin{array}{c}{\beta }_i=O\left(\epsilon \right|ln\epsilon \left|\right)\forall 1\le i\le N\hfill \\ {\wedge }_i=O\left(|{\beta }_i|+{\epsilon }^{2\gamma }\right)\forall 1\le i\le N\hfill \\ D^{2}K\left(y\right)({\zeta }_i,.)-(n-2)\sum _{j\ne i}\frac{{\zeta }_j-{\zeta }_i}{|{\overline{z}}_j-{\overline{z}}_i{|}^n}+n(n-2)\sum _{j\ne i}〈\frac{{\overline{z}}_j-{\overline{z}}_i}{|{\overline{z}}_j-{\overline{z}}_i{|}^2},{\zeta }_j-{\zeta }_i〉\hfill \\ -(n-2)\sum _{j\ne i}\frac{{\overline{z}}_j-{\overline{z}}_i}{|{\overline{z}}_j-{\overline{z}}_i{|}^n}\left(\frac{n-6}{4}{\wedge }_i+\frac{n-2}{4}{\wedge }_j\right)=O\left(R_4\right)\forall 1\le i\le N,\hfill \end{array}\right.$$

where

$$R_4=\sum _{j=1}^N{\wedge }_j^2+\sum _{j=1}^N|{\zeta }_j{|}^2+{\epsilon }^{\gamma }+{\epsilon }^{\frac{1}{2}-\gamma }{|ln\epsilon |}^{4/3}.$$

Proof. 

First, using Proposition 3, Lemma 5 and the fact that $(\alpha ,\lambda ,a)\in V_{N,y,\epsilon }$, we see that $\left(E_{{\alpha }_i}\right)$ is equivalent to

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

Second, using Lemma 6, we write

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

Using Proposition 4 and Lemma 5, we obtain

$$c_2\epsilon -c_1\frac{K\left(a_i\right)}{{\lambda }_i^2}=O\left(\epsilon |{\beta }_i|+{\epsilon }^{2\gamma +1}\right).$$

However, we have

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

This implies that $\left(E_{{\lambda }_i}\right)$ is equivalent to

$$\left(F_{{\lambda }_i}^{\prime }\right){\wedge }_i=O\left(|{\beta }_i|+{\epsilon }^{1-4/n}\right)\forall 1\le i\le N.$$

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

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

(141)

However, combining (139), Proposition 5 and Lemma 5, (141) becomes

$$\begin{array}{cc}\hfill \frac{c_4}{{\lambda }_i^3}\nabla K\left(a_i\right)& -c_5\frac{{\epsilon }^{1+\gamma }}{{\lambda }_i}\sum _{j\ne i}\frac{({\zeta }_j+{\overline{z}}_j-{\zeta }_i-{\overline{z}}_i)}{|{\zeta }_j+{\overline{z}}_j-{\zeta }_i-{\overline{z}}_i{|}^n}\left(1+\frac{n-2}{4}{\wedge }_i+\frac{n-2}{4}{\wedge }_j+O\left({\wedge }_i^2+{\wedge }_j^2\right)\right)\hfill \\ \hfill & =O\left({\epsilon }^2{|ln\epsilon |}^{4/3}\right),\hfill \end{array}$$

(142)

where

$$c_5=\frac{c_3(n-2)}{{\sigma }^{n-1}}{\left(\frac{c_2}{c_{1}K\left(y\right)}\right)}^{\frac{n-2}{2}}.$$

Observe that

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

(143)

Combining (142) and (143), we obtain

$$\begin{array}{cc}\hfill & D^{2}K\left(y\right)({\zeta }_i+{\overline{z}}_i,.)\hfill \\ \hfill & -(n-2)\sum _{j\ne i}\frac{({\zeta }_j+{\overline{z}}_j-{\zeta }_i-{\overline{z}}_i)}{|{\zeta }_j+{\overline{z}}_j-{\zeta }_i-{\overline{z}}_i{|}^n}\left(1+\frac{n-6}{4}{\wedge }_i+\frac{n-2}{4}{\wedge }_j+O\left({\wedge }_i^2+{\wedge }_j^2\right)\right)\hfill \\ \hfill & =O\left({\epsilon }^{\gamma }+{\epsilon }^{\frac{1}{2}-\gamma }{|ln\epsilon |}^{\frac{4}{3}}\right)=O\left({\epsilon }^{\frac{1}{2}-\gamma }{|ln\epsilon |}^{\frac{4}{3}}\right).\hfill \end{array}$$

(144)

However, we have

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

(145)

and

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

(146)

Combining (144)–(146) and the fact that $({\overline{z}}_1,\dots ,{\overline{z}}_N)$ is a critical point of $F_{N,y}$, we see that equation $\left(E_{a_i}\right)$ is equivalent to

$$\begin{array}{cc}\hfill \left(F_{a_i}^{\prime }\right)D^{2}K\left(y\right)({\zeta }_i,.)& -(n-2)\sum _{j\ne i}\frac{{\zeta }_j-{\zeta }_i}{|{\overline{z}}_j-{\overline{z}}_i{|}^n}+n(n-2)\sum _{j\ne i}〈\frac{{\overline{z}}_j-{\overline{z}}_i}{|{\overline{z}}_j-{\overline{z}}_i{|}^2},{\zeta }_j-{\zeta }_i〉\hfill \\ \hfill & -(n-2)\sum _{j\ne i}\frac{{\overline{z}}_j-{\overline{z}}_i}{|{\overline{z}}_j-{\overline{z}}_i{|}^n}\left(\frac{n-6}{4}{\wedge }_i+\frac{n-2}{4}{\wedge }_j\right)=O\left(R_4\right),\hfill \end{array}$$

which completes the proof of Lemma 7. □

Now, we are ready to prove our results related to the construction of clustered bubbling solutions.

Proof of Theorem 3 

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

$$\begin{array}{cc}\hfill \frac{1}{2}{D}^2{F}_{N,y}({\overline{z}}_1,\dots ,{\overline{z}}_N)({\zeta }_1,\dots ,{\zeta }_N)& -(n-2)\sum _{j\ne i}\frac{{\overline{z}}_j-{\overline{z}}_i}{|{\overline{z}}_j-{\overline{z}}_i{|}^n}\left(\frac{n-6}{4}{\wedge }_i+\frac{n-2}{4}{\wedge }_j\right)\hfill \\ \hfill & =O\left(\sum _{j=1}^N{\wedge }_j^2+\sum _{j=1}^N|{\zeta }_j{|}^2+{\epsilon }^{\gamma }+{\epsilon }^{\frac{1}{2}-\gamma }{|ln\epsilon |}^{4/3}\right).\hfill \end{array}$$

As in the proof of Theorem 1, we defined a linear map by taking the left hand side of the system defined by $\left(F_{{\alpha }_i}^{\prime }\right)$, $\left(F_{{\lambda }_i}^{\prime }\right)$, and $\left(F_{a_i}^{\prime }\right)$. Since $({\overline{z}}_1,\dots ,{\overline{z}}_N)$ is a non-degenerate critical point of $F_{N,y}$, we deduced that such a linear map is invertible and arguing as in the proof of Theorem 1, we derive that the system $\left(S^{\prime }\right)$ has a solution $({\beta }_{\epsilon },{\wedge }_{\epsilon },{\zeta }_{\epsilon })$ for $\epsilon$ small. This implies that $\left({\mathcal{P}}_{\epsilon }\right)$ admits a solution $u_{\epsilon ,y}={\sum }_{i=1}^N{\alpha }_{i,\epsilon }{\delta }_{a_{i,\epsilon },{\lambda }_{i,\epsilon }}+v_{\epsilon }.$ and by construction, properties (4)–(7) are satisfied. The proof of Theorem 3 is thereby completed. □

Proof of Theorem 4 

For $y_1$, $y_2$ two non-degenerate critical points of K, we remark that, for $a_{1,\epsilon }$, $a_{2,\epsilon }\in$ such that $|a_{1,\epsilon }-y_1|<<1$ and $|a_{2,\epsilon }-y_2|<<1$, we see that the interaction between the bubbles ${\delta }_{a_{1,\epsilon },{\lambda }_{1,\epsilon }}$ and ${\delta }_{a_{2,\epsilon },{\lambda }_{1,\epsilon }}$ is of the order of $\frac{1}{{\left({\lambda }_{1,\epsilon }{\lambda }_{2,\epsilon }\right)}^{(n-2)/2}}=O\left({\epsilon }^{(n-2))/2}\right)$ which is a negligible term in front of ${\epsilon }^{3/2}$ for $n\ge 6$. This implies that the interaction goes into the remainder term and thus, we may separate each pack alone. Hence, arguing as in the proof of Theorem 3 and taking a new system $(\left(S_1\right)$, ..., $\left(S_m\right))$ with each $\left(S_i\right)$ represents the system studied in the proof of Theorem 3, and the proof of the theorem follows. □