1 Introduction and Main Results

In this paper, we construct the solutions for the following nonlinear Schrödinger system

$$\begin{aligned} {\left\{ \begin{array}{ll} -\epsilon ^{2}\Delta u+P(x)u= \mu _{1} u^{p}+\beta u^{\frac{p-1}{2}}v^{\frac{p+1}{2}} \ \ \ \text {in} \ \ \mathbb {R}^{N},\\ -\epsilon ^{2}\Delta v+Q(x)v= \mu _{2} v^{p}+\beta u^{\frac{p+1}{2}}v^{\frac{p-1}{2}} \ \ \ \text {in} \ \ \mathbb {R}^{N}, \end{array}\right. } \end{aligned}$$
(1.1)

where \(\epsilon >0\) is a small parameter, \(5\le p<+\infty \), \(N\in \{1,2\}\), and the potentials P, Q satisfy \(0<P_{0} \le P(x)\le P_{1}\), Q(x), respectively \(0<Q_{0} \le Q(x)\le Q_{1}\).

The use of the Lyapunov–Schmidt reduction method to construct solutions for the nonlinear Schrödinger equation attracted much attention in the last decade, starting from the pioneering contribution by Floer and Weinstein [11]. Noussair and Yan [21] considered multi-peak solutions for the following problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -\epsilon ^{2} \Delta u+u= Q(x) |u|^{q-2}u,\ \ x \in \mathbb {R}^{N},\\ u\in H^{1}(\mathbb {R}^{N}), \end{array}\right. } \end{aligned}$$
(1.2)

when \(x_{0}\) is a local maximum point of Q(x) and \(\epsilon \) is sufficiently small, they proved that for each positive integer k, problem (1.2) has a positive solution with k-peaks concentrating near \(x_{0}\). Wei and Yan [32] studied the following nonlinear Schroödinger equation

$$\begin{aligned} {\left\{ \begin{array}{ll} - \Delta u+V(|x|)u= u^{p},\ u>0 \ \ x \in \mathbb {R}^{N},\\ u\in H^{1}(\mathbb {R}^{N}), \end{array}\right. } \end{aligned}$$
(1.3)

when \(1<p<(N+2)/(N-2)\) and V(|x|) is positive function with following expansion

$$\begin{aligned} V(|x|)=V_{0}+\frac{a}{r^{m}}+O\left( \frac{1}{r^{m+\theta }}\right) \ \text {as}\ r\rightarrow +\infty . \end{aligned}$$

They proved that problem (1.3) has infinitely many non-radial positive solutions, whose energy can be made arbitrarily large. For more results about the nonlinear Schrödinger equation, we refer the reader to [1, 6, 8,9,10, 12] and the references therein.

Inspired by the work of [21, 32], when \(N=3,\ p=3,\ \epsilon =1\), Peng and Wang [23] considered system (1.1), where P(x) and Q(x) satisfy the following hypotheses:

(P):

There are constants \(a\in \mathbb {R}\), \(m>1,\) and \(\theta >0\) such that as \(r\rightarrow +\infty \)

$$\begin{aligned} P(|x|) = 1 + \frac{a}{|x|^m} + O\left( \frac{1}{|x|^{m+\theta }} \right) , \quad {\text{ as }} \quad |x| \rightarrow \infty . \end{aligned}$$
(H1)
(Q):

There are constants \(b\in \mathbb {R}\), \(n>1,\) and \(\epsilon >0\) such that as \(r\rightarrow +\infty \)

$$\begin{aligned} Q(|x|) = 1 + \frac{b}{|x|^n} + O\left( \frac{1}{|x|^{n+\epsilon }} \right) , \quad {\text{ as }} \quad |x| \rightarrow \infty . \end{aligned}$$
(H2)

By using the number of the bumps of the solutions as a parameter, for the repulsive case, they constructed non-radial positive vector solutions of segregated type, for the attractive case, they constructed non-radial positive vector solutions of synchronized type. When \(N=3, p=3\), Peng and Pi [22] constructed k interacting spikes for u near the local maximum point \(x_{0}\) of P(x) and m interacting spikes for v near the local maximum point \(\overline{x}_{0}\) of Q(x), respectively, when \(x_{0}\ne \overline{x}_{0}\). Tang and Xie in [27] constructed synchronized positive vector solutions for \(\epsilon \) small. Since it seems to be difficult to provide a complete list of references, we just refer the readers to [2,3,4, 7, 13, 14, 17,18,19,20, 22,23,26, 28,29,30, 33, 34] and the references therein.

In this paper, we have been inspired by the analysis developed in [21,22,23, 31, 32] for scalar nonlinear elliptic equations (systems), in particular, by the ideas introduced by Noussair and Yan [21] to deal with nonlinear elliptic equations. Compared with the single scalar equation, we encounter some new difficulties in estimates due to the nonlinear coupling. Firstly, we need to establish non-degenerate results for the solutions of the coupled system, which will be used to prove the invertibility of the operator for the repulsive case. The difficulty is that we need to give an exact integral estimate for the coupled term and the system is more complicated than single equations. We point out that the sign of \(\beta \) has great influence on the structure of the solutions. Roughly speaking, for the repulsive case, the solutions are small perturbations of (U, V), where (U, V) are scaling and translation of the solution of \(-\Delta u+\lambda u=u^{p}\). For the attractive case, the solutions are small perturbations of \((U_{\mu },U_{\nu })\), where \(U_{\mu }\) are scaling and translation of the solution of \(-\Delta u+\lambda u=\mu u^{p}\) and \(U_{\nu }\) are scaling and translation of the solution of \(-\Delta u+\lambda u=\nu u^{p}\).

In this paper, we examine how potentials and the interspecies scattering length \(\beta \) influence the structure of solutions to problem (1.1), which improves the results of [19, 22] for least energy solutions. We study the existence of high energy solutions to problem (1.1) and provide not only the locations of spikes, but also much finer information on the interaction of spikes. Furthermore, we prove the attractive phenomenon for \(\beta <0\) and the repulsive phenomenon for \(\beta >0\).

Define the function space

$$\begin{aligned} {\mathcal {H}}=\bigg \{(u,v)\in H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N}):\int _{\mathbb {R}^{N}}P(x)u^{2}dx<+\infty ,\int _{\mathbb {R}^{N}}Q(x)v^{2}dx<+\infty \bigg \} \end{aligned}$$

endowed with the following norm:

$$\begin{aligned} \Vert (u,v)\Vert ^{2}=\langle (u,v),(u,v)\rangle =\Vert u\Vert ^{2}_{\epsilon ,P}+\Vert v\Vert ^{2}_{\epsilon ,Q},\ \forall \, (u,v)\in {\mathcal {H}}, \end{aligned}$$

where \(H^{1}=H^{1}(\mathbb {R}^{N})\) is the usual Sobolev space,

$$\begin{aligned} \Vert u\Vert _{\epsilon ,P}=\langle u,u\rangle _{\epsilon ,P}=\int _{\mathbb {R}^{N}}\left( \epsilon ^{2}|\nabla u|^{2}+P(x)u^{2}\right) dx \end{aligned}$$

and

$$\begin{aligned} \Vert v\Vert _{\epsilon ,P}=\langle v,v\rangle _{\epsilon ,Q}=\int _{\mathbb {R}^{N}}\left( \epsilon ^{2}|\nabla v|^{2}+Q(x)v^{2}\right) dx. \end{aligned}$$

Define

$$\begin{aligned} {\textbf{E}}_{\epsilon }=\bigg \{(\varphi ,\psi )\in H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N})&,\left\langle (\varphi ,\psi ),\left( \frac{\partial U_{\epsilon ,x_{\epsilon ,j}}}{\partial y_{i}},\frac{\partial V_{\epsilon ,x_{\epsilon ,j}}}{\partial y_{i}}\right) \right\rangle _{\epsilon }=0\\&j=1,2\cdots k;\ \ i=1,2\cdots N\bigg \}, \end{aligned}$$

where

$$\begin{aligned} \langle (u,v),(g,h)\rangle =\int _{\mathbb {R}^{N}}(\epsilon ^{2}\nabla u \nabla g+P(x)ug+\epsilon ^{2}\nabla v \nabla h+Q(x)vh)dx. \end{aligned}$$

Consider the following system

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+\lambda u= \mu _{1} u^{p}+\beta u^{\frac{p-1}{2}}v^{\frac{p+1}{2}} \ \ \ \text {in} \ \ \mathbb {R}^{N},\\ -\Delta v+\lambda v= \mu _{2} v^{p}+\beta u^{\frac{p+1}{2}}v^{\frac{p-1}{2}} \ \ \ \text {in} \ \ \mathbb {R}^{N}. \end{array}\right. } \end{aligned}$$
(1.4)

Let W be the unique solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+ \lambda u =u^{p},\ \text {in}\ \mathbb {R}^{N},\\ u>0 \ \text {in} \ \mathbb {R}^{N}, u(x)\rightarrow 0 \ \text {as} \ |x|\rightarrow +\infty . \end{array}\right. } \end{aligned}$$
(1.5)

Then

$$\begin{aligned} (U,V)=(k_{1}W,\tau _{0}k_{1}W) \end{aligned}$$

in \(H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N})\) is a solution of (1.4), where \(\lambda :=P(x_{0})=Q(x_{0})\) and \(\tau _{0}\) satisfies

$$\begin{aligned} \mu _{1}+\beta \tau ^{\frac{p+1}{2}}_{0}-\mu _{2}\tau ^{p-1}_{0}-\beta \tau ^{\frac{p-3}{2}}_{0}=0, \ k^{p-1}_{1}=\left( \mu _{1}+\beta \tau ^{\frac{p+1}{2}}_{0}\right) ^{-1}. \end{aligned}$$
(1.6)

Let

$$\begin{aligned} \left( U_{\epsilon ,x_{j,\epsilon }}(x),V_{\epsilon ,x_{j,\epsilon }}(x)\right) =\left( U\left( \frac{x-x_{j,\epsilon }}{\epsilon }\right) ,V\left( \frac{x-x_{j,\epsilon }}{\epsilon }\right) \right) . \end{aligned}$$

In the sequel, we will use \(\left( U_{\epsilon ,x_{j,\epsilon }}(x),V_{\epsilon ,x_{j,\epsilon }}(x)\right) \) to build up the solutions of (1.1).

To show our main results, we first recall some known results from [15]. In the following, let

$$\begin{aligned} 2^*={\left\{ \begin{array}{ll} +\infty ,\ {} &{} N=1,2,\\ \frac{2N}{N-2},&{}N\ge 3. \end{array}\right. } \end{aligned}$$

Proposition 1.1

Suppose that \(1<p<2^{*}-1,\ \mu _{1}>\mu _{2}>0,\ \beta >0\) and one of the following conditions holds

\((A_{1})\):

\(3<p<2^{*}-1,\ 0<\beta \le (\frac{p-1}{2})\mu _{1}\);

\((A_{2})\):

\(3<p<2^{*}-1,\ \mu _{1}\ge \frac{\mu _{2}}{2}(\frac{p+1}{p-1})^{\frac{p-1}{2}},\ \beta >(\frac{p-1}{2})\mu _{1}\);

\((A_{3})\):

\(3<p<2^{*}-1,\ \mu _{1}<\frac{\mu _{2}}{2}(\frac{p+1}{p-1})^{\frac{p-1}{2}},\ (\frac{p-1}{2})\mu _{1}\le \beta \le \beta _{0} \) or \(\beta \ge \max \{\beta _{1},(\frac{p-1}{2})\mu _{1}\}.\)

Then problem (1.4) has s positive solution \((U,V)=(k_{1}W,\tau _{0}k_{1}W)\) in \(H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N})\) which is non-degenerate, where \(\tau _{0}\) satisfies

$$\begin{aligned} \mu _{1}+\beta \tau ^{\frac{p+1}{2}}_{0}-\mu _{2}\tau ^{p-1}_{0}-\beta \tau ^{\frac{p-3}{2}}_{0}=0, \ k^{p-1}_{1}=\left( \mu _{1}+\beta \tau ^{\frac{p+1}{2}}_{0}\right) ^{-1}. \end{aligned}$$

Proposition 1.2

Suppose \(1< p<2^{*}-1\), \(\mu _{1}>\mu _{2}>0,\ \beta <0. \) Then there exists a decreasing sequence \(\{\beta _{k}\}\subset (-\sqrt{\mu _{1}\mu _{2}},0)\) such that for \(\beta \in (-\sqrt{\mu _{1}\mu _{2}},0){\setminus }\{\beta _{k}\}\), problem (1.4) has s positive solution \((U,V)=(k_{1}W,\tau _{0}k_{1}W)\) in \(H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N})\) which is non-degenerate, where \(\tau _{0}\) satisfies

$$\begin{aligned} \mu _{1}+\beta \tau ^{\frac{p+1}{2}}_{0}-\mu _{2}\tau ^{p-1}_{0}-\beta \tau ^{\frac{p-3}{2}}_{0}=0, \ k^{p-1}_{1}=\left( \mu _{1}+\beta \tau ^{\frac{p+1}{2}}_{0}\right) ^{-1}. \end{aligned}$$

Remark 1.1

Although the authors in [15] dealt with the existence and the non-degeneracy of positive solutions for the fractional Schrödinger system, the main results are still true for the classical Schrödinger system with \(s=1.\)

The main results in this paper are stated in what follows.

Theorem 1.1

Assume the conditions in Propositions 1.1 or 1.2. If \(x_{0}\) is a local maximum point of P(x), Q(x) and \(P(x_{0})=Q(x_{0})\), then there exists \(\epsilon _{0}>0\) such that for any \(\epsilon \in (0,\epsilon _{0}]\), problem (1.1) has a solution of the form

$$\begin{aligned} u_{\epsilon }=\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}+\varphi _{\epsilon },\ \ v_{\epsilon }=\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}+\psi _{\epsilon }, \end{aligned}$$

for some \(x_{j,\epsilon }\in B_{\delta }(x_{0})\) and \(\Vert (\varphi _{\epsilon },\psi _{\epsilon })\Vert _{\epsilon }=O(\epsilon ^{\frac{N}{2}+1}).\) Moreover, as \(\epsilon \rightarrow 0,\) \(x_{j,\epsilon }\rightarrow x_{0},\ \frac{|x_{i,\epsilon }-x_{j,\epsilon }|}{\epsilon }\rightarrow +\infty \) if \(i\ne j. \)

Set \(\lambda =P(x_{0})\), \(\overline{\lambda }=Q(\overline{x}_{0})\), it is easy to see that \(U_{\lambda ,\mu }=\lambda ^{\frac{1}{p-1}}\mu ^{-\frac{1}{p-1}}W(\sqrt{\lambda }x)\) is a solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+ \lambda u =\mu u^{p},\ \text {in}\ \mathbb {R}^{N},\\ u>0 \ \text {in} \ \mathbb {R}^{N}, u(x)\rightarrow 0 \ \text {as} \ |x|\rightarrow +\infty , \end{array}\right. } \end{aligned}$$

and \(U_{\overline{\lambda },\nu }=\overline{\lambda }^{\frac{1}{p-1}}\nu ^{-\frac{1}{p-1}}W(\sqrt{\overline{\lambda }}x)\) is a solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+ \overline{\lambda } u =\nu u^{p},\ \text {in}\ \mathbb {R}^{N},\\ u>0 \ \text {in} \ \mathbb {R}^{N}, u(x)\rightarrow 0 \ \text {as} \ |x|\rightarrow +\infty . \end{array}\right. } \end{aligned}$$

Let

$$\begin{aligned} \left( U_{\epsilon ,x_{j},\mu }(x),U_{\epsilon ,z_{j},\nu }(x)\right) =\left( U_{\lambda _{j},\mu }\left( \frac{x-x_{j}}{\epsilon }\right) ,U_{\overline{\lambda }_{j},\nu }\left( \frac{x-z_{j}}{\epsilon }\right) \right) , \end{aligned}$$

where \(x_{j}\in B_{\delta }(x_{0}),\ z_{j}\in B_{\delta }(\overline{x}_{0})\),

$$\begin{aligned} \widetilde{{\textbf{E}}}_{\epsilon }=&\bigg \{(\varphi ,\psi )\in H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N}),\left\langle \varphi ,\frac{\partial U_{\epsilon ,x_{j},\mu }}{\partial x_{j,l}}\right\rangle _{\epsilon }=0,\\&\left\langle \psi ,\frac{\partial U_{\epsilon ,z_{j},\nu }}{\partial z_{j,l}}\right\rangle _{\epsilon }=0,\ j=1,2\ldots k;\ \ l=1,2\ldots N\bigg \}. \end{aligned}$$

We will use \(\left( U_{\epsilon ,x_{j},\mu }(x),U_{\epsilon ,z_{j},\nu }(x)\right) \) to build up the solutions of problem (1.1).

Theorem 1.2

Suppose that \(x_{0}\) is a local maximum point of P(x) and \(\overline{x}_{0}\) is a local maximum point of Q(x), with \(x_{0}\ne \overline{x}_{0}\). Then there exists \(\beta ^{*}>0\) depending on \(x_{0}\) and \(\overline{x}_{0}\) such that for all \(\beta <\beta ^{*}\), there exists \(\epsilon _{0}>0\) such that for any \(\epsilon \in (0,\epsilon _{0}]\), problem (1.1) has a solution of the form

$$\begin{aligned} u_{\epsilon }=\sum ^{k}_{j=1}U_{\epsilon ,x_{{j,\epsilon }},\mu }(x)+\overline{\varphi }_{\epsilon },\ \ v_{\epsilon }=\sum ^{m}_{j=1}U_{\epsilon ,z_{{j,\epsilon }},\nu }(x)+\overline{\psi }_{\epsilon } \end{aligned}$$

for some \(x_{{j,\epsilon }}\in B_{\delta }(x_{0})\), \(z_{{j,\epsilon }}\in B_{\delta }(\overline{x}_{0})\), \(\Vert (\overline{\varphi }_{\epsilon },\overline{\psi }_{\epsilon })\Vert _{\epsilon }=O(\epsilon ^{\frac{N}{2}+1}).\) Moreover, as \(\epsilon \rightarrow 0,\) \(x_{j,\epsilon }\rightarrow x_{0},\ \frac{|x_{i,\epsilon }-x_{j,\epsilon }|}{\epsilon }\rightarrow +\infty \) if \(i\ne j,\ i=1,2,\ldots k, \) and \(z_{j,\epsilon }\rightarrow \overline{x}_{0},\ \frac{|z_{i,\epsilon }-z_{j,\epsilon }|}{\epsilon }\rightarrow +\infty \) \(i\ne j,\ i=1,2,\ldots m. \)

Remark 1.2

The conditions in Theorem 1.1 ensure the existence and the non-degeneracy of positive solutions (U, V) to the related limit problem (1.4), hence the reduction procedure can be carried out successfully. The condition \(\beta <\beta ^{*}\) in Theorem 1.2 is necessary to prove that \(Q_{\epsilon }B_{\epsilon }\) (see (2.5)) is invertible and the inverse operator is bounded.

Remark 1.3

Compared with the well-studied case \(N=3,\ p=3\), in order to get an accurate error estimate and use the Contraction Mapping Theorem to prove that problem (2.54) has a unique solution, our situation is much more complicated. In this sense, we complement the main results established by Peng and Wang (Arch Ration Mech Anal 2013) and Peng and Pi (Discrete Contin Dyn Syst 2016), where the authors considered the case \(N=3, p=3\).

The paper is organized as follows. In Sect. 2, we introduce some preliminaries that will be used to prove Theorems 1.1–1.2. In Sect. 3, we prove Theorem 1.1. In Sect. 4, we prove Theorem 1.2. Finally, we give some elementary computations in Appendix A.

Throughout this paper, \(C, C_{i}, i=1,2,\ldots \) will always denote various generic positive constants, while O(t) and o(t) denote \(C_{1}\le \frac{|O(t)|}{|t|}\le C_{2}\) and \(\frac{|o(t)|}{|t|}\rightarrow 0\) as \(t\rightarrow 0,\) respectively.

2 Preliminary Results

We first give the definition of multi-peak solutions of system (1.1).

Definition 2.1

Let \(k\in \mathbb {N}\), \(1\le j\le k.\) We say that \((u_{\epsilon },v_{\epsilon })\) is k-peak solutions of system (1.1) concentrated at \(\{{x_{1},x_{2},\ldots ,x_{k}},\}\) if \((u_{\epsilon },v_{\epsilon })\) satisfies the following properties.

(i) \((u_{\epsilon },v_{\epsilon })\) has k local maximum points \(x_{j,\epsilon }\in \mathbb {R}^{N}, j=1,2,\ldots ,k\) satisfying

$$\begin{aligned} x_{j,\epsilon }\rightarrow x_{j}\ \text {as}\ \epsilon \rightarrow 0 \ \text { for each}\ j. \end{aligned}$$

(ii) For any given \(\tau >0\), there exists \(R\gg 1\), such that

$$\begin{aligned} |u_{\epsilon }(x)|\le \tau ,\ |v_{\epsilon }(x)|\le \tau \ \text {for}\ x\in \mathbb {R}^{N}\setminus \cup ^{k}_{j}B_{R\epsilon }(x_{j,\epsilon }). \end{aligned}$$

(iii) There exists \(C>0\) such that

$$\begin{aligned} \int _{\mathbb {R}^{N}}\epsilon ^{2}(|\nabla u_{\epsilon }|^{2}+|\nabla v_{\epsilon }|^{2})+u^{2}_{\epsilon }+v^{2}_{\epsilon }\le C\epsilon ^{N}. \end{aligned}$$

Let \(x_{0}\) be the local maximum points of P(x), Q(x) and \(P(x_{0})=Q(x_{0})\). We want to construct a solution \((u_{\epsilon },v_{\epsilon })\) of the following form

$$\begin{aligned} u_{\epsilon }=\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}+\varphi _{\epsilon },\ \ v_{\epsilon }=\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}+\psi _{\epsilon } \end{aligned}$$

where \( x_{j,\epsilon }\rightarrow x_{0}\) and \(\Vert (\varphi _{\epsilon },\psi _{\epsilon })\Vert ^{2}=o(\epsilon ^{N})\) as \(\epsilon \rightarrow 0\). Then, \((\varphi _{\epsilon },\psi _{\epsilon })\) satisfies the following equation

$$\begin{aligned} {\left\{ \begin{array}{ll} B_{\epsilon }(\varphi _{\epsilon },\psi _{\epsilon })+l_{\epsilon }=R_{\epsilon }(\varphi _{\epsilon },\psi _{\epsilon }),\ x\in \mathbb {R}^{N},\\ (\varphi _{\epsilon },\psi _{\epsilon })\in H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N}), \end{array}\right. } \end{aligned}$$
(2.1)

where \(B_{\epsilon }\) is a bounded linear operator in \(H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N})\), defined by

$$\begin{aligned}&\langle B_{\epsilon }(\varphi _{\epsilon },\psi _{\epsilon }),(g,h)\rangle \nonumber \\&\quad =\int _{\mathbb {R}^{N}}\left( \epsilon ^{2}\nabla \varphi _{\epsilon }\nabla g+P(x)\varphi _{\epsilon } g-p\mu \left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{p-1}\varphi _{\epsilon } g\right) dx\nonumber \\&\qquad +\int _{\mathbb {R}^{N}}\left( \epsilon ^{2}\nabla \psi _{\epsilon }\nabla h+Q(x)\psi _{\epsilon } h-p\nu \left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}\right) ^{p-1}\psi _{\epsilon } h\right) dx\nonumber \\&\qquad -\beta \int _{\mathbb {R}^{N}}\left( \frac{p-1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-3}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p+1}{2}}\varphi _{\epsilon }g \right. \nonumber \\&\qquad \left. +\frac{p-1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p+1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-3}{2}}\psi _{\epsilon }h\right) dx\nonumber \\&\qquad -\beta \int _{\mathbb {R}^{N}}\left( \frac{p+1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-1}{2}}\psi _{\epsilon }g\right. \nonumber \\&\qquad \left. +\frac{p+1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-1}{2}}\varphi _{\epsilon }h\right) dx \end{aligned}$$
(2.2)

for all \((g,h)\in H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N}).\) \(l_{\epsilon }\in H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N})\) satisfying

$$\begin{aligned}&\langle l_{\epsilon },(g,h)\rangle \nonumber \\ {}&\quad =\sum ^{k}_{j=1}\int _{\mathbb {R}^{N}}(P(x)-\lambda )U_{\epsilon ,x_{j,\epsilon }}gdx+\mu \int _{\mathbb {R}^{N}}(\sum ^{k}_{j=1}U^{p}_{\epsilon ,x_{j,\epsilon }}-\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{p}gdx\nonumber \\&\qquad +\sum ^{k}_{j=1}\int _{\mathbb {R}^{N}}(Q(x)-\lambda )V_{\epsilon ,x_{j,\epsilon }}hdx+\nu \int _{\mathbb {R}^{N}}(\sum ^{k}_{j=1}V^{p}_{\epsilon ,x_{j,\epsilon }}-\left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}\right) ^{p}hdx \nonumber \\&\qquad +\beta \int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}^{\frac{p-1}{2}}V_{\epsilon ,x_{j,\epsilon }}^{\frac{p+1}{2}}-\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p+1}{2}}\right) gdx\nonumber \\&\qquad +\beta \int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}^{\frac{p+1}{2}}V_{\epsilon ,x_{j,\epsilon }}^{\frac{p-1}{2}}-\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p+1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-1}{2}}\right) hdx,\nonumber \\ \end{aligned}$$
(2.3)

for all \((g,h)\in H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N}).\)

$$\begin{aligned}&\langle R_{\epsilon }(\varphi _{\epsilon },\psi _{\epsilon }), (g,h)\rangle \nonumber \\&\quad =\int _{\mathbb {R}^{N}}\left( \mu \left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}+\varphi _{\epsilon }\right) ^{p}+\beta \left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}+\varphi _{\epsilon }\right) ^{\frac{p-1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}+\psi _{\epsilon }\right) ^{\frac{p+1}{2}}\right) gdx\nonumber \\&\qquad +\int _{\mathbb {R}^{N}}\left( \nu \left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}+\psi _{\epsilon }\right) ^{p}+\beta \left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}+\varphi _{\epsilon }\right) ^{\frac{p+1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}+\psi _{\epsilon }\right) ^{\frac{p-1}{2}}\right) hdx\nonumber \\&\qquad -\mu \int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{p}gdx-\beta \int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p+1}{2}}gdx\nonumber \\&\qquad -\nu \int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}\right) ^{p}h-\beta \int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p+1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-1}{2}}hdx\nonumber \\&\qquad -\beta \int _{\mathbb {R}^{N}}\frac{p-1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-3}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p+1}{2}}\varphi _{\epsilon }gdx\nonumber \\&\qquad -\beta \int _{\mathbb {R}^{N}}\frac{p-1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p+1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-3}{2}}\psi _{\epsilon }hdx\nonumber \\&\qquad -\beta \int _{\mathbb {R}^{N}}\frac{p+1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-1}{2}}\psi _{\epsilon }gdx\nonumber \\&\qquad -\beta \int _{\mathbb {R}^{N}}\frac{p+1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-1}{2}}\varphi _{\epsilon }hdx, \end{aligned}$$
(2.4)

for all \((g,h)\in H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N}).\)

From [8, 9], we have following estimates.

Lemma 2.1

For any \(\alpha>0,\beta >0\) and \(l\ne j \), there exists a constant \(\tau >0\) such that

$$\begin{aligned} \int _{\mathbb {R}^{N}}U^{\alpha }_{\epsilon ,x_{j,\epsilon }}U^{\beta }_{\epsilon ,x_{l,\epsilon }}dx\le C\epsilon ^{N}e^{-\tau \frac{|x_{l,\epsilon }-x_{j,\epsilon }|}{\epsilon }}, \int _{\mathbb {R}^{N}}V^{\alpha }_{\epsilon ,x_{j,\epsilon }}V^{\beta }_{\epsilon ,x_{l,\epsilon }}dx\le C\epsilon ^{N}e^{-\tau \frac{|x_{l,\epsilon }-x_{j,\epsilon }|}{\epsilon }}, \end{aligned}$$
$$\begin{aligned} \int _{\mathbb {R}^{N}}U^{\alpha }_{\epsilon ,x_{j,\epsilon }}V^{\beta }_{\epsilon ,x_{l,,\epsilon }}dx\le C\epsilon ^{N}e^{-\tau \frac{|x_{l,\epsilon }-x_{j,\epsilon }|}{\epsilon }},\ \int _{\mathbb {R}^{N}}V^{\alpha }_{\epsilon ,x_{j,\epsilon }}U^{\beta }_{\epsilon ,x_{l,\epsilon }}dx\le C\epsilon ^{N}e^{-\tau \frac{|x_{l,\epsilon }-x_{j,\epsilon }|}{\epsilon }}. \end{aligned}$$

For any \((g,h)\in H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N}) \), we define the projection \(Q_{\epsilon }\) from \(H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N})\) to \({\textbf{E}}_{\epsilon }\) as follows:

$$\begin{aligned} Q_{\epsilon }(u,v)=(u,v)-\sum ^{k}_{j=1}\sum ^{N}_{i=1}b_{\epsilon ,i,j}\left( \frac{\partial U_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}},\frac{\partial V_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}\right) , \end{aligned}$$
(2.5)

where \(b_{\epsilon ,i,j}\) is chosen in such a way that

$$\begin{aligned} \left\langle Q_{\epsilon }(u,v),(\frac{\partial U_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}},\frac{\partial V_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}})\right\rangle _{\epsilon }=0,j=1,2\cdots k; i=1,2\cdots N. \end{aligned}$$

Therefore \(b_{\epsilon ,i,j}\) is determined by the following equations:

$$\begin{aligned}&\sum ^{k}_{j=1}\sum ^{N}_{i=1}b_{\epsilon ,i,j}\left\langle \left( \frac{\partial U_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}},\frac{\partial V_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}\right) ,\left( \frac{\partial U_{\epsilon ,x_{m,\epsilon }}}{\partial x_{l}},\frac{\partial V_{\epsilon ,x_{m,\epsilon }}}{\partial x_{l}}\right) \right\rangle _{\epsilon }\nonumber \\ {}&=\left\langle (u,v),\left( \frac{\partial U_{\epsilon ,x_{m,\epsilon }}}{\partial x_{l}},\frac{\partial V_{\epsilon ,x_{m,\epsilon }}}{\partial x_{l}}\right) \right\rangle _{\epsilon }\nonumber \\ {}&m=1,2\ldots k,l=1,2\ldots N. \end{aligned}$$
(2.6)

We now prove that problem (2.6) is solvable. Since \((U_{\epsilon ,x_{j,\epsilon }},V_{\epsilon ,x_{j,\epsilon }})\) satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} -\epsilon ^{2}\Delta U_{\epsilon ,x_{j,\epsilon }}+\lambda U_{\epsilon ,x_{j,\epsilon }}= \mu U_{\epsilon ,x_{j,\epsilon }}^{p}+\beta U_{\epsilon ,x_{j,\epsilon }}^{\frac{p-1}{2}}V_{\epsilon ,x_{j,\epsilon }}^{\frac{p+1}{2}} \ \ \ \text {in} \ \ \mathbb {R}^{N},\\ -\epsilon ^{2}\Delta V_{\epsilon ,x_{j,\epsilon }}+\lambda V_{\epsilon ,x_{j,\epsilon }}= \nu V_{\epsilon ,x_{j,\epsilon }}^{p}+\beta U_{\epsilon ,x_{j,\epsilon }}^{\frac{p+1}{2}}V_{\epsilon ,x_{j,\epsilon }}^{\frac{p-1}{2}} \ \ \ \text {in} \ \ \mathbb {R}^{N}, \end{array}\right. } \end{aligned}$$

one has

$$\begin{aligned} {\left\{ \begin{array}{ll} -\epsilon ^{2}\Delta \frac{\partial U_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}+\lambda \frac{\partial U_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}= p\mu U_{\epsilon ,x_{j,\epsilon }}^{p-1}\frac{\partial U_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}\\ +\beta \frac{p-1}{2} U_{\epsilon ,x_{j,\epsilon }}^{\frac{p-3}{2}}V_{\epsilon ,x_{j,\epsilon }}^{\frac{p+1}{2}}\frac{\partial U_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}+\beta \frac{p+1}{2} U_{\epsilon ,x_{j,\epsilon }}^{\frac{p-1}{2}}V_{\epsilon ,x_{j,\epsilon }}^{\frac{p-1}{2}}\frac{\partial V_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}} \ \ \ \text {in} \ \ \mathbb {R}^{N},\\ \\ -\epsilon ^{2}\Delta \frac{\partial V_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}+\lambda \frac{\partial V_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}= p\nu V_{\epsilon ,x_{j,\epsilon }}^{p-1}\frac{\partial V_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}\\ +\beta \frac{p+1}{2} U_{\epsilon ,x_{j,\epsilon }}^{\frac{p-1}{2}}V_{\epsilon ,x_{j,\epsilon }}^{\frac{p-1}{2}}\frac{\partial U_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}+\beta \frac{p-1}{2} U_{\epsilon ,x_{j,\epsilon }}^{\frac{p+1}{2}}V_{\epsilon ,x_{j,\epsilon }}^{\frac{p-3}{2}}\frac{\partial V_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}} \ \ \ \text {in} \ \ \mathbb {R}^{N}. \end{array}\right. } \end{aligned}$$
(2.7)

Therefore

$$\begin{aligned}&\left\langle (\frac{\partial U_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}},\frac{\partial V_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}),(\varphi ,\psi )\right\rangle _{\epsilon }\nonumber \\&\quad =\int _{\mathbb {R}^{N}}\left( \epsilon ^{2}\nabla \frac{\partial U_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}\nabla \varphi +P(x)\frac{\partial U_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}\varphi +\epsilon ^{2}\nabla \frac{\partial V_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}\nabla \psi +Q(x)\frac{\partial V_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}\psi \right) dx\nonumber \\&\quad =\int _{\mathbb {R}^{N}}\left( (P(x)-\lambda )\frac{\partial U_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}\varphi +p\mu U_{\epsilon ,x_{j,\epsilon }}^{p-1}\frac{\partial U_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}\varphi \right) \nonumber \\&\qquad +\int _{\mathbb {R}^{N}}\left( (Q(x)-\lambda )\frac{\partial V_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}\psi +p\nu V_{\epsilon ,x_{j,\epsilon }}^{p-1}\frac{\partial V_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}\psi \right) \nonumber \\&\qquad +\int _{\mathbb {R}^{N}}\left( \beta \frac{p-1}{2} U_{\epsilon ,x_{j,\epsilon }}^{\frac{p-3}{2}}V_{\epsilon ,x_{j,\epsilon }}^{\frac{p+1}{2}}\frac{\partial U_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}+\beta \frac{p+1}{2} U_{\epsilon ,x_{j,\epsilon }}^{\frac{p-1}{2}}V_{\epsilon ,x_{j,\epsilon }}^{\frac{p-1}{2}}\frac{\partial V_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}\right) \varphi dx\nonumber \\&\qquad +\int _{\mathbb {R}^{N}}\left( \beta \frac{p+1}{2} U_{\epsilon ,x_{j,\epsilon }}^{\frac{p-1}{2}}V_{\epsilon ,x_{j,\epsilon }}^{\frac{p-1}{2}}\frac{\partial U_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}+\beta \frac{p-1}{2} U_{\epsilon ,x_{j,\epsilon }}^{\frac{p+1}{2}}V_{\epsilon ,x_{j,\epsilon }}^{\frac{p-3}{2}}\frac{\partial V_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}\right) \psi dx. \end{aligned}$$
(2.8)

By Lemma 2.1, (2.8) and the symmetry of \(U_{\epsilon ,x_{j,\epsilon }}, V_{\epsilon ,x_{j,\epsilon }}\), we have

$$\begin{aligned} \left\langle \left( \frac{\partial U_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}},\frac{\partial V_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}\right) ,\left( \frac{\partial U_{\epsilon ,x_{j,\epsilon }}}{\partial x_{h}},\frac{\partial V_{\epsilon ,x_{j,\epsilon }}}{\partial x_{h}}\right) \right\rangle _{\epsilon }=\delta _{h,i}\epsilon ^{N-2}(c_{j}+o(1)), \end{aligned}$$
(2.9)
$$\begin{aligned}&\left\langle \left( \frac{\partial U_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}},\frac{\partial V_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}\right) ,\left( \frac{\partial U_{\epsilon ,x_{m,\epsilon }}}{\partial x_{h}},\frac{\partial V_{\epsilon ,x_{m,\epsilon }}}{\partial x_{h}}\right) \right\rangle _{\epsilon }\nonumber \\ {}&=O\left( e^{-\frac{1}{2}\sqrt{\lambda }\frac{|x_{m}-x_{j}|}{\epsilon }}\right) \epsilon ^{N-2},\quad j\ne m, \end{aligned}$$
(2.10)

where \(\delta _{h,i}=0\) if \(h\ne i\) and \(\delta _{i,i}=1\), \(c_{j}>0\) is a constant.

Hence (2.6) is solvable and we have the following estimate

$$\begin{aligned} |b_{\epsilon ,i,j}|&\le C\epsilon ^{2-N}\sum ^{k}_{j=1}\sum ^{N}_{i=1}\left\langle (u,v),\left( \frac{\partial U_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}},\frac{\partial V_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}\right) \right\rangle _{\epsilon }\nonumber \\&\le C\epsilon ^{2-N}\Vert (u,v)\Vert _{\epsilon }\Vert \left( \frac{\partial U_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}},\frac{\partial V_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}\right) \Vert _{\epsilon }\le C\epsilon ^{-\frac{N}{2}+1}\Vert (u,v)\Vert _{\epsilon }. \end{aligned}$$
(2.11)

In order to prove the invertibility of the operator \(Q_{\epsilon }B_{\epsilon }\), we use the following non-degenerate results for system (1.4), which can be found in [15].

Proposition 2.1

Under the conditions of Propositions 1.1 or 1.2, the system (1.4) has a positive non-degenerate solution for \((U_{\lambda },V_{\lambda })=(k_{1}W,k_{1}\tau _{0}W )\) in \(H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N})\) in the sense that the kernel is given by span \(\{(\theta (\beta )\frac{\partial W}{\partial y_{j}},\frac{\partial W}{\partial y_{j}})\mid j=1,2,\cdots N\},\) where \(\theta (\beta )\ne 0\), \(\tau _{0}\) satisfies \(\mu _{1}+\beta \tau ^{\frac{p+1}{2}}_{0}-\mu _{2}\tau ^{p-1}_{0}-\beta \tau ^{\frac{p-3}{2}}_{0}=0\), \(k^{p-1}_{1}=\left( \mu _{1}+\beta \tau ^{\frac{p+1}{2}}_{0}\right) ^{-1}\).

Proposition 2.2

Suppose \(1< p<2^{*}-1\), \(\mu _{1}>\mu _{2}>0,\ \beta <0, \) then there exists a decreasing sequence \(\{\beta _{k}\}\subset (-\sqrt{\mu _{1}\mu _{2}},0)\) such that for \(\beta \in (-\sqrt{\mu _{1}\mu _{2}},0){\setminus }\{\beta _{k}\}\), the system (1.4) has a positive solution \((U,V)=(k_{1}W,\tau _{0}k_{1}W)\) in \(H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N})\) which is non-degenerate, where \(\tau _{0}\) satisfies

$$\begin{aligned} \mu _{1}+\beta \tau ^{\frac{p+1}{2}}_{0}-\mu _{2}\tau ^{p-1}_{0}-\beta \tau ^{\frac{p-3}{2}}_{0}=0,\quad \ k^{p-1}_{1}=\left( \mu _{1}+\beta \tau ^{\frac{p+1}{2}}_{0}\right) ^{-1}. \end{aligned}$$

In order to carry out the reduction arguments, we give following key lemma.

Lemma 2.2

There exist \(\epsilon _{0},\ \theta _{0}>0,\ \rho >0,\) independent of \(x_{j},j=1,2,\cdots , k\) such that for any \(\epsilon \in (0,\epsilon _{0}]\) and \(x_{j}\in B_{\theta _{0}}(x_{0}),\) \(Q_{\epsilon }B_{\epsilon }(\varphi _{\epsilon },\psi _{\epsilon })\) is bijective in \({\textbf{E}}_{\epsilon }\). Moreover, it holds

$$\begin{aligned} \Vert Q_{\epsilon }B_{\epsilon }(\varphi _{\epsilon },\psi _{\epsilon })\Vert _{\epsilon }\ge \rho \Vert (\varphi _{\epsilon },\psi _{\epsilon })\Vert _{\epsilon },\ \text{ for } \text{ all } \ (\varphi _{\epsilon },\psi _{\epsilon })\in {\textbf{E}}_{\epsilon }. \end{aligned}$$

Proof

Suppose by contradiction that there are \(\epsilon _{n}\rightarrow 0,\ x_{\epsilon _{n},j}\rightarrow x_{0},\ (\varphi _{n},\psi _{n})\in {\textbf{E}}_{\epsilon _{n}}\) such that

$$\begin{aligned} \Vert Q_{\epsilon _{n}}B_{\epsilon _{n}}(\varphi _{n},\psi _{n})\Vert _{\epsilon _{n}}\le \frac{1}{n}\Vert (\varphi _{n},\psi _{n})\Vert _{\epsilon _{n}},\ \text{ for } \text{ all } \ (\varphi _{n},\psi _{n})\in {\textbf{E}}_{\epsilon _{n}}. \end{aligned}$$
(2.12)

We assume \(\Vert (\varphi _{n},\psi _{n})\Vert ^{2}_{\epsilon _{n}}=\epsilon ^{N}_{n}\). By (2.12), for any \((g,h)\in {\textbf{E}}_{\epsilon _{n}}\), we have

$$\begin{aligned}&\int _{\mathbb {R}^{N}}\left( \epsilon _{n}^{2}\nabla \varphi _{n}\nabla g+P(x)\varphi _{n} g-p\mu \left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}\varphi _{n} g\right) dx\nonumber \\&\qquad +\int _{\mathbb {R}^{N}}\left( \epsilon _{n}^{2}\nabla \psi _{n}\nabla h+Q(x)\psi _{n} h-p\nu \left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}\psi _{n} h\right) dx\nonumber \\&\qquad -\beta \int _{\mathbb {R}^{N}}\frac{p-1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-3}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p+1}{2}}\varphi _{n}gdx\nonumber \\&\qquad -\beta \int _{\mathbb {R}^{N}}\frac{p-1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p+1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-3}{2}}\psi _{n}hdx\nonumber \\&\qquad -\beta \int _{\mathbb {R}^{N}}\frac{p+1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-1}{2}}\psi _{n}gdx\nonumber \\&\qquad -\beta \int _{\mathbb {R}^{N}}\frac{p+1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-1}{2}}\varphi _{n}hdx\nonumber \\&\quad =\langle B_{\epsilon _{n}}(\varphi _{n},\psi _{n}),(g,h)\rangle _{\epsilon _{n}}=\langle Q_{\epsilon _{n}}B_{\epsilon _{n}}(\varphi _{n},\psi _{n}),(g,h)\rangle _{\epsilon _{n}}\nonumber \\&\quad =o(1)\Vert (\varphi _{n},\psi _{n})\Vert _{\epsilon _{n}}\Vert (g,h)\Vert _{\epsilon _{n}}=o(\epsilon ^{\frac{N}{2}}_{n})\Vert (g,h)\Vert _{\epsilon _{n}}. \end{aligned}$$
(2.13)

Taking \((g,h)=(\varphi _{n},\psi _{n})\) in (2.13), we have

$$\begin{aligned}&\int _{\mathbb {R}^{N}}\left( \epsilon _{n}^{2}|\nabla \varphi _{n}|^{2}+P(x)\varphi ^{2}_{n} -p\mu \left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}\varphi ^{2}_{n} \right) dx\nonumber \\&\quad +\int _{\mathbb {R}^{N}}\left( \epsilon _{n}^{2}|\nabla \psi _{n}|^{2}+Q(x)\psi ^{2}_{n} -p\nu \left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}\psi ^{2}_{n} \right) dx\nonumber \\&\quad -\beta \int _{\mathbb {R}^{N}}\frac{p-1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-3}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p+1}{2}}\varphi ^{2}_{n}dx\nonumber \\&\quad -\beta \int _{\mathbb {R}^{N}}\frac{p-1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p+1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-3}{2}}\psi ^{2}_{n}dx\nonumber \\&\quad -\beta \int _{\mathbb {R}^{N}}(p+1)\left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-1}{2}}\psi _{n}\varphi _{n}dx=o\left( \epsilon ^{N}_{n}\right) .\nonumber \\ \end{aligned}$$
(2.14)

By Hölder’s inequality and Young’s inequality, we have

$$\begin{aligned}&\beta (p+1)\int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-1}{2}}\psi _{n}\varphi _{n}dx\nonumber \\&\le |\beta |(p+1)\left( \int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}\varphi ^{2}_{n}dx\right) ^{\frac{1}{2}}\left( \int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}\psi ^{2}_{n}dx\right) ^{\frac{1}{2}}\nonumber \\&\le |\beta |\frac{p+1}{2}\left( \int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}\varphi ^{2}_{n}dx+\int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}\psi ^{2}_{n}dx\right) .\nonumber \\ \end{aligned}$$
(2.15)

Since \(U_{\epsilon _{n},x_{\epsilon _{n},j}}=\frac{1}{\tau _{0}}V_{\epsilon _{n},x_{\epsilon _{n},j}},\) we have

$$\begin{aligned}&\beta \frac{p-1}{2}\int _{\mathbb {R}^{N}}\left( \left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-3}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p+1}{2}}\varphi ^{2}_{n}\right. \nonumber \\&\qquad \left. +\left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p+1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-3}{2}}\psi ^{2}_{n}\right) dx\nonumber \\&\quad =\beta \frac{p-1}{2}\tau _{0}^{\frac{p+1}{2}}\int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}\varphi ^{2}_{n}dx+\beta \frac{p-1}{2}\left( \frac{1}{\tau _{0}}\right) ^{\frac{p+1}{2}}\nonumber \\&\qquad \times \int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}\psi ^{2}_{n}dx.\nonumber \\ \end{aligned}$$
(2.16)

On the other hand, we can take a large \(R>0\) such that

$$\begin{aligned}&\left( p\mu +|\beta |\frac{p+1}{2}+\beta \frac{p-1}{2}\tau _{0}^{\frac{p+1}{2}}\right) \left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}\nonumber \\ {}&\le \frac{1}{2}P(x) \ \ \text {in}\ \mathbb {R}^{N}\setminus \bigcup ^{k}_{j=1}B_{\epsilon _{n},R}(x_{\epsilon _{n},j}),\nonumber \\&\left( p\nu +|\beta |\frac{p+1}{2}+\beta \frac{p-1}{2}(\frac{1}{\tau _{0}})^{\frac{p+1}{2}}\right) \left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}\nonumber \\ {}&\le \frac{1}{2}Q(x) \ \ \ \text {in}\ \mathbb {R}^{N}\setminus \bigcup ^{k}_{j=1}B_{\epsilon _{n},R}(x_{\epsilon _{n},j}). \end{aligned}$$
(2.17)

Thus, combining relations (2.15), (2.16) and (2.17) with (2.14), we obtain

$$\begin{aligned} o(\epsilon ^{N}_{n})&=\int _{\mathbb {R}^{N}}\left( \epsilon _{n}^{2}|\nabla \varphi _{n}|^{2}+P(x)\varphi ^{2}_{n} -p\mu \left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}\varphi ^{2}_{n} \right) dx\nonumber \\&\quad +\int _{\mathbb {R}^{N}}\left( \epsilon _{n}^{2}|\nabla \psi _{n}|^{2}+Q(x)\psi ^{2}_{n} -p\nu \left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}\psi ^{2}_{n} \right) dx\nonumber \\&\quad -\beta \int _{\mathbb {R}^{N}}\frac{p-1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-3}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p+1}{2}}\varphi ^{2}_{n}\nonumber \\&\quad -\beta \int _{\mathbb {R}^{N}}\frac{p-1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p+1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-3}{2}}\psi ^{2}_{n}\nonumber \\&\quad -\beta \int _{\mathbb {R}^{N}}(p+1)\left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-1}{2}}\psi _{n}\varphi _{n}\nonumber \\&\ge \frac{1}{2}\Vert (\varphi _{n},\psi _{n})\Vert ^{2}_{\epsilon _{n}}-\left( p\mu +|\beta |\frac{p+1}{2}+\beta \frac{p-1}{2}\tau _{0}^{\frac{p+1}{2}}\right) \nonumber \\&\quad \times \int _{\bigcup ^{k}_{j=1}B_{\epsilon _{n},R}(x_{\epsilon _{n},j})}\left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}\varphi ^{2}_{n}dx\nonumber \\&\quad -\left( p\nu +|\beta |\frac{p+1}{2}+\beta \frac{p-1}{2}\left( \frac{1}{\tau _{0}}\right) ^{\frac{p+1}{2}}\right) \nonumber \\&\quad \times \int _{\bigcup ^{k}_{j=1}B_{\epsilon _{n},R}(x_{\epsilon _{n},j})}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}\psi ^{2}_{n}dx.\nonumber \\ \end{aligned}$$
(2.18)

Therefore,

$$\begin{aligned} \epsilon ^{N}_{n}&\le C\int _{\bigcup ^{k}_{j=1}B_{\epsilon _{n},R}\left( x_{\epsilon _{n},j}\right) }\left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}\varphi ^{2}_{n}dx\nonumber \\&\quad +\int _{\bigcup ^{k}_{j=1}B_{\epsilon _{n},R}(x_{\epsilon _{n},j})}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}\psi ^{2}_{n}dx+o\left( \epsilon ^{N}_{n}\right) \nonumber \\&\le C\sum ^{k}_{j=1}\int _{B_{\epsilon _{n},R}\left( x_{\epsilon _{n},j}\right) }\left( \varphi ^{2}_{n}+\psi ^{2}_{n}\right) dx+o\left( \epsilon ^{N}_{n}\right) .\nonumber \\ \end{aligned}$$
(2.19)

If we can prove that

$$\begin{aligned} \int _{B_{\epsilon _{n},R}(x_{\epsilon _{n},j})}(\varphi ^{2}_{n}+\psi ^{2}_{n})dx=o(\epsilon ^{N}_{n}),\quad \ j=1,2,\cdots ,k, \end{aligned}$$
(2.20)

we get a contradiction. For this purpose, we will discuss the local behaviors near each points \(x_{\epsilon _{n},m}\). We define

$$\begin{aligned} {\widetilde{\varphi }}_{n,m}(x)=\varphi _{n}(\epsilon _{n}x+x_{\epsilon _{n},m}), \end{aligned}$$
$$\begin{aligned} {\widetilde{\psi }}_{n,m}(x)=\psi _{n}(\epsilon _{n}x+x_{\epsilon _{n},m}), \end{aligned}$$

then

$$\begin{aligned}&\int _{\mathbb {R}^{N}}\left( |\nabla {\widetilde{\varphi }}_{n,m}(x) |^{2}+P(\epsilon _{n}x+x_{\epsilon _{n},m})|{\widetilde{\psi }}_{n,m}(x)|^{2}+|\nabla {\widetilde{\varphi }}_{n,m}(x) |^{2}\right. \nonumber \\&\qquad \left. +Q(\epsilon _{n}x+x_{\epsilon _{n},m})|{\widetilde{\psi }}_{n,m}(x)|^{2}\right) dx\nonumber \\&\quad =\epsilon ^{^{-N}}_{n}\int _{\mathbb {R}^{N}}(\epsilon ^{2}_{n}|\nabla \varphi _{n}(x) |^{2}+\lambda |\psi _{n}(x)|^{2}+\epsilon ^{2}_{n}|\nabla \varphi _{n}(x) |^{2}\nonumber \\ {}&\qquad +\lambda |\psi _{n}(x)|^{2})dx=o(1)\le C. \end{aligned}$$
(2.21)

Therefore,

$$\begin{aligned} ({\widetilde{\varphi }}_{n,m},{\widetilde{\psi }}_{n,m})\rightharpoonup (\varphi ,\psi )\ \text {weakly in}\ H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N}), \end{aligned}$$
$$\begin{aligned} ({\widetilde{\varphi }}_{n,m},{\widetilde{\psi }}_{n,m})\rightarrow (\varphi ,\psi )\ \text {strongly in}\ L_{loc}^{2}(\mathbb {R}^{N})\times L_{loc}^{2}(\mathbb {R}^{N}). \end{aligned}$$

Moreover, \((\varphi ,\psi )\) satisfies

$$\begin{aligned}&\int _{\mathbb {R}^{N}}\nabla \frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}\nabla \varphi +\lambda \frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}\varphi \nonumber \\&\quad +\int _{\mathbb {R}^{N}}\nabla \frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}\nabla \psi +\lambda \frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}\psi =0,\quad \ l=1,2,\ldots N. \end{aligned}$$
(2.22)

To prove (2.20), we only need to show that \((\varphi ,\psi )=(0,0)\). Remark that relation (2.13) holds just for \((g,h)\in {\textbf{E}}_{\epsilon _{n}} \) not for all \((g,h)\in H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N})\). For \((g,h)\in H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N})\), we take

$$\begin{aligned} Q_{\epsilon _{n}}(g,h)=(g,h)-\sum ^{k}_{j=1}\sum ^{N}_{i=1}b_{\epsilon _{n},i,j}\left( \frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},j}}}{\partial x_{i}},\frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},j}}}{\partial x_{i}}\right) \in {\textbf{E}}_{\epsilon _{n}}. \end{aligned}$$
(2.23)

Then

$$\begin{aligned} b_{\epsilon _{n},h,m}=\sum ^{k}_{j=1}\sum ^{N}_{i=1}a_{\epsilon _{n},i,j}\left\langle (g,h),\left( \frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},j}}}{\partial x_{i}},\frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},j}}}{\partial x_{i}}\right) \right\rangle _{\epsilon }, \end{aligned}$$

for some constant \(a_{\epsilon ,i,j}\). From (2.23), we have

$$\begin{aligned}&\int _{\mathbb {R}^{N}}(\epsilon _{n}^{2}\nabla \varphi _{n}\nabla g+P(x)\varphi _{n} g-p\mu \left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}\varphi _{n} g)dx\nonumber \\ {}&\qquad +\int _{\mathbb {R}^{N}}\left( \epsilon _{n}^{2}\nabla \psi _{n}\nabla h+Q(x)\psi _{n} h-p\nu \left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}\psi _{n} h\right) dx\nonumber \\&\qquad -\beta \int _{\mathbb {R}^{N}}\frac{p-1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-3}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p+1}{2}}\varphi _{n}gdx\nonumber \\&\qquad -\beta \int _{\mathbb {R}^{N}}\frac{p-1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p+1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-3}{2}}\psi _{n}hdx\nonumber \\&\qquad -\beta \int _{\mathbb {R}^{N}}\frac{p+1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-1}{2}}\psi _{n}gdx\nonumber \\&\qquad -\beta \int _{\mathbb {R}^{N}}\frac{p+1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-1}{2}}\varphi _{n}hdx\nonumber \\&\quad =\langle B_{\epsilon _{n}}(\varphi _{n},\psi _{n}),(g,h)\rangle _{\epsilon _{n}}=\langle B_{\epsilon _{n}}(\varphi _{n},\psi _{n}),Q_{\epsilon _{n}}(g,h)\rangle _{\epsilon _{n}}\nonumber \\&\qquad +\sum ^{k}_{j=1}\sum ^{N}_{i=1}b_{\epsilon _{n},i,j}\left\langle B_{\epsilon _{n}}(\varphi _{n},\psi _{n}),\left( \frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},j}}}{\partial x_{i}},\frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},j}}}{\partial x_{i}}\right) \right\rangle _{\epsilon _{n}}.\nonumber \\ \end{aligned}$$
(2.24)

Since

$$\begin{aligned} \langle B_{\epsilon _{n}}(\varphi _{n},\psi _{n}),Q_{\epsilon _{n}}(g,h)\rangle _{\epsilon _{n}}&=\langle Q_{\epsilon _{n}}B_{\epsilon _{n}}(\varphi _{n},\psi _{n}),Q_{\epsilon _{n}}(g,h)\rangle _{\epsilon _{n}}\nonumber \\&=o(1)\Vert (\varphi _{n},\psi _{n})\Vert _{\epsilon _{n}}\Vert Q_{\epsilon _{n}}(g,h)\Vert _{\epsilon _{n}}=o\left( \epsilon ^{\frac{N}{2}}_{n}\right) \Vert (g,h)\Vert _{\epsilon _{n}}, \end{aligned}$$
(2.25)
$$\begin{aligned}&\sum ^{k}_{j=1}\sum ^{N}_{i=1}b_{\epsilon _{n},i,j}\left\langle B_{\epsilon _{n}}(\varphi _{n},\psi _{n}),\left( \frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},j}}}{\partial x_{i}},\frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},j}}}{\partial x_{i}}\right) \right\rangle _{\epsilon _{n}}\nonumber \\&\quad =\sum ^{k}_{j=1}\sum ^{N}_{i=1}a_{\epsilon ,i,j}\left\langle (g,h),\left( \frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},j}}}{\partial x_{i}},\frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},j}}}{\partial x_{i}}\right) \right\rangle _{\epsilon _{n}}\nonumber \\&\qquad \times \left\langle Q_{\epsilon _{n}}B_{\epsilon _{n}}(\varphi _{n},\psi _{n}),\left( \frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},j}}}{\partial x_{i}},\frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},j}}}{\partial x_{i}}\right) \right\rangle _{\epsilon _{n}}\nonumber \\&\qquad +\sum ^{k}_{j=1}\sum ^{N}_{i=1}c_{\epsilon ,i,j}\left\langle (g,h),\left( \frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},j}}}{\partial x_{i}},\frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},j}}}{\partial x_{i}}\right) \right\rangle _{\epsilon _{n}}\nonumber \\&\qquad \times \left\langle (\frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},j}}}{\partial x_{i}},\frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},j}}}{\partial x_{i}}),(\frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},j}}}{\partial x_{i}},\frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},j}}}{\partial x_{i}})\right\rangle _{\epsilon _{n}}\nonumber \\&\quad =\sum ^{k}_{j=1}\sum ^{N}_{i=1}\gamma _{n,i,j}\left\langle (g,h),\left( \frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},j}}}{\partial x_{i}},\frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},j}}}{\partial x_{i}}\right) \right\rangle _{\epsilon _{n}}. \end{aligned}$$
(2.26)

Substitute (2.25), (2.26) into (2.24), we obtain

$$\begin{aligned}&\int _{\mathbb {R}^{N}}(\epsilon _{n}^{2}\nabla \varphi _{n}\nabla g+P(x)\varphi _{n} g-p\mu \left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}\varphi _{n} g)dx\nonumber \\&\qquad +\int _{\mathbb {R}^{N}}(\epsilon _{n}^{2}\nabla \psi _{n}\nabla h+Q(x)\psi _{n} h-p\nu \left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}\psi _{n} h)dx\nonumber \\&\qquad -\beta \int _{\mathbb {R}^{N}}\frac{p-1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-3}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p+1}{2}}\varphi _{n}gdx\nonumber \\&\qquad -\beta \int _{\mathbb {R}^{N}}\frac{p-1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p+1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-3}{2}}\psi _{n}hdx\nonumber \\&\qquad -\beta \int _{\mathbb {R}^{N}}\frac{p+1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-1}{2}}\psi _{n}gdx\nonumber \\&\qquad -\beta \int _{\mathbb {R}^{N}}\frac{p+1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-1}{2}}\varphi _{n}hdx\nonumber \\&\quad =o(\epsilon ^{\frac{N}{2}}_{n})\Vert (g,h)\Vert _{\epsilon _{n}}+\sum ^{k}_{j=1}\sum ^{N}_{i=1}\gamma _{n,i,j}\left\langle (g,h),\left( \frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},j}}}{\partial x_{i}},\frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},j}}}{\partial x_{i}}\right) \right\rangle _{\epsilon _{n}}. \end{aligned}$$
(2.27)

By (2.27) and \((\varphi _{n},\psi _{n})\in {\textbf{E}}_{\epsilon _{n}}\), we can estimate \(\gamma _{n,i,j}\) as following

$$\begin{aligned}&\sum ^{k}_{j=1}\sum ^{N}_{i=1}\gamma _{n,i,j}\left\langle \left( \frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},j}}}{\partial x_{i}},\frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},j}}}{\partial x_{i}}\right) ,\left( \frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}},\frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}\right) \right\rangle _{\epsilon _{n}}+o(\epsilon ^{N-1}_{n})\nonumber \\&\quad =-\int _{\mathbb {R}^{N}}p\mu \left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}\varphi _{n} \frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}})dx\nonumber \\&\qquad -\int _{\mathbb {R}^{N}}p\nu \left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}\psi _{n} \frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}})dx\nonumber \\&\qquad -\beta \int _{\mathbb {R}^{N}}\frac{p-1}{2}(\sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}})^{\frac{p-3}{2}}(\sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}})^{\frac{p+1}{2}}\varphi _{n}\frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}dx\nonumber \\&\qquad -\beta \int _{\mathbb {R}^{N}}\frac{p-1}{2}(\sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}})^{\frac{p+1}{2}}(\sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}})^{\frac{p-3}{2}}\psi _{n}\frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}dx\nonumber \\&\qquad -\beta \int _{\mathbb {R}^{N}}\frac{p+1}{2}(\sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}})^{\frac{p-1}{2}}(\sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}})^{\frac{p-1}{2}}\psi _{n}\frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}dx\nonumber \\&\qquad -\beta \int _{\mathbb {R}^{N}}\frac{p+1}{2}(\sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}})^{\frac{p-1}{2}}(\sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}})^{\frac{p-1}{2}}\varphi _{n}\frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}dx\nonumber \\&\quad =:A_{1}+A_{2}+A_{3}+A_{4}+A_{5}+A_{6}. \end{aligned}$$
(2.28)

On the other hand, from (2.8) and \((\varphi _{n},\psi _{n})\in {\textbf{E}}_{\epsilon _{n}} \), we have

$$\begin{aligned}&\int _{\mathbb {R}^{N}}p\mu (U_{\epsilon _{n},x_{\epsilon _{n},m}})^{p-1}\varphi _{n} \frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}})dx+\int _{\mathbb {R}^{N}}p\nu (V_{\epsilon _{n},x_{\epsilon _{n},m}})^{p-1}\psi _{n} \frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}})dx\nonumber \\&\qquad +\beta \int _{\mathbb {R}^{N}}\frac{p-1}{2}(U_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p-3}{2}}(V_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p+1}{2}}\varphi _{n}\frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}\nonumber \\&\qquad +\beta \int _{\mathbb {R}^{N}}\frac{p-1}{2}(U_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p+1}{2}}(V_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p-3}{2}}\psi _{n}\frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}\nonumber \\&\qquad +\beta \int _{\mathbb {R}^{N}}\frac{p+1}{2}(U_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p-1}{2}}(V_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p-1}{2}}\psi _{n}\frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}\nonumber \\&\qquad +\beta \int _{\mathbb {R}^{N}}\frac{p+1}{2}(U_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p-1}{2}}(V_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p-1}{2}}\varphi _{n}\frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}\nonumber \\&\quad =\int _{\mathbb {R}^{N}}\left( (\lambda -P(x))\frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}\varphi _{n}+(\lambda -Q(x))\frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}\psi _{n}\right) \nonumber \\&\quad =O\left( \left( \int _{\mathbb {R}^{N}} (\lambda -P(x))^{2}\left( \frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}\right) ^{2}\right) ^{\frac{1}{2}}\right) \Vert (\varphi _{n},\psi _{n})\Vert _{\epsilon }\nonumber \\&\qquad +O\left( \left( \int _{\mathbb {R}^{N}} (\lambda -Q(x))^{2}\left( \frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}\right) ^{2}\right) ^{\frac{1}{2}}\right) \Vert (\varphi _{n},\psi _{n})\Vert _{\epsilon }\nonumber \\&\quad =\varepsilon ^{\frac{N}{2}}O\left( \left( \int _{\mathbb {R}^{N}} (\lambda -P(\epsilon x+x_{\epsilon _{n},m}))^{2}\left( \frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},m}}(\varepsilon x+x_{\epsilon _{n},m})}{\partial x_{l}}\right) ^{2}\right) ^{\frac{1}{2}}\right) \nonumber \\&\qquad \Vert (\varphi _{n},\psi _{n})\Vert _{\epsilon }\nonumber \\&\qquad +\varepsilon ^{\frac{N}{2}}O\left( \left( \int _{\mathbb {R}^{N}} (\lambda -Q(\epsilon x+x_{\epsilon _{n},m}))^{2}\left( \frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},m}}(\epsilon x+x_{\epsilon _{n},m})}{\partial x_{l}}\right) ^{2}\right) ^{\frac{1}{2}}\right) \nonumber \\&\qquad \Vert (\varphi _{n},\psi _{n})\Vert _{\epsilon }\nonumber \\&\quad = \epsilon ^{N}_{n}O(|P(x_{\epsilon _{n},m})-\lambda |+|Q(x_{\epsilon _{n},m})-\lambda |+\epsilon _{n})=o(\epsilon ^{N-1}_{n}). \end{aligned}$$
(2.29)

There is a constant \(\sigma >0\) such that

$$\begin{aligned} \left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}-\left( U_{\epsilon _{n},x_{\epsilon _{n},m}}\right) ^{p-1}=O\left( \sum ^{k}_{j\ne m}U_{\epsilon _{n},x_{\epsilon _{n},j}}^{\sigma }\right) , \end{aligned}$$
(2.30)
$$\begin{aligned}&\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}-\left( V_{\epsilon _{n},x_{\epsilon _{n},m}}\right) ^{p-1}\nonumber \\&\quad =\tau _{0}^{p-1}\left( \left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}-\left( U_{\epsilon _{n},x_{\epsilon _{n},m}}\right) ^{p-1}\right) =O\left( \sum ^{k}_{j\ne m}U_{\epsilon _{n},x_{\epsilon _{n},j}}^{\sigma }\right) , \end{aligned}$$
(2.31)
$$\begin{aligned}&\left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-3}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p+1}{2}}-\left( U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-3}{2}}(V_{\epsilon _{n},x_{\epsilon _{n},j}})^{\frac{p+1}{2}}\nonumber \\&\quad =\tau _{0}^{\frac{p+1}{2}}\left( \left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}-\left( U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}\right) =O\left( \sum ^{k}_{j\ne m}U_{\epsilon _{n},x_{\epsilon _{n},j}}^{\sigma }\right) , \end{aligned}$$
(2.32)
$$\begin{aligned}&\left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-1}{2}}-\left( U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-1}{2}}\left( V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-1}{2}}\nonumber \\&\quad =\tau _{0}^{\frac{p-1}{2}}\left( \left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}-\left( U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}\right) =O\left( \sum ^{k}_{j\ne m}U_{\epsilon _{n},x_{\epsilon _{n},j}}^{\sigma }\right) . \end{aligned}$$
(2.33)

By (2.30)–(2.33) and (2.29), we obtain

$$\begin{aligned} A_{1}&=-\int _{\mathbb {R}^{N}}p\mu \left( \left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}-\left( U_{\epsilon _{n},x_{\epsilon _{n},m}}\right) ^{p-1}\right) \varphi _{n} \frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}})dx\\&\quad -\int _{\mathbb {R}^{N}}p\mu (U_{\epsilon _{n},x_{\epsilon _{n},m}})^{p-1}\varphi _{n} \frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}dx\\&=O(e^{-\frac{\tau }{\epsilon _{n}}})\Vert (\varphi _{n},0)\Vert _{\epsilon _{n}}-\int _{\mathbb {R}^{N}}p\mu (U_{\epsilon _{n},x_{\epsilon _{n},m}})^{p-1}\varphi _{n} \frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}dx, \end{aligned}$$
$$\begin{aligned} A_{2}&=-\int _{\mathbb {R}^{N}}p\nu \left( \left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}-\left( V_{\epsilon _{n},x_{\epsilon _{n},m}}\right) ^{p-1}\right) \psi _{n} \frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}dx\\&\quad -\int _{\mathbb {R}^{N}}p\nu \left( V_{\epsilon _{n},x_{\epsilon _{n},m}}\right) ^{p-1}\psi _{n} \frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}dx\\&=O\left( e^{-\frac{\tau }{\epsilon _{n}}}\right) \Vert (0,\psi _{n})\Vert _{\epsilon _{n}}-\int _{\mathbb {R}^{N}}p\nu (V_{\epsilon _{n},x_{\epsilon _{n},m}})^{p-1}\psi _{n} \frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}dx, \end{aligned}$$
$$\begin{aligned} A_{3}&=-\frac{p-1}{2}\beta \int _{\mathbb {R}^{N}}\bigg (\sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}})^{\frac{p-3}{2}}(\sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}})^{\frac{p+1}{2}}\\&\quad -(U_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p-3}{2}}(V_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p+1}{2}}\bigg )\varphi _{n} \frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}dx\\&\quad -\frac{p-1}{2}\beta \int _{\mathbb {R}^{N}}(U_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p-3}{2}}(V_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p+1}{2}}\varphi _{n} \frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}dx\\&=O(e^{-\frac{\tau }{\epsilon _{n}}})\Vert (\varphi _{n},0)\Vert _{\epsilon _{n}}-\frac{p-1}{2}\beta \\ {}&\quad \times \int _{\mathbb {R}^{N}}(U_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p-3}{2}}(V_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p+1}{2}}\varphi _{n} \frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}dx, \end{aligned}$$
$$\begin{aligned} A_{4}&=-\frac{p-1}{2}\beta \int _{\mathbb {R}^{N}}\bigg ((\sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}})^{\frac{p+1}{2}}(\sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}})^{\frac{p-3}{2}}\\&\quad -(U_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p+1}{2}}(V_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p-3}{2}}\bigg )\psi _{n} \frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}dx\\&\quad -\frac{p-1}{2}\beta \int _{\mathbb {R}^{N}}(U_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p+1}{2}}(V_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p-3}{2}}\psi _{n} \frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}dx\\&=O(e^{-\frac{\tau }{\epsilon _{n}}})\Vert (0,\psi _{n})\Vert _{\epsilon _{n}}-\frac{p-1}{2}\beta \\ {}&\quad \times \int _{\mathbb {R}^{N}}(U_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p+1}{2}}(V_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p-3}{2}}\psi _{n} \frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}dx, \end{aligned}$$
$$\begin{aligned} A_{5}&=-\frac{p+1}{2}\beta \int _{\mathbb {R}^{N}}\bigg ((\sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}})^{\frac{p-1}{2}}(\sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}})^{\frac{p-1}{2}}\\&\quad -(U_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p-1}{2}}(V_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p-1}{2}}\bigg )\psi _{n} \frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}dx\\&\quad -\frac{p+1}{2}\beta \int _{\mathbb {R}^{N}}(U_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p-1}{2}}(V_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p-1}{2}}\psi _{n} \frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}dx\\&=O(e^{-\frac{\tau }{\epsilon _{n}}})\Vert (0,\psi _{n})\Vert _{\epsilon _{n}}-\frac{p+1}{2}\beta \\ {}&\quad \int _{\mathbb {R}^{N}}(U_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p-1}{2}}(V_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p-1}{2}}\psi _{n} \frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}dx, \end{aligned}$$
$$\begin{aligned} A_{6}&=-\frac{p+1}{2}\beta \int _{\mathbb {R}^{N}}\bigg ((\sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}})^{\frac{p-1}{2}}(\sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}})^{\frac{p-1}{2}}\\&\quad -(U_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p-1}{2}}(V_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p-1}{2}}\bigg )\varphi _{n} \frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}dx\\&\quad -\frac{p+1}{2}\beta \int _{\mathbb {R}^{N}}(U_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p-1}{2}}(V_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p-1}{2}}\varphi _{n} \frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}dx\\&=O(e^{-\frac{\tau }{\epsilon _{n}}})\Vert (\varphi _{n},0)\Vert _{\epsilon _{n}}-\frac{p+1}{2}\beta \\ {}&\quad \int _{\mathbb {R}^{N}}(U_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p-1}{2}}(V_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p-1}{2}}\varphi _{n} \frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}dx. \end{aligned}$$

Combining (2.28), (2.29) with above \(A_{1}\) to \(A_{6}\), we have

$$\begin{aligned}&\sum ^{k}_{j=1}\sum ^{N}_{i=1}\gamma _{n,i,j}\left\langle \left( \frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},j}}}{\partial x_{i}},\frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},j}}}{\partial x_{i}}\right) ,\left( \frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}},\frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}\right) \right\rangle _{\epsilon _{n}}=o\left( \epsilon ^{N-1}_{n}\right) . \end{aligned}$$
(2.34)

So, by (2.9) and (2.10), we have

$$\begin{aligned} \gamma _{n,i,j}=o(\epsilon _{n}). \end{aligned}$$
(2.35)

Thus, (2.27) becomes

$$\begin{aligned}&\int _{\mathbb {R}^{N}}\left( \epsilon _{n}^{2}\nabla \varphi _{n}\nabla g+P(x)\varphi _{n} g-p\mu \left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}\varphi _{n} g\right) dx\nonumber \\&\qquad +\int _{\mathbb {R}^{N}}\left( \epsilon _{n}^{2}\nabla \psi _{n}\nabla h+Q(x)\psi _{n} h-p\nu \left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{p-1}\psi _{n} h\right) dx\nonumber \\&\qquad -\beta \int _{\mathbb {R}^{N}}\frac{p-1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-3}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p+1}{2}}\varphi _{n}gdx\nonumber \\&\qquad -\beta \int _{\mathbb {R}^{N}}\frac{p-1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p+1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-3}{2}}\psi _{n}hdx\nonumber \\&\qquad -\beta \int _{\mathbb {R}^{N}}\frac{p+1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-1}{2}}\psi _{n}gdx\nonumber \\&\qquad -\beta \int _{\mathbb {R}^{N}}\frac{p+1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}\right) ^{\frac{p-1}{2}}\varphi _{n}hdx\nonumber \\&\quad =\Vert (\varphi _{n},\psi _{n})\Vert _{\epsilon _{n}}\Vert (g,h)\Vert _{\epsilon _{n}},\ \forall \ (g,h)\in H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N}). \end{aligned}$$
(2.36)

For any \((g,h)\in H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N})\), we let \(\widetilde{g}_{n}(x)=g(\frac{x-x_{\epsilon _{n},m}}{\epsilon _{n}})\). Using (2.36), we have

$$\begin{aligned}&\int _{\mathbb {R}^{N}}\left( \nabla {\widetilde{\varphi }}_{n,m}\nabla g+P(\epsilon _{n}x+x_{\epsilon _{n},m}){\widetilde{\varphi }}_{n,m} g\right. \nonumber \\ {}&\quad \left. -p\mu \left( \sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}(\epsilon _{n}x+x_{\epsilon _{n},m})\right) ^{p-1}{\widetilde{\varphi }}_{n,m} g\right) dx\nonumber \\&+\int _{\mathbb {R}^{N}}\left( \nabla {\widetilde{\psi }}_{n,m}\nabla h+Q(\epsilon _{n}x+x_{\epsilon _{n},m}){\widetilde{\psi }}_{n,m} h\right. \nonumber \\ {}&\quad \left. -p\nu \left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}(\epsilon _{n}x+x_{\epsilon _{n},m})\right) ^{p-1}{\widetilde{\psi }}_{n,m} h\right) dx\nonumber \\&-\beta \int _{\mathbb {R}^{N}}\frac{p-1}{2}\left( (\sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}(\epsilon _{n}x+x_{\epsilon _{n},m})\right) ^{\frac{p-3}{2}}\nonumber \\ {}&\quad \left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}(\epsilon _{n}x+x_{\epsilon _{n},m})\right) ^{\frac{p+1}{2}}{\widetilde{\varphi }}_{n,m}gdx \beta \int _{\mathbb {R}^{N}}\frac{p-1}{2}\nonumber \\&\quad \left( (\sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}(\epsilon _{n}x+x_{\epsilon _{n},m})\right) ^{\frac{p+1}{2}}\nonumber \\&\quad \left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}(\epsilon _{n}x+x_{\epsilon _{n},m})\right) ^{\frac{p-3}{2}}{\widetilde{\psi }}_{n,m}hdx\nonumber \\&-\beta \int _{\mathbb {R}^{N}}\frac{p+1}{2}\left( (\sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}(\epsilon _{n}x+x_{\epsilon _{n},m})\right) ^{\frac{p-1}{2}}\nonumber \\ {}&\quad \left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}(\epsilon _{n}x+x_{\epsilon _{n},m})\right) ^{\frac{p-1}{2}}{\widetilde{\psi }}_{n,m}gdx\nonumber \\&-\beta \int _{\mathbb {R}^{N}}\frac{p+1}{2}\nonumber \\ {}&\quad \left( (\sum ^{k}_{j=1}U_{\epsilon _{n},x_{\epsilon _{n},j}}(\epsilon _{n}x+x_{\epsilon _{n},m})\right) ^{\frac{p-1}{2}}\nonumber \\ {}&\quad \left( \sum ^{k}_{j=1}V_{\epsilon _{n},x_{\epsilon _{n},j}}(\epsilon _{n}x+x_{\epsilon _{n},m})\right) ^{\frac{p-1}{2}}{\widetilde{\varphi }}_{n,m}hdx\nonumber \\&=\epsilon ^{-N}_{n}\Vert (\varphi _{n},\psi _{n})\Vert _{\epsilon _{n}}\Vert (\widetilde{g}_{n},\widetilde{h}_{n})\Vert _{\epsilon _{n}}=o(1),\ \forall \ (g,h)\in H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N}). \end{aligned}$$
(2.37)

Therefore, \((\varphi ,\psi )\) satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta \varphi +\lambda \varphi - p\mu U_{\lambda }(x)^{p-1}\varphi -\beta \frac{p-1}{2}U_{\lambda }(x)^{\frac{p-3}{2}}V_{\lambda }(x)^{\frac{p+1}{2}}\varphi \\ -\beta \frac{p+1}{2}U_{\lambda }(x)^{\frac{p-1}{2}}V_{\lambda }(x)^{\frac{p-1}{2}}\psi =0 \ \ \ \text {in} \ \ \mathbb {R}^{N},\\ -\Delta \psi +\lambda \psi -p\nu V_{\lambda }(x)^{p-1}\psi -\beta \frac{p-1}{2}U_{\lambda }(x)^{\frac{p+1}{2}}V_{\lambda }(x)^{\frac{p-3}{2}}\psi \\ -\beta \frac{p+1}{2}U_{\lambda }(x)^{\frac{p-1}{2}}V_{\lambda }(x)^{\frac{p-1}{2}}\varphi =0 \ \ \ \text {in} \ \ \mathbb {R}^{N}. \end{array}\right. } \end{aligned}$$
(2.38)

From proposition 2.1, the solution of \((U_{\lambda },V_{\lambda })\) gives

$$\begin{aligned} \varphi =\sum ^{N}_{l=1}c_{l}\frac{\partial U_{\lambda }}{\partial x_{l}},\ \psi =\sum ^{N}_{l=1}d_{l}\frac{\partial V_{\lambda }}{\partial x_{l}}. \end{aligned}$$
(2.39)

On the other hand, from \((\varphi _{n},\psi _{n})\in {\textbf{E}}_{\epsilon _{n}} \) and (2.29), we have

$$\begin{aligned}&\int _{\mathbb {R}^{N}}p\mu (U_{\epsilon _{n},x_{\epsilon _{n},m}})^{p-1}\varphi _{n} \frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}dx+\int _{\mathbb {R}^{N}}p\nu (V_{\epsilon _{n},x_{\epsilon _{n},m}})^{p-1}\psi _{n} \frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}dx\nonumber \\&\qquad +\beta \int _{\mathbb {R}^{N}}\frac{p-1}{2}(U_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p-3}{2}}(V_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p+1}{2}}\varphi _{n}\frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}dx\nonumber \\&\qquad +\beta \int _{\mathbb {R}^{N}}\frac{p-1}{2}(U_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p+1}{2}}(V_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p-3}{2}}\psi _{n}\frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}dx\nonumber \\&\qquad +\beta \int _{\mathbb {R}^{N}}\frac{p+1}{2}(U_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p-1}{2}}(V_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p-1}{2}}\psi _{n}\frac{\partial U_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}dx\nonumber \\&\qquad +\beta \int _{\mathbb {R}^{N}}\frac{p+1}{2}(U_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p-1}{2}}(V_{\epsilon _{n},x_{\epsilon _{n},m}})^{\frac{p-1}{2}}\varphi _{n}\frac{\partial V_{\epsilon _{n},x_{\epsilon _{n},m}}}{\partial x_{l}}dx\nonumber \\&\quad =o(\epsilon ^{N-1}_{n}). \end{aligned}$$
(2.40)

Thus,

$$\begin{aligned}&\int _{\mathbb {R}^{N}}p\mu U_{\lambda }^{p-1}\varphi \frac{\partial U_{\lambda }}{\partial x_{l}}dx\nonumber \\ {}&\quad +\int _{\mathbb {R}^{N}}p\nu V_{\lambda }^{p-1}\psi _{n} \frac{\partial V_{\lambda }}{\partial x_{l}}dx +\beta \int _{\mathbb {R}^{N}}\frac{p-1}{2}U_{\lambda }^{\frac{p-3}{2}}V_{\lambda }^{\frac{p+1}{2}}\varphi \frac{\partial U_{\lambda }}{\partial x_{l}}dx\nonumber \\&\quad +\beta \int _{\mathbb {R}^{N}}\frac{p-1}{2}U_{\lambda }^{\frac{p+1}{2}}V_{\lambda }^{\frac{p-3}{2}}\psi \frac{\partial V_{\lambda }}{\partial x_{l}}dx+\beta \int _{\mathbb {R}^{N}}\frac{p+1}{2}U_{\lambda }^{\frac{p-1}{2}}V_{\lambda }^{\frac{p-1}{2}}\psi \frac{\partial U_{\lambda }}{\partial x_{l}}dx\nonumber \\&\quad +\beta \int _{\mathbb {R}^{N}}\frac{p+1}{2}U_{\lambda }^{\frac{p-1}{2}}V_{\lambda }^{\frac{p-1}{2}}\varphi \frac{\partial V_{\lambda }}{\partial x_{l}}dx=0. \end{aligned}$$
(2.41)

By (2.39), (2.41) and \(U_{\lambda }=\frac{1}{\tau _{0}}V_{\lambda },\) we have

$$\begin{aligned} c_{l}=d_{l}=0,\quad l=1,2,\ldots N. \end{aligned}$$

Thus,

$$\begin{aligned} (\varphi ,\psi )=(0,0). \end{aligned}$$

So, we have prove there exist \(\epsilon _{0},\theta _{0}>0,\rho >0,\) independent of \(x_{j},j=1,2,\cdots k\) such that for any \(\epsilon \in (0,\epsilon _{0}]\) and \(x_{j}\in B_{\theta _{0}}(y_{0}),\) \(Q_{\epsilon }B_{\epsilon }(\varphi _{\epsilon },\psi _{\epsilon })\) is bijective in \({\textbf{E}}_{\epsilon }\). Moreover, it holds

$$\begin{aligned} \Vert Q_{\epsilon }B_{\epsilon }(\varphi ,\psi )\Vert _{\epsilon }\ge \rho \Vert (\varphi ,\psi )\Vert _{\epsilon },\quad \ \ \forall \ (\varphi ,\psi )\in {\textbf{E}}_{\epsilon }. \end{aligned}$$

Thus the proof is complete. \(\square \)

Next, we give the error estimate for \(\Vert l_{\epsilon }\Vert _{\epsilon }\) and \(\Vert R_{\epsilon }(\varphi _{\epsilon },\psi _{\epsilon })\Vert _{\epsilon }\).

Lemma 2.3

There is a constant \(C>0\) independent of \(\epsilon \), such that

$$\begin{aligned}{} & {} \Vert l_{\epsilon }\Vert _{\epsilon } \le C\left( \epsilon ^{\frac{N+2}{2}}+\epsilon ^{\frac{N}{2}} \sum ^{k}_{j=1} (|P(x_{j,\epsilon })-\lambda |+|Q(x_{j,\epsilon })-\lambda |)\right) \\ {}{} & {} \quad + \epsilon ^{\frac{N}{2}}\sum _{i\ne j} e^{-\frac{\sqrt{\alpha }|x_{i,\epsilon }-x_{j,\epsilon }|}{\epsilon }}. \end{aligned}$$

Proof

Observe that

$$\begin{aligned}&\langle l_{\epsilon },(g,h)\rangle \\ {}&\quad = \sum ^{k}_{j=1}\int _{\mathbb {R}^{N}}(P(x)-\lambda )U_{\epsilon ,x_{j,\epsilon }}gdx+\mu \int _{\mathbb {R}^{N}}(\sum ^{k}_{j=1}U^{p}_{\epsilon ,x_{j,\epsilon }}-(\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }})^{p}gdx\\&\qquad +\sum ^{k}_{j=1}\int _{\mathbb {R}^{N}}(Q(x)-\lambda )V_{\epsilon ,x_{j,\epsilon }}hdx+\nu \int _{\mathbb {R}^{N}}(\sum ^{k}_{j=1}V^{p}_{\epsilon ,x_{j,\epsilon }}-(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }})^{p}hdx\\&\qquad +\beta \int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}^{\frac{p-1}{2}}V_{\epsilon ,x_{j,\epsilon }}^{\frac{p+1}{2}}-(\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }})^{\frac{p-1}{2}}(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }})^{\frac{p+1}{2}}\right) gdx\\&\qquad +\beta \int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}^{\frac{p+1}{2}}V_{\epsilon ,x_{j,\epsilon }}^{\frac{p-1}{2}}-(\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }})^{\frac{p+1}{2}}(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }})^{\frac{p-1}{2}}\right) hdx.\\ \end{aligned}$$

Firstly, by Hölder’s inequality, we have

$$\begin{aligned}&\sum ^{k}_{j=1}\int _{\mathbb {R}^{N}}(P(x)-\lambda )U_{\epsilon ,x_{j,\epsilon }}gdx+\sum ^{k}_{j=1}\int _{\mathbb {R}^{N}}(Q(x)-\lambda )V_{\epsilon ,x_{j,\epsilon }}hdx\nonumber \\&\quad \le C\sum ^{k}_{j=1}\left( \int _{\mathbb {R}^{N}}(P(x)-\lambda )^{2}U^{2}_{\epsilon ,x_{j,\epsilon }}dx\right) ^{\frac{1}{2}}\Vert g\Vert _{\epsilon }\nonumber \\ {}&\qquad +C\sum ^{k}_{j=1}\left( \int _{\mathbb {R}^{N}}(Q(x)-\lambda )^{2}V^{2}_{\epsilon ,x_{j,\epsilon }}dx\right) ^{\frac{1}{2}}\Vert h\Vert _{\epsilon }\nonumber \\&\quad \le C\sum ^{k}_{j=1}\left( \epsilon ^{N}\int _{\mathbb {R}^{N}}(P(\epsilon x+x_{j,\epsilon })-\lambda )^{2}U^{2}_{\epsilon ,x_{j,\epsilon }}(\epsilon x+x_{j,\epsilon })dx\right) ^{\frac{1}{2}}\Vert g\Vert _{\epsilon }\nonumber \\&\qquad +C\sum ^{k}_{j=1}\left( \epsilon ^{N}\int _{\mathbb {R}^{N}}(Q(\epsilon x+x_{j,\epsilon })-\lambda )^{2}V^{2}_{\epsilon ,x_{j,\epsilon }}(\epsilon x+x_{j,\epsilon })dx\right) ^{\frac{1}{2}}\Vert h\Vert _{\epsilon }\nonumber \\&\quad \le C\sum ^{k}_{j=1}\epsilon ^{\frac{N}{2}}\left( \int _{\mathbb {R}^{N}}\left( \left( P(x_{j,\epsilon })-\lambda )\right) +\epsilon ^{2}|x|^{2})^{2}U^{2}_{\epsilon ,x_{j,\epsilon }}(\epsilon x+x_{j,\epsilon }\right) dx\right) ^{\frac{1}{2}}\Vert g\Vert _{\epsilon }\nonumber \\&\qquad + C\sum ^{k}_{j=1}\epsilon ^{\frac{N}{2}}\left( \int _{\mathbb {R}^{N}}\left( \left( Q(x_{j,\epsilon })-\lambda )\right) +\epsilon ^{2}|x|^{2})^{2}V^{2}_{\epsilon ,x_{j,\epsilon }}(\epsilon x+x_{j,\epsilon }\right) dx\right) ^{\frac{1}{2}}\Vert h\Vert _{\epsilon }\nonumber \\&\quad \le C\left( \epsilon ^{\frac{N+2}{2}}+\sum ^{k}_{j=1}\epsilon ^{\frac{N}{2}}\left( |P(x_{j,\epsilon })-\lambda |+|Q(x_{j,\epsilon })-\lambda |\right) \right) \Vert (g,h)\Vert _{\epsilon }. \end{aligned}$$
(2.42)

Since \(U_{\epsilon ,x_{j,\epsilon }}(x)\le C e^{-\frac{\sqrt{\alpha }|x-x_{j,\epsilon }|}{\epsilon }},V_{\epsilon ,x_{j,\epsilon }}(x)\le C e^{-\frac{\sqrt{\alpha }|x-x_{j,\epsilon }|}{\epsilon }}\), we have if \(p\ge 3\), then

$$\begin{aligned}&\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{p}-\sum ^{k}_{j=1}U^{p}_{\epsilon ,x_{j,\epsilon }}\nonumber \\ {}&\quad =p\sum ^{k}_{i\ne j}U^{p-1}_{\epsilon ,x_{i}}U_{\epsilon ,x_{j,\epsilon }}+O\left( \sum ^{k}_{i\ne j}U^{p-2}_{\epsilon ,x_{i}}U^{2}_{\epsilon ,x_{j}}+\sum ^{k}_{i\ne j}U^{p-2}_{\epsilon ,x_{j}}\right) \nonumber \\ {}&=O\left( \sum ^{k}_{i\ne j}e^{-\frac{\sqrt{\alpha }|x-x_{j,\epsilon }|}{\epsilon }}\right) , \ \forall \ x\in B_{\theta }(x_{j,\epsilon }), \end{aligned}$$
(2.43)
$$\begin{aligned}&\left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}\right) ^{p}-\sum ^{k}_{j=1}V^{p}_{\epsilon ,x_{j,\epsilon }} =p\sum ^{k}_{i\ne j}V^{p-1}_{\epsilon ,x_{i}}V_{\epsilon ,x_{j,\epsilon }}\nonumber \\ {}&\quad +O\left( \sum ^{k}_{i\ne j}V^{p-2}_{\epsilon ,x_{i}}V^{2}_{\epsilon ,x_{j}}+\sum ^{k}_{i\ne j}V^{p-2}_{\epsilon ,x_{j}}\right) \nonumber \\ {}&=O\left( \sum ^{k}_{i\ne j}e^{-\frac{\sqrt{\alpha }|x-x_{j,\epsilon }|}{\epsilon }}\right) , \ \forall \ x\in B_{\theta }(x_{j,\epsilon }), \end{aligned}$$
(2.44)
$$\begin{aligned}&\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p+1}{2}}-\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}^{\frac{p-1}{2}}V_{\epsilon ,x_{j,\epsilon }}^{\frac{p+1}{2}}\nonumber \\&\quad =\left( \frac{b_{\lambda }}{a_{\lambda }}\right) ^{\frac{p+1}{2}}\left( (\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }})^{p}-\sum ^{k}_{j=1}U^{p}_{\epsilon ,x_{j,\epsilon }}\right) \nonumber \\ {}&\quad =O\left( \sum ^{k}_{i\ne j}e^{-\frac{\sqrt{\alpha }|x-x_{j,\epsilon }|}{\epsilon }}\right) , \ \forall \ x\in B_{\theta }(x_{j,\epsilon }), \end{aligned}$$
(2.45)
$$\begin{aligned}&\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p+1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-1}{2}}-\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}^{\frac{p+1}{2}}V_{\epsilon ,x_{j,\epsilon }}^{\frac{p-1}{2}}\nonumber \\&\quad =\left( \frac{b_{\lambda }}{a_{\lambda }}\right) ^{\frac{p-1}{2}}\left( (\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }})^{p}-\sum ^{k}_{j=1}U^{p}_{\epsilon ,x_{j,\epsilon }}\right) \nonumber \\ {}&\quad =O\left( \sum ^{k}_{i\ne j}e^{-\frac{\sqrt{\alpha }|x-x_{j,\epsilon }|}{\epsilon }}\right) ,\quad \ \forall \ x\in B_{\theta }(x_{j,\epsilon }). \end{aligned}$$
(2.46)

From (2.43) to (2.46), we have

$$\begin{aligned}&\mu \int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}U^{p}_{\epsilon ,x_{j,\epsilon }}-(\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }})^{p}\right) gdx+\nu \int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}V^{p}_{\epsilon ,x_{j,\epsilon }}-(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }})^{p}\right) hdx\nonumber \\&\quad +\beta \int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}^{\frac{p-1}{2}}V_{\epsilon ,x_{j,\epsilon }}^{\frac{p+1}{2}}-(\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }})^{\frac{p-1}{2}}(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }})^{\frac{p+1}{2}}\right) gdx\nonumber \\&\quad +\beta \int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}^{\frac{p+1}{2}}V_{\epsilon ,x_{j,\epsilon }}^{\frac{p-1}{2}}-(\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }})^{\frac{p+1}{2}}(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }})^{\frac{p-1}{2}}\right) hdx\nonumber \\&\quad \le C \epsilon ^{\frac{N}{2}}\sum _{i\ne j} e^{-\frac{\sqrt{\alpha }|x_{i,\epsilon }-x_{j,\epsilon }|}{\epsilon }}\Vert (g,h)\Vert _{\epsilon }. \end{aligned}$$
(2.47)

So, by (2.42) and (2.47), we obtain

$$\begin{aligned} \Vert l_{\epsilon }\Vert _{\epsilon }\le & {} C\left( \epsilon ^{\frac{N+2}{2}}+\epsilon ^{\frac{N}{2}} \sum ^{k}_{j=1} (|P(x_{j,\epsilon })-\lambda |+|Q(x_{j,\epsilon })-\lambda |)\right) \\ {}{} & {} + \epsilon ^{\frac{N}{2}}\sum _{i\ne j} e^{-\frac{\sqrt{\alpha }|x_{i,\epsilon }-x_{j,\epsilon }|}{\epsilon }}. \end{aligned}$$

This completes the proof. \(\square \)

Lemma 2.4

There is a constant \(C>0\) independent of \(\epsilon \), such that

$$\begin{aligned} \Vert R_{\epsilon }(\varphi ,\psi )\Vert _{\epsilon }\le C\left( \epsilon ^{-\frac{N}{2}}\Vert (\varphi ,\psi )\Vert ^{2}_{\epsilon }+ \epsilon ^{-N}\Vert (\varphi ,\psi )\Vert ^{3}_{\epsilon }+ \epsilon ^{-\frac{3N}{2}}\Vert (\varphi ,\psi )\Vert ^{4}_{\epsilon }\right) . \end{aligned}$$

Proof

Since

$$\begin{aligned}&\langle R_{\epsilon }(\varphi ,\psi ), (g,h)\rangle =B_{1}+B_{2}+B_{3}-B_{4}-B_{5}, \end{aligned}$$
(2.48)

where

$$\begin{aligned} B_{1}=\int _{\mathbb {R}^{N}}\left( \mu (\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}+\varphi )^{p}-\mu (\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }})^{p}\right) gdx, \end{aligned}$$
$$\begin{aligned} B_{2}=\int _{\mathbb {R}^{N}}\left( \nu (\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}+\psi )^{p}-\nu (\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }})^{p}\right) hdx, \end{aligned}$$
$$\begin{aligned} B_{3}=&\int _{\mathbb {R}^{N}}\beta (\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}+\varphi )^{\frac{p-1}{2}}(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}+\psi )^{\frac{p+1}{2}}gdx\\&+\int _{\mathbb {R}^{N}}\beta (\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}+\varphi )^{\frac{p+1}{2}}(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}+\psi )^{\frac{p-1}{2}}hdx,\\ \end{aligned}$$
$$\begin{aligned} B_{4}=&\beta \int _{\mathbb {R}^{N}}\left( (\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }})^{\frac{p-1}{2}}(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }})^{\frac{p+1}{2}}\right) gdx\\&+\beta \int _{\mathbb {R}^{N}}\left( (\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }})^{\frac{p+1}{2}}(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }})^{\frac{p-1}{2}}\right) hdx, \end{aligned}$$
$$\begin{aligned} B_{5}&=\beta \int _{\mathbb {R}^{N}}\frac{p-1}{2}(\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }})^{\frac{p-3}{2}}(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }})^{\frac{p+1}{2}}\varphi gdx\\&\quad +\beta \int _{\mathbb {R}^{N}}\frac{p-1}{2}(\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }})^{\frac{p+1}{2}}(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }})^{\frac{p-3}{2}}\psi hdx\\&\quad +\beta \int _{\mathbb {R}^{N}}\frac{p+1}{2}(\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }})^{\frac{p-1}{2}}(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }})^{\frac{p-1}{2}}\psi gdx\\&\quad +\beta \int _{\mathbb {R}^{N}}\frac{p+1}{2}(\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }})^{\frac{p-1}{2}}(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }})^{\frac{p-1}{2}}\varphi hdx.\\ \end{aligned}$$

For any \(\xi ,\) we let \({\widetilde{\xi }}(y)=\xi (\epsilon y)\), then

$$\begin{aligned} \int _{\mathbb {R}^{N}}|\xi |^{p+1}dx&=\epsilon ^{N}\int _{\mathbb {R}^{N}}|{\widetilde{\xi }}|^{p+1}dx\le C \epsilon ^{N}\left( \int _{\mathbb {R}^{N}}(|\nabla {\widetilde{\xi }}|^{2}+|{\widetilde{\xi }}|^{2})dx\right) ^{\frac{p+1}{2}}\nonumber \\&\le C\epsilon ^{N(1-\frac{p+1}{2})}\left( \int _{\mathbb {R}^{N}}(\epsilon ^{2}|\nabla \xi |^{2}+|\xi |^{2})dx\right) ^{\frac{p+1}{2}}\nonumber \\&\le C\epsilon ^{N(1-\frac{p+1}{2})}\Vert \xi \Vert ^{p+1}_{\epsilon }. \end{aligned}$$
(2.49)

When \(p\ge 3\), by (2.49) and the Hölder inequality, we have

$$\begin{aligned} B_{1}&\le \mu \int _{\mathbb {R}^{N}}\left( p(p-1)(\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }})^{p-2}\varphi ^{2}+o(|\varphi |^{p-1})\right) gdx\nonumber \\&\le C\left( \int _{\mathbb {R}^{N}}(\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }})^{p+1}dx\right) ^{\frac{p-2}{p+1}}\left( \int _{\mathbb {R}^{N}}|\varphi |^{p+1}dx\right) ^{\frac{2}{p+1}}\left( \int _{\mathbb {R}^{N}}|g|^{p+1}dx\right) ^{\frac{1}{p+1}}\nonumber \\&\le C\epsilon ^{N\times \frac{p-2}{p+1}}\epsilon ^{N(1-\frac{p+1}{2})\frac{3}{p+1}}\Vert \varphi \Vert ^{2}_{\epsilon }\Vert g\Vert _{\epsilon }\le C \epsilon ^{-\frac{N}{2}}\Vert \varphi \Vert ^{2}_{\epsilon }\Vert g\Vert _{\epsilon }. \end{aligned}$$
(2.50)

Similarly, we have

$$\begin{aligned} B_{2}=\int _{\mathbb {R}^{N}}\left( \nu (\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}+\varphi )^{p}-\nu (\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }})^{p}\right) hdx \le C \epsilon ^{-\frac{N}{2}}\Vert \psi \Vert ^{2}_{\epsilon }\Vert h\Vert _{\epsilon }. \end{aligned}$$
(2.51)

Thus

$$\begin{aligned} B_{1}+B_{2} \le C \epsilon ^{-\frac{N}{2}}\Vert (\varphi ,\psi )\Vert ^{2}_{\epsilon }\Vert (g,h)\Vert _{\epsilon }. \end{aligned}$$
(2.52)

We expand \(B_{3}\) as following

$$\begin{aligned} B_{3}&=\int _{\mathbb {R}^{N}}\beta (\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}+\varphi )^{\frac{p-1}{2}}(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}+\psi )^{\frac{p+1}{2}}gdx\nonumber \\&\quad +\int _{\mathbb {R}^{N}}\beta (\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}+\varphi )^{\frac{p+1}{2}}(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}+\psi )^{\frac{p-1}{2}}hdx\nonumber \\&=B_{4}+B_{5}+C_{1}+C_{2}+C_{3}+C_{4}+C_{5}+C_{6}+o\bigg (\int _{\mathbb {R}^{N}}(\varphi ^{2}+\psi ^{2})gdx\bigg )\nonumber \\&\quad +o\bigg (\int _{\mathbb {R}^{N}}(\varphi ^{2}+\psi ^{2})hdx\bigg ), \end{aligned}$$
(2.53)

where

$$\begin{aligned} C_{1}&=\beta \frac{p+1}{2}\frac{p-1}{2}\int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-3}{2}}\psi ^{2}gdx\\&\quad +\beta \frac{p+1}{2}\frac{p-1}{2}\int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-3}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-1}{2}}\varphi ^{2}hdx, \end{aligned}$$
$$\begin{aligned} C_{2}&=\beta \frac{p+1}{2}\frac{p-1}{2}\int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-3}{2}}(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }})^{\frac{p-1}{2}}\varphi \psi gdx\\&\quad +\beta \frac{p+1}{2}\frac{p-1}{2}\int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-1}{2}}(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }})^{\frac{p-3}{2}}\varphi \psi hdx, \end{aligned}$$
$$\begin{aligned} C_{3}&=\beta \frac{p+1}{2}(\frac{p-1}{2})^{2}\int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-3}{2}}(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }})^{\frac{p-3}{2}}\varphi \psi ^{2}gdx\\&\quad +\beta \frac{p+1}{2}(\frac{p-1}{2})^{2}\int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-3}{2}}(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }})^{\frac{p-3}{2}}\varphi ^{2}\psi hdx, \end{aligned}$$
$$\begin{aligned} C_{4}&=\beta \frac{p-1}{2}\frac{p-3}{2}\int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-5}{2}}(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }})^{\frac{p+1}{2}}\varphi ^{2}gdx\\&\quad +\beta \frac{p-1}{2}\frac{p-3}{2}\int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p+1}{2}}(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }})^{\frac{p-5}{2}}\psi ^{2}hdx, \end{aligned}$$
$$\begin{aligned} C_{5}&=\beta \frac{p+1}{2}\frac{p-1}{2}\frac{p-3}{2}\int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-5}{2}}(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }})^{\frac{p-1}{2}}\varphi ^{2}\psi gdx\\&\quad +\beta \frac{p+1}{2}\frac{p-1}{2}\frac{p-3}{2}\int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-1}{2}}(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }})^{\frac{p-5}{2}}\varphi \psi ^{2}hdx, \end{aligned}$$
$$\begin{aligned} C_{6}&=\beta \frac{p+1}{2}\frac{p-3}{2}(\frac{p-1}{2})^{2}\int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-5}{2}}(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }})^{\frac{p-3}{2}}\varphi ^{2}\psi ^{2}gdx\\&\quad +\beta \frac{p+1}{2}\frac{p-3}{2}(\frac{p-1}{2})^{2}\int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-3}{2}}(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }})^{\frac{p-5}{2}}\varphi ^{2}\psi ^{2}hdx. \end{aligned}$$

Since \(V_{\epsilon ,x_{j,\epsilon }}=\tau _{0}U_{\epsilon ,x_{j,\epsilon }}\), we have

$$\begin{aligned} C_{1}&\le C\left( \int _{\mathbb {R}^{N}}(\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }})^{p+1}dx\right) ^{\frac{p-2}{p+1}}\left( \int _{\mathbb {R}^{N}}|\psi |^{p+1}dx\right) ^{\frac{2}{p+1}}\left( \int _{\mathbb {R}^{N}}|g|^{p+1}dx\right) ^{\frac{1}{p+1}}\\&\quad + C\left( \int _{\mathbb {R}^{N}}(\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }})^{p+1}dx\right) ^{\frac{p-2}{p+1}}\left( \int _{\mathbb {R}^{N}}|\varphi |^{p+1}dx\right) ^{\frac{2}{p+1}}\left( \int _{\mathbb {R}^{N}}|h|^{p+1}dx\right) ^{\frac{1}{p+1}}\\&\le C\epsilon ^{N\times \frac{p-2}{p+1}}\epsilon ^{N(1-\frac{p+1}{2})\frac{3}{p+1}}\Vert (\varphi ,\psi )\Vert ^{2}_{\epsilon }\Vert (g,h)\Vert _{\epsilon }\le C \epsilon ^{-\frac{N}{2}}\Vert (\varphi ,\psi )\Vert ^{2}_{\epsilon }\Vert (g,h)\Vert _{\epsilon }. \end{aligned}$$

Similarly, we have

$$\begin{aligned}&C_{2}\le C \epsilon ^{-\frac{N}{2}}\Vert (\varphi ,\psi )\Vert ^{2}_{\epsilon }\Vert (g,h)\Vert _{\epsilon },\\&C_{4}\le C \epsilon ^{-\frac{N}{2}}\Vert (\varphi ,\psi )\Vert ^{2}_{\epsilon }\Vert (g,h)\Vert _{\epsilon }. \end{aligned}$$
$$\begin{aligned} C_{3}&\le C\left( \int _{\mathbb {R}^{N}}(\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }})^{p+1}dx\right) ^{\frac{p-3}{p+1}}\\ {}&\quad \left( \int _{\mathbb {R}^{N}}|\psi |^{p+1}dx\right) ^{\frac{2}{p+1}}\left( \int _{\mathbb {R}^{N}}|\varphi |^{p+1}dx\right) ^{\frac{1}{p+1}}\left( \int _{\mathbb {R}^{N}}|g|^{p+1}\right) ^{\frac{1}{p+1}}\\&\quad + C\left( \int _{\mathbb {R}^{N}}(\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }})^{p+1}dx\right) ^{\frac{p-3}{p+1}}\\ {}&\quad \left( \int _{\mathbb {R}^{N}}|\varphi |^{p+1}dx\right) ^{\frac{2}{p+1}}\left( \int _{\mathbb {R}^{N}}|\psi |^{p+1}dx\right) ^{\frac{1}{p+1}}\left( \int _{\mathbb {R}^{N}}|h|^{p+1}dx\right) ^{\frac{1}{p+1}}\\&\le C \epsilon ^{-N}\Vert (\varphi ,\psi )\Vert ^{3}_{\epsilon }\Vert (g,h)\Vert _{\epsilon }. \end{aligned}$$

By the similar argument as \(C_{3}\), we have

$$\begin{aligned} C_{5}\le C \epsilon ^{-N}\Vert (\varphi ,\psi )\Vert ^{3}_{\epsilon }\Vert (g,h)\Vert _{\epsilon }. \end{aligned}$$

We also have

$$\begin{aligned} C_{6}&\le C\left( \int _{\mathbb {R}^{N}}(\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }})^{p+1}dx\right) ^{\frac{p-4}{p+1}}\\ {}&\quad \left( \int _{\mathbb {R}^{N}}|\psi |^{p+1}dx\right) ^{\frac{2}{p+1}}\left( \int _{\mathbb {R}^{N}}|\varphi |^{p+1}dx\right) ^{\frac{2}{p+1}}\left( \int _{\mathbb {R}^{N}}|g|^{p+1}dx\right) ^{\frac{1}{p+1}}\\&\quad + C\left( \int _{\mathbb {R}^{N}}(\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }})^{p+1}dx\right) ^{\frac{p-4}{p+1}}\\ {}&\quad \left( \int _{\mathbb {R}^{N}}|\varphi |^{p+1}dx\right) ^{\frac{2}{p+1}}\left( \int _{\mathbb {R}^{N}}|\psi |^{p+1}dx\right) ^{\frac{2}{p+1}}\left( \int _{\mathbb {R}^{N}}|h|^{p+1}dx\right) ^{\frac{1}{p+1}}\\&\le C \epsilon ^{-\frac{3N}{2}}\Vert (\varphi ,\psi )\Vert ^{4}_{\epsilon }\Vert (g,h)\Vert _{\epsilon }. \end{aligned}$$

Combining (2.50)–(2.53), the estimates for \(C_{1}\)–\(C_{6}\) and (2.48), we obtain

$$\begin{aligned} \Vert R_{\epsilon }(\varphi ,\psi )\Vert _{\epsilon }\le C\left( \epsilon ^{-\frac{N}{2}}\Vert (\varphi ,\psi )\Vert ^{2}_{\epsilon }+ \epsilon ^{-N}\Vert (\varphi ,\psi )\Vert ^{3}_{\epsilon }+ \epsilon ^{-\frac{3N}{2}}\Vert (\varphi ,\psi )\Vert ^{4}_{\epsilon }\right) . \end{aligned}$$

This finishes the proof. \(\square \)

Now, we consider the following projection problem

$$\begin{aligned} Q_{\epsilon }B_{\epsilon }(\varphi ,\psi )+Q_{\epsilon }l_{\epsilon }=Q_{\epsilon }R_{\epsilon }(\varphi ,\psi ), \end{aligned}$$
(2.54)

by using the contraction mapping theorem, we give the following lemma.

Lemma 2.5

There exists \(\epsilon _{0}>0\), such that for any \(\epsilon \in (0,\epsilon _{0}],\ x_{j}\in B_{\theta }(x_{0})\), then the problem (2.54) has a unique \((\varphi _{\epsilon },\psi _{\epsilon })\in {\textbf{E}}_{\epsilon }\) and

$$\begin{aligned} \Vert (\varphi _{\epsilon },\psi _{\epsilon })\Vert _{\epsilon }{} & {} \le C \Vert l_{\epsilon }\Vert _{\epsilon }\le C\left( \epsilon ^{\frac{N+2}{2}}+\epsilon ^{\frac{N}{2}} \sum ^{k}_{j=1} (|P(x_{j,\epsilon })-\lambda |+|Q(x_{j,\epsilon })-\lambda |)\right) \\{} & {} \quad + \epsilon ^{\frac{N}{2}}\sum _{i\ne j} e^{-\frac{\sqrt{\alpha }|x_{i,\epsilon }-x_{j,\epsilon }|}{\epsilon }}. \end{aligned}$$

Proof

From Lemma 2.2, we can rewrite (2.54) as follows:

$$\begin{aligned} (\varphi ,\psi )={\textbf{B}}(\varphi ,\psi ):=(Q_{\epsilon }B_{\epsilon })^{-1}Q_{\epsilon }l_{\epsilon }+(Q_{\epsilon }B_{\epsilon })^{-1}Q_{\epsilon }R_{\epsilon }(\varphi ,\psi ). \end{aligned}$$

By Lemmas 2.2 and 2.3, we have

$$\begin{aligned}&\Vert (Q_{\epsilon }B_{\epsilon })^{-1}Q_{\epsilon }l_{\epsilon }\Vert _{\epsilon }\le C\Vert Q_{\epsilon }l_{\epsilon }\Vert _{\epsilon }\le C\Vert l_{\epsilon }\Vert _{\epsilon } \nonumber \\&\quad \le C\left( \epsilon ^{\frac{N+2}{2}}+\epsilon ^{\frac{N}{2}} \sum ^{k}_{j=1} (|P(x_{j,\epsilon })-\lambda |+|Q(x_{j,\epsilon })-\lambda |)\right) + \epsilon ^{\frac{N}{2}}\sum _{i\ne j} e^{-\frac{\sqrt{\alpha }|x_{i,\epsilon }-x_{j,\epsilon }|}{\epsilon }}. \end{aligned}$$
(2.55)

Next, we will use the contraction mapping theorem in a ball whose radius is slightly bigger than \(C\Vert l_{\epsilon }\Vert _{\epsilon }\). So we take

$$\begin{aligned} S&:=\bigg \{(\varphi ,\psi ):(\varphi ,\psi )\in {\textbf{E}}_{\epsilon },\\&\quad \Vert (\varphi ,\psi )\Vert _{\epsilon } \le C\left( \epsilon ^{\frac{N+2}{2}-\tau }+\epsilon ^{\frac{N}{2}} \sum ^{k}_{j=1} (|P(x_{j,\epsilon })-\lambda |^{1-\tau }+|Q(x_{j,\epsilon })-\lambda |^{1-\tau })\right) \\&\qquad + \epsilon ^{\frac{N}{2}}\sum _{i\ne j} e^{-\frac{\sqrt{\alpha }|x_{i,\epsilon }-x_{j,\epsilon }|}{\epsilon }}\bigg \}, \end{aligned}$$

where \(\tau >0\) is a fixed small constant.

Step 1 \({\textbf{B}}\) is a map from S to S. In fact, from Lemmas 2.2, 2.3 and 2.4, we have

$$\begin{aligned}&\Vert {\textbf{B}}(\varphi ,\psi )\Vert _{\epsilon }\le C \Vert l_{\epsilon }\Vert _{\epsilon }+C \Vert R_{\epsilon }(\varphi ,\psi )\Vert _{\epsilon }\\&\quad \le C\left( \epsilon ^{\frac{N+2}{2}-\tau }+\epsilon ^{\frac{N}{2}} \sum ^{k}_{j=1} (|P(x_{j,\epsilon })-\lambda |^{1-\tau }+|Q(x_{j,\epsilon })-\lambda |^{1-\tau })\right) + \epsilon ^{\frac{N}{2}}\sum _{i\ne j} e^{-\frac{\sqrt{\alpha }|x_{i,\epsilon }-x_{j,\epsilon }|}{\epsilon }}\\&\qquad + C \left( \epsilon ^{-\frac{N}{2}}\Vert (\varphi ,\psi )|^{2}_{\epsilon }+ \epsilon ^{-N}\Vert (\varphi ,\psi )|^{3}_{\epsilon }+ \epsilon ^{-\frac{3N}{2}}\Vert (\varphi ,\psi )|^{4}_{\epsilon }\right) \\&\quad \le C\left( \epsilon ^{\frac{N+2}{2}-\tau }+\epsilon ^{\frac{N}{2}} \sum ^{k}_{j=1} (|P(x_{j,\epsilon })-\lambda |^{1-\tau }+|Q(x_{j,\epsilon })-\lambda |^{1-\tau })\right) + \epsilon ^{\frac{N}{2}}\sum _{i\ne j} e^{-\frac{\sqrt{\alpha }|x_{i,\epsilon }-x_{j,\epsilon }|}{\epsilon }}\\&\qquad + C\left( \epsilon ^{\frac{N}{2}+2(1-\tau )}+\epsilon ^{\frac{N}{2}} \sum ^{k}_{j=1} (|P(x_{j,\epsilon })-\lambda |^{2(1-\tau )}+|Q(x_{j,\epsilon })-\lambda |^{2(1-\tau )})\right) \\&\qquad + \epsilon ^{\frac{N}{2}}\sum _{i\ne j} e^{-\frac{\sqrt{\alpha }|x_{i,\epsilon }-x_{j,\epsilon }|}{\epsilon }}\\&\qquad + C\left( \epsilon ^{\frac{N}{2}+3(1-\tau )}+\epsilon ^{\frac{N}{2}} \sum ^{k}_{j=1} (|P(x_{j,\epsilon })-\lambda |^{3(1-\tau )}+|Q(x_{j,\epsilon })-\lambda |^{3(1-\tau )})\right) \\&\qquad + \epsilon ^{\frac{N}{2}}\sum _{i\ne j} e^{-\frac{\sqrt{\alpha }|x_{i,\epsilon }-x_{j,\epsilon }|}{\epsilon }}\\&\qquad + C\left( \epsilon ^{\frac{N}{2}+4(1-\tau )}+\epsilon ^{\frac{N}{2}} \sum ^{k}_{j=1} (|P(x_{j,\epsilon })-\lambda |^{4(1-\tau )}+|Q(x_{j,\epsilon })-\lambda |^{4(1-\tau )})\right) \\&\qquad + \epsilon ^{\frac{N}{2}}\sum _{i\ne j} e^{-\frac{\sqrt{\alpha }|x_{i,\epsilon }-x_{j,\epsilon }|}{\epsilon }}\\&\quad \le C\left( \epsilon ^{\frac{N+2}{2}-\tau }+\epsilon ^{\frac{N}{2}} \sum ^{k}_{j=1} (|P(x_{j,\epsilon })-\lambda |^{1-\tau }+|Q(x_{j,\epsilon })-\lambda |^{1-\tau })\right) \\&\qquad + \epsilon ^{\frac{N}{2}}\sum _{i\ne j} e^{-\frac{\sqrt{\alpha }|x_{i,\epsilon }-x_{j,\epsilon }|}{\epsilon }}. \end{aligned}$$

Thus, \({\textbf{B}}\) is a map from S to S.

Step 2 \({\textbf{B}}\) is a contraction map. For any \((\varphi _{1},\psi _{1})\in S\), we have

$$\begin{aligned} \Vert (\varphi _{1},\psi _{1})\Vert _{\epsilon }{} & {} \le C\left( \epsilon ^{\frac{N+2}{2}-\tau }+\epsilon ^{\frac{N}{2}} \sum ^{k}_{j=1} (|P(x_{j,\epsilon })-\lambda |^{1-\tau }+|Q(x_{j,\epsilon })-\lambda |^{1-\tau })\right) \\{} & {} \quad + \epsilon ^{\frac{N}{2}}\sum _{i\ne j} e^{-\frac{\sqrt{\alpha }|x_{i,\epsilon }-x_{j,\epsilon }|}{\epsilon }}, \end{aligned}$$
$$\begin{aligned} \Vert (\varphi _{2},\psi _{2})\Vert _{\epsilon }{} & {} \le C\left( \epsilon ^{\frac{N+2}{2}-\tau }+\epsilon ^{\frac{N}{2}} \sum ^{k}_{j=1} (|P(x_{j,\epsilon })-\lambda |^{1-\tau }+|Q(x_{j,\epsilon })-\lambda |^{1-\tau })\right) \\{} & {} \quad + \epsilon ^{\frac{N}{2}}\sum _{i\ne j} e^{-\frac{\sqrt{\alpha }|x_{i,\epsilon }-x_{j,\epsilon }|}{\epsilon }}. \end{aligned}$$

Since

$$\begin{aligned} \int _{\mathbb {R}^{N}}\langle R_{\epsilon }(\varphi _{1},\psi _{1})-R_{\epsilon }(\varphi _{2},\psi _{2}),(g,h)\rangle dx&=D_{1}+D_{2}+D_{3}-D_{4}+D_{5}\nonumber \\&\quad -D_{6}-D_{7}-D_{8}-D_{9}-D_{10}, \end{aligned}$$
(2.56)

where

$$\begin{aligned}{} & {} D_{1}=\int _{\mathbb {R}^{N}}\left( \mu \left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}+\varphi _{1}\right) ^{p}-\mu \left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}+\varphi _{2}\right) ^{p}\right) gdx, \end{aligned}$$
$$\begin{aligned}{} & {} D_{2}=\int _{\mathbb {R}^{N}}\left( \nu \left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}+\psi _{1}\right) ^{p}-\nu \left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}+\psi _{2}\right) ^{p}\right) hdx, \end{aligned}$$
$$\begin{aligned}{} & {} D_{3}=\int _{\mathbb {R}^{N}}\beta \left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}+\varphi _{1}\right) ^{\frac{p-1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}+\psi _{1}\right) ^{\frac{p+1}{2}}gdx, \end{aligned}$$
$$\begin{aligned}{} & {} D_{4}=\int _{\mathbb {R}^{N}}\beta \left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}+\varphi _{2}\right) ^{\frac{p-1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}+\psi _{2}\right) ^{\frac{p+1}{2}}gdx, \end{aligned}$$
$$\begin{aligned}{} & {} D_{5}=\int _{\mathbb {R}^{N}}\beta \left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}+\varphi _{1}\right) ^{\frac{p+1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}+\psi _{1}\right) ^{\frac{p-1}{2}}hdx, \end{aligned}$$
$$\begin{aligned}{} & {} D_{6}=\int _{\mathbb {R}^{N}}\beta \left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}+\varphi _{2}\right) ^{\frac{p+1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}+\psi _{2}\right) ^{\frac{p-1}{2}}hdx, \end{aligned}$$
$$\begin{aligned}{} & {} D_{7}=\beta \int _{\mathbb {R}^{N}}\frac{p-1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-3}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p+1}{2}}(\varphi _{1} -\varphi _{2}) gdx, \end{aligned}$$
$$\begin{aligned}{} & {} D_{8}=\beta \int _{\mathbb {R}^{N}}\frac{p-1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p+1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-3}{2}}(\psi _{1}-\psi _{2} ) hdx, \end{aligned}$$
$$\begin{aligned}{} & {} D_{9}=\beta \int _{\mathbb {R}^{N}}\frac{p+1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-1}{2}}(\psi _{1}-\psi _{2}) gdx, \end{aligned}$$
$$\begin{aligned}{} & {} D_{10}=\beta \int _{\mathbb {R}^{N}}\frac{p+1}{2}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-1}{2}}\left( \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}\right) ^{\frac{p-1}{2}}(\varphi _{1} -\varphi _{2}) hdx. \end{aligned}$$

Since \(p\ge 3\), we have

$$\begin{aligned} D_{1}&=\int _{\mathbb {R}^{N}}p\mu \left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}+\varphi _{1}+t(\varphi _{1}-\varphi _{2})\right) ^{p-1}(\varphi _{1}-\varphi _{2})gdx\\&\le C \int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}\right) ^{p-2}(|\varphi _{1}|+|\varphi _{2}|)(|\varphi _{1}-\varphi _{2}|)gdx\\&\le C\left( \int _{\mathbb {R}^{N}}(\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }})^{p+1}dx\right) ^{\frac{p-2}{p+1}}(\Vert \varphi _{1}\Vert _{L^{p+1}}+\Vert \varphi _{2}\Vert _{L^{p+1}})\Vert \varphi _{1}-\varphi _{2}\Vert _{L^{p+1}}\Vert g\Vert _{L^{p+1}}\\&\le C \epsilon ^{N\times \frac{p-2}{p+1}\epsilon ^{N(1-\frac{p+1}{2})}\times \frac{3}{p+1}}(\Vert \varphi _{1}\Vert _{L^{p+1}}+\Vert \varphi _{2}\Vert _{L^{p+1}})\Vert \varphi _{1}-\varphi _{2}\Vert _{L^{p+1}}\Vert g\Vert _{L^{p+1}}\\&\le \epsilon ^{-\frac{N}{2}}(\Vert \varphi _{1}\Vert _{\epsilon }+\Vert \varphi _{2}\Vert _{\epsilon })\Vert \varphi _{1}-\varphi _{2}\Vert _{\epsilon }\Vert g\Vert _{\epsilon }. \end{aligned}$$

Similarly, we have

$$\begin{aligned} D_{2}=&\int _{\mathbb {R}^{N}}\left( \nu (\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}+\psi _{1})^{p}-\nu (\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}+\psi _{2})^{p}\right) hdx\\&\le \epsilon ^{-\frac{N}{2}}(\Vert \psi _{1}\Vert _{\epsilon }+\Vert \psi _{2}\Vert _{\epsilon })\Vert \psi _{1}-\psi _{2}\Vert _{\epsilon }\Vert h\Vert _{\epsilon }. \end{aligned}$$

Thus,

$$\begin{aligned} D_{1}+D_{2}\le C \epsilon ^{-\frac{N}{2}}(\Vert (\varphi _{1},\psi _{1})\Vert _{\epsilon }+\Vert (\varphi _{2},\psi _{2})\Vert _{\epsilon })\Vert (\varphi _{1}-\varphi _{2},\psi _{1}-\psi _{2}\Vert _{\epsilon }\Vert (g,h)\Vert _{\epsilon }. \end{aligned}$$
(2.57)

We also have

$$\begin{aligned} D_{3}-D_{4}&=\int _{\mathbb {R}^{N}}\beta \frac{p-1}{2}(\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}+\varphi _{2})^{\frac{p-3}{2}}(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}+\psi _{2})^{\frac{p+1}{2}}(\varphi _{1}-\varphi _{2})gdx\nonumber \\&\quad +\int _{\mathbb {R}^{N}}\beta \frac{p+1}{2}(\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}+\varphi _{2})^{\frac{p-1}{2}}(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}+\psi _{2})^{\frac{p-1}{2}}(\psi _{1}-\psi _{2})gdx\nonumber \\&\quad +o(\int _{\mathbb {R}^{N}}(|\varphi _{1}-\varphi _{2}|^{2})gdx)+o(\int _{\mathbb {R}^{N}}o(|\psi _{1}-\psi _{2}|^{2})gdx)\nonumber \\&:=E_{1}+E_{2}+o(\int _{\mathbb {R}^{N}}(|\varphi _{1}-\varphi _{2}|^{2})gdx)+o(\int _{\mathbb {R}^{N}}o(|\psi _{1}-\psi _{2}|^{2})gdx), \end{aligned}$$
(2.58)
$$\begin{aligned} D_{5}-D_{6}&=\int _{\mathbb {R}^{N}}\beta \frac{p+1}{2}(\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}+\varphi _{2})^{\frac{p-1}{2}}(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}+\psi _{2})^{\frac{p-1}{2}}(\varphi _{1}-\varphi _{2})hdx\nonumber \\&\quad +\int _{\mathbb {R}^{N}}\beta \frac{p-1}{2}(\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}+\varphi _{2})^{\frac{p+1}{2}}(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}+\psi _{2})^{\frac{p-3}{2}}(\psi _{1}-\psi _{2})hdx\nonumber \\&\quad +o(\int _{\mathbb {R}^{N}}(|\varphi _{1}-\varphi _{2}|^{2})hdx)+o(\int _{\mathbb {R}^{N}}o(|\psi _{1}-\psi _{2}|^{2})hdx)\nonumber \\&:=E_{3}+E_{4}+o(\int _{\mathbb {R}^{N}}(|\varphi _{1}-\varphi _{2}|^{2})hdx)+o(\int _{\mathbb {R}^{N}}o(|\psi _{1}-\psi _{2}|^{2})hdx). \end{aligned}$$
(2.59)

Thus,

$$\begin{aligned} E_{1}-D_{7}&=\beta \frac{p-1}{2}\frac{p-3}{2}\int _{\mathbb {R}^{N}}(\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }})^{\frac{p-5}{2}}(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }})^{\frac{p+1}{2}}\varphi _{2}(\varphi _{1}-\varphi _{2})gdx\nonumber \\&\quad +\beta \frac{p-1}{2}\frac{p+1}{2}\int _{\mathbb {R}^{N}}(\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }})^{\frac{p-3}{2}}(\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }})^{\frac{p-1}{2}}\psi _{2}(\varphi _{1}-\varphi _{2})gdx\nonumber \\&\quad +o(\int _{\mathbb {R}^{N}}|\varphi _{2}|^{2}(\varphi _{1}-\varphi _{2})gdx)+o(\int _{\mathbb {R}^{N}}|\psi _{2}|^{2}(\varphi _{1}-\varphi _{2})gdx)\nonumber \\&\le C\left( \int _{\mathbb {R}^{N}}(\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }})^{p+1}\right) ^{\frac{p-2}{p+1}}\nonumber \\&\quad \times \left( \Vert \varphi _{2}\Vert _{L^{p+1}}\Vert \varphi _{1}-\varphi _{2}\Vert _{L^{p+1}}+\Vert \psi _{2}\Vert _{L^{p+1}})\Vert \psi _{1}-\psi _{2}\Vert _{L^{p+1}}\right) \Vert g\Vert _{L^{p+1}}\nonumber \\&\le C \epsilon ^{-\frac{N}{2}}\Vert \varphi _{2},\psi _{2})\Vert _{\epsilon }\Vert (\varphi _{1}-\varphi _{2},\psi _{1}-\psi _{2}\Vert _{\epsilon }\Vert (g,h)\Vert _{\epsilon }. \end{aligned}$$
(2.60)

Similarly, we have

$$\begin{aligned} E_{2}-D_{9}\le C \epsilon ^{-\frac{N}{2}}\Vert \varphi _{2},\psi _{2})\Vert _{\epsilon }\Vert (\varphi _{1}-\varphi _{2},\psi _{1}-\psi _{2}\Vert _{\epsilon }\Vert (g,h)\Vert _{\epsilon }, \end{aligned}$$
(2.61)
$$\begin{aligned} E_{3}-D_{10}\le C \epsilon ^{-\frac{N}{2}}\Vert \varphi _{2},\psi _{2})\Vert _{\epsilon }\Vert (\varphi _{1}-\varphi _{2},\psi _{1}-\psi _{2}\Vert _{\epsilon }\Vert (g,h)\Vert _{\epsilon }, \end{aligned}$$
(2.62)
$$\begin{aligned} E_{4}-D_{8}\le C \epsilon ^{-\frac{N}{2}}\Vert \varphi _{2},\psi _{2})\Vert _{\epsilon }\Vert (\varphi _{1}-\varphi _{2},\psi _{1}-\psi _{2}\Vert _{\epsilon }\Vert (g,h)\Vert _{\epsilon }. \end{aligned}$$
(2.63)

Combining (2.57)–(2.63) with (2.56), we have

$$\begin{aligned}&\int _{\mathbb {R}^{N}}R_{\epsilon }(\varphi _{1},\psi _{1})-R_{\epsilon }(\varphi _{2},\psi _{2}),(g,h)\rangle dx\nonumber \\ {}&\quad \le C \epsilon ^{-\frac{N}{2}}\Vert \varphi _{2},\psi _{2})\Vert _{\epsilon }\Vert (\varphi _{1}-\varphi _{2},\psi _{1}-\psi _{2}\Vert _{\epsilon }\Vert (g,h)\Vert _{\epsilon }.\nonumber \\ \end{aligned}$$
(2.64)

Thus,

$$\begin{aligned} \Vert {\textbf{B}}(\varphi _{1},\psi _{1})-{\textbf{B}}(\varphi _{2},\psi _{2})\Vert _{\epsilon }\le \frac{1}{2}\Vert (\varphi _{1},\psi _{1})-(\varphi _{2},\psi _{2})\Vert _{\epsilon }. \end{aligned}$$

So, \({\textbf{B}}\) is a contraction map.

By the contraction mapping theorem, we conclude that for any \(\epsilon \in (0,\epsilon _{0}],\ \ x_{j}\in B_{\theta }(x_{0})\), there is a \((\varphi _{\epsilon },\psi _{\epsilon })\in {\textbf{E}}_{\epsilon }\) depending only on \(x_{j}\) and \(\epsilon \) such that

$$\begin{aligned} (\varphi _{\epsilon },\psi _{\epsilon })={\textbf{B}}(\varphi _{\epsilon },\psi _{\epsilon }). \end{aligned}$$

Moreover, from Lemma 2.3 and Lemma 2.4, we have

$$\begin{aligned}&\Vert (\varphi _{\epsilon },\psi _{\epsilon })\Vert _{\epsilon }\\ {}&\quad =\Vert {\textbf{B}}(\varphi _{\epsilon },\psi _{\epsilon })\Vert _{\epsilon }\le C \Vert l_{\epsilon }\Vert _{\epsilon }+C \Vert R_{\epsilon }(\varphi ,\psi )\Vert _{\epsilon }\\ {}&\le C \Vert l_{\epsilon }\Vert _{\epsilon }+C \left( \epsilon ^{-\frac{N}{2}}\Vert (\varphi ,\psi )|^{2}_{\epsilon }+ \epsilon ^{-N}\Vert (\varphi ,\psi )|^{3}_{\epsilon }+ \epsilon ^{-\frac{3N}{2}}\Vert (\varphi ,\psi )|^{4}_{\epsilon }\right) \\&\le C \Vert l_{\epsilon }\Vert _{\epsilon }+C \left( \epsilon ^{-\frac{N}{2}}\Vert (\varphi ,\psi )|_{\epsilon }+ \epsilon ^{-N}\Vert (\varphi ,\psi )|^{2}_{\epsilon }+ \epsilon ^{-\frac{3N}{2}}\Vert (\varphi ,\psi )|^{3}_{\epsilon }\right) \Vert (\varphi ,\psi )|_{\epsilon }\\&\le C\left( \epsilon ^{\frac{N+2}{2}}+\epsilon ^{\frac{N}{2}} \sum ^{k}_{j=1} (|P(x_{j,\epsilon })-\lambda |+|Q(x_{j,\epsilon })-\lambda |)\right) + \epsilon ^{\frac{N}{2}}\sum _{i\ne j} e^{-\frac{\sqrt{\alpha }|x_{i,\epsilon }-x_{j,\epsilon }|}{\epsilon }}. \end{aligned}$$

As desired. \(\square \)

Next, we solve equation (2.1). Since

$$\begin{aligned} Q_{\epsilon }B_{\epsilon }(\varphi ,\psi )+Q_{\epsilon }l_{\epsilon }-Q_{\epsilon }R_{\epsilon }(\varphi ,\psi ){} & {} =B_{\epsilon }(\varphi ,\psi )+l_{\epsilon }-R_{\epsilon }(\varphi ,\psi )\\{} & {} \quad -\sum ^{k}_{j=1}\sum ^{N}_{i=1}b_{\epsilon ,i,j}\left( \frac{\partial U_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}},\frac{\partial V_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}\right) . \end{aligned}$$

From Lemma 2.5, we know the following equation

$$\begin{aligned} Q_{\epsilon }B_{\epsilon }(\varphi ,\psi )+Q_{\epsilon }l_{\epsilon }=Q_{\epsilon }R_{\epsilon }(\varphi ,\psi ) \end{aligned}$$

has a unique solution \((\varphi _{\epsilon },\psi _{\epsilon })\). So

$$\begin{aligned} B_{\epsilon }(\varphi _{\epsilon },\psi _{\epsilon })+l_{\epsilon }-R_{\epsilon }(\varphi _{\epsilon },\psi _{\epsilon })=\sum ^{k}_{j=1}\sum ^{N}_{i=1}b_{\epsilon ,i,j}\left( \frac{\partial U_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}},\frac{\partial V_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}\right) \end{aligned}$$
(2.65)

for some constant \(b_{\epsilon ,i,j}.\) Next, we should to choose suitable \(x_{j}\) such that \(b_{\epsilon ,i,j}=0, i=1,2\ldots N. j=1,2,\ldots k.\)

Firstly, it is easy to see that the right hand of (2.65) belongs to \({\textbf{E}}_{\epsilon }\), if the left hand of of (2.65) belongs to \({\textbf{E}}_{\epsilon }\), then the right hand of (2.65) must be zero.

Let

$$\begin{aligned} u_{\epsilon }=\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}+\varphi _{\epsilon },\ \ v_{\epsilon }=\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}+\psi _{\epsilon }, \end{aligned}$$

then

$$\begin{aligned}&\langle B_{\epsilon }(\varphi _{\epsilon },\psi _{\epsilon })+l_{\epsilon }-R_{\epsilon }(\varphi _{\epsilon },\psi _{\epsilon }),(g,h)\rangle _{\epsilon }\\&\quad =\int _{\mathbb {R}^{N}}(\epsilon ^{2}\nabla u_{\epsilon } \nabla g+P(x)u_{\epsilon }g+\epsilon ^{2}\nabla v_{\epsilon } \nabla v+Q(x)v_{\epsilon }h)dx\\&\qquad -\int _{\mathbb {R}^{N}}\left( \mu u_{\epsilon }^{p}g+\beta u_{\epsilon }^{\frac{p-1}{2}}v_{\epsilon }^{\frac{p+1}{2}}g+\nu v_{\epsilon }^{p}h+\beta u_{\epsilon }^{\frac{p+1}{2}}v_{\epsilon }^{\frac{p-1}{2}}h\right) dx \end{aligned}$$

for any \((g,h)\in H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N}).\)

Lemma 2.6

Suppose that \(x_{j,\epsilon }\) satisfies

$$\begin{aligned}&\int _{\mathbb {R}^{N}}(\epsilon ^{2}\nabla u_{\epsilon } \nabla \frac{\partial U_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}+P(x)u_{\epsilon }\frac{\partial U_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}+\epsilon ^{2}\nabla v_{\epsilon } \nabla \frac{\partial V_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}+Q(x)v_{\epsilon }\frac{\partial U_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}})dx\nonumber \\&-\int _{\mathbb {R}^{N}}\left( \mu u_{\epsilon }^{p}\frac{\partial U_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}+\beta u_{\epsilon }^{\frac{p-1}{2}}v_{\epsilon }^{\frac{p+1}{2}}\frac{\partial U_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}+\nu v_{\epsilon }^{p}\frac{\partial V_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}+\beta u_{\epsilon }^{\frac{p+1}{2}}v_{\epsilon }^{\frac{p-1}{2}}\frac{\partial V_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}\right) \nonumber \\&dx=0i=1,2,\ldots N, j=1,2,\ldots k. \end{aligned}$$
(2.66)

then

$$\begin{aligned} b_{\epsilon ,i,j}=0,\ i=1,2,\ldots N, j=1,2,\cdots k. \end{aligned}$$

Proof

If (2.66) holds, then

$$\begin{aligned}&\sum ^{k}_{m=1}\sum ^{N}_{h=1}b_{\epsilon ,h,m}\nonumber \\ {}&\quad \bigg \langle (\frac{\partial U_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}},\frac{\partial V_{\epsilon ,x_{j,\epsilon }}}{\partial x_{i}}),(\frac{\partial U_{\epsilon ,x_{m}}}{\partial y_{h}},\frac{\partial V_{\epsilon ,x_{m}}}{\partial y_{h}})\bigg \rangle =0,\ \ i=1,2,\ldots N, j=1,2,\ldots k. \end{aligned}$$
(2.67)

By (2.9) and (2.10), we obtain

$$\begin{aligned} b_{\epsilon ,i,j}=0,\ i=1,2,\ldots N, j=1,2,\ldots k. \end{aligned}$$

\(\square \)

3 Proof of Theorem 1.1

In this section, we give the proof of Theorem 1.1.

Proof

In order to solve (2.65), we define a function as following

$$\begin{aligned} K({\textbf{x}})=I\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}+\varphi _{\epsilon },\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}+\psi _{\epsilon }\right) , \end{aligned}$$

where

$$\begin{aligned} I(u,v)&=\frac{1}{2}\int _{\mathbb {R}^{N}}(\epsilon ^{2}|\nabla u|^{2}+P(x)u^{2}+\epsilon ^{2}|\nabla v|^{2}+Q(x)v^{2})dx\nonumber \\&\quad -\frac{1}{p+1}\int _{\mathbb {R}^{N}}(\mu u^{p+1}+\nu u^{p+1}\nonumber \\ {}&\quad +2\beta u^{\frac{p+1}{2}}v^{\frac{p+1}{2}})dx,\ \forall \ (u,v)\in \ H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N}). \end{aligned}$$
(3.1)

Then, from Lemmas 2.3, 2.4 and Proposition 5.1, there exists a small constant \(\sigma >0\) such that

$$\begin{aligned} K({\textbf{x}})&=I\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }},\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}\right) +\langle l_{\epsilon },(\varphi _{\epsilon },\psi _{\epsilon })\rangle +O(\Vert l_{\epsilon }\Vert \Vert (\varphi _{\epsilon },\psi _{\epsilon })\Vert +\Vert \varphi _{\epsilon },\psi _{\epsilon }\Vert ^{2})\nonumber \\&=I\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }},\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}\right) +O\left( \epsilon ^{N}\left( (P(x_{j,\epsilon })-\lambda )^{2}+(Q(x_{j,\epsilon })-\lambda )^{2}+\epsilon ^{2}\right) \right) \nonumber \\&=(\frac{1}{2}-\frac{1}{p+1})k\epsilon ^{N}\lambda ^{1-\frac{N}{2}}A -C\epsilon ^{N}\left( (\lambda -P(x_{j,\epsilon }))+(\lambda -Q(x_{j,\epsilon }))+\epsilon \right) \nonumber \\&\quad -C\epsilon ^{N}\sum ^{k}_{i\ne j}e^{-\frac{\sqrt{\lambda }|x_{i,\epsilon }-x_{j,\epsilon }|}{\epsilon }}+O\left( \epsilon ^{N}\sum ^{k}_{i\ne j}e^{-\frac{2(\sqrt{\lambda }-\sigma )|x_{i,\epsilon }-x_{j,\epsilon }|}{\epsilon }}\right) \nonumber \\&\quad +O\left( \epsilon ^{N}\left( (P(x_{j,\epsilon })-\lambda )^{2}+(Q(x_{j,\epsilon })-\lambda )^{2}+\epsilon ^{2}\right) \right) . \end{aligned}$$
(3.2)

We use the ideas introduced in [22] to consider the following maximizing problem

$$\begin{aligned} \max _{x\in D} K(x), \end{aligned}$$

where

$$\begin{aligned} D=\{{\textbf{x}}: x_{j}\in \overline{B_{\theta }(x_{0})}, j=1,2,\ldots k, |x_{m}-x_{j}|\ge \theta \epsilon |\ln \epsilon |^{\frac{1}{2}},\ m\ne j\}. \end{aligned}$$

We prove that if \(\max _{x\in D} K(x) \) is achieved by some \({\textbf{x}}_{\epsilon }\) in \(\overline{D}\), then \({\textbf{x}}_{\epsilon }\) is an interior point of \(\overline{D}\). Taking \(\overline{x}_{i}=x_{0}+\theta \epsilon |\ln \epsilon |e_{j}, (j=1,2,\ldots k)\), then \(|\overline{x}_{i}-\overline{x}_{j}|=\theta \epsilon |\ln \epsilon |\), which means that \(\overline{{\textbf{x}}}=(\overline{x}_{1},\overline{x}_{2},\ldots ,\overline{x}_{k})\in \overline{D}\), thus, we have

$$\begin{aligned}&\left( \frac{1}{2}-\frac{1}{p+1}\right) k\epsilon ^{N}\lambda ^{1-\frac{N}{2}}A -C\epsilon ^{N+1}|\ln \epsilon |\le K(\overline{{\textbf{x}}})\le K({\textbf{x}}_{\epsilon })\\&\quad \le \left( \frac{1}{2}-\frac{1}{p+1}\right) k\epsilon ^{N}\lambda ^{1-\frac{N}{2}}A -C\epsilon ^{N}\\ {}&\quad \left( \sum ^{k}_{j=1} (\lambda -P(x_{j,\epsilon }))+\sum ^{k}_{j=1}(\lambda -Q(x_{j,\epsilon }))+\epsilon \right) \\&\qquad -C\epsilon ^{N}\sum ^{k}_{i\ne j}e^{-\frac{\sqrt{\alpha }|x_{\epsilon ,i}-x_{j,\epsilon }|}{\epsilon }}. \end{aligned}$$

Thus,

$$\begin{aligned} \sum ^{k}_{j=1} (\lambda -P(x_{j,\epsilon }))+\sum ^{k}_{j=1}(\lambda -Q(x_{j,\epsilon }))+\epsilon +\sum ^{k}_{i\ne j}e^{-\frac{\sqrt{\alpha }|x_{\epsilon ,i}-x_{j,\epsilon }|}{\epsilon }}\le C\epsilon |\ln \epsilon |, \end{aligned}$$

so

$$\begin{aligned} \sum ^{k}_{j=1} (\lambda -P(x_{j,\epsilon }))\le C\epsilon |\ln \epsilon |,\ \ \ \ \sum ^{k}_{j=1}(\lambda -Q(x_{j,\epsilon }))\le C\epsilon |\ln \epsilon |, \end{aligned}$$
$$\begin{aligned} \frac{\sqrt{\alpha }|x_{\epsilon ,i}-x_{j,\epsilon }|}{\epsilon }\ge C|\ln \epsilon |\ge |\ln \epsilon |^{\frac{1}{2}}. \end{aligned}$$

Therefore, \({\textbf{x}}_{\epsilon }\) is an interior point of \(\overline{D}\), which implies that

$$\begin{aligned} (u_{\epsilon },v_{\epsilon })=\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon }}+\varphi _{\epsilon }, \ \sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}+\psi _{\epsilon }\right) \end{aligned}$$

is a critical point of \(K({\textbf{x}})\). So, (1.1) has a solution of the form

$$\begin{aligned} u_{\epsilon }=\sum ^{k}_{j=1}U_{\epsilon ,x_{\epsilon },j}+\varphi _{\epsilon },\ \ v_{\epsilon }=\sum ^{k}_{j=1}V_{\epsilon ,x_{j,\epsilon }}+\psi _{\epsilon } \end{aligned}$$

for some \(x_{j,\epsilon }\in B_{\delta }(x_{0})\) and \(\Vert (\varphi _{\epsilon },\psi _{\epsilon })\Vert _{\epsilon }=O(\epsilon ^{\frac{N}{2}+1}).\) \(\square \)

4 Proof of Theorem 1.2

This section is devoted to the proof of Theorem 1.2.

Proof

Set \(\lambda =P(x_{0})\), \(\overline{\lambda }=Q(\overline{x}_{0})\) it is easy to see that \(U_{\lambda ,\mu }=\lambda ^{\frac{1}{p-1}}\mu ^{-\frac{1}{p-1}}W(\sqrt{\lambda }x)\) is a solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+ \lambda u =\mu u^{p},\ \text {in}\ \mathbb {R}^{N}\\ u>0 \ \text {in} \ \mathbb {R}^{N}, u(x)\rightarrow 0 \ \text {as} \ |x|\rightarrow +\infty , \end{array}\right. } \end{aligned}$$

and \(U_{\overline{\lambda },\nu }=\overline{\lambda }^{\frac{1}{p-1}}\nu ^{-\frac{1}{p-1}}W(\sqrt{\overline{\lambda }}x)\) is a solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+ \overline{\lambda } u =\nu u^{p},\ \text {in}\ \mathbb {R}^{N}\\ u>0 \ \text {in} \ \mathbb {R}^{N}, u(x)\rightarrow 0 \ \text {as} \ |x|\rightarrow +\infty . \end{array}\right. } \end{aligned}$$

Let

$$\begin{aligned} \left( U_{\epsilon ,x_{j},\mu }(x),U_{\epsilon ,z_{j},\nu }(x)\right) =\left( U_{\lambda _{j},\mu }(\frac{x-x_{j}}{\epsilon }),U_{\overline{\lambda }_{j},\nu }(\frac{x-z_{j}}{\epsilon })\right) , \end{aligned}$$

where \(x_{j}\in B_{\delta }(x_{0}),\ z_{j}\in B_{\delta }(\overline{x}_{0})\).

$$\begin{aligned} \widetilde{{\textbf{E}}}_{\epsilon }=&\bigg \{(\varphi ,\psi )\in H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N}),\left\langle \varphi ,\frac{\partial U_{\epsilon ,x_{j},\mu }}{\partial x_{j,l}}\right\rangle _{\epsilon }=0,\\&\left\langle \psi ,\frac{\partial U_{\epsilon ,z_{j},\nu }}{\partial z_{j,l}}\right\rangle _{\epsilon }=0,\ j=1,2\cdots k;\ \ l=1,2\cdots N\bigg \}. \end{aligned}$$

Let \(\lambda _{j}=P(x_{j}),\ \overline{\lambda }_{j}=Q(z_{j})\) and \(x_{0}\) is a local maximum point of P(x) and \(\overline{x}_{0}\) is a local maximum point of Q(x), we want to construct a solution \((u_{\epsilon },v_{\epsilon })\) of the following form

$$\begin{aligned} u_{\epsilon }=\sum ^{k}_{j=1}U_{\epsilon ,x_{j},\mu }+{\widetilde{\varphi }}_{\epsilon },\ \ v_{\epsilon }=\sum ^{m}_{j=1}U_{\epsilon ,z_{j},\nu }+{\widetilde{\psi }}_{\epsilon } \end{aligned}$$

where as \(\epsilon \rightarrow 0, x_{j}\rightarrow x_{0},\ \ z_{j}\rightarrow \overline{x}_{0}\) and \(\Vert ({\widetilde{\varphi }}_{\epsilon },{\widetilde{\psi }}_{\epsilon })\Vert ^{2}=o(\epsilon ^{N}).\) Then, \(({\widetilde{\varphi }}_{\epsilon },{\widetilde{\psi }}_{\epsilon })\) satisfies the following equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \overline{B}_{\epsilon }({\widetilde{\varphi }}_{\epsilon },{\widetilde{\psi }}_{\epsilon })+l_{\epsilon }=R_{\epsilon }({\widetilde{\varphi }}_{\epsilon },{\widetilde{\psi }}_{\epsilon }),x\in \mathbb {R}^{N},\\ ({\widetilde{\varphi }}_{\epsilon },{\widetilde{\psi }}_{\epsilon })\in H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N}). \end{array}\right. } \end{aligned}$$

where \(\overline{B}_{\epsilon }\) is a bounded linear operator in \(H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N})\), defined by

$$\begin{aligned}&\langle \overline{B}_{\epsilon }({\widetilde{\varphi }}_{\epsilon },{\widetilde{\psi }}_{\epsilon }),(g,h)\rangle \\&\quad =\int _{\mathbb {R}^{N}}\left( \epsilon ^{2}\nabla {\widetilde{\varphi }}_{\epsilon }\nabla g+P(x){\widetilde{\varphi }}_{\epsilon } g-p\mu (\sum ^{k}_{j=1}U_{\epsilon ,x_{j},\mu })^{p-1}{\widetilde{\varphi }}_{\epsilon } g\right) dx\\&\qquad +\int _{\mathbb {R}^{N}}\left( \epsilon ^{2}\nabla {\widetilde{\psi }}_{\epsilon }\nabla h+Q(x){\widetilde{\psi }}_{\epsilon } h-p\nu (\sum ^{m}_{j=1}U_{\epsilon ,z_{j},\nu })^{p-1}{\widetilde{\psi }}_{\epsilon } h\right) dx\\&\qquad -\beta \int _{\mathbb {R}^{N}}\frac{p-1}{2}(\sum ^{k}_{j=1}U_{\epsilon ,x_{j},\mu })^{\frac{p-3}{2}}(\sum ^{m}_{j=1}U_{\epsilon ,z_{j},\nu })^{\frac{p+1}{2}}{\widetilde{\varphi }}_{\epsilon }g\\ {}&\quad +\frac{p-1}{2}(\sum ^{k}_{j=1}U_{\epsilon ,x_{j},\mu })^{\frac{p+1}{2}}(\sum ^{m}_{j=1}U_{\epsilon ,z_{j},\nu })^{\frac{p-3}{2}}{\widetilde{\psi }}_{\epsilon }h\\&\qquad -\beta \int _{\mathbb {R}^{N}}\frac{p+1}{2}(\sum ^{k}_{j=1}U_{\epsilon ,x_{j},\mu })^{\frac{p-1}{2}}(\sum ^{m}_{j=1}U_{\epsilon ,z_{j},\nu })^{\frac{p-1}{2}}{\widetilde{\psi }}_{\epsilon }g\\ {}&\quad +\frac{p+1}{2}(\sum ^{k}_{j=1}U_{\epsilon ,x_{j},\mu })^{\frac{p-1}{2}}(\sum ^{m}_{j=1}U_{\epsilon ,z_{j},\nu })^{\frac{p-1}{2}}{\widetilde{\varphi }}_{\epsilon }h \end{aligned}$$

for any \((g,h)\in H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N}).\)

\(\overline{l}_{\epsilon }\in H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N})\) satisfying

$$\begin{aligned} \langle \overline{l}_{\epsilon },(g,h)\rangle&= \sum ^{k}_{j=1}\int _{\mathbb {R}^{N}}(P(x)-P(x^{j}))U_{\epsilon ,x_{j},\mu }gdx\\ {}&\quad +\mu \int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}U^{p}_{\epsilon ,x_{j},\mu }-(\sum ^{k}_{j=1}U_{\epsilon ,x_{j},\mu })^{p}\right) gdx\\&\quad +\sum ^{k}_{j=1}\int _{\mathbb {R}^{N}}(Q(x)-Q(z_{j}))U_{\epsilon ,z_{j},\nu }hdx\\ {}&\quad +\nu \int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}U^{p}_{\epsilon ,z_{j},\nu }-(\sum ^{m}_{j=1}U_{\epsilon ,z_{j},\nu })^{p}\right) hdx\\&\quad -\beta \int _{\mathbb {R}^{N}}\left( (\sum ^{k}_{j=1}U_{\epsilon ,x_{j},\mu })^{\frac{p-1}{2}}(\sum ^{m}_{j=1}U_{\epsilon ,z_{j},\nu })^{\frac{p+1}{2}}\right) gdx\\&\quad -\beta \int _{\mathbb {R}^{N}}\left( (\sum ^{k}_{j=1}U_{\epsilon ,x_{j},\mu })^{\frac{p+1}{2}}(\sum ^{m}_{j=1}U_{\epsilon ,z_{j},\nu })^{\frac{p-1}{2}}\right) hdx \end{aligned}$$

for any \((g,h)\in H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N}).\)

$$\begin{aligned}&\langle \overline{R}_{\epsilon }({\widetilde{\varphi }}_{\epsilon },{\widetilde{\psi }}_{\epsilon }), (g,h)\rangle \\&=\int _{\mathbb {R}^{N}}\left( \mu (\sum ^{k}_{j=1}U_{\epsilon ,x_{j},\mu }+{\widetilde{\varphi }}_{\epsilon })^{p}+\beta (\sum ^{k}_{j=1}U_{\epsilon ,x_{j},\mu }+{\widetilde{\varphi }}_{\epsilon })^{\frac{p-1}{2}}(\sum ^{m}_{j=1}U_{\epsilon ,z_{j},\nu }+{\widetilde{\psi }}_{\epsilon })^{\frac{p+1}{2}}\right) gdx\\&\quad +\int _{\mathbb {R}^{N}}\left( \nu (\sum ^{m}_{j=1}U_{\epsilon ,z_{j},\nu }+{\widetilde{\psi }}_{\epsilon })^{p}+\beta (\sum ^{k}_{j=1}U_{\epsilon ,x_{j},\mu }+{\widetilde{\varphi }}_{\epsilon })^{\frac{p+1}{2}}(\sum ^{m}_{j=1}U_{\epsilon ,z_{j},\nu }+{\widetilde{\psi }}_{\epsilon })^{\frac{p-1}{2}}\right) hdx\\&\quad -\beta \int _{\mathbb {R}^{N}}\mu (\sum ^{k}_{j=1}U_{\epsilon ,x_{j},\mu })^{p}g+\left( (\sum ^{k}_{j=1}U_{\epsilon ,x_{j},\mu })^{\frac{p-1}{2}}(\sum ^{m}_{j=1}U_{\epsilon ,z_{j},\nu })^{\frac{p+1}{2}}\right) gdx\\&\quad -\beta \int _{\mathbb {R}^{N}}\nu (\sum ^{m}_{j=1}U_{\epsilon ,z_{j},\nu })^{p}h+\left( (\sum ^{k}_{j=1}U_{\epsilon ,x_{j},\mu })^{\frac{p+1}{2}}(\sum ^{m}_{j=1}U_{\epsilon ,z_{j},\nu })^{\frac{p-1}{2}}\right) hdx\\&\quad -\beta \int _{\mathbb {R}^{N}}\frac{p-1}{2}(\sum ^{k}_{j=1}U_{\epsilon ,x_{j},\mu })^{\frac{p-3}{2}}(\sum ^{m}_{j=1}U_{\epsilon ,z_{j},\nu })^{\frac{p+1}{2}}{\widetilde{\varphi }}_{\epsilon }gdx\\&\quad -\beta \int _{\mathbb {R}^{N}}\frac{p-1}{2}(\sum ^{k}_{j=1}U_{\epsilon ,x_{j},\mu })^{\frac{p+1}{2}}(\sum ^{m}_{j=1}U_{\epsilon ,z_{j},\nu })^{\frac{p-3}{2}}{\widetilde{\psi }}_{\epsilon }hdx\\&\quad -\beta \int _{\mathbb {R}^{N}}\frac{p+1}{2}(\sum ^{k}_{j=1}U_{\epsilon ,x_{j},\mu })^{\frac{p-1}{2}}(\sum ^{m}_{j=1}U_{\epsilon ,z_{j},\nu })^{\frac{p-1}{2}}{\widetilde{\psi }}_{\epsilon }gdx\\&\quad -\beta \int _{\mathbb {R}^{N}}\frac{p+1}{2}(\sum ^{k}_{j=1}U_{\epsilon ,x_{j},\mu })^{\frac{p-1}{2}}(\sum ^{m}_{j=1}U_{\epsilon ,z_{j},\nu })^{\frac{p-1}{2}}{\widetilde{\varphi }}_{\epsilon }hdx \end{aligned}$$

for any \((g,h)\in H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N}).\) \(\square \)

Lemma 4.1

There exist \(\beta ^{*}>0\) and \(\epsilon _{0},\theta _{0}>0,\rho >0,\) independent of \(x_{j},j=1,2,\ldots k\) and \(z_{j},j=1,2,\ldots m\), such that for any \(\epsilon \in (0,\epsilon _{0}]\), \(x_{j}\in B_{\theta _{0}}(x_{0})\) and \(z_{j}\in B_{\theta _{0}}(\overline{x}_{0})\), if \(\beta <\beta ^{*}\), then \(Q_{\epsilon }\overline{B}_{\epsilon }({\widetilde{\varphi }}_{\epsilon },{\widetilde{\psi }}_{\epsilon })\) is bijective in \(\widetilde{{\textbf{E}}}_{\epsilon }\). Moreover, it holds

$$\begin{aligned} \Vert Q_{\epsilon }\overline{B}_{\epsilon }({\widetilde{\varphi }}_{\epsilon },{\widetilde{\psi }}_{\epsilon })\Vert _{\epsilon }\ge \rho \Vert ({\widetilde{\varphi }}_{\epsilon },{\widetilde{\psi }}_{\epsilon })\Vert _{\epsilon },\ \ \forall \ ({\widetilde{\varphi }}_{\epsilon },{\widetilde{\psi }}_{\epsilon })\in \widetilde{{\textbf{E}}}_{\epsilon }. \end{aligned}$$

Proof

The proof is similar as the proof of Lemma 2.2. To get a contradiction, we take the projection to

$$\begin{aligned} \widetilde{{\textbf{E}}}_{\epsilon }=&\bigg \{(\varphi ,\psi )\in H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N}),\left\langle \varphi ,\frac{\partial U_{\epsilon ,x_{j},\mu }}{\partial x_{j,l}}\right\rangle _{\epsilon }=0,\\&\left\langle \psi ,\frac{\partial U_{\epsilon ,z_{j},\nu }}{\partial z_{j,l}}\right\rangle _{\epsilon }=0,\ j=1,2\cdots k;\ \ l=1,2\cdots N\bigg \}. \end{aligned}$$

We need to prove \(\varphi =0\) and \(\psi =0\). When we prove \(\varphi =0\), we only need to set \(h=0,\varphi _{n}=0\) after (2.23) in Lemma 2.2. So we will get \(\varphi \) satisfies \(-\Delta \varphi +\lambda \varphi -p\mu U^{p-1}_{_{r}}=0\). By the non-degenerate of the solution of \(-\Delta u+\lambda u=\mu u^{p}\), we have \(\varphi =\sum ^{k}_{i=1}c_{i}\frac{\partial U_{\mu ,r}}{\partial y_{i}}\). Then, we can get \(c_{i}=0\), thus \(\varphi =0\). To show \(\psi =0\), we need to set \(g=0,\psi _{n}=0\) after (2.23) in Lemma 2.2. So we will get that \(\psi \) satisfies \(-\Delta \psi +\overline{\lambda }\psi -p\nu U^{p-1}_{_{r}}=0\). By the non-degenerate of the solution of \(-\Delta u+\lambda u=\nu u^{p}\), we have \(\psi =\sum ^{k}_{i=1}d_{i}\frac{\partial U_{\nu ,r}}{\partial y_{i}}\). Then, we can get \(d_{i}=0\), thus \(\psi =0\). The rest is similar to the proof of (3.8) in Lemma 3.2 in [23], we can get a contradiction when \(\beta <\beta ^{*}.\) \(\square \)

To carry out the contraction mapping theorem, we give the following error estimate.

Lemma 4.2

There is a constant \(C>0\) independent of \(\epsilon \) such that

$$\begin{aligned} \Vert \overline{l}_{\epsilon }\Vert _{\epsilon }&\le C\left( \epsilon ^{\frac{N+2}{2}}+\epsilon ^{\frac{N}{2}}\left( \sum ^{k}_{i\ne j}(P(x_{i})-P(x_{j}))+\sum ^{m}_{i\ne j}(Q(z_{i})-Q(z_{j}))\right) \right) \\&\quad + C\epsilon ^{\frac{N}{2}}\sum ^{k}_{i=1}\sum ^{m}_{j=1}e^{-\frac{\min \{\sqrt{\lambda _{i}}, \sqrt{\overline{\lambda }_{j}}\}|x_{i}-z_{j}|}{\epsilon }}+C\epsilon ^{\frac{N}{2}}\sum ^{k}_{i=1}\sum ^{m}_{j=1}e^{-\frac{\min \{\sqrt{\lambda _{i}}, \sqrt{\lambda }_{j}\}|x_{i}-x_{j}|}{\epsilon }}\\&\quad +C\epsilon ^{\frac{N}{2}}\sum ^{k}_{i=1}\sum ^{m}_{j=1}e^{-\frac{\min \{\sqrt{\overline{\lambda }_{i}}, \sqrt{\lambda _{j}}\}|z_{i}-x_{j}|}{\epsilon }}. \end{aligned}$$

Proof

The proof is similar to that of Lemma 2.3. First, by the Hölder inequality, we have

$$\begin{aligned}&\sum ^{k}_{j=1}\int _{\mathbb {R}^{N}}(P(x)-P(x_{j}))U_{\epsilon ,x_{j},\nu }gdx+\mu \int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}U^{p}_{\epsilon ,x_{j},\mu }-(\sum ^{k}_{j=1}U_{\epsilon ,x_{j},\mu })^{p}\right) gdx\\&+\sum ^{k}_{j=1}\int _{\mathbb {R}^{N}}(Q(x)-Q(z_{j}))U_{\epsilon ,z_{j},\nu }hdx+\nu \int _{\mathbb {R}^{N}}\left( \sum ^{k}_{j=1}U^{p}_{\epsilon ,z_{j},\nu }-(\sum ^{m}_{j=1}U_{\epsilon ,z_{j},\nu })^{p}\right) hdx\\&\le C\left( \epsilon ^{\frac{N+2}{2}}+\epsilon ^{\frac{N}{2}}\left( \sum ^{k}_{i\ne j}(P(x_{i})-P(x_{j}))+\sum ^{m}_{i\ne j}(Q(z_{i})-Q(z_{j}))\right) \right) \Vert (g,h)\Vert _{\epsilon }, \end{aligned}$$

and

$$\begin{aligned}&\beta \int _{\mathbb {R}^{N}}\left( (\sum ^{k}_{j=1}U_{\epsilon ,x_{j},\mu })^{\frac{p-1}{2}}(\sum ^{m}_{j=1}U_{\epsilon ,z_{j},\nu })^{\frac{p+1}{2}}\right) gdx\\&\quad +\beta \int _{\mathbb {R}^{N}}\left( (\sum ^{k}_{j=1}U_{\epsilon ,x_{j},\mu })^{\frac{p+1}{2}}(\sum ^{m}_{j=1}U_{\epsilon ,z_{j},\nu })^{\frac{p-1}{2}}\right) hdx\\&\le \bigg (C\epsilon ^{\frac{N}{2}}\sum ^{k}_{i=1}\sum ^{m}_{j=1}e^{-\frac{\min \{\sqrt{\lambda _{i}}, \sqrt{\overline{\lambda }_{j}}\}|x_{i}-z_{j}|}{\epsilon }}+C\epsilon ^{\frac{N}{2}}\sum ^{k}_{i=1}\sum ^{m}_{j=1}e^{-\frac{\min \{\sqrt{\lambda _{i}}, \sqrt{\lambda }_{j}\}|x_{i}-x_{j}|}{\epsilon }}\\&\quad +C\epsilon ^{\frac{N}{2}}\sum ^{k}_{i=1}\sum ^{m}_{j=1}e^{-\frac{\min \{\sqrt{\overline{\lambda }_{i}}, \sqrt{\lambda _{j}}\}|z_{i}-x_{j}|}{\epsilon }}\bigg )\Vert (g,h)\Vert _{\epsilon }. \end{aligned}$$

So, we obtain

$$\begin{aligned} \Vert \overline{l}_{\epsilon }\Vert _{\epsilon }&\le C\left( \epsilon ^{\frac{N+2}{2}}+\epsilon ^{\frac{N}{2}}\left( \sum ^{k}_{i\ne j}(P(x_{i})-P(x_{j}))+\sum ^{m}_{i\ne j}(Q(z_{i})-Q(z_{j}))\right) \right) \\&\quad + C\epsilon ^{\frac{N}{2}}\sum ^{k}_{i=1}\sum ^{m}_{j=1}e^{-\frac{\min \{\sqrt{\lambda _{i}}, \sqrt{\overline{\lambda }_{j}}\}|x_{i}-z_{j}|}{\epsilon }}+C\epsilon ^{\frac{N}{2}}\sum ^{k}_{i=1}\sum ^{m}_{j=1}e^{-\frac{\min \{\sqrt{\lambda _{i}}, \sqrt{\lambda }_{j}\}|x_{i}-x_{j}|}{\epsilon }}\\&\quad +C\epsilon ^{\frac{N}{2}}\sum ^{k}_{i=1}\sum ^{m}_{j=1}e^{-\frac{\min \{\sqrt{\overline{\lambda }_{i}}, \sqrt{\lambda _{j}}\}|z_{i}-x_{j}|}{\epsilon }}. \end{aligned}$$

This proof is thus complete. \(\square \)

Lemma 4.3

There is a constant \(C>0\) independent of \(\epsilon \) and a small constant \(\sigma >0\) such that

$$\begin{aligned} \Vert \overline{R}_{\epsilon }(\varphi ,\psi )\Vert _{\epsilon }&\le C\left( \epsilon ^{-\frac{N}{2}}\Vert (\varphi ,\psi )|^{2}_{\epsilon }+ \epsilon ^{-N}\Vert (\varphi ,\psi )|^{3}_{\epsilon }+ \epsilon ^{-\frac{3N}{2}}\Vert (\varphi ,\psi )|^{4}_{\epsilon }\right) \\&\quad +C\epsilon ^{N}\sum ^{k}_{i\ne j}e^{-\frac{\min \{\sqrt{\lambda _{i}}, \sqrt{\lambda _{j}}\}|x_{i}-x_{j}|}{\epsilon }}\\ {}&\quad +O\left( \epsilon ^{N}\sum ^{k}_{i\ne j}e^{-\frac{(2\min \{\sqrt{\lambda _{i}},\sqrt{\lambda _{j}}\}-\sigma )|x_{i}-x_{j}|}{\epsilon }}\right) \\&\quad +C\epsilon ^{N}\sum ^{m}_{i\ne j}e^{-\frac{\min \{\sqrt{\overline{\lambda }_{i}}, \sqrt{\overline{\lambda }_{j}}\}|z_{i}-z_{j}|}{\epsilon }}\\ {}&\quad +O\left( \epsilon ^{N}\sum ^{m}_{i\ne j}e^{-\frac{(2\min \{\sqrt{\overline{\lambda }_{i}},\sqrt{\overline{\lambda }_{j}}\}-\sigma )|z_{i}-z_{j}|}{\epsilon }}\right) \\&\quad +C\epsilon ^{N}\sum ^{k}_{i=1}\sum ^{m}_{j=1}e^{-\frac{\min \{\sqrt{\lambda _{i}}, \sqrt{\overline{\lambda }_{j}}\}|x_{i}-z_{j}|}{\epsilon }}\\ {}&\quad +O\left( \epsilon ^{N}\sum ^{k}_{i=1}\sum ^{m}_{j=1}e^{-\frac{(2\min \{\sqrt{\lambda _{i}}, \sqrt{\overline{\lambda }_{j}}\}-\sigma )|x_{i}-z_{j}|}{\epsilon }}\right) . \end{aligned}$$

Proof

The proof is similar to that of Lemma 2.4, we only need to small changes, so, we omit the details here. \(\square \)

Based on the similar arguments as in Lemma 2.5, we have the following lemma.

Lemma 4.4

There exists \(\epsilon _{0}>0\), such that for any \(\epsilon \in (0,\epsilon _{0}],\ x_{j}\in B_{\theta }(x_{0}),\ z_{j}\in B_{\theta }(\overline{x}_{0})\), then (2.54) has a unique \((\overline{\varphi }_{\epsilon },\overline{\psi }_{\epsilon })\in \overline{{\textbf{E}}}_{\epsilon }\) and

$$\begin{aligned}&\Vert (\overline{\varphi }_{\epsilon },\overline{\psi }_{\epsilon })\Vert _{\epsilon }\le C\Vert \overline{l}_{\epsilon }\Vert _{\epsilon } \\ {}&\quad \le C\left( \epsilon ^{\frac{N+2}{2}}+\epsilon ^{\frac{N}{2}}\left( \sum ^{k}_{i\ne j}(P(x_{i})-P(x_{j}))+\sum ^{m}_{i\ne j}(Q(z_{i})-Q(z_{j}))\right) \right) \\&\qquad + C\epsilon ^{\frac{N}{2}}\sum ^{k}_{i=1}\sum ^{m}_{j=1}e^{-\frac{\min \{\sqrt{\lambda _{i}}, \sqrt{\overline{\lambda }_{j}}\}|x_{i}-z_{j}|}{\epsilon }}+C\epsilon ^{\frac{N}{2}}\sum ^{k}_{i=1}\sum ^{m}_{j=1}e^{-\frac{\min \{\sqrt{\lambda _{i}}, \sqrt{\lambda }_{j}\}|x_{i}-x_{j}|}{\epsilon }}\\&\qquad +C\epsilon ^{\frac{N}{2}}\sum ^{k}_{i=1}\sum ^{m}_{j=1}e^{-\frac{\min \{\sqrt{\overline{\lambda }_{i}}, \sqrt{\lambda _{j}}\}|z_{i}-x_{j}|}{\epsilon }}. \end{aligned}$$

Proof of Theorem 1.2

In order to solve (2.65), we define a function as following

$$\begin{aligned} \overline{K}({\textbf{x}}, {\textbf{z}})=I\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j},\mu }+\overline{\varphi }_{\epsilon },\sum ^{k}_{j=1}U_{\epsilon ,z_{j},\nu }+\overline{\psi }_{\epsilon }\right) \end{aligned}$$

where I(u, v) ia given by (3.1). From Lemma 4.1, 4.2, 4.3, 4.4 and Proposition 5.2, we have

$$\begin{aligned} \overline{K}({\textbf{x}},{\textbf{z}})&=I\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j},\mu },\sum ^{k}_{j=1}U_{\epsilon ,z_{j},\nu }\right) +\langle \overline{l}_{\epsilon },(\varphi _{\epsilon },\psi _{\epsilon })\rangle \\ {}&\quad +O(\Vert \overline{l}_{\epsilon }\Vert \Vert (\overline{\varphi }_{\epsilon },\overline{\psi }_{\epsilon })\Vert +\Vert \overline{\varphi }_{\epsilon },\overline{\psi }_{\epsilon }\Vert ^{2})\\&=\left( \frac{1}{2}-\frac{1}{p+1}\right) A\left( \sum ^{k}_{j=1}\epsilon ^{N}\lambda _{j}^{\frac{p+1}{p-1}-\frac{N}{2}}\mu ^{-\frac{2}{p-1}}+\sum ^{m}_{j=1}\epsilon ^{N}\overline{\lambda }_{j}^{\frac{p+1}{p-1}-\frac{N}{2}}\nu ^{-\frac{2}{p-1}}\right) \\&\quad -C\epsilon ^{N}\sum ^{k}_{i\ne j}e^{-\frac{\min \{\sqrt{\lambda _{i}}, \sqrt{\lambda _{j}}\}|x_{i}-x_{j}|}{\epsilon }}+O\left( \epsilon ^{N}\sum ^{k}_{i\ne j}e^{-\frac{(2\min \{\sqrt{\lambda _{i}},\sqrt{\lambda _{j}}\}-\sigma )|x_{i}-x_{j}|}{\epsilon }}\right) \\&\quad -C\epsilon ^{N}\sum ^{m}_{i\ne j}e^{-\frac{\min \{\sqrt{\overline{\lambda }_{i}}, \sqrt{\overline{\lambda }_{j}}\}-\sigma )|z_{i}-z_{j}|}{\epsilon }}+O\left( \epsilon ^{N}\sum ^{m}_{i\ne j}e^{-\frac{(2\min \{\sqrt{\overline{\lambda }_{i}},\sqrt{\overline{\lambda }_{j}}\}-\sigma )|z_{i}-z_{j}|}{\epsilon }}\right) \\&\quad -C\epsilon ^{N}\sum ^{k}_{i=1}\sum ^{m}_{j=1}e^{-\frac{\min \{\sqrt{\lambda _{i}}, \sqrt{\overline{\lambda }_{j}}\}|x_{i}-z_{j}|}{\epsilon }}+O\left( \sum ^{k}_{i=1}\sum ^{m}_{j=1}e^{-\frac{(2\min \{\sqrt{\lambda _{i}}, \sqrt{\overline{\lambda }_{j}}\}-\sigma )|x_{i}-z_{j}|}{\epsilon }}\right) \\&\quad +O(\epsilon ^{N+1}). \end{aligned}$$

Consider the following maximizing problem

$$\begin{aligned} \max _{(x,y)\in D_{1}\times D_{2}}\overline{ K}({\textbf{x}},{\textbf{z}}), \end{aligned}$$

where

$$\begin{aligned} D_{1}=\{{\textbf{x}}: x_{j}\in \overline{B_{\theta }(x_{0})}, j=1,2,\ldots k, |x_{i}-x_{j}|\ge \theta \epsilon |\ln \epsilon |^{\frac{1}{2}},\ i\ne j\}, \end{aligned}$$
$$\begin{aligned} D_{2}=\{{\textbf{z}}: z_{j}\in \overline{B_{\theta }(\overline{x}_{0})}, j=1,2,\ldots m, |z_{i}-z_{j}|\ge \theta \epsilon |\ln \epsilon |^{\frac{1}{2}},\ i\ne j\}. \end{aligned}$$

We prove that if \(\max _{(x,y)\in D_{1}\times D_{2}}\overline{ K}({\textbf{x}},{\textbf{z}}),\) is achieved by some \(({\textbf{x}}_{\epsilon },{\textbf{z}}_{\epsilon })\) in \(D_{1}\times D_{2}\), then \(({\textbf{x}}_{\epsilon },{\textbf{z}}_{\epsilon })\) is an interior point of \(D_{1}\times D_{2}\). Taking \(\overline{x}_{i}=x_{0}+\theta \epsilon \ln \epsilon |e_{j}, (j=1,2,\ldots k),\ \ \overline{z}_{i}=\overline{x}_{0}+\theta \epsilon \ln \epsilon |e_{j}, (j=1,2,\ldots m)\), then \(|\overline{x}_{i}-\overline{x}_{j}|=\theta \epsilon |\ln \epsilon |,\ \ |\overline{z}_{i}-\overline{z}_{j}|=\theta \epsilon |\ln \epsilon |\), which means that \((\overline{{\textbf{x}}}=(\overline{x}_{1},\overline{x}_{2},\ldots ,\overline{x}_{k}),\overline{{\textbf{z}}}=(\overline{z}_{1},\overline{z}_{2},\ldots ,\overline{z}_{m}))\in D_{1}\times D_{2}\), thus, we have

$$\begin{aligned}&=\left( \frac{1}{2}-\frac{1}{p+1}\right) A\left( \sum ^{k}_{j=1}\epsilon ^{N}\lambda _{j}^{\frac{p+1}{p-1}-\frac{N}{2}}\mu ^{-\frac{2}{p-1}}+\sum ^{m}_{j=1}\epsilon ^{N}\overline{\lambda }_{j}^{\frac{p+1}{p-1}-\frac{N}{2}}\nu ^{-\frac{2}{p-1}}\right) -C\epsilon ^{N+1}|\ln \epsilon |\\&\le \overline{ K}({\textbf{x}}_{\epsilon },{\textbf{z}}_{\epsilon })=(\frac{1}{2}-\frac{1}{p+1})A\left( \sum ^{k}_{j=1}\epsilon ^{N}\lambda _{j,\epsilon }^{\frac{p+1}{p-1}-\frac{N}{2}}\mu ^{-\frac{2}{p-1}}+\sum ^{m}_{j=1}\epsilon ^{N}\overline{\lambda }_{j,\epsilon }^{\frac{p+1}{p-1}-\frac{N}{2}}\nu ^{-\frac{2}{p-1}}\right) \\&\quad -C\epsilon ^{N}\sum ^{k}_{i\ne j}e^{-\frac{\min \{\sqrt{\lambda _{i,\epsilon }}, \sqrt{\lambda _{j,\epsilon }}\}|x_{i,\epsilon }-x_{j,\epsilon }|}{\epsilon }}+O\left( \epsilon ^{N}\sum ^{k}_{i\ne j}e^{-\frac{(2\min \{\sqrt{\lambda _{i,\epsilon }},\sqrt{\lambda _{j,\epsilon }}\}-\sigma )|x_{i,\epsilon }-x_{j,\epsilon }|}{\epsilon }}\right) \\&\quad -C\epsilon ^{N}\sum ^{m}_{i\ne j}e^{-\frac{\min \{\sqrt{\overline{\lambda }_{i,\epsilon }}, \sqrt{\overline{\lambda }_{j,\epsilon }}\}|z_{i,\epsilon }-z_{j,\epsilon }|}{\epsilon }}+O\left( \epsilon ^{N}\sum ^{m}_{i\ne j}e^{-\frac{(2\min \{\sqrt{\overline{\lambda }_{i,\epsilon }},\sqrt{\overline{\lambda }_{j,\epsilon }}\}-\sigma )|z_{i,\epsilon }-z_{j,\epsilon }|}{\epsilon }}\right) . \end{aligned}$$

Thus, we can get

$$\begin{aligned}&\left( \frac{1}{2}-\frac{1}{p+1}\right) A\epsilon ^{N}\left( \sum ^{k}_{j=1}\left( \lambda _{j}^{\frac{p+1}{p-1}-\frac{N}{2}}-\lambda _{j,\epsilon }^{\frac{p+1}{p-1}-\frac{N}{2}}\right) \mu ^{-\frac{2}{p-1}}\right. \\&\qquad \left. +\sum ^{m}_{j=1}\left( \overline{\lambda }_{j}^{\frac{p+1}{p-1}-\frac{N}{2}}-\overline{\lambda }_{j,\epsilon }^{\frac{p+1}{p-1}-\frac{N}{2}}\right) \nu ^{-\frac{2}{p-1}}\right) \le C\epsilon |\ln \epsilon | \end{aligned}$$

and

$$\begin{aligned} \sum ^{m}_{i\ne j}e^{-\frac{\min \{\sqrt{\overline{\lambda }_{i,\epsilon }}, \sqrt{\overline{\lambda }_{j,\epsilon }}\}|z_{i,\epsilon }-z_{j,\epsilon }|}{\epsilon }}\le C\epsilon |\ln \epsilon |, \end{aligned}$$
$$\begin{aligned} \sum ^{k}_{i\ne j}e^{-\frac{\min \{\sqrt{\lambda _{i,\epsilon }}, \sqrt{\lambda _{j,\epsilon }}\}|x_{i,\epsilon }-x_{j,\epsilon }|}{\epsilon }}\le C\epsilon |\ln \epsilon |, \end{aligned}$$

which imply that

$$\begin{aligned} \sum ^{m}_{i\ne j}\frac{|x_{i,\epsilon }-x_{j,\epsilon }|}{\epsilon }\ge C |\ln \epsilon |\ge |\ln \epsilon |^{\frac{1}{2}},\ \ \sum ^{k}_{i\ne j}\frac{|z_{i,\epsilon }-z_{j,\epsilon }|}{\epsilon }\ge C |\ln \epsilon |\ge |\ln \epsilon |^{\frac{1}{2}}. \end{aligned}$$

Therefore, \(({\textbf{x}}_{\epsilon }, {\textbf{z}}_{\epsilon })\) is an interior point of \(D_{1}\times D_{2}\), which implies that

$$\begin{aligned} (u_{\epsilon },v_{\epsilon })=\left( \sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon },\mu }+\overline{\varphi }_{\epsilon },\sum ^{k}_{j=1}U_{\epsilon ,z_{j,\epsilon },\nu }+\overline{\psi }_{\epsilon }\right) \end{aligned}$$

is a critical point of \(\overline{K}({\textbf{x}},{\textbf{z}})\). So, (1.1) has a solution of the form

$$\begin{aligned} u_{\epsilon }=\sum ^{k}_{j=1}U_{\epsilon ,x_{j,\epsilon },\mu }+\overline{\varphi }_{\epsilon },\ \ v_{\epsilon }=\sum ^{k}_{j=1}U_{\epsilon ,z_{j,\epsilon },\nu }+\overline{\psi }_{\epsilon } \end{aligned}$$

for some \(x_{j,\epsilon }\in B_{\theta }(p_{j}),\ \ z_{\epsilon ,j}\in B_{\theta }(\overline{p}_{j})\) and \(\Vert (\overline{\varphi }_{\epsilon },\overline{\psi }_{\epsilon })\Vert _{\epsilon }=O(\epsilon ^{\frac{N}{2}+1}).\) \(\square \)