Appendix
In this section, we are devoted to prove Proposition 4.1. To this end, let us first introduce some notations. Let \(H^1({\mathbb {R}}^N)\) be equipped with the usual scalar product and norm
$$\begin{aligned} \langle u,v\rangle =\int _{{\mathbb {R}}^N}(\nabla u\nabla v+uv)dx \text { and } \Vert u\Vert =\left( \int _{{\mathbb {R}}^N}\left( |\nabla u|^2+u^2\right) dx\right) ^{\frac{1}{2}}, \end{aligned}$$
and \(H^1_{\varepsilon }({\mathbb {R}}^N)\) be equipped with the usual scalar product and norm
$$\begin{aligned} \langle u,v\rangle _{\varepsilon }=\int _{{\mathbb {R}}^N}(\varepsilon ^2\nabla u\nabla v+uv)dx \text { and } \Vert u\Vert _{\varepsilon }=\left( \int _{{\mathbb {R}}^N}\left( \varepsilon ^2|\nabla u|^2+u^2\right) dx\right) ^{\frac{1}{2}}. \end{aligned}$$
Let \(\xi _i\), \(i=1,\cdots ,k\), be the critical points of V(x). We want to construct a solution \(u_\varepsilon \) of the form
$$\begin{aligned} u_{\varepsilon }(x)=\sum _{i=1}^k\left[ U\left( \frac{x-\xi _{i\varepsilon }}{\varepsilon }\right) -\varepsilon ^4W_{\xi _i}\left( \frac{x-\xi _{i\varepsilon }}{\varepsilon }\right) \right] +\phi _\varepsilon (x):=Z+\phi _\varepsilon (x), \end{aligned}$$
where
$$\begin{aligned} \xi _{i\varepsilon }=\varepsilon ^2\tau _i+\xi _i \text { with } \tau _i\in {\mathbb {R}}^N, \end{aligned}$$
\(W_{\xi _i} \in K^\perp \) is defined in (4.4) and \(\phi _\varepsilon \in K^\perp \) is a remainder term with
$$\begin{aligned} K^\perp :=span\left\{ \phi _\varepsilon \in H^1({\mathbb {R}}^N): \left\langle \phi _\varepsilon , \frac{\partial U\left( \frac{x-\xi _{i\varepsilon }}{\varepsilon }\right) }{\partial x_j}\right\rangle _{\varepsilon }=0, i=1,\cdots ,k, j=1,\cdots ,N\right\} . \end{aligned}$$
Then, \(\phi _\varepsilon \) satisfies the following equation
$$\begin{aligned} {\left\{ \begin{array}{ll} L_{\varepsilon ,\varvec{\tau }}\phi _\varepsilon =l_{\varepsilon ,\varvec{\tau }} +R_{\varepsilon ,\varvec{\tau }}\phi _\varepsilon , \ x\in {\mathbb {R}}^N,\\ \phi _\varepsilon \in H^1({\mathbb {R}}^N), \end{array}\right. } \end{aligned}$$
(5.1)
where \(L_{\varepsilon ,\varvec{\tau }}\) is a bounded linear operator in \(H^1({\mathbb {R}}^N)\), defined by
$$\begin{aligned} \langle L_{\varepsilon ,\varvec{\tau }}\phi _\varepsilon , \psi \rangle _{\varepsilon }{} & {} =\int _{{\mathbb {R}}^N}\left( \varepsilon ^2\nabla \phi _\varepsilon \nabla \psi +(1+\varepsilon ^2V(x))\phi _\varepsilon -pZ^{p-1}\phi _\varepsilon \psi \right) dx, \nonumber \\{} & {} \quad \forall \psi \in H^1({\mathbb {R}}^N), \end{aligned}$$
(5.2)
\(l_{\varepsilon ,\varvec{\tau }}\in H^1({\mathbb {R}}^N)\) satisfying
$$\begin{aligned}{} & {} \langle l_{\varepsilon ,\varvec{\tau }}, \psi \rangle _{\varepsilon } =\int _{{\mathbb {R}}^N}\left( \sum _{i=1}^k\left[ -U_i^p +\varepsilon ^4(H_{i}+pU_i^{p-1}W_i)\right] -\varepsilon ^2V(x)Z+Z^p\right) \psi dx, \nonumber \\{} & {} \quad \forall \psi \in H^1({\mathbb {R}}^N), \end{aligned}$$
(5.3)
with
$$\begin{aligned} U_i:=U\left( \frac{x-\xi _{i\varepsilon }}{\varepsilon }\right) ,\ H_{i}:=H_{\xi _i}:=H_{\xi _i}\left( \frac{x-\xi _{i\varepsilon }}{\varepsilon }\right) ,\ W_{i}:=W_{\xi _i}:=W_{\xi _i}\left( \frac{x-\xi _{i\varepsilon }}{\varepsilon }\right) , \end{aligned}$$
and \(R_{\varepsilon ,\varvec{\tau }}\phi _\varepsilon \in H^1({\mathbb {R}}^N)\) satisfying
$$\begin{aligned} \langle R_{\varepsilon ,\varvec{\tau }}\phi _\varepsilon , \psi \rangle _{\varepsilon } =\int _{{\mathbb {R}}^N}\left( (Z+\phi _\varepsilon )^p-Z^p-pZ^{p-1}\phi _\varepsilon \right) \psi dx, \ \forall \psi \in H^1({\mathbb {R}}^N). \end{aligned}$$
(5.4)
We define the projection \(Q_{\varepsilon }\) from \(H^1({\mathbb {R}}^N)\) to \(K^\perp \) as follows
$$\begin{aligned} Q_{\varepsilon }u_{\varepsilon }:=u_{\varepsilon }-\sum _{j=1}^k\sum _{i=1}^Nb_{\varepsilon ,i,j}\frac{\partial U\left( \frac{x-\xi _{j\varepsilon }}{\varepsilon }\right) }{\partial x_i}:=u_{\varepsilon }-\sum _{j=1}^k\sum _{i=1}^Nb_{\varepsilon ,i,j}\frac{\partial U_j}{\partial x_i}. \end{aligned}$$
(5.5)
Then, we have the following result, which plays an essential role in carrying out the reduction argument.
Proposition 5.1
There exist \(\varepsilon _0>0\) and \(C>0\) such that, for any \(0<\varepsilon \le \varepsilon _0\) and any \(\tau _i\) for \(i=1,\cdots ,k\), one has
$$\begin{aligned} \Vert Q_{\varepsilon }L_{\varepsilon ,\varvec{\tau }}\phi _\varepsilon \Vert _{\varepsilon } \ge C\Vert \phi _\varepsilon \Vert _{\varepsilon }, \ \ \ \text{ for } \text{ all }\ \phi _\varepsilon \in K^\perp . \end{aligned}$$
(5.6)
Proof
We argue by contradiction. Assume that there exists \(\varepsilon _n\rightarrow 0\), \(\xi _{i\varepsilon _n}\rightarrow \xi _{i}\), \(\Vert \phi _n\Vert _{\varepsilon _n}^2=\varepsilon _n^N\) with \(\phi _n\in K^\perp \) such that
$$\begin{aligned} \Vert Q_{\varepsilon _n}L_{\varepsilon _n,\varvec{\tau }}\phi _n\Vert _{\varepsilon _n} \le \frac{1}{n} \Vert \phi _n\Vert _{\varepsilon _n}. \end{aligned}$$
(5.7)
By the definition of \(L_{\varepsilon ,\varvec{\tau }}\), we have
$$\begin{aligned} \int _{{\mathbb {R}}^N}\left( \varepsilon _n^2|\nabla \phi _n|^2 +(1+\varepsilon _n^2V(x))\phi _n^2 -pZ^{p-1}\phi _n^2\right) dx&=\langle L_{\varepsilon _n,\varvec{\tau }}\phi _n,\phi _n\rangle _{\varepsilon _n}\nonumber \\&=o(1)\Vert \phi _n\Vert _{\varepsilon _n}^2=o(\varepsilon _n^N). \end{aligned}$$
(5.8)
Let us take \(R>0\) large enough such that
$$\begin{aligned} pZ^{p-1}=p\left( \sum _{i=1}^kU_i-\varepsilon _n^4\sum _{i=1}^kW_i\right) ^{p-1} \le \frac{1}{2}\varepsilon _n^2V(x), \text { in }{\mathbb {R}}^N\setminus \cup _{i=1}^kB_{\varepsilon _nR}(\xi _{i\varepsilon _n}). \end{aligned}$$
Thus,
$$\begin{aligned}&\int _{{\mathbb {R}}^N}\left( \varepsilon _n^2|\nabla \phi _n|^2 +(1+\varepsilon _n^2V(x))\phi _n^2 -pZ^{p-1}\phi _n^2\right) dx \nonumber \\&\quad =\Vert \phi _n\Vert _{\varepsilon _n}^2+\varepsilon _n^2\int _{{\mathbb {R}}^N}V(x)\phi _n^2dx -\int _{{\mathbb {R}}^N}pZ^{p-1}\phi _n^2dx \nonumber \\&\quad \ge \frac{1}{2}\varepsilon _n^N-\int _{\cup _{i=1}^kB_{\varepsilon _nR}(\xi _{i\varepsilon _n})}pZ^{p-1}\phi _n^2dx. \end{aligned}$$
(5.9)
Therefore, combining this with (5.8), we get
$$\begin{aligned} \varepsilon _n^N\le C\int _{\cup _{i=1}^kB_{\varepsilon _nR}(\xi _{i\varepsilon _n})}pZ^{p-1}\phi _n^2dx\le {\tilde{C}}\sum _{i=1}^k\int _{B_{\varepsilon _nR}(\xi _{i\varepsilon _n})}\phi _n^2dx. \end{aligned}$$
(5.10)
If we prove that
$$\begin{aligned} \int _{B_{\varepsilon _nR}(\xi _{i\varepsilon _n})}\phi _n^2dx=o(\varepsilon _n^N). \end{aligned}$$
(5.11)
Then, it is a contradiction with (5.10) and we obtain (5.6).
Next, we prove (5.11). Define
$$\begin{aligned} {\tilde{\phi }}_n(y)=\phi _n(\varepsilon _ny+\xi _{i\varepsilon _n}). \end{aligned}$$
By \(\Vert \phi _n\Vert _{\varepsilon _n}^2=\varepsilon _n^N\), we have \(\int _{{\mathbb {R}}^N}(|\nabla {\tilde{\phi }}_n|^2+{\tilde{\phi }}_n^2)dy\le C\). Thus, there exits a subsequence, still denote \({\tilde{\phi }}_n\), such that \({\tilde{\phi }}_n\rightharpoonup {\tilde{\phi }}\) in \(H^1({\mathbb {R}}^N)\) and \({\tilde{\phi }}_n\rightarrow {\tilde{\phi }}\) in \(L_{loc}^2({\mathbb {R}}^N)\). We first claim that
$$\begin{aligned} -\Delta {\tilde{\phi }}+{\tilde{\phi }}-pU^{p-1}{\tilde{\phi }}=0. \end{aligned}$$
(5.12)
In fact, for any \(\varphi \in H^1({\mathbb {R}}^N)\), we have
$$\begin{aligned} Q_{\varepsilon _n}\varphi =\varphi -\sum _{j=1}^k\sum _{i=1}^Nb_{\varepsilon _n,i,j}\frac{\partial U_j}{\partial x_i}\in K^\perp . \end{aligned}$$
Then,
$$\begin{aligned} \sum _{j=1}^k\sum _{i=1}^Nb_{\varepsilon _n,i,j}\langle \frac{\partial U_j}{\partial x_i},\frac{\partial U_m}{\partial x_l}\rangle _{\varepsilon _n} =\langle \varphi ,\frac{\partial U_m}{\partial x_l}\rangle _{\varepsilon _n}. \end{aligned}$$
Moreover, for some \(\alpha _{i,j,h,m}\), we obtain
$$\begin{aligned} b_{\varepsilon _n,h,m}=\sum _{j=1}^k\sum _{i=1}^N\alpha _{i,j,h,m}\langle \varphi ,\frac{\partial U_j}{\partial x_i}\rangle _{\varepsilon _n}. \end{aligned}$$
(5.13)
Let
$$\begin{aligned} \gamma _{i,j}=\langle L_{\varepsilon _n,\varvec{\tau }}\phi _n,\frac{\partial U_j}{\partial x_i}\rangle _{\varepsilon _n}. \end{aligned}$$
We deduce that for any \(\varphi \in K^\perp \),
$$\begin{aligned}&\int _{{\mathbb {R}}^N}\left( \varepsilon _n^2\nabla \phi _n\nabla \varphi +(1+\varepsilon _n^2V(x))\phi _n\varphi -pZ^{p-1}\phi _n\varphi \right) dx \nonumber \\&\quad =\langle L_{\varepsilon _n,\varvec{\tau }}\phi _n,\varphi \rangle _{\varepsilon _n} \nonumber \\&\quad =\langle L_{\varepsilon _n,\varvec{\tau }}\phi _n,Q_{\varepsilon _n}\varphi +\sum _{j=1}^k\sum _{i=1}^Nb_{\varepsilon _n,i,j}\frac{\partial U_j}{\partial x_i}\rangle _{\varepsilon _n} \nonumber \\&\quad =\langle L_{\varepsilon _n,\varvec{\tau }}\phi _n,Q_{\varepsilon _n}\varphi \rangle _{\varepsilon _n} +\sum _{j=1}^k\sum _{i=1}^Nb_{\varepsilon _n,i,j}\gamma _{i,j}, \end{aligned}$$
(5.14)
where
$$\begin{aligned} \langle L_{\varepsilon _n,\varvec{\tau }}\phi _n,Q_{\varepsilon _n}\varphi \rangle _{\varepsilon _n} =\langle Q_{\varepsilon _n}L_{\varepsilon _n,\varvec{\tau }}\phi _n,Q_{\varepsilon _n}\varphi \rangle _{\varepsilon _n} =o(1)\Vert \phi _n\Vert _{\varepsilon _n}\Vert Q_{\varepsilon _n}\varphi \Vert _{\varepsilon _n} =o(\varepsilon _n^{\frac{N}{2}})\Vert \varphi \Vert _{\varepsilon _n}. \end{aligned}$$
Consequently, from (5.13), we have
$$\begin{aligned}&\int _{{\mathbb {R}}^N}\left( \varepsilon _n^2\nabla \phi _n\nabla \varphi +(1+\varepsilon _n^2V(x))\phi _n\varphi -pZ^{p-1}\phi _n\varphi \right) dx =o(\varepsilon _n^{\frac{N}{2}})\Vert \varphi \Vert _{\varepsilon _n}\nonumber \\&\quad +\sum _{j=1}^k\sum _{i=1}^N\sigma _{i,j}\langle \varphi ,\frac{\partial U_j}{\partial x_i}\rangle _{\varepsilon _n}. \end{aligned}$$
(5.15)
Then, we estimate \(\sigma _{i,j}\). Let us choose \(\varphi =\frac{\partial U_m}{\partial x_h}\), we get \(\Vert \frac{\partial U_m}{\partial x_h}\Vert _{\varepsilon _n}^2=O(\varepsilon _n^{N-2})\) and for \(\phi _n\in K^\perp \),
$$\begin{aligned}&\sum _{j=1}^k\sum _{i=1}^N\sigma _{i,j}\langle \frac{\partial U_m}{\partial x_h},\frac{\partial U_j}{\partial x_i}\rangle _{\varepsilon _n} \nonumber \\&\quad =\int _{{\mathbb {R}}^N}\left( \varepsilon _n^2\nabla \phi _n\nabla \frac{\partial U_m}{\partial x_h} +(1+\varepsilon _n^2V(x))\phi _n\frac{\partial U_m}{\partial x_h} -pZ^{p-1}\phi _n\frac{\partial U_m}{\partial x_h}\right) dx+o(\varepsilon _n^{N-1}) \nonumber \\&\quad =\langle \phi _n,\frac{\partial U_m}{\partial x_h}\rangle _{\varepsilon _n} +\int _{{\mathbb {R}}^N}\left( \varepsilon _n^2V(x)\phi _n\frac{\partial U_m}{\partial x_h} -pZ^{p-1}\phi _n\frac{\partial U_m}{\partial x_h}\right) dx+o(\varepsilon _n^{N-1}) \nonumber \\&\quad =-\int _{{\mathbb {R}}^N}pZ^{p-1}\phi _n\frac{\partial U_m}{\partial x_h}dx+o(\varepsilon _n^{N-1}). \end{aligned}$$
(5.16)
By [7, Lemma 6.1.1, Lemma 2.2.2], we get that for \(\delta >0\) small,
$$\begin{aligned} \int _{{\mathbb {R}}^N}pZ^{p-1}\phi _n\frac{\partial U_m}{\partial x_h}dx =&\int _{{\mathbb {R}}^N}p\left( \sum _{j=1}^kU_j-\varepsilon _n^4\sum _{j=1}^kW_j\right) ^{p-1}\phi _n\frac{\partial U_m}{\partial x_h}dx \nonumber \\ =&p\int _{{\mathbb {R}}^N}\left[ \left( \sum _{j=1}^kU_j-\varepsilon _n^4\sum _{j=1}^kW_j\right) ^{p-1} -(U_m-\varepsilon _n^4W_m)^{p-1}\right] \nonumber \\&\qquad \phi _n\frac{\partial U_m}{\partial x_h}dx \nonumber \\&+p\int _{{\mathbb {R}}^N}(U_m-\varepsilon _n^4W_m)^{p-1}\phi _n\frac{\partial U_m}{\partial x_h}dx \nonumber \\ =&O(e^{-\frac{\delta }{\varepsilon _n}})\Vert \phi _n\Vert _{\varepsilon _n}+p\int _{{\mathbb {R}}^N}(U_m-\varepsilon _n^4W_m)^{p-1}\phi _n\frac{\partial U_m}{\partial x_h}dx. \end{aligned}$$
(5.17)
It holds
$$\begin{aligned}&p\int _{{\mathbb {R}}^N}(U_m-\varepsilon _n^4W_m)^{p-1}\phi _n\frac{\partial U_m}{\partial x_h}dx =p\int _{{\mathbb {R}}^N}U_m^{p-1}\phi _n\frac{\partial U_m}{\partial x_h}dx \\&\quad +{\left\{ \begin{array}{ll} O\left( \int _{{\mathbb {R}}^N}(-\varepsilon _n^4W_m)^{p-1}\phi _n\frac{\partial U_m}{\partial x_h}dx\right) \ {} &{}\text { if }1<p\le 2,\\ p\int _{{\mathbb {R}}^N}(-\varepsilon _n^4W_m)^{p-1}\phi _n\frac{\partial U_m}{\partial x_h}dx\\ \qquad +O\left( \int _{{\mathbb {R}}^N}U_m^{\frac{p-1}{2}}(-\varepsilon _n^4W_m)^{\frac{p-1}{2}}\phi _n\frac{\partial U_m}{\partial x_h}dx\right) \ {} &{}\text { if }2<p\le 3,\\ p\int _{{\mathbb {R}}^N}(-\varepsilon _n^4W_m)^{p-1}\phi _n\frac{\partial U_m}{\partial x_h}dx\\ \qquad +O\left( \int _{{\mathbb {R}}^N}(U_m^{p-2}\varepsilon _n^4W_m+(\varepsilon _n^4W_m)^{p-2}U_m)\phi _n\frac{\partial U_m}{\partial x_h}dx\right) \ {} &{}\text { if }p>3. \end{array}\right. } \end{aligned}$$
For \(\phi _n\in K^\perp \), we have
$$\begin{aligned} p\int _{{\mathbb {R}}^N}U_m^{p-1}\phi _n\frac{\partial U_m}{\partial x_h}dx =\int _{{\mathbb {R}}^N}\left( -\varepsilon _n^2\Delta \frac{\partial U_m}{\partial x_h}+\frac{\partial U_m}{\partial x_h}\right) \phi _ndx=\langle \phi _n,\frac{\partial U_m}{\partial x_h}\rangle _{\varepsilon _n} =0. \end{aligned}$$
(5.18)
It follows from Hölder inequality, the fact that \(W_j\) is even and \(\frac{\partial U_m}{\partial x_h}\) is odd that
$$\begin{aligned}&p\varepsilon _n^{4(p-1)}\int _{{\mathbb {R}}^N}W_m^{p-1}\phi _n\frac{\partial U_m}{\partial x_h}dx \nonumber \\&\quad \le C\varepsilon _n^{4(p-1)}\left( \int _{{\mathbb {R}}^N}W_m^{2(p-1)}\frac{\partial U_m}{\partial x_h}dx\right) ^{\frac{1}{2}}\left( \int _{{\mathbb {R}}^N}\phi _n^2\frac{\partial U_m}{\partial x_h}dx\right) ^{\frac{1}{2}} \nonumber \\&\quad \le C\varepsilon _n^{4(p-1)}\left( \left[ \int _{B_{\varepsilon _nR}(\xi _{m\varepsilon })}+\int _{{\mathbb {R}}^N\setminus B_{\varepsilon _nR}(\xi _{m\varepsilon })}\right] W_m^{2(p-1)}\frac{\partial U_m}{\partial x_h}dx\right) ^{\frac{1}{2}} \nonumber \\&\qquad \times \left( \left[ \int _{B_{\varepsilon _nR}(\xi _{m\varepsilon })}+\int _{{\mathbb {R}}^N\setminus B_{\varepsilon _nR}(\xi _{m\varepsilon }) }\right] \phi _n^2\frac{\partial U_m}{\partial x_h}dx\right) ^{\frac{1}{2}} \nonumber \\&\quad \le C\varepsilon _n^{N+4(p-1)}e^{-\frac{\delta }{\varepsilon _n}}. \end{aligned}$$
(5.19)
Similarly, we have
$$\begin{aligned} \int _{{\mathbb {R}}^N}U_m^{\frac{p-1}{2}}(-\varepsilon _n^4W_m)^{\frac{p-1}{2}}\phi _n\frac{\partial U_m}{\partial x_h}dx=o(\varepsilon _n^N) \end{aligned}$$
(5.20)
and
$$\begin{aligned} \int _{{\mathbb {R}}^N}(U_m^{p-2}\varepsilon _n^4W_m+(\varepsilon _n^4W_m)^{p-2}U_m)\phi _n\frac{\partial U_m}{\partial x_h}dx=o(\varepsilon _n^N). \end{aligned}$$
(5.21)
From (5.17)-(5.21), we get
$$\begin{aligned} p\int _{{\mathbb {R}}^N}(U_m-\varepsilon _n^4W_m)^{p-1}\phi _n\frac{\partial U_m}{\partial x_h}dx =o(\varepsilon _n^N), \end{aligned}$$
which together with (5.16) gives
$$\begin{aligned} \sum _{j=1}^k\sum _{i=1}^N\sigma _{i,j}\Bigg \langle \frac{\partial U_m}{\partial x_h},\frac{\partial U_j}{\partial x_i}\Bigg \rangle _{\varepsilon _n}=o(\varepsilon _n^{N-1}). \end{aligned}$$
Then combining this with (2.2.20), (2.2.21) of [7], we obtain
$$\begin{aligned} \sigma _{i,j}=o(\varepsilon _n). \end{aligned}$$
From (5.15), we then have
$$\begin{aligned} \int _{{\mathbb {R}}^N}\left( \varepsilon _n^2\nabla \phi _n\nabla \varphi +(1+\varepsilon _n^2V(x))\phi _n\varphi +pZ^{p-1}\phi _n\varphi \right) dx =o(\varepsilon _n^{\frac{N}{2}})\Vert \varphi \Vert _{\varepsilon _n}. \end{aligned}$$
(5.22)
Set \({\tilde{\varphi }}(y):=\varphi \left( \frac{y-\xi _{i\varepsilon _n}}{\varepsilon _n}\right) \), we find
$$\begin{aligned}&\int _{{\mathbb {R}}^N}\left( \nabla {\tilde{\phi }}_n\nabla \varphi +(1+\varepsilon _n^2V(\varepsilon _ny+\xi _{i\varepsilon _n}))\phi _n\varphi +p\left( \sum _{j=1}^kU_j-\varepsilon _n^4\sum _{j=1}^kW_j\right) ^{p-1}\right. \nonumber \\&\left. (\varepsilon _ny+\xi _{i\varepsilon _n}){\tilde{\phi }}_n\varphi \right) dy \nonumber \\&\quad =\varepsilon _n^{-N}\int _{{\mathbb {R}}^N}\left( \varepsilon _n^2\nabla \phi _n\nabla {\tilde{\varphi }} +(1+\varepsilon _n^2V(y))\phi _n{\tilde{\varphi }}+p\left( \sum _{j=1}^kU_j- \varepsilon _n^4\sum _{j=1}^kW_j\right) ^{p-1}\phi _n{\tilde{\varphi }}\right) dy \nonumber \\&\quad =o(\varepsilon _n^{-\frac{N}{2}})\Vert {\tilde{\varphi }}\Vert _{\varepsilon _n}=o(1). \end{aligned}$$
(5.23)
Letting \(n\rightarrow \infty \) in (5.23), we see that \({\tilde{\phi }}\) satisfies (5.12). Thus, there exists \(c_j\) such that
$$\begin{aligned} {\tilde{\phi }}=\sum _{j=1}^Nc_j\frac{\partial U}{\partial x_j}. \end{aligned}$$
(5.24)
Since \(\phi _n\in K^\perp \), we have
$$\begin{aligned} 0=\langle \phi _n,\frac{\partial U_i}{\partial x_j}\rangle _{\varepsilon _n} =\int _{{\mathbb {R}}^N}\left( \varepsilon _n^2\nabla \phi _n\nabla \frac{\partial U_i}{\partial x_j}+\phi _n\frac{\partial U_i}{\partial x_j}\right) dx =p\int _{{\mathbb {R}}^N}U_i^{p-1}\phi _n\frac{\partial U_i}{\partial x_j}dx, \end{aligned}$$
this implies that
$$\begin{aligned} p\int _{{\mathbb {R}}^N}U^{p-1}{\tilde{\phi }}\frac{\partial U}{\partial x_j}dx=0 \text { for every } j=1,\cdots ,N. \end{aligned}$$
Thus, \(c_j=0\) for every \(j=1,\cdots ,N\), that is, \({\tilde{\phi }}=0\), and so (5.11) follows. That concludes the proof. \(\square \)
We are now ready to carry out the reduction for (5.1). That is, we consider the following problem
$$\begin{aligned} Q_{\varepsilon }L_{\varepsilon ,\varvec{\tau }}\phi _\varepsilon =Q_{\varepsilon }l_{\varepsilon ,\varvec{\tau }} +Q_{\varepsilon }R_{\varepsilon ,\varvec{\tau }}\phi _\varepsilon , \ x\in {\mathbb {R}}^N. \end{aligned}$$
(5.25)
To this end, we need to estimate \(\Vert l_{\varepsilon ,\varvec{\tau }}\Vert _{\varepsilon }\) and \(\Vert R_{\varepsilon ,\varvec{\tau }}\phi _\varepsilon \Vert _{\varepsilon }\).
Lemma 5.2
We have
$$\begin{aligned} \Vert l_{\varepsilon ,\varvec{\tau }}\Vert _{\varepsilon }\le C\varepsilon ^{\frac{N}{2}+4+\theta }. \end{aligned}$$
Proof
By definition of \(l_{\varepsilon ,\varvec{\tau }}\), we get
$$\begin{aligned} \langle l_{\varepsilon ,\varvec{\tau }}, \psi \rangle _{\varepsilon } =&\int _{{\mathbb {R}}^N}\left( -\varepsilon ^2V(x)Z\psi +\sum _{i=1}^k\varepsilon ^4H_{i}\psi \right) dx\\&+\int _{{\mathbb {R}}^N}\left( Z^p-\sum _{i=1}^kU_i^p +\varepsilon ^4p\sum _{i=1}^kU_i^{p-1}W_i\right) \psi dx\\ =&l_1+l_2. \end{aligned}$$
It follows from (4.2) that,
$$\begin{aligned} \varepsilon ^2V(x) =&\varepsilon ^4\sum _{i=1}^Na^j_i\left( \frac{x-\xi _j}{\varepsilon }\right) ^2+O(\varepsilon ^2|x-\xi _j|^3)\\ =&\varepsilon ^4\sum _{i=1}^Na^j_i\left( \frac{x-\xi _{j\varepsilon } +\varepsilon ^2\tau _j}{\varepsilon }\right) ^2+O\left( \varepsilon ^5\left| \frac{x-\xi _{j\varepsilon } +\varepsilon ^2\tau _j}{\varepsilon }\right| ^3\right) \\ =&\varepsilon ^4\sum _{i=1}^Na^j_i\left( \frac{x-\xi _{j\varepsilon } }{\varepsilon }\right) ^2+O\left( \varepsilon ^5\left( 1+\left| \frac{x-\xi _{j\varepsilon } }{\varepsilon }\right| ^3\right) \right) . \end{aligned}$$
Therefore,
$$\begin{aligned} l_1=&\int _{{\mathbb {R}}^N}\left( -\varepsilon ^2V(x)Z\psi +\sum _{i=1}^k\varepsilon ^4H_{i}\left( \frac{x-\xi _{j\varepsilon } }{\varepsilon }\right) \psi \right) dx \nonumber \\ =&\int _{{\mathbb {R}}^N}\left[ -\varepsilon ^4\sum _{i=1}^Na^j_i\left( \frac{x-\xi _{j\varepsilon } }{\varepsilon }\right) ^2+O\left( \varepsilon ^5\left( 1+\left| \frac{x-\xi _{j\varepsilon } }{\varepsilon }\right| ^3\right) \right) \right] \nonumber \\&\times \left( \sum _{j=1}^kU_j-\varepsilon _n^4\sum _{j=1}^kW_j\right) \psi dx +\int _{{\mathbb {R}}^N}\sum _{i=1}^k\varepsilon ^4H_{i}\left( \frac{x-\xi _{j\varepsilon }}{\varepsilon }\right) \psi dx \nonumber \\ =&O\left( \int _{{\mathbb {R}}^N}\sum _{i,j=1}^k\varepsilon ^5\left( 1+\left| \frac{x-\xi _{i\varepsilon } }{\varepsilon }\right| ^3\right) U_j|\psi |dx\right. \nonumber \\&\left. +\int _{{\mathbb {R}}^N}\sum _{i,j=1}^k\varepsilon ^9\left( 1+\left| \frac{x-\xi _{i\varepsilon } }{\varepsilon }\right| ^3\right) |W_j||\psi |dx\right) \nonumber \\&+O\left( \int _{{\mathbb {R}}^N}\sum _{j=1}^k\varepsilon ^8|W_j||\psi |dx\right) \nonumber \\ =&O\left( \varepsilon ^{\frac{N}{2}+5}\Vert \psi \Vert _{\varepsilon }\right) . \end{aligned}$$
(5.26)
Next, we estimate \(l_2\). Since
$$\begin{aligned} l_2=&\int _{{\mathbb {R}}^N}\left( Z^p-\sum _{i=1}^kU_i^p +\varepsilon ^4p\sum _{i=1}^kU_i^{p-1}W_i\right) \psi dx\\ =&\int _{{\mathbb {R}}^N}\left[ Z^p-\left( \sum _{i=1}^kU_i\right) ^p-p\left( \sum _{i=1}^kU_i\right) ^{p-1} \left( -\varepsilon ^4\sum _{j=1}^kW_j\right) \right] \psi dx\\&+\int _{{\mathbb {R}}^N}\left[ \left( \sum _{i=1}^kU_i\right) ^p-\sum _{i=1}^kU_i^p\right] \psi dx\\&+\int _{{\mathbb {R}}^N}p\varepsilon ^4\left[ \sum _{i=1}^kU_i^{p-1}W_i-\left( \sum _{i=1}^kU_i\right) ^{p-1} \left( \sum _{j=1}^kW_j\right) \right] \psi dx:=l_{21}+l_{22}+l_{23}. \end{aligned}$$
We next compute \(l_{2i}\) for \(i=1,2,3\), respectively. First of all, using the Hölder inequality, we have
$$\begin{aligned} |l_{21}|\le&C {\left\{ \begin{array}{ll} \int _{{\mathbb {R}}^N}\left| \varepsilon ^4\sum _{j=1}^kW_j\right| ^p|\psi |dx,\ {} &{}\text { if }1<p\le 2,\\ \int _{{\mathbb {R}}^N}\left( \left| \varepsilon ^4\sum _{j=1}^kW_j\right| ^p +\left| \varepsilon ^4\sum _{j=1}^kW_j\right| ^2\left| \sum _{i=1}^kU_i\right| ^{p-2}\right) |\psi |dx, &{}\text { if }p>2, \end{array}\right. } \nonumber \\ =&{\left\{ \begin{array}{ll} O\left( \varepsilon ^{\frac{N}{2}+4p}\Vert \psi \Vert _{\varepsilon }\right) ,\ {} &{}\text { if }1<p\le 2,\\ O\left( \varepsilon ^{\frac{N}{2}+8}\Vert \psi \Vert _{\varepsilon }\right) , &{}\text { if }p>2. \end{array}\right. } \end{aligned}$$
(5.27)
As to \(l_{22}\), we have
$$\begin{aligned} |l_{22}|=&\left| \int _{{\mathbb {R}}^N}\left[ \left( \sum _{i=1}^kU_i\right) ^p-\sum _{i=1}^kU_i^p\right] \psi dx\right| \nonumber \\ =&\left| \left[ \int _{\cup _{l=1}^kB_{\epsilon }(\xi _l)}+\int _{{\mathbb {R}}^N\setminus \cup _{l=1}^kB_{\epsilon }(\xi _l) }\right] \left[ \left( \sum _{i=1}^kU_i\right) ^p-\sum _{i=1}^kU_i^p\right] \psi dx\right| \nonumber \\ \le&C\varepsilon ^{\frac{N}{2}}e^{-\frac{\delta }{\varepsilon }}\Vert \psi \Vert _{\varepsilon }. \end{aligned}$$
(5.28)
In the same way,
$$\begin{aligned} |l_{23}|=O\left( \varepsilon ^{\frac{N}{2}+4}e^{-\frac{\delta }{\varepsilon }}\Vert \psi \Vert _{\varepsilon }\right) . \end{aligned}$$
(5.29)
In conclusion,
$$\begin{aligned} \Vert l_{\varepsilon ,\varvec{\tau }}\Vert _{\varepsilon }\le C\varepsilon ^{\frac{N}{2}+4+\theta }. \end{aligned}$$
\(\square \)
Lemma 5.3
We have
$$\begin{aligned} \Vert R_{\varepsilon ,\varvec{\tau }}\phi _\varepsilon \Vert _{\varepsilon }\le C\varepsilon ^{N(1-\frac{\min \{2,p\}+1}{2})}\Vert \phi _\varepsilon \Vert _{\varepsilon }^{\min \{2,p\}}. \end{aligned}$$
Proof
For \(1<p\le 2\), then by the mean value Theorem, there exists \(t\in (0,1)\) such that
$$\begin{aligned} R_{\varepsilon ,\varvec{\tau }}\phi _\varepsilon =&(Z+\phi _\varepsilon )^p-Z^p-pZ^{p-1}\phi _\varepsilon =p(Z+t\phi _\varepsilon )^{p-1}-pZ^{p-1}\phi _\varepsilon =O(|\phi _\varepsilon |^p). \end{aligned}$$
Thus, from (2.2.46) in [7],
$$\begin{aligned} \left| \int _{{\mathbb {R}}^N}R_{\varepsilon ,\varvec{\tau }}\phi _\varepsilon \varphi dx\right|&\le C\int _{{\mathbb {R}}^N}|\phi _\varepsilon |^p|\varphi |dx \nonumber \\&\le C \left( \int _{{\mathbb {R}}^N}|\phi _\varepsilon |^{p+1}dx\right) ^{\frac{p}{p+1}} \left( \int _{{\mathbb {R}}^N}|\varphi |^{p+1}dx\right) ^{\frac{1}{p+1}}\nonumber \\&\le C\varepsilon ^{N(1-\frac{p+1}{2})}\Vert \phi _\varepsilon \Vert _{\varepsilon }^p\Vert \varphi \Vert _{\varepsilon }. \end{aligned}$$
(5.30)
Similarly, as for \(p>2\), we get
$$\begin{aligned} \left| \int _{{\mathbb {R}}^N}R_{\varepsilon ,\varvec{\tau }}\phi _\varepsilon \varphi dx\right| \le C\varepsilon ^{-\frac{N}{2}}\Vert \phi _\varepsilon \Vert _{\varepsilon }^2\Vert \varphi \Vert _{\varepsilon }. \end{aligned}$$
(5.31)
By (5.30) and (5.31), we deduce
$$\begin{aligned} \Vert R_{\varepsilon ,\varvec{\tau }}\phi _\varepsilon \Vert _{\varepsilon }\le C\varepsilon ^{N(1-\frac{\min \{2,p\}+1}{2})}\Vert \phi _\varepsilon \Vert _{\varepsilon }^{\min \{2,p\}}. \end{aligned}$$
\(\square \)
Based on Lemmas 5.2 and 5.3, we have the following proposition.
Proposition 5.4
For any compact set \(S\subset {\mathbb {R}}^N\), there exists \(\varepsilon _0>0\) and \(C > 0\) such that for any \(\varepsilon \in (0,\varepsilon _0)\) and for any \(\tau _i\in S\) there exists a unique \(\phi _\varepsilon \in K^\perp \) which solves equation (5.25) and
$$\begin{aligned} \Vert \phi _\varepsilon \Vert _{\varepsilon }\le C\varepsilon ^{\frac{N}{2}+4+\theta }. \end{aligned}$$
Proof
By Proposition 5.1, we define
$$\begin{aligned} \phi _\varepsilon =T(\phi _\varepsilon ):=(Q_{\varepsilon }L_{\varepsilon ,\varvec{\tau }})^{-1} (l_{\varepsilon ,\varvec{\tau }} +R_{\varepsilon ,\varvec{\tau }}\phi _\varepsilon ), \end{aligned}$$
and by Lemma 5.2,
$$\begin{aligned} \Vert (Q_{\varepsilon }L_{\varepsilon ,\varvec{\tau }})^{-1}l_{\varepsilon ,\varvec{\tau }}\Vert _{\varepsilon } \le C \Vert l_{\varepsilon ,\varvec{\tau }}\Vert _{\varepsilon } \le C\varepsilon ^{\frac{N}{2}+4+\theta }. \end{aligned}$$
Then, we define
$$\begin{aligned} B:=\{\phi _\varepsilon :\phi _\varepsilon \in K^\perp , \Vert \phi _\varepsilon \Vert _{\varepsilon }\le C\varepsilon ^{\frac{N}{2}+4+\theta }\}, \end{aligned}$$
where \(\theta \) is a small positive constant.
Step 1. We claim that T maps B to B. In fact, By Proposition 5.1, Lemma 5.2, and Lemma 5.3, we get
$$\begin{aligned} \Vert T(\phi _\varepsilon )\Vert _{\varepsilon }\le C\Vert l_{\varepsilon ,\varvec{\tau }}\Vert _{\varepsilon } +C\Vert R_{\varepsilon ,\varvec{\tau }}\phi _\varepsilon \Vert _{\varepsilon } \le C\varepsilon ^{\frac{N}{2}+4+\theta }. \end{aligned}$$
(5.32)
Step 2. We prove that T is a contraction map. Given \(\phi _1, \phi _2\in B\), then by Proposition 5.1, we have
$$\begin{aligned} \Vert T(\phi _1)-T(\phi _2)\Vert _{\varepsilon }\le C\Vert R_{\varepsilon ,\varvec{\tau }}\phi _1-R_{\varepsilon ,\varvec{\tau }}\phi _2\Vert _{\varepsilon }. \end{aligned}$$
Using the mean value Theorem, for \(t\in (0,1)\),
$$\begin{aligned}&|R_{\varepsilon ,\varvec{\tau }}\phi _1-R_{\varepsilon ,\varvec{\tau }}\phi _2| =|\left( (Z+\phi _1)^p-(Z+\phi _2)^p)\right) -pZ^{p-1}(\phi _1-\phi _2)|\\&\quad =|p\left( (Z+\phi _1+t(\phi _1-\phi _2))^{p-1}-Z^{p-1}\right) (\phi _1-\phi _2)|\\&\quad \le {\left\{ \begin{array}{ll} C(|\phi _1|^{p-1}+|\phi _2|^{p-1})|\phi _1-\phi _2|, &{}\text { if } 1<p\le 2,\\ CZ^{p-2}(|\phi _1|+|\phi _2|)|\phi _1-\phi _2|+C(|\phi _1|^{p-1}+|\phi _2|^{p-1})|\phi _1-\phi _2|, &{}\quad \text { if } p>2. \end{array}\right. } \end{aligned}$$
Therefore, for \(1<p\le 2\), we deduce
$$\begin{aligned}&\left| \int _{{\mathbb {R}}^N}|R_{\varepsilon ,\varvec{\tau }}\phi _1-R_{\varepsilon ,\varvec{\tau }}\phi _2|\varphi dx\right| \\&\quad \le C\left( \int _{{\mathbb {R}}^N}(|\phi _1|+|\phi _2|)^{p+1}dx\right) ^{\frac{p-1}{p+1}} \left( \int _{{\mathbb {R}}^N}|\phi _1-\phi _2|^{p+1}dx\right) ^{\frac{1}{p+1}} \left( \int _{{\mathbb {R}}^N}|\varphi |^{p+1}dx\right) ^{\frac{1}{p+1}}\\&\quad \le C\varepsilon ^{N\frac{1-p}{2}}\left( \Vert \phi _1\Vert _{\varepsilon }^{p-1}+\Vert \phi _2\Vert _{\varepsilon }^{p-1}\right) \Vert \phi _1-\phi _2\Vert _{\varepsilon }\Vert \varphi \Vert _{\varepsilon }\\&\quad \le \frac{1}{2} \Vert \phi _1-\phi _2\Vert _{\varepsilon }\Vert \varphi \Vert _{\varepsilon }. \end{aligned}$$
Similarly, for \(p>2\), we have
$$\begin{aligned} \left| \int _{{\mathbb {R}}^N}|R_{\varepsilon ,\varvec{\tau }}\phi _1-R_{\varepsilon ,\varvec{\tau }}\phi _2|\varphi dx\right| \le \frac{1}{2} \Vert \phi _1-\phi _2\Vert _{\varepsilon }\Vert \varphi \Vert _{\varepsilon }. \end{aligned}$$
Therefore,
$$\begin{aligned} \Vert T(\phi _1)-T(\phi _2)\Vert _{\varepsilon }\le \frac{1}{2} \Vert \phi _1-\phi _2\Vert _{\varepsilon }\Vert \varphi \Vert _{\varepsilon }. \end{aligned}$$
Using a contraction mapping Theorem, we conclude that there exists unique \(\phi _\varepsilon \in K^\perp \) satisfying \(\phi _\varepsilon =T(\phi _\varepsilon )\). Thus, by (5.32), we get
$$\begin{aligned} \Vert \phi _\varepsilon \Vert _{\varepsilon }=\Vert T(\phi _\varepsilon )\Vert _{\varepsilon }\le C\varepsilon ^{\frac{N}{2}+4+\theta }. \\ \end{aligned}$$
\(\square \)
By Proposition 5.4, we deduce that
$$\begin{aligned} L_{\varepsilon ,\varvec{\tau }}\phi _\varepsilon -l_{\varepsilon ,\varvec{\tau }} -R_{\varepsilon ,\varvec{\tau }}\phi _\varepsilon =\sum _{j=1}^k\sum _{i=1}^Na_{\varepsilon ,i,j}\frac{\partial U_j}{\partial x_i}. \end{aligned}$$
(5.33)
Next, we choose \(\tau _j\) suitably, such that \(a_{\varepsilon ,i,j}= 0\), \(i = 1,\cdots ,N\), \(j = 1,\cdots ,k\). That will conclude the proof.
Proposition 5.5
There exists \(\varepsilon _0>0\) such that for any \(\varepsilon \in (0,\varepsilon _0)\) there exists \(\tau _j\in {\mathbb {R}}^N\) with \(j=1,\cdots ,k\), such that equation (5.33) is satisfied.
Proof
Let us multiply (5.33) by \(\frac{\partial U_j}{\partial x_i}\). On the one hand, by (2.2.20) and (2.2.21) in [7], we get
$$\begin{aligned} \langle L_{\varepsilon ,\varvec{\tau }}\phi _\varepsilon -l_{\varepsilon ,\varvec{\tau }} -R_{\varepsilon ,\varvec{\tau }}\phi _\varepsilon ,\frac{\partial U_j}{\partial x_i}\rangle _{\varepsilon }&=\left\langle \sum _{l=1}^k\sum _{m=1}^Na_{\varepsilon ,m,l}\frac{\partial U_l}{\partial x_m},\frac{\partial U_j}{\partial x_i}\right\rangle _{\varepsilon }\nonumber \\&=\varepsilon ^{N-2}(\sigma _{ij}a_{\varepsilon ,i,j}+o(1)). \end{aligned}$$
(5.34)
On the other hand, using \(-\varepsilon ^2\Delta U_j+U_j=U_j^p\), we obtain
$$\begin{aligned}&\langle L_{\varepsilon ,\varvec{\tau }}\phi _\varepsilon -l_{\varepsilon ,\varvec{\tau }} -R_{\varepsilon ,\varvec{\tau }}\phi _\varepsilon ,\frac{\partial U_j}{\partial x_i}\rangle _{\varepsilon }\nonumber \\&\quad =\int _{{\mathbb {R}}^N}\left( \varepsilon ^2\nabla u_\varepsilon \nabla \frac{\partial U_j}{\partial x_i} +(1+\varepsilon ^2V(x))u_\varepsilon \frac{\partial U_j}{\partial x_i} -u_\varepsilon ^p \frac{\partial U_j}{\partial x_i} \right) dx \nonumber \\&\quad =\int _{{\mathbb {R}}^N}\left( \varepsilon ^2\nabla (u_\varepsilon -U_j) \nabla \frac{\partial U_j}{\partial x_i} +\left( u_\varepsilon -U_j\right) \frac{\partial U_j}{\partial x_i} \right) dx \nonumber \\&\qquad +\int _{{\mathbb {R}}^N}\varepsilon ^2V(x)u_\varepsilon \frac{\partial U_j}{\partial x_i}dx +\int _{{\mathbb {R}}^N}\left( U_j^p-u_\varepsilon ^p\right) \frac{\partial U_j}{\partial x_i}dx, \end{aligned}$$
(5.35)
where by [7, Lemma 2.2.2] and \(W_l\in K^\perp \), we have
$$\begin{aligned}&\int _{{\mathbb {R}}^N}\left( \varepsilon ^2\nabla (u_\varepsilon -U_j) \nabla \frac{\partial U_j}{\partial x_i} +\left( u_\varepsilon -U_j\right) \frac{\partial U_j}{\partial x_i} \right) dx \nonumber \\&\quad =\int _{{\mathbb {R}}^N}\left( \varepsilon ^2\nabla \left( \sum _{l\ne j}U_l\right) \nabla \frac{\partial U_j}{\partial x_i} +\left( \sum _{l\ne j}U_l\right) \frac{\partial U_j}{\partial x_i} \right) dx +\Bigg \langle -\varepsilon ^4\sum _{l=1}^kW_l,\frac{\partial U_j}{\partial x_i}\Bigg \rangle _{\varepsilon } \nonumber \\&\quad =O\left( \varepsilon ^N\sum _{l\ne j}e^{-\frac{\delta }{\varepsilon }|\xi _{l\varepsilon }-\xi _{j\varepsilon }|}\right) . \end{aligned}$$
(5.36)
It follows from (4.2) that
$$\begin{aligned}&\frac{\partial }{\partial y_i}\left( V(\varepsilon y+\varepsilon ^2\tau _j+\xi _j)\right) \\&\quad =a_j^i(\varepsilon y^i+\varepsilon ^2\tau _j^i) +\frac{1}{2}\sum _{l,k=1}^N\frac{\partial ^3V(\xi _j)}{\partial x_k\partial x_l\partial x_i}(\varepsilon ^2y_ly_k)+O\left( \varepsilon ^3(1+|y|^3)\right) . \end{aligned}$$
Thus,
$$\begin{aligned}&\int _{{\mathbb {R}}^N}\varepsilon ^2V(x)u_\varepsilon \frac{\partial U_j}{\partial x_i}dx\nonumber \\&\qquad =\int _{{\mathbb {R}}^N}\varepsilon ^2V(x)(Z+\phi _\varepsilon )\frac{\partial U_j}{\partial x_i}dx \nonumber \\&\qquad =\int _{{\mathbb {R}}^N}\varepsilon ^2V(x)\left( U_j-\varepsilon ^4W_j\right) \frac{\partial U_j}{\partial x_i}dx+O\left( \varepsilon ^{N+5+\theta }\right) \nonumber \\&\qquad =\int _{{\mathbb {R}}^N}\varepsilon ^{N+2}V(\varepsilon y+\varepsilon ^2\tau _j+\xi _j)\left( U-\varepsilon ^4W_j\right) \frac{1}{\varepsilon }\frac{\partial U}{\partial y_i}dy+O\left( \varepsilon ^{N+5+\theta }\right) \nonumber \\&\qquad =\int _{{\mathbb {R}}^N}\varepsilon ^{N+1}V(\varepsilon y+\varepsilon ^2\tau _j+\xi _j)U\frac{\partial U}{\partial y_i}dy+O\left( \varepsilon ^{N+5}\right) \nonumber \\&\qquad =\int _{{\mathbb {R}}^N}\varepsilon ^{N+1}V(\varepsilon y+\varepsilon ^2\tau _j+\xi _j)\frac{\partial }{\partial y_i}\left( \frac{1}{2}U^2\right) dy+O\left( \varepsilon ^{N+5}\right) \nonumber \\&\qquad =-\frac{1}{2}\int _{{\mathbb {R}}^N}\varepsilon ^{N+2}\frac{\partial }{\partial y_i}\left( V(\varepsilon y+\varepsilon ^2\tau _j+\xi _j)\right) U^2dy+O\left( \varepsilon ^{N+5}\right) \nonumber \\&\qquad =-\frac{1}{2}\varepsilon ^{N+4}\left( a_j^i\tau _j^i\int _{{\mathbb {R}}^N}U^2dx +\frac{1}{2N}\sum _{l,k=1}^N\frac{\partial ^3V(\xi _j)}{\partial y_k\partial y_l\partial y_i}\int _{{\mathbb {R}}^N}|x|^2U^2dx\right) \nonumber \\&\qquad +O\left( \varepsilon ^{N+5}\right) . \end{aligned}$$
(5.37)
For \(1<p\le 2\), we have
$$\begin{aligned} u_\varepsilon ^p-U_j^p=pU_j^{p-1}\left( \sum _{l\ne j}U_l-\varepsilon ^4\sum _{l=1}^kW_l\right) +O\left( \left( \sum _{l\ne j}U_l-\varepsilon ^4\sum _{l=1}^kW_l\right) ^p\right) . \end{aligned}$$
Then, from [7, Lemma 2.2.2], for \(l\ne j\),
$$\begin{aligned} p\int _{{\mathbb {R}}^N}U_j^{p-1}\frac{\partial U_j}{\partial x_i}U_ldx =O\left( \varepsilon ^Ne^{-\frac{\delta }{\varepsilon }|\xi _{l\varepsilon }-\xi _{j\varepsilon }|}\right) , \end{aligned}$$
and
$$\begin{aligned} \int _{{\mathbb {R}}^N}U_l^{p}\frac{\partial U_j}{\partial x_i}dx =O\left( \varepsilon ^Ne^{-\frac{\delta }{\varepsilon }|\xi _{l\varepsilon }-\xi _{j\varepsilon }|}\right) . \end{aligned}$$
Similarly,
$$\begin{aligned} p\varepsilon ^4\int _{{\mathbb {R}}^N}U_j^{p-1}\frac{\partial U_j}{\partial x_i}W_ldx=O\left( \varepsilon ^Ne^{-\frac{\delta }{\varepsilon }|\xi _{l\varepsilon }-\xi _{j\varepsilon }|}\right) . \end{aligned}$$
In terms of \(W_j\in K^\perp \), we have
$$\begin{aligned} p\varepsilon ^4\int _{{\mathbb {R}}^N}U_j^{p-1}\frac{\partial U_j}{\partial x_i}W_jdx&=\varepsilon ^4\int _{{\mathbb {R}}^N}\left( -\varepsilon ^2\Delta \frac{\partial U_j}{\partial x_i}+\frac{\partial U_j}{\partial x_i}\right) W_jdx\\&=\varepsilon ^4\langle \frac{\partial U_j}{\partial x_i},W_j\rangle _{\varepsilon } =0. \end{aligned}$$
Since W is even and \(\partial _jU\) is odd, we see
$$\begin{aligned} \varepsilon ^{4p}\int _{{\mathbb {R}}^N}W_j^p\frac{\partial U_j}{\partial x_i}dx =&\varepsilon ^{4p}\left( \int _{B_{\varepsilon R}(\xi _{j\varepsilon })}+\int _{{\mathbb {R}}^N\setminus B_{\varepsilon R}(\xi _{j\varepsilon })}\right) W_j^p\frac{\partial U_j}{\partial x_i}dx\\ =&\varepsilon ^{N+4p-1}\left( \int _{B_{R}(0)}+\int _{{\mathbb {R}}^N\setminus B_{ R}(0)}\right) W^p\frac{\partial U}{\partial x_i}dx\\ =&\varepsilon ^{N+4p-1}\int _{{\mathbb {R}}^N\setminus B_{ R}(0)}W^p\frac{\partial U}{\partial x_i}dx =O\left( \varepsilon ^{N+4p-1}e^{-\frac{\delta }{\varepsilon }}\right) . \end{aligned}$$
In conclusion, for \(1<p\le 2\),
$$\begin{aligned} \int _{{\mathbb {R}}^N}\left( U_j^p-u_\varepsilon ^p\right) \frac{\partial U_j}{\partial x_i}dx=O\left( \varepsilon ^{N}e^{-\frac{\varrho }{\varepsilon }}\right) . \end{aligned}$$
(5.38)
In the same way, we can get the similar result for the case \(p>2\). Then by (5.34)-(5.38), we get
$$\begin{aligned} \sigma _{ij}a_{\varepsilon ,i,j}=-\frac{1}{2}\varepsilon ^{6}\left( a_j^i\tau _j^i\int _{{\mathbb {R}}^N}U^2dx +\frac{1}{2N}\sum _{l,k=1}^N\frac{\partial ^3V(\xi _j)}{\partial y_k\partial y_l\partial y_i}\int _{{\mathbb {R}}^N}|x|^2U^2dx+o(1)\right) . \end{aligned}$$
Since \(\sigma _{ij}\), \(\int _{{\mathbb {R}}^N}U^2dx\) and \(\int _{{\mathbb {R}}^N}|x|^2U^2dx\) are positive constants, \(a_j^i\ne a_k^l\) for \(j\ne k\), \(i\ne l\), this implies that if \(\varepsilon \rightarrow 0\), there exists \(\tau _j^i={\tau _j^i}(\varepsilon )\) such that the right hand of the above identity is zero, thus \(a_{\varepsilon ,i,j}=0\) for every \(i=1,\cdots ,N\), \(j=1,\cdots ,k\). \(\square \)
