Uniqueness of positive solutions with concentration for the Schrödinger–Newton problem

Abstract

We are concerned with the following Schrödinger–Newton problem

$$\begin{aligned} -\varepsilon ^2\Delta u+V(x)u=\frac{1}{8\pi \varepsilon ^2} \left( \int _{\mathbb {R}^3}\frac{u^2(\xi )}{|x-\xi |}d\xi \right) u,~x\in {\mathbb {R}}^3. \end{aligned}$$

For $\varepsilon$ small enough, we show the uniqueness of positive solutions concentrating at the nondegenerate critical points of V(x). The main tools are a local Pohozaev type of identity, blow-up analysis and the maximum principle. Our results also show that the asymptotic behavior of concentrated points to Schrödinger–Newton problem is quite different from those of Schrödinger equations, which is mainly caused by the nonlocal term.

Appendices

Appendix

Some basic estimates

Lemma A.1

(Hardy–Littlewood–Sobolev inequality, c.f. [7]) Let $p,r>1, 0<\lambda <3$, $\frac{1}{p}+\frac{1}{r}+\frac{\lambda }{3}=2$, $f\in L^p(\mathbb {R}^3)$, $h\in L^r(\mathbb {R}^3)$, then there exists $C(\lambda ,p)>0$ such that

$$\begin{aligned} \int _{\mathbb {R}^3}\int _{\mathbb {R}^3}f(x)|x-y|^{-\lambda }h(y)dxdy \le C(\lambda ,p)\Vert f\Vert _{L^p(\mathbb {R}^3)}\Vert g\Vert _{L^r(\mathbb {R}^3)}. \end{aligned}$$

(A.1)

Lemma A.2

(Nash–Moser iteration, c.f. Theorem 8.17 in [23]) If $u\in H^1(\mathbb {R}^3)$ is the solution of $-\Delta u=f(x)$ in $\mathbb {R}^3$ and $f\in L^{q/2}(\mathbb {R}^3)$ for some $q>3$, then for any ball $B_{2R}(y)\subset \mathbb {R}^3$ and $p>1$, there exists $C=C(p,q)$ such that

$$\begin{aligned} \sup _{B_R(y)}u(x)\le C\big (R^{-3/p}\Vert u\Vert _{L^2(B_{2R}(y))}+R^{2(1-3/q)}\Vert f\Vert _{L^{q/2}(\mathbb {R}^3)}\big ). \end{aligned}$$

Lemma A.3

(Decomposition lemma, c.f. [24] or Lemma A.1 in [14]) For $u(x)\in H_\varepsilon$, if there exist $\delta _0>0$, $\varepsilon _0>0$ such that

$$\begin{aligned} \left\| u-\sum ^k_{j=1}U_{a_j} \left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) \right\| _\varepsilon \le \delta _0 \varepsilon ^3,~\text{ and }~|x_{j,\varepsilon }-a_j|\le \delta ,~\text{ for } \text{ all }~\delta \in (0,\delta _0]~\text{ and }~\varepsilon \in (0,\varepsilon _0], \end{aligned}$$

then for all $\delta \in (0,\delta _0]$ and $\varepsilon \in (0,\varepsilon _0]$, the following minimization problem

$$\begin{aligned} \inf \left\{ \varepsilon ^{-3}\left\| u-\sum ^k_{j=1}U_{a_j} \left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) \right\| _\varepsilon ; ~x_{j,\varepsilon }\in B_{4\delta }(a_j)\right\} \end{aligned}$$

has a unique solution which can be written as

$$\begin{aligned} u=\sum ^k_{j=1}\alpha _{j,\varepsilon } U_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) +v_\varepsilon (x), \end{aligned}$$

where $|\alpha _{j,\varepsilon }-1|\le 2\delta$, $v_\varepsilon \in \bigcap ^k_{j=1}E_{\varepsilon ,a_j,x_{j,\varepsilon }}$ and

$$\begin{aligned} \begin{aligned} E_{\varepsilon ,a_j,x_{j,\varepsilon }}&=\left\{ u(x)\in H_\varepsilon : \left( u(x),U_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) \right) _{\varepsilon }=0,\right. \\&\left. \quad ~\left( u(x),\frac{\partial {U_{a_j}\big (\frac{x-x_{j,\varepsilon }}{\varepsilon }\big )}}{\partial {x^i}}\right) _{\varepsilon }=0, ~i=1,2,3 \right\} . \end{aligned} \end{aligned}$$

Lemma A.4

Suppose $f_{\varepsilon }\in L^1({\mathbb {R}^3})\cap C(\mathbb {R}^3)$, for any fixed small ${\bar{d}}>0$ independent of $\varepsilon$ and $x_{\varepsilon }$, there exists a small constant $d_\varepsilon \in ({\bar{d}},2{\bar{d}})$ such that

$$\begin{aligned} \int _{\partial B_{d_\varepsilon }(x_{\varepsilon })}|f_{\varepsilon }(x)|d\sigma \le \frac{1}{{\bar{d}}} \int _{\mathbb {R}^3}|f_{\varepsilon }(x)|dx. \end{aligned}$$

(A.2)

Proof

First, for any fixed small ${\bar{d}}>0$ and $x_{\varepsilon }$, it holds

$$\begin{aligned} \int ^{2{\bar{d}}}_{{\bar{d}}}\int _{\partial B_{r}(x_{\varepsilon })}|f_{\varepsilon }(x)|d\sigma dr= \int _{B_{2{\bar{d}}}(x_{\varepsilon })\setminus B_{{\bar{d}}}(x_{\varepsilon })}|f_{\varepsilon }(x)|dx\le \int _{\mathbb {R}^3}|f_{\varepsilon }(x)|dx. \end{aligned}$$

(A.3)

Also $\int _{\partial B_{r}(x_{\varepsilon })}|f_{\varepsilon }(x)|d\sigma$ is continuous with respect to r. By mean value theorem of integrals, there exists $d_\varepsilon \in ({\bar{d}},2{\bar{d}})$ such that

$$\begin{aligned} \int ^{2{\bar{d}}}_{{\bar{d}}}\int _{\partial B_{r}(x_{\varepsilon })}|f_{\varepsilon }(x)|d\sigma dr= d_\varepsilon \int _{\partial B_{r}(x_{\varepsilon })}|f_{\varepsilon }(x)|d\sigma . \end{aligned}$$

(A.4)

Then (A.3) and (A.4) imply (A.2). $\square$

Lemma A.5

For any $u_1,u_2,u_3,u_4\in H_{\varepsilon }$ and $0<\lambda \le 2$, then

$$\begin{aligned} \int _{\mathbb {R}^3}\int _{\mathbb {R}^3}u_1(\xi )u_2(\xi )u_3(x)u_4(x)\cdot {|x-\xi |^{-\lambda }}d\xi dx\le C\varepsilon ^{-\lambda }\Vert u_1\Vert _{\varepsilon }\Vert u_2\Vert _{\varepsilon }\Vert u_3\Vert _{\varepsilon }\Vert u_4\Vert _{\varepsilon }. \end{aligned}$$

(A.5)

Proof

First, by Hardy–Littlewood–Sobolev inequality in Lemma A.1, we have

$$\begin{aligned} \int _{\mathbb {R}^3}\int _{\mathbb {R}^3}u_1(\xi )u_2(\xi )u_3(x)u_4(x)\cdot {|x-\xi |^{-\lambda }}d\xi dx \le C \Vert u_1\cdot u_2\Vert _{L^{\frac{6}{6-\lambda }}(\mathbb {R}^3)} \cdot \Vert u_3\cdot u_4\Vert _{L^{\frac{6}{6-\lambda }}(\mathbb {R}^3)}. \end{aligned}$$

(A.6)

Next, for $0<\lambda \le 2$, by Hölder’s inequality and Sobolev embedding, we get

$$\begin{aligned} \begin{aligned} \Vert u_1\cdot u_2\Vert _{L^{\frac{6}{6-\lambda }}(\mathbb {R}^3)}&\le \Vert u_1\Vert _{L^2(\mathbb {R}^3)}\Vert u_2\Vert _{L^{\frac{6}{3-\lambda }}(\mathbb {R}^3)}\le \Vert u_1\Vert _{\varepsilon }\Vert u_2\Vert ^{{\frac{3(2-\lambda )}{6}}}_{L^2(\mathbb {R}^3)}\Vert u_2\Vert ^{{\frac{\lambda }{2}}}_{L^6(\mathbb {R}^3)}\\&\le \varepsilon ^{-\frac{\lambda }{2}}\Vert u_1\Vert _{\varepsilon }\Vert u_2\Vert _{\varepsilon }. \end{aligned} \end{aligned}$$

(A.7)

Similarly, for $0<\lambda \le 2$, we have

$$\begin{aligned} \Vert u_3\cdot u_4\Vert _{L^{\frac{6}{6-\lambda }}(\mathbb {R}^3)}\le \varepsilon ^{-\lambda /2}\Vert u_3\Vert _{\varepsilon }\Vert u_4\Vert _{\varepsilon }. \end{aligned}$$

(A.8)

Then (A.6), (A.7) and (A.8) imply (A.5). $\square$

Lemma A.6

For any $u_1,u_2,u_3,u_4\in H^1(\mathbb {R}^3)$, and $0<\lambda \le 2$, then

$$\begin{aligned} \int _{\mathbb {R}^3}\int _{\mathbb {R}^3}\frac{u_1(\xi )u_2(\xi )u_3(x)u_4(x)}{|x-\xi |^{\lambda }}d\xi dx\le C\Vert u_1\Vert _{H^1(\mathbb {R}^3)}\Vert u_2\Vert _{H^1(\mathbb {R}^3)}\Vert u_3\Vert _{H^1(\mathbb {R}^3)}\Vert u_4\Vert _{H^1(\mathbb {R}^3)}. \end{aligned}$$

(A.9)

Proof

Similar to the proof of Lemma A.5, we can obtain (A.9) by Hardy-Littlewood-Sobolev inequality, Hölder’s inequality and Sobolev embedding. $\square$

Lemma A.7

(1) There exist two positive constants $d_1$ and $\eta$ such that, for $~j=1,2,\ldots ,k$,

$$\begin{aligned} U_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) =O(e^{-\eta /\varepsilon }), ~\text{ for }~ x\in \mathbb {R}^3\backslash B_{d}(x_{j,\varepsilon }),~\text{ and }~0<d<d_1. \end{aligned}$$

(A.10)

(2) Let $\{a_1,\ldots ,a_k\}\subset \mathbb {R}^3$ be the different nondegenerate critical points of V(x) with $k\ge 1$, then it holds

$$\begin{aligned} \int _{\mathbb {R}^3}\big (V(a_j)-V(x)\big )U_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) u(x)dx=O\big (\varepsilon ^{7/2}+\varepsilon ^ {3/2}|x_{j,\varepsilon }-a_j|^2\big )\Vert u\Vert _{\varepsilon }, \end{aligned}$$

(A.11)

and

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R}^3}\frac{\partial V(x)}{\partial x^i}U_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) u(x)dx= O\big (\varepsilon ^{5/2}+\varepsilon ^ {3/2}|x_{j,\varepsilon }-a_j|\big )\Vert u\Vert _{\varepsilon }, \end{aligned} \end{aligned}$$

(A.12)

where $u(x)\in H_{\varepsilon }$ and $j=1,2,\ldots ,k$.

Proof

First, the exponential decay of $U_{a_j}(x)$ implies (A.10). Next, since $a_j$ is a nondegenerate critical point of V(x), we know

$$\begin{aligned} V(a_j)-V(x)=-\sum ^3_{i=1}\sum ^3_{l=1}(x^i-a^{i}_{j})(x^l-a^{l}_{j})\frac{\partial ^2 V(a_{j})}{\partial x^i\partial x^l}+o(|x-a_j|^2). \end{aligned}$$

(A.13)

Then using (A.13) and Hölder’s inequality, for any small constant d, we have

$$\begin{aligned} \begin{aligned}&\left| \int _{B_{d}(x_{j,\varepsilon })}\big (V(a_j)-V(x)\big )U_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) u(x)dx\right| \\&\quad \le C\int _{B_{d}(x_{j,\varepsilon })}|x-a_{j}| ^2U_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) |u(x)|dx \\&\quad \le C\left( \int _{B_{d}(x_{j,\varepsilon })}|x-a_j|^{4}U_{a_j}^2\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) dx\right) ^{\frac{1}{2}}\left( \int _{B_{d}(x_{j,\varepsilon })}u^2(x) dx\right) ^{\frac{1}{2}}\\&\quad \le C\varepsilon ^{\frac{3}{2}}\left( \int _{B_{{d}/{\varepsilon }}(0)}|\varepsilon y+(x_{j,\varepsilon }-a_j)|^{4}U_{a_j}^2(y)\mathrm {d}y\right) ^{\frac{1}{2}}\Vert u\Vert _{\varepsilon }\\&\quad \le C\varepsilon ^{\frac{3}{2}}\big (\varepsilon ^2+|x_{j,\varepsilon }-a_j|^2\big )\Vert u \Vert _{\varepsilon }. \end{aligned} \end{aligned}$$

(A.14)

Also, by (A.10), we can deduce that

$$\begin{aligned} \left| \int _{\mathbb {R}^3\backslash B_{d}(x_{j,\varepsilon })}\big (V(a_j)-V(x)\big )U_{a_j} \left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) u(x) dx\right| \le Ce^{-\eta /\varepsilon }\Vert u\Vert _{\varepsilon }. \end{aligned}$$

(A.15)

Then from (A.14) and (A.15), we get (A.11).

Similarly, since $a_j$ is the nondegenerate critical point of V(x), we know

$$\begin{aligned} \frac{\partial V(x)}{\partial x^i}=\frac{\partial V(x)}{\partial x^i}- \frac{\partial V(a_{j})}{\partial x^i}=\sum ^3_{l=1}(x^l- a^l_{j})\frac{\partial ^2 V(a_{j})}{\partial x^i\partial x^l}+o(|x- a_{j}|),~\text{ for }~i=1,2,3. \end{aligned}$$

(A.16)

So similar to (A.14), from (A.16) and Hölder’s inequality, for any small fixed d, we have

$$\begin{aligned} \begin{aligned} \left| \int _{B_{{d}}(x_{j,\varepsilon })}\frac{\partial V(x)}{\partial x^i}U_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) u(x)dx\right|&\le C\int _{B_{{d}}(x_{j,\varepsilon })}|x-a_{j}| U_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) |u(x)|dx \\&\le C\varepsilon ^{\frac{3}{2}}\big (\varepsilon +|x_{j,\varepsilon }-a_j|\big )\Vert u \Vert _{\varepsilon }. \end{aligned} \end{aligned}$$

(A.17)

Also, by (A.10), we know

$$\begin{aligned} \left| \int _{\mathbb {R}^3\backslash B_{{d}}(x_{j,\varepsilon })}\frac{\partial V(x)}{\partial x^i}U_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) u(x)dx\right| \le Ce^{-\eta /\varepsilon }\Vert u\Vert _{\varepsilon }. \end{aligned}$$

(A.18)

Then (A.17) and (A.18) imply (A.12). $\square$

Regularization and some calculations

Let $u^{(1)}_{\varepsilon }(x)$, $u^{(2)}_{\varepsilon }(x)$ be two different positive solutions concentrating at $\{a_1,\ldots , a_k\}$. Set

$$\begin{aligned} \eta _{\varepsilon }(x)=\frac{u_{\varepsilon }^{(1)}(x)-u_{\varepsilon }^{(2)}(x)}{\Vert u_{\varepsilon }^{(1)}-u_{\varepsilon }^{(2)}\Vert _{L^{\infty }(\mathbb {R}^3)}}. \end{aligned}$$

(B.1)

Then we know $\Vert \eta _{\varepsilon }\Vert _{L^{\infty }(\mathbb {R}^3)}=1$ and

$$\begin{aligned} \begin{aligned} -\varepsilon ^2\Delta \eta _{\varepsilon }(x)+V(x)\eta _{\varepsilon }(x)=\,&E_1(x)\eta _{\varepsilon }(x)+E_2(x),~~x\in \mathbb {R}^{3}, \end{aligned} \end{aligned}$$

(B.2)

where

$$\begin{aligned} E_1(x)= \frac{1}{8\pi \varepsilon ^2} \int _{\mathbb {R}^3} \frac{\big (u^{(1)}_{\varepsilon }(\xi )\big )^2}{|x-\xi |}d\xi , ~E_2(x)=\frac{u^{(2)}_{\varepsilon }(x)}{8\pi \varepsilon ^2} \int _{\mathbb {R}^3} \frac{u^{(1)}_{\varepsilon }(\xi )+u^{(2)}_{\varepsilon }(\xi )}{|x-\xi |}\eta _\varepsilon (\xi )d\xi . \end{aligned}$$

(B.3)

Proposition B.1

For $\eta _{\varepsilon }(x)$ defined by (B.1), we have

$$\begin{aligned} \Vert \eta _{\varepsilon }\Vert _{\varepsilon }=O(\varepsilon ^{3/2}). \end{aligned}$$

(B.4)

Proof

From (B.2) we have

$$\begin{aligned} \Vert \eta _{\varepsilon }\Vert ^2_{\varepsilon }=\int _{\mathbb {R}^3}E_1(x) \eta ^2_{\varepsilon }(x)dx+\int _{\mathbb {R}^3}E_2(x) \eta _{\varepsilon }(x)dx. \end{aligned}$$

(B.5)

Next, by Hardy–Littlewood–Sobolev inequality, Hölder’s inequality and the fact $|\eta _{\varepsilon }(x)|\le 1$, we know

$$\begin{aligned} \begin{aligned} \left| \int _{\mathbb {R}^3}E_1(x) \eta ^2_{\varepsilon }(x)dx\right|&\le C \varepsilon ^{-2}\left( \int _{\mathbb {R}^3} \big |u^{(1)}_{\varepsilon }(\xi )\big |^{12/5}d\xi \right) ^{5/6} \cdot \left( \int _{\mathbb {R}^3} \big |\eta _\varepsilon (x)\big |^{12/5} dx\right) ^{5/6}\\&\le C \varepsilon ^{-2}\left( \int _{\mathbb {R}^3} \big |u^{(1)}_{\varepsilon }(\xi )\big |^{12/5}d\xi \right) ^{5/6} \cdot \left( \int _{\mathbb {R}^3} \big |\eta _\varepsilon (x)\big |^{2} dx\right) ^{5/6} \\&\le C\varepsilon ^{1/2}\Vert \eta _{\varepsilon }\Vert ^{5/3}_{\varepsilon }\le C \varepsilon ^{3}+\frac{1}{2}\Vert \eta _{\varepsilon }\Vert ^{2}_{\varepsilon }, \end{aligned} \end{aligned}$$

(B.6)

and

$$\begin{aligned} \begin{aligned}&\left| \int _{\mathbb {R}^3}E_2(x) \eta _{\varepsilon }(x)dx \right| \le C \varepsilon ^{-2}\left( \int _{\mathbb {R}^3} \big |u^{(2)}_{\varepsilon }(x)\big |^{\frac{6}{5}}dx\right) ^{\frac{5}{6}}\\&\quad \cdot \left( \int _{\mathbb {R}^3} \big |(u^{(1)}_{\varepsilon }(\xi )+u^{(2)}_{\varepsilon }(\xi ))\big |^{\frac{6}{5}} d\xi \right) ^{\frac{5}{6}}\le C\varepsilon ^{3}. \end{aligned} \end{aligned}$$

(B.7)

Then (B.5), (B.6) and (B.7) imply (B.4). $\square$

Lemma B.2

For any fixed $R>0$, it holds

$$\begin{aligned} E_1(x)=o(1)\cdot R+O\left( \frac{1}{R}\right) ,~ \text{ for } ~x\in \mathbb {R}^3\backslash \bigcup _{j=1}^k B_{R\varepsilon }(x_{j,\varepsilon }), \end{aligned}$$

(B.8)

and

$$\begin{aligned} E_2(x)=O\big (e^{-\theta ' R}\big ),~ \text{ for } ~x\in \mathbb {R}^3\backslash \bigcup _{j=1}^k B_{R\varepsilon }(x_{j,\varepsilon })~ \text{ and } \text{ some }~\theta '>0. \end{aligned}$$

(B.9)

Proof

First, we know

$$\begin{aligned} \big \{\xi ,|x-\xi |\le R \varepsilon /2\big \}\subset \mathbb {R}^3\backslash \bigcup _{j=1}^k B_{R\varepsilon /2}(x_{j,\varepsilon }), ~\text{ for }~x\in \mathbb {R}^3\backslash \bigcup _{j=1}^k B_{R\varepsilon }(x_{j,\varepsilon }), \end{aligned}$$

and $\Vert u_{\varepsilon }\Vert _{\varepsilon }=O(\varepsilon ^{3/2})$. Then by (2.9), for $x\in \mathbb {R}^3\backslash \bigcup _{j=1}^k B_{R\varepsilon }(x_{j,\varepsilon })$, it holds

$$\begin{aligned} \begin{aligned} E_1(x)=\,&\frac{1}{8\pi \varepsilon ^2}\int _{|x-\xi |\le R\varepsilon /2} \big (u^{(1)}_{\varepsilon }(\xi )\big )^2|x-\xi |^{-1}d\xi \\&+ \frac{1}{8\pi \varepsilon ^2}\int _{|x-\xi |> R\varepsilon /2} \big (u^{(1)}_{\varepsilon }(\xi )\big )^2|x-\xi |^{-1}d\xi \\ =\,&O\left( \varepsilon ^{-2}\int _{|x-\xi |\le R\varepsilon /2} \big (w^{(1)}_{\varepsilon }(\xi )\big )^2|x-\xi |^{-1}d\xi \right) +O\big (e^{-2\theta R}R^{2}\big )+O\left( \frac{1}{R}\right) . \end{aligned} \end{aligned}$$

(B.10)

Also, by Hölder’s inequality, we have

$$\begin{aligned} \begin{aligned}&\int _{|x-\xi |\le R\varepsilon /2} \big (w^{(1)}_{\varepsilon }(\xi )\big )^2|x-\xi |^{-1}d\xi \\&\quad =O\left( \left( \int _{\mathbb {R}^3} \big (w^{(1)}_{\varepsilon }(\xi ))^6d\xi \right) ^{1/3}\cdot \left( \int _{|x-\xi |\le R\varepsilon /2} {|x-\xi |^{-3/2}}d\xi \right) ^{2/3}\right) \\&\quad = R\cdot O\big (\varepsilon ^{-1}\Vert w^{(1)}_{\varepsilon }\Vert _{\varepsilon }^2\big )=o(\varepsilon ^2)\cdot R. \end{aligned} \end{aligned}$$

(B.11)

Then (B.10) and (B.11) imply (B.8).

Next for $x\in \mathbb {R}^3\backslash \bigcup _{j=1}^k B_{R\varepsilon }(x_{j,\varepsilon })$, we have

$$\begin{aligned} E_2(x)=O\big (e^{-\theta R}\big )\cdot \varepsilon ^{-2} \int _{\mathbb {R}^3} \big (u^{(1)}_{\varepsilon }(\xi )+u^{(2)}_{\varepsilon }(\xi )\big ) |x-\xi |^{-1}\cdot |\eta _\varepsilon (\xi )|d\xi , \end{aligned}$$

(B.12)

and

$$\begin{aligned} \begin{aligned}&\int _{\mathbb {R}^3} \big (u^{(1)}_{\varepsilon }(\xi )+u^{(2)}_{\varepsilon }(\xi )\big ) |x-\xi |^{-1}\cdot |\eta _\varepsilon (\xi )|d\xi \\&\quad =\int _{|x-\xi |\le R\varepsilon /2} \big (u^{(1)}_{\varepsilon }(\xi )+u^{(2)}_{\varepsilon }(\xi )\big ) |x-\xi |^{-1}\cdot |\eta _\varepsilon (\xi )|d\xi \\&\qquad + O\big ((R\varepsilon )^{-1}\Vert u^{(1)}_{\varepsilon }+u^{(2)}_{\varepsilon }\Vert _{\varepsilon }\Vert \eta _\varepsilon \Vert _{\varepsilon }\big )\\&\quad =O\big (\Vert (u^{(1)}_{\varepsilon }(\cdot )+u^{(2)}_{\varepsilon }(\cdot )\Vert _\varepsilon \big ) \cdot \left( \int _{|x-\xi |\le R\varepsilon /2} |x-\xi |^{-2}d\xi \right) ^{\frac{1}{2}}+O\big (R^{-1}\varepsilon ^2\big )\\&\quad =O\big ( (R^{\frac{1}{2}}+R^{-1}) \varepsilon ^{2}\big ). \end{aligned} \end{aligned}$$

(B.13)

Then (B.12) and (B.13) imply (B.9). $\square$

Lemma B.3

For any fixed small $d>0$, it holds

$$\begin{aligned} E_1(x)=\frac{1}{8\pi \varepsilon ^2} \left( \int _{\mathbb {R}^3}U_{a_j}^2\left( \frac{\xi -x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) {|x-\xi |}^{-1}d\xi \right) +o(1),~ \text{ in }~B_{d}(x^{(1)}_{j,\varepsilon }), \end{aligned}$$

(B.14)

and

$$\begin{aligned} E_2(x)=\frac{1}{4\pi }\cdot U_{a_j}\left( \frac{x-x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) \left( \int _{\mathbb {R}^3} U_{a_j}\left( \frac{\xi -x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) \eta _{\varepsilon }(\xi )|x-\xi |^{-1}d\xi \right) +o(1), ~\text{ in }~B_{d}(x^{(1)}_{j,\varepsilon }). \end{aligned}$$

(B.15)

Proof

For $x\in B_{d}(x^{(1)}_{j,\varepsilon })$, we have

$$\begin{aligned} \begin{aligned}&\left| E_1(x)-\frac{1}{8\pi \varepsilon ^2} \int _{\mathbb {R}^3}U_{a_j}^2\left( \frac{\xi -x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) {|x-\xi |^{-1}}d\xi \right| \\&\quad =O\left( \varepsilon ^{-2} \int _{\mathbb {R}^3} \big |w^{(1)}_{\varepsilon }(\xi )\big |\cdot \left( u^{(1)}_{\varepsilon }(\xi )+ U_{a_j} \left( \frac{\xi -x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) \right) |x-\xi |^{-1}d\xi \right) + O(e^{-\eta /\varepsilon }) \\&\quad = O\left( \varepsilon ^{-2} \int _{|x-\xi |\le C} \big |w^{(1)}_{\varepsilon }(\xi )\big |\cdot \left( u^{(1)}_{\varepsilon }(\xi )+ U_{a_j} \left( \frac{\xi -x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) \right) |x-\xi |^{-1}d\xi \right) \\&\qquad +O\left( \varepsilon ^{-2}\Vert w^{(1)}_{\varepsilon }(\cdot )\Vert _{\varepsilon }\cdot \left\| u^{(1)}_{\varepsilon }(\cdot )+ U_{a_j} \left( \frac{\cdot -x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) \right\| _{\varepsilon }\right) + O(e^{-\eta /\varepsilon }), \end{aligned} \end{aligned}$$

(B.16)

where C is a fixed constant.

On the other hand, by Hölder’s inequality, we know

$$\begin{aligned} \begin{aligned}&\int _{|x-\xi |\le C} \big |w^{(1)}_{\varepsilon }(\xi )\big |\cdot \left( u^{(1)}_{\varepsilon }(\xi )+ U_{a_j} \left( \frac{\xi -x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) \right) |x-\xi |^{-1}d\xi \\&\quad =O\left( \Vert w^{(1)}_{\varepsilon }(\cdot )\Vert _{L^6(\mathbb {R}^3)}\cdot \left\| u^{(1)}_{\varepsilon }(\cdot )+ U_{a_j} \left( \frac{\cdot -x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) \right\| _{L^2(\mathbb {R}^3)} \left( \int _{|x-\xi |\le C} |x-\xi |^{-3}d\xi \right) ^{1/3}\right) \\&\quad = O\left( \varepsilon ^{-1}\Vert w^{(1)}_{\varepsilon }(\cdot )\Vert _{\varepsilon }\cdot \left\| u^{(1)}_{\varepsilon }(\cdot )+ U_{a_j} \left( \frac{\cdot -x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) \right\| _{\varepsilon }\right) =o\big (\varepsilon ^2\big ). \end{aligned} \end{aligned}$$

(B.17)

Then (B.16) and (B.17) imply (B.14). Similar to the estimates of (B.14), combining Proposition 3.1 and Proposition 4.3, we deduce (B.15). $\square$

Estimates of the term $w_\varepsilon$

For convenience, we define the following notations:

$$\begin{aligned} R_\varepsilon (x)=\sum ^k_{l=1}U_{a_l} \left( \frac{x-x_{l,\varepsilon }}{\varepsilon }\right) ,~R^{(m)}_\varepsilon (x)=\sum ^k_{l=1}U_{a_l} \left( \frac{x-x^{(m)}_{l,\varepsilon }}{\varepsilon }\right) , ~\text{ for }~m=1,2, \end{aligned}$$

(C.1)

and

$$\begin{aligned} W_{j,\varepsilon }(x)=\sum ^k_{l=1,l\ne j}U_{a_l} \left( \frac{x-x_{l,\varepsilon }}{\varepsilon }\right) ,~W^{(m)}_{j,\varepsilon }(x)=\sum ^k_{l=1,l\ne j}U_{a_l} \left( \frac{x-x^{(m)}_{l,\varepsilon }}{\varepsilon }\right) , ~\text{ for }~m=1,2. \end{aligned}$$

(C.2)

Let $M_{\varepsilon }\big (x,w_{\varepsilon }(x)\big )$ as follows:

$$\begin{aligned} \begin{aligned} M_{\varepsilon }\big (x,w_{\varepsilon }(x)\big ):=\,&-\varepsilon ^2\Delta w_{\varepsilon }(x)+G\big (x,w_{\varepsilon }(x)\big ), \end{aligned} \end{aligned}$$

(C.3)

where

$$\begin{aligned} \begin{aligned} G\big (x,w_{\varepsilon }(x)\big )=\,&V(x)w_{\varepsilon }(x)-\frac{1}{8\pi \varepsilon ^2} \left( \int _{\mathbb {R}^3}\frac{\big (R_\varepsilon (\xi )\big )^2}{|x-\xi |}d\xi \right) w_{\varepsilon }(x)\\&+ \frac{1}{4\pi \varepsilon ^2} \left( \int _{\mathbb {R}^3}\frac{R_\varepsilon (\xi ) w_{\varepsilon }(\xi )}{|x-\xi |}d\xi \right) R_\varepsilon (x). \end{aligned} \end{aligned}$$

(C.4)

Let $u_\varepsilon (x)=R_{\varepsilon }(x)+w_\varepsilon (x)$ be the solution of (1.3), then

$$\begin{aligned} \begin{aligned} M_{\varepsilon }\big (x,w_{\varepsilon }(x)\big )=N\big (x,w_{\varepsilon }(x)\big )+ l_{\varepsilon }(x), \end{aligned} \end{aligned}$$

(C.5)

where

$$\begin{aligned} \begin{aligned} N\big (x,w_{\varepsilon }(x)\big )=\,&\frac{1}{8\pi \varepsilon ^2} \left( \int _{\mathbb {R}^3}\frac{w_{\varepsilon }^2(\xi )}{|x-\xi |}d\xi \right) \big (R_\varepsilon (x)+ w_{\varepsilon }(x)\big ) +\frac{w_{\varepsilon }(x)}{4\pi \varepsilon ^{2}}\int _{\mathbb {R}^{3}}\frac{R_\varepsilon (\xi )w_{\xi }(\xi )}{|x-\xi |}d\xi , \end{aligned} \end{aligned}$$

(C.6)

and

$$\begin{aligned} \begin{aligned} l_{\varepsilon }(x)=\,&\frac{W_{j,\varepsilon }(x)}{8\pi \varepsilon ^{2}}\int _{\mathbb {R}^{3}} \frac{W_{j,\varepsilon }(\xi )U_{a_{j}}(\frac{\xi -x_{j,\varepsilon }}{\varepsilon }) }{|x-\xi |}d\xi +\sum _{j=1}^k\big (V(a_j) - V(x)\big )U_{a_j} \big (\frac{x-x_{j,\varepsilon }}{\varepsilon }\big ). \end{aligned} \end{aligned}$$

(C.7)

Proposition C.1

Let $u_\varepsilon (x)=R_{\varepsilon }(x)+w_\varepsilon (x)$ be the solution of (1.3), then there exists a constant $\bar{\rho }>0$ independent of $\varepsilon$ such that

$$\begin{aligned} \int _{\mathbb {R}^3} M_{\varepsilon }\big (x,w_\varepsilon (x)\big )w_\varepsilon (x)dx \ge \bar{\rho } \Vert w_\varepsilon \Vert ^2_{\varepsilon }. \end{aligned}$$

(C.8)

Proof

Similar to the proof of Proposition 3.1 in [25], we can prove (C.8) by the contradiction argument and blow-up analysis. For the more details, one can refer to [15, 25]. $\square$

Proposition C.2

Suppose that $u_\varepsilon (x)=R_{\varepsilon }(x)+w_\varepsilon (x)$ is a positive solution of (1.3) and $\{a_1,\ldots ,a_k\}\subset \mathbb {R}^3$ are the different nondegenerate critical points of V(x) with $k\ge 1$, then it holds

$$\begin{aligned} \Vert w_{\varepsilon }\Vert _{\varepsilon }=O(\varepsilon ^{7/2})+O\left( \varepsilon ^ {3/2}\max _{j=1,\ldots ,k}|x_{j,\varepsilon }-a_j|^{2}\right) . \end{aligned}$$

(C.9)

Proof

First, from Proposition C.1, we know

$$\begin{aligned} \begin{aligned} \Vert w_{\varepsilon }\Vert ^2_{\varepsilon }&\le C\int _{\mathbb {R}^3}N\big (x,w_{\varepsilon }(x)\big )w_{\varepsilon }(x)dx+C \int _{\mathbb {R}^3}l_{\varepsilon }(x)w_{\varepsilon }(x)dx. \end{aligned} \end{aligned}$$

(C.10)

Next, using (2.2) and (A.5), we deduce

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R}^3}N\big (x,w_{\varepsilon }(x)\big )w_{\varepsilon }(x)dx&=\frac{1}{8\pi \varepsilon ^2} \int _{\mathbb {R}^3}\int _{\mathbb {R}^3}\frac{w_{\varepsilon }^2(\xi )}{|x-\xi |} \big (R_\varepsilon (x)+ w_{\varepsilon }(x)\big )w_{\varepsilon }(x)dxd\xi \\&\quad +\frac{1}{4\pi \varepsilon ^{2}}\int _{\mathbb {R}^3}\int _{\mathbb {R}^{3}}\frac{R_\varepsilon (\xi )w_{\xi }(\xi )}{|x-\xi |} w^2_{\varepsilon }(x)dxd\xi \\&=O\big (\varepsilon ^{-3}\Vert w_{\varepsilon }\Vert ^3_\varepsilon \cdot \Vert w_{\varepsilon }+R_\varepsilon \Vert _\varepsilon \big )=o(1)\Vert w_{\varepsilon }\Vert ^2_\varepsilon . \end{aligned} \end{aligned}$$

(C.11)

Also from (A.10) and (A.13), we have

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R}^3}l_{\varepsilon }(x)w_{\varepsilon }(x)dx=\,&\sum ^k_{j=1}\int _{\mathbb {R}^3}\big (V(a_j)-V(x)\big ) U_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) w_\varepsilon (x)d\xi dx\\&-\frac{1}{8\pi \varepsilon ^{2}}\int _{\mathbb {R}^{3}}\int _{\mathbb {R}^{3}} W_{j,\varepsilon }(\xi )U_{a_{j}}\left( \frac{\xi -x_{j,\varepsilon }}{\varepsilon }\right) W_{j,\varepsilon }(x) w_\varepsilon (x){|x-\xi |}^{-1}d\xi \\ =\,&O\left( \varepsilon ^{3/2}\Vert w_\varepsilon \Vert _{\varepsilon }\left( \varepsilon ^2 +\max _{j=1,\ldots ,k}|x_{j,\varepsilon }-a_j|^{2}\right) +e^{-\eta /\varepsilon }\right) . \end{aligned} \end{aligned}$$

(C.12)

Then (C.10), (C.11) and (C.12) imply (C.9). $\square$

Proposition C.3

Let $u_\varepsilon (x)$ be a positive solution of (1.3) as in Proposition C.2, then it holds

$$\begin{aligned} \Vert w_{\varepsilon }\Vert _{\varepsilon }=O(\varepsilon ^{7/2}). \end{aligned}$$

(C.13)

Proof

It follows from the results of Propositions 3.1 and C.2 directly. $\square$

The estimates of $A_{1,1}$ and $A_{1,2}$ in (4.12)

Lemma D.1

It holds

$$\begin{aligned} \begin{aligned} A_{1,1}=\,&\frac{1}{4\pi \varepsilon ^2}\int _{\mathbb {R}^3}\int _{\mathbb {R}^3}U^2_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) U_{a_j}\left( \frac{\xi -x_{j,\varepsilon }}{\varepsilon }\right) w_\varepsilon (\xi ) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx\\&+\frac{1}{8\pi \varepsilon ^2}\sum ^k_{l=1,l\ne j}\int _{\mathbb {R}^3}\int _{\mathbb {R}^3} U^2_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) U^2_{a_l}\left( \frac{\xi -x_{l,\varepsilon }}{\varepsilon }\right) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx+O(\varepsilon ^6). \end{aligned} \end{aligned}$$

(D.1)

Proof

First, $A_{1,1}$ can be written as follows:

$$\begin{aligned} A_{1,1}=A_{1,1,1}+A_{1,1,2}+A_{1,1,3}+A_{1,1,4}+A_{1,1,5}, \end{aligned}$$

(D.2)

where

$$\begin{aligned}&A_{1,1,1}=\frac{1}{8\pi \varepsilon ^2}\int _{ B_{d}(x_{j,\varepsilon })}\int _{\mathbb {R}^3}U^2_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) U^2_{a_j}\left( \frac{\xi -x_{j,\varepsilon }}{\varepsilon }\right) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx,\\&A_{1,1,2}=\frac{1}{4\pi \varepsilon ^2}\int _{ B_{d}(x_{j,\varepsilon })}\int _{\mathbb {R}^3}U^2_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) U_{a_j}\left( \frac{\xi -x_{j,\varepsilon }}{\varepsilon }\right) w_\varepsilon (\xi ) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx,\\&A_{1,1,3}=\frac{1}{8\pi \varepsilon ^2}\int _{ B_{d}(x_{j,\varepsilon })}\int _{\mathbb {R}^3} U^2_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) w^2_{\varepsilon }(\xi ) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx,\\&A_{1,1,4}= \frac{1}{4\pi \varepsilon ^2}\int _{ B_{d}(x_{j,\varepsilon })}\int _{\mathbb {R}^3} U^2_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) W_{j,\varepsilon }(\xi )\left( U_{a_j}\left( \frac{\xi -x_{j,\varepsilon }}{\varepsilon }\right) +w_\varepsilon (\xi )\right) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx,\\&A_{1,1,5}= \frac{1}{8\pi \varepsilon ^2}\int _{ B_{d}(x_{j,\varepsilon })}\int _{\mathbb {R}^3} U^2_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) \big (W_{j,\varepsilon }(\xi )\big )^2 \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx. \end{aligned}$$

Now by symmetry and (A.10), we have

$$\begin{aligned} \begin{aligned} A_{1,1,1}&=-\frac{1}{8\pi \varepsilon ^2}\int _{ \mathbb {R}^3 \backslash B_{d}(x_{j,\varepsilon })}\int _{\mathbb {R}^3}U^2_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) U^2_{a_j}\left( \frac{\xi -x_{j,\varepsilon }}{\varepsilon }\right) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx \\&=O(e^{-\eta /\varepsilon }), \end{aligned} \end{aligned}$$

(D.3)

and

$$\begin{aligned} \begin{aligned} A_{1,1,2}&=\frac{1}{4\pi \varepsilon ^2}\int _{\mathbb {R}^3}\int _{\mathbb {R}^3}U^2_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) U_{a_j}\left( \frac{\xi -x_{j,\varepsilon }}{\varepsilon }\right) w_\varepsilon (\xi ) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx+O(e^{-\eta /\varepsilon }). \end{aligned} \end{aligned}$$

(D.4)

Next, by (A.5) and (C.13), we get

$$\begin{aligned} \begin{aligned} A_{1,1,3}&=O\left( \varepsilon ^{-4}\left\| U_{a_j}\left( \frac{\cdot -x_{j,\varepsilon }}{\varepsilon }\right) \right\| ^2_{\varepsilon }\cdot \Vert w_\varepsilon \Vert ^2_{\varepsilon }\right) =O\big (\varepsilon ^{6} \big ). \end{aligned} \end{aligned}$$

(D.5)

Also, (2.10) and (A.10) imply

$$\begin{aligned}&W_{j,\varepsilon }(x)\left( U_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) +w_\varepsilon (x)\right) =O(e^{-\eta /\varepsilon }),~\text{ for }~x\in \mathbb {R}^3. \end{aligned}$$

(D.6)

This means

$$\begin{aligned} \begin{aligned} A_{1,1,4}&=O(e^{-\eta /\varepsilon }). \end{aligned} \end{aligned}$$

(D.7)

Also, from (A.10), we can deduce

$$\begin{aligned} \begin{aligned} A_{1,1,5}=\,&\frac{1}{8\pi \varepsilon ^2}\sum ^k_{l=1,l\ne j}\int _{\mathbb {R}^3}\int _{\mathbb {R}^3} U^2_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) U^2_{a_l}\left( \frac{\xi -x_{l,\varepsilon }}{\varepsilon }\right) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx+O\big (e^{-\eta /\varepsilon }\big ). \end{aligned} \end{aligned}$$

(D.8)

Then (D.2), (D.3), (D.4), (D.5), (D.7) and (D.8) imply (D.1). $\square$

Lemma D.2

It holds

$$\begin{aligned} \begin{aligned} A_{1,2}=\,&-\frac{1}{4\pi \varepsilon ^2}\int _{\mathbb {R}^3}\int _{\mathbb {R}^3}U^2_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) U_{a_j}\left( \frac{\xi -x_{j,\varepsilon }}{\varepsilon }\right) w_\varepsilon (\xi ) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx+O\big (\varepsilon ^{6}\big ). \end{aligned} \end{aligned}$$

(D.9)

Proof

First, $A_{1,2}$ can be written as follows:

$$\begin{aligned} A_{1,2}=A_{1,2,1}+A_{1,2,2}+A_{1,2,3}+A_{1,2,4}+A_{1,2,5}, \end{aligned}$$

(D.10)

where

$$\begin{aligned}&A_{1,2,1}=\frac{1}{4\pi \varepsilon ^2}\int _{ B_{d}(x_{j,\varepsilon })}\int _{\mathbb {R}^3}U_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) w_\varepsilon (x) U^2_{a_j}\left( \frac{\xi -x_{j,\varepsilon }}{\varepsilon }\right) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx,\\&A_{1,2,2}=\frac{1}{2\pi \varepsilon ^2}\int _{ B_{d}(x_{j,\varepsilon })}\int _{\mathbb {R}^3}U_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) w_\varepsilon (x) U_{a_j}\left( \frac{\xi -x_{j,\varepsilon }}{\varepsilon }\right) w_\varepsilon (\xi ) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx,\\&A_{1,2,3}=\frac{1}{4\pi \varepsilon ^2}\int _{ B_{d}(x_{j,\varepsilon })}\int _{\mathbb {R}^3} U_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) w_\varepsilon (x) w^2_{\varepsilon }(\xi ) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx,\\&A_{1,2,4}=\frac{1}{2\pi \varepsilon ^2}\int _{ B_{d}(x_{j,\varepsilon })}\int _{\mathbb {R}^3} U_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) w_\varepsilon (x) W_{j,\varepsilon }(\xi )\left( U_{a_j}\left( \frac{\xi -x_{j,\varepsilon }}{\varepsilon }\right) +w_\varepsilon (\xi )\right) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx,\\&A_{1,2,5}= \frac{1}{4\pi \varepsilon ^2}\int _{ B_{d}(x_{j,\varepsilon })} \int _{\mathbb {R}^3} U_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) w_\varepsilon (x) \big (W_{j,\varepsilon }(\xi )\big )^2 \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx. \end{aligned}$$

Now similar to the calculations of (D.3) and (D.4), by symmetry and (A.10), we know

$$\begin{aligned} \begin{aligned} A_{1,2,1}&=-\frac{1}{4\pi \varepsilon ^2}\int _{\mathbb {R}^3}\int _{\mathbb {R}^3}U^2_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) U_{a_j}\left( \frac{\xi -x_{j,\varepsilon }}{\varepsilon }\right) w_\varepsilon (\xi ) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx\\&\quad +O(e^{-\eta /\varepsilon }). \end{aligned} \end{aligned}$$

(D.11)

Next, by (A.5) and (C.13), we get

$$\begin{aligned} \begin{aligned} A_{1,2,2}=O\left( \varepsilon ^{-4}\left\| U_{a_j}\left( \frac{\cdot -x_{j,\varepsilon }}{\varepsilon }\right) \right\| ^2_{\varepsilon }\cdot \Vert w_\varepsilon \Vert ^2_{\varepsilon }\right) =O\big (\varepsilon ^{6} \big ), \end{aligned} \end{aligned}$$

(D.12)

and

$$\begin{aligned} A_{1,2,3}=O\left( \varepsilon ^{-4}\left\| U_{a_j}\left( \frac{\cdot -x_{j,\varepsilon }}{\varepsilon }\right) \right\| _{\varepsilon }\cdot \Vert w_\varepsilon \Vert ^3_{\varepsilon }\right) =O\big (\varepsilon ^{8} \big ). \end{aligned}$$

(D.13)

Also, similar to (D.7), we have

$$\begin{aligned} A_{1,2,4}=O(e^{-\eta /\varepsilon }). \end{aligned}$$

(D.14)

On the other hand, for $l\ne j$ and fixed small d, from (A.10) and (C.13), we have

$$\begin{aligned} \begin{aligned} \int _{ B_{d}(x_{j,\varepsilon })}&\int _{\mathbb {R}^3} U_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) w_\varepsilon (x)U^2_{a_l}\left( \frac{\xi -x_{l,\varepsilon }}{\varepsilon }\right) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx\\ =\,&\int _{ B_{d}(x_{j,\varepsilon })} \int _{\mathbb {R}^3\backslash B_{2d}(x_{j,\varepsilon })} U_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) w_\varepsilon (x)U^2_{a_l}\left( \frac{\xi -x_{l,\varepsilon }}{\varepsilon }\right) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx\\&+O\big (e^{-\eta /\varepsilon }\big )\\ =\,&O\left( \int _{ B_{d}(x_{j,\varepsilon })} \int _{\mathbb {R}^3\backslash B_{2d}(x_{j,\varepsilon })} U_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) w_\varepsilon (x)U^2_{a_l}\left( \frac{\xi -x_{l,\varepsilon }}{\varepsilon }\right) d\xi dx\right) +O\big (e^{-\eta /\varepsilon }\big )\\ =\,&O\left( \left\| U_{a_j}\left( \frac{\cdot -x_{j,\varepsilon }}{\varepsilon }\right) \right\| _{\varepsilon }\cdot \Vert w_\varepsilon \Vert _{\varepsilon }\cdot \left\| U_{a_l}\left( \frac{\cdot -x_{l,\varepsilon }}{\varepsilon }\right) \right\| ^2_{\varepsilon }\right) +O\big (e^{-\eta /\varepsilon }\big ) =O\big (\varepsilon ^{8}\big ). \end{aligned} \end{aligned}$$

(D.15)

Then (A.10) and (D.15) imply

$$\begin{aligned} \begin{aligned} A_{1,2,5}=\,&O\big (\varepsilon ^{6}\big )+O\big (e^{-\eta /\varepsilon }\big )=O\big (\varepsilon ^{6}\big ). \end{aligned} \end{aligned}$$

(D.16)

Then (D.10), (D.11), (D.12), (D.13), (D.14) and (D.16) imply (D.9). $\square$

Lemma D.3

For $l\ne j$, it holds

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R}^3}&\int _{\mathbb {R}^3} U^2_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) U^2_{a_l}\left( \frac{\xi -x_{l,\varepsilon }}{\varepsilon }\right) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx\\&=\varepsilon ^3(a^{i}_{j}-a^{i}_{l})\int _{\mathbb {R}^3}\int _{\mathbb {R}^3} U^2_{a_j}(x) U^2_{a_l}\left( \xi +\frac{x_{j,\varepsilon }-x_{l,\varepsilon }}{\varepsilon }\right) |x-\xi |^{-3}d\xi dx+o(\varepsilon ^6). \end{aligned} \end{aligned}$$

(D.17)

Proof

First, we have

$$\begin{aligned} \int _{\mathbb {R}^3}&\int _{\mathbb {R}^3} U^2_{a_j}\left( \frac{x-x_{j,\varepsilon }}{\varepsilon }\right) U^2_{a_l}\left( \frac{\xi -x_{l,\varepsilon }}{\varepsilon }\right) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx =B_{1}+B_{2}, \end{aligned}$$

(D.18)

where

$$\begin{aligned} B_{1}={\varepsilon ^4}\int _{\mathbb {R}^3}\int _{\mathbb {R}^3} U^2_{a_j}(x) U^2_{a_l}\left( \xi +\frac{x_{j,\varepsilon }-x_{l,\varepsilon }}{\varepsilon }\right) \frac{x^i}{|x-\xi |^3}d\xi dx, \end{aligned}$$

and

$$\begin{aligned} B_{2}=-{\varepsilon ^4}\int _{\mathbb {R}^3}\int _{\mathbb {R}^3} U^2_{a_j}(x) U^2_{a_l}\left( \xi +\frac{x_{j,\varepsilon }-x_{l,\varepsilon }}{\varepsilon }\right) \frac{\xi ^i}{|x-\xi |^3}d\xi dx. \end{aligned}$$

Then for small fixed $d>0$, we know

$$\begin{aligned} \begin{aligned} B_{1}=\,&{\varepsilon ^4}\int _{B_{d/\varepsilon }(0)}\int _{\mathbb {R}^3\backslash B_{(2d)/\varepsilon }(0)} U^2_{a_j}(x) U^2_{a_l}\left( \xi +\frac{x_{j,\varepsilon }-x_{l,\varepsilon }}{\varepsilon }\right) \frac{x^i}{|x-\xi |^3}d\xi dx+O\big (e^{-\eta /\varepsilon }\big )\\ =&O\left( \varepsilon ^7 \left( \int _{\mathbb {R}^3} U^2_{a_j}(x)|x|dx\right) \cdot \left( \int _{\mathbb {R}^3} U^2_{a_l}\left( \xi +\frac{x_{j,\varepsilon }-x_{l,\varepsilon }}{\varepsilon }\right) d\xi \right) \right) +O\big (e^{-\eta /\varepsilon }\big )=O\big (\varepsilon ^7\big ). \end{aligned} \end{aligned}$$

(D.19)

Also we have

$$\begin{aligned} \begin{aligned} B_{2}=\,&-{\varepsilon ^4}\int _{\mathbb {R}^3}\int _{\mathbb {R}^3} U^2_{a_j}(x) U^2_{a_l}\left( \xi +\frac{x_{j,\varepsilon }-x_{l,\varepsilon }}{\varepsilon }\right) \frac{\xi ^i+\frac{x^i_{j,\varepsilon }-x^i_{l,\varepsilon }}{\varepsilon }}{|x-\xi |^3}d\xi dx\\&+ (x^i_{j,\varepsilon }-x^i_{l,\varepsilon })\varepsilon ^3\int _{\mathbb {R}^3}\int _{\mathbb {R}^3} U^2_{a_j}(x) U^2_{a_l}\left( \xi +\frac{x_{j,\varepsilon }-x_{l,\varepsilon }}{\varepsilon }\right) |x-\xi |^{-3}d\xi dx. \end{aligned} \end{aligned}$$

(D.20)

Next, similar to (D.19), we deduce

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R}^3}&\int _{\mathbb {R}^3} U^2_{a_j}(x) U^2_{a_l}\left( \xi +\frac{x_{j,\varepsilon }-x_{l,\varepsilon }}{\varepsilon }\right) |x-\xi |^{-3}d\xi dx=O\big (\varepsilon ^3\big ), \end{aligned} \end{aligned}$$

(D.21)

and

$$\begin{aligned} \begin{aligned}&\int _{\mathbb {R}^3}\int _{\mathbb {R}^3} U^2_{a_j}(x) U^2_{a_l}\left( \xi +\frac{x_{j,\varepsilon }-x_{l,\varepsilon }}{\varepsilon }\right) \frac{\xi ^i+\frac{x^i_{j,\varepsilon }-x^i_{l,\varepsilon }}{\varepsilon }}{|x-\xi |^3}d\xi dx=O\big (\varepsilon ^3\big ). \end{aligned} \end{aligned}$$

(D.22)

Then using (2.2), (D.20), (D.21) and (D.22), we obtain

$$\begin{aligned} \begin{aligned} B_{2}=\,&\varepsilon ^3(a^i_{j}-a^i_{l}) \int _{\mathbb {R}^3}\int _{\mathbb {R}^3} U^2_{a_j}(x) U^2_{a_l}\left( \xi +\frac{x_{j,\varepsilon }-x_{l,\varepsilon }}{\varepsilon }\right) |x-\xi |^{-3}d\xi dx+o\big (\varepsilon ^6\big ). \end{aligned} \end{aligned}$$

(D.23)

Then (D.18), (D.19) and (D.23) imply (D.17). $\square$

The estimates of $F_{1,1}$, $F_{1,2}$, $F_{2,1}$ and $F_{2,3}$ in (5.6) and (5.7)

Lemma E.1

It holds

$$\begin{aligned} \begin{aligned} F_{1,1}&=G_1+\frac{1}{8\pi \varepsilon ^2} \int _{\mathbb {R}^3}\int _{\mathbb {R}^3} U_{a_j}^2\left( \frac{x-x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) \big (w^{(1)}_{\varepsilon }(\xi )+w^{(2)}_{\varepsilon }(\xi )\big )\eta _{\varepsilon }(\xi )\\&\quad \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx+o\big (\varepsilon ^4\big ), \end{aligned} \end{aligned}$$

(E.1)

where

$$\begin{aligned} G_1&=\frac{1}{8\pi \varepsilon ^2} \int _{\mathbb {R}^3}\int _{\mathbb {R}^3} U_{a_j}^2\left( \frac{x-x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) \left( U_{a_j}\left( \frac{\xi -x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) \right. \\&\quad \left. + U_{a_j}\left( \frac{\xi -x^{(2)}_{j,\varepsilon }}{\varepsilon }\right) \right) \eta _{\varepsilon }(\xi ) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx. \end{aligned}$$

Proof

$F_{1,1}$ can be written as

$$\begin{aligned} \begin{aligned} F_{1,1}&=F_{1,1,1}+F_{1,1,2}+F_{1,1,3}, \end{aligned} \end{aligned}$$

(E.2)

where

$$\begin{aligned}&F_{1,1,1}=\frac{1}{8\pi \varepsilon ^2} \int _{ B_{\delta }(x_{j,\varepsilon }^{(1)})}\int _{\mathbb {R}^3} U_{a_j}^2\left( \frac{x-x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) \left( U_{a_j}\left( \frac{\xi -x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) + U_{a_j}\left( \frac{\xi -x^{(2)}_{j,\varepsilon }}{\varepsilon }\right) \right) \\&\quad \eta _{\varepsilon }(\xi ) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx,\\&F_{1,1,2}=\frac{1}{8\pi \varepsilon ^2} \int _{ B_{\delta }(x_{j,\varepsilon }^{(1)})}\int _{\mathbb {R}^3} U_{a_j}^2\left( \frac{x-x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) \big (w^{(1)}_{\varepsilon }(\xi )+w^{(2)}_{\varepsilon }(\xi )\big )\eta _{\varepsilon }(\xi ) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx,\\&F_{1,1,3}=\frac{1}{8\pi \varepsilon ^2} \int _{ B_{\delta }(x_{j,\varepsilon }^{(1)})}\int _{\mathbb {R}^3} U_{a_j}^2\left( \frac{x-x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) \big ( W^{(1)}_{j,\varepsilon }(\xi )+W^{(2)}_{j,\varepsilon }(\xi )\big )\eta _{\varepsilon }(\xi ) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx. \end{aligned}$$

Now, by (A.10), we get

$$\begin{aligned} \begin{aligned} F_{1,1,1}&=G_1+O\big (e^{-\eta /\varepsilon }\big ), \end{aligned} \end{aligned}$$

(E.3)

and

$$\begin{aligned} \begin{aligned} F_{1,1,2}&=\frac{1}{8\pi \varepsilon ^2} \int _{\mathbb {R}^3}\int _{\mathbb {R}^3} U_{a_j}^2\left( \frac{x-x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) \big (w^{(1)}_{\varepsilon }(\xi )+w^{(2)}_{\varepsilon }(\xi )\big )\eta _{\varepsilon }(\xi ) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx\\&\quad +O\big (e^{-\eta /\varepsilon }\big ). \end{aligned} \end{aligned}$$

(E.4)

Next, using Proposition 5.1, we can calculate that, for $l\ne j$,

$$\begin{aligned} \begin{aligned}&\frac{1}{8\pi \varepsilon ^2} \int _{ B_{\delta }(x_{j,\varepsilon }^{(1)})}\int _{\mathbb {R}^3} U_{a_j}^2\left( \frac{x-x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) U_{a_l}\left( \frac{\xi -x^{(1)}_{l,\varepsilon }}{\varepsilon }\right) \eta _{\varepsilon }(\xi ) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx\\&\quad = \frac{\varepsilon ^2}{8\pi } \int _{\mathbb {R}^3}\int _{\mathbb {R}^3} U_{a_j}^2\left( x-\frac{x^{(1)}_{j,\varepsilon }-x^{(1)}_{l,\varepsilon }}{\varepsilon }\right) U_{a_l}(\xi )\eta _{l,\varepsilon }(\xi ) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx+O\big (e^{-\eta /\varepsilon }\big )\\&\quad = \frac{\varepsilon ^2}{8\pi } \int _{\mathbb {R}^3}\int _{\mathbb {R}^3} U_{a_j}^2\left( x-\frac{x^{(1)}_{j,\varepsilon }-x^{(1)}_{l,\varepsilon }}{\varepsilon }\right) U_{a_l}(\xi )\left( \sum ^3_{m=1}d_{m,l} \frac{\partial U_{a_l}(\xi )}{\partial \xi _m}\right) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx\\&\qquad +o\left( \varepsilon ^2\int _{\mathbb {R}^3}\int _{\mathbb {R}^3} U_{a_j}^2\left( x-\frac{x^{(1)}_{j,\varepsilon }-x^{(1)}_{l,\varepsilon }}{\varepsilon }\right) U_{a_l}(\xi ) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx\right) +O\big (e^{-\eta /\varepsilon }\big )\\&\quad = \frac{\varepsilon ^2}{16\pi } \int _{\mathbb {R}^3}\int _{\mathbb {R}^3} U_{a_j}^2\left( x-\frac{x^{(1)}_{j,\varepsilon }-x^{(1)}_{l,\varepsilon }}{\varepsilon }\right) U^2_{a_l}(\xi )\left( \sum ^3_{m=1}d_{m,l} \frac{\partial \frac{x^i-\xi ^i}{|x-\xi |^3}}{\partial \xi _m}\right) d\xi dx +o\big (\varepsilon ^4\big ), \end{aligned} \end{aligned}$$

(E.5)

and

$$\begin{aligned} \begin{aligned}&\int _{\mathbb {R}^3}\int _{\mathbb {R}^3} U_{a_j}^2\left( x-\frac{x^{(1)}_{j,\varepsilon }-x^{(1)}_{l,\varepsilon }}{\varepsilon }\right) U^2_{a_l}(\xi )\left( \sum ^3_{m=1}d_{m,l} \frac{\partial \frac{x^i-\xi ^i}{|x-\xi |^3}}{\partial \xi _m}\right) d\xi dx \\&\quad = O\left( \int _{\mathbb {R}^3}\int _{\mathbb {R}^3} U_{a_j}^2\left( x-\frac{x^{(1)}_{j,\varepsilon }-x^{(1)}_{l,\varepsilon }}{\varepsilon }\right) U^2_{a_l}(\xi )|x-\xi |^{-3} d\xi dx\right) = O\big (\varepsilon ^3\big ), \end{aligned} \end{aligned}$$

(E.6)

here we also use the following estimate, which can be found by (A.9),

$$\begin{aligned} \int _{\mathbb {R}^3}\int _{\mathbb {R}^3} U_{a_j}^2\left( x-\frac{x^{(1)}_{j,\varepsilon }-x^{(1)}_{l,\varepsilon }}{\varepsilon }\right) U^2_{a_l}(\xi )|x-\xi |^{-\alpha } d\xi dx=O\big (\varepsilon ^\alpha \big ),~\text{ for }~\alpha >0,~\text{ and }~l\ne j. \end{aligned}$$

Similar to (E.5) and (E.6), we have

$$\begin{aligned} \begin{aligned}&\frac{1}{8\pi \varepsilon ^2} \int _{ B_{\delta }(x_{j,\varepsilon }^{(1)})}\int _{\mathbb {R}^3} U_{a_j}^2\left( \frac{x-x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) U_{a_l}\left( \frac{\xi -x^{(2)}_{l,\varepsilon }}{\varepsilon }\right) \eta _{\varepsilon }(\xi ) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx=o\big (\varepsilon ^4\big ). \end{aligned} \end{aligned}$$

(E.7)

Then (E.5), (E.6) and (E.7) imply

$$\begin{aligned} \begin{aligned} F_{1,1,3}=o\big (\varepsilon ^4\big ). \end{aligned} \end{aligned}$$

(E.8)

Then (E.1) can be deduced by (E.2), (E.3), (E.4) and (E.8). $\square$

Lemma E.2

It holds

$$\begin{aligned} \begin{aligned} F_{1,2}&=\frac{1}{2\pi \varepsilon ^2} \int _{\mathbb {R}^3}\int _{\mathbb {R}^3} U_{a_j}\left( \frac{x-x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) w_\varepsilon ^{(1)}(x) U_{a_j}\left( \frac{\xi -x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) \big )\eta _{\varepsilon }(\xi ) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx +o\big (\varepsilon ^4\big ). \end{aligned} \end{aligned}$$

(E.9)

Proof

First, we write $F_{1,2}$ as follows:

$$\begin{aligned} F_{1,2}=F_{1,2,1}+F_{1,2,2}+F_{1,2,3}, \end{aligned}$$

(E.10)

where

$$\begin{aligned}&F_{1,2,1}=\frac{1}{4\pi \varepsilon ^2} \int _{ B_{\delta }(x_{j,\varepsilon }^{(1)})}\int _{\mathbb {R}^3} U_{a_j}\left( \frac{x-x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) w_\varepsilon ^{(1)}(x) \left( U_{a_j}\left( \frac{\xi -x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) + U_{a_j}\left( \frac{\xi -x^{(2)}_{j,\varepsilon }}{\varepsilon }\right) \right) \\&\quad \eta _{\varepsilon }(\xi ) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx,\\&F_{1,2,2}=\frac{1}{4\pi \varepsilon ^2} \int _{ B_{\delta }(x_{j,\varepsilon }^{(1)})}\int _{\mathbb {R}^3} U_{a_j}\left( \frac{x-x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) w_\varepsilon ^{(1)}(x)\big ( W^{(1)}_{j,\varepsilon }(\xi )+W^{(2)}_{j,\varepsilon }(\xi )\big )\eta _{\varepsilon }(\xi ) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx,\\&F_{1,2,3}=\frac{1}{4\pi \varepsilon ^2} \int _{ B_{\delta }(x_{j,\varepsilon }^{(1)})}\int _{\mathbb {R}^3} U_{a_j}\left( \frac{x-x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) w_\varepsilon ^{(1)}(x) \big (w^{(1)}_{\varepsilon }(\xi )+w^{(2)}_{\varepsilon }(\xi )\big )\eta _{\varepsilon }(\xi ) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx. \end{aligned}$$

Now by direct calculation, we get

$$\begin{aligned} \begin{aligned} U_{a_j}\left( \frac{x-x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) -U_{a_j}\left( \frac{x-x^{(2)}_{j,\varepsilon }}{\varepsilon }\right) =\,&O\left( \left| \frac{x^{(1)}_{j,\varepsilon }-x^{(2)}_{j,\varepsilon }}{\varepsilon }\right| \right) \cdot \left| \nabla U_{a_j}\left( \frac{x-x^{(1)}_{j,\varepsilon }}{\varepsilon }x\right) \right| \\ =\,&O\left( \frac{|x^{(1)}_{j,\varepsilon }-x^{(2)}_{j,\varepsilon }|}{\varepsilon }\right) U_{a_j} \left( \frac{x-x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) . \end{aligned} \end{aligned}$$

(E.11)

Then by (A.10), (E.11), we have

$$\begin{aligned} \begin{aligned} F_{1,2,1}&=\frac{1}{2\pi \varepsilon ^2} \int _{\mathbb {R}^3}\int _{\mathbb {R}^3} U_{a_j}\left( \frac{x-x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) w_\varepsilon ^{(1)}(x) U_{a_j}\\&\left( \frac{\xi -x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) \eta _{\varepsilon }(\xi ) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx+o\big (\varepsilon ^{4}\big ). \end{aligned} \end{aligned}$$

(E.12)

Also, similar to (D.15), we get

$$\begin{aligned} \begin{aligned} F_{1,2,2}=\,&O\left( \varepsilon ^{-2} \left\| U_{a_j}\left( \frac{\cdot -x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) \right\| _{\varepsilon }\cdot \Vert w_\varepsilon ^{(1)} \Vert _{\varepsilon }\cdot \Vert W^{(1)}_{j,\varepsilon }(\cdot )+W^{(2)}_{j,\varepsilon }(\cdot )\Vert _{\varepsilon }\cdot \Vert \eta _{\varepsilon }\Vert _{\varepsilon } \right) \\&+O\big (e^{-\eta /\varepsilon }\big )= O\big (\varepsilon ^6\big ). \end{aligned} \end{aligned}$$

(E.13)

And by (A.5) and (C.13), we obtain

$$\begin{aligned} \begin{aligned} F_{1,2,3}&=O\left( \varepsilon ^{-4}\left\| U_{a_j}\left( \frac{\cdot -x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) \right\| _{\varepsilon }\Vert w^{(1)}_{\varepsilon }\Vert _{\varepsilon } \cdot \Vert w^{(1)}_{\varepsilon }+w^{(2)}_{\varepsilon }\Vert _{\varepsilon }\cdot \Vert \eta _\varepsilon \Vert _{\varepsilon }\right) =O\big (\varepsilon ^6\big ). \end{aligned} \end{aligned}$$

(E.14)

Then (E.9) can be deduced by (E.10), (E.12), (E.13) and (E.14). $\square$

Lemma E.3

It holds

$$\begin{aligned} \begin{aligned} F_{2,1}&=G_2-\frac{1}{2\pi \varepsilon ^2} \int _{\mathbb {R}^3}\int _{\mathbb {R}^3} U_{a_j}\left( \frac{x-x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) w_\varepsilon ^{(1)}(x) U_{a_j}\left( \frac{\xi -x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) \eta _{\varepsilon }(\xi ) \\&\quad \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx +o\big (\varepsilon ^4\big ), \end{aligned} \end{aligned}$$

(E.15)

where

$$\begin{aligned} G_2=\,&-\frac{1}{8\pi \varepsilon ^2} \int _{\mathbb {R}^3}\int _{\mathbb {R}^3}U^2_{a_j}\left( \frac{x-x^{(2)}_{j,\varepsilon }}{\varepsilon }\right) \left( U_{a_j}\left( \frac{\xi -x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) \right. \\&\left. + U_{a_j}\left( \frac{\xi -x^{(2)}_{j,\varepsilon }}{\varepsilon }\right) \right) \eta _{\varepsilon }(x)\frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx. \end{aligned}$$

Proof

First, we write $F_{2,1}$ as follows:

$$\begin{aligned} F_{2,1}=F_{2,1,1}+F_{2,1,2}+F_{2,1,3}+F_{2,1,4}+F_{2,1,5}+F_{2,1,6}, \end{aligned}$$

(E.16)

where

$$\begin{aligned}&F_{2,1,1}=\frac{1}{8\pi \varepsilon ^2} \int _{B_{\delta }(x_{j,\varepsilon }^{(1)})}\int _{\mathbb {R}^3} \left( U_{a_j}\left( \frac{x-x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) + U_{a_j}\left( \frac{x-x^{(2)}_{j,\varepsilon }}{\varepsilon }\right) \right) \eta _{\varepsilon }(x)U^2_{a_j}\left( \frac{\xi -x^{(2)}_{j,\varepsilon }}{\varepsilon }\right) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx,\\&F_{2,1,2}=\frac{1}{4\pi \varepsilon ^2} \int _{B_{\delta }(x_{j,\varepsilon }^{(1)})}\int _{\mathbb {R}^3}\left( U_{a_j}\left( \frac{x-x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) + U_{a_j}\left( \frac{x-x^{(2)}_{j,\varepsilon }}{\varepsilon }\right) \right) \eta _{\varepsilon }(x)U_{a_j}\left( \frac{\xi -x^{(2)}_{j,\varepsilon }}{\varepsilon }\right) w^{(2)}_{\varepsilon }(\xi )\frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx, \\&F_{2,1,3}=\frac{1}{8\pi \varepsilon ^2} \int _{B_{\delta }(x_{j,\varepsilon }^{(1)})}\int _{\mathbb {R}^3}\left( U_{a_j}\left( \frac{x-x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) + U_{a_j}\left( \frac{x-x^{(2)}_{j,\varepsilon }}{\varepsilon }\right) \right) \eta _{\varepsilon }(x)\big (w^{(2)}_{\varepsilon }(\xi )\big )^2\frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx, \\&F_{2,1,4}=\frac{1}{4\pi \varepsilon ^2} \int _{B_{\delta }(x_{j,\varepsilon }^{(1)})}\int _{\mathbb {R}^3} \left( U_{a_j}\left( \frac{x-x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) + U_{a_j}\left( \frac{x-x^{(2)}_{j,\varepsilon }}{\varepsilon }\right) \right) \eta _{\varepsilon }(x)W^{(2)}_{j,\varepsilon }(\xi ) U_{a_j}\left( \frac{\xi -x^{(2)}_{j,\varepsilon }}{\varepsilon }\right) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx, \\&F_{2,1,5}=\frac{1}{4\pi \varepsilon ^2} \int _{B_{\delta }(x_{j,\varepsilon }^{(1)})}\int _{\mathbb {R}^3} \left( U_{a_j}\left( \frac{x-x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) + U_{a_j}\left( \frac{x-x^{(2)}_{j,\varepsilon }}{\varepsilon }\right) \right) \eta _{\varepsilon }(x)w^{(2)}_{\varepsilon }(\xi ) U_{a_j}\left( \frac{\xi -x^{(2)}_{j,\varepsilon }}{\varepsilon }\right) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx, \\&F_{2,1,6}=\frac{1}{8\pi \varepsilon ^2} \int _{B_{\delta }(x_{j,\varepsilon }^{(1)})}\int _{\mathbb {R}^3}\left( U_{a_j}\left( \frac{x-x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) + U_{a_j}\left( \frac{x-x^{(2)}_{j,\varepsilon }}{\varepsilon }\right) \right) \eta _{\varepsilon }(x) \big (W^{(2)}_{j,\varepsilon }(\xi )\big )^2\frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx. \end{aligned}$$

Now, by (A.10) and symmetry, we have

$$\begin{aligned} \begin{aligned} F_{2,1,1}= G_{2}+O\big (e^{-\eta /\varepsilon }\big ). \end{aligned} \end{aligned}$$

(E.17)

Also similar to (E.12), we know

$$\begin{aligned} \begin{aligned} F_{2,1,2} =\,&-\frac{1}{2\pi \varepsilon ^2} \int _{\mathbb {R}^3}\int _{\mathbb {R}^3} U_{a_j}\left( \frac{x-x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) w_\varepsilon ^{(2)}(x) U_{a_j}\left( \frac{\xi -x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) \eta _{\varepsilon }(\xi ) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx +o\big (\varepsilon ^4\big )\\ =\,&-\frac{1}{2\pi \varepsilon ^2} \int _{\mathbb {R}^3}\int _{\mathbb {R}^3} U_{a_j}\left( \frac{x-x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) w_\varepsilon ^{(1)}(x) U_{a_j}\left( \frac{\xi -x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) \eta _{\varepsilon }(\xi ) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx\\&+O\left( \left\| U_{a_j}\left( \frac{\cdot -x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) \right\| ^2_{\varepsilon } \Vert w_\varepsilon ^{(1)}-w_\varepsilon ^{(2)}\Vert _\varepsilon \Vert \eta _{\varepsilon }\Vert _\varepsilon \right) +o\big (\varepsilon ^4\big )\\ =\,&-\frac{1}{2\pi \varepsilon ^2} \int _{\mathbb {R}^3}\int _{\mathbb {R}^3} U_{a_j}\left( \frac{x-x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) w_\varepsilon ^{(1)}(x) U_{a_j}\left( \frac{\xi -x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) \eta _{\varepsilon }(\xi ) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx +o\big (\varepsilon ^4\big ). \end{aligned} \end{aligned}$$

(E.18)

Next, by (A.5) and (C.13), we get

$$\begin{aligned} \begin{aligned} F_{2,1,3}&=O\left( \varepsilon ^{-4} \left\| U_{a_j}\left( \frac{\cdot -x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) + U_{a_j}\left( \frac{\cdot -x^{(2)}_{j,\varepsilon }}{\varepsilon }\right) \right\| _{\varepsilon }\cdot \Vert \eta _{\varepsilon }\Vert _{\varepsilon }\cdot \Vert w^{(2)}_{\varepsilon }\Vert _{\varepsilon }^2\right) = O\big (\varepsilon ^6\big ). \end{aligned} \end{aligned}$$

(E.19)

And by (D.6), we obtain

$$\begin{aligned} \begin{aligned} F_{2,1,4}&=O\big (e^{-\eta /\varepsilon }\big )\quad \text{ and }\quad F_{2,1,5}=O\big (\varepsilon ^6\big ). \end{aligned} \end{aligned}$$

(E.20)

Also, $l\ne j$, similar to (E.5) and (E.6), we have

$$\begin{aligned} \begin{aligned}&\frac{1}{8\pi \varepsilon ^2} \int _{B_{\delta }(x_{j,\varepsilon }^{(1)})}\int _{\mathbb {R}^3} U_{a_j}\left( \frac{x-x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) \eta _{\varepsilon }(x) U^2_{a_l}\left( \frac{\xi -x^{(2)}_{l,\varepsilon }}{\varepsilon }\right) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx\\&\quad = \frac{\varepsilon ^2}{8\pi } \int _{\mathbb {R}^3}\int _{\mathbb {R}^3} U_{a_j}(x)\eta _{j,\varepsilon }(x) U^2_{a_l}\left( \xi +\frac{x^{(1)}_{j,\varepsilon }-x^{(2)}_{l,\varepsilon }}{\varepsilon }\right) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx+O\big (e^{-\eta /\varepsilon }\big ), \end{aligned} \end{aligned}$$

(E.21)

and

$$\begin{aligned} \begin{aligned}&\int _{\mathbb {R}^3}\int _{\mathbb {R}^3} U_{a_j}(x)\eta _{j,\varepsilon }(x) U^2_{a_l}\left( \xi +\frac{x^{(1)}_{j,\varepsilon }-x^{(2)}_{l,\varepsilon }}{\varepsilon }\right) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx\\&\quad = \int _{\mathbb {R}^3}\int _{\mathbb {R}^3}U_{a_j}(x)\left( \sum ^3_{m=1}d_{m,j} \frac{\partial U_{a_j}(x)}{\partial x_m}\right) U^2_{a_l}\left( \xi +\frac{x^{(1)}_{j,\varepsilon }-x^{(2)}_{l,\varepsilon }}{\varepsilon }\right) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx \\&\qquad + o\left( \int _{\mathbb {R}^3}\int _{\mathbb {R}^3} U_{a_j}(x) U^2_{a_l}\left( \xi +\frac{x^{(1)}_{j,\varepsilon }-x^{(2)}_{l,\varepsilon }}{\varepsilon }\right) \frac{1}{|x-\xi |^2}d\xi dx\right) \\&\quad = O\left( \int _{\mathbb {R}^3}\int _{\mathbb {R}^3} U_{a_j}^2(x)U^2_{a_l}\left( \xi +\frac{x^{(1)}_{j,\varepsilon }-x^{(2)}_{l,\varepsilon }}{\varepsilon }\right) |x-\xi |^{-3}d\xi dx\right) +o\big (\varepsilon ^2\big )\\&\quad = O\big (\varepsilon ^3\big )+o\big (\varepsilon ^2\big )=o\big (\varepsilon ^2\big ). \end{aligned} \end{aligned}$$

(E.22)

Similar to (E.21) and (E.22), we obtain

$$\begin{aligned} \begin{aligned} \frac{1}{8\pi \varepsilon ^2} \int _{B_{\delta }(x_{j,\varepsilon }^{(1)})}&\int _{\mathbb {R}^3} U_{a_j}\left( \frac{x-x^{(2)}_{j,\varepsilon }}{\varepsilon }\right) \eta _{\varepsilon }(x) U^2_{a_l}\left( \frac{\xi -x^{(2)}_{l,\varepsilon }}{\varepsilon }\right) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx =o\big (\varepsilon ^4\big ). \end{aligned} \end{aligned}$$

(E.23)

Then (E.21), (E.22) and (E.23) imply

$$\begin{aligned} \begin{aligned} F_{2,1,6}=\,&o\big (\varepsilon ^4\big ). \end{aligned} \end{aligned}$$

(E.24)

Then (E.16), (E.17), (E.18), (E.19), (E.20) and (E.24) imply (E.15). $\square$

Lemma E.4

It holds

$$\begin{aligned} \begin{aligned} F_{2,3}=-\frac{1}{8\pi \varepsilon ^2} \int _{\mathbb {R}^3}\int _{\mathbb {R}^3} U^2_{a_j}\left( \frac{x-x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) \big (w^{(1)}_{\varepsilon }(\xi )+w^{(2)}_{\varepsilon }(\xi )\big ) \eta _{\varepsilon }(\xi )\frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx+o\big (\varepsilon ^{4}\big ). \end{aligned} \end{aligned}$$

(E.25)

Proof

First, we write $F_{2,3}$ as follows:

$$\begin{aligned} \begin{aligned} F_{2,3}&=F_{2,3,1}+F_{2,3,2}+F_{2,3,3}+F_{2,3,4}+F_{2,3,5}+F_{2,3,6}, \end{aligned} \end{aligned}$$

(E.26)

where

$$\begin{aligned}&F_{2,3,1}=\frac{1}{8\pi \varepsilon ^2} \int _{B_{\delta }(x_{j,\varepsilon }^{(1)})}\int _{\mathbb {R}^3} \big (w^{(1)}_{\varepsilon }(x)+w^{(2)}_{\varepsilon }(x)\big ) \eta _{\varepsilon }(x)U^2_{a_j}\left( \frac{\xi -x^{(2)}_{j,\varepsilon }}{\varepsilon }\right) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx,\\&F_{2,3,2}=\frac{1}{4\pi \varepsilon ^2} \int _{B_{\delta }(x_{j,\varepsilon }^{(1)})}\int _{\mathbb {R}^3} \big (w^{(1)}_{\varepsilon }(x)+w^{(2)}_{\varepsilon }(x)\big ) \eta _{\varepsilon }(x)U_{a_j}\left( \frac{\xi -x^{(2)}_{j,\varepsilon }}{\varepsilon }\right) w^{(2)}_{\varepsilon }(\xi )\frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx, \\&F_{2,3,3}=\frac{1}{8\pi \varepsilon ^2} \int _{B_{\delta }(x_{j,\varepsilon }^{(1)})}\int _{\mathbb {R}^3} \big (w^{(1)}_{\varepsilon }(x)+w^{(2)}_{\varepsilon }(x)\big ) \eta _{\varepsilon }(x)\big (w^{(2)}_{\varepsilon }(\xi )\big )^2\frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx,\\&F_{2,3,4}=\frac{1}{4\pi \varepsilon ^2} \int _{B_{\delta }(x_{j,\varepsilon }^{(1)})}\int _{\mathbb {R}^3} \big (w^{(1)}_{\varepsilon }(x)+w^{(2)}_{\varepsilon }(x)\big )\eta _{\varepsilon }(x) W^{(2)}_{j,\varepsilon }(\xi ) U_{a_j}\left( \frac{\xi -x^{(2)}_{j,\varepsilon }}{\varepsilon }\right) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx,\\&F_{2,3,5}=\frac{1}{4\pi \varepsilon ^2} \int _{B_{\delta }(x_{j,\varepsilon }^{(1)})}\int _{\mathbb {R}^3} \big (w^{(1)}_{\varepsilon }(x)+w^{(2)}_{\varepsilon }(x)\big )\eta _{\varepsilon }(x) w^{(2)}_{\varepsilon }(\xi ) U_{a_j}\left( \frac{\xi -x^{(2)}_{j,\varepsilon }}{\varepsilon }\right) \frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx,\\&F_{2,3,6}=\frac{1}{8\pi \varepsilon ^2} \int _{B_{\delta }(x_{j,\varepsilon }^{(1)})}\int _{\mathbb {R}^3} \big (w^{(1)}_{\varepsilon }(x)+w^{(2)}_{\varepsilon }(x)\big )\eta _{\varepsilon }(x) \big (W^{(2)}_{j,\varepsilon }(\xi )\big )^2\frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx. \end{aligned}$$

Now by (A.10), (E.11) and symmetry, we have

$$\begin{aligned} \begin{aligned} F_{2,3,1} =-\frac{1}{8\pi \varepsilon ^2} \int _{\mathbb {R}^3}\int _{\mathbb {R}^3} U^2_{a_j}\left( \frac{x-x^{(1)}_{j,\varepsilon }}{\varepsilon }\right) \big (w^{(1)}_{\varepsilon }(\xi )+w^{(2)}_{\varepsilon }(\xi )\big ) \eta _{\varepsilon }(\xi )\frac{x^i-\xi ^i}{|x-\xi |^3}d\xi dx+o\big (\varepsilon ^{4}\big ). \end{aligned} \end{aligned}$$

(E.27)

Next, by (A.5) and (C.13), we get

$$\begin{aligned} \begin{aligned} F_{2,3,2}&=O\left( \varepsilon ^{-4}\Vert w^{(1)}_{\varepsilon }+w^{(2)}_{\varepsilon }\Vert _{\varepsilon } \left\| U_{a_j}\left( \frac{\cdot -x^{(2)}_{j,\varepsilon }}{\varepsilon }\right) \right\| _{\varepsilon } \Vert \eta _{\varepsilon }\Vert _{\varepsilon }\Vert w^{(2)}_{\varepsilon }\Vert _{\varepsilon }\right) = O\big (\varepsilon ^6\big ), \end{aligned} \end{aligned}$$

(E.28)

and

$$\begin{aligned} \begin{aligned} F_{2,3,3}&=O\big ( \varepsilon ^{-4} \Vert w^{(1)}_{\varepsilon }+w^{(2)}_{\varepsilon }\Vert _{\varepsilon } \Vert \eta _{\varepsilon }\Vert _{\varepsilon }\Vert w^{(2)}_{\varepsilon }\Vert ^2_{\varepsilon }\big )= O\big (\varepsilon ^8\big ). \end{aligned} \end{aligned}$$

(E.29)

Also by (D.6), we know

$$\begin{aligned} \begin{aligned} F_{2,3,4}&=O\big (e^{-\eta /\varepsilon }\big )~\text{ and }~ F_{2,3,5}=O\big (\varepsilon ^6\big ). \end{aligned} \end{aligned}$$

(E.30)

Next, similar to (D.15), we obtain

$$\begin{aligned} \begin{aligned} F_{2,3,6}=O\big (\varepsilon ^{-2}\Vert w^{(1)}_{\varepsilon }+w^{(2)}_{\varepsilon }\Vert _{\varepsilon } \Vert W^{(2)}_{j,\varepsilon }\Vert _{\varepsilon } \Vert \eta _{\varepsilon }\Vert _{\varepsilon } \big )+O\big (e^{-\eta /\varepsilon }\big )=O\big (\varepsilon ^6 \big ). \end{aligned} \end{aligned}$$

(E.31)

Then (E.26), (E.27), (E.28), (E.29), (E.30) and (E.31) imply (E.25). $\square$