Appendix A. Some estimates for the nonlocal term
In this part, we will give the estimates of the nonlocal terms.
Lemma A.1
It holds,
$$\begin{aligned} \int _{B_{d}(\xi _{i, \varepsilon })} \int _{{\mathbb {R}}^{N}} u_{\varepsilon }^{2}(x) u_{\varepsilon }^{2}(y) \frac{ x_{j} - y_{j}}{|x - y|^{N}} dy dx = O\Bigl ( \varepsilon ^{\min \{N+3, N+1+m\}} +\varepsilon ^{N+1} \sum _{i=1}^{l} |\xi _{i, \varepsilon } - \xi _{i}|^{m} \Bigr ).\nonumber \\ \end{aligned}$$
(A.1)
Proof
Firstly, we split the nonlocal term into four parts and denote
$$\begin{aligned} \int _{B_{d}(\xi _{i, \varepsilon })} \int _{{\mathbb {R}}^{N}} u_{\varepsilon }^{2}(x) u_{\varepsilon }^{2}(y) \frac{x_{j} - y_{j}}{|x - y|^{N}} dy dx := F_{1} + F_{2} + F_{3} + F_{4}, \end{aligned}$$
where
$$\begin{aligned} F_{1}= & {} \int _{B_{d}(\xi _{i, \varepsilon })} \int _{{\mathbb {R}}^{N}} W^{2}_{\xi _{i}}\Bigl (\frac{x- \xi _{i, \varepsilon }}{\varepsilon }\Bigr ) u_{\varepsilon }^{2}(y) \frac{x_{j} - y_{j}}{|x - y|^{N}} dy dx; \\ F_{2}= & {} 2 \int _{B_{d}(\xi _{i, \varepsilon })} \int _{{\mathbb {R}}^{N}} W_{\xi _{i}}\Bigl (\frac{x- \xi _{i, \varepsilon }}{\varepsilon }\Bigr ) \phi _{\varepsilon }(x) u_{\varepsilon }^{2}(y) \frac{x_{j} - y_{j}}{|x - y|^{N}} dy dx; \\ F_{3}= & {} \int _{B_{d}(\xi _{i, \varepsilon })} \int _{{\mathbb {R}}^{N}} \phi ^{2}_{\varepsilon }(x) u_{\varepsilon }^{2}(y) \frac{x_{j} - y_{j}}{|x - y|^{N}} dy dx ;\\ F_{4}= & {} \int _{B_{d}(\xi _{i, \varepsilon })} \int _{{\mathbb {R}}^{N}} \Bigl ( \bigl ( \sum _{h \ne i} W_{\xi _{h }} \bigl (\frac{x- \xi _{h, \varepsilon }}{\varepsilon }\bigl )\bigr )^{2} + 2 \sum _{h \ne i} W_{\xi _{h }} \Bigl (\frac{x- \xi _{h, \varepsilon }}{\varepsilon }\Bigr ) W_{\xi _{i }} \Bigl (\frac{x- \xi _{i, \varepsilon }}{\varepsilon }\Bigr ) \\&+ 2\sum _{h \ne i} W_{\xi _{h }} \Bigl (\frac{x- \xi _{h, \varepsilon }}{\varepsilon }\Bigr ) \phi _{\varepsilon } \Bigr ) \times u_{\varepsilon }^{2}(y) \frac{x_{j} - y_{j}}{|x - y|^{N}} dy dx. \end{aligned}$$
Next, we will deal with each term. Indeed, the term \(F_{1}\) is more complicated than others. Expand \(F_{1}\) as follows:
$$\begin{aligned} F_{1} : = F_{11}+ F_{12}+ F_{13}+ F_{14} +F_{15}, \end{aligned}$$
(A.2)
where
$$\begin{aligned} F_{11}= & {} \int _{B_{d}(\xi _{i, \varepsilon })} \int _{{\mathbb {R}}^{N}} W^{2}_{\xi _{i}}\Bigl (\frac{x -\xi _{i, \varepsilon }}{\varepsilon }\Bigr ) W^{2}_{\xi _{i}}\Bigl (\frac{y -\xi _{i, \varepsilon }}{\varepsilon }\Bigr ) \frac{x_{j} - y_{j}}{|x - y|^{N}} dy dx; \\ F_{12}= & {} 2\int _{B_{d}(\xi _{i, \varepsilon })} \int _{{\mathbb {R}}^{N}} W^{2}_{\xi _{i}}\Bigl (\frac{x -\xi _{i, \varepsilon }}{\varepsilon }\Bigr ) W_{\xi _{i}}\Bigl (\frac{y -\xi _{i, \varepsilon }}{\varepsilon }\Bigr ) \bigl (\sum _{ h \ne i} W_{\xi _{h}}\bigl (\frac{y -\xi _{h, \varepsilon }}{\varepsilon }\bigl ) \bigr ) \frac{x_{j} - y_{j}}{|x - y|^{N}} dy dx;\\ F_{13}= & {} \int _{B_{d}(\xi _{i, \varepsilon })} \int _{{\mathbb {R}}^{N}} W^{2}_{\xi _{i}}\Bigl (\frac{x -\xi _{i, \varepsilon }}{\varepsilon }\Bigr ) \bigl (\sum _{ h \ne i} W_{\xi _{h}}\bigl (\frac{y -\xi _{h, \varepsilon }}{\varepsilon }\bigl ) \bigr )^{2} \frac{x_{j} - y_{j}}{|x - y|^{N}} dy dx;\\ F_{14}= & {} 2 \int _{B_{d}(\xi _{i, \varepsilon })} \int _{{\mathbb {R}}^{N}} W^{2}_{\xi _{i}}\Bigl (\frac{x -\xi _{i, \varepsilon }}{\varepsilon }\Bigr ) \bigl (\sum _{h=1}^{l} W_{\xi _{h}}\bigl (\frac{y -\xi _{h, \varepsilon }}{\varepsilon }\bigl ) \bigr ) \phi _{\varepsilon }(y) \frac{x_{j} - y_{j}}{|x - y|^{N}} dy dx;\\ F_{15}= & {} \int _{B_{d}(\xi _{i, \varepsilon })} \int _{{\mathbb {R}}^{N}} W^{2}_{\xi _{i}}\Bigl (\frac{x -\xi _{i, \varepsilon }}{\varepsilon }\Bigr ) \phi ^{2}_{\varepsilon }(y) \frac{x_{j} - y_{j}}{|x - y|^{N}} dy dx. \end{aligned}$$
It follows from symmetry that
$$\begin{aligned} F_{11} = 0. \end{aligned}$$
(A.3)
From Lemma 2.2, we obtain
$$\begin{aligned} \begin{aligned} |F_{12}|&\le C \Vert W_{\xi _{i}}\Bigl (\frac{x - \xi _{i ,\varepsilon }}{\varepsilon }\Bigr ) \Vert _{L^{\frac{4N}{N+1}}({\mathbb {R}}^{N})}^{2} \Bigl ( \sum _{h \ne i }\int _{{\mathbb {R}}^{N}} W^{\frac{2N}{N+1}}_{\xi _{h}}\Bigl (\frac{x - \xi _{h ,\varepsilon }}{\varepsilon }\Bigr ) W^{\frac{2N}{N+1}}_{\xi _{i}}\Bigl (\frac{x - \xi _{i ,\varepsilon }}{\varepsilon }\Bigr ) dx \Bigr )^{\frac{N+1}{2N}} \\&= O\Bigl ( e^{-\frac{\tau }{\varepsilon }} \Bigr ). \end{aligned} \end{aligned}$$
(A.4)
Thus, we have, for \(0< d < \min _{i \ne h, i, h =1, 2, \cdots , l}\{ \frac{|\xi _{i} - \xi _{h}|}{3}, \frac{\tau }{2}\} \)
$$\begin{aligned} |F_{13}|\le & {} C \sum _{ h \ne i} \int _{B_{d}(\xi _{i, \varepsilon })} \int _{{\mathbb {R}}^{N}} W^{2}_{\xi _{i}}\left( \frac{x -\xi _{i, \varepsilon }}{\varepsilon }\right) W^{2}_{\xi _{h}}\left( \frac{y -\xi _{h, \varepsilon }}{\varepsilon }\right) \frac{1}{|x - y|^{N-1}} dy dx \nonumber \\= & {} C \sum _{ h \ne i} \int _{B_{d}(\xi _{i, \varepsilon })} \int _{B_{2d}(\xi _{i, \varepsilon })} W^{2}_{\xi _{i}}(\frac{x -\xi _{i, \varepsilon }}{\varepsilon }) W^{2}_{\xi _{h}}\left( \frac{y -\xi _{h, \varepsilon }}{\varepsilon }\right) \frac{1}{|x - y|^{N-1}} dy dx \nonumber \\&+ C \sum _{ h \ne i} \int _{B_{d}(\xi _{i, \varepsilon })} \int _{{\mathbb {R}}^{N} \setminus B_{2d}(\xi _{i, \varepsilon })} W^{2}_{\xi _{i}}\left( \frac{x -\xi _{i, \varepsilon }}{\varepsilon }\right) W^{2}_{\xi _{h}}\left( \frac{y -\xi _{h, \varepsilon }}{\varepsilon }\right) \frac{1}{|x - y|^{N-1}} dy dx \nonumber \\= & {} C \varepsilon ^{N+1} \sum _{ h \ne i} \int _{B_{d/\varepsilon }(0)} \int _{B_{2d/ \varepsilon }(0)} W^{2}_{\xi _{i}}(x) W^{2}_{\xi _{h}}\left( y + \frac{\xi _{i, \varepsilon } -\xi _{h, \varepsilon }}{\varepsilon }\right) |x - y|^{1-N}dy dx \nonumber \\&+ C \varepsilon ^{N+1} \sum _{ h \ne i} \int _{B_{d/\varepsilon }(0)} \int _{{\mathbb {R}}^{N} \setminus B_{2d/ \varepsilon }(0)} W^{2}_{\xi _{i}}(x) W^{2}_{\xi _{h}}\left( y + \frac{\xi _{i, \varepsilon } -\xi _{h, \varepsilon }}{\varepsilon }\right) |x - y|^{1-N} dy dx.\nonumber \\ \end{aligned}$$
(A.5)
Since \(|\xi _{i} - \xi _{h}| > 3d,\) then
$$\begin{aligned} |y + \frac{\xi _{i, \varepsilon } - \xi _{h, \varepsilon }}{\varepsilon }| \ge |\frac{\xi _{i, \varepsilon } - \xi _{h, \varepsilon }}{\varepsilon } | - |y| \ge \frac{d}{2\varepsilon }, \end{aligned}$$
if \(y \in B_{2d/ \varepsilon }(0)\). Thus, it follows from the exponential decay of \(W_{\xi _{h}}\) at infinity that
$$\begin{aligned} \sum _{ h \ne i} \int _{B_{d/\varepsilon }(0)} \int _{B_{2d/ \varepsilon }(0)} W^{2}_{\xi _{i}}(x) W^{2}_{\xi _{h}}( y + \frac{\xi _{i, \varepsilon } -\xi _{h, \varepsilon }}{\varepsilon }) |x - y|^{1-N}dy dx = O\Bigl (e^{-\frac{\sigma }{\varepsilon }}\Bigr ). \end{aligned}$$
Therefore,
$$\begin{aligned}&|F_{13}| \nonumber \\&\quad \le C \varepsilon ^{N+1} \sum _{ h \ne i} \int _{B_{d/\varepsilon }(0)} \int _{{\mathbb {R}}^{N} \setminus B_{2d/ \varepsilon }(0)} W^{2}_{\xi _{i}}(x) W^{2}_{\xi _{h}}\Bigl ( y + \frac{\xi _{i, \varepsilon } -\xi _{h, \varepsilon }}{\varepsilon }\Bigr ) |x - y|^{1-N} dy dx + O\Bigl (e^{-\frac{\sigma }{\varepsilon }}\Bigr ) \nonumber \\&\quad = C \varepsilon ^{2N} \sum _{ h \ne i} \int _{B_{d/\varepsilon }(0)} \int _{{\mathbb {R}}^{N} \setminus B_{2d/ \varepsilon }(0)} W^{2}_{\xi _{i}}(x) W^{2}_{\xi _{h}}\Bigl ( y + \frac{\xi _{i, \varepsilon } -\xi _{h, \varepsilon }}{\varepsilon }\Bigr ) dy dx + O\Bigl (e^{-\frac{\sigma }{\varepsilon }} \Bigr ) \nonumber \\&\quad = O\Bigl ( \varepsilon ^{2N} \sum _{h \ne i} \int _{{\mathbb {R}}^{N}} W^{2}_{\xi _{i}}(x) dx \cdot \int _{{\mathbb {R}}^{N}} W^{2}_{\xi _{h}}\Bigl ( y + \frac{\xi _{i, \varepsilon } -\xi _{h, \varepsilon }}{\varepsilon }\Bigr ) dy \Bigr ) dy dx + O\Bigl (e^{-\frac{\sigma }{\varepsilon }} \Bigr ) \nonumber \\&\quad = O\bigl (\varepsilon ^{2N}\bigr ). \end{aligned}$$
(A.6)
It is similar with (A.4) to compute that, by Lemmas 2.1 and 2.6,
$$\begin{aligned} \begin{aligned} |F_{14}| \le&C \Vert W_{\xi _{i}}\Bigl (\frac{x - \xi _{i ,\varepsilon }}{\varepsilon }\Bigr ) \Vert _{L^{\frac{4N}{N+1}}({\mathbb {R}}^{N})}^{2} \Bigl (\sum _{h=1}^{l} \int _{{\mathbb {R}}^{N}} W^{\frac{2N}{N+1}}_{\xi _{h}}\Bigl (\frac{x - \xi _{h ,\varepsilon }}{\varepsilon }\Bigr ) \phi _{\varepsilon }^{\frac{2N}{N+1}}(x) dx \Bigr )^{\frac{N+1}{2N}} \\ \le&C \varepsilon ^{\frac{3(N+1)}{4}} \Vert \phi _{\varepsilon }\Vert _{L^{\frac{4N}{N+1}}({\mathbb {R}}^{N})} = O\Bigl ( \varepsilon ^{\min \{N+3, N+1 + m\}} +\varepsilon ^{N+1} \sum _{i=1}^{l} |\xi _{i, \varepsilon } - \xi _{i}|^{m} \Bigr ). \end{aligned} \end{aligned}$$
(A.7)
and
$$\begin{aligned} \begin{aligned} |F_{15}| \le&C \Vert W_{\xi _{i}}\Bigl (\frac{x - \xi _{i ,\varepsilon }}{\varepsilon }\Bigr ) \Vert _{L^{\frac{4N}{N+1}}({\mathbb {R}}^{N})}^{2} \Vert \phi _{\varepsilon } \Vert _{L^{\frac{4N}{N+1}}({\mathbb {R}}^{N})}^{2} \\ \le&C \varepsilon \Vert \phi _{\varepsilon }\Vert _{\varepsilon }^{2} = O\Bigl ( \varepsilon ^{\min \{N+5, N+1+ 2m\}} + \varepsilon ^{N+1} \sum _{i=1}^{l} |\xi _{i, \varepsilon } - \xi _{i}|^{2m} \Bigr ). \end{aligned} \end{aligned}$$
(A.8)
Thus, from (A.2) - (A.6),
$$\begin{aligned} F_{1} = O\Bigl ( \varepsilon ^{\min \{N+3, N+1+m\}} +\varepsilon ^{N+1} \sum _{i=1}^{l} |\xi _{i, \varepsilon } - \xi _{i}|^{m} \Bigr ). \end{aligned}$$
(A.9)
By Hölder inequality, Lemmas 2.1, 2.6 and 2.2, we obtain that
$$\begin{aligned} \begin{aligned} |F_{2}| \le&C \int _{B_{d}(\xi _{i, \varepsilon })} \int _{{\mathbb {R}}^{N}} | W_{\xi _{i}}\Bigl (\frac{x- \xi _{i, \varepsilon }}{\varepsilon }\Bigr ) \phi _{\varepsilon }(x) | u_{\varepsilon }^{2}(y) \frac{1}{|x - y|^{N-1 }} dy dx \\ \le&C \Bigl ( \int _{{\mathbb {R}}^{N}} W^{\frac{2N}{N+1}}_{\xi _{i}}\Bigl (\frac{x- \varepsilon _{i, \varepsilon }}{\varepsilon }\Bigr ) |\phi _{\varepsilon }(x)|^{\frac{2N}{N+1}} dx \Bigr )^{\frac{N+1}{2N}} \Vert u_{\varepsilon }\Vert ^{2}_{L^{\frac{4N}{N+1}}({\mathbb {R}}^{N})} \\ \le&C \varepsilon ^{\frac{3(N+1)}{4}} \Vert \phi _{\varepsilon }\Vert _{L^{\frac{4N}{N+1}}({\mathbb {R}}^{N})} = O\Bigl ( \varepsilon ^{\min \{N+3, N+1+ m\}} + \varepsilon ^{N+1} \sum _{i =1 }^{l} |\xi _{i, \varepsilon } - \xi _{i}|^{m} \Bigr ). \end{aligned} \end{aligned}$$
(A.10)
and
$$\begin{aligned} \begin{aligned} |F_{3}| =&O\Bigl (\Vert \phi _{\varepsilon }\Vert ^{2}_{L^{\frac{4N}{N+1}}({\mathbb {R}}^{N})} \Vert u_{\varepsilon }\Vert _{L^{\frac{4N}{N+1}}({\mathbb {R}}^{N})}^{2}\Bigr ) = O\Bigl ( \varepsilon \Vert \phi _{\varepsilon }\Vert _{\varepsilon }^{2}\Bigr )\\ =&O\Bigl ( \varepsilon ^{\min \{N+5, N+1 + 2m\}} + \varepsilon ^{N+1} \sum _{i=1}^{l} |\xi _{i, \varepsilon } - \xi _{i}|^{2m} \Bigr ). \end{aligned} \end{aligned}$$
(A.11)
By the exponential decay of \(W_{\xi }(x)\) away from the concentrating points, we can get
$$\begin{aligned} |F_{4}| = O\Bigl (e^{-\frac{\sigma }{\varepsilon }}\Bigr ). \end{aligned}$$
(A.12)
The result follows from (A.9) - (A.12). \(\square \)
Furthermore, we discuss with the nonlocal terms in (3.10).
Lemma A.2
We have
$$\begin{aligned} \begin{aligned}&\int _{B_{d}(\xi ^{(1)}_{i, \varepsilon })} \int _{{\mathbb {R}}^{N}} [u_{\varepsilon }^{(1)}(y)]^{2} \bigl ( u_{\varepsilon }^{(1)}(x) + u_{\varepsilon }^{(2)}(x) \bigr ) \eta _{\varepsilon }(x) \frac{(x_{j} - y_{j})}{|x - y|^{N}} dy dx\\&\qquad + \int _{B_{d}(\xi ^{(1)}_{i, \varepsilon })} \int _{{\mathbb {R}}^{N}} \bigl ( u_{\varepsilon }^{(1)}(y) + u_{\varepsilon }^{(2)}(y) \bigr ) \eta _{\varepsilon }(y) [u_{\varepsilon }^{(2)}(x)]^{2} \frac{(x_{j} - y_{j})}{|x - y|^{N}} dy dx \\&\quad = O\bigl (\varepsilon ^{\min \{N+1+ m, N+3\}}\bigr ). \end{aligned} \end{aligned}$$
(3.13)
Proof
Denote
$$\begin{aligned}&\int _{B_{d}(\xi ^{(1)}_{i, \varepsilon })} \int _{{\mathbb {R}}^{N}} [u_{\varepsilon }^{(1)}(y)]^{2} \bigl ( u_{\varepsilon }^{(1)}(x) + u_{\varepsilon }^{(2)}(x) \bigr ) \eta _{\varepsilon }(x) \frac{(x_{j} - y_{j})}{|x - y|^{N}} dy dx \\&\quad = H_{1} + H_{2} + H_{3} + H_{4} + H_{5}, \end{aligned}$$
where
$$\begin{aligned} H_{1}= & {} \int _{B_{d}(\xi ^{(1)}_{i, \varepsilon })} \int _{{\mathbb {R}}^{N}}W^{2}_{\xi _{i}}(\frac{y - \xi ^{(1)}_{i, \varepsilon }}{\varepsilon }) \bigl ( W_{\xi _{i} }\bigl (\frac{x - \xi ^{(1)}_{i, \varepsilon }}{\varepsilon }\bigr ) + W_{\xi _{i } } \bigl (\frac{x - \xi _{i, \varepsilon }^{(2)}}{\varepsilon } \bigr )\bigr ) \eta _{\varepsilon }(x) \frac{(x_{j} - y_{j})}{|x - y|^{N}} dy dx;\\ H_{2}= & {} \int _{B_{d}(\xi ^{(1)}_{i, \varepsilon })} \int _{{\mathbb {R}}^{N}} [u_{\varepsilon }^{(1)}(y)]^{2}\bigl ( \sum _{h \ne i } W_{\xi _{h} }\bigl (\frac{x - \xi ^{(1)}_{h, \varepsilon }}{\varepsilon }\bigr ) + \sum _{h \ne i} W_{\xi _{h} }\bigl (\frac{x - \xi ^{(2)}_{h, \varepsilon }}{\varepsilon }\bigr ) \bigr ) \\&\cdot \eta _{\varepsilon }(x) \frac{(x_{j} - y_{j})}{|x - y|^{N}} dy dx;\\ H_{3}= & {} \int _{B_{d}(\xi ^{(1)}_{i, \varepsilon })} \int _{{\mathbb {R}}^{N}} [u_{\varepsilon }^{(1)}(y)]^{2}\bigl ( \phi _{\varepsilon }^{(1)}(x) +\phi _{\varepsilon }^{(2)}(x)\bigr ) \eta _{\varepsilon }(x) \frac{(x_{j} - y_{j})}{|x - y|^{N}} dy dx;\\ H_{4}= & {} \int _{B_{d}(\xi ^{(1)}_{i, \varepsilon })} \int _{{\mathbb {R}}^{N}} W_{\xi _{i}}\bigl (\frac{y - \xi ^{(1)}_{i, \varepsilon }}{\varepsilon }\bigr )\bigl ( \sum _{h \ne i } W_{\xi _{h}}\bigl (\frac{y - \xi ^{(1)}_{h, \varepsilon }}{\varepsilon }\bigr ) + \phi _{\varepsilon }^{(1)}(y) \bigr ) \\&\cdot \bigl ( W_{\xi _{i} }\bigl (\frac{x - \xi ^{(1)}_{i, \varepsilon }}{\varepsilon }\bigr ) + W_{\xi _{i } } \bigl (\frac{x - \xi _{i, \varepsilon }^{(2)}}{\varepsilon } \bigr )\bigr ) \eta _{\varepsilon } \frac{(x_{j} - y_{j})}{|x - y|^{N}} dy dx;\\ H_{5}= & {} \int _{B_{d}(\xi ^{(1)}_{i, \varepsilon })} \int _{{\mathbb {R}}^{N}}u_{\varepsilon }^{(1)}(y) \bigl ( \sum _{h \ne i } W_{\xi _{h}}\bigl (\frac{y - \xi ^{(1)}_{h, \varepsilon }}{\varepsilon }\bigr ) + \phi _{\varepsilon }^{(1)}(y)\bigr ) \\&\cdot \bigl ( W_{\xi _{i} }\bigl (\frac{x - \xi ^{(1)}_{i, \varepsilon }}{\varepsilon }\bigr ) + W_{\xi _{i } } \bigl (\frac{x - \xi _{i, \varepsilon }^{(2)}}{\varepsilon } \bigr )\bigr ) \eta _{\varepsilon } \frac{(x_{j} - y_{j})}{|x - y|^{N}} dy dx. \end{aligned}$$
It follows from Proposition 2.8 and the properties of \(W_{\xi }(x)\) that
$$\begin{aligned} H_{1}= & {} \int _{{\mathbb {R}}^{N}} \int _{{\mathbb {R}}^{N}}W^{2}_{\xi _{i}}\bigl (\frac{y - \xi ^{(1)}_{i, \varepsilon }}{\varepsilon }\bigr ) \bigl ( W_{\xi _{i} }\bigr (\frac{x - \xi ^{(1)}_{i, \varepsilon }}{\varepsilon }\bigr ) + W_{\xi _{i } } \bigl (\frac{x - \xi _{i, \varepsilon }^{(2)}}{\varepsilon } \bigr )\bigr ) \eta _{\varepsilon }(x) \frac{(x_{j} - y_{j})}{|x - y|^{N}} dy dx \\&+ O\Bigl (e^{-\frac{\sigma }{\varepsilon }}\Bigr ). \end{aligned}$$
By the exponential decay of \(W_{\xi _{i}}\) at infinity, we have
$$\begin{aligned} H_{2}= & {} \varepsilon ^{N+1} \int _{B_{d/\varepsilon }(0)} \int _{{\mathbb {R}}^{N}} [u_{\varepsilon }^{(1)}(\varepsilon y + \xi _{i, \varepsilon }^{(1)})]^{2}\bigl ( \sum _{h \ne i } W_{\xi _{h} }\bigl (x + \frac{\xi _{i, \varepsilon }^{(1)} - \xi _{h, \varepsilon }^{(1)}}{\varepsilon }\bigr ) \\&+ \sum _{h \ne i} W_{\xi _{h}}\bigl ( x + \frac{\xi _{i, \varepsilon }^{(1)} - \xi _{h, \varepsilon }^{(2)}}{\varepsilon } \bigr ) \bigr ) \eta _{i, \varepsilon }(x) \frac{(x_{j} - y_{j})}{|x - y|^{N}} dy dx\\= & {} \varepsilon ^{N+1} \int _{B_{d/\varepsilon }(0)} \int _{{\mathbb {R}}^{N} \setminus B_{\frac{d}{2\varepsilon }} (0)} [u_{\varepsilon }^{(1)}(\varepsilon y + \xi _{i, \varepsilon }^{(1)})]^{2}\bigl ( \sum _{h \ne i } W_{\xi _{h} }(x + \frac{\xi _{i, \varepsilon }^{(1)} - \xi _{h, \varepsilon }^{(1)}}{\varepsilon }) \\&+ \sum _{h \ne i} W_{\xi _{h}}\bigl ( x + \frac{\xi _{i, \varepsilon }^{(1)} - \xi _{h, \varepsilon }^{(2)}}{\varepsilon } \bigr ) \bigr ) \cdot \eta _{i, \varepsilon }(x) \frac{(x_{j} - y_{j})}{|x - y|^{N}} dy dx + O\bigl (e^{-\frac{\tau }{\varepsilon }}\bigr ) \\= & {} O\bigl ( \varepsilon ^{2N}\bigr ). \end{aligned}$$
Moreover, by Hardy-Littlewood-Sobolev inequality and Lemma 2.6, we have
$$\begin{aligned} |H_{3}|\le & {} \int _{B_{d}(\xi ^{(1)}_{i, \varepsilon })} \int _{{\mathbb {R}}^{N}} [u_{\varepsilon }^{(1)}(y)]^{2}\bigl ( |\phi _{\varepsilon }^{(1)}(x)| +|\phi _{\varepsilon }^{(2)}(x)|\bigr ) \frac{|\eta (x)|}{|x - y|^{N-1}} dy dx \\\le & {} C \Vert u_{\varepsilon }^{(1)}\Vert _{L^{\frac{4N}{N+1}}({\mathbb {R}}^{N})}^{2} \bigl (\Vert \phi _{\varepsilon }^{(1)}\Vert _{L^{\frac{4N}{N+1}}({\mathbb {R}}^{N})} + \Vert \phi _{\varepsilon }^{(2)}\Vert _{L^{\frac{4N}{N+1}}({\mathbb {R}}^{N})} \bigr ) \Vert \eta _{\varepsilon }\Vert _{L^{\frac{4N}{N+1}}({\mathbb {R}}^{N})} \\\le & {} C \varepsilon ^{\frac{N+2}{2}} \bigl ( \Vert \phi _{\varepsilon }^{(1)}\Vert _{\varepsilon } + \Vert \phi _{\varepsilon }^{(2)}\Vert _{\varepsilon } \bigr ) =O \bigl (\varepsilon ^{\min \{N+1+m, N+3\}}\bigr ). \end{aligned}$$
Similarly, by Hardy-Littlewood-Sobolev inequality
$$\begin{aligned} |H_{4}|\le & {} C \Bigl ( \int _{{\mathbb {R}}^{N}} \Bigl [ \sum _{h \ne i} W_{\xi _{i}}\Bigl (\frac{x - \xi _{i, \varepsilon }^{(1)}}{\varepsilon }\Bigr ) W_{\xi _{h}}\Bigl (\frac{x - \xi _{h, \varepsilon }^{(1)}}{\varepsilon }\Bigr ) \Bigr ]^{\frac{2N}{N+1}} \Bigr )^{\frac{N+1}{2N}} \Bigl ( \Vert W_{\xi _{i}}\Bigl (\frac{x - \xi _{i, \varepsilon }^{(1)}}{\varepsilon }\Bigr )\Vert _{L^{\frac{4N}{N+1}}({\mathbb {R}}^{N})} \\&+ \Vert W_{\xi _{i}}\Bigl (\frac{x - \xi _{i, \varepsilon }^{(2)}}{\varepsilon }\Bigr ) \Vert _{L^{\frac{4N}{N+1}}({\mathbb {R}}^{N})}\Bigr ) \Vert \eta _{\varepsilon }\Vert _{L^{\frac{4N}{N+1}}({\mathbb {R}}^{N})} \\&+ C \Vert W_{\xi _{i}}\Bigl (\frac{x - \xi _{i, \varepsilon }^{(1)}}{\varepsilon }\Bigr )\Vert _{L^{\frac{4N}{N+1}}({\mathbb {R}}^{N})} \Vert \phi _{\varepsilon }\Vert _{L^{\frac{4N}{N+1}}({\mathbb {R}}^{N})}\bigl ( \Vert W_{\xi _{i}}\bigl (\frac{x - \xi _{i, \varepsilon }^{(1)}}{\varepsilon }\bigr )\Vert _{L^{\frac{4N}{N+1}}({\mathbb {R}}^{N})} \\&+ \Vert W_{\xi _{i}}\bigl (\frac{x - \xi _{i, \varepsilon }^{(2)}}{\varepsilon }\bigr ) \Vert _{L^{\frac{4N}{N+1}}({\mathbb {R}}^{N})}\bigr ) \Vert \eta _{\varepsilon }\Vert _{L^{\frac{4N}{N+1}}({\mathbb {R}}^{N})} \\= & {} O\bigl ( \varepsilon ^{N+1} \sum _{h \ne i} e^{-\frac{\sigma |\xi ^{(1)}_{i, \varepsilon }- \xi ^{(1)}_{h, \varepsilon } |}{\varepsilon }}\bigr ) + O\bigl ( \varepsilon ^{\min \{N+1+m, N+3\}}\bigr ) = O\bigl ( \varepsilon ^{\min \{N+1+m, N+3\}}\bigr ). \end{aligned}$$
On the other hand, let
$$\begin{aligned} H_{5} = H_{51} + H_{52} + H_{53} + H_{54}, \end{aligned}$$
where
$$\begin{aligned} H_{51}= & {} \int _{B_{d}(\xi ^{(1)}_{i, \varepsilon })} \int _{{\mathbb {R}}^{N}}u_{\varepsilon }^{(1)}(y) \phi _{\varepsilon }^{(1)}(y) \bigl ( W_{\xi _{i}}\bigl (\frac{x - \xi _{i, \varepsilon }^{(1)}}{\varepsilon }\bigr )+ W_{\xi _{i}}\bigl (\frac{x - \xi _{i, \varepsilon }^{(2)}}{\varepsilon }\bigr )\bigr ) \eta _{\varepsilon } \frac{(x_{j} - y_{j})}{|x - y|^{N}} dy dx; \\ H_{52}= & {} \int _{B_{d}(\xi ^{(1)}_{i, \varepsilon })} \int _{{\mathbb {R}}^{N}}W_{\xi _{i}}\bigl (\frac{y - \xi _{i, \varepsilon }^{(1)}}{\varepsilon }\bigr ) \sum _{h \ne i } W_{\xi _{h}}\bigl (\frac{x - \xi _{h, \varepsilon }^{(1)}}{\varepsilon }\bigr ) \\&\cdot \bigl ( W_{\xi _{i}}\bigl (\frac{x - \xi _{i, \varepsilon }^{(1)}}{\varepsilon }\bigr ) + W_{\xi _{i}}\bigl (\frac{x - \xi _{i, \varepsilon }^{(2)}}{\varepsilon }\bigr )\bigr ) \eta _{\varepsilon } \frac{(x_{j} - y_{j})}{|x - y|^{N}} dy dx; \\ H_{53}= & {} \int _{B_{d}(\xi ^{(1)}_{i, \varepsilon })} \int _{{\mathbb {R}}^{N}}\phi _{\varepsilon }^{(1)}(y) \sum _{h \ne i } W_{\xi _{h}}\bigl (\frac{y - \xi _{h, \varepsilon }^{(1)}}{\varepsilon }\bigr ) \bigl ( W_{\xi _{i}}\bigl (\frac{x - \xi _{i, \varepsilon }^{(1)}}{\varepsilon }\bigr ) + W_{\xi _{i}}\bigl (\frac{x - \xi _{i, \varepsilon }^{(2)}}{\varepsilon }\bigr )\bigr ) \\&\cdot \eta _{\varepsilon } \frac{(x_{j} - y_{j})}{|x - y|^{N}} dy dx; \\ H_{54 }= & {} \int _{B_{d}(\xi ^{(1)}_{i, \varepsilon })} \int _{{\mathbb {R}}^{N}} \bigl ( \sum _{h \ne i } W_{\xi _{h}}\bigl (\frac{y - \xi _{h, \varepsilon }^{(1)}}{\varepsilon }\bigr ) \bigr )^{2} \bigl ( W_{\xi _{i}}\bigl (\frac{x - \xi _{i, \varepsilon }^{(1)}}{\varepsilon }\bigr ) + W_{\xi _{i}}\bigl (\frac{x - \xi _{i, \varepsilon }^{(2)}}{\varepsilon }\bigr )\bigr ) \eta _{\varepsilon } \frac{(x_{j} - y_{j})}{|x - y|^{N}} dy dx. \end{aligned}$$
As before, by Hardy-Littlewood-Sobolev inequality, we have
$$\begin{aligned} H_{51} =O\bigl ( \varepsilon ^{\min \{N+1+m, N+3\}}\bigr ) , \quad H_{52} =O\bigl (e^{-\frac{\sigma }{\varepsilon }} \bigr ), \quad \text {and } \quad H_{53} = O\bigl ( \varepsilon ^{\min \{N+1+m, N+3\}}\bigr ) . \end{aligned}$$
By direct computations,
$$\begin{aligned}&|H_{54}|\\&\quad = |\int _{B_{d}(\xi ^{(1)}_{i, \varepsilon })} \int _{{\mathbb {R}}^{N}} \sum _{h \ne i } W^{2}_{\xi _{h}}\bigl (\frac{y - \xi _{h, \varepsilon }^{(1)}}{\varepsilon }\bigr ) \bigl ( W_{\xi _{i}}\bigl (\frac{x - \xi _{i, \varepsilon }^{(1)}}{\varepsilon }\bigr ) + W_{\xi _{i}}\bigl (\frac{x - \xi _{i, \varepsilon }^{(2)}}{\varepsilon }\bigr )\bigr ) \eta _{\varepsilon } \frac{(x_{j} - y_{j})}{|x - y|^{N}} dy dx | \\&\qquad +O\bigl (e^{-\frac{\sigma }{\varepsilon }} \bigr ) \\&\quad \le |\int _{B_{d}(\xi ^{(1)}_{i, \varepsilon })} \int _{{\mathbb {R}}^{N} \setminus B_{2d}(\xi ^{(1)}_{i, \varepsilon })} \sum _{h \ne i } W^{2}_{\xi _{h}}\bigl (\frac{y - \xi _{h, \varepsilon }^{(1)}}{\varepsilon }\bigr ) \bigl ( W_{\xi _{i}}\bigl (\frac{x - \xi _{i, \varepsilon }^{(1)}}{\varepsilon }\bigr ) + W_{\xi _{i}}\bigl (\frac{x - \xi _{i, \varepsilon }^{(2)}}{\varepsilon }\bigr )\bigr ) |x - y|^{1 - N}dy dx | \\&\qquad + |\int _{B_{d}(\xi ^{(1)}_{i, \varepsilon })} \int _{ B_{2d}(\xi ^{(1)}_{i, \varepsilon })} \sum _{h \ne i } W^{2}_{\xi _{h}}\bigl (\frac{y - \xi _{h, \varepsilon }^{(1)}}{\varepsilon }\bigr ) \bigl ( W_{\xi _{i}}\bigl (\frac{x - \xi _{i, \varepsilon }^{(1)}}{\varepsilon }\bigr ) + W_{\xi _{i}}\bigl (\frac{x - \xi _{i, \varepsilon }^{(2)}}{\varepsilon }\bigr )\bigr ) |x - y|^{1 - N}dy dx | \\&\qquad +O\bigl (e^{-\frac{\sigma }{\varepsilon }} \bigr ) \\&\quad \le C \varepsilon ^{2N} \sum _{h \ne i} \int _{B_{d/\varepsilon }(0)} \int _{{\mathbb {R}}^{N} \setminus B_{2d/\varepsilon }(0)} W^{2}_{\xi _{h}} (y + \frac{\xi ^{(1)}_{i, \varepsilon } - \xi ^{(1)}_{h, \varepsilon }}{\varepsilon }) \Bigl ( W_{\xi _{i}}(x) + W_{\xi _{i}} \bigl (x + \frac{\xi _{i, \varepsilon }^{(1)} - \xi _{i, \varepsilon }^{(2)}}{\varepsilon }\bigr ) \Bigr ) \\&\qquad + O\bigl (e^{-\frac{\sigma }{\varepsilon }} \bigr ) \\&\quad = O\Bigl ( \varepsilon ^{2N} \sum _{h \ne i} \int _{{\mathbb {R}}^{N}}W^{2}_{\xi _{h}} \bigl (y + \frac{\xi ^{(1)}_{i, \varepsilon } - \xi ^{(1)}_{h, \varepsilon }}{\varepsilon }\bigr ) dy \cdot \int _{{\mathbb {R}}^{N}} \Bigl ( W_{\xi _{i}}(x) + W_{\xi _{i}} \bigl (x + \frac{\xi _{i, \varepsilon }^{(1)} - \xi _{i, \varepsilon }^{(2)}}{\varepsilon }\bigr ) \Bigr ) dx \Bigr ) \\&\qquad + O\bigl (e^{-\frac{\sigma }{\varepsilon }} \bigr )\\&\quad = O\Bigl (\varepsilon ^{2N}\Bigr ). \end{aligned}$$
Therefore,
$$\begin{aligned}&\int _{B_{d}(\xi ^{(1)}_{i, \varepsilon })} \int _{{\mathbb {R}}^{N}} [u_{\varepsilon }^{(1)}(y)]^{2} \bigl ( u_{\varepsilon }^{(1)}(x) + u_{\varepsilon }^{(2)}(x) \bigr ) \eta _{\varepsilon }(x) \frac{(x_{j} - y_{j})}{|x - y|^{N}} dy dx \\&\quad = \int _{{\mathbb {R}}^{N}} \int _{{\mathbb {R}}^{N}}W^{2}_{\xi _{i}}\bigl (\frac{y - \xi ^{(1)}_{i, \varepsilon }}{\varepsilon }\bigr ) \bigl ( W_{\xi _{i} }\bigl (\frac{x - \xi ^{(1)}_{i, \varepsilon }}{\varepsilon }\bigr ) + W_{\xi _{i } } \bigl (\frac{x - \xi _{i, \varepsilon }^{(2)}}{\varepsilon }\bigr )\bigr ) \eta _{\varepsilon }(x) \frac{(x_{j} - y_{j})}{|x - y|^{N}} dy dx \\&\qquad + O\bigl (\varepsilon ^{\min \{N+1+ m, N+3\}}\bigr ). \end{aligned}$$
Through the similar argument, we have
$$\begin{aligned}&\int _{B_{d}(\xi ^{(1)}_{i, \varepsilon })} \int _{{\mathbb {R}}^{N}} \bigl ( u_{\varepsilon }^{(1)}(y) + u_{\varepsilon }^{(2)}(y) \bigr ) \eta _{\varepsilon }(y) [u_{\varepsilon }^{(2)}(x)]^{2} \frac{(x_{j} - y_{j})}{|x - y|^{N}} dy dx \\&\quad = \int _{{\mathbb {R}}^{N}} \int _{{\mathbb {R}}^{N}} \bigl ( W_{\xi _{i} }\bigl (\frac{y - \xi ^{(1)}_{i, \varepsilon }}{\varepsilon }\bigr ) + W_{\xi _{i } } \bigl (\frac{y - \xi _{i, \varepsilon }^{(2)}}{\varepsilon } \bigr )\bigr ) \eta _{\varepsilon }(y) W^{2}_{\xi _{i}}\bigl (\frac{x - \xi ^{(1)}_{i, \varepsilon }}{\varepsilon }\bigr ) \frac{(x_{j} - y_{j})}{|x - y|^{N}} dy dx \\&\qquad + O\bigl (\varepsilon ^{\min \{N+1+ m, N+3\}}\bigr ). \end{aligned}$$
Hence, the result follows. \(\square \)
