\\
=& \delta_{ij h\bar h}\frac
c{|\ln{\varepsilon}|^{p+1}}+O\left(\frac{{\varepsilon}}{|\ln{\varepsilon}|^{p+1}}\right),
\end{split}$$

where $c>0$ is a constant, $\delta_{ijh \bar h}=1$, if $i=j$ and $h=\bar h$; otherwise, $\delta_{ijh \bar h}=0$.

Set

$$L_{\delta}u= -{\delta}^2 \Delta u-\sum_{j=1}^m p\chi_{{\Omega}_j}\left(
P_{{\delta},Z}-\kappa_j-\frac{2\pi q(x)}{|\ln{\varepsilon}|}\right)_+^{p-1}u.$$

We have the following lemma.

\[l31\] There are constants $\rho_0>0$ and ${\delta}_0>0$, such that for any ${\delta}\in (0,{\delta}_0]$, $Z$ satisfying , $ u\in E_{{\delta},Z}$ with $Q_{\delta}L_{\delta}u =0$ in $\Omega\setminus \cup_{j=1}^m
 B_{L s_{{\delta},j}}
(z_j)$ for some $L>0$ large, then

$$\|Q_{\delta}L_{\delta}u\|_{L^p({\Omega})}  \ge
\frac{\rho_0{\delta}^{\frac{2}{p}}}{|\ln{\delta}|^{\frac{(p-1)^2}{p}}}
\|u\|_{L^\infty({\Omega})}.$$

Set $s_{n,j}=s_{{\delta}_n,j}$. We will use $\|\cdot\|_p, \|\cdot\|_\infty$ to denote $\|\cdot\|_{L^p({\Omega})}$ and $ \|\cdot\|_{L^\infty({\Omega})}$ respectively.

We argue by contradiction. Suppose that there are ${\delta}_n\to 0$, $Z_n$ satisfying and $u_n\in E_{{\delta}_n,Z_n}$ with $Q_{{\delta}_n}L_{{\delta}_n} u_n =0$ in $\Omega\setminus \cup_{j=1}^m
 B_{L s_{n,j}}
(z_{j,n})$, $\|u_n\|_\infty =1$, such that $$\|Q_{{\delta}_n} L_{{\delta}_n } u_n\|_{p} \le
\frac{1}{n}\frac{{\delta}_n^{\frac{2}{p}}}{|\ln{\delta}_n|^{\frac{(p-1)^2}{p}}}.$$

Firstly, we estimate $b_{j,h,n}$ in the following formula:

$$\label{3.6}
Q_{{\delta}_n} L_{{\delta}_n } u_n= L_{{\delta}_n } u_n-\sum_{j=1}^m \sum_{h=1}^2
b_{j,h,n} \left(-{\delta}_n^2\Delta\frac{\partial
P_{{\delta}_n,Z_n,j}}{\partial z_{j,h}}\right).$$

For each fixed $i$, multiplying by $
 \frac{\partial
P_{{\delta}_n,Z_n,i}}{\partial z_{i,\bar h}}$, noting that

$$\int_{\Omega}\bigl( Q_{{\delta}_n} L_{{\delta}_n } u_n\bigr)
 \frac{\partial
P_{{\delta}_n,Z_n,i}}{\partial z_{i,\bar h}}=0,$$ we obtain

$$\begin{split}
& \int_{\Omega}u_n\, L_{{\delta}_n} \left(\frac{\partial
P_{{\delta}_n,Z_n,i}}{\partial z_{i,\bar h}} \right)= \int_{\Omega}\bigl(
  L_{{\delta}_n} u_n\bigr)\, \frac{\partial
P_{{\delta}_n,Z_n,i}}{\partial z_{i,\bar h}} \\
&=\sum_{j=1}^m
 \sum_{\bar h=1}^2 b_{j,h,n} \int_{\Omega}\left(-{\delta}_n^2\Delta\frac{\partial P_{{\delta}_n,Z_n,j}}{\partial
 z_{j,h}}\right)
\frac{\partial P_{{\delta}_n,Z_n,i}}{\partial z_{i,\bar h}}\\
\end{split}$$ Using and Lemma 