\[al1\], we obtain

$$\begin{split}
&\int_{\Omega}u_n\, L_{{\delta}_n} \left(\frac{\partial
P_{{\delta}_n,Z_n,i}}{\partial z_{i,\bar h}}\right)\\
&=\int_{\Omega}\left(-{\delta}_n^2\Delta \left(\frac{\partial
P_{{\delta}_n,Z_n,i}}{\partial z_{i,\bar
h}}\right)-\sum_{j=1}^mp\chi_{{\Omega}_j}\left(
P_{{\delta}_n,Z_n}-\kappa_j-\frac{2\pi
q(x)}{|\ln{\varepsilon}_n|}\right)_+^{p-1}\frac{\partial
P_{{\delta}_n,Z_n,i}}{\partial z_{i,\bar h}}\right)u_n\\
&=p\int_{\Omega}\left(W_{{\delta}_n,z_{i,n},a_{{\delta}_n,i}}-a_{{\delta}_n,i}\right)_+^{p-1}\left(\frac{\partial
W_{{\delta}_n,z_{i,n},a_{{\delta}_n,i}}}{\partial z_{i,\bar h}}-\frac{\partial
a_{{\delta}_n,i}}{\partial z_{i,\bar h}}\right)u_n\\
&\quad-\sum_{j=1}^mp\int_{\Omega_j}\left(W_{{\delta}_n,z_{j,n},a_{{\delta}_n,j}}-a_{{\delta}_n,j}+O\left(\frac{s_{n,j}}{|\ln{\varepsilon}_n|}\right)\right)_+^{p-1}\frac{\partial
P_{{\delta}_n,Z_n,i}}{\partial
z_{i,\bar h}}u_n\\
&=O\left(\frac{{\varepsilon}_n^2}{|\ln{\varepsilon}_n|^p}\right).
\end{split}$$

Using , we find that $$b_{i,h,n}=O\left({\varepsilon}_n^2|\ln{\varepsilon}_n|\right).$$ Therefore, $$\begin{split}
&\sum_{j=1}^m\sum_{h=1}^2b_{j,h,n}\left(-{\delta}_n^2\Delta\frac{\partial
P_{{\delta}_n,Z_n,j}}{\partial z_{j,h}}\right)\\
&=p\sum_{j=1}^m\sum_{h=1}^2b_{j,h,n}\left(W_{{\delta}_n,z_{j,n},a_{{\delta}_n,j}}-a_{{\delta}_n,j}\right)_+^{p-1}
\left(\frac{\partial W_{{\delta}_n,z_{j,n},a_{{\delta}_n,j}}}{\partial
z_{j,h}}-\frac{\partial a_{{\delta}_n,j}}{\partial z_{j,h}}\right)\\
&=O\left(\sum_{j=1}^m\sum_{h=1}^2  \frac{{\varepsilon}_n^{\frac{2}{p}-1}|b_{j,h,n}|}{|\ln{\varepsilon}_n|^{p}}\right)\\
&=O\left(\frac{{\varepsilon}_n^{\frac{2}{p}+1}}{|\ln{\varepsilon}_n|^{p-1}}\right)\quad
\text{in}~~L^p({\Omega}).
\end{split}$$

Thus, we obtain

$$L_{{\delta}_n}u_n = Q_{{\delta}_n}L_{{\delta}_n}u_n
+O\left(\frac{{\varepsilon}_n^{\frac{2}{p}+1}}{|\ln{\varepsilon}_n|^{p-1}}\right)
 =O\left(\frac 1n\frac{{\delta}_n^{\frac{2}{p}}}{|\ln{\delta}_n|^{\frac{(p-1)^2}{p}}}\right).$$

For any fixed $i$, define

$$\tilde u_{i,n} (y)= u_n(s_{n,i} y+z_{i,n}).$$

Let $$\tilde L_n u= -\Delta u
-\sum_{l=1}^mp\frac{s_{n,i}^2}{{\delta}_n^2}\chi_{{\Omega}_l}\left(
P_{{\delta}_n,Z_n}(s_{n,i}y+z_{i,n})-\kappa_l-\frac{2\pi
q}{|\ln{\varepsilon}_n|}\right)_+^{p-1}u,$$ Then

$$s_{n,i}^{\frac{2}{p}}\times\frac{{\delta}_n^2}{s_{n,i}^{2}}\|\tilde L_n
\tilde u_{i,n}\|_p= \| L_{{\delta}_n} u_n\|_p.$$

Noting that $$\left(\frac{{\delta}_n}{s_{n,i}}\right)^2=O\left(\frac{1}{|\ln{\delta}_n|^{p-1}}\right),$$ we find that $$L_{{\delta}_n}u_n=o\left(\frac{{\delta}_n^{\frac{2}{p}}}{|\ln{\delta}_n|^{\frac{(p-1)^2}{p}}}\right).$$ As a result,

$$\tilde  L_{n } \tilde u_{i,n}
 =o(1),\quad \text{in}\; L^p(\Omega_n),$$ where $\Omega_n=\bigl\{y: s_{n,i} y+z_{i,n}\in\Omega\bigr\}$.

Since $\|\tilde u_{i,n}\|_\infty=1$, by the regularity theory of elliptic equations, we may assume that

$$\tilde u_{i,n}\to u_i,  \quad \text{in}\; C_{loc}^1(\mathbb{R}^2).$$

It is easy to see that $$\begin{split}
&\sum_{l=1}^m\frac{s_{n,i}^2}{{\delta}_n^2}\chi_{{\Omega}_l}\left(
P_{{\delta}_n.Z_n}(s_{n,i}y+z_{i,n})-\kappa_l-\frac{2\pi q}{|\ln{\varepsilon}_n|}\right)_+^{p-1}\\
&=\frac{s_{n,i}^2}{{\delta}_n^2}\left(W_{{\delta}_n,z_{i,n},a_{{\delta}_n,i}}-a_{{\delta}_n,i}
+O\left(\frac{s_{n,i}}{|\ln{\varepsilon}_n|}\right)\right)_+^{p-1}+o(1)\\
&\rightarrow w_+^{p-1}.
\end{split}$$ Then, by Lemma \[al1\], we find that $u_i$ satisfies

$$-\Delta u_i-pw_+^{p-1} u_i= 0.$$ Now from the Proposition 3.1 in [@DY], we have

$$\label{3.7}
u_i= c_1 \frac{\partial w}{\partial x_1}+ c_2 \frac{\partial
w}{\partial x_2}.$$

Since

$$\int_{\Omega}\Delta \bigl(\frac{\partial P_{{\delta}_n,Z_n,i}}{\partial z_{i,
h}}\bigr) u_n =0,$$ we find that

$$\int_{\mathbb{R}^2}\phi_+^{p-1} \frac{\partial \phi}{\partial z_h}
u_i =0,$$ which, together with 