a good estimate for $\omega_{\delta}$, we need to choose $a_{{\delta},j}$ properly.

By , we have

$$\label{2.4}
\begin{split}
&-{\delta}^2 \Delta  \sum_{j=1}^m PW_{{\delta},z_j,a_{{\delta},j}}
-\sum_{j=1}^m\chi_{{\Omega}_j}\left(\sum_{i=1}^mPW_{{\delta},z_i,a_{{\delta},i}}-\kappa_j-\frac{2\pi q}{|\ln{\varepsilon}|}\right)_+^{p}\\
=&  \sum_{j=1}^m \left(
W_{{\delta},z_j,a_{{\delta},j}}-a_{{\delta},j}\right)_+^{p}-
\sum_{j=1}^m\chi_{{\Omega}_j}\left(\sum_{i=1}^m
\left(W_{{\delta},z_i,a_{{\delta},i}}-\frac{a_{{\delta},i}}{\ln\frac R{s_{\delta}} }
g(y,z_i)\right) -\kappa_j-\frac{2\pi q}{|\ln{\varepsilon}|}\right)_+^{p}.
\end{split}$$

Denote $Z=(z_1,\cdots,z_m)\in \mathbb{R}^{2m}$. In this paper, we always assume that $z_j\in{\Omega}$ satisfies

$$\label{2.5}
d(z_j,\partial{\Omega})\ge \varrho>0,\quad |z_i-z_j|\ge \varrho^{\bar
L},\quad i, j=1,\cdots,m,\; i\ne j,$$

where $\varrho>0$ is a fixed small constant and $\bar L>0$ is a fixed large constant.

\[l2.1\] For ${\delta}>0$ small, there exist $(s_{{\delta},1}(Z),\cdots,s_{{\delta},m}(Z))$ and $(a_{{\delta},1}(Z),\cdots,a_{{\delta},m}(Z))$ satisfying the following system $$\label{2.6}
{\delta}^{2/(p-1)}s_i^{-2/(p-1)}\phi^\prime(1)=\frac{a_{i}}{\ln(s_i/R)},\qquad
i=1,\cdots,m$$ and $$\label{2.7}
a_{i}= \kappa_i + \frac{2\pi
q(z_i)}{|\ln{\varepsilon}|}+\frac{g(z_i,z_i)}{\ln\frac R {s_{i}} } a_{i}
-\sum_{j\ne i}^m \frac{\bar G(z_i,z_j)}{\ln\frac R{s_{j}} }a_{j}
,\qquad i=1,\cdots, m,$$ where for $i=1,\cdots,m$, $$\bar G(x,z_i)=\ln \frac R{|x-z_i|} -g(x,z_i).$$

We will show that system - has a solution $(s_1,\cdots,s_m, a_1,\cdots,a_m)$ in $\mathcal{D}\doteq[\frac{{\delta}}{|\ln{\delta}|},{\delta}|\ln{\delta}|\,]^m\times
\Pi_{i=1}^m[\frac{\kappa_i}{2},\,\frac{3\kappa_i}{2}]$. It is easy to see, for fixed $(s_1,\cdots,s_m)$ with $0<{\delta}<{\delta}^*$ small, that has a solution $(a_1,\cdots,a_m)$ depending on $(s_1,\cdots,s_m)$, such that $\frac{\kappa_i}{2}\leq
a_i\leq\frac{3\kappa_i}{2}$. For such $(a_1,\cdots,a_m)$ and for $i=1,\cdots, m$ define $$\theta_i(s_1,\cdots,s_m)=\frac{s_i^\frac{2}{p-1}}{\ln\frac{R}{s_i}}+\frac{\phi^\prime(1)}{a_i}{\delta}^\frac{2}{p-1},$$ then it is easy to verify that $$\left\{
\begin{array}{ll}
\theta_i(s_1,\cdots,s_m)>0,& s_i={\delta}|\ln{\delta}|, s_j\in [\frac{{\delta}}{|\ln{\delta}|},\,\,{\delta}|\ln{\delta}|\,] \,\,{\rm for}\,\, j=1,\cdots,m, j\neq i, \\
\theta_i(s_1,\cdots,s_m)<0,& s_i=\frac{{\delta}}{|\ln{\delta}|},\,\,s_j\in
[\frac{{\delta}}{|\ln{\delta}|},\,\,{\delta}|\ln{\delta}|\,] \,\,\,\,\,{\rm for}\,\,
j=1,\cdots,m, j\neq i.
\end{array}
\right.$$

By the Poincaré-Miranda Theorem in [@K; @M], we can 