get $(s_{{\delta},1},\cdots,s_{{\delta},m})$ such that $\theta_i(s_{{\delta},1},\cdots,s_{{\delta}, m})=0$. Therefore we have completed our proof of Lemma  \[l2.1\].

For simplicity, for given $Z=(z_1,\cdots,z_m)$, in this paper, we will use $a_{{\delta},i}$,$s_{{\delta},i}$ instead of $a_{{\delta},i}(Z)$,$s_{{\delta},i}(Z)$.

\[remark2.2\] More precisely, we have the following relation $$\label{r2.2.1}
\frac{1}{\ln\frac{R}{s_{{\delta},i}}}=\frac{1}{\ln\frac{R}{{\varepsilon}}}
+O\left(\frac{\ln|\ln{\varepsilon}|}{|\ln{\varepsilon}|^2}\right),\, i=1,\cdots,m,$$

$$\label{r2.2.2}
a_{{\delta},i}=\kappa_i+\frac{2\pi
q(z_i)}{|\ln{\varepsilon}|}+\frac{g(z_i,z_i)}{\ln\frac{R}{{\varepsilon}}}-\sum_{j\neq
i}^m\frac {\bar G(z_i,z_j)}{\ln\frac{R}{{\varepsilon}}}+
O\Bigl(\frac{\ln|\ln{\varepsilon}|}{|\ln{\varepsilon}|^2}\Bigr),\, i=1,\cdots,m,$$

$$\label{2.10}
\frac{\partial a_{{\delta},i}}{\partial
z_{j,h}}=O\left(\frac1{|\ln{\varepsilon}|}\right),\quad~~\frac{\partial
s_{{\delta},i}}{\partial
z_{j,h}}=O\left(\frac{{\varepsilon}}{|\ln{\varepsilon}|}\right),\,i,j=1,\cdots,m,\,h=1,2.$$

Indeed, can be deduced from (see [@DY], for example). can been deduced from and . Differentiating both sides of and with respect to $z_{j,\,h}$ we can get a linear system of $\frac{\partial a_{{\delta},i}}{\partial z_{j,h}}$ and $\frac{\partial s_{{\delta},i}}{\partial z_{j,h}}$, which will deduces .

From now on we will always choose $(a_{{\delta},1},\cdots,a_{{\delta},m})$ and $(s_{{\delta},1},\cdots,s_{{\delta},m})$ such that and are satisfied. For $(a_{{\delta},1},\cdots,a_{{\delta},m})$ and $(s_{{\delta},1},\cdots,s_{{\delta},m})$ chosen in such a way define $$\label{2.8}
P_{{\delta}, Z,j}=
PW_{{\delta},z_j,a_{{\delta},j}},\,\,P_{{\delta},Z}=\sum_{j=1}^mP_{{\delta}, Z,j}.$$

Then, we find that for $x\in
B_{L s_{{\delta},i}}(z_i)$, where $L>0$ is any fixed constant,

$$\begin{split}
& P_{{\delta}, Z,i}(x)-\kappa_i-\frac{2\pi q(x)}{|\ln{\varepsilon}|}= W_{{\delta},z_i,
a_{{\delta},i}}(x) -\frac {a_{{\delta},i}}{\ln \frac{R}{s_{{\delta},i} }}
g(x,z_i)-\kappa_i-\frac{2\pi q(x)}{|\ln{\varepsilon}|}
\\
=&  W_{{\delta},z_i, a_{{\delta},i}}(x)- \kappa_i-\frac {a_{{\delta},i}}{\ln
\frac{R}{s_{ {\delta},i}}}
 g(z_i,z_i)-\frac {a_{{\delta},i}}{\ln \frac{R}{s_{{\delta},i}}}\Bigl(
 \left\langle D g(z_i,z_i), x-z_i\right\rangle +O( |x-z_i|^2)\Bigr)\\
\quad&-\frac{2\pi q(z_i)}{|\ln{\varepsilon}|}-\frac{2\pi}{|\ln{\varepsilon}|}\left(\left\langle Dq(z_i),x-z_i\right\rangle+O(|x-z_i|^2)\right)\\
 =&  W_{{\delta},z_i, a_{{\delta},i}}(x)- \kappa_i-\frac{2\pi q(z_i)}{|\ln{\varepsilon}|}-\frac{2\pi}{|\ln{\varepsilon}|}\left\langle
Dq(z_i),x-z_i\right\rangle\\
\quad&-\frac {a_{{\delta},i}}{\ln \frac{R}{s_{ {\delta},i}}}
 g(z_i,z_i)-\frac {a_{{\delta},i}}{\ln \frac{R}{s_{{\delta},i}}}
 \left\langle D g(z_i,z_i), x-z_i\right\rangle +O\left(\frac{s_{{\delta},i}^2}{|\ln{\varepsilon}|}\right),
\end{split}$$ and for $j\ne i$ and $x\in 