B_{Ls_{{\delta},i}}(z_i)$, by

$$\begin{split}
&  P_{{\delta}, Z,j}(x)=W_{{\delta},
z_j,a_{{\delta},j}}(x)-\frac{a_{{\delta},j}}{\ln\frac R{s_{{\delta},j}} }
  g(x,z_j)=
 \frac{a_{{\delta},j}}{\ln\frac R{s_{{\delta},j}}
}   \bar G(x,z_j)\\
=&
  \frac{a_{{\delta},j}}{\ln\frac R{s_{{\delta},j}}
}     \bar  G(z_i,z_j)+\frac{a_{{\delta},j}}{\ln\frac R{s_{{\delta},j}} }
\left\langle D  \bar G(z_i,z_j),x-z_i\right\rangle +
 O\Bigl( \frac{s_{{\delta},i}^2 }{|\ln{\varepsilon}|
} \Bigr).
\end{split}$$ So, by using , we obtain

$$\label{2.9}
\begin{split}
&  P_{{\delta}, Z}(x)-\kappa_i-\frac{2\pi q(x)}{|\ln{\varepsilon}|}\\
=&
 W_{{\delta},z_i, a_{{\delta},i}}(x)- a_{{\delta},i} -\frac{2\pi}{|\ln{\varepsilon}|}\left\langle Dq(z_i),x-z_i\right\rangle-\frac {a_{{\delta},i}}{\ln \frac{R}{s_{{\delta},i}}}
 \left\langle D g(z_i,z_i), x-z_i\right\rangle \\
& +\sum_{j\ne i}^m \frac{a_{{\delta},j} }{\ln\frac R{s_{{\delta},j}} }
\left\langle D \bar G(z_i,z_j),x-z_i\right\rangle
  +O\left( \frac{s_{{\delta},i}^2 }{|\ln{\varepsilon}| } \right),\quad x\in B_{L
s_{{\delta},i}}(z_i).
\end{split}$$

We end this section by giving the following formula which can be obtained by direct computation and will be used in the next two sections.

$$\label{2.11}
\begin{array}{ll}
 \displaystyle\frac{\partial W_{{\delta},z_i,a_{{\delta},i}}(x)}{\partial z_{i,h}}&\\
 =\left\{
 \begin{array}{ll}
\displaystyle\frac{1}{{\delta}}\Bigl(\frac{a_{{\delta},i}}{|\phi^\prime(1)||\ln\frac{R}{s_{{\delta},i}}|}\Bigr)^{(p+1)/2}
\phi^\prime\bigl(\frac{|x-z_i|}{s_{{\delta},i}}\bigr)
\frac{z_{i,h}-x_h}{|x-z_i|}+ O\left(\frac{1}{|\ln{\varepsilon}|}\right),
 ~~x\in B_{s_{{\delta},i}}(z_i),\\
 \,\\
\displaystyle-\frac{a_{{\delta},i}}{\ln\frac{R}{s_{{\delta},i}}}\frac{z_{i,h}-x_h}{|x-z_i|^2}+O\left(\frac{1}{|\ln{\varepsilon}|}\right),
 \qquad\qquad\qquad\qquad\qquad\quad x\in \Omega\setminus B_{s_{{\delta},i}}(z_i).
\end{array}
\right.\\
\end{array}$$

the reduction
=============

Let

$$w(x)=
\begin{cases}
\phi(|x|), &|x|\le 1,\\
\phi^\prime(1)\ln |x|,  & |x|>1.
\end{cases}$$ Then $w\in C^1(\mathbb{R}^2)$. Since $\phi^\prime(1)<0$ and $\ln
|x|$ is harmonic for $|x|>1$, we see that $w$ satisfies

$$\label{3.1}
-\Delta w= w_+^{p}, \quad \text{in}\; \mathbb{R}^2.$$

Moreover, since $w_+$ is Lip-continuous, by the Schauder estimate, $w\in C^{2,\alpha}$ for any $\alpha\in (0,1)$.

Consider the following problem:

$$\label{3.2}
-\Delta v-  pw_+^{p-1} v=0,\quad v\in L^\infty(\mathbb{R}^2),$$

It is easy to see that $\frac{\partial w}{\partial x_i}$, $i=1,2,$ 