corresponding to $l_{\delta}+ R_{\delta}(\omega)$ satisfies

$$\begin{split}
|b_{j,h}|\le & C|\ln{\delta}|^{p+1} \sum_{i,\,\bar h} \left(\sum_{j=1}^m
\int_{B_{Ls_{{\delta},j} }(z_j)}
 \Bigl|\frac{\partial P_{{\delta},Z,i}}
{\partial z_{i,\bar h}}\Bigr||l_{\delta}+ R_{\delta}(\omega)
|\right)\\
\le & C{\varepsilon}^{1-\frac{2}{p}}|\ln{\varepsilon}|^{p}\|l_{\delta}+ R_{\delta}(\omega) \|_p.
\end{split}$$ As a result,

$$\begin{split}
&\|Q_{\delta}(l_{\delta}+ R_{\delta}(\omega))\|_p \\
\le & \| l_{\delta}+ R_{\delta}(\omega)\|_p+C \sum_{j,\,h} |b_{j,h}| \left\|
-{\delta}^2\Delta\Bigl(\frac{\partial P_{{\delta},Z,j}}{\partial
z_{j,h}}\Bigr)
\right\|_p\\
\le & C \| l_{\delta}\|_p+ C\| R_{\delta}(\omega) \|_p.
\end{split}$$

On the other hand, from Lemma \[al1\] and , we can deduce

$$\begin{split}
\|l_{\delta}\|_p =&\left\|\sum_{j=1}^m\chi_{{\Omega}_j}\left(
P_{{\delta},Z}-\kappa_j-\frac{2\pi q(x)}{|\ln{\varepsilon}|}\right)_+^{p}-\sum_{j=1}^m\bigl(W_{{\delta},z_j,a_{{\delta},j}}-a_{{\delta},j}\bigr)_+^{p}\right\|_p\\
\le&\sum_{j=1}^m\frac{Cs_{{\delta},j}}{|\ln{\varepsilon}|}\Big\|\bigl(W_{{\delta},z_j,a_{{\delta},j}}-a_{{\delta},j}\bigr)_+^{p-1}\Big\|_p\\
=&O\left(\frac{{\delta}^{1+\frac{2}{p}}}{|\ln{\delta}|^{\frac
{p-1}2+\frac{1}p}}\right).
\end{split}$$

For the estimate of $\| R_{\delta}(\omega) \|_p$, we have $$\label{3.16}
\begin{split}
\| R_{\delta}(\omega) \|_p=&\bigg\|\sum_{j=1}^m\chi_{{\Omega}_j}\Big(
P_{{\delta},Z}-\kappa_j+\omega-\frac{2\pi
q(x)}{|\ln{\varepsilon}|}\Big)_+^{p}-\sum_{j=1}^m\chi_{{\Omega}_j}\Big(
P_{{\delta},Z}-\kappa_j-\frac{2\pi q(x)}{|\ln{\varepsilon}|}\Big)_+^{p}\\
&-\sum_{j=1}^mp\chi_{{\Omega}_j}\Big(
P_{{\delta},Z}-\kappa_j-\frac{2\pi q(x)}{|\ln{\varepsilon}|}\Big)_+^{p-1}\omega\bigg\|_p\\
\le&\sum_{j=1}^mC\|\omega\|_\infty^2\left\|\chi_{{\Omega}_j}\left(
P_{{\delta},Z}-\kappa_j-\frac{2\pi q(x)}{|\ln{\varepsilon}|}\right)_+^{p-2}\right\|_p\\
=&O\left(\frac{{\delta}^{\frac{2}{p}}\|\omega\|_\infty^2}{|\ln{\delta}|^{p-3+\frac{1}p}}\right).
\end{split}$$

Thus, we obtain

$$\label{3.17}
\begin{split}
\| G_{\delta}(\omega)\|_\infty \le &
\frac{C|\ln{\delta}|^{\frac{(p-1)^2}{p}}}{{\delta}^{\frac{2}{p}}}\Bigl(\|
l_{\delta}\|_p+\| R_{\delta}(\omega)\|_p
\Bigr)\\
\le  & C|\ln{\delta}|^{\frac{(p-1)^2}{p}}\left(\frac{{\delta}}{|\ln{\delta}|^{\frac
{p-1}2+\frac{1}p}}+\frac{\|\omega\|_\infty^2}{|\ln{\delta}|^{p-3+\frac{1}p}}\right)\\
\le&{\delta}|\ln{\delta}|^{\frac{p-1}2}
\end{split}$$

Thus, $G_{\delta}$ is a map from $M$ to $M$.

Step 2. $G_{\delta}$ is a contraction map.

In fact, for any $\omega_i\in M$, $i=1,2 $, we have

$$G_{\delta}\omega_1-G_{\delta}\omega_2= (Q_{\delta}L_{\delta})^{-1} Q_{\delta}\bigl(
R_{\delta}(\omega_1)-R_{\delta}(\omega_2)\bigr).$$ Noting that

$$R_{\delta}(\omega_1)=R_{\delta}(\omega_2)=0,\quad\text{in}\; {\Omega}\setminus
\cup_{j=1}^m B_{Ls_{{\delta},j}}(z_j),$$ we can deduce as in Step 1 that $$\begin{split}
\|G_{\delta}\omega_1-G_{\delta}\omega_2\|_\infty\le& \frac{C|\ln{\delta}|^{\frac{(p-1)^2}{p}}}{{\delta}^{\frac{2}{p}}}\|R_{\delta}(\omega_1)-R_{\delta}(\omega_2)\|_p\\
\le&C|\ln{\delta}|^{p-1}\left(\frac{\|\omega_1\|_\infty}{|\ln{\delta}|^{p-2}}+\frac{\|\omega_2\|_\infty}{|\ln{\delta}|^{p-2}}\right)\|\omega_1-\omega_2\|_\infty\\
\le&C{\delta}|\ln{\delta}|^{\frac {p+1}2}\|\omega_1-\omega_2\|_\infty \le\frac
12 \|\omega_1-\omega_2\|_\infty.
\end{split}$$

Combining Step 1 and Step 2, we have proved that $G_{\delta}$ is a contraction map from $M$ to $M$. By the contraction mapping theorem, there 