that $Q_{\delta}(-{\delta}^2\Delta )=-{\delta}^2\Delta$ is an one to one and onto map from $ E_{{\delta},Z}$ to $F_{{\delta},Z}$. On the other hand, $Q_{\delta}L_{\delta}u=h$ is equivalent to

$$\label{3.9}
u=p{\delta}^{-2} (-Q_{\delta}\Delta )^{-1}\left[
Q_{\delta}\left(\sum_{j=1}^m\chi_{{\Omega}_j} \left(
P_{{\delta},Z}-\kappa_j-\frac{2\pi q(x)}{|\ln{\varepsilon}|}\right)_+^{p-1}
u\right)\right]+{\delta}^{-2} (-Q_{\delta}\Delta )^{-1} h,\quad u\in
E_{{\delta},Z}.$$

It is easy to check that ${\delta}^{-2} (-Q_{\delta}\Delta )^{-1}\left[
Q_{\delta}\left(\sum_{j=1}^m\chi_{{\Omega}_j} \left(
P_{{\delta},Z}-\kappa_j-\frac{2\pi q(x)}{|\ln{\varepsilon}|}\right)_+^{p-1}
u\right)\right]$ is a compact operator in $E_{{\delta},Z}$. By the Fredholm alternative, is solvable if and only if

$$u=p{\delta}^{-2} (-Q_{\delta}\Delta )^{-1}\left[
Q_{\delta}\left(\sum_{j=1}^m\chi_{{\Omega}_j} \left(
P_{{\delta},Z}-\kappa_j-\frac{2\pi q(x)}{|\ln{\varepsilon}|}\right)_+^{p-1}
u\right)\right]$$ has trivial solution, which is true since $Q_{\delta}L_{\delta}$ is a one to one map. Thus the result follows.

Now consider the equation

$$\label{3.10}
Q_{\delta}L_{\delta}\omega=
 Q_{\delta}l_{\delta}+ Q_{\delta}R_{\delta}(\omega),$$

where

$$\label{3.11}
l_{\delta}=\sum_{j=1}^m \chi_{{\Omega}_j}\left( P_{{\delta},Z}
-\kappa_j-\frac{2\pi q(x)}{|\ln{\varepsilon}|} \right)_+^{p}-\sum_{j=1}^m
\left(W_{{\delta},z_j,a_{{\delta},j}}-a_{{\delta},j}\right)_+^{p},$$

and

$$\label{3.12}
\begin{split}
R_{\delta}(\omega)=& \sum_{j=1}^m \chi_{{\Omega}_j}\left(P_{{\delta},Z}
-\kappa_j+\omega-\frac{2\pi q(x)}{|\ln{\varepsilon}|} \right)_+^{p}
-\sum_{j=1}^m \chi_{{\Omega}_j}
 \left(P_{{\delta},Z}
-\kappa_j-\frac{2\pi q(x)}{|\ln{\varepsilon}|}\right)_+^{p} \\
&-\sum_{j=1}^m \chi_{{\Omega}_j}p\left(P_{{\delta},Z} -\kappa_j-\frac{2\pi
q(x)}{|\ln{\varepsilon}|}\right)_+^{p-1}\omega.
\end{split}$$

Using Proposition \[p32\], we can rewrite as

$$\label{3.13}
\omega =G_{\delta}\omega =: (Q_{\delta}L_{\delta})^{-1} Q_{\delta}\bigl(
  l_{\delta}+   R_{\delta}(\omega)\bigr).$$

The next Proposition enables us to reduce the problem of finding a solution for to a finite dimensional problem.

\[p33\]

There is an ${\delta}_0>0$, such that for any ${\delta}\in (0,{\delta}_0]$ and $Z$ satisfying , has a unique solution $\omega_{\delta}\in
 E_{{\delta},Z}$, with

$$\|\omega_{\delta}\|_\infty =O\Bigl({\delta}|\ln{\delta}|^{\frac{p-1}{2}}\Bigr).$$

It follows from Lemma \[al1\] that if $L$ is large enough, 