becomes $$\label{1}
\begin{cases}
-{\delta}^2 \Delta w=\sum_{j=1}^m\chi_{{\Omega}_j}\left(w-\kappa_j-\frac{2\pi}{ |\ln{\varepsilon}|}q(x)\right)_+^{p},  & \text{in}\;{\Omega},\\
w=0, &\text{on}\; \partial{\Omega}.
\end{cases}$$

We will use a reduction argument to prove Theorem \[th3\] and Theorem \[th1\]. To this end, we need to construct an approximate solution for . For the problem studied in this paper, the corresponding “limit" problem in $\mathbb{R}^2$ has no bounded nontrivial solution. So, we will follow the method in [@CPY; @DY] to construct an approximate solution. Since there are two parameters ${\delta},~{\varepsilon}$ in problem , which causes some difficulty, we must take this influence into careful consideration and give delicate estimates in order to perform the reduction argument. For example we need to consider $(s_{1,{\delta}},\cdots,s_{m,{\delta}})$ and $(a_{1,{\delta}},\cdots,a_{m,{\delta}})$ together in Lemma \[l2.1\].

As a final remark, we point out that problem can be considered as a free boundary problem. Similar problems have been studied extensively. The reader can refer to [@CF; @CPY; @DY; @FW; @LP] for more results on this kind of problem.

This paper is organized as follows. In section 2, we construct the approximate solution for . We will carry out a reduction argument in section 3 and the main results will be proved in section 4. We put some basic estimates in the appendix.

