to $x_i^*\in
\Omega$, as ${\varepsilon}\to 0$.

\[re1.4\] For the case $m=1$ suppose that $x_1$ is a strict local maximum(or minimum) point of Kirchhoff-Routh function $\mathcal{W}(x)$ defined by , statement of Theorem \[th3\] still holds which can be proved by making corresponding modification of the proof of Theorem \[th3\] in obtaining critical point of $K(z)$ defined by (see Propositions 2.3,2.5 and 2.6 [@Cao] for detailed arguments).

For domains which may not be simply-connected, we show in the following result that the topology of the domain plays an important role in the existence of solutions.

\[th1\]

Suppose that the homology of ${\Omega}$ is nontrivial. Then, for any positive integer $m$, there exists ${\varepsilon}_0>0$, such that for each ${\varepsilon}\in (0,{\varepsilon}_0)$, has a solution $u_{\varepsilon}$, such that the set $\Omega_{{\varepsilon},i} =\{x:
u_{\varepsilon}(x)-\frac{\kappa_i}{2\pi}\,\ln\frac{1}{{\varepsilon}}-q(x)>0\}\subset\subset{\Omega}_i,
\,i=1,2,\cdots,m$ and each $\Omega_{{\varepsilon},\,i}$ shrinks to a point $x_i^*\in \Omega$, as ${\varepsilon}\to 0$. Moreover $x_i^*\neq x_j^*$, if $
i\neq j$.

Since $m$ is arbitrary, from Theorem \[th1\], we can see that the number of solutions for is unbounded as ${\delta}\to 0$.

Not as in [@SV] where is investigated directly, we prove Theorem \[nth1\], Theorem \[th3\] and Theorem \[th1\] by considering an equivalent problem of instead. Let $ w=\frac{2\pi}{|\ln{\varepsilon}|}u$ and $\delta={\varepsilon}(\frac{2\pi}{ |\ln{\varepsilon}|})^{\frac{p-1}{2}}$, then 