Planar vortices in a bounded domain with a hole

In this paper, we consider the inviscid, incompressible planar flows in a bounded domain with a hole and construct stationary classical solutions with single vortex core, which is closed to the hole. This is carried out by constructing solutions to the following semilinear elliptic problem\begin{document}$ \begin{equation} \begin{cases} -\Delta \psi = \lambda(\psi-\frac{\kappa}{4\pi}\ln\lambda)_+^p,\quad &\text{in}\; \Omega,\\ \psi = \rho_\lambda,\quad &\text{on}\; \partial O_0,\\ \psi = 0,\quad &\text{on}\; \partial\Omega_0, \end{cases} \;\;\;\;\;\;\;\;(1)\end{equation} $\end{document}where \begin{document}$ p>1 $\end{document} , \begin{document}$ \kappa $\end{document} is a positive constant, \begin{document}$ \rho_\lambda $\end{document} is a constant, depending on \begin{document}$ \lambda $\end{document} , \begin{document}$ \Omega = \Omega_0\setminus \bar{O}_0 $\end{document} and \begin{document}$ \Omega_0 $\end{document} , \begin{document}$ O_0 $\end{document} are two planar bounded simply-connected domains. We show that under the assumption \begin{document}$ (\ln\lambda)^\sigma\leq\rho_\lambda\leq (\ln\lambda)^{1-\sigma} $\end{document} for some \begin{document}$ \sigma>0 $\end{document} small, (1) has a solution \begin{document}$ \psi_\lambda $\end{document} , whose vorticity set \begin{document}$ \{y\in \Omega:\, \psi(y)-\kappa+\rho_\lambda\eta(y)>0\} $\end{document} shrinks to the boundary of the hole as \begin{document}$ \lambda\to +\infty $\end{document} .