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{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} $

where $ p>1 $, $ \kappa $ is a positive constant, $ \rho_\lambda $ is a constant, depending on $ \lambda $, $ \Omega = \Omega_0\setminus \bar{O}_0 $ and $ \Omega_0 $, $ O_0 $ are two planar bounded simply-connected domains. We show that under the assumption $ (\ln\lambda)^\sigma\leq\rho_\lambda\leq (\ln\lambda)^{1-\sigma} $ for some $ \sigma>0 $ small, (1) has a solution $ \psi_\lambda $, whose vorticity set $ \{y\in \Omega:\, \psi(y)-\kappa+\rho_\lambda\eta(y)>0\} $ shrinks to the boundary of the hole as $ \lambda\to +\infty $.