On the supercritical mean field equation on pierced domains

Abstract

We consider the problem $$(P_\eps)\qquad \Delta u +\lambda {e^{u}\over\int\limits_{\Omega\setminus B(\xi,\eps)}e^u}=0\ \hbox{in}\ \Omega\setminus B(\xi,\eps),\quad u =0\ \hbox{on}\ \partial\(\Omega\setminus B(\xi,\eps)\), $$ where $\Omega$ is a smooth bounded open domain in $\rr^2$ which contains the point $\xi.$ We prove that if $\lambda>8\pi,$ problem $(P_\eps)$ has a solutions $u_\eps$ such that $$u_\eps(x)\to {8\pi+ \lambda\over2} G(x,\xi) \ \hbox{uniformly on compact sets of $\Omega\setminus\{\xi\}$}$$ as $\eps$ goes to zero. Here $G$ denotes Green's function of Dirichlet Laplacian in $\Omega.$ If $\lambda\not\in 8\pi \mathbb N$ we will not make any symmetry assumptions on $\Omega,$ while if $\lambda \in 8\pi \mathbb N$ we will assume that $\Omega$ is invariant under a rotation through an angle ${8\pi^2\over \la} $ around the point $\xi.$