On the supercritical mean field equation on pierced domains

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

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