On the supercritical mean field equation on pierced domains

Abstract:

We consider the problem \[ (P_\epsilon )\qquad \Delta u +\lambda { e^{u }\over \int \limits _{\Omega \setminus B(\xi ,\epsilon )}e^u}=0\ \hbox {in}\ \Omega \setminus B(\xi ,\epsilon ),\quad u =0\ \hbox {on}\ \partial \left (\Omega \setminus B(\xi ,\epsilon )\right ), \] where $\Omega$ is a smooth bounded open domain in $\mathbb {R}^2$ which contains the point $\xi$. We prove that if $\lambda >8\pi$, problem $(P_\epsilon )$ has a solutions $u_\epsilon$ such that \[ u_\epsilon (x)\to \frac {8\pi + \lambda }{2} G(x,\xi ) \ \text {uniformly on compact sets of $\Omega \setminus \{\xi \}$ } \] as $\epsilon$ 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 ${ 8\pi ^2\over \lambda }$ around the point $\xi$.