The mean field equation with critical parameter in a plane domain

Abstract

Consider the mean field equation with parameter $\lambda=8\pi$ in a bounded smooth domain $\Omega$. Denote by $E_{8\pi}(\Omega)$ the infimum of the associated functional $I_{8\pi}(\Omega)$. We prove that if $|\Omega|=\pi$, then $E_{8\pi}(\Omega)\ge E_{8\pi}(B_1)$ and equality holds if and only if $\Omega$ is a ball. We also give a sufficient condition for the existence of a minimizer for $I_{8\pi}(\Omega)$.