Existence results for mean field equations

Abstract

Let $\Omega$ be an annulus. We prove that the mean field equation $-Δψ=\frac{e\sp{-βψ}}{\int\sbΩe\sp{-βψ}} $ admits a solution with zero boundary for $\beta \in (-16\pi ,-8\pi )$. This is a supercritical case for the Moser-Trudinger inequality.

Filling a gap in the argument and adding 2 referrences