Existence results for mean field equations

TL;DR

This paper proves the existence of solutions to a mean field equation in an annular domain for certain supercritical parameter values, extending understanding of solutions beyond the critical threshold.

Contribution

It establishes the existence of solutions to the mean field equation in an annulus for supercritical parameters, a case previously not well-understood.

Findings

Solutions exist for $eta$ in (-16π, -8π).

The result applies to a supercritical regime beyond the Moser-Trudinger inequality.

Provides new existence results in geometric PDEs.

Abstract

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