An Existence Result for the Mean Field Equation on Compact Surfaces in a Doubly Supercritical Regime

TL;DR

This paper proves the existence of solutions for a mean field equation with exponential nonlinearities on compact surfaces, in a doubly supercritical regime, using a new variational approach and a specialized inequality.

Contribution

It introduces a novel Moser-Trudinger type inequality applicable to doubly supercritical parameters on compact surfaces, enabling existence results for the mean field equation.

Findings

Existence of solutions in a doubly supercritical regime.

Development of a new Moser-Trudinger type inequality.

Application to equilibrium turbulence with signed vortices.

Abstract

We consider a class of variational equations with exponential nonlinearities on a compact Riemannian surface, describing the mean field equation of the equilibrium turbulance with arbitrarily signed vortices. For the first time, we consider the problem with both supercritical parameters and we give an existence result by using variational methods. In doing this, we present a new Moser-Trudinger type inequality under suitable conditions on the center of mass and the scale of concentration of both e^u and e^{-u}, where u is the unknown function in the equation.