Calculus of Variations and Geometric Measure Theory

A. Jevnikar

An existence result for the mean-field equation on compact surfaces in a doubly supercritical regime

created by jevnikar on 05 Nov 2015


Published Paper

Inserted: 5 nov 2015
Last Updated: 5 nov 2015

Journal: Proc. Royal Soc. Edinburgh A
Volume: 5
Number: 143
Pages: 1021-1045
Year: 2015
Links: arxiv


We consider a class of variational equations with exponential nonlinearities on a compact Riemannian surface, describing the mean field equation of the equilibrium turbulence 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.