New existence results for the mean field equation on compact surfaces via degree theory

TL;DR

This paper establishes new existence results for the mean field equation with exponential nonlinearities on compact surfaces, especially the sphere, using degree theory and Leray-Schauder degree parity analysis.

Contribution

It introduces a novel application of degree theory to prove existence of solutions for the mean field equation on compact surfaces, including the sphere.

Findings

New existence results for the mean field equation on the sphere.

Application of degree theory and Leray-Schauder degree parity analysis.

Recovery of some previously known results using the new method.

Abstract

We consider a class of equations with exponential non-linearities on a compact surface which arises as the mean field equation of the equilibrium turbulence with arbitrarily signed vortices. We prove an existence result via degree theory. This yields new existence results in case of a topological sphere. The proof is carried out by considering the parity of the Leray-Schauder degree associated to the problem. With this method we recover also some known previous results.