A note on a multiplicity result for the mean field equation on compact surfaces

TL;DR

This paper establishes a new multiplicity result for a Liouville-type mean field equation on compact surfaces, relevant to equilibrium turbulence with vortices, using Morse theory techniques.

Contribution

It provides the first multiplicity result for this class of equations, advancing understanding of solutions on compact surfaces.

Findings

Proves multiple solutions exist for the mean field equation.

Uses Morse theory to analyze the solution space.

Enhances theoretical understanding of vortex-related equations.

Abstract

We are concerned with a Liouville-type equation with exponential nonlinearities on a compact surface which describes the mean field equation of the equilibrium turbulence with arbitrarily signed vortices. We provide the first multiplicity result for this class of equations by using Morse theory.