Compactness, existence and multiplicity for the singular mean field problem with sign-changing potentials

TL;DR

This paper studies a mean field problem with sign-changing potentials on compact surfaces, providing new results on the conditions for solution existence, multiplicity, and compactness, relevant to geometry and physics models.

Contribution

It introduces novel results on the compactness, existence, and multiplicity of solutions for mean field equations with sign-changing potentials on surfaces.

Findings

Results on solution compactness for sign-changing potentials

Existence theorems for solutions under certain conditions

Multiplicity results indicating multiple solutions

Abstract

In this paper we consider a mean field problem on a compact surface with conical singularities. This problem appears in the Gaussian curvature prescription problem in Geometry, and also in the Electroweak Theory and in the abelian Chern-Simons-Higgs model in Physics. In this paper we focus on the case of sign-changing potentials, and we give results on compactness, existence and multiplicity of solutions.