Compactness, existence and multiplicity for the singular mean field   problem with sign-changing potentials
  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. 
Francesca De Marchis, Rafael L\'opez-Soriano, David Ruiz
math.AP