EXISTENCE RESULT FOR THE MEAN FIELD PROBLEM ON RIEMANN SURFACES OF ALL GENUSES

Abstract

Given a compact surface (Σ,g), we prove the existence of a solution for the mean field equation on Σ. The problem consists of solving a second-order nonlinear elliptic equation with variational structure and exponential nonlinearity. Since the corresponding Euler functional is, in general, unbounded from above and from below, we employ topological methods and min-max schemes. Our result generalizes previous results by Lin [11], by Chen and Lin [6] and by Ding et al. [8]. The main point here is that, by taking values of the parameter greater than 8π, we make no assumption on the topology of the surface.