Singular mean field equations on compact Riemann surfaces

TL;DR

This paper investigates the existence of solutions with concentrated nonlinear terms in mean field elliptic PDEs on compact Riemann surfaces, including cases with singular sources, advancing understanding of these complex equations.

Contribution

It introduces new methods to establish solution existence for mean field equations with singular sources on compact Riemann surfaces.

Findings

Proved existence of solutions with concentrated nonlinearities.

Addressed the challenges posed by singular Dirac mass sources.

Extended the theory to more degenerate cases.

Abstract

For a general class of elliptic PDE's in mean field form on compact Riemann surfaces with exponential nonlinearity, we address the question of the existence of solutions with concentrated nonlinear term, which, in view of the applications, are physically of definite interest. In the model, we also include the possible presence of singular sources in the form of Dirac masses, which makes the problem more degenerate and difficult to attack.