Weighted Barycentric Sets and Singular Liouville Equations on Compact Surfaces

TL;DR

This paper proves existence results for a class of elliptic PDEs with exponential nonlinearities and singularities on compact surfaces by developing new topological and functional methods, extending previous regular case theories.

Contribution

It introduces a new model space for unbounded functionals and employs topological and min-max methods to address singular Liouville equations on surfaces.

Findings

Established existence of solutions for singular Liouville equations

Developed a new functional inequality in the spirit of previous work

Connected the problem to topological properties of a generalized barycenter space

Abstract

Given a closed two dimensional manifold, we prove a general existence result for a class of elliptic PDEs with exponential nonlinearities and negative Dirac deltas on the right-hand side, extending a theory recently obtained for the regular case. This is done by global methods: since the associated Euler functional is in general unbounded from below, we need to define a new model space, generalizing the so-called space of formal barycenters and characterizing (up to homotopy equivalence) its very low sublevels. As a result, the analytic problem is reduced to a topological one concerning the contractibility of this model space. To this aim, we prove a new functional inequality in the spirit of [16] and then we employ a min-max scheme based on a cone-style construction, jointly with the blow-up analysis given in [5] (after [6] and [8]). This study is motivated by abelian Chern- Simons theory in self-dual regime, or from the problem of prescribing the Gaussian curvature in presence of conical singularities (hence generalizing a problem raised by Kazdan and Warner in [26]).