Weighted barycentric sets and singular Liouville equations on compact surfaces

Given a closed surface, we prove a general existence result for some elliptic PDE with exponential nonlinearities and negative Dirac deltas, extending a theory recently obtained for the regular case. This is done by global methods: since the associated Euler functional might be unbounded from below, we define a new model space, generalizing the so-called space of formal barycenters and characterizing (up to homotopy equivalence) its 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 Chen and Li (1991) [11], and then employ a min–max scheme based on conical construction, jointly with the blow-up analysis in Bartolucci and Montefusco (2007) [4], (after Bartolucci and Tarantello, 2002; Brezis and Merle, 1991 [5,7]). This study is motivated by abelian Chern–Simons theory in self-dual regime, or from the problem of prescribing the Gaussian curvature with conical singularities (generalizing a problem raised in Kazdan and Warner, 1974 [24]).