Given a closed surface, we prove a general existence result for some elliptic PDEs 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  and then employ a min-max scheme, jointly with the blow-up analysis in  (after , ). 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 ).
|Titolo:||Weighted barycentric sets and singular Liouville equations on compact surfaces|
|Data di pubblicazione:||2012|
|Digital Object Identifier (DOI):||http://dx.doi.org/10.1016/j.jfa.2011.09.012|
|Appare nelle tipologie:||1.1 Articolo in rivista|