@article {2012,
title = {Weighted barycentric sets and singular Liouville equations on compact surfaces},
journal = {Journal of Functional Analysis 262 (2012) 409-450},
number = {arXiv:1105.2363;},
year = {2012},
publisher = {Elsevier},
abstract = {Given a closed two dimensional manifold, we prove a general existence result\\r\\nfor a class of elliptic PDEs with exponential nonlinearities and negative Dirac\\r\\ndeltas on the right-hand side, extending a theory recently obtained for the\\r\\nregular case. This is done by global methods: since the associated Euler\\r\\nfunctional is in general unbounded from below, we need to define a new model\\r\\nspace, generalizing the so-called space of formal barycenters and\\r\\ncharacterizing (up to homotopy equivalence) its very low sublevels. As a\\r\\nresult, the analytic problem is reduced to a topological one concerning the\\r\\ncontractibility of this model space. To this aim, we prove a new functional\\r\\ninequality 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\\r\\n[6] and [8]). This study is motivated by abelian Chern- Simons theory in\\r\\nself-dual regime, or from the problem of prescribing the Gaussian curvature in\\r\\npresence of conical singularities (hence generalizing a problem raised by\\r\\nKazdan and Warner in [26]).},
doi = {10.1016/j.jfa.2011.09.012},
url = {http://hdl.handle.net/1963/5218},
author = {Alessandro Carlotto and Andrea Malchiodi}
}
@article {2011,
title = {A class of existence results for the singular Liouville equation},
journal = {Comptes Rendus Mathematique 349 (2011) 161-166},
year = {2011},
publisher = {Elsevier},
abstract = {We consider a class of elliptic PDEs on closed surfaces with exponential nonlinearities and Dirac deltas on the right-hand side. The study arises from abelian Chern{\textendash}Simons theory in self-dual regime, or from the problem of prescribing the Gaussian curvature in presence of conical singularities. A general existence result is proved using global variational methods: the analytic problem is reduced to a topological problem concerning the contractibility of a model space, the so-called space of formal barycenters, characterizing the very low sublevels of a suitable functional.},
doi = {10.1016/j.crma.2010.12.016},
url = {http://hdl.handle.net/1963/5793},
author = {Alessandro Carlotto and Andrea Malchiodi}
}