%0 Journal Article
%J Journal of Functional Analysis 262 (2012) 409-450
%D 2012
%T Weighted barycentric sets and singular Liouville equations on compact surfaces
%A Alessandro Carlotto
%A Andrea Malchiodi
%X 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]).
%B Journal of Functional Analysis 262 (2012) 409-450
%I Elsevier
%G en
%U http://hdl.handle.net/1963/5218
%1 5040
%2 Mathematics
%3 Functional Analysis and Applications
%4 -1
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2012-01-10T11:49:50Z\\nNo. of bitstreams: 1\\n1105.2363v2.pdf: 793763 bytes, checksum: 4996db876d03946c03fa0d4766801974 (MD5)
%R 10.1016/j.jfa.2011.09.012
%0 Journal Article
%J Comptes Rendus Mathematique 349 (2011) 161-166
%D 2011
%T A class of existence results for the singular Liouville equation
%A Alessandro Carlotto
%A Andrea Malchiodi
%X 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 Chernâ€“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.
%B Comptes Rendus Mathematique 349 (2011) 161-166
%I Elsevier
%G en
%U http://hdl.handle.net/1963/5793
%1 5648
%2 Mathematics
%3 Functional Analysis and Applications
%4 -1
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2012-05-07T07:37:16Z\\nNo. of bitstreams: 0
%R 10.1016/j.crma.2010.12.016