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Weighted barycentric sets and singular Liouville equations on compact surfaces
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Carlotto, Alessandro and Malchiodi, A. (2012) Weighted barycentric sets and singular Liouville equations on compact surfaces. Journal of Functional Analysis, Vol.262 (No.2). pp. 409-450. doi:10.1016/j.jfa.2011.09.012 ISSN 0022-1236.
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Official URL: http://dx.doi.org/10.1016/j.jfa.2011.09.012
Abstract
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 and ). 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]).
Item Type: | Journal Article | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Journal of Functional Analysis | ||||
Publisher: | Academic Press | ||||
ISSN: | 0022-1236 | ||||
Official Date: | 2012 | ||||
Dates: |
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Volume: | Vol.262 | ||||
Number: | No.2 | ||||
Page Range: | pp. 409-450 | ||||
DOI: | 10.1016/j.jfa.2011.09.012 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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