Weighted barycentric sets and singular Liouville equations on compact surfaces
By Alessandro Carlotto, A. Malchiodi. 2011
63 citations recorded by the source corpus.
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]).
Subfields: Applied Mathematics, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/81ec5bb1f8493e96d44610e4e1d339a77f0a77de
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In Applied Mathematics
- Daniele Bartolucci
- A. Malchiodi
- Jun-cheng Wei
- C. B. Ndiaye
- Changshou Lin
- Manuel del Pino
- Lei Zhang
- Guofang Wang
In Computational Theory and Mathematics
- A. Malchiodi
- Daniele Bartolucci
- Jun-cheng Wei
- A. Ambrosetti
- Manuel del Pino
- M. Kowalczyk
- Yanyan Li
- Changshou Lin
Citations in the corpus
- An Improved Geometric Inequality via Vanishing Moments, with Applications to Singular Liouville Equations (2012, background): “We also expect that our result, combined with those in [17], might allow to treat the case of α’s with arbitrary sign, more relevant for geometric applications, as well as give precise homological information on the variational structure (about the latter issue see more comments after (10)).”