A class of existence results for the singular Liouville equation

Carlotto, Alessandro; Malchiodi, Andrea

Comptes Rendus. Mathématique, Tome 349 (2011) no. 3-4, pp. 161-166.

Nous considérons une classe d'EDP elliptiques sur une surface compacte et sans bord, avec une nonlinéarité exponentielle et des masses de Dirac dans le membre de droite. Ce travail est motivé par l'étude d'équations de Chern–Simons abéliennes en régime auto-dual, ainsi que par le problème de la courbure gaussienne prescrite pour des surfaces avec singularités coniques. Nous démontrons un résultat général d'existence en utilisant des méthodes variationnels globales : le problème analytique est réduit à un problème topologique concernant la contractilité d'un espace modèle, l'espace des barycentres formels, qui caractérise les sous-niveaux très bas d'une fonctionnelle appropriée.

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–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.

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