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 andMontefusco (2007) [4] (after Bartolucci and Tarantello, 2002; Brezis and
Merle, 1991 [5,7]). 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]).
© 2011 Elsevier Inc. All rights reserved.
