Sharp existence results for mean field equations with singular data
Journal
Journal of Differential Equations
Journal Volume
252
Journal Issue
7
Pages
4115-4137
Date Issued
2012
Author(s)
Bartolucci, D.
Abstract
Let Ω be a simply connected, open and bounded domain in R2. We are concerned with the nonlinear elliptic problem, where α j>0, δpj denotes the Dirac mass with singular point p j and {p 1,..., p m}⊂Ω. We provide necessary and sufficient conditions for the existence of solutions to (0.1). Our result is the two dimensional version of the sharp existence/nonexistence result obtained in Druet (2002) [13] for elliptic equations with critical exponent in dimension 3. In particular, we prove that the set Ω+m(α) is open, where, for a given α=(α1,...,αm)⊂(0,+∞)×⋯×(0,+∞), Ω+m(α)={(p1,...,pm)| problem (0.1) has a solution}⊂Ω×⋯×Ω. © 2012 Elsevier Inc.
Type
journal article