Sharp existence results for mean field equations with singular data

Let Ω be a simply connected, open and bounded domain in R2. We are concerned with the nonlinear elliptic problem(0.1){−Δv=8πev∫Ωev−4π∑j=1mαjδpjin Ω,v=0on ∂Ω, where αj>0, δpj denotes the Dirac mass with singular point pj and {p1,…,pm}⊂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(α̲) is open, where, for a given α̲=(α1,…,αm)⊂(0,+∞)×⋯×(0,+∞), Ω+m(α̲)={(p1,…,pm)| problem (0.1) has a solution}⊂Ω×⋯×Ω.