Multiple Blow-Up Solutions for the Liouville Equation with Singular Data

We study the existence of solutions with multiple concentration to the following boundary value problem −∆u=ε2eu −4π\sum_{p∈Z} α_pδ_p inΩ, u=0 on ∂Ω, where Ω is a smooth and bounded domain in R^2, α_p’s are positive numbers, Z ⊂ Ω is a finite set, δ_p defines the Dirac mass at p, and ε > 0 is a small parameter. In particular we extend the result of Del-Pino-Kowalczyk-Musso ([12]) to the case of several singular sources. More precisely we prove that, under suitable restrictions on the weights α_p, a solution exists with a number of blow-up points ξj ∈Ω\Z up to \sum_{p∈Z}max{n∈N|n<1+α_p}.