Multiple blow-up solutions for the Liouville equation with singular data

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We study the existence of solutions with multiple concentration to the following boundary value problem -Δu=² eᵘ-4π∑ₚ∈ Zαₚ δₚin Ω, u=0 on∂ Ω, where Ω is a smooth and bounded domain in ², αₚ's are positive numbers, Z⊂ Ω is a finite set, δₚ defines the Dirac mass at p, and >0 is a small parameter. In particular we extend the result of Del-Pino-Kowalczyk-Musso (delkomu) to the case of several singular sources. More precisely we prove that, under suitable restrictions on the weights αₚ, a solution exists with a number of blow-up points ξⱼ∈ Ω Z up to ∑ₚ∈ Z{n∈| n<1+αₚ}.