Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains

We are concerned with the existence of blowing-up solutions to the following boundary value problem −Δu=λV(x)eu−4πNδ0 in Ω,u=0 on ∂Ω, where Ω is a smooth and bounded domain in R2 such that 0∈Ω, V is a positive smooth potential, N is a positive integer and λ>0 is a small parameter. Here δ0 defines the Dirac measure with pole at 0. We assume that Ωis (N+1) -symmetric and we find conditions on the potential Vand the domain Ω under which there exists a solution blowing up at N+1 points located at the vertices of a regular polygon with center 0.