Blow-up phenomena for the Liouville equation with a singular source of integer multiplicity

We are concerned with the existence of blowing-up solutions to the following boundary value problem −Δu=λa(x)eu−4πNδ0 in Ω,u=0 on ∂Ω where Ω is a smooth and bounded domain in R2 such that 0∈Ω a(x) is a positive smooth function, N is a positive integer and λ>0 is a small parameter. Here δ0 defines the Dirac measure with pole at 0. We find conditions on the function a and on the domain Ω under which there exists a solution uλ blowing up at 0 and satisfying λ∫Ωa(x)euλ→8π(N+1) as λ→0+.