On the construction of non-simple blow-up solutions for the singular Liouville equation with a potential

We are concerned with the existence of blowing-up solutions to the following boundary value problem −Δu=λV(x)eu−4πNδ0 in B1,u=0 on ∂B1, where B1 is the unit ball in R2 centered at the origin, V(x) is a positive smooth potential, N is a positive integer (N≥1). Here δ0 defines the Dirac measure with pole at 0, and λ>0 is a small parameter. We assume that N=1 and, under some suitable assumptions on the derivatives of the potential V at 0, we find a solution which exhibits a non-simple blow-up profile as λ→0+.