Bubbling solutions for the Liouville equation with a singular source: non-simple blow-up

We are concerned with the existence of blowing-up solutions to the following boundary value problem-Delta u = lambda e(u) - 4 pi N-lambda delta(0) in Omega, u = 0 on partial derivative Omegawhere Omega is a smooth and bounded domain in R-2 such that 0 is an element of Omega, N-lambda is a positive number close to an integer N (N >= 1) from the right side, delta(0 )defines the Dirac measure with pole at 0, and lambda > 0 is a small parameter. We assume that Omega is (N+1)-symmetric and the regular part of the Green's function satisfies a nondegeneracy condition (both assumptions are verified if Omega is the unit ball) and we find a solution which exhibits a non-simple blow-up profile as lambda -> 0(+). (C) 2020 Elsevier Inc. All rights reserved.