Abstract

In this paper we consider the Liouville equation Delta u+lambda(2)e(u) = 0 with Dirichlet boundary conditions in a two dimensional, doubly connected domain Omega. We show that there exists a simple, closed curve gamma subset of Omega such that for a sequence lambda(n) -> 0 and a sequence of solutions u(n) it holds u(n)/log 1/lambda(n) -> H, where H is a harmonic function in Omegagamma and lambda(2)(n)/log 1/lambda(n) integral(Omega)e(un) dx -> 8 pi c(Omega), where c(Omega) is a constant depending on the conformal class of Omega only. (C) 2019 Elsevier Inc. All rights reserved.