Maximal solution of the Liouville equation in doubly connected domains

Abstract

In this paper we consider the Liouville equation Δu+λ2eu=0 with Dirichlet boundary conditions in a two dimensional, doubly connected domain Ω. We show that there exists a simple, closed curve γ⊂Ω such that for a sequence λn→0 and a sequence of solutions un it holds [Formula presented], where H is a harmonic function in Ω∖γ and [Formula presented], where cΩ is a constant depending on the conformal class of Ω only.