Maximal solution of the Liouville equation in doubly connected domains

Abstract In this paper we consider the Liouville equation Δ u + λ 2 e u = 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 λ n → 0 and a sequence of solutions u n it holds u n log ⁡ 1 λ n → H , where H is a harmonic function in Ω ∖ γ and λ n 2 log ⁡ 1 λ n ∫ Ω e u n d x → 8 π c Ω , where c Ω is a constant depending on the conformal class of Ω only.