Maximal solution of the Liouville equation in doubly connected domains

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 $\gamma \subset \Omega$ such that for a sequence ${\lambda }_n\to 0$ and a sequence of solutions $u_n$ it holds $\frac{u_n}{\log \frac{1}{{\lambda }_n}}\to H$, where $H$ is a harmonic function in $\Omega \setminus \gamma$ and $\frac{{\lambda }_n^2}{\log \frac{1}{{\lambda }_n}}{\int }_{\Omega }{e}^{u_n}dx\to 8\pi c_{\Omega }$, where $c_{\Omega }$ is a constant depending on the conformal class of $\Omega$ only.