Maximal solution of the Liouville equation in doubly connected domains

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 $\mathrm{\Omega }$. We show that there exists a simple, closed curve $\gamma \subset \mathrm{\Omega }$ 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 $\mathrm{\Omega }\setminus \gamma$ and $\frac{\lambda_n^2}{\log\frac{1}{\lambda_n}}\int_\Omega e^{,u_n},dx\to 8\pi c_\Omega$,where$c_\Omega$isaconstantdependingontheconformalclassof$\Omega$ only.