Convergence for a Liouville equation

Abstract.

In this paper, we study the asymptotic behavior of solutions of the Dirichlet problem for the Liouville equation ¶¶$-\Delta u= \lambda {{K(x)e^u} \over {\int_{\Omega}K(x)e^u}}$¶¶on a bounded smooth domain $\Omega$ in the plane as $\lambda \to 8m\pi$, where $m=1,2,...$. The equation is also called the Mean Field Equation in Statistical Mechanics. By a result of H. Brezis and F. Merle, any solution sequence may have a finite number of bubbles. We give a necessary condition for the location of the bubble points.