Blow up points and the Morse indices of solutions to the Liouville equation: inhomogeneous case

Abstract

Let us consider the Liouville equation $$ -\Delta u = \lambda V(x) e^u \quad\text{in}\ \Omega,\quad u = 0\quad \text{on}\ \partial\Omega, $$ where $\Omega$ is a smooth bounded domain in $\mathbb{R}^2$, $V(x) \gt 0$ is a given function in $C^1(\overline{\Omega})$, and $\lambda \gt 0$ is a constant. Let $\{u_n\}$ be an $m$-point blowing up solution sequence for $\lambda = \lambda_n \downarrow 0$, in the sense that $$ \lambda_n\int_\Omega V(x)e^{u_n} dx \to 8\pi m\quad \text{as}\ n\to\infty $$ for $m\in\mathbb{N}$. We prove that the number of blow up points $m$ is less than or equal to the Morse index of $u_n$ for $n$ sufficiently large. This extends the main result of the recent paper [13] to an inhomogeneous ($V \not\equiv 1$) case.