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

Futoshi Takahashi

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.

Article information

Dates
Revised: 22 January 2013
First available in Project Euclid: 30 October 2018

https://projecteuclid.org/ euclid.aspm/1540934212

Digital Object Identifier
doi:10.2969/aspm/06410175

Mathematical Reviews number (MathSciNet)
MR3381201

Zentralblatt MATH identifier
1336.35140

Citation

Takahashi, Futoshi. Blow up points and the Morse indices of solutions to the Liouville equation: inhomogeneous case. Nonlinear Dynamics in Partial Differential Equations, 175--182, Mathematical Society of Japan, Tokyo, Japan, 2015. doi:10.2969/aspm/06410175. https://projecteuclid.org/euclid.aspm/1540934212