article

Title: A Harnack inequality for Liouville-type equations with singular sources

Authors: Gabriella Tarantello

Issue: Volume 54 (2005), Issue 2, 599-616

Abstract:

$Let $\Omega \subset \mathbb{R}^2$ be a bounded domain, and $V \in C^{0,1}(\Omega)$ satisfy: $0 < a \le V \le b$, $|\nabla V| \le A$ in $\Omega$. For given $\alpha > 0$ and $0 \in \Omega$, we show that every solution of the equation $-\Delta u = |z|^{2\alpha}Ve^u$ in $\Omega$ satisfies $u(0) + \inf\limits_{\Omega}u \le C$, with a suitable constant $C$ depending only on $a$, $b$, $A$ and $\mbox{dist},(0,\partial\Omega)$. This furnishes a nontrivial extension of an analogous result established by Brezis-Li-Shafrir in [cited work at the end of the abstract], in case $\alpha = 0$. (H. Brezis, Y.Y. Li, and I. Shafrir, \textit{A $\mbox{\upshape sup} + \mbox{\upshape inf}$ inequality for some nonlinear elliptic equations involving exponential nonlinearities}, J. Funct. Anal. \textbf{115} (1993), 344--358.)$