A Harnack type inequality for singular Liouville type equations

TL;DR

This paper establishes a Harnack inequality for solutions to a singular Liouville equation with a weight, extending previous regular case results to include singularities characterized by lpha in (-1,0).

Contribution

It generalizes the Harnack inequality to singular Liouville equations with conical singularities, overcoming challenges due to the lack of translation invariance.

Findings

Proved a Harnack inequality for singular Liouville equations.

Extended Chen-Lin's regular case results to singular cases.

Developed a new approach for handling non-invariance due to singularities.

Abstract

We obtain a Harnack type inequality for solutions of the Liouville type equation, \begin{equation}\nonumber -\Delta u=|x|^{2\alpha}K(x)e^{\displaystyle u} \qquad\text{in} \,\,\, \Omega, \end{equation} where $\alpha\in(-1,0)$, $\Omega$ is a bounded domain in $\mathbb{R}^2$ and $K$ satisfies, \begin{equation}\nonumber 0<a\leq K(x)\leq b<+\infty. \end{equation} This is a generalization to the singular case of a result by C.C. Chen and C.S. Lin [Comm. An. Geom. 1998], which considered the regular case $\alpha=0$. Part of the argument of Chen-Lin can be adapted to the singular case by means of an isoperimetric inequality for surfaces with conical singularities. However, the case $\alpha\in(-1,0)$ turns out to be more delicate, due to the lack of traslation invariance of the singular problem, which requires a different approach.