A Harnack type inequality for singular Liouville type equations

Abstract

We obtain a Harnack type inequality for solutions of the Liouville type equation,$-\Delta u=\left|x{|}^{2\alpha }K\right(x)e^u\text{in}\Omega ,$ where $\alpha \in (-1,0)$, Ω is a bounded domain in $R^2$ and K satisfies,$0<a\le K\left(x\right)\le b<+\infty .$ This is a generalization to the singular case of a result by Chen and Lin (1998) [12], 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 translation invariance of the singular problem, which requires a different approach.