Vanishing Estimates for Liouville equation with quantized singularities

TL;DR

This paper investigates the behavior of solutions to the Liouville equation with quantized singularities, showing that if bubbling solutions violate the spherical Harnack inequality near a singular source, then the derivatives of coefficient functions tend to zero.

Contribution

It establishes a new vanishing estimate for the derivatives of coefficient functions in singular Liouville equations under specific bubbling conditions.

Findings

Derivatives of coefficient functions tend to zero when bubbling solutions violate the spherical Harnack inequality.

Provides conditions linking bubbling behavior to coefficient function regularity.

Extends previous work on singular Liouville equations with quantized singularities.

Abstract

In this article we continue with the research initiated in our previous work on singular Liouville equations with quantized singularity. The main goal of this article is to prove that as long as the bubbling solutions violate the spherical Harnack inequality near a singular source, the first derivatives of coefficient functions must tend to zero.