Profile of bubbling solutions to a Liouville system

TL;DR

This paper proves uniqueness and provides uniform estimates for bubbling solutions to Liouville systems, which are important for understanding solutions on Riemann surfaces across physics, chemistry, and ecology.

Contribution

It establishes a key uniqueness theorem and uniform estimates for bubbling solutions of Liouville systems in the plane, aiding in solution existence analysis.

Findings

Proved a uniqueness result for Liouville systems in .

Established uniform estimates near isolated blowup points.

Facilitated a priori estimates and degree counting for solutions on Riemann surfaces.

Abstract

In several fields of Physics, Chemistry and Ecology, some models are described by Liouville systems. In this article we first prove a uniqueness result for a Liouville system in $\mathbb R^2$. Then we establish an uniform estimate for bubbling solutions of a locally defined Liouville system near an isolated blowup point. The uniqueness result, as well as the local uniform estimates are crucial ingredients for obtaining a priori estimate, degree counting formulas and existence results for Liouville systems defined on Riemann surfaces.