Structure of bubbling solutions of Liouville systems with negative singular sources

TL;DR

This paper analyzes the detailed structure of bubbling solutions in Liouville systems with negative singular sources, providing precise estimates and characterizations crucial for understanding blowup behavior.

Contribution

It extends previous work by offering a priori estimates at critical parameter positions and thoroughly characterizes bubble interactions and blowup configurations.

Findings

Exact heights of bubbling solutions at blowup points

Precise integrals of each component during blowup

Identification of a key leading term near critical hyper-surfaces

Abstract

Liouville systems on Riemann surfaces are instrumental in modeling species growth and particle dynamics in biology and physics. Previously, we established a priori estimates for parameters across regions defined by critical hyper-surfaces. Here, we extend this by giving a priori estimates when parameters are critically positioned. This involves thoroughly characterizing bubble interaction, a key challenge in Liouville systems. During blowup events, we ascertain the exact heights of bubbling solutions about each blowup point, the integrals of each component, and the blowup points' positions. Moreover, as the parameter $\rho$ approaches a critical hyper-surface, we identify a pivotal leading term vital for numerous applications.