Structure of bubbling solutions of Liouville systems with negative singular sources

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 [Formula: see text] approaches a critical hyper-surface, we identify a pivotal leading term vital for numerous applications.