On Liouville systems at critical parameters, Part 1: One bubble

Abstract

In this paper we consider bubbling solutions to the general Liouville system:(0.1)Δguik+∑j=1naijρjk(hjeujk∫hjeujk−1)=0in M,i=1,…,n(n⩾2) where (M,g) is a Riemann surface, and A=(aij)n×n is a constant non-negative matrix and ρjk→ρj as k→∞. Among other things we prove the following sharp estimates.(1)The location of the blowup point.(2)The convergence rate of ρjk−ρj, j=1,…,n. These results are of fundamental importance for constructing bubbling solutions. It is interesting to compare the difference between the general Liouville system and the SU(3) Toda system on estimates (1) and (2).