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

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.