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

In this paper we consider bubbling solutions to the general Liouville system: \label{abeq1} \Delta_g u_i^k+\sum_{j=1}^n a_{ij}\rho_j^k(\frac{h_j e^{u_j^k}}{\int h_j e^{u_j^k}}-1)=0\quad\text{in}M, i=1,...,n (n\ge 2) where $(M,g)$ is a Riemann surface, and $A=(a_{ij})_{n\times n}$ is a constant non-negative matrix and $\rho_j^k\to \rho_j$ as $k\to \infty$. Among other things we prove the following sharp estimates. The location of the blowup point. The convergence rate of $\rho_j^k-\rho_j$, $j=1,..,n$. 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).