---
abstract: |
    In this paper we consider bubbling solutions to the general Liouville system: $$\label{abeq1}
    \Delta_g u_i^k+\sum_{j=1}^n a_{ij}\rho_j^k\left(\frac{h_j e^{u_j^k}}{\int h_j e^{u_j^k}}-1\right)=0\quad\text{in \,}M,\,\, i=1,...,n \,\, (n\ge 2)$$ 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.

    1.  The location of the blowup point.

    2.  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).
address:
- |
    Department of Mathematics\
    Taida Institute of Mathematical Sciences\
    National Taiwan University\
    Taipei 106, Taiwan 
- |
    Department of Mathematics\
    University of Florida\
    358 Little Hall P.O.Box 118105\
    Gainesville FL 32611-8105
author:
- 'Chang-shou Lin'
- Lei Zhang
title: 'On Liouville systems at critical parameters, Part 1: one bubble'
---

[^1]

Introduction
============

Let $(M,g)$ be a compact Riemann surface whose volume is normalized to be $1$, $h_1,...,h_n$ be positive $C^3$ functions on $M$, $\rho_1,..,\rho_n$ be nonnegative constants. In this article we continue our study of the following Liouville system defined on $(M,g)$: $$\label{mainsys}
\Delta_g u_i+\sum_{j=1}^n\rho_j 