临界参数下的Liouville系统，第2部分：多气泡
On Liouville systems at critical parameters, Part 2: Multiple bubbles
摘要：
本文继续考虑广义Liouville系统：$$ Δ_g u_i+\sum_{j=1}^n a_{ij}ρ_j\left(\frac{h_j e^{u_j}}{\int h_j e^{u_j}}- {1} \right)=0\quad\text{in \,}M,\quad i\in I=\{1,\cdots,n\}, $$其中$(M,g)$是具有体积$1$的黎曼曲面$M$，$h_1,..,h_n$是正光滑函数，$ρ_j\in \mathbb R^+$（$j\in I$）.在以前的作品林张确定了家庭的超曲面$Γ_N$和证明了先验估计$ρ=(ρ_1,..,ρ_n)$在地区分开的$Γ_N$。后来张林还计算了$ρ^k-ρ$的首项，其中$ρ\in Γ_1$是$ρ^k$对$Γ_1$的极限，$ρ^k$是冒泡序列的参数。这个引导项在实际应用中特别重要，但如果$ρ^k$趋向于高阶超曲面$Γ_N$（$N>1$）则很难确定.多年来，无数的尝试都失败了，但在这篇文章中，我们克服了所有的绊脚石，并完全解决了最一般的情况下的问题：我们不仅捕获的领先条款的$ρ^k-ρ\in Γ_N$，但也揭示了新的鲁棒性关系的系数函数在不同的爆破点。
英文摘要：
In this paper, we continue to consider the generalized Liouville system: $$ Δ_g u_i+\sum_{j=1}^n a_{ij}ρ_j\left(\frac{h_j e^{u_j}}{\int h_j e^{u_j}}- {1} \right)=0\quad\text{in \,}M,\quad i\in I=\{1,\cdots,n\}, $$ where $(M,g)$ is a Riemann surface $M$ with volume $1$, $h_1,..,h_n$ are positive smooth functions and $ρ_j\in \mathbb R^+$($j\in I$). In previous works Lin-Zhang identified a family of hyper-surfaces $Γ_N$ and proved a priori estimates for $ρ=(ρ_1,..,ρ_n)$ in areas separated by $Γ_N$. Later Lin-Zhang also calculated the leading term of $ρ^k-ρ$ where $ρ\in Γ_1$ is the limit of $ρ^k$ on $Γ_1$ and $ρ^k$ is the parameter of a bubbling sequence. This leading term is particularly important for applications but it is very hard to be identified if $ρ^k$ tends to a higher order hypersurface $Γ_N$ ($N>1$). Over the years numerous attempts have failed but in this article we overcome all the stumbling blocks and completely solve the problem under the most general context: We not only capture the leading terms of $ρ^k-ρ\in Γ_N$, but also reveal new robustness relations of coefficient functions at different blowup points.
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