On Liouville systems at critical parameters, Part 2: Multiple bubbles

TL;DR

This paper advances the understanding of Liouville systems at critical parameters by analyzing multiple bubbling phenomena, providing precise asymptotics and revealing new relations among coefficients at blowup points.

Contribution

It extends previous work by fully characterizing the asymptotic behavior of parameters near higher-order hypersurfaces and uncovering robustness relations among coefficients at multiple blowup points.

Findings

Captured leading terms of parameter differences near higher-order hypersurfaces

Revealed new robustness relations among coefficients at blowup points

Solved longstanding difficulties in asymptotic analysis of Liouville systems

Abstract

In this paper, we continue to consider the generalized Liouville system: $$ \Delta_g u_i+\sum_{j=1}^n a_{ij}\rho_j\left(\frac{h_j e^{u_j}}{\int h_j e^{u_j}}- {1} \right)=0\quad\text{in \,}M,\quad i\in I=\{1,\cdots,n\}, $$ where $(M,g)$ is a Riemann surface $M$ with volume $1$, $h_1,..,h_n$ are positive smooth functions and $\rho_j\in \mathbb R^+$($j\in I$). In previous works Lin-Zhang identified a family of hyper-surfaces $\Gamma_N$ and proved a priori estimates for $\rho=(\rho_1,..,\rho_n)$ in areas separated by $\Gamma_N$. Later Lin-Zhang also calculated the leading term of $\rho^k-\rho$ where $\rho\in \Gamma_1$ is the limit of $\rho^k$ on $\Gamma_1$ and $\rho^k$ is the parameter of a bubbling sequence. This leading term is particularly important for applications but it is very hard to be identified if $\rho^k$ tends to a higher order hypersurface $\Gamma_N$ ($N>1$). 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 $\rho^k-\rho\in \Gamma_N$, but also reveal new robustness relations of coefficient functions at different blowup points.