Construction of blowup solutions for Liouville systems

Zetao Cheng; Haoyu Li; Lei Zhang

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arXiv · 2503.07467

Construction of blowup solutions for Liouville systems

Zetao Cheng·Haoyu Li·Lei Zhang

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Abstract

We study the following Liouville system defined on a flat torus \begin{equation} \left\{ \begin{array}{lr} -\Delta u_i=\sum_{j=1}^n a_{ij}\rho_j\Big(\frac{h_j e^{u_j}}{\int_\Omega h_j e^{u_j}}-1\Big),\nonumber \\ u_j\in H_{per}^1(\Omega)\mbox{ for }i\in I=\{1,\cdots,n\}\nonumber, \end{array} \right. \end{equation} where $h_j\in C^3(\Omega)$, $h_j>0$, $\rho_j>0$ and $u=(u_1,..,u_n)$ is doubly periodic on $\partial\Omega$. The matrix $A=(a_{ij})_{n\times n}$ satisfies certain properties. One central problem about Liouville systems is whether multi-bubble solutions do exist. In this work we present a comprehensive construction of multi-bubble solutions in the most general setting.

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Zetao Cheng, Haoyu Li, Lei Zhang. 2025-03-10. Construction of blowup solutions for Liouville systems. https://arxiv.org/abs/2503.07467

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