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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.28302 |
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| _version_ | 1866914431596756992 |
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| author | Chen, Zhijie Li, Houwang Li, Tuoxin Wei, Juncheng |
| author_facet | Chen, Zhijie Li, Houwang Li, Tuoxin Wei, Juncheng |
| contents | We study the blow-up behavior of solutions to the singular Liouville equation
\[
Δ\tilde u+λe^{\tilde u}=4παδ_0
\quad\text{in }B,\quad
\tilde u=0
\quad\text{on }\partial B,
\] where $α>0$, $λ>0$ and $B\subset\mathbb R^2$ is the unit disk. Our main results give a complete classification of all blow-up solutions and determine the exact number of solutions to the above equation. More precisely, for fixed $α>0$ and $λ\in(0,λ_α)$, the singular Liouville equation has exactly $\lceil α\rceil+2$ solutions (up to rotation): a unique minimal energy solution; a unique singular sequence blowing up at the origin; and for each $1\le m\le\lceil α\rceil$, a unique $m$-peak sequence whose blow-up points are the vertices of a regular $m$-gon centered at the origin. This result answers the questions raised in Bartolucci-Montefusco \cite{Bartolucci-Montefusco06} and Bartolucci \cite{Bartolucci10}. We also prove the non-degeneracy of these solutions. Thus we provide a full description of the blow-up structure for the singular Liouville equation on the disk. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_28302 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Classification of blow-up solutions of a singular Liouville equation on the disk Chen, Zhijie Li, Houwang Li, Tuoxin Wei, Juncheng Analysis of PDEs We study the blow-up behavior of solutions to the singular Liouville equation \[ Δ\tilde u+λe^{\tilde u}=4παδ_0 \quad\text{in }B,\quad \tilde u=0 \quad\text{on }\partial B, \] where $α>0$, $λ>0$ and $B\subset\mathbb R^2$ is the unit disk. Our main results give a complete classification of all blow-up solutions and determine the exact number of solutions to the above equation. More precisely, for fixed $α>0$ and $λ\in(0,λ_α)$, the singular Liouville equation has exactly $\lceil α\rceil+2$ solutions (up to rotation): a unique minimal energy solution; a unique singular sequence blowing up at the origin; and for each $1\le m\le\lceil α\rceil$, a unique $m$-peak sequence whose blow-up points are the vertices of a regular $m$-gon centered at the origin. This result answers the questions raised in Bartolucci-Montefusco \cite{Bartolucci-Montefusco06} and Bartolucci \cite{Bartolucci10}. We also prove the non-degeneracy of these solutions. Thus we provide a full description of the blow-up structure for the singular Liouville equation on the disk. |
| title | On the Classification of blow-up solutions of a singular Liouville equation on the disk |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.28302 |