Second order classification for singular Liouville equations with a coefficient function
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911508514996224 |
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| author | D'Aprile, Teresa Wei, Juncheng Zhang, Lei |
| author_facet | D'Aprile, Teresa Wei, Juncheng Zhang, Lei |
| contents | In this article we are concerned with the existence of blow-up solutions to the following boundary value problem $$-Δv= λV(x) |x|^2e^v\;\mbox{in}\quad B_1,\quad v=0 \;\mbox{ on }\quad \partial B_1,$$ where $B_1$ is the unit ball in $\mathbb R^2$ centered at the origin, $V(x)$ is a positive smooth potential, and $λ>0$ is a small parameter. We find necessary and sufficient conditions on the potential $V$ for the existence of a blow-up sequence of solutions tending to infinity near the origin as $λ\to 0^+$. In particular, we obtain a second-order classification of the coefficient function $V$ for which (simple) blow-up occurs at the origin. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_11735 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Second order classification for singular Liouville equations with a coefficient function D'Aprile, Teresa Wei, Juncheng Zhang, Lei Analysis of PDEs 35J20, 35J57, 35J61 In this article we are concerned with the existence of blow-up solutions to the following boundary value problem $$-Δv= λV(x) |x|^2e^v\;\mbox{in}\quad B_1,\quad v=0 \;\mbox{ on }\quad \partial B_1,$$ where $B_1$ is the unit ball in $\mathbb R^2$ centered at the origin, $V(x)$ is a positive smooth potential, and $λ>0$ is a small parameter. We find necessary and sufficient conditions on the potential $V$ for the existence of a blow-up sequence of solutions tending to infinity near the origin as $λ\to 0^+$. In particular, we obtain a second-order classification of the coefficient function $V$ for which (simple) blow-up occurs at the origin. |
| title | Second order classification for singular Liouville equations with a coefficient function |
| topic | Analysis of PDEs 35J20, 35J57, 35J61 |
| url | https://arxiv.org/abs/2603.11735 |