Conformally invariant equations with negative critical exponents on the three dimensional hyperbolic space
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917341083729920 |
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| author | Ganguly, Debdip Li, Jungang Lu, Guozhen Wang, Jianxiong |
| author_facet | Ganguly, Debdip Li, Jungang Lu, Guozhen Wang, Jianxiong |
| contents | We establish a symmetry result for positive entire solutions with a prescribed growth rate to the following fourth order equation on the 3-dimensional hyperbolic space $\mathbb{H}^3$: \[ P_2 u = - u^{-7}, \] where $P_2$ denotes the fourth-order Paneitz operator. We prove that any positive solution $u$ on $\mathbb{H}^3$ exhibiting exponential growth at infinity must, up to hyperbolic isometries, be radial and strictly decreasing with respect to some point $P \in \mathbb{H}^3$.
Fourth order equations with negative critical growth on 3-dimensional Euclidean space $\mathbb{R}^3$ has been studied by Choi and Xu in \cite{CX09 }, and subsequently by McKenna and Reichel \cite{MR03} and Xu \cite{Xu05}. Unlike the Euclidean case, the behavior of the Green's function of $P_2$ is substantially different, which prevents us from using the moving plane (sphere) method directly. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_13601 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Conformally invariant equations with negative critical exponents on the three dimensional hyperbolic space Ganguly, Debdip Li, Jungang Lu, Guozhen Wang, Jianxiong Analysis of PDEs We establish a symmetry result for positive entire solutions with a prescribed growth rate to the following fourth order equation on the 3-dimensional hyperbolic space $\mathbb{H}^3$: \[ P_2 u = - u^{-7}, \] where $P_2$ denotes the fourth-order Paneitz operator. We prove that any positive solution $u$ on $\mathbb{H}^3$ exhibiting exponential growth at infinity must, up to hyperbolic isometries, be radial and strictly decreasing with respect to some point $P \in \mathbb{H}^3$. Fourth order equations with negative critical growth on 3-dimensional Euclidean space $\mathbb{R}^3$ has been studied by Choi and Xu in \cite{CX09 }, and subsequently by McKenna and Reichel \cite{MR03} and Xu \cite{Xu05}. Unlike the Euclidean case, the behavior of the Green's function of $P_2$ is substantially different, which prevents us from using the moving plane (sphere) method directly. |
| title | Conformally invariant equations with negative critical exponents on the three dimensional hyperbolic space |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.13601 |