Symmetry of Solutions to Fractional Semilinear Equations on Hyperbolic Spaces
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866910058656301056 |
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| author | Wang, Jianxiong |
| author_facet | Wang, Jianxiong |
| contents | We study a semilinear equation involving the fractional Laplacian on the hyperbolic space $\mathbb{H}^n$. Unlike in conformally compact Einstein manifolds, the fractional Laplacian on $\mathbb{H}^n$ does not enjoy conformal covariance. By employing Helgason-Fourier analysis, we explicitly derive the Green's function of the fractional Laplacian on $\mathbb{H}^n$ as well as its asymptotic behaviors. We then apply a direct method of moving planes to the integral form of the equation, and show that nonnegative weak solutions are symmetric. In addition, we extend several maximum principles to hyperbolic space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12140 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Symmetry of Solutions to Fractional Semilinear Equations on Hyperbolic Spaces Wang, Jianxiong Analysis of PDEs We study a semilinear equation involving the fractional Laplacian on the hyperbolic space $\mathbb{H}^n$. Unlike in conformally compact Einstein manifolds, the fractional Laplacian on $\mathbb{H}^n$ does not enjoy conformal covariance. By employing Helgason-Fourier analysis, we explicitly derive the Green's function of the fractional Laplacian on $\mathbb{H}^n$ as well as its asymptotic behaviors. We then apply a direct method of moving planes to the integral form of the equation, and show that nonnegative weak solutions are symmetric. In addition, we extend several maximum principles to hyperbolic space. |
| title | Symmetry of Solutions to Fractional Semilinear Equations on Hyperbolic Spaces |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2601.12140 |