Liouville theorem of the subcritical biharmonic equation on complete manifolds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908496221437952 |
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| author | Ma, Xi-Nan Wu, Tian Wu, Wangzhe |
| author_facet | Ma, Xi-Nan Wu, Tian Wu, Wangzhe |
| contents | In this paper, we study the subcritical biharmonic equation \[Δ^2 u=u^α\] on a complete, connected, and non-compact Riemannian manifold $(M^n,g)$ with nonnegative Ricci curvature. Using the method of invariant tensors, we derive a differential identity to obtain a Liouville theorem, i.e., there is no positive $C^4$ solution if $n\geqslant5$ and $1<α<\frac{n+4}{n-4}$. We establish a crucial second-order derivative estimate, which is established via Bernstein's technique and the continuity method. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_14497 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Liouville theorem of the subcritical biharmonic equation on complete manifolds Ma, Xi-Nan Wu, Tian Wu, Wangzhe Analysis of PDEs In this paper, we study the subcritical biharmonic equation \[Δ^2 u=u^α\] on a complete, connected, and non-compact Riemannian manifold $(M^n,g)$ with nonnegative Ricci curvature. Using the method of invariant tensors, we derive a differential identity to obtain a Liouville theorem, i.e., there is no positive $C^4$ solution if $n\geqslant5$ and $1<α<\frac{n+4}{n-4}$. We establish a crucial second-order derivative estimate, which is established via Bernstein's technique and the continuity method. |
| title | Liouville theorem of the subcritical biharmonic equation on complete manifolds |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2508.14497 |