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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.10757 |
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| _version_ | 1866910700929024000 |
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| author | Jin, Cheng Wang, Youde Zeng, Fanqi |
| author_facet | Jin, Cheng Wang, Youde Zeng, Fanqi |
| contents | In this paper, we consider the nonlinear elliptic equation $$Δ_fv^τ+λv=0$$ on a complete smooth metric measure space with $m$-Bakry-Émery Ricci curvature bounded from below, where $τ>0$ and $λ$ are constant. We obtain some new local gradient estimates for positive solutions to the equation using the Nash-Moser iteration technique. As applications of these estimates, we obtain a Liouville type theorem and a Harnack inequality, and the global gradient estimates for such solutions. Our results generalize and improve the estimates in Wang (J. Differential Equations 260:567-585, 2016) and Zhao (Arch. Math. (Basel) 114:457-469, 2020). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_10757 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cheng-Yau logarithmic gradient estimates for a nonlinear elliptic equation on smooth metric measure spaces Jin, Cheng Wang, Youde Zeng, Fanqi Differential Geometry In this paper, we consider the nonlinear elliptic equation $$Δ_fv^τ+λv=0$$ on a complete smooth metric measure space with $m$-Bakry-Émery Ricci curvature bounded from below, where $τ>0$ and $λ$ are constant. We obtain some new local gradient estimates for positive solutions to the equation using the Nash-Moser iteration technique. As applications of these estimates, we obtain a Liouville type theorem and a Harnack inequality, and the global gradient estimates for such solutions. Our results generalize and improve the estimates in Wang (J. Differential Equations 260:567-585, 2016) and Zhao (Arch. Math. (Basel) 114:457-469, 2020). |
| title | Cheng-Yau logarithmic gradient estimates for a nonlinear elliptic equation on smooth metric measure spaces |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2411.10757 |