Local and Global Log-Gradient estimates of solutions to $Δ_pv+bv^q+cv^r =0$ on manifolds and applications
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| Format: | Preprint |
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2024
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| _version_ | 1866909187609460736 |
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| author | He, Jie Ma, Yuanqing Wang, Youde |
| author_facet | He, Jie Ma, Yuanqing Wang, Youde |
| contents | In this paper, we employ the Nash-Moser iteration technique to study local and global properties of positive solutions to the equation $$Δ_pv+bv^q+cv^r =0$$ on complete Riemannian manifolds with Ricci curvature bounded from below, where $b, c\in\mathbb R$, $p>1$, and $q\leq r$ are some real constants. Assuming certain conditions on $b,\, c,\, p,\, q$ and $r$, we derive succinct Cheng-Yau type gradient estimates for positive solutions, which is of sharp form. These gradient estimates allow us to obtain some Liouville-type theorems and Harnack inequalities. Our Liouville-type results are novel even in Euclidean spaces. Based on the local gradient estimates and a trick of Sung and Wang, we also obtain the global gradient estimates for such solutions. As applications we show the uniqueness of positive solutions to some generalized Allen-Cahn equation and Fisher-KPP equation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_00703 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local and Global Log-Gradient estimates of solutions to $Δ_pv+bv^q+cv^r =0$ on manifolds and applications He, Jie Ma, Yuanqing Wang, Youde Analysis of PDEs In this paper, we employ the Nash-Moser iteration technique to study local and global properties of positive solutions to the equation $$Δ_pv+bv^q+cv^r =0$$ on complete Riemannian manifolds with Ricci curvature bounded from below, where $b, c\in\mathbb R$, $p>1$, and $q\leq r$ are some real constants. Assuming certain conditions on $b,\, c,\, p,\, q$ and $r$, we derive succinct Cheng-Yau type gradient estimates for positive solutions, which is of sharp form. These gradient estimates allow us to obtain some Liouville-type theorems and Harnack inequalities. Our Liouville-type results are novel even in Euclidean spaces. Based on the local gradient estimates and a trick of Sung and Wang, we also obtain the global gradient estimates for such solutions. As applications we show the uniqueness of positive solutions to some generalized Allen-Cahn equation and Fisher-KPP equation. |
| title | Local and Global Log-Gradient estimates of solutions to $Δ_pv+bv^q+cv^r =0$ on manifolds and applications |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2405.00703 |