Local and Global Log-Gradient estimates of solutions to $\Delta_pv+bv^q+cv^r =0$ on manifolds and applications
| Autor: | He, Jie, Ma, Yuanqing, Wang, Youde |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | In this paper, we employ the Nash-Moser iteration technique to study local and global properties of positive solutions to the equation $$\Delta_pv+bv^q+cv^r =0$$ 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. Comment: arXiv admin note: substantial text overlap with arXiv:2311.02568; text overlap with arXiv:2311.13179 |
| Databáze: | arXiv |
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