Local and Global Log-Gradient estimates of solutions to $\Delta_pv+bv^q+cv^r =0$ on manifolds and applications

Jie He; Yuanqing Ma; Youde Wang

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arXiv · 2405.00703

Local and Global Log-Gradient estimates of solutions to $\Delta_pv+bv^q+cv^r =0$ on manifolds and applications

Jie He·Yuanqing Ma·Youde Wang

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Abstract

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.

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Jie He, Yuanqing Ma, Youde Wang. 2024-04-22. Local and Global Log-Gradient estimates of solutions to $\Delta_pv+bv^q+cv^r =0$ on manifolds and applications. https://arxiv.org/abs/2405.00703

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