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arXiv · 2010.00776

Yau Type Gradient Estimates For $Δu + au(\log u)^{p}+bu=0$ On Riemannian Manifolds

Bo Peng·Youde Wang·Guodong Wei
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Abstract

In this paper, we consider the gradient estimates of the positive solutions to the following equation defined on a complete Riemannian manifold $(M, g)$ $$Δu + au(\log u)^{p}+bu=0,$$ where $a, b\in \mathbb{R}$ and $p$ is a rational number with $p=\frac{k_1}{2k_2+1}\geq2$ where $k_1$ and $k_2$ are positive integer numbers. we obtain the gradient bound of a positive solution to the equation which does not depend on the bounds of the solution and the Laplacian of the distance function on $(M, g)$. Our results can be viewed as a natural extension of Yau's estimates on positive harmonic function.

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Bo Peng, Youde Wang, Guodong Wei. 2020-09-30. Yau Type Gradient Estimates For $Δu + au(\log u)^{p}+bu=0$ On Riemannian Manifolds. https://arxiv.org/abs/2010.00776

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