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

Abstract

In this paper, we consider the gradient estimates of the positive solutions to the following equation defined on a complete Riemannian manifold $(M,g)$ $$\hat{I}”u+au(\log u{)}^p+bu=0,$$ where $a,b\in \mathbb{R}$ and $p$ is a rational number with $p=\frac{k_1}{2k_2+1}\ge 2$ 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.

arXiv admin note: text overlap with arXiv:2009.14566