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

In this paper, we consider the gradient estimates of the positive solutions to the following equation defined on a complete Riemannian manifold $(M, g)$ $$Delta u + au(log u)^{p}+bu=0,$$ where $a, bin mathbb{R}$ 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 Yaus estimates on positive harmonic function.