Bulletin of the
Korean Mathematical Society BKMS

In this paper, we consider gradient estimates for positive solutions to the following nonlinear elliptic equation on a complete Riemannian manifold: $$\Delta_{V}u+cu^{\alpha}=0,$$ where $c$, $\alpha$ are two real constants and $c\neq0$. By applying Bochner formula and the maximum principle, we obtain local gradient estimates for positive solutions of the above equation on complete Riemannian manifolds with Bakry-\'{E}mery Ricci curvature bounded from below, which generalize some results of \cite{MHL2017}.

Keywords: gradient estimate, nonlinear elliptic equation, Bakry-\'{E}mery Ricci curvature

MSC numbers: Primary 58J35; Secondary 35B45

Supported by: The research of the author was supported by Nanhu Scholars Program for Young Scholars of XYNU, and Doctoral Scientific Research Startup Fund of Xinyang Normal University (2018)