Gradient estimates of a nonlinear elliptic equation for the $V$-Laplacian
Bull. Korean Math. Soc. 2019 Vol. 56, No. 4, 853-865
https://doi.org/10.4134/BKMS.b180639
Published online July 31, 2019
Fanqi Zeng
Xinyang Normal University
Abstract : 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
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