Gradient estimates of a nonlinear elliptic equation for the $V$-Laplacian on noncompact Riemannian manifolds

Authors: DENG YIHUA

Abstract: In this paper, we consider gradient estimates for positive solutions to the following equation $$\triangle_V u+au^p\log u=0$$ on complete noncompact Riemannian manifold with $k$-dimensional Bakry-Emery Ricci curvature bounded from below. Using the Bochner formula and the Cauchy inequality, we obtain upper bounds of $|\nabla u|$ with respect to the lower bound of the Bakry-Emery Ricci curvature.

Keywords: Gradient estimates, $V$-Laplacian, Riemannian manifolds, Bakry-Emery Ricci curvature

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