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

Abstract

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