Yau’s gradient estimates for a nonlinear elliptic equation

Abstract

Let \({(M^n,g)}\) be an n-dimensional complete Riemannian manifold. We consider Yau’s gradient estimates for positive solutions to the following nonlinear equation $$\Delta u + au {\rm log} u=0$$ where a is a constant. As an application, we obtain the Liouville property for this equation in the case of a < 0. In addition, we illustrate, by giving concrete examples, that our results are sharp.