Gradient estimates and Liouville type theorems for a nonlinear elliptic equation

Abstract

Let (Mn, g) be an n-dimensional complete Riemannian manifold. We consider gradient estimates and Liouville type theorems for positive solutions to the following nonlinear elliptic equation: $$\Delta u+au\log u=0,$$ where a is a nonzero constant. In particular, for a < 0, we prove that any bounded positive solution of the above equation with a suitable condition for a with respect to the lower bound of Ricci curvature must be \({u\equiv 1}\). This generalizes a classical result of Yau.