Gradient estimates and Liouville type theorems for a nonlinear elliptic equation

Let (M (n) , 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 mu + 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 . This generalizes a classical result of Yau.