  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: $$\Delta u+au\log u=0,$$ 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.
