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