Liouville Type Theorems For A Nonlinear Elliptic Equation

Let(Mn,g)be an n-dimensional noncompact complete Riemannian manifold.In this paper,we consider the Liouville type theorems for positive solutions to the follow-ing nonlinear elliptic equation:?fu + au log u = 0,where a is a nonzero constant.By applying Bochner formula and the maximum principle,we obtain local Li-Yau type gra-dient estimates for positive solutions of the above equation on Riemannian manifolds with Bakry-Emery Ricci curvature bounded from below,as a corollary we reproved the Liouville type theorems under the condition that N-Bakry-Emery Ricci curvature is bounded from below which was proved by Dung and Khanh.Meanwhile,we get Liouville type theorems when ?-Bakry-Emery Ricci curvature is bounded from below.