Liouville type theorems of a nonlinear elliptic equation for the V-Laplacian

In this paper, we consider Liouville type theorems for positive solutions to the following nonlinear elliptic equation: Delta(V) u + au log u = 0, where a is a nonzero real constant. By using gradient estimates, we obtain upper bounds of vertical bar del u vertical bar with respect to sup u and the lower bound of Bakry-Emery Ricci curvature. In particular, for complete noncompact manifolds with a < 0, we prove that any positive solution must be u equivalent to 1 under a suitable condition for a with respect to the lower bound of Bakry-Emery Ricci curvature. It generalizes a classical result of Yau.