一类非线性椭圆方程的刘维尔型定理

摘要

Let (Mn, g) be an n-dimensional complete noncompact Riemannian manifold. In this paper, we consider the Liouville type theorems for positive solutions to the following nonlinear elliptic equation: ?f u+au log u=0, where a is a nonzero constant. By applying Bochner formula and the maximum principle, we obtain local gradient estimates of the Li-Yau type for positive solu-tions of the above equation on Riemannian manifolds with Bakry-Emery Ricci curvature bounded from below and some relevant Liouville type theorems, which improve some results of [7].%设(Mn,g)是一个n维非紧的完备黎曼流行.本文考虑有正解的非线性椭圆方程?fu+au log u=0的刘维尔型定理,其中a是一个非零常数.利用Bochner公式和极大值原理,获得了以上方程在Bakry-Emery里奇曲率有下界时正解的Li-Yau型梯度估计和某些有关的刘维尔理论,推广了文献[7]的结果.