f-Laplace非线性方程的梯度估计和Liouville定理

Gradient Estimates and Liouville Theorems of a Nonlinear Equation for f-Laplacian

设(M,g,e^(-f)dv_g)是n维完备光滑的度量测度空间.考虑以下非线性椭圆方程△_f^u+hu~α=0,1<α<(n+m)/(n+m-2)(n+m≥4)和非线性抛物方程(△_f-?/?t)u+hu~α=0,α>0正解的梯度估计.对于经典的Laplace情形,Li (Li J. Gradient estimates and harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds [J]. J Funct Anal,1991, 100:233-256.)证明了正解的梯度估计和Liouville定理.在本文中,对于上述的f-Laplace方程,作者将推导出相应的结果.

Let (M,g,e^-fdvg)be an n-dimensional complete smooth metric measure space. The author considers gradient estimates for the positive solutions to the following nonlinear elliptic equation and nonlinear parabolic equation△fu+hu^α=0,1<α<n+m/n+m-2(n+m≥4) and (△f-■/■t)u+huα=0,α>0 on M.For the classical Laplacian,Li (Li J.Gradient estimates and harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds [J].J Funct Anal,1991,100:233-256.)proved the gradient estimates and Liouville theorems.In this paper,the similar results for f-Laplacian are derived.