f-Laplace非线性方程的梯度估计和Liouville定理

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=0,1α(n+m)/(n+m-2)(n+m≥4)和非线性抛物方程(△_f-?/?t)u+hu~α=0,α0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.

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

Let $(M, g, rme^{-f}rmd v_g)$ be an $n$-dimensional complete smoothmetric measure space. The author considers gradient estimates for thepositive solutions to the following nonlinear elliptic equation andnonlinear parabolic equation$$triangle_{f}u+hu^alpha=0, 10$$on $M$. For the classical Laplacian, Li (Li J. Gradient estimates and harnackinequalities for nonlinear parabolic and nonlinear elliptic equations onRiemannian manifolds [J]. {it 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.