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 $(M, g, \rme^{-f}\rmd v_g)$ 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 \begin{align*} \triangle_{f}u+hu^\alpha=0,\ \ 1<\alpha<\frac{n+m}{n+m-2}\quad (n+m\geq4) \end{align*} 和非线性抛物方程 $$ \Big(\triangle_f-\frac{\partial}{\partial t}\Big)u+hu^{\alpha}=0, \quad \alpha>0 $$ 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]. {\it 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 $(M, g, \rme^{-f}\rmd v_g)$ 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 $$ \triangle_{f}u+hu^\alpha=0,\ \ 1<\alpha<\frac{n+m}{n+m-2}\ \ (n+m\geq4) $$ and $$\Big(\triangle_f-\frac{\partial}{\partial t}\Big)u+hu^{\alpha}=0,\ \ \alpha>0$$ 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]. {\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.