| 朱超娜.f-Laplace非线性方程的梯度估计和 Liouville定理[J].数学年刊A辑,2018,(4):349~366 |
| 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 |
| Received:March 03, 2017 |
| DOI：10.16205/j.cnki.cama.2018.0030 |
| 中文关键词: Gradient estimate, Liouville theorem, $f$-Laplacian |
| 英文关键词:Gradient estimate, Liouville theorem, $f$-Laplacian |
| 基金项目: |
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| 中文摘要: |
| 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. |
| 英文摘要: |
| 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. |
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