完备黎曼流形上方程$\Delta v+v^r-v^s= 0$解的梯度估计

doi:10.3969/j.issn.1006­8074.2023.03.001

数学理论与应用 ›› 2023, Vol. 43 ›› Issue (3): 1-22.doi: 10.3969/j.issn.10068074.2023.03.001

• • 下一篇

完备黎曼流形上方程$\Delta v+v^r-v^s= 0$解的梯度估计

王友德^1,2,3 ,张艾琦¹

1. 广州大学数学与信息科学学院, 广州, 510006; 2. 中国科学院数学与系统科学研究院数学研究所华罗庚数学重点实验室, 北京, 100190; 3. 中国科学院大学数学科学学院, 北京, 100049.

出版日期:2023-09-28 发布日期:2023-10-09

Gradient Estimate for Solutions of $\Delta v+v^r-v^s= 0$ on a Complete Riemannian Manifold

Wang Youde^1,2,3 , Zhang Aiqi¹

1. School of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China; 2. Hua Loo-Keng Key Laboratory of Mathematics, Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China; 3. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China.

Online:2023-09-28 Published:2023-10-09

摘要/Abstract

摘要：

在本文中, 我们讨论定义于完备黎曼流形$(M,\,g)$上的椭圆方程

$\Delta v+v^r-v^s= 0$

正解的梯度估计, 其中$r$和$s$是常数.

当$(M,\,g)$满足$Ric \geq -(n-1)\kappa$时(其中$n\geq2$是$M$的维数, $\kappa$是非负常数),

在适当的几何和分析条件下, 我们采用Nash-Moser迭代技巧导出该方程正解的Cheng-Yau型梯度估计, 并证明当

$(M,\,g)$的Ricci曲率非负时, 若$r<s$, 并且$1<r<\frac{n+3}{n-1}$或$1<s<\frac{n+3}{n-1}$, 则该方程除了$v\equiv1$以外无其它正解.

关键词: 椭圆方程, 黎曼流形 , 梯度估计

Abstract:

In this paper we consider the gradient estimates on the positive solutions to the elliptic equation $\Delta v+v^r-v^s= 0,$ defined on a complete Riemannian manifold $(M,\,g)$,

where $r$ and $s$ are two real constants.

When$(M,\,g)$ satisfies $Ric \geq -(n-1)\kappa$ (where $n\geq2$ is the dimension of $M$ and $\kappa$ is a nonnegative constant), we employ the Nash-Moser iteration technique to derive a Cheng-Yau type gradient estimate for the positive solutions to the above equation under some suitable geometric and analysis conditions.

Moreover, it is shown that when the Ricci curvature of $M$ is nonnegative, this elliptic equation does not admit any positive solutions except for $v\equiv 1$ if\ $r<s$ and $1<r<\frac{n+3}{n-1}~\mbox{or}~1<s<\frac{n+3}{n-1}.$

Key words: Elliptic equation, Riemannian manifold , Gradient estimate

王友德, 张艾琦. 完备黎曼流形上方程$\Delta v+v^r-v^s= 0$解的梯度估计[J]. 数学理论与应用, 2023, 43(3): 1-22.

Wang Youde, Zhang Aiqi. Gradient Estimate for Solutions of $\Delta v+v^r-v^s= 0$ on a Complete Riemannian Manifold[J]. Mathematical Theory and Applications, 2023, 43(3): 1-22.

完备黎曼流形上方程$\Delta v+v^r-v^s= 0$解的梯度估计

Gradient Estimate for Solutions of $\Delta v+v^r-v^s= 0$ on a Complete Riemannian Manifold

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 1

编辑推荐

Metrics

本文评价