THE ASYMPTOTIC BEHAVIOR AND SYMMETRY OF POSITIVE SOLUTIONS TO p-LAPLACIAN EQUATIONS IN A HALF-SPACE

doi:10.1007/s10473-022-0524-y

数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (5): 2149-2164.doi: 10.1007/s10473-022-0524-y

• 论文 • 上一篇

THE ASYMPTOTIC BEHAVIOR AND SYMMETRY OF POSITIVE SOLUTIONS TO p-LAPLACIAN EQUATIONS IN A HALF-SPACE

Yujuan CHEN¹, Lei WEI², Yimin ZHANG³

1. School of Science, Nantong University, Nantong, 226007, China;
2. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, China;
3. Center for Mathematical Sciences, Wuhan University of Technology, Wuhan, 430070, China

收稿日期:2020-01-21 修回日期:2022-04-06 发布日期:2022-11-02
通讯作者: Yimin Zhang,E-mail:zhangym802@126.com E-mail:zhangym802@126.com
基金资助:
L. Wei was supported by NSFC (11871250). Y.M. Zhang was supported by NSFC (11771127, 12171379) and the Fundamental Research Funds for the Central Universities (WUT: 2020IB011, 2020IB017, 2020IB019).

THE ASYMPTOTIC BEHAVIOR AND SYMMETRY OF POSITIVE SOLUTIONS TO p-LAPLACIAN EQUATIONS IN A HALF-SPACE

Yujuan CHEN¹, Lei WEI², Yimin ZHANG³

1. School of Science, Nantong University, Nantong, 226007, China;
2. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, China;
3. Center for Mathematical Sciences, Wuhan University of Technology, Wuhan, 430070, China

Received:2020-01-21 Revised:2022-04-06 Published:2022-11-02
Contact: Yimin Zhang,E-mail:zhangym802@126.com E-mail:zhangym802@126.com
Supported by:
L. Wei was supported by NSFC (11871250). Y.M. Zhang was supported by NSFC (11771127, 12171379) and the Fundamental Research Funds for the Central Universities (WUT: 2020IB011, 2020IB017, 2020IB019).

摘要/Abstract

摘要： We study a nonlinear equation in the half-space with a Hardy potential, specifically, \[ \displaystyle - \Delta_p u= \lambda \frac{u ^{p-1}}{x_1^p}-x_1^\theta f(u)\ \ {\rm in}\ T,\] where $\Delta_p$ stands for the $p$-Laplacian operator defined by $\Delta_p u = {\rm div}(|\nabla u|^{p-2}\nabla u)$, $p>1$, $\theta > -p$, and $T$ is a half-space $\{x_1>0\}$. When $\lambda > \Theta$ (where $\Theta$ is the Hardy constant), we show that under suitable conditions on $f$ and $\theta$, the equation has a unique positive solution. Moreover, the exact behavior of the unique positive solution as $x_1\to 0^+$, and the symmetric property of the positive solution are obtained.

关键词: p-Lapacian, Hardy potential, symmetry, uniqueness, asymptotic behavior

Abstract: We study a nonlinear equation in the half-space with a Hardy potential, specifically, \[ \displaystyle - \Delta_p u= \lambda \frac{u ^{p-1}}{x_1^p}-x_1^\theta f(u)\ \ {\rm in}\ T,\] where $\Delta_p$ stands for the $p$-Laplacian operator defined by $\Delta_p u = {\rm div}(|\nabla u|^{p-2}\nabla u)$, $p>1$, $\theta > -p$, and $T$ is a half-space $\{x_1>0\}$. When $\lambda > \Theta$ (where $\Theta$ is the Hardy constant), we show that under suitable conditions on $f$ and $\theta$, the equation has a unique positive solution. Moreover, the exact behavior of the unique positive solution as $x_1\to 0^+$, and the symmetric property of the positive solution are obtained.

Key words: p-Lapacian, Hardy potential, symmetry, uniqueness, asymptotic behavior

中图分类号:

35J20

Yujuan CHEN, Lei WEI, Yimin ZHANG. THE ASYMPTOTIC BEHAVIOR AND SYMMETRY OF POSITIVE SOLUTIONS TO p-LAPLACIAN EQUATIONS IN A HALF-SPACE[J]. 数学物理学报(英文版), 2022, 42(5): 2149-2164.

Yujuan CHEN, Lei WEI, Yimin ZHANG. THE ASYMPTOTIC BEHAVIOR AND SYMMETRY OF POSITIVE SOLUTIONS TO p-LAPLACIAN EQUATIONS IN A HALF-SPACE[J]. Acta mathematica scientia,Series B, 2022, 42(5): 2149-2164.

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THE ASYMPTOTIC BEHAVIOR AND SYMMETRY OF POSITIVE SOLUTIONS TO p-LAPLACIAN EQUATIONS IN A HALF-SPACE

THE ASYMPTOTIC BEHAVIOR AND SYMMETRY OF POSITIVE SOLUTIONS TO p-LAPLACIAN EQUATIONS IN A HALF-SPACE

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