加权的退化椭圆系统稳定解的Liouville定理

数学物理学报 ›› 2020, Vol. 40 ›› Issue (1): 156-168.

加权的退化椭圆系统稳定解的Liouville定理

吴千秋,胡良根*()

^{宁波大学数学与统计学院浙江宁波 315211}

收稿日期:2018-05-08 出版日期:2020-02-26 发布日期:2020-04-08
通讯作者: 胡良根 E-mail:hulianggen@tom.com
基金资助:
浙江省自然科学基金(LY17A010007);宁波市自然科学基金(2018A610194)

Liouville Type Theorems for Stable Solutions of the Degenerate Elliptic System with Weight

Qianqiu Wu,Lianggen Hu*()

^{School of Mathematics and Statistics, Ningbo University, Zhejiang Ningbo 315211}

Received:2018-05-08 Online:2020-02-26 Published:2020-04-08
Contact: Lianggen Hu E-mail:hulianggen@tom.com
Supported by:
the Natural Science Foundation of Zhejiang Province(LY17A010007);the Natural Science Foundation of Ningbo(2018A610194)

摘要/Abstract

摘要：

该文研究了加权的退化椭圆系统

$\left\{\begin{array}{ll}-\Delta_{G} u=\omega (x) v, \\-\Delta_{G} v=\omega (x) u^q, \end{array}\right.\qquad \mathbb{R} ^N=\mathbb{R} ^{N_1}\times \mathbb{R} ^{N_2}, $

其中$\Delta_{G} u=\Delta_{x} u+(a+1)^2|x|^{2a}\Delta_{y} u$是Grushin算子, $a, \beta\ge0$, $q>1$, $\omega(x)=\left (1+\| x \|^{2(a+1)}\right)^{\frac{\beta}{2(a+1)}}$.超临界指数正稳定解的Liouville定理被建立.

关键词: Grushin算子, 稳定解, Liouville定理, Bootstrap方法

Abstract:

We study the degenerate elliptic system with weight

$\left\{\begin{array}{ll}-\Delta_{G} u=\omega (x) v, \\-\Delta_{G} v=\omega (x) u^q, \end{array}\right.\quad \mbox{in}\;\ \mathbb{R} ^N=\mathbb{R} ^{N_1}\times \mathbb{R} ^{N_2}, $

where $\Delta_{G} u=\Delta_{x} u+(a+1)^2|x|^{2a}\Delta_{y} u$ is the Grushin operator, $a, \beta\ge0$, $q>1$, $\omega(x)=\left (1+\| x \|^{2(a+1)}\right)^{\frac{\beta}{2(a+1)}}$. Liouville type results for positive stable solutions in the supercritical exponent are established.

Key words: Grushin operator, Stable solution, Liouville theorem, Bootstrap method

中图分类号:

O175.25

吴千秋,胡良根. 加权的退化椭圆系统稳定解的Liouville定理[J]. 数学物理学报, 2020, 40(1): 156-168.

Qianqiu Wu,Lianggen Hu. Liouville Type Theorems for Stable Solutions of the Degenerate Elliptic System with Weight[J]. Acta mathematica scientia,Series A, 2020, 40(1): 156-168.

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