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.

参考文献 21

1	Farina A . On the classification of solutions of Lane-Emden equation on unbounded domains of $\mathbb{R} ^N$ . J Math Pures Appl, 2007, 87, 537- 561 doi: 10.1016/j.matpur.2007.03.001
2	Dávila J , Dupaigne L , Wang K L , Wei J C . A monotonicity formula and a Liouville-type theorem for a forth order supercritical problem. Adv Math, 2014, 258, 240- 285 doi: 10.1016/j.aim.2014.02.034
3	Cowan C , Ghoussoub N . Regularity of semi-stable solutions to fourth order nonlinear eigenvalue problems on general domains. Calc Var PDE, 2014, 49, 291- 305 doi: 10.1007/s00526-012-0582-4
4	Dupaigne L , Ghergu M , Goubet O , Warnault G . The Gel'fand problem for the biharmonic operator. Arch Ration Mech Anal, 2013, 208, 725- 752 doi: 10.1007/s00205-013-0613-0
5	Souplet P H . The proof of the Lane-Emden conjecture in four space dimensions. Adv Math, 2009, 221, 1409- 1427 doi: 10.1016/j.aim.2009.02.014
6	Hajlaoui H , Harrabi A , Ye D . On stable solutions of biharmonic problems with polynomial growth. Pacific J Math, 2014, 270, 79- 93 doi: 10.2140/pjm.2014.270.79
7	Hu L G , Zeng J . Liouville type theorems for stable solutions of the weighted elliptic system. J Math Anal Appl, 2016, 437, 882- 901 doi: 10.1016/j.jmaa.2016.01.032
8	Birindelli I , Capuzzo Dolcetta I , Cutri A . Liouville theorems for semilinear equations on the Heisenberg group. Ann Inst Henri Poincare Non Linéaire Anal, 1997, 14, 295- 308 doi: 10.1016/S0294-1449(97)80138-2
9	Birindelli I , Prajapat J . Nonlinear Liouville theorems in the Heisenberg group via the moving plane method. Comm Partial Differential Equations, 1999, 24, 1875- 1890 doi: 10.1080/03605309908821485
10	张书陶, 韩亚洲. Heisenberg群上移动球面法的应用-一类半线性方程的Liouville型定理. 数学物理学报, 2017, 37A (2): 278- 286 doi: 10.3969/j.issn.1003-3998.2017.02.007
	Zhang S T , Han Y Z . An application of the method of moving shpere in the Heiseberg group-Liouville type theorem of a class of semilinear equatoins. Acta Math Sci, 2017, 37A (2): 278- 286 doi: 10.3969/j.issn.1003-3998.2017.02.007
11	Garofalo N , Vassilev D . Symmetry properties of positive entire solutions of Yamabe-type equations on groups of Heisenberg type. Duke Math J, 2001, 106, 411- 448 doi: 10.1215/S0012-7094-01-10631-5
12	Franchi B , Gutiérrez C E , Wheeden R L . Weighted Sobolev-Poincaré inequalities for Grushin type operators. Comm Partial Differential Equations, 1994, 19, 523- 604 doi: 10.1080/03605309408821025
13	Monti R , Morbidelli D . Kelvin transform for Grushin operators and critical semilinear equations. Duke Math J, 2006, 131, 167- 202 doi: 10.1215/S0012-7094-05-13115-5
14	Monticelli D D . Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators. J Eur Math Soc, 2010, 12, 611- 654
15	Yu X H . Liouville type theorem for nonlinear elliptic equation involving Grushin operators. Commun Contemp Math, 2015, 17, 1450050 doi: 10.1142/S0219199714500503
16	Duong A T , Phan Q H . Liouville type theorem for nonlinear elliptic system involving Grushin operator. J Math Anal Appl, 2017, 454, 785- 801 doi: 10.1016/j.jmaa.2017.05.029
17	Zhao X J , Wang X H , Yu X H . Morse index and Liouville type theorems for elliptic equation involving Grushin operator. Sci China: Math, 2014, 44, 711- 718
18	Cowan C . Liouville theorems for stable Lane-Emden systems with biharmonic problems. Nonlinearity, 2013, 26, 2357- 2371 doi: 10.1088/0951-7715/26/8/2357
19	Montenegro M . Minimal solutions for a class of elliptic systems. Bull London Math Soc, 2005, 37, 405- 416 doi: 10.1112/S0024609305004248
20	Hu L G . A monotonicity formula and Liouville-type theorems for stable solutions of the weighted elliptic system. Adv Differential Equations, 2017, 22, 49- 76
21	Cheng Z , Huang G G , Li C M . On the Hardy-Littlewood-Sobolev type systems. Commun Pure Appl Anal, 2016, 15, 2059- 2074 doi: 10.3934/cpaa.2016027

[1]	张建玲. 多总体二阶随机占优对无约束的检验问题[J]. 数学物理学报, 2020, 40(1): 212-220.
[2]	鲁萍萍, 胡良根. 加权椭圆系统稳定解的单调公式的一个注记[J]. 数学物理学报, 2017, 37(4): 698-705.
[3]	胡良根. 非线性Hénon方程解的Liouville定理[J]. 数学物理学报, 2016, 36(4): 639-648.
[4]	郑神州, 卢寒芳. 次椭圆拟线性方程在外区域上的Liouville定理[J]. 数学物理学报, 2012, 32(4): 644-653.
[5]	胡泽军. 一类非线性椭圆型方程的一个Liouville定理[J]. 数学物理学报, 2000, 20(4): 474-479.