非线性Hénon方程解的Liouville定理

数学物理学报 ›› 2016, Vol. 36 ›› Issue (4): 639-648.

非线性Hénon方程解的Liouville定理

胡良根

宁波大学数学系浙江宁波 315211

收稿日期:2015-11-30 修回日期:2016-05-04 出版日期:2016-08-26 发布日期:2016-08-26
作者简介:胡良根,hulianggen@tom.com
基金资助:
国家自然科学基金（11201248）和宁波市自然科学基金（2014A610027）资助

Liouville Type Theorems of Solutions for the Nonlinear Hénon Equations

Hu Lianggen

Department of Mathematics, Ningbo University, Zhejiang Ningbo 315211

Received:2015-11-30 Revised:2016-05-04 Online:2016-08-26 Published:2016-08-26
Supported by:
Supported by the NSFC(11201248)and Ningbo Natural Science Foundation(2014A610027)

摘要/Abstract

摘要：

该文研究了二阶和四阶非线性Hénon-Lane-Emden方程有限Morse指标解的Liouville定理.利用一种新方法，即使用单调公式、Pohozaev恒等式和doubling引理等相结合证明了其结果.

关键词: 有限Morse指标解, 单调公式, Liouville定理, Doubling引理, Pohozaev恒等式

Abstract:

In this paper, Liouville theorem of finite Morse index solutions for the second order and fourth order nonlinear Hénon-Lane-Emden equations is considered. Adopting a new method, i.e., the monotonicity formula, Pohozaev identity combining with doubling lemma, the main results is proved.

Key words: Finite Morse index solution, Monotonicity formula, Liouville theorem, Doubling lemma, Pohozaev identity

中图分类号:

O175.25

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.

Hu Lianggen. Liouville Type Theorems of Solutions for the Nonlinear Hénon Equations[J]. Acta mathematica scientia,Series A, 2016, 36(4): 639-648.

参考文献

[1] Bahri A, Lions P L. Solutions of superlinear elliptic equations and their Morse indicies. Comm Pure Appl Math, 1992, 45: 1205-1215
[2] Dancer E N. Stable and finite Morse index solutions on or on bounded domains with small diffusion. Trans Amer Math Soc, 2004, 357: 1225-1243
[3] Farina A. On the classification of solutions of Lane-Emden equation on unbounded domains of Rⁿ. J Math Pures Appl, 2007, 87: 537-561
[4] Dancer E N, Du Y H, Guo Z M. Finite Morse index solutions of an elliptic equation with supercritical exponent. J Differential Equations, 2011, 250: 3281-3310
[5] Wang C, Ye D. Some Liouville theorems for Hénon type elliptic equations. J Funct Anal, 2012, 262: 1705-1727
[6] Wei J C, Ye D. Liouville Theorems for finite Morse index solutions of Biharmonic problem. Math Ann, 2013, 356: 1599-1612
[7] 梁占平, 苏加宝. 具有凸凹项非齐次拟线性椭圆方程的多解性. 数学物理学报, 2014, 34A(2): 217-226 Liang Z P, Su J B. Solutions to inhomogeneous quasilinear elliptic problems with concave-convex type nonlinearities. Acta Math Sci, 2014, 34A(2): 217-226
[8] 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
[9] Falzy M, Ghoussoub N. On the Hénon-Lane-Emden conjecture. Disc Cont Dyn Syst, 2014, 34: 2513-2533
[10] Souplet Ph. The proof of the Lane-Emden conjecture in four space dimensions. Adv Math, 2009, 221: 1409-1427
[11] Phan Q H, Souplet Ph. Liouville-type theorems and bounds of solutions of Hardy-Hénon equations. J Differential Equations, 2012, 252: 2544-2562
[12] Hu L G. Liouville-type theorems for the fourth order nonlinear elliptic equation. J Differential Equation, 2014, 256: 1817-1846
[13] Polácik P, Quittner P, Souplet Ph. Singularity and decay estimates in superliner problems via Liouville-type theorems I. elliptic equations and systems. Duke Math J, 2007, 139: 555-579