带有扰动项的Hénon方程的多解性研究

数学物理学报 ›› 2012, Vol. 32 ›› Issue (4): 785-796.

带有扰动项的Hénon方程的多解性研究

朱红波¹|王征平²|郭渊斌²

1.广东工业大学应用数学学院广州 510006； 2.中国科学院武汉物理与数学研究所武汉 430071

收稿日期:2011-11-02 修回日期:2012-07-26 出版日期:2012-08-25 发布日期:2012-08-25
基金资助:
国家自然科学基金(11026138, 10801132)资助

Multiple Solutions for the Nonlinear Hénon Equation Under Perturbations

ZHU Hong-Bo¹, WANG Zheng-Ping², GUO Yuan-Bin²

1.Faculty of Applied Mathematics, Guangdong University of Technology, Guangzhou 510006； 2.Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071

Received:2011-11-02 Revised:2012-07-26 Online:2012-08-25 Published:2012-08-25
Supported by:
国家自然科学基金(11026138, 10801132)资助

摘要/Abstract

摘要：

研究了下面带有非齐次扰动项的H\'{e}non方程
u(x)=|x|^α|u|^p-2u+h(x), x∈B,
u=0, x∈∂B
其中B 是全空间R^N, N>4 上的单位球. 应用Bahri-Berestycki (见文献[3])中的扰动方法, 证明了对任意的h(x)=h(y, z)=h(|y|, |z|)L²B, x=(y, z)∈
R^l×R^N-l, 当α> N+2时, 存在常数p_{N, l}>2 使得对任意的p∈(2, p_{N, l}), 方程(P)存在无穷多互异解.

关键词: Hénon 方程, 扰动方法, 无穷多互异解

Abstract:

In this paper, we are concerned with the following nonlinear elliptic problem
u(x)=|x|^α|u|^p-2u+h(x), x∈B,
u=0, x∈∂B
Here Ω( R^N, N>4 is smooth and bounded. Applying the perturbation method introduced by Bahari-Berestycki^[3], for any h(x)=h(y, z)=h(|y|, |z|)L²B, x=(y, z)∈
R^l×R^N-l, when α> N+2, we show that there exists p_{N, l}>2 such that for any p∈(2, p_{N, l}), problem (P) has infinity many distinct solutions.

Key words: Hénon equation, Perturbation method, Infinity many distinct solutions

中图分类号:

60F15

朱红波, 王征平, 郭渊斌. 带有扰动项的Hénon方程的多解性研究[J]. 数学物理学报, 2012, 32(4): 785-796.

ZHU Hong-Bo, WANG Zheng-Ping, GUO Yuan-Bin. Multiple Solutions for the Nonlinear Hénon Equation Under Perturbations[J]. Acta mathematica scientia,Series A, 2012, 32(4): 785-796.

参考文献

[1]} Agmom S. Lectures on Elliptic Boundary Value Probolem. Princeton, NJ: Van Nonstrand, 1965

[2] Badiale M, Serra E. Multiplicity results for the supercritical Hé}non equation. Adv Nonlinear Stud, 2004, 4: 453--467

[3] Bahri A, Berestycki H. A perturbation method in critical theory and applications. Trans Am Math Soc, 1981, 267: 1--32

[4] Byeon J, Wang Z Q. On the Hénon equation: asympotic profile of ground states, I. Ann Inst H Poincaré Anal Non Linéaire, 2006, 23: 803--828

[5] Cao D M, Peng S J. The asymptotic behaviour of ground state solutions for Hénon equation. J Math Anal Appl, 2003, 278: 1--17

[6] Gidas B, Ni W M, Nirenberg L. Symmetry and related properties via the maximun principle. Comm Math Phys, 1979, 68: 209--243

[7] Hénon M. Numerical experiments on the stability of spherical stellar systems. Astronomy and Astrophysics, 1973, 24: 229--238

[8] Li Y Y. Existence and many positive solutions of semilinear elliptic equations on annulus. J Differential Equations, 1990, 83: 348--367

[9] Lin S S. Existence and many positive nonradial solutions for nonlinear elliptic equations on annulus. J Differential Equations, 1993, 103: 338--349

[10] Ni W M. A nonlinear Dirichlet problem on the unit ball and its applications. Indiana Univ Math Jour, 1982, 31: 801--807

[11] Peng S J. Multiple boundary concentrating solutions to Dirichlet problem of Hénon equation. Acta Mathematicae Applicatae (Engilsh Series), 2006, 22: 137--162

[12] Pistoia A, Serra E. Multi-peak solutions for Hénon equation with slightly subcritical growth. Math Z, 2007, 256: 75--97

[13] Serra E. Nonradial positive solutions for Hénon equation with critical growth. Calc Var PDEs, 2005, 23: 301--326

[14] Smete D, Su J B, Willem M. Nonradial ground states for the Hénon equation. Comm Contemp Math, 2002, 4: 467--480

[1]	小巴桑次仁. RELLICH 引理的新证明[J]. 数学物理学报, 2014, 34(6): 1435-1439.
[2]	耿永才. 具有球对称结构的相对论欧拉方程组稳态解的存在性[J]. 数学物理学报, 2014, 34(4): 841-850.
[3]	刘丹红, 许跟起. 糖尿病人群发展状况预警建模研究[J]. 数学物理学报, 2014, 34(3): 495-511.
[4]	邹群, 曾广兴, 漆志鹏. 形式实模的核心[J]. 数学物理学报, 2014, 34(3): 619-625.
[5]	邱德华, 陈平炎. NA随机变量序列加权和的Chover型重对数律[J]. 数学物理学报, 2014, 34(3): 674-683.
[6]	张晶. 具有不连续非线性项和Robin边值条件的半线性椭圆型方程解的多重性[J]. 数学物理学报, 2014, 34(1): 86-94.
[7]	吴永锋. 两两NQD 列的L_p 收敛性和矩完全收敛性[J]. 数学物理学报, 2013, 33(5): 850-859.
[8]	李炜, 甘师信. 重尾随机变量序列滑动平均过程的极限性质[J]. 数学物理学报, 2013, 33(4): 787-792.
[9]	傅可昂. 关于修整和强逼近的一个注记[J]. 数学物理学报, 2013, 33(2): 267-275.
[10]	周慧, 林正炎. U-统计量对数律的精确渐近性[J]. 数学物理学报, 2012, 32(1): 41-54.
[11]	邱德华. &rho\|混合随机变量加权和的收敛性[J]. 数学物理学报, 2011, 31(1): 132-141.
[12]	陈平炎, 李远梅. 鞅差序列滑动和过程的极限结果[J]. 数学物理学报, 2011, 31(1): 179-187.
[13]	张勇, 董志山, 赵世舜. 相依序列加权和的几乎处处中心极限定理[J]. 数学物理学报, 2009, 29(6): 1487-1491.
[14]	邢国东, 杨善朝. 负相协随机变量的指数不等式[J]. 数学物理学报, 2009, 29(6): 1679-1688.
[15]	谭中权, 彭作祥. 部分和乘积的几乎处处中心极限定理[J]. 数学物理学报, 2009, 29(6): 1689-1698.