带有位势的Schrödinger-Poisson系统解的存在性与多解性

数学物理学报 ›› 2016, Vol. 36 ›› Issue (1): 49-64.

带有位势的Schrödinger-Poisson系统解的存在性与多解性

张琦

山西大学数学科学学院太原 030006

收稿日期:2015-07-22 修回日期:2015-11-12 出版日期:2016-02-25 发布日期:2016-02-25
作者简介:张琦,zhangqi@sxu.edu.cn
基金资助:
国家自然科学青年基金(11301313)和山西省自然科学基金(2013021001-4,2015021007)资助

Existence and Multiplicity of Solutions to Schrödinger-Poisson System with Potentials

Zhang Qi

School of Mathematical Sciences, Shanxi University, Taiyuan 030006

Received:2015-07-22 Revised:2015-11-12 Online:2016-02-25 Published:2016-02-25
Supported by:
Supported by the NSFC (11301313) and Science Council of Shanxi Province (2013021001-4, 2015021007)

摘要/Abstract

摘要：

该文讨论以下带有位势V的薛定谔-泊松(Schrödinger-Poisson)系统
-Δu+λVu+φu=f(x,u),x∈R³
-Δφ=u²,x∈R³
其中λ≥1是一个参数,位势函数V∈C(R³,R₊)满足比较一般的假设.当非线性项f在无穷远点是超四次的,并且空间嵌入缺乏紧性时,该文讨论了参数λ≥1充分大时问题解的存在性与多解性.也考虑了非线性性项f满足一般的次线性假设时问题无穷多个解的存在性.

关键词: Schrödinger-Poisson 系统, 位势, 变分法, 超四次, 次线性

Abstract:

In this paper, we discuss the existence and multiplicity solutions to the following Schrödinger-Poisson system with potentials
-Δu+λVu+φu=f(x,u),x∈R³
-Δφ=u²,x∈R³
where λ≥qslant1 is a parameter, the potential V∈C(R³, R₊) and satisfies the more general assumptions. When f is 4-superlinear at infinity, we discuss the existence and multiplicity of solutions for λ≥1 large enough without the compactness of embedding of the work space. We also consider the existence of infinity many solutions with f satisfying the more general sublinear assumptions.

Key words: Schrödinger-Poisson system, Potential, Variational method, 4-Superlinear, Sublinear

中图分类号:

O175.29

张琦. 带有位势的Schrödinger-Poisson系统解的存在性与多解性[J]. 数学物理学报, 2016, 36(1): 49-64.

Zhang Qi. Existence and Multiplicity of Solutions to Schrödinger-Poisson System with Potentials[J]. Acta mathematica scientia,Series A, 2016, 36(1): 49-64.

参考文献

[1] Benci V, Fortunato D. An eigenvalue problem for the Schrödinger-Maxwell equations. Topol Methods Nonlinear Anal, 1998, 11(2):283-293
[2] Ruiz D. On the Schrödinger-Poisson-Slater system:behavior of minimizers, radial and nonradial cases. Archive for rational mechanics and analysis, 2010, 198(1):349-368
[3] Alves C O, Souto M, Soares S. Schrödinger-Poisson equations without Ambrosetti-Rabinowitz condition. J Math Anal Appl, 2011, 377(2):584-592
[4] Ambrosetti A, Ruiz D. Multiple bound states for the Schrödinger-Poisson problem. Commun Contemp Math, 2008, 10(3):391-404
[5] Azzollini A. Concentration and compactness in nonlinear Schrödinger-Poisson system with a general nonlinearity problem. J Differential Equations, 2010, 249(7):1746-1763
[6] Azzollini A, d'Avenia P, Pomponio A. On the Schrödinger-Maxwell equations under the effect of a general nonlinear term. Ann Inst H Poincaré Anal Non Linéaire, 2010, 27(2):779-791
[7] Azzollini A, Pomponi A. Ground state solutions for the nonlinear Schrödinger-Maxwell equations. J Math Anal Appl, 2008, 345(1):90-108
[8] Coclite G M. A multiplicity result for the nonlinear Schrödinger-Maxwell equations. Commun Appl Anal, 2003, 7(2/3):417-423
[9] D'Aprile T, Mugnai D. Non-existence results for the coupled Klein-Gordon-Maxwell equations. Adv Nonlinear Stud, 2004, 4(3):307-322
[10] D'Aprile T, Mugnai D. Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations. Proc Roy Soc Edinburgh Sect A, 2004, 134(5):893-906
[11] Kikuchi H. On the existence of a solution for elliptic system related to the Maxwell-Schrödinger equations. Nonlinear Anal, 2007, 67(5):1445-1456
[12] Li Q D, Han S, Wei Z L. Existence of infinitely many large solutions for the nonlinear Schrödinger-Maxwell equations. Nonlinear Anal, 2010, 72(11):4264-4270
[13] Ruiz D. The Schrödinger-Poisson equation under the effect of a nonlinear local term. J Funct Anal, 2006, 237(2):655-674
[14] Wang Z P, Zhou H S. Positive solution for a nonlinear stationary Schrödinger-Poisson system in R³. Discrete Contin Dyn Syst, 2007, 18(4):809-816
[15] Zhao L G, Zhao F K. On the existence of solutions for the Schrödinger-Poisson equations. J Math Anal Appl, 2008, 346(1):155-169
[16] Li F Y, Zhang Q. Existence of positive solutions to the Schrödinger-Poisson system without compactness conditions. J Math Anal Appl, 2012, 401(2):754-762
[17] Zhang J. On the Schrödinger-Poisson equations with a general nonlinearity in the critical growth. Nonlinear Anal, 2012, 75(18):6391-6401
[18] Zhang Q, Li F Y, Liang Z P. Existence of multiple positive solutions to nonhomogeneous Schrödinger-Poisson system. Applied Mathematics and Computation, 2015, 259:353-363
[19] Mawhin J, Willem M. Critical Point Theory and Hamiltonian Systems. New York:Springer-Verlag, 1989
[20] Berestycki H, Lions P L. Nonlinear scalar field equations, I. Existence of a ground state. Archive for Rational Mechanics and Analysis, 1983, 82(4):313-345
[21] Chen S J, Tang C L. High energy solutions for the superlinear Schrödinger-Maxwell equations. Nonlinear Anal, 2009, 71(10):4927-4934
[22] Chen S J, Tang C L. Multiple solutions for nonhomogeneous Schrödinger-Maxwell and Klein-Gordon-Maxwell equations on R³. NoDEA Nonlinear Differential Equations Appl, 2010, 17(5):559-574
[23] Sun J T. Infinitely many solutions for a class of sublinear Schrödinger-Maxwell equations. J Math Anal Appl, 2012, 390(2):514-522
[24] Bartsch T, Alexander Pankov, Wang Zhiqiang. Nonlinear Schrödinger equations with steep potential well. Commun Contemp Math, 2001, 3(4):549-569
[25] Jiang Y S, Zhou H S. Schrödinger-Poisson system with steep potential well. J Differential Equations., 2011, 251:582-608
[26] Zhao L G, Liu H D, Zhao F K. Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential. Journal of Differential Equations, 2013, 255:1-23
[27] Zou W M. Variant fountain theorems and their applications. Manuscripta Math, 2001, 104(3):343-358
[28] Ye Y W, Tang C L. Existence and multiplicity of solutions for fourth-order elliptic equations in R^N. J Math Anal Appl, 2013, 406(1):335-351
[29] Ekeland I. Convexity methods in Hamiltonian mechanics//Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Berlin:Springer-Verlag, 1990
[30] Rabinowitz Paul H. Minimax Methods in Critical Point Theory with Applications to Differential Equations. Providence:Amer Math Soc, 1986
[31] Kajikiya R. A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations. J Funct Anal, 2005, 225(2):352-370