带有次线性项和超线性项的Klein-Gordon-Maxwell系统多重解的存在性

数学物理学报 ›› 2017, Vol. 37 ›› Issue (4): 663-670.

带有次线性项和超线性项的Klein-Gordon-Maxwell系统多重解的存在性

陈丽珍¹, 李安然², 李刚³

1 山西财经大学应用数学学院太原 030006;
2 山西大学数学科学学院太原 030006;
3 扬州大学数学科学学院江苏扬州 225002

收稿日期:2016-11-17 修回日期:2017-04-09 出版日期:2017-08-26 发布日期:2017-08-26
通讯作者: 李安然,E-mail:lianran@sxu.edu.cn E-mail:lianran@sxu.edu.cn
作者简介:陈丽珍,E-mail:chenlz409@126.com;李刚,E-mail:gli@yzu.edu.cn
基金资助:
国家自然科学基金（11271316）和山西财经大学青年科研基金（Z06045）

Existence of Infinitely Many Solutions to a Class of Klein-Gordon-Maxwell System with Superlinear and Sublinear Terms

Chen Lizhen¹, Li Anran², Li Gang³

1 Department of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006;
2 School of Mathematical Sciences, Shanxi University, Taiyuan 030006;
3 School of Mathematical Sciences, Yangzhou University, Jiangsu Yangzhou 225002

Received:2016-11-17 Revised:2017-04-09 Online:2017-08-26 Published:2017-08-26
Supported by:
Supported by the NSFC (11271316) and the Youth Fund of Shanxi University of Finance and Economics (Z06045)

摘要/Abstract

摘要：

该文研究如下Klein-Gordon-Maxwell系统

多重解的存在性，其中4 < p < 6，1 < q < 2，λ>0.在a（x）、b（x）、参数λ满足一定的假设条件下，通过变分方法证明了系统无穷解的存在性.补充和完善了以上方程解存在性的以往结果.

关键词: Klein-Gordon-Maxwell系统, 超线性和次线性, 无穷多解, 变分方法

Abstract:

In this paper, we establish the multiplicity of solutions for the Klein-GordonMaxwell system

where 4 < p < 6, 1 < q < 2, λ > 0. Under some assumptions on the a(x),b(x),λ, the multiplicity result of solutions for the system was obtained by variational methods. Our result is a complement to some recent works concerning the existence of solutions of the above equation.

Key words: Klein-Gordon-Maxwell system, Superlinear and sublinear, Infinitely many solutions, Variational methods

中图分类号:

O176.3

陈丽珍, 李安然, 李刚. 带有次线性项和超线性项的Klein-Gordon-Maxwell系统多重解的存在性[J]. 数学物理学报, 2017, 37(4): 663-670.

Chen Lizhen, Li Anran, Li Gang. Existence of Infinitely Many Solutions to a Class of Klein-Gordon-Maxwell System with Superlinear and Sublinear Terms[J]. Acta mathematica scientia,Series A, 2017, 37(4): 663-670.

参考文献

[1] Benci V, Fortunato D. Solitary waves of the nonlinear Klien-Gordon equations with the Maxwell equations. Rev Math Phys, 2002, 14(4):409-420
[2] Benci V, Fortunato D. The nonlinear Klein-Gordon equations coupled with the Maxwell equations. Nonlinear Analysis, 2001, 47:6065-6072
[3] D'Aprile T, Mugnai D. Solitary waves for nonlinear Klein-Gorden-Maxwell and Schrödinger-Maxwell equations. Proc Roy Soc Edinburgh Sect A, 2004, 134(5):893-906
[4] Cassani D. Existence and non-existence of solitary waves for the critical Klein-Gordon equations coupled with Maxwell's equations. Nonlinear Anal, 2004, 58(7):733-747
[5] Carriao P C, Cunha P L, Miyagaki O H. Existence results for the Klein-Gordon-Maxwell equations in higher dimensions with critical exponents. Commun Pure Appl Anal, 2011, 10(2):709-718
[6] Carriao P C, Cunha P L, Miyagaki O H. Positive ground state solutions for the critical Klein-GordonMaxwell systems with potentials. Nonlinear Anal, 2012, 75(10):4068-4078
[7] He X. Multiplicity of solutions for a nonlinear Klein-Gordon-Maxwell system. Acta Appl Math, 2014, 130:237-250
[8] Ding L, Li L. Infinitely many standing wave solutions for the nonlinear Klein-Gordon-Maxwell system with sign-changing potential. Computers and Mathematics with Applications, 2014, 68:589-595
[9] 叶一蔚, 唐春雷. 带有超线性项或次线性项的Schrödinger-Poisson系统解的存在性和多重性. 数学物理学报, 2015, 35A(4):668-682 Ye Y W, Tang C L. Existence and multiplicity results for Schrödinger-Poisson system with superlinear or sublinear terms. Acta Math Sci, 2015, 35A(4):668-682
[10] Sun M Z, Su J B, Zhao L G. Solutions of a Schrödinger-Poisson systems with combined nonlinearities. J Math Anal Appl, 2016, 442:385-403
[11] Bartolo P, Benci V, Fortunato D. Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity. Nonlinear Anal, 1983, 7(9):981-1012

[1]	张申贵. 带类p(x)-拉普拉斯算子的双非局部问题的无穷多解[J]. 数学物理学报, 2018, 38(3): 514-526.
[2]	廖家锋, 李红英. 带Sobolev临界指数的超线性Kirchhoff型方程正解的存在性与多解性[J]. 数学物理学报, 2017, 37(6): 1119-1124.
[3]	李琴, 杨作东. 含变号位势的p-Kirchhoff型方程组无穷多个高能量解的存在性[J]. 数学物理学报, 2017, 37(5): 869-876.
[4]	刘健, 赵增勤. 基于变分方法的奇异脉冲方程弱解的存在性[J]. 数学物理学报, 2016, 36(3): 521-530.
[5]	叶一蔚, 唐春雷. 带有超线性项或次线性项的Schrödinger-Poisson系统解的存在性和多重性[J]. 数学物理学报, 2015, 35(4): 668-682.
[6]	邓义华, 罗李平, 周立君. 一类带负位势函数的超线性p-Laplace方程基态解的存在性[J]. 数学物理学报, 2015, 35(3): 578-586.
[7]	李本鸟, 陈晓莉. 带凹凸项的分数阶Laplace方程解的存在性[J]. 数学物理学报, 2014, 34(5): 1228-1235.
[8]	葛斌, 薛小平, 周庆梅. 一类具有非光滑位势p(x)-Kirchhoff 型问题多解性[J]. 数学物理学报, 2011, 31(5): 1431-1438.
[9]	裘良华. 体积元保持变换下的一类特殊预给定数量曲率问题[J]. 数学物理学报, 2011, 31(5): 1317-1322.
[10]	彭艳芳, 张正杰. 具临界指数及奇异性的双调和方程解的存在性[J]. 数学物理学报, 2009, 29(6): 1492-1499.
[11]	刘祥清; 黄毅生; 邓志颖. 双调和方程特征值问题[J]. 数学物理学报, 2009, 29(1): 57-69.
[12]	邓志颖; 黄毅生. 一类非线性椭圆方程正解的存在性[J]. 数学物理学报, 2008, 28(5): 971-976.
[13]	林丽珊;李永青. 带权的 p-Laplacian 非线性特征值问题解的存在性[J]. 数学物理学报, 2007, 27(6): 996-1005.
[14]	刘祥清; 黄毅生. 一类带权函数的拟线性椭圆方程[J]. 数学物理学报, 2007, 27(2): 277-287.
[15]	孙金丽;孙经先. 拟增算子的多个不动点[J]. 数学物理学报, 2006, 26(5): 695-699.