GROUND STATE SOLUTIONS FOR THE CRITICAL KLEIN-GORDON-MAXWELL SYSTEM

doi:10.1007/s10473-019-0521-y

Acta mathematica scientia,Series B ›› 2019, Vol. 39 ›› Issue (5): 1451-1460.doi: 10.1007/s10473-019-0521-y

• Articles • Previous Articles

GROUND STATE SOLUTIONS FOR THE CRITICAL KLEIN-GORDON-MAXWELL SYSTEM

Lixia WANG¹, Xiaoming WANG², Luyu ZHANG³

1. School of Sciences, Tianjin Chengjian University, Tianjin 300384, China;
2. School of Mathematics and Computer Science, Shangrao Normal University, Shangrao 334001, China;
3. School of Mathematical and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450045, China

Received:2016-06-29 Revised:2019-04-29 Online:2019-10-25 Published:2019-11-11
Contact: Lixia WANG E-mail:wanglixia0311@126.com
Supported by:
L.X. Wang was partially supported by the National Natural Science Foundation of China (11801400, 11571187) and the Postdoctoral Science Foundation of China (2017M611159). X.M. Wang was partially supported by the National Natural Science Foundation of China (11861053).

Abstract

Abstract:

In this article, we study the following Klein-Gordon-Maxwell system involving critical exponent
(KGM)

where λ and ω are two positive constants. We found the existence of positive ground state solutions (that is, for solutions which minimizes the action functional among all the solutions) of (KGM) which improves some previous existence result in Carri?o et al.(2012)[8].

Key words: ground state solution, Klein-Gordon-Maxwell system, Ekeland's variational principle, mountain pass theorem

CLC Number:

35B33

Lixia WANG, Xiaoming WANG, Luyu ZHANG. GROUND STATE SOLUTIONS FOR THE CRITICAL KLEIN-GORDON-MAXWELL SYSTEM[J].Acta mathematica scientia,Series B, 2019, 39(5): 1451-1460.

Trendmd

References

[1] Azzollini A, Pomponio A. Ground state solutions for the nonlinear Klein-Gordon-Maxwell equations. Topol Methods Nonlinear Anal, 2010, 35:33-42
[2] Azzollini A, Pisani L, Pomponio A. Improved estimates and a limit case for the electrostatic Klein-GordonMaxwell system. Proc Roy Soc Edinb Sect A, 2011, 141:449-463
[3] Benci V, Fortunato D. The nonlinear Klein-Gordon equation coupled with the Maxwell equations. Nonlinear Anal, 2001, 47:6065-6072
[4] Benci V, Fortunato D. Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations. Rev Math Phys, 2002, 14:409-420
[5] Brézis H, Nirenberg L. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun Pure Appl Math, 1983, 36:437-477
[6] Cassani D. Existence and non-existence of solitary waves for the critical Klein-Gordon equation coupled with Maxwell's equations. Nonlinear Anal, 2004, 58:733-747
[7] Carrião 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
[8] Carrião P C, Cunha P L, Miyagaki O H. Positive ground state solutions for the critical Klein-GordonMaxwell system with potentials. Nonlinear Anal, 2012, 75(10):4068-4078
[9] Chen S J, Tang C L. Multiple solutions for a non-homogeneous Schrödinger-Maxwell and Klein-GordonMaxwell equations on R³. Nonlinear Differ Equa Appl, 2010, 17:559-574
[10] Chen S J, Song S Z. Multiple solutions for nonhomogeneous Klein-Gordon-Maxwell equations on R³. Nonlinear Analysis:Real World Applications, 2015, 22:259-271
[11] Cunha P. Subcritical and supercritical Klein-Gordon-Maxwell equtions without Ambrosetti-Rabinowitz condition. Differ Integral Equ, 2014, 27(3/4):387-399
[12] D'Aprile T. Mugnai D. Non-existence results for the coupled Klein-Gordon-Maxwell equations. Adv Nonlinear Stud, 2004, 4:307-322
[13] D'Aprile T, Mugnai D. Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations. Proc Roy Soc Edinb Sect A, 2004, 134:893-906
[14] D'Avenia P, Pisani L, Siciliano G. Dirichlet and Neumann problems for Klein-Gordon-Maxwell systems. Nonlinear Anal, 2009, 71(12):1985-1995
[15] D'Avenia P, Pisani L, Siciliano G. Klein-Gordon-Maxwell system in a bounded domain. Discrete Contin Dyn Syst, 2010, 26(1):135-159
[16] Ding L, Li L. Infinitely many standing wave solutions for the nonlinear Klein-Gordon-Maxwell system with sign-changing potential. Comput Math Appl, 2014, 68:589-595
[17] Georgiev V, Visciglia N. Solitary waves for Klein-Gordon-Maxwell system with external Coulomb potential. J Math Pures Appl, 2005, 9:957-983
[18] He X M. Multiplicity of solutions for a nonlinear Klein-Gordon-Maxwell system. Acta Appl Math, 2014, 130:237-250
[19] Li L, Tang C L. Infinitely many solutions for a nonlinear Klein-Gordon-Maxwell system. Nonlinear Anal, 2014, 110:157-169
[20] Wang F Z. Solitary waves for the Klein-Gordon-Maxwell system with critical exponent. Nonlinear Anal, 2011, 74:827-835
[21] Wang F Z. Ground-state solutions for the electrostatic nonlinear Klein-Gordon-Maxwell system. Nonlinear Anal, 2011, 74(14):4796-4803
[22] Wang L X, Chen S J. Two solutions for nonhomogeneous Klein-Gordon-Maxwell system with sign-changing potential. Elect J Differ Equ, 2018, 124:1-21
[23] Wang L X, Ma S W, Xu N. Multiple solutions for nonhomogenerous Schrödinger-Possion equations with sign-changing potential. Acta Math Sci, 2017, 37B(2):555-572
[24] Xu L P, Chen H B. Existence and multiplicity of solutions for nonhomogenous Klein-Gordon-Maxwell equations. Elect J Differ Equ, 2015, 102:1-12

GROUND STATE SOLUTIONS FOR THE CRITICAL KLEIN-GORDON-MAXWELL SYSTEM

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 9

Metrics

Comments

Recommended 1

[1]	Lixia WANG, Shiwang MA, Na XU. MULTIPLE SOLUTIONS FOR NONHOMOGENEOUS SCHRODINGER-POISSON EQUATIONS WITHÖSIGN-CHANGING POTENTIAL [J]. Acta mathematica scientia,Series B, 2017, 37(2): 555-572.
[2]	Liping XU, Haibo CHEN. MULTIPLICITY RESULTS FOR FOURTH ORDER ELLIPTIC EQUATIONS OF KIRCHHOFF-TYPE [J]. Acta mathematica scientia,Series B, 2015, 35(5): 1067-1076.
[3]	ZHANG Jian, TANG Xian-Hua, ZHANG Wen. ON GROUND STATE SOLUTIONS FOR SUPERLINEAR DIRAC EQUATION [J]. Acta mathematica scientia,Series B, 2014, 34(3): 840-850.
[4]	WAN Li-Li, TANG Chun-Lei. EXISTENCE OF SOLUTIONS FOR NON-PERIODIC SUPERLINEAR SCHRÖDINGER EQUATIONS WITHOUT (AR) CONDITION [J]. Acta mathematica scientia,Series B, 2012, 32(4): 1559-1570.
[5]	LIANG Zhan-Ping, SU Jia-Bao. EXISTENCE OF SOLITARY WAVES TO A GENERALIZED KADOMTSEV-PETVIASHVILI EQUATION [J]. Acta mathematica scientia,Series B, 2012, 32(3): 1149-1156.
[6]	WAN Li-Li, TANG Chun-Lei. HOMOCLINIC ORBITS FOR A CLASS OF THE SECOND ORDER HAMILTONIAN SYSTEMS [J]. Acta mathematica scientia,Series B, 2010, 30(1): 312-318.
[7]	Liu Yue; Wang Zhengping. BIHARMONIC EQUATIONS WITH ASYMPTOTICALLY LINEAR NONLINEARITIES [J]. Acta mathematica scientia,Series B, 2007, 27(3): 549-560.
[8]	Wang Zhengping; Zhou Huansong. SOLUTIONS FOR A NONHOMOGENEOUS ELLIPTIC PROBLEM INVOLVING CRITICAL SOBOLEV-HARDY EXPONENT IN R^N [J]. Acta mathematica scientia,Series B, 2006, 26(3): 525-536.
[9]	Tang Zhongwei. EXISTENCE AND ASYMPTOTIC BEHAVIOR OF THE SOLUTIONS FOR A NONLINEAR ELLIPTIC EQUATION ARISING IN ASTROPHYSICS [J]. Acta mathematica scientia,Series B, 2006, 26(2): 229-245.