Browse > Article
http://dx.doi.org/10.4134/JKMS.j200463

ON SEMILOCAL KLEIN-GORDON-MAXWELL EQUATIONS  

Han, Jongmin (Department of Mathematics Kyung Hee University)
Sohn, Juhee (College of General Education Kookmin University)
Yoo, Yeong Seok (Department of Mathematics Kyung Hee University)
Publication Information
Journal of the Korean Mathematical Society / v.58, no.5, 2021 , pp. 1131-1145 More about this Journal
Abstract
In this article, we study the Klein-Gordon-Maxwell equations arising from a semilocal gauge field model. This model describes the interaction of two complex scalar fields and one gauge field, and generalizes the classical Klein-Gordon equation coupled with the Maxwell electrodynamics. We prove that there exist infinitely many standing wave solutions for p ∈ (2, 6) which are radially symmetric. Here, p comes from the exponent of the potential of scalar fields. We also prove the nonexistence of nontrivial solutions for the critical case p = 6.
Keywords
Semilocal gauge field model; Klein-Gordon-Maxwell equations; mountain pass solutions; Pohozaev identi;
Citations & Related Records
연도 인용수 순위
  • Reference
1 T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud. 4 (2004), no. 3, 307-322. https://doi.org/10.1515/ans-2004-0305   DOI
2 G. W. Gibbons, M. E. Ortiz, F. Ruiz Ruiz, and T. M. Samols, Semi-local strings and monopoles, Nuclear Phys. B 385 (1992), no. 1-2, 127-144. https://doi.org/10.1016/0550-3213(92)90097-U   DOI
3 C. T. Hill, A. L. Kagan, and L. M. Widrow, Are cosmic strings frustrated?, Phys. Rev. D 38 (1988), 1100-1107.   DOI
4 W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), no. 2, 149-162. http://projecteuclid.org/euclid.cmp/1103900983   DOI
5 T. Vachaspati and A. Achucarro, Semilocal cosmic strings, Phys. Rev. D (3) 44 (1991), no. 10, 3067-3071. https://doi.org/10.1103/PhysRevD.44.3067   DOI
6 F. Wang, Solitary waves for the Klein-Gordon-Maxwell system with critical exponent, Nonlinear Anal. 74 (2011), no. 3, 827-835. https://doi.org/10.1016/j.na.2010.09.033   DOI
7 L. Xu and H. Chen, Existence and multiplicity of solutions for nonhomogeneous KleinGordon-Maxwell equations, Electron. J. Differential Equations 2015 (2015), No. 102, 12 pp.
8 P. C. Carriao, P. L. Cunha, and O. H. Miyagaki, Positive ground state solutions for the critical Klein-Gordon-Maxwell system with potentials, Nonlinear Anal. 75 (2012), no. 10, 4068-4078. https://doi.org/10.1016/j.na.2012.02.023   DOI
9 A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349-381. https://doi.org/10.1016/0022-1236(73)90051-7   DOI
10 S. Chen and X. Tang, Improved results for Klein-Gordon-Maxwell systems with general nonlinearity, Discrete Contin. Dyn. Syst. 38 (2018), no. 5, 2333-2348. https://doi.org/10.3934/dcds.2018096   DOI
11 A. Azzollini, L. Pisani, and A. Pomponio, Improved estimates and a limit case for the electrostatic Klein-Gordon-Maxwell system, Proc. Roy. Soc. Edinburgh Sect. A 141 (2011), no. 3, 449-463. https://doi.org/10.1017/S03082105090018   DOI
12 A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Klein-Gordon-Maxwell equations, Topol. Methods Nonlinear Anal. 35 (2010), no. 1, 33-42.
13 G. Che and H. Chen, Existence and multiplicity of nontrivial solutions for Klein-Gordon-Maxwell system with a parameter, J. Korean Math. Soc. 54 (2017), no. 3, 1015-1030. https://doi.org/10.4134/JKMS.j160344   DOI
14 V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations, Rev. Math. Phys. 14 (2002), no. 4, 409-420. https://doi.org/10.1142/S0129055X02001168   DOI
15 P. C. Carriao, P. L. Cunha, and O. H. Miyagaki, Existence results for the Klein-Gordon-Maxwell equations in higher dimensions with critical exponents, Commun. Pure Appl. Anal. 10 (2011), no. 2, 709-718. https://doi.org/10.3934/cpaa.2011.10.709   DOI
16 D. Cassani, Existence and non-existence of solitary waves for the critical Klein-Gordon equation coupled with Maxwell's equations, Nonlinear Anal. 58 (2004), no. 7-8, 733-747. https://doi.org/10.1016/j.na.2003.05.001   DOI
17 S.-J. Chen and S.-Z. Song, Multiple solutions for nonhomogeneous Klein-GordonMaxwell equations on R3, Nonlinear Anal. Real World Appl. 22 (2015), 259-271. https://doi.org/10.1016/j.nonrwa.2014.09.006   DOI
18 T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrodinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), no. 5, 893-906. https://doi.org/10.1017/S030821050000353X   DOI