Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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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)
  • Received : 2020.08.14
  • Accepted : 2020.10.16
  • Published : 2021.09.01
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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

Acknowledgement

Jongmin Han was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(2018R1D1A1B07042681).

References

  1. 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
  2. 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
  3. 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.
  4. 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
  5. 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
  6. 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
  7. 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
  8. 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
  9. 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
  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
  11. 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
  12. 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
  13. 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
  14. C. T. Hill, A. L. Kagan, and L. M. Widrow, Are cosmic strings frustrated?, Phys. Rev. D 38 (1988), 1100-1107. https://doi.org/10.1103/PhysRevD.38.1100
  15. 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 https://doi.org/10.1007/BF01626517
  16. 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
  17. 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
  18. 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.