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Periodic solutions of an asymptotically linear Dirac equation

  • Yanheng Ding
  • Xiaoying Liu
Article

Abstract

Using the variational method, we investigate periodic solutions of a Dirac equation with asymptotically nonlinearity. The variational setting is established and the existence and multiplicity of periodic solutions are obtained.

Keywords

Dirac equation Periodic solutions Variational method Asymptotically linear 

Mathematics Subject Classification

35Q40 49J35 

Notes

Acknowledgments

The work was supported by the National Science Foundation of China (NSFC11571146, NSFC11331010).

References

  1. 1.
    Ackermann, N.: On a periodic Schrödinger equation with nonlocal superlinear part. Math. Z. 248, 423–443 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alama, S., Li, Y.Y.: Existence of solutions for semilinear elliptic equations with indefinite linear part. J. Differ. Equ. 96, 89–115 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bartsch, T., Ding, Y.H.: Solutions of nonlinear Dirac equations. J. Differ. Equ. 226, 210–249 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bartsch, T., Ding, Y.: On a nonlinear Schrödinger equation with periodic potential. Math. Ann. 313, 15–37 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brüll, L., Kapellen, H.-J.: Periodic solutions of one-dimensional nonlinear Schrödinger equations. Acta Math. Hung. 54, 191–195 (1989)CrossRefzbMATHGoogle Scholar
  6. 6.
    Cerami, G.: Un criterio di esistenza per i punti critici su varieta illimitate. Rend. Acad. Sci. Lst. Ist. Lombardo 112, 332–336 (1978)Google Scholar
  7. 7.
    Chang, K.C.: Critical Point Theory and Its Applications (in Chinese). Shanghai Science and Technology Press, Shanghai (1986)Google Scholar
  8. 8.
    Ding, Y.H.: Variational Methods for Strongly Indefinite Problems. Interdisciplinary Math. Sci, vol. 7. World Scientific Publ, River Edge (2007)CrossRefGoogle Scholar
  9. 9.
    Ding, Y.H.: Infinitely many solutions for a class of nonlinear Dirac equations without symmetry. Nonlinear Anal. 70, 921–935 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ding, Y.H.: Semi-classical ground states concerntrating on the nonlinear potential for a Dirac equation. J. Differ. Equ. 249, 1015–1034 (2010)CrossRefzbMATHGoogle Scholar
  11. 11.
    Ding, Y.H., Liu, X.Y.: Semi-classical limits of ground states of a nonlinear Dirac equation. J. Differ. Equ. 252, 4962–4987 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ding, Y.H., Liu, X.Y.: Periodic waves of nonlinear Dirac equations. Nonlinear Anal. 109, 252–267 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ding, Y.H., Liu, X.Y.: Periodic solutions of a Dirac equation with concave and convex nonlinearities. J. Differ. Equ. 258, 3567–3588 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ding, Y.H., Ruf, B.: Existence and concentration of semiclassical solutions for Dirac equations with critical nonlinearities. SIAM J. Math. Anal. 44, 3755–3785 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ding, Y.H., Ruf, B.: Solutions of a nonlinear Dirac equation with external fields. Arch. Ration. Mech. Anal. 190, 1007–1032 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ding, Y.H., Wei, J.C.: Stationary states of nonlinear Dirac equations with general potentials. Rev. Math. Phys. 20, 1007–1032 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Esteban, M.J., Séré, E.: An overview on linear and nonlinear Dirac equations. Discr. Contin. Dyn. Syst. 8, 281–397 (2002)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Esteban, M.J., Séré, E.: Stationary states of the nonlinear Dirac equation: a variational approach. Commun. Math. Phys. 171, 323–350 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Gentile, G., Procesi, M.: Periodic solutions for the Schrödinger equation with nonlocal smoothing nonlinearities in higher dimension. J. Differ. Equ. 245, 3253–3326 (2008)CrossRefzbMATHGoogle Scholar
  20. 20.
    Li, G.B., Szulkin, A.: An asymptotically periodic Schrödinger equation with indefinite linear part. Commun. Contemp. Math. 4, 763–776 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Merle, F.: Existence of stationary states for nonlinear Dirac equations. J. Differ. Equ. 74, 50–68 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. In: CBMS Regional Conference Series No. 65 AMS, Providence, RI (1986)Google Scholar
  23. 23.
    Ranada, A.F.: Classical nonlinear dirac field models of extended particles. In: Barut, A.O. (ed.) Quantum Theory, Groups, Fields and Particles. Reidel, Amsterdam (1982)Google Scholar
  24. 24.
    Roy Chowdhury, A., Paul, S., Sen, S.: Periodic solutions of the mixed nonlinear Schrödinger equation. Phys. Rev. D 32, 3233–3237 (1985)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Thaller, B.: The Dirac Equation, Texts and Monographs in Physics. Springer, Berlin (1992)Google Scholar
  26. 26.
    Troestler, C., Willem, M.: Nontrivial solution of a semilinear Schrödinger equation. Commun. Partial Differ. Equ. 21, 1431–1449 (1996)CrossRefzbMATHGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institute of Mathematics, AMSS, and Hua Loo-Keng Key Laboratory of MathematicsChinese Academy of SciencesBeijingPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsJiangsu Normal UniversityXuzhouPeople’s Republic of China

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