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Solutions of super linear Dirac equations with general potentials

  • Jian Ding
  • Junxiang Xu
  • Fubao Zhang
Original Research

Abstract

This paper is concerned with solutions to the Dirac equation: −iΣα k k u + aβu + M(x)u = g(x, ‖u‖)u. Here M(x) is a general potential and g(x, ‖u‖) is a self-coupling which grows super-quadratically in u at infinity. We use variational methods to study this problem. By virtue of some auxiliary system related to the “limit equation” of the Dirac equation, we constructed linking levels of the variational functional Φ M such that the minimax value c M based on the linking structure of Φ M satisfies 0 < c M < Ĉ, where Ĉ is the least energy of the limit equation. Thus we can show the (C) c -condition holds true for all c < Ĉ and consequently we obtain one solution of the Dirac equation.

Keywords

Dirac equations The Coulomb-type potential (C)c-condition Super linear Linking 

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References

  1. 1.
    Thaller B., The Dirac Equation, Texts and Monographs in Physics (Springer, Berlin, 1992)Google Scholar
  2. 2.
    Balabane M., Cazenave T., Douady A. and Merle F., Existence of excited states for a nonlinear Dirac field, Comm. Math. Phys., 119, 153–176, (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Balabane M., Cazenave T. and Vazquez L., Existence of standing waves for Dirac fields with singular nonlinearities, Comm. Math. Phys., 133, 53–74, (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Cazenave T. and Vazquez L., Existence of local solutions for a classical nonlinear Dirac field, Comm. Math. Phys., 105, 35–47, (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Merle F., Existence of stationary states for nonlinear Dirac equations, J. Differential Equations, 74, 50–68, (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Esteban M. J. and Séré E., Stationary states of the nonlinear Dirac equation: A variational approach, Comm. Math. Phys., 171, 323–350, (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    del Pino M. and Felmer P., Semi-classical states of nonlinear Schrödinger equations: A variational reduction method, Math. Ann., 324, 1–32, (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Bartsch T. and Ding Y. H., Solutions of nonlinear Dirac equations, J. Differential Equations, 226, 210–249, (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Ding Y. H. and Ruf B., Solutions of a nonlinear Dirac equation with external fields, Arch. Ration. Mech. Anal., in press; doi: 10.1007/s00205-008-0163-z.Google Scholar
  10. 10.
    Ding Y. H. and Wei J. C., Stationary states of nonlinear Dirac equations with general potentials, Rev. Math. Phys., 20, 1007–1032, (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Soler M., Classical stable nonlinear spinor field with positive rest energy, Phys. Rev., D1, 2766–2769, (1970)Google Scholar
  12. 12.
    Ding Y. H., Variational Methods for Strongly Indefinite Problems, Interdisciplinary Math. Sci., Vol. 7 (World Scientific Publ., 2007).Google Scholar
  13. 13.
    Bartsch T. and Ding Y. H., Deformation theorems on non-metrizable vector spaces and applications to critical point theory, Math. Nachr., 279, 1267–1288, (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Alama A. and Li Y. Y., On “multibump” bound states for certain semilinear elliptic equations, Indiana Univ. Math. J., 41, 983–1026, (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    del Pino M. and Felmer P., Semi-classical states of nonlinear Schrödinger equations: A variational reduction method, Math. Ann., 324, 1–32, (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Ding Y. H., Multiple homoclinics in Hamiltonian system with asymptotically or super linear terms, Commu. Contemp. Math., 8, 453–480, (2006)zbMATHCrossRefGoogle Scholar
  17. 17.
    Ackermann N., A nonlinear superposition principle and multibump solutions of peridic Schrödinger equations, J. Funct. Anal., 234, 423–443, (2006)CrossRefMathSciNetGoogle Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2009

Authors and Affiliations

  1. 1.Department of MathematicsSoutheast UniversityNanjingP.R. China

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