Advertisement

Semiclassical states of nonlinear Dirac equations with degenerate potential

  • Xu Zhang
  • Zhi-Qiang WangEmail author
Article
  • 26 Downloads

Abstract

In this paper, we study the following nonlinear Dirac equation
$$\begin{aligned} -\,i\varepsilon \alpha \cdot \nabla u+a\beta u+V(x)u=|u|^{p-2}u,\ x\in {\mathbb {R}}^3, \quad \mathrm{for}\ u\in H^1({\mathbb {R}}^3, {\mathbb {C}}^4), \end{aligned}$$
where \(p\in (2,3)\), \(a > 0\) is a constant, \(\alpha =(\alpha _1,\alpha _2,\alpha _3)\), \(\alpha _1,\alpha _2,\alpha _3\) and \(\beta \) are \(4\times 4\) Pauli–Dirac matrices. Our investigation focuses on the case in which |V(x)| may approach a as \(|x|\rightarrow \infty \). This is a degenerate case as most works in the literature assume a strict gap condition \(\sup _{x\in {\mathbb {R}}^3} |V(x)|< a\), which is a key condition used in setting up an infinitely dimensional topological linking structure as well as in dealing with the compactness issues of the variational formulation. Under the assumption that V has a local trapping potential well, for \(\varepsilon >0\) small, we construct bound state solutions concentrating around the local minimum points of V. As a consequence we construct an infinite sequence of localized bound state solutions as \(\varepsilon \rightarrow 0\).

Keywords

Dirac equation Degenerate potential Semiclassical states Concentration 

Mathematics Subject Classification

35Q40 49J35 

Notes

Acknowledgements

Research was supported by the Specialized Fund for the Doctoral Program of Higher Education of China, NSFC 11771324 and a Simons Collaboration Grant.

References

  1. 1.
    Ambrosetti, A., Malchiodi, A.: Perturbation Methods and Semilinear Elliptic Problems on \({\mathbb{R}}^N\), Progress in Mathematics, vol. 240. Birkhäuser, Basel (2006)zbMATHGoogle Scholar
  2. 2.
    Ambrosetti, A., Felli, V., Malchiodi, A.: Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity. J. Eur. Math. Soc. (JEMS) 7(1), 117–144 (2005)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Balabane, M., Cazenave, T., Douady, A., Merle, F.: Existence of excited states for a nonlinear Dirac field. Commun. Math. Phys. 119(1), 153–176 (1988)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bartsch, T., Ding, Y.H.: Solutions of nonlinear Dirac equations. J. Differ. Equ. 226(1), 210–249 (2006)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bartsch, T., Ding, Y.H.: Deformation theorems on non-metrizable vector spaces and applications to critical point theory. Math. Nachr. 279(12), 1267–1288 (2006)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Byeon, J., Wang, Z.-Q.: Standing waves with a critical frequency for nonlinear Schrödinger equations. Arch. Ration. Mech. Anal. 165(4), 295–316 (2002)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Byeon, J., Wang, Z.-Q.: Standing waves with a critical frequency for nonlinear Schrödinger equations. II. Calc. Var. Partial Differ. Equ. 18(2), 207–219 (2003)zbMATHGoogle Scholar
  8. 8.
    Cazenave, T., Vazquez, L.: Existence of localized solutions for a classical nonlinear Dirac field. Commun. Math. Phys. 105(1), 35–47 (1986)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Cerami, G., Devillanova, G., Solimini, S.: Infinitely many bound states for some nonlinear scalar field equations. Calc. Var. Partial Differ. Equ. 23(2), 139–168 (2005)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Chen, S.W., Wang, Z.-Q.: Localized nodal solutions of higher topological type for semiclassical nonlinear Schrödinger equations. Calc. Var. Partial Differ. Equ. 56(1), 1 (2017)zbMATHGoogle Scholar
  11. 11.
    Dautray, R., Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 3. Springer, Berlin (1990)zbMATHGoogle Scholar
  12. 12.
    Del Pino, M., Felmer, P.: Semi-classical states for nonlinear Schröinger equations. J. Funct. Anal. 149(1), 245–265 (1997)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Del Pino, M., Felmer, P.: Multi-peak bound states for nonlinear Schröinger equations. Ann. Inst. H. Poincaré Anal. Non Lináire 15(2), 127–149 (1998)zbMATHGoogle Scholar
  14. 14.
    Ding, Y.H.: Variational Methods for Strongly Indefinite Problems, vol. 7. World Scientific Publishing, Singapore (2007)zbMATHGoogle Scholar
  15. 15.
    Ding, Y.H.: Semi-classical ground states concentrating on the nonlinear potential for a Dirac equation. J. Differ. Equ. 249(5), 1015–1034 (2010)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Ding, Y.H., Lee, C., Ruf, B.: On semiclassical states of a nonlinear Dirac equation. Proc. R. Soc. Edinb. Sect. A 143(4), 765–790 (2013)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Ding, Y.H., Ruf, B.: Solutions of a nonlinear Dirac equation with external fields. Arch. Ration. Mech. Anal. 190(1), 57–82 (2008)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Ding, Y.H., Ruf, B.: Existence and concentration of semiclassical solutions for Dirac equations with critical nonlinearities. SIAM J. Math. Anal. 44(6), 3755–3785 (2012)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Ding, Y.H., Wei, J.C.: Stationary states of nonlinear Dirac equations with general potentials. J. Math. Phys. 20(8), 1007–1032 (2008)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Ding, Y.H., Wei, J.C., Xu, T.: Existence and concentration of semi-classical solutions for a nonlinear Maxwell–Dirac system. J. Math. Phys. 54(6), 061505 (2013)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Ding, Y.H., Xu, T.: Localized concentration of semi-classical states for nonlinear Dirac equations. Arch. Ration. Mech. Anal. 216(2), 415–447 (2015)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Esteban, M.J., Séré, E.: Stationary states of the nonlinear Dirac equation: a variational approach. Commun. Math. Phys. 171(2), 323–350 (1995)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Esteban, M.J., Lewin, M., Séré, E.: Variational methods in relativistic quantum mechanics. Bull. Am. Math. Soc. (N.S.) 45(4), 535–593 (2008)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Finkelstein, R., LeLevier, R., Ruderman, M.: Nonlinear spinor fields. Phys. Rev. 83(2), 326–332 (1951)zbMATHGoogle Scholar
  25. 25.
    Finkelstein, R., Fronsdal, C., Kaus, P.: Nonlinear spinor field. Phys. Rev. 103(5), 1571–1579 (1956)zbMATHGoogle Scholar
  26. 26.
    Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1997)zbMATHGoogle Scholar
  27. 27.
    Kang, X., Wei, J.C.: On interacting bumps of semi-classical states of nonlinear Schrödinger equations. Adv. Differ. Equ. 5(7–9), 899–928 (2000)zbMATHGoogle Scholar
  28. 28.
    Lions, P.L.: The concentration-compactness principle in the calculus of variations. The locally compact case, parts 1 and 2. In: Annales de l’Institut Henri Poincaré Anual. Non Linéair , vol. 1, pp. 109–145, 223–283 (1984)Google Scholar
  29. 29.
    Merle, F.: Existence of stationary states for nonlinear Dirac equations. J. Differ. Equ. 74(1), 50–68 (1988)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Moroz, V., Van Schaftingen, S.: Semiclassical stationary states for nonlinear Schrödinger equations with fast decaying potentials. Calc. Var. Partial Differ. Equ. 37(1), 1–27 (2010)zbMATHGoogle Scholar
  31. 31.
    Pankov, A.: On decay of solutions to nonlinear Schrödinger equations. Proc. Am. Math. Soc. 136(7), 2565–2570 (2008)zbMATHGoogle Scholar
  32. 32.
    Rabinowitz, P.H.: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43(2), 270–292 (1992)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Wang, Z.-Q., Zhang, X.: An infinite sequence of localized semiclassical bound states for nonlinear Dirac equations. Calc. Var. Partial Differ. Equ. 57(2), 30 (2018). (Art. 56)MathSciNetzbMATHGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsCentral South UniversityChangshaChina
  2. 2.School of Mathematics and InformaticsFujian Normal UniversityFuzhouChina
  3. 3.Department of Mathematics and StatisticsUtah State UniversityLoganUSA

Personalised recommendations