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Localized Concentration of Semi-Classical States for Nonlinear Dirac Equations

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Abstract

The present paper studies concentration phenomena of the semiclassical approximation of a massive Dirac equation with general nonlinear self-coupling:

$$\begin{array}{ll}-i \hbar \alpha \cdot \nabla w+a \beta w + V (x) w = g (|w|) w.\end{array}$$

Compared with some existing issues, the most interesting results obtained here are twofold: the solutions concentrating around local minima of the external potential; and the nonlinearities assumed to be either super-linear or asymptotically linear at the infinity. As a consequence one sees that, if there are k bounded domains \({\Lambda_j \subset \mathbb{R}^3}\) such that \({-a < \min_{\Lambda_j} V=V(x_j) < \min_{\partial \Lambda_j}V}\) , \({x_j\in\Lambda_j}\) , then the k-families of solutions \({w_\hbar^j}\) concentrate around x j as \({\hbar\to 0}\) , respectively. The proof relies on variational arguments: the solutions are found as critical points of an energy functional. The Dirac operator has a continuous spectrum which is not bounded from below and above, hence the energy functional is strongly indefinite. A penalization technique is developed here to obtain the desired solutions.

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References

  1. Ackermann N.: A nonlinear superposition principle and multibump solution of periodic Schrödinger equations. J. Funct. Anal. 234, 423–443 (2006)

    Article  MathSciNet  Google Scholar 

  2. Ambrosetti A., Badiale M., Cignolani S.: Semi-classical states of nonlinear Shrödinger equations. Arch. Rational Mech. Anal. 140, 285–300 (1997)

    Article  MATH  Google Scholar 

  3. Ambrosetti A., Felli V., Malchiodi A.: Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity. J. Eur. Math. Soc. 7, 117–144 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  4. Byeon J., Jeanjean L.: Standing waves for nonlinear Schrödinger equations with a general nonlinearity. Arch. Rational Mech. Anal. 185(2), 185–200 (2007)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  5. Byeon J., Wang Z.Q.: Standing waves with a critical frequency for nonlinear Schrödinger equations. Arch. Rational Mech. Anal. 165(4), 295–316 (2002)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. Dautray R., Lions J.L.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 3. Springer, Berlin (1990)

    Book  Google Scholar 

  7. Del Pino M., Felmer P.: Local mountain passes for semilinear ellipitc problems in unbounded domains. Calc. Var. Partial Differential Equations 4, 121–137 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  8. Del Pino, M., Felmer, P.: Multi-peak bound states for nonlinear Schrödinger equations, Annales de l’Institut Henri Poincare (C) Non Linear Analysis, vol. 15. No. 2. Elsevier, Masson, pp. 127–149, (1998)

  9. Ding Y.H.: Variational Methods for Strongly Indefinite Problems, Interdiscip. Math. Sci., vol. 7. World Scientific Publ, Singapore (2007)

    Google Scholar 

  10. Ding Y.H.: Semi-classical ground states concentrating on the nonlinear potentical for a Dirac equation. J. Differ. Equ. 249, 1015–1034 (2010)

    Article  ADS  MATH  Google Scholar 

  11. Ding, Y.H., Lee, C., Ruf, B.: On semiclassical states of a nonlinear Dirac equation. Proc. R. Soc. Edinburgh Sect. A Math. 143.04, 765–790 (2013)

  12. Ding Y.H., Liu Xiaoying: Semi-classical limits of ground states of a nonlinear Dirac equation. J. Differ. Equ. 252, 4962–4987 (2012)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  13. Ding Y.H., Ruf B.: Existence and concentration of semi-classical solutions for Dirac equations with critical nonlinearities. SIAM J. Math. Anal. 44(6), 3755–3785 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ding Y.H., Wei J.C.: Stationary states of nonlinear Dirac equations with general potentials. Rev. Math. Phys. 20, 1007–1032 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. 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(06), 061505 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  16. Ding Y.H., Xu T.: On semi-classical limits of ground states of a nonlinear Maxwell–Dirac system. Calc. Var. Partial Differ. Equ. 51(1), 17–44 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ding Y.H., Xu T.: On the concentration of semi-classical states for a nonlinear Dirac–Klein–Gordon system. J. Differ. Equ. 256, 1264–1294 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  18. Esteban M.J., Séré E.: Stationary states of the nonlinear Dirac equation: a variational approach. Commun. Math. Phys. 171, 323–350 (1995)

    Article  ADS  MATH  Google Scholar 

  19. Finkelstein R., LeLevier R., Ruderman M.: Nonlinear spinor fields. Phys. Rev. 83(2), 326–332 (1951)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  20. Finkelstein R., Fronsdal C., Kaus P.: Nonlinear spinor field. Phys. Rev. 103(5), 1571–1579 (1956)

    Article  ADS  MATH  Google Scholar 

  21. Floer A., Weinstein A.: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69, 397–408 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  22. Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1998)

    Google Scholar 

  23. Ivanenko D.D.: Notes to the theory of interaction via particles. Zh.Éksp. Teor. Fiz. 8, 260–266 (1938)

    MATH  Google Scholar 

  24. Jeanjean L., Tanaka K.: Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities. Calc. Var. Partial Differ. Equ. 21(3), 287–318 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  25. Lions, P.L.: The concentration-compactness principle in the calculus of variations: the locally compact case, Part II. AIP Anal. non linéaire 1, 223–283

  26. Miguel R., Hugo T.: Solutions with multiple spike patterns for an elliptic system. Calc. Var. Partial Differ. Equ. 31(1), 1–25 (2008)

    MATH  Google Scholar 

  27. Miguel R., Yang J.F.: Spike-layered solutions for an elliptic system with Neumann boundary conditions. Trans. Am. Math. Soc. 357, 3265–3284 (2005)

    Article  MATH  Google Scholar 

  28. Oh Y.G.: Existence of semi-classical bound states of nonlinear Schrödinger equations with potentials of the class (V) a . Commun. Partial Differ. Equ. 13(12), 1499–1519 (1988)

    Article  MATH  Google Scholar 

  29. Oh Y.G.: On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential. Commun. Math. Phys. 131(2), 223–253 (1990)

    Article  ADS  MATH  Google Scholar 

  30. Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. Amer. Math. Soc., Providence, 1986

  31. Rabinowitz P.H.: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43(2), 270–291 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  32. Stein E.M.: Singular integrals and differentiability properties of functions, vol. 2. Princeton University Press, Princeton (1970)

    Google Scholar 

  33. Simon B.: Schrödinger semigroups. Bull. Am. Math. Soc. (N.S.) 7, 447–526 (1982)

    Article  MATH  Google Scholar 

  34. Wang X.: On concentration of positive bound states of nonlinear Schrödinger equations. Commun. Math. Phys. 153(2), 229–244 (1993)

    Article  ADS  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190, Beijing, China

    Yanheng Ding & Tian Xu

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  1. Yanheng Ding
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  2. Tian Xu
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Correspondence to Tian Xu.

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Communicated by G. Dal Maso

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Ding, Y., Xu, T. Localized Concentration of Semi-Classical States for Nonlinear Dirac Equations. Arch Rational Mech Anal 216, 415–447 (2015). https://doi.org/10.1007/s00205-014-0811-4

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