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Admissible solutions for Dirac equations with singular and non-monotone nonlinearity

Part of: Boundary value problems

Published online by Cambridge University Press:  25 February 2011

Yujun Dong  and
Jing Xie
Yujun Dong
Affiliation:
Department of Mathematics, Nanjing Normal University, Nanjing, Jiangsu 210097, People's Republic of China, (yjdong@njnu.edu.cn)
Jing Xie
Affiliation:
Department of Mathematics, Nanjing Normal University, Nanjing, Jiangsu 210097, People's Republic of China, (yjdong@njnu.edu.cn)
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Abstract

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By making use of Merle's general shooting method we investigate Dirac equations of the form

Here it is possible that F(0) = −∞ and that F(s) defined on (0,+∞) is not monotonously nondecreasing. Our results cover some known ones as a special case.

Keywords

admissible solutionsDirac equationssingular and non-monotone nonlinearitygeneral shooting methods

MSC classification

Secondary: 34B15: Nonlinear boundary value problems
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 54 , Issue 2 , June 2011 , pp. 363 - 371
Copyright
Copyright © Edinburgh Mathematical Society 2011

References

1.Balabane, M., Cazenave, T., Douady, A. and Merle, F., Existence of excited states for a nonlinear Dirac field, Commun. Math. Phys. 119 (1988), 153–176.CrossRefGoogle Scholar
2.Balabane, M., Cazenave, T. and Vazquez, L., Existence of standing waves for Dirac fields with singular nonlinearities, Commun. Math. Phys. 133 (1990), 53–74.CrossRefGoogle Scholar
3.Cazenave, T. and Vazquez, L., Existence of localized solutions for a classical nonlinear Dirac field, Commun. Math. Phys. 103 (1986), 35–47.Google Scholar
4.Esteban, M. J., Lewin, M. and Séré, E., Variational methods in relativistic quantum mechanics, Bull. Am. Math. Soc. 45 (2008), 535–593.CrossRefGoogle Scholar
5.Esteban, M. J. and Séré, E., Stationary states of the nonlinear Dirac equation: a variational approach, Commun. Math. Phys. 171 (1995), 323–350.Google Scholar
6.Esteban, M. J. and Séré, E., Solutions of the Dirac–Fock equations for atoms and molecules, Commun. Math. Phys. 203 (1999), 499–530.Google Scholar
7.Esteban, M. J. and Séré, E., An overview on linear and nonlinear Dirac equations, Discrete Contin. Dynam. Syst. 8 (2002), 381–397.Google Scholar
8.Hartman, P., Ordinary differential equations (Birkhäuser, Boston, MA, 1982).Google Scholar
9.Merle, F., Existence of stationary states for nonlinear Dirac equations, J. Diff. Eqns 74 (1988), 50–68.CrossRefGoogle Scholar
10.Paturel, E., A new variational principle for a nonlinear Dirac equation on the Schwarzschild metric, Commun. Math. Phys. 213 (2000), 249–266.CrossRefGoogle Scholar
11.Soler, M., Classical stable nonlinear spinor field with positive rest energy. Phys. Rev. D 1 (1970), 2766–2769.CrossRefGoogle Scholar
12.Wakano, M., Intensely localized solutions of the classical Dirac–Maxwell field equations, Prog. Theor. Phys. 35 (1966), 1117–1141.CrossRefGoogle Scholar