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Stationary states of the nonlinear Dirac equation: A variational approach. (English) Zbl 0843.35114

Using variational techniques, the authors prove the existence of stationary solutions of some nonlinear Dirac equations.
Reviewer: N.Jacob (Erlangen)

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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[1] Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical points theory and applications. J. Funct. Anal.14, 38–349 (1973) · Zbl 0273.49063
[2] Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations. Arch. Rat. Mech. Anal.82, 313–346 (1983) · Zbl 0533.35029
[3] Benci, V., Capozzi, A., Fortunato, D.: Periodic solutions of Hamiltonian systems with superquadratic potential. Ann. Mat. Pura et app. (IV), Vol.CXLIII, 1–46 (1986) · Zbl 0632.34036
[4] Bjorken, J.D., Drell, S.D.: Relativistic quantum fields. New York: McGraw-Hill, 1965 · Zbl 0184.54201
[5] Balabane, M., Cazenave, T., Douady, A., Merle, F.: Existence of excited states for a nonlinear Dirac field. Commun. Math. Phys.119, 153–176 (1988) · Zbl 0696.35158
[6] Balabane, M., Cazenave, T., Vazquez, L.: Existence of standing waves for Dirac fields with singular nonlinearities. Commun. Math. Phys.133, 53–74 (1990) · Zbl 0721.35065
[7] Benci, V., Rabinowitz, P.H.: Critical point theorems for indefinite functionals, Inv. Math.52, 336–352 (1979) · Zbl 0465.49006
[8] Cazenave, T.: On the existence of stationary states for classical non-linear Dirac fields. In: Hyperbolic systems and Mathematical Physics. Textos e Notas4, CMAF, Lisbonne, 1989
[9] Cazenave, T., Vazquez, L.: Existence of localized solutions for a classical nonlinear Dirac field. Commun. Math. Phys.105, 35–47 (1986) · Zbl 0596.35117
[10] Finkelstein, R., Lelevier, R., Ruderman, M.: Phys. Rev.83, 326–332 (1951) · Zbl 0043.21603
[11] Hofer, H., Wysocki, K.: First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems. Math. Ann.288, 483–503 (1990) · Zbl 0702.34039
[12] Lions, P.-L.: The concentration-compactness method in the Calculus of Variations. The locally compact case. Part. I: Anal. non-linéaire, Ann. IHP1, 109–145 (1984); Part. II: Anal. non-linéaire, Ann. IHP1, 223–283 (1984) · Zbl 0541.49009
[13] Merle, F.: Existence of stationary states for nonlinear Dirac equations. J. Diff. Eq.74 (1), 50–68 (1988) · Zbl 0696.35154
[14] Rañada, A.F.: Classical nonlinear Dirac field models of extended particles. In: Quantum theory, groups, fields and particles (editor A.O. Barut). Amsterdam, Reidel: 1982
[15] Séré, E.: Homoclinic orbits on compact hypersurfaces in IR2N , of restricted contact type. Preprint CEREMADE 9238 (1992)
[16] Smale, S.: An infinite dimensional version of Sard’s theorem. Amer. J. Math.87, 861–866 (1965) · Zbl 0143.35301
[17] Soler, M.: Phys. Rev.D1, 2766–2769 (1970)
[18] Strauss, W.A.: Existence of solitary waves in higher dimensions. Commun. Math. Phys.55, 149–162 (1977) · Zbl 0356.35028
[19] Tanaka, K.: Homoclinic orbits in the first order superquadratic Hamiltonian system: convergence of subharmonics. J. Diff. Eq.94, 315–339 (1991) · Zbl 0787.34041
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