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Existence and multiplicity of stationary solutions for a class of Maxwell-Dirac system. (English) Zbl 1326.35305

Summary: This paper is concerned with the following Maxwell-Dirac system \[ \begin{cases} - i \sum_{k = 1}^3 \alpha_k \partial_k u + a \beta u + M(x) u - K(x) \phi u = F_u(x, u), \\ - \Delta \phi = 4 \pi K(x) \mid u \mid^2, \end{cases} \text{ in } \mathbb{R}^3 \] where \(M(x)\) is a external potential and \(F(x, u)\) is an asymptotically quadratic nonlinearity modeling various types of interactions. In view of the effects of the nonlocal term, we use some special techniques to deal with the nonlocal term. Moreover, existence and multiplicity of stationary solutions are obtained for system without any periodicity assumption via variational methods.

MSC:

35Q41 Time-dependent Schrödinger equations and Dirac equations
35Q60 PDEs in connection with optics and electromagnetic theory
81V10 Electromagnetic interaction; quantum electrodynamics
18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)
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[1] Abenda, S., Solitary waves for Maxwell-Dirac and Coulomb-Dirac models, Ann. Inst. Henri. Poincaré, 68, 229-244, (1998) · Zbl 0907.35104
[2] Ackermann, N., On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z., 248, 423-443, (2004) · Zbl 1059.35037
[3] Ackermann, N., A Cauchy-Schwarz type inequality for bilinear integrals on positive measures, Proc. Amer. Math. Soc., 133, 2647-2656, (2005) · Zbl 1066.26013
[4] Ambrosetti, A., On Schrödinger-Poisson systems, Milan J. Math., 76, 257-274, (2008) · Zbl 1181.35257
[5] Azzollini, A.; Pomponio, A., Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345, 90-108, (2008) · Zbl 1147.35091
[6] Bartsch, T.; Ding, Y. H., Deformation theorems on non-metrizable vector spaces and applications to critical point theory, Math. Nachr., 279, 1267-1288, (2006) · Zbl 1117.58007
[7] Bartsch, T.; Ding, Y. H., Solutions of nonlinear Dirac equations, J. Differential Equations, 226, 210-249, (2006) · Zbl 1126.49003
[8] Brezis, H.; Lieb, E., A relation between pointwise convergence of function and convergence of functional, Proc. Amer. Math. Soc., 88, 486-490, (1983) · Zbl 0526.46037
[9] Cazenave, T.; Vazquez, L., Existence of local solutions of a classical nonlinear Dirac field, Comm. Math. Phys., 105, 35-47, (1986) · Zbl 0596.35117
[10] Cerami, G.; Vaira, G., Positive solutions for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations, 248, 521-543, (2010) · Zbl 1183.35109
[11] Chadam, J. M., Global solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac system in one space dimension, J. Funct. Anal., 13, 173-184, (1973) · Zbl 0264.35058
[12] Chadam, J. M.; Glassey, R. T., On the Maxwell-Dirac equations with zero magnetic field and their solutions in two space dimension, J. Math. Anal. Appl., 53, 495-507, (1976) · Zbl 0324.35076
[13] Chen, G. Y.; Zheng, Y. Q., Stationary solutions of non-autonomous maxwell—dirac systems, J. Differential Equations, 255, 840-864, (2013) · Zbl 1281.49040
[14] Ding, Y. H., Variational methods for strongly indefinite problems, (2008), World Scientific Press
[15] Ding, Y. H., Semi-classical ground states concentrating on the nonlinear potential for a Dirac equation, J. Differential Equations, 249, 1015-1034, (2010) · Zbl 1193.35161
[16] Ding, Y. H.; Liu, X. Y., Semi-classical limits of ground states of a nonlinear Dirac equation, J. Differential Equations, 252, 4962-4987, (2012) · Zbl 1236.35133
[17] Ding, Y. H.; Liu, X. Y., On semiclassical ground states of a nonlinear Dirac equation, Rev. Math. Phys., 10, 1250029, (2012) · Zbl 1457.35044
[18] Ding, Y. H.; Ruf, B., Solutions of a nonlinear Dirac equation with external fields, Arch. Ration. Mech. Anal., 190, 57-82, (2008) · Zbl 1161.35041
[19] Ding, Y. H.; Ruf, B., Existence and concentration of semi-classical solutions for Dirac equations with critical nonlinearites, SIAM J. Math. Anal., 44, 3755-3785, (2012) · Zbl 1259.35171
[20] Ding, Y. H.; Wei, J. C., Stationary states of nonlinear Dirac equations with general potentials, Rev. Math. Phys., 20, 1007-1032, (2008) · Zbl 1170.35082
[21] 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, 061505, (2013) · Zbl 1282.81073
[22] Ding, Y. H.; Xu, T., On the concentration of semi-classical states for a nonlinear Dirac-Klein-Gordon system, J. Differential Equations, 256, 1264-1294, (2014) · Zbl 1283.35099
[23] Ding, Y. H.; Xu, T., On semi-classical limits of ground states of a nonlinear Maxwell-Dirac system, Calc. Var. Partial Differential Equations, 51, 17-44, (2014) · Zbl 1297.35197
[24] Esteban, M. J.; Georgiev, V.; Séré, E., Stationary solutions of the Maxwell-Dirac and the Klein-Gordon- Dirac equations, Calc. Var. Partial Differential Equations, 4, 265-281, (1996) · Zbl 0869.35105
[25] Esteban, M. J.; Lewin, M.; Séré, E., Variational methods in relativistic quantum mechanics, Bull. Amer. Math. Soc., 45, 535-593, (2008) · Zbl 1288.49016
[26] Esteban, M. J.; Séré, E., Stationary states of nonlinear Dirac equations: A variational approach, Comm. Math. Phys., 171, 323-350, (1995) · Zbl 0843.35114
[27] Esteban, M. J.; Séré, E., An overview on linear and nonlinear Dirac equations, Discrete Contin. Dyn. Syst., 8, 281-397, (2002) · Zbl 1162.49307
[28] Finkelstein, R.; Fronsdal, C.; Kaus, P., Nonlinear spinor field, Phys. Rev., 103, 1571-1579, (1956) · Zbl 0073.44705
[29] Finkelstein, R.; LeLevier, R.; Ruderman, M., Nonlinear spinor fields, Phys. Rev., 83, 326-332, (1951) · Zbl 0043.21603
[30] Flato, M.; Simon, J.; Taffin, E., On the global solutions of the Maxwell-Dirac equations, Comm. Math. Phys., 113, 21-49, (1987) · Zbl 0641.35064
[31] Garrett Lisi, A., A solitary wave solution of the Maxwell-Dirac equations, J. Phys. A: Math. Gen., 28, 5385-5392, (1995) · Zbl 0868.35121
[32] Georgiev, V., Small amplitude solutions of Maxwell-Dirac equations, Indiana Univ. Math. J., 40, 845-883, (1991) · Zbl 0754.35171
[33] Glassey, R. T.; Chadam, J. M., Properties of the solutions of the Cauchy problem for the classical coupled Maxwell-Dirac equations in one space dimension, Proc. Amer. Math. Soc., 43, 373-378, (1974) · Zbl 0258.35004
[34] Grandy, W. T., (Relativistic Quantum Mechanics of Leptonsand Fields, Fundam. Theor. Phys., vol. 41, (1991), Kluwer Academic Publishers Group Dordrecht)
[35] Gross, L., The Cauchy problem for the coupled Maxwell-Dirac equations, Comm. Pure Appl. Math., 19, 1-5, (1966) · Zbl 0137.32401
[36] Merle, F., Existence of stationary states for nonlinear Dirac equations, J. Differential Equations, 74, 50-68, (1988) · Zbl 0696.35154
[37] Radford, C. J., The stationary Maxwell-dirace quations, J. Phys. A: Math. Gen., 36, 5663-5681, (2003) · Zbl 1053.81022
[38] Ruiz, D., The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237, 665-674, (2006) · Zbl 1136.35037
[39] Sparber, C.; Markowich, P., Semiclassical asymptotics for the Maxwell-Dirac system, J. Math. Phys., 44, 4555-4572, (2003) · Zbl 1062.81059
[40] Tang, X. H., New super-quadratic conditions on ground state solutions for superlinear Schrödinger equation, Adv. Nonlinear Stud., 14, 361-373, (2014) · Zbl 1305.35036
[41] Tang, X. H., Non-Nehari manifold method for asymptotically periodic Schrödinger equations, Sci. China Math., 58, 715-728, (2015) · Zbl 1321.35055
[42] Thaller, B., (The Dirac Equation, Texts and Monographs in Physics, (1992), Springer Berlin)
[43] Willem, M., Minimax theorems, (1996), Birkhäuser Berlin · Zbl 0856.49001
[44] Yang, M. B.; Ding, Y. H., Stationary states for nonlinear Dirac equations with superlinear nonlinearities, Topol. Methods Nonlinear Anal., 39, 175-188, (2012) · Zbl 1269.35035
[45] Zhang, J.; Qin, W. P.; Zhao, F. K., Multiple solutions for a class of nonperiodic Dirac equations with vector potentials, Nonlinear Anal., 75, 5589-5600, (2012) · Zbl 1253.81059
[46] Zhang, J.; Tang, X. H.; Zhang, W., Ground state solutions for nonperiodic Dirac equation with superquadratic nonlinearity, J. Math. Phys., 54, 101502, (2013) · Zbl 1284.81134
[47] Zhang, J.; Tang, X. H.; Zhang, W., On ground state solutions for superlinear Dirac equation, Acta Math. Sci., 34, 840-850, (2014) · Zbl 1313.35268
[48] Zhang, J.; Tang, X. H.; Zhang, W., Ground states for nonlinear Maxwell-Dirac system with magnetic field, J. Math. Anal. Appl., 421, 1573-1586, (2015) · Zbl 1304.35597
[49] Zhang, J.; Tang, X. H.; Zhang, W., Infinitely many large energy solutions for superlinear Dirac equations, Math. Methods Appl. Sci., 38, 1485-1493, (2015) · Zbl 1318.35092
[50] Zhang, W.; Tang, X. H.; Zhang, J., Stationary solutions for a superlinear Dirac equations, Math. Methods Appl. Sci., (2015)
[51] Zhao, F. K.; Ding, Y. H., Infinitely many solutions for a class of nonlinear Dirac equations without symmetry, Nonlinear Anal., 70, 921-935, (2009) · Zbl 1152.35501
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