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Infinitely many solutions for a class of superlinear Dirac-Poisson system. (English) Zbl 1394.35391

Summary: This paper is concerned with the nonlinear Dirac-Poisson system \[ \begin{cases} -i \sum_{k = 1}^3 \alpha_k \partial_k u +(V(x) + a) \beta u + \omega u - \phi u = F_u(x,u), \\ -\varDelta \phi = 4 \pi |u|^2, \end{cases} \quad \text{in} \; \mathbb{R}^3 \] where \(V\) is an external potential and \(F\) is a superlinear nonlinearity modeling various types of interactions. Existence and multiplicity of stationary solutions are obtained for the system with periodicity condition via variational methods.

MSC:

35Q41 Time-dependent Schrödinger equations and Dirac equations
35Q60 PDEs in connection with optics and electromagnetic theory
35A15 Variational methods applied to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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[1] Esteban, M. J.; Lewin, M.; Séré, E., Variational methods in relativistic quantum mechanics, Bull. Amer. Math. Soc., 45, 535-593, (2008) · Zbl 1288.49016
[2] Bartsch, T.; Ding, Y. H., Solutions of nonlinear Dirac equations, J. Differential Equations, 226, 210-249, (2006) · Zbl 1126.49003
[3] 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
[4] 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
[5] Liu, Z.; Zhang, J., Multiplicity and concentration of positive solutions for the fractional Schrödinger-Poisson systems with critical growth, ESAIM Control Optim. Calc. Var., 23, 1515-1542, (2017) · Zbl 06811887
[6] 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
[7] Zhang, J.; Tang, X. H.; Zhang, W., Existence and multiplicity of stationary solutions for a class of Maxwell-Dirac system, Nonlinear Anal., 127, 298-311, (2015) · Zbl 1326.35305
[8] 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
[9] Chen, G. Y.; Zheng, Y. Q., Stationary solutions of non-autonomous Maxwell-Dirac systems, J. Differential Equations, 255, 840-864, (2013) · Zbl 1281.49040
[10] Zhang, J.; Tang, X. H.; Zhang, W., Existence and multiplicity of solutions for nonlinear Dirac-Poisson systems, Electron. J. Differ. Equ., 91, 1-17, (2017)
[11] Zhang, J.; Tang, X. H.; Zhang, W., Ground state solutions for a class of nonlinear Maxwell-Dirac system, Topol. Methods Nonlinear Anal., 46, 785-798, (2015) · Zbl 1375.35425
[12] Ding, Y. H., Variational methods for strongly indefinite problems, (2008), World Scientific Press
[13] Zhang, J.; Zhang, W.; Tang, X. H., Ground state solutions for Hamiltonian elliptic system with inverse square potential, Discrete Contin. Dyn. Syst., 37, 4565-4583, (2017) · Zbl 1370.35111
[14] 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
[15] Lions, P. L., The concentration-compactness principle in the calculus of variations. the locally compact case, part 2, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1, 223-283, (1984) · Zbl 0704.49004
[16] Coti-Zelati, V.; Rabinowitz, P. H., Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4, 693-727, (1991) · Zbl 0744.34045
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