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.


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
Full Text: DOI


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.