Multiplicity of solutions for asymptotically quadratic Dirac-Poisson system with non-periodic potential. (English) Zbl 1475.35008

Summary: In this paper we study a class of Dirac-Poisson system with non-periodic potential. The Dirac operator is unbounded from below and above, so the associated energy functional is strongly indefinite. By using linking theorem for strongly indefinite functionals, we obtain a new multiplicity result of solutions when the nonlinearity has asymptotic quadratic growth.


35A15 Variational methods applied to PDEs
35J61 Semilinear elliptic equations
35Q41 Time-dependent Schrödinger equations and Dirac equations
Full Text: DOI


[1] Thaller, B., (The Dirac Equation. The Dirac Equation, Texts and Monographs in Physics (1992), Springer: Springer Berlin)
[2] Ding, Y., Variational Methods for Strongly Indefinite Problems (2008), World Scientific: World Scientific Press
[3] Esteban, M. J.; Lewin, M.; Séré, E., Variational methods in relativistic quantum mechanics, Bull. Amer. Math. Soc., 45, 535-593 (2008) · Zbl 1288.49016
[4] 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
[5] Abenda, S., Solitary waves for Maxwell-Dirac and Coulomb-Dirac models, Ann. Inst. Henri. Poincaré, 68, 229-244 (1998) · Zbl 0907.35104
[6] Chen, G.; Zheng, Y., Stationary solutions of non-autonomous Maxwell-Dirac systems, J. Differential Equations, 255, 840-864 (2013) · Zbl 1281.49040
[7] Zhang, J.; Tang, X.; 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] Bartsch, T.; Ding, Y., Deformation theorems on non-metrizable vector spaces and applications to critical point theory, Math. Nachr., 279, 1267-1288 (2006) · Zbl 1117.58007
[9] Zhang, J.; Tang, X.; Zhang, W., Ground state solutions for a class of nonlinear Maxwell-Dirac system, Topol. Methods Nonlinear Anal., 46, 785-798 (2015) · Zbl 1375.35425
[10] Zhang, W.; Zhang, J.; Jiang, W., Infinitely many solutions for a class of superlinear Dirac-Poisson system, Appl. Math. Lett., 80, 79-87 (2018) · Zbl 1394.35391
[11] Zhang, J.; Tang, X.; Zhang, W., Existence and multiplicity of solutions for nonlinear Dirac-Poisson systems, Electron. J. Differential Equations, 91, 1-17 (2017) · Zbl 1370.49005
[12] Ding, Y.; Wei, J.; Xu, T., Existence and concentration of semi-classical solutions for a nonlinear Maxwell-Dirac system, J. Math. Phys., 54, Article 061505 pp. (2013) · Zbl 1282.81073
[13] Ding, Y.; 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
[14] Zhang, J.; Zhang, W.; Tang, X., Ground state solutions for Hamiltonian elliptic system with inverse square potential, Discrete Contin. Dyn. Syst., 37, 4565-4583 (2017) · Zbl 1370.35111
[15] Zhang, J.; Zhang, W.; Xie, X., Infinitely many solutions for a gauged nonlinear Schrödinger equation, Appl. Math. Lett., 88, 21-27 (2019) · Zbl 1411.35098
[16] Zhang, J.; Qin, W.; Zhao, F., Multiple solutions for a class of nonperiodic Dirac equations with vector potentials, Nonlinear Anal., 75, 5589-5600 (2012) · Zbl 1253.81059
[17] Ackermann, N., A Cauchy-Schwarz type inequality for bilinear integrals on positive measures, Proc. Amer. Math. Soc., 133, 2647-2656 (2005) · Zbl 1066.26013
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.