Ground state solutions for a class of nonlinear Maxwell-Dirac system

Jian Zhang, Xianhua Tang, Wen Zhang

DOI: http://dx.doi.org/10.12775/TMNA.2015.068

Abstract


This paper is concerned with the following nonlinear Maxwell-Dirac system
\begin{equation*}
\begin{cases}
\displaystyle
-i\sum^{3}_{k=1}\alpha_{k}\partial_{k}u + a\beta u + \omega u-\phi u =F_{u}(x,u),
\\
-\Delta \phi=4\pi|u|^{2,\\
\end{cases}
 \end{equation*}
for $x\in\R^{3}$. The Dirac operator is unbounded from below and above, so the associated energy functional is strongly indefinite. We use the linking and concentration compactness arguments to establish the existence of ground state solutions
for this system with asymptotically quadratic nonlinearity.


Keywords


Maxwell-Dirac system; Ground state solutions; Asymptotically quadratic; Strongly indefinite functionals

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References


S. Abenda, Solitary waves for Maxwell-Dirac and Coulomb-Dirac models, Ann. Inst. H. Poincare 68 (1998), 229-244.

A. Ambrosetti, On Schrodinger-Poisson systems, Milan J. Math. 76 (2008), 257-274.

N. Ackermann, On a periodic Schrodinger equation with nonlocal superlinear part, Math. Z. 248 (2004), 423-443.

N. Ackermann, A Cauchy-Schwarz type inequality for bilinear integrals on positive measures, Proc. Amer. Math. Soc. 133 (2005), 2647-2656.

A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrodinger-Maxwell equations, J. Math. Anal. Appl. 345 (2008), 90-108.

T. Bartsch and Y.H. Ding, Deformation theorems on non-metrizable vector spaces and applications to critical point theory, Math. Nachr. 279 (2006), 1267-1288.

T. Bartsch and Y.H. Ding, Solutions of nonlinear Dirac equations, J. Differential Equations 226 (2006), 210-249.

J. M. Chadam, Global solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac system in one space dimension, J. Funct. Anal. 13 (1973), 173-184.

J. M. Chadam and R.T. Glassey, On the Maxwell-Dirac equations with zero magnetic field and their solutions in two space dimension, J. Math. Anal. Appl. 53 (1976), 495-507.

T. Cazenave and L. Vazquez, Existence of local solutions of a classical nonlinear Dirac field, Comm. Math. Phys. 105 (1986), 35-47.

G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrodinger-Poisson systems, J. Differential Equations 248 (2010), 521-543.

G. Y. Chen and Y. Q. Zheng, Stationary solutions of non-autonomous Maxwell-Dirac systems, J. Differential Equations 255 (2013), 840-864.

Y. H. Ding, Semi-classical ground states concentrating on the nonlinear potential for a Dirac equation, J. Differential Equations 249 (2010), 1015-1034.

Y. H. Ding, Variational Methods for Strongly Indefinite Problems, World Scientific Press, 2008.

Y. H. Ding and X. Y. Liu, Semi-classical limits of ground states of a nonlinear Dirac equation, J. Differential Equations 252 (2012), 4962-4987.

Y. H. Ding and X. Y. Liu, On semiclassical ground states of a nonlinear Dirac equation, Rev. Math. Phys. 10 (2012), 1250029.

Y. H. Ding and B. Ruf, Solutions of a nonlinear Dirac equation with external fields, Arch. Rational Mech. Anal. 190 (2008), 57-82.

Y. H. Ding and B. Ruf, Existence and concentration of semi-classical solutions for Dirac equations with critical nonlinearites, SIAM J. Math. Anal. 44 (2012), 3755-3785.

Y. H. Ding and J. C. Wei, Stationary states of nonlinear Dirac equations with general potentials, Rev. Math. Phys. 20 (2008), 1007-1032.

Y. H. Ding, J. C. Wei and T. Xu, Existence and concentration of semi-classical solutions for a nonlinear Maxwell-Dirac system, J. Math. Phys. 54 (2013), 061505.

Y. H. Ding and T. Xu, On the concentration of semi-classical states for an onlinear Dirac-Klein-Gordon system, J. Differential Equations 256 (2014), 1264-1294.

Y. H. Ding and T. Xu, On semi-classical limits of ground states of a nonlinear Maxwell-Dirac system, Calc. Var. Partial Differentail Equations 51 (2014), 17-44.

M. J. Esteban, V. Georgiev and E. Sere, Stationary solutions of the Maxwell-Dirac and the Klein-Gordon-Dirac equations, Calc. Var. Partial Differential Equations 4 (1996),

-281.

M. J. Esteban, M. Lewin and E. Sere, Variational methods in relativistic quantum mechanics, Bull. Amer. Math. Soc. 45 (2008), 535-593.

M. J. Esteban and E. Sere, Stationary states of nonlinear Dirac equations: A variational approach, Comm. Math. Phys. 171 (1995), 323-350.

M. J. Esteban and E. Sere, An overview on linear and nonlinear Dirac equations, Discrete Cotin. Dyn. Syst. 8 (2002), 281-397.

M. Flato, J. Simon and E. Taffin, On the global solutions of the Maxwell-Dirac equations, Commun. Math. Phys. 113 (1987), 21-49.

A. Garrett Lisi, A solitary wave solution of the Maxwell-Dirac equations, J. Phys. A Math. Gen. 28 (1995), 5385-5392.

V. Georgiev, Small amplitude solutions of Maxwell-Dirac equations, Indiana Univ. Math. J. 40 (1991), 845-883.

R. T. Glassey and J. M. Chadam, Properties of the solutions of the Cauchy problem for the classical coupled Maxwell-Dirac equations in one space dimension, Proc. Amer. Math.

Soc. 43 (1974), 373-378.

W. T. Grandy Jr, Relativistic Quantum Mechanics of Leptonsand Fields, Fundam. Theor. Phys., vol. 41, Kluwer Academic Publishers Group, Dordrecht, 1991.

L. Gross, The Cauchy problem for the coupled Maxwell-Dirac equations, Commun. Pure Appl. Math. 19 (1966), 1-5.

W. Kryszewki and A. Szulkin, Generalized linking theorem with an application to semilinear Schrodinger equation, Adv. Differential Equations 3 (1998), 441-472.

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, part 2, Ann. Inst. H. Poincare Anal. Non Lineaire 1 (1984), 223-283.

F. Merle, it Existence of stationary states for nonlinear Dirac equations, J. Differential Equations 74 (1988), 50-68.

C. J. Radford, The stationary Maxwell-Dirace quations, J. Phys. A Math. Gen. 36 (2003), 5663-5681.

D. Ruiz, The Schrodinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal. 237 (2006), 665-674.

C. Sparber and P. Markowich, Semiclassical asymptotics for the Maxwell-Dirac system, J. Math. Phys. 44 (2003), 4555-4572.

B. Thaller, The Dirac Equation, Texts and Monographs in Physics, Springer, Berlin, 1992.

X. H. Tang, New super-quadratic conditions on ground state solutions for superlinear Schrodinger equation, Adv. Nonlinear Studies 14 (2014), 349-361.

X. H. Tang, New condition on nonlinearity for a periodic Schrodinger equation having zero as spectrum, J. Math. Anal. Appl. 413 (2014), 392-410.

M. Willem, Minimax Theorems, Birkhauser, Berlin, 1996.

M. B. Yang and Y. H. Ding, Stationary states for nonlinear Dirac equations with superlinear nonlinearities, Topol. Methods Nonlinear Anal. 39 (2012), 175-188.

F. K. Zhao and Y. H. Ding, Infinitely many solutions for a class of nonlinear Dirac equations without symmetry, Nonlinear Anal. 70 (2009), 921-935.

J. Zhang, W. P. Qin and F. K. Zhao, Multiple solutions for a class of nonperiodic Dirac equations with vector potentials, Nonlinear Anal. 75 (2012), 5589-5600.

J. Zhang, X. H. Tang and W. Zhang, Ground states for nonlinear Maxwell-Dirac system with magnetic field, J. Math. Anal. Appl. 421 (2015), 1573-1585.

J. Zhang, X. H. Tang and W. Zhang, Ground state solutions for nonperiodic Dirac equation with superquadratic nonlinearity, J. Math. Phys. 54 (2013), 101502.

J. Zhang, X. H. Tang and W. Zhang, On ground state solutions for superlinear Dirac equation, Acta Math. Sci. 34 (2014), 840-850.


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