doi: 10.3934/dcdss.2019132

Solutions of nonlinear periodic Dirac equations with periodic potentials

1. 

Department of Mathematics, Huaihua College, Huaihua, Hunan 418008, China

2. 

School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China

* Corresponding author: Xiaoyan Lin

Received  December 2017 Revised  April 2018 Published  December 2018

Fund Project: This work is partially supported by the NNFC (No: 11471137) of China and by Hunan Provincial Natural Science Foundation (No:2017JJ22) of China

This paper is concerned with the nonlinear Dirac equation $ -i\sum_{k = 1}^{3}\alpha_{k}\partial_{k}u + [V(x)+a]\beta u + \omega u = f(x, u) $ in $ \mathbb{R}^3 $, where $ V(x) $ and $ f(x, u) $ are periodic in $ x $, $ f(x, u) $ is asymptotically linear and superlinear as $ |u|\rightarrow \infty $. Under weaker assumptions on $ f $, we obtain the existence of one nontrivial solution for the above equation.

Citation: Xiaoyan Lin, Xianhua Tang. Solutions of nonlinear periodic Dirac equations with periodic potentials. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2019132
References:
[1]

M. BalabaneT. CazenaveA. Douady and F. Merle, Existence of excited states for a nonlinear Dirac field, Commun. Math. Phys., 119 (1988), 153-176. doi: 10.1007/BF01218265.

[2]

M. BalabaneT. Cazenave and L. Vazquez, Existence of standing waves for Dirac fields with singular nonlinearities, Commun. Math. Phys., 133 (1990), 53-74. doi: 10.1007/BF02096554.

[3]

T. Bartsch and Y. H. Ding, Solutions of nonlinear Dirac equations, J. Differ. Equations, 226 (2006), 210-249. doi: 10.1016/j.jde.2005.08.014.

[4]

J. D. Bjorken and S. D. Drell, Relativistic Quantum Fields, McGraw-Hill, 1965.

[5]

B. Booss-Bavnbek, Unique continuation property for Dirac operator, revisited, Contemp. Math., 258 (2000), 21-32. doi: 10.1090/conm/258/04053.

[6]

T. Cazenave and L. Vazquez, Existence of local solutions of a classical nonlinear Dirac field, Commun. Math. Phys., 105 (1986), 35-47. doi: 10.1007/BF01212340.

[7]

S. T. Chen and X. H. Tang, Improved results for Klein-Gordon-Maxwell systems with general nonlinearity, Discrete Contin Dyn Syst-A, 38 (2018), 2333-2348. doi: 10.3934/dcds.2018096.

[8]

Y. H. Ding, Variational Methods for Strongly Indefinite Problems, World Scientific, Singapore, 2007. doi: 10.1142/9789812709639.

[9]

Y. H. Ding, Semi-classical ground states concentrating on the nonlinear potential for a Dirac equation, J. Differ. Equations, 249 (2010), 1015-1034. doi: 10.1016/j.jde.2010.03.022.

[10]

Y. H. Ding and X. Y. Liu, Semi-classical limits of ground states of a nonlinear Dirac equation, J. Differ. Equations, 252 (2012), 4962-4987. doi: 10.1016/j.jde.2012.01.023.

[11]

Y. H. Ding and X. Y. Liu, Semi-classical limits of ground states of a nonlinear Dirac equation, Rev. Math. Phys., 24 (2012), 1250029, 25pp. doi: 10.1142/S0129055X12500298.

[12]

Y. H. Ding and B. Ruf, Solutions of a nonlinear Dirac equation with external fields, Arch. Ration. Mech. Anal., 190 (2008), 57-82. doi: 10.1007/s00205-008-0163-z.

[13]

Y. H. Ding and B. Ruf, Existence and concentration of semi-classical solutions for Dirac equations with critical nonlinearities, SIAM J. Math. Anal., 44 (2012), 3755-3785. doi: 10.1137/110850670.

[14]

Y. H. Ding and J. C. Wei, Stationary states of nonlinear Dirac equations with general potentials, Rev. Math. Phys., 20 (2008), 1007-1032. doi: 10.1142/S0129055X0800350X.

[15] D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1987.
[16]

M. J. Esteban and E. Séré, Stationary states of nonlinear Dirac equations: A variational approach, Commun. Math. Phys., 171 (1995), 323-350. doi: 10.1007/BF02099273.

[17]

M. J. Esteban and E. Séré, An overview on linear and nonlinear Dirac equations, Discrete Contin. Dyn. Syst., 8 (2002), 381-397. doi: 10.3934/dcds.2002.8.381.

[18]

W. Kryszewski, Generalized linking theorem with an application to semilinear Schrödinger equations, Adv. Differential Equations, 3 (1998), 441-472.

[19]

G. B. Li and A. Szulkin, An asymptotically periodic equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776. doi: 10.1142/S0219199702000853.

[20]

X. Y. Lin and and X. H. Tang, An asymptotically periodic and asymptotically linear Schrödinger equation with indefinite linear part, Comput. Math. Appl., 70 (2015), 726-736. doi: 10.1016/j.camwa.2015.06.013.

[21]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, part 2, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283. doi: 10.1016/S0294-1449(16)30422-X.

[22]

F. Merle, Existence of stationary states for nonlinear Dirac equations, J. Differ. Equations, 74 (1988), 50-68. doi: 10.1016/0022-0396(88)90018-6.

[23]

B. Thaller, The Dirac Equation, Texts and Monographs in Physics, Springer, Berlin, 1992. doi: 10.1007/978-3-662-02753-0.

[24]

X. H. Tang, New Super-quadratic Conditions for asymptotically periodic Schrödinger equations, Canad. Math. Bull., 60 (2017), 422-435. doi: 10.4153/CMB-2016-090-2.

[25]

X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potential, Disc. Contin. Dyn. Syst., 37 (2017), 4973-5002. doi: 10.3934/dcds.2017214.

[26]

X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohožaev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differential Equations, 56 (2017), Art. 110, 25 pp. doi: 10.1007/s00526-017-1214-9.

[27]

X. H. Tang, X. Y. Lin and J. S. Yu, Nontrivial solutions for Schrödinger equation with local super-quadratic conditions, J. Dyn. Differ. Equ, (2018), 1-15. doi: 10.1007/s10884-018-9662-2.

[28]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.

[29]

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

[30]

F. K. Zhao and Y. H. Ding, Stationary states for nonlinear Dirac equations with superlinear nonlinearities, Nonlinear Anal.TMA, 70 (2009), 921-935. doi: 10.1016/j.na.2008.01.022.

[31]

J. ZhangW. P. Qin and F. K. Zhao, Multiple solutions for a class of nonperiodic Dirac equations with vector potentials, Nonlinear Anal.TMA, 75 (2012), 5589-5600. doi: 10.1016/j.na.2012.05.006.

[32]

J. Zhang, X. H. Tang and W. Zhang, Ground state solutions for nonperiodic Dirac equation with superquadratic nonlinearity, J. Math. Phys., 54 (2013), 101502, 10pp. doi: 10.1063/1.4824132.

[33]

J. ZhangX. H. Tang and W. Zhang, On ground state solutions for superlinear Dirac equation, Acta Math. Scientia, 34 (2014), 840-850. doi: 10.1016/S0252-9602(14)60054-0.

[34]

J. ZhangX. H. Tang and W. Zhang, Ground states for nonlinear Maxwell-Dirac system with magnetic field, J. Math. Anal. Appl., 421 (2015), 1573-1586. doi: 10.1016/j.jmaa.2014.08.009.

[35]

J. ZhangX. H. Tang and W. Zhang, Existence and multiplicity of stationary solutions for a class of Maxwell-Dirac system, Nonlinear Anal., 127 (2015), 298-311. doi: 10.1016/j.na.2015.07.010.

[36]

J. ZhangX. H. Tang and W. Zhang, Ground state solutions for a class of nonlinear Maxwell-Dirac system, Topol. Meth. Nonl. Anal., 46 (2015), 785-798.

show all references

References:
[1]

M. BalabaneT. CazenaveA. Douady and F. Merle, Existence of excited states for a nonlinear Dirac field, Commun. Math. Phys., 119 (1988), 153-176. doi: 10.1007/BF01218265.

[2]

M. BalabaneT. Cazenave and L. Vazquez, Existence of standing waves for Dirac fields with singular nonlinearities, Commun. Math. Phys., 133 (1990), 53-74. doi: 10.1007/BF02096554.

[3]

T. Bartsch and Y. H. Ding, Solutions of nonlinear Dirac equations, J. Differ. Equations, 226 (2006), 210-249. doi: 10.1016/j.jde.2005.08.014.

[4]

J. D. Bjorken and S. D. Drell, Relativistic Quantum Fields, McGraw-Hill, 1965.

[5]

B. Booss-Bavnbek, Unique continuation property for Dirac operator, revisited, Contemp. Math., 258 (2000), 21-32. doi: 10.1090/conm/258/04053.

[6]

T. Cazenave and L. Vazquez, Existence of local solutions of a classical nonlinear Dirac field, Commun. Math. Phys., 105 (1986), 35-47. doi: 10.1007/BF01212340.

[7]

S. T. Chen and X. H. Tang, Improved results for Klein-Gordon-Maxwell systems with general nonlinearity, Discrete Contin Dyn Syst-A, 38 (2018), 2333-2348. doi: 10.3934/dcds.2018096.

[8]

Y. H. Ding, Variational Methods for Strongly Indefinite Problems, World Scientific, Singapore, 2007. doi: 10.1142/9789812709639.

[9]

Y. H. Ding, Semi-classical ground states concentrating on the nonlinear potential for a Dirac equation, J. Differ. Equations, 249 (2010), 1015-1034. doi: 10.1016/j.jde.2010.03.022.

[10]

Y. H. Ding and X. Y. Liu, Semi-classical limits of ground states of a nonlinear Dirac equation, J. Differ. Equations, 252 (2012), 4962-4987. doi: 10.1016/j.jde.2012.01.023.

[11]

Y. H. Ding and X. Y. Liu, Semi-classical limits of ground states of a nonlinear Dirac equation, Rev. Math. Phys., 24 (2012), 1250029, 25pp. doi: 10.1142/S0129055X12500298.

[12]

Y. H. Ding and B. Ruf, Solutions of a nonlinear Dirac equation with external fields, Arch. Ration. Mech. Anal., 190 (2008), 57-82. doi: 10.1007/s00205-008-0163-z.

[13]

Y. H. Ding and B. Ruf, Existence and concentration of semi-classical solutions for Dirac equations with critical nonlinearities, SIAM J. Math. Anal., 44 (2012), 3755-3785. doi: 10.1137/110850670.

[14]

Y. H. Ding and J. C. Wei, Stationary states of nonlinear Dirac equations with general potentials, Rev. Math. Phys., 20 (2008), 1007-1032. doi: 10.1142/S0129055X0800350X.

[15] D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1987.
[16]

M. J. Esteban and E. Séré, Stationary states of nonlinear Dirac equations: A variational approach, Commun. Math. Phys., 171 (1995), 323-350. doi: 10.1007/BF02099273.

[17]

M. J. Esteban and E. Séré, An overview on linear and nonlinear Dirac equations, Discrete Contin. Dyn. Syst., 8 (2002), 381-397. doi: 10.3934/dcds.2002.8.381.

[18]

W. Kryszewski, Generalized linking theorem with an application to semilinear Schrödinger equations, Adv. Differential Equations, 3 (1998), 441-472.

[19]

G. B. Li and A. Szulkin, An asymptotically periodic equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776. doi: 10.1142/S0219199702000853.

[20]

X. Y. Lin and and X. H. Tang, An asymptotically periodic and asymptotically linear Schrödinger equation with indefinite linear part, Comput. Math. Appl., 70 (2015), 726-736. doi: 10.1016/j.camwa.2015.06.013.

[21]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, part 2, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283. doi: 10.1016/S0294-1449(16)30422-X.

[22]

F. Merle, Existence of stationary states for nonlinear Dirac equations, J. Differ. Equations, 74 (1988), 50-68. doi: 10.1016/0022-0396(88)90018-6.

[23]

B. Thaller, The Dirac Equation, Texts and Monographs in Physics, Springer, Berlin, 1992. doi: 10.1007/978-3-662-02753-0.

[24]

X. H. Tang, New Super-quadratic Conditions for asymptotically periodic Schrödinger equations, Canad. Math. Bull., 60 (2017), 422-435. doi: 10.4153/CMB-2016-090-2.

[25]

X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potential, Disc. Contin. Dyn. Syst., 37 (2017), 4973-5002. doi: 10.3934/dcds.2017214.

[26]

X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohožaev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differential Equations, 56 (2017), Art. 110, 25 pp. doi: 10.1007/s00526-017-1214-9.

[27]

X. H. Tang, X. Y. Lin and J. S. Yu, Nontrivial solutions for Schrödinger equation with local super-quadratic conditions, J. Dyn. Differ. Equ, (2018), 1-15. doi: 10.1007/s10884-018-9662-2.

[28]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.

[29]

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

[30]

F. K. Zhao and Y. H. Ding, Stationary states for nonlinear Dirac equations with superlinear nonlinearities, Nonlinear Anal.TMA, 70 (2009), 921-935. doi: 10.1016/j.na.2008.01.022.

[31]

J. ZhangW. P. Qin and F. K. Zhao, Multiple solutions for a class of nonperiodic Dirac equations with vector potentials, Nonlinear Anal.TMA, 75 (2012), 5589-5600. doi: 10.1016/j.na.2012.05.006.

[32]

J. Zhang, X. H. Tang and W. Zhang, Ground state solutions for nonperiodic Dirac equation with superquadratic nonlinearity, J. Math. Phys., 54 (2013), 101502, 10pp. doi: 10.1063/1.4824132.

[33]

J. ZhangX. H. Tang and W. Zhang, On ground state solutions for superlinear Dirac equation, Acta Math. Scientia, 34 (2014), 840-850. doi: 10.1016/S0252-9602(14)60054-0.

[34]

J. ZhangX. H. Tang and W. Zhang, Ground states for nonlinear Maxwell-Dirac system with magnetic field, J. Math. Anal. Appl., 421 (2015), 1573-1586. doi: 10.1016/j.jmaa.2014.08.009.

[35]

J. ZhangX. H. Tang and W. Zhang, Existence and multiplicity of stationary solutions for a class of Maxwell-Dirac system, Nonlinear Anal., 127 (2015), 298-311. doi: 10.1016/j.na.2015.07.010.

[36]

J. ZhangX. H. Tang and W. Zhang, Ground state solutions for a class of nonlinear Maxwell-Dirac system, Topol. Meth. Nonl. Anal., 46 (2015), 785-798.

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