November  2019, 12(7): 2051-2061. 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, 2019, 12 (7) : 2051-2061. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[18]

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

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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