# American Institute of Mathematical Sciences

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
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##### References:
 [1] Noboru Okazawa, Kentarou Yoshii. Linear evolution equations with strongly measurable families and application to the Dirac equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 723-744. doi: 10.3934/dcdss.2011.4.723 [2] Alireza Khatib, Liliane A. Maia. A positive bound state for an asymptotically linear or superlinear Schrödinger equation in exterior domains. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2789-2812. doi: 10.3934/cpaa.2018132 [3] Jiaquan Liu, Yuxia Guo, Pingan Zeng. Relationship of the morse index and the $L^\infty$ bound of solutions for a strongly indefinite differential superlinear system. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 107-119. doi: 10.3934/dcds.2006.16.107 [4] Federico Cacciafesta, Anne-Sophie De Suzzoni. Weak dispersion for the Dirac equation on asymptotically flat and warped product spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4359-4398. doi: 10.3934/dcds.2019177 [5] Shiwang Ma. Nontrivial periodic solutions for asymptotically linear hamiltonian systems at resonance. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2361-2380. doi: 10.3934/cpaa.2013.12.2361 [6] Paolo Gidoni, Alessandro Margheri. Lower bound on the number of periodic solutions for asymptotically linear planar Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 585-606. doi: 10.3934/dcds.2019024 [7] D. Motreanu, V. V. Motreanu, Nikolaos S. Papageorgiou. Nonautonomous resonant periodic systems with indefinite linear part and a nonsmooth potential. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1401-1414. doi: 10.3934/cpaa.2011.10.1401 [8] D. Motreanu, V. V. Motreanu, Nikolaos S. Papageorgiou. Two nontrivial solutions for periodic systems with indefinite linear part. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 197-210. doi: 10.3934/dcds.2007.19.197 [9] Guglielmo Feltrin. Positive subharmonic solutions to superlinear ODEs with indefinite weight. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 257-277. doi: 10.3934/dcdss.2018014 [10] Julián López-Gómez, Marcela Molina-Meyer, Andrea Tellini. Spiraling bifurcation diagrams in superlinear indefinite problems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1561-1588. doi: 10.3934/dcds.2015.35.1561 [11] Addolorata Salvatore. Sign--changing solutions for an asymptotically linear Schrödinger equation. Conference Publications, 2009, 2009 (Special) : 669-677. doi: 10.3934/proc.2009.2009.669 [12] Guangze Gu, Xianhua Tang, Youpei Zhang. Ground states for asymptotically periodic fractional Kirchhoff equation with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3181-3200. doi: 10.3934/cpaa.2019143 [13] Yemin Chen. Analytic regularity for solutions of the spatially homogeneous Landau-Fermi-Dirac equation for hard potentials. Kinetic & Related Models, 2010, 3 (4) : 645-667. doi: 10.3934/krm.2010.3.645 [14] M. Gaudenzi, P. Habets, F. Zanolin. Positive solutions of superlinear boundary value problems with singular indefinite weight. Communications on Pure & Applied Analysis, 2003, 2 (3) : 411-423. doi: 10.3934/cpaa.2003.2.411 [15] Andrea Tellini. Imperfect bifurcations via topological methods in superlinear indefinite problems. Conference Publications, 2015, 2015 (special) : 1050-1059. doi: 10.3934/proc.2015.1050 [16] Guglielmo Feltrin. Existence of positive solutions of a superlinear boundary value problem with indefinite weight. Conference Publications, 2015, 2015 (special) : 436-445. doi: 10.3934/proc.2015.0436 [17] Julián López-Gómez, Pavol Quittner. Complete and energy blow-up in indefinite superlinear parabolic problems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 169-186. doi: 10.3934/dcds.2006.14.169 [18] Maria J. Esteban, Eric Séré. An overview on linear and nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 381-397. doi: 10.3934/dcds.2002.8.381 [19] Xiaoyan Lin, Yubo He, Xianhua Tang. Existence and asymptotic behavior of ground state solutions for asymptotically linear Schrödinger equation with inverse square potential. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1547-1565. doi: 10.3934/cpaa.2019074 [20] José Godoy, Nolbert Morales, Manuel Zamora. Existence and multiplicity of periodic solutions to an indefinite singular equation with two singularities. The degenerate case. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4137-4156. doi: 10.3934/dcds.2019167

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