一类带局部非线性项的静态狄拉克方程的多重周期解

Abstract

Abstract:

In this paper, we mainly study multiple periodic solutions for a class of stationary Dirac equations with local nonlinearity via variational methods. Under some fairly mild conditions on the nonlinearity, we show that the existence of a sequence of nontrivial periodic solutions is rather a local phenomenon, which is only forced by the sublinearity of the nonlinearity near the origin. Meanwhile, some recent results from the literature are generalized and essentially improved.

Key words: Dirac equation, Periodic solution, Variational method, Local nonlinearity

CLC Number:

O175

Qingye Zhang,Bin Xu. Multiple Periodic Solutions for a Class of Stationary Dirac Equations with Local Nonlinearity[J].Acta mathematica scientia,Series A, 2021, 41(4): 1013-1023.

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References 23

1	Thaller B . The Dirac Equation. Berlin: Springer, 1992
2	Esteban M , Séré E . Stationary states of the nonlinear Dirac equation: a variational approach. Comm Math Phys, 1995, 171 (2): 323- 350 doi: 10.1007/BF02099273
3	Merle F . Existence of stationary states for nonlinear Dirac equations. J Differential Equations, 1988, 74 (1): 50- 68 doi: 10.1016/0022-0396(88)90018-6
4	Esteban M , Séré E . An overview on linear and nonlinear Dirac equations. Discrete Contin Dyn Syst, 2002, 8 (2): 381- 397 doi: 10.3934/dcds.2002.8.381
5	Bartsch T , Ding Y . Solutions of nonlinear Dirac equations. J Differential Equations, 2006, 226 (1): 210- 249 doi: 10.1016/j.jde.2005.08.014
6	Ding Y . Variational Methods for Strongly Indefinite Problems. Singapore: World Scientific Publ, 2007
7	Ding Y , Ruf B . Solutions of a nonlinear Dirac equation with external fields. Arch Ration Mech Anal, 2008, 190 (1): 57- 82 doi: 10.1007/s00205-008-0163-z
8	Ding Y , Wei J . Stationary states of nonlinear Dirac equations with general potentials. Rev Math Phys, 2008, 20 (8): 1007- 1032 doi: 10.1142/S0129055X0800350X
9	Zhao F , Ding Y . Infinitely many solutions for a class of nonlinear Dirac equations without symmetry. Nonlinear Anal, 2009, 70 (2): 921- 935 doi: 10.1016/j.na.2008.01.022
10	Ding J , Xu J , Zhang F . Solutions of non-periodic super-quadratic Dirac equations. J Math Anal Appl, 2010, 366 (1): 266- 282 doi: 10.1016/j.jmaa.2010.01.022
11	Ding Y , Liu X . Semi-classical limits of ground states of a nonlinear Dirac equation. J Differential Equations, 2012, 252 (9): 4962- 4987 doi: 10.1016/j.jde.2012.01.023
12	Ding Y , Ruf B . Existence and concentration of semiclassical solutions for Dirac equations with critical nonlinearities. SIAM J Math Anal, 2012, 44 (6): 3755- 3785 doi: 10.1137/110850670
13	Yang M , Ding Y . Stationary states for nonlinear Dirac equations with superlinear nonlinearities. Topol Methods Nonlinear Anal, 2012, 39 (1): 175- 188
14	Zhang J , Qin W , Zhao F . Multiple solutions for a class of nonperiodic Dirac equations with vector potentials. Nonlinear Anal, 2012, 75 (14): 5589- 5600 doi: 10.1016/j.na.2012.05.006
15	Zhang J , Tang X , Zhang W . Infinitely many large energy solutions for superlinear Dirac equations. Math Methods Appl Sci, 2015, 38 (8): 1485- 1493 doi: 10.1002/mma.3162
16	Figueiredo G , Pimenta M . Existence of ground state solutions to Dirac equations with vanishing potentials at infinity. J Differential Equations, 2017, 262 (1): 486- 505 doi: 10.1016/j.jde.2016.09.034
17	Lin X , Tang X . Solutions of nonlinear periodic Dirac equations with periodic potentials. Discrete Contin Dyn Syst Ser S, 2019, 12 (7): 2051- 2061
18	Ding Y , Liu X . Periodic waves of nonlinear Dirac equations. Nonlinear Anal, 2014, 109, 252- 267 doi: 10.1016/j.na.2014.06.015
19	Ding Y , Liu X . Periodic solutions of a Dirac equation with concave and convex nonlinearities. J Differential Equations, 2015, 258 (10): 3567- 3588 doi: 10.1016/j.jde.2015.01.013
20	Zhang J , Zhang Q , Tang X , Zhang W . Periodic solutions for nonlinear Dirac equation with superquadratic nonlinearity. Electron J Differ Equ, 2016, 2016 (141): 1- 12
21	Ding Y , Liu X . Periodic solutions of an asymptotically linear Dirac equation. Ann Mat Pura Appl, 2017, 196 (2): 717- 735 doi: 10.1007/s10231-016-0592-5
22	Ding Y , Liu X . Periodic solutions of superlinear Dirac equations with perturbations from symmetry. J Math Phys, 2018, 59 (1): 011504 doi: 10.1063/1.5021688
23	Liu Z , Wang Z Q . On Clark's theorem and its applications to partially sublinear problems. Ann Inst H Poincaré Anal Non Linéaire, 2015, 32 (5): 1015- 1037 doi: 10.1016/j.anihpc.2014.05.002

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Multiple Periodic Solutions for a Class of Stationary Dirac Equations with Local Nonlinearity

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