Solutions of nonlinear periodic Dirac equations with periodic potentials

Xiaoyan Lin; Xianhua Tang

doi:10.3934/dcdss.2019132

Article Contents

2019, Volume 12, Issue 7: 2051-2061. Doi: 10.3934/dcdss.2019132

This issue Previous Article Existence of positive ground state solutions for Choquard equation with variable exponent growth Next Article A Leslie-Gower predator-prey model with a free boundary

Solutions of nonlinear periodic Dirac equations with periodic potentials

Xiaoyan Lin^1, , and
Xianhua Tang^2,

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
* Corresponding author: Xiaoyan Lin

Received: December 2017

Revised: April 2018

Published online: November 2019

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 35J20; Secondary: 35Q55.

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