Electron. J. Differential Equations, Vol. 2017 (2017), No. 91, pp. 1-17.

Existence and multiplicity of solutions for nonlinear Dirac-Poisson systems

Jian Zhang, Wen Zhang, Xianhua Tang

Abstract:
This article concerns the nonlinear Dirac-Poisson system
$$\displaylines{ -i\sum^3_{k=1}\alpha_{k}\partial_{k}u + (V(x)+a)\beta u + \omega u-\phi u =F_u(x,u),\cr -\Delta \phi=4\pi|u|^2, }$$
in $\mathbb{R}^3$, where V(x) is a potential function and F(x,u) is an asymptotically quadratic nonlinearity modeling various types of interaction. Since the effects of the nonlocal term, we use some special techniques to deal with the nonlocal term. Moreover, the existence of infinitely many stationary solutions is obtained for system with periodicity assumption via variational methods.

Submitted December 17, 2016. Published March 29, 2017.
Math Subject Classifications: 49J35, 35Q40, 81V10.
Key Words: Dirac-Poisson system; asymptotically quadratic; variational methods; strongly indefinite functionals.

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Jian Zhang
School of Mathematics and Statistics
Hunan University of Commerce
Changsha, 410205 Hunan, China
email: zhangjian433130@163.com
Wen Zhang
School of Mathematics and Statistics
Hunan University of Commerce
Changsha, 410205 Hunan, China
email: zwmath2011@163.com
Xianhua Tang
School of Mathematics and Statistics
Central South University
Changsha, 410083 Hunan, China
email: tangxh@mail.csu.edu.cn

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