Existence and multiplicity of solutions for nonlinear Dirac-Poisson systems

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