Ground state solutions for a class of nonlinear Maxwell-Dirac system

Abstract

This paper is concerned with the following nonlinear Maxwell-Dirac system \begin{equation*} \begin{cases} \displaystyle -i\sum^{3}_{k=1}\alpha_{k}\partial_{k}u + a\beta u + \omega u-\phi u =F_{u}(x,u), \\ -\Delta \phi=4\pi|u|^{2},\\ \end{cases} \end{equation*} for $x\in\mathbb R^{3}$. The Dirac operator is unbounded from below and above, so the associated energy functional is strongly indefinite. We use the linking and concentration compactness arguments to establish the existence of ground state solutions for this system with asymptotically quadratic nonlinearity.