Existence and exponential decay of ground-state solutions for a nonlinear Dirac equation

In this paper, we study the following nonlinear Dirac equation $$\begin{aligned} -i\sum _{k=1}^{3}\alpha _{k}\partial _{k}u+a\beta u+V(x)u=f(x,|u|)u,~~x\in \mathbb {R}^{3}. \end{aligned}$$ The Dirac operator is unbounded from below and above so the associate energy functional is strongly indefinite. Under some suitable conditions on the potential and nonlinearity, we obtain the existence of ground-state solutions in periodic case and asymptotically periodic case via variational methods, respectively. Moreover, we also explore some properties of these ground-state solutions, such as compactness of set of ground-state solutions and exponential decay of ground-state solutions.