Existence of solutions for nonlinear Dirac equations in the Bopp-Podolsky electrodynamics

In this paper, we study the following nonlinear Dirac-Bopp-Podolsky system \begin{equation*} \left\lbrace \begin{array}{rll} \displaystyle{ -i\sum_{k=1}^{3}α_{k}\partial_{k}u+[V(x)+q]βu+wu-φu}&=f(x,u), \ \ &$\text{in}$\ $\mathbb{R}$^3, \ & \ & \ -\triangleφ+a^2\triangle^2 φ&=4π\vert u\vert^2,\ \ & $\text{in}$\ $\mathbb{R}$^3, \end{array} \right. \end{equation*} where $a,q>0,w\in \mathbb{R}$, $V(x)$ is a potential function, and $f(x,u)$ is the interaction term (nonlinearity). First, we give a physical motivation for this new kind of system. Second, under suitable assumptions on $f$ and $V$, and by means of minimax techniques involving Cerami sequences, we prove the existence of at least one pair of solutions $(u,{\phi }_u)$.