  In this paper, we study the following system \begin{eqnarray*}
  \left\{ \begin{array}{ll}
  -\Delta u + V(x)u-(2\omega+\phi)\phi u=\lambda f(u)+|u|^{4}u, \ & \text{in} \
\mathbb{R}^{3},
  \Delta \phi + \beta\Delta_4\phi = 4\pi(\omega+\phi) u^{2}, \ & \text{in}\
\mathbb{R}^{3},\\ \end{array} \right. \end{eqnarray*} where $f(u)$ without any
growth and Ambrosetti-Rabinowitz conditions. We use cut-off function and Moser
iteration to obtain the existence of nontrivial solution. Finally, as a
by-product of our approaches, we get the same result for Klein-Gordon-Maxwell
system.
