This paper deals with the following system
\begin{equation*}
\left\{\begin{aligned}
&{-\Delta u+ (\lambda
A(x)+1)u-(2\omega+\phi)
\phi u=\mu f(u)+u^{5}}, & &
{\quad x \in
\mathbb{R}^{3}}, \\
&{\Delta
\phi=(\omega+\phi)
u^{2}}, & & {\quad x \in
\mathbb{R}^{3}},
\end{aligned}\right.
\end{equation*} where $\lambda,
\mu>0$ are positive parameters. Under some
suitable conditions on $A$ and $f$, we show the boundedness of
Cerami sequence for the above system by adopting
Poho\v{z}aev identity and then prove the existence of
ground state solution for the above system on Nehari manifold by using
Br\’{e}zis-Nirenberg technique, which improve the
existing result in the literature.