  In this paper, we study the following Schr\"odinger-Poisson system: $$
\left\{\aligned&-\Delta u+V_\lambda(x)u+K(x)\phi u=f(x,u)&\quad\text{in
}\bbr^3,\\ &-\Delta\phi=K(x)u^2&\quad\text{in }\bbr^3,\\
&(u,\phi)\in\h\times\D,\endaligned\right.\eqno{(\mathcal{SP}_{\lambda})} $$
where $V_\lambda(x)=\lambda a(x)+b(x)$ with a positive parameter $\lambda$,
$K(x)\geq0$ and $f(x,t)$ is continuous including the power-type nonlinearity
$|u|^{p-2}u$. By applying the method of penalized functions, the existence of
one nontrivial solution for such system in the less-studied case $3<p\leq4$ is
obtained for $\lambda$ sufficiently large. The concentration behavior of this
nontrivial solution for $\lambda\to+\infty$ are also observed. It is worth to
point out that some new conditions on the potentials are introduced to obtain
this nontrivial solution and the Schr\"odinger operator $-\Delta+V_\lambda(x)$
may be strong indefinite in this paper.
