  We study, with respect to the parameter $q\neq0$, the following
Schr\"odinger-Bopp-Podolsky system in $\mathbb R^{3}$ \begin{equation*} \left\{
\begin{aligned} -&\Delta u+\omega u+q^2\phi u=|u|^{p-2}u, \\ &-\Delta
\phi+a^2\Delta^2 \phi = 4\pi u^2, \end{aligned} \right.
  \end{equation*} where $p\in(2,3], \omega>0, a\geq0$ are fixed. We prove, by
means of the fibering approach, that the system has no solutions at all for
large values of $q's$, and has two radial solutions for small $q's$. We give
also qualitative properties about the energy level of the solutions and a
variational characterization of these extremals values of $q$. Our results
recover and improve some results in the literature.
