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We study, with respect to the parameter $q\neq0$, the followingSchr\"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, bymeans of the fibering approach, that the system has no solutions at all forlarge values of $q's$, and has two radial solutions for small $q's$. We givealso qualitative properties about the energy level of the solutions and avariational characterization of these extremals values of $q$. Our resultsrecover and improve some results in the literature.

The fibering method approach for a non-linear Schr\"odinger equation coupled with the electromagnetic field