  We consider the following Schr\"odinger-Bopp-Podolsky system in $\mathbb
R^{3}$
  $$\left\{ \begin{array}{c} -\varepsilon^{2} \Delta u + V(x)u + \phi u =
f(u)\\ -\varepsilon^{2} \Delta \phi + \varepsilon^{4} \Delta^{2}\phi =
4\pi\varepsilon u^{2}\\ \end{array} \right.$$
  where $\varepsilon > 0$ with $ V:\mathbb{R}^{3} \rightarrow \mathbb{R},
f:\mathbb{R} \rightarrow \mathbb{R}$ satisfy suitable assumptions. By using
variational methods, we prove that the number of positive solutions is
estimated below by the Ljusternick-Schnirelmann category of $M$, the set of
minima of the potential $V$.
