The aim of this notes is to show how the Ljusternick–Schnirelmann Theory can be used to show existence and multiplicity of solutions for the following Schrödinger–Bopp–Podolsky system in \(\mathbb R^{3}\) \(\begin{aligned} \left\{ \begin{array}{ll} -\varepsilon ^{2} \Delta u + V(x)u + \phi u = |u|^{p-2}u \\ -\varepsilon ^{2} \Delta \phi + \varepsilon ^{4} \Delta ^{2}\phi = 4\pi u^{2}, \end{array} \right. \end{aligned}\) when the parameter \(\varepsilon > 0\) is sufficiently small, namely in the so called semiclassical limit. In the system, \(V:\mathbb {R}^{3} \rightarrow \mathbb {R}\) is a given non constant external potential and \(p\in (4, 6)\) . 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. The results in this paper are taken from [17] where a more general nonlinearity is considered. However here more preliminaries are given on the Ljusternick–Schnirelmann category in order to familiarize the reader with this important topological invariant.
