Positive Solutions For a Schrödinger-Bopp-Podolsky system in $\mathbb R^{3}$

Abstract: We consider the following Schr\"odinger-Bopp-Podolsky system in $\mathbb R<sup>{3}$ $$\left{ \begin{array}{c} -\varepsilon^{2} \Delta u + V(x)u + \phi u = f(u)\ -\varepsilon^{2} \Delta \phi + \varepsilon^{4} \Delta^{2}\phi = 4\pi\varepsilon u^{2}\ \end{array} \right.$$ where $\varepsilon &gt; 0$ 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 $V$.