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

TL;DR

This paper proves the existence of multiple positive solutions for a Schr"odinger-Bopp-Podolsky system in three-dimensional space using variational methods, with the number of solutions related to the topology of the potential's minima.

Contribution

It introduces a variational approach to establish multiple positive solutions for the system, linking solution count to the topology of the potential's minima.

Findings

Number of positive solutions estimated by the Ljusternick-Schnirelmann category of minima set

Existence of solutions under suitable assumptions on V and f

Application of variational methods to a coupled PDE system

Abstract

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 $V$.