Positive semiclassical states for a fractional Schr\"odinger-Poisson system

TL;DR

This paper investigates the existence and multiplicity of positive solutions for a fractional Schr"odinger-Poisson system in the semiclassical limit, linking solutions to the topology of the potential's minima.

Contribution

It establishes a lower bound on the number of positive solutions based on the Ljusternick-Schnirelmann category of the potential's minima set.

Findings

Number of positive solutions estimated by topological category

Solutions exist for small positive parameter in the semiclassical limit

Link between potential minima and solution multiplicity

Abstract

We consider a fractional Schr\"odinger-Poisson system in the whole space $\mathbb R^{N}$ in presence of a positive potential and depending on a small positive parameter $\varepsilon.$ We show that, for suitably small $\varepsilon$ (i.e. in the "semiclassical limit") the number of positive solutions is estimated below by the Ljusternick-Schnirelmann category of the set of minima of the potential.