Concentration phenomena for fractional magnetic NLS equations

TL;DR

This paper investigates the existence, multiplicity, and concentration of solutions for fractional magnetic Schrödinger equations, introducing new methods and inequalities to advance understanding in this area.

Contribution

It develops a novel approach combining penalization, Nehari manifold, and Ljusternik-Schnirelman theory for fractional magnetic problems, and proves a new Kato's inequality.

Findings

Established multiplicity and concentration results for solutions.

Proved a new Kato's inequality for fractional magnetic Laplacian.

Applied advanced variational methods to fractional magnetic equations.

Abstract

We study the multiplicity and concentration of complex valued solutions for a fractional magnetic Schr\"odinger equation involving a scalar continuous electric potential satisfying a local condition and a continuous nonlinearity with subcritical growth. The main results are obtained by applying a penalization technique, generalized Nehari manifold method and Ljusternik-Schnirelman theory. We also prove a Kato's inequality for the fractional magnetic Laplacian which we believe to be useful in the study of other fractional magnetic problems.