Existence and concentration results for some fractional Schr\"odinger equations in $\mathbb{R}^{N}$ with magnetic fields

TL;DR

This paper investigates the existence and concentration of solutions for nonlinear fractional Schr"odinger equations with magnetic fields, using minimax methods under certain potential conditions.

Contribution

It introduces new existence and concentration results for fractional Schr"odinger equations with magnetic fields involving various nonlinear growth conditions.

Findings

Existence of nontrivial weak solutions established.

Solutions concentrate under specific potential conditions.

Applicable to subcritical, critical, and supercritical nonlinearities.

Abstract

We consider some nonlinear fractional Schr\"odinger equations with magnetic field and involving continuous nonlinearities having subcritical, critical or supercritical growth. Under a local condition on the potential, we use minimax methods to investigate the existence and concentration of nontrivial weak solutions.