Fractional NLS equations with magnetic field, critical frequency and critical growth

TL;DR

This paper investigates fractional Schrödinger equations with magnetic fields, critical frequency, and growth, establishing existence and multiplicity of solutions using variational methods, including solutions that change sign.

Contribution

It introduces new results on the existence and multiplicity of solutions for fractional Schrödinger equations with critical parameters and magnetic fields, expanding understanding of such equations.

Findings

Existence of mountain pass solutions tending to zero as perturbation vanishes.

Infinitely many solutions including sign-changing solutions without magnetic effects.

Solutions are obtained via variational methods under critical growth conditions.

Abstract

The paper is devoted to the study of a singularly perturbed fractional Schr\"{o}dinger equations involving critical frequency and critical growth in the presence of a magnetic field. By using variational methods, we obtain the existence of mountain pass solutions $u_{\varepsilon}$ which tend to the trivial solution as $\varepsilon\rightarrow0$. Moreover, we get infinitely many solutions and sign-changing solutions for the problem in absence of magnetic effects under some extra assumptions.