Qualitative properties of the fractional magnetic $p$-Laplacian and applications to critical quasilinear problems

TL;DR

This paper studies the fractional magnetic p-Laplacian operator in three dimensions, establishing functional properties, and proving the existence of solutions for related quasilinear equations with critical and subcritical nonlinearities using variational methods.

Contribution

It introduces a new concentration compactness principle for the quasilinear magnetic setting and analyzes the operator's functional framework and solution existence.

Findings

Established functional properties of the fractional magnetic p-Laplacian.

Proved existence of weak solutions for critical and subcritical problems.

Developed a new concentration compactness principle in the magnetic setting.

Abstract

We investigate the fractional magnetic $p$-Laplacian operator in the physical dimension case $N=3$, with $0<s<1<p$ and $sp<3$. Our goal is twofold. First, we define and study suitable functional settings for such operator proving significant properties. Then we get the existence of weak solutions for some quasilinear equations involving a weighted critical and subcritical power type nonlinearity. Our technique relies on variational methods and faces various difficulties: the complex quasilinear framework due to the presence of an external magnetic potential, the nonlocal setting, which entails appropriate tools, and the lack of compactness, which requires concentration compactness arguments. In this direction, we state a new concentration compactness principle in the quasilinear magnetic setting that seems to be missing in the literature.