On a fractional magnetic pseudorelativistic operator: properties and applications

TL;DR

This paper introduces a new fractional magnetic pseudorelativistic operator, explores its fundamental properties, analyzes its behavior as the fractional order approaches 1, and establishes existence results for related semilinear equations.

Contribution

It defines a novel fractional magnetic operator, studies its properties, and proves existence of solutions for associated nonlinear equations.

Findings

Behavior of the operator as s approaches 1 matches classical results

Existence of weak solutions for semilinear equations with power and nonlocal terms

Removal of singularity in the integral definition of the operator

Abstract

We introduce a fractional magnetic pseudorelativistic operator for a general fractional order $s\in(0,1)$. First we define a suitable functional setting and we prove some fundamental properties. Then we show the behavior of the operator as $s \nearrow 1$ obtaining some results \`a la Bourgain-Brezis-Mironescu and removing the singularity from the integral definition. Finally we get existence of weak solutions for some semilinear equations involving a power type nonlinearity or a nonlocal (Choquard type) term.