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On a fractional magnetic pseudorelativistic operator: properties and applications

Federico Bernini, Pietro d'Avenia

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 à 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.

On a fractional magnetic pseudorelativistic operator: properties and applications

Abstract

We introduce a fractional magnetic pseudorelativistic operator for a general fractional order . First we define a suitable functional setting and we prove some fundamental properties. Then we show the behavior of the operator as obtaining some results à 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.

Paper Structure

This paper contains 14 sections, 28 theorems, 218 equations.

Table of Contents

  1. Introduction
  2. Functional setting
  3. The space HsA(R3,C)
  4. Some remarks on Hs and Hsm
  5. Properties of HsA(R3,C)
  6. Behavior as s -> 1
  7. Bourgain-Brezis-Mironescu type formulas
  8. Convergence of the operator
  9. Weak to strong
  10. Applications to some semilinear equations
  11. The pure power nonlinearity
  12. The Choquard equation
  13. Properties of the modified Bessel function
  14. Proof of Proposition \ref{['normHmsC']}

Key Result

Theorem 1.3

Let $\Omega \subset \mathbb{R}^3$ be an open bounded set with Lipschitz boundary and $A \in C^2(\overline{\Omega},\mathbb{R}^3)$. Then, for every $u \in H^1_A(\Omega,\mathbb{C})$

Theorems & Definitions (53)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • ...and 43 more