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

Laura Baldelli, Federico Bernini

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.

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

Abstract

We investigate the fractional magnetic -Laplacian operator in the physical dimension case , with and . 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.
Paper Structure (13 sections, 26 theorems, 191 equations)

This paper contains 13 sections, 26 theorems, 191 equations.

Table of Contents

  1. Introduction
  2. Functional setting
  3. Notations
  4. The non-magnetic case
  5. The non-homogeneous magnetic case
  6. The homogeneous magnetic case
  7. Quasilinear critical problems in R3
  8. Preliminaries
  9. Functional geometry
  10. (PS)-sequences: boundedness and compactness
  11. Proof of Theorem \ref{['main:theorem:q>p']}
  12. Proof of Theorem \ref{['main:theorem:q<p']}
  13. The magnetic concentration compactness

Key Result

Theorem 1.1

Let $0<s<1<p$, $sp<3$, $p<q<p^*_s$, $A:\mathbb R^3 \to \mathbb R^3$ be a vector field with locally bounded gradient, $\lambda>0$, and H:hyp, and K:hyp hold. Moreover, we assume that Then, there exists $\lambda^*>0$ such that for any $\lambda>\lambda^*$, then main:equation admits at least one nontriavial solution with positive energy.

Theorems & Definitions (50)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • proof : Proof of Theorem \ref{['density:theorem']}.
  • ...and 40 more