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Ljusternik-Schnirelmann eigenvalues for the fractional $m-$Laplacian without the $Δ_2$ condition

Julian Fernandez Bonder, Juan F. Spedaletti

Abstract

In this work we analyze the eigenvalue problem associated to the fractional $m-$Laplacian, defined as $$ (-Δ_m)^s u(x):=2\text{p.v.}\int_{{\mathbb R}^n} m\left(\frac{|u(x)-u(y)|}{|x-y|^s}\right)\frac{(u(x)-u(y))}{|u(x)-u(y)|}\frac{dy}{|x-y|^{n+s}}, $$ This operator serves as a model for nonlocal, nonstandard growth diffusion problems. In contrast to previous analyses, we explore the eigenvalue problem without presuming the $Δ_2$ condition on $M$ -- the primitive function of $m$. Our results show the existence of a sequence of eigenvalues $λ_k\to\infty$. This research contributes to advancing our understanding of nonlocal diffusion models, specifically those characterized by the fractional $m-$Laplacian, by relaxing the constraints imposed by the $Δ_2$ condition.

Ljusternik-Schnirelmann eigenvalues for the fractional $m-$Laplacian without the $Δ_2$ condition

Abstract

In this work we analyze the eigenvalue problem associated to the fractional Laplacian, defined as This operator serves as a model for nonlocal, nonstandard growth diffusion problems. In contrast to previous analyses, we explore the eigenvalue problem without presuming the condition on -- the primitive function of . Our results show the existence of a sequence of eigenvalues . This research contributes to advancing our understanding of nonlocal diffusion models, specifically those characterized by the fractional Laplacian, by relaxing the constraints imposed by the condition.
Paper Structure (10 sections, 23 theorems, 143 equations)

This paper contains 10 sections, 23 theorems, 143 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Young functions and Orlicz spaces
  4. Fractional Orlicz-Sobolev spaces
  5. Complementary systems
  6. An abstract result
  7. The fractional $m-$laplacian $(-\Delta_m)^s$
  8. The eigenvalue problem
  9. Density results
  10. A Rellich-Kondrachov type result

Key Result

Theorem 1.1

Under suitable assumptions on $\Omega$, $m(t)$, and $g(t)$without necessitating the $\Delta_2-$condition on $M(t)$ there exists a sequence $\{\lambda_k\}_{k\in\mathbb{N}}$ of eigenvalues for intro.2. Moreover, $\lambda_k\to\infty$ as $k\to\infty$.

Theorems & Definitions (40)

  • Theorem 1.1
  • Lemma 2.1: Gossez, Lemma 1.2
  • Theorem 2.2: Ti, Theorem 3.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 30 more