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Eigenvalue type problem in $s(.,.)$-fractional Musielak-Sobolev spaces

E. Azroul, A. Benkirane, M. Srati

Abstract

In this paper, first we introduce the $s(.,.)$-fractional Musielak-Sobolev spaces $W^{s(x,y)}L_{\varPhi_{x,y}}(Ω)$. Next, by means of Ekeland's variational principal, we show that there exists $λ_*>0$ such that any $λ\in(0, λ_*)$ is an eigenvalue for the following problem $$(\mathcal{P}_a) \left\{ \begin{array}{ll}\left( -Δ\right)^{s(x,.)}_{a_{(x,.)}} u = λ|u|^{q(x)-2}u &\quad {\rm in}\ Ω, \\ \qquad\quad u = 0 &\quad {\rm in }\ \mathbb{R}^N\setminus Ω, \end{array} \right. $$ where $Ω$ is a bounded open subset of $\mathbb{R}^N$ with $C^{0,1}$-regularity and bounded boundary.

Eigenvalue type problem in $s(.,.)$-fractional Musielak-Sobolev spaces

Abstract

In this paper, first we introduce the -fractional Musielak-Sobolev spaces . Next, by means of Ekeland's variational principal, we show that there exists such that any is an eigenvalue for the following problem where is a bounded open subset of with -regularity and bounded boundary.
Paper Structure (5 sections, 19 theorems, 158 equations)

This paper contains 5 sections, 19 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Preliminaries results
  3. $s(.,.)$-fractional Musielak-Sobolev spaces
  4. Existence results and proofs
  5. Examples

Key Result

Theorem 2.1

$($benkirane$)$. Let $\Omega$ be an open subset of $\mathbb{R} ^N$, and let $s\in (0,1)$. The space $W^sL_{\varPhi_{x,y}}(\Omega)$ is a Banach space with respect to the norm $(r2)$, and a separable $($resp. reflexive$)$ space if and only if $\varPhi_{x,y} \in \Delta_2$$($resp. $\varPhi_{x,y}\in \Del

Theorems & Definitions (38)

  • Definition 2.1
  • Remark 2.1: 1
  • Theorem 2.1
  • Definition 2.2
  • Theorem 2.2
  • Lemma 2.1
  • Proposition 2.1
  • Theorem 2.3
  • Theorem 2.4
  • Proposition 2.2
  • ...and 28 more