On a class of nonlocal problems in new fractional Musielak-Sobolev spaces. Issue 6 (13th April 2022)

• Record Type: Journal Article
• Authors: Azroul, E.; Benkirane, A.; Shimi, M.; Srati, M.
• Is Part Of: Applicable analysis. Volume 101:Issue 6(2022)
• Page Start: 1933
• Page End: 1952
• Publication Date: 2022-04-13
• Subjects: Fractional Musielak-Sobolev spaces -- Nonlocal problems -- Dirichlet problem -- Direct variational method; 46E35 -- 35R11 -- 35J20 -- 47G20; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2020.1789601
• Languages: English
• ISSNs: 0003-6811

Abstract

ABSTRACT: In this paper, we first introduce the new fractional Musielak-Sobolev spaces, and we establish some qualitative properties of these spaces. Then, using the direct variational approach, we investigate the existence of a nontrivial weak solution for a class of fractional type problems with Dirichlet boundary data of the following form 1 ( P a ) ( − Δ ) a ( x, . ) s u + a ˆ x ( | u | ) u = f ( x, u ) in Ω, u = 0 in R N ∖ Ω, where s ∈ ( 0, 1 ), Ω is an open bounded subset in R N with Lipschitz boundary ∂ Ω, and f : Ω × R ⟶ R is a Carathéodory function with suitable growth conditions.