General fractional Sobolev Space with variable exponent and applications\n to nonlocal problems

2019/01/17 by Elhoussine Azroul, Azroul, Elhoussine, Abdelmoujib Benkirane +3 · 1 citation
Engineering · Mathematics · #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.1901.05687

openalex publication_date 2019/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we extend the fractional Sobolev spaces with variable\nexponents W^s,p(x,y) to include the general fractional case W^K,p(x,y),\nwhere p is a variable exponent, s\∈ (0,1) and K is a suitable kernel. We\nare concerned with some qualitative properties of the space W^K,p(x,y)\n(completeness, reflexivity, separability, and density). Moreover, we prove a\ncontinuous and compact embedding theorem of these spaces into variable exponent\nLebesgue spaces. As applications, we discuss the existence of a nontrivial\nsolution for a nonlocal p(x,.)-Kirchhoff type problem. Further, we establish\nthe existence and uniqueness of a solution for a variational problem involving\nthe integro-differential operator of elliptic type \L^p(x,.)_K.\n

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• Embedding Results and Fractional p(x,.)-Laplacian Problem in Unbounded Domains

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