General fractional Sobolev space with variable exponent and applications to nonlocal problems

Abstract

In this paper, we extend the fractional Sobolev spaces with variable exponents W s,p(x,y) to include the general fractional case W K,p(x,y) , where p is a variable exponent, s ∈ (0, 1) and K is a suitable kernel. We are concerned with some qualitative properties of the space W K,p(x,y) (completeness, reflexivity, separability and density). Moreover, we prove a continuous and compact embedding theorem of these spaces into variable exponent Lebesgue spaces. As applications, we discus the existence of a nontrivial solution for a nonlocal p(x, .)-Kirchhoff type problem. Further, we establish the existence and uniqueness of a solution for a variational problem involving the integro-differential operator of elliptic type L p(x,.) K . 2010 Mathematics Subject Classification. 46E35, 35R11, 35S05, 35J35.

Embedding and extension results in Fractional Musielak-Sobolev spaces

“…Recently, great attention has been devoted to the study of a new class of fractional Sobolev spaces and related nonlocal problems, in particular, in the fractional Orlicz-Sobolev spaces W s L Φ (Ω) (see [6,7,13,14,17,19,24,25]) and in the fractional Sobolev spaces with variable exponents W s,p(x,y) (Ω) (see [8,9,10,11,12,15,16,33]), in which the authors establish some basic properties of these modular spaces and the associated nonlocal operators, they also obtained certain existence results for nonlocal problems involving this type of integro-differential operators. Furthermore, in that context, the authors in [5] introduced a new functional framework which can be seen as a natural generalization of the above mentioned functional spaces.…”

Local regularity for nonlocal equations with variable exponents

“…The fractional Sobolev spaces with variable exponents were first introduced recently by Kaufmann, Rossi, and Vidal [43], and have been studied in different contexts. See [4, 6–15, 19, 25, 39, 40, 50, 55] and references therein. Note that the Triebel–Lizorkin spaces with variable smoothness and integrability have been introduced in [30], which are isomorphic to $$W^{k, p(\cdot )}(\mathbb {R}^n)$$ if $$k \in \mathbb {N} \cup \lbrace 0 \rbrace$$, respectively, the variable exponent Bessel potential space $$\mathcal {L}^{\alpha , p(\cdot )}(\mathbb {R}^n)$$ for $$\alpha > 0$$ under suitable assumptions on p .…”

Local regularity for nonlocal equations with variable exponents

“…The fractional Sobolev spaces with variable exponents were first introduced recently by Kaufmann, Rossi, and Vidal [43], and have been studied in different contexts. See [25,11,12,4,7,10,39,8,13,15,19,50,6,9,14,40,55] and references therein. Note that the Triebel-Lizorkin spaces with variable smoothness and integrability have been introduced in [31], which are isomorphic to…”