Basic results of fractional Orlicz-Sobolev space and applications to non-local problems

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.…”

Least-energy nodal solutions of nonlinear equations with fractional Orlicz-Sobolev spaces

“…They also introduce a suitable functional space to study an equation in which a fractional g−Laplace operator is present. After that in 2020 some authors, like S. Bahrouni and A. M. Salort have continued these studies see [6,8,9,37]. More precisely, they proved some basic results like embedding problems and some other fundamental properties.…”

“…[see [9], Lemma 3.1 and [27] Lemma 2.1 ] Assume (g 1 ) and (g 2 ) hold, then min{a L , a m }G(t) ≤ G(at) ≤ max{a L , a m }G(t) for a, t ≥ 0,…”

Existence and multiplicity of solutions for a Dirichlet problem in Fractional Orlicz- Sobolev spaces

“…Moreover, in [5] the existence of a nodal solution, that is, a changing-sing solution, was given. Regarding multiple solutions for (−△ g ) s , we quote the works [7], for the existence of two non-trivial solutions, and [6], where infinitely many solutions are obtained for a class of non-local Orlicz-Sobolev Schrödinger equations. Inspired by [9], we obtain an existence result using the sub-supersolution method.…”