Leray-Schauder’s solution for a nonlocal problem in a fractional Orlicz-Sobolev space

Abstract

<jats:title>Abstract</jats:title><jats:p>Via Leray-Schauder’s nonlinear alternative, we obtain the existence of a weak solution for a nonlocal problem driven by an operator of elliptic type in a fractional Orlicz-Sobolev space, with homogeneous Dirichlet boundary conditions.</jats:p>

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.…”

Existence of ground state solutions of elliptic system in Fractional Orlicz-Sobolev Spaces

“…Lemma 3.3. [11] Suppose that A i ( √ t) is convex. Moreover, we assume that the sequence (w k ) converges weakly to w in X s,A i 0 (Ω) and lim sup F i (w k ), w k − w ≤ 0.…”

Multiple Solutions for a Nonlocal Kirchhoff Problem in Fractional Orlicz-Sobolev Spaces

“…If the role played by L p (Ω) in the definition of fractional Sobolev spaces W s,p (Ω) is assigned to an Orlicz L A (Ω) space, the resulting space W s L A (Ω) is exactly a fractional Orlicz-Sobolev space . Many properties of fractional Sobolev spaces have been extended to fractional Orlicz-Sobolev spaces (see [4,5,8,9,12,16,17]). For this, many researchers have studied the existence of solutions for the eigenvalue problems involving nonhomogeneous operators in the divergence form through Orlicz-Sobolev spaces by using variational methods and critical point theory, monotone operator methods, fixed point theory and degree theory (see for instance [14,15,20,32]).…”