Existence of solutions for a nonlocal type problem in fractional Orlicz Sobolev spaces

Abstract

In this paper, we investigate the existence of weak solution for a fractional type problems driven by a nonlocal operator of elliptic type in a fractional Orlicz-Sobolev space, with homogeneous Dirichlet boundary conditions. We first extend the fractional Sobolev spaces W s,p to include the general case W s L A , where A is an N-function and s ∈ (0, 1). We are concerned with some qualitative properties of the space W s L A (completeness, reflexivity and separability). Moreover, we prove a continuous and compact embedding theorem of these spaces into Lebesgue spaces.

A Pólya–Szegö principle for general fractional Orlicz–Sobolev spaces

“…• When M (r) = r s G −1 (r n ) and N (r) = 1, n ≥ 1 we obtain the Orlicz-Slobodetskii spaces defined in [4]. if and only if n s < p − p + − p − , where we have used that G −1 (r) ≤ max{r 1/p + , r 1/p − }.…”

“…and N (r) = 1, n ≥ 1 we obtain the Orlicz-Slobodetskii spaces defined in [4]. Indeed, (P 1 ) and (P 2 ) easily hold and (P 3 ) reads as…”

“…The main goal of this paper is to extend that principle to the more general setting of fractional Orlicz-Sobolev spaces, thus allowing growth laws different than powers. However, several definitions of fractional Orlicz-Sobolev have been proposed in the literature in [4] [9] and [11]. It turns out that essentially the same proof of the Pólya-Szegö principle works for all of them.…”

Neumann and Robin type boundary conditions in Fractional Orlicz-Sobolev spaces

“…It worths to be mention that the local counterpart of (1.2) for Orlicz functions in the Dirichlet case was studied in [12,24,32]. For some existence results in the nonlocal Orlicz case with Dirichlet boundary conditions see [5]. For problems with critical Trudinger-Moser nonlinearities see [28].…”

On the fractional p-Laplacian problems

“…[4]) Assume that Q is a continuous vector field from R N to RN and satisfies Q(x) • x ≥ 0 if |x| = ρ for some ρ > 0. Then there exists a point x ∈ B ρ (0) such that Q(x) = 0, where B ρ (0) denotes a ball centered at the origin with radius ρ in R N .…”