In this paper, we study a nonlocal eigenvalue problem involving variable exponent growth conditions, on a bounded domain Ω ⊂ R n. Using adequate variational techniques, mainly based on Ekeland's variational principle, we establish the existence of a continuous family of eigenvalues lying in a neighborhood at the right of the origin.

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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.

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