An eigenvalue problem for a nonlocal quasilinear anisotropic equation in fractional Orlicz Sobolev spaces without the ${\mathrm{\Delta }}_2$--condition

In this paper we analyze an eigenvalue problem associated to fractional operators of the form $$L_a^{s}u(x)=2\text{p.v.}{\int }_{{\mathbb{R}}^n}a(x,y,D^{s}u(x,y))\frac{dy}{\mathrm{\mid }x-y{\mathrm{\mid }}^{n+s}},$$ which represents a generalization model for nonlocal, nonstandard growth diffusion problems. We study this problem in the context of the fractional Orlicz Sobolev spaces without assuming the so-called ${\mathrm{\Delta }}_2$–condition on the Young functions involved. We show existence of a sequence of eigenpairs $(u_k,{\lambda }_k)\to (0,+\mathrm{\infty })$.