Remarks on a nonlinear nonlocal operator in Orlicz spaces

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We study integral operators Lu(x)=∫ᵣₙψ(u(x)-u(y))J(x-y)dy of the type of the fractional p-Laplacian operator, and the properties of the corresponding Orlicz and Sobolev-Orlicz spaces. In particular we show a Poincaré inequality and a Sobolev inequality, depending on the singularity at the origin of the kernel J considered, which may be very weak. Both inequalities lead to compact inclusions. We then use those properties to study the associated elliptic problem Lu=f in a bounded domain Ω, and boundary condition u≡0 on Ωᶜ; both cases f=f(x) and f=f(u) are considred, including the generalized eigenvalue problem f(u)=λψ(u).