Remarks on a nonlinear nonlocal operator in Orlicz spaces

Abstract

We study integral operators Lu(x)=∫RNpsi(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 Omega, and boundary condition u≡0 on Omegac; both cases f=f(x) and f=f(u) are considred, including the generalized eigenvalue problem f(u)=lambdapsi(u).