Remarks on a nonlinear nonlocal operator in Orlicz spaces

Abstract

We study integral operators $\mathcal{L}u(x)={\int }_{{\mathbb{R}}^{\mathbb{N}}}\psi (u(x)-u(y))J(x-y)dy$ 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 $\mathcal{L}u=f$ in a bounded domain $\Omega$, and boundary condition $u\equiv 0$ on ${\Omega }^c$; both cases $f=f(x)$ and $f=f(u)$ are considred, including the generalized eigenvalue problem $f(u)=\lambda \psi (u)$.