Remarks on a nonlinear nonlocal operator in Orlicz spaces

TL;DR

This paper investigates a class of nonlinear nonlocal operators in Orlicz spaces, establishing fundamental inequalities and applying them to elliptic problems, including eigenvalue analysis, in a generalized fractional setting.

Contribution

It introduces new Poincaré and Sobolev inequalities for nonlinear nonlocal operators in Orlicz spaces, enabling analysis of associated elliptic problems and eigenvalues.

Findings

Established Poincaré inequality for the operator.

Proved Sobolev inequality depending on kernel singularity.

Analyzed elliptic problems and eigenvalues in Orlicz spaces.

Abstract

We study integral operators $\mathcal{L}u(x)=\int_{\mathbb{R^N}}\psi(u(x)-u(y))J(x-y)\,dy$ 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 Poincar\'e 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 $u\equiv0$ on $\Omega^c$; both cases $f=f(x)$ and $f=f(u)$ are considred, including the generalized eigenvalue problem $f(u)=\lambda\psi(u)$.