Remarks on a nonlinear nonlocal operator in Orlicz spaces

Abstract

We study integral operators Lu(χ)=∫ℝℕψ(u(x)−u(y))J(x−y)dy$\mathcal{L}u\left( \chi \right)=\int{_{_{\mathbb{R}}\mathbb{N}}\psi \left( u\left( x \right)-u\left( y \right) \right)J\left( x-y \right)dy}$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$\mathcal{L}u=f$in a bounded domain Ω,$\Omega ,$and boundary condition u ≡ 0 on Ωc;${{\Omega }^{c}};$both cases f = f(x) and f = f(u) are considred, including the generalized eigenvalue problem f(u)=λψ(u).$f\left( u \right)=\lambda \psi \left( u \right).$

Keywords

• nonlocal equations
• integral operators
• p–fractional laplacian
• orlicz spaces
• 45p05
• 46e35
• 45g10