Eigenvalues for a nonlocal pseudo $p-$Laplacian

TL;DR

This paper investigates the eigenvalues of a nonlocal pseudo p-Laplacian operator, establishing their properties, limits, and associated eigenfunctions, and introduces anisotropic fractional Sobolev spaces for analysis.

Contribution

It introduces a nonlocal pseudo p-Laplacian operator, analyzes its eigenvalues, and studies their limits as parameters vary, along with developing new anisotropic fractional Sobolev space theory.

Findings

Existence of an unbounded sequence of eigenvalues for the nonlocal operator.

The first eigenvalue is positive, simple, isolated, with a positive bounded eigenfunction.

Limits of the first eigenvalue as p→∞ and s→1− are characterized, linking to pseudo infinity Laplacian and local p-Laplacian.

Abstract

In this paper we study the eigenvalue problems for a nonlocal operator of order $s$ that is analogous to the local pseudo $p-$Laplacian. We show that there is a sequence of eigenvalues $\lambda_n \to \infty$ and that the first one is positive, simple, isolated and has a positive and bounded associated eigenfunction. For the first eigenvalue we also analyze the limits as $p\to \infty$ (obtaining a limit nonlocal eigenvalue problem analogous to the pseudo infinity Laplacian) and as $s\to 1^-$ (obtaining the first eigenvalue for a local operator of $p-$Laplacian type). To perform this study we have to introduce anisotropic fractional Sobolev spaces and prove some of their properties.