Monotonicity of eigenvalues of the fractional p-Laplacian with singular weights

TL;DR

This paper investigates the eigenvalues of the fractional p-Laplacian with singular weights, establishing their existence, characterizations, and strict monotonicity properties.

Contribution

It introduces new variational characterizations for eigenvalues and proves their strict decreasing monotonicity with respect to the weight function.

Findings

Existence of an unbounded sequence of positive eigenvalues

Characterizations of the first and second eigenvalues

Proof of strict decreasing monotonicity of eigenvalues

Abstract

We study a nonlinear, nonlocal eigenvalue problem driven by the fractional p-Laplacian with an indefinite, singular weight chosen in an optimal class. We prove the existence of an unbounded sequence of positive variational eigenvalues and alternative characterizations of the first and second eigenvalues. Then, by means of such characterizations, we prove strict decreasing monotonicity of such eigenvalues with respect to the weight function.