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

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.