  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.
