  We consider eigenvalue problems for elliptic operators of arbitrary order
$2m$ subject to Neumann boundary conditions on bounded domains of the Euclidean
$N$-dimensional space. We study the dependence of the eigenvalues upon
variations of mass density and in particular we discuss the existence and
characterization of upper and lower bounds under both the condition that the
total mass is fixed and the condition that the $L^{\frac{N}{2m}}$-norm of the
density is fixed. We highlight that the interplay between the order of the
operator and the space dimension plays a crucial role in the existence of
eigenvalue bounds.
