  In this paper we obtain generalized Keller-Osserman conditions for wide
classes of differential inequalities on weighted Riemannian manifolds of the
form $L u\geq b(x) f(u) \ell(|\nabla u|)$ and $L u\geq b(x) f(u) \ell(|\nabla
u|) - g(u) h(|\nabla u|)$, where $L$ is a non-linear diffusion-type operator.
Prototypical examples of these operators are the $p$-Laplacian and the mean
curvature operator. While we concentrate on non-existence results, in many
instances the conditions we describe are in fact necessary for non-existence.
The geometry of the underlying manifold does not affect the form of the
Keller-Osserman conditions, but is reflected, via bounds for the modified
Bakry-Emery Ricci curvature, by growth conditions for the functions $b$ and
$\ell$. We also describe a weak maximum principle related to inequalities of
the above form which extends and improves previous results valid for the
$\vp$-Laplacian.
