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

Keller--Osserman conditions for diffusion-type operators on Riemannian Manifolds