Keller–Osserman conditions for diffusion-type operators on Riemannian manifolds

In this paper we obtain essentially sharp generalized Keller–Osserman conditions for wide classes of differential inequalities of the form Lu⩾b(x)f(u)ℓ(|∇u|) and Lu⩾b(x)f(u)ℓ(|∇u|)−g(u)h(|∇u|) on weighted Riemannian manifolds, where L is a non-linear diffusion-type operator. Prototypical examples of these operators are the p-Laplacian and the mean curvature operator. The geometry of the underlying manifold is reflected, via bounds for the modified Bakry–Emery Ricci curvature, by growth conditions for the functions b and ℓ. A weak maximum principle which extends and improves previous results valid for the φ-Laplacian is also obtained. Geometric comparison results, valid even in the case of integral bounds for the modified Bakry–Emery Ricci tensor, are presented.