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

Abstract

In this paper we obtain generalized Keller-Osserman conditions for wide classes of differential inequalities on weighted Riemannian manifolds of the form $Lu\ge b(x)f(u)\ell (|\nabla u|)$ and $Lu\ge b(x)f(u)\ell (|\nabla u|)-g(u)h(|\nabla u|)$, 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.