Advanced Nonlinear Studies 9 (2009), 429–452 Maximum Principles for Inhomogeneous Elliptic Inequalities on Complete Riemannian Manifolds Dimitri Mugnai, Patrizia Pucci * Dipartimento di Matematica e Informatica Universit`a degli Studi di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy e-mails: mugnai@dmi.unipg.it, pucci@dmi.unipg.it Received 14 July 2008 Communicated by Paul Rabinowitz Abstract We prove some maximum principle results for weak solutions of elliptic inequali- ties, possibly inhomogeneous, on complete Riemannian manifolds. 1991 Mathematics Subject Classification. Primary: 58 J 05, 35 B 50; Secondary: 35 B 45, 35 J 60. Key words. Elliptic inequalities on manifolds, Maximum Principles. 1 Introduction In this paper we are concerned with weak solutions of differential inequalities on a complete Riemannian manifold M of dimension n. More precisely, our aim is to prove maximum principles for inequalities governed by operators which may be inhomogeneous. Let us start with three examples of general interest: the first concerns the mean curvature operator, say with global growth p = 1, while the latter two involve the p–Laplace–Beltrami operator, that is Δ p u :=div(|∇u| p−2 ∇u), p> 1, where ∇u is the Riemannian gradient of u on M. Take the mean curvature equation div  ∇u  1+ |∇u| 2  = nH(x) * Research supported by the National Project Metodi Variazionali ed Equazioni Differenziali Non Lineari. 429