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 formPrototypical 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 ℓ. We also describe a weak maximum principle related to inequalities of the above form which extends and improves previous results valid for the ϕ-Laplacian. F (t)∈ L 1 (+∞),2000 Mathematics Subject Classification. 58J05, 35J60.

Keller–Osserman type conditions for differential inequalities with gradient terms on the Heisenberg group

“…Nonetheless, however interesting these results are, they do not seem to cover the cases that we treated in our main theorems. Furthermore, it should be noted that the approach of the present paper and of [13] can be successfully applied to the case of complete Riemannian manifolds (see [26]), an environment in which the techniques of [11] seem to be unable to yield sharp results.…”

“…Moreover, since the writing of this paper, a number of further contributions to the subject have appeared (see [14,13,11,26]). Some of these use techniques which are similar, and in fact are in some sense sharp evolutions of the present arguments [14,13], while others are based on completely different principles and are more tailored for the case of the Euclidean space or of structures, such as Carnot groups, which generalize the Heisenberg group ( [11]; we also point out the earlier papers of A. Bonfiglioli and F. Uguzzoni [6] and of N. Garofalo and E. Lanconelli [16]).…”

On the role of gradient terms in coercive quasilinear differential inequalities on Carnot groups

“…For instance, given ϕ, b, f, l, if both the properties listed in (P1), (P2) hold and f > 0 on R + , then there are no non-constant, non-negative solutions of (1.12): this because (P1) and (P2) would imply that any non-constant solution u of (1.12) must be bounded and satisfy f (u * ) ≤ 0, hence u * ≤ 0. Our approach to (P1) has its roots in the works [31,30,32,26] by the third author and his collaborators, and in the subsequent improvements in [27,21]. Interesting Liouville theorems for slowly growing solutions have also been shown in [12,28,9] for a broad class of differential inequalities including (1.12).…”

“…To the best of our knowledge, Liouville type theorems for global solutions of (1.12) have mainly been investigated by means of two different approaches: the first rests on radialization techniques and refined comparison theorems [21,20,5,4,28], while the second is directly based on the weak formulation, via a careful choice of test functions, in the spirit of the work of Mitidieri-Pokhozhaev [24] [8,12,9]. Radialization techniques exploit the properties of a homogeneous norm r(x) in order to construct suitable radial supersolutions.…”

“…2 In Corollary A2 in [21], the Euclidean case is achieved by setting β = −2. In this respect, note that there is typo, a missing minus sign in the right-hand side of inequality (1.12) therein, which should be replaced by…”