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

Abstract

In the sub-Riemannian setting of Carnot groups, this work investigates a priori estimates and Liouville type theorems for solutions of coercive, quasilinear differential inequalities of the type ΔGφu≥b(x)f(u)l(|∇u|). Prototype examples of ΔGφ are the (subelliptic) p-Laplacian and the mean curvature operator. The main novelty of the present paper is that we allow a dependence on the gradient l(t) that can vanish both as t→0+ and as t→+∞. Our results improve on the recent literature and, by means of suitable counterexamples, we show that the range of parameters in the main theorems are sharp.