On the role of gradient terms in quasilinear coercive 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 $$ \Delta_{\mathbb{G}}^\varphi u \ge b(x) f(u) l(|\nabla u|). $$ Prototype examples of $\Delta_{\mathbb{G}}^\varphi$ 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 $t \rightarrow 0^+$ and as $t \rightarrow +\infty$. 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.