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In the sub-Riemannian setting of Carnot groups, this work investigatesa-priori estimates and Liouville type theorems for solutions of coercive,quasilinear differential inequalities of the type $$\Delta_{\mathbb{G}}^\varphi u \ge b(x) f(u) l(|\nabla u|). $$ Prototypeexamples of $\Delta_{\mathbb{G}}^\varphi$ are the (subelliptic) $p$-Laplacianand the mean curvature operator. The main novelty of the present paper is thatwe 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 therecent literature and, by means of suitable counterexamples, we show that therange of parameters in the main theorems are sharp.

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