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This article is devoted to the study of the behavior of generalized entiresolutions for a wide class of quasilinear degenerate inequalities modeled onthe following prototype with p-Laplacian in the main part \begin{equation*}{\underset{m}{\overset{i=1}{\sum}}} X_i^*(|\mathbf{X}u|^{p-2} X_i u)\geq|u|^{q-2}u, \ \ x\in {\mathbb{R}}^{n},\ q>1,\ p>1, \end{equation*} where${\mathbb{R}}^{n}$ is a Carnot-Carath\'{e}odory metric space, generated by thesystem of vector fields $\mathbf{X}=(X_1,X_2,..,X_m)$ and $X_i^*$ denotes theadjoint of $X_i$ with respect to Lebesgue measure. For the case where $p$ isless than the homogeneous dimension $Q$ we have obtained a sharp a prioriestimate for essential supremum of generalized solutions from below which implysome Liouville-type results.

Asymptotic behavior of entire solutions for degenerate partial  differential inequalities on Carnot-Carath\'{e}odory metric spaces and  Liouville type results