  This article is devoted to the study of the behavior of generalized entire
solutions for a wide class of quasilinear degenerate inequalities modeled on
the 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 the
system of vector fields $\mathbf{X}=(X_1,X_2,..,X_m)$ and $X_i^*$ denotes the
adjoint of $X_i$ with respect to Lebesgue measure. For the case where $p$ is
less than the homogeneous dimension $Q$ we have obtained a sharp a priori
estimate for essential supremum of generalized solutions from below which imply
some Liouville-type results.

Analysis of PDEs