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

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 $\mathbf{X}=(X_1,X_2,\mathrm{.}\mathrm{.},X_m)$ and $X_i^{\ast }$ 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.