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.
Categories: math.AP