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

TL;DR

This paper investigates the asymptotic behavior of solutions to a class of degenerate quasilinear inequalities involving the p-Laplacian on Carnot-Carathéodory spaces, establishing sharp estimates and Liouville-type results.

Contribution

It provides new sharp a priori estimates and Liouville-type theorems for solutions of degenerate inequalities on Carnot-Carathéodory metric spaces, extending previous results to a broader class of equations.

Findings

Established sharp a priori estimates for solutions.

Proved Liouville-type nonexistence results under certain conditions.

Analyzed solutions when p is less than the homogeneous dimension Q.

Abstract

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.