Nonnegative solutions of some quasilinear elliptic inequalities and applications

TL;DR

This paper establishes conditions under which weak solutions to certain quasilinear elliptic inequalities are nonnegative, extending results to subelliptic operators on Carnot groups.

Contribution

It provides new criteria ensuring nonnegativity of solutions for a class of quasilinear elliptic inequalities, including extensions to subelliptic operators on Carnot groups.

Findings

Weak solutions are nonnegative under specified conditions.

Extensions to subelliptic operators on Carnot groups.

Applicable to a broad class of quasilinear inequalities.

Abstract

Let $f :\R\to\R$ be a continuous function. We prove that under some additional assumptions on $f$ and $A:\R\to\R_{+}$, weak $\Cuno$ solutions of the differential inequality $-\diver(A(\abs{\nabla u})\nabla u)\ge f(u)$ on $\RN$ are nonnegative. Some extensions of the result in the framework of subelliptic operators on Carnot Groups are considered.