Nonnegative solutions of some quasilinear elliptic inequalities and applications

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.