Nonnegative solutions of some quasilinear elliptic inequalities and applications

Let $f\colon \mathbb R\to\mathbb R$ be a continuous function. It is shown that under certain assumptions on $f$ and $A\colon \mathbb R\to\mathbb R_+$ weak $\mathscr C^1$ solutions of the differential inequality $-\operatorname{div}(A(|\nabla u|)\nabla u)\geqslant f(u)$ on $\mathbb R^N$ are nonnegative. Some extensions of the result in the framework of subelliptic operators on Carnot groups are considered.
Bibliography: 19 titles.