On Global Solutions of Second-Order Quasilinear Elliptic Inequalities

Differential inequalities of the form $$- \operatorname{div} A (x, \nabla u)\ge f(u)\quad \text{in}\quad {\mathbb R}^n$$ are considered, where $$n \ge 2$$ and $$A$$ is a Carathéodory function that satisfies the uniform ellipticity conditions $$C_1|\xi|^p\le\xi A (x, \xi), \qquad |A (x, \xi)| \le C_2 |\xi|^{p-1}, \qquad C_1, C_2 > 0, \qquad p > 1,$$ for almost every $$x \in {\mathbb R}^n$$ and all $$\xi \in {\mathbb R}^n$$ . For nonnegative solutions of these inequalities, precise conditions for the absence of nontrivial solutions are obtained.