On global solutions of quasilinear second-order elliptic inequalities

We consider the inequality $$-\mathrm{div}A(x,\mathrm{\nabla }u)\ge f(u)in{\mathbb{R}}^n,$$ where $n\ge 2$ and $A$ is a Caratheodory function such that $$C_1\mathrm{\mid }\xi {\mathrm{\mid }}^p\le \xi A(x,\xi )and\mathrm{\mid }A(x,\xi )\mathrm{\mid }\le C_2\mathrm{\mid }\xi {\mathrm{\mid }}^{p-1}$$ with some constants $C_1>0$, $C_2>0$, and $p>1$ for almost all $x\in {\mathbb{R}}^n$ and for all $\xi \in {\mathbb{R}}^n$. Our aim is to find exact conditions on the function $f$ guaranteeing that any non-negative solution of this inequality is identically zero.