On existence of global solutions of second-order quasilinear elliptic inequalities

We study the existence of global positive solutions of the differential inequalities
$$ -\operatorname{div}A(x,u,\nabla u) \ge f(u)\qquad \text{in}\quad \mathbb R^n, $$
where $n\geqslant 2$ and $A$ is a Carathéodory function such that
\begin{gather*} \bigl(A(x,s,\zeta)-A (x,s,\xi)\bigr)(\zeta-\xi)\ge 0, C_1|\xi|^p \leqslant\xi A(x,s,\xi),\qquad |A(x,s,\xi)| \leqslant C_2|\xi|^{p-1},\qquad C_1,C_2>0,\quad p>1, \end{gather*}
for almost all $x\in\mathbb R^n$ and all $s\in\mathbb R$ and $\zeta,\xi\in\mathbb R^n$.