The behaviour of solutions of elliptic inequalities that are non-linear with respect to the highest derivatives

This paper deals with non-negative solutions of the elliptic inequalities $\operatorname{div} A(x,Du)\ge F(x,u)$ in $\Omega$, where $A\colon\Omega\times\mathbb R^n\to\mathbb R^n$ and $F\colon\Omega\times[0,\infty)\to[0,\infty)$ are functions and $\Omega$ is an unbounded open subset of $\mathbb R^n$, $n\geqslant2$.