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Let $\Omega$ be an unbounded open subset of ${\mathbb R}^n$, $n \ge 2$, and$A : \Omega \times {\mathbb R}^n \to {\mathbb R}^n$ be a function such that $$  C_1  |\zeta|^p  \le  \zeta  A (x, \zeta),  \quad  |A (x, \zeta)|  \le  C_2  |\zeta|^{p-1} $$ with some constants $C_1 > 0$, $C_2 > 0$, and $p > 1$ foralmost all $x \in \Omega$ and for all $\zeta \in {\mathbb R}^n$. We obtainblow-up conditions and priori estimates for inequalities of the form $$  {\rm div} \, A (x, D u)  +  b (x) |D u|^\alpha  \ge  q (x) g (u)  \quad  \mbox{in } \Omega, $$ where $p - 1 \le \alpha \le p$ is a real number and,moreover, $b \in L_{\infty, loc} (\Omega)$, $q \in L_{\infty, loc} (\Omega)$,and $g \in C ([0, \infty))$ are non-negative functions.

On properties of solutions of quasilinear second-order elliptic inequalities