  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 $p > 1$ for
almost all $x \in \Omega$ and for all $\zeta \in {\mathbb R}^n$. We obtain
blow-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.
