  We study systems of the differential inequalities $$
  \left\{
  \begin{aligned}
  &
  - \operatorname{div} A_1 (x, \nabla u_1)
  \ge
  F_1 (x, u_2)
  &
  \mbox{in } {\mathbb R}^n,
  &
  - \operatorname{div} A_2 (x, \nabla u_2)
  \ge
  F_2 (x, u_1)
  &
  \mbox{in } {\mathbb R}^n,
  \end{aligned}
  \right. $$ where $n \ge 2$ and $A_i$ are Caratheodory functions such that $$
  C_1
  |\xi|^{p_i}
  \le
  \xi
  A_i (x, \xi),
  \quad
  |A_i (x, \xi)|
  \le
  C_2
  |\xi|^{p_i - 1},
  \quad i = 1,2, $$ with some constants $C_1, C_2 > 0$ and $p_1, p_2 > 1$ for
almost all $x \in {\mathbb R}^n$ and for all $\xi \in {\mathbb R}^n$, $n \ge
2$. For non-negative solutions of these systems we obtain exact blow-up
conditions.
