On blow-up conditions for solutions of systems of quasilinear second-order elliptic inequalities

We study systems of the differential inequalities $$ \{ {aligned} & - div A_1 (x, ∇ u_1) ≥ F_1 (x, u_2) & in { R}^n, & - div A_2 (x, ∇ u_2) ≥ F_2 (x, u_1) & in { R}^n, {aligned} . $$ where $n ≥ 2$ and $A_i$ are Caratheodory functions such that $$ C_1 |ξ|p_i ≤ ξ A_i (x, ξ), |A_i (x, ξ)| ≤ C_2 |ξ|p_i - 1, i = 1,2, $$ with some constants $C_1, C_2 > 0$ and $p_1, p_2 > 1$ for almost all $x ∈ { R}^n$ and for all $ξ ∈ { R}^n$, $n ≥ 2$. For non-negative solutions of these systems we obtain exact blow-up conditions.