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

We study systems of the differential inequalities $$\{\begin{array}{rlrlr}{\displaystyle }& {\displaystyle -\mathrm{div}{A}_1(x,\mathrm{\nabla }{u}_1)\ge F_1(x,u_2)}& {\displaystyle in{\mathbb{R}}^n,}& {\displaystyle -\mathrm{div}{A}_2(x,\mathrm{\nabla }{u}_2)\ge F_2(x,u_1)}& {\displaystyle in{\mathbb{R}}^n,}\end{array}$$ where $n\ge 2$ and $A_i$ are Caratheodory functions such that $$C_1\mathrm{\mid }\xi {\mathrm{\mid }}^{p_i}\le \xi A_i(x,\xi ),\mathrm{\mid }{A}_i(x,\xi )\mathrm{\mid }\le C_2\mathrm{\mid }\xi {\mathrm{\mid }}^{p_i-1},i=1,2,$$ 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.