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

Abstract

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 $$C_1|\xi {|}^{p_i}\le \xi A_i(x,\xi ),|A_i(x,\xi )|\le C_2|\xi {|}^{p_i-1},i=1,2,$$ 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.