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