On properties of solutions of quasilinear second-order elliptic inequalities

Abstract

Let $\Omega$ be an unbounded open subset of ${\mathbb R}^n$, $n \ge 2$, and $A : \Omega \times {\mathbb R}^n \to {\mathbb R}^n$ be a function such that $$ C_1 |\zeta|^p \le \zeta A (x, \zeta), \quad |A (x, \zeta)| \le C_2 |\zeta|^{p-1} $$ with some constants $C_1 > 0$, $C_2 > 0$, and $p > 1$ for almost all $x \in \Omega$ and for all $\zeta \in {\mathbb R}^n$. We obtain blow-up conditions and priori estimates for inequalities of the form $$ {\rm div} \, A (x, D u) + b (x) |D u|^\alpha \ge q (x) g (u) \quad \mbox{in } \Omega, $$ where $p - 1 \le \alpha \le p$ is a real number and, moreover, $b \in L_{\infty, loc} (\Omega)$, $q \in L_{\infty, loc} (\Omega)$, and $g \in C ([0, \infty))$ are non-negative functions.