On properties of solutions of quasilinear second-order elliptic inequalities

Abstract

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