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 $$C_1|\zeta {|}^p\le \zeta A(x,\zeta ),|A(x,\zeta )|\le C_2|\zeta {|}^{p-1}$$ 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 $$ {\rm div} \, A (x, D u) + b (x) |D u|^α \ge q (x) g (u) \quad \mbox{in } Ω, $$ 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.