On global solutions of quasilinear second-order elliptic inequalities

Abstract

We consider the inequality $$ - \operatorname{div} A (x, \nabla u) \ge f (u) \quad \mbox{in } {\mathbb R}^n, $$ where $n\ge 2$ and $A$ is a Caratheodory function such that $$ C_1 |ξ|^p \le ξ A (x, ξ) \quad \mbox{and} \quad |A (x, ξ)| \le C_2 |ξ|^{p-1} $$ with some constants $C_1>0$, $C_2>0$, and $p>1$ for almost all $x\in {\mathbb{R}}^n$ and for all $\xi \in {\mathbb{R}}^n$. Our aim is to find exact conditions on the function $f$ guaranteeing that any non-negative solution of this inequality is identically zero.