On the existence of global solutions of second-order quasilinear elliptic inequalities

Abstract

We study the existence of global positive solutions of the differential inequalities $$ - \operatorname{div} A (x, u, \nabla u) \ge f (u) \quad \mbox{in } {\mathbb R}^n, $$ where $n\ge 2$ and $A$ is a Carathéodory function such that $$(A(x,s,\zeta )-A(x,s,\xi ))(\zeta -\xi )\ge 0,$$ $$C_1|\xi {|}^p\le \xi A(x,s,\xi ),|A(x,s,\xi )|\le C_2|\xi {|}^{p-1},C_1,C_2>0,p>1,$$ for almost all $x\in {\mathbb{R}}^n$ and for all $s\in \mathbb{R}$ and $\zeta ,\xi \in {\mathbb{R}}^n$.