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

TL;DR

This paper investigates conditions under which positive solutions exist globally for a class of second-order quasilinear elliptic inequalities in Euclidean space, extending understanding of solution behavior for nonlinear PDEs.

Contribution

It establishes new criteria for the existence of global positive solutions to second-order quasilinear elliptic inequalities involving Carathéodory functions.

Findings

Derived sufficient conditions for solution existence.

Extended previous results to broader classes of inequalities.

Provided insights into the structure of solutions in unbounded domains.

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 Carath\'eodory function such that $$ (A (x, s, \zeta) - A (x, s, \xi))(\zeta - \xi) \ge 0, $$ $$ C_1 |\xi|^p \le \xi A (x, s, \xi), \quad |A (x, s, \xi)| \le C_2 |\xi|^{p-1}, \quad C_1, C_2 > 0, \; p > 1, $$ for almost all $x \in {\mathbb R}^n$ and for all $s \in {\mathbb R}$ and $\zeta, \xi \in {\mathbb R}^n$.