On blow-up conditions for solutions of differential inequalities with $\varphi$-Laplacian

TL;DR

This paper establishes conditions under which solutions to certain differential inequalities involving the $ ext{ } ext{$ extphi$-Laplacian} ext{ }$ blow up, extending understanding of solution behavior in unbounded domains.

Contribution

It provides new blow-up criteria for non-negative solutions of inequalities with the $ extphi$-Laplacian, generalizing previous results to broader classes of functions and domains.

Findings

Derived explicit blow-up conditions for solutions.

Extended blow-up analysis to unbounded domains.

Applicable to a wide class of $ extphi$-Laplacian problems.

Abstract

Let $\Omega \ne \emptyset$ be an unbounded open subset of ${\mathbb R}^n$, $n \ge 2$. We obtain blow-up conditions for non-negative solutions of the problem $$ {\mathcal L}_\varphi u \ge F (x, u) \quad \mbox{in } \Omega, \quad \left. u \right|_{ \partial \Omega } = 0, $$ where $\varphi$ and $F$ are some function and $$ {\mathcal L}_\varphi u = \operatorname{div} \left( \frac{ \varphi (|\nabla u|) }{ |\nabla u| } \nabla u \right) $$ is the $\varphi$-Laplace operator.