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

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.