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

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