  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.
