Let $\Omega\in\mathbf{R}^N$ an unbounded domain and $u\in C^2(\Omega)\cap C(\overline\Omega)$, $u>0$ such that $\Delta u + f(u)=0, \ \ \ \mbox{em} \ \ \Omega,$ where $f$ is a bounded lipschitz continuous function. Then $u$ is bounded.
I don't know where I can find this result, and I believe that this assumptions implies $\nabla u$ is bounded too. Someone can help me?