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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?

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This appears to be a counterexample: $\Omega=\{(x,y)\in \mathbb R^2: x>1\}$; $u(x,y)=x$; $f\equiv 0$.

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    @JoséCarlos I think the key term for that is "Alexandrov-Bakelman-Pucci".2012-12-17