I've been given the following BVP: \begin{align*} -\Delta u = u- u^3,\: x\in \Omega \end{align*}\begin{align} u = 0,\: x\in \partial \Omega \end{align} where $\Omega\subset \mathbb{R}^N$ is bounded.

I am supposed to show that $-1< u(x)< 1$ for all $x\in\mathbb{R}^N$.

I have experimented with sub/sup solutions, but this yields something different, and I suspect it is (very) wrong.

Any thoughts/hints?

all$x\in\mathbb{R}^N$? In any case, I think your question is more suitable for MSE. – timur Feb 5 '13 at 15:45