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?



Boundedness of a given boundary value problem.
Multiply your equation times the function $v(x) = \max(u-1,0)$. After you integrate by parts the Laplacian you will get an equality with opposite signs in each side that gives you a contradiction unless $v \equiv 0$.


Then do the same thing with $\min(u+1,0)$.


There are other ways to do it, including using sub/super-solutions.


