|
1
|
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? |
||||||||
|
|
0
|
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. |
||
|
|
