Let $u$ be a $C^2$-solution of $\Delta u=u^3-u$ on a bounded domain $\Omega$ with $u=0$ on $\partial\Omega$. How can one show that $-1\leq u\leq 1$ for every $x\in\Omega$? Also is it possible for $u$ to achieve values $\pm1$? I am think of using maximum principle but have no clue. Could someone help me with this? Thank you in advance.
[Math] Maximum Principle for a Poisson Equation
partial differential equationspoisson's equation
Best Answer
Partial answer: If $u(x_0)=\max u\,.$ Notice that $x_0$ is necessarily in the interior of the domain unless the function is $\le 0\,,$ in which case of course $u(x)\le 1\,.$ Otherwise there are points where $u$ is positive. Then $0\ge\Delta u(x_0)=u(x_0)^3-u(x_0),$ hence $u(x_0)\le1\,.$ An analogous argument shows that $\min u\ge -1\,.$