I have this functional, $W\in C^2(\mathbb{R}^d), u \in C^1(\mathbb{R}^d)$ $$I_\varepsilon[u]:=\int_{\mathbb{R}^d}W(u)+\frac{\varepsilon}{2}||\nabla u||^2dx$$ I derived the Euler-Lagrange eqations which is $$-\varepsilon \Delta u+W'(u)=0$$ Now we have this boundary problem with $u:\mathbb{R}\rightarrow \mathbb{R}$:
$$-\varepsilon u''+W'(u)=0; \ u(\pm \infty)=\pm 1$$ With W having the following properties:
- $W\in C^2(\mathbb{R})$
- $W(u)>0 \ \forall u \in \mathbb{R}\setminus\{-1,1\}$
- $W(\pm 1)=W'(\pm 1)=0$
Show that every solution $u_\varepsilon$ is monotone and $u_\varepsilon(x)\in ]-1,1[$
I tried arguing with the functional or taking the derivative to get $u'$ but it didnt work like I expected.