TITLE: Problem 7 - Chapter 6 - Evans' PDE (Second Edition)
QUESTION [15 upvotes]: Problem 7 in §6.6 states as follows:

Let $u\in H^1(\mathbb{R}^n)$ have compact support and be a weak solution of the semilinear PDE
  $$-\Delta u+c(u)=f\,\,\text{ in } \mathbb{R}^n,$$
  where $f\in L^2(\mathbb{R}^n)$ and $c:\mathbb{R}\to\mathbb{R}$ is smooth, with $c(0)=0$ and $c'\geqslant 0.$ Prove $u\in H^2(\mathbb{R}^n)$.

I am trying to prove $c(u(x))\in L^2(\mathbb{R}^n)$ or $c'(u(x))\in L^{\infty}(\mathbb{R}^n)$ but both failed. I know in fact $u$ can be in $H_0^1(\mathbb{R}^n)$, but I don't know whether this is useful here. And $c(u(x))$ has compact support since $u$ has compact support. However, again, I can't figure out that $u\in L^{\infty}(\mathbb{R}^n)$. And I really don't know how to use the condition $c'\geqslant 0$. 
In conclusion, I've no idea about this problem.
Anyone could help me? Any advice will be appreciated.

REPLY [3 votes]: First step:
Since $u$ has compact support and $c(0)=0$, and $c(x)$ is continuous, $c(u(x))$ is continuous and has compact support and so it is in $L^2(R^n)$.
Second step:
Take Fourier transform of $$-\Delta u+c(u(x))=f(x)$$
Then you can get $$|\xi|^2\hat u(\xi)+\widehat{c(u(x))}=\hat f(\xi)$$
It follows from the first step that $\widehat {c(u(x))} \in L^2$. Thus $|\xi|^2\hat u(\xi) \in L^2$
We also know $\hat u \in L^2$
So $(1+|\xi|^2)\hat u(\xi) \in L^2$
It follows that $u \in H^2$.