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.
Best Answer
Here's my try.
The hint in Evans suggests to mimic the proof of the $H^2$ interior regularity theorem. This seems a very good hint, so let's try to follow it.
What do you think?