I'm reading Evans' PDE book (second edition) and I tried to solve this problem but I'm a little confused
Problem 7, chapter 6:
Let $u\in H^1(\mathbb{R}^n)$ have compact support and be a weak solution of the semilinear PDE \begin{equation} -\Delta u+c(u)=f \text{ in } \mathbb{R}^n \end{equation} where $f\in L^2(\mathbb{R}^n)$ and $c:\mathbb{R}\to\mathbb{R}$ is smooth, with $c(0)=0$ and $c'\geq 0$. Prove $u\in H^2(\mathbb{R}^n).$
Any help is much appreciated!
My suggestion would be: $-\Delta u = f - c(u)$.
Now you are done if you can show that the RHS is in $L^2$ by elliptic regularity. To do this it is of course enough to show that $c(u)$ is in $L^2(\mathbb{R}^n)$. $$\int |c(u)|^2 = \int_{\text{supp}(u)}|c(u)|^2 \le \|c\|^2_{L^{\infty}(\text{supp}(u))}|\text{supp}(u)| < \infty$$
Do you think this work? Let me know! :)