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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.

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    Have you tried a Fourier transform approach? In Fourier space the problem boils down to prove $\mathscr{F}[c(u)] \in L^2(\mathbb{R}^n)$, which looks more manageable. What do you think?2011-12-26
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    @GiuseppeNegro: Thanks. I haven't tried this approach before. Acturally I'm not familiar with Fourier transform, I mean I know some basic facts about it but I've never done any exercises. Here I still do not know how to deal with the $c(u(x))$ under the integral. I think if I can deal with this integral I can also deal with the former approach. Do you have some more details?2011-12-26
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    @GiuseppeNegro Could you please add more details about the Fourier transform approach? Thanks!2015-04-10
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    @Sherry: I am afraid I cannot answer right now. I don't remember what was my idea back in '11, and frankly, there is a big possibility that it was just plain wrong.2015-04-10

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