Let $g: \Omega\times C^2(\Omega)\to\mathbb{R}, (x,u)\mapsto u(x)|u(x)|^{p-2}$, where $\Omega$ is a smooth, bounded domain in $\mathbb{R}^n$ and $2<p<\frac{2n}{n-2}$. I found a proof, that $K:H_0^1(\Omega)\to H^{-1}(\Omega), K(u) = g(\cdot, u)$ is compact, but I have some problems understanding it.
First of all it is said that $u\mapsto g(\cdot, u)$ maps bounded sets in $L^p(\Omega)$ into bounded sets in $L^{p/p-1}(\Omega) \subset H^{-1}(\Omega)$. (1)
Then, because of this and since Rellich-Kondrakov gives that $H_0^1(\Omega)$ embeds into $L^p(\Omega)$ compactly (2), the map $K:H_0^1(\Omega)\to H^{-1}(\Omega), K(u) = g(\cdot, u)$ is compact.
I can't follow that argumentation.
What we want to show is that $K$ maps bounded sets into relative compact sets.
So first of all we know by Rellich-Kondrakov, that a bounded set $M$ in $H_0^1(\Omega)$ is relative compact in $L^p(\Omega)$. Why does (1) then gives me relative compactness of the image of $M$ in $L^{p/p-1}(\Omega)$?
Can someone explain me the steps in between?