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I want to show: Let $N\geq 2$ and $2< q <2^\ast$. Then the embedding \begin{align} H^1_{\text{rad}}(\mathbb{R}^N)\hookrightarrow L^q(\mathbb{R}^N) \end{align} is compact.

I was able to show that \begin{align}|u(r)|\leq C R^{\frac{-(N-1)}{2}} \|\nabla u\|_2^{\frac{1}{2}} \|u\|_2^{\frac{1}{2}}\leq \hat C R^{\frac{-(N-1)}{2}} \|u\|_{H^1} \end{align} holds almost everywhere for $r\geq R$.

How can I conclude now? I think the idea would be to use the above estimate to be able to only restrict on a bounded domain and then use the usual Rellich-Kondrachov embedding. But how to make this rigorous? Do I need some cut-off?

I also posted the question on MSE but received no answers.

I would also be happy about any reference regarding this proof.

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It's Strauss embedding theorem for radially symmetric functions, proven here:

W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys. 55 (1977), 149-162.

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  • $\begingroup$ Maybe I am stupid but let $q=2+\epsilon$ then $|x|^{-q\frac{N-1}{2}} = |x|^{-(N-1)} |x|^{-\epsilon \frac{N-1}{2}}$. Then I think this is only integrable if $\epsilon \frac{N-1}{2} >1$. $\endgroup$
    – Peter
    Feb 21, 2015 at 22:16
  • $\begingroup$ Uhm... I got it mentally and I was sure it was. I'll remove the suggestion then. $\endgroup$
    – Pietro Majer
    Feb 21, 2015 at 22:41
  • $\begingroup$ Thanks anyway, an answer has been posted on MSE: math.stackexchange.com/questions/1157769/… $\endgroup$
    – Peter
    Feb 21, 2015 at 22:44

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