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](https://math.stackexchange.com/questions/1157769/compactness-sobolev-embedding-for-radial-functions-on-mathbbrn) but received no answers.


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



Compact radial Sobolev embedding $H^1_{rad}\hookrightarrow L^p$
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.


