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.