I am studying the Compactness lemma ( on page 570) of the article http://projecteuclid.org/euclid.cmp/1103922134.
The lemma says
(Compactness lemma ): for $0 < \sigma < \frac{2}{N-2}$, $(N \geq 3)$, the imbedding
$$ H^{1}_{rad}(R^n) \rightarrow L^{2 \sigma +2} (R^n)$$ is compact, where $H^{1}_{rad}(R^n)$ is the radial functions of $H^{1}(R^n)$.
Proof of the lemma:
Consider the interpolation estimate
$$ || u ||^{2 \sigma +2}_{2 \sigma +2} \leq C || u||^{\sigma N}_{H^{1}(R^n)} || u||_{L^2(R^n)}^{2 + \sigma(2-N)} , 0 < \sigma < \frac{2}{N-2} $$
(this is the classical Gagliardo - Nirenberg - Sobolev inequality ).
If we can show that a bounded sequence in $H^{1}_{rad}(R^n)$ is uniformly small at infinity , then follows the result (*).
This uniformity follows from the inequality
$$|u(x)| \leq \frac{C}{|x|^{\frac{N-1}{2}}} || u||_{H^1} (**)$$
The inequality of $(**)$ is given in a article. But i am not understanding the affirmation of $(*)$ Someone can help me to understand?
thanks in advance
It suffices to establish that any sequence $\,u_k=u_k(|x|)\to 0\,$ weakly convergent in $\,H_{rad}^1(\mathbb{R}^n)\,$ will be srongly convergent in $\,L^{2\sigma+2}(\mathbb{R}^n)$, which might be ruined by the unboundedness of $\,\mathbb{R}^n$ if it were not for the remarkable inequality $\,(\ast\ast)\,$ that provides cutting off the infinity. Indeed, given any $\varepsilon >0$, choose $R_{\varepsilon}>1$ such that $$ \int\limits_{|x|>R_{\varepsilon}}\Bigl|u_k(|x|)\Bigr|^{2\sigma+2}dx\leqslant \Biggl(\int\limits_{\,\,|x|>R_{\varepsilon}}\Bigl|u_k(|x|)\Bigr|^{2}dx\Biggr)\!\cdot\!\Biggl(\frac{\|u_k\|_{H^1(\mathbb{R}^n)}}{R_{\varepsilon}^{\frac{n-1}{2}}}\Biggr)^{\!2\sigma}\! \leqslant\\ \leqslant R_{\varepsilon}^{-\sigma (n-1)}\!\!\cdot\!\Bigl(\sup\limits_{k\geqslant 1}\|u_k\|_{H^1(\mathbb{R}^n)}\Bigr)^{2\sigma+2}<\frac{\varepsilon}{2} \quad \forall\, k\geqslant 1, $$ and notice that there is $\,m_{\varepsilon}\in \mathbb{N}\,$ such that $$ \int\limits_{|x|<R_{\varepsilon}}\Bigl|u_k(|x|)\Bigr|^{2\sigma+2}dx <\frac{\varepsilon}{2} \quad \forall\, k>m_{\varepsilon} $$ due to the compact embedding $\,H^1(B_{R_{\varepsilon}})\hookrightarrow L^{2\sigma+2}(B_{R_{\varepsilon}})\,$ for a ball $\,B_{R_{\varepsilon}}= \{x\in\mathbb{R}^n\,\colon\,|x|<R_{\varepsilon}\} $.