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Let$\{u_k\}\in W^{1,p}(R^n)$, $p\in[1,n)$, $\sup_k||u_k||_{W^{1,p}(R^n)}<C$, then $\forall r>0$ and $\forall q\in[1,p^*)$,where $p^*=\frac{np}{n-p}$, there exist a subsequence $\{u_{k_i}\}$ of $\{u_k\}$ such that $\{u_{k_i}\}$ is convergence in $L^q(B_r(0))$.

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up vote 2 down vote accepted

If you know that $W^{1,p}(B_r(0))$ is compactly embedded in $L^q(B_r(0))$ for $q\in[1,p^\star)$ then the result follows, because $\|u_k\|_{1,p}(\mathbb{R}^n)\leq C$ implies that $$\|u_k\|_{1,p}(B_r(0))\leq C$$

You can find the proof of the first statement in the book of Adams or Leoni:

R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975.

G. Leoni, A First Course in Sobolev Spaces, Graduated Studies in Mathematics; v.105

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@ Tomás:Thank you! I find the "new" fact for me in the second book you recommended. The new knowledge combining with your hint help me solve the problem! :-) – Darry Nov 26 '12 at 14:10
You are welcome @Darry – Tomás Nov 26 '12 at 20:22

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