Let$\{u_k\}\in W^{1,p}(R^n)$, $p\in[1,n)$, $\sup_k||u_k||_{W^{1,p}(R^n)}
On compact embedding of Sobolev space?
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functional-analysis
sobolev-spaces
compactness
1 Answers
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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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0You are welcome @Darry – 2012-11-26