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Is there a counterexample of Sobolev Embedding Theorem? More precisely, please help me construct a sobolev function $u\in W^{1,p}(R^n),\,p\in[1,n)$ such that $u\notin L^q(R^n)$, where $q>p^*:=\frac{np}{n-p}$.^-^

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up vote 8 down vote accepted

Here is how you can do this on the unit ball $\{x | \|x \| \le 1\}$: Set $u(x) = \|x\|^{-\alpha}$. Then $\nabla u$ is easy to find. Now you can compute $\|u\|_{L^q}$ and $\|u\|_{W^{1,p}}$ using polar coordinates. Play around until the $L^q$ norm is infinite while the $W^{1,p}$ norm is still finite.

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To Hans Engler: Thank you for your help! I have done the problem by your hint! By the way, can you give me some hint to construct the corresponding version on the hole space $R^n$ if it is possible. Thanks again! :-) – Darry Nov 25 '12 at 2:15
To Hans Engler: Maybe we need to use Fourier analysis tool to construct the counterexample in a littile sophisticated form. How do you think? – Darry Nov 25 '12 at 2:58
@Darry: Just modify the argument you have. E.g. extend $u$ to all of $\mathbb{R}^n$ by defining it as $u(x) = 2-\|x\|$ for $1 \le \|x \| \le 2$ and $u(x) = 0$ for $\|x \| > 2$. – Hans Engler Nov 25 '12 at 14:39
@ Hans Engler:Thank you! ^-^ – Darry Nov 26 '12 at 2:00

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