Counterexample of Sobolev Embedding Theorem?
Solution 1:
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.