Let $\Omega \subset \mathbb{R}^d$ be a domain of finite Lebesgue measure, not assumed to be smooth or bounded. Is it true that the embedding of, say, $W^{1,p}_0(\Omega)$ (Sobolev functions with zero boundary value) into $L^q(\Omega)$ is compact for $1/q > 1/p  1/d$?
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You can find a rather general answer in the book of Adams and Fournier, Theorem 6.16. You do have the embedding you wish, under a mild measure theoretical assumption. 

