The following is the Rellich-Kondrachov Compactness Theorem in Evans's Partial Differential Equations
The author gives a remark as follows
I don't understand the first sentence in the Remark. Here are my questions:
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How does one get $W^{1,p}(U)\Subset L^p(U)$ for $p=n$?
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Why does "$p^*\to\infty$ as $p\to n$" imply the desired embedding?
Best Answer
Remember $U$ is bounded, so $W^{1,n}(U) \subset W^{1,p}(U)$ for all $1 \leq p < n$ by Holder. Since $p^*\to \infty$ as $p\to n$, we can choose a fixed $p<n$ close enough to $n$ so that $p^*>n$. Then by the Rellich-Kondrachov compactness thoerem
$$W^{1,n}(U) \subset W^{1,p}(U) \subset\subset L^n(U).$$
Arguing along the same lines, we actually have $W^{1,n}(U) \subset\subset L^q(U)$ for all $1 \leq q < \infty$.