TITLE: A remark about the Rellich-Kondrachov Compactness Theorem in Evans's PDE book
QUESTION [9 upvotes]: 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:  


How does one get $W^{1,p}(U)\Subset L^p(U)$ for $p=n$?


Why does "$p^*\to\infty$ as $p\to n$" imply the desired embedding?

REPLY [5 votes]: 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 $1 \leq q < \infty$.