I am studying compact embedding in Brezis' books and I faced the following problem:
I have done the problem with $p > N$ and $p < N$, but i don't really know why the case $p = N$ reduces to the case $p < N$?
Thanks for your ideas.
2026-04-26 09:09:51.1777194591
Sobolev Compact Embedding in Brezis' books
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For all $s \in [1, p)$, you have $W^{1,p}(\Omega) \hookrightarrow W^{1,s}(\Omega)$. Thus, $W^{1,p}(\Omega) \hookrightarrow L^{s^*}(\Omega)$ and this gives you the desired embeddings.