Let $U \subset \mathbb{R}^n$ be an open ball and $p^* = \frac{pn}{n-p}$ be the Sobolev conjugate. Show that, for $p=2$, $W^{1,p}(U)$ cannot be compactly embedded in $L^{p^*}(U)$.
Now, from the Sobolev inequality, for bounded $U$, we have that $$\|u\|_{L^{p^*}(U)} \leq C\ \|u\|_{W^{1,p}(U)}$$ for $1 \leq p < n$ and $u \in W^{1,p}(U)$. That means that $W^{1,2}(U)$ is continuously embedded in ${L^{p^*}(U)}$, so to prove the above claim we would need to show that a bounded sequence in $W^{1,2}(U)$ is not precompact in ${L^{p^*}(U)}$.
That is the point that I am missing, how do I proceed with this? Do I need a counterexample or a proof? I am missing the intuition of what's happening and I cannot get it started. Any hint would be appreciated.
Also, there is no mention of boundedness on $U$, but I am not sure if it is by mistake or by intention. Even for $U$ bounded, I think it is still not a compact embedding due to the Rellich–Kondrachov theorem (Evans section 5.7 Theorem 1).
Take $u_n(x)=a_n (1-n|x|)$ if $|x|<1/n$ and $u_n(x)=0$ if $|x|\ge 1/n$ and show that you can take $a_n$ is such a way that the sequence in bounded in $W^{1,p}(B(0,1))$ but no subsequence converges in $L^{p^*}(B(0,1))$. So the idea is that the mass is concentrating at the origin.