Let $ U=B_1(0)$ be the unit ball in $ \Bbb{R}^n$ , $ 1≤p<n$ , $ p^∗=\frac{np}{n−p}$ . Consider the sequence:

\begin{equation} \label{eq:aqui-le-mostramos-como-hacerle-la-llave-grande} u_m = \left\{ \begin{array}{ll} m^{\frac{n}{p}-1}(1-m|x|) & \mathrm{if\ } x <1/m \ 0 &\mathrm{if\ } x\geq 1/m\ \end{array} \right. \end{equation} Prove that $ \{u_m\}$ is bounded in $ W^{1, p}(U)$ , but does not possess a (norm-)convergent sequence in $ L^{p^∗}(U)$ .

I’ve managed to prove that $ \{u_m\}$ is bounded in $ W^{1, p}(U)$ , and I’ve tried to prove the non-existance of a convegent subsequence via completeness of $ L^{p^∗}(U)$ and taking a Cauchy series. So far I have gotten nothing. Any hints?