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-existence of a convegent subsequence via completeness of $L^{p^∗}(U)$ and taking a Cauchy series. So far I have gotten nothing. Any hints?
The sequence converges pointwise a.e. to zero. But $\|u_m\|_{L^{p^*}} \not\to0$. Hence $(u_m)$ cannot contain a subsequence that strongly converges in $L^{p^*}$, since pointwise a.e. limits and strong limits coincide (if they exist).