I am studing several results about existence of weak solution for PDE's. Critical points of appropriate functional are used.
In order to prove one of this results I have a sequence $(u_n)_{n \in \mathbb{N}}$ in the space $H^1_0(\Omega)$ such that converges weakly to some $u \in H^1_0(\Omega)$, where $\Omega $ is a bounded open subset of $\mathbb{R}^N$. I need to get that this sequence (or maybe a subsequence if necessary) converges strongly in the space $L^{2^*}(\Omega)$, where $2^*=\frac{2N}{N-2}$ is the critical Sobolev exponent. The book doesn't suggest prove this but if this is true I'll have completed my work. Is it really true?. The problem is that we don't have that the embedding $H^1_0(\Omega)↪L^{2^*}(\Omega)$ is compact.
Thanks in advance.
This is not true.
It is known that the embedding of $H^1_0$ into $L^{2^*}$ is not compact, see for instance Counterexample to Rellich-Kondrachov Compactness Theorem, case $q=p^*$, so there is a sequence $f_n$ which is bounded in $H^1_0$ and has no $L^{2^*}$-convergent subsequence. Since $H^1_0$ is separable, by Alaoglu's theorem norm-bounded sets are weakly compact metrizable, so there is a subsequence $f_{n_k}$ converging weakly in $H^1_0$, but as noted above, neither $f_{n_k}$ nor any further subsequence converges in $L^{2^*}$.
Indeed, one can show that an operator from a separable Hilbert space to a Banach space is compact if and only if it maps weakly convergent sequences to strongly convergent ones. One direction is essentially shown above, and the other follows from the uniform boundedness principle's corollary that weakly convergent sequences are bounded in norm.