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Let $\Omega$ be an open bounded subset of $\mathbb{R}^N$, $n\ge 2$. Let $(u_n)_n\subset W_0^{1, p}(\Omega)$ be a bounded sequence in $W_0^{1, p}(\Omega)$. It implies, in general, that $$u_n\rightharpoonup u \quad\mbox{ in } W_0^{1, p}(\Omega)\\ u_n\to u \quad\mbox{ strongly in } L^{p}(\Omega), \; p\in [p, p^*[,$$ isn't it?

Could anyone please explain me why the strong convergence holds too? Or could you please give some references?

If we set $\Omega$ to be unbounded, does the same convergences hold?

I hope could someone please help. Thank you in advance!

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In general, $u_{n'}\rightharpoonup u$ only for a subsequence, and only for $p\in (1,\infty)$, since this weak compactness result needs reflexivity of the space.

Strong convergence of a subsequence holds if $\Omega$ is bounded due to Rellich theorem (compact embeddings).

If $\Omega$ is unbounded, then there is still a weakly convergent subsequence in $W^{1,p}$ provided $p\in (1,\infty)$. But no strongly convergent subsequences in $L^p$. Counterexample: Fix $u_0\in W^{1,p}(\mathbb R)\setminus\{0\}$, define $u_n(x):=u_0(x+n)$. Then $u_n\rightharpoonup 0$ in $W^{1,p}(\mathbb R)$ but $u_n \not\to 0$ in $L^p(\mathbb R)$.

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  • $\begingroup$ Could you please give me a reference? $\endgroup$
    – C. Bishop
    May 10, 2021 at 8:47
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    $\begingroup$ The textbook by Evans is a good reference. To get a first impression, just use wikipedia ;) $\endgroup$
    – daw
    May 10, 2021 at 10:24

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