# Does these convergences hold true? (In unbounded domains, too?)

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?

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 $$L^p(\mathbb R)$$.