Let $\Omega$ be an open bounded domain in $\mathbb{R}^N$ and $p\geq 2$. Let $(u_n)_n\subset W_0^{1, p}(\Omega)$ such that $u\in W_0^{1, p}(\Omega)$ exists so that $$ u_n\rightharpoonup u \quad\mbox{ in } W_0^{1, p}(\Omega). $$
I don't know if it is a stupid question or not, but in these hypotheses, we can say that $u$ is bounded in $W_0^{1, p}(\Omega)$? I mean, it is true that there exists $M>0$ such that $\| u\|_{W^{1, p}}\leq M$?
Since $u\in W_0^{1, p}(\Omega)$, thus $\| u\|_{W_0^{1, p}}<+\infty$, so I maybe my guess holds. Could anyone please confirm (or not) that?
Thank you in advance!
One even have the following: If there exists $M \ge 0$ with $\|u_n\| \le M$, then $\|u\| \le M$. This follows from the fact that $$B_M := \{ v \in V | \|v\| \le M\}$$ is convex and closed, thus weakly closed.
(One even has $$\|u\| \le \liminf_{n\to\infty} \|u_n\|.$$ )