Let $p>1$ and $\Omega$ be an open bounded domain in $\mathbb{R}^N$. Take $(u_n)\subset W_0^{1, p}(\Omega)\cap L^{\infty}(\Omega)$ and $u\in W_0^{1, p}(\Omega)\cap L^{\infty}(\Omega)$.
If $$u_n\longrightarrow u \quad\mbox{ in } W_0^{1, p}(\Omega),$$ it it true that $u_n\longrightarrow u$ in the classical sense (up to subsequences)?
Could anyone please help me to understand?
Thank you in advance!
If $u_n\rightarrow u$ in $W^{1,p}_0,$ then (in particular) it converges in $L^p$. If $u_n\rightarrow u$ in $L^p$, then there exists a subsequence $(u_{n_j})$ of $(u_n)$ with the property that $u_{n_j}\rightarrow u$ pointwise almost everywhere. The proof of the latter fact comes in when demonstrating that $L^p$ is complete.