Please help me regarding this question: Let $1<p<\infty$ and $\Omega\subset\mathbb{R}^N$ be a smooth bounded domain. Given a uniformly bounded sequence in $W^{1,p}_{loc}(\Omega)$. Then upto a subsequence $ u_n\to u\,\,weakly\,\,in\,\,W^{1,p}_{loc}(\Omega) $
$
u_n\to u\,\,strongly\,\,in\,\,L^p_{loc}(\Omega)
$
and
$
u_n\to u\,\,a.e.\,\,in\,\,\Omega.
$
If this is true , can You give a clear explanation how to prove it?\
Your help is very much appreciated.
If $A\subset\Omega$ is a smooth subset of $\Omega$ with $\overline{A}\subset\Omega$, the injection from $W^{1,p}(A)$ to $L^p(A)$ is compact (by the Rellich-Kondrachov theorem).
You can construct a sequence $(A_j)$ of smooth domains invading $\Omega$. Since $(u_n)$ is bounded in $W^{1,p}(A_j)$, you can extract a subsequence strongly convergent in $L^p(A_j)$ (and also pointwise a.e.). Moreover, up to another subsequence, you have also weak convergence in $W^{1,p}(A_j)$ (since $1<p<\infty$).
Using a diagonal argument, you can construct a subsequence of $(u_n)$ that weakly converges in $W^{1,p}(A_j)$ adn strongly converges in $L^p(A_j)$ for every $j$ (and pointwise a.e. on $\bigcup_j A_j = \Omega$).