I have managed to show that we have a subsequence converges to a function $\bar{u}$ on any compact subsets and it agrees with $u$ a.e. I have some trouble extending the argument to show that $u$ is smooth. More precisely, $u$ agrees with a smooth function a.e. I am wondering if anyone could help me with that?
I thought $\bar{u}$ depends on $\Omega.$ Maybe I was thinking somethign wrong.
I have made another attempt to solve this problem:
