I have asked this question several months ago, I have understood every thing and there are good comments and they have helped me , but only I have a question about tomas comment, how can Calderón-Zygmund estimate prove that $\{u_n \}$ is bounded in $ W^{3,q} $.
And I have another new question, can some one says that why $v$ , $u$ are a positive solutions and why we can consider $u_n >0$ and $u_n \in L^{\infty}$ .
I have not changed the question, and it was :
Can someone see the 10th line of page 9 in this article and give a hint that why $$ \nabla v_n \to \nabla v \ \ (a.e.)$$ and $$ v_n \to v $$ and how with theorem 2.1 we could conclude there exists $ u \in W^{2,q}(\Omega) $ such that $ v=-\Delta u $
Thanks