I need your help to answer this question:
Let $\psi\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)$ and $u\in W_{0}^{1,p}(\Omega)$, with $u\geq\psi$ a.e. in $\Omega$. There exits a sequence $u_{n}\in C_{0}^{\infty}(\Omega)$ such that $u_{n}\geq\psi$ a.e. in $\Omega$, and $u_{n}\to u$ strongly in $W_{0}^{1,p}(\Omega)$?
This is not possible.
Consider $\Omega=(0,1)$ and $u = \psi=x(1-x)$.
Then there is no function $u_n\in C_0^\infty(\Omega)$ with $u_n\geq \psi$ a.e., because $u_n$ needs to have a compact support and has to be equal to $0$ on a set of positive measure.