I need your help to answer this questions:
Let $\Omega$ be a bounded open subset of $\mathbb{R}^{n}$, and let $u\in W^{1,p}_{0} (\Omega)\cap L^{\infty}(\Omega)$. There exists $u_{n}\in C^{\infty}_{0}(\Omega)\cap L^{\infty}(\Omega)$ converging to $u$ strongly in $ W^{1,p}_{0} (\Omega)$?
If $u\in W^{1,p}_{0} (\Omega)$, with $u\geq 0$. There exists a sequence of nonnegative smooth functions $\{u_{n}\}$ converging to $u$ strongly in $ W^{1,p}_{0} (\Omega)$?
For your first question, any function in $C^\infty_0(\Omega)$ is in $L^\infty(\Omega)$ (continuous function on compact sets are bounded). Do you want the sequence to be bounded? In that case, see this question.
For your second question: this has been answered here.