Can $u\in W^{1,2}_0$, $\Delta u\in L^2$, $u\geq 0$ be approximated by a sequence smooth function $u_k\geq 0$

Question

Assume that $\Omega\in R^3$ is a bounded Lipschitz domain. $u\in W^{1,2}_{0}$, $\Delta u\in L^2$, $u\geq 0$. Is it possible to approximate u by a sequence of nonnegative smooth functions $u_k$, $u_k|_{\partial \Omega}=0$ such that $\|u-u_k\|_{L^2}+\|\Delta (u-u_k)\|_{L^2}\to 0$?

Thanks for any assistance.