Assume that $\Omega\subset\mathbb{R}^N$ is a bounded Lipschitz domain and let $p\in [1,\infty)$. Suppose that $u\in W_0^{1,p}(\Omega)\cap L^\infty (\Omega)$.
Is it possible to approximate $u$ by a sequence of function $u_k\in C_0^\infty(\Omega)$ such that $\|u_k\|_\infty\leq M$ for some positive constant $M$?
I was trying to do the following. Extend $u$ by zero outside $\Omega$ and let $\eta_\delta$ be a mollifier sequence. Hence $u_\delta=\eta_\delta\star u$ is such that $u_\delta\in C_0^\infty (\mathbb{R}^N)$ and $u_\delta\to u$ in $W^{1,p}(\Omega)$.
Now for each $\delta>0$ let $\Omega_\delta=\{x\in \Omega:\ \operatorname{d}(x,\partial\Omega)\geq \delta\}$. Take $\lambda_\delta\in C_0^\infty(\Omega)$ such that $\lambda_\delta =1$ in $\Omega_\delta$, $\lambda_\delta\in [0,1]$ in $\Omega_{\delta}\setminus\Omega_{2\delta}$ and $\lambda_\delta =0$ in $\Omega\setminus \Omega_\delta$. Define $g_\delta=\lambda_\delta u_\delta$ and note that $g_\delta\in C_0^\infty(\Omega)$.
Does $g_\delta$ converges to $u$ in $W^{1,p}(\Omega)$?
Update: Note that $$\|g_\delta-u\|_p\leq \|\eta_\delta\star u-u\|_p+\|\lambda_\delta u-u\|_p\tag{1}$$
and $$\left\|\frac{\partial g_\delta}{\partial x_i}-\frac{\partial u}{\partial x_i}\right\|_p\leq \left\| \frac{\partial \lambda_\delta}{\partial x_i}(\eta_\delta\star u)\right\|_p+\left\|\eta_\delta\star \frac{\partial u}{\partial x_i}-\frac{\partial u}{\partial x_i}\right\|_p+\left\|\lambda_\delta \frac{\partial u}{\partial x_i}-\frac{\partial u}{\partial x_i}\right\|_p\tag{2}$$
By using Lebesgue theorem and the definition of $u_\delta$, we have that $(1)$ converges to $0$. By the same argument, we have that the two terms on the right of $(2)$ does converges to $0$. Hence, the question is: Can we choose $\lambda_\delta$ in such a way that $$\left\| \frac{\partial \lambda_\delta}{\partial x_i}(\eta_\delta\star u)\right\|_p\to 0$$
Here is another, simpler, approach. Let $m=\|f\|_{L^\infty}$. Fix a smooth function $\psi:\mathbb R\to\mathbb R$ such that $\psi(t)=t$ when $|t|\le m$, $\psi(t)=m+1$ when $t\ge m+1 $ and $\psi(t)=-m-1$ when $t<-m-1 $. Since $\psi $ is smooth (and $\Omega$ is bounded), the composition operator $g\mapsto \psi\circ g$ is Lipschitz on $W^{1,p}(\Omega)$. (It suffices to check the Lipschitz property on smooth functions, for which it follows from the chain rule.)
Since $u\in W^{1,p}_0(\Omega)$, there is a sequence $v_k$ of $C_c^\infty(\Omega)$ functions that converges to $u$ in $W^{1,p}$. Let $u_k=\psi\circ v_k$. By the above, $u_k\to \psi\circ u=u$ in $W^{1,p}$. And $|u_k|\le m+1$ by construction.