Let $\Omega\subset\mathbb{R}^N$ be a bounded domain of class $C^2$. Take $u\in W_0^{1,p}(\Omega)$ for $p\in (1,\infty)$ and assume that $$\frac{\partial u}{\partial \nu}(x)<0,\ \forall\ x\in \partial\Omega,$$
where $\nu$ is the exterior unitary normal vector through the boundary of $\Omega$ and I am assuming that the normal derivative does exist in the classical sense. For $\delta>0$, define $$\Omega_\delta=\{x\in \Omega:\ \operatorname{dist}(x,\partial\Omega)<\delta\},$$
where $\operatorname{dist}$ is the distance function. My question is the following:
Can we find $\delta>0$ and a sequence $u_n\in W_0^{1,p}(\Omega)$ such that $u_n\to u$ in $W^{1,p}(\Omega_\delta)$ and the functions $u_n$ are $p$-subharmonic in $\Omega_\delta$, that is, $$\int _{\Omega_\delta}|\nabla u_n|^{p-2}\nabla u_n\nabla \varphi\le 0,\ \forall\ \varphi\in C_0^\infty(\Omega_\delta).$$
Any idea is appreciated. I have nothing for this problem by now.