Assume that $\Omega\subset\mathbb{R}^N$ is a bounded regular domain. Let $1<p<\infty$ and take $u\in W_0^{1,p}(\Omega)$. Is true that given $\epsilon>0$ there is a neighbourhood $V$ of $\partial\Omega$ such that $$|u(x)|\leq \epsilon,\ \mbox{a.e. in}\ V \ \ ?$$
If $N=1$ then the above is true, but for $N\ge 2$, I could not prove it. Now I am trying to find a counter example.
Any idea is appreciated.
Here is a sketch of a counterexample:
Let $\{x_n\}$ be a sequence of distinct points in $\Omega$ converging to a point $x \in \partial \Omega$.
For each $n$ let $f_n \in W_0^{1,p}(\Omega)$ be a positive function that is unbounded in a neighborhood of $x_n$ and satisfies $\|f_n\|_{W^{1,p}(\Omega)} < 2^{-n}$. (You can do this if $p < N$).
Let $f = \sum f_n$.