Counter example for Poincare inequality does not hold on unbounded domain The Poincare inequality states that if domain $\Omega$ is bounded in one direction by length $d>0$ then for any $u\in W_0^{1,p}(\Omega)$ we have 
$$ \int_\Omega|u|^p\,dx\leq \frac{d^p}{p}\int_\Omega |\nabla u|^p\,dx $$
Now I assume the domain $U\subset \mathbb R^N$ contains a sequence $B(x_n,r_n)$, where $x_n\in U$ and $r_n\to \infty$, then I want to prove that the previous Poincare's inequality fails on $W_0^{1,p}(U)$.
Yes, of course, if I have such balls in $U$ then the domain $U$ can not be bound in one direction, and hence I done. But honest, is that it? I feel uncomfortable with my argument... Is there something more going on? Could you help me to write a more serious argument?

Find functions $u$ compactly supported inside $B(x,r)$ so that $\int_{B(x,r)} |u|^p \, dx / \int_{B(x,r)} |\nabla u|^p \, dx$ goes to infinity as $r \to \infty$.  Probably something like
$$ u(y) = \prod_{k=1}^N \cos\bigl((y_k-x_k)\sqrt N \pi/(2r)\bigr) I_{|x_k-y_k| < r/\sqrt N} $$
