This is a question related to the PDE book by Evans. Consider $\Omega=B^n(0, 1)$ and the function $$u(x)=\sum_{k=1}^{\infty}\frac{1}{2^k}|x-r_k|^{-\alpha},$$ where $r_k$ is a countable dense subset of $\Omega$. Evans then claims that $u\in W^{1, p}(\Omega)$ iff $\alpha<\frac{n-p}{p}$ and is not bounded in any open subset $U\subset\Omega$ when $0<\alpha<\frac{n-p}{p}$. I think the first claim comes from the fact that $x\mapsto |x|^{-\alpha}$ belongs to $W^{1, p}(\Omega)$ iff $\alpha<\frac{n-p}{p}$. But I don't understand the second one, would someone kindly explain it to me?
So if $n\geq 3$ the function $u$ is in Hilbert space $H^1(\Omega)$ but is not bounded in any open subset. Is there a similar example when $n=2$?
To show $u\in W^{1,p}$ consider the partial sums
$$u_N(x)=\sum_{k=1}^N \frac{1}{2^k}|x-r_k|^{-\alpha}$$
and show they form a Cauchy sequence in $W^{1,p}$. Each term in the sum is unbounded at the rational number $r_k$, so $u(x)$ is unbounded on the rationals, which are dense in the reals (every open subset contains a rational $r_k$).