A $W^{1,p}$ function that is unbounded on any open subset of $B_1(0)$.

Question

I'm currently studying the properties of Sobolev Spaces in calculus of variations and functional analysis and was wondering if there is a function, that is $W^{1,p}$ but is unbounded on any open subset of $B_1(0)$.

Answer 1

One standard example is given in section 5.2.2 of Evans' book (p.260 in the second edition), and is $$u(x) = \sum_{k = 1}^{\infty} \frac{|x - r_k|^{-\alpha}}{2^k}$$ where $\{r_k\}_k$ is a countable, dense subset of your set $U$. This function is in $W^{1, p}(U)$ if $\alpha < (n - p) / p$.