What is an example of a function that is in $W^{1,2}(\mathbb{R}^3)$ but not in $L^1(\mathbb{R}^3)$?
I don't really know how to approach this question. I know there is this result that says that the function $f:B(0,1) \subset \mathbb{R}^n \to \mathbb{R}$ given by $f(x)=|x|^{\alpha}$ for $x \neq 0$ and $f(0)=0$ is weakly differentiable if and only if $\alpha>1-n$. I was thinking that maybe there is a similar way to construct an example for this problem, but I have no idea.