Example of Function $u\in H^{-1}(\Omega)\setminus L^2(\Omega)$

Question

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $H^{-1}(\Omega)$ denote the dual of the Sobolev space $H_0^1(\Omega)$. Note that $$H_0^1\subset L^2\subset H^{-1}$$

How to construct a function (distribution) $u\in H^{-1}$ such that $u\notin L^2$?

Answer 1

Recall that $\Delta$ is a bounded operator from $H_0^1$ into $H^{-1}$ (reference). Thus, the distributional Laplacian of any function $u\in H^1_0\setminus H^2$ will do the job. The example suggested by @Mercy comes from $u=|x-a|^{2-n}$ (or $\log|x-a|$ in two dimensions).

Answer 2

Hint: Dirac $\delta$-function $\delta_a: f \mapsto f(a)$, with $a \in \Omega$.