Solution
= Solution
For smooth compactly supported $u$, multiply by $1-\varphi(nx)$. The removed term has $L^2$ norm $O(n^{-3/2})$, while its <gradient> terms are $O(n^{-1/2})$ in dimension three. Thus it tends to zero in $H^1$. Density then proves $H_0^1(\mathbb R^3\setminus\{0\})=H^1(\mathbb R^3)$.