 Solution

In three dimensions, ${\delta }_0\in H^s$ for every $s<-3/2$. Hence the solution belongs to $H^{s+4}$ for every such $s$; choosing, for example, $s=-2$ gives $u\in H^2({\mathbb{R}}^3)$. Since $2>3/2$, Sobolev embedding supplies a unique continuous representative. It solves the equation distributionally, and uniqueness follows from part (a), so this is the unique continuous solution.