I have the following problem
The definition I have for Green's function is
The ${\bf Green's \; function}$ $G(x)$ for the operator $-\Delta$ and the domain $D$ at $x_0 \in D$ is a function defined for $x \in D$ that satisfies
G(x) possesses continuous second derivatives and $\Delta G=0$ in $D$, except at the point $x=x_0$.
$G(x) =0$ for $x \in \partial D$.
The function $G(x) + \dfrac{1}{4 \pi |x-x_0|}$ is finite at $x_0$ and has continuous second derivatives everywhere and is harmonic at $x_0$.
My confusion is that this problem is asking for green's function on the entire space, but what is the operator here? The wording of the problem is confusing a little bit.