Given a linear operator $L$, a Green's function $G(x,s)$ is any solution of $$\tag{1} LG(x,s) = \delta(x-s)$$ where $\delta(x-s)$ is the Dirac Delta function.
The Green's function can also be used in solving equations of the form $$\tag{2} Lu(x) = f(x)$$
My question is, if we have any linear operator $L$, is it possible that the Green's function for that operator is unique in the sense that any function $u(x)$ and $f(x)$ be used but it would still yield the same Green's function?
$$\\$$ Let's say that we have Poisson's equation for an arbitrary function: $$\nabla^2u(x) = f(x) \tag{3}$$ According to books the Green's function for this operator would be $G(r) = -\frac{1}{4\pi r}$ such that the solution for Poisson's equation would be $$\tag{4} u(x) = \int G(x,s) f(s) \, ds$$
But is that really the case even for any value of $u(x)$ and $f(x)$?