This is somewhat of a curiosity that can hide somewhat deeper. For a Green function of a nonlinear PDE I mean something like
$$
   \partial^2\phi+V(\phi)=\delta^D(x).
$$
I do not know if a real meaning can be attached to something like this. But I can think to a gradient expansion. E.g., consider the nonlinear Klein-Gordon equation
$$
   \partial^2\phi+\lambda\phi^3=\delta^D(x).
$$
A gradient expansion has a leading order
$$
   \partial_t^2\phi_0+\lambda\phi_0=\delta^D(x)
$$
and I know the exact solution to $\partial_t^2\phi'+\lambda\phi'=\delta(t)$. Can I attach a meaning to something like this from the exact solution of this latter equation maybe using Coulombeau functions?

This is more than an exercise as can have some applications in physics. But what I am really interested to is if all this can have a mathematical meaning.

Thanks.