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 $$ \partial^2\phi+\lambda\phi^3=\delta^D(x). $$ We can rewrite this equation as $$ \partial_t^2\phi+\lambda\phi^3=\delta(t)\delta^{D-1}(x) + \epsilon\Delta\phi. $$ and to treat the Laplacian as a perturbation (for this aim I introduced an arbitrary $\epsilon$ that I will set to 1 at the end of computation). Then, I have a gradient expansion in term of powers $\epsilon$. This has for the 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?

There is also a less singular approach. I just rescale time as $\tau=\sqrt{\lambda}t$ and I get $$ \lambda\partial_\tau^2\phi+\lambda\phi^3=\sqrt{\lambda}\delta(\tau)\delta^{D-1}(x) + \Delta\phi $$ that is $$ \partial_\tau^2\phi+\phi^3=\frac{1}{\sqrt{\lambda}}\delta(\tau)\delta^{D-1}(x) + \frac{1}{\lambda}\Delta\phi. $$ This is again a gradient expansion in powers of $\frac{1}{\sqrt{\lambda}}$ but the leading order has the form $$ \partial_\tau^2\phi_0+\phi_0^3=0 $$ that has a known exact solution. Next-to-leading order is no more singular giving a linear equation.

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.