In the case of linear PDE, say $$Lu=0$$ if we have its green function say $G(x,y)$ then using that one can give solution of non homogenous PDE i.e. $Lu\_f=f$ where $u\_f=G\*f$.


Is the same thing hold for non-linear PDE? Even if not, I wanted to know if we have quasilinear PDE is that holds? If this is not true at all, then what is the use of the green function for nonlinear PDE?


Any help or reference will be appreciated.



Application of Green function for non linear PDE
The equality $u\_f= G\ast f$ uses linearity in an essential way since
$$
L\_y G(x,y)= \delta(y-x).
$$
The function $f$ is a superposition of $\delta$'s
$$
f(\bullet)=\int f(x)\delta(\bullet-x) dx.
$$
On the other hand the Green function has indirect uses in nonlinear equations. The solution on the [Yamabe problem](https://www.ams.org/journals/bull/1987-17-01/S0273-0979-1987-15514-5/S0273-0979-1987-15514-5.pdf) is one such situation.


