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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.

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    $\begingroup$ Unfortunately, this does not look to me like a research level question. Forgetting about PDEs, the same question can be asked of any linear equation $Lu=f$. In any such problem (linear or nonlinear) the role of $L^{-1}$ (the "Green function") is fundamental and is discussed in any good basic course on analysis. $\endgroup$ Aug 24, 2021 at 10:57

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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 is one such situation.

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