I have basic knowledge of PDE and know how to use the standard (Banach space) inverse function theorem to solve $$ -\Delta u+ g(u)=f $$ when $g(0)=g'(0)=0$ and $f$ is small:
Define $A(u):=-\Delta u+g\circ u$. Then
- $A(0)=0$
- $A'(0)v=-\Delta v$ is invertible for example if we consider $A\colon C^{2,\alpha}\to C^{0,\alpha}$ and $A'(0)\colon C^{2,\alpha}\to C^{0,\alpha}$.
Therefore, for small enough $\|f\|_{C^{0,\alpha}}$, there is a solution $u\in C^{2,\alpha}$ of $A(u)=f$.
Is there a similarly simple example for the application of the Nash-Moser inverse function theorem?
A side question that is easier to answer than the main question: According to Wikipedia, the Nash-Moser theorem is helpful when the inverse of the derivative loses derivatives. Does this mean that the inverse is for example a map $C^{0,\beta}\to C^{2,\alpha}$ with $\beta>\alpha$ or does it actually mean that derivatives are lost in the sense that the inverse is for example a map $C^{2,\beta}\to C^{2,\alpha}$ with $\beta>\alpha$? In any case, why would this preclude application of the standard inverse function theorem (on smaller spaces)?