Vtyjkyuk

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 $A(u):=-Delta u+gcirc u$. Then

• $A(0)=0$


• $A'(0)v=-Delta v$ is invertible for example if we consider $Acolon C^2,alphato C^0,alpha$ and $A'(0)colon C^2,alphato C^0,alpha$.

Therefore, for small enough $|f|_C^0,alpha$, there is a solution $uin C^2,alpha$ 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 $C^0,betato C^2,alpha$ with $beta>alpha$ or does it actually mean that derivatives are lost in the sense that the inverse is for example a map $C^2,betato C^2,alpha$ with $beta>alpha$? In any case, why would this preclude application of the standard inverse function theorem (on smaller spaces)?