# A version of the Nash-Moser inverse function for unbounded domains?

Do there exist versions of the Nash-Moser inverse function theorem applicable to spaces of unbounded smooth functions on unbounded domains in $$\mathbb{R}^n$$? Any reference would be appreciated but here is more specifically what I'm looking for:

1. I'm interested in a right inverse function theorem, i.e. the existence of right (but not left) inverse of the derivative is assumed.

2. The result should be applicable to semi-linear elliptic operators.

3. Since linear elliptic operators considered on the space of all smooth functions do not have linear right inverse, one expects that the relevant function space should be the space of smooth functions with derivatives of all orders that are of polynomial growth or something similar.

• Nash-Moser is an inverse function theorem for a Fréchet space or a scale of Banach spaces, where the functional and the right inverse to the linearized operator are smooth tame. You get to choose which spaces and norms you want to use. This is usually determined by what kind of a priori estimates you can prove for the linearized PDE. In the situation you describe, it sounds like you need to use some kind of weighted Sobolev norm. Perhaps you could provide more details. Feb 19 at 4:08
• @DeaneYang, I'm interested in finding local right inverse of nonlinear operators $L$ given by $Lf=\Delta f + f^p$ for $f$ in some function space. I don't see how these can be set up in a weighted Sobolev space. The point is that there are no Sobolev multiplication theorems for weighted Sobolev spaces, as far as I know.
– S.Z.
Feb 19 at 15:07
• First, it occurs to me that for an elliptic PDE, you normally do not need Nash-Moser. The standard Banach space inverse function theorem with the right choice of a norm is usually good enough. Feb 19 at 15:20
• @DeaneYang, I'm aware of that and that's what I tried to do for a while. Unfortunately I didn't find any reference in which the Banach space inverse function theorem is used to solve elliptic semilinear equations in unbounded domains or even just in $\mathbb{R^n}$.
– S.Z.
Feb 19 at 15:32
• @DeaneYang, actually it's probably true that what is needed is a generalization of the Banach space inverse function in different direction, rather than Nash-Moser type of theorem. For example, this apparently neglected article looks promising to me: onlinelibrary.wiley.com/doi/abs/10.1002/mana.19861290108
– S.Z.
Feb 19 at 15:36