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:

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

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

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.

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