An inverse function theorem in Sobolev spaces and applications to quasi-linear Schrödinger equations (Q5939790)

An inverse function theorem of Nash-Moser type in Banach spaces with loss of derivatives is proved. The proof uses methods of Hörmanders result for Hölder spaces [\textit{L. Hörmander}, Arch. Ration. Mech. Anal. 62, 1-52 (1976; Zbl 0331.35020)] but this theorem applies to Sobolev spaces. Applications to non-linear evolution equations are given, e.g.~ a well-posedness result in Sobolev spaces for strongly singular quasi-linear Schrödinger equations.