An Inverse Function Theorem in Sobolev Spaces and Applications to Quasi-Linear Schr¨odinger Equations

A Nash–Moser type inverse function theorem in Banach spaces with loss of derivatives is proved, and applications are given to singular quasi-linear Schr¨odinger equations like the superfluid film equation in plasma physics. Based on an implicit function theorem in Sobolev spaces, a linearization method is introduced for the local and global well-posedness of the Cauchy problem for nonlinear evolution equations. The technique compensates for a loss of derivatives in the linearized problem. This is illustrated by an application to strongly singular quasi-linear Schr¨odinger equations where the nonlinearities include derivatives of second order. The local well-posedness of the Cauchy problem is proved in Sobolev spaces for arbitrary space dimension without assuming smallness assumptions on the initial value. Here the linearized problem is solved using hyperbolic semigroup theory, including evolution systems