A surjection theorem for maps with singular perturbation and loss of derivatives

Abstract

In this paper we introduce a new algorithm for solving perturbed nonlinear functional equations which admit a right-invertible linearization, but with an inverse that loses derivatives and may blow up when the perturbation parameter $\epsilon$ goes to zero. These equations are of the form $F_{\epsilon }(u)=v$ with $F_{\epsilon }(0)=0$, $v$ small and given, $u$ small and unknown. The main difference from the by now classical Nash–Moser algorithm is that, instead of using a regularized Newton scheme, we solve a sequence of Galerkin problems thanks to a topological argument. As a consequence, in our estimates there are no quadratic terms. For problems without perturbation parameter, our results require weaker regularity assumptions on $F$ and $v$ than earlier ones, such as those of Hörmander [17]. For singularly perturbed functionals $F_{\epsilon }$, we allow $v$ to be larger than in previous works. To illustrate this, we apply our method to a nonlinear Schrödinger Cauchy problem with concentrated initial data studied by Texier–Zumbrun [26], and we show that our result improves significantly on theirs.