The existence of solutions for nonlinear elliptic equations: Simple proofs and extensions of a paper by Y. Shi

TL;DR

This paper extends previous work on nonlinear elliptic equations by weakening regularity assumptions, broadening parameter analysis, and applying classical methods and a new center manifold theorem to construct solutions.

Contribution

It introduces simpler proofs and extends results of Shi (2019) by relaxing assumptions and analyzing parameter phenomena using classical and modern techniques.

Findings

Weaker regularity assumptions on perturbations.

Description of solution phenomena for all parameters.

Application of a new time-dependent center manifold theorem.

Abstract

The paper [Shi19] uses the Craig-Wayne-Bourgain method to construct solutions of an elliptic problem involving parameters. The results of [Shi19] include regularity assumptions on the perturbation and involve excluding parameters. The paper [Shi19] also constructs response solutions to a quasi-periodically perturbed (ill-posed evolution) problem. In this paper, we use several classical methods (freezing of coefficients, alternative methods for nonlinear elliptic equations) to extend the results of [Shi19]. We weaken the regularity assumptions on the perturbation and we describe the phenomena that happens for all parameters. In the ill-posed problem, we use a recently developed time-dependent center manifold theorem which allows to reduce the problem to a finite-dimensional ODE with quasi-periodic dependence on time. The bounded and sufficiently small solutions of these ODE give solutions of the ill-posed PDE.