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arXiv · 2108.01293

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

Xiaodan Xu·Rafael de la Llave·Fenfen Wang
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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.

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Xiaodan Xu, Rafael de la Llave, Fenfen Wang. 2021-08-03. The existence of solutions for nonlinear elliptic equations: Simple proofs and extensions of a paper by Y. Shi. https://arxiv.org/abs/2108.01293

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