标题： 非线性椭圆方程解的存在性：Y. Shi论文的简单证明和扩展 显示英文标题

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

摘要： 论文[Shi19]使用Craig-Wayne-Bourgain方法构造了一个涉及参数的椭圆问题的解。 [Shi19]的结果包括对扰动的正则性假设，并涉及排除参数。 论文[Shi19]还构造了准周期扰动（不适定演化）问题的响应解。 在本文中，我们使用几种经典方法（系数冻结法，非线性椭圆方程的替代方法）来扩展[Shi19]的结果。 我们减弱了对扰动的正则性假设，并描述了所有参数下发生的现象。 在不适定问题中，我们使用了一种最近开发的时间依赖中心流形定理，该定理允许将问题简化为具有准周期时间依赖性的有限维ODE。 这些ODE的有界且足够小的解给出了不适定PDE的解。 显示英文摘要

摘要： 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.