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

The paper [39] uses the Craig-Wayne-Bourgain method to construct solutions of an elliptic problem involving parameters. results include regularity assumptions on perturbation and involve excluding also constructs response a quasi-periodically perturbed (ill-posed evolution) problem. In this paper, we use several classical methods (freezing coefficients, alternative for nonlinear equations) extend [39]. We weaken describe phenomena that happens all ill-posed problem, recently developed time-dependent center manifold theorem which allows reduce finite-dimensional ODE with quasi-periodic dependence time. bounded sufficiently small these give PDE.