  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.
