Question of using solution of Helmholtz equation to solve Poisson equation

Helmholtz equation stated that

This is being used for Poisson equation with zero boundary:

and

I just don't see how this can work as is a number only.

If which means is only constant numbers depending on !!!

If is a constant number only, that cannot be right?

Please explain. Thanks

Re: Question of using solution of Helmholtz equation to solve Poisson equation

Maybe your question would be answered easily, if you wrote

"...Poisson equation with zero boundary

etc"