Question of using solution of Helmholtz equation to solve Poisson equation

Helmholtz equation stated that

$\displaystyle \nabla^2 u(r,\theta,\phi) =-ku(r,\theta,\phi) = f(r,\theta,\phi) $

This is being used for Poisson equation with zero boundary:

$\displaystyle \nabla^2 u(r,\theta,\phi) = f(r,\theta,\phi) $

and

$\displaystyle u(a,\theta,\phi)=0$

I just don't see how this can work as $\displaystyle k=m^2$ is a number only.

If $\displaystyle \nabla^2 u(r,\theta,\phi)=1$ which means $\displaystyle ku(r,\theta,\phi)$ is only constant numbers depending on $\displaystyle m$!!!

If $\displaystyle u(r,\theta,\phi)$ 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

$\displaystyle \nabla^2 w=f, w(a,\theta,\phi)=0$

etc"