Can solutions of the Poisson's equation be written as linear combinations of Laplace's equation solutions?

Question

Given that the Laplacian operator $\Delta$ acts on the space of functions(at least $C^2$), does the equation $\Delta\phi=0$, define a base of that space such that solutions of $\Delta\psi=f$ can be decomposed in that base.

Answer 1

The linear combination will only solve Laplace's equation. So no.

If each $\phi_i$ solves Laplace's equation, then $\Delta\phi_i=0$, and $$\Delta\left(\sum_ia_i\phi_i\right)=\sum_ia_i\Delta\phi_i=0$$