I know virtually nothing about PDE's. Actually, I'm asking this questions because I'm trying to help out a friend of mine, so, please, excuse any ignorant/nonsensical statement on my part and forgive me if the questions is not in line with the rules.
Consider the function $f:\mathbb{R^3}\rightarrow\mathbb{R} $ such that $$f(x,y,z)=\begin{cases} 1 & \text{if $z=0$} \\ 0 & \text{if $z\neq0$} \end{cases} $$
Is there a function $\phi$ such that $$\Delta\phi=f$$ where $\Delta\phi=\frac{\partial ^2\phi}{\partial x^2}+\frac{\partial ^2\phi}{\partial y^2}+\frac{\partial ^2\phi}{\partial z^2}$?
If so, is the solution unique? And how do I find it?