So, as title says, I'm trying to solve the equation below for $\phi(x, y, z)$
$\Delta\phi=-\frac{k}{r^2}\phi$
Where $\Delta=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}$ and $r^2=x^2+y^2+z^2$
Considering that space is isotropic
$\frac{\partial^2\phi}{\partial x^2}=\frac{\partial^2\phi}{\partial y^2}=\frac{\partial^2\phi}{\partial z^2}$
How do I solve it?
This trick is famous when $\phi$ only depends on $r$: $$\Delta\phi=\frac{1}{r^2}\frac{\partial}{\partial r}(r^2\frac{\partial}{\partial r})\phi=\frac{1}{r^2}(r^2\phi''+2r\phi').$$ I think your equation reduces to the equation, $$r^2\frac{d^2\phi}{dr^2}+2r\frac{d\phi}{dr}+k\phi=0$$ and this is apart from other classifications, a Strum's Liouville Equation. It's solution is luckily simple: $$\phi(r)=c_1r^{-\frac{1}{2}-\frac{\sqrt{1-4k}}{2}}+c_2r^{-\frac{1}{2}+\frac{\sqrt{1-4k}}{2}}.$$ If I interpreted your isotropy condition correctly. Some documents said depends on $\ln r$, the same thing I think.