It is about heat equation in 3D. How to prove, that $$w(M,M_0)=\frac{1}{\lambda r(M,M_0)}$$ is a solution of $$\lambda \nabla^2w(M,M_0)+4\pi \delta(M,M_0)=0$$ ?
I understand that I should integrate it somehow in spherical coordinates.
$$\nabla^2 f = {1 \over r^2} {\partial \over \partial r} \left(r^2 {\partial f \over \partial r} \right) + {1 \over r^2 \sin \theta} {\partial \over \partial \theta} \left(\sin \theta {\partial f \over \partial \theta} \right) + {1 \over r^2 \sin^2 \theta} {\partial^2 f \over \partial \varphi^2}.$$
$$\nabla w =\frac{2}{\lambda r} $$
But the integral $$\iiint \frac{2}{r}r^2sin\theta\,dr\,d\theta\,d\phi$$ diverges.