I have an issue with a basic Poisson's equation:
$$ \frac{1}{r^2} \partial_r r^2 \partial_r V = - f(r)$$
with $r > 0$, $f(r) > 0$ has the following property:
$$ \int_0^{\infty} r^2 f(r) d r = 1 $$
and the boundary condition is of second type:
$$ \partial_r V \vert_{r=0} = 0 $$
Do you have an idea ? I checked a big textbook on PDEs I did not find it.
What I tried
I performed a direct integration, this leads to divergence in $r = 0$ and the same for the Green function integration $\propto 1/r$. My other idea was to sum on spherical Bessel functions ($j_0$ only), using eigenvalue decomposition but it is a long process so I want to make sure that there is no simple trick to do here... Thank you for reading.
Edit
I tried a decomposition with $j_0$ spherical Bessel function, it is not possible because they are not square integrable on $r \geq 0$ domain.