Poisson equation with radial function

Question

while reading the book, I ran into a problem with the partial differential equation. The problem is following: Find positive radially symmetric function $w$ such that $-\Delta w = \Phi (r) \ (r=|x|)$ in $\mathbb{R}^{N}$ and $\lim_{r \to \infty}w(r)=0$. The book says that $$w(r)= K-\int_{0}^{r}\zeta^{1-N} \int_{0}^{\zeta}\sigma^{N-1}\Phi(\sigma)d\sigma d\zeta,$$ where $$ K=\int_{0}^{\infty}\zeta^{1-N} \int_{0}^{\zeta}\sigma^{N-1}\Phi(\sigma)d\sigma d\zeta. $$ I thought the solution would be a convolution of the fundamental solution of the Laplace equation with the right side as in the Poisson equation but it did not work. I need a tip on how to go about it. Thank you in advance.