Radial Laplacian

For radial $u(r)$ in $n$ dimensions, $\Delta u=u\prime \prime +(n-1)u\prime /r$.

For $r>0$, the substitution $y=ru$ removes the first derivative in the three-dimensional radial Laplacian, since $u^{\prime \prime }+2u^{\prime }/r=y^{\prime \prime }/r$. Bounded $u$ at zero implies $y(0)=0$. Bounded $u$ at infinity means $y=O(r)$; it does not require bounded $y$. For example, constant forcing $g=1$ has the bounded particular solution $u=-1$, whose reduced function is $y=-r$. A Dirichlet half-line Green function for d2 minus 1 can still represent this solution when its integral converges, but the resulting reduced function has the growth demanded by the forcing.