In this talk we study the \textbf{positive} solutions $%
\phi =\phi \left( r\right) $ of the differential equation%
\begin{equation*}
\phi ^{\prime \prime }+\frac{2}{r}\phi ^{\prime }=-\frac{r^{\lambda -2}}{%
\left( 1+r^{2}\right) ^{\lambda /2}}\phi ^{p},\qquad p>1,\;\lambda >1,
\end{equation*}%
on their maximal intervals of the positive real line $\mathbb{R}^{+}$. For $%
\lambda =2$, these solutions are the radial solutions of the semilinear
elliptic equation%
\begin{equation*}
\Delta \phi =-\frac{1}{1+x^{2}}\phi ^{p},
\end{equation*}%
on $\mathbb{R}^{3}$, which T.~Matukuma proposed in 1935 for the description
of certain stellar globular clusters in a steady state. They correspond to
time-independent solutions of the Vlasov-Poisson system%
\begin{align*}
& \text{(V)} & \partial _{t}f+v\partial _{x}f-\partial _{x}U\left(
t,x\right) \partial _{v}f& =0 \\
& \text{(P)} & \Delta U\left( t,x\right) & =4\pi \rho \left( t,x\right) \\
& \text{(D)} & \rho \left( t,x\right) & :=\int f\left( t,x,v\right)
\,dv,\qquad x,v\in \mathbb{R}^{3},
\end{align*}%
in the case of spherical symmetry; here $f=f\left( t,x,v\right) \geq 0$ is
the distribution function of the considered system of gravitating mass in
the space-velocity space $\mathbb{R}^{3}\times \mathbb{R}^{3}$, $t\geq 0$
the time, $U=U\left( t,x\right) $ the Newtonian potential and $\rho =\rho
\left( t,x\right) $ the local density.
