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.