Stationary spherically symmetric solutions of the Vlasov–Poisson system in the three-dimensional case

We consider the three-dimensional stationary Vlasov–Poisson system of equations with respect to the distribution function of the gravitating matter $f=f_q(r,u)$, the local density $\rho=\rho(r)$, and the Newtonian potential $U=U(r)$, where $r:=|x|$, $u:=|v|$ ($(x,v)\in\mathbb R^3\times\mathbb R^3$ are the space–velocity coordinates), and $f$ is a function $q$ of the local energy $E:=U(r)+\dfrac{u^2}2$. For a given function $p=p(r)$, we obtain sufficient conditions for $p$ to be “extendable”. This means that there exists a stationary spherically symmetric solution $(f_q,\rho,U)$ of the Vlasov–Poisson system depending on the local energy $E$ such that $\rho=p$.