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On the radius continuity of the models of polytropic gas spheres which correspond to the positive solutions of the generalized Emden-Fowler equation. (English) Zbl 0676.34017
Positive parts of the solutions of the Emden-Fowler equation \[ (*)\quad (1/r^ 2)(r^ 2\Phi !(r))!=-r^{2m}\Phi^{m+n}(r),\quad r>0 \] with coefficients \(n>\), \(m>-1\), \(m+n>0\) induce models of polytropic gas spheres, i.e. special solutions of the Vlasov-Poisson system. We investigate (*) by transforming it into an autonomous system and characterize the solutions which induce models of finite radius or finite mass respectively. Further we consider a sequence \((C_ k)_{k\in {\mathbb{N}}}\) of trajectories of the autonomous system that converges to a trajectory \(C_ 0\) (in the sense that there are points \(z_ k\) in the traces of the \(C_ k\), \(k\in {\mathbb{N}}\cup \{0\}\), so that \(z_ k\to z_ 0)\) and give all implications between the following statements \[ (I)\quad \phi_ k\to \phi_ 0,\quad (II)\quad R_ k\to R_ 0,\quad (III)\quad M_ k\to M_ 0, \] where for \(k\in {\mathbb{N}}\cup \{0\}\) \(\phi_ k\) are solutions representing the trajectories \(C_ k\), \(R_ k\) are the radii of the corresponding models of polytropic gas spheres and \(M_ k\) their masses. We thus complete the investigations of W. J. van den Broek and F. Verhulst: Math. Methods Appl. Sci. 4, 259- 271 (1982; Zbl 0493.35046)].
Reviewer: J.Batt

MSC:
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
34D30 Structural stability and analogous concepts of solutions to ordinary differential equations
85A05 Galactic and stellar dynamics
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