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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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