The positive solutions of the Matukuma equation and the problem of finite radius and finite mass (Q717183)

This paper concerns the 3 different types of positive solutions of the Matukuma equation \[ {\frac{1}{r^{2}}\left( r^{2}\phi^{\prime}\right) ^{\prime}=-{\frac{r^{\lambda-2}}{\left( 1+r^{2}\right)^{\lambda /2}}}\phi^{p}, \quad p >1 ,\lambda >0,} \] namely, \(E\)-solutions (regular at \(r = 0\)), \(M\)-solutions (singular at \(r = 0\)) and \(F\)-solutions (whose existence begins away from \(r = 0\)). An essential tool is a transformation of the equation into a 2-dimensional asymptotically autonomous system, whose limit sets (by a theorem of H. R. Thieme) are the limit sets of Emden-Fowler systems, and serve to characterize the different solutions. The main attention is paid to \(M\)-solutions. The asymptotic expansions obtained make it possible to apply the results to the important question of stellar dynamics, solutions to which lead to galactic models (stationary solutions of the Vlasov-Poisson system) of finite radius and/or finite mass for different \(p, \lambda \).