The Positive Solutions of the Matukuma Equation and the Problem of Finite Radius and Finite Mass

This work is an extensive study of the 3 different types of positive solutions of the Matukuma equation ${\frac{1}{r^{2}}\left( r^{2}\phi^{\prime}\right) ^{\prime}=-{\frac{r^{\lambda-2}}{\left( 1+r^{2}\right)^{\lambda /2}}}\phi^{p},p >1 ,\lambda >0 }${\frac{1}{r^{2}}\left( r^{2}\phi^{\prime}\right) ^{\prime}=-{\frac{r^{\lambda-2}}{\left( 1+r^{2}\right)^{\lambda /2}}}\phi^{p},p >1 ,\lambda >0 }: the E-solutions (regular at r = 0), the M-solutions (singular at r = 0) and the 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 as to characterizate the different solutions. The emphasis lies on the study of the 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, λ.