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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 1r2(r2ϕ′)′=−rλ−2(1+r2)λ/2ϕp,p>1,λ>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 a characterization of 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.

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