Semilinear elliptic equations of Matukuma-type and related topics

We investigate the structure of solutions of some semilinear elliptic equations, which include Matukuma's equation as a special case. It is a mathematical model proposed by Matukuma, an astrophysicist, in 1930 to describe the dynamics of a globular cluster of stars. Equations of this kind have come up both in geometry and in physics, and have been a subject of extensive studies for some time. However, almost all the methods previously developed do not seem to apply to the original Matukuma's equation. Our results cover most of the cases left open by previous works.