Abstract
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.
Similar content being viewed by others
References
H. Berestycki, P. L. Lions and L. A. Peletier, An ODE approach to the existence of positive soluions for semilinear problems inR n. Indiana Univ. Math. J.,30, (1981), 141–157.
S. Chandrasekhar, An Introduction to the Study of Stellar Structure. Univ. Chicago Press, 1939.
W.-Y. Ding and W.-M. Ni, On the elliptic equation Δu+Ku (n+2)/(n−2)=0 and related topics. Duke Math. J.,52, (1985), 485–506.
W.-Y. Ding and W.-M. Ni, On the existence of positive entire solution of a semilinear elliptic equations. Arch. Rational Mech. Anal.,91, (1986), 283–308.
A. S. Eddington, The dynamics of a globular stellar system. Monthly Notices Royal Astronom. Soc.,75, (1915), 366–376.
R. H. Fowler, Further studies of Emden’s and similar differential equations. Quart. J. Math. (Oxford Ser.),2 (1931), 259–287.
M. Hénon, Numerical experiments on the stability of spherical stellar systems. Astronom. and Astrophys.,24, (1973), 229–238.
D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources. Arch Rational Mech. Anal.,49, (1973), 241–269.
N. Kawano, On bounded entire solutions of semilinear elliptic equations. Hiroshima Math. J.,14 (1984), 125–158.
N. Kawano, J. Satsuma and S. Yotsutani, On the positive solutions of an Emden-type elliptic equation. Proc. Japan Acad. Ser. A,61, (1985), 181–189.
N. Kawano, J. Satsuma and S. Yotsutani, Existence of positive entire solutions of an Emden-type elliptic equation. Funkcial. Ekvac., to appear.
T. Kusano and M. Naito, Oscillation theory of entire solutions of second order superlinear elliptic equations. Funkcial. Ekvac., to appear.
T. Matukuma, Dynamics of globular clusters. Nippon Temmongakkai Yoho,1, (1930), 68–89 (In Japanese).
T. Matukuma, Sur la dynamique des amas globulaires stellaires. Proc. Imp. Acad.,6, (1930), 133–136.
T. Matukuma, The Cosmos. Iwanami Shoten, Tokyo, 1938 (In Japanese).
K. McLeod and J. Serrin, Uniquenss of solutions of semilinear Poisson equations. Proc. Nat. Acad. Sci. USA.,78, (1981), 6592–6595.
R. McOwen, Conformal metrics inR 2 with prescribed Gaussian curvature and positive total curvature. Indiana Univ. Math. J.,34, (1985), 97–104.
M. Naito, A note on bounded positive entire solution of semiliner elliptic equations. Hiroshima Math. J.,14, (1984), 211–214.
W.-M. Ni, On the elliptic equation Δu+K(x)u (n+2)/(n−2)=0 its generalization, and applications in geometry. Indiana Univ. Math. J.,31, (1982), 493–529.
W.-M. Ni, On the positive radial solutions of some semilinear elliptic equations onR n. Appl. Math. Optim.,9, (1983), 373–380.
W.-M. Ni, Uniqueness, nonuniqueness and related questions of nonlinear elliptic and parabolic equations. Proc. Symp. Pure Math., 45, Amer. Math. Soc., 1986, 229–241.
W.-M. Ni and R. D. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of Δu+f(u,r)=0. Comm. Pure Appl. Math.,38, (1985), 67–108.
W.-M. Ni and J. Serrin, Existence and non-existence theorems for ground states for quasilinear partial differential equations. Accad. Naz. dei Lincei,77, (1986), 231–257.
W.-M. Ni and S. Yotsutani, On the Matukuma’s equation and related topics. Proc. Japan Acad. Ser. A,62, (1986), 260–263.
N. Kawano, W.-M. Ni and S. Yotsutani, In preparation.
Author information
Authors and Affiliations
About this article
Cite this article
Ni, WM., Yotsutani, S. Semilinear elliptic equations of Matukuma-type and related topics. Japan J. Appl. Math. 5, 1–32 (1988). https://doi.org/10.1007/BF03167899
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF03167899