## Duke Mathematical Journal

### On the positive solutions of the Matukuma equation

Yi Li

#### Article information

Source
Duke Math. J., Volume 70, Number 3 (1993), 575-589.

Dates
First available in Project Euclid: 20 February 2004

https://projecteuclid.org/euclid.dmj/1077290889

Digital Object Identifier
doi:10.1215/S0012-7094-93-07012-3

Mathematical Reviews number (MathSciNet)
MR1224099

Zentralblatt MATH identifier
0801.35024

#### Citation

Li, Yi. On the positive solutions of the Matukuma equation. Duke Math. J. 70 (1993), no. 3, 575--589. doi:10.1215/S0012-7094-93-07012-3. https://projecteuclid.org/euclid.dmj/1077290889

#### References

• [A] P. Aviles, Local behavior of solutions of some elliptic equations, Comm. Math. Phys. 108 (1987), no. 2, 177–192.
• [BFH] J. Batt, W. Faltenbacher, and E. Horst, Stationary spherically symmetric models in stellar dynamics, Arch. Rational Mech. Anal. 93 (1986), no. 2, 159–183.
• [BN] H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, preprint.
• [CGS] L. A. Caffarelli, B. Gidas, and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), no. 3, 271–297.
• [CL1] W. X. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991), no. 3, 615–622.
• [CL2] W. X. Chen and C. Li, Qualitative properties of solutions to some nonlinear elliptic equations in $\mathbfR^2$, preprint.
• [GNN1] B. Gidas, W.-M. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209–243.
• [GNN2] B. Gidas, W. M. Ni, and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\bf R\spn$, Mathematical analysis and applications, Part A ed. L. Nachbin, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York, 1981, pp. 369–402.
• [GT] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983.
• [H] H. Hopf, Differential geometry in the large, Lecture Notes in Mathematics, vol. 1000, Springer-Verlag, Berlin, 1983.
• [KKL] H. G. Kaper, M. K. Kwong, and Y. Li, On the positive solutions of the free-boundary problem for Emden-Fowler type equations, Partial differential equations with minimal smoothness and applications (Chicago, IL, 1990) eds. B. Dahlberg, E. Fabes, and R. Fefferman and D. Jerison and C. Kenig and J. Pipher, IMA Vol. Math. Appl., vol. 42, Springer, New York, 1992, pp. 163–172.
• [K] T. Kato, Growth properties of solutions of the reduced wave equation with a variable coefficient, Comm. Pure Appl. Math. 12 (1959), 403–425.
• [KL] M. K. Kwong and Y. Li, Uniqueness of radial solutions of semilinear elliptic equations, Trans. Amer. Math. Soc. 333 (1992), no. 1, 339–363.
• [KYY] N. Kawano, E. Yanagida, and S. Yotsutani, Structure theorems for positive radial solutions to $\Delta u+K(\vert x\vert)u^p=0$ in $\mathbfR^n$, preprint.
• [Li] C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains, Comm. Partial Differential Equations 16 (1991), no. 4-5, 585–615.
• [L1] Y. Li, Asymptotic behavior of positive solutions of equation $\Delta u+K(x)u\sp p=0$ in $\bf R\sp n$, J. Differential Equations 95 (1992), no. 2, 304–330.
• [L2] Y. Li, Symmetry properties of finite total mass solutions of Matukuma equation, Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989) eds. W.-M. Ni, L. A. Peletier, and J. Serrin, Progr. Nonlinear Differential Equations Appl., vol. 7, Birkhäuser Boston, Boston, MA, 1992, pp. 375–389.
• [LN1] Y. Li and W.-M. Ni, On conformal scalar curvature equations in $\bf R\sp n$, Duke Math. J. 57 (1988), no. 3, 895–924.
• [LN2] Y. Li and W.-M. Ni, On the existence and symmetry properties of finite total mass solutions of the Matukuma equation, the Eddington equation and their generalizations, Arch. Rational Mech. Anal. 108 (1989), no. 2, 175–194.
• [LN3] Y. Li and W.-M. Ni, On the asymptotic behavior and radial symmetry of positive solutions of semilinear elliptic equations in $\bf R\sp n$. I. Asymptotic behavior, Arch. Rational Mech. Anal. 118 (1992), no. 3, 195–222.
• [LN4] Y. Li and W.-M. Ni, On the asymptotic behavior and radial symmetry of positive solutions of semilinear elliptic equations in $\bf R\sp n$. II. Radial symmetry, Arch. Rational Mech. Anal. 118 (1992), no. 3, 223–243.
• [LN5] Y. Li and W.-M. Ni, Radial symmetry of positive solutions of nonlinear elliptic equations in $\mathbfR^n$, to appear in Comm. Partial Differential Equations.
• [M] T. Matukuma, The Cosmos, Iwanami Shoten, Tokyo, 1938.
• [NS] E. S. Noussair and C. A. Swanson, Solutions of Matukuma's equation with finite total mass, Indiana Univ. Math. J. 38 (1989), no. 3, 557–561.
• [NY1] W.-M. Ni and S. Yotsutani, On Matukuma's equation and related topics, Proc. Japan Acad. Ser. A Math. Sci. 62 (1986), no. 7, 260–263.
• [NY2] W.-M. Ni and S. Yotsutani, Semilinear elliptic equations of Matukuma-type and related topics, Japan J. Appl. Math. 5 (1988), no. 1, 1–32.
• [PW] M. H. Protter and H. F. Weinberger, Maximum principles in differential equations, Prentice-Hall Inc., Englewood Cliffs, N.J., 1967.
• [S] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), 304–318.
• [Y] E. Yanagida, Structure of positive radial solutions of Matukuma's equation, Japan J. Indust. Appl. Math. 8 (1991), no. 1, 165–173.