Open Access

Entire positive solution to the system of nonlinear elliptic equations

Boundary Value Problems20062006:32492

DOI: 10.1155/BVP/2006/32492

Received: 8 November 2005

Accepted: 15 May 2006

Published: 18 September 2006

Abstract

The second-order nonlinear elliptic system , with , is considered in . Under suitable hypotheses on functions , , , and , it is shown that this system possesses an entire positive solution such that both and are bounded below and above by positive constant multiples of for all .

[123456789101112131415161718192021]

Authors’ Affiliations

(1)
Department of Mathematics, Tianjin University
(2)
Liu Hui Center for Applied Mathematics, Nankai University and Tianjin University

References

  1. Adams RA: Sobolev Spaces. Academic Press, New York; 1975:xviii+268.MATHGoogle Scholar
  2. Aronson D, Crandall MG, Peletier LA: Stabilization of solutions of a degenerate nonlinear diffusion problem. Nonlinear Analysis 1982,6(10):1001-1022. 10.1016/0362-546X(82)90072-4MathSciNetView ArticleMATHGoogle Scholar
  3. Callegari A, Nachman A: Some singular, nonlinear differential equations arising in boundary layer theory. Journal of Mathematical Analysis and Applications 1978,64(1):96-105. 10.1016/0022-247X(78)90022-7MathSciNetView ArticleMATHGoogle Scholar
  4. de Figueiredo DG, Yang J: Decay, symmetry and existence of solutions of semilinear elliptic systems. Nonlinear Analysis, Theory, Methods & Applications 1998,33(3):211-234. 10.1016/S0362-546X(97)00548-8MathSciNetView ArticleMATHGoogle Scholar
  5. de Mottoni P, Schiaffino A, Tesei A: Attractivity properties of nonnegative solutions for a class of nonlinear degenerate parabolic problems. Annali di Matematica Pura ed Applicata, Serie Quarta 1984, 136: 35-48. 10.1007/BF01773375MathSciNetView ArticleMATHGoogle Scholar
  6. Edelson AL: Entire solutions of singular elliptic equations. Journal of Mathematical Analysis and Applications 1989,139(2):523-532. 10.1016/0022-247X(89)90126-1MathSciNetView ArticleMATHGoogle Scholar
  7. Fukagai N: On decaying entire solutions of second order sublinear elliptic equations. Hiroshima Mathematical Journal 1985,14(3):551-562.MathSciNetMATHGoogle Scholar
  8. Gilbarg D, Trudinger NS: Elliptic Partial Differential Equations of Second Order, Fundamental Principles of Mathematical Sciences. Volume 224. 2nd edition. Springer, Berlin; 1983:xiii+513.View ArticleGoogle Scholar
  9. Gurtin ME, MacCamy RC: On the diffusion of biological populations. Mathematical Biosciences 1977,33(1-2):35-49. 10.1016/0025-5564(77)90062-1MathSciNetView ArticleMATHGoogle Scholar
  10. Kawano N: On bounded entire solutions of semilinear elliptic equations. Hiroshima Mathematical Journal 1984,14(1):125-158.MathSciNetMATHGoogle Scholar
  11. Kawano N, Kusano T: On positive entire solutions of a class of second order semilinear elliptic systems. Mathematische Zeitschrift 1984,186(3):287-297. 10.1007/BF01174883MathSciNetView ArticleMATHGoogle Scholar
  12. Ladyzhenskaya OA, Ural'tseva NN: Linear and Quasilinear Elliptic Equations. Academic Press, New York; 1968:xviii+495.MATHGoogle Scholar
  13. Nachman A, Callegari A: A nonlinear singular boundary value problem in the theory of pseudoplastic fluids. SIAM Journal on Applied Mathematics 1980,38(2):275-281. 10.1137/0138024MathSciNetView ArticleMATHGoogle Scholar
  14. Naito M: A note on bounded positive entire solutions of semilinear elliptic equations. Hiroshima Mathematical Journal 1984,14(1):211-214.MathSciNetMATHGoogle Scholar
  15. Okubo A: Diffusion and Ecological Problems: Mathematical Models, Biomathematics. Volume 10. Springer, Berlin; 1980:xiii+254.Google Scholar
  16. Serrin J, Zou H: Existence of positive entire solutions of elliptic Hamiltonian systems. Communications in Partial Differential Equations 1998,23(3-4):577-599.MathSciNetMATHGoogle Scholar
  17. Shi J, Yao M: On a singular nonlinear semilinear elliptic problem. Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 1998,128(6):1389-1401. 10.1017/S0308210500027384MathSciNetView ArticleMATHGoogle Scholar
  18. Swanson CA: Extremal positive solutions of semilinear Schrödinger equations. Canadian Mathematical Bulletin 1983,26(2):171-178. 10.4153/CMB-1983-028-3MathSciNetView ArticleMATHGoogle Scholar
  19. Teramoto T: Existence and nonexistence of positive entire solutions of second order semilinear elliptic systems. Funkcialaj Ekvacioj 1999,42(2):241-260.MathSciNetMATHGoogle Scholar
  20. Wang LZ, Huang AX: Existence of positive solutions of a kind of reaction-diffusion system. Journal of Xi'an Jiaotong University 2002,36(2):211-213.MathSciNetMATHGoogle Scholar
  21. Yarur CS: Existence of continuous and singular ground states for semilinear elliptic systems. Electronic Journal of Differential Equations 1998,1998(1):1-27.MathSciNetGoogle Scholar

Copyright

© Qiu and Yao 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.