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  • Research Article
  • Open Access

Entire positive solution to the system of nonlinear elliptic equations

Boundary Value Problems20062006:32492

  • Received: 8 November 2005
  • Accepted: 15 May 2006
  • Published:


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 .


  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Equation
  • Elliptic Equation


Authors’ Affiliations

Department of Mathematics, Tianjin University, Tianjin, 300072, China
Liu Hui Center for Applied Mathematics, Nankai University and Tianjin University, Tianjin, 300072, China


  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