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On positive solutions of some pairs of differential equations


Author: E. N. Dancer
Journal: Trans. Amer. Math. Soc. 284 (1984), 729-743
MSC: Primary 35J65; Secondary 47H15, 92A15
DOI: https://doi.org/10.1090/S0002-9947-1984-0743741-4
MathSciNet review: 743741
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Abstract: In this paper, we discuss the existence of solutions, with both components positive, of a Dirichlet problem for a coupled pair of partial differential equations. The main result is proved by using degree theory in cones. We also discuss the asymptotic behaviour of solutions as a parameter tends to zero.


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  • [1] R. A. Adams, Sobolev spaces, Academic Press, New York, 1975. MR 0450957 (56:9247)
  • [2] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), 620-709. MR 0415432 (54:3519)
  • [3] N. Anselone, Collectively compact operator approximation theory and applications to integral equations, Prentice-Hall, Englewood Cliffs, N. J., 1971. MR 0443383 (56:1753)
  • [4] H. Berestycki, Le nombre de solutions de certains problèmes semilinéaires elliptiques, J. Funct. Anal. 40 (1981), 1-29. MR 607588 (82k:35033)
  • [5] P. N. Brown, Decay to uniform states in ecological interactions, SIAM J. Appl. Math. 38 (1980), 22-37. MR 559078 (81a:35047)
  • [6] E. Conway, R. Gardner and J. Smoller, Stability and bifurcation of steady state solutions for predator-prey equations, Adv. in Appl. Math. 3 (1982), 288-334. MR 673245 (83m:35023)
  • [7] E. N. Dancer, Global solution branches for positive mappings, Arch. Rational Mech. Anal. 52 (1973), 181-192. MR 0353077 (50:5563)
  • [8] -, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl. 91 (1983), 131-151. MR 688538 (84d:58020)
  • [9] A. Friedman, Partial differential equations, Holt, Rinehart and Winston, New York, 1969. MR 0445088 (56:3433)
  • [10] P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations 10 (1980), 999-1030. MR 588690 (81m:35102)
  • [11] T. Kato, Perturbation theory for linear operators, Springer-Verlag, Berlin, 1966. MR 0203473 (34:3324)
  • [12] M. A. Krasnosel'skii, Positive solutions of operator equations, Noordhoff, Groningen, 1964. MR 0181881 (31:6107)
  • [13] P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev. 24 (1982), 441-448. MR 678562 (84a:35093)
  • [14] G. Pimbley, Eigenfunction branches of nonlinear operators and their bifurcations, Lecture Notes in Math., vol. 104, Springer-Verlag, Berlin, 1969. MR 0266007 (42:916)
  • [15] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487-513. MR 0301587 (46:745)
  • [16] W. Rudin, Real and complex analysis, McGraw-Hill, New York, 1970.
  • [17] H. H. Schaefer, Topological vector spaces, Macmillan, New York, 1966. MR 0193469 (33:1689)
  • [18] J. Smoller and H. Wasserman, Global bifurcation of steady state solutions, J. Differential Equations 39 (1981), 269-290. MR 607786 (82d:58056)
  • [19] A. Leung, Equilibria and stabilities for competing species reaction-diffusion equations with Dirichlet boundary data, J. Math. Anal. Appl. 73 (1980), 204-218. MR 560943 (81e:35067)
  • [20] C. V. Pao, Coexistence and stability of a competition diffusion system in population dynamics, J. Math. Anal. Appl. 83 (1981), 54-76. MR 632326 (82m:35077)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0743741-4
Article copyright: © Copyright 1984 American Mathematical Society

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