On positive solutions of some pairs of differential equations
HTML articles powered by AMS MathViewer
- by E. N. Dancer
- Trans. Amer. Math. Soc. 284 (1984), 729-743
- DOI: https://doi.org/10.1090/S0002-9947-1984-0743741-4
- PDF | Request permission
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.References
- Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
- Herbert Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), no. 4, 620–709. MR 415432, DOI 10.1137/1018114
- Philip M. Anselone, Collectively compact operator approximation theory and applications to integral equations, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. With an appendix by Joel Davis. MR 0443383
- Henri Berestycki, Le nombre de solutions de certains problèmes semi-linéaires elliptiques, J. Functional Analysis 40 (1981), no. 1, 1–29 (French, with English summary). MR 607588, DOI 10.1016/0022-1236(81)90069-0
- Peter N. Brown, Decay to uniform states in ecological interactions, SIAM J. Appl. Math. 38 (1980), no. 1, 22–37. MR 559078, DOI 10.1137/0138002
- E. Conway, R. Gardner, and J. Smoller, Stability and bifurcation of steady-state solutions for predator-prey equations, Adv. in Appl. Math. 3 (1982), no. 3, 288–334. MR 673245, DOI 10.1016/S0196-8858(82)80009-2
- E. N. Dancer, Global solution branches for positive mappings, Arch. Rational Mech. Anal. 52 (1973), 181–192. MR 353077, DOI 10.1007/BF00282326
- E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl. 91 (1983), no. 1, 131–151. MR 688538, DOI 10.1016/0022-247X(83)90098-7
- Avner Friedman, Partial differential equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. MR 0445088
- Peter Hess and Tosio Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations 5 (1980), no. 10, 999–1030. MR 588690, DOI 10.1080/03605308008820162
- Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
- M. A. Krasnosel′skiĭ, Positive solutions of operator equations, P. Noordhoff Ltd., Groningen, 1964. Translated from the Russian by Richard E. Flaherty; edited by Leo F. Boron. MR 0181881
- P.-L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev. 24 (1982), no. 4, 441–467. MR 678562, DOI 10.1137/1024101
- George H. Pimbley Jr., Eigenfunction branches of nonlinear operators, and their bifurcations. , Lecture Notes in Mathematics, Vol. 104, Springer-Verlag, Berlin-New York, 1969. MR 0266007
- Paul H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis 7 (1971), 487–513. MR 0301587, DOI 10.1016/0022-1236(71)90030-9 W. Rudin, Real and complex analysis, McGraw-Hill, New York, 1970.
- Helmut H. Schaefer, Topological vector spaces, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1966. MR 0193469
- J. Smoller and A. Wasserman, Global bifurcation of steady-state solutions, J. Differential Equations 39 (1981), no. 2, 269–290. MR 607786, DOI 10.1016/0022-0396(81)90077-2
- Anthony Leung, Equilibria and stabilities for competing-species reaction-diffusion equations with Dirichlet boundary data, J. Math. Anal. Appl. 73 (1980), no. 1, 204–218. MR 560943, DOI 10.1016/0022-247X(80)90028-1
- C. V. Pao, Coexistence and stability of a competition-diffusion system in population dynamics, J. Math. Anal. Appl. 83 (1981), no. 1, 54–76. MR 632326, DOI 10.1016/0022-247X(81)90246-8
Bibliographic Information
- © Copyright 1984 American Mathematical Society
- 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