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Parameter ranges for the existence of solutions whose state components have specified nodal structure in coupled multiparameter systems of nonlinear Sturm–Liouville boundary value problems

Published online by Cambridge University Press:  14 November 2011

Robert Stephen Cantrell
Robert Stephen Cantrell
Affiliation:
Department of Mathematics and Computer Science, The University of Miami, Coral Gables, Florida 33124, U.S.A.
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Synopsis

The set of solutions to the two-parameter system

has been shown in a preceding paper of the author to exhibit a topological-functional analytic structure analogous to the structure of solution sets for nonlinear Sturm–Liouville boundary value problems. As the parameter λ and µ are varied, transitions in the solution set occur, first from trivial solutions to solutions (u, 0) with u having n nodes on (a, b) or solutions (0, v) with v having m nodes on (a, b), and then to solutions of the form (u, v), where u has n nodes on (a, b) and v has m nodes on (a, b), with n possibly different from m. Moreover, each transition is global in an appropriate bifurcation theoretic sense, with preservation of nodal structure. This paper explores these phenomena more closely, focusing on the range of parameters (λ, µ) for the existence of solutions (u, v) with u having n nodes on (a, b) and v having m nodes on (a, b) and its dependence on the assumptions placed on the coupling functions f and g. The principal tools of the analysis are the Alexander–Antman Bifurcation Theorem and a priori estimate techniques based on the maximum principle.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 119 , Issue 3-4 , 1991 , pp. 347 - 365
Copyright
Copyright © Royal Society of Edinburgh 1991

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References

1Alexander, J. C. and Antman, S. S.. Global and local behavior of bifurcating multidimensional continua of solutions for multiparameter nonlinear eigenvalue problems. Arch. Rational Mech. Anal. 76 (1981), 339–354.CrossRefGoogle Scholar
2Blat, J. and Brown, K. J.. Bifurcation of steady-state-solutions in predator-prey and competition systems. Proc. Roy. Soc. Edinburgh Sect. A 97 (1984), 21–34.CrossRefGoogle Scholar
3Cantrell, R. S.. Global higher bifurcations in coupled systems of nonlinear eigenvalue problems. Proc. Roy. Soc. Edinburgh Sect. A 106 (1987), 113–120.CrossRefGoogle Scholar
4Cantrell, R. S.. Global preservation of nodal structure in coupled systems of nonlinear Sturm-Liouville boundary value problems. Proc. Amer. Math. Soc. 107 (1989), 633–644.CrossRefGoogle Scholar
5Cantrell, R. S. and Cosner, C.. On the steady-state problem for the Volterra-Lotka competition model with diffusion. Houston J. Math. 13 (1987), 337–352.Google Scholar
6Cantrell, R. S. and Cosner, C.. On the uniqueness and stability of positive solutions in the Lotka-Volterra competition model with diffusion. Houston J. Math. 15 (1989), 341–361.Google Scholar
7Cosner, C. and Lazer, A. C.. Stable coexistence states in the Volterra-Lotka competition model with diffusion. SIAM J. Appl. Math. 44 (1984), 1112–1132.CrossRefGoogle Scholar
8Dancer, E. N.. On positive solutions of some pairs of differential equations, I. Trans. Amer. Math. Soc. 284 (1984), 729–743.CrossRefGoogle Scholar
9Dancer, E. N.. On positive solutions of some pairs of differential equations, II. J. Differential Equations 60 (1985), 236–258.CrossRefGoogle Scholar
10Dancer, E. N.. On the existence and uniqueness of positive solutions for competing species models with diffusion. Trans. Amer. Math. Soc., to appear.Google Scholar
11Dancer, E. N.. On the structure of solutions of non-linear eigenvalue problems. Indiana Univ. Math. J. 23 (1974), 1069–1076.Google Scholar
12Leung, A.. Equilibria and stabilities for competing species, reaction-diffusion equations with Dirichlet boundary data. J. Math. Anal. Appl. 73 (1980), 204–218.Google Scholar
13Li, L.. Global positive coexistence of a nonlinear elliptic biological interacting model. Math. Biosci. 97 (1989), 1–15.Google ScholarPubMed
14McKenna, P. J. and Walter, W.. On the Dirchlet problem for elliptic systems. Appl. Anal. 21 (1986), 207–224.Google Scholar
15Rabinowitz, P. H.. Some global results for non-linear eigenvalue problems. J. Funct. Anal. 7 (1971), 487–513.Google Scholar
16Schiaffino, A. and Tesei, A.. Competition systems with Dirichlet boundary conditions. J. Math. Biol. 15 (1982), 93–105.CrossRefGoogle Scholar