Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Global bifurcation in generic systems
of nonlinear Sturm-Liouville problems

Author: Bryan P. Rynne
Journal: Proc. Amer. Math. Soc. 127 (1999), 155-165
MSC (1991): Primary 34B15; Secondary 34B24, 58E07
MathSciNet review: 1487336
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the system of coupled nonlinear Sturm-Liouville boundary value problems $ \begin{array}{c} L_1 u := -(p_1 u')' + q_1 u = \mu u + u f(\cdot,u,v), \quad \text{in}\ (0,1),\\ [1 ex] a_{10} u(0) + b_{10} u'(0) = 0, \quad a_{11} u(1) + b_{11} u'(1) = 0, \end{array} $ $ \begin{array}{c} L_2 v := -(p_2 v')' + q_2 v = \nu v + v g(\cdot,u,v), \quad \text{in}\ (0,1),\\ [1 ex] a_{20} v(0) + b_{20} v'(0) = 0, \quad a_{21} v(1) + b_{21} v'(1) = 0, \end{array} $

where $\mu$, $\nu$ are real spectral parameters. It will be shown that if the functions $f$ and $g$ are `generic' then for all integers $m,\,n \ge 0$, there are smooth 2-dimensional manifolds $ {\mathcal S}_m^1$, $ {\mathcal S}_n^2$, of `semi-trivial' solutions of the system which bifurcate from the eigenvalues $\mu _m$, $\nu _n$, of $L_1$, $L_2$, respectively. Furthermore, there are smooth curves $ {\mathcal B}_{mn}^1 \subset {\mathcal S}_m^1$, $ {\mathcal B}_{mn}^2 \subset {\mathcal S}_n^2$, along which secondary bifurcations take place, giving rise to smooth, 2-dimensional manifolds of `non-trivial' solutions. It is shown that there is a single such manifold, $ {\mathcal N}_{mn}$, which `links' the curves $ {\mathcal B}_{mn}^1$, $ {\mathcal B}_{mn}^2$. Nodal properties of solutions on $ {\mathcal N}_{mn}$ and global properties of $ {\mathcal N}_{mn}$ are also discussed.

References [Enhancements On Off] (What's this?)

  • 1. J. C. ALEXANDER AND S. S. ANTMAN, Global and local behaviour of bifurcating multidimensional continua of solutions for multiparameter nonlinear eigenvalue problems, Arch. Rat. Mech. Anal. 76 (1981), 339-354. MR 82k:58030
  • 2. J. BLAT AND K. J. BROWN, Bifurcation of steady-state solutions in predator-prey and competition systems, Proc. Roy. Soc. Edin. 97A (1984), 21-34. MR 85k:92053
  • 3. R. S. CANTRELL, Global higher bifurcations in coupled systems of nonlinear eigenvalue problems, Proc. Roy. Soc. Edin. 106A (1987), 113-120. MR 88h:92035
  • 4. R. S. CANTRELL, Global preservation of nodal structure in coupled systems of nonlinear Sturm-Liouville boundary value problems, Proc. Amer. Math. Soc. 107 (1989), 633-644. MR 90i:34037
  • 5. R. S. CANTRELL, 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, Proc. Roy. Soc. Edin. 119A (1991), 347-365. MR 93d:34031
  • 6. R. S. CANTRELL AND C. COSNER, On the steady-state problem for the Volterra-Lotka competition model with diffusion, Houston J. Math. 13 (1987), 337-352. MR 89d:92052
  • 7. M. G. CRANDALL AND P. H. RABINOWITZ, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321-340. MR 44:5836
  • 8. E. N. DANCER, On the structure of solutions of non-linear eigenvalue problems, Indiana Univ. Math. J. 23 (1974), 1069-1076. MR 50:1065
  • 9. D. GILBARG AND N. S. TRUDINGER, Elliptic Partial Differential Equations of Second Order, Springer, 1983. MR 86c:35035
  • 10. P. BRUNOVSKY AND P. POLACIK, The Morse-Smale structure of a generic reaction-diffusion equation in higher space dimension, J. Diff. Equns. 135 (1997), 129-181. CMP 97:08
  • 11. P. H. RABINOWITZ, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487-513. MR 46:745
  • 12. B. P. RYNNE, Genericity of hyperbolicity and saddle-node bifurcations in reaction-diffusion equations depending on a parameter, J. Appl. Math. Phys. (ZAMP) 47 (1996), 730-739. MR 97h:35105
  • 13. A. E. TAYLOR AND D. C. LAY, Introduction to Functional Analysis, Second Edition, Wiley, 1980. MR 81b:46001
  • 14. J. C. SAUT AND R. TEMAM, Generic properties of nonlinear boundary value problems, Comm. in PDE 4 (1979), 293-319. MR 80i:35080
  • 15. E. ZEIDLER, Nonlinear Functional Analysis and its Applications, Vol I, Springer, 1986. MR 87f:47083
  • 16. E. ZEIDLER, Nonlinear Functional Analysis and its Applications, Vol IV, Springer, 1988. MR 90b:00004

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 34B15, 34B24, 58E07

Retrieve articles in all journals with MSC (1991): 34B15, 34B24, 58E07

Additional Information

Bryan P. Rynne
Affiliation: Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, Scotland

Keywords: Global bifurcation, genericity, Sturm-Liouville systems
Received by editor(s): May 2, 1997
Communicated by: Hal L. Smith
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society