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On the exactness of an S-shaped
bifurcation curve

Authors: Philip Korman and Yi Li
Journal: Proc. Amer. Math. Soc. 127 (1999), 1011-1020
MSC (1991): Primary 34B15
MathSciNet review: 1610804
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Abstract | References | Similar Articles | Additional Information

Abstract: For a class of two-point boundary value problems we prove exactness of an S-shaped bifurcation curve. Our result applies to a problem from combustion theory, which involves nonlinearities like $e^{au/(u+a)}$ for $a>0$.

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  • 1. A. Ambrosetti and P. Hess, Positive solutions of asymptotically linear elliptic eigenvalue problems, J. Math. Anal. Applic. 73, 411-422 (1980). MR 81j:35092
  • 2. J. Bebernes and D. Eberly, Mathematical Problems from Combustion Theory, Springer-Verlag (1989). MR 91d:35165
  • 3. K.J. Brown, M.M.A. Ibrahim and R. Shivaji, S-shaped bifurcation curves, Nonlinear Anal. TMA 5 (5), 475-486 (1981). MR 82h:35007
  • 4. A. Castro and R. Shivaji, Uniqueness of positive solutions for elliptic boundary value problems, Proc. Royal Soc. Edinburgh 98A, 267-269 (1984). MR 86d:35051
  • 5. M.G. Crandall and P.H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal. 52, 161-180 (1973). MR 49:5962
  • 6. E.N. Dancer, On the structure of solutions of an equation in catalysis theory when a parameter is large, J. Differential Equations 37, 404-437 (1980). MR 82b:35018
  • 7. Y. Du and Y. Lou, S-Shaped global bifurcation curve of positive solutions to a prey-predator model, MSRI Preprint No. 1996-044 (1996).
  • 8. P. Korman, Y. Li and T. Ouyang, Exact multiplicity results for boundary-value problems with nonlinearities generalizing cubic, Proc. Royal Soc. Edinburgh 126A, 599-616 (1996). MR 97c:34038
  • 9. N. Mizoguchi and T. Suzuki, Equations of gas combustion: s-shaped bifurcation and mushrooms, J. Differential Equations 134, 183-215 (1997). MR 97m:35090
  • 10. R. Shivaji, Remarks on an S-shaped bifurcation curve, J. Math. Anal. Applic. 111, 374-387 (1985). MR 87c:35015
  • 11. S.-H. Wang, On S-shaped bifurcation curves, Nonlinear Analysis TMA 22, 1475-1485 (1994). MR 96d:34021

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Additional Information

Philip Korman
Affiliation: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025

Yi Li
Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242

Keywords: S-shaped bifurcation curve, Crandall-Rabinowitz theorem
Received by editor(s): July 8, 1997
Communicated by: Hal L. Smith
Article copyright: © Copyright 1999 American Mathematical Society

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