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Bifurcation and stability of positive solutions of a two-point boundary value problem

Part of: Boundary value problems

Published online by Cambridge University Press:  09 April 2009

Shin-Hwa Wang  and
Nicholas D. Kazarinoff
Shin-Hwa Wang
Affiliation:
Department of MathematicsNational Tsing Hua UniversityHsinchu, Taiwan 300 R.O.C.
Nicholas D. Kazarinoff
Affiliation:
Department of MathematicsNational Tsing Hua UniversityHsinchu, Taiwan 300 R.O.C.
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Abstract

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We consider the existence of multiple positive solutions of a nonlinear two-point boundary value problem by modifying a “time map” technique introduced by J. Smoller and A. Wasserman. We count the number of positive solutions and find their Conley indices and thus determine their stabilities.

Keywords

bifurcationstabilitypositive solutiontwo-point boundary value problemand Conley index.

MSC classification

Secondary: 34B15: Nonlinear boundary value problems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 52 , Issue 3 , June 1992 , pp. 334 - 342
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Ambrosetti, A., ‘On the exact number of positive solutions of convex nonlinear problems’, Bulletino, U.M.I. (5) 15–A (1978), 610–615.Google Scholar
[2]Ambrosetti, A. and Hess, P., ‘Positive solutions of asymptotically linear elliptic eigenvalue problems’, J. Math. Anal. Appl. 73 (1980), 411–422.CrossRefGoogle Scholar
[3]Berestycki, H., ‘Le nombre de solutions de certain problèmes semilinéaires elliptiques’, J. Funct. Anal. 40 (1980), 1–29.CrossRefGoogle Scholar
[4]Bramson, M., Convergence of solutions of the Kolmogorov equation to travelling waves, Memoirs Amer. Math. Soc. 285, 1983.Google Scholar
[5]Dancer, E. N., ‘On the number of positive solutions of weakly non-linear elliptic equations when a parameter is large’, Proc. London Math. Soc. 53 (1987), 429–452.Google Scholar
[6]de Figueiredo, D. G., ‘On the uniqueness of positive solutions of the Dirichlet problem –Δu = λ sin u,’ pp. 80–83 in Nonlinear Partial Differential Equations and Applications vol. 7, Brezis, H. and Lions, J. L. (editors) (Pitman, London, 1984).Google Scholar
[7]Gelfand, I., ‘Some problems in the theory of quasilinear equations’, Amer. Math. Soc. Translations 29 (1963), 295–381.CrossRefGoogle Scholar
[8]Kolmogoroff, A., Petrovsky, I., and Piscounoff, N., ‘Étude de l'équations de la diffusion avec croissance de la quantité de matière et son application à un problème biologique’, Bull. Univ. Moscow. Ser. Internat., Sec. A 1 (1937), 1–25.Google Scholar
[9]Sevryuk, M. B., Reversible Systems (Lecture Notes in Math. 1211, Springer-Verlag, Berlin, Heidelberg, 1986).CrossRefGoogle Scholar
[10]Seidman, T. I., ‘Asymptotic growth of solutions of –Δu = λf(u) for large λ,’ Indiana Univ. Math. J. 30 (1981), 305–311.CrossRefGoogle Scholar
[11]Smoller, J., Shock Waves and Reaction-Diffusion Equations (Springer-Verlag, New York, 1983).CrossRefGoogle Scholar
[12]Smoller, J. and Wasserman, A., ‘Global bifurcation of steady-state solutions’, J. Differential Equations 39 (1981), 269–290.CrossRefGoogle Scholar
[13]Smoller, J. and Wasserman, A., ‘Generic bifurcation of steady-state solutions’, J. Differential Equations 52 (1984), 432–438.CrossRefGoogle Scholar
[14]Wang, S. H., ‘A correction for a paper by J. Smoller and A. Wasserman’, J. Differential Equations 77 (1989), 199–202.CrossRefGoogle Scholar
[15]Wang, S. H. and Kazarinoff, N. D., ‘Bifurcation of steady-state solutions of a scalar reaction-diffusion equation in one space variable’, J. Austral. Math. Soc. (Series A) 52 (1992), 343–355.CrossRefGoogle Scholar