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A global branch of solutions to a semilinear equation on an unbounded interval

Published online by Cambridge University Press:  14 November 2011

C. A. Stuart
C. A. Stuart
Affiliation:
Département de Mathématiques, EPFL, CH-1015 Lausanne, Switzerland
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Synopsis

For a semilinear second order differential equation on (0, ∞), conditions are given for the bifurcation and asymptotic bifurcation in Lp of solutions to the Neumann problem. Bifurcation occurs at the lowest point of the spectrum of the linearised problem. Under stronger hypotheses, there is a global branch of solutions. These results imply similar conclusions for the same equation on R with appropriate symmetry.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 101 , Issue 3-4 , 1985 , pp. 273 - 282
Copyright
Copyright © Royal Society of Edinburgh 1985

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References

1Brézis, H.. Analyse Functionnelle, Théorie et Applications (Paris: Masson, 1983).Google Scholar
2Kantorovitch, L. and Akilov, G.. Analyse Fonctionnelle Vol. II (Moscow: Mir, 1981).Google Scholar
3Demay, Y., Bifurcation d'un soliton pour une équation de la physique des plasmas. C.R. Acad. Sci. Paris 285 (1977), 769–772.Google Scholar
4Robert, M. and Stuart, C. A.. Intrinsic structure of the critical liquid-gas interface. Phys. Rev. Letters 49 (1982), 1434–1437.CrossRefGoogle Scholar
5Magnus, R. J.. The transformation of vector-functions scaling and bifurcation. Trans. Amer. Math. Soc. 286 (1984), 689–714.CrossRefGoogle Scholar
6. Stuart, C. A.. Bifurcation pour des problémes de Dirichlet et de Neumann sans valeurs propres. C. R. Acad. Sci. Paris 288 (1979), 761–764.Google Scholar
7Stuart, C. A.. Bifurcation for Neumann problems without eigenvalues. J. Differential Equations 36 (1980), 391–407.Google Scholar
8Stuart, C. A.. Bifurcation in Lv(R) for a semilinear equation, to appear.Google Scholar
9Toland, J. F., Global bifurcation for Neumann problems without eigenvalues. J. Differential Equations 44 (1982), 82–101.CrossRefGoogle Scholar
10Toland, J. F.. Singular elliptic eigenvalue problems for equations and systems. In Systems of Nonlinear Partial Differential Equations (ed. Ball, J. M.) (New York: Reidel, 1983).Google Scholar
11Toland, J. F.. Uniqueness of positive solutions of some Sturm-Liouville problems on the half-line. Proc. Roy. Soc. Edinburgh Sect. A 97 (1984), 259–263.CrossRefGoogle Scholar
12Küpper, T. and Reimer, D.. Necessary and sufficient conditions for bifurcation from the essential spectrum. Nonlinear Anal. T.M.A. 3 (1979), 555–561.CrossRefGoogle Scholar