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A bifurcation theorem for critical points of variational problems. (English) Zbl 0659.58007
The authors generalize bifurcation theorems for potential operators by admitting a nonlinear dependence on the bifurcation parameter. The main result is the following: Let E be a real Hilbert space (a generalization to some special Banach spaces is possible), let $$F: E\times {\mathbb{R}}\to {\mathbb{R}}$$ be $$C^ 2$$ and denote $$f(x,\lambda):=D_ xF(x,\lambda),\quad A(\lambda):=D_ xf(0,\lambda).$$ Suppose $$f(0,\lambda)\equiv 0$$ and let 0 be an eigenvalue of A(0) of a finite multiplicity and an isolated point in the spectrum $$\sigma(A(0))$$. Suppose, moreover, that for $$\lambda\neq 0$$ small enough all eigenvalues of $$A(\lambda)$$ are different from zero and the limits $$r^{\pm}:=\lim_{\lambda \to 0\pm}r(A(\lambda))$$ exist, where $$r(A(\lambda)$$) denotes the number of negative eigenvalues of $$A(\lambda)$$ converging to zero as $$\lambda\to 0$$. Then (0,0) is a bifurcation point of the equation $$f(x,\lambda)=0$$. This result is proved by the use of the center manifold theorem and the Conley index.
Reviewer: P.Quittner

##### MSC:
 58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces 35B32 Bifurcations in context of PDEs
##### Keywords:
bifurcation; center manifold; Conley index
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##### References:
  Bates, P.W.; Jones, C.K.R.T., The center manifold theorem with applications, (1985), preprint · Zbl 0696.35142  Berger, M.S., Bifurcation theory and the type number of marston Morse, Proc. natn. acad. sci., 69, 1737-1738, (1972) · Zbl 0237.58013  Böhme, R., Die Lösung der verzweigungsgleichung für nichtlineare eigenwertprobleme, Math. Z., 127, 105-126, (1972) · Zbl 0254.47082  Carr, J., Applications of centre manifold theory, (1981), Springer New York · Zbl 0464.58001  Chow, S.-N.; Hale, J.K., Methods in bifurcation theory, Grundl. der athe. wiss., 250, (1983), Springer Berlin  Clark, D.C., Eigenvalue perturbation for odd, gradient operators, Rocky mt. J. math., 5, 317-336, (1975) · Zbl 0312.47053  Conley, C., Isolated invariant sets and the Morse index, Regional conference series in mathematics, 38, (1976), Conference Board of the Mathematics Society  Fadell, E.R.; Rabinowitz, P.M., Bifurcation for odd potential operators and an alternative index, J. funct. analysis, 26, 48-67, (1977) · Zbl 0363.47029  Henry, D., Geometric theory of semilinear parabolic equations, Lecture notes in mathematics, 840, (1982), Springer New York  Kato, T., Perturbation theory for linear operators, Grundl. der mathe. wiss, 132, (1980), Springer Berlin, corrected printing to 2nd edition  Marino, A., La biforcazione nel caso variationale, Conf. sem. mat., (1973), Bari  Marsden, J.E.; Hughes, T.J.R., Mathematical foundations of elasticity, (1983), Prentice Hall Englewood · Zbl 0545.73031  Palmer, K., Qualitative behaviour of a system of ODE near an equilibrium point—a generalization of the hartman – grobman theorem, (1980), preprint  Rabinowitz, P.H., Variational methods for non-linear eigenvalue problems, (1974), C.I.M.E Varenna  Rabinowitz, P.H., A bifurcation theorem for potential operators, J. funct. analysis, 25, 412-424, (1977) · Zbl 0369.47038  Takens, F., Some remarks on the Böhme – berger bifurcation theorem, Math. Z., 129, 359-364, (1972) · Zbl 0237.47032  Vanderbauwhede, A.; von Gils, S.A., Center manifolds and contractions on a scale of Banach spaces, (1985), preprint · Zbl 0621.47050  Kielhöfer, H., A bifurcation theorem for potential operators, (1985), preprint  Abraham, R.; Marsden, J.E.; Ratiu, T., Manifolds, tensor analysis, and applications, (1983), Addison Wesley London · Zbl 0875.58002
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