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PERSISTENCE OF THE NORMALIZED EIGENVECTORS OF A PERTURBED OPERATOR IN THE VARIATIONAL CASE

Published online by Cambridge University Press:  25 February 2013

RAFFAELE CHIAPPINELLI ,
MASSIMO FURI  and
MARIA PATRIZIA PERA
RAFFAELE CHIAPPINELLI
Affiliation:
Dipartimento di Scienze Matematiche ed Informatiche, Pian dei Mantellini 44, I-53100 Siena, Italy e-mail: [email protected]
MASSIMO FURI
Affiliation:
Dipartimento di Matematica Applicata ‘G. Sansone’, Via S. Marta 3, I-50139 Florence, Italy e-mails: [email protected], [email protected]
MARIA PATRIZIA PERA
Affiliation:
Dipartimento di Matematica Applicata ‘G. Sansone’, Via S. Marta 3, I-50139 Florence, Italy e-mails: [email protected], [email protected]
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Abstract

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Let H be a real Hilbert space and denote by S its unit sphere. Consider the nonlinear eigenvalue problem Ax + ε B(x) =δ x, where A: H → H is a bounded self-adjoint (linear) operator with nontrivial kernel Ker A, and B: H → H is a (possibly) nonlinear perturbation term. A unit eigenvector x0 ∈ S∩ Ker A of A (thus corresponding to the eigenvalue δ=0, which we assume to be isolated) is said to be persistent, or a bifurcation point (from the sphere S∩ Ker A), if it is close to solutions x ∈ S of the above equation for small values of the parameters δ ∈ ℝ and ε ≠ 0. In this paper, we prove that if B is a C1 gradient mapping and the eigenvalue δ=0 has finite multiplicity, then the sphere S∩ Ker A contains at least one bifurcation point, and at least two provided that a supplementary condition on the potential of B is satisfied. These results add to those already proved in the non-variational case, where the multiplicity of the eigenvalue is required to be odd.

Keywords

Primary 47A55Secondary 47J0547J1047J15
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 55 , Issue 3 , September 2013 , pp. 629 - 638
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

References

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