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

Let H be a real Hilbert space and denote by S its unit sphere. Consider the nonlinear eigenvalue problem Ax+εB(x)=δx, where A:H→H is a bounded self-adjoint (linear) operator with nontrivial kernel KerA, and B:H→H is a (possibly) nonlinear perturbation term. A unit eigenvector x0∈S∩KerA 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∩KerA), 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∩KerA 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.