On the persistence of the eigenvalues of a perturbed Fredholm operator of index zero under nonsmooth perturbations

Let H be a real Hilbert space and denote by S its unit sphere. Consider the nonlinear eigenvalue problem Lx + εN (x) = λx, where ε, λ ∈ R, L : H → H is a bounded self-adjoint (linear) operator with nontrivial kernel and closed image, and N : H → H is a (possibly) nonlinear perturbation term. A unit eigenvector x* ∈ S ∩ Ker L of L (corresponding to the eigenvalue λ = 0) is said to be persistent if it is close to solutions x ∈ S of the above equation for small values of the parameters ε and λ, with ε different from 0. We give an affirmative answer to a conjecture formulated by R. Chiappinelli and the last two authors in an article published in 2008. Namely, we prove that if N is Lipschitz continuous and the eigenvalue λ = 0 has odd multiplicity, then the sphere S ∩ Ker L contains at least one persistent eigenvector. We provide examples in which our results apply, as well as examples showing that if the dimension of Ker L is even, then the persistence phenomenon may not occur.