On the Persistence of the Eigenvalues of a Perturbed Fredholm Operator of Index Zero under Nonsmooth Perturbations

Abstract

Let $H$ be a real Hilbert space and denote by $S$ its unit sphere. Consider the nonlinear eigenvalue problem $Lx+ϵN(x)=\lambda x$, where $ϵ,\lambda \in \mathbb{R}$, $L:H\to H$ is a bounded self-adjoint (linear) operator with nontrivial kernel and closed image, and $N:H\to H$ is a (possibly) nonlinear perturbation term. A unit eigenvector $\bar{x}\in S\cap \mathrm{Ker}L$ of $L$ (corresponding to the eigenvalue $\lambda =0$) is said to be persistent if it is close to solutions $x\in S$ of the above equation for small values of the parameters $ϵ\ne 0$ and $\lambda$. 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 $\lambda =0$ has odd multiplicity, then the sphere $S\cap \mathrm{Ker}L$ contains at least one persistent eigenvector. We provide examples in which our results apply, as well as examples showing that, if the dimension of $\mathrm{Ker}L$ is even, then the persistence phenomenon may not occur.