Variational Methods for NLEV Approximation Near a Bifurcation Point

We review some more and less recent results concerning bounds on nonlinear eigenvalues (NLEV) for gradient operators. In particular, we discuss the asymptotic behaviour of NLEV (as the norm of the eigenvector tends to zero) in bifurcation problems from the line of trivial solutions, considering perturbations of linear self-adjoint operators in a Hilbert space. The proofs are based on the Lusternik-Schnirelmann theory of critical points on one side and on the Lyapounov-Schmidt reduction to the relevant finite-dimensional kernel on the other side. The results are applied to some semilinear elliptic operators in bounded domains of . A section reviewing some general facts about eigenvalues of linear and nonlinear operators is included. 1. Introduction and Examples The term “nonlinear eigenvalue” (NLEV) is a frequent shorthand for “eigenvalue of a nonlinear problem,” see, for instance [1, 3]. While for the estimation of eigenvalues of linear operators there is wealth of abstract and computational methods (see, e.g., Kato's [4] and Weinberger's [5] monographs), for NLEV, the question is relatively new and there is not much literature available. In this paper, we review some abstract methods which allow for the computation of upper and lower bounds of NLEV near a bifurcation point of the linearized problem. Moreover, as one of our aims is to stimulate further research on the subject, we spend some effort in presenting it in a sufficiently general context and emphasize the question of the existence of eigenvalues for a nonlinear operator. In fact, Section 2 is entirely devoted to this, and to a parallel consideration of similar facts for linear operators. Thus, generally speaking, consider two nonlinear (= not necessarily linear) operators ( real Banach spaces) such that . If for some the equation has a solution , then we say that is an eigenvalue of the pair and is an eigenvector corresponding to . This definition is a word-by-word copy of the standard one for pairs of linear operators, where most frequently one takes and , and of course it may be of very little significance in general. However, it goes back at least to Krasnosel'skii [6] the demonstration of the importance of this concept for operator equations such as (1.1), with a view in particular to nonlinear integral equations of Hammerstein or Urysohn type. In this paper, we consider (1.1) under the following qualitative assumptions:(A) (1.1) possesses infinitely many eigenvalues ;(B) (1.1) has a linear reference problem which also possesses infinitely many eigenvalues . It is then natural to try to