Variational Methods for Nonlinear Eigenvalue Problems

Abstract

The goal of these lectures is to present an introduction to variational methods for nonlinear eigenvalue problems both in an abstract setting and as applied to nonlinear partial differential equations. Several different situations will be treated. Our study begins with “Theorems on Ljustemik-Schnirelmann type”. The simplest such result, which is due to Ljusternik [1] states: If f is an even continuously differentiable real valued function on ℝⁿ, then ${\rm{f}}|_{_{{\rm{S}}^{{\rm{n - 1}}}}}$ possesses at least n distinct pairs of critical points (i.e. points at which f′(x) = λx, λ = (f′(x),x)). This theorem serves as a prototype for more general situations where one has a real valued function (usually even) on a manifold (usually “spherelike”) and uses topological invariants associated with the manifold to obtain lower bounds for the number of critical points the functional possesses. Morse theory treats similar questions and indeed there are many ideás in common. However smoothness requirements for theorems of Ljusternik-Schnirelmann type are less stringent (C¹ rather than C²) than in Morse theory and critical points need not be nondegenerate. These facts combine to make the Ljusternik-Schnirelmann theory more applicable to nonlinear partial differential equations. On the other hand when it can be used, Morse theory gives more detailed information on critical points and their types (see e.g. [2]).