The purpose of this chapter is to introduce the reader to Morse theoretic methods used in variational problems. General references are Milnor , Mawhin and Willem , Chang , and Benci ; see also Perera et al. . We begin by briefly collecting some basic results of Morse theory. These include the Morse inequalities, Morse lemma and its generalization splitting lemma, the shifting theorem of Gromoll and Meyer, and the handle body theorem. Results that are needed later in the text will be proved in subsequent sections.
Let G be a real-valued function defined on a real Banach space E. We say that G is Fréchet differentiable at u E if there is an element G′(u) of the dual E′, called the Fréchet ...