Morse Theory in Differential Equations

In the study of closed geodesics, Marston Morse developed his theory on the calculus of variations in the large. The Morse inequalities, which link on one hand, the numbers of critical points in various types of a function, and on the other hand, the topological invariants of the underlying manifold, play an important role in Morse theory. Naturally, they provide an estimate for the number of critical points of a function by using the topology of the manifold. Hopefully, this topological method would deal with the existence and the multiplicity of solutions of certain nonlinear differential equations. However, in this theory, the manifold is compact, and the functions are assumed to be C2 and to have only nondegenerate critical points; all of these restrict the applications seriously. In contrast, Leray-Schauder degree theory has become a very useful topological method. In 1946, at the bicentennial conferences of Princeton University, there was much discussion of their contrast. M. Shiffman hoped that the two methods could be brought closer together “so that they may alter and improve each other, and also so that each may fill out the gaps in the scope of the other” [Pr]. Since then, great efforts have been made to extend the Morse theory. We only mention a few names of the pioneers as follows: R. Bott, E. Rothe, R. S. Palais, S. Smale, D. Gromoll, W. Meyer, A. Marino, and G. Prodi.