Critical Point Theory and Applications to Differential Equations: A Survey

Abstract

The purpose of this paper is to survey developments in the field of critical point theory and its applications to differential equations that have occurred during the past 20–25 years. This is too broad a theme for a single survey and we will focus on three particular areas. First we will examine contributions to the minimax approach to critical point theory. In particular the Mountain Pass Theorem, the Saddle Point Theorem, and variants thereupon will be discussed in Part 1. Then in Part 2, applications of critical point theory to the existence of periodic solutions of Hamiltonian systems of differential equations will be surveyed. Many of the different questions that have been studied will be described and a variety of results will be presented. Lastly, Part 3 deals with connecting orbits of Hamiltonian systems, mainly homoclinic and heteroclinic orbits. This last set of material represents the least developed subject matter treated here and therefore it can be described more completely than the earlier topics.