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Topological Nonlinear Analysis pp 464–513Cite as

Critical Point Theory and Applications to Differential Equations: A Survey

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Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 15))

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.

This research was sponsored in part by the National Science Foundation under Grant #MCS-8110556 and by the US Army under contract #DAAL03-87- k-0043. Any reproduction for the pruposes of the US Government is permitted.

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  1. Mathematics Department, University of Wisconsin, Madison, WI, 53706, USA

    Paul H. Rabinowitz

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  1. Department of Mathematics, University of Rome, Tor Vergata, 00133, Rome, Italy

    Michele Matzeu  & Alfonso Vignoli  & 

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Rabinowitz, P.H. (1995). Critical Point Theory and Applications to Differential Equations: A Survey. In: Matzeu, M., Vignoli, A. (eds) Topological Nonlinear Analysis. Progress in Nonlinear Differential Equations and Their Applications, vol 15. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2570-6_6

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  • DOI: https://doi.org/10.1007/978-1-4612-2570-6_6

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