Introduction

Summary

The methods we will take up here are all variations on a basic result known to everyone who has done any walk in the hills: the mountain pass lemma.

L. Nirenberg, Variational and topological methods in nonlinear problems, Bull. Am. Math. Soc., 4, 1981

Why a Book on the Mountain Pass Theorem?

The mountain pass theorem (henceforth abbreviated as MPT) is a “phenomenal result” that marks the beginning of a new approach to critical point theory. It constitutes a particularly interesting model for the abstract minimax principle known since the pioneering work of Ljusternik and Schnirelman in the 1940s. It is also the grandfather and the prototype of all the “postmodern” critical point results from the linking family. As early as it appeared, it attracted attention by raising up a lot of theoretical development and serving to solve a very large number of problems in many areas of nonlinear analysis.

The MPT has been intensively investigated. Indeed, there is actually a huge amount of references specifically devoted to its study or presenting one of its variants, generalizations, or applications. Its influence can be measured by the fact that you will rarely find a recent paper or book dealing with variational methods that do not cite it.