Measuring instruments always have limited resolution. For this reason, in real world applications of mathematics, approximate solutions are often as good as, and frequently indistinguishable from, true solutions. Since mathematical laws governing real world situations are often in variational form, it is natural to develop a theory of “almost critical points”, and the paper under review can be regarded as an interesting and successful step in that direction.

In a substantial part of modern analysis characterized by the tendency to avoid differentiability assumptions, this principle will likely play at least the same role as, say, the contraction mapping principle plays in “smooth” analysis. The elegance of the proofs and the natural way the principle appears in them lend much support to this belief.

This principle discovered in 1972 has found a multitude of applications in different fields of Analysis. It has also served to provide simple and elegant proofs of known results. And as we see, it is a tool that unifies many results where the underlying idea is some sort of approximation.

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.

In this chapter, we are interested in some methods used to find multiplicity results, as in the symmetric MPT, when the symmetry is broken by a “little” perturbation added to the functional. We focus on two methods closely related to the MPT.

… we shall develop methods, employing ideas contained in some of L.A. Ljusternik's work, which allow us to establish the existence of a denumerable number of stable critical values of an even functional – they do not disappear under small perturbations by odd functionals.

It is with the help of the methods of topology, that we shall seek answers to the fundamental question connected with the study of nonlinear equations. However, this last assertion does not imply a negative role for other methods of investigations. In fact, topological methods become powerful only by virtue of their combination with other approaches.

This chapter is devoted to a pure topological version of the MPT due to Katriel [516]. It constitutes a natural “upgrade” of the finite dimensional MPT presented in the former chapter to locally compact topological spaces. It will also help us to clarify more our vision of the situation and will make us see the MPT under another angle. In fact, the topological considerations constitute an inherent part of the different faces of the MPT as will be clarified later. This is the case for all variational analysis results.

Mathematical discoveries, small or great are never born of spontaneous generation. They always presuppose a soil seeded with preliminary knowledge and well prepared by labour, both conscious and subconscious.

Going further in nonsmoothness, we present now a variant of the MPT for continuous functionals on metric spaces. Appropriate notions of critical point and Palais-Smale condition are defined to handle this more general situation that still contains as particular cases the previous results stated when more smoothness and regularity on the functional were supposed.

Intrinsicadj inherent, born, built-in, congenital, connate, constitutional, deepseated, elemental, essential, inborn, inbred, indwelling, ingenerate, ingrained, innate, intimate.

Using the concept of linking of two subsets A and B, seen in Chapter 19, Schechter proved an intrinsic version of the MPT where an estimate for ||Φ′(u)|| appears, as a function of the difference between the supremum of Φ on A and its infimum on B, and of the distance between B and the proper subset of A where Φ assumes greater values than on B.

We will present Schechter's result and some of its immediate consequences, but we will focus on its metric extension due to Corvellec, which presents nicely and clearly its principles and basic ideas.

Locating critical values for a smooth functional / on a manifold X essentially reduces to capturing the changes in the topology of the sublevel sets Ia = {x ∊ X; I(x) < a} as a varies in ℝ.

This chapter is a continuation of the previous one. The “deformation lemma” we will study is very important because, as we will see later, an important part of the inherent topological aspects of critical point theorems is always expressed in terms of deformations.

Geometry may sometimes appear to take the lead over analysis, but in fact precedes it only as a servant goes before his master to clear the path and light him on the way.

The notion of linking is very important in critical point theory. It expresses in an elegant way the geometric conditions that appear in all the abstract results seen until now. Various definitions in many contexts (homotopical, homological, local, isotopic, etc.) were given. Some linking notions are presented and their respective forms of linking theorems, containing the MPT, are stated.

… it was Riemann who aroused great interest in them [problems of the calculus of variations] by proving many interesting results in function theory by assuming Dirichlet's principle …

The mountain pass lemma of Ambrosetti and Rabinowitz is a result of great intuitive appeal as well as practical importance in the determination of critical points of functionals, particularly those which occur in the theory of ordinary and differential equations.

One would expect that it is virtually impossible to find critical points which are not extrema. The first to show that this is not the case are Ambrosetti and Rabinowitz.

This chapter is devoted to the MPT, the source of inspiration for all the results that constitute the material of this monograph. We will also see two of its direct applications to boundary value problems: a superlinear Dirichlet problem and a problem of Ambrosetti-Prodi type.

Plus ça change, plus c'est la même chose.

[The more that changes, the more it's the same.]

When nonlinear problems have a variational structure but do not present enough compactness to use the MPT in its classical form, we have to study carefully the behavior of the Palais-Smale sequence for the inf max level in the statement of the MPT. In this chapter, we will review some of the results that refined the MPT to treat this situation. We will discuss some variants of the MPT where “weighted” Palais-Smale conditions, weaker than the classical one, appear. Then, we will see a very interesting procedure, attributable to Corvellec [257], for deducing new critical point theorems with weighted Palais-Smale conditions from older ones with the standard Palais-Smale condition just by performing a change of the metric of the underlying space X.