ABSTRACT
Variational methods are a valuable tool in the analysis of nonlinear problems. According to these methods, we are trying to find solutions of a given nonlinear equation, by looking for critical (stationary) points of a functional defined on the function space in which we want the solution of our problem to lie. The Euler-Lagrange equation satisfied by a critical point is the nonlinear equation that we are trying to solve. The functional, whose critical points we are trying to determine, in many cases is unbounded (both from above and below – indefinite functional) and so we cannot expect to have global maxima or minima. Instead we look for local extrema or for saddle points using minimax arguments. So critical point theory is the main ingredient in the variational methods. In this chapter we present some aspects of critical point theory which are useful in the study of nonlinear boundary value problems. A fruitful technique in obtaining critical points of a C1-functional is based
on deformation arguments along the gradient flow or a substitute of it when due to the geometry of the space or the lack of regularity of the functional it is impossible to use the gradient flow. For this reason in Section 5.1, we introduce the so-called pseudogradient vector field and the compactness-type conditions that this vector field must satisfy and we derive various deformation results, which describe the deformations of the sublevel sets of the functionals near a critical point where topologically interesting things may occur. In Section 5.2 we use the deformation results and the geometric notion of
linking sets in order to obtain minimax expressions for the critical values of the functionals. We prove a general minimax principle which generates as special cases the classical mountain pass theorem, saddle point theorem and generalized mountain pass theorem. In Section 5.3 we prove strong forms of the mountain pass theorem, which,
besides an existence statement for critical points, give in addition information about the fine structure of the functional near them. In Section 5.4 we prove results establishing the existence of multiple critical
points. For this purpose we introduce the notion of local linking, we impose symmetry conditions on the functional and we use the Krasnoselskii’s genus of a set. The Krasnoselskii’s genus is an example of a topological index. Historically the first topological index was introduced by Lusternik-Schnirelman in order to extend to nonlinear eigenvalue problems the theory of eigenvalues of quadratic forms developed by Courant, Weyl and others. It turns out that
invariant under homeomorphisms and satisfies certain properties (see Proposition 5.5.5). In Section 5.6 we introduce the Lusternik-Schnirelman category and prove
a basic multiplicity result for critical points of a C1-functional. Also we develop the method of Lagrange multipliers for infinite dimensional constrained optimization problems, which provides the analytical framework for the study of nonlinear eigenvalue problems.
