Minimax Theorems for Indefinite Functionals

Abstract

The dual least action principle has provided sharp existence theorems for the periodic solutions of Hamiltonian systems when the Hamiltonian is convex in u. When it is not the case, the existence of critical points of saddle point type can be proved by using some minimax arguments. To motivate them, we can consider the following intuitive situation. If φ ∈ C ¹(R ²,R), we can view φ(x, y) as the altitude of the point of the graph of φ having (x,y) as projection on R². Assume that there exists points u ₀ ∈ R ², u ₁ ∈ R² and a bounded open neighborhood Ω of u ₀ such that u ₁ ∈ R ² \ Ω and φ(u) > max(φ(u ₀), φ(u ₁)) whenever u ∈ ∂Ω (that is the case for example if u ₀ and u ₁ are two isolated local minimums of φ).