Existence of multiple critical points for an asymptoticallyquadratic functional with applications

Shujie Li,Jiabao Su

doi:10.1155/s1085337596000140

Abstract

Morse theory for isolated critical points at infinity is used for the existence of multiple critical points for an asymptotically quadratic functional. Applications are also given for the existence of multiple nontrivial periodic solutions of asymptotically Hamiltonian systems.

Highlights

Introduction and preliminariesIt is known that the objective of the Morse theory is the relation between the topological type of critical points of a function f and the topological structure of the manifold on which the function is defined
The topological type of a critical point x is described by the critical groups Ck(f, x) for which there have been many known results, cf ([3], [8], [7], etc.)
The topological structure of the manifold M is described by its Betti number βk = dim Hk(M )

Summary

Introduction and preliminaries

It is known that the objective of the Morse theory is the relation between the topological type of critical points of a function f and the topological structure of the manifold on which the function is defined. H∗(·, ·) denotes a singular relative homology group with the abelian coefficient group G From this definition we see that the topology of the pair (X, f a) contains all the information about the critical points of f because we require that f (K) be bounded from below by a ∈ R1. A) Ck(f, ∞) ∼= δkμG provided f satisfies the following angle condition at infinity:. B) Ck(f, ∞) ∼= δk,μ+νG provided f satisfies the following condition at infinity:. A) Ck(f, θ) ∼= δkμ0G provided f satisfies the following angle condition at θ:. B) Ck(f, θ) ∼= δkμ0+ν0G provided f satisfies the following angle condition at θ:. This paper is organized in the following way: In section 2 we prove some abstract critical point theorems by means of Proposition 1.2 and 1.3 and the Morse theory. As we will see that the main difficulty is to verify the strong angle conditions which can be guaranteed by the so-called “pinching” condition

Some abstract critical point theorems

Applications to a second order hamiltonian system