Homoclinic Orbits for a Class of Nonperiodic Hamiltonian Systems

Wenping Qin,Fukun Zhao,Jian Zhang

doi:10.1155/2012/769232

Abstract

We study the following nonperiodic Hamiltonian system , where H ∈ C1(ℝ × ℝ2N, ℝ) is the form H(t, z) = (1/2)B(t)z · z + R(t, z). We introduce a new assumption on B(t) and prove that the corresponding Hamiltonian operator has only point spectrum. Moreover, by applying a generalized linking theorem for strongly indefinite functionals, we establish the existence of homoclinic orbits for asymptotically quadratic nonlinearity as well as the existence of infinitely many homoclinic orbits for superquadratic nonlinearity.

Highlights

Introduction and Main ResultsIn this paper, we are interested in the existence of homoclinic orbits of the Hamiltonian system z JHz t, z, HS where z p, q ∈ RN × RN R2N, J, and H ∈ C1 R × R2N, R is the form H t, z B t z· z R t, z1.1 with B t ∈ C R, R4N2 being a 2N × 2N symmetric matrix valued function, and R ∈ C1 R × R2N, R
Motivated by 9, 11, in this paper, we introduce a new nonperiodic assumption on B t as the following
The assumptions R1, R4 - R5 imply that R t, z → ∞ uniformly in t as |z| → ∞

Summary

Introduction and Main Results

We are interested in the existence of homoclinic orbits of the Hamiltonian system z JHz t, z , HS where z p, q ∈ RN × RN R2N, J. By a homoclinic orbit of HS , we mean a solution of the equation satisfying z t /≡ 0 and z t → 0 as |t| → ∞. Establishing the existence of homoclinic orbits for system like HS is one of the most important problems in the theory of Hamiltonian systems. Suppose that R t, z and B t depend periodically on t, the existence of homoclinic orbit for HS was considered in 7, 8, 12, 16, 18. In , Ding and Li first obtained one homoclinic orbit for the nonperiodic system HS , see 9, and the references therein for recent works on this direction. The assumptions R1 , R4 - R5 imply that R t, z → ∞ uniformly in t as |z| → ∞

Variational Framework and Linking Structure

The C c Condition

Proof of Theorems