Homoclinic Orbits for a Class of Nonperiodic Hamiltonian Systems

Abstract

We study the following nonperiodic Hamiltonian systemż JH z t, z , whereWe 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.

Homoclinic orbits for first order Hamiltonian systems with some twist conditions

“…In this sense, our main results extend the Theorem 1.1 in [21]. We point out that we do not consider the case that H is resonant at the infinity or at origin, it is not our purpose to give a survey in this paper.…”

“…Many authors treated the case where L(t) and H(t, u) are either independent of t or periodic in t. Coti-Zelati and Rabinowitz first consider the system (1.1) in [5] using variational methods, and they obtained a homoclinic orbit for strictly convex Hamiltonian system. The result was deepened in [21,22] when Sére established the existence of infinitely many homoclinic orbits. Subsequently, Hofer and Wysocki removed the convexity assumptions in [14].…”

“…It is quite different in nature if A on L 2 has essential spectrum or not. Recently, in [21], under some assumptions on L, the spectrum of A consists of a sequence of eigenvalues with finite multiplicity which is unbounded from above and below. More precisely, L satisfies…”

“…Generally, the initial index can be any prescribed integer and the index depends on B 0 and the initial index. [21], under the assumption (L 1 ) on L, we know that the operator A has a sequence of eigenvalues…”