Infinitely many homoclinic orbits for the second-order Hamiltonian systems with super-quadratic potentials

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Abstract

We study the existence of infinitely many homoclinic orbits for some second-order Hamiltonian systems: üL(t)u(t)+F(t,u(t))=0,tR, by the variant fountain Theorem, where F(t,u) satisfies the super-quadratic condition F(t,u)/|u|2 as |u| uniformly in t, and need not satisfy the global Ambrosetti–Rabinowitz condition.

Section snippets

Introduction and main results

In this paper, we consider the second-order Hamiltonian system: üL(t)u(t)+F(t,u(t))=0,tR, where LC(R,RN2) is a symmetric matrix valued function, FC1(R×RN,R), and F(t,u)=(F/u)(t,u). We say that a solution of (1.1) is homoclinic (to 0) if uC2(R,RN),u0,u(t)0 and u̇(t)0 as |t|. With the aid of the variational methods, many authors extensively investigated the existence of (multiple) homoclinic solutions for problem (1.1), see [2], [3], [4], [5], [6]. Most of them treated the

Proof of Theorem 1

We denote by A the self-adjoint extension of the operator d2dt2+L(t) with the domain DL2(R,RN). Let E=D(|A|1/2), the domain of |A|1/2, and define on E the inner product and norm (u,w)0=(|A|1/2u,|A|1/2w)2+(u,w)2,u0=(u,u)01/2, where (,)2 denotes the inner product of L2. Then E is a Hilbert space.

It is easy to prove that the spectrum σ(A) consists of eigenvalues numbered by λ1λ2λn (counted in their multiplicities) and a corresponding system of eigenfunctions {en},Aen=λnen which forms

Acknowlegements

The authors of this paper would like to express their sincere thanks to the referee for useful suggestions. Especially, they would also like to thank Guihua Fei and Ou-Tang whose papers drew their attention to this problem.

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    The study of homoclinic orbits of Hamiltonian systems dates back to the time of Poincaré, but mainly by perturbation techniques. It is only relatively the last twenty years that variational methods have been extensively applied to find homoclinic orbits of Hamiltonian systems, see [1–27] and the references therein. We summarize the findings as follows.

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