Journal of Dynamics and Differential Equations

, Volume 24, Issue 4, pp 857–871

Homoclinic Orbits for second order Hamiltonian Equations in \({\mathbb{R}}\)

Article

DOI: 10.1007/s10884-012-9275-0

Cite this article as:
Makita, P.D. J Dyn Diff Equat (2012) 24: 857. doi:10.1007/s10884-012-9275-0

Abstract

We are concerned with the existence and multiplicity of homoclinic solutions for the second order Hamiltonian equation
$$-\ddot{u}+\omega(t)u=F_u(t,u) \quad t \in \mathbb{R}, \quad\quad\quad(1)$$
where \({\omega \in \mathcal{C}(\mathbb{R})}\) is positive and bounded, and \({F\in \mathcal{C}^1(S^1\times\mathbb{R})}\) . Under some growth condition on F, we prove that (1) admits at least two solutions which are homoclinic to zero and do not change sign. We also prove that for every integer k ≥  1, (1) possesses at least two solutions homoclinic to zero changing sign exactly k times, and for k ≥  2 these solutions have at least k and at most k + 2 zeros which are isolated, or ‘isolated from the left’, or ‘isolated from the from right’.

Keywords

Homoclinic solutionSolution changing sign a prescribed number of timesNehari manifold

Mathematics Subject Classification

37J4535B4535J55

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of GiessenGiessenGermany