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

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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

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## Abstract

We are concerned with the existence and multiplicity of homoclinic solutions for the second order Hamiltonian equation
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

$$-\ddot{u}+\omega(t)u=F_u(t,u) \quad t \in \mathbb{R}, \quad\quad\quad(1)$$

*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