Homoclinic orbits for a class of Hamiltonian systems      with superquadratic or asymptotically quadratic potentials

Jun Wang; Junxiang Xu; Fubao Zhang

doi:10.3934/cpaa.2011.10.269

Article Contents

2011, Volume 10, Issue 1: 269-286. Doi: 10.3934/cpaa.2011.10.269

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Homoclinic orbits for a class of Hamiltonian systems with superquadratic or asymptotically quadratic potentials

1.
Department of Mathematics, Southeast University, Nanjing 210096

Received: November 2009

Revised: August 2010

Published: January 2011

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Abstract

In this paper we study the following nonperiodic second order Hamiltonian system

$-\ddot{u}(t)+L(t)u(t)=\nabla_u R(t,u(t)), \forall (t,u)\in R\times R^N, $

where the matrix $L(t)\in C(R,R^{N^2})$ and $R(t,u)$ is asymptotically quadratic or super quadratic in $u$ as $|u|\rightarrow\infty$. Under more general assumptions on the matrix $L(t)$, if $R$ is superquadratic and even in $u$, we obtain infinitely many homoclinic orbits. On the other hand, if $R$ is asymptotically quadratic, we also prove the existence and multiplicity of homoclinic orbits for the above system.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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