# American Institute of Mathematical Sciences

January  2011, 10(1): 269-286. doi: 10.3934/cpaa.2011.10.269

## 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  November 2010

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.

Citation: Jun Wang, Junxiang Xu, Fubao Zhang. Homoclinic orbits for a class of Hamiltonian systems with superquadratic or asymptotically quadratic potentials. Communications on Pure & Applied Analysis, 2011, 10 (1) : 269-286. doi: 10.3934/cpaa.2011.10.269
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