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

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.