Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems

Abstract

In this paper, we consider a class of second-order Hamiltonian systems of the form \begin{document}$ \ddot{u}(t)-L(t) u(t)+\nabla W(t,u(t)) = 0 $\end{document} where \begin{document}$ L:R\rightarrow R^{N^2} $\end{document} and \begin{document}$ W \in C^1(R\times R^N, R) $\end{document} are asymptotically periodic in \begin{document}$ t $\end{document} at infinity. Under the reformative perturbation conditions and weaker superquadratic conditions on the nonlinearity, the existence of a ground state homoclinic orbit is established. The main tools employed here are the local mountain pass theorem and the concentration-compactness principle.