Existence of even homoclinic orbits for a class of asymptotically quadratic Hamiltonian systems

Abstract

In this work we study the existence of a nontrivial even homoclinic orbit for second-order nonautonomous Hamiltonian systems with symmetric potentials $\ddot{x}(t)+V'(t,x(t))=0,$ where $t\in \mathbb{R},$  $ x\in\mathbb{R}^{N}$ and $V\in C^{1}(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R}),$ $ V(t,x)=-K(t,x)+W(t,x),$ under a kind of new super quadratic conditions. For the proof we use the Mountain Pass Theore