Homoclinic Solutions for a Class of Hamiltonian Systems with Small External Perturbations

Abstract

This paper is concerned with the existence of nontrivial homoclinic solutions for a class of second order Hamiltonian systems with external forcing perturbations $\ddot{q}+A\dot{q}+V_q(t,q)= f(t),$ where $q= (q_1,q_2,···,q_N)∈\mathbb{R}^N,$ $A$ is an antisymmetric constant $N×N$ matrix, $V(t,q) = −K(t,q)+W(t,q)$ with $K,W ∈ C^1 (\mathbb{R},\mathbb{R}^N)$ and satisfying $b_1|q|^2 ≤ K(t,q) ≤ b_2|q|^2$ for some positive constants $b_2 ≥b_1 >0$ and external forcing term $f ∈C(\mathbb{R},\mathbb{R}^N)$ being small enough. Under some new weak superquadratic conditions for $W,$ by using the mountain pass theorem, we obtain the existence of at least one nontrivial homoclinic solution.