Existence of Homoclinic Solutions for a Class of Second-Order Hamiltonian Systems with Locally Subquadratic Potentials

Abstract

In this paper, we mainly consider the existence of homoclinic orbits for the following second-order Hamiltonian systems $$\begin{aligned} \ddot{u}(t)-L(t)u(t)+\nabla W\bigl (t,u(t)\bigr )=f(t), \end{aligned}$$where L(t) is a positive definite and symmetric matrix for all $$t\in \mathbb {R}$$ and the potential function W(t, u) is locally subquadratic. Here, the coefficient of the upper bound for W is a positive constant, whereas in the previous literature the corresponding coefficient need to be some integrable functions a(t) on $$\mathbb {R}$$.