Homoclinic Solutions for a Class of Second Order Hamiltonian Systems

Abstract

In this paper we consider the existence of homoclinic solutions for the following second order non-autonomous Hamiltonian system

$${\ddot q}-L(t)q+\nabla W(t,q)=0, \quad\quad\quad\quad\quad\quad\quad (\rm HS)$$

where ${L\in C({\mathbb R},{\mathbb R}^{n^2})}$ is a symmetric and positive definite matrix for all ${t\in {\mathbb R}}$, W(t, q) = a(t)U(q) with ${a\in C({\mathbb R},{\mathbb R}^+)}$ and ${U\in C^1({\mathbb R}^n,{\mathbb R})}$. The novelty of this paper is that, assuming L is bounded from below in the sense that there is a constant M > 0 such that (L(t)q, q) ≥ M |q|² for all ${(t,q)\in {\mathbb R}\times {\mathbb R}^n}$, we establish one new compact embedding theorem. Subsequently, supposing that U satisfies the global Ambrosetti–Rabinowitz condition, we obtain a new criterion to guarantee that (HS) has one nontrivial homoclinic solution using the Mountain Pass Theorem, moreover, if U is even, then (HS) has infinitely many distinct homoclinic solutions. Recent results from the literature are generalized and significantly improved.