Results in Mathematics

, Volume 61, Issue 1, pp 195–208

Homoclinic Solutions for a Class of Second Order Hamiltonian Systems

Article

DOI: 10.1007/s00025-010-0088-3

Cite this article as:
Yuan, R. & Zhang, Z. Results. Math. (2012) 61: 195. doi:10.1007/s00025-010-0088-3

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.

Mathematics Subject Classification (2010)

Primary 34C37 Secondary 35A15 37J45 

Keywords

Homoclinic solutions critical point variational methods 

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences Ministry of EducationBeijing Normal UniversityBeijingPeople’s Republic of China
  2. 2.Department of MathematicsTianjin Polytechnic UniversityTianjinPeople’s Republic of China