Abstract
In this paper we consider the existence of homoclinic solutions for the following second order non-autonomous Hamiltonian system
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.
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This work was supported by National Natural Science Foundation of China and RFDP.
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Yuan, R., Zhang, Z. Homoclinic Solutions for a Class of Second Order Hamiltonian Systems. Results. Math. 61, 195–208 (2012). https://doi.org/10.1007/s00025-010-0088-3
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DOI: https://doi.org/10.1007/s00025-010-0088-3