FRACTIONAL HAMILTONIAN SYSTEMS WITH POSITIVE SEMI-DEFINITE MATRIX

We study the existence of solutions for the following fractional Hamiltonian systems $ \left\{ \begin{array}{ll} - _tD^{\alpha}_{\infty}(_{-\infty}D^{\alpha}_{t}u(t))-\lambda L(t)u(t)+\nabla W(t,u(t)) = 0,\\[0.1cm] u\in H^{\alpha}(\mathbb{R},\mathbb{R}^n), \end{array} \right. \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(\mbox{FHS})_\lambda $ where $ \alpha\in (1/2,1) $, $ t\in \mathbb{R} $, $ u\in \mathbb{R}^n $, $ \lambda>0 $ is a parameter, $ L\in C(\mathbb{R},\mathbb{R}^{n^2}) $ is a symmetric matrix, $ W\in C^1(\mathbb{R} \times \mathbb{R}^n,\mathbb{R}) $. Assuming that $ L(t) $ is a positive semi-definite symmetric matrix, that is, $ L(t)\equiv 0 $ is allowed to occur in some finite interval $ T $ of $ \mathbb{R} $, $ W(t,u) $ satisfies some superquadratic conditions weaker than Ambrosetti-Rabinowitz condition, we show that (FHS)$ _\lambda $ has a solution which vanishes on $ \mathbb{R}\setminus T $ as $ \lambda \to \infty $, and converges to some $ \tilde{u}\in H^{\alpha}( \mathbb R, \mathbb R^n) $. Here, $ \tilde{u}\in E_{0}^{\alpha} $ is a solution of the Dirichlet BVP for fractional systems on the finite interval $ T $. Our results are new and improve recent results in the literature even in the case $ \alpha = 1 $.