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- Publisher
- Springer Journals
- Copyright
- Copyright © 2017 by Springer International Publishing AG
- Subject
- Mathematics; Analysis; Mathematical Methods in Physics
- ISSN
- 1664-2368
- eISSN
- 1664-235X
- DOI
- 10.1007/s13324-017-0197-1
- Publisher site
- See Article on Publisher Site
Abstract
This paper concerns the existence of infinitely many solutions for the following fractional Hamiltonian systems: $$\begin{aligned} \left\{ \begin{array}{lllll} -_{t}D^{\alpha }_{\infty }(_{-\infty }D^{\alpha }_{t}x(t))-L(t).x(t)+\nabla W(t,x(t))=0,\\ x\in H^{\alpha }({\mathbb R}, {\mathbb R}^{N}), \end{array} \right. \end{aligned}$$ - t D ∞ α ( - ∞ D t α x ( t ) ) - L ( t ) . x ( t ) + ∇ W ( t , x ( t ) ) = 0 , x ∈ H α ( R , R N ) , where $$\alpha \in \left( {1\over {2}}, 1\right) ,\ t\in {\mathbb R},\ x\in {\mathbb R}^N,\ _{-\infty }D^{\alpha }_{t}$$ α ∈ 1 2 , 1 , t ∈ R , x ∈ R N , - ∞ D t α and $$_{t}D^{\alpha }_{\infty }$$ t D ∞ α are left and right Liouville–Weyl fractional derivatives of order $$\alpha $$ α on the whole axis $${\mathbb R}$$ R respectively, the matrix L(t) is not necessarily positive definite for all $$t\in {\mathbb R}$$ t ∈ R nor coercive and the nonlinearity $$W \in C^{1}(\mathbb {R}\times \mathbb {R}^{N},\mathbb {R})$$ W ∈ C 1 ( R × R N , R ) involves a combination of superquadratic and subquadratic terms and is allowed to be sign-changing. Some examples will be given to illustrate our main theoretical results. We also give the proof of new version of Theorem 2.4 in Bartolo et al. (Nonlinear Anal 7(9):981–1012, 1983).
Journal
Analysis and Mathematical Physics – Springer Journals
Published: Oct 11, 2017