Infinitely Many Solutions for Fractional Hamiltonian Systems with Locally Defined Potentials

Abstract

In this paper, we prove the existence of infinitely many solutions for the following nonperiodic fractional Hamiltonian system

$$\begin{aligned} \left\{ \begin{array}{l} _{t}D_{\infty }^{\alpha }(_{-\infty }D_{t}^{\alpha }u)(t)+L(t)u(t)=\nabla W(t,u(t)),\ t\in {\mathbb {R}}\\ u\in H^{\alpha }({\mathbb {R}}), \end{array}\right. \end{aligned}$$

where \(_{-\infty }D_{t}^{\alpha }\) and \(_{t}D^{\alpha }_{\infty }\) are left and right Liouville–Weyl fractional derivatives of order \(\frac{1}{2}<\alpha <1\) on the whole axis, respectively, \(L\in C({\mathbb {R}},{\mathbb {R}}^{N^{2}})\) is a symmetric matrix valued function unnecessary coercive and \(W(t,x)\in C^{1}({\mathbb {R}}\times {\mathbb {R}}^{N},{\mathbb {R}})\) is only locally defined and superquadratic near the origin with respect to x. Our results extend and improve some existing results in the literature.

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Acknowledgements

The author would like to express sincere thanks to the anonymous referee for his/her carefully reading the paper and valuable comments and suggestions.

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Correspondence to Mohsen Timoumi.

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Communicated by Majid Gazor.

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Timoumi, M. Infinitely Many Solutions for Fractional Hamiltonian Systems with Locally Defined Potentials. Bull. Iran. Math. Soc. (2021). https://doi.org/10.1007/s41980-021-00588-6

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Keywords

  • Fractional Hamiltonian systems
  • Infinitely many solutions
  • Variational methods
  • Local conditions

Mathematics Subject Classification

  • 34C37
  • 35A15
  • 35B38