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Ricerche di Matematica

, Volume 68, Issue 2, pp 727–743 | Cite as

Ground state solutions for a class of fractional Hamiltonian systems

  • Abderrazek BenhassineEmail author
Article
  • 34 Downloads

Abstract

In this paper we study the following fractional Hamiltonian systems
$$\begin{aligned} \left\{ \begin{array}{lllll} -_{t}D^{\alpha }_{\infty }(_{-\infty }D^{\alpha }_{t}x(t))- L(t).x(t)+\nabla W(t,x(t))=0, \\ x\in H^{\alpha }(\mathbb {R}, \mathbb {R}^{N}), \end{array} \right. \end{aligned}$$
where \(\alpha \in \left( {1\over {2}}, 1\right] ,\ t\in \mathbb {R}, x\in \mathbb {R}^N,\ _{-\infty }D^{\alpha }_{t}\) and \(_{t}D^{\alpha }_{\infty }\) are the left and right Liouville–Weyl fractional derivatives of order \(\alpha \) on the whole axis \(\mathbb {R}\) respectively, \(L:\mathbb {R}\longrightarrow \mathbb {R}^{2N}\) and \(W: \mathbb {R}\times \mathbb {R}^{N}\longrightarrow \mathbb {R}\) are suitable functions. One ground state solution is obtained by applying the monotonicity trick of Jeanjean and the concentration-compactness principle in the case where the matrix L(t) is positive definite and \(W \in C^{1}(\mathbb {R}\times \mathbb {R}^{N},\mathbb {R})\) is superquadratic but does not satisfy the usual Ambrosetti–Rabinowitz condition.

Keywords

Fractional Hamiltonian systems Fractional Sobolev space Ground state solution Critical point theory Concentration phenomena 

Mathematics Subject Classification

34C37 35A15 35B38 

Notes

Acknowledgements

The author would like to thank the handling editors and the anonymous reviewers.

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Copyright information

© Università degli Studi di Napoli "Federico II" 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Higher Institute of Informatics and Mathematics of MonastirUniversity of MonastirMonastirTunisia

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