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


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.


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

Mathematics Subject Classification

34C37 35A15 35B38 



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


  1. 1.
    Cruz, G.A.M., Ledesma, C.E.T.: Multiplicity of solutions for fractional Hamiltonian systems with Liouville–Weyl fractional derivatives. Fract. Calc. Appl. Anal. 18, 875–890 (2015)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ambrosetti, A., Coti Zelati, V.: Multiple homoclinic orbits for a class of conservative systems. Rend. Semin. Mat. Univ. Padova 89, 177–194 (1993)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bahri, A.: Critical Points at Infinity in Some Variational Problems. Pitman Research Notes in Mathematics Series, vol. 182. Longman House, Harlow (1989)zbMATHGoogle Scholar
  5. 5.
    Bai, Z.B., Lu, H.S.: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 311, 495–505 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Benhassine, A.: Multiplicity of solutions for nonperiodic perturbed fractional Hamiltonian systems. Elect. J. of Diff. Eq., 93, 1–15 (2017)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Benhassine, A.: Multiple of homoclinic solutions for a perturbed dynamical systems with combined nonlinearities. Medit. J. Math. 14:132 (2017).
  8. 8.
    Benhassine, A.: Existence and multiplicity of periodic solutions for a class of the second order Hamiltonian systems. Nonlinear Dyn. Syst. Theory 14(3), 257–264 (2014)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Benhassine, A: Existence and infinitely of many solutions for a nonperiodic fractional Hamiltonian systems Diff. Int. Eq. 33(9/10) (Forthcoming)Google Scholar
  10. 10.
    Benhassine, A.: Fractional Hamiltonian systems with locally defined potentials. Theor. Math. Phys. 195(1), 563–571 (2018)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bucur, C., Valdinoci, E.: Nonlocal Diffusion and Applications. Lecture Notes of the Unione Matematica Italiana, vol. 20, p. xii+155. Springer, Cham (2016)CrossRefGoogle Scholar
  12. 12.
    Carriao, P.C., Miyagaki, O.H.: Existence of homoclinic solutions for a class of time dependent Hamiltonian systems. J. Math. Anal. Appl. 230, 157–172 (1999)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ding, Y.H.: Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems. Nonlinear Anal. 25, 1095–1113 (1995)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Dipierro, S., Patrizi, S., Valdinoci, E.: Chaotic orbits for systems of nonlocal equations. Commun. Math. Phys. 2, 583–626 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Fall, M., Mahmoudi, F., Valdinoci, E.: Ground states and concentration phenomena for the fractional Schrödinger equation. Nonlinearity 28, 1937–1961 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Jiang, W., Zhang, Q.: Multiple homoclinic soluions for superquadratic Hamiltonian systems. Elect. J. of Differ. Equ. 2016, 1–12 (2016)CrossRefGoogle Scholar
  17. 17.
    Hilfer, R.: Applications of Fractional Calculus in Physics. World Science, Singapore (2000)CrossRefGoogle Scholar
  18. 18.
    Izydorek, M., Janczewska, J.: Homoclinic solutions for a class of the second order Hamiltonian systems. J. Differ. Equ. 219, 375–389 (2005)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Jeanjean, L.: On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on \(\mathbb{R}^{N}\). Proc. R. Soc. Edinb. A 129, 787–809 (1999)CrossRefGoogle Scholar
  20. 20.
    Jiang, W.H.: The existence of solutions for boundary value problems of fractional differential equatios at resonance. Nonlinear Anal. 74, 1987–1994 (2011)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Jiao, F., Zhou, Y.: Existence results for fractional boundary value problem via critical point theory. Int. J. Bif. Chaos 22, 1–17 (2012)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Kilbas, A., Bonilla, B., Trujillo, J.J.: Existence and uniqueness theorems for nonlinear fractional differential equations. Demonst. Math. 33, 583–602 (2000)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Singapore (2006)CrossRefGoogle Scholar
  24. 24.
    LI, G., YE, H.: Existence of positive solutions to semilinear elliptic systems in \({\mathbb{R}}^{N}\) with zero mass. Act. Math. Sci. 33, 913–928 (2013)CrossRefGoogle Scholar
  25. 25.
    Lions, P.L.: The concentration-compactness principle in the calculus of variations. The locally compact cases, part II. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1(4), 223–283 (1984)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian Systems. Springer, New York (1989)CrossRefGoogle Scholar
  27. 27.
    Miller, K., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)zbMATHGoogle Scholar
  28. 28.
    Poincaré, H.: Les méthodes nouvelles de la mécanique céleste, Gauthier-Villars, Paris (1897–1899)Google Scholar
  29. 29.
    Rabinowitz, P. H.: Minimax methods in critical point theory with applications to differential equations. In: CBMS Regional Conference Series in Mathematics, vol. 65. American Mathematical Society, Provodence (1986)Google Scholar
  30. 30.
    Rabinowitz, P.H., Tanaka, K.: Some results on connecting orbits for a class of Hamiltonian systems. Math. Z. 206, 473–499 (1991)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Schechter, M.: Linking Methods in Critical Point Theory. Birkhauser, Boston (1999)CrossRefGoogle Scholar
  32. 32.
    Servadei, R., Valdinoci, E.: Mountain pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389, 887–898 (2012)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Servadei, R., Valdinoci, E.: Variational methods for non-local operators of elliptic type. Disc. Cont. Dyn. Syst. 33, 2105–2137 (2013)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Torres, C.: Existence of solution for a class of fractional Hamiltonian systems. Electron. J. Differ. Equ. 2013, 1–12 (2013)CrossRefGoogle Scholar
  35. 35.
    Willem, M.: Minimax Thorems. Birkhauser, Boston (1996)CrossRefGoogle Scholar
  36. 36.
    Wu, X., Zhang, Z.: Solutions for perturbed fractional Hamiltonian systems without coercive conditions. Bound. Value Probl. 2015, 149 (2015)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Xu, J., O’Regan, D., Zhang, K.: Multiple solutions for a calss of fractional Hamiltonian systems. Fract. Calc. Appl. Anal. 18, 48–63 (2015)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Zaslavsky, G.M.: Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, Oxford (2005)zbMATHGoogle Scholar
  39. 39.
    Zhang, S.Q.: Existence of a solution for the fractional differential equation with nonlinear boundary conditions. Comput. Math. Appl. 61, 1202–1208 (2011)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Zou, W.: Variant fountain theorems and their applications. Manuscr. Math. 104, 343–358 (2001)MathSciNetCrossRefGoogle Scholar

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

Personalised recommendations