Multiple solutions for a class of superquadratic fractional Hamiltonian systems

Year 2018, Volume: 1 Issue: 3, 186 - 195, 30.09.2018

Mohsen Timoumi

https://doi.org/10.32323/ujma.388067

Cited By: 2

Abstract

In this paper, we are concerned with the existence of solutions for a class of fractional Hamiltonian systems \[\left\{ \begin{array}{l} _{t}D_{\infty}^{\alpha}(_{-\infty}D_{t}^{\alpha}u)(t)+L(t)u(t)=\nabla W(t,u(t)),\ t\in\mathbb{R}\\ u\in H^{\alpha}(\mathbb{R},\ \mathbb{R}^{N}), \end{array}\right. \] where $_{t}D_{\infty}^{\alpha}$ and $_{-\infty}D^{\alpha}_{t}$ are the Liouville-Weyl fractional derivatives of order $\frac{1}{2}<\alpha<1$, $L\in C(\mathbb{R},\mathbb{R}^{N^{2}})$ is a symmetric matrix-valued function and $W(t,x)\in C^{1}(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R})$. Applying a Symmetric Mountain Pass Theorem, we prove the existence of infinitely many solutions for (1) when $L$ is not required to be either uniformly positive definite or coercive and $W(t,x)$ satisfies some weaker superquadratic conditions at infinity in the second variable but does not satisfy the well-known Ambrosetti-Rabinowitz superquadratic growth condition.

Keywords

Fractional Hamiltonian systems, Variational methods, Symmetric Mountain Pass Theorem

References

• [1] O. Agrawal, J. Tenreiro Machado, J. Sabatier, Fractional derivatives and their applications, Springer-Verlag, Berlin, 2004;
• [2] T. Bartolo, V. Benci, D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlinear Analysis, Vol. 7 , No. 9 (1983) 981-1012;
• [3] N. Nyamoradi, A. Alsaedi, B. Ahmad, Y. Zou, Multiplicity of homoclinic solutions for fractional Hamiltonian systems with subquadratic potential, Entropy 2017, 19,50,1-24;
• [4] N. Nyamoradi, A. Alsaedi, B. Ahmad, Y. Zou, Variational approach to homoclinic solutions for fractional Hamiltonian systems, J. Optim. Theory Appl. 2017;
• [5] Z. Bai, H. L ¨ u, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl. (2005), 311, 495-505;
• [6] Z. Bai, Y. Zhang, The existence of solutions for a fractional multi-point boundary value problem, Computers and Mathematics with Applications 2010, 69, 2364-2372;
• [7] P. Chen, X. He, X.H. Tang, Infinitely many solutions for a class of Hamiltonian systems via critical point theory, Math. Meth. Appl. Sci. 2016, 39, 1005-1019;
• [8] Y. Li, B. Dai, Existence and multiplicity of nontrivial solutions for Liouville-Weyl fractional nonlinear Schr ¨ odinger equation, RA SAM (2017);
• [9] W. Jiang, The existence of solutions for boundary value problems of fractional differential equations at resonance, Nonlinear Analysis (2011), 74, 1987-1994;
• [10] F. Jiao, Y. Zhou, Existence results for fractional boundary value problem via critical point theory, Intern. Journal of Bif. and Chaos, 22, No. 4 (2012), 1-17;
• [11] R. Hiffer, Applications of fractional calculus in physics, World Science, Singapore, 2000;
• [12] S. G. Samko, A.A Kilbas, O.I. Marichev, Fractional integrals and derivatives, Theory and applications, Gordon and Breach, Switzerland 1993;
• [13] A.A. Kilbas, H.M. Srivastawa, J.J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematical Studies; Vol. 204, Singapore 2006;
• [14] S. Liang, J. Zhang, Positive solutions for boundary value problems of nonlinear fractional differential equations, Nonlinear Analysis, 2009, 71, 5545-5550;
• [15] J. Mawhin, M. Willem, Critical point theory and Hamiltonian systems, Applied Mathematical Sciences, Springer, Berlin, 1989;
• [16] A. M`endez, C. Torres, Multiplicity of solutions for fractional Hamiltonian systems with Liouville-Weyl fractional derivative, arXiv: 1409.0765v1[mathph] 2 Sep. 2014;
• [17] K. Miller, B. Ross, An introduction to differential equations, Wiley and Sons, New York, 1993;
• [18] I. Pollubny, Fractional differential equations, Academic Press, 1999;
• [19] P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Reg. Conf. Ser. in Math., Vol. 65, American Mathematical Society, Providence, RI, 1986;
• [20] K. Tang, Multiple homoclinic solutions for a class of fractional Hamiltonian systems, Progr. Fract. DIff. Appl. 2, , No. 4 (2016), 265-276;
• [21] C. Torres, Existence of solutions for fractional Hamiltonian systems, Electr. J. DIff. Eq., Vol. 2013 (2013), No. 259, 1-12;
• [22] C. Torres Ledesma, Existence of solutions for fractional Hamiltonian systems with nonlinear derivative dependence in R, J. Fractional Calculus and Applications; Vol. 7 (2) (2016) 74-87;
• [23] X. Wu, Z. Zhang, Solutions for perturbed fractional Hamiltonian systems without coercive conditions, Boundary Value Problems (2015) 2015: 149, 1-12;
• [24] S. Zhang, Existence of solutions for the fractional equations with nonlinear boundary conditions, Computers and Mathematics with Applications (2011), 61, 1202-1208;
• [25] S. Zhang, Existence of solutions for a boundary value problems of fractional differential equations at resonance, Nonlinear Analysis (2011): 74, 1987-1994;
• [26] Z. Zhang, R. Yuan, Existence of solutions to fractional Hamiltonian systems with combined nonlinearities, Electr. J. Diff. Eq., Vol. 2016 (2016) No. 40, 1-13;
• [27] Z. Zhang, R. Yuan, Solutions for subquadratic fractional Hamiltonian systems without coercive conditions, Math. Meth. Appl. Sci. (2014) 37, 2934-2945;
• [28] Z. Zhang, R. Yuan, Variational approach to solutions for a class of fractional Hamiltonian systems, Math. Meth. Appl. Sci. (2014) 37, 1873-1883;