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# Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia

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Volume 18, Issue 1

# Multiple Solutions for a Class of Fractional Hamiltonian Systems

Jiafa Xu
/ Donal O'Regan
/ Keyu Zhang
Published Online: 2015-02-10 | DOI: https://doi.org/10.1515/fca-2015-0005

## Abstract

In this paper, we establish two existence theorems for multiple solutions for the following fractional Hamiltonian system

$\left\{\begin{array}{c}{D}_{\infty }^{\alpha }\left({D}_{\infty }^{\alpha }u\left(t\right)\right)+L\left(t\right)u\left(t\right)=\nabla W\left(t,u\left(t\right)\right),\\ u\in {H}^{}\left(ℝ,{ℝ}^{N}\right),\end{array}$

where $\alpha \in \left(1/2,1\right),t\in ℝ,u={\left({u}^{1},\dots ,{u}^{N}\right)}^{T}\in {ℝ}^{N}$, and $L\in C\left(ℝ,{ℝ}^{{N}^{2}}\right)$ is a symmetric and positive definite matrix for all $t\in ℝ,W\in {C}^{1}\left(ℝ×{ℝ}^{N}×ℝ\right)$ and ∇W is the gradient of W about u.

MSC 2010: Primary 34C37; Secondary 35A15; 35B38

Key Words and Phrases: fractional Hamiltonian system; critical point; variational methods; multiple solutions

## References

• [1] M. Al-Refai, Y. Luchko, Maximum principle for the fractional diffusion equations with the Riemann-Liouville fractional derivative and its applications. Fract. Calc. Appl. Anal. 17, No 2 (2014), 483-498; DOI: 10.2478/s13540-014-0181-5, http://link.springer.com/article/10.2478/s13540-014-0181-5.

• [2] A. Ambrosetti, P. Rabinowitz, Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, No 4 (1973), 349-381.

• [3] G. Bin, Multiple solutions for a class of fractional boundary value problems. Abstr. Appl. Anal. 2012 (2012), Article ID 468980.Google Scholar

• [4] C. Bai, Existence of three solutions for a nonlinear fractional boundary value problem via a critical points theorem. Abstr. Appl. Anal. 2012 (2012), Article ID 963105.Google Scholar

• [5] J. Chen, X. Tang, Infinitely many solutions for a class of fractional boundary value problem. Bull. Malays. Math. Sci. Soc. 36, No 4 (2013), 1083-1097.Google Scholar

• [6] X. Chang, Ground state solutions of asymptotically linear fractional Schrodinger equations. J. Math. Phys. 54 (2013), Article ID 061504.

• [7] X. Chang, Z. Wang, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity. Nonlinearity 26 (2013), 479-494.

• [8] W. Dong, J. Xu, Z. Wei, Infinitely many weak solutions for a fractional Schrodinger equation. Boundary Value Problems 2014 (2014), Article ID 159, 14 p.Google Scholar

• [9] F. Jiao, Y. Zhou, Existence of solutions for a class of fractional boundary value problems via critical point theory. Comput. Math. Appl. 62, No 3 (2011), 1181-1199.

• [10] F. Jiao, Y. Zhou, Existence results for fractional boundary value problem via critical point theory. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 22, No 4 (2012), Article ID 1250086.

• [11] Y. Li, H. Sun, Q. Zhang, Existence of solutions to fractional boundary value problems with a parameter. Electron. J. Diff. Equ. 141 (2013), 1-12.Google Scholar

• [12] Y. Luchko, Maximum principle and its application for the timefractional diffusion equations. Fract. Calc. Appl. Anal. 14, No 1 (2011), 110-124; DOI: 10.2478/s13540-011-0008-6; http://link.springer.com/article/10.2478/s13540-011-0008-6.

• [13] X. Lin, X. Tang, Existence of infinitely many solutions for p-Laplacian equations in RN. Nonlinear Anal. 92, (2013), 72-81.Google Scholar

• [14] Q. Li, H. Su, Z. Wei, Existence of infinitely many large solutions for the nonlinear Schrodinger-Maxwell equations. Nonlinear Anal. 72, No 11 (2010), 4264-4270.Google Scholar

• [15] W. Liu, X. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations. J. Appl. Math. Comput. 39, No 1-2 (2012), 473-487.Google Scholar

• [16] N. Nyamoradi, Infinitely many solutions for a class of fractional boundary value problems with Dirichlet boundary conditions. Mediterr. J. Math. 11, No 1 (2014), 75-87.

• [17] J. Sun, H. Chen, J. Nieto, Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems. J. Math. Anal. Appl. 373, No 1 (2011), 20-29.

• [18] M. Struwe, Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. 3rd Ed., Springer-Verlag, New York (2000).Google Scholar

• [19] C. Torres, Mountain pass solution for a fractional boundary value problem. J. Fract. Calc. Appl. 5 (2014), 1-10.Google Scholar

• [20] C. Torres, Existence of solution for a class of fractional Hamiltonian systems. arXiv:1212.5811v1.Google Scholar

• [21] M. Willem, Minimax Theorems. Birkhauser, Boston (1996).Google Scholar

• [22] Y. Ye, C. Tang, Multiple solutions for Kirchhoff-type equations in RN. J. Math. Phys. 54 (2013), Article ID 081508.

• [23] Z. Zhang, R. Yuan, Variational approach to solutions for a class of fractional Hamiltonian systems. Math. Methods Appl. Sci. 37, No 13 (2014), 1873-1883.

Published Online: 2015-02-10

Citation Information: Fractional Calculus and Applied Analysis, Volume 18, Issue 1, Pages 48–63, ISSN (Online) 1314-2224,

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