Multiplicity of Solutions for Fractional Hamiltonian Systems under Local Conditions

Under some local superquadratic conditions on ( ) , W t u with respect to u, the existence of infinitely many solutions is obtained for the nonperiodic fractional Hamiltonian systems ( ) ( ) ( ) ( ) ( ) ( ) , , α α ∞ −∞ + = ∇ ∀ ∈ t t D D u t L t u t W t u t t , where ( ) L t is unnecessarily coercive.

where ( ) L t is unnecessarily coercive.

Keywords
Fractional Hamiltonian Systems, Local Conditions, Variational Methods

Introduction
In this paper, we consider the fractional Hamiltonian system Fractional calculus has received increased popularity and importance in the past decade, which is mainly due to its extensive applications in many engineering and scientific disciplines such as physics, chemistry, biology, economics, control theory, signal and image processing, biophysics, blood flow phenomena, aerodynamics, etc. (see [1]- [6]). Models containing left and right fractional differential operators have been recognized as best tools to describe long-memory processes and hereditary properties. However, compared with classical theories for integer-order differential equations, researches on fractional differential equations are only on their initial stage of development.
Recently, the critical point theory and variational methods have become effec-tive tools in studying the existence of solutions to fractional differential equations with variational structures. In [7], for the first time, Jiao  assuming that W satisfies the (AR) condition and L satisfies the following coercive condition: (L) ( ) L t is a positive definite symmetric matrix for all ∈  t , and there ex- Subsequently, the existence and multiplicity of solutions for the fractional Hamiltonian system (1) have been extensively investigated in many papers; see [9]- [15] and the references therein. However, it is worth noting that in most of these papers, L is required to satisfy the coercivity condition (L). Recently, the authors in [16] proved the existence of one nontrivial solution for (1), where L does not necessarily satisfy the condition (L) and W satisfies some kind of local superquadratic condition: (W) There exist ( ) Here W is only required to be superquadratic at infinitely with respect to x when the first variable t belongs to some finite interval.
Motivated by the above papers, in this note, we will consider the multiplicity of solutions for the fractional Hamiltonian system (1), where L is not necessarily coercive and W satisfies some local growth condition. The exact assumptions on L and W are as follows: Theorem 1. Assume the following conditions hold:  t n c I n t n n n L t n c I n t n n n n n n t c I n t n n Note that W is superquadratic near the origin and there are no conditions assumed on W for x large. As far as the authors know, there is little research concerning the multiplicity of solutions for problem (1) simultaneously under local conditions and non-coercivity conditions, so our result is different from the previous results in the literature.
The proof is motivated by the argument in [17]. We will modify and extend W to an appropriate  W and show for the associated modified functional I the existence of a sequence of solutions converging to zero in ∞ L norm, therefore to obtain infinitely many solutions for the original problem.

Preliminary Results
In this section, for the reader's convenience, we introduce some basic definitions of fractional calculus. The left and right Liouville-Weyl fractional integrals of order 0 1 α < < on the whole axis  are defined as The left and right Liouville-Weyl fractional derivatives of order 0 1 α < < on the whole axis  are defined as The definitions of (2) and (3) may be written in an alternative form as follows: To establish the variational structure which enables us to reduce the existence of solutions of (1), it is necessary to construct appropriate function spaces. In what follows, we introduce some fractional spaces, for more details see [8] and [18]. Denote by I are equivalent with equivalent semi-norm and norm (see [18]).
Then we obtain the following lemma.
In what follows, we introduce the fractional space in which we will construct the variational framework of (1). Let and the corresponding norm is Proof. First, by (L 1 ) and the Hölder inequality, one has By (L 1 ), for any 0 Since by Lemma 2 α X is continuously embedded into Combining (4)-(6) and the Hölder inequality, for each and ( ) , Proof. By (W 1 ) and (W 2 ) one has Next we modify Define a cut-off function . It follows from (W 1 ) and (W 2 ) that (16) Then by (11), (14), (W 2 ) and the choice of the cut-off function ρ , we have Finally, we prove (9) and (10). On one hand, using (16) we know that On the other hand, assume that 2 < < r x r . By (12), (15), (W 4 ) and the choice of the cut-off function ρ , we obtain The above estimates imply that Thus (9) and (10) are verified. The proof is completed. □ We now consider the modified problem whose solutions correspond to critical points of the functional ( ) ( for all α ∈ u X . By (11) and (13) Thus, I is well defined. Rewrite I as follows: In the following, c will be used to denote various positive constants where the exact values are different.
Lemma 5. Let (L 1 ), (W 1 ) and (W 2 ) be satisfied. Then u v X . Moreover, nontrivial critical points of I in α X are solutions of problem (17).
Proof. It is easy to check that Combining the above arguments, we have that 2 ′ I is weakly continuous.
Therefore, 2 ′ I is compact and Proof. By (W 1 ), (W 2 ) and Lemma 5, we know that 0 is a critical point of I with where Ŵ is defined in (9). This together with (ii) of Lemma 4 implies that The proof is completed. □

Proof of Theorem 1
The following lemma is due to Bartsch Since X α is compactly embedded into 1 L θ + , there holds (see [20]) 0 as .
For each k N ∈ , it follows from (7), (19), (20)  Then by Lemma 6, u must be 0. Thus 0 k u → in X α as k → ∞ . By (7), we further have 0 k u → in ( ) , N L ∞   as k → ∞ . Therefore, for k large enough, they are solutions of problem (1). The proof is completed.

Conclusions and Remarks
Let us conclude this paper with some open questions whose answers might largely improve the applicability of the results in this present paper.
Question. Whether or not can we improve the non-coercivity condition (L 1 ): There is 1 0 l > such that ( ) 1 ,

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Author's Contributions
The authors read and approved the final manuscript.