In this paper, we consider the fractional Hamiltonian system
where , , is a symmetric and positive definite matrix for all , and is the gradient of at u. In the following, denotes the standard inner product in and is the induced norm.
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  -  ). 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 effective tools in studying the existence of solutions to fractional differential equations with variational structures. In  , for the first time, Jiao and Zhou used the critical point theory to tackle the existence of solutions to the following fractional boundary value problem
Jiao and Zhou studied the problem by establishing corresponding variational structure in some suitable fractional space and applying the least action principle and Mountain Pass theorem. Then in  , Torres proved the existence of solutions for the fractional Hamiltonian system (1) by using the Mountain Pass theorem. The author showed that (1) possesses at least one nontrivial solution by assuming that W satisfies the (AR) condition and L satisfies the following coercive condition:
(L) is a positive definite symmetric matrix for all , and there exists an such that as and
Subsequently, the existence and multiplicity of solutions for the fractional Hamiltonian system (1) have been extensively investigated in many papers; see  -  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  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 such that uniformly with respect to .
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:
(L1) There exists such that
where is the smallest eigenvalue of ;
(W1) is even in x and , where denotes the ball in centered at 0 with radius ;
(W2) There are constants and such that
(W3) There exists a constant such that
(W4) for all and ;
(W5) There exists a constant such that
Then problem (1) has a sequence of solutions such that as .
Remark 1. There exist L and W that satisfy all assumptions in Theorem 1. For example, let
with . Note that W is superquadratic near the origin and there are no conditions assumed on W for 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  . We will modify and extend W to an appropriate and show for the associated modified functional I the existence of a sequence of solutions converging to zero in norm, therefore to obtain infinitely many solutions for the original problem.
2. 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 on the whole axis are defined as
The left and right Liouville-Weyl fractional derivatives of order on the whole axis are defined as
The definitions of (2) and (3) may be written in an alternative form as follows:
Moreover, recall that the Fourier transform of is defined by
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  and  . Denote by ( ) the Banach spaces of functions on with values in under the norms
and is the Banach space of essentially bounded functions from into equipped with the norm
For , define the semi-norm
and the norm
where denotes the space of infinitely differentiable functions from into with vanishing property at infinity.
Now we can define the fractional Sobolev space in terms of the Fourier transform. Choose , define the semi-norm
and the norm
Moreover, we note that a function belongs to if and only if
Especially, we have
Therefore, and are equivalent with equivalent semi-norm and norm. Analogous to , we introduce . Define the semi-norm
and the norm
Then and are equivalent with equivalent semi-norm and norm (see  ).
Let denote the space of continuous functions from into . Then we obtain the following lemma.
Lemma 1. (  , Theorem 2.1) If , then and there is a constant such that
Remark 2. From Lemma 1, we know that if with , then for all , since
In what follows, we introduce the fractional space in which we will construct the variational framework of (1). Let
then is a Hilbert space with the inner product
and the corresponding norm is
Lemma 2. If satisfies (L1), then is continuously embedded in .
Proof. By (L1) we have
It implies that
where . □
Lemma 3. If satisfies (L1), then is compactly embedded in for .
Proof. First, by (L1) and the Hölder inequality, one has
This implies that is continuously embedded into .
Next, we prove that is compactly embedded into . Let be a bounded sequence such that in . We will show that in . Obviously, there exists a constant such that
By (L1), for any there exists such that
Since by Lemma 2 is continuously embedded into , the Sobolev embedding theorem implies in . Then for the above, there exists such that
Combining (4)-(6) and the Hölder inequality, for each , we have
This means that in and hence is compactly embedded into .
Last, since for one has
it is easy to verify that the embedding of in is also continuous and compact for . The proof is completed. □
Remark 3. By Lemma 1 - 3 we see that there exists a constant such that
Lemma 4. Assume that (W1)-(W4) are satisfied. There is and such that
where is a constant;
Proof. By (W1) and (W2) one has
Next we modify for x outside a neighborhood of the origin 0. Choose
where is the constant given in (7). By (W3), there is a constant such that
Define a cut-off function satisfying
and for . Using , we define
where . Then by direct computation we get
for . It follows from (W1) and (W2) that
Then by (11), (14), (W2) and the choice of the cut-off function , we have
Therefore, (8) is satisfied if .
Finally, we prove (9) and (10). On one hand, using (16) we know that whenever . On the other hand, assume that . By (12), (15), (W4) and the choice of the cut-off function , we obtain
The above estimates imply that if . Besides, when , by (15) we have
when , by (W4) we get
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 . By (11) and (13) we have
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 (L1), (W1) and (W2) be satisfied. Then and is compact with
for . Moreover, nontrivial critical points of I in are solutions of problem (17).
Proof. It is easy to check that and
For any , by (8) we have
where c is independent of . Hence, for any , by the mean value theorem and Lebesgue’s dominated convergence theorem, we get
where depends on . Moreover, it follows from (8) and (9) that
Therefore, is linear and bounded in h, and is the Gateaux derivative of at u.
Next we prove that is weakly continuous. Set . There exist such that , where is bounded and continuous from to and is bounded and continuous from to . For any ,
which implies that
Now suppose in , then by Lemma 3, in and . Combining the above arguments, we have that is weakly continuous. Therefore, is compact and .
Finally, by a standard argument, it is easy to show that the critical points of I in are solutions of problem (18) with . The proof is completed. □
Lemma 6. Assume that (L1), (W1)-(W4) are satisfied. Then 0 is the only critical point of I such that .
Proof. By (W1), (W2) and Lemma 5, we know that 0 is a critical point of I with . Now let be a critical point of I with . Then we have
where is defined in (9). This together with (ii) of Lemma 4 implies that for all . The proof is completed. □
3. Proof of Theorem 1
The following lemma is due to Bartsch and Willem  .
Lemma 7. Let E be a Banach space with the norm and , where are all finite dimensional subspaces of E. Let be an even functional and satisfy
(F1) For every , there exists such that for every with , and as . Here ;
(F2) For every , there exist and such that for every with ;
(F3) I satisfies condition with respect to , i.e. every sequence with bounded and as has a subsequence which converges to a critical point of I.
Then for each , I has a critical value , hence and as .
Let be the standard orthogonal basis of and define for each . Now we show that the functional I has the geometric property of Lemma 7 under the conditions of Theorem 1.
Lemma 8. Assume that (L1), (W1) and (W2) hold. Then there exist a positive integer and a sequence as such that
where and for all .
Proof. By (18) we obtain
Since is compactly embedded into , there holds (see  )
For each , it follows from (7), (19), (20) and the choice of that
For each , choose
then by (20) one has
and hence there exists a positive integer such that
Now by (22), (23) and (25), we have
Noting that and
which combined with (21) and (24) implies that
The proof is completed. □
Lemma 9. Assume that (L1), (W1) and (W5) hold. Then for every , there exist and such that for every with .
Proof. For a fixed , since is finitely-dimensional, there is a constant such that
Set . Then by (W5), there exists a constant such that
where . Now by (7), (26), (27) and Lemma 3, for with , we get
If with , we have
The proof is completed. □
Lemma 10. Assume that (L1), (W1), (W2) and (W4) hold. Then I satisfies condition with respect to .
Proof. Let be a sequence, that is,
Then we claim that is bounded. If not, passing to a subsequence if necessary, we may assume that
From (13), (14), (15), we have
for all . From (28), (29) and (30), it follows that
as . By (8) we get
which combined with (7) implies that
From this and (31) it follows that
which is a contradiction. Hence is bounded. Noting that by Lemma 5 has a subsequence converging to a critical point of I (see  ). Hence, I satisfies the condition. The proof is completed. □
Proof of Theorem 1. It follows from Lemma 8 - 10 that the functional I satisfies the conditions (F1)-(F3) of Lemma 7. Therefore, by Lemma 7, there exists a sequence of critical values with as . Let be a sequence of critical points of I corresponding to these critical values, i.e. and for all k. Then by Lemma 5, is a sequence of solutions of problem (17). By Lemma 10 and Remark 3.19 in  , I satisfies condition and hence we may assume without loss of generality that in as . Evidently, u is a critical point of I with . Then by Lemma 6, u must be 0. Thus in as . By (7), we further have in as . Therefore, for k large enough, they are solutions of problem (1). The proof is completed.
4. 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 (L1): There is such that and , in order to obtain similar results?
Availability of Data and Material
The authors read and approved the final manuscript.
The authors would like to thank the referees for their pertinent comments and valuable suggestions.
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