Multiplicity of Solutions for Fractional Hamiltonian Systems under Local Conditions ()
1. Introduction
In this paper, we consider the fractional Hamiltonian system
(1)
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 [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 effective tools in studying the existence of solutions to fractional differential equations with variational structures. In [7] , 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 [8] , 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 [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
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
and
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
and
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 [17] . 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
(2)
(3)
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 [8] and [18] . 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
Let
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
Set
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
Let
Then
and
are equivalent with equivalent semi-norm and norm (see [18] ).
Let
denote the space of continuous functions from
into
. Then we obtain the following lemma.
Lemma 1. ( [8] , 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
Then
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
(4)
By (L1), for any
there exists
such that
(5)
Since by Lemma 2
is continuously embedded into
, the Sobolev embedding theorem implies
in
. Then for the
above, there exists
such that
(6)
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
(7)
Lemma 4. Assume that (W1)-(W4) are satisfied. There is
and
such that
i)
(8)
where
is a constant;
ii)
(9)
and
(10)
Proof. By (W1) and (W2) one has
(11)
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
(12)
Define a cut-off function
satisfying
and
for
. Using
, we define
(13)
where
. Then by direct computation we get
(14)
(15)
for
. It follows from (W1) and (W2) that
(16)
Then by (11), (14), (W2) and the choice of the cut-off function
, we have
and
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
and
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
(17)
whose solutions correspond to critical points of the functional
for all
. By (11) and (13) we have
(18)
Thus, I is well defined.
Rewrite I as follows:
where
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 [19] .
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
and
where
and
for all
.
Proof. By (18) we obtain
(19)
Set
(20)
Since
is compactly embedded into
, there holds (see [20] )
(21)
For each
, it follows from (7), (19), (20) and the choice of
that
(22)
For each
, choose
(23)
then by (20) one has
(24)
and hence there exists a positive integer
such that
(25)
Now by (22), (23) and (25), we have
Noting that
and
we have
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
(26)
Set
. Then by (W5), there exists a constant
such that
(27)
where
. Now by (7), (26), (27) and Lemma 3, for
with
, we get
Choose
and let
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,
(28)
Then we claim that
is bounded. If not, passing to a subsequence if necessary, we may assume that
(29)
From (13), (14), (15), we have
(30)
for all
. From (28), (29) and (30), it follows that
(31)
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 [21] ). 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 [20] , 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?
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Funding
Not applicable.
Author’s Contributions
The authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank the referees for their pertinent comments and valuable suggestions.