1. Introduction
In this article, we investigate the existence and multiplicity of nontrivial solutions for the following fractional
p-Laplacian system:
where
with
,
are two parameters,
and
is the fractional critical Sobolev exponent, and
is the fractional
p-Laplacian operator, which is defined as
where
. The weight function
f is positive continuous function on
satisfying the following condition:
The function
satisfies the following assumptions:
In recent years, a great deal of attention has been paid to the study of fractional problems, from the view of pure mathematics and concrete applications, since this type of operator arises in many different applications, such as the thin obstacle problem, stratified materials, anomalous diffusion, finance, phase transitions, water waves, and many others. For more details, we refer to [
1,
2,
3].
On the one hand, the fractional elliptic problems for
have been studied by many researchers, see [
4,
5,
6,
7,
8,
9] for the subcritical case, [
10,
11,
12,
13] for the critical case, and [
14,
15] for the fractional Kirchhoff-type problem. On the other hand, for the case
, many interesting results have also been obtained; see [
16,
17,
18,
19] and references therein. In particular, Chen and Deng [
20] considered the following system with a subcritical concave–convex nonlinearity:
where
is a smooth bounded set in
. They obtained the existence of at least two nontrivial solutions for (2) by variational methods. Furthermore, Chen and Squassina [
21] obtained the multiplicity of solutions for the critical fractional
p-Laplacian system (2). In [
22], Zhen and Zhang discussed the following system involving concave–convex nonlinearities and sign-changing weight functions:
where
is a smooth bounded domain in
. Under suitable assumptions, they proved that (3) has at least two nontrivial solutions by using the Nehari manifold together with Ekeland’s variational principle.
In a recent paper, the authors of [
23] considered the following fractional
p-Laplacian elliptic system:
where
is a smooth bounded set in
. The existence of multiple solutions were obtained by the Nehari manifold and Lusternik–Schnirelmann category.
Recently, some researchers have focused on fractional
p-Laplacian systems with homogeneous nonlinearities of critical growth; for example, Lu and Shen [
24] considered the following system with homogeneous nonlinearities:
where
is a bounded domain in
with Lipschitz boundary, and
Q and
H are homogeneous functions. Under some proper conditions, they concluded that (5) provides a nontrivial solution by variational methods.
In [
25], Shen studied the following type of fractional
p-Laplacian elliptic systems:
where
and
are the partial derivatives of the 2-variable
-functions
and
,
denotes the completion of
. Under some suitable assumptions, the author proved that system (6) has a weak solution by variational methods.
However, as far as we know, there are few results on fractional p-Laplacian elliptic systems with homogeneous nonlinearities in . Motivated by the aforementioned work, in this paper, we discuss the existence and multiplicity of nontrivial solutions for system (1) in by variational methods. The first nontrivial solution can be obtained by using the same argument as that in the subcritical case. In order to obtain the second nontrivial solution, we have to add restrictions on the functions to obtain the compactness of the extraction of the Palais–Smale sequences in the Nehari manifold.
The main results of this paper are as follows.
Theorem 1. Let for . Assume that and hold. Then, system (1) has at least one nontrivial solution for all , where .
Theorem 2. Let for . Assume that and hold. Then, system (1) has at least two nontrivial solutions for all , where will be given in Section 5. In order to obtain the results of Theorems 1 and 2, we face two main difficulties: how to determine the range of dual parameters
and ensure the existence of solutions for the problem (1). To overcome this difficulty, we will adopt the Nehari manifold and Fibering maps, which are widely used to deal with problems with concave–convex terms [
26,
27,
28]. On the other hand, for
, the minimizers of
S (see (7)) are not yet known, so this remains an open question currently. As in [
29], we can overcome this difficulty through the optimal asymptotic behavior of minimizers, which was obtained in [
30]. We give some useful estimates on auxiliary functions
(see
Section 2 for their definition), which help us to deal with the study in the critical case.
Remark 1. (i) Compared with the fractional p-Laplacian problem (2)–(4) with a bounded domain, system (1) is in and the nonlinearity is more general. Theorems 1 and 2 extend the results in [27] to a class of fractional p-Laplacian elliptic systems with homogeneous nonlinearities. (ii) In [20,22], the authors study the subcritical growth, and we further discuss the existence of solutions with the critical growth. We note that if and , system (1) reduces to system (2) in bounded domains. Consequently, our results extend and complement the existing relevant results in the literature. The structure of this paper is as follows. In
Section 2, we introduce some preliminaries. In
Section 3, we give some properties of the Nehari manifold and fibering maps. In
Section 4 and
Section 5, we prove Theorem 1 and Theorem 2, respectively.
2. Preliminaries
Firstly, we recall some facts about the fractional Sobolev space. For 0
, the fractional Sobolev space
is defined by
where the term
is the Gagliardo norm. With the induced norm
, the space
is a uniformly convex Banach space. Let
denote Lebesque space with norm
. Let
be the best fractional Sobolev constant.
denote various positive constants.
For system (1), we work in the product space
with the norm
For the convenience of the reader, we list the following homogeneous properties; see [
25,
31]. Let
be a
-homogeneous differential function with
.
(i) There exists
, such that
(ii)
is attained at some
, where
(iv) and are (-1)-homogeneous.
Based on the above homogeneous properties and assumption
, we have the so-called Euler identity:
and
where
.
Definition 1. A pair of functions is said to be a weak solution of system (1), if for all , there holds
The corresponding energy functional of system (1) is defined by
It is easy to see that and the critical points of the functional are equivalent to the weak solutions of system (1).
In the following, we fix a radially symmetric non-negative decreasing minimizer
for
S. Multiplying
U by a positive constant if necessary, we may assume that
For any
, we know that
is also a minimizer for
S satisfying (10).
Lemma 1 ([
30])
. There exist constants and such that for all , In what follows, if
is the above constant, then for
, as in [
18], set
and
The functions
and
are nondecreasing and absolutely continuous. Consider the radially symmetric non-increasing function
which satisfies
Then, we have the following estimates for .
Lemma 2 ([
29,
32])
. There exists such that for any , there holds Moreover, there exists
, such that
3. Nehari Manifold and Fibering Maps
We consider the system (1) on the Nehari manifold. Define the Nehari manifold as follows:
Note that
if and only if
The Nehari manifold
is closely linked to the behavior of fibering maps
for
, defined by
Such maps were first introduced by Drabek and Pohozaev in [
33] and were discussed by Brown and Wu in [
34].
For
, we note that
and
which implies that
if and only if
. Hence, for
, by (15), we obtain
Now, we split
into three parts.
Lemma 3. Assume that holds. Then, there exists a constant such that ifwe have , where . Proof. We argue by contradiction. Assume that there exist
with
such that
. Then, for
, by (16), we have
Using
, (7) and the Hölder inequality, we obtain
Utilizing (17) and (19), we have
By (7), (8) and the Minkowski inequality, we obtain
From (18) and (21), we have
By (20) and (22), we obtain
which is a contradiction with the assumption. Therefore, if
, then we have
. □
Lemma 4. The energy functional is coercive and bounded below on for .
Proof. If
, then by (9) and (14), we have
Combining (19) with (23), we have
Due to
, we see that
is coercive and bounded from below on
. □
By Lemmas 3 and 4, we know that
for any
. Therefore, we may define
Lemma 5. Assume that is a local minimizer of on and , then is a critical point of .
Proof. The proof is almost the same as [
22,
35], so we omit it here. □
Lemma 6. The following facts hold:
(i) Suppose , then .
(ii) Suppose , then there exists a such that , where is a positive constant depending on .
Proof. (i) Let
, and by (16), we have
Thus,
Therefore, by the definition of
, we can obtain
.
(ii) Let
, and we have
. By (16), we have
From (21), we obtain
By (26) and (27), we have
Taking (28) into (24), we deduce that
where
. We complete the proof. □
Next, we consider the function
defined by
By simple computations, we have the following results.
Lemma 7. Suppose , then the function satisfies the following properties:
(i) has a unique critical point at (ii) is strictly increasing on and strictly decreasing on .
(iii) .
Lemma 8. if and only if .
Proof. It is easy to see that for
if and only if
Moreover,
and if
, then
Thus,
(or
) if and only if
.
Lemma 9. Suppose , then for any satisfying , where given in Lemma 3, there exist unique such that andAlso, □
Proof. Since
, we see that (29) has no solution if and only if
and
satisfy the following condition:
By Lemma 7, we obtain
Because of
and (19), we have
On the other hand, from (21), we obtain
For any
and
satisfying
, with
given in Lemma 3, we can obtain
Hence, by (32) and (33), if
and
satisfy
, we deduce that
Then, there are unique
and
, with
, such that
In addition, from (15) and (29), we can find that
. Due to (31), we have that
and
. Thus, it is not hard to find that
has a local minimum at
and a local maximum at
such that
and
. Because of
, we can find that
for each
and
for each
. Therefore,
We complete the proof.
4. Proof of Theorem 1
In this section, we establish the existence of a local minimum for on . First, we state some preliminary results.
We assert that a sequence is a sequence at level c, if and as . is said to satisfy the condition if any sequence contains a convergent subsequence.
Lemma 10. (i) If , then the functional has a -sequence ,
(ii) If , then the functional has a -sequence .
Proof. The proof is similar to that in [
21,
22]; we have omitted the details here. □
Lemma 11. If is a -sequence for , then is bounded in W.
Proof. If
is a
-sequence for
, then we have
That is,
According to
, (7) the Hölder inequality and the
-Young inequality, we obtain
Therefore,
It follows that
Let
; we find that
is bounded in
W. □
Proof of Theorem 1. From Lemma 6-(i) and Lemma 10, there exists a minimizing sequence
such that
By Lemma 11, we know that
is bounded in
W. Then, there exists
and a subsequence, still denoted by
, such that
Thus, according to the Dominated Convergence Theorem, we have
From
, we have
That is,
Let
; by Lemma 6-(i), we have
Thus, this indicates that
and
cannot both be zero.
Next, we prove that strongly in W.
Indeed, by Fatou’s lemma and
, we have
which implies that
Then according to the Brézis–Lieb lemma, one has
Hence,
strongly in
W.
Finally, we claim that and .
In fact, if
, by Lemma 9, there are unique
and
such that
Since
there exists a
such that
. By Lemma 9, we have
which is a contradiction. This shows that
. By Lemma 6, we obtain that
. Hence, from Lemma 5, we have that
is a nontrivial solution of system (1). □
5. Proof of Theorem 2
In this section, we establish the existence of a local minimum for on . For this, we give some necessary lemmas.
Lemma 12. If is a -sequence for with in W, then , and there exists a positive constant depending on , such that Proof. Let
be a
-sequence for
with
in
W, then we have
For all
, by standard argument [
10], we obtain that
which yields
. In particular, we have
. That is,
Then,
By (11), the Hölder inequality and the Young inequality, we obtain
where
. We complete the proof. □
Set
By (21), we have
which implies that
Lemma 13. The functional satisfies the condition with c satisfyingwhere is the positive constant given in Lemma 12. Proof. Let
be a
-sequence for
with
. From Lemma 11, we know that
is bounded in
W. Then, there exists a subsequence still denoted by
and
, such that
Thus, we obtain that
and
Now, we set
; then, according to the Brézis–Lieb lemma [
35] and arguing as in [
31], we obtain
From (40)–(42), we deduce that
Thus, we may assume that
If
, the proof is completed. Assume that
; then, from (38), we have
which implies that
. Therefore, by (43) and Lemma 12, we have
which is a contradiction. Hence,
strongly in
W. The proof of Lemma 15 is completed. □
Lemma 14. Let for . Assume that and hold. Then there exist and , such thatfor . Proof. We can follow the approach presented in [
21] to complete the proof of Lemma 15. Now, we consider the functional
defined by
Set
, where
is defined by (11),
,
, such that
. Define
Using the following fact,
Combining (12) with (39), we deduce that
We can choose
, such that for all
we have
Then,
which implies that there exists
satisfying
for all
By
, and (44), we have
where
. In view of (13) and (45), we have
For
, we have
We observe that
Thus, if
, we can choose
, such that
satisfying
, we can obtain
where
is the positive constant defined in Lemma 12.
In addition, if
, we can choose
such that
satisfying
, we can obtain
when
for
and
Therefore, we take
, for all
satisfying
, we obtain
This completes the proof. □
Lemma 15. Let . Assume and . Then, for all , there exists a such that
(i) .
(ii) is a nontrivial solution of (1).
Proof. If , then by Lemma 10-(ii), there exists a minimizing sequence for . From Lemma 13, Lemma 14, and Lemma 10-(ii), we know that satisfies the condition and . By Lemma 11, we obtain that is bounded in W. Therefore, there exists a subsequence, still denoted by and , such that strongly in W and for all . Finally, by the same arguments as in the proof of Theorem 1, we obtain that is a nontrivial solution of system (1). □
Proof of Theorem 2. Applying Theorem 1, we know that system (1) has a nontrivial solution for all . On the other hand, taking , from Lemma 15, we obtain the second nontrivial solution for . Since , then those two solutions are distinct.
Now we prove that
and
are not semi-trivial solutions. By Theorem 1 and Lemma 15, we have
We know that if
(or
) is a semi-trivial solution of problem (1), then (1) becomes
Then, the corresponding energy functional of Equation (
47) is given by
By (46) and (48), we know that
is not a semi-trivial solution.
Next, we show that
is not a semi-trivial solution. Without loss of generality, we may assume that
. Then,
is a nontrivial solution of Equation (
47) and
Additionally, we may choose
, such that
From Lemma 7, we know that there exists a unique
, such that
, where
Moreover,
By means of the fact that
, we have that
which implies a contradiction. Thus,
is not a semi-trivial solution. We complete the proof. □