1. Introduction
In this paper, for given
, we shall investigate the existence of
which fulfill the following problem,
where
,
, and
,
is the fractional Sobolev critical exponent.
is the pseudo-differential operator defined by
with a positive constant
, and
is the Cauchy principal value on the integral.
Equation (
1) can be regarded as the fractional version of the following classical Kirchhoff-type equation
which is a variant type of the Dirichlet problem of the Kirchhoff type:
Equation (
3) is related to the steady-state analogues of the following equation,
which was firstly proposed by Kirchhoff [
1] in 1883 as an extension of the classical D’Alembert’s wave equations
for free vibration of elastic strings, where
denotes the mass density,
u the lateral displacement,
h the cross section area,
the initial axial tension,
E the Young modulus,
L the length of the string, and
f the external force. Equation (
5) takes into account the changes in the length of the string produced by transverse vibrations.
There are two very different points of views as far as frequency
is concerned in (
1). One is to treat the frequency
as a given constant. At this time,
u solves (
1) if it is a critical point of the corresponding energy functional on the working space (see, e.g., [
2,
3]). The other point of view is to regard
as an unknown quantity to problem (
1). At this time,
can be interpreted as the Lagrange multiplier.
The operator
can be seen as the infinitesimal generators of Lévy stable diffusion processes; see [
4] for example. This operator arises in several areas such as physics, biology, chemistry, and finance (see, e.g., [
4,
5]). Recently, by employing the constraint variational method, the Poho
aev method based on the Br
zis–Lieb, and monotonicity trick, Kong and Chen [
6] studied the existence and asymptotic behavior of positive ground states for the following fractional Kirchhoff equation,
In addition, Liu and Jin [
7] replaced
with
and proved the existence of solutions to (
6) without the Ambrosetti–Rabinowitz condition (see, e.g, [
8]) by using a perturbation approach. Furthermore, they also studied the asymptotic behavior of solutions for (
6). Kong et al. [
9] obtained the equivalent system of (
6) by using appropriate transform. With the equivalence result, they obtained the nonexistence, existence, and multiplicity of normalized solutions when
. By replacing
with
, He and Zou [
10] proved that (
6) has positive solutions by adapting the indirect approach and finding a minimum mountain threshold level. Liu et al. [
11] considered Equation (
6) when
and
, respectively. In [
12], Li et al. obtained some similar existence results and derived some asymptotic results on the obtained normalized solutions.
In [
13], Zhang, Tang, and Chen studied the following equation,
where
By assuming that
f satisfies appropriate conditions and
satisfies the following conditions,
(V1)
satisfies
;
(V2) There exists
such that
for all
, and by considering the mountain pass level
they proved that there exists a ground state solution for (
7). Li, Zhang, Wang, and Teng [
14] obtained the existence of non-trivial solutions to (
7), under the assumption (V1) and
are coercive. Bartsch and Wang [
15] first proposed the condition (V2) to overcome the lack of compactness and proved the existence and multiplicity for a kind of elliptic equation.
For the case
and
, problem (
1) turns into a fractional Schr
dinger equation, which has also been widely studied. By the constrained minimization method, Cazenave and Lions [
16] demonstrated the existence of solutions of the following equation in the subcritical case.
When
is replaced by
, Zou and Liu [
17] obtained the existence of the normalized solution for (
8). When (
8) exhibited critical nonlinearity
, by using iterative techniques, Zhang and Han [
18] proved the existence of ground state solutions in the
-subcritical perturbation case,
-critical perturbation case, and
-supercritical perturbation case, respectively.
When
, i.e., for the classical Laplacian, there have been many works on this problem. Ye [
19] obtained global constraint minimizers for the following equation,
where
and
is the
-critical exponent. Later, in [
20], Ye studied the existence of normalized solutions to (
9) with
and
. Li et al. [
21] have studied the existence and asymptotic of normalized solutions for a kind of Kirchhoff equation with Sobolev critical growth by using the Sobolev subcritical approximation method. For more results on this topics, we refer the readers to [
22,
23,
24] for normalized solutions in
. Moreover, [
25] is for coupled fractional Kirchhoff-type systems and [
26] is for a fractional Choquard system.
2. Preliminaries
In this section, we introduce some notations and useful preliminary conclusions. To treat the problem in (
1), we will use a method by He, Lv, Zhang, and Zhong [
27] to study an extension problem.
In this paper, the work space
is defined by
equipped with the norm
denotes the space of radial functions in
, i.e.,
(see [
28]). For
, the fractional Sobolev space
is defined by
which is the completion of
under the norm
Let
and
; there exists an optimal constant
such that the following fractional Gagliardo–Nirenberg–Sobolev inequality holds
The best Sobolev constant is given by
We introduce the following fiber map which is used in this paper:
By direct calculation, we have
It is meaningful to study the solution of problem
with a given
norm, that is, for a given
, to study the solution of problem (
1) under the
-norm constrained manifold
Physically, these kinds of solutions are called the normalized solution of
, which are the critical points of the energy functional
restricted to the manifold
.
Consider the following functionals
and
,
3. Main Results
In this part, we present the principal results of this work. To perform this, we make the following assumptions.
(F1)
is continuous and odd.
(F2) There exists some
that fulfill
such that
(F3) The function defined by
is of class
and
where
is defined in (F2).
We are now prepared to present and demonstrate the key results.
Remark 1. Under the assumptions (F1) and (F2), we can deduce that for arbitrary
and for all
, we haveFurthermore, there exists a non-negative constant
such that for arbitrary
, and we have We are now in a position to state the main results of this paper.
Theorem 1. Let
and
. Assuming (F1)–(F3) hold, problem
possesses at least a couple of solutions
with
Remark 2. In the present paper, we consider the case of non-homogeneous and mass supercritical general nonlinearities. It is also blank even for the existence. For
, the embedding
is compact. Nevertheless, the Sobolev term makes the problem more difficult, and the f leads to the relevant functional being unbounded on S(m). Motivated by [29,30], we prove that the corresponding functions satisfy mountain geometry and, using compactness argument and minimax approaches, we prove the existence of ground state solutions. Now, we shall introduce the Pohozaev manifold and aim to prove that it is a natural constraint.
Lemma 1. If
is a solution to problem (1), then
, whereand Remark 3. From Lemma 1, we can obtain that
. Particularly,
.
Lemma 2. For arbitrary
is a critical point of
iff
, where Proof. From Remark 3, by the fact
, we can directly obtain Lemma 2. □
Lemma 3. For arbitrary critical point of
, if
, there exists
such that Proof. Take
be a critical point for
. Then, there exist
such that
We claim that
. Define the energy functional of (
16):
Given that
u solves (
16), it must fulfill the corresponding Poho
aev identity,
Observing that
hence
Consequently,
Since
, so we conclude that
. □
Lemma 4. Assume that (F1) and (F2) hold. Then, for arbitrary
, there exists some
such thatwhere
and
. That is, Proof. From
, we obtain that
. By
, we obtain
From (F2), we can assume that there exists some suitable constant
such that
According to (
11), we can obtain
Then, by fractional Gagliardo–Nirenberg–Sobolev inequality, there exists a non-negative
such that
By (F2), for
, we have
Then, by (
18)–(
21), we obtain
Since
, we imply the existence of lower bound
. □
Lemma 5. Assume that (F1)–(F3) hold. Then, for arbitrary
, we have
and ρ is a natural constraint of
.
Proof. For arbitrary
, we obtain
Hence
By (F3), (
22), and Lemma 4, we obtain
Therefore, according to Lemma 3, we conclude that
is a natural constraint of
. □
Lemma 6. Assume that (F1)–(F3) hold. Then, for arbitrary
, there exists a unique
such that
. Moreover, Proof. By (
18), we have
For
, it can be inferred from (F2) and (
13) that
It can be easily determined that
From (
25) and (
26), one can see that there exists sufficiently small
such that
, and large enough
such that
. Hence, there exists
such that
, and combining Lemma 2, one notes that
. On the other hand, assume that there exists
such that
. Then by Lemma 5, we derive that
are the strict local maximum of
. Next, we suppose that
, then there exists
such that
In other words,
is a local minimum of
. Consequently, we obtain that
and hence
with
. This contradicts with Lemma 5. □
Remark 4. Assume that (F1)–(F3) hold. Then, for arbitrary
, according to Lemma 6, we obtain Next, we give characterizations of the mountain pass levels for
and
. Let
denote the closed set
.
Lemma 8. Assume that (F1), (F2) hold. Then, there exists a sequence
such that Proof. By Ekeland’s variational principle [
31], the first two properties hold. By Lemma 9 in [
32], we can obtain the third property in (
27). □
For given
, set
Since the solution
u to problem (
1) must belong to
, it can be inferred that if
u satisfies
, then
u is indeed a normalized solution.
Lemma 9. Assume that (F1)–(F3) hold. Then, for arbitrary
, we have
Proof. For any
, we obtain
and
It can be seen from (F2) that
Combining (
28) and (
29), we can conclude that
Therefore, from (
29) and (
30), we derive that
Since
and
, we can conclude that
,
and
. By Lemma 4, we have
□
Define
Then, it is easy to see that
Proof. For any
, let
be the symmetric decreasing rearrangement of
u. Then we obtain
. It can be inferred from the Schwarz rearrangement that
. It is clear that
. In addition, for arbitrary
, we have
Consequently,
Then, for arbitrary
,
Then, we derive that
. Hence,
. □
Define
where
Then combining (F2), (
10), (
11), and (
18) by the scaling technique, we conclude that for
, there exists a suitable
k such that
Lemma 11. .
Proof. First, we claim that
can deduce
. By Lemma 6, we have that for any
, there exists a unique
such that
and
is decreasing on
. Since
and
we infer that
Again by Lemma 6, we obtain the conclusion that
Next, let
. Take
and
such that
and
, respectively. Then define
According to the definition of
, we can determine that
Hence, we obtain
. Furthermore, for arbitrary
, consider the following function,
By (F2), (
10), (
11), and (
18), we derive
Since
, we can take a suitable
k such that
Then, since
, we derive that
Since
, one has
Hence, there exists
such that
, one has
Then,
Hence, we conclude that
. We complete the proof. □
Proof of Theorem 1. Take a sequence
such that
. By (
31), we know that
is bounded in
. Hence, up to a subsequence, we may assume that
in
. Noting that
is an open constraint, there exist
such that
Applying a similar argument as Lemma 3, we have
By (
23), we obtain
Furthermore, from Lemma 4, we derive that
. Therefore,
. Since
is bounded, we have
It can be concluded from (
35) and
that
On the other hand,
Since
, by Lemma 4,
is bounded. Then, we assume up to a subsequence,
Next, we claim
. Assume by contradiction that
. Under (F2), by the compact embedding
for
, we derive that
as
Then by
, we have
and
. By
, we deduce that
a contradiction to Lemma 4.
We may assume that (up to a subsequence)
Then, one can see that
satisfies
Then, we have
, where
Then, we deduce
Hence,
Since
, we obtain
. Consequently, we derive that
in
. Hence, by (
36), we determine that
u satisfies
Furthermore, we have
Then, we imply that
We deduce that
and
. Therefore,
That is to say,
is a couple of solutions to (
1). □