Multiple Solutions for a Class of Variable-Order Fractional Laplacian Equations with Concave-Convex Nonlinearity ()
1. Introduction
In this paper, we consider the following variable-order fractional Laplacian equations
(1.1)
where
and
for
,
is a bounded domain in
,
,
is the variable-order fractional Laplacian operator,
are two parameters,
is a continuous function,
and
. The Laplacian operator
is defined by
for each
and any
, where
denotes the Cauchy principal value.
When
and
,
becomes to the usual fractional Laplacian operator and problem (1.1) reduces to fractional Schrödinger equation. This kind of equation is introduced by Laskin [1] [2] as a result of expanding the Feynman path integral from the Brownian-like to the Lévy-like quantum mechanical paths. The fractional Schrödinger equation is studied by many researchers. For example, Zhang et al. [3] investigated fractional Schrödinger equation with critical exponents by using variational methods. They used Pohožaev identity and Jeanjean’s monotonicity trick to obtain a radially symmetric weak solution. Another example is [4], the multiplicity and concentration of solutions for fractional Schrödinger equation with concave-convex nonlinearity are studied by Gao et al. For more results about fractional Schrödinger equation, please see [5] [6] [7] and the references therein. Particularly, when
and
, problem (1.1) becomes to the Schrödinger equation with concave-convex nonlinearity.
If
, (1.1) becomes the following second order elliptic equation with variable growth nonlinearity
(1.2)
Some interesting phenomena can be described by this type of model. For example, Ružička [8] showed the application in the modeling of electrorheological fluids involving variable exponent Laplacian operator. It happens that there is a similar case, Ayazoglu and Ekincioglu [9] obtained a positive solution for electrorheological fluids equations with variable exponent via mountain pass technique. For other applications of these similar models, we refer the readers to [10] [11] [12] [13] [14].
In this paper, we consider the variable-order fractional Laplacian operator case with variable growth. The fractional variable order derivatives are introduced by Lorenzo and Hartley [15] to better describe some diffusion processes reacting to temperature changes. In fact, the literature involving the variable-order fractional Laplacian operator cases is few. Specially, Xiang et al. [16] obtained multiple solutions for the following elliptic equations with variable-order fractional Laplacian operator involving variable exponents by using variational methods,
(1.3)
where
. Another example is [17], Biswas and Tiwari studied a type of Choquard problem with variable-order nonlocal term and variable exponents and obtained some results for the above mention problem by employing Hardy-Sobolev-Littlewood-type inequality. Very recently, Xiang et al. [18] investigated variable-order fractional Kirchhoff equations with nonstandard growth and obtained multiple solutions for these equations by applying the Nehari manifold approach. For other results on variable-order fractional Kirchhoff equations, please see [19] [20] and the references therein.
Inspired mainly by the aforementioned results, we proved the existence of solutions for (1.1) with concave-convex nonlinearity. Compared to [16], we deal with a general case, i.e., the general nonlinearity f with variable growth conditions. To show our result, we make the following assumptions first:
(H1)
.
(H2)
,
.
(H3)
is a nonempty bounded domain and
.
(H4) there exists a nonempty open domain
such that
for all
.
For the nonlinearity term f and the variable exponents q, we assume that
and the following assumptions hold:
(H5)
.
(H6)
and there exist a positive constant c and a continuous function
with
such that
.
(H7)
as
uniformly for
.
(H8) there exists
such that
for every
, where
and
.
Let
and
be the Sobolev embedding constants which will be defined in the next section, set
We assume that
is a positive parameter satisfying the following assumption:
(H9) there holds
where
is a positive constant depending on
.
Based on the hypothesis (H2), we can give the following definition of weak solutions for problem (1.1).
Definition 1.1. We say that
is a (weak) solution of problem (1.1), if for any
, there holds
where
is a variable exponent Banach space which will be defined in the next section.
Theorem 1.2. Suppose (H1)-(H9) hold. Let
, then problem (1.1) admits at least two distinct solutions for all
.
Remark 1.3. In fact, the multiple solutions for variable-order fractional Laplacian equation involving general nonlinearity with critical growth of variable exponent have not been investigated. It is very interesting and full of challenge for us to deal with this problem.
This paper is organized as follows. In Section 1, we give some reviews on the topic of variable-order fractional Lapla-cian equations and give the main result of this paper. In Section 2, some preliminary results are presented. In Section 3, we give the proof of Theorem 1.2.
2. Preliminaries
Some preliminary results of variable exponent Lebesgue spaces will be given in this section which come from [21].
A variable exponent is a measurable function
. The exponent p is said to be bounded if
is finite. Let
then
is a variable exponent Banach space with the following Luxemburg norm
If p is bounded, there holds
(2.1)
From (2.1), we know that the norm convergence and the convergence is equivalent with respect to
when p is bounded. Moreover, the dual space of
can be written as
for bounded exponent, where
. It is obvious that
is separable and reflexive since
.
The following inequality is Hölder’s inequality in variable exponent Lebesgue space
for all
,
and
.
Next, the variational setting for problem (1.1) will be given. Let
be a nonempty open subset of
and let
be a measurable function, there are two constants
with
such that
Set
where
Let
be equipped with the norm
Especially,
becomes to the usual fractional Sobolev space if
is a constant function.
The space
is defined as
where
The norm on
is given as
The following lemma implies that
is an equivalent norm of
.
Lemma 2.1. [16]. The embedding
↪
↪
are continuous. Moreover, if
, for any fixed constant exponent
,
can be continuously embedded into
.
Using the same argument as the proof of ( [22], Lemma 7), one can easily prove that
is a Hilbert space. From ( [22], Theorem 2.1), we know that the embedding
↪
is continuous and compact. Moreover, there exists
such that
(2.2)
and
(2.3)
Let
the inner product on E is defined as
and the corresponding norm is
. The following inner product
and the corresponding norm
are also used in this paper. Obviously, for all
, one has
. Set
. Moreover, for
, from (2.3), one has
Evidently, problem (1.1) has a variational formulation and the corresponding functional is defined in
by
(2.4)
Actually,
is well-defined and
(2.5)
Hence, u is a solution to problem (1.1) if
is a critical point of
.
3. Proof of Theorem 1.2
It is well known that a
functional
satisfying Palais-Smale ((PS) for short) condition at level c if for any sequence
such that
and
, there exists a convergent subsequence in
, which is called a (PS)c sequence.
First, we shall verify the mountain pass geometry of
.
Lemma 3.1. Assume that (H1) and (H3)-(H9) hold. Then for all
, the functional
satisfies
1) There exists
such that
if
;
2) There exists
with
such that
.
Proof. For any
, it follows from the condition (H5) and (H6) that there exists
such that
(3.1)
Thus, from (3.1) and the fractional Sobolev inequality, one has
(3.2)
For
and
, it follows from (2.4) and (3.2) that
Let
where
By (H9) and an easy computation, for
, one has
Since
, we have
Let
and
, then (1) of Lemma 3.1 is satisfied by.
By (3.1) and (H7), there exists
such that
Then we choose a function
such that
By (2.4), for all
, we obtain
Since
and
, there exists
large enough such that
. The proof is completed.
Let
and
Notice that
Obviously,
is independent of
. It is clear that the mountain pass geometry of
is proved by Lemma 3.1. Since
for all
, we have
for all
. Evidently, for all
,
. Hence, it follows from
that there exists
such that
Then, for all
, we have
(3.3)
By Lemma 3.1 and the mountain pass theorem, we derive that there exists
such that
(3.4)
Lemma 3.2. Assume that (H1) and (H3)-(H9) hold. Then the (PS)c sequence
given in (3.4) is bounded for all
.
Proof. By (2.4), (2.5), (3.4) (H7) and the Hölder inequality, one obtains
(3.5)
Arguing indirectly, we assume that
is not bounded in
. Then there exists a subsequence still denoted by
such that
as
. Then, by (3.5), we obtain
(3.6)
which contradicts to
. Hence, the boundedness of
in
is obtained for all
.
Lemma 3.3. Assume that (H1) and (H3)-(H9) hold. Then
satisfies the (PS)c condition in
for all
and
.
Proof. Let
be a (PS)c sequence with
. From Lemma 3.2, we know that
is bounded in
and there exists
such that
. Hence, there exists a subsequence of
still denoted by
and
in
such that
(3.7)
Now we prove that
in
. From ( [22], Theorem 2.1), we have
in
and
, respectively. Hence,
(3.8)
By (H5) and (H6), we have
Thus,
(3.9)
It follows from (2.5) and (3.4) that
(3.10)
By (3.8), (3.9) and (3.10), we have
The proof is complete.
Proof of Theorem 1.2. By Lemmas 3.1 - 3.2 and the mountain pass theorem, for all
, there exists a
sequence
for
on
. From Lemma 3.2 and
, there exists a subsequence of
still denoted by
and
such that
in
. Moreover,
and
is a solution to problem (1.1).
Next, we verify that problem (1.1) has another solution. Let
where
and
is given by Lemma 3.1. Then
for all
. Let
such that
. By (2.4), (H7) and (H8), one has that
(3.11)
for
small enough.
Hence, there is
such that
for all
small enough.
It follows from Lemma 3.1 and the Ekeland’s variational principle that there exists a sequence
such that
(3.12)
and
(3.13)
Now we should show that
for k large enough. Indirectly, we suppose that
for infinitely many k. Without loss of generality, one may assume that
for any
. By Lemma 3.1, one has
(3.14)
From (3.12) and (3.14), we have
, which contradicts to
. Now we prove that
in
. Let
where
,
is small enough such that
for fixed k large. Thus
which implies that
. Hence, from (3.13), one has
that is,
Letting
, one has
for any fixed k large. Similarly, by repeating the process above, choosing
and
small enough, one gets
Hence, we obtain
which implies that
in
as
. Thus,
is a
sequence for the functional
. Using the same argument as Lemma 3.3, there exists
such that
in
. Hence, one obtains a nontrivial solution
of (1.1) satisfying
Hence, we conclude that
which completes the proof.
4. Conclusion
This paper considers the existence of solutions for a kind of variable-order fractional Laplacian equations. By employing the mountain pass theorem and Ekeland’s variational principle, two solutions are obtained under some suitable conditions on f. Specially speaking, we first prove the mountain pass geometry of the function for this kind of variable-order fractional Laplacian equations. Secondly, we verify the boundedness of (PS)c sequences. Finally, we prove that
satisfies the (PS)c condition in
for all
and
. The result obtained in this paper generalizes the related ones in the literature, which can be used in similar kinds of variable-order fractional Laplacian equations. We hope that this result can be widely used in variable-order fractional Laplacian equations and other systems.
Funding
This work is supported by the National Natural Science Foundation of China (No. 11961014) and Guangxi Natural Science Foundation (2021GXNSFAA196040).
Authors’ Contributions
The authors have the same contributions in writing this paper.