Ground State Solutions for the Fractional Klein-Gordon-Maxwell System with Steep Potential Well ()
1. Introduction and Main Results
In the present paper, we are concerned the following fractional Klein-Gordon-Maxwell system
(1.1)
where
,
denotes the fractional Laplacian,
is a parameter,
are functions. Recently, a great attention has been focused
on the study of nonlinear problems involving the fractional Laplacian, in view of concrete real-world applications. For instance, this type of operators arises in the thin obstacle problem, optimization, finance, phase transitions, stratified materials, crystal dislocation, soft thin films, semipermeable membranes, flame propagation, materials science and water waves, see [1]. The study of existence of positive solutions for problems related to the fractional Laplacian operator has been vigorous in the past three decades, see [1] [2] [3] [4] and references therein. In [5], Miyagaki, de Moura and Ruviaor studied the fractional Klein-Gordon-Maxwell system
(1.2)
and the existence of positive ground state solutions was obtained.
The Klein-Gordon-Maxwell system has been introduced in [6] as a model describing solitary waves for the non-linear stationary Klein-Gordon equation coupled with Maxwell equation in the three-dimensional space interacting with the electrostatic field. Some existence results for Klein-Gordon-Maxwell system have been investigated extensively. In [7] [8] [9], the authors studied the existence of ground stated solutions for Klein-Gordon-Maxwell system with periodic potential. Moura [10] obtained the same result involving zero mass potential.
In [11], Liu, Chen and Tang studied the following Klein-Gordon-Maxwell system
(1.3)
and the existence of a ground state solution was proved by using variational methods. Later, In [12], Zhang, Gan, Xiao and Jia expand the range of
, and the existence of a ground state solution for the above system is established under suitable conditions on
and f.
We are going to explore problem (1) showing the existence of the ground state solution with steep potential well. Moreover, we will treat the problem Klein-Gordon-Maxwell using the fractional Laplacian operator instead of classical Laplacian operator. The interest in this kind of problem is: the vast range of applications; the mathematical challenge the nonlocal problem; the challenge when working in the domain like
and also fractional Laplacian.
In case
, our problem becomes doubly nonlocal because of the term
and the fractional operator. Classical compactness arguments are not available and the equation cannot be treated point wisely. We overcome these difficulties using the reduction method introduced by Caffarelli and Silvestre [3], for the fractional Laplacian. When
a process of plugging the
into the main equation is used, allowing look at the system as a single equation. This technique was also employed in [6] [13] [14], and so on.
Inspired by the works in the above references, our main purpose in this paper is to study the existence of ground state solution for problem (1.1). In order to state our main results, we assume that
(a1)
,
for all
.
(a2) There is
such that
.
(a3) The set
is nonempty and has smooth boundary with
.
(f1)
and
for some
and
, where
.
(f2)
uniformly in
.
(f3) There exists
such that
,
.
(f4)
.
Remark 1.1. The conditions (a1)-(a3) were first introduced in [15] and
was called a steep potential well when
was large.
Now we state our main results as following.
Theorem 1.1 Suppose
and assume that (a1)-(a3) and (f1)-(f4) hold. There exists
such that problem (1.1) has a ground state solution for
if one of the following conditions is satisfied:
1)
;
2)
and
.
The plan of the paper is as follows. In Section 2, we give the variational framework for problem (1.1) and some preliminary results. In Section 3, we prove some basic lemmas. In Section 4, we complete the proof of Theorem 1.1.
Throughout the paper, we give the following notations:
・ C and
for psositive constants.
・
(
) denote the strong (weak) convergence.
・
.
・ The integral
is represented by
.
・
for
.
2. Variational Setting and Preliminaries
We reformulate the nonlocal Klein-Gordon-Maxwell system (1.1) into a local system using the local reduction due to Caffarelli and Silvestre [3], that is,
(2.1)
Here
, such that
where
,
and the outward normal derivative should be understood as
Similar definition is given for
.
The fractional Laplacian
with
of a function
is defined by
where
is the Fourier transform, that is
i is the imaginary unit. If
is smooth enough,
can be computed by the following singular integral
where
is a normalization constant and P.V. stands the principal value. For any
. About fractional Sobolev space a very complete introduction can be found in [1].
The spaces
is defined as the completion of
, under the norms (which actually coincide, see ( [16], Lemma A.2))
The Sobolev space
is defined by
where
is the usual Fourier transforms of u, which is the completion of
under the norm
We are looking for a solution in the Hilbert space E defined by
endowed with norm
It follows from Poincaré inequality and (a1)-(a3) that the embedding
↪
is continuous (Its proof is similar to [17]). Thus there exists
for any
such that
(2.2)
E is a Hilbert space. In the following, for convenience, for any u, let
, furthermore
for any
. By a standard argument, solution
of problem (1.1) is a critical point of the energy functional
defined as
We need the following lemma to reduce the functional
in the only variable u.
Lemma 2.1. For every
, there exists a unique
which solves
(2.3)
Furthermore, in the set
we have
if
.
Proof. Its proof is the same as ( [5], Lemma 2.1), and we omit it.
We rewrite
as a
functional
defined as
where
.
From (f1), I is well defined
-functional with derivative given by
Lemma 2.2. If
in E, as
, then passing to a subsequence if necessary,
weakly in
, as
. As a consequence
in the sense of distributions.
Proof. The proof of this lemma is similar to ( [5], Lemma 2.3], but we exhibit it here for completeness. Consider
such that
in
, as
. It follows that
Since that
↪
it is compact for bounded domain, we have
We denote by
the function
. From Lemma 2.1, note that for any
we get
It means that
is bounded in
. Since that
is a Hilbert space, there is a
such that
and
We want to prove the following equality
. To this end, it is necessary to show, in the sense of distributions,
(2.4)
and use the uniqueness of the solution given from Lemma 2.1.
Consider a test function
and
. We know by Lemma 2.1 the following equality
(2.5)
Then, we just need to see how each term of the equality above converges. To verify that
it follows from the weak convergence, also
By the strong convergence in
, as
,
, we have
Whereas
Now we pass to prove the second part of the Lemma. Consider a test function
. Using boundedness of
, the strong convergences in
,
and the Sobolev embeddings follow that as
, it has
Analogously, we prove that
as
. For density,
we infer that,
converges to
and
converges to
as
, thus
in the sense of distributions.
3. Basic Lemmas
In this section, we first begin proving that I satisfies the assumptions of the mountain pass theorem.
Lemma 3.1. Suppose that (a1)-(a3) and (f1)-(f4) are satisfied. Then the functional I satisfies the mountain pass geometry, that is,
1) There exist
such that
for any
such that
;
2) There exists
with
such that
.
Proof. From (f1) and (f2), given
there exists
such that
By sobolev embedding, we have
(3.1)
then we can choose
such that
for
. On the other hand, from (f3), for any
, there exists
such that
, for all
. Hence
, by Lemma 2.1, one has
(3.2)
It is obvious that
as
. Thus, there exists
such that
. This completes the proof of Lemma 3.1.
So, there is a Cerami sequence
such that
(3.3)
where
is the Mountain-Pass level, with
.
Lemma 3.2. Under the assumptions of Theorem 1.1, the Cerami sequence
given in (3.3) is bounded.
Proof. With the fact that (a1) and (a3) hold, there exists
such that v has support in
. Then,
If
, by (3.3), (f3) and Lemma 2.1, we have
(3.4)
Thus,
(3.5)
If
and
. Using Lemma 2.1 and (f3), it obtains
(3.6)
Similar to (3.5), one sees
(3.7)
In light of (3.5) and (3.7) we conclude that
is bounded. £
Lemma 3.3. For any boundedness of (C)c sequence
, there exists
such that
.
Proof. Inspired by [15]. Let
be a bounded (C)c sequence, there exists
such that
. Lemma 2.1 implies that
. By (f1) and (f2), there exists
such that
(3.8)
If
and
. Let
, it follows from (3.9) and (3.10) that
(3.9)
Thus,
(3.10)
If
. Let
, as in proof of (3.9), we can also get (3.10).
For
, let
Since
, from (a2), (2.2) and (3.7), for
, one sees
(3.11)
and using the Hölder inequality and (3.7), we obtain
where
is the measure of
. Hence
(3.12)
When
and r are large enough, we can obtain that
is small enough. Hölder inequality and (2.2) imply that
Since
is bounded which is independent of n and
, there are
and
such that
,
,
. From (3.10), we obtain
,
,
, which implies
. £
4. Proof of Theorem 1.1
In this section, we prove that problem (1.1) has a ground state solution.
Proof of Theorem 1.1. By the previous discussion of Lemmas 3.1-3.3, it follows that there exists a sequence
and
such that
and
. Define
(4.1)
where
.
Clearly,
is not an empty set. Let
be a minimizing sequence for I, that is,
and
. It follows from Lemmas 3.2 and 3.3,
is bounded in E and there exists
such that
in E,
in
for
,
a.e. in
. Using the definition of m together with the fact that
and
, it has
and
. Next, we will claim
.
If
. Combining (f3) and Fatou’s Lemma, it gets
(4.2)
If
and
. It follows from (a2) that
is a bounded domain. Therefore,
(4.3)
From (f3), (4.3) and Fatou’s Lemma, we have
(4.4)
Obviously, this proves that
is a ground state solution of problem (1.1). Hence, Theorem 1.1 is proved. £