Existence of Solutions for Klein-Gordon-Maxwell Equations Involving Hardy-Sobolev Critical Exponents ()
1. Introduction and Preliminaries
In recent years, great attention has been given to problems driven by the Laplacian. One of the reasons for this comes from the fact that this operator appears in several applications in different subjects, such as flame propagation, free boundary obstacle problems, and ultrarelativistic limits of quantum mechanics. In particular, from a probabilistic point of view, the Laplace operator is the infinitesimal generator of a Lévy process. For more details and applications, see other references [1] [2].
Problems with two nonlinearities recently have been studied by several authors. In particular, such problems were considered in [3] [4] for the Laplacian, the p-Laplacian, the Biharmonic operator and the fractional Laplacian. In [5], Ghoussoub, Robert and Shakerian investigated problems with doubly critical nonlinear terms, with either critical Sobolev term or critical Hardy-Sobolev term, for the Laplacian and the fractional Laplacian.
Solutions of critical Sobolev problems were found in [6], as critical points of a suitable functional, by the Mountain-Pass lemma without the
condition. In this case, the
condition only holds true for c in certain intervals related to the best Sobolev constant. In the control of the Mountain-Pass level, the extremal function of the best Sobolev constant plays an important role.
For example, Jannelli (see [7] ) considered that the problem
(1)
they proved the existence of nontrivial solutions for the preceding equations involving a critical Hardy-Sobolev exponent in a bounded domain.
Then Kang and Peng in [8] considered the following problem based on the above equations
(2)
They established the problem above has at least a pair of sign-changing solutions with
and
in this reference [8]. Kang and Peng in [9] proved that problem above has at least one positive solution under some conditions for
and
. They also proved in [10] that as
, the problem above has at least a pair of sign-changing solutions for
,
and
.
Motivated by the study of solitary waves of the nonlinear Klein-Gordon equation interacting with an electromagnetic field, Benci and Fortunato derived in [11] a model that is described by the following elliptic system
(3)
where
and
are real constants. They proved existence of infinitely many radially symmetric solutions
for the above system when
and for sub-critical exponents p satisfying
. As in [12], they derived a variational identity to prove the non-existence of nontrivial weak solutions for the system above. In [13], Cassani investigated the critical case i.e.
, the critical Sobolev exponent. Moreover, Cassani used a Pohozaev-type argument, which points out an invariance property for the problem (3), to prove non-existence of solutions with a suitable decay at infinity and in particular it turns out to be the case of radially symmetric solutions. In [13], Cassani replaced the first equation of the System (3), adding a lower order perturbation, by the following
(4)
where
and
. In this case, they recovered a Mountain-Pass type solution for Equation (4) and the second equation of System (3).
The above-mentioned equations with Hardy-Sobolev critical exponents are restricted to bounded regions, and the two Klein-Gordon-Maxwell systems mentioned above involve Sobolev critical exponents, not Hardy-Sobolev critical exponents. Moreover, some parts of the certification process did not give a specific certification process. We are inspired by the proof methods of a nontrivial solution and infinitely many solutions in the above-mentioned literatures, and investigate the existence of solutions for subcritical equations and critical equations with Hardy-Sobolev critical exponents in
.
In this paper, firstly we study existence of solutions for the following Klein-Gordon-Maxwell equations involving Hardy-Sobolev critical exponents
(5)
where m and
are real constants,
,
.
Then we study existence of solutions for the following equations
(6)
where m,
and
are real constants,
,
,
.
In Section 2, we first give main results for the Systems (5) and (6). In Section 3, we prove the existence of solutions of System (5). In Section 4, we establish existence of solutions of System (6).
2. Main Results
Throughout this paper, we denote the
norm by
and
norm by
. For simplicity, set
, since
.
is a Sobolev space with norm
Moreover,
is the usual Sobolev space with norm
continuously embedded in
for
. Define Hardy-Sobolev best constant as follows
Theorem 2.1. If one of the following conditions is satisfied:
1)
and
, or
2)
and
.
Then System (5) admits at least a nontrivial solution.
Theorem 2.2. If one of the following conditions is satisfied:
1)
and
, or
2)
and
.
Then there exists a constant
such that System (6) admits at least two different solutions
satisfying
,
when
.
We define the functional of System (5)
(7)
Define the functional of System (6) as follows
(8)
Remark 2.1. The functional F and
are strongly indefinite i.e. unbounded from below and from above on infinite dimensional subspaces. In order to avoid this indefiniteness, which rules out many of the usual tools of critical point theory, a reduction method is performed in [11] which we now recall. For u and
defined above, we have the following lemmas.
3. The Proof of Theorem 2.1
Lemma 3.1. For every
,
1) there exists a unique function
that solves the second equation of System (5);
2) if u is radially symmetric, then
is radial too;
3)
, moreover,
, if
and
.
Proof. The first result is proved in Lemma 3 of [14]. While the second one, though not explicitly stated, is proved in Lemma 5 of [14]. The third result can be found in Lemma 2.3 of [15]. □
Lemma 3.2. The map
is
and
.
Proof. Noticing that
is a solution of the second equation in System (5), we have
(9)
In addition,
According to (9), one gets
for any
. Thus
. □
Define
. If
, then one has
(10)
Lemma 3.3. The following statements are equivalent:
1)
is a solution of System (5);
2) u is a critical point for I and
.
Proof. 2)
1) Obviously.
1)
2) Suppose
and
denote the partial derivatives of F at
. Then for every
and
, one gets
(11)
(12)
By the standard computations, we can prove that
and
are continuous. From (11) and (12), it is easy to obtain that its critical points are solutions of System (5), by 1) of Lemma 3.1, one has
. □
Lemma 3.4. For
, if
, then there exist some constants
such that
.
Proof. From (7), one obtains
(13)
Substituting (9) into (13), we have
Thus
and the proof is completed. □
Lemma 3.5. Under the assumptions of Theorem 2.1, there exists a function
with
such that
.
Proof. It is easy to obtain
which implies that
, as
.
The lemma is proved by taking
with
large enough and
. Therefore we know that there exists
,
such that
. □
Therefore, there exists a sequence
, so-called Palais-Smale sequence, such that
(14)
where
with
Since System (5) is set on
, it is well known that the Sobolev embedding
↪
is not compact and then it is usually difficult to prove that a Palais-Smale sequence is strongly convergent when we seek solutions of System (5) by variational methods. A standard tool to overcome the problem is to restrict ourselves to radial functions, namely we look at the functional I on the subspace
and
compactly embedded in
for
and
for
. By standard arguments, one sees that if a critical point
for the functional
is also a critical point of I.
Lemma 3.6. The PS sequence
given in (14) is bounded. Moreover,
is bounded, too.
Proof. Case 1.
. There exists a constant
, then by (7), (9) and (10), we get
for n large enough. Therefore, it follows that
is bounded in
.
Case 2.
. We have that there is a positive constant
such that
According to (9), (10) and (13),
, one has
for n large enough. It follows that
is bounded in
.
According to Equation (9), one has
then by Hölder inequality
and Sobolev inequality
One obtains
Thus
is bounded in
by the boundedness of
. □
Up to subsequence, we may assume that there exists
and
such that
(15)
(16)
(17)
Lemma 3.7.
and
in
.
Proof. First we prove the uniqueness. For every fixed
, we consider the following minimizing problem
, where
defined as energy functional of the second equation in System (5).
In fact, by the proof of Lemma 2.1 in [16], one can know
so we obtain
From the weak lower semicontinuity of the norm in
and the convergence above, one has
then by 1) of Lemma 3.1,
.
Next, we prove that
converges strongly in
. Since
satisfies the following equation
(18)
Let us take the difference between (18) and the corresponding equation for
to have
(19)
Testing with
, by the Hölder inequality, the following holds
(20)
according to (16), one has
strongly in
. □
Lemma 3.8.
has a strongly convergent subsequence in
.
Proof. Consider a sequence
in
, which satisfies
,
, and
. Going if necessary to a subsequence, since the embedding
↪
is compact for any
, we have
(21)
According to (10), one obtains
Similarly, one gets
By (10), we easily get that
(22)
It is clear that
(23)
Furthermore, in view of (21), we have
(24)
Similarly, we also obtain that
(25)
Thus combining (24) and (25), one gets that
(26)
By Hölder inequality and Sobolev inequality, one has
According to (16), one gets
, as
.
And
Thus we get that
(27)
We observe that the sequence
is bounded in
, since
so that
By (16), one has
(28)
Therefore according to (23)-(28) and
, we obtain that
Thus
has a strongly convergent subsequence in
.
Consequently, we conclude that
□
Next we begin to prove Theorem 2.1.
Proof. We only need to prove that
. Suppose by contradiction that
, and hence
. Since as
,
,
in
and
. Thus we get
We may assume
Set
obviously,
in
. As a consequence we obtain that
According to
, we get
which implies that
is impossible, i.e., which contradicts with
. Therefore, u is a nontrivial solution of System (5).
This theorem is mainly based on satisfying the conditions of the Mountain Pass Theorem, and then there is a (PS) sequence, proving that the (PS) sequence is bounded and
has a strongly convergent sub-sequence in
, so as to prove that the system of Equation (5) has at least a nontrivial solution. □
4. The Proof of Theorem 2.2
Similarly, we also have the following lemmas.
Lemma 4.1. For every
,
1) there exists a unique function
that solves the second equation of System (6);
2) if u is radially symmetric, then
is radial too;
3)
, moreover,
, if
and
.
Lemma 4.2. The map
is
and
.
Likewise, define
.
Lemma 4.3. The pair
is a weak solution of System (6) if and only if it is a critical point of J in
.
Lemma 4.4 For
, if
, then there exist some constants
such that
for all
satisfying
.
Proof. From (8), one obtains
(29)
Substituting (9) into (29), we have
Set
.
Since
Evidently, when
is small enough,
is greater than 0, and
increases monotonically. When
is large enough,
is less than 0, and
decreases monotonically. Therefore there exists a maximum point
such that
Obviously, we observe that
or
, while
is impossible. Then
, next we obtain
. Choosing that
, we deduce, for all
satisfying
,
and the proof is completed. □
Lemma 4.5. Under the assumptions of Theorem 2.2, there exists a function
with
such that
.
Similarly, a standard tool is to restrict ourselves to radial functions, namely
and
compactly embedded in
for
and
for
. Moreover, one sees that if a critical point
for the functional
is also a critical point of J.
Lemma 4.6. Under the assumptions of Theorem 2.2, if
is a bounded Palais-Smale sequence of J, then
has a strongly convergent subsequence in
.
Proof. Consider a sequence
in
, which satisfies
,
, and
. Going if necessary to a subsequence, we assume
Since the embedding
↪
is compact for any
, we have
(30)
Moreover, likewise, for
, we also get
(31)
According to
(32)
one obtains
Similarly, one gets
By (32), we easily get that
(33)
Then we use similar method in the proof of Lemma 3.8, we obtain that
has a strongly convergent subsequence in
. □
Next we begin to prove Theorem 2.2.
Proof. The proof is divided into two steps.
Step 1. There exists
such that
and
.
We choose a function
. Since
, one has
(34)
for
small enough. Thus we have
, where
is given by Lemma 4.4,
. By the Ekeland’s variational principle [17], Let
,
for all
, then there exists a sequence
such that
and
Then we obtain that
Obviously, in view of Lemma 4.4,
, for n large enough. Thus for any
with
, we can take
such that
for n large enough (see [18] ). Then we have
Letting
, we get
We replace
by
in the above inequality, then it follows that
i.e.,
Thus one obtains
which implies
as
.
Hence we conclude that
is bounded PS sequence of J for
. Therefore, by Lemma 4.6, we get that there exists a function
such that
and
.
Step 2. There exists
such that
and
.
From Lemma 4.4, Lemma 4.5 and the Mountain Pass Theorem [19], there is a sequence
such that
and
where
. From Lemma 4.6, we only need to prove that
is bounded in
.
Case 1):
. From (8) and (9), one has
Then by (32), one obtains
for n large enough. Therefore, it follows that
is bounded in
.
Case 2):
. From (8) and (9), one has
Then by (32) and
and
,
, we get
for n large enough. It follows that
is bounded in
.
This theorem is mainly based on the Mountain Pass Theorem and the Ekeland variational principle to prove that the System (6) has at least two different solutions. □
Remark 4.1. Here a simple infimum definition is given as follows. Suppose S is a set of numbers in
. If the number
satisfies:
1)
, that is,
is a lower bound of S.
2)
, there exists a
that satisfies
, that is,
is maximum lower bound of S, then the number
is the infimum of the number set. Referred to as:
.