The Existence of Ground State Solutions for Schrödinger-Kirchhoff Equations Involving the Potential without a Positive Lower Bound ()
1. Introduction
In this paper, we study the existence of ground state solutions for the following Schrödinger-Kirchhoff equation:
(1.1)
where
and the potential
satisfying:
(V1)
at
and
in
for some
.
(V2) There holds true;
(1.2)
and the nonlinear term
is a continuous function on
. Moreover, we impose the following conditions on the nonlinearity
;
(f0)
for all
;
(f1) critical exponential growth; there exists
such that
(1.3)
(f2) there exists
such that
(1.4)
(f3) there exist
and
such that
for any
;
(f4)
and
;
(f5)
and
is increasing.
Without losing generality, we suppose that
. So we may rewrite problem (1.1) in the following form:
(1.5)
Remark 1.1 The condition (f2) implies that
as
. Indeed, the condition (f2) implies that
(1.6)
from which we can promptly obtain
as
. From the condition (f1), (f2) and (f4), we can get the following growth condition for
; for any
and
, there exists
such that
(1.7)
From the condition (f5), we can also easily check that the function
is increasing.
The corresponding Dirichlet problem for (1.1) on a smooth domain
,
(1.8)
is related to the stationary analogue of the Kirchhoff equation
(1.9)
which was first proposed by Kirchhoff [1] as an extension of the classical D’Alembert’s wave equation for free vibrations of elastic strings. Problem (1.9) has attracted considerable attention after Lions [2] introduced an abstract framework to the problem.
The above problem is nonlocal as the appearance of the term
implies that (1.5) is not a pointwise identity. This phenomenon provokes some mathematical difficulties, which make the study of such class of problems particularly interesting. For more details on the physical and mathematical background of this problem to [2] [3] [4] [5].
Since we will work with critical exponential growth, we need to review the Trudinger-Moser inequality. For one thing, let
denote a smooth bounded domain in
. N. Trudinger [6] proved that there exists
such that
is embedded in the Orlicz space
determined by the
Young function
. It was sharpened by J. Moser [7] who found the best exponent
, where
is the surface measure of the unit
sphere in
. For another, the Trudinger-Moser inequality was extended for unbounded domains by D. M. Cao [8] in
and for any dimension
by J. M. do Ó [9]. Moreover, J. M. do Ó et al. [10] established a sharp Concentration-compactness principle associated with the singular Trudinger-Moser inequality in
.
Many significant research results about (1.1) have been obtained. For example, in [11], X. Wu studied the nontrivial solutions and high energy solutions of problem (1.1) if
has a positive constant lower bound and the nonlinearities term with 4-superlinear growth at infinity. In [12], the authors studied the following Schrödinger-Kirchhoff type equation
(1.10)
where M is a Kirchoff-type function and
is a continuous function, A is locally bounded and the function f has critical exponential growth. Applying variational methods beside a new Trudinger-Moser type inequality, they get the of ground state solution. Moreover, in the the local case
, they also get some relevant results. We emphasize that in these papers, the potential
have a positive constant lower bound. Some studies of the Kirchhoff equation with critical exponential growth may refer [13] [14] [15] [16].
In [17], the author establishes a class of Trudinger-Moser inequality and proves the existence of the ground state solution to a class of Schrödinger equation with critical exponential growth. In addition, a class of quasilinear n-Laplace Schrödinger equations with degenerate potentials and of exponential growth is also studied. But to the best of our knowledge, the Schrödinger-Kirchhof equation that satisfies condition (V1), (V2) doesn’t seem to have been studied. Different from the first two results, the appearance of the term
, Some proof methods in the original text are invalid, so we have to find other methods, for the details see Lemmas 3.2 and 3.9.
Motivated by [17], we can prove the existence of the ground state solution to problem (1.5) as in [17]. In order to get the result we want, we use a version of Trudinger-Moser inequality.
Lemma 1.2. (Trudinger-Morse inequality [17] ) Assume that the potential
satisfies that
at the ball
centered at the origin with the radius
and
in
for some
. Then
(1.11)
Lemma 1.2 will be used to obtain the existence of ground state solution of the following Schrödinger-Kirchhof equation;
(1.12)
Lemma 1.3. (Fatou’s Lemma) Let
be a measure space, and
be a sequence of non-negative measurable functions. Then the function
is measurable and
(1.13)
Now, we are ready to state the main results of this paper.
Theorem 1.4. Suppose that (V1), (V2) and f0 - f5 hold. If we further assume that
(1.14)
then (1.12) admits a positive ground state solution.
2. Preliminaries
In this section, we give some useful notions and lemmas, which are used to prove our results.
Now, we introduce some notations. For any
,
is the usual Lebesgue space with the norm
is the usual Sobolev space with the norm
Lemma 2.1. ( [17] ) Assume that
such that
where
satisfies the assumption (V1). Then there exists some constant
depending on
and
such that
which was proved in [17].
Remark 2.2. If we define
as the completion of
under the norm
then Lemma 2.1 implies an result;
The problem (1.5) associated functional is
where
, and its Nehari manifold is
where
In order to study the problem (1.5) under the assumptions (V1) and (V2), we introduce the following limiting equation;
(2.1)
where we recall from (V2) that
The corresponding functional and Nehari manifold associated with (2.1) are
and
where
We can easily verify that if
, then
and if
, then
3. The Proof of Theorem 1.3
In this section, we want to show that (1.5) has the existence of ground state solutions.
Lemma 3.1.
and
are not empty.
Proof. we only prove that
is not empty since the proof of
is similar. Let
be positive and compactly supported in a bounded domain
. Define
being not empty is a direct result of the fact;
for
small enough and
for
sufficiently large.
We first prove that
for
small enough. From Remark 1.1, we conclude that for any
, there exist
such that
(3.1)
for any
. Using this estimate, we can write
(3.2)
Which implies that
for small
since
.
Next, we prove that
for
sufficiently large. The condition (f2) implies that
when
. Then it follows that there exists
such that
for all
. this together with the condition (f2) yields that there exists
such that
for all
. Noticing that
is compactly supported in the bounded domain
, we can write
(3.3)
which implies that
is negative for sufficiently large
.
Now, we set
and we claim the following lemma.
Lemma 3.2. It holds that
(3.4)
Proof. To show that
, it is enough to find u satisfying
such that
. From [18], we know that if
then
is attained by some
. By the definition of
, it is easy to check that
Hence there exists
such that
which implies that
. then it follows that
Next, we show
. We prove this by contradiction. Assume that there exists some sequence
such that
, then we have
which implies that
. From
and (3.1), we know that
(3.5)
By the Trudinger-Morse inequality (1.11) and
, we get for any
,
(3.6)
From (3.5) and (3.6), there exist
, such that
Therefore
Since
, there exist
such that
which is a contradicion to
.
We now consider a minimizing sequence
for
. Since
we can assume that
. The (A-R) condition (f2),
and Remark 2.2 give that
is bounded in
, and then up to a subsequence, there exists
, such that
By extracting a subsequence, if necessary, we define
as
By the weak convergence, it is obvious that
.
Lemma 3.3. It holds that
.
Proof. We proof this by contradiction. Assume that
. Then we have
which contradicts (3.4).
Lemma 3.4 The case
cannot occur.
Proof. We prove this by contradiction. If
, then
, and
in
. we first claim that
(3.7)
For any fixed
, we take
such that
combining this and the boundedness of
in
, we derive that
where
. This together with
in
as
yields that
which implies that (3.7) hold.
Similarly to the proof of ( [19]. Proposition 6.1), we can get there exists some sequence
such that
and
converges to 1 as
.
Now, by (3.7), we can write
This together with the monotonicity of
and
gives
which contradicts (3.4). This accomplishes the proof of Lemma 3.4.
Note that (f5) implies the following inequality;
(3.8)
Lemma 3.5. If
, then
and
.
Proof. If
, then
Then we can get
If the above equality holds, then
, and the lemma is proved. Therefore, it remains to show that the case where
(3.9)
cannot occur. In fact, if (3.9) holds, we can take some
such that
. Indeed, let
Obviously,
is positive for small t. This together with
implies that there exists
such that
, i.e.,
.
Using (f5), we can obtain
(3.10)
From (3.8) we know that
(3.11)
Combining (3.10) and (3.11), we derive
(3.12)
Since
, by the define of
, (3.12) and Fatou’s lemma, we deduce that
Which is a contradiction.
In the following, we consider the case
. If
, then
in
. We can choose an increasing sequence
such that
,
(3.13)
and
for any
. We define
Lemma 3.6. For the
given above, we have
(3.14)
and
(3.15)
Proof. We prove (3.14) by contradiction. If there exists some subsequence
of
such that (3.14) fails, then we must have
However, we have
which arrives at a contradiction. Similarly, we can also prove (3.15).
Lemma 3.7. ( [17] ) It holds that
which was proved in [17].
Lemma 3.8. ( [20] ) Let
be a domain in
. Suppose
and
. If
and
then
.
Lemma 3.9. It hold that
provided j is large enough.
Proof. Since
is bounded in
, there is
, and for any
, we have
(3.16)
According to (ii) in definition 6.1 in reference [21], when
, it can be obtained
(3.17)
where E stands for measurable set.
From (3.16) and (3.17), we can deduce that
which implies that
is bounded in
.
We can let
, using (1.4), we can derive
Because
is bounded in
, then there is
, we have
where
.
Using
in
and lemma 3.8, we can get
which implies that
(3.18)
The proof is completed.
From (3.18) and Lemma 3.6, since
, we can extract a subsequence
such that for every
,
and
Now, we take
as a new minimizing sequence renaming it
.
Lemma 3.10. It cannot be
(3.19)
Proof. If (3.19) is true, then there exists some
such that
. Since
, by the define of
, (3.12) and Fatou’s lemma, we deduce that
which is a contradiction.
Lemma 3.11. It cannot be
(3.20)
Proof. We prove (3.20) by contradiction. If
(3.21)
Since
weakly in
, from Lemma 3.9, we can deduce
This implies that
which is a contradiction.
End of the proof of Theorem 1.3. Lemma 3.10 and Lemma 3.11 imply
. Hence
which implies that u is a minimum point for
on
since
. Therefore u is a ground state solution of the Equation (1.5) through the definition of the ground state.
4. Conclusion
In this paper, we use the Nehari manifold technique to prove the existence of ground state solutions for a class of Schrödinger-Kirchhoff equations with vanishing potential and exponential growth.
Acknowledgements
This work is supported by the Natural Science Foundation of China (11961081).