Multiple Solutions to the Problem of Kirchhoff Type Involving the Critical Caffareli-Kohn-Niremberg Exponent, Concave Term and Sign-Changing Weights ()
1. Introduction
In this paper, we consider the multiplicity results of positive solutions of the following Kirchhoff problem
(1.1)
where
,
is a real parameter,
,
,
,
is
the critical Caffareli-Kohn-Niremberg exponent and
are continuous and sign-changing functions which we will specify later.
The original one-dimensional Kirchhoff equation was introduced by Kirchhoff [1] in 1883 as an generalization of the well-known d’Alembert’s wave equation:
His model takes into account the changes in length of the strings produced by transverse vibrations. Here, L is the length of the string, h is the area of the cross section, E is the Young modulus of the material,
is the mass density and
is the initial tension.
In recent years, the existence and multiplicity of solutions to the nonlocal problem
(1.2)
has been studied by various researchers and many interesting and important results can be found. In [2] , it was pointed out that the problem (1.2) models several physical systems, where u describes a process which depend on the average of itself. Nonlocal effect also finds its applications in biological systems. The movement, modeled by the integral term, is assumed to be dependent on the energy of the entire system with u being its population density. Alternatively, the movement of a particular species may be subject to the total population density within the domain (for instance, the spreading of bacteria) which gives rise to equations of the type
For instance, positive solutions could be obtained in [2] [3] [4] [5] . Especially, Chen et al. [6] discussed a Kirchhoff type problem when
, where
if
,
if
,
and
with some proper conditions are sign-changing weight functions. And they have obtained the existence of two positive solutions if
.
Researchers, such as Mao and Zhang [7] , Mao and Luan [8] , found sign-changing solutions. As for in nitely many solutions, we refer readers to [9] [10] . He and Zou [11] considered the class of Kirchhoff type problem when
with some conditions and proved a sequence of positive weak solutions tending to zero in
.
In the case of a bounded domain of
with
, Tarantello [12] proved, under a suitable condition on
, the existence of at least two solutions to (1.2)
for
and
.
Before formulating our results, we give some definitions and notation.
The space
is equiped with the norm
Let
be the best Sobolev constant, then
(2.1)
Since our approach is variational, we define the functional J on
by
(2.2)
A point
is a weak solution of the Equation (1.1) if it is the critical point of the functional J. Generally speaking, a function u is called a solution of (1.1) if
and for all
it holds
Throughout this work, we consider the following assumptions:
(F) f is a continuous function satisfies:
(G) h is a continuous function and there exist
and
positive such that:
Here,
denotes the ball centered at a with radius r.
In our work, we research the critical points as the minimizers of the energy functional associated to the problem (1.1) on the constraint defined by the Nehari manifold, which are solutions of our problem.
Let
be real number such that
where
Now we can state our main results.
Theorem 1 Assume that
,
,
, and (F)
satisfied and
verifying
, then the problem (1.1) has at least one positive solution.
Theorem 2 In addition to the assumptions of the Theorem 1, if (G) hold, then there exists
such that for all
verifying
the problem (1.1) has at least two positive solutions.
Theorem 3 In addition to the assumptions of the Theorem 2, assuming
, then the problem (1.1) has at least two positive solution and two opposite solutions.
This paper is organized as follows. In Section 2, we give some preliminaries. Section 3 and 4 are devoted to the proofs of Theorems 1 and 2. In the last Section, we prove the Theorem 3.
2. Preliminaries
Definition 1 Let
, E a Banach space and
.
1)
is a Palais-Smale sequence at level c ( in short
) in E for I if
where
tends to 0 as n goes at infinity.
2) We say that I satisfies the
condition if any
sequence in E for I has a convergent subsequence.
Lemma 1 Let X Banach space, and
verifying the Palais-Smale condition. Suppose that
and that:
1) there exist
,
such that if
, then
;
2) there exist
such that
and
;
let
where
then c is critical value of J such that
.
Nehari Manifold
It is well known that the functional J is of class
in
and the solutions of (1.1) are the critical points of J which is not bounded below on
. Consider the following Nehari manifold
Thus,
if and only if
(2.3)
Define
Then, for
(2.4)
Now, we split
in three parts:
Note that
contains every nontrivial solution of the problem (1.1). Moreover, we have the following results.
Lemma 2 J is coercive and bounded from below on
.
Proof. If
, then by (2.3) and the Hölder inequality, we deduce that
Thus, J is coercive and bounded from below on
.
We have the following results.
Lemma 3 Suppose that
is a local minimizer for J on
. Then, if
,
is a critical point of J.
Proof. If
is a local minimizer for J on
, then
is a solution of the optimization problem
Hence, there exists a Lagrange multipliers
such that
Thus,
But
, since
. Hence
. This completes the proof.
Lemma 4 There exists a positive number
such that, for all
we have
.
Proof. Let us reason by contradiction.
Suppose
such that
. Moreover, by the Hölder inequality and the Sobolev embedding theorem, we obtain
(2.5)
and
(2.6)
with
From (2.5) and (2.6), we obtain
, which contradicts an hypothesis.
Thus
. Define
For the sequel, we need the following Lemma.
Lemma 5
1) For all
such that
, one has
.
2) There exists
such that for all
, one has
Proof. 1) Let
. By (2.4), we have
and so
We conclude that
.
2) Let
. By (2.4) and the Hölder inequality we get
Thus, for all
such that
, we have
.
We define:
and for each
with
, we write
Lemma 6 Let
real parameters such that
. For each
we have:
1) If
then there exists unique
such that
and
2) If
then there exist unique
and
such that
,
,
and
3) If
, then does not exist
such that
.
4) If
,then there exists unique
such that
and
Proof. With minor modifications, we refer to [13] .
Proposition 1 (see [13] )
1) For all
such that
, there exists a
sequence in
.
2) For all
such that
, there exists a a
sequence in
.
3. Proof of Theorems 1
Now, taking as a starting point the work of Tarantello [12] , we establish the existence of a local minimum for J on
.
Proposition 2 For all
such that
, the functional J has a minimizer
and it satisfies:
1)
2)
is a nontrivial solution of (1.1).
Proof. If
, then by Proposition 1 (i) there exists a
sequence in
, thus it bounded by Lemma 2. Then, there exists
and we can extract a subsequence which will denoted by
such that
(3.1)
Thus, by (3.1),
is a weak nontrivial solution of (1.1). Now, we show that
converges to
strongly in
. Suppose otherwise. By the lower semi-continuity of the norm, then either
and we obtain
We get a contradiction. Therefore,
converge to
strongly in
. Moreover, we have
. If not, then by Lemma 6, there are two numbers
and
, uniquely defined so that
and
. In particular, we have
. Since
there exists
such that
. By Lemma 6, we get
which contradicts the fact that
. Since
and
, then by Lemma 6, we may assume that
is a nontrivial nonnegative solution of (1.1). By the Harnack inequality, we conclude that
, see for exanmple [14] .
4. Proof of Theorem 2
Next, we establish the existence of a local minimum for J on
. For this, we require the following Lemma.
Lemma 7 Assume that
then, for all
such that
,
the functional J has a minimizer
in
and it satisfies:
1)
2)
is a nontrivial solution of (1.1) in
.
Proof. If
, then by Proposition 1 (ii) there exists a
,
sequence in
, thus it bounded by Lemma 2. Then, there exists
and we can extract a subsequence which will denoted by
such that
This implies that
Moreover, by (G) and (2.4) we obtain
if
we get
(4.1)
This implies that
Now, we prove that
converges to
strongly in
. Suppose otherwise. Then, either
. By Lemma 6 there is a unique
such that
. Since
we have
and this is a contradiction. Hence,
Thus,
Since
and
, then by (4.1) and Lemma 3, we may assume that
is a nontrivial nonnegative solution of (1.1). By the maximum principle, we conclude that
.
Now, we complete the proof of Theorem 2. By Propositions 2 and Lemma 7, we obtain that (1.1) has two positive solutions
and
. Since
, this implies that
and
are distinct.
5. Proof of Theorem 3
In this section, we consider the following Nehari submanifold of
Thus,
if and only if
Firsly, we need the following Lemmas
Lemma 8 Under the hypothesis of theorem 3, there exist
such that
is nonempty for any
and
.
Proof. Fix
and let
Clearly
and
as
. Moreover, we have
If
for
, then there exists
such that
. Thus,
and
is nonempty for any
.
Lemma 9 There exist M positive real such that
for
and any
.
Proof. Let
, then by (2.3), (2.4) and the Holder inequality, allows us to write
Thus, if
then we obtain that
(5.1)
Lemma 10 There exist
and
positive constants such that
1) we have
2) there exists
when
, with
, such that
.
Proof. We can suppose that the minima of J are realized by
and
. The geometric conditions of the mountain pass theorem are satisfied. Indeed, we have
1) By (2.4), (5.1), the Holder inequality, we get
Thus, for
there exist
,
such that
2) Let
, then we have for all
By the fact that
we have
and letting
for t large enough, we obtain
. For t large enough we can ensure
.
Let
and
defined by
and
Proof of Theorem 3.
If
then, by the Lemmas 2 and Proposition 1 2), J verifying the Palais -Smale condition in
. Moreover, from the Lemmas 3, 9 and 10, there exists
such that
Thus
is the third solution of our system such that
and
. Since (1.1) is odd with respect
, we obtain that
is also a solution of (1.1).
Acknowledgements
The author gratefully acknowledges Qassim University, represented by the Deanship of Scientific Research, on the material support for this research under the number (1026) during the academic year 1438AH/2017AD.