In this paper, we consider the multiplicity results of positive solutions of the following Kirchhoff problem
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  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
has been studied by various researchers and many interesting and important results can be found. In  , 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     . Especially, Chen et al.  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  , Mao and Luan  , found sign-changing solutions. As for in nitely many solutions, we refer readers to   . He and Zou  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  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
Since our approach is variational, we define the functional J on by
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
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.
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 ;
then c is critical value of J such that .
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
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
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
From (2.5) and (2.6), we obtain , which contradicts an hypothesis.
Thus . Define
For the sequel, we need the following Lemma.
1) For all such that , one has .
2) There exists such that for all , one has
Proof. 1) Let . By (2.4), we have
We conclude that .
2) Let . By (2.4) and the Hölder inequality we get
Thus, for all such that , we have .
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
Proof. With minor modifications, we refer to  .
Proposition 1 (see  )
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  , 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:
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
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  .
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:
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
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
and this is a contradiction. Hence,
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
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
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).
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.