In this paper, we consider the multiplicity results of nontrivial solutions of the following Kirchhoff problem
where , is a smooth bounded domain of , , , with , , , is a real parameter, with is the topological dual of satisfying suitable conditions and continuous function.
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 depends on the average of it self. 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 the multiplicity, certain chemical reactions in tubular reactors can be mathematically described by a nonlinear two-point boundary-value problem and one is interested if multiple steady-states exist, for a recent treatment of chemical reactor theory and multiple solutions and the references therein. Bonanno in  established the existence of two intervals of positive real parameters for which the functional has three critical points whose norms are uniformly bounded in respect to belonging to one of the two intervals and he obtained multiplicity results for a two point boundary-value problem.
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 i.e. positive weak solutions tending to zero in .
In the case of a bounded domain of with , Tarantello  proved, under a suitable condition on f, 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
wich equivalent to the norm
Let be the best constant of the weighted Caffarelli-Kohn-Nirenberg type,
Since our approach is variational, we define the functional I on by
A point is a weak solution of the Equation (1.1) if it is the critical point of the functional I. Generally speaking, a function u is called a solution of (1.1) if and for all it holds
with M is incresing and verifying
Throughout this work, we consider the following assumption:
(K) There exist and such that , for all x in .
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 system.
Let be positive number such thatwhere
Now we can state our main results.
Theorem 1. Assume that , and (K) 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 (K) hold and 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, there exists such that for all verifying , 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 I such that .
Nehari Manifold 
It is well known that the functional I is of class in and the solutions of (1.1) are the critical points of I which is not bounded below on . Consider the lowing 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. I is coercive and bounded from below on .
Proof. If , then by (2.3) and the Hölder inequality, we deduce that
Thus, I is coercive and bounded from below on .
We have the following results.
Lemma 3. Suppose that is a local minimizer for I on . Then, if , is a critical point of I.
Proof. If is a local minimizer for I 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.
Lemma 5. If then,
1) For all such that , one has .
2) There exists such that for all , one has
1) Let . By (2.4), we have
If , we conclude that .
2) Let . By (2.4) and the Hölder inequality we get
Thus, for all such that
we have .
For each , we write
Lemma 6. Let real parameters such that . For each , there exist unique and 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 Theorem 1
Now, taking as a starting point the work of Tarantello  , we establish the existence of a local minimum for I on .
Proposition 2. For all such that , the functional I has a minimizer and it satisfies:
2) is a nontrivial solution of (1.1).
Proof. If , then by Proposition 1 (1) 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 3, we may assume that is a nontrivial nonnegative solution of (1.1). By the Harnack inequality, we conclude that and , see for exanmple  .
4. Proof of Theorem 2
Next, we establish the existence of a local minimum for I on . For this, we require the following Lemma.
Lemma 7. Assume that then for all such that , the functional I has a minimizer in and it satisfies:
2) is a nontrivial solution of (1.1) in .
Proof. If , then by Proposition 1 (2) 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 (K) 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 exist and such that . Thus, and is nonempty for any .
Lemma 9. There exist and 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 with , then we obtain that
Lemma 10. There exist r and positive constants such that if ,
1) We have
2) There exists when , with , such that .
Proof. We can suppose that the minima of I are realized by and . The geometric conditions of the mountain pass theorem are satisfied. Indeed, we have
a) By (2.4), (5.1), the Holder inequality, we get
Thus, for there exist such that
b) Let , then we have for all
Letting for t large enough, we obtain . For t large enough we can ensure .
Let and c defined by
Proof of Theorem 3.
If then, by the Lemmas 2 and Proposition 1 (2), I 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 u, we obtain that is also a solution of (1.1).
Conclusion 1. In our work, we have searched the critical points as the minimizers of the energy functional associated to the problem on the constraint defined by the Nehari manifold , which are solutions of our problem. Under some sufficient conditions on coefficients of equation of (1.1), we split in two disjoint subsets and thus we consider the minimization problems on and respectively. In the Sections 3 and 4 we have proved the existence of at least two nontrivial solutions on for all .
In the perspectives we will try to find more nontrivil solutions by splliting again the sub varieties of Nehari.
The author gratefully acknowledge Qassim University, represented by the Deanship of Scientific Research, on the material support for this research under the number (1027) during the academic year 1440AH/2019AD.
 Alves, C.O., Correa, F.J.S.A. and Ma, T.F. (2005) Positive Solutions for a Quasilinear Elliptic Equation of Kirchhoff Type. Computers & Mathematics with Applications, 49, 85-93.
 Cheng, C.T. and Wu, X. (2009) Existence Results of Positive Solutions of Kirchhoff Type Probleme. Nonlinear Analysis: Theory, Methods & Applications, 71, 4883-4892.
 Ma, T.F. and Rivera, J.E.M. (2003) Positive Solutions for a Nonlinear Nonlocal Elliptic Transmission Problem. Applied Mathematics Letters, 16, 243-248.
 Chen, C., Kuo, Y. and Wu, T. (2011) The Nehari Manifold for a Kirchhoff Type Problem Involving Sign-Changing Weight Functions. Journal of Differential Equations, 250, 1876-1908.
 Mao, A.M. and Zhang, Z.T. (2009) Sign-Changing and Multiple Solutions of Kirchhoff Type Problems without the P.S. Condition. Nonlinear Analysis: Theory, Methods & Applications, 70, 1275-1287.
 Mao, A.M. and Luan, S.X. (2011) Sign-Changing Solutions of a Class of Nonlocal Quasilinear Elliptic Boundary Value Problems. Journal of Mathematical Analysis and Applications, 383, 239-243.
 Jin, J.H. and Wu, X. (2010) Infinitely Many Radial Solutions for Kirchhoff-Type Problems in RN. Journal of Mathematical Analysis and Applications, 369, 564-574.
 Ambrosetti, A. and Rabinowitz, P.H. (1973) Dual Variational Methods in Critical Points Theory and Applications. Journal of Functional Analysis, 14, 349-381.
 Brown, K.J. and Zhang, Y. (2003) The Nehari Manifold for a Semilinear Elliptic Equation with a Sign-Changing Weight Function. Journal of Differential Equations, 193, 481-499.
 He, X.M. and Zou, W.M. (2009) Infinitely Many Positive Solutions for Kirchhoff-Type Problems. Nonlinear Analysis: Theory, Methods & Applications, 70, 1407-1414.