Multiple Solutions to the Problem of Kirchhoff Type Involving the Critical Caffareli-Kohn-Niremberg Exponent , Concave Term and Sign-Changing Weights

In this paper, we establish the existence of at least four distinct solutions to an Kirchhoff type problems involving the critical Caffareli-Kohn-Niremberg exponent, concave term and sign-changing weights, by using the Nehari manifold and mountain pass theorem.


Introduction
In this paper, we consider the multiplicity results of positive solutions of the following Kirchhoff problem ( ) 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, ρ is the mass density and 0 P is the initial tension.
In recent years, the existence and multiplicity of solutions to the nonlocal problem ( ) ( )

2
; in , 0 on 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 ( ) 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 N  with 3 N ≥ , Tarantello [12] proved, under a suitable condition on f , the existence of at least two solutions to (1.2)   for 0, 1 ; Before formulating our results, we give some definitions and notation.The space ( ) Let S µ be the best Sobolev constant, then ( ) { } ( ) Since our approach is variational, we define the functional J on ( ) 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 ( ) Throughout this work, we consider the following assumptions: (F) f is a continuous function satisfies: h is a continuous function and there exist 0 g and 0 ρ positive such that: ( ) ( ) Here, ( ) , B a r 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 0 µ be real number such that Now we can state our main results.
, and (F) satisfied and µ verifying 0 µ µ < , 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 1 0 µ > such that for all µ verifying ( ) the problem (1.1) has at least two positive solutions.Theorem 3 In addition to the assumptions of the Theorem 2, assuming 0 µ < , 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.

Preliminaries
Definition 1 Let c ∈  , E a Banach space and ( ) where ( ) tends to 0 as n goes at infinity.
2) We say that I satisfies the ( ) c PS condition if any ( ) c PS sequence in E for I has a convergent subsequence.

Nehari Manifold
It is well known that the functional J is of class 1 C in ( ) 0   and the solutions of (1.1) are the critical points of J which is not bounded below on ( ) 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 u ∈  , 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 0 u is a local minimizer for J on  .Then, if Hence, there exists a Lagrange multipliers θ ∈  such that ( ) ( ) and the Sobolev embedding theorem, we obtain From (2.5) and (2.6), we obtain 0 µ µ ≥ , which contradicts an hypothesis. Thus : inf , : inf and : inf .
2) There exists 1 0 µ > such that for all 1 0 µ µ < < , one has ( ) and so ( ) We conclude that 0 2) Let u − ∈  .By (2.4) and the Hölder inequality we get ( ) Thus, for all µ such that We define: and for each u ∈  with u F + ∈ , we write ( ) ( ) Lemma 6 Let µ real parameters such that 0 0 µ µ < < .For each u ∈  we have: ∈  then there exists unique then there exist unique t + and t − such that 0 for and for 0, J t u J tu t t J t u J tu t t ,then there exists unique 0 t − < < +∞ such that ( ) Proof.minor modifications, we refer to [13].Proposition 1 (see [13]) 1) For all µ such that 0 0 µ µ < < , there exists a ( ) 2) For all µ such that 1 0 µ µ < < , there exists a a ( )

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 µ such that 0 0 µ µ < < , the functional J has a minimizer 0 u + + ∈  and it satisfies: 2) ( ) u + is a nontrivial solution of (1.1).
Proof.If 0 0 µ µ < < , then by Proposition 1 (i) there exists a ( ) , thus it bounded by Lemma 2.Then, there exists 0 u + ∈  and we can extract a subsequence which will denoted by ( ) ( ) ( ) Thus, by (3.1), 0 u + is a weak nontrivial solution of (1.1).Now, we show that n u converges to 0 u + strongly in ( ) We get a contradiction.Therefore, n u converge to 0 u + strongly in ( ) which contradicts the fact that ( ) = and 0 u + + ∈  , then by Lemma 6, we may assume that 0 u + is a nontrivial nonnegative solution of (1.1).By the Harnack inequality, we conclude that 0 0 u + > , see for exanmple

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 ( ) ( ) the functional J has a minimizer 0 u − in −  and it satisfies: Proof.If 1 0 δ δ < < , then by Proposition 1 (ii) there exists a ( )  , thus it bounded by Lemma 2.Then, there exists ( ) and we can extract a subsequence which will denoted by ( ) n n u such that ( ) Moreover, by (G) and (2.4) we obtain , this implies that 0 u + and 0 u − are distinct.

Proof of Theorem 3
In this section, we consider the following Nehari submanifold of  ( ) { } ( ) { } some proper conditions are sign-changing weight functions.And they have obtained the existence of two positive solutions if

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There exists a positive number 0 µ such that, for all Moreover, by the Hölder inequality q then by(4.1)  and Lemma 3, we may assume that 0 u − is a nontrivial nonnegative solution of (1.1).By the maximum principle, we conclude that 0 0 u − > .Now, we complete the proof of Theorem 2. By Propositions 2 and Lemma 7, we obtain that (1.1) has two positive solutions 0