On Kirchhoff Problems Involving Critical Exponent and Critical Growth

In this paper, we establish the existence of multiple solutions to a class of Kirchhoff type equations involving critical exponent, concave term and critical growth. Our main tools are the Nehari manifold and mountain pass theorem.


Introduction
In this paper, we consider the multiplicity results of nontrivial solutions of the ( ) 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 0 P 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 [1], 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 [2] 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 [3] [4] [5].Especially, Chen et al. [6] discussed a Kirchhoff type problem when ; where ( ) and ( ) g x 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 [8] [9].He and Zou [9] 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 N  with 3 N ≥ , Tarantello [10] proved, under a suitable condition on f, the existence of at least two solutions to (1.Since our approach is variational, we define the functional I on ( ) A point ( ) with M is incresing and verifying ( ) ( ) ( ) Throughout this work, we consider the following assumption: (K) There exist 0 0 ν > and 0 0 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 system.Let 0 Λ be positive number such thatwhere ( ) ( ) ( ) Now we can state our main results.
Theorem 1. Assume that 1 2 q < < , and (K) satisfied and λ verifying 0 0 λ < < Λ , then the problem (1.1) has at least one positive solution.Journal of Applied Mathematics and Physics 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 ( )

PS
sequence in E for I has a convergent subsequence.Lemma 1. [11] Let X Banach space, and ( ) Palais-Smale condition.Suppose that ( ) 2) there exist ( ) then c is critical value of I such that c r ≥ .

Nehari Manifold [12]
It is well known that the functional I is of class 1 C in ( ) and the solutions of (1.1) are the critical points of I which is not bounded below on ( ) . Consider the lowing Nehari manifold 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 u ∈  , then by (2.3) and the Hölder inequality, we deduce that Thus, I is coercive and bounded from below on  .
For the sequel, we need the following Lemma.
2) Let u − ∈  .By (2.4) and the Hölder inequality we get ( p q p p p q p p q p p a p Thus, for all λ such that ( ) , there exist unique t + and t − such that 0 m t t t Proof.With minor modifications, we refer to [13].

Proof of Theorem 1
Now, taking as a starting point the work of Tarantello [10], we establish the existence of a local minimum for I on Proposition 2. For all λ such that 0 0 λ < < Λ , the functional I has a minimizer 0 u + + ∈  and it satisfies: 1) ( ) Proof.If 0 0 λ < < Λ , then by Proposition 1 (1) there exists a ( ) , thus it bounded by Lemma 2.Then, there exists ( ) and we can extract a subsequence which will denoted by ( ) We get a contradiction.Therefore, n u converge to 0 u + strongly in ( )  u + is a nontrivial nonnegative solution of (1.1).By the Harnack inequality, we conclude that Journal of Applied Mathematics and Physics 0 0 u + > and 0 0 v + > , see for exanmple [12].

Proof of Theorem 2
Next, we establish the existence of a local minimum for I on −  .For this, we require the following Lemma.
Proof.If 1 0 λ < < Λ , then by Proposition 1 (2) there exists a ( )  , thus it bounded by Lemma 2.Then, there exists ( ) and we can extract a subsequence which will denoted by ( )  , this implies that 0 u + and 0 u − are distinct.

Proof of Theorem 3
In this section, we consider the following Nehari submanifold of  \ 0 : , 0 and 0 .Proof.We can suppose that the minima of I are realized by ( ) 0 u + and 0 u − 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 In the perspectives we will try to find more nontrivil solutions by splliting again the of Nehari.

Theorem 2 . 1 ) 2 0Λ 1 )
In addition to the assumptions of the Theorem 1, if (K) hold and has at least two positive solutions.Theorem 3. In addition to the assumptions of the Theorem 2, there exists > such that for all λ verifying has at least two positive solution and two opposite solutions.

Lemma 3 .
Suppose that 0 u is a local minimizer for I on  .Then, if 0 0 u ∉  , 0 u is a critical point of I. Proof.If 0 u is a local minimizer for I on  , then 0 u is a solution of the optimization problem . Moreover, by the Hölder inequality and the Sobolev embedding theorem, we obtain (3.1), 0 u + is a weak nontrivial solution of (1.1).Now, we show that n u converges to 0 u + strongly in

∈
 , then by Lemma 3, we may assume that 0

1
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


enough, we obtain ( ) 0 I σ ≤ .For t large enough we can ensure r σ > .Let Γ and c defined by Thus c u is the third solution of our system such that0 c u u + ≠ and 0 c u u − ≠ .Since (1.1) is odd with respect u, we obtain that c u − 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 + respectively.In the Sections 3 and 4 we have proved the existence of at least two nontrivial solutions on )