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In this paper, we consider the existence of multiple solutions to the Kirchhoff problems with critical potential, critical exponent and a concave term. Our main tools are the Nehari manifold and mountain pass theorem.

In this paper, we consider the multiplicity results of nontrivial solutions of the following Kirchhoff problem

where

The original one-dimensional Kirchhoff equation was introduced by Kirchhoff [

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. For instance, positive solutions could be obtained in [

Researchers, such as Mao and Zhang [

In the case of a bounded domain of

the existence of at least two solutions to (1.2) for

Before formulating our results, we give some definitions and notation.

The space

wich equivalent to the norm

with

for all

Let

Since our approach is variational, we define the functional

A point

Throughout this work, we consider the following assumptions:

(F) There exist

Here,

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

where

Now we can state our main results.

Theorem 1. Assume that

Theorem 2. In addition to the assumptions of the Theorem 1, if (H) hold and

Theorem 3. In addition to the assumptions of the Theorem 2, assuming

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

i)

where

ii) We say that I satisfies the

Lemma 1. Let X Banach space, and

i) there exist

ii) there exist

let

then c is critical value of J such that

It is well known that the functional

Thus,

Define

Then, for

Now, we split

Note that

Lemma 2.

Proof. If

Thus,

We have the following results.

Lemma 3. Suppose that

Proof. If

Hence, there exists a Lagrange multipliers

Thus,

But

Lemma 4. There exists a positive number

Proof. Let us reason by contradiction.

Suppose

and

with

From (2.5) and (2.6), we obtain

Thus

For the sequel, we need the following Lemma.

Lemma 5. i) For all

ii) There exists

Proof. i) Let

and so

We conclude that

ii) Let

Thus, for all

For each

Lemma 6. Let

Proof. With minor modifications, we refer to [

Proposition 1. (see [

i) For all

ii) For all

Now, taking as a starting point the work of Tarantello [

Proposition 2. For all

i)

ii)

Proof. If

Thus, by (3.1),

in

We get a contradiction. Therefore,

there exists

which contradicts the fact that

Next, we establish the existence of a local minimum for

Lemma 7. Assume that

i)

ii)

Proof. If

This implies that

Moreover, by (H) and (2.4) we obtain

if

This implies that

Now, we prove that

we have

and this is a contradiction. Hence,

Thus,

Since

Now, we complete the proof of Theorem 2. By Propositions 2 and Lemma 7, we obtain that (1.1) has two positive solutions

In this section, we consider the following Nehari submanifold of

Thus,

Firsly, we need the following Lemmas.

Lemma 8. Under the hypothesis of theorem 3, there exist

Proof. Fix

Clearly

If

Lemma 9. There exist M positive real such that

for

Proof. Let

Thus, if

Lemma 10. There exist r and

i) we have

ii) there exists

Proof. We can suppose that the minima of

i) By (2.4), (5.1), the Holder inequality and the fact that

Thus, for

ii) Let

Letting

Let

and

Proof of Theorem 3.

If

Thus

Mohammed El Mokhtar Ould ElMokhtar, (2015) Four Nontrivial Solutions for Kirchhoff Problems with Critical Potential, Critical Exponent and a Concave Term. Applied Mathematics,06,2248-2256. doi: 10.4236/am.2015.614198