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In this paper, we establish the existence of at least four distinct solutions to an elliptic problem with singular cylindrical potential, a concave term, and critical Caffarelli-Kohn-Nirenberg exponent, by using the Nehari manifold and mountain pass theorem.

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

where

Some results are already available for

therein. Wang and Zhou [

existence of two solutions of

Concerning existence results in the case

where

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

We denote by

and

respectively, with

From the Hardy-Sobolev-Maz’ya inequality, it is easy to see that the norm

for all

We list here a few integral inequalities.

The starting point for studying

for any

The second one that we need is the Hardy inequality with cylindrical weights [

It is easy to see that (1.1) hold for any

where

Since our approach is variational, we define the functional J on

with

A point

with

Here

Let

From [

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

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. Sections 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 J is of class

Thus,

Note that

Lemma 2. J is coercive and bounded from below on

Proof. If

Thus, J is coercive and bounded from below on

Define

Then, for

Now, we split

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

we have

Proof. Let us reason by contradiction.

Suppose

Moreover, by the Hölder inequality and the Sobolev embedding theorem, we obtain

and

From (2.5) and (2.6), we obtain

Thus

For the sequel, we need the following Lemma.

Lemma 5.

i) For all

ii) For all

Proof. i) Let

and so

We conclude that

ii) Let

Moreover, by (H) and Sobolev embedding theorem, we have

This implies

By (2.2), we get

Thus, for all

For each

Lemma 6. Let

i) If

ii) If

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),

We get a contradiction. Therefore,

there exists

which contradicts the fact that

Next, we establish the existence of a local minimum for J on

Lemma 7. For all

i)

ii)

Proof. If

This implies

Moreover, by (H) and (2.3) we obtain

where,

such that

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

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

where

and

and there exists

Lemma 9. There exist M,

and any

Proof. Let

where

and choosing

Lemma 10. Suppose

i) we have

ii) there exists

Proof. We can suppose that the minima of J are realized by

i) By (2.3), (5.1) and the fact that

Exploiting the function

ii) Let

Letting

we obtain

Let

and

Proof of Theorem 3.

If

then, by the Lemmas 2 and Proposition 1 ii), J verifying the Palais-Smale condition in

Thus

Mohammed El MokhtarOuld El Mokhtar, (2015) On Elliptic Problem with Singular Cylindrical Potential, a Concave Term, and Critical Caffarelli-Kohn-Nirenberg Exponent. Applied Mathematics,06,1891-1901. doi: 10.4236/am.2015.611166