Medical, Pharma, Engineering, Science, Technology and Business

**Mohamed El Mokhtar Ould El Mokhtar ^{*}**

Departement of Mathematics, College of Science, Qassim University, Kingdom of Saudi Arabia

- *Corresponding Author:
- Mohamed El Mokhtar Ould El Mokhtar

Departement of Mathematics, College of Science

Qassim University, Kingdom of Saudi Arabia, BO 6644, Buraidah: 51452

**Tel:**+966 16 380 0050

**E-mail:**[email protected]

**Received Date:** July 18, 2015 **Accepted Date:** October 19, 2015 **Published Date:** October 30, 2015

**Citation:** Ould El Mokhtar MEM (2015) Existence and Nonexistence for Elliptic Equation with Cylindrical Potentials, Subcritical Exponent and Concave Term. J Astrophys Aerospace Technol 3:126. doi:10.4172/2329-6542.1000126

**Copyright:** © 2015 Ould El Mokhtar MEM. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Visit for more related articles at** Journal of Astrophysics & Aerospace Technology

In this paper, we consider the existence and nonexistence of non-trivial solutions to elliptic equations with cylindrical potentials, concave term and subcritical exponent. First, we shall obtain a local minimizer by using the Ekeland’s variational principle. Secondly, we deduce a Pohozaev-type identity and obtain a nonexistence result.

Existence; Nonexistence; **Elliptic equation**; Nontrivial solutions

In this paper we study the **existence**, multiplicity and nonexistence of **nontrivial solutions **of the following problem

where *k* and *N* be integers such that and k belongs to , Sobolev exponent, *μ* > 0,,,1 < *q* < 2, g is a bounded function on , *λ* and *β* are parameters which we will specify later.

We denote point *x *in by the pair , and , the closure of with respect to the norms

We define the weighted Sobolev space with *b* = *aγ*, which is a Banach space with respect to the norm defined by

My motivation of this study is the fact that such equations arise in the search for solitary waves of nonlinear evolution equations of the Schrödinger or Klein-Gordon type [1-3]. Roughly speaking, a solitary wave is a nonsingular solution which travels as a localized packet in such a way that the physical quantities corresponding to the invariances of the equation are finite and conserved in time. Accordingly, a solitary wave preserves intrinsic properties of particles such as the energy, the angular momentum and the charge, whose finiteness is strictly related to the finiteness of the L2- norm. Owing to their particle-like behavior, solitary waves can be regarded as a model for extended particles and they arise in many problems of mathematical physics, such as classical and quantum field theory, nonlinear optics, fluid mechanics and plasma physics [4].

Several existence and nonexistence result are available in the case *k* = *N*, we quote for example [5,6] and the reference therein. When *μ* = 0 *g*(*x*) ≡1 , problem has been studied in the famous paper by Brézis and Nirenberg [7] and B. Xuan [8] which consider the existence and **nonexistence** of nontrivial solutions to quasilinear Brézis- Nirenberg-type problems with singular weights.

Concerning existence result in the case *k* < *N* we cite [9,10], and the reference therein. As noticed in [11], for a = 0 and λ = 0, M. Badiale et al. has considered the problem . She established the nonexistence of nonzero classical solutions when and the pair (*β*, *γ*) belongs to the region. i.e: where

Since our approach is variational, we define the functional on * by*

We say that is a weak solution of the problem if it is a nontrivial nonnegative function and satisfies

=0, for .

Concerning the perturbation *g* we assume

In our work, we prove the existence of at least one critical points of by the Ekeland’s variational in [12]. By the Pohozaev type identities in [12], we show the nonexistence of positive solution for our problem.

We shall state our main result

**Theorem 1** *Assume , β=2, 0<a<1<q<2 and (G) hold.*

If *,*then there exist Λ_{0} and Λ^{∗} such that the problem has at least one nontrivial solution for any *.*

**Theorem 2 ,**

If ** with ****, ***λ* < 0 and* 1<q<2*, then ** has no positive solutions.**

This paper is organized as follows. In Section 2, we give some preliminaries. Section 3 is devoted to the proof of Theorem 1. Finally in the last section, we give a nonexistence result by the proof of Theorem 2.

We list here a few integrals inequalities. The first inequality that we need is the weighted Hardy inequality [13]

The starting point for studying is the Hardy-Sobolev- Maz’ya inequality that is peculiar to the cylindrical case *k* < *N* and that was proved by Maz’ya in [14]. It state that there exists positive constant C_{γ} such that

for *μ* = 0 equation of**is related to a family of inequalities given by Caffarelli, Kohn and Nirenberg [15], for any **** .** The embedding

**Definition 1 **Assume ** ** and

is achieved on *.*

**Lemma 1** *Let* *be a Palais-Smale sequence ( (PS ) _{δ} for short) of I_{2,λ,μ} such that*

**(1)**

for some **.Then if ****,** in and **.**

*Proof*. From (*??*), we have

and

Where *o _{n}*

If **then, (***u _{n}*) is bounded in .Going if necessary to a subsequence, we can assume that there exists such that

Consequently, we get for all

which means that

Firstly, we require following Lemmas

**Lemma 2 ***Let be a (PS ) _{δ} sequence of I_{2,λ,μ} for some .Then,*

and either

Proof. We know that (*u _{n}*) is bounded in . Up to a subsequence if necessary, we have that

Denote *, then **.* As in Brézis and Lieb [16], we have

and

From Lebesgue theorem and by using the assumption (*G*), we obtain

Then, we deduce that

and

From the fact that * in ,we can assume that*

Assume α > 0, we have by definition of ,

and so

Then, we get

Therefore, if not we obtain *α* = 0. i.e *u _{n}* →

**Lemma 3** *Suppose *2* < k ≤ N , and (G) hold. There exist Λ ^{∗} > 0 such that if λ > Λ^{∗} , then there exist ρ and v positive constants such that,*

*i)* there exist *such that *

*ii)* we have

Proof. *i)* Let *t _{0}* > 0,

* Choosing **then, if *

< 0

Thus, if *, we obtain that *

ii) By the Holder inequality and the definition of* S _{μ γ}* and since

If λ > 0, then there exist *v* > 0 and *ρ _{0}>0* small enough such that

We also assume that *t _{0}* is so small enough such that

Thus, we have

Using the Ekeland’s variational principle, for the complete metric space * with respect to the norm of , we can prove that there exists a **sequence **such that **u _{n} →u_{1}* for some

Now, we claim that *u _{n} →u_{1}*. If not, by Lemma 2, we have

*>c _{1}*,

which is a contradiction.

Then we obtain a critical point *u _{1}* of

**Proof of Theorem 1**

*Proof.* From Lemmas 2 and 3, we can deduce that there exists at least a nontrivial solution *u _{1}* for our problem

By a Pohozaev type identity we show the nonexistence of positive solution of*when **β* ∈(2,3), *with*

*, **λ* < 0, 1<*q*<2 and (G) hold with 0<*a*<1.

First, we need the following Lemma

**Lemma 4** *Let be a positive solution of and. Then the following identity holds*

*Proof.* [we shall state the similar proof of proposition 30 and Lemma 31 in [11]].

1) Multiplying the equation of *by the inner product **and integrating on , we obtain*

*(2)*

2) By multiplying the equation of*by **u*, using the identity

in *and applying the divergence theorem on , we obtain*

*(3)*

From (3), we have

*(4)*

Combining (*??*) and (*??*), we obtain

**Proof of Theorem 2**.We proceed by contradictions.

From Lemma 4, since (G) hold and 1<*q*<2 therefore, if *β *∈ (2,3),

*with **we obtain that λ > 0 what contradicts the fact that λ > 0.*

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