ISSN: 2329-6542
Journal of Astrophysics & Aerospace Technology
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Existence and Nonexistence for Elliptic Equation with Cylindrical Potentials, Subcritical Exponent and Concave Term

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
Received: July 18, 2015 Accepted: October 19, 2015 Published: 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.

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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 image k and N be integers such that image and k belongs to image , image Sobolev exponent, μ > 0,image,image,1 < q < 2, g is a bounded function on image, λ and β are parameters which we will specify later.
We denote point x in image by the pair image, image and image, the closure of imagewith respect to the norms
We define the weighted Sobolev space image with b = , which is a Banach space with respect to the norm defined by image
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 imagehas 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 image. She established the nonexistence of nonzero classical solutions when image and the pair (β, γ) belongs to the region. i.e: imageimage where
Since our approach is variational, we define the functional image on image by
We say that image is a weak solution of the problem image if it is a nontrivial nonnegative function and satisfies
=0, for image.
Concerning the perturbation g we assume
In our work, we prove the existence of at least one critical points of image 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 image, imageβ=2, 0<a<1<q<2 and (G) hold.
If image,then there exist Λ0 and Λ such that the problem image has at least one nontrivial solution for any image.
Theorem 2 image, 0<a<1 and (G) hold.
If imageimage with image, λ < 0 and 1<q<2, then image 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 image 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 Cγ such that
for μ = 0 equation ofimageis related to a family of inequalities given by Caffarelli, Kohn and Nirenberg [15], for any image. The embedding image is compact where b=aγ and image is the weighted Lγ space with respect to the norm
Definition 1 Assume image image and image. Then the infimum Sμ,γ defined by
is achieved on image.
Lemma 1 Letimage be a Palais-Smale sequence ( (PS )δ for short) of I2,λ,μ such that
image (1)
for some image.Then if image,image in imageand image.
Proof. From (??), we have
Where on (1) denotes on(1) →0 as n →. Then,
If imagethen, (un) is bounded in image.Going if necessary to a subsequence, we can assume that there exists image such that
Consequently, we get for all image
which means that
Existence Result
Firstly, we require following Lemmas
Lemma 2 Let image be a (PS )δ sequence of I2,λ,μ for some image.Then,
and either
Proof. We know that (un) is bounded in image. Up to a subsequence if necessary, we have that
Denote image, then image. As in Brézis and Lieb [16], we have
From Lebesgue theorem and by using the assumption (G), we obtain
Then, we deduce that
From the fact that image in image,we can assume that
Assume α > 0, we have by definition of ,image
and so
Then, we get
Therefore, if not we obtain α = 0. i.e unu in image.
Lemma 3 Suppose 2 < k ≤ N , image 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 imagesuch that image
ii) we have
Proof. i) Let t0 > 0, t0 small and imagesuch that
image Choosing image then, if image
< 0
Thus, if image, we obtain that image
ii) By the Holder inequality and the definition of Sμ γ and since γ > 2 , we get for all image
If λ > 0, then there exist v > 0 and ρ0>0 small enough such that
We also assume that t0 is so small enough such that image
Thus, we have
Using the Ekeland’s variational principle, for the complete metric space image with respect to the norm of image, we can prove that there exists a imagesequence imagesuch that un →u1 for some u1 with image
Now, we claim that un →u1. If not, by Lemma 2, we have
which is a contradiction.
Then we obtain a critical point u1 of image for all image
Proof of Theorem 1
Proof. From Lemmas 2 and 3, we can deduce that there exists at least a nontrivial solution u1 for our problem image with positive energy [17-19].
Nonexistence Result
By a Pohozaev type identity we show the nonexistence of positive solution ofimagewhen β ∈(2,3), image with
image, λ < 0, 1<q<2 and (G) hold with 0<a<1.
First, we need the following Lemma
Lemma 4 Let image be a positive solution of image 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 image by the inner product image and integrating on image, we obtain
image (2)
2) By multiplying the equation ofimageby u, using the identity
in imageand applying the divergence theorem on image, we obtain
image (3)
From (3), we have
image (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),
image with image we obtain that λ > 0 what contradicts the fact that λ > 0.

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