Mohamed El Mokhtar Ould El Mokhtar*
Departement of Mathematics, College of Science, Qassim University, Kingdom of Saudi Arabia
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.
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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
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 .
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  and B. Xuan  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 , 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 . By the Pohozaev type identities in , 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 Λ0 and Λ∗ such that the problem has at least one nontrivial solution for any .
Theorem 2 , 0<a<1 and (G) hold.
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 
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 . It state that there exists positive constant Cγ such that
for μ = 0 equation ofis related to a family of inequalities given by Caffarelli, Kohn and Nirenberg , for any . The embedding is compact where b=aγ and is the weighted Lγ space with respect to the norm
Definition 1 Assume and . Then the infimum Sμ,γ defined by
is achieved on .
Lemma 1 Let be a Palais-Smale sequence ( (PS )δ for short) of I2,λ,μ such that
for some .Then if , in and .
Proof. From (??), we have
Where on (1) denotes on(1) →0 as n →. Then,
If then, (un) 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 I2,λ,μ for some .Then,
Proof. We know that (un) is bounded in . Up to a subsequence if necessary, we have that
Denote , then . As in Brézis and Lieb , we have
From Lebesgue theorem and by using the assumption (G), we obtain
Then, we deduce that
From the fact that in ,we can assume that
Assume α > 0, we have by definition of ,
Then, we get
Therefore, if not we obtain α = 0. i.e un → u in .
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 t0 > 0, t0 small and such that
Choosing then, if
Thus, if , we obtain that
ii) By the Holder inequality and the definition of Sμ γ and since γ > 2 , we get for all
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
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 un →u1 for some u1 with
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 for all
Proof of Theorem 1
By a Pohozaev type identity we show the nonexistence of positive solution ofwhen 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 ].
1) Multiplying the equation of by the inner product and integrating on , we obtain
2) By multiplying the equation ofby u, using the identity
in and applying the divergence theorem on , we obtain
From (3), we have
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.