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In this paper, we consider the existence and nonexistence of nontrivial 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 Pohozaevtype identity and obtain a nonexistence result.
Keywords 
Existence; Nonexistence; Elliptic equation; Nontrivial solutions 
Introduction 
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 KleinGordon type [13]. 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 particlelike 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 Nirenbergtype 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 , 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. 
Preliminaries 
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 HardySobolev 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 ofis related to a family of inequalities given by Caffarelli, Kohn and Nirenberg [15], 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 S_{μ,γ }defined by 
is achieved on . 
Lemma 1 Let be a PalaisSmale 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}_{ }(1) denotes o_{n}(1) →0 as n →. Then, 
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 
Existence Result 
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} → 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 t_{0} > 0, t_{0} small and such that 
Choosing then, if 
< 0 
Thus, if , we obtain that 
ii) By the Holder inequality and the definition of S_{μ γ} and since γ > 2 , we get for all 
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 u_{1} with 
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 for all 
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 with positive energy [1719]. 
Nonexistence Result 
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 [11]]. 
1) Multiplying the equation of by the inner product and integrating on , we obtain 
(2) 
2) By multiplying the equation ofby 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. 
References 
