Existence and Nonexistence for Elliptic Equation with Cylindrical Potentials, Subcritical Exponent and Concave Term

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 L2norm. 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].


Introduction
In this paper we study the existence, multiplicity and nonexistence of nontrivial solutions of the following problem , 1 < q < 2, g is a bounded function on  N , λ and β are parameters which we will specify later.
We denote point x in  N by the pair ( ) 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][2][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 L 2 -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 = β λ µ  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.
Since our approach is variational, we define the functional 2, , We say that u∈ is a weak solution of the problem ( ) Concerning the perturbation g we assume In our work, we prove the existence of at least one critical points of 2, , I λ µ 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 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 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 ii) By the Holder inequality and the definition of , S µ γ and since > 2 γ , we get for all

Nonexistence Result
By a Pohozaev type identity we show the nonexistence of positive solution of ( ) 2) By multiplying the equation of ( ) we obtain that λ > 0 what contradicts the fact that λ > 0.