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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
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.

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Abstract

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.

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

equation

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

We denote point x in equation by the pair equation, equation and equation, the closure of equationwith respect to the norms

equation

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

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 equationhas 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 equation. She established the nonexistence of nonzero classical solutions when equation and the pair (β, γ) belongs to the region. i.e: equationequation where

equation

equation

equation

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

equation

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

equation

=0, for equation.

Concerning the perturbation g we assume

equation

In our work, we prove the existence of at least one critical points of equation 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 equation, equationβ=2, 0<a<1<q<2 and (G) hold.

If equation,then there exist Λ0 and Λ such that the problem equation has at least one nontrivial solution for any equation.

Theorem 2 equation, 0<a<1 and (G) hold.

If equationequation with equation, λ < 0 and 1<q<2, then equation 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]

equation

The starting point for studying equationis 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

equation

for μ = 0 equation ofequationis related to a family of inequalities given by Caffarelli, Kohn and Nirenberg [15], for any equation. The embedding equationis compact where b=aγ and equationis the weighted Lγ space with respect to the norm

equation

Definition 1 Assume equation equation and equation. Then the infimum Sμ,γ defined by

equation

is achieved on equation.

Lemma 1 Letequation be a Palais-Smale sequence ( (PS )δ for short) of I2,λ,μ such that

equation(1)

for some equation.Then if equation,equation in equationand equation.

Proof. From (??), we have

equation

and

equation

Where on (1) denotes on(1) →0 as n →. Then,

equation

equation

equation

If equationthen, (un) is bounded in equation.Going if necessary to a subsequence, we can assume that there exists equation such that

equation

equation

equation

equation

Consequently, we get for all equation

equation

which means that

equation

Existence Result

Firstly, we require following Lemmas

Lemma 2 Let equation be a (PS )δ sequence of I2,λ,μ for some equation.Then,

equation

and either

equation

Proof. We know that (un) is bounded in equation. Up to a subsequence if necessary, we have that

equation

equation

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

equation

and

equation

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

equation

Then, we deduce that

equation

and

equation

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

equation

Assume α > 0, we have by definition of ,equation

equation

and so

equation

Then, we get

equation

Therefore, if not we obtain α = 0. i.e unu in equation.

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

i) there exist equationsuch that equation

ii) we have

equation

Proof. i) Let t0 > 0, t0 small and equationsuch that

equation Choosing equationthen, if equation

equation

equation

< 0

Thus, if equation, we obtain that equation

ii) By the Holder inequality and the definition of Sμ γ and since γ > 2 , we get for all equation

equation

equation

If λ > 0, then there exist v > 0 and ρ0>0 small enough such that

equation

We also assume that t0 is so small enough such that equation

Thus, we have

equation

Using the Ekeland’s variational principle, for the complete metric space equation with respect to the norm of equation, we can prove that there exists a equationsequence equationsuch that un →u1 for some u1 with equation

Now, we claim that un →u1. If not, by Lemma 2, we have

equation

equation

>c1,

which is a contradiction.

Then we obtain a critical point u1 of equation for all equation

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 equation with positive energy [17-19].

Nonexistence Result

By a Pohozaev type identity we show the nonexistence of positive solution ofequationwhen β ∈(2,3), equationwith

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

First, we need the following Lemma

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

equation

equation

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

1) Multiplying the equation of equationby the inner product equationand integrating on equation, we obtain

equation(2)

equation

2) By multiplying the equation ofequationby u, using the identity

equation

in equationand applying the divergence theorem on equation, we obtain

equation(3)

From (3), we have

equation(4)

equation

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

equation

equation

Proof of Theorem 2.We proceed by contradictions.

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

equationwith equationwe obtain that λ > 0 what contradicts the fact that λ > 0.

References

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