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ISSN: 2168-9679
Journal of Applied & Computational Mathematics
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Existence of Solutions for a Fractional and Non-Local Elliptic Operator

Mourad Chamekh*

Mathematics Department, Faculty of Sciences and Arts-Alkamel, University of Jeddah, Jeddah, Saudi Arabia

*Corresponding Author:
Chamekh M
Mathematics Department
Faculty of Sciences and Arts-Alkamel
University of Jeddah, Jeddah, Saudi Arabia
Tel: +216 98562268
E-mail: chamekhmourad1@gmail.com

Received Date: October 12, 2016 Accepted Date: December 26, 2016 Published Date: December 30, 2016

Citation: Chamekh M (2016) Existence of Solutions for a Fractional and Non-Local Elliptic Operator. J Appl Computat Math 5: 335. doi: 10.4172/2168-9679.1000335

Copyright: © 2016 Chamekh M. 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 a fractional and p-laplacian elliptic equation. In order to study this problem, we apply the technique of Nehari manifold and fibering map, which permit treating the existence of nontrivial solutions of a fractional and nonlocal equation, satisfies the homogeneous Dirichlet boundary conditions.

Keywords

Nontrivial solutions; Fractional p-laplacian equation; Nehari manifold

Classification: 35J35, 35J50, 35J60

Introduction

Consider the fractional and p-laplacian elliptic problem

Equation (1)

We assume that the Ω is a bounded domain in Equation and ∂Ω its smooth boundary, Equation andEquation

and the fractional p-laplacian operator may be defined for p∈(1,∞) as

Equation (2)

Over the recent years, numerous scientists have been attracted by the fractional and or p-laplacian equations. In fact, a few great models have been upgraded considerably for satisfactory answers to the modelling issues. We mention as examples the fractional Navies Stokes equations [1], fractional transport equations [2] and fractional Schrödinger equations [3], integral equations of fractional order [4,5]. Generally, a large variety of applications leads to these types of equations in ecology, elasticity and finance [6-8]. Despite significant progress in the field, and because of the difficulty to find an exact solution, research projects are still ongoing.

In this paper, we will think about the partial and p-laplacian elliptic equation (1). A considerable measure has been given for to explore this type of problems as of late. We can discover comparative equations in the many works where the issue of the existence of solutions has been dealt with. For instance, in [9], a local operator issue has been treated with φ(t) = Ψ(t) = 1. In addition, in [10] we have comes a class of Kirchhoff sort having a similar right-hand-side term that in the problem (1). See likewise [11] for a late consideration of the fractional and p-laplacian elliptic issue with φ = Ψ = 0. In this case, the solution u called a γ-p-harmonic function. Partial Laplacian equations satisfy the homogeneous Dirichlet boundary has been as of late considered in [9,11-13], using variational techniques. The existence of solutions has been considered at Ghanmi [14] utilizing a right-hand-side term of the treated condition comprises a homogeneous map, yet at the same time positive. Moreover, Xiang et al. in [15], use non-negative weight functions with the same issue. Here, we have treated the issue with sign-changing weight functions, and we proposed another proof for the existence of solutions. In view of the disintegration of the Nehari manifold is by all accounts less demanding. The remainder of this paper is organized as follows. In section 2, a few preliminaries are presented, in section 3 we explore the principle comes about.

Preliminaries

We start with some preliminaries on the notation we will use in this report. See Ghanmi A, Nezza ED, Brown KJ, [16-19] for further detail.

For all h∈C(Ω), we consider the following properties

Equation

Equation

For r∈[1∞], we consider Equation the norm of Ly(Ω). For all measurable functions Equation we define the Gagliardo seminorm, by

Following Di Nezza [16], we consider the fractional Sobolev space

Equation

with the norm defined by

Equation

We consider, thereafter, the closed subspace

Equation

with the norm Equation It is easy to verify thatEquation is a uniformly convex Banach space and that the embedding Equation is continuous for all Equation and compact for allEquation The dual space of Equation is denoted byEquation andEquation denotes the usual duality between S and S*

We define a weak solutions by,

Definition 2.1: A function u is a weak solution of (1) in S; if for every v∈S we have:

Equation

The energy functional associated to the problem (1) is given by

Equation

The functional εα is frechet differentiable. We have Equation if u is a weak solution in S of (1). Then, the weak solutions of (1) are critical points of the functional ɛα. The energy functional ɛα is unbounded below on the space S. Besides, this will certainly require the construction of an additional subset 𝓕α of S, where the functional ɛ is bounded. To accomplish this end, we will study the following Nehari manifold to ensure that a solution exists

Equation

Then, u ∈𝓕αif and only if

Equation (3)

The aim in the following to provide an existence result.

Theorem 2.2: If f and g satisfying (Ƥ1 – Ƥ2). Then, there exists α0 > 0 such that for all 0 < α < α0, problem (1) has at least two nontrivial solutions.

The proof of the last theorem comprises basically of a simple few stages.

Lemma 2.3:ɛα is coercive and bounded bellow on 𝓕α.

Proof: Let u ∈𝓕α, then, we have

Equation

Equation

Equation

Hence, ɛαis bounded bellow and coercive on 𝓕α.

We define fiber maps Fu:[0,∞)→R according Drabek P and Brown [17,20] by,

Equation

These fiber maps Fu Act as an important use in the proof because the Nehari manifold is closely linked to the behavior for them.

For u∈S, we can denote that tu∈𝓕α if and only if Equation Thus, we consider the follow parts 𝓕αinto three parts corresponding to relative minima, relative maxima and points of inflection.

Equation and Equation

We need to define mu:[0,∞)→R by

Equation

Clearly, for s > 0, su∈𝓕αif and only if s is a solution of

Equation

We consider the following subsets

Equation

and Equation

with Equation

For studying the fiber map Fu correspond to the sign of Iφ and IΨ, then, four possible cases can occur:

Equation then, Fu(0) = 0 and Equation which implies that Fu is strictly increasing, this resulting the absence of critical points.

Equation As we have mu(s1) = 0. Here the only critical point of Fu is s1, which is a absolute minimum point. Hence Equation

Equation exists μ0 > 0 such that for α ∈ (0, μ0), Fu has exactly one relative minimum s1and one relative maxima s2. Thus Equation and Equation

We have the following result:

Corollary 2.4: If α < μ0, then, there exists δ1 > 0 such that α (u) > δ1 for all u Equation

Proof. Let Equation then Fu has a positive absolute maximum at

Equation then we have

Equation

the value of δ is given in Lemma 2.5.

Lemma 2.5: There exists 0 > 0 such that for α∈(0,μ00, Fusub> take positive value for all non-zero u∈S. Moreover, if

Equation then, Fu has exactly two critical points. Proof. Let u∈S, define

Equation

Then,

Equation

If Equation reaches its maximum value at

Equation

For Equation we denoted by Sv be the Sobolev constant of embedding Equation then, by 3 we have

Equation

which is independent of u. We now show that there exists μ0 > 0 such that Fu(T) > 0. Using condition g satisfying (Ƥ1Ƥ2) and the Sobolev imbedding, we get

Equation

Thus

Equation

where δ is the constant given in (4).

Equation (5)

Then, choice of such μ0 completes the proof.

Lemma 2.6: There exists 1 such that if 0 < α < μ1, thenEquation

Proof: Let

Equation

where K is given by (3).

Suppose otherwise, that 0 < α < μ1 such that Equation Then, forEquation we have

Equation

So, it follows from (3) that

Equation

and so

Equation (6)

On the other hand, by (3) we get

Equation

Equation

then

Equation (7)

Combining (6) and (7) we obtain α≥1, which is a contradiction.

Lemma 2.7: Let u be a relative minimizer for &jukcy; єα on subsets Equation orEquation then u is a critical point of єα.

Proof: Since u is a minimizer for єα under the constraint Equation by the theory of Lagrange multipliers, there exists μ∈Rsuch thatEquation Thus:

Equation

but Equation and soEquation Hence μ = 0 completes the proof.

In the remain of this section, we assume that the parameter α satisfies 0 < α < α0, where α0 is constant. That leads us consequently to the following results on the existence of minimizers in Equation for α ∈(0,α0).

Theorem 2.8: We have the following results

єαhas reached its minimum on Equation and its maxima onEquation

Proof: To prove the theorem we proceed in two steps

Step 1: Since єα is bounded below on 𝓕α and so on Equation there exists a minimizing sequence Equation such that

Equation

As єαis coercive on Equation is a bounded sequence in S. Therefore, for all Equation we have

Equation

If we choose u ∈ S such that Equation then, there exists s1 > 0 such that Equation andEquation Hence, infEquation

On the other hand, since Equation then we have

Equation

and so

Equation

Letting k to infinity, we get

Equation (8)

Next we claim that Equation Suppose this is not true, then

Equation

Since Equation it follows thatEquation for sufficiently large k.So, we must have s1 > 1 but Equation and so

Equation

which is a contradiction. It leads to Equation and soEquation since Equation Finally, uα is a minimizer for Equation

Step 2: Let Equation then from corollary 2.4, there exists δ1 > 0 such thatEquation So, there exists a minimizing sequenceEquation such that

Equation (9)

On the other hand, since ɛα is coercive, Equation is a bounded sequence in S. Therefore, for all Equation we have

Equation

Since u∈𝓕α then we have

Equation (10)

Letting k to infinity, it follows from (9) and (10) that

Equation (11)

Conclusion

Hence, Equation and soEquation has a absolute maximum at some point T and consequently, Equation on the other hand,Equation implies that 1 is a absolute maximum point for Equation i.e.

Equation (12)

Next we claim that Equation Suppose it is not true, then

Equation

it follows from (12) that

Equation

which is a contradiction. Hence, Equation and soEquation sinceEquation

Now, Let us proof Theorem 1.1: By the Lemmas 2.5, 2.6, 2.7 and the theorem 2.8 the problem (1) has two weak solution Equation andEquation On the other hand, from (8) and (11), this solutions are nontrivial. Since Equation then, uα and vα are distinct.

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