Medical, Pharma, Engineering, Science, Technology and Business

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:**[email protected]

**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.

**Visit for more related articles at** Journal of Applied & Computational Mathematics

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.

Nontrivial solutions; Fractional p-laplacian equation; Nehari manifold

**Classification:** 35J35, 35J50, 35J60

Consider the fractional and p-laplacian elliptic problem

(1)

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

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

(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.

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

For r∈[1∞], we consider the norm of L^{y}(Ω). For all measurable functions we define the Gagliardo seminorm, by

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

with the norm defined by

We consider, thereafter, the closed subspace

with the norm It is easy to verify that is a uniformly convex Banach space and that the embedding is continuous for all and compact for all The dual space of is denoted by and 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:*

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

The functional &_{e}psilon;α is frechet differentiable. We have if u is a weak solution in S of (1). Then, the weak solutions of (1) are critical points of the functional ?_{α}. The energy functional ?_{α} is unbounded below on the space S. Besides, this will certainly require the construction of an additional subset ?_{α} of S, where the functional ? is bounded. To accomplish this end, we will study the following Nehari manifold to ensure that a solution exists

Then, u ∈?_{α}if and only if

(3)

The aim in the following to provide an existence result.

**Theorem 2.2:** If f and g satisfying (*?*1 – *?*2). Then, there exists α0 > 0 such that for all 0 < α < α_{0}, problem (1) has at least two nontrivial solutions.

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

**Lemma 2.3:**?_{α} is coercive and bounded bellow on ?_{α}.

Proof: Let u ∈?_{α}, then, we have

Hence, ?_{α}is bounded bellow and coercive on ?_{α}.

We define fiber maps *F _{u}*:[0,∞)→R according Drabek P and Brown [17,20] by,

These fiber maps F_{u} 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∈?_{α} if and only if Thus, we consider the follow parts ?_{α}into three parts corresponding to relative minima, relative maxima and points of inflection.

and

We need to define m_{u}:[0,∞)→R by

Clearly, for s > 0, s_{u}∈?_{α}if and only if s is a solution of

We consider the following subsets

and

with

For studying the fiber map F_{u} correspond to the sign of I_{φ} and I_{Ψ}, then, four possible cases can occur:

then, F_{u}(0) = 0 and which implies that F_{u} is strictly increasing, this resulting the absence of critical points.

As we have m_{u}(s_{1}) = 0. Here the only critical point of F_{u} is s_{1}, which is a absolute minimum point. Hence

exists μ_{0} > 0 such that for α ∈ (0, μ_{0}), Fu has exactly one relative minimum s_{1}and one relative maxima s_{2}. Thus and

We have the following result:

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

**Proof**. Let then F_{u} has a positive absolute maximum at

then we have

the value of δ is given in Lemma 2.5.

**Lemma 2.5:** *There exists _{0} > 0 such that for α∈(0,μ0_{0}, F_{u}sub> take positive value for all non-zero u∈S. Moreover, if*

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

Then,

If reaches its maximum value at

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

which is independent of u. We now show that there exists μ0 > 0 such that *F _{u}*(T) > 0. Using condition g satisfying (

Thus

where δ is the constant given in (4).

(5)

Then, choice of such μ_{0} completes the proof.

**Lemma 2.6: ***There exists _{1} such that if 0 < α < μ_{1}, then*

**Proof:** Let

where K is given by (3).

Suppose otherwise, that 0 < α < μ_{1} such that Then, for we have

So, it follows from (3) that

and so

(6)

On the other hand, by (3) we get

then

(7)

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

**Lemma 2.7:*** Let u be a relative minimizer for є ? _{α} on subsets* or then u is a critical point of

*Proof:* Since *u* is a minimizer for *? _{α}* under the constraint by the theory of Lagrange multipliers, there exists μ∈Rsuch that Thus:

but and so Hence *μ* = 0 completes the proof.

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

**Theorem 2.8: ***We have the following results*

*? _{α}*has reached its minimum on and its maxima on

**Proof:** To prove the theorem we proceed in two steps

Step 1: Since *? _{α}* is bounded below on ?

As *? _{α}*is coercive on is a bounded sequence in S. Therefore, for all we have

If we choose *u* ∈ S such that then, there exists s_{1} > 0 such that and Hence, inf

On the other hand, since then we have

and so

Letting k to infinity, we get

(8)

Next we claim that Suppose this is not true, then

Since it follows that for sufficiently large k.So, we must have s_{1} > 1 but and so

which is a contradiction. It leads to and so since Finally, *u*_{α} is a minimizer for

**Step 2:** Let then from corollary 2.4, there exists δ_{1} > 0 such that So, there exists a minimizing sequence such that

(9)

On the other hand, since ?_{α} is coercive, is a bounded sequence in S. Therefore, for all we have

Since u∈?_{α} then we have

(10)

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

(11)

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

(12)

Next we claim that Suppose it is not true, then

it follows from (12) that

which is a contradiction. Hence, and so since

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 and On the other hand, from (8) and (11), this solutions are nontrivial. Since then, *u*_{α} and *v*_{α} are distinct.

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