Existence of Solutions for a Fractional and Non-Local Elliptic Operator

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. Citation: Chamekh M (2016) Existence of Solutions for a Fractional and Non-Local Elliptic Operator. J Appl Computat Math 5: 335. doi: 10.4172/21689679.1000335


Introduction
Consider the fractional and p-laplacian elliptic problem We assume that the Ω is a bounded domain in  n and ∂Ω its smooth boundary, p≥2, (0,1),1 < < < < = np q p r p n p γ γ * ∈ − if n > p and p * = ∞ else, φ(t),Ψ(t) ∈ C(Ω). α > 0 and the fractional p-laplacian operator may be defined for p∈(1,∞) as 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][7][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][12][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][17][18][19] for further detail.
For all h∈C(Ω), we consider the following properties For r∈[1∞], we consider ||.|| r the norm of L y (Ω). For all measurable functions u: n →, we define the Gagliardo seminorm, by The energy functional associated to the problem (1) is given by (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 The aim in the following to provide an existence result.
The proof of the last theorem comprises basically of a simple few stages. 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 ( ) = 0 u F s ′ . Thus, we consider the follow parts  α into three parts corresponding to relative minima, relative maxima and points of inflection.  For studying the fiber map F u correspond to the sign of I φ and I Ψ , then, four possible cases can occur: • If u∈ − ∩ − , then, F u (0) = 0 and ( ) > 0, > 0 u F t t ′ ∀ which implies that F u is strictly increasing, this resulting the absence of critical points.
We have the following result: Proof. Let u α − ∈  , then F u has a positive absolute maximum at  the value of δ is given in Lemma 2.5. which is independent of u. We now show that there exists µ 0 > 0 such that F u (T) > 0. Using condition g satisfying ( 1 - 2 ) and the Sobolev imbedding, we get Proof: Let where K is given by (3).

Theorem 2.8: We have the following results
 α has reached its minimum on α +  and its maxima on α −  Proof: To prove the theorem we proceed in two steps Step 1: Since  α is bounded below on  α and so on α  On the other hand, since {u k }⊂ α , then we have Next we claim that u k →u α . Suppose this is not true, then || || < lim || || . inf