Mathematics Department, Faculty of Sciences and Arts-Alkamel, University of Jeddah, Jeddah, Saudi Arabia
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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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
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
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 , fractional transport equations  and fractional Schrödinger equations , 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 , a local operator issue has been treated with φ(t) = Ψ(t) = 1. In addition, in  we have comes a class of Kirchhoff sort having a similar right-hand-side term that in the problem (1). See likewise  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  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 , 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.
For all h∈C(Ω), we consider the following properties
For r∈[1∞], we consider the norm of Ly(Ω). For all measurable functions we define the Gagliardo seminorm, by
Following Di Nezza , 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 εα 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 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
Then, u ∈𝓕αif and only if
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
Hence, ɛαis bounded bellow and coercive on 𝓕α.
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 Thus, we consider the follow parts 𝓕αinto three parts corresponding to relative minima, relative maxima and points of inflection.
We need to define mu:[0,∞)→R by
Clearly, for s > 0, su∈𝓕αif and only if s is a solution of
We consider the following subsets
For studying the fiber map Fu correspond to the sign of Iφ and IΨ, then, four possible cases can occur:
then, Fu(0) = 0 and which implies that Fu is strictly increasing, this resulting the absence of critical points.
As we have mu(s1) = 0. Here the only critical point of Fu is s1, which is a absolute minimum point. Hence
exists μ0 > 0 such that for α ∈ (0, μ0), Fu has exactly one relative minimum s1and one relative maxima s2. 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 uProof. Let then Fu 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,μ00, Fusub> take positive value for all non-zero u∈S. Moreover, if
then, Fu has exactly two critical points. Proof. Let u∈S, define
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 Fu(T) > 0. Using condition g satisfying (Ƥ1 – Ƥ2) and the Sobolev imbedding, we get
where δ is the constant given in (4).
Then, choice of such μ0 completes the proof.
Lemma 2.6: There exists 1 such that if 0 < α < μ1, then
where K is given by (3).
Suppose otherwise, that 0 < α < μ1 such that Then, for we have
So, it follows from (3) that
On the other hand, by (3) we get
Combining (6) and (7) we obtain α≥1, which is a contradiction.
Lemma 2.7: Let u be a relative minimizer for є єα on subsets or then u is a critical point of єα.
Proof: Since u is a minimizer for єα 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 < α0, where α0 is constant. That leads us consequently to the following results on the existence of minimizers in for α ∈(0,α0).
Theorem 2.8: We have the following results
єαhas reached its minimum on and its maxima on
Proof: To prove the theorem we proceed in two steps
Step 1: Since єα is bounded below on 𝓕α and so on there exists a minimizing sequence such that
As єαis coercive on is a bounded sequence in S. Therefore, for all we have
If we choose u ∈ S such that then, there exists s1 > 0 such that and Hence, inf
On the other hand, since then we have
Letting k to infinity, we get
Next we claim that Suppose this is not true, then
Since it follows that for sufficiently large k.So, we must have s1 > 1 but and so
which is a contradiction. It leads to and so since Finally, uα is a minimizer for
Step 2: Let then from corollary 2.4, there exists δ1 > 0 such that So, there exists a minimizing sequence such that
On the other hand, since ɛα is coercive, is a bounded sequence in S. Therefore, for all we have
Since u∈𝓕α then we have
Letting k to infinity, it follows from (9) and (10) that
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.
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, uα and vα are distinct.