UNIQUENESS AND EXISTENCE OF SOLUTIONS FOR A SINGULAR SYSTEM WITH NONLOCAL OPERATOR VIA PERTURBATION METHOD

In this work, we investigate the existence and the uniqueness of solutions for the nonlocal elliptic system involving a singular nonlinearity as follows:  (−∆p)u = a(x)|u|q−2u+ 1−α 2−α−β c(x)|u| −α|v|1−β , in Ω, (−∆p)v = b(x)|v|q−2v + 1−β 2−α−β c(x)|u| 1−α|v|−β , in Ω, u = v = 0, in R \ Ω, where Ω is a bounded domain in R with smooth boundary, 0 < α < 1, 0 < β < 1, 2 − α − β < p < q ≤ ps = Np N−sp , a, b, c ∈ C(Ω) are non-negative weight functions with compact support in Ω, and (−∆)p is the fractional p-laplacian operator. We use a perturbation method combine with some variationals methods in order to show the existence of a solution to the above system. We also prove the uniqueness of the solution to the system for some additional condition.


Introduction
The purpose of this paper is to study the following elliptic system involving nonlocal operator and singular nonlinearity: where Ω is an open bounded domain of R N with smooth boundary, N > ps, s ∈ (0, 1), 1 < p < ∞ and (−∆ p ) s is the fractional p-laplacian operator which is defined as (−∆ p ) s u(x) = C N,s P.V.
During the past few decades, the study of nonlocal elliptic PDEs involving singularity with Dirichlet boundary condition has drawn interest by many researchers, both from a pure mathematical point of view and for concrete applications, since the problems of this type are important in many fields of sciences, notably the fields of physics, probability, finance, electromagnetism, astronomy, and fluid dynamics, it also they can be used to accurately describe the jump Lévy processes in probability theory and fluid potentials for more details see [1,6] and references therein.
Set α = β, α + β = γ and u = v. Then the problem (1.1) becomes the following nonlocal singular problem where N > ps, M ≥ 0, c : Ω → R is a nonnegative bounded function. When M = 0 and p = 2 (the purely singular problem), Fang [8] had shown that the problem (1.2) has a unique solution u ∈ C 2,α (Ω) for 0 < α < 1. In [12], the multiplicity result for the problem (1.2) is proved by converting the nonlocal problem to a local problem. For 1 < p < ∞, M = 0 and λ = 1 the problem (1.2) was studied by Canino et al. [5]. In [9,10], the authors established the existence and the multiplicity of weak solutions to the problem (1.2) by using the Nehari manifold method. Recently, Saoudi et al. in [14] has guaranteed the existence of at least two solutions by using min-max method with the help of modified Mountain Pass theorem. In particular, in [13], the author considered the following nonlocal problem ps is the fractional Sobolev exponent, λ, µ are two parameters, a, b, c ∈ C(Ω) are non-negative weight functions with compact support in Ω. With the help of the Nehari manifold and the fibering maps (appropriately modified), the authors proved the existence of at least two non-negatives solutions of problem (1.3).
Motivated by above results, in the present work, we are interested in the existence and the uniqueness of solutions for nonlocal system (1.1) via perturbed method's.
In order to state our result, let us introduce some notations. We define For a detailed account on the properties of W s,p (R N ) we refer the reader to [11].
and we define the space with the norm Through this paper we consider the space with the norm It is readily seen that (X 0 , ||.||) is a uniformly convex Banach space and that the embedding X 0 → L q (Ω) is continuous for all 1 ≤ q ≤ p * s , and compact for all 1 ≤ q < p * s . The dual space of (X 0 , ||.||) is denoted by (X * , ||.|| * ), and ⟨., .⟩ denotes the usual duality between X 0 and X * .
Let X 0 = X 0 × X 0 be the Cartesian product of two Hilbert spaces, which is (In [6] it is claimed that X 0 is a Hilbert space) a reflexive Banach space endowed with the norm Before stating our main results, we make the following assumptions throughout this paper: in Ω, α = β and 1 2 < α < 1. We now list out the results that we will prove in this work.  This paper is organized as follows: The Section 2 is devoted to study approximated system. While, existence of solution (Theorem 1.1) and uniqueness of solution (Theorem 1.2) will be presented in Section 3 and in Section 4 respectively.

The approximated fractional system
In this section, we introduce the following approximated system, for a fixed n > 1   Associated to the approximated problem (2.1), we define the functional E n : X 0 → R by Notice that E n is a C 1 functional and obviously, any critical point of E n is a weak solution of the problem (2.1).
. Before, given the first result in this section let us recall the following inequality for all d, e ∈ R + and m ∈ (0, 1). On the other hand, using Hölder's inequality and Sobolev inequalities, one has The functional E n is coercive and bounded below in X 0 .
Proof. From the Hölder's and Sobolev inequalities combine with Eq. (2.2) and Eq. (2.3), we obtain On the other hand, it is very simple to see that 1 ≤ max(r 2 , r 3 ) ≤ r 1 ≤ N sp . Hence, E n is coercive and bounded below on X 0 .
Consider the following minimization problem Hence, from Lemma 2.1 combine with the Ekeland's variational principle, we obtain the existence of the sequences Proof. At first, it is simple to see from the coerviness of the functional energy E n , the boundedness of the sequence By dominated convergence theorem, we claim that in Ω. Therefore, by Dominated convergence theorem, our claim is true. Similarly, On the other hand, note that {u k } is bounded, by the Sobolev embedding theorem, so there exists a constant C > 0 such that (2.7) Thus, using Eq. (2.7), there exists l 1 ∈ L (1−α) (Ω), and up to a subsequence, Now, apply the Lebesque's dominated convergence theorem, we conclude that, Moreover, let p ′ the Hölder conjugate of p given by (2.10) Using the same argument, as above, we obtain Therefore, from Eqs. (2.5), (2.6), (2.8), (2.9), (2.10) and (2.11), we deduce that Now, we will prove that the sequence (u k , v k ) converge strongly to (U n , V n ) in X 0 . Indeed, since

Proof of Theorem 1.1
This section is devoted to prove the existence of a solution to the problem (1.1). The proof is done in servals Steps.
Step 1: The solutions (U n ) and (V n ) of the problem (2.1) are uniformly bounded.
Firstly, we have proved in section 2 that (U n , V n ) is a solution to the problem 2.1. Then, taking (U n , 0) as test function in the weak formulation of the problem (2.1), we obtain In the same way, taking (0, V n ) as test function in the weak formulation of the problem (2.1), we get Now, combine Eq. (3.1) with Eq. (3.2) and using Hölder and Sobolev inequalities, we deduce that which yields that (U n ) and (V n ) are uniformly bounded in X 0 . Now, let us prove a priori estimate in L ∞ (Ω) of the solution U n and V n . Similarly, for sufficiently small M > 0. Moreover, Since |s| q−2 s + |s| + 1 n −α |s| + 1 n 1−β is a uniformly nonincreasing function with respect to x ∈ Ω for sufficiently small s > 0. Also from the monotonicity of (−∆ p ) s we have, for sufficiently small M > 0, 0 ≤ ξ ′ (1) − ξ ′ (t). From the Taylor series expansion, we have ∃ 0 < θ < 1 such that .
Similarly, by using (0, (M − V n ) + ) as test function, we obtain V n ≤ M.
Step 2: The solutions (U n ) and (V n ) of the problem (2.1) are positive almost every where in Ω.
At first, taking into account that (U n , V n ) is a solution of the problem (2.1) and taking (U − n , 0) as test function, we obtain Therefore, since the right hand side of the Eq. (3.5) is nonnegative and not equivalently to 0, we can deduce from (3.6) that ||U − n || = 0. Which implies that U n is nonnegative and by the strong maximum principle we conclude that U n is positive almost every where in Ω. In the same manner, we can prove that V n is positive almost every where in Ω.
Step 3: The solutions (U n ) and (V n ) of the problem (2.1) are bounded from below.
From step 1, we know that (U n ) and (V n ) are bounded in X 0 . Therefore, by using a standard comparison argument (see Lemma 2.1 in [2]) combine with the strong maximum principle version of the fractional p-laplace operator in [4] we conclude that for all K ⊂⊂ Ω there exists C K such that U n (x) ≥ C K > 0 for a. e.
x ∈ K and for any n ∈ N. The boundedness from below of V n follows by the same manner.
Step 4: The problem (1.1) has a positive weak solution. Firstly, by Step 1, (U n ) and (V n ) are bounded in X 0 . Thus, since the space X 0 is reflexive, there exists a subsequence, still denoted by {U n } and {V n }, which weakly converges to, say, U, V ∈ X 0 such that U n → U and V n → V weakly in X 0 U n → U and V n → V strongly in L r (Ω) for 1 ≤ r < p * s U n → U and V n → V pointwise a.e. in Ω. (3.7) Now, since (U n , V n ) is a positive solution of the problem (2.1), ones has for all (φ, ψ) ∈ X 0 . Then, by the weak convergence of U n and V n to U and V respectively, we have |U n (x)−U n (y)| p−2 (U n (x)−U n (y))/|x−y| (N +sp)/p ′ is bounded in L p ′ (R 2N ) where p ′ is the Hölder conjugate of p given by p ′ = p p−1 , and converges On the other hand, since U n → U in L r (supp(φ)) and V n → V in L r ( supp(ψ)). We deduce, using the dominated convergence theorem, that Applying, Lebsgue's dominated convergence theorem  That is, and for all Φ in C ∞ c (Ω). Now, taking Φ = (u 1 − u 2 ) as a test function, we have So, from equation (4.9) and according to Lemma 4.1, we must have ||u 1 − u 2 || = 0. Therefore, we get u 1 = u 2 almost everywhere in Ω. Hence, (u 1 , u 1 ) is the unique solution to the system (4.6). This completes the proof of the Theorem 1.2.