Existence and Uniqueness of Positive Solution for a Fractional Dirichlet Problem with Combined Nonlinear Effects in Bounded Domains

and Applied Analysis 3 In the elliptic case (i.e., α = 2), problems related to (14) have been studied by several authors (see, e.g., [34–39] and references therein). Using the subsupersolution method, the authors in [36] have established the existence and uniqueness of a positive continuous solution to (14) for α = 2, σ 1 , σ 2 < 1, where the functions a 1 , a 2 are required to satisfy some adequate assumptions related to the Karamata classK. Here, our goal is to study problem (14) for 0 < α < 2. To this end, we assume that the potential functions a 1 , a 2 satisfy the following hypothesis. (H) for i ∈ {1, 2}, a i ∈ C γ loc(D), 0 < γ < 1, and satisfies, for x ∈ D, a i (x) ≈ (δ (x)) −λ iL i (δ (x)) , (15) where λ i < (α/2)(1+σ i )+1−σ i and L i ∈K defined on (0, η] with η > diam(D). As it turns out, estimates (12) depend closely on min(α/2, (α−λ)/(1−σ)). Also, as it will be seen, the numbers β 1 := min( 2 , α − λ 1 1 − σ 1 ) , β 2 := min( 2 , α − λ 2


Introduction
In the last two decades, several studies have been performed for the so-called fractional Laplacian, (−Δ) /2 , 0 <  < 2, which can be defined by the integral representation
They have proved that problem (3) has a positive continuous solution  in  satisfying, for each  ∈ , where    (, ) denotes the Green function of the fractional Laplacian (−Δ) /2 in .However they have not investigated the asymptotic behavior of such solution.
As a typical example of function  satisfying (A 1 ) and (A 2 ), we quote (, ) = ()  , where  ≤ 0 and  is a positive measurable function in  such that the function belongs to the Kato class   () defined as follows.
Definition 1 (see [26] It has been proved in [26] that the function For more examples of functions belonging to   (), we refer to [26].Note that for the classical case (i.e.,  = 2) the class  2 () was introduced and studied in [27].
In the present paper, we aim at studying the following fractional nonlinear problem involving both singular and sublinear nonlinearities with the reformulated Dirichlet boundary condition: where 0 <  < 2 and  1 ,  2 ∈ (−1, 1).We will address the question of existence, uniqueness, and global behavior of a positive continuous solution to problem (14).
Here, our goal is to study problem (14) for 0 <  < 2. To this end, we assume that the potential functions  1 ,  2 satisfy the following hypothesis.
(H) for  ∈ {1, 2},   ∈   loc (), 0 <  < 1, and satisfies, for  ∈ , where As it turns out, estimates ( 12) depend closely on min(/2, ( − )/(1 − )).Also, as it will be seen, the numbers play an important role in the combined effect of singular and superlinear nonlinearities in ( 14) and lead to a competition.It is not obvious which wins, essentially in the estimates of solution.From here on and without loss of generality, we may assume that ( −  1 )/(1 −  1 ) ≤ ( −  2 )/(1 −  2 ) and we introduce the function  defined on (0, ) by For an explicit form of the function , see (36).Throughout this paper, we define the potential kernel    by where  + () denotes the set of the nonnegative Borel measurable functions in .
Our main results are the following.
) and assume ().Then one has, for  ∈ , Using Theorem 3 and the Schauder fixed-point theorem, we will prove the following.
In particular, we generalize the result obtained in [36] to the fractional setting and we recover the result obtained in [28].
The content of this paper is organized as follows.In Section 2, we collect some properties of functions belonging to the Karamata class K and the Kato class   (), which are useful to establish our results.In Section 3, we prove our main results.
As usual, we denote by  0 () the set of continuous functions in  vanishing continuously on .Note that  0 () is a Banach space with respect to the uniform norm ‖‖ ∞ = sup ∈ |()|.As in the elliptic case, if  ∈  + () satisfies ∫  (()) /2 () < ∞, then the functions  and     are in  1 loc () and we have in the distributional sense 2. The Karamata Class K and the Kato Class   () We collect in this paragraph some properties of the Karamata class K and the Kato class   ().We recall that a function  defined on (0, ] belongs to the class K if where  > diam(),  > 0, and  ∈ ([0, ]) such that (0) = 0.
(iii) Let  ∈ K and  > 0. Then one has Applying Karamata's theorem (see [30,31]), we get the following.Lemma 6.Let  ∈ R and let  be a function in K.One has the following: Lemma 7 (see [36]).Let  be a function in K. Then one has In particular Proposition 8 (see [40,41]).For (, ) ∈  × , one has is uniformly bounded and equicontinuous in .Consequently Λ  is relatively compact in  0 ().

Proofs of the Main Results
In this section we aim at proving Theorems 3 and 4. To this end, we need the following lemmas.
Now we are ready to prove our main results.