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

Abstract

We prove the existence and uniqueness of a positive continuous solution to the following singular semilinear fractional Dirichlet problem (−Δ)^𝛼/2𝑢=𝑎₁(𝑥)𝑢^𝜎1+𝑎₂(𝑥)𝑢^𝜎2, in 𝐷lim_{𝑥→𝑧∈𝜕𝐷}(𝛿(𝑥))^1−(𝛼/2)𝑢(𝑥)=0, where 0<𝛼<2,𝜎₁,𝜎₂∈(−1,1),𝐷 is a bounded 𝐶^1,1-domain in ℝ^𝑛,𝑛≥2, and 𝛿(𝑥) denotes the Euclidian distance from 𝑥 to the boundary of 𝐷. The nonnegative weight functions 𝑎₁,𝑎₂ are required to satisfy certain hypotheses related to the Karamata class. We also investigate the global behavior of such solution.

1. Introduction

In the last two decades, several studies have been performed for the so-called fractional Laplacian, (−Δ)^𝛼/2,  0<𝛼<2, which can be defined by the integral representation (−Δ)^𝛼/2𝑢(𝑥)=𝑐_𝑛,𝛼lim_{𝜀↘0(|𝑥−𝑦|>𝜀)}𝑢(𝑥)−𝑢𝑦𝑥−𝑦^𝑛+𝛼𝑑𝑦, (1) where 𝑐_𝑛,𝛼=(𝛼2^𝛼−1/𝜋^𝑛/2)(Γ((𝑛+𝛼)/2)/Γ(1−(𝛼/2))) is a normalization constant; see, for instance, [1, 2]. From a probabilistic point of view, the fractional Laplacian appears as the infinitesimal generator of the stable Lévy process [3, 4]; see also [5]. The fractional powers of the Laplacian arise in a numerous variety of equations in mathematical physics and related fields (see, for instance, [6–11] and the references therein). Motivation from mechanics appears in the Signorini problem (cf. [12, 13]). And there are applications in fluid mechanics, (cf. [14]). The systematic study of the corresponding PDE models is more recent and many of the results have arisen in the last decade. The linear or quasilinear elliptic theory has been actively studied recently in the works of Caffarelli and collaborators [15, 16], Kassmann [17], Silvestre [18], and many others. The standard linear evolution equation involving fractional diffusion is 𝜕𝑢+()𝜕𝑡−Δ^𝛼/2𝑢=0. (2) This is a model of the so-called anomalous diffusion, a much studied topic in physics, probability, and finance (see [19–23] and their references). For more applications, we refer the reader to the survey papers [24, 25].

Throughout this paper, we consider a bounded 𝐶^1,1-domain 𝐷 in ℝ^𝑛, 𝑛≥2, and we denote by 𝛿(𝑥) the Euclidian distance from 𝑥 to the boundary of 𝐷. For two nonnegative functions 𝑓 and 𝑔 defined on a set 𝑆, the notation 𝑓(𝑥)≈𝑔(𝑥),  𝑥∈𝑆, means that there exists 𝑐>0 such that (1/𝑐)𝑓(𝑥)≤𝑔(𝑥)≤𝑐𝑓(𝑥), for all 𝑥∈𝑆.

Recently, in [26], the authors considered the following problem: (−Δ)^𝛼/2𝑢=𝜑(⋅,𝑢)in𝐷(inthesenseofdistributions),𝑢>0in𝐷,lim_{𝑥→𝑧∈𝜕𝐷}(𝛿(𝑥))^1−(𝛼/2)𝑢(𝑥)=0, (3) where 0<𝛼<2 and 𝜑 is a positive measurable function in 𝐷×(0,∞) satisfying the following:(A₁) the map 𝑡→𝜑(𝑥,𝑡) is continuous and nonincreasing in (0,∞), for 𝑥∈𝐷;(A₂) for each 𝑐>0, the function 𝑥→(𝛿(𝑥))^1−(𝛼/2)𝜑(𝑥,𝑐(𝛿(𝑥))^(𝛼/2)−1) is in 𝐾^𝛼(𝐷) (see Definition 1 below).

They have proved that problem (3) has a positive continuous solution 𝑢 in 𝐷 satisfying, for each 𝑥∈𝐷, 𝑢(𝑥)=_𝐷𝐺^𝛼_𝐷𝑥,𝑦𝜑𝑦,𝑢𝑦𝑑𝑦, (4) where 𝐺^𝛼_𝐷(𝑥,𝑦) denotes the Green function of the fractional Laplacian (−Δ)^𝛼/2 in 𝐷. However they have not investigated the asymptotic behavior of such solution.

As a typical example of function 𝜑 satisfying (A₁) and (A₂), we quote 𝜑(𝑥,𝑢)=𝑎(𝑥)𝑢^𝜎, where 𝜎≤0 and 𝑎 is a positive measurable function in 𝐷 such that the function 𝑥⟶(𝛿(𝑥))^{((𝛼/2)−1)(𝜎−1)}𝑎(𝑥) (5) belongs to the Kato class 𝐾^𝛼(𝐷) defined as follows.

Definition 1 (see [26]). A Borel measurable function 𝑞 in 𝐷 belongs to the Kato class 𝐾^𝛼(𝐷) if lim_𝑟→0sup_{𝑥∈𝐷(|𝑥−𝑦|≤𝑟)∩𝐷}𝛿𝑦𝛿(𝑥)^𝛼/2𝐺^𝛼_𝐷𝑥,𝑦𝑞𝑦𝑑𝑦=0. (6)

It has been proved in [26] that the function 𝑥⟶(𝛿(𝑥))^−𝜆belongsto𝐾^𝛼(𝐷)iff𝜆<𝛼. (7) For more examples of functions belonging to 𝐾^𝛼(𝐷), we refer to [26]. Note that for the classical case (i.e., 𝛼=2) the class 𝐾²(𝐷) was introduced and studied in [27].

On the other hand, Chemmam et al. considered in [28] the following semilinear fractional Dirichlet problem: (−Δ)^𝛼/2𝑢=𝑎(𝑥)𝑢^𝜎in𝐷(inthesenseofdistributions),𝑢>0in𝐷,lim_{𝑥→𝑧∈𝜕𝐷}(𝛿(𝑥))^1−(𝛼/2)𝑢(𝑥)=0, (8) where 0<𝛼<2, 𝜎<1, and 𝑎 satisfies the following hypothesis:

(H₀)  𝑎∈𝐶^𝛾_loc(𝐷), 0<𝛾<1, satisfying 𝐷, 𝑎(𝑥)≈(𝛿(𝑥))^−𝜆𝐿(𝛿(𝑥)), (9) where 𝜆<(𝛼/2)(1+𝜎)+1−𝜎 and 𝐿 belongs to the Karamata class 𝒦 defined as follows.

Definition 2. The class 𝒦 is the set of all the Karamata functions 𝐿 defined on (0,𝜂] by 𝐿(𝑡)∶=𝑐exp^𝜂_𝑡𝑧(𝑠)𝑠𝑑𝑠, (10) where 𝜂>diam(𝐷),  𝑐>0, and 𝑧∈𝐶([0,𝜂]) such that 𝑧(0)=0.

As a typical example of a function belonging to the class 𝒦 (see [29–31]), we quote 𝐿(𝑡)=^𝑚_𝑘=1log_𝑘𝜔𝑡^−𝜉𝑘, (11) where 𝜉_𝑘 are real numbers, log_𝑘𝑥=log∘log∘⋯log𝑥(𝑘times), and 𝜔 is a sufficiently large positive real number such that 𝐿 is defined and positive on (0,𝜂].

Using a fixed-point argument, the authors have proved in [28] the existence and uniqueness of a positive continuous solution 𝑢 for (8) satisfying, for 𝑥∈𝐷, 𝑢(𝑥)≈(𝛿(𝑥))^{min(𝛼/2,(𝛼−𝜆)/(1−𝜎))}Ψ_{𝐿,𝜆,𝜎}(𝛿(𝑥)), (12) where the function Ψ_{𝐿,𝜆,𝜎} is defined on (0,𝜂) by Ψ_{𝐿,𝜆,𝜎}(𝑡)(𝐿(𝑡∶=))^1/(1−𝜎),𝛼2()<𝛼1+𝜎<𝜆2()∫1+𝜎+1−𝜎,_𝜂𝑡𝐿(𝑠)𝑠𝑑𝑠^1/(1−𝜎),𝛼if𝜆=2(),𝛼1+𝜎1,if𝜆<2(1+𝜎). (13) In particular, they have extended the results of [32, 33].

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: (−Δ)^𝛼/2𝑢=𝑎₁(𝑥)𝑢^𝜎1+𝑎₂(𝑥)𝑢^𝜎2(),in𝐷inthesenseofdistributions𝑢>0in𝐷,lim_{𝑥→𝑧∈𝜕𝐷}(𝛿(𝑥))^1−(𝛼/2)𝑢(𝑥)=0, (14) where 0<𝛼<2 and 𝜎₁, 𝜎₂∈(−1,1). We will address the question of existence, uniqueness, and global behavior of a positive continuous solution to problem (14).

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,𝜎₁,  𝜎₂<1, where the functions 𝑎₁, 𝑎₂ are required to satisfy some adequate assumptions related to the Karamata class 𝒦.

Here, our goal is to study problem (14) for 0<𝛼<2. To this end, we assume that the potential functions 𝑎₁, 𝑎₂ 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))^−𝜆𝑖𝐿_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 𝛽₁𝛼∶=min2,𝛼−𝜆₁1−𝜎₁,𝛽₂𝛼∶=min2,𝛼−𝜆₂1−𝜎₂ (16) 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 (𝛼−𝜆₁)/(1−𝜎₁)≤(𝛼−𝜆₂)/(1−𝜎₂) and we introduce the function 𝜃 defined on (0,𝜂) by 𝜃(𝑡)=𝑡^𝛽1Ψ_{𝐿1,𝜆1,𝜎1}(𝑡)if𝛽₁<𝛽₂,𝑡^𝛽1Ψ_{𝐿1,𝜆1,𝜎1}(𝑡)+Ψ_{𝐿2,𝜆2,𝜎2}(𝑡)if𝛽₁=𝛽₂. (17) For an explicit form of the function 𝜃, see (36).

Throughout this paper, we define the potential kernel 𝐺^𝛼_𝐷 by 𝐺^𝛼_𝐷𝑓(𝑥)∶=_𝐷𝐺^𝛼_𝐷𝑥,𝑦𝑓𝑦𝑑𝑦,for𝑥∈𝐷,𝑓∈𝐵⁺(𝐷), (18) where 𝐵⁺(𝐷) denotes the set of the nonnegative Borel measurable functions in 𝐷.

Our main results are the following.

Theorem 3. Let 𝜎₁, 𝜎₂∈(−1,1) and assume (𝐻). Then one has, for 𝑥∈𝐷, 𝐺^𝛼_𝐷𝑎₁𝜃^𝜎1(𝛿(⋅))+𝑎₂𝜃^𝜎2(𝛿(⋅))(𝑥)≈𝜃(𝛿(𝑥)). (19) Using Theorem 3 and the Schauder fixed-point theorem, we will prove the following.

Theorem 4. Let 𝜎₁, 𝜎₂∈(−1,1) and assume (𝐻). Then problem (14) has a unique positive continuous solution 𝑢 in 𝐷 satisfying, for 𝑥∈𝐷, 𝑢(𝑥)≈𝜃(𝛿(𝑥)). (20) 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 𝐾^𝛼(𝐷), which are useful to establish our results. In Section 3, we prove our main results.

As usual, we denote by 𝐶₀(𝐷) the set of continuous functions in 𝐷 vanishing continuously on 𝜕𝐷. Note that 𝐶₀(𝐷) is a Banach space with respect to the uniform norm ‖𝑢‖_∞=sup_𝑥∈𝐷|𝑢(𝑥)|. As in the elliptic case, if 𝑓∈𝐵⁺(𝐷) satisfies ∫_𝐷(𝛿(𝑦))^𝛼/2𝑓(𝑦)𝑑𝑦<∞, then the functions 𝑓 and 𝐺^𝛼_𝐷𝑓 are in 𝐿¹_loc(𝐷) and we have in the distributional sense (−Δ)^𝛼/2𝐺^𝛼_𝐷𝑓=𝑓,in𝐷. (21)

2. The Karamata Class 𝒦 and the Kato Class 𝐾^𝛼(𝐷)

We collect in this paragraph some properties of the Karamata class 𝒦 and the Kato class 𝐾^𝛼(𝐷). We recall that a function 𝐿 defined on (0,𝜂] belongs to the class 𝒦 if 𝐿(𝑡)∶=𝑐exp^𝜂_𝑡𝑧(𝑠)𝑠𝑑𝑠, (22) where 𝜂>diam(𝐷),  𝑐>0, and 𝑧∈𝐶([0,𝜂]) such that 𝑧(0)=0.

Proposition 5 (see [30, 31]). (i) A function 𝐿 is in 𝒦 if and only if 𝐿 is a positive function in 𝐶¹((0,𝜂]) such that lim_𝑡→0+𝑡𝐿^′(𝑡)𝐿(𝑡)=0. (23)
(ii) Let 𝐿₁, 𝐿₂∈𝒦, 𝑝∈ℝ. Then one has 𝐿₁+𝐿₂∈𝒦,𝐿₁𝐿₂∈𝒦,𝐿^𝑝₁∈𝒦. (24)
(iii) Let 𝐿∈𝒦 and 𝜀>0. Then one has lim_𝑡→0+𝑡^𝜀𝐿(𝑡)=0. (25)

Applying Karamata’s theorem (see [30, 31]), we get the following.

Lemma 6. Let 𝜇∈ℝ and let 𝐿 be a function in 𝒦. One has the following:(i)if 𝜇<−1, then ∫^𝜂₀𝑠^𝜇𝐿(𝑠)𝑑𝑠 diverges and ∫^𝜂_𝑡𝑠^𝜇𝐿(𝑠)𝑑𝑠∼_𝑡→0⁺(−𝑡^1+𝜇𝐿(𝑡))/(𝜇+1);(ii)if 𝜇>−1, then ∫^𝜂₀𝑠^𝜇𝐿(𝑠)𝑑𝑠 converges and ∫^𝑡₀𝑠^𝜇𝐿(𝑠)𝑑𝑠∼_𝑡→0⁺(𝑡^1+𝜇𝐿(𝑡))/(𝜇+1).

Lemma 7 (see [36]). Let 𝐿 be a function in 𝒦. Then one has lim_𝑡→0+𝐿(𝑡)∫^𝜂_𝑡(𝐿(𝑠)/𝑠)𝑑𝑠=0. (26) In particular 𝑡→^𝜂_𝑡𝐿(𝑠)𝑠𝑑𝑠∈𝒦. (27)

Proposition 8 (see [40, 41]). For (𝑥,𝑦)∈𝐷×𝐷, one has 𝐺^𝛼_𝐷𝑥,𝑦≈𝑥−𝑦^𝛼−𝑛min1,𝛿(𝑥)𝛿(𝑦)^𝛼/2𝑥−𝑦^𝛼. (28)

Proposition 9 (see [26, Corollary 6]). Let 𝑞 be a nonnegative function in 𝐾^𝛼(𝐷); then the family of functions Λ_𝑞=𝑥⟶_𝐷𝛿𝑦𝛿(𝑥)^(𝛼/2)−1𝐺^𝛼_𝐷𝑥,𝑦𝑓𝑦𝑑𝑦,𝑓≤𝑞 (29) is uniformly bounded and equicontinuous in 𝐷. Consequently Λ_𝑞 is relatively compact in 𝐶₀(𝐷).

3. Proofs of the Main Results

In this section we aim at proving Theorems 3 and 4. To this end, we need the following lemmas.

3.1. Technical Lemmas

Lemma 10. For 𝑟, 𝑠>0, one has 2^{−max(1−𝜎1,1−𝜎2)}(𝑟+𝑠)≤𝑟^1−𝜎1(𝑟+𝑠)^𝜎1+𝑠^1−𝜎2(𝑟+𝑠)^𝜎2≤2(𝑟+𝑠). (30)

Proof. Let 𝑟, 𝑠>0 and put 𝑡=𝑟/(𝑟+𝑠). Since 0≤𝑡≤1, then we get obviously 2^{−max(1−𝜎1,1−𝜎2)}≤𝑡^1−𝜎1+(1−𝑡)^1−𝜎2≤2. (31)

Lemma 11 provides sharp estimates on some Riesz potential functions.

Lemma 11 (see [28, Proposition 3.1]). Let 𝜇≤(𝛼/2)+1 and let 𝐿 be a function in 𝒦 such that ∫^𝜂₀𝑡^{(𝛼/2)−𝜇}𝐿(𝑡)𝑑𝑡<∞. Let 𝑞 be a positive measurable function in 𝐷 such that, for 𝑥∈𝐷, 𝑞(𝑥)≈(𝛿(𝑥))^−𝜇𝐿(𝛿(𝑥)). (32) Then, for 𝑥∈𝐷, one has 𝐺^𝛼_𝐷𝑞(𝑥)≈𝜓(𝛿(𝑥)), (33) where 𝜓 is the function defined on (0,𝜂) by 𝜓(𝑡)𝑡∶=^(𝛼/2)−1∫^𝑡₀𝐿(𝑠)𝑠𝛼𝑑𝑠,if𝜇=2𝑡+1,_𝛼−𝜇𝐿(𝑡)𝛼,if2𝛼<𝜇<2𝑡+1,_𝛼/2∫^𝜂_𝑡𝐿(𝑠)𝑠𝛼𝑑𝑠,if𝜇=2,𝑡^𝛼/2𝛼,if𝜇<2. (34)

Lemma 12. Assume (𝐻). Let 𝑢 be a continuous function in 𝐷 such that, for 𝑥∈𝐷, 𝑢(𝑥)≈𝜃(𝛿(𝑥)). Then 𝑢 is a solution of problem (14) if and only if 𝑢(𝑥)=_𝐷𝐺^𝛼_𝐷𝑥,𝑦𝑎₁𝑦𝑢^𝜎1𝑦+𝑎₂𝑦𝑢^𝜎2𝑦𝑑𝑦,𝑥∈𝐷. (35)

Proof. Assume (H). First we will give an explicit form of the function 𝜃. We recall that, for 𝑖∈{1,2},𝜆_𝑖<(𝛼/2)(1+𝜎_𝑖)+1−𝜎_𝑖 and 𝛽_𝑖∶=min(𝛼/2,(𝛼−𝜆_𝑖)/(1−𝜎_𝑖)). Since 𝛽₁<𝛽₂ is equivalent to (𝛼−𝜆₁)/(1−𝜎₁)<(𝛼−𝜆₂)/(1−𝜎₂) and (𝛼/2)(1+𝜎₁)<𝜆₁, we deduce that, for 𝑡∈(0,𝜂), we have 𝜃(𝑡)=𝑡^{(𝛼−𝜆1)/(1−𝜎1)}𝐿₁(𝑡)^{1/(1−𝜎1)},if𝛼−𝜆₁1−𝜎₁<𝛼−𝜆₂1−𝜎₂,𝛼21+𝜎₁<𝜆₁,𝑡^{(𝛼−𝜆1)/(1−𝜎1)}𝐿(𝑡),if𝛼−𝜆₁1−𝜎₁=𝛼−𝜆₂1−𝜎₂,𝛼21+𝜎₁<𝜆₁,𝑡^𝛼/2𝑀(𝑡),if𝜆₁=𝛼21+𝜎₁,𝜆₂=𝛼21+𝜎₂,𝑡^𝛼/2∫^𝜂_𝑡𝐿₁(𝑠)𝑠𝑑𝑠^{1/(1−𝜎1)},if𝜆₁=𝛼21+𝜎₁,𝜆₂<𝛼21+𝜎₂,𝑡^𝛼/2if𝜆₁<𝛼21+𝜎₁, (36) where 𝐿(𝑡)∶=𝐿₁(𝑡)^{1/(1−𝜎1)}+𝐿₂(𝑡)^{1/(1−𝜎2)},𝑀(𝑡)∶=^𝜂_𝑡𝐿₁(𝑠)𝑠𝑑𝑠^{1/(1−𝜎1)}+^𝜂_𝑡𝐿₂(𝑠)𝑠𝑑𝑠^{1/(1−𝜎2)}. (37) Now using the fact that 𝑢(𝑥)≈𝜃(𝛿(𝑥)), we deduce by simple computation from hypothesis (H), (36), and Proposition 5 that 𝑎₁(𝑥)𝑢^𝜎1(𝑥)+𝑎₂(𝑥)𝑢^𝜎2(𝑥)≈(𝛿(𝑥))^(𝛼/2)−1ℎ(𝛿(𝑥)), (38) where ℎ is defined in (0,𝜂) byℎ(𝑡)𝑡∶=^{1−(𝛼/2)−((𝜆1−𝛼𝜎1)/(1−𝜎1))}𝐿₁(𝑡)^{1/(1−𝜎1)},if𝛼−𝜆₁1−𝜎₁<𝛼−𝜆₂1−𝜎₂,𝛼21+𝜎₁<𝜆₁,𝑡^{1−(𝛼/2)−((𝜆1−𝛼𝜎1)/(1−𝜎1))}𝐿₁𝐿^𝜎1+𝐿₂𝐿^𝜎2(𝑡),if𝛼−𝜆₁1−𝜎₁=𝛼−𝜆₂1−𝜎₂,𝛼21+𝜎₁<𝜆₁,𝑡^1−𝛼𝐿₁𝑀^𝜎1+𝐿₂𝑀^𝜎2(𝑡),if𝜆₁=𝛼21+𝜎₁,𝜆₂=𝛼21+𝜎₂,𝑡^1−𝛼𝐿₁(𝑡)∫^𝜂_𝑡𝐿₁(𝑠)𝑠𝑑𝑠^{𝜎1/(1−𝜎1)},if𝜆₁=𝛼21+𝜎₁,𝜆₂<𝛼21+𝜎₂,𝑡^{1+(𝛼/2)(𝜎1−1)−𝜆1}𝐿₁(𝑡)+𝑡^{1+(𝛼/2)(𝜎2−1)−𝜆2}𝐿₂(𝑡),if𝜆₁<𝛼21+𝜎₁. (39)We point out that for each case, the function ℎ(𝑡) can be written as a sum of terms of the form 𝑡^−𝜇𝐿(𝑡), where 𝜇<𝛼. By Proposition 5 and Lemma 7, we have 𝐿∈𝒦. On the other hand, since by Proposition 5, the function 𝑥→(𝛿(𝑥))^{(𝛼−𝜇)/2}𝐿(𝛿(𝑥)) is positive and belongs to 𝐶₀(𝐷), then there exists 𝑐>0 such that for each 𝑥∈𝐷0<(𝛿(𝑥))^−𝜇𝐿(𝛿(𝑥))≤𝑐(𝛿(𝑥))^{−(𝛼+𝜇)/2}. (40) Hence we deduce from (7) that the function 𝑥→ℎ(𝛿(𝑥)) is in 𝐾^𝛼(𝐷).
Now using Proposition 9, we obtain that 𝑥→(𝛿(𝑥))^1−(𝛼/2)𝐺^𝛼_𝐷[𝑎₁𝑢^𝜎1+𝑎₂𝑢^𝜎2](𝑥) is in 𝐶₀(𝐷). In particular, we have lim_{𝑥→𝑧∈𝜕𝐷}(𝛿(𝑥))^1−(𝛼/2)𝐺^𝛼_𝐷𝑎₁𝑢^𝜎1+𝑎₂𝑢^𝜎2𝑥)()(=0,−Δ^𝛼/2𝐺^𝛼_𝐷𝑎₁𝑢^𝜎1+𝑎₂𝑢^𝜎2=𝑎₁(𝑥)𝑢^𝜎1+𝑎₂(𝑥)𝑢^𝜎2in𝐷(inthesenseofdistributions). (41) Consequently, it follows by (41) that 𝑢 is a weak continuous solution of problem (14) if and only if 𝑢 satisfies (−Δ)^𝛼/2𝑢−𝐺^𝛼_𝐷𝑎₁𝑢^𝜎1+𝑎₂𝑢^𝜎2=0in𝐷lim_{𝑥→𝑧∈𝜕𝐷}(𝛿(𝑥))^1−(𝛼/2)𝑢−𝐺^𝛼_𝐷𝑎₁𝑢^𝜎1+𝑎₂𝑢^𝜎2=0. (42) We deduce by [26, Theorem 6] that 𝑢−𝐺^𝛼_𝐷(𝑎₁𝑢^𝜎1+𝑎₂𝑢^𝜎2)=0 in 𝐷. The proof is complete.

Lemma 13. For 𝑖∈{1,2}, let 𝐿_𝑖∈𝒦 defined on (0,𝜂] with 𝜂>diam(𝐷) and let 𝑀 be the function given by (37). Then one has, for 𝑡∈(0,𝜂), ^𝜂_𝑡𝐿₁𝑀^𝜎1+𝐿₂𝑀^𝜎2(𝑠)𝑠𝑑𝑠≈𝑀(𝑡). (43)

Proof. The proof can be found in [36].

Now we are ready to prove our main results.

3.2. Proof of Theorem 3

Assume (H). For 𝑖∈{1,2}, let 𝐿_𝑖∈𝒦 defined on (0,𝜂] with 𝜂>diam(𝐷) and define the nonnegative functions 𝑏_𝑖 in (0,𝜂) by 𝑏_𝑖(𝑡)=^𝜂_𝑡𝐿_𝑖(𝑠)𝑠𝑑𝑠^{1/(1−𝜎𝑖)}. (44) Let 𝜃 be the function given by (36). To prove Theorem 3, we distinguish the following cases.

Case 1. (𝛼−𝜆₁)/(1−𝜎₁)<(𝛼−𝜆₂)/(1−𝜎₂) and (𝛼/2)(1+𝜎₁)<𝜆₁<(𝛼/2)(1+𝜎₁)+1−𝜎₁.
Since 𝜃(𝑡)=𝑡^{(𝛼−𝜆1)/(1−𝜎1)}(𝐿₁(𝑡))^{1/(1−𝜎1)}, then we have 𝑎₁(𝑥)𝜃^𝜎1(𝛿(𝑥))≈(𝛿(𝑥))^{(𝛼𝜎1−𝜆1)/(1−𝜎1)}𝐿₁(𝛿(𝑥))^{1/(1−𝜎1)},𝑎₂(𝑥)𝜃^𝜎2(𝛿(𝑥≈(𝛿(𝑥))))^{((𝛼−𝜆1)/(1−𝜎1))𝜎2−𝜆2}𝐿₂𝐿^{𝜎2/(1−𝜎1)}₁(𝛿(𝑥)). (45) Using the fact that (𝛼𝜎₁−𝜆₁)/(1−𝜎₁)<((𝛼−𝜆₁)/(1−𝜎₁))𝜎₂−𝜆₂, we deduce by Proposition 5 that 𝑎₁(𝑥)𝜃^𝜎1(𝛿(𝑥))+𝑎₂(𝑥)𝜃^𝜎2(𝛿(𝑥≈(𝛿(𝑥))))^{(𝛼𝜎1−𝜆1)/(1−𝜎1)}𝐿₁(𝛿(𝑥))^{1/(1−𝜎1)}. (46) Since, for 𝜇=(𝜆₁−𝛼𝜎₁)/(1−𝜎₁)∈(𝛼/2,(𝛼/2)+1), we have ∫^𝜂₀𝑡^{(𝛼/2)−𝜇}(𝐿₁(𝑡))^{1/(1−𝜎1)}𝑑𝑡<∞, then applying Lemma 11, we deduce that 𝐺^𝛼_𝐷𝑎₁𝜃^𝜎1(𝛿(⋅))+𝑎₂𝜃^𝜎2(𝛿(⋅))(𝑥)≈𝐺^𝛼_𝐷(𝛿(⋅))^{(𝛼𝜎1−𝜆1)/(1−𝜎1)}𝐿₁(𝛿(⋅))^{1/(1−𝜎1)}𝑥)≈(𝛿(𝑥())^{(𝛼−𝜆1)/(1−𝜎1)}𝐿₁(𝛿(𝑥))^{1/(1−𝜎1)}=𝜃(𝛿(𝑥)). (47)

Case 2. (𝛼−𝜆₁)/(1−𝜎₁)=(𝛼−𝜆₂)/(1−𝜎₂) and (𝛼/2)(1+𝜎₁)<𝜆₁<(𝛼/2)(1+𝜎₁)+1−𝜎₁.
In this case 𝜃(𝑡)=𝑡^{(𝛼−𝜆1)/(1−𝜎1)}𝐿(𝑡). Therefore 𝑎₁(𝑥)𝜃^𝜎1(𝛿(𝑥))≈(𝛿(𝑥))^{(𝛼𝜎1−𝜆1)/(1−𝜎1)}𝐿₁𝐿^𝜎1(𝛿(𝑥)). (48) So we obtain by Proposition 5 and Lemma 11 with 𝜇=(𝜆₁−𝛼𝜎₁)/(1−𝜎₁)∈(𝛼/2,(𝛼/2)+1), 𝐺^𝛼_𝐷𝑎₁𝜃^𝜎1(𝛿(⋅))(𝑥)≈(𝛿(𝑥))^{(𝛼−𝜆1)/(1−𝜎1)}𝐿₁𝐿^𝜎1(𝛿(𝑥)). (49) Similarly, since (𝛼/2)(1+𝜎₂)<𝜆₂<(𝛼/2)(1+𝜎₂)+1−𝜎₂, we obtain 𝐺^𝛼_𝐷𝑎₂𝜃^𝜎2(𝛿(⋅𝑥)≈(𝛿(𝑥))())^{(𝛼−𝜆2)/(1−𝜎2)}𝐿₂𝐿^𝜎2(𝛿(𝑥≈(𝛿(𝑥))))^{(𝛼−𝜆1)/(1−𝜎1)}𝐿₂𝐿^𝜎2(𝛿(𝑥)). (50) Hence by using (30), we deduce that 𝐺^𝛼_𝐷𝑎₁𝜃^𝜎1(𝛿(⋅))+𝑎₂𝜃^𝜎2(𝛿(⋅𝑥)≈(𝛿(𝑥))())^{(𝛼−𝜆1)/(1−𝜎1)}𝐿₁𝐿^𝜎1+𝐿₂𝐿^𝜎2(𝛿(𝑥≈(𝛿(𝑥))))^{(𝛼−𝜆1)/(1−𝜎1)}𝐿(𝛿(𝑥))=𝜃(𝛿(𝑥)). (51)

Case 3. If 𝜆₁=(𝛼/2)(1+𝜎₁) and 𝜆₂=(𝛼/2)(1+𝜎₂) and since 𝜃(𝑡)=𝑡^𝛼/2𝑀(𝑡), then we have 𝑎₁(𝑥)𝜃^𝜎1(𝛿(𝑥))+𝑎₂(𝑥)𝜃^𝜎2(𝛿(𝑥≈(𝛿(𝑥))))^−𝛼/2𝐿₁𝑀^𝜎1+𝐿₂𝑀^𝜎2(𝛿(𝑥)). (52) So by Proposition 5, Lemma 11 with 𝜇=𝛼/2, and Lemma 13, we deduce that 𝐺^𝛼_𝐷𝑎₁𝜃^𝜎1(𝛿(⋅))+𝑎₂𝜃^𝜎2(𝛿(⋅𝑥)≈(𝛿(𝑥))())^𝛼/2𝜂_𝛿(𝑥)𝐿₁𝑀^𝜎1+𝐿₂𝑀^𝜎2(𝑠)𝑠≈(𝛿(𝑥𝑑𝑠))^𝛼/2𝑀(𝛿(𝑥))=𝜃(𝛿(𝑥)). (53)

Case 4. 𝜆₁=(𝛼/2)(1+𝜎₁) and 𝜆₂<(𝛼/2)(1+𝜎₂).
In this case 𝜃(𝑡)=𝑡^𝛼/2𝑏₁(𝑡). Since 𝜆₂−(𝛼𝜎₂/2)<(𝛼/2), we deduce by Proposition 5 that 𝑎₁(𝑥)𝜃^𝜎1(𝛿(𝑥))+𝑎₂(𝑥)𝜃^𝜎2(𝛿(𝑥≈(𝛿(𝑥))))^−𝛼/2𝐿₁𝑏^𝜎1₁(𝛿(𝑥+(𝛿(𝑥))))^{(𝛼𝜎2/2)−𝜆2}𝐿₂𝑏^𝜎2₁𝛿(𝑥≈(𝛿(𝑥())))^−𝛼/2𝐿₁𝑏^𝜎1₁(𝛿(𝑥)). (54) Hence applying Lemma 11 with 𝜇=𝛼/2, we obtain 𝐺^𝛼_𝐷𝑎₁𝜃^𝜎1(𝛿(⋅))+𝑎₂𝜃^𝜎2(𝛿(⋅))(𝑥)≈𝐺^𝛼_𝐷(𝛿(⋅))^−𝛼/2𝐿₁𝑏^𝜎1₁(𝛿(⋅𝑥)≈(𝛿(𝑥))())^𝛼/2𝜂_𝛿(𝑥)𝐿₁𝑏^𝜎1₁(𝑠)𝑠≈(𝛿(𝑥𝑑𝑠))^𝛼/2𝑏₁(𝛿(𝑥))=𝜃(𝛿(𝑥)). (55)

Case 5. 𝜆₁<(𝛼/2)(1+𝜎₁).
We have 𝜃(𝑡)=𝑡^𝛼/2. So 𝑎₁(𝑥)𝜃^𝜎1(𝛿(𝑥))≈(𝛿(𝑥))^{−(𝜆1−(𝛼𝜎1/2))}𝐿₁(𝛿(𝑥)). (56) Applying again Lemma 11 with 𝜇=𝜆₁−(𝛼𝜎₁/2)<(𝛼/2), we obtain 𝐺^𝛼_𝐷𝑎₁𝜃^𝜎1(𝛿(⋅))(𝑥)≈(𝛿(𝑥))^𝛼/2. (57) On the other hand, since (𝛼/2)<(𝛼−𝜆₁)/(1−𝜎₁)≤(𝛼−𝜆₂)/(1−𝜎₂), then 𝜆₁<(𝛼/2)(1+𝜎₁) and therefore 𝐺^𝛼_𝐷𝑎₂𝜃^𝜎2(𝛿(⋅))(𝑥)≈(𝛿(𝑥))^𝛼/2. (58) Hence 𝐺^𝛼_𝐷𝑎₁𝜃^𝜎1(𝛿(⋅))+𝑎₂𝜃^𝜎2(𝛿(⋅))(𝑥)≈(𝛿(𝑥))^𝛼/2=𝜃(𝛿(𝑥)). (59) The proof is complete.

3.3. Proof of Theorem 4

Let 𝜎₁, 𝜎₂∈(−1,1), assume (H), and consider 𝑣∶=𝐺^𝐷_𝛼[𝑎₁𝜃^𝜎1(𝛿(⋅))+𝑎₂𝜃^𝜎2(𝛿(⋅))]. Using Theorem 3, there exists 𝑚>1 such that 1𝑚𝑣(𝑥)≤𝜃(𝛿(𝑥))≤𝑚𝑣(𝑥). (60) Put 𝜎∶=max(|𝜎₁|,|𝜎₂|), 𝑐∶=𝑚^{𝜎/(1−𝜎)} and consider the set Γ∶=𝜔∈𝐶₀(𝐷)∶1𝑐(𝛿(𝑥))^1−(𝛼/2)𝑣(𝑥)(𝑥)(𝛿(𝑥≤𝜔≤𝑐))^1−(𝛼/2)𝑣(𝑥),𝑥∈𝐷. (61) Let ℎ be the function given by (39). Since 𝑎₁(𝑥)𝜃^𝜎1(𝛿(𝑥))+𝑎₂(𝑥)𝜃^𝜎2(𝛿(𝑥))≈(𝛿(𝑥))^(𝛼/2)−1ℎ(𝛿(𝑥)) and the function 𝑥→ℎ(𝛿(𝑥)) is in 𝐾_𝛼(𝐷), it follows by Proposition 9 that 𝑥→(𝛿(𝑥))^1−(𝛼/2)𝑣(𝑥) is in 𝐶₀(𝐷). So Γ is a nonempty, closed, bounded, and convex set in 𝐶₀(𝐷). Define the operator 𝑇 on Γ by 𝑇𝜔(𝑥)∶=(𝛿(𝑥))^1−(𝛼/2)𝐺^𝐷_𝛼(𝛿(⋅))^(𝛼/2)−1𝜔^𝜎1𝑎₁+(𝛿(⋅))^(𝛼/2)−1𝜔^𝜎2𝑎₂(𝑥). (62) We will prove that 𝑇 has a fixed point. Since there exists a constant 𝑐>0 such that for all 𝜔∈Γ we have 𝑎₁(𝑥)(𝛿(𝑥))^(𝛼/2)−1𝜔(𝑥)^𝜎1+𝑎₂(𝑥)(𝛿(𝑥))^(𝛼/2)−1𝜔(𝑥)^𝜎2≤𝑐(𝛿(𝑥))^(𝛼/2)−1ℎ(𝛿(𝑥)), (63) where the function 𝑥→ℎ(𝛿(𝑥)) is in 𝐾_𝛼(𝐷), it follows that 𝑇(Γ)⊂Λ_{ℎ(𝛿(⋅))}, where Λ_{ℎ(𝛿(⋅))} is given by (29). Therefore by Proposition 9, the family of functions {𝑥→𝑇𝜔(𝑥),𝜔∈Γ} is relatively compact in 𝐶₀(𝐷).

Next, we will prove that 𝑇 maps Γ into itself.

Indeed, by using (60) we have for all 𝜔∈Γ𝐺^𝐷_𝛼𝑎₁(𝛿(⋅))^(𝛼/2)−1𝜔^𝜎1+𝑎₂(𝛿(⋅))^(𝛼/2)−1𝜔^𝜎2≤𝐺^𝐷_𝛼𝑎₁𝑐^𝜎𝑚^𝜎𝜃^𝜎1(𝛿(⋅))+𝑎₂𝑐^𝜎𝑚^𝜎𝜃^𝜎2(𝛿(⋅))=𝑐𝑣. (64) On the other hand, we have 𝐺^𝐷_𝛼𝑎₁(𝛿(⋅))^(𝛼/2)−1𝜔^𝜎1+𝑎₂(𝛿(⋅))^(𝛼/2)−1𝜔^𝜎2≥𝐺^𝐷_𝛼𝑎₁𝑐^−𝜎𝑚^−𝜎𝜃^𝜎1(𝛿(⋅))+𝑎₂𝑐^−𝜎𝑚^−𝜎𝜃^𝜎2(𝛿(⋅=1))𝑐𝑣. (65) This implies that 𝑇(Γ)⊂Γ.

Now, we will prove the continuity of the operator 𝑇 in Γ in the supremum norm. Let (𝜔_𝑘)_𝑘∈ℕ be a sequence in Γ which converges uniformly to a function 𝜔 in Γ. Then, for each 𝑥∈𝐷, we have 𝑇𝜔_𝑘(𝑥)(𝑥≤(𝛿(𝑥−𝑇𝜔)))^1−(𝛼/2)𝐺^𝐷_𝛼𝑎₁(𝛿(.))^{((𝛼/2)−1)𝜎1}𝜔^𝜎1_𝑘−𝜔^𝜎1+𝑎₂(𝛿(⋅))^{((𝛼/2)−1)𝜎2}𝜔^𝜎2_𝑘−𝜔^𝜎2(𝑥). (66) On the other hand, by similar arguments to the previous ones, we have 𝑎₁(𝑥)(𝛿(𝑥))^{((𝛼/2)−1)𝜎1}𝜔^𝜎1_𝑘−𝜔^𝜎1(𝑥)+𝑎₂(𝑥)(𝛿(𝑥))^{((𝛼/2)−1)𝜎2}𝜔^𝜎2_𝑘−𝜔^𝜎2𝑥)(𝛿(𝑥(≤𝑐))^(𝛼/2)−1ℎ(𝛿(𝑥)). (67) We conclude by Proposition 9 and the dominated convergence theorem that, for all 𝑥∈𝐷, 𝑇𝜔_𝑘(𝑥)⟶𝑇𝜔(𝑥)as𝑘⟶+∞. (68) Consequently, as 𝑇(Γ) is relatively compact in 𝐶₀(𝐷), we deduce that the pointwise convergence implies the uniform convergence; namely, 𝑇𝜔_𝑘−𝑇𝜔_∞⟶0as𝑘⟶+∞. (69) Therefore, 𝑇 is a compact operator from Γ into itself. So the Schauder fixed-point theorem implies the existence of 𝜔∈Γ such that 𝜔(𝑥)=(𝛿(𝑥))^1−(𝛼/2)𝐺^𝐷_𝛼𝑎₁(𝛿(⋅))^(𝛼/2)−1𝜔^𝜎1+𝑎₂(𝛿(⋅))^(𝛼/2)−1𝜔^𝜎2(𝑥). (70) Put 𝑢(𝑥)=(𝛿(𝑥))^(𝛼/2)−1𝜔(𝑥). Then 𝑢 is continuous and satisfies 𝑢(𝑥)=𝐺^𝐷_𝛼𝑎₁𝑢^𝜎1+𝑎₂𝑢^𝜎2(𝑥). (71) Hence by Lemma 12 and Theorem 3, 𝑢 is a required solution.

Next, we aim at proving the uniqueness in the cone 𝑆∶={𝑢∈𝐶(𝐷)∶𝑢(𝑥)≈𝜃(𝛿(𝑥))}. (72) Let 𝑢 and 𝑣 be two solutions of (14) in 𝑆. Then there exists a constant 𝑚>1 such that 1𝑚≤𝑢𝑣≤𝑚. (73) This implies that the set (),1𝐽=𝑡∈1,∞𝑡𝑣≤𝑢≤𝑡𝑣 (74) is not empty. Let 𝑐₀∶=inf𝐽 and put 𝑤=𝑣−𝑐₀^−𝜎𝑢 with 𝜎=max(|𝜎₁|,|𝜎₂|).

We claim that 𝑐₀=1. Indeed, assume that 𝑐₀>1; then by using Lemma 12, we deduce that 𝑤=𝐺^𝐷_𝛼𝑎₁𝑣^𝜎1−𝑐₀^−𝜎𝑢^𝜎1+𝑎₂𝑣^𝜎2−𝑐₀^−𝜎𝑢^𝜎2≥0in𝐷, (75) which implies that 𝑣≥𝑐₀^−𝜎𝑢. (76) By symmetry, we deduce that 𝑣≤𝑐^𝜎₀𝑢. (77) So 𝑐^𝜎₀∈𝐽. Since 𝜎<1, then we have 𝑐^𝜎₀<𝑐₀. This is a contradiction to the fact that 𝑐₀∶=inf𝐽. Hence 𝑐₀=1 and so 𝑢=𝑣. This completes the proof.

Example 14. Let 𝜎₁∈(−1,0),let𝜎₂∈(0,1), and put 𝑑=diam(𝐷). For 𝑖∈{1,2}, let 𝑎_𝑖∈𝐶^𝛾_loc(𝐷),0<𝛾<1, satisfying 𝑎₁(𝑥)≈(𝛿(𝑥))^−𝜆1log3𝑑𝛿(𝑥)⁻¹,𝑎₂(𝑥)≈(𝛿(𝑥))^−𝜆2, (78)
where 𝜆_𝑖<(𝛼/2)(1+𝜎_𝑖)+1−𝜎_𝑖, such that (𝛼−𝜆₁)/(1−𝜎₁)≤(𝛼−𝜆₂)/(1−𝜎₂). Then using Theorem 4, problem (14) has a unique positive continuous solution 𝑢 in 𝐷 satisfying the following estimates:(i) if (𝛼−𝜆₁)/(1−𝜎₁)<(𝛼−𝜆₂)/(1−𝜎₂) and (𝛼/2)(1+𝜎₁)<𝜆₁, then, for 𝑥∈𝐷, 𝑢(𝑥)≈(𝛿(𝑥))^{(𝛼−𝜆1)/(1−𝜎1)}log3𝑑𝛿(𝑥)^{−1/(1−𝜎1)}; (79) (ii)if (𝛼−𝜆₁)/(1−𝜎₁)=(𝛼−𝜆₂)/(1−𝜎₂) and (𝛼/2)(1+𝜎₁)<𝜆₁, then, for 𝑥∈𝐷, 𝑢(𝑥)≈(𝛿(𝑥))^{(𝛼−𝜆1)/(1−𝜎1)}; (80) (iii)if 𝜆₁=(𝛼/2)(1+𝜎₁) and 𝜆₂=(𝛼/2)(1+𝜎₂), then, for 𝑥∈𝐷, 𝑢(𝑥)≈(𝛿(𝑥))^𝛼/2log3𝑑𝛿(𝑥)^{1/(1−𝜎2)}; (81)(iv)if 𝜆₁=(𝛼/2)(1+𝜎₁) and 𝜆₂<(𝛼/2)(1+𝜎₂), then, for 𝑥∈𝐷, 𝑢(𝑥)≈(𝛿(𝑥))^𝛼/2log∘log3𝑑𝛿(𝑥)^{1/(1−𝜎1)}; (82) (v)if 𝜆₁<(𝛼/2)(1+𝜎₁), then, for 𝑥∈𝐷, 𝑢(𝑥)≈(𝛿(𝑥))^(𝛼/2). (83)

Acknowledgments

The authors are thankful to the referees for their careful reading of the paper and for their helpful comments and suggestions. The research of Imed Bachar is supported by NPST Program of King Saud University, Project no. 11-MAT1716-02.