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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 295480, 9 pages
http://dx.doi.org/10.1155/2013/295480
Research Article

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

1Mathematics Department, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
2Mathematics Department, College of Sciences and Arts, King Abdulaziz University, Rabigh Campus, P.O. Box 344, Rabigh 21911, Saudi Arabia

Received 20 February 2013; Accepted 10 July 2013

Academic Editor: Daniel C. Biles

Copyright © 2013 Imed Bachar and Habib Mâagli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We prove the existence and uniqueness of a positive continuous solution to the following singular semilinear fractional Dirichlet problem , in where is a bounded -domain in and denotes the Euclidian distance from to the boundary of The nonnegative weight functions 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, ,  , which can be defined by the integral representation where 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, [611] 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 This is a model of the so-called anomalous diffusion, a much studied topic in physics, probability, and finance (see [1923] and their references). For more applications, we refer the reader to the survey papers [24, 25].

Throughout this paper, we consider a bounded -domain in , , and we denote by the Euclidian distance from to the boundary of . For two nonnegative functions and defined on a set , the notation ,  , means that there exists such that , for all .

Recently, in [26], the authors considered the following problem: where and is a positive measurable function in satisfying the following: the map is continuous and nonincreasing in , for ; for each , the function is in (see Definition 1 below).

They have proved that problem (3) has a positive continuous solution in satisfying, for each , where denotes the Green function of the fractional Laplacian in . However they have not investigated the asymptotic behavior of such solution.

As a typical example of function satisfying and , we quote , where and is a positive measurable function in such that the function belongs to the Kato class defined as follows.

Definition 1 (see [26]). A Borel measurable function in belongs to the Kato class if

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., ) the class was introduced and studied in [27].

On the other hand, Chemmam et al. considered in [28] the following semilinear fractional Dirichlet problem: where , , and satisfies the following hypothesis:

  , , satisfying , where and belongs to the Karamata class defined as follows.

Definition 2. The class is the set of all the Karamata functions defined on by where ,  , and such that .

As a typical example of a function belonging to the class (see [2931]), we quote where are real numbers, , and is a sufficiently large positive real number such that is defined and positive on .

Using a fixed-point argument, the authors have proved in [28] the existence and uniqueness of a positive continuous solution for (8) satisfying, for , where the function is defined on by 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: where and , . 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., ), problems related to (14) have been studied by several authors (see, e.g., [3439] 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 ,  , where the functions , are required to satisfy some adequate assumptions related to the Karamata class .

Here, our goal is to study problem (14) for . To this end, we assume that the potential functions , satisfy the following hypothesis.

for , , , and satisfies, for , where and defined on with .

As it turns out, estimates (12) depend closely on . 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 and we introduce the function defined on 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.

Theorem 3. Let , and assume . Then one has, for , Using Theorem 3 and the Schauder fixed-point theorem, we will prove the following.

Theorem 4. Let , and assume . Then problem (14) has a unique positive continuous solution in satisfying, for , 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 and the Kato class , which are useful to establish our results. In Section 3, we prove our main results.

As usual, we denote by the set of continuous functions in vanishing continuously on . Note that is a Banach space with respect to the uniform norm As in the elliptic case, if satisfies , then the functions and are in and we have in the distributional sense

2. The Karamata Class and the Kato Class

We collect in this paragraph some properties of the Karamata class and the Kato class . We recall that a function defined on belongs to the class if where ,  , and such that .

Proposition 5 (see [30, 31]). A function is in if and only if is a positive function in such that
Let , , . Then one has
Let and . Then one has

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

Lemma 6. Let and let be a function in . One has the following:(i)if , then diverges and ;(ii)if , then converges and .

Lemma 7 (see [36]). Let be a function in . Then one has In particular

Proposition 8 (see [40, 41]). For , one has

Proposition 9 (see [26, Corollary 6]). Let be a nonnegative function in ; then the family of functions is uniformly bounded and equicontinuous in . Consequently is relatively compact in .

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 , , one has

Proof. Let , and put . Since , then we get obviously

Lemma 11 provides sharp estimates on some Riesz potential functions.

Lemma 11 (see [28, Proposition 3.1]). Let and let be a function in such that . Let be a positive measurable function in such that, for , Then, for , one has where is the function defined on by

Lemma 12. Assume . Let be a continuous function in such that, for , . Then is a solution of problem (14) if and only if

Proof. Assume . First we will give an explicit form of the function . We recall that, for and . Since is equivalent to and , we deduce that, for , we have where Now using the fact that , we deduce by simple computation from hypothesis , (36), and Proposition 5 that where is defined in byWe point out that for each case, the function can be written as a sum of terms of the form , where . By Proposition 5 and Lemma 7, we have . On the other hand, since by Proposition 5, the function is positive and belongs to , then there exists such that for each Hence we deduce from (7) that the function is in .
Now using Proposition 9, we obtain that is in . In particular, we have Consequently, it follows by (41) that is a weak continuous solution of problem (14) if and only if satisfies We deduce by [26, Theorem 6] that in . The proof is complete.

Lemma 13. For , let defined on with and let be the function given by (37). Then one has, for ,

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

Now we are ready to prove our main results.

3.2. Proof of Theorem 3

Assume . For , let defined on with and define the nonnegative functions in by Let be the function given by (36). To prove Theorem 3, we distinguish the following cases.

Case 1. and .
Since , then we have Using the fact that , we deduce by Proposition 5 that Since, for , we have , then applying Lemma 11, we deduce that

Case 2. and .
In this case . Therefore So we obtain by Proposition 5 and Lemma 11 with , Similarly, since , we obtain Hence by using (30), we deduce that

Case 3. If and and since , then we have So by Proposition 5, Lemma 11 with , and Lemma 13, we deduce that

Case 4. and .
In this case . Since , we deduce by Proposition 5 that Hence applying Lemma 11 with , we obtain

Case 5. .
We have . So Applying again Lemma 11 with , we obtain On the other hand, since , then and therefore Hence The proof is complete.

3.3. Proof of Theorem 4

Let , , assume , and consider . Using Theorem 3, there exists such that Put , and consider the set Let be the function given by (39). Since and the function is in , it follows by Proposition 9 that is in . So is a nonempty, closed, bounded, and convex set in . Define the operator on by We will prove that has a fixed point. Since there exists a constant such that for all we have where the function is in , it follows that , where is given by (29). Therefore by Proposition 9, the family of functions is relatively compact in .

Next, we will prove that maps into itself.

Indeed, by using (60) we have for all On the other hand, we have This implies that .

Now, we will prove the continuity of the operator in in the supremum norm. Let be a sequence in which converges uniformly to a function in . Then, for each , we have On the other hand, by similar arguments to the previous ones, we have We conclude by Proposition 9 and the dominated convergence theorem that, for all , Consequently, as is relatively compact in , we deduce that the pointwise convergence implies the uniform convergence; namely, Therefore, is a compact operator from into itself. So the Schauder fixed-point theorem implies the existence of such that Put . Then is continuous and satisfies Hence by Lemma 12 and Theorem 3, is a required solution.

Next, we aim at proving the uniqueness in the cone Let and be two solutions of (14) in . Then there exists a constant such that This implies that the set is not empty. Let and put with .

We claim that . Indeed, assume that ; then by using Lemma 12, we deduce that which implies that By symmetry, we deduce that So . Since , then we have . This is a contradiction to the fact that . Hence and so . This completes the proof.

Example 14. Let , and put . For , let , satisfying
where , such that . Then using Theorem 4, problem (14) has a unique positive continuous solution in satisfying the following estimates:(i) if and , then, for , (ii)if and , then, for , (iii)if and , then, for , (iv)if and , then, for , (v)if , then, for ,

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.

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