Positive Solutions for Some Competitive Fractional Systems in Bounded Domains

Using some potential theory tools and the Schauder fixed point theorem, we prove the existence and precise global behavior of positive continuous solutions for the competitive fractional system (−Δ |D) α/2 u + p(x)uV = 0, (−Δ |D) α/2 V + q(x)uV = 0 in a bounded C-domain D in R (n ≥ 3), subject to some Dirichlet conditions, where 0 < α < 2, σ, β ≥ 1, s, r ≥ 0.The potential functions p, q are nonnegative and required to satisfy some adequate hypotheses related to the Kato class of functions Kα(D).


Introduction and Statement of Main Results
Let  be a bounded  1,1 -domain in R  ( ≥ 3) and Δ | be the Dirichlet Laplacian in .The fractional power −(−Δ | ) /2 , 0 <  < 2, of the negative Dirichlet Laplacian is a very useful object in analysis and partial differential equations; see, for instance, [1,2].There is a Markov process    corresponding to −(−Δ | ) /2 which can be obtained as follows: we first kill the Brownian motion  at   , the first exit time of  from the domain , and then we subordinate this killed Brownian motion using the /2-stable subordinator   starting at zero.For more description of the process    and the development of its potential theory, we refer to [3][4][5][6].
In this paper, we will exploit these potential theory tools to study the existence of positive solutions for some nonlinear systems of fractional differential equations.More precisely, we fix two positive continuous functions  and  on , and we will deal with the existence of positive continuous solutions (in the sense of distributions) for the following competitive fractional system: where ,  ≥ 1, ,  ≥ 0, 0 <  < 2 and the nonnegative potential functions ,  are required to satisfy some adequate hypotheses related to the Kato class of functions   () (see Definition 1).The function    1() is defined by where (   ) >0 is the semigroup corresponding to the killed Brownian motion upon exiting .
We recall that in [6, Remark 3.3], the authors have proved the existence of a constant  > 0 such that for each  ∈ , 1  ( () where () denotes the Euclidian distance from  to the boundary of .
In the classical case (i.e.,  = 2), there is a large amount of literature dealing with the existence, nonexistence, and qualitative analysis of positive solutions for the problems related Journal of Function Spaces and Applications to (1); see for example, the papers of Cîrstea and Rȃdulescu [7], Ghanmi et al. [8], Ghergu and Rȃdulescu [9], Lair and Wood [10,11], Mu et al. [12], and references therein.In these works, various existence results of positive bounded solutions or positive blowing up ones (called also large solutions) have been established, and a precise global behavior is given.We note also that several methods have been used to treat these systems such as sub-and supersolutions method, variational method, and topological methods.In [11], the authors studied the system (1) with  = 2 in the case  =  = 0,  > 0,  > 0, and ,  are nonnegative continuous and not necessarily radial.They showed that entire positive bounded solutions exist if  and  satisfy the following condition: for some positive constant  and || large. (4) These results have been extended recently by Alsaedi et al. in [13], in the case  = 2, ,  ≥ 1,  > 0,  > 0, where the authors established the existence of a positive continuous bounded solution for (1).
In this paper, first, we aim at proving the existence and uniqueness of a positive continuous solution (in the sense of distributions) for the following scalar equation: where  ≥ 1 and  0 is a nonnegative Borel measurable function in  satisfying the following.
The class of functions   (), is defined by means of the Green function    of    as follows.
For more examples of functions belonging to   (), we refer to [14].Note that for the classical case (i.e.,  = 2), the class of functions  2 () was introduced and studied in [15].In order to state our existence result, we denote by     (see [3]) the unique positive continuous solution of Using some potential theory tools and an approximating sequence, we establish the following.
(H 2 ) The functions ,  are two nonnegative Borel measurable functions such that Then, by using Schauder's fixed point theorem, we prove the following.
We note that contrary to the classical case  = 2, in our situation, the solution blows up on the boundary of .
The content of this paper is organized as follows.In Section 2, we collect some properties of functions belonging to the Kato class of functions   (), which are useful to establish our results.Our main results are proved in Section 3.
As usual, let  + () be the set of nonnegative Borel measurable functions in .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 ∈ |()|.When two positive functions  and  are defined on a set , we write  ≈  if the twosided inequality (1/) ≤  ≤  holds on .We define the potential kernel    of    by Finally, let us recall some potential theory tools that are needed, and we refer to [14,16,17] for more details.For  ∈  + (), we define the kernel   on  + () by with  0 :=  =    , where Ẽ stands for the expectation with respect to    starting from .If  satisfies  < ∞, we have the following resolvent equation: In particular, if  ∈  + () is such that () < ∞, then we have

The Kato Class of Functions 𝐾 𝛼 (𝐷)
Proposition 4 (see [14]).Let  be a function in   (), then we have the following.
(ii) Let ℎ be a positive excessive function on  with respect to    .Then, we have Furthermore, for each  0 ∈ , we have (iii) The function  → (() The next two lemmas will play a special role.
Lemma 5. Let  be a nonnegative function in   () and ℎ be a positive finite excessive function on  with respect to    .Then, for all  ∈ , we have Proof.Let ℎ be a positive finite excessive function on  with respect to    .Then, by [18, Chapter II, proposition 3.11], there exists a sequence (  )  of nonnegative measurable functions in  such that ℎ = sup    .Let  ∈  and  ∈ N such that 0 <   < ∞.Consider () =     (), for  ≥ 0.Then, by (14), the function  is completely monotone on [0, ∞), and so from the Hölder inequality and [19, Theorem 12a], the function log  is convex on [0, ∞).This implies that That is Hence, it follows from (17) that Consequently, from (15) we obtain The result holds by letting  → ∞.

Lemma 6.
Let  be a nonnegative function in   (), then the family of functions So, the family Λ  is uniformly bounded.Next, we aim at proving that the family Λ  is equicontinuous in .
First, we recall the following interesting sharp estimates on    , which is proved in [5]: Consequently, by Ascoli's theorem, we deduce that Λ  is relatively compact in  0 ().

Proofs of Theorems 2 and 3
The next Lemma will be used for uniqueness.