# Positive Solutions for Some Competitive Fractional Systems in Bounded Domains

Authors: Imed Bachar; Habib Mâagli; Noureddine Zeddini
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

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/2u+p(x)uσvr=0, (−Δ|D)α/2v+q(x)usvβ=0 in a bounded C1,1-domain D in ℝn(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).


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## Body

## 1. Introduction and Statement of Main Results

LetD be a bounded C1,1-domain in ℝn(n≥3) and Δ∣D be the Dirichlet Laplacian in D. The fractional power -(-Δ∣D)α/2, 0<α<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 ZαD corresponding to -(-Δ∣D)α/2 which can be obtained as follows: we first kill the Brownian motion X at τD, the first exit time of X from the domain D, and then we subordinate this killed Brownian motion using the α/2-stable subordinator Tt starting at zero. For more description of the process ZαD and the development of its potential theory, we refer to [3–6].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 functionsφ and ψ on ∂D, and we will deal with the existence of positive continuous solutions (in the sense of distributions) for the following competitive fractional system:
(1)(-Δ∣D)α/2u+p(x)uσvr=0inD,(-Δ∣D)α/2v+q(x)usvβ=0inD,limx→z∈∂Du(x)MαD1(x)=φ(z),limx→z∈∂Dv(x)MαD1(x)=ψ(z),
where σ,β≥1, s,r≥0, 0<α<2 and the nonnegative potential functions p, q are required to satisfy some adequate hypotheses related to the Kato class of functions Kα(D) (see Definition 1). The function MαD1(x) is defined by
(2)MαD1(x)=1-α/2Γ(α/2)∫0∞t-2+(α/2)(1-PtD1(x))dt,
where (PtD)t>0 is the semigroup corresponding to the killed Brownian motion upon exiting D.We recall that in [6, Remark 3.3], the authors have proved the existence of a constant C>0 such that for each x∈D,
(3)1C(δ(x))α-2≤MαD1(x)≤C(δ(x))α-2,
where δ(x) denotes the Euclidian distance from x to the boundary of D.In the classical case (i.e.,α=2), there is a large amount of literature dealing with the existence, nonexistence, and qualitative analysis of positive solutions for the problems related to (1); see for example, the papers of Cîrstea and Rădulescu [7], Ghanmi et al. [8], Ghergu and Rădulescu [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 α=2 in the case σ=β=0, s>0, r>0, and p, q are nonnegative continuous and not necessarily radial. They showed that entire positive bounded solutions exist if p and q satisfy the following condition:
(4)p(x)+q(x)≤C|x|-(2+δ),forsomepositiveconstantδand|x|large.
These results have been extended recently by Alsaedi et al. in [13], in the case α=2, σ,β≥1, s>0, r>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:(5)(-Δ∣D)α/2u+p0(x)uγ=0inD,u>0inD,limx→z∈∂Du(x)MαD1(x)=φ(z),
where γ≥1 and p0 is a nonnegative Borel measurable function in D satisfying the following.(
H
1
) The function x→(δ(x))(α-2)(γ-1)p0(x)∈Kα(D).The class of functionsKα(D), is defined by means of the Green function GαD of ZαD as follows.Definition 1 (see [14]).
A Borel measurable functionq in D belongs to the Kato class of functions Kα(D) if
(6)limr→0(supx∈D∫(|x-y|≤r)∩Dδ(y)δ(x)GαD(x,y)|q(y)|dy)=0.It has been shown in [14], that
(7)x⟶(δ(x))-λ∈Kα(D),forλ<α.
For more examples of functions belonging to Kα(D), we refer to [14]. Note that for the classical case (i.e., α=2), the class of functions K2(D) was introduced and studied in [15].In order to state our existence result, we denote byMαDφ (see [3]) the unique positive continuous solution of
(8)(-Δ∣D)α/2u=0inD(inthesenseofdistributions).limx→z∈∂Du(x)MαD1(x)=φ(z).
Using some potential theory tools and an approximating sequence, we establish the following.Theorem 2.
Under hypothesis(H1), the problem (5) has a unique positive continuous solution satisfying for each x∈D,
(9)c0MαDφ(x)≤u(x)≤MαDφ(x),
where the constant c0∈(0,1].Using (7), hypothesis (H1) is satisfied if p0 verifies the following condition: there exists a constant C>0, such that for each x∈D,
(10)p0(x)≤C(δ(x))τ,withτ+(2-α)(γ-1)<α.
Next, we exploit the result of Theorem 2, to prove the existence of a positive continuous solution (u,v) to the system (1). To this end, we assume the following hypothesis.(
H
2
) The functions p, q are two nonnegative Borel measurable functions such that
(11)x⟶(δ(x))(α-2)(σ+r-1)p(x)∈Kα(D),x⟶(δ(x))(α-2)(β+s-1)q(x)∈Kα(D).
Then, by using Schauder’s fixed point theorem, we prove the following.Theorem 3.
Under assumption(H2), the problem (1) has a positive continuous solution (u,v) satisfying for each x∈D,
(12)c1MαDφ(x)≤u(x)≤MαDφ(x),c2MαDψ(x)≤v(x)≤MαDψ(x),
where c1∈(0,1] and c2∈(0,1].We note that contrary to the classical caseα=2, in our situation, the solution blows up on the boundary of D.The content of this paper is organized as follows. In Section2, we collect some properties of functions belonging to the Kato class of functions Kα(D), which are useful to establish our results. Our main results are proved in Section 3.As usual, letB+(D) be the set of nonnegative Borel measurable functions in D. 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∥∞=supx∈D|u(x)|. When two positive functions f and g are defined on a set S, we write f≈g if the two-sided inequality (1/C)g≤f≤Cg holds on S. We define the potential kernel GαD of ZαD by
(13)GαDf(x):=∫D‍GαD(x,y)f(y)dy,forf∈B+(D),x∈D.
Finally, let us recall some potential theory tools that are needed, and we refer to [14, 16, 17] for more details. For q∈B+(D), we define the kernel Vq on B+(D) by
(14)Vqf(x):=∫0∞E~x(e-∫0t‍q(ZαD(s))dsf(ZαD(t)))dt,x∈D,
with V0:=V=GαD, where E~x stands for the expectation with respect to ZαD starting from x. If q satisfies Vq<∞, we have the following resolvent equation:
(15)V=Vq+Vq(qV)=Vq+V(qVq).
In particular, if u∈B+(D) is such that V(qu)<∞, then we have
(16)(I-Vq(q.))(I+V(q.))u=(I+V(q.))(I-Vq(q.))u=u.


## 2. The Kato Class of FunctionsKα(D)

Proposition 4 (see [14]).
Let
q
be a function in
K
α
(
D
)
, then we have the following.
(i)
a
α
(
q
)
:
=
sup
x
,
y
∈
D
∫
D
(
G
α
D
(
x
,
z
)
G
α
D
(
z
,
y
)
/
G
α
D
(
x
,
y
)
)
|
q
(
z
)
|
d
z
<
∞.
(ii)
Let
h
be a positive excessive function on
D
with respect to
Z
α
D. 									Then, we have(17)∫DGαD(x,y)h(y)|q(y)|dy≤aα(q)h(x).
Furthermore, for each
x
0
∈
D
¯,
we have(18)limr→0(supx∈D1h(x)∫B(x0,r)∩DGαD(x,y)h(y)|q(y)|dy)=0.
(iii)
The function  x→(δ(x))α-1q(x)is inL1(D).The next two lemmas will play a special role.Lemma 5.
Letq be a nonnegative function in Kα(D) and h be a positive finite excessive function on D with respect to ZαD. Then, for all x∈D, we have
(19)exp(-aα(q))h(x)≤h(x)-Vq(qh)(x)≤h(x).Proof.
Leth be a positive finite excessive function on D with respect to ZαD. Then, by [18, Chapter II, proposition 3.11], there exists a sequence (fk)k of nonnegative measurable functions in D such that h=supkVfk. Let x∈D and k∈ℕ such that 0<Vfk<∞. Consider θ(t)=Vtqfk(x), for t≥0. Then, by (14), the function θ is completely monotone on [0,∞), and so from the Hölder inequality and [19, Theorem 12a], the function logθ is convex on [0,∞). This implies that
(20)θ(0)≤θ(1)exp(-θ′(0)θ(0)).
That is
(21)Vfk(x)≤Vqfk(x)exp(V(qVfk)(x)Vfk(x)).
Hence, it follows from (17) that
(22)Vfk(x)≤Vqfk(x)exp(aα(q)).
Consequently, from (15) we obtain
(23)exp(-aα(q))Vfk(x)≤Vfk(x)-Vq(qVfk)(x)≤Vfk(x).
The result holds by letting k→∞.Lemma 6.
Letq be a nonnegative function in Kα(D), then the family of functions
(24)Λq={1MαDφ(x)∫DGαD(x,y)MαDφ(y)f(y)dy,|f|≤q}
is uniformly bounded and equicontinuous in D¯. Consequently, Λq is relatively compact in C0(D).Proof.
Takingh≡MαDφ in (17), we deduce that for |f|≤q and x∈D, we have
(25)|∫DGαD(x,y)MαDφ(x)MαDφ(y)f(y)dy|≤∫DGαD(x,y)MαDφ(x)MαDφ(y)q(y)dy≤aα(q)<∞.
So, the family Λq is uniformly bounded.
Next, we aim at proving that the familyΛq is equicontinuous in D¯.
First, we recall the following interesting sharp estimates onGαD, which is proved in [5]:
(26)GαD(x,y)≈|x-y|α-nmin(1,δ(x)δ(y)|x-y|2).
Let x0∈D¯ and ε>0. By (18), there exists r>0 such that
(27)supz∈D1MαDφ(z)∫B(x0,2r)∩DGαD(z,y)MαDφ(y)q(y)dy≤ε2.
If x0∈D and x,x′∈B(x0,r)∩D, then for |f|≤q, we have
(28)|∫DGαD(x,y)MαDφ(x)MαDφ(y)f(y)dy-∫DGαD(x′,y)MαDφ(x′)MαDφ(y)f(y)dy|≤∫D|GαD(x,y)MαDφ(x)-GαD(x′,y)MαDφ(x′)|MαDφ(y)q(y)dy≤2supz∈D∫B(x0,2r)∩D1MαDφ(z)GαD(z,y)MαDφ(y)q(y)dy+∫(|x0-y|≥2r)∩D|GαD(x,y)MαDφ(x)-GαD(x′,y)MαDφ(x′)|MαDφ(y)q(y)dy≤ε+∫(|x0-y|≥2r)∩D|GαD(x,y)MαDφ(x)-GαD(x′,y)MαDφ(x′)|MαDφ(y)q(y)dy.
On the other hand, for every y∈Bc(x0,2r)∩D and x,x′∈B(x0,r)∩D, by using (26) and the fact that MαDφ(z)≈(δ(z))α-2, we have
(29)|1MαDφ(x)GαD(x,y)-1MαDφ(x′)GαD(x′,y)|MαDφ(y)≤C(δ(y))α-1.
Now since x→(1/MαDφ(x))GαD(x,y) is continuous outside the diagonal and q∈Kα(D), we deduce by the dominated convergence theorem and Proposition 4 (iii) that
(30)∫(|x0-y|≥2r)∩D|GαD(x,y)MαDφ(x)-GαD(x′,y)MαDφ(x′)|MαDφ(y)q(y)dy⟶0as|x-x′|⟶0.
If x0∈∂D and x∈B(x0,r)∩D, then we have
(31)|∫DGαD(x,y)MαDφ(x)MαDφ(y)f(y)dy|≤ε2+∫(|x0-y|≥2r)∩DGαD(x,y)MαDφ(x)MαDφ(y)q(y)dy.
Now, since GαD(x,y)/MαDφ(x)→0 as |x-x0|→0, for |x0-y|≥2r, then by the same argument as above, we get
(32)∫(|x0-y|≥2r)∩DGαD(x,y)MαDφ(x)MαDφ(y)q(y)dy⟶0as|x-x0|⟶0.
Consequently, by Ascoli’s theorem, we deduce that Λq is relatively compact in C0(D).


## 3. Proofs of Theorems2 and 3

The next Lemma will be used for uniqueness.Lemma 7 (see [14, Proposition 6]).
Leth∈B+(D) and υ be a nonnegative excessive function on D with respect to ZαD. Let z be a Borel measurable function in D such that V(h|z|)<∞ and υ=z+V(hz). Then, z satisfies
(33)0≤z≤υ.Proof of Theorem2.
Letφ be a positive continuous function on ∂D. We recall that on D, we have
(34)MαDφ(x)≈MαD1(x)≈(δ(x))α-2.
Let p0~=γ(MαDφ)γ-1p0 and put c0=e-aα(p0~), where aα(p0~) is given by Proposition 4(i). Since by (H1), p0~∈Kα(D), it follows from Proposition 4 that V(p0~)≤aα(p0~)<∞. Define the nonempty closed bounded convex Λ by
(35)Λ={ω∈B+(D):c0≤ω≤1}.
Let T be the operator defined on Λ by
(36)Tω:=1-1MαDφVp0~(p0~MαDφ)+1MαDφVp0~(p0~ωMαDφ-p0(ωMαDφ)γ).
We claim that T maps Λ to itself. Indeed, for each ω∈Λ, we have
(37)Tω≤1-1MαDφVp0~(p0(ωMαDφ)γ)≤1.
On the other hand, since the function p0~ωMαDφ-p0(ωMαDφ)γ≥0, we deduce by Lemma 5 with h=MαDφ, that Tω≥1-(1/MαDφ)Vp0~(p0~MαDφ)≥c0. Hence, TΛ⊂Λ. Next, we aim at proving that T is nondecreasing on Λ. To this end, we let ω1, ω2∈Λ such that ω1≤ω2. Using the fact that the function t→γt-tγ is nondecreasing on [0,1], we deduce that
(38)Tω2-Tω1=1MαDφVp0~(p0~ω2MαDφ-p0(ω2MαDφ)γ)-1MαDφVp0~(p0~ω1MαDφ-p0(ω1MαDφ)γ)=1MαDφVp0~(p0(MαDφ)γ[(γω2-ω2γ)-(γω1-ω1γ)])≥0.
Next, we define the sequence (ωk)k≥0 by
(39)ω0=1-1MαDφVp0~(p0~MαDφ),ωk+1=Tωk.
Clearly ω0∈Λ and ω1=Tω0≥ω0. Thus, from the monotonicity of T, we deduce that
(40)c0≤ω0≤ω1≤⋯≤ωk≤1.
So, the sequence (ωk)k≥0 converges to a measurable function ω∈Λ. Therefore, by applying the monotone convergence theorem, we obtain
(41)ω=1-1MαDφVp0~(p0~MαDφ)+1MαDφVp0~(p0~ωMαDφ-p0(ωMαDφ)γ).
Put u=ωMαDφ. Then, we have
(42)u=MαDφ-Vp0~(p0~MαDφ)+Vp0~(p0~u-p0uγ)
or equivalently
(43)u-Vp0~(p0~u)=MαDφ-Vp0~(p0~MαDφ)-Vp0~(p0uγ).
Observe that by Proposition 4(ii), we have V(p0~u)<∞. So, applying the operator (I+V(p0~.)) on both sides of (43), we deduce by using (15) and (16) that
(44)u=MαDφ-V(p0uγ).
Now, using (H1) and similar argument as in the proof of Lemma 6, we prove that x→(1/MαDφ)V(p0uγ)∈C0(D). So, u is a continuous function in D, and u is a solution of (5). It remains to prove the uniqueness of such a solution. Let u be a continuous solution of (5). Since the function x→u(x)/MαD1(x) is continuous and positive in D such that limx→z∈∂D(u(x)/MαD1(x))=φ(z), it follows that u(x)≈MαD1(x)≈MαDφ(x). Then, by using this fact and Lemma 6, we have
(45)(-Δ∣D)α/2(u+V(p0uγ))=0inD,limx→z∈∂D(u+V(p0uγ))(x)MαD1(x)=φ(z)inD.
So, from the uniqueness of the problem (8) (see [3]), we deduce that
(46)u+V(p0uγ)=MαDφinD.
It follows that if u and v are two continuous solution of (5), then z=v-u satisfies
(47)z+V(p0hz)=0inD,
where h is the nonnegative measurable function defined in D by
(48)h(x)={vγ-uγv-u,ifu(x)≠v(x),0,ifu(x)=v(x).
Since V(p0h|z|)<∞, we deduce by Lemma 7 that z=0, and so u=v.Proof of Theorem3.
Letp~=σ(MαDφ)σ-1(MαDψ)rp and q~=β(MαDψ)β-1(MαDφ)sq.
Putc1=e-aα(p~), c2=e-aα(q~). Note that from (H2) and Proposition 4, we have aα(p~)<∞ and aα(q~)<∞. Consider the nonempty closed convex set Γ defined by
(49)Γ={(y,z)∈C(D¯)×C(D¯):c1≤y≤1,c2≤z≤1}.
Let T be the operator defined on Γ by T(y,z):=(ω,θ), such that (u~=ωMαDφ,v~=θMαDψ) is the unique positive continuous solution of the following problem:
(50)(-Δ∣D)α/2u~+((MαDψ)rzrp)(x)u~σ=0inD,(-Δ∣D)α/2v~+((MαDφ)sysq)(x)v~β=0inD,limx→z∈∂Du~(x)MαD1(x)=φ(z),limx→z∈∂Dv~(x)MαD1(x)=ψ(z).
According to Theorem 2, we have
(51)ω=1-1MαDφV(zrωσ(MαDψ)r(MαDφ)σp)θ=1-1MαDψV(ysωβ(MαDφ)s(MαDψ)βq).
Moreover, we have c1≤ω≤1 and c2≤θ≤1 and by Lemma 6, T(Γ) is equicontinuous on D¯. Since T(Γ) is also bounded, then we deduce that T(Γ) is relatively compact in C(D¯)×C(D¯). This implies in particular that T(Γ)⊂Γ.
Next, we shall prove the continuity of the operatorT in Γ in the supremum norm. Let (yk,zk)k be a sequence in Γ which converges uniformly to a function (y,z) in Γ. Put (ωk,θk)=T(yk,zk) and (ω,θ)=T(y,z). Then, we have
(52)|ωk-ω|=|1MαDφV(zrωσ(MαDψ)r(MαDφ)σp)-1MαDφV(zkrωkσ(MαDψ)r(MαDφ)σp)|≤1σMαDφV(|zrωσ-zkrωkσ|(MαDφ)p~).
Using the fact that |zrωσ-zkrωkσ|≤2 and that p~∈Kα(D), we deduce by Proposition 4 and the dominated convergence theorem, that ωk→ω as k→∞. Similarly, we prove that θk→θ as k→∞. So, T(yk,zk)→T(y,z) as k→∞. Since T(Γ) is relatively compact in C(D¯)×C(D¯), we deduce that
(53)∥T(yk,zk)-T(y,z)∥∞⟶0ask⟶∞.
From the Schauder fixed point theorem there exists (y,z)∈Γ such that T(y,z)=(y,z) or equivalently
(54)u=MαDφ-V(puσvr),v=MαDψ-V(qusvβ),
where (u,v)=(yMαDφ,zMαDψ). The pair (u,v) is a required solution of (1) in the sense of distributions. This completes the proof.



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*Source: 140130-2013-05-07.xml*