Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 208, pp. 1–11.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE OF POSITIVE SOLUTIONS FOR NONLINEAR
FRACTIONAL SYSTEMS IN BOUNDED DOMAINS
IMED BACHAR
Abstract. We prove the existence of positive continuous solutions to the
nonlinear fractional system
(−∆|D )α/2 u + λg(., v) = 0,
(−∆|D )α/2 v + µf (., u) = 0,
in a bounded C 1,1 -domain D in Rn (n ≥ 3), subject to Dirichlet conditions,
where 0 < α ≤ 2, λ and µ are nonnegative parameters. The functions f and g
are nonnegative continuous monotone with respect to the second variable and
satisfying certain hypotheses related to the Kato class.
1. Introduction and statement of main results
Let χ = (Ω, F, Ft , Xt , θt , P x ) be a Brownian motion in Rn , n ≥ 3 and π =
(Ω, G, Tt ) be an α2 -stable process subordinator starting at zero, where 0 < α ≤ 2
and such that χ and π are independent. Let D be a bounded C 1,1 -domain in Rn and
ZαD be the subordinate killed Brownian motion process. This process is obtained
by killing χ at τD , the first exit time of χ from D giving the process χD and then
subordinating this killed Brownian motion using the α/2-stable subordinator Tt .
For more description of the process ZαD we refer to [7, 9, 14, 15]. Note that the
infinitesimal generator of the process ZαD is the fractional power(−∆|D )α/2 of the
negative Dirichlet Laplacian in D, which is a prototype of non-local operator and
a very useful object in analysis and partial differential equations, see, for instance
[13, 16].
In this article, we will deal with the existence of positive continuous solutions
for the nonlinear fractional system
(−∆|D )α/2 u + λg(., v) = 0
in D, in the sense of distributions
(−∆|D )α/2 v + µf (., u) = 0
in D, in the sense of distributions
lim
x→z∈∂D
u(x)
= ϕ(z),
MαD 1(x)
lim
x→z∈∂D
(1.1)
v(x)
= ψ(z),
MαD 1(x)
2000 Mathematics Subject Classification. 35J60, 34B27, 35B44.
Key words and phrases. Fractional nonlinear systems; Green function; positive solutions;
blow-up solutions.
c
2012
Texas State University - San Marcos.
Submitted September 8, 2012. Published November 25, 2012.
1
2
I. BACHAR
EJDE-2012/208
where λ, µ are nonnegative parameters, ϕ, ψ are positive continuous functions on
∂D and MαD 1 is the nonnegative harmonic function with respect to ZαD given by
the formula (see [7, Theorem 3.1],
Z
1 − α2 ∞ −2+ α
2 (1 − P D 1(x))dt,
t
(1.2)
MαD 1(x) =
t
Γ( α2 ) 0
where (PtD )t>0 is the semi-group corresponding to the killed Brownian motion χD .
Note that from [15, remark 3.3], there exists a constant C > 0 such that
α−2
α−2
1
δ(x)
≤ MαD 1(x) ≤ C δ(x)
,
C
for all x ∈ D,
(1.3)
where δ(x) denotes the Euclidian distance from x to the boundary of D.
In the classical case (i.e. α = 2), there exist a lot of work related to the existence
and nonexistence of solutions for the problem (1.1); see for example, the papers
of Cirstea and Radulescu [3], Ghanmi et al [6], Ghergu and Radulescu [8], Lair
and Wood [10, 11] and references therein. Most of the studies of these papers turn
about the existence or the nonexistence of positive radial ones. In [11], the authors
studied the system (1.1) with α = 2, in the case µf (., u) = pus , λg(., v) = qv r ,
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
p(x) + q(x) ≤ C|x|−(2+γ)
for some positive constant γ and |x| large.
D
Throughout this article, we denote by GD
α the Green function of Zα . We recall
D
the following interesting sharp estimates on Gα due to [14]. Namely, there exists a
positive constant C > 0 such that for all x, y in D, we have
1
H(x, y) ≤ GD
α (x, y) ≤ CH(x, y),
C
(1.4)
where
δ(x)δ(y) 1
min
1,
.
|x − y|n−α
|x − y|2
H(x, y) =
We also denote by MαD ϕ the unique positive continuous solution of
(−∆|D )α/2 u = 0
in D, in the sense of distributions
lim
x→z∈∂D
u(x)
= ϕ(z),
MαD 1(x)
(1.5)
which is given (see [7]) by
MαD ϕ(x) =
α
1
−1
E x (ϕ(XτD )τD2 ).
Γ(α/2)
(1.6)
We aim at giving two existence results for (1.1) as f and g are nondecreasing or
nonincreasing with respect to the second variable. More precisely, to state our first
existence result, we assume that f, g : D × [0, ∞) → [0, ∞) are Borel measurable
functions satisfying
(H1) The functions f and g are continuous and nondecreasing with respect to
the second variable.
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FRACTIONAL SYSTEMS IN BOUNDED DOMAINS
(H2) The functions
1
pe(y) := D
f (y, MαD ϕ(y))
Mα ψ(y)
and qe(y) :=
3
1
g(y, MαD ψ(y))
MαD ϕ(y)
belong to the Kato class Kα (D), defined below.
Definition 1.1 ([5]). A Borel measurable function q in D belongs to the Kato class
Kα (D) if
Z
δ(y) D
Gα (x, y)|q(y)|dy = 0.
lim sup
r→0 x∈D (|x−y|≤r(∩D δ(x)
This class is quite rich, it contains for example any function belonging to Ls (D),
with s > n/α (see Example 2.1 below). On the other hand, it has been shown in
[5], that
−γ
∈ Kα (D), for γ < α.
(1.7)
x → δ(x)
For more examples of functions belonging to Kα (D), we refer to [5]. Note that for
the classical case (i.e. α = 2), the class K2 (D) was introduced and studied in [12].
Our first existence result is the following.
Theorem 1.2. Assume that (H1), (H2) are satisfied. Then there exist two constants λ0 > 0 and µ0 > 0 such that for each λ ∈ [0, λ0 ) and each µ ∈ [0, µ0 ),
problem (1.1) has a positive continuous solution such that
λ
)MαD ϕ ≤ u ≤ MαD ϕ
λ0
µ
(1 −
)MαD ψ ≤ v ≤ MαD ψ
µ0
(1 −
in D,
in D.
In particular limx→z∈∂D u(x) = ∞ and limx→z∈∂D v(x) = ∞.
We note that in [6], the authors studied a problem similar to (1.1) for the case
α = 2. They have obtained positive continuous bounded solution (u, v). Here, we
are interesting in the fractional setting.
As second existence result, we aim at proving the existence of blow-up positive
continuous solutions for the system
(−∆|D )α/2 u + p(x)g(v) = 0
in D, in the sense of distributions
(−∆|D )α/2 v + q(x)f (u) = 0
in D, in the sense of distributions
lim
x→z∈∂D
u(x)
= ϕ(z),
MαD 1(x)
lim
x→z∈∂D
(1.8)
v(x)
= ψ(z),
MαD 1(x)
where ϕ, ψ are positive continuous functions on ∂D and p, q are nonnegative Borel
measurable functions in D. To this end, we fix φ a positive continuous functions
on ∂D, we put h0 = MαD φ and we assume the following:
(H3) The functions f, g : (0, ∞) → [0, ∞) are continuous and nonincreasing.
g(h0 )
0)
(H4) The functions p0 := p f (h
h0 and q0 := q h0 belongs to the class Kα (D).
As a typical example of nonlinearity f and p satisfying (H3)-(H4), we have
f (t) = t−ν , for ν > 0, and p a nonnegative Borel measurable function such that
C
r , for all x ∈ D,
p(x) ≤
δ(x)
for some C > 0 and r + (1 + ν)(α − 2) < α.
4
I. BACHAR
EJDE-2012/208
Indeed, since there exists a constant c > 0, such that for all x ∈ D, h0 (x) ≥
α−2
0)
c δ(x)
, we deduce by (1.7), that the function p0 := p f (h
∈ Kα (D). Using
h0
the Schauder’s fixed point theorem, we prove the following result.
Theorem 1.3. Under the assumptions (H3), (H4), there exists a constant c > 1
such that if ϕ ≥ cφ and ψ ≥ cφ on ∂D, then problem (1.8) has a positive continuous
solution (u, v) satisfying for each x ∈ D,
h0 ≤ u ≤ MαD ϕ
in D,
MαD ψ
in D.
h0 ≤ v ≤
In particular limx→z∈∂D u(x) = ∞ and limx→z∈∂D v(x) = ∞.
This result extends the one of Athreya [1], who considered the problem
∆u = g(u),
u=ϕ
in Ω
on ∂Ω,
(1.9)
where Ω is a simply connected bounded C 2 -domain and g(u) ≤ max(1, u−α ), for
f0
0 < α < 1. Then he proved that there exists a constant c > 1 such that if ϕ ≥ ch
f
on ∂Ω, where h0 is a fixed positive harmonic function in Ω, problem (∗) has a
f0 .
positive continuous solution u such that u ≥ h
The content of this article is organized as follows. In Section 2, we collect some
properties of functions belonging to the Kato class Kα (D), which are useful to
establish our results. Our main results are proved in Section 3.
As usual, let B + (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
kuk∞ = sup |u(x)|. The letter C will denote a generic positive constant which may
x∈D
vary from line to line. When two positive functions ρ and θ are defined on a set S,
we write ρ ≈ θ if the two sided inequality C1 θ ≤ ρ ≤ Cθ holds on S. For ρ ∈ B + (D),
D
we define the potential kernel GD
α of Zα by
Z
GD
GD
for x ∈ D
α ρ(x) :=
α (x, y)ρ(y)dy,
D
and we denote by
Z
aα (ρ) := sup
x,y∈D
D
D
GD
α (x, z)Gα (z, y)
ρ(y)dy.
D
Gα (x, y)
(1.10)
2. The Kato class Kα (D)
n
Example 2.1. For s > α
, we have Ls (D) ⊂ Kα (D). Indeed, let 0 < r < 1 and
n
s
q ∈ L (D) with s > α . Using (1.4), there exists a constant C > 0, such that for
each x, y ∈ D
δ(y) D
1
G (x, y) ≤ C
.
(2.1)
δ(x) α
|x − y|n−α
This fact and the Hölder inequality imply that
Z
δ(y) GD
α (x, y)|q(y)|dy
δ(x)
B(x,r)∩D
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FRACTIONAL SYSTEMS IN BOUNDED DOMAINS
|q(y)|
dy
n−α
B(x,r)∩D |x − y|
Z
1/s Z
s
|q(y)| dy
≤C
5
Z
≤C
D
r
≤C
Z
s
|x − y|(α−n) s−1 dy
s−1
s
B(x,r)
t
s
+n−1
(α−n) s−1
dt
s−1
s
→ 0,
0
s
+ n − 1 > −1 when s >
as r → 0, since (α − n) s−1
n
α.
Proposition 2.2 ([5]). Let q be a function in Kα (D), then we have
(i) aα (q) < ∞.
(ii) Let h be a positive excessive function on D with respect to ZαD . Then we
have
Z
GD
(2.2)
α (x, y)h(y)|q(y)|dy ≤ aα (q)h(x).
D
Furthermore, for each x0 ∈ D, we have
Z
1
GD
(x,
y)h(y)|q(y)|dy
= 0.
lim sup
α
r→0 x∈D h(x) B(x ,r)∩D
0
α−1
(iii) The function x → δ(x)
q(x) is in L1 (D).
(2.3)
Lemma 2.3. Let q be a nonnegative function in Kα (D), then the family of functions
Z
n
o
1
D
D
Λq =
G
(x,
y)M
ϕ(y)ρ(y)dy,
|ρ|
≤
q
α
MαD ϕ(x) D α
is uniformly bounded and equicontinuous in D. Consequently Λq is relatively compact in C0 (D).
Proof. Taking h ≡ MαD ϕ in (2.2), we deduce that for ρ such that |ρ| ≤ q and x ∈ D,
we have
Z
Z
GD
GD
α (x, y)
α (x, y)
D
M
ϕ(y)ρ(y)dy
≤
MαD ϕ(y)q(y)dy ≤ aα (q) < ∞. (2.4)
α
D ϕ(x)
D
M
M
D
D
α
α ϕ(x)
So the family Λq is uniformly bounded.
Next we aim at proving that the family Λq is equicontinuous in D. Let x0 ∈ D
and ε > 0. By (2.3), there exists r > 0 such that
Z
1
ε
D
sup D
GD
α (z, y)Mα ϕ(y)q(y)dy ≤ .
2
z∈D Mα ϕ(z) B(x0 ,2r)∩D
If x0 ∈ D and x, x0 ∈ B(x0 , r) ∩ D, then for ρ such that |ρ| ≤ q, we have
Z
Z GD (x, y)
0
GD
α
α (x , y)
D
D
M
ϕ(y)ρ(y)dy
−
M
ϕ(y)ρ(y)dy
α
α
D
D
0
D Mα ϕ(x)
D Mα ϕ(x )
Z
D 0
GD
α (x, y) − Gα (x , y) MαD ϕ(y)q(y)dy
≤
D
D
0
Mα ϕ(x )
D Mα ϕ(x)
Z
1
GD (z, y)MαD ϕ(y)q(y)dy
≤ 2sup
D ϕ(z) α
M
z∈D B(x0 ,2r)∩D
α
Z
D 0
GD
α (x, y) − Gα (x , y) MαD ϕ(y)q(y)dy
+
D
D
0
Mα ϕ(x )
(|x0 −y|≥2r)∩D Mα ϕ(x)
6
I. BACHAR
EJDE-2012/208
D 0
GD
α (x, y) − Gα (x , y) MαD ϕ(y)q(y)dy.
D
D
0
Mα ϕ(x )
(|x0 −y|≥2r)∩D Mα ϕ(x)
Z
≤ε+
On the other hand, for every y ∈ B c (x0 , 2r) ∩ D and x, x0 ∈ B(x0 , r) ∩ D, by
using (1.4) and the fact that MαD ϕ(z) ≈ (δ(z))α−2 , we have
D
1
1
0
GD
GD
α (x, y) −
α (x , y) Mα ϕ(y)
D
D
0
Mα ϕ(x)
Mα ϕ(x )
MαD ϕ(y) D 0
M D ϕ(y) D
Gα (x, y) + D
G (x , y)
≤ αD
Mα ϕ(x)
Mα ϕ(x0 ) α
3−α
α−1 i
h δ(x)3−α δ(y)α−1
δ(x0 )
δ(y)
≤C
+
|x − y|n+2−α
|x0 − y|n+2−α
h
i
1
1
≤C
(δ(y))α−1
+
|x − y|n+2−α
|x0 − y|n+2−α
α−1
≤ C δ(y)
.
Now since x 7→ M D1ϕ(x) GD
α (x, y) is continuous outside the diagonal and q ∈
α
Kα (D), we deduce by the dominated convergence theorem and Proposition 2.2
(iii), that
Z
D 0
GD
α (x, y) − Gα (x , y) MαD ϕ(y)q(y)dy → 0 as |x − x0 | → 0.
0
D
D
Mα ϕ(x )
(|x0 −y|≥2r)∩D Mα ϕ(x)
If x0 ∈ ∂D and x ∈ B(x0 , r) ∩ D, then
Z
Z
ε
GD
GD
α (x, y)
α (x, y)
D
M
ϕ(y)ρ(y)dy
≤
+
MαD ϕ(y)q(y)dy.
α
D
D
2
D Mα ϕ(x)
(|x0 −y|≥2r)∩D Mα ϕ(x)
GD (x,y)
Now, since MαD ϕ(x) → 0 as |x − x0 | → 0, for |x0 − y| ≥ 2r, then by same argument
α
as above, we obtain
Z
GD
α (x, y)
MαD ϕ(y)q(y)dy → 0 as |x − x0 | → 0.
D
(|x0 −y|≥2r)∩D Mα ϕ(x)
So the family Λq is equicontinuous in D. Therefore by Ascoli’s theorem, the family
Λq becomes relatively compact in C0 (D).
3. Proofs of Theorems 1.2 and 1.3
Proof of Theorem 1.2. Put
MαD ϕ(x)
,
D
D ψ))(x)
x∈D Gα (g(., Mα
λ0 := inf
MαD ψ(x)
.
D
D ϕ))(x)
x∈D Gα (f (., Mα
µ0 := inf
Using (H2) and (2.2) we deduce that λ0 > 0 and µ0 > 0.
Let λ ∈ [0, λ0 ) and µ ∈ [0, µ0 ). Then for each x ∈ D, we have
D
D
λ 0 GD
α (g(., Mα ψ))(x) ≤ Mα ϕ(x)
D
D
µ0 GD
α (f (., Mα ϕ))(x) ≤ Mα ψ(x).
So we define the sequences (uk )k≥0 and (vk )k≥0 by
v0 = 1,
uk (x) = 1 −
λ
MαD ϕ(x)
Z
D
D
GD
α (x, y)g y, vk (y)Mα ψ(y) dy,
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FRACTIONAL SYSTEMS IN BOUNDED DOMAINS
µ
D
Mα ψ(x)
By induction, we can see that
Z
vk+1 (x) = 1 −
7
D
GD
α (x, y)f y, uk (y)Mα ϕ(y) dy.
D
λ
) ≤ uk ≤ 1,
λ0
µ
0 < (1 −
) ≤ vk+1 ≤ 1.
µ0
Next, we prove that the sequence (uk )k≥0 is nondecreasing and the sequence
(vk )k≥0 is nonincreasing. Indeed, we have
µ
(f (., u0 MαD ϕ)) ≤ 0
v1 − v0 = − D GD
Mα ψ α
0 < (1 −
and therefore by (H1), we obtain that
λ
[g(., v0 MαD ψ) − g(., v1 MαD ψ)] ≥ 0.
u 1 − u 0 = D GD
Mα ϕ α
By induction, we assume that uk ≤ uk+1 and vk+1 ≤ vk . Then we have
µ
[f (., uk MαD ϕ) − f (., uk+1 MαD ϕ)] ≤ 0
vk+2 − vk+1 = D GD
Mα ψ α
and
λ
uk+2 − uk+1 = D GD
[g(., vk+1 MαD ψ) − g(., vk+2 MαD ψ)] ≥ 0.
Mα ϕ α
Therefore, the sequences (uk )k≥0 and (vk )k≥0 converge respectively to two functions
u
e and ve satisfying
λ
e ≤ 1,
0 < (1 − ) ≤ u
λ0
(3.1)
µ
0 < (1 −
) ≤ ve ≤ 1.
µ0
On the other hand, using (H1), Proposition 2.2 and the dominate convergence
theorem, we deduce that
Z
λ
GD (x, y)g(y, ve(y)MαD ψ(y))dy,
u
e(x) = 1 − D
Mα ϕ(x) D α
Z
µ
GD (x, y)f (y, u
e(y)MαD ϕ(y))dy.
ve(x) = 1 − D
Mα ψ(x) D α
Now by using (H1), (H2) and similar arguments as in the proof of Lemma 2.3, we
deduce that u
e and ve belongs to C(D).
Put u = u
eMαD ϕ and v = veMαD ψ. Then u and v are continuous in D and satisfy
Z
u(x) = MαD ϕ(x) − λ
GD
α (x, y)g(y, v(y))dy
D
Z
(3.2)
D
D
v(x) = Mα ψ(x) − µ
Gα (x, y)f y, u(y) dy.
D
α−2
In addition, since for each x ∈ D, f y, u(y) ≤ C δ(y)
pe(y) and g y, u(y) ≤
α−2
C δ(y)
qe(y), we deduce by Proposition 2.2 (iii) that the map y → f y, u(y) ∈
L1loc (D) and y → g y, u(y) ∈ L1loc (D). On the other hand, by (3.2), we have that
1
D
1
α/2
GD
on both
α f (., u) ∈ Lloc (D) and Gα g(., v) ∈ Lloc (D). Hence, applying (−∆|D )
sides of (3.2), we conclude by [9, p. 230] that (u, v) is the required solution.
8
I. BACHAR
EJDE-2012/208
Example 3.1. Let ν ≥ 0, σ ≥ 0, r + (1 − σ)(α − 2) < α and β + (1 − ν)(α − 2) < α.
Let p and q be two positive Borel measurable functions such that
−r
−β
p(x) ≤ C δ(x)
, q(x) ≤ C δ(x)
for all x ∈ D.
Let ϕ and ψ be positive continuous functions on ∂D. Therefore by Theorem 1.2,
there exist two constants λ0 > 0 and µ0 > 0 such that for each λ ∈ [0, λ0 ) and each
µ ∈ [0, µ0 ), the problem
(−∆|D )α/2 u + λp(x)v σ = 0
α/2
(−∆|D )
lim
ν
v + µq(x)u = 0
x→z∈∂D
in D, in the sense of distributions
in D, in the sense of distributions
u(x)
= ϕ(z),
MαD 1(x)
lim
x→z∈∂D
v(x)
= ψ(z),
MαD 1(x)
has a positive continuous solution (u, v) such that
λ
)MαD ϕ ≤ u ≤ MαD ϕ
λ0
µ
(1 −
)MαD ψ ≤ v ≤ MαD ψ
µ0
(1 −
in D,
in D.
In particular, limx→z∈∂D u(x) = ∞ and limx→z∈∂D v(x) = ∞.
Proof of Theorem 1.3. Let c := 1+aα (p0 )+aα (q0 ), where aα (p0 ) and aα (q0 ) are the
constant defined by the formula (1.10). We recall that from (H4) and Proposition
2.2 (i), we have aα (p0 ) < ∞ and aα (q0 ) < ∞. Let ϕ, ψ be positive continuous
functions on ∂D such that ϕ ≥ cφ and ψ ≥ cφ on ∂D. It follows from the integral
representation of MαD ϕ(x) and MαD ψ(x) (see [5, p. 265]), that for each x ∈ D we
have
MαD ϕ(x) ≥ ch0 (x) and MαD ψ(x) ≥ ch0 (x).
(3.3)
Let Λ be the nonempty closed convex set given by
h0
Λ = ω ∈ C(D) : D ≤ ω ≤ 1 .
Mα ϕ
We define the operator T on Λ by
T (ω) = 1 −
1
MαD ϕ
D
D
D
GD
α (pf Mα ψ − Gα (qg(ωMα ϕ)) ).
We will prove that T has a fixed point. Since for ω ∈ Λ, we have ω ≥
we deduce from hypotheses (H3), (H4) and (2.2) that
D
D
D
GD
α (qg(ωMα ϕ)) ≤ Gα (qg(h0 )) = Gα (q0 h0 ) ≤ aα (q0 )h0 .
(3.4)
h0
Dϕ ,
Mα
then
(3.5)
So by using (3.3) and (3.5), we obtain
D
D
MαD ψ − GD
α (qg(ωMα ϕ)) ≥ Mα ψ − aα (q0 )h0
≥ ch0 − aα (q0 )h0
= (1 + aα (p0 ))h0
≥ h0 > 0.
Hence, by using again (H3), (H4) and (2.2), we deduce that
D
D
D
D
D
GD
α (pf Mα ψ − Gα (qg(ωMα ϕ)) ) ≤ Gα (pf (h0 )) = Gα (p0 h0 ) ≤ aα (p0 )h0 . (3.6)
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FRACTIONAL SYSTEMS IN BOUNDED DOMAINS
9
Using the fact that MαD ϕ ≈ h0 and Lemma 2.3, we deduce that the family of
functions
1
D
D
D
GD
α (pf Mα ψ − Gα (qg(ωMα ϕ)) ) : ω ∈ Λ
D
Mα ϕ
is relatively compact in C0 (D). Therefore, the set T Λ is relatively compact in
C(D).
Next, we shall prove that T maps Λ into it self.
D
Since MαD ψ − GD
α (qg(ωMα ϕ)) ≥ h0 > 0, we have for all ω ∈ Λ, T ω ≤ 1.
0
≥ MhD0 ϕ , which proves that
Moreover, form (3.6), we obtain T ω ≥ 1 − aαM(pD0 )h
α ϕ
α
T (Λ) ⊂ Λ.
Now, we shall prove the continuity of the operator T in Λ in the supremum norm.
Let (ωk )k∈N be a sequence in Λ which converges uniformly to a function ω in Λ.
Then, for each x ∈ D, we have
h 1
D
GD
pf (MαD ψ − GD
|T ωk (x) − T ω(x)| ≤ D
α (qg(ωk Mα ϕ)))
Mα ϕ(x) α
i
D
− f (MαD ψ − GD
(qg(ωM
ϕ)))
(x).
α
α
On the other hand, by similar arguments as above, we have
D
D
D
D
pf (MαD ψ − GD
(qg(ω
M
ϕ)))
−
f
(M
ψ
−
G
(qg(ωM
ϕ)))
k α
α
α
α
α
h
i
D
D
D
D
≤ p f (MαD ψ − GD
α (qg(ωk Mα ϕ))) + f (Mα ψ − Gα (qg(ωMα ϕ)))
≤ 2p0 h0 .
By the fact that MαD ϕ ≈ h0 , (2.2) and the dominated convergence theorem, We
conclude that for all x ∈ D,
T ωk (x) → T ω(x)
as k → +∞.
Consequently, as T (Λ) is relatively compact in C(D), we deduce that the pointwise
convergence implies the uniform convergence, namely,
kT ωk − T ωk∞ → 0
as k → +∞.
Therefore, T is a continuous mapping from Λ into itself. So, since T (Λ) is relatively
compact in C(D), it follows that T is compact mapping on Λ.
Finally, the Schauder fixed-point theorem implies the existence of a function
ω ∈ Λ such that ω = T ω. Put
u(x) = ω(x)MαD ϕ(x)
and
υ(x) = MαD ψ(x) − GD
α (qg(u))(x),
for x ∈ D.
Then (u, υ) satisfies
u(x) = MαD ϕ(x) − GD
α (pf (υ))(x),
υ(x) = MαD ψ(x) − GD
α (qg(u))(x).
Finally, we verify that (u, υ) is the required solution.
Example 3.2. Let ν > 0, σ > 0, r + (1 + ν)(α − 2) < α and β + (1 + σ)(α − 2) < α.
Let p and q be two nonnegative Borel measurable functions such that
−r
−β
p(x) ≤ C δ(x)
, q(x) ≤ C δ(x)
for all x ∈ D.
10
I. BACHAR
EJDE-2012/208
Let ϕ, ψ and φ be positive continuous functions on ∂D. Then there exists a constant
c > 1 such that if ϕ ≥ cφ and ψ ≥ cφ on ∂D, then the problem
(−∆|D )α/2 u + p(x)v −σ = 0
α/2
(−∆|D )
lim
v + q(x)u
x→z∈∂D
−ν
=0
in D, in the sense of distributions
in D, in the sense of distributions
u(x)
= ϕ(z),
MαD 1(x)
lim
x→z∈∂D
v(x)
= ψ(z),
MαD 1(x)
has a positive continuous solution (u, v) satisfying that for each x ∈ D,
MαD φ ≤ u ≤ MαD ϕ
in D,
MαD φ
in D.
≤v≤
MαD ψ
α−2
≈ v(x) in D.
In particular u(x) ≈ δ(x)
Acknowledgements. The research is supported by NPST Program of King Saud
University; project number 11-MAT1716-02. The author is thankful to the referees for their careful reading of the paper, and for their helpful comments and
suggestions.
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EJDE-2012/208
FRACTIONAL SYSTEMS IN BOUNDED DOMAINS
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Imed Bachar
King Saud University College of Science Mathematics Department P.O. Box 2455 Riyadh
11451 Kingdom of Saudi Arabia
E-mail address: abachar@ksu.edu.sa