Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 33, pp. 1–10.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
POSITIVE BLOWUP SOLUTIONS FOR SOME FRACTIONAL
SYSTEMS IN BOUNDED DOMAINS
RAMZI ALSAEDI
Abstract. Using some potential theory tools and the Schauder fixed point
theorem, we prove the existence of a positive continuous weak solution for the
fractional system
(−∆)α/2 u + p(x)uσ v r = 0,
C 1,1 -domain
(−∆)α/2 v + q(x)us v β = 0
Rn
in a bounded
D in
(n ≥ 3), subject to 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 Kα (D). We also investigate the global behavior of such solution.
1. Introduction and statement of main results
Let D be a bounded C 1,1 -domain in Rn , (n ≥ 3) and 0 < α < 2. This paper is
devoted to the study of the following system involving the fractional Laplacian
(−∆)α/2 u + p(x)uσ v r = 0
(−∆)
α/2
s β
v + q(x)u v = 0
lim
x→z∈∂D
lim
x→z∈∂D
in D,
in D,
u(x)
α 1(x) = ϕ(z),
MD
v(x)
α 1(x) = ψ(z),
MD
(1.1)
where σ, β ≥ 1, r, s ≥ 0 , the functions ϕ and ψ are positive continuous on ∂D
and the nonnegative potential functions p, q are required to satisfy some adequate
hypotheses related to the Kato class Kα (D) (see Definition 1.1 below). The function
α
MD
1(x) is defined on D by
Z
α
α
MD
1(x) =
MD
(x, z)ν(dz).
(1.2)
∂D
Here, ν is an appropriate measure on ∂D which will be defined later in (1.7) and
α
MD
(x, z) is the Martin kernel of the killed symmetric α-stable process X D =
D
(Xt )t>0 in D associated to (−∆)α/2 .
2000 Mathematics Subject Classification. 26A33, 34B27, 35B44, 35B09.
Key words and phrases. Fractional nonlinear systems, Green function, positive solutions,
maximum principle.
c
2013
Texas State University - San Marcos.
Submitted November 2, 2012. Published January 30, 2013.
1
2
R. ALSAEDI
EJDE-2013/33
For the reader convenience, we recall the definition of the fractional Laplacian
−(−∆)α/2 which is a nonlocal operator and can be defined by the formula
−(−∆)α/2 u(x) = cn , α lim
&0
Z
(|x−y|>)
u(y) − u(x)
,
|x − y|n+α
where cn,α is a dimensional constant that depends on n and α (see [5, 4, 14] for
more details).
Fractional Laplacian is of interest in many branches of sciences such as physics,
biologists, queuing theory, operation research, mathematical finance and risk estimation. The fractional powers of the Laplacian in all of Rn are useful to describe
anomalous diffusions in plasmas, flames propagation and chemical reactions in liquids, population dynamics, geophysical fluid dynamics, and American options in
finance, see [3, 18, 19].
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
problems related to (1.1); see for example, the papers of Cirstea and Radulescu
[13], Ghanmi et al [16], Ghergu and Radulescu [17], Lair and Wood [20], [21], Mu
et al [24] and references therein. In these works various existence results of positive
bounded solutions or positive blow-up solutions (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 super-solutions
method, variational method and topological methods. These results have been
extended recently by Alsaedi et al in [2] for n ≥ 3 and by Alsaedi in [1] for n = 2, 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 (1.1) in the case n ≥ 3 and positive
continuous solution having logarithm growth at infinity in an exterior domain of
R2 . Recently, there has been intensive interest in studying the fractional Laplacian
(−∆)α/2 , the development of its potential theory and the global behavior of its
Green function Gα
D , see [8, 9, 10, 11].
In this article, we will exploit these potential theory tools and the properties of
the Kato class Kα (D), defined and studied in [7], to study the existence of positive
continuous solutions (in the sense of distributions) for (1.1). More precisely, we aim
first at proving the existence and uniqueness of a positive continuous solution (in
the sense of distributions) for the scalar equation
(−∆)α/2 u + p0 (x)uγ = 0
u>0
lim
x→z∈∂D
in D,
in D,
(1.3)
u(x)
α 1(x) = ϕ(z),
MD
where γ ≥ 1 and p0 is a nonnegative Borel measurable function in D satisfying the
condition
α
(H1) The function x → (δ(x))( 2 −1)(γ−1) p0 (x) ∈ Kα (D),
where δ(x) denotes the Euclidian distance from x to the boundary of D and the
α/2
class Kα (D) is defined by means of the Green’s function Gα
as follows.
D of (−∆)
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POSITIVE BLOWUP SOLUTIONS
3
Definition 1.1 ([7]). A Borel measurable function ϕ in D belongs to the Kato
class Kα (D) if
Z
δ(y) α/2 α
lim sup
GD (x, y)|ϕ(y)|dy = 0.
r→0 x∈D (|x−y|≤r)∩D δ(x)
It has been shown in [7], that the function
x → (δ(x))−λ belongs Kα (D) if and only if λ < α.
(1.4)
For more examples of functions belonging to Kα (D), we refer to [7]. Note that for
the classical case (i.e. α = 2), the class K2 (D) was introduced and studied in [23].
Using (1.4), hypothesis (H1) is satisfied if p0 verifies the following condition:
There exists a constant C > 0, such that for each x ∈ D,
p0 (x) ≤
C
,
(δ(x))τ
with τ + (1 −
α
)(γ − 1) < α.
2
α
To state our existence result for (1.1), we denote by MD
ϕ (see [7]), the unique
positive continuous solution of
(−∆)α/2 u = 0
in D, (in the sense of distributions)
lim
x→z∈∂D
(1.5)
u(x)
α 1(x) = ϕ(z) .
MD
We recall also that in [9], the authors have proved the existence of a constant C > 0
such that for each x ∈ D,
α
α
1
α
(δ(x)) 2 −1 ≤ MD
1(x) ≤ C(δ(x)) 2 −1 .
C
(1.6)
Using some potential theory tools and an approximating sequence, we establish the
following result.
Theorem 1.2. Under hypothesis (H1), problem (1.3) has a unique positive continuous solution satisfying for each x ∈ D
α
α
c0 MD
ϕ(x) ≤ u(x) ≤ MD
ϕ(x),
where the constant c0 ∈ (0, 1].
Next we exploit the result of Theorem 1.2 to prove the existence of a positive
continuous solution (u, v) to the system ((1.1). To this end, we assume the following
hypothesis:
(H2) The functions p, q are nonnegative Borel measurable functions such that
α
x 7→ (δ(x))( 2 −1)(σ+r−1) p(x) ∈ Kα (D),
α
x 7→ (δ(x))( 2 −1)(β+s−1) q(x) ∈ Kα (D).
Then by using the Schauder’s fixed point theorem, we prove the following result.
Theorem 1.3. Under assumption (H2), system (1.1) has a positive continuous
solution (u, v) satisfying: for each x ∈ D,
α
α
α
α
c1 MD
ϕ(x) ≤ u(x) ≤ MD
ϕ(x) and c2 MD
ψ(x) ≤ v(x) ≤ MD
ψ(x),
where c1 , c2 constants in (0, 1].
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R. ALSAEDI
EJDE-2013/33
We note that contrary to the classical case α = 2 and n ≥ 3, in our situation the
solution blows up on the boundary of D.
The content of this paper 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)|. When two positive functions f and g are defined on a set S, we
x∈D
write f ≈ g if the two sided inequality C1 g ≤ f ≤ Cg holds on S.
Let GD be the Green function of the Dirichlet Laplacian in D. The Martin
kernel MD (., .) of the killed Brownian motion is defined by
MD (x, z) =
lim
D3y→z
GD (x, y)
GD (x, z)
for x ∈ D and z ∈ ∂D.
Similarly, the Martin Kernel of the killed process X D is defined by
α
MD
(x, z) =
lim
D3y→z
Gα
D (x, y)
Gα
D (x, z)
for x ∈ D and z ∈ ∂D.
Using, the Hergoltz theorem, there exists a positive measure ν in ∂D such that
Z
1=
MD (x, z) ν(dz).
(1.7)
∂D
α
1. We define the potential kernel Gα
This measure ν is used in (1.2) to define MD
D
D
of X by
Z
+
Gα
f
(x)
:=
Gα
(1.8)
D
D (x, y)f (y)dy, for f ∈ B (D) and x ∈ D.
D
Finally, let us recall some potential theory tools that will be needed in section 3
and we refer to [7, 12, 22] for more details. For q ∈ B + (D), we define the kernel Vq
on B + (D) by
Z ∞
Rt
D
Vq f (x) :=
E x (e− 0 q(Xs )ds f (XtD ))dt, x ∈ D,
(1.9)
0
x
with V0 := V = Gα
D and E stands for the expectation with respect to the symD
metric α-stable process X starting from x. If q satisfies V q < ∞, we have the
following resolvent equation
V = Vq + Vq (qV ) = Vq + V (qVq ).
(1.10)
It follows that for each each measurable function u in D such that V (q|u|) < ∞,
we have
(I − Vq (q.))(I + V (q.))u = (I + V (q.))(I − Vq (q.))u = u.
(1.11)
2. The Kato class Kα (D)
Proposition 2.1. [[7]] Let q be a function in Kα (D), then we have
R Gα (x,z)GαD (z,y)
(i) aα (q) := supx,y∈D D D Gα (x,y)
|q(z)|dz < ∞.
D
EJDE-2013/33
POSITIVE BLOWUP SOLUTIONS
5
(ii) Let h be a positive α-superharmonic function with respect to X D . Then,
for all x ∈ D we have
Z
Gα
(2.1)
D (x, y)h(y)|q(y)|dy ≤ aα (q)h(x).
D
Furthermore, for each x0 ∈ D, we have
Z
1
lim sup
Gα
(x,
y)h(y)|q(y)|dy
= 0.
D
r→0 x∈D h(x) B(x ,r)∩D
0
(2.2)
(iii) The function x → (δ(x))α−1 q(x) is in L1 (D).
The next two Lemmas will play a special role.
Lemma 2.2 ([7]). Let q be a nonnegative function in Kα (D) and h be a positive
finite α-superharmonic function with respect to X D . Then for all x ∈ D, such that
0 < h(x) < ∞, we have
exp(−aα (q))h(x) ≤ h(x) − Vq (qh)(x) ≤ h(x).
Lemma 2.3. Let q be a nonnegative function in Kα (D), then the family of functions
Z
1
α
Λq =
Gα
D (x, y)MD ϕ(y)f (y)dy, |f | ≤ q
α ϕ(x)
MD
D
is uniformly bounded and equicontinuous in D. Consequently Λq is relatively compact in C0 (D).
α
Proof. Taking h ≡ MD
ϕ in (2.1), we deduce that for |f | ≤ q and x ∈ D, we have
Z
Z
α
GD (x, y) α
Gα
α
D (x, y)
|
M
ϕ(y)f
(y)dy|
≤
MD
ϕ(y)q(y)dy ≤ aα (q) < ∞. (2.3)
D
α
α
D MD ϕ(x)
D MD ϕ(x)
So the family Λq is uniformly bounded.
Next we aim at proving that the family Λq is equicontinuous in D. First, we
recall the following interesting sharp estimates on Gα
D , which is proved in [8]:
(δ(x)δ(y))α/2 α−n
Gα
min 1,
.
(2.4)
D (x, y) ≈ |x − y|
|x − y|α
Let x0 ∈ D and ε > 0. By (2.2), there exists r > 0 such that
Z
1
ε
α
sup α
Gα
D (z, y)MD ϕ(y)q(y)dy ≤ .
M
2
ϕ(z)
z∈D
B(x0 ,2r)∩D
D
If x0 ∈ D and x, x0 ∈ B(x0 , r) ∩ D, then for |f | ≤ q, we have
Z
Z Gα (x, y)
0
Gα
α
α
D
D (x , y)
M
ϕ(y)f
(y)dy
−
M
ϕ(y)f
(y)dy
D
D
α
α
0
D MD ϕ(x )
D MD ϕ(x)
Z
α
0
Gα
α
D (x, y) − GD (x , y) MD
≤
ϕ(y)q(y)dy
α
α
0
MD ϕ(x )
D MD ϕ(x)
Z
1
α
α
≤ 2sup
α ϕ(z) GD (z, y)MD ϕ(y)q(y)dy
M
z∈D B(x0 ,2r)∩D
D
Z
α
0
Gα
α
D (x, y) − GD (x , y) MD
+
ϕ(y)q(y)dy
α
α
0
MD ϕ(x )
(|x0 −y|≥2r)∩D MD ϕ(x)
Z
α
0
α
Gα
D (x, y) − GD (x , y) MD
ϕ(y)q(y)dy.
≤ε+
α
α
0
MD ϕ(x )
(|x0 −y|≥2r)∩D MD ϕ(x)
6
R. ALSAEDI
EJDE-2013/33
On the other hand, for every y ∈ B c (x0 , 2r) ∩ D and x, x0 ∈ B(x0 , r) ∩ D, by
α
α
using (2.4) and the fact that MD
ϕ(z) ≈ (δ(z)) 2 −1 , we have
α
1
1
0
α−1
Gα
Gα
.
D (x, y) −
D (x , y) MD ϕ(y) ≤ C(δ(y))
α
α
0
MD ϕ(x)
MD ϕ(x )
Now since x → M α1ϕ(x) Gα
D (x, y) is continuous outside the diagonal and q ∈ Kα (D),
D
we deduce by the dominated convergence theorem and Proposition 2.1 (iii), that
Z
Gα (x, y) Gα
(x0 , y) α
| Dα
− Dα
|M ϕ(y)q(y)dy → 0 as |x − x0 | → 0.
MD ϕ(x0 ) D
(|x0 −y|≥2r)∩D MD ϕ(x)
If x0 ∈ ∂D and x ∈ B(x0 , r) ∩ D, then we have
Z
Z
ε
Gα
Gα
α
α
D (x, y)
D (x, y)
MD ϕ(y)f (y)dy ≤ +
MD
ϕ(y)q(y)dy.
α
α
M
ϕ(x)
2
M
D
(|x0 −y|≥2r)∩D
D
D ϕ(x)
Gα (x,y)
Now, since MDα ϕ(x) → 0 as |x − x0 | → 0, for |x0 − y| ≥ 2r, then by same argument
D
as above, we obtain
Z
Gα
α
D (x, y)
α ϕ(x) MD ϕ(y)q(y)dy → 0 as |x − x0 | → 0.
M
(|x0 −y|≥2r)∩D
D
Consequently, by Ascoli’s theorem, we deduce that Λq is relatively compact in
C0 (D).
3. Proofs of Theorems 1.2 and 1.3
The next Lemma will be used for uniqueness.
Lemma 3.1 ([7, Lemma 4]). Let h ∈ B + (D) and υ be a nonnegative α-superharmonic function on D with respect to X D . Let z be a Borel measurable function in
D such that V (h|z|) < ∞ and υ = z + V (hz). Then z satisfies
0 ≤ z ≤ υ.
Proof of Theorem 1.2. Let ϕ be a positive continuous function on ∂D. We recall
that on D we have
α
α
α
MD
ϕ(x) ≈ MD
1(x) ≈ (δ(x)) 2 −1 .
p0 )
α
ϕ)γ−1 p0 and put c0 = e−aα (f
, where aα (pe0 ) is given by Proposition
Let pe0 = γ(MD
2.1(i). Since by (H1), pe0 ∈ Kα (D), it follows from Proposition 2.1 that V (pe0 ) ≤
aα (pe0 ) < ∞. Define the nonemty closed bounded convex Λ by
Λ = {ω ∈ B + (D) : c0 ≤ ω ≤ 1}.
Let T be the operator defined on Λ by
1
1
α
α
α
T ω := 1 − α Vpf0 (pe0 MD
ϕ) + α Vpf0 (pe0 ωMD
ϕ − p0 (ωMD
ϕ)γ ).
MD ϕ
MD ϕ
We claim that T maps Λ to itself. Indeed, for each ω ∈ Λ we have
1
α
T ω ≤ 1 − α Vpf0 (p0 (ωMD
ϕ)γ ) ≤ 1.
MD ϕ
α
α
On the other hand, since the function pe0 ωMD
ϕ − p0 (ωMD
ϕ)γ ≥ 0, we deduce
1
α
α
by Lemma 2.2 with h = MD ϕ, that T ω ≥ 1 − M α ϕ Vpf0 (pe0 MD
ϕ) ≥ c0 . Hence
D
T Λ ⊂ Λ. Next, we aim at proving that T is nondecreasing on Λ. To this end, we
EJDE-2013/33
POSITIVE BLOWUP SOLUTIONS
7
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
T ω2 − T ω1
1
1
α
α
α
α
= α Vpf0 (pe0 ω2 MD
ϕ − p0 (ω2 MD
ϕ)γ ) − α Vpf0 (pe0 ω1 MD
ϕ − p0 (ω1 MD
ϕ)γ )
MD ϕ
MD ϕ
1
α
ϕ)γ [(γω2 − ω2γ ) − (γω1 − ω1γ )]) ≥ 0.
= α Vpf0 (p0 (MD
MD ϕ
Next we define the sequence (ωk )k≥0 by
1
α
(pe0 MD
ϕ),
f
0
α ϕ Vp
MD
ωk+1 = T ωk .
ω0 = 1 −
Clearly ω0 ∈ Λ and ω1 = T ω0 ≥ ω0 . Thus, from the monotonicity of T , we deduce
that
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
1
1
α
α
α
ϕ) + α Vpf0 (pe0 ωMD
ϕ − p0 (ωMD
ϕ)γ )
ω = 1 − α Vpf0 (pe0 MD
MD ϕ
MD ϕ
α
ϕ. Then we have
Put u = ωMD
α
α
ϕ) + Vpf0 (pe0 u − p0 uγ )
u = MD
ϕ − Vpf0 (pe0 MD
(3.1)
α
α
ϕ) − Vpf0 (p0 uγ ).
ϕ − Vpf0 (pe0 MD
u − Vpf0 (pe0 u) = MD
(3.2)
or equivalently
Observe that by Proposition 2.1 (ii), we have V (pe0 u) < ∞. So applying the operator
(I + V (pe0 .)) on both sides of (3.2), we deduce by using (1.10) and (1.11) that
α
u = MD
ϕ − V (p0 uγ ).
Now using (H1) and similar argument as in the proof of Lemma 2.3, we prove that
x 7→ M1α ϕ V (p0 uγ ) ∈ C0 (D). So u is a continuous function in D and u is a solution
D
of (1.3). It remains to prove the uniqueness of such a solution. Let u be a continuous
solution of (1.3). Since the function x 7→ Mu(x)
α 1(x) is continuous and positive in D
D
such that
u(x)
α
x→z∈∂D MD 1(x)
lim
α
α
= ϕ(z), it follows that u(x) ≈ MD
1(x) ≈ MD
ϕ(x). Then
by using this fact and Lemma 2.3, we have
(−∆)α/2 (u + V (p0 uγ )) = 0
in D,
γ
lim
x→z∈∂D
(u + V (p0 u ))(x)
= ϕ(z).
α 1(x)
MD
So from the uniqueness of problem (1.5) (see [7]]), we deduce that
α
u + V (p0 uγ ) = MD
ϕ in D.
It follows that if u and v are two continuous solution of (1.3), then z = v − u
satisfies
z + V (p0 hz) = 0 in D,
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R. ALSAEDI
EJDE-2013/33
where h is the nonnegative measurable function defined in D by
( γ γ
v −u
v−u , if u(x) 6= v(x),
h(x) =
0,
if u(x) = v(x).
Since V (p0 h|z|) < ∞, we deduce by Lemma 3.1 that z = 0, and so u = v.
α
α
σ(MD
ϕ)σ−1 (MD
ψ)r p,
α
α
β(MD
ψ)β−1 (MD
ϕ)s q.
Proof of Theorem 1.3. Let pe =
qe =
Then by hypothesis (H2 ), pe and qe are Kα (D).
Put c1 = e−aα (ep) , c2 = e−aα (eq) . Note that by Proposition 2.1, we have aα (e
p) < ∞
and aα (e
q ) < ∞. Consider the nonemty closed convex set Γ defined by
Γ = {(y, z) ∈ C(D) × C(D) : c1 ≤ y ≤ 1 and c2 ≤ z ≤ 1}.
Let T be the operator defined on the set Γ by T (y, z) := (ω, θ), such that
α
α
(e
u = ωMD
ϕ, ve = θMD
ψ) is the unique positive continuous solution of the problem
α
(−∆)α/2 u
e + ((MD
ψ)r z r p)(x)e
uσ = 0
(−∆)
α/2
ve +
α
((MD
ϕ)s y s q)(x)e
vβ
lim
x→z∈∂D
lim
x→z∈∂D
=0
in D,
in D,
u
e(x)
α 1(x) = ϕ(z),
MD
ve(x)
α 1(x) = ψ(z),
MD
According to Theorem 1.2, we have
1
α
α
ω = 1 − α V (z r ω σ (MD
ψ)r (MD
ϕ)σ p),
MD ϕ
1
α
α
ϕ)s (MD
ψ)β q).
θ = 1 − α V (y s ω β (MD
MD ψ
Moreover we have c1 ≤ ω ≤ 1 and c2 ≤ θ ≤ 1 and by Lemma 2.3, 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 operator T 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
1
1
α
α
α
α
|ωk − ω| = α V (z r ω σ (MD
ψ)r (MD
ϕ)σ p) − α V (zkr ωkσ (MD
ψ)r (MD
ϕ)σ p)
MD ϕ
MD ϕ
1
r σ
r σ
α
≤
p).
α ϕ V (|z ω − zk ωk |(MD ϕ)e
σMD
Using the fact that |z r ω σ − zkr ωkσ | ≤ 2 and that pe ∈ Kα (D), we deduce by Proposition 2.1 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
kT (yk , zk ) − T (y, z)k∞ → 0
as k → ∞.
Now the Schauder fixed point theorem implies that there exists (y, z) ∈ Γ such that
T (y, z) = (y, z). Which is equivalent to
α
u = MD
ϕ − V (puσ v r ),
α
v = MD
ψ − V (qus v β ),
EJDE-2013/33
POSITIVE BLOWUP SOLUTIONS
9
α
α
where (u, v) = (yMD
ϕ, zMD
ψ). The pair (u, v) is a required solution of (1.1) in
the sense of distributions. This completes the proof.
References
[1] R. S. Alsaedi; Positive Solutions for Some Nonlinear Elliptic Systems in Exterior Domains
of R2 , Abstract and Applied Analysis, Volume 2012, Article ID 273017, 13 pages.
[2] R.S. Alsaedi, H. Mâagli, N. Zeddini: Positive Solutions for Some Competetive Elliptic Systems, to appear in Mathematica Slovaca.
[3] D. Applebaum; Lévy processes-from probability to finance and quantum groups, Notices
Amer. Math. Soc., 51 (2004), 1336-1347.
[4] K. Bogdan; Representation of α-harmonic functions in Lipschitz domains, Hiroshima Math.
J. 29 (1999), 227-243.
[5] K. Bogdan, T. Byczkowski; Potential theory for the α-stable Schrödinger operator on bounded
Lipschitz domains, Studia Math. 133 (1999), 53-92.
[6] J. Bliedtner, W. Hansen; Potential Theory. An Analytic and Probabilistic Approach to Balayage, Springer Berlin, 1986.
[7] R. Chemmam, H. Mâagli, S. Masmoudi; On a new Kato class and positive solutions of Dirichlet problems for the fractional Laplacian in bounded domains, Nonlinear Anal. 74 (2011),
1555-1576.
[8] Z. Q. Chen, R. Song; Estimates on Green functions and Poisson kernels for symmetric stable
processes, Math. Ann. 312 (1998) 465-501.
[9] Z. Q. Chen, R. Song; Martin boundary and integral representation for harmonic function of
symmetric processes, J. Funct. Anal. 159 (1998) 267-294.
[10] Z. Q. Chen, R. Song; General gauge and conditional gauge theorems, Ann. Probab. 30 (2002)
1313-1339.
[11] Z. Q. Chen, P. Kim; Stability of Martin boundary under nonlocal FeynmanKac perturbations,
Probab. Theory Related Fields 128 (2004) 525-564.
[12] K.L. Chung, Z. Zhao; From Brownian Motion to Schrodinger’s Equation, Springer Verlag
(1995).
[13] F .C. Cirstea, V. D. Radulescu; Entire solutions blowing up at infinity for semilinear elliptic
systems, J. Math. Pures Appl. 81 (2002), 827–846.
[14] E. Di Nezza, G. Palatucci, E. Valdinoci; Hitchhikers guide to the fractional Sobolev spaces,
Bull. Sci. Math. 136 (2012), 521-573.
[15] A. Ghanmi, H. Mâagli, V. Radulescu, N. Zeddini; Large and bounded solutions for a class of
nonlinear Schrödinger stationary Systems, Analysis and Applications 7 (2009), 391-401.
[16] A. Ghanmi, H. Mâagli, S. Turki, N. Zeddini; Existence of Positive Bounded Solutions for
Some Nonlinear Elliptic Systems, J. Math. Anal. Appl. 352 (2009), 440–448.
[17] M. Ghergu, V. D. Radulescu; On a class of singular Gierer–Meinhart systems arising in
morphogenesis, C. R. Math. Acad. Sci. Paris. Ser. I 344 (2007), 163–168.
[18] A. Janicki, A. Weron: Simulation and Chaotic Behavior of α-Stable Processes, Dekker, 1994
[19] J. Klater, M. F. Shlesinger, G. Zumofen; Beyond Brownian motion, Phys. Today 49 (2)
(1996) 3339.
[20] A. V. Lair, A. W. Wood; Large solutions of semilinear elliptic equations with nonlinear
gradient terms, Int. J. Math. Sci. 22 (1999), 869-883.
[21] A. V. Lair, A. W. Wood; Existence of entire large positive solutions of semilinear elliptic
systems, J. Differential Equations (2) 164 (2000), 380–394.
[22] H. Mâagli; Perturbation Semi-linéaire des Résolvantes et des Semi-groupes, Potential Analysis. vol. 3 (1994), 61-87.
[23] H. Mâagli, M. Zribi; On a new Kato class and singular solutions of a nonlinear elliptic
equation in bounded domains of Rn , Positivity. 9 (2005), 667-686.
[24] C. Mu, S. Huang, Q. Tian, L. Liu; Large solutions for an elliptic system of competetive type:
Existence, uniqueness and asymptotic behavior, Nonlinear Anal. 71 (2009) 4544-4552.
[25] A. Pazy; Semigroups of Linear Operators and Applications to Partial Differential Equations,
Springer, New York, 1983.
[26] D. V. Widder; The Laplace Transform, Princeton Univ. Press, Princeton, NJ, 1941.
[27] K. Yosida; Functional Analysis, 6th Edition, Springer, Berlin, 1980.
10
R. ALSAEDI
EJDE-2013/33
Ramzi Alsaedi
Department of Mathematics, College of Science and Arts, King Abdulaziz University,
Rabigh Campus, P. O. Box 344, Rabigh 21911, Saudi Arabia
E-mail address: ramzialsaedi@yahoo.co.uk