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J. Math. Anal. Appl. 418 (2014) 476–491
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Journal of Mathematical Analysis and Applications
www.elsevier.com/locate/jmaa
Singular solutions of nonlinear Schrödinger problem in boundedpiecewise Dini-smooth Jordan domain
Mohamed Amine Ben BoubakerInstitut préparatoire aux études d’ingénieur de Nabeul, Campus Universitaire Merazka, 8000 Nabeul,Tunisia
a r t i c l e i n f o a b s t r a c t
Article history:Received 22 December 2012Available online 12 April 2014Submitted by P.J. McKenna
Keywords:Green functionPiecewise Dini-smooth JordandomainConformal mappingSchauder fixed point theoremSuperharmonic function
We prove the existence of positive singular solutions of the nonlinear equation
Δu− μu = f(., u) on Ω \ {0},
where Ω is a bounded simply connected piecewise Dini-smooth Jordan domain in R2,μ and f are, respectively, a signed measure on Ω and a Borel measurable function,required to satisfy suitable assumptions related to a new functional class J .
© 2014 Elsevier Inc. All rights reserved.
1. Introduction
In [13], Zhang and Zhao studied the following problem
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
Δu(x) + V (x)up(x) = 0, x ∈ D \ {0},u(x) > 0, x ∈ D \ {0},
u(x) ∼ 1|x|n−2 , near x = 0,
u(x) = 0, x ∈ ∂D,
where D ⊂ Rn (n ≥ 3) is a bounded Lipschitz domain containing 0, p > 1, Δ is the Laplacian operator and
V is in the Kato class Kn introduced in [1]. In [1] Aizenman and Simon identified the Kato class Kn as thenatural class of functions so that the weak solutions of the equation Δu+ϕu = 0, are continuous. We recallthat a function ϕ ∈ L1
loc(Rn) is said to be belong to the Kato class Kn provided that
E-mail address: [email protected].
http://dx.doi.org/10.1016/j.jmaa.2014.04.0030022-247X/© 2014 Elsevier Inc. All rights reserved.
M.A. Ben Boubaker / J. Math. Anal. Appl. 418 (2014) 476–491 477
limr−→0
supx∈D
∫(|x−y|≤r)∩D
|ϕ(y)||x− y|n−2 dy = 0, for n ≥ 3
and
limr−→0
supx∈D
∫(|x−y|≤r)∩D
Log(
1|x− y|
)∣∣ϕ(y)∣∣ dy = 0, for n = 2.
In [4] K. Hirata, studied the following problem
(P ) =
⎧⎨⎩
Δu− μu = f(., u) in Γ,
u > 0 in Γ,
u = 0 on ∂rΓ,
where Γ is the uniform cone in Rn (n ≥ 3), ∂rΓ denotes the set of all Dirichlet regular points of ∂Γ ,
μ a signed measure on Γ and f a Borel measurable function in Γ × ]0,+∞[, satisfying some appropriatehypotheses. By applying sharp estimates for the Green function, he proved the existence of solutions withthe growth as the Martin kernel at infinity.
Definition 1.1. (See [10, p. 43 and p. 51].) A Jordan curve C is said to be Dini-smooth if it has a parametriza-tion ω(t), 0 ≤ t ≤ 2π, such that ω′(t) is Dini continuous and �= 0.
Let Ω be any simply connected domain in C with locally connected boundary. Let φ be a conformalmapping from Ω onto the unit disk D of R2. We say that ∂Ω has a Dini-smooth corner of opening angle π
α
(12 < α < ∞) at a = φ−1(eiϕ) if there are closed arcs A+, A− ⊂ C(0, 1) ending at ϕ and lying on opposite
sides of ϕ that are mapped by φ−1 onto Dini-smooth Jordan domain arcs C+ and C−, forming the angle πα
at a = φ−1(eiϕ). That means
arg[φ−1(eit)− a
]−→
{α, as t −→ ϕ−
α + πα , as t −→ ϕ+.
Let Ω be a bounded simply connected domain in R2. We say that Ω is a piecewise Dini-smooth Jordan
domain if and only if ∂Ω is a closed piecewise Dini-smooth Jordan curve, having finite Dini-smooth cornersof vertices a1, . . . , an, where the opening angle at ai is π
αi, with αi ∈ ]12 ,+∞[\{1}. In this paper, we consider
a bounded simply connected piecewise Dini-smooth Jordan domain Ω in R2 containing 0. The purpose of
this paper is two-folded. One is to introduce a new functional class J on Ω and to study its properties. Theother is to investigate the existence of positive singular solutions for the following nonlinear Schrödingerproblem:
(PΩ)
⎧⎪⎪⎨⎪⎪⎩
Δu− μu = f(., u), on Ω \ {0},u > 0, in Ω \ {0},u(x) ∼ G(x, 0), near x = 0,u = 0, on ∂Ω,
(1.1)
where G denotes the Laplacian Green function with Dirichlet boundary condition on Ω, μ and f are,respectively, a signed measure on Ω and a Borel measurable function in Ω × ]0,+∞[, required to satisfysuitable assumptions related to the class J . The notation u(x) ∼ G(x, 0), near x = 0 means that for somec > 0, lim|x|−→0
u(x)G(x,0) = c. The solutions in this problem are understood as distributional solutions in Ω.
In [8], Mâagli and Mâatoug studied (PΩ) when ∂Ω has no corners, μ = 0 and f is a Borel measurablefunction satisfying some hypothesis related to the Kato class K(Ω) defined by:
478 M.A. Ben Boubaker / J. Math. Anal. Appl. 418 (2014) 476–491
Definition 1.2. (See [8].) We say that a Borel measurable function ϕ on Ω belongs to the Kato class K(Ω)if it satisfies the following condition
limr−→0
supx∈Ω
∫Ω∩(|x−y|<r)
δ(y)δ(x)G(x, y)
∣∣ϕ(y)∣∣ dy = 0,
where δ(x) is the distance from x to ∂Ω.
In [11], L. Riahi studied (PΩ) in the case that Ω is bounded non-tangentially accessible (NTA) domain(see definition in [5]) with μ = 0 and f(x, u) = V (x)up. He introduced and studied a new Kato class Kloc(Ω),that strictly contains K(Ω) and J .
Definition 1.3. A signed Radon measure μ on Ω belongs to the Kato class Kloc(Ω) if the following conditionsare satisfied:
(i) supx∈Ω
∫Ω
ϕ(y)ϕ(x)G(x, y)|μ|(dy) < ∞.
(ii) For any compact subset E ∈ Ω, limr−→0 supx∈E
∫Ω∩(|x−y|<r) G(x, y)|μ|(dy) = 0, where ϕ(.) =
min(1, G(., 0)).
The following notations will be adopted:
(i) D is the unit disk of R2.(ii) Let f and g be two positive functions on a set Ω. We say that f is equivalent to g on Ω and we denote
f g, if there exists C ≥ 0, such that, 1C g(x) ≤ f(x) ≤ Cg(x), ∀x ∈ Ω.
(iii) Let a, b in R we denote by a ∨ b = max(a, b) and a ∧ b = min(a, b).(iv) For x ∈ Ω, δ(x) is the distance from x to ∂Ω.(v) |μ| denotes the total variational measure of a signed measure μ.(vi) ‖|μ‖|G = supx∈Ω
1G(x,0)
∫ΩG(x, y)G(y, 0) d|μ|(y).
Throughout this paper, C denotes a generic positive constant and Ωc denotes the complementary of Ω
in R2.
The following hypotheses on f and μ are adopted:
(H1) f is a measurable function in Ω×]0,+∞[, continuous with respect to the second variable and satisfying
∣∣f(x, t)∣∣ ≤ tq(x, t), for (x, t) ∈ Ω × ]0,∞[,
where q is a nonnegative measurable function on Ω × ]0,∞[ such that the function t −→ q(x, t) isnondecreasing on ]0,+∞[ and limt→0 q(x, t) = 0.
(H2) The function ψ defined on Ω by ψ(x) = q(x,G(x, 0)) belongs to the class J .(H3) μ is a signed measure on Ω such that |μ| ∈ J and ‖|μ‖|G < 1
2 .
As important examples of functions satisfying (H1) are f(x, t) = V (x)g(t), where g is a positive nonde-creasing function on ]0,+∞[ such that limt−→0
g(t)t = 0. If we take μ = 0 and g(t) = tp, p > 1, we get the
semilinear elliptic equation Δu(x) = V (x)up(x) studied by several authors (see [6,7,9,11]).
M.A. Ben Boubaker / J. Math. Anal. Appl. 418 (2014) 476–491 479
Our main result is the following:
Theorem 1.1. The problem (PΩ) has infinitely many solutions. More precisely, there exists λ0 > 0 such thatfor each λ ∈ ]0, λ0], there exists a positive solution u of (PΩ) continuous on Ω \ {0} and satisfying
2(1 − 2‖|μ‖|G)3 − 2‖|μ‖|G
λG(x, 0) ≤ u(x) ≤ 43 − 2‖|μ‖|G
λG(x, 0) ∀x ∈ Ω \ {0},
lim|x|−→0
u(x)G(x, 0) = λ.
The outline of this paper is as follows. In Section 2 we recall some results in [2] that will be necessarythroughout this paper. In Section 3, we introduce the Kato class J , we study some of its properties, wegive typical examples of their functions and we prove that the classical Kato class K2 ⊂ J ⊂ Kloc(Ω). InSection 4 we prove Theorem 1.1.
2. Preliminary results
In this section, we recall some results in [2].
Theorem 2.1. Let Ω be a bounded simply connected piecewise Dini-smooth Jordan domain in R2 having n
Dini-smooth corners at a1, . . . , an of opening angles respectively πα1
, . . . , παn
, αi ∈ ]12 ,+∞[ \ {1} and φ aconformal mapping from Ω onto D. Then, there exists a constant C > 0 such that, for all x, y, z ∈ Ω,
G(x, z)G(z, y)G(x, y) ≤ C
[n∏
k=1
∣∣∣∣ z − akx− ak
∣∣∣∣αk−1δ(z)δ(x)G(x, z) +
n∏k=1
∣∣∣∣z − aky − ak
∣∣∣∣αk−1δ(z)δ(y)G(z, y)
]. (2.1)
Theorem 2.2. Let Ω be a bounded simply connected piecewise Dini smooth Jordan domain in R2 having n
Dini-smooth corners at a1, a2, . . . , an of opening angles respectively πα1
, . . . , παn
, αi ∈ ]12 ,+∞[ \ 1 and φ aconformal mapping from Ω onto D. Then,
G(x, y) Log(
1 +n∏
k=1
((|x− ak| ∧ |y − ak|)(|x− ak| ∨ |y − ak|)
)αk−1δ(x)δ(y)|x− y|2
), ∀x, y ∈ Ω, (2.2)
∣∣φ′(x)∣∣ n∏
i=1|x− ai|αi−1, ∀x ∈ Ω, (2.3)
δ(φ(x)
)
n∏i=1
|x− ai|αi−1δ(x), ∀x ∈ Ω, (2.4)
where δ(φ(x)) is the distance from φ(x) to ∂D(0, 1).
Theorem 2.3. Let Ω be a bounded simply connected piecewise Dini-smooth Jordan domain in R2 having n
Dini-smooth corners at a1, . . . , an of opening angles respectively πα1
, . . . , παn
, αi ∈ ]12 ,+∞[ \ {1} and φ aconformal mapping from Ω onto D. Then, for all x, y ∈ Ω,
∣∣∣∣φ(x) − φ(y)x− y
∣∣∣∣2 ∣∣φ′(x)
∣∣∣∣φ′(y)∣∣ n∏i=1
((|x− ai| ∨ |y − ai|)(|x− ai| ∧ |y − ai|)
)αi−1
n∏
i=1
(|x− ai| ∨ |y − ai|
)2(αi−1)
480 M.A. Ben Boubaker / J. Math. Anal. Appl. 418 (2014) 476–491
3. Kato class
Let us put, for all x, y ∈ Ω,
Gα(x, y) = Log(
1 +n∏
i=1
((|x− ai| ∧ |y − ai|)(|x− ai| ∨ |y − ai|)
)αi−1δ(x)δ(y)|x− y|2
),
Kα(x, y) =n∏
i=1
(|y − ai||x− ai|
)αi−1δ(y)δ(x)Gα(x, y),
and
‖x− y‖α = |x− y|n∏
i=1
(|x− ai| ∨ |y − ai|
)αi−1 ∣∣φ(x) − φ(y)
∣∣. (3.1)
Definition 3.1. We say that a measure ν on Ω belongs to the Kato class J if it satisfies the following condition
limr−→0
(supx∈Ω
∫(‖x−y‖α≤r∩Ω)
Kα(x, y) dν(y))
= 0. (3.2)
We also say that a Borel measurable function ϕ in Ω belongs to the Kato class J if the measure dν = |ϕ| dybelongs to J .
Proposition 3.1. Let Ω be a bounded simply connected piecewise Dini-smooth Jordan domain in R2 having
n Dini-smooth corners at a1, . . . , an of opening angles respectively πα1
, . . . , παn
, αi ∈ ]12 ,+∞[ \ {1}. Then,there exists CΩ > 1, depending only on Ω, such that, for all x, y ∈ Ω,
1CΩ
|x− y|γ1 ≤ ‖x− y‖α ≤ CΩ |x− y|γ0 , (3.3)
where γ0 = min(1, α1, . . . , αn) and γ1 = max(1, α1, . . . , αn).
Proof. We have to discuss three cases.Let r = mini�=j
|ai−aj |2 .
(i) If x, y ∈ B(ai0 , r2), then maxj �=i0(|x− aj |, |y − aj |) ≥ r
2 . Hence
‖x− y‖α |x− y|(|x− ai0 | ∨ |y − ai0 |
)αi0−1.
Assume that (|x− ai0 | ∨ |y − ai0 |) = |x− ai0 |, then
12 |x− y| ≤
(|x− ai0 | ∨ |y − ai0 |
)≤ r
2 .
It follows that
21−αi0 |x− y|αi0 ≤ |x− y|(|x− ai0 | ∨ |y − ai0 |
)αi0−1 ≤ dαi0−1|x− y| if αi0 > 1,
dαi0−1|x− y| ≤ |x− y|(|x− ai0 | ∨ |y − ai0 |
)αi0−1 ≤ 21−αi0 |x− y|αi0 if 12 < αi0 < 1.
M.A. Ben Boubaker / J. Math. Anal. Appl. 418 (2014) 476–491 481
(ii) If x ∈ B(ai, r2 ) and y /∈ B(ai, r
2 ) then
(|x− aj | ∨ |y − aj |
)≥ r
2, ∀j ∈ {1, 2, . . . , n}.
Thus,
‖x− y‖α |x− y|
and the result follows.(iii) If x, y /∈ B(aj , r
2 ), ∀j ∈ {1, 2, . . . , n}, then in the same way as the second case, we deduce theresult. �Proposition 3.2. Let ν be a measure on Ω such that ν ∈ J , then
∫Ω
n∏i=1
|y − ai|2(αi−1)δ2(y) dν(y) < +∞.
Proof. Since Log(1 + t) ≥ t1+t , ∀t ≥ 1. Then, ∀(x, y) ∈ Ω2,
Gα(x, y) ≥∏n
i=1(|x− ai| ∧ |y − ai|)αi−1δ(x)δ(y)|x− y|2
∏ni=1(|x− ai| ∨ |y − ai|)αi−1 +
∏ni=1(|x− ai| ∧ |y − ai|)αi−1δ(x)δ(y)
.
As, for all a, b ∈ R, we have ab = (a ∨ b)(a ∧ b), then
Gα(x, y) ≥∏n
i=1 |x− ai|αi−1|y − ai|αi−1δ(x)δ(y)‖x− y‖2
α +∏n
i=1 |x− ai|αi−1δ(x)∏n
i=1 |y − ai|αi−1δ(y).
This implies, by relation (2.4), that for all x, y ∈ Ω,
Kα(x, y) ≥ C
∏ni=1 |y − ai|2(αi−1)δ2(y)
‖x− y‖2α + Cδ(φ(x))δ(φ(y)) .
Now, let x1, x2, . . . , xn ∈ Ω, such that
Ω ⊂n⋃
i=1B(xi, r).
Then, for all j ∈ {1, 2, . . . , n} and y ∈ B(xj , r) ∩Ω, one gets by relation (3.3)
‖xj − y‖α ≤ cΩrγ0 = r0.
So,
n∏i=1
|y − ai|2(αi−1)δ2(y) ≤ CKα(xj , y).
It follows that
482 M.A. Ben Boubaker / J. Math. Anal. Appl. 418 (2014) 476–491
∫Ω
n∏i=1
|y − ai|2(αi−1)δ2(y)dν(y) ≤ Cn∑
j=1
∫(|xj−y|≤r)∩Ω
Kα(xj , y) dν(y),
≤ C
n∑j=1
∫(‖y−xj‖α≤r0)∩Ω
Kα(xj , y) dν(y),
≤ nC supx∈Ω
∫(‖x−y‖α≤r0)∩Ω
Kα(x, y) dν(y).
Since ν ∈ J , then, by (3.2), there exists r ∈ (0, 1) such that
supx∈Ω
∫(‖x−y‖α≤r0)∩Ω
Kα(x, y) dν(y) ≤ 1.
Hence,∫Ω
n∏i=1
|y − ai|2(αi−1)δ2(y)dν(y) ≤ Cn < ∞. �
In the sequel, we use the notation
‖ν‖Ω = supx∈Ω
∫Ω
Kα(x, y) dν(y). (3.4)
Proposition 3.3. If ν ∈ J , then ‖ν‖Ω < ∞.
Proof. Let r > 0. Then, we have∫Ω
Kα(x, y) dν(y) ≤∫
(‖x−y‖α≤r)∩Ω
Kα(x, y) dν(y)
︸ ︷︷ ︸I1
+∫
(‖x−y‖α≥r)∩Ω
Kα(x, y) dν(y)
︸ ︷︷ ︸I2
Besides
I2 ≤ C
∫(‖x−y‖α≥r)
n∏i=1
(|y − ai||x− ai|
)αi−1
Log(
1 +∏n
i=1 |x− ai|δ(x)∏n
i=1 |y − ai|αi−1δ(y)‖x− y‖2
α
)dν(y),
≤ C
∫(‖x−y‖α≥r)
∏ni=1 |y − ai|2(αi−1)δ2(y)
‖x− y‖2α
dν(y),
≤ C
r2
∫Ω
n∏i=1
|y − ai|2(αi−1)δ2(y) dν(y).
The result follows immediately from relation (3.2) and Proposition 3.2. �Proposition 3.4. There exists C > 0 depending only on Ω such that, for all x, y in Ω,
n∏i=1
(|y − ai||x− ai|
)αi−1δ(y)δ(x)G(x, y) ≤ C
(1 + G(x, y)
). (3.5)
M.A. Ben Boubaker / J. Math. Anal. Appl. 418 (2014) 476–491 483
Proof. Let φ be the conformal mapping defined in Theorem 2.2. By [3] and [12] there exists C > 0 suchthat, for all x, y ∈ D,
δ(y)δ(x)GD(x, y) ≤ C
(1 + GD(x, y)
). (3.6)
Since, for all x, y ∈ Ω,GΩ(x, y) = GD(φ(x), φ(y)), it follows that
δ(φ(y))δ(φ(x))GD
(φ(x), φ(y)
)≤ C
(1 + GD
(φ(x), φ(y)
)).
The result follows immediately from relation (2.4). �Proposition 3.5. There exists a constant C > 0, depending only on Ω such that, for any measure ν belongingto J , any nonnegative superharmonic function h in Ω and all x ∈ Ω,
∫Ω
G(x, y)h(y) dν(y) ≤ CΩ‖ν‖Ωh(x). (3.7)
Proof. Let h be a nonnegative superharmonic function in Ω. Then there exists a sequence (fn)n of nonneg-ative measurable functions in Ω such that
h(y) = supn
∫Ω
G(y, z)fn(z)dz.
Hence we need only to verify the relation (3.7) for h(y) = G(y, z), for all z ∈ Ω.By using Theorem 2.1 and relation (2.2), there exists C > 0 such that
1G(x, z)
∫Ω
G(x, y)G(y, z)dν(y) ≤ C
∫Ω
(Kα(x, y) + Kα(z, y)
)dν(y)
≤ C‖ν‖Ω �Corollary 3.1. Let ν be a measure in J . Then
supx∈Ω
∫Ω
G(x, y)dν(y) < ∞ and ‖ν‖G < +∞
Corollary 3.2. Let ν be a measure in J . Then
∫Ω
n∏i=1
|y − ai|αi−1δ(y) dν(y) < +∞.
Proof. Since Ω is bounded and Log(1+ t) ≥ t1+t ∀t ≥ 0, then by relation (2.2), there exists C > 0 such that
n∏i=1
|x− ai|αi−1n∏
i=1|y − ai|αi−1δ(x)δ(y) ≤ CG(x, y), ∀x, y ∈ Ω.
Let x0 ∈ Ω. Then
484 M.A. Ben Boubaker / J. Math. Anal. Appl. 418 (2014) 476–491
∫Ω
n∏i=1
|y − ai|αi−1δ(y) dν(y) ≤ C∏ni=1 |x0 − ai|αi−1δ(x0)
∫Ω
G(x0, y) dν(y).
The result follows from Corollary 3.1. �Proposition 3.6. Let φ be a conformal mapping from Ω into D. Then there exists C > 0, such that
Kα(x, y) ≤ C
(C + Log
(1
|φ(x) − φ(y)|
)), ∀x, y ∈ Ω (3.8)
Proof. By using relations (3.5) and (2.2), we get
Kα(x, y) ≤ C(1 + CGα(x, y)
), ∀x, y ∈ Ω.
It follows by relations (2.4) and (3.1) that
Kα(x, y) ≤ C
(1 + CLog
(1 + δ(φ(x))δ(φ(y))
|φ(x) − φ(y)2|
)),
≤ C
(1 + CLog
(1 + 1
|φ(x) − |φ(y)|2|
)),
≤ C
(C + Log
(1
|φ(x) − φ(y)|
)). �
Proposition 3.7. Let p < 2. Then the function defined on Ω by
V (y) = 1∏ni=1 |y − ai|(p−2)(αi−1)(δ(y))p
belongs to the Kato class J .
Proof. Let φ be the conformal mapping defined in Theorem 2.2. For x ∈ Ω, we put
Ix =∫
(‖x−y‖α≤r)∩Ω
Kα(x, y)V (y) dy.
It follows by relations (3.8) and (2.4) that
Ix ≤ C
∫(‖x−y‖α≤r)∩Ω
(C + Log
(1
|φ(x) − φ(y)|
))∏ni=1 |y − ai|2(αi−1)
(δ(φ(y)))p dy.
Moreover, by relation (3.1), we obtain
Ix ≤ C
∫(|φ(x)−φ(y)|≤cr)∩Ω
(C + Log
(1
|φ(x) − φ(y)|
))∏ni=1 |y − ai|2(αi−1)
(δ(φ(y)))p dy. (3.9)
1. Assume p ≤ 0. Let us consider the variable change u = φ(y). Since φ is a conformal mapping, thenJφ(y) = |φ′(y)|2. Thus by relation (2.3) we obtain Jφ(y)
∏ni=1 |y − ai|2(αi−1). This implies that
Ix ≤ C
∫ (C + Log
(1
|φ(x) − u|
))1
(δ(u))p du.
Ω∩(|φ(x)−u|≤cr)
M.A. Ben Boubaker / J. Math. Anal. Appl. 418 (2014) 476–491 485
Since Ω is bounded, then
Ix ≤ C
∫Ω∩(|φ(x)−u|≤cr)
(C + Log
(1
|φ(x) − u|
))du,
≤ C
cr∫0
(C + Log 1
t
)t dt −→
r−→00.
2. Assume 0 < p < 2, we have to discuss two cases:
Case 1: If δ(φ(y)) ≥ |φ(x) − φ(y)|, then by relation (3.9), we obtain
Ix ≤ C
∫(|φ(x)−φ(y)|≤cr)∩Ω
(C + Log
(1
|φ(x) − φ(y)|
))∏ni=1 |y − ai|2(αi−1)
|φ(x) − φ(y)|p dy.
By using again the variable change u = φ(y), we get
Ix ≤ C
cr∫0
(C + Log 1
t
)t1−p dt −→
r−→00.
Case 2: If δ(φ(y)) ≤ |φ(x) − φ(y)|, then
Ix ≤ C
∫Ω∩(|φ(x)−φ(y)|≤cr)
δ[φ(y)]δ[φ(x)]Log
(1 + δ[φ(x)]δ[φ(y)]
|φ(x) − φ(y)|
)∏ni=1 |y − ai|2(αi−1)
(δφ(y))p dy,
≤ C
∫Ω∩(|φ(x)−φ(y)|≤cr)
[δφ(y)]2−p
|φ(x) − φ(y)|2n∏
i=1|y − ai|2(αi−1) dy,
≤ C
∫Ω∩(|φ(x)−φ(y)|≤cr)
∣∣φ(x) − φ(y)∣∣−p
n∏i=1
|y − ai|2(αi−1) dy,
≤ C
cr∫0
t1−p dt −→r−→0
0. �
Now, we shall prove that the class J contains the classical Kato class K2. Let us recall the definition ofthe class K2.
Definition 3.2. (See [1] and [3].) A Borel measurable function ϕ in Ω belongs to the Kato class K2 if
limr−→0
(supx∈R2
∫(|x−y|≤r)∩Ω
Log(
1|x− y|
)∣∣ϕ(y)∣∣ dy) = 0.
Remark 3.1. Let Ω be a bounded simply connected piecewise Dini-smooth Jordan domain in R2 having n
Dini-smooth corners and let p ∈ ]1, 2[. Then V (y) /∈ K2. In fact, K2 ⊂ L1(Ω) and for p > 1,
∫Ω
V (y) dy 1∫
0
r
(1 − r)p dr = ∞.
486 M.A. Ben Boubaker / J. Math. Anal. Appl. 418 (2014) 476–491
Proposition 3.8. The class J properly contains the classical Kato class K2.
Proof. Let ϕ ∈ K2 and r > 0. By using relations (3.8), (3.1) and (3.3), there exists C > 0 such that for allx, y ∈ Ω
Kα(x, y) ≤ C
(C + Log 1
|x− y|
).
This gives
Ix =∫
(‖x−y‖α≤r)∩Ω
Kα(x, y)∣∣ϕ(y)
∣∣ dy,≤ C
∫(‖x−y‖α≤r)∩Ω
(C + Log 1
|x− y|
)∣∣ϕ(y)∣∣ dy.
By relation (3.3), we get
Ix ≤ C
∫|x−y|<crγ0
(C + Log 1
|x− y|
)∣∣ϕ(y)∣∣ dy.
Since ϕ ∈ K2, then for all ε > 0, there exists r0 > 0 such that
supx∈Ω
∫|x−y|≤cr
γ00
Log(
1|x− y|
)∣∣ϕ(y)∣∣ dy ≤ ε.
Let r ≤ (crγ00 ∧ 1
e ). Then, for x ∈ Ω,
Ix ≤ C supx∈Ω
∫(|x−y|≤r)∩Ω
Log(
1|x− y|
)∣∣ϕ(y)∣∣ dy ≤ Cε. �
Proposition 3.9. The class J is contained in the Kato class Kloc(Ω).
Proof. Let us note that, by [11, Proposition 4.2], a Borel measurable function ψ belongs to the Kato classK(D) if and only if the function
Pψ(x) =∫D
δ(y)δ(x)G(x, y)
∣∣ψ(y)∣∣ dy is in C(D).
Let φ be a conformal mapping from Ω onto the unit disk D. Then, we can deduce that a Borel measurablefunction Ψ(y) = ψ((φ(y))) on Ω belongs to the Kato class Kloc(Ω) if and only if
PΨ (x) =∫Ω
δ(φ(x))δ(φ(y))G
(φ(x), φ(y)
)Ψ(y)
∣∣φ′(y)∣∣2 dy is in C(D).
As below, we can prove that Ψ ∈ Kloc(Ω) if and only if PΨ (x) is in Cb(D). �
M.A. Ben Boubaker / J. Math. Anal. Appl. 418 (2014) 476–491 487
4. Proof of Theorem 1.1
Lemma 4.1. Let r > 0. Then there exists C > 0 such that for all x, y ∈ Ω satisfying ‖x− y‖α ≥ r, we have
Kα(x, y) ≤ C
n∏i=1
|y − ai|2(αi−1)δ2(y). (4.1)
Moreover, if ‖0 − y‖α ≥ r, then
G(x, y)G(y, 0)G(x, 0) ≤ C
n∏i=1
|y − ai|2(αi−1)δ2(y). (4.2)
Proof. This lemma follows immediately from relation (2.2) and Theorem 2.1. �Lemma 4.2. Let x0 ∈ Ω. Then for any measure ν belonging to J and any positive superharmonic functionh in Ω,
limδ−→0
(supx∈Ω
1h(x)
∫Ω∩(‖x0−y‖α≤δ)
G(x, y)h(y) dν(y))
= 0. (4.3)
Proof. Let h be a positive superharmonic function in Ω, then there exists a sequence (fn)n∈N of nonnegativemeasurable functions in Ω such that h(y) = supn
∫G(y, z)fn(z)dz. Hence it is enough to verify relation
(4.3) for h(y) = G(y, z) uniformly for z ∈ Ω. Let δ > 0. By using Theorem 2.1, we have
1G(x, z)
∫Ω∩(‖x0−y‖α≤δ)
G(x, y)G(y, z)∣∣ϕ(y)
∣∣ dy ≤ 2C supx∈Ω
∫Ω∩(‖y−x0‖α≤δ)
Kα(x, y) dν(y).
Let r > 0. Then using (4.1), we have
∫(‖y−x0‖α≤δ)∩Ω
Kα(x, y) dν(y)
≤∫
Ω∩(‖x−y‖α≤r)
Kα(x, y) dν(y) +∫
(‖y−x0‖α≤δ)∩Ω∩(‖x−y‖α≥r)
Kα(x, y) dν(y)
≤∫
Ω∩(‖x−y‖α≤r)
Kα(x, y) dν(y)
︸ ︷︷ ︸I1
+C
∫(‖y−x0‖α≤δ)∩Ω
n∏i=1
|y − ai|2(αi−1)δ2(y) dν(y)
︸ ︷︷ ︸I2
.
Since ν is in J , it follows from relation (3.2) that for all ε > 0 we can find some r > 0 such that I1 ≤ ε,∀x ∈ Ω. Fixing this r > 0, letting δ −→ 0, we obtain (4.3) by Proposition 3.2. �Proposition 4.1. There exists C0 > 0 depending only on Ω such that, for all x, y, z in Ω,
‖x− y‖α ≤ C0(‖x− z‖α + ‖z − y‖α
)(4.4)
Proof. Let φ be a conformal mapping from Ω to D. Since
488 M.A. Ben Boubaker / J. Math. Anal. Appl. 418 (2014) 476–491
∣∣φ(x) − φ(y)∣∣ ≤ ∣∣φ(x) − φ(z)
∣∣+ ∣∣φ(z) − φ(y)∣∣,
then the result follows from relation (3.1). �For x0 in Ω and δ > 0, we put
Ωx0,δ ={x ∈ Ω : ‖x− x0‖α ≤ δ
}.
Let C(Ω) denote the space of all continuous functions in Ω endowed with the uniform norm ‖ ‖∞ and let
C1(Ω) ={w ∈ C(Ω) : 0 < w(x) ≤ 1 for x ∈ Ω
}.
For w ∈ C1(Ω), we define
Tw(x) = 1G(x, 0)
∫Ω
G(x, y)G(y, 0)w(y) dμ(y) + 1G(x, 0)
∫Ω
G(x, y)f(y, w(y)G(y, 0)
)dy.
In order to simplify the notation, in the following, we write
dν(y) = d|μ|(y) + ψ(y) dy.
Note, from (H2) and (H3) that ν ∈ J .
Lemma 4.3. The class {Tw : w ∈ C1(Ω)} is equicontinuous in Ω.
Proof. Let x0 ∈ Ω and r > 0, x, x′ ∈ B(x0, r) ∩ Ω and w ∈ C1(Ω). From relation (3.3), we deduce thatx, x′ ∈ Ωx0,
δ2C0
, where δ2C0
= CΩrγ0 and C0 is the constant defined in Proposition 4.1. We have
∣∣Tw(x) − Tw(x′)∣∣ ≤ 2 sup
x∈Ω
1G(x, 0)
∫(‖y−0‖α≤δ)∩Ω
G(x, y)G(y, 0) dν(y)
+ 2 supx∈Ω
1G(x, 0)
∫(‖y−0‖α≥δ)∩Ω∩Ωx0,δ
G(x, y)G(y, 0) dν(y)
+∫
(‖y−0‖α≥δ)∩Ω∩Ωcx0,δ
∣∣∣∣G(x, y)G(x, 0) − G(x′, y)
G(x′, 0)
∣∣∣∣G(y, 0) dν(y).
Since x ∈ B(x0, r) and y ∈ D0 = ((‖y−0‖α ≥ δ)∩Ω∩Ωcx0,δ
), then from relation (4.4) we get ‖x−y‖α ≥ δ2C0
,which implies, by Lemma 4.1 and relation (4.2), that for all (x, y) ∈ B(x0, r) × D0 = ((‖y − 0‖α ≥δ) ∩Ω ∩Ωc
x0,δ), we have
G(x, y)G(x, 0)G(y, 0)ψ(y) ≤ C|y − a|2(α−1)δ2(y)ψ(y).
Since, for all (x, y) ∈ B(x0, r)∩Ω×D0, we have ‖x−y‖α ≥ δ2C0
, it follows by relation (3.3) that |x−y| ≥ r > 0.Then, we deduce that G(x,y)
G(x,0) is continuous on (x, y) ∈ B(x0, r) ∩ Ω × D0 and G(x, 0) has no singularitieswhen x ∈ (‖x− 0‖α ≥ δ) ∩Ω. So using Proposition 3.2 and Lebesgue’s theorem, we have∫ ∣∣∣∣G(x, y)
G(x, 0) − G(x′, y)G(x′, 0)
∣∣∣∣G(y, 0)ψ(y)dy −→ 0, as∣∣x− x′∣∣ −→ 0.
D0
M.A. Ben Boubaker / J. Math. Anal. Appl. 418 (2014) 476–491 489
It follows from Lemma 4.2 that as |x− x′| −→ 0,
∣∣Tw(x) − Tw(x′)∣∣ −→ 0, uniformly for all w ∈ C1(Ω). �
Let ‖|μ‖|G < 12 . For λ > 0, we define
Sλ ={w ∈ C(Ω) : 2(1 − 2‖|μ‖|G)
3 − 2‖|μ‖|Gλ ≤ w(x) ≤ 4
3 − 2‖|μ‖|Gλ for x ∈ Ω
}.
Obviously, Sλ is a nonempty bounded closed convex set in C(Ω). Note that for λ ≤ 12 , Sλ ⊂ C1(Ω). We
define the operator Fλ on Sλ by
Fλw(x) = λ− Tw(x) ∀x ∈ Ω
and write Fλ(Sλ) = {Fλw : w ∈ Sλ}.
Lemma 4.4. There exists a positive constant λ0 ≤ 12 such that if 0 < λ ≤ λ0, then Fλ(Sλ) ⊂ Sλ. Moreover,
Fλ(Sλ) is relatively compact in C(Ω).
Proof. Let 0 < λ ≤ 12 and let w ∈ Sλ. Then, by Lemma 4.3, we deduce that Fλ(w) ∈ C(Ω). Now, we will
show that there exists a positive constant λ0 ≤ 12 such that if 0 < λ ≤ λ0, then
2(1 − 2‖|μ‖|G)3 − 2‖|μ‖|G
λ ≤ Fλw(x) ≤ 43 − 2‖|μ‖|G
λ, ∀x ∈ Ω. (4.5)
Let σ ∈ ]0, 1[. Then, the function
Tσ(x) = 1G(x, 0)
∫Ω
G(x, y)G(y, 0)q(y, σG(y, 0)
)dy
is continuous on Ω and satisfies
limσ−→0
Tσ(x) = 0, ∀x ∈ Ω.
Moreover, the function σ �−→ Tσ(x) is nondecreasing in ]0, 1[. By Dini’s Lemma, we have
limσ−→0
(supx∈Ω
Tσ(x) = 0).
Thus, there exists 0 < σ0 < 12 such that
supx∈Ω
Tσ0 ≤ 1 − 2‖|μ‖|G4 .
Let λ0 = 3−2‖|μ‖|G4 σ0 and 0 < λ ≤ λ0. Then
43 − 2‖|μ‖|G
− λ ≤ σ0.
It follows from (H1) that ∀x ∈ Ω,
490 M.A. Ben Boubaker / J. Math. Anal. Appl. 418 (2014) 476–491
∣∣Fλw(x) − λ∣∣ =
∣∣Tw(x)∣∣ ≤ 4λ‖|μ‖|G
3 − 2‖|μ‖|G+ 4λ
3 − 2‖|μ‖|GTσ0(x)
≤ 1 + 2‖|μ‖|G3 − 2‖|μ‖|G
.
Hence, we obtain (4.5), and thus Fλ(Sλ) ⊂ Sλ. In the other hand it follows, by Lemma 4.3, that Fλ(Sλ)is uniformly bounded and is equicontinuous in Ω. This implies, by Ascoli–Arzelà theorem, that Fλ(Sλ) isrelatively compact in C(Ω). �Lemma 4.5. Let 0 < λ ≤ λ0. Then Fλ is continuous on Sλ.
Proof. Let (wk)k∈N be a sequence in Sλ which converges uniformly to w ∈ Sλ. Then for each x ∈ Ω, itfollows from (H1)–(H3) and Lebesgue’s convergence theorem that
limk−→+∞
1G(x, 0)
∫Ω
G(x, y)G(y, 0)∣∣wk(y) − w(y)
∣∣ d|μ|(y) = 0,
limk−→+∞
1G(x, 0)
∫Ω
G(x, y)∣∣f(y, wk(y)G(y, 0)
)− f
(y, w(y)G(y, 0)
)∣∣ dy = 0.
Hence, Fλ(wk) converges pointwisely to Fλ(w) in Ω as k −→ +∞. Since Fλ(Sλ) is relatively compactin C(Ω), the pointwise convergence implies the uniform convergence. Therefore, limk−→+∞ ‖Fλ(wk) −Fλ(w)‖∞ = 0. Thus, Fλ is continuous on Sλ. �Proof of Theorem 1.1. Let 0 < λ < λ0, where λ0 is the positive constant in Lemma 4.4. Note that Fλ is acompact mapping from Sλ into itself. Since Sλ is a nonempty bounded closed convex set in C(Ω), it follows,from Schauder’s fixed point theorem, that there exists w ∈ Sλ such that w = Fλ(w) = λ − T (w). That is,for x ∈ Ω,
w(x) = λ− 1G(x, 0)
∫Ω
G(x, y)G(y, 0)w(y) dμ(y) + 1G(x, 0)
∫Ω
G(x, y)f(y, w(y)G(y, 0)
)dy.
Let u(x) = w(x)G(x, 0). Then
u(x) = λG(x, 0) −∫Ω
G(x, y)u(y) dμ(y) −∫Ω
G(x, y)f(y, u(y)
)dy, ∀x ∈ Ω.
Since G(x, 0) Log 1|x| near x = 0, then it is clear that u is a solution of (PΩ), continuous on Ω \ {0} and
satisfying
2(1 − 2‖|μ‖|G)3 − 2‖|μ‖|G
λG(x, 0) ≤ u(x) ≤ 43 − 2‖|μ‖|G
λG(x, 0), ∀x ∈ Ω
and
lim|x|−→0
u(x)G(x, 0) = λ. �
Example. Let Ω be a bounded simply connected piecewise Dini smooth Jordan domain in R2 containing 0 and
having n Dini-smooth corners at a1, a2, . . . , an of opening angles respectively πα1
, . . . , παn
, αi ∈ ]12 ,+∞[\{1}.Let 0 < p < 2 and q > 1. Then, there exists λ0 > 0 such that for each λ ∈ ]0, λ0], the problem
M.A. Ben Boubaker / J. Math. Anal. Appl. 418 (2014) 476–491 491
(PΩ)
⎧⎪⎪⎨⎪⎪⎩
Δu(x) + (u(x))q∏ni=1 |x− ai|(p−2)(αi−1)(δ(x))p
= 0, x ∈ Ω \ {0},
u(x) > 0, x ∈ Ω \ {0},u(x) = 0, x ∈ ∂Ω,
has a solution u continuous on Ω \ {0} such that
lim|x|−→0
u(x)Log 1
|x|= λ.
Acknowledgments
I want to thank my PhD advisor Professor Mohamed Selmi for stimulating discussions and useful sug-gestions.
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