1 Introduction
This paper is concerned with the boundary behavior of the unique classical convex solution to the following singular Dirichlet problem for the Monge–Ampère equation:
(1.1)
det
D
2
u
=
b
(
x
)
g
(
-
u
)
,
u
<
0
,
x
∈
Ω
,
u
|
∂
Ω
=
0
,
where Ω is a strictly convex and bounded smooth domain in ℝN, with N≥2, D2u(x)=(∂2u(x)∂xi∂xj) denotes the Hessian of u(x), detD2u is the so-called Monge–Ampère operator, and b and g have the following properties:
b
∈
C
∞
(
Ω
)
is positive in Ω.
g
∈
C
1
(
(
0
,
∞
)
,
(
0
,
∞
)
)
is decreasing in (0,∞) and lims→0+g(s)=∞.
The problem arises from the construction of a Riemannian metric on Ω that is invariant under projective transformations. When g(s)=s-(N+2), s>0 and b(x)≡1 in Ω, this problem was investigated by Nirenberg [13], Loewner and Nirenberg [9] for N=2, and Cheng and Yau [3] for N≥2. Cheng and Yau established that if Ω is bounded convex, then there exists a unique solution u∈C∞(Ω)∩C(Ω¯) to problem (1.1). Later Lazer and McKenna [8] considered problem (1.1) with g(s)=s-γ (s>0) for γ>0 and b∈C∞(Ω¯), which is positive on Ω¯. They showed that if γ>1, then there exists a unique convex solution u∈C2(Ω)∩C(Ω¯) to problem (1.1). Moreover, u satisfies
c
1
(
d
(
x
)
)
(
N
+
1
)
/
(
N
+
γ
)
≤
-
u
(
x
)
≤
c
2
(
d
(
x
)
)
(
N
+
1
)
/
(
N
+
γ
)
,
x
∈
Ω
,
where c1 and c2 are positive constants, d(x)=dist(x,∂Ω), x∈Ω.
In [11] and [12], Mohammed extended the results in [8] for more general g and b. He showed the following results. Let b satisfy (b1) and g be a positive nonincreasing smooth function in (0,∞).
Problem (1.1) admits a convex solution u∈C∞(Ω)∩C(Ω¯) if and only if the problem
det
D
2
u
(
x
)
=
b
(
x
)
,
x
∈
Ω
,
u
|
∂
Ω
=
0
,
admits a convex solution u0∈C∞(Ω)∩C(Ω¯). In particular, when b∈C∞(Ω¯) is positive on Ω¯, problem (1.1) admits a convex solution u∈C∞(Ω)∩C(Ω¯).
If lims→0+g(s)=∞, then problem (1.1) has a unique convex solution u∈C∞(Ω)∩C(Ω¯). Moreover, u satisfies the estimates
c
1
φ
(
d
(
x
)
)
≤
-
u
(
x
)
≤
c
2
φ
(
d
(
x
)
)
and
|
∇
u
(
x
)
|
≤
c
2
φ
(
d
(
x
)
)
d
(
x
)
near
∂
Ω
,
where c1 and c2 are positive constants, and φ is the solution to the following problem:
∫
0
φ
(
t
)
(
G
(
s
)
)
-
1
/
(
N
+
1
)
𝑑
s
=
t
,
t
∈
(
0
,
t
0
)
,
where G(s)=∫ss0g(τ)𝑑τ<∞, s∈(0,s0), s0∈(0,∞].
For the case where Ω is the unit ball, Goncalves and Santos [6] showed the existence, uniqueness and regularity of convex radially symmetric classical solutions to a more general problem than (1.1). For the case where the Monge–Ampère operator is replaced by the Laplace operator, many papers have been dedicated to resolving asymptotic behavior issues for solutions, see, for instance, [17, 18, 22, 20, 21] and the references therein. For the existence, regularity and other properties of solutions to the Monge–Ampère equations, see, for instance, [2, 4, 5, 7, 16, 19] and the references therein.
For convenience, we introduce three kinds of functions. Firstly, we denote K the set of all functions L^ which are defined on (0,η] as follows:
L
^
(
s
)
=
c
0
exp
(
∫
s
η
y
(
τ
)
τ
d
τ
)
,
s
∈
(
0
,
η
]
, for some
η
>
0
,
where c0>0 and y∈C[0,η], with y(0)=0. Secondly, let Λ denote the set of all positive monotonic functions θ in C1(0,δ0)∩L1(0,δ0) (δ0>0) which satisfy
(1.2)
lim
t
→
0
+
d
d
t
(
Θ
(
t
)
θ
(
t
)
)
:=
C
θ
∈
[
0
,
∞
)
,
Θ
(
t
)
:=
∫
0
t
θ
(
s
)
𝑑
s
.
Finally, we denote by κ1(x¯),…,κN-1(x¯) the principal curvatures of ∂Ω at x¯ and set
(1.3)
M
0
:=
max
x
¯
∈
∂
Ω
∏
i
=
1
N
-
1
κ
i
(
x
¯
)
,
m
0
:=
min
x
¯
∈
∂
Ω
∏
i
=
1
N
-
1
κ
i
(
x
¯
)
.
Then we can see that m0>0, provided that Ω is a strictly convex, bounded smooth domain in ℝN, with N≥2.
In this paper, we show a new boundary behavior of the unique convex solution to problem (1.1) under the following structure condition on g:
there exists Cg≥0 such that
lim
s
→
0
+
H
′
(
s
)
∫
0
s
d
τ
H
(
τ
)
=
-
C
g
,
H
(
s
)
:=
(
N
g
(
s
)
)
1
/
N
,
s
>
0
.
A complete characterization of g in (g2) is provided in Lemma 2.10.
Our main results are summarized as follows.
Theorem 1.1.
Let g satisfy (g1) and (g2), and b satisfy (b1) and the following condition:
There exist
θ
∈
Λ
and positive constants
b
i
(
i
=
1
,
2
) such that
b
1
:=
lim
d
(
x
)
→
0
inf
b
(
x
)
θ
N
+
1
(
d
(
x
)
)
≤
b
2
:=
lim
d
(
x
)
→
0
sup
b
(
x
)
θ
N
+
1
(
d
(
x
)
)
.
If
(1.4)
N
C
θ
+
(
1
+
N
)
C
g
>
1
+
N
,
then, for the unique classical convex solution u to problem (1.1), we have
ξ
1
1
-
C
g
≤
lim
d
(
x
)
→
0
inf
-
u
(
x
)
ψ
(
(
Θ
(
d
(
x
)
)
)
(
N
+
1
)
/
N
)
≤
lim
d
(
x
)
→
0
sup
-
u
(
x
)
ψ
(
(
Θ
(
d
(
x
)
)
)
(
N
+
1
)
/
N
)
≤
ξ
2
1
-
C
g
,
where m0,M0 are given in (1.3), ψ is the solution to the following problem:
∫
0
ψ
(
t
)
(
N
g
(
τ
)
)
-
1
/
N
𝑑
τ
=
t
for all
t
>
0
,
and
(1.5)
ξ
1
N
=
(
N
N
+
1
)
N
b
1
M
0
(
(
1
+
N
)
C
g
+
N
C
θ
-
1
-
N
)
,
(1.6)
ξ
2
N
=
(
N
N
+
1
)
N
b
2
m
0
(
(
1
+
N
)
C
g
+
N
C
θ
-
1
-
N
)
.
In particular, (i1) holds when Cg=1 and u verifies
lim
d
(
x
)
→
0
-
u
(
x
)
ψ
(
(
Θ
(
d
(
x
)
)
)
(
N
+
1
)
/
N
)
=
1
,
and (i2) holds when Ω=BR, which is a ball of radius R centered at the origin, Cg<1 and b1=b2=b0 in (b2), r=|x|, x∈BR, and u verifies
lim
r
→
R
-
u
(
x
)
ψ
(
(
Θ
(
R
-
r
)
)
(
N
+
1
)
/
N
)
=
ξ
0
1
-
C
g
,
where
ξ
0
N
=
(
N
N
+
1
)
N
b
0
R
N
-
1
(
1
+
N
)
C
g
+
N
C
θ
-
1
-
N
.
Remark 1.2.
In Lemma 2.10, one can see that Cg∈[0,1]. Then (1.4) implies Cθ>0.
Theorem 1.3.
Let g satisfy (g1) and (g2), and b satisfy (b1) and the following condition:
There exist positive constants
b
i
(
i
=
1
,
2
) and
L
^
∈
K
, with
∫
0
η
L
^
(
τ
)
τ
𝑑
τ
<
∞
,
such that
b
1
:=
lim
d
(
x
)
→
0
inf
b
(
x
)
(
d
(
x
)
)
-
N
-
1
(
L
^
(
d
(
x
)
)
)
N
≤
b
2
:=
lim
d
(
x
)
→
0
sup
b
(
x
)
(
d
(
x
)
)
-
N
-
1
(
L
^
(
d
(
x
)
)
)
N
.
Then, for the unique classical convex solution u to problem (1.1), we have
ξ
3
1
-
C
g
≤
lim
d
(
x
)
→
0
inf
-
u
(
x
)
ψ
(
h
(
d
(
x
)
)
)
≤
lim
d
(
x
)
→
0
sup
-
u
(
x
)
ψ
(
h
(
d
(
x
)
)
)
≤
ξ
4
1
-
C
g
,
where
(1.7)
h
(
t
)
=
∫
0
t
L
^
(
τ
)
τ
𝑑
τ
,
t
∈
(
0
,
η
]
,
and
(1.8)
ξ
3
N
=
b
1
M
0
N
,
ξ
4
N
=
b
2
m
0
N
.
In particular, (i1) holds when Cg=1 and u verifies
lim
d
(
x
)
→
0
-
u
(
x
)
ψ
(
h
(
d
(
x
)
)
)
=
1
,
and (i2) holds when Ω=BR, Cg<1 and b1=b2=b0 in (b3), and u verifies
lim
r
→
R
-
u
(
x
)
ψ
(
h
(
R
-
r
)
)
=
(
b
0
R
N
-
1
N
)
(
1
-
C
g
)
/
N
.
Remark 1.4.
In Lemma 2.9, one can see that θ1(t)=t-1(L^(t))N/(1+N) does not belong to Λ.
Corollary 1.5.
Let Ω=BR, let b satisfy (b1) and let g(s)=s-γ, with γ>0.
If
b
satisfies (b2) with b1=b2=b0, and
C
θ
>
N
+
1
N
+
γ
,
then for the unique classical convex solution
u
to problem (
1.1
), we have
lim
r
→
R
-
u
(
x
)
(
Θ
(
R
-
r
)
)
(
N
+
1
)
/
(
N
+
γ
)
=
(
(
N
+
γ
N
+
1
)
N
b
0
(
N
+
γ
)
R
N
-
1
C
θ
(
N
+
γ
)
-
(
N
+
1
)
)
1
/
(
N
+
γ
)
.
If
b
satisfies (b3) with b1=b2=b0, then for the unique classical convex solution u to problem (1.1), we have
lim
r
→
R
-
u
(
x
)
(
h
(
R
-
r
)
)
N
/
(
N
+
γ
)
=
(
b
0
R
N
-
1
(
N
+
γ
N
)
N
)
1
/
(
N
+
γ
)
.
For NCθ+(1+N)Cg=1+N, we have the following theorem.
Theorem 1.6.
Let g(s)=s-γ, s∈(0,∞), with γ>0, and let b satisfy (b1) and the following condition:
There exist positive constants
b
i
(
i
=
1
,
2
) and
L
^
∈
K
, with
∫
0
η
L
^
(
τ
)
τ
𝑑
τ
=
∞
,
such that
b
1
:=
lim
d
(
x
)
→
0
inf
b
(
x
)
(
d
(
x
)
)
γ
-
1
L
^
(
d
(
x
)
)
≤
b
2
:=
lim
d
(
x
)
→
0
sup
b
(
x
)
(
d
(
x
)
)
γ
-
1
L
^
(
d
(
x
)
)
.
Then, for the unique classical convex solution u to problem (1.1), we have
ξ
5
≤
lim
d
(
x
)
→
0
inf
-
u
(
x
)
Φ
(
d
(
x
)
)
≤
lim
d
(
x
)
→
0
sup
-
u
(
x
)
Φ
(
d
(
x
)
)
≤
ξ
6
,
where
Φ
(
t
)
=
t
(
∫
t
η
L
^
(
τ
)
τ
𝑑
τ
)
1
/
(
N
+
γ
)
,
t
∈
(
0
,
η
)
,
and
(1.9)
ξ
5
N
+
γ
=
b
1
(
N
+
γ
)
M
0
,
ξ
6
N
+
γ
=
b
2
(
N
+
γ
)
m
0
.
Remark 1.7.
In Theorem 1.6, we have Cg=γN+γ, Cθ=N+1N+γ and NCθ=(N+1)(1-Cg).
The outline of this paper is as follows. In Section 2, we give some basics of the Karamata regular variation theory and some preliminaries. The proofs of Theorems 1.1, 1.3 and 1.6 are provided in Section 3.
2 Some Basics of the Karamata Regular Variation Theory
Our approach relies on the Karamata regular variation theory, established by Karamata in 1930, which is a basic tool in stochastic processes (see [1, 10, 14, 15] and the references therein).
In this section, we present some basics of the Karamata regular variation theory and some preliminaries.
Definition 2.1 ([15, Definition 1.1]).
A positive continuous function f, defined on (0,η] for some η>0, is called regularly varying at zero with index ρ∈ℝ, denoted by f∈RVZρ, if for each ξ>0,
(2.1)
lim
s
→
0
+
f
(
ξ
s
)
f
(
s
)
=
ξ
ρ
.
In particular, when ρ=0, f is called slowly varying at zero.
Clearly, if f∈RVZρ, then L(s):=f(s)/sρ is slowly varying at zero.
Proposition 2.2 (Uniform Convergence Theorem, see [14, Proposition 0.5]).
If f∈RVZρ, then (2.1) holds uniformly for ξ∈[c1,c2], with 0<c1<c2.
Proposition 2.3 (The Karamata Representation Theorem, see [15, Theorem 1.2] and [14, Corollary, p. 17]).
A function L is slowly varying at zero if and only if it can be written in the form
L
(
s
)
=
l
(
s
)
exp
(
∫
s
η
y
(
τ
)
τ
d
τ
)
,
s
∈
(
0
,
η
]
,
where the functions l and y are continuous, and y(s)→0 and l(s)→c0 as s→0+, with c0>0.
Definition 2.4 ([10, Definition 0.3]).
The function
L
^
(
s
)
=
c
0
exp
(
∫
s
η
y
(
τ
)
τ
d
τ
)
,
s
∈
(
0
,
η
]
,
is normalized slowly varying at zero and
f
(
s
)
=
s
ρ
L
^
(
s
)
,
s
∈
(
0
,
η
]
,
is normalized regularly varying at zero with index ρ (denoted by f∈NRVZρ).
Equivalently, f∈NRVZρ if and only if
(2.2)
f
∈
C
1
(
0
,
η
]
for some
η
>
0
and
lim
s
→
0
+
s
f
′
(
s
)
f
(
s
)
=
ρ
.
Proposition 2.5 ([1, Proposition 1.3.6]).
If the functions L,L1 are slowly varying at zero, then the following hold:
The functions
L
ρ
(for every
ρ
∈
ℝ
),
c
1
L
+
c
2
L
1
(
c
1
≥
0
, c2≥0, with c1+c2>0), L⋅L1, and L∘L1 (if L1(s)→0 as s→0+), are also slowly varying at zero.
For every
ε
>
0
, sεL(s)→0 and s-εL(s)→∞ as s→0+.
For
ρ
∈
ℝ
, ln(L(s))/lns→0 and ln(sρL(s))/lns→ρ as s→0+.
Proposition 2.6 ([1, Proposition 1.5.7]).
If f1∈RVZρ1, f2∈RVZρ2, with lims→0f2(s)=0, then f1∘f2∈RVZρ1ρ2.
Proposition 2.7 (Asymptotic Behavior, see [1, Propositions 1.5.8 and 1.5.10]).
If a function L is slowly varying at zero, then, for η>0 and as t→0+, we have
∫
0
t
s
ρ
L
(
s
)
𝑑
s
≅
(
1
+
ρ
)
-
1
t
1
+
ρ
L
(
t
)
for
ρ
>
-
1
,
∫
t
η
s
ρ
L
(
s
)
𝑑
s
≅
(
-
ρ
-
1
)
-
1
t
1
+
ρ
L
(
t
)
for
ρ
<
-
1
.
Proposition 2.8 ([17, Lemma 2.3]).
Let L^∈K be defined on (0,η]. Then we have
lim
t
→
0
+
L
^
(
t
)
∫
t
η
L
^
(
τ
)
τ
𝑑
τ
=
0
.
If, further, ∫0ηL^(τ)τ𝑑τ converges, then we have
lim
t
→
0
+
L
^
(
t
)
∫
0
t
L
^
(
τ
)
τ
𝑑
τ
=
0
.
Our results in this section are summarized as follows.
Lemma 2.9.
Let θ∈Λ.
We have
lim
t
→
0
+
Θ
(
t
)
θ
(
t
)
=
0
.
We have
lim
t
→
0
+
Θ
(
t
)
θ
′
(
t
)
θ
2
(
t
)
=
1
-
lim
t
→
0
+
d
d
t
(
Θ
(
t
)
θ
(
t
)
)
=
1
-
C
θ
.
When θ is nondecreasing, Cθ∈[0,1], and Cθ≥1, provided that θ is nonincreasing.
When
C
θ
>
0
, θ∈NRVZ(1-Cθ)/Cθ. In particular, when Cθ=1, θ is normalized slowly varying at zero.
When
C
θ
=
0
, θ grows faster than any tp (p>1) near zero.
Proof.
Let θ∈Λ.
(i) When θ is nondecreasing, we have that 0<Θ(t)≤tθ(t) for all t∈(0,δ0) and the result holds. When θ is nonincreasing, since θ∈L1(0,δ0), it follows that
lim
t
→
0
+
Θ
(
t
)
θ
(
t
)
=
lim
t
→
0
+
Θ
(
t
)
lim
t
→
0
+
1
θ
(
t
)
=
0
.
(ii) By (1.2), we see that
lim
t
→
0
+
Θ
(
t
)
θ
′
(
t
)
θ
2
(
t
)
=
1
-
lim
t
→
0
+
d
d
t
(
Θ
(
t
)
θ
(
t
)
)
=
1
-
C
θ
.
So, when θ is nondecreasing, Cθ∈[0,1], and Cθ≥1, provided that θ is nonincreasing.
(iii) Equation (1.2), part (i) and L’Hospital’s rule imply that
lim
t
→
0
+
Θ
(
t
)
t
θ
(
t
)
=
lim
t
→
0
+
Θ
(
t
)
θ
(
t
)
t
=
lim
t
→
0
+
d
d
t
(
Θ
(
t
)
θ
(
t
)
)
=
C
θ
.
So, when Cθ>0, Θ∈NRVZCθ-1 and it follows that
lim
t
→
0
+
t
θ
′
(
t
)
θ
(
t
)
=
lim
t
→
0
+
Θ
(
t
)
θ
′
(
t
)
θ
2
(
t
)
lim
t
→
0
t
θ
(
t
)
Θ
(
t
)
=
1
-
C
θ
C
θ
,
i.e., θ∈NRVZ(1-Cθ)/Cθ.
(iv) When Cθ=0, it follows from (iii) that
lim
t
→
0
t
θ
(
t
)
Θ
(
t
)
=
+
∞
.
Then, for an arbitrary q>0, there exists t0>0 such that
(2.3)
θ
(
t
)
Θ
(
t
)
>
(
q
+
1
)
t
-
1
for all
t
∈
(
0
,
t
0
]
.
Integrating (2.3) from t to t0, we obtain
ln
(
Θ
(
t
0
)
)
-
ln
(
Θ
(
t
)
)
>
(
q
+
1
)
(
ln
t
0
-
ln
t
)
for all
t
∈
(
0
,
t
0
]
,
i.e.,
0
<
Θ
(
t
)
t
q
<
Θ
(
t
0
)
t
0
q
+
1
t
,
t
∈
(
0
,
t
0
]
.
Letting t→0, we see that Θ grows faster than any tp (p>1) near zero and so does θ. ∎
Lemma 2.10.
Let g satisfy (g1).
If
g
satisfies (g2), then Cg≤1.
(g2) holds with Cg∈(0,1) if and only if g∈NRVZ-γN, with γ>0. In this case, γ=Cg/(1-Cg).
(g2) holds with Cg=0 if and only if g is normalized slowly varying at zero.
If (g2) holds with Cg=1, then g grows faster than any s-p (p>1) near zero.
If
g
∈
C
2
(
0
,
s
0
)
for some
s
0
>
0
and
(2.4)
g
′′
(
s
)
>
0
for all
s
∈
(
0
,
s
0
)
,
lim
s
→
0
+
g
(
s
)
g
′′
(
s
)
(
g
′
(
s
)
)
2
=
1
,
then
g
satisfies (g2) with Cg=1.
Proof.
Recalling that H(s)=(Ng(s))1/N, s>0, and g satisfies (g1), we see that
0
<
∫
0
s
d
τ
H
(
τ
)
≤
s
H
(
s
)
for all
s
>
0
,
i.e.,
0
<
H
(
s
)
∫
0
s
d
τ
H
(
τ
)
≤
s
for all
s
>
0
,
and
(2.5)
lim
s
→
0
+
H
(
s
)
∫
0
s
d
τ
H
(
τ
)
=
0
.
(i) Let
I
(
s
)
=
-
H
′
(
s
)
∫
0
s
d
τ
H
(
τ
)
for all
s
>
0
.
Integrating I(t) from 0 to s and using integration by parts, by (2.5), we obtain
∫
0
s
I
(
t
)
𝑑
t
=
-
H
(
s
)
∫
0
s
d
τ
H
(
τ
)
+
s
for all
s
>
0
,
i.e.,
0
<
H
(
s
)
∫
0
s
d
τ
H
(
τ
)
s
=
1
-
∫
0
s
I
(
t
)
𝑑
t
s
for all
s
>
0
.
From L’Hospital’s rule, it follows that
(2.6)
0
≤
lim
s
→
0
+
H
(
s
)
∫
0
s
d
τ
H
(
τ
)
s
=
1
-
lim
s
→
0
+
I
(
s
)
=
1
-
C
g
.
So (i) holds.
(ii) When (g2) holds with Cg∈(0,1), from (2.6), it follows that
lim
s
→
0
+
H
(
s
)
s
H
′
(
s
)
=
lim
s
→
0
+
H
(
s
)
∫
0
s
d
τ
H
(
τ
)
s
H
′
(
s
)
∫
0
s
d
τ
H
(
τ
)
=
-
1
C
g
lim
s
→
0
+
H
(
s
)
∫
0
s
d
τ
H
(
τ
)
s
=
-
1
-
C
g
C
g
,
i.e., H∈NRVZ-Cg/(1-Cg). Then g∈NRVZ-NCg/(1-Cg).
Conversely, when g∈NRVZ-γ with γ>0, i.e., lims→0+sg′(s)g(s)=-γ, there exist a positive constant η and L^∈K such that g(s)=s-γL^(s), s∈(0,η], and H(s)=N1/Ns-γ/NL^1(s), with L^1(s)=L^1/N(s). From (2.2) and Proposition 2.7 (i), it follows that
-
lim
s
→
0
+
H
′
(
s
)
∫
0
s
d
τ
H
(
τ
)
=
-
lim
s
→
0
+
s
H
′
(
s
)
H
(
s
)
lim
s
→
0
+
H
(
s
)
∫
0
s
d
τ
H
(
τ
)
s
=
γ
N
lim
s
→
0
+
s
-
(
1
+
γ
)
/
(
N
+
γ
)
L
^
1
(
s
)
∫
0
s
τ
γ
/
N
(
L
^
1
(
τ
)
)
-
1
𝑑
τ
=
γ
N
+
γ
=
C
g
.
(iii) Since Cg=0, from the proof of (i), one can see that
lim
s
→
0
+
s
H
′
(
s
)
H
(
s
)
=
lim
s
→
0
+
s
H
′
(
s
)
∫
0
s
d
τ
H
(
τ
)
H
(
s
)
∫
0
s
d
τ
H
(
τ
)
=
(
lim
s
→
0
+
H
(
s
)
s
∫
0
s
d
τ
H
(
τ
)
)
-
1
lim
s
→
0
+
H
′
(
s
)
∫
0
s
d
τ
H
(
τ
)
=
0
,
i.e., H is normalized slowly varying at zero, and so is g. Conversely, when H is normalized slowly varying at zero, i.e., lims→0+sH′(s)H(s)=0, it follows, by (2.6), that
lim
s
→
0
+
H
′
(
s
)
∫
0
s
d
τ
H
(
τ
)
=
lim
s
→
0
+
s
H
′
(
s
)
H
(
s
)
H
(
s
)
∫
0
s
d
τ
H
(
τ
)
s
=
0
.
(iv) Since Cg=1, from the proof of (ii), we see that lims→0+H(s)sH′(s)=0, i.e., lims→0+sH′(s)H(s)=-∞. Similar to the proof of Lemma 2.9 (iv), we can show that H grows faster than any t-p (p>1) near zero, and so does g.
(v) By a direct calculation and L’Hospital’s rule, we see that
lim
s
→
0
+
H
′
(
s
)
∫
0
s
d
τ
H
(
τ
)
=
lim
s
→
0
+
∫
0
s
d
τ
H
(
τ
)
(
H
′
(
s
)
)
-
1
=
-
lim
s
→
0
+
(
H
′
(
s
)
)
2
H
(
s
)
H
′′
(
s
)
=
-
lim
s
→
0
+
(
g
′
(
s
)
)
2
N
g
(
s
)
g
′′
(
s
)
-
(
N
-
1
)
(
g
′
(
s
)
)
2
=
-
lim
s
→
0
+
1
N
g
(
s
)
g
′′
(
s
)
(
g
′
(
s
)
)
2
-
(
N
-
1
)
=
-
1
.
The proof is completed. ∎
Recall that ψ is the solution to
∫
0
ψ
(
t
)
d
τ
H
(
τ
)
=
t
,
H
(
τ
)
=
(
N
g
(
τ
)
)
1
/
N
,
t
∈
[
0
,
∞
)
.
Lemma 2.11.
Let g satisfy (g1) and (g2). Then the following hold:
We have
ψ
(
t
)
>
0
for
t
>
0
,
ψ
(
0
)
=
0
,
ψ
′
(
t
)
=
H
(
ψ
(
t
)
)
=
(
N
g
(
ψ
(
t
)
)
)
1
/
N
,
ψ
′
(
0
)
:=
lim
t
→
0
+
ψ
′
(
t
)
=
lim
t
→
0
+
(
N
g
(
ψ
(
t
)
)
)
1
/
N
=
∞
,
ψ
′′
(
t
)
=
H
(
ψ
(
t
)
)
H
′
(
ψ
(
t
)
)
=
g
′
(
ψ
(
t
)
)
(
N
g
(
ψ
(
t
)
)
)
(
N
-
2
)
/
N
,
t
>
0
.
We have
lim
t
→
0
+
t
H
(
ψ
(
t
)
)
=
0
𝑎𝑛𝑑
lim
t
→
0
+
t
H
′
(
ψ
(
t
)
)
=
lim
t
→
0
+
t
g
′
(
ψ
(
t
)
)
(
N
g
(
ψ
(
t
)
)
)
(
N
-
1
)
/
N
=
-
C
g
.
ψ
∈
NRVZ
1
-
C
g
and
ψ
′
∈
NRVZ
-
C
g
.
If
θ
∈
Λ
and
(
1
+
N
)
C
g
+
N
C
θ
>
1
+
N
, then
ψ
∘
Θ
(
1
+
N
)
/
N
∈
NRVZ
ρ
0
,
ρ
0
=
(
1
+
N
)
(
1
-
C
g
)
N
C
θ
∈
[
0
,
1
)
,
and
lim
t
→
0
+
t
ψ
(
ξ
Θ
(
1
+
N
)
/
N
(
t
)
)
=
0
uniformly for
ξ
∈
[
c
1
,
c
2
]
, with
0
<
c
1
<
c
2
, where Θ is given as in (1.2).
lim
t
→
0
+
t
ψ
(
ξ
h
(
t
)
)
=
0
uniformly for
ξ
∈
[
c
1
,
c
2
]
, with
0
<
c
1
<
c
2
, where
h
is given in (
1.7
).
Proof.
(i) This is easily obtained.
(ii) By (g2) and the definitions of ψ and H, we see that (s=ψ(t))
H
′
(
ψ
(
t
)
)
=
g
′
(
ψ
(
t
)
)
(
N
g
(
ψ
(
t
)
)
)
(
N
-
1
)
/
N
and
lim
t
→
0
+
t
g
′
(
ψ
(
t
)
)
(
N
g
(
ψ
(
t
)
)
)
(
N
-
1
)
/
N
=
lim
t
→
0
+
t
H
′
(
ψ
(
t
)
)
=
lim
s
→
0
+
H
′
(
s
)
∫
0
s
d
ν
H
(
ν
)
=
-
C
g
.
(iii) From (i), (ii) and (2.6), it follows that
lim
t
→
0
+
t
ψ
′
(
t
)
ψ
(
t
)
=
lim
t
→
0
+
t
H
(
ψ
(
t
)
)
ψ
(
t
)
=
lim
t
→
0
+
H
(
ψ
(
t
)
)
∫
0
ψ
(
t
)
d
τ
H
(
τ
)
ψ
(
t
)
=
lim
s
→
0
+
H
(
s
)
∫
0
s
d
τ
H
(
τ
)
s
=
1
-
C
g
,
i.e., ψ∈NRVZ1-Cg. Moreover, by (i), we have
lim
t
→
0
+
t
ψ
′′
(
t
)
ψ
′
(
t
)
=
lim
t
→
0
+
t
H
′
(
ψ
(
t
)
)
=
-
C
g
,
i.e., ψ′∈NRVZ-Cg.
(iv) When Cg=1, the result follows from (iii) and Proposition 2.5.
When Cg<1, since Cθ>0 (see Remark 1.2), by Lemma 2.9 and Proposition 2.6, we see that Θ∈NRVZ1/Cθ and ψ∘Θ(1+N)/N∈NRVZρ0. So there exist c0>0 and a function L^, which is normalized slowly varying at zero, such that
ψ
(
Θ
(
1
+
N
)
/
N
(
t
)
)
=
t
ρ
0
L
^
(
t
)
.
Since (1+N)Cg+NCθ>1+N, by Proposition 2.5 (ii), we have
1
-
ρ
0
=
N
C
θ
+
(
1
+
N
)
C
g
-
1
-
N
N
C
θ
>
0
and
lim
t
→
0
+
t
ψ
(
Θ
(
1
+
N
)
/
N
(
t
)
)
=
lim
t
→
0
+
t
1
-
ρ
0
(
L
^
(
t
)
)
-
1
=
0
.
(v) Since h∈NRVZ0, by Definition 2.1 and Proposition 2.6, we can show that ψ∘h∈NRVZ0, and thus the final result follows from Proposition 2.5. ∎
3 Boundary Behavior
In this section, we prove Theorems 1.1, 1.3 and 1.6.
Lemma 3.1 (The Comparison Principle, see [12, Lemma 4.1]).
Let Ω be a bounded convex domain in RN, with N≥2, and let f∈C1(Ω×(0,∞),(0,∞)) be decreasing in s for each x∈Ω. In addition, assume that u1,u2∈C(Ω¯)∩C2(Ω) satisfy the following conditions:
det
D
2
u
1
(
x
)
≥
f
(
x
,
-
u
1
)
and
det
D
2
u
2
(
x
)
≤
f
(
x
,
-
u
2
)
, x∈Ω,
Then we have u2≥u1 in Ω.
For any δ>0, let
Ω
δ
=
{
x
∈
Ω
:
0
<
d
(
x
)
<
δ
}
.
When Ω is Cm-smooth for m≥2, choose δ1∈(0,δ0) (δ0 is given as in Λ) such that (see [5, Lemmas 14.16 and 14.17])
d
∈
C
m
(
Ω
δ
1
)
and
|
∇
d
(
x
)
|
=
1
for all
x
∈
Ω
δ
1
.
Moreover, let x¯ be the projection of the point x∈Ωδ1 to ∂Ω, and κi(x¯) (i=1,…,N-1) be the principal curvatures of ∂Ω at x¯. Then we have
D
2
(
d
(
x
)
)
=
diag
[
-
κ
1
(
x
¯
)
1
-
d
(
x
)
κ
1
(
x
¯
)
,
…
,
-
κ
N
-
1
(
x
¯
)
1
-
d
(
x
)
κ
N
-
1
(
x
¯
)
,
0
]
.
Lemma 3.2 (See the proof of [8, Proposition 2.4] and [4, Proposition 2.1 and Corollary 2.3]).
Let ϕ be a C2-function on (0,δ). Then we have
det
D
2
ϕ
(
d
(
x
)
)
=
(
-
ϕ
′
(
d
(
x
)
)
)
N
-
1
ϕ
′′
(
d
(
x
)
)
(
∏
i
=
1
N
-
1
κ
i
(
x
¯
)
1
-
d
(
x
)
κ
i
(
x
¯
)
)
,
x
∈
Ω
δ
1
.
Proof of Theorem 1.1.
For convenience, let
(3.1)
σ
(
d
(
x
)
)
=
(
Θ
(
d
(
x
)
)
)
(
N
+
1
)
/
N
,
x
∈
Ω
δ
1
.
For an arbitrary ε∈(0,min{1/4,b1/4}), let
ζ
1
N
=
(
N
N
+
1
)
N
(
1
-
ε
)
(
b
1
-
2
ε
)
M
0
(
(
1
+
N
)
C
g
+
N
C
θ
-
1
-
N
)
,
ζ
2
N
=
(
N
N
+
1
)
N
(
1
+
ε
)
(
b
2
+
2
ε
)
m
0
(
(
1
+
N
)
C
g
+
N
C
θ
-
1
-
N
)
,
where m0 and M0 are given in (1.3), and b1 and b2 are given in (b2). It follows that
ξ
1
N
/
4
≤
ζ
1
N
≤
ζ
2
N
≤
4
ξ
2
N
,
where ξ1 and ξ2 are given as in (1.5) and (1.6).
Recalling (1.2) and using Lemma 2.9 (ii) and Lemma 2.11 (ii), we find that
Θ
∈
C
1
(
0
,
δ
0
)
∩
C
[
0
,
δ
0
)
,
Θ
(
0
)
=
0
,
ζ
Θ
(
d
(
x
)
)
=
∫
0
ψ
(
ζ
Θ
(
d
(
x
)
)
)
d
τ
H
(
τ
)
,
ζ
>
0
,
x
∈
Ω
,
lim
d
(
x
)
→
0
Θ
(
d
(
x
)
)
θ
′
(
d
(
x
)
)
θ
2
(
d
(
x
)
)
=
1
-
C
θ
,
lim
d
(
x
)
→
0
ζ
j
σ
(
d
(
x
)
)
g
′
(
ψ
(
ζ
j
σ
(
d
(
x
)
)
)
)
(
N
g
(
ψ
(
ζ
j
σ
(
d
(
x
)
)
)
)
)
(
N
-
1
)
/
N
=
-
C
g
for
j
=
1
,
2
,
lim
d
(
x
)
→
0
(
∏
i
=
1
N
-
1
(
1
-
d
(
x
)
κ
i
(
x
¯
)
)
)
=
1
,
m
0
ζ
2
N
(
N
+
1
N
)
N
(
(
1
+
N
)
C
g
+
N
C
θ
-
1
-
N
)
-
(
1
+
ε
)
(
b
2
+
ε
)
=
ε
(
1
+
ε
)
,
M
0
ζ
1
N
(
N
+
1
N
)
N
(
(
1
+
N
)
C
g
+
N
C
θ
-
1
-
N
)
-
(
1
-
ε
)
(
b
1
-
ε
)
=
-
ε
(
1
-
ε
)
.
It follows from (b2) that there exists a sufficiently small δε∈(0,δ1) corresponding to ε such that, for x∈Ωδε and j=1,2,
(
b
1
-
ε
)
θ
N
+
1
(
d
(
x
)
)
<
b
(
x
)
<
(
b
2
+
ε
)
θ
N
+
1
(
d
(
x
)
)
,
and
m
0
(
(
1
+
N
)
C
g
+
N
C
θ
-
1
-
N
)
1
+
ε
≤
(
∏
i
=
1
N
-
1
κ
i
(
x
¯
)
1
-
d
(
x
)
κ
i
(
x
¯
)
)
(
-
(
N
+
1
)
ζ
j
σ
(
d
(
x
)
)
g
′
(
ψ
(
ζ
j
σ
(
d
(
x
)
)
)
)
(
N
g
(
ψ
(
ζ
j
σ
(
d
(
x
)
)
)
)
)
(
N
-
1
)
/
N
-
(
1
+
N
Θ
(
d
(
x
)
)
θ
′
(
d
(
x
)
)
θ
2
(
d
(
x
)
)
)
)
≤
M
0
(
(
1
+
N
)
C
g
+
N
C
θ
-
1
-
N
)
1
-
ε
.
Let
u
¯
ε
=
-
ψ
(
ζ
2
σ
(
d
(
x
)
)
)
and
u
¯
ε
=
-
ψ
(
ζ
1
σ
(
d
(
x
)
)
)
,
x
∈
Ω
δ
ε
,
where σ is given in (3.1).
Let ϕ(t)=-ψ(ζ2(σ(t)) in Lemma 3.2. Then, by a direct computation, for x∈Ωδε, we have
det
D
2
u
¯
ε
(
x
)
-
(
b
2
+
ε
)
(
θ
(
d
(
x
)
)
)
N
+
1
g
(
-
u
¯
ε
(
x
)
)
=
(
∏
i
=
1
N
-
1
κ
i
(
x
¯
)
1
-
d
(
x
)
κ
i
(
x
¯
)
)
(
ζ
2
N
+
1
N
θ
(
d
(
x
)
)
Θ
1
/
N
(
d
(
x
)
)
(
N
g
(
ψ
(
ζ
2
σ
(
d
(
x
)
)
)
)
)
1
/
N
)
N
-
1
×
(
-
(
ζ
2
N
+
1
N
θ
(
d
(
x
)
)
Θ
1
/
N
(
d
(
x
)
)
)
2
g
′
(
ψ
(
ζ
2
σ
(
d
(
x
)
)
)
)
(
N
g
(
ψ
(
ζ
2
σ
(
d
(
x
)
)
)
)
)
(
2
-
N
)
/
N
-
ζ
2
N
+
1
N
(
N
g
(
ψ
(
ζ
2
σ
(
d
(
x
)
)
)
)
)
1
/
N
(
1
N
(
Θ
(
d
(
x
)
)
)
(
1
-
N
)
/
N
θ
2
(
d
(
x
)
)
+
Θ
1
/
N
(
d
(
x
)
)
θ
′
(
d
(
x
)
)
)
)
-
(
b
2
+
ε
)
(
θ
(
d
(
x
)
)
)
N
+
1
g
(
ψ
(
ζ
2
σ
(
d
(
x
)
d
(
x
)
)
)
)
=
(
1
+
ε
)
-
1
(
θ
(
d
(
x
)
)
)
N
+
1
g
(
ψ
(
ζ
2
σ
(
d
(
x
)
)
)
)
(
(
ζ
2
N
+
1
N
)
N
(
∏
i
=
1
N
-
1
κ
i
(
x
¯
)
1
-
d
(
x
)
κ
i
(
x
¯
)
)
×
(
-
(
N
+
1
)
ζ
2
σ
(
d
(
x
)
)
g
′
(
ψ
(
ζ
2
σ
(
d
(
x
)
)
)
)
(
N
g
(
ψ
(
ζ
2
σ
(
d
(
x
)
)
)
)
)
(
N
-
1
)
/
N
-
(
1
+
N
Θ
(
d
(
x
)
)
θ
′
(
d
(
x
)
)
θ
2
(
d
(
x
)
)
)
)
(
1
+
ε
)
(
b
2
+
ε
)
)
≥
0
,
i.e., u¯ε is a subsolution to equation (1.1) in Ωδε.
In a similar way, we can show that u¯ε=-ψ(ζ1σ(d(x))) is a supersolution to equation (1.1) in Ωδε. Let u∈C(Ω¯)∩C2(Ω) be the unique convex solution to problem (1.1). We assert that there exists a large C such that
(3.2)
u
¯
ε
-
C
d
(
x
)
≤
u
≤
u
¯
ε
+
C
d
(
x
)
,
x
∈
Ω
δ
ε
.
In fact, since u¯ε|∂Ω=u¯ε|∂Ω=u|∂Ω=0, we can choose a large positive constant C such that
u
¯
ε
-
C
d
(
x
)
≤
u
≤
u
¯
ε
+
C
d
(
x
)
on
d
(
x
)
=
δ
ε
.
Let ϕ(t)=-ψ(ζ2σ(t))-Ct in Lemma 3.2. Then, by (g1) and a direct computation, for x∈Ωδε, we have
det
(
D
2
(
u
¯
ε
(
x
)
-
C
d
(
x
)
)
)
=
(
-
ϕ
′
(
d
(
x
)
)
)
N
-
1
ϕ
′′
(
d
(
x
)
)
(
∏
i
=
1
N
-
1
κ
i
(
x
¯
)
1
-
d
(
x
)
κ
i
(
x
¯
)
)
=
(
(
ψ
(
ζ
2
σ
(
t
)
)
)
′
|
t
=
d
(
x
)
+
C
)
N
-
1
(
-
ψ
(
ζ
2
σ
(
t
)
)
)
′′
|
t
=
d
(
x
)
(
∏
i
=
1
N
-
1
κ
i
(
x
¯
)
1
-
d
(
x
)
κ
i
(
x
¯
)
)
≥
det
(
D
2
(
u
¯
ε
(
x
)
)
)
≥
b
(
x
)
g
(
-
u
¯
ε
(
x
)
)
≥
b
(
x
)
g
(
-
u
¯
ε
(
x
)
+
C
d
(
x
)
)
.
In a similar way, we can show that
det
D
2
(
u
¯
ε
(
x
)
+
C
d
(
x
)
)
≤
b
(
x
)
g
(
-
u
¯
ε
(
x
)
-
C
d
(
x
)
)
,
x
∈
Ω
δ
ε
.
Thus, (3.2) follows from Lemma 3.1.
Consequently, by Lemma 2.11 (iv),
1
≤
lim
d
(
x
)
→
0
inf
-
u
(
x
)
ψ
(
ζ
1
σ
(
d
(
x
)
)
)
and
lim
d
(
x
)
→
0
sup
-
u
(
x
)
ψ
(
ζ
2
σ
(
d
(
x
)
)
)
≤
1
.
Thus, letting ε→0, we see that
1
≤
lim
d
(
x
)
→
0
inf
-
u
(
x
)
ψ
(
ξ
1
σ
(
d
(
x
)
)
)
and
lim
d
(
x
)
→
0
sup
-
u
(
x
)
ψ
(
ξ
2
σ
(
d
(
x
)
)
)
≤
1
.
By Lemma 2.11 (iii), we have
lim
d
(
x
)
→
0
ψ
(
ξ
2
σ
(
d
(
x
)
)
)
ψ
(
σ
(
d
(
x
)
)
)
=
ξ
2
1
-
C
g
and
lim
d
(
x
)
→
0
ψ
(
ξ
1
σ
(
d
(
x
)
)
)
ψ
(
σ
(
d
(
x
)
)
)
=
ξ
1
1
-
C
g
.
The proof is completed. ∎
Proof of Theorem 1.3.
Let ε∈(0,min{1/4,b1/4}),
ζ
3
N
=
(
1
-
ε
)
(
b
1
-
2
ε
)
N
M
0
and
ζ
4
N
=
(
1
+
ε
)
(
b
2
+
2
ε
)
N
m
0
.
It follows that
b
1
4
N
M
0
<
ζ
3
N
<
ζ
4
N
<
4
b
2
N
m
0
,
where ξ3 and ξ4 are given as in (1.8).
By (b3), (2.2), Proposition 2.8 and Lemma 2.11, we derive, for j=3,4, that
lim
d
(
x
)
→
0
(
-
ζ
j
h
(
d
(
x
)
)
g
′
(
ψ
(
ζ
j
h
(
d
(
x
)
)
)
)
(
g
(
ψ
(
ζ
j
h
(
d
(
x
)
)
)
)
)
(
N
-
1
)
/
N
)
=
C
g
,
lim
d
(
x
)
→
0
L
^
(
d
(
x
)
)
h
(
d
(
x
)
)
=
0
,
lim
d
(
x
)
→
0
d
(
x
)
L
^
′
(
d
(
x
)
)
L
^
(
d
(
x
)
)
=
0
.
It follows from (b3) that there exists a sufficiently small δε∈(0,δ1) corresponding to ε such that, for x∈Ωδε and j=3,4,
(
b
1
-
ε
)
(
d
(
x
)
)
-
N
-
1
(
L
^
(
d
(
x
)
)
)
N
<
b
(
x
)
<
(
b
2
+
ε
)
(
d
(
x
)
)
-
N
-
1
(
L
^
(
d
(
x
)
)
)
N
,
and
m
0
1
+
ε
≤
lim
d
(
x
)
→
0
(
(
∏
i
=
1
N
-
1
κ
i
(
x
¯
)
1
-
d
(
x
)
κ
i
(
x
¯
)
)
(
-
ζ
j
h
(
d
(
x
)
)
g
′
(
ψ
(
ζ
j
h
(
d
(
x
)
)
)
)
(
g
(
ψ
(
ζ
j
h
(
d
(
x
)
)
)
)
)
(
N
-
1
)
/
N
L
^
(
d
(
x
)
)
h
(
d
(
x
)
)
-
d
(
x
)
L
^
′
(
d
(
x
)
)
-
L
^
(
d
(
x
)
)
L
^
(
d
(
x
)
)
)
)
≤
M
0
1
-
ε
.
Thus, u¯ε=-ψ(ζ4h(d(x))) (x∈Ωδε) satisfies
det
D
2
u
¯
ε
(
x
)
-
(
b
2
+
ε
)
(
d
(
x
)
)
-
N
-
1
(
L
^
(
d
(
x
)
)
)
N
g
(
-
u
¯
ε
(
x
)
)
=
(
N
g
(
ψ
(
ζ
4
h
(
d
(
x
)
)
)
)
)
(
N
-
1
)
/
N
ζ
4
N
-
1
(
L
^
(
d
(
x
)
)
d
(
x
)
)
N
-
1
∏
i
=
1
N
-
1
κ
i
(
x
¯
)
1
-
d
(
x
)
κ
i
(
x
¯
)
×
(
-
ζ
4
2
g
′
(
ψ
(
ζ
4
h
(
d
(
x
)
)
)
)
(
N
g
(
ψ
(
ζ
4
h
(
d
(
x
)
)
)
)
)
-
(
N
-
2
)
/
N
(
L
^
(
d
(
x
)
)
d
(
x
)
)
2
-
ζ
4
(
N
g
(
ψ
(
ζ
4
h
(
d
(
x
)
)
)
)
)
1
/
N
d
(
x
)
L
^
′
(
d
(
x
)
)
-
L
^
(
d
(
x
)
)
d
2
(
x
)
)
-
(
b
2
+
ε
)
(
d
(
x
)
)
-
N
-
1
(
L
^
(
d
(
x
)
)
)
N
g
(
ψ
(
ζ
4
h
(
d
(
x
)
)
)
)
=
(
1
+
ε
)
-
1
(
d
(
x
)
)
-
N
-
1
(
L
^
(
d
(
x
)
)
)
N
g
(
ψ
(
ζ
4
h
(
d
(
x
)
)
)
)
(
N
ζ
4
N
(
∏
i
=
1
N
-
1
κ
i
(
x
¯
)
1
-
d
(
x
)
κ
i
(
x
¯
)
)
×
(
-
ζ
4
h
(
d
(
x
)
)
g
′
(
ψ
(
ζ
4
h
(
d
(
x
)
)
)
)
(
g
(
ψ
(
ζ
4
h
(
d
(
x
)
)
)
)
)
(
N
-
1
)
/
N
L
^
(
d
(
x
)
)
h
(
d
(
x
)
)
-
d
(
x
)
L
^
′
(
d
(
x
)
)
-
L
^
(
d
(
x
)
)
L
^
(
d
(
x
)
)
)
-
(
1
+
ε
)
(
b
2
+
ε
)
)
≥
0
,
i.e., u¯ε is a subsolution to equation (1.1) in Ωδε.
In a similar way, we can show that u¯ε=-ψ(ζ3h(d(x))) is a supersolution to equation (1.1) in Ωδε.The rest of the proof is similar to that of Theorem 1.1 and thus it is omitted. ∎
Proof of Theorem 1.6.
Let ε∈(0,min{1/4,b1/4}),
ζ
5
N
+
γ
=
(
1
-
ε
)
(
b
1
-
2
ε
)
(
N
+
γ
)
M
0
and
ζ
6
N
+
γ
=
(
1
+
ε
)
(
b
2
+
2
ε
)
(
N
+
γ
)
m
0
.
It follows that
ξ
5
N
+
γ
4
<
ζ
5
N
+
γ
<
ζ
6
N
+
γ
<
4
ξ
6
N
+
γ
,
where ξ5 and ξ6 are given in (1.9).
Combining the fact that L^ is slowly varying at zero with (2.2) and Proposition 2.8, we see that
lim
d
(
x
)
→
0
(
(
1
-
L
^
(
d
(
x
)
)
(
N
+
γ
)
∫
d
(
x
)
η
L
^
(
τ
)
τ
𝑑
τ
)
N
-
1
(
∏
i
=
1
N
-
1
1
1
-
d
(
x
)
κ
i
(
x
¯
)
)
(
1
+
N
+
γ
-
1
N
+
γ
L
^
(
d
(
x
)
)
∫
d
(
x
)
η
L
^
(
τ
)
τ
𝑑
τ
+
d
(
x
)
L
^
′
(
d
(
x
)
)
L
^
(
d
(
x
)
)
)
)
=
1
.
It follows from (b4) that there exists a sufficiently small δε∈(0,δ1) corresponding to ε such that, for x∈Ωδε,
(
b
1
-
ε
)
(
d
(
x
)
)
γ
-
1
L
^
(
d
(
x
)
)
<
b
(
x
)
<
(
b
2
+
ε
)
(
d
(
x
)
)
γ
-
1
L
^
(
d
(
x
)
)
,
and
m
0
1
+
ε
≤
(
1
-
L
^
(
d
(
x
)
)
(
N
+
γ
)
∫
d
(
x
)
η
L
^
(
τ
)
τ
𝑑
τ
)
N
-
1
(
∏
i
=
1
N
-
1
κ
i
(
x
¯
)
1
-
d
(
x
)
κ
i
(
x
¯
)
)
(
1
+
N
+
γ
-
1
N
+
γ
L
^
(
d
(
x
)
)
∫
d
(
x
)
η
L
^
(
τ
)
τ
𝑑
τ
+
d
(
x
)
L
^
′
(
d
(
x
)
)
L
^
(
d
(
x
)
)
)
≤
M
0
1
-
ε
.
Thus, u¯ε=-ζ6d(x)(∫d(x)ηL^(τ)τ𝑑τ)1/(N+γ) (x∈Ωδε) satisfies
det
D
2
u
¯
ε
(
x
)
-
(
b
2
+
ε
)
(
d
(
x
)
)
γ
-
1
L
^
(
d
(
x
)
)
(
-
u
¯
ε
(
x
)
)
-
γ
=
(
N
+
γ
)
-
1
ζ
6
-
γ
L
^
(
d
(
x
)
)
d
(
x
)
(
∫
d
(
x
)
η
L
^
(
τ
)
τ
𝑑
τ
)
-
γ
/
(
N
+
γ
)
×
(
ζ
6
N
+
γ
(
(
1
-
L
^
(
d
(
x
)
)
(
N
+
γ
)
∫
d
(
x
)
η
L
^
(
τ
)
τ
𝑑
τ
)
N
-
1
(
∏
i
=
1
N
-
1
κ
i
(
x
¯
)
1
-
d
(
x
)
κ
i
(
x
¯
)
)
×
(
1
+
N
+
γ
-
1
N
+
γ
L
^
(
d
(
x
)
)
∫
d
(
x
)
η
L
^
(
τ
)
τ
𝑑
τ
+
d
(
x
)
L
^
′
(
d
(
x
)
)
L
^
(
d
(
x
)
)
)
)
-
(
N
+
γ
)
(
b
2
+
ε
)
)
≥
(
N
+
γ
)
-
1
(
1
+
ε
)
-
1
ζ
6
-
γ
L
^
(
d
(
x
)
)
d
(
x
)
(
∫
d
(
x
)
η
L
^
(
τ
)
τ
𝑑
τ
)
-
γ
/
(
N
+
γ
)
(
m
0
ζ
6
N
+
γ
-
(
1
+
ε
)
(
N
+
γ
)
(
b
2
+
ε
)
)
≥
0
,
i.e., u¯ε is a subsolution to equation (1.1) in Ωδε.
In a similar way, we can show that u¯ε=-ζ5d(x)(∫d(x)ηL^(τ)τ𝑑τ)1/(N+γ) is a supersolution to equation (1.1) in Ωδε.
The rest of the proof is similar to that of Theorem 1.1 and thus it is omitted. ∎
