1 Introduction and main results
This presentation is to establish the second boundary behavior of the unique strictly convex solution to a singular Dirichlet problem for the Monge–Ampère equation
(1.1)
det(D2u)=b(x)g(−u),u<0 in Ω and u=0 on ∂Ω,
where Ω is a bounded, smooth and strictly convex domain in ℝN (N ≥ 2), and
D2u(x)=(∂2u(x)∂xi∂xj)N×N
denotes the Hessian of u and D2u is the so called Monge–Ampère operator. The nonlinearity g satisfies
(g1) g ∈ C1((0, ∞), (0, ∞)) is decreasing on (0, ∞) and
lim t → 0 + g ( t ) = ∞
;
(g2) there exist a constant γ>1 and some function f ∈ C1(0, a1)∩ C[0, a1) for a sufficiently small constant a1 > 0 such that
−tg′(t)g(t):=γ+f(t) withlimt→0+f(t)=0,
i.e.,
g(t)=c0t−γexp∫ta1f(s)sds,c0=g(a1)a1γ,
where the function f satisfies
(S1) f ≡ 0 on (0, a1] (or (S2) f(t)≠0, ∀ t ∈ (0, a] for some a ≤ a1).
If (S2) holds in (g2) , then we suppose
(g3) there exists θ ≥ 0 such that
lim t → 0 + t f ′ ( t ) f ( t ) = θ ≥ 0.
If θ=0 in (g3), then we further suppose
(g4) there exist β ∈ ℝ+ and
σ∈R
such that
limt→0+(−lnt)βf(t)=σ.
The weight b satisfies
(b1) b ∈ C∞(Ω) is positive in Ω
and one of the following two conditions
(b2) there exist k ∈ Λ,
B0∈R
and μ ∈ ℝ+ such that
b(x)=kN+1(d(x))(1+B0(d(x))μ+o((d(x))μ)),d(x)→0,
where d(x)≔ dist(x, ∂Ω), x ∈ Ω, Λ denotes the set of all of positive monotonic functions in C1(0, δ0) ∩ L (0, δ0) which satisfy
limt→0+ddt(K(t)k(t))=Dk≥0,K(t)=∫0tk(s)ds
and
(b3) there exist
L˜∈L
,
B0∈R
and μ ∈ ℝ+ such that
b(x)=(d(x))−(N+1)L˜N(d(x))(1+B0(d(x))μ+o((d(x))μ)),d(x)→0,
where
L
denotes the set of all of positive functionsdefined on (0, t0] by
L ~ ( t ) := c exp ( ∫ t t 0 y ( s ) s d s ) , t ∈ ( 0 , t 0 ] ,
where c ∈ ℝ+, y ∈ C (0, t0] and
limt→0+y(t)=0
.
The set Λ in (b2) was first introduced by Cîrstea and Rădulescu [6]- [8] for non-decreasing functions and by Mohammed [33] for non-increasing functions to study the exact boundary behavior and uniqueness of boundary blow-up elliptic problems. When b satisfies (b2) with Dk > 0, we see by Lemma 3.1 (iii) -(iv) that b may be singular on the boundary with the index
(1−Dk)(N+1)Dk>−(N+1).
The condition (b3) implies that b is critical singular with the index −(N + 1).
problem (1.1) has a wide range of applications in Riemannian geometry and optical physics and one important geometric application is to structure a Riemannian metric in Ω that is invariant under projective transformations. When g(t)=t−(N+2), t > 0 and b ≡ 1 in Ω, Nirenberg [38], Loewner and Nirenberg [31] for N = 2, Cheng and Yau [5] for N ≥ 2 studied the existence and uniqueness of solutions to problem (1.1). In particular, Cheng and Yau [5] showed that if Ω is convex and bounded but not necessarily strictly convex then problem (1.1) possesses a unique solution
u∈C∞(Ω)∩C(Ωˉ)
which is negative in Ω. When g(t)=t−γ (t > 0) with γ>1 and
b∈C∞(Ωˉ)
with b(x) > 0 for all x ∈ Ω, Lazer and McKenna [27] proved the existence and uniqueness of solutions to problem (1.1). Moreover, they also obtained the following global estimate
c1(d(x))N+1N+γ≤u(x)≤c2(d(x))N+1N+γ,x∈Ω.
When b satisfies (b1) and g:(0, ∞) → (0, ∞) is a non-increasing, smooth function, Mohammed [34] showed that problem (1.1) has a strictly convex solution
u∈C∞(Ω)∩C(Ωˉ)
if and only if the problem
(1.2)
det(D2u)=b(x) in Ω and u=0 on ∂Ω
has a strictly convex solution
v∈C∞(Ω)∩C(Ωˉ)
, where b may be singular or may vanish on ∂Ω. In particular, the author showed that
(i1) if
b∈C∞(Ωˉ)
is positive in
Ωˉ
, then problem (1.1) has a strictly convex solution
u∈C∞(Ω)∩C(Ωˉ)
;
(i2) if
limt→0+g(t)=∞
,then problem (1.1) has a unique strictly convex solution
u∈C∞(Ω)∩C(Ωˉ)
and the solution usatisfies
c1ϕ(d(x))≤−u(x)≤c2ϕ(d(x)) and |∇u(x)|≤c2ϕ(d(x))d(x) near ∂Ω,
where c1, c2 are positive constants and ϕ is uniquely determined by
∫ 0 ϕ ( t ) ( G ( s ) ) − ( N + 1 ) d s = t , G ( t ) = ∫ t t ^ g ( s ) d s , t ∈ ( 0 , t ^ ) , t ^ ∈ ( 0 , ∞ ] ;
(i3) if b(x) ⩽ C(d (x))δ−N−1 for somepositive constants δ and C, then problem (1.2) has a strictly convex solution;
(i4) if b(x)=C(d (x))−(N+1) for somepositive constant C, then problem (1.2) has no strictlyconvex solution.
Later, Yang and Chang [47] extended the above results (i3) -(i4) to the following cases:
(i5) if b(x) ⩽ C(d (x))−(N+1)(−ln d(x))−qnear ∂Ω for some q > N and C > 0, then problem (1.2) has a strictly convex solution;
(i6) if b(x)=C(d (x))−(N+1)(−lnd(x))−N near ∂Ω for some C > 0, then problem (1.2) has no strictly convex solution.
Let
P∈C1(0,∞)
satisfy
P′(t)<0
and
limt→0+P(t)=∞
and define
P(t)=∫t1P(s)ds
. Recently, under the hypothesis of (b1), Zhang and Du [49] obtain the following results:
(i7) if
b(x)≤P(d(x))
near∂Ω and
∫01(P(s))1/Nds<∞
, then problem (1.2) has a strictly convex solution;
(i8) if
b(x)≥P(d(x))
and
∫01(P(s))1/Nds=∞
, then problem (1.2) has no strictly convex solution.
The above facts imply that problem (1.2) has a strictly convex solution if b satisfies (b1) and
b(x)≤CkN+1(d(x)) near ∂Ω orb(x)≤C(d(x))−(N+1)L˜N(d(x)) near ∂Ω,
where C is a positive constant, k ∈ Λ in (b2) and
L˜∈L
in (b3) with
(1.3)
∫ 0 t L ~ ( s ) s d s < ∞ .
In [28], Li and Ma studied the existence and the firstboundary behavior of the strictly convex solutions to problem (1.1) by using regularity theory and sub-supersolution method.In particular, when b ∈ C3(Ω) is positive in Ω and satisfies
(b01) there exist k ∈ Λ and positiveconstants b1 and b2 such that
b1:=lim infd(x)→0b(x)kN+1(d(x))≤lim supd(x)→0b(x)kN+1(d(x))=:b2,
g satisfies (g1) and
(g01)
lim t → 0 + ( G ( t ) ) 1 / ( N + 1 ) ∫ 0 t ( G ( s ) ) − 1 / ( N + 1 ) d s = D g , G ( t ) = ∫ t a 1 g ( s ) d s for some a 1 > 0 ,
they showed that the unique strictly convex solution u to problem (1.1) satisfies
1≤lim infd(x)→0−u(x)ψ(ϑ1K(d(x)))≤lim supd(x)→0−u(x)ψ(ϑ2K(d(x)))≤1,
where ψ is uniquely determined by
(1.4)
∫ 0 ψ ( t ) ( ( N + 1 ) G ( s ) ) − 1 / ( N + 1 ) d s = t ,
ϑ1=(b1mˆ−(1−Dg−1(1−Dk)))1/(N+1) and ϑ2=(b2mˆ+(1−Dg−1(1−Dk)))1/(N+1)
with
(1.5)
mˆ−:=maxxˉ∈∂ΩωN−1(xˉ) and mˆ+:=minxˉ∈∂ΩωN−1(xˉ),
where
ωN−1(xˉ)=∏i=1N−1κi(xˉ)
denotes the (N − 1)-th curvature at
xˉ
and
κ1(xˉ),⋅⋅⋅,κN−1(xˉ)
denote the principal curvatures of ∂Ω at
xˉ
. In [51], Zhang showed that if b satisfies (b1) and (b01), g satisfies (g1) and
(g02)
lim t → 0 + ( ( g ( t ) ) 1 / N ) ′ ∫ 0 t ( g ( s ) ) − 1 / N d s = − C g ,
and NDk + (1 + N)Cg > 1+N, then the unique strictly convex solution u to problem (1.1) satisfies
ϑ31−Cg:=lim infd(x)→0−u(x)ϕg((K(d(x)))(N+1)/N)≤lim supd(x)→0−u(x)ϕg((K(d(x)))(N+1)/N)=:ϑ41−Cg,
where ϕg is uniquely determined by
∫ 0 ϕ g ( t ) ( N g ( s ) ) − 1 / N d s = t , t > 0 ,
ϑ3=((NN+1)Nb1mˆ−((1+N)Cg+NDk−1−N))1/N
and
ϑ4=((NN+1)Nb2mˆ+((1+N)Cg+NDk−1−N))1/N.
Especially, if (b01) is replaced by the following condition
(b02) there exist
L˜∈L
with (1.3) and positive constants b1 and b2 such that
b1:=lim infd(x)→0b(x)(d(x))−(N+1)L˜N(d(x))≤lim supd(x)→0b(x)(d(x))−(N+1)L˜N(d(x))=:b2,
Zhang [51] showed that the unique strictly convex solution u to problem (1.1) satisfies
ϑ51−Cg≤lim infd(x)→0−u(x)ϕg(∫0d(x)L˜(s)sds)≤lim supd(x)→0−u(x)ϕg(∫0d(x)L˜(s)sds)≤ϑ61−Cg,
where
ϑ5=(b1mˆ−N)1/N and ϑ6=(b2mˆ+N)1/N.
Then, Sun and Feng [43] and Li and Ma [29] generalized the above boundary behavior results to the case of the following Hessian equation for i=1, ···, N
Si(D2u)=b(x)g(−u),u<0 in Ω and u=0 on ∂Ω,
where
S i ( D 2 u ) = S i ( λ 1 , ⋅ ⋅ ⋅ , λ N ) = ∑ 1 ≤ j 1 < ⋅ ⋅ ⋅ < j i ≤ N λ j 1 ⋅ ⋅ ⋅ λ j i
and λ1, ···, λN are the eigenvalues of D2 u. Furthermore S0(λ) ≡ 1 for λ ∈ ℝN. Espically, Li and Ma [29] also studied the existence and uniqueness of viscosity solution to the problem. For related insights on the existence, regularity and asymptotic behavior of solutions to the Monge-Ampère equations, please refer to [4], [11], [19], [21]- [25], [30], [35]- [37], [44]- [45] and the references therein. When the Monge-Ampère operator (det(D2u)) is replaced by the Laplace operator (Δ), many papers have been dedicated to resolving existence, uniqueness and asymptotic behavior issues for solutions, please refer to [1]- [2], [10], [12]- [17], [26], [39], [46], [48], [50] and the references therein.
In this paper, by making a complete and detailed analysis to some indexes in various cases, we establish the exact second boundary behavior of the unique strictly convex solution to problem (1.1), which is quite different from the first behavior of this solution. For all we know, in literature there aren't articles on the second boundary behavior of the strictly convex solution to problem (1.1).
To our aims, we define the following subclasses of Λ and
L
as follows:
Λ 1 := { k ∈ Λ : lim t → 0 + t − 1 [ d d t ( K ( t ) k ( t ) ) − D k ] = E 1 , k } ; Λ 2 , β := { k ∈ Λ : lim t → 0 + ( − ln t ) β [ d d t ( K ( t ) k ( t ) ) − D k ] = E 2 , k } ; L β := { L ~ ∈ L : lim t → 0 + ( − ln t ) β y ( t ) = E 3 } ,
where β is a positive constant and the relation between
L˜
and y is given in (b3). Our results are summarized as follows and
mˆ±
(given in Theorems 1.1-1.3) are defined by (1.5).
Theorem 1.1
Let b satisfy (b1) -(b2) with (γ+N)Dk>N + 1, g satisfy (g1) -(g2).
(I) When (S1) holds (or (S2) and (g3) -(g4) hold with θ=0 in (g3)), we have
(i) If k ∈ Λ1, and then the unique strictly convex solution u to problem (1.1) satisfies
(1.6)
ξ−ψ(K(d(x)))(1+C−(−lnd(x))−β+o((−lnd(x))−β))≤−u(x)≤ξ+ψ(K(d(x)))(1+C+(−lnd(x))−β+o((−lnd(x))−β)),d(x)→0,
where ψ is uniquely determined by (1.4) and
(1.7)
ξ±=(((γ+N)Dk−(N+1))mˆ±γ−1)−1/(γ+N),
C ± = − ( ( γ + N ) D k N + 1 ) β C ± , i f ( S 2 ) a n d ( g 3 ) h o l d w i t h θ = 0 , 0 , i f ( S 1 ) h o l d s ,
where
C ± = A ± γ + N , i f A + ≤ 0 a n d A − ≤ 0 , C + = A + γ + N m ^ − m ^ + a n d C − = A − γ + N m ^ + m ^ − , i f A + ≥ 0 a n d A − ≥ 0 , C + = A + γ + N m ^ − m ^ + a n d C − = A − γ + N , i f A + > 0 a n d A − < 0 , C + = A + γ + N a n d C − = A − γ + N m ^ + m ^ − , i f A + < 0 a n d A − > 0
with
(1.8)
A±=(C0+1γ+Nlnmˆ±−1)σ,
and
(1.9)
C0=(N+1)(1−Dk)((γ+N)Dk−(N+1))(γ−1)−(γ+N)−1ln(γ+N)Dk−(N+1)γ−1.
(ii) If k ∈ Λ2,β (β is the same as the one in(g4)), then (1.6) still holds, where
C ± = − C ^ ± , i f ( S 2 ) a n d ( g 3 ) h o l d w i t h θ = 0 , − D ^ ± , i f ( S 1 ) h o l d s ,
and
C ^ ± = A ± + B γ + N , i f A + + B ≤ 0 a n d A − + B ≤ 0 , C ^ + = A + + B γ + N m ^ − m ^ + a n d C ^ − = A − + B γ + N m ^ + m ^ − , i f A + + B ≥ 0 a n d A − + B ≥ 0 , C ^ + = A + + B γ + N m ^ − m ^ + a n d C ^ − = A − + B γ + N , i f A + + B > 0 a n d A − + B < 0 , C ^ + = A + + B γ + N a n d C ^ − = A − + B γ + N m ^ + m ^ − , i f A + + B < 0 a n d A − + B > 0 ,
(1.10)
D ^ + = B γ + N m ^ − m ^ + a n d D ^ − = B γ + N m ^ + m ^ − , i f B ≥ 0 , D ^ + = D ^ − = B γ + N , i f B < 0
with
(1.11)
A ± = ( ( γ + N ) D k N + 1 ) β A ± a n d B = ( γ + N ) E 2 , k ( γ + N ) D k − ( N + 1 ) ,
where A ± are given by (1.8).
(II) When (S2)
and (g3) hold with θ > 0 in (g3)
and k ∈ Λ2,β, then the unique strictly convex
solution u to problem (1.1) satisfies (1.6) with
C±=−Dˆ±,
where
Dˆ±
are given by
(1.10).
Corollary 1.1
In Theorem 1.1, if Ω is a ball with radius R and center x0, then
(I) When (S1) holds (or(S2) and (g3) -(g4) hold with θ = 0 in (g3)), we have
(i) If k ∈ Λ1, then the unique strictly convex solution u to problem (1.1) satisfies
(1.12)
−u(x)=ξR1ψ(K(R−r))(1+CR1(−ln(R−r))β+o((−ln(R−r))β)),r→R,
where r=∣x − x0∣,
(1.13)
ξR1=((γ+N)Dk−(N+1)(γ−1)RN−1)−1/(γ+N)
and
C R 1 = C ^ R 1 , i f ( S 2 ) a n d ( g 3 ) h o l d w i t h θ = 0 , 0 , i f ( S 1 ) h o l d s ,
where
(1.14)
CˆR1=−((γ+N)DkN+1)β(C0+N−1γ+NlnR)σγ+N
and C0 is given by (1.9).
(ii) If k ∈ Λ2,β (β is the same as the one in (g4)), then (1.12) still holds, where
C R 1 = C ^ R 1 − E 2 , k ( γ + N ) D k − ( N + 1 ) , i f ( S 2 ) a n d ( g 3 ) h o l d w i t h θ = 0 , − E 2 , k ( γ + N ) D k − ( N + 1 ) , i f ( S 1 ) h o l d s ,
where
CˆR1
is given by (1.14) and C0 isgiven by (1.9).
(II) When (S2) and (g3) hold with θ > 0 in (g3) and k ∈ Λ2,β, then the unique strictly convex solution u to problem (1.1) satisfies (1.12), where
CR1=−E2,k(γ+N)Dk−(N+1).
Theorem 1.2
Let b satisfy (b1) -(b2) with μ ∈ (0, 1), g satisfy (g1) -(g2) with (1−μ)(γ+N)Dk>N + 1, and if (S2) holds in (g2),we further suppose that (g3) with θ > 0 and the following hold
(1.15)
θ ( N + 1 ) μ − N > γ > N + 1 ( 1 − μ ) D k − N , i f D k ∈ ( 0 , 1 ) , θ ( N + 1 ) − N > γ > max { N + 1 ( 1 − μ ) D k − N , 2 N + 1 N } a n d μ D k ∈ ( 0 , 1 ) , i f D k ∈ [ 1 , ∞ ) .
If k ∈ Λ1, then the unique strictly convex solution u to problem (1.1) satisfies
(1.16)
ξ−ψ(K(d(x)))(1+C˜−(d(x))μ+o((d(x))μ))≤−u(x)≤ξ+ψ(K(d(x)))(1+C˜+(d(x))μ+o((d(x))μ)),d(x)→0,
where ψ is uniquely determined by (1.4), ξ ± are given by (1.7) and
C ~ ± = B 0 ( N + 1 ) ( ( γ + N ) D k − ( N + 1 ) ) H + γ ( N + 1 ) ( ( γ + N ) D k − ( N + 1 ) ) , i f B 0 ≥ 0 , C ~ + = B 0 ( N + 1 ) ( ( γ + N ) D k − ( N + 1 ) ) H m ^ − m ^ + + γ ( N + 1 ) ( ( γ + N ) D k − ( N + 1 ) ) , i f B 0 < 0 , C ~ − = B 0 ( N + 1 ) ( ( γ + N ) D k − ( N + 1 ) ) H m ^ + m ^ − + γ ( N + 1 ) ( ( γ + N ) D k − ( N + 1 ) ) , i f B 0 < 0
with
H=((1−μ)(γ+N)Dk−(N+1))(N(N+1)+μ(N−1)(γ+N)Dk)+μ(γ+N)Dk((N−2)(N+1)+μ(N−1)(γ+N)Dk)+μ(1−μ)(γ+N)2Dk2.
Corollary 1.2
In Theorem 1.2, if Ω is a ball with radius R and center x0, then the unique strictly convex solution u to problem (1.1) satisfies
−u(x)=ξR1ψ(K(R−r))(1+CR2(R−r)μ+o((R−r)μ)),r→R,
where
ξR1
is given by (1.13), r=∣x − x0∣ and
CR2=B0(N+1)((γ+N)Dk−(N+1))H+γ(N+1)((γ+N)Dk−(N+1)).
Remark 1.1
In Theorem 1.2, if we replace k ∈ Λ1 by
k∈Λ′:={k∈Λ:limt→0+t−μ(ddt(K(t)k(t))−Dk)=E1,k}
and other conditions still hold, then (1.16) holds, where
C ~ ± = ( N + 1 ) [ B 0 ( ( γ + N ) D k − ( N + 1 ) ) − ( γ + N ) E 1 , k ] H + γ ( N + 1 ) ( ( γ + N ) D k − ( N + 1 ) ) , i f B 0 ( ( γ + N ) D k − ( N + 1 ) ) ≥ ( γ + N ) E 1 , k , C ~ + = ( N + 1 ) [ B 0 ( ( γ + N ) D k − ( N + 1 ) ) − ( γ + N ) E 1 , k ] H m ^ − m ^ + + γ ( N + 1 ) ( ( γ + N ) D k − ( N + 1 ) ) , i f B 0 ( ( γ + N ) D k − ( N + 1 ) ) < ( γ + N ) E 1 , k , C ~ − = ( N + 1 ) [ B 0 ( ( γ + N ) D k − ( N + 1 ) ) − ( γ + N ) E 1 , k ] H m ^ + m ^ − + γ ( N + 1 ) ( ( γ + N ) D k − ( N + 1 ) ) , i f B 0 ( ( γ + N ) D k − ( N + 1 ) ) < ( γ + N ) E 1 , k .
When b is critical singular on ∂Ω, we have the following second boundary behavior.
Theorem 1.3
Let b satisfy (b1) and (b3), g satisfy (g1) -(g2), and if (S2) holds in (g2), we further suppose that (g3) -(g4) hold with θ = 0 in (g3) and with β ∈ (0, 1] in (g4). If
L˜∈Lβ
with
(1.17)
E3<0,β∈(0,1),−1,β=1,
then the unique strictly convex solution u to problem (1.1) satisfies
η − ψ ( ( ∫ 0 d ( x ) L ~ ( s ) s d s ) N N + 1 ) ( 1 + C − ∗ M ( d ( x ) ) + o ( M ( d ( x ) ) ) ) ≤ − u ( x ) ≤ η + ψ ( ( ∫ 0 d ( x ) L ~ ( s ) s d s ) N N + 1 ) ( 1 + C + ∗ M ( d ( x ) ) + o ( M ( d ( x ) ) ) ) , d ( x ) → 0 ,
where ψ is uniquely determined by (1.4),
η±=((NN+1)Nγ+Nγ−1mˆ±)−1/(γ+N),
M ( d ( x ) ) = ( − ln ψ ( ( ∫ 0 d ( x ) L ~ ( s ) s d s ) N N + 1 ) ) − β ,
and
C ± ∗ = D ± , i f ( S 2 ) a n d ( g 3 ) h o l d w i t h θ = 0 , 0 , i f ( S 1 ) h o l d s ,
where
D ± = X ± γ + N , i f X + ≥ 0 a n d X − ≥ 0 , D + = X + γ + N m ^ − m ^ + a n d D − = X − γ + N m ^ + m ^ − , i f X + ≤ 0 a n d X − ≤ 0 , D + = X + γ + N a n d D − = X − γ + N m ^ + m ^ − , i f X + > 0 a n d X − < 0 , D + = X + γ + N m ^ − m ^ + a n d D − = X − γ + N , i f X + < 0 a n d X − > 0
with
X±=(γ+N)−1(N+1γ−1+ln((NN+1)Nγ+Nγ−1)+lnmˆ±)σ.
Corollary 1.3
In Theorem 1.3, if Ω is a ball with radius R and center x0, then the unique strictly convex solution u to problem (1.1) satisfies
− u ( x ) = η R 1 ψ ( ( ∫ 0 R − r L ( s ) s d s ) N N + 1 ) ( 1 + C R 3 M ( R − r ) + o ( M ( R − r ) ) ) , r → R ,
where
ηR1=((NN+1)Nγ+N(γ−1)RN−1)−1/(γ+N),r=|x−x0|
and
CR3=(N+1γ−1+ln((NN+1)Nγ+Nγ−1)−(N−1)lnR)σ(γ+N)2.
Remark 1.2
In Theorem 1.3, we obtain by Lemma 3.2 (i) that if
L˜∈Lβ
with (1.17), then (1.3) holds. But, when
L˜∈Lβ
with β > 1, by a direct calculation we see that
(1.18)
∫ 0 t 0 L ~ ( s ) s d s = ∞ .
Since
limt→0+(−lnt)βy(t)=E3
, for any ε > 0 we can choose a small enough constant
t˜<min{t0,1}
such that
E3−ε(−lnt)β≤y(t),t∈(0,t˜].
A simple calculation shows that
exp ( ∫ t t ~ y ( s ) s d s ) ≥ exp ( ∫ t t ~ E 3 − ε s ( − ln s ) β d s ) = exp ( E 3 − ε β − 1 ( ( − ln t ~ ) 1 − β − ( − ln t ) 1 − β ) ) .
Moreover, we have
(1.19)
lim t → 0 + exp ( ∫ t t ~ E 3 − ε s ( − ln s ) β d s ) = lim t → 0 + exp ( E 3 − ε β − 1 ( ( − ln t ~ ) 1 − β − ( − ln t ) 1 − β ) ) = exp ( E 3 − ε β − 1 ( − ln t ~ ) 1 − β ) > 0.
It follows by the definition of
L˜
in (b3) that there exists a positive constant
C˜
such that
(1.20)
C ~ exp ( ∫ t t 0 E 3 − ε s ( − ln s ) β d s ) ≤ L ~ ( t ) , t ∈ ( 0 , t 0 ] .
Combining (1.19) with (1.20), we obtain (1.18) holds.
Remark 1.3
If
y ( t ) = ± ( − ln t ) − 1 , 0 < t ≤ t ~ < min { t 0 , 1 }
in (b3), then we have
L ~ ( t ) = exp ( ∫ t t 0 y ( s ) s d s ) = exp ( ∫ t ~ t 0 y ( s ) s d s ) exp ( ± ∫ t t ~ d s s ( − ln s ) ) = exp ( ∫ t ~ t 0 y ( s ) s d s ) ( ln t ln t ~ ) ± 1 ∈ L β ,
where β ∈ (0, 1) with E3=0 or β=1 with E3=± 1. It’s clear that (1.8) holds here. This implies that when
L˜∈Lβ
with β ≤ 1, (1.3) is not always true if (1.17) fails.
Remark 1.4
In Theorem 1.3, if we substitute
L˜∈Lβ0
(β0>β) and
E3<0,β0∈(0,1),−1,β0=1,
for
L˜∈Lβ
with (1.17), then the conclusion of Theorem 1.3 still holds.
Remark 1.5
L˜∈L
is normalized slowly varying at zero and
limt→0+tL˜′(t)L˜(t)=0
.
The rest of the paper is organized as follows. In Section 2, we give some bases of Karamata regular variation theory. In Section 3, we show some auxiliary lemmas. The proofs of Theorems 1.1-1.3 are given in Section 4.
3 Auxiliary results
In this section, we show some useful results, which are necessary in the proofs of our results.
Lemma 3.1
(Lemma 2.9 in [51]). Let k ∈ Λ, then
when k is non-decreasing, Dk ∈ [0, 1]; and when k is non-increasing, Dk ∈ [1, ∞);
limt→0+K(t)k(t)=0
,
limt→0+K(t)tk(t)=Dk
and
limt→0+K(t)k′(t)k2(t)=1−Dk;
when Dk > 0 and Dk≠1,
k∈NRVZ(1−Dk)/Dk;
when Dk=1, k is normalized slowly
varying at zero.
Lemma 3.2
Let
L˜∈Lβ
and (1.17) hold, then
limt→0+(−lnt)βL˜(s)∫0tL˜(s)sds=−E3,β∈(0,1),−(E3+1),β=1;
limt→0+(−lnt)βL˜′(t)tL˜(t)=E3.
Proof. (i) By
limt→0+(−lnt)βy(t)=E3<0,β∈(0,1),−1,β=1,
we see that for any
0<ε<−E3+τ2
with
τ=0,β∈(0,1),1,β=1,
there exists a small enough positive constant t* ∈ (0, 1) such that
y(t)≤E3+ε(−lnt)β,t∈(0,t∗] with E3+ε<0,β∈(0,1),−1,β=1.
A straightforward calculation shows that for any
t∈0,t⋆,
exp(∫tt∗y(s)sds)≤exp(∫tt∗E3+εs(−lns)βds)=exp(E3+ε1−β((−lnt)1−β−(−lnt∗)1−β)),β∈(0,1),(lntlnt∗)E3+ε,β=1.
So, we see that there exists a large constant C > 0 such that for any
t∈(0,t∗],
(3.1)
L˜(t)≤Cexp(E3+ε1−β(−lnt)1−β),β∈(0,1),C(−lnt)E3+ε,β=1.
Case 1. If β∈(0, 1), it is clear that that
exp(−E3+ε1−β(−lnt)1−β)=∑n=0∞(−E3+ε1−β)n(−lnt)n(1−β)n!.
This together with (3.1) implies that we can choose a positive integer n* >(1-β)-1 such that
(3.2)
Cexp(E3+ε1−β(−lnt)1−β)=C(exp(−E3+ε1−β(−lnt)1−β))−1=C(∑n=0∞(−E3+ε1−β)n(−lnt)n(1−β)n!)−1≤C((−E3+ε1−β)n∗(−lnt)n∗(1−β))−1=C1(−lnt)−n∗(1−β),t∈(0,t∗],
where
C1=C(−E3+ε1−β)−n∗
. So, we have
∫0t∗L˜(s)sds≤C∫0t∗exp(E3+ε1−β(−lns)1−β)sds≤C1∫0t∗dss(−lns)n∗(1−β)=C1(−lnt∗)1−n∗(1−β)n∗(1−β)−1<∞.
Case 2. If β=1, then by (3.1) we obtain
∫0t∗L˜(s)sds≤∫0t∗C(−lns)E3+εsds=−C(−lnt∗)1+E3+ε1+E3+ε<∞.
Combining Cases 1-2, we obtain (i) holds.
(ii) It follows by (3.1)-(3.2) that
limt→0+(−lnt)βL˜(t)=0.
By the l'Hospital’s rule, we obtain
limt→0+(−lnt)βL˜(s)∫0tL˜(s)sds=limt→0+(−(−lnt)βy(t)−β(−lnt)β−1)=−E3,β∈(0,1),−(E3+1),β=1.
(iii) A direct calculation shows that
limt→0+(−lnt)βL˜′(t)tL˜(t)=limt→0+(−lnt)βy(t)=E3.
□
Define
(3.3)
φ(t):=∫0t((N+1)G(s))−1/(N+1)ds,t>0,
where φ is the inverse of solution ψ to (1.4).
Lemma 3.3
Let g satisfy (g1)-(g2), then
φ(t)∼(γ−1N+1)1/(N+1)N+1γ+Ntγ+NN+1(Lˆ0(t))−1/(N+1),t→0+,
where
Lˆ0(t)=c0exp(∫ta1f(s)sds),c0=g(a1)a1γ;
limt→0+φ(t)tφ′(t)=limt→0+φ(t)((N+1)G(t))1/(N+1)t=N+1γ+N,i.e., φ∈NRVZγ+NN+1;
limt→0+((N+1)G(t))N/(N+1)g(t)φ(t)=γ+Nγ−1.
proof. (i) It follows by (g2) that (i) holds.
(ii)-(iii) By (g2) we see that
(3.4)
g(t)=t−γLˆ0(t),t∈(0,a1].
It follows by Proposition 2.8 (i) that (ii) holds. Furthermore, we have
((N+1)G(t))−1/(N+1)∼(γ−1N+1)1/(N+1)t(γ−1)/(N+1)(Lˆ0(t))−1/(N+1),t→0+.
It follows by Proposition 2.8 (ii) that (iii) holds.
(iv)-(v) (3.3) implies that
(3.5)
φ′(t)=((N+1)G(t))−1/(N+1),t>0.
By (3.4)-(3.5) and (i)-(ii), we obtain (iv)-(v) hold.
Lemma 3.4
Suppose that g satisfies (g1)-(g2). In particular, if (S2) holds in (g2), we suppose that (g3) holds. And if θ = 0 in (g3), we further suppose that (g4) holds. Then,
limt→0+υ(t)(tg′(t)g(t)+γ)=χ1,
where
χ1=−σ,if(S2)and(g3)−(g4)holdwithθ=0in(g3),υ(t)=(−lnt)β,0,if(S1)holds(or(S2)and(g3)holdwithθ>0),υ(t)=(−lnt)β,0,if(S1)holds(or(S2)and(g3)holdwithθ(N+1)>(γ+N)ρ),υ(t)=(φ(t))−ρwithρ∈(0,1];
limt→0+υ(t)(G(t)tg(t)−1γ−1)=χ2,
where
χ2=−σ(γ−1)2,if(S2)and(g3)−(g4)holdwithθ=0in(g3),υ(t)=(−lnt)β,0,if(S1)holds(or(S2)and(g3)holdwithθ>0),υ(t)=(−lnt)β,0,if(S1)holdsandγ>(1+ρ)N+1N+1−ρin(g2),υ(t)=(φ(t))−ρ,ρ∈(0,1],0,if(S2)and(g3)holdwithθ(N+1)ρ−N>γ>(1+ρ)N+1N+1−ρ,υ(t)=(φ(t))−ρwithρ∈(0,1];
limt→0+υ(t)(((N+1)G(t))N/(N+1)g(t)φ(t)−γ+Nγ−1)=χ3,
where
χ3=−(N+1)σ(γ−1)2,if(S2)and(g3)−(g4)holdwithθ=0in(g3),υ(t)=(−lnt)β,0,if(S1)holds(or(S2)and(g3)holdwithθ>0),υ(t)=(−lnt)β,0,if(S1)holdsandγ>(1+ρ)N+1N+1−ρin(g2),υ(t)=(φ(t))−ρ,ρ∈(0,1],0,if(S2)and(g3)holdwithθ(N+1)ρ−N>γ>(1+ρ)N+1N+1−ρ,υ(t)=(φ(t))−ρwithρ∈(0,1];
let ξ be a positive constant, then
limt→0+υ(t)g(ξt)ξNg(t)−ξ−(γ+N)=χ4,
where
χ4=−σlnξξγ+N,ifS2andg3−g4holdwithθ=0ing3,v(t)=(−lnt)β,0,ifS1holdsorS2andg3holdwithθ>0,v(t)=(−lnt)β,0,ifS1holdsorS2andg3holdwithθ(N+1)>(γ+N)ρ,v(t)=(φ(t))−ρwithρ∈(0,1].
Proof. (i) When (S2) and (g3)-(g4) hold with θ = 0 in (g3), we have
(3.6)
limt→0+(−lnt)β(tg′(t)g(t)+γ)=−limt→0+(−lnt)βf(t)=−σ.
When (S1) holds, we have
tg′(t)g(t)=−γ,t∈(0,a1].
This implies that for υ (t)=(-ln t)β or
υ (t)=(φ (t))-ρ, we obtain
(3.7)
limt→0+υ(t)(tg′(t)g(t)+γ)=0.
When (S2) and (g3)-(g4) hold with θ > 0, we conclude by Proposition 2.5 (ii) that
(3.8)
limt→0+(−lnt)β(tg′(t)g(t)+γ)=−limt→0+(−lnt)βf(t)=−limt→0+tθ(−lnt)βL(t)=0,
where L ∈ NRVZ0. Moreover, it follows by Lemma (3.3) (iv) and Proposition (2.7) that
(3.9)
1/φρ∈NRVZ−(γ+N)ρN+1.
When (g3) holds with θ (N + 1) > (γ + N)ρ, we conclude by Proposition 2.6 and Proposition 2.5 (ii) that
(3.10)
limt→0+(φ(t))−ρ(tg′(t)g(t)+γ)=−limt→0+(φ(t))−ρf(t)=−limt→0+tθ−(γ+N)ρN+1L(t)=0,
where L ∈ NRVZ0. So, (i) follows by (3.6)-(3.10).
(ii) By (g2), we have
−tg′(t)=γg(t)+g(t)f(t),t∈(0,a1].
Integrating it from t to a1 and integration by parts, we obtain
tg(t)=(γ−1)G(t)+∫ta1g(s)f(s)ds+c,t∈(0,a1],
i.e.,
(3.11)
G(t)tg(t)−1γ−1=−1γ−1∫ta1g(s)f(s)dstg(t)−c(γ−1)g(t)t,t∈(0,a1],
where c is a constant. The condition (g2) implies that g ∈ NRVZ-γ with γ > 1. Moreover, by t ↦ (-ln t)β ∈ NRVZ0 and (3.9), we see that
υ∈NRVZ0, if υ(t)=(−lnt)β,NRVZ−(γ+N)ρN+1, if υ(t)=(φ(t))−ρ.
We conclude by Proposition 2.7 and Proposition 2.6 that
t↦tg(t)(υ(t))−1∈NRVZ1−γ, if υ(t)=(−lnt)β,NRVZ(1−γ)(N+1)+(γ+N)ρN+1, if υ(t)=(φ(t))−ρ.
So, we can take L ∈ NRVZ0 such that
tg(t)(υ(t))−1=tςL(t)
with
ς=1−γ, if υ(t)=(−lnt)β,(1−γ)(N+1)+(γ+N)ρN+1, if υ(t)=(φ(t))−ρ.
It’s clear that
ς<0, if γ>1 in (g2) and υ(t)=(−lnt)β,ς<0, if γ>(1+ρ)N+1N+1−ρ in (g2) and υ(t)=(φ(t))−ρ.
By Proposition 2.5 (ii), we have
limt→0+tg(t)(υ(t))−1=∞,
i.e.,
(3.12)
limt→0+−c(γ−1)tg(t)(υ(t))−1=0.
Combining with (3.6)-(3.10), by the l'Hospital’s rule, we can obtain
(3.13)
limt→0+υ(t)−∫ta1g(s)f(s)ds(γ−1)tg(t)=limt→0+∫a1tg(s)f(s)ds(γ−1)tg(t)(υ(t))−1=limt→0+1γ−1g(t)f(t)g(t)(υ(t))−1+tg′(t)(υ(t))−1+tg(t)[(υ(t))−1]′=limt→0+1γ−1υ(t)f(t)1+tg′(t)g(t)+t[(υ(t))−1]′(υ(t))−1=limt→0+(−lns)βf(s)−(γ−1)2, if γ>1 in (g2) and υ(t)=(−lnt)βlimt→0+(N+1)(φ(t))−ρf(t)(γ−1)((1−γ)(N+1)+(γ+N)ρ), if γ>(1+ρ)N+1N+1−ρ in (g2) and υ(t)=(φ(t))−ρ=χ2.
(3.11)-(3.13) imply that
limt→0+υ(t)G(t)tg(t)−1γ−1=−limt→0+∫ta1g(s)f(s)ds(γ−1)tg(t)(υ(t))−1−limt→0+c(γ−1)tg(t)(υ(t))−1=χ2.
(iii) By Lemma 3.3 (iv), Proposition 2.7 and Proposition 2.6, we have
t↦(υ(t))−1φ(t)∈NRVZ(γ+N)ρN+1, if υ(t)=(−lnt)β,NRVZ(γ+N)(ρ+1)N+1, if υ(t)=(φ(t))−ρ.
This implies that
(υ(t))−1φ(t)→0 as t→0+.
Moreover, it follows by Lemma 3.3 (v) that
((N+1)G(t))1/(N+1)g(t)−γ+Nγ−1φ(t)→0 as t→0+.
If ν(t) = (− ln t)β, then we have by the l’Hospital’s rule, (3.5) and (i)-(ii) that
(3.14)
limt→0+υ(t)(((N+1)G(t))N/(N+1)g(t)φ(t)−γ+Nγ−1)=limt→0+((N+1)G(t))N/(N+1)g(t)−γ+Nγ−1φ(t)(−lnt)−βφ(t)=limt→0+−N((N+1)G(t))−1/(N+1)g2(t)−((N+1)G(t))N/(N+1)g′(t)g2(t)−γ+Nγ−1φ′(t)(−lnt)−βφ′(t)+β(−lnt)−β−1φ(t)t−1=limt→0+−(N+1)(−lnt)β(G(t)tg(t)tg′(t)g(t)+γγ−1)=limt→0+−(N+1)(−lnt)β[(G(t)tg(t)−1γ−1)(tg′(t)g(t)+γ)−γ(G(t)tg(t)−1γ−1)+1γ−1(tg′(t)g(t)+γ)]=−(N+1)σ(γ−1)2, if (S2) and (g3)-(g4) hold with θ=0 in (g3),0, if (S1) holds (or (S2) and (g3) hold with θ>0).
If ν(t) = (φ(t))−ρ, then we have by the l'Hospital’s rule, (3.5) and (i)-(ii) that
(3.15)
limt→0+υ(t)(((N+1)G(t))N/(N+1)g(t)φ(t)−γ+Nγ−1)=limt→0+((N+1)G(t))N/(N+1)g(t)−γ+Nγ−1φ(t)(φ(t))ρ+1=limt→0+−N((N+1)G(t))−1/(N+1)g2(t)−((N+1)G(t))N/(N+1)g′(t)g2(t)−γ+Nγ−1φ′(t)(ρ+1)(φ(t))ρφ′(t)=limt→0+−N+1ρ+1(φ(t))−ρ(G(t)tg(t)tg′(t)g(t)+γγ−1)=limt→0+N+1ρ+1(φ(t))−ρ[−(G(t)tg(t)−1γ−1)(tg′(t)g(t)+γ)+γ(G(t)tg(t)−1γ−1)−1γ−1(tg′(t)g(t)+γ)]=0, if (S1) holds and γ>(1+ρ)N+1N+1−ρ in (g2),0, if (S2) and (g3) hold with θ(N+1)ρ−N>γ>(1+ρ)N+1N+1−ρ.
(3.14)-(3.15) imply that (iii) holds.
(iv) When (S1) holds in (g2), we have
g(ξt)ξNg(t)=ξ−(γ+N),t∈(0,a1].
In this case, for υ(t) = (−ln t)β or υ(t) = (φ(t))−ρ, we obtain
(3.16)
limt→0+υ(t)(g(ξt)ξNg(t)−ξ−(γ+N))=0.
Now, we investigate the case that (S2) holds in (g2). If ξ = 1, then the result is obvious. If ξ ≠ 1, then we have
g(ξt)ξNg(t)−ξ−(γ+N)=ξ−(γ+N)(exp(∫ξttf(s)sds)−1).
It follows by Proposition 2.2 that
limt→0+f(st)s=0andlimt→0+f(st)sf(t)=sθ−1
uniformly with respect to s ∈ [1, ξ] or s ∈ [ξ, 1]. So, we have
lim t → 0 + ∫ ξ t t f ( τ ) τ d τ τ := t s _ _ lim t → 0 + ∫ ξ 1 f ( t s ) s d s = 0
and
limt→0+∫ξ1f(st)sf(t)ds=∫ξ1sθ−1ds=−lnξ, if g3 holds with θ=0,1−ξθθ, if g3 holds with θ>0.
Since er−1 ∼ r as r → 0+, we conclude by (3.6), (3.8)-(3.10) that
limt→0+υ(t)(g(ξt)ξNg(t)−ξ−(γ+N))=limt→0+ξ−(γ+N)υ(t)f(t)∫ξ1f(st)sf(t)ds=−σlnξξγ+N, if (g3)-(g4) hold with θ=0 in (g3),υ(t)=(−lnt)β,0, if (g3) hold with θ>0,υ(t)=(−lnt)β,0, if (g3) hold with θ(N+1)>(γ+N)ρ,υ(t)=(φ(t))−ρ.
This, combined with (3.16), shows that (iv) holds. □
Lemma 3.5
(Lemma 2.3 in [48]) Let
L˜∈L
, then
limt→0+L˜(t)∫tt0L˜(s)sds=0.
If further
∫0t0L˜(s)sds<∞
, then
limt→0+L˜(t)∫0tL˜(s)sds=0.
Lemma 3.6
Suppose that g satisfies (g1)-(g2). In particular, if (S2) holds in (g2), we suppose that (g3) holds, and if θ = 0 in (g3), we further suppose that (g4) holds. ψ is
uniquely determined by (1.4). Then,
ψ′(s)=((N+1)G(ψ(t)))1N+1,ψ(t)>0,ψ(0)=0
and
−ψ′′(t)=((N+1)G(ψ(t)))1−NN+1g(ψ(t)),t∈(0,a1);
−(ψ′(t))N−1ψ′′(t)=g(ψ(t));
limt→0+tψ′(t)ψ(t)=N+1γ+N,i.e.,ψ∈NRVZN+1γ+N;
limt→0+ψ′(t)tψ′′(t)=−γ+Nγ−1,i.e.,ψ′∈NRVZ−γ−1γ+N;
if
k∈Λ,thenlimt→0+lntlnψ(K(t))=(γ+N)DkN+1;
if
L˜∈Land(1.3)holds,thenlimt→0+lnψ((∫0tL˜(s)sds)NN+1)lnt=0
;
limt→0+(−lnψ(t))β(ψ′(t)tψ′′(t)+γ+Nγ−1)=(N+1)σ(γ−1)2,if(S2)and(g3)−(g4)holdwithθ=0in(g3),0,if(S1)holds(or(S2)and(g3)holdwithθ>0);
let ρ ∈ (0, 1], then
lim t → 0 + t − ρ ( ψ ′ ( t ) t ψ ″ ( t ) + γ + N γ − 1 ) = 0 , i f ( S 1 ) h o l d s a n d γ > ( 1 + ρ ) N + 1 N + 1 − ρ i n ( g 2 ) , 0 , i f ( S 2 ) a n d ( g 3 ) h o l d w i t h θ ( N + 1 ) ρ − N > γ > ( 1 + ρ ) N + 1 N + 1 − ρ ;
let ξ be a positive constant, then
limt→0+(−lnψ(t))β(g(ξψ(t))ξNg(ψ(t))−ξ−(γ+N))=−σξ−(γ+N)lnξ,if(S2)and(g3)−(g4)holdwithθ=0in(g3),0,if(S1)holds(or(S2)and(g3)holdwithθ>0);
let (S1) hold (or (S2) and (g3) hold with θ(N + 1)>(γ+N)ρ), ξ be a positive constant and ρ ∈ (0, 1], then
limt→0+t−ρ(g(ξψ(t))ξNg(ψ(t))−ξ−(γ+N))=0.
Proof. (i)-(ii) By the definition of ψ, we obtain that (i)-(ii) hold.
(iii)-(iv) It follows by (i) and Lemma 3.3 (iv)-(v) that
lim t → 0 + t ψ ′ ( t ) ψ ( t ) = lim t → 0 + t ( ( N + 1 ) G ( ψ ( t ) ) ) 1 N + 1 ψ ( t ) z := ψ ( t ) _ _ lim z → 0 + φ ( z ) ( ( N + 1 ) G ( z ) ) 1 N + 1 z = N + 1 γ + N
and
limt→0+ψ′(t)tψ′′(t)=−limz→0+((N+1)G(z))NN+1g(z)φ(z)=−γ+Nγ−1.
(v) We conclude by (iii), Lemma 3.1 (ii) and Lemma 2.5 (iii) that (v) holds.
(vi) By using Lemma 3.5, we have
limt→0+t((∫0tL˜(s)sds)NN+1)′(∫0tL˜(s)sds)NN+1=NN+1limt→0+L˜(t)∫0tL˜(s)sds=0.
This implies that
(3.17)
t↦(∫0tL˜(s)sds)NN+1∈NRVZ0.
So, by (3.17), (iii) and Proposition 2.7 we further obtain
(3.18)
t↦ψ((∫0tL˜(s)sds)NN+1)∈NRVZ0.
We have by Proposition 2.5 (iii) that (vi) holds.
(vii)-(viii) It follows by (i) and Lemma 3.4 (iii) that
lim t → 0 + ( − ln ψ ( t ) ) β ( ψ ′ ( t ) t ψ ″ ( t ) + γ + N γ − 1 ) = lim t → 0 + ( − ln ψ ( t ) ) β ( − ( ( N + 1 ) G ( ψ ( t ) ) ) N N + 1 g ( ψ ( t ) ) t + γ + N γ − 1 ) z := ψ ( t ) _ _ lim z → 0 + ( − ln z ) β ( − ( ( N + 1 ) G ( z ) ) N N + 1 g ( z ) φ ( z ) + γ + N γ − 1 ) = ( N + 1 ) σ ( γ − 1 ) 2 , if ( S 2 ) and ( g 3 ) - ( g 4 ) hold with θ = 0 in ( g 3 ) , 0 , if ( S 1 ) holds (or ( S 2 ) and ( g 3 ) hold with θ > 0 ) ,
and
lim t → 0 + t − ρ ( ψ ′ ( t ) t ψ ″ ( t ) + γ + N γ − 1 ) = lim t → 0 + t − ρ ( − ( ( N + 1 ) G ( ψ ( t ) ) ) N N + 1 g ( ψ ( t ) ) t + γ + N γ − 1 ) z := ψ ( t ) _ _ lim z → 0 + ( φ ( z ) ) − ρ ( − ( ( N + 1 ) G ( z ) ) N N + 1 g ( z ) φ ( z ) + γ + N γ − 1 ) = 0 , if ( S 1 ) holds and γ > ( 1 + ρ ) N + 1 N + 1 − ρ in ( g 2 ) , 0 , if ( S 2 ) and ( g 3 ) hold with θ ( N + 1 ) ρ − N > γ > ( 1 + ρ ) N + 1 N + 1 − ρ .
(ix)-(x) It follows by Lemma 3.4 (iv) that
lim t → 0 + ( − ln ψ ( t ) ) β ( g ( ξ ψ ( t ) ) ξ N g ( ψ ( t ) ) − ξ − ( γ + N ) ) z := ψ ( t ) _ _ lim z → 0 + ( − ln z ) β ( g ( ξ z ) ξ N g ( z ) − ξ − ( γ + N ) ) = − σ ξ − ( γ + N ) ln ξ , if ( S 2 ) and ( g 3 ) - ( g 4 ) hold with θ = 0 in ( g 3 ) , 0 , if ( S 1 ) holds (or ( S 2 ) and ( g 3 ) hold with θ > 0 ) .
Moreover, if (S1) holds or (S2) and (g3) hold with θ(N + 1)>(γ+N)ρ, then by Lemma 3.4 (iv), we have
limt→0+t−ρg(ξψ(t))ξNg(ψ(t))−ξ−(γ+N)=z:=ψ(t)limz→0+(φ(z))−ρg(ξz)ξNg(z)−ξ−(γ+N)=0.
□
4 The Second Boundary Behavior
In this section, we prove Theorems 1.1-1.3. We first introduce some lemmas as follows.
Lemma 4.1
(Lemma 2.1 in [27]) Let Ω be a bounded domain in ℝN with N ≥ 2, and let
u,v∈C(Ωˉ)∩C2(Ω)
. Suppose h(x, t) is defined for x ∈ Ω and t in some interval containing
the ranges of u and v. If the following hold:
(i) h is strictly increasing in t for all x ∈ Ω,
(ii) the matrix D2v is positive definite in Ω,
(iii) det(D2v)=h(x, v) and det (D2u) ⩽ h(x, u), x ∈ Ω,
(iv) u=v on ∂Ω,
then, we have u≥v in Ω.
For any δ>0, let
Ωδ={x∈Ω:0<d(x)<δ}.
Since Ω is Cm-smooth for m≥2, we can always take
δ1 > 0 such that (see Lemmas 14.16 and 14.17 in
[18])
d∈Cm(Ωδ1)and|∇d(x)|=1,x∈Ωδ1.
Let
xˉ∈∂Ω
be the projection of the point
x∈Ωδ1
to ∂Ω, and
κi(xˉ)
(i=1, ···, N − 1) be the principal
curvatures of ∂Ω at
xˉ
, then
D2(d(x))=diag(−κ1(xˉ)1−d(x)κ1(xˉ),⋅⋅⋅,−κN−1(xˉ)1−d(x)κN−1(xˉ),0).
Lemma 4.2
(See the proof of Proposition 2.4 in [27], Proposition 2.1 and Corollary 2.3 in [9]) Let h be a C2-function on (0, δ1), then
det(D2h(d(x)))=(−h′(d(x)))N−1h′′(d(x))∏i=1N−1κi(xˉ)1−d(x)κi(xˉ),x∈Ωδ1.
Lemma 4.3
(Corollary 2.3 in [20]) Let h be a C2-function on (0, δ1) and Ω be a bounded domain with ∂Ω ∈ Cm for m=2, then, for i=1, ···, N,
Si(D2(h(d(x))))=(−h′(d(x)))iSi(ϵ1,⋅⋅⋅,ϵN−1)+(−h′(d(x)))i−1h′′(d(x))Si−1(ϵ1,⋅⋅⋅,ϵN−1),x∈Ωδ1
where
ϵj=κj(xˉ)1−κj(xˉ)d(x),j=1,⋅⋅⋅,N−1.
Definition 4.4
(Definition 1.1 in [42]) Let i ∈ {1, ···, N} and Ω be an open bounded subset of ℝN; a function u ∈ C2(Ω) is (strictly) i-convex if Sl(D2u)(>)⩾0 in Ω for l=1, ···, i. In particular, if i=N, then we say that u is (strictly) convex in Ω.
4.1 Proof of Theorem 1.1
Next, we prove Theorem 1.1 and we first show some preliminaries as follows.
Fix ε>0 and let
w±(d(x))=ξ±ψ(K(d(x)))(1+(C±±ε)(−lnd(x))−β),x∈Ωδ1,
where ξ ± and C ± are in Theorem 1.1. By the Lagrange’s mean value theorem, we obtain that there exist λ ± ∈ (0, 1) and
(4.1)
Θ±(d(x))=ξ±ψ(K(d(x)))(1+λ±(C±±ε)(−lnd(x))−β)
such that for any
x∈Ωδ1
g(w±(d(x)))=g(ξ±ψ(K(d(x))))+ξ±ψ(K(d(x)))g′(Θ±(d(x)))(C±±ε)(−lnd(x))−β.
Since g ∈ NRVZ−γ, we have by Proposition 2.2 that
(4.2)
limd(x)→0g(ξ±ψ(K(d(x))))g(Θ±(d(x)))=limd(x)→0g′(ξ±ψ(K(d(x))))g′(Θ±(d(x)))=1.
Moreover, by the definitions of ξ ± and C ± , we can take a sufficiently small positive constant still denoted by δ1 such that
(4.3)
w+(d(x))≥w−(d(x)),x∈Ωδ1.
Proof. Our proof is divided into two steps and the outline of the proof is given as below.
• In Step 1, for fixed ε>0 and
x∈Ωδ1
, we first give some functions I1 ± (d(x)), I2 ± (d(x)) and I3 ± (d(x)) (which are corresponding to ε>0), and then by detailed calculation we will show that there exists a sufficiently small positive constant δε<δ1 such that
(4.4)
I1+(d(x))+I2+(d(x))+I3+(d(x))>0,x∈Ωδε
and
(4.5)
I1−(d(x))+I2−(d(x))+I3−(d(x))<0,x∈Ωδε.
• In Step 2, we will define two functions
u_ε,u‾ε
in
Ωδε
and show they are sub-and super-solutions of Eq. (1.1) in
Ωδε
by (4.4) and (4.5), respectively. In particular, we will show
u_ε
is strictly convex in
Ωδε
. Finally, we will establish the second boundary behavior of the unique strictly convex solution to problem (1.1) by using Lemma 4.1.
By the above analysis, we see that how to get (4.4)-(4.5) is the key of the research.
Step 1.We first define functions I1 ± (d(x)), I2 ± (d(x)) and I3 ± (d(x)) as follows.
For fixed ε>0 and
∀x∈Ωδ1
, we define
I1±(d(x))=(−lnd(x))β[(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)mˆ±−g(ξ±ψ(K(d(x))))ξ±Ng(ψ(K(d(x))))];I2±(d(x))=C±[N(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)A±∏i=1N−1(1−d(x)κi(xˉ))−1−ξ±ψ(K(d(x)))g′(ξ±ψ(K(d(x))))g(ξ±ψ(K(d(x))))g′(Θ±(d(x)))g′(ξ±ψ(K(d(x))))g(ξ±ψ(K(d(x))))ξ±Ng(ψ(K(d(x))))]±ε[N(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)mˆ+∏i=1N−1(1−d(x)κi(xˉ))−1−ξ±ψ(K(d(x)))g′(ξ±ψ(K(d(x))))g(ξ±ψ(K(d(x))))g′(Θ±(d(x)))g′(ξ±ψ(K(d(x))))g(ξ±ψ(K(d(x))))ξ±Ng(ψ(K(d(x))))];I3±(d(x))=(−lnd(x))β(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)×mˆ±∑i=1N−1CN−1i(−1)i+1(m±d(x))i(1−d(x)m±)N−1+ζ±(d(x))(−lnd(x))−β∏i=1N−1κi(xˉ)1−d(x)κi(xˉ)−(B0±ε)g(ξ±ψ(K(d(x))))ξ±Ng(ψ(K(d(x))))(d(x))μ(−lnd(x))β−(B0±ε)(C±±ε)ξ±ψ(K(d(x)))g′(ξ±ψ(K(d(x))))g(ξ±ψ(K(d(x))))g′(Θ±(d(x)))g′(ξ±ψ(K(d(x))))g(ξ±ψ(K(d(x))))ξ±Ng(ψ(K(d(x))))(d(x))μ,
where
(S2) holds and θ=0k∈Λ1A±=mˆ±, if A+≤0 and A−≤0,A±=mˆ∓, if A+≥0 and A−≥0,A+=A−=mˆ−, if A+>0 and A−<0,A+=A−=mˆ+, if A+<0 and A−>0;k∈Λ2,βA±=mˆ±, if A++B≤0 and A++B≤0,A±=mˆ∓, if A++B≥0 and A++B≥0,A+=A−=mˆ−, if A++B>0 and A++B<0,A+=A−=mˆ+, if A++B<0 and A++B>0;(S1) holdsk∈Λ1,A±=mˆ±;k∈Λ2,β,A+=mˆ−, if B≥0,A−=mˆ+, if B<0;(S2) holds and θ>0k∈Λ2,β,A+=mˆ−, if B≥0,A−=mˆ+, if B<0,
(4.6)
m+:=min{minxˉ∈∂Ωκi(xˉ):i=1,⋅⋅⋅,N−1} and m−:=max{maxxˉ∈∂Ωκi(xˉ):i=1,⋅⋅⋅,N−1}
and
ζ±(d(x))=(C±±ε)2(N−1)(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)(−lnd(x))−2β+2β(C±±ε)ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k(d(x))d(x)(−lnd(x))−β−1×(1+(C±±ε)(N−1)(−lnd(x))−β+ν±(d(x)))−β(C±±ε)ψ(K(d(x)))ψ′(K(d(x)))K(d(x))ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))(K(d(x))d(x)k(d(x)))2(−lnd(x))−β−1×(1−(β+1)(−lnd(x))−1)(1+(C±±ε)(N−1)(−lnd(x))−β+ν±(d(x)))+ν±(d(x))(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)(1+(C±±ε)(−lnd(x))−β),
where A ± are given by (1.8),
A±
and
B
are given by (1.11) and
ν±(d(x))=R2±(d(x))+∑i=2N−1CN−1i(R1±(d(x))+R2±(d(x)))i,
where
(4.7)
CN−1i=(N−1)!i!(N−1−i)!,R1±(d(x))=(C±±ε)(−lnd(x))−β,
and
R2±(d(x))=(N−1)β(C±±ε)ψ(K(d(x)))ψ′(K(d(x)))K(d(x))K(d(x))k(d(x))d(x)(−lnd(x))−β−1.
Next, we prove
(4.8)
limd(x)→0(I1±(d(x))+I2±(d(x))+I3±(d(x)))=±ε(γ+Nmˆ+mˆ±)ξ±−(γ+N).
To prove (4.8), we calculate the limits of I1 ± (d(x)),
I2 ± (d(x)) and I3 ± (d(x)) as d(x) → 0.
• First, by Lemma 3.1 (ii), Lemma 3.6 (v), (vii) and (ix), we obtain that
(4.9)
limd(x)→0I1±(d(x))=ϖ1±+ϖ2±, if (S2) and (g3) hold with θ=0,k∈Λ1,ϖ1±+ϖ2±+(γ+N)E2,kmˆ±γ−1, if (S2) and (g3) hold with θ=0,k∈Λ2,β,γ+Nγ−1E2,kmˆ±, if (S2) and (g3) hold with θ>0,k∈Λ2,β,γ+Nγ−1E2,kmˆ±, if (S1) holds,k∈Λ2,β,0, if (S1) holds,k∈Λ1,
where
ϖ1±=((γ+N)DkN+1)β(N+1)(1−Dk)σmˆ±(γ−1)2andϖ2±=((γ+N)DkN+1)βσlnξ±ξ±γ+N.
• Second, by (4.2), Lemma 3.1 (ii), Lemma 3.6 (iv) and the
choices of ξ ± , C ± in Theorem 1.1, we obtain
(4.10)
limd(x)→0I2±(d(x))=C±[(γ+N)Dk−(N+1)]Nγ−1A±+γξ±−(γ+N)±ε(γ+Nmˆ+mˆ±)ξ±−(γ+N).
• Third, by Lemma 3.1 (ii), Lemma 3.6 (iii)-(iv) and a straightforward calculation, we obtain
(4.11)
limd(x)→0I3±(d(x))=0.
Combining (4.9) and (4.10)-(4.11), we obtain (4.8) holds. By (b1)-(b2) and (4.8), we see that there exists a
sufficiently small constant
δε<δ1
such that
(4.12)
kN+1(d(x))1+B0−ε(d(x))μ≤b(x)≤kN+1(d(x))1+B0+ε(d(x))μ,x∈Ωδε
and (4.4)-(4.5) hold.
Step 2. Let
u_ε(x)=−ξ+ψ(K(d(x)))(1+(C++ε)(−lnd(x))−β),x∈Ωδε.
Then
g(−u_ε(x))=g(ξ+ψ(K(d(x))))+ξ+ψ(K(d(x)))g′(Θ+(d(x)))(C++ε)(−lnd(x))−β,x∈Ωδε,
where
Θ+(d(x))
is given by (4.1). By Lemma 4.2, we have for any
x∈Ωδε
(4.13)
det D 2 u _ ε ( x ) − b ( x ) g − u _ ε ( x ) ≥ − ξ + N ψ ′ ( K ( d ( x ) ) ) N − 1 ψ ′ ′ ( K ( d ( x ) ) ) k N + 1 ( d ( x ) ) × 1 + C + + ε ( N − 1 ) ( − ln d ( x ) ) − β + v + ( d ( x ) ) × ψ ′ ( K ( x ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) k ′ ( d ( x ) ) k 2 ( d ( x ) ) + 1 1 + C + + ε ( − ln d ( x ) ) − β + 2 β C + + ε × ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) k ( d ( x ) ) d ( x ) ( − ln d ( x ) ) − β − 1 − β C + + ε ψ ( K ( d ( x ) ) ) ψ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) 2 ( − ln d ( x ) ) − β − 1 × 1 − ( β + 1 ) ( − ln d ( x ) ) − 1 ∏ i = 1 N − 1 κ i ( x ¯ ) 1 − d ( x ) κ i ( x ¯ ) − k N + 1 ( d ( x ) ) 1 + B 0 + ε ( d ( x ) ) μ × g ξ + ψ ( K ( d ( x ) ) ) + ξ + ψ ( K ( d ( x ) ) ) g ′ Θ + ( d ( x ) ) C + + ε ( − ln d ( x ) ) − β ≥ − ξ + N ψ ′ ( K ( d ( x ) ) ) N − 1 ψ ′ ′ ( K ( d ( x ) ) ) k N + 1 ( d ( x ) ) ( − ln d ( x ) ) − β ( − ln d ( x ) ) β l × ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) k ′ ( d ( x ) ) k 2 ( d ( x ) ) + 1 ∏ i = 1 N − 1 m ^ + 1 − d ( x ) m + N − 1 − m ^ + + m ^ + − g ( ξ + ψ ( K ( d ( x ) ) ) ) ξ + N g ( ψ ( K ( d ( x ) ) ) ) + C + + ε N ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) k ′ ( d ( x ) ) k 2 ( d ( x ) ) + 1 × ∏ i = 1 N − 1 κ i ( x ¯ ) 1 − d ( x ) κ i ( x ¯ ) − ξ + ψ ( K ( d ( x ) ) ) g ′ ξ + ψ ( K ( d ( x ) ) ) g ξ + ψ ( K ( d ( x ) ) ) × g ′ Θ + ( d ( x ) ) g ′ ξ + ψ ( K ( d ( x ) ) ) g ξ + ψ ( K ( d ( x ) ) ) ξ + N g ( ψ ( K ( d ( x ) ) ) ) + ζ + ( d ( x ) ) ( − ln d ( x ) ) − β ∏ i = 1 N − 1 κ i ( x ¯ ) 1 − d ( x ) κ i ( x ¯ ) − B 0 + ε g ξ + ψ ( K ( d ( x ) ) ) ξ + N g ( ψ ( K ( d ( x ) ) ) ) ( d ( x ) ) μ ( − ln d ( x ) ) β − B 0 + ε C + + ε × ξ + ψ ( K ( d ( x ) ) ) g ′ ξ + ψ ( K ( d ( x ) ) ) g ( ξ + ψ ( K ( d ( x ) ) ) ) g ′ Θ + ( d ( x ) ) g ′ ξ + ψ ( K ( d ( x ) ) ) g ξ + ψ ( K ( d ( x ) ) ) ξ + N g ( ψ ( K ( d ( x ) ) ) ) ( d ( x ) ) μ ≥ − ξ + N ψ ′ ( K ( d ( x ) ) ) N − 1 ψ ′ ′ ( K ( d ( x ) ) ) k N + 1 ( d ( x ) ) ( − ln d ( x ) ) − β ∑ i = 1 3 I i + ( d ( x ) ) > 0 ,
i.e.,
u_ε
is a subsolution of Eq. (1.1) in
Ωδε
. Moreover, it follows by Lemma 4.3 that for i = 1, ..., N
SiD2u_ε(x)=ξ+iψ′(K(d(x)))i1+C++ε(−lnd(x))−β+βC++εψ(K(d(x)))ψ′(K(d(x)))K(d(x))×K(d(x))k(d(x))d(x)(−lnd(x))−β−1iki(d(x))Siϵ1,⋯,ϵN−1−ξ+iψ′(K(d(x)))i−1ψ′′(K(d(x)))×ki+1(d(x))1+C++ε(−lnd(x))−β+βC++εψ(K(d(x)))ψ′(K(d(x)))K(d(x))K(d(x))k(d(x))d(x)×(−lnd(x))−β−1i−1ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1×1+C++ε(−lnd(x))−β+2βC++εψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k(d(x))d(x)(−lnd(x))−β−1−βC++εψ(K(d(x)))ψ′(K(d(x)))K(d(x))ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))d(x)k(d(x))2(−lnd(x))−β−1×1−(β+1)(−lnd(x))−1Si−1ϵ1,⋯,ϵN−1
This implies that we can adjust the above positive constant
δε
such that for any
x∈Ωδε
S i ( D 2 u _ ε ( x ) ) > 0 for i = 1 , ⋅ ⋅ ⋅ , N .
We obtain by Definition 4.4 that
D2u_ε
is positive definite in
Ωδε.
Let
u‾ε(x)=−ξ−ψ(K(d(x)))(1+(C−−ε)(−lnd(x))−β),x∈Ωδε.
By the same calculation as (4.13), we obtain
detD2uˉε(x)−b(x)g−uˉε(x)≤−ξ−Nψ′(K(d(x)))N−1ψ′′(K(d(x)))kN+1(d(x))(−lnd(x))−β∑i=13Ii−(d(x))<0,
i.e.
uˉε
is a supersolution of Eq. (1.1) in
Ωδε.
Let u be the unique strictly convex solution to problem (1.1). Then, there exists a large constant M > 0 such that
u_ε(x)−Md(x)≤u(x)≤u‾ε(x)+Md(x),x∈{x∈Ω:d(x)=δε}.
We assert that
(4.14)
u(x)≥u_ε(x)−Md(x),x∈Ωδε
and
(4.15)
u(x)≤u‾ε(x)+Md(x),x∈Ωδε.
Since
D2u_ε
is positive definite in
Ωδε
and D2(−Md (x)) is positive semidefinite in
Ωδε
, we have by the Minkowski inequality that
D2(u_ε(x)−Md(x))
is positive definite in
Ωδε
and
det(D2(u_ε(x)−Md(x)))≥det(D2u_ε(x))≥b(x)g(−u_ε(x))≥b(x)g(−u_ε(x)+Md(x)),x∈Ωδε.
Similarly, we have D2(u(x)−Md (x)) is positive definite in
Ωδε
and
det(D2u(x)−Md(x))≥det(D2u(x))=b(x)g(−u(x))≥b(x)g(−u(x)+Md(x)).
By Lemma 4.1, we see that (4.14)-(4.15) hold. Hence, for any
x∈Ωδε
C++ε+Md(x)(−lnd(x))βξ+ψ(K(d(x)))≥−u(x)ξ+ψ(K(d(x)))−1(−lnd(x))β;C−−ε−Md(x)(−lnd(x))βξ−ψ(K(d(x)))≤−u(x)ξ−ψ(K(d(x)))−1(−lnd(x))β.
Since (γ+N)Dk−(N + 1)>0, we conclude from Lemma 3.1 (ii), Lemma 3.6 (iii), Proposition 2.7 and Proposition 2.5 (ii) that
C++ε≥lim supd(x)→0−u(x)ξ+ψ(K(d(x)))−1(−lnd(x))β;C−−ε≤lim infd(x)→0−u(x)ξ−ψ(K(d(x)))−1(−lnd(x))β.
Letting ε → 0, the proof is finished. □
4.2 Proof of Theorem 1.2
Now, we prove Theorem 1.2. For fixed ε>0, we define
w±(d(x))=ξ±ψ(K(d(x)))(1+(C˜±±ε)(d(x))μ),x∈Ωδ1,
where ξ ± and
C˜±
are given in Theorem 1.2. By the Lagrange’s mean value theorem, we see that there exist λ ± ∈ (0, 1) and
(4.16)
Θ±(d(x))=ξ±ψ(K(d(x)))(1+λ±(C˜±±ε)(d(x))μ)
such that for any
x∈Ωδ1
g(w±(d(x)))=g(ξ±ψ(K(d(x))))+ξ±ψ(K(d(x)))g′(Θ±(d(x)))(C±±ε)(d(x))μ.
By Proposition 2.2, we see that (4.2) still holds. Moreover, we can adjust δ1 > 0 such that (4.3) holds here.
Proof. Similar to the proof of Theorem 1.1, the proof is divided into the following two steps.
Step 1. We first define functions I1 ± (d(x)), I2 ± (d(x)) and I3 ± (d(x)) as follows.
For fixed ε > 0 and
∀x∈Ωδ1
, we define
I 1 ± ( d ( x ) ) = 1 ( d ( x ) ) μ ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) k ′ ( d ( x ) ) k 2 ( d ( x ) ) + 1 m ^ ± − g ξ ± ψ ( K ( d ( x ) ) ) ξ ± N g ( ψ ( K ( d ( x ) ) ) ) ; I 2 ± ( d ( x ) ) = C ~ ± ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) k ′ ( d ( x ) ) k 2 ( d ( x ) ) + 1 × N + μ ( N − 1 ) ψ ( K ( d ( x ) ) ) ψ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) + 2 μ ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) + μ × ( μ − 1 ) ψ ( K ( d ( x ) ) ) ψ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) 2 B ± ∏ i = 1 N − 1 1 − d ( x ) κ i ( x ¯ ) − 1 − ξ ± ψ ( K ( d ( x ) ) ) g ′ ξ ± ψ ( K ( d ( x ) ) ) g ξ ± ψ ( K ( d ( x ) ) ) g ′ Θ ± ( d ( x ) ) g ′ ξ ± ψ ( K ( d ( x ) ) ) g ξ ± ψ ( K ( d ( x ) ) ) ξ ± N g ( ψ ( K ( d ( x ) ) ) ) ± ε ψ ′ ( x ) ( x ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) k ′ ( d ( x ) ) k 2 ( d ( x ) ) + 1 × N + μ ( N − 1 ) ψ ( K ( d ( x ) ) ) ψ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) + 2 μ ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) + μ × ( μ − 1 ) ψ ( K ( d ( x ) ) ) ψ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) 2 × m ^ + ∏ i = 1 N − 1 1 − d ( x ) κ i ( x ¯ ) − 1 − ξ ± ψ ( K ( d ( x ) ) ) g ′ ξ ± ψ ( K ( d ( x ) ) ) g ξ ± ψ ( K ( d ( x ) ) ) g ′ Θ ± ( d ( x ) ) g ′ ξ ± ψ ( K ( d ( x ) ) ) × g ξ ± ψ ( K ( d ( x ) ) ) ξ ± N g ( ψ ( K ( d ( x ) ) ) ) − B 0 ± ε g ξ ± ψ ( K ( d ( x ) ) ) ξ ± N g ( ψ ( K ( d ( x ) ) ) ) ; I 3 ± ( d ( x ) ) = ( d ( x ) ) − μ ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) k ′ ( d ( x ) ) k 2 ( d ( x ) ) + 1 × m ^ ± ∑ i = 1 N − 1 C N − 1 i ( − 1 ) i + 1 m ± d ( x ) i 1 − d ( x ) m ± N − 1 − B 0 ± ε C ~ ± ± ε ξ ± ψ ( K ( d ( x ) ) ) g ′ ξ ± ψ ( K ( d ( x ) ) ) g ξ ± ψ ( K ( d ( x ) ) ) × g ′ Θ ± ( d ( x ) ) g ′ ξ ± ψ ( K ( d ( x ) ) ) g ξ ± ψ ( K ( d ( x ) ) ) ξ ± N g ( ψ ( K ( d ( x ) ) ) ) ( d ( x ) ) μ + ζ ~ ± ( d ( x ) ) ( d ( x ) ) μ ∏ i = 1 N − 1 κ i ( x ¯ ) 1 − d ( x ) k ( x ¯ ) ,
where
B±=mˆ±,if B0≥0,B±=mˆ∓, if B0<0,
m ± are defined by (4.6) and
ζ ~ ± ( d ( x ) ) = C ~ ± ± ε 2 ( N − 1 ) 1 + ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) k ′ ( d ( x ) ) k 2 ( d ( x ) ) × 1 + μ ψ ( K ( d ( x ) ) ) ψ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) ( d ( x ) ) 2 μ + 2 μ C ~ ± ± ε 2 ( N − 1 ) 1 + μ ψ ( K ( d ( x ) ) ) ψ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) × K ( d ( x ) ) k ′ ( d ( x ) ) k 2 ( d ( x ) ) ( d ( x ) ) 2 μ + μ ( μ − 1 ) C ~ ± ± ε 2 ( N − 1 ) 1 + μ ψ ( K ( d ( x ) ) ) ψ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) × K ( d ( x ) ) d ( x ) k ( d ( x ) ) ψ ( K ( d ( x ) ) ) ψ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) 2 ( d ( x ) ) 2 μ + v ~ ± ( d ( x ) ) × 1 + ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) k ′ ( d ( x ) ) k 2 ( d ( x ) ) 1 + C ~ ± ± ε ( d ( x ) ) μ + 2 μ C ~ ± ± ε ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) ( d ( x ) ) μ + μ ( μ − 1 ) C ~ ± ± ε ψ ( K ( d ( x ) ) ) ψ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) 2 ( d ( x ) ) μ
with
ν˜±(d(x))=∑i=2N−1CN−1i(R˜±(d(x)))i,
where
CN−1i
is given in (4.7) and
R˜±(d(x))=(C˜±±ε)(1+μψ(K(d(x)))ψ′(K(d(x)))K(d(x))K(d(x))d(x)k(d(x)))(d(x))μ.
Next, we prove
(4.17)
limd(x)→0(I1±(d(x))+I2±(d(x))+I3±(d(x)))=±εϱ±mˆ±,
where
(4.18)
ϱ±=(1−μ)(γ+N)Dk−(N+1)γ−1N+μ(N−1)γ+NN+1Dk+μγ+Nγ−1DkN+μ(N−1)γ+NN+1Dk−2+μ(1−μ)(γ+N)2(γ−1)(N+1)Dk2mˆ+mˆ±+(γ+N)Dk−(N+1).
To prove (4.17), we calculate the limits of I1 ± (d(x)), I2 ± (d(x)) and I3 ± (d(x)) as
d(x)→0.
• First, we investigate the limits of
I1±(d(x)) as d(x)→0.
As the preliminaries, we prove some limits as below.
Case 1. When Dk ∈ (0, 1), by a direct calculation, we have
N+1(1−μ)Dk−N>N+1(1−μ)−N>(1+μ)N+1N+1−μ.
So, by (1.15), Lemma 3.1 (ii) and Lemma 3.6 (viii) and (x) with ρ=μ ∈ (0, 1), we obtain
(4.19)
limd(x)→0+(d(x))−μψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))+γ+Nγ−1K(d(x))k′(d(x))k2(d(x))mˆ±=limd(x)→0+K(d(x))d(x)μ(K(d(x)))−μψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))+γ+Nγ−1K(d(x))k′(d(x))k2(d(x))mˆ±=0, if S2 and g3 hold with θ(N+1)μ−N>γ>N+1(1−μ)Dk−N,0, if s1 holds with γ>N+1(1−μ)Dk−N in g2
and
(4.20)
limd(x)→0(d(x))−μ(ξ±−(γ+N)−g(ξ±ψ(K(d(x))))ξ±Ng(ψ(K(d(x)))))=limd(x)→0(K(d(x))d(x))μ(K(d(x)))−μ(ξ±−(γ+N)−g(ξ±ψ(K(d(x))))ξ±Ng(ψ(K(d(x)))))=0, if (S2) and (g3) hold with θ(N+1)μ−N>γ,0, if (S1) holds in (g2).
Case 2. When Dk ∈ [1, ∞), by (1.15), Lemma 3.1 (ii) and Lemma 3.6 (viii) and (x) with ρ=1, we obtain
(4.21)
limd(x)→0(d(x))−μ(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))+γ+Nγ−1)K(d(x))k′(d(x))k2(d(x))mˆ±=limd(x)→0K(d(x))(d(x))μ(K(d(x)))−1(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))+γ+Nγ−1)K(d(x))k′(d(x))k2(d(x))mˆ±=0, if (S2) and (g3) hold with θ(N+1)−N>γ>2N+1N,0, if (S1) holds with γ>2N+1N in (g2)
and
(4.22)
limd(x)→0(d(x))−μ(ξ±−(γ+N)−g(ξ±ψ(K(d(x))))ξ±Ng(ψ(K(d(x)))))=limd(x)→0K(d(x))(d(x))μ(K(d(x)))−1(ξ±−(γ+N)−g(ξ±ψ(K(d(x))))ξ±Ng(ψ(K(d(x)))))=0, if (S2) and (g3) hold with θ(N+1)−N>γ,0, if (S1) holds in (g2).
On the other hand, a simple calculation shows that
(4.23)
−limd(x)→0γ+Nγ−1(d(x))−μ(K(d(x))k′(d(x))k2(d(x))−(1−Dk))mˆ±=−limd(x)→0γ+Nγ−1(d(x))1−μ(d(x))−1(K(d(x))k′(d(x))k2(d(x))−(1−Dk))mˆ±=0
We obtain by combining (4.19)-(4.20) (or (4.21)-(4.22)) and (4.23) that
(4.24)
limd(x)→0I1±(d(x))=0.
• Second, by (4.2), Lemma 3.1 (ii), Lemma 3.6 (iii)-(iv) and the choices of ξ ± ,
C˜±
in Theorem 1.2, we obtain
(4.25)
limd(x)→0I2±(d(x))=C˜±[((1−μ)(γ+N)Dk−(N+1)γ−1(N+μ(N−1)γ+NN+1Dk)+μγ+Nγ−1Dk(N+μ(N−1)γ+NN+1Dk−2)+μ(1−μ)(γ+N)2(γ−1)(N+1)Dk2)B±+γ((γ+N)Dk−(N+1))mˆ±γ−1]−B0((γ+N)Dk−(N+1))mˆ±γ−1±εϱ±mˆ±=±εϱ±mˆ±, where ϱ± are given by (4.18).
• Third, we conclude by Lemma 3.1 (ii) Lemma 3.6 (iii)-(iv) and a straightforward calculation that
(4.26)
limd(x)→0I3±(d(x))=0.
Combining (4.24) and (4.25)-(4.26), we obtain (4.17) holds. Moreover, it follows by (1.15) that thatϱ ± >(γ+N)Dk−(N + 1)>0. By (b1)-(b2) and (4.17), we see that there exists a sufficiently small constant δε > 0 such that (4.12) and (4.4)-(4.5) hold here.
Step 2. Let
u_ε(x)=−ξ+ψ(K(d(x)))(1+(C˜++ε)(d(x))μ),x∈Ωδε.
Then
g(−u_ε(x))=g(ξ+ψ(K(d(x))))+ξ+ψ(K(d(x)))g′(Θ+(d(x)))(C˜++ε)(d(x))μ,x∈Ωδε,
where Θ+(d(x)) is given by (4.16). By Lemma 4.2, we have for any x∈Ωδε
(4.27)
det(D2u_ε(x))−b(x)g(−u_ε(x))≥−ξ+N(ψ′(K(d(x))))N−1ψ′′(K(d(x)))kN+1(d(x))×(1+(C˜++ε)(N−1)(d(x))μ+μ(C˜++ε)(N−1)ψ(K(d(x)))ψ′(K(d(x)))K(d(x))K(d(x))d(x)k(d(x))(d(x))μ+ν˜+(d(x)))[(1+ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x)))(1+(C˜++ε)(d(x))μ)+2μ(C++ε)ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))d(x)k(d(x))(d(x))μ+μ(μ−1)(C˜++ε)ψ(K(d(x)))ψ′(K(d(x)))K(d(x))ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))(K(d(x))d(x)k(d(x)))2(d(x))μ]×∏i=1N−1κi(xˉ)1−d(x)κi(xˉ)−kN+1(d(x))(1+(B0+ε)(d(x))μ)[g(ξ+ψ(K(d(x))))+ξ+ψ(K(d(x)))g′(Θ+(d(x)))(C˜++ε)(d(x))μ]=−ξ+N(ψ′(K(d(x))))N−1ψ′′(K(d(x)))kN+1(d(x))(d(x))μ×{(d(x))−μ(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)∏i=1N−1κi(xˉ)1−d(x)κi(xˉ)+(C˜++ε)[(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)(N+μ(N−1)ψ(K(d(x)))ψ′(K(d(x)))K(d(x))×K(d(x))d(x)k(d(x)))+2μψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))d(x)k(d(x))+μ(μ−1)ψ(K(d(x)))ψ′(K(d(x)))K(d(x))×ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))(K(d(x))d(x)k(d(x)))2]∏i=1N−1κi(xˉ)1−d(x)κi(xˉ)+ζ˜+(d(x))(d(x))μ∏i=1N−1κi(xˉ)1−d(x)κi(xˉ)}−kN+1(d(x))g(ξ+ψ(K(d(x))))−(B0+ε)kN+1(d(x))g(ξ+ψ(K(d(x))))(d(x))μ−(C˜++ε)kN+1(d(x))ξ+ψ(K(d(x)))g′(Θ+(d(x)))(d(x))μ−(B0+ε)(C˜++ε)kN+1(d(x))×ξ+ψ(K(d(x)))g′(Θ+(d(x)))(d(x))2μ=−ξ+N(ψ′(K(d(x))))N−1ψ′′(K(d(x)))kN+1(d(x))(d(x))μ×{(d(x))−μ[(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)mˆ+−g(ξ+ψ(K(d(x))))ξ+Ng(ψ(K(d(x))))]+(C˜++ε)[((ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)(N+μ(N−1)×ψ(K(d(x)))ψ′(K(d(x)))K(d(x))K(d(x))d(x)k(d(x)))+2μψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))d(x)k(d(x))+μ(μ−1)ψ(K(d(x)))ψ′(K(d(x)))K(d(x))ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))(K(d(x))d(x)k(d(x)))2)∏i=1N−1κi(xˉ)1−d(x)κi(xˉ)−ξ+ψ(K(d(x)))g′(ξ+ψ(K(d(x))))g(ξ+ψ(K(d(x))))g′(Θ+(d(x)))g′(ξ+ψ(K(d(x))))g(ξ+ψ(K(d(x))))ξ+Ng(ψ(K(d(x))))]−(B0+ε)g(ξ+ψ(K(d(x))))ξ+Ng(ψ(K(d(x))))−(B0+ε)(C˜++ε)ξ+ψ(K(d(x)))g′(ξ+ψ(K(d(x))))g(ξ+ψ(K(d(x))))×g′(Θ+(d(x)))g′(ξ+ψ(K(d(x))))g(ξ+ψ(K(d(x))))ξ+Ng(ψ(K(d(x))))(d(x))μ+(d(x))−μ(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)mˆ+∑i=1N−1CN−1i(−1)i+1(m+d(x))i(1−d(x)m+)N−1+ζ˜+(d(x))(d(x))μ∏i=1N−1κi(xˉ)1−d(x)κi(xˉ)}≥−ξ+N(ψ′(K(d(x))))N−1ψ′′(K(d(x)))kN+1(d(x))(d(x))μ∑i=13Ii+(d(x))>0,
i.e., uε is a subsolution of Eq. (1.1) in Ωδε. Moreover, it follows by Lemma 4.3 that for i=1, ... ,N
Si(D2u_ε(x))=ξ+i(ψ′(K(d(x))))iki(d(x))(1+(C˜++ε)(d(x))μ+μ(C˜++ε)K(d(x))d(x)k(d(x))×ψ(K(d(x)))ψ′(K(d(x)))K(d(x))(d(x))μ)iSi(ϵ1,⋅⋅⋅,ϵN−1)−ξ+i(ψ′(K(d(x))))i−1ψ′′(K(d(x)))ki+1(d(x))(1+(C˜++ε)(d(x))μ+μ(C˜++ε)×ψ(K(d(x)))ψ′(K(d(x)))K(d(x))K(d(x))d(x)k(d(x))(d(x))μ)i−1[(1+ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x)))×(1+(C˜++ε)(d(x))μ)+2μ(C++ε)ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))d(x)k(d(x))(d(x))μ+μ(μ−1)(C˜++ε)ψ(K(d(x)))ψ′(K(d(x)))K(d(x))ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))(K(d(x))d(x)k(d(x)))2(d(x))μ]×Si−1(ϵ1,⋅⋅⋅,ϵN−1).
This implies that we can adjust the above positive constant δε such that for any x∈Ωδε
S i ( D 2 u _ ε ( x ) ) > 0 for i = 1 , ⋅ ⋅ ⋅ , N .
We obtain by Lemma 4.4 that D2uε is positive definite in Ωδε.
Let
u‾ε(x)=−ξ−ψ(K(d(x)))(1+(C˜−−ε)(d(x))μ),x∈Ωδε
By the same calculation as (4.27), we obtain
det(D2u‾ε(x))−b(x)g(−u‾ε(x))≤−ξ−N(ψ′(K(d(x))))N−1ψ′′(K(d(x)))kN+1(d(x))(d(x))μ∑i=13Ii−(d(x))<0,
i.e. uε is a supersolution of Eq. (1.1) in Ωδε.
Let u be the unique strictly convex solution to problem (1.1). Through the same argument as Theorem 1.1, we see that there exists a large constant M > 0 such that
u_ε(x)−Md(x)≤u(x)≤u‾ε(x)+Md(x),x∈Ωδε.
Hence, for any x∈ Ωδε
C˜++ε+M(d(x))1−μξ+ψ(K(d(x)))≥(−u(x)ξ+ψ(K(d(x)))−1)(d(x))−μ;C˜−−ε−M(d(x))1−μξ−ψ(K(d(x)))≤(−u(x)ξ−ψ(K(d(x)))−1)(d(x))−μ.
Since (1-μ)(γ+N)Dk-(N+1)>0, we conclude from Lemma 3.1 (ii) , Lemma 3.6 (iii), Proposition 2.7 and Proposition 2.5 (ii) that
C˜++ε≥lim supd(x)→0(−u(x)ξ+ψ(K(d(x)))−1)(d(x))−μ;C˜−−ε≤lim infd(x)→0(−u(x)ξ−ψ(K(d(x)))−1)(d(x))−μ.
Letting ε → 0, the proof is finished. □
4.3 Proof of Theorem 1.3
Now, we prove Theorem 1.3. As before, for fixed ε > 0, we define
w ± ( d ( x ) ) = η ± ψ ( ( ∫ 0 d ( x ) L ~ ( s ) s d s ) N N + 1 ) ( 1 + ( C ± ∗ ± ε ) M ( d ( x ) ) ) , x ∈ Ω δ 1 ,
where η ± , C ± * and
M
are given in Theorem 1.3. By the Lagrange’s mean value theorem, it is clear that there exist λ ± ∈(0, 1) and
(4.28)
Θ ± ( d ( x ) ) = η ± ψ ( ( ∫ 0 d ( x ) L ~ ( s ) s d s ) N N + 1 ) ( 1 + λ ± ( C ± ∗ ± ε ) M ( d ( x ) ) )
such that for x∈Ωδ1
g ( w ± ( d ( x ) ) ) = g ( η ± ψ ( ( ∫ 0 d ( x ) L ~ ( s ) s d s ) N N + 1 ) ) + η ± ψ ( ( ∫ 0 d ( x ) L ~ ( s ) s d s ) N N + 1 ) g ′ ( Θ ± ( d ( x ) ) ) ( C ± ∗ ± ε ) M ( d ( x ) ) .
We obtain by Proposition 2.2 that (4.2) still holds. Moreover, we can adjust δ1 such that (4.3) holds here.
Proof. The proof is still divided into the following two steps.
Step 1. As before, for fixed ε> 0 and ∀ x∈Ωδ1, we define
(4.29)
r ( d ( x ) ) = ( ∫ 0 d ( x ) L ~ ( s ) s d s ) N N + 1
and
I 1 ± ( d ( x ) ) = ( M ( d ( x ) ) ) − 1 { [ ( N N + 1 ) N ( N N + 1 − 1 N + 1 ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + ( N N + 1 ) N ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) ] m ^ ± − g ( ξ ± ψ ( r ( d ( x ) ) ) ) ξ ± N g ( ψ ( r ( d ( x ) ) ) ) } ; I 2 ± ( d ( x ) ) = C ± ∗ [ ( N N + 1 ) N N ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) C ± ∏ i = 1 N − 1 ( 1 − d ( x ) κ i ( x ¯ ) ) − 1 − η ± ψ ( r ( d ( x ) ) ) g ′ ( η ± ψ ( r ( d ( x ) ) ) ) g ( η ± ψ ( r ( d ( x ) ) ) ) g ′ ( Θ ± ( d ( x ) ) ) g ′ ( η ± ψ ( r ( d ( x ) ) ) ) g ( η ± ψ ( r ( d ( x ) ) ) ) η ± N g ( ψ ( r ( d ( x ) ) ) ) ] ± ε [ ( N N + 1 ) N N ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) m ^ + ∏ i = 1 N − 1 ( 1 − d ( x ) κ i ( x ¯ ) ) − 1 − η ± ψ ( r ( d ( x ) ) ) g ′ ( η ± ψ ( r ( d ( x ) ) ) ) g ( η ± ψ ( r ( d ( x ) ) ) ) g ′ ( Θ ± ( d ( x ) ) ) g ′ ( η ± ψ ( r ( d ( x ) ) ) ) g ( η ± ψ ( r ( d ( x ) ) ) ) η ± N g ( ψ ( r ( d ( x ) ) ) ) ] ; I 3 ± ( d ( x ) ) = − ( B 0 ± ε ) g ( η ± ψ ( r ( d ( x ) ) ) ) η ± N g ( ψ ( r ( d ( x ) ) ) ) ( d ( x ) ) μ ( M ( d ( x ) ) ) − 1 + ( M ( d ( x ) ) ) − 1 [ ( N N + 1 ) N ( N N + 1 − 1 N + 1 ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + ( N N + 1 ) N ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) ] m ^ ± ∑ i = 1 N − 1 C N − 1 i ( − 1 ) i + 1 ( m ± d ( x ) ) i ( 1 − d ( x ) m ± ) N − 1 − ( B 0 ± ε ) ( C ± ∗ ± ε ) η ± ψ ( r ( d ( x ) ) ) g ′ ( η ± ψ ( r ( d ( x ) ) ) ) g ( η ± ψ ( r ( d ( x ) ) ) ) g ′ ( Θ ± ( d ( x ) ) ) g ′ ( η ± ψ ( r ( d ( x ) ) ) ) g ( η ± ψ ( r ( d ( x ) ) ) ) η ± N g ( ψ ( r ( d ( x ) ) ) ) ( d ( x ) ) μ + ( N N + 1 ) N ζ ± ∗ ( d ( x ) ) ( M ( d ( x ) ) ) − 1 ∏ i = 1 N − 1 κ i ( x ¯ ) 1 − d ( x ) κ i ( x ¯ ) ,
where
if(S2) holds and θ=0,C±=mˆ±, if X+≥0 and X−≥0,C∓=mˆ∓, if X+≤0 and X−≤0,C+=C−=mˆ+, if X+>0 and X−<0,C+=C−=mˆ−, if X+<0 and X−>0; if (S1)holds,C±=mˆ±,
m ± are defined by (4.6) and
ζ ± ∗ = ( C ± ∗ ± ε ) ( N N + 1 − 1 N + 1 ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ) M ( d ( x ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s × ( N + ( C ± ∗ ± ε ) ( N − 1 ) M ( d ( x ) ) ) + ( C ± ∗ ± ε ) 2 ( N − 1 ) ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ′ ( d ( x ) ) d ( x ) L ( d ( x ) ) − 1 ) × ( M ( d ( x ) ) ) 2 + [ β N N + 1 ( C ± ∗ ± ε ) ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ψ ′ ( r ( d ( x ) ) ) r ( d ( x ) ) ψ ( r ( d ( x ) ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s × ( M ( d ( x ) ) ) β + 1 β + β N N + 1 ( C ± ± ε ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s ( M ( d ( x ) ) ) β + 1 β + β ( β + 1 ) N ( C ± ± ε ) N + 1 × ψ ′ ( r ( d ( x ) ) ) r ( d ( x ) ) ψ ( r ( d ( x ) ) ) ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s ( M ( d ( x ) ) ) β + 2 β + β ( C ± ± ε ) ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + L ~ ′ ( d ( x ) ) d ( x ) L ( d ( x ) ) − 1 ) ( M ( d ( x ) ) ) β + 1 β ] × ( 1 + ( C ± ∗ ± ε ) ( N − 1 ) M ( d ( x ) ) + ν ± ∗ ( d ( x ) ) ) + ν ± ∗ ( d ( x ) ) [ ( N N + 1 − 1 N + 1 ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ) ( 1 + ( C ± ∗ ± ε ) M ( d ( x ) ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) ( 1 + ( C ± ± ε ) M ( d ( x ) ) ) ]
with
ν±∗(d(x))=R2±∗(d(x))+∑i=2N−1CN−1i(R1±∗(d(x))+R2±∗(d(x)))i,
where CiN-1 is given in (4.7) and
R1±∗(d(x))=(C±∗±ε)M(d(x))andR2±∗(d(x))=β(C±∗±ε)(M(d(x)))β+1β.
Next, we prove
(4.30)
limd(x)→0(I1±(d(x))+I2±(d(x))+I3±(d(x)))=±ε(γ+Nmˆ+mˆ±)η±−(γ+N).
To prove (4.30), we calculate the limits of I1 ± (d(x)), I2 ± (d(x)) and I3 ± (d(x)) as d(x)→0.
∙ First, by Lemma 3.2 (ii)−(iii), Lemma 3.6 (iv), (vi)−(vii) and (ix), we obtain
(4.31)
limd(x)→0I1±(d(x))=−NM+1N(N+1)σmˆ±(γ−1)2+σlnη±η±γN, if S2 and g3−g4 hold with θ=0,0, if S1 holds ing2.
∙ Second, by (4.2), Lemma 3.6 (iv) and the choices of η ± and C* ± in Theorem 1.3, we obtain
(4.32)
limd(x)→0I2±(d(x))=C±∗[(NN+1)NN(γ+N)γ−1C±+γη±−(γ+N)]±ε(γ+Nmˆ+mˆ±)η±−(γ+N).
∙ ,Third, we conclude by Lemma 3.5, Lemma 3.6 (iii)−(iv) and a straightforward calculation that
(4.33)
limd(x)→0I3±(d(x))=0.
Combining (4.31)-(4.33), we obtain (4.30) holds. By (b1), (b3) and (4.30), we see that there exists a sufficiently small positive constant δεδ1 such that for any x∈Ωδε
(d(x))−(N+1)L˜N(d(x))(1+(B0−ε)(d(x))μ)≤b(x)≤(d(x))−(N+1)L˜N(d(x))(1+(B0+ε)(d(x))μ)
and (4.4)-(4.5) hold here.
Step 2. Let
u_ε(x)=−η+ψ(r(d(x)))(1+(C+∗+ε)M(d(x))),x∈Ωδε,
where r(d(x)) is given by (4.29). Then
g(−u_ε(x))=g(η+ψ(r(d(x))))+η+ψ(r(d(x)))g′(Θ+(d(x)))(C+∗+ε)M(d(x)),x∈Ωδε,
where Θ+(d(x)) is given by (4.28). By Lemma 4.2, we have for any x∈Ωδε
(4.34)
det ( D 2 u _ ε ( x ) ) − b ( x ) g ( − u _ ε ( x ) ) ≥ − η + N ( N N + 1 ) N ( ψ ′ ( r ( d ( x ) ) ) ) N − 1 ψ ″ ( r ( d ( x ) ) ) ( d ( x ) ) − ( N + 1 ) × L ~ N ( d ( x ) ) ( 1 + ( C + ∗ + ε ) ( N − 1 ) M ( d ( x ) ) + ν + ∗ ( d ( x ) ) ) [ ( N N + 1 − 1 N + 1 ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ) × ( 1 + ( C + ∗ + ε ) M ( d ( x ) ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) × ( 1 + ( C + ∗ + ε ) M ( d ( x ) ) ) + β N N + 1 ( C + ∗ + ε ) ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ψ ′ ( r ( d ( x ) ) ) r ( d ( x ) ) ψ ( r ( d ( x ) ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s × ( M ( d ( x ) ) ) β + 1 β + β N N + 1 ( C + ∗ + ε ) ( M ( d ( x ) ) ) β + 1 β L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + β ( β + 1 ) N ( C + ∗ + ε ) N + 1 × ψ ′ ( r ( d ( x ) ) ) r ( d ( x ) ) ψ ( r ( d ( x ) ) ) ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( M ( d ( x ) ) ) β + 2 β L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + β ( C + ∗ + ε ) × ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( M ( d ( x ) ) ) β + 1 β ( L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) ] × ∏ i = 1 N − 1 κ i ( x ¯ ) 1 − d ( x ) κ i ( x ¯ ) − ( d ( x ) ) − ( N + 1 ) L ~ N ( d ( x ) ) ( 1 + ( B 0 + ε ) ( d ( x ) ) μ ) × [ g ( η + ψ ( r ( d ( x ) ) ) ) + η + ( C + ∗ + ε ) g ′ ( Θ + ( d ( x ) ) ) ψ ( r ( d ( x ) ) ) M ( d ( x ) ) ] = − η + N ( ψ ′ ( r ( d ( x ) ) ) ) N − 1 ψ ″ ( r ( d ( x ) ) ) ( d ( x ) ) − ( N + 1 ) L ~ N ( d ( x ) ) M ( d ( x ) ) { ( M ( d ( x ) ) ) − 1 × [ ( ( N N + 1 ) N ( N N + 1 − 1 N + 1 ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + ( N N + 1 ) N ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) ) m ^ + − g ( η + ψ ( r ( d ( x ) ) ) ) η + N g ( ψ ( r ( d ( x ) ) ) ) ] + ( C + ∗ + ε ) [ ( N N + 1 ) N N ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) ∏ i = 1 N − 1 κ i ( x ¯ ) 1 − d ( x ) κ i ( x ¯ ) − η + ψ ( r ( d ( x ) ) ) g ′ ( η + ψ ( r ( d ( x ) ) ) ) g ( η + ψ ( r ( d ( x ) ) ) ) g ′ ( Θ + ( d ( x ) ) ) g ′ ( η + ψ ( r ( d ( x ) ) ) ) g ( η + ψ ( r ( d ( x ) ) ) ) η + N g ( ψ ( r ( d ( x ) ) ) ) ] − ( B 0 + ε ) g ( η + ψ ( r ( d ( x ) ) ) ) η + N g ( ψ ( r ( d ( x ) ) ) ) ( d ( x ) ) μ ( M ( d ( x ) ) ) − 1 + ( M ( d ( x ) ) ) − 1 [ ( N N + 1 ) N ( N N + 1 − 1 N + 1 × ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + ( N N + 1 ) N ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) ] × m ^ + ∑ i = 1 N − 1 C N − 1 i ( − 1 ) i + 1 ( m + d ( x ) ) i ( 1 − d ( x ) m + ) N − 1 − ( B 0 + ε ) ( C + ∗ + ε ) η + ψ ( r ( d ( x ) ) ) g ′ ( η + ψ ( r ( d ( x ) ) ) ) g ( η + ψ ( r ( d ( x ) ) ) ) g ′ ( Θ + ( d ( x ) ) ) g ′ ( η + ψ ( r ( d ( x ) ) ) ) g ( η + ψ ( r ( d ( x ) ) ) ) η + N g ( ψ ( r ( d ( x ) ) ) ) ( d ( x ) ) μ + ( N N + 1 ) N ζ + ∗ ( d ( x ) ) ( M ( d ( x ) ) ) − 1 ∏ i = 1 N − 1 κ i ( x ¯ ) 1 − d ( x ) κ i ( x ¯ ) } ≥ − η + N ( ψ ′ ( r ( d ( x ) ) ) ) N − 1 ψ ″ ( r ( d ( x ) ) ) ( d ( x ) ) − ( N + 1 ) L ~ N ( d ( x ) ) M ( d ( x ) ) ∑ i = 1 3 I i + ( d ( x ) ) > 0 ,
i.e. uε is a subsolution of Eq. (1.1) in Ωδε. Moreover, it follows by Lemma 4.3 that for i=1, ... ,N
S i ( D 2 u _ ε ( x ) ) = η + i ( N N + 1 ) i ( ψ ′ ( r ( d ( x ) ) ) ) i ( d ( x ) ) − i L ~ i ( d ( x ) ) ( ∫ 0 d ( x ) L ~ ( s ) s d s ) − i N + 1 × ( 1 + ( C + ∗ + ε ) M ( d ( x ) ) + β ( C + ∗ + ε ) ( M ( d ( x ) ) ) β + 1 β ) i S i ( ϵ 1 , ⋅ ⋅ ⋅ , ϵ N − 1 ) − η + i ( N N + 1 ) i ( ψ ′ ( r ( d ( x ) ) ) ) i − 1 ψ ″ ( r ( d ( x ) ) ) ( d ( x ) ) − ( i + 1 ) L ~ i ( d ( x ) ) ( ∫ 0 d ( x ) L ~ ( s ) s d s ) N − i N + 1 × ( 1 + ( C + ∗ + ε ) M ( d ( x ) ) + β ( C + ∗ + ε ) ( M ( d ( x ) ) ) β + 1 β ) i − 1 [ ( N N + 1 − 1 N + 1 ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ) × ( 1 + ( C + ∗ + ε ) M ( d ( x ) ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) × ( 1 + ( C + ∗ + ε ) M ( d ( x ) ) ) + β N N + 1 ( C + ∗ + ε ) ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ψ ′ ( r ( d ( x ) ) ) r ( d ( x ) ) ψ ( r ( d ( x ) ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s × ( M ( d ( x ) ) ) β + 1 β + β N N + 1 ( C + ∗ + ε ) ( M ( d ( x ) ) ) β + 1 β L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + β ( β + 1 ) N ( C + ∗ + ε ) N + 1 × ψ ′ ( r ( d ( x ) ) ) r ( d ( x ) ) ψ ( r ( d ( x ) ) ) ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( M ( d ( x ) ) ) β + 2 β L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + β ( C + ∗ + ε ) × ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( M ( d ( x ) ) ) β + 1 β ( L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) ] S i − 1 ( ϵ 1 , ⋅ ⋅ ⋅ , ϵ N − 1 ) .
This implies that we can adjust the above positive constant δε such that for any x∈Ωδε
S i ( D 2 u _ ε ( x ) ) > 0 for i = 1 , ⋅ ⋅ ⋅ , N .
We obtain by Lemma 4.4 that D2uε is positive definite in Ωδε.
Let
u‾ε(x)=−η−ψ(r(d(x)))(1+(C−∗−ε)M(d(x))),x∈Ωδε
By the same calculation as (4.34), we obtain
det ( D 2 u ¯ ε ( x ) ) − b ( x ) g ( − u ¯ ε ( x ) ) ≤ − η − N ( ψ ′ ( r ( d ( x ) ) ) ) N − 1 ψ ″ ( r ( d ( x ) ) ) ( d ( x ) ) − ( N + 1 ) L ~ N ( d ( x ) ) M ( d ( x ) ) ∑ i = 1 3 I i − ( d ( x ) ) < 0 ,
i.e. uε is a supersolution of Eq. (1.1) in Ωδε.
Let u be the unique strictly convex solution to problem (1.1). Through the same argument as Theorem 1.1, we see that there exists a large constant M > 0 such that
u_ε(x)−Md(x)≤u(x)≤u‾ε(x)+Md(x),x∈Ωδε.
Hence, we have for any x∈ Ωδε
C+∗+ε+M(M(d(x)))−1d(x)η+ψ(r(d(x)))≥(−u(x)η+ψ(r(d(x)))−1)(M(d(x)))−1;C−∗−ε−M(M(d(x)))−1d(x)η−ψ(r(d(x)))≤(−u(x)η−ψ(r(d(x)))−1)(M(d(x)))−1.
We conclude from (3.17)-(3.18), Proposition 2.5 (i)−(ii) that
C+∗+ε≥lim supd(x)→0(−u(x)η+ψ(r(d(x)))−1)(M(d(x)))−1;C−∗−ε≤lim infd(x)→0(−u(x)η−ψ(r(d(x)))−1)(M(d(x)))−1.
Letting ε → 0 , the proof is finished.
