Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 33, pp. 1–13.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu

HÖLDER ESTIMATES AND ASYMPTOTIC BEHAVIOR FOR

DEGENERATE ELLIPTIC EQUATIONS IN THE HALF SPACE

XIAOBIAO JIA, SHANSHAN MA

Abstract. In this article we investigate the asymptotic behavior at infinity
of viscosity solutions to degenerate elliptic equations. We obtain Hölder es-

timates, up to the flat boundary, by using the rescaling method. Also as a

byproduct we obtain a Liouville type result on Baouendi-Grushin type opera-
tors.

1. Introduction

In this article we study the asymptotic behavior at infinity of viscosity solutions
to the degenerate non-divergence elliptic equation

Lu = x2α
n

n−1∑
i,j=1

aij(x)Diju(x) + 2xαn

n−1∑
i=1

ain(x)Dinu(x) +Dnnu(x) = 0 (1.1)

in Rn+\B
+

1 , where n ≥ 2, α > 0, Rn+ = Rn ∩ {xn > 0}, B+
1 = Rn+ ∩ {|x| < 1}.

To ensure the ellipticity of operator L, we assume that aij(x), ain(x) ∈ C(Rn+)
(i, j = 1, . . . , n − 1) and that there exist constants 0 < λ ≤ Λ < ∞ such that for
each ξ ∈ Rn−1,

λ|ξ|2 ≤ ξT
n−1∑
i,j=1

aij(x)ξ ≤ Λ|ξ|2, ∀x ∈ Rn+, (1.2)

and for some 0 < δ < 1,

1− λ−1
n−1∑
i=1

‖ain‖2L∞(Rn+) > δ. (1.3)

In this article, solutions always indicate viscosity solutions (see [3] for definition).
For α = 0, by (1.2) and (1.3), L is uniformly elliptic. The asymptotic behavior at
infinity was considered in [7]. Note that the crucial key to obtain the asymptotic
behavior is the boundary Hölder estimates, which is classical for uniformly elliptic
equations (see [3, 5]).

For α > 0, aij ≡ 1 and ain ≡ 0 (i, j ≤ n − 1), L is a Baouendi-Grushin type
operator,

Lu := x2α
n

n−1∑
i=1

Diiu(x) +Dnnu(x) = 0, (1.4)

2020 Mathematics Subject Classification. 35B40, 35J70, 35B65.

Key words and phrases. Asymptotic behavior; degenerate elliptic equation; Hölder estimate.
©2023. This work is licensed under a CC BY 4.0 license.

Submitted September 17, 2022. Published April 5, 2023.
1



2 X. B. JIA, S. S. MA EJDE-2023/33

which was introduced in [1, 6]. There have been extensive works on the studies
of the Baouendi-Grushin type operators (see [2, 4, 8, 10] and references therein).
For α > 0 and aij satisfies (1.2), Le and Savin [9] obtained the boundary Schauder
estimates for solutions of the degenerate elliptic equation

xαn

n−1∑
i,j=1

aij(x)Diju(x) +Dnnu(x) = xαnf(x) in B+
1 .

In this article, we study the asymptotic behavior at infinity of solutions of (1.1)
with the coefficients satisfying (1.2) and (1.3).

By rescaling method similar to the one in [9], we establish the Hölder estimates
up to the flat boundary of solutions of (1.1).

Theorem 1.1. Let u ∈ C(B
+

1 ) be a solution of

Lu(x) = 0 in B+
1

u(x) = 0 on B1 ∩ {xn = 0},
(1.5)

where L is given by (1.1) with the coefficients satisfying (1.2) and (1.3). Then

u ∈ C
1

1+α (B
+

1/2).

Theorem 1.1, Harnack inequalities, and the comparison principle yield our main
theorem as follows.

Theorem 1.2. Let u ∈ C1(Rn+\B+
1 ) be a solution of

Lu = 0 in Rn+\B
+

1 ,

u = 0 on {xn = 0, |x| ≥ 1},
(1.6)

where L is given by (1.1) with the coefficients satisfying (1.2) and (1.3); and for
some s > 0,

|aij(x)− δij |+ |ain(x)| ≤
(
|x′|+ x1+α

n

)−s
in Rn+\B

+

1 , i, j < n. (1.7)

Assume that |u| ≤ 1 on ∂B1 ∩ {xn > 0}, |Du| ≤ 1 in Rn+\B+
1 and |Du| → 0 as

|x| → ∞. Then

|u(x)| ≤ Cxn(
|x′|2 + 1

(1+α)2x
2+2α
n

)n−1
2 + 1

2(1+α)

in Rn+\B+
R , (1.8)

where C > 0 and R ≥ 1 depend only on α, δ, s and n.

Remark 1.3. When α = 0, Theorem 1.2 still holds (see [7]).

By Theorem 1.2 and the comparison principle, we have the following Liouville
type theorem.

Theorem 1.4. Let u ∈ C1(Rn+) be a solution of

Lu = 0 in Rn+,
u = 0 on {xn = 0}, (1.9)

where L is as in (1.4). If |Du| → 0 as |x| → ∞. Then u(x) must be zero.



EJDE-2023/33 HÖLDER ESTIMATES AND ASYMPTOTIC BEHAVIOR 3

The rescaling method is a classical one to show the boundary Schauder/Hölder
estimates in (degenerate) linear elliptic equations (see [9]). Similarly, one can also
show the boundary Hölder estimates

x2α
n

n−1∑
i,j=1

aij(x)Diju(x) + 2xαn

n−1∑
i=1

ain(x)Dinu(x) +Dnnu(x) = x2α
n f(x).

The asymptotic result (Theorem 1.2) may push forward the study on asymptotic
behavior of solutions of the following degenerate Monge-Ampère equation

detD2u = f(x)x2α
n on {xn > 0},

where α > 0, and f(x) is positive and continuous.
This article is organized as follows. In Section 2, we show the boundary Hölder

estimates, which can be approached by the interior Hölder estimates via rescaling.
In Section 3, a supersolution is constructed according to the fundamental solution
of one Baouendi-Grushin type operator in the half space. Then it together with
the Hölder estimates up to the flat boundary implies that Theorem 1.2 holds.

2. Proof of Theorem 1.1

First, we show that (1.2) and (1.3) ensure the ellipticity of L.

Lemma 2.1. Let the coefficients of L in (1.1) satisfy (1.2) and (1.3). Then L

is elliptic in B
+

1 . Furthermore, for each fixed ε0 > 0, L is uniformly elliptic in

B
+

1 ∩ {xn ≥ ε0}.

The proof of the above lemma is standard, and is shown in the Appendix. To
show the Hölder estimates up to the flat boundary, we need to give some notion
(see [9]).

Definition 2.2. We define a distance dα between point y and point z by

dα(y, z) := |y′ − z′|+
∣∣y1+α
n − z1+α

n

∣∣ .
Observe that the relation between dα and the Euclidean distance,

c|y − z|1+α ≤ dα(y, z) ≤ C|y − z|, (2.1)

dα(y, z) ∼ |y − z| if y, z ∈ B+

1 ∩
{
xn ≥

1

8

}
. (2.2)

For each h > 0 and each x̃ ∈ Rn, we denote

Eh(x̃) =
{
x ∈ Rn : |x′ − x̃′|2 + |xn − x̃n|2(1+α) < h

}
, (2.3)

and Fh = diag
(
h

1
2 , h

1
2 , . . . , h

1
2 , h

1
2(1+α)

)
. For simplicity, we denote

Eh = Eh(0) =
{
x ∈ Rn : |x′|2 + |xn|2(1+α) < h

}
; E+

h = Eh ∩ {xn > 0}.
A simple calculation gives

FhEα′
(1

2
en

)
= Eα′h

(1

2
h

1
2(1+α) en

)
, FhE

+
1 = E+

h , (2.4)

where en = (0, . . . , 0, 1), α′ = 4−2(1+α).
Note that (1.1) and dα keep their forms under the transformation x → Fhx.

Precisely, let
ũ(x) = u(Fhx), x ∈ E1 . (2.5)



4 X. B. JIA, S. S. MA EJDE-2023/33

Then it solves

L̃ũ = x2α
n

n−1∑
i,j=1

ãij(x)Dij ũ(x) + xαn

n−1∑
i=1

2ãin(x)Dinũ(x) +Dnnũ(x) = 0 (2.6)

with

ãij(x) = aij(Fhx), ãin(x) = ain(Fhx), i, j ≤ n− 1, (2.7)

dα(y, z) = h−1/2dα(Fhy, Fhz). (2.8)

If function w is γ-Hölder continuous in Ω ⊂ B
+

1 with respect to dα, we write
w ∈ Cγα(Ω) and define

[w]Cγα(Ω) = sup
y,z∈Ω,y 6=z

|w(y)− w(z)|
(dα(y, z))γ

, ‖w‖Cγα(Ω) = ‖w‖L∞(Ω) + [w]Cγα(Ω).

Prrof of Theorem 1.1. We divided this proof into two cases.

Case 1. u ∈ C
1

1+α
(
B

+

1/2 ∩ {xn > 1
8}
)
. By Lemma 2.1, L is uniformly elliptic

in B
+

1/2 ∩ {xn > 1/8}. Applying the classical Hölder estimates to u, there exists

C > 0, depending only on λ, Λ, α, δ, n and ‖u‖L∞ , such that

[u]
C

1
1+α

(
Eα′( 1

2 en)
) ≤ C‖u‖L∞ ≤ C.

Case 2. u ∈ C
1

1+α
(
B

+

1/2 ∩ {xn ≤ 1/8}
)
. We show this case by four steps.

Step 1. There exists C > 0, depending only on λ, Λ, α, δ, n and ‖u‖L∞ , such that

|u(x)| ≤ Cxn in B+
3
4

. (2.9)

We only need to show that for each x0 ∈ {xn = 0, |x′| < 3
4},

|u(x0, xn)| ≤ Cxn.
Let

u(x) = Cxn +B|x′ − x′0|2 −
C

2
x2+α
n

with B = 16‖u‖L∞ . One can choose C > 0, depending only on Λ, α, n, and ‖u‖L∞ ,
such that

Lu ≤ 0 in B+
1 , u ≥ ‖u‖L∞ ≥ u on ∂B+

1 , (2.10)

by taking

2(n− 1)ΛB − (2 + α)(1 + α)C/2 ≤ 0,
C

2
xn +B|x′ − x′0|2 > ‖u‖L∞ on ∂B+

1 .

Therefore, (2.10) and the comparison principe (see [11, Theorem 6]) yield (2.9).

Step 2. For any fixed h ∈ (0, 1],

[u]
C

1
1+α
α

(
Eα′h

(
1
2h

1
2(1+α) en

)) ≤ C. (2.11)

In fact, let ũ be as in (2.5), and then ũ solves (2.6) in B+
1 . By (2.5) and (2.9),

ũ ≤ Ch
1

2(1+α) in B+
1 . (2.12)

Similar to Case 1, applying the Hölder estimates to ũ in E 1
4

(
1
2en
)
, we have

[ũ]
C

1
1+α

(
Eα′( 1

2 en)
) ≤ C‖ũ‖L∞(B+

1 ) ≤ Ch
1

2(1+α) .



EJDE-2023/33 HÖLDER ESTIMATES AND ASYMPTOTIC BEHAVIOR 5

By (2.2), we see

[ũ]
C

1
1+α
α

(
Eα′( 1

2 en)
) ≤ Ch 1

2(1+α) .

This together with (2.4), (2.5), and (2.8) yields (2.11), since

|ũ(y)− ũ(z)|
(dα(y, z))

1
1+α

=
|u(Fhy)− u(Fhz)|

h−
1

2(1+α) (dα(Fhy, Fhz))
1

1+α

.

Step 3. We prove that u ∈ C
1

1+α
α at 0 along en direction, that is,

sup
0<h<1

∣∣u( 1
2h

1
2(1+α) en

)
− u(0)

∣∣(
( 1

2h
1

2(1+α) )1+α
) 1

1+α

≤ C

for some C > 0 depending only on λ, Λ, α and n. It suffices to prove that∣∣∣u(1

2
h

1
2(1+α) en

)
− u(0)

∣∣∣ ≤ Ch 1
2(1+α) ,

where C > 0 independents on h.
Indeed, Step 2 yields that for each k = 1, 2, . . . ,∣∣∣u( 1

2k
h

1
2(1+α) en

)
− u
( 1

2k+1
h

1
2(1+α) en

)∣∣∣ ≤ C2−k−1h
1

2(1+α) ,

This implies that∣∣u( 1
2h

1
2(1+α) en

)
− u(0)

∣∣
h

1
2(1+α)

≤
∞∑
k=1

∣∣u( 1
2k
h

1
2(1+α) en

)
− u
(

1
2k+1h

1
2(1+α) en

)∣∣
h

1
2(1+α)

≤
∞∑
k=1

C2−k−1 ≤ C.

Therefore, u ∈ C
1

1+α
α at 0 along en direction.

Step 4. We show Case 2. Similar to Step 3, we have that u ∈ C
1

1+α
α at any

x ∈ B+
1/2 ∩ {xn = 0} along en direction.

Let y, z ∈ B+

1/2 ∩ {xn ≤ 1
8} and denote by yn, zn the nth component of y and z,

respectively. If z ∈ E
2−2(1+α)y

2(1+α)
n

(yn) or y ∈ E
2−2(1+α)z

2(1+α)
n

(zn), by (2.11), we

are done. Otherwise, z /∈ E
2−2(1+α)y

2(1+α)
n

(yn) and y /∈ E
2−2(1+α)z

2(1+α)
n

(zn), which

yields

|y − z|2 ≥ max
{

2−2(1+α)z2(1+α)
n , 2−2(1+α)y2(1+α)

n

}
. (2.13)

By Step 3 and the boundary value condition, we obtain

|u(y)− u(z)| ≤ |u(y)− u(y′, 0)|+ |u(y′, 0)− u(z′, 0)|+ |u(z′, 0)− u(z)|

≤ C|yn|+ C|zn| ≤ C|y − z|
1

1+α (by (2.13)).
(2.14)

It follows that u ∈ C
1

1+α
(
B

+

1/2 ∩ {xn ≤ 1
8}
)
. Therefore, by Case 1 and Case 2, we

complete the proof of Theorem 1.1. �



6 X. B. JIA, S. S. MA EJDE-2023/33

3. Proof of Theorem 1.2

In this section we divide the proof of Theorem 1.2 into two steps as the following.
In fact, Subsection 3.1 gives the convergence at infinity of the solutions in Theorem
1.2, and then Subsection 3.2 shows its asymptotic behavior at infinity. Recall that
the symbols Fh, Eh and E+

h are defined in Section 2.

3.1. Convergence at infinity. In the subsection we apply Hölder estimates up
to the flat boundary to show that the solution in Theorem 1.2 converges at infinity.
Hereinafter, we say a constant is universal if it depends only on λ, Λ, α, δ and n.
The universal constant may change from line to line if necessary. A straightforward
corollary of the boundary Hölder estimates is the following result.

Corollary 3.1. Let u ∈ C(E+
4R\E

+
R ) be a solution of

Lu = 0 in E+
4R\E

+

R,

u ≤ 1 on ∂(E+
4R\E

+

R) ∩ {xn > 0},

u ≤ 1

2
on ∂(E+

4R\E
+

R) ∩ {xn = 0},

(3.1)

where L is given by (1.1) with coefficients satisfying (1.2) and (1.3) in E+
4R\E

+

R for
some R > 0. Then there exists a universal constant c0 > 0 such that

u(x) ≤ 1− c0 on ∂E2R ∩ {xn ≥ 0}.

Proof. We only need to set u(x) = 1/2 on ∂(E+
4R\E

+

R) ∩ {xn = 0}. Otherwise, one

can consider a supersolution v with v(x) = 1
2 on ∂(E+

4R\E
+

R) ∩ {xn = 0}, and if it
holds for v, by the comparison principle, so does for u. Let

û(x) = u(FRx), x ∈ E+
4 \E

+

1 .

By the definitions of FR and E+
R in Section 2, we have FR(E+

4 \E
+

1 ) = E+
4R\E

+

R.
Then

L̃û = 0 in E+
4 \E

+

1 ,

û ≤ 1 on ∂(E+
4 \E

+

1 ) ∩ {xn > 0},

û =
1

2
on ∂(E+

4 \E
+

1 ) ∩ {xn = 0},

(3.2)

where L̃ is given by (2.6). Clearly, the coefficients of L̃ also satisfy (1.2) and (1.3)

in E+
4 \E

+

1 . Then by the third equality in (3.2) and Theorem 1.1, there exists a
universal constant 0 < τ ≤ 1 such that

û(x) ≤ 2

3
on ∂E2 ∩ {0 ≤ xn ≤ τ}. (3.3)

By the comparison principle, we have û ≤ 1 in E+
4 \E

+

1 . Then 1− û satisfies

L̃(1− û) = 0 in E+
4 \E

+

1 .

By the interior Harnack inequality for 1− û, there exists a universal constant C ≥ 1
such that

C inf
∂E2∩{xn≥τ}

(1− û) ≥ sup
∂E2∩{xn≥τ}

(1− û) ≥ sup
∂E2∩{xn=τ}

(1− û) ≥ 1

3
.



EJDE-2023/33 HÖLDER ESTIMATES AND ASYMPTOTIC BEHAVIOR 7

This implies

û(x) ≤ 1− 1

3C
on ∂E2 ∩ {xn ≥ τ}. (3.4)

This, the definition of û ,and (3.3) implies the conclusion, via taking c0 = 1
3C . �

Applying Corollary 3.1, we have the following convergence result.

Theorem 3.2. Let u ∈ C(Rn+\E+
1 ) be a solution of Lu = 0 in Rn+\E

+

1 , where L is

given by (1.1) with the coefficients satisfying (1.2) and (1.3) in Rn+\E
+

1 . If

• |u| ≤ 1 on (∂E1 ∩ {xn > 0}) ∪ {xn = 0, |x| ≥ 1},
• u(x′, 0)→ β as |x′| → ∞
• |Du(x)| → 0 as |x| → ∞.

Then u(x)→ β as |x| → ∞.

Proof. The proof of this theorem is divided into two steps as follows.

Step 1. |u| ≤ 1 in Rn+\E
+

1 . For any ε > 0, since |Du| → 0 as |x| → ∞, there exists
Rε ≥ 1 such that

|Du| ≤ ε in Rn+\Q+
Rε
, (3.5)

where Q+
Rε

:= {(x′, xn) : |x′| < Rε, 0 < xn < Rε} is a cylinder. By |u| ≤ 1 on
{xn = 0, |x| ≥ 1}, (3.5) and Newton-Leibniz formula, we have

|u(x)| ≤ 1 + 2εxn on ∂Q+
Rε
∩ {xn > 0}.

Since |u| ≤ 1 on (∂E1 ∩ {xn > 0}) ∪ {xn = 0, |x| ≥ 1}, we obtain

|u(x)| ≤ 1 + 2εxn on ∂(Q+
Rε
\E+

1 ).

Obviously, 1 + 2εxn solves (1.1) in Q+
Rε
\E+

1 . Then by the comparison principle,

|u(x)| ≤ 1 + 2εxn in Q+
Rε
\E+

1 .

Letting ε→ 0, it completes the proof of step 1.

Step 2. u(x)→ β as |x| → ∞. We only need to set β = 0. Otherwise, we consider
u(x)−β
1+|β| .

Now we argue by contradiction. If this step is not true, by Step 1, u has finite
superior limit u > 0 or inferior limit u < 0 at infinity. It suffices to assume that
u > 0.

By the definition of u and u(x′, 0) → β as |x′| → ∞, there exists large R1 ≥ 1
such that for all R ≥ R1,

u(x) ≤
(

1 +
c0
2

)
u in Rn+\E

+

R

and

u(x′, 0) ≤ 1

2

(
1 +

c0
2

)
u if |x′| ≥ R,

where c0 is given by Corollary 3.1. Then applying Corollary 3.1 to u(x)

(1+
c0
2 )u

in

E+
4R\E

+

R, we obtain for all R ≥ R1,

u(x) ≤ (1− c0)
(
1 +

c0
2

)
u ≤

(
1− c0

2

)
u on ∂E2R ∩ {xn ≥ 0}.

This implies

u(x) ≤
(
1− c0

2

)
u in Rn+\E

+

2R1
,



8 X. B. JIA, S. S. MA EJDE-2023/33

which reaches a contradiction. �

Theorem 3.2 implies the following corollary.

Corollary 3.3. Let u be as in Theorem 1.2. Then u(x)→ 0 as |x| → ∞.

The proofs is obvious and thus we omit it here.

3.2. Asymptotic behavior at infinity. In this subsection we obtain the asymp-
totic behavior at infinity of solutions in Theorem 1.2, through constructing a barrier
function.

To get the barrier function, we first let

w(x′, xn) =
xn(

|x′|2 + βx2+2α
n

)γ , (3.6)

where β = 1
(1+α)2 , γ = n−1

2 + 1
2(1+α) . Simple calculations deduce that

Diw = − 2γxixn(
|x′|2 + βx2+2α

n

)γ+1 , i < n;

Dnw =
1(

|x′|2 + βx2+2α
n

)γ − γβ(2 + 2α)x2+2α
n(

|x′|2 + βx2+2α
n

)γ+1 ;

Dijw = − 2γxnδij(
|x′|2 + βx2+2α

n

)γ+1 +
4γ(γ + 1)xixjxn(
|x′|2 + βx2+2α

n

)γ+2 , i, j < n;

Dinw = − 2γxi(
|x′|2 + βx2+2α

n

)γ+1 +
2γβ(2 + 2α)xix

2+2α
n(

|x′|2 + βx2+2α
n

)γ+2 , i < n;

Dnnw = − γβ(2 + 2α)x1+2α
n(

|x′|2 + βx2+2α
n

)γ+1 −
γβ(2 + 2α)2x1+2α

n(
|x′|2 + βx2+2α

n

)γ+1

+
γ(γ + 1)β2(2 + 2α)2x3+4α

n(
|x′|2 + βx2+2α

n

)γ+2 .

(3.7)

Then

Lw = − 2γ(n− 1)x1+2α
n(

|x′|2 + βx2+2α
n

)γ+1 +
4γ(γ + 1)|x′|2x1+2α

n(
|x′|2 + βx2+2α

n

)γ+2 −
γβ(2 + 2α)x1+2α

n(
|x′|2 + βx2+2α

n

)γ+1

− γβ(2 + 2α)2x1+2α
n(

|x′|2 + βx2+2α
n

)γ+1 +
γ(γ + 1)β2(2 + 2α)2x3+4α

n(
|x′|2 + βx2+2α

n

)γ+2

=
{−2γ(n− 1)− γβ(2 + 2α)(3 + 2α)}x1+2α

n(
|x′|2 + βx2+2α

n

)γ+1

+
4γ(γ + 1){|x′|2 + β2(1 + α)2x2+2α

n }x1+2α
n(

|x′|2 + βx2+2α
n

)γ+2

=
{−2γ(n− 1)− γβ(2 + 2α)(3 + 2α)}x1+2α

n(
|x′|2 + βx2+2α

n

)γ+1 +
4γ(γ + 1)x1+2α

n(
|x′|2 + βx2+2α

n

)γ+1

=
2γ{−(n− 1)− (1 + α)−1(3 + 2α) + 2(γ + 1)}x1+2α

n(
|x′|2 + βx2+2α

n

)γ+1



EJDE-2023/33 HÖLDER ESTIMATES AND ASYMPTOTIC BEHAVIOR 9

=
2γ{−n+ 1− (1 + α)−1 + 2γ}x1+2α

n(
|x′|2 + βx2+2α

n

)γ+1 = 0

where γ = n−1
2 + 1

2(1+α) , and L is given by (1.4). Using w, we can construct a

supersolution of (1.1) as follows.

Lemma 3.4. Let L be given by (1.1) with coefficients satisfying (1.2), (1.3) and
(1.7). Then for each ρ ∈

(
0,min{ s

n−1 , 1}
)
, there exists R0 ≥ 1 depending only on

ρ, s, α and n such that

L(w − w1+ρ) ≤ 0 in Rn+\E
+

R0
. (3.8)

Proof. For i, j < n, we have

|Dij(w
1+ρ)| =

∣∣(1 + ρ)wρDijw + ρ(1 + ρ)wρ−1DiwDjw
∣∣

≤ (1 + ρ)wρ
{ 2γxn(
|x′|2 + βx2+2α

n

)γ+1 +
4γ(γ + 1)|x′|2xn(
|x′|2 + βx2+2α

n

)γ+2

}
+ ρ(1 + ρ)wρ−1 4γ|x′|2x2

n(
|x′|2 + βx2+2α

n

)2(γ+1)

≤ C(ρ, α, n)wρxn(
|x′|2 + βx2+2α

n

)γ+1 +
C(ρ, α, n)wρ−1x2

n(
|x′|2 + βx2+2α

n

)2γ+1

≤ C(ρ, α, n)wρ−1x2
n(

|x′|2 + βx2+2α
n

)2γ+1 ,

(3.9)

and

|Din(w1+ρ)| =
∣∣ρ(1 + ρ)wρ−1DiwDnw + (1 + ρ)wρDinw

∣∣
≤ 2γρ(1 + ρ)wρ−1|x′|xn(
|x′|2 + βx2+2α

n

)γ+1

{ 1(
|x′|2 + βx2+2α

n

)γ +
γβ(2 + 2α)x2+2α

n(
|x′|2 + βx2+2α

n

)γ+1

}
+ (1 + ρ)wρ

{ 2γ|x′|(
|x′|2 + βx2+2α

n

)γ+1 +
2γβ(2 + 2α)|x′|x2+2α

n(
|x′|2 + βx2+2α

n

)γ+2

}
≤ C(ρ, α, n)wρ−1|x′|xn(
|x′|2 + βx2+2α

n

)2γ+1 +
C(ρ, α, n)wρ|x′|(
|x′|2 + βx2+2α

n

)γ+1

≤ C(ρ, α, n)wρ−1|x′|xn(
|x′|2 + βx2+2α

n

)2γ+1 ,

(3.10)
where C(ρ, α, n) is positive, depending only on ρ, α and n, and may change from
line to line. Thus,

L(w1+ρ) = x2α
n

n−1∑
i=1

(1 + ρ)wρDiiw + ρ(1 + ρ)wρ−1DiwDiw + (1 + ρ)wρDnnw

+ ρ(1 + ρ)wρ−1(Dnw)2

= +ρ(1 + ρ)wρ−1(Dnw)2

= ρ(1 + ρ)wρ−1
{
x2α
n

n−1∑
i=1

(
− 2γxixn(
|x′|2 + βx2+2α

n

)γ+1

)2



10 X. B. JIA, S. S. MA EJDE-2023/33

+
( 1(
|x′|2 + βx2+2α

n

)γ − γβ(2 + 2α)x2+2α
n(

|x′|2 + βx2+2α
n

)γ+1

)2}
= ρ(1 + ρ)wρ−1

{ 4γ2|x′|2x2+2α
n(

|x′|2 + βx2+2α
n

)2(γ+1)
+

1(
|x′|2 + βx2+2α

n

)2γ
− γβ(2 + 2α)x2+2α

n(
|x′|2 + βx2+2α

n

)2γ+1 +
γ2β2(2 + 2α)2x4+4α

n(
|x′|2 + βx2+2α

n

)2(γ+1)

}
= ρ(1 + ρ)wρ−1

{ 4γ2x2+2α
n(

|x′|2 + βx2+2α
n

)2γ+1 −
2γ(1 + α)−1x2+2α

n(
|x′|2 + βx2+2α

n

)2γ+1

+
1(

|x′|2 + βx2+2α
n

)2γ .}
=

(n− 1)
(
n− 1 + 1

1+α

)
ρ(1 + ρ)wρ−1x2+2α

n(
|x′|2 + βx2+2α

n

)2γ+1 +
ρ(1 + ρ)wρ−1(
|x′|2 + βx2+2α

n

)2γ ,
where γ = n−1

2 + 1
2(1+α) , and L is given by (1.4). This, (3.9), and (3.10) imply that

L
(
w1+ρ

)
≥ L

(
w1+ρ

)
−

n−1∑
i,j=1

|aij(x)− δij‖Dij(w
1+ρ)|x2α

n −
n−1∑
i=1

|ain(x)‖Din(w1+ρ)|

≥ ρ(1 + ρ)wρ−1(
|x′|2 + βx2+2α

n

)2γ − (|x′|+ x1+α
n

)−s C(ρ, α, n)wρ−1x2+2α
n(

|x′|2 + βx2+2α
n

)2γ+1

−
(
|x′|+ x1+α

n

)−s C(ρ, α, n)wρ−1|x′|xn(
|x′|2 + βx2+2α

n

)2γ+1

≥ ρ(1 + ρ)wρ−1(
|x′|2 + βx2+2α

n

)2γ − C(ρ, α, n)wρ−1(
|x′|2 + βx2+2α

n

)2γ+ s
2

− C(ρ, α, n)wρ−1(
|x′|2 + βx2+2α

n

)2γ+ s
2 + 1

2−
1

2(1+α)

≥
1
2ρ(1 + ρ)wρ−1(
|x′|2 + βx2+2α

n

)2γ in Rn+\E
+

R0

(3.11)

for some R0 ≥ 1 depending only on ρ, s, α, and n. Similarly,

Lw ≤ Lw +

n−1∑
i,j=1

|aij(x)− δij‖Dijw|x2α
n +

n−1∑
i=1

|ain(x)‖Dinw|

≤ C(ρ, α, n)x1+2α
n(

|x′|2 + βx2+2α
n

)γ+1+ s
2

+
C(ρ, α, n)|x′|(

|x′|2 + βx2+2α
n

)γ+1+ s
2

≤ C(ρ, α, n)(
|x′|2 + βx2+2α

n

)γ+1+ s
2−

1+2α
2(1+α)

. (3.12)



EJDE-2023/33 HÖLDER ESTIMATES AND ASYMPTOTIC BEHAVIOR 11

Since ρ ∈
(
0,min{ s

n−1 , 1}
)
, we obtain

wρ−1(
|x′|2 + βx2+2α

n

)2γ =
xρ−1
n(

|x′|2 + βx2+2α
n

)2γ+γ(ρ−1)

≥
(
|x′|2 + βx2+2α

n

)−2γ−γ(ρ−1)− 1−ρ
2(1+α) ,

(3.13)

(
− 2γ − γ(ρ− 1)− 1− ρ

2(1 + α)

)
+
(
γ + 1 +

s

2
− 1 + 2α

2(1 + α)

)
= −n− 1

2
ρ+

s

2
> 0.

(3.14)

By (3.11), (3.12), (3.13), and (3.14), we have

L(w − w1+ρ) ≤ 0 in Rn+\E
+

R0

for larger R0 ≥ 1 depending only on ρ, s, α and n. �

Proof of Theorem 1.2. By Lemma 3.4, for each fixed ρ ∈
(
0,min

{
s

n−1 , 1
})

, there
exists R > 1 depending only on s, α and n such that

L
(
w − w1+ρ

)
≤ 0 in Rn+\E

+

R.

By u(x) = 0 on {xn = 0}, |Du(x)| ≤ 1 in Rn+\E
+

1 and Newton-Leibniz formula,

|u(x)| ≤ 2xn on ∂ER ∩ {xn ≥ 0}.
On ∂ER ∩ {xn ≥ 0}, it is clear that

w − w1+ρ = w(1− wρ) ≥ c(R,α, n)xn.

The above two inequalities imply that for some C > 0 depending only on s, δ, α
and n,

|u(x)| ≤ C(w − w1+ρ), on ∂ER ∩ {xn ≥ 0}. (3.15)

For any ε > 0, by Corollary 3.3, there exists Rε > R such that

|u(x)| ≤ ε, x ∈ ∂ERε ∩ {xn ≥ 0}. (3.16)

It follows from (3.15), (3.16) and u(x) = 0 on (ERε\ER) ∩ {xn = 0} that

|u(x)| ≤ C(w − w1+ρ) + ε on ∂(E+
Rε
\E+

R).

By the comparison principle,

|u(x)| ≤ C(w − w1+ρ) + ε in E+
Rε
\E+

R.

Then (1.8) is immediate by letting ε→ 0. �

4. Appendix

Proof of Lemma 2.1. We denote

A′(x) =

 a11(x) . . . a1,n−1(x)
...

. . .
...

an−1,1(x) . . . an−1,n−1(x)

 ,

Ã(x) =


a1,n(x)xαn

A′(x)x2α
n

...
an−1,n(x)xαn

an,1(x)xαn . . . an,n−1(x)xαn 1

 ,



12 X. B. JIA, S. S. MA EJDE-2023/33

where aij(x) and ain(x) are given by (1.1). It suffices to show that eigenvalues of

Ã(x) are positive in B
+

1 and have uniformly bound (depending on the fixed number

ε0) in B
+

1 ∩ {xn ≥ ε0}.
When A′(x) has eigenvalues λ1(x), . . . , λn−1(x), by (1.2), we obtain λ ≤ λi(x) ≤

Λ, i = 1, 2, . . . , n − 1. Then there exists a orthogonal matrix P ′(n−1)×(n−1) such
that

(P ′)TA′P ′ = diag{λ1(x), . . . , λn−1(x)}.
Observe that eigenvalues of Ã(x) are that of the matrix

B(x) := PTAP =


λ1(x)x2α

n ã1,n(x)xαn
. . .

...
λn−1(x)x2α

n ãn−1,n(x)xαn
ãn,1(x)xαn . . . ãn,n−1(x)xαn 1


with

ãi,n(x) =

n−1∑
j=1

P ′ijaj,n(x), i = 1, . . . , n− 1; P =

(
P ′ 0
0 1

)
.

Thus, we only need to show that all eigenvalues of B(x) are positive in B
+

1 and

have uniformly bound in B
+

1 ∩ {xn ≥ ε0}. For any i = 1, 2, . . . , n, let ei ∈ Rn be
the unit vector with its ith component is 1. Then

eTi B(x)ei = λix
2α
n , eTi B(x)en = ãinx

α
n, eTi B(x)ej = 0,

for i, j ≤ n− 1, and eTnB(x)en = 1.
For each ξ ∈ Rn with |ξ| = 1, there exists a unique sequence {bi}ni=1 such that

ξ =
∑n
i=1 biei and

∑n
i=1 b

2
i = 1. Then, by (1.2),

ξTB(x)ξ =

n∑
i,j=1

(biei)
TBij(x)(bjej) ≥ λx2α

n

n−1∑
i=1

b2i +

n−1∑
i=1

2bibnãi,nx
α
n + b2n.

Applying Cauchy’s inequality to 2bibnãi,nx
α
n, we have that for each τ ∈ (0, 1),∣∣ n−1∑

i=1

2bibnãi,nx
α
n

∣∣ ≤ τ n−1∑
i=1

{
λ

1
2 bix

α
n

}2

+ τ−1
n−1∑
i=1

{
λ−1/2bnãi,n

}2

= τλx2α
n

n−1∑
i=1

b2i + τ−1b2nλ
−1

n−1∑
i=1

ã2
i,n.

Therefore, for each τ ∈ (1− δ, 1),

ξTB(x)ξ ≥ λx2α
n

n−1∑
i=1

b2i + b2n − τλx2α
n

n−1∑
i=1

b2i − τ−1b2nλ
−1

n−1∑
i=1

ã2
i,n

≥ (1− τ)λx2α
n

n−1∑
i=1

b2i + b2n
{

1− τ−1(1− δ)
}

(by (1.3)),

which implies that L is elliptic in B
+

1 . And if {xn ≥ ε0}, then

ξTA(x)ξ ≥ (1− τ)λε2α
0

n−1∑
i=1

b2i + b2n{1− τ−1(1− δ)}



EJDE-2023/33 HÖLDER ESTIMATES AND ASYMPTOTIC BEHAVIOR 13

≥ min
{

(1− τ)λε2α
0 , 1− τ−1(1− δ)

}
.

In particular, taking τ = 1− 1
2δ, we have that for each x ∈ B+

1 ∩ {xn ≥ ε0},

ξTA(x)ξ ≥ min
{1

2
δλε2α

0 , 1−
(
1− 1

2
δ
)−1

(1− δ)
}
> 0.

Therefore, eigenvalues of B(x) have uniformly below bound in B
+

1 ∩ {xn ≥ ε0}.
Similarly, one can obtain the uniformly upper bound of eigenvalues of B(x) in

B
+

1 ∩ {xn ≥ ε0}. �

Acknowledgments. The authors are grateful to the referees for their valuable
comments. Also the authors want to thank the journal editor for kind help. The first
author was supported by the Natural Science Foundation of Henan Province (Grant
No. 222300420232). The second author was supported by the Natural Science
Foundation of Henan Province (Grant No. 222300420321).

References

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Xiaobiao Jia (corresponding author)

School of Mathematics and Statistics, North China University of Water Resources and
Electric Power, Zhengzhou 450046, China

Email address: jiaxiaobiao@ncwu.edu.cn

Shanshan Ma (corresponding author)

School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China

Email address: mass1210@zzu.edu.cn


	1. Introduction
	2. Proof of Theorem 1.1
	3. Proof of Theorem 1.2
	3.1. Convergence at infinity
	3.2. Asymptotic behavior at infinity

	4. Appendix
	Acknowledgments

	References