# Local Integral Estimates for Quasilinear Equations with Measure Data

Authors: Qiaoyu Tian; Shengzhi Zhang; Yonglin Xu; Jia Mu
Journal: The Scientific World Journal
 (2016)

---

## Abstract

Local integral estimates as well as local nonexistence results for a class of quasilinear equations-
Δ
p
u
=
σ
P
(
u
)
+
ω for p
>
1 and Hessian equations F
k
-
u
=
σ
P
(
u
)
+
ω were established, where σ is a nonnegative locally integrable function or, more generally, a locally finite measure, ω is a positive Radon measure, and P
(
u
)
~
exp
⁡
α
u
β with α
>
0 and β
≥
1 or P
(
u
)
=
u
p
-
1.


---

## Body

## 1. Introduction and Main Results

LetΩ
⊂
R
N be a bounded domain, let σ be a nonnegative locally integrable function or, more generally, a locally finite measure on Ω, and let ω be a nonnegative Borel measure. In this paper, we consider the following nonlinear partial differential equations with measure data:(1)
-
Δ
p
u
=
σ
P
u
+
ω
,
u
≥
0
,
u
is
p
-superharmonic  in
Ω
,where P
(
u
)
=
P
l
,
α
,
β
(
u
)
∈
L
σ
,
l
o
c
1
(
Ω
) is defined, following [1, 2], as(2)
P
l
,
α
,
β
u
=
H
l
α
u
βand l-truncated exponential function H
l
(
r
) is given by(3)
H
l
r
=
∑
j
=
l
∞
r
j
j
!
.Here, Δ
p
u is the p-Laplacian of u defined by Δ
p
u
≔
div
⁡
(
D
u
p
-
2
D
u
)
(
p
>
1
). For convenience, here and elsewhere in the paper, we assume that l
β
>
p
-
1. We will understand (1) in the following potential-theoretic sense using p-superharmonic functions (see Section 2).Quasilinear equations (1) where σ
P
(
u
) term is replaced by σ
u
q for superlinear case q
>
p
-
1 are well studied (see [3–5] and references therein, see [6, 7] for natural growth terms q
=
p
-
1, and see [8, 9] for sublinear problems q
<
p
-
1). In particular, it was shown in [3] that if u
∈
L
σ
,
l
o
c
q
(
Ω
) is a solution of (1) with nonlinear source terms σ
u
q
,
q
>
p
-
1 in Ω and B
4
R
⊂
Ω, then there is a constant M
=
M
(
N
,
p
,
q
) such that (4)
∫
0
r
σ
B
t
t
N
-
p
1
/
p
-
1
d
t
t
p
-
1
/
q
-
p
+
1
·
∫
r
R
/
2
∫
B
t
u
q
d
σ
t
N
-
p
1
/
p
-
1
d
t
t
≤
M
, for all 0
<
r
≤
R
/
4; (4) improves the related results of [10, 11].Recently, Quoc-Hung and Véron [2] obtained two-sided estimates on the solutions in terms of T-truncated η-fractional maximal potential of ω, which is suitable for dealing with exponential nonlinearities: (5)
M
α
,
T
η
=
sup
⁡
ω
B
t
t
N
-
α
h
η
t
:
0
<
t
≤
T
,where (6)
h
η
t
=
-
ln
⁡
t
-
η
χ
0,1
/
2
t
+
ln
⁡
2
-
η
χ
1
/
2
,
+
∞
t
,
η
>
0
,
1
,
η
=
0
. Some analogous estimates for Hessian equations also are given in this paper.In this paper, firstly, we will establish a priori estimates of (1) with exponential reaction P
l
,
α
,
β
(
u
), defined by (2) and (3). One of our main results are the following theorems.Theorem 1.
Letu
(
x
) be a solution of (1) in Ω with p
>
1 and l
β
>
p
-
1. Suppose that B
4
R
(
x
0
)
⊂
Ω. Then, there exists a constant M
=
M
(
N
,
p
,
α
,
β
,
l
) such that(7)
∫
0
r
σ
B
t
t
N
-
p
1
/
p
-
1
d
t
t
p
-
1
/
l
β
-
p
+
1
·
∫
r
R
/
2
ω
B
t
x
0
t
N
-
p
1
/
p
-
1
d
t
t
≤
M
,
(8)
∫
0
r
σ
B
t
t
N
-
p
1
/
p
-
1
d
t
t
p
-
1
/
l
β
-
p
+
1
∫
r
R
/
2
∫
B
t
P
l
,
α
,
β
u
d
σ
t
N
-
p
1
/
p
-
1
d
t
t
≤
M
,for all 0
<
r
≤
R
/
4.As a consequence of Theorem1, we have the following nonexistence results of local solutions to quasilinear equations.Theorem 2.
Letu be a solution of (1) in an open connected set Ω
⊂
R
N. Suppose that 1
<
p
<
n, d
σ
=
x
-
r
d
x with r
>
p, and 0
∈
Ω. Then, u
≡
0.Now, we consider (1) with natural growth terms; that is, the σ
P
(
u
) term in (1) is replaced by σ
u
p
-
1. It is worthwhile to point out that this problem turns out to be more complex than the supercritical case. The interaction between the differential operator -
Δ
p
u and the lower order term σ
u
p
-
1 was investigated by Jaye and Verbitsky [6, 7].Theorem 3.
Letu
(
x
) be a solution of (1) in Ω with p
>
1 and P
(
u
)
=
σ
u
p
-
1. Suppose that B
4
R
(
x
0
)
⊂
Ω. Then, there exists a constant M
=
M
(
N
,
p
,
α
,
β
,
l
) such that(9)
∫
0
r
σ
B
t
t
N
-
p
1
/
p
-
1
d
t
t
∫
r
R
/
2
ω
B
t
x
0
t
N
-
p
1
/
p
-
1
d
t
t
≤
M
,
(10)
∫
0
r
σ
B
t
t
N
-
p
1
/
p
-
1
d
t
t
∫
r
R
/
2
∫
B
t
u
p
-
1
d
σ
t
N
-
p
1
/
p
-
1
d
t
t
≤
M
,for all 0
<
r
≤
R
/
4.Similarly, we have the following.Theorem 4.
Letu
∈
L
l
o
c
p
-
1
(
d
σ
) be a solution of (1) with P
(
u
)
=
σ
u
p
-
1 in an open connected set Ω
⊂
R
N. Suppose that 1
<
p
<
n, d
σ
=
x
-
p
d
x, and 0
∈
Ω. Then, u
≡
0.Remark 5.
The four previous theorems are particular case of the more general class of nonlinear Wolff integral equations:(11)
u
x
=
W
α
,
p
R
P
u
d
σ
x
+
f
x
,where f
(
x
)
=
W
α
,
p
R
[
ω
]
(
x
), which includes fractional Laplacian -
Δ
γ. Therefore, we also can obtain similar results of these integral equations.The plan of the paper is as follows. In Section2, we collect some elements notions and potential estimates for p-superharmonic. Theorems 1 and 2 will be proved in Section 3. In this section, we also discuss the relations of σ and ω provided that there exist solutions of (1). After this, Section 4 presents the proof of Theorems 3 and 4 by a new iteration scheme. Section 5 is devoted to considering fully nonlinear analogues of the Dirichlet problem (1) for Hessian equations without proof.


## 2. Preliminaries

In this section, we first recall some notations and definitions. In the following, we denote byC a general constant, possibly varying from line to line, to indicate a dependence of C on the real parameters N
,
p
,
α
,
β
,
l; we will write C
=
C
(
N
,
p
,
α
,
β
,
l
). We also denote by B
(
x
0
,
r
)
=
{
x
∈
R
n
:
x
-
x
0
<
r
} the open ball with center x
0 and radius r
>
0; when it is not important or clear from the context, we shall omit denoting the center as B
r
=
B
(
x
0
,
r
).Letp
>
0 and σ be a nonnegative Borel measures in Ω which are finite on compact subsets of Ω. The σ-measure of a measurable set E
⊂
Ω is denoted by σ
(
E
)
≔
∫
E
d
σ. We denote by L
p
(
Ω
,
d
σ
)
(
L
l
o
c
p
(
Ω
,
d
σ
), resp.) the space of measurable functions f such that f
p is integrable (locally integrable) with respect to σ. When d
σ
=
d
x, we write L
p
(
Ω
) (L
l
o
c
p
(
Ω
), resp.).Forα
>
0,p
>
1, such that α
p
<
n, the R-truncated Wolff’s potential W
1
,
p
R
μ
(
x
) of a nonnegative Borel measure μ on R
N is defined by (12)
W
1
,
p
R
μ
x
=
∫
0
R
μ
B
t
t
N
-
p
1
/
p
-
1
d
t
t
. We also denote by W
1
,
p
[
μ
]
(
x
) the ∞-truncated Wolff’s potential.In this paper, all solutions are understood in the potential-theoretic sense. A lower semicontinuous functionu
:
Ω
→
(
-
∞
,
+
∞
] is called p-superharmonic if u is not identically infinite in each component of Ω, and if for all open sets D such that D
¯
⊂
Ω, and all functions h
∈
C
(
D
), p-harmonic in D, the implication holds: h
≤
u on ∂
D implies h
≤
u in D. Note that p-superharmonic function u does not necessarily belong to W
l
o
c
1
,
p
(
Ω
), but its truncation T
k
(
u
)
=
min
⁡
{
u
,
k
} does for every integer k; therefore, we will need the generalized gradient of a p-superharmonic function u defined by D
u
=
l
i
m
k
→
∞
∇
(
T
k
(
u
)
). For more properties of p-superharmonic, see [12].The following lower pointwise estimates forp-superharmonic functions play an important role in our estimate.Proposition 6 (see [13]).
There exists a positive constantC
=
C
(
N
,
p
) such that if u
≥
0 is p-superharmonic on Ω and μ
=
-
Δ
p
u, then(13)
μ
B
R
R
N
-
p
1
/
p
-
1
≤
C
inf
B
R
⁡
u
,for all balls B
R such that B
2
R
⊂
Ω.The following lemma was also proved in [13].Proposition 7.
LetB
R
(
x
) be a ball such that B
2
R
(
x
)
⊂
Ω. Then, there exists a positive constant C
=
C
(
N
,
p
) such that if u
≥
0 is p-superharmonic on Ω and μ
=
-
Δ
p
u, then(14)
C
W
1
,
p
R
μ
x
≤
u
x
,where W
1
,
p
R
[
μ
]
(
x
) is the Wolff potential of μ.Givenr
>
0
,we consider a ball B
(
x
,
2
r
)
⊂
Ω and shrinking balls B
j
≔
B
r
j
(
x
0
), where r
j
=
r
2
-
j with j
≥
0 is an integer.Proposition 8 (see Lemma2.5 in [3]).
Letμ be locally finite nonnegative measures on Ω. Then, there exists a constant C
=
C
(
N
,
p
)
>
0 such that for any s
>
0 we have(15)
ϕ
s
+
1
≤
s
+
1
W
1
,
p
R
ϕ
s
p
-
1
d
σ
,where ϕ
=
∑
j
=
0
∞
c
j
χ
B
j with c
j
=
C
σ
(
B
j
)
/
(
r
2
-
j
)
N
-
p
1
/
(
p
-
1
).The following theorem is an analogue of the above theorems fork-Hessian equations. For more details, see [14].Proposition 9.
Ifu
≥
0 is such that -
u is k-convex in Ω, then(16)
μ
B
R
/
8
R
N
-
2
k
1
/
k
≤
C
n
,
k
inf
B
R
/
8
⁡
u
,for all balls B
R, provided that B
3
R
⊂
Ω; here, μ
=
F
k
[
-
u
] is the corresponding k-Hessian measure associated with the k-convex function -
u.Proposition 10.
Letu
≥
0 be such that -
u is k-convex in Ω. Then, there is a constant C
=
C
(
n
,
k
)
>
0 such that if μ
=
F
k
[
-
u
] then(17)
C
W
2
k
/
k
+
1
,
k
+
1
R
/
8
μ
x
≤
u
x
, whenever the ball B
3
R
⊂
Ω.


## 3. Proof of Theorems1 and 2

In this section, we will give the proof of our main theorem. Firstly, we prove the following integral estimate for solutions of quasilinear equations (1), which shows that if (1) has a nontrivial p-superharmonic supersolution, then ω is absolutely continuous with respect to σ. The fact can be used to obtain a characterization of removable singularities for the homogeneous quasilinear equation: (18)
-
Δ
p
u
=
σ
P
u
,
u
≥
0
,
u
i
s
p
-superharmonic  in
Ω
,in terms of Hausdorff measures. For more details, see Theorem  2.18 in [4] and Theorem  3.1 in [3].Lemma 11.
Letσ and ω be locally finite nonnegative measures on Ω and p
>
1. There exists a constant C
=
C
(
N
,
p
,
α
,
β
,
l
)
>
0 such that if u
(
x
) is a solution to (1), then(19)
∫
B
R
P
l
,
α
,
β
u
d
σ
+
ω
B
R
≤
C
σ
B
R
p
-
1
/
l
β
-
p
+
1
R
l
β
N
-
p
/
l
β
-
p
+
1
,for all balls B
R such that B
2
R
⊂
Ω.Proof.
Define(20)
d
μ
=
σ
P
l
,
α
,
β
u
+
ω
.According to Proposition 6 and the definitions of P
l
,
α
,
β, we know that, for all x
∈
B
R,(21)
α
l
l
!
μ
B
R
R
N
-
p
l
β
/
p
-
1
≤
P
l
,
α
,
β
μ
B
R
x
R
N
-
p
1
/
p
-
1
≤
C
P
l
,
α
,
β
u
x
.Integrating both sides of (21) against d
σ over B
R, we find (22)
α
l
l
!
σ
B
R
μ
B
R
R
N
-
p
l
β
/
p
-
1
≤
C
∫
B
R
P
l
,
α
,
β
u
x
d
σ
,which combined with (20) implies that (23)
α
l
l
!
σ
B
R
μ
B
R
R
N
-
p
l
β
/
p
-
1
≤
C
μ
B
R
. This inequality is equivalent to (24)
μ
B
R
≤
C
σ
B
R
p
-
1
/
l
β
-
p
+
1
R
l
β
N
-
p
/
l
β
-
p
+
1
, which, together with (20), leads to (19).Proof of Theorem1.
For fixedx
0
∈
Ω, let R
>
0 be such that B
4
R
(
x
0
)
⊂
Ω. Suppose that u is a positive solution of (1). In view of the lower pointwise potential estimate (14), we find that, for all x
∈
B
R
(
x
0
),(25)
u
x
≥
C
W
1
,
p
R
ω
x
,
u
x
≥
C
W
1
,
p
R
P
l
,
α
,
β
u
d
σ
x
≥
C
W
1
,
p
R
α
l
l
!
u
l
β
d
σ
x
=
C
α
l
l
!
1
/
p
-
1
W
1
,
p
R
u
l
β
d
σ
x
,where C depends on N
,
p.
Restrict the integrationσ on B
R
(
x
0
) and let d
σ
′
=
χ
B
R
(
x
0
)
d
σ; thus, taking into account (25), we obtain(26)
u
x
≥
C
α
l
l
!
1
/
p
-
1
W
1
,
p
R
C
W
1
,
p
R
ω
l
β
d
σ
x
=
C
1
+
l
β
/
p
-
1
α
l
l
!
1
/
p
-
1
∫
0
R
1
t
N
-
p
∫
B
t
x
∩
B
r
x
0
W
1
,
p
R
ω
y
l
β
d
σ
y
1
/
p
-
1
d
t
t
,in view of (27)
W
1
,
p
R
ω
y
≥
C
0
N
,
p
χ
B
r
x
0
y
∫
r
R
/
2
ω
B
s
x
0
s
N
-
p
1
/
p
-
1
d
s
s
,which combined with (26) leads to the fact that, for all x
∈
B
R
(
x
0
),(28)
u
x
≥
C
1
+
l
β
/
p
-
1
α
l
l
!
1
/
p
-
1
C
0
M
x
0
,
ω
l
β
/
p
-
1
W
1
,
p
R
χ
B
R
x
0
d
σ
x
,where M
(
x
0
,
ω
) is defined as(29)
M
x
0
,
ω
=
∫
r
R
/
2
ω
B
s
x
0
s
N
-
p
1
/
p
-
1
d
s
s
.Thus, taking into account (26) and (28) and arguing by induction, we find(30)
u
x
0
≥
C
N
,
p
,
α
,
β
,
l
M
x
0
,
ω
l
β
/
p
-
1
n
M
n
1
x
0
,where M is a nonlinear integral operator defined by M
f
=
W
1
,
p
R
(
f
l
β
d
σ
′
). The iterates of M are denoted by M
i
f
=
M
(
M
i
-
1
f
). It is then easy to see from Proposition 8 that, for all s
>
0, (31)
M
1
x
0
=
W
1
,
p
R
d
σ
′
≥
ϕ
y
,
M
ϕ
s
p
-
1
y
≥
ϕ
s
l
β
+
1
y
s
l
β
+
1
,where ϕ
(
y
) appears in Proposition 8. Consequently, (32)
M
n
1
x
0
≥
∏
j
=
1
n
-
1
∑
i
=
1
j
l
β
p
-
1
i
-
l
β
/
p
-
1
n
-
j
-
1
ϕ
x
0
∑
i
=
0
n
-
1
l
β
/
p
-
1
i
,and this yields(33)
limsup
n
→
∞
⁡
M
n
1
x
0
p
-
1
/
l
β
n
≥
C
N
,
p
,
l
,
α
,
β
ϕ
x
0
p
-
1
/
l
β
-
p
+
1
.Here, we use the fact that (34)
∑
i
=
1
∞
p
-
1
l
β
i
=
p
-
1
l
β
-
p
+
1
,
if
l
β
>
p
-
1
.
In the following, we will divide the proof into two cases.
Case
1 (u
(
x
0
)
<
∞). In this case, (33), combined with (30), implies that (35)
1
≥
C
N
,
p
,
l
,
α
,
β
M
x
0
,
ω
ϕ
x
0
p
-
1
/
l
β
-
p
+
1
.Recalling that,(36)
ϕ
x
0
=
∑
j
=
0
∞
c
j
=
∑
j
=
0
∞
C
σ
B
r
2
-
j
x
0
r
2
-
j
N
-
p
1
/
p
-
1
≥
C
2
N
,
p
∫
0
r
σ
B
t
x
0
t
N
-
p
1
/
p
-
1
d
t
t
,which leads to (7).Case
2 (u
(
x
0
)
=
∞). According to [12], we know that u
<
∞ a.e. in Ω. Therefore, choose a sequence {
x
n
}
n
≥
1
⊂
B
R
/
10
(
x
0
), such that l
i
m
n
→
∞
x
n
=
x
0 and u
(
x
n
)
<
∞. Then, (7) holds with x
n in place of x
0, for all n
≥
1. Then, (7) holds by the lower semicontinuity of Wolff potentials inequality.The proof of inequality (8) is completely similarly and more details are omitted.Proof of Theorem2.
Letu be a nonnegative p-superharmonic of (1). Then, u satisfies (7), while it is well known that (37)
∫
0
r
σ
B
t
t
N
-
p
1
/
p
-
1
d
t
t
=
∞
,Provided that d
σ
=
x
-
r
d
x with r
>
p, which contradicts (7).


## 4. Proof of Theorems3 and 4

In this section, we will prove Theorem3. It is interesting to note that, in order to prove this theorem, we should give a new iterative process.Proof of Theorem3.
This proof will be divided into two parts according to the value ofp.
Case
1 (1
<
p
≤
2). For nonnegative measurable functions f, define(38)
N
f
x
=
W
1
,
p
R
f
p
-
1
d
σ
′
x
.Obviously, N is a homogeneous superlinear operator acting on nonnegative functions. Assume that u is a solution of (1); then, for all x
∈
B
R
(
x
0
),(39)
u
x
≥
C
W
1
,
p
R
u
p
-
1
d
σ
′
x
+
C
W
1
,
p
R
ω
x
=
C
N
u
x
+
C
W
1
,
p
R
ω
x
≥
C
2
N
W
1
,
p
R
ω
x
+
C
W
1
,
p
R
ω
x
,where C depends on N
,
p.
Iterating (39) n times yields(40)
u
x
≥
∑
i
=
1
i
=
n
C
i
+
1
N
i
W
1
,
p
R
ω
x
+
C
W
1
,
p
R
ω
x
.Here, we use the fact that N is a homogeneous superlinear operator and ith iterate of N is defined by N
i
(
u
)
=
N
(
N
i
-
1
(
u
)
) for i
>
1.
In the following, we will estimate the iterates ofN. Recall d
σ
′
=
χ
B
R
(
x
0
)
d
σ; thus, (41)
N
W
1
,
p
R
ω
x
=
∫
0
R
1
t
N
-
p
∫
B
t
x
∩
B
r
x
0
W
1
,
p
R
ω
y
p
-
1
d
σ
y
1
/
p
-
1
d
t
t
, in view of (42)
W
1
,
p
R
ω
y
≥
C
0
N
,
p
M
x
0
,
ω
, where M
(
x
0
,
ω
) is defined in (29). Consequently, for all x
∈
B
R
x
0, (43)
N
W
1
,
p
R
ω
x
≥
C
0
M
x
0
,
ω
W
1
,
p
R
χ
B
R
x
0
d
σ
x
≥
C
0
M
x
0
,
ω
ϕ
y
, where ϕ
(
y
) appears in Proposition 8. Obviously,(44)
N
i
W
1
,
p
R
ω
x
≥
C
0
M
x
0
,
ω
i
ϕ
i
y
i
!
.Here, the following fact has been used in this inequality: (45)
N
ϕ
s
p
-
1
y
≥
ϕ
s
p
-
1
+
1
y
s
p
-
1
+
1
.
Note thati here is arbitrary; this fact, together with (40) and (44), leads to(46)
u
x
0
≥
∑
i
=
0
i
=
∞
C
C
0
M
x
0
,
ω
ϕ
y
i
i
!
=
exp
⁡
C
C
0
M
x
0
,
ω
ϕ
y
,which, combined with (36), leads to (9) provided that u
(
x
0
)
<
∞. In a similar way, we can prove (9) if u
(
x
0
)
=
∞; more details are omitted.
Case  2 (p
>
2). A point worth emphasizing is that the operator N defined by (38) does not fall within this framework since it is not a superlinear operator. Therefore, define (47)
N
f
x
=
W
1
,
p
R
f
d
σ
′
p
-
1
x
.In this case, we have (48)
u
x
p
-
1
≥
C
p
-
1
W
1
,
p
R
ω
p
-
1
x
≥
C
C
0
M
x
0
,
ω
p
-
1
,
u
x
≥
C
W
1
,
p
R
u
p
-
1
d
σ
x
+
C
W
1
,
p
R
ω
x
. Thus, by Minkowski’s inequality,(49)
u
x
p
-
1
≥
C
p
-
1
W
1
,
p
R
u
p
-
1
d
σ
p
-
1
x
+
C
W
1
,
p
R
ω
p
-
1
=
C
p
-
1
N
u
p
-
1
x
+
C
W
1
,
p
R
ω
p
-
1
x
≥
C
2
p
-
1
N
W
1
,
p
R
ω
p
-
1
x
+
C
W
1
,
p
R
ω
p
-
1
,where C depends on N
,
p. It is clear that(50)
N
W
1
,
p
R
ω
p
-
1
x
=
∫
0
R
1
t
N
-
p
∫
B
t
x
∩
B
r
x
0
W
1
,
p
R
ω
p
-
1
d
σ
y
1
/
p
-
1
d
t
t
p
-
1
≥
C
0
M
x
0
,
ω
W
1
,
p
R
χ
B
R
x
0
d
σ
p
-
1
x
≥
C
0
M
x
0
,
ω
ϕ
y
p
-
1
,where C
0
,
M
(
x
0
,
ω
) appears in (29). Using (49) and (50), we find (51)
u
x
p
-
1
≥
C
2
C
0
M
x
0
,
ω
ϕ
y
p
-
1
+
C
C
0
M
x
0
,
ω
ϕ
y
p
-
1
.Therefore, (52)
u
x
p
-
1
≥
∑
i
=
0
i
=
∞
C
2
C
0
M
x
0
,
ω
ϕ
y
i
i
!
p
-
1
.By reverse Hölder inequality, we get (53)
u
x
≥
∑
i
=
0
i
=
∞
C
2
C
0
M
x
0
,
ω
ϕ
y
i
i
!
=
exp
⁡
C
2
C
0
M
x
0
,
ω
ϕ
y
.The following proof is similar to that of (46), so it is clear.
This finishes the proof of Theorem3.The proof of Theorem4 is standard and will be omitted.


## 5. A Fully Nonlinear Analogue: Thek-Hessian

We now move tok-Hessian operator and present fully nonlinear counterparts of the results obtained in the previous theorems. More precisely, consider fully nonlinear k-Hessian operator F
k, introduced by Trudinger and Wang [15–17]:(54)
F
k
-
u
=
σ
P
u
+
ω
,
u
≥
0
,
-
u
is
k
-convex  in
Ω
,where F
k
[
u
] denotes the k-Hessian (
k
=
1,2
,
…
,
n
),(55)
F
k
u
=
∑
1
≤
i
1
<
i
2
<
⋯
<
i
k
≤
n
λ
i
1
⋯
λ
i
k
,where λ
i
1
⋯
λ
i
k are the eigenvalues of the Hessian matrix D
2
u; that is, F
k
[
u
] is the sum of the k
×
k principal minors of D
2
u, which coincides with the Laplacian F
1
[
u
]
=
Δ
p
u if k
=
1, and the Monge-Ampère operator F
n
[
u
]
=
det
⁡
(
D
2
u
) if k
=
n.The proof of the following theorems is completely analogous to that of (1). One only needs to use Propositions 9 and 10 in place of Propositions 6 and 7, respectively, and argue as in Sections 3 and 4 with W
2
k
/
(
k
+
1
)
,
k
+
1
R in place of W
1
,
p
R. Therefore, the proof is omitted.Theorem 12.
Letu
(
x
) be a solution of (54) in Ω with p
>
1 and l
β
>
k. Suppose that B
4
R
(
x
0
)
⊂
Ω. Then, there exists a constant M
=
M
(
N
,
p
,
α
,
β
,
l
) such that (56)
∫
0
r
σ
B
t
t
N
-
p
1
/
k
d
t
t
k
/
l
β
-
k
∫
r
R
/
16
ω
B
t
x
0
t
N
-
p
1
/
k
d
t
t
≤
M
,
∫
0
r
σ
B
t
t
N
-
p
1
/
k
d
t
t
k
/
l
β
-
k
∫
r
R
/
16
∫
B
t
P
l
,
α
,
β
u
d
σ
t
N
-
p
1
/
k
d
t
t
≤
M
,for all 0
<
r
≤
R
/
32.Theorem 13.
Letu
(
x
) be a solution of (54) in Ω with p
>
1 and P
(
u
)
=
σ
u
p
-
1. Suppose that B
4
R
(
x
0
)
⊂
Ω. Then, there exists a constant M
=
M
(
N
,
p
,
α
,
β
,
l
) such that (57)
∫
0
r
σ
B
t
t
N
-
p
1
/
k
d
t
t
∫
r
R
/
16
ω
B
t
x
0
t
N
-
p
1
/
k
d
t
t
≤
M
,
∫
0
r
σ
B
t
t
N
-
p
1
/
k
d
t
t
∫
r
R
/
16
∫
B
t
u
p
-
1
d
σ
t
N
-
p
1
/
k
d
t
t
≤
M
, for all 0
<
r
≤
R
/
32.



---

*Source: 5742063-2016-05-17.xml*