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J. Nonlinear Sci. Appl. 9 (2016), 342–349
Research Article
Existence of viscosity solutions with asymptotic
behavior of exterior problems for Hessian equations
Xianyu Meng, Yongqiang Fu∗
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, P. R. China.
Abstract
The Perron method is used to establish the existence of viscosity solutions of exterior problems for a class
c
of Hessian type equations with prescribed behavior at infinity. 2016
All rights reserved.
Keywords: Hessian equation, viscosity solution, asymptotic behavior, exterior problem coincidence.
2010 MSC: 35J40, 35J60.
1. Introduction
In this paper, we study the Hessian equation
F (λ(D2 u)) = σ > 0
u = −β
x ∈ Rn \ ∂Ω,
x ∈ ∂Ω,
(1.1)
(1.2)
where σ is a constant, Ω ⊂ Rn (n ≥ 3) is a bounded domain, β is a constant, λ(D2 u) = (λ1 , λ2 , · · · , λn ) are
eigenvalues of the Hessian matrix D2 u. F is assumed to be defined in the symmetric open convex cone Γ,
with vertex at the origin, containing
Γ+ = {λ ∈ Rn : each component of λ, λi > 0, i = 1, 2, · · · , n},
and satisfies the fundamental structure conditions:
Fi (λ) =
∗
∂F
>0
∂λi
in
Γ,
1≤i≤n
Corresponding author
Email addresses: mcauchy@163.com (Xianyu Meng), fuyongqiang@hit.edu.cn (Yongqiang Fu)
Received 2015-02-26
(1.3)
X. Y. Meng, Y. Q. Fu, J. Nonlinear Sci. Appl. 9 (2016), 342–349
343
and F is a continuous concave function. In addition, F will be assumed to satisfy some more technical
assumptions, such as
F > 0 in Γ, F = 0 on ∂Γ
(1.4)
and for any r ≥ 1, R > 0
1
, r, · · · , r)) ≥ F (R(1, 1, · · · , 1)).
rn−1
For every C > 0 and every compact set K in Γ there is Λ = Λ(C, K) such that
F (R(
F (Λλ) ≥ C for all λ ∈ K.
(1.5)
(1.6)
There exists a number Λ sufficiently large such that at every point x ∈ ∂Ω, if x1 , · · · , xn−1 represent the
principal curvatures of ∂Ω, then
(x1 , · · · , xn−1 , Λ) ∈ Γ.
(1.7)
n
P
It is easy to verify that Γ ⊂ {λ ∈ Rn :
λi > 0}.
i=1
Equation (1.5) is satisfied by each kth root of elementary symmetric function (1 ≤ k ≤ n) and the
k − lth root of each quotient of kth elementary symmetric function and lth elementary symmetric function
(1 ≤ l < k ≤ n).
The Hessian equation (1.1) is an important class of fully nonlinear elliptic equations. There exist many
excellent results in the case of bounded domains, see for examples [2, 3, 7, 13, 16] and the references therein.
Caffarelli, Nirenberg and Spruck [2, 3] and Trudinger [15] established the classical solvability of the Dirichlet
problems under various hypothesis. In [11] Ivochkina, Trudinger and Wang provided a simple approach
the estimation of second derivatives of solutions. In [7] Guan studied the Dirichlet problems in bounded
domains of Riemanian manifolds. Other boundary value problems have also been considered. In [13],
Trudinger treated the Dirichlet and Neumann problems in balls for the degenerate case and in [16] Urbas
studied nonlinear oblique boundary value problems in two dimensions. But for unbounded domains there
are few results in this directions.
The study on this kind of fully nonlinear elliptic equations is close to the investigation on prescribed
curvature equations and hypersurfaces of constant curvature with boundary, see for example [8, 9, 14] and
the references therein.
When
F (λ(D2 u)) = σk (λ(D2 u)), Γ = Γk = {λ ∈ Rn : σj > 0, j = 1, 2, · · · , k},
where the kth elementary symmetric function
X
σk (λ) =
λi1 · · · λik
i1 <···<ik
for λ = (λ1 , · · · , λn ), in [6] Dai obtained the following result:
Theorem 1.1. Let k ≥ 3. Then for any C ∈ R, there exists a constant β0 ∈ R such that for any β > β0
there exists a k−convex viscosity solution u ∈ C 0 (Rn \ ∂Ω) of
σk (λ(D2 u)) = 1, x ∈ Rn \ ∂Ω
satisfying
lim sup |x|
|x|→∞
C∗ 2
u(x) − 2 |x| + C < ∞,
k−2 u = −β
where C∗ =
1
k
Cn
1
k
.
x ∈ ∂Ω
X. Y. Meng, Y. Q. Fu, J. Nonlinear Sci. Appl. 9 (2016), 342–349
344
The following theorem, which is the main result in this paper, is a generalization of Theorem 1.1 for
k-Hessian equations.
Theorem 1.2. Let k ≥ 3. Then for any C ∈ R, there exists a constant β0 ∈ R such that for any β > β0
there exists an admissible viscosity solution u ∈ C 0 (Rn \ ∂Ω) of (1.1) satisfying
R 2
n−2 lim sup |x|
u(x) − 2 |x| + C < ∞,
|x|→∞
u = −β
x ∈ ∂Ω,
where R is the constant satisfying F (R, R, · · · , R) = σ.
This paper is arranged as follows. In section 2, we give some preliminary facts which will be used later.
In section 3, we prove the main result of this paper.
2. Preliminaries
The notion of viscosity solutions was introduced by Crandall and Lions [5]. Now viscosity solution is a
rather standard concept in partial differential equations. For the completeness of this paper, we first recall
the notion of viscosity solutions.
Definition 2.1. A function u ∈ C 2 (Rn \ ∂Ω) is called admissible if λ(D2 u) ∈ Γ for every x ∈ Rn \ ∂Ω.
Definition 2.2. A function u ∈ C 0 (Rn \ ∂Ω) is called a viscosity subsolution(supersolution) to (1.1), if for
any y ∈ Rn \ ∂Ω and any admissible function ξ ∈ C 2 (Rn \ ∂Ω) satisfying
u(x) ≤ (≥)ξ(x),
x ∈ Rn \ ∂Ω,
u(y) = ξ(y),
we have
F (λ(D2 ξ(y)) ≥ (≤)σ.
Definition 2.3. A function u ∈ C 0 (Rn \ ∂Ω) is called a viscosity solution to (1.1) if it is both a viscosity
subsolution and a viscosity supersolution to (1.1).
Definition 2.4. A function u ∈ C 0 (Rn \ ∂Ω) is called a viscosity subsolution (supersolution, solution) to
(1.1)-(1.2), if u is a viscosity subsolution (supersolution, solution) to (1.1) and u ≤ (≥, =)ϕ(x) on ∂Ω.
Definition 2.5. A function u ∈ C 0 (Rn \ ∂Ω) is called admissible if for any y ∈ Rn \ ∂Ω and any function
ξ ∈ C 2 (Rn \ ∂Ω) satisfying u(x) ≤ (≥)ξ(x), x ∈ Rn \ ∂Ω, u(y) = ξ(y), we have λ(D2 ξ(y)) ∈ Γ.
It is obvious that if u is a viscosity subsolution, then u is admissible.
Lemma 2.6. Let Ω be a bounded strictly convex domain in Rn , ∂Ω ∈ C 2 , ϕ ∈ C 2 (Ω). Then there exists a
constant C only dependent on n, ϕ and Ω such that for any ξ ∈ ∂Ω, there exists x(ξ) ∈ Rn such that
|x(ξ)| ≤ C, wξ (x) < ϕ(x) for x ∈ Ω \ {ξ},
where wξ (x) = ϕ(ξ)+ R2 (|x−x(ξ)|2 −|ξ−x(ξ)|2 ) for x ∈ Rn and R is the constant satisfying F (R, R, · · · , R) =
σ.
This is a modification of Lemma 5.1 in [1].
X. Y. Meng, Y. Q. Fu, J. Nonlinear Sci. Appl. 9 (2016), 342–349
345
Lemma 2.7. Let Ω be a domain in Rn and f ∈ C 0 (Rn ) be nonnegative. Assume that the admissible
functions v ∈ C 0 (Ω), u ∈ C 0 (Rn ) satisfy, respectively,
F (λ(D2 v)) ≥ f (x)
F (λ(D2 u)) ≥ f (x)
x ∈ Ω,
x ∈ Rn .
Moreover
u ≤ v,
u = v,
Set
w(x) =
x ∈ Ω;
x ∈ ∂Ω.
v(x)
u(x)
x ∈ Ω,
x ∈ Rn \ Ω.
Then w ∈ C 0 (Rn ) is an admissible function and satisfies in the viscosity sense
F (λ(D2 w(x))) ≥ f (x),
x ∈ Rn .
Lemma 2.8. Let B be a Ball in Rn and f ∈ C 0,α (B) be positive. Suppose that u ∈ C 0 (B) satisfies in the
viscosity sense
F (λ(D2 u)) ≥ f (x), x ∈ B.
Then the Dirichlet problem
F (λ(D2 u)) = f (x)
u = u(x)
x ∈ B.
x ∈ ∂B
admits a unique admissible viscosity solution u ∈ C 0 (B).
We refer to [12] for the proof of Lemma 2.7 and 2.8.
3. Proof of Main Result
We divide the proof of Theorem 1.2 into two steps.
Step 1. By [2], there is an admissible solution Φ ∈ C ∞ (Ω) of the Dirichlet problem:
F (λ(D2 Φ)) = C0 > σ,
Φ = 0,
x ∈ Ω,
x ∈ ∂Ω.
By the comparison principles in [4], Φ ≤ 0 in Ω. Further by Lemma 2.6, for each ξ ∈ ∂Ω, there exists
x(ξ) ∈ Rn such that
Wξ (x) < Φ(x), x ∈ Ω \ {ξ},
where
Wξ (x) =
R
|x − x(ξ)|2 − |ξ − x(ξ)|2 , ξ ∈ Rn
2
and sup |x(ξ)| < ∞. Therefore
ξ∈∂Ω
Wξ (ξ) = 0, Wξ (x) ≤ Φ(x) ≤ 0, x ∈ Ω,
F (λ(D2 Wξ (x))) = F (R, R, · · · , R) = σ, ξ ∈ Rn .
Denote
W (x) = sup Wξ (x).
ξ∈∂Ω
X. Y. Meng, Y. Q. Fu, J. Nonlinear Sci. Appl. 9 (2016), 342–349
346
Then
W (x) ≤ Φ(x),
x ∈ Ω,
and by [10]
F (λ(D2 W )) ≥ σ,
Define
x ∈ Rn .
Φ(x), x ∈ Ω,
W (x), x ∈ Rn \ Ω.
V (x) =
Then V ∈ C 0 (Rn ) is an admissible viscosity solution of
F (λ(D2 V )) ≥ σ,
x ∈ Rn .
Fix some R1 > 0 such that Ω ⊂ BR1 (0) where BR1 (0) is the ball centered at the origin with radius R1 .
1
Let R2 = 2R1 R 2 . For a > 1, define
1
Z
|R 2 x|
BR1
Then
Dij Wa = (|y|n + a)
1
−1
n
1
(sn + a) n ds,
Wa (x) = inf V +
x ∈ Rn .
2R2
"
a
|y|n−1 +
|y|
#
2
aR xi xj
,
Rδij −
|y|3
|x| > 0,
1
where y = R 2 x. By rotating the coordinates we may set x = (r, 0, · · · 0). Therefore
1
a
a
,
D2 Wa = (Rn + a) n −1 Rdiag Rn−1 , Rn−1 + , · · · , Rn−1 +
R
R
where R = |y|. Consequently λ(D2 Wa ) ∈ Γ for |x| > 0 and by (1.5)
F (λ(D2 Wa )) ≥ F (R, R, · · · , R) = σ,
|x| > 0.
Moreover
Wa (x) ≤ V (x),
Fix some R3 > 3R2 satisfying
|x| ≤ R1 .
(3.1)
1
R3 R 2 > 3R2 .
We choose a1 > 1 such that for a ≥ a1 ,
Z
3R2
Wa (x) > inf V +
B R1
1
(sn + a) n ds ≥ V (x),
|x| = R3 .
2R2
Then by (3.1), R3 ≥ R1 . According to the definition of Wa ,
1
Z |R 21 x|
a n1
Wa (x) = inf V +
s
1+ n
− 1 ds +
sds
BR1
s
2R2
2R2
Z +∞ R 2
a n1
= |x| + C + inf V +
− 1 ds − C−
s
1+ n
BR1
2
s
2R2
Z +∞ a n1
2
− 2R2 −
s
1+ n
− 1 ds, x ∈ Rn .
1
s
|R 2 x|
Z
|R 2 x|
X. Y. Meng, Y. Q. Fu, J. Nonlinear Sci. Appl. 9 (2016), 342–349
Let
Z
+∞
s
µ(a) = inf V +
BR1
2R2
347
a n1
1+ n
− 1 ds − C − 2R22 .
s
Then µ(a) is continuous and monotonic increasing for a and when a → ∞, µ(a) → ∞. Moreover,
Wa (x) =
R 2
|x| + C + µ(a) − O(|x|2−n ), when
2
|x| → ∞.
(3.2)
Define, for a ≥ a1 , set β0 = µ(a) and define, for any β > β0 ,
max{V (x), Wa (x)} − β, |x| ≤ R3 ,
ua (x) =
Wa − β,
|x| ≥ R3 .
Then by (3.2),
ua (x) =
R 2
|x| + C − O(|x|2−n ), when |x| → ∞
2
and by the definition of V ,
ua (x) = −β,
x ∈ ∂Ω.
Choose a2 ≥ a1 large enough such that when a ≥ a2 ,
V (x) − β ≤ V (x) − β0
Z
+∞
= V (x) − inf V −
BR1
s
2R2
a n1
− 1 ds + C + 2R22
1+ n
s
≤C
≤
R 2
|x| + C,
2
|x| ≤ R3 .
Therefore
R 2
|x| + C, a ≥ a2 , x ∈ Rn .
2
By Lemma 2.7, ua ∈ C 0 (Rn ) is admissible and satisfies in the viscosity sense
ua (x) ≤
F (λ(D2 ua )) ≥ σ,
x ∈ Rn .
Step 2. We define the solution of (1.1) by Perron method.
For a ≥ a2 , let Sa denote the set of admissible function V ∈ C 0 (Rn ) which satisfies
F (λ(D2 V )) ≥ σ, x ∈ Rn \ ∂Ω,
V (x) = −β, x ∈ ∂Ω,
R 2
|x| + C, x ∈ Rn .
2
It is obvious that ua ∈ Sa . Hence Sa 6= ∅. Define
V (x) ≤
ua (x) = sup{V (x) : V ∈ Sa }, x ∈ Rn .
Next we prove that ua is a viscosity solution of (1.1). From the definition of ua , it is a viscosity subsolution
of (1.1) and satisfies
R
ua (x) ≤ |x|2 + C, x ∈ Rn .
2
X. Y. Meng, Y. Q. Fu, J. Nonlinear Sci. Appl. 9 (2016), 342–349
348
So we need only to prove that ua is a viscosity supersolution of (1.1) satisfying (1.2).
For any x0 ∈ Rn \ ∂Ω, fix ε > 0 such that B = Bε (x0 ) ⊂ Rn \ ∂Ω. Then by Lemma 2.8, there exists an
admissible viscosity solution u
e ∈ C 0 (B) to the princelet problem
F (λ(D2 u
e)) = σ, x ∈ B,
u
e = ua , x ∈ ∂B.
By the comparison principle in [4],
ua ≤ u
e, x ∈ B.
Define
ψ(x) =
(3.3)
u
e(x), x ∈ B,
ua (x), x ∈ Rn \ {B ∪ ∂Ω}.
By Lemma 2.7,
F (λ(D2 ψ(x))) ≥ σ, x ∈ Rn .
As
F (λ(D2 u
e)) = σ = F (λ(D2 g)), x ∈ B,
u
e = ua ≤ g, x ∈ ∂B,
where g(x) =
R
2
2 |x|
+ C, we have
u
e ≤ g, x ∈ B,
by the comparison principle in [4]. Therefore ψ ∈ Sa .
By the definition of ua , ua ≥ ψ in Rn . Consequently u
e ≤ ua in B and further u
e = ua , x ∈ B in view of
(3.3). Since x0 is arbitrary, we conclude that ua is an admissible viscosity solution of (1.1).
By the definition of ua ,
ua ≤ ua ≤ g, x ∈ Rn ,
so ua satisfies (1.2) and we complete the proof of Theorem 1.2.
Acknowledgements:
This work was supported by the National Natural Science Foundation of China (Grant No. 11371110).
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