ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO THE
CONFORMAL QUOTIENT EQUATION
BY YUNPENG WANG
A dissertation submitted to the
Graduate SchoolNew Brunswick
Rutgers, The State University of New Jersey
in partial fulllment of the requirements
for the degree of
Doctor of Philosophy
Graduate Program in Mathematics
Written under the direction of
Professor YanYan Li
and approved by
New Brunswick, New Jersey
May, 2013
ABSTRACT OF THE DISSERTATION
Asymptotic behavior of solutions to the conformal quotient
equation
by Yunpeng Wang
Dissertation Director: Professor YanYan Li
We classify all radial admissible solutions to the conformal quotient equation on the
punctured Euclidean space and prove that an admissible solution to the conformal
quotient equation with an isolated singular point is asymptotic to a radial solution. We
also provide an alternative proof to obtain higher order expansion of solutions using
analysis of the linearized operators.
This dissertation is based on a preprint of the
author [89].
ii
Acknowledgements
I would like to thank my advisor, Professor YanYan Li, for his unfailing inspiration
and patience in helping me advance.
I am very grateful for his insightful guidance
and enormous suggestions throughout my study. His expertise in mathematics, dedication to work, cordiality with people, and easygoing attitude to life have inuenced me
profoundly, and I feel very fortunate to work with him.
I would like to thank my former advisor, Professor Daomin Cao, for his continued
encouragement and help all along. I am very grateful to my thesis committee members
Professor Zheng-Chao Han, Professor Natasa Sesum, and Professor Meijun Zhu for their
help on my research and eorts to work with me. I am indebted to Professor Marcello
Lucia and Professor Xiaosen Han for their interest in this work and helpful discussions.
I would like to express my deep gratitude to all other professors who have ever taught
me mathematics.
I want to acknowledge to my fellows and friends Biao Yin, Luc Nguyen, Tianlin
Jin, Yuan Zhang, Longzhi Lin, Jian Song, Po Lam Yung, Yuan Yuan, Yu Wang, Ke
Wang, Haigang Li, Hui Wang, Jinwei Yang, Tian Yang, Zhan Li, Min Xiao, Jinggang
Xiong, Xiaoshan Li, Xukai Yan, Bin Guo, Jianguo Xiao, Liming Sun, Xiaohao Guo,
Lihua Huang, Li Li, Chunyun Li and many others at or outside Rutgers, for their help
and friendship.
I also wish to express my gratitude to my reliable friends Sharon and Stephen for
helping me get used to the life in United States and improve the writing of my thesis.
I owe a great deal to Rutgers University and the mathematics department for their
nancial support and technical assistance during my graduate study.
Finally I would like to thank my family and relatives, especially my father and
mother for always believing in me, for their endless love and their support, without
iii
which I could never have made it here. Special thanks also go to my girlfriend Michelle
for her love.
iv
Dedication
To my parents Gangqiang Wang and Huanxi Peng
v
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Classication of radial solutions to the conformal quotient equation
ii
iii
v
1
11
2.1.
Phase plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.2.
Classication of radial solutions . . . . . . . . . . . . . . . . . . . . . . .
14
3. Exploiting the ODE satised by the radial average . . . . . . . . . . .
20
3.1.
Several preliminary properties . . . . . . . . . . . . . . . . . . . . . . . .
20
3.2.
Perturbed ODEs
23
3.3.
Asymptotic behavior of singular solutions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4. Analysis of the linearized operator
. . . . . . . . . . . . . . . . .
30
. . . . . . . . . . . . . . . . . . . . .
37
4.1.
Linearization of the conformal quotient equation
. . . . . . . . . . . . .
37
4.2.
Decay rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
4.3.
Expansion in terms of Wronskian . . . . . . . . . . . . . . . . . . . . . .
46
Appendix A. Proof for the phase plane . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
54
65
71
1
Chapter 1
Introduction
One of the most important problems in conformal geometry is the Yamabe Problem,
which is to determine whether there exists a conformal metric with constant scalar
curvature on any closed Riemannian manifold. In what follows, let
be an
n-dimensional
curvature of metric
compact Riemannian manifold with metric
g0 .
some positive function
∆ g0
g0 , Rg0
and
(M n , g0 )
be the scalar
4
Denoting the conformal change of metric as
u, Rg
and
−∆g0 u +
where
n≥3
Rg0
g = u n−2 g0
are related by the equation:
n+2
n−2
n−2
Rg0 u =
Rg u n−2 ,
4(n − 1)
4(n − 1)
(1.0.1)
is the Laplace-Beltrami operator. The Yamabe problem is to solve
for a positive function
u
with
Rg ≡ c
for
for some constant
(1.0.1)
c.
The Yamabe problem has been solved through the works of Yamabe [90], Trudinger
[83], Aubin [1], and Schoen [77].
A crucial ingredient in their solution is to verify a
criteria for compactness of a minimizing sequence of the functional associated with
(1.0.1)
and in the solution by Aubin, the non-vanishing of the Weyl tensor (a local
conformal invariant) plays the role in dimensions
n≥6
for
(M n , g)
which is not locally
conformally at. The remaining cases require a global invariant and Schoen used the
Positive Mass Theorem to complete the solution. An important fact is that
the group of conformal transformations of the
non-compact.
In [79], Schoen proved that if
n-sphere Sn
(M, g)
SO(n+1, 1),
with the round metric, is
is locally conformally at and is
not conformally dieomorphic to standard spheres, then the space of all solutions to
(1.0.1)
is compact. When
(M n , g)
is not locally conformally at, the same conclusion
has been proved to hold by Li-Zhang [63] and Marques [68] independently in dimensions
n ≤ 7.
For
n = 3, 4, 5,
see works of Druet [22, 23], Li-Zhu [65] and Li-Zhang [62]. For
8 ≤ n ≤ 24, it has been proved that this compactness result holds under the assumption
2
that the Positive Mass Theorem holds in these dimensions; see Li-Zhang [63, 64] for
8 ≤ n ≤ 11,
and Khuri-Marques-Schoen [45] for
exist counterexamples in dimensions
and Marques [7] for
25 ≤ n ≤ 50.
n ≥ 25;
12 ≤ n ≤ 24.
On the other hand, there
see Brendle [6] for
n ≥ 52,
and Brendle
For the corresponding problem to prescribe scalar
curvature, see, e.g., [3, 12, 17, 18, 19, 20, 41, 46, 55, 56, 57, 75, 80].
In recent years, there are many works on studying a fully nonlinear Yamabe Problem.
We recall the Schouten tensor
1
Ag =
n−2
where
Ricg
Ricg −
g.
denotes the Ricci tensor of
Rg
g ,
2(n − 1)
This tensor arises naturally in the decompo-
sition of the full Riemannian curvature tensor
Rm = Wg + Ag g,
where
Wg
is the Weyl tensor and
Wg
denotes the Kulkari-Nomizu product [4]. Since
is conformally invariant, the behavior of the full curvature tensor under a conformal
change of metric is entirely determined by the Schouten tensor.
Let
F
denote any
symmetric function of the eigenvalues, which is homogeneous of degree one, and consider
the equation
F (g) := F g −1 Ag = constant.
Under the conformal change
g = e2ω g0 ,
(1.0.2)
the Schouten tensor transforms as
1
Ag = ∇2 ω + dω ⊗ dω − |∇ω|2 g0 + Ag0 ,
2
so equation
(1.0.2)
is equivalent to
|∇ω|2
−1
2
F g
∇ ω + dω ⊗ dω −
g 0 + Ag 0
= constant.
2
Let
for
λ = (λ1 , λ2 , · · · , λn )
1 ≤ k ≤ n, σk
be the set of eigenvalues of a symmetric
denote the
k th
n×n
matrix
A
and
elementary symmetric function of the eigenvalues
σk (λ) =
X
i1 <···<ik
λ i1 · · · λ ik ,
3
1
for the case of
F = σkk ,
the equation
(1.0.2)
has become known as the
σk -Yamabe
equation :
σk (g) = constant.
(1.0.3)
Let
n
Γ+
k = {Λ = (λ1 , λ2 , · · · , λn ) ∈ R | σj (Λ) > 0, ∀1 ≤ j ≤ k},
and we say
the
Γ+
k
g ∈ Γ+
k
if
g −1 · Ag ∈ Γ+
k
class in some region if
the linearization of
(1.0.3)
the algebraic fact that
at
e−2ω g0 ∈ Γ+
k
g
x ∈ M.
We call
there. If the metric
g
ω k -admissible
or in
k -admissible,
then
is
is elliptic, and Guan-Viaclovsky-Wang[34] also proved
λ(Ag ) ∈ Γ+
k
(see also [14, 38]). When
for every point
for
k≥
k = 1, σ1 (Ag ) =
n
2 implies the positivity of the Ricci tensor
1
2(n−1) Rg and
(1.0.3) is the Yamabe equation.
k = n, σn (Ag ) = determinant of Ag . Fully nonlinear elliptic equations involving
F λ ∇2 u have been investigated in the classic paper of Caarelli-Nirenberg-Spruck
When
[8]. Fully nonlinear elliptic equations involving the Schouten tensor and applications to
geometry and topology have been studied extensively in and after the pioneering works
of Viaclovsky [86, 87, 88] and Chang-Gursky-Yang [13, 14, 15, 16].
When
(1.0.3)
k 6=
n
2 and the manifold
(M, g)
is the Euler-Lagrange equation of the variational functional
in the exceptional case
k =
n
2 , the integral
viaclovsky [86]. We remark that when
R
σ2 (Ag )dvg
8π χ(M ) =
In the case that
σk
R
n=4
R
σk (Ag )dvg ,
and
σk (Ag )dvg
is a conformal invariant, see
k = 2,
the invariance of the integral
and
is a reection of the Chern-Gauss-Bonnet formula [4]
2
of the
is locally conformally at, the equation
(M n , g)
Z
1
( |W |2 + 4σ2 (Ag )dvg .
M 4
is locally conformally at, there is much progress on the study
equation when the conformal structure admits metrics whose Schouten tensor
belongs to the cone
Γ+
k.
This is largely due to the result of Schoen-Yau [81] which as-
sures that the developing map of locally conformally at manifolds having nonnegative
Sn ,
and in addition,
dim(Sn \Ω) ≤
n−2
2 , so that the
Yamabe invariants realizes the holonomy cover as a domain in
the complement of
Ω
has small Hausdor dimension:
method of moving planes may be used to derive a priori estimates for such equations.
In particular Li-Li [52] classied all of the solutions of the equation
(1.0.3)
on
Rn ,
thus
4
providing a priori estimates for this equation in the locally conformally at case. Subsequently, their results has been generalized to much more general classes of symmetric
functions
F,
see Li-Li [51, 53, 54]. In addition, Guan-Wang [36] applied the heat ow
associated to the
σk (k 6=
n
2 ) equation to derive the conformally invariant Sobolev in-
equality for locally conformally at manifolds.
at, the existence of solution to
When
(M n , g)
is locally conformally
(1.0.3) was obtained by Guan-Wang and Li-Li indepen-
dently in the above mentioned papers. In the case of general manifold, the existence
of solution to
(1.0.3)
was given by Chang-Gursky-Yang [14] rst for
for the existence result in arbitrary dimension
and Sheng-Trudinger-Wang [82]; When
[84]; When
k>
k=
n ≥ 3,
when
k = 2,
k = 2, n = 4.
As
see Ge-Wang [27]
n
2 , see Li-Nguyen [61] and Trudinger-Wang
n
2 , see Gursky-viaclovsky [39, 40].
The proof of existence and compactness of solutions to
(1.0.3)
involves a bubbling
analysis, and the applications of various estimate techniques to determine the growth
rate of solutions at isolated singular points. Recall that a classical theorem of Bôcher [5]
asserts that any positive harmonic function in the punctured ball
B1 \{0} ⊂ Rn
can be
expressed as the sum of a multiple of the fundamental solution of the Laplace equation
and a harmonic function in the whole unit ball
B1 .
This can be viewed as a statement on
the asymptotic behavior of a positive harmonic function near its isolated singularities.
And a remarkable work related to the Yamabe problem by Caarelli-Gidas-Spruck [9]
proved the asymptotic radial symmetry of positive singular solutions to the conformal
scalar curvature equation
−∆u(x) =
n+2
n(n − 2) n−2
u
(x)
4
(1.0.4)
on a punctured ball, and further proved that such solutions are asymptotic to radial
singular solutions to
(1.0.4)
in
and some
B1 \{0},
α>0
(1.0.4)
on
Rn \{0}.
Specically, for any singular solution
there exists a radial singular solution
u∗ (|x|)
to
(1.0.4)
on
u(x)
to
Rn \{0}
such that
u(x) = u∗ (|x|) (1 + O(|x|α ))
as
|x| → 0.
In the same paper, they also proved that any positive solution to
−∆u(x) =
n+2
n(n − 2) n−2
u
(x)
4
on
Rn
(1.0.5)
5
is of the form
ua (x) =
for some
x̄ ∈ Rn
are constant in
and
a > 0.
W 1,2 (Rn ),
2a
2
1 + a |x − x̄|2
We note
ua (x)
n−2
2
is a sequence of functions whose norm
and the weak limit of the sequence is zero; hence it does not
2n
have a convergent subsequence in
L n−2 .
This lack of compactness turns out to be at
the heart of many problems. A key ingredient in the proof of [9] uses a measure theoretic variation of the moving plane technique, while Korevaar-Mazzeo-Pacard-Schoen
[47] proved
(1.0.5)
through an analysis of linearized operators at those global singular
(1.0.1)
solutions. For the study of singular solutions to
in the positive scalar curvature
case, Schoen-Yau [81] proved that if a complete conformal metric
Ω ⊂ Sn
∂Ω
σ1 (Ag )
with
Sn \Λ
when
Λ
exists on a domain
having a positive lower bound, then the Hausdor dimension of
≤ (n − 2)/2.
has to be
g
In [78] Schoen constructed complete conformal metrics on
is either a nite discrete set on
Sn
containing at least two points or a set
arising as the limit set of a Kleinian group action. Later Mazzeo-Pacard gave another
proof of this result in [70]; they also proved in [69] that if
that
Sn \Ω
Ω ⊂ Sn
is a domain such
consists of a nite number of disjoint smooth submanifolds of dimension
1 ≤ k ≤ (n − 2)/2,
then one can nd a complete conformal metric
scalar curvature identical to
+1.
g
on
Ω
with its
For the negative scalar curvature case, the results
of Loewner-Nirenberg [66], Aviles [2], and Veron [85] imply that if
Ω ⊂ Sn
admits a
complete, conformal metric with negative constant scalar curvature, then the Hausdor
dimension of
∂Ω > (n − 2)/2.
main with smooth boundary
conformal metric
g
on
Finn [24] to the case of
Ω
Loewer-Nirenberg [66] also proved that if
∂Ω
with
∂Ω
of dimension>
σ1 (Ag ) = −1.
(n − 2)/2,
Ω ⊂ Sn
is a do-
then there exists a complete
This result was later generalized by D.
consisting of smooth submanifolds of dimension>
(n − 2)/2
and with boundary, see also [48, 49, 67, 74, 72] and the references therein.
Advancing beyond the singular Yamabe Problem case, there are some relevant research on the singular solution to
radial solutions to
(1.0.3)
in
Γ±
k
(1.0.3).
Chang-Han-Yang [10] classied all possible
class on an annular domain including punctured ball
and punctured Euclidean space. In [58], Li proved that an admissible solution with an
isolated singularity at
0 ∈ Rn
to
(1.0.2)
is asymptotically radially symmetric.
Later
6
Han-Li-Teixeira [43] studied the singular solution of
2 ≤ k ≤ n.
t = − ln r
(r, θ)
Let
(1.0.3)
be the polar coordinates such that
on a punctured ball when
x = rθ = |x|θ
with
θ ∈ Sn−1 ,
be the cylindrical variable, and
4
g = u n−2 (x)|dx|2 = e−2ω(t,θ) (dt2 + dθ2 ).
Then with the exibility of treating
or as an equation for
ω(t, θ)
the background metric
(1.0.3)
on a cylinder
dt2 + dθ2 ,
either as an equation for
{(t, θ) : t > − ln R, θ ∈ Sn−1 }
they were able to prove
|ω(t, θ) − ω ∗ (t)| ≤ Ce−αt
Also Chang-Hang-Yang [11] proved that if
metric
g
n \Ω)
< (n − 4)/2.
and
t → ∞.
and
|Rg | + |∇g Rg | ≤ c0 ,
2 < k < n/2 :
if
Ω ⊂ Sn
(1.0.6)
admits a complete, conformal
with
σ1 (Ag ) ≥ c1 > 0, σ2 (Ag ), · · · σk (Ag ) ≥ 0,
n \Ω)
then dim(S
C3
with respect to
This has been generalized by M. Gonzalez [29] and Guan-
Lin-Wang [32] to the case of
of
BR \{0}
with
then dim(S
g
for
(1.0.5)
on
Ω ⊂ Sn (n ≥ 5) admits a complete, conformal
σ1 (Ag ) ≥ c1 > 0, σ2 (Ag ) ≥ 0,
metric
u(x)
< (n − 2k)/2.
solutions of
(1.0.3)
(1.0.6)
Gonzalez also showed in [30] that isolated singularities
with nite volume are bounded, among other statements.
For the degenerate case of
solution in
and
(1.0.2),
Li [59] proved that a locally Lipschitz viscosity
Rn \{0} must be radially symmetric about {0}, see Li-Nguyen [60] for recent
development in this direction.
We studied a special case of
(1.0.2)
with
F =
σk
σl
1
k−l
,
1 ≤ l < k ≤ n,
i.e.,
(1.0.2)
is the conformal quotient equation:
σk (g)
=c
σl (g)
(1.0.7)
or, equivalently,
σk
σl
|∇ω|2
2
∇ ω + dω ⊗ dω −
g0 + Ag0 = ce2(l−k)ω ,
2
(1.0.8)
7
where
c
is a positive constant. Similar to the work [10] of Chang, Han and Yang, we
prove Theorem 2.2.1 in [89], which is the a classication result on radial solutions of
(1.0.8)
in the
Γ+
k
class on the entire
which omits the results for the case
R × Sn−1 .
2k ≥ n
We state below a part of Theorem 2.2.1,
and
h > 0.
Theorem 1.0.1. Any radial solution ξ(t) := ω(t, θ) of (1.0.8) in the Γ
+
k
class on
the entire R × Sn−1 , when 1 ≤ l < k ≤ n and c is a positive constant, normalized
to be 2l−k (nk ) / (nl ) , has the property that 1 − ξt2 > 0 for all t. Furthermore, h :=
e(2k−n)ξ(t) (1 − ξt2 (t))k − e(2l−n)ξ (1 − ξt2 (t))l is a nonnegative constant. Moreover, if
h = 0, these solutions give rise to the round spherical metric on Rn ∪ {∞} = Sn . If
h > 0 and 2k < n, then h ≤ h∗ := ( n−2k
n−2l )
gives rise to metric g =
When
l = 0, σl (g) = 1,
[10] for the
Γ+
k
dened for all
of
e−2ξ(ln|x|)
|dx|2
|x|2
then
n−2k
2
− ( n−2k
n−2l )
n−2l
2
and ξ(t) is periodic and
on Rn \{0} which is complete.
(1.0.8)
is
(1.0.3),
so our result is a generalization of
class part. We note that [10] just need the condition that
t∈R
to exclude
h<0
ξ(t)
is well
case, however, the phase plane of radial solutions
(1.0.8) is dierent from (1.0.3) and we also use the ξ(t) ∈ Γ+
k
condition to prove
h is
nonnegative in Theorem 1.0.1. This theorem also exemplies the result in Li [58] which
says any solution
ω(t, θ)
in the
Γ+
k
class to (1.0.2) must have a lower bound. Although
the analysis involved here is of elementary nature, these results provide useful guidance
in studying the behavior of singular solutions in punctured balls. Actually with some a
priori estimates and asymptotically radially symmetric properties for general conformal
equation established by Li [58] and the local rst and second order gradient estimates by
Guan-Lin-Wang [33] and Li [59], we nd that any admissible solution with an isolated
singularity at
0 ∈ Rn
is asymptotic to a radial solution of the above theorem, i.e., we
prove the following theorem in [89], which is the generalization of the results in [43].
Theorem 1.0.2. Let ω(t, θ) be a smooth solution to (1.0.8) on {t > t } × S
0
the Γ+
k class, where n ≥ 3, 1 ≤ l < k ≤ n, and c is normalized to be
2l−k (n
k)
(n
l )
n−1
in
. Then
there exist a radial solution ω ∗ (t) to (1.0.8) on R × Sn−1 in the Γ+
k class, and constants
α > 0, C > 0 such that
8
|ω(t, θ) − ω ∗ (t)| ≤ Ce−αt for t > t0 + 1.
(1.0.9)
First a linearization analysis and integral estimate showed that the radial average
of
ω(t, θ)
(denoted as
γ(t))
solves some perturbation forms of both
(2.2.1)
and
e(2k−n)ξ(t) (1 − ξt2 (t))k − e(2l−n)ξ (1 − ξt2 (t))l = h.
Note that
(1.0.10)
satised by
is the rst integral of
(2.2.1).
(1.0.10)
By exploiting the perturbed ODE
γ(t), we prove that the radial average γ(t) is approximated by a (translated)
radial solution to
(2.2.1)
as
t → ∞,
moreover, Li [58] showed that the radial average
of the solution is a good approximation to the solution as
t → ∞,
thus we arrive at
Theorem 1.0.2. We note that when the radial average is bounded, a key ingredient is
a general asymptotic approximation result for solutions of certain ODEs (see Theorem
B in Section 3.1) by Han-Li-Teixeira [43]; when the radial average is unbounded from
above and
2k > n,
a new estimate technique was provided to dig out the information
lim γt (t) = 1
t→∞
which leads to the solution of Theorem 1.0.2; also a Pohozeav type identity established
by Han [42] plays a important role in the proof especially for the
2k < n and h = 0 case
(see proof of Claim 3.2.1). The study of singular solutions of equations of the above
type is related to the characterization of the size of the limit of the image domain in
of the developing map of a locally conformally at
n-manifold.
is led to look for necessary/sucient conditions on a domain
a metric
g
More specically, one
Ω ⊂ Sn
which is pointwisely conformal to the standard metric on
with its Weyl-Schouten tensor
Ag
in the
Γ±
k
Sn
so that it admits
Sn ,
complete, and
class. Theorem 1.0.2 would also be helpful
to study the blow-up phenomenon of solutions to more general equations to prescribe
arbitrary function
f (x)
n = 4, k = 2
l=1
and
for
σk /σl .
We note that for such prescribing problem, the case
was solved in [14]. When the underlying manifold
(M, g0 )
is
locally conformally at, thanks to the work of Schoen-Yau on the developing map, one
can even solve a more general equation, see [31, 54, 37]. When
k>
n
2 , the work of [40]
and the local estimates of [33] implies the existence. See also [25, 26] for the existences
on general compact Riemannian manifold.
9
Inspired by the work of Korevaar-Mazzeo-Pacard-Schoen [47] and Han-Li-Teixeira
[43], we obtain higher order expansions for solutions to
(1.0.8)
in [89]:
Theorem 1.0.3. Let ω(t, θ) be a solution to (1.0.8) on {t > t } × S
0
n−1
in the Γ+
k class,
where n ≥ 3, 0 < l < k ≤ n2 , and the constant c is normalized to be 2l−k (nk ) / (nl ), and
let ω ∗ (t) = ξh (t + τ ) be the radial solution to (1.0.8) on R × Sn−1 in the Γ+
k class for
which (1.0.9) holds. Let {Yj (θ) : j = 0, 1, · · · } denote the set of normalized spherical
harmonics. In the case k <
n
2
and h > 0 let ρ be the inmum of the positive character-
istic exponents dened through Floquent theory to the linearized equation of (1.0.8) at
ω ∗ (t) corresponding to higher order spherical harmonics Yj (θ), j > n; in the case k =
n
2
or h = 0, a similar notion of ρ can also be dened in a straightforward fashion. Then
ρ > 1 and there is a
ω1 (t, θ) =
n
X
0
aj e−t−τ 1 + ξh (t + τ ) Yj (θ)
j=1
which is a solution to the linearized equation of (1.0.8) at ω ∗ (t), such that
|ω(t, θ) − ω ∗ (t) − ω1 (t, θ)| ≤ Ce− min{2,ρ}t for t > t0 + 1
(1.0.11)
This theorem requires some knowledge on the spectrum of the linearized operator
of
(1.0.8).
We rst establish the linearization equation and then a long algebraic com-
putation helps us to prove
ρj > 1
for all
λj ≥ 2n
as in Lemma
4.2.4
in Section 4.2.
After this, for a non-homogeneous form of the linearization equation, we apply a decomposition of the solutions with Wronskian function and Maximum principle to get
the higher order estimate, and then an elaborated iteration argument leads to Theorem
1.0.3. When we apply those analysis on the conformal quotient equation, the computation is highly more involved than the case in [43]. However, we introduce a crucial
quantity
P (ξ, ξt ) :=
e−2ξ
1−ξt2
k−l
to a quadratic function of
that if
l=0
and
q(ξ, ξt )
q(ξ, ξt ) =
P −1
to simplify the algebraic expression
1−P kl
and then the computation is straightforward. Note
as in [43], the algebraic expression would be a linear function of
q(ξ, ξt ),
and it is easy to verify Lemma 4.2.4. Also due to some new observations, we obtain the
anticipated estimate with a dierent approach (see Section 4.2 for more description).
The analysis of linearized operator should be useful in constructing solutions to
(1.0.8)
on
Sn \Λ,
and in analyzing the moduli space of solutions to
(1.0.8)
on
Sn \Λ,
10
when
Λ is a nite set.
positive
on
Actually Mazzieri-Ndiaye claimed in [73] the existence of constant
σk -curvature metrics which are complete and conformal to the standard metric
Sn \Λ, where Λ ⊂ Sn is a nite number of symmetrically balanced points of cardinality
at least 2, and
n, k
the same result for
are positive integers such that
σk /σl
2 ≤ 2k < n.
So we expect to obtain
using this linearization analysis. Also our knowledge of the
spectrum of the linearized operator to
(1.0.8)
immediately yields Fredholm mapping
properties of these operators on appropriately dened spaces as in [71, 72].
This thesis is organized as follows.
In Chapter 2, we establish a classication of
radial solutions to the conformal quotient equation. In Chapter 3, we prove Theorem
1.0.2 by exploiting the ODE satised by the radial average.
In Chapter 4, we give
proofs for Theorem 1.0.2 and Theorem 1.0.3 by an analysis of the linearized operator.
We provide a proof for the phase plane results in Appendix A.
11
Chapter 2
Classication of radial solutions to the conformal quotient
equation
2.1 Phase plane
We begin with a phase plane result and will provide a proof in the Appendix A. We
consider the level set given by
G(x, y) = e(2k−n)x (1 − y 2 )k − e(2l−n)x (1 − y 2 )l = h
in the region
Ω ≡ (x, y) ∈ R2 | |y| < 1 .
We also set
(2.1.1)
Ω− ≡ (x, y) ∈ R2 | − 1 < y < 0 ,
then
∀(x, y) ∈ Ω− .
G(x, y) = G(x, −y)
Let
Sh
denote the level set given by
(2.1.1),
then we have the following phase plane
result:
Case h=0
Set
The level set
Sh
is
1
(2.1.2)
1 − y 2 = e−2x ⇐⇒ x = − ln(1 − y 2 ).
2
W ≡ (x, y) ∈ Ω | x < − 12 ln 1 − y 2 and V ≡ (x, y) ∈ Ω | x > − 21 ln 1 − y 2 ,
then obviously
∀(x, y) ∈ Ω
solving
(2.1.1),
(x, y) ∈ W ⇐⇒ h < 0
Case h>0
(P1)
If
The level set
1≤l<k<
curve in
V.
Sh
n
2 &
we have
while
(x, y) ∈ V ⇐⇒ h > 0.
is classied in the following subcases:
0 < h < h∗ := ( n−2k
n−2l )
n−2k
2
− ( n−2k
n−2l )
n−2l
2 ,
then
Sh
is a closed
12
(P2)
If
1≤l<k<
n
2 &
h = h∗ ,
then
(P3)
If
1≤l<k<
n
2 &
h > h∗ ,
there is no solution for
(P4)
If
1≤l<k=
x
in
Ω− ,
Roughly
and
Sh
n
2 &
y(x)
is a
h < 1,
U -shaped
If
1≤l<k=
n
2 &
h ≥ 1,
(P6)
If
1 ≤ l < k,
n
2
<k
&
x
in
Roughly
Case h<0
and
Sh
y(x)
is a
is the point
(2.1.1)
1
2(k−l)
(2.1.1) denes
y := y(x) for
in
Ω−
as
large
x → ∞.
V.
We introduce the notations below:
1
1
ln
2

x̃(y) =
 v
u
u
1

u
l k−l 
∗ ∗

t
(κh , ζh ) ≡ − 1 −
k
!
( kl ) k−l
1 − y2
2l−n
2k−n
(2.1.3)
|y| < 1
(2.1.4)
1
k−l
1 − y2
2
k
k−l

n
 , − 1 ln 
l
n
− kl k−l
h
l
k
|y| < 1
v
u
 
u
−l
u

2l−n k−l
 1  h 2k−n
1
 u
u

2l−n k−l


(ζ̃h , ±κ̃h ) = − ln 
1 − 2k−n
 , ±u
t
2l−2k
 n

2k−n

2l−n
2k−n
l 2(k−l)
l 2(k−l)
~=
−
,
k
k
and the level set
If
,0 .
V.
y(x) → −1
curve that opens right in
1
x(y) =
ln
2
(N1)
y := y(x) for large
p
√
y → − 1 − k h as x → ∞.
a function
is strictly decreasing. Besides,
U -shaped
n−2l
n−2k
(2.1.1).
there is no solution for
then
(2.1.1).
curve that opens right in
h>0,
ln
denes a function
is strictly decreasing. Besides,
(P5)
Ω− ,
then
Sh
Sh
℘=
n − 2l
n − 2k
2k−n
−
k
k−l

2
 
−l
n

2l−n k−l
 h 2k−n
 

 

 
2l−2k


2k−n
2(k−l)
l
k



h

l

− kl k−l
n − 2l
n − 2k
2l−n
2(k−l)
is classied in the following subcases:
1 ≤ l < k, l <
n
2 &
h < ~,
then
Sh
intersects with the
x-axis
only
(x∗h (0), 0), and (2.1.1) denes a function y := y(x) in {(x, y) | x <
T −
l
1
Ω , which is strictly increasing. Moreover, y(x) → −1 in Ω− as
2(k−l) ln( k )}
at point
x → −∞.
Roughly
Sh
is an
U -shaped
curve that opens left in
W.
13
(N2)
If
1 ≤ l < k, l <
at point
(x∗h (0), 0)
n
2
& ~ ≤ h < 0,
satises
(N3)
If
1<l=
(N4)
If
1 < l =
n
2
<k
n
2
&
< k
axis only at point
1
2(k−l)
in
(N5)
Ω−
n
2
denes a function



 ∂x(y) > 0
∂y
if


 ∂x(y) < 0
∂y
if
h ≤ −1,
&
at
only
(ζh∗ , ±κ∗h ).
x := x(y) − 1 < y < 0
and
x(y)
− 1 < y < κ∗h
kh∗ < y < 0
l
k
−1 < h ≤ ~ =
− 1,
then
Roughly
Sh
is a
If
n
2
x(y)
If
n
2
with
&
~ < h < 0,
then



 ∂x(y) > 0
∂y
if


 ∂x(y) < 0
∂y
if
Sh
U -shaped
x-
y = y(x)
in
curve that opens left in
W.
then
−
p
√
1 − l −h < y < κ∗h
kh∗ < y < 0
Sh
is a closed curve, which intersects with
(ζh∗ , ±κ∗h ).
< l < k & ℘ < h < ~,
(2.1.3),
and all
Sh
then
(x, y) ∈ Ω
it intersects with (2.1.4) at
y = y(x),
intersects with the
satises
< l < k & ~ ≤ h < 0,
at
Sh
intersects with the
{(x, y) | x <
p
√
Moreover, y(x) → −
1 − l −h
denes a function
Ω− , which is strictly increasing there.
<k
(2.1.3)
x-axis
(2.1.1) has no solution.
(x∗h (0), 0). (2.1.1)
x → −∞.
1<l=
(2.1.3)
(N7)
as
intersects with the
x-axis only at point
p
√
(x∗h (0), 0), and in Ω− (2.1.1) denes a function x := x(y) − 1 − l −h < y < 0
If
and
(N6)
ln( kl )}
T
Sh
and it also intersects with the curve
Ω− , (2.1.1)
Moreover, in
then
is a closed curve, which doesn't intersect
solving
(ζ̃h , ±κ̃h ),
(2.1.1)
satises
x<
and in
Ω− (2.1.1)
which satises



 ∂y < 0
∂x
if
x < ζ̃h


 ∂y > 0
∂x
if
x > ζ̃h
(N8)
If
n
2
< l < k & h = ℘, Sh
(N9)
If
n
2
< l < k & h < ℘, (2.1.1)
is a point
(x̃(0), 0).
has no solution.
(x, y) ∈ Ω− .
1
2
1
ln( kl ) k−l .
Also
denes a function
14
2.2 Classication of radial solutions
ω(t, θ) =: ξ(t) is a function of t,
1 k
2
1−k n
2 k−1 k
(1 − ξt ) e2kξ .
σk (g) = 2
(k ) (1 − ξt )
ξtt +
−
n
2 n
Chang, Han and Yang [10] showed that if
then
See Chapter 1 for notations. As a computation in [43], the conformal quotient equation
σk (g)
=c
σl (g)
in the radial case becomes
21−k (nk ) (1 − ξt2 )k−1 nk ξtt +
21−l nl (1 − ξt2 )l−1 nl ξtt +
1
2
1
2
−
−
k
2
2kξ
n (1 − ξt ) e
l
2
2lξ
n (1 − ξt ) e
= c.
2l−k (nk ) / (nl ), (2.2.1) is
1 k
2
2 k−1 k
ξtt +
−
(1 − ξt ) e2kξ
(1 − ξt )
n
2 n
1
l
2 l−1 l
2
= (1 − ξt )
ξtt +
−
(1 − ξt ) e2lξ ,
n
2 n
When the positive constant
c
and multiplying both sides by
is normalized to
ne−nξ ξt ,
we get
(1.0.10),
(2.2.1)
just
i.e.,
e(2k−n)ξ(t) (1 − ξt2 (t))k − e(2l−n)ξ (1 − ξt2 (t))l = h
where
h
is constant. Since
1 − ξt2 < 0
for all
t ∈ R,
σk
σl has a xed sign on
R × Sn−1 ,
then either
1 − ξt2 > 0
or
and actually we can claim
1 − ξt2 > 0
by a contradiction argument. Indeed
2
ξ(t) ∈ Γ+
t and k ≥ 2 imply σ1 , σ2 > 0, if 1−ξt < 0,
then
2
1 2
ξtt + ( − )(1 − ξt2 ) < 0,
n
2 n
1
1 1
σ1 > 0 =⇒ ξtt + ( − )(1 − ξt2 ) > 0.
n
2 n
σ2 > 0 =⇒
Combing these two inequalities,
(1 −
n
n
)(1 − ξt2 ) > ξtt > (1 − )(1 − ξt2 ),
4
2
a contradiction. Furthermore, as a work [10] of Chang, Han and Yang, we proved the
following Theorem , which is a classication result on radial solutions of
Γ+
k
class on the entire
R × Sn−1 .
(2.2.1)
in the
15
Theorem 2.2.1. Any radial solution ξ(t) := ω(t, θ) of (2.2.1) in the Γ
+
k
class on
the entire R × Sn−1 , when 1 ≤ l < k ≤ n and c is a positive constant, normalized
to be 2l−k (nk ) / (nl ) , has the property that 1 − ξt2 > 0 for all t. Furthermore, h :=
e(2k−n)ξ(t) (1 − ξt2 (t))k − e(2l−n)ξ (1 − ξt2 (t))l is a nonnegative constant. Moreover,
h = 0,
1. If
4
u n−2 (|x|) = ( |x|22ρ+ρ2 )2
then
for some positive parameter
solutions give rise to the round spherical metric on
h > 0,
2. If
and
•
n
and
ξ(t)
Rn \{0}
metric
•
u
then
4
n−2
Rn ∪ {∞} = Sn .
is classied according to the relation between
h
has the further restriction
is a periodic function of
t,
h ≤ h∗ := ( n−2k
n−2l )
giving rise to a metric
which is complete. Note that the case
|dx|2
|x|2
2k = n,
If
So these
2k
:
2k < n,
If
u
then the behavior of
ρ.
g=
− ( n−2k
n−2l )
n−2l
2
e−2ξ(ln|x|)
|dx|2 on
|x|2
gives rise to the cylindrical
Rn \{0}.
on
then
(|x|)|dx|2
h = h∗
n−2k
2
h
satises the further restriction
h < 1
|x| → 0, g =
and as
has the asymptotic
g ∼ |x|
√
√
k
−2 1− 1− h
√
√
k
− 2 1− h t
2
|dx| = e
(dt2 + dθ2 ),
4
and as
|x| → ∞,g = u n−2 (|x|)|dx|2 has
√
√
k
−2 1+
g ∼ |x|
Thus
g
1−
h
the asymptotic
|dx|2 = e
Rn \{0}
gives rise to a metric on
√
√
k
2 1− h t
singular at
0
(dt2 + dθ2 ).
and
∞
which behaves like
the cone metric, is incomplete with nite volume.
•
If
2k > n,
4
then
u
as
|x| → 0,
limit, but
u n−2 (|x|)
4
n−2
where
(
(|x|) = ρ−2
ρ>0
urr (|x|)
has an asymptotic expansion of the form
1−
√
k
k
h·
2k − n
|x|
ρ
is a positive parameter, thus
blows up at
|x| → 0.
2− n
)
k
+ ···
u(|x|)
The behavior of
has a positive, nite
u
as
be described similarly. Putting them together, we conclude that
extends to a
n
C 2− k
metric on
Sn .
|x| → ∞
u
4
n−2
can
(|x|)|dx|2
16
Proof.
We consider
e(2k−n)ξ (1 − ξt2 )k − e(2l−n)ξ (1 − ξt2 )l = h
h as a constant, and rst show h ≥ 0 in our setting with a contradiction argument.
with
1≤l<k
When
l<
&
n
2 , let
ξ
be a solution satisfying
(2.2.1)
x∗h (0) and (−Th , Th ) be the maximal open interval of ξ .
If
and (1.0.10) with
h ≤ ( kl )
2k−n
2(k−l)
− ( kl )
then by the description for the level set in the above section, we have
limt→T ξ = −∞,
that
ξtt ≤ 0
thus
t.
for all
Th = ∞
ξt
is always decreasing as
t
|ξt | < 1
< 0,
and
increases, which implies
σk e−2ξ (dt2 + dθ2 ) ≤ 0 everywhere and
−2ξ (dt2 + dθ 2 ) > 0 for some point
Assume σk e
Now we can show that
the solution is not in the
(t0 , θ0 ) ∈ R × Sn−1 ,
and
ξ(0) =
2l−n
2(k−l)
Γ+
k
then at
class.
(t0 , θ0 ),
k
1 k
ξtt + ( − )(1 − ξt2 ) > 0.
n
2 n
However in the equality
k
1
n ξtt + ( 2
l
1
n ξtt + ( 2
21−k (nk )(1 − ξt2 )k−1
2−k (nk )
=
n
2−l (l )
21−l (nl )(1 − ξt2 )l−1
we have
e2kξ < e2lξ
and
(1 − ξt2 )k−1 < (1 − ξt2 )l−1 ,
− nk )(1 − ξt2 ) e2kξ
,
− nl )(1 − ξt2 ) e2lξ
so
k
1 k
l
1
l
ξtt + ( − )(1 − ξt2 ) > ξtt + ( − )(1 − ξt2 ),
n
2 n
n
2 n
hence
ξtt > 1 − ξt2 > 0,
2k−n
a contradiction. If
set
of
Sh
ξtt
2l−n
( kl ) 2(k−l) − ( kl ) 2(k−l) < h < 0,
intersects with
will approach
0
ξ → ζh∗ ,
and
while the other parts will approach
1
2 (1
(2.1.3)
(ζh∗ , ±κ∗h ).
then from the above section , the level
at
And as
ξt → κ∗h ,
the coecient
− kl )(1 − ξt2 )
in
(2.2.1),
thus
lim ξtt =
ξ→ζh∗
lim
1
2
−
ξ→ζh∗
k
n
1
l
2 − n
l 2(l−k)ξ
(1
ne
−
e2(l−k)ξ (1 − ξt2 )l−k
− ξt2 )l−k −
k
n
= ∞,
which implies that
0 < Th < ∞
the solution is dened all over
and
limt→T ξ(t) = ζh∗ .
R × Sn−1 .
If
ξ
This contradicts the fact that
is a solution dened on other pieces of
17
t
the level set, the maximal interval of
other cases with
(1.0.10) with
1. when
h<0
h≥0
h=0
is still nite, a contradiction. The exclusion of
is similar. Thus the solutions
ξ(t)
in our theorem must satisfy
and
we have
1 − ξt2 = e−2ξ ,
4
so the solution is
r−2 e−2ξ = u n−2 (|x|) = ( |x|22ρ+ρ2 )2
for some positive parameter
and these solutions give rise to the round spherical metric on
2. when
(a) If
h > 0,
ρ
Rn ∪ {∞} = Sn .
we consider it in the following subcases:
2k < n,
then by the phase plane results
h ≤ h∗ := ( n−2k
n−2l )
n−2k
2
− ( n−2k
n−2l )
n−2l
2 .
If
(P1)(P2)(P3)
0 < h < h∗ := ( n−2k
n−2l )
then we can verify that the solution is in the
σk e−2ξ (dt2 + dθ2 ) ≤ 0,
in Section 2.1,
Γ+
k
class.
n−2k
2
− ( n−2k
n−2l )
0 <
n−2l
2 ,
By contradiction if
then
k
1 k
ξtt + ( − )(1 − ξt2 ) ≤ 0.
n
2 n
However in the equality
21−k (nk )(1 − ξt2 )k−1
2−k (nk )
=
2−l (nl )
21−l (nl )(1 − ξt2 )l−1
we have
1 − ξt2 > e−2ξ
k
1
n ξtt + ( 2
l
1
n ξtt + ( 2
− nk )(1 − ξt2 ) e2kξ
,
− nl )(1 − ξt2 ) e2lξ
which implies that
k
1 k
l
1
l
ξtt + ( − )(1 − ξt2 ) ≥ ξtt + ( − )(1 − ξt2 ),
n
2 n
n
2 n
hence
ξtt ≥ 1 − ξt2 > 0,
which implies
1 ≤ i < k,
σk e−2ξ (dt2 + dθ2 ) > 0,
a contradiction.
Meanwhile, for any
we have
=
i
i
1
ξtt + ( − )(1 − ξt2 )
n
2 n
1
i k
1 k
i
2
ξtt + ( − )(1 − ξt ) +
−
1 − ξt2
k n
2 n
2 2k
> 0,
18
so
σi e−2ξ (dt2 + dθ2 ) > 0.
periodic function of t, giving rise to a metric
is complete. Note that the case
on
(b) If
ξ
Thus the solution
h = h∗
is in the
class and
e−2ξ(ln|x|)
|dx|2 on
|x|2
g=
ξ(t)
is a
Rn \{0} which
gives rise to the cylindrical metric
|dx|2
|x|2
Rn \{0}.
2k = n,
(P4)(P5)
then by the phase plane results
h < 1. Furthermore Th = ∞
p
√
ξt → 1 − k h, thus
q
√
k
ξ∼
1− h t
the further restriction
as
Γ+
k
t → ∞, ξ → ∞
&
which implies when
|x| ∼ 0,
in Section 2.1,
ξ
and
is the
Γ+
k
h
satises
class. Also
we have
√ √
k
g ∼ |x|−2(1− 1− h) |dx|2 .
Similarly when
|x| ∼ ∞,
we have
√
−2(1+
g ∼ |x|
1−
√
k
h)
These are incomplete, nite volume metrics on
metrics on
(c) If
2k > n,
is in the
then by the phase plane result (P6) in Section 2.1,
Γ+
k
>0
Rn \{0} , corresponding to conical
Sn \{0, ∞}.
class. Also as
t → ∞, ξ → ∞
ζ(t) := e(2l−n)ξ (1 − ξt2 )l → 0 as t → ∞,
some
|dx|2 .
such that as
&
ξt → 1 .
or else there exists sequences
ζ(tj ) = e(2l−n)ξ(tj ) (1 − ξt2 (tj ))l > .
ζ(t),
we get
1
n
e2ξ(tj ) (1 − ξt2 (tj )) = ζ(tj ) l e l ξ(tj ) ,
plugging it into the curve
h
(1.0.10),
then as
k
=
ζ(tj ) l e(
>
l e(
k
→ ∞,
tj → ∞
nk
−n)ξ(tj )
l
nk
−n)ξ(tj )
l
−
− ζ(tj )
and
ξ
Moreover, we can show
tj → ∞,
However, from the denition of
Th = ∞
{tj }1≤j
and
19
a contradiction. Thus as
t→∞
1 − ξt2 ∼
√
k
√
k
1 − ξt ∼
he
n−2k
ξ
k
n−2k
ξ
k
he
2
,
,
t → ∞,
√
k
n−2k
k
h
·
e k ξ + h.o.t.
ξ−t=c+
2 2k − n
and taking integral, we get as
4
u n−2 (|x|) has
(
√
4
k
u n−2 (|x|) = ρ−2 1 − h ·
for some constant
as
|x| → 0,
where
c.
Then
ρ = ec > 0
an asymptotic expansion of the form
k
2k − n
|x|
ρ
2− n
)
k
+ ···
is a positive parameter. The analysis near
can be carried out in a similar way. Thus we conclude that
a
C
2− n
k
metric on
Sn .
v −2 |dx|2
x
at
∞
extends to
20
Chapter 3
Exploiting the ODE satised by the radial average
3.1 Several preliminary properties
In this section we present some needed preliminary properties for solutions derived
in [58]. Recall that under the Euclidean coordinates, we introduce
conformal change of metric
g
u(x)
to write the
as
4
g = u n−2 (x)|dx|2 = e−2ω(t,θ) (dt2 + dθ2 ).
Then (1.0.7) is equivalent to
σk −(n − 2)u(x)∇2 u(x) + n∇u(x) ⊗ ∇u(x) − |∇u(x)|2 Id
σl (−(n − 2)u(x)∇2 u(x) + n∇u(x) ⊗ ∇u(x) − |∇u(x)|2 Id)
2kn
=c·
2k · (n − 2)−2k · u n−2 (x)
2ln
.
(3.1.1)
2l · (n − 2)−2l · u n−2 (x)
The following theorem is drawn from Theorem
0
1.1
and
1.3
in [58].
Theorem A (Y. Y. Li). Suppose that u ∈ C (B \{0}) is a positive solution to (3.1.1)
2
2
on BR \{0}, then
lim sup |x|
n−2
2
u(x) < ∞,
(3.1.2)
x→0
and there exists some constant C > 0 such that
|u(x) − ū(|x|)| ≤ C|x| · ū(|x|)
for all 0 < |x| ≤ 1, where
ū(|x|) =
1
|∂B|x| (0)|
Z
u(y)dσ(y)
∂B|x| (0)
is the spherical average of u(x) over ∂B|x| (0).
(3.1.3)
21
As in the previous section, in terms of
U (t, θ) = r
β(t) = |S
n−2
2
t = − ln r = − ln |x|,
u(rθ) = e−
n−1 −1
let
n−2
ω(t,θ)
2
Z
|
U (t, θ)dθ
S n−1
γ(t) = |S
n−1 −1
Z
|
ω(t, θ)dθ,
(3.1.4)
e−2ω(t,θ) ≤ C,
(3.1.5)
S n−1
then the inequality
(3.1.2)
is reformulated as
U (t, θ) ≤ C
and
plus
|U (t, θ) − β(t)| ≤ Cβ(t)e−t ,
and
|ω̂(t, θ)| := |ω(t, θ) − γ(t)| ≤ Ce−t .
We also have the gradient estimate for positive singular solutions
to
(3.1.1)
on
(3.1.6)
u(x)
in the
Γ+
k
class
BR (0)\{0}.
Proposition 3.1.1. Let u(x) be a positive singular solution to (3.1.1) on B \{0} in
R
the Γ+
k class, U (t, θ), β(t), ω(t, θ) and γ(t) be dened above. Then for any 0 < δ small,
there exists a constant C > 0 depending on δ such that
|∇jt,θ (U (t, θ) − β(t))| ≤ Cβ(t)e−(1−δ)t
|∇jt,θ (ω(t, θ) − γ(t))| ≤ Ce−(1−δ)t
(3.1.7)
for all t ≥ 0 and j = 1, 2.
We now provide an argument for (3.1.7). First,
for solutions to
(3.1.7),
(1.0.8),
see [33], give a bound
(3.1.5)
B > 0
and the gradient estimates
depending on
j > 0
and
C
in
such that
|∇jt,θ ω(t, θ)| ≤ B,
(3.1.8)
22
which leads to
|∇jt,θ γ(t)| ≤ B.
(3.1.9)
Combining them together, we get
|∇jt,θ (ω(t, θ) − γ(t))| ≤ 2B
This estimate, together with
(3.1.6)
and interpolation, proves
(3.1.7).
We also quote Theorem 3 in [43] here, which is an elaborated inductive argument
and plays a key role in proving Theorem 1.0.2.
Theorem B (Z.-C. Han-Y. Y. Li-E. V. Teixeira). Consider a solution β(t) to
β 00 (t) = f (β 0 (t), β(t)) + e1 (t) t ≥ 0
where f is locally Lipschitz, and ei (t) is considered as a perturbation term with e1 (t) → 0
as t → ∞ at a suciently fast rate to be specied later. Suppose that |β(t)| + |β 0 (t)| is
bounded over t ∈ [0, ∞). Then by a compactness argument there exists a sequence of
ti → ∞ and a solution ψ(t) to
ψ 00 (t) = f (ψ 0 (t), ψ(t))
(3.1.10)
which exists for all t ∈ R such that
1
(−∞, ∞) as i → ∞
β(ti + ·) → ψ in Cloc
(3.1.11)
Suppose ψ(t) is a periodic solution with T ≥ 0, thus for some nite m ≤ M
ψ(R) = [m, M ].
We may do a time translation for ψ(t) so that ψ(0) = m, ψ 0 (0) = 0, then the approximation property (3.1.11) can be reformulated as, for some s,
1
β(ti + ·) − ψ(−s + ·) → 0 in Cloc
(−∞, ∞) as i → ∞
Suppose that (3.1.10) has a rst integral in the form of
H(ψ 0 (t), ψ(t)) = 0
for some continuous function H(x, y), where H satises the following non-degeneracy
condition, depending on
23
1. (ψ(t) := m is a constant ): there exist some 1 > 0, A > 0 and l > 0
H(x, y) ≥ A(|x|l + |y − m|l )
(3.1.12)
for any (x, y) with |x| + |y − m| ≤ 1 ;
2. (ψ(t) is non-constant): there exists some 1 > 0, A > 0 and l > 0,
|H(0, y)| = |H(0, y) − H(0, m)| ≥ A|y − m|l
(3.1.13)
for any y with |y − m| ≤ 1 .
Suppose also that β(t) has H as an approximate rst integral
|H(β 0 (t), β(t))| ≤ e2 (t) for t ≥ 0
where e2 (t) → 0 as t → ∞. Without loss of generality, we may suppose that e2 (t) is
monotone non-increasing in t. Finally suppose that
Z
∞
1
((e2 (t)) l + supτ ≥t |e1 (τ )| dt < ∞
0
Then there exists some s∞ and C > 0 such that
|β(t) − ψ(t − s∞ )| + |β 0 (t) − ψ 0 (t − s∞ )|
Z ∞
1
≤ C
(e2 (t0 )) l + supτ ≥t0 |e1 (τ )| dt0 → 0 as t → ∞.
t−(T +2)
3.2 Perturbed ODEs
Let
c
ω(t, θ) be a positive solution of (1.0.8) on BR \{0} in the Γ+
k class, where the constant
is normalized to be
mathn
3.2.1
h
where
.
If
2l−k (nk ) / (nl ),
and
γ(t)
is dened in
(3.1.4).
We can make
1 ≤ l < k ≤ n, there exists some constant h such that
n − 2k
2 k−1 k
2
γtt +
(1 − γt ) + η1 (t) e2kγ
2(1 − γt )
n
2n
n − 2l
2
2 l−1 l
= 2(1 − γt )
γtt +
(1 − γt ) + η2 (t) e2lγ ,
n
2n
i
h
i
(1 − γt2 )k + η3 (t) · e(2k−n)γ − (1 − γt2 )l + η4 (t) · e(2l−n)γ = h
ηi (t),
for
arbitrarily small
i = 1, · · · 4,
δ>0
have the decay rate
as in (3.1.7).
ηi (t) = O(e−2(1−δ)t )
(3.2.1)
(3.2.2)
as
t → ∞,
for
24
Proof.
We rst prove
2k ·
1
n
k
(3.2.1).
By equation
(1.0.7)
σk (g0−1 · Ag ) · e2kω(t,θ) = 2l ·
we have
1
n
l
σl (g0−1 · Ag ) · e2lω(t,θ) .
Let
σk (Aω ) := σk (g0−1 · Ag )
be a functional of
ω(t, θ),
then with
ω̂(t, θ) = ω(t, θ) − γ(t),
we have the following
expansion
σk (Aω ) = σk (Aγ(t) ) + Lγ(t) [ω̂(t, θ))] + η̂1 (t, θ)
where
Lγ(t)
denotes the linearized operator for
|η̂1 (t, θ)| = O(e−2(1−δ)t )
as
t→∞
by
(3.1.7)
σk (Aω(t,θ) )
and
(3.1.9).
at
γ(t),
and
η̂1 (t, θ)
satises
Next,
e2kω(t,θ) = e2kγ(t) · e2kω̂(t,θ)
and
e2kω̂(t,θ) = 1 + 2k ω̂(t, θ) + η̂2 (t, θ)
where
|η̂2 (t, θ)| = O(e−2t )
as
t→∞
since
(3.1.6).
Putting them together, we have
σk (g0−1 · Ag ) · e2kω(t,θ)
σk (Aγ(t) ) + Lγ(t) [ω̂(t, θ))] + η̂1 (t, θ) (1 + 2k ω̂(t, θ) + η̂2 (t, θ)) · e2kγ(t)
(
=
= e2kγ(t) σk (Aγ(t) ) + Lγ(t) [ω̂(t, θ)] + η̂1 (t, θ) + σk (Aγ(t) ) · 2k ω̂(t, θ)
+Lγ(t) [ω̂(t, θ)] · 2k ω̂(t, θ) + η̂1 (t, θ) · 2k ω̂(t, θ) + σk (Aγ(t) )η̂2 (t, θ)
)
+Lγ(t) [ω̂(t, θ)]η̂2 (t, θ) + η̂1 (t, θ)η̂2 (t, θ) .
Integrating over
1
|Sn−1 |
θ ∈ Sn−1 ,
we have
Z
Sn−1
σk (Aγ(t) )dθ =
1
21−k (nk ) (1
−
γt2 )k−1
k
n − 2k
γtt +
(1 − γt2 )
n
2n
Z
L [ω̂(t, θ))]dθ = 0
|Sn−1 | Sn−1 γ(t)
Z
1
σk (Ar(t) ) · 2k ω̂(t, θ)dθ = 0
|Sn−1 | Sn−1
1
|Sn−1 |
Z
Sn−1
ˆ θ)dθ = O(e−(2−δ)t )
Lγ(t) [ω̂(t, θ)] · 2k ω(t,
25
1
Z
η̂1 (t, θ) · 2k ω̂(t, θ)dθ = O(e−(3−2δ)t )
|Sn−1 | Sn−1
Z
1
σk (g0−1 · Aγ(t) )η̂2 (t, θ)dθ = O(e−2(1−δ)t )
|Sn−1 | Sn−1
Z
1
L [ω̂(t, θ)]η̂2 (t, θ)dθ = O(e−3(1−δ)t )
|Sn−1 | Sn−1 γ(t)
Z
1
η̂1 (t, θ)η̂2 (t, θ)dθ = O(e−4(1−δ)t )
|Sn−1 | Sn−1
as
t → ∞.
Similarly we have the same estimate for the right side, and thus we have
(3.2.1).
Next we are ready to prove
(3.2.2)
follows from
(3.2.1).
(3.2.2).
We use
C
First we show that, when
or
h 6= 0,
to denote a positive constant, which may vary
according to the context. Multiplying both sides of
h
2k ≥ n
e(2k−n)γ(t) (1 − γt2 )k − e(2l−n)γ(t) (1 − γt2 )l
i
t
(3.2.1)
by
ne−nγ(t) γt (t),
one has
h
i
= ne−nγ(t) γt (t) e2kγ(t) η1 (t) − e2lγ(t) η2 (t) .
(3.2.3)
By the estimates
(3.1.5) and (3.1.6), γ(t) ≥ −C ∀t ∈ [0, ∞).
are two possibilities:
γ(t) ≤ C , ∀t ∈ [0, ∞),
or
For its upper bound, there
supt∈[0,∞) γ(t) = ∞.
We consider them
separately:
•
If
γ(t) ≤ C , ∀t ∈ [0, ∞), then the right hand of (3.2.3) is of the order O(e−2(1−δ)t )
from the gradient estimates
that for some constant
h,
(3.1.9),
and the decay rates of
η(t).
It then follows
we have
e(2k−n)γ(t) (1 − γt2 )k − e(2l−n)γ(t) (1 − γt2 )l = h + O(e−2(1−δ)t ),
thus (3.2.2) holds.
•
If
supt∈[0,∞) γ(t) = ∞ and 2k > n, we claim that γ(t) is strictly increasing when t
is large enough, or else there must exist an increasing sequence
tj → ∞ & γ(tj ) → ∞
as
j → ∞,
γ(tj ) = max γ(t), & γ(tj ) > 2γ(tj−1 )
t∈[0,tj ]
and
γt (tj ) = 0 ∀j ∈ N.
{tj }j∈N
∀j ≥ 2,
such that
26
By taking the integral on
[tj−1 , tj ]
for both sides of
(3.2.3),
we have
k
l t
e(2k−n)γ(t) 1 − γt2 (t) − e(2l−n)γ(t) 1 − γt2 (t) |tjj−1
Z tj
h
i
=
ne−nγ(s) γs (s) e2kγ(s) η1 (s) − e2lγ(s) η2 (s) ds.
(3.2.4)
tj−1
However, as
j → ∞,
the left side of equation (3.2.4) satises
h
k
l i
e(2k−n)γ(tj ) 1 − γt2 (tj ) − e(2l−n)γ(tj ) 1 − γt2 (tj )
h
k
l i
− e(2k−n)γ(tj−1 ) 1 − γt2 (tj−1 ) − e(2l−n)γ(tj−1 ) 1 − γt2 (tj−1 )
h
i h
i
= e(2k−n)γ(tj ) − e(2l−n)γ(tj ) − e(2k−n)γ(tj−1 ) − e(2l−n)γ(tj−1 )
>
1 (2k−n)γ(tj )
e
,
2
(3.2.4)
while the right hand side of
Z
tj
satises
h
i
ne−nγ(s) γs (s) e2kγ(s) η1 (s) − e2lγ(s) η2 (s) ds
tj−1
Z tj
≤ C
e(2k−n)γ(s) |η1 (s)| + e(2l−n)γ(s) |η2 (s)| ds
tj−1
≤ Ce(2k−n)γ(tj )
Z
tj
e−2(1−δ)s ds
tj−1
(2k−n)γ(tj ) −2(1−δ)tj−1
≤ Ce
<
e
1 (2k−n)γ(tj )
e
,
2
a contradiction. Hence when
to
∞,
t is large enough, γ(t) is strictly increasing and goes
and with a similar argument, we can show that
Indeed, if there exists
>0
and an increasing sequence
t̄j
1 − γt2 (t̄j )
1 − γt2 → 0
{t̄j }j∈N
∀j ∈ N
γ(t̄j ) → ∞,
then by taking integral on
[t̄j−1, t̄j ],
we get
k
l t̄
e(2k−n)γ(t) 1 − γt2 (t) − e(2l−n)γ(t) 1 − γt2 (t) |t̄jj−1
Z t̄j
h
i
=
ne−nγ(s) γs (s) e2kγ(s) η1 (s) − e2lγ(s) η2 (s) ds,
t̄j−1
t → ∞.
such that
→ ∞
>
as
27
which leads to
k (2k−n)γ(t̄j )
e
< Ce(2k−n)γ(t̄j ) e−2(1−δ)t̄j
2
as
j → ∞,
a contradiction. Thus
1 − γt2 → 0
as
t → ∞,
consequently,
lim γt (t) = 1.
t→∞
Now let
(n−2k)γ(t)
Z
t
ne
η3 (t) = e
−nγ(s)
h
2kγ(s)
γs (s) e
2lγ(s)
η1 (s) − e
i
η2 (s) ds,
0
then clearly
k
l i
d h (2k−n)γ(t) = 0,
e
1 − γt2 (t) + η3 (t) − e(2l−n)γ(t) 1 − γt2 (t)
dt
so there exists a constant
e(2k−n)γ(t)
Moreover, as
h
such that
k
l
1 − γt2 (t) + η3 (t) − e(2l−n)γ(t) 1 − γt2 (t) = h.
t → ∞,
(n−2k)γ(t)
t
Z
|η3 (t)| ≤ Ce
e(2k−n)γ(s) · e−2(1−δ)s ds
0
≤ Ce(n−2k)(1−)t
Z
t
e(2k−n)(1+)s · e−2(1−δ)s ds
0
−2[1−(δ+(2k−n))]t
≤ Ce
where
•
If
δ
and
,
could be arbitrarily small, so we have
supt∈[0,∞) γ(t) = ∞,
and
2k < n,
(3.2.2)
with
η4 = 0.
it then follows that for some constant
h,
we
have
e(2k−n)γ(t) (1 − γt2 )k − e(2l−n)γ(t) (1 − γt2 )l = h + O(e−2(1−δ)t ).
If
h 6= 0,
then
γ(t)
is strictly increasing and
an increasing sequence
{tj }j∈N
limt→∞ γ(t) = ∞,
such that
tj
γt (tj )
→ ∞ & γ(tj ) → ∞
=
0
∀j ∈ N.
(3.2.5)
or else there exists
28
Then as
j → ∞,
the left side of
(3.2.5)
e(2k−n)γ(t) (1 − γt2 )k − e(2l−n)γ(t) (1 − γt2 )l → 0
while the right side of
(3.2.5)
argument as above we can get
(n−2k)γ(t)
Z
∞
η3 = −e
is
h,
a contradiction.
(3.2.2)
Consequently by a similar
with
h
i
ne−nγ(s) γs (s) e2kγ(s) η1 (s) − e2lγ(s) η2 (s) ds
t
and
•
If
η4 = 0.
2k = n,
it is a similar argument as the case
However we are unable to get
(3.2.2)
from
γ(t) ≤ C
(3.2.1)
on
[0, ∞).
directly when
2k < n
and
and have to utilize an approach with Pohozaev identity from [42]. Let us assume
n 6= 2l
&
and during this proof we use
rate
η(t) = O(e−2(1−δ)t )
and
T̄ab
as
t → ∞,
η(t)
h=0
n 6= 2k
to denote a quantity which has the decay
and it may vary according to the context. Let
denote the components of Newton Tensor
Tk−1
and
Tl−1
Tab
respectively, then by
[42], we have the following identity:
1
∇a Tba ∇b (divX) + 2kσk X a
2k − n
1
< X, ∇σl >=
∇a T̄ba ∇b (divX) + 2lσl X a ,
2l − n
< X, ∇σk >=
so let
φt denote the local one-parameter family of conformal dieomorphism of (M, g)generated
X,
by
then from the equation (1.0.7)
d
|t=0 (σk − cσl )
dt
0 =
< X, ∇(σk − cσl ) >
= < X, ∇σk > −c < X, ∇σl >
1
c
=
∇a Tba ∇b (divX) + 2kσk X a −
∇a T̄ba ∇b (divX) + 2lσl X a .
2k − n
2l − n
Now taking
Z
Sn−1
M = R × Sn−1 , g = e−2ω(t,θ) (dt2 + dθ2 )
and
X = ∂t ,
we get
1 c 2kω
2lω
2kσk − nT1b ωtb e
−
2lσl − nT̄1b ωtb e
e−nω dθ = h
2k − n
2l − n
(3.2.6)
29
h is a constant. Let L̃γ(t) and L̄γ(t) denote the linearized operators for
Pn
b=1 T̄b1 ωbt at γ(t), then a computation in [43] shows that
where
and
n
X
b=1 Tb1 ωbt
Tb1 ωbt =
2k −k n
· 2 (k )(1 − γt2 )k−1 γtt + L̃γ [ω̂(t, θ)] + O(e−2(1−δ)t ),
n
T̄b1 ωbt =
2l −l n
· 2 (l )(1 − γt2 )l−1 γtt + L̄γ [ω̂(t, θ)] + O(e−2(1−δ)t ).
n
b=1
n
X
Pn
b=1
So we can write
(3.2.6)
as
h=E+F +G
where
Z
E =
1 (1 − γt2 )k−1 γtt e2kω e−nω · 2k · 2−k (nk )dθ
2k − n
n−1
SZ
c +
(1 − γt2 )l−1 γtt e2lω · e−nω · 2l · 2−l (nl )dθ,
Sn−1 2l − n
−
Z
F
=
Sn−1
2k
2l
−
2k − n 2l − n
· σk e−nω dθ,
−n L̃γ [ω̂(t, θ)] + η(t) e(2k−n)ω dθ
n−1 2k − n
SZ
cn
+
L̄γ [ω̂(t, θ)] + η(t) e(2l−n)ω dθ.
Sn−1 2l − n
Z
G =
Clearly
G = η(t) · e(2k−n)γ(t) + η(t) · e(2l−n)γ(t) ,
and
n(2l − 2k)
σk e−nω dθ
Sn−1 (2k − n)(2l − n)
Z
2n(l − k)2−k (nk )
=
Sn−1 (2k − n)(2l − n)
n − 2k
2 k−1 k
2
× 2(1 − γt )
γtt +
(1 − γt ) + η(t) e(2k−n)γ dθ,
n
2n
Z
F
and by
=
(3.2.1),
−2k
E =
e
·
(1 − γt2 )k−1 γtt e2kγ dθ
2k
−
n
n−1
S
Z
2l
−nγ
−k n
+
e
· 2 (k )
(1 − γt2 )l−1 γtt e2lγ dθ
2l
−
n
n−1
ZS
+
η(t) · e(2k−n)γ(t) + η(t) · e(2l−n)γ(t) dθ
Z
−nγ
Sn−1
2−k (nk )
30
−2n
e−nγ · 2−k (nk )
2k − n
Sn−1
n − 2k
k
2 k−1
2 k
1 − γt
γtt +
1 − γt
e2kγ dθ
×
n
2n
Z
k
−
e−nγ · 2−k (nk ) · 1 − γt2 e2kγ dθ
n−1
ZS
2n
−nγ
−k n
e
· 2 (k )
+
2l − n
Sn−1
2lγ
n − 2l
l
2 l−1
2 l
1 − γt
γtt +
1 − γt
e dθ
×
n
2n
Z
l
+
e−nγ · 2−k (nk ) · 1 − γt2 e2lγ dθ
Z
=
Sn−1
+ η(t) · e(2k−n)γ(t) + η(t) · e(2l−n)γ(t)
Z
2n
2n
(2k−n)γ
−k n
=
e
· 2 (k ) ·
−
2l − n 2k − n
Sn−1
n − 2k
2 (k−1) k
2
× 1 − γt
γtt +
(1 − γt ) dθ
n
2n
Z
i
h
+
2−k (nk ) − 1 − γt2 k e(2k−n)γ + 1 − γt2 e(2l−n)γ dθ
Sn−1
+ η(t) · e(2k−n)γ(t) + η(t) · e(2l−n)γ(t)
Z
= −F +
2−k (nk )
n−1
i
h S
× (1 − γt2 )k + η(t) e(2k−n)γ − (1 − γt2 )l + η(t) e(2l−n)γ dθ.
Combining them together, we have
h
i
h
i
(1 − γt2 )k + η3 (t) · e(2k−n)γ − (1 − γt2 )l + η4 (t) · e(2l−n)γ = h
for some constant
h,
where
η3 (t)
and
η4 (t)
have the decay rate
O e−2(1−δ)t
as
t → ∞.
Combing them together, we have (3.2.2) in all cases.
3.3 Asymptotic behavior of singular solutions
Now we are ready to prove Theorem 1.0.2. We rst establish the following claim, which
help us to exclude the
2k < n
&
h<0
case, then we apply Theorem B to (3.2.1) and
(3.2.2) to describe asymptotic behavior of singular solutions.
31
mathn
ball
.
3.3.1
BR \{0},
u(x)
Let
n−2
2
x→0
2k < n,
Especially in the case of
If
in the
Γ+
k
class in a punctured
then
lim inf |x|
•
(3.1.1)
be a positive solution to
h > 0,
we always have
lim inf x→0 |x|
then
u(x) > 0 =⇒ 2k < n
n−2
2
u(x) > 0
and
h ≥ 0,
h > 0.
and
and for some
> 0,
we have
1 − γt2 ≥ for all suciently large
•
If
Proof.
h = 0,
If
γt (t) > 0
then
lim inf x→0 |x|
n−2
2
t.
for
t
large, and
u(x) > 0,
then by
limt→∞ γ(t) = ∞.
(3.1.5)
and
(3.1.6),
we have
−C ≤ γ(t) ≤ C.
So by a rescaling and compactness argument on the translations to
of
(3.1.8)
t∈R
and
(3.2.2),
One produces a limiting
(2.2.1)
and satises the equation
γ̂(t)
γ(t),
with the help
which exists and is bounded for all
with
e(2k−n)γ̂(t) (1 − γ̂t2 (t))k − e(2l−n)γ̂(t) (1 − γ̂t2 )l = h
for some
h.
But according to Theorem 2.2.1, the classication result says that no
bounded admissible solution of
implies
that
h>0
and
1 − γt2 > 2k < n.
(2.2.1)
And from
exists for all
(3.2.2),
lim inf x→0 |x|
with
there is some
h≤0
>0
or
2k ≥ n,
depending on
n−2
2
u(x) = 0
lim sup γ(t) = ∞.
2k < n,
it follows from
follows from above that
for suciently large
we see that
γt (t) > 0
(3.1.8)
lim inf x→0 |x|
t, γt (t) = 0
h
such
then
(3.3.1)
x→0
So when
which
t.
for all suciently large
On the other hand, if
t∈R
and
n−2
2
(3.2.2)
u(x) = 0.
that
t
and
Conversely if
In addition, it follows
can occur only near
for suciently large
h = 0.
γ(t) = 0.
h = 0,
(3.2.2)
Together with
limt→∞ γ(t) = ∞.
it
that,
(3.3.1),
32
The above claim leads to:
Proof of Theorem 1.0.2.
dierently: Case
Case
By Claim 3.3.1, we just need to handle the four cases slightly
(a), 2k < n
(d), 2k = n,
and
and
h 6= 0.
h>0
(b), h = 0;
; Case
The proof of case
Case (a) 2k < n and h > 0:
h, γ(t)
For each
(a)
Case
(c), 2k > n
and
h 6= 0;
needs the help of Theorem B.
is bounded, thus equation
(3.2.2)
is
transformed into
e(2k−n)γ (1 − γt2 )k − e(2l−n)γ (1 − γt2 )l = h + η(t)
where
η(t) = O(e−2(1−δ)t )
as
t → ∞,
h
so
here is subject to the same upper bound as
in Theorem 1.0.1:
∗
h≤h =
n − 2k
n − 2l
n−2k
2
−
n − 2k
n − 2l
n−2l
2
.
Let
i
h
H(x, y) = h − e(2k−n)y (1 − x2 )k − e(2l−n)y (1 − x2 )l .
For each
h ∈ (0, h∗ ],
bounded over
we have proved that
t ∈ [0, ∞),
and by
(3.2.1)
γ(t)
and
satises that
(3.2.2),
|γ(t)| and |γ 0 (t)|
are
it is a solution to the following
condition
where
f
γ 00 (t) = f (γ 0 (t), γ(t)) + e1 (t)
t≥0
|H(γ 0 (t), γ(t))| ≤ e2 (t)
t≥0
is a locally Lipschitz function and
e1 (t), e2 (t) = O(e−2(1−δ)t ).
•
When the solution of
H(x, y) = 0
is a point , we take
•
When the solution of
H(x, y) = 0
is not a point, we take
So we can apply Theorem B and
Case (b) h = 0:
Using
(3.1.3)
to get Theorem
limt→∞ γ(t) = ∞
back into
l=2
Furthermore,
to satisfy
l=1
(3.1.12)
.
to satisfy (3.1.13).
1.0.2.
(3.2.2),
which now takes the form
n
o
n
o
e2kγ (1 − γt2 )k + η3 (t) − e2lγ (1 − γt2 )l + η4 (t) = 0,
33
we see that
1 − γt2 (t) := η(t) → 0
conclude that
t
1 − γt (t) → 0
>0
and some
small and
t → ∞.
as
t → ∞.
as
γ0 .
Since
As a consequence,
C>0
and for large
t,
or as
for suciently large
γ(t) ≥ (1 − )t + γ0
This would imply through
|η(t)| ≤ Ce−
for some constant
γt (t) > 0
(3.2.2)
t,
we
for large
that
2(1−δ)
t
k
t → ∞,
η l η k O(e−2(1−δ)t ).
However
(3.2.2)
shows
η k−l ≈ e2(l−k)γ(t)
η ≈ e−2γ(t) ,
a contradiction. Finally, we have
|γt (t) − 1| = |
p
2(1−δ)
1 − η(t) − 1| ≤ Ce− k t ,
from which we conclude that
γ(t) − t = τ + O(e−
t → ∞.
for some
τ
to which
(3.2.1)
also
as
Note that
2(1−δ)t
k
ξ0 (t) = ln cosh(t),
is a perturbation, satises
)
the solution to
ξ0 (t) = t − ln 2 + O(e−2t ).
with
t → ∞,
which is
(1.0.5).
u(x) = e
2(1−δ)t
k
)
Furthermore,
− n−2
(ω(t,θ)−t)
2
O
2(1−δ)t − n−2
ξ
(t+τ
+ln
2)−t+O
e− k
0
2
=e
2(1−δ)t
e− k
= u∗ (|x|)e
2(1−δ)t = u∗ (|x|) 1 + O e− k
where
n−2
u∗ (|x|) = e− 2 (ξ0 (t+τ +ln 2)−t)
n−2
2
4eτ
=
2τ
2
4e + |x|
h = 0,
Therefore, using
(3.1.6),
ω(t, θ) = γ(t) + ω̂(t, θ) = ξ0 (t + τ + ln 2) + O(e−
as
(2.2.1)
34
is a positive radial solution to
(3.1.1)
on
Rn \{0}.
lim u(x) = e−
n−2
τ
2
x→0
We also nd in this case that
:= u(0) > 0
exists, with
|u(x) − u(0)| ≤ |u(0)||e
Case (c) 2k > n and h 6= 0:
2(1−δ)t
)
− n−2
ω̂(t,θ)+O(e− k
2
≤ Ce−
2(1−δ)t
k
≤ C|x|
2(1−δ)
k
By Claim
− 1|
.
3.3.1, lim inf x→0 |x|
n−2
2
u(x) = 0
which implies
lim γ(t) = ∞
(3.3.2)
t→∞
Equation
(3.2.2)
implies that, for large
certain nite value. Together with
t, γ 0 (t) = 0
(3.3.2),
conclusion of
h=0
(1.0.5)
(3.2.2),
implies that
γ(t)
is near
this implies
γt > 0
which, together with
can occur only when
for
t
large
(1 − γt2 )k → 0
as
t → ∞.
Then the
is proved in an almost identical way as was done above for the
case.
Case (d) 2k = n and h 6= 0:
First from Claim
3.3.1,
lim γ(t) = ∞.
t→∞
In [7], they proved: If
e−2ω(t,θ) (dt2 + dθ2 ) ∈ Γ+
2
1−
and in
(3.2.2),
γt2
+
1
|S n−1 |
Z
for all
θ ∈ Sn−1
at some
|∇ω̂|2 dθ ≥ 0,
S n−1
∀>0,



− < (1 − γt2 )k + η3 < 1 + we have


0 < e−(2l−n)γ < when
.
Thus
0 < h ≤ 1.
t
is large enough
t,
then
35
Moreover, if
implies
h = 1,
(1 − γt2 )l >
t
then when
is large enough,
(1 − γt2 )k + η3 >
1
2 by
(3.2.2),
which
1
4 , thus
(1 − γt2 )l + η4 > 0.
Still using
(3.2.2),
(1 − γt2 )k + η3 > 1,
from which we obtain
γt = O(e−(1−δ)t )
and
Z
∞
γt dt < ∞,
lim γ(t) = γ(1) +
t→∞
a contradiction. So
h
1
must satisfy
0 < h < 1.
Arguing as before, we can also show
implies that
(1 − γt2 )k → h
depending on
0 < h < 1.
as
γt > 0
t → ∞;
Now with
and
when
t
is large enough. Then
e−nγ(t) = O(e−αt )
η(t) := 1 − γt2 (t),
as
t→∞
(3.2.2)
for some
α>0
we nd
η k (t) = h + e(2l−n)γ(t) (1 − γt2 )l + η4 (t) − η3 (t)
= h + O(e−αt )
as
t → ∞,
and
√
k
γt (t) = 1 − η(t) = 1 − h + O(e−αt )
p
√
which implies that γ(t) =
1 − k ht + γ0 + O(e−αt ), for some γ0 . Similarly, ξh (t)
p
√
satises ξh (t) =
1 − k ht + ξ0 + O(e−αt ) for some ξ0 . Thus for some τ , we have
q
p
γ(t) = ξh (t + τ ) + O(e−αt )
and
u(x) = e−
n−2
(ω(t,θ)−t)
2
= e−
n−2
(γ(t)−t+ω̂(t,θ))
2
,
from which we nd that
|x|
extends to a
n−2
2
C α (BR )
√
√
k
1− 1− h
u(x) = e
√ √
k
− n−2
γ(t)−
1−
ht+ω̂(t,θ)
2
positive function for some
α > 0.
36
37
Chapter 4
Analysis of the linearized operator
4.1 Linearization of the conformal quotient equation
In this section we provide a proof of Theorem 1.0.3 by an analysis of the linearized
operator of
2l−k (nk )/(nl ).
(1 −
Let
ξt2 )k−1
(1.0.8)
at a radial solution
Clearly
ξ(t)
ξ(t)
to
(1.0.8),
where
c
is normalized to be
satises
k
1 k
1
l
2
2kξ
2 l−1 l
2
ξtt + ( − )(1 − ξt ) e
= (1 − ξt )
ξtt + ( − )(1 − ξt ) e2lξ .
n
2 n
n
2 n
P (ξ, ξt ) :=
e−2ξ
1−ξt2
k−l
, and we use
P
to denote the function
P (ξ, ξt ) all through the
paper and we obtain
ξtt
Since
ξ(t)
satises


k−l
−2ξ

e
l
n
n


· ( 2k − k ) − ( 2k − 1) 
1−ξt2
=
(1 − ξt2 )
k−l


l
e−2ξ


1 − k · 1−ξ2
t
!
l
n
n
P ( 2k − k ) − ( 2k − 1)
(1 − ξt2 )
=
1 − P kl
!
n P −1
=
1+
·
(1 − ξt2 ).
2k 1 − P kl
1 − ξt2 > 0
and the rst integral
e(2k−n)ξ (1 − ξt2 )k − e(2l−n)ξ (1 − ξt2 )l = h,
P
must satisfy



0 < P < 1
if
h>0


P = 1
if
h = 0.
(4.1.1)
38
Next we have the linearization of
σk (Ag )
at
g = e−2ξ(t) (dt2 + dθ2 )
(for reference see
[43]),
(1 − |ξt2 |2 )k−2
2k−2
Lξ (φ) =
n−1
k−1
× [A(t)φtt (t, θ) + B(t)φt (t, θ) + C(t)∆θ φ(t, θ)] ,
where
A(t) =
1 − |ξt |2
2 n − 2k
2
B(t) = −ξt (t) (k − 1)ξtt (t) +
(1 − |ξt | )
2
n − 2k + 1 1 − |ξt |2
k−1
ξtt (t) +
·
.
C(t) =
n−1
n−1
2
We use
and
L̇(φ)
Ċ(t)
When
to denote the linearization of
σl (Ag )
at
g = e−2ξ(t) (dt2 + dθ2 )
while
Ḃ(t)
is dened correspondingly as well.
c
is normalized to be
2l−k (nk )/(nl ),
the linearization of PDE
(1.0.8)
Lξ (φ) + 2kσk φ − (ce−2(k−l)ξ L̇ξ (φ) + 2lσk φ) = 0.
A direct computation shows
2−k (nk ) −2(k−l)ξ (1 − |ξt |2 )l−1 n−1
·e
·
(l−1 )
2−l (nl )
2l−1
"
#
Ḃ(t)
Ċ(t)
× φtt +
φt +
∆θ φ(t, θ)
A(t)
A(t)
−2ξ k−l
(1 − |ξt |2 )k−1 n−1 l
e
=
(k−1 ) · ·
k−1
k
1 − |ξt |2
2
"
#
Ċ(t)
Ḃ(t)
× φtt +
φt +
∆θ φ(t, θ) ,
A(t)
A(t)
ce−2(k−l)ξ L̇ξ (φ) =
and also
σk φ =
=
=
k
1 k
2
−
ξtt + ( − )(1 − ξt ) φ
n
2 n
h
i
2
k−1
(1 − |ξt | )
n
n−1
2
(
)
ξ
+
(
−
1)(1
−
|ξ
|)
φ
tt
t
k−1
2k
2k−1
"
!
#
(1 − |ξt |2 )k−1 n−1
n
P −1
2
(k−1 )
+ 1 (1 − ξt ) φ.
2k 1 − P kl
2k−1
21−k (nk )(1
ξt2 )k−1
at
ξ(t)
is
39
Let
of
Yj (θ) be the normalized eigenfunctions of ∆θ
∆θ
on
L2 (Sn−1 )
associated with
Yj (θ).
on
L2 (Sn−1 ) and λj
Thus
λ0 = 0, λ1 = · · · = λn = n − 1, λj ≥ 2n,
If we take the projection of
φ(t, θ)
be the eigenvalues
for j
> n.
into spherical harmonics:
φ(t, θ) =
X
φj (t)Yj (θ),
j
then
φj (t)
satises the ODE
"
!
# )
C(t)
P −1
B(t) 0
φ + −λj
+n
+ 1 (1 − ξt2 ) φj
Fλj (φj (t)) : =
φj +
A(t) j
A(t)
1 − P kl
(
"
!
# )
l
Ḃ(t) 0
Ċ(t)
P −1
n
00
− · P φj +
φ + −λj
+
+ 1 (1 − ξt2 ) φj
k
A(t) j
A(t) P 1 − P kl
(
00
= 0
(4.1.2)
4.2 Decay rate
Now we try to gure out the decay rate of the solutions of
Fλj (φj (t)),
B(t) 0
A(t) φj
the term
"
ξt
and when
2k < n,
l
k
·
l
− ·P
k
it can always alter its sign as
l=0
φ0j ,
Note that in
Ḃ(t) 0
A(t) φj is
C(t)
1 − (n − 1)
A(t)
computation. In [43],
this term involving
−
(4.1.2).
Ċ(t)
1 − (n − 1)
A(t)
t → ∞,
!#
0
φj
and hard to be controlled in
and they introduced an auxiliary function
V (t)
to remove
more specically, they obtained
V (t)Fλj V −1 φj = φtt + E(t)φ(t)
to go through the computation. But when
l 6= 0,
results by their method, however, we observe
it is hard to recover the anticipated
(4.2.1)
and with the estimate
have
"
!#
C(t)
l
Ċ(t)
ξt
1 − (n − 1)
− · P 1 − (n − 1)
A(t)
k
A(t)
!
C(t)
l
Ċ(t)
<
1 − (n − 1)
,
− · P 1 − (n − 1)
A(t)
k
A(t)
|ξt | < 1
we
40
and this upper bound help us to have the following properties for the decay rate of
solutions:
Fact 4.2.1. When 2k ≤ n and h ≥ 0
Fλj [e±t ] < 0
for all λj ≥ 2n and t ∈ R+ .
Proof.
First we state the following claim and provide a proof in the end,
#
"
C(t)
l
Ċ(t)
1 − (n − 1)
− · P 1 − (n − 1)
< 0.
A(t)
k
A(t)
By this we have
(
−t
Fλj [e ] = e
−t
1−
"
B(t)
A(t)
!
#)
P −1
C(t)
+n
+ 1 (1 − ξt2 )
+ −λj
A(t)
1 − P kl
(
Ḃ(t)
l
−e−t · · P 1 −
k
A(t)
"
!
#)
n
P −1
Ċ(t)
+
+ −λj
+ 1 (1 − ξt2 )
A(t) P 1 − P kl
(
C(t)
−t
= e
1 − ξt 1 − (n − 1)
A(t)
"
!
#)
C(t)
P −1
2
+ −λj
+n
+ 1 (1 − ξt )
A(t)
1 − P kl
(
!
Ċ(t)
−t l
−e · · P 1 − ξt 1 − (n − 1)
k
A(t)
!
#)
"
n
P −1
Ċ(t)
+
+ 1 (1 − ξt2 )
+ −λj
A(t) P 1 − P kl
(
C(t)
−t
≤ e
1 − 1 − (n − 1)
A(t)
"
!#)
C(t)
P −1
+ −λj
+n
+1
A(t)
1 − P kl
(
!
Ċ(t)
−t l
−e · · P 1 − 1 − (n − 1)
k
A(t)
"
!#)
Ċ(t)
n
P −1
+ −λj
+
+1
A(t) P 1 − P kl
(4.2.1)
41
λj
If we replace
by
2n
and use
F2n (t)
to denote the right side of the last inequality,
then the above expression becomes
("
!#
C(t)
P −1
F2n [e ] ≤ F2n (t) := −e
(n + 1)
+1
−n
A(t)
1 − P kl
!#)
"
l
P −1
Ċ(t)
n
− · P (n + 1)
+1
−
k
A(t) P 1 − P kl
−t
−t
("
= −e−t (n + 1)
C(t)
n
−
A(t) n + 1
"
l
Ċ(t)
n
− ·P
−
k
A(t) (n + 1)P
Let
q(ξ, ξt ) =
P −1
and use
1−P kl
q
to denote
q(ξ, ξt )
!#
P −1
+1
1 − P kl
!#)
P −1
+1
1 − P kl
all through the paper, then
(4.2.2)
P =
q+1
1+q kl
−1 < q ≤ 0 since 0 < P ≤ 1. By (4.1.1) we have
1
k−1
n − 2k + 1 1 − ξt2
=
·
ξtt +
·
A(t)
n−1
n−1
2
!
"
#
2
1
k−1
n P −1
n
−
2k
+
1
1
−
ξ
t
=
·
1+
·
(1 − ξt2 ) +
·
A(t)
n−1
2k 1 − P kl
n−1
2
and obviously
C(t)
A(t)
k−1 n P −1
· ·
n − 1 k 1 − P kl
k−1 n
· · q.
= 1+
n−1 k
= 1+
Now we plug it into
(4.2.2)
to get
−t k−1 n
n
−t
1+
F2n e
≤ F2n (t) = −e (n + 1)
· ·q −
(q + 1)
n−1 k
n+1
l
l−1 n
n
1+
· ·q P −
(1 + q)
−
k
n−1 k
(n + 1)
1
(k − 1)n
l
l
−t
= −e (n + 1)
1− +
q(1 + q)
l
k
(n
−
1)k
k
1 + kq
n
l
l
nl(l − 1)
−
(q + 1)(1 − )(1 + q) − 2
q(q + 1)
n+1
k
k
k (n − 1)
n+1
nl(k − l)
n(k − l)l
= −e−t
−
q2
2
l
k (n − 1) (n + 1)k 2
1 + kq
n(k − l)(k + l − 1) n(k − l)(k + l)
1
l
−
q+
1−
+
k 2 (n − 1)
(n + 1)k 2
n+1
k
k−l
2nl
n (2(k + l) − n − 1)
= −e−t
q2 +
q+1 .
l
k(n − 1)
k(1 + k q) k(n − 1)
42
Next we consider two cases:
•
If
2(k + l) − n − 1 ≤ 0,
for all
•
If
then the parabola
2nl
n (2(k + l) − n − 1)
q2 +
q+1≥1
k(n − 1)
k(n − 1)
q ∈ (−1, 0], which implies that F2n e−t ≤ F2n (t) < 0.
2(k + l) − n − 1 > 0,
we let
a=
n
2
− k,
and
a≥0
by assumption, then the
discriminant of the parabola is
n(2(k + l) − n − 1) 2
2nl
−4·
k(n − 1)
k(n − 1)
n
2
n
(2(k
+
l)
−
n
−
1)
−
8lk(n
−
1)
k 2 (n − 1)2
n
n(2l − 2a − 1)2 − 4l(n − 2a)(n − 1)
2
2
k (n − 1)
n(2l − 2a − 1)
(n(2l − 2a − 1) − 2(n − 2a)(n − 1))
k 2 (n − 1)2
n(2l − 2a − 1)
(n(n − 4a − 1) − 2(n − 2a)(n − 1))
k 2 (n − 1)2
n(2l − 2a − 1)
(−2an − 2a)
k 2 (n − 1)2
∆ =
=
=
≤
≤
<
< 0.
We still have
2nl
n (2(k + l) − n − 1)
q2 +
q + 1 > 0,
k(n − 1)
k(n − 1)
and thus
F2n [e−t ] ≤ F2n (t) < 0.
Also we nd the coecient of
−t
−e
λj
in
C(t)
l Ċ(t)
− P
A(t) k A(t)
Fλj e−t
!
=
is
F2n (t) − n(1 − kl )(q + 1)e−t
< 0,
n+1
which implies
Fλj e−t < F2n e−t < 0,
similarly we have
Fλj et < 0.
43
Finally it remains to verify
(4.2.1),
"
indeed,
#
C(t)
l
Ċ(t)
1 − (n − 1)
− · P 1 − (n − 1)
A(t)
k
A(t)
k−1 n
l−1 n
l q+1
= 1 − (n − 1) 1 +
1 − (n − 1) 1 +
· q
− ·
· q
n−1 k
k 1 + q kl
n−1 k
1
{[(2 − n)k − (k − 1)nq] (k + ql) − l(q + 1)[(2 − n)k − (l − 1)nq}
=
k(k + ql)
1
=
(2 − n)k 2 + ((2 − n)kl − (k − 1)nk)q − (k − 1)nlq 2
k(k + ql)
−(2 − n)kl + ((l − 1)nl − (2 − n)kl)q + (l − 1)nlq 2
=
=
Since the axis
1
(2 − n)k(k − l) + (k − l)(1 − k − l)nq + nl(l − k)q 2
k(k + ql)
l−k 2
nlq + n(k + l − 1)q + (n − 2)k ,
k(k + ql)
2
l−k
of symmetry of parabola
nlq + n(k + l − 1)q + (n − 2)k is
k
x=−
n(k + l − 1)
2nl
and obviously
n(k + l − 1)
,
2nl
2
l−k
nlq + n(k + l − 1)q + (n − 2)k is decreasing when q > −1,
k
"
#
l
Ċ(t)
C(t)
− · P 1 − (n − 1)
1 − (n − 1)
A(t)
k
A(t)
l−k <
nl(−1)2 + n(k + l − 1)(−1) + (n − 2)k
k
l−k
=
{n − 2k}
k
−1 > −
we have
thus
≤ 0
for all
q ∈ (−1, 0].
Fact 4.2.2. When
n
2
< k < n and h > 0 we have
Fλj e−t < 0
for all λj ≥ 2n and t large enough. There also exists some small λ > 0 such that
h i
Fλj eλt < 0
44
for all λj ≥ 2n and t large enough.
Proof.
First as
t → ∞, ξ → ∞
and
1
n
1 − ξt2 → h k e( k −2)ξ → 0,
and also
(1 − ξt2 )k = he(n−2k)ξ + (1 − ξt2 )l e(2l−2k)ξ ,
we get
P
thus
q → −1
=
e−2ξ
1 − ξt2
k−l
→ h−
k−l
k
e−
n(k−l)
ξ
k
→ 0,
and
−t k−1 n
k−1 n
−t
· (−1) − 2n 1 +
· (−1)
F2n e
→ e
1 − 1 − (n − 1) 1 +
n−1 k
n−1 k
n−k
= −e−t (n + 1) ·
k(n − 1)
<
Moreover, as
0,
l Ċ(t)
t → ∞, − C(t)
A(t) − P k A(t) →
k−n
k(n−1)
< 0,
which implies
Fλj e−t < F2n e−t
< 0
as
t → ∞.
Similarly there exist some
λ>0
such that
h i
Fλj eλt < 0
as
t → ∞.
Using the same argument as in the proof of Proposition 2 in [43], we nd out the
the solution basis
−
φ+
0 (t), φ0 (t)
of
(4.1.2)
for
λ0 = 0
and
λ j = n − 1:
−
• λ0 = 0 φ+
0 (t) = ∂t ξ(t) φ0 (t) = ∂h ξh (t)



t
−t
φ+
φ−
0 (t) = [1 − ∂t ξ(t)] e ,
0 (t) = [1 + ∂t ξ(t)] e
• λj = n − 1
R1


−
−1
−1
n
φ+
0 (t) = cosh (t) 0 (cosh(s)) ds, φ0 (t) = cosh (t)
and when
t→∞
if
h >
k < n.
0, φ+
0 (t) grows unbounded and
if
h>0
if
h=0
φ−
0 (t) decays exponentially as
45
For
λj ≥ 2n,
it is not easy to write down the solution basis
φ+
0
and
φ−
0.
However by
Fact 4.2.1 and 4.2.2 we can still use the comparison principle to prove that
φ−
0 (t)
unbounded and
decays faster than
establish the following proposition.
e−t
as
t → ∞
if
k < n.
φ+
0
grows
Consequently we
For details of the comparison principle proof see
proposition 2 of [43].
Proposition 4.2.3. For all solutions ξ (t) to (4.1.2) with h ≥ 0, k < n and j ≥ 1, the
h
following holds:
1. The basis of solution space to
solutions on
as
Fλj [φ] = 0
contains a pair of linearly independent
R, of which one grows unbounded and the other decays exponentially
t → ∞;
R+
2. Any solution of
Fλj [φ] = 0
which is bounded for
3. Any solution of
Fλj [φ] = 0
which is bounded for all of
4. Any non-zero solution of
unbounded on
Fλj [φ] = 0
must decay exponentially;
R
must be identically
which is bounded for all of
R+
0;
must be
R− .
While the asymptotic behavior of the solution basis is sucient for providing a proof
for Theorem 1.0.2, Theorem 1.0.3 requires some more detailed knowledge about the
linearized operator to
(1.0.8).
More specically, the decay rates of bounded solutions to
Fj [φ] = 0 on R+ need to be faster than e−t when λj ≥ 2n.
Lj
In the case
2k < n and h > 0,
is an ordinary dierential operator with periodic coecient, so by Floquent Theory,
it has a set of well dened characteristic roots which give the exponential decay/grow
rates to solution
applied to
of
eρj t p1 (t)
When
Fj
φ
of
Fj [φ] = 0
implies that
and
e−ρj t p2 (t)
on
R,
Fj [φ] = 0
Indeed Theorem 5.1 in Chap. 3 of [21] which
has a set of fundamental solutions in the form
for some periodic functions
h = 0, ξ(t) = ln cosh(t),
and
(4.1.2)
p1 (t)
and
p2 (t),
where
ρj 6= 0.
take the form of
00
0
Fj [φj ] = φj + (2 − n) tanh(t)φj + −λj + n cosh−2 (t) φj ,
(4.2.3)
so a similar notion for characteristic roots can be dened, which is also the case when
2k = n.
Fact 4.2.1 implies the following
46
Lemma 4.2.4. When 2k ≤ n and h ≥ 0, there is a β
∗
> 1 such that for all λj ≥ 2n,
the associated ρj satises ρj ≥ β∗ .
4.3 Expansion in terms of Wronskian
With the knowledge from the previous section, we can now establish
Proposition 4.3.1. When 2k ≤ n, and suppose that φ(t, θ) → 0 as t → ∞ uniformally
in θ ∈ Sn−1 and also satises
Lξ (φ) + 2kσk φ − ce−2(k−l)ξ L̇ξ (φ) + 2lσk φ = r(t, θ)
(4.3.1)
for all t > t0 and θ ∈ Sn−1 . Suppose that for some 0 < β < β∗ and β 6= 1, |r(t, θ)| .
e−βt . Then there exist constants aj for j = 1, · · · , n such that
|φ(t, θ) −
n
X
aj e−t (1 + ξt (t))Yj (θ)| . e−βt .
(4.3.2)
j=1
In fact, when β∗ ≤ β < ρn+1 , it continues to hold and when β > ρn+1 ,we will have
|φ(t, θ) −
n
X
aj e−t (1 + ξt (t))Yj (θ)| . e−ρn+1 t ;
(4.3.3)
j=1
when β = 1 ,
|φ(t, θ) −
n
X
aj e−t (1 + ξt (t))Yj (θ)| . te−t ;
(4.3.4)
j=1
and when β = ρn+1 ,
|φ(t, θ) −
n
X
aj e−t (1 + ξt (t))Yj (θ)| . te−ρn+1 t .
j=1
Proof.
Dene
φ̂(t, θ) = φ(t, θ) −
n
X
πj [φ(t, θ] Yj (θ)
j=0
where
φj (t) := πj [φ(t, θ)]
is the
L2
φ(t, θ)
onto span{Yj (θ)} .
∇φ̂(t, θ) · ∇Yj (θ)dθ
(4.3.5)
orthogonal projection of
Then
Z
Z
φ̂(t, θ)Yj (θ)dθ =
Sn−1
n−1
ZS
∆θ Yj (θ)φ̂(t, θ)dθ = 0
=
Sn−1
47
j = 0, · · · , n. As a consequence,

R
R



∆ φ(t, θ)φ̂(t, θ)dθ = −
for
θ
Sn−1
R



Sn−1
φt (t, θ)φ̂(t, θ)dθ
=
R
Sn−1
|∇θ φ̂(t, θ)dθ
(4.3.6)
φ̂t (t, θ)φ̂(t, θ)dθ =
Sn−1
1 d
2 dt
R
Sn−1
In the following we will prove separately the expected decays for
πj [φ(t, θ)],
for
j = 0, 1, · · · , n.
Multiplying both sides of
and integrating over
We rst estimate
(4.3.1)
φ̂(t, θ)
φj (t) = πj [φ(t, θ)]
for
and
φj (t) :=
j = 0, · · · , n.
by
(1 − |ξt |2 )k−1
2k−1
θ ∈ Sn−1 ,
|φ̂(t, θ)|2 dθ.
n−1
k−1
−1
l
1− P
Yj (θ),
k
we obtain
00
0
φj (t) + Φφj (t) + Ψφj (t) = r̂j (t)
(4.3.7)
where

2 2


k −P l
P −1

−
1
Φ
=
−ξ
(n
−
2)
+
n
·

t
k−P l
k−P l



o
n
(n+1)(k+l)−2n
(n+1)l 2
1
q
+
q
+
Ψ
=
2n
2 −
2k(n−1)
2(n−1)k




R


r̂j (t) = n−1 r̂(t, θ)Yj (θ)dθ.
S
We will write out the details for the
of
(4.2.3)
as
σk = c
h>0
n
( P −1
1−P kl 1−P kl
case; the
h=0
case, so it was solved by [43]. For
0
+ 1)(1 − kl )ξt2
has the same linearization
j = 1, · · · , n, λj = n − 1,
and
0
+
−t
t
φ−
1 (t) := e (1 + ξt (t)), φ1 := e (1 − ξt (t)) form a basis of solutions to the homogeneous
equation
00
0
φj (t) + Φφj (t) + Ψφj (t) = 0.
Since
φj (t) is a solution to (4.3.4) and φj (t) → 0 as t → ∞, by the variation of constant
formula,
φj (t) =
cφ−
1 (t)
for some constant
c,
+
φ−
1 (t)
t
Z
0
φ+
1 (s)r̂j (s)
ds + φ+
1 (t)
W1 (s)
t
Z
0
φ−
1 (s)r̂j (s)
ds
W1 (s)
where
0
0
−
−
+
W1 (s) = φ+
1 (s)φ1 (s) − φ1 (s)φ1 (s)
is the Wronskian of
+
φ−
1 (t), φ1 (t) ,
0
and satises
W1 (s) = −Φ(t)W1 (s).
(4.3.8)
48
Integrating this equation out, using
P − 1 k 2 − P l2
Φ = −ξt (n − 2) + n ·
−1
,
k − Pl
k − Pl
we nd
Rt
W1 (t) = (cosst.)e
0
Rt
= (cosst.)e
0
Rt
0
= (cosst.)e
= (cosst.)e
If
2k < n,
then
ξ(t)
P (k−l)
P l(k−l)
ξs (n−2)+n k(k−P l) − k1
+(k−1) ds
k−P l
ξs
2k = n,
R ξ(t)
W1 (t)
ξ(0)
1≤e
2
P (k−l)
P (k−l)(k−l−1)
+n
+
( n−2k
k )
k(k−P l)
k(k−P l)2
n−2k
ξ(t)
k
is bounded and
upper and lower bound. If
Consequently
P −1 k2 −P l2
ξs (n−2)+n· k−P
−1 ds
l
k−P l
n
R ξ(t)
·e
ξt (t)
then
ξ(0)
n
ds
P (k−l)(k−l−1)
P 2 (k−l)
+
k(k−P l)
k(k−P l)2
is a function of
P ∼h
− k−l
k
ξ(t),
thus
e−2(k−l)ξ → 0
P (k−l)(k−l−1)
P 2 (k−l)
+
k(k−P l)
k(k−P l)2
r(t, θ),
as
dξ(s)
t
we have
Z ∞
φ−
1 (s)r̂j (s)
ds .
e−(1+β)s ds . e−(1+β)t
W1 (s)
t
from which we deduce that
∞
Z
+
φ (t)
1
t
When
β 6= 1,
φ−
1 (s)r̂j (s)
ds . e−βt .
W1 (s)
we also have
Z
0
t
Z t
φ+
1 (s)r̂j (s)
ds .
e(1−β)s ds . e(1−β)t
W1 (s)
0
from which we deduce that
Z
−
φ (t)
1
t
0
Putting these estimates into
(4.3.8),
has a positive
t → ∞,
thus
≤ C.
Thus
∞
.
|rj (s)| ≤ Ce−βs .
Z
dξ(s)
W1 (t)
also has a positive upper and lower bound.
assumption on the decay rate of
φ+
1 (s)r̂j (s)
ds . e−βt .
W1 (s)
we have
φj (t) − ce−t (1 + ξ 0 (t)) . e−βt .
According to our
49
When
β = 1, (4.3.8)
For
gives the modied estimate.
0
j = 0 , φ+
0 (t) = ξh (t)
and
0
φ0 (t) = ∂h ξh (t)
also form a solution basis to the
homogeneous equation
00
0
φj (t) + Φφj (t) + Ψφj (t) = 0.
Since
φ0 (t)
φ0 (t) =
−φ−
0 (t)
t → ∞, a variant of (4.3.8)
Z ∞ −
φ+
φ0 (s)r̂0 (s)
+
0 (s)r̂0 (s)
ds + φ0 (t)
ds,
W0 (s)
W0 (s)
t
(4.3.7)
is a solution to
∞
Z
t
and
φ0 (t) → 0
as
gives:
(4.3.9)
where
0
0
−
−
+
W0 (s) = φ+
0 (t)φ0 (t) − φ0 (t)φ0 (t)
is the Wronskian of
+
φ−
0 (t), φ0 (t) ,
and also satises
0
W0 (s) = −
Thus, as for
bound. Let
ξh (t).
W1 (s), W0 (s)
T (h)
B(s)
W0 (s).
A(s)
is a periodic function, having a positive upper and lower
denote the minimal period of the solution
Dierentiating in
h,
(4.3.10)
ξh (t).
Then
ξh (t + T (h)) =
we obtain
0
0
−
φ−
0 (t + T (h)) + T (h)ξh (t + T (h)) = φ0 (t),
which implies that
|φ0 (t)| . te−βt .
φ−
0 (t)
grows in
(4.3.11)
at most linearly. Then
(1.0.9).
(4.3.2),
note
implies that
p(t) :=
T (h)
would imply that
To obtain the more precise estimate
0
is
(4.3.10)
This is not quite as claimed, but is good enough to be used in our
iterative argument in proving
that
t
(4.3.11)
φ+
0 (t)
0
T (h) 0
T (h) +
+
tξh (t) = φ−
tφ (t)
0 (t) +
T (h)
T (h) 0
periodic. Thus we can express
0
0
φ0 (t)
0
as
p(t) −
!Z
T (h) +
T (h) tφ0 (t) in
(4.3.9)
∞ +
T (h) +
φ0 (s)r̂0 (s)
tφ0 (t)
ds
φ0 (t) = − p(t) −
T (h)
W0 (s)
t
0
T (h)
+
Z ∞ p(s) − T (h) sφ0 (s)r̂0 (s)
+
+φ0
ds
W0 (s)
t
Z ∞ +
Z ∞
φ0 (s)r̂0 (s)
p(s)r̂0 (s)
= −p(t)
ds + φ+
(s)
ds
0
W
(s)
W0 (s)
0
t
t
Z ∞Z ∞ +
0
T (h) +
φ0 (τ )r̂0 (s)
−
φ0 (t)
dτ ds,
T (h)
W0 (s)
t
s
to obtain
50
from which
|φ0 (t)| . e−βt
follows.
Finally, we estimate the decay rate of
integrating over
(1 − |ξt |2 )k−1
2k−1
θ ∈ Sn−1
and using
n−1
k−1
φ̂(t, θ).
Multiplying both sides of
(4.3.1)
by
−1
l
1− P
φ̂(t, θ),
k
(4.3.5)
and
(4.3.6),
we nd
(
Z
φ̂tt (t, θ)φ̂(t, θ) + Φφ̂t (t, θ)φ̂(t, θ)
(4.3.12)
Sn−1
)
2
k−l n
· (1 + q) (1 − |ξt2 |)| φ̂(t, θ) dθ
+
1 − kl P k
(
#
)
"
Z
2
λj
C(t)
l Ĉ(t) −
− P
φ̂θ (t, θ) dθ
l
A(t)
k
A(t)
n−1
1
−
P
S
k
Z
=
r̂(t, θ)φ̂(t, θ)dθ,
Sn−1
where
r̂(t, θ) =
(1 − |ξt |2 )k−1
2k−1
n−1
k−1
−1
l
1− P
r(t, θ) ≈ r(t, θ)
k
Dening
sZ
y(t) =
Sn−1
2
φ̂(t, θ) dθ,
then
Z
0
y (t) =
φ̂t (t, θ)φ̂(t, θ)dθ/y(t)
whenever y(t)
> 0,
Sn−1
and
Z
00
y(t)y (t) =
Sn−1
2 0 2
ˆ
φtt (t, θ)φ̂(t, θ) + φ̂t (t, θ) dθ − y (t) .
The Cauchy-Schwarz inequality implies that
0 2
y (t) ≤
Z
Sn−1
2
φ̂t (t, θ) dθ.
Using these relations and plugging
Z
Sn−1
into
(4.3.12)
Z
2
∇θ φ̂(t, θ) dθ ≥ 2n
Sn−1
2
φ̂(t, θ) dθ
, we obtain
00
0
y(t)y (t) + Φy(t)y (t) + Ψy 2 (t) ≥ − ||r̂(t, ·)||L2 (Sn−1 ) y(t),
(4.3.13)
51
whenever
y(t) > 0,
from which we deduce
00
y (t) + Φy(t) + Ψy(t) ≥ − ||r̂(t, ·)||L2 (Sn−1 )
whenever
y(t) > 0.
According to our assumption on
r(t, θ),
we have
||r̂(t, ·)||L2 (Sn−1 ) ≤ Ce−βt
for some constant
C > 0.
By Lemma 4.2.4,
(∂tt + Φ∂t + Ψ) e−βt ≤ −e−βt
for some
>0
when
β < β∗ .
So
z(t) :=
C −βt
satises
e
(∂tt + Φ∂t + Ψ) (z(t) − y(t)) ≤ 0
whenever
y(t) > 0.
large so that
z(0) ≥ y(0).
min (z(t) − y(t)) < 0
holds at
t = t∗ ,
contradicts
We also know that
and
(4.3.14).
y(t) → 0
Then we claim that
as
t → ∞.
z(t) − y(t) ≥ 0
is nite, and is attained at some
∂t (z(t) − y(t))|t=t∗ = 0,
(4.3.14)
t∗ ,
as well as
so
We may choose
for all
t ≥ 0,
C > 0
for if not,
y(t∗ ) > z(t∗ ) > 0, (4.3.14)
∂tt (z(t) − y(t))|t=t∗ ≥ 0.
This
Thus we conclude
sZ
2
C
φ̂(t, θ) dθ = y(t) ≤ e−βt .
Sn−1
We can now bootstrap this integral estimate to obtain a pointwise decay estimate
φ̂(t, θ) . e−βt .
When
φ̂(t, θ),
β ≥ β∗ ,
we can simply split those components
and estimate them as we did for
φj
φj , j = 0, · · · , n,
of
φ
with
and estimate
λj = 2n
φ̂(t, θ)
from
with an
exponential decay rate.
With Proposition 4.3.1 and a iteration argument from[43], we proved Theorem 1.0.3.
Proof of Theorem 1.0.3.
Our starting point is still
Lξh (t+τ ) (φ) + 2kσk φ − ce−2(k−l)ξh (t+τ ) L̇ξh (t+τ ) (φ) + 2lσk φ + Q(φ) = 0,
52
and our premise is
|Q(φ)| . e−2αt
|φ, ∂φ, ∂ 2 φ| . e−αt .
whenever we have
In Theorem 1.0.2 we have established
Step 1:
proved
For some
α0 > 0, |φ, ∂φ, ∂ 2 φ| . e−α0 t .If α0 ≥ ρn+1 ,
|ω(t, θ) − ξh (t + τ )| = |φ(t, θ)| . e−ρn+1 t ,
we jump to Step 3; if
Step 2:
α0 ≤ 1,
where
if
1 ≤ α0 < ρn+1 ,
|Q(φ)| . e−2α0 t . If 2α0 > ρn+1 , then we can apply
Recall that we now have
|Q(φ)| . e−2αt
2;
we move to
1 < 2α0 ≤ ρn+1 ,
Proposition 4.3.1 directly to conclude our proof; If
still have
ρn+1 >
we stop and have now
√
1 < 2α < ρn+1
for some
then we certainly
and can apply Proposition 4.3.1 to
imply that
|ω(t, θ) − ξh (t + τ ) −
n
X
0
aj e−(t+τ ) (1 + ξh (t + τ ))Yj (θ)| . e−2αt ,
j=1
for some constants
to satisfy
2α0 < 1
aj
for
j = 1, · · · , n,
and jump to Step 3; if
2α0 ≤ 1,
we may take
α0
and apply Proposition 4.3.1 to imply that
|φ(t, θ) −
n
X
0
aj e−(t+τ ) (1 + ξh (t + τ ))Yj (θ)| . e−2α0 t ,
j=1
aj
for some constant
for
j = 1, · · · , n.
This certainly implies that
|φ(t, θ)| . e−2α0 t .
Next we use higher derivative estimates for
(4.3.15)
ω(t, θ)
(4.3.15)
and
ξh (t + τ )
and interpolation with
to obtain
0
|φ, ∂φ, ∂ 2 φ| . e−2α t
for any
α0 < α0 .
with a new
Now we go back to the beginning of Step 2 and repeat the process
α1 > α 0
to replace the
steps, we will reach a stage where
Step 3:
α0
2α > 1
At this stage, we have
to
whether
|Q(φ)| . e−αt ,
ρn+1 ≥ 2
with
α
α1 = 1.8α0 .
After a nite number of
and ready to move onto
|φ(t, θ)| . e−t .
involving the derivative estimates for
Q(φ)
there, say,
Repeating the last part of Step 2
ω(t, θ) and ξh (t + τ ) to bootstrap the estimate for
can be as close to
2
as one needs. Then, depending on
or otherwise, one can apply Proposition 4.3.1 to obtain
(4.3.2)
and
53
(4.3.3).
In the rst case, we can continue the iteration until
presence of
2α > 2.
But due to the
0
e−(t+τ ) (1+ξh (t+τ ))Yj (θ) in the estimate for φ, the estimate for Q(φ) cannot
be better than
e−2t .
This explains the appearance of
min{2, ρn+1 }
in
(1.0.11).
54
Appendix A
Proof for the phase plane
Now we prove the result in Section 2.1. Remember that we consider the level set given
by
G(x, y) = e(2k−n)x (1 − y 2 )k − e(2l−n)x (1 − y 2 )l = h
(2.1.1)
(x, y) ∈ Ω ≡ (x, y) ∈ R2 | |y| < 1 and h is a constant. We also recall that
Ω− ≡ (x, y) ∈ R2 | − 1 < y < 0 and the level set is symmetric with respect to x-axis,
where
i.e.,
G(x, y) = G(x, −y)
for all
(x, y) ∈ Ω.
Moreover, we set
(x, y) ∈ Ω | x > − 21 ln 1 − y
2
(A.0.1)
W := (x, y) ∈ Ω | x < − 12 ln 1 − y 2
, and
∀(x, y) ∈ Ω
(x, y) ∈ W ⇐⇒ h < 0
while
∂G
∂y
V :=
(2.1.1),
(x, y) ∈ V ⇐⇒ h > 0.
Proof of (P1) (P2) (P3) in Case h > 0
Lemma A.0.2. If 1 < l < k < & h > 0, then
n
2
solving
and
(A.0.2)
> 0 in V ∩ Ω− . Besides, the set
< ≡ (x, y) ∈ Ω | ∂G
=
0
is the curve
∂x
x̄(y) =
n−2l
n−2k
1
k−l
1
ln
2
(1 − y 2 )
|y| < 1
(A.0.3)
and it stays in V . Moreover, for all (x, y) ∈ Ω we have



 ∂G > 0
∂x
if x < x̄(y)


 ∂G < 0
∂x
if x > x̄(y)
(A.0.4)
55
Proof.
First if
∂G
∂y
1<l<k<
n
2 , we have that
= (−2y) e(2k−n)x k(1 − y 2 )k−1 − e(2l−n)x l(1 − y 2 )l−1
= (−2y)(1 − y 2 )−1 ke(2k−n)x (1 − y 2 )k − le(2l−n)x (1 − y 2 )l
> 0
in
V ∩ Ω− .
Secondly we solve
∂G
= (2k − n)e(2k−n)x (1 − y 2 )k − (2l − n)e(2l−n)x (1 − y 2 )l = 0
∂x
in
Ω
to obtain the curve
n−2l
n−2k
1
k−l
1
ln
2
(1 − y 2 )
x̄(y) =
|y| < 1,
and obviously
1
x̄(y) > − ln 1 − y 2
2
so
∂G
∂x
<
must stay in
<0
if
V.
Consequently for all
(x, y) ∈ Ω,
we have
∂G
∂x
>0
if
x < x̄(y)
and
x > x̄(y).
Lemma A.0.3. If 1 < l < k <
( n−2k
n−2l )
|y| < 1,
if
n−2k
2
− ( n−2k
n−2l )
n−2l
2
only at
n
2
, then G(x, y) achieves its maximum h∗ :=
1
2(k−l)
n−2l
ln n−2k
,0
in V . Moreover, if 0 < h < h∗ ,
the level set given by (2.1.1) is a closed curve in V ; and if h = h∗ , it is the point
(x̄(0), 0) . All the sets {(x, y) | G(x, y) = h}, 0 < h < h∗ foliate V and there is no
solution to (2.1.1) in Ω if h > h∗ .
Proof.
By Lemma A.0.2
∂G
∂y
>0
in
V ∩ Ω−
G(x, y) < G(x, 0)
Also
and (A.0.1), we have
∀(x, y) ∈ V



 ∂G (x, 0) > 0
if


 ∂G (x, 0) < 0
if
∂x
∂x
with y
6= 0.
(A.0.5)
x < x̄(0)
(A.0.6)
x̄(0) < x,
which implies
G(x, 0) < G (x̄(0), 0) ≡ h∗
∀x > 0 and x 6= x̄(0).
(A.0.7)
56
Combining
(A.0.5)
(A.0.7),
and
∀(x, y) ∈ V & (x, y) 6= (x̄(0), 0).
G(x̄(0), 0) = max G(x, y) > G(x, y)
(x,y)∈V
Besides,
G(0, 0) = lim G(x, 0) = 0,
x→∞
so when
0 < h < h∗ ,
x̄(0) < a2 (h) < ∞
by
(A.0.6),
there are just
a1 (h)
&
a2 (h)
such that
0 < a1 (h) <
and
G(a1 (h), 0) = G(a2 (h), 0) = h.
If
x ∈ (a1 (h), a2 (h)),
one unique
y(x)
√
G(x, 0) > h & G(x, − 1 − e−2x ) = 0,
√
− 1 − e−2x < y(x) < 0 solving
we have
such that
so there must exist
G(x, y(x)) = h.
However, if
(x, y) ∈ V
solution of
(2.1.1)
(A.0.1),
point
with
0 < x < a1 (h)
must be the set
x > a2 (h),
then
G(x, y) > h,
{(x, ±y(x)) | a1 (h) < x < a2 (h)}
whose graph is a closed curve in
(x̄(0), 0) .
or
There is no solution to
V.
When
(2.1.1)
if
by symmetry
the graph of
(2.1.1)
is the
h > h∗ .
Proof of (P4) (P5) in Case h > 0
Lemma A.0.4. If 1 ≤ l < k = & h > 0, then
n
2
h = h∗ ,
so the
∂G
∂y
> 0 in V ∩ Ω− and
∂G
∂x
> 0 in
Ω. Consequently in V ∩ Ω− the function y(x) solving (2.1.1) is well dened and strictly
decreasing.
Proof.
Similarly as Lemma
∂G
∂y
A.0.2,
we have
= (−2y)(1 − y 2 )−1 k(1 − y 2 )k − le(2l−2k)x (1 − y 2 )l
= (−2y)(1 − y 2 )l−1 e(2l−2k)x (ke(2k−2l)x (1 − y 2 )k−l − l)
> 0
in
V ∩ Ω− ,
and also in
Ω,
∂G
= (n − 2l)e(2l−n)x (1 − y 2 ) > 0.
∂x
(A.0.8)
57
Thus
∂G
∂y
∂x
<0
= − ∂G
∂x
∂y
in
V ∩ Ω− , which implies that y(x) is well dened and strictly decreasing in V ∩ Ω− .
Lemma A.0.5. If 1 ≤ l < k =
n
2
& 0 < h < 1, the level set given by (2.1.1) is a
U -shaped curve which opens right. Moreover the function y(x) solving (2.1.1) is strictly
p
√
decreasing and y → − 1 − k h as x → ∞ in Ω− . If 1 ≤ l < k = n2 & h ≥ 1, there is
no solution for (2.1.1) in Ω.
Proof.
By Lemma
A.0.4,
if
1≤l<k=
n
2 , then
G(x, y) ≤ G(x, 0) < lim G(x, 0) = 1
x→∞
for all
(x, y) ∈ V .
Thus when
h ≥ 1,
there is no solution for
y(x) solving (2.1.1) is well dened and
p
√
y → − 1 − k h as x → ∞ in Ω− , and by
(2.1.1)
in
Ω.
If
0 < h < 1,
V ∩ Ω− .
the function
strictly decreasing in
Besides,
symmetry the level set is a U-
shaped curve that opens right in
Proof of (P6) in Case h > 0
Lemma A.0.6. If 1 ≤ l < k,
n
2
V.
< k & h > 0, then
∂G
∂y
> 0 in V ∩ Ω− and
∂G
∂x
> 0 in
V , thus the function y(x) solving (2.1.1) is well dened and strictly decreasing in Ω− .
Moreover, y(x) → −1 as x → ∞ in Ω− and the level set given by (2.1.1) is a U -shaped
curve that opens right.
Proof.
If
1 ≤ l < k,
∂G
∂x
n
2
<k
&
h > 0,
then
= (2k − n)e(2k−n)x (1 − y 2 )k − (2l − n)e(2l−n)x (1 − y 2 )l
= e(2l−n)x (1 − y 2 )l ((2k − n)e(2k−2l)x (1 − y 2 )k−l − (2l − n))
> 0
in
V
and same as above
∂G
∂y
>0
in
Ω− ∩ V .
Thus in
∂G
∂y
∂x
= − ∂G
< 0,
∂x
∂y
Ω− ∩ V ,
58
so the function
y(x)
solving
We claim
y(x) → −1
y(x) > c,
then
(2.1.1)
x→∞
as
in
is well dened and strictly decreasing in
Ω− ,
x→∞
given by
(2.1.1)
Hence
is a
y(x) → −1
U -shaped
as
0 > c > −1
such that
1
lim e(2k−n)x (1 − c2 )k = ∞
2 x→∞
h ≡ lim G(x, y(x)) >
a contradiction.
or else there exists one
V ∩ Ω− .
x → ∞
in
Ω−
and by symmetry the level set
curve that opens right in
V.
Proof of (N1) (N2) in Case h < 0
Lemma A.0.7. If 1 ≤ l < k, l < & h < 0, then
n
2
∂G
>0
∂x
(x, y) ∈ W,
and ∀ y ∈ (−1, 1), there exists a unique x∗h (y) such that
G(x∗h (y), y) = h.
Moreover, x∗h (0) < 0 and
lim x∗h (y) = −∞.
(A.0.9)
y→−1
Proof.
If
1≤l<
∂G
∂x
n
2
≤ k,
then in
Ω,
= e(2l−n)x (1 − y 2 )l ((2k − n)e(2k−2l)x (1 − y 2 )k−l − (2l − n))
> 0.
With (A.0.6),
Besides,
(A.0.8),
we obtain that if
1 ≤ l < k, l <
∂G
>0
∂x
(x, y) ∈ W.
n
2 , then
∀y ∈ (−1, 1)



G(x, y) = 0


limx→∞ G(x, y) = −∞
Thus there is one unique
x∗h (y)
such that
∈ ∂V
if (x, y)
(x∗h (y), y) ∈ W
G(x∗h (y), y) = h.
.
by (A.0.2) and
59
Specically
x∗h (0) < 0.
Also we claim
lim x∗h (y) = −∞,
y→−1
or else there exist
b
and a sequence
{yk }k∈Z
such that
k→∞
yk −−−→ −1
&
x∗h (yk ) > b,
it
implies
lim G(x∗h (yk ), yk )
h =
k→∞
lim −e(2l−n)b (1 − yk2 )l
>
k→∞
= 0,
contradiction.
Lemma A.0.8. If 1 ≤ l < k, l <
& h < 0, set ℵ ≡ {(x(y), y) ∈ Ω} where
1 !
( kl ) k−l
1
x(y) = ln
|y| < 1,
2
1 − y2
n
2
(A.0.10)
then ℵ stays in W , and also {(x, y) ∈ Ω | ∂G
∂y = 0} = ℵ ∪ {(x, 0) | x ∈ R}. Moreover, for
all (x, y) ∈ Ω− ,
Proof.
If
1 ≤ l < k, l <
∂G
∂y
by solving
∂G
∂y
n
2



 ∂G < 0
∂y
if x < x(y)


 ∂G > 0
∂y
if x > x(y).
& h < 0,
= e(2k−n)x k(1 − y 2 )k−1 (−2y) − e(2l−n)x l(1 − y 2 )l−1 (−2y)
= (−2y)e(2l−n)x (1 − y 2 )l−1 ke(2k−2l)x (1 − y 2 )k−l − l ,
= 0,
we obtain
y=0
or the curve
1
1
x(y) =
ln
2
Thus
( kl ) k−l
1 − y2
!
|y| < 1.
{(x, y) ∈ Ω | ∂G
∂y = 0} = ℵ ∪ {(x, 0) | x ∈ R}
x(y)
so it stays in
W.
(A.0.11)
1
< − ln(1 − y 2 )
2
and obviously
|y| < 1,
(x, y) ∈ Ω− with y 6= 0,



 ∂G < 0 if x < x(y)
∂y
Moreover for all


 ∂G > 0
∂y
if
x > x(y).
60
Lemma A.0.9. Let ~ := ( )
l
k
∂G ∗
∂y (xh (y), y)
2k−n
2(k−l)
2l−n
− ( kl ) 2(k−l) and if 1 ≤ l < k, l <
n
2
& h < ~, then
< 0 and consequently x∗h (y) satises
∂G
∗
∂x∗h (y)
∂y (xh (y), y)
> 0.
= − ∂G ∗
∂y
∂x xh (y), y
Thus the level set given by (2.1.1)vis an U -shaped curve that opens left in W . If 1 ≤ l <
u
u
k, l < n2 & ~ ≤ h < 0, let κ∗h ≡ −t1 −
!2
n
1
l k−l
, then x∗h (y) satises
h
k
( kl )
k
k−l −
( kl )

∗


 ∂xh (y) > 0
∂y
if − 1 < y < κ∗h
∗


 ∂xh (y) < 0
∂y
if kh∗ < y < 0.
Ω,



G(x, y) = h
l
k−l
First we consider the equation system in
.
(A.0.12)


x = x(y)
If
1 ≤ l < k, l <
n
2
& h < ~,
there is no solution to
(A.0.12),
and
x∗h (0) < x(0),
thus
x∗h (y) < x(y).
By (A.0.11),
∂G ∗
∂y (xh (y), y)
<0
and by
(A.0.4)
&
(A.0.8),
∂G ∗
∂x (xh (y), y)
> 0,
so
∂G
∗
∂x∗h (y)
∂y (xh (y), y)
> 0.
= − ∂G ∗
∂y
∂x xh (y), y
!
If
1 ≤ l < k, l <
(A.0.12)
n
2
& ~ ≤ h < 0,
let
ζh∗ ≡ x(κ∗h ) = − n1 ln
h
k
l
( kl ) k−l −( kl ) k−l
,
has the solutions
(ζh∗ , ±κ∗h ).
When
−1 < y < kh∗ ,
since
(A.0.9),
we have
x∗h (y) < x(y),
(A.0.13)
thus
∂G ∗
∂y (xh (y), y)
∂G
∗
∂x∗h (y)
∂y (xh (y), y)
> 0.
= − ∂G ∗
∂y
∂x xh (y), y
When
then
κ∗h < y < 0,
we have
x∗h (y) > x(y),
thus
∂G ∗
∂y (xh (y), y)
∂G
∗
∂x∗h (y)
∂y (xh (y), y)
< 0.
= − ∂G ∗
∂y
∂x xh (y), y
Proof of (N3) (N4) (N5) in Case h < 0
>0
and
<0
and
61
Lemma A.0.10. If 1 < l =
n
2
−
< k , then ∂G
∂x > 0 in Ω and also in Ω



 ∂G < 0 if x < x(y)
∂y
if x > x(y)


 ∂G > 0
∂y
and then
G(x, y) >
G(x, y) = −1
inf
(x, y) ∈ Ω.
(x,y)∈Ω
So if 1 ≤ l < k ≤
n
2
& − 1 < h < ~, then
satises
∂G ∗
∂y (xh (y), y)
(A.0.14)
< 0 and consequently x∗h (y)
∂G
∗
∂x∗h (y)
∂y (xh (y), y)
> 0.
= − ∂G ∗
∂y
x
(y),
y
h
∂x
Thus the level set given by (2.1.1)
v is an U -shaped curve that opens left in Ω. If 1 ≤ l <
u
u
n
∗
k ≤ 2 & ~ ≤ h < 0, let κh ≡ −t1 −
If 1 < l =
Proof.
If
n
2
!2
n
1
l k−l
h
k
l
k
( )
k
k−l −
l
k
( )
l
k−l

∗


 ∂xh (y) > 0
∂y
if − 1 < y < κ∗h
∗


 ∂xh (y) < 0
∂y
if kh∗ < y < 0.
, then x∗h (y) satises
< k & h ≤ −1, there is no solution for (2.1.1).
1<l=
n
2
< k,
then for
(x, y) ∈ Ω,
∂G
= (2k − n)e(2k−n)x (1 − y 2 )k > 0
∂x
and similar to Lemma A.0.8, we have that in
thus for any
(x, y) ∈ Ω, ln
(A.0.15)
(A.0.16),
and
l
k



 ∂G < 0
∂y
if


 ∂G > 0
∂y
if
1
k−l
(A.0.15)
Ω−
x < x(y)
(A.0.16)
x > x(y),
< miny∈(−1,1) x(y) < x(y),
consequently by
we obtain
1 ) !
l k−l
G(x, y) ≥ G min x, ln
,y
k
(
1 ) !
l k−l
≥ G min x, ln
,0
k
(
>
lim G(x, 0)
x→−∞
= −1,
(A.0.1),
62
so we have
(A.0.14).
The proof for
n
2
& ( kl )
2k−n
2(k−l)
If
1 < l =
1 ≤ l < k ≤
− ( kl )
2l−n
2(k−l)
n
2
n
2
< k
h ≤ −1,
&
& − 1 < h < ( kl )
≤h<0
there is no solution to
2k−n
2(k−l)
− ( kl )
and
(2.1.1).
1 ≤ l < k ≤
is same as Lemma A.0.9.
Proof of (N6) (N7) (N8) (N9) in Case h < 0
Lemma A.0.11. If < l < k, the set O = {(x, y) ∈ Ω |
n
2
x̃(y) =
2l−n
2(k−l)
1
ln
2
2l−n
2k−n
∂G
∂x
= 0} is the curve
1
k−l
|y| < 1,
1 − y2
(A.0.17)
and x̃(y) < x(y) < − 21 ln(1 − y 2 ) ∀|y| < 1, which is dened in (A.0.10). Consequently
we have



∂G
∂G


∂x < 0 & ∂y < 0



if x < x̃(y)
if x̃(y) < x < x(y)
∂G
∂G
∂x > 0 & ∂y < 0






 ∂G > 0 & ∂G > 0
∂x
∂y
Proof.
If
n
2
< l < k,
0 =
if x(y) < x.
then we solve
∂G
∂x
= (2k − n)e(2k−n)x (1 − y 2 )k − (2l − n)e(2l−n)x (1 − y 2 )l
= e(2l−n)x (1 − y 2 )l (2k − n)e(2k−2l)x (1 − y 2 )k−l − (2l − n)
in
Ω
to obtain the solution
x̃(y) =
1
ln
2
2l−n
2k−n
1
k−l
|y| < 1.
1 − y2
Obviously
1
x̃(y) < x(y) < − ln(1 − y 2 ) ∀|y| < 1
2
and consequently



∂G
∂G


∂x < 0 & ∂y < 0



∂G
∂G
∂x > 0 & ∂y < 0





 ∂G > 0 & ∂G > 0

∂x
∂y
if x
< x̃(y)
if x̃(y)
< x < x(y)
if x(y)
< x.
(A.0.18)
63
Lemma A.0.12. If
n−2l
n−2k
2l−n
2(k−l)
n
2
< l < k , then G(x, y) achieves it is minimum ℘ :=
n−2l
n−2k
2k−n
2(k−l)
−
at point (x̃(0), 0). Also
1. If ~ ≤ h < 0, the level set given by (2.1.1) is a closed curve, and it has intersection
points (A.0.13) with (A.0.10).
2. If ℘ < h < ~, the level set given by (2.1.1) is a closed curve. This curve and
(A.0.17) have two intersection points
v
u
 
u
−l
u

k−l
2l−n
 1  h 2k−n
1
 u

2l−n k−l
 , ±u
u
(ζ̃h , ±κ̃h ) = − ln 
1
−
2k−n
 t
2l−2k
 n 

2k−n

Besides, ∀(x, y) ∈ Ω solving (2.1.1) satises x <
well dened,



 ∂y < 0
∂x
if x < ζ̃h


 ∂G > 0
∂y
if x > ζ̃h
1
2

2
 
−l n 
2l−n k−l
h

 
 2k−n  
,

 
2l−2k


2k−n
1
ln( kl ) k−l , and in Ω− y(x) is
(x, y) ∈ Ω− .
3. If h = ℘, (2.1.1) is a point (x̃(0), 0).
4. If h < ℘, (2.1.1) has no solution.
Proof.
If
n
2
< l < k , then




G(x, y) = 0




G(x, y) = 0






limx→−∞ G(x, y) = 0
The critical point of
G(x, y)
in
W
if (x, y)
∈ ∂V
if (x, y)
∈ ∂Ω
uniformally for
y ∈ (−1, 1).
is only
(x̃(0), 0),
G(x̃(0), 0) = ℘ < 0, thus G(x, y) achieves it is
2l−n
n−2l 2(k−l)
at point (x̃(0), 0). If ~ ≤ h < 0, consider
n−2k
and
minimum
℘ :=
the equation system
2k−n
2(k−l)
−
(A.0.12)
in
n−2l
n−2k
64
Ω, and we obtain the solutions (A.0.13).
there exist only two points
a3 (h)
&
Since
a4 (h)
(A.0.18) and h > G(x(0), 0) > G(x̃(0), 0),
such that
G(a3 (h), 0) = G(a4 (h), 0) = h,
and also
0 > a4 (h) > x(0) > x̃(0) > a3 (h).
see the level set given by
(2.1.1)
Repeating using
is a closed curve with
The proof of the remaining cases is similar.
x
(A.0.18)
ranging from
and
(A.0.1),
a3 (h)
to
ζh∗ .
we
65
References
[1] Aubin,T: Équations diérentielles non linéaires et problème de Yamabe
concernant la courbure scalaire, J. Math. Pures Appl. 55(3) 269296 (1976)
[2] Aviles, P.: A study of the singularities of solutions of a class of nonlinear
elliptic partial dierential equations. Commun. Partial Dier. Equ. 7, 609643 (1982)
[3] Bahri, A., Coron, J. M.: The scalar curvature problem on the standard
three dimensional sphere, J. Funct. Anal., 95 106-172 (1991)
[4] Besse, A. L.: Einstein Manifolds, Springer-Verlag, Berlin, 1987
[5] Bôcher, M.: Singular points of functions which satisfy partial dierential
equations of the elliptic type. Bull. Amer. Math. Soc., 9, 455-465 (1903)
[6] Brendle, S., Blow-up phenomena for the Yamabe equation I, J. Amer.
Math. Soc. 21, 951-979 (2008)
[7] Brendle, S., Marques, F. C.: Blow-up phenomena for the Yamabe equation
II, J. Dier. Geom. 81, no. 2, 225-250 (2009)
[8] Caarelli, L., Nirenberg, L., Spruck, J.: The Dirichlet problem for nonlinear second-order elliptic equations. III Functions of the eigenvalues of the
Hessian, Acta Math., 155, 261-301 (1985)
[9] Caarelli, L., Gidas, B., Spruck, J.: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Commun. Pure Appl. Math. 42, 271-297 (1989)
[10] Chang, S.-Y.A., Han, Z., Yang, P.: Classication of singular radial solutions to the
σk
Yamabe equation on annular domains. J. Dier. Equ. 216,
482501 (2005)
[11] Chang, S.-Y.A., Hang, F., Yang, P.: On a class of locally conformally at
manifolds. Int. Math. Res. Not. 2004(4), 185209 (2004)
[12] Chang, S.-Y. A., Gursky, M., Yang, P.: The scalar curvature equation on
2-
and
3-
spheres, Calculus of Variation, 1 205-229 (1993)
[13] Chang, S.-Y. A., Gursky, M., Yang, P.: An equation of Monge-Ampère
type in conformal geometry, and four-manifolds of positive Ricci curvature,
Ann. of Math. (2), 155, 709-787 (2002)
[14] Chang, S.-Y. A., Gursky, M., Yang, P.: An a prior estimate for a fully
nonlinear equation on four-manifolds, J. D'Analyse Math.; Thomas Wol
memorial issue, 87 151-186 (2002)
66
[15] Chang, S.-Y. A., Gursky, M., Yang, P.: A conformally invariant sphere
theorem in four dimensions, Publ. Math. Inst. Hautes Études Sci., 105143 (2003)
[16] Chang, S.-Y. A., Gursky, M., Yang, P.: Entire solutions of a fully nonlinear
equation, in Lectures on partial dierential equations, vol. 2 of New Stud.
Adv. Math., Int. Press, Somerville, MA, 43-60 (2003)
[17] Chang, S.-Y., Yang, P.: Prescribing Gaussian curvature on
S2 . Acta Math.
159, 215-259 (1987)
[18] Chang, S.-Y., Yang, P.: Conformal deformation of metrics on
S2 . J. Dier.
Geom. 27, 259-296 (1988)
[19] Chang, S.-Y., Yang, P.: A perturbation result in prescribing scalar curvature on
Sn ,
Duke Math. J. 64 no.1, 27-69 (1991)
[20] Chen, C.-C., Lin C.-S.: Prescribing scalar curvature on
Sn , Part I, A priori
estimates, J. Dier. Geom., 57 67-171 (2002)
[21] Coddington, E., Levison, N.: Theory of Ordinary Dierential Equations.
McGraw-Hill, New York (1955)
[22] Druet, O.: From one bubble to several bubbles: the low-dimensional case,
J. Dier. Geom., 63, 399-473 (2003)
[23] Druet, O.: Compactness for Yamabe metrics in low dimensions, Int. Math.
Res. Not., 1143-1191 (2004)
[24] Finn, D.: Positive solutions to nonlinear elliptic equations with prescribed
singularities . Ph.D. thesis, Northeastern University (1995)
[25] Ge,Y.,Lin, C.S., Wang, G.-F.: On the
σ2 -scalar
curvature. J. di. G. 84,
45-86 (2010)
[26] Ge,Y.,Wang, G.-F.:
On a conformal quotient equation.Int. Math. Res.
Not. 2007, no. 6, Art. ID rnm019, (2007)
[27] Ge, Y., Wang, G.: On a fully nonlinear Yamabe problem, Ann. Sci. Ecole
Norm. Sup, 39 no. 4, 569-598 (2006)
[28] Ge, Y., Wang, G.: On a conformal quotient equation. Int. Math. Res. Not.
IMRN no. 6, Art. ID rnm019, 32 pp. (2007)
[29] González, M.: Singular sets of a class of locally conformally at manifolds.
Duke Math. J. 129(3), 551572 (2005)
[30] González, M.: Removability of singularities for a class of fully non-linear
elliptic equations. Calc. Var. Partial Dierential Equations. 27, 439-466
(2006)
[31] Guan, P., Lin, C.S., Wang, G.:
Application of the Method of Moving
Planes to conformally Invariant Equations, Math. Z.,247, 1-19, (2004)
67
[32] Guan, P., Lin, C.S., Wang, G.:
Schouten tensor and some topological
properties. Commun. Anal. Geom. 13(5), 887902 (2005)
[33] Guan, P., Lin, C.S., Wang, G.:
Local gradient estimates for conformal
quotient equations, Int. J. Math. 18(4), 349-362, (2007)
[34] Guan, P., viaclovsky J., Wang, G.: Some properties of the Schouten tensor
and applications to conformal geometry, Trans. Amer. Math. Soc., 355 no.
3, 925-933 (2003)
[35] Guan, P., Wang, G.: Local estimates for a class of fully nonlinear equations
arising from conformal geometry. Int. Math. Res. Not. 2003(26), 14131432
(2003)
[36] Guan, P., Wang, G.: A fully nonlinear conformal ow on locally conformally at manifolds, J. Reine Angew. Math., 557 219-238 (2003)
[37] Guan, P., Wang, G.:
Geometric inequalities on locally conformally at
manifolds, Duke Math. J.124, 177-212, (2004)
[38] Gursky, M., viaclovsky, J.: A new variational characterization of threedimensional space forms, Invent. Math., 145 251-278 (2001)
[39] Gursky, M., viaclovsky, J.: Prescribing symmetric functions of the eigenvalues of the Ricci tensor, Ann. of Math. (2) 166 no.2, 447-487 (2004)
[40] Gursky, M., Viaclovsky, J.: Prescribing symmetric functions of the eigenvalues of the Ricci tensor, Ann. of Math. (2) 166, 47553 (2007)
[41] Han, Z.-C..:
Prescribing Gaussian curvature on
S2 ,
Duke Math. J. 61,
for
the
679-703 (1990)
[42] Han,
Z.-C.:
A
Kazdan-Warner
type
identity
σk
curvature.
C.R.Math. 342(7),475-478 (2006)
[43] Han, Z.-C., Li,Y.Y., Teixeira, E.: Asymptotic behavior of solutions to the
σk -Yamabe equation near isolated singularities. Invent math. 182,
635-684
(2010)
[44] Kazdan, J., Warner, F.:
Curvature functions for compact
2-manifolds,
Ann. of Math. (2) 99, 14-47 (1974)
[45] Khuri, M., Marques, F. C., Schoen, R.: A compactness theorem for the
Yamabe problem, J. Dier., Geom. 81, no.1, 143-196 (2009)
[46] Koutrouotis, D.: Gaussian curvature and conformal mapping, J. Dier.
Geom. 7, 479-488 (1972)
[47] Korevaar, N., Mazzeo, R., Pacard, F., Schoen, R.: Rened asymptotics for
constant scalar curvature metrics with isolated singularities. Invent. Math.
135. 233-272 (1999)
[48] Labutin, D.: Wiener regularity for large solutions of nonlinear equations.
Ark. Mat. 41(2), 307339 (2003)
68
[49] Le Gall, J.:
Spatial Branching Processes, Random Snakes and Partial
Dierential Equations. Lectures in Mathematics ETH Zürich. Birkhäuser,
Basel (1999)
[50] Lelong-Ferrand, J.: Transformations conformes et quasi-conformes des variétés riemanniennes compactes (démonstration de la conjecture de A. Lichnerowicz), Acad. Roy. Belg. Cl. Sci. Mém. Coll. in-8deg (2) 39 no. 5, 44
(1971)
[51] Li, A. and Li, Y.Y.: A fully nonlinear version of the Yamabe problem and
a Harnack type inequality, C. R. Math. Acad. Sci. Paris, 336 no. 4, 319-324
(2003)
[52] Li, A. and Li, Y.Y.: On some conformally invariant fully nonlinear equations, Comm. Pure Appl. Math., 56 no. 10 1416-1464 (2003)
[53] Li, A. and Li, Y.Y.: On some conformally invariant fully nonlinear equations. II. Liouville, Harnack and Yamabe, Acta Math. 195, 117-154 (2005)
[54] Li, A., Li, Y.Y.: A Liouville type theorem for some conformally invariant
fully nonlinear equations, Geometric analysis of PDE and several complex
variables, Contemp. Math., 368, Amer. math. Soc., Providence, RI,321328, (2005)
[55] Li, Y.Y.: Prescribing scalar curvature on
S3 , S4
and related problems. J.
Funct. Anal. 118, 43-118 (1993)
[56] Li, Y.Y.: Prescribing scalar curvature on
Sn
and related problems.
I.,
J.
Dier. Equ. 120, 319-410 (1995)
[57] Li, Y.Y.: Prescribing scalar curvature on
Sn
and related problems, Part II:
Existence and compactness, Comm. Pure and Appl. Math., XLIX 541-587
(1996)
[58] Li, Y.Y.: Conformally invariant fully nonlinear elliptic equations and isolated singularities. J. Funct. Anal. 233, 380425 (2006)
[59] Li, Y.Y.: Local gradient estimates of solutions to some conformally invariant fully non-linear equations, Comm. Pure Appl. Math., 62, 1293-1326
(2009) (C. R. Math. Acad. Sci. Paris 343, no. 4, 249-252 (2006)
[60] Li,Y.Y., Nguyen, L.:
Harnack inequalities and Bôcher-type theorems
for conformally invariant fully nonlinear degenerate elliptic equations.
arXiv:1206.6264v1 (2012)
[61] Li,Y.Y., Nguyen, L.: A compactness theorem for a fully nonlinear Yamabe
problem under a lower Ricci curvature bound. arXiv:1212.0460v1
[62] Li, Y.Y., Zhang, L.: A Harnack type inequality for the Yamabe equation
in low dimensions, Cal. Var. Partial Dierential Equations, 20, 133-151
(2004)
69
[63] Li, Y.Y., Zhang, L.: Compactness of solutions to the Yamabe problem. II,
Calc. Var. Partial Dierential Equations, 24, 185-237 (2005)
[64] Li, Y.Y., Zhang, L.: Compactness of solutions to the Yamabe Problem.
III, J. Funct. Anal., 245, 438-474 (2007)
[65] Li, Y.Y., Zhu, M.: Yamabe type equations on three-dimensional Riemannian manifolds, Commun. Contemp. Math., 1, 1-50 (1999)
[66] Loewner, C., Nirenberg, L.: Partial dierential equations invariant under
conformal and projective transformation. In: Contributions to Analysis,
245-275. Academic Press, New York (1975)
[67] Marcus, M., Veron, L.: Boundary trace of positive solutions of semilinear
elliptic equations in Lipschitz domains. arXiv:0907.1006 [math.AP] (2009)
[68] Marques, F. C., A priori estimates for the Yamabe problem in the nonlocally conformally at cases, J. Dier. Geom., 71, 315-346 (2005)
[69] Mazzeo, R., Pacard, F.: A construction of singular solutions for a semilinear elliptic equation using asymptotic analysis. J. Dier. Geom. 44(2),
331370 (1996)
[70] Mazzeo, R., Pacard, F.: Constant scalar curvature metrics with isolated
singularities. Duke Math. J. 99(3), 353418 (1999)
[71] Mazzeo, R., Smale, N.: Conformally at metrics of constant positive scalar
curvature on subdomains of the sphere. J. Dier. Geom. 34, 581-621 (1988)
[72] Mazzeo, R., Pollack, D., Uhlenbeck, K. Moduli spaces of singular Yamabe
metrics. J. of the A.M.S., 9, 303-344 (1996)
[73] Mazzieri, L., Ndiaye, C., Existence of solutions for the singular
σk -Yamabe
Problem. (preprint)
[74] McOwen, R.: Singularities and the conformal scalar curvature equation. In:
Geometric Analysis and Nonlinear Partial Dierential Equations, Denton,
TX, 1990. Lecture Notes in Pure and Appl. Math., vol. 144, pp. 221233.
Dekker, New York (1993)
[75] Moser, J.:
On a nonlinear problem in dierential geometry. Dynamical
systems (M.Peixoto, Editor), Academic Press, New York (1973)
[76] Obata, M.: The conjectures on conformal transformations of Riemannian
manifolds, J. Dier. Geom. 6, 247-258 (1971/72)
[77] Schoen, R.: Conformal deformation of a Riemannian metric to constant
scalar curvature, J. Dier. Geom. 20, 479-495 (1984)
[78] Schoen, R.: The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation. Commun. Pure Appl.
Math. 41, 317392 (1988)
70
[79] Schoen, R.: on the number of constant scalar curvature metrics in a conformal class, in Dierential Geometry, vol.52 of Pitman Monogr. Surveys
Pure Appl. Math., Longman Sci. Tech., Harlow, 311-320 (1991)
[80] Schoen, R., Zhang, D., Prescribed scalar curvature on the
n-sphere,
Cal.
of Vari., 4 1-25 (1996).
[81] Schoen, R., Yau, S.: Conformally at manifolds, Klein groups and scalar
curvature. Invent. Math. 92, 47-72 (1988)
[82] Sheng, W., Trudinger, N., Wang, X.:
The Yamabe problem for higher
order curvatures. J. Dier. Geom. 77(3), 515-553 (2007)
[83] Trudinger, N.: Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Sc. Norm. Super. Pisa, Cl. Sci.
22, 265-274 (1968)
[84] Trudinger, N., Wang, X.: The intermediate case of the Yamabe problem
for higher order curvatures, Int. Math. Res. Not. IMRN, no. 13, 2437-2458
(2010)
[85] Veron, L.: Singularités éliminables d'équations elliptiques non linéaires. J.
Dier. Equ. 41, 225-242 (1981)
[86] viaclovsky, J.: Conformal geometry, contact geometry, and the calculus of
variations, Duke Math. J. 101 no.2 283-316 (2000)
[87] viaclovsky, J.: Some fully nonlinear equations in conformal geometry, Differential equations and mathematical physics (Birmingham, AL, 1999),
Amer. Math. Soc. Providence, RI, 425-433 (2000)
[88] viaclovsky, J.:
Estimates and existence results for some fully nonlinear
elliptic equations on Riemannian manifolds, Comm. Anal. Geom., 10, 815846 (2002)
[89] Wang, Y.:
Asymptotic behavior of solutions to the conformal quotient
equation near isolated singularities. (preprint)
[90] Yamabe, H,: On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 21-37 (1960)
71
Vita
Yunpeng Wang
2007-2013
2004-2007
2000-2004
2007-2013
Ph. D. in Mathematics, Rutgers University
M. Sc. in Mathematics, Graduate University of Chinese Academy of Sciences.
B. Sc. in Mathematics from Wuhan University.
Teaching assistant, Department of Mathematics, Rutgers University