1 Introduction
This work aims to prescribe the scalar curvature on Cauchy–Riemann spheres. The prescription of the scalar curvature in the Riemannian settings is known to be the Kazdan–Warner problem and it was extensively studied by many authors for the dimension 2, 3 and 4 as well as in high dimensions. This problem was treated in many papers, as well as the related differential equation, like in [16, 12, 14, 15, 13, 31, 32, 34, 35]. For the CR settings we cite [22, 36].
Here, we will merely refer to [3, 5, 11, 10] and recently [17] which are the closest works to ours, using Abbas Bahri’s method, based on the theory of critical points at infinity. For Cauchy–Riemannian manifolds we cite [24], which is the pioneer work and more recently [20, 19, 18, 25, 29, 28, 27, 26].
Let 𝕊2n+1 be the unit sphere of ℂn+1 endowed with its standard contact form θ1, and let K:𝕊2n+1→ℝ be a given C2 positive function. The problem of finding a contact form θ on 𝕊2n+1 conformal to θ1 admitting the function K as Webster scalar curvature is equivalent to the resolution of the following semi-linear equation:
(1.1)
{
L
θ
1
u
=
K
u
1
+
2
n
on
𝕊
2
n
+
1
,
u
>
0
,
where Lθ1 is the conformal Laplacian of 𝕊2n+1, Lθ1=(2+2n)Δθ1+Rθ1, where Δθ1=Δ𝕊2n+1 and Rθ1=n(n+1)2 are respectively the sub-Laplacian operator and the Webster scalar curvature of (𝕊2n+1,θ1).
Problem (1.1) has a nice variational structure, the associated Euler functional is
J
(
u
)
=
∫
𝕊
2
n
+
1
L
θ
1
u
u
θ
1
∧
d
θ
1
n
(
∫
𝕊
2
n
+
1
K
u
2
+
2
n
θ
1
∧
d
θ
1
n
)
n
n
+
1
,
u
∈
S
1
2
(
𝕊
2
n
+
1
)
,
where S12(𝕊2n+1) is the completion of C∞(𝕊2n+1) by means of the norm ∥u∥2=∫𝕊2n+1Lθ1uuθ1∧dθ1n and is called a Folland–Stein space, cf. [23].
Let
Σ
=
{
u
∈
S
1
2
(
𝕊
2
n
+
1
)
∣
∥
u
∥
=
1
}
and
Σ
+
=
{
u
∈
Σ
∣
u
≥
0
}
.
The functional J fails to satisfy the Palais–Smale condition denoted by (PS) on Σ+, which is: there exist noncompact sequences along which the functional J is bounded and its gradient goes to zero. The failure of the (PS) condition has been analyzed for the Riemannian case throughout the works of [1, 3, 10, 9, 34, 35, 37, 38]. For the Cauchy–Riemannian case, a complete description of sequences that fail to satisfy (PS) is given in [30]. A solution u of (1.1) is a critical point of J subject to the constraint u∈Σ+.
Using the CR equivalence F induced by the Cayley transform (see Definition 2.1 below) between 𝕊2n+1 minus a point and the Heisenberg group ℍn, equation (1.1) is equivalent up to an influent constant to
(1.2)
{
(
2
+
2
n
)
Δ
ℍ
n
u
=
K
~
u
1
+
2
n
on
ℍ
n
,
u
>
0
where Δℍn is the sub-Laplacian of ℍn and K~=K∘F-1.
In this work, we focus on the case n=1. In order to give new existence results for problem (1.1), where the prescribed function K satisfies a β-flatness condition near its critical points, we will use the same techniques given in [24]. These techniques are based on the work of Bahri [1, 3]. More precisely, the method we will use consists on studying the critical points at infinity of the associated variational problem, by computing their total Morse index and then comparing this total index to the Euler characteristic of the space of variation.
To state our results, we set up the following conditions and notations.
Let G(a,⋅) be a Green’s function for L at a∈𝕊3.
We denote by
𝒦
=
{
(
ξ
i
)
(
1
≤
i
≤
r
)
∣
∇
K
(
ξ
i
)
=
0
}
the set of all critical points of K. We say that K satisfies the β-flatness condition if for all ξi∈𝒦, there exist
2
≤
β
=
β
(
ξ
i
)
<
4
and
b
1
=
b
1
(
ξ
i
)
,
b
2
=
b
2
(
ξ
i
)
,
b
0
=
b
0
(
ξ
i
)
∈
ℝ
∗
such that in some pseudo-hermitian normal coordinates system centered at ξi, we have
(1.3)
K
(
x
)
=
K
(
0
)
+
b
1
|
x
1
|
β
+
b
2
|
x
2
|
β
+
b
0
|
t
|
β
2
+
ℛ
(
x
)
,
where ∑k=12bk+κb0≠0 and ∑k=12bk+κ′b0≠0 with
κ
=
∫
ℍ
1
|
t
|
β
2
1
-
|
|
z
|
2
-
i
t
|
2
|
1
+
|
z
|
2
-
i
t
|
6
θ
0
∧
d
θ
0
∫
ℍ
1
|
x
1
|
β
1
-
|
|
z
|
2
-
i
t
|
2
|
1
+
|
z
|
2
-
i
t
|
6
θ
0
∧
d
θ
0
,
κ
′
=
∫
ℍ
1
|
t
|
β
2
|
1
+
|
z
|
2
-
i
t
|
4
θ
0
∧
d
θ
0
∫
ℍ
1
|
x
1
|
β
|
1
+
|
z
|
2
-
i
t
|
4
θ
0
∧
d
θ
0
.
We have
∑
p
=
0
[
β
]
|
∇
p
ℛ
(
x
)
|
∥
x
∥
ℍ
1
-
β
-
r
=
o
(
1
)
as x approaches ξi, where ∇r denotes all possible partial derivatives of order r and [β] the integer part of β.
Let
𝒦
1
:=
{
ξ
i
∈
𝒦
∣
(1.3) is satisfied with
β
=
β
(
ξ
i
)
=
2
and
∑
k
=
1
2
b
k
+
κ
′
b
0
<
0
}
,
𝒦
2
:=
{
ξ
i
∈
𝒦
∣
(1.3) is satisfied with
β
=
β
(
ξ
i
)
>
2
and
b
1
+
b
2
+
κ
′
b
0
<
0
}
.
The index of the function K at ξi∈𝒦, denoted by m(ξi), is the number of strictly negative coefficients bk(ξi):
m
(
ξ
i
)
=
#
{
b
k
(
ξ
i
)
∣
b
k
(
ξ
i
)
<
0
}
.
With each p-tuple (ξi1,…,ξip)∈(𝒦1)p (ξil≠ξij if l≠j), we associate the matrix M(ξi1,…,ξip)=(Mst)1≤s,t≤p given by
(1.4)
{
M
s
s
=
-
c
∑
k
=
1
2
b
k
+
κ
′
b
0
2
K
2
(
ξ
s
)
,
M
s
t
=
-
c
′
G
(
ξ
s
,
ξ
t
)
[
K
(
ξ
s
)
K
(
ξ
t
)
]
1
2
,
for
s
≠
t
,
where
c
=
∫
ℍ
1
|
x
1
|
2
|
1
+
|
z
|
2
-
i
t
|
4
,
c
′
=
2
π
ω
3
and ω3 is the volume of the unit Koranyi’s ball.
We say that K satisfies condition (C) if for each p-tuple (ξi1,…,ξip)∈(𝒦1)p the corresponding matrix (Mst) is non-degenerate. In this case, we denote by ϱ(ξi1,…,ξip) the least eigenvalue of the matrix M(ξi1,…,ξip).
Next, we define the sets
𝒦
1
+
:=
⋃
p
{
(
ξ
i
1
,
…
,
ξ
i
p
)
∈
(
𝒦
1
)
p
∣
ϱ
(
ξ
i
1
,
…
,
ξ
i
p
)
>
0
}
,
l
+
:=
max
{
p
∈
ℕ
∣
there exists
(
ξ
i
1
,
…
,
ξ
i
p
)
∈
𝒦
1
+
}
.
For (ξi1,…,ξip)∈𝒦1+, let
i
(
ξ
i
1
,
…
,
ξ
i
p
)
:=
4
p
-
1
-
∑
j
=
1
p
m
(
ξ
i
j
)
.
The following theorem is the main result of this paper.
Theorem 1.1
Let K be a C2 positive function on S3 satisfying the β-flatness condition and condition (C). If
∑
ξ
∈
𝒦
2
(
-
1
)
3
-
m
(
ξ
)
+
∑
p
=
1
l
+
∑
(
ξ
i
1
,
…
,
ξ
i
p
)
∈
𝒦
1
+
(
-
1
)
i
(
ξ
i
1
,
…
,
ξ
i
p
)
≠
1
,
then there exists at least one solution of equation (1.1).
The proof of Theorem 1.1 will be obtained by a contradiction argument. Therefore we assume that equation (1.1) has no solution. Our approach involves a Morse lemma at infinity, it relies on the construction of a suitable pseudo-gradient for the functional J. The Palais–Smale condition is satisfied along the decreasing flow lines of this pseudo-gradient, as long as these flow lines do not enter the neighborhood of a finite number of critical points of K where the related matrix given in (1.4) is positive definite.
This paper is organized as follows: In Section 2, the problem framework is introduced, we begin by recalling the local structure of the Heisenberg group, the extremals for the Yamabe functional on ℍ1 and the Cayley transform which gives, when restricted to the sphere minus a point, a CR diffeomorphism with the Heisenberg group. In Section 3, the Morse lemma at infinity is detailed, we begin by constructing the pseudo-gradient for the functional J, then we refine the expansion of J near the sets of its critical points at infinity using a change of variables. The proof of our main result, i.e. Theorem 1.1, is presented in Section 4. The last section is an appendix, where some technical estimates used in the different sections of this paper are developed.
2 Preliminary Tools
The Heisenberg group ℍn is the Lie group whose underlying manifold is ℂn×ℝ, with coordinates g=(z,t)=(z1,…,zn,t) and whose law group is given by
g
⋅
g
′
=
(
z
,
t
)
⋅
(
z
′
,
t
′
)
=
(
z
+
z
′
,
t
+
t
′
+
2
Im
z
.
z
¯
′
)
,
where z.z¯′=∑j=1nzj.z¯j′. We define a norm in ℍn by
∥
g
∥
ℍ
n
=
∥
(
z
,
t
)
∥
ℍ
n
=
(
∥
z
∥
4
+
t
2
)
1
4
,
and dilations by g=(z,t)→λg=(λz,λ2t), λ>0. The CR structure on ℍn is given by the left invariant vector fields
Z
j
=
∂
∂
z
j
+
i
z
¯
∂
∂
t
,
Z
¯
j
=
∂
∂
z
¯
j
-
i
z
∂
∂
t
(
1
≤
j
≤
n
)
,
which are homogenous of degree -1 with respect to the dilations, the associated contact form is
θ
0
=
d
t
+
∑
j
=
1
n
(
i
z
j
d
z
¯
j
-
i
z
¯
j
d
z
j
)
.
We denote by Δθ0 the sub-Laplacian operator:
Δ
θ
0
=
-
1
2
∑
j
=
1
n
(
Z
j
Z
¯
j
+
Z
¯
j
Z
j
)
.
Since the Webster scalar curvature Rθ0 is zero, the conformal Laplacian L0 is equal to (2+2n)Δθ0.
In [33] Jerison and Lee showed that all solutions of (1.2) are obtained from
w
(
0
,
1
)
(
z
,
t
)
=
c
0
|
1
+
|
z
|
2
-
i
t
|
n
,
c
0
>
0
,
by left translations and dilatations on ℍn. That is, for g0=(z0,t0), g=(z,t) in ℍn and λ>0, we have
w
(
g
0
,
λ
)
(
z
,
t
)
=
c
0
λ
n
|
1
+
λ
2
|
z
-
z
0
|
2
-
i
λ
2
(
t
-
t
0
-
2
Im
z
0
z
¯
)
|
n
.
Next, we will introduce the Cayley transform. Let Bn+1={z∈ℂn+1∣|z|<1} be the unit ball in ℂn+1 and 𝒟n+1={(z,w)∈ℂn×ℂ∣Im(w)>|z|2} the Siegel domain, where ∂𝒟n+1={(z,w)∈ℂn×ℂ∣Im(w)=|z|2}.
Definition 2.1
Definition 2.1 ([21])
The Cayley transform is the correspondence between the unit ball Bn+1 in ℂn+1 and the Siegel domain 𝒟n+1, given by
𝒞
(
ζ
)
=
(
ζ
′
1
+
ζ
n
+
1
,
i
1
-
ζ
n
+
1
1
+
ζ
n
+
1
)
,
ζ
=
(
ζ
′
,
ζ
n
+
1
)
,
1
+
ζ
n
+
1
≠
0
.
The Cayley transform gives a biholomorphism of the unit ball Bn+1 in ℂn+1 onto the Siegel domain 𝒟n+1. Moreover, when restricted to the sphere minus a point, 𝒞 gives a CR diffeomorphism.
𝒞
:
𝕊
2
n
+
1
∖
(
0
,
…
,
0
,
-
1
)
→
∂
𝒟
n
+
1
.
Let us recall the CR diffeomorphism
f
:
ℍ
n
→
∂
𝒟
n
+
1
,
(
z
,
t
)
↦
f
(
z
,
t
)
=
(
z
,
t
+
i
|
z
|
2
)
,
with the obvious inverse f-1(z,w)=(z,Re(w)), z∈ℂn, w∈ℂ. We obtain the CR equivalence with the mapping
F
:
𝕊
2
n
+
1
∖
(
0
,
…
,
0
,
-
1
)
→
ℍ
n
,
ζ
=
(
ζ
1
,
…
,
ζ
n
+
1
)
↦
(
z
,
t
)
=
(
ζ
1
1
+
ζ
n
+
1
,
…
,
ζ
n
1
+
ζ
n
+
1
,
i
2
Im
ζ
n
+
1
|
1
+
ζ
n
+
1
|
2
)
whose inverse is
F
-
1
:
ℍ
n
→
𝕊
2
n
+
1
∖
(
0
,
…
,
0
,
-
1
)
,
(
z
,
t
)
↦
ζ
=
(
2
z
1
1
+
|
z
|
2
-
i
t
,
…
,
2
z
n
1
+
|
z
|
2
-
i
t
,
i
1
-
|
z
|
2
+
i
t
1
+
|
z
|
2
-
i
t
)
.
If we choose the standard contact form of 𝕊2n+1 as
θ
1
=
i
∑
j
=
1
n
+
1
(
ζ
j
d
ζ
¯
j
-
ζ
¯
j
d
ζ
j
)
,
then we have
F
*
(
4
(
c
0
-
1
w
(
0
,
1
)
)
2
n
θ
0
)
=
θ
1
.
Let us differentiate and take into account that w(0,1)(F(ζ))=c0|1+ζn+1|2. We obtain
d
θ
1
=
(
d
ζ
n
+
1
1
+
ζ
n
+
1
+
d
ζ
¯
n
+
1
1
+
ζ
¯
n
+
1
)
∧
θ
1
+
|
1
+
ζ
n
+
1
|
2
F
*
(
d
θ
0
)
and
θ
1
∧
d
θ
1
n
=
|
1
+
ζ
n
+
1
|
2
(
n
+
1
)
F
*
(
θ
0
∧
d
θ
0
n
)
.
We introduce the following function for each (ζ0,λ) on 𝕊2n+1×]0,+∞[:
δ
(
ζ
0
,
λ
)
(
ζ
)
=
|
1
+
ζ
n
+
1
|
-
n
w
(
F
(
ζ
0
)
,
λ
)
∘
F
(
ζ
)
.
We have
L
θ
1
δ
(
ζ
0
,
λ
)
=
δ
(
ζ
0
,
λ
)
1
+
2
n
,
i.e., δ(ζ0,λ) is a solution of the Yamabe problem on 𝕊2n+1.
We also have
∫
𝕊
2
n
+
1
L
θ
1
δ
(
ζ
0
,
λ
)
δ
(
ζ
0
,
λ
)
θ
1
∧
d
θ
1
n
=
∫
ℍ
n
L
θ
0
w
(
g
0
,
λ
)
w
(
g
0
,
λ
)
θ
0
∧
d
θ
0
n
and
∫
𝕊
2
n
+
1
|
δ
(
ζ
0
,
λ
)
|
2
+
2
n
θ
1
∧
d
θ
1
n
=
∫
ℍ
n
|
w
(
g
0
,
λ
)
|
2
+
2
n
θ
0
∧
d
θ
0
n
,
where g0=F(ζ0) and g=F(ζ).
As a consequence, the variational formulation for (1.1) is equivalent to the variational formulation for (1.2).
From now on, we focus on the case n=1. Here, we have the presence of multiple blow-up points. In fact, looking at the possible formations of blow-up points, it comes out that the interaction of two different bubbles given by 〈δai,λi,δaj,λj〉Lθ1 with i≠j dominates the self-interaction 〈δai,λi,δai,λi〉Lθ1 (see Section 2 for definitions), in the case where 2<β<4, whereas in the case where β=2 we have a balance phenomenon, that is, any interaction of two bubbles is of the same order with respect to the self-interaction.
We begin by defining the sets of potential critical points at infinity of the functional J.
For any ε>0 and p∈ℕ+, let
V
(
p
,
ε
)
=
{
u
∈
Σ
+
∣
there exist
(
a
1
,
…
,
a
p
)
∈
𝕊
3
,
α
1
,
…
,
α
p
>
0
and
(
λ
1
,
…
,
λ
p
)
∈
(
ε
-
1
,
∞
)
p
such that
∥
u
-
∑
i
=
1
p
α
i
δ
a
i
,
λ
i
K
(
a
i
)
1
2
∥
S
1
2
(
𝕊
3
)
<
ε
and
ε
i
j
<
ε
,
|
α
i
2
K
(
a
i
)
α
j
2
K
(
a
j
)
-
1
|
<
ε
for all
1
≤
i
≠
j
≤
p
}
,
where
ε
i
j
=
(
λ
i
λ
j
+
λ
j
λ
i
+
λ
i
λ
j
(
d
(
a
i
,
a
j
)
2
)
)
-
1
.
Then we proceed as in [30, Proposition 8] to characterize the sequences which violate the (PS) condition as follows:
Proposition 2.2
Proposition 2.2 ([30])
Let {uk} be a sequence such that ∂J(uk)→0 and J(uk) is bounded. Then there exist an integer p∈N*, a sequence εk→0 (εk>0) and an extracted subsequence of {uk}, again denoted by {uk}, such that uk∈V(p,εk).
We consider the following minimization problem for a function u∈V(p,ε) with ε small:
(2.1)
min
α
i
>
0
,
λ
i
>
0
,
a
i
∈
𝕊
3
∥
u
-
∑
i
=
1
p
α
i
δ
a
i
,
λ
i
∥
S
1
2
(
𝕊
3
)
.
As shown in [3, 24], we obtain the following parametrization of the set V(p,ε).
Proposition 2.3
Proposition 2.3 ([24])
For any p∈N*, there exists εp>0 such that, for any 0<ε<εp, u∈V(p,ε), the minimization problem (2.1) has a unique solution (α¯1,…,α¯p,λ¯1,…,λ¯p,a¯1,…,a¯p) (up to permutation on the set of indices {1,…,p}). In particular, we can write u∈V(p,ε) as
u
=
∑
i
=
1
p
α
¯
i
δ
a
¯
i
,
λ
¯
i
+
v
,
where v∈S12(S3) satisfies
(V0)
{
〈
v
,
δ
a
i
,
λ
i
〉
S
1
2
(
𝕊
3
)
=
0
,
〈
v
,
∂
δ
a
i
,
λ
i
∂
a
i
〉
S
1
2
(
𝕊
3
)
=
0
,
〈
v
,
∂
δ
a
i
,
λ
i
∂
λ
i
〉
S
1
2
(
𝕊
3
)
=
0
,
i
=
1
,
2
,
…
,
p
.
Here 〈⋅,⋅〉 denotes the L-scalar product defined on S12(S3) by
〈
u
,
v
〉
=
∫
𝕊
3
L
θ
1
u
v
θ
1
∧
d
θ
1
.
Next, we will focus on the behavior of the functional J with respect to the variable v. We will prove the existence of a unique v¯ which minimizes J(∑i=1pαiδ+ai,λiv) with respect to v∈Hεp(a,λ), where
H
ε
p
(
a
,
λ
)
=
H
ε
p
(
δ
a
1
,
λ
1
,
…
,
δ
a
p
,
λ
p
)
=
{
v
∈
S
1
2
(
𝕊
3
)
∣
v
satisfies (V0) and
∥
v
∥
<
ε
p
}
.
Proposition 2.4
Proposition 2.4 ([24])
There exists a C1-map which associates to each u∈V(p,ε), ε small, v¯=v¯(α,a,λ) such that v¯ is unique and minimizes J(∑i=1pαiδai,λi+v) with respect to v∈Hεp(a,λ). Furthermore, we have the following estimate:
∥
v
¯
∥
≤
c
1
(
∑
i
≤
p
(
|
∇
K
(
a
i
)
|
λ
i
+
1
λ
i
2
)
+
∑
k
≠
r
ε
k
r
Log
(
ε
k
r
-
1
)
)
.
3 Morse Lemma at Infinity
The Morse lemma at infinity establishes near the set of critical points at infinity of the functional J a change of variables in the space (ai,αi,λi,v), 1≤i≤p to (a~i,α~i,λ~i,V), (α~i=αi), where V is a variable completely independent of a~i and λ~i such that J(∑αiδai,λi) behaves like J(∑αiδa~i,λ~i)+∥V∥2. We define also a pseudo-gradient for the V variable in order to make this variable disappear by setting ∂V∂s=-νV, where ν is taken to be a very large constant. Then at s=1, V(s)=exp(-νs)V(0) will be as small as we wish. This shows that, in order to define our deformation, we can work as if V was zero. The deformation will be extended immediately with the same properties to a neighborhood of zero in the V variable.
We begin by characterizing the critical points at infinity of J in the sets V(p,ε), p≥1 under condition (1.3). This characterization is obtained through the construction of a suitable pseudo-gradient at infinity for the functional J for which the Palais–Smale condition is satisfied along the decreasing flow lines as long as these flow lines do not enter in the neighborhood of a finite number of critical points ξi, 1≤i≤p in 𝒦2 or such that (ξi,…,ξp)∈𝒦1+. Notice that the deformation lemmas in Morse theory are realized by using the gradient flow-lines or the flow-lines of any decreasing pseudo-gradient vector field.
We first introduce some definitions and notations due to Bahri [1, 3]. Let ∂J denote the gradient of the functional J.
Definition 3.1
A critical point at infinity of J on Σ+ is a limit of a flow line u(s) of the equation
{
∂
u
∂
s
=
-
∂
J
(
u
)
,
u
(
0
)
=
u
0
such that u(s) remains in V(p,ε(s)) for s≥s0, and ε(s) satisfies lims→∞ε(s)=0.
One can write
u
(
s
)
=
∑
i
=
1
p
α
i
(
s
)
δ
(
a
i
(
s
)
,
λ
i
(
s
)
)
+
v
(
s
)
.
Let ai:=lims→∞ai(s) and αi:=lims→∞αi(s). We denote such a critical point at infinity by
ξ
∞
or
(
a
1
,
…
,
a
p
)
∞
or
∑
i
=
1
p
α
i
δ
(
a
i
,
∞
)
.
To a critical point at infinity ξ∞ are associated stable and unstable manifolds Ws(ξ∞) and Wu(ξ∞). Those manifolds allow to compare critical points at infinity by what we call a “domination property”. For a detailed description of these manifolds, we refer to [3, 24].
Definition 3.2
A critical point at infinity ξ∞ is said to be dominated by another critical point at infinity ξ∞′ if
W
s
(
ξ
∞
)
∩
W
u
(
ξ
∞
′
)
≠
∅
;
we write ξ∞′>ξ∞.
If we assume that the intersection Ws(ξ∞)∩Wu(ξ∞′) is transverse, then we obtain
index
(
ξ
∞
′
)
≥
index
(
ξ
∞
)
+
1
.
3.1 Construction of the Pseudo-Gradient
In the set V(1,ε), we have the following result:
Proposition 3.3
Assume that K satisfies the β-flatness condition and condition (C) and let
β
:=
max
{
β
(
ξ
i
)
∣
ξ
i
verifying (1.3)
}
.
Then there exist a pseudo-gradient W and a constant c>0 independent of u=αδ(a,λ)∈V(1,ε), ε small enough, such that, if we write u¯=u+v¯, we have
-
J
′
(
u
)
(
W
)
≥
c
(
|
∇
K
(
a
)
|
λ
+
1
λ
β
)
.
-
J
′
(
u
¯
)
(
W
+
∂
v
¯
∂
(
α
,
a
,
λ
)
(
W
)
)
≥
c
(
|
∇
K
(
a
)
|
λ
+
1
λ
β
)
.
|
W
|
is bounded. Furthermore, λ is an increasing function along the flow lines generated by W, only if a is close to a critical point ξi∈𝒦1∪𝒦2.
In the set V(p,ε), p≥2, we obtain:
Proposition 3.4
Let K and β be as in Proposition 3.3. For any p≥2, there exists a pseudo-gradient W so that the following holds: there is a positive constant c independent of u=∑i=1pαiδai,λi∈V(p,ε), ε small enough, such that, if we write u¯=u+v¯, we have
-
J
′
(
u
)
(
W
)
≥
c
(
∑
i
=
1
p
|
∇
K
(
a
i
)
|
λ
i
+
∑
i
=
1
p
1
λ
i
β
+
∑
i
≠
j
ε
i
j
)
.
-
J
′
(
u
¯
)
(
W
+
∂
v
¯
∂
(
α
,
a
,
λ
)
(
W
)
)
≥
c
(
∑
i
=
1
p
|
∇
K
(
a
i
)
|
λ
i
+
∑
i
=
1
p
1
λ
i
β
+
∑
i
≠
j
ε
i
j
)
.
|
W
|
is bounded. Furthermore, the only cases where the maximum of the
λ
i
is not bounded is when the concentration points
(
a
1
,
…
,
a
p
)
satisfy the following: each point
a
j
is close to a critical point
ξ
i
j
of
K
in the set
𝒦
1
with
i
j
≠
i
k
for
j
≠
k
and
ϱ
(
ξ
i
1
,
…
,
ξ
i
p
)
>
0
, where
ϱ
(
ξ
i
1
,
…
,
ξ
i
p
)
is the least eigenvalue of
M
(
ξ
i
1
,
…
,
ξ
i
p
)
.
3.2 Critical Points at Infinity
Once the pseudo-gradient is constructed, following [3, 24], we establish our Morse lemma at infinity: we can find a change of variables which gives the normal form of the functional J on the subsets V(p,ε). We obtain the following result.
Proposition 3.5
For ξ∈K1∪K2, there exists a change of variables in the set {αδ(a,λ)+v∣a is close to ξ}, v-v¯↦V and (a,λ)↦(a~,λ~), such that in these new variables the functional J behaves as
J
(
α
δ
(
a
,
λ
)
+
v
)
=
S
K
(
a
~
)
1
2
(
1
+
c
(
1
-
μ
)
Γ
(
ξ
)
λ
~
γ
(
ξ
)
)
+
∥
V
∥
2
,
where μ is a small positive constant and
γ
(
ξ
)
=
{
2
if
ξ
∈
𝒦
1
,
β
if
ξ
∈
𝒦
2
,
Γ
(
ξ
)
=
-
∑
k
=
1
2
b
k
+
κ
′
b
0
if
ξ
∈
𝒦
1
∪
𝒦
2
.
The proof is similar to the one given in [3, 9, 24], so we omit it here.
As a consequence of Proposition 3.3, we obtain the following corollary.
Corollary 3.6
Let K be a positive function on S3 satisfying the β-flatness condition and condition (C). The only critical points at infinity in V(1,ε) are ξ∞ where ξ∈K1∪K2. The Morse index i(ξ∞) of such a critical point is equal to
i
(
ξ
∞
)
=
3
-
m
(
ξ
)
.
If p≥2, we have the following result.
Proposition 3.7
Proposition 3.7 ([24])
For any u=∑i=1pαiδai,λi∈V(p,ε1) with ε1<ε2, each ai close to a critical point ξ∈K1, we find a change of variables in the space (ai,αi,λi,v), 1≤i≤p, to (a~i,α~i,λ~i,V), (α~i=αi), such that
J
(
∑
i
=
1
p
α
i
δ
a
i
,
λ
i
+
v
¯
(
α
,
a
,
λ
)
)
=
J
(
∑
i
=
1
p
α
i
δ
a
~
i
,
λ
~
i
)
with
(3.1)
∑
i
≠
j
ε
~
i
j
+
∑
i
1
λ
~
i
2
→
0
⟺
∑
i
≠
j
ε
i
j
+
∑
i
1
λ
i
2
→
0
,
and
(3.2)
∥
a
~
i
-
a
i
∥
→
0
as
∑
i
≠
j
ε
i
j
+
∑
i
1
λ
i
2
→
0
.
Proof.
We only give the key idea of the proof, and refer to [24, Lemma 4.4] for the complete proof. Since the vector field W constructed in the next section is Lipschitz, there is a one-parameter group ηs generated by the solution W of the equation
∂
∂
s
η
s
(
∑
i
=
1
p
α
i
δ
a
i
,
λ
i
)
=
W
(
η
s
(
∑
i
=
1
p
α
i
δ
a
i
,
λ
i
)
)
with initial condition
η
0
(
∑
i
=
1
p
α
i
δ
a
i
,
λ
i
)
=
∑
i
=
1
p
α
i
δ
a
i
,
λ
i
,
where J(ηs(∑i=1pαiδai,λi)) and J(ηs(∑i=1pαiδai,λi))+v¯(s) are decreasing functions of s. As v¯(s) is a minimizer, we have
J
(
∑
i
=
1
p
α
i
δ
a
i
,
λ
i
+
v
¯
(
s
)
)
≤
J
(
η
0
(
∑
i
=
1
p
α
i
δ
a
i
,
λ
i
)
)
.
Regarding the construction of the vector field W the flow line ηs(∑i=1pαiδai,λi) satisfies the Palais–Smale condition if it does not approach the critical points at infinity. Since the maximum of the λi(s) is a decreasing function, and the flow line started far away from these critical points at infinity, it will take an infinite time to this flow line to go to infinity. During this trip, we would be down the level J(∑i=1pαiδai,λi)+v¯(s). In any case, as long as we do not cut the lower bound level, the speed of decay is at least -c. Hence, we are forced to cut the level J(∑i=1pαiδai,λi)+v¯(s) unless the flow line exits V(p,ε) which means there is at most one solution of the equation
(3.3)
J
(
η
s
(
∑
i
=
1
p
α
i
δ
a
i
,
λ
i
)
)
=
J
(
∑
i
=
1
p
α
i
δ
a
i
,
λ
i
)
+
v
¯
(
s
)
.
Does the flow line ηs exit V(p,ε)? We assume that ∑i=1pαiδai,λi∈V(p,ε1), ε1<ε2. Then we have
-
∂
J
(
η
s
∑
i
=
1
p
α
i
δ
a
i
,
λ
i
)
W
(
η
s
∑
i
=
1
p
α
i
δ
a
i
,
λ
i
)
≥
C
(
∑
i
=
1
p
|
∇
K
(
a
i
)
|
λ
i
+
1
λ
i
2
+
∑
i
≠
j
ε
i
j
)
≥
c
(
ε
)
>
0
,
|
W
|
≤
c
during the trip between the boundaries of V(p,ε1) and V(p,ε), which we suppose are of length l(ε). If we denote by Δs the corresponding time to travel on this portion of the flow trajectory, we have l(ε)≤cΔs. Let δ(ε)=-c(ε)l(ε)/c. Then J(ηs∑i=1pαiδai,λi) decreases at least -δ(ε) during the trip from the boundary of V(p,ε1) to the boundary of V(p,ε). To prove the result, we have to show that
(3.4)
J
(
η
s
∑
i
=
1
p
α
i
δ
a
i
,
λ
i
)
W
(
η
s
∑
i
=
1
p
α
i
δ
a
i
,
λ
i
+
v
¯
)
>
J
(
η
s
∑
i
=
1
p
α
i
δ
a
i
,
λ
i
)
-
δ
(
ε
)
.
To this end, we know from [24] that
J
(
η
s
∑
i
=
1
p
α
i
δ
a
i
,
λ
i
)
-
J
(
η
s
∑
i
=
1
p
α
i
δ
a
i
,
λ
i
+
v
¯
)
→
0
as
ε
1
→
0
.
Hence by choosing ε1 small enough, we have (3.4), and therefore equation (3.3) has a unique solution that we denote by ηs0(∑i=1pαiδai,λi).
Next, we are going to prove (3.1). Set
∑
i
=
1
p
α
i
(
s
)
δ
a
i
(
s
)
,
λ
i
(
s
)
=
η
s
(
∑
i
=
1
p
α
i
δ
a
i
,
λ
i
)
.
Since the vector field W has no action on the variables αi, we have
W
=
∑
i
=
1
p
α
i
1
λ
i
(
s
)
∂
δ
a
i
(
s
)
,
λ
i
(
s
)
∂
a
i
(
s
)
(
λ
i
(
s
)
a
˙
i
(
s
)
)
+
∑
i
=
1
p
α
i
λ
i
(
s
)
∂
δ
a
i
(
s
)
,
λ
i
(
s
)
∂
λ
i
(
s
)
(
λ
˙
i
(
s
)
λ
i
(
s
)
)
,
where a˙i(s) and λ˙i(s) denote the actions of W on the variables ai and λi. Since W is bounded, both
1
λ
i
(
s
)
∂
δ
a
i
(
s
)
,
λ
i
(
s
)
∂
a
i
(
s
)
and
λ
i
(
s
)
∂
δ
a
i
(
s
)
,
λ
i
(
s
)
∂
λ
i
(
s
)
are nearly orthogonal and bounded (both are O(δai(s),λi(s))), so |λi(s)a˙i(s)|+|λ˙i(s)λi(s)|≤C′, i=1,…,p. Both
1
λ
i
∂
ε
i
j
∂
a
i
and
λ
i
∂
ε
i
j
∂
λ
i
are O(εij) since εij=o(1). We obtain ∂εij∂s≤Cεij. Therefore
exp
-
c
s
≤
ε
i
j
(
s
)
ε
i
j
(
0
)
exp
c
s
and
exp
-
c
s
≤
λ
i
(
s
)
λ
i
(
0
)
exp
c
s
,
which establishes (3.1).
Now, we will prove (3.2): We have |a˙i(s)|≤C′λi(s)≤C′expcsλi(0), thus
|
a
i
(
s
)
-
a
i
|
≤
C
′
s
exp
c
s
λ
i
(
0
)
.
Since s0 satisfies equation (3.3), |ai(s)-ai| is bounded, hence we have (3.2). ∎
3.3 Proofs of Propositions 3.3 and 3.4
Proof of Proposition 3.3.
We define the following partition for the set V(1,ε):
V
(
1
,
ε
)
1
=
{
α
δ
(
a
,
λ
)
∣
λ
|
∇
K
(
a
)
|
≥
C
}
,
V
(
1
,
ε
)
2
=
{
α
δ
(
a
,
λ
)
∣
λ
|
∇
K
(
a
)
|
≤
2
C
,
where
a
is close to
ξ
i
in
𝒦
1
}
,
V
(
1
,
ε
)
3
=
{
α
δ
(
a
,
λ
)
∣
λ
|
∇
K
(
a
)
|
≤
2
C
,
where
a
is close to
ξ
i
in
𝒦
2
}
,
where C is a large positive constant.
For 2≤β<4, we set
(3.5)
W
β
1
=
∑
k
=
1
2
1
λ
(
D
a
)
k
δ
(
a
,
λ
)
Ξ
k
1
+
|
Ξ
k
|
+
1
λ
(
D
a
)
0
δ
(
a
,
λ
)
Ξ
0
1
+
|
Ξ
0
|
,
W
2
=
(
-
∑
k
=
1
2
b
k
-
κ
b
0
+
τ
K
(
ξ
i
)
A
ξ
i
)
λ
∂
δ
(
a
,
λ
)
∂
λ
ψ
(
λ
|
a
|
)
,
(3.6)
W
3
=
(
-
∑
k
=
1
2
b
k
-
κ
b
0
)
λ
∂
δ
(
a
,
λ
)
∂
λ
ψ
(
λ
|
a
|
)
,
where
(
D
a
)
1
=
∂
∂
(
a
)
1
+
2
(
a
)
2
∂
∂
(
a
)
0
,
(
D
a
)
2
=
∂
∂
(
a
)
2
-
2
(
a
)
1
∂
∂
(
a
)
0
and
(
D
a
)
0
=
∂
∂
(
a
)
0
,
(
a
)
1
(
a
)
2
, (a)0 are the coordinates of a in some pseudo-hermitian normal coordinates, k∈{1,2},
k
′
=
{
1
if
k
=
2
,
2
if
k
=
1
,
Ξ
k
=
b
k
∫
ℍ
1
|
x
k
+
λ
(
a
)
k
|
β
|
1
+
|
z
|
2
-
i
t
|
6
x
k
(
1
+
|
z
|
2
)
θ
0
∧
d
θ
0
+
b
0
∫
ℍ
1
|
t
+
λ
2
(
a
)
0
+
2
λ
(
x
2
(
a
)
1
-
x
1
(
a
)
2
)
|
β
2
|
1
+
|
z
|
2
-
i
t
|
6
(
x
k
(
1
+
|
z
|
2
)
+
(
-
1
)
k
′
x
k
′
t
)
θ
0
∧
d
θ
0
,
Ξ
0
=
b
0
∫
ℍ
1
|
t
+
λ
2
(
a
)
0
+
2
λ
(
x
2
(
a
)
1
-
x
1
(
a
)
2
)
|
β
2
|
1
+
|
z
|
2
-
i
t
|
6
t
θ
0
∧
d
θ
0
,
and ψ is a cut-off function defined by ψ(t)=1 if t≤μ and ψ(t)=0 if t≥2μ, where μ is a small positive constant.
In V(1,ε)1, we define
W
1
=
∇
K
(
a
)
|
∇
K
(
a
)
|
1
λ
∂
δ
(
a
,
λ
)
∂
a
and we use Proposition A.1 to get
(3.7)
-
J
′
(
u
)
(
W
1
)
≥
c
|
∇
K
(
a
)
|
λ
+
O
(
1
λ
2
)
≥
c
|
∇
K
(
a
)
|
λ
+
c
1
λ
2
.
In V(1,ε)2, we use the vector field W2 and Proposition A.2 to obtain
(3.8)
-
J
′
(
u
)
(
W
2
)
≥
c
|
Δ
K
(
a
)
|
2
λ
2
+
o
(
1
λ
2
)
≥
c
′
|
∇
K
(
a
)
|
λ
+
c
′
1
λ
2
.
In V(1,ε)3, the vector field will be defined by W3:=Wβ1+W3. Using again Proposition A.2, we get
(3.9)
-
J
′
(
u
)
(
W
3
)
≥
c
λ
β
+
c
|
∇
K
(
a
)
|
λ
.
In V(1,ε2), the pseudo-gradient W will be defined as the convex combination of the partial vector fields W1,W2 and W3 and be equal to -∂J outside V(1,ε2). Using (3.7), (3.8) and (3.9) claim (i) follows. Claim (ii) follows from claim (i), the estimates of ∥v¯∥2 and ∥∇J(u¯)∥∥v¯∥. Finally, claim (iii) follows from the definition of W. The proof of Proposition 3.3 is thereby complete. ∎
Before giving the proof of Proposition 3.4, we give the following results related to the interactions between the bubbles ai, i∈{1,2,…,p}.
Lemma 3.8
Lemma 3.8 ([26])
Suppose that, for all
i
∈
{
1
,
2
,
…
,
p
}
,
(3.10)
∑
j
≠
i
ε
i
j
≤
1
λ
i
2
.
Then
(3.11)
|
J
′
(
u
)
(
1
λ
i
∂
δ
i
∂
a
i
)
|
≥
C
|
∇
θ
K
(
a
i
)
|
λ
i
-
1
c
′
1
λ
i
2
.
If there are indices for which (
3.10
) is not satisfied, two cases may occur:
If
i
is an index having the maximal concentration, we obtain
(3.12)
N
|
J
′
(
u
)
(
1
λ
i
∂
δ
i
∂
a
i
)
|
+
1
N
J
′
(
u
)
(
λ
i
∂
δ
i
∂
λ
i
)
≥
C
N
|
∇
θ
1
K
(
a
i
)
|
λ
i
-
1
c
λ
i
2
+
o
(
∑
k
≠
r
ε
k
r
)
,
where
N
and
C
are positive constants.
Otherwise, if we denote
J
i
=
{
k
∣
λ
k
>
λ
i
and
∑
j
≠
k
ε
k
j
≤
1
λ
k
2
}
, we have
C
|
J
′
(
u
)
(
1
λ
i
∂
δ
i
∂
a
i
)
|
+
∑
k
∈
J
i
2
k
J
′
(
u
)
(
λ
k
∂
δ
k
∂
λ
k
)
≥
C
|
∇
θ
K
(
a
i
)
|
λ
i
-
N
c
λ
i
2
+
o
(
∑
k
≠
r
ε
k
r
)
.
Proof of Proposition 3.4.
Without loss of generality, we can assume that λ1≤⋯≤λp. Given a large positive constant N, we define
(3.13)
I
1
:=
{
1
}
∪
{
i
≤
p
∣
λ
k
≤
N
λ
k
-
1
for all
k
≤
i
}
,
I
2
:=
{
i
∈
I
1
∣
a
i
is close to a critical point
ξ
k
i
satisfying (1.3) with
β
>
2
}
.
I
1
contains the indices i such that λi and λ1 are of the same order.
We divide the set V(p,ε) into five subsets:
V
(
p
,
ε
)
1
:=
{
u
=
∑
i
=
1
p
α
i
δ
a
i
,
λ
i
∣
λ
1
|
∇
θ
K
(
a
1
)
|
≥
C
′
}
,
V
(
p
,
ε
)
2
:=
{
u
=
∑
i
=
1
p
α
i
δ
a
i
,
λ
i
∣
∑
j
≠
i
ε
i
j
≥
C
λ
1
2
}
,
V
(
p
,
ε
)
3
:=
{
u
=
∑
i
=
1
p
α
i
δ
a
i
,
λ
i
∣
λ
i
|
∇
θ
K
(
a
i
)
|
≤
2
C
′
for all
i
∈
I
1
,
∑
j
≠
k
ε
j
k
≤
C
λ
1
2
and
I
2
=
∅
}
,
V
(
p
,
ε
)
4
:=
{
u
=
∑
i
=
1
p
α
i
δ
a
i
,
λ
i
∣
λ
i
|
∇
θ
K
(
a
i
)
|
≤
2
C
′
for all
i
∈
I
1
,
♯
(
I
1
)
≥
2
,
∑
j
≠
k
ε
j
k
≤
C
λ
i
2
and
I
2
≠
∅
}
,
V
(
p
,
ε
)
5
:=
{
u
=
∑
i
=
1
p
α
i
δ
a
i
,
λ
i
∣
I
1
=
I
2
=
{
1
}
and
λ
1
|
∇
θ
K
(
a
1
)
|
≤
2
C
′
,
∑
j
≠
i
ε
i
j
≤
C
λ
1
2
}
,
where C and C′ are large positive constants.
In each subset given above, we will define a partial pseudo-gradient for the functional J.
For 1≤i≤p, let
X
i
:=
∇
K
(
a
i
)
|
∇
K
(
a
i
)
|
1
λ
i
∂
δ
i
∂
a
i
and
Z
i
:=
λ
i
∂
δ
i
∂
λ
i
.
In V(p,ε)1, two cases may occur, J1=∅ or J1≠∅, which are case (i) and case (ii) of Lemma 3.8. If J1≠∅, we set i=1 in (3.12) and, using the hypothesis and [24], we obtain
|
J
′
(
u
)
(
1
λ
1
∂
δ
1
∂
a
1
)
|
+
C
∑
k
∈
J
1
2
k
J
′
(
u
)
(
λ
k
∂
δ
k
∂
λ
k
)
≥
C
λ
1
2
+
C
|
∇
θ
K
(
a
1
)
|
λ
1
+
o
(
∑
k
≠
r
ε
k
r
)
.
For i∈{1,2,…,p}, define
Γ
i
=
-
C
J
′
(
u
)
(
∇
K
(
a
1
)
|
∇
K
(
a
1
)
|
1
λ
1
∂
δ
1
∂
a
1
)
+
1
C
C
′
∑
k
∈
J
1
2
k
J
′
(
u
)
(
λ
k
∂
δ
k
∂
λ
k
)
-
C
J
′
(
u
)
(
∇
K
(
a
i
)
|
∇
K
(
a
i
)
|
1
λ
i
∂
δ
i
∂
a
i
)
+
1
C
C
′
∑
k
∈
J
i
2
k
J
′
(
u
)
(
λ
k
∂
δ
k
∂
λ
k
)
+
N
J
′
(
u
)
(
λ
i
∂
δ
i
∂
λ
i
)
,
where N,C,c are positive constants. Using Lemma 3.8 and [24], we obtain
(3.14)
Γ
i
≥
C
λ
i
2
+
C
|
∇
θ
K
(
a
i
)
|
λ
i
+
o
(
∑
k
≠
r
ε
k
r
)
-
∑
j
≠
i
c
i
j
λ
i
∂
ε
i
j
∂
λ
i
(
1
+
o
(
1
)
)
.
We define the partial vector field as
W
1
:=
∑
i
≤
p
M
i
X
i
-
C
1
∑
i
≤
p
N
i
Z
i
,
where Mi,Ni, i∈{1,2,…,p} and C1 are appropriate positive constants. Then by using (3.14) and the fact that
2
λ
k
∂
ε
k
j
∂
λ
k
+
λ
j
∂
ε
j
k
∂
λ
j
≤
-
1
2
ε
j
k
(
1
+
o
(
1
)
)
for
λ
k
>
λ
i
,
we derive the estimate of claim (i) in this case.
If J1=∅, having in hand (3.11), an analogous combination of the vector fields Xi and Zi, i∈{1,2,…,p} as for the case J1≠∅ gives the result.
In V(p,ε)2, we notice that the proof given in the subset V(p)1 is still valid in this subset even if λ1|∇θK(a1)|≤C′. So, we use in this case the vector field
W
2
:=
∑
i
≤
p
M
i
′
X
i
-
C
2
∑
i
≤
p
N
i
′
Z
i
,
where Mi′,Ni′, i∈{1,2,…,p} and C2 are appropriate positive constants. We obtain
-
J
′
(
u
)
(
W
2
)
≥
c
(
∑
i
≤
p
|
∇
θ
K
(
a
i
)
|
λ
i
+
∑
i
≤
p
1
λ
i
2
+
∑
k
≠
r
ε
k
r
)
.
In V(p,ε)3, we can write u as u=u1+u2 with
u
1
=
∑
i
∈
I
1
α
i
δ
i
and
u
2
=
∑
i
∉
I
1
α
i
δ
i
.
Observe that u1∈V(♯(I1),ε). Since I2=∅, we can apply the vector field defined in [24] in this set, we denote it by G(u1). Hence, the partial vector field in this set will be defined by
W
3
:=
G
(
u
1
)
-
N
∑
i
∉
I
1
2
i
Z
i
+
∑
i
∉
I
1
X
i
,
where N is given in (3.13). We obtain
(3.15)
-
J
′
(
u
)
(
W
3
)
≥
c
(
∑
i
≤
p
|
∇
θ
K
(
a
i
)
|
λ
i
+
∑
i
≤
p
1
λ
i
2
+
∑
k
≠
r
ε
k
r
)
.
Thus, the estimate of claim (i) follows in this case.
In V(p,ε)4, we observe that 1/λi2≤cNεij for each i,j∈I1, i≠j, and ∇K(ai)=o(1) for i∈I2. We define the partial vector field in this set as
W
4
:=
-
∑
i
∈
I
2
Z
i
-
m
′
∑
i
∈
I
1
∖
I
2
Z
i
-
∑
i
∉
I
1
2
i
Z
i
+
m
′′
∑
i
≤
p
X
i
,
where m′ and m′′ are small positive constants satisfying m′cM and m′′/m′ are small. Since 1/λi2=o(εij) for i∉I1 and j∈I1, we obtain, using Propositions A.3 and A.4,
-
J
′
(
u
)
(
W
4
)
≥
c
(
∑
i
≤
p
|
∇
θ
K
(
a
i
)
|
λ
i
+
∑
i
≤
p
1
λ
i
2
+
(
∑
k
≠
r
ε
k
r
)
)
.
Claim (i) follows in this case.
In V(p,ε)5, we have I1=I2={1}. Let ξk1 be the critical point which is close to a1.
In this case, we use the vector fields Wβ1 and W3 defined by (3.5) and (3.6). Then, following the lines of the proof of (3.9), we obtain
-
J
′
(
u
)
(
W
β
1
+
W
3
)
≥
c
(
|
∇
θ
K
(
a
1
)
|
λ
1
+
1
λ
1
β
)
+
O
(
∑
k
≠
r
ε
k
r
)
.
In this set, we use the pseudo-gradient W5 defined by
(3.16)
W
5
:=
W
β
1
+
W
3
-
C
∑
i
≥
2
2
i
Z
i
+
∑
i
≤
p
X
i
,
where C is a positive constant. Since 1/λi2=o(εij) for i∉I1 and j∈I1, we obtain, using (3.15) and (3.16),
-
J
′
(
u
)
(
W
5
)
≥
c
(
∑
i
≤
p
|
∇
θ
K
(
a
i
)
|
λ
i
+
∑
i
≥
2
1
λ
i
2
+
1
λ
1
β
+
∑
k
≠
r
ε
k
r
)
,
which implies claim (i) in this case.
Since all the constructions that we set are compatible with the convex combinations, we define W on V(p,ε2) to be the convex combination of {Wi}1≤i≤5 and -∂J outside V(p,ε2).
As shown in [24], claim (ii) follows directly from claim (i) and the estimates of ∥v¯∥2 and ∥∇J(u¯)∥∥v¯∥. For claim (iii), it is easy to see that W is bounded. We remark also that the maximum of the λi is a decreasing function on the sets V(p,ε)1,V(p,ε)2,V(p,ε)4 and V(p,ε)5. However, in V(p,ε)3, if I1≠{1,…,p}, the maximum of the λi is a decreasing function. But if I1={1,…,p}, we have the same case as in [24]. Thus claim (iii) follows. ∎
As a consequence of Proposition 3.4, we obtain:
Corollary 3.9
The only critical points at infinity in V(p,ε), p≥2 are ξ∞=(ξi1,…,ξip)∞ such that the matrix M(ξi1,…,ξip) defined in (1.4) is positive definite, where the ξij are critical points of K in the set K1 and ij≠ik for j≠k. Such a critical point at infinity has a Morse index equal to
i
(
ξ
∞
)
=
i
(
ξ
i
1
,
…
,
ξ
i
p
)
∞
=
4
p
-
1
-
∑
j
=
1
p
m
(
ξ
i
j
)
.
4 Existence Results
For ε0>0 small enough, we introduce the following neighborhood of Σ+:
V
ε
0
(
Σ
+
)
=
{
u
∈
Σ
∣
∥
u
-
∥
L
4
≤
ε
0
}
,
where u-=max(0,-u) denotes the negative part of u and
∥
u
-
∥
L
4
=
(
∫
𝕊
3
|
u
-
|
4
θ
∧
d
θ
)
1
4
.
We will build a global vector field Z on Vε0(Σ+). We know from the previous section that there exists a vector field W defined in V(p,ε) for p≥1 in the new variables such that Propositions 3.3 and 3.4 are satisfied respectively for p=1 and p≥2. For the V-part, we construct a pseudo-gradient T by setting ∂V∂s=-νV, locally on the base space of the bundle V(p,ε) where ν is taken to be a very large constant. Define Z on Vε0(Σ+) to be Z=W+T. Thus, the defined vector field Z is a pseudo-gradient vector field for the functional -J on Vε0(Σ+) which is invariant under the flow generated by Z (the proof of this claim is similar to the one given in [10]).
Lemma 4.1
The critical points at infinity of the functional J in the set V(p,ε), p≥2, ε small, lie in the subset V(p,ε)3.
Proof.
Denote by
μ
(
s
,
u
0
)
=
∑
i
=
1
p
α
i
δ
a
i
(
s
)
,
λ
i
(
s
)
+
v
¯
(
s
)
the flow line of the vector field Z with initial condition u0. We have μ(s,u0)∈V(p,ε2). Suppose that in the new variables ∑i=1pα~i(s)δa~i(s),λ~i(s) is outside V(p,ε2)3, then we derive from the construction of Z that the maximum of the λi(s) and the maximum of the λ~i(s) are bounded by a constant c. Since -J′(u)Z(u)>0 and ∑i=1pαiδai,λi is in the compact set {αi≤1,λi≤c,ai∈𝕊3}, the minimum is achieved, hence -J′(u)Z(u)>ν>0. Therefore
J
(
μ
(
s
,
u
0
)
)
=
J
(
μ
(
0
,
u
0
)
)
+
∫
0
s
J
′
(
u
)
(
t
)
Z
(
u
)
(
t
)
𝑑
t
≤
J
(
μ
(
0
,
u
0
)
)
-
ν
(
s
-
s
0
)
,
which gives that J is not bounded, hence a contradiction. ∎
Lemma 4.2
For any u=∑i=1pαiδai,λi in V(p,ε)3 close to a critical point at infinity of J, we obtain the following expansion of J in the new variables:
J
(
u
)
=
S
(
∑
i
=
1
p
1
K
(
ξ
i
)
)
1
2
(
1
-
|
Q
|
2
+
∑
i
=
1
p
(
|
a
i
s
|
2
-
|
a
i
u
|
2
)
+
c
∑
i
=
1
p
1
λ
i
2
)
,
where (ais,aiu) are the coordinates of ai near ξi along the manifolds Ws(ξi) and Wu(ξi), and Q∈Rp-1 is the coordinate of (α1,…,αp).
Proof.
Using Proposition A.2, we obtain the following expansion of the functional J in V(p,ε)3 in the new variables (v=0):
J
(
∑
i
=
1
p
α
i
δ
a
i
,
λ
i
)
=
∑
i
=
1
p
α
i
2
S
[
∑
i
=
1
p
α
i
4
K
(
a
i
)
]
1
/
2
[
1
-
c
S
2
∑
i
=
1
p
α
i
4
∑
l
=
1
p
α
l
4
K
(
a
l
)
∑
k
=
1
2
b
k
+
κ
′
b
0
λ
i
2
+
c
0
4
ω
3
4
S
2
∑
i
≠
j
ε
i
j
(
α
i
α
j
∑
l
=
1
p
α
l
2
-
2
α
i
3
α
j
K
(
a
i
)
∑
l
=
1
p
α
l
4
K
(
a
l
)
)
+
o
(
∑
i
≠
j
ε
i
j
)
]
.
We can refine the expansion of J, since in this set, we have αi2K(ai)≃αj2K(aj) and εij=(λiλjd2(ai,ai))-1. Hence, we obtain
J
(
∑
i
=
1
p
α
i
δ
a
i
,
λ
i
)
=
∑
i
=
1
p
α
i
2
[
∑
i
=
1
p
α
i
4
K
(
a
i
)
]
1
/
2
S
[
1
+
1
S
2
∑
k
=
1
p
1
K
(
a
k
)
Λ
(
M
+
o
(
1
)
)
Λ
t
]
,
where Λ=(1λ1,…,1λp).
Let us turn now to the term
𝒢
(
α
1
,
…
,
α
p
)
=
∑
i
=
1
p
α
i
2
(
∑
i
=
1
p
α
i
4
K
(
a
i
)
)
1
/
2
.
The function 𝒢 is homogeneous and (1K(a1),…,1K(ap)) is a critical point (a maximum) with critical value ∑j=1p1K(ai).
By performing a Morse lemma for 𝒢, we obtain in the new variables
J
(
u
)
=
S
(
∑
i
=
1
p
1
K
(
ξ
i
)
)
[
1
-
|
h
|
2
+
∑
i
=
2
p
(
|
a
i
+
|
2
-
|
a
i
-
|
2
)
+
c
∑
i
=
1
p
1
λ
i
2
]
,
since ΛMΛt≥c|Λ|2=∑j=1pcλi2, and the lemma follows. ∎
4.1 Topological Argument
For a critical point at infinity of J, (ξi)∞, ξi=(ξi1,…,ξip)∈𝒦1+ or ξi∈𝒦2, 1≤p≤l+, we denote by
c
(
ξ
i
)
∞
=
S
∑
j
=
1
p
K
(
ξ
i
j
)
the associated critical value. Here, we choose to consider a simplified situation, where for (ξi)∞≠(ξi′)∞ we have c(ξi)∞≠c(ξi′)∞, and thus order the c(ξi)∞ as c(ξ0)∞<c(ξ1)∞<⋯<c(ξN)∞.
By using a deformation lemma (see [1]), we derive the existence of a positive constant σ0(ε) such that for any 0<σ<σ0, the set Jc(ξk)∞-σ∪Wu∞(ξk)∞ is a retract by deformation of Jc(ξk)∞+σ, where Ja denotes the level set for the functional, Ja={u∈Σ+∣J(u)≤a}, and Wu∞(ξk)∞ is the unstable manifold of the critical point at infinity (ξk)∞.
Lemma 4.3
If c(ξk-1)∞<a<c(ξk)∞<b<c(ξk+1)∞, then for any coefficient group G, we have
H
q
(
J
b
,
J
a
)
=
{
0
if
q
≠
i
(
ξ
k
)
∞
,
G
if
q
=
i
(
ξ
k
)
∞
.
We are now ready to state the proof of our main result.
Proof of Theorem 1.1.
Let
b
0
<
c
(
ξ
1
)
∞
=
inf
u
∈
Σ
+
J
(
u
)
<
b
1
<
c
(
ξ
2
)
∞
<
⋯
<
b
N
-
1
<
c
(
ξ
N
)
∞
<
b
N
.
Since we assume that problem (1.1) has no solution, JbN is a retract by deformation of the set Σ+, which is a retract by deformation of Vε0(Σ+) and hence they have the same Euler–Poincaré characteristic,
χ
(
V
ε
0
(
Σ
+
)
)
=
χ
(
J
b
N
)
.
By Lemma 4.3, we obtain
χ
(
J
b
k
+
1
)
=
χ
(
J
b
k
)
+
(
-
1
)
i
(
ξ
k
)
∞
.
Recalling that χ(Jb0)=χ(∅)=0, we then derive from Corollaries 3.6 and 3.9 that
∑
ξ
∈
𝒦
2
(
-
1
)
3
-
m
(
ξ
)
+
∑
p
=
1
l
+
∑
(
ξ
i
1
,
…
,
ξ
i
p
)
∈
𝒦
1
+
(
-
1
)
i
(
ξ
i
1
,
…
,
ξ
i
p
)
=
1
.
Therefore (1.1) has a solution u0 in Vε0(Σ+) if the equality above is not true.
We claim that u0>0, when ε0 is small enough. Otherwise, by multiplying (1.1) by u0- and integrating, using the fact that u0 is in Vε0(Σ+), we obtain
∥
u
0
-
∥
2
≤
C
∥
u
0
-
∥
L
4
4
≤
C
′
∥
u
0
-
∥
2
.
Hence, either u0-=0 or ∥u0-∥≥C0, where C0>0. Thus we have a contradiction if ε0 is small enough. Therefore u0-=0 and u0>0. ∎
A Appendix
Following the work done in [24], we obtain the following expansion of the functional J in the subset V(p,ε)3 of V(p,ε), p≥2.
Proposition A.1
There exists ε0>0 such that, for any
u
=
∑
i
=
1
p
α
i
δ
a
i
,
λ
i
+
v
∈
V
(
p
,
ε
)
3
,
ε
<
ε
0
, v satisfying (V0), we have
J
(
u
)
=
∑
i
=
1
p
α
i
2
[
∑
α
i
4
K
(
a
i
)
]
1
/
2
S
[
1
-
c
2
S
2
∑
i
=
1
p
α
i
4
∑
k
=
1
p
α
k
4
K
(
a
k
)
∑
k
=
1
2
b
k
+
κ
′
b
0
λ
i
2
+
S
-
2
∑
i
≠
j
c
0
4
ω
3
4
ε
i
j
(
α
i
α
j
∑
k
=
1
p
α
k
2
-
2
α
i
3
α
j
K
(
a
i
)
∑
k
=
1
p
α
k
4
K
(
a
k
)
)
+
f
(
v
)
+
Q
(
v
,
v
)
+
o
(
∑
i
≠
j
ε
i
j
)
+
o
(
∥
v
∥
θ
1
2
)
]
,
with
f
(
v
)
=
-
1
γ
1
∫
𝕊
3
K
(
∑
i
=
1
p
α
i
δ
a
i
,
λ
i
)
3
v
θ
1
∧
d
θ
1
,
Q
(
v
,
v
)
=
1
γ
2
∥
v
∥
L
θ
1
2
-
3
γ
1
∫
𝕊
3
K
∑
i
=
1
p
α
i
2
δ
a
i
,
λ
i
2
v
2
θ
1
∧
d
θ
1
,
γ
1
=
S
2
∑
i
=
1
p
α
i
4
K
(
a
i
)
,
γ
2
=
S
2
∑
i
=
1
p
α
i
2
.
Furthermore, if K satisfies (1.3), then ∥f∥θ1 is bounded by
∥
f
∥
θ
1
=
O
(
∑
i
=
1
p
(
|
∇
K
(
a
i
)
|
λ
i
+
1
λ
i
2
)
+
∑
i
≠
j
ε
i
j
(
log
ε
i
j
-
1
)
1
2
)
.
Proof.
We have
J
(
u
)
=
∫
𝕊
3
L
θ
1
u
u
θ
1
∧
d
θ
1
[
∫
𝕊
3
K
u
4
θ
1
∧
d
θ
1
]
1
/
2
=
N
D
,
where u=∑i=1pαiδai,λi+v, v satisfies conditions (V0).
From the expansions of N and D given in the appendix of [24] we derive the following estimates:
J
(
u
)
=
∑
i
=
1
p
α
i
2
S
2
[
∑
i
α
i
4
K
(
a
i
)
S
2
]
1
/
2
[
1
+
∑
i
≠
j
α
i
α
j
∑
k
=
1
p
α
k
2
S
2
(
c
0
4
ω
3
4
ε
i
j
(
1
+
o
(
1
)
)
)
+
o
(
∑
i
=
1
p
1
λ
i
2
)
+
∥
v
∥
θ
1
2
∑
k
=
1
p
α
k
2
S
2
]
×
[
1
+
c
0
4
c
∑
i
=
2
p
α
i
4
∑
l
=
2
p
α
l
4
K
(
a
l
)
S
2
∑
k
=
1
2
b
k
+
κ
′
b
0
λ
i
2
+
4
∑
i
≠
j
α
i
3
α
j
K
(
a
i
)
∑
k
=
1
p
α
k
4
K
(
a
k
)
S
2
(
c
0
4
ω
3
4
ε
i
j
(
1
+
o
(
1
)
)
)
+
O
(
∑
i
=
1
p
(
|
∇
K
(
a
i
)
|
λ
i
+
1
λ
i
2
)
+
∑
i
≠
j
ε
i
j
(
log
ε
i
j
-
1
)
1
2
)
+
O
(
∫
|
∇
v
|
2
)
3
/
2
+
6
(
∑
i
=
1
p
α
i
2
K
(
a
i
)
∑
k
=
1
p
α
k
4
K
(
a
k
)
S
2
∫
𝕊
3
δ
a
i
,
λ
i
2
v
2
)
+
O
(
∑
i
=
1
p
1
λ
i
2
)
+
O
(
∑
i
≠
j
1
λ
i
3
λ
j
)
]
-
1
2
.
∎
Notice that for ε>0 very small, there is α0>0 such that, for all v∈Hεp(a,λ),
Q
(
v
,
v
)
≥
α
1
∥
v
∥
2
and we have
(
f
,
v
)
+
Q
(
v
,
v
)
+
o
(
∥
v
∥
θ
1
2
)
=
Q
(
v
-
v
¯
,
v
-
v
¯
)
+
o
(
∥
v
¯
∥
θ
1
2
)
since
(
f
,
v
¯
)
+
Q
(
v
¯
,
v
¯
)
+
o
(
∥
v
¯
∥
θ
1
2
)
=
0
.
Therefore, we obtain the following result:
Proposition A.2
There exists ε0>0 (ε0<ε) such that, for any
u
=
∑
i
=
1
p
α
i
δ
a
i
,
λ
i
+
v
,
v
∈
H
ε
p
(
a
,
λ
)
,
we have
J
(
∑
i
=
1
p
α
i
δ
a
i
,
λ
i
+
v
)
=
∑
i
=
1
p
α
i
2
[
∑
i
=
1
p
α
i
4
K
(
a
i
)
]
1
/
2
S
[
1
-
c
2
S
2
∑
i
=
1
p
α
i
4
∑
l
=
1
p
α
l
4
K
(
a
l
)
∑
k
=
1
2
b
k
+
κ
′
b
0
λ
i
2
+
ω
3
4
S
2
∑
i
≠
j
ε
i
j
(
α
i
α
j
∑
l
=
1
p
α
l
2
-
2
α
i
3
α
j
K
(
a
i
)
∑
l
=
1
p
α
l
4
K
(
a
l
)
)
+
Q
(
v
-
v
¯
,
v
-
v
¯
)
+
o
(
∥
v
¯
∥
θ
1
2
)
+
o
(
∑
i
≠
j
ε
i
j
)
]
.
Next, we will give the expansions of the gradient of the functional J which is the key of the Morse lemma. Since the vector field W is a variation of ∑i=1pαiδi∈V(p,ε) (p≥2), we will expand
J
′
(
u
)
(
λ
j
∂
δ
j
∂
λ
j
)
,
J
′
(
u
)
(
1
λ
j
∂
δ
j
∂
a
j
)
,
J
′
(
u
)
(
1
λ
j
(
D
j
)
k
δ
j
)
(
for
k
=
1
,
2
)
,
J
′
(
u
)
(
1
λ
j
2
(
D
j
)
0
δ
j
)
in the case where the concentration point aj, j∈{1,2,…,p} is close to a critical point ξj of K verifying (1.3). We follow the lines of the method used in [24] and [26]. Some of the following results are extracted from [27].
For the sake of simplicity, we will use the notation δj instead of δaj,λj. Let u=∑i=1pαiδi∈V(p,ε).
Proposition A.3
Proposition A.3 ([24])
We have
-
J
′
(
u
)
(
λ
j
∂
δ
j
∂
λ
j
)
=
2
J
(
u
)
[
∑
i
≠
j
c
α
i
λ
j
∂
ε
i
j
∂
λ
j
(
1
+
o
(
1
)
)
-
ω
3
24
α
j
△
K
(
a
j
)
K
(
a
j
)
λ
j
2
(
1
+
o
(
1
)
)
+
o
(
∑
i
≠
j
ε
i
j
)
]
,
-
J
′
(
u
)
(
1
λ
j
∂
δ
j
∂
a
j
)
=
2
J
(
u
)
[
α
j
K
(
a
j
)
ω
3
48
∇
K
(
a
j
)
λ
j
(
1
+
o
(
1
)
)
+
O
(
∑
i
≠
j
ε
i
j
+
1
λ
j
2
)
]
.
If there exists a point aj, j∈{1,2,…,p}, close to a critical point ξj of K verifying (1.3), then the estimates in the above proposition can be improved as follows:
Proposition A.4
For k∈{1,2}, we have
J
′
(
u
)
(
1
λ
j
(
D
j
)
k
δ
j
)
=
-
4
J
(
u
)
3
α
j
4
c
0
4
λ
j
β
[
b
k
∫
ℍ
1
|
x
k
+
λ
j
(
a
j
)
k
|
β
|
1
+
|
z
|
2
-
i
t
|
6
x
k
(
1
+
|
z
|
2
)
θ
0
∧
d
θ
0
+
b
0
∫
ℍ
1
|
t
+
λ
j
2
(
a
j
)
0
+
2
λ
j
(
x
2
(
a
j
)
1
-
x
2
(
a
j
)
1
)
|
β
2
|
1
+
|
z
|
2
-
i
t
|
6
(
x
k
(
1
+
|
z
|
2
)
+
(
-
1
)
k
′
x
k
′
t
)
θ
0
∧
d
θ
0
]
+
o
(
1
λ
j
β
)
+
O
(
∑
i
≠
j
ε
i
j
)
,
J
′
(
u
)
(
1
λ
j
2
(
D
j
)
0
δ
j
)
=
-
4
J
(
u
)
3
α
j
4
c
0
4
λ
j
β
b
0
∫
ℍ
1
|
t
+
λ
j
2
(
a
j
)
0
+
2
λ
j
(
x
2
(
a
j
)
1
-
x
2
(
a
j
)
1
)
|
β
2
|
1
+
|
z
|
2
-
i
t
|
6
t
θ
0
∧
d
θ
0
+
o
(
1
λ
j
β
)
+
O
(
∑
i
≠
j
ε
i
j
)
.
If we assume that λj|aj|≤μ, where μ is a small positive constant, then
J
′
(
u
)
(
1
λ
j
∂
δ
j
∂
λ
j
)
=
-
2
c
4
J
(
u
)
∑
i
≠
j
α
i
λ
j
∂
ε
i
j
∂
λ
j
+
c
5
∑
i
=
1
2
b
i
+
κ
b
0
λ
j
β
+
o
(
∑
k
≠
r
ε
k
r
+
1
λ
j
β
)
.
Proof.
If φj denotes λj∂δj∂λj or 1λj(Dj)kδj for k∈{1,2} and 1λj2(Dj)0δj for k=0, then we have
J
′
(
u
)
(
φ
j
)
=
2
J
(
u
)
[
〈
u
,
φ
j
〉
-
J
(
u
)
2
∫
M
K
u
3
φ
j
]
=
2
J
(
u
)
[
∑
i
=
1
p
α
i
〈
δ
i
,
φ
j
〉
L
-
J
(
u
)
2
∫
M
K
(
∑
i
=
1
p
α
i
δ
)
i
3
φ
j
]
.
Following Bahri [1] to expand J′(u)(φj) for u=∑i=1pαiδi∈V(p,ε), we have to estimate all the terms involved in the expression of the gradient of the functional. These estimates are the purposes of Lemmas A.5–A.12. Thus, the proof of the proposition is obtained by using these lemmas combined with the fact that if u=∑i=1pαiδi∈V(p,ε), then J2(u)αi2K(ai)→1 for i∈{1,2,…,p}. ∎
Lemma A.5
For k∈{1,2}, we have
1
λ
j
∫
𝕊
3
K
δ
j
3
(
D
j
)
k
δ
j
θ
∧
d
θ
=
2
c
0
4
λ
j
β
[
b
k
∫
ℍ
1
|
x
k
+
λ
j
(
a
j
)
k
|
β
|
1
+
|
z
|
2
-
i
t
|
6
x
k
(
1
+
|
z
|
2
)
θ
0
∧
d
θ
0
+
b
0
∫
ℍ
1
|
t
+
λ
j
2
(
a
j
)
0
+
2
λ
j
(
x
2
(
a
j
)
1
-
x
1
(
a
j
)
2
)
|
β
2
|
1
+
|
z
|
2
-
i
t
|
6
×
(
x
k
(
1
+
|
z
|
2
)
+
(
-
1
)
k
′
x
k
′
t
)
θ
0
∧
d
θ
0
]
+
o
(
1
λ
j
β
)
.
For k=0, we have
1
λ
j
2
∫
𝕊
3
K
δ
j
3
(
D
j
)
0
δ
j
θ
∧
d
θ
=
2
c
0
4
λ
j
β
b
0
∫
ℍ
1
|
t
+
λ
j
2
(
a
j
)
0
+
2
λ
j
(
x
2
(
a
j
)
1
-
x
1
(
a
j
)
2
)
|
β
2
|
1
+
|
z
|
2
-
i
t
|
6
t
θ
0
∧
d
θ
0
+
o
(
1
λ
j
β
)
.
Lemma A.6
If λj|aj| is very small, we have
∫
𝕊
3
K
δ
j
3
λ
j
∂
δ
j
∂
λ
j
θ
∧
d
θ
=
c
0
4
1
λ
j
β
(
∫
ℍ
1
|
x
1
|
β
1
-
|
|
z
|
2
-
i
t
|
2
|
1
+
|
z
|
2
-
i
t
|
6
θ
0
∧
d
θ
0
)
(
∑
i
=
1
2
b
i
+
κ
b
0
)
+
o
(
1
λ
j
β
)
.
For a proof of Lemma A.5 and Lemma A.6, we refer to [27] and [26].
Lemma A.7
Lemma A.7 ([26])
We have
〈
δ
j
,
1
λ
j
(
D
j
)
k
δ
j
〉
L
=
0
,
k
=
1
,
2
,
〈
δ
i
,
1
λ
j
(
D
j
)
k
δ
j
〉
L
=
O
(
ε
i
j
)
,
k
=
1
,
2
,
i
≠
j
,
〈
δ
j
,
1
λ
j
2
(
D
j
)
0
δ
j
〉
L
=
0
,
〈
δ
i
,
1
λ
j
2
(
D
j
)
0
δ
j
〉
L
=
O
(
ε
i
j
)
,
k
=
1
,
2
,
i
≠
j
.
Lemma A.8
Lemma A.8 ([26])
For i≠j, we have
∫
𝕊
3
K
δ
i
3
1
λ
j
(
D
j
)
k
δ
j
θ
∧
d
θ
=
c
i
j
K
(
a
i
)
1
λ
j
(
D
j
)
k
ε
i
j
(
1
+
o
(
1
)
)
+
o
(
ε
i
j
)
,
k
=
1
,
2
,
∫
𝕊
3
K
δ
i
3
1
λ
j
2
(
D
j
)
0
δ
j
θ
∧
d
θ
=
c
i
j
K
(
a
i
)
1
λ
j
2
(
D
j
)
k
ε
i
j
(
1
+
o
(
1
)
)
+
o
(
ε
i
j
)
,
∫
𝕊
3
K
δ
j
2
1
λ
j
(
D
j
)
k
δ
j
δ
l
θ
∧
d
θ
=
1
3
c
j
l
K
(
a
j
)
1
λ
j
(
D
j
)
k
ε
j
l
(
1
+
o
(
1
)
)
+
o
(
ε
j
l
)
,
j
≠
l
,
∫
𝕊
3
K
δ
j
2
1
λ
j
(
D
j
)
0
δ
j
δ
l
θ
∧
d
θ
=
1
3
c
j
l
K
(
a
j
)
1
λ
j
2
(
D
j
)
0
ε
j
l
(
1
+
o
(
1
)
)
+
+
o
(
ε
j
l
)
,
j
≠
l
.
Lemma A.9
Lemma A.9 ([26])
For i≠j, we have
∫
𝕊
3
K
δ
j
|
1
λ
j
(
D
j
)
k
δ
j
|
δ
l
2
θ
∧
d
θ
=
O
(
ε
j
l
2
log
ε
j
l
-
1
)
,
j
≠
l
,
∫
𝕊
3
K
δ
j
|
1
λ
j
2
(
D
j
)
0
δ
j
|
δ
l
2
θ
∧
d
θ
=
O
(
ε
j
l
2
log
ε
j
l
-
1
)
,
j
≠
l
,
∫
𝕊
3
K
δ
i
δ
l
2
|
1
λ
j
(
D
j
)
k
δ
j
|
θ
∧
d
θ
=
O
(
ε
i
j
2
log
ε
i
j
-
1
)
+
O
(
ε
j
l
2
log
ε
j
l
-
1
)
+
O
(
ε
i
l
2
log
ε
i
l
-
1
)
,
j
≠
l
,
l
≠
i
,
∫
𝕊
3
K
δ
i
δ
l
2
|
1
λ
j
2
(
D
j
)
0
δ
j
|
θ
∧
d
θ
=
O
(
ε
i
j
2
log
ε
i
j
-
1
)
+
O
(
ε
j
l
2
log
ε
j
l
-
1
)
+
O
(
ε
i
l
2
log
ε
i
l
-
1
)
,
j
≠
l
,
l
≠
i
.
Lemma A.10
We have
〈
δ
j
,
λ
j
∂
δ
j
∂
λ
j
〉
L
=
c
0
4
∫
ℍ
1
1
-
|
|
z
|
2
-
i
t
|
2
|
1
+
|
z
|
2
-
i
t
|
6
θ
0
∧
d
θ
0
=
0
.
Lemma A.11
Lemma A.11 ([24])
We have
〈
δ
i
,
λ
j
∂
δ
j
∂
λ
j
〉
L
=
c
i
j
λ
j
∂
ε
i
j
∂
λ
j
(
1
+
o
(
1
)
)
+
o
(
ε
i
j
)
.
Lemma A.12
Lemma A.12 ([24])
We have
∫
𝕊
3
K
δ
i
3
λ
j
∂
δ
j
∂
λ
j
θ
∧
d
θ
=
c
i
j
λ
j
K
(
a
i
)
∂
ε
i
j
∂
λ
j
(
1
+
o
(
1
)
)
+
o
(
ε
i
j
)
,
j
≠
i
,
∫
𝕊
3
K
δ
j
2
λ
∂
δ
j
∂
λ
j
j
δ
k
θ
∧
d
θ
=
1
3
K
(
a
j
)
c
j
k
λ
j
∂
ε
j
k
∂
λ
j
(
1
+
o
(
1
)
)
+
o
(
ε
j
k
)
,
j
≠
k
,
∫
𝕊
3
K
δ
j
|
λ
j
∂
δ
j
∂
λ
j
|
δ
k
2
θ
∧
d
θ
=
O
(
ε
j
k
2
log
ε
j
k
-
1
)
,
j
≠
k
,
∫
𝕊
3
K
δ
i
δ
k
|
λ
j
∂
δ
j
∂
λ
j
|
θ
∧
d
θ
=
O
(
ε
i
j
2
log
ε
i
j
-
1
)
+
O
(
ε
j
k
2
log
ε
j
k
-
1
)
+
O
(
ε
i
k
2
log
ε
i
k
-
1
)
,
j
≠
k
,
k
≠
i
,
j
≠
i
.
Lemma A.13
For each ξ∈R and k∈{1,2}, we have
∫
ℍ
1
|
x
k
+
ξ
|
β
x
k
|
1
+
|
z
|
2
-
i
t
|
6
(
1
+
|
z
|
2
)
θ
0
∧
d
θ
0
=
0
⟺
ξ
=
0
.
Furthermore, for each ε>0, there exists a positive constant c¯>0 such that
|
∫
ℍ
1
|
x
k
+
ξ
|
β
x
k
|
1
+
|
z
|
2
-
i
t
|
6
(
1
+
|
z
|
2
)
θ
0
∧
d
θ
0
|
≥
c
¯
for each
|
ξ
|
≥
ε
.
Lemma A.14
For each ξ0,ξ1,ξ2∈R, we have
∫
ℍ
1
|
t
+
ξ
0
+
ξ
1
x
1
+
ξ
2
x
2
|
β
|
1
+
|
z
|
2
-
i
t
|
6
t
θ
0
∧
d
θ
0
=
0
⟺
ξ
0
=
0
.
Furthermore, for each ε>0, there exists a positive constant c¯>0 such that
|
∫
ℍ
1
|
t
+
ξ
0
+
ξ
1
x
1
+
ξ
2
x
2
|
β
|
1
+
|
z
|
2
-
i
t
|
6
t
θ
0
∧
d
θ
0
|
≥
c
¯
for each
|
ξ
0
|
≥
ε
and
ξ
1
,
ξ
2
bounded
.
Acknowledgements
The first author is grateful to the Laboratory of Signals and Systems, Centrale Supélec, Université Paris-Sud 11, where most of this work was done while visiting the second author. She also would like to express her thanks to the staff of Centrale Supélec for their hospitality.
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and
Boutheina Hafassa
