Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 568245, 11 pages
http://dx.doi.org/10.1155/2013/568245
Research Article
Existence of Conformal Metrics with Prescribed Q-Curvature
Mohammed Ali Al-Ghamdi,1 Hichem Chtioui,1 and Afef Rigane2
1
2
Department of Mathematics, King Abdulaziz University, P.O. Box 80230, Jeddah, Saudi Arabia
Department of Mathematics, Faculty of Sciences of Sfax, Road Soukra, Sfax, Tunisia
Correspondence should be addressed to Hichem Chtioui; hichem.chtioui@fss.rnu.tn
Received 13 October 2012; Revised 5 February 2013; Accepted 8 February 2013
Academic Editor: Svatoslav Staněk
Copyright © 2013 Mohammed Ali Al-Ghamdi et al. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
We consider the problem of existence of conformal metrics with prescribed Q-curvature on standard sphere 𝑆𝑛 , 𝑛 ≥ 5. Under the
assumption that the order of flatness at critical points of prescribed Q-curvature function 𝐾(𝑥) is 𝛽 ∈ ]1, 𝑛 − 4], we give precise
estimates on the losses of the compactness, and we prove new existence and multiplicity results through an Euler-Hopf type formula.
1. Introduction and Main Result
The Paneitz operator on 𝑛-Riemannian manifold (𝑀, 𝑔0 ) is a
fourth-order differential operator which arises in conformal
geometry and satisfies a certain covariance property (see [1]).
For 𝑛 ≥ 5 it is defined by
𝑃𝑔𝑛0 𝑢 = Δ2𝑔0 𝑢 − div𝑔0 (𝑎𝑛 𝑆𝑔0 𝑔0 + 𝑏𝑛 Ric𝑔0 ) 𝑑𝑢 +
𝑛−4 𝑛
𝑄𝑔0 𝑢,
2
(1)
where
(𝑛 − 2)2 + 4
𝑎𝑛 =
,
2 (𝑛 − 1) (𝑛 − 2)
𝑄𝑔𝑛0 = −
−
4
𝑏𝑛 = −
,
𝑛−2
1
𝑛3 − 4𝑛2 + 16𝑛 − 16 2
Δ 𝑔0 𝑆𝑔0 +
𝑆𝑔0
2 (𝑛 − 1)
8(𝑛 − 1)2 (𝑛 − 2)2
(2)
2 󵄨󵄨
󵄨2
󵄨Ric𝑔0 󵄨󵄨󵄨 .
2 󵄨󵄨
󵄨
(𝑛 − 2)
Such a 𝑄𝑔𝑛0 is a fourth-order invariant called Q-curvature or
Paneitz curvature. See [2, 3] for details about the properties
of 𝑃𝑔𝑛0 .
If 𝑔 = 𝑢4/(𝑛−4) 𝑔0 is a conformal metric to 𝑔0 , where 𝑢 is
a smooth positive function, then the conformal covariance
property of the Paneitz operator reads as follows:
∀𝜓 ∈ 𝐶∞ (𝑀) : 𝑃𝑔𝑛0 (𝑢𝜓) = 𝑢(𝑛+4)/(𝑛−4) 𝑃𝑔𝑛 (𝜓) .
(3)
If one prescribes the Q-curvature for the metric 𝑔 by a
function 𝐾, this leads to the following equation:
𝑃𝑔𝑛0 (𝑢) = 𝑢(𝑛+4)/(𝑛−4) 𝑃𝑔𝑛 (1) =
𝑛 − 4 (𝑛+4)/(𝑛−4)
.
𝐾𝑢
2
(4)
The literature for the existence of solutions of the prescribed
Q-curvature problem on compact manifolds is considerably bigger. In [4–7], existence results for the constant Qcurvature problem in 4-dimensional manifolds are given. On
manifolds of dimension greater than 4, existence results were
given for Einstein manifolds in [3]. On the sphere 𝑆𝑛 , we refer
to results of [8–14] and the references therein.
In this paper we continue to study the problem of
prescribing Q-curvature on the standard sphere (𝑆𝑛 , 𝑔0 ), 𝑛 ≥
5. The Paneitz operator in this case is coercive on the Sobolev
space 𝐻22 (𝑆𝑛 ) and has the following expression:
P (𝑢) := 𝑃𝑔𝑛0 (𝑢) = Δ2𝑔0 𝑢 − 𝑐𝑛 Δ 𝑔0 𝑢 + 𝑑𝑛 𝑢,
(5)
2
Abstract and Applied Analysis
where
1 2
𝑛−4
(6)
𝑑𝑛 =
(𝑛 − 2𝑛 − 4) ,
𝑛 (𝑛2 − 4) .
2
16
According to problem (4), the problem can be formulated
as follows. Given a smooth function 𝐾 on 𝑆𝑛 , we look for
solutions of
𝑛 − 4 (𝑛+4)/(𝑛−4)
,
P (𝑢) =
𝐾𝑢
2
(7)
𝑐𝑛 =
𝑢>0
on 𝑆𝑛 .
The special case where the manifold is a sphere endowed with
its standard metric deserves particular attention. Indeed due
to Kazdan-Warner type obstructions, see [3], conditions have
to be imposed on the function 𝐾.
To state our result we set up the following conditions and
notation.
(𝑓)𝛽 Assume that 𝐾 : 𝑆𝑛 → R, 𝑛 ≥ 5 be a 𝐶1
positive function such that for any critical point 𝑦 of 𝐾,
there exists some real number 𝛽 = 𝛽(𝑦) such that in some
geodesic normal coordinate system centered at 𝑦 (through
the exponential map), we have
and 𝐺 is the Green function for the operator P(𝑢) on 𝑆𝑛 .
Here 𝑥1 is the first component of 𝑥 in some geodesic normal
coordinate system.
Let 𝜌(𝜏𝑝 ) be the least eigenvalue of 𝑀(𝜏𝑝 ). It was first
pointed out by Bahri [15] that when the interaction between
the different bubbles is of the same order as the self interaction, the least eigenvalue of some matrices like (11) plays
a fundamental role in the existence of solutions to problems
like (7). Regarding problem (7), such kind of phenomenon
appears under (𝑓)𝛽 condition in K𝑛−4 ; see [11].
(𝐴 1 ) Assume that 𝜌(𝜏𝑝 ) ≠ 0 for each distinct point
𝑦𝑙1 , . . . , 𝑦𝑙𝑝 ∈ K+ ∩ K𝑛−4 .
Now, we introduce the following set:
𝑝
C∞ := {𝜏𝑝 = (𝑦𝑙1 , . . . , 𝑦𝑙𝑝 ) ∈ (K+ ) , 𝑝 ≥ 1,
s.t. 𝑦𝑖 ≠ 𝑦𝑗 ∀𝑖 ≠ 𝑗, and if
{𝑦𝑙1 , . . . , 𝑦𝑙𝑝 } ∩ K𝑛−4 ≠ 0,
we denote by 𝑦𝑖1 , . . . , 𝑦𝑖𝑞 all elements of
𝑛
󵄨
󵄨𝛽
𝐾 (𝑥) = 𝐾 (𝑦) + ∑ 𝑏𝑘 󵄨󵄨󵄨(𝑥 − 𝑦)𝑘 󵄨󵄨󵄨 + 𝑅 (𝑥 − 𝑦) ,
{𝑦𝑙1 , . . . , 𝑦𝑙𝑝 } with 𝛽 (𝑦𝑖𝑗 ) = 𝑛 − 4
(8)
𝑘=1
where 𝑏𝑘 = 𝑏𝑘 (𝑦) ≠ 0, for all 𝑘 = 1, . . . , 𝑛, ∑𝑛𝑘=1 𝑏𝑘 ≠ 0 and
[𝛽]
∑𝑠=0 |∇𝑠 𝑅(𝑥 − 𝑦)||𝑥 − 𝑦|−𝛽+𝑠 = 𝑜(1) as 𝑥 tends to 𝑦. Here
𝑠
∇ denotes all possible derivatives of order 𝑠 and [𝛽] is the
integer part of 𝛽. For each critical point 𝑦 of 𝐾, we denote
̃𝑖 (𝑦) =: ♯ {𝑏𝑘 , 𝑘 = 1, . . . , 𝑛, such that 𝑏𝑘 < 0} .
(9)
Let,
for each 𝑗 = 1, . . . , 𝑞 then
𝜌 (𝑦𝑖1 , . . . , 𝑦𝑖𝑞 ) > 0} .
The main result of this paper is the following.
Theorem 1. Assume that 𝐾 satisfies (𝐴 1 ) and (𝑓)𝛽 , with
1 < 𝛽 ≤ 𝑛 − 4.
K = {𝑦 ∈ 𝑆𝑛 , ∇𝑔0 𝐾 (𝑦) = 0}
K = {𝑦 ∈ K, − ∑ 𝑏𝑘 > 0} ,
(10)
𝑘=1
For each 𝑝-tuple, 𝑝 ≥ 1 of distinct points 𝜏𝑝 := (𝑦𝑙1 , . . . , 𝑦𝑙𝑝 )
such that for all 𝑖 = 1, . . . , 𝑝, 𝑦𝑙𝑖 ∈ K+ ∩ K𝑛−4 , we define a
𝑝 × 𝑝 symmetric matrix 𝑀(𝜏𝑝 ) = (𝑚𝑖𝑗 ) by
𝑛
𝑛 − 4 − ∑𝑘=1 𝑏𝑘 (𝑦𝑙𝑖 )
̃𝑐1
,
𝑛/4
𝑛
𝐾(𝑦 )
−𝐺 (𝑦𝑙𝑖 , 𝑦𝑙𝑗 )
(𝑛−4)/8
[𝐾 (𝑦𝑙𝑖 ) 𝐾 (𝑦𝑙𝑗 )]
(11)
,
where
𝑐1 = ∫
R𝑛
̃𝑐1 =
𝑑𝑥
(1 + |𝑥|2 )
𝑐02𝑛/(𝑛−4)
(𝑛+4)/2
,
󵄨󵄨 󵄨󵄨𝑛−4
󵄨󵄨𝑥1 󵄨󵄨
∫
𝑛 𝑑𝑥,
R𝑛 (1 + |𝑥|2 )
∑
𝑝
̃
(−1)𝑝−1+∑𝑘=1 𝑛−𝑖(𝑦𝑘 ) − 1 ≠ 0.
(15)
Then (7) has at least one solution.
Moreover if (𝑛 − 4)/2 < 𝛽 ≤ 𝑛 − 4, for generic 𝐾 one has
♯𝑆 ≥ |𝑑| ,
(16)
where 𝑆 denotes the set of solutions to (7).
𝑙𝑖
𝑚𝑖𝑗 = 2(𝑛−4)/2 𝑐1
𝑑=
(𝑦1 ,...,𝑦𝑝 )∈C∞
K𝑛−4 = {𝑦 ∈ K, 𝛽 = 𝛽 (𝑦) = 𝑛 − 4} .
𝑚𝑖𝑖 =
(14)
If
𝑛
+
(13)
(12)
Our argument uses a careful analysis of the lack of
compactness of the Euler Lagrange functional 𝐽 associated
with problem (7). Namely, we study the noncompact orbits
of the gradient flow of 𝐽 the so-called critical points at
infinity following the terminology of Bahri [15]. These critical
points at infinity can be treated as usual critical points
once a Morse lemma at infinity is performed from which
we can derive just as in the classical Morse theory the
difference of topology induced by these noncompact orbits
and compute their Morse index. Such a Morse lemma at
infinity is obtained through the construction of suitable pseudogradient for which the Palais-Smale condition is satisfied
along the decreasing flow lines, as long as these flow lines do
Abstract and Applied Analysis
3
not enter the neighborhood of a finite number 𝑦1 , . . . , 𝑦𝑝 of
critical points of 𝐾 such that (𝑦1 , . . . , 𝑦𝑝 ) ∈ C∞ .
Similar Morse lemma at infinity has been established for
the problem (7) on the sphere 𝑆𝑛 , 𝑛 ≥ 5, under the hypothesis
that the order of flatness at critical points of 𝐾 is 𝛽 ∈ [𝑛−4, 𝑛[;
see [11].
The rest of this paper is organized as follows. In Section 2,
we set up the variational problem and we recall the expansion
of the gradient of the associated Euler-Lagrange functional
near infinity. In Section 3, we characterize the critical points
at infinity of the associated variational problem. Section 4 is
devoted to the proof of the main result Theorem 1, while we
give in Section 5 a more general statement than Theorem 1.
2. General Framework and Some Known Facts
2.1. Variational Structure and Lack of Compactness. Our
problem (7) enjoys a variational structure. Indeed, solutions
to (7) correspond to positive critical points of the functional
𝐽 (𝑢) =
∫𝑆𝑛 P𝑢𝑢𝑑V𝑔0
2𝑛/(𝑛−4)
(∫𝑆𝑛 𝐾|𝑢|
𝑑V𝑔0 )
(𝑛−4)/2𝑛
,
(17)
defined on
=
[(𝜆 𝑖 /𝜆 𝑗 ) + (𝜆 𝑗 /𝜆 𝑖 ) + (𝜆 𝑖 𝜆 𝑗 /2)(1 −
Here, 𝜀𝑖𝑗
cos 𝑑(𝑎𝑖 , 𝑎𝑗 ))]−(𝑛−4)/2 .
For 𝑤 a solution to (7) we also define 𝑉(𝑝, 𝜀, 𝑤) as
{
𝑉 (𝑝, 𝜀, 𝑤) = {
{
𝑢 ∈ Σ/∃ 𝛼0 > 0 s.t. 𝑢 − 𝛼0 𝑤 ∈ 𝑉 (𝑝, 𝜀) ,
󵄨󵄨 8/(𝑛−4)
󵄨
󵄨󵄨𝛼0
𝐽(𝑢)𝑛/(𝑛−4) − 1󵄨󵄨󵄨󵄨 < 𝜀.
󵄨
(22)
If 𝑢 is a function in 𝑉(𝑝, 𝜀, 𝑤), one can find an optimal
representation, following the ideas introduced in Proposition
5.2 of [15] and [16, pages 348–350]. Namely, we have the
following.
Proposition 2. For any 𝑝 ∈ N∗ , there is 𝜀𝑝 > 0 such that if
𝜀 ≤ 𝜀𝑝 and 𝑢 ∈ 𝑉(𝑝, 𝜀, 𝑤), then the following minimization
problem
󵄩󵄩
󵄩󵄩
𝑝
󵄩
󵄩󵄩
󵄩󵄩𝑢 − ∑𝛼𝑖 𝛿(𝑎 ,𝜆 ) − 𝛼0 (𝑤 + ℎ)󵄩󵄩󵄩
min
󵄩󵄩
󵄩
𝑖
𝑖
(23)
󵄩
󵄩󵄩
𝛼𝑖 >0,𝜆 𝑖 >0,𝑎𝑖 ∈𝑆𝑛 , 󵄩
𝑖=1
󵄩
ℎ∈𝑇𝑤 (𝑊𝑢 (𝑤)).
has a unique solution (𝛼, 𝜆, 𝑎, ℎ), up to a permutation.
In particular, we can write 𝑢 as follows:
Σ = {𝑢 ∈
𝐻22
𝑛
2
(𝑆 ) , ‖𝑢‖ := ∫ P𝑢𝑢𝑑V𝑔0 = 1} .
(18)
𝑆𝑛
However, it is delicate from a variational viewpoint
because the functional 𝐽 does not satisfy the Palais-Smale
condition. This means that there exist sequences along which
𝐽 is bounded, its gradient goes to zero and which not converge. The analysis of sequences failing Palais-Smale condition can be analyzed along the ideas introduced in [16, pages
325 and 334]. In order to describe such a characterization in
our case, we need to introduce some notations. For 𝑎 ∈ 𝑆𝑛
and 𝜆 > 0, let
1
𝛿(𝑎,𝜆) (𝑥) = 𝑐0 (𝑛−4)/2
2
×
𝜆(𝑛−4)/2
(𝑛−4)/2
(1 + ((𝜆2 − 1) /2) (1 − cos (𝑑 (𝑥, 𝑎))))
,
(19)
𝑛
where 𝑑 is the geodesic distance on (𝑆 , 𝑔) and 𝑐0 is chosen so
that 𝛿(𝑎,𝜆) is the family of solutions of the problem
P (𝑢) = 𝑢
(𝑛+4)/(𝑛−4)
,
𝑛
𝑢 > 0, on 𝑆
(20)
(see [17]).
We define now the set of potential critical points at
infinity associated with the function 𝐽. For 𝜀 > 0 and 𝑝 ∈ N∗ ,
let us define
𝑉 (𝑝, 𝜀)
𝑢 ∈ Σ/∃ 𝑎1 , . . . , 𝑎𝑝 ∈ 𝑆𝑛 , ∃𝜆 1 , . . . , 𝜆 𝑝 > 𝜀−1 ,
{
{
{
󵄩󵄩
󵄩󵄩
𝑝
{
{
󵄩
󵄩󵄩
󵄩
= { ∃𝛼1 , . . . , 𝛼𝑝 > 0 with 󵄩󵄩𝑢 − ∑𝛼𝑖 𝛿(𝑎𝑖 ,𝜆 𝑖 ) 󵄩󵄩󵄩󵄩 < 𝜀,
{
󵄩󵄩
󵄩
{
󵄩󵄩
𝑖=1
{
󵄩
{󵄨
󵄨
8/(𝑛−4)
𝑛/(𝑛−4)
󵄨
𝛼𝑖
𝐾 (𝑎𝑖 ) − 1󵄨󵄨󵄨󵄨 < 𝜀 ∀𝑖, 𝜀𝑖𝑗 < 𝜀 ∀𝑖 ≠ 𝑗.
{ 󵄨󵄨󵄨𝐽(𝑢)
(21)
𝑝
𝑢 = ∑𝛼𝑖 𝛿(𝑎𝑖 ,𝜆 𝑖 ) + 𝛼0 (𝑤 + ℎ) + V,
(24)
𝑖=1
where V belongs to 𝐻22 (𝑆𝑛 ) ∩ 𝑇𝑤 (𝑊𝑠 (𝑤)) and it satisfies (𝑉0 ),
𝑇𝑤 (𝑊𝑢 (𝑤)) and 𝑇𝑤 (𝑊𝑠 (𝑤)) are the tangent spaces at 𝑤 of
the unstable and stable manifolds of 𝑤 for a decreasing
pseudogradient of 𝐽, and (𝑉0 ) is the following:
⟨V, 𝜓⟩ = 0 for 𝜓 ∈ {𝛿𝑖 ,
𝜕𝛿𝑖 𝜕𝛿𝑖
,
, 𝑖 = 1, . . . , 𝑝}
𝜕𝜆 𝑖 𝜕𝑎𝑖
⟨V, 𝑤⟩ = 0,
⟨V, ℎ⟩ = 0
(𝑉0 )
∀ℎ ∈ 𝑇𝑤 𝑊𝑢 (𝑤) .
Here, 𝛿𝑖 = 𝛿(𝑎𝑖 ,𝜆 𝑖 ) and ⟨⋅, ⋅⟩ denotes the scalar product defined
on 𝐻22 (𝑆𝑛 ) by
⟨𝑢, V⟩ = ∫ Δ 𝑔0 𝑢Δ 𝑔0 V𝑑V𝑔0 + 𝑐𝑛 ∫ ∇𝑔0 𝑢∇𝑔0 V𝑑V𝑔0
𝑆𝑛
𝑆𝑛
(25)
+ 𝑑𝑛 ∫ 𝑢V𝑑V𝑔0 .
𝑆𝑛
Notice that Proposition 2 is also true if we take 𝑤 = 0 and,
therefore, ℎ = 0 and 𝑢 in 𝑉(𝑝, 𝜀).
We are ready now to state the characterization of the
sequences failing the Palais-Smale condition. For technical
reasons, we introduce the following subsets of Σ. Let Σ+ =
{𝑢 ∈ Σ, 𝑢 ≥ 0} and let for 𝜂 positive very small,
󵄩 󵄩8/(𝑛−4)
𝑉𝜂 (Σ+ ) = {𝑢 ∈ Σ, 𝐽(𝑢)(2𝑛−4)/(𝑛−4) 𝑒2𝐽(𝑢) 󵄩󵄩󵄩𝑢− 󵄩󵄩󵄩𝐿2𝑛/(𝑛−4) < 𝜂} ,
(26)
where 𝑢− = max(0, −𝑢). Using the idea introduced in [16, 18],
see also [19], we have the following proposition.
4
Abstract and Applied Analysis
Proposition 3. Let (𝑢𝑘 ) be a sequence in 𝑉𝜂 (Σ+ ) such that
𝐽(𝑢𝑘 ) is bounded and 𝜕𝐽(𝑢𝑘 ) goes to zero. Then there exists an
integer 𝑝 ∈ N∗ , a sequence (𝜀𝐾 ) > 0, 𝜀𝑘 tends to zero, and
an extracted subsequence of 𝑢𝑘 ’s, again denoted (𝑢𝑘 ), such that
𝑢𝑘 ∈ 𝑉(𝑝, 𝜀𝑘 , 𝑤) where 𝑤 is zero or a solution to (7).
Now arguing as in [16] (pages 326, 327, and 334), we
have the following Morse lemma which completely gets rid
of the V-contributions and shows that it can be neglected with
respect to the concentration phenomenon.
Proposition 4. There is a C1 -map which to each (𝛼𝑖 , 𝑎𝑖 , 𝜆 𝑖 , ℎ)
𝑝
such that ∑𝑖=1 𝛼𝑖 𝛿(𝑎𝑖 ,𝜆 𝑖 ) + 𝛼0 (𝑤 + ℎ) belongs to 𝑉(𝑝, 𝜀, 𝑤)
associates V = V(𝛼, 𝑎, 𝜆, ℎ) such that V is unique and satisfies
𝑝
𝐽 (∑𝛼𝑖 𝛿(𝑎𝑖 ,𝜆 𝑖 ) + 𝛼0 (𝑤 + ℎ) + V)
𝑖=1
(27)
𝑝
= min {𝐽 (∑𝛼𝑖 𝛿(𝑎𝑖 ,𝜆 𝑖 ) + 𝛼0 (𝑤 + ℎ) + V)} .
V∈(𝑉0 )
𝑖=1
Moreover, there exists a change of variables V − V → 𝑉 such
that
𝑝
𝐽 (∑𝛼𝑖 𝛿(𝑎𝑖 ,𝜆 𝑖 ) + 𝛼0 (𝑤 + ℎ) + V)
positive function tending to zero when 𝑠 → +∞. Using
Proposition 2, 𝑢(𝑠) can be written as
𝑝
𝑢 (𝑠) = ∑ 𝛼𝑖 (𝑠) 𝛿(𝑎𝑖 (𝑠),𝜆 𝑖 (𝑠)) + 𝛼0 (𝑠) (𝑤 + ℎ (𝑠)) + V (𝑠) . (31)
𝑖=1
̃ 𝑖 := lim𝑠 → +∞ 𝑎𝑖 (𝑠), we
̃ 𝑖 := lim𝑠 → +∞ 𝛼𝑖 (𝑠), 𝑦
Denoting 𝛼
denote by
𝑝
̃ 0 𝑤 or (̃
̃ 𝑝 , 𝑤)
̃ 𝑖 𝛿(̃𝑦𝑖 ,∞) + 𝛼
𝑦1 , ..., 𝑦
∑𝛼
∞
𝑖=1
such a critical point at infinity. If 𝑤 ≠ 0, it is called of 𝑤-type
or mixed type.
Notice that 𝑉𝜂 (Σ+ ) remains invariant under the flow
generated by (−𝜕𝐽) (see Lemma 5.1 of [20]; see also Lemma
4.1 of [18]).
2.2. Expansion of the Gradient of the Functional. In this
subsection, we recall the expansion of the gradient of the
functional 𝐽 in 𝑉(𝑝, 𝜀), 𝑝 ≥ 1.
𝑝
𝑖=1
(28)
𝑝
= 𝐽 (∑𝛼𝑖 𝛿(𝑎𝑖 ,𝜆 𝑖 ) + 𝛼0 (𝑤 + ℎ) + V) + ‖𝑉‖2 .
Proposition 7 (see [11]). For any 𝑢 = ∑𝑗=1 𝛼𝑗 𝛿𝑗 in 𝑉(𝑝, 𝜀), the
following expansion holds
(i)
𝑖=1
The following proposition gives precise estimates of V.
⟨𝜕𝐽 (𝑢) , 𝜆 𝑖
𝜕𝜀𝑖𝑗
𝜕𝛿𝑖
⟩ = − 2𝑐2 𝐽 (𝑢) ∑ 𝛼𝑗 𝜆 𝑖
𝜕𝜆 𝑖
𝜕𝜆 𝑖
𝑖 ≠ 𝑗
(33)
𝑝
Proposition 5 (see [11, Lemma 3.1]). Let 𝑢 = ∑𝑖=1 𝛼𝑖 𝛿𝑖 +
𝛼0 (𝑤 + ℎ) ∈ 𝑉(𝑝, 𝜀, 𝑤) and let V be defined in Proposition 4.
One has the following estimates: there exists 𝑐 > 0 independent
of 𝑢 such that the following holds:
𝑝
‖V‖ ≤ 𝑐∑ [
𝑖=1
1
𝜆𝑛/2
𝑖
(𝑛+4)/2𝑛
, 𝑖𝑓 𝑛 ≥ 12
𝑖𝑓 𝑛 < 12.
(29)
At the end of this subsection, we give the following
definition.
Definition 6 (see [16, 18]). A critical point at infinity of 𝐽 on
𝑉𝜂 (Σ+ ) is a limit of a flow line 𝑢(𝑠) of the equation
𝜕𝑢
= −𝜕𝐽 (𝑢 (𝑠)) ,
𝜕𝑠
1
+ 𝑜 (∑ 𝜀𝑖𝑗 ) + 𝑜 ( ) ,
𝜆𝑖
𝑖 ≠ 𝑗
where 𝑐2 = 𝑐02𝑛/(𝑛−4) ∫R𝑛 𝑑𝑦/(1 + |𝑦|2 )(𝑛+4)/2 .
(ii) If 𝑎𝑖 ∈ 𝐵(𝑦𝑗𝑖 , 𝜌), 𝑦𝑗𝑖 ∈ K and 𝜌 is a positive constant
small enough, one has
(𝑛+4)/2𝑛
󵄨
󵄨󵄨
󵄨∇𝐾 (𝑎𝑖 )󵄨󵄨󵄨 (log 𝜆 𝑖 )
+ 𝛽+󵄨
+
]
𝜆𝑖
𝜆(𝑛+4)/2
𝜆𝑖
𝑖
1
−1
)
∑ 𝜀(𝑛+4)/2(𝑛−4) (log 𝜀𝑘𝑟
{
{
{𝑘 ≠ 𝑟 𝑘𝑟
+ 𝑐{
−1 (𝑛−4)/𝑛
{
{ ∑ 𝜀𝑘𝑟 (log 𝜀𝑘𝑟
)
,
{𝑘 = ̸ 𝑟
(32)
(30)
𝑢 (0) = 𝑢0 ,
such that 𝑢(𝑠) remains in 𝑉(𝑝, 𝜀(𝑠), 𝑤) for 𝑠 ≥ 𝑠0 . Here
𝑤 is either zero or a solution to (7) and 𝜀(𝑠) is some
⟨𝜕𝐽 (𝑢) , 𝜆 𝑖
𝜕𝛿𝑖
⟩
𝜕𝜆 𝑖
𝜕𝜀𝑖𝑗 𝑛 − 4 2𝑛/(𝑛−4)
𝛼
+
𝛽 𝑖
= 2𝐽 (𝑢) [−𝑐2 ∑ 𝛼𝑗 𝜆 𝑖
𝑐0
𝜕𝜆 𝑖
2𝑛
𝐾 (𝑎𝑖 )
[ 𝑗 ≠ 𝑖
×
∑𝑛𝑘=1 𝑏𝑘
𝛽
𝜆𝑖
∫
R𝑛
signe(𝑥𝑘 + 𝜆 𝑖 (𝑎𝑖 − 𝑦𝑗𝑖 )𝑘 )
(34)
󵄨󵄨𝛽−1
󵄨
× 󵄨󵄨󵄨󵄨𝑥𝑘 + 𝜆 𝑖 (𝑎𝑖 − 𝑦𝑗𝑖 )𝑘 󵄨󵄨󵄨
×
𝑥𝑘
𝑛 𝑑𝑥
(1 + |𝑥|2 )
𝑝
+𝑜 (∑ 𝜀𝑖𝑗 + ∑
𝑗 ≠ 𝑖
1
𝛽
𝑗=1 𝜆 𝑗
)] .
]
Abstract and Applied Analysis
5
(iii) Furthermore if 𝜆 𝑖 |𝑎𝑖 − 𝑦𝑗𝑖 | < 𝛿, for 𝛿 very small, one
then has
⟨𝜕𝐽 (𝑢) , 𝜆 𝑖
𝜕𝛿𝑖
⟩
𝜕𝜆 𝑖
𝑛
𝜕𝜀𝑖𝑗
𝛼𝑖 ∑𝑘=1 𝑏𝑘
𝑛−4
− 𝑐2 ∑ 𝛼𝑗 𝜆 𝑖
= 2𝐽 (𝑢) × [
𝛽𝑐3
𝛽
2𝑛
𝜕𝜆 𝑖
𝐾 (𝑎𝑖 ) 𝜆
𝑗 ≠ 𝑖
𝑖
[
𝑝
+𝑜 (∑ 𝜀𝑖𝑗 + ∑
1
𝛽
𝑗=1 𝜆 𝑗
𝑗 ≠ 𝑖
)] ,
]
enter in the neighborhood of finite number of critical points
𝑦𝑖 , 𝑖 = 1, . . . , 𝑝 of 𝐾 such that (𝑦1 , . . . , 𝑦𝑝 ) ∈ C∞ .
Now we introduce the following main result.
Theorem 9. Assume that 𝐾 satisfies (𝐴 1 ) and (𝑓)𝛽 , 1 < 𝛽 ≤
𝑛 − 4.
Let 𝛽 := max{𝛽(𝑦)/𝑦 ∈ K}. For 𝑝 ≥ 1, there exists a
pseudogradient 𝑊 in 𝑉(𝑝, 𝜀) so that the following holds.
There exists a constant 𝑐 > 0 independent of 𝑢 =
𝑝
∑𝑖=1 𝛼𝑖 𝛿(𝑎𝑖 𝜆 𝑖 ) ∈ 𝑉(𝑝, 𝜀) such that
(𝑖)
󵄨
𝑝 󵄨󵄨
󵄨󵄨∇𝐾 (𝑎𝑖 )󵄨󵄨󵄨
+
+ ∑ 𝜀𝑖𝑗 ) .
∑
𝛽
𝜆𝑖
𝑖=1 𝜆 𝑖
𝑖=1
𝑗 ≠ 𝑖
(35)
𝑝
≤ −𝑐 (∑
where 𝑐3 = 𝑐02𝑛/(𝑛−4) ∫𝑆𝑛 |𝑥1 |𝛽 /(1 + |𝑥|2 )𝑛 𝑑𝑥.
𝑝
Proposition 8 (see [11]). Letting 𝑢 = ∑𝑗=1 𝛼𝑗 𝛿𝑗 ∈ 𝑉(𝑝, 𝜀), one
has
(i)
= −𝑐5 (𝐽 (𝑢))2(𝑛−2)/(𝑛−4) 𝛼𝑖(𝑛+4)/(𝑛−4)
1
𝜆
𝑖 ≠ 𝑗 𝑖
(𝑖𝑖)
∇𝐾 (𝑎𝑖 )
𝜆𝑖
(36)
󵄨󵄨 𝜕𝜀 󵄨󵄨
1
󵄨󵄨 𝑖𝑗 󵄨󵄨
󵄨󵄨
󵄨󵄨) + 𝑜 (∑ 𝜀𝑖𝑗 + ) ,
𝜆
󵄨󵄨󵄨 𝜕𝑎𝑖 󵄨󵄨󵄨
𝑖
𝑖 ≠ 𝑗
1
⟨𝜕𝐽 (𝑢 + V) , 𝑊 (𝑢) +
𝜕V
(𝑊 (𝑢))⟩
𝜕 (𝛼𝑖 , 𝑎𝑖 , 𝜆 𝑖 )
(38)
󵄨
𝑝 󵄨󵄨
󵄨󵄨∇𝐾 (𝑎𝑖 )󵄨󵄨󵄨
+
+ ∑ 𝜀𝑖𝑗 ) .
∑
𝛽
𝜆𝑖
𝑖=1 𝜆 𝑖
𝑖=1
𝑗 ≠ 𝑖
𝑝
≤ −𝑐 (∑
1 𝜕𝛿𝑖
⟩
⟨𝜕𝐽 (𝑢) ,
𝜆 𝑖 𝜕𝑎𝑖
+ 𝑂 (∑
⟨𝜕𝐽 (𝑢) , 𝑊 (𝑢)⟩
1
Furthermore |𝑊| is bounded and the only case where the
maximum of the 𝜆 𝑖 ’s is not bounded is when 𝑎𝑖 ∈ 𝐵(𝑦𝑙𝑖 , 𝜌) with
𝑦𝑙𝑖 ∈ K, for all 𝑖 = 1, . . . , 𝑝, (𝑦𝑙1 , . . . , 𝑦𝑙𝑝 ) ∈ C∞ .
We will prove Theorem 9 at the end of the section. Now
we state two results which deal with two specific cases of
Theorem 9. Let
where 𝑐5 = ∫R𝑛 𝑑𝑦/(1 + |𝑦|2 )𝑛 .
(ii) If 𝑎𝑖 ∈ 𝐵(𝑦𝑗𝑖 , 𝜌), 𝑦𝑗𝑖 ∈ K, one has
𝑝
𝑉1 (𝑝, 𝜀) = {𝑢 = ∑𝛼𝑖 𝛿𝑖 ∈ 𝑉 (𝑝, 𝜀) s.t., 𝑎𝑖 ∈ 𝐵 (𝑦𝑙𝑖 , 𝜌) ,
𝑖=1
1 𝜕𝛿𝑖
⟨𝜕𝐽 (𝑢) ,
⟩
𝜆 𝑖 𝜕(𝑎𝑖 )𝑘
= −2 (𝑛 − 4) 𝑐02𝑛/(𝑛−4) 𝛼𝑖(𝑛+4)/(𝑛−4) (𝐽 (𝑢))2(𝑛−2)/(𝑛−4)
󵄨
󵄨𝛽
× ∫ 𝑏𝑘 󵄨󵄨󵄨󵄨𝑥𝑘 + 𝜆 𝑖 (𝑎𝑖 − 𝑦𝑗𝑖 )𝑘 󵄨󵄨󵄨󵄨
R𝑛
𝑥𝑘
2 𝑛+1
(1 + |𝑥| )
1
𝛽
𝜆𝑖
𝑑𝑦
󵄨
󵄨
1
1 󵄨󵄨 𝜕𝜀𝑖𝑗 󵄨󵄨󵄨
󵄨󵄨) ,
+ 𝑜 (∑ 𝜀𝑖𝑗 ) + 𝑜 (∑ 𝛽 ) + 𝑂 (∑ 󵄨󵄨󵄨󵄨
󵄨
𝑖 ≠ 𝑗
𝑖=1 𝜆 𝑖
𝑖 ≠ 𝑗 𝜆 𝑖 󵄨󵄨 𝜕𝑎𝑖 󵄨󵄨
(37)
𝑝
where 𝑘 = 1, . . . , 𝑛 and (𝑎𝑖 )𝑘 is the 𝑘th component of 𝑎𝑖 in some
geodesic normal coordinates system.
3. Characterization of the Critical
Points at Infinity
This section is devoted to the characterization of the critical
points at infinity in 𝑉(𝑝, 𝜀), 𝑝 ≥ 1, under 𝛽-flatness condition
with 1 < 𝛽 ≤ 𝑛 − 4. This characterization is obtained
through the construction of a suitable pseudogradient at
infinity for which the Palais-Smale condition is satisfied along
the decreasing flow-lines as long as these flow lines do not
𝑦𝑙𝑖 ∈ K \ K𝑛−4 ∀𝑖 = 1, . . . , 𝑝} ,
𝑝
𝑉2 (𝑝, 𝜀) = {𝑢 = ∑𝛼𝑖 𝛿𝑖 ∈ 𝑉 (𝑝, 𝜀) s.t., 𝑎𝑖 ∈ 𝐵 (𝑦𝑙𝑖 , 𝜌) ,
𝑖=1
𝑦𝑙𝑖 ∈ K𝑛−4 ∀𝑖 = 1, . . . , 𝑝} .
(39)
We then have the following.
Proposition 10 (see [11], Proposition 3.7). For 𝑝 ≥ 1 there
exists a pseudogradient 𝑊2 in 𝑉2 (𝑝, 𝜀) such that for all 𝑢 =
𝑝
∑𝑖=1 𝛼𝑖 𝛿𝑖 ∈ 𝑉2 (𝑝, 𝜀), one has
𝑝
⟨𝜕𝐽 (𝑢) , 𝑊2 (𝑢)⟩ ≤ −𝑐 (∑
1
𝑛−4
𝑖=1 𝜆 𝑖
󵄨
𝑝 󵄨󵄨
󵄨∇𝐾 (𝑎𝑖 )󵄨󵄨󵄨
+ ∑ 𝜀𝑖𝑗 + ∑ 󵄨
),
𝜆𝑖
𝑖 ≠ 𝑗
𝑖=1
(40)
where 𝑐 is a positive constant independent of 𝑢. Furthermore,
one has |𝑊2 | which is bounded and the only case where the
maximum of 𝜆 𝑖 ’s is not bounded is when 𝑎𝑖 ∈ 𝐵(𝑦𝑙𝑖 , 𝜌), 𝑦𝑙𝑖 ∈
K+ , for all 𝑖 = 1, . . . , 𝑝, with 𝜌(𝑦𝑙1 , . . . , 𝑦𝑙𝑝 ) > 0.
6
Abstract and Applied Analysis
Proposition 11. For 𝑝 ≥ 1, there exists a pseudogradient 𝑊1
in 𝑉1 (𝑝, 𝜀) so that the following holds.
𝑝
There exists 𝑐 > 0 independent of 𝑢 = ∑𝑖=1 𝛼𝑖 𝛿𝑖 ∈ 𝑉1 (𝑝, 𝜀)
such that
󵄨
𝑝 󵄨󵄨
󵄨󵄨∇𝐾 (𝑎𝑖 )󵄨󵄨󵄨
+
𝜀
+
∑
).
∑
𝑖𝑗
𝛽
𝜆𝑖
𝑖=1 𝜆 𝑖
𝑖 ≠ 𝑗
𝑖=1
(41)
𝑝
⟨𝜕𝐽 (𝑢) , 𝑊1 (𝑢)⟩ ≤ −𝑐 (∑
1
Now, we introduce the following two lemmas.
𝑝
Lemma 12. Let 𝑢 = ∑𝑖=1 𝛼𝑖 𝛿(𝑎𝑖 ,𝜆 𝑖 ) ∈ 𝑉(𝑝, 𝜀), such that 𝑎𝑖 ∈
𝐵(𝑦𝑙𝑖 , 𝜌), 𝑦𝑙𝑖 ∈ K, for all 𝑖 = 1, . . . , 𝑝. One then has
⟨𝜕𝐽 (𝑢) , 𝑍𝑖 (𝑢)⟩ = − 2𝑐2 𝐽 (𝑢) ∑ 𝛼𝑖 𝛼𝑗
𝑗 ≠ 𝑖
𝑍𝑖 (𝑢) = 𝛼𝑖 𝜆 𝑖
𝜕𝛿(𝑎𝑖 ,𝜆 𝑖 )
𝜕𝜆 𝑖
󵄨󵄨
󵄨𝛽
󵄨󵄨𝑥𝑘 + 𝜆 𝑖 (𝑎𝑖 − 𝑦𝑙𝑖 ) 󵄨󵄨󵄨 𝑖
1 𝜕𝛿(𝑎𝑖 ,𝜆 𝑖 )
󵄨
𝑘󵄨
𝑋𝑖 = 𝛼𝑖 ∑
∫ 𝑏
𝜆 𝜕(𝑎𝑖 )𝑘 R𝑛 𝑘 (1 + 𝜆 󵄨󵄨󵄨(𝑎 − 𝑦 ) 󵄨󵄨󵄨)𝛽𝑖 −1
𝑘=1 𝑖
𝑖 󵄨󵄨 𝑖
𝑙𝑖 𝑘 󵄨󵄨
𝑥𝑘
×
𝑑𝑥.
𝑛+1
(1 + |𝑥|2 )
(42)
𝛽
)
1
),
where 𝑘𝑖 is defined in (45).
Proof. Using Proposition 7, we have
⟨𝜕𝐽 (𝑢) , 𝑍𝑖 (𝑢)⟩
𝜕𝜀𝑖𝑗
𝜕𝜆 𝑖
+
2
𝑛 − 4 2𝑛/(𝑛−4) 𝛼𝑖
𝛽
𝑐0
2𝑛
𝐾 (𝑎𝑖 )
󵄨𝛽−1
󵄨
×∫ signe (𝑥𝑘 + 𝜆 𝑖 (𝑎𝑖 − 𝑦𝑗𝑖 )𝑘 ) 󵄨󵄨󵄨󵄨𝑥𝑘 + 𝜆 𝑖 (𝑎𝑖 − 𝑦𝑗𝑖 )𝑘 󵄨󵄨󵄨󵄨
R𝑛
𝑥𝑘
𝑝
1
𝑑𝑥 + 𝑜 (∑ 𝜀𝑖𝑗 ) + 𝑜 ( ∑ 𝛽 ) .
2 𝑛
𝑗
(1 + |𝑥| )
𝑗 ≠ 𝑖
𝑗=1 𝜆 𝑗
(47)
Observe that for 𝑘 ∈ {1, . . . , 𝑛}, if 𝜆 𝑖 |(𝑎𝑖 − 𝑦𝑙𝑖 )𝑘 | > 𝑀1 /√𝑛, we
have
∫ signe (𝑥𝑘 + 𝜆 𝑖 (𝑎𝑖 − 𝑦𝑗𝑖 )𝑘 )
󵄨󵄨
󵄨𝛽
󵄨󵄨𝑥𝑘 + 𝜆 𝑖 (𝑎𝑖 − 𝑦𝑙𝑖 ) 󵄨󵄨󵄨 𝑖 𝑥𝑘
󵄨
𝑘󵄨
𝑑𝑥
∫
𝑛+1
R𝑛
(1 + |𝑥|2 )
R𝑛
(43)
󵄨
󵄨 𝛽𝑖 −1
= 𝑐 (signe 𝜆 𝑖 (𝑎𝑖 − 𝑦𝑙𝑖 )𝑘 ) (𝜆 𝑖 󵄨󵄨󵄨󵄨(𝑎𝑖 − 𝑦𝑙𝑖 )𝑘 󵄨󵄨󵄨󵄨)
󵄨󵄨
󵄨𝛽 −1
󵄨󵄨𝑥𝑘 + 𝜆 𝑖 (𝑎𝑖 − 𝑦𝑙𝑖 ) 󵄨󵄨󵄨 𝑖 𝑥𝑘
󵄨
𝑘󵄨
×
𝑑𝑥
𝑛
(1 + |𝑥|2 )
(48)
= 𝑐 signe (𝜆 𝑖 (𝑎𝑖 − 𝑦𝑗𝑖 )𝑘 )
× (1 + 𝑜 (1)) .
󵄨
󵄨 𝛽𝑖 −2
× (𝜆 𝑖 󵄨󵄨󵄨󵄨(𝑎𝑖 − 𝑦𝑙𝑖 )𝑘 󵄨󵄨󵄨󵄨)
(1 + 𝑜 (1)) ,
Hence our claim follows. Next, we will say that
𝑖 ∈ 𝐿2
1
𝜆𝑖 𝑖
(46)
×
We claim that 𝑋𝑖 is bounded. Indeed, the claim is trivial if
𝜆 𝑖 |𝑎𝑖 −𝑦𝑙𝑖 | ≤ 𝑀1 . If 𝜆 𝑖 |𝑎𝑖 −𝑦𝑙 | > 𝑀1 , for any 𝑘, 1 ≤ 𝑘 ≤ 𝑛, such
𝑖
that 𝜆 𝑖 |(𝑎𝑖 − 𝑦𝑙𝑖 )𝑘 | > 𝑀1 /√𝑛, we have the following estimate:
+ 𝑂(
𝛽𝑗
𝑗=1 𝜆 𝑗
𝑗 ≠ 𝑖
𝑗 ≠ 𝑖
𝑛
𝑖 ∈ 𝐿1
𝑝
+ 𝑜 (∑ 𝜀𝑖𝑗 ) + 𝑜 (∑
= −2𝑐2 𝐽 (𝑢) ∑ 𝛼𝑖 𝛼𝑗 𝜆 𝑖
,
𝜕𝜆 𝑖
󵄨󵄨
󵄨𝛽 −2
󵄨󵄨(𝑎𝑖 − 𝑦𝑙 ) 󵄨󵄨󵄨 𝑖
󵄨󵄨
𝑖 𝑘𝑖 󵄨
[
]
󵄨
+ [𝑂 (
) , if 𝑖 ∈ 𝐿 2 ]
𝜆2𝑖
[
]
Furthermore |𝑊1 | is bounded and the only case where the
maximum of the 𝜆 𝑖 ’s is not bounded is when 𝑎𝑖 ∈ 𝐵(𝑦𝑙𝑖 , 𝜌) with
𝑦𝑙𝑖 ∈ K+ , for all 𝑖 = 1, . . . , 𝑝, and 𝑦i ≠ 𝑦𝑗 for all 𝑖 ≠ 𝑗.
Observe that in 𝑉1 (𝑝, 𝜀) the interaction of two bubbles
is negligible with respect to the self-interaction. Similar
phenomena occur for the scalar curvature problem, see [21],
so the proof of Proposition 11 is similar to the corresponding
statement in [21].
Before giving the proof of Theorem 9, we state the
following notations extracted from [11].
Let 𝑀1 be a fixed positive constant large enough and let
𝑝
𝑢 = ∑𝑖=1 𝛼𝑖 𝛿(𝑎𝑖 ,𝜆 𝑖 ) ∈ 𝑉(𝑝, 𝜀) such that 𝑎𝑖 ∈ 𝐵(𝑎𝑖 , 𝜌), 𝑦𝑙𝑖 ∈ K,
for all 𝑖 = 1, . . . , 𝑝. For any index 𝑖, 1 ≤ 𝑖 ≤ 𝑝, we define the
following vector fields:
𝜕𝜀𝑖𝑗
󵄨
󵄨
if 𝜆 𝑖 󵄨󵄨󵄨󵄨𝑎𝑖 − 𝑦𝑙𝑖 󵄨󵄨󵄨󵄨 ≤ 𝑀1 ,
󵄨
󵄨
if 𝜆 𝑖 󵄨󵄨󵄨󵄨𝑎𝑖 − 𝑦𝑙𝑖 󵄨󵄨󵄨󵄨 > 𝑀1 ,
(44)
󵄨󵄨
󵄨𝛽 −1
󵄨󵄨𝑥𝑘 + 𝜆 𝑖 (𝑎𝑖 − 𝑦𝑙𝑖 ) 󵄨󵄨󵄨 𝑖 󵄨󵄨󵄨󵄨𝑥𝑘 󵄨󵄨󵄨󵄨
󵄨
𝑘󵄨
𝑑𝑥 = 𝑂 (1) .
∫
𝑛
R𝑛
(1 + |𝑥|2 )
and we will denote by 𝑘𝑖 the index satisfying
󵄨
󵄨󵄨
󵄨󵄨(𝑎𝑖 − 𝑦𝑙 ) 󵄨󵄨󵄨 = max 󵄨󵄨󵄨󵄨(𝑎𝑖 − 𝑦𝑙 ) 󵄨󵄨󵄨󵄨 .
󵄨󵄨
𝑖 𝑘𝑖 󵄨
𝑖 𝑘󵄨
󵄨 1≤𝑘≤𝑛 󵄨
It easy to see that if 𝑖 ∈ 𝐿 2 , then 𝜆 𝑖 |(𝑎𝑖 − 𝑦𝑙𝑖 )𝑘𝑖 | > 𝑀1 /√𝑛.
taking 𝑀1 large enough. If not, we have
(49)
(45)
Using the fact that 𝑘𝑖 defined in (45) satisfies 𝜆 𝑖 |(𝑎𝑖 − 𝑦𝑙𝑖 )𝑘𝑖 | >
𝑀1 /√𝑛, if 𝑖 ∈ 𝐿 2 , Lemma 12 follows.
Abstract and Applied Analysis
7
𝑝
Lemma 13. For 𝑢 = ∑𝑖=1 𝛼𝑖 𝛿(𝑎𝑖 ,𝜆 𝑖 ) ∈ 𝑉(𝑝, 𝜀), such that 𝑎𝑖 ∈
𝐵(𝑦𝑙𝑖 , 𝜌), 𝑦𝑙𝑖 ∈ K, for all 𝑖 = 1, . . . , 𝑝, one has
𝑝
Subset 1. We consider here the case of 𝑢 = ∑𝑖=1 𝛼𝑖 𝛿(𝑎𝑖 𝜆 𝑖 ) =
∑𝑖∈𝐼1 𝛼𝑖 𝛿(𝑎𝑖 𝜆 𝑖 ) + ∑𝑖∈𝐼2 𝛼𝑖 𝛿(𝑎𝑖 𝜆 𝑖 ) such that
⟨𝜕𝐽 (𝑢) , 𝑋𝑖 (𝑢)⟩
1
𝑗 ≠ 𝑖 𝜆 𝑖
≤ 𝑂 (∑
󵄨󵄨 𝜕𝜀 󵄨󵄨
1
󵄨󵄨 𝑖𝑗 󵄨󵄨
󵄨󵄨
󵄨
󵄨󵄨 𝜕𝑎𝑖 󵄨󵄨󵄨) + 𝑂 [( 𝜆𝛽𝑖 ) , if 𝑖 ∈ 𝐿 1 ]
󵄨
󵄨
𝑖
𝐼1 ≠ 0, 𝐼2 ≠ 0,
󵄨󵄨
󵄨𝛽 −1
󵄨󵄨(𝑎𝑖 − 𝑦𝑙 ) 󵄨󵄨󵄨 𝑖
󵄨󵄨
𝑖 𝑘𝑖 󵄨
1
[
] (50)
󵄨
+ [−𝑐 ( 𝛽 +
) , if 𝑖 ∈ 𝐿 2 ]
𝜆𝑖
𝜆𝑖 𝑖
[
]
𝑝
+ 𝑜 (∑
remains (using Propositions 10 and 11) to focus attention on
the two following subsets of 𝑉(𝑝, 𝜀).
1
𝛽𝑗
𝑗=1 𝜆 𝑗
∑ 𝛼𝑖 𝛿(𝑎𝑖 𝜆 𝑖 ) ∈ 𝑉1
𝑖∈𝐼1
∑ 𝛼𝑖 𝛿(𝑎𝑖 𝜆 𝑖 ) ∈ 𝑉2
𝑖∈𝐼2
(♯𝐼1 , 𝜀) ,
(52)
(♯𝐼2 , 𝜀) .
Without loss of generality, we can assume that
𝜆1 ≤ ⋅ ⋅ ⋅ ≤ 𝜆𝑝.
),
(53)
We distinguish three cases.
where 𝑘𝑖 is defined in (45).
Case 1. 𝑢1 := ∑𝑖∈𝐼1 𝛼𝑖 𝛿(𝑎𝑖 𝜆 𝑖 ) ∉ 𝑉11 (♯𝐼1 , 𝜀) = {𝑢 =
♯𝐼
∑𝑗=11 𝛼𝑗 𝛿(𝑎𝑗 ,𝜆 𝑗 ) , 𝑎𝑗 ∈ 𝐵(𝑦𝑙𝑗 , 𝜌), 𝑦𝑙𝑗 ∈ K+ for all 𝑗 = 1, . . . , ♯𝐼1
and 𝑦𝑙𝑗 ≠ 𝑦𝑙𝑘 for all 𝑗 ≠ 𝑘}.
̃1 be the pseudogradient on 𝑉(𝑝, 𝜀) defined by
Let 𝑊
̃1 (𝑢) = 𝑊1 (𝑢1 ), where 𝑊1 is the vector filed defined by
𝑊
Proposition 11 in 𝑉1 (♯𝐼1 , 𝜀). Note that if 𝑢1 ∉ 𝑉11 (♯𝐼1 , 𝜀), then
the pseudo-gradient 𝑊1 (𝑢1 ) does not increase the maximum
of the 𝜆 𝑖 ’s, 𝑖 ∈ 𝐼1 . Using Proposition 11, we have
Proof. Using Proposition 8, we have
⟨𝜕𝐽 (𝑢) , 𝑋𝑖 (𝑢)⟩
= −2 (𝑛 − 4) 𝑐02𝑛/(𝑛−4) 𝛼𝑖2𝑛/(𝑛−4)
1
𝜆 𝑖 𝛽𝑖
󵄨󵄨
󵄨𝛽
󵄨󵄨𝑥𝑘 + 𝜆 𝑖 (𝑎𝑖 − 𝑦𝑙𝑖 ) 󵄨󵄨󵄨 𝑖
󵄨
𝑘󵄨
× ∑ (∫ 𝑏𝑘
󵄨
󵄨 (𝛽𝑖 −1)/2
R𝑛
𝑘=1
(1 + 𝜆 𝑖 󵄨󵄨󵄨󵄨(𝑎𝑖 − 𝑦𝑙𝑖 )𝑘 󵄨󵄨󵄨󵄨)
𝑛
×
1
+ 𝑂 (∑
𝜆
𝑗 ≠ 𝑖 𝑖
𝑥𝑘
̃1 (𝑢)⟩
⟨𝜕𝐽 (𝑢) , 𝑊
2
2 𝑛+1
(1 + |𝑥| )
𝛽𝑖
𝑖∈𝐼1 𝜆 𝑖
𝑑𝑥)
𝑝
󵄨󵄨 𝜕𝜀 󵄨󵄨
1
󵄨󵄨 𝑖𝑗 󵄨󵄨
󵄨󵄨
󵄨󵄨) + 𝑜 ( ∑
)
𝛽𝑗
󵄨󵄨󵄨 𝜕𝑎𝑖 󵄨󵄨󵄨
𝑗=1 𝜆 𝑗
1
+ 𝑂 (∑
𝜆
𝑗 ≠ 𝑖 𝑖
𝑛+1
(1 + |𝑥|2 )
󵄨󵄨
󵄨
󵄨∇𝐾 (𝑎𝑖 )󵄨󵄨󵄨
)
∑ 𝜀𝑖𝑗 + ∑ 󵄨
𝜆𝑖
𝑗 ≠ 𝑖,𝑖,𝑗∈𝐼1
𝑖∈𝐼1
(54)
𝑖∈𝐼1 ,𝑗∈𝐼2
(51)
2
×
+
+ 𝑂 ( ∑ 𝜀𝑖𝑗 ) .
󵄨󵄨
󵄨𝛽
󵄨󵄨𝑥𝑘 + 𝜆 𝑖 (𝑎𝑖 − 𝑦𝑙 ) 󵄨󵄨󵄨 𝑖
󵄨
𝑖
1
𝑘𝑖 󵄨󵄨
󵄨
≤ −𝑐 𝛽 (∫ 𝑏𝑘𝑖
󵄨
󵄨
󵄨
󵄨 (𝛽𝑖 −1)/2
R𝑛
𝜆𝑖 i
(1 + 𝜆 𝑖 󵄨󵄨󵄨(𝑎𝑖 − 𝑦𝑙𝑖 )𝑘 󵄨󵄨󵄨)
󵄨
𝑖󵄨
𝑥𝑘𝑖
1
≤ −𝑐 (∑
𝑑𝑥)
𝑝
󵄨󵄨 𝜕𝜀 󵄨󵄨
1
󵄨󵄨 𝑖𝑗 󵄨󵄨
󵄨󵄨
󵄨󵄨) + 𝑜 ( ∑
).
𝛽𝑗
󵄨󵄨󵄨 𝜕𝑎𝑖 󵄨󵄨󵄨
𝑗=1 𝜆 𝑗
An easy calculation yields
𝜀𝑖𝑗 = 𝑂 (
1
(𝜆 𝑖 𝜆 𝑗 )
(𝑛−4)/2
) = 𝑜(
1
𝛽
𝜆𝑖 𝑖
) + 𝑜(
1
𝛽
𝜆𝑗𝑗
),
(55)
Fix 𝑖0 ∈ 𝐼1 , we denote by
1 𝛽
≥ 𝜆 𝑖0𝑖0 } ,
𝐽1 = {𝑖 ∈ 𝐼2 , s.t, 𝜆𝑛−4
𝑖
2
𝐽2 = 𝐼2 \ 𝐽1 .
(56)
Using (54) and (55), we find that
̃1 (𝑢)⟩
⟨𝜕𝐽 (𝑢) , 𝑊
󵄨󵄨
󵄨
󵄨󵄨∇𝐾 (𝑎𝑖 )󵄨󵄨󵄨
+
+ ∑ 𝜀𝑖𝑗 )
∑
𝛽𝑖
𝜆𝑖
𝑖∈𝐼1 ∪𝐽1 𝜆 𝑖
𝑖∈𝐼1
𝑗 ≠ 𝑖∈𝐼1
Using (43) and the fact that 𝜆 𝑖 |(𝑎𝑖 − 𝑦𝑙𝑖 )𝑘𝑖 | > 𝑀1 /√𝑛, if 𝑖 ∈ 𝐿 2 ,
Lemma 13 follows.
≤ −𝑐 ( ∑
Proof of Theorem 9. In order to complete the construction
of the pseudogradient 𝑊 suggested in Theorem 9, it only
+ 𝑜 (∑
𝑝
1
1
𝛽𝑖
𝑖=1 𝜆 𝑖
).
(57)
8
Abstract and Applied Analysis
Let 𝑚1 > 0 be small enough; using Lemma 13, (63), and
(55), we get
From another part, by Lemma 12 we have
⟨𝜕𝐽 (𝑢) , ∑ − 2𝑖 𝑍𝑖 (𝑢)⟩
⟨𝜕𝐽 (𝑢) , ∑ − 2𝑖 𝑍𝑖 (𝑢) + 𝑚1 ∑ 𝑋𝑖 (𝑢)⟩
𝑖∈𝐽1
𝑖
≤ 𝑐 ∑ 2 𝜆𝑖
𝑗 ≠ 𝑖,𝑖∈𝐽1
𝜕𝜀𝑖𝑗
𝜕𝜆 𝑖
+ 𝑂 (∑
𝑖∈𝐽1
1
𝛽𝑖
𝑖∈𝐽1 𝜆 𝑖
)
(58)
󵄨󵄨
󵄨𝛽 −2
󵄨󵄨(𝑎𝑖 − 𝑦𝑙 ) 󵄨󵄨󵄨 𝑖
󵄨󵄨
𝑖 𝑘𝑖 󵄨
󵄨
+ 𝑂( ∑
).
2
𝜆
𝑖
𝑖∈𝐽1 ∩𝐿 2
𝜕𝜀𝑖𝑗
𝜕𝜆 𝑖
≤ − 𝑐𝜀𝑖𝑗 ,
𝛽𝑖
𝑖∈𝐽1 𝜆 𝑖
if 𝜆 𝑖 ≥ 𝜆 𝑗 or 𝜆 𝑖 ∼ 𝜆 𝑗
𝜕𝜆 𝑖
+ 2𝑗 𝜆 𝑗
𝜕𝜀𝑖𝑗
𝜕𝜆 𝑗
(60)
),
𝑖∈𝐽1 ∩𝐿 2
𝑖 ≠ 𝑗∈𝐼1
󵄨󵄨
󵄨
󵄨∇𝐾 (𝑎𝑖 )󵄨󵄨󵄨
𝜀𝑖𝑗 + ∑ 󵄨
)
𝜆𝑖
𝑗 ≠ 𝑖,𝑖∈𝐽1 ,𝑗∈𝐽1 ∪𝐽2
𝑖∈𝐼1 ∪𝐽1
𝑝
+ 𝑜 (∑
∑
1
𝛽𝑖
𝑖=1 𝜆 𝑖
).
(65)
̃ :=
We need to add the remainder indices 𝑖 ∈ 𝐽2 . Note that 𝑢
∑𝑗∈𝐽2 𝛼𝑗 𝛿𝑗 ∈ 𝑉2 (♯𝐽2 , 𝜀). Thus using Proposition 10, we apply
̃2 . We
the associated vector field which we will denote by 𝑊
then have the following estimate:
𝑖∈𝐽1
𝑗 ≠ 𝑖,𝑖∈𝐽1 ,𝑗∈𝐽1 ∪𝐽2
𝛽𝑖
𝑖=1 𝜆 𝑖
+ ∑ 𝜀𝑖𝑗
𝛽𝑖
𝑖∈𝐼1 ∪𝐽1 𝜆 𝑖
+
≤ −𝑐𝜀𝑖𝑗 ,
1
≤ −𝑐 ( ∑
⟨𝜕𝐽 (𝑢) , ∑ − 2𝑖 𝑍𝑖 (𝑢)⟩
∑
1
𝑖∈𝐽1
(59)
and for 𝑖 ∈ 𝐽1 and 𝑗 ∈ 𝐽2 we have 𝜆 𝑗 ≤ 𝜆 𝑖 , we obtain
𝜆 𝑖 (𝜕𝜀𝑖𝑗 /𝜕𝜆 𝑖 ) ≤ −𝑐𝜀𝑖𝑗 . These estimates yield
≤ −𝑐
𝑝
) + 𝑜 (∑
̃1 (𝑢) + 𝑚1 (∑ − 2𝑖 𝑍𝑖 (𝑢) + 𝑚1 ∑ 𝑋𝑖 (𝑢))⟩
⟨𝜕𝐽 (𝑢) , 𝑊
Since for 𝑖 < 𝑗, we have
𝜕𝜀𝑖𝑗
1
(64)
and by (57) we obtain
󵄨
󵄨
or 󵄨󵄨󵄨󵄨𝑎𝑖 − 𝑎𝑗 󵄨󵄨󵄨󵄨 ≥ 𝛿0 > 0.
2𝑖 𝜆 𝑖
󵄨󵄨
󵄨
󵄨∇𝐾 (𝑎𝑖 )󵄨󵄨󵄨
𝜀𝑖𝑗 + ∑ 󵄨
)
𝜆𝑖
𝑗 ≠ 𝑖,𝑖∈𝐽1 ,𝑗∈𝐽1 ∪𝐽2
𝑖∈𝐽1
∑
+ 𝑂 (∑
Observe that using a direct calculation, we have
𝜆𝑖
≤ −𝑐 (
𝑖∈𝐽1 ∩𝐿 2
𝜀𝑖𝑗 + 𝑂 (∑
1
𝛽𝑖
𝑖∈𝐽1 𝜆 𝑖
)
󵄨󵄨
󵄨𝛽 −2
󵄨󵄨(𝑎𝑖 − 𝑦𝑙 ) 󵄨󵄨󵄨 𝑖
󵄨󵄨
𝑖 𝑘𝑖 󵄨
󵄨
+ 𝑂( ∑
)
2
𝜆
𝑖
𝑖∈𝐽1 ∩𝐿 2
̃2 (𝑢)⟩
⟨𝜕𝐽 (𝑢) , 𝑊
(61)
≤ −𝑐 ( ∑
1
𝛽𝑗
𝑗∈𝐽2 𝜆 𝑗
󵄨󵄨
󵄨
󵄨󵄨∇𝐾 (𝑎𝑗 )󵄨󵄨󵄨
󵄨
󵄨)
+ ∑ 𝜀𝑖𝑗 + ∑
𝜆𝑗
𝑖 ≠ 𝑗,𝑖,𝑗∈𝐽2
𝑗∈𝐽2
𝑝
+ 𝑂 ( ∑ 𝜀𝑖𝑗 ) + 𝑜 (∑
𝛽𝑖
𝑖=1 𝜆 𝑖
𝑗∈𝐽2 ,𝑖∈𝐽1
+ 𝑂 ( ∑ 𝜀𝑖𝑗 ) .
It is easy to see that for any index 𝑖 ∈ 𝐿 2 , we have
(62)
for any 𝑖 ∈ 𝐿 2 .
󵄨
𝑝 󵄨󵄨
󵄨󵄨∇𝐾 (𝑎𝑖 )󵄨󵄨󵄨
+
+ ∑ 𝜀𝑖𝑗 ) .
∑
𝛽𝑖
𝜆𝑖
𝑖=1 𝜆 𝑖
𝑖=1
𝑖 ≠ 𝑗
(67)
𝑝
⟨𝜕𝐽 (𝑢) , 𝑊 (𝑢)⟩ ≤ −𝑐 (∑
where 𝑘𝑖 is defined in (45) and 𝑀1 is large enough. Thus, we
derive that
󵄨󵄨
󵄨𝛽 −2
󵄨𝛽 −1
󵄨󵄨
󵄨󵄨(𝑎𝑖 − 𝑦𝑙 ) 󵄨󵄨󵄨 𝑖
󵄨󵄨(𝑎𝑖 − 𝑦𝑙 ) 󵄨󵄨󵄨 𝑖
󵄨
󵄨
󵄨󵄨
𝑖 𝑘𝑖 󵄨
𝑖 𝑘𝑖 󵄨
󵄨
= 𝑜 (󵄨
),
2
𝜆𝑖
𝜆𝑖
),
since |𝑎𝑖 − 𝑎𝑗 | ≥ 𝜌 for 𝑗 ∈ 𝐽2 and 𝑖 ∈ 𝐼1 .
̃1 + 𝑚1 (𝑊
̃2 + ∑𝑖∈𝐽 −2𝑖 𝑍𝑖 +
Let in this case 𝑊 = 𝑊
1
𝑚1 ∑𝑖∈𝐽1 ∩𝐿 2 𝑋𝑖 ).
From (65) and (66) we obtain
𝑖∈𝐽1 ,𝑗∈𝐼1
󵄨󵄨
󵄨𝛽 −2
󵄨󵄨
󵄨󵄨𝛽𝑖 −1
󵄨󵄨(𝑎𝑖 − 𝑦𝑙 ) 󵄨󵄨󵄨 𝑖
󵄨
󵄨
󵄨󵄨
𝑖 𝑘𝑖 󵄨
√𝑛 󵄨󵄨󵄨(𝑎𝑖 − 𝑦𝑙𝑖 )𝑘𝑖 󵄨󵄨󵄨
󵄨
≤
,
𝑀1
𝜆𝑖
𝜆2𝑖
1
(66)
(63)
1
Case 2. 𝑢1 := ∑𝑖∈𝐼1 𝛼𝑖 𝛿(𝑎𝑖 𝜆 𝑖 ) ∈ 𝑉11 (♯𝐼1 , 𝜀) and 𝑢2 :=
♯𝐼
∑𝑖∉𝐼2 𝛼𝑖 𝛿(𝑎𝑖 𝜆 𝑖 ) ∉ 𝑉21 (♯𝐼2 , 𝜀) := {𝑢 = ∑𝑗=12 𝛼𝑗 𝛿(𝑎𝑗 ,𝜆 𝑗 ) , 𝑎𝑗 ∈
𝐵(𝑦𝑙𝑗 , 𝜌), 𝑦𝑙𝑗 ∈ K+ , for all 𝑗 = 1, . . . , ♯𝐼2 and 𝜌(𝑦𝑙1 , . . . , 𝑦♯𝐼2 ) >
0}.
Abstract and Applied Analysis
9
Since 𝑢2 ∈ 𝑉2 (♯𝐼2 , 𝜀), by Proposition 10, we can apply the
associated vector field which we will denote by 𝑉1 . We get
Using the previous estimate and Lemma 13, we find that
󵄨
𝑝 󵄨󵄨
󵄨󵄨∇𝐾 (𝑎𝑖 )󵄨󵄨󵄨
+
+ ∑ 𝜀𝑖𝑗 ) .
∑
𝛽𝑖
𝜆𝑖
𝑖=1 𝜆 𝑖
𝑖=1
𝑖 ≠ 𝑗
(74)
𝑝
⟨𝜕𝐽 (𝑢) , 𝑊 (𝑢)⟩ ≤ −𝑐 (∑
⟨𝜕𝐽 (𝑢) , 𝑉1 (𝑢)⟩
󵄨󵄨
󵄨󵄨
󵄨󵄨∇𝐾 (𝑎𝑖 )󵄨󵄨 + ∑ 𝜀 )
+
∑
𝑖𝑗
𝛽𝑖
𝜆𝑖
𝑖∈𝐼2 𝜆 𝑖
𝑖∈𝐼2
𝑖 ≠ 𝑗,𝑖,𝑗∈𝐼2
1
≤ −𝑐 (∑
(68)
+ 𝑂 ( ∑ 𝜀𝑖𝑗 ) .
𝑖∈𝐼2 , 𝑗∈𝐼1
Observe that 𝑉1 (𝑢) does not increase the maximum of the 𝜆 𝑖 ’s,
𝑖 ∈ 𝐼2 , since 𝑢2 ∉ 𝑉21 (♯𝐼2 , 𝜀). Fix 𝑖0 ∈ 𝐼2 and let
1
𝛽
},
𝐽̃1 = {𝑖 ∈ 𝐼1 , s.t, 𝜆 𝑖 𝑖 ≥ 𝜆𝑛−4
2 𝑖0
𝐽̃2 = 𝐼1 \ 𝐽̃1 .
Case 3. 𝑢1 ∈ 𝑉11 (♯𝐼1 , 𝜀) and 𝑢2 ∈ 𝑉21 (♯𝐼2 , 𝜀).
̃2 ) be the pseudo-gradient in 𝑉(𝑝, 𝜀) defined
̃1 (resp., 𝑉
Let 𝑉
̃
̃2 (𝑢) = 𝑊2 (𝑢2 )) where 𝑊1
by 𝑉1 (𝑢) = 𝑊1 (𝑢1 ) (resp., 𝑉
(resp., 𝑊2 ) is the vector field defined by Proposition 11 (resp.,
Proposition 10) in 𝑉11 (♯𝐼1 , 𝜀) (resp., 𝑉21 (♯𝐼2 , 𝜀)) and let in this
case
̃2 .
̃1 + 𝑉
𝑊=𝑉
(69)
󵄨
𝑝 󵄨󵄨
󵄨󵄨∇𝐾 (𝑎𝑖 )󵄨󵄨󵄨
+
+ ∑ 𝜀𝑖𝑗 ) .
∑
𝛽𝑖
𝜆𝑖
𝑖=1 𝜆 𝑖
𝑖=1
𝑖 ≠ 𝑗
(76)
𝑝
⟨𝜕𝐽 (𝑢) , 𝑊 (𝑢)⟩ ≤ −𝑐 (∑
⟨𝜕𝐽 (𝑢) , 𝑉1 (𝑢)⟩
1
𝑝
+ 𝑜 (∑
1
𝛽𝑖
𝑖=1 𝜆 𝑖
(70)
).
̃ :=
We need to add the indices 𝑖, 𝑖 ∈ 𝐽̃2 . Letting 𝑢
̃ ∈ 𝑉1 (♯̃𝐽2 , 𝜀), we can apply the
∑𝑗∈̃𝐽2 𝛼𝑗 𝛿(𝑎𝑗 𝜆 𝑗 ) , since 𝑢
associated vector field given by Proposition 10. Let 𝑉2 be this
vector field. By Proposition 11 we have
⟨𝜕𝐽 (𝑢) , 𝑉2 (𝑢)⟩
≤ −𝑐 ( ∑
1
𝛽𝑗
𝑗∈𝐽̃2 𝜆 𝑗
󵄨󵄨
󵄨
󵄨󵄨∇𝐾 (𝑎𝑗 )󵄨󵄨󵄨
󵄨
󵄨+
+∑
𝜆
𝑗
̃
𝑗∈𝐽2
∑
𝑖 ≠ 𝑗, 𝑖,𝑗∈𝐽̃2
𝜀𝑖𝑗 )
(71)
+ 𝑂 ( ∑ 𝜀𝑖𝑗 ) .
𝑗∈𝐽̃2 ,𝑖∉𝐽̃2
Observe that 𝐼1 = 𝐽̃1 ∪ 𝐽̃2 and we are in the case where for all
𝑖 ≠ 𝑗 ∈ 𝐼1 , we have |𝑎𝑖 − 𝑎𝑗 | ≥ 𝜌. Thus by (55), we get
𝑝
𝑂 ( ∑ 𝜀𝑖𝑗 ) = 𝑜 (∑
𝑗∈𝐽̃2 ,𝑖∉𝐽̃2
1
𝛽𝑖
𝑖=1 𝜆 𝑖
),
(72)
and hence
󵄨󵄨
󵄨󵄨
󵄨󵄨∇𝐾 (𝑎𝑖 )󵄨󵄨 + ∑ 𝜀 ) .
+
∑
𝑖𝑗
𝛽𝑖
𝜆𝑖
𝑖=1 𝜆 𝑖
𝑖 ≠ 𝑗
𝑖∈𝐼2 ∪𝐽̃2
𝑝
1
1
Notice that in the first and second cases, the maximum
of the 𝜆 𝑖 ’s, 1 ≤ 𝑖 ≤ 𝑝, is a bounded function and hence the
Palais-Smale condition is satisfied along the flow lines of 𝑊.
However in the third case all the 𝜆 𝑖 ’s, 1 ≤ 𝑖 ≤ 𝑝, will increase
and go to +∞ along the flow lines generated by 𝑊.
𝑝
Subset 2. We consider the case of 𝑢 = ∑𝑖=1 𝛼𝑖 𝛿𝑖 ∈ 𝑉(𝑝, 𝜀), such
that there exist 𝑎𝑖 satisfying 𝑎𝑖 ∉ ∪𝑦∈K 𝐵(𝑦, 𝜌).
In this region, the construction of the pseudo-gradient 𝑊
proceeds exactly as the proof of Theorem 3.2, of subset 2, of
[11].
Finally, observe that our pseudogradient 𝑊 in 𝑉(𝑝, 𝜀)
satisfies claim (i) of Theorem 9 and it is bounded, since
‖𝜆 𝑖 (𝜕𝛿(𝑎𝑖,𝜆 ) /𝜕𝜆 𝑖 )‖ and ‖(1/𝜆 𝑖 )(𝜕𝛿(𝑎𝑖,𝜆 ) /𝜕𝑎𝑖 )‖ are bounded.
𝑖
𝑖
From the definition of 𝑊, the 𝜆 𝑖 ’s, 1 ≤ 𝑖 ≤ 𝑝, decrease along
the flow lines of 𝑊 as long as these flow lines do not enter
in the neighborhood of finite number of critical points 𝑦𝑙𝑖 ,
𝑖 = 1, . . . , 𝑝, of K such that (𝑦𝑙1 , . . . , 𝑦𝑙𝑝 ) ∈ C∞ .
Now, arguing as in Appendix 2 of [16], see also Appendix
B of [18], claim (ii) of Theorem 9 follows from (i) and
Proposition 5. This completes the proof of Theorem 9.
Corollary 14. Let 𝑝 ≥ 1. The critical points at infinity of 𝐽 in
𝑉(𝑝, 𝜀) correspond to
𝑝
(𝑦𝑙1 , . . . , 𝑦𝑙𝑝 )
∞
:= ∑
1
𝑖=1 𝐾(𝑦𝑙
)
𝑖
𝛿
,
(𝑛−4)/2 (𝑦𝑙𝑖 ,∞)
(77)
where (𝑦𝑙1 , . . . , 𝑦𝑙𝑝 ) ∈ C∞ . Moreover, such a critical point at
⟨𝜕𝐽 (𝑢) , 𝑉1 (𝑢) + 𝑉2 (𝑢)⟩
≤ −𝑐 (∑
(75)
Using Proposition 10, Proposition 11, and (55) we get
Using (68) and (55), we get
󵄨󵄨
󵄨
󵄨󵄨∇𝐾 (𝑎𝑖 )󵄨󵄨󵄨
+ ∑ 𝜀𝑖𝑗 )
≤ −𝑐 ( ∑ 𝛽 + ∑
𝑖
𝜆𝑖
𝑖∈𝐼2
𝑖 ≠ 𝑗,𝑖,𝑗∈𝐼2
𝑖∈𝐼2 ∪𝐽̃1 𝜆 𝑖
1
𝑝
(73)
Let in this case 𝑊 = 𝑉1 + 𝑉2 + 𝑚1 ∑𝑖∈𝐽̃1 𝑋𝑖 (𝑢), 𝑚1 small
enough.
infinity has an index equal to 𝑖(𝑦𝑙1 , . . . , 𝑦𝑙𝑝 )∞ = 𝑝−1+∑𝑖=1 𝑛−
̃𝑖(𝑦).
4. Proof of Theorem 1
We prove the existence result by contradiction. Assume that
𝐽 has no critical point in 𝑉𝜂 (Σ+ ). It follows from Corollary 14
10
Abstract and Applied Analysis
that the critical points at infinity of the associated variational
problem are in one to one correspondence with the elements
of C∞ defined in (13).
Notice that, just like for usual critical points, it is associated with each critical point at infinity 𝑤∞ of 𝐽 stable and
unstable manifolds 𝑊𝑠∞ (𝑤∞ ) and 𝑊𝑢∞ (𝑤∞ ) (see [16, pages
356-357]). These manifolds can be easily described once a
finite-dimensional reduction like the one we performed in
Section 3 is established.
For any 𝑤∞ = (𝑦𝑖1 , . . . , 𝑦𝑖𝑝 ) ∈ C∞ , let 𝑐(𝑤)∞ =
𝑝
𝑆𝑛 (∑𝑗=1
(𝑛−4)/2 4/𝑛
1/𝐾(𝑦𝑖𝑗 )
)
denote the associated critical
value. Here we choose to consider a simplified situation,
󸀠
, 𝑐(𝑤)∞ ≠ 𝑐(𝑤󸀠 )∞ , and thus order the
where for any 𝑤∞ ≠ 𝑤∞
𝑐(𝑤)∞ ’s, 𝑤∞ ∈ C∞ as
𝑐(𝑤1 )∞ < ⋅ ⋅ ⋅ < 𝑐(𝑤𝑘0 )∞ .
(78)
By using a deformation lemma (see [22, proposition 7.24 and
Theorem 8.2]), we know that if 𝑐(𝑤𝑘−1 )∞ < 𝑎 < 𝑐(𝑤𝑘 )∞ <
𝑏 < 𝑐(𝑤𝑘+1 )∞ , then
𝐽𝑏 ≃ 𝐽𝑎 ∪ 𝑊𝑢∞ (𝑤𝑘 )∞ ,
(79)
where 𝐽𝑏 = {𝑢 ∈ 𝑉𝜂 (Σ+ ), 𝐽(𝑢) ≤ 𝑏} and ≃ denotes retracts by
deformation.
We apply the Euler-Poincaré characteristic of both sides
of (79); we find that
𝜒 (𝐽𝑏 ) = 𝜒 (𝐽𝑎 ) + (−1)𝑖(𝑤𝑘 )∞ ,
(80)
where 𝑖(𝑤𝑘 )∞ denotes the index of the critical point at infinity
(𝑤𝑘 )∞ . Let
𝑏1 < 𝑐(𝑤1 )∞ =
min 𝐽 (𝑢) < 𝑏2 < 𝑐(𝑤2 )∞
𝑢∈𝑉𝜂 (Σ+ )
(81)
< ⋅ ⋅ ⋅ < 𝑏𝑘0 < 𝑐(𝑤𝑘0 )∞ < 𝑏𝑘0 +1 .
Since we have assumed that (4) has no solution, 𝐽𝑏𝑘 +1 is a
0
retard by deformation of 𝑉𝜂 (Σ+ ). Therefore 𝜒(𝐽𝑏𝑘 +1 ) = 1, since
0
𝑉𝜂 (Σ+ ) is a contractible set. Now using (80), we derive after
recalling that 𝜒(𝐽𝑏1 ) = 𝜒(0) = 0
𝑘0
1 = ∑(−1)𝑖(𝑤𝑗 )∞ .
(82)
𝑗=1
Hence if (82) is violated, 𝐽 has a critical point in 𝑉𝜂 (Σ+ ).
Now, arguing as in the proof of theorem of [18, pages 659660], we prove that such a critical point is positive.
To prove the multiplicity part of the statement, we observe
that it follows from Sard-Smale theorem that for generic
𝐾’s, the solutions to (7) are all nondegenerate, in the sense
that the associated linearized operator does not admit zero
as eigenvalue. We need to introduce the following lemma
extracted from [11].
Lemma 15 (see [11, Section 3.2]). Let 𝑤 be a solution to (7).
Assume that the function 𝐾 satisfies condition (𝑓)𝛽 , with (𝑛 −
4)/2 < 𝛽 ≤ 𝑛 − 4; then for each 𝑝 ∈ N∗ , there are neither
critical points nor critical points at infinity in 𝑉(𝑝, 𝜀, 𝑤).
Once the existence of mixed critical points at infinity is
ruled out, it follows from the previous arguments that
𝑘0
𝑉𝜂 (Σ+ ) ≃ ⋃𝑊𝑢∞ (𝑤𝑗 )∞ ∪
𝑗=1
⋃
𝑊𝑢 (𝑤) .
(83)
𝑤,𝜕𝐽(𝑤)=0
Now using the Euler-Poincaré theorem, we derive that
𝑘0
1 = ∑(−1)𝑖(𝑤𝑗 )∞ +
𝑗=1
(−1)morse(𝑤) .
∑
(84)
𝑤,𝜕𝐽(𝑤)=0
Hence our theorem follows.
5. A General Existence Result
In this last section of this paper, we give a generalization of
Theorem 1. Namely, in view of the result of Theorem 1, one
may think about the situation where the degree 𝑑 given in
Theorem 1 is equal to zero; that is, the total sum in 𝑑 is equal
to 1, but a partial one is not equal to 1. A natural question
arises: is it possible in this case to use such information to
derive an existence result? In the following theorem we give a
partial answer to this question.
Theorem 16. Let 𝐾 be a function satisfying (𝐴 1 ) and (𝑓)𝛽 ,
with 1 < 𝛽 ≤ 𝑛 − 4. If there exists 𝑘 ∈ N such that
(1)
∀𝑤∞ ∈ C∞ ,
(2)
∑
𝑖(𝑤)∞ ≠ 𝑘 + 1
(−1)𝑖(𝑤∞ ) ≠ 1,
(85)
𝑤∞ ∈C∞ ,𝑖(𝑤∞ )≤𝑘
then the problem (7) has at least one solution.
Moreover for generic 𝐾’s, if (𝑛 − 4)/2 < 𝛽 ≤ 𝑛 − 4, then the
number of solutions is lower bounded by
󵄨󵄨
󵄨󵄨
󵄨
󵄨󵄨
󵄨󵄨
𝑖(𝑤∞ ) 󵄨󵄨󵄨
∑
(−1)
󵄨󵄨 .
󵄨󵄨1 −
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨
𝑤∞ ∈C∞ ,𝑖(𝑤∞ )≤𝑘
(86)
Let 𝑙♯ be the maximal index over all elements of C∞ .
Please observe that the integer 𝑘 = 𝑙♯ satisfies (1) of
Theorem 16; it follows that Theorem 1 is a corollary of
Theorem 16.
Proof of Theorem 16. We set
𝑋𝑘∞ =
⋃
𝑥∞ ∈C∞ ,𝑖(𝑥)∞ ≤𝑘
𝑊𝑢∞ (𝑥∞ ),
(87)
where 𝑊𝑢∞ (𝑥∞ ) is the closure of the unstable manifolds of
𝑥∞ . Observe that 𝑋𝑘∞ is a stratified set of top dimension
𝑘, which is contractible set in 𝑉𝜂 (Σ+ ), since 𝑉𝜂 (Σ+ ) is a
contractible set. Let 𝑈 denote the image of such a contraction. To prove the first part of Theorem 16, arguing by
contradiction, we assume that (7) has no solution. Using the
pseudo-gradient constructed in Theorem 9, we can deform
𝑈. By transversality arguments, we can assume that such
a deformation avoids all critical points at infinity of index
Abstract and Applied Analysis
11
greater or equal to 𝑘 + 1. Note that, from assumption (1) of
Theorem 16, there is no critical point at infinity with index
𝑘 + 1. It follows then from a theorem of Bahri and Rabinowitz
[22] that
𝑈 ≃ 𝑋𝑘∞ .
(88)
Hence from the Euler-Poincaré theorem, we get
1 = 𝜒 (𝑋𝑘∞ ) =
∑
(−1)𝑖(𝑤∞ ) ,
𝑤∞ ∈𝐶∞ ,𝑖(𝑤∞ )≤𝑘
(89)
which is a contradiction with assumption (2) of Theorem 16.
Regarding the multiplicity result, we observe that for
generic 𝐾’s the functional 𝐽 admits only nondegenerate critical points. Hence by Lemma 15, the set 𝑈 will be deformed
into
𝑈 ≃ 𝑋𝑘∞ ∪ ⋃ 𝑊𝑢 (𝑤) ,
𝑤∈𝐶𝑘
(90)
where 𝐶𝑘 denotes the set of the critical points of 𝐽 of Morse
index less than or equal to 𝑘, which are dominated by 𝑈.
Finally by using the Euler-Poincaré theorem, the proof of
Theorem 16 follows.
References
[1] S. M. Paneitz, “A quartic conformally covariant differential
operator for arbitrary pseudo-Riemannian manifolds,” SIGMA.
Symmetry, Integrability and Geometry. Methods and Applications, vol. 4, article 036, 2008.
[2] T. P. Branson, “Differential operators canonically associated to
a conformal structure,” Mathematica Scandinavica, vol. 57, no.
2, pp. 293–345, 1985.
[3] Z. Djadli, E. Hebey, and M. Ledoux, “Paneitz-type operators
and applications,” Duke Mathematical Journal, vol. 104, no. 1, pp.
129–169, 2000.
[4] Sun-Y.A. Chang and P. C. Yang, “Extremal metrics of zeta function determinants on 𝑆𝑛 -manifolds,” Annals of Mathematics, vol.
142, no. 1, pp. 171–212, 1995.
[5] Z. Djadli and A. Malchiodi, “Existence of conformal metrics
with constant 𝑄-curvature,” Annals of Mathematics, vol. 168, no.
3, pp. 813–858, 2008.
[6] M. J. Gursky, “The principal eigenvalue of a conformally
invariant differential operator, with an application to semilinear
elliptic PDE,” Communications in Mathematical Physics, vol.
207, no. 1, pp. 131–143, 1999.
[7] J. Wei and X. Xu, “On conformal deformations of metrics on 𝑆𝑛 ,”
Journal of Functional Analysis, vol. 157, no. 1, pp. 292–325, 1998.
[8] W. Abdelhedi and H. Chtioui, “On the prescribed Paneitz curvature problem on the standard spheres,” Advanced Nonlinear
Studies, vol. 6, no. 4, pp. 511–528, 2006.
[9] A. Bensouf and H. Chtioui, “Conformal metrics with prescribed
𝑄-curvature on 𝑆𝑛 ,” Calculus of Variations and Partial Differential Equations, vol. 41, no. 3-4, pp. 455–481, 2011.
[10] H. Chtioui and A. Rigane, “On the prescribed 𝑄-curvature
problem on 𝑆𝑛 ,” Comptes Rendus Mathématique. Académie des
Sciences. Paris, vol. 348, no. 11-12, pp. 635–638, 2010.
[11] H. Chtioui and A. Rigane, “On the prescribed Q-curvature
problem on 𝑆𝑛 ,” Journal of Functional Analysis, vol. 261, no. 10,
pp. 2999–3043, 2011.
[12] Z. Djadli, A. Malchiodi, and M. O. Ahmedou, “Prescribing
a fourth order conformal invariant on the standard sphere.
I. A perturbation result,” Communications in Contemporary
Mathematics, vol. 4, no. 3, pp. 375–408, 2002.
[13] Z. Djadli, A. Malchiodi, and M. O. Ahmedou, “Prescribing a
fourth order conformal invariant on the standard sphere. II.
Blow up analysis and applications,” Annali della Scuola Normale
Superiore di Pisa, vol. 1, no. 2, pp. 387–434, 2002.
[14] V. Felli, “Existence of conformal metrics on 𝑆𝑛 with prescribed
fourth-order invariant,” Advances in Differential Equations, vol.
7, no. 1, pp. 47–76, 2002.
[15] A. Bahri, Critical Points at Infinity in Some Variational Problems,
vol. 182 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, UK, 1989.
[16] A. Bahri, “An invariant for Yamabe-type flows with applications
to scalar-curvature problems in high dimension,” Duke Mathematical Journal, vol. 81, no. 2, pp. 323–466, 1996.
[17] C.-S. Lin, “A classification of solutions of a conformally invariant fourth order equation in R𝑛 ,” Commentarii Mathematici
Helvetici, vol. 73, no. 2, pp. 206–231, 1998.
[18] M. Ben Ayed, Y. Chen, H. Chtioui, and M. Hammami, “On
the prescribed scalar curvature problem on 𝑆𝑛 -manifolds,” Duke
Mathematical Journal, vol. 84, no. 3, pp. 633–677, 1996.
[19] E. Hebey and F. Robert, “Asymptotic analysis for fourth order
Paneitz equations with critical growth,” Advances in Calculus of
Variations, vol. 4, no. 3, pp. 229–275, 2011.
[20] M. Ben Ayed and K. El Mehdi, “The Paneitz curvature problem
on lower-dimensional spheres,” Annals of Global Analysis and
Geometry, vol. 31, no. 1, pp. 1–36, 2007.
[21] R. Ben Mahmoud and H. Chtioui, “Prescribing the scalar
curvature problem on higher-dimensional manifolds,” Discrete
and Continuous Dynamical Systems A, vol. 32, no. 5, pp. 1857–
1879, 2012.
[22] A. Bahri and P. H. Rabinowitz, “Periodic solutions of Hamiltonian systems of three body type,” Annales de l’Institut Henri
Poincaré. Analyse Non Linéaire, vol. 8, no. 6, pp. 561–649, 1991.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014