On The Number of Interior Peak Solutions for A
Singularly Perturbed Neumann Problem
Fang-Hua Lin
Courant Institute of Mathematical Sciences
Wei-Ming Ni
University of Minnesota
AND
Jun-Cheng Wei
Chinese University of Hong Kong
Abstract
We consider the following singularly perturbed Neumann problem:
where "#%! $'&)(+*
(-, *.
in in and on is the Laplace operator, / is a constant, is a
with its unit outward normal , and is superlinear and subcritical. A typical is 1243 where 57698:6 <; when
= :>
! @ when =DCE .
and 5<6?86 !BA GF
6
and for each
We show that there exists an GFHI such that for 6
integer J bounded by
!N ON P
5<KJLK
M
! GQSRUTVWQ !
=
where !N ON P is a constant depending on X and only, there exists a solution
M
with J interior peaks. (An explicit formula for !N ON P is also given.) As a conM
sequence, we obtain that for sufficiently small, there exists at least Y[
a Z]\'^ _]a `
\cbedf g d h%\ji
n = number of solutions. Moreover, for each kml
there exist solutions with
o
!
q
A
p
energies in the order of
.
r
bounded smooth domain in 0 !
c 2000 Wiley Periodicals, Inc.
1 Introduction
The main theme of this paper is the concentration phenomena of the following
singularly perturbed elliptic problem
s
t uwvWxzy|{}~{|€{‚?ƒD„
(1.1)
{ˆ
Šz‹ ‡‰„
ŠŒ ƒL„
in …†
in …†
on Ž
Communications
on Pure and Applied Mathematics, Vol. 000, 0001–0027 (2000)

c 2000 Wiley Periodicals, Inc.
2
F.-H. LIN, W.-M. NI, J.-C. WEI
Š
"“n’ ”V• Šz– #
’1š
satisfies ˜7™~—I™ ’œ› x
x
smooth domain in ¥ ’
where y‘ƒ
is the Laplace operator,
is a constant, the exponent —
v‡/„
for ŸžD and ˜?™~—¡™/¢ for  ƒ¤£ , and … is a bounded
with its unit outward normal ¦ .
For the last fifteen years, problem (1.1) has received considerable attention as it
has been shown that its solutions have rich and interesting structures. In particular,
the various concentration phenomena exhibited by the solutions of (1.1) seem both
mathematically intriguing and scientifically useful.
Although problem (1.1) takes a classical form of singular perturbations, the
traditional techniques in that area did not seem helpful as the error terms appeared
in the inner and outer expansions are exponentially small in vœ‡§„ .
In•the
papers [22, 28, 29], the authors studied the following “energy” functional
©
in ¨
ǻ associated with (1.1) via a variational approach
(1.2)
˜
¬H­¯® {±°²ƒ
£
³µ´
©
{ ¶ x ¹{ x ª }
v x·¶ ¸
³µ´
˜
—

˜
{ š
š
•
†
where { š ƒDº¼»¾½¿{ † „À .
It is easily seen that ¬«­ is not bounded above nor bounded below. However, Ni
and Takagi observed in [28] that, among all possible solutions of (1.1), there is a
“least-energy” solution { ­ ; that is, a solution { ­ with minimal energy. Furthermore,
they showed in [28, 29] that, for each v?‡m„ sufficiently small, { ­ has exactly one
located on Ž… and near the “most
(local) maximum point Á­ in … , and Á­ must be
´
curved” part of the Ž… . More
precisely, { ­ must © tend to „ , as v«ÃĄ , everywhere
©
on … except at Á­ , and ¨ Á­ª à º»¾½'ÅÆ Š ¨ Áª where ¨ denotes the mean
curvature of the boundary Ž… . (Such points Á­ will be referred to as peaks or
spikes.) The crucial idea in [28, 29] is to establish the following energy estimate
for v small
(1.3)
¬H­
• †-Ç
©
{ ­ª ƒLv
’
®UÇ
•
}
Ç
x
¨
©
Á­ª v€È
©
v ª °
where Ç
are two positive constants. From (1.3), the above conclusion on the
x
location of the peak ÁV­ is derived.
Since the publication of [29], problem (1.1) has received a great deal of attention and significant progress has been made. More specifically, solutions with
multiple boundary peaks as well as multiple interior peaks have been established.
It turns out that a general guideline is that while multiple boundary spikes tend
© to
cluster around the local minimum points of the boundary mean curvature ¨ Áɪ ,
the location of the interior spikes are governed by the distance between the peaks
as well as the boundary of Ž… . (See [3, 4, 6-10, 12-15, 19, 20, 29-36] and the
references therein.) In particular, it was established in Gui and Wei [14] that for
any two given integers Ê˞ „ †W̜ž „ and Ê  Ì ‡Í„ , problem (1.1) has a solution
with exactly Ê interior spikes and Ì boundary spikes for every v sufficiently small.
This solution has its “energy” still at the level v ’ and again has the property that it
tends to „ everywhere in … except at those Ê  Ì points. Thus, in general we call
such spiky solutions as solutions with „'} dimensional concentration sets.
MULTI-PEAK SOLUTIONS
3
It seems natural to ask if problem (1.1) has solutions which “concentrate” on
higher dimensional sets, e.g. curves, or surfaces. In this regards, we mention that it
has been conjectured for a long time that problem (1.1) actually possesses solutions
which have Î } dimensional concentration sets for every „¼Ï Î Ï2Ð7} ˜ . (See e.g.
[27].) Furthermore, it is intuitively clear that the “energy” levels of many solutions
with Î } dimensional concentration sets would be at the order v ’œ›±Ñ . Progress in
this direction, although still limited, has also been made in [2, 23, 24, 25, 26].
In this paper we shall explore the question of the maximal number of spikes,
in terms of the small parameter v9‡Ò„ , a solution of (1.1) could possibly have.
Our
Ï
© main result, Theorem 1.1 below, asserts that for every positive integer Ó
›’ , where Ç is a suitable constant depending only on — and … , problem
Ç
v ¶ÕÔnÖ v ¶ ª
(1.1) has a solution with exactly Ó peaks. As a corollary, one derives immediately
that problem (1.1) already possesses spiky solutions { ­ (i.e. solutions with only
„'} dimensional concentration sets), with “energy” levels
(1.4)
¬
­ ® { ­ °×/v
’œ›±Ñ
where Î ranges from „ to  } ˜ . (In this paper we use “ØÙ­ ×ÛÚ­ Ü Ü to denote that
there exist positive constants Ç)Ý and ÇÝ Ý such that Ç)Ý Ï ØÙ­ßÞ Ú ­ Ï ÇÝ Ý for v small.)
Thus it seems difficult to use different energy levels to characterize solutions with
concentration sets of different dimensions. This further illustrates how complicated
the simple problem (1.1) could be, and its richness seems somewhat surprising.
Our proof uses a “localized energy” method as in [13], but we now need to
obtain accurate estimates of the “interactions” between two peaks as well as the
“interactions” between a peak and the boundary of the domain … . This is not an
easy task – it requires a thorough understanding of various approximations of the
solutions of (1.1) we are looking for.
To state our main result, we shall include a slightly more general equation than
(1.1), namely,
©
s
t u"v+xzy|{}~{àâá { ª ƒD„
{¡
Š‹ ‡§„ in …)†
(1.5)
ŠŒ
ƒD„
in …†
on Ž…ã
We will always assume that áwä ¥ Ã ¥ is of class
and satisfies the following conditions (f1)-(f2):
©
(f1) á { ª<ç „ for {¡Ï§„ † á
(f2) The following equation
(1.6)
has a
(1.7)
©
©
„ ª ƒLá Ý „ ª ƒD„
Ç
•
š²å
for some „
.
s
©
t u·èêé } é âá é ª ƒD„ † é
in ¥ ’ †
‡§„
©
©Sí
é
é
ª-† é
at ¢[†
„ ª ƒ‘ë º¼»¾½
Ãî„
ƍì !
©Sí
unique solution é
ª and é is nondegenerate, i.e.
Kernel
©
yÛ}
©
˜ âá Ý é
ªª ƒ
span ï
Ž
Ž é
í
Ž é
í
• †zããㆠŽ
’ ð
9
ã
™/æ
Ï
˜
4
F.-H. LIN, W.-M. NI, J.-C. WEI
Occasionally, we assume further that á satisfies (f3):
©
(f3) The principal eigenvalue of yD} ˜ Ëá Ý é ª is positive. That is, there exists
an eigenvalue ñ • ‡D„ and the corresponding eigenfunction òôó (which can
be made positive and radially symmetric) satisfying
(1.8)
y
ò
ó
©
ó âá Ý é
ò
}
One typical example of
¢
if  ƒ/£ú-ƒ
©
á
©
’1š
ª
ƒ
’œ› x š
x
and [18].
is:
’ š
1
’œ› x
x
©
á
ªò
ó
ñ • ò
{ ª
ƒ
ó †-ò
• ©
¨
óÙõ
’
¥
ª-ã
}2ö·{÷ , where ö ž
„ †˜™ùø¡™L—§™
. For the uniqueness of é , see [5], [17]
ƒ{ 
if 
‡ £ ª
‰
The energy functional associated with (1.5) is
(1.9)
ü
¬«­¯® {û°²ƒ
‹
©
³µ´
˜
©Xþ
£
³µ´7ü
©
v x ¶¸
©
{ ¶ x ¹{ x ª }
{ ª-† {
¨
õ
• ©
Hª
þ
where { ª ƒDý ó á ªÕÿ .
We now state our main result in this paper.
Then there exists an v
Theorem 1.1. Let á satisfy assumptions (f1)-(f2).
´
that for „ ™ v ™ v ó and any positive integer Ó satisfying
(1.10)
˜
´
Ï
Ó
Ï
© ’
¶ÕÔnÖ
’
v
ó
‡§„
such
†
’
v ¶ª
where ’ is a constant depending on ¡†+… and á only, problem (1.5) has a
­
solution { which possesses exactly Ó local maximum points • †zããã†
such that
(1.11)
{ ­
©
ª ƒ
é
¯”V•
©
­
}
v
and we have the following energy estimate
(1.12)
where ® é
°
¬H­-® { ­ °BƒLv
Ó
® é
°Bƒ
³
˜
˜
©
€È
˜]ªª
ì !
£
³
©
é
¶¸
(1.14)
Î
õ
¬H­¯® { ­ °²×Lv
ƒ

, we have a solution {
(1.15)
If we further assume that
á
¬«­¯® { ­ °×
é
¶x 
As a consequence, for each integer
(1.5) with the following energy bound
Ó
°
˜]ª
is the energy of é :
(1.13)
When Î
©
® é
’
©
ª ¹È
­
©
x ª }
ì !
® „ †W9ª
’›±Ñ
ü
©
é
ª-ã
, there exists a solution
{ ­
to
ã
with the following energy estimate
¶ÕÔnÖ
v ¶ª
›’
ã
satisfies (f3), then the Morse index of {
­
is at least
.
A simple corollary is the following result which gives a lower bound on the
number of positive solutions to (1.5).
MULTI-PEAK SOLUTIONS
Theorem 1.2. For v sufficiently small, problem (1.5) has at least
!N ON P
­ ! ­ !
5
num-
ber of positive solutions.
Remarks: 1. It seems that the upper bound for Ó is “almost” best possible. Note
that when  ƒ ˜ , the upper bound is ­ . When  ž £ , because of the boundary
mean curvature, it seems that the best upper bound could be ­ !
­ ! , where the
´
´
best constant Ç should depend on ¡† á , and the domain geometry.
©
´
Let Ï[º¼»¾½ Æ ÿ |†+Ž…Hª
2. The constant ’ can be made more precise.
©
be a small positive number. We denote by Ó
ª , the maximum number of nonoverlapping balls with equal radius packed in … . By checking the computations
of the proof of Theorem 1.1 (see (5.4)), we can take
"!
´
(1.16)
˜
Ó
Ï
$# &
 %
©
Ó
Ï
æ
£ æ

˜

ª v ¶ÕÔnÖ
£µ„q„
v ¶
'
†
where
is the Hölder exponent of á
æ
´
Let
(1.17)
Ç
Then we may take
( +-, (nó Ö*) Ô º
ƒ
’
©
Ç
ƒ
’
©
Ó
ª-ã
!
´
´
(1.18)
Ý ã
´
# .
 %
’
£ æ
æcªÕ

• å óó
'
ã
©
ª can be
3. Unlike [13], where the limiting location of the spikes • †zããã†
identified as sphere-packing positions, we can not say much more about the locations of the spikes. This remains an interesting question.
To conclude the Introduction, we include a brief description of the background
of (1.1). Problem (1.1) arises in many models concerning biological pattern formations. For instance, it gives rise to steady states in the Keller-Segel model of the
chemotactic aggregation of the cellular slime molds ([19, 22]) and it also plays an
important role in the Gierer-Meinhardt model describing the regeneration phenomena of hydra.
By a straightforward scaling argument, we can easily construct positive steady
state solutions of the following system from solutions of (1.1)
/
(1.19)
where ÿ
•
ƒ/vWx † ¶
©
3
43 57´ 6 in+ …48 „
1
0
2
0
0
/ 9;: ƒm} :  : ›< ý
Š 2
0 †
©
Š3 Œ ƒD„ on Ž…48
„ †  ¢2ª-†
þ
† †9
¶ is the measure of … , and —c†Wø†=¾
— } ˜
ã
þ
™
™
s
tu
ƒ
´
} •
ÿ • y
„
ø


˜
† 
­
­
¢2ª-†
are nonnegative and satisfy
6
F.-H. LIN, W.-M. NI, J.-C. WEI
Problem (1.19) is, in turn, the shadow-system of the well-known activatorinhibitor system proposed by Gierer and Meinhardt following Turing’s idea of
diffusion-driven instability, in modelling the regeneration phenomenon of hydra
in morphogenesis ([11, 32]), see also the survey [27]. More precisely, (1.19) is
obtained formally by letting ÿ Ã  ¢ in the following system
/
x
s
(1.20)
3
3> 6
ƒ ÿ y
}

2
0
0
0
/ 9;Š @ ƒ Š ÿ > y @ } @ 43B> CA
2 ŠŒ x
Š3 Œ ƒ
ƒD„ on Ž…48
tu
•
©
in …?8
in …48
©
„ † 
„© † 
„ † 
¢2ª-†
¢§ª-†
¢§ª-ã
Indeed, the system (1.20) was motivated by biological experiments on hydra in
morphogenesis. Hydra, an animal of a few millimeters in length, is made up of
approximately 100, 000 cells of about 15 different types. It consists of a “head”
region located at one end along its length. Typical experiments on hydra involve
removing part of its “head” region and transplanting it to other parts of the body
column. Then, a new “head” will form if and only of the transplanted area is sufficiently far from the (old) “head”. These observation have led to the assumption
of the existence of two chemical substances – a slowly diffusing (short-range) activator and a rapidly diffusing (long-range) inhibitor. In 1952, A. Turing argued,
although diffusion is a smoothing and trivializing process in a single chemical, for
systems of two or more chemicals, different diffusion rates could force the uniform steady states to become unstable and lead to nonhomogeneous distributions
for such reactants. This is now known as the “diffusion-driven instability”. Exploring this idea further, in 1972, Gierer and Meinhardt proposed the system (1.20) to
model the above regeneration phenomenon of hydra.
The paper is organized as follows. Notation, preliminaries and some useful estimates are explained in Section 2. Section 3 contains the study of a linear problem
which is the first step in the Liapunov-Schmidt reduction process. In Section 4 we
solve a nonlinear problem which sets up a maximization problem in Section 5. Finally in Section 6 we show that the solution to the maximization problem is indeed
a solution of (2.2) and satisfies all the properties of Theorem 1.1.
Throughout this paper, unless otherwise stated, the letter Ç will always denote
various generic constants which are independent of v and Ó , for v sufficiently
small.
2 Notation and Some Preliminary Analysis
In this section we introduce some notation and present some preliminary analysis on approximate solutions.
Without loss of generality, we may assume that „ õ … . By the following rescaling:
(2.1)
-D † D
ƒLv
õ
-D
­ ä ƒm¿4v
õ
À †
MULTI-PEAK SOLUTIONS
7
equation (1.5) becomes
©
E
(2.2)
For {
©
¨
õ
x
y|{}~{àâá
in …­+†
{ˆ‡§„
{ ª ƒL„ Š‹ in …­W†
and ŠŒ ƒD„
, we put
­ª
©
F²­-® {û°²ƒLy|{}~{|âá
(2.3)
in Ž…œ­+ã
{ ª-ã
Then (2.2) is equivalent to
FB­¯® {û°²ƒD„
(2.4)
† {
©
¨
õ
­ª-† {I‡§„
x
Ž {
in …­-†
on Ž…­+ã
ƒL„
Ž¦
Associated with problem (2.2) is the following energy functional
G
(2.5)
˜
­ ® {û°ƒ
³ ´
³ ´
©
{ ¶ x w{ x ª }
¶¸
a
£
Note that G ­ ƒDv ›’ ¬ ­ .
We define two inner products:
(2.6)
=H
™
©
(2.7)
{ †
³q´
=H
³ ´
­ ƒ
‡
H
a
a
¸
H
{ ¸
K
(2.8)
H
#
‡
{ ª-†
⣠æ
æ
©
†
†
­ ª-ã
­ª ú
x
=H
for {
• ©
¨
õ
{
=H J
õ I
for {
¹{ 'ª-†
Let
©
a
{ û†
©
{ † 'ªÕ­ ƒ
ü
• ©
¨
õ
­ª-ã

be a fixed positive constant. Now we define a configuration space:
©
L
(2.9)
E äƒ
where
©
(2.10)
O
• †zããã†
• †zããã†
ª ƒ
ª
S
“
NMM
©
MM MO
õ
©
”Vº •
PK
• †zããã†
Tn( TÖ T V“ ”*U ¶ “
ªž
}
RQ
v ¶ÕÔ Ö
¶† £ ÿ
©
v ¶
S
†+Ž…Hªª-ã
Let é be the unique solution of (1.6). By the
result of Gidas, Ni and
©Sí well-known
© í
é is radially symmetric: é
é
ª ƒ
ª and strictly decreasing:
Nirenberg
[12],
¶
¶
©
í
é Ý ·ª"™
¶ ¶ . Moreover, we have the following asymptotic
„ for ‡
„ †=
ƒ
behavior of é :
(2.11)
&
&
é
©
ª
Ø
ƒ
’
?W › +
! A
› B
©
© ˜
YX ˜ 
é
ªª-†
for large, where Ø ’ ‡D„ is a constant.
}Ùy/ ˜ centered at „ . Then we have
(2.12)
é
©
ª
©
ƒ
Ø
© ˜
PX ó 
ªªÕÓ
©
·ª-†
é
©
Ý ·ª ƒÍ}
©
Let Ó ·ª
©
Ý ·ª ƒ
©
}
Ø
Ø
’
?W › +
! A
› o
©
© ˜
PX ˜ 
ªª-†
be the fundamental solution of
© ˜
YX ó 
ªªÕÓ
©
·ª-†
for |ž¤˜
8
F.-H. LIN, W.-M. NI, J.-C. WEI
where Ø
For is a positive constant.
õ … , we define é ­ to be the unique solution of
ó
(2.13)
H
y
©
H
}
We first analyze é
­
[Z
©
é
âá
}
v
H
Ž
in …­W†
ªª ƒD„
. To this end, set
­
O
©
© ¶
é
ª ƒ
}
¶
é
ª }
v
­
on Ž…­-ã
ƒD„
Ž¦
©
ª-ã
v
We state the following useful lemma on the properties of
\
Lemma 2.1. Assume that
We have
(2.14)
v ¶ÕÔnÖ
x
©
­ ~ƒÍ}
Ø ó €È
©
⣠ÿ |†+Ž…Hªß¦
O
ÿ
v ¶ Ï
©
©
© ¶
˜]ªªÕÓ
|†+Ž…Hª 4
Ï ]
"^
}
where
©
YX v _ x \
¶
ª 
v
]
­
O
.
is sufficiently small.
šB’š
•
ª
where ^ ƒ a ` , ¦ ` denotes
the unit© outer normal at
©
 is the unique point on Ž… such that ÿ | †?ª ƒ ÿ à†+Ž…Hª .
Proof. Let b­
©
be the unique solution of
ª
(2.15)
b­
v x y
b)­
}
It is easy to see that
(2.16)
db­
©
­ ª
™
„
On the other hand,
H
v x è
O
H
}
©
b
ª Ï
©
•
ª Ï
eH
Ž
Ž
ƒ
Ž¦
© ¶
Ž¦
é
}
¶
ª ƒ
v
©
ƒÍ}
Ø
©
’
€È
˜
é
v
Ü
© ¶
Ahg
W
[f !B?
˜]ªª
¶
jk i
d ™
v
›
d, A
ª
}
¶
ª
and
on Ž…ã
} à†+¦
¶ } ¶
†+¦
|
} !@ & ‡
¶ } ¶ ™
Ž
™m˜µã
v
}
© ¶
é
Ž¦
Using (2.11), we see that on Ž…†
Ž
for v
Çن
õ
on Ž…ã
˜
ƒ
Ž¦
satisfies
in …)†
ƒD„
cb
Ž ­
in …†
ƒD„
Â
‡
†
which implies, by a comparison principle, that
©ml
à†+Ž…«ª1ž
£ K
 
€£ ª v ¶ÕÔnÖ v ¶ ã
©
© l
O
Therefore it remains to consider the case when \ v ¶ÕÔnÖ v ¶ Ï ÿ |†+Ž…Hª Ï
£ K
x
⣠ª v ¶ÕÔnÖ v ¶ . In this case, we use the following comparison function
&
•
©
©
© ¶
} "^ ¶
n
šB’1š
•
ª ƒÛ}
Ø ó }¹v ªÕÓ
ª âv _ x \
b ­Wã
v
O For õ Ž…† ¶ } ¶ Ï[v-o-p[q , we have
©
© &
™
†+¦ ‡
™
} à
} "]^ †+¦ ‡ †
˜ YX
ƒ
v ¶ÕÔ Ö v ¶ ªª
^¶
¶
¶
¶
(2.17)

¶
­
©
ª ¶ Ï
Ç
}
v
_ x\
šB’1š
•
b­
©
ª-†
if ÿ
©
}

MULTI-PEAK SOLUTIONS
¶
}
©
¶
ƒ
v
and hence
Ž
For
O Ž¦
Ž…† ¶
õ
!
­ ©
}
ª
Ï
Ž
O Ž¦
Ï
Summarizing we have for
ó âv
W
Ç
For
­
O
©
v
ª1ž
• †zããã†
“
é
(2.18)
©
©
n&
Ø«ó âv
}
ƒ
­
O
©
D
ª
é
(2.19)
©
where
O
D
ƒÍ¿ ¶ 7}
vª
©
O Ž¦
é
(2.20)
(2.21)
õ
­
ª
Ï
Ž
Ï
¶
'
ª
•
šB’1š
Ž
Ï
•
O
ã
Ž¦
in
©
˜
O
£qv
“
v
w
­
•
O
Ž¦
ã
ã
for
ª-†
"^
}
O
¶
ª }wv
v
D
é
Ž
Ï
•
Ž¦
©
•
O
© ¶
¶ É}
¶ Ï
v
†
ã
õ
_ x\
é
¶ ª-†
­
vª À V† y
©
w D
­
b
for
­
é
x
­
˜µ†zããã†WÓ¡†
õ
w
š
­
•
ƒ
­
õ
é
­
V† y
˜µ†zããã†WÓ
ƒ
©
 X
x P
, we have
~}
©
é
v ª ƒ
Ó
D
¶ É}
, we have
é
­
ƒ
X
w €
©
Ó
~}
v ª-ã
v
u
ã
ã
. To this end, we divide
ƒ
¯”V•
ª ƒ
•
šB’1š
­
š
•
­
YX
¶ª 
©
ƒ
Ó
into
Ó
­[z|{ ”V•
is defined at (2.10).
Lemma 2.2. For D
For D
^
}
, we define
The next lemma analyzes
parts:
­ ¶
v
v
_ x\
줣qv
­
ªÕÓ
L
õ
ª ƒ
r^
}
,
Ž
õ
Rstn
By a comparison principle, we have
©
© ¶
ªÕÓ
&
› ­ A
Ž
Similarly, we obtain
n&
¶
v ¶ ªª
, we have
v
­
Ø
}
Ž¦
-o-p[q
¶ ž

©
Ž
&
¶ Ô Ö
Õ
v
©
YX
˜
9
} v
ª-ã
˜

­
-
10
F.-H. LIN, W.-M. NI, J.-C. WEI
Proof. For ʂƒ 
y
and D
­
õ
©
, we have
„ƒ D ª
©
iƒ
ƒ
€
X W › † › a é
and so
­
©
é
ƒ
S
D
¶ É}
v
;i ƒ‡
W › † › a 
é
S *” U ©
©
„ƒ …v D
­
O
\
âv
„ƒHƒ?X
­
¶ª }
•
šB’š
ª
©
€X ~v } ª ƒ
ª
u
~}
v ª
Ó
which proves (2.20). The proof of (2.21) is similar.
Next we state a useful lemma about the interactions of two é ’s.
Lemma 2.3. For
³
& ›
©
(2.22)
ì !
©
é
á
­
large, it holds
D
¶ 7}
•
v
where
ˆ
(2.23)
ˆ
Remark: Note that
é

•
}
©
x ¶ª ƒ
v
ì !
í
¶ ™7™
¶v
v
© í
¶ ¶ ªª
é
Ø
•
› ë &
W
}
©
˜]ªª
©
ÿ
í
© ¶
˜]ªª é
€È
©
ì !
á
©
ƒ
˜ È
©
©
}
v
D
é
¶ ?}
© ¶
˜]ªª é
•
v
•
}
©
¶ ªª é
v
x
¶,
x
v
í
¶
D
¶ }
¶v

ª
©
©
é
á
ì !
© í
¶ ¶ ªª
é
á
ì !
³
x ¶ª ƒ
v
³
Ú
³ ´
©
­
v
Lemma 2.4. For
(2.25)
(2.26)
Ú
­
©
Ú
­
©
©
ª ƒÍ}
Then we have
©
ƒ
ª ƒ
R© ˆ
“ † ¶
x
ª
&
W › ë š i Aha i &
!oA
ª
© í
¶ ¶ ªª é
ë i & Aji W B› ‰ d i & Aji d
Š
á
a
é
• †zããã†
©
âÈ
ª ƒ
Rˆ
ª
ª õ
© £ ÿ
˜]ªª é
©
x
† Ú
©
­
˜]ªª é
© ¶
ª €È
“
}
v
¶

©
ƒ
©
©
é
ª €È
©
a
K
©
¶ÕÔnÖ
é
©
K
}
)È
³ ´
ª ƒ
©
•
x ¶ª
v
Rˆ
í
ÿ
“ † L © , it holds
†+Ž…Hª
v
©
€È
O
­
© í
¶
Let us define several quantities for later use:
(2.24)
}
ã
Thus by Lebesgue’s Dominated Convergence Theorem
³
•
v
• } x¶
x ¶ ª W ›B‰ ë d ii && AhAhii d
Š šc‹ Œ ë ã
€È
’
•
¶
©
© ¶
é
ƒ
á
Rˆ
©
x ¶ª ƒ
. See Lemma 4.7 of [31].
‡§„
©
ƒ
D
¶ }
³
Proof. By (2.11), we have for
© í
¶
©
¶ ªª é
á
é
“ ª é
v ¶ ªª-†
¶ÕÔnÖ
© ¶
˜]ªª é
v ¶ ªª-ã
•
}
u ã
v
x
¶
ª-ã
MULTI-PEAK SOLUTIONS
› ‡ ª
­
© –
Proof. Note that ØÙózÓ
Rˆ
©
ƒ
©
­
Ú
©
© ¶
˜]ªª é
È
©
ª ƒ
³´
©
˜ €È
©
˜]ªª
r ^
}
©
¶
a
©
ª È
v
©
é
é
á
©
˜ ¹È
ƒ
K
› ‡ ª
­
^ } © –
˜]ªª é
©
ª é
D
¶ É}
and by Lemma 2.1
Rˆ
©
©
È
©
YX
¶ª 
v
v ¶ ªª ƒ
¶ÕÔnÖ
11
©
© £ ÿ
˜]ªª é
_ x\
v
šB’1š
†+Ž…Hª
•
ª
ª È
v
©
©
é
K
(2.25) follows from Lemma 2.1. To prove (2.26), we note that
Ú
³
“ † ©
­
ª ƒ
©
á
ì !
Rˆ
©
ƒ
©
© í
¶ ¶ ªª é
é
Rˆ
“
}
v
©
ƒ
© ¶
˜]ªª é
€È
© í
¶ }
©
© ¶
˜]ªª é
€È
“
}
´
³
¶ª }
v
©
Ž
a N #
•
©
bi
YX W › š - ‘
¶
©
©
} ª €È é
K
¶
ixN
W› ‘b a(
# N O
a ( h
ª 
“
á
i
ì !
O h
© í
¶ }
“
}
¶ª
v
ª
u
v ¶ ªª-ã
¶ÕÔnÖ
v
© í
¶ ¶ ªª é
é
v ¶ ªª-ã
¶ÕÔnÖ
Finally we state the following which provides the key estimates on the energy
expansion and error estimates. The proof of it is delayed to the appendix.
Lemma 2.5. For any
(2.27)
G·­
®
“”V•
é
­
©
v
Ó ® é
x-°²ƒ ‚
ƒ
’“FB­¯® ¯”V•
’
‡
x
• †zããã†
“n”V•
£
for any ø
˜
°G}
and
(2.28)
Ú
­
©
L
ª õ
˜
“ ª
}
and v sufficiently small we have
“ ¯”V•
£
TTT “V”*U ´
­ x-°
’” 6 a
é
Ï
Ç)Ó
6 @6
&
š²å
Ú
­
©
&
b @
}
v
“ † 
!
©
é
ª )È
K
¶ÕÔnÖ
v ¶ª
'
†
h
.
3 An Auxiliary Linear Problem
In this section we study a linear theory which allows us to perform the finitedimensional reduction procedure.
The key to our argument is to show that the
©
constants are independent of v †WÓ~ª .
Fix v õ L . We define the following functions
(3.1)
•
!
“
©
ƒ
where
™ –
for ¶ ¶ ‡
y
˜]ª
}
™
©
\
ª
\
Ž é
Ž
“
©
D ª
D —–
“
†
where –
“
©
D
ª ƒ
•
–
©
K
D
}
˜]ª v ¶ÕÔnÖ
£ ¶ v 7}
™
©
“
¶
is a smooth cut-off function such that – ª ƒ ˜ for ¶
“ belongs to
A
Ú
› • . Note that the support of
v ¶
™
'
¶ ™¤˜
­
} } š &
©
˜
and –
­# ª.
ey
˜µ†zããã†Wӈ†
ƒ
™
©
ª ƒD„
ƒ
˜µ†zããã†W¡†
12
F.-H. LIN, W.-M. NI, J.-C. WEI
In this section, we consider the following linear problem: Given
•
find a function œ satisfying
/
s
©
I ­-®œ °c• ä ƒLy œ } œ âáÜ é ­ w ª7œ ƒ ›
/
™PV
œ † “ ‡ ­ ƒD„ † ˜ ƒ ˜µ†zããã†WÓ¡† y ƒ
(3.2)
S ž †+Ê
for some constants Ÿ
norms
’œ’£
ƒ
N
¢¡
Š
and
˜µ†zããã†W¡†
ŠŒ
©
x
­Õª
,
on Ž…­+†
ƒD„
. To this purpose, we define two
˜µ†zããã†W
´
´
’œ’ ^
(3.3)
˜µ†zããã†WÓ¡†WÌ ƒ
ƒ
S ž$Ÿ S ž S ž ú

tu
› .
õ I
6 a †&’ á ’
=^ ^
’á ’” 6 a†
ƒ
where ø ‡ ’ is a fixed number.
We havex the following result:
œ
Proposition 3.1. Let
have
v
satisfy (3.2). Then for v sufficiently small and
’œ’ ^
(3.4)
’›’ ^=^
L
õ
, we
Ç
Ï
where Ç is a positive constant independent of v
and
†WÓ
v
L
õ
.
Proof. Arguing by contradiction, assume that
©
’œ’ ^ ƒ ˜ ú
’›’ ^=^ ƒ[È ˜]ª-ã
Š$¤ #
©
We multiply (3.2) by Š † x – “ D ª and integrate over …œ­ to obtain
•
©
©
Ž é “
Ž é “
“
“
™
†
ª
­
§
™
B
›
†
D
‡
ƒ
}
Dª ‡ ­
S ž Ž D ¦–
Ÿ S ž
¦
–
Ž D
S¥ ž
©
©
Ž é “
“
(3.6)
™ y œ } œ âá Ü é ­ w ª7œV†

– D
Ž D ¦
(3.5)
ª
‡
­ã
From the exponential decay of é one finds
Ž é
d›B†
™
$¤
Š
Observe that
(3.7)
“
© Ž é
y
©
Ž
“
D – D
© Ž é
ªª }
©
#
†x – “ D
Š
Ž
“
Ž
“
©
D ¦– D
“
ª
­ ƒDÈ
‡
©
˜]ª-ã
satisfies
ª
©
©
ªª :á Ý é
“
D Ž– D
© Ž é
“ ª
Ž
“
©
“
D ¦– D
ªª ƒL£ ¸
†
Ž é
Ž
D
“
¸
†–
“
©

–
y
é
“ ª Ž
Ž
D
“
ã
Integrating by parts and using Lemma 2.2, we deduce
™
y
œ
}
œ
©
 á Ü é
â
­
w 7œ
ª V†
Ž é
Ž
“
©
D
D ¦–
“
ª
‡
­
©
™
ƒ
¨X
ƒ
©
©
Ó
å
©
ª }¹á Ý é
w

v} ’œ’
^
á Ü é
­
ª ƒDÈ
é
“ ªª Ž
Ž
©
“
©
D …œ
D Ž–
©
’œ’ ^ ª ƒDÈ ˜]ª
“
ª-†
‡
rX
­ 
©
v
}
A
&
’œ’ ^
ª
MULTI-PEAK SOLUTIONS
where we have used the fact that K
’
©
©
á Ý é
©
ª }wá Ý é
w
­
“
é
“ ªª Ž
Ž
It is easy to see
• that
“
Ž é
“ ’
D – =^ ^
™
†
“
(3.9)
and for Ê
­ ƒÍ}
‡
and ̐ƒ 
˜
y
“
é
“ å Ž
¶ ¶
Ž
é
}
©
á Ý é
ì !
“
Ž é
™
w
­
©
S¥ ž † Ž D – “ D
, we • have
(3.10)
¶é
© Ž é
í
Ž
ª
“
“ ’
D – ^
ª x ÿ
í
©
å
Ó
Ï
v
} 
ã
˜]ª-ã
¹È
we have
™
ƒ
’
Ç
Ï
and that

³
©
Dª
Ž D –
On the other hand, for ʂƒ  ˜ •
(3.8)
“
ª© š å x å
‡
13
“
Ž é
©
ž † Ž D – “ D
ª
ª
­ ƒD„
‡
©
?X
­ ƒ
‡
\
v
ª-ã
The left hand side of (3.6) becomes
Ÿ “ and hence
Ÿ ©
“
(3.11)

©
\ Ÿ “ ž ª
v
˜
˜]ª-†
ƒDÈ
©
X
ž ”*U Vy
˜µ†zããã†Wӈ†
ƒ
˜]ª
ƒLÈ
˜µ†zããã†Wˆã
ƒ
To obtain a contradiction, we define the following cut-off functions:
!
œ
(3.12)
Note that –
“
œ –
ݓ †
ݓ ƒ
for D
õ
ƒ
˜
where –
•
A &
Ú
}
Then the conditions ™§œV† } “ (3.13)
y
œ
“
}
“
œ
©
­# ª
•„
­ ƒL
‡
§œ
“ †
becomes
“
©
âá Ý é
w 7ª œ
­
“
“
ƒ
-D
K
£ ¶ v 7}
K
©
š ­ ™
The equation for œ
(3.14)
–
ݓ ƒ
}
›
•
“
¶
ª v ¶ÕÔnÖ
v ¶
' †˜
˜µ†zããã†Wӈã
ƒ
and the support of œ belongs to Ú
is equivalent to
­ ƒL„ ã
‡
“ Ÿ }
©
#
š ­ ­
•
“
› –

“Ý
œ
⣠¸
¸
–
“Ý
©

y
– “Ý ª7œ
Lemma 2.2 yields
©
á Ý é
(3.15)
­
w 7ª œ
“
©
ƒ
©
á
é
“ ª
€È
©
v
\ pßx
7œ
›’
ªª
“ ã
Using (3.13) and (3.15), a contradiction argument similar to that of Proposition
3.2 of [13] gives
´
´
´
(3.16)
’œ “ +’ ÷£
N
6 a
Ï
’› – ݓ +’ ÷” 6 a Ç

’
Ç
£ ¸
œ
¸
–
ݓ 
©
y
– ݓ ª7œ’+÷” 6 a ã
ª
.
14
F.-H. LIN, W.-M. NI, J.-C. WEI
Next, we decompose
(3.17)
where ò
(3.18)
©
œ
ƒ
ò
y
˜
âá Ý é
©
Ý é
By Lemma 2.2, á
(3.19) ´
’¯ò«’+÷£
N
6 a
w
­
©
’›
Ç
Ï
–
“n”V•
©
ò
}
}
œ
“Ý ª
˜
}
ªò
œ
“n”V•
©
›
ƒ
©
ªò
ƒ
“
ò

. Then the equation for ò becomes
w
­
}
–
“n”V•
“Ý ª
}w£
´
– “Ý ª¥’+÷” 6 a “n”V•
œ
¸
“n”V•
–
¸
“Ý
}
©
y
“n”V•
– “Ý ª7œVã
. Standard regularity theory gives
˜]ªò
ƒDÈ
˜
¬’
Ç

£
œ
¸
“n”V•
–
¸
“Ý

“n”V•
´
©
y
– “Ý ª7œ’+÷” 6 a ã
(Observe that the constant Ç in the I ÷ -estimates of (3.19) is independent of
. Inequality (3.19) in the case of Dirichlet boundary condition has been
proved in Lemma 6.4 of [31]. Inequality (3.19) in the case of Neumann boundary
condition can be proved similarly.)
Combining (3.17), (3.16) and (3.19), we obtain
™Ä˜
v
´
’œ’ ÷£
Ï
Ç
´
6 a Ï Ç’
œ “ ’ ÷£
“”V•
! ´
’› – ݓ ’ ÷” 6 a  ’›
“”V•
N
Ï
since
(3.20)
©
“n”V•
–
ݓ ª ÷
´
´
6 a  ǒ¯ò«’ ÷£ N 6 a Ï Ç
’œ “ ’ ÷£ N 6 a  ǒ¯ò«’ ÷£ N 6 a “”V•
´
´
©
©
˜ }
– ݓ ª¥’ ÷” 6 a '  Ç “n”V• ’ £ ¸ ´ œ ¸ – ݓ  y – ݓ ª7œ’ ÷” 6 a ´“”V•
•
©
›
Ǭ’›Ž’+÷” 6 a YX
ª¥’œ’+÷£ N 6 a ¶ÕÔ Ö v ¶
•
©
©
›
ݓ ª ÷ ω£ † ¶ ¸
Ý ¶ 
˜ }
ã
¶y – Ý ¶ Ï Ç
¶ÕÔnÖ v ¶ ª

–
–
“n”V•
N
´
This gives
’œ’£
(3.21)
N
6 a
©
ƒDÈ
˜]ª-ã
u
A contradiction to (3.5).
Proposition 3.2. There exists v ó ‡Û„ such
that for any „ ™ v ™ v ó the following
©
©
property
holds
true.
Given
›
ß
­
ª
, there exists a unique pair œV†-¯‚ª ƒ
®
õ
­
÷
x
©
•
œV† ¿ Ÿ “ À “n”V• TTT ¯”V• TTT ’ ª such that
I ­ ®œ
(3.22)
•
(3.23)
§œV†
™
“
´
‡
­ ƒD„ †
˜
ƒ
°Bƒ
›

“
“ Ÿ y
˜µ†zããã†Wӈ†
ƒ
“
†
˜µ†zããã†W¡†
œ
Ž
Ž¦
ƒL„
on Ž…­-ã
MULTI-PEAK SOLUTIONS
15
Moreover, we have
’œ’ ^
(3.24)
’›’ ^=^
Ç
Ï
for some positive constant Ç .
Proof. The bound in (3.24) follows from Proposition 3.1 and (3.11). Let us now
•
prove the existence part. Set
°
¨±'{
• ©
¨
õ
ƒ
h²
­ßª ¶
µ
³µ´
=H
{ †
y
•]©
where we define the inner product on ¨
©
©
{ †
¸
a
H
{ ¸
õ
PµH†
if and only if
“
™
•]©
¨
õ
Observe that œ solves (3.22) and (3.23) if and only if œ
³ ´
©
œ
a
¸
µ
¸
œµôª

©
á Ü é
™
}
This equation can be rewritten as
­
w 7ª œV†=µ
‡
©
œ &
œ ª ƒ ›Â
B
 ·
°
°
where › is defined by duality and §
· ä Ã
(3.25)
e´
­ ƒL„
H
­ ƒL„ †
‡
$³
€{ 'ª-ã
• for µ
Note that, integrating by parts, one has
°
“
as
­ª
©
ªÕ­ ƒ
•
›
˜]ª
}
˜
ƒ
õ
Vy
˜µ†zããã†WÓ¡†
°
™
°
‡
¶µ
­ †
˜µ†zããã†W¡ã
ƒ
satisfies
§›B†=µ
­ ƒ
in
­ª
õ
°
ã
†
is a linear compact operator.
Using Fredholm’s alternative, showing that equation (3.25) has a unique soluu
tion for each ›Â , is equivalent to showing that the equation has a unique solution for
›  ƒD„ , which in turn follows from Proposition 3.1 and our proof is complete.
In the following, if œ is the unique solution given in Proposition 3.2, we set
œ d
ƒ ¸
(3.26)
Note that (3.24) implies
’¸
(3.27)
©
­
›ª¥’ ^
­
Ï
©
›ª-ã
’›’ ^=^
Ç
ã
4 Liapunov-Schmidt Reduction: A Nonlinear Problem
In this section we reduce problem (2.4) to a finite-dimensional one.
For v small and for v õ L , we are going to find a function œ­ w such that for
some constants Ÿ “ †Vy ƒ ˜µ†zããã†W , the following equation holds true •
/
s
(4.1)
©
tu
/
y
é
§œV†
™
­
•w
œBª
“

‡
©
}
é
Vy
­ ƒD„ †
w
­

œBª
©
âá
¢¡
Š
ƒ
˜µ†zããã†W¡†
é
Š]Œ
­
w
ƒD„

œBª
ƒ
on Ž…­-ã
S¥ ž$Ÿ S ž S¥ ž
in …­W†
16
F.-H. LIN, W.-M. NI, J.-C. WEI
The first equation in (4.1) can be written as
y
œ
©
œ
}
âá Ü é
(4.2)
x
(4.3)
œ
°²ƒÛ}
v
õ
­¯®
Lemma 4.1. For
’œ ’ Ï ˜ ,
^
©
é
á
L
ƒ
and
w
­
F²­¯® é ­ w
}
©
œBª

œ
° ª 
}wá
é
É­ ®
°'
“
w 7ª œ ­
sufficiently small, we have for
v
•
“ Ÿ “
©
ª }wá Ü é
w
­
†
ã
’œ’ ^

’œ • ’ ^
•
­

°}
’+É­ ®œ ° ’ =^ ^ Ï Ç’œ’ ^ š²å ú
©
®œ ° ’
’ œ • ’ å^  ’ œ x ’ å^ ¥ª ’œ
Ï Ç
^=^
x
•
œ x’^
}
ã
Proof. Inequality (4.3) follows from the mean-value theorem. In fact, for all
œ­ there holds
©
é
á
w
­
©
œBª

é
}wá
©
ª ƒLá Ü é
w
­
­
©
é
­
w
©
œ²ª

é
}¹á
w
­
©
ª }¹á Ü é
­
D
õ
 ¹ œ²ª7œVã
w .
Since áÜ is Hölder continuous with exponent æ , we deduce
¶á

•
’+ ­ ®œ
(4.4)
w 7ª œ
­
where
©
w 7ª œ
Ç
¶ Ï
¶
•
œ
¶
š²å
u
†
which implies (4.3). The proof of (4.4) goes along the same way.
Proposition 4.2. For v õ L and v sufficiently small, there exists• a unique œ ƒ
œ­ w ©such that
° (4.1) holds. Moreover, v»ºÃ œ­ w is of class Ç as a map into
­ x ÷ …­ßªB¼ , and we have
(4.5)
for some constant Proof. Let ¸
­
.
‡§„
’œ ­ w ’ ^
6 @6
qÓ
Ï
&
š²å
v
}
&
b @
h
be as defined in (3.26). Then (4.1) can be written as
œ d
ƒ ¸
(4.6)
©
­
F
}
­ ®é
w
­
° ª 

Let be a positive (large) number, and set
½ +

ƒ
„œ
ï
°
õ
Define now the map ¾­
ä
½ +
¼ ­
Ã
¾­ ® œ °Bƒ§¸
°
©
x ÷
­ ª ä
¼ ­
­
©
}
x ÷
’œ’ ^ ™YqÓ
©
­ª
F²­¯® é ­ w
œ ° ­ ®
6 @6
ã
&
Ï
¬’“F²­¯® é ­ w ° ’ =^ ^
Ç

¬’+É­
Ç
v
}
&
b @

h
ð
ã
as
° ª 
œ ° ­ ®
ã
¾­ . By Lemma 2.5 and
&
6@ &
@
® œ ° ’ =^ ^ ™YqÓ 6 š²å v } b  h †
Solving (4.1) is equivalent to finding a fixed point for
Lemma 4.1, for v sufficiently small and large we have
’¾­z®œ ° ’ ^
š²å
MULTI-PEAK SOLUTIONS
17
˜
À¿ ­¯®œ x ° ’ ^ Ï Ç+’ ­¯®œ • °±} ­ ®œ x ° ’ ^ ™ £ ’œ • } œ x ’ ^ †
½
which shows that 
¾ ­ is a contraction mapping on + . Hence there exists a unique
½
œ ƒ œ ­ w õ + such that (4.1) holds.
Now we come to© the differentiability
of 
œ © ­ w . Consider
map
the following
•
°
°
’
’
of class Ç •
¨­ ä Ld8
¼ ­ x ÷ …­ª81Á
¼ ­ x ÷ …­Õª8ÂÁ
Ã
(4.7)
©
©
• ©
•
•
›
F²­ ® é © ­ w ©  œ ° ª } • “ Ÿ “ y } ˜]ª › “ DžÈÈÈ
yÛ} ˜]ª
©
œ † y } ˜]ª › • • ªÕ­
V
¨¼­ v †…œV†-¯‚ª ƒÄÅÃ
.. •
É †
ÅÅÆ
. •
©
©
œ † y } ˜]ª › ’ ªÕ­
V
©
Equation (4.1) is equivalent to ¨¼­ v †…Vœ †-‚¯ ª ƒÍ„ . We know that, given v õ L ,
there is a unique local solution œ ­ w † Ÿ ­ w obtained with the above procedure. We
’¾­ ® œ
•
°±}
prove that the linear operator
­
Ž¨
Ž
©
©
MM
v †…œV†-¯‚ª
œ †-¯ ª
c
MM
M w ¡ a N Ê Ë a N Ê °
ä
¼ ­
÷
x
©
81Á
’
­ª
Ã
•
­
Ž¨
Ž
©
©
v †…œV†-¯‚ª
œ †-¯ ª
c
y
MM
MM
µ -Ì
M w ¡ a N Ê Ë a N Ê ® H†
•4©
˜]ª ›
}
©
v®ºÃ
is invertible for v small. Then the Ç -regularity of
the Implicit Function Theorem. Indeed we have
©
F
Ý ®é
ÃÅÅ
°<ƒ
°
ÅÅÆ
­
¼ ­
÷
x
œ­ w † Ÿ ­ w
w ©
©

µ«†
µH†
y
follows from
©
}
©
y
81Á
}
’
­ª
ª
œ­ w
©
©
µ
° ôªª • }
˜]ª ›
..
.
˜]ª ›
•
•
•
“
• • ªÕ­
’
ÿ ͓ ªÕ­
’œ ­ ’ is small, the same proof as in that of Proposition 3.1 shows that
Î a Since
¡ w¡ ^
Š ¡ w Ë ¶ Ë w a N Ê Ë a N Ê is invertible for v small.
u
This concludes the proof of Proposition 4.2.
5 The reduced problem: A Maximization Procedure
In this section, we study a maximization problem.
Fix v õ L . Let œ­ w be the solution given by Proposition 4.2. We define a new
functional
Ï
­
©
vª
ƒ
G·­¯® é ­ w

œ­ w
°oä
L
Ã
Á|ã
We shall prove
Proposition 5.1. For v small, the following maximization problem
(5.2)
has a solution
v
­
º¼»¾½¿
õ
Ï
­
LÐ -the interior of L
©
vª ä v
.
õ
L
À
y
}
˜]ª ›
•
“
DžÈ
ÈÈÈ
É
Š
(5.1)
•
©
ã
18
F.-H. LIN, W.-M. NI, J.-C. WEI
Proof. Since G·­ ® é © ­ w  œ­ w ° is continuous in v , the maximization problem has a
­
­
solution. Let Ï ­ v ª be the maximum where v õ L .
­
We claim that v must stay in the interior of L © .
We first obtain an asymptotic formula for Ï ­ vª . In fact for any v õ L , we
have
©
³µ´
Ï
©
G·³µ­ ´® é ­ w
ƒ
w
­
¸
é

w œ ­ w
°
©
©
’ œ­ w ’ x^
 X
w ª7œ­ w Y
a
a
³µ´
©
©
G·­¯® é ­ w °
’œ ­ w ’ x^ ª
ƒ
} F²­¯® é ­ w ° ª7œ­ w YX
a
©
©
G·­ ® é ­ w °PX ’“F²­ ® é ­ w ° ’ ^=^ ’œ­ w ’ ^ ª PX ’œ ­ w ’ x^ ª
ƒ
•
©
©
©
G ­ ® é ­ w °YX Ó x š 6 š x å v \ š²å ª ƒ G ­ ® é ­ w °€È é K ¶ÕÔnÖ v ¶ ªª
ƒ
by Lemma 2.5, Proposition 4.2 and the choice of K at (2.8).
é
¸

œ­ w
vª
­
­
ª }
By Lemmas 2.4 and 2.5, we obtain
(5.3)
Ï
©
vª
­
‚® é
Ó
ƒ
Rˆ
˜ ©
°G}
©
˜]ªª
È
£
é
“n”V•
©
© £ ÿ
é
á
“ †+Ž…«ª
­
Rˆ
˜ ©
ª }
v
©
)È
£
˜]ªª
´
t*U “”
© ¶
é
ª
“
}
¶
ª )È
v
©
é
©
K
©
·ª is the maximum
Next, we obtain a lower bound for Ï ­ ä Recall that Ó
number of non-overlapping balls with equal radius packed in … . Now we choose
´
Ó
such that
(5.4)
˜
Let v
•
ƒ
balls. Certainly v
© £ ÿ
é
©
Ó
Ï
ó
©
v
­
v ¶ ª-ã
v ¶ÕÔnÖ
£
Ï
PÏ
ª1ž
©
­
v
® é
ó
ªž‰Ó
‚® é
ž§Ó
°û}
Rˆ
£
©
Ó
x
Rˆ
©
Ó
°G}
©
©
\
˜]ªª v
\
˜]ªª v
È
€È
´
š
x
š
’
x
“ó
Rˆ
x ©
Ó
©
é
âÈ
ó
}
£
K
©
(5.6)
Ï
­
©
v
­
ª
Ï
‚® é
Ó
°}
Rˆ
˜ ©
©
˜]ªª é
€È
£
©
K
˜ †Vyª
Combining (5.6) and (5.5), we obtain
(5.7)
é
©
K
¶ÕÔ Ö
v ¶ª
Ï[£ Ó
x v
\
š
x
’
Ï
Ç
v
\
©
¶ÕÔnÖ
v ¶ª
©
›
\
©
x
é
’
š
x
K
x
’
©
È
­
¶ÕÔnÖ
’
x
v ¶ÕÔnÖ
v ¶ª
¶ÕÔnÖ
v ¶ ªª
’
x
š
\
é
such that ¶ “ }
v ¶ . In both cases
v ¶ÕÔnÖ
v ¶ ª €È
¶ÕÔnÖ
˜]ªª v
š
v ¶ ªª-ã
¶ÕÔ Ö
©
­
\
ª Ï[v
È
On the other hand, if v
õ ŽeL , then either there exists
©
­
,
or
there
exists
a Ê such that ÿ S †+Ž…Hª ƒ \
¶ ƒ K v ¶ÕÔnÖ v ¶
x
we have
­
¶
v
}
©
’
©
balls among those Ó
x
ωv
and hence
(5.5)
­
⣠
†zããã†
ó
õ
v
Ï
K
ó ª be the centers of arbitrary Ó
L . Then we have
F
©
ó“
© †+Ž…Hª ª W › ‘ b i a# N ( O h
\ š ’ † é ¶
©
ó
Ó
Ï
v ¶ ªª-ã
©
K
¶ÕÔ Ö
v ¶ ªª-ã
MULTI-PEAK SOLUTIONS
which is impossible.
We conclude that
Remark: Since K
is exactly (1.16).
v
­
L
õ
19
u
ª© š å x å
. This completes the proof of Proposition 5.1.
, we may choose K

‡
Ñ© š å x å
•

ƒ

q
, such that (5.4)
óó
6 Proof of Theorem 1.1
In this section, we apply results in Section 4 and Section 5 to prove Theorem
1.1.
Proof of Theorem
1.1. By Proposition 4.1, there exists v ó such that for „
•
• that
we have a Ç map which, to any v õ L , associates œ ­ w such
F²­-® é ­ w
(6.1)
œ­ w

ÓÒ
°ƒ
Õ
# Nx ¶ Hence we have
³ ´
{ ­ ¸
¸
a
a# Ï
# ”
Ž
©
é
­
©
}«á
©
­
v
­
Ž
{ ­ßª
é
(6.2)
„
S
”V•
TTT ž ”V• TTT ’
Ÿ SԞ
MM
M
˜
Vy
˜µ†zããã†Wӈ†
ƒ
# ”
•
³µ´
S¥ ž
a
Ž
©
Ž
Ù{ ­
a
é
­
#
w  œ ­ w
cŽ “ which gives
Ò
Ó
, achieving the maximum of the maximizaé ­ w a  œ­ w a . Then we have
MM
ª
­
©
MM
MM
ª
M
é
­
# ”
#
a
˜µ†zããã†Wˆã
ƒ
w  œ­ w
cŽ “ ª
MM
MM
M
# ”
a
#
ƒL„ †
w  œ­ w
cŽ “ ª
MM
MM
M
# ”
ƒD„ ã
a
œ w a
­

S
ž
a
Žc “ ˜ , we have
•
If Ê
ƒ
Ž ­
³µ´
³ ´
ƒÍ}
ל w a
S ž Žc ­ S
a
Ž ­
a
Ž
œ­ w a Žc S“­ ž
•
³µ´
ƒÍ}
S ž œ­ w
­
Žc S Ž
a
ƒL„
if Êփ 
a
•
#
We claim that (6.2) is a diagonally dominant system. In fact, since ™§œ­
•
, we have that
•
³ ´
v ó
­ ƒ
ª ƒ[„ †
w  œ­ w
cŽ “ ©
L
™
v
Ÿ SԞ S ž †
S •”V• TTT ž ”V• TTT ’
“ ™§œ­ w †
‡ ­ ƒD„
for some constants Ÿ SԞ õ Á ’ .
­
By Proposition 5.1, we have v
õ
tion problem in Proposition 5.1. Let {
™
˜ã
w
†
“
‡
­ ƒ
•
20
S¥ ž ’ ’ œ­ w a ’ ƒ€X
­
^=^
^=^
Ž× S¥ For ʃ  ˜ , we
• have
³µ´
Ž é ­ a#
ƒ

S
ž
­
a
Žc “ For Ê ƒ ˜ , we have
•
©
Ž
’
ƒ
F.-H. LIN, W.-M. NI, J.-C. WEI
Ž é
a
©
X v~} &
b @
}
v

h › •
©
?X
ª ƒ
&
b @
}
v

6 @6
h ›
&
’›
š²å
•
?X
ª ƒ
©
v
}
ª-ã
, the off-diagonal term gives
ʏ†WÌXª
©
] ž
•
›
š²å
©
Ž é ­ a#
a
?
ƒ
X
v \
S

ž
­
i
ƒ
atØ~Ù
Žc “ a a n
d
f
g
d
}
•
³ ´
Ž é ­ aƒ
a
ƒ
i ƒ S¥ ž Ž× ­ atØÙ
a a S
} df g d
³
©
© Ž é
©
í
ª x PX
˜]ª-ã
á Ý é ª
Ž ì !
­
ƒÍ}Ùv
&
•
³q´
aƒ
S ž cŽ ­ S
³ ´
For each
6 @6
Ó
ª 
\
S ”U “
by our choice of K ‡ª© š å x å
v
©
X
S ”“ ž ”*U 
©
?X v} v ª ƒ
Ó

\
v
©
âv ª ƒDÈ
˜]ª
.
Thus equation (6.2) becomes a system of homogeneous equations for Ÿ SԞ and
the matrix of the system
is nonsingular. So Ÿ SԞ ç „ †+Ê ƒ ˜µ†zããã†Wӈ†WÌ ƒ ˜µ†zãããU .
“n”V• é
­ a#  œ­ a & TTT aÚ is a solution of (2.2).
Hence { ­ ƒ
By our construction and Maximum Principle, it is easy to see that { ­ ‡§„ in … .
Moreover by (5.3) and Lemma 2.2
G
(6.3)
®{
©
® é °'PX
©
­ °²ƒ ӂ® é ° ˜
­ ® { ­ °²ƒ
and hence ¬H­

Ó
Ó
\
x v
©
D
Furthermore, by Lemma 2.2, for
{ ­ ƒ
é
w
­
a
X
Ó
œ­ w
´
ƒ
é
©
D
}
©
rX
ª 
v
©
¶ 7}
w a …’ ”ÜÛ a ƒ?X ’œ­ w a ’ £
since ’œ­
©

a
­
}
.
˜]ªª
€È
©
ª €È
Ó
N
´ }
v
©
é
K
x a ™®\
­ ¶
6 @6 x &

Ó
š²å
6 a ª . For D
Ó
v ¶,
v ¶ÕÔ Ö
&
b @
}
v
©
‚® é
v ¶ ªª ƒ
¶ÕÔnÖ
°
we also have

h
ª ƒ
é
©
[z { - ”V• Ú
­
õ
©
˜ €È
v ª-ã
}
©
D
­
}
š ­ ˜]ªª-†
v
x
©
rX
ª 
a
©
Ó
ªª-† { ­ ƒ
­
Note that at a local maximum point of { ­ , the value of { ­ must be great than
© xa
¯
V
”
•
some fixed positive value { ó . So { ­ can only have maximum points in {
ÚÞÝ
­ ª
for some Á۞¤˜ . For each y ƒ ˜µ†zããã†WÓ , { ­ satisfies
y|{ ­ }~{ ­ âá
x
­
a
Let
ª 2È
àx a
©
­
˜]ª
©
{ ­Õª ƒD„ †
be a local maximum
point of
a
for
the difference œ­
D
©
D
õ
ÞÝ
©
Ú
ª ƒD{ ­
x
©
­
D
ª
{ ­
D
ßÝ
õ
in
Ú
ª }
é
D}
eà x a
­
ª
©
ÞÝ
Ú
, it is easy to see that
©
©
¶
xa
­
­
ª
v
ª-ã
. Since
}â á ­
­
¶ ƒ
È
{ ­
©
©
v ª
D
ª
ƒ
é
©
D}
. We look at
. Then similar to the proof of Step 2 of
}
v ª-†
ª-ã
MULTI-PEAK SOLUTIONS
21
© ڬ㠩 ­ x ªª for „ ™
™
˜
©
a
x
and hence { ­ can have only one local maximum point in ÚÞÝ ­ ª . This shows that
©
­
­
­
­
local maximum points á • †zããㆠá such that á } ƒDÈ v ª-†Vy ƒ
{ ­ has exactly Ó
Theorem 1.1 in [28], one can show that œ­
.
(6.3) also shows that if we take Ó
(Hence ¬H­ ×Ív ’œ›±Ñ .) If we take Ó
˜µ†zããã†WÓ
Ã
in
„
Ç
a
x
š G
› Ñ , then ·­¯® ­
›±Ñ , where Îҙ§ .
®v ±
°
{ °B×Lv
® ­ ! !N O­ N P ! ° , then we have a solution { ­ with
ƒ
©
›’ .)
¬H­ ×
¶ÕÔnÖ v ¶ ª
•
ƒ
energy G·­ ® { ­ °²× ­ !š ­ ! . (Hence
Finally, if á also satisfies (f3), we show that the Morse index of {
In fact, let ò ó be given by (1.8). Set
(6.4)
ò
©D
where –
³q´
©
ò
¶¸
a
ª ƒ
ò
©
ó
­
D7}
v
©
– D ª-†Vy
ª
˜µ†zããã†WÓ
ƒ
is defined at (3.1). It is easy to see that for v sufficiently small
ª
ó
©
D
ó
is at least Ó .
­
¶x 
ò
xó
ª }«á Ý { ­ªò
(6.5)
xó
³ ´
ñ •
Ï/}
a
£
³q´
©
ƒ
©
¶¸
a
ò
xó
™
ò
Vy
„ †
ò
¶x 
xó
©
ª }Ùá Ý é
ªò
xó
³µ´

©
a
˜µ†zããã†WÓ¡ã
ƒ
This, together with the fact that the supports of ò
implies that the Morse index of { ­ is at least Ó .
ó
ó • †zããã ò
ó
are mutually disjoint,
u
Appendix: Proof of Lemma 2.5
We prove the energy expansion formula and error estimates given in Lemma
2.5. The main concern is that we have to make precise the dependence of the error
terms on Ó .
We decompose
G·­¯® ¯”V•
é
­
x¯°ƒ ¯”V• G
·­¯® é
ü
³ ´
}
­
a
x
©
°
é
³ ´
V*U “ ”
£
-”V•
˜
­
x
©
¸
a
ª }
-”V•
é
ü
­
©
# ¸
é
é
­
x ª ­
x
é
­
# é
­
ex
ª
©
á
é
©
ª }«á Ý { ­ªªò
xó
22
F.-H. LIN, W.-M. NI, J.-C. WEI
(1.1)
G·­
¯”V•
ƒ
˜
ex-°'
®é
­
³µ´
©
V*U “ ”
£
á
a
Observe that
G·­
³ ´
ƒ
³µ´
ƒ
a
a
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á
£
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é
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a
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é
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ì !
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a
é
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é
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á
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³ ´
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(1.3)
ƒ
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v
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é
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}
# ° ã
é
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•
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é
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a
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# ª }
ì !
a
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³
³ ´
ü
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x
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é
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é
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M
O
• š ã
Ç v \
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ü
a
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ü
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é
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Next we have
ü
a
­
ª
³µ´
©
a
³ ´
­
ü
# }
“ ª
}
“ ª
}
® é ° YX
Ï
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£
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ƒ
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é
é
-”V•
³q´
­
©
a
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é
á
a
³
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}
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³ ´
“
xô}
­
³q´
˜
# °ƒ
“ ª é
é
á
£
­
ü
©
˜
®é
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é
ü
³q´
á
“ ª
é
­
O
“ † ª YX
x
©
_ x \
Ó
x v
ª-ã
It© remains to compute the last term in (1.1). To this end, we divide the domain
into Ó  ˜]ª } parts as in (2.19). On …­ š • , we have
³ ´
a N
Ú
ü
©
&
@
³q´
a N
x
©
é
­
w
­
ª }
ü
¯”V•
©
é
!
©
?X
w
ª }
¯”V•
ü
Ó
a N
š²å
x
, we have
˜µ†zããã†WÓ
ƒ
©
é
­
³ ´
ƒ
X
x ª ?
­
ƒ
V† y
On …­
ü
é
x ª
ƒ
a N
x
©
@
&
©
é
­
é
¯”V•
¶
• š  ª-ã
v \
ü
³q´
Ú
ex] ”*U ž
é
­
­
x
Bä
¶x
š²å
ü
ª }
¯”V•

©
é
­
ex
é
ª }
š²å
x ª '
­x
ü
ž ”*U ©
é
­
Bä ª MULTI-PEAK SOLUTIONS
where the last term can be estimated as
³ ´
ü
ž *U ”
a N
ü
³µ´
x
a N
ƒ
©
ž *U ”
a N
é
x
á
©
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x
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é
x
a N
³ ´
é
ü
³´
ƒ
©
ª é
Bä ª
­
x ”*U ž
­ x 
ž ”*U ©
é
é
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ü
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é
ü
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x
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}
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•
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v
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Bä ¶ x š²å
¶é
x
a N
Ç)Ó
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ª }
ü
a N
¶é
©
x
x
­
©
ž ”*U  X
x ª Y
!
­
³ ´
•
\ š  ª 
v
!
ž *U }
ä ª
­
³ ´
”
h
ä ª
­
Ç
¶ Ï
•
W › š å b a Ê
ä rX
­
é
¶
x
Ç)Ó
Ï
So
©
23
Ó
ä ª ­
•
\ š  ª
v
©
å
¶
é
é
ž *U ”
©
ä ª x
­

é
ž *U ”
­
ä ª x š²å '
where the last term can be estimated by
³µ´
x
a N
Ç)Ó
Ï
x v
¶é
­
x
•
\ š ©
a N
x
á
é
­
ex
ª é
ä
­
ü
³´
©
a
(1.4)
ƒ
é
ž *U ”
ƒ
©
a N
­
©
á
x
é
©
„ä ª x
­
š²å
x
³q´

ª é
† ž ª Y
 X
ü
©
v
©
Bä ª x š²å '
­
Ç)Ó
Ï
x
š²å
_ x\
¶é
x
a N
Ú
é
­
ª }
é
­
•
\ š ã
v
³´
©
 X
ä Y
­
é
ž *U ”
•
\ š v
¶
å
¶
­
O
x
¶é
­
ä ª
ª-ã
ü
ex ”*U Bä
x
ž
ž ”*U ©
©
• š  Ó
š²å
­ ž † ª YX
Ó o
v \
­
ž ”*U é
Ç7Ó
ƒDÚ
Thus
©
å

Note that
³µ´
¶
ª }
©
x v
é
­
_ x\
„ä ª ª-ã
Combining the estimates (1.2), (1.3) and (1.4) together, we arrive
G·­
®é
­
w
°²ƒ
‚® é
Ó
°G}
•
˜
“”V•
£
Ú
­
©
˜
“ ª
}
t*U “”
£
­
Ú
©
“ † ª rX
©
•
o š²å v \ š  
Ó
š  › o š²å ’ )v
Since X Ó o š²å v \
 Ó xzv _ x \ ª ƒ€X
v \ v \ by our choice of K at (2.8), we obtain (2.27) of Lemma 2.5.
Finally we prove (2.28). Observe that
©
F²­¯® é ­ w
©
©
°BƒLá
-”V•
é
­
x
ª }
-”V•
©
á
é
ª
_ x\
›
x
’
ª ƒDÈ
Ó
_ x\
©
x v
é
©
K
ª
¶ÕÔnÖ
v ¶ ªª
24
F.-H. LIN, W.-M. NI, J.-C. WEI
In …­
•
š
, we have
¶
F ­ ® ¯ ”V•
é
ex
­
•
©
dX
Hence
V† y
In …­
F
¶ B­-®
é
“”V•
é
¶á
ƒ
x-° ¶
­
©
©
­
ex
é
¶á
ƒ
©
ª }wá
é
©
ª }«á


•
é
•
@
é
¶á
Ó
x
­
©
ª }wá
!
¶é
ž *U ”
&
b @
}
v
©
ç
Ï X
&
ª ¶ rX
é
!
ª ¶ PX
é
ƒ
x ° ’ ” 6 a NÚ
­
ex
­
©
(1.6)
š²å
ex
­
, we have
˜µ†zããã†WÓ
ƒ
•
é
š²å
¶

&
b @
}
v
ª ¶ YX
é
•
©
å é
¶

h
ª
ª-ã
&
b @
©
Bä 
­
é
ž *U ”
h' Y
 X
&
b @
W }
š²å
Ó
š²å
&
\ @  ª-ã
v
W }
š²å
Ó

š²å
x
­
•
h
•
•
š²å
š²å
Ó
é
´
’“F²­¯® -”V•
(1.5)
•
x
­
¯”V•
&
©
W › @ a| æ w ª ÏçX Ó
š²å
Ó
Ï
é
¶
å
Ç)Ó
Ï
©
Ç
° ¶ Ï

h
©
Ó
©
F²­ ® ¯ ”V•
a Nx
é
³µ´
ex
­
-° ª ÷
Ï
a N
÷ å
x
é
and
(1.7)
³µ´
¯”V•
©
a N
x
F²­¯® ¯”V•
é
ƒ
X
x-° ª ÷ ?
­
x
&
b @

6 O
}
v
­
¶
©
dX
Ï
©
¶á
x
a N
³µ´
Ï
©
Ç
6
Ó
}
v
é
x
­
©
YX
¶÷ 
h
Ó

&
b @

š
÷
h
©
ª }"á
Ó
÷
÷
š
÷
徚
÷
©
èX
6
W }
&
b @
•
&
b @

h

h
ž
ž ”*U é
ª
ª-ã
ª ¶÷ 
å
6
W }
å
÷
Ó

š
é

©
' r
 X
š²å
&
b @
}
v
Therefore, we obtain
³´
•
Bä ª
­
h
Ó
÷
š
÷
6
W }
å
&
b @

h
ª
ª
ª
6
W }
&
b @

h
?X
ª ƒ
©
Ó
š
÷
÷
å4š
6
•
v
&
b @
}
Combining (1.5), (1.6) and (1.7), we obtain
’“FB­-® ¯”V•
é
´
­
ex¯° ’ ” 6 a ÏçX
©
•
Ó
š²å
v
which proves (2.28) of Lemma 2.5.
}
&
b @

h

•
Ó
š
6&
š²å
W }
&
b @

h
•
ª Ï
Ç)Ó
š
6&
š²å
W }
&
b @
u

h
ª

h
ª-ã
•
š²å
ª
MULTI-PEAK SOLUTIONS
25
Acknowledgment. Research supported in part by the NSF. This project was
initiated while the second author was on sabbatical leave at Courant Institute. He
wishes to thank Courant Institute and Louis Nirenberg for the generous support
and the hospitality during his stay in New York. The research of the third author
is partially supported by an Earmarked Grant from RGC of Hong Kong (RGC
402503).
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éŽê
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FANG-HUA LIN
Courant Institute of Mathematical Sciences,
251 Mercer Street,
New York, N.Y. 10012, USA.
Email:linf@cims.nyu.edu
WEI-MING NI
School of Mathematics
MULTI-PEAK SOLUTIONS
University of Minnesota
Minneapolis, MN 55455, USA.
Email: ni@math.umn.edu
JUN-CHENG WEI
Department of Mathematics
The Chinese University of Hong Kong,
Shatin, Hong Kong.
Email: wei@math.cuhk.edu.hk
Received Month 200X.
27