Full text of DOI:10.1112/S002461070000898X

ON POSITIVE MULTIPEAK SOLUTIONS OF A NONLINEAR
ELLIPTIC PROBLEM
EZZAT S. NOUSSAIR  SHUSEN YAN
1
. Introduction
In this paper we continue our investigation in [5, 7, 8] on multipeak solutions to
the problem
kε#∆uju l Q(x) QuQ u, x ? ꢀ ,
q−#
N
u ? H"(ꢀ ),
N
(1.1)
N
#
#
N
where ∆ l ꢀ c \cx is the Laplace operator in ꢀ , 2 ! q ! _ for N l 1, 2, 2 !
i="
satisfying the following conditions.
i
N
q ! 2N\(Nk2) for N & 3, and Q(x) is a bounded positive continuous function on ꢀ
N
(
Q ) Q has a strict local minimum at some point x ? ꢀ , that is, for some δ " 0
"
!
Q(x) " Q(x )
!
for all 0 ! Qxkx Q ! δ.
!
(
Q ) There are constants C, θ " 0 such that
#
θ
QQ(x)kQ(y)Q % CQxkyQ
for all Qxkx Q % δ, Qyky Q % δ.
!
!
Our aim here is to show that corresponding to each strict local minimum point x
N
of Q(x) in ꢀ , and for each positive integer k, (1.1) has a positive solution with k-
!
peaks concentrating near x , provided ε is sufficiently small, that is, a solution with
!
k-maximum points converging to x , while vanishing as ε ! 0 everywhere else
N
in ꢀ .
!
Problem (1.1) arises in various applications, such as chemotaxis, population
genetics, chemical reactor theory, and the study of standing wave solutions of certain
$
nonlinear Schrodinger equations.
The study of single and multi-peak solutions to (1.1) and related problems has
attracted considerable attention in recent years, and there are several results in the
literature on the existence of such solutions.
However, to the best of the authors’ knowledge, all previous results on this
problem are restricted to solutions with at most one positive peak near x , where x
!
!
is assumed to be a nondegenerate critical point of Q, a strict local maximum
point, or some other classes of ‘topologically nontrivial critical points’ (see [11]). We
mention the recent results by the authors and D. Cao [5, 8] for the case of a
nondegenerate critical point, and [7] for the degenerate case. For the related
$
Schrodinger equation
kε#∆ujV(x) u l QuQ u, x ? ꢀ ,
q−#
N
u ? H"(ꢀ ),
N
Received 19 August 1997; revised 24 June 1999.
000 Mathematics Subject Classification 35J60.
2
Both authors were supported by the Australian Research Council.
J. London Math. Soc. (2) 62 (2000) 213–227

2
14
 .    
there are various results by several authors. We mention the works of Floer and
Weinstein [13], Oh [22, 23], X. Wang [26], Z. Q. Wang [27], P. Rabinowitz [24], M. del
Pino and P. Felmer [11, 12], C. Gui [15], and Y. Y. Li [17], and the references therein.
N
When Q(x) is a positive constant and ꢀ is replaced by an arbitrary domain Ω,
problem (1.1) has been considered by several authors. In these studies both the
topology of Ω (see, for example, Benci and Cerami [3, 4]) and the geometry of Ω (see
[6, 9, 20, 21]) play an important role in the existence and multiplicity of positive
solutions of (1.1), and on the location of their peaks (see [6, 25]).
To state our results we introduce some notations first. Let w denote the ground
state solution of the problem
q−"
N
k∆wjw l w
, x ? ꢀ ,
N
w ? H"(ꢀ ),
w(0) l max w(x).
(1.2)
?
ꢀN
It is well known [16] that w is unique and satisfies
x
1
N
w(x) l w(QxQ),
wh(r) ! 0,
B x ? ꢀ ,
B r " 0, wd(0) ! 0,
(
N−")/# r
2
3
lim r
!
e w(r) l λ " 0,
!
_
r
wh(r)
w(r)
lim
!
lk1.
_
4
r
Let
N
Šε (:) l Š((:ky)\ε), y ? ꢀ ,
,
y
fŠ, Šg l ε# ]u:]Šj uŠ,
ε
& &
ε
RuR# l fu, ug ,
ε
and we write R R l R R for all u, Š ? H"(ꢀ ), where all our integrals are Lebesgue
N
"
N
integrals over ꢀ , unless otherwise stated. The main results of this paper may be
stated as follows.
T A. Assume that conditions (Q ) and (Q ) hold. Then, for each k l
"
#
1
the form
, 2,…, there exists ε l ε(k) such that for all ε ? (0, ε ) problem (1.1) has a solution u of
ε
!
!
k
j="
j
u l ꢀ α w j jŠ
ε
ε
ε
ε
x
ε
,
j
j
N
for some positiŠe constants α , j l 1,…, k, points x ? ꢀ , j l 1, 2,…, k, and Š ?
ε
ε
ε
H"(ꢀ ), satisfying
N
j
x ! x ,
ε
!
i
j
Qx kx Q
ε
ε
!
j_, i ꢀ j,
ε
j
−"/(q−#)
α ! Q(x )
,
ε
!
N/#
RŠ R l o(ε ),
ε
ε
as ε ! 0.


  
215
Our procedure is based on local reduction methods like those by O. Rey [24], A.
Bahri [1], and Y. Li [17]. We should mention that the reduction procedure has been
modified here to allow for the degeneracy of the critical point of Q.
We mention that there is a similarity between problem (1.1) and the singularly
perturbed elliptic problem with Neumann boundary condition (see [10]), where the
mean curvature function on cΩ plays a similar role to that of Q(x) here.
We also mention the following example which shows that when x is a local
maximum point of Q, one cannot expect, in general, to have a positive solution with
more than one peak concentrating at x .
!
!
E 1.1. Let u be any positive solution of
kε#∆uju l Q(QxQ) u
q−"
,
N
in ꢀ ,
u ? H"(ꢀ ),
N
where Q is a non-increasing continuous positive function, which is radically
symmetric. It is well known [14] that u is radially symmetric and non-increasing, and
N
hence has a single peak in ꢀ .
This paper is organized as follows. In Section 2, we introduce some notations and
establish two basic results; one is a decomposition lemma for functions in H"(ꢀ ),
N
and the other is a spectrum result. In Section 3 we prove Theorem A.
R 1.2. Recently we learnt from the referee that similar results were
obtained by G. Lu and J. Wei [18] for similar equations, although not including the
one considered here.
2
. Notations and preliminary results
Without loss of generality, we assume that x l 0 and Q(0) l 1. Set
!
1
I (u) l " RuR#k Q(x) QuQ dx, u ? H (ꢀ ).
q
"
N
(2.1)
ε
ε
&
#
q
N
`
Let B l ox ? ꢀ , QxQ ! rq, and B be its closure.
r
r
For δ, R " 0 and any positive integer k l 1, 2,…,
k
"
k
kN
j
`
i
j
D
l o(x ,…, x ) ? ꢀ , x ? B , j l 1,…, k, Qx kx Q\ε & R for i ꢀ jq
ε
δ
δ
,
R,
cw j
ε
Eε l Š ? H"(ꢀ ):fŠ, w j g l Š,
N
x ,
l 0, j l 1,…, k, i l 1,…, N
ε
ε
- .
*
,
k
(
x ,
j
cx
ε
i
Mε
l o(α",…, α , x ,…, x , Š):(x ,…, x ) ? D
k
"
k
"
k
k
, Qα k1Q % δ,
i
δ
ε
δ
,
R,
,R,
i l 1,…, k, Š ? E , RŠR# % δε q.
N
(2.2)
ε
ε
,
k
Set
k
J (α",…, α ; x ,…, x , Š) l I ꢀ α w j jŠ
k
"
k
j
ε
ε
0
ε
1
x ,
j="
α ,…, α , x",…, x , Š) ? M
k
.
(2.3)
(
ε
δ
"
k
,R,

2
16
 .    
L 2.1. There exist R " 0, δ " 0 and ε " 0 such that for ε ? (0, ε ], R & R ,
!
!
!
!
δ ? (0, δ ], we haŠe the equiŠalence (α",…, α , x ,…, x , Š) is a critical point of the
k
"
k
!
functional
J :M
,- R
ε
ε
δ
,
R,
k
j="
α",…, α , x ,…, x , Š) ,- I ꢀ α w j jŠ
k
"
k
j
(
ε
0
ε
1
x ,
if and only if
k
j="
j
u l ꢀ α w j jŠ
(2.4)
ε
x ,
is a critical point of I in H"(ꢀ ).
N
ε
L 2.2. There exist ρ " 0, R " 0, δ " 0 and ε " 0 such that for ε ? (0, ε ],
!
!
!
!
δ ? (0, δ ] and R & R , we haŠe
!
!
k
j="
q−#
RŠR#k(qk1)
ꢀ w j
Š# & ρRŠR#
(2.5)
ε
&0
ε
1
ε
x ,
for all (x",…, x ) ? D
k
k
and Š ? E .
ε
δ
ε
,
The proofs of Lemmas 2.1 and 2.2 are given in Appendix A.
R,
,k
We notice that (α",…, α , x ,…, x , Š) ? M
k
"
k
is a critical point of J if and only
ε
ε
δ
,
R,
if there are scalars A , B , % l 1,…, k, h l 1,…, N, such that
%
%
,
h
N
h="
#
cJε
c w j
i
ε
x ,
l ꢀ B
, Š
(2.6)
(2.7)
(2.8)
j,h
-
j
j
.
j
cx
cx cx
ε
i
h
cJε
l 0
i
cα
k
=
k
N
=" h="
%
cJε
cw
ε
x ,
,
} l ꢀ A (w
%
, })jꢀ ꢀ B
, }
0cŠ
1
%
x ,ε
%,h 0 cx
%
1
%
%
"
h
for all } ? H"(ꢀ ).
N
In order to prove Theorem A, we show first that for (x", x#,…, x ) given, ε small
k
i
enough, there exist α , i l 1,…, k, Š ? E , and scalars A , B , % l 1,…, k,
ε
ε
%
%
,
k
,h
). We then show that for
h l 1,…, N, such that (2.7) and (2.8) are satisfied, and the mappings (x",…, x ) ,-
k
i
"
k
"
k
"
k
"
k
α (x ,…, x ), (x ,…, x ) ,- Š(x ,…, x ) ? E are C (D
ε
ε
δ
,
k
,R,
sufficiently small ε, there exists a point (x",…, x ) ? D
k
k
"
k
such that (α ,…, α ,
ε
δ
,
R,
x",…, x , Š) ? M
k
and satisfies (2.6).
δ
ε
,
R,
3
. Existence of a multi-peak solution
In this section we fix k l 1, 2,…, and write
α l (α",…, α ) ? ꢀ
k
k
x l (x",…, x ) ? ꢀ
k
kN
.
k
k
We define inner products in ꢀ and ꢀ iE , respectively, by
ε
,
k
k
(
α, γ) l ε# ꢀ α γ , α, γ ? ꢀ ,
j j
k
(3.1)
(3.2)
ε
j="
(
(α, Š), (γ, u)) l (α, γ) j(Š, u) ,
ε
ε
ε
k
for (α, Š), (γ, u) ? ꢀ iE . The corresponding norms are denoted by
ε
,
k
RαRε, R(α, Š)Rε.


  
217
Let
k
j="
Hε l ꢀ w j .
(3.3)
ε
,
x
x ,
P 3.1. Assume that x l 0 and conditions (Q ) and (Q ) hold. Then for
!
"
#
any giŠen integer k l 1, 2,…, there exist constants ε , R , δ " 0 such that for ε ? (0, ε ],
!
! !
!
R " R , δ ? (0, δ ], there is a C"-mapping
!
!
k
k
(
α , Š ):D
,- R iEε
ε
ε
ε
δ
,
R,
,k
such that (2.7) and (2.8) hold. MoreoŠer,
k
j="
θ
Q i jQ ε
N/#
ε
i
N/#
−min((q−")/#,") x −x /
ꢀ Qα k1QjRŠ R l ε O(ε jꢀ e
ε
ε ε
ꢀ
i j
k
j
jꢀ QQ(x )k1Q).
j="
(3.4)
Proof. We follow an argument due to A. Bahri [1]. We expand J in a
ε
i
i
"
k
neighbourhood of (α, Š) l (1, 0). Let β l α k1, β l (β ,…, β ) and u l (β, Š) ?
k
ꢀ
iE .
ε
,
k
Define
J$(x, u) l J (α, x, Š).
ε
ε
Then
J$(x, u) l J*(x, 0)jf (u)jQ (u)jR (u)
ε
ε
ε
ε
,
x
,x
,x
where u l (β, Š),
q−"
f (u) lk Q(y) H (y) Š(y) dy
ε
&
ε
,
x
,x
k
q−"
jꢀ (H , w j ) k Q(y) H w j
β
(3.5)
(3.6)
9
ε,x x ,ε ε
&
ε,x x ,ε
:
j
j="
(
")
(#)
($)
Q (u) l Q (β)jQ (Š)jQ (β, Š)
ε
ε
ε
ε
,
x
,x
,x
,x
k
(
")
q−#
i j
Q (β) l ꢀ (w i , w j ) k(qk1) ꢀ Q(y) (H ) w i w j β β (3.7)
ε
9
ε
ε ε
&
ε
ε
ε
:
,
x
x ,
x ,
,x
x , x ,
i,j="
(
#)
#
q−# #
Q (Š) l RŠR k(qk1) Q(y) (H )
Š
(3.8)
ε
ε
&
ε
,
x
,x
k
(
$)
q−#
j
Q (β, Š) lkꢀ Q(y) (H )
w
Šβ
(3.9)
ε
&
and Rε denotes all the higher order terms, and it satisfies
ε
j ε
,
x
,x
x ,
j="
,
x
Min($,q)
R (u) l O(RuR
)
ε
ε
,
x
Min(#,q−")
R! (u) l O(RuR
)
ε
ε
,
x
Min(",q−#)
R" (u) l O(RuR
).
(3.10)
ε
ε
,
x
k
Now fε is a continuous linear form over ꢀ iE equipped with the scalar product
ε
,
f , u) .
x
,k
k
defined in (3.2). Therefore there exists a unique f ? ꢀ iEε such that f (u) l
ε
ε
,
x
,k
k
,x
(
ε
ε
,
x
In the same way Qε is a continuous quadratic form over ꢀ iE , and therefore
there exists a continuous linear operator A from ꢀ iEε onto itself such that
ε
,
x
,k
k
ε
,
x
,x
Q (u) l (A u, u)ε,
ε
ε
,
x
,x
(
")
(#)
($)
(
Aε u, u) l Q (β)jQ (Š)jQ (β, Š).
(3.11)
ε
ε
ε
ε
,
x
,x
,x
,x

2
18
 .    
We show next that for R sufficiently large, and δ and ε sufficiently small, the
operator Aε has a bounded inverse.
,
x
From Lemma 2.2 there exists ρ, R , δ , ε " 0 such that for ε ? (0, ε ], δ ? (0, δ ] and
!
! !
!
!
(3.12)
R & R , we have
!
(
#)
#
Q (Š) & ρRŠR
ε
ε
,
x
k
for all x ? Dε δ.
,
R,
On the other hand, if R " 0 is sufficiently large and δ " 0 is sufficiently small, we
!
!
have
k
(
")
#
q
i #
N −R
#
Q (β) l ꢀ (Rw i R k(qk1) Q(y) w i )Q β Q jO(ε e ) QβQ
ε
ε
ε
&
ε
,
x
x ,
x ,
i="
N
#
−R
#
l ε o(2kq) RwR jo (1)jO(e )q QβQ
δ
N
#
#
%
kC ε QβQ lkC RβR
(3.13)
ε
N
N
for some constant C " 0, which depends only on N, where o (1) ! 0 as δ ! 0,
δ
N
and
k
(
$)
q−#
i
QQ (β, Š)Q % ꢀ Q(y) Hε w i Š Qβ Q
ε
)&
ε
)
,
x
,x x ,
q−"
i="
k
N/# −R
i
l ꢀ Q(y) wε ŠjO(ε e ) RŠR QβQ
0
0
&
&
i
ε
1
,
x
i="
k
q−"
N/#
N/# −R
i
l ꢀ
w
Šjo (1) ε RŠR jO(ε e ) RŠR Qβ Q
ε
i
δ
ε
ε
1
,
x
i="
−
R
l (O(e )jo(δ)) RβR RŠR .
(3.14)
ε
ε
From the above estimates, there is, for ε ? (0, ε ), R " R , δ ? (0, δ ), a unique linear
!
!
!
k
operator from ꢀ iEε to itself such that
,
k
(
")
(#)
(
Bε u, u) l Q (β)jQ (Š).
ε
ε
ε
,
x
,x
,x
Furthermore, from (3.12) and (3.13), B is invertible and
ε
,
x
x
−
"
RB R % C
ε
,
for some constant C " 0, independently of ε and x.
From (3.14) we have
(
$)
−R
RA kB R l RQ R % O(e )jo (1).
ε
ε
ε
δ
,
x
,x
,x
Hence we can choose R , δ , ε such that for ε ? (0, ε ], R & R , δ ? (0, δ ), A is
ε
!
! !
!
!
!
,x
invertible and
−
"
RA R % C
ε
,
We now follow the argument in Rey [24].
x
for some constant C " 0.
Since
cJ$
ε
(
x, u) l f j2A ujR! (u)
)
ε,x
ε,x
ε,x
cu ꢀk
×
E
ε
,
k
there is an equivalence between the existence of u l (α, Š) such that (2.7) and (2.8) are
satisfied and
f j2A ujR! (u) l 0.
(3.15)
ε
ε
ε
,
x
,x
,x


  
219
As in [24] we employ the implicit function theorem to conclude that for some
ε , R , δ , we have a C"-mapping
!
! !
k
k
u l (α , Š ):D
,- ꢀ iEε
ε
ε
ε
ε
δ
,
R,
,k
for ε ? (0, ε ], R & R , δ ? (0, δ ], satisfying (3.15), and
!
!
!
Ru R % CR f R.
(3.16)
(3.17)
ε
ε
ε
,
x
We estimate next R f R.
We claim that
ε
,
x
k
i="
Q i jQ ε
q−"
q−"
N/#
−Min((q−")/#,") x −x /
Q(y) H Š l ꢀ Q(y) w i ŠjO(ε ꢀ e
) RŠRε.
&
ε,x
&
x ,ε
ꢀ
i j
In fact, if 2 ! q ! 3, from the inequality
1
q−#
CQaQ QbQ
CQbQ QaQ
if QaQ % QbQ
if QbQ % QaQ
q−"
q−"
q−"
2
3
Q QajbQ kQaQ kQbQ Q %
q−#
4
(
q−")/# (q−")/#
%
CQaQ
QbQ
,
for some constant C " 0, we obtain
k
i="
q−"
q−"
(q−")/# (q−")/#
Q(y) H kꢀ w i Š % C ꢀ w
w
QŠQ
)
& 0 ε,x
x ,ε1 ) &
i ε
i ε
x ,
x ,
ꢀ
i j
q/(q−") "−"/q
N/#RŠRε
(q−")/# (q−")/#
%
Cε
ꢀ w
i j
w
0&0
Q
i
j
Q ε
1 1
"
, x −x /
ꢀ
Q i jQ
ε
N/#
−(q−") x −x /#
l O(ε ꢀ e
) RŠRε ;
(3.18)
ꢀ
i j
if q " 3, then
k
i="
Q i jQ ε
q−"
q−"
q−#
N/#
− x −x /
Q(y) H kꢀ w i Š % C ꢀ w i w j QŠQ l O(ε ꢀ e
) RŠRε. (3.19)
)
& 0 ε,x
x ,ε1 ) & x ,ε x ,ε
ꢀ
ꢀ
i j
i j
Combining (3.18) and (3.19), we obtain (3.17).
However,
q−"
q−"
Q(y) w i ŠjO
x ,ε
q−"
w i QŠQ
x ,ε
Q(y) w i Š l
&
x ,ε &B
0&
1
i
Q
iQ&δ
δ
(x )
y−x
θ
i
q−"
i
q−"
l Q(x )
w i ŠjO
Qykx Q w i QŠQ
&B x ,ε 0&B
x ,ε
1
i
i
δ
(x )
δ
(x )
q−"
w i QŠQ
x ,ε
jO
0
&
1
Q
iQ&δ
y−x
θ
i
q−"
q−"
l O
Qykx Q w i QŠQ jO
w i QŠQ
0&B
x ,ε 1 0&
x ,ε
1
i
Q
iQ&δ
δ
(x )
y−x
θ
θ
N+
l O(ε
q−"
i
N
q−"
w
i
QyQ w QŠ(εyjx )Q)jO ε
QŠ(εyjx )Q1
&
0 &
Q
Q&δ ε
y
/
θ
δ ε
N/#+
N/# − /
l O(ε
l O(ε
jε
e
) RŠRε
θ
N/#+
) RŠR .
(3.20)
ε

2
20
 .    
Using (3.17) and (3.20), we obtain
θ
Q i jQ ε
q−"
Q(y) H l O(ε
ε,x
N/#+
N/#
−Min((q−")/#,") x −x /
jε ꢀ e
) RŠRε.
(3.21)
&
ꢀ
i j
We also have
"
/q
q−"
q−"
N
q
w
Q(y) H w i
Q(y) H w i jO ε
&
ε,x x ,ε &B
ε,x x ,ε 0 0& 1 1
i
Q
Q&δ ε
δ
(x )
y
/
θ
δ ε
i
q−"
i
q−"
N − /
l Q(x )
H
w i jO
Qykx Q H w i jO(ε e
)
&B ε,x x ,ε 0&B
ε,x x ,ε
1
i
i
δ
(x )
δ
(x )
θ
Q i jQ ε
i
q
N+
N
− x −x /
l Q(x )
w i jO (ε jε ꢀ e
)
&B
N
(x )
x ,ε
i j
i
ꢀ
δ
θ
Q i jQ ε
i
#
N+
N
− x −x /
l Q(x ) ε RwR jO (ε jε ꢀ e
).
(3.22)
ꢀ
i j
Thus
(
Q i jQ ε
q−"
N
#
− x −x /
H , w i ) k Q(y) H w i l ε (RwR jO (ꢀ e
))
ε
ε
ε
&
ε
ε
,
x
x ,
,x x ,
ꢀ
i j
θ
Q i jQ ε
i
N
#
N+
N
− x −x /
k(Q(x ) ε RwR jO (ε jε ꢀ e
))
ꢀ
i j
k
i="
θ
Q i jQ ε
i
− x −x /
N
ε .
l O ε jꢀ Q1kQ(x )Qjꢀ e
(3.23)
0
1
ꢀ
i j
From (3.21), (3.23) and (3.5), we have
k
i="
θ
Q i jQ ε
N/#
i
−Min((q−")/#,") x −x /
Q( f , u) Q l ε O ε jꢀ Q1kQ(x )Qjꢀ e
RŠRε
ε
ε
0
1
,
x
ꢀ
i j
which implies that
k
i="
θ
Q i jQ ε
N/#
−Min((q−")/#,") x −x /
i
R f R l ε O ε jꢀ e
jꢀ QQ(x )k1Q .
(3.24)
ε
ε
0
Combining the above estimate and (3.16), (3.4) follows. This completes the proof of
1
,
x
ꢀ
i j
Proposition 3.1.
*
Let ε , R , δ be as in Proposition 3.1. For ε ? (0, ε ], R & R , δ ? (0, δ ], let
!
! !
!
!
!
(
α (x), Š (x)) be the C"-mapping established in Proposition 3.1. Define
ε
ε
g
J (x) l J (α (x), x, Š (x)),
ε
ε
ε
ε
k
x ? Dε δ.
,
be any point for which
R,
Let x l (x",…, x ) ? D
k
ε
ε
ε
ε
δ
,
R,
g
g
k
J (x ) l MaxoJ (x):x ? D δq.
(3.25)
ε
ε
ε
ε
,
R,


  
221
In the following proposition we show that x for small ε is an interior point of Dε δ,
ε
,
R,
g
and hence a critical point of J .
ε
P 3.2. Let x satisfy (3.25). Then as ε ! 0,
ε
i
x ! 0,
i l 1, 2,…, k,
ε
i
j
Qx kx Q\ε ! _, i ꢀ j.
ε
ε
Proof. With the notations of Proposition 3.1 and from Appendix B and the
estimates (3.4) and (3.24), we have
J (α (x ), x , Š (x )) l J$(x , u )
ε
ε
ε
ε
ε
ε
ε
ε
ε
l J$(x , 0)jO(R f R#jRuR#)
ε
ε
ε
ε
ε
,
x
ε
k
θ
Q i jQ ε
N
#
−Min(q−",#) x −x /
j
#
l J$(x , 0)jO ε ε jꢀ e
jꢀ QQ(x )k1Q
j="
ε
ε
0 0
11
ꢀ
i j
k
k−"
k
q−"
k
1
j
# N
l
k ꢀ Q(x ) RwR ε k Q(y) ꢀ w i
ꢀ w j
02
1 &
x ,ε
0
x ,ε
1
q
j="
j="
j=i+"
k
θ
Q i jQ ε
N
#
j #
−Min(q−",#) x −x /
jO ε ε jꢀ Q1kQ(x )Q jꢀ e
.
(3.26)
0
0
11
ꢀ
j="
i j
Let
τ
i
z l ε e , i l 1, 2,…, k,
ε
i
for some τ ? (", 1) and some vectors e ,…, e with e ꢀ e (i ꢀ j). Then
#
"
k
i
j
τ
i
j
"−
Qz kz Q\ε l Qe ke Q\ε ! _
ε
ε
i
j
as ε ! 0.
Thus z l (z",…, z ) ? D
k
k
for ε sufficiently small, and by (3.25) and (3.26) we
δ
ε
ε
ε
ε
,
R,
obtain
J (α (x ), x , Š (x )) & J (α (z ), z , Š (z ))
ε
ε
ε
ε
ε
ε
ε
ε
εh
ε
ε
ε
k
k
1
i
# N
&
k ꢀ Q(z ) RwR ε
02
ε
1
q
i="
k
j="
τ
θ
σ Q
Q ε"−
N
#
i #
−("+ ) e −e /
jO ε ε jꢀ Q1kQ(z )Q jꢀ e
i
j
0
0
ε
11
ꢀ
i j
1
1
θ
kRwR# ε jO(ε ")
N
N+
(3.27)
l
k
02
q1
where θ l Min(τ, 2θ) and 1jσ l Min(qk1, 2).
Using (3.26) and (3.27), we obtain
"
k
i="
k−"
k
q−"
k
1
i
# N
k ꢀ Q(x ) RwR ε k Q(y) ꢀ w i
ꢀ w j
02
ε
1 &
x
ε
,ε
0
x
ε
,ε
1
q
i="
j=i+"
k
θ
σ Q i jQ ε
N
#
i #
−("+ ) x
ε
−x
ε
/
jO ε ε jꢀ Q1kQ(x )Q jꢀ e
0
k
0
ε
11
ꢀ
i="
i j
1
1
θ
kRwR# ε jO(ε ").
N
N+
&
02
q1

2
22
 .    
Thus
k
i="
k−"
k
q−"
1
i
# N
ꢀ
(Q(x )k1) RwR ε j Q(y) ꢀ w
ꢀ w
ε
&
i ε
0
j ε
1
x
ε
,
x
ε
,
q
i="
j=i+"
k
i="
θ
σ Q i jQ ε
N
#
i #
−("+ ) x
ε
−x
ε
/
%
O ε ε jꢀ Q1kQ(x )Q jꢀ e
(3.28)
0
0
ε
11
ꢀ
i j
but
k−"
k
q−"
k−"
k
q−"
Q(y) ꢀ w i
ꢀ w i
&
ꢀ w i
ꢀ w j
&
x
ε
,ε
0
x
ε
,ε1 &
x
ε
,ε
0
x
ε
,ε
1
ꢀ
N
i="
j=i+"
i="
j=i+"
k−"
k
q−"
q−"
j&
j&
(Q(y)k1) ꢀ w i
ꢀ w j
,ε
,ε
0
0
,ε
1
1
x
ε
x
ε
Q
Q
Q%δ
y
i="
j=i+"
k−"
k
(Q(y)k1) ꢀ w i
ꢀ w j
(3.29)
,ε
x
ε
x
ε
Q&δ
y
i="
j=i+"
and
k−"
k
q−"
(
Q(y)k1) ꢀ w i
ꢀ w j
dy
,ε1 )
)
&
x
ε
,ε
0
x
ε
Q
Q&δ
y
i="
j=i+"
%
N
q−"
j
i
Cε ꢀ
w(y) w (yk(x kx )\ε) dy
&
&
ε
ε
ꢀ
Q
Q&δ ε
j i
y
/
Q
Q
Q
i
j Q ε
N
− y −(q−") y−(x
ε
−x
ε
) /
%
Cε ꢀ
e
e
dy
ꢀ
Q
Q&δ ε
i j
y
/
δ ε
N − /
l O(ε e ).
(3.30)
(3.31)
From (3.29), (3.30), and assumptions (Q ), (Q ), we have
"
#
k−"
k
q−"
σ
Q i jQ ε
ε δ
−
("+ /#) x
ε
−x
ε
/ N
N − /
Q(y) ꢀ w i
ꢀ w i
& C ꢀ e
ε jO(ε e
)
&
where C is a positive constant.
x
ε
,ε
0
x
ε
,ε
1
ꢀ
i="
j=i+"
i j
From (3.28) and (3.31) we have
k
i="
σ
Q i jQ ε
θ
N
i
−("+ /#) x
ε
−x
ε
/ N
N+#
ε ꢀ (Q(x )k1)jC ꢀ e
ε % O(ε
),
ε
ꢀ
i j
which implies that
i
Q(x ) ! 1 l Q(0),
ε
i
x ! 0,
ε
i
j
Qx kx Q\ε !_, i ꢀ j,
ε
ε
as ε ! 0, i, j l 1, 2,…, k.
This completes the proof of Proposition 3.2.
*
Proof of Theorem A. From Proposition 3.1, for each k, there exist ε , R , δ , and
a C"-mapping (α (x ), Š (x )):D
!
! !
k
,- ꢀ iEε for each ε ? (0, ε ], R " R , δ ? (0, δ ],
ε
ε
ε
ε
ε
δ
,
R,
,k
!
!
!


  
223
i
such that (2.7) and (2.8) hold. By Proposition 3.2 we can choose x such that x ! 0,
ε
ε
i
j
g
j
Qx kx Q\ε ! _, as ε ! 0, i, j l 1, 2,…, k, i ꢀ j, and cJ (x )\cx l 0. That is,
ε
ε
ε
ε
i
%
k
l ꢀ
%
cJ cα cJε
cJ cŠ
ε
ε
ε
0
j
j
,
%
- .
j
j
j
cα cx cx
cŠ cx
ε
=
"
i
i
i
k
cJ
cŠ
ε
ε
l
jꢀ A
w
%
ε,
%
- .
j
x
ε
,
j
cx
cx
%
ε
i
="
i
k
=
N
cw
%
ε
cŠ
ε
x
ε
,
jꢀ ꢀ B
,
(3.32)
%
-
%
.
%
,
,
h
j
cx cx
%
ε
" h="
h
i
by (2.7) and (2.8). However fw
Therefore, fw ε
%
, Š g l 0, fcw
%
\cx , Š g l 0, since Š ? E .
ε
ε
ε
ε
ε
ε
ε
ε
x
ε
,
x
ε
h
,k
%
, cŠ \cx g l 0, and from (3.32) we obtain
j
ε
ε
ε
x ,
i
N
h="
c#w
%
ε
cJε
ε
x
,
l ꢀ B
, Šε ,
.
j
j,h
-
j
j
cx
cx cx
ε
i
i
h
which is (2.6). Theorem A follows easily.
*
R 3.3. If Q(x) has several strict local minimum points, say a ,…, a , the
above arguments may be used to show that (1.1) has, for any given integers k , % l
"
m
%
1
,…, m, a solution u of the form
ε
m k
%
u l ꢀ ꢀ α
w i εjŠε
ε
ε %
,
,i
x
ε
%
,
,
%
=
" i="
for sufficiently small ε, and as ε ! 0,
i
x
! a%,
i l 1, 2,…, k%,
i ꢀ j,
ε %
,
i
j
Qx kx Q\ε ! _,
ε %
,
ε %
,
! 1\(Q(a ))"
/
(q−#)
,
i l 1, 2,…, k%,
α
ε %
,
%
,i
RŠ R# l o(ε ).
N
ε
R 3.4. The requirement that Q(x) has a strict local minimum may be
N
replaced by the weaker condition that there is a set Λ 9 ꢀ such that
Min Q(x) ! Min Q(x).
Λ
-
cΛ
We may use the same arguments to construct a solution u of the form
ε
k
i="
u l ꢀ α w i jŠ
ε
ε
ε
ε
,
i
x
ε
,
for any given positive integer k, provided ε l ε(k) is sufficient small, and as ε ! 0,
i
i
x ! x
ε
!
i
Q(x ) l Min
Λ
-
Q(x)
!
i "/(q−#)
α ! 1\(Q(x ))
ε
,
i
!
RŠ R# l o(ε ).
N
ε
ε
Appendix A
We give here a sketch of the proofs of Lemmas 2.1 and 2.2. The results are
essentially known, and have been used in one form or another by several authors.

2
24
 .    
To prove Lemma 2.1, we need the following decomposition lemma.
Let k l 1, 2,… be fixed. Let x l (x",…, x ) ? ꢀ , α l (α ,…, α ) ? ꢀ ,
k
kN
i
"
k
k
k
"
k
i
ꢀ
l o(α", α#,…, α , x ,…, x ):Qx Q, Qα k1Q % δ, i l 1, 2,…, kq
δ
k
i="
W(δ, R, ε) l u:u ? H", ukꢀ α w i % δε for some x ? D
i
N
k
.
(
))
x ,ε))
ε,R,δ
*
x
L A.1. There are R , ε , δ " 0 such that for R & R , ε ? (0, ε ], δ ? (0, δ ] and
!
! !
!
!
!
u ? W(δ, R, ε), the minimization problem
k
#
i
inf ukꢀ α w i
:(α, x) ? ꢀ
(A.1)
(
))
x
ε
,ε)) *
ε
δ
i="
%
is achieŠed in ꢀ and not in ꢀ Bꢀ . Furthermore, the aboŠe minimization problem
δ
δ
δh
#
%
#
admits a unique solution.
For the proof of the above lemma we refer the reader to [7, 8].
R A.2. Let (α, x) be the minimizer of (A.1), as given by Lemma A.1. Set
k
i="
i
Š
l ukꢀ α w i .
ε
x ,
Then Š satisfies
cw j
ε
x ,
fŠ, w j g l Š,
l 0,
ε
ε
- .
x ,
j
cx
ε
i
j l 1,…, k, i l 1,…, N. Therefore, (α, x, Š) ? Mε
for ε ? (0, ε ], R & R , δ ? [0, δ ].
δ
,
The proof of Lemma 2.1 then follows as in [24, Proposition 3]. See also [8].
R,
!
!
!
L A.3. There exist R , ε , δ " 0 such that for ε ? (0, ε ], δ ? (0, δ ], R & R ,
!
! !
!
!
!
k
j="
q−#
RŠR#k(qk1)
ꢀ w j
Š# & ρRŠR#
ε
& 0
ε
1
ε
x
ε
,
ꢀ
N
for all x ? Dδ , Š ? E , where ρ " 0 is a positiŠe constant.
ε
ε
,
This lemma can be proved in exactly the same way as [8, Lemma B.2].
R,
,k
Appendix B
We establish here an estimate for J$(x, 0).
ε
L B.1. For any x ? Dδ , x l (x",…, x ) we haŠe
k
ε
,
R,
k
k
1
i
# N
J*(x, 0) l k ꢀ Q(x ) RwR ε
02
1
q
i="
k−"
k
q−"
k Q(y) ꢀ w i
ꢀ w j
&
x ,ε
k
0
x ,ε
1
i="
j=i+"
θ
σ Q i jQ ε
N
i #
−("+ ) x −x /
jO ε ε jꢀ Q1kQ(x )Q jꢀ e
.
0
0
11
ꢀ
i="
i j


  
225
k
Proof. Let Hε l ꢀ w j . Then
ε
,
x
j=" x ,
RH R# l kε RwR jꢀ (w i , w j ) l kε RwR j2 ꢀ w i w j .
N
#
N
#
q−"
(B.1)
ε
ε
ε
ε
&
ε
ε
,
x
x ,
x ,
x , x ,
ꢀ
!
i j
i j
We also have
k
j="
q
j=#
q
k
j=#
q
q
q
Q(y) H k Q(y) ꢀ w j l Q(y) ꢀ w j k Q(y) ꢀ w j
&
ε,x
&
x ,ε & 0 x ,ε1 &
x ,ε
k
q−"
jq Q(y) w
ꢀ w j
&
&
x"
0
x ,ε
1
j=#
k
q−"
jq Q(y) w ꢀ w j
x"
0
x ,ε
1
j=#
σ Q i jQ ε
N
−("+ ) x −x /
jO (ε ꢀ e
)
(B.2)
ꢀ
i j
for some σ " 0, as ε ! 0, where we have used the following inequalities:
(
i) For 2 ! q % 3
q−"
CQbQ QaQ, QbQ % QaQ,
q
q
q
q−"
q−"
QQajbQ ka kb kqa bkqab Q %
(
q−"
CQaQ QbQ, QbQ " QaQ,
q/# q/#
CQaQ QbQ .
%
(
ii) For p " 3
q
q
q
q−"
q−"
q−# #
# q−#
QQajbQ ka kb kqa bkqab Q % C(a b ja b ).
By repeated application of the above inequalities to (B.2), we obtain
k
j="
q
q
q−"
Q(y) H l Q(y) ꢀ w j jq Q(y) ꢀ w i w j
&
ε,x
&
x ,ε
&
0
x ,ε x ,ε
q−"
!
i j
k−"
k
jq Q(y) ꢀ w i
ꢀ w j
&
x ,ε
x ,ε
1
i="
j=i+"
σ Q i jQ ε
N
−("+ ) x −x /
jO (ε ꢀ e
).
(B.3)
ꢀ
i j
Using the estimates (1.3) we also have
δ ε
q
q
N −q /
Q(y) w j l
Q(y) w j jO(ε e
)
&
x ,ε &B
x ,ε
j
δ
(x )
θ
j
q
N+
l Q(x )
w j jO(ε
)
&
x ,ε
j
B
δ
(x )
θ
j
# N
N+
l Q(x ) RwR ε jO(ε
)
(B.4)
(B.5)
δ ε
)
q−"
q−"
N −(q−") /
Q(y) w i w j l
x ,ε x ,ε &B
Q(y) w i w j jO(ε e
&
x ,ε x ,ε
i
δ
(x )
θ
i
q−"
N+
l Q(x )
w i w j jO(ε
)
&
x ,ε x ,ε
j
B
δ
(x )
θ
i
q−"
N+
l Q(x ) w i w j jO(ε ).
&
x ,ε x ,ε

2
26
 .    
Combining (B.1)–(B.5), we obtain
1
J*(x",…, x , 0) l RH R k Q(y) Hε
k
"
#
q
ε
ε
&
#
,x
q
,x
k
j="
k
1
j
# N
l
k ꢀ Q(x ) RwR ε
0
1
&
2
q
i
q−"
jꢀ 1kQ(x ) w i w i
0
x ,ε x ,ε
1
!
i j
k−"
k
q−"
k Q(y) ꢀ w i
ꢀ w j
&
x ,ε
0
x ,ε
1
i="
j=i+"
θ
σ Q i jQ ε
N+
N
−("+ ) x −x /
jO (ε jε ꢀ e
).
*
ꢀ
i j
References
1
2
3
4
5
6
. A. B, Critical points at infinity in some Šariational problems, Research Notes in Mathematics 182
(
. A. B and J. M. C, ‘On a nonlinear elliptic equation involving the Sobolev exponent: the
effect of the topology of the domain’, Comm. Pure Appl. Math. 41 (1988) 253–294.
. V. B and G. C, ‘The effect of the domain topology on the number of positive solutions of
nonlinear elliptic problems’, Arch. Rational Mech. Anal. 114 (1991) 79–93.
. V. B and G. C, ‘The multiple positive solutions of some elliptic problems via the Morse
theory and the domain topology’, Calc. Var. 2 (1994) 29–48.
. D. C and E. S. N, ‘Multiplicity of positive and nodal solutions for nonlinear elliptic
Pitman, 1989).
problems in ꢀN’, Ann. Inst. Henri Poincare
T
13 (1996).
. D. C, N. E. D, E. S. N and S. Y, ‘On the existence and profile of multi-peaked
solutions to singularly perturbed semilinear Dirichlet problems’, Discrete Contin. Dynam.
Systems 2 (1996) 221–236.
7
8
9
. D. C, E. S. N and S. Y, ‘Existence and uniqueness results on single-peaked solutions of
a semilinear problem’, Ann. Inst. H. Poincare 15 (1998) 73–111.
T
. D. C, E. S. N and S. Y, ‘Solutions with multiple ‘‘peaks’’ for nonlinear elliptic
equations’, Proc. Roy. Soc. Edinburgh. Sect. A (1999) 235–264.
. E. N. D, ‘The effect of the domain shape on the number of positive solutions of certain
nonlinear equations II’, J. Differential Equations 87 (1990) 120–156.
0. N. D and S. Y, ‘Multi-peaks solutions for a singularly perturbed Neumann problem’. Pacific
1
1
1
1
1
1
1
J. Math. 189 (1999) 241–262.
1. M.  P and P. F, ‘Multi-peak bound states for nonlinear Schro
Inst. H. Poincare Anal. Non Lineaire 15 (1998) 127–149.
2. M.  P and P. L. F, ‘Semi-classical states for nonlinear Schro
Anal. 149 (1997) 245–265.
3. A. F and A. W, ‘Nonspreading wave packets for the cubic Schro
a bounded potential’, J. Funct. Anal. 69 (1986) 397–408.
$
dinger equations’. Ann.
dinger equations’, J. Funct.
dinger equations with
T
T
$
$
4. B. G, W. M. N and L. N, ‘Symmetry of positive solutions of nonlinear elliptic
equations in ꢀN’, AdŠ. Math. Supplementary Studies 7A (1981) 369–402.
$
5. C. G, ‘Existence of multi-bump solutions for nonlinear Schrodinger equations via variational
method’, Comm. Partial Differential Equations 21 (1996) 787–820.
6. M. K. K, ‘Uniqueness of positive solutions for ∆ukujup l 0 in ꢀN’, Arch. Rational Mech.
Anal. 105 (1989) 243–266.
7. Y. Y. L, ‘On a singularly perturbed elliptic equation’, AdŠ. Differential Equations 2 (1997) 955–980.
1
1
8. G. L and J. W, ‘On nonlinear Schro
$
dinger equations with totally degenerate potentials’, C. R.
Acad. Sci. Paris Ser. 1 326 (1998) 1–7.
T
1
2
2
2
9. W. M. N and J. W, ‘On the location and profile of spike-layer solutions to singularly perturbed
semilinear Dirichlet problems’, Comm. Pure. Appl. Math. 48 (1995) 731–768.
0. E. S. N and S. Y, ‘The effect of the domain geometry in singular perturbation problems’,
Proc. London Math. Soc. 76 (1998) 427–452.
1. E. S. N and S. Y, ‘On the existence and multiplicity of nodal solutions in singular
perturbation problems’, preprint.
2. Y. J. O, ‘Existence of semi-classical bound states of nonlinear Schro
$
dinger equations with potentia
of the class (V) ’, Comm. Partial Differential Equations 13 (1988) 731–768.
α


  
227
2
2
2
2
2
3. Y. J. O, ‘On positive multi-lump bounded states of nonlinear Schro
well potential’, Comm. Math. Phys. 131 (1990) 223–253.
4. P. H. R, ‘On a class of nonlinear Schrodinger equations’, Z. Angew Math. Phys. 43 (1992)
70–291.
5. O. R, ‘The role of the Green’s function in a nonlinear elliptic equation involving the critical Sobolev
exponent’, J. Funct. Anal. 89 (1990) 1–52.
$
dinger equations under multiple
$
2
$
6. X. W, ‘On the concentration of positive bound states of nonlinear Schrodinger equations’, Comm.
Math. Phys. 153 (1993) 229–244.
7. Z. Q. W, ‘On the existence of multiple, single-peaked solutions of a semilinear Neumann
problem’, Arch. Rational Mech. Anal. 120 (1992) 375–399.
School of Mathematics
UniŠersity of New South Wales
Sydney
School of Mathematics and Statistics
UniŠersity of Sydney
Sydney
NSW 2052
Australia
NSW 2006
Australia