ARTICLE IN PRESS
Journal of Functional Analysis 212 (2004) 472–499
Blowing up solutions for an elliptic Neumann
problem with sub- or supercritical nonlinearity
Part I: N ¼ 3
Olivier Reya, and Juncheng Weib,1
a
Centre de Mathématiques de l’Ecole Polytechnique, UMR 7640 du CNRS, 91128 Palaiseau Cedex, France
b
Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong
Received 3 June 2003; accepted 20 June 2003
Communicated by H. Brezis
Abstract
We consider the sub- or supercritical Neumann elliptic problem Du þ mu ¼ u5þe ; u40 in
3
O; @u
@n ¼ 0 on @O; O being a smooth bounded domain in R ; m40 and ea0 a small number. Hm
denoting the regular part of the Green’s function of the operator D þ m in O with Neumann
1
boundary conditions, and jm ðxÞ ¼ m2 þ Hm ðx; xÞ; we show that a nontrivial relative homology
between the level sets jcm and jbm ; boco0; induces the existence, for e40 small enough, of
a solution to the problem, which blows up as e goes to zero at a point aAO such that
bpjm ðaÞpc: The same result holds, for eo0; assuming that 0oboc: It is shown that,
Mm ¼ supxAO jm ðxÞo0 (resp. 40) for m small (resp. large) enough, providing us with cases
where the above assumptions are satisfied.
r 2003 Elsevier Inc. All rights reserved.
1. Introduction
In this paper we consider the nonlinear Neumann elliptic problem
8
< Du þ mu ¼ uq u40 in O;
ðPq;m Þ
@u
:
¼0
on @O;
@n
where 1oqo þ N; m40 and O is a smooth and bounded domain in R3 :
Corresponding author.
E-mail addresses: rey@math.polytechnique.fr (O. Rey), wei@math.cuhk.edu.hk (J. Wei).
1
Research supported in part by an Earmarked Grant from RGC of HK.
0022-1236/$ - see front matter r 2003 Elsevier Inc. All rights reserved.
doi:10.1016/j.jfa.2003.06.006
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473
Equation ðPq;m Þ arises in many branches of the applied sciences. For example, it
can be viewed as a steady-state equation for the shadow system of the Gierer–
Meinhardt system in biological pattern formation [13,26] or of parabolic equations
in chemotaxis, e.g. Keller–Segel model [24].
When q is subcritical, i.e. qo5; Lin, Ni and Takagi proved that the only solution,
for small m; is the constant one, whereas nonconstant solutions appear for large m
[24] which blow up, as m goes to infinity, at one or several points. The least energy
solution blows up at a boundary point which maximizes the mean curvature of the
frontier [28,29]. Higher-energy solutions exist which blow up at one or several points,
located on the boundary [8,14,18,22,38], in the interior of the domain
[5,7,11,12,16,21,37,40], or some of them on the boundary and others in the interior
[17]. (A good review can be found in [26].) In the critical case, i.e. q ¼ 5; Zhu [41]
proved that, for convex domains, the only solution is the constant one for small m
(see also [39]). For large m; nonconstant solutions exist [1,33]. As in the subcritical
case the least energy solution blows up, as m goes to infinity, at a unique point which
maximizes the mean curvature of the boundary [3,27]. Higher-energy solutions have
also been exhibited, blowing up at one [2,15,31,34] or several boundary points
[19,25,35,36]. The question of interior blow-up is still open. However, in contrast
with the subcritical situation, at least one blow-up point has to lie on the boundary
[32]. Very few is known about the supercritical case, save the uniqueness of the radial
solution on a ball for small m [23].
Our aim, in this paper, is to study the problem for fixed m; when the exponent q is
close to the critical one, i.e. q ¼ 5 þ e and e is a small nonzero number. Whereas the
previous results, concerned with peaked solutions, always assume that m goes to
infinity, we are going to prove that a single peak solution may exist for finite m;
provided that q is close enough to the critical exponent. Such a solution blows up, as
q goes to 5, at one point which may be characterized.
In order to state a precise result, some notations have to be introduced. Let
Gm ðx; yÞ denote the Green’s function of the operator D þ m in O with Neumann
boundary conditions. Namely, for any yAO; x/Gm ðx; yÞ is the unique solution of
DGm þ mGm ¼ 4pdy ; xAO;
@Gm
¼ 0; xA@O:
@n
ð1:1Þ
Gm writes as
1=2
Gm ðx; yÞ ¼
em jxyj
Hm ðx; yÞ;
jx yj
where Hm ðx; yÞ; regular part of the Green’s function, satisfies
!
1=2
@Hm
1 em jxyj
DHm þ mHm ¼ 0; xAO;
¼
; xA@O:
@n jx yj
@n
We set
1
jm ðxÞ ¼ m2 þ Hm ðx; xÞ:
ð1:2Þ
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O. Rey, J. Wei / Journal of Functional Analysis 212 (2004) 472–499
It is to be noticed that
Hm ðx; xÞ- N as dðx; @OÞ-0
ð1:3Þ
implying that
Mm ¼ sup jm ðxÞ
xAO
is achieved in O: (See (A.10) in Proposition 5.2 for the proof of (1.3).) Denoting
f a ¼ fxAO; f ðxÞpag
the level sets of a function f defined in O; we have
Theorem 1.1. Assume that there exist b and c; boco0; such that c is not a critical
value of jm and the relative homology H ðjcm ; jbm Þa0: ðP5þe;m Þ has a nontrivial solution,
for e40 close enough to zero, which blows up as e goes to zero at a point aAO; such
that bojm ðaÞoc:
The same result holds, for eo0; assuming that 0oboc:
We notice that, Mm o0 (resp. 40) when m is small (resp. large) enough
(see (A.12) and (A.13) of Proposition 5.2). Consequently, we deduce from the
previous result:
Theorem 1.2. There exist m0 and m1 ; 0om0 pm1 ; such that:
(1) If 0omom0 ; ðP5þe;m Þ has a nontrivial solution, for e40 close enough to zero, which
blows up as e goes to zero at a maximum point a of Hm ða; aÞ:
(2) If m4m1 ; ðP5þe;m Þ has a nontrivial solution, for eo0 close enough to zero, which
blows up as e goes to zero at a maximum point a of Hm ða; aÞ:
Remarks. (1) In the critical case, i.e. e ¼ 0; further computations suggest that a
nontrivial solution should exist for m4m0 close enough to m0 ; such that Mm 40 and
Mm0 ¼ 0: This solution would blow up, as previously, at a maximum point of
Hm0 ða; aÞ as m goes to m0 : (This contrasts to previous results for ðP5;0 Þ on the
nonexistence of solutions for m small [39,41] and nonexistence of interior bubble
solutions for m large [10,31].)
(2) In a forthcoming paper, we shall treat the case NX4; which appears to be
qualitatively different.
The scheme of the proof is the following. In the next section, we define a twoparameter set of approximate solutions to the problem, and we look for a true
solution in a neighborhood of this set. Considering in Section 3 the linearized
problem at an approximate solution, and inverting it in suitable functional spaces,
the problem reduces to a finite-dimensional one, which is solved in Section 4. Some
useful facts and computations are collected in Appendix.
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475
2. Approximate solutions and rescaling
For sake of simplicity, we consider in the following the supercritical case, i.e. we
assume that e40: The subcritical case may be treated exactly in the same way.
For normalization reasons, we consider throughout the paper the equation
Du þ mu ¼ 3u5þe ;
ð2:1Þ
u40
instead of the original one. The solutions are identical, up to the multiplicative
1
constant 34þe : We recall that, according to [6], the functions
1
l2
Ul;a ðxÞ ¼
2
ð1 þ l jx 1
aj2 Þ2
; l40; aAR3
ð2:2Þ
are the only solutions to the problem
Du ¼ 3u5 ; u40
in R3 :
As aAO and l goes to infinity, these functions provide us with approximate solutions
to the problem that we are interested in. However, in view of the additional linear
term mu which occurs in ðP5þe;m Þ; the approximation needs to be improved. Actually,
we define in O the following functions:
0
1
1
Ũl;a;m ðxÞ ¼ Ul;a ðxÞ 1 B1 em2 jxaj
C
þ Hm ða; xÞA
1@
jx
aj
l2
which satisfy
0
5
DŨl;a;m þ mŨl;a;m ¼ 3Ul;a
þ m@Ul;a 1
1
1
A:
ð2:3Þ
l2 jx aj
We are going to seek for solutions in a neighborhood of such functions, with the a
priori assumption that a remains far from the boundary of the domain, that is there
exists some number d40 such that
dða; @OÞ4d:
ð2:4Þ
Moreover, integral estimates (see Appendix) suggest to make the additional a priori
assumption that l behaves as 1=e as e goes to zero. Namely, we set
l¼
1
;
Le
with d0 some strictly positive number.
1
oLod0
d0
ð2:5Þ
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O. Rey, J. Wei / Journal of Functional Analysis 212 (2004) 472–499
In fact, in order to avoid the singularity which appears in the right-hand side of
(2.3), and to cancel the normal derivative on the boundary, we modify slightly the
definition of Ũl;a;m ; setting
2
VL;a;m;e ðxÞ ¼ Ũ 1
Le;a;m
ðxÞ e
1
m
2
ðLeÞ2 jx ajð1 e jxaj Þ þ yL;a;m;e ðxÞ
2
ð2:6Þ
yL;a;m;e ¼ y being the unique solution to the problem
8
Dy þ my ¼ 0
>
>
0
1
<
1
e2
1
@y
@@
ðLeÞ2
m
2
>
¼
U 1 ðxÞ þ
þ ðLeÞ2 jx ajð1 e jxaj ÞA
>
: @n @n
jx aj 2
Le;a
in O;
on @O:
From the above assumption (2.4) we know that
5
Hm ða; xÞ ¼ Oð1Þ;
yl;a;m;e ¼ Oðe2 Þ
ð2:7Þ
in C 2 ðOÞ: We note that VL;a;m;e ¼ V satisfies
0
1
8
1
e2
>
2
>
>
> DV þ mV ¼ 3U 51 þ m@U 1 ðLeÞ e jxaj2 A
>
>
>
jx aj
Le;a
>
Le;a
>
>
>
>
!
1 5
>
>
e2
>
mL2 e2
2e2
2
<
jxaj
1þ
;
e
3
2
jx aj
jx aj
>
>
>
>
e2
>
1
>
m2 e2
>
jxaj2 Þ
2
>
ðLeÞ
jx
ajð1
e
in O;
>
>
2
>
>
>
>
>
: @V ¼ 0
on @O:
@n
ð2:8Þ
The VL;a;m;e ’s are the suitable approximate solutions in the neighborhood of which we
shall find a true solution to the problem. In order to make further computations
easier, we proceed to a rescaling. We set
Oe ¼
O
e
and we define in Oe the functions
1
WL;x;m;e ðxÞ ¼ e2 VL;a;m;e ðexÞ; x ¼
a
e
ð2:9Þ
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477
which write as
0
1
m2 ejxxj
1B 1 e
WL;x;m;e ðxÞ ¼ U 1 ðxÞ L2 @
;x
jx xj
L
1
C
þ Hm;e ðx; xÞA
1
me2 1
2
L2 jx xjð1 e jxxj Þ þ y* L;x;m;e ðxÞ
2
ð2:10Þ
where Hm;e denotes the regular part of the Green’s function of the operator D þ me2
1
with Neumann boundary conditions in Oe ; and y* L;x;m;e ðxÞ ¼ e2 yL;a;m;e ðexÞ: We notice
that, taking account of (2.7)
Hm;e ðx; xÞ ¼ OðeÞ;
y* L;x;m;e ðxÞ ¼ Oðe3 Þ
ð2:11Þ
in C 2 ðOe Þ: We notice also that assumption (2.4) is equivalent to
dðx; @Oe Þ4
d
e
ð2:12Þ
and that WL;x;m;e ¼ W satisfies the uniform estimate jWL;x;m;e jpCU 1
L;x
over, we have
8
0
1
1
1
>
>
2
L
2
>
>
DW þ me2 W ¼ 3U 15 þ me2 @U 1 e jxxj A
>
>
>
;x
jx xj
L;a
>
L
>
>
>
>
!
>
1
>
1
>
mL2 e2
2
<
jxxj2
1
þ
e
jx xj3
jx xj2
>
>
>
>
1
>
1
m2 e4
>
2
>
> ðLeÞ2 jx xjð1 e jxxj Þ
>
>
2
>
>
>
>
>
: @W ¼ 0
@n
in Oe : More-
ð2:13Þ
in Oe ;
on @Oe :
Finding a solution to ðP5þe;m Þ in a neighborhood of the functions VL;a;m;e is
equivalent, through the rescaling, to solving the problem
8
< Du þ me2 u ¼ 3u5þe u40 in Oe ;
0
ð2:14Þ
ðP5þe;m Þ
@u
:
¼0
on @Oe
@n
in a neighborhood of the functions WL;x;m;e : For that purpose, we have to use some
local inversion procedure. Namely, we are going to look for a solution to ðP0e;m Þ
writing as
w ¼ WL;x;m;e þ o
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O. Rey, J. Wei / Journal of Functional Analysis 212 (2004) 472–499
with o small and orthogonal at WL;x;m;e ; in a suitable sense, to the manifold
M ¼ fWL;x;m;e ; ðL; xÞ satisfying ð2:5Þ ð2:12Þg:
ð2:15Þ
The general strategy consists in finding first, using an inversion procedure, a smooth
map ðL; xÞ/oðL; xÞ such that WL;x;m;e þ oðL; x; m; eÞ solves the problem in an
orthogonal space to M: Then, we are left with a finite-dimensional problem, for
which a solution may be found using the topological assumption of the theorem. In
the subcritical or critical case, the first step may be performed in H 1 (see e.g.
[4,30,31]). However, this approach is not valid any more in the supercritical case, for
H 1 does not inject into Lq as q46: Following [9], we use instead weighted Hölder
spaces to reduce the problem to a finite-dimensional one.
3. The finite-dimensional reduction
3.1. Inversion of the linearized problem
We first consider the linearized problem at a function WL;x;m;e ; and we invert it in
an orthogonal space to M: From now on, we omit for sake of simplicity the indices
in the writing of WL;x;m;e : Equipping H 1 ðOe Þ with the scalar product
Z
ðru rv þ me2 uvÞ
ðu; vÞe ¼
Oe
orthogonality to the functions
Y0 ¼
@W
;
@L
Yi ¼
@W
; 1pip3
@xi
ð3:1Þ
in that space is equivalent, setting
Z0 ¼ D
@W
@W
þ me2
;
@L
@L
Zi ¼ D
@W
@W
þ me2
; 1pip3
@xi
@xi
ð3:2Þ
to the orthogonality in L2 ðOe Þ; equipped with the usual scalar product /; S; to the
functions Zi ; 0pip3: Then, we consider the following problem : hALN ðOe Þ being
given, find a function f which satisfies
P
8
Df þ me2 f 3ð5 þ eÞWþ4þe f ¼ h þ i ci Zi in Oe ;
>
>
<
@f
ð3:3Þ
¼0
on @Oe ;
>
@n
>
:
0pip3
/Zi ; fS ¼ 0
for some numbers ci :
Existence and uniqueness of f will follow from an inversion procedure in
suitable functional spaces. Namely, for f a function in Oe ; we define the following
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479
weighted LN -norms:
1
jj f jj ¼ sup jð1 þ jx xj2 Þ2 f ðxÞj
xAOe
and
jj f jj ¼ sup jð1 þ jx xj2 Þ2 f ðxÞj:
xAOe
Writing U instead of U 1 ; the first norm is equivalent to jjU 1 f jjN and the second
L;x
one to jjU 4 f jjN ; uniformly with respect to x and L:
We have the following result:
Proposition 3.1. There exists e0 40 and a constant C40; independent of e and x; L
satisfying (2.12) (2.15), such that for all 0oeoe0 and all hALN ðOe Þ; problem (3.5) has
a unique solution f Le ðhÞ: Besides,
jjLe ðhÞjj pCjjhjj ;
jci jpCjjhjj :
ð3:4Þ
Moreover, the map Le ðhÞ is C 2 with respect to L; x and the LN
-norm, and
jjDðL;xÞ Le ðhÞjj pCjjhjj ;
jjD2ðL;xÞ Le ðhÞjj pCjjhjj :
ð3:5Þ
Proof. The argument follows closely the ideas in [9]. We repeat it for convenience of
the reader. The proof relies on the following result:
Lemma 3.1. Assume that fe solves (3.3) for h ¼ he : If jjhe jj goes to zero as e goes to
zero, so does jjfe jj :
Proof. For 0oro1; we define
1
jj f jjr ¼ sup jð1 þ jx xj2 Þ2ð1rÞ f ðxÞj
xAOe
and we first prove that jjfe jjr goes to zero. Arguing by contradiction, we may assume
that jjfe jjr ¼ 1: Multiplying the first equation in (3.3) by Yj and integrating in Oe
we find
X
ci /Zi ; Yj S ¼ / DYj þ me2 Yj 3ð5 þ eÞWþ4þe Yj ; fe S /he ; Yj S:
i
On one hand we check, in view of the definition of Zi ; Yj
/Z0 ; Y0 S ¼ jjY0 jj2e ¼ g0 þ oð1Þ;
/Zi ; Yi S ¼ jjYi jj2e ¼ g1 þ oð1Þ; 1pip3;
ð3:6Þ
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480
where g0 ; g1 are strictly positive constants, and
/Zi ; Yj S ¼ oð1Þ; iaj:
ð3:7Þ
On the other hand, in view of the definition of Yj and W ; straightforward
computations yield
/ DYj þ me2 Yj 3ð5 þ eÞWþ4þe Yj ; fe S ¼ oðjjfe jjr Þ
and
/he ; Yj S ¼ Oðjjhe jj Þ:
Consequently, inverting the quasi-diagonal linear system solved by the ci ’s, we find
ci ¼ Oðjjhe jj Þ þ oðjjfe jjr Þ:
ð3:8Þ
In particular, ci ¼ oð1Þ as e goes to zero. The first equation in (3.3) may be written as
!
Z
X
4þe
fe ðxÞ ¼ 3ð5 þ eÞ
Ge ðx; yÞ Wþ fe þ he þ
ci Zi dy
ð3:9Þ
Oe
i
for all xAOe ; Ge denoting the Green’s function of the operator ðD þ me2 Þ in Oe with
Neumann boundary conditions.
We notice that by scaling and (A.11) of Proposition 5.2,
x y
C
ð3:10Þ
Ge ðx; yÞ ¼ eGm ; p
e e
jx yj
and hence we obtain
Z
Z
4þe
p Cjjf jj
G
ðx;
yÞW
f
dy
e
e
e r
þ
Oe
Oe
1
jx yj
1
1
dy
ð1 þ jx xj2 Þ2ð3þeþrÞ
1
Z
p Cjjfe jjr ð1 þ jx xj2 Þ2 ;
Z
1
1
Ge ðx; yÞhe dyp Cjjhe jj
dy
2 2
Oe jx yj ð1 þ jx xj Þ
Z
p Cjjhe jj ð1 þ jx xj2 Þ2 ;
Z
1
1
Ge ðx; yÞZi dyp C
dy
5
Oe jx yj ð1 þ jx xj2 Þ2
Oe
1
Oe
1
p Cð1 þ jx xj2 Þ2
ð3:11Þ
from which we deduce
1
r
ð1 þ jx xj2 Þ2ð1rÞ jfe ðxÞjpCð1 þ jx xj2 Þ2 :
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481
jjfe jjr ¼ 1 implies the existence of R40; g40 independent of e such that
4g: Then, elliptic theory shows that along some subsequence, f* ðxÞ ¼
jjf jj N
e L ðBR ðxÞÞ
e
fe ðx xÞ converges uniformly in any compact subset of R3 to a nontrivial solution of
* ¼ 15U 4* f*
Df
L;0
*
* writes as
*
for some L40:
Moreover, jfðxÞjpC=jxj:
As a consequence, f
* ¼ a0
f
3
X
@UL;0
@UL;0
*
*
ai
þ
*
@ai
@L
i¼1
(see e.g. [30]). On the other hand, equalities /Zi ; fe S ¼ 0 provide us with the equalities
Z
Z
@UL;0
@UL;0
*
*
4
*
* ¼ 0;
D
UL;0
f¼
f
*
*
*
3
3
@L
@L
R
R
Z
Z
@UL;0
@UL;0
*
*
4
* ¼
* ¼ 0; 1pip3:
D
UL;0
f
f
*
3
3
@ai
@ai
R
R
As we have also
2
Z @UL;0
* r
¼ g0 40;
* 3
@L
2
Z @UL;0
* r
¼ g1 40; 1pip3
3
@a Z
Z
R
i
R
and
R3
r
@UL;0
@UL;0
*
*
¼
:r
*
@ai
@L
r
R3
@UL;0
*
@aj
:r
@UL;0
*
@ai
¼ 0; iaj
the aj ’s solve a homogeneous quasi-diagonal linear system, yielding aj ¼ 0; 0paj p3;
* ¼ 0; hence a contradiction. This proves that jjf jj ¼ oð1Þ as e goes to zero.
and f
e r
Furthermore, (3.9), (3.11) and (3.8) show that
jjfe jj pCðjjhe jj þ jjfe jjr Þ
whence also jjfe jj ¼ oð1Þ as e goes to zero.
&
Proof of Proposition 3.1 (Conclusion). We set
H ¼ ffAH 1 ðOe Þ; /Zi ; fS ¼ 0; 0pip3g
equipped with the scalar product ð; Þe : Problem (3.3) is equivalent to finding fAH
such that
ðf; yÞe ¼ /3ð5 þ eÞWþ4þe f þ h; yS
8yAH
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482
that is
f ¼ Te ðfÞ þ h̃
ð3:12Þ
h̃ depending linearly on h; and Te being a compact operator in H: Fredholm’s
alternative ensures the existence of a unique solution, provided that the kernel of
Id Te is reduced to 0. We notice that fe AKerðId Te Þ solves (3.3) with h ¼ 0:
Thus, we deduce from Lemma 3.1 that jjfe jj ¼ oð1Þ as e goes to zero. As KerðId Te Þ is a vector space, KerðId Te Þ ¼ f0g: Inequalities (3.4) follow from Lemma 3.1
and (3.8). This completes the proof of the first part of Proposition 3.1.
The smoothness of Le with respect to L and x is a consequence of the smoothness
of Te and h̃; which occur in the implicit definition (3.12) of f Le ðhÞ; with respect to
these variables. Inequalities (3.5) are obtained differentiating (3.3), writing the
derivatives of f with respect L and x as a linear combination of the Zi ’ and an
orthogonal part, and estimating each term using the first part of the proposition—
see [9,20] for detailed computations. &
3.2. The reduction
In view of (2.13), a first correction between the approximate solution W and a true
solution to ðP0e;m Þ writes as
ce ¼ Le ðRe Þ
ð3:13Þ
with
Re ¼ 3Wþ5þe ðDW þ me2 W Þ
0
1
!
1
1
1
1
2
L
mL2 e2
1
2
5þe
5
2@
jxxj
jxxj2
A
¼ 3Wþ 3U 1 me U 1 e
þ
1
þ
e
jx xj
L;a
jx xj3
jx xj2
L;x
!
1
1
m2 e4
2
jxxj
ðLeÞ2 jx xj 1 e
þ
:
ð3:14Þ
2
We have:
Lemma 3.2. There exists C; independent of x; L satisfying (2.12) (2.5), such that
jjRe jj pCe;
jjDðL;xÞ Re jj pCe;
jjD2ðL;xÞ Re jj pCe:
Proof. According to (2.10), W ¼ U þ OðeÞ uniformly in Oe : Consequently, noticing
that UXCe in Oe ; C independent of e
U 5 Wþ5þe ¼ OðeU 5 jln Uj þ eU 4 Þ
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483
uniformly in Oe ; whence
jjU 5 Wþ5þe jj pCjjU 4 ðU 5 Wþ5þe ÞjjN ¼ OðeÞ:
On the other hand
2
0
1
!
1
1
1
1
2
L
mL2 e2
1
2
2 24 2@
jxxj
jxxj2
A
e
ð1 þ jx xj Þ me U 1 e
1
þ
jx xj
L;a
jx xj3
jx xj2
3
1
1
m2 e4
2
ðLeÞ2 jx xjð1 e jxxj Þ5 ¼ OðeÞ
2
uniformly for xAOe ; since
1
1
L2
2
e jxxj ¼ Oðjx xj3 Þ
U1 ;a
jx
xj
L
as jx xj goes to infinity, and jx xj ¼ Oð1=eÞ in Oe : The first estimate of the lemma
follows. The others are obtained in the same way, differentiating (3.14) and
estimating each term as previously. &
Lemma 3.2 and Proposition 3.1 yield:
Lemma 3.3. There exists C; independent of x; L satisfying (2.12) (2.5), such that
jjce jj pCe;
jjDðL;xÞ ce jj pCe;
jjD2ðL;xÞ ce jj pCe:
We consider now the following nonlinear problem: finding f such that, for some
numbers ci
8
DðW þ c þ fÞ þ me2 ðW þ c þ fÞ
>
>
>
P
>
>
< 3ðW þ c þ fÞ5þe
in Oe ;
þ ¼
i ci Z i
ð3:15Þ
@f
>
>
¼0
on @Oe ;
>
>
>
: @n
0pip3:
/Zi ; fS ¼ 0
Setting
5þe
4þe
Ne ðZÞ ¼ ðW þ ZÞ5þe
þ Wþ ð5 þ eÞWþ Z
ð3:16Þ
the first equation in (3.15) writes as
Df þ me2 f 3ð5 þ eÞWþ4þe f ¼ 3Ne ðc þ fÞ þ
X
i
ci Zi
ð3:17Þ
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for some numbers ci : Assuming that jjZjj is bounded, say jjZjj pM for some
constant M; we have
jjNe ðZÞjj pCjjZjj2
whence, assuming that jjfjj p1 and using Lemma 3.3
jjNe ðc þ fÞjj pCðjjfjj2 þ e2 Þ:
ð3:18Þ
We state the following result:
Proposition 3.2. There exists C; independent of e and x; L satisfying (2.12) (2.5), such
that for small e problem (3.15) has a unique solution f ¼ fðL; x; m; eÞ with
jjfjj pCe2 :
ð3:19Þ
Moreover, ðL; xÞ/fðL; x; m; eÞ is C 2 with respect to the LN
-norm, and
jjDðL;xÞ fjj pCe2 ;
jjD2ðL;xÞ fjj pCe2 :
ð3:20Þ
Proof. Following [9], we consider the map Ae from F ¼ ffAH 1 -LN ðOe Þ :
jjfjj peg to H 1 -LN ðOe Þ defined as
Ae ðfÞ ¼ Le ð3Ne ðf þ cÞÞ
and we remark that finding a solution f to problem (3.15) is equivalent to finding a
fixed point of Ae : One the one hand we have, for fAF
jjAe ðfÞjj pjjLe ð3Ne ðf þ cÞÞCjj pjjNe ðf þ cÞjj pCe2 pe
for e small enough, implying that Ae sends F into itself. On the other hand Ae is a
contraction. Indeed, for f1 and f2 in F; we write
jjAe ðf1 Þ Ae ðf2 Þjj p jjNe ðc þ f1 Þ Ne ðc þ f2 Þjj
p jjU 4 ðNe ðc þ f1 Þ Ne ðc þ f2 ÞÞjjN :
In view of (3.16) we have
4þe
@Z Ne ðZÞ ¼ ð5 þ eÞ ððW þ ZÞ4þe
þ Wþ ÞÞ
whence
jNe ðc þ f1 Þ Ne ðc þ f2 ÞjpCU 3 jc þ tf1 þ ð1 tÞf2 j jf1 f2 j
ð3:21Þ
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for some tAð0; 1Þ: Then
jjU 4 ðNe ðc þ f1 Þ Ne ðc þ f2 ÞÞjjN p CjjU 1 ðc þ tf1 þ ð1 tÞf2 Þðf1 f2 ÞjjN
p Cðjjcjj þ jjf1 jj þ jjf2 jj Þjjf1 f2 jj
p ejjf1 f2 jj :
This implies that Ae has a unique fixed point in F; that is problem (3.15) has a
unique solution f such that jjfjj pe: Furthermore, the definition of f as a fixed
point of Ae yields
jjfjj ¼ jjLe ð3Ne ðf þ cÞÞjj pCjjNe ðf þ cÞjj pCe2
using (3.18), whence (3.19).
In order to prove that ðL; xÞ/fðL; xÞ is C 2 ; we remark that setting for ZAF
BðL; x; ZÞ Z Le ð3Ne ðZ þ cÞÞ
f is defined as
BðL; x; fÞ ¼ 0:
ð3:22Þ
We have
@Z BðL; x; ZÞ½y ¼ y 3Le ðy ð@Z Ne ÞðZ þ cÞÞ
and, using (3.21)
jjLe ðyð@Z Ne ÞðZ þ cÞÞjj p Cjjyð@Z Ne ÞðZ þ cÞjj
p CjjU 3 ð@Z Ne ÞðZ þ cÞjjN jjyjj
p CjjZ þ cjj jjyjj
p Cejjyjj :
Consequently, @Z BðL; x; fÞ is invertible in LN
with uniformly bounded inverse.
Then, the fact that ðL; xÞ/fðL; xÞ is C 2 follows from the fact that
ðL; x; ZÞ/Le ðNe ðZ þ cÞÞ is C 2 and the implicit functions theorem.
Finally, let us show how estimates (3.20) may be obtained. Derivating (3.22) with
respect to L; we have
@L f ¼ 3ð@Z BðL; x; fÞÞ1 ðð@L Le Þ ðNe ðf þ cÞÞ þ Le ðð@L Ne Þðf þ cÞÞ þ Le ðð@Z Ne Þðf þ cÞ@L cÞÞ
whence, according to Proposition 3.1
jj@L fjj pCðjjNe ðf þ cÞjj þ jjð@L Ne Þðf þ cÞjj þ jjð@Z Ne Þðf þ cÞ@L cjj Þ:
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486
From (3.18) and (3.19) we know that
jjNe ðf þ cÞjj pCe2 :
Concerning the next term, we notice that according to definition (3.16) of Ne
4þe
3þe
jð@L Ne Þðf þ cÞj ¼ ð5 þ eÞjðW þ f þ cÞ4þe
þ Wþ ð4 þ eÞWþ ðf þ cÞjj@L W j
p CU 5 jjf þ cjj2
p CU 5 e2
using again (3.18) and (3.19), whence
jjð@L Ne Þðf þ cÞjj pCe2 :
Lastly, from (3.21) we deduce
jð@Z Ne Þðf þ cÞ@L cjpU 5 jjf þ cjj jj@L cjj
yielding
jjð@Z Ne Þðf þ cÞ@L cjj pCe2 :
Finally we obtain
jj@L fjj pCe2 :
The other first and second derivatives of f with respect to L and x may be estimated
in the same way (see [20] for detailed computations concerning the second
derivatives). This concludes the proof of Proposition 3.2. &
3.3. Coming back to the original problem
We introduce the following functional defined in H 1 ðOÞ-L6þe ðOÞ:
Z
Z
1
3
Je ðuÞ ¼
ðjruj2 þ mu2 Þ u6þe
2 O
6þe O þ
ð3:23Þ
whose nontrivial critical points are solutions to ðP5þe;m Þ (up to the multiplicative
1
constant 34þe ). We consider also the rescaled functions defined in O
1
ŴðL; aÞðxÞ ¼ ez WL;x ðe1 xÞ ¼ e2z VL;a ðxÞ
with
z¼
1
;
2 þ 12e
a ¼ ex:
ð3:24Þ
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487
We define also
#
cðL;
aÞðxÞ ¼ ez cðL; xÞðe1 xÞ;
#
fðL;
aÞðxÞ ¼ ez fðL; xÞðe1 xÞ
ð3:25Þ
and we set
# þ fÞðL;
#
Ie ðL; aÞ Je ððŴ þ c
aÞÞ:
ð3:26Þ
We have:
1
# is a solution to problem ðP5þe;m Þ if
Proposition 3.3. The function u ¼ 34þe ðŴ þ c# þ fÞ
and only if ðL; aÞ is a critical point of Ie :
Proof. For v in H 1 ðOe Þ-L6þe ðOe Þ; we set
Z
Z
1
3
2
2 2
Ke ðvÞ ¼
ðjrvj þ me v Þ v6þe
2 Oe
6 þ e Oe þ
ð3:27Þ
whose nontrivial critical points are solutions to ðP05þe;m Þ: According to the definition
Ie we have
Ie ðL; aÞ ¼ e12z Ke ððW þ c þ fÞðL; xÞÞ:
ð3:28Þ
1
# þ fÞ
# being a solution to ðP5þe;m Þ is equivalent to
We notice that u ¼ 34þe ðÛ þ c
W þ c þ f being a solution to ðP05þe;m Þ; that is a critical point of Ke : It is also
equivalent to the cancellation of the ci ’s in (3.15) or, in view of (3.6) (3.7)
Ke0 ðW þ c þ fÞ½Yi ¼ 0; 0pip3:
ð3:29Þ
On the other hand, we deduce from (3.28) that Ie0 ðL; aÞ ¼ 0 is equivalent to the
cancellation of Ke0 ðW þ c þ fÞ applied to the derivatives of W þ c þ f with respect
to L and x: According to definition (3.1) of the Yi ’s, Lemma 3.3 and Proposition 3.2
we have
@ðW þ c þ fÞ
¼ Y 0 þ y0 ;
@L
@ðW þ c þ fÞ
¼ Yj þ yj ; 1pjp3
@xj
¼ oð1Þ; 0pip3: Writing
with jjyi jjLN
yi ¼ y0i þ
3
X
aij Yj ;
/y0i ; Zj S ¼ ðy0i ; Yj Þe ¼ 0; 0pi; jp3
j¼0
and
Ke0 ðW þ c þ fÞ½Yi ¼ ai
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it turns out that Ie0 ðL; aÞ ¼ 0 is equivalent, since Ke0 ðW þ c þ pÞ½y ¼ 0 for /y; Zj S ¼
ðy; Yj Þe ¼ 0; 0pjp3; to
ðId þ ½aij Þ½ai ¼ 0:
As aij ¼ Oðjjyi jj Þ ¼ oð1Þ; we see that Ie0 ðL; aÞ ¼ 0 means exactly that (3.29) is
satisfied. &
4. Proof of Theorem 1.1
In view of Proposition 3.3 we have, for proving the theorem, to find critical points
of Ie : We establish first a C 2 -expansion of Ie :
4.1. Expansion of Ie
Proposition 4.1. There exist A; B; C; strictly positive constants such that
Ie ðL; aÞ ¼ A þ
A
1
A
3BL 1=2
e lnðeLÞ þ C þ
ðm þ Hm ða; aÞÞe þ ese ðL; aÞ
eþ
4
2
6
2
with se ; DðL;aÞ se and D2ðL;aÞ se going to zero as e goes to zero, uniformly with respect to
a; L satisfying (2.4) and (2.5).
Proof. In view of definition (3.26) of Ie ; we first estimate Je ðŴÞ: We have
1
e2z1 Je ðŴÞ ¼ e2z1 Je ðe2z V Þ
e Z
1 e2
V 6þe
¼ Je ðV Þ þ 3
6þe O þ
Z
1 e
¼ Je ðV Þ þ ln e þ oðeÞ
Vþ6þe
2
2
O
from which we deduce, using the integral estimates (A.8), (A.9) and Proposition 5.1
in Appendix, that
A
1
A
3BL 1=2
ðm þ Hm ða; aÞÞe þ oðeÞ:
Je ðŴÞ ¼ A þ e lnðeLÞ þ C þ
eþ
4
2
6
2
ð4:1Þ
Then, we prove that
# ¼ oðeÞ:
Ie ðL; aÞ Je ðŴ þ cÞ
ð4:2Þ
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489
#
Indeed, from a Taylor expansion and the fact that Je0 ðŴ þ c# þ fÞ½f
¼ 0; we have
#
IðL; aÞ Je ðŴ þ cÞ
# þ fÞ
# Je ðŴ þ cÞ
#
¼ Je ðŴ þ c
¼
Z
0
1
# þ tfÞ½
# f;
# ft
# dt
Je00 ðŴ þ c
¼ e12z
Z
0
12z
¼e
¼ e12z
1
Ke00 ðW þ c þ fÞ½f; ft dt
Z 1 Z
0
Oe
0
Oe
Z 1 Z
2
2
2
ðjfj þ me f 3ð5 þ eÞðW þ c þ
2
fÞ4þe
þ f Þ
t dt
2
ðNe ðf þ cÞf þ 3ð5 þ eÞ ½Wþ4þe ðW þ c þ tfÞ4þe
f
Þ
t dt:
þ
The desired result follows from (3.18), Lemma 3.3 and (3.19). Similar computations
show that estimate (4.2) is also valid for the first and second derivatives of Ie ðL; aÞ # with respect to L and a: Then, the proposition will follow from an
Je ðŴ þ cÞ
# Je ðŴÞ: We have
estimate of Je ðŴ þ cÞ
# Je ðŴÞ ¼ e12z ðKe ðW þ cÞ Ke ðW ÞÞ
Je ðŴ þ cÞ
¼ e12z ðKe0 ðW Þ½c þ
By definition of c and Re
Ke0 ðW Þ½c ¼ Z
1
0
Z
ð1 tÞKe00 ðW þ tcÞ½c; cÞ:
Re c
Oe
and we have
Ke00 ðW
þ tcÞ½c; c ¼
Z
2
2
2
ðjrcj þ me c Þ 3ð5 þ eÞ
Oe
Oe
Then, integration by parts and c ¼ Le ðRe Þ yield
Z
Z
Ke00 ðW þ tcÞ½c; c ¼
Re c 3ð5 þ eÞ
Oe
Z
Oe
2
ðW þ tcÞ4þe
þ c :
2
4þe
ððW þ tcÞ4þe
þ Wþ Þc :
Consequently
# Je ðŴÞ
Je ðŴ þ cÞ
Z
Z
Z 1
1
4þe 2
Re c 3ð5 þ eÞ
ð1 tÞ
½ðW þ tcÞ4þe
W
c
¼ e12z dt
þ
þ
2 Oe
Oe
0
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O. Rey, J. Wei / Journal of Functional Analysis 212 (2004) 472–499
and Lemmas 3.2 and 3.3 yield
# Je ðŴÞ ¼ oðeÞ:
Je ðŴ þ cÞ
The same estimate holds for the first and second derivatives with respect to L and a;
obtained similarly with more delicate computations—see Proposition 3.4 in [20].
This concludes the proof of Proposition 4.1. &
4.2. Proof of Theorem 1.1 (Conclusion)
According to the statement of Theorem 1.1, we assume the existence of b and c;
1
boco0; such that c is not a critical value of jm ðxÞ ¼ m2 þ Hm ðx; xÞ and the relative
homology H ðjcm ; jbm Þa0: In view of Proposition 3.3, we have to prove the existence
of a critical point of Ie ðL; aÞ: According to Proposition 4.1, we have
@Ie
Ae 3B
þ
j ðaÞe þ oðeÞ
ðL; aÞ ¼
4L
2 m
@L
and
@ 2 Ie
Ae
ðL; aÞ ¼ 2 þ oðeÞ
@L2
4L
uniformly with respect to a and L satisfying (2.4) (2.5). For d40; Z40; we define
Od;g ¼ faAO s:t: dða; @OÞ4d; jm ðaÞo gg:
The implicit functions theorem provides us, for e small enough, with a C 1 -map
aAOd;g /LðaÞ such that
@Ie
ðLðaÞ; aÞ ¼ 0;
@L
LðaÞ ¼ A
ðj ðaÞÞ1 þ oð1Þ:
6B m
Then, finding a critical point of ðL; aÞ/Ie ðL; aÞ reduces to finding a critical point of
a/I˜e ðaÞ; with
I˜e ðaÞ ¼ Ie ðLðaÞ; aÞ:
We deduce from Proposition 4.1 the C 1 -expansion
A
1
A A A
A
I˜e ðaÞ ¼ A þ e ln e þ C þ ln
e e lnjjm ðaÞj þ oðeÞ:
4
2
3 2 6B
4
Therefore, up to an additive and to a multiplicative constant, we have to look for
critical points in Od;g of
Ie ðaÞ ¼ lnjjm ðaÞj þ te ðaÞ
ð4:3Þ
with te ðaÞ ¼ oð1Þ; rte ðaÞ ¼ oð1Þ as e goes to zero, uniformly with respect to aAOd;g :
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491
Arguing by contradiction, we assume
ðHÞ Ie has no critical point aAOd;g such that bojm ðaÞoc:
We are going to use the gradient of Ie to build a continuous deformation of jcm onto
jbm ; a contradiction with the assumption H ðjcm ; jbm Þa0:
We first remark that jm has isolated critical values, since jm is analytic in O and
jm ¼ N on the boundary of O: Therefore, the assumption that c is not a critical
value of jm implies the existence of Z40 such that jm has no critical value in
bþZ
ðb; b þ Z,ðc Z; c: Moreover, jcm retracts by deformation onto jcZ
retracts
m ; jm
b
cZ
bþZ
by deformation onto jm ; and H ðjm ; jm Þa0:
Secondly, we choose d40 such that jm ðxÞob for dðx; @OÞpd: We choose also
g40 such that g4c: Then, a point x in the complementary of Od;g in O is either in
onto jbþZ
is equivalent to deforming
jbm ; or not in jcm : Consequently, deforming jcZ
m
m
cZ
bþZ
cZ
jm -Od;g onto jm : To this end we set, for a0 Aðjm -Od;g Þ
d
aðtÞ ¼ rIe ðaðtÞÞ; að0Þ ¼ a0 :
dt
aðtÞ is defined as long as the boundary of Od;g is not achieved. Ie ðaðtÞÞ being
decreasing, (4.3) shows that for e small enough, aðtÞ remains in jcm : Then, the
boundary of Od;g may only be achieved by aðtÞ in jbm : This means that aðtÞ is well
defined as long as bojm ðaðtÞÞoc; and according to assumption ðHÞ; Ie ðaðtÞÞ is
strictly decreasing in that region. Therefore (4.3) proves, for e small enough, the
existence of t40 such that jm ðaðtÞÞ ¼ b þ Z: Composing the flow with a retraction
cZ
of jcm onto jcZ
onto jbþZ
m ; we obtain a continuous deformation of jm
m ; a
cZ
bþZ
contradiction with H ðjm ; jm Þa0:
The previous arguments prove the existence, for e small enough, of a nontrivial
solution ue to the problem
Du þ mu ¼ u5þe
in O;
þ
@u
¼ 0 on @O:
@n
Then, the strong maximum principle shows that ue 40 in O: The fact that ue blows
up, as e goes to zero, at a point a such that bojm ðaÞoc; rjm ðaÞ ¼ 0; follows from
the construction of ue : In particular, rjm ðaÞ ¼ 0 is a straightforward consequence of
(4.3) as e goes to zero. This concludes the proof of the theorem.
Appendix A
A.1. Integral estimates
In this subsection, we collect the integral estimates which are needed
in the previous section. We recall that according to the definitions of Section 2,
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492
we have
0
1
m2 jxaj
1 B1 e
VL;a;m;e ðxÞ ¼ U 1 ;a ðxÞ ðLeÞ2 @
jx aj
Le
1
C
þ Hm ða; xÞA þ rL;a;m;e ðxÞ
ðA:1Þ
with
3
rL;a;m;e ¼ Oðjej2 Þ
ðA:2Þ
uniformly in O and with respect to a and L satisfying (2.4) (2.5), and the same
estimate holds for the derivatives of rL;a;m;e with respect to a and L: We recall also
that VL;a;m;e satisfies
0
1
8
1
>
2
>
ðLeÞ
> DV
5
@
A þ r0
>
in O;
<
L;a;m;e þ mVL;a;m;e ¼ 3U 1 þ m U 1 L;a;m;e
jx aj
Le;a
Le;a
ðA:3Þ
>
>
>
@V
>
: L;a;m;e ¼ 0
on @O
@n
with
r0L;a;m;e
!
1
e2
e2
2
4
1
7
ðLeÞ2
e
2e
2
jxaj2 þ Oðjej2 Þ
ð1 e jxaj Þ mðLeÞ2
¼m
þ
e
5
3
jx aj
jx aj
jx aj
ðA:4Þ
and such an expansion holds for the derivatives of r0L;a;m;e with respect to a and L:
Omitting, for sake of simplicity, the indices L; a; m; e; we state:
Proposition 5.1. Assuming that a and L satisfy (2.4) (2.5), we have the uniform
expansions as e goes to zero
A
1
A
3BL 1=2
e lnðjejLÞ þ C þ
ðm þ Hm ða; aÞÞjej þ Oðe2 ð ln jejÞ2 Þ;
eþ
4
2
6
2
Ae 3B 1=2
þ
ðm þ Hm ða; aÞÞjej þ Oðe2 ð ln jejÞ2 Þ;
¼
4L
2
3BL @
ðHm ða; aÞÞjej þ Oðe2 ð ln jejÞ2 Þ;
¼
2 @a
Je ðV Þ ¼ A þ
@Je
@L
@Je
@a
@ 2 Je
Ae
¼ 2 þ Oðe2 ð ln jejÞ2 Þ;
@L2
4L
@ 2 Je
3B @
ðHm ða; aÞÞjej þ Oðe2 ð ln jejÞ2 Þ
¼
2 @a
@L@a
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493
with
A¼
Z
R3
p2
;
4
6
U1;0
¼
B¼
Z
5
U1;0
¼
R3
4p
;
3
1
2
C¼
Z
6
U1;0
ln U1;0 40:
R3
Proof. For sake of simplicity, we assume that e40 (the computations are equivalent
as eo0), and we set r ¼ jx aj: In view of (A.3), we write
Z
ðjrV j2 þ mV 2 Þ ¼
Z
O
1
1
1
2
ðLeÞ
A þ r 0 AV :
ðDV þ mV ÞV ¼ @3U 5 þ m@U r
O
Z
O
0
0
ðA:5Þ
From (A.1), (A.2) we deduce
Z
U 5V ¼
O
Z
U6 0
Z
1
ðLeÞ2
O
O
B1 e
U 5@
r
1
m2 r
1
C
þ Hm ða; xÞA þ Oðe2 Þ
noticing that
Z
1
U 5 ¼ Oðe2 Þ:
ðA:6Þ
O
One one hand
Z
Z
U 6 ¼ A þ Oðe3 Þ with A ¼
U 6 ¼ 4p
R3
O
Z
N
0
r2 dr
ð1 þ
r2 Þ3
¼
p2
:
4
On the other hand, since dða; @OÞXd40
0
1
1
Z
2
m r
B1 e
C
þ Hm ða; xÞA
U 5@
r
O
¼
¼
1
Z
1
1
ðLeÞ2
ðOaÞ=ðLeÞ
Z
4p
1
ðLeÞ2
1 em2 Ler
dx þ
U
r
5
R=ðLeÞ
1e
1
Bða;RÞ
r dr þ Hm ða; aÞ
Z
Bða;RÞ
ð1 þ r2 Þ2
1
5
U 5 Hm ða; xÞ þ Oðe2 Þ
1
m2 Ler
5
0
Z
3
¼ 4pBðLeÞ2 ðm2 þ Hm ða; aÞÞ þ Oðe2 Þ
5
U þO
Z
Bða;RÞ
5 2
U r þ
5
e2
!
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494
with
B¼
Z
R3
5
U1;0
¼ 4p
Z
0
N
r2 dr
ð1 þ
5
r2 Þ2
¼
4p
:
3
Concerning the second term in the right-hand side of (A.5), denoting by R0 the
diameter of O and using (2.6), we have
1
0 0
1
1
1
Z
Z 2
2
ðLeÞ
ðleÞ
AU
AV ¼ O @ U m@U r
r O
O
0
0
1
1
Z R0 =ðLeÞ
2
1
1
r
dr
@ A
A
¼ O @e 2
1
1
r
0
2
2
ð1 þ r Þ2 ð1 þ r Þ2
¼ Oðe2 Þ:
Lastly, noticing that V ¼ OðUÞ uniformly in O and with respect to the parameters
a; L satisfying (2.4) and (2.5), we have, using (A.4)
0 0 1
1
1
2
2
Z
Z
4
e2
e
1
2
e
e
e
r0 V ¼ O@ @ ð1 er2 Þ þ e2 3 þ 5 er2 AU þ e4 A
r
r
r
O
O
0
1
Z R0 1
e
1
1
dr
þ
¼ O @e 2
rð1 er2 Þ þ
þ e4 A
1
r r2
0
2
2
ð1 þ r Þ
¼ Oðe2 Þ
whence finally
Z
ðjrV j2 þ mV 2 Þ ¼ 3A 3BLðm1=2 þ Hm ða; aÞÞe þ Oðe2 Þ:
ðA:7Þ
O
In the same way we have
Z
Vþ6 ¼ A 6BLðm1=2 þ Hm ða; aÞÞe þ Oðe2 Þ:
ðA:8Þ
O
Namely, from (A.1) (A.2) and V ¼ OðUÞ we derive
0
1
1
Z
Z
Z
Z
Z
m2 r
1
3
B1 e
C
þ Hm ða; xÞA þ O e2
Vþ6 ¼
U 6 6ðLeÞ2
U 5@
U5 þ e
U4
r
O
O
O
O
O
and the conclusion follows from the previous computations, noticing that
Z
U 4 ¼ OðeÞ:
O
ARTICLE IN PRESS
O. Rey, J. Wei / Journal of Functional Analysis 212 (2004) 472–499
Then, we write
Z
O
Vþ6þe ¼
Z
O
Vþ6 þ
Z
O
495
Vþ6 ðVþe 1Þ:
1
Noticing that 0pVþ p2e2
Vþe 1 ¼ e ln Vþ þ Oðe2 ðln eÞ2 Þ
1
and using the fact that Vþ ¼ U þ Oðe2 Þ we have
Vþ6 ¼ U 6 þ
1
Oðe2 U 5 Þ;
0 11
e2
ln Vþ ¼ ln U þ O@ A
U
1
(note that UXRe20 in O) whence
1
1
Vþ6 ln Vþ ¼ U 6 ln U þ Oðe2 U 5 þ e2 U 5 j ln UjÞ:
We find easily
Z
U 6 ln U ¼ O
and noticing that
Z
O
R
A
ln ðLeÞ C þ Oðe3 j ln ejÞ
2
1
U 5 jln Uj ¼ Oðe2 jln ejÞ; we obtain
Z
A
Vþ6þe ¼
Vþ6 e lnðjejLÞ Ce þ Oðe2 ðlnjejÞ2 Þ:
2
O
O
ðA:9Þ
The first expansion of Proposition 5.1 follows from (A.7)–(A.9) and definition (3.23)
of Je :
The expansions for the derivatives of Je are obtained exactly in the same way. &
A.2. Green’s function
We study the properties of Green’s function Gm ðx; yÞ and its regular part Hm ðx; yÞ:
We summarize their properties in the following proposition.
Proposition 5.2. Let Gm ðx; yÞ and Hm ðx; yÞ be defined in (1.1) and (1.2), respectively.
Then we have
Hm ðx; xÞ- N as dðx; @OÞ-0;
jGm ðx; yÞjp
1
C
;
jx yj
m2 þ max Hm ðx; xÞ- N;
xAO
ðA:10Þ
ðA:11Þ
as m-0;
ðA:12Þ
ARTICLE IN PRESS
O. Rey, J. Wei / Journal of Functional Analysis 212 (2004) 472–499
496
1
m2 þ max Hm ðx; xÞ- þ N;
xAO
as m- þ N:
ðA:13Þ
Proof. Eq. (A.10) follows from standard argument. Let xAO be such that d ¼
dðx; @OÞ is small. So there exists a unique point xA@O
such that d ¼ jx xj:
%
%
Without loss of generality, we may assume x% ¼ 0 and the outer normal at x% is
pointing toward xN -direction. Let x be the reflection point x ¼ ð0; y; 0; dÞ and
consider the following auxiliary function:
1
em2 jyx j
H ðy; xÞ ¼
:
jy x j
Then H satisfies Dy H mH ¼ 0 in O and on @O
0
1
m2 jyxj
1
@
@ Be
C
ðH ðy; xÞÞ ¼ @
A þ Oð1Þ:
@n
@n jy xj
Hence we derive that
Hðy; xÞ ¼ H ðy; xÞ þ Oð1Þ
ðA:14Þ
1
þ Oð1Þ
dðx; @OÞ
ðA:15Þ
which implies that
Hðx; xÞ ¼ hence (A.10).
From (A.14), we see that as dðx; @OÞ-0; we have
1
em2 jyxj
C
Gm ðy; xÞ ¼
:
þ H ðy; xÞ þ Oð1Þp
jx yj
jy xj
ðA:16Þ
On the other hand, if dðx; @OÞ4d0 40; then jHm ðy; xÞjpC and (A.11) also holds.
We now prove (A.12). For m small, we can decompose Hm as follows:
Hm ðx; yÞ ¼ c þ H0 ðx; yÞ þ Ĥðx; yÞ;
where
c¼
1
jOj
Z
Hm ðx; yÞ ¼
O
1
mjOj
0
Z
@O
1
m2 jyxj
ðA:17Þ
1
@ Be
4p
C
þ Oð1Þ
@
A¼
@n jy xj
mjOj
ðA:18Þ
ARTICLE IN PRESS
O. Rey, J. Wei / Journal of Functional Analysis 212 (2004) 472–499
and H0 satisfies
4p
;
DH0 ¼
jOj
Z
H0 ¼ 0;
O
@
@
1
H0 ¼
@n
@n jy xj
497
on @O
and Ĥ is the remainder term. By simple computations, Ĥ satisfies
Z
@
Ĥ ¼ Oð1Þ on @O
Ĥ ¼ 0;
DĤ mĤ þ OðmH0 ðx; yÞÞ þ Oð1Þ ¼ 0 in O;
@n
O
which shows that Ĥ ¼ Oð1Þ: Thus
1
m2 þ max Hm ðx; xÞp xAO
4p
þ Oð1Þ- N
mjOj
as m-0: (A.12) is thus proved.
To prove (A.13), we choose a point x0 AO such that dðx0 ; @OÞ ¼ maxxAO dðx; @OÞ:
Then, since
1
m2
1
@ em2 jx0 xj
@nð jx0 xj Þ
þ max
xAO
¼
1
m2
2 dðx0 ;@OÞ
Oðe
Þ
1
Hm ðx; xÞXm2
þ
as m- þ N; which proves (A.13).
on @O; for m large enough we see that
1
Hðx0 ; x0 ÞXm2
þ
1
m2
2 dðx0 ;@OÞ
Oðe
Þ-
þN
&
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