Hiroshima Math. J.
33 (2003), 87–111
Asymptotic behaviors of least-energy solutions to
Matukuma type equations with an inverse square potential
Dedicated to Professors Mitsuru Ikawa and Sadao Miyatake on the occasion
of their sixtieth birthdays
Yoshitsugu Kabeya*yz
(Received November 30, 2001)
(Revised July 9, 2002)
Abstract. Behavior of least-energy solutions to Matukuma type equations with an
inverse square potential are discussed. The di¤erence of the behavior of solutions are
obtained. We also consider the behavior of scaled solutions and obtain a limiting
function.
1.
Introduction
This is a note on the behavior of least-energy solutions to
ð1:1Þ
Du þ KðxÞu 1þe ¼ 0
in R n
as e # 0 under the condition
8
>
KðxÞ A C 1 ðR n Þ; KðxÞ > 0
>
<
x ‘KðxÞ þ 2KðxÞ b 0;D 0
ðKÞ
2
>
>
: lim jxj KðxÞ ¼ c0 > 0:
in R n ;
on R n ;
jxj!y
In some cases, we assume further that KðxÞ satisfies
ðK:1Þ
KðxÞ ¼ jxj2 ðc0 þ c1 jxj1 þ k1 ðxÞÞ
on jxj b R
with R > 0, c1 A R, where k1 satisfies k1 ðjxjÞ ¼ Oðjxjm Þ and x ‘k1 ðxÞ ¼
Oðjxjm Þ for some m > 1.
A typical example of KðxÞ which satisfies (K) with c0 ¼ 1 is that
* Supported in part by the Grant-in-Aid for Encouragement of Young Scientists (No. 13740116),
Japan Society for the Promotion of Science.
y
2000 Mathematics Subject Classification. Primary 35J60, Secondary 35J20, 35B40.
z
Key words and phrases. Semilinear elliptic equations, least-energy solutions, the Hardy inequality, Asymptotic behavior.
88
Yoshitsugu Kabeya
KðxÞ ¼
1
1 þ jxj 2
;
which is exactly the same KðxÞ appearing in the original Matukuma equation. Moreover, this KðxÞ also satisfies (K.1) with c1 ¼ 0, since KðxÞ ¼
jxj2 ð1 1=fð1 þ jxj 2 Þjxj 2 g.
By the terminology ‘‘least-energy’’ solution, we mean a positive solution to
(1.1) determined by the minimization problem
Ð
2
R n j‘uj dx
Se :¼ inf
ð1:2Þ
;
Ð
u A D; uD0 ð n KðxÞu 2þe dxÞ 2=ð2þeÞ
R
where D is the space which is the completion of C0y ðR n Þ with respect to the
norm k‘ k2 . Then a function ue which attains (1.2) is a solution to
ð1:3Þ
Se
Due þ Ð
KðxÞue1þe ¼ 0:
2þe e=ð2þeÞ
ð R n KðxÞjue j Þ
Thus, by setting
1=e
Se
ue ;
u~e :¼ Ð
ð R n KðxÞjue j 2þe dxÞ 1=ð2þeÞ
u~e satisfies (1.1) and
ð1:4Þ
ð
R
n
j‘~
ue j 2 dx ¼ Seð2þeÞ=e :
Related to (1.2), we introduce the value
Ð
2
R n j‘uj dx
SðlÞ :¼ inf
ð1:5Þ
Ð
u A D; uD0 ð n jxjl juj 2ðnlÞ=ðn2Þ dxÞðn2Þ=ðnlÞ
R
with 0 < l a 2. As is known by Egnell [2] or Horiuchi [5, 6], SðlÞ is the best
constant of the embedding
ð
p
l
p
D ,! Ll :¼ u j
jxj juj dx < y
Rn
with p ¼ 2ðn lÞ=ðn 2Þ. The extremal function is
ðn2Þ=ð2lÞ
1
UðxÞ ¼ 1 þ
jxj 2l
ð1:6Þ
:
ðn 2Þðn lÞ
However, the pointwise limit of UðxÞ as l " 2 is
Asymptotic behaviors of least-energy solutions
lim UðxÞ ¼
l"2
1;
0;
89
x ¼ 0;
x 0 0;
which is never a minimizer for Sð2Þ. Note that Sð2Þ ¼ ðn 2Þ 2 =4 in case of
l ¼ 2 in view of the Hardy inequality
ð
ð
ðn 2Þ 2
u2
dx
a
j‘uj 2 dx:
2
n
n
4
jxj
R
R
We also note that there exists no minimizer for Sð2Þ (see, e.g., [2]).
In this paper, we investigate the behavior of Se and solutions ue as well as
their scaled properties as e # 0.
Theorem 1.1.
Under (K), the behavior of Se is as follows:
lim Se ¼
e#0
ðn 2Þ 2
:
4c0
Similar to Kabeya [7] for slowly decaying KðxÞ (KðxÞ @ jxjl with
0 < l < 2), the behavior of the norm of a least-energy solution is obtained. In
view of the Pohozaev identity (see Lemma 2.2 of [7] or Proposition 1 of Naito
[12]) yields
ð n2
n
x ‘KðxÞ
ð1:7Þ
KðxÞjue j 2þe dx ¼ 0:
n
2
2
þ
e
ð2
þ
eÞKðxÞ
R
Ð
Under (K), if R n KðxÞu 2 dx < y (especially u A D), then u 1 0 for e ¼ 0.
However, depending on c0 , any least-energy solution blows up in this case.
This is di¤erent from the case where KðxÞ is slowly decaying as studied in [7].
One explanation is that the limiting problem for the slowly decaying case is
still a nonlinear one, while this one is a linear one. The limiting problem
(linear problem) in this case does not admit any scalings which erase c0 . Thus
the dependence on c0 arises.
The case for the faster decaying KðxÞ will be discussed in Kabeya and
Yanagida [8].
For radial solutions, the blowup or vanishing behaviors are obtained in
Theorem 2.5 of Yanagida and Yotsutani [15]. We also see more precise
behaviors of solutions than those obtained in [15].
Theorem 1.2. Under (K), the norms k‘ue k2 and kue ky of any least-energy
solution ue to (1.1) blow up if 0 < c0 < ðn 2Þ 2 =4 and vanish if c0 > ðn 2Þ 2 =4.
Moreover, in either case, ue satisfies
lim k‘ue k2e ¼
e#0
ðn 2Þ 2
:
4c0
90
Yoshitsugu Kabeya
Note that the blowup and vanishing are determined by the limit of Se in
Theorem 1.1.
In the ‘‘critical case’’ c0 ¼ ðn 2Þ 2 =4, we need a careful calculation and we
impose further assumptions on KðxÞ.
Theorem 1.3. In the case where c0 ¼ ðn 2Þ 2 =4, suppose that (K) and
(K.1) hold. Then the norms k‘ue k2 and kue ky of any least-energy solution blow
up.
Using Theorems 1.2 and 1.3, by a scaling, we see a limiting behavior of a leastenergy solution. Unfortunately, the scaling is only valid for any domain that
are the exterior of a ball centered at the origin.
Theorem 1.4. Suppose that (K) and (K.1) hold.
solution ue ðxÞ and for any R > 0, let
ð1:8Þ
ve ðyÞ :¼
ue ðxÞ
max ue ðxÞ
with x ¼
jxjbR=e
For any least-energy
y
:
e
Then there exists a subsequence fej g such that the maximum point Qej of vej
converges to Q and vej ðyÞ converges locally uniformly to V ð yÞ on f y A R n j
R a jyj a R 0 g, where V is a positive solution to
8
c~
>
>
< Dy V þ 2 V ¼ 0;
jyj
ð1:9Þ
>
>
: V ðQ Þ ¼ 1; lim V ðyÞ ¼ 0;
jyj!y
with any R 0 > R and some 0 < c~ a ðn 2Þ 2 =4 and Dy being the Laplacian with
respect to y.
Remark 1.1. In Theorem 1.2, we see that k‘ue k2 blows up or vanishes as
e # 0. By the scaling (1.8), despite the di¤erence of the behavior of k‘ue k2 , we
can extract a special limiting function, to which the scaled solution converges.
If we scale
ve ð yÞ :¼
ue ðxÞ
;
kue ky
then we only have the limiting function
1;
V ð yÞ ¼
0;
x¼
y
;
e
y ¼ 0;
y 0 0:
Thus we have used the scaling as in Theorem 1.4 to derive a useful information.
Asymptotic behaviors of least-energy solutions
91
If KðxÞ is radial and Kr a 0, then ue must be radial by Gidas, Ni and
Nirenberg [3]. In this case, we can take jQ j ¼ R without extracting a subsequence and the limiting solution is a radial one since the local uniform limit
of a radial function is also radial. However, we do not know whether positive
solutions of (1.9) are necessarily radially symmetric or not.
In Section 2, we give a proof of Theorem 1.1. Fundamental Lemmas
for proofs of Theorems 1.2, 1.3 and 1.4 are given in Section 3. Proofs of
Theorems 1.2, 1.3 and 1.4 are given in Section 4. In Section 5, we give a
proof of Lemma 2.1 for the sake of the reader’s convenience as an appendix.
2.
Proof of Theorem 1.1
First we note that SðlÞ is expressed in terms of the gamma functions and
the exact value is obtained in Lemma 3.1 of Horiuchi [6] and Theorem 1.1 of
Catrina and Wang [1]. The continuity of SðlÞ at l ¼ 2 is shown also in [1].
We summarize their results as below. We remark that they studied wider class
of the weighted Sobolev type embeddings.
Lemma 2.1 (Horiuchi [6], Catrina and Wang [1]). The explicit form of
SðlÞ is given by
on ð2lÞ=ðnlÞ
n 2l þ 2
n2
G
G
2l
2l
2l
SðlÞ ¼ ðn 2Þ 2 2ðn2Þ=ðnlÞ ð2lÞ=ðnlÞ
nl
2ðn lÞ
G
G
2l
2l
for l < 2 and SðlÞ ! ðn 2Þ 2 =4 ¼ Sð2Þ as l " 2, where on is the surface area of
the unit sphere in R n .
For the sake of self-containedness, we will give a proof in the Appendix.
The following is the estimate of the supremum norm, which will be useful
for the uniform estimate. The estimate is essentially due to Lemma B.3 of
Struwe [13] (see also Lemma 7 of Han [4]).
Lemma 2.2. For any classical solution ue A D to (1.1), there exists a constant C^e ¼ C^ ðk‘ue k2e Þ > 0 such that kue ky a C^e k‘ue k2 .
Proof. We regard (1.1) as
Due þ ðKðxÞuee Þue ¼ 0:
Since KðxÞ A Ly ðR n Þ and since e > 0 is small, we see that KðxÞuee A L a ðBðQe ; 1ÞÞ
with a > n=2. Then as in Lemma B.3 (pp. 244–245) of [13] or as in Lemma 7
of [4], we have
92
Yoshitsugu Kabeya
max ue a CðkKðxÞuee ka ; a; nÞk‘ue k2 ;
ð2:1Þ
BðQe ; 1=2Þ
where C depends on L a -norm of KðxÞuee , a and n. For L a -norm of KðxÞuee , by
the Hölder and Sobolev inequalities, for su‰ciently small e > 0, we have
ð
ðKðxÞuee Þ a dx
BðQe ; 1Þ
a
a kKky
ð
BðQe ; 1Þ
ueea dx
!ðn2Þea=ð2nÞ
ð
a
a kKky
jBðQe ; 1Þjf2nðn2Þeag=ð2nÞ
BðQe ; 1Þ
ue2n=ðn2Þ
dx
a
a CkKky
jBðQe ; 1Þj k‘ue k2ea
where C is a constant independent of e. Since ue A D by assumption, we have
the desired estimate (The dependence of C^e on e comes from the e-dependence
of k‘ue k2 and note that a and n are independent of e).
r
Using Lemma 2.2, we prove Theorem 1.1.
Proof of Theorem 1.1. Before proving the equality, we easily see that Se
is uniformly bounded. Then as in the proof of Lemma 2.4 in Kabeya [7], we
prove
lim inf Se b
e#0
1
Sð2Þ
c0
and
lim sup Se a
e#0
1
Sð2Þ:
c0
1
First we Ðprove lim inf e#0 Se b ðc0 Þ Sð2Þ. Let ue ðxÞ be a function which attains
Se with R n KðxÞue2þe dx ¼ 1 and let ve ð yÞ ¼ ue ðxÞ with x ¼ y=e. Then, ue is a
solution to
Due þ Se KðxÞue1þe ¼ 0
and we have
ð
R
n
j‘x ue j dx ¼
ð
1
2
e n2
R
n
j‘y ve j 2 dy ¼ Se
and
ð
Rn
KðxÞue2þe dx ¼
1
e n2
1
y 2þe
K
dy ¼ 1;
v
n e2
e e
R
ð
where ‘x , ‘y denote the gradient with respect to x and y, respectively.
we get
Hence
Asymptotic behaviors of least-energy solutions
93
Ð
2
R n j‘x ue j dx
Se ¼ Ð
ð R n KðxÞue2þe dxÞ 2=ð2þeÞ
Ð
eðn2Þ R n j‘y ve j 2 dy
¼
2=ð2þeÞ
Ð 1
y 2þe
ve dy
e2ðn2Þ=ð2þeÞ R n 2 K
e
e
Ð
2
R n j‘y ve j dy
b
2=ð2þeÞ 2e=ð2þeÞ :
Ð 1
y 2
ðn2Þe=ð2þeÞ
e
max
ve
v dy
R n 2K
e
e e
Rn
ð2:2Þ
1=2
By Lemma 2.2 and since k‘ue k2 ¼ Se is uniformly bounded, we see that the
right-hand side of (2.1) for KðxÞ replaced by Se KðxÞ is uniformly bounded.
Hence, in view of kue ky ¼ kve ky , we have lim supe#0 ðmaxR n ve Þ 2e=ð2þeÞ a 1.
Thus, taking a limit infimum, we obtain
Ð
2
1
ðn 2Þ 2
n j‘y ve j dy
lim inf Se b lim inf ÐR
b
Sð2Þ
¼
e#0
e#0
c0
4c0
c0 n j yj2 ve2 dy
R
in view of
1
y
lim 2 K
¼ c0 j yj2 ;
e#0 e
e
where the convergence is locally uniformly in R n nf0g by (K), and the Hardy
inequality.
To prove lim supe#0 Se a ð1=c0 ÞSð2Þ, we set wðxÞ ¼ jxjðn2Þ=2 je ðjxjÞ.
Here, je ðb0Þ A C0y ðð0; yÞÞ satisfies supp je H ½1; 2e1 , max½0; yÞ j ¼ 1,
supp j 0 H ½1; 2 U ½e1 ; 2e1 , je ðxÞ 1 f ðxÞ on ½1; 2 such that f A C y ð0; yÞ, fulfills f ð1Þ ¼ 0, f ðxÞ > 0 in ð1; 2 and f ð2Þ ¼ 1, and je ðxÞ 1 gðexÞ on ½e1 ; 2e1 ,
where gðxÞ A C y ð0; yÞ satisfies gð1Þ ¼ 1, gðxÞ > 0 in ½1; 2Þ and gð2Þ ¼ 0.
Since w 0 ¼ fðn 2Þ=2grn=2 je þ rðn2Þ=2 je0 , we have
ð
j‘wj 2 dx
Rn
)
ð y (
n 2 2 n 2
2
¼ on
r je ðn 2Þrðn1Þ je je0 þ rðn2Þ ðje0 Þ r n1 dr
2
0
ðy
ðy
ðy
n2 2
on
r1 je2 dr ðn 2Þon
je je0 dr þ on
rðje0 Þ 2 dr
2
0
0
0
ðy
ðy
ð 2 ð 2e1 !
n2 2
1 2
0
¼
on
r je dr ðn 2Þon
je je dr þ on
þ
rðje0 Þ 2 dr:
2
e1
0
0
1
¼
94
Yoshitsugu Kabeya
The second and third terms yield
ðy
0
je je0
1 2 y
dr ¼ je
¼0
2
0
and
ð2
þ
e1
1
respectively.
ð2:3Þ
ð 2e1 !
rðje0 Þ 2 dr a 6 þ 4e 2
ð 2e1
r dr ¼ 12;
e1
Thus we get
n 2 2 Ð 2e1 1 2
on 1 r je dr þ 12on
2
Se a
:
Ð
ð R n KðxÞw 2þe dxÞ 2=ð2þeÞ
As for the denominator, for any h > 0, take R > 0 so large that
jxj 2 KðxÞ b c0 h
holds for any x A R n nBR .
ð
KðxÞw 2þe dx
Rn
¼
ð
B2e1 nB1
KðxÞjxjðn2Þð2þeÞ=2 je2þe dx
ð
b
B2e1 nBR
b
Then we have
jxj
2
KðxÞjxjnðn2Þe=2 je2þe
dx þ
ð
KðxÞjxjðn2Þð2þeÞ=2 dx
BR nB2
ðn2Þe=2
ð 2e1
ð
2
ðc0 hÞon
r1 je2þe dr þ
KðxÞjxjðn2Þð2þeÞ=2 dx
e
R
BR nB2
for any e1 > R. The second term is uniformly bounded. Since jee ! 1
locally uniformly in ð1; yÞ by the definition of je , we see that
Ð 2e1
r1 je2 dr
¼ 1:
lim Ð 2e1 1
e#0 ð
r1 je2þe drÞ 2=ð2þeÞ
R
Thus taking a limit supremum of (2.3) as e # 0, we have
1
n2 2
lim sup Se a
:
c0 h
2
e#0
Asymptotic behaviors of least-energy solutions
95
Since Sð2Þ ¼ ðn 2Þ 2 =4 and since h > 0 is arbitrary, we obtain lime#0 Se ¼
ð1=c0 ÞSð2Þ.
r
3.
Fundamental properties of solutions
To prove Theorems 1.3 and 1.4, we need to estimate the location of
the maximum point of ue ðxÞ and kuky from above. First we need a
uniform a priori estimate for ue satisfying (1.4), almost identical to Lemma
2.2.
Lemma 3.1. For any least-energy solution ue to (1.1), there exists a cone
stant C~ > 0 independent of e such that kue ky
a C~ .
Proof. The proof is almost identical to that of Lemma 2.2.
of Lemma 2.2, we just note here that
ð
a
ðKðxÞuee Þ a dx a CkKky
jBðQe ; 1Þj k‘ue k2ea
In the proof
BðQe ; 1Þ
a
a 2CkKky
jBðQe ; 1Þj
!a
ðn 2Þ 2
;
4c0
with a > n=2 by Theorem 1.1, where C is a constant independent of e. Thus,
the L a norm kKðxÞuee kL a ðBðQe ; 1=2ÞÞ is bounded independent of e. Hence, the
constant in (2.1) is independent of e. By Theorem 1.1 and (1.4), k‘ue k2e is
uniformly bounded. Thus, we have the desired estimate.
r
Next, we show a decay property of ue .
Lemma 3.2.
Under (K), there exists R1 > 0 independent of e such that
ue ðxÞ ¼ C~e jxjðn2Þ þ he ðxÞ
holds on jxj b R1 with C~e > 0 and a higher order term he ðxÞ ¼ Oðjxjðn2Þð1þeÞ Þ
at infinity.
Proof. As in Lemma 3.5 of Kabeya [7] (if KðxÞ is radial, the decay order
is obtained by Li and Ni [9, 11]), we deduce the decay order of ue . Using the
Green function of D in R n , we have
ð
1
Kð yÞue ð yÞ 1þe
ue ðxÞ ¼
ð3:1Þ
dy:
ðn 2Þon R n jx yj n2
By the Hölder inequality, we have
96
Yoshitsugu Kabeya
ð
Kð yÞue ð yÞ 1þe
Rn
jx yj n2
ð
Kð yÞue ð yÞ 1þe
dy a
jx yj n2
jxyja1
þ
dy
8
<ð
KðyÞ
:
jx yj n2
jxyjb1
!2n=ðnþ2ðn2ÞeÞ
dy
9ðnþ2ðn2ÞeÞ=ð2nÞ
=
;
!ðn2Þð1þeÞ=ð2nÞ
ð
jxyjb1
ue2n=ðn2Þ
dy
:
Since KðyÞ @ jyj2 at jyj ¼ y, kue ky is finite and since ue A D, the right-hand
side is finite.
From now on, R 0 > 0 is supposed to be large so that
ð3:2Þ
c0
a jxj 2 KðxÞ a 2c0
2
on jxj b R 0 , and we take x so that jxj b R1 :¼ maxf2R 0 ; R 0 þ 1g, and C
denotes the generic constant independent of e. x may be taken even larger if
necessary.
For jxj b R1 , first we note that
maxjxyja1 KðyÞ
aC
KðxÞ
Indeed, jx yj a 1 implies jxj 1 a jyj a
with C > 0 independent of x.
jxj þ 1. Thus we have
Kð yÞ a
in view of (3.2) for jxj b R1 .
2c0
j yj
2
a
2c0
ðjxj 1Þ 2
Again from (3.2), there holds
KðxÞ b
c0
2jxj 2
and we get
maxjxyja1 KðyÞ
4jxj 2
a
KðxÞ
ðjxj 1Þ 2
for jxj b R1 . Note that the right-hand side is uniformly bounded for jxj b R1 .
Thus we see that
ð
ð
Kð yÞue ðyÞ 1þe
1
dy
a
Cku
k
KðxÞ
dy a Ck‘ue k2 jxj2
e
y
n2
n2
jxyja1 jx yj
jxyja1 jx yj
Asymptotic behaviors of least-energy solutions
97
e
holds by (2.1) and Lemma 3.1 (kue ky
is uniformly bounded).
decompose
!2n=ðnþ2ðn2ÞeÞ
ð
KðyÞ
dy ¼ I1 þ I2 þ I3 þ I4
n2
jxyjb1 jx yj
Next, we
with
I1 ¼
ð
ð
KðyÞ
I4 ¼
ð
ð
dy;
jx yj n2
!2n=ðnþ2ðn2ÞeÞ
KðyÞ
jxyjb2jxj
jyjb2jxj
dy;
!2n=ðnþ2ðn2ÞeÞ
Kð yÞ
jxyjb2jxj
2jxjbjyjbjxj
!2n=ðnþ2ðn2ÞeÞ
jx yj n2
jxj=2ajxyja2jxj
I3 ¼
dy;
jx yj n2
1ajxyjajxj=2
I2 ¼
!2n=ðnþ2ðn2ÞeÞ
Kð yÞ
dy:
jx yj n2
On 1 a jx yj a jxj=2, we see j yj b jxj=2 b R 0 and get
I1 a Cjxj4n=fnþ2ðn2Þeg
ð jxj=2
r2nðn2Þ=ðnþ2ðn2ÞeÞþn1 dr
1
4n=ðnþ2ðn2ÞeÞ
a Cðjxj
þ jxjnðn2Þð1þeÞ=ðnþ2ðn2ÞeÞ Þ:
For I2 , we have
I2 a Cjxj
2nðn2Þ=ðnþ2ðn2ÞeÞ
ð
þ
jyjaR1
ð
!
KðyÞ 2n=ðnþ2ðn2ÞeÞ dy
R1 ajyja3jxj
a Cðjxj2nðn2Þ=ðnþ2ðn2ÞeÞ þ jxjnðn2Þð1þeÞ=ðnþ2ðn2ÞeÞ Þ;
since jx yj a 2jxj implies j yj a 3jxj.
Similarly, for I3 , we get
ð
I3 a Cjxj2nðn2Þ=ðnþ2ðn2ÞeÞ jxj4n=ðnþ2ðn2ÞeÞ
dy
jxjajyja2jxj
a Cjxjnðn2Þð1þeÞ=ðnþ2ðn2ÞeÞ :
Finally, for I4 , we note that jx yj b 2jxj with j yj b 2jxj implies jx yj b jyj=2
(indeed, jx yj b jyj jxj b jyj jyj=2). Thus we have
98
Yoshitsugu Kabeya
ð
I4 a C
jxyjb2jxj
jyjb2jxj
Hence we obtain
0
ð
@
j yj2n
2
=ðnþ2ðn2ÞeÞ
KðyÞ
jxyjb1
dy a Cjxjnðn2Þð1þeÞ=ðnþ2ðn2ÞeÞ :
1ðnþ2ðn2ÞeÞ=ð2nÞ
!2n=ðnþ2ðn2ÞeÞ
dyA
jx yj n2
¼ ðI1 þ I2 þ I3 þ I4 Þðnþ2ðn2ÞeÞ=ð2nÞ
a Cðjxj2 þ jxjðn2Þð1þeÞ=2 þ jxjðn2Þ Þ:
Thus we have
ð3:3Þ
ue ðxÞ a Ck‘ue k2 jxj2 þ Ck‘ue k2 jxjminf2; ðn2Þð1þeÞ=2g
¼ Ce jxjminf2; ðn2Þð1þeÞ=2g
for jxj b R1 , with Ce ¼ Ck‘ue k2 , again by (2.1) and Lemma 3.1 for the estimate of the second term.
By (3.3), we can estimate the right-hand side of (3.1) directly. We again
decompose
ð
KðyÞue ð yÞ 1þe
dy ¼ J1 þ J2 þ J3 þ J4 þ J5
n2
R n jx yj
with
J1 ¼
ð
KðyÞue ð yÞ 1þe
jxyja1
J2 ¼
jx yj n2
ð
KðyÞue ð yÞ 1þe
jx yj n2
1ajxyjajxj=2
J3 ¼
ð
J5 ¼
dy;
Kð yÞue ðyÞ 1þe
jxj=2ajxyja2jxj
J4 ¼
dy;
ð
jx yj n2
Kð yÞue ð yÞ 1þe
jxyjb2jxj
2jxjbjyjbjxj
ð
jx yj n2
Kð yÞue ðyÞ 1þe
jxyjb2jxj
jyjb2jxj
jx yj n2
dy;
dy;
dy:
Then, denoting the generic constant (may dependent on e) by Ce , we have, via
the step similar to the previous one,
Asymptotic behaviors of least-energy solutions
J1 a Ce jxj2minf2; ðn2Þð1þeÞ=2gð1þeÞ
ð
1
n2
jxyja1 jx yj
99
dy
a Ce jxj2minf2; ðn2Þð1þeÞ=2gð1þeÞ ;
J2 a Ce jxj2minf2; ðn2Þð1þeÞ=2gð1þeÞ
ð jxj=2
r dr
1
a Ce jxjminf2; ðn2Þð1þeÞ=2gð1þeÞ ;
J3 a Ce jxj
ðn2Þ
ð
þ
jyjaR1
!
ð
Kð yÞue ð yÞ 1þe dy
R1 ajyja3jxj
a Ce ðjxjðn2Þ þ jxjminf2; ðn2Þð1þeÞ=2gð1þeÞ Þ;
ð
ðn2Þ
2minf2; ðn2Þð1þeÞ=2gð1þeÞ
jxj
J4 a Ce jxj
dy
2jxjbjyjbjxj
a Ce jxjminf2; ðn2Þð1þeÞ=2gð1þeÞ ;
ðy
J5 a C e
jyjminf2; ðn2Þð1þeÞ=2gð1þeÞ1 dy
jxj
a Ce jxjminf2; ðn2Þð1þeÞ=2gð1þeÞ :
Thus we obtain
ue ðxÞ a Ce jxjminf2; ðn2Þð1þeÞ=2gð1þeÞ
for jxj b R1 if minf2; ðn 2Þð1 þ eÞ=2gð1 þ eÞ a n 2. If minf2; ðn 2Þ ð1 þ eÞ=2gð1 þ eÞ > n 2, then we are done. For fixed e > 0, repeating this
process l times so that minf2; ðn 2Þð1 þ eÞ=2gð1 þ eÞl > n 2, we have
ue ðxÞ a Ce jxjðn2Þ
ð3:4Þ
for jxj b R1 . We should note here that the decay rate jxjðn2Þ is never
improved in view of the estimate in J3 .
Then as in Theorem 2.4 of Li and Ni [10] (the estimate of the second order
of the expansion), we obtain
ue ðxÞ ¼ C~e jxjðn2Þ þ he ðxÞ:
ð3:5Þ
with
ð3:6Þ
C~e :¼ lim jxj n2 ue ðxÞ ¼
jxj!y
ðn2Þð1þeÞ
1
ðn 2Þon
ð
R
n
KðxÞue1þe dx;
Þ being a higher order term. Since (3.4) holds for
and he ðxÞ ¼ Oðjxj
jxj b R1 with R1 independent of e, (3.5) holds also for jxj b R1 .
r
100
Yoshitsugu Kabeya
Remark 3.1. The reason why we have obtained the exact decay rate is
that the dominant term in the estimate of J3 is jxjðn2Þ and that the iteration is
no longer e¤ective to gain the decay rate due to this term.
Ce may go to infinity as e # 0 because k‘ue k2 may go to infinity and the
iteration needs more times (unbounded) as e # 0.
Remark 3.2. Under (K) and (K.1), according to Theorem 2.16 of Li and
Ni [10], ue is expanded as follows:
ð3:7Þ
ue ðxÞ ¼
1
2k
e þ1
X
jxj n2
m¼1
Ce
þ
n2
C1; m; e
þ
ðn2Þme
1
ke
X
C2; m; e
jxj n1 m¼1 jxjðn2Þme
!
!
ke
X
ae x
C3; m; e
x
1
1þ
þ
þ Re
n
2
ðn2Þme
jxj
jxj jxj n1
m¼1 jxj
jxj
jxj
for jxj b R1 , where ke is an integer such that ðn 2Þke e a 1 < ðn 2Þðke þ 1Þe,
C1; m; e , C2; m; e , C3; m; e are constants (not necessarily positive), ae A R n is a
constant vector, and Re ðtÞ is a Lipschitz continuous function near t ¼ 0 with
Re ð0Þ ¼ 0. Note that R1 can be taken larger than R . The assumption (K.1)
is needed to have the exact expansion as above. Without (K.1), it is hard to
obtain the expansion (3.7).
Carefully following the proof of Theorem 2.16 of [10], we can obtain the
constants C1; m; e .
Lemma 3.3.
The constant C1; m; e in (3.7) satisfies
ð3:8Þ
C1; m; e ¼
ð1þeÞ m m
c0
Qm
2m m
2Þ e m! l¼1 ð1
ð1Þ m Ce
ðn þ leÞ
for m ¼ 1; 2; . . . ; ke .
Proof. A proof is done by following the proof of Theorem 2.16 of [10].
So we give a sketchy proof. The essential part is to express (1.1) as
!1þe
Ce1þe
jxj n2 ue
2
1
Due þ jxj ðc0 þ c1 jxj þ k1 ðxÞÞ ðn2Þð1þeÞ
ð3:9Þ
¼0
Ce
jxj
on jxj b R1 and expand Ce1 jxj n2 ue step by step.
In what follows, fi and ui; e represent the remainder terms.
that the equation (1.1) is expressed as
First, we note
Due þ Ce1þe c0 jxjðn2Þð1þeÞ þ f1 ðue Þ ¼ 0
on jxj b R1 .
Note that the original proof is done via the Kelvin transforma-
Asymptotic behaviors of least-energy solutions
101
tion. However, we can prove this lemma directly since we know the decay
order. Then, as in (2.22) of [10] (p. 203), we have
ð1þeÞ
ue ¼ Ce jxjðn2Þ Ce
c0
2
ðn 2Þ ð1 þ eÞe
jxjðn2Þð1þeÞ þ u1; e :
The expression is seemingly a contradictory one since the coe‰cient of the
second term is apparently larger than that of the first term. However, the
second term is eventually almost cancelled out by u1; e since ue is always
positive.
Next, using the expansion, we see that ue satisfies
ð3:10Þ Due þ
Ce1þe c0 jxjðn2Þð1þeÞ
ð1þeÞ 2 2
c0
2
Ce
ðn 2Þ ð1 þ eÞe
jxjðn2Þð1þ2eÞ þ f2 ðue Þ ¼ 0:
Then we have via the method of the deduction of (2.22) in [10],
ð1þeÞ
ue ¼ Ce jxjðn2Þ þ
Ce
c0
2
ðn 2Þ ð1 þ eÞe
ð1þeÞ 2 2
c0
Ce
2ðn 2Þ 4 ð1 þ eÞð1 þ 2eÞe 2
jxjðn2Þðn2Þe
jxjðn2Þð1þ2eÞ þ u2; e :
Indeed, three terms from the top satisfy (3.10) with f2 1 0 (calculate Dðue u2; e Þ
as a radial function). Moreover, the ‘‘uniqueness’’ of the top three terms are
verified as in the proof of Theorem 2.16 of [10] (p. 203). Thus C1; 2; e is
obtained. To obtain C1; 3; e , we again repeat the argument. Thus ue satisfies
Due þ
Ce1þe c0 jxjðn2Þð1þeÞ
þ
ð1þeÞ 2 2
c0
2
Ce
ðn 2Þ ð1 þ eÞe
ð1þeÞ 3 3
c0
Ce
2ðn 2Þ 4 ð1 þ eÞð1 þ 2eÞe 2
jxjðn2Þð1þ2eÞ
jxjðn2Þð1þ3eÞ þ f3 ðue Þ ¼ 0:
Then we have
C1; 3; e ¼ ð1þeÞ 3 3
c0
Ce
6ðn 2Þ 6 ð1 þ eÞð1 þ 2eÞð1 þ 3eÞe 3
:
Inductively, we obtain the conclusion.
r
As for the other terms in (3.7), we have the following estimate.
define
CðeÞ :¼ max Ce ; max jC1; m; e j :
1amake
Let us
102
Yoshitsugu Kabeya
Lemma 3.4. The constants in (3.7) of C1; m; e ðke þ 1 a m a 2ke þ 1Þ,
C2; m; e , jae jC3; m; e and Re ðxÞ for fixed x are at most of order CðeÞ in e.
Proof. We again consider the process of the proof of Lemma 3.3. Also
confer to the deduction of (2.22) in [10]. It is easy to see that ð1 þ eÞ m a
ð1 þ eÞ ke a ð1 þ eÞ 1=fðn2Þeg a 2e 1=ðn2Þ for any su‰ciently small e > 0.
For coe‰cients C1; m; e with 1 a m a ke , e m appears in the denominator to
cancel out the previous term as mentioned in the deduction of C1; 2; e in the
proof of Lemma 3.3. The powers in x of the terms in the coe‰cient C1; m; e
ð1 a m a ke Þ converge to ðn 2Þ as e # 0. These terms induce the higher e
dependence.
C1; m; e with ke þ 1 a m a 2ke þ 1 is determined in the same way as in the
proof of Lemma 3.3. But a new e power does not appear since ðn 2Þð1 þ meÞ
with ke þ 1 a m a 2ke þ 1 never converges to n 2 as e # 0.
C2; m; e , ae , C3; m; e and Re ðxÞ are determined by the terms in (3.9) which are
products of jxjðn2Þð1þmeÞ term and c1 jxj1 or k1 ðxÞ inductively. Indeed, when
the terms up to C1; ke ; e jxjðn2Þð1þke eÞ are obtained, (3.9) can be written as
!
ke
c0 þ c1 jxj1 þ k1 ðxÞ X
C1; m; e
Due þ
þ f4 ðue Þ ¼ 0
ðn2Þð1þmeÞ
jxj 2
k¼1 jxj
with f4 being a remainder term. Thus we see that C2; m; e , ae , C3; m; e and Re ðxÞ
are determined from the product of jxj1 or ke ðxÞ with C1; m; e jxjðn2Þð1þmeÞ .
Since the powers of other terms never converge to jxjðn2Þ , the above
process shows that the coe‰cients and Re cannot create the higher e dependence as in CðeÞ. To keep ue positive on jxj b R1 , they must be at most of the
order CðeÞ.
r
Using (3.7) and Lemma 3.4, we show the boundedness of the maximum
point of ue .
Lemma 3.5. Suppose that (K) and (K.1) hold. Then the maximum point
of the least-energy solution ue is uniformly bounded.
Proof. By (K.1), using the expansion (3.7), we can express ue as
!
Ce
f1; e ðxÞ f2; e ðxÞ ðae xÞ f3; e ðxÞ
x
1
ð3:11Þ ue ðxÞ ¼ n2 þ
þ
þ
þ Re
;
n
n2
n1
2
jxj
jxj
jxj
jxj jxj n1
jxj
with
f1; e ðxÞ ¼
2k
e þ1
X
m¼1
C1; m; e
jxj
;
ðn2Þme
f2; e ðxÞ ¼
ke
X
C2; m; e
m¼1 jxj
;
ðn2Þme
f3; e ðxÞ ¼ 1 þ
ke
X
C3; m; e
m¼1 jxj
ðn2Þme
:
Asymptotic behaviors of least-energy solutions
103
Note that f1; e ðxÞ, f2; e , f~3; e ðxÞ :¼ ðae xÞ f3; e ðxÞ and Re ðx=jxj 2 Þ in terms of e are
at most of the order of CðeÞ by Lemma 3.4. Then we can express
(
)
f~3; e ðxÞ
R~e ðxÞ
CðeÞ
Ce
f1; e ðxÞ f2; e ðxÞ
þ
þ
þ
þ
ð3:12Þ ue ðxÞ ¼ n2
;
CðeÞ
CðeÞjxj CðeÞjxj 2 CðeÞjxj
CðeÞ
jxj
with R~e ðxÞ :¼ Re ðx=jxj 2 Þ.
Suppose that the maximum point Qj of uej (ej # 0 as j ! y) tends to
infinity. We may suppose that jQj j > R1 for any j and fix x0 so that
jx0 j b R1 . Then we have
ð3:13Þ
1<
uej ðQj Þ
uej ðx0 Þ
9
8
f~3; ej ðQj Þ
>
R~ej ðQj Þ >
f1; ej ðQj Þ
f2; ej ðQj Þ
C
e
>
>
>
>
þ
þ
þ
þ
>
>
Cðej Þ
Cðej ÞjQj j Cðej ÞjQj j 2 Cðej ÞjQj j =
jx0 j ðn2Þ < CðeÞ
¼
>
jQj j
f~3; ej ðx0 Þ
R~ej ðx0 Þ >
f1; ej ðx0 Þ
f2; ej ðx0 Þ
>
>
Ce
>
>
>
>
þ
þ
þ
þ
;
:
2
Cðej Þ
Cðej Þjx0 j Cðej Þjx0 j
Cðej Þjx0 j
CðeÞ
jx0 j n2
¼:
Lj ;
jQj j
We consider the behavior of Lj . Here we note that constants in the
denominator and those in the numerator are the same and that the remainder
terms f~3; ej ðQj Þ=ðCðeÞjQj j 2 Þ and R~ej ðQj Þ=ðCðeÞjQj jÞ are negligible compared with
three terms in the numerator due to their decay properties as jQj j ! y.
In view of (3.4), since each term has a decay order and since the absolute
value of each coe‰cient in (3.7) are bounded by Cðej Þ, the case where the
numerator goes to infinity while the denominator stays bounded is impossible.
In the case where the denominator converges to 0, if we can find suitable
point x ðjx j b R1 Þ independent of e so that uej ðx Þ stay uniformly away from
zero, we can replace x0 by x .
If this is not the case, then the denominator converges locally uniformly to
0. In this case, the decay property of the expanded functions in (3.7) shows
that the numerator decays faster than the denominator.
Similarly, if the both of the denominator and the numerator go to infinity,
the decay order shows that the slower growth of the numerator. Thus, the
inequality (3.13) is violated if jQj j ! y. We complete the proof.
r
4.
Proofs of Theorems 1.2, 1.3 and 1.4
Now we are in a position to prove Theorems 1.2, 1.3 and 1.4.
104
Yoshitsugu Kabeya
Proof of Theorem 1.2.
Since
lim Se ¼
e#0
ðn 2Þ 2
4c0
by Theorem 1.1, we immediately see that
(
lim k‘ue k22
e#0
¼ lim
e#0
!
Seð2þeÞ=e
0;
y;
ðn 2Þ 2
¼ lim
e#0
4c0
)ð2þeÞ=e
if c0 > ðn 2Þ 2 =4;
if 0 < c0 < ðn 2Þ 2 =4:
Thus we have the desired limiting behavior.
k‘ue k2e ¼ Seð2þeÞ=2 !
Moreover, by (1.4), we have
ðn 2Þ 2
4c0
as e # 0.
Now, we consider the behavior of kue ky . If lime#0 k‘ue k2 ¼ 0, then by
(2.1) and the proof of Lemma 3.1, we see that kue ky ! 0 as e # 0.
When lime#0 k‘ue k2 ¼ y, suppose that lim sup#0 kue ky < y. Letting
ue ðxÞ
¼ We ðyÞ;
kue ky
x¼
y
;
e
we see that We ðyÞ is a solution to
1
y
e
Dy We þ 2 K
We 1þe ¼ 0:
kue ky
e
e
Since lime#0 e2 Kð y=eÞ ¼ c0 j yj2 locally uniformly in R n nf0g, by choosing a
subsequence if necessary (still denoted by e), We converges locally uniformly in
R n nf0g to W, where W is a solution to
Dy W þ
C
jyj 2
W ¼0
in R n
e
with C :¼ c0 lime#0 kue ky
< ðn 2Þ 2 =4. The limiting equation does not have
any positive solution which is bounded near the origin unless C ¼ 0. Thus
W 1 0 if 0 < C < ðn 2Þ 2 =4. However, concerning the constant C~e in (3.5),
since we have
y
ue ðxÞ ¼ kue ky We ðyÞ ¼ C~e e n2 jyjðn2Þ þ he
;
e
we obtain lime#0 e n2 C~e ¼ 0 in view of the local uniform convergence of We to
Asymptotic behaviors of least-energy solutions
105
W 1 0. Since the boundedness of ue implies the boundedness of ‘ue in view
of the equation, we have
ð
2
lim k‘ue k2 ¼ lim
j‘ue j 2 dx ¼ y;
e#0
e#0
R n nBR
for any R > 0. In view of (3.5), we see that
ð
j‘ue j 2 dx a C C~e2
R n nBR
as e # 0 with C > 0.
Thus we have
ð
j‘ue j 2 dx ¼ oðe2ðn2Þ Þ
R n nBR
as e # 0.
However, we have seen
lim k‘ue k22
e#0
ðn 2Þ 2
¼ lim
e#0
4c0
!ð2þeÞ=e
¼ y;
thus the growth order of k‘ue k2 is faster than e2ðn2Þ . Hence we get a contradiction for 0 < C < ðn 2Þ 2 =4.
If C ¼ 0, then there is a possibility of W 1 1. In this case, as in the last
part of the proof of Lemma 3.2, we have
We ð2yÞ
¼ 2ðn2Þ þ oð1Þ
We ð yÞ
as e # 0. This contradicts the uniform convergence of We to 1. The case
where W 1 0 is proved as in 0 < C < ðn 2Þ 2 =4.
Thus we have reached a contradiction for 0 a C < ðn 2Þ 2 =4, that is, we
have proved kue ky ! y as e # 0 if 0 < c0 < ðn 2Þ 2 =4.
r
Proof of Theorem 1.3. In this case, since lime#0 k‘ue k2e ¼ 1, we need
careful calculations. Suppose that lim supe#0 k‘ue k2 < y. Then by (2.1), we
see that kue ky is bounded. Then, choosing a subsequence if necessary, we see
that uej converges to U~ A D locally uniformly in R n , where U~ is a nonnegative
solution to
DU~ þ KðxÞU~ ¼ 0:
Note that this equation has only U~ 1 0 as a nonnegative solution by the
Pohozaev identity (1.7). Thus, the convergence does not depend on subsequences. Moreover, by Lemma 3.5, the maximum point of ue is bounded.
Thus, kue ky ! 0 as e # 0.
106
Yoshitsugu Kabeya
Let v^e ðxÞ :¼ ue ðxÞ=kue ky .
Then v^e satisfies
e 1þe
D^
ve þ KðxÞkue ky
v^e ¼ 0;
k^
ve ky ¼ 1:
e
kue ky
! c > 0 as e # 0. Then, since the maximum point is
Suppose that
bounded, v^e converges to a nontrivial function (along a subsequence).
However, again by the Pohozaev identity, the limiting equation has only v^ 1 0
as a solution, which is a contradiction.
e
Suppose that kue ky
! 0 as e # 0. Then, v^e converges to 1 locally unin
formly in R . However, as in the last part of the proof of Theorem 1.2 for
0 < c0 < ðn 2Þ 2 =4, there exists large R > 0 independent of e such that
ð4:1Þ
v^e ð2xÞ
¼ 2ðn2Þ þ oð1Þ
v^e ðxÞ
for jxj b R. Hence, the local uniform convergence of v^e to 1 is impossible.
Thus we see that lime#0 kue ky ¼ y and lime#0 k‘ue k2 ! y.
2e=ð2þeÞ
¼ Se ! 1 as e # 0, i.e., k‘ue k2e ! 1 as e # 0.
By Theorem 1.1, k‘ue k2
Thus the proof is complete for any case.
r
Using Lemmas 3.1, 3.2 and 3.5, we prove Theorem 1.4.
It is easy to see that ve ð yÞ satisfies
e
1
y
Dy ve ðyÞ þ 2 K
max ue ðxÞ ve1þe ¼ 0:
e
e
jxjbR=e
Proof of Theorem 1.4.
Since maxjyjbR ve ðyÞ ¼ 1 and since lime#0 e2 Kðy=eÞ ¼ c0 =jyj 2 locally uniformly
on fy j jyj b Rg, by choosing a subsequence if necessary, ve ð yÞ converges to
V ðyÞ locally uniformly on f y j jyj b Rg, where V ð yÞ is a solution to
ð4:2Þ
DV þ
c0 c
jyj 2
V ¼0
with c being an accumulation point of ðmaxjxjbR=e ue ðxÞÞ e as e # 0.
Suppose that c can be taken as c ¼ 0. As in the proof of Lemma
~ of ve ðyÞ (maxjxjbR=e ue ðxÞ ¼ ve ðQ
~ Þ) is uniformly
3.5, the maximum point Q
e
e
bounded in view of the decay (3.5). Thus there exists y0 ðj y0 j b RÞ such that
V ðy0 Þ ¼ 1. Since (4.2) yields DV ¼ 0 in this case, V might satisfy V 1 1.
However, by (3.5), the local uniform convergence to 1 is absurd as in (4.1)
(consider the ratio ve ð2y1 Þ=ve ð y1 Þ with su‰ciently large j y1 j). Thus V cannot
be a positive constant in this case. Hence c must be positive.
As for the estimate of the upper bound of c , we use (2.1) and Lemma
3.1. By them, we have
kue ky a C~ k‘ue k2
Asymptotic behaviors of least-energy solutions
107
with C~ > 0 independent of e. Combining Theorem 1.1 with (1.4), we see
e
that k‘ue k2e ! ðn 2Þ 2 =ð4c0 Þ as e # 0. Thus we obtain lim supe#0 kue ky
a
2
ðn 2Þ =ð4c0 Þ. Since
1
y
c0
lim 2 K
¼ 2;
e#0 e
e
jyj
choosing a subsequence, we see that ve ð yÞ converges locally uniformly in
f y j j yj b Rg to V, which is a solution to
8
c~
>
>
< DV þ 2 V ¼ 0;
j yj
ð4:3Þ
>
>
: V ðQ Þ ¼ 1; lim V ðyÞ ¼ 0;
jyj!y
where 0 < c~ a ðn 2Þ 2 =4 and jQ j b R. Note that (4.3) has a solution (at
least a radial one). Thus we obtain the desired conclusion.
r
5.
Appendix
Here we give a proof of Lemma 2.1 for the sake of self-containedness.
Proof of Lemma 2.1. Since SðlÞ is attained by UðxÞ ¼
ð1 þ jxj 2l Þðn2Þ=ð2lÞ , we have
ð
ðy
2
2
j‘Uj dx ¼ ðn 2Þ on
r n1 r 22l ð1 þ r 2l Þ2ðnlÞ=ð2lÞ dr
Rn
0
2l ðn2Þ=ð2lÞ1
since Ur ¼ ðn 2Þð1 þ jxj Þ
jxj 1l .
ðy
r nþ12l ð1 þ r 2l Þ2ðnlÞ=ð2lÞ dr
Letting r ¼ r 2=ð2lÞ , we get
0
2
¼
2l
ðy
rð2n3lþ2Þ=ð2lÞ ð1 þ r 2 Þ2ðnlÞ=ð2lÞ dr:
0
Letting r ¼ tan y, we have
ðy
r nþ12l ð1 þ r 2l Þ2ðnlÞ=ð2lÞ dr
0
¼
2
2l
ð p=2
sin 2ðn2lþ2Þ=ð2lÞ1 y cos 2ðn2Þ=ð2lÞ1 y dy:
0
Since the beta function Bð p; qÞ is defined as
ð p=2
Bð p; qÞ ¼ 2
sin 2p1 y cos 2q1 y dy
0
108
Yoshitsugu Kabeya
and since Bð p; qÞ ¼ GðpÞGðqÞ=Gð p þ qÞ, we obtain
ð
ðn 2Þ 2 on
n 2l þ 2 n 2
;
B
j‘Uj 2 dx ¼
2l
2l
2l
Rn
n 2l þ 2
n2
2
ðn 2Þ on G
G
2l
2l
¼
:
2ðn lÞ
ð2 lÞG
2l
Similarly, we have
ð
jxjl U 2ððnlÞ=ðn2ÞÞ dx
Rn
¼ on
ðy
r n1l ð1 þ r 2l Þ2ðnlÞ=ð2lÞ dr
0
2on
¼
2l
¼
2on
2l
ðy
rð2ðn1ÞlÞ=ð2lÞ ð1 þ r 2 Þ2ðnlÞ=ð2lÞ dr
0
ð p=2
sin 2ðnlÞ=ð2lÞ1 y cos 2ðnlÞ=ð2lÞ1 y dy
0
on
nl nl
;
B
2l 2l
2l
on
nl 2
G
2l
2l
¼
;
2ðn lÞ
G
2l
¼
where r ¼ r 2=ð2lÞ and r ¼ tan y as before. Thus we obtain
Ð
2
R n j‘Uj dx
SðlÞ ¼ Ð
ð R n jxjl U 2ðnlÞ=ðn2Þ dxÞðn2Þ=ðnlÞ
n 2l þ 2
n2
G
G
ðn 2Þ 2 on
2l
2l
¼
2ðn lÞ
2l
G
2l
8
1
2ðn2Þ=ðnlÞ 9
>
>
>
>
n
l
>
>
>
>
G
<
=
on ðn2Þ=ðnlÞ
2l
>
2ðn lÞ ðn2Þ=ðnlÞ >
> 2l
>
>
>
>
>
G
:
;
2l
Asymptotic behaviors of least-energy solutions
¼ ðn 2Þ
2
on
2l
109
ð2lÞ=ðnlÞ
n 2l þ 2
n2
G
G
2l
2l
:
n l 2ðn2Þ=ðnlÞ
2ðn lÞ ð2lÞ=ðnlÞ
G
G
2l
2l
Thus the first part is proved.
Since our aim is to let l " 2, we need the asymptotic expansion of the
gamma function. The expansion formula (Stirling’s formula, see e.g., Taylor
[14], p. 267, (A.39)) yields
pffiffiffiffiffiffi
GðzÞ ¼ 2pez z z1=2 þ LðzÞ;
near z ¼ y with limjzj!y e z zðz1=2Þ LðzÞ ¼ 0. The formula yields
pffiffiffiffiffiffi ðn2lþ2Þ=ð2lÞ n 2l þ 2 ðn2lþ2Þ=ð2lÞ1=2
n 2l þ 2
þ L1 ðlÞ;
¼ 2pe
G
2l
2l
pffiffiffiffiffiffi ðn2Þ=ð2lÞ n 2 ðn2Þ=ð2lÞ1=2
n2
G
þ L 2 ðlÞ;
¼ 2pe
2l
2l
pffiffiffiffiffiffi
nl
n l ðnlÞ=ð2lÞ1=2
G
þ L 3 ðlÞ;
¼ 2peðnlÞ=ð2lÞ
2l
2l
pffiffiffiffiffiffi
2ðn lÞ
2ðn lÞ 2ðnlÞ=ð2lÞ1=2
G
þ L4 ðlÞ;
¼ 2pe2ðnlÞ=ð2lÞ
2l
2l
where Li ðlÞ ði ¼ 1; 2; 3; 4Þ are lower order terms as l " 2. Thus we have
n 2l þ 2
n2
G
G
¼ 2pe2ðnlÞ=ð2lÞ ðn 2l þ 2Þðn2lþ2Þ=ð2lÞ1=2
2l
2l
ðn 2Þðn2Þ=ð2lÞ1=2 ð2 lÞ 12ðnlÞ=ð2lÞ þ L5 ðlÞ
and
n l 2ðn2Þ=ðnlÞ
2ðn lÞ ð2lÞ=ðnlÞ
G
G
2l
2l
¼ ð2pÞð2nl2Þ=ð2ðnlÞÞ 2 2ð2lÞ=ð2ðnlÞÞ e22ðn2Þ=ð2lÞ
n l 2þ2ðn2Þ=ð2lÞð2nl2Þ=ð2ðnlÞÞ
þ L6 ðlÞ
2l
2ðnlÞ=ð2lÞð2nl2Þ=ð2ðnlÞÞ
2ðnlÞ=ð2lÞ n l
¼ 8pe
þ L 7 ðlÞ;
2l
110
Yoshitsugu Kabeya
where Li ðlÞ ði ¼ 5; 6; 7Þ are lower order terms.
on ð2lÞ=ðnlÞ
2
SðlÞ ¼ ðn 2Þ
2l
Hence we see that
e2ðnlÞ=ð2lÞ ðn 2l þ 2Þðn2lþ2Þ=ð2lÞ1=2 ðn 2Þðn2Þ=ð2lÞ1=2 ð2 lÞ 12ðnlÞ=ð2lÞ
n l 2ðnlÞ=ð2lÞð2nl2Þ=ð2ðnlÞÞ
2ðnlÞ=ð2lÞ
4e
2l
þ L8 ðlÞ
(
)1=ð2lÞ
n 2 on ð2lÞ=ðnlÞ ðn 2l þ 2Þ n2lþ2 ðn 2Þ n2
¼
4
2l
ðn lÞ 2ðnlÞ
ð2 lÞð2lÞ=ð2ðnlÞÞ ðn lÞð2nl2Þ=ð2ðnlÞÞ þ L 9 ðlÞ;
where L8 ðlÞ and L 9 ðlÞ are lower order terms with liml"2 Li ðlÞ ¼ 0 ði ¼ 8; 9Þ.
Note that
(
)
1
ðn 2l þ 2Þ n2lþ2 ðn 2Þ n2
log
lim
l"2 2 l
ðn lÞ 2ðnlÞ
¼ lim
l"2
ðn 2l þ 2Þ logðn 2l þ 2Þ þ ðn 2Þ logðn 2Þ 2ðn lÞ logðn lÞ
2l
¼ limf2 logðn 2l þ 2Þ 2 þ 2 logðn lÞ þ 2g ¼ 0
l"2
by l’Hospital’s rule.
lim
(
Thus we have
ðn 2l þ 2Þ n2lþ2 ðn 2Þ n2
)1=ð2lÞ
ðn lÞ 2ðnlÞ
l"2
¼ 1:
Moreover, by limx!þ0 x x ¼ 1, we see that
lim SðlÞ ¼
l"2
ðn 2Þ 2
¼ Sð2Þ
4
as desired.
r
Remark 5.1. Since the area of the unit sphere is given by on ¼
2p n=2 ðGðn=2ÞÞ1 , SðlÞ coincides with SR ð p; q; a; b; nÞ in Lemma 3.1 of [6]
with p ¼ 2, q ¼ 2ðn lÞ=ðn 2Þ, a ¼ 0, b ¼ ðn 2Þl=f2ðn lÞg and g ¼
nð2 lÞ=f2ðn lÞg.
Acknowledgment
The author thanks the referee for pointing out several gaps in the original
manuscript.
Asymptotic behaviors of least-energy solutions
111
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Yoshitsugu Kabeya
Department of Applied Mathematics
Faculty of Engineering
Miyazaki University
Kibana, Miyazaki, 889-2192
Japan
e-mail: [email protected]