Energy concentration of the focusing energy-critical FNLS

0.3 MB·English

by  Yonggeun Cho

ENERGY CONCENTRATION OF THE FOCUSING ENERGY-CRITICAL FNLS
YONGGEUN CHO,GYEONGHA HWANG,ANDYONG-SUNSHIM
5
Abstract. WeconsiderthefractionalnonlinearSchr¨odingerequation(FNLS)withgeneraldisper-
1
0 sion |∇|α and focusing energy-critical nonlinearities −|u|d2−ααu and −(|x|−2α∗|u|2)u. By adopting
2
Kenig-Tsutsumi [24], Kenig-Merle [19] and Killip-Visan [22] arguments, we show the energy con-

n
centration of radial solutions near the maximal existence time. For this purpose we use Sobolev
a
J inequalities for radial functions and establish strong energy decoupling of profiles. And we also

1
show that when the kinetic energy is confined the maximal existence time is finite for some large
3
class of initial data satisfying the initial energy E(ϕ) is less than energy of ground state E(W )
α
] α α
P butk|∇|2ϕkL2 ≥k|∇|2WαkL2.
A
.
h
t
a
m
[
1. Introduction

1
v In this paper we consider the Cauchy problem of the focusing fractional nonlinear Schro¨dinger
0
0 equations:
1
0 i∂ u= αu V(u)u, in R1+d d 2,
0 (1.1) t |∇| − ≥
2. ( u(x,0) = ϕ(x) in Rd,
0
5 where
1 u d2−αα (1 < α < 2),
: V(u) = | |
v ( (x −2α u2) (1 < α < min(2, d)).
i | | ∗| | 2
X
r The equation (1.1) is of H˙ α2-scaling invariance (so-called energy-critical). That is, if u is a solution
a
of (1.1), then for any λ > 0 the scaled function u , given by
λ
uλ(t,x) = λd2−α2u(λαt,λx),
is also a solution to (1.1).
α
The problem (1.1) can be easily shown to be well-posedness in C(( T ,T∗);H2 ) for α
− ∗ rad ∈
[ 2d ,2) and d > α in the case of power type (d > 2α in the case of Hartree type) through the
2d−1
radialStrichartzestimate. SeeLemma2.1 belowfor Strichartzestimate, andalso seeTheorem 4.10
of [17] and Theorem 5.2 of [5] for LWP and small data GWP. Here T ,T∗ [ ,+ ] are the
∗
− ∈ −∞ ∞
2010 Mathematics Subject Classification. M35Q55, 35Q40.
Key words and phrases. blowup, energy concentration, focusing energy-critical FNLS,Strichartzestimate, profile
decomposition, virial argument, Sobolev inequality for radial functions.
This research was supported byOverseas Research Grants of LG Yonam Foundation.
1
2 Y.CHO,G.HWANG,ANDY.-S.SHIM
maximal existence times and X denotes the Banach space X of radially symmetric functions.
rad
The solution u satisfies the mass and energy conservation laws: for t ( T ,T∗)
∗
∈ −
m(u(t)) = u(t) 2 = m(ϕ),
k kL2
(1.2)
E(u(t)) = (u(t))+ (u(t)) = E(ϕ),
K V
where
1 1
(u) = α2u(x)2dx, (u) = V(u)u2dx,
K 2 ||∇| | V −µ | |
Z Z
µ = 2d for power type and µ = 4 for Hartree type.
d−α
At this point due to the dependency on the profile which is the critical nature of (1.1) we do
α
not know that limsupt→T∗k|∇|2u(t)kL2 is infinity or not. In this paper we want to address some
energy concentration phenomena for both cases. We first consider the concentration in the case of
unconfined kinetic energy.
Theorem 1.1 (Unconfined case). Let d > α+1 in the case of power type (d > 2α in the case of
α α
Hartree type). Assume that ϕ H2 and u is the unique solution to (1.1) in C([0,T∗);H2 ) with
∈ rad rad
α
limsup 2u(t) L2 = + .
t→T∗ k|∇| k ∞
Then for any R > 0 we have
α
limsup 2u(t) L2(|x|≤R) = + .
t→T∗ k|∇| k ∞
Moreover, if for t < T∗ u(t) L∞, then
∈
limsup u(t) L∞(|x|≤R) = + .
t→T∗ k k ∞
The same result holds near T .
∗
−
Next we deal with the confined case for which it is necessary to implement subtle estimate
associated with ground state. The ground state of (1.1) plays an important role. It is a unique
positive radial solution of
(1.3) αW V(W)W.
|∇| −
In [4, 23, 14] the authors showed that any solution of the elliptic equation (1.3) is a constant
multiple, dilation and translation of the function Wα(x) = C1(1+C2 x 2)−d−2α which is in H˙ α2 for
| |
0 < α < d, where C ,C depend on d,α. The solution W is closely related to the best constant
2 1 2 α
problem of the inequality
V(u)|u|2dx ≤ Cd,αk|∇|α2ukµL2.
Z
Indeed, the maximizer u= 0 of the above inequality, that is,
6
(1.4) V(u)|u|2dx = Cd,αk|∇|α2ukµL2,
Z
ENERGY CONCENTRATION OF THE FOCUSING ENERGY-CRITICAL FNLS 3
is characterized as u = eiθλd−2αWα(λ(x x0)) for some θ [ π,π], λ > 0 and x0 Rd. See
− ∈ − ∈
[1, 27, 13] for power type. We will treat this problem for Hartree type in the appendix. Since W
α
is a solution of (1.3), k|∇|α2Wαk2L2 = V(Wα)|Wα|2dx. Thus k|∇|α2Wαk2L2 = Cd−,αµ−22.
Let us denote by S (I) for an inteRrval I the spaces L2(dd−+αα) for power type and L6Ld−2d43α for
α I,x I x
Hartree type. Then Lemma 2.4 below states that LWP of (1.1) in H˙ α2 holds for some α and
d and also shows the blowup criterion that kukSα((−T∗,0]) = +∞ and kukSα([0,T∗)) = +∞ when
T ,T∗ < + . Furthermorebyfollowingtheargumentsin[19,22,26,16]withprofiledecomposition
∗
∞
developed in Section 3 below one can readily get the following theorem.
Theorem 1.2. Let d 2, 2d < α < 2, α < d 2α for power type (d > 2α for Hartree type)
≥ 2d−1 ≤
α
and let ϕ H˙ 2 . Assume that
∈ rad
α α
sup 2u(t) L2 < 2Wα L2.
t∈(−T∗,T∗)k|∇| k k|∇| k
Then T∗,T∗ = +∞ and kukSα(R) < +∞.
As a corollary one can show that T∗ = T∗ = +∞ and kukSα(R) < +∞ if E(ϕ) < E(Wα) and
α α
2ϕ L2 < 2Wα L2. The same result also holds for the defocusing case. The restriction
k|∇| k k|∇| k
α ( 2d ,2) comes from the optimal range of Strichartz estimates (see Lemma 2.1). The con-
∈ 2d−1
dition α ≤ 2α for power type is necessary to estimate perturbation like k| Jj fj|d2−αα( Jj fj) −
Jj |fj|d2−ααfjkH˙ α2. For this see the arguments below (3.6). P P
PAt this point one may expect the sharpness of Theorem 1.2 and the blowup (kukSα((−T∗,T∗)) =
α α
+ ) when E(ϕ) < E(Wα) and 2ϕ L2 2Wα L2. Unfortunately we do not know the
∞ k|∇| k ≥ k|∇| k
completeanswers. Wethinkthisisjustatechnicalproblemduetonon-localityarisingwhentreating
α. However, in case when kinetic energy is confined we can show the energy concentration near
|∇|
the maximal existence time and also find some class of initial data guaranteeing the finite time
blowup. We first introduce the energy concentration.
Theorem 1.3 (Confined case). Let d 2, 2d < α < 2, α < d 2α for power type (d > 2α for
≥ 2d−1 ≤
α
Hartree type) and let ϕ H˙ 2 . Assume that
∈ rad
α
kukSα([0,T∗)) = +∞, t∈s[0u,Tp∗)k|∇|2u(t)kL2 < +∞.
If T∗ is finite, then there exists a sequence t T∗ such that for any sequence R (0, ) obeying
n n
→ ∈ ∞
(T∗ tn)−α1Rn ,
− → ∞
limsup ||∇|α2u(tn,x)|2dx ≥ k|∇|α2Wαk2L2.
n→∞ Z|x|≤Rn
The same result also holds near T if T < + .
∗ ∗
− ∞
4 Y.CHO,G.HWANG,ANDY.-S.SHIM
The Scho¨dinger case was treated by Killip and Visan in [22]. Here we adapt their arguments
to fractional case with nonlinear profile approximation. We want to mention that due to the lack
of pseudo-conformal symmetry of the equation (1.1) we could not get the similar result when the
solution blowup at time infinity.
From now on we try to demonstrate some evidence of the finite time blowup. Based on the virial
argument the finite time blowup was shown for mass-critical Hartree type fractional Schro¨dinger
equations in [6]and for fourthorder power typeNLS [11], wherethe mass-critical natureandradial
symmetry are playing a crucial role in the proof of blowup. Those arguments cannot be applied
to the power type mass-critical fractional NLS because of the lack of enough cancelation property
of nonlinearity for virial argument to proceed. Since we do not know whether the kinetic energy
is confined, it is hard to apply them to energy subcritical and mass supercritical or energy critical
problem. However, if we are involved in energy critical problem and the energy is confined, then
by using Sobolev inequality for radial functions [10] it is plausible to establish the virial argument
to get finite time blowup for both power type and Hartree type. The following theorem leads us
off the finite time blowup.
α α
Theorem 1.4. Let ϕ H2 and u be the unique solution of (1.1) in C([0,T∗);H2 ) for the
∈ rad rad
maximal existence time T∗ (0,+ ]. Suppose that d 2, 4 α < 2, α < d < 3α for power type
∈ ∞ ≥ 3 ≤
and d> 2α+2, 2d < α < 2 for Hartree type. Then for any ϕ satisfying that
2d−1
(1.5) x √1 ∆ϕ + x 2ϕ < + ,
L2 L2
k| | − k k| | k ∞
α α
(1.6) E(ϕ) <E(Wα), 2ϕ L2 2Wα L2,
k|∇| k ≥ k|∇| k
if sup α2u(t) L2 < + , then T∗ < .
0≤t<T∗k|∇| k ∞ ∞
The rest of paper is organized as follows: In Section 2 we gather some preliminary lemmas
necessary for the proof of confined energy concentration including the profile decomposition in
energy space. In Section 3 we show the energy concentration, Theorems 1.1 and 1.3. Section 4 is
devoted to proving finite time blowup under energy confinement. In the last section we consider
the best constant problem (1.4) for Hartree equation.
Notations. We will use the notations:
= √ ∆, H˙s = −sLr, H˙s = H˙s, Hs = (1 ∆)−s/2Lr, Hs = Hs, Lr = Lr(Rd) for some
• |∇| − r |∇| 2 r − 2 x
s R and 1 r .
∈ ≤ ≤ ∞
We use the following mixed-norm notations LqLr = Lq(I;Lr(Rd)), Lq = LqLq and LqLr =
• I t x I,x I t
LqLr.
R
f(ξ) = e−ix·ξf(x)dx.
• Rd
For anyRdyadic number N we denote frequency localization of function f by fN, which is defined
• b
by f (ξ) = P f(ξ) = β(ξ/N)f for a fixed Littlewood-Paley function β C∞ with ββ = β and
N N ∈ 0,rad
c d b e
ENERGY CONCENTRATION OF THE FOCUSING ENERGY-CRITICAL FNLS 5
P P = P , where β(ξ) = β(ξ/2)+β(ξ)+β(2ξ) and P = P +P +P .
N N N N N/2 N 2N
As usual different positive constants are denoted by the same letter C, if not specified.
• e e e
[A,B] denotes the commutator AB BA for any operators A and B defined on suitable Banach
• −
spaces.
u,v = uvdx and f; g = f ,g for f =(f , ,f ),g = (g , ,g ).
• Rd 1≤j≤d j j 1 ··· d 1 ··· d
(cid:10) (cid:11) R (cid:10) (cid:11) P (cid:10) (cid:11)
2. Preliminary lemmas
We define the linear propagator U(t) of the linear equation iu = αu with initial datum f.
t
|∇|
Then it is formally given by
1
(2.1) U(t)f = e−it|∇|αf = ei(x·ξ−t|ξ|α)f(ξ)dξ.
(2π)d Rd
Z
We have Strichartz estimates for radial functions (see [9] and [17, b18]) as follows.
Lemma 2.1. Suppose that d 2, 2d α < 2 and f,F are radial. Then there hold
≥ 2d−1 ≤
t
kU(t)fkLqILrx ≤ CkfkL2, kZ0 U(t−t′)F(t′)dt′kLqILrx ≤ CkFkLqIe′Lrxe′
for the pairs (q,r) and (q,r) such that
α d d 4d 2
e e+ = , 2 q,r , (q,r) = (2, − ).
q r 2 ≤ ≤ ∞ 6 2d 3
−
Such pairs are said to be α-admissible.
Then we have the following inverse Strichartz estimate.
α
Lemma 2.2. Fix d 2 for power type and d> 2α for Hartree type. Let f H˙ 2 and η >0 such
≥ ∈ rad
that
U(t)f η
k kSα(I) ≥
for some interval I R. Then there exists C = C( α2f L2,η), and J I so that
⊂ k|∇| k ⊂
2
U(t) eα2f edx C−1 for all t J.
1 |∇| ≥ ∈
Z|x|≤Ce|J|α (cid:12) (cid:12)
(cid:12) (cid:12)
e
Here C does not depend on I or J.(cid:12) (cid:12)
Proof eof Lemma 2.2. For simplicity we only consider the Hartree type, the power type can be
treated similarly to [22]. We will show that
(2.2) kU(t)fMkSα(I) ≥C−1η43αdk|∇|α2fkL1−243αd
for some dyadic M ≥ A|I|−α1 and some (depending on d and α). Here A = Ck|∇|α2fkL−2D1ηD2 and
the constants D ,D will be specified later.
1 2
6 Y.CHO,G.HWANG,ANDY.-S.SHIM
We assume that (2.2) is true. By Strichartz estimate we have
kU(t)fMkL6(dd−α)L2d(−d−43αα) ≤ CM−α2k|∇|α2fkL2.
I
Combining this with (2.2), we get by H¨older’s inequality that
kU(t)fMkL∞I,x ≥ C−1k|∇|α2fkL1−243αd22η43αd22Md−2α.
From this with the fact that the kernel of Mα2 −α2PM is integrable and its value is independent
|∇|
of M we deduce that
e
C−1k|∇|α2fkL1−243αd22η43αd22Md2 ≤ k(Mα2|∇|−α2PM)(U(t)|∇|α2fM)kL∞I,x ≤ CkU(t)|∇|α2fMkL∞I,x.
Thus there exist (t ,x ) I Rd so that e
0 0
∈ ×
α d
(2.3) (U(t0) 2fM(x0) A0M2,
| |∇| | ≥
where A0 = C−1k|∇|α2fkL1−243αd22η43αd22. Let A1 = 2Ck|∇A|0α2fkL2. Then for |x − x0| ≤ A1M−1 and
t t A M−α we have
0 1
| − |≤
1
α α d
U(t0) 2fM(x0) U(t) 2fM(x) A0M2
| |∇| − |∇| | ≤ 2
and thus
1
α d
U(t) 2fM(x) A0M2.
| |∇| | ≥ 2
This yields for all t J = t I : t t A M−α
0 1
∈ { ∈ | − |≤ }
s
Z|x−x0|≤A1M−1|U(t)|∇|α2fM(x)|2dx ≥ 4dA20Ad1,
where s is the measure of the unit ball. By convexity we have
d
|U(t)|∇|α2fM|2 ≤ C0|U(t)|∇|α2f|2∗βM∗ ,
where β∗ (x) = Md β(Mx) and C = β∗dx. And also
M | | 0
Z|x−x0|≤A1Mb−1|U(t)|∇|α2fM(xR)|2dx ≤ C0 |U(t)|∇|α2f|2,βM∗ ∗χ{|x−x0|≤A1M−1} .
(cid:10) (cid:11)
We divide inner product into two parts as follows:
|U(t)|∇|α2f|2,βM∗ ∗χ{|x−x0|≤A1M−1} ≤ I +II,
(cid:10) (cid:11)
where
I = |U(t)|∇|α2f|2,χ{|x−x0|≤A2A1M−1}βM∗ ∗χ{|x−x0|≤A1M−1} ,
II =(cid:10) |U(t)|∇|α2f|2,χ{|x−x0|>A2A1M−1}βM∗ ∗χ{|x−x0|≤A1M−1(cid:11)} .
(cid:10) (cid:11)
ENERGY CONCENTRATION OF THE FOCUSING ENERGY-CRITICAL FNLS 7
Now we can find a constant A2 = A2(k|∇|α2fkL2,η) > 11 such that C0II ≤ s8dA20Ad1. Then
I ≤ C0 |x−x0|≤A2A1M−1|U(t)|∇|α2f(x)|2dx and thus we obtain
R s
(2.4) Z|x−x0|≤A2A1M−1|U(t)|∇|α2f(x)|2dx ≥ 8Cd02A20Ad1.
On the other hand, since f is radial, we use the Sobolev inequality [10] that
d−α α
(2.5) sup x 2 f(x) C 2f L2 a.e.
x∈Rd| | | | ≤ k|∇| k
together with (2.3) to get
A0(M x0 )d−2α x0 d−2αM−α2 U(t0) α2fM(x0) C α2f L2,
| | ≤ | | | |∇| | ≤ k|∇| k
which means that
− 2
(2.6) x A d−αM−1.
| 0| ≤ 0
Combining (2.6) with (2.4), we have
s
Z|x|≤(A−0 d−2α+A2A1)M−1|U(t)|∇|α2f(x)|2dx ≥ 8Cd02A20Ad1.
− 2
Since I AαM−α, we deduce that J 1min(Aα,A )M−α. By letting C = max(2(A d−α +
| | ≥ | | ≥ 2 1 0
A A )max(A−1,A−α1), 8C02 ) we get the desired result.
1 2 1 sdA20Ad1 e
Now we show (2.2). By Littlewood-Paley theory and H¨older’s inequality we have
η6 U(t)f 6
≤ k kSα(I)
3d−4α
d
d
C ( U(t)fN 2)d−4α/3 dx dt
≤ | |
ZI Z N !
X
3d−4α
d d d
C U(t)fM d−4α/3 U(t)fN d−4α/3 dx dt
≤ | | | |
M≤NZI(cid:18)Z (cid:19)
X
3d−4α 4α 4α 3d−4α
C U(t)f d U(t)f d U(t)f d U(t)f d dt
≤ M≤NZIk MkLd−2d2αk MkLd−24dα/3k NkLd−24dα/3k NkLd−22dα/3
X
M α(3d−4α) α α 3d−4α 4α
≤ C (N ) 3d (k|∇|2fMkL2k|∇|2fNkL2) d (kU(t)fMkSα(I)kU(t)fNkSα(I)) d
M≤N
X
3d−4α
d
8α M α α α
≤ CsMupkU(t)fMkSdα(I) (N )3k|∇|2fMkL2k|∇|2fNkL2
M≤N
X
8α  α 6d−8α 
≤ CsupkU(t)fMkSdα(I)k|∇|2fkL2d .
M
1WemaychooseA asA >1+max(A−1,C0CβAd1−2),whereC istheconstantsatisfyingβ∗(x)≤C (1+|x|)−1.
2 2 1 8A20 β β
8 Y.CHO,G.HWANG,ANDY.-S.SHIM
From the Sobolev inequality it follows that
kU(t)fMkSα(I) ≤ C(|I|Mα)16k|∇|α2fkL2.
Thus we conclude that there exists M ≥ A|I|−α1 so that (2.2) holds with A= Ck|∇|α2fk−L229αd2η29αd2.
(cid:3)
Next we introduce the tightness of trajectories of solution. The proof is almost same as the one
of Proposition 2.13 in [22] and so we omit it.
Lemma 2.3. Let v :I Rd C be a radial solution to (1.1) with v < . Suppose that
× → k kSα(I) ∞
U(tk) α2v(τk)2dx ε
| |∇| | ≥
Z|x|≤rk
for some ε >0, r > 0, and bounded sequences t R and τ I. Then
k k k
∈ ∈
k|∇|α2v(τk)k2L2 − |U(tk)|∇|α2v(τk)|2dx → 0
(cid:12) Z|x|≤akrk (cid:12)
(cid:12) (cid:12)
for any sequence ak (cid:12)+ . (cid:12)
→ ∞
We close this section by introducing local well-posedness and stability. Since the proof is quite
standard, we omit the details (for instance see [19, 5]).
Lemma 2.4. Let α ( 2d ,2) and α < d < 3α for power type (d > 2α for Hartree type), and let
∈ 2d−1
α
ϕ ∈ H˙r2ad, kϕkH˙ α2 ≤ A. Then there exists δ = δ(A) such that if kU(t−t0)ϕkSα(I) ≤ δ, t0 ∈ I, there
α
exists a unique solution u C(I;H˙ 2 ) to (1.1) with
∈ rad
α
supku(t)kH˙ α2 +k|∇|2ukXα(I) ≤ C(A), kukSα(I) ≤ 2δ.
I
2(d+α) 2d(d+α) 2d
Here Xα(I) = LId−α L d2+α2 for power type and L6ILd−α/3 for Hartree type. Moreover, ϕ 7→ u ∈
C(I;H˙ α2) is Lipschitz. If A is sufficiently small, then I =R.
Lemma 2.5. Assume that α ( 2d ,2) and α < d< 3α for power type (d >2α for Hartree type).
∈ 2d−1
Let I = [0,L),L + , and let u be radial and defined on Rd I be such that
≤ ∞ ×
α
supku(t)kH˙eα2 ≤ A, kukSα(I) ≤ M, k|∇|2ukXα(I) < ∞
t∈I
for some constants A and Me , and u verifiees in the sense of inteegral equation
iu = αu V(u)u+e
e t
|∇| −
α
for some function e. Let ϕ ∈ H˙r2ad bee such thaet kϕ−eue(0)kH˙ α2 ≤ A′. Suppose there exists ε0 =
ε (M,A,A′) such that if 0 <ε ε and
0 0
≤
e
α
k|∇|2ekYα(I) ≤ ε, kU(t)(ϕ−u(0))kSα(I) ≤ ε,
e
ENERGY CONCENTRATION OF THE FOCUSING ENERGY-CRITICAL FNLS 9
then there exists a unique radial solution u on Rd I to (1.1) such that
×
kukSα(I)+supku(t)−u(t)kH˙ α2 ≤ C(M,A,A′).
I
2d
Here Y (I) = L2Ld+α for both power type and foer Hartree type.
α I x
Now we consider the profile decomposition in energy space. Most of them are standard and thus
we only show the energy decoupling of Hartree case.
Lemma 2.6 (see Theorem 1 of [12]). Let t be sequence in R. Suppose lim t = , then
n n→∞ n
{ } | | ∞
for any f C∞
∈ 0
U(t )f 0 as n ,
n Lp
k k → → ∞
when p > 2.
The profile decomposition of U(t) for mass critical case was already verified for radial data in [7]
(see also [8]). From that decomposition, one can easily prove the following profile decomposition
for the energy critical case:
Lemma 2.7. Let d 2, 2d < α < 2, and (q,r) be α-admissible pairs with 2 < q,r < .
≥ 2d−1 ∞
Suppose that un n≥1 is a sequence of complex-valued radial functions bounded in H˙ α2. Then up to
{ }
a subsequence, for any J ≥ 1, there exist a sequence of radial functions {φj}1≤j≤J ⊂ H˙ α2, ωnJ ∈ H˙ α2
j j
and a family of parameters (h ,t ) such that
n n 1≤j≤J,n≥1
(2.7) u (x) = U(tj)[(hj)−d/2+α/2φj(/hj)](x)+ωJ(x)
n n n · n n
1≤j≤J
X
and the following properties are satisfied:
(2.8) Jl→im∞linm→s∞upk|∇|α2U(·)ωnJkLqtLrx = 0,
and for j = k, (hj,tj) and (hk,tk) are asymptotically orthogonal in the sense that
6 n n n≥1 n n n≥1
hj hk
either limsup n + n = ,
n→∞ hkn hjn! ∞
(2.9)
tj tk
or (hj)= (hk) and limsup| n− n| = ,
n n n→∞ (hjn)α ∞
and for each J
(2.10) lim u 2 ( φj 2 + ωJ 2 ) = 0.
n→∞ k nkH˙ α2 − k kH˙ α2 k nkH˙ α2
h 1≤Xj≤J i
Remark 1. Since the space and frequency translations do not appear in the profile decomposition
α
dueto the radial symmetry, it is possible to get the strong convergence of remainder term in LqH˙ 2
t r
rd
as in (2.8) not in LqLd−r2α norm. It plays a crucial role in the proof of Theorem 1.3.
t x
10 Y.CHO,G.HWANG,ANDY.-S.SHIM
From energy critical profile decomposition, we prove some useful corollaries.
Corollary 2.8. Suppose that u is a sequence of complex-valued radial functions bounded in
n n≥1
{ }
H˙ α2. Let {φj}1≤j≤J ⊂ H˙ α2, ωnJ ∈ H˙ α2 and a family of parameters (hjn,tjn)1≤j≤J,n≥1 from Lemma
2.7. Define group operator Gj as Gj(f)= U(tj)[(hj)−d/2+α/2f(/hj)](x). Then we have
n n n n n
·
(Gjn)−1(ωnJ)⇀ 0 weakly in H˙ α2 as n → ∞,
(2.11)
(Gjn)−1(un)⇀ φj weakly in H˙ α2 as n → ∞.
Proof. We first prove
2(2+α)
(Gjn)−1U(t)(|∇|α2un) ⇀ U(t)|∇|α2φj weakly in Lt,xd as n → ∞.
Applying (Gj)−1U(t) to (2.7), we obtain
n
J
(Gjn)−1(U(t)|∇|α2un)= U(t)|∇|α2φj + (Gjn)−1(Gjn′)(U(t)|∇|α2φi)+(Gjn)−1U(t)|∇|α2ωnJ.
j′6=j
X
From the pairwise orthogonality of the family of parameters, we have
(Gjn)−1(Gjn′)(U(t)|∇|α2φj)⇀ 0 weakly in Lt2,(x2d+α) as n → ∞
for every j′ 6= j. Let ωJ be the weak limit of {(Gjn)−1U(t)|∇|α2ωnJ}. Then
(Gjn)−1(U(t)|∇|α2un)⇀ U(t)|∇|α2φj +ωJ.
Since the weak limit is unique, ωJ does not depend on J. And from
k(Gjn)−1U(t)|∇|α2ωnJk 2(2+α) ≤ limsupkU(t)|∇|α2ωnJk 2(2+α) −→ 0 as J → ∞,
L d n→∞ L d
t,x t,x
we have ωJ = 0 for every J 1. So we have
≥
2(2+α)
(Gjn)−1U(t)(|∇|α2un) ⇀ U(t)|∇|α2φj weakly in Lt,xd as n → ∞.
Then following lemma gives the conclusion.
Lemma 2.9 (Lemma 3.63 in [25]). Let v and v be in L2. The following statements are equiva-
n
{ }
lent.
(1) v ⇀ v weakly in L2.
n
2(2+α)
(2) U(t)v ⇀ U(t)v weakly in L d .
n t,x
(cid:3)
Proposition 2.10. Let u be a sequence of complex-valued radial functions satisfying
n n≥1
{ }
kunkH˙ α2 ≤ A and kU(t)unkSα(R) ≥ δ.