Scattering for the focusing energy-subcritical NLS

0.39 MB·English

by  Daoyuan Fang

SCATTERING FOR THE FOCUSING ENERGY-SUBCRITICAL
NLS
1
1 DAOYUANFANG1,JIANXIE1,ANDTHIERRYCAZENAVE2
0
2
Abstract. For the 3D focusing cubic nonlinear Schro¨dinger equation, Scat-
n tering of H1 solutions inside the (scale invariant) potential well was estab-
a lished by Holmer and Roudenko [13] (radial case) and Duyckaerts, Holmer
J andRoudenko [6](general case). Inthis paper,weextend this resulttoarbi-
6 traryspacedimensionsandfocusing,mass-supercriticalandenergy-subcritical
1 powernonlinearities,byadaptingthemethodof[6].

]
P
A
1. Introduction
.
h
t In this paper, we study the Cauchy problem for the focusing, mass supercritical
a
m and energy subcritical nonlinear Schr¨odinger equation in RN, N ≥1
[ iu +∆u+|u|αu=0,
t (NLS)

1 (u(0)=ϕ,
v
8 where
5 4
4 if N ≥3,
0 <α< N −2 (1.1)
3 N 
∞ if N =1,2.
. 
1
As is well-known, the Cauchy problem (NLS) is locally well-posed in H1(RN).
0 
1 (See e.g. [9, 14].) More precisely, given ϕ ∈ H1(RN), there exists T > 0 and a
1 unique solution u∈C([0,T],H1(RN)) of (NLS). This solution can be extended to
: a maximal existence interval [0,T ), and either T = ∞ (global solution) or
v max max
Xi else Tmax <∞ and ku(t)kH1 →∞ as t↑Tmax (finite time blowup). Moreover,the
massM(u(t)), energyE(u(t)) andmomentum P(u(t)) are independent of t, where
r
a
M(w)= |w(x)|2dx, (1.2)
ZRN
1 1
E(w)= |∇w(x)|2dx− |w(x)|α+2dx, (1.3)
2ZRN α+2ZRN
P(w)=Im w(x)∇w(x)dx, (1.4)
ZRN
for all w ∈H1(RN). Throughout this paper, we will only consider solutions in the
above sense (H1 solutions).
2010 Mathematics Subject Classification. 35Q41,35B40.
Key words and phrases. NonlinearSchro¨dingerequation, scattering.
1ResearchsupportedbyNSFC10871175, 10931007, andZhejiangNSFCZ6100217.
2ResearchsupportedinpartbyZhejiangUniversity’sPaoYu-KongInternational Fund.
1
2 DAOYUANFANG,JIANXIE,ANDTHIERRYCAZENAVE
In the defocusing case, i.e. when the nonlinearity |u|p−1u in (NLS) is replaced
by−|u|p−1u,it is well-known[10,20]that allsolutions scatteras t→∞, i.e. there
exists a scattering state u+ ∈H1(RN) such that
ku(t)−eit∆u+k −→ 0. (1.5)
H1
t→∞
Inthefocusingcase,thesituationismuchricher: solutionswithsmallinitialvalues
scatter [22, Theorem 17], but some solutions blow up in finite time [11], and some
solutionsareglobalanddo notscatter. Atypicalexampleofsolutions ofthe latter
type are standing waves,i.e. solutions of the form u(t,x)=eiωtϕ(x). When ω >0
such solutions exist [21] and correspond to ϕ(x) =ωα1ψ(ω12x) where ψ ∈ H1(RN)
is a nontrivial solution of the elliptic problem
−∆w+w =|w|αw. (1.6)
Standing waves of particular interest are given by the ground states, i.e. the solu-
tions of (1.6) which minimize the energy (1.3) among all nontrivial H1 solutions
of (1.6). Ground states exist and are all of the form eiθQ(·−y) where θ ∈ R,
y ∈ RN and Q is the unique, positive and radially decreasing solution of (1.6).
Moreover, E(Q)>0. (See [3, 8, 19].)
No necessary and sufficient condition on the initial value ϕ is known for the
solutionuof(NLS)tobeglobalorblowupinfinitetime. HolmerandRoudenko[12,
Theorem 2.1] have obtained the following sufficient condition. Let
4−(N −2)α
σ = >0, (1.7)
Nα−4
and set
K={u∈H1(RN); E(u)M(u)σ <E(Q)M(Q)σ
and k∇uk kukσ <k∇Qk kQkσ }, (1.8)
L2 L2 L2 L2
whereQisthegroundstatedefinedabove. ItfollowsthatKisinvariantbytheflow
of (NLS) and that every initial value ϕ ∈ K produces a global solution of (NLS)
which is bounded in H1(RN) as t → ∞. This condition is optimal in the sense
that if E(ϕ)M(ϕ)σ < E(Q)M(Q)σ and k∇ϕk kϕkσ > k∇Qk kQkσ , then the
L2 L2 L2 L2
resulting solution of (NLS) blows up in finite time provided |·|ϕ∈L2(RN). (This
observationapplies forexample to ϕ(·)=λα2Q(λ·) withλ>1.) Note that K is the
scale-invariant version of the potential well constructed in [2], with respect to the
natural scaling of the equation
uλ =λα2u(λ2t,λx). (1.9)
SinceKcontainsaneighborhoodof0,andsolutionswithsmallinitialvaluesscatter,
it is naturalto ask whether or not solutions with initial values in K also scatter. A
positive answer to a similar question in the energy-critical case α= 4 has been
N−2
givenbyKenigandMerle[16]forradialsolutionsandN =3,4,5. (Thatresultwas
extended by Killip and Visan [18] to general solutions for N ≥ 5.) For the cubic
3D equation (N = α+1= 3), a positive answer is given in [13] (radial case) then
in [6] (general case). In this paper, we extend the result of [6] to arbitrary N ≥ 1
and α satisfying (1.1). Our main result is the following.
Theorem 1.1. Let N ≥1, assume (1.1) and let K be defined by (1.8). Given any
ϕ∈K, it follows that the corresponding solution u of (NLS) is global and scatters,
i.e. there exists a scattering state u+ ∈H1(RN) such that (1.5) holds.
SCATTERING FOR THE FOCUSING ENERGY-SUBCRITICAL NLS 3
TheconclusionofTheorem1.1isthatforallinitialvalueinK,thecorresponding
solution of (NLS) is global and scatters as t →∞. We note that for every ϕ ∈K,
the solution is also globalfor negative times and scatters as t→−∞. This follows
from the fact that K is invariant by complex conjugation and that if u(t) is a
solutionof (NLS)fort≥0,thenu(−t)isasolutionfort≤0(withthe initialvalue
ϕ).
The rest of the paper is organized as follows. In Section 2, we give the sketch
of the proof of Theorem 1.1, assuming all the technical results. In Section 3 we
recall some energy inequalities related to the set K and in Section 4 we recall
some results on the Cauchy problem (NLS). Section 5 is devoted to a profile
decomposition theorem and Section 6 to the construction of a particular “critical”
solution. Finally,weproveinSection7arigiditytheorem. Theappendix(Section8)
contains a Gronwall-type lemma, which we use in Section 4.
Notation: Throughout this paper, we use the following notation. We denote
byp′theconjugateoftheexponentp∈[1,∞]definedby 1+ 1 =1. Wewillusethe
p p′
(complex valued) Lebesgue space Lp(RN) (with norm k·kLp) and Sobolev spaces
Hs(RN), H˙s(RN) (with respective norms k·k and k·k ). Given any interval
Hs H˙s
I ⊂R, the norm of the mixed space La(I,Lb(RN)) is denoted by k·k . We
La(I,Lb)
denote by (eit∆)t∈R the Schr¨odinger group, which is isometric on Hs and H˙s for
every s ≥ 0. We will use freely the well-known properties of (eit∆)t∈R. (See e.g.
Chapter 2 of [4] for an account of these properties.) We define the following set of
indices, depending only on N and α,
2α(α+2) 2α(α+2) 2(N +2)
a= , b= , γ = ,
4−(N −2)α Nα2+(N −2)α−4 N
4(α+2) N 2 N
q = , r =α+2, s= − ∈ 0,min 1, ,
Nα 2 α 2
and we note for further use that (α+1)r′ =r, and (α(cid:16)+1)b′ =n a. o(cid:17)
2. Sketch of the proof of Theorem 1.1
Inthis shortsection,we givethe proofofTheorem1.1, assumingall preliminary
results. Our proof is adapted from the proof of [6] concerning the cubic 3D equa-
tion, but follows more closely the proof of [16] of the energy-critical case. Note,
however, that the argument in the present situation is considerably simpler than
theargumentin[16]foratleasttworeasons. Theequation(NLS)isenergysubcrit-
ical;andweconsidersolutionsnotonlywithfiniteenergy,butalsowithfinitemass.
ThisrestrictionisessentialsothatthesetKmakessense. Themainingredientsare
a profile decomposition (Theorem 5.1) and a Liouville-type (Theorem 7.1) result,
and the construction of a “critical” solution (Proposition 6.1). These results, as
well as all other technical results, are stated and proved in the following sections.
We define
L={ϕ∈K; u∈La((0,∞),Lr(RN))}, (2.1)
where u is the solution of (NLS) with initial value ϕ. It is well-known (see Propo-
sition 4.2) that if ϕ ∈ L, then u scatters. Therefore, we need only show that
K=L.
Given 0<ω <1, we set
K ={u∈K; E(u)M(u)σ ≤ωE(Q)M(Q)σ}. (2.2)
ω
4 DAOYUANFANG,JIANXIE,ANDTHIERRYCAZENAVE
In particular, K = ∪0<ω<1Kω. Since by (3.6), k∇ϕkL2kϕkσL2 ≤ ω21k∇QkL2kQkσL2
for all ϕ ∈ K , we deduce from the elementary interpolation inequality kϕk1+σ ≤
ω H˙s
k∇ϕkL2kϕkσL2 that kϕk1H˙+sσ ≤ ω21k∇QkL2kQkσL2 for all ϕ ∈ Kω. Applying Propo-
sition 4.3, we conclude that K ⊂L for all sufficiently small ω >0. Therefore, we
ω
may define the number
ω =sup{ω ∈(0,1); K ⊂L}∈(0,1], (2.3)
0 ω
and we need only show that ω =1.
0
We argue by contradiction and assume 0 < ω < 1. The first crucial step of
0
the proof (Proposition 6.1), which is based on the profile decomposition theorem,
showsthatthereexistsaninitialvalueϕ ∈K whichisnotinL. Moreover,the
crit ω0
correspondingsolutionu of (NLS)hasthefollowingcompactnessproperty: there
crit
exists a function x ∈ C([0,∞),RN) such that ∪ {u (t,·−x(t))} is relatively
t≥0 crit
compact in H1(RN). It follows (see Lemma 4.5) that
sup {|∇u (t,x)|2+|u (t,x)|α+2+|u (t,x)|2} −→ 0.
crit crit crit
t≥0Z{|x+x(t)|>R} R→∞
The second crucial step (Theorem 7.1) shows that necessarily ϕ = 0. Since
crit
ϕ 6∈L, we obtain a contradiction. Thus ω =1 and the proof is complete.
crit 0
3. Energy inequalities
In this section, we collect certain energy inequalities that are related to the set
K. We recall that the best constant in the Gagliardo-Nirenberginequality
kfkα+2 ≤C kfk4−(N2−2)αk∇fkN2α, (3.1)
Lα+2 GN L2 L2
is given by
kQkα+2
C = Lα+2 , (3.2)
GN 4−(N−2)α Nα
kQk 2 k∇Qk 2
L2 L2
where Q is the ground state of (1.6). (See Corollary 2.1 in [24].) We also recall
that by Pohozaev’s identity,
4−(N −2)α 4−(N −2)α
kQk2 = k∇Qk2 = kQkα+2
L2 Nα L2 2(α+2) Lα+2
(3.3)
8−2(N −2)α
= E(Q),
Nα−4
(see e.g. Corollary 8.1.3 in [4]) and we note that by (3.2) and (3.3)
CGN = 2(αN+α2)[k∇QkL2kQkσL2]−Nα2−4. (3.4)
Lemma 3.1. If u∈H1(RN) and k∇uk kukσ ≤k∇Qk kQkσ , then
L2 L2 L2 L2
Nα−4
E(u)≥ k∇uk2 , (3.5)
2Nα L2
E(u)M(u)σ 1
k∇uk kukσ ≤ 2k∇Qk kQkσ , (3.6)
L2 L2 E(Q)M(Q)σ L2 L2
(cid:16) (cid:17)
4Nα E(u)M(u)σ Nα−4
8k∇uk2 − kukα+2 ≥8 1− 4 k∇uk2 . (3.7)
L2 α+2 Lα+2 E(Q)M(Q)σ L2
h (cid:16) (cid:17) i
In particular, the above estimates hold for all u∈K, where K is defined by (1.8).
SCATTERING FOR THE FOCUSING ENERGY-SUBCRITICAL NLS 5
Proof. It follows from Gagliardo-Nirenberg’sinequality (3.1) that
1 1
E(u)M(u)σ = [k∇uk kukσ ]2− kukα+2 kuk2σ
2 L2 L2 α+2 Lα+2 L2 (3.8)
≥f(k∇uk kukσ ),
L2 L2
where
1 C
f(x)= x2− GN xN2α, (3.9)
2 α+2
for x≥0. Since equality holds in (3.1) for f =Q, we also have
E(Q)M(Q)σ =f(k∇Qk kQkσ ). (3.10)
L2 L2
Note that f is increasing on (0,x ) and decreasing on (x ,∞), where
1 1
x = CGN Nα −Nα2−4 =k∇Qk kQkσ . (3.11)
1 α+2 2 L2 L2
(cid:16) (cid:17)
(We used (3.4) in the last identity.) An elementary calculation shows that
Nα−4
f(x)≥ x2, (3.12)
2Nα
for 0≤x≤x , with equality for x=x , i.e.
1 1
Nα−4
f(k∇Qk kQkσ )= (k∇Qk kQkσ )2. (3.13)
L2 L2 2Nα L2 L2
Therefore, we see that (3.5) follows from (3.11), (3.8) and (3.12) and that (3.6)
follows from (3.8), (3.12), (3.10) and (3.13). Set now
4Nα
R(u)=8k∇uk2 − kukα+2 .
L2 α+2 Lα+2
If x=k∇uk kukσ , it follows from (3.1) and (3.11) that
L2 L2
1 Nα
M(u)σR(u)≥x2− CGNxN2α
8 2(α+2)
=x2−x1−Nα2−4xN2α =x2[1−(x/x1)Nα2−4].
Applying (3.6), we deduce that
1 E(u)M(u)σ Nα−4
M(u)σR(u)≥x2 1− 4 ,
8 E(Q)M(Q)σ
h (cid:16) (cid:17) i
from which (3.7) follows. (cid:3)
Remark 3.2. Let 0<ω <1 and let K be defined by (2.2). It follows from (3.6)
ω
that K is a closed subset of H1(RN).
ω
4. The Cauchy problem
We collect in this section some results concerning the Cauchy problem (NLS).
We will use in particular the Strichartz estimates
kei·∆ϕkLq(R,Lr)+kei·∆ϕkLγ(R,Lγ) ≤CkϕkL2, (4.1)
kei·∆ϕkLa(R,Lr) ≤CkϕkH˙s. (4.2)
6 DAOYUANFANG,JIANXIE,ANDTHIERRYCAZENAVE
(4.1) is a standard Strichartz estimate. (See [23, 25].) (4.2) follows by applying
(−∆)s2 tothestandardStrichartzestimateandusingSobolev’sinequality. Weneed
thefollowingStrichartz-typeestimatefornonadmissiblepairs(see[5,Lemma2.1])
t
ei(·−t′)∆f(t′)dt′ ≤CkfkLb′((0,∞),Lr′). (4.3)
(cid:13)Z0 (cid:13)La((0,∞),Lr)
Next, using G(cid:13)(cid:13)agliardo-Nirenberg’(cid:13)(cid:13)s inequality kukLr ≤ Ck∇ukηL2kuk1L−γη (for an
appropriate value of η ∈(0,1)) and Sobolev’s embedding H1(RN)⊂Lγ(RN), it is
not difficult to show that
a−γ γ
kuk ≤Ckuk a kuka , (4.4)
La((0,∞),Lr) L∞((0,∞),H1) Lγ((0,∞),Lγ)
for all u∈Lγ((0,∞),Lγ(RN))∩L∞((0,∞),H1(RN)).
It follows from [12, Theorem 2.1] that for every initial value ϕ ∈ K (defined
by (1.8)), the corresponding solution u of (NLS) is global, for both positive and
negative time. Thus we may define the flow (S(t))t∈R by
S(t)ϕ=u(t), (4.5)
for ϕ ∈ K and t ∈ R, where u is the solution of (NLS). The map (t,ϕ) 7→ S(t)ϕ
is continuous R×K→H1(RN), where K is equipped with the H1 topology. Note
that
S(−t)ϕ=S(t)ϕ, (4.6)
for all t ∈ R and ϕ∈ K. It is proved in [12, Theorem 2.1] that S(t)K ⊂ K, for all
t∈R, from which it follows by conservation of energy that
S(t)K ⊂K , (4.7)
ω ω
for all t∈R and ω ∈(0,1), where K is defined by (2.2).
ω
Remark 4.1. Let 0 < ω < 1 and F ⊂ K be a relatively compact subset of
ω
H1(RN). It follows that for every R,T >0,
E ={S(t)ϕ(·−y); ϕ∈F,0≤t≤T,|y|≤R},
isarelativelycompactsubsetofH1(RN). Indeed,suppose(ϕ ) ⊂F,(t ) ⊂
n n≥1 n n≥1
[0,T]and(y ) ⊂{|y|≤R}. Bypossiblyextracting(andapplyingRemark3.2),
n n≥1
there exist ϕ ∈ K , t ∈ [0,T] and y ∈ RN such that ϕ → ϕ in H1(RN), t → t
ω n n
andy →y as n→∞. It follows that S(t )ϕ (·−y )→S(t)ϕ(·−y) in H1(RN),
n n n n
which proves the claim.
The next result is a sufficient condition for scattering.
Proposition 4.2. Let ϕ∈H1(RN) and u be the corresponding solution of (NLS).
If u is global and u ∈ La((0,∞),Lr(RN)), then u scatters, i.e. there exists a
scattering state u+ ∈H1(RN) such that (1.5) holds.
Proof. It follows from Proposition2.3 in [5] that u∈Lη((0,∞),Lµ(RN)) for every
admissible pair (η,µ). Scattering then follows by standard calculations. (See e.g.
Section 7.8 in [4].) (cid:3)
We next recall a small data global existence property.
SCATTERING FOR THE FOCUSING ENERGY-SUBCRITICAL NLS 7
Proposition 4.3. There exist 0 < δ ≤ 1 and C such that if ϕ ∈ H1(RN) and
sd
kϕk ≤δ , then the corresponding solution u of (NLS) is global for both positive
H˙s sd
and negative time and
kukLγ(RN+1)+kukLa(R,Lr)+kukL∞(R,H1) ≤CsdkϕkH1. (4.8)
Proof. It follows from Propositions 2.4 and 2.3 in [5] that if ϕ ∈ H1(RN) and
kei·∆ϕkLa(R,Lr) is sufficiently small, then the conclusions of the proposition hold.
The result then follows from Strichartz estimate (4.2). (cid:3)
Corollary 4.4. Let ϕ ,ϕ ∈ H1(RN) satisfy kϕ k ,kϕ k ≤ δ where δ is
1 2 1 H1 2 H1 sd sd
given by Proposition 4.3 and let u ,u be the corresponding solutions of (NLS). If
1 2
(t1) ,(t2) ⊂R and (x1) ,(x2) ⊂RN satisfy |t1 −t2|+|x1 −x2|→∞
n n≥1 n n≥1 n n≥1 n n≥1 n n n n
as n→∞, then
sup|(u (t−t1,·−x1),u (t−t2,·−x2)) | −→ 0, (4.9)
1 n n 2 n n H1
t∈R n→∞
where (·,·) is the scalar product in H1(RN).
H1
Before proving Corollary 4.4, we state the following lemma, which we will also
use later on. Its elementary proof is left to the reader.
Lemma 4.5. Let ν ≥ 1, 1 ≤ q < ∞ and let E be a relatively compact subset of
Lp(Rν). It follows that
sup |u|q −→ 0. (4.10)
u∈EZ{|x|>R} R→∞
Consequently, if (y1) ,(y2) ⊂Rν and |y1 −y2|→∞ as n→∞, then
n n≥1 n n≥1 n n
sup |u(·−yn1)|q2|v(·−yn2)|2q −→ 0. (4.11)
u,v∈EZRν n→∞
Proof of Corollary 4.4. WenotethatbyPropositions4.3and4.2andformula(4.6),
u1andu2scatterat±∞. Supposebycontradictionthereexistε>0andasequence
(t ) ⊂ R such that |(u (t −t1,·−x1),u (t −t2,·−x2)) | ≥ ε. Without
n n≥1 1 n n n 2 n n n H1
loss of generality, we may replace tj by t −tj for j =1,2, so we obtain
n n n
|(u (t1,·−x1 +x2),u (t2,·)) |≥ε. (4.12)
1 n n n 2 n H1
Bypossiblyextracting,wemayassumethatoneofthefollowingholds: |t1|+|t2|is
n n
bounded, or |t1|→∞ and |t2|→∞, or |t1|→∞ and |t2| is bounded. In the first
n n n n
case, u (t1) and u (t1) belong to a relatively compact subset of H1(RN) and we
1 n 2 n
obtainacontradictionby applying Lemma 4.5, since |x1 −x2|→∞. Suppose now
n n
t1 → ∞ and t2 → ±∞. Since u and u scatter we may approximate u (tj) by
n n 1 2 j n
eitjn∆ψj, j =1,2,whereψj is the scatteringstate ofuj, andwe deduce from(4.12)
that for n large,
ε
|(ei(t1n−t2n)∆ψ1(·−x1n+x2n),ψ2)H1|≥ 2.
Since |t1n−t2n|+|x1n−x2n|→∞, we seethat ei(t1n−t2n)∆ψ1(·−x1n+x2n)⇀0, andwe
obtain again a contradiction. Finally, |t2| is bounded and, say, t1 → ∞ a similar
n n
argument yields a contradiction. (cid:3)
We now construct the wave operator at −∞ on a certain subset of H1(RN).
8 DAOYUANFANG,JIANXIE,ANDTHIERRYCAZENAVE
Proposition 4.6. Let 0 < ω < 1 and K be defined by (2.2). If ψ ∈ H1(RN)
ω
satisfies
1
k∇ψk2 M(ψ)σ ≤ωE(Q)M(Q)σ, (4.13)
2 L2
then there exists ϕ∈K such that
ω
kS(−t)ϕ−e−it∆ψk −→0.
H1
t→∞
Proof. Given any ψ ∈ H1(RN), is well-known [22, Theorem 17] that there exist
T ∈R and a sulution u∈C((−∞,T],H1(RN)) of (NLS) such that
ku(−t)−e−it∆ψk −→ 0. (4.14)
H1
t→∞
It follows from (4.14), (4.13), and (3.3) that
lim k∇u(−t)k2 ku(−t)k2σ =k∇ψk2 kψk2σ ≤2ωE(Q)M(Q)σ
L2 L2 L2 L2
t→∞
Nα−4
=ω k∇Qk2 kQk2σ.
Nα L2 L2
Therefore, if t is sufficiently large, then
k∇u(−t)k ku(−t)kσ <k∇Qk kQkσ . (4.15)
L2 L2 L2 L2
It also follows from (4.14) that ku(−t)−e−it∆ψk → 0. Since ke−it∆ψk → 0
Lr Lr
(see e.g. Corollary 2.3.7 in [4]), we see that ku(−t)kLr →0, so that
1
E(u(−t)) −→ k∇ψk2 .
t→∞2 L2
Nota also that M(u(−t))→kψk2 , so that
L2
1
E(u(−t))M(u(−t))σ −→ k∇ψk2 kψk2σ ≤ωE(Q)M(Q)σ, (4.16)
t→∞2 L2 L2
where we used (4.13) inthe lastinequality. (4.15), (4.16) andconservationofmass
and energy imply that u(−t) ∈ K for all t ≤ T. In particular, the solution u is
ω
global and, since K is invariant by the flow S(t) (see (4.7)), u(t) ∈ K for all
ω ω
t∈R. The result follows by setting ϕ=u(0). (cid:3)
Finally, we prove a perturbation result. It is analogous to Theorem 2.14 in [16]
(for the energy-critical equation) and Proposition 2.3 in [13] (for the 3D cubic
equation). Theproofsoftheaboveresultswouldapplywithobviousmodifications,
but we use a slightly more direct argument, based on a Gronwall-type inequality
(Lemma 8.1).
Proposition 4.7. Given any A ≥ 0, there exist ε(A) > 0 and C(A) > 0 with
the following property. If u ∈ C([0,∞),H1(RN)) is a solution of (NLS), if u ∈
C([0,∞),H1(RN)) and e∈L1 ([0,∞),H−1(RN)) satisfy
loc
iu +∆u+|u|p−1u=e, e
t
for a.a. t>0, and if
e e e e
kukLa([0,∞),Lr) ≤A, kekLb′([0,∞),Lr′) ≤ε≤ε(A),
(4.17)
kei·∆(u(0)−u(0))k ≤ε≤ε(A),
La([0,∞),Lr)
e
then u∈La((0,∞),Lr(RN)) and ku−uk ≤Cε.
La([0,∞),Lr)
e
e
SCATTERING FOR THE FOCUSING ENERGY-SUBCRITICAL NLS 9
Proof. We let w=u−u, so that
iw +∆w+|u+w|α(u+w)−|u|αu+e=0. (4.18)
t e
Since
e e e e
|u+w|α(u+w)−|u|αu ≤C(|u|α+|w|α)|w|=C(|u|α|w|+|w|α+1), (4.19)
(cid:12) (cid:12)
we (cid:12)(cid:12)deeduce froem the equeatieo(cid:12)(cid:12)n (4.18e), Strichartz-type esetimate estimate (4.3) and
the assumption (4.17) that there exists M >0 such that
kwkLa((0,t),Lr) ≤ε+Mk|u|α|w|+|w|α+1kLb′((0,t),Lr′)+Mε, (4.20)
for every t > 0. Since k|u|α|w|keLr′ ≤ ku(t)kαLrkw(t)kLr, we deduce from (4.20)
that
kϕkLa(0,t) ≤e(M +1)ε+MkeϕkαL+a(10,t)+MkfϕkLb′(0,t), (4.21)
where we have set
ϕ(t)=kw(t)k , f(t)=ku(t)kα .
Lr Lr
Let
e
ε(A)≤2−p−11[(2M +1)Φ(Aα)]−αα+1, (4.22)
where Φ is given by Lemma 8.1. Observe that
kfkL(α+α1)b′(0,∞) =kfkLαa(0,∞) =kukαLa((0,∞),Lr) ≤Aα. (4.23)
Given any 0 < T ≤ ∞ such that kϕkα+1 ≤ eε ≤ ε(A), we deduce from (4.21),
La(0,T)
(4.23) and Lemma 8.1 that kϕk ≤ (2M +1)εΦ(Aα). Applying (4.22), we
La(0,T)
see that kϕkα+1 < εα+1ε(A)−α/2 ≤ ε/2. It easily follows that we may let
La(0,T)
T →∞, so that kϕk ≤(2M +1)Φ(Aα)ε. This is the desired estimate with
La(0,∞)
C(A)=(2M +1)Φ(Aα). (cid:3)
5. Profile decomposition
The following profile decomposition property is an essential ingredient in the
proof of Theorem 1.1. A quite similar property is established in [17], and applied
in [16] to the study of the energy critical NLS. (An analogous result for the wave
equationisprovedin[1].) Aresultsimilarto Theorem5.1isprovedin[6], adapted
to the 3D cubic NLS.
Theorem 5.1. Let (φ ) be a bounded sequence of H1(RN). There is a sub-
n n≥1
sequence, which we still denote by (φ ) , and sequences (ψj) ⊂ H1(RN),
n n≥1 j≥1
(Wj) ⊂H1(RN), (tj) ⊂[0,∞), (tj) ⊂[0,∞)∪{∞} and (xj) ⊂
n n,j≥1 n n,j≥1 j≥1 n n,j≥1
RN such that for every ℓ≥1
ℓ
φn = e−itjn∆ψj(·−xjn)+Wnℓ, (5.1)
j=1
X
10 DAOYUANFANG,JIANXIE,ANDTHIERRYCAZENAVE
and
tℓ −→ tℓ, (5.2)
n
n→∞
ℓ
kφ k2 − kψjk2 −kWℓk2 −→ 0, ∀0≤λ≤1, (5.3)
n Hλ Hλ n Hλn→∞
j=1
X
ℓ
E(φn)− E(e−itjn∆ψj(·−xjn))−E(Wnℓ) −→ 0. (5.4)
n→∞
j=1
X
Furthermore, there exists J ∈N∪{∞} such that ψj 6=0 for all j <J and ψj =0
for all j ≥J, and
lim |ti −tj|+|xi −xj|=∞, (5.5)
n n n n
n→∞
for all ℓ≥1 and 1≤i6=j <J. In addition,
limsupkei·∆Wℓk −→ 0. (5.6)
n La((0,∞),Lr)
n→∞ ℓ→∞
Theorem 5.1 is proved by iterative application of the following lemma.
Lemma 5.2. Let a>0 and let (v ) ⊂H1(RN) satisfy
n n≥1
limsupkv k ≤a<∞. (5.7)
n H1
n≥1
If
kei·∆vnkL∞((0,∞),Lr) −→ A, (5.8)
n→∞
then there exist a subsequence, which we still denote by (v ) , and sequences
n n≥1
(t ) ⊂[0,∞), (x ) ⊂RN, ψ ∈H1(RN) and (W ) ⊂H1(RN) such that
n n≥1 n n≥1 n n≥1
vn =e−itn∆ψ(·−xn)+Wn, (5.9)
with
eitn∆v (·+x ) ⇀ ψ, (5.10)
n n
n→∞
or, equivalently,
eitn∆Wn(·+xn) ⇀ 0, (5.11)
n→∞
in H1(RN),
kv k2 −kψk2 −kW k2 −→ 0, (5.12)
n H˙λ H˙λ n H˙λn→∞
for all 0≤λ≤1 and
kvnkαL+r2−ke−itn∆ψ(·−xn)kαL+r2−kWnkαL+r2 −→ 0. (5.13)
n→∞
Moreover,
kψkH1 ≥νA2Nλ−(12−λλ2)a−2λN(−1−2λλ), (5.14)
where
Nα N
λ= ∈ 0,min 1, , (5.15)
2(α+2) 2
(cid:16) n o(cid:17)
and the constant ν >0 is independent of a, A and (v ) . Finally, if A=0, then
n n≥1
for every sequences satisfying (5.9) and (5.11), we must have ψ =0.