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Scattering for NLS with a potential on the line

Abstract

We show the H1 scattering for a one dimensional nonlinear Schrödinger equation with a non-negative, repulsive potential V such that V,xVW1,1, and a mass-supercritical non-linearity. We follow the approach of concentration-compacity/rigidity first introduced by Kenig and Merle.

1.Introduction

We consider the following one dimensional defocusing, non linear Schrödinger equation with a potential

(1.1)itu+ΔuVu=u|u|α,u(0)=φH1(R).
If VL1, Δ+V is essentially self-adjoint, so by Stones theorem the equation is globally well posed in L2(R) and eit(Δ+V) is an L2-isometry. Goldberg and Schlag obtained in [7] the dispersive estimate
eit(Δ+V)ψL1|t|12ψL1
under the assumption that V belongs to L11(R), i.e. |V(x)|(1+|x|)dx<, and that Δ+V has no resonance at zero energy. In particular, we will consider a non-negative potential, which always verifies this no-resonance hypothesis as we will see in Section 2. This estimate gives us usual Strichartz estimates described below in the paper. Because of the energy conservation law
E(u(t)):=12|u(t)|2+V|u(t)|2+1α+2|u(t)|α+2=E(u(0))
the L2-well-posedness result extends to the global well-posedness of the problem (1.1) in H1(R): for every φH1(R), there exists a unique, global solution uC(R,H1(R)) of (1.1). Finally, let us recall that the mass M(u(t)):=|u(t)|2 is conserved too.

For the mass-supercritical (α>4) homogeneous equation

(1.2)itu+Δu=u|u|α,u(0)=φH1(R)
it is well known since Nakanishi’s paper [12] that the solutions scatter in H1(R), that is, for every solution uC(R,H1(R)) of (1.2), there exists a unique couple of data ψ±H1(R) such that
u(t)eitΔψ±H1(R)t±0.
Alternative proofs of this result can be found in [2,5,13] and [14].

We prove the scattering of solutions of (1.1) in dimension one for sufficiently regular, non-negative and repulsive potential V.

Theorem 1.

Let α>4 and VL11(R) be such that VL11(R). We suppose moreover that V is non-negative and repulsive: V0 and xV0. Then, every solution uC(R,H1(R)) of (1.1) with potential V scatters in H1(R).

We use the strategy of concentration-compacity/rigidity first introduced by Kenig and Merle in [11], and extented to the intercritical case by Holmer and Roudenko in [8], Duyckaerts, Holmer and Roudenko in [3]. In the case of a potential, the main difficulty is the lack of translation invariance of the equation. Notice that Hong obtained in [9] the same result in the three dimensional case for the focusing equation. However, his approach cannot be extended to lower dimensions, as it requires endpoint Strichartz estimates which are not available. Banica and Visciglia treated in [1] the case of the non linear Schrödinger equation with a Dirac potential on the line, and we follow their approach. The Dirac potential is more singular, but it allows the use of explicit formulas that are not available in the present more general framework.

Remark 1.

In dimension one or two, assume that V is smooth and compactly supported, and such that V<0. Then the operator Δ+V has a negative eigenvalue: as a consequence, the hypothesis of positivity of V cannot be relaxed as in dimension three, where [9] only supposes that the potential has a small negative part, and, in the same way, the hypothesis of repulsivity, which is needed for the rigidity, cannot be relaxed to xV having a small positive part.

Remark 2.

The hypothesis V,VL1 are needed to show that the operator A=Δ+V verifies the hypothesis of the abstract profile decomposition of [1], whereas the hypothesis xVL1 and xV0 are needed in the rigidity part.

Remark 3.

The same proof holds in dimension two up to the numerology and some changes in the Hölder inequalities used in Propositions 6, 7, and 8 to deal with the fact that H1(R2) is not embedded in L(R2).

Remark 4.

In the focusing, mass-supercritical case

itu+ΔuVu+u|u|α=0
the same arguments could be used to prove the scattering up to the natural threshold given by the ground state associated to the equation, in the spirit of [5].

1.1.Notations

We will denote by V a potential on the line satisfying the hypothesis of Theorem 1, α will be a real number such that α>4. We set

H1=H1(R),C(H1)=C(R,H1(R)),LpLr=Lp(R,Lr(R)),Lp(I)Lr=Lp(I,Lr(R))
for any interval I of R. We will denote by τy the translation operator defined by τyu=u(·y). Finally, we will use AB for inequalities of the type ACB where C is a universal constant.

2.Preliminaries

From now on, we will fix the four following Strichartz exponents

r=α+2,q=2α(α+2)α2α4,p=2α(α+2)α+4,γ=2αα2.

2.1.Strichartz estimates

Recall that we assume all along the paper that V is in L11(R) and non negative. Goldberg and Schlag obtained in particular in [7] the dispersive estimate for the Schrödinger operator Δ+V under these assumptions.

Indeed, they require the hypothesis of absence of resonances at zero energy. We claim that for V0 this hypothesis is satisfied: by the definition of [7], if there is a resonance at zero, the solutions u± of

(2.1)u=Vu
such that u±(x)1 as x± have a null Wronskian. Therefore u± are proportional, so they are both non trivial bounded solutions of (2.1). But such solutions cannot exist: indeed, if u is such a solution, integrating (2.1) one deduces that u has limits at ±. These limits are both zero otherwise u is not bounded. Now, multiplying (2.1) by u, integrating it on [R,R], and letting R going to infinity, we obtain R|u|2+V|u|2=0. Therefore u=0, a contradiction.

Proposition 1

Proposition 1(Dispersive estimate [7]).

Let VL11(R) be such that V0. Then, for all ψL1(R), we have

(2.2)eit(Δ+V)ψL1|t|12ψL1.

Note that, interpolating the previous dispersive estimate (2.2) with the mass conservation law, we obtain immediately for all a[2,]

(2.3)eit(Δ+V)ψLa1|t|12(1a1a)ψLa.

Because of (2.2), we obtain by the classical TT method (see for example [10]) the Strichartz estimates

(2.4)eit(Δ+V)φLq1Lr1+0tei(ts)(Δ+V)F(s)dsLq2Lr2φL2+FLq3Lr3
for all pairs (qi,ri) satisfying the admissibility condition in dimension one, that is
2qi+1ri=12.
We will need moreover the following Strichartz estimates associated to non admissible pairs:
Proposition 2

Proposition 2(Strichartz estimates).

For all φH1, all FLqLr, all GLqLr and all HLγL1

(2.5)eit(Δ+V)φLpLrφH1(2.6)eit(Δ+V)φLαLφH1(2.7)0tei(ts)(Δ+V)F(s)dsLαLFLqLr(2.8)0tei(ts)(Δ+V)G(s)dsLpLrGLqLr(2.9)0tei(ts)(Δ+V)H(s)dsLpLrHLγL1.

Proof.

The estimates (2.5)–(2.8) are exactly the same as (3.1)–(3.4) of [14], with the operator Δ+V instead of Hq. As the proof of [14] relies only on the admissible Strichartz estimates (2.4) that are given by Proposition 1, the same proof holds here. Finally, (2.9) enters on the frame of the non-admissible inhomogeneous Strichartz estimates of Theorem 1.4 of Foschi’s paper [6]. □

2.2.Perturbative results

We will need the three following classical perturbative results, which follow immediately from the previous Strichartz inequalities:

Proposition 3.

Let uC(H1) be a solution of (1.1). If uLpLr, then u scatters in H1.

Proposition 4.

There exists ϵ0>0, such that, for every data φH1 such that φH1ϵ0, the corresponding maximal solutions of (1.1) and (1.2) both scatter in H1.

Proof of Propositions 3and 4.

The proof is the same as for Propositions 3.1 and 3.2 of [1], using the Strichartz estimates of our Proposition 2 instead of their estimates (3.1), (3.2), (3.3), (3.4). □

Proposition 5.

For every M>0 there exists ϵ>0 and C>0 such that the following occurs. Let vC(H1)LpLr be a solution of the following integral equation with source term e(t,x)

v(t)=eit(ΔV)φi0tei(ts)(ΔV)(v(s)|v(s)|α)ds+e(t)
with vLpLr<M and eLpLr<ϵ. Assume moreover that φ0H1 is such that eit(ΔV)φ0LpLr<ϵ. Then, the solution uC(H1) to (1.1) with initial condition φ+φ0 satisfies
uLpLr,uvLpLr<C.

Proof.

It is the same as for Proposition 4.7 in [5], using Strichartz estimates (2.8) instead of Strichartz-type inequality (4.3) of their paper. □

3.Profile decomposition

The aim of this section is to show that we can use the abstract profile decomposition obtained by [1], and inspired by [4]:

Theorem

Theorem(Astract profile decomposition [1]).

Let A:L2D(A)L2 be a self adjoint operator such that:

  • for some positive constants c,C and for all uD(A),

    (3.1)cuH12(Au,u)+uL22CuH12,

  • let B:D(A)×D(A)(u,v)(Au,v)+(u,v)L2(u,v)H1C. Then, as n goes to infinity

    (3.2)B(τxnψ,τxnhn)0ψH1
    as soon as
    xn±,suphnH1<
    or
    xnx¯R,hnH10,

  • let (tn)n1, (xn)n1 be sequences of real numbers, and t¯,x¯R. Then

    (3.3)|tn|eitnAτxnψLp0,2<p<,ψH1,(3.4)tnt¯,xn±ψH1,φH1,τxneitnAτxnψH1φ,(3.5)tnt¯,xnx¯ψH1,eitnAτxnψH1eit¯Aτx¯ψ.

And let (un)n1 be a bounded sequence in H1. Then, up to a subsequence, the following decomposition holds
un=j=1JeitjnAτxnjψj+RnJJN,
where
tjnR,xjnR,ψjH1
are such that
  • for any fixed j,

    (3.6)tjn=0n, or tnjn±(3.7)xjn=0n, or xnjn±,

  • orthogonality of the parameters:

    (3.8)|tjntkn|+|xjnxkn|n,jk,

  • decay of the reminder:

    (3.9)ϵ>0,JN,lim supneitARnJLLϵ,

  • orthogonality of the Hilbert norm:

    (3.10)unL22=j=1JψjL22+RnJL22+on(1),JN(3.11)unH2=j=1JτxnjψjH2+RnJH2+on(1),JN
    where (u,v)H=(Au,v), and
    (3.12)unLpp=j=1JeitjnAτxnjψjLpp+RnJLpp+on(1),2<p<,JN.

We will see that the self-adjoint operator A:=Δ+V verifies the hypothesis of the previous theorem.

Proposition 6.

Let A:=Δ+V. Then A satisfies the assumptions (3.1), (3.2), (3.3), (3.4), (3.5).

Proof.

Assumption (3.1). Because V is positive and by the Sobolev embedding H1(R)L,

uH12(Au,u)+uL2=|u|2+V|u|2+|u|2(1+VL1)uH12
and (3.1) holds.

Assumption (3.2). We have

B(τxnψ,τxnhn)=Vτxnψτxnhn.
If xnx¯R,hnH10, then τxnψτx¯ψ strongly in L2 and Vτxnhn0 weakly in L2 (indeed, note that VW1,1(R)L2), so B(τxnψ,τxnhn)0. Now, let us assume that xn± and suphnH1<. For example assume that xn+. ψH1(R) and therefore decays at infinity: ϵ>0 been fixed, we can choose Λ>0 large enough so that
sup|x|Λ|ψ(x)|ϵ.
Because VL1, Λ can also be chosen large enough so that
|x|Λ|V|ϵ.
Then, by the Cauchy–Schwarz inequality, and because of the Sobolev embedding H1(R)L
|B(τxnψ,τxnhn)|hnL|Vτxnψ|supj1hjH1(|xxn|Λ|Vψ(·xn)|+|xxn|Λ|Vψ(·xn)|).
Now, let n0 be large enough so that for all nn0, xn2Λ. Then, for all nn0
|xxn|Λ|x|Λ
and, for all nn0
|B(τxnψ,τxnhn)|M(ϵVL1+ϵψL)
so (3.2) holds.

Assumption (3.3). It is an immediate consequence of the dispersive estimate and the translation invariance of the Lp norms. Indeed, because H01(R)=H1(R), if ϵ>0, there exists a C, compactly supported function ψ˜ such that

(3.13)ψ˜ψH1ϵ.
But ψ˜Lp, so by the dispersive estimate (2.3)
eitnAτxnψ˜Lp1|tn|12(1p1p)τxnψ˜Lp=1|tn|12(1p1p)ψ˜Lp0
as n. Therefore, for n big enough
(3.14)eitnAτxnψ˜Lpϵ.
To achieve the proof, note that eitAf verifies
(3.15)eitAfH1fH1.
Indeed, as V is positive and in L1, by the Sobolev embedding H1(R)L we get
fL22(Δ+V)12fL22=|u|2+V|u|2fH1.
So, as eitA commute with (Δ+V)12 and is an isometry on L2,
eitAfH12eitAfL22+(Δ+V)12eitAfL22=eitAfL22+eitA(Δ+V)12fL22=fL22+(Δ+V)12fL22fH12.
Now, because of the Sobolev embedding H1Lp we obtain using (3.13), (3.14) and (3.15), for n big enough
eitnAτxnψLpeitnAτxn(ψψ˜)Lp+eitnAτxnψ˜LpeitnAτxn(ψψ˜)H1+eitnAτxnψ˜Lpψψ˜H1+eitnAτxnψ˜Lp2ϵ
which achieves the proof of (3.3).

Assumption (3.4). We will show that

tnt¯,xn±τxneitn(Δ+V)τxnψeit¯ΔψH10
and hence (3.4) will hold with φ=eit¯Δψ. As τxn is an H1 isometry and commute with eit¯Δ, it is sufficient to show that, if tnt¯ and xn±, we have
eitn(Δ+V)τxnψeit¯ΔτxnψH10.
For example, if xn+. Let us first remark that, as τxn commutes with eit¯Δ and eitnΔ, is an H1 isometry, and because eitΔψC(H1)
eit¯ΔτxnψeitnΔτxnψH1=eit¯ΔψeitnΔψH10.
Hence, decomposing
eitn(Δ+V)τxnψeit¯Δτxnψ=(eitn(Δ+V)τxnψeitnΔτxnψ)+(eitnΔτxnψeit¯Δτxnψ)
we see that it is sufficient to show that
(3.16)eitn(Δ+V)τxnψeitnΔτxnψH10.

Note that eitΔτxnψeit(Δ+V)τxnψ is a solution of the following linear Schrödinger equation with zero initial data

ituΔu+Vu=VeitΔτxnψ.
Therefore, by the inhomogenous Strichartz estimates, as (4,) is admissible in dimension one, and because the translation operator commutes with eitΔ, we have for n large enough so that tn(0,t¯+1)
eitn(Δ+V)τxnψeitnΔτxnψL2eit(Δ+V)τxnψeitΔτxnψL(0,t¯+1)L2VeitΔτxnψL43(0,t¯+1)L1=(τxnV)eitΔψL43(0,t¯+1)L1(t¯+1)34(τxnV)eitΔψL(0,t¯+1)L1.
Hence, estimating in the same manner the gradient of these quantities, it is sufficient to obtain (3.16) to show that, as n goes to infinity
(3.17)(τxnV)eitΔψL(0,t¯+1)W1,10.

Let us fix ϵ>0. eitΔψC([0,t¯+1],H1) and the functions of H1(R) vanish at infinity, so, using the compacity in time, there exists Λ>0 such that

eitΔψL(0,t¯+1)L(|x|Λ)ϵ.
On the other hand, as VL1, Λ can also be taken large enough so that
|x|Λ|V(x)|dxϵ.
Let n0 be large enough so that for all nn0, xn2Λ. Then, for nn0
|x+xn|Λ|x|Λ
and for all t(0,t¯+1) and all nn0 we obtain
(τxnV)eitΔψL1=|x+xn|Λ|V(·+xn)eitΔψ|+|x+xn|Λ|V(·+xn)eitΔψ|ϵeitΔψL(0,t¯+1)L+ϵVL1C(t¯,ψ,V)ϵ
thus (τxnV)eitΔψL(0,t¯+1)L10. With the same argument, because VL1, we can show that (τxnV)eitΔψL(0,t¯+1)L10. To obtain (3.17), it only remain to show that
τxnV(eitΔψ)L(0,t¯+1)L10.
To this purpose, let ψ˜ be a C, compactly supported function such that (recall that we are in dimension one)
ψψ˜H1ϵ.
We have, by the Cauchy–Schwarz inequality
τxnV(eitΔψ)L1τxnV(eitΔψ˜)L1+τxnV(eitΔ(ψψ˜))L1τxnV(eitΔψ˜)L1+VL2(eitΔ(ψψ˜))L2τxnV(eitΔψ˜)L1+ϵVL2,
where VL2 because of the Sobolev embedding W1,1(R)L2(R). Then, as (eitΔψ˜)H1, τxnV(eitΔψ˜)L(0,t¯+1)L1 can be estimated as (τxnV)eitΔψL(0,t¯+1)L1, so (3.17) holds and the proof of (3.4) is completed.

Assumption (3.5). We decompose

eitnAτxnψeit¯Aτx¯ψ=(eitnAτxnψeitnAτx¯ψ)+(eitnAτx¯ψeit¯Aτx¯ψ).
On the one hand, using the estimate (3.15)
eitnAτxnψeitnAτx¯ψH1τxnψτx¯ψH10n
by the Lebesgue’s dominated convergence theorem. On the other hand,
eitnAτx¯ψeit¯Aτx¯ψH10n
because ei·Aτx¯ψC(H1), and the last assumption is verified. □

4.Non linear profiles

In this section, we will see that for a data which escapes to infinity, the solutions of (1.1) and (1.2) are the same, in the sense given by the three following Propositions.

Propositions 7, 8 and 9 are the analogous of Propositions 3.4 and 3.6 of [1]. The non linear Schrödinger equation with a Dirac potential is more singular, but it allows the use of explicit formulas that are not available in the present more general framework.

Proposition 7.

Let ψH1, (xn)n1RN be such that |xn|. Then, up to a subsequence

(4.1)eitΔτxnψeit(ΔV)τxnψLpLr0
as n.

Proof.

Up to a subsequence, we can assume that xn+ or xn. Let us assume for example xn+.

As a first step, we will show that

(4.2)supnNeit(Δ+V)τxnψLp(T,)Lr0
as T. Pick ϵ>0. There exists a C, compactly supported function ψ˜ such that
ψ˜ψH1ϵ.
By Strichartz estimates
eit(Δ+V)(τxnψ˜τxnψ)LpLrτxnψ˜τxnψH1=ψ˜ψH1ϵ.
On the other hand, as τxnψ˜Lr the dispersive estimate (2.3) gives us
eit(Δ+V)τxnψ˜Lr1|t|12(1r1r)τxnψ˜Lr=1|t|12(12r)ψ˜Lr
but p2(12r)=α2α+4>1 and t1|t|12(12r)Lq(1,). So, there exists T>0 such that
supnNeit(Δ+V)τxnψ˜Lp(|t|T)Lrϵ.
Taking τxnψ=τxnψ˜+(τxnψτxnψ˜), we then obtain for T>0 large enough
supnNeit(Δ+V)τxnψLp(|t|T)Lrϵ
and (4.2) holds.

To obtain (4.1), we are now reduced to show that for T>0 fixed

eitΔτxnψeit(Δ+V)τxnψLp(0,T)Lr0
as n. Let ϵ>0. eitΔτxnψeit(Δ+V)τxnψ is a solution of the following linear Schrödinger equation with zero initial data
ituΔu+Vu=VeitΔτxnψ.
So, by the inhomogenous Strichartz estimate (2.9)
eitΔτxnψeit(Δ+V)τxnψLtp(0,T)LrVeitΔτxnψLtγ(0,T)L1T1γVeitΔτxnψL(0,T)L1=T1γ(τxnV)eitΔψL(0,T)L1
because the translation operator τxn commutes with the propagator eitΔ. But
(τxnV)eitΔψL(0,T)L1n0
as seen in the proof of Proposition 6, point (3.4). □

Proposition 8.

Let ψH1, (xn)n1RN be such that |xn|, UC(H1)LpLr be the unique solution to (1.2) with initial data ψ, and Un(t,x):=U(t,xxn). Then, up to a subsequence

(4.3)0tei(ts)Δ(Un|Un|α)(s)ds0tei(ts)(ΔV)(Un|Un|α)(s)dsLpLr0
as n.

Proof.

We follow the same spirit of proof as for Proposition 7. We begin to show that

(4.4)supnN0tei(ts)(ΔV)(Un|Un|α)(s)dsLp([T,))Lr0
as T goes to infinity.

We decompose

0tei(ts)(ΔV)(Un|Un|α)(s)dsLp([T,))Lr0Tei(ts)(ΔV)(Un|Un|α)(s)dsLp([T,))Lr+Ttei(ts)(ΔV)(Un|Un|α)(s)dsLp([T,))Lr
where, by the inhomogenous Strichartz estimates
Ttei(ts)(ΔV)(Un|Un|α)(s)dsLp([T,)Lr)Un|Un|αLq([T,)Lr)=U|U|αLq([T,)Lr)
and, by the Hölder inequality
U|U|αLq([T,)Lr)ULp([T,)Lr)α+1T0
independently of n. On the other hand, by the dispersive estimate (2.3)
0Tei(ts)(ΔV)(Un|Un|α)(s)dsLp[T,)Lr0Tei(ts)(ΔV)(Un|Un|α)(s)LrdsLp([T,))0T(ts)12(12r)(Un|Un|α)(s)LrdsLp([T,))=0T(ts)12(12r)(U|U|α)(s)LrdsLp([T,))R|ts|12(12r)(U|U|α)(s)LrdsLp([T,))0
as T goes to infinity. Indeed, note that by the Hardy–Littlewood–Sobolev inequality
R|ts|12(12r)(U|U|α)(s)LrdsLpU|U|αLqLrULpLrα+1<
so (4.4) holds. The same estimate is obviously valid for the propagator eitΔ.

It remains to show that for T>0 fixed,

0tei(ts)Δ(Un|Un|α)ds0tei(ts)(ΔV)(Un|Un|α)dsLp(0,T)Lr0
as n. The difference
0tei(ts)Δ(Un|Un|α)ds0tei(ts)(ΔV)(Un|Un|α)ds
is the solution of the following linear Schrödinger equation, with zero initial data
ituΔu+Vu=V0tei(ts)Δ(Un|Un|α)ds.
As a consequence, by the Strichartz estimate (2.9)
0tei(ts)Δ(Un|Un|α)ds0tei(ts)(ΔV)(Un|Un|α)dsLp(0,T)LrV0tei(ts)Δ(Un|Un|α)dsLγ(0,T)L1T1γ(τxnV)0tei(ts)Δ(U|U|α)dsL(0,T)L1.
But 0tei(ts)Δ(U|U|α)dsC([0,T],H1) and the functions of H1(R) vanish at infinity, so there exists Λ>0 such that
0tei(ts)Δ(U|U|α)dsL(0,T)L(|x|Λ)ϵ
so
(τxnV)0tei(ts)Δ(U|U|α)dsL(0,T)L1n0
in the same way as in the proof of Proposition 6, point (3.4). □

Proposition 9.

Let ψH1, (xn)n1, (tn)n1RN be such that |xn| and tn±, U be a solution to (1.2) such that

U(t)eitΔψH1t±0
and Un(t,x):=U(ttn,xxn). Then, up to a subsequence
(4.5)ei(ttn)Δτxnψei(ttn)(ΔV)τxnψLpLr0
and
(4.6)0tei(ts)Δ(Un|Un|α)ds0tei(ts)(ΔV)(Un|Un|α)dsLpLr0
as n.

Proof.

The proof is the same as for Proposition 7 and Proposition 8, decomposing the time interval in {|ttn|>T} and its complementary. □

Finally, we will need the following Proposition of non linear scattering:

Proposition 10.

Let φH1. Then there exists W±C(H1)LR±pLr, solution of (1.1) such that

(4.7)W±(t,·)eit(ΔV)φH1t±0
moreover, if tn and
(4.8)φn=eitn(ΔV)φ,W±,n(t)=W±(ttn)
then
(4.9)W±,n(t)=eit(ΔV)φn+0tei(ts)(ΔV)(W±,n|W±,n|α)(s)ds+f±,n(t)
where
(4.10)f±,nLR±pLrn0.

Proof.

The same proof as [1], Proposition 3.5, holds, as it involves only the analogous Strichartz estimates. □

5.Construction of a critical element

We have now all the tools to extract a critical element following the approach of [5]. Let

Ec=sup{E>0|φH1,E(φ)<Ethe solution of (1.1) with data φ is in LpLr}.
We will suppose that the critical energy Ec is finite, and deduce the existence of a solution of (1.1) with a relatively compact flow in H1.
Proposition 11.

If Ec<, then there exists φcH1, φc0, such that the corresponding solution uc of (1.1) verifies that {uc(t),t0} is relatively compact in H1.

Proof.

Because of Proposition 4, Ec>0. Therefore, if Ec<, there exists a sequence φn of non-zero elements of H1, such that, if we denote by unC(H1) the corresponding solution of (1.1), we have

E(φn)nEc
and
unLqLr.
Thanks to the Proposition 6, we can apply the abstract profile decomposition of [1] to the H1-bounded sequence φn and the operator A=Δ+V. Up to a subsequence, φn writes, for all JN:
φn=j=1Jeitjn(Δ+V)τxjnψj+RnJ,
where tjn, xjn, ψj, RnJ verifies (3.6)–(3.12). From (3.11) and (3.12), we have
Eclim supnj=1JE(eitjn(Δ+V)τxjnψj).
We show that there is exactly one non trivial profile, that is J=1. By contradiction, assume that J>1. To each profile ψj we associate family of non linear profiles (Uj,n)n0. Let j{1,,J}. We are in exactly one of the following situations:
  • (1) If (tjn,xjn)=(0,0). By the orthogonality condition, notice that this can happen only for one profile. Because J>1, we have E(ψj)<Ec, so the solution of (1.1) with data ψj scatters. If this case happens, let NC(H1)LpLr be this solution, otherwise, we set N=0.

  • (2) If tjn=0 and |xjn|. Let UjC(H1)LpLr be the unique solution to (1.2) with initial data ψj. We set Un,j(x,t):=U(xxjn,t).

  • (3) If xjn=0 and tjn±. By Proposition 10, there exists UjCR±(H1)LR±pLr a solution to (1.1) such that

    Uj(t)eit(ΔV)ψjH1t±0
    and verifying (4.7), (4.8), (4.9), (4.10). We have
    E(Uj)=limnE(eitjn(Δ+V)τxjnψj)<Ec
    so UjLqLr. We set Uj,n(t,x):=Uj(ttjn,x).

  • (4) If |xjn| and tjn±. Let UjC(H1)LpLr be a solution to (1.2) such that

    Uj(t)eitΔψjH1t±0.
    We set Uj,n(t,x):=Uj(ttjn,xxjn).

Now, let
Zn,J:=N+jUn,j.
By the results of the non linear profiles section – Propositions 7 and 8 in situation (2), Proposition 9 in situation (3) and Proposition 10 in situation (4), we have
Zn,J=eit(ΔV)(φnRn,J)+0tei(ts)(ΔV)(N|N|α)(s)ds(5.1)+j0tei(ts)(ΔV)(Uj,n|Uj,n|α)(s)ds+rn,J
with
rn,JLpLr0
as n. The decomposition (5.1) is the same as obtained in the proof of Proposition 4.1 of [1], and we therefore obtain the critical element following their proof, using our perturbative result of Proposition 5 instead of their Proposition 3.3, and the Strichartz inequalities of our Proposition 2 instead of estimates (3.1), (3.2), (3.3), (3.4) of their paper. □

6.Rigidity

In this section, we will show that the critical solution constructed in the previous one assuming the fact that Ec< cannot exist.

We will need the following classical result concerning the compact families of H1

Proposition 12.

Suppose that {u(t),t0} is relatively compact in H1. Then, for any ϵ>0, there exists R>0 such that

supt0|x|R(|u(t,x)|2+|u(t,x)|2+|u(t,x)|α+2)dxϵ.

Proof.

Classic, see e.g. [5]. □

Now, we can show the rigidity Proposition needed to end the proof:

Proposition 13.

Suppose that uC(H1) is a solution of (1.1) such that {u(t),t0} is relatively compact in H1. Then u=0.

Proof.

By a classical elementary computation, we get the following virial identities:

Lemma 1.

Let uC(H1) be a solution to (1.1) and χ be a compactly supported, regular function. Then

(6.1)tχ|u|2=2Imχuu¯(6.2)t2χ|u|2=4χ|u|2+2αα+2χ|u|α+22χV|u|2χ(4)|u|2.

Now, we assume by contradiction that u0. Let χCc be such that χ(x)=x2 for |x|1 and χ(x)=0 for |x|2, set χR:=R2χ(·R) and

zR(t)=χR|u(t)|2
we have, by (6.1), the Cauchy–Schwarz inequality and the conservation of energy
(6.3)|zR(t)|2|χR||u||u¯|CE(u)12M(u)12R.
Moreover, by (6.2)
zR(t)=4χR|u|2+2αα+2χR|u|α+22χRV|u|2χR(4)|u|28|x|R|u|2+4αα+2|x|R|u|α+2C|x|>R(|u|2+|u|α+2+|u|2)(6.4)2χRV|u|2χ(4)|u|2
but, because of conservation of the mass
(6.5)|χ(4)|u|2|CR2u(0)L2
and, because V is repulsive (i.e. xV0), using the Cauchy–Schwarz inequality, the Sobolev injection H1L and the conservation laws
2χRV|u|2=2|x|RxV|u|2+2|x|>RχRV|u|2C|x|>R|xV||u|2CxVL1(|x|>R)uL2(6.6)CxVL1(|x|>R)uH12C(u(0))xVL1(|x|>R).
Let R0 be large enough so that
(6.7)|x|R0|u|α+212|u|α+2:=δ.
We have δ>0 because we suppose that u is non zero. For RR0, we obtain combining (6.4) with (6.5), (6.6), and (6.7)
(6.8)zR(t)C(δ|x|>R(|u|2+|u|α+2+|u|2)1R2u(0)L2xVL1(|x|>R)).
Because xVL1 and using the compacity hypothesis combined with Proposition 12, there exists RR0 large enough so that
|x|>R(|u|2+|u|α+2+|u|2)+1R2u(0)L2+xVL1(|x|>R)δ2
then, (6.8) gives
zR(t)Cδ2>0.
Integrating this last inequality contradicts (6.3) as t. □

We are now in position to end the proof of Theorem 1:

Proof of Theorem 1.

If Ec<, then the Proposition 11 allows us to extract a critical element φcH1, φc0, such that the corresponding solution uc of (1.1) verifies that {uc(t),t0} is relatively compact in H1. By Proposition 13, such a critical solution cannot exist, so Ec= and by Proposition 3, all the solutions of (1.1) scatter in H1. □

Acknowledgements

The author thanks N. Visciglia for having submitted him this problem, enlighting discussions and his warm welcome in Pisa, J. Zheng for his helpfull comments, his Ph.D. advisor F. Planchon for his disponibility and advices, and the referee for his careful reading and his constructive criticism.

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