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Scattering for NLS with a potential on the line

Article type: Research Article

Authors: Lafontaine, David

Affiliations: Laboratoire de Mathématiques J.A. Dieudonné, UMR CNRS 7351, Université de Nice – Sophia Antipolis, 06108 Nice Cedex 02, France. E-mail: [email protected]

Keywords: nonlinear Schrödinger equation, scattering, potential perturbation

DOI: 10.3233/ASY-161384

Journal: Asymptotic Analysis, vol. 100, no. 1-2, pp. 21-39, 2016

Published: 7 November 2016
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Abstract

We show the H1 scattering for a one dimensional nonlinear Schrödinger equation with a non-negative, repulsive potential V such that V,xV∈W1,1, 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)i∂tu+Δu−Vu=u|u|α,u(0)=φ∈H1(R).
If V∈L1, −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
‖e−it(−Δ+V)ψ‖L∞≲1|t|12‖ψ‖L1
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 φ∈H1(R), there exists a unique, global solution u∈C(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)i∂tu+Δu=u|u|α,u(0)=φ∈H1(R)
it is well known since Nakanishi’s paper [12] that the solutions scatter in H1(R), that is, for every solution u∈C(R,H1(R)) of (1.2), there exists a unique couple of data ψ±∈H1(R) such that
‖u(t)−e−itΔψ±‖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 V∈L11(R) be such that V′∈L11(R). We suppose moreover that V is non-negative and repulsive: V⩾0 and xV′⩽0. Then, every solution u∈C(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,V′∈L1 are needed to show that the operator A=−Δ+V verifies the hypothesis of the abstract profile decomposition of [1], whereas the hypothesis xV′∈L1 and xV′⩽0 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

i∂tu+Δu−Vu+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 A≲B for inequalities of the type A⩽CB 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)α2−α−4,p=2α(α+2)α+4,γ=2αα−2.

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 V⩾0 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 V∈L11(R) be such that V⩾0. Then, for all ψ∈L1(R), we have

(2.2)‖e−it(−Δ+V)ψ‖L∞≲1|t|12‖ψ‖L1.

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)ψ‖La≲1|t|12(1a′−1a)‖ψ‖La′.

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

(2.4)‖e−it(−Δ+V)φ‖Lq1Lr1+‖∫0te−i(t−s)(−Δ+V)F(s)ds‖Lq2Lr2≲‖φ‖L2+‖F‖Lq3′Lr3′
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 φ∈H1, all F∈Lq′Lr′, all G∈Lq′Lr′ and all H∈Lγ′L1

(2.5)‖e−it(−Δ+V)φ‖LpLr≲‖φ‖H1(2.6)‖e−it(−Δ+V)φ‖LαL∞≲‖φ‖H1(2.7)‖∫0te−i(t−s)(−Δ+V)F(s)ds‖LαL∞≲‖F‖Lq′Lr′(2.8)‖∫0te−i(t−s)(−Δ+V)G(s)ds‖LpLr≲‖G‖Lq′Lr′(2.9)‖∫0te−i(t−s)(−Δ+V)H(s)ds‖LpLr≲‖H‖Lγ′L1.

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 u∈C(H1) be a solution of (1.1). If u∈LpLr, then u scatters in H1.

Proposition 4.

There exists ϵ0>0, such that, for every data φ∈H1 such that ‖φ‖H1⩽ϵ0, 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 v∈C(H1)∩LpLr be a solution of the following integral equation with source term e(t,x)

v(t)=e−it(Δ−V)φ−i∫0te−i(t−s)(Δ−V)(v(s)|v(s)|α)ds+e(t)
with ‖v‖LpLr<M and ‖e‖LpLr<ϵ. Assume moreover that φ0∈H1 is such that ‖e−it(Δ−V)φ0‖LpLr<ϵ. Then, the solution u∈C(H1) to (1.1) with initial condition φ+φ0 satisfies
u∈LpLr,‖u−v‖LpLr<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:L2⊃D(A)→L2 be a self adjoint operator such that:

  • for some positive constants c,C and for all u∈D(A),

    (3.1)c‖u‖H12⩽(Au,u)+‖u‖L22⩽C‖u‖H12,

  • let B:D(A)×D(A)∋(u,v)→(Au,v)+(u,v)L2−(u,v)H1∈C. Then, as n goes to infinity

    (3.2)B(τxnψ,τxnhn)→0∀ψ∈H1
    as soon as
    xn→±∞,sup‖hn‖H1<∞
    or
    xn→x¯∈R,hn⇀H10,

  • let (tn)n⩾1, (xn)n⩾1 be sequences of real numbers, and t¯,x¯∈R. Then

    (3.3)|tn|→∞⟹‖eitnAτxnψ‖Lp→0,∀2<p<∞,∀ψ∈H1,(3.4)tn→t¯,xn→±∞⟹∀ψ∈H1,∃φ∈H1,τ−xneitnAτxnψ→H1φ,(3.5)tn→t¯,xn→x¯⟹∀ψ∈H1,eitnAτxnψ→H1eit¯Aτx¯ψ.

And let (un)n⩾1 be a bounded sequence in H1. Then, up to a subsequence, the following decomposition holds
un=∑j=1JeitjnAτxnjψj+RnJ∀J∈N,
where
tjn∈R,xjn∈R,ψj∈H1
are such that

  • for any fixed j,

    (3.6)tjn=0∀n, or tnj→n→∞±∞(3.7)xjn=0∀n, or xnj→n→∞±∞,

  • orthogonality of the parameters:

    (3.8)|tjn−tkn|+|xjn−xkn|→n→∞∞,∀j≠k,

  • decay of the reminder:

    (3.9)∀ϵ>0,∃J∈N,lim supn→∞‖e−itARnJ‖L∞L∞⩽ϵ,

  • orthogonality of the Hilbert norm:

    (3.10)‖un‖L22=∑j=1J‖ψj‖L22+‖RnJ‖L22+on(1),∀J∈N(3.11)‖un‖H2=∑j=1J‖τxnjψj‖H2+‖RnJ‖H2+on(1),∀J∈N
    where (u,v)H=(Au,v), and
    (3.12)‖un‖Lpp=∑j=1J‖eitjnAτxnjψj‖Lpp+‖RnJ‖Lpp+on(1),∀2<p<∞,∀J∈N.

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∞,

‖u‖H12⩽(Au,u)+‖u‖L2=∫|∇u|2+∫V|u|2+∫|u|2⩽(1+‖V‖L1)‖u‖H12
and (3.1) holds.

Assumption (3.2). We have

B(τxnψ,τxnhn)=∫Vτxnψτxnhn‾.
If xn→x¯∈R,hn⇀H10, then τxnψ→τx¯ψ strongly in L2 and Vτxnhn⇀0 weakly in L2 (indeed, note that V∈W1,1(R)↪L2), so B(τxnψ,τxnhn)→0. Now, let us assume that xn→±∞ and sup‖hn‖H1<∞. For example assume that xn→+∞. ψ∈H1(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 V∈L1, 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)|⩽‖hn‖L∞∫|Vτxnψ|⩽supj⩾1‖hj‖H1(∫|x−xn|⩾Λ|Vψ(·−xn)|+∫|x−xn|⩽Λ|Vψ(·−xn)|).
Now, let n0 be large enough so that for all n⩾n0, xn⩾2Λ. Then, for all n⩾n0
|x−xn|⩽Λ⇒|x|⩾Λ
and, for all n⩾n0
|B(τxnψ,τxnhn)|⩽M(ϵ‖V‖L1+ϵ‖ψ‖L∞)
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˜−ψ‖H1⩽ϵ.
But ψ˜∈Lp′, so by the dispersive estimate (2.3)
‖eitnAτxnψ˜‖Lp≲1|tn|12(1p′−1p)‖τxnψ˜‖Lp′=1|tn|12(1p′−1p)‖ψ˜‖Lp′→0
as n→∞. Therefore, for n big enough
(3.14)‖eitnAτxnψ˜‖Lp⩽ϵ.
To achieve the proof, note that eitAf verifies
(3.15)‖eitAf‖H1≲‖f‖H1.
Indeed, as V is positive and in L1, by the Sobolev embedding H1(R)↪L∞ we get
‖∇f‖L22⩽‖(−Δ+V)12f‖L22=∫|∇u|2+∫V|u|2≲‖f‖H1.
So, as eitA commute with (−Δ+V)12 and is an isometry on L2,
‖eitAf‖H12⩽‖eitAf‖L22+‖(−Δ+V)12eitAf‖L22=‖eitAf‖L22+‖eitA(−Δ+V)12f‖L22=‖f‖L22+‖(−Δ+V)12f‖L22≲‖f‖H12.
Now, because of the Sobolev embedding H1↪Lp we obtain using (3.13), (3.14) and (3.15), for n big enough
‖eitnAτxnψ‖Lp⩽‖eitnAτxn(ψ−ψ˜)‖Lp+‖eitnAτxnψ˜‖Lp≲‖eitnAτxn(ψ−ψ˜)‖H1+‖eitnAτxnψ˜‖Lp≲‖ψ−ψ˜‖H1+‖eitnAτxnψ˜‖Lp⩽2ϵ
which achieves the proof of (3.3).

Assumption (3.4). We will show that

tn→t¯,xn→±∞⇒‖τ−xneitn(−Δ+V)τxnψ−e−it¯Δψ‖H1→0
and hence (3.4) will hold with φ=e−it¯Δψ. As τxn is an H1 isometry and commute with e−it¯Δ, it is sufficient to show that, if tn→t¯ and xn→±∞, we have
‖eitn(−Δ+V)τxnψ−e−it¯Δτxnψ‖H1→0.
For example, if xn→+∞. Let us first remark that, as τxn commutes with e−it¯Δ and e−itnΔ, is an H1 isometry, and because e−itΔψ∈C(H1)
‖e−it¯Δτxnψ−e−itnΔτxnψ‖H1=‖e−it¯Δψ−e−itnΔψ‖H1→0.
Hence, decomposing
eitn(−Δ+V)τxnψ−e−it¯Δτxnψ=(eitn(−Δ+V)τxnψ−e−itnΔτxnψ)+(e−itnΔτxnψ−e−it¯Δτxnψ)
we see that it is sufficient to show that
(3.16)‖eitn(−Δ+V)τxnψ−e−itnΔτxnψ‖H1→0.

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

i∂tu−Δu+Vu=Ve−itΔτxnψ.
Therefore, by the inhomogenous Strichartz estimates, as (4,∞) is admissible in dimension one, and because the translation operator commutes with e−itΔ, we have for n large enough so that tn∈(0,t¯+1)
‖eitn(−Δ+V)τxnψ−e−itnΔτxnψ‖L2⩽‖eit(−Δ+V)τxnψ−e−itΔτxnψ‖L∞(0,t¯+1)L2⩽‖Ve−itΔτxnψ‖L43(0,t¯+1)L1=‖(τ−xnV)e−itΔψ‖L43(0,t¯+1)L1⩽(t¯+1)34‖(τ−xnV)e−itΔψ‖L∞(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)‖(τ−xnV)e−itΔψ‖L∞(0,t¯+1)W1,1→0.

Let us fix ϵ>0. e−itΔψ∈C([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

‖e−itΔψ‖L∞(0,t¯+1)L∞(|x|⩾Λ)⩽ϵ.
On the other hand, as V∈L1, Λ can also be taken large enough so that
∫|x|⩾Λ|V(x)|dx⩽ϵ.
Let n0 be large enough so that for all n⩾n0, xn⩾2Λ. Then, for n⩾n0
|x+xn|⩽Λ⇒|x|⩾Λ
and for all t∈(0,t¯+1) and all n⩾n0 we obtain
‖(τ−xnV)e−itΔψ‖L1=∫|x+xn|⩾Λ|V(·+xn)e−itΔψ|+∫|x+xn|⩽Λ|V(·+xn)e−itΔψ|⩽ϵ‖e−itΔψ‖L∞(0,t¯+1)L∞+ϵ‖V‖L1⩽C(t¯,ψ,V)ϵ
thus ‖(τ−xnV)e−itΔψ‖L∞(0,t¯+1)L1→0. With the same argument, because V′∈L1, we can show that ‖(τ−xnV)′e−itΔψ‖L∞(0,t¯+1)L1→0. To obtain (3.17), it only remain to show that
‖τ−xnV(e−itΔψ)′‖L∞(0,t¯+1)L1→0.
To this purpose, let ψ˜ be a C∞, compactly supported function such that (recall that we are in dimension one)
‖ψ−ψ˜‖H1⩽ϵ.
We have, by the Cauchy–Schwarz inequality
‖τ−xnV(e−itΔψ)′‖L1⩽‖τ−xnV(e−itΔψ˜)′‖L1+‖τ−xnV(e−itΔ(ψ−ψ˜))′‖L1⩽‖τ−xnV(e−itΔψ˜)′‖L1+‖V‖L2‖(e−itΔ(ψ−ψ˜))′‖L2⩽‖τ−xnV(e−itΔψ˜)′‖L1+ϵ‖V‖L2,
where V∈L2 because of the Sobolev embedding W1,1(R)↪L2(R). Then, as (e−itΔψ˜)′∈H1, ‖τ−xnV(e−itΔψ˜)′‖L∞(0,t¯+1)L1 can be estimated as ‖(τ−xnV)e−itΔψ‖L∞(0,t¯+1)L1, so (3.17) holds and the proof of (3.4) is completed.

Assumption (3.5). We decompose

eitnAτxnψ−eit¯Aτx¯ψ=(eitnAτxnψ−eitnAτx¯ψ)+(eitnAτx¯ψ−eit¯Aτx¯ψ).
On the one hand, using the estimate (3.15)
‖eitnAτxnψ−eitnAτx¯ψ‖H1≲‖τxnψ−τx¯ψ‖H1⟶0n→∞
by the Lebesgue’s dominated convergence theorem. On the other hand,
‖eitnAτx¯ψ−eit¯Aτx¯ψ‖H1⟶0n→∞
because ei·Aτx¯ψ∈C(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 ψ∈H1, (xn)n⩾1∈RN be such that |xn|→∞. Then, up to a subsequence

(4.1)‖e−itΔτxnψ−e−it(Δ−V)τxnψ‖LpLr→0
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)supn∈N‖eit(−Δ+V)τxnψ‖Lp(T,∞)Lr→0
as T→∞. Pick ϵ>0. There exists a C∞, compactly supported function ψ˜ such that
‖ψ˜−ψ‖H1⩽ϵ.
By Strichartz estimates
‖eit(−Δ+V)(τxnψ˜−τxnψ)‖LpLr≲‖τxnψ˜−τxnψ‖H1=‖ψ˜−ψ‖H1⩽ϵ.
On the other hand, as τxnψ˜∈Lr′ the dispersive estimate (2.3) gives us
‖eit(−Δ+V)τxnψ˜‖Lr≲1|t|12(1r′−1r)‖τxnψ˜‖Lr′=1|t|12(1−2r)‖ψ˜‖Lr′
but p2(1−2r)=α2α+4>1 and t→1|t|12(1−2r)∈Lq(1,∞). So, there exists T>0 such that
supn∈N‖eit(−Δ+V)τxnψ˜‖Lp(|t|⩾T)Lr⩽ϵ.
Taking τxnψ=τxnψ˜+(τxnψ−τxnψ˜), we then obtain for T>0 large enough
supn∈N‖eit(−Δ+V)τxnψ‖Lp(|t|⩾T)Lr≲ϵ
and (4.2) holds.

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

‖e−itΔτxnψ−eit(−Δ+V)τxnψ‖Lp(0,T)Lr→0
as n→∞. Let ϵ>0. e−itΔτxnψ−eit(−Δ+V)τxnψ is a solution of the following linear Schrödinger equation with zero initial data
i∂tu−Δu+Vu=Ve−itΔτxnψ.
So, by the inhomogenous Strichartz estimate (2.9)
‖e−itΔτxnψ−eit(−Δ+V)τxnψ‖Ltp(0,T)Lr≲‖Ve−itΔτxnψ‖Ltγ′(0,T)L1≲T1γ′‖Ve−itΔτxnψ‖L∞(0,T)L1=T1γ′‖(τ−xnV)e−itΔψ‖L∞(0,T)L1
because the translation operator τxn commutes with the propagator e−itΔ. But
‖(τ−xnV)e−itΔψ‖L∞(0,T)L1⟶n→∞0
as seen in the proof of Proposition 6, point (3.4). □

Proposition 8.

Let ψ∈H1, (xn)n⩾1∈RN be such that |xn|→∞, U∈C(H1)∩LpLr be the unique solution to (1.2) with initial data ψ, and Un(t,x):=U(t,x−xn). Then, up to a subsequence

(4.3)‖∫0te−i(t−s)Δ(Un|Un|α)(s)ds−∫0te−i(t−s)(Δ−V)(Un|Un|α)(s)ds‖LpLr→0
as n→∞.

Proof.

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

(4.4)supn∈N‖∫0te−i(t−s)(Δ−V)(Un|Un|α)(s)ds‖Lp([T,∞))Lr→0
as T goes to infinity.

We decompose

‖∫0te−i(t−s)(Δ−V)(Un|Un|α)(s)ds‖Lp([T,∞))Lr⩽‖∫0Te−i(t−s)(Δ−V)(Un|Un|α)(s)ds‖Lp([T,∞))Lr+‖∫Tte−i(t−s)(Δ−V)(Un|Un|α)(s)ds‖Lp([T,∞))Lr
where, by the inhomogenous Strichartz estimates
‖∫Tte−i(t−s)(Δ−V)(Un|Un|α)(s)ds‖Lp([T,∞)Lr)⩽‖Un|Un|α‖Lq′([T,∞)Lr′)=‖U|U|α‖Lq′([T,∞)Lr′)
and, by the Hölder inequality
‖U|U|α‖Lq′([T,∞)Lr′)⩽‖U‖Lp([T,∞)Lr)α+1⟶T→∞0
independently of n. On the other hand, by the dispersive estimate (2.3)
‖∫0Te−i(t−s)(Δ−V)(Un|Un|α)(s)ds‖Lp[T,∞)Lr⩽‖∫0T‖e−i(t−s)(Δ−V)(Un|Un|α)(s)‖Lrds‖Lp([T,∞))≲‖∫0T(t−s)−12(1−2r)‖(Un|Un|α)(s)‖Lr′ds‖Lp([T,∞))=‖∫0T(t−s)−12(1−2r)‖(U|U|α)(s)‖Lr′ds‖Lp([T,∞))⩽‖∫R|t−s|−12(1−2r)‖(U|U|α)(s)‖Lr′ds‖Lp([T,∞))⟶0
as T goes to infinity. Indeed, note that by the Hardy–Littlewood–Sobolev inequality
‖∫R|t−s|−12(1−2r)‖(U|U|α)(s)‖Lr′ds‖Lp≲‖U|U|α‖Lq′Lr′⩽‖U‖LpLrα+1<∞
so (4.4) holds. The same estimate is obviously valid for the propagator e−itΔ.

It remains to show that for T>0 fixed,

‖∫0te−i(t−s)Δ(Un|Un|α)ds−∫0te−i(t−s)(Δ−V)(Un|Un|α)ds‖Lp(0,T)Lr→0
as n→∞. The difference
∫0te−i(t−s)Δ(Un|Un|α)ds−∫0te−i(t−s)(Δ−V)(Un|Un|α)ds
is the solution of the following linear Schrödinger equation, with zero initial data
i∂tu−Δu+Vu=V∫0te−i(t−s)Δ(Un|Un|α)ds.
As a consequence, by the Strichartz estimate (2.9)
‖∫0te−i(t−s)Δ(Un|Un|α)ds−∫0te−i(t−s)(Δ−V)(Un|Un|α)ds‖Lp(0,T)Lr≲‖V∫0te−i(t−s)Δ(Un|Un|α)ds‖Lγ′(0,T)L1≲T1γ′‖(τ−xnV)∫0te−i(t−s)Δ(U|U|α)ds‖L∞(0,T)L1.
But ∫0te−i(t−s)Δ(U|U|α)ds∈C([0,T],H1) and the functions of H1(R) vanish at infinity, so there exists Λ>0 such that
‖∫0te−i(t−s)Δ(U|U|α)ds‖L∞(0,T)L∞(|x|⩾Λ)⩽ϵ
so
‖(τ−xnV)∫0te−i(t−s)Δ(U|U|α)ds‖L∞(0,T)L1⟶n→∞0
in the same way as in the proof of Proposition 6, point (3.4). □

Proposition 9.

Let ψ∈H1, (xn)n⩾1, (tn)n⩾1∈RN be such that |xn|→∞ and tn→±∞, U be a solution to (1.2) such that

‖U(t)−e−itΔψ‖H1⟶t→±∞0
and Un(t,x):=U(t−tn,x−xn). Then, up to a subsequence
(4.5)‖e−i(t−tn)Δτxnψ−e−i(t−tn)(Δ−V)τxnψ‖LpLr→0
and
(4.6)‖∫0te−i(t−s)Δ(Un|Un|α)ds−∫0te−i(t−s)(Δ−V)(Un|Un|α)ds‖LpLr→0
as n→∞.

Proof.

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

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

Proposition 10.

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

(4.7)‖W±(t,·)−e−it(Δ−V)φ‖H1⟶t→±∞0
moreover, if tn→∓∞ and
(4.8)φn=e−itn(Δ−V)φ,W±,n(t)=W±(t−tn)
then
(4.9)W±,n(t)=e−it(Δ−V)φn+∫0te−i(t−s)(Δ−V)(W±,n|W±,n|α)(s)ds+f±,n(t)
where
(4.10)‖f±,n‖LR±pLr⟶n→∞0.

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|∀φ∈H1,E(φ)<E⇒the 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 φc∈H1, φc≠0, such that the corresponding solution uc of (1.1) verifies that {uc(t),t⩾0} 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 un∈C(H1) the corresponding solution of (1.1), we have

E(φn)⟶n→∞Ec
and
un∉LqLr.
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 J∈N:
φn=∑j=1Je−itjn(−Δ+V)τxjnψj+RnJ,
where tjn, xjn, ψj, RnJ verifies (3.6)–(3.12). From (3.11) and (3.12), we have
Ec⩾lim supn→∞∑j=1JE(e−itjn(−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)n⩾0. 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 N∈C(H1)∩LpLr be this solution, otherwise, we set N=0.

  • (2) If tjn=0 and |xjn|→∞. Let Uj∈C(H1)∩LpLr be the unique solution to (1.2) with initial data ψj. We set Un,j(x,t):=U(x−xjn,t).

  • (3) If xjn=0 and tjn→±∞. By Proposition 10, there exists Uj∈CR±(H1)∩LR±pLr a solution to (1.1) such that

    ‖Uj(t)−e−it(Δ−V)ψj‖H1⟶t→±∞0
    and verifying (4.7), (4.8), (4.9), (4.10). We have
    E(Uj)=limn→∞E(e−itjn(−Δ+V)τxjnψj)<Ec
    so Uj∈LqLr. We set Uj,n(t,x):=Uj(t−tjn,x).

  • (4) If |xjn|→∞ and tjn→±∞. Let Uj∈C(H1)∩LpLr be a solution to (1.2) such that

    ‖Uj(t)−e−itΔψj‖H1⟶t→±∞0.
    We set Uj,n(t,x):=Uj(t−tjn,x−xjn).

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=e−it(Δ−V)(φn−Rn,J)+∫0te−i(t−s)(Δ−V)(N|N|α)(s)ds(5.1)+∑j∫0te−i(t−s)(Δ−V)(Uj,n|Uj,n|α)(s)ds+rn,J
with
‖rn,J‖LpLr→0
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),t⩾0} is relatively compact in H1. Then, for any ϵ>0, there exists R>0 such that

supt⩾0∫|x|⩾R(|∇u(t,x)|2+|u(t,x)|2+|u(t,x)|α+2)dx⩽ϵ.

Proof.

Classic, see e.g. [5]. □

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

Proposition 13.

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

Proof.

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

Lemma 1.

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

(6.1)∂t∫χ|u|2=2Im∫χ′u′u¯(6.2)∂t2∫χ|u|2=4∫χ″|u′|2+2αα+2∫χ″|u|α+2−2∫χ′V′|u|2−∫χ(4)|u|2.

Now, we assume by contradiction that u≠0. Let χ∈Cc∞ 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)=4∫χR″|u′|2+2αα+2∫χR″|u|α+2−2∫χR′V′|u|2−∫χR(4)|u|2⩾8∫|x|⩽R|u′|2+4αα+2∫|x|⩽R|u|α+2−C∫|x|>R(|u|2+|u|α+2+|u′|2)(6.4)−2∫χR′V′|u|2−∫χ(4)|u|2
but, because of conservation of the mass
(6.5)|∫χ(4)|u|2|⩽CR2‖u(0)‖L2
and, because V is repulsive (i.e. xV′⩽0), using the Cauchy–Schwarz inequality, the Sobolev injection H1↪L∞ and the conservation laws
−2∫χR′V′|u|2=−2∫|x|⩽RxV′|u|2+2∫|x|>RχR′V′|u|2⩾−C∫|x|>R|xV′||u|2⩾−C‖xV′‖L1(|x|>R)‖u‖L∞2(6.6)⩾−C‖xV′‖L1(|x|>R)‖u‖H12⩾−C(u(0))‖xV′‖L1(|x|>R).
Let R0 be large enough so that
(6.7)∫|x|⩽R0|u|α+2⩾12∫|u|α+2:=δ.
We have δ>0 because we suppose that u is non zero. For R⩾R0, 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)−1R2‖u(0)‖L2−‖xV′‖L1(|x|>R)).
Because xV′∈L1 and using the compacity hypothesis combined with Proposition 12, there exists R⩾R0 large enough so that
∫|x|>R(|u|2+|u|α+2+|u′|2)+1R2‖u(0)‖L2+‖xV′‖L1(|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 φc∈H1, φc≠0, such that the corresponding solution uc of (1.1) verifies that {uc(t),t⩾0} 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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