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Scattering threshold for the focusing energy-critical generalized Hartree equation

  • Saleh Almuthaybiri , Congming Peng and Tarek Saanouni EMAIL logo
Published/Copyright: April 19, 2024
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Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

This work investigates the asymptotic behavior of energy solutions to the focusing nonlinear Schrödinger equation of Choquard type

i ∂ t u + Δ u + ∣ u ∣ p − 2 ( I α * ∣ u ∣ p ) u = 0 , p = 1 + 2 + α N − 2 , N ≥ 3 .

Indeed, in the energy-critical spherically symmetric regime, one proves a global existence and scattering versus finite time blow-up dichotomy. Precisely, if the data have an energy less than the ground state one, two cases are possible. If the kinetic energy of the radial data is less than the ground state one, then the solution is global and scatters. Otherwise, if the data have a finite variance or is spherically symmetric and have a finite mass, then the solution is nonglobal. The main difficulty is to deal with the nonlocal source term. The argument is the concentration-compactness-rigidity method introduced by Kenig and Merle (Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math. 166 (2006), no. 3, 645–675). This note naturally complements the work by Saanouni (Scattering theory for a class of defocusing energy-critical Choquard equations, J. Evol. Equ. 21 (2021), 1551–1571), where the scattering of the defocusing energy-critical generalized Hartree equation was obtained.

Keywords: energy-critical Hartree equation; nonlinear equations; scattering; blow-up
MSC 2010: 35Q55

1 Introduction

This article studies the Cauchy problem for an energy-critical focusing generalized Hartree equation

(NLS) i ∂ t u + Δ u + ∣ u ∣ p − 2 ( I α * ∣ u ∣ p ) u = 0 ; u ∣ t = 0 = u 0 .

In this note, the space dimension is N ≥ 3 , the wave function is u : R × R N → C . Moreover, 0 < α < N and the Riesz-potential stands for

I α ( x ) ≔ Γ N − α 2 Γ α 2 π N 2 2 α ∣ x ∣ N − α , 0 ≠ x ∈ R N .

Introduced by Ph. Choquard in 1976, the Choquard equation models a one-component plasma [1]. It appears also in Fröhlich and Pekar’s model of the polaron, which describes the interaction of an electron with its own hole [2–4].

If u is a solution to (NLS), one gets a family of solutions

λ 2 + α 2 ( p − 1 ) u ( λ 2 ⋅ , λ ⋅ ) , λ > 0 .

The equation (NLS) is called H ˙ s c critical, where

s c ≔ N 2 − 2 + α 2 ( p − 1 )

is the unique index keeping the following Sobolev norm invariant

λ 2 + α 2 ( p − 1 ) u ( λ 2 t , λ ⋅ ) H ˙ s c = ‖ u ( λ 2 t ) ‖ H ˙ s c .

This note treats the energy-critical case corresponding to s c = 1 , equivalently

p = p * ≔ 1 + α + 2 N − 2 .

The particular case p * = 2 in (NLS) corresponds to the repulsive energy-critical nonlinear equations of Hartree type. The global existence and scattering versus finite time blow-up of solutions were established for radial setting in the study by Miao et al. [5]. This result was extended to the nonradial case [6]. For p * > 2 , it seems that the study by Arora and Roudenko [7] is the only work dealing with the energy-critical generalized Hartree equation (NLS). Indeed, the global well-posedness in H s c , s c ≥ 0 was established for small data. A natural complementing of the study by Arora and Roudenko [7] is to establish the scattering of global solutions with a precise data size in H ˙ 1 . This is the goal of this note. The argument is the concentration-compactness-rigidity method used first by [8] in the energy-critical focusing NLS. Note that (NLS) is a part of the nonlinear evolution equations with nonlocal source term [9,10].

This article naturally complements the work [11], where the scattering of the defocusing energy-critical generalized Hartree equation was obtained by the third author in the spirit of the study by Miao et al. [5]. This helps to understand the asymptotic behavior of the energy-critical Choquard problem in different regimes. Note also that the intercritical regime was investigated by Saanouni [12]. It was also revisited with an alternative proof in the study by Arora [13], using a new method suggested by Dodson and Murphy [14], which relies on a scattering criterion [15], combined with the radial Sobolev and Morawetz-type estimates.

This article is organized as follows: Section 2 contains some technical estimates and the main result. Section 3 is devoted to prove the linear profile decomposition of bounded sequences in H ˙ 1 . In Section 4, one collects some variational estimates about energy solutions to (NLS). The goal of Section 5 is to obtain the nonexistence of global solutions. In Section 6, one proves the global existence and scattering of solutions. Finally, in the Appendix, we prove a scattering criteria and give a second proof of the profile decomposition.

Here and hereafter, for simplicity, one denotes the spaces

L r ≔ L r ( R N ) , H ˙ 1 ≔ H ˙ 1 ( R N ) ; H ˙ r d 1 ≔ { f ∈ H ˙ 1 , f ( x ) = f ( ∣ x ∣ ) , ∀ x ∈ R N } ,

and the norms

‖ ⋅ ‖ r ≔ ‖ ⋅ ‖ L r , ‖ ⋅ ‖ ≔ ‖ ⋅ ‖ 2 .

Finally, T * ≔ T * ( u 0 ) > 0 denotes the lifespan for a possible solution to the Schrödinger problem (NLS).

2 Background and main result

This section contains the contribution of this note and some useful estimates.

2.1 Notations

Here and hereafter, define, for u ∈ H ˙ 1 ,

N ≔ N ( u ) ≔ ∫ R N ∣ u ∣ p * ( I α * ∣ u ∣ p * ) d x ; ℱ ≔ ℱ ( u ) ≔ ( I α * ∣ u ∣ p * ) ∣ u ∣ p * − 2 u ; E ( u ) ≔ ‖ ∇ u ‖ 2 − 1 p * N .

Definition 2.1

A ground state of (NLS) resolves the elliptic problem

(1) Δ Q + ( I α * ∣ Q ∣ p * ) ∣ Q ∣ p * − 2 Q = 0 , 0 ≠ Q ∈ H ˙ 1

and minimizes the problem

inf 0 ≠ u ∈ H ˙ 1 E ( u ) s.t. ‖ ∇ u ‖ 2 = ∫ R N ∣ u ∣ p * ( I α * ∣ u ∣ p * ) d x 1 p * .

According to [16], the following explicit function is a ground state of (NLS),

(2) Q ≔ S α ( 2 − N ) 4 ( 2 + α ) C * 2 − N 4 ( 2 + α ) 1 + ∣ ⋅ ∣ 2 N ( N − 2 ) − N N − 2 ,

where S is the best Sobolev injection constant in ‖ ∇ u ‖ 2 ≥ C ‖ u ‖ 2 N N − 2 2 and

C * ≔ 1 2 π α Γ N − α 2 Γ ( N + α 2 ) Γ ( N ) Γ ( N 2 ) α N .

Note that global solutions to (NLS) such as this ground state may not scatter. Finally, let us denote, for an interval I ⊂ R , the Strichartz spaces

S ( I ) ≔ L 2 p * ( I , L 2 N p * N p * − 2 ) , S 1 ( I ) ≔ L 2 p * ( I , L 2 N p * ( p * − 1 ) 2 + α p * ) .

2.2 Preliminary

Solutions to the Schrödinger problem (NLS) will be considered with an equivalent way as a fixed point of the integral equation:

e i t Δ u 0 + i ∫ 0 t e i ( t − s ) Δ [ ( I α * ∣ u ∣ p * ) ∣ u ∣ p * − 2 u ] d s ,

where

e i t Δ u ≔ ℱ − 1 ( e − i t ∣ ⋅ ∣ 2 ) * u .

In the study by Du and Yang [16], a sharp Gagliardo-Nirenberg type inequality related to (NLS) was established.

Proposition 2.1

Let N ≥ 3 and 0 < α < N . Then,

  1. ∫ R N ∣ u ∣ p * ( I α * ∣ u ∣ p * ) d x ≤ C N , α ‖ ∇ u ‖ 2 p * , for any u ∈ H ˙ 1 ;

  2. the best constant in the previous inequality is

    C N , α = C * S α + N N − 2 .

The problem (NLS) has a local solution in the energy space, which is global for small data [7].

Proposition 2.2

Let N ≥ 3 , 0 < α < N ≤ 4 + α , and u 0 ∈ H ˙ 1 . Then, there exists T * > 0 such that (NLS) admits a unique maximal solution

u ∈ C T * ( H ˙ 1 ) .

Moreover,

  1. u ∈ L loc 2 p * ( ( 0 , T * ) , W ˙ 1 , 2 N p * N p * − 2 ) ;

  2. the energy is conserved E ( u ( t ) ) = E ( u 0 ) ;

  3. if T * < ∞ , then, ‖ u ‖ S 1 ( 0 , T * ) = ∞ .

Furthermore, there exists δ ( ‖ ∇ u 0 ‖ ) > 0 such that if ‖ e i ⋅ Δ u 0 ‖ S 1 ( R ) ≤ δ , then, T * = ∞ and

  1. ‖ u ‖ S 1 ( R ) ≤ 2 ‖ e i ⋅ Δ u 0 ‖ S 1 ( R ) ;

  2. ‖ ∇ u ‖ S ( R ) ≲ ‖ ∇ u 0 ‖ S ( R ) .

For the reader convenience, the next scattering criterion will be proved in the Appendix.

Proposition 2.3

Let N ≥ 3 , 0 < α < N ≤ 4 + α , and u ∈ C ( R , H ˙ 1 ) be a global solution to (NLS) satisfying

‖ u ‖ S 1 ( R ) < ∞ .

Then, u scatters in H ˙ 1 .

The following long-time perturbation theory [12] will be useful.

Proposition 2.4

Let N ≥ 3 , 0 < α < N ≤ 4 + α . Let T > 0 , u ∈ C T ( H ˙ 1 ) be a solution to (NLS) and u ˜ ∈ L T ∞ ( H ˙ 1 ) satisfying for some ε , A > 0 ,

‖ u ˜ ‖ L T ∞ ( H ˙ 1 ) ∩ S 1 ( 0 , T ) ≤ A ; i ∂ t u ˜ + Δ u ˜ + ℱ ( u ˜ ) = e ; ‖ ∇ e ‖ S ′ ( 0 , T ) ≤ ε , ‖ e i ⋅ Δ [ u 0 − u ˜ 0 ] ‖ S 1 ( 0 , T ) ≤ ε .

Thus, there exists ε 0 ≔ ε 0 ( A ) satisfying for any 0 < ε < ε 0 ,

‖ u ‖ S 1 ( 0 , T ) ≤ C ( A ) .

The next linear profile decomposition for bounded radial sequences in H ˙ 1 is a key tool for the scattering proof [5,17].

Proposition 2.5

Let N ≥ 3 , 0 < α < N such that p * is an integer. Take ( u n ) be a bounded sequence in H ˙ r d 1 such that ‖ e i ⋅ Δ u n ‖ S 1 ( R ) ≥ δ > 0 . Then, for any M ∈ N , there exist a subsequence denoted also ( u n ) and

  1. for any 1 ≤ j ≤ M , a profile ψ j ∈ H ˙ 1 ;

  2. for any 1 ≤ j ≤ M , a sequence ( t n j , λ n j ) ∈ R 2 satisfying for 1 ≤ i ≠ j ≤ M and n → ∞ ,

    λ n j λ n i + λ n i λ n j + ∣ t n j − t n i ∣ ( λ n j ) 2 → ∞ ;

  3. a sequence of remainders W n M ∈ H ˙ r d 1 , such that, for ψ l j ≔ e i ⋅ Δ ψ j ,

    u n = ∑ j = 1 M 1 ( λ n j ) N − 2 2 e − i t n j ( λ n j ) 2 Δ ψ j ⋅ λ n j + W n M ≔ ∑ j = 1 M 1 ( λ n j ) N − 2 2 ψ l j − t n j ( λ n j ) 2 , ⋅ λ n j + W n M .

    Moreover, ‖ ∇ ψ 1 ‖ > 0 and

    lim M → ∞ [ lim n → ∞ ‖ e i ⋅ Δ W n M ‖ S 1 ( R ) ] = 0 .

    For fixed M, one has the next Pythagorean expansions

    ‖ ∇ u n ‖ 2 = ∑ j = 1 M ‖ ∇ ψ j ‖ 2 + ‖ ∇ W n M ‖ 2 + o n ( 1 ) ; E ( u n ) = ∑ j = 1 M E ψ l j − t n j ( λ n j ) 2 + E ( W n M ) + o n ( 1 ) .

Finally, one recalls the notion of nonlinear profile.

Definition 2.2

Let a sequence of real numbers ( t n ) such that lim n t n ≔ t ¯ ∈ [ − ∞ , + ∞ ] and v 0 ∈ H ˙ 1 . One says that u ( t , x ) is a nonlinear profile associated to ( v 0 , t n ) if there exists an interval I ∋ t ¯ , ( I = [ a , ∞ ) or I = ( ∞ , a ] , if t ¯ = ∞ ) and u ( t , x ) a solution to (NLS) on I such that

lim n ‖ u ( t n ) − e i t n Δ v 0 ‖ H ˙ 1 = 0 .

The main result proved in this article is given in the following subsection.

2.3 Main result

The novelty in this work is the next dichotomy of global existence and scattering versus nonexistence of global energy solutions to the Schrödinger problem (NLS).

Theorem 2.1

Let N ≥ 3 , 0 < α < N ≤ 4 + α , Q be the ground state solution to (1) given in (2) and u 0 ∈ H ˙ 1 satisfying

(3) E [ u 0 ] < E [ Q ] .

Take u ∈ C T * ( H ˙ 1 ) be the unique maximal solution to (NLS). Then,

  1. T * < ∞ if x u 0 ∈ L 2 and

    (4) ‖ ∇ u 0 ‖ > ‖ ∇ Q ‖ .

  2. T * = ∞ and u scatters if u 0 is radial and

    (5) ‖ ∇ u 0 ‖ < ‖ ∇ Q ‖ .

In view of the result stated in the aforementioned theorem, some comments are in order.

Remarks 2.1

  • The condition α ≥ N − 4 avoids a singular source term if p * < 2 ;

  • the range of space dimensions N containing 3, is { 3 , 4 , 5 , 6 } ;

  • the scattering means the existence of ψ ∈ H ˙ 1 such that

    lim t → ∞ ‖ u ( t ) − e i t Δ ψ ‖ H ˙ 1 = 0 ;

  • in the proof of the finite time blow-up, we establish also that there is no global solution to (NLS) in the energy space if u 0 ∈ H r d 1 and N 2 − ( 5 + α ) N − α ≥ 0 . This extra assumption is related to the method and is needed in order to apply Young estimate;

  • the major part of the proof is devoted to prove the energy scattering;

  • the scattering in the defocusing regime was proved by Saanouni [11];

  • the local well-posedness of (NLS) for 1 < p < 2 was investigated for a class of data in weighted Sobolev spaces in [18];

  • the proof of the scattering follows the concentration-compactness-rigidity method due to the pioneering work [8], which has a deep influence in the NLS context.

  • the new method of Dodson and Murphy [14], used by Arora [13] in the intercritical regime fails in the energy critical case.

Section 2.4 presents some technical estimates needed in the sequel.

2.4 Tools

The next consequence of Hardy-Littlewood-Sobolev inequality [19] is adapted to estimate a possible solution to the Hartree equation (NLS).

Proposition 2.6

Let 0 < α < N ≥ 1 and 1 < s , r < ∞ . Then,

  1. if 1 + α N = 1 r + 1 s ,

    (6) ∫ R N ( I α * g ) ( x ) f ( x ) d x ≤ C N , α , s ‖ f ‖ r ‖ g ‖ s , ∀ ( f , g ) ∈ L r × L s .

  2. If 1 + α N = 1 q + 1 r + 1 s ,

    ‖ ( I α * f ) g ‖ r ′ ≤ C N , α , s ‖ f ‖ s ‖ g ‖ q , ∀ ( f , g ) ∈ L s × L q .

Definition 2.3

A couple of real numbers ( q , r ) is said to be H ˙ s admissible (admissible if s = 0 ) if

N 1 2 − 1 r = 2 q + s ,

where

2 N N − 2 s < r ≤ 2 N N − 2 − , if N ≥ 3 ; 2 1 − s < r ≤ 2 1 − s + ′ , if N = 2 ; 2 1 − 2 s < r ≤ ∞ , if N = 1 .

Here, ( a + ) ′ = a + a a + − a . Finally, one says that ( q , r ) is said to be H ˙ − s admissible if

N 1 2 − 1 r = 2 q − s ,

where

2 N N − 2 s + < r ≤ 2 N N − 2 − , if N ≥ 3 ; 2 1 − s + < r ≤ 2 1 − s + ′ , if N = 2 ; 2 1 − 2 s + < r ≤ ∞ , if N = 1 .

For simplicity, one denotes Γ s the set of s admissible pairs.

A standard tool to control solutions of (NLS) is the Strichartz estimate [20,21].

Proposition 2.7

Let N ≥ 1 and s ∈ R . Then, there exists C > 0 such that

  1. sup ( q , r ) ∈ Γ s ‖ e i ⋅ Δ u ‖ L q ( L r ) ≤ C ‖ ∣ ∇ ∣ s u ‖ ;

  2. sup ( q , r ) ∈ Γ s ∫ 0 . e i ( ⋅ − τ ) Δ f ( τ ) d τ L q ( L r ) ≤ C inf ( a , b ) ∈ Γ − s ‖ f ‖ L a ′ ( L b ′ ) .

To investigate the nonexistence of global solutions to the Schrödinger problem (NLS), one will use the next variance identity [22].

Proposition 2.8

Let N ≥ 3 , 0 < α < N ≤ 4 + α , and u ∈ C T ( H ˙ 1 ) be a solution to (NLS) satisfying x u 0 ∈ L 2 . Then, the variance defined on [ 0 , T ) by

V : t ↦ ∫ R N ∣ x u ( t , x ) ∣ 2 d x

satisfies V ∈ C 2 ( − T , T ) and

V ″ = 8 ‖ ∇ u ‖ 2 − ∫ R N ∣ u ∣ p * ( I α * ∣ u ∣ p * ) d x .

3 Profile decomposition

In this section, one proves Proposition 2.5. Taking account of [17], it is sufficient to prove the last equation, which reads

(7) N ( u n ) = ∑ j = 1 M N ψ l j − t n j ( λ n j ) 2 + N ( W n M ) + o n ( 1 ) .

For this, one needs to establish that

(8) 0 = lim n ( N ( u n ) − N ( u n − ( ψ ˜ l 1 ) n ) − N ( ( ψ ˜ l 1 ) n ) ) ≔ lim n I n ,

where, one denotes the sequence

( ψ ˜ l 1 ) n ≔ 1 ( λ n 1 ) N − 2 2 ψ l 1 − t n 1 ( λ n 1 ) 2 , ⋅ λ n 1 .

By re-indexing and adjusting profiles, one can assume that one of the two scenarios happens. The first one is t n 1 ( λ n 1 ) 2 → ∞ . The second one is t n 1 ( λ n 1 ) 2 → − t ˜ .

Take the first case and recall the useful inequality [23], for any m ≥ 2 ,

(9) ∣ ∣ x ∣ m − ∣ x − y ∣ m − ∣ y ∣ m ∣ ≤ m 2 m − 1 ( ∣ x − y ∣ m − 1 ∣ y ∣ + ∣ x − y ∣ ∣ y ∣ m − 1 ) .

Let us write

(10) I n = ∫ R N ( I α * ∣ u n ∣ p * ) ( ∣ u n ∣ p * − ∣ u n − ( ψ ˜ l 1 ) n ∣ p * − ∣ ( ψ ˜ l 1 ) n ∣ p * ) d x + ∫ R N ∣ u n − ( ψ ˜ l 1 ) n ∣ p * [ I α * ( ∣ u n ∣ p * − ∣ u n − ( ψ ˜ l 1 ) n ∣ p * ) ] d x + ∫ R N ∣ ( ψ ˜ l 1 ) n ∣ p * [ I α * ( ∣ u n ∣ p * − ∣ ( ψ ˜ l 1 ) n ∣ p * ) ] d x ≔ ( I n ) 1 + ( I n ) 2 + ( I n ) 3 .

Now, by (9), via Sobolev embeddings, Hölder estimate and (6), one writes for 2 N N − 2 ≔ 2 * ,

(11) ( I n ) 1 ≲ ‖ u n ‖ 2 * ‖ ∣ u n ∣ p * − ∣ u n − ( ψ ˜ l 1 ) n ∣ p * − ∣ ( ψ ˜ l 1 ) n ∣ p * ‖ 2 N α + N ≲ ‖ ∣ u n ∣ p * − ∣ u n − ( ψ ˜ l 1 ) n ∣ p * − ∣ ( ψ ˜ l 1 ) n ∣ p * ‖ 2 N α + N ≲ ‖ ∣ u n − ( ψ ˜ l 1 ) n ∣ p * − 1 ∣ ( ψ ˜ l 1 ) n ∣ + ∣ u n − ( ψ ˜ l 1 ) n ∣ ∣ ( ψ ˜ l 1 ) n ∣ p * − 1 ‖ 2 N α + N ≲ ‖ u n − ( ψ ˜ l 1 ) n ‖ 2 * p * − 1 ‖ ( ψ ˜ l 1 ) n ‖ 2 * + ‖ u n − ( ψ ˜ l 1 ) n ‖ 2 * ‖ ( ψ ˜ l 1 ) n ‖ 2 * p * − 1 .

Using the dispersive estimate [24] of the free Schrödinger operator, ‖ e i t Δ ⋅ ‖ r ≤ C t N 1 2 − 1 r ‖ ⋅ ‖ r ′ , for all r ≥ 2 , via (11) and (10), one gets lim n I n = 0 .

Now, take the second case and denote

(12) s n ≔ t n 1 ( λ n 1 ) 2 → − t ˜ ;

(13) v n ≔ ℒ 1 u n ≔ ( λ n 1 ) N − 2 2 u n ( λ n 1 ⋅ ) ⇀ e i t ˜ Δ ψ 1 in H ˙ 1 ;

(14) ψ l 1 ( − s n ) → e i t ˜ Δ ψ 1 in L 2 * .

Let us write

(15) I n = N ( u n ) − N ( u n − ( ψ ˜ l 1 ) n ) − N ( ( ψ ˜ l 1 ) n ) = N ( v n ) − N ( v n − ψ l 1 ( − s n ) ) − N ( ψ l 1 ( − s n ) ) ≔ ( I n ) 1 + ( I n ) 2 + ( I n ) 3 .

Here, one takes

( I n ) 1 ≔ N ( v n ) − N ( v n − e i t ˜ Δ ψ 1 ) − N ( e i t ˜ Δ ψ 1 ) ; ( I n ) 2 ≔ N ( v n − e i t ˜ Δ ψ 1 ) − N ( v n − ψ l 1 ( − s n ) ) ; ( I n ) 3 ≔ N ( e i t ˜ Δ ψ 1 ) − N ( ψ l 1 ( − s n ) ) .

Moreover, the aforementioned first term reads

(16) ( I n ) 1 = ∫ R N ( I α * ∣ v n ∣ p * ) ( ∣ v n ∣ p * − ∣ v n − e i t ˜ Δ ψ 1 ∣ p * − ∣ e i t ˜ Δ ψ 1 ∣ p * ) d x + ∫ R N ∣ v n − e i t ˜ Δ ψ 1 ∣ p * [ I α * ( ∣ v n ∣ p * − ∣ v n − e i t ˜ Δ ψ 1 ∣ p * ) ] d x + ∫ R N ∣ e i t ˜ Δ ψ 1 ∣ p * [ I α * ( ∣ v n ∣ p * − ∣ e i t ˜ Δ ψ 1 ∣ p * ) ] d x ≔ ( I n ) 11 + ( I n ) 12 + ( I n ) 13 .

By (6) via Hölder estimate and (9), one obtains

(17) ( I n ) 11 = ∫ R N ( I α * ∣ v n ∣ p * ) ( ∣ v n ∣ p * − ∣ v n − e i t ˜ Δ ψ 1 ∣ p * − ∣ e i t ˜ Δ ψ 1 ∣ p * ) d x ≲ ‖ v n ‖ 2 * ‖ ∣ v n − e i t ˜ Δ ψ 1 ∣ p * − 1 ∣ e i t ˜ Δ ψ 1 ∣ + ∣ v n − e i t ˜ Δ ψ 1 ∣ ∣ e i t ˜ Δ ψ 1 ∣ p * − 1 ‖ 2 N α + N ≔ ( I n ) 111 + ( I n ) 112 .

By density argument, one takes χ ∈ C 0 ∞ ( R N ) such that ‖ χ − e i t ˜ Δ ψ 1 ‖ H ˙ 1 ⪡ 1 . Thus, by (13), via Hölder estimate, it follows that

( I n ) 111 ≲ ‖ v n − e i t ˜ Δ ψ 1 ‖ 2 * p * − 1 ‖ e i t ˜ Δ ψ 1 − χ ‖ 2 * + ‖ v n − e i t ˜ Δ ψ 1 ‖ 2 * p * − 2 ‖ χ ( v n − e i t ˜ Δ ψ 1 ) ‖ 2 * 2 → 0 .

Moreover, by Hölder estimate, one has

(18) ( I n ) 112 = ‖ ∣ v n − e i t ˜ Δ ψ 1 ∣ ∣ e i t ˜ Δ ψ 1 − χ + χ ∣ p * − 1 ‖ 2 N α + N ≲ ‖ v n − e i t ˜ Δ ψ 1 ‖ 2 * ‖ e i t ˜ Δ ψ 1 − χ ‖ 2 * p * − 1 + ‖ χ ( v n − e i t ˜ Δ ψ 1 ) ‖ 2 * 2 ‖ χ ‖ 2 * p * − 2 → 0 .

Now, one considers the second term. Arguing as previously and taking account of (14), one obtains

(19) ( I n ) 12 = ∫ R N ∣ v n − e i t ˜ Δ ψ 1 ∣ p * [ I α * ( ∣ v n ∣ p * − ∣ v n − e i t ˜ Δ ψ 1 ∣ p * ) ] d x ≲ ∫ R N ∣ v n − e i t ˜ Δ ψ 1 ∣ p * [ I α * ( ∣ e i t ˜ Δ ψ 1 − ψ l 1 ( − s n ) ∣ ( ∣ v n ∣ p * − 1 + ∣ e i t ˜ Δ ψ 1 ∣ p * − 1 + ∣ ψ l 1 ( − s n ) ∣ p * − 1 ) ) ] d x ≲ ( ‖ v n ‖ 2 * p * − 1 + ‖ e i t ˜ Δ ψ 1 ‖ 2 * p * − 1 + ‖ ψ l 1 ( − s n ) ‖ 2 * p * − 1 ) ‖ e i t ˜ Δ ψ 1 − ψ l 1 ( − s n ) ‖ 2 * → 0 .

The third term is controlled similarly. This finishes the proof.

4 Variational analysis

In this section, one prepares the proof of the main result by collecting some useful estimates. Taking account of [16], the ground state Q given in (2) is a minimizer for

S N , α ≔ inf { 0 ≠ u ∈ H 1 } ‖ ∇ u ‖ 2 ∫ R N ∣ u ∣ p * ( I α * ∣ u ∣ p * ) d x 1 p * .

This ground state is a global solution to (NLS), which does not scatter. Thus, global solutions may not scatter. Moreover, the equation (1) gives

‖ ∇ Q ‖ 2 = ∫ R N ( I α * ∣ Q ∣ p * ) ∣ Q ∣ p * d x .

Thus,

‖ ∇ Q ‖ 2 = S N , α α + N α + 2 , E ( Q ) = α + 2 α + N S N , α α + N α + 2 .

Lemma 4.1

For δ ∈ ( 0 , 1 ) , there exists δ ¯ ∈ ( 0 , 1 ) such that if u ∈ H ˙ 1 satisfies

‖ ∇ u ‖ < ‖ ∇ Q ‖ ; E ( u ) < ( 1 − δ ) E ( Q ) < E ( Q ) ,

then,

(20) ‖ ∇ u ‖ 2 < ( 1 − δ ¯ ) ‖ ∇ Q ‖ 2 ;

(21) ‖ ∇ u ‖ 2 − N ≥ δ ¯ ‖ ∇ u ‖ 2 ;

(22) E ( u ) > 0 .

Proof

Take the real function f ( x ) ≔ x − 1 p * S N , α p * x p * . Then,

f ( ‖ ∇ u ‖ 2 ) = ‖ ∇ u ‖ 2 − 1 p * S N , α p * ‖ ∇ u ‖ 2 p * ≤ ‖ ∇ u ‖ 2 − 1 p * N ≤ E ( u ) ≤ ( 1 − δ ) E ( Q ) .

The equation f ′ ( x ) = 0 is equivalent to

x = x * = S N , α p * p * − 1 = ‖ ∇ Q ‖ 2 .

Moreover, f ( x * ) = E ( Q ) . Now, since f is positive and strictly increasing on [ 0 , x * ] , one has (20) and (22). Now, let the real function g ( x ) ≔ x − ( x S N , α ) p * . Then,

‖ ∇ u ‖ 2 − N ≥ ‖ ∇ u ‖ 2 − ‖ ∇ u ‖ 2 S N , α p * = g ( ‖ ∇ u ‖ 2 ) .

Moreover, g ( x ) = 0 if and only if x = 0 or x = x * . Thus, g ( x ) ≳ x on [ 0 , ( 1 − δ ¯ ) x * ] . So, since 0 ≤ ‖ ∇ u ‖ 2 < ( 1 − δ ¯ ) x * , one obtains (21).□

Corollary 4.1

If u ∈ H ˙ 1 satisfies ‖ ∇ u ‖ < ‖ ∇ Q ‖ , then E ( u ) ≥ 0 .

Proof

The case E ( u ) ≥ E ( Q ) = α + 2 α + N S N , α α + N α + 2 > 0 is clear. Otherwise, the Lemma 4.1 gives the result.□

With a continuity argument via the conservation of the energy and Lemma 4.1, one has the following energy trapping.

Proposition 4.1

For δ ∈ ( 0 , 1 ) , there exists δ ¯ ∈ ( 0 , 1 ) such that if u 0 ∈ H ˙ 1 satisfies

‖ ∇ u 0 ‖ < ‖ ∇ Q ‖ ; E ( u 0 ) < ( 1 − δ ) E ( Q ) < E ( Q ) ,

then, the maximal solution to (NLS) satisfies for any t ∈ [ 0 , T * ) ,

(23) ‖ ∇ u ( t ) ‖ 2 < ( 1 − δ ¯ ) ‖ ∇ Q ‖ 2 ;

(24) ‖ ∇ u ( t ) ‖ 2 − N ( u ( t ) ) ≥ δ ¯ ‖ ∇ u ( t ) ‖ 2 ;

(25) E ( u ) > 0 .

Moreover,

E ( u ( t ) ) ≃ ‖ ∇ u ( t ) ‖ 2 ≃ ‖ ∇ u 0 ‖ 2 .

Proof

For the last point, since E ( u ( t ) ) ≤ ‖ ∇ u ( t ) ‖ 2 , by (23), one has

E ( u ( t ) ) ≥ 1 − 1 p * ‖ ∇ u ( t ) ‖ 2 + 1 p * ( ‖ ∇ u ( t ) ‖ 2 − N ( u ( t ) ) ) ≥ 1 − 1 p * ‖ ∇ u ( t ) ‖ 2 .

Taking account of Lemma 4.1, it is sufficient to prove that

‖ ∇ u ( t ) ‖ < ‖ ∇ Q ‖ , ∀ t ∈ ( 0 , T * ) .

Assume that there exists t 0 ∈ ( 0 , T * ) such that

‖ ∇ u ( t ) ‖ < ‖ ∇ Q ‖ , ∀ t ∈ ( 0 , t 0 ) and ‖ ∇ u ( t 0 ) ‖ = ‖ ∇ Q ‖ .

Thus, by Lemma 4.1, one obtains

‖ ∇ u ( t ) ‖ 2 < ( 1 − δ ¯ ) ‖ ∇ Q ‖ 2 , ∀ t ∈ ( 0 , t 0 ) .

With a continuity argument, it follows that

‖ ∇ u ( t 0 ) ‖ < ‖ ∇ Q ‖ .

This contradiction finishes the proof.□

5 Nonexistence of global solutions

In this section, one proves the first part of the main result dealing with the nonexistence of global solutions to (NLS). The next result follows like Proposition 4.1.

Lemma 5.1

For δ ∈ ( 0 , 1 ) , there exists δ ¯ ∈ ( 0 , 1 ) such that if the maximal solution to (NLS) denoted by u ∈ C T * ( H ˙ 1 ) satisfies (4) and

E ( u ) < ( 1 − δ ) E ( Q ) < E ( Q ) .

Then, holds on [ 0 , T * ) ,

‖ ∇ u ‖ 2 > ( 1 + δ ¯ ) ‖ ∇ Q ‖ 2 .

Let us discuss two cases.

  1. x u 0 ∈ L 2 . Compute

    ‖ ∇ u ‖ 2 − ∫ R N ∣ u ∣ p * ( I α * ∣ u ∣ p * ) d x = p * E ( u ) − ( p * − 1 ) ‖ ∇ u ‖ 2 ≤ p * E ( Q ) − ( p * − 1 ) ( 1 + δ ¯ ) ‖ ∇ Q ‖ 2 ≤ α + N N − 2 2 + α N + α − 2 + α N − 2 ( 1 + δ ¯ ) S N , α N + α 2 + α ≤ − δ ¯ ( p * − 1 ) S N , α p * p * − 1 .

Thus, by the variance identity in Proposition 2.8,

∂ t 2 ‖ ∣ ⋅ ∣ u ( t ) ‖ 2 = 8 ‖ ∇ u ‖ 2 − ∫ R N ∣ u ∣ p * ( I α * ∣ u ∣ p * ) d x ≤ − 8 δ ¯ ( p * − 1 ) S N , α p * p * − 1 .

Thus, integrating twice, one obtains

‖ ∣ ⋅ ∣ u ( t ) ‖ 2 ≤ − 4 δ ¯ ( p * − 1 ) S N , α p * p * − 1 t 2 + 4 t ∫ R N ( x . ∇ u 0 ) u 0 d x + ‖ x u 0 ‖ 2 .

This quantity becomes negative for large time. Thus, T * < ∞ .

  1. Second case u 0 ∈ H r d 1 .

In the rest of this section, take a smooth radial function ψ ∈ C 0 ∞ ( R n ) satisfying ψ ″ ≤ 1 and

ψ : x ↦ ∣ x ∣ 2 2 , if ∣ x ∣ ≤ 1 ; 0 , if ∣ x ∣ ≥ 2 .

Then, the truncated function ψ R ≔ R 2 ψ ( . R ) satisfies

ψ R ″ ≤ 1 , ψ R ′ ( r ) ≤ r and Δ ψ R ≤ N .

Denote the localized virial

V ψ [ u ( t ) ] ≔ ∫ R N ψ ∣ u ( t ) ∣ 2 d x and V R ≔ V ψ R .

The first derivative reads using the convention of summed repeated index,

∂ t V ψ [ u ( t ) ] ≔ M ψ [ u ( t ) ] = 2 ℑ ∫ R N u ¯ ( t ) ∂ k ψ ∂ k u ( t ) d x .

Compute using the equation (NLS),

∂ t ℑ ( ∂ k u u ¯ ) = ℑ ( ∂ k ∂ t u u ¯ ) + ℑ ( ∂ k u ∂ t u ¯ ) = ℜ ( i ∂ t u ∂ k u ¯ ) − ℜ ( i ∂ k ∂ t u u ¯ ) = ℜ ( ∂ k u ¯ ( − Δ u − ℱ ) ) − ℜ ( u ¯ ∂ k ( − Δ u − ℱ ) ) = ℜ ( u ¯ ∂ k Δ u − ∂ k u ¯ Δ u ) + ℜ ( u ¯ ∂ k ℱ − ∂ k u ¯ ℱ ) .

Recall the identity

1 2 ∂ k Δ ( ∣ u ∣ 2 ) − 2 ∂ l ℜ ( ∂ k u ∂ l u ¯ ) = ℜ ( u ¯ ∂ k Δ u − ∂ k u ¯ Δ u ) .

Then,

∫ R N ∂ k ψ ℜ ( u ¯ ∂ k Δ u − ∂ k u ¯ Δ u ) d x = ∫ R N ∂ k ψ 1 2 ∂ k Δ ( ∣ u ∣ 2 ) − 2 ∂ l ℜ ( ∂ k u ∂ l u ¯ ) d x = − 1 2 ∫ R N Δ 2 ψ ∣ u ∣ 2 d x + 2 ∫ R N ∂ l ∂ k ψ ℜ ( ∂ k u ∂ l u ¯ ) d x .

On the other hand,

∫ R N ∂ k ψ ℜ ( u ¯ ∂ k ℱ − ∂ k u ¯ ℱ ) d x = ∫ R N ∂ k ψ ℜ ( ∂ k [ u ¯ ℱ ] − 2 ∂ k u ¯ ℱ ) d x = − ∫ R N ( Δ ψ u ¯ ℱ + 2 ℜ ( ∂ k ψ ∂ k u ¯ ℱ ) ) d x = − ∫ R N Δ ψ ∣ u ∣ p * ( I α * ∣ u ∣ p * ) d x − 2 ∫ R N ∂ k ψ ℜ ( ∂ k u ¯ ℱ ) d x = − ∫ R N Δ ψ ∣ u ∣ p * ( I α * ∣ u ∣ p * ) d x − 2 p * ∫ R N ∂ k ψ ∂ k ( ∣ u ∣ p * ) ( I α * ∣ u ∣ p * ) d x .

By using the identity ∇ I α = − ( N − α ) ⋅ ∣ ⋅ ∣ 2 I α , one obtains

∫ R N ∂ k ψ ∂ k ( ∣ u ∣ p * ) ( I α * ∣ u ∣ p * ) d x = − ∫ R N Δ ψ ∣ u ∣ p * ( I α * ∣ u ∣ p * ) d x − ∫ R N ∂ k ψ ( ∂ k I α * ∣ u ∣ p * ) ∣ u ∣ p * d x = − ∫ R N Δ ψ ∣ u ∣ p * ( I α * ∣ u ∣ p * ) d x + ( N − α ) ∫ R N ∂ k ψ x k ∣ ⋅ ∣ 2 I α * ∣ u ∣ p * ∣ u ∣ p * d x .

Thus,

∫ R N ∂ k ψ ℜ ( u ¯ ∂ k ℱ − ∂ k u ¯ ℱ ) d x = − ∫ R N Δ ψ ∣ u ∣ p * ( I α * ∣ u ∣ p * ) d x − 2 p * ∫ R N ∂ k ψ ∂ k ( ∣ u ∣ p * ) ( I α * ∣ u ∣ p * ) d x = 2 p * − 1 ∫ R N Δ ψ ∣ u ∣ p * ( I α * ∣ u ∣ p * ) d x − 2 p * ( N − α ) ∫ R N ∂ k ψ x k ∣ ⋅ ∣ 2 I α * ∣ u ∣ p * ∣ u ∣ p * d x .

Regrouping previous computation, one obtains

∂ t M ψ [ u ( t ) ] = − ∫ R N Δ 2 ψ ∣ u ∣ 2 d x + 4 ∫ R N ∂ l ∂ k ψ ℜ ( ∂ k u ∂ l u ¯ ) d x + 2 2 p * − 1 ∫ R N Δ ψ ∣ u ∣ p * ( I α * ∣ u ∣ p * ) d x − 4 p * ( N − α ) ∫ R N ∂ k ψ x k ∣ ⋅ ∣ 2 I α * ∣ u ∣ p * ∣ u ∣ p * d x

= − ∫ R N Δ 2 ψ ∣ u ∣ 2 d x + 4 ∫ R N ∂ l ∂ k ψ ℜ ( ∂ k u ∂ l u ¯ ) d x + 2 2 p * − 1 ∫ R N Δ ψ ∣ u ∣ p * ( I α * ∣ u ∣ p * ) d x − 2 p * ( N − α ) ∫ R N ∫ R N ( ∂ k ψ ( x ) − ∂ k ψ ( y ) ) ( x k − y k ) I α ( x − y ) ∣ x − y ∣ 2 ∣ u ( x ) ∣ p * ∣ u ( y ) ∣ p * d x d y .

Denote, for simplicity, M R ≔ M ψ R . By using the properties of ψ , one has

∂ t M R [ u ( t ) ] ≤ C R 2 + 4 ‖ ∇ u ‖ 2 + 2 2 p * − 1 ∫ R N Δ ψ R ( I α * ∣ u ∣ p * ) ∣ u ∣ p * d x − 2 p * ( N − α ) ∫ R N ∫ R N ( ∂ k ψ R ( x ) − ∂ k ψ R ( y ) ) ( x k − y k ) I α ( x − y ) ∣ x − y ∣ 2 ∣ u ( x ) ∣ p * ∣ u ( y ) ∣ p * d x d y .

Following lines in [11], one has

( M ) ≔ ∫ R N × R N ( ∇ ψ R ( x ) − ∇ ψ R ( y ) ) x − y ∣ x − y ∣ 2 I α ( x − y ) ∣ u ( y ) ∣ p * ∣ u ( x ) ∣ p * d y d x = ∫ R N ∣ u ∣ p * ( I α * ∣ u ∣ p * ) d x + O ∫ { ∣ x ∣ > R } ( I α * ∣ u ∣ p * ) ∣ u ∣ p * d x .

Thus, using the equality Δ ψ R ( r ) − N = 0 if r ≤ R ,

( N ) ≔ − 2 ( N − α ) p * ( M ) + 2 2 p * − 1 ∫ R N Δ ψ R ( I α * ∣ u ∣ p * ) ∣ u ∣ p * d x = − 2 N − α p * + N 1 − 2 p * ∫ R N ∣ u ∣ p * ( I α * ∣ u ∣ p * ) d x + O ∫ { ∣ x ∣ > R } ( I α * ∣ u ∣ p * ) ∣ u ∣ p * d x + 2 2 p * − 1 ∫ { ∣ x ∣ > R } ( Δ ψ R − N ) ( I α * ∣ u ∣ p * ) ∣ u ∣ p * d x = − 4 ∫ R N ∣ u ∣ p * ( I α * ∣ u ∣ p * ) d x + O ∫ { ∣ x ∣ > R } ( I α * ∣ u ∣ p * ) ∣ u ∣ p * d x .

By using Hardy-Littlewood-Sobolev inequality, one has

∫ { ∣ x ∣ > R } ( I α * ∣ u ∣ p * ) ∣ u ∣ p * d x ≲ ‖ u ‖ L 2 N N − 2 ( ∣ x ∣ > R ) p * ‖ u ‖ L 2 N N − 2 p * ≲ ‖ u ‖ L 2 N N − 2 p * ‖ u ‖ L ∞ ( ∣ x ∣ > R ) p * − 1 − α N ‖ u ‖ α + N N .

Thanks to Hardy-Littlewood-Sobolev and Strauss inequalities via the mass conservation and Sobolev injections, one obtains

∫ { ∣ x ∣ > R } ( I α * ∣ u ∣ p * ) ∣ u ∣ p * d x ≲ ‖ ∇ u ‖ p * ‖ u ‖ L ∞ ( ∣ x ∣ > R ) p * − 1 − α N ≲ ‖ ∇ u ‖ p * 1 R N − 1 ‖ ∇ u ‖ L 2 ( ∣ x ∣ > R ) ‖ u ‖ 1 2 ( p * − 1 − α N ) ≲ 1 R ( N − 1 ) ( α + N ) N ( N − 2 ) ‖ ∇ u ‖ ( α + N ) ( 1 + N ) N ( N − 2 ) .

So, by Young inequality, since N 2 − ( 5 + α ) N − α ≥ 0 , there exists C ( R ) → 0 when R → ∞ , such that

∂ t M R [ u ( t ) ] ≤ 4 ‖ ∇ u ‖ 2 − 4 ∫ R N ∣ u ∣ p * ( I α * ∣ u ∣ p * ) d x + C R 2 + C R ( N − 1 ) ( α + N ) N ( N − 2 ) ‖ ∇ u ‖ ( 1 + N ) ( α + N ) N ( N − 2 ) ≤ 4 ‖ ∇ u ‖ 2 − 4 ∫ R N ∣ u ∣ p * ( I α * ∣ u ∣ p * ) d x + C ( R ) ( 1 + ‖ ∇ u ‖ 2 ) .

Compute using the previous lemma

( 1 + ε ) ‖ ∇ u ‖ 2 − ∫ R N ∣ u ∣ p * ( I α * ∣ u ∣ p * ) d x = p * E ( u ) − ( p * − 1 − ε ) ‖ ∇ u ‖ 2 ≤ p * E ( Q ) − ( p * − 1 − ε ) ( 1 + δ ¯ ) ‖ ∇ Q ‖ 2 ≤ α + N N − 2 2 + α N + α − 2 + α N − 2 − ε ( 1 + δ ¯ ) S N , α N + α 2 + α ≤ − ( δ ¯ ( p * − 1 ) − ε ( 1 + δ ¯ ) ) S N , α p * p * − 1 .

Finally, taking R ≫ 1 , there exists ε > 0 such that

∂ t M R [ u ( t ) ] < − ε for all 0 < t < T * .

By integrating twice the previous estimate, one obtains

0 < V R < V R ( 0 ) + t V R ′ ( 0 ) − ε 2 t 2 for all 0 < t < T * .

This implies that T * < ∞ and the proof is achieved.

6 Global existence and scattering of radial solutions

Let us recall for any time slab I ⊂ R , the Strichartz spaces

S ( I ) ≔ L 2 p * ( I , L 2 N p * N p * − 2 ) , S 1 ( I ) ≔ L 2 p * ( I , L 2 N p * ( p * − 1 ) 2 + α p * ) .

Remarks 6.1

  1. ( 2 p * , 2 N p * N p * − 2 ) ∈ Γ 0 and ( 2 p * , 2 N p * ( p * − 1 ) 2 + α p * ) ∈ Γ 1 ;

  2. S ≔ S ( R ) and S 1 ≔ S 1 ( R ) .

One says that the statement ( S C ) ( u 0 ) holds if:

For u 0 ∈ H ˙ 1 with ‖ ∇ u 0 ‖ < ‖ ∇ Q ‖ and E ( u 0 ) < E ( Q ) , the corresponding solution to (NLS) is global and satisfies

u ∈ S 1 ( R ) .

By using Sobolev injection and Strichartz estimate, one has

‖ e i ⋅ Δ u 0 ‖ S 1 ≲ ‖ ∇ e i ⋅ Δ u 0 ‖ S ≲ ‖ ∇ u 0 ‖ .

Thus, if ‖ ∇ u 0 ‖ ≪ 1 , by the small data theory in Proposition 2.2, ( S C ) ( u 0 ) holds. Now, for each δ > 0 , define the set

S δ ≔ { u 0 ∈ H ˙ r d 1 , E [ u 0 ] < δ and ‖ ∇ u 0 ‖ < ‖ ∇ Q ‖ } .

Define also

E c ≔ sup { δ > 0 s.t. u 0 ∈ S δ ⇒ ( S C ) ( u 0 ) holds } .

The goal is to prove that E c = E [ Q ] . By contradiction, assume that

(26) E c < E [ Q ] .

Then, there is a sequence u n of solutions to (NLS) such that the data u n , 0 ∈ H ˙ 1 satisfies ‖ ∇ u n , 0 ‖ < ‖ ∇ Q ‖ and E [ u n , 0 ] → E c as n → ∞ and ( S C ) ( u n , 0 ) does not hold for any n .

The goal in this section is to show the existence of a critical solution u c to (NLS) with data u c , 0 such that

‖ ∇ u c , 0 ‖ < ‖ ∇ Q ‖ , E [ u c ] = E c and ( S C ) ( u c , 0 ) does not hold .

In the next result, the constant δ is given by Proposition 2.2.

Lemma 6.1

Let a sequence u n ∈ H ˙ r d 1 satisfying

‖ ∇ u n ‖ < ‖ ∇ Q ‖ a n d E ( u n ) → E c .

Assume that ‖ e i ⋅ Δ u n ‖ S 1 ( R ) ≥ δ > 0 and that the profile decomposition of u n satisfies one of the two following hypothesis

  1. liminf n E ( ψ l 1 ( − t n 1 ( λ n 1 ) 2 ) ) < E c ;

  2. liminf n E ( ψ l 1 ( − t n 1 ( λ n 1 ) 2 ) ) = E c . Moreover, if s n 1 ≔ − t n 1 ( λ n 1 ) 2 → s * ∈ [ − ∞ , + ∞ ] and U 1 is the nonlinear profile associated to ( ψ 1 , s n 1 ) , then U 1 is global and ‖ U 1 ‖ S 1 ( R ) < ∞ .

Then, ( S C ) ( u n ) holds, for a subsequence and large n.

Proof

Thanks to the profile decomposition, one has

u n = ∑ j = 1 M 1 ( λ n j ) N − 2 2 e − i t n j ( λ n j ) 2 Δ ψ j ⋅ λ n j + W n M = ∑ j = 1 M 1 ( λ n j ) N − 2 2 ψ l j − t n j ( λ n j ) 2 , ⋅ λ n j + W n M .

Moreover,

‖ ∇ Q ‖ 2 > ‖ ∇ u n ‖ 2 = ∑ j = 1 M ‖ ∇ ψ j ‖ 2 + ‖ ∇ W n M ‖ 2 + o n ( 1 ) = ∑ j = 1 M ‖ ∇ ψ l j − t n j ( λ n j ) 2 ‖ 2 + ‖ ∇ W n M ‖ 2 + o n ( 1 ) ;

and

E c ← E ( u n ) = ∑ j = 1 M E ψ l j − t n j ( λ n j ) 2 + E ( W n M ) + o n ( 1 ) .

By Corollary 4.1, one obtains, for 1 ≤ j ≤ M ,

min E ψ l j − t n j ( λ n j ) 2 , E ( W n M ) ≥ 0 .

One discusses two cases.

(1) Take for a subsequence lim n E ( ψ l 1 ( − t n 1 ( λ n 1 ) 2 ) ) < E c . Write

∇ ψ l 1 − t n 1 ( λ n 1 ) 2 = ‖ ∇ ψ 1 ‖ < ‖ ∇ Q ‖ ; E ( ψ l 1 − t n 1 ( λ n 1 ) 2 ) ≤ E c + o n ( 1 ) < E ( Q ) .

Lemma 4.1 implies that

E ψ l 1 − t n 1 ( λ n 1 ) 2 = ∇ ψ l 1 − t n 1 ( λ n 1 ) 2 2 − 1 p * ∫ R N I α * ψ l 1 − t n 1 ( λ n 1 ) 2 p * ψ l 1 − t n 1 ( λ n 1 ) 2 , x p * d x = 1 p * ∇ ψ l 1 − t n 1 ( λ n 1 ) 2 2 − ∫ R N I α * ψ l 1 − t n 1 ( λ n 1 ) 2 p * ψ l 1 − t n 1 ( λ n 1 ) 2 , x p * d x + 1 − 1 p * ∇ ψ l 1 − t n 1 ( λ n 1 ) 2 2 ≳ ‖ ∇ ψ 1 ‖ 2 .

Thus,

E c ← E ( u n ) ≥ C ‖ ∇ ψ 1 ‖ 2 + ∑ j = 2 M E ψ l j − t n j ( λ n j ) 2 + E ( W n M ) + o n ( 1 ) .

So,

liminf n E ψ l j − t n j ( λ n j ) 2 < E c , ∀ 2 ≤ j ≤ M .

Let U j be the nonlinear profile associated to ( ψ j , − t n j ( λ n j ) 2 ) . Then, for large n ,

(27) ‖ ∇ U j ‖ = ∇ ψ l j − t n j ( λ n j ) 2 + o n ( 1 ) < ‖ ∇ Q ‖ ; E ( U j ) = E ψ l j − t n j ( λ n j ) 2 + o n ( 1 ) < E ( Q ) .

Then, with Proposition 4.1, one has

E ( U j ) ≃ ‖ ∇ U j ( t ) ‖ 2 ≃ ‖ ∇ U j ( 0 ) ‖ 2 .

By the aforementioned estimates, one has

∑ j = 1 M ‖ ∇ ψ j ‖ 2 ≤ ‖ ∇ u n ‖ 2 + o n ( 1 ) ≤ ‖ ∇ Q ‖ 2 .

So, there exists j 0 > 0 such that for j ≥ j 0 , ‖ ∇ ψ j ‖ is small enough. With Strichartz estimate, it follows that ‖ e i ⋅ Δ ψ j ‖ S 1 ( R ) < δ given by the small data theory. Then, the integral formula via the small data theory gives

‖ U j ‖ S 1 ( R ) ≲ ‖ ∇ ψ j ‖ , ∀ j ≥ j 0 .

Now, take for ε 0 > 0 , the near solution

H n , ε 0 ≔ ∑ j = 1 M ( ε 0 ) 1 ( λ n j ) N − 2 2 U j ⋅ − t n j ( λ n j ) 2 , ⋅ λ n j , ‖ e i ⋅ Δ W n M ( ε 0 ) ‖ S 1 ( R ) < ε 0 ,

and the rest

R n , ε 0 ≔ ∑ j = 1 M ( ε 0 ) ℱ 1 ( λ n j ) N − 2 2 U j ⋅ − t n j ( λ n j ) 2 , ⋅ λ n j − ℱ ( H n , ε 0 ) .

Thus,

u n = H n , ε 0 + R n , ε 0 , ‖ e i ⋅ Δ R n , ε 0 ‖ S 1 ( R ) < 2 ε 0 .

Using (27), one writes

‖ ∇ H n , ε 0 ‖ 2 ≲ ‖ ∇ Q ‖ 2 .

Moreover,

‖ H n , ε 0 ‖ S ( R ) 2 p * ≤ ∫ R N ∑ j = 1 M ( ε 0 ) 1 ( λ n j ) N − 2 2 U j t − t n j ( λ n j ) 2 , ⋅ λ n j 2 N p * N p * − 2 2 p * d t ≤ ∑ j = 1 M ( ε 0 ) ∫ R N 1 ( λ n j ) N − 2 2 U j t − t n j ( λ n j ) 2 , ⋅ λ n j 2 N p * N p * − 2 2 p * d t + ∑ I ⊂ [ 1 , 2 p * ] , ∣ I ∣ < 2 p * ∫ R N ∏ j k ∈ I 1 ( λ n j k ) N − 2 2 U j k t − t n j k ( λ n j k ) 2 , ⋅ λ n j k 2 N p * N p * − 2 d t ≔ ( A n ) + ( B n ) .

Now, there exists C 0 independent of M ( ε 0 ) such that

( A n ) ≤ ∑ j = 1 j 0 + ∑ 1 + j 0 M ( ε 0 ) ‖ U j ‖ S 1 ( R ) 2 p * ≤ ∑ j = 1 j 0 ‖ U j ‖ S 1 ( R ) 2 p * + ∑ 1 + j 0 M ( ε 0 ) ‖ ∇ ψ j ‖ 2 p * ≤ C 0 .

Arguing as previously, the orthogonality condition gives

lim n ( B n ) = 0 .

Thus,

‖ H n , ε 0 ‖ L 2 p * ( R , L 2 N p * N p * − 2 ) ≲ C 0 .

Moreover, for large n ,

‖ e i ⋅ Δ ( u n − H n , ε 0 ( 0 ) ) ‖ S 1 ( R ) ≤ ∥ e i ⋅ Δ W ˜ n M ( ε 0 ) ∥ S 1 ( R ) → 0 .

Now,

‖ ∇ R n , ε 0 ‖ S ′ ( R ) = ∇ ℱ ∑ j = 1 M ( ε 0 ) 1 ( λ n j ) N − 2 2 U j ⋅ − t n j ( λ n j ) 2 , ⋅ λ n j − ∑ j = 1 M ( ε 0 ) ℱ 1 ( λ n j ) N − 2 2 U j ⋅ − t n j ( λ n j ) 2 , ⋅ λ n j S ′ ( R ) = ∇ I α * ∑ j = 1 M ( ε 0 ) 1 ( λ n j ) N − 2 2 U j ⋅ − t n j ( λ n j ) 2 , ⋅ λ n j p * ∑ j = 1 M ( ε 0 ) 1 ( λ n j ) N − 2 2 U j ⋅ − t n j ( λ n j ) 2 , ⋅ λ n j p * − 2 × ∑ j = 1 M ( ε 0 ) 1 ( λ n j ) N − 2 2 U j ⋅ − t n j ( λ n j ) 2 , ⋅ λ n j − ∑ j = 1 M ( ε 0 ) 1 ( λ n j ) ( N − 2 ) ( 2 p * − 1 ) 2 I α * U j ⋅ − t n j ( λ n j ) 2 , ⋅ λ n j p * U j ⋅ − t n j ( λ n j ) 2 , ⋅ λ n j p * − 2 U j ⋅ − t n j ( λ n j ) 2 , ⋅ λ n j S ′ ( R ) .

Take a term of the previous quantity expansion

( I ) = I α * ∑ I ⊂ [ 1 , M ( ε 0 ) ] , ∣ I ∣ = p * − 2 ∏ j ∈ I 1 ( λ n j ) N − 2 2 U j ⋅ − t n j ( λ n j ) 2 , ⋅ λ n j ℜ 1 ( λ n k ) N − 1 ∇ U k ⋅ − t n k ( λ n k ) 2 , ⋅ λ n k U ¯ k ⋅ − t n k ( λ n k ) 2 , ⋅ λ n k × ∑ I ⊂ [ 1 , M ( ε 0 ) ] , ∣ I ∣ = p * − 2 ∏ j ∈ I 1 ( λ n j ) N − 2 2 U j ⋅ − t n j ( λ n j ) 2 , ⋅ λ n j 1 ( λ n k ) N − 2 2 U k ⋅ − t n k ( λ n k ) 2 , ⋅ λ n k − I α * 1 ( λ n k ) α k ( N − 2 ) 2 U k ⋅ − t n k ( λ n k ) 2 , ⋅ λ n k p * − 2 ℜ 1 ( λ n k ) N − 1 ∇ U k ⋅ − t n k ( λ n k ) 2 , ⋅ λ n k U ¯ k ⋅ − t n k ( λ n k ) 2 , ⋅ λ n k × 1 ( λ n k ) N − 2 2 U k ⋅ − t n k ( λ n k ) 2 , ⋅ λ n k p * − 2 1 ( λ n k ) N − 2 2 U k ⋅ − t n k ( λ n k ) 2 , ⋅ λ n k = I α * ∑ I ⊂ [ 1 , M ( ε 0 ) ] , ∣ I ∣ = p * − 2 ∏ j ∈ I 1 ( λ n j ) N − 2 2 U j ⋅ − t n j ( λ n j ) 2 , ⋅ λ n j − 1 ( λ n k ) α k ( N − 2 ) 2 U k ⋅ − t n k ( λ n k ) 2 , ⋅ λ n k p * − 2 × ℜ 1 ( λ n k ) N − 1 ∇ U k ⋅ − t n k ( λ n k ) 2 , ⋅ λ n k U ¯ k ⋅ − t n k ( λ n k ) 2 , ⋅ λ n k 1 ( λ n k ) N − 2 2 U k ⋅ − t n k ( λ n k ) 2 , ⋅ λ n k p * − 2 × 1 ( λ n k ) N − 2 2 U k ⋅ − t n k ( λ n k ) 2 , ⋅ λ n k + ( I I ) ,

where ( I I ) is a mixed term. Now, with the triangular and Hardy-Littlewood-Paley inequalities, one obtains

‖ ( I ) − ( I I ) ‖ S ′ ( R ) ≤ ∑ I ⊂ [ 1 , M ( ε 0 ) ] , ∣ I ∣ = p * − 2 I α * ∏ j ∈ I , j ≠ k 1 ( λ n j ) N − 2 2 U j ⋅ − t n j ( λ n j ) 2 , ⋅ λ n j ℜ 1 ( λ n k ) N − 1 ∇ U k ⋅ − t n k ( λ n k ) 2 , ⋅ λ n k × U ¯ k ⋅ − t n k ( λ n k ) 2 , ⋅ λ n k × 1 ( λ n k ) N − 2 2 U k ⋅ − t n k ( λ n k ) 2 , ⋅ λ n k p * − 1 ∣ ‖ S ′ ( R ) ≲ 1 ( λ n j ) N − 2 2 U j ⋅ − t n j ( λ n j ) 2 , ⋅ λ n j 1 ( λ n k ) N − 2 2 U k ⋅ − t n k ( λ n k ) 2 , ⋅ λ n k L p * ( R , L r 1 2 ) , j ≠ k ≲ λ n k λ n j 2 ( N − 1 ) ( N − 2 ) α + N + N 2 − 1 U j ( ⋅ − t n j ) ( λ n k ) 2 ( λ n j ) 2 , λ n k λ n j U k ( ⋅ − t n k ) L p * R , L r 1 2 , j ≠ k .

Here, r 1 ≔ 2 N p * ( p * − 1 ) 2 + α p * and one used the estimate

‖ ∇ ℱ ⋅ ‖ S ′ ≲ ‖ ⋅ ‖ S 1 2 ( p − 1 ) ‖ ∇ ⋅ ‖ S .

Thus, with the orthogonality property of the profile decomposition, it follows that

‖ ∇ R n , ε 0 ‖ S ′ ( R ) → 0 .

Then, applying the perturbation theory in Proposition 2.4, the property S C ( u n ) holds.

(2) By Proposition 4.1, one obtains, for 2 ≤ j ≤ M ,

ψ j = 0 , ‖ ∇ W n M ‖ → 0 .

Thus, ‖ ∇ W n 1 ‖ → 0 and

u n = 1 ( λ n 1 ) N − 2 2 ψ l 1 s n 1 , ⋅ λ n 1 + W n 1 .

Taking the scaling

z n ≔ ( λ n 1 ) N − 2 2 u n ( λ n 1 ⋅ ) , W ¯ n ≔ ( λ n 1 ) N − 2 2 W n ( λ n 1 ⋅ ) .

Then,

‖ ∇ z n ‖ = ‖ ∇ u n ‖ < ‖ ∇ Q ‖

and

z n = ψ l 1 ( s n 1 , ⋅ ) + W ¯ n 1 , ‖ ∇ W ¯ n 1 ‖ → 0 .

Moreover,

‖ ∇ U 1 ( s n 1 ) − ∇ ψ l 1 ( s n 1 ) ‖ → 0 .

Then,

z n = ψ l 1 ( s n 1 , ⋅ ) + W ¯ n 1 ; E ( U 1 ( s n 1 ) ) = E ( ψ l 1 ( s n 1 ) ) + o n ( 1 ) → E c ; ‖ ∇ U 1 ( s n 1 ) ‖ = ‖ ∇ ψ l 1 ( s n 1 ) ‖ + o n ( 1 ) = ‖ ∇ ψ 1 ‖ + o n ( 1 ) < ‖ ∇ Q ‖ .

This case follows by applying the long time perturbation theory in Proposition 2.4 with e = 0 , u ˜ = U 1 and u 0 = z n .□

The proof of the next result, about the existence of a critical solution with a pre-compact flow, is omitted because it follows like in he study by Miao et al. [5].

Proposition 6.1

Assume that E c < E [ Q ] . Then, there exists a global solution u c to (NLS) with data u c , 0 such that

‖ ∇ u c , 0 ‖ < ‖ ∇ Q ‖ , E [ u c ] = E c a n d ( S C ) ( u c , 0 ) d o e s n o t h o l d .

Moreover, there exists a real function λ : ( T * , T * ) → R + such that the following set is pre-compact in H ˙ 1 ,

K ≔ 1 λ ( t ) N − 2 2 u c t , x λ ( t ) , t ∈ ( T * , T * ) .

6.1 Rigidity theorem

The aim of this subsection is to establish the following Liouville-type result.

Theorem 6.1

Let N ≥ 3 and 0 < α < N ≤ 4 + α . Let u 0 ∈ H ˙ r d 1 satisfying (3) and (5) and u ∈ C ( ( T * , T * ) , H ˙ r d 1 ) be the maximal solution to (NLS). If there exists a real function λ : ( T * , T * ) → R + such that the following set is pre-compact in H ˙ 1 ,

K ≔ 1 λ ( t ) N − 2 2 u c t , x λ ( t ) , t ∈ ( T * , T * ) .

Then u 0 = 0 .

Proof

Let u ≔ u c given in the previous theorem and v ≔ 1 λ N − 2 2 u c ( ⋅ , ⋅ λ ) . One discusses two cases.

(1) First case: inf λ ( t ) > λ 0 > 0 . Following lines in the study by Miao et al. [5], one has T * = ∞ . Moreover, by Hardy inequality

‖ u ∣ ⋅ ∣ ‖ ≤ 2 N − 2 ‖ ∇ u ‖ , ∀ N ≥ 3

and the variational analysis, one has, for any ε > 0 , there exists R ( ε ) > 0 such that

∫ ∣ x ∣ > R ( ε ) ∣ u ∣ 2 ∣ x ∣ 2 + ∣ ∇ u ∣ 2 d x + ∫ ∣ x ∣ > R ( ε ) ∪ ∣ y ∣ > R ( ε ) ( I α * ∣ u ∣ p * ) ∣ u ∣ p * d x d y < ε .

Indeed, with Hardy-Littlewood-Sobolev inequality, via the fact that ∇ u ∈ L ∞ ( R , L 2 ) , one has

∫ ∣ x ∣ > R ( ε ) ∪ ∣ y ∣ > R ( ε ) ( I α * ∣ u ∣ p * ) ∣ u ∣ p * d x d y ≤ C ‖ u ‖ L 2 N N − 2 ( ∣ x ∣ > R ( ε ) ) p * ‖ u ‖ L 2 N N − 2 p * ≤ C ‖ ∇ u ‖ L 2 ( ∣ x ∣ > R ( ε ) ) p * ‖ ∇ u ‖ p * < ε 2 .

Take the truncated variance defined in Section 5,

V R ≔ ∫ R N ψ R ( x ) ∣ u ( . , x ) ∣ 2 d x .

Taking account of computation in section 5 via Hardy inequality, one writes

1 4 V R ″ = ‖ ∇ u ‖ 2 − ∫ R N ∣ u ∣ p * ( I α * ∣ u ∣ p * ) d x + o R ( 1 ) .

By computing using the variational estimates, we obtain

F ‖ ∇ u ‖ 2 ‖ ∇ Q ‖ 2 ≔ ‖ ∇ u ‖ ‖ ∇ Q ‖ 2 − ‖ ∇ u ‖ ‖ ∇ Q ‖ 2 p * ≤ ‖ ∇ u ‖ ‖ ∇ Q ‖ 2 − 1 ‖ ∇ Q ‖ 2 p * S N , α p * ∫ R N ∣ u ∣ p * ( I α * ∣ u ∣ p * ) d x ≤ 1 ‖ ∇ Q ‖ 2 ‖ ∇ u ‖ 2 − ∫ R N ∣ u ∣ p * ( I α * ∣ u ∣ p * ) d x .

Now, since p * > 1 , there exists C δ > 0 such that F ( x ) > C δ x for 0 < x < 1 − δ . Thus, by Proposition 4.1, one obtains

V R ″ > C δ ‖ ∇ u 0 ‖ 2 .

On the other hand, by Hardy and Hölder inequalities,

∣ V R ′ ∣ = 2 R ∣ ℑ ∫ R N ∇ ψ x R u ¯ ∇ u d x ∣ ≲ R ∫ { ∣ x ∣ < 2 R } ∣ u ∇ u ∣ d x ≲ R 2 u ∣ ⋅ ∣ ∇ u 1 ≲ R 2 ‖ ∇ u 0 ‖ 2 .

Then, integrating in time, one obtains

t ‖ ∇ u 0 ‖ 2 ≲ ∣ V R ′ ( t ) − V R ′ ( 0 ) ∣ ≲ R 2 ‖ ∇ u 0 ‖ 2 .

This gives u 0 = 0 .

(2) The case inf t λ ( t ) = 0 follows similarly as in the study by Kenig and Merle [8].□

6.2 Proof of the global existence and scattering

Now, we are ready to establish the second part of Theorem 2.1. Theorem 6.1 applied to u c , via Proposition 6.1, gives u c , 0 = 0 . This contradicts the equality ‖ u c ‖ S 1 ( R ) = ∞ . This absurdity means that (26) is false. Thus, the H ˙ 1 scattering holds by Proposition 2.3.

Acknowledgements

Authors would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC-2024-9/1).

  1. Funding information: The second author is partially supported by NSF of Gansu Province (China) (Nos. 20JR5RA498 and 21JR7RE176) and Innovation Ability Promotion Foundation of Universities in Gansu Province (No. 2020B-185).

  2. Conflict of interest: Authors state no conflict of interest.

  3. Data availability statement: No datasets were generated or analyzed during the current study.

Appendix

This section contains two parts. The first one is about a proof of the scattering criteria in Proposition 2.3, and the second one gives an elementary proof of the linear profile decomposition in Proposition 2.5.

A.1 Proof of a scattering criteria

Let us prove Proposition 2.3. By Duhamel formula, one writes

u − e i ⋅ Δ u 0 = i ∫ 0 ⋅ e i ( ⋅ − s ) Δ ℱ d s ; ϕ − u 0 = i ∫ 0 ∞ e − i s Δ ℱ d s ; u ( t ) − e i ⋅ Δ ϕ = − i ∫ ⋅ ∞ e i ( ⋅ − s ) Δ ℱ d s .

Take the admissible pair ( q , r ) ≔ ( 2 p * , 2 N p * N p * − 2 ) ∈ Γ . Thanks to Strichartz estimate and Hardy-Littlewood-Sobolev inequality, it follows that

‖ ∇ ( u − e i t Δ ϕ ) ‖ S ( R ) ≲ ‖ ( I α * ∣ u ∣ p * ) ∣ u ∣ p * − 2 ∇ u ‖ L q ′ ( ( t , ∞ ) , L r ′ ) + ‖ ( I α * ∣ u ∣ p * − 2 ℜ ( u ¯ ∇ u ) ) ∣ u ∣ p * − 2 u ‖ L q ′ ( ( t , ∞ ) , L r ′ ) ≲ ‖ u ‖ L 2 p * ( ( t , ∞ ) , L 2 N p * ( p * − 1 ) 2 + α p * ) 2 ( p * − 1 ) ‖ ∇ u ‖ L 2 p * ( ( t , ∞ ) , L 2 N p * N p * − 2 ) ≲ ‖ u ‖ S 1 ( ( t , ∞ ) ) 2 ( p * − 1 ) ‖ ∇ u ‖ S ( ( t , ∞ ) ) .

Letting t ≫ 1 so that ‖ u ‖ S 1 ( t , ∞ ) ≪ 1 , one obtains

∇ u ∈ S ( R ) .

Then,

‖ ∇ ( u − e i t Δ ϕ ) ‖ S ( R ) ≲ ‖ u ‖ S 1 ( t , ∞ ) 2 ( p * − 1 ) ‖ ∇ u ‖ S ( t , ∞ ) → 0 , if t → ∞ .

This achieves the proof.

A.2 Proof of the profile decomposition

This subsection contains a variant proof of Proposition 2.5. This elementary proof is available for an integer p * . Taking account of [17], the last equality is the only point to prove. It is sufficient to prove that

N ( u n ) = ∑ j = 1 M N ψ l j − t n j ( λ n j ) 2 + N ( W n M ) + o n ( 1 ) .

First step: one proves that

N ∑ j = 1 M 1 ( λ n j ) N − 2 2 ψ l j − t n j ( λ n j ) 2 , ⋅ λ n j = ∑ j = 1 M N ψ l j − t n j ( λ n j ) 2 + o n ( 1 ) .

By re-indexing and adjusting profiles, one can assume that there exists 1 ≤ M 0 ≤ M such that

  1. t n j ( λ n j ) 2 = 0 , for any 1 ≤ j ≤ M 0 ;

  2. ∣ t n j ∣ ( λ n j ) 2 → ∞ , for any M 0 < j ≤ M .

Now, for 1 ≤ j < k ≤ M 0 , we computed with some integrations by parts

( A n ) ≔ N ∑ j = 1 M 0 1 ( λ n j ) N − 2 2 ψ j ⋅ λ n j − ∑ j = 1 M 0 N ( ψ j ) = ∫ R N ∫ R N I α ( x − y ) ∑ j = 1 M 0 1 ( λ n j ) N − 2 2 ψ j y λ n j p * ∑ k = 1 M 0 1 ( λ n k ) N − 2 2 ψ k x λ n k p * − ∑ j = 1 M 0 ∣ ψ j ( y ) ∣ p * ∣ ψ j ( x ) ∣ p * d x d y ≤ ∑ l 1 + ⋯ + l M 0 = p * , h 1 + ⋯ + h M 0 = p * , ( l j , h j ) ≠ ( p * , p * ) ∫ R N ∫ R N I α ( x − y ) ∏ j , k = 1 M 0 1 ( λ n j ) N − 2 2 ψ j y λ n j l j 1 ( λ n k ) N − 2 2 ψ k x λ n k h k d x d y .

Thus, with Hardy-Littlewood-Sobolev inequality and denoting for short ∫ ⋅ ≔ ∫ R N × R N ⋅ ,

( A n ) ≤ ∑ l 1 + ⋯ + l M 0 = p * , h 1 + ⋯ + h M 0 = p * , ( l j , h j ) ≠ ( p * , p * ) ∫ I α ( x − y ) ( λ n 1 ) N − α ( λ n 1 ) 2 N ∏ j , k = 1 M 0 1 ( λ n j ) N − 2 2 ψ j λ n 1 λ n j y l j 1 ( λ n k ) N − 2 2 ψ k λ n 1 λ n k x h k d x d y ≤ ∑ l 1 + ⋯ + l M 0 = p * , h 1 + ⋯ + h M 0 = p * , ( l j , h j ) ≠ ( p * , p * ) ∫ I α ( x − y ) ∏ j , k = 1 M 0 λ n 1 λ n j N − 2 2 ψ j λ n 1 λ n j y l j λ n 1 λ n k N − 2 2 ψ k λ n 1 λ n k x h k d x d y ,

where without loss of generality, one assumes that , for 1 < j ≤ M 0 , yield λ n j λ n 1 → ∞ and l 1 ≠ 0 . Then,

N ∑ j = 1 M 0 1 ( λ n j ) N − 2 2 ψ j ⋅ λ n j = ∑ j = 1 M 0 N ( ψ j ) + o n ( 1 ) .

Now, an expansion gives

( B n ) ≔ N ∑ j = 1 M 0 1 ( λ n j ) N − 2 2 ψ j ⋅ λ n j + ∑ j = 1 + M 0 M 1 ( λ n j ) N − 2 2 e − i t n j ( λ n j ) 2 Δ ψ l j ⋅ λ n j − N ∑ j = 1 M 0 ψ j = ∫ R N I α * ∑ j = 1 M 0 1 ( λ n j ) N − 2 2 ψ j ⋅ λ n j + ∑ j = 1 + M 0 M 1 ( λ n j ) N − 2 2 e − i t n j ( λ n j ) 2 Δ ψ j ⋅ λ n j p * × ∑ j = 1 M 0 1 ( λ n j ) N − 2 2 ψ j x λ n j + ∑ j = 1 + M 0 M 1 ( λ n j ) N − 2 2 e − i t n j ( λ n j ) 2 Δ ψ j x λ n j p * d x − ∫ R N I α * ∑ j = 1 M 0 ψ j p * ∑ j = 1 M 0 ψ j p * d x .

Thus,

( B n ) ≤ ∫ I α ( x − y ) ∑ j = 1 M 0 1 ( λ n j ) N − 2 2 ψ j x λ n j + ∑ j = 1 + M 0 M 1 ( λ n j ) N − 2 2 e − i t n j ( λ n j ) 2 Δ ψ j x λ n j p * ⋅ ∑ j = 1 M 0 1 ( λ n j ) N − 2 2 ψ j y λ n j + ∑ j = 1 + M 0 M 1 ( λ n j ) N − 2 2 e − i t n j ( λ n j ) 2 Δ ψ j y λ n j p * − ∑ j = 1 M 0 ψ j ( x ) p * ∑ j = 1 M 0 ψ j ( y ) p * d x d y ≤ ∑ k 1 + ⋯ + k M = p * , h 1 + ⋯ + h M = p * , k i + h i ≠ 2 p * ∫ I α ( x − y ) ∏ j = 1 M 0 1 ( λ n j ) N − 2 2 ψ j x λ n j k j ∏ j ′ = 1 + M 0 M 1 ( λ n j ′ ) N − 2 2 e − i t n j ′ ( λ n j ′ ) 2 Δ ψ j ′ x λ n j ′ k j ′ × ∏ i = 1 M 0 1 ( λ n i ) N − 2 2 ψ i y λ n i h i ∏ i ′ = 1 + M 0 M 1 ( λ n i ′ ) N − 2 2 e − i t n i ′ ( λ n i ′ ) 2 Δ ψ i ′ y λ n i ′ h i ′ d x d y .

Thus, taking account of Corollary 2.6, one obtains

( B n ) ≲ ∑ k 1 + ⋯ + k M = p * , h 1 + ⋯ + h M = p * , k i + h i ≠ 2 p * ∏ j = 1 M 0 1 ( λ n j ) k j ( N − 2 ) 2 ψ j ⋅ λ n j 2 N p * α + N k j ∏ j ′ = 1 + M 0 M 1 ( λ n j ′ ) k j ′ ( N − 2 ) 2 e − i t n j ′ ( λ n j ′ ) 2 Δ ψ j ′ ⋅ λ n j ′ 2 N p * α + N k j ′ × ∏ i = 1 M 0 1 ( λ n j ) h i ( N − 2 ) 2 ψ i ⋅ λ n i 2 N p * α + N h i ∏ i ′ = 1 + M 0 M 1 ( λ n i ′ ) h i ′ ( N − 2 ) 2 ∥ e − i t n i ′ Δ ψ i ′ ∥ 2 N p * α + N h i ′ ≲ ∑ k 1 + ⋯ + k M = p * , h 1 + ⋯ + h M = p * , k i + h i ≠ 2 p ∏ j = 1 M 0 ‖ ψ j ‖ H ˙ 1 k j ∏ j ′ = 1 + M 0 M e − i t n j ′ ( λ n j ′ ) 2 Δ ψ j ′ 2 N N − 2 k j ′ ∏ i = 1 M 0 ‖ ψ i ‖ H ˙ 1 h i ∏ i ′ = 1 + M 0 M e − i t n i ′ ( λ n i ′ ) 2 Δ ψ i ′ 2 N N − 2 h i ′ .

For M 0 < j ≤ M , via Hardy-Littlewood-Sobolev inequality, the free kernel Schrödinger decay and a density argument [25], one obtains

‖ ψ l j − t n j ( λ n j ) 2 ‖ 2 N N − 2 ≲ ( λ n j ) 2 t n j ‖ ψ j ‖ 2 N N + 2 → 0 .

Thus, ( B n ) → 0 . This finishes the first step. Now, one writes using Hardy-Littlewood-Sobolev injection

N [ W n M 1 ] ≤ sup t N [ e i t Δ W n M 1 ] ≤ sup t ∫ R N ( I α * ∣ e i t Δ W n M 1 ∣ p * ) ∣ e i t Δ W n M 1 ∣ p * d x ≤ sup t ‖ e i t Δ W n M 1 ‖ 2 N N − 2 2 p * .

Then, by the asymptotic smallness,

lim M 1 → ∞ ( lim n → ∞ N [ W n M 1 ] ) = 0 .

Take M ≥ 1 and ε > 0 . By Sobolev injections, Hölder inequality and Proposition 2.1, one obtains

sup n N [ u n ] + sup n N [ W n M ] ≲ 1 .

Thus, one can choose M 1 ≥ M and N 1 > 0 such that for n ≥ N 1 ,

( C n ) ≔ ∣ N [ u n − W n M 1 ] − N [ u n ] ∣ + ∣ N [ W n M − W n M 1 ] − N [ W n M ] ∣ ≲ ∫ I α ( x − y ) ∣ ( u n ( x ) − W n M 1 ( x ) ) p * ( u n ( y ) − W n M 1 ( y ) ) p * − ( u n ( x ) u n ( y ) ) p * ∣ d x d y + ∫ I α ( x − y ) ∣ ( W n M ( x ) − W n M 1 ( x ) ) p * ( W n M ( y ) − W n M 1 ( y ) ) p * − ( W n M ( x ) W n M ( y ) ) p * ∣ d x d y ≲ ∑ i j + l j = i k + l k = p * , i j , i k < p * ∫ R N ∫ R N I α ( x − y ) ∏ 1 ≤ j , l ≤ p ∣ u n ( x ) ∣ i j ∣ u n ( y ) ∣ i k ∣ W n M 1 ( x ) ∣ l j ∣ W n M 1 ( y ) ∣ l k d x d y + ∑ i j + l j = i k + l k = p , i j , i k < p ∫ R N ∫ R N I α ( x − y ) ∏ 1 ≤ j , l ≤ p ∣ W n M ( x ) ∣ i j ∣ W n M ( y ) ∣ i k ∣ W n M 1 ( x ) ∣ l j ∣ W n M 1 ( y ) ∣ l k d x d y ≲ ∑ i j + l j = i k + l k = p * , i j , i k < p * ‖ u n ‖ 2 N p * α + N i j + i k ‖ W n M 1 ‖ 2 N p * α + N l j + l k + ∑ i j + l j = i k + l k = p * , i j , i k < p * ‖ W n M ‖ 2 N p * α + N i j + i k ‖ W n M 1 ‖ 2 N p * α + N l j + l k .

Thus,

( C n ) ≲ ∑ i j + l j = i k + l k = p * , i j , i k < p * ‖ u n ‖ 2 N N − 2 i j + i k ‖ W n M 1 ‖ 2 N N − 2 l j + l k + ∑ i j + l j = i k + l k = p , i j , i k < p ‖ W n M ‖ 2 N N − 2 i j + i k ‖ W n M 1 ‖ 2 N N − 2 l j + l k ≲ ∑ i j + l j = i k + l k = p * , i j , i k < p * ‖ u n ‖ H ˙ 1 i j + i k ‖ W n M 1 ‖ 2 N N − 2 l j + l k + ∑ i j + l j = i k + l k = p , i j , i k < p ‖ W n M ‖ H ˙ 1 i j + i k ‖ W n M 1 ‖ 2 N N − 2 l j + l k ≲ ∑ i j + l j = i k + l k = p * , i j , i k < p * ‖ W n M 1 ‖ 2 N N − 2 l j + l k ≤ ε .

By the first step and the profile expansion, one takes N 2 ≥ N 1 such that for n ≥ N 2 ,

N [ u n − W n M 1 ] − ∑ j = 1 M 1 N e − i t n j ( λ n j ) 2 Δ ψ j ⋅ λ n j < ε .

By the profile expansion,

W n M − W n M 1 = ∑ j = 1 + M M 1 1 ( λ n j ) N − 2 N e − i t n j ( λ n j ) 2 Δ ψ j ⋅ λ n j .

So, with the first step, one takes N 3 ≥ N 2 such that for n ≥ N 3 ,

N [ W n M − W n M 1 ] − ∑ j = 1 + M M 1 N 1 ( λ n j ) N − 2 N e − i t n j ( λ n j ) 2 Δ ψ j ⋅ λ n j < ε .

Taking account of the last three inequalities with right hand side ε , one obtains

∣ N [ u n ] − ∑ j = 1 M N 1 ( λ n j ) N − 2 N e − i t n j ( λ n j ) 2 Δ ψ j ⋅ λ n j − N [ W n M ] ∣ ≲ ε .

This finishes the proof.

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Received: 2023-10-24
Revised: 2024-02-06
Accepted: 2024-02-27
Published Online: 2024-04-19

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  4. Zagreb connection indices on polyomino chains and random polyomino chains
  5. On the multiplicative sum Zagreb index of molecular graphs
  6. The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two
  7. Special Issue on Convex Analysis and Applications - Part I
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  10. (p, q)-Compactness in spaces of holomorphic mappings
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https://doi.org/10.1515/math-2024-0002
Keywords for this article
energy-critical Hartree equation; nonlinear equations; scattering; blow-up
Creative Commons
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Articles in the same Issue

  1. Special Issue on Contemporary Developments in Graph Topological Indices
  2. On the maximum atom-bond sum-connectivity index of graphs
  3. Upper bounds for the global cyclicity index
  4. Zagreb connection indices on polyomino chains and random polyomino chains
  5. On the multiplicative sum Zagreb index of molecular graphs
  6. The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two
  7. Special Issue on Convex Analysis and Applications - Part I
  8. Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function
  9. Scattering threshold for the focusing energy-critical generalized Hartree equation
  10. (p, q)-Compactness in spaces of holomorphic mappings
  11. Characterizations of minimal elements of upper support with applications in minimizing DC functions
  12. Some new Hermite-Hadamard-type inequalities for strongly h-convex functions on co-ordinates
  13. Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void
  14. Extension of Fejér's inequality to the class of sub-biharmonic functions
  15. On sup- and inf-attaining functionals
  16. Regularization method and a posteriori error estimates for the two membranes problem
  17. Rapid Communication
  18. Note on quasivarieties generated by finite pointed abelian groups
  19. Review Articles
  20. Amitsur's theorem, semicentral idempotents, and additively idempotent semirings
  21. A comprehensive review of the recent numerical methods for solving FPDEs
  22. On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating
  23. Pullback and uniform exponential attractors for non-autonomous Oregonator systems
  24. Regular Articles
  25. On certain functional equation related to derivations
  26. The product of a quartic and a sextic number cannot be octic
  27. Combined system of additive functional equations in Banach algebras
  28. Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices
  29. Local and global solvability for the Boussinesq system in Besov spaces
  30. Construction of 4 x 4 symmetric stochastic matrices with given spectra
  31. A conjecture of Mallows and Sloane with the universal denominator of Hilbert series
  32. The uniqueness of expression for generalized quadratic matrices
  33. On the generalized exponential sums and their fourth power mean
  34. Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions
  35. Computing the determinant of a signed graph
  36. Two results on the value distribution of meromorphic functions
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  39. About j-Noetherian rings
  40. Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model
  41. On the distribution of powered numbers
  42. Almost periodic dynamics for a delayed differential neoclassical growth model with discontinuous control strategy
  43. A new distributionally robust reward-risk model for portfolio optimization
  44. Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results
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  47. Mutually unbiased bases via complex projective trigonometry
  48. Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three
  49. On second-order linear Stieltjes differential equations with non-constant coefficients
  50. Complex dynamics of a nonlinear discrete predator-prey system with Allee effect
  51. The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains
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  53. Boundary value problems for integro-differential and singular higher-order differential equations
  54. Existence and properties of soliton solution for the quasilinear Schrödinger system
  55. Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension
  56. Endpoint boundedness of toroidal pseudo-differential operators
  57. Matrix stretching
  58. A singular perturbation result for a class of periodic-parabolic BVPs
  59. On Laguerre-Sobolev matrix orthogonal polynomials
  60. Pullback attractors for fractional lattice systems with delays in weighted space
  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
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  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
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