Thomas Duyckaerts

Phan Van Tin

MASS-ENERGY SCATTERING CRITERION FOR DOUBLE POWER SCHRÖDINGER EQUATIONS

Keywords: February 19, 2024. 2020 Mathematics Subject Classication. 35Q55 Nonlinear Schrödinger equations, double power, prole decomposition, stability theory, scattering

 for all L 2 -supercritical powers, specially, our results adapt to the cases of energy-supercritical nonlinearity. Contents 1. Introduction and main resuls Acknowledgment 2. Preliminaries 2.1. Cauchy theory 2.2. Prole decomposition 2.3. A uniform bound on the scattering norm 2.4. Global well posedness 3. Variational properties 3.1. Preliminaries 3.2. Positivity of virial functional below critical mass 3.3. Positivity of virial functional below a critical energy, at xed mass 4. Scattering 4.1. Proof of Theorem 1.4 in the energy-supercritical case 4.2. Sketch of proof of Theorem 1.2. Appendix A. Elliptic properties

Introduction and main resuls

This article concerns the nonlinear Schrödinger (NLS) equation (1.1) i∂ t u + ∆u = |u| p0 u -|u| p1 u, in space dimension d ⩾ 1, where 4 d < p 1 < p 0 . The equation (1.1) has 3 conserved quantities: the mass,

M (u(t)) = R d |u(t, x)| 2 dx,
the energy

E(u(t)) = R d |∇u(t, x)| 2 dx + 2 p 0 + 2 R d |u(t, x)| p0+2 dx - 2 p 1 + 2 R d |u(t, x)| p1+2 dx.
and the momentum

P (u(t)) = Im R d ∇u(t, x) u(t, x) dx.
We recall that a solution of (1.1) with initial data in a Sobolev space H s where the equation (1.1) is well-posed is said to scatter in H s forward in time when it is dened on [0, ∞), and there exists v 0 ∈ H s such that lim t→∞ e it∆ v 0 -u(t) H s = 0.

Our goal is to give sucient conditions of scattering for solutions of (1.1).

The equation (1.1) has stationary wave solutions, of the form e iωt Q ω , where Q ω is a solution of the equation (1.2) -∆φ = -|φ| p0 φ + |φ| p1 φ -ωφ, see [START_REF] Lewin | The double-power nonlinear Schrödinger equation and its generalizations: uniqueness, non-degeneracy and applications[END_REF]Theorem 2] and references therein. It has positive solutions for some values of ω. More precisely, there exists an explicit ω * (depending only on p 0 and p 1 ) such that for 0 < ω < ω * , the equation (1.2) has exactly one positive solution (up to translation) Q ω ∈ (H 1 ∩ L ∞ )(R d ) when 0 < ω < ω * , and no such solution for ω ⩾ ω * .

Some of these solutions are related to the minimization problem:

(1.3)

I(m) = inf φ ∈ H 1 (R d ) ∩ L p0+2 (R d ) R d |φ| 2 dx = m E(u).
(see [START_REF] Lewin | The double-power nonlinear Schrödinger equation and its generalizations: uniqueness, non-degeneracy and applications[END_REF]Theorem 6]). There is a critical mass m c such that for m ⩽ m c , I(m) = 0 and for m > m c , I(m) < 0. The minimum is not attained for m < m c . In the case p 1 > 4 d that we consider here, the minimum is attained for m ⩾ m c by a positive, radial solution of (1.2), that we will denote by Q ω . The solution Q ωc corresponding to the mass m c has minimal mass among all solutions of the family of equations (1.2).

Obviously, the stationary wave solutions do not scatter. Other examples of nonscattering solutions are travelling waves that are exactly the Galilean transforms of stationary waves. According to the soliton resolution conjecture, these stationary and travelling waves should be the only obstruction, leading to the conjecture that solutions with mass below m c scatter to a positive solution. A weaker result was proved in [START_REF] Killip | Solitons and scattering for the cubic-quintic nonlinear Schrödinger equation on R 3[END_REF] in the case p 0 = 4, p 1 = 2, d = 3, namely that scattering holds for solutions with mass M (u) ⩽ 4 3 √ 3 mc , and with an additional bound on the energy in the range of mass 4 3 √ 3 mc < M < m c . The goal of this article is to show that this picture is not specic to the case studied in [START_REF] Killip | Solitons and scattering for the cubic-quintic nonlinear Schrödinger equation on R 3[END_REF], but holds for all exponents 4 d < p 1 < p 0 . We will rely on the work [START_REF] Lewin | The double-power nonlinear Schrödinger equation and its generalizations: uniqueness, non-degeneracy and applications[END_REF] on the stationary problem (1.2), and on our previous article [START_REF] Duyckaerts | Prole decomposition and scattering for general nonlinear schrödinger equations[END_REF], where preliminary results were obtained for a family of nonlinear Schrödinger equations that includes (1.1).

As in [START_REF] Killip | Solitons and scattering for the cubic-quintic nonlinear Schrödinger equation on R 3[END_REF], we will use the standard compactness/rigidity scheme (as initiated in [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case[END_REF] for the energy-critical problem). One of the key ingredient of this scheme is the positivity of the functional:

Φ(u) = |∇u| 2 - dp 1 2(p 1 + 2) |u| p1+2 + dp 0 2(p 0 + 2) |u| p0+2 ,
that appears in the virial identity (1.4)

d dt Im x • ∇u u = 2Φ(u).
In Section 3, we prove that there is a critical value mc ∈ (0, m c ) such that Φ(u) > 0 for every nonzero u such that M (u) < mc , and that for mc ⩽ M (u) < m c , we have Φ(u) > 0 with an additional condition on the energy. Namely, let:

(1.5)

mc = p 0 p 1 p 1 d-4 2(p 0 -p 1 )
R := φ ∈ H s0 ∩ H 1 , 0 < |φ| 2 < m c , E(φ) < e |φ| 2 ,
one has

∀u ∈ R, Φ(u) > 0.
The function e is strictly decreasing on [ mc , m c ] if d ∈ {2, 3, 4} or p 0 ⩽ 4 d-2 . We next state our scattering results. Throughout of this paper, we will make the following additional assumption on the exponent: Assumption A. 4 d < p 1 < p 0 and for all j ∈ {1, 2}, ⌈s 0 ⌉ < p j or p j is an even integer. 1This type of assumption is classical to ensure a minimal regularity of the nonlinearity, however we have not tried to obtain the optimal conditions. It is certainly possible to improve this condition using techniques as in [START_REF] Visan | The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions[END_REF].

We rst consider the case where the higher order nonlinearity is energy-subcritical or energycritical, that is d = 2 or d ⩾ 3 and p 0 ⩽ 4 d-2 . Equivalently, the critical Sobolev exponent

s 0 = d 2 -2 p0
for the usual (NLS) equation with a single power nonlinearity (1.7)

i∂ t u + ∆u = |u| p0 u satises s 0 ⩽ 1.
We recall that this equation (1.7) is well-posed in H 1 (see [START_REF] Kato | On nonlinear Schrödinger equations[END_REF], [START_REF] Cazenave | The Cauchy problem for the critical nonlinear Schrödinger equation in H s[END_REF]), One can prove in this case that the equation is well-posed in H 1 . It follows from conservation laws and the standard Gagliardo-Nirenberg inequality that all solutions of (1.1) are bounded in H 1 . Using the well-posedness theory when s 0 < 1, and a more rened argument ultimately relying on the fact that all solutions of (1.7) scatter (see [START_REF] Colliander | Global well-posedness and scattering for the energy-critical Schrödinger equation in R 3[END_REF], [START_REF] Ryckman | Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in R 1+4[END_REF] and [START_REF] Visan | The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions[END_REF]) for s 0 = 1, one can deduce from this bound that all solutions of (1.1) are global (see [START_REF] Zhang | On the Cauchy problem of 3d energy-critical Schrödinger equations with subcritical perturbations[END_REF], [START_REF] Tao | Minimal-mass blowup solutions of the mass-critical NLS[END_REF]). Theorem 1.2. Assume d = 2 and 2 < p 1 < p 0 , or d ⩾ 3 and p 0 ⩽ 4 d-2 . Suppose that assumption A holds. Let u 0 ∈ R. Then u scatters in both time direction. Theorem 1.2 was proved in the case d = 3, p 0 = 4 (thus s 0 = 1), p 1 = 2 in [START_REF] Killip | Solitons and scattering for the cubic-quintic nonlinear Schrödinger equation on R 3[END_REF]. See also [START_REF] Tao | The nonlinear Schrödinger equation with combined power-type nonlinearities[END_REF]Theorem 1.3] where a scattering result for small mass was obtained for general double powernonlinearities with s 0 ⩽ 1.

We note that the ground state solution e itωc Q ωc provides a non-scattering solution of (1.1) which is bounded in the space H s0 ∩ H 1 , with mass m c . However, as in [START_REF] Killip | Solitons and scattering for the cubic-quintic nonlinear Schrödinger equation on R 3[END_REF], the rst critical mass mc is not given by this minimal ground state mass, but by a multiple of it (corresponding to the mass of a rescaled solution of (1.2)). It is not clear whether this restriction is due the method, based on the positivity of Φ(u), or if there exists nonscattering, bounded solutions of (1.1) with mc < |u 0 | 2 < m c .

In the case s 0 > 1, the equations (1.1) and (1.7) are no longer well-posed in H 1 , but rather in the Sobolev space H s0 (see [START_REF] Duyckaerts | Prole decomposition and scattering for general nonlinear schrödinger equations[END_REF]Section 2]), at least under Assumption A. The homogeneous equation (1.7) is also well-posed in the homogeneous Sobolev space Ḣs0 (see [START_REF] Cazenave | The Cauchy problem for the critical nonlinear Schrödinger equation in H s[END_REF] and [START_REF] Duyckaerts | Prole decomposition and scattering for general nonlinear schrödinger equations[END_REF]Remark 2.20]).

Furthermore the global well-posedness is not guaranteed. Indeed Merle, Raphaël, Rodnianski and Szeftel [START_REF] Merle | On blow up for the energy super critical defocusing nonlinear Schrödinger equations[END_REF] have constructed solutions of the homogeneous equation (1.7) for some values of p 0 , d (with s 0 > 1), that blow up in nite time. It is very likely that such a phenomenon persists in the case of a double power nonlinearity.

It is known however that for many values of p 0 a solution of (1.7) that remains bounded in the critical Sobolev space are global and scatter. We thus consider the following property:

Property 1.3. Let A 0 ∈ (0, ∞]. For any solution u of (1.7) with initial data in Ḣs0 , if (1.8) lim sup t→T+(u) ∥u(t)∥ Ḣs 0 < A 0 .
Then T + (u) = +∞ and u scatters for positive times in Ḣs0 , i.e. there exists v 0 ∈ Ḣs0 such that lim t→∞ u(t) -e it∆ v 0 Ḣs 0 = 0. Property 1.3 holds for small A 0 by standard small data theory. It is conjectured that it holds for A 0 = ∞. It was proved [START_REF] Colliander | Global well-posedness and scattering for the energy-critical Schrödinger equation in R 3[END_REF], [START_REF] Ryckman | Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in R 1+4[END_REF] and [START_REF] Visan | The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions[END_REF] in the case s 0 = 1 (where the bound (1.8) is automatically given by conservation of the energy). The study of Property 1.3 in for other critical exponents was initiated in [START_REF] Kenig | Scattering for H 1/2 bounded solutions to the cubic, defocusing NLS in 3 dimensions[END_REF] where it was proved when d = 3, p 0 = 2 (thus s 0 = 1/2), for radial solutions. It was later proved in many other cases: see e.g. [START_REF] Killip | Energy-supercritical NLS: critical Ḣs -bounds imply scattering[END_REF], [START_REF] Miao | The defocusing energy-supercritical NLS in four space dimensions[END_REF], [START_REF] Lu | The radial defocusing energy-supercritical NLS in dimension four[END_REF], [START_REF] Dodson | The defocusing quintic NLS in four space dimensions[END_REF]

for s 0 > 1 in dimension d ⩾ 4 (with technical restriction if d ⩾ 7). Theorem 1.4. Assume d ⩾ 3, p 0 > 4 d-2
and assumption A holds. Let A 0 > 0 such that Property 1.3 holds. Let u be a solution of (1.1) such that u 0 ∈ R. Then u scatters forward in time.

The proofs of Theorem 1.2 and 1.4 is by contradiction. One of the tool needed for this method is a prole decomposition adapted to equation (1.1), which was constructed in [START_REF] Duyckaerts | Prole decomposition and scattering for general nonlinear schrödinger equations[END_REF] in a more general context. Using this prole decomposition, one can show that if the scattering does not hold, there exists a nonzero global solution u c , of (1.1) such that there exists x(t) such that

K = {u c (t, • + x(t)), t ∈ R} has compact closure in H s0 ∩ H 1 ,
and ∀t, u c (t) ∈ R. The second step of the argument is to prove that 0 is the only solution u c (with mass and energy in the considered domain) such that K dened above has compact closure in H 1 ∩ H s0 . For this we use a general rigidity results, proved in [START_REF] Duyckaerts | Prole decomposition and scattering for general nonlinear schrödinger equations[END_REF], and that extends the standard rigidity result (see [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case[END_REF] for radial (NLS) equation, [START_REF] Duyckaerts | Scattering for the non-radial 3D cubic nonlinear Schrödinger equation[END_REF] for extension to nonradial cases, and [START_REF] Killip | Solitons and scattering for the cubic-quintic nonlinear Schrödinger equation on R 3[END_REF] for the cubic/quintic double power nonlinearity): Proposition 1.5. With the assumptions above, let u be a solution of (1.1) dened on [0, ∞) such that there exists x(t) dened for t ∈ [0, ∞) with (1.9)

K = {u(t, x + x(t)); t ⩾ 0} has compact closure in H s0 ∩ H 1 . Then (1.10) min t⩾0 Φ(u(t)) - |P (u)| 2 M (u) = 0.
To conclude the proof of Theorems 1.2 and 1.4, it remains to prove that

Φ(u) - |P (u)| 2 M (u) > 0,
for u ∈ R. This follows easily from Theorem 1.1 considering all the e ix•ξ0 u and optimizing in ξ 0 ). The mathematical study of (NLS) equation with a double-power nonlinearity was initiated in [START_REF] Zhang | On the Cauchy problem of 3d energy-critical Schrödinger equations with subcritical perturbations[END_REF], in dimension 3, where the author investigated the global well-posedness, scattering and blowup phenomena in the case p 0 = 4. This includes in particular a scattering result for small mass, in the spirit of Theorem 1.4, but without explicit mass. Similar results were obtained in [START_REF] Tao | The nonlinear Schrödinger equation with combined power-type nonlinearities[END_REF], in general dimension d in the energy-critical and supercritical setting s 0 ⩽ 1. As already mentioned, Theorem 1.4 was proved in [START_REF] Killip | Solitons and scattering for the cubic-quintic nonlinear Schrödinger equation on R 3[END_REF] in the particular case d = 3, p 0 = 4, p 1 = 2. The scattering was extended to a slightly larger region in [START_REF] Killip | Scattering for the cubic-quintic NLS: crossing the virial threshold[END_REF].

The problem with p 1 = 4 d , p 0 < 4 d-2 was considered in [START_REF] Cheng | Scattering for the mass super-critical perturbations of the mass critical nonlinear Schrödinger equations[END_REF], [START_REF] Murphy | Threshold scattering for the 2d radial cubic-quintic NLS[END_REF], where the author investigated scattering below or at the mass of the ground-state for the mass-critical homogeneous equation. See also [START_REF] Carles | Orbital stability vs. scattering in the cubic-quintic Schrödinger equation[END_REF] which considers the case (p 0 , p 1 ) = (4, 2) in space dimensions 1, 2 and 3.

We conclude this introduction with an outline of the paper. In Section 2 we recall results from [START_REF] Duyckaerts | Prole decomposition and scattering for general nonlinear schrödinger equations[END_REF] in the particular case of equation (1.1). This includes a well-posedness theory, as well as a prole decomposition adapted to (1.1) and a quantitative version of Property 1.3. In Section 3, we prove Theorem 1.1 on the positivity of the virial functional. In Section 4 we prove our scattering results Theorems 1.2 and 1.4.

Acknowledgment

The second author is supported by Post-doc fellowship from Labex MME-DII: SAIC/2022 No 10078.

Preliminaries

2.1. Cauchy theory. This section is concerned with the Cauchy and stability theory for the equation (1.1). We refer to [START_REF] Duyckaerts | Prole decomposition and scattering for general nonlinear schrödinger equations[END_REF] for the proofs. We start with a few notations.

For each q ⩾ 1, we let q ′ be the conjugate exponent given by 1

q + 1 q ′ = 1.
If m is a complex valued function on R d , we dene by m(∇) the Fourier multiplier with symbol m(ξ), i.e. m(∇)u = m(ξ)û(ξ), where û is the Fourier transform of u. For a multi-index α = (α 1 , α 2 , ..., α d ), denote

D α = ∂ α1 x1 • • • ∂ α d x d and |α| = d i=1 |α i |.
If X is a vector space, (u, v) ∈ X 2 , we will make a small abuse of notation, denoting ∥(u, v)∥ X = ∥u∥ X + ∥v∥ X .

We will consider the following function spaces:

S 0 (I) =        L ∞ I, L 2 (R d ) ∩ L 2 I, L 2d d-2 (R d ) if d ⩾ 3 L ∞ I, L 2 (R 2 ) ∩ L q2 I, L r2 (R 2 ) if d = 2 L ∞ I, L 2 (R) ∩ L 4 (I, L ∞ (R)) if d = 1,
where when d = 2, (q 2 , r 2 ) is an admissible pair with q 2 > 2 close to 2, and

N 0 (I) =        L 1 I, L 2 (R d ) + L 2 I, L 2d d+2 (R d ) if d ⩾ 3 L 1 I, L 2 (R 2 ) + L q ′ 2 I, L r ′ 2 (R 2 ) if d = 2 L 1 I, L 2 (R) + L 4/3 I, L 1 (R) if d = 1,
By Strichartz estimates ( [START_REF] Strichartz | Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations[END_REF], [10] [12]),

∥u∥ S 0 (I) ≲ ∥u 0 ∥ L 2 + ∥f ∥ N 0 (I)
for any solution C 0 (I, R) of i∂ t u + ∆u = f on I ∋ 0 with initial data u 0 at t = 0.

We also dene the following Strichartz spaces, and dual Strichartz spaces:

W 0 (I) = L 2(2+d) d I × R d Z 0 (I) = W 0 (I) ′ = L 2(d+2) d+4 (I × R d ),
so that we have S 0 (I) ⊂ W 0 (I) and Z 0 (I) ⊂ N 0 (I) (with continous embedding). We denote (2.1)

X(I) = L p0(d+2)/2 t,x (I × R d ) ∩ L p1(d+2)/2 t,x (I × R d ).
For s ⩾ 0, we denote

∥u∥ S s (I) = ∥⟨∇⟩ s u∥ S 0 (I) , ∥u∥ Ṡs (I) = ∥|∇| s u∥ S 0 (I)
and dene similarly W s (I), Ẇ s (I), N s (I), Ṅ s (I), Z s (I) and Żs (I).

By Sobolev inequalities, S s0 (I) is continuously embedded in X(I).

Denition 2.1. Let 0 ∈ I be an interval. By denition, a solution u to (1.1) on I, with initial data in H s0 is a function u ∈ C(I, H s0 ) such that for all K ⊂ I compact, u ∈ S s0 (K) and u satises the following Duhamel formula (2.2)

u(t) = e it∆ u 0 -i t 0 e i(t-τ )∆ g(u)(τ ) dτ,
for all t ∈ I, where g(u) = |u| p0 u -|u| p1 u.

Noting that the assumption p 0 > 4 d implies p 0 (d + 2)/2 > 2 (and p 0 (d + 2)/2 ⩾ 6 if d = 1), we can choose q 0 such that (p 0 (d + 2)/2, q 0 ) is an admissible pair. By Sobolev inequality and the denitions of s 0 , q 0 , one can check

(2.3) ∥u∥ L p 0 (d+2) 2 x ≲ ∥|∇| s0 u∥ L q 0 and thus (2.4) ∥u∥ L p 0 (d+2) 2 (I×R d ) ≲ ∥|∇| s0 u∥ S 0 (I) , ∥u∥ X(I) ≲ ∥u∥ S s 0 (I) .
The following local well-posedness statement is contained in [9, Section 2]. See in particular Proposition 2.12, Remarks 2.13, 2.14 and 2.15 and Lemma 2.16 there: Theorem 2.2. Let u 0 ∈ H s0 . Let p 0 , p 1 such that Assumption A holds. Then there exists a unique maximal solution of u to (2.2), dened on an interval (T -, T + ) ∋ 0, such that for any interval I such that I ⊂ (T -, T + ), one has u ∈ S s0 (I). Furthermore we have the following:

• T + < ∞ =⇒ u / ∈ X([0, T + )). • If u ∈ X([0, ∞)), then u scatters as t → ∞. Remark 2.3. Let I be an interval such that 0 ∈ I. If e i•∆ u 0 X(I) < δ is small enough, then u is dened on I. By Strichartz, e i•∆ u 0 X (R) ≲ ∥u 0 ∥ H s 0 < ∞ which implies local well posedness of (2.2).
We next state the existence of wave operators for equation (1.1):

Proposition 2.4. Let u 0 ∈ H s0 . Let p 0 , p 1 such that Assumption A holds. Let v 0 ∈ H s0 and v L (t) = e it∆ v 0 . Then there exist a unique solution u ∈ C 0 ((T -(u), ∞), H s0 ) of (1.1) such that lim t→∞ ∥u(t) -v L (t)∥ H s 0 = 0.
We recall the following long time perturbation theory: Theorem 2.5. (see [START_REF] Duyckaerts | Prole decomposition and scattering for general nonlinear schrödinger equations[END_REF][Theorem 2.19]) Let A > 0. There exists constants ε(A) ∈ (0, 1], C(A) > 0 with the following properties. Let 0 ∈ I be an compact interval of R and w be a solution of the following equation

Lw = g(w) + e,
and u 0 ∈ H s0 such that

∥w 0 ∥ H s 0 + ∥w∥ X(I) ⩽ A, ∥e∥ N s 0 (I) + ∥u 0 -w(0)∥ H s 0 = ε ⩽ ε(A).
There the solution u of (1.1) with initial data u 0 is dened on I and satises

(2.5) ∥u -w∥ S s 0 (I) ⩽ C(A)ε, ∥u∥ S s 0 (I) ⩽ C(A).
Remark 2.6. The analog of Theorem 2.2 for the equation (1.7) where all the spaces are replaced by homogeneous spaces hold. Precisely, in the statement of Theorem 2.2, one can replace X(I) by

L p 0 (d+2) 2 (I × R d ), H s0 by Ḣs0 , S s0 (I) by Ṡs0 (I).
2.2. Prole decomposition. In this subsection we recall results on prole decomposition for the linear Schrödinger equation (2.6)

i∂ t u L + ∆u L = 0.
and the nonlinear Schrödinger equation (1.1). Prole decompositions for Schrödinger equations go back to [START_REF] Merle | Compactness at blow-up time for L 2 solutions of the critical nonlinear Schrödinger equation in 2[END_REF] (see also, among other works, [START_REF] Keraani | On the defect of compactness for the Strichartz estimates of the Schrödinger equations[END_REF], [START_REF] Bégout | Mass concentration phenomena for the L 2 -critical nonlinear Schrödinger equation[END_REF] and [START_REF] Shao | Maximizers for the Strichartz and the Sobolev-Strichartz inequalities for the Schrödinger equation[END_REF]). Here, we follow the theory developed in Section 3 of our previous article [START_REF] Duyckaerts | Prole decomposition and scattering for general nonlinear schrödinger equations[END_REF] which is adapted to nonlinearities of the form ±|u| p0 u+l.o.t.. We start by recalling the prole decomposition for sequences of solutions of (2.6) such that the corresponding of initial data is bounded in H s0 . We denote by u L (t) = e it∆ u 0 the solution to this equation, with initial data u L (0) = u 0 ∈ H s0 Denition 2.7. A linear H s0 -prole, in short prole, is a sequence (φ Ln ) n , of solutions of (2.6), of the form (2.7)

φ Ln (t, x) = 1 λ 2 p 0 n φ L t -t n λ 2 n , x -x n λ n ,
where φ L is a xed solution of (2.6) and

Λ n = (λ n , t n , x n ) n is a sequence in (0, 1] × R × R d (called sequence of transformations) such that (2.8) lim n→∞ -t n λ 2 n = τ ∈ R ∪ {±∞}.
Denition 2.8. We say that two sequence of transformations Λ n = (λ n , t n , x n ) and M n = (µ n , s n , y n ) are orthogonal when they satisfy

lim n→∞ |t n -s n | λ 2 n + |x n -y n | λ n + log λ n µ n = ∞.
We say that two H s0 -proles are orthogonal when one of the two proles is identically 0 or when the corresponding sequences of transformations are orthogonal.

Let

q 0 = 2d d-2s0 = d
2 p 0 be the Lebesgue exponent such that the Sobolev embedding Ḣs0 ⊂ L q0 holds. Then we have: Proposition 2.9. For any bounded sequence (u 0,n ) n in H s0 , there exists a subsequence (that we still denote by (u 0,n ) n ), a sequence φ j Ln n j⩾1 of H s0 proles such that (2.9)

lim J→∞ lim sup n→∞ w J Ln X(R) + |∇| s0 w J Ln W 0 (R) + ∥w j Ln ∥ L ∞ L q 0 = 0,
where (2.10)

w J Ln = u Ln - J j=1 φ j Ln ,
u L,n = e it∆ u 0,n . Furthermore, we have the Pythagorean expansions:

∀J ⩾ 1, ∥u 0,n ∥ 2 Ḣs 0 = J j=1 φ j L (0) 2 Ḣs 0 + w J Ln (0) 2 Ḣs 0 + o(1), n → ∞ (2.11) ∀J ⩾ 1, ∥u 0,n ∥ 2 Ḣs = j∈J N C 1⩽j⩽J φ j L (0) 2 Ḣs + w J Ln (0) 2 Ḣs + o(1), n → ∞; 0 ⩽ s < 1.
(2.12)

We say that the sequence (φ j Ln ) n j⩾1 is a prole decomposition of (u Ln ) n See Denition 3.4, Proposition 3.5, Lemma 3.7 in [START_REF] Duyckaerts | Prole decomposition and scattering for general nonlinear schrödinger equations[END_REF], and also Remark 3.10 there for the Pythagorean expansion (2.12). The norm X(R) does not appear in the analog of (2.9) in [START_REF] Duyckaerts | Prole decomposition and scattering for general nonlinear schrödinger equations[END_REF] (see (3.5) there). The fact that this norm also goes to 0 follows from the boundedness of the sequence (u 0,n ) n is bounded in H s0 and Strichartz estimates.

We next construct a nonlinear analog, adapted to the equation (1.1), of the preceding prole decomposition. With the notations of Proposition 2.9, extracting subsequences if necessary, we see that for all j, one of the following holds:

• φ j L ≡ 0.
In this case we say that (φ j Ln ) n is a null prole, and denote j ∈ J 0 .

• φ j L ̸ ≡ 0 and lim n→∞ λ j n = λ j ∈ (0, 1]. In this case we say that (φ j Ln ) n is a nonconcentrating prole, and denote j ∈ J N C . We have φ j 0 = φ j L (0) ∈ H s0 . Rescaling, we see that we can assume λ j = 1. As a consequence, we can assume λ j n = 1 for all n.

• φ j L ̸ ≡ 0 and lim n→∞ λ j n = 0. In this case we say that (φ j Ln ) n is a concentrating prole, and denote j ∈ J C . Remark 2.10. If the sequence (u 0,n ) n is bounded in H s for some s > s 0 , then J C = ∅.

Remark 2.11. Let 2 < q < q 0 = 2d d-2s0 , and

(2.13)

τ j = lim n→∞ -t j n (λ j n ) 2 .
Extracting subsequences and translating the proles in time, we can assume τ j ∈ {0, ±∞}. Then if τ j ∈ {±∞} or j ∈ J C

(2.14)

lim n→∞ ∥φ j Ln (0)∥ L q = 0.
Furthermore, we have the following expansion:

lim n→∞ ∥u 0,n ∥ q L q = j∈J N C τ j =0 ∥φ j L (0)∥ q L q .
To each (linear) prole φ j Ln , we associate a nonlinear prole φ j n and a modied nonlinear prole φj n in the following way:

• If j ∈ J 0 , φj n and φ j n are both equal to the constant null function. • If j ∈ J N C , the modied nonlinear prole and the nonlinear prole are equal, and dened by

φj n (t, x) = φ j n (t, x) = φ j t -t j n , x -x j n ,
where φ j is the unique solution of (1.1) such that (2.15)

lim t→τ j φ j (t) -φ j L (t) H s 0 = 0, τ j = lim n→∞ -t j n .
• If j ∈ J C , the nonlinear prole φ j n is dened by (2.16)

φ j n (t, x) = 1 (λ j n ) 2 p 0 φ j t -t j n (λ j n ) 2 , x -x j n λ j n
where φ j is the unique solution of the homogeneous equation (1.7) such that

lim t→τ j φ j (t) -φ j L (t) Ḣs 0 = 0, τ j = lim n→∞ -t j,n (λ j n ) 2 .
To dene the modied prole, we x σ j in the maximal interval of existence of φ j (if τ j is nite, we can take σ j = τ j , if τ j = ±∞, |σ j | large and with the same sign than τ j ). We let

s j n = (λ j n ) 2 σ j + t j n
and denote by φj n the solution of (1.7) such that (2.17)

φj n (s j n ) = χ x -x j n φ j n (s j n ) = χ x -x j n 1 (λ j n ) 2 p 0 φ j σ j , x -x j n λ j n , where χ ∈ C ∞ 0 (R N ) is radially symmetric, χ(x) = 1 for |x| < 1, χ(x) = 0 for |x| > 2.
When j ∈ J C , the modied proles φj n are close to the proles φ j n in Ḣs0 , Lemma 2.12. Let j ∈ J C . Then (2.18)

lim n→∞ φj n (0) -φ j n (0) Ḣs 0 = 0.
These modied proles are also approximate solutions of (1.1) (see Lemma 3.13 in [START_REF] Duyckaerts | Prole decomposition and scattering for general nonlinear schrödinger equations[END_REF]). We next give an approximation result. We let wJ L,n be the solution of the linear wave equation with initial data

(2.19) wJ L,n (0) = u 0,n - J j=1 φj n (0).
One can prove that for all s with 0 < s ⩽ s 0 ,

lim J→∞ lim sup n→∞ wJ L,n X(R) + wJ L,n Ẇ s (R) = 0.
Then we have: Theorem 2.13 (Approximation by proles). Let (u 0,n ) n be a sequence bounded in H s0 that admits a prole decomposition (φ j Ln ) n j . Let φ j n , φj n and wJ L,n be as above. Let I n be a sequence of intervals such that 0 ∈ I n , and assume that for each j ⩾ 1, for large n,

0 ∈ I n ⊂ I max (φ j n ), j ∈ J C =⇒ lim sup n→∞ |∇| s0 φ j n S 0 (In) < ∞, (2.20)
j ∈ J N C =⇒ lim sup n→∞ φ j n S s 0 (In) < ∞.
(2.21)

Let u n be the solution of (1.1) with initial data u 0,n . Then for large n, I n ⊂ I max (u n ), We will also need the fact that Pythagorean expansions of the Sobolev norms hold in the setting of the preceding Theorem: Lemma 2.14. With the same assumptions and notations as in Theorem 2.13, if (t n ) n is a sequence of time with t n ∈ I n for all time, then for all J ⩾ 1,

u n (t) = 1⩽j⩽J φj n (t, x) + wJ L,n (t) + r J n (t), t ∈ I n ,
∥u n (t n )∥ 2 Ḣs 0 = J j=1 ∥ φj n (t n )∥ 2 Ḣs 0 + ∥ wJ n (t n )∥ 2 Ḣs 0 + o J,n (1)
,

where lim J→∞ lim sup n→∞ o J,n (1) = 0. Furthermore, for all s with 0 ⩽ s < s 0 ,

∥u n (t n )∥ 2 Ḣs = 1⩽j⩽J j∈J N C ∥ φj n (t n )∥ 2 Ḣs + ∥ wJ n (t n )∥ 2 Ḣs + o J,n (1)
][Theorem 1.2]) Let d ⩾ 2, p 0 > 4 d , s 0 = d 2 -2
p0 . Suppose that assumption A holds. Assume Property 1.3. Let u be a solution of (1.1), with initial data in H s0 , such that (1.8) holds. Then T + (u) = ∞.

Variational properties

In this section, we x d ⩾ 2 and consider a nonlinearity g of the form

g(u) = |u| p0 u -|u| p1 u, with 4 d < p 1 < p 0 , such that d 2 -2 p0 = s 0 ⩾ 1.
We also assume that p 0 and p 1 are even integers, or that ⌈s 0 ⌉ ⩽ p 1 .

We recall the denition of the energy and the virial functional:

E(u) = R d |∇u| 2 - 2 p 1 + 2 |u| p1+2 + 2 p 0 + 2 |u| p0+2 dx, Φ(u) = R d |∇u| 2 - dp 1 2(p 1 + 2) |u| p1+2 + dp 0 2(p 0 + 2) |u| p0+2 dx,
In this subsection, we prove positivity properties of E and Φ in some subsets of H s0 , see Theorems 3.2, Corollary 3.5 and Theorem 3.3 below. These properties are obtained essentially as consequences of [START_REF] Lewin | The double-power nonlinear Schrödinger equation and its generalizations: uniqueness, non-degeneracy and applications[END_REF].

3.1. Preliminaries. As in [START_REF] Lewin | The double-power nonlinear Schrödinger equation and its generalizations: uniqueness, non-degeneracy and applications[END_REF], we have the following Gagliardo-Nirenberg inequality (3.1)

∥u∥ p1+2 p1+2 ⩽ p1,p0,d ∥u∥ p1-θp0 L 2 ∥∇u∥ 2(1-θ) L 2 ∥u∥ θ(p0+2) p0+2
, where

θ = p 1 -4 d p 0 -4 d ∈ [0, 1).
We note that this implies the following:

Claim 3.1. Let u ∈ H 1 (R d ) ∩ L p0+2 (R d ) be such that M (u) = m > 0 and Φ(u) = 0 Then, ∥∇u∥
Proof. Using Φ(u) = 0 and Gagliardo-Nirenberg inequality (3.1), we have

dp 0 2(p 0 + 2) ∥u∥ p0+2 p0+2 + ∥∇u∥ 2 2 = dp 1 2(p 1 + 2) ∥u∥ p1+2 p1+2 ≲ d,p1,p0,m ∥∇u∥ 2(1-θ) 2 ∥u∥ θ(p0+2) p0+2
.

Thus,

∥u∥ p0+2 p0+2 ≈ ∥u∥ p1+2 p1+2 ≈ ∥∇u∥ 2 2 . If p 1 < 4 d-2 then ∥u∥ p1+2 ≲ ∥u∥ 1- dp 1 2(p 1 +2) 2 ∥∇u∥ dp 1 2(p 1 +2) 2 .
Then,

∥∇u∥ 2 2 ≈ ∥u∥ p1+2 p1+2 ≲ ∥∇u∥ dp 1 2 2 . It implies that ∥∇u∥ 2 ≳ d,p1,p0,m 1 since p 1 > 4/d. If p 1 = 4 d-2 then ∥∇u∥ 2 p 1 +2 2 ≲ ∥u∥ p1+2 ≲ ∥∇u∥ 2 . It implies that ∥∇u∥ 2 ≳ d,p1,p0,m 1. If p 1 > 4 d-2 then ∥u∥ p1+2 ≲ ∥u∥ θ 2d d-2 ∥u∥ 1-θ p0+2 ≲ ∥∇u∥ θ 2 ∥u∥ 1-θ p0+2 ,
where θ ∈ (0, 1) satises

1 p 1 + 2 = θ 2d/(d -2) + 1 -θ p 0 + 2 .
Thus,

∥∇u∥ 2 p 1 +2 2 ≲ ∥∇u∥ θ 2 ∥∇u∥ 2(1-θ) p 0 +2 2 .
We see that 2

p1+2 < θ + 2(1-θ)
p0+2 . Thus, ∥∇u∥ 2 ≳ d,p1,p0,m 1. This completes the proof.

□ Dene I(m) = inf u ∈ H 1 (R d ) ∩ L p0+2 (R d ) R d |u| 2 dx = m E(u).
The following Theorem is essentially contained in [START_REF] Lewin | The double-power nonlinear Schrödinger equation and its generalizations: uniqueness, non-degeneracy and applications[END_REF].

-∆u = -u p0+1 + u p1+1 -ωu,
for some ω ∈ (0, ω * ), hence must be equal to Q ω , where ω * , Q ω are dened in the introduction. The minimizer in the case m = m c is an optimizer for the Gagliardo-Nirenberg inequality (3.1).

The inmum is not attained for m < m c . For 0 < m < m c , there exists δ(m) > 0 such that for any

u ∈ H 1 (R d ) ∩ L p0+2 (R d ) such that R d |u| 2 dx = m < m c , one has E(u) ⩾ δ(m)∥∇u∥ 2 2 .
For each minimizer u of the problem I(m) satises the Pohozaev identity: Φ(u) = 0.

Proof. This is [START_REF] Lewin | The double-power nonlinear Schrödinger equation and its generalizations: uniqueness, non-degeneracy and applications[END_REF]Theorem 6], except for the claim that there exists δ(m) such that E(u) ⩾ δ(m)∥∇u∥ 2 2 if |u| 2 dx = m < m c . To prove this point, we let v(x) = u(βx), where

β d = m mc < 1. Then |v| 2 = m c and E(v) = mc m ((β 2 -1) ∥|∇u|∥ 2 2 + E(u)). Since E(v) ⩾ 0, we obtain E(u) ⩾ (1 -β 2 ) ∥∇u∥ 2 2 .
Let u be a minimizer of I(m). Then u solves the following Euler-Lagrange equation:

-∆u = -|u| p0 u + |u| p1 u -λu,
for some λ ∈ R. By Claim A.1, we have Φ(u) = 0, which completes the proof.

□

Recall from (1.5) the denition of mc . In the remainder of this subsection, we will prove: Theorem 3.3. There exists a nonincreasing function e : (0, 3.2. Positivity of virial functional below critical mass. Dene

m c ) → [0, ∞] such that e(m) = ∞ if 0 < m < mc ,

and, for all

u ∈ H 1 (R d ) ∩ L p0+2 (R d ) such that M (u) = m ∈ (0, m c ), 0 < E(u) < e(m), one has Φ(u) -|P (u)| 2 /M (u) > 0.
Ĩ(m) = inf u ∈ H 1 (R d ) ∩ L p0+2 (R d ) R d |u| 2 dx = m Φ(u). Let u ∈ H 1 (R d ) ∩ L p0+2 (R d ) and a, b > 0. Dene u a,b (x) = au(bx).
We have

R d |u a,b (x)| 2 dx = a 2 b -d R d |u| 2 dx, E(u a,b ) = a 2 b 2-d E a,b (u),
where

E a,b (u) := R d |∇u| 2 dx - 2 p 1 + 2 a p1 b -2 R d |u| p1+2 dx + 2 p 0 + 2 a p0 b -2 R d |u| p0+2 dx.
We have

I(m) = inf u ∈ H 1 (R d ) ∩ L p0+2 (R d ) R d |u a,b (x)| 2 dx = m E(u a,b ) = a 2 b 2-d inf u ∈ H 1 (R d ) ∩ L p0+2 (R d ) R d |u| 2 dx = ma -2 b d E a,b (u), for all a, b > 0. We chose a = a 0 , b = b 0 such that 2 p 1 + 2 a p1 0 b -2 0 = dp 1 2(p 1 + 2) , 2 p 0 + 2 a p0 0 b -2 0 = dp 0 2(p 0 + 2) ,
that is

a 0 = p 0 p 1 1 p 0 -p 1 , b 2 0 = 4 dp 1 p 0 p 1 p 1 p 0 -p 1 .
By the denition of a 0 , b 0 , E a0,b0 ≡ Φ. By the above, we have

I a 2 0 b -d 0 m = a 2 0 b 2-d 0 inf u ∈ H 1 (R d ) ∩ L p0+2 (R d ) R d |u| 2 dx = m Φ(u) = a 2 0 b 2-d 0 Ĩ(m). Dene mc = (a 2 0 b -d 0 ) -1 m c . We have mc < m c ⇔ a 2 0 > b d 0 ⇔ p 0 p 1 2 p 0 -p 1 > 4 dp 1 d 2 p 0 p 1 dp 1 2(p 0 -p 1 ) ⇔ p 0 p 1 p 1 p 0 -p 1 < dp 1 4 dp 1 dp 1 -4 (3.3) mc < m c ⇐⇒ f 1 p 0 -p 1 p 1 < f 2 dp 1 4 ,
where f 1 (x) = (1 + x)

1
x and f 2 (x) = x x x-1 . We see that f 1 is strictly decreasing on (0, ∞) and f 2 is strictly increasing on (1, ∞). Thus,

f 1 p 0 -p 1 p 1 < lim x→0 f 1 (x) = e = lim x→1 f 2 (x) < f 2 dp 1 4
Then, (3.3) holds when p 0 > p 1 > 4/d, and mc < m c . From Theorem 3.2, we have the following result. The problem Ĩ(m) admits at least one positive radial-decreasing minimizer ũ for every m ⩾ mc . For 0 < m < mc , there exists δ(m) such that for any u ∈ L p0+2 ∩ H 1 with |u| 2 = m,

Φ(u) ⩾ δ(m)∥∇u∥ 2 L 2 .
3.3. Positivity of virial functional below a critical energy, at xed mass. As before, we assume 4 d < p 1 < p 0 and thus mc < m c . Dene

I Φ (m) = inf u ∈ H 1 (R d ) ∩ L p0+2 (R d ) M (u) = m, Φ(u) = 0 E(u).

If there is no function

u ∈ H 1 (R d ) ∩ L p0+2 (R d ) such that M (u) = m and Φ(u) = 0 then we let I Φ = ∞. Dene R = {(m, e) : 0 < m < m c , 0 < e < I Φ (m)}.
As in [17, Theorem 5.2], we have the following result. Theorem 3.6.

If (M (u), E(u)) ∈ R for some u ∈ H 1 (R d ) ∩ L p0+2 (R d ) then Φ(u) > 0. • I Φ (m) = ∞ when 0 < m < mc . • 0 < I Φ (m) < ∞ when mc ⩽ m < m c . • I Φ (m) = I(m) when m ⩾ m c . Moreover, I Φ (m) is nonincreasing as a function of m for m ⩾ mc .
If we further assume that d = 2, 3, 4 or p 0 ⩽ 4 d-2 , then the inmum I Φ (m) achieved and I Φ (m) is strictly decreasing for m ⩾ mc .

Before proving Theorem 3.6, we prove that it is equivalent to consider the minimum of E on the set {Φ(u) = 0} and on the larger set {Φ(u) ⩽ 0}. Lemma 3.7. Suppose u ∈ H 1 (R d ) ∩ L p0+2 (R d ) is not identically zero and that Φ(u) < 0 Then there exists λ > 1 so that u λ (x) = λ d 2 u(λx) obeys Φ(u λ ) = 0 and E(u λ ) < E(u). Note that M (u λ ) = M (u) and

R d |∇u λ | 2 dx = λ 2 R d |∇u| 2 dx > R d |∇u| 2 dx.
Proof. By direct computation, we have

(3.4) Φ(u λ ) = λ 2 dE(u λ ) dλ = λ 2 R d |∇u| 2 dx - dp 1 λ dp 1 2 2(p 1 + 2) R d |u| p1+2 dx + dp 0 λ dp 0 2 2(p 0 + 2) R d |u| p0+2 dx.
Suppose Φ(u) < 0. By (3.4), lim λ→∞ Φ(u λ ) = ∞. Thus, there exists λ > 1 such that Φ(u λ ) = 0. Let λ 0 be smallest such λ. It implies that from (3.4), E(u λ ) is decreasing on [1, λ 0 ). Thus,

E(u λ0 ) < E(u). □ Lemma 3.8. ([17, Lemma 5.4]) Assume d = 1, 2 or d ⩾ 3 and p 0 ⩽ 4 d-2 . Let m > 0 and u ∈ H 1 (R d ) be such that 0 < M (u) < m, Φ(u) = 0. Then there exists v ∈ H 1 (R d ) satisfying: M (v) = m, E(v) < E(u) and Φ(v) = 0.
Proof. The proof is similar as in [START_REF] Killip | Solitons and scattering for the cubic-quintic nonlinear Schrödinger equation on R 3[END_REF][ Lemma 5.4]. Remark that H 1 → L p0+2 since p 0 ⩽ 4 d-2 . □ We divide the proof of Theorem 3.6 into three steps.

u ∈ H 1 (R d )∩L p0+2 (R d ) such that (M (u), E(u)) ∈ R. If Φ(u) = 0 then by the denition of the function I Φ (λ), I Φ (M (u)) < E(u).
If Φ(u) < 0 then using Lemma 3.7, there exists u λ such that Φ(u λ ) = 0 and E(u λ ) < E(u). Thus,

I Φ (M (u)) = I Φ (M (u λ )) ⩽ E(u λ ) < E(u).
In all cases, this gives a contradiction with the denition of R. Thus, Φ(u) > 0.

If M (u) = m < mc then by Theorem 3.2, Φ(u) > 0.

Thus I Φ (m) = ∞.
If mc ⩽ m < m c . By Claim 3.1 and Theorem 3.2,

I Φ (m) > 0. Let u ∈ H 1 (R d ) ∩ L p0+2 (R d ) be such that M (u) = m. By Corollary 3.5, there exists u 0 ∈ H 1 (R d ) ∩ L p0+2 (R d ) such that M (u 0 ) = m and Φ(u 0 ) = Ĩ(m) < 0. By Lemma 3.7, there exists λ 0 > 1 such that Φ(u λ0 0 ) = 0, E(u λ0 0 ) < E(u 0 ), M (u λ0 0 ) = m.
Combining all the above, we have

0 < I Φ (m) ⩽ E(u λ0 0 ) < E(u 0 ) < ∞.
If m ⩾ m c then by Theorem 3.2, there exists

u 0 ∈ H 1 (R d ) ∩ L p0+2 (R d ) such that M (u 0 ) = m, E(u 0 ) = I(m) ⩽ 0.
Since u 0 is a minimizer of problem I(m), from Theorem 3.2, we have Φ(u 0 ) = 0. Thus, I(m) = I Φ (m) and I Φ (m) is achieved. Moreover, using Theorem 3.2, I Φ (m) is negative, strictly decreasing on (m c , ∞).

Monotonicity. We next prove that

m → I ϕ (m) is nonincreasing on [ mc , m c ]. Let m ∈ [ mc , m c ]
and m ′ > m such that m ′ ⩽ m c . We will prove that for all ε > 0, there exists

U ε ∈ H 1 ∩ L p0+2 such that M (U ε ) = m ′ , Φ(U ε ) ⩽ 0 and E(U ε ) ⩽ I ϕ (m) + ε.
Using Lemma 3.7, we see that it would impliy I ϕ (m ′ ) ⩽ I ϕ (m) + ε, and thus, letting ε goes to 0, I ϕ (m ′ ) ⩽ I ϕ (m), which yields that I ϕ is nonincreasing.

To prove the existence of U ε satisfying the desired property, we rst consider a function v ∈ H 1 ∩ L p0+2 , given by the denition of I ϕ , such that

(3.5) M (v) = m, Φ(v) = 0 and E(v) ⩽ I ϕ (m) + ε 8 .
We let v b (x) = v(bx), for some b < 1. Then

(3.6) |v b | 2 = b -d |v| 2 = m + o(1) as b → 1. Furthermore Φ(v b ) = b -d b 2 R d |∇v| 2 dx -dp1 2(p1+2) R d |v| p1+2 dx + dp0 2(p0+2) R d |v| p0+2 dx which implies, using that Φ(v) = 0, (3.7) Φ(v b ) = b -d (b 2 -1) |∇v| 2 < 0, when b < 1.
We also have

(3.8) E(v b ) = E(v) + o(1) as b → 1.
By (3.5), (3.6) and (3.8) one can choose b < 1 such that (3.9)

M (v b ) ⩽ m ′ , Φ(v b ) < 0, E(v b ) ⩽ I ϕ (m) + ε 4 . If M (v b ) = m ′ , we can take U ε = v b and we are done. If M (v b ) < m ′ , we x ψ ∈ C ∞ 0 (R d ) such that M (ψ) = m ′ -M (v b
), and we let

ũλ (x) = v b (x) + λ d/2 ψ(λx), u λ = √ m ′ ∥ũ λ ∥ 2 ũλ .
Then

lim λ→∞ ∥ũ λ ∥ 2 2 = M (v b ) + M (ψ) = m ′ (3.10) ∥u λ ∥ 2 2 = m ′ . (3.11) Furthermore, since 4 d < p 1 < p 0 , j=0,1 lim λ→∞ ∥ũ λ -v b ∥ L p j + ∥∇(ũ λ -v b )∥ L 2 = 0,
and thus, by (3.10) and the denition of u λ , j=0,1

lim λ→∞ ∥u λ -v b ∥ L p j + ∥∇(u λ -v b )∥ L 2 = 0,
which implies, using also (3.9), (3.12)

lim λ→∞ Φ(u λ ) = Φ(v b ) < 0, lim λ→∞ E(u λ ) = E(v b ) ⩽ I Φ (m) + ε 2 .
By (3.12), we can choose λ large such that

Φ(u λ ) ⩽ 0, E(u λ ) ⩽ I ϕ (m) + ε.
Combining with (3.11) we see that U ε = u λ satises the desired properties. . We start by proving that I ϕ (m) is achieved. Note that it is already known in the case m = m c . We thus assume m ⩽ m < m c . We consider a minimizing sequence u n for I ϕ (m):

Φ(u n ) = 0, E(u n ) ⩽ I ϕ (m) + 1 2 n , |u n | 2 = m.
Let v n be the symmetric decreasing rearrangement of u n . Then (3.13)

Φ(v n ) ⩽ 0, E(v n ) ⩽ E(u n ) ⩽ I ϕ (m) + 1 2 n , |v n | 2 = m. Let a = max s⩾0 p 1 d 2(p 1 + 2) s p1 - p 0 d 2(p 0 + 2) s p0 ∈ (0, ∞).
Then

- p 1 d 2(p 1 + 2) |v n | p1+2 + p 0 d 2(p 0 + 2) |v n | p0+2 + aM (v n ) ⩾ 0,
and

Φ(v n ) ⩽ 0, |v n | 2 = m imply |∇v n | 2 ⩽ am. Thus v n is bounded in Ḣ1 . Since Φ(v n ) ⩽ 0,
using Gagliardo-Nirenberg inequality and the boundedness in L 2 , we deduce that v n is bounded in

H 1 ∩ L p0+2 . Let v ∈ H 1 ∩ L p0+2 such that (after extraction) v n n→∞ ----⇀ v weakly in H 1 ∩ L p0+2 .
Since v n is radial, using Strauss Lemma, we obtain that the convergence is strong in

L q , 2 < q < 2d d-2 (for all q > 2 is d = 2). Also, since (v n ) n is bounded in L p0+2
, the convergence is strong in L q for all 2 < q < p 0 + 2. In particular, it is strong in L p1+2 . We thus obtain:

Φ(v) ⩽ lim inf n Φ(v n ) ⩽ 0 E(v) ⩽ lim inf n E(v n ) ⩽ I ϕ (m) M (v) ⩽ lim inf n M (v n ) ⩽ m.
By Lemma 3.7 and the fact that Φ(v) is nonpositive, we obtain E(v) ⩾ I ϕ (M (v)). Since by 3.3.2, I ϕ is nonincreasing, we obtain

I ϕ (m) ⩽ I ϕ (M (v)) ⩽ E(v) ⩽ I ϕ (m),
which proves that all these quantities must be equal. Thus

lim n E(v n ) = E(v)
, which shows that v n converges strongly to v in Ḣ1 and in L p0+2 . Thus we have found

v ∈ L p0+2 ∩ H 1 such that Φ(v) ⩽ 0 and (3.14)
E(v) = I ϕ (m) = I ϕ (M (v)), M (v) ⩽ m.
Note that since m < m c , we have E(v) > 0 and thus v is not identically 0. By Lemma 3.7, we see that we cannot have Φ(v) < 0. Thus (3.15)

Φ(v) = 0
If M (v) = m we are done. To conclude the proof that I ϕ (m) is attained, we will assume M (v) < m and obtain a contradiction. First assume d ∈ {2, 3, 4}. Denoting m ′ = M (v) , we see by the above that I ϕ (m) = I ϕ (m ′ ) and since I ϕ is nonincreasing, we deduce

I ϕ (µ) = I ϕ (m), m ⩽ µ ⩽ m ′ .
By the denition of I ϕ we obtain (3.16)

E(v) = min 0<M (u)⩽m ′ Φ(u)=0 E(u).
Since M (v) = m < m ′ , we see that v is a minimum of E(u) with the sole constraint Φ(u) = 0.

Thus there exists a Lagrange multiplier α such that (3.17)

-∆v -v p1+1 + v p0+1 = α -∆v - dp 1 4 v p1+1 + dp 0 4 v p0+1 .
If α = 1, we obtain

dp 1 4 -1 v p1+1 = dp 0 4 -1 v p0+1 , which shows, since v ∈ H 1 , that v = 0, a contradiction. If α ̸ = 1, we see that v is a H 1 solution of an equation of the form -∆v + a 1 v p1+1 + a 0 v p0+1 = 0,
where (a 0 , a 1 ) ∈ R 2 . Since we have assumed d ∈ {2, 3, 4}, this proves again by Proposition A.2 that v = 0, a contradiction which concludes the proof if d ∈ {2, 3, 4}.

We next assume that d ⩾ 5 and p 0 ⩽ 4 d-2 . By Lemma 3.8, there exists ṽ ̸ = 0 such that M (ṽ) = m, Φ(ṽ) = 0 and E(ṽ) < E(v) = I Φ (m), this gives a contradiction to the denition of I Φ (m). Thus, the achievement of I Φ (m) is proved.

Now, we prove

I Φ (m) is strictly decreasing on [ mc , m c ]. Let mc ⩽ m 1 < m 2 ⩽ m c . Since I Φ (m 1 ) is achieved, let u ∈ H 1 (R d ) ∩ L p0+2 (R d ) be such that M (u) = m 1 , Φ(u) = 0, E(u) = I Φ (m 1 ). By Lemma 3.8, there exists ũ such that M (ũ) = m 2 , Φ(ũ) = 0, E(ũ) < E(u) = I Φ (m 1 )
. Thus, by the denition of I Φ (m 2 ), we have I Φ (m 1 ) > I Φ (m 2 ), which proves that I Φ is strictly decreasing on

[ mc , m c ].
It remains to prove that I Φ is lower semicontinuous on [ mc , ∞). Since, I Φ is nonincreasing, it suces to show that I Φ is right continuous. We argue by contradiction, assume that there exists m n decreasing to m with I Φ (m) > lim n→∞ I Φ (m n ). Let u n be radial functions such that M (u n ) = m n , Φ(u n ) = 0, E(u n ) = I Φ (m n ). We see that u n is bounded in H 1 (R d ). Thus, u n weakly converges to v in H 1 (R d ) and strongly converges to v in L p1+2 (R d ). This implies that

M (v) ⩽ m, Φ(v) ⩽ 0, E(v) ⩽ lim n→∞ E(u n ) < I Φ (m). If Φ(v) = 0 then I Φ (m) ⩽ I Φ (M (v)) ⩽ E(v) < I Φ (m)
, which gives a contradiction. If Φ(v) < 0 then by Lemma 3.7, there exists ṽ such that M (ṽ) = M (v) ⩽ m, Φ(ṽ) = 0, E(ṽ) < E(v) < I Φ (m). Similar as the above, we gives a contradiction. This completes the proof.

Proof of Theorem 3.3. For m ∈ (0, m c ), let e(m) = I Φ (m). Theorem 3.6 implies that e(m) is nonincreasing on (0, m c ). For each u ∈ H 1 (R d ) ∩ L p0+2 (R d ) such that M (u) = m ∈ (0, m c ) and 0 < E(u) < e(m), let ξ = P (u) M (u) and u ξ (x) = e ixξ u(x). We have M (u ξ ) = M (u) ∈ (0, m c ) and

0 < E(u ξ ) = E(u) -|P (u)| 2
M (u) ⩽ E(u) < e(m) ⩽ I Φ (m). Using Theorem 3.6, Φ(u ξ ) > 0. This implies that

Φ(u) - |P (u)| 2 M (u) > 0.
This proves the desired result. We consider the set

R := φ ∈ H s0 , 0 < |φ| 2 < m c , E(φ) < e |φ| 2 ,
where m c , e are dened in Section 3. We will prove Theorem 1.4 as a consequence of Theorem 4.1. Suppose that Assumption A holds and that s 0 > 1. Let A 0 such that Property 1.3 holds. For η > 0, we denote

R η = φ ∈ H s0 , 0 < |φ| 2 ⩽ m c -η, E(φ) ⩽ R(φ) -η ,
where R(φ) = e |φ| 2 . Then for all A ∈ (0, A 0 ) there exists F(A, η) > 0 such that for any interval I, for any solution u ∈ C 0 (I, H s0 ) of (1.1) such that Proof of Theorem 4.1. We argue by contradiction, following the compactness/rigidity scheme as in [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case[END_REF]. We x η > 0 throughout the argument. In all the proof, we will endow H s0 with the norm dened by (4.2)

∥u∥ 2 H s 0 = ∥u∥ 2 Ḣs 0 + η∥u∥ 2 2 .
We will denote by P(A) the property that there exists F(A) such that for any interval I, for any solution u ∈ C 0 (I, H s0 ) such that (4.1) holds, one has u ∈ S s0 and ∥u∥ S s 0 (I) ⩽ F(A).

By the small data theory for (1.1), if A > 0 is small and ∥u(t)∥ H s 0 ⩽ A for some t ∈ I max (u), then u is globally dened, scatters and ∥u∥ S s 0 (R) ≲ A. This implies that P(A) holds for small

A > 0.
Thus if the conclusion of Theorem 4.1 does not hold, there exists A c ∈ (0, A 0 ) such that for all A < A c , P(A) holds, and P(A c ) does not hold, i.e. there exists a sequence of intervals We will prove Claim 4.2. For any sequences (a n ) n , (b n ) n with a n < 0 < b n , for any sequence (u n ) n of solutions of (1.1) satisfying (4.3), (4.4), there exist, after extraction of subsequences, a sequence (x n ) n ∈ (R d ) N and φ ∈ H s0 such that (4.5)

((a n , b n )) n , a sequence (u n ) n of solutions of (1.1) on (a n , b n ),
lim n→∞ ∥u n (0, • -x n ) -φ∥ H s 0 = 0.
We rst assume the claim and conclude the proof of Theorem 4.1. By the claim, there exist (after extraction of subsequences) φ ∈ H s0 and (x n ) n such that (4.5) holds. Let u be the solution of (1.1) such that u(0

) = φ. Since R η is closed in H s0 , we have φ ∈ R η .
We next prove by contradiction By (4.4) and stability theory for equation (1.1), one has ∥u∥ S s 0 ((-∞,0)) = ∥u∥ S s 0 ((0,+∞)) = +∞.

If (t n ) n is any sequence of times, Claim 4.2 and the preceding properties imply that one can extract subsequence such that u(t n , • -x n ) converges in H s0 for some sequence (x n ) n ∈ (R d ) N . This is classical that it implies that one can nd a function x(t), t ∈ R such that K dened by (1.9) has compact closure in H s0 . We give a sketch of proof of this fact. Using the compactness and the fact that the solution u is not identically 0, we rst notice that there exists R 0 > 0, η > 0 such that

inf t∈R sup X∈R d |x|<R0 |u(t, x -X)| 2 dx ⩾ η.
Thus for all t, there exists

x(t) ∈ R d such that |x|<R0 |u(t, x -x(t))| 2 dx ⩾ η/2.
For this choice of x(t), one can check that K dened by (1.9) is compact.

By Proposition 1.5, we have (4.7)

min t⩾0 Φ(u(t)) - |P (u)| 2 M (u) = 0.
By (4.7), there exists a sequence of times (t n ) n such that (4.8)

lim tn→∞ Φ(u(t n )) - |P (u(t n ))| 2 M (u(t n )) = 0.
By the claim, extracting subsequences, there exists (x n ) n such that u(t n , • -x n ) convergences to φ 0 in H s0 (up to extract subsequence). By (4.8), (4.9)

Φ(φ 0 ) - |P (φ 0 )| 2 M (φ 0 ) = 0. Since R η is closed in H s0 , φ 0 ∈ R η . This implies, by Theorem 3.3, that Φ(φ 0 ) - |P (φ 0 )| 2 M (φ 0 ) > 0.
This contradicts to (4.9). This completes the proof.

□

We are left with proving Claim 4.2.

Proof of Claim 4.2.

Step 1. Prole decomposition. Extracting subsequences, we can assume that u 0,n = u n (0) has a prole decomposition as in Subsection 2.2:

u 0,n = J j=1 φ j Ln (0) + w J n (0), where φ j Ln = 1 λ j n φ j L t-t j n λ j n , x-x j n λ j n
. We denote by φ j n the corresponding nonlinear proles, and φj n the modied nonlinear proles.

Our goal is to prove that there is a unique j 0 ⩾ 1 such that φ j0 is not identically zero, and that j 0 ∈ J N C . We rst note that there is at least one j such that φ j is not identically zero. If not, by the small data local well-posedness, we would have lim n→∞ ∥u n ∥ S s 0 (In) = 0, a contradiction with our assumptions.

In the remaining step, we will prove that there is at most one nonzero prole. Arguing by contradiction, we assume that there is at least two nonzero proles, say φ 1 and φ 2 . By the small data theory, there exists ε 0 > 0 such that (4.10)

inf t∈In φj n H s 0 ⩾ ε 0 , j ∈ {1, 2}.
Step 2. Bound of the H s0 norm. We prove that for all j ⩾ 1 we have I n ⊂ I max ( φj n ) and, for large n, (4.11)

sup t∈In ∥ φj n (t)∥ H s 0 ⩽ A 2 c - 1 4 ε 2 0 .
By (4.3), and the Pythagorean expansions (2.11), (2.12), we obtain the bounds, for J ⩾ 1

j⩾1 ∥φ j L (0)∥ 2 Ḣs 0 ⩽ A 2 c , j∈J N C ∥φ j L (0)∥ 2 H s 0 ⩽ A 2 c
Fixing a small ε > 0, we obtain, by the small data theory for equations (1.1) and (1.7) and Lemma 2.12, that there exists J 0 ⩾ 1 such that, for j ⩾ J 0 + 1, I max (φ j ) = R, (4.12)

∀j ⩾ J 0 + 1, ∥φ j ∥ Ṡs 0 (R) < ∞ if j ∈ J C ∥φ j ∥ S s 0 (R) < ∞ if j ∈ J N C . and (4.13) ∀j ⩾ J 0 + 1, lim sup n→∞ sup t∈R ∥ φj n (t)∥ H s 0 ⩽ ε.
We next prove by contradiction that for j ∈ 1, J 0 , I n ⊂ I max ( φj n ) and (4.11) holds. If not, we can assume (inverting time if necessary, and using Theorem 2.16) that for large n, there exists

′ n ∈ (0, b n ] such that [0, b ′ n ] ⊂ 1⩽j⩽J0 I max ( φj n ) and (4.14) sup 1⩽j⩽J0 sup 0⩽t⩽b ′ n φj n (t) 2 H s 0 = A 2 c - 1 2 ε 2 0 .
This implies that for large n

∀j ∈ J C ∩ 1, J 0 , sup 0⩽t∈b ′ n ∥φ j n (t)∥ Ḣs 0 < A c < A 0 .
Thus by Proposition 2.15, we obtain a constant C > 0 such that for large n, Going back to (4.14), we see also that for large n ∀j ∈ 1, J 0 ∩ J N C , sup

0⩽t⩽b ′ n ∥φ j n (t)∥ H s 0 ⩽ A 2 c - 1 4 ε 2 0 .
Also, using the Pythagorean expansion of the mass we see that

∀j ∈ J N C , M (φ j ) ⩽ m c -η.
This implies that E(φ j ) ⩾ 0 for j ∈ J N C . By the Pythagorean expansion (2.12) of the Ḣ1 norm and Remark 2.11, we obtain where we used the fact that e is nonincreasing. Thus φ j n (0) ∈ R η . Using that P(A) holds for Step 3. Uniqueness of the nonzero prole.

A = A 2 c -
In this step we still assume that φ 1 and φ 2 are nonzero proles. Using (4.3) and (4.11), and arguing as in Step 2, we see that the assumptions of Theorem 2.13 are satised on [a n , b n ]. This proves that u n scatters for large n, contradicting (4.4). This concludes the proof that there is only one nonzero prole.

Step 4. End of the proof. We assume that φ 1 is the only nonzero prole. By the same argument as before, we obtain that for large n, I n ⊂ I max ( φ1 n ) and

lim n→∞ sup t∈In ∥ φ1 n ∥ H s 0 ⩽ A c .
If 1 ∈ J C , we obtain by Proposition 2.15 that lim sup n→∞ ∥φ 1 n ∥ Ṡs 0 (In) < ∞. Thus the assumptions of Theorem 2.13 are satised on I n , a contradiction with (4.4). Thus 1 ∈ J N C . By the same argument, we obtain lim sup n→∞ sup t∈In ∥ w1 Ln (t)∥ H s 0 = 0. Indeed, if not, we would have by the conservation of the H s0 norm for the linear Schrödinger equation (and after extraction of a subsequence) lim n→∞ sup t∈In ∥ w1

Ln (t)∥ H s 0 = ε 0 > 0, and the same strategy as in Steps 2,3 would yield that u n scatters for large n, a contradiction. We have proved u n (0, x) = φ 1 L (-t 1,n , x -x 1,n ) + o(1) in H s0 . By (4.4), t 1,n must be bounded, and we can assume t 1,n = 0 for all n, i.e. (4.17)

u n (0, x) = φ 1 (0, x -x 1,n ) + o (1) in H s0 , which concludes the proof of the claim. Theorem 4.3. Suppose that Assumption A holds and that 0 < s 0 ⩽ 1. For η > 0, we denote

R η = φ ∈ H 1 , 0 < |φ| 2 ⩽ m c -η, E(φ) ⩽ R(φ) -η .
Then for all A > 0 there exists F(A, η) > 0 such that for any interval I, for any solution u ∈ C 0 (I, H 1 ) of (1.1) such that Note that in this case, by conservation of the energy, any solution of (1.1) is bounded in H 1 , so that assumption (4.18) is always satised for some A.

The proof of Theorem 4.3 goes along the same lines as the proof of Theorem 4.1. Not however that if s 0 < 1, in the H s0 prole decomposition of u 0,n in the proof of Claim 4.2, the set J c is empty since the sequence (u 0,n ) is bounded in H 1 . Indeed, since |r d-1 ∂ r u| and u(r) go to 0 as r → ∞, (A.6) holds for a = d -2. Next, assuming that (A.6) holds for some a ⩾ d-2, one has, integrating (A.4) between r and ∞, |r d-1 ∂ r u| ≲ r d-(p1+1)a , and thus |u(r)| ≲ r 2-(p1+1)a . This shows that (A.6) holds with a replaced by a ′ = (p 1 + 1)a -2 ⩾ a + ap 1 -2. Since ap 1 -2 ⩾ p 1 -2 > 0, we deduce that (A.6) holds for all a > 0. Integrating twice the equation (A.4) as above we obtain that M (r) ≲ 1 r M (r) for large r, which shows again that u is identically 0. □

1 .

 1 Assume 4/d < p 1 < p 0 . There exists a nonincreasing function e(m): [0, m c ) → (0, ∞], with e(m) = ∞ for 0 < m < mc , and e(m) ∈ (0, ∞) for mc ⩽ m < m c , such that, letting(1.6)

  ∥u n ∥ S s 0 (In) < ∞ and (2.22) lim J→∞ lim sup n→∞ r J n S s 0 (In) = 0.

Theorem 3 . 2 .

 32 Let d ⩾ 2 and p 0 > p 1 > 4 d . The function m → I(m) is concave non-increasing over [0, ∞). There exists m c > 0 such that it satises I(m) = 0 for all 0 ⩽ m ⩽ m c , m → I(m) is negative and strictly decreasing on (m c , ∞). The problem I(m) admits at least one positive, radial, decreasing minimizer u for every m ⩾ m c . Any minimizer u solves the Euler-Lagrange equation (3.2)

Remark 3 . 4 .

 34 If d ∈ {2, 3, 4} or d ⩾ 5 and p 0 ⩽ 4 d-2 , the function e is strictly decreasing on [ mc , m c ]. The case d ⩾ 5, p 0 > 4 d-2 is open.

Corollary 3 . 5 .

 35 Let d ⩾ 2 and p 0 > p 1 > 4 d . The function m → Ĩ(m) is concave non-increasing over [0, ∞). It satises Ĩ(m) = 0 for all 0 ⩽ m ⩽ mc , m → Ĩ(m)is negative and strictly decreasing on ( mc , ∞).

3. 3 . 1 .

 31 Finiteness. We argue as in [17, Proof of Theorem 5.2]. Assume

3. 3 . 3 .

 33 Strict monotonicity. Now, we prove the last statement in Theorem 3.6. From Theorem 3.2 and I Φ (m) = I(m) for m ⩾ m c , we now that I Φ is strictly decreasing and achieved on [m c , ∞). Now, consider I Φ on [ mc , m c ]. Assume that d = 2, 3, 4 or d ⩾ 5 and p 0 ⩽ 4 d-2

□ 4 . 4 . 4 . 1 .

 4441 ScatteringIn this subsection, we conclude the proof of Theorem 1.Proof of Theorem 1.4 in the energy-supercritical case. We prove here Theorem 1.4 in the case where s 0 > 1.

(4. 1 ) 2 ⩽ A 2 ,

 122 ∃t ∈ I, u(t) ∈ R η and sup t∈I ∥u(t)∥ 2 Ḣs 0 + η∥u(t)∥ 2one has u ∈ S s0 (I) and ∥u∥ S s 0 (I) ⩽ F(A, η).

Theorem 4 . 1 implies

 41 Theorem 1.4 by the scattering criterion of Theorem 2.2, the conservation of the L 2 norm and the fact that R = ∪ η>0 R η .

( 4 . 3 )

 43 u n ∈ C 0 ((a n , b n ), H s0 ), ∃t ∈ I n , u n (t) ∈ R η , lim n→∞ sup an<t<bn ∥u n (t)∥ H s 0 = A c .and lim n→∞ ∥u n ∥ S s 0 ((an,bn)) = ∞. Time translating u n , we can assume (4.4) a n < 0 < b n , lim n→∞ ∥u n ∥ S s 0 ((an,0)) = lim n→∞ ∥u n ∥ S s 0 ((0,bn)) = +∞.

  Assume to x ideas, and after extraction of subsequences limn→∞ b n = b ∈ [0, ∞). Using that lim n ∥u n ∥ S s 0 ([0,bn)) = ∞, we must have T + (u) < ∞ and b ⩾ T + (u).

  b

(4. 15 )

 15 sup j∈J C ∩ 1,J0 ∥φ j n ∥ Ṡs 0 ([0,b ′ n ]) ⩽ C.

  j ) ⩽ lim inf n E(u 0,n ) ⩽ I ϕ (m c ) -η and thus ∀j ∈ J N C , E(φ j ) ⩽ R(φ j ) -η

□ 4 . 2 .

 42 Sketch of proof of Theorem 1.2. The proof of Theorem 1.2 in the case s 0 ⩽ 1 is essentially the same Theorem 1.4, although a little simpler than the proof in the case s 0 > 1. One can prove the following analog of Theorem 4.1:

(4. 18 ) 2 ⩽ A 2 ,

 1822 ∃t ∈ I, u(t) ∈ R η and sup t∈I ∥u(t)∥ 2 Ḣ1 + η∥u(t)∥ 2one has u ∈ S 1 (I) and ∥u∥ S 1 (I) ⩽ F(A, η).

  , 2.3. A uniform bound on the scattering norm. We recall here [9, Proposition 4.1], which gives a quantitative version of Property 1.3. The proof uses prole decomposition: Proposition 2.15. Let d ⩾ 2, p 0 > 4 d , s 0 = d 2 -2 p0 . Assume Property 1.3. Then for all A ∈ (0, A 0 ), there exists F(A) > 0 such that for all interval 0 ∈ I, for all solution u ∈ C 0 (I, Ḣs0 ) of (1.7) such that (2.23) Ṡs 0 (I) ⩽ F(A).2.4. Global well posedness. Using Proposition 2.15, we obtain that if Property 1.3 holds then solutions satisfying (1.8) is global forward in time. More precisely, we have the following result: Theorem 2.16. (see [9

	sup	∥u(t)∥ Ḣs 0 ⩽ A
	t∈I
	we have
	(2.24)

∥u∥

  This implies by Theorem 2.16 that u is global and sup

		By the last assertion of (4.3),
	we obtain
	sup	∥u(t)∥ H s 0 ⩽ A c .
	0⩽t<T+(u)

This implies by Theorem 2.16 that T + (u) = +∞, a contradiction. Hence

(4.6)

. Next, we see that (4.5), perturbation theory for equation (1.1) and the last assertion in (4.3) implies that for any compact interval

I ⊂ I max (u), sup t∈I ∥u(t)∥ H s 0 ⩽ A c . t∈R ∥u(t)∥ H s 0 ⩽ A c .

  1 4 ε 2 0 we obtain that there exists a constant C > 0 such that for large n (4.16) Combining (4.12), (4.15) and (4.16), we obtain that the assumptions of Theorem 2.13 are satised on [0, b ′ n ]. Using the Pythagorean expansion of Lemma 2.14 together with the limit in (4.3), we obtain contradicting(4.14). This proves that (4.11) holds for all j ⩾ 1, for large n.

			J0
	lim sup n→∞	sup 0⩽t⩽b ′ n	j=1	φj n (t)	2 H s 0 ⩽ A 2 c .
	By (4.10), we deduce
	∀j ∈ 1, J 0 , lim sup n→∞	sup 0⩽t⩽b ′ n	φj n (t)	2 H s 0 ⩽ A 2 c -ε 2 0 ,

sup j∈J N C ∩ 1,J0 ∥φ j n ∥ S s 0 ([0,b ′ n ]) ⩽ C.

⌈s⌉ denotes the smallest integer number larger or equal s.

≳ d,p1,p0,m 1.

Appendix A. Elliptic properties

Claim A.1 (Pohozaev identity). Let u ∈ H 1 ∩ L p0+2 (R d ), 0 < p 1 < p 0 . Assume that u is radial or u ∈ L ∞ loc (R d ) and, for some real numbers α 1 , α 2 , µ (A.1)

Sketch of proof. Multiply by x • ∇u + d 2 u, take the real part and integrate by parts on R d . It is easy to check that the formal integration by parts are rigorous when u ∈ L p0+2 ∩ H 1 with u radial or L ∞ loc . See [START_REF] Pohozaev | On the eigenfunctions of the equation ∆u + λf (u) = 0[END_REF] and [2, Proposition 1].

□

We next prove a uniqueness result for the elliptic equation with double power nonlinearity:

Then u ≡ 0.

Proof. We have (A.4)

Since |u| 2 r d-1 dr < ∞, the right-hand side of the preceding inequality is integrable, which proves that the following limit exists: [START_REF] Cazenave | The Cauchy problem for the critical nonlinear Schrödinger equation in H s[END_REF], contradicting our assumptions. Thus ℓ = 0, i.e. (A.5)

We rst assume d = 2 and let M (r) = sup s⩾r |u(s)|. Then by (A.4), for large r, we have

Integrating between r = R and ∞, using that M (r) is decreasing, that ∞ 0 |u| 2 rdr < ∞, and (A.5) we obtain

Since M (R) ≲ R -1/2 , we deduce, letting α = p 1 -2 > 0, that for r large,

and thus integrating again between r = R and ∞,

for large r. Thus M (r) = 0 for large r. This implies u(r) = 0 for large r and thus, by standard ODE theory, that u is identically 0.

We next assume d ∈ {3, 4}. We rst show that for all a > 0, there exists a constant C > 0 such that (A.6) ∃C > 0, r > C =⇒ |u(r)| ⩽ Cr -a .


Oscar Yamfwa
email: oscaryamfwa@yahoo.fr

B Kafola

Arsène M Muswaswa

Dieudonné E Musibono

Réné V Gizanga

Emmanuel M Biey

Problem of artisanal gold mining in the mining area of Mangi, Tshopo Province in the Democratic Republic of Congo

Keywords: Risks, artisanal mining, gold, Mercury, Mangi, Tshopo, DR Congo

Objective: assess the environmental and health risks generated by artisanal gold mining in the community of MANGI.

Methods: This is a descriptive cross-sectional study carried out from October 2021 to June 2022. Data collection was done through direct observations and survey questionnaires. ArcGIS software was used to carry out our cartographic analyses. Landsat images from 2003 and 2023 were used to assess the deforestation rate of the mining area. Of the 362 questionnaires distributed to various artisanal miners and other stakeholders, 229 responses were collected, representing a response rate of 63.26%. Data analysis is done with Epi-info 7.2.5 software and Excel 2016 software.

Results:

The area generally suffered deforestation of 2.3% and particularly forest degradation of around 10.9% due to artisanal gold mining over a period of twenty years. Mercury is the only chemical element used by gold miners in the processing of gold. A presence of minors on operating sites estimated at 9.2%. The level of ignorance of operators on the health impacts of the use of mercury was estimated at 96.5% and that of knowledge on the environmental impacts was at 64.6%. Deforestation was 100% certified. It was also noted the pollution of waterways and a disappearance of fish criticized by 92.6%; also the presence of rock debris in all sites studied with abandoned gaping holes. 91% of respondents experienced accidents/incidents during the activity. 94% of operators complained of lower back pain due to activity, 67% of chest pain, 62% of chronic cough, 21% of chronic digestive disorders, 81% of insomnia, 67% of persistent headaches, 13% of cases of memory loss, 39% of respondents presenting limb tremors. This study also reports 86% of cases of permanent conflicts linked to the activity with excessive alcohol consumption noted among the majority of operators (86%).

Conclusion:

Gold panning has promoted the economic stability of several households in the region, but unfortunately, it is the basis of several health and environmental risks that it is essential to manage effectively. Raising awareness among operators and rigorous control over the application of texts are alternatives to be considered by the political-administrative authorities in this sector.

INTRODUCTION

Worldwide, the Artisanal and Small-Scale Gold Mining (ASM) sector provides direct livelihoods to around 16 million artisanal miners and other sector actors and around 100 million people in indirect livelihoods [START_REF] Seccatore | An Estimation of the Artisanal Small-Scale Production of Gold in the World[END_REF]. Although a provider of employment and the main source of income for gold mining communities, the sector generates various environmental impacts. In its report on biodiversity in the DRC, USAID (2010) shows that there are considerable overlaps between mining reserves and tropical forests, which presents a risk for the protection of this biodiversity. The potential environmental impacts of mining, according to this report, are: deforestation, siltation of waterways, land degradation, dust pollution, mortality of wild animals (poaching) and water pollution by acid, copper, lead, arsenic, cyanide or mercury. Certainly, one of the particularities of this activity is the use of chemicals, the most common of which are cyanide and mercury. Mercury is a "heavy" metal which several studies prove to be the cause of harmful effects on the neurological, digestive and immune systems as well as on the lungs, kidneys and eyes, particularly among unborn children and newborns (Grasmick et al.1998). The transport mechanisms of mercury in the environment are one of the main reasons why it was decided that global action was needed to tackle the problem of mercury pollution. Thus, United Nations member states negotiated the Minamata Convention on Mercury, a global treaty aimed at protecting human health and the environment from anthropogenic emissions and releases of mercury and mercury compounds (PAN, 2019). However, this sector represents 37.7% of anthropogenic atmospheric emissions of mercury into the environment. This is even more serious in Sub-Saharan Africa where gold ASM contributes up to 80% of mercury emissions (UNEP, 2018). In the DRC, artisanal mining is not informal. It has existed for years and has followed a regulatory formalization process. It is with the economic liberalization measures decided by the Zairian Government in 1982 that artisanal mining has grown to this day. The Government has decided to liberalize research, possession, transport and trade in diamonds and gold from artisanal mining. The North-East of DR Congo, formerly called Orientale Province, which brought together the current Provinces of Tshopo, Bas-Uélé, Haut-Uélé and Ituri contains the main gold resources of the country. These resources make this part of the country a pole of attraction for international and national mining companies, but also for artisanal miners. Gold mining has grown significantly in the area for decades with the discovery of gold veins and the increase in poverty. In this region, gold panning feeds several families and has taken an important place in the local economy [START_REF] Bedidjo | Study on gold panning and mercury use in artisanal mining in Ituri 6[END_REF]. Artisanal gold extraction is an essential reality in rural areas, an occupation in the same way as agriculture and livestock breeding. The agglomeration of MANGI in the territory of BANALIA is today one of the most active artisanal mining areas in the province of TSHOPO. This activity constitutes the main source of income in this part of the country. In the various mining sites that are the subject of this study, mercury is the main chemical used for the processing of gold. However, this metal causes several negative impacts on the environment and health risks for the population. This reality motivated our study with a view to contributing to the improvement of operating conditions in this mining sector.

STUDY ENVIRONMENT, MATERIALS AND METHODS

Study environment

This study on the exploitation of gold and the use of mercury is carried out in the twenty-three operating sites of the MANGI mining zone located mainly in the BABOA DE KOLE Sector, Territory of BANALIA, Province of TSHOPO in the Democratic Republic of Congo

Methodology

This is a descriptive cross-sectional study carried out from October 2021 to June 2022, i.e. 9 months, in twenty-three operating sites in the area. To collect the information necessary for this study, practices were observed using observation grids and a survey based on a questionnaire. We also used a certain number of materials in particular: the Garmin brand GPS to collect the geographical coordinates of the mining sites in the study area, the ArcGIS software with its ArcMap extension made it possible to carry out our analyzes cartographic which were presented in the form of maps, the Google Earth software which made it possible to visualize the different mining sites on satellite image backgrounds, the SRTM image with a resolution of 90 m pixel was used to generate a model digital terrain (DTM) as well as to extract the topography of the area under study, Landsat images from 2003 and 2023 for the evaluation of plant cover and the rate of deforestation and/or forest degradation. The study population consisted of all operators and stakeholders working on the 23 sites in the area. The sampling technique was stratified probability with an estimated sample size of 362 individuals. After collection, the data underwent statistical processing using Epi Info and Excel software. The results from these data processing revealed information which was analyzed and then interpreted.

RESULTS

I. OBSERVATION RESULTS

I.1. Mining technique and method of processing gold ore

In the Mangi mining area, four main types of gold mining have been categorized. Namely alluvial mining along watercourses, open-cast eluvial mining, deep eluvial mining, open-cast hard rock mining. The ore is reduced into small pieces of at least 1 cm3 using 2kg hammers. It is then put in the sun to be dried on tarpaulins. Once dried, the crushed ore is poured into a mechanical crusher tank containing several hundred kilograms of steel balls which transform it into a fine powder. The powder is poured into water and leached in a sluice box to obtain the pulps which will be amalgamated to recover the gold. The pulp obtained after leaching is poured into a basin then mixed with a small quantity of water. A person designated by his teammates begins to stir this mixture of mercury and gold concentrate contained in the pulp with his bare hand in order to obtain the amalgam. The amalgam is pressed in a cloth to remove the unamalgamated mercury. The pure and dry amalgam is finally heated over a fire in a pan to recover only the raw gold. Another part of the Hg evaporates during the operation. In the study environment, artisanal workers do not know of any other methods of recovering gold apart from amalgamation. They all use mercury as the only means of processing the ore to obtain gold.

I.2. Impacts of alluvial and eluvial extraction on forest resources and soil in the area

 Deforestation and forest degradation

Deforestation is one of the unavoidable elements observed on gold sites. It constitutes one of the most visible impacts. Over a period of twenty years, the dynamics of deforestation occurs mainly in the exploitation zones as illustrated in the following figures. The results are as follows:

 Presence of numerous excavations and soil erosion

Gold mining in the area has left thousands of abandoned pits and excavations. This imbalance caused over-siltation of valleys and sedimentation of rivers.

I.3. Impacts of extraction on surface water resources

The presence of mercury in freshwater is mainly due to mine discharges after amalgamation, runoff through dissolved contents in the soil, but also through atmospheric deposition.

II.1. General informations a)

Response rate

Age of operators

The different age groups found in the operating sites are presented as follows:

Fig. 4: Distribution by age

c) Socio-professional categories

It appears that 77.73% of those interviewed were artisanal operators followed by 13.54% of agents of SAEMAPE (State service responsible for the supervision and assistance of artisanal exploitation), 4.8% of land chiefs and 3.93% of AFMs/mining site managers. The level of knowledge of the health impacts of gold mining with the use of mercury is as follows.

d) Gender

A predominantly male rate reached 90% during the study on the different sites.

II.2. Level of knowledge of the impacts of exploitation by operators

In terms of knowledge of the impacts of artisanal gold mining, the results obtained from the various stakeholders in this sector are as shown below.

a) Knowledge of health impacts

The various stakeholders in the sector show a high level of ignorance at 96.5% on the health impacts of the use of mercury in gold mining.

Figure 06: Level of knowledge of health impacts

b) Level of knowledge of environmental impacts

In terms of knowledge of the impacts of artisanal gold mining, the results obtained show that 64.6% of operators recognize that their activity creates impacts on their environment.

b) Cases of insomnia and memory loss

Surveys report that 81% of operators experience insomnia and 13% say they do not remember some of their actions.

Figure 11: (i) Complaints of insomnia (ii) Cases of memory loss c) Consumption of narcotics in mining sites

Among artisanal miners, 86% drink alcoholic beverages for different reasons.

Figure 12: Rate of alcohol use

d) Cases of conflicts and complaints in operations

Cases of conflicts of various causes within artisanal industries have been reported with a rate of 86%. Among mining operators, 94% of cases of lower back pain, 67% of chest pain, 62% of chronic cough, 21% of digestive disorders, 67% of persistent headaches and 39% of cases of head tremor were noted. upper members In the DRC, the principle of environmental responsibility is regulated by the Mining Code and Regulations, which require industrial operators to assess the impact of their activities on the environment and to submit a project to restore it. The Code clearly explains that industrial operators have a responsibility to ensure the restoration of the environment by requiring 0.5% of their turnover as collateral. As for artisanal diggers, the Mining Code strictly prohibits them from using mercury or any other substance harmful to health or the environment.

The Mining Code stipulates that environmental protection is one of the reasons that may lead the Prime Minister to declare a zone prohibited from mining (Article 6). The Mining Regulations also stipulate that any processing of mining products using mercury or other hazardous substances may only be carried out in a workshop or factory approved by the Minister of Mines (Article 238). An artisanal miner who undertakes the processing of his products on his own using mercury or other harmful products may lose his operator card. Article 575 prohibits the use of mercury in artisanal mining and specifies that only gravimetric separation and processes using reagents that do not cause serious harm to ecosystems are permitted. Annex IV of the Mining Regulations provides a comprehensive overview of the Code of Conduct for Artisanal Miners with regard to the protection of the environment. As is often the case in the DRC, the application of the Code and the regulations is practically not taken into consideration, as was an observation made during our study.

However, this study reports an estimated deforestation rate of 2.3% in the area and particularly a forest degradation of around 10.9% due to artisanal gold mining between 2003 and 2023. Operators say they are 100% aware of the negative impact of their activity on the forest ecosystem, 92.6% say they are aware of the negative impact of fish and 91.27% of their water pollution. Rock debris was present at all sites. We have recorded 100% of the cases of abandoned holes NDIAYE (2020) notes that 30% of respondents say that the discharge of mercury into the river's waters kills fish. Some of these respondents say that "mercury contaminates fish, and the people who eat those fish."

In the study carried out by Bossissi Nkuba et al (2016), all of them state that the real environmental problems caused by mining activities are deforestation and the pollution of rivers by clay and sandy particles released during gold extraction. While deforestation and river disturbance are clearly visible, the effects of mercury are not directly visible and can be difficult to identify.

In terms of health impacts, 62% of farmers have a chronic cough during their activity. 81% of farmers with insomnia due to the activity. 13% of cases of memory loss collected from artisanal miners who use mercury in their farms. Surveys conducted by Sène et al., (2019) on the locality of Tinkoto showed that 43% of respondents claim to have experienced respiratory problems, headaches, health problems that are related to mercury exposure.

Although mercury is not visible to the naked eye and its harmfulness is not universally recognized, it is a chemical time bomb. It does pose a real danger to humans (its effects on the central nervous system and on the growth of foetuses are particularly deplorable) and aquatic organisms (which are subject to physiological disturbances). It would therefore be dangerous to limit ourselves to directly visible impacts and ignore those that hide in the shadows, without being any less dangerous, at the risk of experiencing a case similar to that of Minamata, where mercury poisoned more than 2000 people, half of whom died. In the gold mining industry, there are several cases of permanent conflicts and fights (86%) and cases (86%) of excessive alcohol consumption before, during and after the activity, which hinder the maintenance of good and lasting social relations.

(b) Presence of minors

Our study reports the presence of minors (9.2%) on the exploitation sites studied, which is prohibited by Congolese law. Bossissi Nkuba et al, also report that children are present in all mines in South Kivu, despite the prohibition by national and international laws. In the study area, they are involved in the transport of minerals, processing, water supply, etc. They are often there for various reasons: the disappearance of their father (death) and/or the poverty of their families. Many drop out of school to work permanently (thus reducing the chances of being able to work and thus perpetuating an unskilled and very unproductive workforce in the region), thus indulging in alcohol or drugs, etc. But a few use the same opportunity to finance their schooling or to keep themselves busy rather than engaging in criminal activities.

(c) Level of knowledge

The gold miners surveyed showed a 96.5% level of ignorance about the health impacts of the use of mercury in gold mining. And a level of knowledge on the environmental impacts of gold mining at 64.6%. NDIAYE (2020) reports that 81% of respondents say that water will be undrinkable when it is contaminated with mercury. The rest of the respondents reported that they did not know the effects of mercury on water resources. It found that 49% of respondents were unaware of the health effects of mercury and 39% said that mercury degrades the soil, seeps and persists in the soil. The remaining 12% say mercury has no effect on the soil. The results of Bossissi Nkuba et al (2016) also showed that diggers are unaware of the health and environmental risks posed by mercury. This is due to the fact that no organization has so far organized an awareness campaign.

(d) Chemicals used

The chemical element used for gold processing is mercury in 100% of cases. In Senegal, Ndiaye reveals that 88% of respondents use mercury for gold recovery. However, the DRC's mining code strictly prohibits the use of mercury or any other substance that is harmful to health or the environment.

CONCLUSION

Artisanal gold mining has favored the development of several localities in the territory of Banalia, including that of Mangi. This locality is currently one of the most active gold sites in the Tshopo province. Gold panning is practiced intensively in the study area with the use of mercury for the processing of gold. This study made it possible to carry out an analysis of certain environmental and health problems caused by artisanal gold mining in general and by mercury in particular. In addition to the elemental mercury used to form the gold-mercury alloy, the population of the Mangi mining area is exposed to methylmercury through the consumption of fish which would be contaminated by mercurial emissions. This exposure may cause adverse health effects. In addition, in the area there is a lack of qualified health personnel to diagnose the effects of exposure to mercury. Despite the difficulties of the diggers' lives and the damage caused to the environment, artisanal gold mining (EMA) remains the best and sometimes the only means of survival for several families in the territory of Banalia and the engine of the economy of several mining villages and neighboring towns.

Protecting the environment through law enforcement and increasing pressure on their source of income is neither realistic nor socially just given the widespread poverty in the study area. We recommend that authorized state services and civil society can support them in formalizing their activities, facilitate their access to finances to better produce and train them in gold processing techniques without the use of mercury and economically viable in their situation. At the local level among artisanal gold miners in the Mangi mining area, it currently seems difficult, if not impossible, to do without mercury because the miners have been using it for years. And they don't seem to know any other techniques that would allow them to extract gold without using mercury. However, gold miners can be made aware of taking specific measures to reduce and if possible eliminate the worst practices in the use of mercury, for example the washing of ore during processing at surface water level, the release of mercury onto the ground. in order to limit the contamination of groundwater and surface water, the burning of amalgam in the open air and in areas inhabited by the population. The Democratic Republic of Congo has developed a national action plan aimed at gradually banning the use of mercury by 2036 at gold mining sites through the application of national regulations in relation to the convention of Minamata. It is important to strengthen these measures to be able to resolve this problem of mercury use throughout the national territory. It is important that the State tries to initiate an awareness policy among gold panning communities on the impact of mercury. These awareness-raising measures could focus on the different impacts that mercury causes on aquatic resources, soil and air. Focus this awareness on certain key people such as mercury sellers, who through them, mercury is present on the sites, people in charge of site security, the Mining Services (SAEMAPE and Mines Division) as well as the village chief who represents the highest authority on the operating sites. It is also possible to train certain local opinion leaders on the harmful effects of mercury and its various environmental impacts and health risks that it can cause, and to integrate awarenessraising measures at school level to reach students from a young age.

Figure 1 .

 1 Figure 1. Study area and its hydrography (Geomatics Laboratory -UNIKIN 2023)

Figure 2 .

 2 Figure 2. Land cover maps (Geomatics Laboratory -UNIKIN 2023)

Figure 3 :

 3 Figure 3: Abandoned shafts and excavations (Zuwa Idée and Loningisa site)

Figure 4 :

 4 Figure 4: Presence of sediments and mining discharges containing Hg on the banks and in the watercourse II. SURVEY RESULTS

Figure 05 :

 05 Figure 05: Distribution by gender

Figure 07 :

 07 Figure 07: Knowledge of the environmental impacts of gold mining c) The use of chemical substancesThe results report that the chemical element used for the treatment of gold is mercury in 100% of cases. The use of mercury in gold processing is confirmed by all operators.

Figure 08 :

 08 Figure 08: chemical substances used in gold processing

Figure 09 :

 09 Figure 09: Pollution of rivers

Figure 10 :

 10 Figure 10: Accident/incident rate

Figure 13 :

 13 Figure 13: Conflicts and Complaints

Table 1 :

 1 Land Cover ClassificationThe total area of the study area being 12,722 km2 while the area occupied respectively in 2002 and 2023 by forest vegetation is 12,556,614 and 12,264,008 km2. Thus, the percentage of vegetation cover is respectively 98.7% and 96.4% in 2003 and 2023. The results obtained

	N° Classes	Percentage	Percentage
		2003	2023
	01. Water	0, 3	0,3
	02. Primary Forest	69,7	59,3
	03. Secondary Forest	28,3	25,5
	04. Degraded Forest	0,7	11,6
	05. Bare Soil	1	3,4
	Total	100	100

show that the mining area of Mangi has generally suffered deforestation of 2.3% and particularly forest degradation of around 10.9% due to artisanal gold mining between 2003 and 2023.

 Soil degradation

Table 2 :

 2 Distribution of respondents by site

	N°	Mining site	Workforce	Samples	Respondents	%
	01. VICTOIRE KPOROGBOZA	175	11	6	54,54
	02. OKAPI		200	12	8	66,67
	03. TALA YESU	190	11	5	45,45
	04. BAKOKO	250	15	6	40
	05. VICTOIRE	199	12	5	41,67
	06. KITAMATA	300	18	10	55,56
	07. MALEKOMBE	290	18	11	61,11
	08. VATICAN	250	15	10	66,67
	09. GBADO		350	21	15	71,43
	10. DIEU VOIT TOUT	250	15	9	60
	11. NDINDA	179	11	5	45,45
	12. KANUA		300	18	12	66,67
	13. MOTUKA MUNENE	270	16	10	62,5
	14. ZUA IDEE	350	21	13	61,90
	15. PARADISO	220	13	8	61,54
	16. TIKAA		233	14	7	50
	17. MOTEMA NA MWANA	260	16	9	56,25
	18. TOMATE	240	14	8	57,14
	19. MOKOKA	290	17	11	64,71
	20. MOKOKA I	330	20	15	75
	21. LONINGISA	245	15	13	86,67
	22. MON VILLAGE	305	18	18	100
	23. MOSEKA	350	21	15	71,43
	Total		6026	362	229	63,26

Table 03 :

 03 Socio-professional categories

	Position occupied	Frequency Percent
	AFM/site manager	9	3,93
	State agent (SAEMAPE)	31	13,54
	Lands Chiefs	11	4,8
	Artisanal miner/digger	178	77,73
	Total	229	100