On stability of rotational 2D binary Bose-Einstein condensates

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On stability of rotational 2D binary Bose-Einstein condensates

Rémi Carles, Van Duong Dinh, Hichem Hajaiej

To cite this version:

Rémi Carles, Van Duong Dinh, Hichem Hajaiej. On stability of rotational 2D binary Bose-Einstein

condensates. 2021. �hal-02965508v2�

CONDENSATES

R´EMI CARLES, VAN DUONG DINH, AND HICHEM HAJAIEJ

Abstract. We consider a two-dimensional nonlinear Schr¨odinger equation proposed in Physics to model rotational binary Bose-Einstein condensates.

The nonlinearity is a logarithmic modification of the usual cubic nonlinearity.

The presence of both the external confining potential and rotating frame makes it difficult to apply standard techniques to directly construct ground states, as we explain in an appendix. The goal of the present paper is to analyze the orbital stability of the set of energy minimizers under mass constraint, according to the relative strength of the confining potential compared to the angular frequency. The main novelty concerns the critical case (lowest Landau Level) where these two effects compensate exactly, and orbital stability is established by using techniques related to magnetic Schr¨odinger operators.

1. Introduction

The formation of self-bound droplets is a well-known macroscopic phenomenon.

Recent experiments with ultracold quantum gases of bosonic atoms revealed a novel type of quantum liquid: dilute self-bound Bose-Einstein condensate (BEC) having orders of magnitude lower density than air (see e.g. [25, 35, 19, 20] for Bose gases of dysprosium and [36, 12] for binary Bose gases of potassium). Since these droplets form out of a BEC, there is good reason to assume that they have superfluid properties. One remarkable feature of a superfluid is its response to rotation, in particular the occurrence of quantized vortices (see [1] for a broad introduction to these phenomena). In [37], binary BEC droplets carrying angular momentum were considered. Using weak first-order corrections to the mean field energy, often referred to as the Lee-Huang-Yang correction [27], a binary BEC droplet with angular momentum is well described by the wave function ψ : R

₊

× R

²

→ C whose evolution is governed by the Gross-Pitaevskii equation (GPE) with angular momentum

i∂

t

ψ + 1

2 ∆ψ = V ψ + | ψ |

²

ln( | ψ |

²

)ψ − iK

3

| ψ |

⁴

ψ − ΩL

z

ψ, (1.1) where the scaling invariances have been used to bring the equation into its dimen- sionless form. Here the external potential V is of the form

V (x) = γ

²

2 | x |

²

+ V

0

e

^−γ|x|2

, (1.2) where γ > 0 is the harmonic trap frequency and V

0

> 0 is the amplitude of the Gaussian. The parameter K

3

> 0 is the rate of three-body losses. The angular momentum operator L

z

is of the form

L

z

= i(x

2

∂

x1

− x

1

∂

x2

), x = (x

1

, x

2

) ∈ R

²

(1.3)

2010Mathematics Subject Classification. 35Q55, 35A01.

Key words and phrases. Nonlinear Schr¨odinger equation; Bose-Einstein condensate; Harmonic potential; Rotation; Standing waves; Global existence; Stability.

RC is supported by Rennes M´etropole through its AIS program. VDD is supported by the Labex CEMPI (ANR-11-LABX-0007-01).

1

and Ω > 0 is the rotational speed. The fact that the constants in the harmonic trap and the Gaussian part of the potential are equal stems from [37], but is not crucial in our analysis, so we consider more generally

V (x) = γ

²

2 | x |

²

+ V

0

e

^−γ0|x|2

, γ, γ

0

> 0. (1.4) The main purpose of this paper is to study the existence/nonexistence and or- bital stability of mass-constraint standing waves for (1.1). We consider the Cauchy problem for (1.1) with initial data ψ

0

∈ Σ, where

Σ :=

f ∈ H

¹

( R

²

), x 7→ | x | f (x) ∈ L

²

( R

²

) is equipped with the norm

k f k

²Σ

= k f k

²H¹

+ k xf k

²L²

.

Due to the presence of the harmonic potential, this space is rather natural (see e.g.

[14]). In the case K

3

= 0, there are three physical quantities which are formally conserved along the flow of (1.1)

M (ψ(t)) = k ψ(t) k

²L²

= M (ψ

0

), (Mass)

L(ψ(t)) = Z

R²

ψ(t, x)L ¯

z

ψ(t, x)dx = L(ψ

0

), (Angular momentum) E

Ω

(ψ(t)) = 1

2 k∇ ψ(t) k

²L²

+ Z

R²

V (x) | ψ(t, x) |

²

dx (Energy) + 1

2 Z

R²

| ψ(t, x) |

⁴

ln

| ψ(t, x) √ |

²

e

dx − ΩL(ψ(t)) = E

Ω

(ψ

0

).

Here we note that the angular momentum is real-valued, but has no definite sign.

Similarly, the term involving the natural logarithm also has no definite sign.

In the case K

3

6 = 0, solutions to (1.1) formally satisfy 1

2 d

dt k ψ(t) k

²L²

+ K

3

k ψ(t) k

⁶L⁶

= 0. (1.5) This shows that for K

3

6 = 0, the equation is irreversible. This is the reason why, since K

3

> 0, we consider only positive time in the present paper. In the case K

3

= 0, the equation is reversible, and ψ( − t, x) solves (1.1): considering the case t > 0 suffices to describes the dynamics for all time.

When γ = V

0

= K

3

= Ω = 0, the equation (1.1) was recently studied in [16].

More precisely, the global well-posedness for H

¹

data, the existence and uniqueness of positive ground state solutions for (1.1) were shown there, along with the orbital stability of prescribed mass standing waves.

Our first result is the following global existence for (1.1).

Theorem 1.1 (Global well-posedness). Let γ, γ

0

> 0, Ω > 0, V

0

> 0, K

3

> 0, and ψ

0

∈ Σ. Then there exists a unique global-in-time solution to (1.1) satisfying ψ ∈ C( R

₊

; Σ) ∩ L

³_loc

( R

₊

; L

⁶

( R

²

)).

• If K

3

= 0, the conservation laws of mass, angular momentum, and energy hold.

• If K

3

> 0, the solution asymptotically vanishes in the sense that k ψ(t) k

²L²

= O

t

^−1/4

as t → ∞ . (1.6)

The Cauchy problem is addressed by resuming the approach from [3]. In pass-

ing, we fix a small flaw present in this paper regarding dispersive estimates. The

asymptotic extinction (1.6) is then established like in [2], thanks to a suitable uni-

form bound which makes it possible to control the L

⁶

-norm in (1.5) from below in

terms of the L

²

-norm.

Remark 1.2. The potential V in (1.4) is radially symmetric, since it is the model given in [37]. We will see in the proof of Theorem 1.1 that the result still holds true in the more general case of smooth potentials which are at most quadratic (and thus need not be radial); see Remarks 2.2 and 2.3.

In the rest of the introduction, we are interested in the absence of three-body losses, i.e. K

3

= 0. In this case, (1.1) admits standing waves, i.e. solutions of the form

ψ(t, x) = e

^iωt

φ(x), ω ∈ R , (1.7) where φ solves

− 1

2 ∆φ + V φ + φ | φ |

²

ln( | φ |

²

) − ΩL

z

φ + ωφ = 0, x ∈ R

²

. (1.8) Note that in the case K

3

> 0, there is no such solution in view of the asymptotic extinction (1.6).

The existence of standing waves for (1.8) can be achieved by several ways. The first way is to minimize the energy functional

E

Ω

(f ) = 1

2 k∇ f k

²L²

+ Z

R²

V | f |

²

dx + 1 2 | f |

⁴

ln

| √ f |

²

e

dx − ΩL(f ),

with prescribed mass constraint, i.e. k f k

²L²

= ρ > 0, an strategy which is often adopted in Physics. In this case, the parameter ω in (1.8) appears as a Lagrange multiplier associated to the minimization problem. Another way is to look for critical points of the action functional

S

ω

(f ) = E

Ω

(f ) + ωM (f )

= 1

2 k∇ f k

²L²

+ ω k f k

²L²

+ Z

R²

V | f |

²

dx + 1 2 Z

R²

| f |

⁴

ln | f |

²

√ e

dx − ΩL(f ), with ω being given and fixed. However, this approach seems difficult to apply in the present context. More precisely, (1.8) has two features which make it difficult to characterize the range of ω’s allowed to find a non-trivial solution to (1.8). The presence of the external potential V and the rotation L

z

introduces an x-dependence which makes it impossible to invoke the results from [9] (the 2D counterpart of [10]), or even adapt easily the proof. On the other hand, the fact that the nonlinearity is not homogeneous in φ makes it impossible to reproduce the arguments from [34]

(see also [22] in the case of a harmonic potential). In an appendix, we collect some information regarding the possible range for ω in the radial case, where the rotating term is absent from (1.8), and explain in more details why minimizing the action seems difficult here.

We therefore consider the following minimization problem: for ρ > 0, I

Ω

(ρ) := inf

E

Ω

(f ) : f ∈ Σ, k f k

²L²

= ρ . (1.9) Our next result concerns the existence and stability of prescribed mass standing waves for (1.8) in the case of low rotational speed.

Theorem 1.3. Let K

3

= 0, γ, γ

0

> 0, V

0

> 0 and 0 < Ω < γ. Then for any ρ > 0, there exists φ ∈ Σ such that E

Ω

(φ) = I

Ω

(ρ) and k φ k

²L²

= ρ. Moreover, the set

G

Ω

(ρ) :=

φ ∈ Σ : E

Ω

(φ) = I

Ω

(ρ), k φ k

²L²

= ρ

is orbitally stable under the flow of (1.1) in the sense that for any ǫ > 0, there exists δ > 0 such that for any initial data u

0

∈ Σ satisfying

φ∈G

inf

Ω(ρ)

k u

0

− φ k

^Σ

< δ,

the corresponding solution to (1.1) exists globally in time and satisfies sup

t∈R

φ∈G

inf

Ω(ρ)

k u(t) − φ k

^Σ

< ǫ.

The proof of Theorem 1.3 is based on a standard variational argument using the following observation (see Lemma 3.1): for 0 < Ω < γ,

k∇ f k

²L²

+ 2 Z

R²

V | f |

²

dx − 2ΩL(f ) ≃ k∇ f k

²L²

+ k xf k

²L²

which enables us to use the standard compact embedding Σ ֒ → L

^r

( R

²

) for all 2 6 r < ∞ . We also make use of the log-type inequality

Z

R²

| f |

⁴

ln | √ f |

²

e

dx

.

ǫ

k f k

⁴L^−4−ǫǫ

+ k f k

^4+ǫL^4+ǫ

(1.10) for any ǫ > 0. For more details, we refer to Subsection 3.1.

Next we consider the critical rotational speed Ω = γ (lowest Landau level, see e.g. [1] and references therein), which constitutes the main novelty of this paper.

In this case, the energy functional can be rewritten as E

γ

(f ) = 1

2 k∇

^A

f k

²L²

+ V

0

Z

R²

e

^−γ0|x|2

| f (x) |

²

dx + 1 2

Z

R²

| f |

⁴

ln | f |

²

√ e

dx, (1.11) where ∇

^A

:= ∇ − iA with A(x) = γ( − x

2

, x

1

). Thanks to this observation, it is convenient to consider the above functional on the magnetic Sobolev space

H

_A¹

( R

²

) :=

f ∈ L

²

( R

²

) : (∂

j

− iA

j

)f ∈ L

²

( R

²

), j = 1, 2 (1.12) endowed with the norm

k f k

²H_A¹

:= k f k

²L²

+ k∇

A

f k

²L²

.

Note that the energy functional (1.11) is well-defined on H

_A¹

( R

²

) by using (1.10) and the magnetic Gagliardo-Nirenberg inequality (see e.g., [18]): for 2 < r < ∞ ,

k f k

^rL^r

6 C

r

k∇

^A

f k

^rL⁻²²

k f k

²L²

, ∀ f ∈ H

_A¹

( R

²

).

We also have Σ ⊂ H

_A¹

( R

²

). Indeed, for f ∈ Σ, we have k f k

²H_A¹

= k f k

²L²

+ k∇ f k

²L²

+ γ

²

k xf k

²L²

− γ

Z

R²

f L ¯

z

f dx 6 C k f k

²Σ

, where the last inequality follows from

| L(f ) | 6 k xf k

L²

k∇ f k

L²

6 1

2γ k∇ f k

²L²

+ γ

2 k xf k

²L²

.

To show the existence and stability of standing waves to (1.1) in the critical rotational case, it is convenient to consider separately two cases: V

0

= 0 and V

0

> 0.

When V

0

= 0, we denote I

_γ⁰

(ρ) := inf

E

_γ⁰

(f ) : f ∈ H

_A¹

( R

²

), k f k

²L²

= ρ , (1.13) where

E

⁰_γ

(f ) := 1

2 k∇

A

f k

²L²

+ 1 2

Z

R²

| f |

⁴

ln | f |

²

√ e

dx, (1.14)

and the superscript 0 is here to emphasize the assumption V

0

= 0. Let us first introduce the cubic ground state, that is, the unique positive, radially symmetric solution to

− 1

2 ∆Q + Q = Q

³

, x ∈ R

²

. (1.15)

By making use of a variant of the celebrated concentration-compactness principle of Lions adapted to the magnetic Sobolev space H

_A¹

( R

²

) (see Lemma 3.8), we prove the following result.

Theorem 1.4. Let K

3

= V

0

= 0, 0 < γ <

_2e¹3/2

and Ω = γ. Let 0 < ρ 6 k Q k

²L²

. Then there exists φ ∈ H

_A¹

( R

²

) such that E

γ⁰

(φ) = I

γ⁰

(ρ) and k φ k

²L²

= ρ. Moreover, the set of minimizers for I

_γ⁰

(ρ) is orbitally stable under the flow of (1.1).

The assumption ρ 6 k Q k

²L²

is probably technical, due to our argument (see after (3.18)). The proof also relies on the property I

_γ⁰

(ρ) < 0 (see Proposition 3.5). In the case K

3

= V

0

= γ = Ω = 0, the latter condition was shown in [16] for any ρ > 0. In our setting, showing that I

_γ⁰

(ρ) < 0 is more complicated since the scaling argument used in [16] does not work because of the presence of magnetic potential. However, by using a trial function of the form λe

^−b|x|2

with suitable positive constants λ and b, we are able to show (see Lemma 3.10) that both conditions ρ 6 k Q k

²L²

and I

_γ⁰

(ρ) < 0 are fulfilled by some data f ∈ H

_A¹

( R

²

). This also leads to a restriction on the validity of γ. We refer the reader to Subsection 3.2 for more details.

When V

0

> 0, the above-mentioned concentration-compactness argument does not work due to the lack of spatial translation of the term

Z

R²

e

^−γ0|x|2

| f (x) |

²

dx.

To overcome the difficulty, we restrict our consideration on H

_A,rad¹

( R

²

) the space of radially symmetric functions of H

_A¹

( R

²

). Note that this restriction has a drawback since we no longer see the effect of rotation to the equation as L

z

f = 0 for a radial function f . We consider

I

γ,rad

(ρ) := inf

E

γ

(f ) : f ∈ H

_A,rad¹

( R

²

), k f k

²L²

= ρ , (1.16) where E

γ

(f ) is as in (1.11). We have the following result.

Theorem 1.5. Let K

3

= 0, V

0

> 0, γ, γ

0

> 0, and Ω = γ. Then for any ρ > 0, there exists φ ∈ H

_A,rad¹

( R

²

) such that E

γ

(φ) = I

γ,rad

(ρ) and k φ k

²L²

= ρ. Moreover, the set of minimizers for I

γ,rad

(ρ) is orbitally stable under the flow of (1.1).

Remark 1.6. As mentioned above that the rotation term vanishes for radially sym- metric functions, the result in Theorem 1.5 holds for any Ω ≥ 0. In particular, it gives another (radial) solution, besides the one obtained in Theorem 1.4, to (1.8).

The proof of Theorem 1.5 relies on the following compact embedding

H

_A,rad¹

( R

²

) ∋ f 7→ | f | ∈ L

^r

( R

²

) (1.17) for all 2 6 r < ∞ . This compact embedding follows from the well-known compact embedding H

_rad¹

( R

²

) ֒ → L

^r

( R

²

) for all 2 6 r < ∞ and the fact that

H

_A,rad¹

( R

²

) ∋ f 7→ | f | ∈ H

_rad¹

( R

²

) is continuous due to the diamagnetic inequality (see e.g. [28])

|∇| f | (x) | 6 |∇

A

f (x) | a.e. x ∈ R

²

.

When the rotational speed exceeds the critical value, i.e. Ω > γ, we have the following nonexistence of minimizers for the constrained variational problem I

Ω

(ρ).

Theorem 1.7. Let K

3

= 0, γ, γ

0

> 0, V

0

> 0, and Ω > γ . Then for any ρ > 0,

there is no minimizer for I

Ω

(ρ), i.e., I

Ω

(ρ) = −∞ .

The proof of Theorem 1.7 is based on an idea of Bao, Wang, and Markowich [7, Section 3.2], using the central vortex state with winding number m, namely

f

m

(x) = f

m

(r, θ) =

r ργ

^m+1

πm! r

^m

e

^−γ|x|

2

2

e

^imθ

, m ∈ N ,

where (r, θ) are the polar coordinates in R

²

. Physically, when the angular velocity of rotation exceeds the trapping frequency, the harmonic potential cannot provide enough necessary centripetal force that counteracts the centrifugal force caused by the rotation, and the gas may fly apart. In the classical rotating BEC with purely power-type nonlinearity, the nonexistence of prescribed mass standing waves was proved by Bao, Wang, and Markowich [7].

This work is organized as follows. In Section 2, we study the local well-posedness for (1.1) and prove the global existence given in Theorem 1.1. In Section 3, we investigate the existence/nonexistence and the orbital stability of prescribed mass standing waves for (1.1) given in Theorems 1.3, 1.4, 1.5, and 1.7. Finally, we give some information on stationary solutions of (1.8) in the radial case in Appendix A, and characterize prescribed mass minimizers in Appendix B.

2. Cauchy problem

In the case K

3

= 0, Theorem 1.1 is reminiscent of the results from [3] for the local and global well-posedness. We present an alternative proof, to show that local solutions are actually global, which simplifies the approach, and yields better bounds. The asymptotic extinction in the case K

3

> 0 is obtained by adapting the arguments from [4] based on the introduction of a suitable pseudo-energy (see also [2] for generalizations).

2.1. Local well-posedness. Local well-posedness follows from a fixed point argu- ment on Duhamel’s formula

ψ(t) = U (t)ψ

0

− iλ Z

t

0

U (t − s)f (u)(s) ds, (2.1) where here and in the following, we denote

f (z) = z | z |

²

ln( | z |

²

) − iK

3

| z |

⁴

z, z ∈ C .

The notation U ( · ) stands for the linear propagator, that is, U (t)ψ

0

= ψ

lin

(t) is the solution to

i∂

t

ψ

lin

+ 1

2 ∆ψ

lin

= V (x)ψ

lin

− ΩL

z

ψ

lin

, ψ

lin

(0, x) = ψ

0

. (2.2) As noticed in [3], U (t) = e

^−itH(x,Dx)

, where

H(x, ξ) = 1

2 | ξ |

²

+ V (x) + Ω(x

1

ξ

2

− x

2

ξ

1

),

enters into the general framework of [26], where the fundamental solution, that is, the kernel associated to U ( · ), is constructed. However, this does not yield directly local in time dispersive properties, since unlike in [21], the solution to

¹

i∂

t

ψ + H(t, x, − i∂

x

)ψ = 0 ; ψ

_|t=0

= ϕ,

for H = H (t, x, ξ) smooth, real-valued and at most quadratic in (x, ξ) (locally in time), is represented as

U(t)ϕ(x) = Z

R²

e

^iφ(t,x,ξ)

a(t, x, ξ) ˆ ϕ(ξ)dξ,

1We do not include here the semi-classical parameter present in [21, 26].

that is, an oscillatory integral involving the Fourier transform of ϕ, and not ϕ directly. Consequently, dispersive L

¹

− L

^∞

estimates cannot be inferred in general, as shown by the trivial case H = 0: it enters the framework of [26], the above representation holds with a = constant (whose value depends on the definition of the Fourier transform), but no Strichartz estimate is available, of course.

In the present case, local dispersive estimates are available though, as we show by relying on the isotropy of V and L

z

; this corresponds to the trick presented in [3] in the 3D case. This alternative approach not only yields dispersive estimates, but also makes it possible to study the Cauchy problem as if the rotation term was absent. Define the function ϕ by

ϕ(t, x) = ψ (t, x

1

cos(Ωt) + x

2

sin(Ωt), − x

1

sin(Ωt) + x

2

cos(Ωt)) . (2.3) We check that since V depends only on | x |

²

(and is therefore invariant under rota- tion), ψ solves (1.1) if and only if ϕ solves







i∂

t

ϕ + 1

2 ∆ϕ = V ϕ + ϕ | ϕ |

²

ln( | ϕ |

²

) − iK

3

| ϕ |

⁴

ϕ, (t, x) ∈ R

₊

× R

²

, ϕ(0, x) = ψ

0

,

(2.4) that is, (1.1) with Ω = 0. Now for the perturbed harmonic oscillator H = −

¹₂

∆+V , local in time dispersive estimates are available: from [21], there exists δ > 0 such that

k e

^−itH

ψ

0

k

^L∞

. 1

| t | k ψ

0

k

L¹

, | t | 6 δ. (2.5) Note that δ is necessarily finite, as H possesses eigenvalues (as a consequence from e.g., [33, Theorem XIII.67]). This implies local in time Strichartz estimates (see e.g., [17]) and so, as long as bounded time intervals only are involved, (2.1) can be studied like in the case where U (t) = e

^it∆

. Note that since (2.3) preserves the Lebesgue norms, we infer

k e

^−itHΩ

ψ

0

k

L^∞

. 1

| t | k ψ

0

k

L¹

, | t | 6 δ, (2.6) for all Ω ∈ R , with δ and an implicit multiplicative constant independent of Ω.

In the case K

3

= 0, the analysis meets essentially the one presented in [16]. In the general case K

3

> 0, f ∈ C

¹

( R

²

; R

²

) satisfies f (0) = 0,

| f (u) | . | u |

^3−ε

+ | u |

⁴

, ∀ ε > 0, as well as

|∇ f (u) | . ( | u |

^2−ε

+ | u |

³

) |∇ u | . We then obtain:

Lemma 2.1 (Local well-posedness) . Let γ > 0, Ω > 0, V

0

> 0, K

3

> 0, and ψ

0

∈ Σ.

Then there exist T > 0 and a unique solution

ϕ ∈ C([0, T ]; Σ) ∩ L

³

((0, T ); L

⁶

( R

²

)), to (2.4). Equivalently, in view of (2.3),

ψ ∈ C([0, T ]; Σ) ∩ L

³

((0, T ); L

⁶

( R

²

)),

solves (1.1). Moreover, it satisfies (1.5). Either the solution is global in time, ϕ, ψ ∈ C( R

₊

; Σ) ∩ L

³_loc

( R

₊

; L

⁶

( R

²

)),

or there exists T

^∗

< ∞ such that

t→

lim

T∗

k∇ ϕ(t) k

L²

= lim

t→T∗

k∇ ψ(t) k

L²

= ∞ .

In the case K

3

= 0, we also have the conservation laws of mass, angular momentum,

and energy.

Sketch of the proof. We present the elements of the proof which require a little modification due to the presence of the damping term (K

3

> 0).

We recall that the standard blow-up alternative involves the condition

t→

lim

T^∗

k ϕ(t) k

^Σ

= lim

t→T^∗

k ψ(t) k

^Σ

= ∞ .

In view of (1.5), the L

²

-norm of ψ remains bounded for positive time, and thus, so does the L

²

-norm of ϕ. On the other hand, we compute

d

dt k xϕ(t) k

²L²

= − Im Z

R²

| x |

²

ϕ∆ϕdx ¯ − 2K

3

Z

R²

| x |

²

| ϕ |

⁶

dx

= 2 Im Z

R²

¯

ϕx · ∇ ϕdx − 2K

3

Z

R²

| x |

²

| ϕ |

⁶

dx.

Using Cauchy–Schwarz and Young’s inequalities, we infer that d

dt k xϕ(t) k

²L²

6 k xϕ(t) k

²L²

+ k∇ ϕ(t) k

²L²

. (2.7) Therefore, if ∇ ϕ remains bounded in L

²

, then so does the Σ-norm, and the blow-up criterion can be reduced to the one stated in the lemma.

In the case K

3

= 0, the proof of conservation laws follows, for instance, from

[31].

Remark 2.2 (Non-radial potential). The case of a non-radial potential can be treated along essentially the same lines. Indeed, for a potential V ∈ C

^∞

( R

²

; R ), which is at most quadratic in the sense of [21], that is,

∂

_x^α

V ∈ L

^∞

( R

²

), ∀ α ∈ N

²

, | α | > 2, the function ϕ, given by (2.3), solves

i∂

t

ϕ + 1

2 ∆ϕ = ˜ V (t, x)ϕ + ϕ | ϕ |

²

ln( | ϕ |

²

) − iK

3

| ϕ |

⁴

ϕ,

with ˜ V (t, x) = V (x

1

cos(Ωt) + x

2

sin(Ωt), − x

1

sin(Ωt) + x

2

cos(Ωt)). The time dependent potential ˜ V is smooth in (t, x), and at most quadratic in space (in the same sense as above), uniformly in time. It follows from [21] that (2.5) re- mains valid with e

^−itH

replaced by the evolution group U(s + t, s) associated to H ˜ (t) = −

¹2

∆ + ˜ V (t, x), hence (2.6) holds in this case too. The rest of the proof of Lemma 2.1 can be repeated exactly then.

2.2. Global well-posedness. In the case K

3

= 0, the following energy is inde- pendent of time,

E

0

(ϕ) := 1

2 k∇ ϕ k

²L²

+ Z

R²

V | ϕ |

²

dx + 1 2 Z

R²

| ϕ |

⁴

ln | √ ϕ |

²

e

dx, as well as the mass of ϕ. The positive part of the energy satisfies E

+

(ϕ(t)) : = 1

2 k∇ ϕ(t) k

²L²

+ Z

R²

V | ϕ(t) |

²

dx + 1 2

Z

|ϕ|²>√e

| ϕ(t, x) |

⁴

ln

| ϕ(t, x) |

²

√ e

dx

= E

0

(ψ

0

) + 1 2

Z

|ϕ|²<√

e

| ϕ(t, x) |

⁴

ln √

e

| ψ(t, x) |

²

dx 6 E

0

(ψ

0

) +

√ e 2

Z

R²

| ϕ(t, x) |

²

dx,

where we have used the easy bound ln a 6 a for a > 1. Using the conservation of mass, this yields

E

+

(ϕ(t)) . 1,

and thus global existence (in the past as well, since for K

3

= 0, the equation is reversible), since k ϕ k

Σ

. E

+

(ϕ). Note that even the L

²

-norm of ϕ is controlled by E

+

, in view of the uncertainty principle (see e.g., [32, § 3.2])

k f k

²L²

6 k∇ f k

L²

k xf k

L²

, f ∈ Σ. (2.8) In particular, there exists C > 0 such that

Z

R²

| x |

²

| ϕ(t, x) |

²

dx = Z

R²

| y |

²

| ψ(t, y) |

²

dy 6 C, ∀ t > 0. (2.9) In the case K

3

> 0, following the strategy presented in [4, Section 3.1], we introduce a pseudo-energy, for k > 0,

E(k, ϕ) = E

0

(ϕ) + k k ϕ k

⁶L⁶

. Adapting slightly the computations from [2], we find:

d

dt E(k, ϕ(t)) = K

3

Z

R²

| ϕ |

⁴

Re ( ¯ ϕ∆ϕ) dx − 3k Z

R²

| ϕ |

⁴

Im ( ¯ ϕ∆ϕ) dx

− 2K

3

Z

R²

V | ϕ |

⁶

dx − 2K

3

Z

R²

| ϕ |

⁸

ln( | ϕ |

²

)dx − 6kK

3

Z

R²

| ϕ |

¹⁰

dx.

Again, following the computations from [4, Section 3.1], we introduce the polar factor related to ϕ,

θ(t, x) :=

(

_ϕ(t,x)

|ϕ(t,x)|

if ϕ(t, x) 6 = 0, 0 if ϕ(t, x) = 0, and the above time derivative can be rewritten as

d

dt E(k, ϕ(t)) = − (K

3

− 6k) Z

R²

| ϕ |

⁴

|∇ ϕ |

²

dx − 4K

3

Z

R²

| ϕ |

⁴

|∇| ϕ ||

²

dx

− 6k Z

R²

| ϕ |

⁴

Re(¯ θ ∇ ϕ) − Im(¯ θ ∇ ϕ)

2

dx − 2K

3

Z

R²

V | ϕ |

⁶

dx

− 2K

3

Z

R²

| ϕ |

⁸

ln( | ϕ |

²

)dx − 6kK

3

Z

R²

| ϕ |

¹⁰

dx.

Picking 0 < k < K

3

/6, the above relation implies d

dt E (k, ϕ(t)) 6 2K

3

Z

|ϕ|<1

| ϕ |

⁸

ln 1

| ϕ |

²

dx 6 2K

3

Z

R²

| ϕ |

⁶

dx.

Since the change of unknown (2.3) preserves the Lebesgue norms, (1.5) yields 2K

3

Z

R²

| ϕ |

⁶

= − d dt k ϕ k

²L²

, hence

d

dt E (k, ϕ(t)) + k ϕ k

²L²

6 0,

and we conclude like in the case K

3

= 0. In particular, (2.9) holds.

Remark 2.3 (Non-radial potential, continued). For a general potential V like in Remark 2.2, the natural energy associated to ϕ becomes

E

0

(ϕ(t)) = 1

2 k∇ ϕ(t) k

²L²

+ Z

R²

V ˜ (t, x) | ϕ(t, x) |

²

dx+ 1 2

Z

R²

| ϕ(t, x) |

⁴

ln

| ϕ(t, x) √ |

²

e

dx, and even for K

3

= 0, it is not constant in time if V is not radial (∂

t

V ˜ 6≡ 0).

However, following either the virial computation from [13] or the pseudo-energy

argument from [15], we can easily adapt the above argument and infer that the

global well-posedness in Theorem 1.1 remains valid.

2.3. Large time extinction in the presence of loss. In view of [2, Lemma 4.1], k ϕ k

L²

. k ϕ k

^3/5L⁶

k xϕ k

^2/5L²

. k ϕ k

^3/5L⁶

,

where we have used (2.9). This, together with (1.5), yields d

dt k ϕ(t) k

²L²

+ C k ϕ(t) k

¹⁰L²

6 0,

for some uniform C > 0. Recalling that the (nonnegative) solution of the ODE

˙

y + Cy

⁵

= 0 with y(0) = k ψ

0

k

²L²

satisfies y(t) = O (t

^−1/4

), this implies k ϕ(t) k

²L²

. 1

t

^1/4

,

hence the decay announced in Theorem 1.1, since k ϕ(t) k

L²

= k ψ(t) k

L²

. Again, this conclusion remains true when V satisfies the assumptions of Remark 2.2 and is not necessarily radial.

3. Orbital stability

In this section, we study the existence/nonexistence and stability of constraint mass standing waves associated to (1.8). Recall that K

3

= 0 throughout this section. We will consider separately three cases: low rotational speed (0 < Ω < γ), critical rotational speed (Ω = γ), and high rotational speed (Ω > γ).

3.1. Low rotational speed. In this subsection, we consider the low rotational speed 0 < Ω < γ. Let us start with the following observation, mimicking the proof of [5, Proposition 2.1] (see also [8, Theorem 7.8]):

Lemma 3.1. Let γ, γ

0

> 0 and V

0

> 0. If 0 < Ω < γ, then for any f ∈ Σ, k∇ f k

²L²

+ 2

Z

R²

V | f |

²

dx − 2ΩL(f ) ≃ k∇ f k

²L²

+ k xf k

²L²

. (3.1) Proof. We first observe that by Cauchy-Schwarz and Young inequalities, we have for any δ > 0,

Ω | L(f ) | 6 Ω ( k x

1

f k

L²

k ∂

x2

f k

L²

+ k x

2

f k

L²

k ∂

x1

f k

L²

) 6 Ω k xf k

L²

k∇ f k

L²

6 δ k∇ f k

²L²

+ Ω

²

4δ k xf k

²L²

. (3.2)

Using (3.2) with δ =

¹₂

, we have B(f ) := k∇ f k

²L²

+ 2

Z

R²

V | f |

²

dx − 2ΩL(f ) (3.3) 6 2 k∇ f k

²L²

+ (γ

²

+ Ω

²

) k xf k

²L²

+ V

0

k f k

²L²

6

2 + V

0

2

k∇ f k

²L²

+

γ

²

+ Ω

²

+ V

0

2

k xf k

²L²

6 C

1

k∇ f k

²L²

+ k xf k

²L²

,

where we have used (2.8) to get the third line. On the other hand, by (3.2), we have for any δ > 0,

B(f ) > (1 − 2δ) k∇ f k

²L²

+

γ

²

− Ω

²

2δ

k xf k

²L²

+ 2V

0

Z

R²

e

^−γ0|x|2

| f (x) |

²

dx.

We choose δ > 0 so that γ

²

− Ω

²

2δ = γ

²

− Ω

²

2 or δ = Ω

²

γ

²

+ Ω

²

.

It follows that

B(f ) > γ

²

− Ω

²

γ

²

+ Ω

²

k∇ f k

²L²

+ γ

²

− Ω

²

2 k xf k

²L²

> C

2

k∇ f k

²L²

+ k xf k

²L²

.

The proof is complete.

Proof of Theorem 1.3. The proof is divided into two steps.

Step 1. We show the existence of minimizers for I

Ω

(ρ). Let ρ > 0 and f ∈ Σ satisfy k f k

²L²

= ρ.

We first show that I

Ω

(ρ) is well-defined. We have E

Ω

(f ) = 1

2 B(f ) + 1 2

Z

|f|²>√ e

| f |

⁴

ln

| √ f |

²

e

dx − 1

2 Z

|f|²<√ e

| f |

⁴

ln

√ e

| f |

²

dx

> 1 2 B(f ) −

√ e 2

Z

|f|²<√e

| f |

²

dx

> 1 2 B(f ) −

√ e

2 ρ, (3.4)

where B(f ) is as in (3.3). Here we have used the fact that ln(1/λ) < 1/λ for 0 < λ < 1. Using (3.1), we see that I

Ω

(ρ) > −∞ .

Next let (f

n

)

n>1

be a minimizing sequence for I

Ω

(ρ). By (3.4) and (3.1), we see that (f

n

)

n>1

is a bounded sequence in Σ. Thus there exists φ ∈ Σ such that (up to a subsequence) f

n

→ φ weakly in Σ and strongly in L

^r

( R

²

) for all 2 6 r < ∞ (see e.g., [33, Theorem XIII.67] for the compactness of the embedding Σ ֒ → L

^r

( R

²

)). It follows that

k φ k

²L²

= lim

n→∞

k f

n

k

²L²

= ρ. (3.5) By the strong convergence, H¨ older’s inequality, and the fact that

| f |

⁴

ln

| f |

²

√ e

− | g |

⁴

ln | g |

²

√ e

6 C | f − g | | f |

²

+ | f |

⁴

+ | g |

²

+ | g |

⁴

, (3.6) we infer that

Z

R²

| φ |

⁴

ln | φ |

²

√ e

dx = lim

n→∞

Z

R²

| f

n

|

⁴

ln | f

n

|

²

√ e

dx.

Moreover, by the lower semicontinuity of the weak convergence and (3.1), we have B(φ) 6 lim inf

n→∞

B(f

n

).

It follows that

E

Ω

(φ) 6 lim inf

n→∞

E

Ω

(f

n

) = I

Ω

(ρ)

which together with (3.5) show that φ is a minimizer for I

Ω

(ρ). In view of Lemma 3.1, we also have that (up to a subsequence) f

n

converges to φ strongly in Σ.

Step 2. We show the orbital stability. We argue by contradiction. Suppose that G

^Ω

(ρ) is not orbitally stable. There exist ǫ

0

> 0 and φ

0

∈ G

^Ω

(ρ) and a sequence u

0,n

∈ Σ satisfying

n

lim

→∞

k u

0,n

− φ

0

k

Σ

= 0 (3.7) and a sequence of time (t

n

)

n>1

⊂ R such that

φ∈G

inf

Ω(ρ)

k u

n

(t

n

) − φ k

^Σ

> ǫ

0

, (3.8) where u

n

is the solution to (1.1) with initial data u

n

(0) = u

0,n

.

Since φ

0

∈ G

Ω

(ρ), we have E

Ω

(φ

0

) = I

Ω

(ρ). By (3.7), (3.1) and Sobolev embed- ding, we have

k u

0,n

k

²L²_n

−→

→∞

k φ

0

k

²L²

= ρ, E

Ω

(u

0,n

)

_n

−→

→∞

E

Ω

(φ

0

) = I

Ω

(ρ).

By the conservation of mass and energy, we get k u

n

(t

n

) k

²L²_n

−→

→∞

ρ, E

Ω

(u

n

(t

n

))

_n

−→

→∞

I

Ω

(ρ).

This shows that (u

n

(t

n

))

n>1

is a minimizing sequence for I

Ω

(ρ). By Step 1, we see that up to a subsequence, u

n

(t

n

) → φ strongly in Σ for some φ ∈ G

Ω

(ρ) which

contradicts (3.8). The proof is now complete.

3.2. Critical rotational speed. In this subsection, we study the existence and stability of standing waves related to (1.8) with a critical rotational speed Ω = γ (lowest Landau level). To this end, we first collect some basic properties of the magnetic Sobolev space H

_A¹

( R

²

) (see (1.12)) in the following lemma.

Lemma 3.2 ([18, 28]) . We have:

• H

_A¹

( R

²

) is a Hilbert space.

• C

₀^∞

( R

²

) is dense in H

_A¹

( R

²

).

• H

_A¹

( R

²

) is continuously embedded in L

^r

( R

²

) for all 2 6 r < ∞ .

• H

_A¹

( R

²

) ⊂ H

_loc¹

( R

²

).

• Diamagnetic inequality:

|∇| f | (x) | 6 |∇

A

f (x) | a.e. x ∈ R

²

. (3.9)

• Magnetic Gagliardo-Nirenberg inequality. For 2 6 r < ∞ ,

k f k

^rL^r

6 C

r

k∇

A

f k

^r_L⁻²²

k f k

²L²

, ∀ f ∈ H

_A¹

( R

²

). (3.10) Lemma 3.3. The best constant in (3.10) is the same as the optimal constant in Gagliardo-Nirenberg inequality for the case A = 0. In particular, when r = 4, the best constant in (3.10) can be taken as

C

4

= k Q k

⁻_L²²

,

where Q is the cubic (positive, radial) ground state given by (1.15). Moreover, the equality in (3.10) cannot be attained.

Proof. We recall the standard Gagliardo-Nirenberg inequality

k f k

^rL^r

= k| f |k

^rL^r

6 C

r^GN

k f k

²L²

k∇| f |k

^rL⁻²²

, (3.11) where C

_r^GN

stands for the best constant in the “regular” Gagliardo-Nirenberg in- equality. This best constant was first characterized in [38], in terms of a suitable ground state solution of some elliptic equation. In particular, C

₄^GN

is given by

C

₄^GN

= k Q k

⁻L²²

,

where Q is the (only) positive, radially symmetric solution to (1.15) (recall that unlike in [38], there is a

¹₂

factor in front of the Laplacian in (1.15), hence a slightly different formula). As a consequence of the diamagnetic inequality (3.9), we infer

k f k

^rL^r

6 C

_r^GN

k f k

²L²

k∇

^A

f k

^rL⁻²²

.

This shows that (3.10) holds and C

r

6 C

_r^GN

. To see C

_r^GN

6 C

r

, we follow the argument of [11, Proposition 3.1]. Note that the argument given [11] does not hold in two dimensions. Denote A

^σ

(x) := σA(σx) with σ > 0. By (3.10), we have for any smooth compactly supported function f ,

k f k

^rL^r

6 C

r

k∇

A^σ

f k

^r_L⁻²²

k f k

²L²

, ∀ σ > 0.

On the other hand, by the triangle inequality, we have

k∇

A^σ

f k

L²

6 k∇ f k

L²

+ k A

^σ

f k

L²

.

We estimate Z

R²

| A

^σ

(x)f (x) |

²

dx 6 Z

|x|6R

| A

^σ

(x) |

²

dx

! k f k

^L∞

= Z

|x|6Rσ

| A(x) |

²

dx

!

k f k

^L∞

→ 0 as σ → 0, where R > 0 is such that supp(f ) ⊂ { x ∈ R

²

: | x | 6 R } . This shows that

k f k

^rL^r

6 C

r

k∇ f k

^rL⁻²²

k f k

²L²

, hence C

_r^GN

6 C

r

.

Finally we show that the equality in (3.10) cannot be achieved. Assume by contradiction that (3.10) is attained by some function φ. By (3.9) and (3.11), we have

k φ k

^rL^r

= C

r

k∇

A

φ k

^r_L⁻²²

k φ k

²L²

> C

_r^GN

k∇| φ |k

^r_L⁻²²

k φ k

²L²

> k φ k

^rL^r

which implies that

k∇

A

φ k

L²

= k∇| φ |k

L²

.

This together with the following estimate of [28, Theorem 7.21]:

|∇| φ || =

Re

∇ φ φ

| φ |

=

Re

( ∇ − iA)φ φ

| φ |

6 | ( ∇ − iA)φ | imply that

Im

( ∇ − iA)φ φ

| φ |

= 0 ⇐⇒ A = Im ∇ φ

φ

.

In particular, B

12

(x) := ∂

1

A

2

(x) − ∂

2

A

1

(x) = 0, where A(x) = (A

1

(x), A

2

(x)) = γ( − x

2

, x

1

). This is a contradiction since B

12

(x) = 2. The proof is complete.

Note that in view of [23, Lemma 1] (recall again the

¹₂

factor (1.15)),

π 6 k Q k

²L²

6 π ln 2. (3.12) We also have the following useful remark.

Remark 3.4. Let (f

n

)

n>1

be a bounded sequence in H

_A¹

( R

²

). Then up to a sub- sequence, f

n

⇀ φ weakly in H

_A¹

( R

²

), f

n

→ φ a.e. in R

²

and f

n

→ φ strongly in L

^r_loc

( R

²

) for all 2 6 r < ∞ . Moreover,

k∇

^A

f

n

k

²L²

= k∇

^A

φ k

²L²

+ k∇

^A

(f

n

− φ) k

²L²

+ o

n

(1).

We have the following conditional result on the existence and stability of pre- scribed mass standing waves to (1.1).

Proposition 3.5. Let K

3

= V

0

= 0, γ > 0 and Ω = γ. Let 0 < ρ 6 k Q k

²L²

and assume that I

_γ⁰

(ρ) < 0. Then there exists φ ∈ H

_A¹

( R

²

) such that E

_γ⁰

(φ) = I

_γ⁰

(ρ) and k φ k

²L²

= ρ. Moreover, the set

G

γ⁰

(ρ) :=

φ ∈ H

_A¹

( R

²

) : E

_γ⁰

(φ) = I

_γ⁰

(ρ), k φ k

²L²

= ρ

is orbitally stable under the flow of (1.1) in the sense that for any ǫ > 0, there exists δ > 0 such that for any initial data u

0

∈ H

_A¹

( R

²

) satisfying

φ∈G

inf

⁰γ(ρ)

k u

0

− φ k

H_A¹

< δ,

the corresponding solution to (1.1) exists globally in time and satisfies

φ∈G

inf

γ⁰(ρ)

k u(t) − φ k

H_A¹

< ǫ

for all t ∈ R .

Remark 3.6. In the case A ≡ 0 and γ = 0, it was shown in [16] that I(ρ) := I

₀⁰

(ρ) <

0 for any ρ > 0. This is done by using the scaling f

λ

(x) := λf(λx) and taking λ > 0 sufficiently small. In our case, showing I

_γ⁰

(ρ) < 0 is more complicated. The above scaling argument does not work due to the presence of magnetic potential A.

Remark 3.7. The assumption ρ 6 k Q k

²L²

is probably only technical, due to our argument (see after (3.18)).

Before giving the proof of Proposition 3.5, let us recall the following version of concentration-compactness lemma.

Lemma 3.8 (Concentration-compactness lemma [29, 30]). Let (f

n

)

n>1

be a bounded sequence in H

_A¹

( R

²

) satisfying

k f

n

k

²L²

= ρ

for all n > 1 with some fixed constant ρ > 0. Then there exists a subsequence – still denoted by (f

n

)

n>1

– satisfying one of the following three possibilities:

• Vanishing. lim

n→∞

sup

y∈R²

Z

B(y,R)

| f

n

(x) |

²

dx = 0 for all R > 0.

• Dichotomy. There exist a ∈ (0, ρ) and sequences (g

n

)

n>1

, (h

n

)

n>1

bounded in H

_A¹

( R

²

) such that



 

 

 

 

k f

n

− g

n

− h

n

k

L^r

→ 0 as n → ∞ for all 2 6 r < ∞ , k g

n

k

²L²

→ a, k h

n

k

²L²

→ ρ − a as k → ∞ ,

dist(supp(g

n

), supp(h

n

)) → ∞ as n → ∞ ,

lim inf

n→∞

k∇

^A

f

n

k

²L²

− k∇

^A

g

n

k

²L²

− k∇

^A

h

n

k

²L²

> 0.

(3.13)

• Compactness. There exists a sequence (y

n

)

n>1

⊂ R

²

such that for all ǫ > 0, there exists R(ǫ) > 0 such that for all k > 1,

Z

B(yn,R(ǫ))

| f

n

(x) |

²

dx > ρ − ǫ.

The proof of this result is very similar to that of [29, 30], see also [17, Proposi- tion 1.7.6]. The only difference is the last inequality in (3.13) which is proved by using the fact that

k∇

^A

(ϕ

R

f

n

) k

²L²

− k ϕ

R

∇

^A

f

n

k

²L²

6 CM R

⁻¹

,

where ϕ

R

(x) = ϕ(x/R) with a suitable function ϕ ∈ C

^∞

( R

²

) and M := sup

_n>1

k f

n

k

²_H¹

A

. Remark 3.9. • If vanishing occurs, then we infer that f

n

→ 0 strongly in

L

^r

( R

²

) for all 2 < r < ∞ . Indeed, by the magnetic inequality (3.9), we see that | f

n

| is a bounded sequence in H

¹

( R

²

). The result follows by applying [30, Lemma 1.1] to ( | f

n

| )

n>1

.

• If compactness occurs, then (f

n

)

n>1

is relatively compact in L

^r

( R

²

) for all 2 6 r < ∞ up to a translation and change of gauge, i.e. there exists (y

n

)

n>1

⊂ R

²

such that up to subsequence

e

^iA¯n

f

n

( · + y

n

) → φ

strongly in L

^r

( R

²

) for all 2 6 r < ∞ , where ¯ A

n

(x) = A(y

n

) · x and φ ∈ H

_A¹

( R

²

). In fact, denote

f ˜

n

(x) := e

^iA(yn)·x

f

n

(x + y

n

).

We see that

| f ˜

n

(x) | = | f

n

(x + y

n

) | , k∇

A

f ˜

n

k

²L²

= k∇

A

f

n

k

²L²

.

It follows that ( ˜ f

n

)

n>1

is a bounded sequence in H

_A¹

( R

²

) satisfying for all ǫ > 0, there exists R(ǫ) > 0 such that for all n > 1,

Z

B^c(0,R(ǫ))

| f ˜

n

(x) |

²

dx = Z

B^c(0,R(ǫ))

| f

n

(x + y

n

) |

²

dx < ǫ.

By the standard diagonalization argument and the compact embedding H

_A¹

( R

²

) ֒ → L

^r

(B(0, R)) for all R > 0 and all 2 6 r < ∞ , we show that f ˜

n

→ φ strongly in L

²

( R

²

), hence strongly in L

^r

( R

²

) for all 2 6 r < ∞ by Sobolev embedding.

Proof of Proposition 3.5. The proof is divided into three steps.

Step 1. We first show that for each ρ > 0, I

_γ⁰

(ρ) is well-defined, i.e. I

_γ⁰

(ρ) > −∞ . Indeed, recalling (1.14) and arguing as in (3.4), we have directly

E

_γ⁰

(f ) > 1

2 k∇

^A

f k

²L²

−

√ e

2 ρ (3.14)

for any f ∈ H

_A¹

( R

²

) satisfying k f k

²L²

= ρ.

Step 2. We show the existence of minimizers for I

γ⁰

(ρ), under the assumption I

_γ⁰

(ρ) < 0. It is done by using the concentration-compactness argument as in [18].

Let (f

n

)

n>1

be a minimizing sequence for I

_γ⁰

(ρ). By (3.14), we see that (f

n

)

n>1

is a bounded sequence in H

_A¹

( R

²

) satisfying

k f

n

k

²L²

= ρ, ∀ n > 1, E

_γ⁰

(f

n

) → I

_γ⁰

(ρ) as n → ∞ .

By Lemma 3.8, there exists a subsequence still denoted by (f

n

)

n>1

satisfying one of the following three possibilities: vanishing, dichotomy and compactness.

No vanishing. Suppose that vanishing occurs. Passing to a subsequence, we infer that f

n

→ 0 strongly in L

^r

( R

²

) for all 2 < r < ∞ . Thanks to (1.10), we infer that

Z

R²

| f

n

|

⁴

ln | f

n

|

²

√ e

dx −→

n→∞

0, hence

I

_γ⁰

(ρ) = lim

n→∞

E

_γ⁰

(f

n

) = lim

n→∞

k∇

^A

f

n

k

²L²

> 0, which contradicts the assumption I

_γ⁰

(ρ) < 0.

No dichotomy. If dichotomy occurs, then there exist a ∈ (0, ρ) and sequences (g

n

)

n>1

, (h

n

)

n>1

bounded in H

_A¹

( R

²

) such that (3.13) holds. To rule out the dichotomy, we first claim that there exists δ > 0 such that

lim inf

n→∞

k f

n

k

⁴L⁴

> δ > 0. (3.15) Since E

_γ⁰

(f

n

) → I

_γ⁰

(ρ) < 0, we see that for n sufficiently large, E

_γ⁰

(f

n

) 6

^I

0 γ(ρ)

2

. It follows that

I

_γ⁰

(ρ)

2 > E

_γ⁰

(f

n

) = 1

2 k∇

A

f

n

k

²L²

+ 1 2

Z

|fn|²>√e

| f

n

|

⁴

ln | f

n

|

²

√ e

dx

− 1 2

Z

|fn|²<√

e

| f

n

|

⁴

ln √ e

| f

n

|

²

dx which implies, since 0 < ln z . √ z for z > 1,

k f

n

k

³L³

&

Z

|fn|²<√e

| f

n

|

⁴

ln √ e

| f

n

|

²

dx > − I

_γ⁰

(ρ) 2 > 0,

for n sufficiently large. The claim (3.15) follows from the H¨ older inequality k f k

³L³

6

k f k

²L⁴

k f k

L²

and k f

n

k

²L²

= ρ.