Gross-Pitaevskii方程之行波解

國立臺灣大學理學院數學系 碩士論文

Department of Mathematics College of Science National Taiwan University

Master Thesis Dissertation

Gross-Pitaevskii方程 之行波解

Traveling Waves for the Gross-Pitaevskii Equation

葉冠廷 Kuan-Ting Yeh

指導教授： 陳俊全 教授 Advisor: Prof. Chiun-Chuan Chen

中華民國 104 年 6 月

June 2015

致謝

沒想到這麼快就畢業了，兩年真的很快就過去了。雖然一路上蠻顛簸的，但還

是順利的畢業了，感謝一路上陪伴我的人們。

首先，我要感謝台大的老師們。感謝劉豐哲老師，讓我在實分析中看到了數 學家應有的態度及風範。 感謝王振男、余正道老師給我機會當分析導論優助教，

讓我在複習高微之餘，也藉機練習穩健的台風。另外，特別感謝兩位口委：夏俊 雄、林太家老師，在我口試中給了我很多建議以及改進的空間。還有最重要的就

是我的指導教授陳俊全老師，老師除了指導我寫論文、也教導我做學問的態度，

當我多次碰到瓶頸時，給了突破的方向，謝謝老師。

然後，感謝同一師門的簡鴻宇、洪立昌、黃志強師兄、林俊良師叔，提供我

論文思考的方向，彼此討論讓我成長了不少。同研究室的有慶、梓彥、幼鈞、居

謙boy、豐哥，感謝他們跟我切磋討論數學，讓我了解到不同層面的思考方式。也

感謝大學的同學們，凱衛、延安、瑋柏、宇翔，提供我各種建議。感謝其他研究

室的邦哥、小胡、揚智兄，經常一起吃飯和討論。另外，特別感謝天文所的之藩 兄，經常分享數學與物理上的不同，也分享工作上遇到的事情。最後，謝謝同窗 六年的好友正偉以及經研所的方方土，雖然是不同系所但經常給我鼓勵與協助、

傾聽與分享日常間的大小事，伴我走過許多困難。

最後，我要感謝我的家人，幫我打理好家裡的事情、還有常跟我講電話散散 心，讓我一個人可以安心地在台北唸書。

要謝的人實在太多，怎麼也寫不完，那就感謝天吧，我會永遠記得這份感動，

去面對嶄新的未來。

數學所 碩二 葉冠廷 6/11/2015

中文摘要

本文的目標是探討Fabrice Bethuel，Philippe Gravejat，和 Jean-Claude Saut 在Gross-

Pitaevskii方程式中關於二，三維度的行波解

c∂1u + ∆u + u(1 − |u|²) = 0

前四章節我們透過最小化能量在動量固定下來探討解之存在性，並提供一些構造

這些定理的動機。

最後一章節我們討論Gross-Pitaevskii equation未來的研究方向。

Abstract

In this thesis, Fabrice Bethuel, Philippe Gravejat, and Jean-Claude Saut discussed the existence of travelling wave solutions to the Gross-Pitaevskii equation in R^N, where N = 2, 3.

c∂₁u + ∆u + u(1 − |u|²) = 0

In the first four sections, we survey the theorems based on minimizing energy under momentum constraint. Also, we give some motivations about how the theorems are constructed.

In the final section, we discuss the future works of Gross-Pitaevskii equation.

目 目

目錄 錄 錄 Contents

致

致致謝謝謝 i

中 中

中文文文摘摘摘要要要 ii

英 英

英文文文摘摘摘要要要 iii

1 Introduction 1

1.1 Background . . . 1

1.2 Statement of the main results . . . 3

1.3 Starting point of the proofs . . . 5

2 Preliminaries 12 2.1 Finite energy solutions for (TWc) . . . 12

2.2 Alternate definitions of the momentum . . . 17

2.3 Decay properties for (TWc) . . . 19

2.4 Pohozaev’s type identities . . . 23

2.5 Solutions without vortices . . . 27

2.6 Subsonic vortexless solutions . . . 30

2.7 Estimates of Fourier transform . . . 34

3 Properties for the function E_min(p) 39 3.1 Proof of Theorem 3 : The Lipschitz condition for Emin(p) . . . 39

3.2 Proof of Lemma 1 : Control the speed c(u_p) . . . 49

3.3 Proof of Lemma 2 : The property of affine energy Emin(p) . . . 50

3.4 An upper bound for E_min(p) and speed c(uⁿ_p) . . . 53

4 Proofs for the main results 54 4.1 Proof of Proposition 1 : The existence of a minimizer on T^Nn . . . 54

4.2 Proof of Proposition 2 : The existence of a finite energy solution . . . 58

4.3 Proof of Proposition 3 : The concentration-compactness principle . . 61

4.4 Proof of Theorem 4 : The existence of E_min(p) in W (R^N) . . . 65

4.5 Proof of Main Theorem 1 : The existence results of (TWc) for N = 2 65 4.6 Proof of Main Theorem 2 : The existence results of (TWc) for N = 3 66

5 Future Study on Gross-Pitaevskii equation 70

參 參

參考考考書書書目目目 73

1 Introduction

1.1 Background

First, take a look at the Gross-Pitaevskii equation

i∂_tΨ = ∆Ψ + Ψ(1 − |Ψ|²) on x ∈ R^N, t ∈ R. (GP) From physics, we know that it is a nonlinear Schr¨odinger equation.

In classical quantum physics, the Schr¨odinger equation has its own Energy and Momentum. So, in this paper, we also want to define the Energy and Momentum for the Gross-Pitaevskii equation.

It is natural to think that the Energy part comes from the variation of the PDE.

But the question is, “ What is the momentum ? ”.

In order to answer this question, we need to focus on the classical Schr¨odinger equation

i~∂Ψ

∂t = −~² 2m∆Ψ.

Now for the Schr¨odinger equation, we want to find it’s Momentum. We could formally compute it by the following strategy.

(a) First, we notice that the Lagrangian of Schr¨odinger.

L(∇xΨ, Ψt) = − ~² 2m

Z

R^N

|∇xΨ|²dx + Z

R^N

Ψ(i~Ψ^t)dx.

(b) Second, we find the Legendre transform of L with respect to Ψ_t= Φ.

L^∗(∇_xΨ, Λ) = sup

Φ

{hΛ, Φi − L(∇_xΨ, Φ)}, where hΛ, Φi = ReR

R^N ΛΦdx.

(c) Third, we find it’s Hamiltonian.

H(∇_xΨ, Λ) = L^∗(∇_xΨ, Λ) = ~² 2m

Z

R^N

|∇_xΨ|²dx L(∇xΨ, Ψt) = −H(∇xΨ, Ψt) − hΨ, −i~∂^tΨi.

(d) Replace _∂t^∂ = −c · ∇, where c is the wave speed.

L(∇_xΨ, Ψ_t) = −H(∇_xΨ, Ψ_t) + c · hΨ, p(Ψ)i,

where p(Ψ) = −i~∇Ψ is the classical momentum in Schr¨odinger equation.

Since we always hope that the solution is a “ T ravelling wave solution ”. That is, we hope that the solution has the form : Ψ(x, t) = eΨ(x − ct).

Just like wave equation, travelling wave Ψ(x, t) = eΨ(x − ct) must be a solution of

∂²Ψ

∂t² = c²∂²Ψ

∂x².

But, the traveling wave is NOT always a solution for Schr¨odinger equation, so we need to add more assumptions.

∂

∂tΨ(x, t) = −c · ∇ eΨ(x − ct) = −c · ∇Ψ(x, t).

In other words, we will assume : _∂t^∂ = −c · ∇.

This is exactly the analogue of the classical equation : L(q, ˙q) = −H(p, q) + p · ˙q, where q means the position and p means the momentum.

Thus, this suggests to us the way to define “ General M omentum Operator ”.

Now for Gross-Pitaevskii equation, we want to find it’s Momentum. We could formally compute it by the similar strategy.

(a) First, we recall the Lagrangian of Gross-Pitaevskii.

L(Ψ, ∇_xΨ, Ψ_t) = −[ 1 2

Z

R^N

|∇Ψ|²dx + 1 4

Z

R^N

(1 − |Ψ|²)²dx ]

− 1 2Re

Z

R^N

iΨ_t(1 − Ψ)dx.

(b) Second, we find the Legendre transform of L with respect to Ψ_t= Φ.

L^∗(Ψ, ∇_xΨ, Λ) = sup

Φ

{hΛ, Φi − L(Ψ, ∇_xΨ, Φ)}, where hΛ, Φi = ReR

R^N ΛΦdx.

(c) Third, we find it’s Hamiltonian.

H(Ψ, ∇_xΨ, Λ) = L^∗(Ψ, ∇_xΨ, Λ) = 1 2

Z

R^N

|∇Ψ|²dx +1 4

Z

R^N

(1 − |Ψ|²)²dx

L(∇_x, Ψ_t) = −H(∇_xΨ, Ψ_t) −1 2

Z

R^N

hi∂_tΨ, Ψ − 1idx, where hi∂_tΨ, Ψ − 1i ≡ Re(i∂_tΨ(Ψ − 1)).

(d) Replace _∂t^∂ = −c · ∇ ,where c is the wave speed.

L(∇_xΨ, Ψ_t) = −H(∇_xΨ, Ψ_t) + c · 1 2

Z

R^N

hi∇Ψ, Ψ − 1idx.

Thus, the momentum operator for Gross-Pitaevskii equation is : P (Ψ) = 1

2 Z

R^N

hi∇Ψ, Ψ − 1idx.

The Gross-Pitaevskii equation appears in several models in various areas of physics :

non-linear optics, fluid mechanics, and Bose-Einstein condensation...(see for instance [5,25,30,31,32,38]).On a formal level, the Gross-Pitaevskii equation is Hamiltonian.

Also, the conserved Hamiltonian is a Ginzburg-Landau energy, namely (in previous observations)

E(Ψ) = 1 2

Z

R^N

|∇Ψ|²dx +1 4

Z

R^N

(1 − |Ψ|²)²dx.

1.2 Statement of the main results

We investigate the existence of travelling waves to the Gross-Pitaevskii equation i∂_tΨ = ∆Ψ + Ψ(1 − |Ψ|²) on x ∈ R^N, t ∈ R, (GP) where Ψ(x, t) : R^N × R −→ C is an unknown function, x is the space variable and t is the time variable. N = 2, 3, and ∆ is the standard Laplace operator. We are interested in the solution of the form Ψ(x, t) = u(x₁− ct, x₂, ..., x_N), called the travelling wave solution. Here u is called the profile and c is called the wave speed.

The equation of profile u is given by

ic∂₁u + ∆u + u(1 − |u|²) = 0. (TWc) Here, the parameter c ∈ R corresponding to the speed of the travelling waves, may be restricted to the case c ≥ 0, since we may consider the complex conjugation.

E(u) = 1 2

Z

R^N

|∇u|²dx +1 4

Z

R^N

(1 − |u|²)²dx = Z

R^N

e(u). (Energy)

Since our travelling wave speed is only considered in the first dimension, so the momentum P in Gross-Pitaevskii equation in the first dimension will be denoted as p, which is hence a scalar

p(u) = 1 2

Z

R^N

hi∂₁u, u − 1idx, (Momentum) where the notation h , i stands for the canonical scalar product of the complex plane C identified to R², that is

hz₁, z₂i ≡ Re(z₁)Re(z₂) + Im(z₁)Im(z₂) = Re(¯z₁z₂).

In order to make the previous definition well-defined, we introduce the space

V (R^N) = {u : R^N → C|∇u ∈ L²(R^N), Re(u) ∈ L²(R^N), Im(u) ∈ L⁴(R^N), ∇Re(u) ∈ L⁴³(R^N)}

W (R^N) = 1 + V (R^N).

We check : hi∂₁v, v − 1i is integrable for any v ∈ W (R^N), so p(v) is well-defined.

First observe that :

hi∂₁v, v − 1i = ∂₁(Re(v))Im(v) − ∂₁(Im(v))(Re(v) − 1).

Let u = v − 1 ∈ V (R^N) and apply Holder’s inequality Z

R^N

|hi∂₁v, v − 1i|dx ≤ Z

R^N

|∂₁Re(v)||Im(v)| + |∂₁Im(v)||Re(v) − 1|

≤ Z

R^N

|∇Re(v)||Im(v)| + |∇v||Re(v) − 1|

= Z

R^N

|∇Re(u)||Im(u)| + |∇u||Re(u)|.

The following two main theorem are from B´ethuel [4].

Theorem 1. For N = 2, let p > 0. There exists a non-constant finite energy solution u_p ∈ W (R^N) to equation (TWc) , with c = c(u_p) s.t.

p(u_p) = 1 2

Z

R^N

hi∂₁u_p, u_p− 1idx = p, and such up is the solution of the minimization problem

E(u_p) = E_min(p) = inf{E(u)|u ∈ W (R^N), p(u) = p}.

Theorem 2. For N = 3, there exists some constant p0 > 0 such that :

For p ≥ p₀, there exists a non-constant finite energy solution u_p ∈ W (R^N) to equa- tion (TWc), with c = c(up) s.t.

p(u_p) = p, E(u_p0) = E_min(p₀) =√ 2p₀, and, for p > p₀, we have

E(u_p) = E_min(p) <√ 2p.

Moreover, for 0 < p < p₀,

Emin(p) = inf{E(u)|u ∈ W (R^N), p(u) = p} =√ 2p, and the infimum is not achieved in W (R³).

1.3 Starting point of the proofs

The starting point of the proofs was due to the analysis of the curve p 7→ Emin(p).

First, we do some linearization of E_min(p) by using Taylor formula. For p > ˆp> 0, Emin(p) ' Emin(ˆp) + d⁺

dpEmin(ˆp)(p − ˆp).

Assume E_min is achieved by some map uˆp, by Euler-Lagrange equation and scaling on ψ₁, we have

c = cDp(uˆp)(ψ₁) = DE(uˆp)(ψ₁).

Consider the curve γ : R 7→ W (R^N) defined by γ(t) = uˆp+ tψ₁, since the functions E and p are smooth on W (R^N), using Taylor formula, we have

p(γ(t)) = ˆp+ s, where s = t + p(ψ₁)t², E(γ(t)) = E(uˆp) + DE(uˆp)(tψ1) + O

t−→0(t²) = Emin(ˆp) + ct + O

t−→0(t²).

Since p(γ(t)) = ˆp+ s =⇒ E_min(ˆp+ s) ≤ E(γ(t)), so we obtain E_min(ˆp+ s) − E_min(ˆp) ≤ E(γ(t)) − E_min(ˆp) ≤ cs + O

s−→0(s²).

Taking s −→ 0⁺,

d⁺

dp E_min(ˆp)
≤ c,

also by Gravejat’s paper [19], there is no travelling wave solution existing for c >√ 2, so we may hope for c ≤√

2 (This is exactly the idea of the proof in Lemma 1).

Taking ˆp−→ 0⁺, we have the following approximate inequality E_min(p) .√

2p. (1.1)

This inequality corresponds in some sense to a linearization of the equation.

We performed some observation to obtain some feeling for its proof, first, considered a map v ∈ {1} + C_c^∞(R^N) such that

δ = inf

x∈R^N|v(x)| ≥ 1 2,

so that we may write v = ρ exp iϕ. To obtain (1.1), we need to construct v so that E(v) '√

2|p(v)|, thus Emin(p) .√ 2p.

From the formula (2.2) for another type of momentum and δ ≤ ρ, we have

|p(v)| =   1 2

Z

R^N

(1 − ρ²)∂₁ϕ  ≤ 1

2δ Z

R^N

1 − ρ²     ρ∂₁ϕ

 . From the inequality ab ≤ ¹₂(a² + b²), we set a = √⁴

2|ρ∂1ϕ|, b = ^|1−ρ√4 ^2|

2 , and also viewed the formula (2.1) for another type of energy

|p(v)| ≤ 1

√2δ

 1 2

Z

R^N

ρ²|∇ϕ|²+ 1 4

Z

R^N

 1 − ρ²

2

≤ 1

√2δE(v)

i.e. √

2δ|p(v)| ≤ E(v).

In order to obtain a map, such that E(v) '√

2|p(v)|, we will need to make δ close to 1, and the inequality ab ≤ ¹₂(a²+ b²) close to an equality, that is a ' b or in other words,

∂₁ϕ ' 1 − ρ²

√2 . This idea introduced the following (see Lemma 3.4).

Let s > 0, there exists (γ_n)_n∈N in {1} + C_c^∞(R^N) s.t. p(γ_n) = s and E(γ_n) −→√ 2s .

From the density of V (R^N), we may suppose that Emin has the following form E_min(p) = inf{E(1 + v)|v ∈ C_c^∞(R^N), p(1 + v) = p},

this lead us to the Lemma 3.2. Combining all observations above, we can proof Theorem 3.

Theorem 3. Let N = 2, 3. For any p, q ≥ 0 we have :

|E_min(p) − E_min(q)| ≤ √

2|p − q|,

that is, the real-valued function p 7−→ Emin(p) is Lipschitz and thus is concave, increasing on R+ .

Set Ξ(p) = √

2p − Emin(p), then the function p 7−→ Ξ(p) is non-negative, convex, increasing on R+, tending to +∞ as p → +∞.

In particular, there exists p0 ≥ 0 such that Ξ(p) = 0, if p ≤ p0, and Ξ(p) > 0, as otherwise .

Let us define Σ(u) = √

2p(u) − E(u) for u ∈ W (R^N), then Ξ(p) = sup{Σ(u)|u ∈ W (R^N), p(u) = p}.

An important consequence of the concavity to the function E_min(p), is the fol- lowing inequality.

Corollary 1. The function E_min is subadditive,

that is, for any non-negative numbers p₁, . . . , p_`, we have the inequality,

`

X

i=1

Emin(pi) ≥ Emin

X^`

i=1

pi

 .

Moreover, if ` ≥ 2 and the previous equation is an equality, then the function E_min, will be linear on (0, p), where p ≡

`

P

i=1

p_i.

Proof. Notice E_min(0) = 0 and E_min is concave, so its graph lies above the line segment joining (0, 0) and (p, E_min(p)).

In particular, for any 0 ≤ q ≤ p, we have E_min(q) − E_min(0)

q− 0 ≥ E_min(p) − E_min(0)

p− 0 =⇒ E_min(q) ≥ qE_min(p) p .

For any 0 ≤ pi ≤ p,

E_min(p_i) ≥ p_iE_min(p)

p =⇒

`

X

i=1

E_min(p_i) ≥

`

X

i=1

p_iE_min(p)

p = E_min(p)

Now, if it is an equality, then E_min(p_i) = p_i^Emin_p^(p) and the graph needs to be linear, that is, the function E_min will be linear on (0, p).

Since the function E_min is Lipschitz, non-decreasing and concave, its left and right derivatives exist for any p ≥ 0 and will be equal on R+, except a countable subset Q (Lipschitz =⇒ differentiable almost everywhere), are non-negative and non-increasing and will satisfy the inequality,

0 ≤ d⁺

dp E_min(p)
≤ d⁻

dp E_min(p)
≤√ 2, where we let

d^±

dp E_min(p)
≡ lim

∆p−→0⁺

E_min(p ± ∆p) − E_min(p)

±∆p .

Lemma 1 (Control the speed c(u_p)). Let p > 0 and assume that E_min(p) is achieved by a solution u_p of (TWc) of speed c(u_p). Then we have,

d⁺

dp E_min(p)
≤ c(u_p) ≤ d⁻

dp E_min(p)
.

This is the main control for speed c, the derivatives are related to the speed c(u_p). Also, using the Lipschitz constant we have a bound for speed c(u_p),

0 ≤ c(up) ≤√ 2.

Lemma 2 (The property of affine energy E_min(p)). Let 0 ≤ p₁ < p₂ and assume the function Emin is affine on (p1, p2). Then, for any p1 < p < p2, the infimum Emin(p) will not be achieved in W (R^N).

Lemma 3. Let v ∈ W (R^N) and assume p(v) > 0. Then we have, inf

x∈R^N

|v(x)| ≤ maxn1

2, 1 − Σ(v)

√2p(v) o

.

Proof. Now define δ as previous,

δ ≡ inf

x∈R^N

|v(x)|.

If δ ≤ ¹₂, the result holds. Otherwise, |v| ≥ δ > ¹₂, v has a lifting, i.e. we may write v = ρ exp iϕ, and recall in previous, we have

|p(v)| ≤ 1

√2δ

 1 2

Z

R^N

ρ²|∇ϕ|²+ 1 4

Z

R^N



1 − ρ²2

≤ 1

√2δE(v)

√

2δp(v) ≤√

2δ|p(v)| ≤ E(v) = √

2p(v) − Σ(v), and hence,

1 − δ ≥ Σ(v)

√2p(v) =⇒ δ ≤ 1 − Σ(v)

√2p(v), that is,

inf

x∈R^N

|v(x)| = δ ≤ maxn1

2, 1 − Σ(v)

√2p(v) o

.

Lemma 3 is the main tool to make the finite energy solutions of (TWc) into nontrivial solutions .

Previously, we haven’t talked about the existence of the solution of (TWc). In order to overcome the difficulties of finding minimizing sequences in whole space R^N, there are several ways to proceed. In this thesis, we consider the corresponding minimization problem on expanding tori. This choice has several advantages.

First, the torus is compact, so that, the existence of minimizers, have no difficulty (see Proposition 1).

Second, it has no boundary, so that, the elliptic theory, is essentially the local one and concentration near the boundary is avoided. The torus also captures some of the translation invariance for the problem on R^N.

Finally, the Pohozaev’s identities will give bounds for the Lagrange multipliers, which provide a uniform control on the ellipticity of (TWc). Our strategy, to obtain compactness for the sequence of approximate minimizers, is then to develop the elliptic theory for the equation on tori, derive several estimates which do not rely on

the size of the torus and then to pass to the limit, when the size of the torus tends to infinity (see Proposition 1,2).

More precisely, we introduce the flat torus, for N = 2 and N = 3, defined by

T^Nn ' Ω^N_n ≡ [−πn, πn]^N.

Now, introduce the Energy and Momentum on flat torus : T^Nn, we define the energy E_n and p_n on X_n^N = H¹(T^Nn, C) as follow

E_n(u) = 1 2

Z

T^Nn

|∇u|²dx +1 4

Z

T^Nn

(1 − |u|²)²dx, as well,

p_n(u) = 1 2

Z

T^Nn

hi∂₁u, uidx,

which defines a quadratic functional on X_n^N, and the discrepancy term Σ_n(v) = √

2p_n(v) − E_n(v).

We introduce the set Γ^N_n(p), defined in dimension three by Γ³_n(p) ≡ {u ∈ X_n³, pn(u) = p}, but in dimension two, its definition is a little different and is given by

Γ²_n(p) ≡ {u ∈ X_n², pn(u) = p} ∩ S_n⁰.

The set S_n⁰, corresponds to a topological sector of the energy E_n, following the approach of Almeida [1]. Let us just mention that we introduce the set S_n⁰, to have appropriate lifting properties far from the possibly vorticity set. For more detail, see B´ethuel [4, Section 4].

Similarly, we consider the minimization problem on torus,

E_minⁿ (p) = inf{E_n(u)|u ∈ Γ^N_n(p)}. (P_n^N(p)) Proposition 1. Assume N = 2, 3, and n ≥ en(p), where en(p) is some integer depending on E_min(p) then there exists a minimizer uⁿ_p ∈ Γ^N_n(p) for E_minⁿ (p) and some constant cⁿ_p ∈ R, such that uⁿp satisfies (TWcⁿ_p) i.e.

icⁿ_p∂₁uⁿ_p + ∆uⁿ_p + uⁿ_p(1 − |uⁿ_p|²) = 0 on T^Nn.

In particular, uⁿ_p is smooth.

Moreover, if Ξ(p) > 0, then there exists a constant K(p) and an integer n(p) s.t.

|cⁿ_p| ≤ K(p), for any n ≥ n(p).

In particular, for any k ∈ N, there exists some constant Kk(p), such that kuⁿ_pk_Ck(T^Nn)≤ K_k(p).

Proposition 2. For N = 2, 3, p > 0, and assume Ξ(p) > 0,

then there exists a non-trivial finite energy solution u_p to (TWc), such that ,by passing to a subsequence, we have

uⁿ_p −→ u_p in C^k(K) as n → +∞,

for any k ∈ N and compact K in R^N. Moreover, we have E(u_p) ≤ E_min(p) and

|u_p(0)| ≤ sup{1

2, 1 − Ξ(p)

√2p} < 1.

Proposition 3. For N = 2, 3, p > 0 and assume Ξ(p) > 0. Let u_p and uⁿ_p be in proposition 2. Then there exists ` finite energy solutions u1 = u<sub>p</sub>, u2, u3, ..., u` to (TWc), such that

Emin(p) =

`

X

i=1

E(ui) p=

`

X

i=1

p(ui).

Also, u_i are minimizers of E_min(p_i), where p_i = p(u_i) and 0 < c(u_p) <√ 2.

Moreover,

lim sup

n−→∞

E_n(uⁿ_p) = lim sup

n−→∞

E_minⁿ (p) = E_min(p).

Theorem 4. Assume N = 2, 3, if p > p₀, where p₀ ≥ 0 is defined in Theorem 3.

Then E_min(p) is achieved by the map, u_p ∈ W (R^N), constructed in Proposition 2.

2 Preliminaries

2.1 Finite energy solutions for (TWc)

Lemma 2.1. Let n ∈ N and let v be a finite energy solution to (TWc) on R^N. There exist some constants K(N ) and K(c,k,N) s.t.

k1 − |v|k_L^∞_((R^N₎ ≤ max{1, c 2} k∇vk_L^∞_(R^N₎ ≤ K(N )(1 + c²

4)³², and more generally,

kvk_C^k_(R^N₎ ≤ K(c, k, N ), ∀k ∈ N.

Proof. In paper [3,13,20,40], a finite energy solution v to (TWc) is a smooth bounded function on R^N, such that,

|v| −→ 1, as |x| −→ ∞,

thus, kvk_L^∞_(R^N₎ ≥ 1, (otherwise, for any x ∈ R^N, |v(x)| ≤ kvk_L^∞_(R^N₎ < 1).

We compute for ∆|v|²,

∆|v|² = 2hv, ∆vi + 2|∇v|²

= 2|∇v|²− 2chi∂₁v, ∆vi − 2|v|²(1 − |v|²)

≥ 2|∇v|²− 2|∂₁v|²− c²

2|v|²− 2|v|²(1 − |v|²).

Since |2chi∂₁v, vi| ≤ 2c|∂₁v||v| ≤ ^(2|∂₂^1v|)2 +^(c|v|₂²⁾, by Cauchy and Young’s inequality, we have,

∆|v|² + 2|v|²(1 + c²

4 − |v|²) ≥ 2(|∇v|²− |∂₁v|²) ≥ 0.

Assume kvk_L^∞_(R^N₎ > 1, let u = |v|² and kuk_L^∞_(R^N₎ > 1.

Using maximum principle argument :

For  = ^{kukL∞(RN )}₂ ⁻¹ > 0, there exists M > 0, such that, u(x) < 1 +  for all |x| > M . Thus,

sup

|x|>M

|u| ≤ 1 +  < kuk_L^∞_(R^N₎ =⇒ kuk_L^∞_(R^N₎ = sup

|x|≤M

|u|.

Say sup

|x|≤M

|u| = |u(x0)| = u(x0) and we claim that :

u = |v|² ≤ (1 + c² 4).

If not, ∃z, u(z) > 1 +^c₄² =⇒ u(x₀) = kuk_L^∞_(R^N₎ ≥ |u(z)| > 1 +^c₄², by

∆u + 2u(1 + c²

4 − u) ≥ 0 =⇒ ∆u(x₀) > 0, contradicts to the maximum.

(The idea comes from : if ∃z, u(z) > 1 +^c₄² =⇒ kuk_L^∞_(R^N₎ > 1 + ^c₄² and |u| −→

1, as |x| −→ ∞, so |u| must have max in some ball ¯B_M(0).)

Assume kvk_L^∞_(R^N₎ = 1, the above inequality also holds, so it holds in any case.

In particular,

k1 − |v|k_L^∞_(RN) ≤ max{1, ( r

1 + c²

4 − 1)} ≤ max{1,c 2}.

Now consider, w(x) = v(x) exp (i₂^cx₁) by (TWc) w satisfies,

∆w + w(1 + c²

4 − |w|²) = 0.

Combine with previous

|∆w| ≤ |w||(1 +c²

4) − |w|²| ≤ |w|((1 +c²

4) + |w|²) ≤ 2(1 + c² 4)³², k∆wk_L^∞_(B(x0,1)) ≤ 2(1 +c²

4)³².

By standard elliptic theory in Gilbarg Trudinger [29], there exists constant K(N ), such that

|∇w(x₀)| ≤ K(N )(k∆wk_L^∞_(B(x0,1))+ kwk_L^∞_(B(x0,1))).

Moreover,

|∇w(x₀)| ≤ 2K(N )(1 + c² 4)³². Also, by definition of w,

|∇v(x₀)| ≤ |∇w(x₀)| + c

2|v(x₀)| ≤ (2K(N ) + 1)(1 + c² 4)³².

Since the estimate holds for any x0 ∈ R^N, so we have the following;

k∇vk_L^∞_(R^N₎ ≤ K(N )(1 +c² 4)³² k∇wk_L^∞_(R^N₎ ≤ K(N )(1 + c²

4)³².

Finally, by standard estimates for elliptic equation in Gilbarg Trudinger [29] with bootstrap argument, we obtain,

kvk_C^k_(R^N₎ ≤ K(c, k, N ) for any k ∈ N.

Lemma 2.2. Let r > 0 and let v be a finite energy solution to (TWc) on R^N. There exist some constants K(N ) s.t. for any x0 ∈ R^N,

k1 − |v|k_L^∞_(B(x0,^r

2)) ≤ max{K(N )(1 +c²

4)²E(v, B(x₀, r))^{N +21} ,K(N )

r^N/2 E(v, B(x₀, r))¹²}, where E(v, B(x₀, r)) =R

B(x0,r)e(v).

Proof. Let η = 1 − |v|², by Lemma 2.1, the function η is smooth on R^N and satisfies k∇ηk_L^∞_(R^N₎ ≤ 2kv∇vk_L^∞_(R^N₎≤ K(N )(1 + c²

2)². Let ¯x ∈ ¯B(x0,^r₂), such that the sup is attached,

|η(¯x)| = sup

y∈ ¯B(x0,^r₂)

|η(y)|,

we have,

|η(¯x)| − |η(y)| ≤ |η(y) − η(¯x)| ≤ |∇η(ζ)||y − ¯x| ≤ K(N )(1 + c²

2)²|y − ¯x|.

Let µ = ^|η(¯x)|

2K(N )(1+^c2₂)².

For any y ∈ B(¯x, µ), |y − ¯x| < µ =⇒ |η(¯x)| − |η(y)| ≤ K(N )(1 + ^c₂²)²µ ≤ ^|η(¯₂^x)|, that is,

|η(y)| ≥ |η(¯x)|

2 E(v, B(x0, r)) = 1

2 Z

B(x0,r)

|∇u|²dx + 1 4

Z

B(x0,r)

(1 − |u|²)²dx.

Let l = min(µ,^r₂), 1

4 Z

B(x0,r)

(1−|u|²)²dx ≥ 1 4

Z

B(¯x,l)

(η(y))²dy ≥ 1 4

Z

B(¯x,l)

(|η(¯x)|

2 )²dy ≥ 1

16η(¯x)²|B(¯x, l)|, the first inequality follows on by B(¯x, l) ⊂ B(x₀, r), also see Figure 1.

By definition of l = min(µ,^r₂) and the measure of ball, 1

16η(¯x)²|B(¯x, l)| = min{ |η(¯x)|^{N +2}|B(x₀, 1)|

2^{N +4}K(N )^N(1 + ^c₄²)^2N,|η(¯x)|²r^N|B(x₀, 1)|

2^{N +4} }, also, notice that, (1 − |v|)(1 + |v|) = η =⇒ |1 − |v|| = _|1+|v||^|η| ≤ |η|,

we obtain,

k1 − |v|kL^∞(B(x0,^r₂)) ≤ kηkL^∞(B(x0,^r₂))= |η(¯x)|.

Finally, we have the estimate, k1 − |v|k_L^∞_(B(x0,^r

2)) ≤ max{K(N )(1 +c²

4)²E(v, B(x₀, r))^{N +21} ,K(N )

r^N/2 E(v, B(x₀, r))¹²}.

Figure 1: Energy estimate

Now, we introduce a new function space,

Λ(R^N) = {v ∈ C⁰(R^N)|E(v) < ∞ and ∃R(v) > 0 s.t. |v(x)| ≥ 1

2, ∀|x| ≥ R(v)}.

Corollary 2.3. Let v be a finite energy solution to (TWc) on R^N. Then v belongs to space Λ(R^N).

Proof. R

R^Ne(v) < ∞ =⇒ lim

R−→∞

R

R^N\B(0,R−1)e(v) = 0 by Lebesgue’s Theorem.

For ¹

(2K(N )(1+^c2₄)²)^{N +2} > 0, there exists R = R(v) > 1, such that Z

R^N\B(0,R−1)

e(v) ≤ 1

(2K(N )(1 + ^c₄²)²)^{N +2}, where K(N ) is the constant in Lemma 2.2.

For any |x0| ≥ R, also see Figure 2 E(v, B(x₀, 1)) =

Z

B(x0,1)

e(v) ≤ Z

R^N\B(0,R−1)

e(v)

Figure 2: Energy estimate

E(v, B(x₀, 1)) ≤ 1

(2K(N )(1 +^c₄²)²)^{N +2} < 1 =⇒ (E(v, B(x₀, 1)))^{N +21} ≥ (E(v, B(x₀, 1)))¹², thus, K(N )(1 + ^c₄²)²(E(v, B(x₀, 1)))^{N +21} is larger than K(N )(E(v, B(x₀, 1)))¹², then by Lemma 2.2,

k1 − |v|k_L∞(B(x0,¹₂))≤ 2K(N )(1 + c²

4)²(E(v, B(x₀, 1)))^{N +21} , so, we have,

1 − |v(x₀)| ≤ K(N )(1 +c²

4)²(E(v, B(x₀, 1)))^{N +21} ≤ 1

2 =⇒ |v(x₀)| ≥ 1 2, that is, v ∈ Λ(R^N).

2.2 Alternate definitions of the momentum

If v ∈ Λ(R^N), we may write, for |x| > R(v) v = ρ exp(iϕ),

where ϕ is a real function on R^N\B(0, R(v)), defined modulo a multiple of 2π.

Also, we have,

∂jv = (iρ∂jϕ+∂jρ) exp(iϕ) =⇒ |∂jv|² = |iρ∂jϕ|²+|∂jρ|²+2Re(iρ∂jϕ)(∂jρ) = |iρ∂jϕ|²+|∂jρ|². Moreover,

hi∂₁v, vi = −ρ²∂₁ϕ and e(v) = 1

2(|∇ρ|²+ ρ²|∇ϕ|²) + 1

4(1 − ρ²)².

Lemma 2.4. Let ρ and ϕ be C¹ scalar functions on a domain U in R^N, such that ρ is positive. Let v = ρ exp(iϕ). Then, we have the pointwise bound

|(ρ²− 1)∂₁ϕ| ≤

√2 ρ e(v).

Proof.

e(v) = 1

2(|∇ρ|²+ ρ²|∇ϕ|²) + 1

4(1 − ρ²)² ≥ 1

2(ρ²|∂₁ϕ|²) + 1 2(1

2(1 − ρ²)²).

Let a = ^√1₂(ρ²− 1), b = ρ∂1ϕ, using Young’s inequality |ab| ≤ ¹₂a²+¹₂b²,

| 1

√2(ρ²− 1)ρ∂₁ϕ| ≤ 1

2(ρ²|∂₁ϕ|²) + 1 2(1

2(1 − ρ²)²) ≤ e(v).

Now, we can have an alternative definition of the momentum on the space Λ(R^N).

We consider the function,

g(v) = hi∂₁v, vi + ∂₁((1 − χ)ϕ),

where v = ρ exp(iϕ) on R^N\B(0, R(v)) and χ is an arbitrary smooth function with compact support, such that χ = 1 on B(0, R(v)) and 0 ≤ χ ≤ 1.

Lemma 2.5. If v belongs to Λ(R^N), then g(v) belongs to L¹(R^N). Moreover, the integral

p(v) ≡e 1 2

Z

R^N

g(v) = 1 2

Z

R^N

(hi∂₁v, vi + ∂₁((1 − χ)ϕ)), for any smooth function χ with compact support.

Proof. v ∈ Λ(R^N) =⇒ ρ = |v| ≥ ¹₂ on R^N\B(0, R(v)). Notice that, g(v) = −ρ²∂1ϕ + ∂1((1 − χ)ϕ) = (1 − ρ²)∂1ϕ on R^N\ supp(χ), so, by Lemma 2.4 :

If x ∈ R^N\ supp(χ) ⊂ R^N\B(0, R(v)) (since B(0, R(v)) ⊂ supp(χ))

|g(v)| = |(1 − ρ²)∂₁ϕ| ≤

√2

ρ e(v) ≤ 2√ 2e(v), thus we can obtain,

Z

R^N\ supp(χ)

|g(v)| ≤ 2√ 2

Z

R^N\ supp(χ)

e(v) ≤ 2√

2E(v) < ∞,

since v is smooth on R^N, so, g(v) is also smooth on compact set in R^N Z

supp(χ)

|g(v)| < ∞,

that is, the function g(v) is integrable on R^N. Now, using integration by parts,

Z

R^N

g(v) = Z

R^N

(hi∂₁v, vi + ∂₁((1 − χ)ϕ))

= Z

R^N

hi∂₁v, vi + ∂₁ϕ − Z

R^N

∂₁(χϕ)

= Z

R^N

hi∂₁v, vi + ∂₁ϕ − Z

supp(χ)

∇ · (χϕ)n₁

= Z

R^N

hi∂₁v, vi + ∂₁ϕ − Z

∂ supp(χ)

(χϕ)n₁· n dS_x

= Z

R^N

hi∂₁v, vi + ∂₁ϕ,

where n₁ = (1, 0, 0) and n is the outer normal vector on ∂(supp(χ)), so, the integral does not depend on the choice of χ.

2.3 Decay properties for (TWc)

Proposition 2.6. Let v be a finite energy solution to (TWc).

(1) There exists a constant v∞, such that |v∞| = 1 and v(x) −→ v∞, as |x| −→ ∞.

Without loss of generality, we may assume v∞= 1.

(2) Assume c(v) < √

2. Then, there exists some constant K > 0, depending only on c(v), E(v) and the dimension N , such that the following estimates hold for any x ∈ R^N,

|Im(v(x))| ≤ K

1 + |x|^{N −1} |Re(v(x)) − 1| ≤ K 1 + |x|^N

|∇Im(v(x))| ≤ K

1 + |x|^N |∇Re(v(x))| ≤ K 1 + |x|^{N +1}. (3) Assume N = 3 and c(v) =√

2. Then, Re(v) − 1 and ∇Im(v) belongs to L^p(R³) for any p > ⁵₃, ∇Re(v) belongs to L^p(R³) for any p > ⁵₄, whereas, Im(v) belongs to L^p(R³) for any p > ¹⁵₄.

Corollary 2.7. Let v be a finite energy solution to (TWc) and assume v∞= 1.

Then v belongs to W (R^N).

Remark 2.8. Since any finite energy solution v to (TWc) has a limit v∞at infinity, we may write v = ρ exp(iϕ) outside some ball B(0, R), for some R > 0, where ϕ is smooth function on R^N\B(0, R), which is defined up to an integer multiple of 2π.

Moreover, the function ϕ has a limit at infinity ϕ∞, which we may take as equal to 0, if we assume that v∞ = 1. The statements given in [18,20,21], are actually expressed in terms of the real function ρ and ϕ, as follow,

(1) Assume 0 ≤ c(v) < √

2. Then, there exists some constant K > 0, depending only on c(v), E(v) and the dimension N , such that the following estimates hold for any x ∈ R^N,

|ϕ(x)| ≤ K

1 + |x|^{N −1} |1 − ρ(x)| ≤ K 1 + |x|^N

|∇ϕ(x)| ≤ K

1 + |x|^N |∇ρ(x)| ≤ K

1 + |x|^{N +1}. (2) If c(v) =√

2. Then, the function ϕ belongs to L^p(R³\B(0, R)) for any p > ⁵₃, whereas, ∇ρ belongs to L^p(R³\B(0, R)) for any p > ⁵₄.

Proof. since v = ρ cos ϕ + iρ sin ϕ, so Re(v) − 1 = ρ cos ϕ − 1, Im(v) = ρ sin ϕ

|Re(v) − 1| ≤ K(|ρ − 1| + ϕ²) |Im(v)| ≤ K|ϕ|

|∇Re(v)| ≤ K(|∇ρ| + |ρ||∇ϕ|) |∇Im(v)| ≤ K(|∇ϕ| + |ϕ||∇ρ|) where K > 0 is a constant.

Proposition 2.9. Let v be a finite energy solution to (TWc) on R^N. Then, we have

ep(v) = p(v).

Proof. Since v is a finite energy solution, so v ∈ Λ(R^N), that is, exists R(v) > 0 s.t.

|v| ≥ 1

2 on R^N\B(0, R(v)).

By Remark 2.8, without loss of generality, we may assume, v∞ = 1 and ϕ∞= 0.

Observe that,

Re(v) + iIm(v) = v = ρ exp iϕ = ρ(cos ϕ + i sin ϕ) =⇒ sin ϕ = Im(v(x)) ρ(x) .

Using Taylor expansion of sin, we have, if x is sufficiently large

|Im(v(x))

ρ(x) − ϕ(x)| = | sin ϕ − ϕ(x)| ≤ |ϕ(x)|³ 3! , we want to estimate, |p(v) −ep(v)| = ¹₂|R

R^N(hi∂₁v, v − 1i − g(v))|.

For R > R(v) is sufficiently large. Using integration by parts, Z

B(0,R)

hi∂₁v, 1i = Z

B(0,R)

Re(i∂₁v) = Z

B(0,R)

−Im(∂₁v)

= − Z

B(0,R)

∇ · (Im(∂₁v), 0, 0.., 0)

= − Z

∂B(0,R)

(Im(∂₁v), 0, 0.., 0) · ndS_x

= − Z

∂B(0,R)

Im(∂₁v)x₁ RdS_x, where, the outer unit normal n = (^x_R¹,^x_R², ...).

Similarly,

Z

B(0,R)

∂1((1 − χ)ϕ) = Z

∂B(0,R)

ϕ(x)x1

RdSx, so, we have,

Z

B(0,R)

(hi∂₁v, v − 1i − g(v)) = 1 R

Z

∂B(0,R)

(Im(v(x)) − ϕ(x))x₁dS_x.

Let Im(v(x)) − ϕ(x) = (^Im(v(x))_ρ − ϕ(x)) + Im(v)^ρ−1_ρ , so that,

| Z

B(0,R)

(hi∂₁v, v − 1i − g(v))| = 1 R|

Z

∂B(0,R)

(Im(v(x)) − ϕ(x))x₁dS_x|

= 1 R|

Z

∂B(0,R)

[(Im(v(x))

ρ − ϕ(x)) + Im(v)ρ − 1

ρ ]x₁dS_x|

≤ 1 R

Z

∂B(0,R)

|(Im(v(x))

ρ − ϕ(x)) + Im(v)ρ − 1

ρ ||x₁|dS_x

≤ 1 R

Z

∂B(0,R)

|(Im(v(x))

ρ − ϕ(x)) + Im(v)ρ − 1 ρ |RdS_x

= Z

∂B(0,R)

|Im(v(x))

ρ − ϕ(x)| + |Im(v)ρ − 1 ρ |dSx

≤ Z

∂B(0,R)

|ϕ(x)|³

3! + |Im(v)ρ − 1 ρ |dS_x

≤ Z

∂B(0,R)

|ϕ(x)|³

3! + 2|Im(v)||ρ − 1|dSx,

the last inequality follows by, ρ = |v| ≥ ¹₂ =⇒ ¹_ρ ≤ 2.

Now, we have two cases.

Case 1. N = 2 or 3 and c(v) ≤√ 2

by Proposition 2.6 and Remark 2.8, we have,

|ϕ(x)|³

3! + 2|Im(v)||ρ − 1| ≤ K 1 + |x|^N, we conclude that,

| Z

B(0,R)

(hi∂₁v, v−1i−g(v))| ≤ Z

∂B(0,R)

K

1 + |x|^NdS_x = K|∂B(0, R)|

1 + R^N −→ 0 as R −→ ∞.

Case 2. N = 3 and c(v) =√ 2

by Remark 2.8 : ϕ ∈ L^q(R³\B(0, R(v))) for any q > ¹⁵₄ and ρ−1 ∈ L^q(R³\B(0, R(v))) for any q > ⁵₃ and by Proposition 2.6, the function f ≡ ^|ϕ(x)|_3! ³ + 2|Im(v)||ρ − 1| ∈ L^q(R³\B(0, R(v))) for any q > ⁵₄.

We claim that : ∀R > R(v), ∀q > ⁵₄ there ∃R⁰, R ≤ R⁰ ≤ 2R, ∃K(q) > 0, such that Z

∂B(0,R⁰)

f^qdS_x ≤ K(q) R .

If not, ∃R > R(v), ∃q > ⁵₄ s.t. ∀R⁰, R ≤ R⁰ ≤ 2R, ∀K(q) > 0,R

∂B(0,R⁰)f^qdSx > ^K(q)_R , so,

Z 2R R

( Z

∂B(0,R⁰)

f^qdS_x)dR⁰ ≥ Z 2R

R

(K(q)

R )dR⁰ = K(q) =⇒

Z

C

f^qdx ≥ K(q), where C = B(0, 2R)\B(0, R), also, see Figure 3. Since f ∈ L^q(R³\B(0, R(v))) =⇒

f ∈ L^q(C). Taking q −→ ∞ and taking K(q) = q, we obtain a contradiction.

∞ >

Z

C

f^qdx ≥ K(q) −→ ∞.

Now, by Holders inequality : Z

∂B(0,R⁰)

f dS_x ≤ ( Z

∂B(0,R⁰)

f^qdS_x)^1q( Z

∂B(0,R⁰)

1^pdS_x)^1p ≤ (K(q)

R )^1q|∂B(0, R⁰)|^1p ≤ ¯K(q)R^2−3q, where ¹_p + ¹_q = 1 and |∂B(0, R⁰)|^p1 = (4πR⁰²)^1p ≤ (4π)^1−1q(2R)^2(1−1q) = ˜K(q)R^2−2q,

that is, we take q = ⁴₃

| Z

B(0,R⁰)

(hi∂1v, v − 1i − g(v))| ≤ ˜K(q)R^2−2q −→ 0 as R −→ ∞

Figure 3: Energy estimate

which yields the conclusion, since the integrand is integrable by Lemma 2.5 and Corollary 2.7.

2.4 Pohozaev’s type identities

Lemma 2.10. Let v be a finite energy solution to (TWc) on R^N, with speed c = c(v).

Then, we have the identities

E(v) = Z

R^N

|∂₁v|², and for any 2 ≤ j ≤ N ,

E(v) = Z

R^N

|∂_jv|²+ c(v)p(v).

Moreover, if c(v) > 0 and v is not constant, then p(v) > 0.

Proof. The first identity was established in [19], and the second identity was proved there for any 2 ≤ j ≤ N ,

E(v) = Z

R^N

|∂jv|²+ c(v)˜p(v).

By Proposition 2.9 we have,

p(v) = ˜p(v) =⇒ E(v) = Z

R^N

|∂_jv|²+ c(v)p(v).

Notice, adding the identities in previous, we obtain Z

R^N

|∇v|²+ (N − 1)c(v)p(v)

= Z

R^N

|∂₁v|²+Z

R^N

|∂₂v|²+ c(v)p(v)

+ ... +Z

R^N

|∂_Nv|²+ c(v)p(v)

= E(v) + (N − 1)E(v) = N E(v)

= N 2

Z

R^N

|∇v|²+ N 4

Z

R^N

(1 − |v|²)², combine all calculations, we get

N − 2 2

Z

R^N

|∇v|² +N 4

Z

R^N

(1 − |v|²)²− c(v)(N − 1)p(v) = 0.

Assume c(v) > 0 and p(v) ≤ 0, if N = 3, using the previous identity N − 2

2 Z

R^N

|∇v|²+N 4

Z

R^N

(1−|v|²)² = c(v)(N −1)p(v) ≤ 0 =⇒ |v| = 1 and ∇v = 0, thus, v is constant, a contradiction.

If N = 2, we only have |v| = 1, that is p(v) = 0, also we may write v = ρ exp iϕ, by Lemma 2.13 (the proof of Lemma 2.13 doesn’t depend on the Lemma 2.10)

c(v)p(v) = Z

R^N

ρ²|∇ϕ|² = 0 =⇒ ∇ϕ = 0, ρ = 0, thus, v is constant, a contradiction.

Now, introduce the quantities Σ(v) = √

2p(v) − E(v) and ε(v) = p2 − c(v)², combine with the second identity in Lemma 2.10, we have, for 2 ≤ j ≤ N

Z

R^N

|∂_jv|²+Σ(v) =√

2−c(v)

p(v) = √

2−p

2 − ε(v)²

p(v) = ε(v)²

√2 +p2 − ε(v)²p(v).

Corollary 2.11. Let v be a finite energy solution to (TWc) on R^N, with speed c =√

2 and such that Σ(v) ≥ 0. Then, v is a constant.

Proof. Since ε(v) = 0 and Σ(v) ≥ 0, by previous, for any 2 ≤ j ≤ N , Z

R^N

|∂_jv|² = ε(v)²

√2 +p2 − ε(v)²p(v) − Σ(v) = −Σ(v) ≤ 0 =⇒

Z

R^N

|∂_jv|² = 0,

so that v depends only on x₁ variable, say v = f (x₁), but, E(v) =

Z

R^N

|∂1v|²dx = Z

R^N

|f⁰(x1)|²dx1dx2...dxN =

Z

R

|f⁰(x1)|²dx1

Z

R

1dx2

 ...

Z

R

1dxN

 . This implies that |f⁰(x₁)|² = 0, since v is a finite energy solution (i.e. E(v) < ∞),

so, v is a constant.

Moreover, if Σ(v) > 0, then identity in previous gives, for any 2 ≤ j ≤ N Z

R^N

|∂_jv|² ≤ Z

R^N

|∂_jv|²+ Σ(v) = ε(v)²

√2 +p2 − ε(v)²p(v) ≤ ε(v)p(v).

Combine with this inequality, the next result gives another version of Corollary 2.11.

Lemma 2.12. Let v be a finite energy solution to (TWc) on R^N. Then, there exists a constant K(c) > 0, depending on c, such that

kηk^{N +1}_L∞(R^N)≤ K(c) Z

R^N



λ|∂_jv|²+ η² λ

 , for any 2 ≤ j ≤ N , and for any λ > 0.

In particular, we have,

kηk^{N +1}_L∞(R^N)≤ K(c) λ

ε(v)p(v) − Σ(v)

+ E(v) λ

 .

Proof. Let η∞ = kηk_L^∞_(R^N₎. We may assume without loss of generality, say that

|η(0)| = η∞. In view of the uniform bound for |∇v|, there exists some constant K(N, c), such that

η∞−|η(x)| = |η(0)|−|η(x)| ≤ |η(0)−η(x)| ≤ |∇η(ζ)||x−0| ≤ 2|v∇v(ζ)||x| ≤ K(N, c)|x|, so, we have, if |x| < _2K(N,c)^η∞ ,

η_∞− |η(x)| ≤ 1

2η_∞ =⇒ η∞

2 ≤ |η(x)|.

In other words,

|η(x)| ≥ η∞

2 , ∀x ∈ B

0, η∞

2K(N, c)

 .

We next consider for any point a = (a₁, . . . , a_j−1, 0, a_j+1, . . . , a_N), the line D_j(a) parallel to the axis x_j, that is, D_j(a) = {a_j(x) ≡ (a₁, . . . , a_j−1, x, a_j+1, . . . , a_N), x ∈ R}. We claim that,

|η∞|² ≤ 4 Z

Dj(a)



λ(∂_jη)²+η² λ

 , for any a = (a₁, . . . , a_j−1, 0, a_j+1, . . . , a_N) ∈ B(0,_2K(c)^η∞ ).

Given a = (a1, . . . , aj−1, 0, aj+1, . . . , aN) ∈ B(0,_2K(c)^η∞ ), we have |η(a)| ≥ ^η∞₂ , since η(x) −→ 0, as |x| −→ +∞, and by Fundamental Theorem of Calculus,

|η∞|² ≤ 4η(a)² = 4 Z 0

−∞

∂_j(η²(a_j(x)))dx = 8 Z 0

−∞

∂_jη(a_j(x))η(a_j(x))dx Using Young’s inequality, ab = (a√

λ)(^√b

λ) ≤ ^λa₂² +_2λ^b2 for any λ > 0, where we take a =√

2∂_jη(a_j(x)) and b =√

2η(a_j(x))², 2

Z 0

−∞

∂_jη(a_j(x))η(a_j(x))dx ≤ Z 0

∞



λ ∂_jη(a_j(x))
2

+ η(a_j(x))² λ

 dx

≤ Z

R



λ ∂_jη(a_j(x))
2

+η(a_j(x))² λ

 dx.

= Z

Dj(a)



λ ∂_jη(z)
2

+ η²(z) λ

 dz.

Now, we integrate the inequality on a = (a1, . . . , aj−1, 0, aj+1, . . . , aN) ∈ B(0,_2K(N,c)^η∞ ).

Z

B(0,_2K(N,c)^η∞ )

|η∞|²da ≤ 4 Z

B(0,_2K(N,c)^η∞ )

Z

Dj(a)



λ ∂_jη(z)
2

+ η²(z) λ

 dzda

≤ 4 Z

R^N



λ ∂_jη
2

+ η² λ

 , that is,

|η∞|²

ω_{N −1}| η∞

2K(N, c)|^{N −1}

≤ 4 Z

R^N



λ ∂_jη
2

+η² λ

 where ω_{N −1} is the volume of unit ball in R^N − 1.

Finally,

kηk^{N +1}_L∞(R^N) ≤ K(N, c) Z

R^N



λ|∂_jv|²+η² λ

 , for any 2 ≤ j ≤ N , and for any λ > 0.

2.5 Solutions without vortices

In this subsection, we consider solutions v to (TWc) on R^N which do not vanish.

Moreover, we will assume that

|v| ≥ 1 2,

so that, v may be written as v = ρ exp iϕ. Also, the energy can be written as variables ρ and ϕ,

E(v) = E(ρ, ϕ) ≡ 1 2

Z

R^N



|∇ρ|²+ ρ²|∇ϕ|²+(1 − ρ²)² 2



, (2.1)

whereas, for the momentum, we have hi∂₁v, vi = −ρ²∂₁ϕ. Therefore, it follows from Proposition 2.9 that,

p(v) = ˜p(v) = 1 2

Z

R^N

− ρ²∂₁ϕ + ∂₁ (1 − χ)ϕ


= 1 2

Z

R^N

(1 − ρ²)∂₁ϕ. (2.2) The system for ρ and ϕ is written







c

2∂₁ρ² + ∇ ·

ρ²∇ϕ

= 0, cρ∂₁ϕ − ∆ρ − ρ

1 − ρ²

+ ρ|∇ϕ|² = 0.

Notice that the quantity η = 1 − ρ² satisfies the equation,

∆²η − 2∆η + c²∂₁²η = −2∆(|∇v|²+ η²− cη∂₁ϕ) − 2c∂₁∇ · (η∇ϕ), where the L.H.S is linear with respect to η, whereas, the R.H.S is quadratic.

A first elementary result is,

Lemma 2.13. Let v be a finite energy solution to (TWc) on R^N, satisfying |v| ≥ ¹₂. Then, we have the identity

cp(v) = Z

R^N

ρ²|∇ϕ|². Proof. The system for ρ and ϕ is written







c

2∂₁ρ² + ∇ ·

ρ²∇ϕ

= 0, cρ∂₁ϕ − ∆ρ − ρ

1 − ρ²

+ ρ|∇ϕ|² = 0.

Multiplying the first equation by ϕ and integrating by parts, also, use the decay properties.

c 2

Z

R^N

ϕ∂1ρ²+ Z

R^N

ϕ∇ ·



ρ²∇ϕ

= 0,

∇ ·

ϕρ²∇ϕ

= ρ²|∇ϕ|²+ ϕ∇ ·

ρ²∇ϕ ,

−c 2

Z

R^N

ϕ∂₁ρ² = Z

R^N

ϕ∇ ·

ρ²∇ϕ

= Z

R^N

∇ ·

ϕρ²∇ϕ

− Z

R^N

ρ²|∇ϕ|². Now we claim,

Z

R^N

∇ ·

ϕρ²∇ϕ

= 0.

We may assume ρ² ≤ 2, since ρ −→ 1 as R = |x| −→ ∞ and by divergent theorem,

| Z

BR

∇ ·

ϕρ²∇ϕ

| = | Z

∂BR



ϕρ²∇ϕ

· ndS_x| = Z

∂BR

|ϕ||ρ²||∇ϕ|dS_x

≤ 2 Z

∂BR

|ϕ||∇ϕ|dS_x ≤ K

(1 + R^{N −1})(1 + R^N)|∂B_R| −→ 0, where n is the unit outer normal vector on ∂B_R, the last inequality follows by the decay property, that is,

c 2

Z

R^N

ϕ∂₁ρ² = Z

R^N

ρ²|∇ϕ|², and similarly,

c 2

Z

R^N

ϕ∂₁ρ² = c 2

Z

R^N

∂₁(ϕρ²) − c 2

Z

R^N

(∂₁ϕ)ρ², again, we claim,

Z

R^N

∂₁(ϕρ²) = Z

R^N

∂₁ϕ, thus,

c 2

Z

R^N

ϕ∂₁ρ² = c 2

Z

R^N

∂₁ϕ − c 2

Z

R^N

(∂₁ϕ)ρ² = c 2

Z

R^N

∂₁ϕ(1 − ρ²).

We may say ρ ≤ 2, since ρ −→ 1 as R = |x| −→ ∞ and by divergent theorem,

| Z

BR

∂₁(ϕρ²− ϕ)| ≤ | Z

∂BR

|ϕρ²− ϕ|dS_x = Z

∂BR

|ϕ||ρ − 1||ρ + 1|dS_x

≤ K

(1 + R^{N −1})(1 + R^N)|∂B_R| −→ 0,

the last inequality also follows by the decay property.

Finally,

cp(v) = c 2

Z

R^N

(1 − ρ²)∂1ϕ = Z

R^N

ρ²|∇ϕ|².

Lemma 2.14. Let v be a finite energy solution to (TWc) on R^N satisfying |v| ≥ ¹₂. Then

E(v) ≤ 7c(v)² Z

R^N

η². Proof. By Lemma 2.13 and Cauchy-Schwarz’s inequality,

Z

R^N

ρ²|∇ϕ|² = c 2

Z

R^N

(1 − ρ²)∂1ϕ ≤ c

 Z

R^N

η²

¹₂ Z

R^N

|∇ϕ|²

¹₂

≤ 2c

 Z

R^N

η²

¹₂ Z

R^N

ρ²|∇ϕ|²

¹₂ , the last inequality follows by ρ = |v| ≥ ¹₂ =⇒ 4ρ² ≥ 1.

Hence, we get,

Z

R^N

ρ²|∇ϕ|² ≤ 4c² Z

R^N

η².

It remains to bound the integral of |∇ρ|², recall the equations







c

2∂₁ρ² + ∇ ·

ρ²∇ϕ

= 0, cρ∂₁ϕ − ∆ρ − ρ

1 − ρ²

+ ρ|∇ϕ|² = 0.

For that purpose, we multiply the second equation by 1 − ρ² and integrate by parts on R^N, using the decay properties in Remark 2.8.

Z

R^N



(∆ρ)(1 − ρ²) + ρ(1 − ρ²)²



= c Z

R^N

ρ(1 − ρ²)∂₁ϕ + Z

R^N

ρ(1 − ρ²)|∇ϕ|², Z

R^N

(∆ρ)(1 − ρ²) = Z

R^N

∇ · (∇ρ)(1 − ρ²)
− ∇ρ · ∇(1 − ρ²) = Z

R^N

2ρ|∇ρ|², thus, we have,

Z

R^N



2ρ|∇ρ|²+ ρ(1 − ρ²)²



= c Z

R^N

ρ(1 − ρ²)∂₁ϕ + Z

R^N

ρ(1 − ρ²)|∇ϕ|²,

also, by 1 ≤ 2ρ and (1 − ρ²) ≤ 2ρ and Cauchy-Schwarz’s inequality, we deduced Z

R^N



|∇ρ|²+ 1

2(1 − ρ²)²



≤ Z

R^N



(2ρ)|∇ρ|²+ (2ρ)1

2(1 − ρ²)²



= c Z

R^N

ρ(1 − ρ²)∂₁ϕ + Z

R^N

ρ(1 − ρ²)|∇ϕ|²

≤ c

 Z

R^N

ρ²|∂₁ϕ|²

¹₂ Z

R^N

(1 − ρ²)²

¹₂ + 2

Z

R^N

ρ²|∇ϕ|²

≤ c

 Z

R^N

ρ²|∇ϕ|²

¹₂ Z

R^N

η²

¹₂ + 8c²

Z

R^N

η²

≤ 2c²

 Z

R^N

η²

¹₂ Z

R^N

η²

¹₂ + 8c²

Z

R^N

η²

= 10c² Z

R^N

η². Finally,

E(v) = 1 2

Z

R^N



ρ²|∇ϕ|²+ |∇ρ|²+1

2(1 − ρ²)²



≤ 1 2(4c²

Z

R^N

η²+ 10c² Z

R^N

η²).

2.6 Subsonic vortexless solutions

We next assume that the solution v satisfies the additional condition, 0 < c(v) <√

2.

For such a solution, we let

ε(v) =p

2 − c(v)².

Proposition 2.15. Let v be a non-trivial finite energy solution to (TWc) on R^N satisfying 0 < c(v) <√

2. Then, 1 − |v|

L^∞(R^N)

≥ ε(v)² 10 .

Proof. Let δ = k1 − |v|k_L^∞_(R^N₎. If δ ≥ ¹₂, then the proof is straightforward. Other- wise, δ < ¹₂ =⇒ |v| ≥ ¹₂, we can use Lemma 2.4,

δ = k1 − |v|k_L^∞_(R^N₎ ≥ 1 − |v| =⇒ ρ ≥ 1 − δ =⇒ 1

ρ ≤ 1 1 − δ,