R N and W is a real-valued even distribution.

ANDRÉDELAIRE

Abstrat. Weonsidera GrossPitaevskii equation witha nonloal inter-

ationpotential. Weprovidesuientonditions onthepotential suhthat

thereexistsarangeofspeedsinwhihnontrivialtravelingwavesdonotexist.

1. Introdution

1.1. The problem. We onsider nite energy traveling waves for the nonloal

GrossPitaevskiiequation

(1.1)

i∂ t u − ∆u − u(W ∗ (1 − | u | ² )) = 0, u(x, t) ∈ C , x ∈ R ^N , t ∈ R .

Here

∗

^{denotes the onvolutionin}

R ^N

and

W

^{is a} real-valued even distribution.

The aim of this work is to provide suient onditionson the potential

W

^suh

that these traveling waves are neessarilyonstant for a ertain rangeof speeds.

Equation(1.1)isHamiltoniananditsenergy

E(u(t)) = 1 2

Z

R ^N

|∇ u(t) | ² dx + 1 4

Z

R ^N

(W ∗ (1 − | u(t) | ² ))(1 − | u(t) | ² ) dx

isformallyonserved. Atravelingwaveofspeed

c

^{thatpropagatesalongthe}

x 1

^-axis

isasolutionoftheform

u c (x, t) = v(x 1 − ct, x _⊥ ), x _⊥ = (x 2 , . . . , x N ).

Henetheprole

v

^satises

(NTW

c

⁾

ic∂ 1 v + ∆v + v(W ∗ (1 − | v | ² )) = 0

ⁱⁿ

R ^N

andbyusingomplexonjugation,weanrestritusto thease

c ≥ 0

^{. Note that}

anyonstant(omplex-valued)funtion

v

^{ofmodulusoneveries(NTW}

c

^{) ,sothat}

werefertothemasthetrivialsolutions.

Notiethat,intheasethat

W

^{oinideswiththeDiradeltafuntion,(NTW}

c

⁾

reduesto thelassialGrossPitaevskiiequation

(TW

c

⁾

ic∂ 1 v + ∆v + v(1 − | v | ² ) = 0

ⁱⁿ

R ^N .

Equation(TW

c

^)hasbeenintensivelystudiedin thelastyears. Wereferto[3℄for asurvey. Fromnowonwesupposethat

N ≥ 2

^{andwereallthefollowingresults.}

2000 Mathematis Subjet Classiation. 35Q55; 35Q40; 35Q51; 35B65; 37K40; 37K05;

81Q99.

Keywordsandphrases. NonloalShrödingerequation,GrossPitaevskiiequation,Traveling

waves,Pohozaevidentities,Nonzeroonditionsatinnity.

TheauthorisgratefultoF.BéthuelforinterestingandhelpfuldisussionsandtoP.Gravejat

forhisollaborationinprovingProposition5.3.

Theorem 1.1 ([8, 6, 16, 17℄). Let

v ∈ H _loc ¹ ( R ^N )

^{be a nite energy solution of}

(TW

c

^{). Assumethat oneof the following aseshold}

(i)

c = 0

^.

(ii)

c > √ 2

^.

(iii)

N = 2

^and

c = √ 2

^.

Then

v

^{isaonstant funtion ofmodulusone.}

Theorem 1.2 ([6, 5,10, 4, 24℄). There is some nonempty set

A ⊂ (0, √ 2)

^suh

that for all

c ∈ A

^{there exists a nononstant nite energy solution of (TW}

c

^{) .}

Furthermore, assume that

N ≥ 3

^{. Then there exists a nononstant nite energy}

solution of (TW

c

^{)for all}

0 < c < √ 2

^.

Itwouldbereasonabletoexpettogeneralizeinsomewaythesetheoremstothe

nonloal equation (NTW

c

^{) . The aim of this paper is to} investigate the analogue ofTheorem1.1in theases(i)and(ii). Beforestatingourpreise results,wegive

somemotivation abouttheritialspeed.

1.2. Physial motivation. As explained in [13℄, (1.1) an be onsidered as a

generalizationof theequation

(1.2)

i ~ ∂ t Ψ(x, t) = ~ ²

2m ∆Ψ(x, t) + Ψ(x, t) Z

R ^N | Ψ(y, t) | ² V (x − y) dy,

ⁱⁿ

R ^N × R ,

introdued by Gross [18℄ and Pitaevskii [27℄ to desribe the kineti of a weakly

interatingBosegasofbosonsofmass

m

^,where

Ψ

^isthewavefuntiongoverningthe ondensateintheHartreeapproximationand

V

^{desribestheinterationbetween}

bosons.

In the most typial approximation,

V

^{is onsidered asa Dira delta funtion.}

Thenthismodelhasappliationsinseveralareasofphysis,suh assuperuidity,

nonlinear optis and BoseEinstein ondensation [21, 20, 23, 11℄. It seems then

natural to analyze equation (1.2) for more general interations. Indeed, in the

studyofsuperuidity,supersolidsandBoseEinsteinondensation,dierenttypes

ofnonloalpotentialshavebeenproposed [7,2,14,29,28,22,30,12, 9,1℄.

Let us now proeed formally and onsider a onstant funtion

u 0

^{of modulus}

one. Sine(1.1)isinvariantbyahangeofphase,weanassume

u 0 = 1

^{. Thenthe}

linearizedequationof (1.1)at

u 0

^isgivenby

(1.3)

i∂ t u ˜ − ∆˜ u + 2W ∗ Re(˜ u) = 0.

Writing

u ˜ = ˜ u 1 + i˜ u 2

^{andtakingrealandimaginarypartsin(1.3),weget}

− ∂ t u ˜ 2 − ∆˜ u 1 + 2W ∗ u ˜ 1 = 0,

∂ t u ˜ 1 − ∆˜ u 2 = 0,

fromwhere wededuethat

(1.4)

∂ _tt ² ˜ u − 2W ∗ (∆˜ u) + ∆ ² u ˜ = 0.

Byimposing

u ˜ = e ^{i(ξ.x − wt)}

^,

w ∈ R

,

ξ ∈ R ^N

,asasolutionof (1.4), weobtainthe dispersionrelation

(1.5)

(w(ξ)) ² = | ξ | ⁴ + 2c W (ξ) | ξ | ² ,

where

W c

^{denotesthe Fouriertransformof}

W

^{. Supposing that}

c W

^{is positiveand}

ontinuousattheorigin,wegetin thelongwaveregime,i.e.

ξ ∼ 0

^,

w(ξ) ∼ (2c W (0)) ^1/2 | ξ | .

Consequently,inthisregimeweanidentify

(2c W (0)) ^1/2

^{asthespeedofsoundwaves}

(alsoalledsonispeed),sothatweset

c s (W ) = (2c W (0)) ^1/2 .

Thedispersionrelation(1.5)wasrstobservedbyBogoliubov[7℄onthestudyof

BoseEinsteingasandunder somephysialonsiderationsheestablishedthatthe

gasshouldmovewithaspeedlessthan

c s (W )

^{topreserveitssuperuid}properties.

FromamathematialpointofviewandomparingwithTheorems1.1and1.2,this

enouragesustothink thatthenonexisteneofanontrivialsolutionof (NTW

c

^)is

relatedtotheondition

(1.6)

c > c s (W ).

Atually,inSubsetion1.4weprovideresultsinthisdiretionandinSubsetion1.5

wespeifythedisussionforsomeexpliitpotentials

W

^{whiharephysially rele-}

vant.

1.3. Hypotheses on

W

^{. Letusintroduethespaes}

M p,q ( R ^N )

^{oftempereddis-}

tributions

W

^{suhthat thelinearoperator}

f 7→ W ∗ f

^{isboundedfrom}

L ^p ( R ^N )

^to

L ^q ( R ^N )

^{. Wewillusethefollowinghypotheseson}

W

^.

(H1)

W

^isareal-valuedeventemperateddistribution.

(H2)

W ∈ M 2,2 ( R ^N )

^{. Moreover,if}

N ≥ 4

^,

(1.7)

W ∈ M N/(N − 1), ∞ ( R ^N ) ∩ M 2N/(N − 2), ∞ ( R ^N ) ∩ M 2N/(N − 2),2N/(N − 2) ( R ^N ).

(H3)

c W

^isdierentiablea.e. on

R ^N

andfor all

j, k ∈ { 1, . . . , N }

^{the map}

ξ → ξ j ∂ k W c (ξ)

^{isboundedandontinuousa.e. on}

R ^N

.

(H4)

c W ≥ 0

^{a.e. on}

R ^N

.

(H5)

c W

^isoflass

C ²

^inaneighborhoodoftheoriginand

W c (0) > 0.

Reall that theondition

W ∈ M ^2,2 ( R ^N )

^{isequivalentto}

c W ∈ L ^∞ ( R ^N )

^{(see e.g.}

[15℄). Therefore (H4) makessense provided that (H2) holds. It is provedin [13℄

that under the assumptions (H1), (H2) and (H4) the Cauhy problem for (1.1)

withnonzeroonditionat innityisglobally well-posed. Atually,ondition(1.7)

ismorerestritivethat theoneusedin[13℄in dimension

N ≥ 4

^{,butweneeditto}

ensuretheregularityofsolutions. Morepreisely,inSetion2weprovethatunder

thehypothesis(H2), thesolutionsof (NTW

c

^{)aresmoothandsatisfy}

| v(x) | → 1, ∇ v(x) → 0,

^as

| x | → ∞ .

On the other hand, by Lemma 2.3, (1.7) is at least fullled for

W ∈ L ¹ ( R ^N ) ∩ L ^N ( R ^N )

^.

Assumption(H2)alsoimpliesthat

E(v)

^{isnitein theenergyspae}

E ( R ^N ) = { ϕ ∈ H _loc ¹ ( R ^N ) : 1 − | ϕ | ² ∈ L ² ( R ^N ), ∇ ϕ ∈ L ² ( R ^N ) } .

Furthermore,if (H4)alsoholds,thenbythePlanherelidentity

E(v) = 1 2

Z

R ^N

|∇ v | ² + 1 4(2π) ^N

Z

R ^N

W c | 1 \ − | v | ² | ² ≥ 0.

InSubsetion1.5weshowseveralexamplesofdistributions

W

^{satisfyingtheon-}

ditions(H1)(H5).

1.4. Statementof the results.

Theorem 1.3. Assume that

W

^{satises (H1)(H5). Let}

c > c s (W )

^{and suppose}

that thereexistonstants

σ 1 , . . . , σ N ∈ R

suhthat (1.8)

W c (ξ) + α c

X N k=2

σ k ξ k ∂ k c W (ξ) − σ 1 ξ 1 ∂ 1 W c (ξ) ≥ 0,

^fora.a.

ξ ∈ R ^N ,

and

(1.9)

X N k=2

σ k + min

− σ 1 − 1, σ 1 − 1

α c + 2 , 2α c σ j + σ 1 − 1

≥ 0,

for all

j ∈ { 2, . . . , N }

^,where

α c := c ² /(c s (W )) ² − 1

^{. Then nontrivial solutions of}

(NTW

c

⁾ⁱⁿ

E ( R ^N )

^donotexist.

ToapplyTheorem1.3weneedtoverifytheexisteneoftheonstants

σ 1 , . . . , σ N

satisfying(1.8)and(1.9). Toavoidthistask,weprovidetwoorollarieswherethe

onditions for thenonexistene of travelingwaves are expressed only in terms of

W

^.

Corollary 1.4. Assumethat

W

^{satises (H1)(H5) andalsothat}

(1.10)

W c (ξ) ≥ max

1, 2 N − 1

X ^N

k=2

| ξ k ∂ k W c (ξ) | + | ξ 1 ∂ 1 c W (ξ) | ,

^{for a.a.}

ξ ∈ R ^N .

Suppose that

c > c s (W )

^{. Then nontrivial solutions of (NTW}

c

^{) in}

E ( R ^N )

^{do not}

exist.

Corollary 1.5. Assumethat

W

^{satises (H1)(H5). Supposethat}

(1.11)

c s (W ) < c ≤ c s (W ) 1 + inf

ξ ∈R ^N

(N − 1)c W (ξ) P N

k=2 | ξ k ∂ k c W (ξ) |

! 1/2

,

^{for a.a.}

ξ ∈ R ^N .

Then nontrivial solutionsof (NTW

c

^{) in}

E ( R ^N )

^{do notexist.}

Conerningthestatiwaves,wehavethefollowingresult.

Theorem1.6. Assumethat

W

^{satises (H1)(H4). Supposethat}

c = 0

^andthat

(1.12)

ξ j ∂ j W c (ξ) ≤ 0,

^{for a.a.}

ξ ∈ R ^N ,

forall

j ∈ { 1, . . . , N } .

^{Thennontrivialsolutionsof (NTW}

c

⁾ⁱⁿ

E ( R ^N )

^donotexist.

Note that in the ase

W = aδ

^,

a > 0

^,

c W = a

^{and so that}

∇ c W = 0

^{. Then}

onditions(1.10), (1.11)and (1.12)hold. Therefore,invokingCorollary1.4or1.5

andTheorem1.6weobtainthenonexisteneofnontrivialsolutionsforall

(1.13)

c ∈ { 0 } ∪ ( √

2a, ∞ ).

Inpartiular,onsidering

a = 1

^{,wereoverTheorem1.1intheases(i)and(ii).}

Sofar, in viewof (H5), wehaveassumed that

c W

^{is regularin a}neighborhood of the origin, whih in partiular allows us to dene

c s (W )

^{. However there are}

interesting examples of kernels provided by the physial literature suh that

c W

is not ontinuous at the originand then

c s (W )

^{is not properly dened. For this}

reasonwewillworkwithamoregeneralgeometrionditionon

W c

^{. Morepreisely,}

denotingby

{ e k } k ∈{ 1,...,N }

^{the anonialunitaryvetorsof}

R ^N

, weintroduethe funtion

(1.14)

w _j (ν 1 , ν 2 ) := c W (ν 1 e 1 + ν 2 e j ), (ν 1 , ν 2 ) ∈ R ² , j ∈ { 2, . . . , N } ,

andtheset

Γ j,c := { ν = (ν 1 , ν 2 ) ∈ R ² : | ν | ⁴ + 2 w _j (ν ) | ν | ² − c ² ν ₁ ² = 0 } .

ThenTheorem1.3anbegeneralizedifwereplae(H5)bytheondition

(H6) Forall

j ∈ { 2, . . . , N }

^and

c > 0

^{, thereexist}

δ > 0

^{andtwofuntions}

γ ⁺ _j,c

and

γ _j,c ⁻

^{,dened ontheinterval}

(0, δ)

^{, suh thattheset}

Γ j,c ∩ B(0, δ)

^has

Lebesguemeasurezero,

γ _j,c ^± ∈ C ¹ ((0, δ))

^,and

γ _j,c ⁺ (t) > 0, γ _j,c ⁻ (t) < 0, (t, γ _j,c ^± (t)) ∈ Γ j,c ,

^forall

t ∈ (0, δ).

Moreover,thefollowinglimitsexist andareequal

t lim → 0 ⁺

γ _j,c ⁺ (t) t

! ²

= lim

t → 0 ⁺

γ _j,c ⁻ (t) t

! ²

=: ℓ j,c .

γ ⁺ _j,c (t)

γ ⁻ _j,c (t) p ℓ j,c t

− p ℓ j,c t

0 t

Figure 1. Theurves

γ ^± _j,c

^{ofondition(H6).}

Figure 1 illustrates ondition (H6). The fat that (H5) and (1.6) atually imply

(H6)isprovedinSetion4(seeLemma4.1). Wealsonotethatfrom(H6)weinfer

that

lim t → 0 ⁺ γ _j,c ^± (t) = 0

^{. Moreover,if}

W c

^{isevenin eahomponent,thatis}

W c (( − 1) ^{m 1} x 1 , ( − 1) ^{m 2} x 2 , . . . , ( − 1) ^{m N} x N ) = c W (x 1 , x 2 , . . . , x N ),

forall

(m 1 , . . . , m N ) ∈ { 0, 1 } ^N ,

^then

γ _j,c ⁻ = − γ _j,c ⁺

^,forall

j ∈ { 2, . . . , N }

^.

Onthe otherhand,ifthe values

ℓ j,c

^{arepositive,aneessaryonditionforthe}

existeneofanontrivialniteenergysolutionof (NTW

c

^{)isthat theyareequal.}

Lemma 1.7. Let

c > 0

^{. Assume that}

W

^{satises (H1)(H4) and (H6) with}

ℓ j,c > 0

^,forall

j ∈ { 2, . . . , N }

^{. Let}

v ∈ E ( R ^N )

^{beanontrivialsolutionof (NTW}

c

⁾

in

E ( R ^N )

^{. Then}

ℓ 1,c = ℓ 2,c = · · · = ℓ N,c .

Nowwearereadytostateourmain resultinitsgeneralform.

Theorem1.8. Let

c > 0

^{. Assumethat}

W

^{satises (H1)(H4)and (H6),with}

(1.15)

ℓ c := ℓ 1,c = ℓ 2,c = · · · = ℓ N,c > 0.

Supposethatthere existonstants

σ 1 , . . . , σ N ∈ R

suhthat (1.16)

c W (ξ) + ℓ c

X N k=2

σ k ξ k ∂ k W c (ξ) − σ 1 ξ 1 ∂ 1 W c (ξ) ≥ 0,

^{for a.a.}

ξ ∈ R ^N ,

and

(1.17)

X N k=2

σ k + min

− σ 1 − 1, σ 1 − 1

ℓ c + 2 , 2ℓ c σ j + σ 1 − 1

≥ 0,

for all

j ∈ { 2, . . . , N }

^{. Then nontrivial solutions of (NTW}

c

^{) in}

E ( R ^N )

^{do not}

exist.

Finally,wegivetheorrespondinganalogueofCorollaries1.41.5.

Corollary 1.9. Let

c > 0

^{. Assumethat}

W

^{satises (H1)(H4), (H6)and (1.15).}

Supposethateither (1.10)or

l c ≤ inf

ξ ∈R ^N

(N − 1)c W (ξ) P N

k=2 | ξ k ∂ k W c (ξ) |

hold. Then nontrivial solutionsof (NTW

c

⁾ⁱⁿ

E ( R ^N )

^{do notexist.}

1.5. Examples. Inthissubsetionweprovidesomepotentialsofphysialinterest

forwhih theCauhyproblemfor(1.1)isgloballywell-posed(see[13℄).

(I)Giventhe spheriallysymmetri interation ofpartiles, in physial models

itisusualtosupposethat

W

^{isradialandthensoisitsFouriertransform,namely}

W c (ξ) = ρ( | ξ | ),

forsomefuntion

ρ : [0, ∞ ) → R

. Assumingthat

ρ

^isdierentiable,weompute (1.18)

ξ k ∂ k c W (ξ) = ρ ^′ ( | ξ | ) ξ _k ²

| ξ | ,

^forall

ξ ∈ R ^N \ { 0 } .

Then,usingthat

P N

k=2 ξ ² _k = | ξ | ² − ξ ₁ ²

^andthat

| ξ k | ≤ | ξ |

^{,weobtainthatonditions}

(1.10)and(1.11)arerespetivelysatisedif

(1.19)

max

1, 2

N − 1

≤ inf

r>0

ρ(r)

| ρ ^′ (r) | r ,

and

(1.20)

2ρ(0) < c ² ≤ 2ρ(0)

1 + inf

r>0

ρ(r)

| ρ ^′ (r) | r

.

We onsider now a generalization of the model proposed by Shhesnovih and

Kraenkel[29℄

ρ(r) = 1

(1 + ar ² ) ^b/2 , a, b > 0,

sothat

c s := c s (W ) = √ 2.

Itisimmediateto verifythat hypotheses(H1),(H3)(H5)aresatised. Also,sine

W c ∈ L ^∞ ( R ^N )

^{,(H2)isfullledfor}

N = 2, 3

^{. Moreover,by}Proposition6.1.5in[15℄, weonludethat

W ∈ L ¹ ( R ^N ) ∩ L ^N ( R ^N )

^for

N ≥ 4

^providedthat

b > N − 1

^{. On}

theotherhand,

(1.21)

inf

r>0

ρ(r)

| ρ ^′ (r) | r = inf

r>0

1 + ar ² abr ² = 1

b .

Therefore, using (1.18)(1.21) and invoking Corollaries 1.4,1.5 and Theorem 1.6,

weonludethatinthefollowingasesthereisnonexisteneofnontrivialsolutions

of (NTW

c

⁾ⁱⁿ

E ( R ^N )

(a)

N = 2

^,

b ≤ 1/2

^,

c ∈ (c s , ∞ )

^.

(b)

N = 2

^,

b > 1/2

^,

c ∈ (c s , p

2 + 2/b)

^.

()

N = 3

^,

b ≤ 1

^,

c ∈ (c s , ∞ )

^.

(d)

N = 3

^,

b > 1

^,

c ∈ (c s , p

2 + 2/b)

^.

(e)

N ≥ 4

^,

b > N − 1

^,

c ∈ (c s , p

2 + 2/b)

^.

(f)

N = 2

^or3,

c = 0

^.

(g)

N ≥ 4

^,

b > N − 1

^,

c = 0

^.

Weremarkthat if

b → 0

^,

W c → 1

^andthen

W → δ

^{in a}distributional sense. Thus theases(a)and()ouldbeseenasageneralizationofTheorem1.1in theases

(i)and(ii).

(II) Let

N = 2, 3

^and

W ε = δ + εf, ε ≥ 0,

where

f

^isanevenreal-valuedfuntion,suhthat

f, | x | ² f, | x |∇ f ∈ L ¹ ( R ^N )

^{. Then}

W c ε = 1 + ε f b ∈ C ² ( R ^N )

^{. Sine}

(1.22)

x \ j ∂ k f = − (δ j,k f b + ξ k ∂ j f b ),

wehave

k f b k L ^∞ ( R ^N ) ≤ k f k L ¹ ( R ^N ) , k ξ k ∂ j f b k L ^∞ ( R ^N ) ≤ k f k L ¹ ( R ^N ) + k x j ∂ k f k L ¹ ( R ^N ) .

Then we see that

W

^{satises onditions(H1)(H5) provided that}

ε < k f k ⁻ _L ^{1 1} _{( R} ^N ₎

andthatthesonispeedgivenby

c s := c s (W ) =

2 + 2ε Z

R ^N

f 1/2

,

iswell-dened. Moreover(1.10)is fullledif

(1.23)

ε < 4 k f k L ¹ ( R ^N ) + X N k=1

k x k ∂ k f k L ¹ ( R ^N )

! − 1

.

Therefore, under ondition(1.23), Corollary1.4 implies thenonexistene of non-

trivialsolutionsof (NTW

c

⁾ⁱⁿ

E ( R ^N )

^forany

c ∈ (c s , ∞ ).

(III)Thefollowingpotentialusedin[9,30℄tomodeldipolarforesinaquantum

gasyieldsanexamplein

R ³

wherethespeedofsoundisnotproperlydened. Let

W = aδ + bK, a, b ∈ R ,

where

K

^{isthesingularkernel}

K(x) = x ² ₁ + x ² ₂ − 2x ² ₃

| x | ⁵ , x ∈ R ³ \{ 0 } .

In the sequel, we will dedue from Lemma 1.7 and Theorem 1.8 that there is

nonexisteneofnontrivialniteenergysolutionsof (NTW

c

⁾ⁱⁿ

E ( R ^N )

^forall

(1.24)

(2 max { a − ˜ b, a } ) ^1/2 < c < ∞ ,

with

˜ b = (4πb)/3

^{,providedthat}

a > 0

^andeither

(1.25)

a ≥ ˜ b ≥ 0

^or

a > − 2˜ b ≥ 0.

Wenowturntotheproofofondition(1.24). Infat,sine(see[9℄)

W c (ξ) = a + ˜ b 3ξ ² ₃

| ξ | ² − 1

, ξ ∈ R ³ \{ 0 } ,

W

^{satises(H1)(H4)ifoneofthe onditionsin (1.25) holds. However,}

W c

^isnot

ontinuousattheorigin. Morepreisely,intermsofthefuntiondened in(1.14),

wehavethat

w ₂

is onstantequalto

a > 0

^{and by Lemma 4.1there exist urves}

γ ₂ ^±

^with

ℓ 2,c = c ² /(2a) − 1

^{. Ontheother hand,}

w ₃

is notontinuousat theorigin butassuming (1.24)weanexpliitlysolvethealgebraiequation

(x ² + y ² ) ² + 2 w ₃ (x, y)(x ² + y ² ) − c ² x ² = 0

anddeduethat

γ ^± _3,c (t) = ± r

− t ² − a − 2˜ b + q

6˜ bt ² + (a + 2˜ b) ² + c ² t ² ,

for

| t | < c ² − 2(a − ˜ b)

^{. Therefore(H6)holdsand}

ℓ 3,c = − 1+(6˜ b+c ² )/(2(a+2˜ b))

^{. Note}

thatby (1.25),

ℓ 3,c

^{isawell-denedpositiveonstant. ByLemma1.7,aneessary}

onditionsothattheequation(NTW

c

^{)hasnontrivialsolutionsis}

ℓ 3,c = ℓ 2,c

^,whih

leadsusto

(c ² − 3a)b = 0.

Thease

b = 0

^{hasalreadybeenanalyzed(see(1.13)). If}

b 6 = 0

^,weobtain

c ² = 3a

^.

Hene

ℓ c := ℓ 2,c = ℓ 3,c = 1/2.

^{Then, taking}

σ 1 = 0

^and

σ 2 = σ 3 = 1/2

^{, (1.17) is}

satisedandthel.h.s. of (1.16)reads

a + ˜ b

3 ξ ₃ ²

| ξ | ²

1 − ξ ² ₂ 2 | ξ | ²

− 1

+ 3˜ b 2

ξ ² ₃

| ξ | ²

1 − ξ ₃ ²

| ξ ² |

,

whihis nonnegativeby (1.25). Therefore, byTheorem 1.8, there isnonexistene

ofnontrivialsolutionsof (NTW

c

⁾ⁱⁿ

E ( R ^N )

^{, providedthat(1.24) and(1.25)hold.}

As provedin [13℄, the Cauhyproblem is alsoglobally well-posed for other in-

terationssuhasthesoftorepotential

W (x) =

( 1,

^if

| x | < a, 0,

^otherwise

,

with

a > 0

^{. However,ourresultsdonotapplytothis kernel,sinethehangesof}

signof

c W

^{willpreventthataninequalitysuhas(1.16)anbesatised. Moreover,}

in thisasetheenergyouldbenegativemakingmorediulttheanalysis. Nev-

ertheless,

W c

^{ispositivenear theoriginand thesonispeedisstillwelldened,so}

thatit isanopenquestionto establishwhiharetheexatimpliations ofhange

ofsignoftheFouriertransforminthenonexisteneresults.

1.6. Outline of the proofs and organization of the paper. We reall that

Theorem1.1-(i)followsfrom alassialPohozaevidentity. Gravejatin[16℄proves

Theorem 1.1-(ii) by ombining therespetive Pohozaev identity with an integral

equalityobtainedfromtheFourieranalysisoftheequationsatisedby

1 −| v | ²

^{. Our}

resultsarederivedin thesamespirit. Inthenextsetionweprovethat onditions

(H1)and(H2)implytheregularityofsolutionsof (NTW

c

^{) . InSetion3weprove}

that ondition (H6) allows us to generalize the arguments in [16℄ sothat we an

derivethe integral identity (3.1). The fat that the set

Γ j,c

^{is desribed by the}

urves

γ ^± _j,c

^{isaonsequeneoftheMorselemma,asexplainedinSetion4.}

In Setion 5 we establish a Pohozaev identity for (NTW

c

^{) with a remainder}

term depending onthederivativesof

W c

^{. Although this identityanbeformally}

obtainedforrapidlydeayingfuntions,itsproofforfuntionsin

E ( R ^N )

^isthemajor

tehnialdiultyofthispaperandreliesonFourieranalysisandthefatthat

W

iseven. Asin[8℄, wethenseein Setion6that Theorem1.6isasstraightforward

onsequeneofthisrelation.

InSetion 6 wealso show that we anreasttheidentities desribed aboveas

a suitable linear system of equations for whih we an invokethe Farkas lemma

to obtain thenonexistene onditionsgiven in Theorems 1.8 and 1.3. Theorol-

laries stated in Subsetion 1.4 then follow by hoosing the values of

σ 1 , . . . , σ N

appropriately.

Notations.Weadoptthestandardnotation

C( · , · , . . . )

^{torepresentagenerion-}

stant that depends only oneah of its arguments. Forany

x, y ∈ R ^N

,

z, w ∈ C

, wedenotethe inner produtsin

R ^N

and

C

, respetively,by

x.y = P N

i=1 x i y i

^and

h z, w i = Re(zw)

^{. The Kronekerdelta}

δ k,j

^{takes thevalue oneif}

k = j

^andzero

otherwise.

F (f )

^or

f b

^{standfortheFouriertransformof}

f

^,namely

F (f )(ξ) = f b (ξ) =

Z

R ^N

f (x)e ^{− ix.ξ} dx,

and

F ^{− 1}

^{foritsinverse.}

From nowon wex

c ≥ 0

^{. We denote by}

v = v 1 + iv 2

⁽

v 1

^,

v 2

real-valued) a solutionof (NTW

c

⁾ⁱⁿ

E ( R ^N )

^{. Wealsosetthe}real-valuedfuntions

ρ := | v | = (v ² ₁ + v ² ₂ ) ^1/2 , η := 1 − | v | ² .

2. Regularity of solutions

Lemma 2.1. Assume that

W ∈ M ^2,2 ( R ^N )

^{. Then}

v ∈ W _loc ^{2, 4/3} ( R ^N )

^{. Suppose}

further that

2 ≤ N ≤ 3

^{. Then}

v

^{is smooth and bounded. Moreover,}

η

^and

∇ v

belong to

W ^k,p ( R ^N )

^{,for all}

k ∈ N

,

2 ≤ p ≤ ∞

^.

Proof. Let

x ¯ ∈ R ^N

and

B r := B(¯ x, r)

^{theballofenter}

x ¯

^andradius

r

^{. Then}

(2.1)

k v k L ⁴ (B 1 ) = k| v | ² k L ² (B 1 ) ≤ k| v | ² − 1 k L ² ( R ^N ) + k 1 k L ² (B 1 ) ≤ E(v) + C(N ).

Onthe other hand,weandeompose

v

^as

v = z 1 + z 2 + z 3

^{, where}

z 1 , z 2

^and

z 3

arethesolutionsofthefollowingequations

(2.2)

− ∆z 1 = 0,

ⁱⁿ

B 1 , z 1 = v,

^on

∂B 1 ,

(2.3)

− ∆z 2 = ic∂ 1 v,

ⁱⁿ

B 1 , z 2 = 0,

^on

∂B 1 ,

(2.4)

( − ∆z 3 = v(W ∗ η),

ⁱⁿ

B 1 , z 3 = 0,

^on

∂B 1 .

Sine

z 1

^{isaharmonifuntion,}

k z 1 k C ^k (B 1/2 ) ≤ C(N, k, E(v)),

forall

k ∈ N

. Using theHölder inequality, (2.1)and elliptiregularityestimates, wealsohave

k z 2 k W ^2,2 (B 1 ) ≤ C(N, E (v)), k z 3 k W ^2,4/3 (B 1 ) ≤ C(N, E (v)) k W c k L ^∞ (R ^N ) k η k L ² (R ^N ) .

Therefore

k v k W ^2,4/3 (B 1/2 ) ≤ C(N, E(v), η, W )

^. Furthermore, by the Sobolev em- beddingtheorem wededuethat

k v k L ^∞ (B 1/2 )

^{is bounded for}

N = 2

^{and thenthis}

bound holds uniformly in

R ²

. If

N = 3

^{, we onlude that}

k v k L ¹² (B _1/2 )

^{is uni-}

formly bounded. Then usingthesamedeomposition(2.2)(2.4) intheball

B 1/4

^,

idential argumentsprovethat

k v k W ^2,12/7 (B _1/4 ) ≤ C(N, E(v), η, W )

^{, whih bythe}

Sobolev embedding theorem in dimension three implies that

k v k L ^∞ (B 1/4 )

^{is uni-}

formlybounded. Consequently,

v ∈ L ^∞ ( R ^N )

^for

N = 2, 3

^.

Finally,usingagain(2.2)(2.4)andastandardbootstrapargument,weonlude

that

v ∈ W ^{k, ∞} ( R ^N )

^forall

k ∈ N

.

Now,setting

w = ∂ j v

^,

j ∈ { 1, . . . , N }

^,anddierentiating(NTW

c

^)withrespet

to

x j

^{,weobtainforany}

λ ∈ R

L λ (w) := − ∆w − ic∂ 1 w + λw = ∂ j v(W ∗ η) + v(W ∗ ∂ j η) + λw,

ⁱⁿ

R ^N .

Sine

∇ v ∈ L ^∞ ( R ^N ) ∩ L ² ( R ^N )

^{, we dedue that the r.h.s. belongs to}

L ² ( R ^N )

^.

Then, for

λ > 0

^{large enough, we an apply the LaxMilgram theorem to the}

operator

L λ

^{to dedue that}

w ∈ H ² ( R ^N )

^{. Thus}

∇ v ∈ H ² ( R ^N )

^{and a bootstrap}

argumentshowsthat

∇ v ∈ H ^k ( R ^N )

^{, forall}

k ∈ N

andtherefore,byinterpolation,

∇ v, η ∈ W ^k,p ( R ^N )

^,forall

p ≥ 2

^and

k ∈ N

.

InLemma 2.1,weneeded todierentiatetheequation (NTW

c

^{)to improvethe}

regularity, whih required that

W ∗ ∇ η

^was well-dened. If

N ≥ 4

^{, proeeding}

as in Lemma 2.1, we an only infer that

∇ η ∈ L ^4/3 _loc ( R ^N )

^{so that it is not lear}

that wean givea sense to theterm

W ∗ ∇ η

^{. Onthe other hand,if}

N ≥ 3

^{, the}

fat that

∇ v ∈ L ² ( R ^N )

^{impliesthat there exists}

z 0 ∈ C

with

| z 0 | = 1

^{suh that}

v − z 0 ∈ L ^{N−2 2N} ( R ^N )

^{(see e.g. [19, Theorem 4.5.9℄). Moreover, sine (NTW}

c

^{) is}

invariantbyahangeofphase,weanassumethat

v − 1 ∈ L ^{N 2N − 2} ( R ^N )

^{. Therefore,}

(2.5)

∇ η = − 2 h v − 1, ∇ v i − 2 h 1, ∇ v i ∈ L ^{N/(N − 1)} ( R ^N ) + L ² ( R ^N ).

Then itwouldbereasonableto suppose that

W ∈ M N/N − 1,q ( R ^N )

^{,for some}

q ≥ N/N − 1

^{. However,thisisnotenoughtoinvoketheelliptiregularityestimatesand}

thatisreasonwhyweworkwiththeassumption(1.7)in(H2)if

N ≥ 4

^{. Weremark}

thattoestablishpreiseonditionson

W

^{thatensuretheregularityofsolutionsof}

(NTW

c

^{)in higherdimensionsgoesbeyondthesopeofthispaper.}

Lemma2.2. Let

N ≥ 4

^{. Assumethat}

W

^{satises (H2). Then}

v

^isboundedand

smooth. Moreover,

η

^and

∇ v

^{belong to}

W ^k,p ( R ^N )

^{,for all}

k ∈ N

,

2 ≤ p ≤ ∞

^.

Proof. From (1.7), by duality (see e.g. [15℄) we infer that

W ∈ M ^1,N ( R ^N ) ∩ M 1,2N/(N +2) ( R ^N )

^{. Then, from the} RieszThorin interpolation theorem and the fatthat

(1/2, (N − 2)/(2N))

^and

((N − 1)/N, (N − 2)/(2N ))

^{belongtotheonvex}

hullof

1 2 , 1

2

,

N − 1 N , 0

,

N − 2 2N , 0

,

1, 1

N

,

1, N + 2 2N

,

weonludethat

(2.6)

W ∈ M 2,2N/(N − 2) ( R ^N ) and W ∈ M N/(N − 1),2N/(N − 2) ( R ^N ).

As mentioned before, we anassume that

˜ v := v − 1 ∈ L ^{N 2N − 2} ( R ^N )

^{. Then using}

(H2), (2.5)and(2.6),weareledto

(2.7)

W ∗ η, W ∗ ∇ η ∈ L ^∞ ( R ^N ) ∩ L ^{2N/(N − 2)} ( R ^N ).

Nowwereast(NTW

c

^)as

(2.8)

L λ (˜ v) := − ∆˜ v − ic∂ 1 v ˜ + λ˜ v = ˜ v((W ∗ η) + λ) + W ∗ η,

ⁱⁿ

R ^N ,

for some

λ > 0

^{. By (2.7), the r.h.s. of (2.8) belongs to}

L ^{2N/(N − 2)} ( R ^N )

^{. Then}

hoosing

λ

^{largeenough,weanapplyelliptiregularityestimatestotheoperator}

L λ

^toonludethat

v ˜ ∈ W ^{2,2N/(N − 2)} ( R ^N )

^{. Then}

∂ j,k η = − 2( h v − 1, ∂ j,k v i + h ∂ j v, ∂ k v i + h 1, ∂ j,k v i ) ∈ L ^{N/(N − 1)} ( R ^N )+L ^{2N/(N − 2)} ( R ^N ),

for any

1 ≤ j, k ≤ N

^{. Therefore, by (1.7) and (2.6),}

W ∗ ∂ j,k η ∈ L ^∞ ( R ^N ) ∩ L ^{2N/(N − 2)}

^{. Thus the r.h.s. of (2.8) belongs to}

W ^{2,2N/(N − 2)} ( R ^N )

^{, so that}

˜ v ∈ W ^{4,2N/(N − 2)} ( R ^N )

^{. A bootstrap argument yields that}

v ˜ ∈ W ^{k,2N/(N − 2)} ( R ^N )

^{, for}

any

k ∈ N

. BytheSobolevembeddingtheorem, weonludethat

v ∈ W ^{k, ∞} ( R ^N )

forany

k ∈ N

. ThentheonlusionfollowsasinLemma2.1.

Lemma2.3. Let

W ∈ L ¹ ( R ^N )

^if

2 ≤ N ≤ 3

^and

W ∈ L ¹ ( R ^N ) ∩ L ^N ( R ^N )

^if

N ≥ 4

^.

Then

W

^{fullls (H2).}

Proof. Sine

W ∈ L ¹ ( R ^N )

^{,bytheYounginequalitywehave}

k W ∗ f k L ^p ( R ^N ) ≤ k W k L ¹ ( R ^N ) k f k L ^p ( R ^N ) ,

^forany

p ∈ [1, ∞ ].

Then, taking

p = 2

^{, weonlude that (H2) holdsfor}

2 ≤ N ≤ 3

^{. For}

N ≥ 4

^{, we}

have

W ∈ L ¹ ( R ^N ) ∩ L ^N ( R ^N )

^{. Inpartiular,}

W ∈ L ^{2N/(N +2)} ( R ^N )

^{andthe Young}

inequalityimpliesthat

k W ∗ f k L ^∞ ( R ^N ) ≤ k W k L ^N ( R ^N ) k f k L ^{N/(N − 1)} ( R ^N ) , k W ∗ f k L ^∞ (R ^N ) ≤ k W k L ^2N/(N+2) ( R ^N ) k f k L ^2N/(N−2) ( R ^N ) .

Therefore(H2)issatised.

Corollary 2.4. Assume that

W

^{satises (H2). Then}

v

^{is smooth and bounded.}

Moreover,

η

^and

∇ v

^{belong to}

W ^k,p ( R ^N )

^{,for all}

k ∈ N

,

2 ≤ p ≤ ∞

^,and

(2.9)

ρ(x) → 1, ∇ v(x) → 0,

^as

| x | → ∞ .

Furthermore, there existsasmooth liftingof

v

^{. Morepreisely, there exist}

R 0 > 0

andasmoothreal-valuedfuntion

θ

^denedon

B(0, R 0 ) ^c

^,with

∇ θ ∈ W ^k,p (B(0, R 0 ) ^c )

^,

for all

k ∈ N

,

2 ≤ p ≤ ∞

^,suhthat

(2.10)

ρ ≥ 1

2

^and

v = ρe ^iθ

^on

B(0, R 0 ) ^c .

Proof. Therst partisexatlyLemmas 2.1 and2.3. Inpartiular,

v

^and

∇ v

^are

uniformly ontinuous on

R ^N

. Then, sine

1 − | v | ² ∈ L ² ( R ^N )

^and

∇ v ∈ L ² ( R ^N )

^,

we obtain (2.9). The existene of the lifting satisfying (2.10) follows as in [25,

Proposition2.5℄. From(2.10)wealsodeduethat

|∇ v | ² = |∇ ρ | ² + ρ ² |∇ θ | ²

^on

B(0, R 0 ) ^c .

Sine

ρ ≥ 1/2

^on

B(0, R 0 ) ^c

^{, we infer that}

∇ θ ∈ W ^k,p (B(0, R 0 ) ^c )

^{, for all}

k ∈ N

,

2 ≤ p ≤ ∞

^.

InvirtueofCorollary2.4,weintroduethefuntion

φ ∈ C ^∞ ( R ^N )

^,

| φ | ≤ 1

^,suh

that

φ = 0

^on

B(0, 2R 0 )

^and

φ = 1

^on

B(0, 3R 0 ) ^c

^{. Inthisway,weanassumethe}

funtion

φθ

^{iswell-dened on}

R ^N

. Thiswill beusefulin thenextsetiontowork withglobalfuntions intermsof

θ

^{. Infat,weend thissetionwiththefollowing}

result.

Lemma2.5. Assumethat

W

^{satises (H2). Then}

(2.11)

G := v 1 ∇ v 2 − v 2 ∇ v 1 − ∇ (φθ),

^on

R ^N ,

belongsto

W ^k,p ( R ^N )

^{,for all}

k ∈ N

and

1 ≤ p ≤ ∞

^.

Proof. ByCorollary2.4,

G ∈ C ^∞ ( R ^N )

^andmoreover

G = − η ∇ θ

^on

B(0, 3R 0 ) ^c .

Sine

∇ θ ∈ W ^k,p (B(0, R 0 ) ^c )

^and

η ∈ W ^k,p (B(0, R 0 ) ^c )

^{, for all}

k ∈ N

,

2 ≤ p ≤ ∞

^,

theonlusionfollows.

3. An integral identity

Theaimofthissetionistoprovethefollowingintegralidentity.

Proposition3.1. Let

c > 0

^{. Supposethat (H2) and (H6) hold with}

ℓ j,c > 0

^{, for}

some

j ∈ { 2, . . . , N }

^{. Then}

(3.1)

Z

R ^N

( |∇ v | ² + η(W ∗ η)) = − c ℓ j,c

1 + ℓ j,c

Z

R ^N

(v 1 ∂ 1 v 2 − v 2 ∂ 1 v 1 − ∂ 1 (φθ)).

We note that sine

W

^{satises (H2), all the resultsof Setion 2 hold. On the}

otherhand,from(NTW

c

^)wededuethat

η = 1 − | v | ²

^satises

(3.2)

∆η = − F + 2W ∗ η − 2c∂ 1 (φθ).

where

F := 2 |∇ v | ² + 2η(W ∗ η) + 2cG 1 .

and

G = (G 1 , . . . , G N )

^{wasdenedin(2.11).} Consideringrealandimaginaryparts in(NTW

c

^)andmultiplyingthemby

v 2

^and

v 1

^,respetively,itfollowsthat (3.3)

div(G) = v 1 ∆v 2 − v 2 ∆v 1 − ∆(φθ) = c

2 ∂ 1 η − ∆(φθ).

Therefore,from(3.2)and(3.3),weonludethat

(3.4)

∆ ² η − 2∆(W ∗ η) + c ² ∂ ₁₁ ² η = − ∆F + 2c∂ 1 (div G),

ⁱⁿ

R ^N .

Sineweareassuming(H2), byCorollary2.4andLemma2.5,wehavethat

F, G ∈ W ^k,1 ( R ^N ) ∩ W ^k,2 ( R ^N )

^{,for all}

k ∈ N

,so that (3.4)standsin

L ² ( R ^N )

^{. Takingthe}

Fouriertransformin equation(3.4)andsetting

R(ξ) := | ξ | ⁴ + 2c W (ξ) | ξ | ² − c ² ξ ² ₁

^and

H (ξ) := | ξ | ² F(ξ) b − 2c X N j=1

ξ 1 ξ j G b j (ξ),

weget

(3.5)

R(ξ) η(ξ) = b H(ξ),

ⁱⁿ

L ² ( R ^N ).

Lemma 3.2. Let

c > 0

^{. Suppose that (H2) and (H6) hold. Then for all}

j ∈ { 2, . . . , N }

^,

(3.6)

H (te 1 + γ ^± _j,c (t)e j ) = 0,

^{for all}

t ∈ (0, δ),

where

δ

^{isgiven by (H6).}

Proof. Wex

j ∈ { 2, . . . , N }

^{andweprove(3.6)for}

γ _j,c ⁺

^{,sinetheprooffor}

γ _j,c ⁻

^is

analogous. To simplify the notation, we put

γ := γ _j,c ⁺

^{. As statedbefore,}

F, G ∈ W ^k,1 ( R ^N ) ∩ W ^k,2 ( R ^N )

^{, for all}

k ∈ N

. In partiular

F, G ∈ L ¹ ( R ^N )

^{, so that}

F b

^,

G b ∈ C( R ^N )

^{. Thus}

H

^{isaontinuousfuntion on}

R ^N

.

Let

δ > 0

^{givenby(H6). Arguingby}ontradition,wesupposethatthere exist

t 0 ∈ (0, δ)

^andaonstant

A > 0

^suhthat

| H ( ˜ ξ) | ≥ A

^{, where}

ξ ˜ = t 0 e 1 + γ(t 0 )e j

^.

Bythe ontinuity of

H

^{, there exists}

r > 0

^{suh that}

| H (ξ) | ≥ A

^{, for all}

ξ ∈ V r

^,

where

V r = B( ˜ ξ, r) ∩ { αe 1 + βe j : α, β ∈ R } .

Thus

V r

^{is a}two-dimensionalset andsine

t 0 > 0

^{, weanhoose}

r

^smallenough

suhthat

0 ∈ / V r

^{. Then(3.5)yields}

(3.7)

|b η(ξ) | ² ≥ A ²

(R(ξ)) ² ,

^forall

ξ ∈ V r \ Γ j,c .

Welaimthat

(3.8)

I :=

Z

V r \ Γ j,c

dξ 1 dξ j

(R(ξ)) ² = + ∞ .

Sinebyhypothesis

Γ j,c ∩ B (0, δ)

^{hasmeasurezero,(3.7)and(3.8)ontraditthat}

b

η ∈ L ² ( R ^N )

^.

Toprove(3.8),sine

V r

^isatwo-dimensionalset, weidentifyitasasubsetof

R ²

andsothat wewrite

e 2

^insteadof

e j

^{. Then,sine}

Γ j,c ∩ B(0, δ)

^{hasmeasurezero,}

I =

Z

V r

dξ 1 dξ 2

(R(ξ)) ² .

Toomputetheintegralwestraightenouttheurve

γ

^{. Namely,weintroduethe}

hangeof variables

ξ 1 = ν 1 =: Φ 1 (ν 1 , ν 2 ),

ξ 2 = ν 2 + γ(ν 1 ) =: Φ 2 (ν 1 , ν 2 ).

Sine

γ

^{is a}

C ¹

^{-funtion, sois}

Φ

^{. Moreover,there is someset}

U r

^{suh that}

V r = Φ(U r )

^and

| det(JΦ(ν)) | = 1

^{for all}

ν ∈ U r

^{. Setting}

F (ν) := R(Φ(ν))

^,

ν ∈ U r

^,the

hangeof variablestheorem yields

(3.9)

I =

Z

U r

dν 1 dν 2

(F (ν)) ² .

Furthermore,sine

F ∈ C ¹ (U r )

^and

F (ν 1 , 0) = 0

^{for all}

(ν 1 , 0) ∈ U r

^{, the Taylor}

theoremimpliesthatforany

(ν 1 , ν 2 ) ∈ U r

^,thereissome

¯ ν ∈ U r

^suhthat

(3.10)

F(ν 1 , ν 2 ) = F(ν 1 , 0) + ∂F

∂ν 2 (¯ ν)ν 2 = ∂F

∂ν 2 (¯ ν)ν 2 ,

Onthe otherhand,by (H2),

c W ∈ L ^∞ ( R ^N )

^{andby (H3),}

∇ W ∈ L ^∞ (V r )

^{, so that}

k c W k W ^1,∞ (V r ) < ∞

^{. Thus}

k∇ F k L ^∞ (U r ) ≤ C(r, γ)(1+ k c W k W ^1,∞ (V r ) )

^{andfrom(3.10)}

weonludethat

(3.11)

| F (ν) | ≤ C(r, γ)(1 + k W c k W ^{1, ∞} (V r ) ) | ν 2 | ,

^forall

ν ∈ U r .

From(3.9)and(3.11),taking

ν ˜ = (˜ ν 1 , ˜ ν 2 ) ∈ U r

^suhthat

ξ ˜ = Φ(˜ ν)

^and

ε > 0

^small

enough,weonludethat

I ≥ C(r, γ, W c ) Z

U r

dν 1 dν 2

ν ₂ ² ≥ C(r, γ, W c ) Z ν ˜ 1 +ε

˜ ν 1 − ε

Z ε

− ε

dν 2 dν 1

ν ₂ ² = + ∞ ,

whihonludestheproof.

Finally,wegivetheproofofidentity (3.1).

Proofof Proposition3.1. ByLemma 3.2,setting

ξ ^± (t) = te 1 + γ _j,c ^± (t)e j ,

^wehave

(t ² + (γ _j,c ^± (t)) ² ) F b (ξ ^± (t)) − 2ct ² G b 1 (ξ ^± (t)) − 2ctγ _j,c ^± (t) G b j (ξ ^± (t)) = 0, t ∈ (0, δ).

Dividingby

t ²

^{andpassingtothelimit}

t → 0 ⁺

^,

(1+ℓ j,c ) F b (0) − 2c G b 1 (0) − 2c p

ℓ j,c G b j (0) = (1+ℓ j,c ) F b (0) − 2c G b 1 (0)+2c p

ℓ j,c G b j (0) = 0.

Therefore,sine

ℓ j,c > 0

^,

G b j (0) = 0

^and

(1 + ℓ j,c ) F b (0) = 2c G b 1 (0)

^{,whihispreisely}

(3.1).

AsaonsequeneofProposition3.1, weobtainLemma1.7.

Proofof Lemma 1.7. From(3.1),setting

J (v) = Z

R ^N

( |∇ v | ² + η(W ∗ η))

^and

P (v) = Z

R ^N

(v 1 ∂ 1 v 2 − v 2 ∂ 1 v 1 − ∂ 1 (φθ)),

weinferthat

(3.12)

ℓ j,c (J (v) + cP (v)) = − J (v).

Sine

v

^{is nononstantand}

W c ≥ 0

^,wehavethat

J (v) > 0

^{. Thenwededue from}

(3.12)that

J(v) + cP (v) 6 = 0

^and

ℓ j,c = − J(v) J(v) + cP (v) .

Sinether.h.s. oftheequalitydoesnotdepend on

j

^{, theonlusion follows.}

4. The set

Γ j,c

^{underthe ondition (H5)}

InSetion3wehaveseenthatidentity(3.1)isaonsequeneofthestrutureof

theset

Γ j,c

^{. Morepreisely,itreliesonthefatthat(H6)providestheexisteneof}

δ > 0

^andtwourves

γ _j,c ^±

^{suh that}

{ (t, y ^± (t)) : t ∈ ( − δ, δ) } ⊆ Γ j,c .

If

W c

^isoflass

C ²

^inaneighborhoodoftheoriginand

α c := c ²

(c s (W )) ² − 1 > 0,

we an use the Morse lemma to justify the existene of the urves

γ _j,c ^±

^{and to}

onludethatset

Γ j,c

^{onsistsofexatlythesetwourvesneartheorigin. Therefore}

theset

Γ j,c

^{lookslikeFigure2andondition(H6)isfullled.}

Lemma4.1. Assumethat (H1) and (H5) hold. Assumealso that

α c > 0

^{. Then,}

for eah

j ∈ { 2, . . . , N }

^{, there exist}

δ > 0

^{and funtions}

y ^± ∈ C ¹ (( − δ, δ)) ∩ C ² (( − δ, δ) \ { 0 } )

^suhthat

(4.1)

Γ j,c ∩ B(0, δ) = { (t, y ^± (t)) : t ∈ ( − δ, δ) } .

Moreover,

(4.2)

lim

t → 0 ⁺ y ^± (t)/t = ± √ α c ,

y ⁺

^{isstritlyinreasingand}

y ⁻

^{isstritlydereasing. Inpartiular,(H6)issatised}

with

l j,c = α c

^.

γ _j,c ⁺ (t)

γ _j,c ⁻ (t) t

Figure 2. Theset

Γ j,c

^{neartheoriginfor}

W c

^oflass

C ²

^.

Proof. Letusset

R j (ν) := | ν | ⁴ + 2 w _j (ν ) | ν | ² − c ² ν ₁ ² , ν = (ν 1 , ν 2 ) ∈ R ² .

Inviewof(H5),

R j ∈ C ² (B(0, δ 0 ))

^,forsome

δ 0 > 0

^{. Sine}

w _j

iseven,wehavethat

∂ 1 w _j (0, 0) = ∂ 2 w _j (0, 0) = 0

^{. Thenweobtain}

R j (0, 0) = 0

^,

∇ R j (0, 0) = 0

^,

(4.3)

∂ ² R j

∂ν ₁ ² (0, 0) = − 4α c w _j (0, 0) < 0, ∂ ² R j

∂ν ₂ ² (0, 0) = 4 w _j (0, 0) > 0, ∂ ² R j

∂ν 1 ∂ν 2

(0, 0) = 0.

Therefore bythe Morselemma (see e.g. [26, Theorem II℄) there exist twoneigh-

borhoods of the origin

U, V ⊂ R ²

and a loal dieomorphism

Φ : U → V

^suh

that

(4.4)

R j (Φ ^{− 1} (z)) = − 2α c w _j (0, 0)z ² ₁ + 2 w _j (0, 0)z ₂ ² ,

^forall

z = (z 1 , z 2 ) ∈ V.

Moreover,denoting

Φ = (Φ 1 , Φ 2 )

^wehavefor

1 ≤ j, k ≤ 2

(4.5)

∂Φ j

∂ν k

(ν) → δ j,k ,

^as

| ν | → 0.

From(4.4)wededue that near theoriginthe set ofsolutions of

R j = 0

^{is given}

bythelines

{ (t, ± √

α c t) : t ∈ ( − δ, δ) } ,

wherewetake

δ > 0

^{suhthatthesetisontainedin}

V

^{. Sine}

Φ

^isadieomorphism weonludethat

(4.6)

Γ j,c ∩ B(0, δ) = { (x ^± ₁ (t), x ^± ₂ (t)) : t ∈ ( − δ, δ) } ,

where

Φ 1 (x ^± ₁ , x ^± ₂ ) = t,

(4.7)

Φ 2 (x ^± ₁ , x ^± ₂ ) = ± √ α c t.

(4.8)

Moreover,dierentiatingrelation (4.7)with respet to

t

^{andusing (4.5), weinfer}

that

(x ^± ₁ ) ^′ (t) → 1

^as

t → 0

^{. Therefore we anreast(4.6) asin (4.1) with}

y ^± ∈

C ¹ (( − δ, δ)) ∩ C ² (( − δ, δ) \ { 0 } )

^{. F}urthermore,dierentiating(4.8)andusingagain (4.5)weonludethat

(y ^± ) ^′ (0) = ± √ α c .

Sine

y ^± ∈ C ¹ (( − δ, δ))

^{, taking a possible smaller value}

δ

^{, this implies (4.2) and}

that

y ⁺

^and

y ⁻

^{arestritlyinreasinganddereasingon}

( − δ, δ)

^,respetively.

5. A Pohozaevidentity

InthissetionweestablishthefollowingPohozaevidentity.

Proposition5.1. Assumethat (H1)(H3)hold. Then

E(v) = Z

R ^N

| ∂ 1 v | ² + 1 4(2π) ^N

Z

R ^N

ξ 1 ∂ 1 W c |b η | ² dξ,

(5.1)

E(v) = Z

R ^N

| ∂ j v | ² − c 2

Z

R ^N

(v 1 ∂ 1 v 2 − v 2 ∂ 1 v 1 − ∂ 1 (φθ)) + 1 4(2π) ^N

Z

R ^N

ξ j ∂ j W c |b η | ² dξ,

(5.2)

for all

j ∈ { 2, . . . , N } .

Notethat byLemma 2.5,

G 1 = v 1 ∂ 1 v 2 − v 2 ∂ 1 v 1 − ∂ 1 (φθ) ∈ L ¹ ( R ^N )

^{, thus every}

integralin(5.1)and(5.2)isnite. AsmentionedinSetion1,intheasethat

W

^is

theDiradeltafuntionthisresultiswell-known(see[8,6,16,25℄). Thestandard

tehniqueis to introdue afuntion

χ ∈ C ^∞ ( R )

^{, with}

χ(x) = 1

^for

| x | < 1

^and

χ(x) = 0

^for

| x | > 2

^,and

χ n (x) := χ(x/n)

^{. Then,}multiplying(NTW

c

^)by

x j χ n ∂ j v ¯

andtakingrealpart,weareledto

(5.3)

h ic∂ 1 v + ∆v, x j χ n ∂ j v i − 1

2 (W ∗ η)x j χ n ∂ j η = 0,

^on

R ^N ,

wherewehaveusedthat

h v, ∂ j v i = − 1 2 ∂ j η.

Conerning(5.3),wereallthefollowingresult.

Lemma5.2 ([8,6, 16,25℄). Let

ϕ = ϕ 1 + ϕ 2 ∈ E ( R ^N ) ∩ C ^∞ ( R ^N )

^{. Assumethat}

there exist

R ^∗ > 0

^{and asmooth} real-valuedfuntion

θ ˜

^{dened on}

B(0, R ^∗ ) ^c

^{, with}

∇ θ ˜ ∈ L ² (B(0, R ^∗ ) ^c )

^,suhthat

| ϕ | ≥ 1

2

^and

ϕ = | ϕ | e ^{i θ ˜}

^on

B(0, R 0 ) ^c .

Let

φ ˜ ∈ C ^∞ ( R ^N )

^{,suh that}

φ ˜ = 0

^on

B(0, 2R ^∗ )

^and

φ ˜ = 1

^on

B (0, 3R ^∗ ) ^c

^{. Then}

for all

j ∈ { 1, . . . , N }

^,wehave

n lim →∞

Z

R ^N

h i∂ 1 ϕ, x j χ n ∂ j ϕ i = 1

2 (1 − δ 1,j ) Z

R ^N

(ϕ 1 ∂ 1 ϕ 2 − ϕ 2 ∂ 1 ϕ 1 − ∂ 1 ( ˜ φ θ)), ˜

(5.4)

n lim →∞

Z

R ^N

h ∆ϕ, x j χ n ∂ j ϕ i = − Z

R ^N

| ∂ j ϕ | ² + 1 2 Z

R ^N

|∇ ϕ | ² ,

(5.5)

n lim →∞ − 1 2 Z

R ^N

x j χ n (1 − | ϕ | ² )∂ j (1 − | ϕ | ² ) = 1 4 Z

R ^N

(1 − | ϕ | ² ) ² .

(5.6)