A refined stability result for standing waves of the Schr¨ odinger-Maxwell system

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A refined stability result for standing waves of the Schr¨ odinger-Maxwell system

Mathieu Colin · Tatsuya Watanabe

Received: date / Accepted: date

Abstract In this paper, we are interested in standing waves of the nonlinear Schr¨odinger equation coupled with the Maxwell equation. Our aim is to for- mulate the orbital stability of standing waves in the full gauge invariant form.

For this purpose, we study a new constraint minimization problem.

Keywords Schr¨odinger-Maxwell system, constraint minimization problem, orbital stability.

Mathematics Subject Classification (2010) 35J20, 35B35, 35Q55

1 Introduction

In this paper, we consider the following nonlinear Schr¨odinger equation coupled with Maxwell equation stated inR+×R³:

iψ_t+∆ψ=eϕψ+e²|A|²ψ+ 2ie∇ψ·A+ieψdivA− |ψ|^p−1ψ, (1.1) Att−∆A=eIm( ¯ψ∇ψ)−e²|ψ|²A− ∇ϕt− ∇divA, (1.2)

−∆ϕ= e

2|ψ|²+ divAt, (1.3)

where ψ : R+×R³ → C, A : R+×R³ → R³, ϕ : R+×R³ → R, e > 0 and i denotes the unit complex number, that is, i² = −1. We recall that System (1.1)-(1.3) describes the interaction of the Schr¨odinger wave function ψwith the gauge potential (A, ϕ) and the constante represents the strength of the interaction. We refer to [18] for more physical backgrounds. Our aim

M. Colin

INRIA CARDAMOM, 200 Avenue de la Vieille Tour, 33405 Talence, Cedex-France, Bor- deaux INP, UMR 5251, F-33400,Talence, France E-mail: [email protected] T. Watanabe

Department of Mathematics, Faculty of Science, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-ku, Kyoto-City, 603-8555, Japan E-mail: [email protected]

is to complete the study of orbital stability of standing waves of (1.1)-(1.3) initiated in [15] by focusing on the particular casep= 2.

It is known that System (1.1)-(1.3) has a so-calledgauge ambiguity. Namely if (ψ,A, ϕ) is a solution of (1.1)-(1.3), then (exp(ieχ)ψ,A+∇χ, ϕ−χ_t) is also a solution of (1.1)-(1.3) for any smooth function χ : R+×R³ →R. To rule out this ambiguity, we adopt the Coulomb gauge:

divA= 0, (1.4)

which is propagated by the set of Equations (1.1)-(1.3). (See e.g. [14,?] for the proof.) In this setting, the last Equation (1.3) can be solved explicitly and the solution is given by

ϕ= e

2(−∆)⁻¹|ψ|²= e

8π|x|∗ |ψ|².

Moreover from (1.4), one can observe that (1.1) can be written as

iψ_t+L_Aψ−V(x)ψ+|ψ|^p−1ψ= 0, (1.5) whereV is the non-local potential:V(x) = ^e₂²(−∆)⁻¹|ψ|² andLA is themag- netic Schr¨odinger operatorwhich is defined byA= (A1, A2, A3) and

LAψ:=

∑3

j=1

( ∂

∂x_j −ieAj(x) )2

ψ=∆ψ−2ie∇ψ·A−e²|A|²ψ.

WhenA ≡0, Equation (1.5) is also called the Schr¨odinger-Poisson(-Slater) equation:

iψ_t+∆ψ− ( e²

8π|x| ∗ |ψ|² )

ψ+|ψ|^p−1ψ= 0 in R³. (1.6) The existence of ground states related with (1.6) as well as their orbital sta- bility have been widely studied. (See [3], [7], [10], [20], [21], [28] and references therein.) The orbital stability of standing waves for the magnetic Schr¨odinger equation (1.5) with givenmagnetic potential has been considered in [1], [12], [17], [19]. Finally in [9], [15], the orbital stability of standing waves for the full system (1.1)-(1.3) has been studied. Our main result will enable us to generalize these previous results.

In the study of the stability of standing waves of (1.1)-(1.3), the following two conserved quantities play an fundamental role:

d dt

∫

R³|ψ|²dx= 0, (Charge conservation) (1.7) d

dtE(ψ,A) = 0, (Energy conservation) (1.8)

E(ψ,A) =1 2

∫

R³|∇ψ−ieAψ|²+|rotA|²+|At|²dx +e²

8

∫

R³|ψ|²(−∆)⁻¹|ψ|²dx− 1 p+ 1

∫

R³|ψ|^p+1dx.

One can observe that the conserved energyE(ψ,A) still has thegeneral gauge invariance:

E(

exp(ieχ)ψ,A+∇χ) =E(ψ,A) for any χ=χ(x)∈C²(R³,R). (1.9) In this point of view, we have to take account of this strong invariance in order to obtain the complete orbital stability for (1.1)-(1.3).

By the standing wave of (1.1)-(1.3), we mean a global solution (ψ,A, ϕ) of the form:

ψ(t, x) = exp(iωt)u(x), A(t, x) =0andϕ(t, x) =ϕ(x), where whereω >0 and (u, ϕ) solves

{−∆u+ωu+eϕu=|u|^p−1u

−∆ϕ=^e₂|u|². (1.10)

Following the reduction method performed in [3], [27], we introduce the func- tionalS as

S(u) :=1

2(−∆)⁻¹|u|²∈D^1,2(R³,R)

whereD^1,2(R³,R) denotes the completion ofC₀^∞(R³) with respect to the norm

∥u∥²_D1,2 =∫

R³|∇u|²dx. As a consequence, System (1.10) can be reduced to the single equation:

−∆u+ωu+e²S(u)u=|u|^p−1u. (1.11) Solutions to (1.11) are no more than the critical points of the energy functional

I_e,ω(u) =1 2

∫

R³|∇u|²+ω|u|²dx+e² 4

∫

R³

S(u)|u|²dx− 1 p+ 1

∫

R³|u|^p+1dx The set of ground statesGe(ω) can be defined as

Ge(ω) ={

u∈H¹(R³,C) ; I_e,ω^′ (u) = 0, Ie,ω(u) =me(ω)} , where

me(ω) = inf{

Ie,ω(u) ; I_e,ω^′ (u) = 0, u∈H¹(R³,C)\ {0}} .

In [3] and [15], the existence of ground states of (1.11) has been established in the situation 2≤p <5. Moreover, if 0< e < e0 is small enough, wheree0

depends also onωandp, the ground state of (1.11) is unique up to translation and phase shift. We note that in the particular casep= 2,e0does not depend

on ω and that, a posteriori, one can prove that the unique ground state of (1.11) is real-valued, positive and radially symmetric up to translations.

In view of introducing some notions of orbital stability, one first needs an initial value condition associated with System (1.1)-(1.3):

ψ(0, x) =ψ₍₀₎(x), A(0, x) =A₍₀₎(x), A_t(0, x) =A₍₁₎(x),

divA₍₀₎= 0, divA₍₁₎ = 0. (1.12) In [15], the standing wave

(ψe,ω,Ae,ω, ϕe,ω) :=

(

exp(iωt)ue,ω, 0, e

2(−∆)⁻¹|ue,ω|²) ,

where u_e,ω is the unique ground state of (1.11), is proved to be stable in the following weak sense: For every ε > 0, there exists δ(ε) >0 such that if an initial value (ψ₍₀₎,A₍₀₎,A₍₁₎) satisfies (1.12) and

∥ψ₍₀₎−ue,ω∥H¹+∥∇A₍₀₎∥L²+∥A₍₁₎∥L² < δ, (1.13) then the corresponding solution (ψ,A, ϕ) of (1.1)-(1.3) satisfies

sup

t>0

{ inf

y∈R³∇|ψ(t,·)| − ∇ue,ω(·+y)

L²+∥∇A(t,·)∥L²+∥At(t,·)∥L² (1.14)

+ inf

y∈R³,θ∈[0,2π)∥ψ(t,·)−exp(iθ)ue,ω(·+y)∥L²

+ inf

y∈R³

ϕ(t,·)−e

2(−∆)⁻¹|u_e,ω(·+y)|²

D^1,2

}

< ε.

The proof is based on the study of the following minimization problem:

˜

c_e(µ) := inf

u∈B(µ)

J˜(u), (1.15)

whereµ >0,B(µ) ={u∈H¹(R³,C), ∥u∥²_L2(R³)=µ} and J˜(u) := 1

2

∫

R³|∇u|²dx+e² 4

∫

R³

S(u)|u|²dx− 1 p+ 1

∫

R³|u|^p+1dx.

As one may sees in (1.14), the main issue concerning the stability result of [15] lies in the fact that, in the previous definition of stability, one has to take the absolute value ofψ. This problem comes from the use of the diamagnetic inequality:

∇|u|(x)≤ |(∇ −ieA)u(x)| a.e.x∈R³ (1.16) for u ∈ L²(R³,C) with

( ∂

∂xj −ieA_j )

u ∈ L²(R³,C) and A ∈ L²_loc(R³,R³).

(See e.g. [22] for the proof.) The main object of this article is to get rid of this constraint in order to take into account the full gauge invariance.

For that purpose, we define the energy functional involving the static mag- netic potential:

J(u,A) :=1 2

∫

R³|∇u−ieAu|²dx+1 2

∫

R³|∇A|²dx +e²

4

∫

R³

S(u)|u|²dx− 1 p+ 1

∫

R³|u|^p+1dx for (u,A)∈X, X=

{

(u,A)∈H¹(R³,C)×(

D^1,2(R³,R³))3

,divA= 0 }

, and consider the following new modified constraint minimization problem:

c_e(µ) := inf

(u,A)∈X,∥u∥²_L2=µ

J(u,A). (1.17)

One finds that the energy functionalJ(u,A) fulfills the general gauge invari- ance:

J(

exp(ieχ)u,A+∇χ) =J(u,A) for anyχ=χ(x)∈C²(R³,R).

We also define the set of minimizersMe(µ) by Me(µ) :={

(u,A)∈X ; ∥u∥²L² =µ, J(u,A) =c_e(µ)} .

In this setting,ω in (1.11) appears as a Lagrange multiplier. It is known that one of the key point for the stability is to establish a link betweenGe(ω) and Me(µ), see [11], [13].

Our first result concerns the existence of solutions to the minimization problem (1.17) and can be stated as follows.

Theorem 1.1 Let µ > 0 be given and suppose that 1 < p < ⁷₃. Then there existse^∗=e^∗(µ, p)>0 such that the following properties hold.

(i) If 2 < p < ⁷₃, the minimization problem (1.17) has a minimizer for 0 <

e≤e^∗ and no minimizer for e > e^∗.

(ii) If p= 2, it holds that e^∗(µ,2) =e^∗(2)≤²₃ for anyµ >0 and(1.17) has a minimizer for 0< e < e^∗. Moreover(1.17) has no minimizer fore > ²₃. (iii) If 1< p <2,(1.17) admits a minimizer for0< e < e^∗.

Furthermore in any cases, the minimizer of (1.17) has the form(u_e,0).

We now focus on the particular casep= 2. In this situation, one can prove that there exists e0 > 0 such that (1.11) has a unique (real-valued) ground state for anyω > 0 and 0 < e < e0. (See Proposition 4.1 below.) Then our refined stability result can be described as follows.

Theorem 1.2 Let ω >0be given. Supposep= 2,e <min(e0, e^∗)andue,ωis the unique real-valued ground state of (1.11). Then the standing wave

(ψe,ω,Ae,ω, ϕe,ω) :=

(

exp(iωt)ue,ω,0, e

2(−∆)⁻¹|ue,ω|²)

of (1.1)-(1.3) is stable in the following sense: For every ε > 0, there exists δ(ε)>0 such that if an initial value(ψ(0),A(0),A(1)) satisfies(1.12)and

∥∇ψ₍₀₎−ieA₍₀₎ψ₍₀₎−∇ue,ω∥L²+∥ψ₍₀₎−ue,ω∥L²+∥rotA₍₀₎∥L²+∥A₍₁₎∥L² < δ, (1.18) then the corresponding solution(ψ,A, ϕ)of (1.1)-(1.3)satisfies

sup

t>0

{ inf

y∈R³,χ∈C²

∇ψ(t,·)−ieA(t,·)ψ(t,·)−exp(

ieχ(t,·))

∇ue,ω(·+y)

L²

+ inf

y∈R³,χ∈C²

ψ(t,·)−exp(

ieχ(t,·))

ue,ω(·+y)

L²

+ inf

χ∈C²

(rot(

A(t,·)− ∇χ(t,·))

L²+∥(

A(t,·)− ∇χ(t,·))

t∥L²

)

+ inf

y∈R³,χ∈C²

ϕ(t,·)−ϕe,ω(·+y) +χt(t,·)

D^1,2

}

< ε.

We note that the restriction p= 2 is due to the necessity of proving the one-to-one correspondence betweenGe(ω) andMe(µ), which can be obtained, for the moment, only forp= 2. In the case 1< p < ⁷₃,p̸= 2 and 0< e < e^∗, we are able to show thestability of the minimizer set Me(µ). For this result, we refer to Remark 4.2 below.

Compared to our previous result in [15], we have replaced the modulus of ψby making use of the gauge invariant norm:

inf

y∈R³,χ∈C²(R+×R³,R)

∇ψ(t,·)−ieA(t,·)ψ(t,·)−exp(

ieχ(t,·))

∇ue,ω(·+y)

L². This new term is consistent with the use of the energy J(u,A) and the fact that our proof of Theorem 1.2 is based on the classical variational approach developed in [11], [13]. Moreover, as we will see in the proof of Theorem 1.2,

∥ψ₍₀₎−u_e,ω∥H¹ can be controlled in terms of

∥∇ψ(0)−ieA(0)ψ(0)− ∇ue,ω∥L²+∥ψ(0)−ue,ω∥L²+∥∇A(0)∥L². Recalling that rotA₍₀₎ =∇A₍₀₎ when divA₍₀₎ = 0, one observes that (1.18) implies (1.13). This yields that Theorem 1.2 is a complete extension of our previous stability result in [15], as well as the ones for (1.6) by takingA≡0 andχ≡θ.

We remark that our stability result, as the one exposed in [15], is still conditional in the following sense. In order to obtain the complete stability result, one requires a global well-posedness theory in the energy space. In this direction, we refer to [16] in which a local existence theory in higher order Sobolev spaces is proposed for system (1.1)-(1.3). To the best of our knowledge, it is still an open problem to obtain global solutions of (1.1)-(1.13) in the energy spaceH¹×H¹×L². Let us also introduce results concerning the solvability of the Cauchy problem related to (1.1)-(1.3). In [6], [26], the linear Schr¨odinger equation coupled with the Maxwell equations (namely, without

|ψ|^p−1ψin (1.1)) has been studied. Using the Strichartz estimate, the authors obtained the global well-posedness in the energy space. However, it is not clear that their argument can be applied to the nonlinear case. On the other hand, a huge attention has been paid in the magnetic Schr¨odinger equation, see [4], [12], [25]. Especially in [25], the local well-posedness for (1.5) in the energy space has been established in the case V ≡0. However, in this situation, the magnetic potentialA is givenand was assumed to be C^∞, which cannot be expected a priori for system (1.1)-(1.3).

Finally we explain the main difficulties and ideas for the study of ce(µ).

Firstly, we need to establish the sub-additivity condition force(µ) in order to prove the existence of a minimizer. Since our functionalJ(u,A) involves not only the non-local term S(u) but also the magnetic potential A, this cannot be obtained directly. However, using the diamagnetic inequality (1.16), we are able to prove the following fact:

ce(µ) = ˜ce(µ). (1.19)

Relation (1.19) enables us to conclude that we have only to study the sub- additivity of ˜c_e(µ), which has been already established in [15].

The second difficulty is the main part of this paper. When we study the relative compactness of a minimizing sequence (uj,Aj) ⊂ X, we are led to the situation that∥∇Aj∥L² is bounded, which shows that, passing to a subse- quence,∇Aj⇀∇AinL²(R³). In order to complete the proof of the compact- ness of the minimizing sequence (u_j,A_j), we need to establish that divA= 0.

Although divA_j = 0 for all j ∈ N, we cannot say that divA = 0 a priori, because the weak convergence provides us no information about the pointwise estimate.

At first sight, this problem can be avoided by restricting the function uj

to real-valued one. Indeed one can easily see that if the minimizing sequence ujis real-valued, then (uj,0) is also a minimizing sequence force(µ). However this approach does not work for our purpose, because solution ψ of (1.1) is complex-valued in general. Another approach is to assume some symmetry on the magnetic potential A. This kind of arguments has been performed in [2], [5], [8], where stationary problems for the Maxwell equation was studied and A was supposed to be cylindrically symmetric. However this approach is not suitable for the study of the stability of standing waves, because the symmetry ofAfor allt >0 requires a restriction of the initial values, causing the corresponding stability result to be weak.

The key idea is rather simple. Although the conserved energy E(ψ,A) in (1.8) is given in terms of |rotA|, we define the functional J(u,A) which is associated with the stationary problem by using |∇A|. If (uj,Aj) ⊂X is a minimizing sequence forc_e(µ), one can show that the weak limit (u,A) satisfies c_e(µ)≥J(u,A) by applying the concentration compactness principle due to [23], [24]. On the other hand, by the standard identity of vector calculus:

|∇A|²=|rotA|²+|divA|²,

we are able to show thatJ(u,A)> ce(µ) if divA̸= 0, yielding that a contra- diction occurs. Once we could prove that divA= 0, one obtains the relative compactness of the minimizing sequence (uj,Aj)⊂X.

This paper is organized as follows. In Section 2, we collect some known results concerning the non-local term S(u). We investigate the existence of minimizers forc_e(µ) and give the proof of Theorem 1.1 in Section 3. Especially the key result will be shown in Lemma 3.5, where the relative compactness of minimizing sequences for c_e(µ) is established. Finally in Section 4, we study the orbital stability of standing waves and complete the proof of Theorem 1.2.

2 Preliminaries

In this section, we present two lemmas which will be used later on. To this end, we denote

D(u) =

∫

R³

S(u)|u|²dx= 1 8π

∫

R³

∫

R³

|u(x)|²|u(y)|²

|x−y| dx dy.

Lemma 2.1 Foru∈H¹(R³,C),S(u)andD(u)satisfy the following proper- ties.

(i) S(u)(x)≥0 andD(u)≥0.

(ii) Forλ >0,a∈R,b∈R, denotinguλ(x) =λ^au(λ^bx), one has S(u_λ)(x) =λ^2a−2bS(u)(λ^bx), D(u_λ) =λ^4a−5bD(u).

(iii) There existsC >0 such thatD(u)≤C∇|u|

L²∥u∥³_L2. (iv) If un→uinL¹²⁵(R³,C), thenD(un)→D(u).

Proof See e.g. [27] for the proof. ⊓⊔

Lemma 2.2 Suppose2≤p≤ ⁷₃. Then it holds (i) There existsC=C(p)>0 such that

∥u∥^p+1_Lp+1≤CD(u)^7−3p2 ∇|u|^3p−5

L² ∥u∥^4(p_L2⁻²⁾ for allu∈H¹(R³,C).

(ii) Let C^∗=C^∗(p)>0 be the quantity defined by

C^∗= sup

u∈H¹(R³,C), u̸=0

∥u∥^p+1_Lp+1

D(u)^7−3p2 ∇|u|^3p−5

L² ∥u∥^4(p_L2⁻²⁾

.

Then C^∗ is well-defined, namely C^∗ < +∞. Moreover, for any C < C˜ ^∗ andµ >0, there existsu˜∈H¹(R³,C)such that ∥u˜∥²_L2 =µand

∥u˜∥^p+1_Lp+1 >Cµ˜ ^2(p−2)D(˜u)^7−3p2 ∇|u˜|^3p−5

L² . (iii) If p= 2, then it follows thatC^∗(2)≤√

2.

Proof We refer to [15] for the proof. ⊓⊔

3 Existence of minimizers of the new constraint minimization problem

In this section, we establish the existence of minimizers for (1.17). First we begin with the following standard result.

Lemma 3.1 Suppose that 1 < p < ⁷₃, e > 0 and µ > 0. Then J(u,A) is bounded from below on {(u,A) ∈ X;∥u∥²_L2 = µ}, and hence ce(µ) is well- defined.

Proof By Lemma 2.1 (i), one has J(u,A)≥ 1

2∥∇u−ieAu∥²L²+1

2∥∇A∥²L²− 1

p+ 1∥u∥^p+1_Lp+1. (3.1) Moreover by the diamagnetic inequality:

∥∇u−ieAu∥L² ≥∇|u|

L², (3.2)

the Gagliardo-Nirenberg inequality, the Young inequality and from ³₂(p−1)<

2, we find that

J(u,A)≥ 1

2∇|u|²

L²−C∇|u|^32(p−1)

L² ∥u∥_L^5−p22

≥ 1

4∇|u|²

L²−Cµ^7−3p5−p ≥ −Cµ^7−3p5−p,

from which we conclude. ⊓⊔

Next we establish the following fact, which will be crucially used to obtain the sub-additivity ofc_e(µ), relative compactness of minimizing sequences and the existence of a minimizer.

Lemma 3.2 Suppose that 1< p < ⁷₃, e >0 andµ >0. Let ˜ce(µ) and ce(µ) be the minimum energies of (1.15)and(1.17) respectively. Then it holds that

c_e(µ) = ˜c_e(µ).

Moreover let (u,A)∈ X with ∥u∥²L² =µ be a minimizer for ce(µ). Then it follows thatA=0.

Proof By the definitions ofce(µ) and ˜ce(µ), it is clear that ˜ce(µ)≥ce(µ). On the other hand, by the diamagnetic inequality (3.2), one finds that

J(u,A)≥J(|u|,0)≥˜c_e(µ) for allu∈H¹(R³,C) with∥u∥²L² =µ, yielding thatce(µ)≥˜ce(µ), which ends the proof of the first part.

Next let (u,A) ∈ X with ∥u∥²_L2 = µ be a minimizer of ce(µ). Then by using the diamagnetic inequality again, we obtain

ce(µ) =J(u,A)≥J(|u|,0) +1 2

∫

R³|∇A|²dx≥ce(µ) +1 2

∫

R³|∇A|²dx,

from which one concludes thatA=0. ⊓⊔

By Lemma 3.2, we are able to obtain the following results.

Lemma 3.3 Suppose that 1< p < ⁷₃,e >0 andµ >0.

(i) c_e(µ)is non-positive and non-increasing with respect toµ.

(ii) When 2≤p < ⁷₃, lete^∗=e^∗(µ, p)>0 be the constant defined by e^∗=

√2(7−3p)¹²(3p−5)^{2(7−3p)3p−5} (p+ 1)^7−3p1

µ^{2(p7−3p−2)}(C^∗)^7−3p1 , (3.3) where C^∗=C^∗(p)is the constant in Lemma 2.2 (ii). Then it follows that

{

ce(µ)<0 for0< e < e^∗,

c_e(µ) = 0 fore≥e^∗. (3.4)

Moreover if p= 2,e^∗ is independent ofµand e^∗(2)≤²₃. (iii) If 1< p <2, then ce(µ)<0 for alle >0.

Proof By Lemma 3.2, it suffices to consider ˜c_e(µ). Then the results follow by

[15, Lemma 4.1, Lemma 4.2]. ⊓⊔

Lemma 3.4 Let µ >0 be given and assume thatce(µ)<0.

(i) If 2≤p < ⁷₃, then it holds

ce(λµ)< λce(µ) for any λ >1. (3.5) Especially,ce(µ)satisfies the sub-additivity condition:

ce(µ)< ce(µ^′) +ce(µ−µ^′) for allµ^′∈(0, µ). (3.6) Moreover in the case2< p < ⁷₃ andce(µ) = 0, we assume that there exists a minimizer (u,A)∈X such thatce(µ) =J(u,A). Then (3.5)also holds true.

(ii) If 1< p <2, then there exists e^∗>0 such that (3.6)holds for0< e < e^∗. Proof (i) Let (u,A)∈X be fixed and consideru^λ(x) :=λ²u(λx), A^λ(x) :=

λA(λx). Then by applying Lemma 2.1 (ii) with a = 2 and b = 1, one has

∥u^λ∥²_L2=λ∥u∥²_L2 and J(u^λ,A^λ)

= λ³ 2

∫

R³|∇u−ieAu|²dx+λ

2∥∇A∥²L²+λ³e²

4 D(u)−λ^2p−1

p+ 1∥u∥^p+1_Lp+1

=λ³J(u,A) +λ−λ³

2 ∥∇A∥²L²+λ³−λ^2p−1

p+ 1 ∥u∥^p+1_Lp+1.

Since 2 ≤ p < ⁷₃ and λ > 1, it follows that λ < λ³ and λ³ ≤ λ^2p−1, from which we deduce that J(u^λ,A^λ) < λ³J(u,A). Choosing u ∈ H¹(R³,C) so that∥u∥²_L2 =µ, one gets fromce(µ)<0 that

ce(λµ)< λ³ce(µ)< λce(µ).

Moreover the second assertion follows from (3.5). (See [23, Lemma II.1, P.

120].) In the case 2< p < ⁷₃ andce(µ) = 0, we choose (u,A) as a minimizer ofce(µ). Then one can see that (3.5) holds.

(ii) By Lemma 3.2, it is sufficient to study ˜ce(µ). Then the claim follows

by the result in [15, Lemma 4.4]. ⊓⊔

The next lemma deals with the compactness of any minimizing sequence force(µ), which plays an essential role in the study of the orbital stability of standing waves.

Lemma 3.5 Suppose1< p < ⁷₃ andµ >0. Assume thatce(µ)<0 andce(µ) satisfies (3.6). Let {(uj,Aj)} ⊂X be a sequence such that ∥uj∥²_L2 →µ and J(uj,Aj)→ce(µ).

Then there exist a subsequence of {(u_j,A_j)} which is still denoted by the same, a sequence{y_j} ⊂R³ andu∈H¹(R³,C)such that

∥∇u_j(· −y_j)−ieA_j(· −y_j)u_j(· −y_j)− ∇u∥L²(R³)→0,

∥uj(· −yj)−u∥L²(R³)→0, ∥∇Aj(· −yj)∥L²(R³)→0 and J(u,0) =ce(µ).

As a consequence, Problem(1.17) admits a minimizer.

Proof First we observe that from (3.1) and (3.2) that ce(µ) +o(1)≥1

4∥∇uj−ieAjuj∥²L²+1

2∥∇Aj∥²L²−Cµ^7−3p5−p

≥1

4∥∇|uj|∥²L²+1

2∥∇Aj∥²L²−Cµ^7−3p5−p, yielding that the sequences

∥∇uj−ieAjuj∥L², ∥∇Aj∥L², ∥uj∥L^q (2≤q≤6) are bounded. (3.7) Moreover by replacinguj by _∥u^√µ

j∥L2uj, we may assume that {(uj,Aj)} ⊂ X is a minimizing sequence ofce(µ).

Now we apply the concentration compactness principle [23, Lemma I.1, P.115] to the sequence

ρ_j(x) =|u_j(x)|²+|∇u_j−ieA_ju_j|²+|∇A_j|². Then from (3.7), there existsC >0 such that

∫

R³

ρj(x)dx≤C for allj∈N. Without loss of generality, one may assume that

∫

R³

ρj(x)dx→λas j→ ∞for someλ >0.

It is well-known that the behavior of the sequence (ρj)j∈N is governed by the three possibilities: Compactness, Vanishing and Dichotomy. Our goal is to show that Compactness occurs.

If Vanishing occurs, there exists a subsequence of {ρj}, still denoted by {ρj}, such that

lim

j→∞sup

y∈R³

∫

BR(y)

ρj(x)dx= 0 for allR >0.

Here B_R(y) describes a ball of radiusR with the center at y ∈R³. Then by [24, Lemma I.1, P. 231], it follows that u_j →0 in L^q(R³) for any q∈ (2,6).

On the other hand since{u_j}is a minimizing sequence forc_e(µ), one has c_e(µ) +o(1) =J(u_j,A_j)

= 1

2∥∇uj−ieAjuj∥²L²+e²

4D(uj) +1

2∥∇Aj∥²L²− 1

p+ 1∥uj∥^p+1_Lp+1

≥ − 1

p+ 1∥uj∥^p+1_Lp+1.

Passing a limitj → ∞, we get 0> ce(µ)≥0. This is a contradiction, which rules out Vanishing.

Next we assume that Dichotomy occurs. Then by a standard argument (see [24, Section I.2] or [11, Proposition 1.7.6, P. 23]), there exist µ^′ ∈ (0, µ) and {(uj,1,Aj,1)},{(uj,2,Aj,2)} ⊂X such that

∥uj,1∥²L² →µ^′, ∥uj,2∥²L² →µ−µ^′,

supp(u_j,1)∩supp(u_j,2) =∅, δ_j := dist(

supp(u_j,1),supp(u_j,2))

→ ∞, (3.8)

∥u_j−u_j,1−u_j,2∥L^q →0 for all 2≤q <6, (3.9) lim inf

j→∞

∫

R³|∇Aj|²− |∇Aj,1|²− |∇Aj,2|²dx≥0, (3.10) lim inf

j→∞

∫

R³|∇uj−ieAjuj|²−|∇uj,1−ieAj,1uj,1|²−|∇uj,2−ieAj,2uj,2|²dx≥0.

(3.11) Moreover replacinguj,1,uj,2by _∥u^√µ′

j,1∥_L2uj,1, _∥^√_u^µ−µ′

j,2∥_L2uj,2 respectively, we may assume that∥uj,1∥²_L2 =µ^′,∥uj,2∥²_L2=µ−µ^′. Now from (3.8), one has

∫

R³

∫

R³

|uj,1(x)|²|uj,2(y)|²

|x−y| dx dy

=

∫

supp(uj,2)

∫

supp(uj,1)

|uj,1(x)|²|uj,2(y)|²

|x−y| dx dy

≤ 1

δ_j∥u_j,1∥²L²∥u_j,2∥²L² →0 as j→ ∞.

Using (3.9) and arguing as in the proof of Lemma 2.2 in [29], a direct compu- tation furnishes

D(uj)−D(uj,1)−D(uj,2)

=

∫

R³

S(u_j)|u_j|²−S(u_j,1)|u_j,1|²−S(u_j,2)|u_j,2|²dx

=

∫

R³

{(S(u_j)|u_j|+S(u_j,1)|u_j,1|+S(u_j,2)|u_j,2|)(

|u_j| − |u_j,1| − |u_j,2|) +|u_j|(

|u_j,1|+|u_j,2|)(

S(u_j)−S(u_j,1)−S(u_j,2)) +|uj,1||uj,2|(

S(uj,1) +S(uj,2)) +|uj|(

|uj,1|S(uj,2) +|uj,2|S(uj,1))}

dx

→0 asj→ ∞.

Thus from (3.9)-(3.11) and by the diamagnetic inequality, we obtain ce(µ) = lim inf

j→∞ J(uj,Aj)

≥lim inf

j→∞ J(u_j,1,A_j,1) + lim inf

j→∞ J(u_j,2,A_j,2)

≥lim inf

j→∞ J(|uj,1|,0) + lim inf

j→∞ J(|uj,2|,0)

≥c_e(µ^′) +c_e(µ−µ^′),

which contradicts (3.6). Thus Dichotomy does not occur.

The only remaining possibility is Compactness, that is, there exists{y_j} ⊂ R³such that for allε >0, there existsR_ε>0 satisfying

∫

B_Rε(yj)

ρj(x)dx≥λ−ε. (3.12)

From (3.7), one has

∥∇u_j∥L² ≤ ∥∇u_j−ieA_ju_j∥L²+∥ieA_ju_j∥L²

≤ ∥∇uj−ieAjuj∥L²+e∥Aj∥L⁶∥uj∥L³,

showing that ∥u_j∥H¹ is bounded. Thus passing to a subsequence, we may assume that

u_j(· −y_j)⇀ uinH¹(R³,C) and A_j(· −y_j)⇀AinD^1,2(R³,R³).

Then from (3.12), it holds thatuj(· −yj)→uinL^q(R³,C) for any 2≤q <6.

Next we claim that A=0and (u,0) is a minimizer for c_e(µ). Indeed by the weak lower semi-continuity of∥∇ · ∥L² and by Lemma 2.1 (iv), one finds

∥u∥²L²= lim

j→∞∥uj(· −yj)∥²L²=µ, ce(µ) = lim inf

j→∞ J(

uj(· −yj),Aj(· −yj))

≥J(u,A). (3.13)

At this stage, we don’t know whether divA= 0 and hence cannot conclude thatJ(u,A)≥ce(µ). To conclude, one observes that

∫

R³|∇A|²dx=

∫

R³|rotA|²dx+

∫

R³

( divA)²dx.

Thus if divA̸= 0, it follows that∫

R³|∇A|²dx >∫

R³|rotA|²dxand hence ce(µ)≥J(u,A)

>1 2

∫

R³

(|∇u−ieAu|²+|rotA|²)

dx+e²

4 D(u)− 1 p+ 1

∫

R³|u|^p+1dx

≥ inf

(v,B)∈X,∥v∥²_L2=µ,divB=0

J(v,B) =ce(µ),

yielding that divA= 0. Then from (3.13), we obtain J(u,A) =c_e(µ). Thus by Lemma 3.2, it holds that A = 0 and (u,0) is a minimizer for c_e(µ), as claimed.

Finally using (3.13) again, one has from divA=0that lim

j→∞

(∥∇u_j(· −y_j)−ieA_j(· −y_j)u_j(· −y_j)∥²L²+∥∇A_j(· −y_j)∥²L²

)

=∥∇u∥²L². (3.14)

This implies that

∥∇uj(· −yj)−ieAj(· −yj)uj(· −yj)− ∇u∥²L²

=∥∇u_j(· −y_j)−ieA_j(· −y_j)u_j(· −y_j)∥²L²

−2

(∇uj(· −yj)−ieAj(· −yj)uj(· −yj),∇u )

L²+∥∇u∥²L²

≤ ∥∇uj(· −yj)−ieAj(· −yj)uj(· −yj)∥²L²+∥∇Aj(· −yj)∥²L²

−2

(∇uj(· −yj)−ieAj(· −yj)uj(· −yj),∇u )

L²+∥∇u∥²L²→0 asj→ ∞, and hence

∇uj(· −yj)−ieAj(· −yj)uj(· −yj)→ ∇uin L².

Then from (3.14), we infer that∇Aj(· −yj)→0inL², which ends the proof.

⊓

⊔ By Lemmas 3.2-3.5, one can see that for 1< p < ⁷₃ and 0< e < e^∗, there exists a minimizer (ue,0) of (1.17). Next we prove the existence of a minimizer in the borderline casee=e^∗ when 2< p < ⁷₃.

Lemma 3.6 Suppose2< p < ⁷₃ and let e^∗=e^∗(µ, p)be the constant defined in Lemma 3.3. Then for e = e^∗, the minimization problem (1.17) admit a minimizer.