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The Nonexistence of Vortices for Rotating Bose-Einstein

Condensates in Non-Radially Symmetric Traps

Yujin Guo∗

October 13, 2020

Abstract

We consider ground states of rotating Bose-Einstein condensates with attractiveinteractions in a homogeneous trap V (x) of degree 2 in R2, which can be non-radiallysymmetric. For any fixed rotating velocity 0 ≤ Ω < Ω∗, where 0 < Ω∗ := Ω∗(V ) ≤∞ denotes the critical rotational velocity, it is known that ground states exist ifand only if a < a∗ for some critical constant 0 < a∗ < ∞, where a > 0 denotesthe absolute product for the number of particles times the scattering length. If0 < Ω < Ω∗ is fixed, we prove that ground states do not have any vortex in theregion R(a) := {x ∈ R2 : |x| ≤ C(a∗ − a)− 13 } as a ր a∗ for some constant C > 0,which is independent of 0 < a < a∗. As a byproduct, we also obtain the refinedlimit profiles of ground states as aր a∗.

Keywords: Bose-Einstein condensate; rotational velocity; nonexistence of vortices; limitprofiles

1 Introduction

Bose-Einstein condensate (BEC) is a state of matter, in which atoms or particles arecooled to the sufficiently low temperature that a large fraction of them “condense” intoa single quantum state. The BECs in magneto-optical traps present remarkable phenom-ena, once the traps are set in rotational motion. Actually, starting from the first physicalachievement of rotating BECs in the late 1990s, various interesting quantum phenomenahave been observed in the experiments of rotational BECs, including the critical-masscollapse [15, 22, 33], the center-of-mass rotation [1, 24, 43], and the appearance of quan-tized vortices [2,17,24]. Therefore, the numerical simulations and mathematical theoriesof rotating BECs have been a focus of international interest in physics and mathematicsover the past two decades, see [1, 2, 6, 21,22,24,34,35,40–42].

The interactions between the cold atoms in the condensates can be either repulsiveor attractive, cf. [2, 14,22,24]. For the repulsive case, the complex structures, includingthe quantized vortices, of trapping BECs under rotation were analyzed and simulatedextensively in the past few years, see [1–4, 9, 21, 22, 24, 34, 35, 46, 47] and the references

∗School of Mathematics and Statistics, and Hubei Key Laboratory of Mathematical Sciences, Central

China Normal University, P.O. Box 71010, Wuhan 430079, P. R. China. Email: yguo@ccnu.edu.cn. Y.

J. Guo is partially supported by NSFC under Grants No. 11671394 and 11931012.

1

http://arxiv.org/abs/2010.05592v1

therein. However, the attractive condensates under rotation behave different extremelyfrom those of the well-understood repulsive case. Typically, the vortices are generallyunstable in the rotating BECs with attractive interactions (cf. [17,43]), even though thevortices may form stable lattice configurations in the repulsive case, cf. [2, 24]. Becauseof the distinct mechanisms, the existing physical observations and numerical simula-tions show that the rotating BECs with attractive interactions present more complicatedphenomena and structures, see [9, 17, 22, 24, 43], only few of which have however beeninvestigated analytically so far.

As derived rigorously in [38] by a mean-field approximation, the ground state of thetwo-dimensional attractive BECs in a rotating trap can be described equivalently by acomplex constraint minimizer of the following Gross-Pitaevskii (GP) energy functional

Fa(u) :=

∫

R2

(|∇u|2+V (x)|u|2

)dx− a

2

∫

R2

|u|4dx−Ω∫

R2

x⊥ · (iu, ∇u)dx, u ∈ H, (1.1)

under the mass constraint

eF (a) := inf{u∈H, ‖u‖22=1}

Fa(u), a > 0, (1.2)

where x⊥ = (−x2, x1) with x = (x1, x2) ∈ R2, (iu, ∇u) = i(u∇ū − ū∇u)/2, and thecomplex space H is defined as

H :={u ∈ H1(R2,C) :

∫

R2

V (x)|u|2dx 0 in eF (a) denotes the absolute product of the scattering length ν ofthe two-body interaction times the number N of particles in the condensates, and whileΩ ≥ 0 describes the rotational velocity of the rotating trap V (x) ≥ 0. We commentthat one may impose eF (a) a different constraint

∫R2

|u(x)|2dx = N > 0, but thesetwo different forms can be reduced equivalently to each other, see [30]. In this paperwe therefore focus on the form of eF (a) instead. It also deserves to remark that eventhough the mass-subcritical version of eF (a), where the nonlinear term |u|4 of Fa(u) isreplaced by |u|p for 2 < p < 4, was studied as early as in the pioneering work of Esteban-Lions [23], the mass-critical constraint variational problem eF (a) in the complex rangewas not addressed until the recent years, see [7,9,13,29,30,38] and the references therein.

The non-rotational case Ω = 0 of eF (a) was studied recently in [28,31,32,45,52] andthe references therein, where the existence, uniqueness, symmetry breaking and otheranalytic properties of real-valued minimizers were investigated widely. For a class oftrapping potentials V (x), specially it was proved there that eF (a) with Ω = 0 admitsreal-valued minimizers if and only if a < a∗, where a∗ = ‖w‖2

L2(R2) and w = w(|x|) > 0 isthe unique (cf. [37, 51]) positive solution of the following nonlinear scalar field equation

∆u− u+ u3 = 0 in R2, u ∈ H1(R2,R). (1.4)

Following the analytic approach of [20, Theorem II.1], this further implies that eF (a)with Ω = 0 admits complex-valued minimizers (i.e., ground states), if and only if a < a∗.

Starting from the earlier works [9, 38], the rotational case Ω > 0 of eF (a) was an-alyzed more recently. More precisely, considering the special trapping potentials, such

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as typically V (x) = |x|2, the existence and nonexistence, stability and some other prop-erties of complex-valued minimizers for eF (a) with Ω > 0 were studied in [7, 9, 13, 38].Generally, if the trapping potential 0 ≤ V (x) ∈ L∞loc(R2) satisfies

lim|x|→∞V (x)

|x|2 > 0, (1.5)

then one can define as in [30] the following critical rotational velocity Ω∗ := Ω∗(V ):

Ω∗ := sup{Ω > 0 : V (x)− Ω

2

4|x|2 → ∞ as |x| → ∞

}. (1.6)

Note that if V (x) satisfies the assumption (1.5), then Ω∗ ∈ (0,+∞] exists, and VΩ(x) :=V (x) − Ω24 |x|2 ≥ 0 holds in R2 for any 0 ≤ Ω < Ω∗. Under the assumption (1.5), weestablished in [30, Theorem 1.1] the following existence and non-existence of complex-valued minimizers:

Theorem A ( [30, Theorem 1.1]). Assume V (x) ∈ L∞loc(R2) satisfies (1.5) such thatΩ∗ ∈ (0,+∞] in (1.6) exists. Then we have

1. If 0 ≤ Ω < Ω∗ and 0 ≤ a < a∗ := ‖w‖22, then there exists at least one minimizer ofeF (a).

2. If 0 ≤ Ω < Ω∗ and a ≥ a∗ := ‖w‖22, then there is no minimizer of eF (a).

3. If Ω > Ω∗, then for any a ≥ 0, there is no minimizer of eF (a).

The proof of Theorem A needs the following Gagliardo-Nirenberg inequality∫

R2

|u(x)|4dx ≤ 2‖w‖22

∫

R2

|∇u(x)|2dx∫

R2

|u(x)|2dx, u ∈ H1(R2,R), (1.7)

where the identity is attained at w, and the following diamagnetic inequality

|∇u|2−Ωx⊥·(iu, ∇u) = |(∇−iA)u|2−Ω2

4|x|2|u|2 ≥

∣∣∇|u|∣∣2−Ω

2

4|x|2|u|2, u ∈ H1(R2,C),

(1.8)where A = Ω2 x⊥, see [39, 51] for more details on such inequalities. By the variationaltheory, if eF (a) admits a minimizer ua, then ua is a ground state of the following Euler-Lagrange equation

−∆ua + V (x)ua + iΩ (x⊥ · ∇ua) = µua + a|ua|2ua in R2,∫

R2

|ua|2dx = 1, (1.9)

where µ = µ(a,Ω, ua) ∈ R is a suitable Lagrange multiplier. We comment that thereexist many interesting progresses on the normalized solutions of the elliptic problem(1.9), see [8, 10–12,16,23,27,36,45,49] and the references therein.

By employing the energy estimates and elliptic PDE theory, it was proved in [30,38]that the minimizer ua of eF (a) concentrates at a global minimum point of VΩ(x) asaր a∗, in the sense that

‖ua‖∞ → ∞ and∫

R2

VΩ(x)|ua|2dx→ VΩ(x0) := infx∈R2

VΩ(x) as aր a∗. (1.10)

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Based on (1.10), the L∞ uniform convergence of ua after rescaling and translation wasalso obtained in [30]. By developing the method of inductive symmetry, we further provedin [30, Theorem 1.3] (see also [44]) the absence of vortices of minimizers ua for eF (a) asa ր a∗ for the case V (x) = |x|2, where the imaginary part Im(ua) ≡ 0 as a ր a∗. Weshould emphasize that the arguments and results of [30, Theorem 1.3] cannot however beextended to the non-radially symmetric case of V (x), since one cannot expect generallyIm(ua) ≡ 0 as aր a∗, see also Remark 2.1 below.

On the other hand, we should mention that the following typical non-radially sym-metric trapping potential

V (x) = x21 + Λx22, where x = (x1, x2) ∈ R2, Λ > 0, (1.11)

the so-called harmonic trap, was already used in BEC experiments, see [3, 18,34,46,47]and the references therein. A little more general than (1.11), we next introduce thefollowing homogeneous functions:

Definition 1.1. A function h(x) : R2 7−→ R is called homogeneous of degree p ∈ R+(about the origin), if

h(tx) = tph(x) for any t ∈ R+ and x ∈ R2. (1.12)

We note from Definition 1.1 that if 0 ≤ V (x) ∈ C2(R2) is homogeneous of degree 2 andsatisfies lim|x|→∞ V (x) = ∞, then x = 0 is the unique minimum point of V (x), and

0 ≤ VΩ(x) := V (x)−Ω2

4|x|2 ∈ C2(R2) is also homogeneous of degree 2 (1.13)

for any fixed 0 < Ω < Ω∗, where 0 < Ω∗ 0 be defined as in (1.6). For any fixed 0 < Ω < Ω∗,suppose 0 ≤ VΩ(x) := V (x)− Ω4 |x|2 ∈ C2(R2) also satisfies

y0 is a unique and non-degenerate critical point of HΩ(y) :=

∫

R2

VΩ(x+ y)w2(x)dx,

(1.14)and let ua be a complex-valued minimizer of eF (a) as a ր a∗. Then there exists aconstant C > 0, independent of 0 < a < a∗, such that

|ua(x)| > 0 in the region R(a) :={x ∈ R2 : |x| ≤ C(a∗ − a)− 13

}as aր a∗, (1.15)

i.e., ua does not admit any vortex in the region R(a) as aր a∗.

Under the assumptions of Theorem 1.1, recall from [29, Theorem 1.1] that up tothe constant phase, there exists a unique complex-valued minimizer of eF (a) as aր a∗.Note that the nonexistence of vortices for rotating BECs were studied earlier by different

4

arguments, including jacobian estimates, vortex ball constructions, the inductive sym-metry, and so on, see [3–5,18,21,30,34,35,44] and the references therein. We also referthe celebrated monograph [50] to various kinds of techniques involved in analyzing thevortices of defocusing nonlinear problems. As far as we know, it however seems that theabove mentioned methods of studying the nonexistence of vortices are mainly applicableto the case where the trap is radially symmetric. The main contribution of Theorem1.1 lies in the fact that our nonexistence of vortices holds for a class of non-radiallysymmetric traps. Moreover, we guess that our proof strategy might be applicable for theopen problems proposed in [2, 4], which concern the nonexistence of vortices for repul-sive BECs in rotating non-radially symmetric traps. Of course, one may further wonderwhether the nonexistence of vortices in Theorem 1.1 holds in the whole plane R2 asaր a∗, especially if the trap V (x) satisfying (1.11) is harmonic, for which investigatingnew skills seems necessary.

As outlined in Subsection 1.1, the proof of Theorem 1.1 relies heavily on, insteadof the energy analysis, the refined limit profiles of minimizers ua as a ր a∗, which isa byproduct of proving Theorem 1.1. Towards this introduction, let y0 ∈ R2 be now aunique global minimum point of HΩ(y) :=

∫R2VΩ(x+ y)w

2(x)dx, and define

λ =[ ∫

R2

(VΩ(x+ y0) +

Ω2

4|x|2

)w2(x)dx

] 14> 0, (1.16)

where VΩ(x) ≥ 0 is as in (1.13), and w = w(|x|) > 0 is the unique positive solution of(1.4). For convenience, we also denote ψ1(x) ∈ C2(R2) ∩ L∞(R2) the unique solution of

(−∆+ 1− 3w2

)ψ1(x) = −

λ4

a∗w3(x)−

[Ω24|x|2 + VΩ

(x+ y0

)]w(x) in R2,

∇ψ1(0) = 0,(1.17)

and define ψI(x) ∈ C2(R2) ∩ L∞(R2) to be the unique solution of

(−∆+ 1− w2

)ψI(x) = −

(x⊥ · ∇ψ1

)in R2,

∫

R2

ψIwdx = 0, (1.18)

where ψ1 satisfies (1.17). Applying above notations, the refined limit profiles of mini-mizers as aր a∗ can be stated as the following theorem.

Theorem 1.2. Under the assumptions of Theorem 1.1, let ua be a complex-valued min-imizer of eF (a). Then there exist constants θa ∈ [0, 2π) and C∗ 6= 0 such that

εa√a∗ ua

(εax+ xa

)e−i

(Ω2εax·x⊥a −θa

)− w = ε4a

{ψ1 + C

∗(w + x · ∇w)}[

1 + o(1)]

+i ε6aΩψI[1 + o(1)

]as aր a∗,

(1.19)

where εa :=(a∗−a)

14

λ> 0, xa is the unique global maximum point of |ua| as a ր a∗,

and ψ1(x), ψI (x) ∈ C2(R2) ∩ L∞(R2) are the unique solutions of (1.17) and (1.18),respectively.

As mentioned in Subsection 1.1 below, even though the idea of proving Theorem 1.2 isstimulated by [28, Theorem 1.4], we emphasize that there appear some extra challenging

5

difficulties in the proof of Theorem 1.2, for which one needs carry out the more involvedand more delicate analysis of ua and x

⊥ · ∇ua, due to the rotating term, and make fulluse of the non-degenerancy of w as well. We also remark that the explicit formula of theconstant C∗ in (1.19) can be obtained from (3.25). Moreover, the proof of Theorem 1.2yields actually the first two leading terms of Ra by combining Lemma 3.2 with (3.25).One can also note from (1.17) and (1.18) that the function ψI of (1.19) does not vanishgenerally, if the trapping potential VΩ(x) is not radially symmetric.

We finally focus on the physically relevant case where V (x) ≥ 0 satisfying (1.11) isa harmonic trap, so that Theorem 1.2 can be improved. We remark that it suffices toconsider the following harmonic form

V (x) = x21 + Λ2x22 for x = (x1, x2) ∈ R2, where 0 < Λ < 1, (1.20)

since the case Λ > 1 can be established similarly, and while the case Λ = 1 was alreadyaddressed in Theorem 1.3 and (1.18) of [30]. If V (x) satisfies (1.20), then it follows from(1.6) that the critical velocity Ω∗ of eF (a) satisfies Ω

∗ := 2Λ. Under the assumption(1.20), letting 0 < Ω < Ω∗ := 2Λ be fixed, in Section 4 we shall improve Theorem 1.2 toobtain the following much more refined limit profiles of minimizers ua as aր a∗:

ǫa√a∗ ua

(ǫax+ xa

)e−i

(Ω2ǫax·x⊥a −θa

)= w + ǫ4a

{ϕ1 + CΛ(w + x · ∇w)

}[1 + o(1)

]

+i ǫ6aΩϕI[1 + o(1)

]as aր a∗,

(1.21)

where ǫa :=(a∗−a)

14

λ0> 0 for λ0 > 0 being given by (4.4), θa ∈ [0, 2π) is as in Theorem

1.2, and xa is the unique global maximum point of |ua| which satisfies∣∣xa

∣∣ = o(ǫ5a)

as aր a∗, (1.22)

see Theorem 4.1 for more details. As commented in Remark 4.1, the function ϕ1 andthe constant CΛ 6= 0 of (1.21) are independent of Ω. In contrast to Theorem 1.2, weare thus able to show that the first two leading terms of (1.21) are independent of0 < Ω < Ω∗ := 2Λ, which seems false for more general trapping potentials. On the otherhand, (1.22) gives an improved estimate of xa as a ր a∗, comparing with Lemma 3.1.The limit profile (1.21) is therefore an improvement of Theorem 1.2 in the harmonic caseof trapping potentials.

1.1 Proof strategy of Theorem 1.1

In this subsection, we follow four steps to explain the general strategy of proving Theorem1.1, for which the key component part is to establish Theorem 1.2.

As the first step, under the assumptions of Theorem 1.1, we proved in [30, Theorem1.2] that the complex-valued minimizer ua of eF (a) satisfies

va(x) := εa√a∗ ua

(εax+ xa

)e−i

(Ω2εax·x⊥a −θa

)→ w(x) (1.23)

uniformly in L∞(R2,C) as a ր a∗, where and below εa > 0 and xa ∈ R2 are as inTheorem 1.2, see also Proposition 2.1 below for more details. Here θa ∈ [0, 2π) can be

6

chosen suitably such that∫R2wIadx ≡ 0. Recall from [30, Section 3] that the Lagrange

multiplier µa of (1.9) and xa satisfy

µaε2a → −1 and

xaεa

= y0 as aր a∗, (1.24)

where y0 ∈ R2 is as in (1.14).The second step is to establish the leading terms of va − w, in terms of εa and

µaε2a + 1, by following (1.23) and (1.24). However, we remark that it seems difficult to

reach this aim by investigating directly the Euler-Lagrange equation (1.9). To overcomethis difficulty, motivated by (1.9) and (1.23) we shall consider carefully the followingcoupled system of real-valued parts Ra(x) and I(x) for va := (Ra + w) + iIa:

L̃aRa = ε2aΩ (x⊥ · ∇Ia) + F̃a(x) in R2, ∇Ra(0) 6≡ 0,

LaIa = −ε2aΩ (x⊥ · ∇Ra) in R2,∫

R2

wIadx ≡ 0,(1.25)

where F̃a(x) is an inhomogeneous term containing µaε2a + 1, and the operator L̃a :=

La −w2 − w(Ra + w) for

La := −∆+(ε4aΩ2

4|x|2 + ε2aVΩ(εax+ xa)− µaε2a −

a

a∗|va|2

)in R2. (1.26)

One can note from (1.23) that Ra and Ia satisfy

Ra(x) → 0 and Ia(x) → 0 uniformly in R2 as aր a∗. (1.27)Even though the rest part of the second step is motivated by [28, Theorem 1.4] and

(1.27), unfortunately, there appear extra challenging difficulties in the present analysis.Essentially, we first note from (1.25) that we have Ra(0) 6≡ 0 in R2, which leads to a newdifficulty in studying the expansion of Ra. More challengingly, the coupled system (1.25)contains the coupled rotating terms (x⊥ ·∇Ia) and (x⊥ ·∇Ra), which makes the analysismore delicate than that of [28, Theorem 1.4]. To overcome these extra difficulties, weneed proceed with the very delicate analysis of Ra and Ia, and make full use of thenon-degenerancy of w as well. By overcoming above difficulties, we shall finally obtainin Lemmas 2.3 and 3.2 the leading terms of Ia and Ra, respectively, in terms of εa andµaε

2a + 1. This therefore finishes the second step.The third step is to apply Lemma 3.2 for establishing Theorem 1.2, for which the

key is to address the refined estimate of µaε2a +1 in terms of εa > 0. We shall achieve it

by taking full advantage of the mass constraint∫R2

|ua|2dx = 1, as well as the analyticalresults of the second step.

Following Theorem 1.2, the last step is to derive the upper estimate of the termψ1 +C

∗(w+ x · ∇w) in (1.19). We shall reach this purpose by the comparison principle.This finally yields Theorem 1.1 on the nonexistence of vortices in the sufficiently largeregion R(a) as aր a∗.

This paper is organized as follows: In Section 2 we analyze the refined estimatesof minimizers ua for eF (a) as a ր a∗. Applying the analytical results of the previoussection, Section 3 is concerned with the proof of Theorem 1.1 on the nonexistence ofvortices for ua, for which we need establish Theorem 1.2 at first. The main purpose ofSection 4 is to prove Theorem 4.1 on the much more refined limit profiles of ua in theharmonic case of trapping potentials. In Appendix A we shall address the proof of theclaims (4.24) and (4.29) used in Section 4.

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2 Refined Estimates of Minimizers

Suppose the trapping potential 0 ≤ V (x) ∈ C2(R2) satisfies the assumptions of Theorem1.1, the purpose of this section is to address the refined expansions of the complex-valuedminimizers ua for eF (a) as aր a∗, where the rotating speed Ω ∈ (0,Ω∗) is fixed. By thevariational theory, ua solves the following Euler-Lagrange equation:

−∆ua + V (x)ua + iΩ(x⊥ · ∇ua) = µaua + a|ua|2ua in R2, (2.1)

where µa := µa(ua) ∈ R is a suitable Lagrange multiplier satisfying

µa = eF (a)−a

2

∫

R2

|ua|4dx. (2.2)

Recall from [51] that the unique positive radial solution w of (1.4) is an optimizer ofthe following Gagliardo-Nirenberg inequality

∫

R2

|u(x)|4dx ≤ 2a∗

∫

R2

|∇u(x)|2dx∫

R2

|u(x)|2dx, u ∈ H1(R2,R). (2.3)

Note also from [19, Lemma 8.1.2] and [25, Proposition 4.1] that w = w(|x|) > 0 satisfies∫

R2

|∇w|2dx =∫

R2

w2dx =1

2

∫

R2

w4dx, (2.4)

andw(x) , |∇w(x)| = O(|x|− 12 e−|x|) as |x| → ∞. (2.5)

Under the assumptions of Theorem 1.1, we define for 0 < a < a∗,

εa :=(a∗ − a) 14

λ> 0, λ =

[ ∫

R2

(VΩ(x+ y0) +

Ω2

4|x|2

)w2(x)dx

] 14> 0, (2.6)

where VΩ(x) ≥ 0 is defined by (1.13), and y0 ∈ R2 is a unique global minimum point ofHΩ(y) :=

∫R2VΩ(x+ y)w

2(x)dx. Define

va(x) := εa√a∗ ua

(εax+ xa

)e−i(

εaΩ2x·x⊥a −θa) = R̃a(x) + iIa(x), (2.7)

where xa is a global maximal point of |ua(x)|, R̃a(x) and Ia(x) denote the real andimaginary parts of va(x), respectively, and the constant phase θa ∈ [0, 2π) is chosen suchthat ∥∥va − w

∥∥L2(R2)

= minθ∈[0,2π)

∥∥eiθva − w∥∥L2(R2)

. (2.8)

The above property gives the following orthogonality condition on Ia(x):

∫

R2

w(x)Ia(x)dx = 0. (2.9)

Using above notations, we proved in [30] the following L∞−uniform convergence asaր a∗.

8

Proposition 2.1. ( [30, Theorem 1.2]) Assume 0 ≤ V (x) ∈ C2(R2) satisfying lim|x|→∞

V (x) =

+∞ is homogeneous of degree 2, and let y0 ∈ R2 be a unique global minimum point ofHΩ(y) :=

∫R2VΩ(x+ y)w

2(x)dx. For any fixed 0 < Ω < Ω∗, where Ω∗ > 0 is defined asin (1.6), suppose ua is a minimizer of eF (a). Then we have

va(x) := εa√a∗ ua

(εax+ xa

)e−i

(Ω2εax·x⊥a −θa

)→ w(x) (2.10)

uniformly in L∞(R2,C) as a ր a∗, where εa > 0 is as in (2.6), θa ∈ [0, 2π) is chosensuch that (2.8) holds true, and the unique global maximal point xa ∈ R2 of |ua| satisfies

limaրa∗

xaεa

= y0. (2.11)

Note from (2.1) that the function va satisfies the following elliptic equation

−∆va + i ε2aΩ(x⊥ · ∇va

)+

[ε4aΩ2|x|24

+ ε2aVΩ(εax+ xa)]va

=µaε2ava +

a

a∗|va|2va in R2.

(2.12)

The analysis of [30, Theorem 1.2] gives that the above Lagrange multiplier µa satisfies

limaրa∗

µaε2a = −1. (2.13)

In order to obtain the refined limit profile of ua as aր a∗, one can note from (2.10) and(2.12) that a more refined estimate than (2.13) is needed for µa as aր a∗, which is oneof the main difficulties in this paper. Towards this purpose, it however seems difficultto handle directly with the single equation (2.12), instead of which we next consider thecoupled system of Re(va) and Im(va) in the coming subsection.

2.1 Refined estimates of va as a ր a∗

Based on Proposition 2.1, in this subsection we shall derive the refined estimates of vadefined in (2.7) as a ր a∗. Towards this purpose, we introduce the following linearizedoperator

L := −∆+ 1− w2 in R2. (2.14)It then obtains from [39, Theorem 11.8] and [39, Corollary 11.9] that

kerL = {w} and 〈Lv, v〉 ≥ 0 for all v ∈ L2(R2). (2.15)

We also denote the linearized operator L̃ by

L̃ := −∆+ 1− 3w2 in R2. (2.16)

It follows from [37,48] that

kerL̃ ={ ∂w∂x1

,∂w

∂x2

}. (2.17)

For convenience, we denote for 0 < a < a∗,

εa :=(a∗ − a) 14

λ> 0, αa := (λεa)

4 = a∗ − a > 0, and βa := 1 + µaε2a, (2.18)

9

where λ > 0 is defined by (2.6) and the Lagrange multiplier µa satisfies (2.13). We thenhave

αa → 0 and βa → 0 as aր a∗.Following (2.7), we now rewrite va as

va(x) := R̃a(x) + iIa(x) =[Ra(x) + w(x)

]+ iIa(x), (2.19)

so thatRa(x) → 0 and Ia(x) → 0 uniformly in L∞(R2,R) as aր a∗,

due to Proposition 2.1. Since ∇|va(0)| ≡ 0 holds for all 0 < a < a∗, we derive from(2.19) that

∇Ra(0) = −Ia(0)∇Ia(0)w(0) +Ra(0)

→ 0 uniformly in L∞(R2,R2) as aր a∗. (2.20)

For simplicity, we denote the operator La by

La := −∆+(ε4aΩ2

4|x|2 + ε2aVΩ(εax+ xa)− µaε2a −

a

a∗|va|2

)in R2. (2.21)

It then follows from (2.9), (2.12) and (2.19) that Ia satisfies

LaIa = −ε2aΩ (x⊥ · ∇Ra) in R2,∫

R2

wIadx ≡ 0, (2.22)

and while Ra satisfies (2.20) and

L̃aRa :=[La − w2 − wR̃a

]Ra = Fa(x) in R

2, (2.23)

where Fa(x) is defined by

Fa(x) :=ε2aΩ (x

⊥ · ∇Ia)−[ε4aΩ2

4|x|2 + ε2aVΩ(εax+ xa)− βa −

a

a∗I2a +

αaa∗R̃2a

]w

=− ε4a[Ω24|x|2 + VΩ

(x+

xaεa

)]w + βaw −

αaa∗R̃2aw

+ ε4aΩ(x⊥ · ∇Ia

ε2a

)+

a

a∗I2aw.

(2.24)

Similar to [30, Lemma 4.2 (1)], where εa > 0 is defined in a slightly different way, onecan deduce from (2.12) that there exists a constant C > 0, independent of 0 < a < a∗,such that as aր a∗,

|∇R̃a(x)|, |∇Ia(x)| ≤ Ce−23|x| uniformly in R2. (2.25)

Applying (2.20) and (2.25), the argument of [30, Lemma 4.3] then yields from (2.22)that there exists a constant C > 0, independent of 0 < a < a∗, such that as aր a∗,

|∇Ia(x)|, |Ia(x)| ≤ Cε2ae−18|x| uniformly in R2. (2.26)

For ψ2(|x|) := −12(w + x · ∇w

), one can check that

∇ψ2(0) = 0, L̃ψ2(|x|) = w(x) in R2, (2.27)

where the operator L̃ is defined by (2.16). Having above estimates, we next establishthe following “rough” limit profiles in terms of εa and βa.

10

Lemma 2.2. Under the assumptions of Proposition 2.1, we have

1. There exist constants C > 0 and 0 < δ < 1, independent of 0 < a < a∗, such thatthe imaginary part Ia of (2.19) satisfies

|∇Ia(x)|, |Ia(x)| ≤ Cε6ae−δ|x| uniformly in R2 as aր a∗. (2.28)

2. The real part Ra of (2.19) satisfies

Ra(x) := ε4aψ1(x) + βaψ2(|x|) + o(ε4a + |βa|) in R2 as aր a∗, (2.29)

where ψ2(|x|) := −12(w + x · ∇w

)is radially symmetric, and ψ1(x) ∈ C2(R2) ∩

L∞(R2) solves uniquely

∇ψ1(0) = 0, L̃ψ1(x) = −λ4

a∗w3(x)−

[Ω24|x|2 + VΩ

(x+ y0

)]w(x) in R2, (2.30)

where y0 ∈ R2 is a unique global minimum point of HΩ(y) :=∫R2VΩ(x+y)w

2(x)dx.

Proof. We shall finish the proof by the following three steps:Step 1. Denote

Ra(x) := Ra(x)− βaψ2(|x|), (2.31)which yields from (2.20), (2.23) and (2.27) that Ra(x) satisfies

∇Ra(0) → 0 uniformly in L∞(R2,R2) as aր a∗, (2.32)

andL̃aRa = Fa(x)− βaw in R2, (2.33)

where L̃a and Fa(x) are defined by (2.21) and (2.24), respectively. Under the assumptionsof Proposition 2.1, we shall prove that

∣∣Ra(x)∣∣,

∣∣∇Ra(x)∣∣ ≤ Cε4ae−

116

|x| uniformly in R2 as aր a∗, (2.34)

where the constant C > 0 is independent of 0 < a < a∗.To prove (2.34), we first claim that there exists a constant C > 0, independent of

0 < a < a∗, such that

∣∣Ra(x)∣∣ ≤ Cε4a uniformly in L∞(R2) as aր a∗. (2.35)

Instead, assume that the above claim (2.35) is false, i.e., limaրa∗‖Ra‖L∞(R2)

ε4a= ∞. Denote

R̄a := Ra‖Ra‖L∞(R2) , so that ‖R̄a‖L∞(R2) = 1. Applying (2.20) and (2.26), we derive from(2.31) that

|∇R̄a(0)| =|Ra(0)|

‖Ra‖L∞(R2)=

1

|w(0) +Ra(0)||Ia(0)||∇Ia(0)|‖Ra‖L∞(R2)

≤ Cε4a

‖Ra‖L∞(R2)as aր a∗,

which then implies that

∇R̄a(0) → 0 uniformly in L∞(R2,R2) as aր a∗. (2.36)

11

We also deduce from (2.33) that

L̃aR̄a =Fa(x)− βaw‖Ra‖L∞(R2)

:= Fa(x) in R2. (2.37)

Note from (2.24) and (2.26) that Fa(x) satisfies

∣∣Fa(x)∣∣ := ε

4a

‖Ra‖L∞(R2)

∣∣Fa(x)− βaw∣∣

ε4a

≤ Cε4a

‖Ra‖L∞(R2)e−

116

|x| uniformly in R2 as aր a∗,(2.38)

where the constant C > 0 is independent of 0 < a < a∗. Suppose ya is a global maximumpoint of |R̄a(x)|, so that |R̄a(ya)| = maxx∈R2 |Ra(x)|‖Ra‖L∞(R2) = 1. In view of the definition ofthe operator L̃a, by the maximum principle one can deduce from (2.37) and (2.38) that|ya| ≤ C uniformly in 0 < a < a∗.

On the other hand, the usual elliptic regularity theory yields that there exist a subse-quence, still denoted by {R̄a}, of {R̄a} and a function R̄0 ∈ H1(R2) such that R̄a → R̄0weakly in H1(R2) and strongly in Lqloc(R

2) for all q ∈ [2,∞) as a ր a∗. Note from(2.36)–(2.38) that R̄0 satisfies

∇R̄0(0) = 0, L̃R̄0(x) = 0 in R2,

which yields that R̄0 =∑2

i=1 ci∂w∂yi

in view of (2.17). Since ∇R̄0(0) = 0, we deduce thatc1 = c2 = 0, which further implies that R̄0(x) ≡ 0 in R2. This however contradicts tothe above fact that by passing to a subsequence if necessary, 1 = R̄0(ya) → R̄0(ȳ0) asaր a∗ for some ȳ0 ∈ R2. Therefore, the claim (2.35) holds true.

Following (2.25), (2.26) and (2.35), the comparison principle applied to (2.33) yieldsthat Ra satisfies

|Ra(x)| ≤ Cε4ae−116

|x| in R2, (2.39)

where the exponential decay of w is also used. By the gradient estimates of (3.15) in [26],we further derive from (2.33) and (2.39) that

|∇Ra| ≤ Cε4ae−120

|x| in R2, (2.40)

which therefore implies that the conclusion (2.34) is true in view of (2.39).Step 2. Since ψ2(|x|) := −12

(w+x ·∇w

)is radially symmetric, we deduce from (2.31)

thatx⊥ · ∇Ra = x⊥ · ∇

[Ra + βaψ2

]= x⊥ · ∇Ra in R2.

This then implies from (2.34) that

|x⊥ · ∇Ra| ≤ Cε4ae−130

|x| uniformly in R2 as aր a∗, (2.41)

where the constant C > 0 is independent of 0 < a < a∗. By the similar argument ofproving (2.35), one can deduce from (2.22) and (2.41) that

∣∣Ia(x)∣∣ ≤ Cε6a uniformly in L∞(R2) as aր a∗, (2.42)

12

where the property (2.15) is used in view of the fact that Ia satisfies∫R2wIadx ≡ 0.

Similar to (2.39) and (2.40), one can further reduce from (2.42) that Lemma 2.2(1) holdstrue.

Step 3. In view of (2.24), we now define

R1,a(x) = Ra(x)− ε4aψ1(x)− βaψ2(|x|), (2.43)

where ψ2(|x|) := −12(w + x · ∇w

)is radially symmetric, and ψ1(x) ∈ C2(R2) ∩ L∞(R2)

is a solution of (2.30). Note that the uniqueness of ψ1(x) follows from (2.17). It thenfollows from (2.20) and (2.23) that R1,a satisfies

∇R1,a(0) → 0 uniformly in L∞(R2,R2) as aր a∗, (2.44)

andL̃aR1,a = −ε4afa(x) in R2, (2.45)

where L̃a is defined by (2.23), and the term fa(x) satisfies

fa(x) =[VΩ

(x+

xaεa

)− VΩ(x+ y0)

]w +

λ4

a∗[R̃2a − w2

]w + o(1) as aր a∗,

where the estimate (2.28) is used. We then get from Proposition 2.1 that fa(x) → 0uniformly in L∞(R2) as aր a∗. By the argument of proving (2.35), we can reduce thatR1,a(x) = o(ε4a+|βa|) uniformly in R2 as aր a∗, and the proof of (2.29) is then completein view of (2.43). This completes the proof of Lemma 2.2(2), and we are done.

Before ending this subsection, we derive the refined estimate of Ia as aր a∗.

Lemma 2.3. Under the assumptions of Proposition 2.1, the imaginary part Ia of (2.19)satisfies

Ia(x) := ε6aΩψI(x) + o(ε

6a + ε

2a|βa|) in R2 as aր a∗, (2.46)

where ψI(x) ∈ C2(R2) ∩ L∞(R2) solves uniquely∫

R2

ψIwdx = 0, LψI(x) = −(x⊥ · ∇ψ1

)in R2, (2.47)

where the operator L is defined by (2.14), and ψ1(x) ∈ C2(R2) ∩ L∞(R2) is given by(2.30).

Proof. Following Lemma 2.2(2), we obtain from (2.22) that∫R2Iawdx ≡ 0, and

LaIa = −ε2aΩ(x⊥ · ∇Ra

)= −ε6aΩ

(x⊥ · ∇ψ1

)+ o(ε6a + ε

2a|βa|) as aր a∗, (2.48)

where ψ1(x) ∈ C2(R2) ∩ L∞(R2) is given by (2.30). Set

I1,a(x) = Ia(x)− ε6aΩψI(x),

where ψI(x) ∈ C2(R2) ∩ L∞(R2) is defined in (2.47).Similar to Step 3 in the proof of Lemma 2.2, by the estimate (2.28) one can deduce

from (2.48) that I1,a(x) = o(ε6a + ε

2a|βa|) uniformly in R2 as aր a∗, which then implies

directly that (2.46) holds true. Finally, since L is a linear operator, the constriction∫R2ψIwdx = 0 and the property (2.15) lead to the uniqueness of ψI(x) ∈ C2(R2) ∩

L∞(R2) defined by (2.47).

13

Remark 2.1. Under the general assumptions on VΩ(x) of Proposition 2.1, the upperbound (2.28) of Ia as a ր a∗ is optimal. However, if the trap VΩ(x) has a bettersymmetry, then the leading term ψI(x) of Ia as aր a∗ may vanish and hence the upperbound (2.28) of Ia as a ր a∗ is not optimal. Specially, if the trap VΩ(x) is radiallysymmetric, we have proved in [30, Theorem 1.3] that Ia(x) ≡ 0 as aր a∗.

3 Nonexistence of Vortices

This section is devoted to the proof of Theorem 1.1 on the nonexistence of vortices forminimizers ua as a ր a∗. Towards this purpose, we first need to establish Theorem 1.2on the refined limiting profiles of ua. Applying the estimates of previous section, westart with the following refined estimate of the unique blow-up point xa for |ua(x)| asaր a∗.Lemma 3.1. Under the assumptions of Proposition 2.1, assume that y0 ∈ R2 is a uniqueand non-degenerate global minimum point of HΩ(y) :=

∫R2VΩ(x+y)w

2(x)dx. Then thereexists a point ȳ0 ∈ R2 such that the unique maximum point xa of |ua(x)| satisfies∣∣∣ε4a

(xaεa

− y0)− ε4aβa

y02

∣∣∣ = ε8aO(|ȳ0|) + o([ε4a + |βa|]2

)as aր a∗. (3.1)

Proof. We calculate from (2.23) and (2.24) that∫

R2

∂w

∂x1L̃aRa =

∫

R2

∂w

∂x1Fa(x)

= −ε4a[ ∫

R2

∂w

∂x1VΩ

(x+

xaεa

)w +

λ4

a∗

∫

R2

∂w

∂x1R̃2aw

]

+ε4aΩ

∫

R2

∂w

∂x1

(x⊥ · ∇Ia

ε2a

)+

a

a∗

∫

R2

∂w

∂x1I2aw

:= A1 +A2,

(3.2)

where the part A2 satisfies

A2 : = ε4aΩ

∫

R2

∂w

∂x1

(x⊥ · ∇Ia

ε2a

)+

a

a∗

∫

R2

∂w

∂x1I2aw

= ε4a[ε4a + o(ε

4a + |βa|)

]Ω2

∫

R2

∂w

∂x1

(x⊥ · ∇ψI

)as aր a∗

(3.3)

in view of Lemmas 2.2 and 2.3, where ψI(x) ∈ C2(R2)∩L∞(R2) is given by Lemma 2.3.As for the part A1, we observe from Proposition 2.1 that

∫

R2

∂w2

∂x1VΩ(x+ y0)dx = 0,

∫

R2

∂w

∂x1w2ψ2(|x|)dx = 0,

where the radial symmetry of ψ2(|x|) is also used. We then have

− 1ε4aA1 : =

∫

R2

∂w

∂x1VΩ

(x+

xaεa

)w +

λ4

a∗

∫

R2

∂w

∂x1R̃2aw

=

∫

R2

∂w

∂x1

[VΩ

(x+

xaεa

)− VΩ(x+ y0)

]w +

λ4

a∗

∫

R2

∂w

∂x1

(R̃2a − w2

)w

=

∫

R2

∂w

∂x1

[(xaεa

− y0)· ∇VΩ(x+ y0)

]w +

2λ4

a∗ε4a

∫

R2

∂w

∂x1w2ψ1

+o(∣∣∣xaεa

− y0∣∣∣+ ε4a + |βa|

)as aր a∗,

(3.4)

14

where the expansion of Lemma 2.2(2) is used in the last identity. We deduce from(3.2)–(3.4) that

−ε4a∫

R2

∂w

∂x1

[(xaεa

− y0)· ∇VΩ(x+ y0)

]w

= ε8a

{2λ4a∗

∫

R2

∂w

∂x1w2ψ1 − Ω2

∫

R2

∂w

∂x1

(x⊥ · ∇ψI

)}

+

∫

R2

∂w

∂x1L̃aRa + o(ε8a + ε4a|βa|) as aր a∗,

(3.5)

where ψI(x) ∈ C2(R2) ∩ L∞(R2) is given by Lemma 2.3.On the other hand, by the definition of L̃, we have

∫R2

∂w∂x1

L̃Radx = 0. It then followsfrom Lemma 2.2(2) that

∫

R2

∂w

∂x1L̃aRa =

∫

R2

∂w

∂x1

(L̃a − L̃

)Ra

=

∫

R2

∂w

∂x1

{ε4a

(Ω24|x|2 + λ

4

a∗w2

)+ ε4aVΩ

(x+

xaεa

)− βa

−wRa −a

a∗(2w +Ra)Ra −

a

a∗I2a

}Ra

= ε8a

∫

R2

∂w

∂x1

(Ω24|x|2 + λ

4

a∗w2

)ψ1 + ε

8a

∫

R2

∂w

∂x1VΩ(x+ y0)ψ1

+ε4aβa

∫

R2

∂w

∂x1VΩ(x+ y0)ψ2 − ε4aβa

∫

R2

∂w

∂x1ψ1 −

ε8a2

∫

R2

∂w2

∂x1ψ21

−ε4aβa∫

R2

∂w2

∂x1ψ1ψ2 −

a

a∗

∫

R2

∂w2

∂x1

(ε8aψ

21 + 2ε

4aβaψ1ψ2

)

+o([ε4a + |βa|]2) as aր a∗,

due to the radial symmetry of ψ2(|x|). The above estimate thus gives that∫

R2

∂w

∂x1L̃aRa = ε8a

{∫

R2

∂w

∂x1

[λ4a∗w2 +

(Ω24|x|2 + VΩ(x+ y0)

)]ψ1 −

3

2

∫

R2

∂w2

∂x1ψ21

}

+ε4aβa

{∫

R2

∂w

∂x1VΩ(x+ y0)ψ2 −

∫

R2

∂w

∂x1ψ1 − 3

∫

R2

∂w2

∂x1ψ1ψ2

}

+o([ε4a + |βa|]2) as aր a∗.(3.6)

We now derive from (3.5) and (3.6) that

−ε4a

2

∫

R2

∂w2

∂x1

[(xaεa

− y0)· ∇VΩ(x+ y0)

]

= ε8a

{∫

R2

∂w

∂x1

[3λ4a∗

w2 +(Ω2

4|x|2 + VΩ(x+ y0)

)]ψ1

−3∫

R2

∂w

∂x1wψ21 − Ω2

∫

R2

∂w

∂x1

(x⊥ · ∇ψI

)}

−ε4aβa{∫

R2

∂w

∂x1ψ1 + 6

∫

R2

∂w

∂x1wψ1ψ2 −

∫

R2

∂w

∂x1VΩ(x+ y0)ψ2

}

+o([ε4a + |βa|]2) as aր a∗,

(3.7)

15

where ψI(x) ∈ C2(R2) ∩ L∞(R2) is given by Lemma 2.3.Similar to (3.23) of [28], one can obtain that

∫

R2

∂w

∂x1ψ1 + 6

∫

R2

∂w

∂x1wψ1ψ2 −

∫

R2

∂w

∂x1VΩ(x+ y0)ψ2

=1

4

∫

R2

∂w2

∂x1

[y0 · ∇VΩ(x+ y0)

].

(3.8)

We now conclude from (3.7) and (3.8) that there exists some ȳ0 = (ȳ10, ȳ20) ∈ R2 suchthat

1

2

∫

R2

∂w2

∂xj

[ε4a

(xaεa

− y0)− ε4aβa

y02

]· ∇VΩ(x+ y0)

= ε8aO(|ȳj0|) + o([ε4a + |βa|]2) as aր a∗, j = 1, 2.(3.9)

Since y0 ∈ R2 is a non-degenerate global minimum point of HΩ(y) :=∫R2VΩ(x +

y)w2(x)dx, we finally derive from (3.9) that (3.1) holds for some ȳ0 ∈ R2, and thelemma is therefore proved.

Since the limit estimates of Lemma 2.2 are not enough for establishing Theorem 1.1,we next employ Lemma 3.1 to derive the following more leading terms of Ra in terms ofεa and βa.

Lemma 3.2. Under the assumptions of Theorem 1.1, the real part Ra of (2.19) satisfies

Ra(x) : = ε4aψ1(x) + βaψ2(x) + ε

8aψ3(x) + β

2aψ4(x) + ε

4aβaψ5(x)

+o([ε4a + |βa|]2

)in R2 as aր a∗,

(3.10)

where ψ1(x) and ψ2(x) are as in Lemma 2.2, and ψi(x) ∈ C2(R2) ∩ L∞(R2) solvesuniquely

∇ψi(0) = 0, L̃ψi(x) = Fi(x) in R2, i = 3, 4, 5, (3.11)and Fi(x) satisfies

Fi(x) =

3wψ21 −(3λ4a∗

w2 +[Ω24|x|2 + VΩ(x+ y0)

])ψ1

−[ȳ0 · ∇VΩ(x+ y0)

]w +Ω

(x⊥ · ∇ψI

), if i = 3;

ψ2 + 3wψ22 , if i = 4;

6wψ1ψ2 + ψ1 −(3λ4a∗

w2 +[Ω24|x|2 + VΩ(x+ y0)

])ψ2

−[y02

· ∇VΩ(x+ y0)]w, if i = 5.

(3.12)

Here the point ȳ0 ∈ R2 is the same as that of (3.1), and ψI(x) ∈ C2(R2) ∩ L∞(R2) isgiven by Lemma 2.3.

Proof. DenoteMa := Ra − ε4aψ1 − βaψ2,

which then yields from (2.23) that

L̃aMa = L̃aRa − L̃a(ε4aψ1 + βaψ2)=

{L̃aRa − L̃(ε4aψ1 + βaψ2)

}−

(L̃a − L̃

)(ε4aψ1 + βaψ2)

:= II1 + II2.

(3.13)

16

Direct calculations give that the term II2 of (3.13) satisfies

II2 : = −(L̃a − L̃

)(ε4aψ1 + βaψ2)

= −(ε4aψ1 + βaψ2){ε4a

[Ω24|x|2 + VΩ

(x+

xaεa

)]− βa − wRa

+αaa∗w2 − a

a∗(2w +Ra)Ra −

a

a∗I2a

}.

(3.14)

Applying Lemmas 2.2 and 2.3, we thus get that

II2 : = ε8a

{3wψ21 −

(λ4a∗w2 +

[Ω24|x|2 + VΩ(x+ y0)

])ψ1

}

+ε4aβa

{6wψ1ψ2 + ψ1 −

(λ4a∗w2 +

[Ω24|x|2 + VΩ(x+ y0)

])ψ2

}

+β2a(ψ2 + 3wψ

22

)+ o([ε4a + |βa|]2) as aր a∗.

(3.15)

As for the term II1 of (3.13), we derive from (2.23) that

II1 : = L̃aRa − L̃(ε4aψ1 + βaψ2)

= L̃aRa − βaw + ε4a{λ4a∗w3 +

[Ω24|x|2 + VΩ(x+ y0)

]w}

= −ε4a[VΩ

(x+ xa

εa

)− VΩ(x+ y0)

]w − αa

a∗w(2w +Ra)Ra

+ε4aΩ(x⊥ · ∇Ia

ε2a

)+

a

a∗I2aw.

(3.16)

Applying Lemma 2.3 yields that

ε4aΩ(x⊥ · ∇Ia

ε2a

)= ε8aΩ

(x⊥ · ∇ψI

)+ o(ε8a + ε

4a|βa|) as aր a∗,

where ψI(x) ∈ C2(R2) ∩ L∞(R2) is given by Lemma 2.3. By Lemma 3.1, we have

−ε4a[VΩ

(x+

xaεa

)− VΩ(x+ y0)

]w

= −ε4a[(xaεa

− y0)· ∇VΩ(x+ y0)

]w(x)

(1 + o(1)

)

= −ε4aβa[y02

· ∇VΩ(x+ y0)]w − ε8a

[ȳ0 · ∇VΩ(x+ y0)

]w

+o([ε4a + |βa|]2

)as aր a∗,

(3.17)

where the point ȳ0 ∈ R2 is the same as that of (3.1). Applying Lemmas 2.2 and 2.3, wenow deduce from above that

II1 : = ε8a

{−

[ȳ0 · ∇VΩ(x+ y0)

]w − 2λ

4

a∗w2ψ1 +Ω

(x⊥ · ∇ψI

)}

+ε4aβa

{−

[y02

· ∇VΩ(x+ y0)]w − 2λ

4

a∗w2ψ2

}

+o([ε4a + |βa|]2

)as aր a∗,

(3.18)

where ψI(x) ∈ C2(R2) ∩ L∞(R2) is given by Lemma 2.3.

17

We now calculate from (3.15) and (3.18) that

L̃aMa = ε8a{3wψ21 −

(3λ4a∗

w2 +[Ω24|x|2 + VΩ(x+ y0)

])ψ1

−[ȳ0 · ∇VΩ(x+ y0)

]w +Ω

(x⊥ · ∇ψI

)}

+ε4aβa

{6wψ1ψ2 + ψ1 −

(3λ4a∗

w2 +[Ω24|x|2 + VΩ(x+ y0)

])ψ2

−[y02

· ∇VΩ(x+ y0)]w}

+β2a(ψ2 + 3wψ

22

)+ o([ε4a + |βa|]2) as aր a∗,

(3.19)

where the point ȳ0 ∈ R2 is the same as that of (3.1). Following (3.19), the argument ofLemma 2.2(2) then yields the estimate (3.10). Moreover, the property (2.17) implies theuniqueness of ψi for i = 3, 4, 5, and the proof of Lemma 3.2 is therefore complete.

Lemma 3.3. Under the assumptions of Theorem 1.1, we have

∫

R2

wψ1 = 0,

∫

R2

wψ2 = 0, T1 =

∫

R2

(2wψ4 + ψ

22

)= 0, (3.20)

and

T2 = 2

∫

R2

wψ5 + 2

∫

R2

ψ1ψ2 = −2λ4 < 0, (3.21)

where ψ1(x), · · · , ψ5(x), ψI(x) ∈ C2(R2) ∩ L∞(R2) be given by Lemmas 2.2, 3.2, and2.3, respectively.

Proof. The assumptions of Lemma 3.3 imply that

(x+ y0) · ∇VΩ(x+ y0) = 2VΩ(x+ y0),∫

R2

w2[y0 · ∇VΩ(x+ y0)

]= 0,

from which we obtain that∫

R2

w2(x · ∇

[Ω24|x|2 + VΩ(x+ y0)

])

=2

∫

R2

(Ω24|x|2 + VΩ(x+ y0)

)w2 = 2λ4 > 0.

(3.22)

Applying (3.22), Lemma 3.3 can be proved in a similar way of [28, Lemma 3.5], and thedetailed proof is omitted for simplicity.

Proof of Theorem 1.2. Applying Proposition 2.1, we derive from (2.19) that Rasatisfies∫

R2

w2 =

∫

R2

|va|2 =∫

R2

[(w+Ra

)2+I22

], i.e., 0 = 2

∫

R2

wRa+

∫

R2

R2a+

∫

R2

I2a . (3.23)

18

Following Lemmas 2.3 and 3.2, we then derive from (3.23) that

0 = 2

∫

R2

wRa +

∫

R2

R2a +

∫

R2

I2a

= 2

∫

R2

w(ε4aψ1 + βaψ2 + ε8aψ3 + β

2aψ4 + ε

4aβaψ5)

+

∫

R2

(ε4aψ1 + βaψ2 + ε8aψ3 + β

2aψ4 + ε

4aβaψ5)

2 + o([ε4a + |βa|]2)

= 2ε4a

(∫

R2

wψ1

)+ 2βa

(∫

R2

wψ2

)+ β2a

(2

∫

R2

wψ4 +

∫

R2

ψ22

)

+2ε4aβa

(∫

R2

wψ5 +

∫

R2

ψ1ψ2

)+ ε8a

(2

∫

R2

wψ3 +

∫

R2

ψ21

)+ o([ε4a + |βa|]2)

= −2λ4ε4aβa + ε8a(2

∫

R2

wψ3 +

∫

R2

ψ21

)+ o([ε4a + |βa|]2) as aր a∗,

(3.24)

where Lemma 3.3 is also used. One can derive from (3.24) that

2

∫

R2

wψ3 +

∫

R2

ψ21 6= 0,

and

− 2λ4βa + ε4a(2

∫

R2

wψ3 +

∫

R2

ψ21

)= 0, i.e., βa = C

∗ε4a, (3.25)

for some constant C∗ 6= 0. Applying (3.25), the refined limit profile (1.19) follows directlyfrom Proposition 2.1 and Lemma 3.2. This completes the proof of Theorem 1.2.

Applying Theorem 1.2, we are now ready to establish Theorem 1.1 on the nonexis-tence of vortices.

Proof of Theorem 1.1. Under the assumptions of Theorem 1.1, we note that

Ω2

4|x|2 + VΩ

(x+ y0

)=

Ω2

4

(|x|2 − |x+ y0|2

)+ V (x+ y0) ≤ C|x|2 as |x| → ∞,

where y0 ∈ R2 is a unique global minimum point of HΩ(y) :=∫R2VΩ(x+ y)w

2(x)dx. Wethen note from (1.17) and (2.5) that the function ψ1 of Theorem 1.2 satisfies

∣∣(−∆+ 1)ψ1∣∣ =

∣∣∣3w2ψ1 −λ4

a∗w3(x)−

[Ω24|x|2 + VΩ

(x+ y0

)]w∣∣∣

≤ C0|x|32 e−|x| as |x| → ∞.

(3.26)

Applying the comparison principle to (3.26) thus yields that

|ψ1(x)| ≤ C1|x|52 e−|x| as |x| → ∞,

which further implies that

|ψ1(x)| ≤ C2(1 + |x|)52 e−|x| in R2, (3.27)

19

where the constant C2 > 0 is independent of 0 < a < a∗. Applying Theorem 1.2, we

then obtain from (2.5) and (3.27) that as aր a∗,∣∣∣εa

√a∗ ua

(εax+ xa

)e−i

(Ω2εax·x⊥a −θa

)∣∣∣ ≥ w2− ε4a|ψ1|

≥ (1 + |x|)− 12 e−|x|[C3 − C2ε4a(1 + |x|)3

]> 0, if 1 + |x| ≤

(2C33C2

) 13ε− 4

3a ,

(3.28)

which implies that

|ua(x)| > 0 in |x| ≤( C32C2

) 13ε− 4

3a as aր a∗.

Since εa :=(a∗−a)

14

λ> 0, we now conclude from (3.28) that (1.15) holds true, i.e., ua does

not admit any vortex in the region R(a) :={x ∈ R2 : |x| ≤ C(a∗ − a)− 13

}as a ր a∗,

where the constant C > 0 is independent of 0 < a < a∗. This completes the proof ofTheorem 1.1.

4 Refined Limit Profiles in Harmonic Traps

In this section we shall improve Theorem 1.2 to obtain much more refined limit profilesof minimizers ua in harmonic traps as a ր a∗. Due to the physically relevant (cf.[3, 18,34,46,47]), in this section we focus on the harmonic trap V (x) ≥ 0 of the form

V (x) = x21 + Λ2x22 for x = (x1, x2) ∈ R2, where 0 < Λ < 1, (4.1)

since the case Λ > 1 can be established similarly, and while the case Λ = 1 was alreadyaddressed in Theorem 1.3 and (1.18) of [30].

Under the assumption (4.1), one can note from (1.6) that eF (a) admits the criticalvelocity Ω∗ := 2Λ. Then for any 0 < Ω < Ω∗ := 2Λ, one can check that VΩ(x) :=

V (x)− Ω24 |x|2 is also homogeneous of degree 2, and

y0 = (0, 0) is the unique and non-degenerate critical point

of HΩ(y) :=

∫

R2

VΩ(x+ y)w2(x)dx.

(4.2)

We then deduce from (4.2) that for any 0 < Ω < Ω∗ := 2Λ,

VΩ(x+ y0) +Ω2

4|x|2 = V (x) = x21 + Λ2x22, (4.3)

which then implies that for any 0 < Ω < Ω∗ := 2Λ, the constant λ > 0 defined in (2.6)is independent of Ω and can be simplified as

λ0 =(∫

R2

V (x)w2dx) 1

4> 0. (4.4)

For convenience, we denote ϕi(x) ∈ C2(R2) ∩ L∞(R2) to be the unique solution of

∇ϕi(0) = 0,(−∆+ 1− 3w2

)ϕi(x) = f̃i(x) in R

2, i = 1, 2, (4.5)

20

where f̃i(x) satisfies

f̃i(x) =

−λ40

a∗w3(x)−

(x21 + Λ

2x22)w(x), if i = 1;

−(1− Λ2)x21w(x), if i = 2.(4.6)

We also denote ϕI(x) ∈ C2(R2) ∩ L∞(R2) to be the unique solution of(−∆+ 1− w2

)ϕI(x) = −

(x⊥ · ∇ϕ2

)in R2,

∫

R2

ϕIwdx = 0, (4.7)

where ϕ2 satisfies (4.5). Following above notations, our main result of this section canbe stated as follows:

Theorem 4.1. Consider the harmonic potential V (x) = x21 + Λ2x22, where 0 < Λ < 1.

Then for any fixed 0 < Ω < Ω∗ := 2Λ, the complex-valued minimizer ua of eF (a) satisfies

ǫa√a∗ ua

(ǫax+ xa

)e−i

(Ω2ǫax·x⊥a −θa

)= w + ǫ4a

{ϕ1 + CΛ(w + x · ∇w)

}[1 + o(1)

]

+i ǫ6aΩϕI[1 + o(1)

]as aր a∗,

(4.8)

where ǫa :=(a∗−a)

14

λ0> 0, xa is the unique global maximum point of |ua| which satisfies

∣∣xa∣∣ = o

(ǫ5a)

as aր a∗, (4.9)

and the constant CΛ 6= 0 is independent of Ω and satisfies

CΛ =1

2λ40

[ ∫

R2

(3w2 − 1)ϕ21 − 4∫

R2

(x21 + Λ

2x22)wϕ1

]6= 0. (4.10)

Remark 4.1. (1). We comment that Theorem 4.1 can be established similarly in thecase Λ > 1 of V (x) = x21 + Λ

2x22, and while a simplified version of Theorem 4.1 wascovered by Theorem 1.3 and (1.18) in [30] for the case Λ = 1 of V (x) = x21+Λ

2x22, whereIm(ua) ≡ 0 in R2 as aր a∗.

(2). One can note from Theorem 4.1 that the function ϕ1 and the constant CΛ 6= 0of (4.8) are independent of 0 < Ω < Ω∗ := 2Λ, if V (x) = x21 + Λ

2x22 is a harmonictrapping potential. In contrast to Theorem 1.2, Theorem 4.1 thus implies that the firsttwo leading terms of (4.8) are independent of 0 < Ω < Ω∗ := 2Λ, which seems false formore general trapping potentials. On the other hand, (4.9) gives an improved estimateof xa as a ր a∗, in view of Lemma 3.1. Theorem 4.1 is therefore an improvement ofTheorem 1.2 in the harmonic case of trapping potentials.

To prove Theorem 4.1, we now consider the harmonic trap V (x) of the form (4.1),and let 0 < Ω < Ω∗ := 2Λ be fixed. Let Ra(x) and Ia(x) be defined by (2.7) and (2.19),respectively, where ua denotes a complex-valued minimizer of eF (a) as aր a∗. Applying(4.2), it then follows from Lemma 2.2 that the real part Ra of (2.19) satisfies

Ra(x) := ε4aψ1(x) + βaψ2(|x|) + o(ε4a + |βa|) in R2 as aր a∗, (4.11)

where εa > 0 and βa ∈ R are defined by (2.18). Here ψ2(|x|) ∈ C2(R2) ∩ L∞(R2)satisfying

ψ2(|x|) := −1

2

(w + x · ∇w

)(4.12)

21

is radially symmetric, and ψ1(x) ∈ C2(R2) ∩ L∞(R2) solves uniquely

∇ψ1(0) = 0, L̃ψ1(x) = −λ40a∗w3(x)−

(x21 +Λ

2x22)w(x) in R2, (4.13)

where 0 < Λ < 1 is as in (4.1), and the operator L̃ is defined by (2.16).We next denote ψ11(x) ∈ C2(R2) ∩ L∞(R2) to be the unique solution of

∇ψ11(0) = 0, L̃ψ11(x) = −λ40a∗w3(x)− Λ2|x|2w(x) in R2, (4.14)

where 0 < Λ < 1 is as above, and ψ(x) ∈ C2(R2) ∩ L∞(R2) to be the unique solution of∇ψ(0) = 0, L̃ψ(x) = −(1− Λ2)x21w(x) in R2, where x = (x1, x2) ∈ R2. (4.15)

One can check from (4.13) that

ψ1(x) = ψ11(|x|) + ψ(x), where ψ11(|x|) is radially symmetric in R2, (4.16)which further implies from Lemma 2.3 that

LψI(x) = −(x⊥ · ∇ψ1

)= −

(x⊥ · ∇ψ

)in R2, (4.17)

where the operator L is defined by (2.14). Applying (4.17), we thus conclude fromLemma 2.3 that the imaginary part Ia of (2.19) satisfies

Ia(x) := ε6aΩψI(x) + o(ε

6a + ε

2a|βa|) in R2 as aր a∗, (4.18)

where ψI(x) ∈ C2(R2) ∩ L∞(R2) solves uniquely∫

R2

ψIwdx = 0, LψI(x) = −(x⊥ · ∇ψ

)in R2. (4.19)

Here ψ(x) ∈ C2(R2)∩L∞(R2) is given uniquely by (4.15), and the operator L is definedby (2.14).

Following (4.1), we next address the following more refined estimate of the uniqueblow-up point xa for |ua(x)| as aր a∗.Proposition 4.2. Under the assumptions of Theorem 4.1, the unique maximum pointxa of |ua(x)| satisfies

∣∣ε3axa∣∣ = o

([ε4a + |βa|]2

)as aր a∗, (4.20)

i.e., the estimate (3.1) holds for y0 = (0, 0) and ȳ0 = (0, 0).

Proof. Applying (4.3), we follow from (3.7) that the unique maximum point xa of|ua(x)| satisfies

−ε4a

2

∫

R2

∂w2

∂x1

[(xaεa

− y0)· ∇VΩ(x+ y0)

]

= ε8a

{∫

R2

∂w

∂x1

[3λ40a∗

w2 +(x21 + Λ

2x22)]ψ1

−3∫

R2

∂w

∂x1wψ21 − Ω2

∫

R2

∂w

∂x1

(x⊥ · ∇ψI

)}

−ε4aβa{∫

R2

∂w

∂x1ψ1 + 6

∫

R2

∂w

∂x1wψ1ψ2 −

∫

R2

∂w

∂x1VΩ(x+ y0)ψ2

}

+o([ε4a + |βa|]2) as aր a∗,

(4.21)

22

where ψI(x) ∈ C2(R2)∩L∞(R2) is given by (4.19). Using (4.2), which implies y0 = (0, 0),we obtain from (3.8) that

∫

R2

∂w

∂x1ψ1 + 6

∫

R2

∂w

∂x1wψ1ψ2 −

∫

R2

∂w

∂x1VΩ(x+ y0)ψ2 = 0.

Moreover, one can note from (4.13) that ψ1(x) is even in x ∈ R2, which thus implies that∫

R2

∂w

∂x1

[3λ40a∗

w2 +(x21 + Λ

2x22)]ψ1 − 3

∫

R2

∂w

∂x1wψ21 = 0.

Applying (4.21), we then derive from above that the unique maximum point xa of |ua(x)|satisfies

−ε4a

2

∫

R2

∂w2

∂x1

(xaεa

)· ∇VΩ(x)

= −ε8aΩ2∫

R2

∂w

∂x1

(x⊥ · ∇ψI

)+ o([ε4a + |βa|]2) as aր a∗.

(4.22)

Similarly, one can obtain that the unique maximum point xa of |ua(x)| also satisfies

−ε4a

2

∫

R2

∂w2

∂x2

(xaεa

)· ∇VΩ(x)

= −ε8aΩ2∫

R2

∂w

∂x2

(x⊥ · ∇ψI

)+ o([ε4a + |βa|]2) as aր a∗.

(4.23)

In the appendix, we shall prove that the following claim is true:

II1 =

∫

R2

∂w

∂x1

(x⊥ · ∇ψI

)= 0, II2 =

∫

R2

∂w

∂x2

(x⊥ · ∇ψI

)= 0, (4.24)

where ψI(x) ∈ C2(R2) ∩ L∞(R2) solves uniquely (4.19). Recall from (4.2) that y0 =(0, 0) is the unique and non-degenerate critical point of HΩ(y) :=

∫R2VΩ(x+ y)w

2(x)dx.Applying (4.24), we then conclude from (4.22) and (4.23) that (4.20) holds true, and theproof of Proposition 4.2 is therefore complete.

Proof of Theorem 4.1. Under the assumptions of Theorem 4.1, since (4.2) givesthat y0 = (0, 0) is the unique and non-degenerate critical point HΩ(y) :=

∫R2VΩ(x +

y)w2(x)dx, the uniqueness, up to the constant phase, of minimizers for eF (a) as aր a∗follows directly from [29, Theorem 1.1]. Because Proposition 4.2 gives that the estimate(3.1) holds for y0 = (0, 0) and ȳ0 = (0, 0), we can conclude directly from Lemma 3.1 and(4.3) that the following refined result holds true: the rest real part Ra of (2.19) satisfies

Ra(x) : = ε4aψ1(x) + βaψ2(x) + ε

8aψ3(x) + β

2aψ4(x) + ε

4aβaψ5(x)

+o([ε4a + |βa|]2

)in R2 as aր a∗,

(4.25)

where ψ1(x), ψ2(x) and ψI(x) are as in (4.12), (4.13) and (4.19), respectively. However,ψi(x) ∈ C2(R2) ∩ L∞(R2) solves uniquely

∇ψi(0) = 0, L̃ψi(x) = fi(x) in R2, i = 3, 4, 5, (4.26)

23

and fi(x) satisfies

fi(x) =

3wψ21 −[3λ40a∗

w2 +(x21 +Λ

2x22)]ψ1 +Ω

(x⊥ · ∇ψI

), if i = 3;

ψ2 + 3wψ22 , if i = 4;

6wψ1ψ2 + ψ1 −[3λ40a∗

w2 +(x21 + Λ

2x22)]ψ2, if i = 5.

(4.27)

Therefore, the refined limit profile (3.24) is still true, from which we have 2

∫

R2

wψ3 +∫

R2

ψ21 6= 0, and

− 2λ40βa + ε4a(2

∫

R2

wψ3 +

∫

R2

ψ21

)= 0, (4.28)

where ψ1(x) and ψ3(x) are as in (4.13) and (4.26), respectively. In the appendix, weshall prove the following claim that

I := 2

∫

R2

wψ3 +

∫

R2

ψ21 =

∫

R2

(3w2 − 1)ψ21 − 4∫

R2

(x21 + Λ

2x22)wψ1. (4.29)

Together with (4.29), we conclude from (4.28) that the constant βa satisfies

βa = C∗ε4a, where C

∗ =1

2λ40

[ ∫

R2

(3w2 − 1)ψ21 − 4∫

R2

(x21 + Λ

2x22)wψ1

]6= 0. (4.30)

Applying (4.11), (4.18) and (4.30), we finally conclude from (2.7) and (2.19) that therefined limit profile (4.8) holds true. Moreover, the much more refined estimate (4.9)follows directly from (4.30) and Proposition 4.2. This completes the proof of Theorem4.1.

A Appendix

In this appendix, we follow those notations of Section 4 to address the proof of the claims(4.24) and (4.29).

Proof of (4.24). Consider the polar coordinate (r, θ) in R2, where θ ∈ [0, 2π]. Rewritew(x) = w(r), ψ(x) = ψ(r, θ) and ψI(x) = ψI(r, θ), where ψ and ψI are given by (4.15)and (4.19), respectively. We then have

∂w

∂x1= w′(r) cos θ,

∂w

∂x2= w′(r) sin θ. (A.1)

Since ∇ψ = xrψr +

x⊥

r2ψθ, we have

x⊥ · ∇ψ = ∂ψ(r, θ)∂θ

, x⊥ · ∇ψI =∂ψI(r, θ)

∂θ. (A.2)

By the symmetry of the linear inhomogeneous equation (4.15), we deduce that

ψ(r, θ) = ψ(r, 2π − θ), ψ(r, θ) = ψ(r, θ − π), θ ∈ [π, 2π], (A.3)

24

where ψ(x) ∈ C2(R2) ∩ L∞(R2) is given by (4.15) as before. It then yields from (A.3)that

∂ψ(r, θ)

∂θ= −∂ψ(r, 2π − θ)

∂(2π − θ) ,∂ψ(r, θ)

∂θ=∂ψ(r, θ − π)∂(θ − π) , θ ∈ [π, 2π]. (A.4)

Note from (4.19) and (A.2) that ψI(x) ∈ C2(R2) ∩ L∞(R2) is the unique solution of∫

R2

ψIwdx = 0, LψI(x) = −(x⊥ · ∇ψ

)= −∂ψ(r, θ)

∂θin R2. (A.5)

We then derive from (A.4) and (A.5) that the unique solution ψI(x) satisfies

ψI(r, θ) = −ψI(r, 2π − θ), ψI(r, θ) = ψI(r, θ − π), θ ∈ [π, 2π]. (A.6)

Applying (A.2), we now deduce that

II1 =

∫

R2

∂w

∂x1

(x⊥ · ∇ψI

)

=

∫ ∞

0

∫ 2π

0rw′(r) cos θ

∂ψI(r, θ)

∂θdθdr

=

∫ ∞

0

∫ 2π

0rw′(r) sin θψI(r, θ)dθdr

=

∫ ∞

0

∫ π

0rw′(r) sin θψI(r, θ)dθdr +

∫ ∞

0

∫ 2π

π

rw′(r) sin θψI(r, θ)dθdr

:= A1 +A2.

(A.7)

As for the term A2, we derive from (A.6) that

A2 : =

∫ ∞

0

∫ 2π

π

rw′(r) sin θψI(r, θ)dθdr

= −∫ ∞

0

∫ 2π

π

rw′(r) sin(θ − π)ψI(r, θ − π)dθdr

= −∫ ∞

0

∫ π

0rw′(r) sin δ ψI(r, δ)dδdr = −A1,

(A.8)

where we denote δ := θ − π. We hence obtain from (A.7) and (A.8) that

II1 =

∫

R2

∂w

∂x1

(x⊥ · ∇ψI

)= A1 +A2 = 0. (A.9)

Similar to (A.7), one can conclude from (A.2) and (A.6) that

II2 =

∫

R2

∂w

∂x2

(x⊥ · ∇ψI

)

= −∫ ∞

0

∫ π

0rw′(r) cos θψI(r, θ)dθdr −

∫ ∞

0

∫ 2π

π

rw′(r) cos θψI(r, θ)dθdr

:= −(B1 +B2),

(A.10)

25

where the term B2 satisfies

B2 : =

∫ ∞

0

∫ 2π

π

rw′(r) cos θψI(r, θ)dθdr

= −∫ ∞

0

∫ 2π

π

rw′(r) cos(2π − θ)ψI(r, 2π − θ)dθdr

=

∫ ∞

0

∫ 0

π

rw′(r) cos δ ψI(r, δ)dδdr = −B1,

(A.11)

where we denote δ := 2π − θ. The above estimates yield that

II2 =

∫

R2

∂w

∂x2

(x⊥ · ∇ψI

)= −(B1 +B2) = 0,

together with (A.9), which thus implies that (4.24) holds true, and we are done.

In the rest of this appendix, we establish the claim (4.29) as follows.

Proof of (4.29). Since ψ2 is radially symmetric, we first note that for ψ′2 =

dψ2dr

,

Ω

∫

R2

ψ2(x⊥ · ∇ψI

)= Ω

∫

R2

ψ2[− x2(ψI)x1 + x1(ψI)x2

]

= Ω

∫

R2

[ψIx2(ψ2)x1 − ψIx1(ψ2)x2

]

= Ω

∫

R2

ψIx1x2r

(ψ′2 − ψ′2

)= 0.

Following (4.13) and (4.26), we then derive from above that for V (x) = x21 + Λ2x22,

I = 2

∫

R2

ψ3L̃ψ2 +∫

R2

ψ21

= 2

∫

R2

ψ2L̃ψ3 − 2Ω∫

R2

ψ2(x⊥ · ∇ψI

)+

∫

R2

ψ21

= −∫

R2

(w + x · ∇w

){3wψ21 −

[3λ40a∗

w2 + V (x)]ψ1

}+

∫

R2

ψ21

:= A+B,

(A.12)

where the part A satisfies

A = −∫

R2

w{3wψ21 −

[3λ40a∗

w2 + V (x)]ψ1

}+

∫

R2

ψ21

= −∫

R2

|∇ψ1|2 +2λ40a∗

∫

R2

w3ψ1.

(A.13)

Since ∫

R2

(x · ∇ψ1)∆ψ1 = −2∫

R2

|∇ψ1|2 −∫

R2

(x · ∇ψ1)∆ψ1,

i.e., −∫R2

|∇ψ1|2 =∫R2(x · ∇ψ1)∆ψ1, it follows from (A.13) that the part A can be

rewritten as

A =

∫

R2

(x · ∇ψ1)∆ψ1 +2λ40a∗

∫

R2

w3ψ1. (A.14)

26

We rewrite the part B of (A.12) as

B = −∫

R2

(x · ∇w){3wψ21 −

[3λ40a∗

w2 + V (x)]ψ1

}= B1 +B2 +B3, (A.15)

where the term B2 satisfies

B2 =3λ4

a∗

∫

R2

(x · ∇w)w2ψ1 = −2λ40a∗

∫

R2

w3ψ1 −λ4

a∗

∫

R2

w3(x · ∇ψ1).

Together with (A.14), we obtain from above that

A+B2 =

∫

R2

(x · ∇ψ1)∆ψ1 −λ40a∗

∫

R2

w3(x · ∇ψ1). (A.16)

The term B1 of (A.15) satisfies

B1 = −∫

R2

(x · ∇w)3wψ21 = 3∫

R2

w2ψ21 + 3

∫

R2

w2ψ1(x · ∇ψ1)

=

∫

R2

(3w2 − 1)ψ21 +∫

R2

(3w2 − 1)ψ1(x · ∇ψ1),(A.17)

due to the fact that

−∫

R2

ψ1(x · ∇ψ1) = 2∫

R2

ψ21 +

∫

R2

ψ1(x · ∇ψ1).

But the term B3 of (A.15) satisfies

B3 =

∫

R2

ψ1V (x)(x · ∇w)

= −2∫

R2

V (x)wψ1 −∫

R2

wψ1[x · ∇V (x)]−∫

R2

wV (x)(x · ∇ψ1)

= −4∫

R2

V (x)wψ1 −∫

R2

wV (x)(x · ∇ψ1),

(A.18)

since V (x) satisfies x · ∇V (x) = 2V (x). Applying (A.16)–(A.18), we now obtain from(4.13) and (A.12) that

I = (A+B2) +B1 +B3

=

∫

R2

(3w2 − 1)ψ21 − 4∫

R2

V (x)wψ1

+

∫

R2

(x · ∇ψ1){(∆− 1 + 3w2)ψ1 −

λ40a∗w3 − V (x)w

}

=

∫

R2

(3w2 − 1)ψ21 − 4∫

R2

V (x)wψ1,

(A.19)

and the claim (4.29) is therefore proved in view of (4.1).

Acknowledgements: The author thanks Dr. Yong Luo very much for his fruitfuldiscussions on the present paper.

27

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31

1 Introduction1.1 Proof strategy of Theorem 1.1

2 Refined Estimates of Minimizers2.1 Refined estimates of va as aa*

3 Nonexistence of Vortices4 Refined Limit Profiles in Harmonic TrapsA Appendix