Amandine Aftalion

AND Benedetta Noris

Christos Sourdis
email: csourdis@tem.uoc.gr

THOMAS-FERMI APPROXIMATION FOR COEXISTING TWO COMPONENT BOSE-EINSTEIN CONDENSATES AND NONEXISTENCE OF VORTICES FOR SMALL ROTATION

We study minimizers of a Gross-Pitaevskii energy describing a two-component Bose-Einstein condensate confined in a radially symmetric harmonic trap and set into rotation. We consider the case of coexistence of the components in the Thomas-Fermi regime, where a small parameter ε conveys a singular perturbation. The minimizer of the energy without rotation is determined as the positive solution of a system of coupled PDE's for which we show uniqueness. The limiting problem for ε = 0 has degenerate and irregular behavior at specific radii, where the gradient blows up. By means of a perturbation argument, we obtain precise estimates for the convergence of the minimizer to this limiting profile, as ε tends to 0. For low rotation, based on these estimates, we can show that the ground states remain real valued and do not have vortices, even in the region of small density.

1. Introduction 1.1. The problem. In this paper, we study the behavior of the minimizers of the following energy functional describing a two component Bose-Einstein condensate

E Ω ε (u 1 , u 2 ) = 2 j=1 R 2 |∇u j | 2 2 + |x| 2 2ε 2 |u j | 2 + g j 4ε 2 |u j | 4 -Ωx ⊥ • (iu j , ∇u j ) dx + g 2ε 2 R 2 |u 1 | 2 |u 2 | 2 dx (1.1)
in the space

H = (u 1 , u 2 ) : u j ∈ H 1 (R 2 , C), R 2 |x| 2 |u j | 2 dx < ∞, u j L 2 (R 2 ) = 1, j = 1, 2 . (1.
2)

The parameters g 1 , g 2 , g, ε and Ω are positive: Ω is the angular velocity corresponding to the rotation of the condensate, x ⊥ = (-x 2 , x 1 ) and • is the scalar product for vectors, whereas ( , ) is the complex scalar product, so that we have

x ⊥ • (iu, ∇u) = x ⊥ • iu∇ū -iū∇u 2 = -x 2 iu∂ x 1 ū -iū∂ x 1 u 2 + x 1 iu∂ x 2 ū -iū∂ x 2 u 2 .
Here, g j is the self interaction of each component (intracomponent coupling) while g measures the effect of interaction between the two components (intercomponent coupling). We are interested in studying the existence and behavior of the minimizers in the limit when ε is small, describing strong interactions, also called the Thomas-Fermi limit. We assume the condition g 2 < g 1 g 2 , (1.3)

Date: March 18, 2014.

1 which implies that the two components u 1 and u 2 of the minimizers can coexist, as opposed to the segregation case g 2 > g 1 g 2 . Additionally, we can assume without loss of generality that 0 < g 1 ≤ g 2 .

(1.4)

We start with the analysis of the minimizers of the energy functional E 0 ε without rotation, namely with Ω = 0. Up to multiplication by a complex number of modulus 1, the minimizers (η 1,ε , η 2,ε ) of E 0 ε in H are positive solutions of the following system of coupled Gross-Pitaevskii equations:

-ε 2 ∆η 1,ε + |x| 2 η 1,ε + g 1 |η 1,ε | 2 η 1,ε + gη 1,ε |η 2,ε | 2 = λ 1,ε η 1,ε in R 2 ,
(1.5a)

-ε 2 ∆η 2,ε + |x| 2 η 2,ε + g 2 |η 2,ε | 2 η 2,ε + g|η 1,ε | 2 η 2,ε = λ 2,ε η 2,ε in R 2 , (1.5b
)

η i,ε (x) → 0 as |x| → ∞, i = 1, 2, (1.5c)
where λ 1,ε , λ 2,ε are the Lagrange multipliers due to the constraints. We will also refer to the positive minimizers as ground state solutions. Formally setting ε = 0 in (1.5) gives rise to the nonlinear algebraic system

   |x| 2 η 1 + g 1 η 3 1 + gη 1 η 2 2 = λ 1,0 η 1 in R 2 , |x| 2 η 2 + g 2 η 3 2 + gη 2 1 η 2 = λ 2,0 η 2 in R 2 , (1.6)
where η i ≥ 0 satisfy η i L 2 (R 2 ) = 1, i = 1, 2. In the region where neither η i is identically zero, this yields the system

   g 1 η 2 1 + gη 2 2 = λ 1,0 -|x| 2 , gη 2 1 + g 2 η 2 2 = λ 2,0 -|x| 2 .
(1.7)

This leads to the condition (1.3) and the fact that the supports of η i are compact sets: more precisely, the supports of η i are either 2 disks or a disk and an annulus. The limiting geometry is two disks when 0 < g < g 1 + g 2 1 + 8g 1 g 2 4 .

(1.8)

This condition is more restrictive than (1.3), in the sense that (1.4) and (1.8) together imply (1.3). If, on the contrary, we assume that

g > g 1 + g 2 1 + 8g 1 g 2 4 , (1.9)
then the limiting configuration consists of a disk and an annulus; in this case, the assumption g 1 ≤ g 2 implies g > g 1 . It is helpful to introduce the following quantities:

Γ 1 = 1 - g g 1 , Γ 2 = 1 - g g 2 , Γ = 1 - g 2 g 1 g 2 .
(1.10)

Under the assumptions (1.3)- (1.4) and (1.8), we prove that

η 2 i,ε → a i uniformly in R 2 as ε → 0, i = 1, 2, (1.11)
where √ a 1 , √ a 2 are the solutions to (1.6) with L 2 constraint 1, so that

a 1 (x) =    a 1,0 (x), |x| ≤ R 1,0 , 0, |x| ≥ R 1,0 , a 2 (x) =            a 2,0 (x), |x| ≤ R 1,0 , a 2,0 (x) + g g 2 a 1,0 (x), R 1,0 ≤ |x| ≤ R 2,0 , 0, |x| ≥ R 2,0 , (1.12
) with R 1,0 ≤ R 2,0 determined explicitly in terms of g, g 1 , g 2 (see (3.5)), and

a 1,0 (x) = Γ 2 g 1 Γ (R 2 1,0 -|x| 2 ), a 2,0 (x) = R 2 2,0 -R 2 1,0 g 2 + Γ 1 g 2 Γ (R 2 1,0 -|x| 2 ), (1.13)
a 2,0 (x) + g g 2 a 1,0 (x) = 1 g 2 (R 2 2,0 -|x| 2 ).

(1. [START_REF] Brezis | Semilinear equations in R n without conditions at infinity[END_REF] We note that R 1,0 < R 2,0 if g 1 < g 2 and a 1 ≡ a 2 if g 1 = g 2 . Moreover, we show that λ i,ε → λ i,0 , i = 1, 2, where

λ 1,0 = g g 2 R 2 2,0 + Γ 2 R 2 1,0 , λ 2,0 = R 2 2,0 . (1.15)
Because of (1.3)-(1.4), we always have that Γ and Γ 2 are positive. On the other hand, Γ 1 can have either sign: if g < g 1 , the singular limits a i consist of two decreasing functions, and in the case g > g 1 , a 2 is increasing near the origin and up to R 1,0 (though it remains strictly positive under assumption (1.8)) and then decreasing, while a 1 is decreasing. If g = g 1 , we have that a 2 is constant on the ball of radius R 1,0 . We remark that the first derivatives of √ a 1 and √ a 2 have an infinite jump discontinuity across the circles |x| = R 1,0 and |x| = R 2,0 respectively, while the first derivative of √ a 2 has a finite jump discontinuity across |x| = R 1,0 (if g 1 < g 2 ). In particular, neither function belongs to the Sobolev space H 1 (R 2 ). Actually, their maximal regularity is that of the Hölder space C 1 2 (R 2 ). In the case of (1.3)- (1.4) and (1.9), that is when a 1 is supported in a disk and a 2 in an annulus, we also define the corresponding functions a i and prove (1.11).

Based on the estimates for the convergence in (1.11), we will show that for a large range of velocities Ω, the minimizers of E Ω ε in H coincide with the minimizers of E 0 ε , provided that ε > 0 is sufficiently small. The aim of this paper is threefold:

(1) prove the uniqueness of the positive solution (η 1,ε , η 2,ε ) of (1.5) (given any λ 1,ε and λ 2,ε ), and of the minimizer of E 0 ε in H (modulo a constant complex phase), (2) get precise estimates on the convergence, as ε → 0, of (η 1,ε , η 2,ε ), the positive minimizer of E 0 ε in H, to the singular limit ( √ a 1 , √ a 2 ) defined in (1.12), (3) prove that for Ω below a critical velocity, the minimizers of E Ω ε in H have no vortices in R 2 , provided that ε > 0 is sufficiently small. Point 1 relies on the division of two possible positive solutions componentwise, and proving that each quotient is equal to a constant of modulus 1.

Point 2 is the extension to the system of the results of [START_REF] Karali | The ground state of a Gross-Pitaevskii energy with general potential in the Thomas-Fermi limit[END_REF] for a single equation. The idea is to apply a perturbation argument to construct a positive solution to (1.29), "near" ( √ a 1 , √ a 2 ). Then, the uniqueness result in Point 1 allows us to conclude that the constructed solution is indeed the ground state. Therefore, we are able to obtain precise asymptotic estimates for the behavior of the ground state as ε → 0. We emphasize that, even though the system (1.5) is coupled, we are going to reduce it, at leading order, to two independent Gross-Pitaevskii equations. The proof of Point 2, when both condensates are disks, with different techniques, is the topic of a paper in preparation by C. Gallo [START_REF] Gallo | The ground state of two coupled Gross-Pitaevskii equations in the Thomas-Fermi limit[END_REF].

Point 3 relies on fine estimates for the Jacobian from [START_REF] Jerrard | Local minimizers with vortex filaments for a Gross-Pitaevsky functional[END_REF]. It consists in extending the proof of [START_REF] Aftalion | Non-existence of vortices in the small density region of a condensate[END_REF] for a single equation to the system, which works well once the difficult results of points 1 and 2 have been established. 1.2. Motivation and known results. Two component condensates can describe a single isotope in two different hyperfine spin states, two different isotopes of the same atom or isotopes of two different atoms. We refer to [START_REF] Aftalion | Classification of the ground states and topological defects in a rotating two-component Bose-Einstein condensate[END_REF] for more details on the modeling and the experimental references.

According to the respective values of g 1 , g 2 , g and Ω, the minimizers exhibit very different properties in terms of shape of the bulk, defects and coexistence of the components or spatial separation, as ε → 0. In a recent paper, Aftalion and Mason [START_REF] Aftalion | Classification of the ground states and topological defects in a rotating two-component Bose-Einstein condensate[END_REF] have produced phase diagrams to classify the types of minimizers according to the parameters of the problem. Below, we summarize their findings.

• Coexisting condensates with vortex lattices: each condensate is a disk, and, for sufficiently large rotation, displays a vortex lattice. The specificity is that each vortex in component 1 creates a peak in component 2 and vice versa. It is this interaction between peaks and vortices which governs the shape of the vortex lattice. For some parameter regimes, the square lattice gets stabilized because it has less energy than the triangular lattice [START_REF] Aftalion | Vortex-peak interaction and lattice shape in rotating twocomponent Bose-Einstein condensates[END_REF]. • phase separation with radial symmetry: component 1 is a disk while component 2 is an annulus. New defects emerge such as giant skyrmions and the presence of peaks inside the annulus, corresponding to vortices in the disk. • phase separation and complete breaking of symmetry with either droplets or vortex sheets.

It turns out that the sign of the parameter Γ defined in (1.10) plays an important role: if Γ > 0, the two components coexist while if Γ < 0, they separate or segregate (case of droplets, vortex sheets). In the case of no rotation, the segregation behavior in two component condensates has been studied by many authors: regularity of the wave function [START_REF] Noris | Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition[END_REF], regularity of the interface [START_REF] Caffarelli | Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries[END_REF], asymptotic behavior near the interface [START_REF] Berestycki | On Phase-Separation Model: Asymptotics and Qualitative Properties[END_REF][START_REF] Berestycki | On Entire Solutions of an Elliptic System Modeling Phase Separations[END_REF][START_REF] Dancer | The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture[END_REF], Γconvergence to a Modica-Mortola type energy [START_REF] Aftalion | A minimal interface problem arising from a two component Bose-Einstein condensate via Gamma-convergence[END_REF] in the case of a trapped condensate. On the other hand, the case of coexistence is the topic of emerging works in terms of vortices: [START_REF] Aftalion | Vortex-peak interaction and lattice shape in rotating twocomponent Bose-Einstein condensates[END_REF] for a trapped condensate and [START_REF] Alama | On compound vortices in a two-component Ginzburg-Landau functional[END_REF] for a homogeneous condensate. The results that are the core of this paper are totally new and will allow a much better description of vortices. Indeed, in order to understand the behaviour of vortices in a trapped condensate, one has first to understand the effect of the trap at leading order on the profile. Therefore, one requires very precise estimates on the ground state at Ω = 0 for small ε. This is the analogue of what has been obtained for the single component case that we now recall. Many papers [START_REF] Aftalion | Vortices in Bose-Einstein Condensates[END_REF][START_REF] Aftalion | Non-existence of vortices in the small density region of a condensate[END_REF][START_REF] Gallo | On the Thomas-Fermi ground state in a harmonic potential[END_REF][START_REF] Ignat | The critical velocity for vortex existence in a two-dimensional rotating Bose-Einstein condensate[END_REF][START_REF] Karali | The ground state of a Gross-Pitaevskii energy with general potential in the Thomas-Fermi limit[END_REF] have studied the one component analogue of the energy functional (1.1), namely the functional

J Ω ε (u) = R 2 |∇u| 2 2 + |x| 2 2ε 2 |u| 2 + γ 4ε 2 |u| 4 -Ωx ⊥ • (iu, ∇u) dx (1.16)
under the constraint R 2 |u| 2 dx = 1, where γ is some positive constant.

In the following theorem we have collected various results from [START_REF] Aftalion | Non-existence of vortices in the small density region of a condensate[END_REF][START_REF] Gallo | On the Thomas-Fermi ground state in a harmonic potential[END_REF][START_REF] Karali | The ground state of a Gross-Pitaevskii energy with general potential in the Thomas-Fermi limit[END_REF] (see also Appendix A herein) concerning the minimizers of the energy J 0 ε without rotation. Theorem 1.1. For every ε > 0, there exists a unique positive minimizer η ε of J 0 ε with L 2 constraint 1, and any minimizer has the form e iα η ε for some α ∈ R. The minimizer η ε is radial and there is a unique pair (η ε , λ ε ) which is a solution of

-ε 2 ∆η + |x| 2 η + γη 3 = λ ε η, x ∈ R 2 , η(x) → 0 as |x| → ∞, (1.17)
with η positive. Let

a 0 (x) =    λ 0 -|x| 2 γ , |x| ≤ R 0 , 0, |x| ≥ R 0 ,
where λ 0 > 0 is uniquely determined from the condition R 2 a 0 (x)dx = 1 and R 0 = √ λ 0 . There exist constants c, C, δ > 0, with δ < R 0 4 , such that the following properties hold:

|λ ε -λ 0 | ≤ C| log ε|ε 2 , η ε - √ a 0 L ∞ (R 2 ) ≤ Cε 1 3 , η ε (r) -a 0 (r) ≤ Cε 1 3 a 0 (r), 0 ≤ r = |x| ≤ R 0 -ε 1 3 , η ε - √ a 0 L ∞ (|x|≤R 0 -δ) ≤ C| log ε|ε 2 ,
and

η ε (r) ≤ Cε 1 3 exp -c r -R 0 ε 2 3 , r ≥ R 0 ,
for sufficiently small ε > 0.

In fact, given λ > 0, the assertions of the above theorem hold for the unique positive solution of equation (1.17) with λ in place of λ ε . Using these estimates and a Jacobian estimate from [START_REF] Jerrard | Local minimizers with vortex filaments for a Gross-Pitaevsky functional[END_REF], the following theorem is proven in [START_REF] Aftalion | Non-existence of vortices in the small density region of a condensate[END_REF]:

Theorem 1.2. Assume that u ε , η ε minimize respectively J Ω ε , J 0 ε under the constraint of L 2 norm 1. There exist ε0 , ω0 , ω1 > 0 such that if 0 < ε < ε0 and Ω ≤ ω0 | log ε| -ω1 log | log ε| then u ε = e iα η ε in R 2 for some constant α.
This leads, in particular, to the uniqueness of the ground state for small Ω. 1.3. Main results. Our first result concerns uniqueness and radial symmetry for the problem without rotation.

Theorem 1.3. Assume that (1.3) holds.

(1) Let us fix some λ i,ε > 0, i = 1, 2. Then, the positive solution of (1.5) is unique, if it exists.

(2) The positive minimizer (η 1,ε , η 2,ε ) of E 0 ε in H is unique and radially symmetric. Every other minimizer has the form (e iα 1 η 1,ε , e iα 2 η 2,ε ), where α 1 , α 2 are constants. If

g 1 = g 2 then η 1,ε ≡ η 2,ε .
In the case g 1 = g 2 , the system reduces to a single equation. Therefore, for the next result, we can assume that g 1 < g 2 .

(1.18) Our main result provides estimates on the convergence of the positive minimizer (η 1,ε , η 2,ε ) to its limiting profile √ a 1 , √ a 2 as ε → 0, first in the case of two disks.

Theorem 1.4. Assume that (1.3), (1.8) and (1.18) hold. Recall that the a i are defined by (1.12)-(1.13) and λ i,0 by (1.15). Let (η 1,ε , η 2,ε ) be the positive minimizer of E 0 ε in H. Then, (η 1,ε , η 2,ε ) is a solution of (1.5), for some positive Lagrange multipliers λ 1,ε , λ 2,ε , and there exist constants c, C, δ > 0, with δ < min

R 1,0 4 , R 2,0 -R 1,0 4
, such that the following estimates hold:

|λ i,ε -λ i,0 | ≤ C| log ε|ε 2 , (1.19) η i,ε - √ a i L ∞ (R 2 ) ≤ Cε 1 3 , (1.20)
η i,ε (r) -a i (r) ≤ Cε 1 3 a i (r), |x| ≤ R i,0 -ε 1 3 , (1.21) 2 i=1 η i,ε - √ a i L ∞ (|x|≤R 1,0 -δ) + η 2,ε - √ a 2 L ∞ (R 1,0 +δ≤|x|≤R 2,0 -δ) ≤ C| log ε|ε 2 , (1.22)
and

η i,ε (r) ≤ Cε 1 3 exp -c r -R i,0 ε 2 3 , r ≥ R i,0 , i = 1, 2, (1.23)
for sufficiently small ε > 0.

This theorem is the natural extension of what is known for a single condensate described in Theorem 1.1. The fine behavior of the minimizer near R 1,0 and R 2,0 , as ε → 0, is established in Theorem 4.1 in Section 4. It is based on a perturbation argument which proves very powerful in this system case, where we have not managed to extend the sub and super solutions techniques of [START_REF] Aftalion | Giant vortex and the breakdown of strong pinning in a rotating Bose-Einstein condensate[END_REF][START_REF] Aftalion | Non-existence of vortices in the small density region of a condensate[END_REF].

Based on the above, we can show the absence of vortices for the minimizers of E Ω ε with small rotation. Theorem 1.5. Assume that (1.3), (1.4) and (1.8) hold. Let (u 1,ε , u 2,ε ) be a minimizer of

E Ω ε in H. There exist ε 0 , ω 0 , ω 1 > 0 such that if 0 < ε < ε 0 and Ω ≤ ω 0 | log ε| -ω 1 log | log ε| then u i,ε = e iα i η i,ε in R 2 for some constants α i , i = 1, 2.
In the case of a disk and an annlus, that is when (1.9) holds instead of (1.8), we can prove the equivalent of Theorem 1.4. Generalizing Theorem 1.5 is harder, because vortices may exist in the central hole of component 2 and this has to be tackled by other techniques than the ones in this paper.

Theorem 1.6. Assume that (1.3), (1.9) and (1.18) hold. We define the functions a i by

a 1 (r) =            a 1,0 (r) + g g 1 a 2,0 (r), 0 ≤ r ≤ R - 2,0 , a 1,0 (r), R - 2,0 ≤ r ≤ R 1,0 , 0, r ≥ R 1,0 ,
where

a i,0 (r) = 1 g i Γ λ i,0 - g g i+1 λ i+1,0 -Γ i+1 r 2 , (1.24) a 2 (r) =                    0, 0 ≤ r ≤ R - 2,0 , a 2,0 (r), R - 2,0 ≤ r ≤ R 1,0 , a 2,0 (r) + g g 2 a 1,0 (r), R 1,0 ≤ r ≤ R + 2,0 , 0 r ≥ R + 2,0 .
(1.25)

and λ 2,0 = (R + 2,0 ) 2 , λ 1,0 - g g 2 λ 2,0 = Γ 2 R 2 1,0 and λ 2,0 - g g 1 λ 1,0 = Γ 1 (R - 2,0 ) 2 (1.26)
where λ 1,0 , λ 2,0 will be given by (3.8). Let (η 1,ε , η 2,ε ) be the positive minimizer of E 0 ε in H. Then there exist constants c, C, δ > 0, such that, for sufficiently small ε > 0, (1.19), (1.20) hold, (1.21) holds for i = 1 and is replaced, for i = 2 by

η 2,ε (r) -a 2 (r) ≤ Cε 1 3 a 2 (r), for R - 2,0 + ε 1 3 < |x| ≤ R + 2,0 -ε 1 3 , (1.27)
(1.23) holds with R + 2,0 instead of R 2,0 , and on fixed compact sets away from |x| = R 1,0 and

|x| = R ± 2,0 , (η 1,ε , η 2,ε ) is close to ( √ a 1 , √ a 2 ) with an error of order O(| log ε|)ε 2 .
We point out that if g 2 < g 1 , then an analogous theorem holds exchanging the subscript 1 and 2 in the formulae.

1.4. Methods of proof and outline of the paper. Theorem 1.3 is proved by assuming that there are two solutions, studying their ratio componentwise and writing the system satisfied by the ratio, as inspired by [START_REF] Brezis | Remarks on sublinear elliptic equations[END_REF][START_REF] Lassoued | Ginzburg-Landau type energy with discontinuous constraint[END_REF]. For the first part of the theorem, we need decay properties at infinity of the solutions of (1.5), that we prove in a similar way to a Liouville theorem in [START_REF] Berestycki | Further qualitative properties for elliptic equations in unbounded domains[END_REF]. We point out that the system is non-cooperative, and the usual moving plane method does not seem to apply easily in this case to derive radial symmetry of positive solutions. Nevertheless, our result implies that since positive solutions of (1.5) are unique, they are thus radial.

For the second part of the theorem, we use the decay of finite energy solutions and extra estimates for radial functions. A key relation is the following splitting of energy: if (η 1 , η 2 ) is a ground state among radial functions, then for any (u

1 , u 2 ), E 0 ε (u 1 , u 2 ) = E 0 ε (η 1 , η 2 ) + F 0 ε (v 1 , v 2 )
, where v i = u i /η i and

F 0 ε (v 1 , v 2 ) = 2 i=1 R 2 η 2 i 2 |∇v i | 2 + g i 4ε 2 η 4 i (|v i | 2 -1) 2 dx+ g 2ε 2 R 2 η 2 1 η 2 2 (1 -|v 1 | 2 )(1 -|v 2 | 2 ) dx. (1.28) The condition Γ > 0, that is g > √ g 1 g 2 , implies that F 0 ε (v 1 , v 2 ) ≥ 0.
If we assume that (u 1 , u 2 ) is a ground state of E 0 ε , then using the sign of F 0 ε , we find that (u 1 , u 2 ) is equal, up to a multiplication by a complex number of modulus 1, to (η 1 , η 2 ). Thus, any ground state is radially symmetric and equal, up to a multiplication by a complex number of modulus 1, to (η 1 , η 2 ). Theorem 1.4 contains a fine asymptotic behaviour of the ground state (η 1,ε , η 2,ε ) as ε tends to zero. The difficulty is especially in the regions near R 1,0 , R 2,0 where the approximate inverted parabola matches an exponentially small function in a region of size ε 2/3 . The general procedure is to first construct a sufficiently good approximate solution to the problem (1.5), for small ε > 0, with coefficients λ 1,ε and λ 2,ε being equal to the unique Lagrange multipliers that are provided by Theorem 1.3. Then, using the invertibility properties of the linearized operator about this approximate solution, we perturb it to a genuine solution. The first uniqueness result of Theorem 1.3 implies that this constructed solution coincides with the positive minimizer of E ε 0 in H. The method is a generalization to the system case of the tools developed in [START_REF] Karali | The ground state of a Gross-Pitaevskii energy with general potential in the Thomas-Fermi limit[END_REF] for the single equation.

In order to construct this approximate solution, we rewrite the system (1.5) as

-ε 2 ∆η 1 + g 1 η 1 η 2 1 -a 1,ε (x) + gη 1 η 2 2 -a 2,ε (x) = 0, (1.29a) -ε 2 ∆η 2 + g 2 η 2 η 2 2 -a 2,ε (x) + gη 2 η 2 1 -a 1,ε (x) = 0, (1.29b) η i (x) → 0 as |x| → ∞, i = 1, 2. (1.29c)
where a 1,ε , a 2,ε are the ε equivalent to (1.13), that is

a 1,ε (x) = 1 g 1 Γ λ 1,ε - g g 2 λ 2,ε -Γ 2 |x| 2 , a 2,ε (x) = 1 g 2 Γ λ 2,ε - g g 1 λ 1,ε -Γ 1 |x| 2 , (1.30) and R 2 1,ε = 1 Γ 2 λ 1,ε - g g 2 λ 2,ε , R 2 2,ε = λ 2,ε . (1.31)
At leading order, in the regions where neither η i is close to zero, we expect that the ε 2 ∆η i terms are negligible so that at leading order,

g 1 η 2 1 -a 1,ε (x) + g η 2 2 -a 2,ε (x) = 0, (1.32a) g 2 η 2 2 -a 2,ε (x) + g η 2 1 -a 1,ε (x) = 0. (1.32b)
Near R 1,0 , the term ε 2 ∆η 1 cannot be neglected so that we use (1.32b) to express η 2 2a 2,ε and insert it into (1.29a) to find a scalar equation for η 1 . In the region of coexistence, that is in the disk of radius R 1,0 , η 2 is obtained from η 1 by (1.32b), while outside this disk, η 1 is small and can be neglected in (1.29b). This reduces the system (1.29) to two independent approximate scalar problems:

-ε 2 ∆η 1 + g 1 - g 2 g 2 η 1 η 2 1 -a 1,ε (x) = 0, x ∈ R 2 ; η 1 (x) → 0 as |x| → ∞, (1.33a) -ε 2 ∆η 2 + g 2 η 2 η 2 2 -a 2,ε (x) - g g 2 a 1,ε (x) = 0, x ∈ R 2 ; η 2 (x) → 0 as |x| → ∞, (1.33b)
whose unique positive solutions are called η1,ε and η2,ε . The properties of η1,ε and η2,ε can be deduced from an analogue of Theorem A.1 (see Proposition 4.2 below). We point out that they are linearly nondegenerate, which implies that the spectrum of the associated linearized operators to (1.33a) and (1.33b) consists only of positive eigenvalues. The main features of a 1,ε and a 2,ε that are used for studying η1,ε and η2,ε are that a 1,ε and a 2,ε + g g 2 a 1,ε change sign once, from positive to negative as |x| crosses R 1,ε and R 2,ε respectively and that

a ′ 1,ε (R 1,ε ) → -c < 0 and a 2,ε + g g 2 a 1,ε ′ (R 2,ε ) → -c ′ < 0 as ε → 0. (1.34)
In particular, η2 1,ε and η2 2,ε converge to (a 1,0 ) + and a 2,0 + g g 2 a 1,0 + , uniformly on R 2 , as ε → 0.

Equation (1.33a) provides an effective approximation for (1.29a) up to a neighborhood of R 2,0 , where the term ε 2 ∆η 2 is expected to be of equal or higher order than ε 2 ∆η 1 as ε → 0. The approximate solutions to (1.29) that are constructed in this way match in C k , in fixed intervals contained between R 1,0 and R 2,0 . Therefore, we can pick any point in (R 1,0 , R 2,0 ), for instance the middle point and we can smoothly glue the solutions together, via a standard interpolation argument to create a global approximate solution to the problem (1.29). We remark that, in order to estimate the remainder that this approximation leaves in (1.29), we have to prove some new estimates for the behavior of the derivatives of the ground states of (1.33a) and (1.33b) near R 1,0 and R 2,0 respectively, as ε → 0.

The next step is to study the associated linearized operator to the system (1.29) about this approximation (see (4.41) below). To this end, as before, our approach is to reduce the corresponding coupled linearized system to the two independent scalar linearized problems that are associated to (1.33a) and (1.33b). As we have already remarked, the spectrum of the latter scalar operators consists only of strictly positive eigenvalues. This property allows us to apply a domain decomposition argument to handle the "two center" difficulty of the problem. We are able to show that the associated linearized operator to the system (1.29) about the approximate solution is invertible, for small ε > 0, and estimate its inverse in various suitable Sobolev L 2 -norms. We point out that this is in contrast to the scalar case, where it is more convenient to estimate the inverse in uniform norms (see [START_REF] Gallo | On the Thomas-Fermi ground state in a harmonic potential[END_REF][START_REF] Karali | The ground state of a Gross-Pitaevskii energy with general potential in the Thomas-Fermi limit[END_REF]). Armed with these estimates, we can apply a contraction mapping argument to prove the existence of a true solution to (1.29), near the approximate one with respect to a suitable ε-dependent Sobolev norm, for small ε > 0. Finally, we can show that the solution is positive and obtain the uniform estimates of Theorem 1.4 by building on the Sobolev estimates and making use of the equation. In particular, we also make use of some carefully chosen weighted uniform norms, see (4.109), which are partly motivated by [START_REF] Pacard | Linear and nonlinear aspects of vortices[END_REF].

We note that our approach can be extended to cover the more complicated case of the disk and annulus configuration, that is when (1.9) is assumed.

In order to prove Theorem 1.5, we need to study some auxiliary functions ξ i,ε , involving the primitive of sη 2 i,ε (s), where (η 1,ε , η 2,ε is the positive minimizer of E 0 ε . For this purpose, we use the estimates that link (1.29) to (1.33), obtained in the proof of Theorem 1.4, to derive the estimates for ξ i,ε as perturbations of those for the scalar equation. Then, we use a division trick which splits the energy as the sum of the energy of the minimizer without rotation plus a reduced energy (see Lemma 7.5). The reduced energy bears similarities to a weighted coupled Ginzburg-Landau energy and, after integrating by parts, includes a Jacobian (see Lemma 7.6). Assumption (1.3) allows us to treat this coupled energy as two uncoupled ones of analogous form, and conclude by following the arguments of [START_REF] Aftalion | Non-existence of vortices in the small density region of a condensate[END_REF] which are based on the control of the auxiliary functions and a Jacobian estimate due to [START_REF] Jerrard | Local minimizers with vortex filaments for a Gross-Pitaevsky functional[END_REF].

The organization of the paper is as follows: in Section 2, we prove Theorem 1.3. In Section 3, we obtain the first rough estimates for the asymptotic behavior of (η 2 1,ε , η 2 2,ε ) as ε → 0. In Section 4, we apply the perturbation argument to prove Theorem 1.4. In Section 5, we prove Theorem 1.6. In Section 6, we study some auxiliary functions that will be useful for the estimates with rotation. In Section 7, we prove Theorem 1.5. We close the paper with two appendixes. In Appendix A, we summarize some known results about the scalar ground states of Theorem 1.1, in a more general setting, and derive estimates for their derivatives. In Appendix B, we have postponed the proof of a technical estimate from Section 4 that is related to our use of the weighted norms. 1.5. Notation. By c/C we denote small/large positive generic constant, whose values may decrease/increase from line to line. By O(•) and o(•), we denote the standard Landau symbols. We write r = |x| to denote the Euclidean distance of a point x from the origin. By B R we denote the Euclidean ball of radius R and center 0.

Uniqueness issues

In this section, we prove Theorem 1.3. Since the result holds for every ε > 0, we often omit the subscript ε.

2.1. Uniqueness of positive solutions of (1.5). Given positive λ 1,ε , λ 2,ε , we want to prove the uniqueness of the positive solutions of (1.5). We use some ideas from [START_REF] Brezis | Remarks on sublinear elliptic equations[END_REF] which deals with a class of scalar equations in bounded domains. In order to extend it to the entire space, we have to establish some control on the decay of positive, possibly non-radial, solutions.

Lemma 2.1. Let (u 1 , u 2 ) be a positive solution of (1.5a)-(1.5b), then

u 2 i ≤ λ i,ε /g i , ∇u i L ∞ (R 2 ) ≤ C λ i,ε (λ i,ε + λ j,ε + 1) ε , i = 1, 2, j = i.
The proof is adapted from [START_REF] Aftalion | Non-existence of vortices in the small density region of a condensate[END_REF] and [START_REF] Ignat | The critical velocity for vortex existence in a two-dimensional rotating Bose-Einstein condensate[END_REF].

Proof. Let w i = ( √ g i u i -λ i,ε )/ε, then Kato's inequality yields ∆w + i ≥ χ {w i ≥0} ∆w i ≥ χ {w i ≥0} √ g i ε 3 u i (g i u 2 i -λ i,ε ),
where χ is the characteristic function of a set. Then we obtain

∆w + i ≥ χ {w i ≥0} εw i + λ i,ε ε 2 w i (εw i + 2 λ i,ε ) ≥ (w + i ) 3 .
A non-existence result by Brezis [START_REF] Brezis | Semilinear equations in R n without conditions at infinity[END_REF] implies that w + i ≡ 0, so that the first bound is proved. In fact, since u i is bounded, it follows by a standard barrier argument that (1.5c) is also satisfied. Now fix x ∈ R 2 , L > 0 and for y ∈ B 2L (x), let z i (y) = u i (ε(yx)). Then

-∆z i = -z i (ε 2 |y -x| 2 + g i z 2 i + gz 2 j -λ i,ε ) =: h i,ε (y)
, (i = j). We have proved above that there exists C > 0 independent of ε and of x such that h i,ε L ∞ (B 2L (x)) ≤ C λ i,ε (λ i,ε + λ j,ε + 1). Standard regularity theory for elliptic equations implies ∇z i L ∞ (B L (x)) ≤ C λ i,ε (λ i,ε + λ j,ε + 1), and in turn the second part of the statement.

This implies in particular uniform bounds for the solutions of (1.5). In the following lemma, we prove that positive solutions of (1.5) decay super-exponentially fast as |x| → ∞. Lemma 2.2. Let (u 1 , u 2 ) be a positive solution of (1.5). For every k > 0, let r i = (1 + k)λ i,ε and

W i (s) = max ∂Br u i • exp - 1 2ε k 1 + k (s 2 -r 2 ) for s ≥ r ≥ r i , i = 1, 2.
Then we have u i (x) ≤ W i (|x|) for |x| ≥ r ≥ r i , i = 1, 2. Moreover,

|u i (x)| + |∇u i (x)| ≤ C ε e -cε|x| 2 , x ∈ R 2 , i = 1, 2.
(2.1)

Proof. Given k > 0, let r ≥ r i = (1 + k)λ i,ε . For |x| ≥ r we have λ i,ε ≤ |x| 2 1+k
, and hence |x| 2λ i,ε ≥ k 1+k |x| 2 , so that u i satisfies

-∆u i + u i ε 2 k 1 + k |x| 2 ≤ -∆u i + u i ε 2 (|x| 2 -λ i,ε ) = - u i ε 2 (g i u 2 i + gu 2 j ) ≤ 0
for every |x| ≥ r (here j = i). On the other hand, it is easy to check that

-∆W i + W i ε 2 k 1 + k |x| 2 > 0 for |x| ≥ r, W i (r) ≥ u i (x) for |x| = r.
Suppose by contradiction that W iu i is negative somewhere in the exterior of Br . Since both functions go to zero at infinity, there exists x, with |x| > r, where W iu i reaches its minimum: W i (|x|)-u i (x) < 0 and ∆W i (|x|)-∆u i (x) ≥ 0. By subtracting the two differential inequalities satisfied by u i and W i , and then evaluating at x, we obtain a contradiction. Lemma 2.1 implies a uniform bound for max u i . Using (1.5), we obtain that

|∆u i (y)| ≤ C ε e -cε|x| 2 , y ∈ B 1 (x), x ∈ R 2 .
By standard interior elliptic estimates, we deduce (2.1).

Proposition 2.3. Assume (1.3). Given λ i,ε > 0, then problem (1.5) has at most one positive solution.

Proof. Let (η 1 , η 2 ) and (u 1 , u 2 ) be two positive solutions of (1.5) with the same λ 1,ε and λ 2,ε . Then, the function ψ i = u i /η i solves the following equation with j = i:

-∇ • (η 2 i ∇ψ i ) = u i ∆η i -η i ∆u i = η 2 i ψ i ε 2 g i η 2 i (1 -ψ 2 i ) + gη 2 j (1 -ψ 2 j ) .
(2.2)

We want to show that ψ i is identically equal to 1. To this end, we multiply equation (2.2) by (ψ 2 i -1)/ψ i in a ball of radius R to obtain

B R η 2 i |∇ψ i | 2 1 + 1 ψ 2 i + η 2 i ε 2 g i η 2 i (ψ 2 i -1) 2 + gη 2 j (ψ 2 i -1)(ψ 2 j -1) dx = ∂B R u i - η 2 i u i ∇u i - u 2 i η i -η i ∇η i • ν dσ,
where ν denotes the outer unit normal vector to ∂B R . We sum the previous identities for i = 1, 2 and then we use the following inequality

|2gη 2 1 η 2 2 (ψ 2 1 -1)(ψ 2 2 -1)| ≤ (g 1 -γ)η 4 1 (ψ 2 1 -1) 2 + (g 2 -γ)η 4 2 (ψ 2 2 -1) 2 , (2.3)
where 0 < γ < min{g 1 , g 2 } is such that

g ≤ √ g 1 -γ √ g 2 -γ, (2.4)
which exists by (1.3). We get

2 i=1 B R η 2 i |∇ψ i | 2 1 + 1 ψ 2 i + γ ε 2 η 4 i (ψ 2 i -1) 2 dx ≤ 2 i=1 ∂B R u i - η 2 i u i ∇u i - u 2 i η i -η i ∇η i • ν dσ.
(2.5)

To conclude that ψ i ≡ 1, it is enough to show that there exist R k → ∞ such that the right-hand side of (2.5), with R = R k , tends to zero as k → ∞. This task will take up the rest of the proof. Let χ be a smooth cutoff function in R 2 which is identically equal to 1 in the unit ball and identically equal to 0 outside of the ball of radius 2. For all R ≥ 1, we define χ R = χ(•/R).

In the remaining part of this proof, we denote by C ε a positive generic constant which is independent of R ≥ 1. We multiply (2.2) by r α χ 2 R ψ i , where α > 2 (r = |x|), and integrate the resulting identity by parts over R 2 to find that

B 2R χ 2 R r α ∇ψ i + αr α-1 x r ψ i η 2 i ∇ψ i dx + B 2R r α ψ i 2χ R ∇χ R η 2 i ∇ψ i dx ≤ C ε ,
where we have also made use of (2.1) and of the definition of ψ i . Our motivation for including χ 2 R comes from [START_REF] Berestycki | Further qualitative properties for elliptic equations in unbounded domains[END_REF]Thm. 1.8]. Thanks to the elementary inequalities

χ 2 R r α-1 x r ψ i η 2 i ∇ψ i ≤ dχ 2 R r α η 2 i |∇ψ i | 2 + 1 2d χ 2 R r α-2 ψ 2 i η 2 i ∀ d > 0, and 2r α ψ i χ R ∇χ R η 2 i ∇ψ i ≤ dχ 2 R r α η 2 i |∇ψ i | 2 + 1 d r α ψ 2 i |∇χ R | 2 η 2 i ∀ d > 0,
choosing a sufficiently small d > 0 (independent of R), via (2.1), we infer that

R 2 r α χ 2 R η 2 i |∇ψ i | 2 dx ≤ C ε .
By Lebesgue's monotone convergence theorem, letting R → ∞ in the above relation, we obtain that

R 2 r α η 2 i |∇ψ i | 2 dx ≤ C ε .
Replacing ψ i by its value and using (2.1), we find that

R 2 r α u 2 i |∇η i | 2 η 2 i dx ≤ C ε .
Reversing the roles of u i and η i , and summing, we reach

R 2 r α 2 i=1 u 2 i |∇η i | 2 η 2 i + η 2 i |∇u i | 2 u 2 i dx ≤ C ε .
Therefore, by the co-area formula, there exists a sequence R k → ∞ such that

R α k ∂B R k 2 i=1 u 2 i |∇η i | 2 η 2 i + η 2 i |∇u i | 2 u 2 i dσ → 0 as k → ∞.
To conclude, we note that the above relation, the Cauchy-Schwarz inequality, and (2.1), imply that the right-hand side of (2.5) at R = R k tends to zero as k → ∞, as desired. We remark that, in the case of radial symmetry, one can argue directly, analogously to [START_REF] Aftalion | Non-existence of vortices in the small density region of a condensate[END_REF], by making use of Lemma 2.8 below.

Remark 2.4. If g 1 = g 2 and λ 1,ε = λ 2,ε , then η 1 = η 2 . Indeed, (η 1 , η 2 ) and (η 2 , η 1 ) are both positive solutions to (1.5) and we can apply Proposition 2.3.

Remark 2.5. The uniqueness result of Proposition 2.3 yields radial symmetry of u 1 and u 2 .

We observe that the proof of Proposition 2.3 applies to provide also the following local uniqueness result, since in this case, the boundary terms vanish.

Proposition 2.6. Assume (1.3). Given λ i,ε > 0, i = 1, 2, and a bounded domain B ⊂ R 2 with Lipschitz continuous boundary, if (η 1 , η 2 ) and (u 1 , u 2 ) are positive solutions to the elliptic system in (1.5) on B such that

u i = η i on ∂B for i = 1, 2, then u i ≡ η i in B.
2.2. Uniqueness and radial symmetry of the ground state. We now turn to the uniqueness and radial symmetry of the positive minimizer of the energy without rotation

E 0 ε (u 1 , u 2 ) = 2 i=1 R 2 |∇u i | 2 2 + |x| 2 2ε 2 |u i | 2 + g i 4ε 2 |u i | 4 dx + g 2ε 2 R 2 |u 1 | 2 |u 2 | 2 dx (2.6) in the space H which is defined in (1.2). If (u 1 , u 2 ) minimizes E 0 ε in H, then the diamagnetic inequality implies E 0 ε (|u 1 |, |u 2 |) ≤ E 0 ε (u 1 , u 2 )
, so that u i differs from |u i | by a constant complex phase. In fact, by the strong maximum principle, we can assume that u i are positive functions. By elliptic regularity, positive minimizers of E 0 ε in H lead to smooth solutions of (1.5a)-(1.5b) for some positive Lagrange multipliers λ 1,ε , λ 2,ε . Nevertheless, we have to prove that (1.5c) holds too. A priori, we only know that it holds for radial functions by the Strauss lemma [START_REF] Strauss | Existence of solitary waves in higher dimensions[END_REF]. In the subsequent lemma, we provide a lower bound for the decay rate of positive solutions to (1.5) as |x| → ∞. The following proof is adapted from [START_REF] Aftalion | Non-existence of vortices in the small density region of a condensate[END_REF] and [START_REF] Ignat | The critical velocity for vortex existence in a two-dimensional rotating Bose-Einstein condensate[END_REF].

Lemma 2.7. Let (u 1 , u 2 ) be a positive solution of (1.5). Let

w i (s) = min ∂Br u i • exp - α i 2 (s 2 -r 2 ) for s ≥ r > λ i,ε ,
where, for i, j = 1, 2 and j = i, we have defined

α i = 1 λ i,ε + 1 λ 2 i,ε + 1 ε 2 1 + gλ j,ε g j λ i,ε . Then u i (x) ≥ w i (|x|) for |x| ≥ r > λ i,ε .
Proof. We know from Lemma 2.1 that

u 2 i ≤ λ i,ε /g i for i = 1, 2. Thus, u i satisfies -ε 2 ∆u i + u i |x| 2 + g λ j,ε g j ≥ 0, x ∈ R 2 , (j = i).
On the other hand, our choice of α i implies that

-ε 2 ∆w i + w i |x| 2 + g λ j,ε g j < w i 2α i ε 2 + g g j λ j,ε + (1 -ε 2 α 2 i )λ i,ε = 0, |x| > r,
where we have used that 1ε 2 α 2 i < 0. The maximum principle can now be applied, as in Lemma 2.2, to yield the desired lower bound.

In the case of radial solutions, we can show the following lemma analogous to the one in [START_REF] Aftalion | Non-existence of vortices in the small density region of a condensate[END_REF].

Lemma 2.8. Let (u 1 , u 2 ) be a positive radial solution of (1.5). There exists C ε > 0 independent of r such that, for i = 1, 2, we have

|u ′ i (r)| ≤ C ε ru i (r) for r > 2 λ i,ε .
Proof. Since u i is radial, and r > 2 λ i,ε , an application of Lemma 2.2 (with k = 3) yields that

u i (r) = W i (r) and u i (s) ≤ W i (s) for s > r. It follows that u ′ i (r) ≤ W ′ i (r) ≤ 0.
Similarly, an application of Lemma 2.7 yields that

u ′ i (r) ≥ w ′ i (r) = -α i ru i (r), completing the proof.
In order to proceed, we need the following splitting of the energy:

Proposition 2.9. Assume (1.3). Let (η 1 , η 2 ) be a minimizer among radial functions in H with η i > 0. Let (u 1 , u 2 ) ∈ H.
Then the splitting of energy (1.28) holds.

Proof. We test the equation for η i by η i (|v i | 2 -1) in a ball of radius R and then integrate by parts. As a result, we get the term

B R |∇η i | 2 (|v i | 2 -1) + 2η i ∇η i • (v i , ∇v i ) dx = B R -∆η i η i (|v i | 2 -1) dx + ∂B R |u i | 2 η i -η i ∇η i • ν dσ,
Lemmas 2.1 and 2.8 apply to η i to provide

∂B R |u i | 2 η i -η i ∇η i • ν dσ ≤ C ε ∂B R R|u i | 2 dσ + C ε ∂B R Rη 2 i dσ,
Note that the conditions (η 1 , η 2 ), (u 1 , u 2 ) ∈ H imply the existence of a sequence R k → ∞ such that the integrals above vanish along R k via the co-area formula. Therefore, we have

R 2 (|∇η i | 2 (|v i | 2 -1) + 2η i ∇η i • (∇v i , v i )) dx = - 1 ε 2 R 2 η 2 i (|v i | 2 -1)(|x| 2 + g i η 2 i + gη 2 j ) dx,
where the Lagrange multiplier term has disappeared because R 2 η 2 i (|v i | 2 -1) dx = 0. We replace the last equality into the definition of E 0 ε (u 1 , u 2 ) to find

E 0 ε (u 1 , u 2 ) = E 0 ε (η 1 v 1 , η 2 v 2 ) = 2 i=1 R 2 |∇η i | 2 2 |v i | 2 + η i ∇η i • (∇v i , v i ) + η 2 i |∇v i | 2 2 + |x| 2 2ε 2 η 2 i |v i | 2 + g i 4ε 2 η 4 i |v i | 4 dx + g 2ε 2 R 2 η 2 1 η 2 2 |v 1 | 2 |v 2 | 2 dx = 2 i=1 R 2 |∇η i | 2 2 + η 2 i |∇v i | 2 2 + |x| 2 2ε 2 η 2 i + g i 4ε 2 η 4 i + g i 4ε 2 η 4 i (|v i | 2 -1) 2 dx + g 2ε 2 R 2 η 2 1 η 2 2 (|v 1 | 2 |v 2 | 2 -|v 1 | 2 -|v 2 | 2 + 2) dx.
By collecting the term E 0 ε (η 1 , η 2 ) in the previous expression, the result follows.

Proof of Theorem 1.3.

Proof. Given λ i,ε > 0, i = 1, 2, the first assertion of the theorem is proven in Proposition 2.3. Now, let (η 1 , η 2 ) be a minimizer of E 0 ε in H among radial functions, and let (u 1 , u 2 ) be a minimizer of E 0 ε in H. Since (η 1 , η 2 ) is an admissible test function, we have

E 0 ε (u 1 , u 2 ) ≤ E 0 ε (η 1 , η 2 ). Consequently, the quantity F 0 ε (v 1 , v 2 ), defined in (1.28), satisfies F 0 ε (v 1 , v 2 ) = E 0 ε (u 1 , u 2 ) -E 0 ε (η 1 , η 2 ) ≤ 0.
On the other hand, recalling (2.4), we find that

F 0 ε (v 1 , v 2 ) ≥ 2 i=1 R 2 η 2 i 2 |∇v i | 2 + γ 4ε 2 η 4 i (|v i | 2 -1) 2 dx ≥ 0.
This implies that F 0 ε (v 1 , v 2 ) = 0 and that |v i | ≡ 1 for i = 1, 2, which implies the second assertion of the theorem. If g 1 = g 2 then η 1 ≡ η 2 , as (η 1 , η 2 ) and (η 2 , η 1 ) are both minimizers.

Preliminary estimates for the energy minimizer without rotation

In this section we prove that, under assumptions (1.3), (1.8) and (1.18), the positive minimizer (η 1 , η 2 ) provided by Theorem 1.3 satisfies

η 2 i → a i in L 2 (R 2
) and λ i,ε → λ i,0 as ε → 0. This result is achieved through the estimate of the energy of the minimizer.

3.1. Limiting profiles. We recall briefly how to calculate the limiting configuration (1.12). We first assume the case of two disks

D i = {x ∈ R 2 : |x| < R i,0 }.
where R 1,0 < R 2,0 to be determined later. If x ∈ D 1 , formally let ε = 0 in (1.5) and solve the resulting algebraic system in η 2 1 , η 2 2 . This provides, for x ∈ D 1 ,

a 1,0 (x) = 1 g 1 Γ λ 1,0 - g g 2 λ 2,0 -Γ 2 |x| 2 , (3.1)
a 2,0 (x) = 1 g 2 Γ λ 2,0 - g g 1 λ 1,0 -Γ 1 |x| 2 , (3.2)
and also the value of R 1,0 , which is the radius at which a 1,0 vanishes:

R 2 1,0 = 1 Γ 2 λ 1,0 - g g 2 λ 2,0 . (3.3) If x ∈ D 2 \ D 1
, then η 1 = 0 and formally with ε = 0 in (1.5), we solve the resulting equation for η 2 2 , to obtain the following limiting behavior for η 2 2 :

λ 2,0 -|x| 2 g 2 , with R 2 2,0 = λ 2,0 . (3.4)
Notice that (λ 2,0 -|x| 2 )/g 2 = a 2,0 + g g 2 a 1,0 , in agreement with our definition of a 2 in (1.12). Finally, by imposing the normalization conditions a

1 L 2 (R 2 ) = a 2 L 2 (R 2 ) = 1, we obtain λ 2 2,0 = 2(g 2 + g) π , λ 1,0 - g g 2 λ 2,0 = 2g 1 ΓΓ 2 π ,
and hence

R 4 1,0 = 2g 1 Γ πΓ 2 , R 4 2,0 = 2(g 2 + g) π . (3.5)
Notice that, in our setting, the condition (1.18) is equivalent to R 1,0 < R 2,0 , as can be deduced from (3.5). Next observe that the monotonicity of a 2,0 depends on the sign of Γ 1 . If Γ 1 > 0, then a 2,0 is decreasing and

a 2,0 (x) ≥ a 2,0 (R 1,0 ) = (R 2 2,0 -R 2 1,0 )/g 2 > 0, x ∈ D 1 . (3.6)
If Γ 1 = 0 then a 2,0 is constant, whereas it is increasing when Γ 1 < 0. In this last case, we have, for

x ∈ R 2 , a 2,0 (x) ≥ a 2,0 (0) = 1 g 2 Γ λ 2,0 - g g 1 λ 1,0 , (3.7)
which is a positive constant thanks to (1.8). Condition (1.8) is thus equivalent to having two disks.

In the case of a disk plus annulus, we assume that a 1 is supported in a disk D 1 of radius R 1,0 and a 2 on an annulus

D 2 = {x ∈ R 2 : R - 2,0 < |x| < R + 2,0 } with R - 2,0 < R 1,0 < R + 2,0 . Other rearrangements of R 1,0 , R - 2,0 , R + 2,0 can be excluded, see [5]. In the coexistence region, that is R - 2,0 < |x| < R 1,0 , ( √ a 1,0 , √ a 2,0 ) given by (3.1)-(3.
2) is the solution of (1.7). The fact that a 1 vanishes at R 1,0 and a 2 at R - 2,0 and R + 2,0 yields (1.26). If

r ≤ R - 2,0 ≤ R + 2,0 , a 2 = 0 and a 1 = λ 1,0 -r 2 g 1 R 1,0 ≤ r, a 1 = 0 and a 2 = λ 2,0 -r 2 g 2 which are consistent with (1.24)-(1.25). The computations of the L 2 norms provide λ 1,0 = 2 g 1 (1 + g 2 2 g 2 (1 -Γ 2 ) 2 ) π and λ 2,0 -λ 1,0 = -Γ 1 Γ 2 2 g 1 g 2 2 (1 -Γ 2 ) πg 2 . (3.8)
3.2. Energy estimates. In order to obtain some energy estimates, we first rewrite the energy functional in a different form.

Lemma 3.1. Assume (1.3), (1.8) and (1.18). Let (u 1 , u 2 ) ∈ H, then E 0 ε (u 1 , u 2 ) = Ẽ0 ε (u 1 , u 2 )+ K, where Ẽ0 ε (u 1 , u 2 ) = 2 i=1 R 2 |∇u i | 2 2 + g i 4ε 2 (|u i | 2 -a i ) 2 dx + g 2ε 2 R 2 (|u 1 | 2 -a 1 )(|u 2 | 2 -a 2 ) dx + g 1 Γ 2ε 2 R 2 \D 1 |u 1 | 2 a - 1,0 dx + 1 2ε 2 R 2 \D 2 (g|u 1 | 2 + g 2 |u 2 | 2 ) a 2,0 + g g 2 a 1,0 - dx and K is the following constant (depending on ε) K = λ 1,0 + λ 2,0 2ε 2 - 1 4ε 2 R 2 (g 1 a 2 1 + g 2 a 2 2 + 2ga 1 a 2 ) dx.
Proof. We note that

|x| 2 + g 1 a 1 + ga 2 =    λ 1,0 , x ∈ D 1 , Γ 2 |x| 2 + g g 2 λ 2,0 , x ∈ D 2 \ D 1 , |x| 2 , x ∈ R 2 \ D 2 ,
and

|x| 2 + g 2 a 2 + ga 1 = λ 2,0 , x ∈ D 2 , |x| 2 , x ∈ R 2 \ D 2 .
Therefore, we have:

g 1 |u 1 | 4 + 2|x| 2 |u 1 | 2 + g 2 |u 2 | 4 + 2|x| 2 |u 2 | 2 + 2g|u 1 | 2 |u 2 | 2 = = g 1 (|u 1 | 2 -a 1 ) 2 + g 2 (|u 2 | 2 -a 2 ) 2 + 2g(|u 1 | 2 -a 1 )(|u 2 | 2 -a 2 ) + 2|u 1 | 2 (|x| 2 + g 1 a 1 + ga 2 ) + 2|u 2 | 2 (|x| 2 + g 2 a 2 + ga 1 ) -g 1 a 2 1 -g 2 a 2 2 -2ga 1 a 2 .
Inserting the above in the definition of E 0 ε , and rearranging the terms, gives the statement.

The following proposition provides some estimates for the minimizer which will be used in the sequel for estimating the associated Lagrange multipliers. Proposition 3.2. Assume (1.3), (1.8) and (1.18). Let (η 1 , η 2 ) be the positive minimizer of E 0 ε in H that is provided by Theorem 1.3. If ε > 0 is sufficiently small, for i = 1, 2, we have

R 2 |∇η i | 2 dx ≤ C| log ε|, (3.9) R 2 (η 2 i -a i ) 2 dx ≤ Cε 2 | log ε|, (3.10)
R 2 \D 1 η 2 1 a - 1,0 dx + R 2 \D 2 (gη 2 1 + g 2 η 2 2 ) a 2,0 + g g 2 a 1,0 - dx ≤ Cε 2 | log ε|. (3.11) In particular, η 2 i → a i in L 2 (R 2 ) as ε → 0. Proof. First, we claim that, for small ε > 0, we have Ẽ0 ε (η 1 , η 2 ) ≤ C| log ε|, (3.12)
with Ẽ0 ε defined in Lemma 3.1. This is proved as in [START_REF] Aftalion | Giant vortex and the breakdown of strong pinning in a rotating Bose-Einstein condensate[END_REF] and [START_REF] Aftalion | Non-existence of vortices in the small density region of a condensate[END_REF], therefore we only give a sketch here. Consider the competitor functions

ηi = h ε (a i ) h ε (a i ) L 2 (R 2 )
, where

h ε (s) = s/ε if 0 ≤ s ≤ ε 2 , √ s if s > ε 2 .
It is proved in the aforementioned papers that

1 -Cε 2 ≤ R 2 h ε (a i ) 2 dx ≤ 1 R 2 |∇h ε (a i )| 2 dx ≤ C| log ε| R 2 (h ε (a i ) 2 -a i ) 2 dx ≤ Cε 2 .
Here we are implicitly using assumption (1.8) which ensures that a i are positive (recall (3.6),(3.7)). In addition notice that 2

R 2 (h ε (a 1 ) 2 -a 1 )(h ε (a 2 ) 2 -a 2 ) dx ≤ 2 i=1 R 2 (h ε (a i ) 2 -a i ) 2 dx
and that

R 2 \D 1 η2 1 a - 1,0 dx = R 2 \D 2 (g η2 1 + g 2 η2 2 ) a 2,0 + g g 2 a 1,0 - dx = 0.
Therefore, we have obtained Ẽ0 ε (η 1 , η2 ) ≤ C| log ε|. Finally, let (η 1 , η 2 ) be the positive minimizer, the decomposition proved in the previous lemma provides

Ẽ0 ε (η 1 , η 2 ) + K ≤ Ẽ0 ε (η 1 , η2 ) + K, so that (3.12) is proved.
On the other hand, relation (2.4) implies 2g

R 2 (η 2 1 -a 1 )(η 2 2 -a 2 ) dx ≤ R 2 (g 1 -γ)(η 2 1 -a 1 ) 2 + (g 2 -γ)(η 2 2 -a 2 ) 2 dx.
The result follows by combining this inequality with (3.12).

In the following proposition, we derive a preliminary estimate for the Lagrange multipliers. Even though this estimate is far from optimal, its form will play an important role when we improve it in Proposition 4.18.

Proposition 3.3. Assume (1.3), (1.8) and (1.18). Let (η 1 , η 2 ) be the positive minimizer of E 0 ε in H. Let λ i,ε be the associated Lagrange multipliers in (1.5). There exists C > 0 independent of ε such that, for i = 1, 2,

|λ i,ε -λ i,0 | ≤ Cε| log ε| 1/2 (3.13)
where λ i,0 are defined in (1.15). Given (1.31), this implies

|R i,ε -R i,0 | ≤ Cε| log ε| 1 2 , i = 1, 2. (3.14)
Proof. We test the equation for η 1 in (1.5) by η 1 itself, integrate by parts (since

η 1 ∈ H 1 (R 2 )),
and then subtract λ 1,0 from both sides to obtain

λ 1,ε -λ 1,0 = R 2 ε 2 |∇η 1 | 2 + η 2 1 (|x| 2 + g 1 η 2 1 + gη 2 2 -λ 1,0 ) dx.
With calculations similar to the one used in the proof of Lemma 3.1, we rewrite the right hand side of the previous expression in the following form:

R 2 ε 2 |∇η 1 | 2 + g 1 (η 2 1 -a 1 ) 2 + g(η 2 1 -a 1 )(η 2 2 -a 2 ) + g 1 a 1 (η 2 1 -a 1 ) +ga 1 (η 2 2 -a 2 ) dx + g 1 Γ R 2 \D 1 a - 1,0 η 2 1 dx + g R 2 \D 2 a 2,0 + g g 2 a 1,0 - η 2 1 dx.
We notice that

R 2 a 1 g 1 (η 2 1 -a 1 ) + g(η 2 2 -a 2 ) dx ≤ a 1 L 2 (R 2 ) g 1 η 2 1 -a 1 L 2 (R 2 ) +g η 2 2 -a 2 L 2 (R 2 )
. Hence by applying Proposition 3.2 we obtain the convergence of λ 1,ε . The convergence of λ 2,ε can be proved similarly.

Remark 3.4. The equivalent of Proposition 3.2 and 3.3 hold when (1.9) is assumed instead of (1.8). The only difference is that (3.11) has to be replaced by

{|x|≥R 1,0 } η 2 1,ε a - 1,0 dx + {|x|≤R - 2,0 } η 2 2,ε a - 2,0 dx + {|x|≥R + 2,0 } (gη 2 1,ε + g 2 η 2 2,ε ) a 2,0 + g g 2 a 1,0 - dx ≤ Cε 2 | log ε|.

Refined estimates for the energy minimizer without rotation

In this section we capture the fine behavior of the minimizer (η 1,ε , η 2,ε ), as ε → 0, by means of a perturbation argument. Since this type of approach is in principle not applicable to problems with integral constraints, we argue indirectly as follows. First, given (λ 1,ε , λ 2,ε ) as in the previous section, for small ε > 0, we construct a positive radial solution of (1.5) "near" (a 1 , a 2 ) by a perturbation argument. Then, the uniqueness result in Theorem 1.3 will imply that this solution coincides with the unique positive minimizer of E 0 ε in H.

4.1.

The main result concerning the minimizer without rotation.

Theorem 4.1. Assume that (1.3), (1.8) and (1.18) hold. Let (η 1 , η 2 ) be the positive minimizer of E 0 ε in H. Let λ i,ε be the associated Lagrange multipliers in (1.5). There exist constants c, C, D > 0 and δ ∈ 0, 1 4 min{R 1,0 , R 2,0 -R 1,0 } such that the following estimates hold: Estimates for the Lagrange multipliers

|λ i,ε -λ i,0 | ≤ C| log ε|ε 2 , which implies that |R i,ε -R i,0 | ≤ C| log ε|ε 2 , i = 1, 2; (4.1)
Outer estimates

η 1,ε - √ a 1,ε L ∞ (|x|≤R 1,ε -δ) + η 2,ε - √ a 2,ε L ∞ (|x|≤R 1,ε -δ) ≤ Cε 2 , (4.2)
and

η 2,ε -a 2,ε + g g 2 a 1,ε L ∞ (R 1,ε +δ≤|x|≤R 2,ε -δ) ≤ Cε 2 , (4.3)
uniformly as ε → 0; Algebraic decay estimates

η 1,ε (r) -a 1,ε (r) = O ε 2 |r -R 1,ε | -5 2 , η 2,ε (r) -a 2,ε (r) = O ε 2 |r -R 1,ε | -2 , (4.4) if r ∈ [R 1,ε -δ, R 1,ε -Dε 2 3
], and

η 2,ε (r) -a 2,ε (r) + g g 2 a 1,ε (r) = O ε 2 |r -R 2,ε | -5 2 if r ∈ [R 2,ε -δ, R 2,ε -Dε 2 3 ], (4.5)
uniformly as ε → 0;

Exponential decay estimates

η i,ε (r) ≤ Cε 1 3 exp c R i,ε -r ε 2 3 , r ≥ R i,ε , i = 1, 2, (4.6)
and

η 2,ε (r) = a 2,ε (r) + g g 2 a 1,ε (r)+O(ε 2 3 ) exp c R 1,ε -r ε 2 3 +O(ε 2 ) if r ∈ [R 1,ε , R 1,ε +δ], (4.7)
uniformly as ε → 0; Inner estimates

η 1,ε (r) = ε 1 3 (g 1 Γ) -1 6 β 1,ε V (g 1 Γ) 1 3 β 1,ε r -R 1,ε ε 2 3 +        O ε + |r -R 1,ε | 3 2 if r ∈ [R 1,ε -δ, R 1,ε ], O(ε) exp c R 1,ε -r ε 2 3 if r ∈ [R 1,ε , R 1,ε + δ], (4.8)
η 2,ε (r) = a 2,ε (r) + g g 2 a 1,ε (r) -g g 2 (g 1 Γ) -1 3 ε 2 3 β 2 1,ε V 2 (g 1 Γ) 1 3 β 1,ε r-R 1,ε ε 2 3 +        O ε 4 3 + |r -R 1,ε | 2 + ε|r -R 1,ε | 1 2 + ε 1 3 |r -R 1,ε | 3 2 if r ∈ [R 1,ε -δ, R 1,ε ], O(ε 4 3 ) exp c R 1,ε -r ε 2 3 + O(ε 2 ) if r ∈ [R 1,ε , R 1,ε + δ], (4.9
) and

η 2,ε (r) = ε 1 3 g -1 6 2 β 2,ε V g 1 3 2 β 2,ε r -R 2,ε ε 2 3 +        O ε + |r -R 2,ε | 3 2 if r ∈ [R 2,ε -δ, R 2,ε ], O(ε) exp c R 2,ε -r ε 2 3 if r ∈ [R 2,ε , R 2,ε + δ], (4.10
) uniformly as ε → 0, where V is the Hastings-McLeod solution [START_REF] Hastings | A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation[END_REF] to the Painlevé-II equation [START_REF] Fokas | Painlevé Transcendents, The Riemann-Hilbert Approach[END_REF], namely the unique solution of the boundary value problem

v ′′ = v(v 2 + s), s ∈ R; v(s) - √ -s → 0 as s → -∞, v(s) → 0 as s → ∞, (4.11)
and

β 1,ε = -a ′ 1,ε (R 1,ε ) 1 3 , β 2,ε = -a ′ 2,ε (R 2,ε ) - g g 2 a ′ 1,ε (R 2,ε ) 1 3
.

The proof of this theorem will be completed in Subsection 4.8. Note that in the case g 1 = g 2 , then an analogous of this theorem holds. It is simpler since

η 1,ε = η 2,ε so that R 1,ε = R 2,ε .
The result is just a consequence of Theorem A.1 in the appendix.

An approximate solution.

In this subsection we construct a sufficiently good approximate solution (η 1,ε , η2,ε ) to the problem (1.29) such that ηi,ε > 0 and ηi,ε → √ a i , uniformly on R 2 , as ε → 0, i = 1, 2. The building blocks of our construction will be the unique positive solutions η1,ε and η2,ε of the reduced problems (1.33a) and (1.33b) respectively.

4.2.1.

The reduced problems. The asymptotic behavior of ηi,ε , i = 1, 2, as ε → 0, can be deduced, after a proper constant re-scaling, from the following proposition which is a special case of the more general result that we prove in Appendix A.

Proposition 4.2. Suppose that λ ε satisfy λ ε → λ 0 as ε → 0, for some λ 0 > 0, and let

A ε (x) = λ ε -µ|x| 2 , x ∈ R 2 ,
with µ > 0 independent of ε.

There exists a unique positive solution u ε of the problem

ε 2 ∆u = u u 2 -A ε (x) , x ∈ R 2 ; u(x) → 0 as |x| → ∞.
This solution is radially symmetric and, for small ε > 0, satisfies the following properties:

u ε -A + ε L ∞ (R 2 ) ≤ Cε 1 3 , u ε -A ε C 2 (|x|≤rε-δ) ≤ Cε 2 , (4.12)
where

r ε = (µ -1 λ ε ) 1 2
, for some δ ∈ 0, 1 4 r 0 , r 0 = (µ -1 λ 0 ) 1/2 , and

u ε (r) ≤ Cε 1 3 exp cε -2 3 (r ε -r) , r ≥ r ε . (4.13)
In fact, the potential of the associated linearized operator satisfies the lower bound

3u 2 ε (r) -A ε (r) ≥    c|r -r ε | + cε 2 3 if |r -r ε | ≤ δ, c otherwise. (4.14)
More precisely, we have

u ε (r) = ε 1 3 β ε V β ε r -r ε ε 2 3 +        O ε + |r -r ε | 3 2 if r ε -δ ≤ r ≤ r ε , O(ε) exp -c |r-rε| ε 2 3 if r ε ≤ r ≤ r ε + δ, (4.15)
where V is the Hastings-McLeod solution, as described in (4.11), and

β ε = (-A ′ ε (r ε )) 1 3
.

Furthermore, we have

u ′ ε (r) = ε -1 3 β 2 ε V ′ β ε r -r ε ε 2 3 + O ε 1 3 + |r -r ε | 1 2 if |r -r ε | ≤ δ, (4.16)
uniformly, as ε → 0. Moreover, there exists D > 0 such that the following estimates hold for r ∈ [r εδ, r ε -Dε

u ε (r) -A ε (r) = ε 2 O(|r -r ε | -5 2 ), u ′ ε - √ A ε ′ = ε 2 O(|r -r ε | -7 2 ), ∆u ε -∆ √ A ε = ε 2 O(|r -r ε | -9 2 ), (4.17)
uniformly, as ε → 0.

(t) = 1 if t ≤ R ε -δ; ζ(t) = 0 if t ≥ R ε , (4.18
) where, for convenience, we have denoted

R ε = R 1,ε + R 2,ε 2 , (4.19)
and δ > 0 is a small number that is independent of small ε > 0. Note that ζ can be chosen independent of ε > 0 as well. We recall that η1,ε , η2,ε are the solutions of (1.33). In view of (4.13), let η1,ε (x

) = ζ(|x|)η 1,ε (x), x ∈ R 2 . (4.20) Then, let η2,ε (x) = a 2,ε (x) + g g 2 a 1,ε (x) - g g 2 η2 1,ε 1 2 , |x| ≤ R ε + δ. (4.21)
The motivation for this comes from neglecting the term ε 2 ∆η 2 in (1.29b), since it is expected to be of higher order, compared to the other terms, in the region |x| ≤ R 2,εδ. From (4.12), it follows that

η2,ε (x) -a 2,ε (x) + g g 2 a 1,ε (x) 1 2 = O(ε 2 ), in C 2 (R ε ≤ |x| ≤ R ε + δ) , as ε → 0.
In other words, recalling (4.18) and (4.21), we have that

η2,ε (x) -η2,ε (x) = O(ε 2 ), in C 2 (R ε ≤ |x| ≤ R ε + δ) , as ε → 0. ( 4.22)
Thus, we can smoothly interpolate between η2,ε and η2,ε to obtain a new approximation η2,ε such that

η2,ε (x) =            η2,ε (x), |x| ≤ R ε , η2,ε (x) + O C 2 (ε 2 ), R ε ≤ |x| ≤ R ε + δ, η2,ε (x), |x| ≥ R ε + δ. (4.23)
To conclude, we define our approximate solution of the system (1.29), for small ε > 0, to be the pair (η 1,ε , η2,ε ), as described by (4.20) and (4.23). We point out that this approximation satisfies the desired limiting behavior

(η 1,ε , η2,ε ) → ( √ a 1 , √ a 2 ), uniformly in R 2 , as ε → 0, ( 4.24)
where a 1 and a 2 are as in (1.12). Moreover, estimates that quantify this convergence can be derived easily from the corresponding ones that are available for the ground states η1,ε and η2,ε from Proposition 4.2.

4.3.

Estimates for the error on the approximate solution. The remainder that is left when substituting the approximate solution (η 1,ε , η2,ε ) to the system (1.29) is

E(η 1,ε , η2,ε ) ≡   E 1 E 2   ≡   -ε 2 ∆η 1,ε + g 1 η1,ε η2 1,ε -a 1,ε + g η1,ε η2 2,ε -a 2,ε -ε 2 ∆η 2,ε + g 2 η2,ε η2 2,ε -a 2,ε + g η2,ε η2 1,ε -a 1,ε   . (4.25)
The next proposition provides estimates for the L 2 -norms of E i , i = 1, 2, which follow from some delicate pointwise estimates that will be established in the process.

Proposition 4.3. The following estimates hold for small ε > 0:

E 1 L 2 (R 2 ) ≤ Ce -cε -2 3 , E 2 L 2 (|x|<Rε) ≤ Cε 5 3 and E 2 L 2 (|x|>Rε) ≤ Cε 2 . (4.26)
Proof. It follows from the construction of η1,ε and η2,ε , via (4.13), that

E 1 = 0 if |x| ≤ R ε -δ or |x| ≥ R ε ; |E 1 | ≤ Ce -cε -2 3 if R ε -δ ≤ |x| ≤ R ε . (4.27)
On the other side, we have

E 2 = -ε 2 ∆η 2,ε if |x| ≤ R ε -δ; (4.28) E 2 = -ε 2 ∆η 2,ε + O(e -cε -2 3 ) uniformly if R ε -δ ≤ |x| ≤ R ε , as ε → 0; E 2 = 0 if |x| ≥ R ε + δ; |E 2 | ≤ Cε 2 if R ε ≤ |x| ≤ R ε + δ (recall (4.22)
). In view of the previous observations, we only have to show the second relation in (4.26). In fact, since ∆η 2,ε remains uniformly bounded if

|r -R 1,ε | ≥ δ as ε → 0, it suffices to show that ∆η 2,ε L 2 (|r-R 1,ε |<δ) ≤ Cε -1 3 for small ε > 0. ( 4
|∆η 2,ε | ≤ C η2 1,ε |∇η 1,ε | 2 + C η1,ε ∆η 1,ε + |∇η 1,ε | 2 + C if |r -R 1,ε | < δ. ( 4
1 2 + 1), s ∈ R, setting ε = (g 1 Γ) -1
2 ε, after a tedious calculation, we arrive at

η1,ε ∆η 1,ε = ε-2 3 β 4 ε V 2 β ε r-R 1,ε ε 2 3 V 2 β ε r-R 1,ε ε 2 3 + β ε r-R 1,ε ε 2 3 + +O ε -2 |r -R 1,ε | 3 + ε -4 3 |r -R 1,ε | 2 + ε -1 |r -R 1,ε | 3 2 + +O ε -1 3 |r -R 1,ε | 1 2 + ε -2 3 |r -R 1,ε | + 1 + ε -5 3 |r -R 1,ε | 5 2
, (4.31)

uniformly if |r-R 1,ε | ≤ δ, as ε → 0, where β 3 ε = -a ′ 1,ε (R 1,ε ).
Similarly, but with considerably less effort, it follows from (4.16), and the bound

|V ′ (s)| ≤ C (|s| + 1) -1 2 , that |∇η 1,ε (r)| 2 = ε-2 3 β 4 ε V ′ β ε r -R 1,ε ε 2 3 2 + O(1), uniformly if |r -R 1,ε | ≤ δ, as ε → 0.
(4.32) It follows readily from the estimates in (4.17) that

η2 1,ε |∇η 1,ε | 2 ≤ C, (4.33)
and η1,ε ∆η

1,ε + |∇η 1,ε | 2 = O 1 + ε 2 |r -R 1,ε | -4 , (4.34) uniformly in -δ ≤ r -R 1,ε ≤ -Cε 2 3
, as ε → 0. Keep in mind that our eventual goal is to show (4.29). In view of (4.30), the above relations imply the partial estimate

∆η 2,ε L ∞ (-δ≤r-R 1,ε ≤-ε 7 12 ) ≤ Cε -1 3 for small ε > 0. (4.35)
On the other side, by the exponential decay of η1,ε for r > R 1,ε , we certainly have that ∆η 2,ε L ∞ (ε

7 12 ≤r-R 1,ε ≤δ) ≤ Cε -1 3
for small ε > 0. (4.36)

In the remaining interval (R 1,εε 7 12 , R 1,ε + ε 7 12 ) we use the inner estimates (4.15), (4.31), and (4.32), which in particular imply that

η2 1,ε |∇η 1,ε | 2 ≤ C, (4.37)
and η1,ε ∆η 12 , as ε → 0 (for obtaining the last relation, we have also used (4.11)). In order to proceed, we need the following easy estimate:

1,ε + |∇η 1,ε | 2 = ε-2 3 β 4 ε V β ε r-R 1,ε ε 2 3 V ′′ β ε r-R 1,ε ε 2 3 + V ′ β ε r-R 1,ε ε 2 3 2 + +O ε -1 4 , (4.38) uniformly if |r -R 1,ε | ≤ ε 7
V (s)V ′′ (s) + [V ′ (s)] 2 = O(|s| -4 ) as |s| → ∞, (4.39)
which follows from the asymptotic behavior V (s) = (-s)

1 2 + O |s| -5 2
as s → -∞, and from the super-exponential decay of V and its derivatives as s → ∞. Now, by (4.37), (4.38), and (4.39), via (4.30), we deduce that ∆η 2,ε L 2 (|r-R 1,ε |≤ε 4.4. Linear analysis. In this part of the paper we are going to study the linearization of (1.29) about the approximate solution (η 1,ε , η2,ε ), namely the linear operator

L(ϕ, ψ) ≡   -ε 2 ∆ϕ + g 1 (3η 2 1,ε -a 1,ε ) + g(η 2 2,ε -a 2,ε ) ϕ + 2g η1,ε η2,ε ψ -ε 2 ∆ψ + g 2 (3η 2 2,ε -a 2,ε ) + g(η 2 1,ε -a 1,ε ) ψ + 2g η1,ε η2,ε ϕ   , (4.41) for (ϕ, ψ) ∈ D(L) = (u, v) ∈ H 2 (R 2 ) × H 2 (R 2 ) : R 2 |x| 2 (u 2 + v 2 )dx < ∞ . By Friedrichs extension, the operator L is self-adjoint in L 2 (R 2 ) × L 2 (R 2 ) with domain D(L).
4.4.1. Energy estimates for L. We estimate from below the quotient

(L(ϕ, ψ), (ϕ, ψ)) (ϕ, ψ), (ϕ, ψ) , (ϕ, ψ) ∈ D(L) \ (0, 0),
which turns out to be positive, where (•, •) symbolizes the usual inner product in

L 2 (R 2 ) × L 2 (R 2 ) while < •,
• > is a suitably weighted one. In turn, these lower bounds provide a-priori estimates for the problem L(ϕ, ψ) = (f 1 , f 2 ). Energy estimates in B Rε . In the sequel, we carry out this plan in detail in the domain B Rε . Analogous results can be deduced in R 2 \ B Rε which we will describe later.

In view of (4.21) and (4.23), which imply that η2

2,εa 2,ε = -g g 2 (η 2 1,εa 1,ε ) in B Rε , we can conveniently rewrite (4.41) in B Rε as

L(ϕ, ψ) =    -ε 2 ∆ϕ + g 1 -g 2 g 2 (3η 2 1,ε -a 1,ε ) ϕ + 2 g 2 g 2 η2 1,ε ϕ + 2g η1,ε η2,ε ψ -ε 2 ∆ψ + 2g 2 η2 2,ε ψ + 2g η1,ε η2,ε ϕ    , (4.42)
for ϕ, ψ ∈ H 2 (B Rε ) (the reason for not adding the similar terms in the first row is to keep the linearization of (1.33a) about η1,ε in the beginning).

Proposition 4.4. The following a-priori estimates hold: Suppose that

ϕ, ψ ∈ H 2 N (B Rε ) ≡ v ∈ H 2 (B Rε ) : ν • ∇v = 0 on ∂B Rε
, where ν denotes the outer unit normal vector to ∂B Rε , satisfy

L(ϕ, ψ) = λ(ε 2 3 ϕ, ψ) and B Rε ε 2 3 ϕ 2 + ψ 2 dx = 1, (4.43) or L(ϕ, ψ) = λ(η 2 1,ε ϕ, ψ) and B Rε η2 1,ε ϕ 2 + ψ 2 dx = 1, (4.44) then λ ≥ c.
Proof. We use the following estimates:

3η 2 1,ε -a 1,ε ≥    c|r -R 1,ε | + cε 2 3 , if |r -R 1,ε | ≤ δ, c , otherwise, (4.45)
and

η2 1,ε ≤    Cε 2 3 + C|r -R 1,ε |, if |r -R 1,ε | ≤ δ, C , otherwise, (4.46)
which are inherited from (4.12), (4.14) and (4.15). In particular, observe that

3η 2 1,ε -a 1,ε ≥ cη 2 1,ε , x ∈ R 2 . (4.47)
Note also that 2g 2 η2 2,ε ≥ c on BRε . (4.48) Suppose that (4.43) holds. Testing by (ϕ, ψ), in the usual sense of L 2 (B Rε ) × L 2 (B Rε ), and integrating by parts the resulting identity, we obtain that

B Rε ε 2 |∇ϕ| 2 + ε 2 |∇ψ| 2 + g 1 - g 2 g 2 (3η 2 1,ε -a 1,ε )ϕ 2 + 2 g √ g 2 η1,ε ϕ + √ g 2 η2,ε ψ 2 dx = λ.
(4.49) In turn, using (4.45) and (4.47), we find that

B Rε ε 2 |∇ϕ| 2 + ε 2 |∇ψ| 2 + cε 2 3 ϕ 2 + cη 2 1,ε ϕ 2 dx ≤ λ. (4.50)
On the other side, the second equation of the system in (4.43) can be written as

-ε 2 ∆ψ + 2g 2 η2 2,ε ψ = λψ -2g η2,ε η1,ε ϕ.
Then, testing the above relation by ψ, integrating by parts, using (4.48) and Young's inequality, we obtain

B Rε (ε 2 |∇ψ| 2 + cψ 2 ) dx ≤ B Rε λψ 2 dx + C η2 1,ε ϕ 2 dx
(L(ϕ, ψ), (ϕ, ψ)) ≥ c B Rε ε 2 |∇ϕ| 2 + ε 2 |∇ψ| 2 + ε 2 3 ϕ 2 + η2 1,ε ϕ 2 + ψ 2 dx ∀ϕ, ψ ∈ D(L), (4.52) where (•, •) denotes the usual inner product in L 2 (R 2 ) × L 2 (R 2 ). Energy estimates in R 2 \ B Rε . Since η1,ε = 0 in R 2 \ B Rε , the operator L in R 2 \ B Rε has the simple "decoupled" form L(ϕ, ψ) =    -ε 2 ∆ϕ + g(η 2 2,ε -a 2,ε ) -g 1 a 1,ε ϕ -ε 2 ∆ψ + g 2 3η 2 2,ε -a 2,ε -g g 2 a 1,ε ψ    , for ϕ, ψ ∈ H 2 (R 2 \ B Rε ). Note that g(η 2 2,ε -a 2,ε ) -g 1 a 1,ε = g η2 2,ε -a 2,ε - g g 2 a 1,ε + g 2 g 2 -g 1 a 1,ε ≥ c, in R 2 \ B Rε , because a 1,ε ≤ -c and η2 2,ε -a 2,ε -g g 2 a 1,ε ≤ Cε 2 3
therein. In analogy to (4.52), for small ε > 0, one can show rather straightforwardly that

(L(ϕ, ψ), (ϕ, ψ)) ≥ c R 2 \B Rε ε 2 |∇ϕ| 2 + ε 2 |∇ψ| 2 + ϕ 2 + ε 2 3 ψ 2 + η2 2,ε ψ 2 dx, (4.53)
for every (ϕ, ψ) ∈ D(L). Energy estimates in R 2 . It follows at once from (4.52) and (4.53) that, for small ε > 0, we have

(L(ϕ, ψ), (ϕ, ψ)) ≥ cε 2 R 2 (|∇ϕ| 2 + |∇ψ| 2 ) dx + c B Rε ε 2 3 ϕ 2 + η2 1,ε ϕ 2 + ψ 2 dx+ +c R 2 \B Rε ϕ 2 + ε 2 3 ψ 2 + η2 2,ε ψ 2 dx, (4.54
) for every (ϕ, ψ) ∈ D(L).

Invertibility properties of L.

We are now in position to obtain estimates for the solution of the inhomogeneous problem L(ϕ, ψ) = (f 1 , f 2 ) in R 2 , with suitable right-hand side.

Proposition 4.6. Let f i ∈ L 2 (R 2 ), i = 1, 2. The equation L(ϕ, ψ) = (f 1 , f 2 ) in R 2 , (4.55) has a unique solution (ϕ, ψ) ∈ H 2 (R 2 ) × H 2 (R 2 )
, provided that ε > 0 is sufficiently small, independently of f i . Moreover, that solution satisfies

|||(ϕ, ψ)||| 2 ≤ C B Rε ε -2 3 f 2 1 + f 2 2 dx + C R 2 \B Rε f 2 1 + ε -2 3 f 2 2 dx, (4.56)
with C > 0 independent of ε and f i , where the norm

|||(•, •)||| in H 1 (R 2 ) × H 1 (R 2 ) is defined by |||(ϕ, ψ)||| 2 = ε 2 ∇ϕ 2 L 2 (R 2 ) + ∇ψ 2 L 2 (R 2 ) + ε 2 3 ϕ 2 L 2 (B Rε ) + ψ 2 L 2 (B Rε ) + + ϕ 2 L 2 (R 2 \B Rε ) + ε 2 3 ψ 2 L 2 (R 2 \B Rε ) .
(4.57)

If ε > 0 is sufficiently small, there exist c, C > 0 such that, for any

f ∈ L 2 (R 2 ), the solution of L(ϕ, ψ) = (η 1,ε f, 0) in R 2 , (4.58) satisfies |||(ϕ, ψ)||| 2 ≤ C B Rε f 2 dx + Ce -cε -2 3 R 2 f 2 dx. (4.59)
If ε > 0 is sufficiently small, there exists C > 0 such that, for any f ∈ L 2 (R 2 ), the solution of

L(ϕ, ψ) = (0, η2,ε f ) in R 2 , (4.60) satisfies |||(ϕ, ψ)||| ≤ C f L 2 (R 2 ) . (4.61)
Proof. As we have already discussed, the linear operator

L is self-adjoint in L 2 (R 2 ) × L 2 (R 2 )
with domain D(L). Relation (4.54) certainly implies that the kernel of L is empty for small ε > 0. Hence, the existence and uniqueness of a solution (ϕ, ψ) ∈ H 2 (R 2 ) × H 2 (R 2 ) to (4.55) are clear. We now turn our attention to establishing estimate (4.56). Testing (4.55) by (ϕ, ψ), and using part of (4.54), we find that

ε 2 R 2 |∇ϕ| 2 + |∇ψ| 2 dx + B Rε ε 2 3 ϕ 2 + ψ 2 dx + R 2 \B Rε ϕ 2 + ε 2 3 ψ 2 dx ≤ ≤ C B Rε (|f 1 ϕ| + |f 2 ψ|) dx + C R 2 \B Rε (|f 1 ϕ| + |f 2 ψ|) dx.
Using Young's inequality, we can bound the first integral in the right-hand side by

B Rε ε 2 3 2 ϕ 2 + Cε -2 3 f 2 1 + 1 2 ψ 2 + Cf 2 2 dx,
and analogously we can bound the second integral. By absorbing into the left-hand side the terms that involve ϕ or ψ, we get (4.56).

Suppose now that (4.58) holds. As before, but this time making more use of (4.54), we find that

ε 2 R 2 |∇ϕ| 2 + |∇ψ| 2 dx + B Rε ε 2 3 ϕ 2 + η2 1,ε ϕ 2 + ψ 2 dx + R 2 \B Rε ϕ 2 + ε 2 3 ψ 2 dx ≤ ≤ C B Rε η1,ε |f ϕ|dx + C R 2 \B Rε η1,ε |f ϕ|dx.
The desired estimate (4.59) follows readily as before, using Young's inequality to absorb a term of the form 1 2 B Rε η2 1,ε ϕ 2 dx into the left-hand side, and recalling the exponential decay (4.13) of η1,ε for r > R 1,ε .

Finally, suppose that (4.60) holds. As before, making use of (4.54) once more, we arrive at

ε 2 R 2 |∇ϕ| 2 + |∇ψ| 2 dx + B Rε ε 2 3 ϕ 2 + ψ 2 dx + R 2 \B Rε ϕ 2 + ε 2 3 ψ 2 + η2 2,ε ψ 2 dx ≤ ≤ C B Rε η2,ε |f ψ|dx + C R 2 \B Rε η2,ε |f ψ|dx.
The desired estimate (4.61) follows readily as before, using Young's inequality to absorb terms of the form 1 2 R 2 \B Rε η2 2,ε ψ 2 dx and 1 2 B Rε ψ 2 dx into the left-hand side. 4.5. Existence and properties of a positive solution of the system (1.29). We seek a true solution of (1.29) in the form

(η 1,ε , η 2,ε ) = (η 1,ε , η2,ε ) + (ϕ, ψ), (4.62)
with ϕ, ψ ∈ H 2 rad (R 2 ). In terms of (ϕ, ψ), system (1.29) becomes

-L(ϕ, ψ) = N (ϕ, ψ) + E(η 1,ε , η2,ε ), (4.63)
where L is the linear operator in (4.41), the nonlinear operator N is

N (ϕ, ψ) =   N 1 (ϕ, ψ) N 2 (ϕ, ψ)   =   g 1 ϕ 3 + 3g 1 η1,ε ϕ 2 + g η1,ε ψ 2 + 2g η2,ε ψϕ + gψ 2 ϕ g 2 ψ 3 + 3g 2 η2,ε ψ 2 + g η2,ε ϕ 2 + 2g η1,ε ϕψ + gϕ 2 ψ   , (4.64)
and the remainder E(η 1,ε , η2,ε ) is as in (4.25).

In view of Proposition 4.6, for small ε > 0, we can define a nonlinear operator T :

H 2 rad (R 2 ) × H 2 rad (R 2 ) → H 2 rad (R 2 ) × H 2 rad (R 2 ) via the relation T (ϕ, ψ) = ( φ, ψ), where ( φ, ψ) ∈ H 2 rad (R 2 ) × H 2 rad (R 2 ) is uniquely determined from the equation -L( φ, ψ) = N (ϕ, ψ) + E(η 1,ε , η2,ε ). (4.65)
Note that Sobolev's inequality implies that functions in H 2 (R 2 ) are bounded, in particular

N (ϕ, ψ) ∈ L 2 (R 2 ) × L 2 (R 2 ) for every (ϕ, ψ) ∈ H 2 (R 2 ) × H 2 (R 2 ). For ε > 0, M > 1, let B ε,M = (ϕ, ψ) ∈ H 2 rad (R 2 ) × H 2 rad (R 2 ) : |||(ϕ, ψ)||| ≤ Mε 5 3
.

The following proposition contains the main properties of the operator T .

Proposition 4.7. If M > 1 is sufficiently large, the operator T maps B ε,M into itself, and its restriction to B ε,M is a contraction with respect to the |||(•, •)||| norm, provided that ε > 0 is sufficiently small.

Proof. Let (ϕ, ψ) ∈ B ε,M , and ( φ, ψ) = T (ϕ, ψ). In view of (4.64) and (4.65), we have

( φ, ψ) = 4 i=1 ( φi , ψi ), (4.66)
where ( φi , ψi )

∈ H 2 rad (R 2 ) × H 2 rad (R 2 ), i = 1, • • • , 4, satisfy -L( φ1 , ψ1 ) =   g 1 ϕ 3 + 2g η2,ε ψϕ + gψ 2 ϕ g 2 ψ 3 + 2g η1,ε ϕψ + gϕ 2 ψ   , -L( φ2 , ψ2 ) =   η1,ε (3g 1 ϕ 2 + gψ 2 ) 0   , -L( φ3 , ψ3 ) =   0 η2,ε (3g 2 ψ 2 + gϕ 2 )   , and -L( φ4 , ψ4 ) = E(η 1,ε , η2,ε ).
Using Proposition 4.6, and the Gagliardo-Nirenberg interpolation inequality in order to estimate the L 2 -norms of the nonlinear terms, it follows readily that ( φ1 , ψ1 ) ≤ CM 3 ε

7 3 + CM 2 ε 11 6 and ( φi , ψi ) ≤ CM 2 ε 2 , i = 2, 3, (4.67)
where C > 0 is independent of both ε and M, provided that ε > 0 is sufficiently small. In order to illustrate the procedure, let us present in detail the proof of the second bound (i = 2): Estimate (4.59) implies that

( φ2 , ψ2 ) 2 ≤ C B Rε (ϕ 4 + ψ 4 )dx + Ce -cε -2 3 R 2 (ϕ 4 + ψ 4 )dx,
with constants c, C > 0 independent of ε, M, provided that ε > 0 is sufficiently small. Since (ϕ, ψ) ∈ B ε,M , it follows that

ϕ L 2 (B Rε ) ≤ Mε 4 3 , ϕ H 1 (R 2 ) ≤ Mε 2 3 ; ψ L 2 (B Rε ) ≤ Mε 5 3 , ψ H 1 (R 2 ) ≤ Mε 2 3 . (4.68)
Now, the desired bound follows via the Gagliardo-Nirenberg inequality

u L p (Ω) ≤ C p u 1-2 p H 1 (Ω) u 2 p L 2 (Ω) , p ≥ 2, (4.69)
for Ω ⊆ R 2 regular, which implies that

ϕ L 4 (B Rε ) ≤ C ϕ 1 2 H 1 (B Rε ) ϕ 1 2 L 2 (B Rε ) ≤ CMε, (4.70)
with constant C > 0 independent of ε, M, provided that ε > 0 is sufficiently small, and analogous estimates can be derived for ψ L 4 (B Rε ) , ϕ L 4 (R 2 ) and ψ L 4 (R

M 3 ε 2 3 + M 2 ε 1 6 + 1 ,
with C > 0 independent of ε, M, provided that ε > 0 is sufficiently small. Consequently, if we choose M = 2C, and fix it from now on, decreasing ε > 0 further if necessary so that

M 3 ε 2 3 + M 2 ε 1 6 ≤ 1, it follows that ( φ, ψ) = |||T (ϕ, ψ)||| ≤ Mε 5 3
. We conclude that, if ε > 0 is sufficiently small, the operator T maps B ε,M into itself, as asserted.

It remains to show that the restriction of T to B ε,M is a contraction with respect to the |||(•, •)||| norm, provided that ε > 0 is sufficiently small. To this end, let

(ϕ i , ψ i ) ∈ B ε,M , i = 1, 2, and ( φi , ψi ) = T (ϕ i , ψ i ), i = 1, 2. (4.72) Then, set ( w, z) = ( φ1 -φ2 , ψ1 -ψ2 ). (4.73) As before, it is convenient to write ( w, z) = 5 i=1 ( wi , zi ), (4.74)
where ( wi

, zi ) ∈ H 2 rad (R 2 ) × H 2 rad (R 2 ), i = 1, • • • , 5, satisfy -L( w1 , z1 ) =   g 1 (ϕ 2 1 + ϕ 1 ϕ 2 + ϕ 2 2 )(ϕ 1 -ϕ 2 ) g 2 (ψ 2 1 + ψ 1 ψ 2 + ψ 2 2 )(ψ 1 -ψ 2 )   , -L( w2 , z2 ) = 2g [ψ 2 (ϕ 1 -ϕ 2 ) + ϕ 1 (ψ 1 -ψ 2 )]   η2,ε η1,ε   , -L( w3 , z3 ) = g   ψ 2 2 (ϕ 1 -ϕ 2 ) + ϕ 1 (ψ 1 + ψ 2 )(ψ 1 -ψ 2 ) ϕ 2 2 (ψ 1 -ψ 2 ) + ψ 1 (ϕ 1 + ϕ 2 )(ϕ 1 -ϕ 2 )   , -L( w4 , z4 ) =   η1,ε [3g 1 (ϕ 1 + ϕ 2 )(ϕ 1 -ϕ 2 ) + g(ψ 1 + ψ 2 )(ψ 1 -ψ 2 )] 0   , and
-L( w5 , z5 ) =   0 η2,ε [3g 2 (ψ 1 + ψ 2 )(ψ 1 -ψ 2 ) + g(ϕ 1 + ϕ 2 )(ϕ 1 -ϕ 2 )]   .
As before, using Proposition 4.6, the Gagliardo-Nirenberg inequality (4.69), and the inequalities

ϕ L 2 (B Rε ) ≤ ε -1 3 |||(ϕ, ψ)|||, ϕ L 2 (R 2 \B Rε ) ≤ |||(ϕ, ψ)|||, ϕ H 1 (R 2 ) ≤ ε -1 |||(ϕ, ψ)|||, ψ L 2 (B Rε ) ≤ |||(ϕ, ψ)|||, ψ L 2 (R 2 \B Rε ) ≤ ε -1 3 |||(ϕ, ψ)|||, ψ H 1 (R 2 ) ≤ ε -1 |||(ϕ, ψ)|||, (4.75)
for every ϕ, ψ ∈ H 1 (R 2 ), we can show that

|||( w1 , z1 )||| ≤ Cε 2 3 |||(ϕ 1 -ϕ 2 , ψ 1 -ψ 2 )|||, |||( w2 , z2 )||| ≤ Cε 1 6 |||(ϕ 1 -ϕ 2 , ψ 1 -ψ 2 )|||, |||( w3 , z3 )||| ≤ Cε 5 6 |||(ϕ 1 -ϕ 2 , ψ 1 -ψ 2 )|||, |||( wi , zi )||| ≤ Cε 1 3 |||(ϕ 1 -ϕ 2 , ψ 1 -ψ 2 )|||, i = 4, 5, (4.76
) provided that ε > 0 is sufficiently small. In order to illustrate the procedure, let us present in detail the proof of the bound for |||( w2 , z2 )|||: From Proposition 4.6, we obtain that

|||( w2 , z2 )||| ≤ Cε -1 3 ψ 2 (ϕ 1 -ϕ 2 ) L 2 (B Rε ) + Cε -1 3 ϕ 1 (ψ 1 -ψ 2 ) L 2 (B Rε ) + +Cε -1 3 ψ 2 (ϕ 1 -ϕ 2 ) L 2 (R 2 \B Rε ) + Cε -1 3 ϕ 1 (ψ 1 -ψ 2 ) L 2 (R 2 \B Rε )
. The second term in the right-hand side of the above relation can be estimated as follows: By the Cauchy-Schwarz inequality, we have

ε -1 3 ϕ 1 (ψ 1 -ψ 2 ) L 2 (B Rε ) ≤ ε -1 3 ϕ 1 L 4 (B Rε ) ψ 1 -ψ 2 L 4 (B Rε ) (4.70) ≤ Cε 2 3 ψ 1 -ψ 2 L 4 (B Rε ) (4.69), (4.75)
≤ Cε

1 6 |||(ϕ 1 -ϕ 2 , ψ 1 -ψ 2 )
|||. The remaining terms can be estimated in a similar fashion, giving the desired bound for |||( w2 , z2 )|||. The other bounds in (4.76) can be verified analogously. Consequently, combining relations (4.72), (4.73), (4.74), and (4.76), we infer that

|||T (ϕ 1 , ψ 1 ) -T (ϕ 2 , ψ 2 )||| ≤ Cε 1 6 |||(ϕ 1 , ψ 1 ) -(ϕ 2 , ψ 2 )||| ∀ (ϕ i , ψ i ) ∈ B ε,M , i = 1, 2.
We therefore conclude that, for sufficiently small ε > 0, the restriction of T to B ε,M is a contraction with respect to the |||(•, •)||| norm, as asserted.

The above proposition implies the main result of this section: Proposition 4.8. There exists a constant M > 0, such that the system (1.29) has a unique solution (η 1,ε , η 2,ε ) such that

|||(η 1,ε -η1,ε , η 2,ε -η2,ε )||| ≤ Mε 5 3 , (4.77)
if ε > 0 is sufficiently small, where the above norm is as in (4.57).

Proof. In view of Proposition 4.7, for small ε > 0, we can define iteratively a sequence

(ϕ n , ψ n ) ∈ B ε,M such that (ϕ n+1 , ψ n+1 ) = T (ϕ n , ψ n ), n ≥ 0, (ϕ 0 , ψ 0 ) = (0, 0). (4.78)
Moreover, the same proposition implies that (ϕ n , ψ n ) is a Cauchy sequence in

H 1 rad (R 2 ) × H 1 rad (R 2 ). Hence, we infer that (ϕ n , ψ n ) → (ϕ ∞ , ψ ∞ ) in H 1 rad (R 2 ) × H 1 rad (R 2 ), as n → ∞, for some (ϕ ∞ , ψ ∞ ) ∈ H 1 rad (R 2 ) × H 1 rad (R 2 ) such that |||(ϕ ∞ , ψ ∞ )||| ≤ Mε 5 3 .
In turn, letting n → ∞ in the weak form of (4.78) (recall (4.65)) yields that (ϕ ∞ , ψ ∞ ) is a weak solution of (4.63). Then, by standard elliptic regularity theory, we deduce that

(ϕ ∞ , ψ ∞ ) ∈ H 2 rad (R 2 ) × H 2 rad (R 2 ) (i.e. (ϕ ∞ , ψ ∞ ) ∈ B ε,M
) and is smooth (i.e. a classical solution). The point being that B ε,M is not closed in the |||(•, •)||| norm. In fact, equation (4.63) has a unique solution in B ε,M , as the restriction of T to B ε,M is a contraction with respect to the |||(•, •)||| norm, provided that ε > 0 is sufficiently small. Consequently, recalling the equivalence of (1.29) to (4.63) via (4.62), we conclude that the assertions of the proposition hold.

A direct consequence of (4.77) is the following Corollary 4.9. If ε > 0 is sufficiently small, the solutions η 1,ε and η 2,ε of the system (1.29) that are provided by Proposition 4.8 satisfy

η i,ε -ηi,ε L ∞ (|x|≥δ) ≤ Cε, and η i,ε (x) → 0 as |x| → ∞, i = 1, 2.
(4.79)

Proof. Consider the fluctuations

ϕ = η 1,ε -η1,ε and ψ = η 2,ε -η2,ε .
It follows from (4.57) and (4.77) that

∇ϕ L 2 (R 2 ) ≤ Cε 2 3 , ϕ L 2 (R 2 ) ≤ Cε 4 3 and ∇ψ L 2 (R 2 ) ≤ Cε 2 3 , ψ L 2 (R 2 ) ≤ Cε 4
3 , (4.80) (note that we did not make full use of (4.77)). In order to transform the above into uniform estimates, we need the following inequality which can be traced back to [START_REF] Strauss | Existence of solitary waves in higher dimensions[END_REF]: There exists a constant C > 0 such that

|x| 1 2 |v(x)| ≤ C ∇v 1 2 L 2 (R 2 ) v 1 2 L 2 (R 2 ) for a.e x ∈ R 2 , (4.81)
and all v ∈ H 1 rad (R 2 ) ≡ {v ∈ H 1 (R 2 ) : v is radial}. The desired asymptotic behavior in (4.79) follows at once. Making use of this inequality for |x| ≥ δ, we obtain that ϕ L ∞ (|x|≥δ) ≤ Cε and ψ L ∞ (|x|≥δ) ≤ Cε, which are exactly the desired uniform estimates in (4.79).

We now turn our attention to establishing uniform estimates on Bδ . The following lemma will come in handy in the proof of Corollary 4.11 below. Lemma 4.10. There exists a constant C > 0 such that the solutions that are provided by Proposition 4.8 satisfy

η i,ε L ∞ (B δ ) ≤ C, i = 1, 2, if ε > 0 is sufficiently small.
Proof. Suppose that the assertion is false. We use a blow-up argument to arrive at a contradiction (see also [START_REF] Gidas | A priori bounds for positive solutions of nonlinear elliptic equations[END_REF]). Without loss of generality, we may assume that there exist ε n → 0 and

x n ∈ B δ such that η 1,εn (x n ) = η 1,εn L ∞ (B δ ) = M n → ∞.
We may further assume that x n → x ∞ ∈ Bδ . Now, we re-scale η 1,εn by setting

v n (y) = µ n η 1,εn (x n + ε n µ n y) with µ n = M -1 n → 0. The function v n satisfies -∆v n + g 1 v 3 n -g 1 µ 2 n a 1,εn (x n + ε n µ n y)v n + gµ 2 n η 2 2,εn (x n + ε n µ n y) -a 2,εn (x n + ε n µ n y) v n = 0.
By using elliptic L p estimates and standard imbeddings, exploiting the bound

η 2 2,ε -a 2,ε L p (B δ ) ≤ C p ε 2 3 + 1
p , p ≥ 2, (readily derivable from Proposition 4.8), we deduce that a subsequence of v n converges uniformly on compact sets to a bounded nontrivial solution v ∞ of the problem

∆v = g 1 v 3 , v(0) = 1, (4.82)
in the entire space R 2 or in an open half-space H containing the origin, with zero boundary conditions on ∂H. Actually, in the latter scenario, one has to perform a rotation and stretching of coordinates in the resulting limiting equation to get (4.82), see [START_REF] Gidas | A priori bounds for positive solutions of nonlinear elliptic equations[END_REF]. In any case, by reflecting v ∞ oddly across ∂H if necessary, we have been led to a nontrivial solution of (4.82) in the whole space R 2 . On the other hand, this contradicts a well known Liouville type theorem of Brezis [START_REF] Brezis | Semilinear equations in R n without conditions at infinity[END_REF].

The following corollary provides additional information to Corollary 4.9, but will be considerably improved in Proposition 4.14.

Corollary 4.11. If ε > 0 is sufficiently small, the solutions provided by Proposition 4.8 satisfy

η i,ε - √ a i,ε L ∞ (B δ ) ≤ Cε 1 3 , i = 1, 2. (4.83) Proof. Let φ = η 1,ε - √ a 1,ε , x ∈ B2δ . (4.84)
Observe that estimates (4.12) and (4.77) imply that From the first equation in (1.29), by rearranging terms, in B 2δ we obtain that

φ L 2 (B 2δ ) ≤ Cε
-ε 2 ∆φ = -g 1 η 1,ε η 1,ε + √ a 1,ε φ + ε 2 ∆ √ a 1,ε -gη 1,ε (η 2 2
,εa 2,ε ) =: f. By interior elliptic regularity theory, we deduce that

φ H 2 (B δ ) ≤ C ε -2 f L 2 (B 2δ ) + φ L 2 (B 2δ ) ≤ Cε -2 3 , (4.86)
where we also used Lemma 4.10 and (4.85). Now, by the two-dimensional Agmon inequality [7, Lem. 13.2], we infer that

φ L ∞ (B δ ) ≤ C φ 1 2 H 2 (B δ ) φ 1 2 L 2 (B δ ) (4.85),(4.86) ≤ Cε 1 3 .
The desired bound for η 1,ε -√ a 1,ε follows at once from (4.84) and the above relation. The corresponding bound for η 2,ε -√ a 2,ε can be shown analogously.

We are now in position to show that the solutions in Proposition 4.8 are in fact positive.

Proposition 4.12. If ε > 0 is sufficiently small, the solutions in Proposition 4.8 satisfy

η i,ε > 0 in R 2 , i = 1, 2.
Proof. By virtue of (4.12), (4.15), (4.79), and Corollary 4.11, given D ≥ 1, we deduce that

η 1,ε ≥ c D ε 1 3 > 0 if |x| ≤ R 1,ε + Dε 2 3 , (4.87)
provided that ε > 0 is sufficiently small; where throughout this proof, unless specified otherwise, the generic constants c, C > 0 are also independent of D ≥ 1. From (1.29a), we observe that η 1,ε satisfies a linear equation of the form

-ε 2 ∆η 1,ε + Q(x)η 1,ε = 0, (4.88)
where

Q(x) = g 1 (η 2 1,ε -a 1,ε ) + g(η 2 2,ε -a 2,ε ). If R 1,ε + Dε 2 3 ≤ |x| ≤ R 2,
ε , recalling (4.13) and (4.79), we find that

Q(x) = g 1 (η 2 1,ε -η2 1,ε + η2 1,ε -a 1,ε ) + g(η 2 2,ε -η2 2,ε + η2 2,ε -a 2,ε ) ≥ -g 1 a 1,ε + g(η 2 2,ε -a 2,ε ) -Cε ≥ g 2 g 2 -g 1 a 1,ε + g η2 2,ε -a 2,ε - g g 2 a 1,ε -Cε.
In particular, if R 1,ε + Dε

2 3 ≤ |x| ≤ R ε
, where η2,ε = η2,ε , via (4.21), which implies that the second term in the right-hand side is -g 2 g -1 2 η2

1,ε , and the exponential decay of the ground state η1,ε for r = |x| > R 1,ε , we obtain that

Q(r) ≥ g 2 g 2 -g 1 a 1,ε -Ce -cD ε 2 3 (1.3),(1.30),(1.34),(3.13) ≥ c(r -R 1,ε ) 2 + cDε 2 3 -Ce -cD ε 2 3 , ≥ c(r -R 1,ε ) 2 + cDε 2 3
, increasing the value of D if needed, provided that ε > 0 is sufficiently small. Clearly, in view of (4.23), the same lower bound holds if R ε ≤ |x| ≤ R ε + δ. On the other side, if

R ε + δ ≤ |x| ≤ R 2,ε , where a 1,ε = -c(r 2 -R 2 1,ε ), η2,ε = η2,ε and η2 2,ε -a 2,ε -g g 2 a 1,ε ≤ Cε 2 3 , we deduce that Q(r) ≥ c(r -R 1,ε ) 2 .
The latter lower bound also holds if |x| ≥ R 2,ε . So far, we have shown that

Q(x) ≥ c(r -R 1,ε ) 2 + cDε 2 3 , |x| ≥ R 1,ε + Dε 2 3 , (4.89)
provided that ε > 0 is sufficiently small. By (4.79), (4.87), (4.88), (4.89), and the maximum principle, we deduce that

η 1,ε ≥ 0 if |x| ≥ R 1,ε + Dε 2 3
. The desired strict positivity of η 1,ε follows immediately from (4.87), (4.88), the above relation, and the strong maximum principle. The corresponding property for η 2,ε can be proven analogously.

The following lemma is motivated from Lemma 2.2, and will be used in the next section. Lemma 4.13. Given D > 1 sufficiently large, we have

η i,ε (s) ≤ η i,ε (r) exp - D 1 3 ε 2 3 (s 2 -r 2 ) for s ≥ r ≥ R i,ε + Dε 2 3 , i = 1, 2,
provided that ε > 0 is sufficiently small.

Proof. Throughout this proof, the generic constants c, C > 0 are independent of both small ε > 0 and large D > 1. Abusing notation slightly, let

u(x) = u(s) = exp - D 1 3 ε 2 3 s 2 , x ∈ R 2 , s = |x|.
It is easy to see that

-ε 2 ∆u + c (s -R 1,ε ) 2 + Dε 2 3 u ≥ -4D 2 3 ε 2 3 s 2 + c(s -R 1,ε ) 2 + cDε 2 3 u ≥ 0, (4.90) if s = |x| ≥ R 1,ε + Dε 2 3
, for sufficiently large D > 1, provided that ε > 0 is sufficiently small. Here c > 0 is as in (4.89). For such D, ε, and any r ≥ R 1,ε + Dε 2 3 , it follows that the function

v(s) = η 1,ε (r) exp - D 1 3 ε 2 3 (s 2 -r 2 ) , s = |x| ≥ r,
is an upper solution of the linear elliptic equation that is defined by the left-hand side of (4.90). On the other side, by virtue of (4.88) and (4.89), the function η 1,ε (s) is a lower solution of the same equation for |x| > r, which clearly coincides with the upper solution v on ∂B r . Hence, by the maximum principle, and (4.79), we deduce that

η 1,ε (s) ≤ v(s), ∀ s = |x| ≥ r ≥ R 1,ε + Dε 2 3
, for any large D > 1, provided that ε > 0 is sufficiently small. The validity of the asserted estimate for η 1,ε now follows immediately, while that for η 2,ε follows analogously. 4.6. Improved uniform estimates away from R 1,ε and R 2,ε . In this subsection we show that the uniform estimates in (4.79), for the difference η i,εηi,ε , can be improved outside of an O(ε

2 3 )-neighborhood of R i,ε , i = 1, 2.
The results of this subsection, as well as those of the following one, are not essential for the proof of Theorem 1.5, and, depending on the reader's preference, can be skipped on a first reading. Proposition 4.14. If ε > 0 is sufficiently small, there exist C, δ > 0, with δ < min

R 1,0 4 , R 2,0 -R 1,0 4 , such that |η i,ε (r) -ηi,ε (r)| ≤ Cε 2 if |r -R 1,ε | ≥ δ and |r -R 2,ε | ≥ δ, i = 1, 2.
Proof. We prove the assertion in the case where r ∈ [0, R 1,εδ], which reduces to show that

η i,ε (r) -a i,ε (r) ≤ Cε 2 if r ∈ [0, R 1,ε -δ], i = 1, 2, (4.91)
(recall the construction of ηi,ε and also see (4.12)). In the remaining intervals the proof carries over analogously. We first require a rough uniform bound for the radial derivatives of the functions

u ≡ η 1,ε - √ a 1,ε and v ≡ η 2,ε - √ a 2,ε , (4.92)
say over the interval δ, R

δ > 0), that ∆ϕ L 2 ( δ 2 ,R 1,ε -δ 100 ) = ∆(η 1,ε -η1,ε ) L 2 ( δ 2 ,R 1,ε -δ 100 ) ≤ Cε -2 3 .
In turn, by interior elliptic regularity theory and (4.77), we deduce that

ϕ H 2 (δ,R1,ε-δ 50 ) ≤ Cε -2 3 .
Hence, from (4.80), via (4.81) with ϕ ′ in place of v, we get that

|ϕ ′ (r)| ≤ C, r ∈ δ, R 1,ε - δ 50 .
Thanks to (3.13) and (4.12), we infer that

u ′ L ∞ (δ,R1,ε-δ 50 ) ≤ C. (4.93) Similarly we have v ′ L ∞ (δ,R1,ε-δ 50 ) ≤ C. (4.94)
Observe that u, v satisfy

M(u, v) ≡   -ε 2 ∆u + g 1 η 1 (η 1 + √ a 1,ε )u + gη 1 (η 2 + √ a 2,ε )v -ε 2 ∆v + g 2 η 2 (η 2 + √ a 2,ε )v + gη 2 (η 1 + √ a 1,ε )u   =   ε 2 ∆ √ a 1,ε ε 2 ∆ √ a 2,ε   , (4.95)
x ∈ B (R1,εδ 50 ) , having dropped some ε subscripts for convenience. By virtue of (1.3), there exists a unique solution (u 0 , v 0 ) to the linear algebraic system

   2g 1 a 1,ε u + 2g √ a 1,ε √ a 2,ε v = ε 2 ∆ √ a 1,ε , 2g √ a 1,ε √ a 2,ε u + 2g 2 a 2,ε v = ε 2 ∆ √ a 2,ε , x ∈ B (R1,ε-δ 50 ) . It follows readily that u 0 C 2 B( R 1,ε -δ 50 ) ≤ Cε 2 and v 0 C 2 B( R 1,ε -δ 50 ) ≤ Cε 2 , (4.96) (keep in mind that a i,ε ≥ c in B (R1,ε-δ 50 ) , i = 1, 2)
. We can write (u, v) = (u 0 , v 0 ) + (ũ, ṽ), (4.97) where ũ, ṽ satisfy

M(ũ, ṽ) =   ε 2 ∆u 0 ε 2 ∆v 0   + g 1 (2a 1,ε -η 2 1 - √ a 1,ε η 1 )u 0 + g(2 √ a 1,ε √ a 2,ε -η 1 η 2 - √ a 2,ε η 1 )v 0 g 2 (2a 2,ε -η 2 2 - √ a 2,ε η 2 )v 0 + g(2 √ a 1,ε √ a 2,ε -η 1 η 2 - √ a 1,ε η 2 )u 0 , (4.98) x ∈ B ( R 1,ε -δ 50 ) . Consider any ρ ∈ 0, R 1,ε -δ 50 .
By testing the above equation by (ũ, ṽ) in the L 2 (B ρ ) × L 2 (B ρ ) sense, making use of (2.4), Proposition 4.8, (4.96), and Young's inequality, it follows readily that (4.99) provided that ε > 0 is sufficiently small. Setting in this relation ρ = R 1,ε -δ 50 , using (4.79), (4.93), (4.94), and (4.96), we obtain that

Bρ ε 2 |∇ũ| 2 + ε 2 |∇ṽ| 2 + ũ2 + ṽ2 dx ≤ Cε 20 3 + Cε 2 |ũ ′ (ρ)ũ(ρ)| + Cε 2 |ṽ ′ (ρ)ṽ(ρ)|,
B ( R 1,ε -δ 50 ) ε 2 |∇ũ| 2 + ε 2 |∇ṽ| 2 + ũ2 + ṽ2 dx ≤ Cε 20 3 + Cε 3 .
Thus, there exists Doing the same procedure with ρ = r 1 , using the above estimates instead of (4.79) when estimating the boundary terms in (4.99), yields that

r 1 ∈ R 1,ε -δ 49 , R 1,ε -δ 50 such that |ũ(r 1 )| + |ṽ(r 1 )| ≤ Cε
Br 1 ε 2 |∇ũ| 2 + ε 2 |∇ṽ| 2 + ũ2 + ṽ2 dx ≤ Cε 20 3 + Cε 4 .
Thus, there exists

r 2 ∈ R 1,ε -δ 48 , r 1 such that |ũ(r 2 )| + |ṽ(r 2 )| ≤ Cε 10 3 + Cε 2 and |ũ ′ (r 2 )| + |ṽ ′ (r 2 )| ≤ Cε.

Iterating this scheme a finite number of times provides us with an r

* ∈ R 1,ε -δ 2 , R 1,ε -δ 3 such that Br * (ũ 2 + ṽ2 )dx ≤ Cε 20 3 . (4.100)
Now, via (4.96), (4.98), and interior elliptic regularity theory, we find that

ũ H 2 B (R 1,ε -δ) ≤ Cε 2 and ṽ H 2 B (R 1,ε -δ) ≤ Cε 2 . (4.101)
By the two-dimensional Agmon inequality [7, Lem. 13.2], we infer that

ũ L ∞ B (R 1,ε -δ) ≤ C ũ 1 2 H 2 B (R 1,ε -δ) ũ 1 2 L 2 B (R 1,ε -δ) (4.100),(4.101) ≤ Cε 3 .
Analogously, we have ṽ

L ∞ B (R 1,ε -δ) ≤ Cε 3 .
The desired estimate (4.91) follows at once from (4.92), (4.96), (4.97), and the above two relations.

In the following proposition, we prove estimates in the intermediate zones, bridging the estimates (4.79) and those provided by Proposition 4.14, on the left side of R i,ε , i = 1, 2.

Proposition 4.15. The following estimates hold:

|η 1,ε (r) -η1,ε (r)| ≤ Cε 2 |r-R 1,ε | -3 2 , |η 2,ε (r) -η2,ε (r)| ≤ Cε 2 |r-R 1,ε | -1 , r ∈ [R 1,ε -δ, R 1,ε -Dε 2 3 ],
and

|η 2,ε (r) -η2,ε (r)| ≤ Cε 2 |r -R 2,ε | -3 2 if r ∈ [R 2,ε -δ, R 2,ε -Dε 2 3
], for some constants C, δ, D > 0 (δ as in Proposition 4.14), provided that ε > 0 is sufficiently small. Proof. We only prove the assertions of the proposition that are related to R 1,ε , since those related to R 2,ε follow analogously and are in fact considerably simpler to verify because η 1,ε is small beyond all orders for r ≥ R 1,ε + δ.

From (4.57) and (4.77), there exists C > 0 and a sequence D j → ∞ such that

(η 2,ε -η2,ε )(R 1,ε -D j ε 2 3 ) ≤ Cε 4 3 , (4.102)
for sufficiently small ε > 0

, j ≥ 1. For r ∈ [R 1,ε -δ, R 1,ε -D j ε 2 3
], we can write

ϕ = η 1,ε -η1,ε = η 1,ε -η1,ε , ψ = η 2,ε -η2,ε = η 2,ε -η2,ε , (4.103)
where ϕ, ψ satisfy (4.63). Let ϕ 0 , ψ 0 be determined from the problems

     -ε 2 ∆ϕ 0 + g 1 -g 2 g 2 (3η 2 1,ε -a 1,ε ) + 2 g 2 g 2 η2 1,ε ϕ 0 = E 1 , r ∈ (R 1,ε -δ, R 1,ε -D j ε 2
3 ),

ϕ 0 (R 1,ε -δ) = ϕ(R 1,ε -δ), ϕ 0 (R 1,ε -D j ε 2 3 ) = ϕ(R 1,ε -D j ε 2 3 ), and  
 -ε 2 ∆ψ 0 + 2g 2 η2 2,ε ψ 0 = N 2 (ϕ, ψ) + E 2 , r ∈ (R 1,ε -δ, R 1,ε -D j ε 2
3 ),

ψ 0 (R 1,ε -δ) = ψ(R 1,ε -δ), ψ 0 (R 1,ε -D j ε 2 3 ) = ψ(R 1,ε -D j ε 2 3 )
,

where E i , N i (•, •), i = 1, 2,
|ϕ 0 (r)| ≤ Cε 2 + Cε exp c r -R 1,ε + D j ε 2 3 ε 2 3 , (4.104)
|ψ 0 (r)| ≤ Cε 2 + Cε 4 |r -R 1,ε | -4 + Cε 4 3 exp c r -R 1,ε + D j ε 2 3 ε , (4.105) if r ∈ [R 1,ε -δ, R 1,ε -D j ε 2 3 ]. We can write ϕ = ϕ 0 + φ, ψ = ψ 0 + ψ, where φ, ψ satisfy            L( φ, ψ) =   Ẽ1 Ẽ2   , r ∈ (R 1,ε -δ, R 1,ε -D j ε 2 3 ), φ(R 1,ε -δ) = ψ(R 1,ε -δ) = 0, φ(R 1,ε -D j ε 2 3 ) = ψ(R 1,ε -D j ε 2 3 ) = 0, (4.106)
with L as in (4.42), for some functions Ẽi , i = 1, 2, satisfying the following pointwise estimates:

| Ẽ1 | ≤ C|η 1,ε ψ 0 | + |N 1 (ϕ, ψ)| via (4.46), (4.64), (4.79), (4.105) ≤ Cε 2 + Cε 4 |r -R 1,ε | -7 2 + C j ε 5 3 exp c r-R 1,ε +D j ε 2 3 ε , (4.107)
| Ẽ2 | ≤ C|η 1,ε ϕ 0 | (4.46),(4.104) ≤ Cε 2 + C j ε 4 3 exp c r -R 1,ε + D j ε 2 3 ε 2 3 , (4.108) for r ∈ [R 1,ε -δ, R 1,ε -D j ε 2 3
]. Our plan is to solve the second equation in (4.106) for ψ and substitute into the first, thus reducing the system to one scalar equation for φ. Then, we derive estimates for φ in some carefully chosen weighted norms that we define afterwards. Let

I = (R 1,ε -δ, R 1,ε -D j ε 2 3 ) and ρ(r) = R 1,ε -r, r ∈ I, for φ ∈ C 2 (I), we define φ * = ε 2 ρ 1 2 φ rr L ∞ (I) + ε 2 ρ -1 2 φ r L ∞ (I) + ρ 3 2 φ L ∞ (I) , (4.109)
(for related weighted norms we refer the interested reader to the monograph [START_REF] Pacard | Linear and nonlinear aspects of vortices[END_REF] and the references therein). In particular, we rely on the following a-priori estimate: there exist

ε 0 , j 0 , K > 0 such that if h ∈ C( Ī) and φ ∈ C 2 r |x| ∈ Ī satisfy -ε 2 ∆φ + g 1 - g 2 g 2 (3η 2 1,ε -a 1,ε )φ = h in I; φ = 0 on ∂I, (4.110)
with 0 < ε < ε 0 , j ≥ j 0 , then

φ * ≤ K ρ 1 2 h L ∞ (I) .
(4.111) We stress that the above constant K is also independent of δ (i.e. ε 0 = ε 0 (δ)). In the remainder of this proof, we denote by k/K a small/large generic constant that is independent of large j and small δ, ε. The proof of this estimate proceeds in two steps. Firstly, similarly to [27, Prop. 3.5], using the following consequence of (4.17)

k|r -R 1,ε | ≤ 3η 2 1,ε -a 1,ε ≤ K|r -R 1,ε |, r ∈ I, (4.112)
and the maximum principle (in the equation for ρ 3 2 φ), one obtains the partial estimate

ρ 3 2 φ L ∞ (I) ≤ K ρ 1 2 h L ∞ (I) . (4.113)
Then, the full a-priori estimate follows by going back to the equation for φ and using the upper bound in (4.112). The details are given in Appendix B. From now on, we fix such a large j and drop the subscript from D j . In view of the second row in (4.42) and (4.106), we can write

ψ = - g g 2 η1,ε η2,ε φ + w + z, r ∈ I, (4.114)
where

-ε 2 ∆w + 2g 2 η2 2,ε w = -ε 2 g g 2 ∆ η1,ε η2,ε φ in I; w = 0 on ∂I, (4.115)
and

-ε 2 ∆z + 2g 2 η2 2,ε z = Ẽ2 in I; z = 0 on ∂I. (4.116)
Using the pointwise estimates

   0 < η1,ε ≤ Kρ 1 2 , |∇η 1,ε | ≤ Kρ -1 2 , |∆η 1,ε | ≤ Kρ -3 2 , k ≤ η2,ε ≤ K, |∇η 2,ε | ≤ K, |∆η 2,ε | ≤ K + Kε 2 ρ -4 ,
for r ∈ I, which follow readily from (1.33a), (4.17) (having increased j if needed), (4.21), (4.30), (4.33), (4.34), and (4.46), we can bound pointwise the right-hand side of (4.115) as

ε 2 g g 2 ∆ η1,ε η2,ε φ ≤ Kε 2 ρ 1 2 |∆ φ| + ρ -1 2 |∇ φ| + ρ -3 2 | φ| ≤ K ε 2 ρ 1 2 | φrr | + ε 2 ρ -1 2 | φr | + ρ 3 2 | φ| = K φ * .
Hence, by the maximum principle, we deduce that

w L ∞ (I) ≤ K φ * . (4.117)
On the other side, from (4.108), (4.116) and a standard comparison argument, it follows that

|z(r)| ≤ Cε 2 + Cε 4 3 exp c r -R 1,ε + Dε 2 3 ε 2 3 , r ∈ I. (4.118)
Substituting (4.114) into the first equation of (4.106), recalling (4.42), we arrive at

-ε 2 ∆ φ + g 1 - g 2 g 2 (3η 2 1,ε -a 1,ε ) φ = Ẽ1 -2g η1,ε η2,ε (w + z) in I; φ = 0 on ∂I.
Making use of the a-priori estimate (4.111), bound (4.117), and the easy estimates

ρ 1 2 Ẽ1 L ∞ (I) ≤ Cε 2 (recall (4.107)), ρ 1 2 η1,ε z L ∞ (I) ≤ Cε 2 (recall (4.46), (4.118)), we obtain that φ * ≤ Cε 2 + δK φ * ,
where we also exploited that 0 < η1,ε ≤ Kρ

1 2 ≤ Kδ 1 2 in I.
Consequently, choosing a sufficiently small δ, and fixing it from now on, we infer that

φ * ≤ Cε 2 .
In particular, for small ε, we have that

| φ(r)| ≤ Cε 2 |r -R 1,ε | -3 2 , r ∈ [R 1,ε -δ, R 1,ε -Dε 2 3
]. In turn, from the second equation in (4.106), recalling (4.46) and (4.108), via a standard barrier argument, we find that

ψ(r) ≤ Cε 2 |r -R 1,ε | -1 , r ∈ [R 1,ε -δ, R 1,ε -Dε 2 3 ].
The desired assertion of the proposition now follows at once from (4.103), (4.104), (4.105), and the above two relations. Lemma 4.16. Given D > 0, we have that

|η 2,ε (r) -η2,ε (r)| ≤ Cε 4 3 , |r -R 1,ε | ≤ Dε 2 3
, provided that ε > 0 is sufficiently small. Proof. As in the beginning of the proof of Proposition 4.15, given D > 0, if ε > 0 is sufficiently small, there exist

r -∈ R 1,ε -(D + 1)ε 2 3 , R 1,ε -Dε 2 3 and r + ∈ R 1,ε + Dε 2 3 , R 1,ε + (D + 1)ε 2 3 such that |ψ(r ± )| ≤ Cε 4 3
, where ψ = η 2,ε -η2,ε . Keeping in mind the proof of Proposition 4.3, (4.42), (4.46) and (4.79)), it follows readily from the second equation of (4.63) that ψ satisfies

-ε 2 ∆ψ + 2g 2 η2 2,ε ψ = O(ε 4
3 ), uniformly on [r -, r + ], as ε → 0. The assertion of the lemma follows directly from the above two relations and the maximum principle, since η2,ε ≥ c in this region. Lemma 4.17. If ε > 0 is sufficiently small, we have

|η 1,ε (r) -η1,ε (r)| ≤ Cε exp c R 1,ε -r ε 2 3 , R 1,ε ≤ r ≤ R 1,ε + δ, |η 2,ε (r) -η2,ε (r)| ≤ Cε 2 + Cε 4 3 exp c R 1,ε -r ε 2 3 , R 1,ε ≤ r ≤ R 1,ε + δ, and
|η 2,ε (r) -η2,ε (r)| ≤ Cε exp c R 2,ε -r ε 2 3 , R 2,ε ≤ r ≤ R 2,ε + δ.
Proof. We only prove the estimates that concern R 1,ε because those concerning R 2,ε follow analogously. As in the proof of Proposition 4.15, let ϕ = η 1,ε -η1,ε and ψ = η 2,ε -η2,ε . In view of (4.42), (4.63), (4.64), the relations

E 1 = 0, |E 2 | ≤ Cε 4 3 exp c R 1,ε -r ε 2 3 +Cε 2 , r ∈ [R 1,ε , R
     -ε 2 ∆ϕ + p(r)ϕ = O(η 1,ε ψ), -ε 2 ∆ψ + q(r)ψ = O(ε 2 ) + O(ε 4 3 ) exp c R 1,ε -r ε 2 3 , (4.119)
uniformly on [R 1,ε , R 1,ε + 3δ], as ε → 0, for some smooth functions p, q satisfying p ≥ cε

2 3
and q ≥ c (recall (4.45) and (4.48)). (4.120) Note that, from (4.79) and Lemma 4.16, we have

   ϕ(R 1,ε ) = O(ε), ϕ(R 1,ε + 2δ) = O(ε), ψ(R 1,ε ) = O(ε 4 3 ), ψ(R 1,ε + 3δ) = O(ε), (4.121)
as ε → 0. A standard barrier argument yields that

|ψ(r)| ≤ Cε 4 3 exp c R 1,ε -r ε 2 3 + Cε 2 + Cε exp c r -R 1,ε -3δ ε , r ∈ [R 1,ε , R 1,ε + 3δ],
which implies that

|ψ(r)| ≤ Cε 4 3 exp c R 1,ε -r ε 2 3 + Cε 2 , r ∈ [R 1,ε , R 1,ε + 2δ],
provided that ε > 0 is sufficiently small, as asserted. Now, via (4.119) and (4.13), we arrive at

-ε 2 ∆ϕ + p(r)ϕ = O(ε 5 3 ) exp c R 1,ε -r ε 2 3 , uniformly on [R 1,ε , R 1,ε + 2δ],
as ε → 0. Keeping in mind (4.120) and (4.121), a standard barrier argument yields that

|ϕ(r)| ≤ Cε exp c R 1,ε -r ε 2 3 + Cε exp c r -R 1,ε -2δ ε 2 3 , r ∈ [R 1,ε , R 1,ε + 2δ],
which implies that

|ϕ(r)| ≤ 2Cε exp c R 1,ε -r ε 2 3 , r ∈ [R 1,ε , R 1,ε + δ],
as asserted.

4.7.

Improved estimate for the Lagrange multipliers. In the sequel, building on Proposition 3.3, via the results of the previous subsection, we are able to considerably improve the estimate for λ i,ελ i,0 of the aforementioned proposition.

Proposition 4.18. If ε > 0 is sufficiently small, we have

|λ i,ε -λ i,0 | ≤ C| log ε|ε 2 , i = 1, 2.
Proof. Motivated by the proof of the corresponding estimate for the scalar equation, as given in [27, Thm. 1.1], we first show that

R 2 (η 2 1,ε -a + 1,ε )dx = O(| log ε|ε 2 ), (4.122)
B R 1,ε (η 2 2,ε -a 2,ε )dx + R 2 \B R 1,ε η 2 2,ε -a 2,ε + g g 2 a 1,ε + dx = O(| log ε|ε 2 ), (4.123)
as ε → 0, and then exploit that

R 2 η 2 i,ε dx = R 2 a i dx = 1, i = 1, 2. ( 4
η 2 2,ε -a 2,ε = η2 2,ε -a 2,ε + O (ε 2 |r -R 1,ε | -1 ) = g g 2 (a 1,ε -η2 1,ε ) + O (ε 2 |r -R 1,ε | -1 ) , uniformly on [R 1,ε -δ, R 1,ε -Dε 2 3
], as ε → 0. Analogously, making use of Lemma 4.16, we see that

η 2 2,ε -a 2,ε = g g 2 (a 1,ε -η2 1,ε ) + O(ε 4 3 ), uniformly on [R 1,ε -Dε 2 3 , R 1,ε + Dε 2 3
], as ε → 0. Hence, via Proposition 4.14, we find that

B R 1,ε (η 2 2,ε -a 2,ε )dx = g g 2 B R 1,ε (a 1,ε -η2 1,ε )dx + O(| log ε|ε 2 ) (4.12) = g g 2 R 1,ε -δ<|x|<R 1,ε (a 1,ε -η2 1,ε )dx + O(| log ε|ε 2 ), (4.125)
as ε → 0. Similarly, keeping in mind (4.13), we have ], we find that

R 1,ε <|x|<R 2,ε -δ η 2 2,ε -a 2,ε - g g 2 a 1,ε dx = - g g 2 R 1,ε <|x|<R 1,ε +δ η2 1,ε dx + O(ε 2 ), ( 4
η 2 2,ε -a 2,ε + g g 2 a 1,ε = η2 2,ε -a 2,ε + g g 2 a 1,ε + 2η 2,ε (η 2,ε -η2,ε ) + (η 2,ε -η2,ε ) 2 = η2 2,ε -a 2,ε + g g 2 a 1,ε + O (ε 2 |r -R 2,ε | -1
) , uniformly, as ε → 0. Analogously, making use of (4.12) and (4.79), we see that

η 2 2,ε -a 2,ε + g g 2 a 1,ε = η2 2,ε -a 2,ε + g g 2 a 1,ε + O(ε 4 3 ), uniformly on [R 2,ε -Dε 2 3 , R 2,ε + Dε 2 3
], as ε → 0. Thus, we get that

R 2,ε -δ<|x|<R 2,ε η 2 2,ε -a 2,ε - g g 2 a 1,ε dx = R 2,ε -δ<|x|<R 2,ε η2 2,ε -a 2,ε - g g 2 a 1,ε dx+O(| log ε|ε 2 ), ( 4
||x|-R 1,ε |<δ (η 2 1,ε -a + 1,ε )dx = O(| log ε|ε 2 ), ||x|-R 2,ε |<δ η2 2,ε -a 2,ε + g g 2 a 1,ε + dx = O(| log ε|ε 2 ),
as ε → 0, which follow from the proof of Theorem 1.1 in [START_REF] Karali | The ground state of a Gross-Pitaevskii energy with general potential in the Thomas-Fermi limit[END_REF]. The proof of relation (4.123) is complete. By virtue of (3.14), increasing the value of D, if needed, we may assume that |R i,ε -R i,0 | ≤ D| log ε|

2 ε (a 1,ε -a 1,0 )dx = O(| log ε|ε 2 ), |x|<R 1,0 -D| log ε| 1 2 ε (a 2,ε -a 2,0 )dx+ + R 1,0 +D| log ε| 1 2 ε<|x|<R 2,0 -D| log ε| 1 2 ε (a 2,ε -a 2,0 ) + g g 2 (a 1,ε -a 1,0 ) dx = O(| log ε|ε 2 ),
as ε → 0. In view of (1.30), (1.31), and (3.13), this leads to the following system:

(λ 1,ε -λ 1,0 ) - g g 2 (λ 2,ε -λ 2,0 ) = O(| log ε|ε 2 ), 1 Γ (λ 2,ε -λ 2,0 ) - g g 1 (λ 1,ε -λ 1,0 ) λ 1,ε - g g 2 λ 2,ε + (λ 2,ε -λ 1,ε )(λ 2,ε -λ 2,0 ) = O(| log ε|ε 2 ),
as ε → 0. Now, recalling that g < g 2 , the assertion of the proposition follows straightforwardly.

4.8. Proof of Theorem 4.1.

Proof. Let (η 1 , η 2 ) be the unique positive minimizer of E 0 ε in H provided by Theorem 1.3 [START_REF] Aftalion | Giant vortex and the breakdown of strong pinning in a rotating Bose-Einstein condensate[END_REF]. We saw in Proposition 3.3 that the associated Lagrange multipliers λ i,ε satisfy |λ i,ελ i,0 | ≤ ε| log ε| 1/2 , i = 1, 2. In view of (4.79) and Proposition 4.12, the solution (η 1,ε , η 2,ε ) that is provided by Proposition 4.8 also fashions a positive radial solution of the system (1.5), with the same Lagrange multipliers λ i,ε . Therefore, by Theorem 1.

Estimates for the annulus case

In this section, we explain how to extend the previous section to prove Theorem 1.6.

Construction of an approximate solution.

5.1.1. Outer approximations. As before, we work with the equivalent problem (1.29), where a 1,ε , a 2,ε are the same as in (1.30), and λ 1,ε , λ 2,ε are provided by Theorem 1.3 in the case of (1.9). This time, the problem with both diffusion terms neglected has a unique continuous, nonnegative solution given by

η 1 = a 1,ε + g g 1 a 2,ε 1 2 , η 2 = 0, 0 ≤ r ≤ R - 2,ε , η 1 = a 1 2 1,ε , η 2 = a 1 2 2,ε , R - 2,ε ≤ r ≤ R 1,ε , η 1 = 0, η 2 = a 2,ε + g g 2 a 1,ε 1 2 , R 1,ε ≤ r ≤ R + 2,ε , η 1 = 0, η 2 = 0, r ≥ R + 2,ε , where (R - 2,ε ) 2 = 1 Γ 1 λ 2,ε - g g 1 λ 1,ε , (R + 2,ε ) 2 = λ 2,ε , and
R 2 1,ε = 1 Γ 2 λ 1,ε - g g 2 λ 2,ε .
In view of (3.13), and Remark 3.4, we have that

|R 1,ε -R 1,0 | + |R ± 2,ε -R ± 2,0 | ≤ C| log ε| 1 2 ε. ( 5
point R - 2,0 < R 1,0 < R + 2,0 . On [0, R 1,0 -δ],
where √ a 1 is away from zero and has bounded gradient, we neglect only the term ε 2 ∆η 1,ε from (1.29), and get the following problem:

   g 1 η 1 (η 2 1 -a 1,ε (r)) + gη 1 (η 2 2 -a 2,ε (r)) = 0, -ε 2 ∆η 2 + g 2 η 2 (η 2 2 -a 2,ε (r)) + gη 2 (η 2 1 -a 1,ε (r)) = 0.
From the first equation, we find that

η 2 1 = a 1,ε + g g 1 (a 2,ε -η 2 2 ). (5.2)
Then, from the second equation, we obtain that

-ε 2 ∆η 2 + g 2 - g 2 g 1 η 2 (η 2 2 -a 2,ε ) = 0.
The function a 2,ε is negative in [0, R - 2,ε ) and positive in (R - 2,ε , ∞). We consider a function A 2,ε which coincides with a 2,ε on [0, R 1,0 + δ], changes sign once in (R 1,0 + δ, ∞), and diverges to -∞ as r → ∞. We then take as an approximation for η 2 on [0, R 1,0δ] the restriction of the unique positive solution η-2,ε of the problem

ε 2 ∆η = g 2 - g 2 g 1 η η 2 -A 2,ε (r) in R 2 , η → 0 as r → ∞.
The properties of η-2,ε which we require are contained in Appendix A. Accordingly, we take as an approximation for η 1 on [0, R 1,0δ] the one given by (5.2) with η-2,ε in place of η 2 . The approximations for η 1,ε and η 2,ε on [R 1,0 + δ, R + 2,0δ] are the same ones as in the case of two disks, namely those given by (1.33a) and (4.21) respectively. Analogously, if r ≥ R 1,0 + δ, we take as an approximation for η 1 the trivial solution, while for η 2 the unique positive solution of the problem (1.33b) which we now call η+ 2,ε .

5.1.3. Gluing approximate solutions. Let

r ε = R - 2,ε + R 1,ε 2 , R ε = R 1,ε + R + 2,ε 2 .
Analogously to Subsection 4.2, we can define a smooth global approximate solution (η 1,ε , η2,ε ) such that

η1,ε =                a 1,ε + g g 1 a 2,ε -g g 1 (η - 2,ε ) 2 1 2 , 0 ≤ r ≤ r ε , a 1 2 1,ε + O C 2 (ε 2 ), r ε ≤ r ≤ r ε + δ, η1,ε , r ε + δ ≤ r, (5.3) η2,ε =                                  η- 2,ε , 0 ≤ r ≤ r ε , a 1 2 2,ε + O C 2 (ε 2 ), r ε ≤ r ≤ r ε + δ, a 2,ε + g g 2 a 1,ε -g g 2 η2 1,ε 1 2 , r ε + δ ≤ r ≤ R ε , a 2,ε + g g 2 a 1,ε 1 2 + O C 2 (ε 2 ), R ε ≤ r ≤ R ε + δ, η+ 2,ε , R ε + δ ≤ r.
(5.4) 5.2. Estimates for the error on the approximate solution. The remainder E(η 1,ε , η2,ε ) that is left when substituting the approximate solution (η 1,ε , η2,ε ) to the system (1.29) is as in (4.25). For convenience, we set

A ε = {x ∈ R 2 : r ε < |x| < R ε }. (5.5)
Analogously to Proposition 4.3 for the case of two disks, we have Proposition 5.1. The following estimates hold for small ε > 0:

E 1 L 2 (Aε) ≤ Cε 2 , E 2 L 2 (Aε) ≤ Cε 5 3 ,
and

E 1 L 2 (R 2 \Aε) ≤ Cε 5 3 , E 2 L 2 (R 2 \Aε) ≤ Cε 2 .
5.3. Linear analysis. In the sequel, we consider the linearization of (1.29) about the approximate solution (η 1,ε , η2,ε ), namely the linear operator that is given by (4.41) for this choice of (η 1,ε , η2,ε ). As in the case of two disks, using that

3η 2 2,ε -a 2,ε ≥    c max{ε 2 3 , η2 2,ε }, |r -R - 2,ε | ≤ δ, c, r ∈ [0, R - 2,ε -δ] ∪ [R - 2,ε + δ, r ε ], 3η 2 1,ε -a 1,ε ≥    c max{ε 2 3 , η2 1,ε }, |r -R 1,ε | ≤ δ, c, r ∈ [r ε , R 1,ε -δ] ∪ [R 1,ε + δ, R ε ],
and

3η 2 2,ε -a 2,ε - g g 2 a 1,ε ≥    c max{ε 2 3 , η2 2,ε }, |r -R + 2,ε | ≤ δ, c, r ∈ [R ε , R + 2,ε -δ] ∪ [R + 2,ε + δ, ∞)
, we can establish an analog of Proposition 4.6.

Proposition 5.2. The assertions of Proposition 4.6 are valid, provided that in (4.56) and in the definition of the |||•|||-norm in (4.57), B Rε is replaced by A ε defined in (5.5).

5.4.

Existence and properties of a positive solution to the system (1.29). As in Subsection 4.5, using the properties of the linearized operator that we discussed above, we construct a positive, radial solution (η 1,ε , η 2,ε ) to (1.29), near the approximate one (η 1,ε , η2,ε ), for small ε > 0. As before, the first part of the uniqueness Theorem 1.3 guarantees that this solution is the desired minimizer.

Using the |||•|||-norm, as redefined in Proposition 5.2, we can show that Propositions 4.7, 4.8 and Corollary 4.9 remain unchanged. We still denote the corresponding solution by (η 1,ε , η 2,ε ). The assertion of the Lemma 4.10 also remains the same. The only difference in the proof is that, say in the equation for v n , we rearrange the terms differently, namely write

-∆v n + g 1 v 3 n -[g 1 a 1,εn (x n + ε n µ n y) + ga 2,εn (x n + ε n µ n y)] µ 2 n v n + gµ 2 n η 2 2,εn (x n + ε n µ n y)v n = 0, with η 2,ε L p (B δ ) ≤ C p ε 2 3 + 4 3p , p ≥ 2.
Then, the analog of Corollary 4.11 is

η 1,ε -a 1,ε + g g 1 a 2,ε L ∞ (B δ ) + η 2,ε L ∞ (B δ ) ≤ Cε 1 3 .
The positivity of the constructed solution, namely the analog of Proposition 4.12, requires some additional considerations, since η 2,ε is also small in the disk |x| < R - 2,ε : Proposition 5.3. If ε > 0 is sufficiently small, the constructed solutions satisfy

η i,ε > 0 in R 2 , i = 1, 2.
Proof. The main difference with the previous case is in the domain |x| < R - 2,ε , which we describe below.

We know that

η 1,ε = λ 1,ε -r 2 g 1 + O(ε 2
3 ), r ∈ [0, R - 2,ε -Dε 2 3 ], (5.6)
where O(ε

3 ) in dependent of D > 1 (this follows directly from the analog of Proposition 4.14 or from the analogs of (4.22) and Corollary 4.9), and

η 2,ε (R - 2,ε -Dε 2 3 ) ≥ cε 1 3 > 0. The function η 2,ε satisfies the elliptic equation -ε 2 ∆η 2,ε + (r 2 + g 2 η 2 2,ε + gη 2 1,ε -λ 2,ε )η 2,ε = 0.
In view of the above, the desired positivity of η 2,ε follows directly from the maximum principle once we show that

r 2 + g 2 η 2 2,ε + gη 2 1,ε -λ 2,ε > 0, r ∈ [0, R - 2,ε -Dε
Note that, thanks to (5.6), the left-hand side equals

Γ 1 r 2 + g g 1 λ 1,ε -λ 2,ε + g 2 η 2 2,ε + O(ε 2
3 ),

where O(ε

3 ) in dependent of D > 1. In view of (5.1), it suffices to show that

Γ 1 r 2 + g g 1 λ 1,0 -λ 2,0 ≥ cDε 2 3 , r ∈ [0, R - 2,0 -Dε 2 3 ], (5.7)
for some constant c > 0 that is independent of D, ε, provided that D is sufficiently large and ε sufficiently small. Observe that, since r ≤ R - 2,0 -Dε 2 3 , we have

r 2 ≤ (R - 2,ε ) 2 + D 2 ε 4 3 -2DR - 2,0 ε 2 3 ≤ (R - 2,ε ) 2 -DR - 2,0 ε 2 3 , provided that ε ≤ ε(D). Now, recalling that Γ 1 < 0,
we can bound the left-hand side of (5.7) from below by

Γ 1 (R - 2,0 ) 2 + cDε 2 3 + g g 1 λ 1,0 -λ 2,0 . (5.8)
We use (1.26) to find that the quantity (5.8) equals cDε

2 3 with c = -Γ 1 R - 2,0 > 0.
5.5. Proof of Theorem 1.6. The proof for the case where (1.9) holds, instead of (1.8), proceeds along the same lines as the proof of Theorem 1.4. This time, we have to decompose [0, ∞) into four intervals with boundary points R - 2,0 < R 1,0 < R + 2,0 . We point out that the reduced problem near R - 2,0 is a scalar equation of the form (A.1) where a(r) < 0 for r ∈ [0, R - 2,0 ), a(R - 2,0 ) = 0, and a(r 2 ) = 0 for some r 2 > R - 2,0 , which is covered in Theorem A.1.

The auxiliary functions

F 1,ε , F 2,ε
Assume that (1.3), (1.8) and (1.18) hold. In this section, we consider the auxiliary functions

F i,ε (r) = ξ i,ε (r) η 2 i,ε (r) , with ξ i,ε (r) = ∞ r sη 2 i,ε (s)ds, r ≥ 0, i = 1, 2, (6.1)
which will play an important role when analyzing the energy with rotation. In particular, we will link them to the limiting functions

F i,0 (r) =    ξ i,0 (r) a i (r) , 0 ≤ r < R i,0 , 0, otherwise, with ξ i,0 (r) = ∞ r sa i (s)ds, (6.2)
where a i is as in (1.12). Note that F i,0 is bounded in R 2 since a i > 0 for r < R i,0 , as observed in (3.6), (3.7), and F i,0 (R i,0 ) = 0. Note also that F i,0 is merely continuous at R i,0 , as F ′ i,0 has a finite jump discontinuity across that point.

This section is devoted to proving the following.

Proof. From now on, let us fix a large D > 1 such that Lemma 4.13 is valid. The latter lemma, similarly to [3, Lem. 2.2], implies that

0 < F 1,ε (r) ≤ Cε 2 3 , r ≥ R 1,ε + Dε 2 3 . (6.8)
Since the above estimate also holds for f 1,ε , by virtue of Lemma 6.2, we infer that (6.6) is valid for r ≥ R 1,ε + Dε

2 3 . If r ≤ R 1,ε + Dε 2 3
, via (4.12), (4.15), and Corollaries 4.9, 4.11, we have

η 2 1,ε (R 1,ε + Dε 2 3 ) η 2 1,ε (r) ≤ C.
Therefore, we can write

F 1,ε (r) = 1 η 2 1,ε (r) R 1,ε +Dε 2 3 r sη 2 1,ε (s)ds + η 2 1,ε (R 1,ε +Dε 2 3 ) η 2 1,ε (r) F 1,ε (R 1,ε + Dε 2 3 ) (6.8) = 1 η 2 1,ε (r) R 1,ε +Dε 2 3 r sη 2 1,ε (s)ds + O(ε 2
3 ), (6.9)

uniformly in r ≥ 0, as ε → 0. After rearranging terms, we find that

F 1,ε (r) -f 1,ε (r) = 1 η 2 1,ε (r) R 1,ε +Dε 2 3 r s η 2 1,ε (s) -η2 1,ε (s) ds + η2 1,ε (r)-η 2 1,ε (r) η 2 1,ε (r)η 2 1,ε (r) R 1,ε +Dε 2 3 r sη 2 1,ε (s)ds+ +O(ε 2 3 ), uniformly in r ≤ R 1,ε + Dε 2 3
, as ε → 0. Since on this interval we can set ϕ = η 1,ε -η1,ε = η 1,ε -η1,ε , we obtain that

F 1,ε (r) -f 1,ε (r) = 1 η 2 1,ε (r) R 1,ε +Dε 2 3 r s(ϕ 2 + 2η 1,ε ϕ)ds- - ϕ 2 (r) η 2 1,ε (r)η 2 1,ε (r) + 2ϕ(r) η 2 1,ε (r)η 1,ε (r) R 1,ε +Dε 2 3 r sη 2 1,ε (s)ds + O(ε 2
3 ), (6.10)

uniformly in r ≤ R 1,ε + Dε 2 3
, as ε → 0. The above terms can be estimated by first decomposing the interval [0, R 1,ε + Dε

3 ] as [0, R 1,ε -δ] ∪ [R 1,ε -δ, R 1,ε -ε 1 3 ] ∪ [R 1,ε -ε 1 3 , R 1,ε + Dε 2 3 ]
(with δ > 0 fixed small), then making use of the uniform estimates in (4.79) and Proposition 4.14 for ϕ, and those in (4.15) and (4.17) for η1,ε . To illustrate the procedure, let us estimate in detail the term 6 , using (4.79) to bound ϕ, we deduce that

1 η 2 1,ε (r) R 1,ε +Dε 2 3 r sη 1,ε (s)ϕ(s)ds. If R 1,ε -ε 1 3 ≤ r ≤ s ≤ R 1,ε + Dε
1 η 2 1,ε (r) R 1,ε +Dε 2 3 r sη 1,ε (s)ϕ(s)ds ≤ Cε -2 3 (R 1,ε + Dε 2 3 -r)ε 1 6 ε ≤ Cε -2 3 ε 1 3 ε 1 6 ε = Cε 2 3 + 1 6 .
where f i,0 are as in (6.4). In view of Proposition 6.3 and (6.5), we infer that the assertion of the lemma is valid for i = 1 and that there exists some C > 0 such that

|F 2,ε (r) -F 2,0 (r)| ≤ Cε 1 3 , r ≥ R 1,ε + ε 2 3 , (6.12)
(recall also (3.14)). So, for the proof to be completed, it remains to show that there exists some C > 0 such that

|F 2,ε (r) -F 2,0 (r)| ≤ Cε 1 3 , 0 ≤ r ≤ R 1,ε + ε 2 3 . (6.13)
To this end, for 0 3 ). Now, estimate (6.13) follows readily from (6.12) and the property that

≤ r ≤ R 1,ε + ε 2 3 , we write F 2,ε (r) = 1 η 2 2,ε (r) R 1,ε +ε 2 3 r sη 2 2,ε (s)ds + η 2 2,ε (R 1,ε + ε 2 3 ) η 2 2,ε (r) F 2,ε (R 1,ε + ε 2 3 ), and
F 2,0 (r) = 1 η 2 2,0 (r) R 1,ε +ε 2 3 r sη 2 2,0 (s)ds + η 2 2,0 (R 1,ε + ε 2 3 ) η 2 2,0 (r) F 2,0 (R 1,ε + ε 2
|η 2,ε (r) -η 2,0 (r)| ≤ Cε 2 3 , 0 ≤ r ≤ R 1,ε + ε 2 3 .
The latter estimate is a consequence of (4.20)-(4.21), the fact that η2

1,ε -a + 1,ε L ∞ (R 2 ) ≤ Cε 2 3
(see Proposition 4.2), Corollary 4.9 and Proposition 4.14.

Finally, we have another estimate which will be used later.

Lemma 6.6. There exists C > 0 such that ∇ξ i,ε L ∞ (R 2 ) ≤ C, i = 1, 2.

Proof. We have

ξ ′ i,ε (r) = -rη 2 i,ε (r) (6.14)
so that the result follows from Lemma 2.2.

The energy minimizer with rotation

In this section, we study the behavior of the minimizers of the energy functional E Ω ε in the space H, defined in (1.1) and (1.2) respectively, as ε → 0. In the following we assume Ω ≤ C| log ε|,

for some constant C independent of ε. Any minimizer (u 1 , u

2 ) = (u 1,ε , u 2,ε ) of E Ω ε in H solves the following system -ε 2 ∆u 1 + u 1 (|x| 2 + g 1 |u 1 | 2 + g|u 2 | 2 ) + 2ε 2 iΩx ⊥ • ∇u 1 = µ 1,ε u 1 in R 2 , -ε 2 ∆u 2 + u 2 (|x| 2 + g 2 |u 2 | 2 + g|u 1 | 2 ) + 2ε 2 iΩx ⊥ • ∇u 2 = µ 2,ε u 2 in R 2 ,
for some Lagrange multipliers µ 1,ε , µ 2,ε . The existence of a minimizer when Ω satisfies (7.1) is a consequence of Lemma 7.1 below and of the compactness induced by the fact that the harmonic potential |x| 2 diverges as |x| → ∞. Lemma 7.1. We have

Ω 2 j=1 R 2 x ⊥ • (iu j , ∇u j ) dx ≤ 2 j=1 R 2 |∇u j | 2 4 dx + 2Ω 2 (R 2 1,0 + R 2 2,0 )+ 2Ω 2 g 1 Γ Γ 2 R 2 \D 1 a - 1,0 |u 1 | 2 dx + 2Ω 2 g 2 R 2 \D 2 a 2,0 + g g 2 a 1,0 - |u 2 | 2 dx. Proof. We have Ω R 2 x ⊥ • (iu i , ∇u i ) dx ≤ R 2 |∇u i | 2 4 + Ω 2 |x| 2 |u i | 2 dx.
We need to estimate the second term in the right hand side. Let us start with i = 1. Notice that

|x| 2 ≤ - 2g 1 Γ Γ 2 a 1,0 (x) = 2g 1 Γ Γ 2 a - 1,0 (x) for |x| ≥ √ 2R 1,0 . This implies R 2 |x| 2 |u 1 | 2 dx = {|x|≤ √ 2R 1,0 } |x| 2 |u 1 | 2 dx + {|x|> √ 2R 1,0 } |x| 2 |u 1 | 2 dx ≤ 2R 2 1,0 + 2g 1 Γ Γ 2 R 2 \D 1 a - 1,0 |u 1 | 2 dx.
Similarly, for i = 2, we have

|x| 2 ≤ 2g 2 a 2,0 + g g 2 a 1,0 - (x) for |x| ≥ √ 2R 2,0 , so that R 2 |x| 2 |u 2 | 2 dx ≤ 2R 2 2,0 + 2g 2 R 2 \D 2 a 2,0 + g g 2 a 1,0 - |u 2 | 2 dx,
which concludes the proof of the lemma.

In analogy to Proposition 3.2, we have the following lemma.

Lemma 7.2. Let (u 1 , u 2 ) be a minimizer of E Ω ε in H. There exists C > 0 independent of ε such that, for i = 1, 2, we have

R 2 |∇u i | 2 dx ≤ C| log ε| 2 , R 2 |u i | 2 -a i 2 dx ≤ Cε 2 | log ε| 2 , R 2 \D 1 |u 1 | 2 a - 1,0 dx + R 2 \D 2 (|u 1 | 2 + |u 2 | 2 ) a 2,0 + g g 2 a 1,0 - dx ≤ Cε 2 | log ε| 2 .
Proof. On the one hand, by the definition of minimizers, by Lemma 3.1 and Proposition 3.2, we have

E Ω ε (u 1 , u 2 ) ≤ E Ω ε (η 1 , η 2 ) = E 0 ε (η 1 , η 2 ) ≤ C| log ε| + K, (7.2)
with K as in Lemma 3.1. On the other hand, we have

E Ω ε (u 1 , u 2 ) = Ẽ0 ε (u 1 , u 2 ) + K -Ω 2 j=1 R 2 x ⊥ • (iu j , ∇u j ) dx.
The right hand side can be bounded from below by means of (2.4) and of Lemma 7.1 as follows:

E Ω ε (u 1 , u 2 ) ≥ 2 i=1 R 2 |∇u i | 2 4 + γ 4ε 2 (|u i | 2 -a i ) 2 dx +g 1 Γ 1 2ε 2 - 2Ω 2 Γ 2 R 2 \D 1 a - 1,0 |u 1 | 2 dx + g 2ε 2 R 2 \D 2 |u 1 | 2 a 2,0 + g g 2 a 1,0 - dx +g 2 1 2ε 2 -2Ω 2 R 2 \D 2 a 2,0 + g g 2 a 1,0 - |u 2 | 2 dx -2Ω 2 (R 2 1,0 + R 2 2,0 ) + K.
The result follows by combining the last inequality with (7.2) and by using (7.1).

In analogy to Lemma 2.1, we have the following Lemma 7.3. Let (u 1 , u 2 ) be a minimizer of E Ω ε in H. For ε sufficiently small, we have

|u i | 2 ≤ µ i,ε /g i , ∇u i L ∞ (R 2 ) ≤ C √ µ i,ε (µ i,ε + µ j,ε + 1) + C ε for some C > 0, i = 1, 2, j = i.
Proof. For ε sufficiently small, the following holds

∆(|u j | 2 ) = 2|∇u j | 2 + 2(u j , ∆u j ) ≥ 2 ε 2 |u j | 2 (g j |u j | 2 -|µ j,ε |)
, where we use -2Ωx ⊥ •(iu j , ∇u j ) ≥ -Ω 2 |x| 2 |u j | 2 -|∇u j | 2 and condition (7.1). We can proceed very similarly to Lemma 2.1: let

w i = g i |u i | 2 -|µ i,ε | ε 2
we have ∆w + i ≥ 2(w + i ) 2 so that we conclude again with the non-existence result by Brezis [START_REF] Brezis | Semilinear equations in R n without conditions at infinity[END_REF]. Note that by testing the equation of u i by u i itself, and working as above, yields that µ i,ε > 0.

To prove the second part, fix x ∈ R 2 , L > 0 and for y ∈ B 2L (x), let z i (y) = u i (ε(yx)). Then

-∆z i = -z i (ε 2 |y -x| 2 + g i |z i | 2 + g|z j | 2 -µ i,ε ) -2ε 2 iΩ(y -x) ⊥ • ∇z i =: h i,ε (y).
We have, by Lemma 7.2 and by (7.1),

ε 2 Ω (y -x) ⊥ • ∇z i L 2 (B 2L (x)) = εΩ x ⊥ • ∇u i L 2 (B 2εL (0)) ≤ C
for a constant C independent of x. Therefore, using also the L ∞ -bound above, we have h i,ε L 2 (B 2L (x)) ≤ C √ µ i,ε (µ i,ε + µ j,ε + 1) + C. We deduce that z i H 2 (B L (x)) ≤ C √ µ i,ε (µ i,ε + µ j,ε + 1) + C and we conclude by a bootstrap argument.

Lemma 7.4. Let (u 1 , u 2 ) be a minimizer of E Ω ε in H and denote by µ i,ε the associated Lagrange multipliers. There exists C > 0 independent of ε such that, for i = 1, 2,

|µ i,ε | ≤ C.
Proof. We test the equation for u i by u i itself and integrate by parts, which is possible since u i ∈ H 1 (R 2 , C). The term containing Ω can be bounded by means of Lemma 7.1, whereas the other terms can be rewritten as in Proposition 3.3. Finally, the desired bound follows from the energy estimates of Lemma 7.2. 7.2. Non-existence of vortices. The proof presented here is an adaptation of the proof of the main theorem in [START_REF] Aftalion | Non-existence of vortices in the small density region of a condensate[END_REF]. Let us start with the following splitting of the energy, which is introduced in [START_REF] Aftalion | Vortex-peak interaction and lattice shape in rotating twocomponent Bose-Einstein condensates[END_REF].

Lemma 7.5. Let (u 1 , u 2 ) be any minimizer of E Ω ε in H and let (η 1 , η 2 ) be the unique positive minimizer of E 0 ε in H provided by Theorem 1.3. Let v i = u i η i , for i = 1, 2.

Then E Ω ε (u 1 , u 2 ) = E 0 ε (η 1 , η 2 ) + F Ω ε (v 1 , v 2 )
, where

F Ω ε (v 1 , v 2 ) = 2 j=1 R 2 η 2 j 2 |∇v j | 2 + g j 4ε 2 η 4 j (|v j | 2 -1) 2 -η 2 j Ωx ⊥ • (iv j , ∇v j ) dx + g 2ε 2 R 2 η 2 1 η 2 2 (1 -|v 1 | 2 )(1 -|v 2 | 2 ) dx.
We skip the proof since it is similar to the one of Proposition 2.9. An integration by parts and assumption (1.3) yield Lemma 7.6. Let F i,ε be the auxiliary functions introduced in (6.1) and let γ be as in (2.4). Then we have

F Ω ε (v 1 , v 2 ) = 2 i=1 R 2 η 2 i 2 |∇v i | 2 -4ΩF i,ε Jv i + γ 4ε 2 η 4 i (|v i | 2 -1) 2 dx ≤ 0,
where Jv j = (i∂ x 1 v j , ∂ x 2 v j ) stands for the Jacobian of v j .

Proof. First we prove that we can rewrite F Ω ε in terms of F i,ε as follows

F Ω ε (v 1 , v 2 ) = 2 i=1 R 2 η 2 i 2 |∇v i | 2 -4ΩF i,ε Jv i + g i 4ε 2 η 4 i (|v i | 2 -1) 2 dx + g 2ε 2 R 2 η 2 1 η 2 2 (1 -|v 1 | 2 )(1 -|v 2 | 2 ) dx. (7.3)
Indeed, by (6.14), the following holds

∇ ⊥ ξ i = (-∂ x 2 ξ i , ∂ x 1 ξ i ) = -η 2 i
x ⊥ , and Stokes theorem yields ∂B R ξ j (iv j , ∇v j ) ⊥ • ν dσ = B R {-ξ j ∇ × (iv j , ∇v j ) + η 2 j x ⊥ • (iv j , ∇v j )} dx,

where ∇ × (iv j , ∇v j ) = ∂ x 1 (iv j , ∂ x 2 v j ) -∂ x 2 (iv j , ∂ x 1 v j ) = 2Jv j . The boundary term vanishes because, by Corollary 6.4, for R large, we have η 2 j F j,ε Jv j dx, and (7.3) is proved. Then, reasoning as in (2.3), we deduce that F Ω ε (v 1 , v 2 ) ≥ F Ω ε (v 1 , v 2 ). On the other hand, since (u 1 , u 2 ) is a minimizer and (η 1 , η 2 ) is real valued, we have

E Ω ε (u 1 , u 2 ) ≤ E Ω ε (η 1 , η 2 ) = E 0 ε (η 1 , η 2 )
, which, by Lemma 7.5, implies that F Ω ε (v 1 , v 2 ) ≤ 0. The rest of the section is devoted to proving that v i = 1. Let 0 ≤ χ i ≤ 1 be regular cut-off functions with the property that χ i (r) = 1 for r ≤ R i,0 -2| log ε| -3/2 and χ i (r) = 0 for r ≥ R i,0 -| log ε| -3/2 , and moreover ∇χ i L ∞ (R 2 ) ≤ 2| log ε| 3/2 . We estimate F Ω ε (v 1 , v 2 ) according to the following splitting

F Ω ε (v 1 , v 2 ) = A 1 + B 1 -C 1 + A 2 + B 2 -C 2
, where

A i = R 2 χ i η 2 i 2 |∇v i | 2 + γ 4ε 2 η 4 i (|v i | 2 -1) 2 dx B i = R 2 (1 -χ i ) η 2 i 2 |∇v i | 2 -4ΩF i,ε Jv i + γ 4ε 2 η 4 i (|v i | 2 -1) 2 dx C i = 2Ω R 2 χ i ξ i Jv i dx.
Lemma 7.6 immediately provides

A 1 + B 1 + A 2 + B 2 ≤ C 1 + C 2 .
(7.4) Proposition 7.7. With the notation above, for ε small, we have B i ≥ 0 for i = 1, 2 so that

A 1 + A 2 ≤ C 1 + C 2 .
Proof. Due to the definition of B i , we can restrict our attention to the set supp(1χ i ) = {x : |x| > R i,0 -2| log ε| -3/2 }.

Corollary 6.4 implies that in such a set we have F i,ε ≤ C| log ε| -3/2 . Hence assumption (7.1) implies that ΩF i,ε ≤ 1/4, for ε sufficiently small. Recalling that

|Jv i | ≤ |∇v i | 2 /2, we deduce that |∇v i | 2 -4ΩF i,ε Jv i ≥ 1 2 |∇v i | 2 ,
and as a consequence,

B i ≥ R 2 (1 -χ i ) η 2 i 4 |∇v i | 2 + γ 4ε 2 η 4 i (|v i | 2 -1) 2 dx. (7.5)
The second part of the statement is obtained by combining with (7.4).

Lemma 7.8. Let εi = εγ -1/2 inf

{suppχ i } η i -1
.

There exists C > 0 such that εi ≤ Cε| log ε| 3/4 .

Proof. Clearly a i ≥ c| log ε| -3/2 in {suppχ i }. Hence property (1.20) implies that η i ≥ √ a i -Cε 1/3 ≥ c| log ε| -3/4 in {suppχ i }, (7.6) which provides the statement.

Lemma 7.9. There exists C independent of ε such that, for small ε, On the other side, estimate (7.5) implies that

R 2 η 2 i 2 |∇v i | 2 + γ 4ε 2 η 4 i (|v i | 2 -1) 2 dx ≤ A i + 2B i .
The result follows by summing the above for i = 1, 2 and combining with (7.4). for a suitable α > 0, where F i,0 are as in (6.2). There exists C > 0 independent of ε such that A i + B i + |C i | ≤ C| log ε| -11 , for i = 1, 2.

Proof. We use a result by Jerrard [START_REF] Jerrard | Local minimizers with vortex filaments for a Gross-Pitaevsky functional[END_REF], as it is stated in [START_REF] Aftalion | Non-existence of vortices in the small density region of a condensate[END_REF]. Following the last mentioned paper, we let As in [START_REF] Aftalion | Non-existence of vortices in the small density region of a condensate[END_REF], we can write [START_REF] Jerrard | Local minimizers with vortex filaments for a Gross-Pitaevsky functional[END_REF]Lemma 8] as

2 i=1 |C i | ≤ 2Ωk 2 i=1 R 2 χ i ξ i | log εi | |∇v i | 2 2 + 1 4ε 2 i (|v i | 2 -1) 2 dx + Cε β (1 + 2 i=1 |C i |),
where εi is defined in Lemma 7.8. This formulation only makes use of the estimates in Lemmas 6.6 and 7.9, so that it holds also in our case. Now, recalling that ξ i = F i,ε η 2 i and that ε2

i ≥ ε 2 /(γη 2 i ) in {suppχ i }, we deduce

2 i=1 |C i | ≤ 2Ωk 2 i=1 F i,ε L ∞ (R 2 ) | log εi | A i + Cε β (1 + 2 i=1 |C i |), so that (1 -Cε β ) 2 i=1 |C i | ≤ 2Ωk 2 i=1 F i,ε L ∞ (R 2 ) | log εi | A i + Cε β .
We estimated F i,ε in Lemma 6.5, which provides

F i,ε L ∞ (R 2 ) ≤ (1 + Cε 1/3 ) F i,0 L ∞ (R 2 ) ≤ (1 + Cε β ) F i,0 L ∞ (R 2 ) ,
where the last inequality holds for ε sufficiently small by virtue of (7.9). Also, Lemma 7.8 implies that, for every K > 0, we have

| log εi | ≥ (| log ε| -log | log ε|)(1 + Kε β )
for ε sufficiently small with respect to K. By combining these facts with assumption (7.7) and with our choice of k in (7.8) we obtain

|C 1 | + |C 2 | ≤ 1 -α log | log ε| | log ε| -log | log ε| k(A 1 + A 2 ) + Cε β ≤ 1 -α 2 log 2 | log ε| | log ε| 2 (A 1 + A 2 ) + Cε β .
(7.10) Recalling Proposition 7.7 we deduce

A 1 + A 2 ≤ |C 1 | + |C 2 | ≤ 1 -α 2 log 2 | log ε| | log ε| 2 (A 1 + A 2 ) + Cε β , so that A 1 + A 2 ≤ Cε β α 2 | log ε| 2 log 2 | log ε| ≤ C| log ε| -11 ,
where in the last step we replaced relation (7.9). Being A i non-negative quantities, the last estimate holds for both terms. In turn we deduce from (7.10) that |C i | ≤ C| log ε| -11 , and from (7.4) that

B 1 + B 2 ≤ A 1 + B 1 + A 2 + B 2 ≤ C 1 + C 2 ≤ C| log ε| -11 .
Being B i non-negative by Proposition 7.7, the estimate holds for both terms.

We can now derive a "clearing-out" property (see also [START_REF] Béthuel | Ginzburg-Landau vortices[END_REF]).

Proposition 7.11. Suppose that (7.7) holds. For ε sufficiently small, we have

|v i | ≥ 1 2 in {suppχ i }.
Proof. We shall prove that

|v i | > 1 -| log ε| -1 in {suppχ i }, (7.11)
for i = 1, 2, which implies the statement. By combining Lemma 7.9 with Proposition 7.10 we obtain This comes from the fact that ∇u i L ∞ (R 2 ) ≤ C/ε, as can be seen by combining Lemmas 7.3 and 7.4, and that ∇v i = ∇u i /η iu i ∇η i /η 2 i , together with estimate (7.6). Suppose by contradiction that (7.11) does not hold, i.e. there exists x 0 ∈ {suppχ i } such that which contradicts (7.12) for ε sufficiently small. Therefore (7.11) is proved.

7.3. Proof of Theorem 1.5. We are now in position to give the proof of Theorem 1.5.

Proof of Theorem 1.5. We take Ω ≤ ω 0 | log ε|ω 1 log | log ε| with ω 0 , ω 1 such that (7.7) holds (recall that F i,0 is bounded in R 2 ). Thanks to the previous proposition the quantity w i = v i /|v i | is well defined in {suppχ i } and satisfies Jw i = 0 (see [START_REF] Aftalion | Non-existence of vortices in the small density region of a condensate[END_REF]). Hence, we find that

C j = 2Ω
{suppχ j } χ j ξ j (Jv j -Jw j ) dx

= 2Ω

{suppχ j } ∇ ⊥ (χ j ξ j )[(iv j , ∇v j ) -(iw j , ∇w j )] dx.

Writing v j = ρ j e iφ j in {suppχ j } we see that (iv j , ∇v j ) = ρ 2 j ∇φ j and (iw j , ∇w j ) = ∇φ j , so that Proposition 7.11 implies |(iv j , ∇v j ) -(iw j , ∇w j )| = |ρ 2 j -1| ρ j |ρ j ∇φ j | ≤ 2|ρ 2 j -1||ρ j ∇φ j | ≤ 2 |v j | 2 -1 |∇v j |.

We insert it in the previous estimate to obtain

C j ≤ 2Ω ∇(χ j ξ j ) L ∞ ({suppχ j }) {suppχ j } 2 |v j | 2 -1 • |∇v j | dx ≤ 4 √ 2Ω ∇(χ j ξ j ) L ∞ (R 2 ) {suppχ j } εj 2 |∇v j | 2 + 1 4ε j |v j | 2 -1 2 dx ≤ CΩε| log ε| 9/4 {suppχ j } |∇v j | 2 2 + 1 4ε 2 j |v j | 2 -1 2 dx,
where we used Lemma 7.8 and the estimate ∇(χ i ξ i ) L ∞ (R 2 ) ≤ C| log ε| 3/2 . We sum for i = 1, 2 and then we use the assumption (7.1), and Lemma 7.9, to obtain C 1 + C 2 ≤ Cε| log ε| 19/4 (C 1 + C 2 ), so that C 1 + C 2 ≤ (C 1 + C 2 )/2 for ε sufficiently small. Since C 1 + C 2 is non-negative by Proposition 7.7, we conclude that C 1 + C 2 = 0. In turn, relation (7.4) and Proposition 7.7 imply also A i = B i = 0 for i = 1, 2, that is

A i = R 2 χ i η 2 i 2 |∇v i | 2 + γ 4ε 2 η 4 i (|v i | 2 -1) 2 dx = 0

. 30 )

 30 Next, we estimate the terms in the right-hand side by making use of Proposition 4.2. To this end, we need to derive a relation for ∆η 1,ε in terms of the Hastings-McLeod solution near R 1,ε . Making use of (1.33a), (4.15) and the natural bound |V (s)| ≤ C(|s|

7 12 ) ≤ Cε - 1 3

 71 for small ε > 0.(4.40)Finally, the desired estimate (4.29) follows directly from (4.35), (4.36) and (4.40).

4 . 5 .

 45 adding (4.50) and (4.51), recalling the integral constraint in (4.43), we deduce that λ ≥ c, as desired. The case where (4.44) holds can be treated analogously.A direct consequence of Proposition 4.4, and of (4.49), is the following Corollary If ε is sufficiently small, there exists c > 0 such that

3 2 and

 32 |ũ ′ (r 1 )| + |ṽ ′ (r 1 )| ≤ Cε

1 2 ε

 2 , i = 1, 2, for small ε > 0. It follows from (4.122), (4.123) and (4.124), recalling (1.30) and (3.13), that |x|<R 1,0 -D| log ε|

1

 1

17

 17

  .

4. 9 .

 9 Proof of Theorem 1.4. The desired minimizer (η 1,ε , η 2,ε ) is that of Theorem 4.1. Clearly, estimate (1.19) is the same as (4.1). Estimate (1.20) follows readily by combining (1.30), (4.4), (4.5), (4.6), (4.7), (4.9), and (4.10). Estimate (1.21) follows readily from (4.2), (4.3), (4.4) and (4.5). In view of (1.12), (4.2) and (4.3), we infer that (1.22) holds. Finally, the decay estimate (1.23) follows immediately from (4.6).

2

 2

2 3 , 3

 33 since (4.15) and (4.79) imply that η 1,ε (r) ≥ cε 1 and η1,ε (s) ≤ Cε

1

 1

7. 1 .

 1 Energy estimates. The following proof uses some ideas from[START_REF] Ignat | The critical velocity for vortex existence in a two-dimensional rotating Bose-Einstein condensate[END_REF] Lem. 3.1].

2 η 2 j

 22 ∂B R ξ j (iv j , ∇v j ) ⊥ • ν dσ = ∂B R F j,ε (iu j , ∇u j ) dσ ≤ Cε 2/3 ∂B R (|∇u j | 2 + |u j | 2 ) dσ which vanishes along a sequence R k → ∞. Hence we have obtained R x ⊥ • (iv j , ∇v j ) dx = 2 R 2

2 {

 2 F i,0 L ∞ (R 2 ) } ≤ | log ε| -(α + 1) log | log ε|, (7.7)

α

  = 1300, k = 1 + α log | log ε| | log ε| , β = kε β = | log ε| -α/100 = | log ε| -13 .(7.9)

that 1 ε 2

 2 |v i (x 0 )| ≤ 1 -| log ε| -1 as ε → 0.Then(7.13) implies|v i (x)| ≤ 1 -C| log ε| -1 in B r 0 (x 0 ) with r 0 = ε| log ε| -5/2 , so Br 0 (x 0 )∩{suppχ i } (|v i | 2 -1) 2 dx ≥ C| log ε| -7 ,

  4.2.2. Gluing approximate solutions. Consider a one-dimensional smooth cutoff function ζ such that ζ

  .29) It follows readily from (4.21) and (4.23) that

  .126) as ε → 0, where we use Lemmas 4.16-4.17 instead of Proposition 4.15. On the other side, thanks to (4.12) and Proposition 4.15, for r ∈ [R 2,εδ, R 2,ε -Dε

	2
	3

  3 (1), it coincides with (η 1 , η 2 ). Estimate (4.1) is proven in Proposition 4.18. Estimates (4.2)-(4.3) follow from Proposition 4.14, the definition of ηi,ε , and the second estimate in (4.12). Estimates (4.4)-(4.5) follow readily from Proposition 4.15, the definition of ηi,ε (especially recall (4.21) for the second estimate in (4.4)), and (4.17). Estimate (4.6) follows readily from Lemma 4.13, (4.12) and (4.79); estimate (4.7) follows from (4.13), (4.21) and Lemma 4.17. Finally, relations (4.8), (4.9) and (4.10) are consequences of (4.15), (4.21), (4.79) and Lemma 4.

  .1) 5.1.2. Inner approximations. Here, we define approximate solutions of the problem in overlapping intervals around each

  (|v i | 2 -1) 2 dx ≤ C| log ε| 3/2 (C 1 + C 2 ).Proof. Recalling that ε2i ≥ ε 2 /(γη 2 i ) in {suppχ i } and relation (7.6), we deduce

	2 i=1 {suppχ i }	|∇v i | 2 2	+	1 4ε 2 i
	{suppχ i }	|∇v i | 2 2 ≤ C| log ε| 3/2 + 1 4ε 2 i	(|v i | 2 -1) 2 dx {suppχ i } η 2 i 2 |∇v i | 2 +	γ 4ε 2 η 4

i (|v i | 2 -1) 2 dx.

  (|v i | 2 -1) 2 dx ≤ C| log ε| 3/2 | log ε| -11 . (|v i | 2 -1) 2 dx ≤ C| log ε| -8(7.12) for i = 1, 2. Next we observe that∇v i L ∞ ({suppχ i }) ≤ C | log ε| 3/2 ε . (7.13)

	2 i=1 {suppχ i }	|∇v i | 2 2	+	1 4ε 2 i
	Then Lemma 7.8 provides
		1
		ε 2	{suppχ i }

].

, r ≥ R 1,ε . (6.7)

Proposition 6.1. Assume that (1.3) and (1.8) hold. Let F i,ε be given by (6.1) and F i,0 by (6.2). Then

and F i,ε -F i,0 L ∞ (R 2 ) ≤ Cε 1 3 , i = 1, 2, provided that ε > 0 is sufficiently small. This proposition follows from Corollary 6.4 and Lemma 6.5. The proof is made under the additional assumption (1.18). If g 1 = g 2 , a simpler proof holds since F 1,ε = F 2,ε and the property is that for a single equation [START_REF] Aftalion | Non-existence of vortices in the small density region of a condensate[END_REF]. The scalar counterparts

and their convergence to the corresponding limiting functions

and f 2,0 (r) = ∞ r s a 2,0 + g g 2 a 1,0 + (s)ds

have been studied in [START_REF] Aftalion | Non-existence of vortices in the small density region of a condensate[END_REF]Lem. 2.2]. We have

Actually, the ground states in the latter lemma had unit L 2 -norm but its proof carries over to the above case, yielding the following lemma.

and

, where

provided that ε > 0 is sufficiently small.

The main task in this section is to show the following proposition.

and

, arguing similarly, this time noting that η 1,ε (r) ≥ cε 1 6 , we find that

Lastly, if 0 ≤ r ≤ s ≤ R 1,εδ, where η 1,ε ≥ c, via Proposition 4.14, we get that

The remaining terms in (6.10) can be estimated analogously to complete the proof of (6.6).

We point out that a rather delicate term is

], which can be estimated as follows: Since in this interval we have η1,ε (r) ≥ c(R 1,εr) 1 2 ≥ Cε 1/6 (from (4.17) and (1.34)), and the same holds for η 1,ε via (4.79), it follows that

The validity of estimate (6.7) can be verified analogously, using (4.21) to show that

The assertion of the following corollary is analogous to the first assertion of Lemma 6.2 for the scalar case. Corollary 6.4. If ε > 0 is sufficiently small, we have

Proof. The desired estimate (6.11) follows readily from the fact that it holds with f i,ε in place of F i,ε (see Lemma 6.2), via Theorem 4.1 and Proposition 6.3.

The next lemma is a natural extension of the second assertion of Lemma 6.2. Lemma 6.5. If ε > 0 is sufficiently small, we have

where F i,0 are as in (6.2).

Proof. The proof is based on the fact that

and (see (7.5))

2 dx = 0. Therefore, we infer that v i are both constants of modulus 1 as we wanted to prove.

Appendix A. The scalar ground state Throughout Subsections 4.2-4.3, we have referred to the following

Let A ε = a + µ ε . For sufficiently small ε > 0, by the implicit function theorem, there exist

to the problem

for some δ ∈ 0, 1 4 min i=1,••• ,n-1 {r i+1r i } . More precisely, we have

, and V is the Hastings-McLeod solution, as described in (4.11). Estimate (A.4) can be differentiated once to give

uniformly, as ε → 0. On the other side, we have

uniformly, as ε → 0. Furthermore,

and

), (A.9)

uniformly, as ε → 0, and

Proof. All the assertions up to (A.8) are essentially contained in [START_REF] Karali | The ground state of a Gross-Pitaevskii energy with general potential in the Thomas-Fermi limit[END_REF]Thm. 1.1], where in fact no radial symmetry is imposed on a(•). Actually, relation (A.5) can be proven by combining the proof of Corollary 4.1 in [START_REF] Karali | The ground state of a Gross-Pitaevskii energy with general potential in the Thomas-Fermi limit[END_REF] with relation (3.40) therein. In passing, we note that V ′ (s) < 0, s ∈ R.

Let us further assume that a(x) = a(|x|) ∈ C 4 (R 2 ). In order to establish relations (A.9)-(A.10), we need a refinement of (A.6). Motivated from the identity

for some fluctuation function φ. Pushing further the analysis in [17, Thm. 2.1] or [29, Thm. 1.1], it can be shown that

Making use of (A.6), and recalling that

it follows readily that

uniformly, as ε → 0. Since .6) and (A.14)), a standard comparison argument yields that

where we have also used that φ r i,ε + (-1) i δ ≤ Cε 4 and φ r i,ε + (-1) i Cε ). Consequently, recalling (A.12), we have shown the following refinement of (A.6):

) ≤ δ, as ε → 0, which complements (A.13). In turn, via (A.1) and some straightforward calculations, this can be shown to imply (A.10). Equivalently, we have that

Integrating the above identity from r i,ε + (-1) i δ to r i,ε + (-1) i Cε 2 3 , and using that η

where

Observe that, thanks to the lower bound in (4.112), we have

provided that D j is sufficiently large. We may assume, without loss of generality, that Φ, h ≥ 0 (by writing h = h +h -if necessary). If Φ attains its maximum value at a point r 0 ∈ I, then Φ rr (r 0 ) ≤ 0 and Φ r (r 0 ) = 0. So, letting ρ 0 = R 1,εr 0 , via (B.1) and (B.2), we obtain that kρ 0 Φ(r 0 ) ≤ ρ