On planar selfdual electroweak vortices

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ARTICLE IN PRESS S0294-1449(03)00034-9/FLA AID:46 Vol.•••(•••) ELSGMLTM(ANIHPC):m3SC v 1.165 Prn:5/09/2003; 8:23

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Ann. I. H. Poincaré – AN ••• (••••) •••–••• www.elsevier.com/locate/anihpc

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On planar selfdual Electroweak vortices Sur les vortex Électrofaibles autoduaux dans le plan

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Dongho Chae a , Gabriella Tarantello b,∗

a Department of Mathematics, Seoul National University, Seoul 151-742, South Korea b Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy

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Received 29 May 2002; accepted 30 January 2003

Abstract

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By perturbation techniques, we obtain a new class of selfdual Electroweak vortices in R2 , whose asymptotic behavior we control in an “optimal” way to yield a sort of “quantization” property for the corresponding flux. Our class of vortex-solutions complements that constructed by Spruck and Yang [Comm. Math. Phys. 144 (1992) 215–234].  2003 Published by Éditions scientifiques et médicales Elsevier SAS. Résumé

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En utilisant des techniques de perturbation, nous obtenons une nouvelle classe de vortex Électrofaibles autoduaux dans R2 , dont nous pouvons contrôler de façon « optimale » le comportement asymptotique en donnant une propriété de « quantisation » pour le flot corréspondant. Notre classe de solutions complète celles construites par Spruck et Yang [Comm. Math. Phys. 144 (1992) 215–234].  2003 Published by Éditions scientifiques et médicales Elsevier SAS. MSC: 35J45; 35J60; 37K40; 70S15

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Introduction

* Corresponding author.

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We shall be concerned with planar vortex-type (self-dual) solutions for the celebrated SU (2) × U (1)Electroweak theory of Glashow, Salam and Weinberg [10]. As observed by Ambjorn and Olesen [2–4], if the physical parameters satisfy a suitable critical condition, it is possible to determine Bogomol’nyi type equations (also known as self-dual equations) to be satisfied by Electroweak-vortices when expressed in terms of the unitary gauge variables. The selfdual equations include a gauge invariant version of the Cauchy–Riemann equation, which makes it reasonable to assume that the W -boson field admits a finite number N of zeroes (counted with

E-mail addresses: [email protected] (D. Chae), [email protected] (G. Tarantello). 0294-1449/$ – see front matter  2003 Published by Éditions scientifiques et médicales Elsevier SAS. doi:10.1016/j.anihpc.2003.01.001

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multiplicity). Such an integer N is called the vortex number, and the corresponding vanishing points of W are called the vortex points. Moreover, by means of an approach introduced by Taubes [15,16] in the context of Ginzburg–Landau vortices, it is possible to reduce the analysis of self-dual Electroweak vortices to the study of the following elliptic system:  m   2 u1 2 u2   −u nk δ(z − zk ), + g e − 4π = 4g e 1  k=1 (P)  2   u    u = g e 2 − φ02 + 2g 2 eu1 2 2 cos2 θ where φ0 is a given positive parameter, g is the SU (2)-coupling constant, and θ ∈ (0, π/2) is the so called “Weinberg-angle”, related with the U (1)-coupling constant g∗ via the relation, g cos θ = 2 . (g + g∗2 )1/2

ϕ 2 = e u2 ,

DP

We refer to the recent monograph of Y. Yang [17] and [5] for details, and, in particular, on how to recover the full vortex configuration out of solutions of (P). We only mention that, by virtue of the vortex ansatz and accordingly to the unitary gauge variables, the vortex solution is specified by the (complex valued) massive field W , the (scalar) field ϕ and the real valued 2-vector fields P = (Pµ )µ=1,2 and Z = (Zµ )µ=1,2 . So that, u1 and u2 determine the magnitude of W and ϕ respectively, as follows: |W |2 = eu1 .

(0.1) m

P12 = ∂1 P2 − ∂2 P1

CT E

Thus, zk corresponds to a vortex point, the integer nk ∈ N to its multiplicity k = 1, . . . , m, and N = k=1 nk is the vortex number. Furthermore, while the 2-vector fields P and Z are defined only up to gauge transformations, their gauge invariant curls, and Z12 = ∂1 Z2 − ∂2 Z1

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are determined by the relations: g P12 = φ 2 + 2g sin θ |W |2 , 2 sin θ 0 (0.2) g Z12 = (ϕ 2 − φ02 ) + 2g cos θ |W |2 2 cos θ (see [17]). Following the numerical evidence provided by Ambjorn and Olesen, the first rigorous results concerning Electroweak-vortices have been obtained by Spruck and Yang in [13,14]. In [13], the authors aim to obtain configurations of the type described in physics literature as the “Abrikosov’s mixed states” (see [1]), so they consider the selfdual equations subject to ’t Hooft boundary conditions [9]. In turn, this amounts to solve (P) under periodic boundary conditions, and Spruck and Yang in [13] obtain necessary and sufficient conditions on the given physical parameters, in order to ensure the presence of periodic N -vortices in the theory. Their results are sharp for N = 1, 2 and recently have been improved by Bartolucci and Tarantello in [5] to a wider range of parameters. In this paper, we shall be concerned with planar vortex-type configurations, and consider (P) over R2 subject to appropriate decay assumptions at infinity. This situation has been considered in [14], where it has been observed that necessarily, the corresponding vortex-solutions must carry infinite energy. As a consequence, solutions of (P) over R2 appear in abundance, and are distinguished according to their rate of decay at infinity. In view of (0.1), also the flux of the fields P and Z is infinite, while this may not be necessarily the case for the field: (sin θ )P + (cos θ )Z, which bares informations about the gauge potential in the original variables. We shall be concerned with such “finite-flux” selfdual solutions, by solving (P) under the condition:   (0.3) eu1 , eu2 ∈ L1 R2 .

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In fact, by setting: 2g 2 eu1 , σ1 = π

σ2 =

g2 2π

R2



eu2

P.3 (1-21) by:violeta p. 3 3

(0.4)

R2

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in view of (0.1), (0.2) we see that the flux  of the field (sin θ )P + (cos θ )Z is given by, π  = (sin θ P12 + cos θ Z12 ) = (σ1 + σ2 ). g R2

As a first fact, we show that necessarily

(0.6)

RO

σ1 + σ2 > 2(N + 1)

(0.5)

as σ1 and σ2 determine the rate of decay at infinity of u1 and u2 (see Theorem 1.1). The existence results for (P)–(0.3) derived by Spruck and Yang in [14] imply that the lower bound (0.6) is “sharp”, as they construct solutions satisfying: σ1 + σ2

any assigned value in (2(N + 1), +∞).

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σ2 > 2N;

σ2 = o(1) and σ1 = 4(N + 1) + o(1).

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Notice that, in contrast to the periodic case, this class of solutions lacks any sort of “quantization” property for the corresponding flux (0.5). A first goal of this paper is to show that, actually, this is no longer the case, if we consider solutions of (P)–(0.3) with σ2
2N . In this case, and when all vortex points coincide, we show that the value σ2 also controls from above the value σ1 . This implies that, in certain regime, (e.g., σ2 small) the lower bound 2(N + 1) in (0.6) is no longer sharp, as σ1 + σ2 is now forced to remain close to the value 4(N + 1), and any other value is ruled out. See Lemma 1.6. Thus, for σ2 small, a sort of “quantization” property is restored as σ1 “approximately” equals to 4(N + 1). Our second goal, is to show that, in fact, such situation does occur. So, in Section 2, we shall focus our attention in constructing several families of solutions of (P)–(0.3) so that (0.7)

R2

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To this end, we provide a priori estimates (see Theorem 1.1), so that the L1 -norm of eu2 uniformly estimates its L∞ -norm. Thus, requiring a small σ2 , implies that we must have eu2 also small in L∞ -norm. Therefore, the first equation in (P) may be viewed as a “perturbation” of the following singular Liouville equation:  m    2 u  −u = 4g nk δ(z − zk ) in R2 , e − 4π    k=1    eu < +∞,   

UN

whose solutions may be classified according to Liouville formula (e.g., see [12]). When all vortex points coincide, we are able to adapt a perturbation approach introduced by Chae and Imanuvilov in [7] to construct “nontopological” Chern–Simons vortices, and obtain solutions of (P), “bifurcating” out of solutions of the singular Liouville equation mentioned above. In this way, we are able to provide rather accurate pointwise estimates on our solutions to ensure (0.7), together with a sharp control on their decay rate at infinity. We refer to Theorem 2.1 and 2.2 for the precise statements. Clearly, our result furnishes a new class of solutions for (P)–(0.3) which complements those constructed in [14].

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1. Asymptotic behavior

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Consider the problem,  m   2 u1 2 u2   −u nk δ(z − zk ), = 4g e + g e − 4π 1     k=1     g 2  u2 (P ) e − φ02 + 2g 2 eu1 in R2 , u2 = 2  2 cos θ      u  eu2 < +∞, e 1 < +∞,    R2

R2

m 

nk ,

k=1

c0 =

g 2 φ02 8 cos2 θ

with g, φ0 given positive constants and θ ∈ (0, π/2). Set g2 2g 2 eu1 , σ2 = eu2 . σ1 = π 2π R2

R2

DP

N=

RO

where {z1 , . . . , zm } ⊂ R2 are given points, {n1 , . . . , nm } ⊂ N are given integers, and δ(z − p) denotes the Dirac measure with pole at p ∈ R2 . Let

(1.1)

We devote this section to obtain the asymptotic behavior, as |x| → +∞, of a solution pair (u1 , u2 ) for (P ) in terms of σ1 and σ2 , and to establish some interesting relation for such values. We have:

for suitable C > 0, u1 (x) → 2N − (σ1 + σ2 ), ln |x| and so necessarily

(1.3)

(1.4)

∈ L∞ (R2 )

In addition, if

CO R

and for suitable C0 > 0 (depending on c0 only)



u

e 2 ∞ 2
C0 eu2 1 2 . L (R ) L (R )   u2 (x) + c0 |x − x0 |2 ∈ L2 R2 1+α/2 1 + |x|

for some x0 ∈ R2 and α ∈ (0, 1), then   u1 (x) = 2N − (σ1 + σ2 ) ln |x| + O(1),  σ2 1 u2 (x) = −c0 |x − x0 |2 + σ1 + ln |x| + O(1) 2 cos2 θ as |x| → +∞.

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(ii)

(1.2)

as |x| → +∞,

σ1 + σ2 > 2(N + 1). u+ 2

as |x| → +∞

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∞ 2 (i) u+ 1 ∈ L (R ),   u1 (x)  2N − (σ1 + σ2 ) ln |x| − C,

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Theorem 1.1. Let (u1 , u2 ) be a solution pair for (P ), then

(1.5)

(1.6)

(1.7) (1.8)

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In order to derive Theorem 1.1 we are going to recall some basically well known facts about the function:   1 ln |x − y| − ln(1 + |y|) f (y) dy (1.9) w(x) = 2π R2

Lemma 1.1. Let   +   f ln |f | + 1 ∈ L1 R2

and β =

1 2π



OF

with f ∈ L1 (R2 ).

f (y) dy

R2

(i) if f  0, then

for suitable C > 0; w(x) (ii) → β, as |x| → +∞; ln |x| (iii) if     ln 1 + |x| f ∈ L1 R2 , then

w(x) = β ln |x| + O(1),

as |x| → +∞.

DP

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  w(x)
β ln |x| + 1 + C

(1.10)

(1.11)

(1.12)

(1.13)

(1.14)

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We notice that properties (1.11), (1.12) and (1.14) remain valid for any other solution v of the equation: v = f

(1.15)

provided it admits a “slow” growth at infinity as expressed by one of the following conditions:

  v ∈ L2 R2 , 1+α/2 1 + |x|

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or

  v+ ∈ L∞ R2 , ln |x| + 1

for some α ∈ (0, 1).

(1.16)

(1.17)

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Corollary 1.1. Assume (1.10), and let v ∈ L1loc (R2 ) be a solution of (1.15) which satisfies (1.16) or (1.17). Then, properties (i), (ii) and (iii) of Lemma 1.2 hold for v. Proof of Corollary 1.1. Notice that the function u = v − w is harmonic over R2 and, in view of Lemma 1.2, satisfies (1.16) or (1.17). In either case u must be constant. Indeed, this is a well known fact in case (1.16) holds (e.g., see Lemma 4.6.1 in [17]), while it is a consequence of Proposition 1.1 in [7] in case u satisfies (1.17). ✷ Proof of Lemma 1.1. Let us start by observing that     ln |x − y| − ln 1 + |y|
ln |x| + 1 , ∀x, y ∈ R2 .

R2

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Consequently, for f  0, property (i) can be easily derived. To obtain (ii), let |x| > 1 and consider,     1 ln |x − y| − ln 1 + |y| − ln |x| f (y) dy. σ (x) = ln |x|

(1.18)

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Thus, we must show that σ (x) → 0 as |x| → +∞. For this purpose, write σ (x) = σ1 (x) + σ2 (x) + σ3 (x),

{|x−y| 1 and |y| > ρ} respectively, with ρ > 1 a fixed constant. We estimate,



 

σ1 (x)
1

ln |x − y| − ln 1 + |y| − ln |x|

f (y) dy ln |x|

1 and b = |f (y)|. Consequently, σ1 (x) → 0 as |x| → +∞. Furthermore, for x ∈ R2 |x| > 2ρ, we with a = ln |x−y| have  





ln |x − y| + ln(ρ + 1) f (y) dy

σ2 (x)
1

ln |x| |x| {|x−y|>1,|y|1, ρ2|x|}


4



{|y|>ρ}

as |x| → +∞.

RE

Finally,

  1 ln 2(ρ + 1) f L1 (R2 ) → 0, ln |x|

    ln |x − y| + ln 1 + |y| + ln |x| f (y) dy

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ln |x − y|

f (y) +

1 + |y|

{|y|>ρ}



f (y) dy + 3 ln 3 f  1 2 → 4 L (R ) ln |x|



{|y|>ρ}



f (y) dy



f (y) dy

R2

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as |x| → +∞, for any fixed ρ > 1. Thus, letting ρ → +∞, we obtain the desired conclusion. Now, suppose that (1.13) holds, then to establish (iii) it remains to show that







ln |x − y| − ln |x| f (y) dy

= O(1) (1.19)

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as |x| → +∞. As above, we are going to estimate the integral (1.19) over various regions. Firstly, note that for |x| > 1 we have:







  1



f (y) dy + ln |x| f (y) dy ln |x − y| − ln |x| f (y) dy
ln

|x − y| {|x−y| 0, and (1.19) follows.

✷

Proof of Theorem 1.1. We start with the following: Claim 1.

 ln |x − y| + ln |x| f (y) dy

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1

c0 u2

e L1 . + ln sup u2
2 π R2



DP








ln |x − y|

f (y) dy +

|x|

R2

  ln 1 + |y| f (y) dy := C1 .

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{|x−y|>1}





OF


π +



(1.20)

Note that (1.20) immediately implies (1.5). To obtain (1.20), let us use the second equation in (P ) together with Green’s representation formula, to derive that,  r 1 4c0 ln u2 (y) dσ dy + u2 (x)
2π |x − y| 2πr 1 = c0 r 2 + 2πr

{|x−y|=r}



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{|x−y| 0. Multiply both sides of (1.21) by 2r and integrate over [0, 1] to obtain: 1 c0 u2 (x)
+ u2 (y) dy. 2 π At this point, we can use Jensen’s inequality to estimate,  

1 1 1

eu2 1 , u2 (y) dy
ln eu2 (y) dy
ln L π π π

and conclude (1.20). Claim 2.

 2 ∞ u+ R . 1 ∈L

B1 (x)

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B1 (x)

(1.22)

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To establish (1.22) note that u1 (x) → −∞ as x → zk , k = 1, . . . , m. So, we need to show that u1 is bounded above outside a large ball which contains all points zk ’s, k = 1, . . . , m. To this end note that, if x: |x| > maxk=1,...,m |zk | + 2, then + (1.23) u1
e u1 B1 (x)

R2

−u1 = 4g 2 eu1 + g 2 eu2

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and in B1 (x)

with eu2 ∈ L1 (R2 ) ∩ L∞ (R2 ). So the argument given by Brezis and Merle [6] in the proof of Theorem 2 applies word by word and yields to the estimate: max u1
C

R2

Hence, by Lemma 1.1, we have   w2 (x)
σ2 ln |x| + 1 , ∀x ∈ R2 , w2 (x) → σ2 , ln |x|

CT E

and

as |x| → +∞.

Define, u0 (x) =

m 

ln |x − zk |2nk ,

k=1

RE

and note that  2 u1 − u0 ∈ L∞ loc R .

Decompose:

v1 ∈ L∞ loc

2



R ;

2 u1 (x)

−v1 (x) = 4g e

  v1+ (x) ∈ L∞ R2 . ln(|x| + 1)

UN

and



CO R

u1 (x) = u0 (x) − w2 (x) + v1 (x) so that v1 (x) satisfies:

DP

with a suitable constant C > 0 independent of x. To proceed further, let   g2  w2 (x) = ln |x − y| − ln 1 + |y| eu2 (y) dy. 2π

RO

B1/4 (x)

= 4g

Therefore, we may use Corollary 1.1 to derive that    −v1 (x)
σ1 ln |x| + 1 + C

2

m 

(1.24)

(1.25)

(1.26)

(1.27)

(1.28)

(1.29)

|x − zk |2nk e−w2 (x) ev1 ,

(1.30)

k=1

(1.31)

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for suitable C > 0, and v1 (x) → −σ1 , ln |x|

as |x| → +∞.

(1.32)

In virtue of (1.25), (1.26) and (1.27), we conclude:   u1 (x)  2N − (σ1 + σ2 ) ln |x| − C, as |x| → +∞ u1 (x) → 2N − (σ1 + σ2 ) ln |x|

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and

as |x| → +∞.

RO

So, (1.2) and (1.3) are established. Consequently, we must have that σ1 + σ2 > 2(N + 1) and   ln 1 + |y| eu1 (y) dy < +∞. R2

(1.33)

Thus, we can use part (iii) of Lemma 1.1 together with Corollary 1.1 to conclude that, 1 + O(1), |x|

as |x| → +∞.

(1.34)

DP

v1 (x) = σ1 ln

w2 (x) = σ2 ln |x| + O(1),

as |x| → +∞,

and

CT E

Finally, suppose that (1.6) holds, then we are in position to apply Corollary 1.1 to the function v2 (x) = u2 (x) + g2 u2 + 2g 2 eu1 , and conclude that c0 |x − x0 |2 with f (x) = 2 cos 2θe  u2 (x) + c0 |x − x0 |2 1 σ2 → σ1 + as |x| → +∞. ln |x| 2 cos2 θ  In particular, this implies that u2 admits exponential decay at infinity. Thus, R2 ln(1 + |y|)eu2(y) dy < +∞, and taking into account (1.33), we derive: (1.35)

 σ2 1 u2 (x) = −c0 |x − x0 | + σ1 + ln |x| + O(1) 2 cos2 θ

RE

2

as |x| → +∞. Moreover, from (1.29), (1.34) and (1.35) we conclude m  k=1

ln |z − zk |2nk − (σ1 + σ2 ) ln |x| + O(1) as |x| → ∞,

CO R

u1 (x) =

and Theorem 1.1 is established.

✷

Remark 1.1. We point out that, as in Lemma 1.3 of [8], it is possible to sharpen the asymptotic behavior (1.7) and (1.8) and prove that

UN

∂u1 ∂u1 → 2N − (σ1 + σ2 ), → 0, ∂r ∂θ   ∂  σ2 1 r u2 + c0 |x − x0 |2 → σ1 + , ∂r 2 cos2 θ

 ∂  u2 + c0 |x − x0 |2 → 0 ∂θ

uniformly, as r = |x| → +∞, with (r, θ ) polar coordinates in R2 .

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In the framework of Theorem 1.1, Spruck and Yang in [14] have constructed two families of solutions satisfying the asymptotic behavior (1.7), (1.8) (with x0 = 0) and corresponding values σ1 , σ2 free to vary as follows: 1. for the first family: σ1 ∈ (0, 4) and σ1 + σ2 ∈ (2(N + 2), +∞); 2. for the second family: σ1 ∈ (0, 2) and 2σ1 + σ2 ∈ (2(N + 1), +∞).

DP

RO

OF

In particular, Spruck–Yang’s construction allows one to obtain a solution pair for (P ) with σ1 + σ2 equals to any prescribed value in the interval (2(N + 1), +∞). So, (1.4) is sharp. On the other hand, their construction always gives, σ2 > 2N , while σ1 can take values as small as wanted. For instance, according to Spruck–Yang’s result, it is possible to exhibit a family of solutions for (P ) such that σ1 = o(1), while σ2 can take any fixed value in (2(N + 1), +∞). Clearly, those solutions deny any sort of “quantization” property to the corresponding flux (0.5). The aim of this paper is to show that the situation is quite different in case we consider solutions of (P ) that admit the asymptotic behavior (1.7), (1.8) with σ2 small. We see that in this case σ1 may not be free to take any value in the interval (2(N + 1), +∞), as (1.4) would imply. In fact, in case all vortex points coincide, we see that σ1 must remain close to the value 4(N + 1) and no other value is allowed. Thus solution of (P ) with σ2 small are very interesting, as they seem to restore a sort of “quantization” property for the flux in (0.5). We are going to construct such class of solutions. To this purpose, we consider the following function spaces introduced by Chae and Imanuvilov in [7]. For given α > 0, let  

2    2 2+α 2

Xα = u ∈ Lloc R : u(x) dx < ∞ , 1 + |x| R2

(u, v) =



R2



CT E

which defines a Hilbert space equipped with the scalar product  1 + |x|2+α uv.

Denote with  · Xα the corresponding norm on Xα . Also let

  2 u 2 R . ∈ L (1 + |x|)1+α/2

RE

 2,2  2  R : u ∈ Xα , Yα = u ∈ Wloc

It defines a Hilbert space with corresponding natural scalar product and norm: = u2Xα

2



u(x)



+

2 1+α/2 1 + |x|

CO R

u2Yα

.

L (R2 )

We refer to [7] for the relevant properties of those spaces. In particular, from [7] we recall,

UN

0 Proposition 1.1. (a) Xα ֒→ L1 (R2 ) continuously, Yα ֒→ Cloc (R2 ). (b) If α ∈ (0, 1) and u ∈ Yα is harmonic, then u is constant. (c) There exists a constant C1 > 0 such that for all u ∈ Yα

  

u(x)
C1 uY ln |x| + + 1 , α

∀x ∈ R2 .

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P.11 (1-21) by:violeta p. 11 11

Notice that assumption (1.6) is motivated by considering solutions of (P ) in the space Yα × Yα . In fact, as an easy consequence of Theorem 1.1, it follows: Corollary 1.2. Under the assumptions of Theorem 1.1 we have (i) U1 := u1 − u0 ∈ Yα , U2 := u2 + c0 |x − x0 |2 ∈ Yα and    eu1 ∈ Xα for every α ∈ 0, 2 σ1 + σ2 − 2(N + 1) 1 2 U1

OF

(ii)

(1.36)

+ U2 ∈ Yα and eu2 ∈ Xα , ∀α > 0.

RO

To complement the situation analyzed by Spruck and Yang in [14], we consider solutions of (P ) with σ2
2N . In this case, the following interesting estimate holds, when all vortex points coincide.

DP

Lemma 1.2. Let (u1 , u2 ) be solutions of (P ) with z1 = z2 = · · · = zm and such that eu2 ∈ Xα with α > 2 and σ2
2N . For any q ∈ (2, α+4 3 ) there exists a constant Cq = C(q, c0 , α) > 0 such that



σ1 + 2σ2 − 4(N + 1)
Cq σ (q−2)/q eu2 2/q . (1.37) 2 Xα

Remark 1.2. Clearly, estimate (1.37) bears interesting consequences in case eu2 Xα (and hence σ2 ) is small, as it forces σ1 to remain close to the value 4(N + 1). Such a situation can actually occur, as we are going to construct family of solutions (u1,ε , u2,ε ) for (P ) (when all vortex points coincide) such that, as ε → 0, eu2,ε Xα = o(1), so g2 2π



eu2,ε = o(1),

R2

σ1,ε =

2g 2 π



eu1,ε = 4(N + 1) + o(1).

R2

CT E

σ2,ε =

(1.38)

(1.39)

CO R

RE

Proof. Without loss of generality suppose that z1 = · · · = zm = 0. Thus, by (1.30) we have:  2 2 2N −w2 (x) v1 v1 ∈ L∞ e loc R , −v1 = 4g |x| e  with w2 as defined in (1.24). Since eu2 ∈ Xα , then we also have R2 ln(1 + |x|)eu2 (x) dx < +∞, and in view of Lemma 1.2, we derive   w2 (x) = σ2 ln |x| + 1 + O(1). Consequently, setting R(x) = 4g 2 |x|2N e−w2 (x) , we see that   R(x) = O |x|2N−σ2 , and, by assumption, 2N − σ2  0. Furthermore, v1 (x) 2 eu1 = 2πσ1 R(x)e dx = 4g

(1.41)

R2

UN

R2

(1.40)

 and, from (1.40), we get R2 (1 + |x|2N−σ2 )ev1 < +∞. Thus, we are in position to apply Theorem 1 (when 2N = σ2 ), or Theorem 2 (when 2N > σ2 ) of Chen and Li [8], to conclude:

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πσ1 (σ1 − 4) =



x · ∇R(x)ev1 (x) dx

R2

 v1 v1 = 4g 2N R(x)e − x · ∇w2 (x)R(x)e 2

R2

= 4πNσ1 − 2πσ2 σ1 − 4g

2



R2

  x · ∇w2 (x) − σ2 R(x)ev1 .

That is,

R2

On the other hand, by (1.24) we have: 2

x · ∇w2 (x) − σ2
g 2π

g2
2π





x · ∇w2 (x) − σ2 R(x)ev1 (x) dx.

|y| u2 (y) g2 e dy
|x − y| 2π

R2



R2



R2



1 + |y|

2+α

e

qu2



|x − y|

{|x−y| 0 independent of x ∈ R2 . Hence, we can use (1.43) in (1.42), and by (1.41), obtain (1.37). ✷

We devote this section to the construction of a four-parameter family of solutions for (P ) in case all vortex points coincide, such that (1.38) and (1.39) hold. Without loss of generality we take all vortex points to coincide with the origin and obtain,

(i)

UN

Theorem 2.1. Let z1 = z2 = · · · = zm = 0, there exists ε0 > 0 sufficiently small such that problem (P) admits a five-parameters family of solutions (uε1,α , uε2,α ) with ε ∈ (−ε0 , ε0 ), α ∈ R4 and |α| < ε0 satisfying:

1

euε2,α = O(1) ∀α > 0, Xα ε

2g 2 π



R2

ε

eu1,α = 4(N + 1) + εO(1)

as ε → 0,

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13

(ii)     1 (N + 1)! ε − 1 + β1,α ln |x| + O(1), 2(N + 2) + ε 2c0 c0N+1   ε uε2,α (x) = −c0 |x|2 + 2(N + 1) + β2,α ln |x| + ln ε + O(1) as |x| → ∞,

uε1,α (x) = −

OF

ε → 0, β ε → 0 as ε → 0, |α| → 0, and O(1) denotes a quantity bounded uniformly in (ε, α). where β1,α 2,α

RO

In order to obtain Theorem 2.1, we shall establish a more general existence result concerning the problem:  m   2 u1 2 u2   −u nk δ(z − εzk ), = 4g e + g e − 4π 1     k=1     g 2  u2 (Pε ) e − φ02 + 2g 2 eu1 , u = 2 2  2 cos θ      eu1 < ∞,  eu2 < ∞    R2

R2

fε (z) = (N + 1)

m 

nk

(z − εzk ) ,

and Fε (z) =

k=1

z 0

8|fε (z)|2 , (1 + |Fε (z) + a|2 )2 and consider the radial function ρ = ρ(r) so that, ηε,a (z) =

fε (ξ ) dξ.

CT E

For any a ∈ C, define,

DP

with z1 , . . . , zm arbitrarily given points in R2 and small ε ∈ R. For this purpose, it is convenient to introduce complex notation and identify the pair x = (x1 , x2 ) ∈ R2 with the complex number z = x + iy ∈ C. Let

8(N + 1)2 |z|2N . (1 + |z|2N+2 )2 As well known (e.g., [12]), ∀ε  0 and ∀a ∈ C we have

RE

ρ(|z|) := ηε=0,a=0 (z) =

R2

CO R

m  − ln ηε,a = ηε,a − 4π nk δ(z − εzk ), k=1 ηε,a = 8π(N + 1).

u2 (z) = −c0 |z|2 + w2 (z) + ln ε − 2 ln g + v2 (z),

UN

and so, it satisfies ρ w2 = , 2

(2.2)

(2.3)

in R2 ,

We shall look for solutions of (Pε ) with a specific structure. Namely, we set   u1 (z) = ln ηε,a (z) + ε w1 (z) + v1 (z) − 2 ln 2g, where w2 is the radial function given by   w2 (r) = ln 1 + r 2N+2 ,

(2.1)

(2.4)

(2.5)

(2.6)

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while w1 is the radial solution in Yα , α ∈ (0, 21 ) of the equation:  2 w + ρw + e−c0 |z| 1 + |z|2N+2 = 0 in R2 ,

(2.7)

OF

as constructed in Lemma 2.1 of [7]. Consequently, relative to the unknowns (v1 , v2 ), problem (Pε ) becomes:  (eε(v1 +w1 ) − 1) 2 2    −v1 = e−c0 |z| +w2 +v2 + ηε,a − ρw1 − e−c0 |z| +w2 , ′ ε (Pε )  ρ 1 ε 2 +w +v  −c |z| 0 2 2  v2 = e + ηε,a eε(v1+w1 ) − , 2 2 2 2 cos θ

Lemma 2.1. Problem (2.7) admits a radial solution w1 such that w1 ∈ Yα ,

(ii)

 1 ∀α ∈ 0, ; 2

DP

(i)

RO

as it follows by direct computations. Notice that, by continuity problem (P′ε ) may be considered also at ε = 0, and for a = 0, (P′ε=0 ) admits the (trivial) solution v1 = v2 = 0. For ε small we aim to construct solutions for (P′ε ) which “bifurcate” from such trivial solution. To this purpose, we start by collecting some useful properties of the function w1 , which will be established in Appendix A.

 1 (N + 1)! ln r + O(1), as r → +∞; 1 − N+1 2c0 c0  rdw1 1 (N + 1)! , as r → +∞; → 1 − N+1 dr 2c0 c0

(iii)

+∞ −2ρ(r)w1 (r) 0

CT E

w1 (r) =

2 r 2(N+1) 1 e−c0 r r 2(N+1) r dr = + . 2(N+1) 2 2(N+1) 2c0 (1 + r ) 1+r

P : Yα 2 × C × R → Xα 2 by setting P = (P1 , P2 ) with

RE

From now on, the function w1 in (2.5) is chosen according to Lemma 2.1. For fixed α > 0 define the operator

(eε(v1+w1 ) − 1) 2 − ρw1 − e−c0 |z| +w2 , ε 1 ρ ε 2 e−c0 |z| +v2 +w2 − ηε,a eε(v1+w1 ) − , P2 (v1 , v2 , a, ε) = v2 − 2 2 cos θ 2 2 2 +w +v 2 2

+ ηε,a

CO R

P1 (v1 , v2 , a, ε) = v1 + e−c0 |z|

and extended by continuity at ε = 0. Thus, P (0, 0, 0, 0) = (0, 0), and finding a solution for problem (P′ε ) is now reduced to finding a small ε0 > 0 and an implicit function ε → (v1,ε , v2,ε , aε ) : (−ε0, ε0 ) → Yα 2 × C

UN

satisfying P (v1,ε , v2,ε , aε , ε) = 0, ∀ε ∈ (−ε0 , ε0 ). To this end, let a = a1 + ia2 and observe that,

∂ηε,a (z)

∂ηε,a (z)

= −4ρϕ , = −4ρϕ− , + ∂a1 a=0,ε=0 ∂a2 a=0,ε=0

(2.8)

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15

where, in polar coordinates, ϕ+ (r, θ ) =

r N+1 cos(N + 1)θ , 1 + r 2N+2

ϕ− (r, θ ) =

r N+1 sin(N + 1)θ . 1 + r 2N+2

′ ′ ′ = (P1,(v , P2,(v ) of P at (0, 0, 0, 0) may be easily computed Therefore, the linearized operator P(v 1 ,v2 ,a) 1 ,v2 ,a) 1 ,v2 ,a) as given by,

′ P2,(v (0, 0, 0, 0)[ψ1, ψ2 , b] = ψ2 1 ,v2 ,a)

ψ2 − 4ρw1 ϕ+ b1 − 4ρw1 ϕ− b2 ,

+ 2ρϕ+ b1 + 2ρϕ− b2 .

′ Set P(v (0, 0, 0, 0) = A, we have: 1 ,v2 ,a)

OF

2 +w 2

′ P1,(v (0, 0, 0, 0)[ψ1, ψ2 , b] = ψ1 + ρψ1 + e−c0 |z| 1 ,v2 ,a)

RO

Proposition 2.1. Let α ∈ (0, 12 ), then the operator A : Yα2 × C → Xα2 is onto, and   Ker A = Span (ϕ+ , 0), (ϕ− , 0), (ϕ0 , 0), (ω1 , 1) × {0}, where ϕ0 is the radial function: 1 − r 2(N+1) , 1 + r 2(N+1)

r = |z|.

DP

ϕ0 (r) =

Proof. In order to prove Proposition 2.1, we recall the following result established in [7] for the operator L =  + ρ : Yα → Xα .

CT E

Namely, if α ∈ (0, 21 ), then Ker L = Span{ϕ+ , ϕ− , ϕ0 }, (see Lemma 2.4 in [7]) and   f ϕ± = 0 Im L = f ∈ Xα : R2

(2.9)

(2.10)

2 +w 2

ψ1 + ρψ1 + e−c0 |z|

RE

(see Proposition 2.2 in [7]). Now, let (f1 , f2 ) ∈ Xα2 , we seek (ψ1 , ψ2 , b) ∈ Yα2 × C, b = b1 + ib2 such that ψ2 − 4ρw1 ϕ+ b1 − 4ρw1 ϕ− b2 = f1 ,

and

CO R

ψ2 + 2ρϕ+ b1 + 2ρϕ− b2 = f2 .

(2.11)

(2.12)

We start by considering (2.12). Decompose ψ2 = φ2 + 2d1 ϕ+ + 2d2ϕ− ,

R2

UN

where the constants d1 , d2 will be specified later, and φ2 ∈ Yα : φ2 ϕ± = 0.

(2.13)

(2.14)

Therefore, recalling that ϕ± + ρϕ± = 0, (2.12) holds if and only if φ2 = 2(d1 − b1 )ρϕ+ + 2(d2 − b2 )ρϕ− + f2 .

(2.15)

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In order to meet the orthogonality condition (2.14), necessarily: 2 2 = 0. f2 ϕ− + 2(d2 − b2 ) ρϕ− f2 ϕ+ + 2(d1 − b1 ) ρϕ+ = 0;

R2

3 4π(N + 1)

R2



R2

f2 ϕ− .

(2.17)

RO

d2 = b2 −

OF

By elementary calculations, we see that 2 2 ρϕ± = π(N + 1). 3 So (2.16) requires the choise: 3 d1 = b1 − f1 ϕ+ , 4π(N + 1)

(2.16)

R2

R2

R2

R2

Hence, inserting (2.17) into (2.15), and letting   3 3 f = f2 − f2 ϕ+ ρϕ+ − f2 ϕ− ρϕ− ∈ Xα , 2π(N + 1) 2π(N + 1) R2

R2

DP

it suffices to take 1 φ2 (x) = ln |x − y|f (y) dy, 2π R2

(2.18)

CT E

in order to obtain a solution for (2.14)–(2.15) in Yα . Thus ψ2 in (2.13) is determined by (2.17) and (2.18). Now, insert such ψ2 into (2.11) to obtain:   2 ψ1 + ρψ1 = g1 − 2 e−c0 |z| +w2 − 2ρw1 (b1 ϕ+ + b2 ϕ− )

with

g1 = f1 − e

−c0 |z|2 +w2

 φ2 −

(2.19)

  3 ϕ+ f2 ϕ+ + ϕ− f2 ϕ− , 2π(N + 1) R2

R2

R2

g1 ϕ− − 2b2



R2

R2



2 +w 2

e−c0 |z|

 2 = 0. − 2ρw1 ϕ−

CO R

R2



RE

and φ2 given in (2.18). Next, we are going to choose b1 and b2 in order to insure that the right hand side of (2.19) satisfies to the orthogonality conditions required by (2.10). Namely, we impose:   −c |z|2 +w 2 − 2ρw ϕ 2 = 0, g1 ϕ+ − 2b1 (2.20) e 0 1 + (2.21)

Recalling (2.6), we can use part (iii) of Lemma 2.1 to obtain:

R2

  −c |z|2 +w 2 − 2ρw ϕ 2 dx = π e 0 1 ±

+∞    −c r 2  e 0 1 + r 2(N+1) − 2ρ(r)w1 (r) 0

R2

UN

Consequently, we may choose: c0 c0 g1 ϕ+ , b2 = g1 ϕ− , b1 = π π R2

π r 2(N+1) . r dr = 2c0 (1 + r 2(N+1) )2

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17

OF

and obtain that the right-hand side of (2.19) belongs to the image of the operator L =  + ρ when defined over Yα (see (2.10)). Thus we derive ψ1 and the desired conclusion follows. Next, to determine Ker A, take f1 = f2 = 0 in the above computations. We find immediately, that d1 = b1 = 0, d2 = b2 = 0, and φ2 = 0. Hence, by Proposition 1.1(b), φ2 must be a constant. If φ2 = 0, then g1 = 0 and we must take ψ1 ∈ ker L. If φ2 = 0, say 2 2 φ2 = 1, then g1 = −e−c0 |z| +ω2 and ψ1 must satisfy: ψ1 + ρψ1 + e−c0 |z| +ω2 = 0. Thus, ψ1 ∈ ω1 + ker L and we conclude that ker A = W × {0} with W ⊂ Yα2 given by   W = Span ker L × {0}, (ω1 , 1) (2.22) as claimed. ✷

Denote by Vα ⊂ Yα 2 the space orthogonal to W defined in (2.22) with respect to the scalar product in Yα 2 . Hence, we may write:

RO

Yα 2 = Vα ⊕ W.

Furthermore, for given r > 0 denote by:   Qr = (ε, α) ∈ R5 : ε ∈ (−r, r) and α = (s, α0 , α1 , α2 ) ∈ R4 , |α| < r .

DP

By a direct application of the Implicit Function Theorem (see Theorem 2.7.5 in [11]) we may conclude: Theorem 2.2. Let α ∈ (0, 1/2) there exists ε0 > 0 sufficiently small and continuous functions   v(ε, α) = (v1 (ε, α)), v2 (ε, α) : Qε0 → Vα , a(ε, α) : Qε0 → C,

CT E

all vanishing at ε = 0 and α = 0, such that ∀ε ∈ (−ε0 , ε0 ) problem (Pε ) admits a four-parameter family of solutions (uε1,α , uε2,α ), α = (s, α0 , α1 , α2 ) ∈ R4 , |α| < ε0 , which decompose as follows:   uε1,α = ln ηε,a(ε,α) + ε (1 + s)w1 + α0 ϕ0 + α1 ϕ+ + α2 ϕ− + v1 (ε, α) − 2 ln 2g, uε2,α = −c0 |z|2 + w2 (z) + ln ε − 2 ln g + s + v2 (ε, α).

RE

At this point, if we take z1 = z2 = · · · = zm = 0, then problem (P) and (Pε ) are one and the same, and in view of (2.2), (2.4), (2.6), Lemma 2.1(ii), and Proposition 1.1(c), we easily derive Theorem 2.1. Final remark. It would be interesting to know whether or not a result similar to Theorem 2.1 remains valid even when the vortex points do not coincide.

CO R

Acknowledgements

The authors wish to thank M. Nolasco for useful comments. This research is supported partially by the grant no.2000-2-10200-002-5 from the basic research program of the KOSEF Korea, and by M.U.R.S.T. project: Variational Methods and Non Linear Differential Equation, Italy.

Appendix A

L1 =

UN

In order to establish Lemma 2.1, let us consider the operator 1 d d2 +ρ + 2 r dr dr

(A.1)

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given by the restriction of the operator L =  + ρ over the radial functions. Thus, we wish to solve L1 w = f

(A.2) −c0 r 2 +w2 (r)

r2

OF

with f (r) = −e = −e−c0 (1 + r 2(N+1) ). Chae and Imanuvilov in [7] have obtained an integral representation for solutions of (A.2) in case  1 1 + . (A.3) f ∈ C (R ) ∩ Xα and α ∈ 0, 2

0

with φf (r) = ϕ0 (r)

−2 (1 − r)2 r

r

ϕ0 (t)tf (t) dt,

ϕ0 (r) =

1 − r 2(N+1) , 1 + r 2(N+1)

0

where φf (1) and w(r) are extended by continuity at r = 1.

DP



RO

Lemma A.1 (see Lemma 2.1 in [7]). Assume (A.3), then (A.2) admits a solution w ∈ Yα given by the formula   r φf (1)r φf (s) − φf (1) (A.4) w(r) = ϕ0 (r) ds + 1−r (1 − s)2

In (r) =

r 0

  2 1 − t 2n te−c0 t dt,

and notice that  −2 (1 − r)2 IN+1 (r). φf (r) = − ϕ0 (r) r

CT E

In order to obtain the solution w1 as claimed in Lemma 2.1, we shall use such a representation formula with 2 f (r) = −e−c0 r (1 + r 2(N+1) ). To this purpose, for given n ∈ N, let

RE

Lemma A.2. The following identity holds:    n 2(n−k)  n! r n! 1 2 . 1 − n − e−c0 r 1 − In (r) = 2c0 c0 c0k (n − k)! k=0

In=1 (r) =

r 0

=−

CO R

Proof. We shall proceed by induction. For n = 1,   2 1 − t 2 te−c0 t dt

r 1 −c0 t 2

t =r d −c0 t 2 2 1 e e t dt +

2c0 2c dt 0 t =0

UN

0



1 2 2 = 1 − e−c0 r + r 2 e−c0 r − 2 2c0

r 0

te

−c0 t 2

dt



(A.5)

(A.6)

(A.7)

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P.19 (1-21) by:violeta p. 19 19

   1 1  −c0 r 2 2 −c0 r 2 −c0 r 2 e 1−e = −1 + +r e 2c0 c0    1 1 1 2 1 − − e−c0 r 1 − r 2 − , = 2c0 c0 c0

In+1 (r) =

r 0

OF

and (A.7) is established for n = 1. Now assume that (A.7) holds for n  1, and we proceed to prove it for n + 1. To this end, notice that   2 1 − t 2(n+1) te−c0 t dt

0

RO

  r 1 2n −c0 t 2 2(n+1) −c0 r 2 −c0 r 2 1−e dt − 2(n + 1) t te +r e = 2c0

 r 1 −c0 t 2 2(n+1) −c0 r 2 −c0 r 2 dt + 2(n + 1)In (r) − 2(n + 1) te +r e 1−e = 2c0 0     1 n + 1 −c0 r 2 2 2 1 − e−c0 r + r 2(n+1) e−c0 r + 2(n + 1)In (r) + = −1 . e 2c0 c0

DP



RE

CT E

Thus, if we apply the inductive assumption (A.7), substituting above, we find   n! 2(n + 1) 1 2 2 In+1 (r) = 1 − e−c0 r + r 2(n+1) e−c0 r + 1− n 2c0 2c0 c0    n  n + 1 −c0 r 2 n + 1 r 2(n−k) n! −c0 r 2 1− + −e − e c0 c0 c0k (n − k)! k=0    n  (n + 1)! r 2(n−k) (n + 1)! 1 2(n+1) −c0 r 2 1 − n+1 − e 1−r − = 2c0 c0 c0k+1 (n − k)! k=0    n+1 2[(n+1)−k]  r (n + 1)! (n + 1)! 1 2 1 − n+1 − e−c0 r 1 − r 2(n+1) − = 2c0 (n + 1 − k)! c0k c0 k=1    n+1 2[(n+1)−k]  r (n + 1)! (n + 1)! 1 −c0 r 2 1 − n+1 − e 1− = 2c0 (n + 1 − k)! c0k c0

CO R

k=0

and the desired identity is established.

✷

An immediate consequence of Lemma 3.2 gives  1 (N + 1)! IN+1 (r) → 1 − N+1 := γ0 as r → +∞. 2c0 c0

w1 (r) = −ϕ0 (r)

UN

Furthermore, for r > 2, inserting (A.7) into (A.4), we find r  2

1 + t 2(N+1) 1 − t 2(N+1)

2

IN+1 (t) dt + O(1) t

(A.8)

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D. Chae, G. Tarantello / Ann. I. H. Poincaré – AN ••• (••••) •••–•••

= −ϕ0 (r)

r 

1 + t 2(N+1) 1 − t 2(N+1)

2

2

γ0 dt + O(1) t

= −γ0 ϕ0 (r) ln r + O(1). Consequently, w1 (r) = γ0 ln r + O(1),

as r → +∞.

(A.9)

OF

On the other hand, for r → +∞,

RO

  −1  −1 r dw1 1 ′ (r) = −rϕ0 (r) ϕ0 (r) w1 (r) − ϕ0 (r) IN+1 (r) + O dr r    2(N+1)  −1 1 4(N + 1)r = − ϕ0 (r) w1 (r) + O . IN+1 (r) + 2(N+1) 2 r (1 + r )

So, taking into account (A.9), we immediately conclude that

DP

r dw1 (r) → γ0 as r → +∞ (A.10) dr and, part (ii) of Lemma 2.1 follows by (3.9) and (3.10). In order to establish (iii), notice that, by direct computation, we find  1 ρ(r)r 2(N+1) 8(N + 1)2 r 4N+2 1 L1 = = , 2 (1 + r 2(N+1) )2 (1 + r 2(N+1) )4 (1 + r 2(N+1) )2 2

+∞ ρ(r)

r 2(N+1) w1 (r)r dr (1 + r 2(N+1) )2

0

=

1 2 1 2

0

+∞

 +∞ d 1 1 1 d ρ(r)w1 (r) dr + r dr r w1 (r) dr dr 2 (1 + r 2(N+1) )2 (1 + r 2(N+1) )2

0 +∞

Consequently,

0

w1 (r)r dr

 +∞ d 1 1 d 1 w1 (r) dr + ρ(r)w1 (r) r r dr 2(N+1) 2 2(N+1) dr dr (1 + r 2 ) (1 + r )2

L1 w1

0

+∞

(1 + r 2(N+1) )2



0 +∞

1

(1 + r 2(N+1) )

r dr = − 2

0

0

1 2

+∞

2

e−c0 r r dr. 1 + r 2(N+1)

0

+∞ 2 1 e−c0 r r 2(N+1) 2ρ(r)w1 (r)r 2(N+1) 2 e−c0 r r dr = r dr = − . 2c0 1 + r 2(N+1) (1 + r 2(N+1) )2

UN

=

1

RE

1 = 2

+∞  L1

CO R

1 = 2

CT E

while, by definition, L1 w1 = −e−c0 r (1 + r 2(N+1) ). Therefore, in view of the asymptotic behaviors (A.9) and (A.10) of w1 as r → +∞, we can use integration by parts, to obtain:

0

✷

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References

[15] [16] [17]

OF

RO

[13] [14]

DP

[8] [9] [10] [11] [12]

CT E

[7]

RE

[6]

CO R

[5]

A.A. Abrikosov, On the magnetic properties of superconductors of second group, Sov. Phys. JETP, 5 (1957) 1174–1182. J. Ambjorn, P. Olesen, A magnetic condensate solution of the classical electroweak theory, Phys. Lett. B 218 (1989) 67–71. J. Ambjorn, P. Olesen, On electroweak magnetis, Nucl. Phys. B 315 (1989) 606–614. J. Ambjorn, P. Olesen, A condensate solution of the electroweak theory which interpolates between the broken and symmetry phase, Nucl. Phys. B 330 (1990) 193–204. D. Bartolucci, G. Tarantello, The Liouville equations with singular data and their applications to electroweak vortices, Comm. Math. Phys. 229 (2002) 3–47. H. Brezis, F. Merle, Uniform estimates and blow-up behaviour for solutions of −u = V (x)eu in two dimensions, Comm. Partial Differential Equations 16, (8,9), 1223–1253. D. Chae, O.Yu. Imanuvilov, The existence of non-topological multivortex solutions in the relativistic self-dual Chern–Simons theory, Comm. Math. Phys. 215 (2000) 119–142. W. Chen, C. Li, Qualitative properties of solutions to some nonlinear elliptic equations in R2 , Duke Math. J. 71 (2) (1993) 427–439. G. ’t Hooft, A property of electric and magnetic flux in nonabelian gauge theories, Nucl. Phys. B 153 (1979) 141–160. C.H. Lai (Ed.), Selected Papers on Gauge Theory of Weak and Electromagnetic Interactions, World Scientific, Singapore. L. Nirenberg, Topics in Nonlinear Analysis, in: Courant Lecture Notes in Math., American Mathematical Society, 2001. J. Prajapat, G. Tarantello, On a class of elliptic problems in R2 : symmetry and uniqueness results, Proc. Royal Soc. Edinburgh 131 (4) (2001) 967–985. J. Spruck, Y. Yang, On multivortices in the electroweak theory I: existence of periodic solutions, Comm. Math. Phys. 144 (1992) 1–16. J. Spruck, Y. Yang, On multivortices in the electroweak theory II: existence of Bogomol’nyi solutions in R2 , Comm. Math. Phys. 144 (1992) 215–234. C.H. Taubes, Arbitrary N -vortex solutions to the first order Ginzburg–Landau equation, Comm. Math. Phys. 72 (1980) 277–292. C.H. Taubes, On the equivalence of first order and second order equations for gauge theories, Comm. Math. Phys. 75 (1980) 207–227. Y. Yang, Solitons in Field Theory and Nonlinear Analysis, in: Springer Monographs in Math., Springer-Verlag, New York, 2001.

UN

[1] [2] [3] [4]