Sharp existence theorems for multiple vortices induced from magnetic impurities

Nonlinear Analysis 115 (2015) 117–129

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Sharp existence theorems for multiple vortices induced from magnetic impurities Ruifeng Zhang a,b,∗ , Huijuan Li b a

Institute of Contemporary Mathematics, Henan University, Kaifeng 475001, PR China

b

School of Mathematics and Statistics, Henan University, Kaifeng 475001, PR China

article

abstract

info

Article history: Received 21 August 2014 Accepted 15 December 2014 Communicated by S. Carl

In a recent work Tong and Wong derived self-dual equations governing multiple vortices in a product Abelian Higgs model which may be regarded as a generalized Ginzburg–Landau theory. The purpose of this paper is to establish some sharp existence and uniqueness theorems for these multiple vortex solutions in a doubly periodic domain and on the full plane using the methods of calculus of variations. © 2014 Elsevier Ltd. All rights reserved.

MSC: 35J20 35J50 35Q 58E15 81T13 Keywords: Abelian Higgs model Vortices Self-dual equations Calculus of variations

1. Introduction Let φ be a complex-valued scalar field defined over R2 . The gauge-covariant derivative is then given by Dj φ = ∂j φ − iAj φ, j = 1, 2, where Aj is a real-valued vector gauge field which induces a magnetic or vorticity field Fjk through Fjk = ∂j Ak − ∂k Aj , j, k = 1, 2. The associated total energy is E (φ, A) =

 

1

R2

2

2 F12

1

1

+ |D1 φ| + |D2 φ| + 2

2

2

2

λ 8

(|φ| − 1) 2

2



dx,

(1.1)

where λ > 0. The Euler–Lagrange equations, or the static Abelian Higgs equations, also known as the Ginzburg–Landau equations, associated with the energy (1.1) are Dk Dk φ =

∂k Fjk =

∗

i 2

λ

(|φ|2 − 1)φ,

(1.2)

(φ Dj φ − φ Dj φ),

(1.3)

2

Corresponding author at: Institute of Contemporary Mathematics, Henan University, Kaifeng 475001, PR China. E-mail address: [email protected] (R. Zhang).

http://dx.doi.org/10.1016/j.na.2014.12.009 0362-546X/© 2014 Elsevier Ltd. All rights reserved.

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R. Zhang, H. Li / Nonlinear Analysis 115 (2015) 117–129

which are still impossible to fully understand. In the special situations λ = 1 that borders type-I and type-II superconductivity, these equations enjoy a dramatic reduction into the so-called self-dual system of equations of the form D1 φ + iD2 φ = 0,

(1.4)

1

(1 − |φ|2 ), (1.5) 2 which are also known as the BPS equations, named after Bogomol’nyi [8] and Prasad–Sommerfield [37]. Existence and uniqueness of solutions of these equations have been established by Taubes [26,44,45] on the full plane and by Wang and Yang [49] on a doubly periodic domain to resemble Abrikosov’s vortices [1]. Since these works, a lot of results have been obtained and developed in numerous areas including the Chern–Simons models [9–14,16–18,22,25,30,31,34,35,40–43], electroweak theory [2–5,38,39,51], and cosmic strings [19,27,28,48], which have greatly enriched our understanding of field-theoretical vortices. Recently there have been a lot of activities in the study and development of field theoretical models governed by the supersymmetric defect dynamics. Supersymmetric gauge theories have been used as tractable, toy models to explore strongly coupled phenomena in high energy physics [21,46,47]. It is shown that one can add charged defects into field theories in d = 2 + 1 dimensions, preserving some amount of supersymmetry [23]. This provides the prospect of using supersymmetric methods to study strongly coupled phenomena in the presence of doped impurities or lattices [7]. As in the study of the classical Abelian Higgs model described above, with some special choices of the Higgs potentials in (2 + 1) dimensional gauge models, one can obtain interesting limiting BPS structures [8,37]. The appearance of such structures for certain special Higgs potentials may be ascribed either to extended supersymmetry [20,50] or to suitable dimensional reductions of some 4D selfdual Yang–Mills systems [45]. The solutions of these systems of equations are of the characteristics of topological defects resembling magnetic vortices in a type-II superconductor and have long played an important role in particle physics and condensed matter physics, and arise as a consequence of spontaneous symmetry breaking [53]. In the context of Abelian theories in d = 2 + 1 dimensions, the addition of electric and magnetic impurity has been explored recently in [23,24]. In [46], Tong and Wong study the dynamics of the monopoles in the presence of the spin impurities. A very similar problem is solved in [47], giving rise to an onset of the Abelian vortices in the presence of electric impurities. In [47], Tong and Wong give an analysis of more general spatially dependent impurities and their effects on vortices. They argued that a moduli space of solitons survives the addition of both electric and magnetic impurities. In the case of electric impurities, the metric remains unchanged but the dynamics is accompanied by a connection term, acting as an effective magnetic field over the moduli space. In contrast, magnetic impurities distort the metric on the moduli space. It is shown that magnetic impurities can be viewed as vortices associated to a separate, frozen, gauge group. It is also seen [53] that multiple distributed cosmic strings arise in the product Abelian gauge field theory of [47]. In particular, in [53], asymptotic behavior of the string solutions has been precisely described to allow the derivation of a necessary and sufficient condition for the gravitational metric to be geodesically complete and an explicit calculation of the deficit angle proportional to the string tension, both stated in terms of string numbers, energy levels of broken symmetries, and the universal gravitational constant. In the present paper, we aim at developing a complete existence theory for the multiple BPS vortices in the product Abelian gauge theory model of Tong and Wong [47]. Extending the methods of [15,29,30,32,33,52], we shall establish a series of existence and uniqueness theorems for these multiple vortex solutions under sharp conditions. The rest of the paper is organized as follows. In Section 2 we present the equations governing the multiple vortices in the product Abelian gauge field theory model of Tong and Wong [47]. In Section 3 we prove the existence and uniqueness of a multiple vortex solution over a doubly periodic domain by a constrained variational method under an explicitly stated necessary and sufficient condition. In Section 4 we prove the existence and uniqueness of a multiple vortex solution realizing an arbitrarily prescribed vortex distribution over R2 . F12 =

2. Vortex equations with magnetic impurities Following Tong and Wong [47], we consider some U (1) gauge theories in d = 2 + 1 dimensions. In the theory unspoilt by the presence of dirt, the action is given by

 S=−

d3 x



1 4e

F F jk + 2 jk

1 2e

(∂j φ)2 + |Dj q|2 + 2

e2 2

 (|q|2 − ξ )2 + φ 2 |q|2 ,

(2.1)

where the electric field is Ei = Fi0 , the magnetic field is B = F12 , a scalar q carry charge +1, φ is a neutral scalar, and ξ > 0 is a parameter. The simplest vortex solutions do not involve the neutral scalar φ . Setting Ei = 0, we can derive first order vortex equations by the BPS trick [8,37] B = e2 (|q|2 − ξ ) and 1

(D1 + iD2 )q ≡ Dz q = 0,

(2.2)

2

with z = x + ix . Then we consider the effect of magnetic impurities on vortices. The impurities are comprised of a fixed, static source term σ (x) for the magnetic field,

 Simpurity = −

d3 xσ (x)B.

(2.3)

R. Zhang, H. Li / Nonlinear Analysis 115 (2015) 117–129

119

It is shown in [23] that such an impurity preserves half of the supersymmetry if the auxiliary D field is similarly sourced. After integrating out this D term, we are left with the action [47]



d3 x

S=−



1 4e

F F jk + |Dj q|2 + 2 jk

e2 2

 (|q|2 − ξ − σ (x))2 + σ (x)B ,

(2.4)

where we have omitted the neutral scalar φ . Similarly, we can also derive the first-order BPS vortex equations. They are given by B = e2 (|q|2 − ξ − σ (x)),

Dz q = 0.

(2.5)

Fields obeying these  equations solve the full second order equations of motion and describe an object with mass M = 2π κξ , where κ = − 21π B ∈ Z+ is the winding number. Now by introducing two, charged scalar fields: q of charge (+1, −1) and p of charge (0, +1), we consider a theory with product gauge group Uˆ (1) × U˜ (1). The action is



d3 x

S=−



1 4e

Fˆ Fˆ jk + 2 jk

2 ˜2 ˜jk F˜ jk + |Dj q|2 + |Dj p|2 + e (|q|2 − ξ )2 + e (−|q|2 + |p|2 − ξ˜ )2 , F 4e˜ 2 2 2

1



(2.6)

where Dj q = ∂j q − iAˆ j q + iA˜ j q,

(2.7)

Dj p = ∂j p − iA˜ j p.

(2.8)

In [47], it is shown how they can take a particular limit so that the translationally invariant theory (2.6) reduces to the impurity theory (2.4). In the following, we suppose that ξ > 0 and ξ˜ > −ξ . The vacuum of the theory is |p|2 = ξ + ξ˜ and |q|2 = ξ and both U (1) factors of the gauge group are spontaneously broken. The theory has two types of vortices, one for each gauge group. The static vortex equations can be derived using the now-familiar BPS trick. They read Bˆ = e2 (|q|2 − ξ ),

(2.9)

Dz q = 0 ,

(2.10)

B˜ = e˜ 2 (−|q|2 + |p|2 − ξ˜ ),

(2.11)

Dz p = 0,

(2.12)

where Bˆ = Fˆ12 = ∂1 Aˆ 2 − ∂2 Aˆ 1 ,

(2.13)

B˜ = F˜12 = ∂1 A˜ 2 − ∂2 A˜ 1 .

(2.14)

To facilitate our computation, it will be convenient to adopt the derivatives

∂=

1

1 ∂¯ = (∂1 + i∂2 ),

(∂1 − i∂2 ),

2 and the notation

2

1

1 (Aˆ1 − iAˆ2 ), A˜ = (A˜1 − iA˜2 ). 2 2 As a consequence, Eqs. (2.7) and (2.10) become Aˆ =

∂ q = i(Aˆ − A˜ )q,

q ̸= 0.

(2.15)

(2.16)

(2.17)

Using (2.8) and (2.12), we have

˜ , ∂ p = iAp

p ̸= 0.

(2.18)

From (2.17), (2.18) we have

∂ ln q = i(Aˆ − A˜ ),

(2.19)

∂ ln p = iA˜ .

(2.20)

On the other hand, using (2.15), (2.16) and (2.20), we have B˜ = F˜12 = ∂1 A˜2 − ∂2 A˜1 = 2i(∂¯ A˜ − ∂ A˜ ∗ )

¯ ln p + ∂ ∂¯ ln p¯ ) = 2(∂∂ ¯ ln |p|2 = 1 1 ln |p|2 . = 2∂∂ 2

(2.21)

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R. Zhang, H. Li / Nonlinear Analysis 115 (2015) 117–129

From (2.19), we have similarly that Fˆ12 − F˜12 = Bˆ − B˜ =

1 2

1 ln |q|2 .

(2.22)

So, according to (2.9), (2.11), (2.21) and (2.22), we have 1 2 1 2

1 ln |q|2 = e2 (|q|2 − ξ ) + e˜ 2 (|q|2 − |p|2 + ξ˜ ),

(2.23)

1 ln |p|2 = e˜ 2 (|p|2 − |q|2 − ξ˜ ).

(2.24)

The complex-valued scalar fields p and q are allowed to independently generate vortices with their respectively prescribed zero sets Zp = {p1 , p2 , . . . , pn },

Zq = {q1 , q2 , . . . , qm }.

(2.25)

Solutions of (2.23)–(2.24) with the prescribed zero set (2.25) are to be obtained either over a doubly periodic lattice cell domain Ω induced from the ‘t Hooft type boundary condition [30,36] or over the full plane R2 satisfying the boundary condition

 |p| →

ξ + ξ˜ ,

|q| →



ξ as |x| → ∞.

(2.26)

Concerning these solution, our existence theorem may be stated as follows. Theorem 2.1. For the BPS vortex equations (2.9)–(2.12) defined over a doubly periodic cell domain Ω or over R2 , we have the following existence results. (i) When the equations are formulated over a doubly periodic cell domain Ω , for any given zero-sets date (2.25), there is a smooth solution (Aˆ j , A˜ j , p, q) with p and q realizing these zeros if and only if the coupling parameters e, e˜ , ξ , ξ˜ , the domain volume |Ω |, and the total vortex number m, n satisfy the conditions 4π (m + n) < µξ |Ω |,

(2.27)

4π (µn + λm + λn) < µλ(ξ + ξ˜ )|Ω |

(2.28)

where λ = 2e˜ , µ = 2e . Furthermore, if there exists a solution, it is unique up to gauge transformations. (ii) For any e, e˜ , ξ , ξ˜ and any given zero sets stated in (2.25), these equations have a finite-energy unique smooth solution (Aˆ j , A˜ j , p, q) over R2 so that (2.25) are the zero sets of p and q and that the boundary condition (2.26) is realized exponentially rapidly. (iii) In both doubly periodic and full-plane cases, there hold the magnetic fluxes and minimum energy of a multiple vortex solutions are given explicitly by 2

ˆ = Φ



˜ = Φ



2

Bˆ dx = 2π (m + n), B˜ dx = 2π n,

  E =

(2.29)

1

Bˆ 2 + 2

(2.30) 1

B˜ 2 + |Dj q|2 + |Dj p|2 + 2

2e 2e˜ = 2π (ξ m + [ξ + ξ˜ ]n),

e2 2

(|q|2 − ξ )2 +

e˜ 2 2

(−|q|2 + |p|2 − ξ˜ )2

 dx (2.31)

where the integrals are evaluated over the doubly periodic domain Ω or R , respectively. 2

Set u = ln |p|2 , v = ln |q|2 , and λ = 2e˜ 2 , µ = 2e2 . We arrive at the coupled system

1u = λ(eu − ev − ξ˜ ) + 4π

n 

δps1 (x),

(2.32)

s 1 =1

1v = −λeu + (µ + λ)ev + λξ˜ − µξ + 4π

m 

δqs2 (x),

(2.33)

s 2 =1

where we have included our consideration of the zero sets Zp , Zq of p, q as given in (2.25). In the subsequent sections, developing and extending the methods of [15,29,30,32,33,52], we study the existence and uniqueness of the system of (2.32) and (2.33).

R. Zhang, H. Li / Nonlinear Analysis 115 (2015) 117–129

121

3. Solution on doubly periodic domain In this section, we consider solutions of (2.32) and (2.33) defined over a doubly periodic domain Ω . We introduce two background functions u0 , v0 satisfying

1u0 = − 1v0 = −

4π n

+ 4π

|Ω |

n 

δps1 (x),

(3.1)

s1 =1

4π m

|Ω |

+ 4π

m 

δqs2 (x).

(3.2)

s 2 =1

Using the new variables w1 and w2 so that u = u0 + w1 , v = v0 + w2 , we can modify (2.32) and (2.33) into

1w1 = λ(eu0 +w1 − ev0 +w2 − ξ˜ ) +

4π n

|Ω |

,

(3.3)

1w2 = −λeu0 +w1 + (µ + λ)ev0 +w2 + λξ˜ − µξ +

4π m

|Ω |

.

(3.4)

Using (3.3) and (3.4), we have

∆(w1 + w2 ) = µev0 +w2 − µξ +

4π n

|Ω |

+

4π m

|Ω |

.

(3.5)

To proceed further, we take w1 + w2 = g, w1 = f . Then the governing system of equations becomes

1f = λ(eu0 +f − ev0 +g −f − ξ˜ ) + 1g = µev0 +g −f

4π n

, |Ω | 4π n 4π m − µξ + + . |Ω | |Ω |

(3.6) (3.7)

Integrating (3.7) and (3.6), we have

 Ω



ev0 +g −f dx = ξ |Ω | −



eu0 +f dx = Ω

Ω

4π n

µ

−

4π m

µ

ev0 +g −f dx + ξ˜ |Ω | −

≡ C1 > 0, 4π n

λ

(3.8)

= C1 + ξ˜ |Ω | −

4π n

λ

≡ C2 > 0.

(3.9)

Of course, the conditions (3.8) and (3.9) imply that the existence of an (n + m)-vortex solution requires that C1 > 0 and C2 > 0, which is 4π (m + n) < µξ |Ω |,

(3.10)

4π (µn + λm + λn) < µλ(ξ + ξ˜ )|Ω |.

(3.11)

We can prove that (3.10) and (3.11) is in fact sufficient for existence as well. We use W 1,2 (Ω ) to denote the usual Sobolev space of scalar-valued or vector valued Ω -periodic L2 -functions whose derivatives belong to L2 (Ω ). Lemma 3.1. Consider the constrained minimization problem min{I (f , g )|(f , g ) ∈ W 1,2 (Ω ), Jk (f , g ) = Ck , Ck > 0, k = 1, 2},

(3.12)

where I (f , g ) =

 

1

Ω

2

J1 (f , g ) = J2 (f , g ) =

 Ω

 Ω

1

4π n

2

|Ω |

µ|∇ f |2 + λ|∇ g |2 − λµξ˜ f − λµξ g +

µf +

4π λ

|Ω |

 (m + n)g dx,

(3.13)

ev0 eg −f dx,

(3.14)

eu0 ef dx.

(3.15)

Then a solution of (3.12) is a solution of Eqs. (3.6) and (3.7).

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R. Zhang, H. Li / Nonlinear Analysis 115 (2015) 117–129

Proof. It is clear that the Fréchet derivatives dJ1 , dJ2 of the constraint functional are linearly independent. According to (3.6) and (3.7), we have

µ1f = λµeu0 +f − λµev0 +g −f − λµξ˜ +

4π n

µ, |Ω | 4π n 4π m − λµξ + λ+ λ. |Ω | |Ω |

λ1g = λµev0 +g −f

(3.16) (3.17)

Let (f , g ) be a solution of (3.12). Then by standard elliptic regularity theory (f , g ) must be smooth and there exist Lagrange multipliers λ1 , λ2 ∈ R such that

µ1f = λ1 eu0 +f − λ2 ev0 +g −f − λµξ˜ +

4π n

µ, |Ω | 4π m 4π n λ+ λ. − λµξ + |Ω | |Ω |

λ1g = λ2 ev0 +g −f

(3.18) (3.19)

Integrating equation (3.19) and using J1 (f , g ) = C1 , we have λ2 = λµ. Integrating equation (3.18) and using J2 (f , g ) = C2 , we have λ1 = λµ. So, (f , g ) is a solution of Eqs. (3.16) and (3.17), namely, (f , g ) is a solution of Eqs. (3.6) and (3.7). The lemma is proven. The admissible set of the variational problem (3.12) will be denoted by

A = {(f , g ) ∈ W 1,2 (Ω )|Jk (f , g ) = Ck , k = 1, 2}.

(3.20)

When (3.8) and (3.9) are satisfied, C1 > 0, C2 > 0. Thus A ̸= ∅. Lemma 3.2. If the condition (3.10) and (3.11) holds, then (3.12) has a solution. In other words, the system (2.32) and (2.33) has a solution if and only if (3.10) and (3.11) is fulfilled. Proof. By virtue of Lemma 3.1, it is sufficient to show the existence of a minimizer of the constrained optimization problem (3.12). First, we proved that under the condition (3.10) and (3.11), the objective functional I is bounded from below on A. For this purpose, we rewrite each η ∈ W 1,2 (Ω ) as follows:

η = η + η′ , where η denotes the integral mean of η, η = |Ω1 | I (f , g ) =

 

1

Ω

2



Ω

ηdx and



Ω

η′ dx = 0. Hence, I may be put for (f , g ) ∈ A in the form

 1 µ|∇ f ′ |2 + λ|∇ g ′ |2 dx + Λ(f , g ), 2

(3.21)

where

Λ(f , g ) = −λµ|Ω |(ξ˜ f + ξ g ) + 4π nµf + 4π λ(m + n)g .

(3.22)

We can derive from (3.8) and (3.9) the expressions

 f = ln C2 − ln

′

Ω



g = ln(C1 C2 ) − ln



eu0 +f dx , ′

(3.23)



eu0 +f dx Ω

  ′ ′ − ln ev0 +g −f dx . Ω

Inserting (3.23) and (3.24) into (3.22), we have

  u 0 +f ′ ˜ Λ(f , g ) = [λµ|Ω |(ξ + ξ ) − 4π (µn + λm + λn)] ln e dx Ω   ′ ′ + [λµ|Ω |ξ − 4π λ(m + n)] ln ev0 +g −f dx + C3 , Ω

where C3 = (4π nµ − λµ|Ω |ξ˜ ) ln C2 + [4π λ(m + n) − λµ|Ω |ξ ] ln(C1 C2 ).

(3.24)

R. Zhang, H. Li / Nonlinear Analysis 115 (2015) 117–129

123

Using Jensen’s inequality, we get



    1 ≥ ln |Ω | exp (u0 + f ′ )dx Ω    | Ω | Ω 1 u0 dx , = ln |Ω | exp |Ω | Ω       1 ′ ′ ln ev0 +g −f dx ≥ ln |Ω | exp v0 dx . |Ω | Ω Ω ln

′

eu0 +f dx



So, we have

    1 ˜ Λ(f , g ) ≥ [λµ|Ω |(ξ + ξ ) − 4π (µn + λm + λn)] ln |Ω | exp u0 dx |Ω | Ω     1 + [λµ|Ω |ξ − 4π λ(m + n)] ln |Ω | exp v0 dx + C3 . |Ω | Ω

(3.25)

Inserting (3.25) into (3.21), we arrive at the coercive lower estimate I (f , g ) ≥

 

1

Ω

2

1

µ|∇ f | + λ|∇ g | ′ 2

′ 2



2

dx − |C4 |,

(3.26)

where |C4 | > 0 is an irrelevant constant. From (3.26), we know that the existence of solution of (3.12) follows. In fact, let {(fj , gj )} ⊂ A be a minimizing sequence of the variational problem (3.12) and set f = j



1

|Ω |

Ω

fj dx,

g = j

1



|Ω |

Ω

gj dx.

(3.27)

Then, with fj′ = fj − f and gj′ = gj − g , we have f ′ = 0 and g ′ = 0. In view of (3.26), we see that {(fj′ , gj′ )} is bounded j

j

j

j

in W 1,2 (Ω ). Without loss of generality, we may assume that {(fj′ , gj′ )} converges weakly in W 1,2 (Ω ) to an element (f ′ , g ′ ). The compact embedding W 1,2 (Ω ) ↩→ Lp (Ω ),

p ≥ 1,

(3.28)

then implies (fj , gj ) → (f , g ) in L (Ω ) (p ≥ 1) as j → ∞. In particular, f = 0 and g = 0. Recall the Trudinger–Moser inequality [6] ′

′

′

′



 Ω

eU dx ≤ C (ε) exp

′

p

1 16π

+ε

 Ω

 |∇ U |2 dx ,

′

U ∈ W 1,2 (Ω ), U = 0,

(3.29)

where C (ε) > 0 is a constant. In view of (3.23) and (3.24), we see that the functionals defined by the right-hand side of (3.23) and (3.24) are continuous in f ′ , g ′ with respect to the weak topology of W 1,2 (Ω ). Therefore, f → f , g → g as j → ∞, as given in (3.23) and (3.24). j

j

In other words, (f , g ) = (f + f ′ , g + g ′ ) satisfies the constraints (3.8) and (3.9), and solves the constrained minimization problem (3.12). Thus, Lemma 3.2 is proven. Lemma 3.3. If system (3.6) and (3.7) has a solution, then the solution must be unique. Proof. Consider the following functional, J (f , g ) =

1

1

µ∥∇ f ∥22 + λ∥∇ g ∥22 − λµξ˜ f |Ω | − λµξ g |Ω | + 4π nµf + 4π λ(m + n)g

2 

2

v 0 +g −f

+

{λµe Ω

+ λµeu0 +f }dx.

It is straightforward to check by calculating the Hessian that J is strictly convex in W 1,2 (Ω ). Thus, J has at most one critical point. However, any solution of (3.6) and (3.7) must be a critical point of J. This proves the lemma. We may summarize our results as follows. Theorem 3.4. The system of the product Abelian Higgs vortex Eqs. (3.6) and (3.7) has a solution if and only if (3.10) and (3.11) hold. Furthermore, if a solution exists, it must be unique.

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R. Zhang, H. Li / Nonlinear Analysis 115 (2015) 117–129

4. Solution on full plane In this section, we prove the existence and uniqueness of the solution of the system of equations (2.32) and (2.33) over R2 satisfying the boundary condition u → ln(ξ + ξ˜ ),

v → ln ξ as |x| → ∞.

(4.1)

It will be convenient to work with the redefined variables and parameters u → u + ln(ξ + ξ˜ ),

v → v + ln ξ ,

λξ → λ,

λξ˜ → ν,

µξ → µ,

(4.2)

in this section. Thus, (2.32) and (2.33) become n 

1u = (λ + ν)eu − λev − ν + 4π

δps1 (x),

(4.3)

s1 =1

1v = −(λ + ν)eu + (λ + µ)ev + ν − µ + 4π

m 

δqs2 (x),

(4.4)

s2 =1

and the boundary condition (4.1) is translated into u → 0,

v → 0 as |x| → ∞.

(4.5)

To proceed further, we introduce the background functions [26] u0 (x) = −

n 

ln(1 + τ |x − ps1 |−2 ),

τ > 0,

(4.6)

ln(1 + τ |x − qs2 |−2 ),

τ > 0.

(4.7)

s1 =1

v 0 ( x) = −

m  s 2 =1

Then, we have

1u0 = −h1 (x) + 4π

n 

δps1 (x),

h1 (x) = 4

s1 =1

1v0 = −h2 (x) + 4π

m 

s1

δqs2 (x),

h2 (x) = 4

s 2 =1

τ , (τ + | x − ps1 |2 )2 =1

(4.8)

τ . (τ + |x − qs2 |2 )2 =1

(4.9)

n 

m  s2

Using the substitution u = u0 + w1 , v = v0 + w2 , we have

1w1 = (λ + ν)eu0 +w1 − λev0 +w2 − ν + h1 (x), 1w2 = −(λ + ν)e

u0 +w1

v0 +w2

+ (λ + µ)e

(4.10)

+ ν − µ + h2 (x).

(4.11)

By (4.10) and (4.11), we have

∆(w1 + w2 ) = µev0 +w2 − µ + h1 (x) + h2 (x).

(4.12)

Taking f = w1 ,

g = w1 + w2 ,

(4.13)

we have

1f = (λ + ν)eu0 +f − λev0 +g −f − ν + h1 (x), v 0 +g −f

1g = µe

(4.14)

− µ + h1 (x) + h2 (x).

(4.15)

Firstly, we consider the existence and uniqueness of the solution of the system of equations (4.14) and (4.15) over R2 satisfying the boundary condition f → 0,

g →0

as |x| → ∞.

(4.16)

It is clear that (4.14) and (4.15) are the Euler–Lagrange equations of the action functional I (f , g ) =

  R2

 ν |∇ f | + |∇ g | + 1 + (eu0 +f − eu0 ) 2λ 2µ λ 1

2

1

+ (ev0 +g −f − ev0 ) +

2

1

λ



(h1 − ν)f − g +

1

µ

 (h1 + h2 )g dx,

(4.17)

R. Zhang, H. Li / Nonlinear Analysis 115 (2015) 117–129

125

then the functional I is a C 1 -functional for f , g ∈ W 1,2 (R2 ) and its Fréchet derivative satisfies

  ν |∇ g |2 + 1 + (eu0 +f − 1)f + ev0 (eg −f − 1)(g − f ) µ λ R2 λ  1 1 v0 + (e − 1)(g − f ) + h1 f + (h1 + h2 )g dx. λ µ  

DI (f , g )(f , g ) =

1

|∇ f |2 +

1

(4.18)

Since

|∇ f |2 + |∇ g |2 = 2|∇w1 |2 + |∇w2 |2 + 2(∇w1 , ∇w2 ),

(4.19)

|∇ f |2 + |∇ g |2 ≤ 3|∇w1 |2 + 2|∇w2 |2 ≤ 3(|∇w1 |2 + |∇w2 |2 ).

(4.20)

hence

On the other hand, we have

|∇ f |2 + |∇ g |2 ≥ 2|∇w1 |2 + |∇w2 |2 − 2|(∇w1 , ∇w2 )|   1 ≥ 2− |∇w1 |2 + (1 − ε)|∇w2 |2 , ε for any ε ∈ ( 21 , 1). Taking ε =

|∇ f |2 + |∇ g |2 ≥

1 2

2 , 3

(4.21)

we get 1

1

3

3

|∇w1 |2 + |∇w2 |2 ≥

(|∇w1 |2 + |∇w2 |2 ).

(4.22)

So, we have 1

(w1 2 + w2 2 ) ≤ f 2 + g 2 ≤ 3(w1 2 + w2 2 ). 3 As a consequence of (4.18), (4.22) and (4.23), with σ = max{λ, µ}, we get DI (f , g )(f , g ) −

1 3σ



2

2

 

{|∇w1 | + |∇w2 | }dx ≥ R2

R2



w1 e 

u0 +w1

(4.23)

 − 1 + h1

+ w2 ev0 +w2 − 1 +

1

µ

1

λ

+

( h1 + h2 )

1



µ 

+

1

µ

 h2

dx

≡ M1 (w1 ) + M2 (w2 ),

(4.24)

where

    1 1 1 w1 eu0 +w1 − 1 + h1 + + h2 dx, λ µ µ R2    1 M2 (w2 ) = w2 ev0 +w2 − 1 + (h1 + h2 ) dx. µ R2 Using the elementary inequality et − 1 ≥ t , t ∈ R, we have     1 1 1 1 1 1 + + h 2 ≥ u0 + w 1 + h 1 + + h2 . (4.25) eu0 +w1 − 1 + h1 λ µ µ λ µ µ We decompose w1 , w2 into their positive and negative parts, w1 = w1+ − w1− , w2 = w2+ − w2− , where q+ = max{q, 0} and q− = − min{q, 0} for q ∈ R [26], which leads to       1 1 1 2 M1 (w1+ ) ≥ (w1+ ) dx + w 1 + u0 + h 1 + + h2 dx λ µ µ R2 R2   2    1 1 1 1 1 2 ≥ (w1+ ) dx − u0 + h 1 + + h2 dx. (4.26) 2 R2 2 R2 λ µ µ M1 (w1 ) =



t , t ≥ 0, we have On the other hand, using the inequality 1 − e−t ≥ 1+ t





1

1



1



w1− 1 − e − h1 + − h2 λ µ µ     w1− 1 1 1 u0 u0 − e − h1 + − h2 ≥ w1− 1 + e 1 + w1− λ µ µ         2 (w1− ) 1 1 1 w1− 1 1 1 = 1 − h1 + − h2 + 1 − eu0 − h1 + − h2 . 1 + w1− λ µ µ 1 + w1− λ µ µ u0 −w1−

(4.27)

126

R. Zhang, H. Li / Nonlinear Analysis 115 (2015) 117–129

In view of (4.8) and (4.9), we see that we may choose τ > 0 large enough so that



1

λ

+

1



µ

1

h1 <

4

1

,

µ

h2 <

1 4

,

x ∈ R2 .

(4.28)

On the other hand, since 1 − eu0 , h1 and h2 both lie in L2 (R2 ), we have

 R2

     w1−  1 1 1  (w1− )2 u0 dx ≤ ε dx + C (ε), − h + − h 1 − e 1 2 1 + w1−  λ µ µ  R2 1 + w1−

(4.29)

where ε > 0 may be chosen to be arbitrarily small. Combining (4.27) and (4.29), we get 1

M1 (−w1− ) ≥ provided that ε < M1 (w1 ) ≥

4

1 . 4

1

 R2

(4.30)

From (4.26) and (4.30), we get

w1 2 dx − C , 1 + |w1 |



4

(w1− )2 dx − C1 (ε), 1 + w1−

R2

(4.31)

where and in the sequel we use C to denote an irrelevant generic positive constant. Similar estimates may be made for M2 (w2 ). Thus, (4.24) gives us



1

DI (f , g )(f , g ) −

3σ

{|∇w1 |2 + |∇w2 |2 }dx ≥ R2

1

 

4

R2

 w1 2 w2 2 + dx − C . 1 + |w1 | 1 + |w2 |

(4.32)

We now recall the well-known Gagliardo–Nirenberg inequality





4

2

u dx ≤ 2 R2



u dx R2

R2

|∇ u|2 dx,

u ∈ W 1,2 (R2 ).

(4.33)

Consequently, we have



u2 dx

2

2 |u| (1 + |u|)|u|dx R 2 1 + | u|   u2 ≤2 dx (u2 + u4 )dx 2 2 2 ( 1 + | u |) R R      u2 2 2 ≤4 dx u dx 1 + |∇ u | dx 2 R2 (1 + |u|) R2 R2   2  4  4  1 u2 2 2 ≤ u dx +C 1+ dx + |∇ u| dx . 2 2 R2 R2 (1 + |u|) R2 

=

R2

(4.34)

As a result of (4.34), we have



2

u dx R2

 21





2

≤C 1+

u2



|∇ u| dx + R2

R2

(1 + |u|)2



dx .

(4.35)

Applying (4.35) in (4.32), we arrive at DI (f , g )(f , g ) ≥ C1 (∥w1 ∥W 1,2 (R2 ) + ∥w2 ∥W 1,2 (R2 ) ) − C2 .

(4.36)

Thus, using (4.20), (4.22) and (4.23) in (4.36), we conclude with the coercive lower bound DI (f , g )(f , g ) ≥ C1 (∥f ∥W 1,2 (R2 ) + ∥g ∥W 1,2 (R2 ) ) − C2 .

(4.37)

With (4.37), we can now show that the existence of a critical point of the action functional (4.17) follows using a standard argument as in [52]. In fact, from (4.37), we can choose R > 0 large enough such that inf{DI (f , g )(f , g )|f , g ∈ W 1,2 (R2 ), ∥f ∥W 1,2 (R2 ) + ∥g ∥W 1,2 (R2 ) = R} ≥ 1.

(4.38)

Now consider the minimization problem

η = min{I (f , g )|∥f ∥W 1,2 (R2 ) + ∥g ∥W 1,2 (R2 ) ≤ R}. (4.39) Let {(fk , gk )} be a minimizing sequence of (4.39). Without loss of generality, we may assume that {(fk , gk )} weakly converges to an element (f , g ) in W 1,2 (R2 ). The weakly lower semi-continuity of I implies that (f , g ) solves (4.39). To show that (f , g ) is a critical point of I, it suffices to see that it is an interior point. That is, ∥f ∥W 1,2 (R2 ) + ∥g ∥W 1,2 (R2 ) < R.

(4.40)

R. Zhang, H. Li / Nonlinear Analysis 115 (2015) 117–129

127

Suppose otherwise that ∥f ∥W 1,2 (R2 ) + ∥g ∥W 1,2 (R2 ) = R. Then for s ∈ (0, 1) the point (1 − s)(f , g ) is interior which gives us I ((1 − s)f , (1 − s)g ) ≥ η = I (f , g ).

(4.41)

On the other hand, we have I ((1 − s)(f , g )) − I (f , g ) d lim = I ((1 − s)(f , g ))|s=0 s→0 s ds

= −DI (f , g )(f , g ) ≤ −1.

(4.42)

Consequently, if s > 0 is sufficiently small, (4.42) leads to I ((1 − s)f , (1 − s)g ) < I (f , g ) = η,

(4.43)

which contradicts (4.41). Therefore, the existence of a critical point of I follows. Note that the part in the integrand of I which does not involve the derivatives of f and g may be rewritten as Q (f , g ) =

 1+

 1 ν 1 (eu0 +f − eu0 ) + (ev0 +g −f − ev0 ) + (h1 − ν)f − g + (h1 + h2 )g , λ λ µ

(4.44)

whose Hessian is easily checked to be positive definite. Thus, the functional I is strictly convex. As a consequence, I can have at most one critical point (f , g ) in the space W 1,2 (R2 ). Using the methods in [15], it can be shown that the right-hand side of (4.14) and (4.15) belongs to L2 (R2 ). We may now apply the standard elliptic theory to (4.14) and (4.15) to infer f , g ∈ W 2,2 (R2 ). In particular, f and g approach zero as |x| → ∞, which renders the validity of the boundary condition (4.16). Finally, we derive the decay rates for u, v and |∇ u|, |∇v|. Consider (4.3) and (4.4) outside the disk DR = {x ∈ R2 | |x| < R}, where R > max{|ps1 |, |qs2 | |s1 = 1, 2, . . . , n, s2 = 1, 2, . . . , m}. We rewrite (4.3) and (4.4) in R2 \ DR as

1u = (λ + ν)eu − λev − ν,

(4.45) v

1v = −(λ + ν)e + (λ + µ)e + ν − µ. u

(4.46)

By computation, we have

∆(u2 + v 2 ) = 2(|∇ u|2 + |∇v|2 ) + 2(λ + ν)u(eu − 1) − 2(λ + ν)v(eu − 1)

− 2λu(ev − 1) + 2v(λ + µ)(ev − 1), x ∈ R2 \ DR . (4.47) Noting u, v → 0 as |x| → ∞, for any ε : 0 < ε < 1, we can find a suitably large Rε > R and take t = min(λ + ν, λ + µ) so that

∆(u2 + v 2 ) ≥ (1 − ε)[2(λ + ν)u2 + 2(λ + µ)v 2 ] − 2(1 + ε)(2λ + ν)|uv|

≥ (1 − ε)t (u2 + v 2 ),

x ∈ R 2 \ D Rε .

(4.48)

Thus, using a comparison function argument and the property u2 + v 2 = 0 at infinity, we can obtain a constant C (ε) > 0 to make √

u2 (x) + v 2 (x) ≤ C (ε)e− (1−ε)t |x|

(4.49)

valid. Let ∂ denote any of the two partial derivatives, ∂1 and ∂2 . Then (4.45) and (4.46) yield

∆(∂ u) = (λ + ν)(∂ u)eu − λ(∂v)ev ,

(4.50) v

∆(∂v) = −(λ + ν)(∂ u)e + (λ + µ)(∂v)e . u

(4.51)

By computation and then using the Cauchy inequality, we get

∆(|∇ u|2 + |∇v|2 ) = 2[|∇(∂1 u)|2 + |∇(∂2 u)|2 + |∇(∂1 v)|2 + |∇(∂2 v)|2 ] + 2(λ + ν)|∇ u|2 eu + 2(λ + µ)|∇v|2 ev − 2λ∂1 u∂1 v ev − 2λ∂2 u∂2 v ev − 2(λ + ν)∂1 u∂1 v eu − 2(λ + ν)∂2 u∂2 v eu ≥ 2(λ + ν)|∇ u|2 eu + 2(λ + µ)|∇v|2 ev − λ(|∇ u|2 + |∇v|2 e2v )

− (λ + ν)(|∇v|2 + |∇ u|2 e2u ),

x ∈ R2 \ DR .

(4.52)

Therefore, as before, we conclude that for any ε : 0 < ε < 1, there is a R˜ε > R, so that

∆(|∇ u|2 + |∇v|2 ) ≥ (1 − ε)t (|∇ u|2 + |∇v|2 ),

x ∈ R2 \ DR˜ε .

(4.53)

Noting the property |∇ u|2 + |∇v|2 = 0 at infinity, applying the comparison principle, we arrive at

|∇ u|2 + |∇v|2 ≤ C (ε)e− where C (ε) > 0 is a constant.

√ (1−ε)t |x|

,

|x| > R,

(4.54)

128

R. Zhang, H. Li / Nonlinear Analysis 115 (2015) 117–129

We may summarize our results as follows. Theorem 4.1. For any distribution of the points p1 , p2 , . . . , pn , q1 , q2 , . . . , qm ∈ R2 , the system of nonlinear elliptic equations (4.3) and (4.4) subject to the boundary condition (4.5) has a unique solution. Furthermore, the solution satisfies the decay estimates √

u2 (x) + v 2 (x) ≤ C (ε)e− (1−ε)t |x| , √ − (1−ε)t |x|

|∇ u| + |∇v| ≤ C (ε)e 2

2

(4.55)

,

(4.56)

for x ∈ R2 near infinity, where t = min(λ + ν, λ + µ), ε ∈ (0, 1) is arbitrary and C (ε) > 0 is a constant. Finally, it can be seen that part (iii) of Theorem 2.1 follows from the estimates obtained above and the calculation in [53]. Acknowledgments This work was supported in part by the National Natural Science Foundation of China (11471099, 11271052) and the Basic Science and Frontier Technology Program Funds of Henan Province (142300410119). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41]

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