for the Chern-Simons System with Two Higgs Particles

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Digital Object Identifier (DOI) 10.1007/s00220-010-1021-z

Mathematical Physics

Uniqueness of Topological Solutions and the Structure of Solutions

for the Chern-Simons System with Two Higgs Particles

Jann-Long Chern¹^,, Zhi-You Chen¹, Chang-Shou Lin²

1 Department of Mathematics, National Central University, Chung-Li 32001, Taiwan. E-mail: chern@math.ncu.edu.tw; zhiyou@math.ncu.edu.tw 2 Department of Mathematics, Taida Institute for Mathematical Sciences,

National Taiwan University, Taipei 10617, Taiwan. E-mail: cslin@math.ntu.edu.tw

Received: 3 October 2008 / Accepted: 7 January 2010 Published online: 4 March 2010 – © Springer-Verlag 2010

Abstract: The existence of topological solutions for the Chern-Simons equation with two Higgs particles has been proved by Lin, Ponce and Yang [16]. However, both the uniqueness problem and the existence of non-topological solutions have been left open.

In this paper, we consider the case of one vortex at origin. Among others, we prove the uniqueness of topological solutions and give a complete study of the radial solutions, in particular, the existence of some non-topological solutions.

1. Introduction and Main Results

In this paper, we will consider the nonlinear elliptic system

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

u +λe^v(1−e^u)=4π ^N

s=1

α_sδp_s in R², v+λe^u(1−e^v)=4π ^N

s=1

αsδp_s in R²,

(1.1)

where = ₂

i=1 ∂²

∂x²_i , λis a positive constant,Nand N are two positive constants which are called the vortex numbers,αs >0 andαs > 0 are constants, andδpis the Dirac measure at p. Equation (1.1) arises from a relativistic Abelian Chern-Simons model with two Higgs particles. For any solution(u, v)to Eq. (1.1), we let z=x¹+ i x²

Work partially supported by National Science Council of Taiwan.

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and defineφ, χ,Ar⁽¹⁾and Ar⁽²⁾,r=1,2, in the following:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

θ1(z)= − ^N

s=1

arg(z−p_s), θ2(z)= − ^N

s=1

arg(z−p_s), φ(z)=e¹²^u⁽^z⁾⁺ⁱ^θ¹⁽^z⁾, χ(z)=e¹²^v(^z⁾⁺ⁱ^θ²⁽^z⁾,

A⁽₁¹⁾(z)= −Re{2i∂lnφ(z)},A⁽₂¹⁾(z)= −Im{2i∂lnφ(z)}, A⁽₁²⁾(z)= −Re{2i∂lnχ(z)},A⁽₂²⁾(z)= −Im{2i∂lnχ(z)};

(1.2)

hereφandχ are interpreted as two complex scalar fields in R²representing two Hi- ggs particles, and Ar⁽¹⁾ and A⁽r²⁾,r = 1,2, are two gauge fields. Then (φ, χ,A⁽r^I⁾), I = 1,2,r = 1,2, satisfy the self-dual equation for the Chern-Simons-Higgs model with two Higgs particles. For details of computations, we refer the readers to [8,15,16]

and the references therein.

For the past twenty years, the equation of Chern-Simons with one Higgs particle has been intensively studied, e.g., see [2–4,6–8,10–15,17–20,22,23] and references therein.

However, the study for the system (1.1) only recently began with the paper [16].

For Eq. (1.1), there are two natural boundary conditions for solutions at∞, namely, (i) lim

|x|→∞u(x)= lim

|x|→∞v(x)=0, or (ii) lim

|x|→∞u(x)= lim

|x|→∞v(x)= −∞. (1.3) We note that if(u, v)is a solution with the boundary condition either (i) or (ii), then, by the maximum principle, we have u(x) <0 andv(x) <0 for all x ∈ R². In physics literature, a solution(u, v)satisfying boundary condition (i) is called a topologi- cal solution. Since the nonlinear term e^v(1−e^u)= −e^vu + O(|u|²)for u small and u(x), v(x)→0 as|x| →+∞, by the estimates of elliptic PDE, we know that if(u, v) is a topological solution of (1.1), then both|u|and|v|decay exponentially at∞.

To solve (1.1), one may consider a regularized form:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

u +λe^v(1−e^u)= ^N

s=1 4αsε (ε+|x−ps|²)²

v+λe^u(1−e^v)= ^N

s=1 4αsε (ε+|x−p_s|²)²,

(1.4)

whereεis a small positive number, and introduce the background functions u^ε₀(x)=

N

s=1

4αsln

ε+|x−p_s|² 1 +|x−p_s|²

, v^ε₀(x)=

N

s=1

4αs ln

ε+|x−p_s|² 1 +|x−p_s|²

. Then

u^ε₀(x)= −h1(x)+

N

s=1

4αsε

(ε+|x−p_s|²)², v^ε₀= −h2(x)+

N

s=1

4αsε (ε+|x−p_s|²)², where h1,h2∈ W¹^,²do not depend onε >0. By letting u=u^ε₀+ f, v=v₀^ε+ g, the regularized form of (1.1) becomes

f +λe^v^ε⁰^+g(1−e^u^ε⁰^{+ f})=h1

g +λe^u^ε⁰^{+ f}(1−e^v^ε⁰^+g)=h2. (1.5)

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It is clear that (1.5) is the Euler-Lagrange equations of the nonlinear functional:

I(f,g)=

(∇f · ∇g +λe^u^ε⁰⁺^v^ε⁰^{+ f +g}−λe^u^ε⁰^{+ f}

−λe^v^ε⁰^+g+ h2f + h1g)d x.

(1.6)

We refer to [16] for the details of arguments. From (1.6), we see that Eq. (1.1) is the so-called skew gradient system in the literature, see [21]. Clearly, the indefinite form of I presents a lot of difficulties for solving Eq. (1.1). Hence, it is remarkable that in Lin-Ponce-Yang [16], they are able to show the existence of topological solutions for Eq. (1.1) for any given set of singularities.

Theorem A. [16] For any given sets{p₁, . . . ,p_N}and{p₁, . . . ,p_N}andαs, αs>0, Eq. (1.1) possesses a topological solution(u, v).

After TheoremA, it is natural to ask the question about the uniqueness of topological solutions for Eq. (1.1). For the single Chern-Simons-Higgs model, the uniqueness result was proved in [6] with only one singularity, and in [3and19] for multi-singularity in R²and largeλas well as in the periodic case. In this article, we consider the topological solution(u, v)for the case N=N=1 and p₁ and p₁to be the origin O. Then(u, v) satisfies

u + e^v(1−e^u)=4πN1δ0

v+ e^u(1−e^v)=4πN2δ0 in R², (1.7) with the boundary condition

u(x)→0, v(x)→0 as|x| → ∞. (1.8)

By noting u(x) < 0, v(x) < 0 for x ∈ R², and by applying the standard method of moving planes, we can show that(u, v)is radially symmetric with respect to the origin O. The proof is standard, and will be omitted here. We refer to [1] for the details of the proof.

To a single nonlinear elliptic equation, the uniqueness problem has been extensively studied for the last decades. It is well-known that the uniqueness problem is closely related to non-degeneracy of its linearized equation. See [4,5] and references therein.

In this paper, we also want to prove the uniqueness by studying the non-degeneracy of linearized equations. The linearized equation at (u, v)of (1.7) is called degenerate if there exists a nonzero bounded solution pair(A(r),B(r))of

A + e^v(1−e^u)B−e^u+^vA=0

B + e^u(1−e^v)A−e^u+^vB=0 in R². (1.9) Comparing to the case of a single equation, there are additional difficulties to be over- come for (1.9). In the proof of the uniqueness for a single equation, some standard techniques such as Sturm-Liouville comparison theorem play important roles. See [4]

and [5]. However, these standard tools are no longer available for a system of Eqs. (1.9).

Hence, we have to develop new ideas to work out for (1.9), which will be presented in Sect.2. We believe that the method developed here should be helpful for a general class of nonlinear elliptic systems.

After the non-degeneracy of (1.9) is established, we can prove the following uniqueness theorem.

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Theorem 1.1. Let(u, v)be a topological solution of (1.7). Then the linearized equa- tion (1.9) of (1.7) at(u, v)is non-degenerate. Moreover, Eq. (1.7) possesses one and only one topological solution.

Now we come back to discuss the case of the boundary condition (ii) of (1.11). In the Abelian Chern-Simons-Higgs model with one particle, a solution u(x)satisfies

u + e^u(1−e^u)=4πNδ0 in R². (1.10) Suppose u =u(|x|)is a non-topological solution of (1.10), i.e., u(r)→ −∞as r → +∞. Then it can be proved that u satisfies

R²

e^u(1−e^u)d x<+∞. (1.11)

But, for the system (1.1), (1.11) might not hold even for the radial solution(u(r), v(r)).

Actually, in Sect.5, we will show that there exists a solution pair(u, v)of (1.7) satis- fying both u(r)andv(r)tend to−∞as r → ∞with

R²e^v(1−e^u)d x <+∞and

R²e^u(1−e^v)d x=+∞. Thus, while compared with (1.10), the structure of solutions for (1.7) could be more complicated. One of our purposes in this paper is to classify solutions according to their behaviors at infinity.

In this paper, we call a solution to be non-topological if(u, v)satisfies the boundary condition (ii) in (1.11), and both e^u(1−e^v)and e^v(1−e^u)are in L¹(R²). For an entire solution(u, v)of (1.1), we set

β1= 1 2π

R²

e^v(1−e^u)d x, β2= 1 2π

R²

e^u(1−e^v)d x. (1.12) In order to investigate the structure of all radial solutions of (1.7), we consider the following ODE system:

⎧⎪

⎨

⎪⎩

u(r)+ 1

ru(r)+ e^v(^r⁾(1−e^u⁽^r⁾)=0, v(r)+1

rv(r)+ e^u⁽^r⁾(1−e^v(^r⁾)=0, r >0 (1.13) with the initial value

u(r)=2N1log r +α1+ o(1),

v(r)=2N2log r +α2+ o(1) as r →0⁺. (1.14) According to the behaviors at∞, all entire solutions of (1.13) can be classified into the following five types:

Type (I): lim

r→∞(u(r), v(r))=(0,0), i.e.,(u, v)is the topological solution.

Type (II): lim

r→∞(u(r), v(r))=(−∞,−∞)withβ1<∞andβ2<∞, i.e., (u, v)is a non-topological solution.

Type (III): lim

r→∞u(r)= −∞, lim

r→∞v(r)= −∞, and

either 2N1< β1≤2N1+ 2, β2= ∞orβ1= ∞,2N2< β2≤2N2+ 2.

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Type (IV): lim

r→∞(u(r), v(r))=(−cu,−∞)or lim

r→∞(u(r), v(r))=(−∞,−c_v) for some constants cu>0 and c_v>0.

Type (V): lim

r→∞(u(r), v(r))=(+∞,−∞)or lim

r→∞(u(r), v(r))=(−∞,+∞).

Our second result is the asymptotic behaviors of all entire solutions.

Theorem 1.2. Let(u, v)be a solution of (1.13)–(1.14). Then(u, v)must be one of the above five types. Conversely, solutions of all types do exist.

Letα=(α1, α2), and(u(r, α), v(r, α))denote the solution of (1.13)-(1.14). Accord- ing to the behavior of(u, v), the set of initial data could be classified into the following regions:

= {α|(u(r, α), v(r, α))is a solution with lim

r→∞(u(r, α), v(r, α))=(−∞,−∞)}, T = {α|(u(r, α), v(r, α))is the unique topological solution},

N T = {α|(u(r, α), v(r, α))is a non-topological solution}, Su= {α|(u(r, α), v(r, α))is a Type (IV) solution with lim

r→∞u(r)= −cu}, S_v= {α|(u(r, α), v(r, α))is a Type (IV) solution with lim

r→∞v(r)= −c_v}, Wu= {α|(u(r, α), v(r, α))is a Type (V) solution with lim

r→∞u(r)= ∞}, W_v= {α|(u(r, α), v(r, α))is a Type (V) solution with lim

r→∞v(r)= ∞}.

Then the structure of solutions sets is described as follows:

Theorem 1.3. Bothand N T are non-empty and open simply connected. Further- more, all sets\N T,Su,S_v,Wuand W_vare non-empty, and the following statements are valid.

(i) =_v N T

uis a non-empty and simple connected set, where u = {α∈|(u(r, α), v(r, α))is a Type (III) solution withβ1<∞}, _v= {α∈|(u(r, α), v(r, α))is a Type (III) solution withβ2<∞}.

(ii) ∂=Su

T

S_vand Su

S_v=∂

∂N T =T. (iii) For anyα∈N T the corresponding(β1, β2)satisfies

(β1−2(N1+ 1))(β2−2(N2+ 1)) >4(N1+ 1)(N2+ 1). (1.15) (iv) Wu is open. Furthermore, for each (θ, η) ∈ Su there exists > 0 such that

(α1, η)∈Wu∀θ < α1< θ+.

(v) W_v is open. Furthermore, for each (µ, ν) ∈ S_v there exists δ > 0 such that (µ, α2)∈W_v∀ν < α2< ν+δ.

We remark that the uniqueness of topological solutions implies the simple-connectedness of bothandN T. The simple-connectedness is important itself, because it allows us to study the linearized equation of (1.7) at any non-topological solution through the argument of continuation. An important question about non-topological solutions arises:

given any pair of(β1, β2)satisfying (1.15) of Theorem1.3, is there an unique non-topological solution(u, v)which satisfies (1.12)? We will come back to this issue in a coming paper.

From Theorem1.3, we note that there are drastic differences between the solutions of (1.10) and (1.7). For Eq. (1.10), if a solution is positive somewhere, then it will blow

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up in finite|x|. But the situations do change for the system of equations. For example, the solution of Type (V) depicts that u might be positive somewhere, but both u andv do not blow up in finite|x|. Another consequence of Theorem1.2is that if both u and vare positive at some|x0|, then u andvmust blow up in finite|x|.

The paper is organized as follows. First we investigate the monotone and non-degenerate properties of the linearized equations on the negative solutions of (1.13) in Sect.2.

Based on the results of Sect.2and applying the Implicit Function Theorem, we prove the uniqueness of topological solution for (1.13) in Sect.3. In Sect. 4, we will give the asymptotic behaviors of all entire solutions. Finally, we prove the existences and classification of solutions of all types, Theorems1.2and1.3, in Sect.5.

2. The Non-Degeneracy of Linearized Equations

In this section, we give the proof about the non-degeneracy of the linearized equation on the topological solution of (1.7). Before going to our proof, we need to state some properties concerning solutions. First, we have the Pohozaev identity as follows.

Lemma 2.1. (Pohozaev identity). Let (u(r), v(r))be a solution of (1.13)–(1.14) in (0,R]for some R>0. Then we have the following identity:

[r u(r)·rv(r)+ r²(e^u⁽^r⁾+ e^v(^r⁾)−r²e^u⁽^r⁾⁺^v(^r⁾] −2 r

0

s(e^u⁽^s⁾+ e^v(^s⁾)ds +2

_r

0

se^u⁽^s⁾⁺^v(^s⁾ds =4N1N2 ∀r∈(0,R]. (2.1) Proof. By multiplying rvand r uon both sides of the first and second equation of (1.13) respectively, we obtain

rv(r u)+ rvr e^v(1−e^u)=0

r u(rv)+ r ur e^u(1−e^v)=0 ∀r∈(0,R]. (2.2) Then adding these two equations together and taking the integration from 0 to r , we get

[r u(r)·rv(r)− lim

r→0⁺(r u(r)·rv(r))]+_r

0 s²d(e^u⁽^s⁾)+_r

0 s²d(e^v(^s⁾)

−r

0s²d(e^u⁽^s⁾⁺^v(^s⁾)=0 ∀r∈(0,R].

By the above equality, and using the initial value (1.14) and the integration by parts, we can easily obtain (2.1).

Secondly, we have the following property for solutions with zero boundary value.

Lemma 2.2. Let(u(r), v(r))be a solution of (1.13)–(1.14) satisfying u(R0)=v(R0)= 0 for some R0>0(or R0=+∞). Then the following are valid:

(i) u < 0, v < 0,u > 0 andv > 0 on(0,R0). Furthermore, if R0 = ∞, i.e., (u, v)is a topological solution of (1.7), then the corresponding(β1, β2)satisfies β1=2N1andβ2=2N2, where(β1, β2)is defined in (1.12).

(ii) If N1<N2, then u > von(0,R0).

(iii) If N1>N2, then u < von(0,R0).

(iv) If N1=N2, then u ≡v.

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Proof. We shall apply the maximum principle to prove (i). Suppose u(r0)= max

(0,R₀]u >0.

Thenu(r0)≤0 and thus

0=u(r0)+ e^v(^r⁰⁾(1−e^u⁽^r⁰⁾) <0,

which yields a contradiction. Hence, u(r)≤0 on(0,R0). The strong maximum principle implies u(r) <0 in(0,R0). Similarly, it holds forv.

Since u(r) < 0 andv(r) < 0 in(0,R0), the maximum principle also implies that both u andv can not attain their local minima inside (0,R0). Since u(r) > 0 and v(r) >0 for r near 0, we obtain u(r) >0, v(r) >0 on(0,R0).

If R0= ∞then, by u(r) <0 on(0,∞)and (1.7), we have(r u(r))=r e^v(^r⁾(e^u⁽^r⁾− 1) < 0 ∀r ∈ (0,∞). Thus, by u(r) > 0 on (0,∞), we get 0 ≤ lim

r→∞r u(r) = 2N1−_∞

0 r e^v(1−e^u)dr exists(≡cu). If cu >0 then we easily have u(r) >0 for large r . This contradiction proves cu =0. From this we getβ1=2N1. The case ofβ2

is similar. Hence (i) holds.

By (1.15), we have

(u−v)=4π(N1−N2)δ0+(e^u−e^v).

If (u −v)(r0) < 0 at some r0 ∈ (0,R0), then we can let r0 satisfy(u−v)(r0) =

(min0,R₀](u−v) <0, and we have

0≤(u−v)(r0)=e^u⁽^r⁰⁾−e^v(^r⁰⁾<0,

a contradiction. Hence u(r)≥v(r). By the strong maximum principle, the strict inequal- ity u(r) > v(r)holds for r ∈(0,R0). This proves (ii). Obviously, (iii) and (iv) follow easily.

In the following, we investigate the monotone property of the negative solution of (1.13)–(1.14). Let, for i =1,2,

⎧⎪

⎨

⎪⎩

φi(r)= ∂U

∂αi, ψi(r)= ∂V

∂αi, (2.3)

where U(r;α1, α2) = u(r;α1, α2)−2N1log r and V(r;α1, α2) = v(r;α1, α2)− 2N2log r . Then(φi, ψi),i=1,2,satisfy the linearized equations

⎧⎨

⎩

φi−eû+^vφi+ e^v(1−eû)ψi =0, r ∈(0,R0), ψi −eû+^vψi+ eû(1−e^v)φi =0, r∈(0,R0),

φ1(0)=1=ψ2(0), φ2(0)=0=ψ1(0), φ_i(0)=0=ψ_i(0). (2.4) The monotone property ofφi andψi is as follows:

Lemma 2.3. Let(u(r), v(r))be a solution of (1.13)–(1.14). If u(r) <0 andv(r) <0 for r ∈(0,R0)for some R0>0(or R0= ∞), then the corresponding(φi, ψi)satisfy

φ1(r) >0, φ₁(r) >0, φ2(r) <0, φ₂(r) <0,

ψ1(r) <0, ψ₁(r) <0, ψ2(r) >0, ψ₂(r) >0 ∀r∈(0,R0). (2.5)

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Proof. By (2.4) and (1.14), we obtain there exists r0∈(0,R0)such that rψ1(r)= −

r 0

s[e^u⁽^s⁾(1−e^v(^s⁾)φ1(s)−e^u⁽^s⁾⁺^v(^s⁾ψ1(s)]ds ∀r>0

≤ − _r

0

s[C1s^2N¹(1−C2s^2N²)φ1(s)−C3s^2N¹^+2N²ψ1(s)]ds ∀r∈(0,r0)

≤ −Cr^2N¹⁺²<0 ∀r ∈(0,r0). (2.6)

Byψ1(0)=0, ψ₁(0)=0 and (2.6), we haveψ1(r) <0 andψ₁(r) <0∀r ∈ (0,r0).

On the other hand, by (2.4), (1.14), and the above result, we get rφ₁(r)=

_r

0

s[e^u⁽^s⁾⁺^v(^s⁾φ1(s)+ e^v(^s⁾(e^u⁽^s⁾−1)ψ1(s)]ds ∀r>0

≥ r

0

C4s·s^2N¹^+2N²φ1(s)ds ∀r ∈(0,r0)

≥Cr^2N¹^+2N²⁺²>0 ∀r∈(0,r0). (2.7) Byφ1(0)= 1, φ₁(0)=0 and (2.7), we haveφ1(r) >0 andφ₁(r) >0∀r ∈ (0,r0). These prove that the first inequality of (2.5) holds for r ∈ (0,r0). However (2.6) and (2.7) hold as long as the first inequality of (2.5) is true. This shows that the first inequality of (2.5) holds. The proof for the second inequality of (2.5) is similar. The proof is complete.

Finally, we state and prove the non-degenerate property of the linearized equation at a topological solution in the following:

Lemma 2.4. Let(u(r), v(r))be a solution of (1.13)–(1.14) satisfying u(R0)=v(R0)= 0 for some R0>0(or R0=+∞). If(φi(r), ψi(r)), i =1,2, is the respective solution pair of (2.4), then the following statements are valid.

(i) If R0 = ∞, i.e.,(u, v)is a topological solution, then there exist constants c1 >

0,c2<0,d1<0 and d2>0 such that,

rlim→∞

φi(r)

r⁻²¹e^r =ci and lim

r→∞

ψi(r)

r⁻¹²e^r =di, i =1,2.

(ii) Let MA(r) = −^φ_φ¹₂⁽₍^r_r⁾₎ and MB(r)= −^ψ_ψ¹₂⁽₍^r_r⁾₎. Then MA(r) > MB(r) > 0 ∀r ∈ [0,R0](resp.,[0,∞)if R0= ∞)and M_A(r) <0,M_B(r) >0∀r ∈(0,R0).

(iii) det

φ1(r) φ2(r) ψ1(r) ψ2(r)

=0∀r ∈ [0,R0](resp.,[0,∞)if R0= ∞).

(iv) The corresponding linearized equation (1.9) is non-degenerate.

Proof. (i) We prove the asymptotic behavior ofφ1. The cases of ψ1, φ2 andψ2 are similar. Letw(r)=φ1(r)−ψ1(r)−e^r. Then by (2.4),wsatisfies

w(r)=(e^uφ1−e^vψ1)−(1 +¹_r)e^r w(0)=0, w(0)= −1.

Since u(r) < 0, v(r) < 0, φ1(r) > 0 andψ1(r) < 0∀r > 0, it follows thatw ≤ (φ1(r)−ψ1(r))−(1 +¹_r)e^r =w(r)−¹_re^r ∀r∈(0,∞). Thus we obtainw(r) <0∀r∈ (0,∞), i.e.,

φ1(r)−ψ1(r) <e^r on (0,∞). (2.8)

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Let z(r)=φ1(r)r¹². Then z satisfies

z(r)+[−1 + q(r)]z(r)=0, (2.9) where

q(r)=1−e^u+^v+e^v(1−e^u)ψ1

φ1

+ 1 4r². Since lim

r→∞

e^u−1

u =1 andψ1<0, we have

(e^u−1)ψ1≤C·u(r)ψ1(r)for large r and some C>0. (2.10) By|u(r)|,|v(r)| ≤Cr⁻¹²e⁻^r for large r , (2.8) and (2.10), we easily obtain

(e^u−1)ψ1≤Cr⁻¹² for large r.

From this andφ1(r) > r for large r , we deduce ⁻ê^v⁽¹_φ⁻êû^)ψ¹

1 ∈ L¹(R,∞)for R >0 large. Moreover, since_∞

R (1−e^u+^v)dr<∞, we get

q(r)∈ L¹[R,∞). (2.11)

By (2.11) and applying Corollary 9.2 of [9] to (2.9), we finally obtain

rlim→∞

z(r)

e^r =c1>0, and hence

rlim→∞

φ1(r) r⁻¹²e^r =c1. This proves the case ofφ1. Thus (i) holds.

(ii) By (2.4), we have limr→0⁺MA(r)= ∞,limr→0⁺MB(r)=0, and thus MA(r) >

MB(r) ∀r ∈(0,r1)for some r1∈(0,R0]. We divide the proof of (ii) into the following two steps.

Step 1. If MA(r) > MB(r) ∀r ∈ (0,r0)for some r0 ≤ R0, then M_A(r) < 0 and M_B(r) >0∀r∈(0,r0).

We prove Step 1 by contradiction. Suppose M_A(r) < 0 ∀r ∈ (0,r0)is not true.

Then there exist 0<r1<r2≤r0such that

M_A(r1) <0,M_A(r2) >0,MA(r1)=MA(r2)(≡C0),and

0<MB(r) <MA(r) <C0 ∀r ∈(r1,r2). (2.12) For any c>0 and r∈(0,R0], we define

Ac(r)=φ1(r)+ c·φ2(r)and Bc(r)=ψ1(r)+ c·ψ2(r). (2.13) Then Acand Bcsatisfy

⎧⎨

⎩

Ac−eû+^vAc=e^v(eû−1)Bc ∀r∈(0,R0], Bc−eû+^vBc=eû(e^v−1)Ac ∀r∈(0,R0],

Ac(0)=1, Bc(0)=c>0. (2.14)

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From (2.12) and (2.13), we easily obtain

AC0(r) <0<BC0(r)∀r ∈(r1,r2) and AC0(r1)=0=AC0(r2), (2.15) which imply that AC₀has a local minimum at somer¯∈(r1,r2)andAC₀(¯r)≥0. But, from (2.14) and (2.15), we get

AC0(¯r)=e^u^(¯^r⁾⁺^v(¯^r⁾AC0(¯r)+ e^v(^r^¯⁾(e^u^(¯^r⁾−1)BC0(¯r) <0. (2.16) This contradiction proves M_A(r) <0∀r ∈(0,r0).

Similarly, suppose M_B(r) >0∀r ∈ (0,r0)is not true. Then there exist 0 <r1 <

r2≤r0such that

M_B(r1) >0,M_B(r2) <0,MB(r1)=MB(r2)(≡C0), and

C0<MB(r) < MA(r)∀r ∈(r1,r2). (2.17) By (2.17) and (2.13), we easily obtain

BC0(r) <0<AC0(r)∀r ∈(r1,r2) and BC0(r1)=0=BC0(r2), (2.18) and hence BC0 has a local minimum at somer¯∈(r1,r2)withBC0(¯r)≥0. However, from (2.14) and (2.15) we get

BC0(¯r)=e^u⁽^r^¯⁾⁺^v(¯^r⁾BC0(¯r)+ e^u^(¯^r⁾(e^v(¯^r⁾−1)AC0(¯r) <0. (2.19) This contradiction proves Step 1.

Step 2. There does not exist R∈(0,R0)such that MA(R)=MB(R).

Suppose Step 2 is not true. Then there exists a smallest R ∈ (0,R0] such that MA(R) = MB(R)(≡C)and MA(r) > MB(r) > 0 ∀r ∈ (0,R). Let Ac and Bc be defined in (2.13). Then, in this case, by Step 1 we obtain

AC(r) >0,BC(r) >0 ∀r ∈(0,R),

AC(R)=BC(R)=0, (2.20)

A_C(R) <0,B_C(R) <0 if R<∞.

Taking the differentiation w.r.t.αi, i = 1,2, on both sides of the Pohozaev identity, (2.1), then for any c>0 and r ∈(0,R0], we obtain

r²A_c(r)v(r)+ r²B_c(r)u(r)+ r²[e^u⁽^r⁾Ac(r)+ e^v(^r⁾Bc(r)]

−r²e^u⁽^r⁾⁺^v(^r⁾(Ac(r)+ Bc(r))2 r

0

s[e^uAc+ e^vBc]ds + 2 r

0

se^u+^v(Ac+ Bc)ds =0.

(2.21) If R <∞then, by replacing c and r with C and R in (2.21) respectively, we easily have

0=

R²A_C(R)v(R)+ R²B_C(R)u(R) +

R²BC(R)e^v(^R⁾(1−e^u⁽^R⁾)+ R²AC(R)e^u⁽^R⁾(1−e^v(^R⁾) +2

_R

0

r ACe^u(e^v−1)dr + _R

0

r BCe^v(e^u−1)dr

. (2.22)

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Then, combining (i) of Lemma2.2, (2.20) and (2.22), we deduce 0> R²A_C(R)v(R)+ R²B_C(R)u(R)

=2 R

0

r ACe^u(1−e^v)dr + R

0

r BCe^v(1−e^u)dr

>0, which yields a contradiction.

If R= ∞then we first claim that

one of ACand BCis unbounded. (†)

Suppose(†)is not true. Then ACand BCare bounded. By (2.21) we have 0= lim

r→∞

r A_C(r)·rv(r)+ r B_C(r)·r u(r) + lim

r→∞

BC(r)·r²e^v(^r⁾(1−e^u⁽^r⁾)+ AC(r)·r²e^u⁽^r⁾(1−e^v(^r⁾) +2

_∞

0

r ACe^u(e^v−1)dr + _∞

0

r BCe^v(e^u−1)dr

. (2.23)

Moreover, (i) of Lemma2.2implies that

rlim→∞r u(r)=0= lim

r→∞rv(r),

rlim→∞[r²e^u⁽^r⁾(1−e^v(^r⁾)] =0= lim

r→∞[r²e^v(^r⁾(1−e^u⁽^r⁾)]. (2.24) Since|u|and|v|decay exponentially at∞, and(AC,BC)is bounded, by (2.14) and (i) of Lemma2.2, we get the limits

rlim→∞r A_C(r)= _∞

0

[r e^u+^vAC]dr− _∞

0

[r e^v(1−e^u)Bc]dr,

rlim→∞r B_C(r)= _∞

0

[r e^u+^vBC]dr− _∞

0

[r e^u(1−e^v)Ac]dr

(2.25)

all exist. Hence, due to (2.20) and (2.23)–(2.25), we finally obtain 0=2

_∞

0

r ACe^u(1−e^v)dr + _∞

0

r BCe^v(1−e^u)dr

>0.

This contradiction shows that ACor BCis unbounded, and the claim is proved.

Secondly, suppose ACis unbounded. From (2.14) we have (AC−BC)−e^u(AC−BC)=(e^u−e^v)BC,

and hence, by Lemma2.2and the strong maximum principle, we obtain that AC(r) intersects BC(r)at most one point on[0,∞). Thus, w.l.o.g., we may assume that there exists r1>0 such that

A_C(r1)≥0,

AC(r) >BC(r) >0 on [r1,∞). (2.26) Since(u, v)is a topological solution of (1.7), there exists r2>r1such that

eû⁽^r⁾⁺^v(^r⁾≥max{e^v(^r⁾(1−eû⁽^r⁾),eû⁽^r⁾(1−e^v(^r⁾)} ∀r ≥r2. (2.27)

(12)

Therefore, by (2.14) and (2.26)–(2.27), we get A_C(r) > 0 on (r1,∞) and thus

rlim→∞AC(r) = ∞. Now, by applying the same arguments in the proof of (i), we can obtain

rlim→∞

AC(r)

r⁻¹²e^r =CA=c1+ C·c2>0,

where c1and c2are constants in (i). Then there exists >0 such that C =c1+(C + )·c2>0 and lim

r→∞

AC+(r)

r⁻¹²e^r =C. But, by Step 1 we have AC+(r) <0 for large r . We get a contradiction. The case of unboundedness for BCis similar. This shows Step 2. According to Steps 1 and 2, we easily obtain (ii).

(iii) Suppose det

φ1(R) φ2(R) ψ1(R) ψ2(R)

= 0 for some R ∈ [0,R0](resp., R ∈ [0,∞)if R0= ∞). Then, w.l.o.g., there exists C0>0 such that

φ1(R) ψ1(R)

+ C0

φ2(R) ψ2(R)

= 0

0

. (2.28)

By (2.28) we obtain MA(R)=C0=MB(R)which contradicts the result of (ii). Hence we prove (iii).

(iv) Let(u, v)be a topological solution of (1.7). Then any solution pair(A(r),B(r))of the linearized equations (1.9) can be written as

A(r)=c1φ1(r)+ c2ψ1(r) and B(r)=c1φ2(r)+ c2ψ2(r) for some c1,c2∈R.

By the result of(†)in the proof of (ii), we easily obtain the non-degeneracy result if c=c2/c1>0. When c≤0 or c1=0, then by (i), we can also get that both Ac(r)and Bc(r)are unbounded. This proves (iv).

3. Uniqueness of Topological Solution

In this section, we will use a continuation argument and Lemma2.4to establish the uniqueness of topological solutions. As we have seen in Sect. 2, if N1 = N2, then u ≡v, and the uniqueness follows from the case of scalar equation1.10. Concerning the uniqueness for the scalar equation, we refer readers to [6 or 8].

Proof of Theorem1.1. Suppose that for some pair(N₁⁰,N₂⁰), Eq. (1.15) possesses at least two topological solutions. Without loss of generality, we may assume 0 ≤ N₁⁰ < N₂⁰. Let

N₁^∗=inf{0≤N1|(1.15)possesses a unique topological solution for all(Nˆ1,N₂⁰)where N1≤ ˆN1≤N₂⁰}.

Clearly, N₁^∗≥ N₁⁰. To yield a contradiction, we claim the following:

(∗) Suppose (u0, v0) is a topological solution of (1.15) with respect to (N₁,N₂).

Let U0(r) = u0(r)−2N₁log r and V0(r) = v0(r)−2N₂log r . Then there is a neighborhood B of(N₁,N₂)such that for any pair of(N1,N2)in B, there exists the corresponding (U,V)with respect to (N1,N2), which is close to (U0,V0) in C²

BR(0)

×C²

BR(0)

for any R > 0, where (u(r), v(r)) = (U(r)+ 2N1log r,V(r)+ 2N2log r)is a topological solution of (1.15) with respect to (N1,N2).