On the solutions to a Liouville-type system involving singularity

DOI 10.1007/s00526-011-0403-1

Calculus of Variations

On the solutions to a Liouville-type system involving singularity

Zhi-You Chen· Jann-Long Chern · Yong-Li Tang

Received: 24 December 2009 / Accepted: 31 January 2011 / Published online: 17 February 2011

© Springer-Verlag 2011

Abstract In this paper, we consider a Liouville-type system with singularity in the plane.

The existence and uniqueness of solutions to the Dirichlet problem are proved. In addition, the structure of solutions in terms of analogues of the so-called total curvature in geometry or total mass in physics will be offered as well.

Mathematics Subject Classification (2010) Primary 35J47·Secondary 35A20

1 Introduction

This paper is concerned with the nonlinear elliptic system u+K₁(|x|)e^v =4πm1δ0,

v+K₂(|x|)e^u =4πm₂δ0, in R², (1.1) where = ₂

i=1∂²/∂x_i² is the Laplacian operator in R²,m1,m2 > 0, δ0 is the Dirac measure at the origin, and K1(r),K2(r)are positive functions for r>0 satisfying

r^2miK_i∈L¹([0,1))∩C¹[(0,∞)], i=1,2, (1.2)

r→0limr^−βiK_i(r), lim

r→∞r^−γiK_i(r)are finite and positive, i =1,2 (1.3)

Communicated by J. Jost.

Z.-Y. Chen (

B

Department of Mathematics, National Tsing Hua University, Hsin-Chu 30013, Taiwan e-mail: [email protected]

J.-L. Chern·Y.-L. Tang

Department of Mathematics, National Central University, Chung-Li 32001, Taiwan J.-L. Chern

e-mail: [email protected] Y.-L. Tang

e-mail: [email protected]

for someβi, γi ∈R(i=1,2)with

β1, γ1>−(2+2m₂), β2, γ2>−(2+2m₁). (1.4) We call the system in the form of (1.1) the Liouville-type system, which is a natural extension of the so-called Liouville equation

u+K e^u =0 in⊂R². (1.5)

It is well-known that the Liouville equation is related to many applications in a variety of fields in mathematics and physics. In the aspect of the differential geometry, the Liouville equation stands for the problem of finding a metric whose Gaussian curvature is prescribed [5]. In physics, just to name but a few, it represents the electric potential induced by charge carriers in the theory of electrolytes [24], and the Newtonian potential of a cluster of self-gravitating mass distribution [1,4,25,26]. Moreover, it is also induced by a mean field equation which comes from the spherical Onsager vortex theory, bridging the gap between statistical mechan- ics of classical vortices and the random surface problem [7,21], and is considered to deal with topics closely related to the abelian model in the Chern–Simons theories [9,10,15,27].

As an extension of the single case, Liouville-type systems have also been used to describe models in the physics of charged particle beams [3,14,20], in the theory of semi-conductors [23], in the theory of chemotaxis [11,19], and other issues in fields of physics, chemistry and ecology. We also remark that another significant extension of the Liouville equation is the Toda system which is closely concerned with the non-abelian Chern–Simons theory. For more details of applications of Liouville-type systems, see for example [2,6,12,13,16–18,22]

and references therein.

Throughout this article, we consider the radial case of (1.1), i.e., the following ODE system:

⎧⎪

⎨

⎪⎩ u+1

ru+K₁(r)e^v=0, v+1

rv+K2(r)e^u =0, r>0 (1.6)

with(u(r), v(r))to be the specific form:

u(r)=2m1log r+α1+o(1),

v(r)=2m2log r+α2+o(1), as r→0⁺, (1.7) whereα1, α2 ∈R. Conventionally, we denote the solution of (1.6)–(1.7) by(u(r;α1, α2), v(r;α1, α2))or simply(u(r), v(r))if there is no confusion. Here we call(α1, α2)in (1.7) the normalized initial data of solution(u(r;α1, α2), v(r;α1, α2))for (1.6)–(1.7). In fact, if we set

U(r)=u(r;α1, α2)−2m₁log r, V(r)=v(r;α1, α2)−2m₂log r, (1.8) then(U(r),V(r))satisfies

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

U(r)+1

rU(r)+r^2m2K₁(r)e^V =0, r>0, V(r)+1

rV(r)+r^2m1K2(r)e^U =0, r>0, U(0)=α1, V(0)=α2,

U(0)=0, V(0)=0.

(1.9)

Note that every solution(u(r), v(r))of (1.6) is defined for all r ∈(0,∞). Indeed, from (1.9), if(U(r),V(r))is defined on[0,R)and lim_r→RU(r)= −∞for some finite R>0, then lim_r→RU(r)= −∞also. Hence,

−∞ = lim

r→RrU(r)= lim

r→R

⎛

⎝− r 0

s^2m2+1K1e^Vds

⎞

⎠= − R 0

s^2m2+1K1e^Vds

which is impossible because V(r)is decreasing in[0,R). For the structure of solutions for the single equation related to (1.6), see for example [7].

In [12], the following system was considered:

u+V1(|x|)e^au+bv =0,

v+V2(|x|)e^bu+cv =0, in, (1.10)

whereis either B_R(0)or R²,V₁,V₂ are positive functions on , and a,b,c are con- stants. If = B_R(0),0 < k₁ ≤ V₁,V₂ ≤ k₂ < ∞on B_R(0)and a,c ≥ 0, then for any 0<M₁,M₂ <∞satisfying certain assumptions (Theorem 1.1 in [12]), there exists a radially symmetric solution(u(r), v(r))of (1.10) so that u(R)=v(R)=0 and

V1e^au+bvd x=M1,

V2e^bu+cvd x=M2. (1.11) For=R²,V1≡V2≡1 and a,b,c≥0, some sufficient and necessary conditions for exis- tence of an entire radial solution of (1.10) associated with prescribed finite “flux”(M1,M2) (i.e., a solution(u, v)satisfying (1.11)) were established. In this paper, we consider the case of a=c=0 in (1.10) with singularity at the origin. The existence and uniqueness of solu- tions to the Dirichlet problem will be proved. Furthermore, unbounded flux is possible and really exists, and all solutions can be classified completely in terms of corresponding fluxes.

It is worth mentioning that in the Chern–Simons systems [10], the “flux” associated with certain solutions can be unbounded, which is quite different from the situation in single equations. Here, we also investigate such phenomena that occur in the Liouville equation and system.

Now we state the main result on the existence and uniqueness of solutions to the Dirichlet problem of (1.6)–(1.7).

Theorem 1.1 Suppose that r K_i(r)

K_i(r) +2≥0, r>0,i=1,2. (1.12) Then for any R>0, (1.6)–(1.7) possesses one and only one solution(u(r), v(r))satisfying

u(r) <0, v(r) <0, 0<r<R,

u(R)=v(R)=0. (1.13)

Remark 1.1 In fact, for the existence result, the condition (1.12) can be removed from The- orem1.1.

Let(u(r), v(r))be a solution of (1.6)–(1.7). We define 1(u, v)=

∞ 0

r K₁(r)e^v(r)dr, 2(u, v)= ∞ 0

r K₂(r)e^u(r)dr. (1.14)

Clearly, 0< i(u, v)≤ ∞. We sometimes denote it by(1, 2)if no confusion arises. In the case of u≡vin (1.6),=1(u, v)=2(u, v)is called, for example, the total curva- ture coming from the prescribed Gaussian curvature equation or the flux in physics. Here we call1(u, v)and2(u, v)the K1-mass and K2-mass with respect to solution(u, v)respec- tively. From standard arguments,1(u, v)and2(u, v)cannot be infinite simultaneously for any solution(u(r), v(r))of (1.6)–(1.7).

For convenience, we classify a solution(u(r), v(r))of (1.6)–(1.7) in terms of its corre- sponding K_i-masses pair(1, 2)as follows:

Type (I): lim

r→∞(u(r), v(r))=(−∞,−∞)with1 <∞and2= ∞.

Type (II): lim

r→∞(u(r), v(r))=(−∞,−∞)with1 = ∞and2<∞. Type (III): lim

r→∞(u(r), v(r))=(−∞,−∞)with1 <∞and2<∞.

Our second result shows that the above classification exhausts all possible situations of solutions for (1.6)–(1.7).

Theorem 1.2 Let(u(r), v(r))be a solution of (1.6)–(1.7). Then(u(r), v(r))must be one of the above three types. Conversely, solutions of all types do exist.

In the following, we conclude that the corresponding Ki-masses with respect to solutions of types stated above range exactly over certain intervals related to m_i andγi.

Theorem 1.3 Consider (1.6)–(1.7). Then

(a) For anyθ1 ∈(0,2+2m₁+γ2], there exist infinitely many solutions(u(r), v(r))of Type (I) such that

(1(u, v), 2(u, v))=(θ1,∞).

Furthermore, if(u(r), v(r))is a solution of Type (I), then1(u, v)≤2+2m₁+γ2and 2(u, v)= ∞.

(b) For anyθ2 ∈(0,2+2m₂+γ1], there exist infinitely many solutions(u(r), v(r))of Type (II) such that

(1(u, v), 2(u, v))=(∞, θ2).

Furthermore, if(u(r), v(r))is a solution of Type (II), then1(u, v)= ∞and2(u, v)≤ 2+2m₂+γ1.

For the special case of K1≡K2 ≡1 in (1.6), i.e.,

⎧⎪

⎨

⎪⎩ u+1

ru+e^v=0, v+1

rv+e^u =0, r>0, (1.15)

we have some further consequences. Before stating our final result, we first introduce the linearized system of (1.15)–(1.7) with respect to solution(u(r), v(r)):

⎧⎪

⎨

⎪⎩ A+1

rA+r^2m2e^V(r)B=0, B+1

rB+r^2m1e^U(r)A=0, r>0 (1.16) where(U(r),V(r))is defined in (1.8). The linearized system (1.16) is degenerate if it pos- sesses a nonconstant bounded solution(A(r),B(r))on(0,∞).

Fig. 1 Structure of solutions of (1.15)–(1.7)

Theorem 1.4 Consider (1.15)–(1.7). Then

(a) All conclusions in Theorems1.2and1.3hold (γ1=γ2 =0 in this case).

(b) For anyθ1 >2+2m₁andθ2>2+2m₂satisfying

θ1θ2−(2m₂+2)θ1−(2m₁+2)θ2=0, (1.17) there exist infinitely many solutions(u(r), v(r))of Type (III) such that

(1(u, v), 2(u, v))=(θ1, θ2).

Furthermore, if(u(r), v(r))is a solution of Type (III), then(θ1, θ2)=(1(u, v), 2(u, v)) satisfies (1.17), and its corresponding linearized system (1.16) is degenerate.

Example 1.1 In the specific case of K₁≡K₂≡1, solutions of (1.15)–(1.7), to the Dirichlet problem and related to associated Ki-masses, are all clarified by Theorem1.1and Theo- rem1.4. By adopting the notations introduced in (2.2), Sect.2, we illustrate the structure of solutions of (1.15)–(1.7) below.

We organize this article as follows. In Sect.2, some results concerning with K_i-masses will be made. Theorems1.2and1.3will be proved in Sect.3. We give a complete verification of Theorem1.4in Sect.4. Finally, Sect.5is devoted to the proof of Theorem1.1.

2 Ki-masses associated with solutions

First of all, we note that for any solution(u(r), v(r))of (1.6)–(1.7) and from (1.14), 1(u, v)=

∞ 0

r^2m2+1K₁e^V(r)dr, 2(u, v)= ∞ 0

r^2m1+1K₂e^U(r)dr, (2.1) where (U(r),V(r)) is defined in (1.8) and satisfies (1.9). Occasionally, we denote i(u(r;α1, α2), v(r;α1, α2))byi(α1, α2)simply to indicate the dependence of(α1, α2).

Our first lemma in this section gives the fact that if one of K_i-masses is finite, then the other one is bounded from below by a positive constant.

Lemma 2.1 Let(u(r), v(r))=(u(r;α1, α2), v(r;α1, α2))be a solution of (1.6)–(1.7). If 1(u, v) <∞(r esp., 2(u, v) <∞), then2(u, v) >2+2m2+γ1(r esp., 1(u, v) >

2+2m₁+γ2).

Proof Let1(u, v) <∞and(U(r),V(r))be defined as in (1.8). Without loss of generality, we assume2(u, v) <∞. Then by (1.9) and (2.1), we have that r V(r) >−2for r ≥0.

Hence

V(r) >−2log r+c1, r≥R and, by (1.3),

K1(r)≥c2r^γ1, r≥R

for some R>0 and c₁,c₂ >0. Therefore, by combining the results above, we obtain that 1 >

∞ R

r^2m2+1K1(r)e^V(r)dr

≥c₃ ∞ R

s^{2m2+1+γ1−2}ds

for some c3 > 0, which implies2 >2+2m2+γ1 because of the finiteness of1. We

complete this proof.

From the above lemma, any solution(u(r;α1, α2), v(r;α1, α2))of (1.6)–(1.7) can be classified into the following regions in terms of normalized initial data(α1, α2)depending on its corresponding Ki-masses(1, 2):

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

S= {(α1, α2):1(u, v) <∞, 2(u, v) <∞}, S1= {(α1, α2):0< 1(u, v) < κ1, 2(u, v)= ∞}, S2= {(α1, α2):0< 2(u, v) < κ2, 1(u, v)= ∞}, C1= {(α1, α2):1(u, v)=κ1, 2(u, v)= ∞}, C2= {(α1, α2):2(u, v)=κ2, 1(u, v)= ∞},

(2.2)

where

κ1=2+2m1+γ2, κ2=2+2m2+γ1. (2.3) It is clear that R²=S∪S1∪S2∪C1∪C2.

Let(u(r), v(r))be a solution of (1.6)–(1.7), and(U(r),V(r))be defined as in (1.8). Then from (1.9), it is easy to see that

(rU)+r^2m2+1K1(r)e^V =0, r>0, (r V)+r^2m1+1K₂(r)e^U =0, r>0.

Multiplying r V(r)and rU(r)on the first and second equation of the above relation respec- tively, adding them together, and then integrating from 0 to r , we have

r 0

(sU)(sV) ds+

r 0

s^2m2+2K1e^VV+s^2m1+2K2e^UU

ds=0 (2.4) and hence obtain the Pohozaev-type identity as follows:

r²U(r)V(r)+r^2m2+2K1(r)e^V+r^2m1+2K2(r)e^U

= r 0

s^2m2+1

(2m2+2)K1+s K₁

e^V+s^2m1+1

(2m1+2)K2+s K₂ e^U

ds.(2.5)

For finite K_i-masses, they satisfy a relation as shown in the following.

Lemma 2.2 If(α1, α2)∈S, then(1, 2)=(1(α1, α2), 2(α1, α2))satisfies 12−(2m₂+2)1−(2m₁+2)2=

∞ 0

r^2m2+2K₁e^V+r^2m1+2K₂e^U

dr.

Proof Let(U(r),V(r))be defined in (1.8) associated with the solution(u(r;α1, α2), v(r;α1, α2)). Since1+2is finite, then by (2.1), we have that

r_k^2m2+2K₁(rk)e^V(rk)+r_k^2m1+2K₂(rk)e^U(rk)→0 as k→ ∞

for some sequence{r_k}with r_k →0 as k → ∞. Hence by taking r =r_k on both sides of (2.5) and then letting k → ∞, we complete the proof of this lemma.

Remark 2.1 (i) In the case of K₁ ≡K₂≡1, Lemma2.2implies 12−(2m2+2)1−(2m1+2)2 =0

for any solution(u(r), v(r))of (1.15)–(1.7) with both1(u, v)and2(u, v)finite, i.e., solution of Type (III).

(ii) For any(α2, α2)∈S, we conclude that

rU(r) <−(2+2m1+γ2), r V(r) <−(2+2m2+γ1) for large r,

where(U(r),V(r))is defined in (1.8) associated with(u(r;α1, α2), v(r;α1, α2)). This is due to Lemma2.1and the facts that rU(r)and r V(r)decrease to−1(α1, α2)and−2(α1, α2) respectively as r→ ∞by (1.9).

Actually,(U(r),V(r)) behave logarithmically at infinity, related to its corresponding K_i-masses if both are finite.

Lemma 2.3 Let(α1, α2)∈Sand(u(r;α1, α2), v(r;α1, α2))be the solution of (1.6)–(1.7).

Then

U(r)= −1(α1, α2)log r+O(1) at r= ∞,

V(r)= −2(α1, α2)log r+O(1) at r= ∞, (2.6) where(U(r),V(r))is defined in (1.8).

Proof From (1.9), we see that

rU(r)= − r 0

s^2m2+1K1e^Vds, r V(r)= − r 0

s^2m1+1K2e^Uds, r≥0.

Then

rlim→∞rU(r)= −1(α1, α2), lim

r→∞r V(r)= −2(α1, α2).

By Remark2.1(ii), we may choose some p>0 such that r V(r)+p<−(2+2m2+γ1) for r ≥R if R is large. Then V(r) <−(p+2+2m2+γ1)log r+c1for r ≥R and some c₁>0, which implies that

e^V(r)<c2r^{−(p+2+2m2+γ1)}, r≥R, where c₂=e^c1.

Let W(r)=U(r)+1(α1, α2)log r for r>0. Then by the above result, (1.3) and (2.1), we obtain that for r≥R,

r W(r)=rU(r)+1(α1, α2)

= − r 0

s^2m2+1K₁e^V(s)ds+1(α1, α2)

= ∞ r

s^2m2+1K1e^V(s)ds

<c3

∞ r

s^2m2+1·s^γ1·s^{−(p+2+2m2+γ1)}ds

=c₃ ∞ r

s^−(p+1)ds

=c4r^−p

for some c₃,c₄ >0. Therefore, we assure that W(r)is bounded on[R,∞), and hence the expression of U(r)in (2.6) holds. The situation for V(r)is similar, and then this lemma is

proved.

3 Existence of solutions of Types (I)–(III)

In this section, we prove the existence of solutions of all types stated in Sect.1. Additionally, the structure of solutions in terms of normalized initial data depending on corresponding Ki-masses will be established as well.

We first show that System (1.6)–(1.7) possesses solutions with one of associated Ki-masses being infinite, i.e., solutions of Types (I) and (II).

Proposition 3.1 Consider (1.6)–(1.7) and let

λ1=max{−(2m2+β1),−(2m1+γ2)}, λ2=max{−(2m1+β2),−(2m2+γ1)}.

Then

(a) For anyξ ∈ [λ1,2), there exist two constantsα¯1(ξ) >0 andα¯2(ξ) <0 such that 1(α1, α2) <2−ξ, 2(α1, α2)= ∞

for allα1≥ ¯α1(ξ)andα2≤ ¯α2(ξ).

(b) For anyζ ∈ [λ2,2), there exist two constantsαˆ2(ζ ) >0 andαˆ1(ζ ) <0 such that 1(α1, α2)= ∞, 2(α1, α2) <2−ζ

for allα1≤ ˆα1(ζ )andα2≥ ˆα2(ζ ).

Proof Let (u(r), v(r)) = (u(r;α1, α2), v(r;α1, α2)) be a solution of (1.6)–(1.7) and (U(r),V(r)) = (U(r;α1, α2),V(r;α1, α2)) be defined as in (1.8) with respect to the solution(u(r), v(r)). We note thatλ1, λ2 <2 by the assumptions in (1.4).

(a) We split this proof into the following steps.

Step 1. Forξ ∈ [λ1,2)andη <−(2m₁+β2), we define

w(r)=w(r;α1, α2)=U(r)+log(1+r^2−ξ), r≥0, z(r)=z(r;α1, α2)=V(r)+log(1+r^2−η), r≥0. Thenw(r)and z(r)satisfy

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

(rw)(r)= r^1−ξ (1+r^2−ξ)²

(2−ξ)²−r^ξ+2m2(1+r^2−ξ)² 1+r^2−η K₁e^z

, r>0, (r z)(r)= r^1−η

(1+r^2−η)²

(2−η)²−r^η+2m1(1+r^2−η)² 1+r^2−ξ K₂e^w

, r>0, w(0)=α1,z(0)=α2;lim

r→0rw(r)=0,lim

r→0r z(r)=0.

(3.1)

Since 2m1+ξ+γ2≥0 and 2−η >0, there existα₁^∗>0 and R0>1 such that for r ≥R0

andα1≥α^∗₁, we have r^1−η (1+r^2−η)²

(2−η)²−r^η+2m1(1+r^2−η)²

1+r^2−ξ K2(r)e^α1

≤ −k2e^α1

r (3.2)

where k2 =limr→∞r^−γ2K2(r) >0. In addition, from the behavior of K1(r)at infinity and (3.2), we get

r^{ξ+2m2−(k2eα}

1∗log R1)·(1+r^2−ξ)²

1+r^2−η K₁(r) < (2−ξ)²

2 , r≥R₁ (3.3)

if R₁ ≥ R₀ is sufficiently large. Moreover, by (3.1) and (3.2), we also get z(r) ≤

−(k₂e^α∗1log R₂)log r forα1≥α₁^∗and r ≥R₃for some large R₃≥R₂≥R₁. Hence e^z(r)≤r^−k2eα

1∗log R2, r≥R₃, α1≥α1^∗. (3.4) By combining (3.1)–(3.4), we obtain

⎧⎪

⎪⎨

⎪⎪

⎩

(2−ξ)²−r^ξ+2m2(1+r^2−ξ)²

1+r^2−η K1(r)e^z(r)>0, (2−η)²−r^η+2m1(1+r^2−η)²

1+r^2−ξ K₂(r)e^w(r)<0,

r≥R₃, α1 ≥α₁^∗. (3.5)

On the other hand, because ofξ+2m2+β1≥0 andη+2m1+β2≤0, we have that (3.5) holds for r ∈(0,R3], α1 ≥ ¯α1andα2 ≤ ¯α2 if we chooseα¯1 ≥α^∗₁ sufficiently large and

¯

α2<0 sufficiently small. Therefore, from (3.1) and the above results, we conclude that (rw)(r) >0, (r z)(r) <0, r>0, α1 ≥ ¯α1, α2≤ ¯α2, (3.6) and hencew(r;α1, α2)is positive for all r ≥0,α1≥ ¯α1andα2≤ ¯α2.

Step 2. From Step 1, we see that forα1 ≥ ¯α1 andα2 ≤ ¯α2, U(r) = U(r;α1, α2) >

−log(1+r^2−ξ)and then e^U(r)>1/(1+r^2−ξ)for r≥0. Hence 2(α1, α2)=

∞ 0

r^2m1+1K2(r)e^U(r)dr ≥C ∞ R

r^2m1+1·r^γ2·r^−(2−ξ)dr

=C ∞ R

r^{2m1+γ2+ξ−1}dr= ∞ for some C>0 and large R since 2m₁+γ2+ξ−1≥ −1.

Step 3. By (3.6), we obtain that 0< lim

r→∞rw(r)= lim

r→∞

rU(r)+(2−ξ)r^2−ξ 1+r^2−ξ

= −1(α1, α2)+(2−ξ), and then1(α1, α2) <2−ξ. Hence (a) is proved.

(b) Forζ ∈ [λ2,2)andη <−(2m₂+β1), we define

w(r)=w(r;α1, α2)=U(r)+log(1+r^2−η), r≥0, z(r)=z(r;α1, α2)=V(r)+log(1+r^2−ζ), r≥0.

Then by the similar arguments as in the proof of (a), we omit the details and hence (b) is

proved.

The next result gives us the existence of solutions of Type (III) to (1.6)–(1.7).

Proposition 3.2 The regionS is nonempty, i.e., System (1.6)–(1.7) possesses solutions of Type (III).

Proof We prove this proposition by contradiction. Suppose that, without loss of generality, there exist(α1, α2) ∈S1∪C1and a sequence{(αⁱ₁, αⁱ₂)}in_S2∪C2 such that(αⁱ₁, α₂ⁱ) → (α1, α2)as i→ ∞. Since(α1, α2)∈S1∪C1,2(α1, α2)= ∞and hence

R 0

r^2m1+1K₂(r)e^U(r;α1,α2)dr >2+2m₂+γ1

if R is large, where U(r;α1, α2)is defined in (1.8) with respect to(α1, α2). By the continuity of solutions with respect to initial data and applying the bounded convergence theorem, we obtain that

2+2m₂+γ1<

R 0

r^2m1+1K₂(r)e^U(r;α1,α2)dr

= lim

i→∞

R 0

r^2m1+1K2(r)e^{U(r;αi1,α2i)}dr

< lim

i→∞2

αⁱ₁, α₂ⁱ

≤2+2m2+γ1

since(αⁱ₁, α₂ⁱ)∈S2∪C2, which leads to a contradiction. Hence this proof is complete.

Remark 3.1 From the proof of Proposition3.2, we know thatS1∪C1andS2∪C2are disjoint.

Proof of Theorem1.2 We recall that for any solution of (1.6)–(1.7), the corresponding Ki-masses cannot be infinite simultaneously. For existence parts, Propositions3.1and3.2 fulfill these requirements. Hence we complete the proof of Theorem1.2.

To obtain further geometric properties of the regions defined in (2.2), the follow- ing concept related to linearized systems needs to be introduced. Let (u(r), v(r)) =

(u(r;α1, α2), v(r;α1, α2))be a solution of (1.6)–(1.7) and(U(r;α1, α2),V(r;α1, α2))be defined in (1.8). We define

⎧⎪

⎨

⎪⎩

φi(r)=φi(r;α1, α2)= ∂U(r;α1, α2)

∂αi ,

ψi(r)=ψi(r;α1, α2)= ∂V(r;α1, α2)

∂αi , i=1,2. (3.7)

Thenφi(r)andψi(r) (i=1,2)satisfy the linearized systems of (1.9)

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

φ_i(r)+1

rφ_i(r)+r^2m2K1(r)e^Vψi =0, r>0, ψ_i(r)+1

rψ_i(r)+r^2m1K₂(r)e^Uφi =0, r>0, φ1(0)=1, φ2(0)=0, ψ1(0)=0, ψ2(0)=1, φ₁(0)=φ₂(0)=ψ₁(0)=ψ₂(0)=0.

(3.8)

The linearized systems play a significant role in deriving the structure of solutions of (1.6)–(1.7). We first present the monotone properties ofφiandψiin the following lemma.

Lemma 3.1 Let(u(r), v(r))be a solution of (1.6)–(1.7), andφi(r), ψi(r) (i = 1,2)be defined as in (3.7). Then for any r>0,

φ1(r) >0, φ₁(r) >0; ψ1(r) <0, ψ₁(r) <0;

φ2(r) <0, φ₂(r) <0; ψ2(r) >0, ψ₂(r) >0. (3.9) Proof We refer the reader to [8,9] for the proof of this lemma. In fact, from (3.8), it is easy to see thatψ1(r)decreases strictly and henceφ1(r)increases strictly near r =0. We omit

the details here.

In the following, some geometric properties of the regions defined in (2.2) are offered.

Proposition 3.3 The regionsS,S1andS2are nonempty simply connected open subsets of R². Furthermore, the following properties hold.

(a) 1(α1, α2)is continuous for(α1, α2)∈S1∪C1∪S, and2(α1, α2)is continuous for (α1, α2)∈S2∪C2∪S.

(b) If(α1, α21), (α1, α22) ∈ S (resp.,S1,S2) with α21 < α22, then(α1, α2) ∈ S (resp., S1,S2) forα2 ∈ (α21, α22). Similarly, if(α11, α2), (α12, α2) ∈ S (resp.,S1,S2) with α11< α12, then(α1, α2)∈S(resp.,S1,S2) forα1∈(α11, α12).

(c) _C1and_C2are curves in R².

Proof (a) Let(α10, α20) ∈ S1 ∪C1 ∪S, i.e.,1(α10, α20) < ∞ and2(α10, α20) >

2+2m₂+γ1. Then2(α10, α20) >2+2m₂+γ1+εfor someε >0. Since r V(r;α10, α20) decreases to−2(α10, α20)as r → ∞, we may select r₀ >0 sufficiently large to assure that

r V(r;α10, α20) <−(2+2m₂+γ1+ε), r≥r₀.

Moreover, by the continuity of solutions with respect to initial data, there existsδ >0 such that for(α1, α2)∈B_δ((α10, α20)), we have r V(r0;α1, α2)≤ −(2+2m2+γ1+ε). Then

r V(r;α1, α2)≤ −(2+2m₂+γ1+ε), r≥r₀ for(α1, α2)∈B_δ((α10, α20)), which implies that

V(r;α1, α2)≤c₁−(2+2m₂+γ1+ε)log r, r≥r₀

for(α1, α2)∈B_δ((α10, α20))with some constant c₁. Hence,

e^V(r;α1,α2)≤c₂r^{−(2+2m2+γ1+ε)}, r≥r₀ (3.10) for(α1, α2)∈B_δ((α10, α20)), where c2=e^c1. Due to the above results, we obtain that

1(α1, α2) <∞, (α1, α2)∈B_δ((α10, α20)),

which implies that the setS1∪C1∪Sis open. In addition, by (3.10) and applying the Lebesgue dominated convergence theorem, we conclude that1(α1, α2)is continuous at(α10, α20). The situation for2is similar, and then we complete the proof of (a).

(b) Let(α1, α21), (α1, α22) ∈ S andα21 < α22. Then 1(α1, α2i) and2(α1, α2i), i =1,2, are all finite. By Lemma3.1, we see thatψ2(r) >0 andφ2(r) < 0. This means that for any fixed r>0,V(r;α1, α2)and U(r;α1, α2)are strictly increasing and decreasing with respect toα2respectively. Hence for anyα2∈(α21, α22), we have that both1(α1, α2) and2(α1, α2)are also finite, and then(α1, α2)∈S. The proofs of other cases are similar, and we omit the details. We complete the proof of (b).

Define

ρ1(α1)=sup{α2:(α1, α2)∈S1}, ρ2(α1)=inf{α2:(α1, α2)∈S2} (3.11) and

σ1(α1)=inf{α2:(α1, α2)∈S}, σ2(α1)=sup{α2:(α1, α2)∈S}. (3.12) Then

S1= {(α1, α2):α1∈R, α2 < ρ1(α1)}, S2= {(α1, α2):α1∈R, α2> ρ2(α1)}

and

S = {(α1, α2):α1∈R, σ1(α1) < α2< σ2(α1)}.

It is easy to see thatS,S1andS2are nonempty simply connected open subsets of R². (c) First we have

C1 = {(α1, α2):α1∈R, ρ1(α1)≤α2≤σ1(α1)}, and

C2= {(α1, α2):α1∈R, σ2(α1)≤α2≤ρ2(α1)}.

To claim that_C1 is a curve, i.e.,ρ1 = σ1, it suffices to show that for anyε > 0,(α1+ ε, α2), (α1, α2+ε) /∈C1whenever(α1, α2)∈C1. If(α1, α2)∈C1, then1(α1, α2)=κ1, whereκ1is defined in (2.3). For anyε >0, we have V(r;α1+ε, α2) <V(r;α1, α2)and then e^{V(r;α1+ε,α2)}<e^V(r;α1,α2)for all r >0 by Lemma3.1. Hence1(α1+ε, α2) < 1(α1, α2) and then(α1+ε, α2) /∈C1(in fact,(α1+ε, α2)∈S1). Similarly, from Lemma3.1again, we also have V(r;α1, α2+ε) >V(r;α1, α2)which implies1(α1, α2+ε) > 1(α1, α2). Therefore(α1, α2+ε) /∈C1(in fact,(α1, α2+ε)∈S∪S2). The situation for_C2can be

done in the same way, and hence (c) is proved.

Now we are in the position to prove Theorem1.3.

Proof of Theorem1.3 To prove (a), it is enough to show that the range of1(α1, α2)over S1∪C1is exactly the interval(0,2+2m1+γ2]. Letθ1∈(0,2+2m1+γ2). Then we can choose someξ∈ [λ1,2), whereλ1is defined as in Proposition3.1, so that 2−ξ < θ1. For

suchξ, Proposition3.1(a) assures that1(α^∗₁, α^∗₂) <2−ξ < θ1for some(α₁^∗, α₂^∗) ∈S1. In addition, we also know that1(α^∗₁, ρ1(α₁^∗))=2+2m₁+γ2 > θ1, whereρ1is defined in (3.11). Therefore, by virtue of Proposition 3.3(a) and (b), we obtain that there exists α2∈(α₂^∗, ρ1(α^∗₁))satisfying(α^∗₁, α2)∈S1and1(α₁^∗, α2)=θ1. Then (a) is proved.

The proof of (b) is similar, and we omit the details. Hence the proof of Theorem1.3is

complete.

4 The case of K₁ ≡K₂≡1

Throughout this section, we consider the case of K1 ≡ K2 ≡ 1 in (1.6). In this case, βi=γi=0(i=1,2)in (1.3). The logarithmic behaviors ofφiandψiat infinity are proved below, and the differentiation properties of K_i-masses with respect to normalized initial data follow.

Lemma 4.1 Let(α1, α2)∈Sand(φi(r), ψi(r))=(φi(r;α1, α2), ψi(r;α1, α2)),i =1,2, be defined as in (3.7). Then

(a) φi(r)=C_φⁱ log r+μi+o(1)at r = ∞for some C_φ¹ >0,C_φ²<0 andμi ∈R,i =1,2.

Furthermore, C_φⁱ =Cⁱ_φ(α1, α2)is continuous with respect to(α1, α2)onS,i =1,2.

(b) ψi(r)=Cⁱ_ψlog r+νi+o(1)at r = ∞for some C_ψ² >0,C_ψ¹ <0 andνi ∈R,i =1,2.

Furthermore, C_ψⁱ =C_ψⁱ (α1, α2)is continuous with respect to(α1, α2)onS,i=1,2.

Proof We only prove the results involvingφ1andψ1. The others involvingφ2andψ2are similar, and we omit the details. We divide the proof into the following steps.

Step 1. We first show that

φ1(r)=C_φ¹log r+μ1+o(1),

ψ1(r)=C¹_ψlog r+ν1+o(1), at r= ∞

for some C_φ¹ =C_φ¹(α1, α2) >0,C_ψ¹ =C_ψ¹(α1, α2) <0 andμ1, ν1∈R. Let(U(r),V(r)) be defined in (1.8) associated with the solution(u(r;α1, α2), v(r;α1, α2))of (1.15)–(1.7).

Since(α1, α2)∈Sand from Remark 2.1(ii), then for R0large, η≡ min

r≥R₀{−rU(r)−(2+2m₁),−r V(r)−(2+2m₂)}>0. Choose 0< ε < ηand C0>0. Then for r ≥R0, we have

[C₀r^1+ε−(φ1−ψ1)] =(1+ε)²C₀r^ε−1+r^2m2e^Vψ1−r^2m1e^Uφ1

≥(1+ε)²C0r^ε−1−r^−2−ε(φ1−ψ1)

≥(1+ε)²C0r^ε−1−(1+ε)²r⁻²(φ1−ψ1)

=(1+ε)²r⁻²[C₀r^1+ε−(φ1−ψ1)], r≥R₁ for some R1≥R0sinceφ1>0, ψ1<0 and the definition ofη. Hence

−ψ1(r) < φ1(r)−ψ1(r)≤C0r^1+ε, r≥R1

if C₀is large.

Now, from (3.8) and the above results, we have

φ1= −r^2m2e^Vψ1(r)≤ ˆC₀r⁻¹, r≥R₁