On the Solutions to a Liouville-type System
Involving Singularity
Zhi-You Chen
∗Department of Mathematics, National Tsing Hua University Hsin-Chu 30013, Taiwan
Jann-Long Chern
†, Yong-Li Tang
Department of Mathematics, National Central UniversityChung-Li 32001, Taiwan
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|)ev= 4πm1δ0, ∆v + K2(|x|)eu= 4πm2δ0, in R2, (1.1) where ∆ =P2i=1∂2/∂x2
i is the Laplacian operator in R2, m1, m2> 0, δ0 is the Dirac measure at the origin, and K1(r), K2(r) are positive for r > 0 satisfying
r2miK
i∈ L1([0, 1)) ∩ C1[(0, ∞)], i = 1, 2,
(1.2)
∗Corresponding author.
†Work partially supported by National Science Council of Taiwan.
E-mail addresses : [email protected] (Z.-Y. Chen), [email protected] (J.-L. Chern), [email protected] (Y.-L. Tang).
lim
r→0r −βiK
i(r), limr→∞r−γiKi(r) are finite and positive, i = 1, 2
(1.3)
for some βi, 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 + Keu= 0 in Ω ⊂ R2. (1.5)
It is well-known that the Liouville equation is related to many applications in 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 [13]. 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 mechanics of classical vortices and the random surface problem [6, 21], and is considered to deal with topics closely related to the abelian model in the Chern-Simons theories [5, 8, 14, 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 [2, 15, 19], in the theory of semi-conductors [23], in the theory of chemotaxis [10, 20], 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 [3, 9, 11, 12, 16, 17, 18, 22] and references therein.
Throughout this article, we consider the radial case of (1.1), i.e., the follow-ing ODE system:
u00+1 ru 0+ K 1(r)ev= 0, v00+1 rv 0+ K 2(r)eu= 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)
where α1, 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 U00(r) +1 rU 0(r) + r2m2K 1(r)eV = 0, r > 0, V00(r) +1 rV 0(r) + r2m1K 2(r)eU = 0, r > 0, U (0) = α1, V (0) = α2, U0(0) = 0, V0(0) = 0. (1.9)
Notice 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 limr→RU (r) = −∞
for some finite R > 0, then limr→RU0(r) = −∞ also. Hence,
−∞ = lim r→RrU 0(r) = lim r→R µ − Z r 0 s2m2+1K 1eVds ¶ = − Z R 0 s2m2+1K 1eVds 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 [6].
In [11], the following system was considered: (
∆u + V1(|x|)eau+bv = 0, ∆v + V2(|x|)ebu+cv= 0,
in Ω, (1.10)
where Ω is either BR(0) or R2, V1, V2 are positive functions on Ω, and a, b, c are constant. If Ω = BR(0), 0 < k1≤ V1, V2≤ k2< ∞ on BR(0) and a, c ≥ 0,
then for any 0 < M1, M2 < ∞ satisfying certain assumptions (Theorem 1.1 in [11]), there exists a radially symmetric solution (u(r), v(r)) of (1.10) so that
u(R) = v(R) = 0 and Z Ω V1eau+bvdx = M1, Z Ω V2ebu+cvdx = M2. (1.11) For Ω = R2, V
1 ≡ V2 ≡ 1 and a, b, c ≥ 0, some sufficient and necessary condi-tions for existence 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 solutions 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 flux.
It is worth mentioning that in the Chern-Simons systems [5], 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 rK0 i(r) Ki(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 re-moved from Theorem 1.1.
Let (u(r), v(r)) be a solution of (1.6)-(1.7). We define
Θ1(u, v) = Z ∞ 0 rK1(r)ev(r)dr, Θ2(u, v) = Z ∞ 0 rK2(r)eu(r)dr. (1.14)
Clearly, 0 < Θi(u, v) ≤ ∞. We sometimes denote it by (Θ1, Θ2) if no confusion arises. In the case of u ≡ v in (1.6), Θ = Θ1(u, v) = Θ2(u, v) is called, for example, the total curvature coming from the prescribed Gaussian curvature equation or the flux in physics. Here we call Θ1(u, v) and Θ2(u, v) the K1
-mass and K2-mass with respect to solution (u, v) respectively. From standard arguments, Θ1(u, v) and Θ2(u, v) can not 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 corresponding Ki-masses pair (Θ1, Θ2) as follows:
Type (I): lim
r→∞(u(r), v(r)) = (−∞, −∞) with Θ1< ∞ and Θ2= ∞.
Type (II): lim
r→∞(u(r), v(r)) = (−∞, −∞) with Θ1= ∞ and Θ2< ∞.
Type (III): lim
r→∞(u(r), v(r)) = (−∞, −∞) with Θ1< ∞ and Θ2< ∞.
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 sated above range exactly over certain intervals related to
mi and γi.
(a) For any θ1 ∈ (0, 2 + 2m1 + γ2], there exist infinitely many solutions (u(r), v(r)) of Type (I) such that
(Θ1(u, v), Θ2(u, v)) = (θ1, ∞).
Furthermore, if (u(r), v(r)) is a solution of Type (I), then Θ1(u, v) ≤ 2 + 2m1+ γ2 and Θ2(u, v) = ∞.
(b) For any θ2 ∈ (0, 2 + 2m2 + γ1], there exist infinitely many solutions (u(r), v(r)) of Type (II) such that
(Θ1(u, v), Θ2(u, v)) = (∞, θ2).
Furthermore, if (u(r), v(r)) is a solution of Type (II), then Θ1(u, v) = ∞
and Θ2(u, v) ≤ 2 + 2m2+ γ1.
For the special case of K1≡ K2≡ 1 in (1.6), i.e., u00+1 ru 0+ ev= 0, v00+1 rv 0+ eu= 0, r > 0, (1.15)
we have some further consequence. Before stating our final result, we first intro-duce the linearized system of (1.15)-(1.7) with respect to solution (u(r), v(r)):
A00+1 rA 0+ r2m2eV (r)B = 0, B00+1 rB 0+ r2m1eU (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 possesses a nonconstant bounded solution (A(r), B(r)) on (0, ∞).
Theorem 1.4 Consider (1.15)-(1.7). Then
(a) All conclusions in Theorems 1.2 and 1.3 hold (γ1= γ2= 0 in this case). (b) For any θ1> 2 + 2m1 and θ2> 2 + 2m2 satisfying
θ1θ2− (2m2+ 2)θ1− (2m1+ 2)θ2= 0, (1.17)
there exist infinitely many solutions (u(r), v(r)) of Type (III) such that
(Θ1(u, v), Θ2(u, v)) = (θ1, θ2).
Furthermore, if (u(r), v(r)) is a solution of Type (III), then (θ1, θ2) = (Θ1(u, v), Θ2(u, v)) satisfies (1.17), and its corresponding linearized
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
Theorem 1.1 and Theorem 1.4. By adopting the notations introduced in (2.2), Section 2, we illustrate the structure of solutions of (1.15)-(1.7) below.
Figure: Structure of solutions of (1.15)-(1.7).
We organize this article as follows. In Section 2, some results concerning with Ki-masses will be made. Theorems 1.2 and 1.3 will be proved in Section
3. We give a complete verification of Theorem 1.4 in Section 4. Finally, Section 5 is devoted to the proof of Theorem 1.1.
2
K
i-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), Θ1(u, v) = Z ∞ 0 r2m2+1K 1eV (r)dr, Θ2(u, v) = Z ∞ 0 r2m1+1K 2eU (r)dr, (2.1)
where (U (r), V (r)) is defined in (1.8) and satisfies (1.9). Occasionally, we denote Θi(u(r; α1, α2), v(r; α1, α2)) by Θi(α1, α2) simply to indicate the dependence of (α1, α2).
Our first lemma in this section gives the fact that if one of Ki-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 Θ1(u, v) < ∞ (resp., Θ2(u, v) < ∞), then Θ2(u, v) > 2+2m2+γ1 (resp., Θ1(u, v) > 2 + 2m1+ γ2).
Proof. Let Θ1(u, v) < ∞ and (U (r), V (r)) be defined as in (1.8). Without loss of generality, we assume Θ2(u, v) < ∞. Then by (1.9) and (2.1), we have that rV0(r) > −Θ
2 for r ≥ 0. Hence
V (r) > −Θ2log r + c1, r ≥ R and, by (1.3),
K1(r) ≥ c2rγ1, r ≥ R
for some R > 0 and c1, c2> 0. Therefore, by combining the results above, we obtain that Θ1 > Z ∞ R r2m2+1K 1(r)eV (r)dr ≥ c3 Z ∞ R s2m2+1+γ1−Θ2 ds
for some c3 > 0, which implies Θ2 > 2 + 2m2+ γ1 because of the finiteness of Θ1. 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 (Θ1, Θ2):
S = {(α1, α2) : Θ1(u, v) < ∞, Θ2(u, v) < ∞}, S1= {(α1, α2) : 0 < Θ1(u, v) < κ1, Θ2(u, v) = ∞}, S2= {(α1, α2) : 0 < Θ2(u, v) < κ2, Θ1(u, v) = ∞}, C1= {(α1, α2) : Θ1(u, v) = κ1, Θ2(u, v) = ∞}, C2= {(α1, α2) : Θ2(u, v) = κ2, Θ1(u, v) = ∞}, (2.2) where κ1= 2 + 2m1+ γ2, κ2= 2 + 2m2+ γ1. (2.3) It is clear that R2= S ∪ S 1∪ 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
(
(rU0)0+ r2m2+1K
1(r)eV = 0, r > 0, (rV0)0+ r2m1+1K
2(r)eU = 0, r > 0.
Multiplying rV0(r) and rU0(r) on the first and second equation of the above
relation respectively, adding them together, and then integrating from 0 to r, we have Z r 0 £ (sU0)(sV0)¤0ds + Z r 0 ¡ s2m2+2K 1eVV0+ s2m1+2K2eUU0 ¢ ds = 0 (2.4)
and hence obtain the Pohozaev type identity as follows: r2U0(r)V0(r) + r2m2+2K 1(r)eV + r2m1+2K2(r)eU = Z r 0 n s2m2+1£(2m 2+ 2)K1+ sK10 ¤ eV +s2m1+1£(2m 1+ 2)K2+ sK20 ¤ eU o ds. (2.5)
For finite Ki-masses, they satisfy a relation as shown in the following.
Lemma 2.2 If (α1, α2) ∈ S, then (Θ1, Θ2) = (Θ1(α1, α2), Θ2(α1, α2))
satis-fies Θ1Θ2− (2m2+ 2)Θ1− (2m1+ 2)Θ2= Z ∞ 0 ³ r2m2+2K0 1eV + r2m1+2K20eU ´ dr.
Proof. Let (U (r), V (r)) be defined in (1.8) associated with the solution (u(r; α1, α2), v(r; α1, α2)). Since Θ1+ Θ2 is finite, then by (2.1), we have that
r2m2+2
k K1(rk)eV (rk)+ rk2m1+2K2(rk)eU (rk)→ 0 as k → ∞
for some sequence {rk} with rk → 0 as k → ∞. Hence by taking r = rk 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, lemma 2.2 implies
Θ1Θ2− (2m2+ 2)Θ1− (2m1+ 2)Θ2= 0
for any solution (u(r), v(r)) of (1.15)-(1.7) with both Θ1(u, v) and Θ2(u, v) finite,
i.e., solution of Type (III).
(ii) For any (α2, α2) ∈ S, we conclude that
rU0(r) < −(2 + 2m
1+ γ2), rV0(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 Lemma 2.1 and the facts that rU0(r) and rV0(r) decrease to
−Θ1(α1, α2) and −Θ2(α1, α2) respectively as r → ∞ by (1.9).
Actually, (U (r), V (r)) behave logarithmically at infinity, related to its cor-responding Ki-masses if both are finite.
Lemma 2.3 Let (α1, α2) ∈ S and (u(r; α1, α2), v(r; α1, α2)) be the solution of (1.6)-(1.7). Then ( U (r) = −Θ1(α1, α2) log r + O(1) at r = ∞, V (r) = −Θ2(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 rU0(r) = − Z r 0 s2m2+1K 1eVds, rV0(r) = − Z r 0 s2m1+1K 2eUds, r ≥ 0. Then lim r→∞rU 0(r) = −Θ 1(α1, α2), r→∞lim rV0(r) = −Θ2(α1, α2).
By Remark 2.1(ii), we may choose some p > 0 such that rV0(r) + p < −(2 +
2m2+ γ1) for r ≥ R if R is large. Then V (r) < −(p + 2 + 2m2+ γ1) log r + c1 for r ≥ R and some c1> 0, which implies that
eV (r)< c2r−(p+2+2m2+γ1), r ≥ R, where c2= ec1.
Let W (r) = U (r) + Θ1(α1, α2) log r for r > 0. Then by the above result, (1.3) and (2.1), we obtain that for r ≥ R,
rW0(r) = rU0(r) + Θ 1(α1, α2) = − Z r 0 s2m2+1K 1eV (s)ds + Θ1(α1, α2) = Z ∞ r s2m2+1K 1eV (s)ds < c3 Z ∞ r s2m2+1· sγ1· s−(p+2+2m2+γ1)ds = c3 Z ∞ 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 Section 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 associ-ated 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 any ξ ∈ [λ1, 2), there exist two constants ¯α1(ξ) > 0 and ¯α2(ξ) < 0
such that
Θ1(α1, α2) < 2 − ξ, Θ2(α1, α2) = ∞
for all α1≥ ¯α1(ξ) and α2≤ ¯α2(ξ).
(b) For any ζ ∈ [λ2, 2), there exist two constants ˆα2(ζ) > 0 and ˆα1(ζ) < 0
such that
Θ1(α1, α2) = ∞, Θ2(α1, α2) < 2 − ζ
for all α1≤ ˆα1(ζ) and α2≥ 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 that λ1, λ2< 2 by the assumptions in (1.4).
(a) We split this proof into the following steps.
Step 1. For ξ ∈ [λ1, 2) and η < −(2m1+ β2), we define (
w(r) = w(r; α1, α2) = U (r) + log(1 + r2−ξ), r ≥ 0,
z(r) = z(r; α1, α2) = V (r) + log(1 + r2−η), r ≥ 0. Then w(r) and z(r) satisfy
(rw0)0(r) = r 1−ξ (1 + r2−ξ)2 · (2 − ξ)2−r ξ+2m2(1 + r2−ξ)2 1 + r2−η K1e z ¸ , r > 0, (rz0)0(r) = r1−η (1 + r2−η)2 · (2 − η)2−rη+2m1(1 + r2−η)2 1 + r2−ξ K2e w ¸ , r > 0, w(0) = α1, z(0) = α2; lim r→0rw 0(r) = 0, lim r→0rz 0(r) = 0. (3.1)
Since 2m1+ ξ + γ2≥ 0 and 2 − η > 0, there exist α∗1> 0 and R0> 1 such that for r ≥ R0 and α1≥ α1∗, we have
r1−η (1 + r2−η)2 · (2 − η)2−rη+2m1(1 + r2−η)2 1 + r2−ξ K2(r)e α1 ¸ ≤ −k2eα1 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
rξ+2m2−(k2eα∗1log R1)·(1 + r 2−ξ)2 1 + r2−η K1(r) < (2 − ξ)2 2 , r ≥ R1 (3.3)
if R1≥ R0is sufficiently large. Moreover, by (3.1) and (3.2), we also get z(r) ≤
−(k2eα
∗
1log R2) log r for α1 ≥ α∗1 and r ≥ R3 for some large R3 ≥ R2 ≥ R1.
Hence ez(r)≤ r−k2eα∗1log R2, r ≥ R 3, α1≥ α∗1. (3.4) By combining (3.1)-(3.4), we obtain (2 − ξ)2−rξ+2m2(1 + r2−ξ)2 1 + r2−η K1(r)e z(r)> 0, (2 − η)2−rη+2m1(1 + r2−η)2 1 + r2−ξ K2(r)e w(r)< 0, r ≥ R3, α1≥ α∗1. (3.5)
On the other hand, because of ξ + 2m2+ β1 ≥ 0 and η + 2m1+ β2 ≤ 0, we have that (3.5) holds for r ∈ (0, R3], α1≥ ¯α1and α2≤ ¯α2if we choose ¯α1≥ α∗1 sufficiently large and ¯α2 < 0 sufficiently small. Therefore, from (3.1) and the above results, we conclude that
(rw0)0(r) > 0, (rz0)0(r) < 0, r > 0, α
1≥ ¯α1, α2≤ ¯α2, (3.6)
and hence w(r; α1, α2) is positive for all r ≥ 0, α1≥ ¯α1 and α2≤ ¯α2.
Step 2. From Step 1, we see that for α1 ≥ ¯α1 and α2 ≤ ¯α2, U (r) =
U (r; α1, α2) > − log(1 + r2−ξ) and then eU (r)> 1/(1 + r2−ξ) for r ≥ 0. Hence
Θ2(α1, α2) = Z ∞ 0 r2m1+1K 2(r)eU (r)dr ≥ C Z ∞ R r2m1+1· rγ2· r−(2−ξ)dr = C Z ∞ R r2m1+γ2+ξ−1dr = ∞
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 0(r) = lim r→∞ · rU0(r) +(2 − ξ)r2−ξ 1 + r2−ξ ¸ = −Θ1(α1, α2) + (2 − ξ), and then Θ1(α1, α2) < 2 − ξ. Hence (a) is proved.
(b) For ζ ∈ [λ2, 2) and η < −(2m2+ β1), we define (
w(r) = w(r; α1, α2) = U (r) + log(1 + r2−η), r ≥ 0,
z(r) = z(r; α1, α2) = V (r) + log(1 + r2−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 region S 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∪ C1 and a sequence {(αi1, αi2)} in
S2∪ C2 such that (αi1, αi2) → (α1, α2) as i → ∞. Since (α1, α2) ∈ S1∪ C1, Θ2(α1, α2) = ∞ and hence
Z R 0
r2m1+1K
2(r)eU (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 < Z R 0 r2m1+1K 2(r)eU (r;α1,α2)dr = lim i→∞ Z R 0 r2m1+1K 2(r)eU (r;α i 1,αi2)dr < lim i→∞Θ2(α i 1, αi2) ≤ 2 + 2m2+ γ1 since (αi
1, αi2) ∈ S2∪ C2, which leads to a contradiction. Hence this proof is complete.
Remark 3.1. From the proof of Proposition 3.2, we know that S1∪ C1 and
S2∪ C2 are disjoint.
Proof of Theorem 1.2. We recall that for any solution of (1.6)-(1.7), the cor-responding Ki-masses can not be infinite simultaneously. For existence parts,
Propositions 3.1 and 3.2 fulfill these requirements. Hence we complete the proof of Theorem 1.2.
To obtain further geometric properties of the regions defined in (2.2), the following 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)
Then φi(r) and ψi(r) (i = 1, 2) satisfy the linearized systems of (1.9) φ00 i(r) + 1 rφ 0 i(r) + r2m2K1(r)eVψi= 0, r > 0, ψ00 i(r) + 1 rψ 0 i(r) + r2m1K2(r)eUφi= 0, r > 0, φ1(0) = 1, φ2(0) = 0, ψ1(0) = 0, ψ2(0) = 1, φ0 1(0) = φ02(0) = ψ01(0) = ψ20(0) = 0. (3.8)
The linearized systems play a significant role in deriving the structure of solu-tions of (1.6)-(1.7). We first present the monotone properties of φi and ψi in
the following lemma.
Lemma 3.1 Let (u(r), v(r)) be a solution of (1.6)-(1.7), and φi(r), ψi(r) (i =
1, 2) be defined as in (3.7). Then for any r > 0, (
φ1(r) > 0, φ01(r) > 0; ψ1(r) < 0, ψ01(r) < 0;
φ2(r) < 0, φ02(r) < 0; ψ2(r) > 0, ψ02(r) > 0. (3.9)
Proof. We refer the reader to [7, 8] for the proof of this lemma. In fact, from (3.8), it is easy to see that ψ1(r) decreases strictly and hence φ1(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 regions S, S1 and S2 are nonempty simply connected
open subsets of R2. Furthermore, the following properties hold.
(a) Θ1(α1, α2) is continuous for (α1, α2) ∈ S1∪ C1∪ S, and Θ2(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) for α2 ∈ (α21, α22). Similarly, if (α11, α2), (α12, α2) ∈ S (resp., S1, S2) with α11 < α12, then (α1, α2) ∈ S (resp., S1, S2) for α1∈ (α11, α12).
(c) C1 and C2 are curves in R2.
Proof. (a) Let (α10, α20) ∈ S1∪C1∪S, i.e., Θ1(α10, α20) < ∞ and Θ2(α10, α20) > 2 + 2m2+ γ1. Then Θ2(α10, α20) > 2 + 2m2+ γ1+ ε for some ε > 0. Since
rV0(r; α
10, α20) decreases to −Θ2(α10, α20) as r → ∞, we may select r0 > 0 sufficiently large to assure that
rV0(r; α
Moreover, by the continuity of solutions with respect to initial data, there exists
δ > 0 such that for (α1, α2) ∈ Bδ((α10, α20)), we have rV0(r0; α1, α2) ≤ −(2 + 2m2+ γ1+ ε). Then
rV0(r; α
1, α2) ≤ −(2 + 2m2+ γ1+ ε), r ≥ r0 for (α1, α2) ∈ Bδ((α10, α20)), which implies that
V (r; α1, α2) ≤ c1− (2 + 2m2+ γ1+ ε) log r, r ≥ r0 for (α1, α2) ∈ Bδ((α10, α20)) with some constant c1. Hence,
eV (r;α1,α2)≤ c
2r−(2+2m2+γ1+ε), r ≥ r0 (3.10)
for (α1, α2) ∈ Bδ((α10, α20)), where c2 = ec1. Due to the above results, we obtain that
Θ1(α1, α2) < ∞, (α1, α2) ∈ Bδ((α10, α20)),
which implies that the set S1∪C1∪S is open. In addition, by (3.10) and applying the Lebesgue dominated convergence theorem, we conclude that Θ1(α1, α2) is continuous at (α10, α20). The situation for Θ2is similar, and then we complete the proof of (a).
(b) Let (α1, α21), (α1, α22) ∈ S and α21 < α22. Then Θ1(α1, α2i) and Θ2(α1, α2i), i = 1, 2, are all finite. By Lemma 3.1, we see that ψ2(r) > 0 and
φ2(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 to α2 respectively. Hence for any α2 ∈ (α21, α22), we have that both Θ1(α1, α2) and Θ2(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 that S, S1and S2are nonempty simply connected open subsets of R2.
(c) First we have
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 any
ε > 0, (α1+ ε, α2), (α1, α2+ ε) /∈ C1 whenever (α1, α2) ∈ C1. If (α1, α2) ∈ C1, then Θ1(α1, α2) = κ1, where κ1 is defined in (2.3). For any ε > 0, we have
V (r; α1+ ε, α2) < V (r; α1, α2) and then eV (r;α1+ε,α2)< eV (r;α1,α2)for all r > 0 by Lemma 3.1. Hence Θ1(α1+ ε, α2) < Θ1(α1, α2) and then (α1+ ε, α2) /∈ C1 (In fact, (α1+ ε, α2) ∈ S1). Similarly, from Lemma 3.1 again, we also have
V (r; α1, α2+ ε) > V (r; α1, α2) which implies Θ1(α1, α2 + ε) > Θ1(α1, α2). Therefore (α1, α2 + ε) /∈ C1 (In fact, (α1, α2+ ε) ∈ S ∪ S2). The situation for C2 can be done in the same way, and hence (c) is proved.
Now we are in the position to prove Theorem 1.3.
Proof of Theorem 1.3. To prove (a), it is enough to show that the range of Θ1(α1, α2) over S1∪ C1 is exactly the interval (0, 2 + 2m1+ γ2]. Let θ1 ∈ (0, 2 + 2m1+ γ2). Then we can choose some ξ ∈ [λ1, 2), where λ1 is defined as in Proposition 3.1, so that 2 − ξ < θ1. For such ξ, Proposition 3.1(a) assures that Θ1(α∗1, α∗2) < 2 − ξ < θ1 for some (α∗1, α2∗) ∈ S1. In addition, we also know that Θ1(α∗1, ρ1(α1∗)) = 2 + 2m1+ γ2 > θ1, where ρ1 is defined in (3.11). Therefore, by virtue of Proposition 3.3(a) and (b), we obtain that there exists
α2 ∈ (α∗2, ρ1(α∗1)) satisfying (α∗1, α2) ∈ S1 and Θ1(α∗1, α2) = θ1. Then (a) is proved.
The proof of (b) is similar, and we omit the details. Hence the proof of Theorem 1.3 is complete.
4
The Case of K
1≡ K
2≡ 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 of φi and
ψi at infinity are proved below, and the differentiation properties of Ki-masses
with respect to normalized initial data follow.
Lemma 4.1 Let (α1, α2) ∈ S and (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) = Cφi log r + µi+ o(1) at r = ∞ for some Cφ1 > 0, Cφ2 < 0 and
µi ∈ R, i = 1, 2. Furthermore, Cφi = Cφi(α1, α2) is continuous with
respect to (α1, α2) on S, i = 1, 2.
(b) ψi(r) = Cψi log r + νi+ o(1) at r = ∞ for some Cψ2 > 0, Cψ1 < 0 and
νi ∈ R, i = 1, 2. Furthermore, Cψi = Cψi(α1, α2) is continuous with
Proof. We only prove the results involving φ1and ψ1. The others involving
φ2 and ψ2 are similar, and we omit the details. We divide the proof into the following steps.
Step 1. We first show that
( φ1(r) = Cφ1log r + µ1+ o(1), ψ1(r) = Cψ1log r + ν1+ o(1), at r = ∞ for some C1 φ = Cφ1(α1, α2) > 0, Cψ1 = Cψ1(α1, α2) < 0 and µ1, 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) ∈ S and from Remark 2.1(ii), then for R0 large,
η ≡ min
r≥R0
{−rU0(r) − (2 + 2m1), −rV0(r) − (2 + 2m2)} > 0. Choose 0 < ε < η and C0> 0. Then for r ≥ R0, we have
∆[C0r1+ε− (φ1− ψ1)] = (1 + ε)2C0rε−1+ r2m2eVψ1− r2m1eUφ1 ≥ (1 + ε)2C 0rε−1− r−2−ε(φ1− ψ1) ≥ (1 + ε)2C 0rε−1− (1 + ε)2r−2(φ1− ψ1) = (1 + ε)2r−2[C 0r1+ε− (φ1− ψ1)], r ≥ R1 for some R1≥ R0 since φ1> 0, ψ1< 0 and the definition of η. Hence
−ψ1(r) < φ1(r) − ψ1(r) ≤ C0r1+ε, r ≥ R1 if C0 is large.
Now, from (3.8) and the above results, we have
∆φ1= −r2m2eVψ1(r) ≤ ˆC0r−1, r ≥ R1 for some ˆC0> 0, which implies
φ1(r) ≤ C1r, r ≥ R1 for some C1> 0. Moreover,
∆ψ1= −r2m1eUφ1(r) ≥ − ˆC1r−1−ε, r ≥ R1, for some ˆC1> 0, and hence we also obtain
ψ1(r) ≥ −C2r1−ε, r ≥ R1
for some C2> 0. By repeating the process as above if necessary, we conclude lim r→∞rφ 0 1= − Z ∞ 0 r2m2+1eVψ 1dr = Cφ1, r→∞lim rψ10 = − Z ∞ 0 r2m1+1eUφ 1dr = Cψ1
for some finite C1
Finally, set w(r) = φ1(r) − Cφ1log r. Then w(r) satisfies ∆w = −r2m2eVψ 1, lim r→∞rw 0(r) = 0, which implies rw0(r) = Z ∞ r s2m2+1eVψ 1ds < 0, r > 0.
By means of the choice of ε and the behavior of ψ1(r) at infinity, we get
rw0(r) ≥ −Cr−ε/2 for large r
for some C > 0, and then w(r) is bounded from below. Therefore limr→∞w(r)
exists and finite since w0(r) < 0. The situation for ψ
1(r) is similar, and then we finish this step.
Step 2. Next we prove that C1
φ(α1, α2) and Cψ1(α1, α2) are continuous with respect to (α1, α2) on S. Let (ˆα1, ˆα2) ∈ S. Then by Remark 2.1(ii), there exist constants ε, δ > 0 and r0> 1 such that
rU0(r; α 1, α2) < −(2 + 2m1) − ε, rV0(r; α1, α2) < −(2 + 2m2) − ε (4.1) for r ≥ r0 and (α1, α2) ∈ Bδ ¡ (ˆα1, ˆα2) ¢ . Define X(r; α1, α2) = |φ1(r; α1, α2) − φ1(r; ˆα1, ˆα2)| log r and Y (r; α1, α2) =|ψ1(r; α1, α2) − ψ1(r; ˆα1, ˆα2)| log r .
Then from (3.8), we get
φ1(r) = φ1(r0) + r0φ01(r0)(log r − log r0) − Z r r0 (log r − log s)s2m2+1eVψ 1ds and ψ1(r) = ψ1(r0) + r0ψ10(r0)(log r − log r0) − Z r r0 (log r − log s)s2m1+1eUφ 1ds. By subtracting with respect to (α1, α2) and (ˆα1, ˆα2), divided by log r, and using (4.1), we obtain that for (α1, α2) ∈ Bδ((ˆα1, ˆα2)),
X(r; α1, α2) ≤ X(r0; α1, α2) + r0|φ01(r0; α1, α2) − φ01(r0; ˆα1, ˆα2)|
+C Z r
r0
and Y (r; α1, α2) ≤ Y (r0; α1, α2) + r0|ψ01(r0; α1, α2) − ψ10(r0; ˆα1, ˆα2)| +C Z r r0 s−1−ε/2X(s; α 1, α2)ds, r ≥ r0 for some C > 0. Hence
(X + Y )(r; α1, α2) ≤ I(α1, α2) + C Z r r0 s−1−ε/2[(X + Y )(s; α1, α2)]ds, r ≥ r0 for (α1, α2) ∈ Bδ((ˆα1, ˆα2)), where I(α1, α2) = (X + Y )(r0; α1, α2) + r0|φ01(r0; α1, α2) − φ01(r0; ˆα1, ˆα2)| +r0|ψ10(r0; α1, α2) − ψ01(r0; ˆα1, ˆα2)|. Therefore, by the Gronwall inequality, we have
(X + Y )(r; α1, α2) ≤ I(α1, α2) exp ½µ 2C ε ¶ r0−ε/2 ¾ , r ≥ r0
for (α1, α2) ∈ Bδ((ˆα1, ˆα2)). Since I(α1, α2) → 0 as (α1, α2) → (ˆα1, ˆα2), we deduce that X(r; α1, α2) and Y (r; α1, α2) converge uniformly to 0 on [r0, ∞) as (α1, α2) → (ˆα1, ˆα2). Then Step 2 and hence this lemma is proved.
Lemma 4.2 Both Θ1(α1, α2) and Θ2(α1, α2) are continuously differentiable
on S. Furthermore, ∂Θ1(α1, α2) ∂αi = Z ∞ 0 r2m2+1eVψ i(r; α1, α2)dr = − limr→∞rφ0i(r; α1, α2), ∂Θ2(α1, α2) ∂αi = Z ∞ 0 r2m1+1eUφ i(r; α1, α2)dr = − lim r→∞rψ 0 i(r; α1, α2) (4.2) for (α1, α2) ∈ S and i = 1, 2.
Proof. From Proposition 3.3, we see that S is open and Θ1(α1, α2), Θ2(α1, α2) are continuous on S. Let (ˆα1, ˆα2) ∈ S. We prove that Θ1, Θ2are differentiable at (ˆα1, ˆα2). By Remark 2.1(ii), (1.15) and the continuity with respect to initial data, there exist positive constants R, δ and ε such that
rU0(r; α 1, α2) < −(2 + 2m1) − ε, rV0(r; α1, α2) < −(2 + 2m2) − ε (4.3) for r ≥ R and (α1, α2) ∈ Bδ ¡ ( ˆα1, ˆα2) ¢ . Set gh(r) = r2m2+1 · eV (r; ˆα1+h, ˆα2)− eV (r; ˆα1, ˆα2) h ¸ , fh(r) = r2m1+1 · eU (r; ˆα1+h, ˆα2)− eU (r; ˆα1, ˆα2) h ¸ ,
and g(r) = r2m2+1eV (r; ˆα1, ˆα2)ψ 1(r; ˆα1, ˆα2), f (r) = r2m1+1eU (r; ˆα1, ˆα2)φ 1(r; ˆα1, ˆα2). Then lim h→0gh(r) = g(r), limh→0fh(r) = f (r) and |gh(r)| ≤ r2m2+1eV (r;α 1 1, ˆα2)¯¯ψ 1(r; α21, ˆα2) ¯ ¯, |fh(r)| ≤ r2m1+1eU (r;α 3 1, ˆα2)¯¯φ 1(r; α41, ˆα2) ¯ ¯ (4.4) for some αi 1, i = 1, 2, 3, 4, between ˆα1+ h and ˆα1. Moreover, by Lemma 4.1, we have
C1 φ(ˆα1, ˆα2) −ε 2φ1(r0; ˆα1, ˆα2) < −2C 1 φ(ˆα1, ˆα2), and Cψ1(ˆα1, ˆα2) −ε 2ψ1(r0; ˆα1, ˆα2) > −2C 1 ψ(ˆα1, ˆα2)
for some large r0≥ R. Then by Lemma 4.1 again, (3.8) and the above inequal-ities, there exists δ1∈ (0, δ) such that
rφ01(r; α1, α2) −ε 2φ1(r; α1, α2) < −C 1 φ(ˆα1, ˆα2) < 0, and rψ0 1(r; α1, α2) −ε 2ψ1(r; α1, α2) > −C 1 ψ(ˆα1, ˆα2) > 0 for r ≥ r0 and (α1, α2) ∈ Bδ1((ˆα1, ˆα2)). Hence
£ r−ε/2φ1(r; α1, α2) ¤0 < 0, £r−ε/2ψ1(r; α1, α2) ¤0 > 0, r ≥ r0 (4.5)
for (α1, α2) ∈ Bδ1((ˆα1, ˆα2)). By combining (4.3), (4.4) and (4.5), we get
max{|gh(r)|, |fh(r)|} ≤ M r−1−ε/2, r ≥ r0, |h| < δ1
for some M > 0. Since gh(r) and fh(r) are bounded on [0, r0] and |h| < δ1, the Lebesgue dominated convergence theorem implies
lim h→0 Z ∞ 0 fh(r)dr = Z ∞ 0 f (r)dr, lim h→0 Z ∞ 0 gh(r)dr = Z ∞ 0 g(r)dr.
Finally, ∂Θ1/∂α1and ∂Θ2/∂α1 are continuous on S by (4.3), (4.5) and the Lebesgue dominated convergence theorem again. The proofs of other parts are similar, and we omit the details. Hence this lemma is proved.
In addition, the following gives us the fact that all K1-masses equal identi-cally along a unique curve passing a given normalized initial data in R2, and so do K2-masses.
Proposition 4.1 (a) For any α∗ = (α∗
1, α∗2) ∈ R2, there exists a unique
smooth curve Γ(α∗) in R2 such that α∗∈ Γ(α∗) and
Θi(α∗) = Θi(α), α ∈ Γ(α∗), i = 1, 2.
(4.6)
Moreover, Θ1(α1, α2) (resp., Θ2(α1, α2)) is increasing (resp., decreasing) as α2
increasing and (α1, α2) ∈ S; Θ1(α1, α2) (resp., Θ2(α1, α2)) is decreasing (resp.,
increasing) as α1 increasing and (α1, α2) ∈ S.
(b) The linearized system (1.16) associated with (α1, α2) ∈ S is degenerate. Proof. (a) Let (u(r), v(r)) = (u(r; α∗
1, α∗2), v(r; α∗1, α∗2)) be the solution of (1.15)-(1.7), and (U (r), V (r)) be defined in (1.8). We define
Γ(α∗) = {(Γ
1(t), Γ2(t)) : t > 0} by
Γ1(t) = α∗1+ (2m1+ 2) log t, Γ2(t) = α∗2+ (2m2+ 2) log t, t > 0. (4.7)
Then for any t > 0,
(Ut(r), Vt(r)) ≡ (U (tr) + (2m1+ 2) log t, V (tr) + (2m2+ 2) log t) is the unique solution of (1.9) with initial data (Γ1(t), Γ2(t)). Moreover, we also
have Z ∞ 0 r2m2+1eVt(r)dr = Z ∞ 0 r2m2+1eV (r)dr, and Z ∞ 0 r2m1+1eUt(r)dr = Z ∞ 0 r2m1+1eU (r)dr.
Hence Θi(α∗1, α∗2) = Θi(Γ1(t), Γ2(t)) for all t > 0, i = 1, 2. By Proposition 3.3, we see that such curve Γ(α∗) is unique for any α∗∈ R2 fixed. The remaining parts in (a) can be proved by Lemmas 4.1 and 4.2.
(b) Let (α∗
1, α∗2) ∈ S and M = (2m2+ 2)/(2m1+ 2). Then we have
∂Θi ∂α1(α ∗ 1, α∗2) + M ∂Θi ∂α2(α ∗ 1, α∗2) = 0, i = 1, 2.
In fact, from the proof of (a), we know that (dΘi/dt)(Γ1(t), Γ2(t)) = 0 and then
∂Θi
∂α1
(Γ1(t), Γ2(t)) + M∂Θi
∂α2
(Γ1(t), Γ2(t)) = 0, t > 0, i = 1, 2
and Γ1(1) = α∗1, Γ2(1) = α2∗, where Γ1(t), Γ2(t) are defined in (4.7). By Lemma 4.2, we have lim r→∞r[φ 0 1(r; α∗1, α∗2) + M φ02(r; α∗1, α∗2)] = 0, and lim r→∞r[ψ 0 1(r; α∗1, α∗2) + M ψ20(r; α∗1, α∗2)] = 0.
Hence φ1(r; α∗1, α∗2) + M φ2(r; α∗1, α∗2) and ψ1(r; α∗1, α∗2) + M ψ2(r; α∗1, α∗2), which satisfy (1.16) with respect to (α∗
1, α∗2), are both bounded on [0, ∞) by Lemma 4.1. The proof of (b) is complete.
Before finishing this section, a verification of Theorem 1.4 is given.
Proof of Theorem 1.4. By Proposition 3.3 and Lemma 4.2, we have that the ranges of Θ1(α1, α2) and Θ2(α1, α2) over the region S are (2 + 2m1, ∞) and (2 + 2m2, ∞) respectively. The remaining parts of Theorem 1.4 can be obtained by virtue of Remark 2.1(i) and Proposition 4.1(b). Hence we complete the proof of Theorem 1.4.
Remark 4.1. In Theorem 1.4, we consider (1.15)-(1.7) with m1, m2 > 0. In fact, by applying similar arguments used in this section, the conclusions of The-orem 1.4 also hold for (1.15)-(1.7) with m1, m2∈ (−1, 0].
5
Dirichlet Problem
We consider (1.9) throughout this section, i.e., U00(r) +1 rU 0(r) + r2m2K 1(r)eV = 0, r > 0, V00(r) +1 rV 0(r) + r2m1K 2(r)eU = 0, r > 0, U (0) = α1, V (0) = α2, U0(0) = 0, V0(0) = 0. (5.1) where α1, α2> 0.
Remark 5.1. If (U (r), V (r)) is a solution of (5.1), then it is easy to see that
U (r) and V (r) must vanish on (0, ∞).
Due to Remark 5.1, any solution of (5.1) can be categorized into the following types: a solution (U (r), V (r)) is a Dirichlet-type solution if U (r) and V (r) are both positive before some finite point but vanish at that point (vanishing point); it is a U -crossing (resp., V -crossing) solution if U (r) (resp., V (r)) vanishes first at some finite point where V (r) (resp., U (r)) is still positive.
For convenience, we use the following notations for the regions of initial data corresponding to various types of solutions of (5.1):
D = {(α1, α2) : (U (r; α1, α2), V (r; α1, α2)) is Dirichlet-type}, D1 = {(α1, α2) : (U (r; α1, α2), V (r; α1, α2)) is U -crossing}, D2 = {(α1, α2) : (U (r; α1, α2), V (r; α1, α2)) is V -crossing}. (5.2)
Lemma 5.1 Consider the initial value problem (5.1). Then
(a) For all α1 > 0 (resp., α2 > 0), there exists ˜α2 > 0 (resp., ˜α1 > 0) such
that (α1, α2) ∈ D1 (resp., (α1, α2) ∈ D2) for all α2> ˜α2 (resp., α1> ˜α1). (b) For all α1 > 0 (resp., α2 > 0), there exists ˜α2 > 0 (resp., ˜α1 > 0) such
that (α1, α2) ∈ D2 (resp., (α1, α2) ∈ D1) for all α2< ˜α2 (resp., α1< ˜α1). Proof. First from (5.1), we obtain that
U (r; α1, α2) = α1− Z r 0 (log r − log s)s2m2+1K 1(s)eVds, V (r; α1, α2) = α2− Z r 0 (log r − log s)s2m1+1K 2(s)eUds, (5.3)
where (U (r; α1, α2), V (r; α1, α2)) is the solution of (5.1). To prove (a), let α1> 0 be fixed and define
A = max{r−β2K
2(r) : r ∈ (0, 1]} > 0, B = min{r−β1K1(r) : r ∈ (0, 1]} > 0. Then for r ∈ (0, 1], we get that
Z r 0 (log r − log s)s2m1+1+β2ds ≤ Z 1 0 (− log s)s2m1+1+β2ds = 1 2m1+ 2 + β2 Z 1 0 s2m1+1+β2ds = 1 (2m1+ 2 + β2)2, and hence by (5.3), V (r; α1, α2) ≥ α2− Aeα1 Z r 0 (log r − log s)s2m1+1+β2ds ≥ α2− Aeα1 (2m1+ 2 + β2)2 ≥ α2 2 , r ∈ (0, 1] if α2> 2Aeα1/(2m1+ 2 + β2)2.
Additionally, we also have
U (1; α1, α2) ≤ α1− Beα2/2 Z 1
0
(− log s)s2m2+1+β1ds.
Therefore, if we choose α2 sufficiently large, then results above deduce that
V (r; α1, α2) > 0 on [0, 1] but U (1; α1, α2) < 0. This means that (α1, α2) ∈ D1 for large α2, and hence (a) is established. The proof of (b) is similar, and we omit the details. Then this lemma is proved.
The following results consist of the geometric structure of solutions to (5.1) in terms of initial data.
Lemma 5.2 Consider the initial value problem (5.1) and notations defined in
(5.2). Then
(a) D1 and D2 are simply connected open sets and
D ∪ D1∪ D2= (0, ∞) × (0, ∞).
(b) There exists a strictly increasing function τ : (0, ∞) → (0, ∞) satisfying limα1→0+τ (α1) = 0, limα1→∞τ (α1) = ∞ such that
D = {(α1, τ (α1)) : α1> 0}.
Proof. It is not difficult to see that (a) holds by Lemma 5.1 and Remark 5.1. To prove (b), it suffices to show that (α∗
1+ ε, α∗2), (α∗1, α∗2 + ε) /∈ D for any ε > 0 whenever (α∗
1, α∗2) ∈ D. In fact, let R > 0 be the point so that
U (r; α∗
1, α∗2), V (r; α∗1, α∗2) > 0 for r ∈ [0, R) and U (R; α∗1, α∗2) = V (R; α∗1, α∗2) = 0. Then from Lemma 3.1, we have that
U (r; α∗
1+ ε, α∗2) − U (r; α∗1, α∗2) = εφ1(r; ˆα1(r), α2∗) > 0, r ∈ (0, R), and
V (R; α∗
1+ ε, α∗2) − V (R; α∗1, α2∗) = εψ1(R; ˜α1, α∗2) < 0,
for some ˆα1(r), ˜α1∈ (α∗1, α∗1+ε). This deduces that (α∗1+ε, α∗2) ∈ D2. Similarly, we also have (α∗
1, α∗2+ ε) ∈ D1 for any ε > 0, and hence (b) is proved.
We now give the existence of solutions to the Dirichlet problem of (5.1) below.
Proposition 5.1 For any R > 0, (5.1) possesses a Dirichlet-type solution which is positive on [0, R) while vanishes at R.
Proof. Let τ be defined as in Lemma 5.2(b). We introduce the func-tion R : (0, ∞) → (0, ∞) to be the point so that the Dirichlet-type solufunc-tion
U (r; α1, τ (α1)), V (r; α1, τ (α1)) > 0 for r ∈ [0, R(α1)) and U (R(α1); α1, τ (α1)) =
V (R(α1); α1, τ (α1)) = 0. We will claim that R(α1) is a continuous and onto function satisfying lim α1→0+ R(α1) = 0, lim α1→∞ R(α1) = ∞. (5.4)
By the continuity of solutions with respect to initial data, it is not difficult to get that R(α1) is continuous and limα1→0+R(α1) = 0. Hence it remains to
prove that limα1→∞R(α1) = ∞.
We apply the scaling arguments to achieve our goal. Let d > 0 be given and consider initial data (α1, α2) in the form (s, ds), s > 0 specifically. Define
ˆ Us(r) = U ¡ e−(cd)s2 r; s, ds¢− s, ˆ Vs(r) = V ¡ e−(cd)s2 r; s, ds¢− ds,
where c > max ½ 2 (2 + 2m2+ β1)d, 2 (2 + 2m1+ β2)d ¾ > 0. Then ( ˆUs(r), ˆVs(r)) satisfies ∆ ˆUs+ e £ 1−cd¡1+2m2+β12 ¢¤sr2m2e(cd)β1s2 K1¡e− (cd)s 2 r¢eVˆs= 0, r > 0, ∆ ˆVs+ e £ 1−cd¡1+2m1+β22 ¢¤sr2m1e(cd)β2s2 K2¡e− (cd)s 2 r¢eUˆs = 0, r > 0, ˆ Us(0) = ˆVs(0) = 0, lim r→0r ˆU 0 s(r) = limr→0r ˆVs0(r) = 0. (5.5)
Note that ˆUs(r), ˆVs(r) are negative for all r > 0. Let {(sj, dsj)} be a sequence in
R2
+ satisfying sj → ∞ as j → ∞. Set ( ˆUj, ˆVj) = ( ˆUsj, ˆVsj). Since e
ˆ
Uj, eVˆj ≤ 1
and r−β1K
1(r), r−β2K2(r) are bounded on [0, R] for any R > 0 , we have that r−(1+2m2+β1)| ˆU0
j(r)|, r−(1+2m1+β2)| ˆVj0(r)| and hence | ˆUj(r)|, | ˆVj(r)| are all
bounded on [0, R] for any R > 0. By using standard elliptic estimates, we obtain that ( ˆUj, ˆVj) converges to some ( ˆU , ˆV ) (passing to a subsequence if necessary)
in C2([0, R]) × C2([0, R]) for any R > 0. Then ( ˆU
j, ˆVj) converges to ( ˆU , ˆV )
pointwisely on [0, ∞) and ( ˆU , ˆV ) satisfies
∆ ˆU (r) = 0, ∆ ˆV (r) = 0, ˆ U (0) = ˆV (0) = 0, lim r→0r ˆU 0(r) = lim r→0r ˆV 0(r) = 0,
which implies that ( ˆU (r), ˆV (r)) ≡ (0, 0).
Assume that limj→∞R(sj) = R∗. Then R∗ > 0, and from (5.5), we obtain
that for r ≥ 0,
| ˆUj(r)| =
Z r 0
(log r−log t)e £ 1−cd¡1+2m2+β12 ¢¤sjt1+2m2e(cd)β1sj2 K 1 ¡ e−(cd)sj2 t¢eVˆjdt, | ˆVj(r)| = Z r 0
(log r−log t)e £ 1−cd¡1+2m1+β22 ¢¤sjt1+2m1e(cd)β2sj2 K2¡e− (cd)sj 2 t¢eUˆjdt. Note that 1 − cd µ 1 + 2m2+ β1 2 ¶ < 0, 1 − cd µ 1 + 2m1+ β2 2 ¶ < 0
due to the choice of c, and
e(cd)βisj2 Ki¡e− (cd)sj
2 t¢ is bounded, i = 1, 2, j ∈ N,
for t on any compact subset of [0, ∞) by the behaviors of K1(r), K2(r) at the origin. If R∗ is finite, then by the pointwise convergence of ( ˆU
j, ˆVj) on [0, ∞)
and applying Fatou’s lemma, we get lim sup j→∞ ¯ ¯ ˆUj¡ecdsj2 r¢¯¯ = lim sup j→∞ ¯
Therefore,
V (r; sj, dsj) → ∞ as j → ∞ for any d > 0 and r > 0.
However, by Lemma 5.2(b), without loss of generality, we may choose d to be small enough so that dsj ≤ τ (sj) for large j, then a contradiction occurs since
for any ε > 0, R(sj) < R∗+ ε for large j and thus V (R∗+ ε; sj, dsj) < 0 for
large j by Lemma 3.1. Consequently, (5.4) holds and we complete the proof of Proposition 5.1.
To attain our uniqueness result of the Dirichlet problem, we introduce the following auxiliary functions:
( Φ(r; α1, α2, C) = φ1(r; α1, α2) + Cφ2(r; α1, α2), r > 0, Ψ(r; α1, α2, C) = ψ1(r; α1, α2) + Cψ2(r; α1, α2), r > 0, (5.6) and CΦ(r; α1, α2) = −φ1 φ2(r; α1, α2), r > 0, CΨ(r; α1, α2) = −ψ1 ψ2 (r; α1, α2), r > 0, (5.7)
where φi and ψi (i = 1, 2) are defined in (3.7) with respect to the solution
(U (r; α1, α2), V (r; α1, α2)) of (5.1) and C ∈ R. For simplification, we leave out the symbol of initial data (α1, α2) in the functions defined by (5.6) and (5.7) if no confusion arises. Then Φ(r; C) and Ψ(r; C) satisfy
Φ00(r; C) +1 rΦ 0(r; C) + r2m2K 1(r)eV (r)Ψ(r; C) = 0, r > 0, Ψ00(r; C) +1 rΨ 0(r; C) + r2m1K 2(r)eU (r)Φ(r; C) = 0, r > 0. Φ(0; C) = 1, Ψ(0; C) = C, Φ0(0; C) = Ψ0(0; C) = 0, (5.8)
Remark 5.2. (i) By Lemma 3.1, it is easy to see that CΦ(r) → +∞ and
CΨ(r) → 0 as r → 0.
(ii) CΦ(r) and CΨ(r) cannot be constant on an interval. Indeed, if CΦ(r) ≡
K for r ∈ [a, b], then Φ(r; K) = 0 for r ∈ [a, b] which is impossible by (5.8).
The following assertions are crucial to proving the uniqueness of Dirichlet-type solutions of (5.1).
Lemma 5.3 Let (u(r), v(r)) be a Dirichlet-type solution of (1.6) − (1.7) with vanishing point R > 0. Then CΦ(r) is strictly decreasing and CΨ(r) is strictly
increasing on (0, R], where CΦ(r), CΨ(r) are defined by (U (r), V (r)) associated
Proof. First of all, by Remark 5.2(i), we have CΦ(r) > CΨ(r) for r ∈ (0, r0) for some 0 < r0≤ R.
Claim. C0
Φ(r) < 0 and CΨ0 (r) > 0 for r ∈ (0, r0).
Proof of Claim. Suppose that CΦ(r) is not strictly decreasing on (0, r0). Then by Remark 5.2(ii), there exist 0 < r1< r2≤ r0 such that
C0
Φ(r1) < 0, CΦ0(r2) > 0, CΦ(r1) = CΦ(r2) ≡ C0 and
0 < CΨ(r) < CΦ(r) < C0, r ∈ (r1, r2). By combining (5.6), (5.7) and Lemma 3.1, we obtain
(
Φ(r; C0) < 0 < Ψ(r; C0), r ∈ (r1, r2), Φ(r1; C0) = Φ(r2; C0) = 0,
(5.9)
which implies that Φ(r; C0) has a local minimum at some ¯r ∈ (r1, r2) and Φ00(¯r; C
0) ≥ 0. However, from (5.8) and (5.9), we have Φ00(¯r; C
0) = −¯r2m2K1(¯r)eV (¯r)Ψ(¯r; C0) < 0.
This is a contradiction. The proof for CΨ(r) is similar and we complete the proof of this claim.
Now, suppose there exists R0∈ (0, R] such that CΦ(R0) = CΨ(R0) ≡ C and
CΦ(r) > CΨ(r) > 0 for r ∈ (0, R0). Then from the claim above, we obtain Φ(r; C) > 0, Ψ(r; C) > 0, r ∈ (0, R0), Φ(R0; C) = Ψ(R0; C) = 0, Φ0(R 0; C) < 0, Ψ0(R0; C) < 0. (5.10)
By taking the differentiation with respect to αi, i = 1, 2, on both sides of (2.5)
and definitions of Φ(r; C) and Ψ(r; C), we get
r2V0(r)Φ0(r; C) + r2U0(r)Ψ0(r; C) + r2K 1evΨ(r; C) + r2K2euΦ(r; C) = Z r 0 n s£(2m2+ 2)K1+ sK10 ¤ evΨ(s; C) + s£(2m 1+ 2)K2+ sK20 ¤ euΦ(s; C)ods, and hence r2v0(r)Φ0(r; C) + r2u0(r)Ψ0(r; C) + r2K1evΨ(r; C) + r2K2euΦ(r; C) = Z r 0 n s£2K1+ sK10 ¤ evΨ(s; C) + s£2K2+ sK20 ¤ euΦ(s; C) o ds. Consequently, we deduce 0 ≥ R2 0v0(R0)Φ0(R0; C) + R20u0(R0)Ψ0(R0; C) = Z R0 0 n s£2K1+ sK10 ¤ evΨ(s; C) + s£2K2+ sK20 ¤ euΦ(s; C) o ds > 0
by (1.4), (1.12) and (5.10), which is a contradiction. Therefore the graphs of
CΦ and CΨ do not intersect on [0, R]. The proof of this lemma is complete. Based on Lemma 5.3, it is easy to obtain the following consequences. Lemma 5.4 If (u(r), v(r)) is a Dirichlet-type solution of (1.6)-(1.7) and define
C∗= −φ1
φ2(R), C∗= −
ψ1
ψ2(R),
then Φ(r; C) and Ψ(r; C) satisfy the following properties.
(a) If C > C∗, then Ψ(r; C) > 0 on [0, R] and Φ(R; C) < 0.
(b) If C = C∗, then Φ(r; C), Ψ(r; C) > 0 on [0, R), Φ(R; C) = 0 and Ψ(R; C) >
0.
(c) If C∗< C < C∗, then Φ(r; C), Ψ(r; C) > 0 on [0, R].
(d) If C = C∗, then Φ(r; C), Ψ(r; C) > 0 on [0, R), Φ(R; C) > 0 and Ψ(R; C) =
0.
(e) If 0 < C < C∗, then Φ(r; C) > 0 on [0, R] and Ψ(R; C) < 0.
(f) If C ≤ 0, then Φ(r; C) > 0 and Ψ(r; C) < 0 on [0, R]. Finally, we prove Theorem 1.1.
Proof of Theorem 1.1. Let R > 0 be given and (u(r), v(r)) = (u(r; α1, α2),
v(r; α1, α2)) be a solution of (1.6)-(1.7) satisfying u(R) = v(R) = 0. If we de-fine ( ˜U (r), ˜V (r)) = (U (r) + 2m1log R, V (r) + 2m2log R), where (U (r), V (r)) is defined in (1.8) associated with (u(r), v(r)). Then ( ˜U (r), ˜V (r)) satisfies (5.1)
replacing K1(r) and K2(r) by R−2m2K1(r) and R−2m1K2(r), respectively. By Proposition 5.1 combining the monotone property (Lemma 3.1) and the conti-nuity of solutions with respect to normalized initial data, there exists a strictly increasing function ˜τ : (−∞, ∞) → (−∞, ∞) satisfying limα1→−∞τ (α˜ 1) =
−∞, limα1→∞τ (α˜ 1) = ∞ such that u(r; α1, ˜τ (α1)), v(r; α1, ˜τ (α1)) < 0 for r ∈
(0, R(α1)), u(R(α1); α1, ˜τ (α1)) = v(R(α1); α1, ˜τ (α1)) = 0, and limα1→−∞R(α1)
= ∞, limα1→∞R(α1) = 0. Then the existence result is proved. To prove the
uniqueness result, it suffices to show that the function R(α1) is strictly de-creasing on (0, ∞). We prove this by contradiction arguments. Without loss of generality, suppose there exists an ˆα1∈ R such that R(ˆα1) is a local maximum value, and two sequences {αi
1} and {¯αi1} such that for i ∈ N ,
αi1< ¯α1i, R(αi1) = R(¯αi1) and limi→∞αi1= limi→∞α¯i1= ˆα1.
Let (ui(r), vi(r)), (¯ui(r), ¯vi(r)) be solutions of (1.6)-(1.7) associated with
nor-malized initial data (αi
1, ˜τ (αi1)), (¯αi1, ˜τ (¯αi1)) respectively. Without loss of gener-ality, we may assume for some ε > 0 fixed,
