DYNAMICAL SYSTEMS
Volume 33, Number 6, June 2013 pp. 2299–2318
STRUCTURE OF SOLUTIONS TO A SINGULAR LIOUVILLE SYSTEM ARISING FROM MODELING DISSIPATIVE
STATIONARY PLASMAS
Jann-Long Chern
Department of Mathematics, National Central University Chung-Li 32001, Taiwan
Mathematics Division, National Center for Theoretical Sciences Hsinchu 30013, Taiwan
Zhi-You Chen
Department of Mathematics, National Central University Chung-Li 32001, Taiwan
Yong-Li Tang
Mathematics Division, National Center for Theoretical Sciences Hsinchu 30013, Taiwan
(Communicated by Hirokazu Ninomiya)
Abstract. Arising from one-particle distribution functions of stationary dis-sipative plasmas, we consider a coupled elliptic system with singular data in the plane. The existence and uniqueness of solutions to the Dirichlet bound-ary value problem are proved. In addition, the structure of other solutions, including blow-up solutions, is also clarified.
1. Introduction. We consider a classical, non-relativistic N -component plasma having charges qk and mass mk. The time evolution of the one-particle distribution
functions fk(x, v, t), x, v ∈ R3, t > 0, k = 1, · · · , N , are given by
∂tfk+ v · ∂xfk+ Ak· ∂vfk = ∂v· λk mk ∂vfk+ γkvfk + Ck(fk, fk), (1)
where λkand γk are velocity-independent, spatially constant coefficients, Ck(fk, fk)
represents a Boltzmann type collision term, and Ak is the acceleration satisfying
mkAk(x, t) = qk E(x, t) +1 cv × B(x, t) (2) with the electric field E and magnetic induction B.
Set the relations of E and B as follows
E = (Ex, Ey, Ez) = −∇φ + Ezez, B = (Bx, By, Bz) = ∇ψ × ez+ Bzez,
2010 Mathematics Subject Classification. Primary: 35J47; Secondary: 35A20.
Key words and phrases. Liouville system, structure of solutions, dissipative stationary plasma. The first author is partially supported by National Science Council of Taiwan.
where (ex, ey, ez) is the standard basis of R3. We would like to find stationary
normalizable solutions of (1)-(2) with φ, Ezand ψ, Bz independent of z, and hence,
in particular, φ and ψ satisfy
∆φ + 4πρ = 0, ∆ψ +4π
c ez· J = 0, (3)
with ρ and J the charge and current densities, respectively. If we consider the spatial variable, also denoted by x, in z⊥ direction, then (3) turns into, after a delicate process, the following integrodifferential system, which is also called the Bennett system ∆φ + 4π N X k=1 Dkqk Zk e−βkqk(φ−νkψ)= 0, ∆ψ + 4π N X k=1 Dkqkνk Zk e−βkqk(φ−νkψ)= 0 (4) for some specific parameters Dk, βk and νk, and
Zk= Zk[φ, ψ] =
Z
R2
e−βkqk[φ(x)−νkψ(x)]dx.
Additionally, in the high temperature or low density limit, a semi-conformal system of Bennett equations like (4) is a reduced form of a nonlinear system of so-called finite-temperature Thomas-Fermi equations, which describes the relativistic quan-tum mechanics of a stationary beam of counter-streaming, negatively charged elec-trons and one species of positively charged ions in the semi-classical limit. Bennett equations also constitute a Liouville-type system, which is generally associated with an asymmetric coefficient matrix with some negative entries and always rank 2. See, e.g., [14,15] for details.
In this paper, we focus our attention on the following coupled elliptic system with singularity at the origin:
( ∆u + aeu+ bev= 4πk 1δ0, ∆v + ceu+ dev = 4πk 2δ0, in R2, (5) where ∆ = P2 i=1∂ 2/∂x2
i is the Laplace operator in R2, a, d ≤ 0, b, c > 0 and
k1, k2> 0 are constants, and δ0is the Dirac measure at the origin. For the Dirichlet
problem of (5), we would like to look for solutions (u, v) which satisfy
u < 0, v < 0 in BR and u = v = 0 on ∂BR. (6)
Here BRis the ball of radius R centered at the origin. Besides, we also investigate
the structure of various types of radial solutions for (5) in which blow-up phenomena occur. Let H be the 2 × 2 matrix consisting of the coefficients of nonlinearities in (5), i.e., H = a b c d . (7)
Throughout this article, we always assume that H is invertible. We remark that if (u, v) is a solution of (5) and let (w, z) be defined as
w = du − bv det H , z =
av − cu
det H . (8)
Then (w, z) will satisfy
where m1= dk1− bk2 det H , m2= ak2− ck1 det H . (10)
Note that m1and m2 are negative if det H > 0; positive if det H < 0.
System (5), in addition to being natural extensions of the well-known classical Liouville equation, is used to describe models in a variety of fields such as topics related to semi-conductors, chemotaxis, and the physics of charged particle beams. For other applications of (5), see, for instance, [1,3,6,7,8,9,10,14,15,16,18,19,
20] and references therein. We note that the condition that H is non-negative, i.e., each entry is non-negative, is always assumed throughout in [8, 18], while we deal with other cases here. Additionally, (5) associated with some specific H is related to the Toda system [11,12,13].
In the smooth case at the origin, i.e., k1 = k2 = 0 in (5), the existence of
solu-tions for the Dirichlet boundary value problem to (5) with non-negative coefficient matrix H has been proved in [8]. Here, we provide the existence result for (5) with singularity, and furthermore, under certain conditions on H, the uniqueness result is also derived. We state the details in our first main result as follows.
Theorem 1.1. Let R > 0 be given. Then the following statements are true. (i) (5)-(6) possesses a radial solution (u, v). Furthermore, if det H > 0 (resp.,
det H < 0) and (w, z) is defined as in (8), then (w, z) is a solution of (9) satisfying w(R) = z(R) = 0 and w > 0, z > 0 (resp., w < 0, z < 0) in (0, R). (ii) If b = c, det H < 0 and
min{a + b, c + d} < 0, (11)
then the solution of (5)-(6) is unique.
Remark 1.1. We note that standard arguments of the method of moving planes, along with assumption (11), guarantee the radial symmetry of solutions of (5)-(6). In fact, let Σσ= {x = (x1, x2) ∈ BR: x1> σ} for σ ∈ (0, R) and xσ= (2σ − x1, x2)
for (x1, x2) ∈ Σσ. Then zσ(x) = v(x) − v(xσ) satisfies
∆zσ+ dC(x)zσ= −ceu(x)− eu(x
σ)
− 4πk2δ(2σ, 0), x ∈ Σσ; zσ(x) ≥ 0, x ∈ ∂Σσ,
where (u, v) is a solution of (5)-(6) and C(x) = [ev(x) − ev(xσ)
]/[v(x) − v(xσ)] if v(x) 6= v(xσ); ev(x) otherwise. If a + b < 0, then ∆u(x) ≥ 0 for |x| near R, and
hence u(x) > u(xσ), x ∈ Σ
σ, for σ sufficiently close to R by the Hopf’s lemma. It
turns out that zσ(x) > 0 in Σσ for σ sufficiently close to R because of dC(x) ≤ 0.
For the details, see, e.g., [5,17] and related references therein.
From Remark 1.1, throughout this article, we only consider the radial case of (5)-(6), i.e., the following system of ordinary differential equations:
u00+1 ru 0+ aeu+ bev= 0, r > 0, v00+1 rv 0+ ceu+ dev = 0, r > 0 (12)
with (u(r), v(r)) behaving at the origin to be
u(r) = 2k1log r + θ1+ o(1),
v(r) = 2k2log r + θ2+ o(1),
where θ1, θ2are real numbers, and satisfying the boundary condition
u(R) = v(R) = 0. (14)
We denote the solution pair of (12)-(13) by (u(r; θ1, θ2), v(r; θ1, θ2)) or simply (u(r),
v(r)) if no confusion arises.
Remark 1.2. Let (u(r), v(r)) be a solution of (12)-(14).
(i) According to the arguments described in Remark 1.1, we have that u0(R) > 0 if a + b < 0; v0(R) > 0 if c + d < 0.
(ii) We also have that u0(r) > 0 and v0(r) > 0 for r ∈ (0, R). Indeed, (13) implies that u0(r) > 0 and v0(r) > 0 for near r = 0. Suppose that both u(r) and v(r) are not increasing in (0, R). Then u(r) and v(r) have local maximum and minimum points in (0, R). We first assume that det H > 0. Let ru and rv be the
first (smallest) local maximum points of u(r) and v(r) in (0, R), respectively. Then aeu(ru)+ bev(ru) ≥ 0 by (12). From Lemma 2.2, which is presented in Section 2, we get that ceu(ru)+ dev(ru) ≤ 0 and hence ceu(rv)+ dev(rv) < 0 since u(r
v) <
u(ru), v(rv) > v(ru) and c > 0, d ≤ 0. It turns out that v00(rv) > 0, which is a
contradiction. The situation for det H < 0 is similar by considering the first local minimum points of u(r) and v(r). Now, assume that, without loss of generality, u(r) is increasing in (0, R), We will show that v(r) is also increasing in (0, R). On the contrary, if there exists ˜r ∈ (0, R) such that v0(˜r) < 0 and v00(˜r) ≤ 0, then
ceu(˜r)+ dev(˜r)> 0. Moreover, we also have that
[ceu+ dev]0(˜r) = cu0(˜r)eu(˜r)+ dv0(˜r)ev(˜r)≥ 0.
Therefore, v0(r) < 0 for r ∈ [˜r, R), which is impossible because of v(R) = 0. (iii) From (13), we conclude that
ru0(r) = 2k1+ o(1) and rv0(r) = 2k2+ o(1) as r → 0+.
In fact, rewriting u(r) = 2k1log r + θ1+ g(r) near r = 0, then limr→0+g(r) = 0 and for given ¯r ∈ (0, R), lim r→0+ru 0(r) = 2k 1+ lim r→0+rg 0(r) = ¯ru0(¯r) + lim r→0+ Z r¯ r s[aeu+ bev]ds < ∞ by (13). Therefore, it deduces that limr→0+rg0(r) = 0; otherwise, limr→0+|g(r)| = +∞.
For single equations, it is known that any solution to ∆u + keu = 0 with nega-tive constant k will blow up at a finite point. For systems like (12)-(13), blow-up phenomena really occur as described in Theorem 1.2 below, while it is not easy to investigate due to the coupled sign-changing nonlinearities. According to facts that will be shown later, the only possible entire solution (u(r), v(r)) of (12)-(13) must be in the form of the following type:
Type I: lim
r→∞(u(r), v(r)) = (−∞, −∞).
In addition to entire solutions, we see that other solutions must blow up at a finite point. Various types of blow-up solutions (u(r), v(r)) for (12)-(13) are introduced as follows.
Type II: lim
r→R−u(r) = ∞, r→Rlim−v(r) = −∞ for some R ∈ (0, ∞). Type III: lim
Type IV: lim
r→R−u(r) = ∞, r→Rlim−v(r) = ∞ for some R ∈ (0, ∞).
Besides, it is clear that for any solution (u, v) of (12)-(13) and from the transfor-mations introduced in (8), the corresponding pair (w, z) is a solution of
w00+1 rw 0+ eaw+bz= 0, r > 0, z00+1 rz 0+ ecw+dz = 0, r > 0 (15)
with behaviors at the origin
w(r) = 2m1log r + η1+ o(1),
z(r) = 2m2log r + η2+ o(1),
as r → 0+, (16)
where m1, m2are defined as in (10) and
η1=
dθ1− bθ2
det H , η2=
aθ2− cθ1
det H . (17)
Now, the structure of solutions of (12)-(13) in terms of θ1 and θ2, along with
relative results for (15)−(16), can be provided below.
Theorem 1.2. Consider (12)-(13) and suppose that a, d < 0 and det H < 0 (resp., det H > 0). Then there exist strictly increasing functions %1, %2 defined on R with
%2> %1 so that the followings hold.
(i) (u(r; θ1, θ2), v(r; θ1, θ2)) is of Type I (resp., Type IV) for %1(θ1) ≤ θ2≤ %2(θ1).
Furthermore, the corresponding pair (w, z) defied in (8) is also a solution of Type I for (15)-(16) (resp., limr→R−(w(r), z(r)) = (−∞, −∞) for some finite R > 0).
(ii) (u(r; θ1, θ2), v(r; θ1, θ2)) is of Type II for θ2 < %1(θ1). Furthermore, the
corresponding pair (w, z) defied in (8) is a solution of (15)-(16) satisfying limr→R−(w(r), z(r)) = (−∞, c) for some finite R > 0 and c.
(iii) (u(r; θ1, θ2), v(r; θ1, θ2)) is of Type III for θ2 > %2(θ1). Furthermore, the
corresponding pair (w, z) defied in (8) is a solution of (15)-(16) satisfying limr→R−(w(r), z(r)) = (c, −∞) for some finite R > 0 and c.
This article is organized as follows. Section 2 is devoted to proving the existence result of Theorem 1.1, and then its uniqueness result is shown via linearization arguments in Section 3. In the last section, Section 4, blow-up solutions will be discussed.
2. Existence for Dirichlet problem. In this section, we present the existence result of Theorem 1.1. Let (u(r), v(r)) = (u(r; θ1, θ2), v(r; θ1, θ2)) be a solution pair
of (12)-(13) and set
U (r) = u(r; θ1, θ2) − 2k1log r, V(r) = v(r; θ1, θ2) − 2k2log r. (18)
Then (U (r), V(r)) satisfies U00(r) +1 rU 0(r) + ar2k1eU+ br2k2eV= 0, r ≥ 0, V00(r) +1 rV 0(r) + cr2k1eU+ dr2k2eV= 0, r ≥ 0, U (0) = θ1, V(0) = θ2, U0(0) = 0, V0(0) = 0. (19)
Define ξi(r) = ξi(r; θ1, θ2) = ∂U (r; θ1, θ2) ∂θi =∂u(r; θ1, θ2) ∂θi , ζi(r) = ζi(r; θ1, θ2) = ∂V(r; θ1, θ2) ∂θi = ∂v(r; θ1, θ2) ∂θi , i = 1, 2. (20)
Then ξi(r) and ζi(r) (i = 1, 2) satisfy the linearized systems of (19):
ξ00i(r) +1 rξ 0 i(r) + ar 2k1eUξ i(r) + br2k2eVζi(r) = 0, r ≥ 0, ζi00(r) +1 rζ 0 i(r) + cr 2k1eUξ i(r) + dr2k2eVζi(r) = 0, r ≥ 0, ξ1(0) = 1, ξ2(0) = 0, ζ1(0) = 0, ζ2(0) = 1, ξ01(0) = ξ20(0) = ζ10(0) = ζ20(0) = 0. (21)
The linearized systems are significant for us to investigate solutions of (12)-(13). First, the monotone properties of ξi and ζi are presented in the following.
Lemma 2.1. Let (u(r), v(r)) be a solution of (12)-(13), and ξi(r), ζi(r) (i = 1, 2)
defined as in (20). Then for any r > 0,
ξ01(r) > 0, ξ20(r) < 0 and ζ10(r) < 0, ζ20(r) > 0. (22) Proof. We refer the reader to [4,5] for the proof of this lemma. In fact, from (21), it is easy to see that ζ1(r) decreases strictly and hence ξ1(r) increases strictly near
r = 0. We omit the details here.
Remark 2.1. Let (u(r), v(r)) be a solution of (12)-(13) defined on a maximal interval (0, RM) (maybe RM = ∞). If there exists r0∈ (0, RM) so that
u0(r0) ≥ 0, aeu(r0)+ bev(r0)< 0 and v0(r0) ≤ 0, ceu(r0)+ dev(r0)> 0,
then it is not difficult to see that u(r) increases and v(r) decreases strictly on [r0, RM). Similar results hold if we switch the roles of u(r) and v(r).
The following observation is easy but useful for us to exclude some kind of solu-tions.
Lemma 2.2. Suppose that a, d ≤ 0, b, c > 0 and H is defined in (7). (i) If there exist M1, M2> 0 such that
aM1+ bM2< 0 and cM1+ dM2< 0,
then det H > 0.
(ii) If there exist N1, N2> 0 such that
aN1+ bN2> 0 and cN1+ dN2> 0,
then det H < 0.
Proof. In (i), the assumptions imply that a < 0 and M1 > −(b/a)M2. Then
(d − bc/a)M2 < cM1+ dM2 < 0 and hence d − bc/a < 0, which deduces that
ad − bc > 0. Similar to above, (ii) can be proved easily.
By using the observations mentioned in Remark 2.1 and Lemma 2.2 above, any solution of (12)-(13) which oscillates near r = ∞ cannot exist.
Proposition 2.1. Suppose that H is invertible and (u(r), v(r)) is a solution pair of (12)-(13) in (0, ∞). Then neither u(r) nor v(r) oscillates near r = ∞.
Proof. We first show the following assertion.
Claim. If one of u(r) and v(r) is monotone near r = ∞, then the other can not be oscillatory near r = ∞.
Proof of Claim. Suppose that v(r) increases on [R, ∞) and u0(r0) = 0, u00(r0) ≤ 0
for some r0≥ R (i.e., r0 is a local maximum point of u(r)). Then aeu(r)+ bev(r)≥
aeu(r0)+ bev(r0) ≥ 0 for r near r
0 with r > r0 due to a ≤ 0 and b > 0. Hence,
u0(r) ≤ 0 for r near r0 with r ≥ r0 since
u0(r) = −r−1 Z r
r0
s(aeu+ bev)ds ≤ 0 for r near r0, r ≥ r0.
Additionally, since v0(r) ≥ 0 and a ≤ 0, b > 0, we have [aeu(r) + bev(r)]0 =
au0(r)eu(r) + bv0(r)ev(r) ≥ 0 as long as u0(r) ≤ 0. Therefore, we conclude that
u0(r) is always non-positive, i.e., u(r) decreases for r ≥ r0. The proofs for other
cases are similar, and then the claim is proved.
We prove this proposition by contradiction. From the above claim, suppose that both of u(r) and v(r) are oscillatory. First, according to Remark 2.1, it is impossible that u(r) achieves a local maximum (resp., minimum) and v(r) achieves a local minimum (resp., maximum) simultaneously at some point. Next, without loss of generality, we may assume that u(r) and v(r) increase near r = R for some R > 0, and t0and r0are the first local maximum points of u(r) and v(r) respectively
in (R, ∞) with t0≤ r0. If t0 = r0, then it is easy to see that aeu(r0)+ bev(r0)≥ 0
and ceu(r0)+ dev(r0)≥ 0, which implies that det H < 0 by Lemma 2.2(ii). If t
0< r0,
then aeu(t0)+ bev(t0)≥ 0 and [aeu(r)+ bev(r)]0 = au0(r)eu(r)+ bv0(r)ev(r)≥ 0, 6≡ 0 for r ∈ [t0, r0] by the proof of the claim above. Hence u0(r0) < 0 and aeu(r0)+ bev(r0)>
0. Besides, we also have ceu(r0)+ dev(r0)≥ 0 and then det H < 0 still holds. Now, let t1 and r1 be the subsequent critical points (local minimum points) of
u(r) and v(r) after t0 and r0, respectively. Note that t1 > r0 by the above result.
We consider all possible situations about t1 and r1 as follows.
Case 1. t1 ≤ r1: If t1 = r1, then aeu(r1)+ bev(r1) ≤ 0 and ceu(r1)+ dev(r1) ≤ 0
which deduces det H > 0 from Lemma 2.2(i). If t1< r1, then aeu(t1)+ bev(t1)≤ 0
and [aeu(r)+ bev(r)]0 = au0(r)eu(r) + bv0(r)ev(r) ≤ 0, 6≡ 0 for r ∈ [t1, r1]. Hence
aeu(r1)+ bev(r1)< 0. Also, ceu(r1)+ dev(r1)≤ 0 and then we still obtain det H > 0. Case 2. t1 > r1: Then aeu(t1)+ bev(t1) ≤ 0 and ceu(r1)+ dev(r1) ≤ 0. Since
[ceu(r)+ dev(r)]0 = cu0(r)eu(r)+ dv0(r)ev(r)≤ 0, 6≡ 0 for r ∈ [r
1, t1], we conclude that
ceu(t1)+ dev(t1)< 0. Therefore, det H > 0 in this case.
All cases discussed above are in contradiction with the fact det H < 0. Conse-quently, we conclude that neither u(r) nor v(r) can oscillate near r = ∞, and hence this proposition is proved.
Remark 2.2. From the proof of Proposition 2.1, it is also true that if the solution (u(r), v(r)) is only defined on a maximal interval (0, RM) for some finite RM, then
both u(r) and v(r) cannot oscillate near r = RM.
Proposition 2.2. Let τ (s) be a strictly increasing function with lims→∞τ (s) = ∞.
Then there exists a constant sτ > 0 such that for s ≥ sτ, (12)-(13) does not possess a
Proof. For any non-positive solution (u(r; s, τ (s)), v(r; s, τ (s))) satisfying (u(r; s, τ (s)), v(r; s, τ (s))) → (−∞, −∞) as r → ∞,
we define γ(s) to be the first critical point so that (u0v0)(γ(s); s, τ (s)) = 0. We will apply the scaling arguments and prove this proposition by contradiction. Consider (θ1, θ2) to be the form (s, τ (s)), s > 0 specifically. Define
Us(r) = U e− c[τ (s)+s] 2 r; s, τ (s) − s, Vs(r) = V e− c[τ (s)+s] 2 r; s, τ (s) − τ (s), (23) where c > max 1 1 + k2 , 1 1 + k1 > 0. (24) Then (Us(r), Vs(r)) satisfies ∆Us+ ae[1−c(1+k1)]se−cτ (s)(1+k1)r2k1eUs +be[1−c(1+k2)]τ (s)e−cs(1+k2)r2k2eVs= 0, r ≥ 0, ∆Vs+ ce[1−c(1+k1)]se−cτ (s)(1+k1)r2k1eUs +de[1−c(1+k2)]τ (s)e−cs(1+k2)r2k2eVs = 0, r ≥ 0, Us(0) = Vs(0) = 0, Us0(0) = V 0 s(0) = 0. (25)
Let {(sj, τ (sj))} be a sequence in R2+so that sj→ ∞ as j → ∞ and (usj(r), vsj(r)) ≡ (u(r; sj, τ (sj)), v(r; sj, τ (sj))) is a non-positive solution of (12)-(13) with
(u(r; sj, τ (sj)), v(r; sj, τ (sj))) → (−∞, −∞) as r → ∞
for all j. Without loss of generality, we may assume u0(γ(sj); sj, τ (sj)) = 0 for all j.
Set (Uj, Vj) = (Usj, Vsj). Since e
usj(r)
, evsj(r)are uniformly bounded above by 1 on [0, ∞), for any R > 0, we have that r−1|Uj0(r)|, r−1|Vj0(r)| are uniformly bounded on
[0, R], and hence |Uj(r)|, |Vj(r)| are uniformly bounded on [0, R] as well. Then by
using standard elliptic estimates, we obtain that (Uj, Vj) converges to some (U, V )
(passing to a subsequence if necessary) in C2([0, R]) × C2([0, R]) for any R > 0.
Hence (Uj, Vj) converges to (U, V ) pointwisely on [0, ∞) and (U, V ) satisfies
∆U (r) = 0, ∆V (r) = 0,
U (0) = V (0) = 0, U0(0) = V0(0) = 0, (26) which implies that (U (r), V (r)) ≡ (0, 0).
Moreover, from (25), we have that for r ≥ 0, |rUj0(r)| ≤ Z r 0 tn− ae[1−c(1+k1)]sje−cτ (sj)(1+k1)t2k1eUj +be[1−c(1+k2)]τ (sj)e−csj(1+k2)t2k2eVj o dt, |rVj0(r)| ≤ Z r 0 tnce[1−c(1+k1)]sje−cτ (sj)(1+k1)r2k1eUj −de[1−c(1+k2)]τ (sj)e−csj(1+k2)r2k2eVj o dt. By the pointwise convergence of (Uj, Vj) on [0, ∞) and applying Fatou’s lemma, we
get that lim sup j→∞ e c[τ (sj )+sj ] 2 rU0 j e c[τ (sj )+sj ] 2 r = lim sup j→∞ e c[τ (sj )+sj ] 2 rV0 j e c[τ (sj )+sj ] 2 r = 0 uniformly on [0, ∞). Therefore, rU0(r; sj, τ (sj)) → 0, rV0(r; sj, τ (sj)) → 0 uniformly on [0, ∞) as j → ∞,
which is in contradiction with
γ(sj)U0(γ(sj); sj, τ (sj)) = γ(sj)u0(γ(sj); sj, τ (sj)) − 2k1= −2k1< 0.
We complete the proof of Proposition2.2.
Remark 2.3. If (u(r), v(r)) is a solution of (12)-(13), then neither u(r) nor v(r) converges finitely as r → ∞ except the case bc = ad, i.e., det H = 0.
Due to Proposition2.2and Remark 2.3, any solution of (12)-(13) can be catego-rized into the following types: a solution (u(r), v(r)) is a Dirichlet-type solution if u(r) and v(r) are both negative 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 negative.
For convenience, we use the following notations for the regions of initial data corresponding to various types of solutions of (12)-(13):
D = {(θ1, θ2) : (u(r; θ1, θ2), v(r; θ1, θ2)) is Dirichlet-type}, Du = {(θ1, θ2) : (u(r; θ1, θ2), v(r; θ1, θ2)) is u-crossing}, Dv = {(θ1, θ2) : (u(r; θ1, θ2), v(r; θ1, θ2)) is v-crossing}. (27)
Lemma 2.3. Consider the initial value problem (12)-(13). Then the followings hold.
(i) For any θ1∈ R (resp., θ2∈ R), there exists ˜θ2∈ R (resp., ˜θ1∈ R) such that
(θ1, θ2) ∈ Dv (resp., (θ1, θ2) ∈ Du) for all θ2> ˜θ2 (resp., θ1> ˜θ1).
(ii) For any θ1∈ R (resp., θ2∈ R), there exists ˜θ2∈ R (resp., ˜θ1∈ R) such that
(θ1, θ2) ∈ Du (resp., (θ1, θ2) ∈ Dv) for all θ2< ˜θ2 (resp., θ1< ˜θ1).
Proof. First, from (12), we obtain that ru0(r; θ1, θ2) = 2k1− Z r 0 sas2k1eU+ bs2k2eV ds, rv0(r; θ1, θ2) = 2k2− Z r 0 scs2k1eU+ ds2k2eVds, (28) and U (r; θ1, θ2) = θ1− Z r 0 s logr s as2k1eU+ bs2k2eVds, V(r; θ1, θ2) = θ2− Z r 0 s logr s cs2k1eU+ ds2k2eV ds. (29)
To prove (i), let θ1, ˆθ2∈ R be fixed. Then there exists r0> 0 such that
u(r; θ1, ˆθ2) < 0, U (r; θ1, ˆθ2) < 2θ1, r ∈ (0, r0). (30)
By (22), we get that u(r; θ1, θ2) and U (r; θ1, θ2) satisfy (30) for all θ2≥ ˆθ2, which
imply V(r; θ1, θ2) ≥ θ2− ce2θ1 Z r0 0 s2k1+1logr sds, r ∈ (0, r0].
Hence V(r0; θ1, θ2) > θ2/2 and v(r0; θ1, θ2) > 0 for large θ2 > ˆθ2, and (i) is
estab-lished. The proof of (ii) is similar, and we omit the details.
The following results consist of the geometric structure of solutions to (12)-(13) in terms of θ1 and θ2.
Lemma 2.4. Consider the initial value problem (12)-(13) and notations defined in (27). Then
(i) If (θ1, θ21), (θ1, θ22) ∈ Dv(resp., Du) with θ21< θ22, then (θ1, θ2) ∈ Dv (resp.,
Du) for θ2 ∈ (θ21, θ22). Similarly, if (θ11, θ2), (θ12, θ2) ∈ Dv (resp., Du) with
θ11< θ12, then (θ1, θ2) ∈ Dv (resp., Du) for θ1∈ (θ11, θ12).
(ii) There exists a strictly increasing function τ∗: R → R satisfying lim θ1→∞ τ∗(θ1) = ∞ and lim θ1→−∞ τ∗(θ1) = −∞ such that D = {(θ1, τ∗(θ1)) : θ1∈ R}.
(iii) Du and Dv are simply connected open sets and
D ∪ Du∪ Dv= R2.
Proof. It is not difficult to see that (i) holds by Lemma2.1. Next, to prove (ii), let D0= R2\ (D
u∪ Dv). We clam that (θ∗1+ ε, θ2∗), (θ∗1, θ∗2+ ε) /∈ D0for any ε > 0
when-ever (θ∗1, θ2∗) ∈ D0. In fact, let R > 0 be the point so that u(r; θ1∗, θ2∗), v(r; θ∗1, θ∗2) < 0 for r ∈ (0, R) and u(R; θ∗1, θ∗2) = v(R; θ1∗, θ2∗) = 0. Then by Lemma 2.1, we have that
u(R; θ1∗+ ε, θ2∗) − u(R; θ1∗, θ2∗) = εξ1(R; ˆθ1, θ∗2) > 0
and
v(R; θ1∗+ ε, θ2∗) − v(R; θ∗1, θ2∗) = εζ1(R; ˜θ1, θ∗2) < 0
for some ˆθ1, ˜θ1∈ (θ∗1, θ∗1+ ε). This deduces that (θ∗1+ ε, θ∗2) ∈ Du. Similarly, we also
have (θ∗1, θ∗2+ ε) ∈ Dvfor any ε > 0. Hence there exists a strictly increasing
contin-uous function τ∗ : R → R satisfying limθ1→∞τ
∗(θ
1) = ∞ and limθ1→−∞τ
∗(θ 1) =
−∞ such that D0 = {(θ
1, τ∗(θ1)) : θ1∈ R}. Then by combining Propositions 2.1,
2.2 and Remark 2.3, we have that u(r; θ1, τ∗(θ1)), v(r; θ1, τ∗(θ1))) is a Dirichlet-type
solution for all large θ1. Since any non-positive solution (u(r), v(r)) of (12)-(13) with
(u(r), v(r)) → (−∞, −∞) as r → ∞ cannot be obtained by the uniform convergence of a sequence of Dirichlet-type solutions of (12)-(13), and vice versa, we conclude that D0 = D. Finally, from (i) and (ii), it is easy to obtain (iii).
We now give a complete proof of the existence of solutions to the Dirichlet prob-lem of (12)-(13) below.
Proof of Theorem 1.1(i). It suffices to show that for any R > 0, (12)-(13) pos-sesses a Dirichlet-type solution (u(r), v(r)) with R as the first vanishing point of u(r) and v(r). Let τ∗ be defined as in Lemma 2.4(ii). We introduce the func-tion R : (−∞, ∞) → (0, ∞) to be the point so that the Dirichlet-type solufunc-tion u(r; θ1, τ∗(θ1)), v(r; θ1, τ∗(θ1)) < 0 for r ∈ (0, R(θ1)) while u(R(θ1); θ1, τ∗(θ1)) =
v(R(θ1); θ1, τ∗(θ1)) = 0. Note that R(θ1) is continuous on (−∞, ∞), and we are
going to claim that lim
θ1→∞
R(θ1) = 0, lim θ1→−∞
R(θ1) = ∞. (31)
First, we prove that limθ1→∞R(θ1) = 0 by applying scaling arguments. Consider initial data (θ1, θ2) to be the form (s, τ∗(s)), s ∈ R specifically. Define (Us(r), Vs(r))
as in (23) by replacing τ by τ∗. Then (Us(r), Vs(r)) satisfies (25) by replacing τ by
τ∗.
Let {(sj, τ∗(sj))} be a sequence in R2+ satisfying sj → ∞ as j → ∞ and we
set (Uj, Vj) = (Usj, Vsj). Since e
on [0, ˆR] for any ˆR ∈ [0, limj→∞R(sj)) and for large j, we have that r−1|Uj0(r)|,
r−1|Vj0(r)| are uniformly bounded on [0, R] for R ∈ [0, ∞), and hence |Uj(r)|, |Vj(r)|
are also uniformly bounded on [0, R]. By using standard elliptic estimates, we obtain that (Uj, Vj) converges to some (U, V ) (passing to a subsequence if necessary)
in C2([0, R]) × C2([0, R]) for any R ∈ (0, ∞). Then (U
j, Vj) converges to (U, V )
pointwisely on [0, R] and (U, V ) satisfies (26). Note that (U (r), V (r)) ≡ (0, 0). Assume that limj→∞R(sj) = R∗. Then R∗ ≥ 0, and from (25), we obtain that
for r ≥ 0, |Uj(r)| = Z r 0 t logr t n − ae[1−c(1+k1)]sje−cτ∗(sj)(1+k1)t2k1eUj +be[1−c(1+k2)]τ∗(sj)e−csj(1+k2)t2k2eVj o dt, |Vj(r)| = Z r 0 t logr t n ce[1−c(1+k1)]sje−cτ∗(sj)(1+k1)t2k1eUj −de[1−c(1+k2)]τ∗(sj)e−csj(1+k2)t2k2eVj o dt. Due to the choice of c given in (24), and by the pointwise convergence of (Uj, Vj)
on [0, ∞) and applying Fatou’s lemma, we obtain that for any fixed R ∈ (0, R∗), lim sup j→∞ Uj e c[τ ∗ (sj )+sj ] 2 r = lim sup j→∞ Vj e c[τ ∗ (sj )+sj ] 2 r = 0 uniformly on [0, R].
Therefore, for any R ∈ (0, R∗),
U (r; sj, τ∗(sj)) → ∞, V(r; sj, τ∗(sj)) → ∞ uniformly on [0, R] as j → ∞.
Nevertheless, for ε ∈ (0, R∗), there exists J
ε∈ N such that v(R∗− ε; sj, τ∗(sj)) < 0
and
V(R∗− ε; sj, τ∗(sj)) = v(R∗− ε; sj, τ∗(sj)) − 2m2log(R∗− ε) < −2m2log(R∗− ε)
for j ≥ Jε, which implies R∗ = 0. We prove limθ1→−∞R(θ1) = ∞ by using similar arguments as above while choosing c < 0 instead of that in (24). The proof of Theorem 1.1(i) is finished.
3. Uniqueness for Dirichlet problem. In this section, the consequence of unique-ness in Theorem 1.1 will be proved via the linearization arguments developed in [2,5]. First, we derive a useful identity stated below.
Lemma 3.1. Let b = c and (u(r), v(r)) be a solution pair of (12)-(13). Then br2u0(r)v0(r) −d 2r 2[u0(r)]2 −a 2r 2[v0(r)]2+ 2(dk2 1− 2bk1k2+ ak22) = (ad − b2) r2 eu(r)+ ev(r) − 2 Z r 0 s eu(s)+ ev(s)ds , r > 0. (32)
Proof. By (12)-(13) and applying the integration by parts, (32) can be obtained by calculating directly the integration of
−d(su0)0(su0) + b(su0)0(sv0) − a(sv0)0(sv0) + b(sv0)0(su0) from 0 to r.
Now, in order to achieve our goal to attain the uniqueness result, the implicit function theorem will be applied, and therefore the necessity that
det ξ1(R) ζ1(R) ξ2(R) ζ2(R)
!
6= 0, R > 0 (33)
plays an essential role. For this purpose, we define for r > 0, ξ(r; θ1, θ2, M ) = ξ1(r; θ1, θ2) + M ξ2(r; θ1, θ2), ζ(r; θ1, θ2, M ) = ζ1(r; θ1, θ2) + M ζ2(r; θ1, θ2), (34) and Nξ(r; θ1, θ2) = − ξ1 ξ2 (r; θ1, θ2), Nζ(r; θ1, θ2) = − ζ1 ζ2 (r; θ1, θ2), r > 0, (35)
where ξiand ζi(i = 1, 2) are defined in (20) with respect to the solution (U (r; θ1, θ2),
V(r; θ1, θ2)) of (19) and M > 0. Occasionally, we leave out the symbol of initial
data (θ1, θ2) in the functions defined in (34) and (35) if no confusion arises. Then
ξ(r; M ) and ζ(r; M ) satisfy ξ00(r; M ) +1 rξ 0(r; M ) + ar2k1eU (r)ξ(r; M ) + br2k2eV(r)ζ(r; M ) = 0, ζ00(r; M ) + 1 rζ 0(r; M ) + br2k1eU (r)ξ(r; M ) + dr2k2eV(r)ζ(r; M ) = 0, ξ(0; M ) = 1, ζ(0; M ) = M, ξ0(0; M ) = ζ0(0; M ) = 0. (36)
The following assertions are crucial to proving the uniqueness of solutions to the Dirichlet problem of (12)-(13).
Lemma 3.2. Suppose that b = c, k1, k2 > 0 and det H < 0. Let R > 0 be given
and (u(r), v(r)) be a solution pair of (12)-(13) with u(r), v(r) < 0 in (0, R) and u(R) = v(R) = 0. Then Nξ(r) decreases strictly and Nζ(r) increases strictly on
(0, R]. Furthermore, Nξ(R) > Nζ(R).
Proof. First of all, by Lemma 2.1, it is easy to see that Nξ(r) → +∞ and Nζ(r) → 0
as r → 0+. Hence N
ξ(r) > Nζ(r) for r ∈ (0, r0) for some 0 < r0 ≤ R. We claim
that Nξ0(r) < 0 and Nζ0(r) > 0 for r ∈ (0, r0).
Suppose that Nξ(r) does not decrease strictly on (0, r0). Then there exist r1, r2∈
(0, r0] and r1< r2such that
Nξ0(r1) < 0, Nξ0(r2) > 0, Nξ(r1) = Nξ(r2) ≡ N0, and 0 < Nζ(r) < Nξ(r) < N0
for r ∈ (r1, r2) because Nξ(r) and Nζ(r) can not be constant on an interval, or
otherwise, ξ(r; K) = 0 or ζ(r; K) = 0 on an interval for some K which is impossible by (36). Hence from (34), (35) and Lemma 2.1, we obtain
ξ(r; N0) < 0 < ζ(r; N0), r ∈ (r1, r2), and ξ(r1; N0) = ζ(r2; N0) = 0, (37)
which implies that ξ(r; N0) has a local minimum at some ¯r ∈ (r1, r2) and ξ00(¯r; N0) ≥
0. Nevertheless, from (36) and (37), we have ξ00(¯r; N0) = − h ar2k1eU (r)ξ(¯r; N 0) + br2k2eV(r)ζ(¯r; N0) i < 0.
This is a contradiction. The proof for Nζ(r) is similar and then the above claim is
Now, suppose there exists R0 ∈ (0, R] such that Nξ(R0) = Nζ(R0) ≡ N and
Nξ(r) > Nζ(r) > 0 for r ∈ (0, R0). Then from the claim above, we obtain
ξ(r; N ) > 0, ζ(r; N ) > 0, r ∈ (0, R0), ξ(R0; N ) = ζ(R0; N ) = 0
and ξ0(R0; N ) < 0, ζ0(R0; N ) < 0.
(38) By taking the differentiation with respect to θi, i = 1, 2, on both sides of (32) and
definitions of ξ(r; N ) and ζ(r; N ), we get
r2bv0(r)ξ0(r; N ) + bu0(r)ζ0(r; N ) − du0(r)ξ0(r; N ) − av0(r)ζ0(r; N ) = (ad − b2) r2euξ(r; N ) + evζ(r; N ) − 2Z r 0 seuξ(s; N ) + evζ(s; N )ds . Consequently, since det H < 0 and by (38), we deduce
0 ≥ R20 bv0(R0) − du0(R0)ξ0(R0; N ) + bu0(R0) − av0(R0)ζ0(R0; N ) = −2(ad − b2) Z R0 0 seuξ(s; N ) + evζ(s; N )ds > 0,
which is a contradiction. Therefore the graphs of Nξ and Nζ do not intersect on
(0, R]. The proof of this lemma is complete.
By applying the implicit function theorem, the uniqueness result can be proved. Proof of Theorem 1.1(ii). We define a map Ξ : R+× R2→ R2 by
Ξ(r; θ1, θ2) = (u(r; θ1, θ2), v(r; θ1, θ2)) ≡ (Ξ1(r; θ1, θ2), Ξ2(r; θ1, θ2)), r > 0,
where (u(r; θ1, θ2), v(r; θ1, θ2)) is the solution pair of (12)-(13). Let
Υ =(r, θ1, θ2) : Ξ(r; θ1, θ2) = (0, 0) and Ξi(t; θ1, θ2) < 0, t ∈ (0, r), i = 1, 2 .
Then for any ( ˆR, ˆθ1, ˆθ2) ∈ Υ, Lemma3.2implies that
det ∂Ξi ∂θj ( ˆR, ˆθ1, ˆθ2) = det ξ1( ˆR) ξ2( ˆR) ζ1( ˆR) ζ2( ˆR) 6= 0.
By applying the implicit function theorem to Ξ, we obtain that there exist δ > 0 and a unique C1 map (ω
1, ω2) : ( ˆR − δ, ˆR + δ) → R2 so that (ω1( ˆR), ω2( ˆR)) = (ˆθ1, ˆθ2)
and {(R, ω1(R), ω2(R)) : R ∈ ( ˆR − δ, ˆR + δ)} ⊂ Υ.
In addition, for any R ∈ ( ˆR − δ, ˆR + δ), Ξi(R, ω1(R), ω2(R)) = 0 and then
u0(R; ω1(R), ω2(R)) + ξ1(R; ω1(R), ω2(R)) · ω10(R) + ξ2(R; ω1(R), ω2(R)) · ω02(R) = 0
and
v0(R; ω1(R), ω2(R)) + ζ1(R; ω1(R), ω2(R)) · ω01(R) + ζ2(R; ω1(R), ω2(R)) · ω02(R) = 0.
Hence, it is easy to see that ω01(R) · ω02(R) > 0, i.e., ω10(R) and ω20(R) have the same sign, because u0(R; ω1(R), ω2(R)), v0(R; ω1(R), ω2(R)) > 0 and by Lemma
2.1. Moreover, from the relations above, we also have u0(R; ω
1(R), ω2(R))
ω01(R) + ξ(R; ω1(R), ω2(R), ω
0
2(R)/ω10(R)) = 0
and thus ω01(R) < 0 due to ξ > 0. Therefore, we get that ω01(R), ω02(R) < 0 for
R ∈ ( ˆR − δ, ˆR + δ).
By applying the implicit function theorem as above repeatedly, let I = (R∗, R∗)
be the maximal interval so that {(R, ω1(R), ω2(R)) : R ∈ I} ⊂ Υ and limR→R+ ∗ ωi(R)
= θi, i = 1, 2. Then θ21+ θ 2
2 = ∞ because otherwise the interval I can be
ex-tended further left by applying the implicit function theorem to Ξ at the point (R∗, θ1, θ2). Consequently, Theorem 1.1(i), Lemma 2.1 and arguments described
above confirm that the monotone curve {(ω1(R), ω2(R)) : R ∈ I} in (θ1, θ2
)-plane must be the unique set on which all solution pairs of (12)-(14) occur, i.e., I = (0, ∞). In fact, from Lemma 2.1, we see that for any (θ10, θ20) such that
(u(r; θ10, θ20), v(r; θ10, θ20)) being a solution pair of (12)-(14) with respect to some
R0 > 0, (u(r; θ1, θ20), v(r; θ1, θ20)) and (u(r; θ10, θ2), v(r; θ10, θ2)) can not be such
kind of solution pairs if θ1∈ (θ10, ∞) and θ2∈ (θ20, ∞).
Finally, according to Remark 1.1 and discussions stated above, Theorem 1.1(ii) is proved.
4. Blow-up phenomena and structure of solutions. In this section, we explore the blow-up phenomena of (12)-(13), i.e., the possibility of solutions which blow up at a finite point. We assume that a, d < 0 throughout this section. Let (u(r), v(r)) be a solution of (12)-(13). Define
Ru,v= sup{¯r : |u(r)|, |v(r)| < ∞ for all r ∈ (0, ¯r)}. (39)
Remark 4.1. Let (u(r), v(r)) be a solution of (12)-(13). If (u(r), v(r)) → (−∞, −∞) as r → R−u,v, then it is easy to see that Ru,v = ∞, i.e., such solution is defined on
(0, ∞). Also, if limr→Ru,v|u(r)| = ∞, then limr→Ru,v|v(r)| = ∞, and vice versa. We first introduce two lemmas which are helpful for us to deal with solutions of Type IV.
Lemma 4.1. Let det H < 0 and (u(r), v(r)) be a solution of (12)-13). Suppose that there exists r0∈ (0.Ru,v) such that u0(r0), v0(r0) > 0. Then the followings hold.
(i) If
aeu(r0)+ bev(r0)> 0, ceu(r0)+ dev(r0)= 0, (ceu+ dev)0(r
0) ≤ 0,
then
aeu(r)+ bev(r)> 0, (ceu+ dev)0(r) < 0, r ∈ (r0, Ru,v).
(ii) If
ceu(r0)+ dev(r0)> 0, aeu(r0)+ bev(r0)= 0, (aeu+ bev)0(r
0) ≤ 0,
then
ceu(r)+ dev(r)> 0, (aeu+ bev)0(r) < 0, r ∈ (r0, Ru,v).
Proof. We only show (i). The proof of (ii) is similar. First, from the assumptions, we have that (ceu+ dev)0(r0) = u0(r0) ceu(r0)+ d v 0(r 0) u0(r 0) ev(r0) ≤ 0. Because ceu(r0)+ dev(r0) = 0 and d < 0, v0(r
0)/u0(r0) ≥ 1, i.e., (u − v)0(r0) ≤ 0. Then [(c − a)eu+ (d − b)ev]0(r0) = u0(r0) (c − a)eu(r0)+ (d − b) v 0(r 0) u0(r 0) ev(r0) < 0
since (c − a)eu(r0)+ (d − b)ev(r0)< 0, v0(r
0)/u0(r0) ≥ 1 and d − b < 0. Note that
(u − v)(r) satisfies
r(u − v)0(r) = r0(u − v)0(r0) +
Z r
r0
s(c − a)eu(s)+ (d − b)ev(s)ds, r ∈ [r
0, Ru,v),
which implies that (u − v)0(r) < 0 for r near r0with r > r0due to the results above.
Therefore, by using the arguments above repeatedly, we conclude that (u − v)0(r) < 0, (c − a)eu(r)+ (d − b)ev(r)< 0, r ∈ (r0, Ru,v),
and then eu(r)−v(r)< (b − d)/(c − a) for r ∈ (r
0, Ru,v). Hence, it follows that
aeu(r)+ bev(r)=aeu(r)−v(r)+ bev(r)> bc − ad
c − a
ev(r)> 0, r ∈ (r0, Ru,v)
since det H < 0.
In addition, since u(r) − v(r) < u(r0) − v(r0) for r ∈ (r0, Ru,v), eu(r)−v(r) <
eu(r0)−v(r0) = −d/c or equivalently, ceu(r) < −dev(r) for r ∈ (r
0, Ru,v). Then
ceu(r)u0(r) < −dev(r)v0(r) for r ∈ (r
0, Ru,v) because of u0(r) < v0(r) for r ∈
(r0, Ru,v), which deduces that (ceu+ dev)0(r) < 0 for r ∈ (r0, Ru,v). We complete
the proof of (i).
Analogous to Lemma 4.1, we also have the following results for det H > 0. Lemma 4.2. Let det H > 0 and (u(r), v(r)) be a solution of (12)-(13). Suppose that there exists r0 ∈ (0, Ru,v) such that u0(r0), v0(r0) > 0. Then the followings
hold. (i) If
aeu(r0)+ bev(r0)< 0, ceu(r0)+ dev(r0)= 0, (ceu+ dev)0(r
0) ≥ 0,
then
aeu(r)+ bev(r)< 0, (ceu+ dev)0(r) > 0, r ∈ (r0, Ru,v).
(ii) If
ceu(r0)+ dev(r0)< 0, aeu(r0)+ bev(r0)= 0, (aeu+ bev)0(r
0) ≥ 0,
then
ceu(r)+ dev(r)< 0, (aeu+ bev)0(r) > 0, r ∈ (r0, Ru,v).
Proposition 4.1. Let (u(r), v(r)) be a solution of (12)-(13). Then the followings are true.
(i) If there exists r0> 0 such that
u0(r0) ≥ 0, aeu(r0)+ bev(r0)< 0; v0(r0) ≤ 0, ceu(r0)+ dev(r0)> 0,
then (u(r), v(r)) is of Type II. (ii) If there exists r0> 0 such that
u0(r0) ≤ 0, aeu(r0)+ bev(r0)> 0; v0(r0) ≥ 0, ceu(r0)+ dev(r0)< 0,
Proof. We only prove (i), and (ii) can be shown in a similar way. First, we show that (u(r), v(r)) cannot be defined on the whole interval (0, ∞). Otherwise, it follows that, by assumptions,
u(r) → ∞, v(r) → −∞ as r → ∞, which implies that
u00(r) +n − 1
r u
0(r) ≥ −a
2e
u(r), u0(r) > 0 for large r.
Then u(r) must blow up at a finite point, which yields a contradiction. Therefore, there exists R0> 0 such that u(r) → ∞ as r → R−0 and v(r) is bounded from above
in (r0, R0). Note that (u(r), v(r)) satisfies
u(r) = 2k1log r + θ1− Z r 0 s log r s [aeu(s)+ bev(s)]ds, v(r) = 2k2log r + θ2− Z r 0 s log r s [ceu(s)+ dev(s)]ds, (40) and then Z r 0 s log r s eu(s)ds → ∞ as r → R−0,
which deduces that v(r) → −∞ as r → R−0. The proof of (i) is complete.
The following lemma depicts the structure of solutions of Types II and III in terms of (θ1, θ2).
Lemma 4.3. Consider the initial value problem (12)-(13). Then the followings hold.
(i) For all θ1∈ R (resp., θ2∈ R), there exists ˜θ2 ∈ R (resp., ˜θ1 ∈ R) such that
solution (u(r; θ1, θ2), v(r; θ1, θ2)) is of Type III (resp., Type II) for all θ2> ˜θ2
(resp., θ1> ˜θ1).
(ii) For all θ1∈ R (resp., θ2∈ R), there exists ˜θ2 ∈ R (resp., ˜θ1 ∈ R) such that
solution (u(r; θ1, θ2), v(r; θ1, θ2)) is of Type II (resp., Type III) for all θ2< ˜θ2
(resp., θ1< ˜θ1).
Proof. We only prove (i), and the proof of (ii) is similar. Let θ1∈ R be fixed. Then,
similar to the proof of Lemma 2.3, we have u(r; θ1, ˆθ2) < 0, U (r; θ1, ˆθ2) < 2θ1, r ∈ (0, r0), v(r0; θ1, θ2) > 0, V(r0; θ1, θ2) > θ2 2 for large θ2> ˆθ2 for some r0 and ˆθ2.
Additionally, from (28) and above results, we have that, for large θ2> ˆθ2,
r0u0(r0; θ1, θ2) < 2k1− Z r0 0 sas2k1e2θ1+ bs2k2eθ2/2ds < 0, r0v0(r0; θ1, θ2) > 2k2− Z r0 0 scs2k1e2θ1+ ds2k2eθ2/2ds > 0, and ( aeu(r0;θ1,θ2)+ bev(r0;θ1,θ2)≥ a + bev(r0;θ1,θ2)> a + br2k2 0 e θ2/2> 0, ceu(r0;θ1,θ2)+ dev(r0;θ1,θ2)≤ c + dev(r0;θ1,θ2)< c + dr2k2 0 eθ2/2< 0.
Therefore, (u(r; θ1, θ2), v(r; θ1, θ2)) is of Type III for large θ2by Proposition 4.1(ii),
and then (i) is proved.
By virtue of Lemmas 4.1 and 4.2, we can obtain the following consequence related to solutions of Type IV.
Proposition 4.2. Suppose that H is invertible and (u(r), v(r)) is a solution of (12)-(13). If there exists r1 > 0 such that u0(r), v0(r) > 0 on [r1, Ru,v), then Ru,v
is finite and (u(r), v(r)) is a solution of Type IV.
Proof. We only prove this proposition for the case det H > 0. The proof for det H < 0 is similar. First, we have the following fact.
Claim. There exists R0∈ [r1, Ru,v) such that both of aeu(r)+ bev(r) and ceu(r)+
dev(r)do not change sign on [R0, Ru,v).
Proof of Claim. If ceu(r)+ dev(r)changes sign infinitely many times near r = Ru,v,
then
ceu(r2)+ dev(r2)= 0, [ceu(r)+ dev(r)0(r
2) ≥ 0 (41)
for some r2∈ (r1, Ru,v). Moreover, since det H > 0 and by (41),
aeu(r2)+ bev(r2)= bc − ad
c e
v(r2)< 0. Therefore Lemma 4.2(i) implies that
aeu(r)+ bev(r)< 0, ceu(r)+ dev(r)> 0, r ∈ (r2, Ru,v),
which is a contradiction. The situation for aeu(r)+ bev(r)can be shown in a similar way, and we complete the proof of this claim.
Now we come back to show that Ru,v is finite. Suppose that Ru,v = ∞. By
combining the assumption of det H > 0, Lemma 2.2(ii) and the above claim, we may assume, without loss of generality, that aeu(r)+ bev(r) < 0 for all large r.
Then ceu(r)+ dev(r) < 0 for all large r. Indeed, otherwise, by the claim above,
ceu(r) + dev(r) > 0 for large r, which implies that ev(r)−u(r) < −c/d for large r.
Then
∆u = (−a − bev−u)eu> − a + b · c d eu= bc − ad d eu
for large r. Since det H = ad − bc > 0 and d < 0, (bc − ad)/d > 0 and hence u(r) must blow up at a finite point, which leads to a contradiction.
From the discussion above, we have that
aeu(r)+ bev(r)< 0, ceu(r)+ dev(r)< 0 for large r, which deduces that
−b a< e
u(r)−v(r)< −d
c for large r. The following situations are considered.
Case 1. lim supr→∞eu(r)−v(r) < −d/c: Then −c[lim sup
r→∞eu(r)−v(r)] − d > 0,
and thus
∆v = (−ceu−v− d)ev ≥ Kev for large r for some K > 0. Hence v(r) must blow up at a finite point.
Case 2. lim supr→∞eu(r)−v(r) = −d/c: First, suppose that lim infr→∞eu(r)−v(r)
that ad − bc + (a − c)ε > 0. Then there exists r3> 0 such that (eu−v)0(r3) > 0 and eu(r3)−v(r3)> −(d + ε)/c. Hence (u − v)0(r 3) > 0. Furthermore, (c − a)eu(r3)−v(r3)+ (d − b) > −(c − a) d + ε c + (d − b) = ad − bc + (a − c)ε c > 0
and then ∆(u − v)(r3) = (c − a)eu(r3)+ (d − b)ev(r3) > 0, which implies that
(u − v)0(r) > 0, and thus (eu−v)0(r) > 0 for r near r
3 with r ≥ r3. Since c − a > 0,
we conclude that (u − v)0(r) > 0 for r ≥ r3by applying the argument stated above
repeatedly. Therefore, eu(r)−v(r)> −(d + ε)/c for r ≥ r3.
Now, because ad − bc + a > cε by the choice of ε above, we obtain that ∆u = (−a−bev−u)eu≥
−a−b· c −(d + ε) eu= ad − bc + aε −(d + ε) eu> c −(d + ε) eu for large r, which implies that u(r) must blow up at a finite point. Next, if limr→∞eu(r)−v(r) = −d/c, then it is not difficult to see that u(r) must also blow
up at a finite point by similar discussions as above.
Both of cases discussed above yield a contradiction, which is due to the assump-tion of Ru,v= ∞. Therefore, we conclude that Ru,vis finite and u(r) and v(r) blow
up to +∞ at r = Ru,v.
Particularly, under the assumption of det H < 0, finite-point blow-up solutions of (12)-(13) can only be of Types II or III, as stated below.
Proposition 4.3. Let (u(r), v(r)) be a solution of (12)-(13) which blows up at some finite R > 0. If det H < 0, then (u(r), v(r)) belongs to Type II or Type III.
Proof. Suppose that (u(r), v(r)) does not belong to Type II and Type III. Then it is not difficult to see that (u(r), v(r)) → (∞, ∞) as r → R−. Hence, from (40), there exists R0∈ (0, R) such that
Z R0 0 s log R0 s aeu(s)+ bev(s)ds = a Z R0 0 s log R0 s eu(s)ds +b Z R0 0 s log R0 s ev(s)ds < 0, Z R0 0 s log R0 s ceu(s)+ dev(s)ds = c Z R0 0 s log R0 s eu(s)ds +d Z R0 0 s log R0 s ev(s)ds < 0. Since det H < 0, the above result cannot occur by Lemma 2.2(i), and we complete this proof.
Finally, we are in a position to provide a complete demonstration of Theorem 1.2. Proof of Theorem 1.2. First, we note that if H is invertible and a, d < 0, all possible solutions of (12)-(13) must belong to one of Types I to IV by Remarks 2.2, 4.1, Lemma 4.3 and Propositions 4.2, 4.3. Next, for any θ1∈ R, we define
%1(θ1) = sup{θ2: (u(r; θ1, θ2), v(r; θ1, θ2)) is of Type II},
%2(θ1) = inf{θ2: (u(r; θ1, θ2), v(r; θ1, θ2)) is of Type III}.
Then by Lemma 4.3 and Lemma 2.1, %1 and %2 are well-defined and strictly
that any solution of Type III cannot be obtained by the uniform convergence of a sequence of solutions of Type II, and vice versa.
Now, let (u(r; θ10, θ20), v(r; θ10, θ20)) be a solution of Type II. Then there exists
r0> 0 such that
(
u0(r0; θ10, θ20) > 0, aeu(r0;θ10,θ20)+ bev(r0;θ10,θ20)< 0,
v0(r0; θ10, θ20) < 0, ceu(r0;θ10,θ20)+ dev(r0;θ10,θ20)> 0.
(42) By the continuity of solutions with respect to (θ1, θ2), the relation (42) still holds
if we replace (θ10, θ20) by any (θ1, θ2) near (θ10, θ20). From Proposition 4.1(i), it
follows that the region of all (θ1, θ2) corresponding to solutions of Type II is open
in R2, and it also holds for the region related to solutions of Type III. Besides, the existence of solutions of Type I will deduces that, from (40), aN1+ bN2 > 0 and
cN1+ dN2 > 0 for some N1, N2 > 0, which implies det H < 0 by Lemma 2.2(ii).
Therefore, solutions of Type I cannot exist while det H > 0.
Finally, according to all discussions mentioned above, Theorem 1.2 can be proved due to Propositions 4.2 and 4.3 again.
Acknowledgments. The authors would like to thank Professor Chang-Shou Lin for giving the motivation to consider the problems presented in this article. They also thank the anonymous referee(s) for helpful suggestions for improving the ex-position of the article.
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Received October 2011; revised October 2012.
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