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˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

耦合 Schrodinger 系统变号解的渐进行为

王友军，杨俊

华南理工大学数学学院，广州　 510640

摘要：我们研究了下列耦合 Schrodinger 系统变号解的渐进性质











−ε

∆u + λ

u = µ

+ βuv

, x ∈ Ω,

−ε

∆v + λ

v = µ

+ βvu

, x ∈ Ω,

u = v = 0, x ∈ ∂Ω,

其中 Ω ⊂ R

(N ≤ 3) 是有光滑边界的有界区域，λ

, � λ

, � µ

，µ

> 0，β < 0。当 ε → 0 时，

证明了半变号解（即，解的一个分支变号，另一个分支大于零）变号的分支有唯一的正和负尖

峰，它们都收敛到 Ω 中不同的点，不变号的分支有唯一的正尖峰，它也收敛到一点。我们同

样也证明了当 ε → 0 时，这些最大值和最小值点与边界的距离除以 ε 是发散的。

关键词：耦合非线性 Schrodinger 系统；变号解；渐进行为

中图分类号： O175

The asymptotic behavior of nodal solutions for

two coupled Schrödinger system

WANG You-Jun , YANG Jun

Department of Mathematics, South China University of Technology, Guangzhou 510640

Abstract: We study the asymptotic behavior of nodal solution for the coupled nonlinear

Schrödinger system











−ε

∆u + λ

u = µ

+ βuv

, x ∈ Ω,

−ε

∆v + λ

v = µ

+ βvu

, x ∈ Ω,

u = v = 0, x ∈ ∂Ω,

where Ω ⊂ R

(N ≤ 3) is a smooth and bounded domain, λ

, λ

, µ

> 0 and β < 0. It is

shown that the semi-nodal solution (u, v), that is, one component changes sign and another

one is positive, has exactly one positive and one negative peaks, which converge to two

distinct points of Ω as ε → 0, and another component has exactly one positive peak, which

Foundations: NSFC (11201154)，SRFDP(No.20120172120040)

Author Introduction: WANG You-Jun（1981-），male，associate professor，major research direction：Nonlinear analysis

and partial diﬀerential equations. Correspondence author：YANG Jun （1979-），female，lecturer，major research direction：

Partial diﬀerential equations.

- 1 -

˖ڍመ᝶஠ڙጲ

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converge to a point of Ω as ε → 0. We also prove the distances between the local maximum

points, the local minimum points and the boundary of Ω divided by ε diverge.

Key words: coupled nonlinear Schrödinger system ; nodal solution; the asymptotic behavior

0 Introduction

In this paper, we consider the existence of nodal solution and its asymptotic behavior for

the coupled nonlinear Schrödinger system











−i

∂Φ

∂t

∆Φ

− W

(x)Φ

+ µ

|Φ

+ β|Φ

, x ∈ Ω, t > 0,

−i

∂Φ

∂t

∆Φ

− W

(x)Φ

+ µ

|Φ

+ β|Φ

, x ∈ Ω, t > 0,

= Φ

(x, t) ∈, j = 1, 2,

(x, t) = 0, x ∈ ∂Ω, t > 0, j = 1, 2.

This system models a binary mixture of Bose-Einstein condensates in two diﬀerent hyperﬁne

states |1⟩ and |2⟩ (cf. [1]). Physically, Φ

and Φ

are the corresponding condensate amplitudes,

Ω ⊆ R

is the domain for condensate dwelling, h is the Planck constant, m is atom mass

and W

is the trapping potential for the j-th hyperﬁne state. µ

= −(N

− 1)U

is a ﬁxed

number of atoms in the hyperﬁne state |j⟩, N

, N

≥ 1, β = −N

and U

= 4π

where a

’s and a

are the intraspecies and interspecies scattering lengths. The sign of the

scattering length a

determines whether the interactions of states |1⟩ and |2⟩ are repulsive or

attractive. When a

> 0, i.e., β < 0, the interactions of states |1⟩ and |2⟩ are repulsive. In

contrast, when a

< 0, i.e., β > 0, the interactions of states |1⟩ and |2⟩ are attractive. For

atoms of the single state |j⟩, when a

< 0, i.e., µ

> 0, the interactions of the single state

|j⟩ are attractive. Spikes may appear in a single condensate when the s-wave scattering length

is negative and large. Due to Feshbach resonance, the s-wave scattering length of a single

condensate can be tuned over a very large range by adjusting the externally applied magnetic

ﬁeld. As the s-wave scattering length of a single condensate is negative and large enough, the

interactions of atoms are strongly attractive and the associated condensate tends to contract

(see [2] and the references therein). It seems spikes may occur in a double condensate when the

intraspecies scattering lengths a

’s are negative and large enough, i.e. µ

’s are positive and

large enough. On the other hand, the interspecies scattering length a

plays an important role

in a double condensate. Without the eﬀect of trap potentials, when the interspecies scattering

length a

is positive and large enough, the states |1⟩ and |2⟩ may repel each other and form

segregated domains called phase separation, see [2] and the references cited therein. In [2], T.

C. Lin and J. Wei showed that spikes of states |1⟩ and |2⟩ repel each other and behave like

two separate spikes as the interspecies scattering length a

is positive and large enough. In

- 2 -

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contrast, if the interspecies scattering length a

is negative and large enough, spikes of states

|1⟩ and |2⟩ attract each other and behave like one single spike. More precisely, they studied

solitary wave solutions of the following elliptic system











∆u − λ

u + µ

+ βuv

= 0, x ∈ Ω,

∆v − λ

v + µ

+ βu

v = 0, x ∈ Ω,

u > 0, v > 0, x ∈ Ω,

u = v = 0, x ∈ ∂Ω,

(1)

where Ω ⊂ R

(N ≤ 3) is a smooth and bounded domain; ε > 0 is a small parameter;

, λ

, µ

are positive constants; β = 0 is a coupling constant. They proved that there

exists a β

∈ (0,

√

) such that for any β ∈ (−∞, β

) and ε small enough, (1) has a least

energy positive solution (u

, v

) concentrates respectively in the local maximum point P

of u

and local minimum point Q

of v

. The result of [2] seems to be the ﬁrst attempt in studying

the positive solutions for systems of singularly perturbed problems.

When ε = 1, (1) turns into the following system:











−∆u + λ

u = µ

+ βuv

, x ∈ Ω,

−∆v + λ

v = µ

+ βu

v, x ∈ Ω,

u > 0, v > 0, x ∈ Ω,

u = v = 0, x ∈ ∂Ω.

(2)

For this case, (2) has also been studied by N. Dancer, J. We and T. Weth in [3], where a priori

bounds and multiple existence of positive solutions to (2) were obtained. Their main tools are

Liouville type theorems and a variant of Lyusternik-Schnirelmann theory on a submanifold of

Nehari type.

When Ω is a unit ball B, λ

= λ

= µ

= 1, β ≤ −1, for any k ∈ N, J. Wei

and T. Weth [4] established the existence of radially symmetric positive solution (u, v) to (2)

such that u − v changing sign precisely k − 1 times in the radial variable. The genus theory

and corresponding parabolic equation theory are applied. Their results provides a theoretical

indications of phase separation into many nodal domains for Bose-Einstein double condensates

with strong repulsion. When λ

= λ

∈ R, µ

= µ

= 1, for −β large enough, the systems (2)

has an unbounded sequence of positive solutions, this was proved by B. Noris and M. Ramos

[5]. The main ideas of [5] is a suitable minimax framework and the Morse index estimates.

When Ω = R

(N ≤ 3), system (1) have been widely investigated by several authors. In

Sirakov [6], the least energy solitary waves for (1) were established when ε = 1. By using the

ﬁxed point index, when ε = 1, T. Bartsch et al. [7] proved a global bifurcation theorem for

positive solutions. They also obtained further information on the set of all positive bound states

- 3 -

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by applying Morse theory to the associated energy functional constrained to a nonnegative

cone. A. Ambrosetti and E. Colorado [8] showed that there exist Λ

′

> Λ > 0, depending

upon λ

and µ

such that (1) has a radially symmetric positive solution provided β ∈ (0, Λ) ∪

(Λ

′

, ∞). Moreover, for β > Λ

′

, these solutions are ground states, in the sense that they have

minimal energy and their Morse index is 1. In A. Pomponio, [9], with suitable assumptions on

the potentials and for ε suﬃciently small, by using ﬁnite dimensional reduction, one positive

solutions were obtained which concentrates respectively on P

and Q

tending toward the same

point and the distance between them divided by ε diverges. In [10], Montefusco, Pellacci and

Squassina studied the case β > 0. They showed for small ε > 0, problem (1) has a non-zero

solution (u

, v

) such that u

(x) + v

(x) has exactly one global maximum point. We refer the

readers to [11, 12, 13, 14, 15, 16] and the references cited therein for more results for the case

of Ω = R

In this paper, we consider the nodal solutions (u, v) to (3) below:











−ε

∆u + λ

u = µ

+ βuv

= 0, x ∈ Ω,

−ε

∆v + λ

v = µ

+ βu

v = 0, x ∈ Ω,

u = v = 0, x ∈ ∂Ω,

(3)

where Ω is bounded or unbounded. Here we say nodal solution (u, v), it means either u or v

or both are sing-changing. Recently, Chen and zou in [17] proved there exists β

> 0 such that

for β ∈ (0, β

), there exists a nodal solution of (3) with ε = 1. Thus, for each ε > 0, we can get

a nodal solution (u

, v

) for the Schrödinger system (3). The purpose of the current paper is

to study their proﬁles.

Theorem 0.1. For β ∈ (−

√

, 0) and ε suﬃciently small, (3) has a semi-nodal solution

, v

) satisfying the following properties:

(1) u

has exactly one local maximum point P

and one minimum point P

, v

has exactly

one local maximum point Q

(2) u

, v

→ 0 in C

loc

(Ω \ {P

, P

, Q

}). Moreover, there exist some C > 0 and 0 < σ < 1

such that

(x)| ≤ Ce

−(1−σ)

√

min{|x−P

|,|x−P

|}/ε

(x)| ≤ Ce

−(1−σ)

√

|x−Q

|/ε

For single equation case, the existence of nodal solutions and its asymptotic behavior for

singularly perturbed Neuman or Dirichlet problems have been studied by some authors. We

refer the readers to [18, 19, 20, 21] and the references therein. For such case, it is easy to

check that if u belongs to the classical Nehari manifold N corresponding to the functional,

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