On Phase-Separation Model: Asymptotics and Qualitative Properties

PREPRINT

國立臺灣大學 數學系 預印本 Department of Mathematics, National Taiwan University

www.math.ntu.edu.tw/ ~ mathlib/preprint/2012- 14.pdf

On Phase-Separation Model: Asymptotics and Qualitative Properties

Henri Berestycki, Tai-Chia Lin, Juncheng Wei, and Chunyi Zhao

November 12, 2012

On Phase-Separation Model: Asymptotics and Qualitative Properties

Henri Berestycki

, Tai-Chia Lin

, Juncheng Wei

, Chunyi Zhao

Abstract

In this paper we study bound state solutions of a class of two-component nonlinear elliptic systems with a large parameter tending to infinity. The large parameter giving strong intercomponent repulsion induces phase separation and forms segregated nodal domains divided by an interface. To obtain the profile of bound state solutions near the interface, we prove the uniform Lipschitz continuity of bound state solutions when the spa- tial dimension is N = 1. Furthermore, we show that the limiting nonlinear elliptic system that arises has unbounded solutions with symmetry and monotonicity. These unbounded solutions are useful to derive rigorously the asymptotic expansion of the minimizing en- ergy which is consistent with the hypothesis of [23]. When the spatial dimension is N = 2, we establish the De Giorgi type conjecture for the blow-up nonlinear elliptic system under suitable conditions at infinity on bound state solutions. These results naturally lead us to formulate De Giorgi type conjectures for this type of systems in higher dimensions.

1 Introduction

In a binary fluid like a mixture of oil and water, the two components of the fluid may spontaneously separate and form two segregated domains divided by an interface. Such a phenomenon called phase separation can be observed as well in cooling binary alloys, glasses and polymer mixtures. The well-known Cahn-Hilliard equation has been proposed as a model to describe the process of phase separation (cf. [10]). It is written in the form:

φ_t = ∆δF_ε δφ = ∆[

ε²∆φ + (1− φ²)φ]

for x∈ Ω , t > 0 ,

∗Ecole des hautes etudes en sciences sociales, CAMS, 54, boulevard Raspail, F - 75006 - Paris, France. Email:

hb@ehess.fr.

†Department of Mathematics & National Center for Theoretical Sciences at Taipei, National Taiwan Uni- versity, Taipei, 10617, Taiwan. Email: tclin@math.ntu.edu.tw.

‡Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong. Email:

wei@math.cuhk.edu.hk.

§Department of Mathematics, East China Normal University, Shanghai, 200241, China. Email:

cyzhao@math.ecnu.edu.cn.

¶Department of Mathematics & Taida Institute of Mathematical Sciences (TIMS), National Taiwan Univer- sity, Taipei, 10617, Taiwan.

with no-flux boundary condition

∂_νφ = ∂_ν[

ε²∆φ + (1− φ²)φ]

= 0 on ∂Ω , and mass conservation

1

|Ω|

∫

Ω

φ(x, t)dx = m.

Here φ = φ(x, t)∈ R is the order parameter, Ω ⊂ R^N is the region occupied by the fluid, ∂ν is the exterior normal derivative on the boundary ∂Ω and ε > 0 is a small parameter giving the length of transition regions between the domains. The Cahn-Hilliard free energy functional F_ϵ is defined by

F_ε(u) =

∫

Ω

ε²|∇u|²+ 1

2(1− u²)² for u∈ H¹(Ω) , (see [15]). Stationary solutions with interfaces of the above equations satisfy

ε²∆φ + (1− φ²)φ = λ_ϵ in Ω , and ∂_νφ = 0 on ∂Ω.

It is well-known that as ϵ → 0 and λϵ → 0, the profile of the solution φ near the interface approaches to a solution of the following Allen-Cahn (AC) equation

∆Φ + (1− Φ²)Φ = 0 in R^N.

For AC equation as above, De Giorgi [27] formulated in 1978 the following celebrated con- jecture:

Let Φ be a bounded solution of AC equation such that ∂_xNΦ > 0. Then the level sets {Φ = λ}

are hyperplanes, at least for dimension N ≤ 8. The conjecture has been investigated exten- sively over the recent years and has been essentially settled by now (see Section 7 for detailed discussions).

Indeed, phase separation is known to occur in a double condensate (cf. [29], [37], [38]).

In general, however, phase separation models between two components involve a system of partial differential equations. The aim of this paper is to investigate questions analogous to the previous one in the more general framework. One such system of particular interest arises in a binary mixture of Bose-Einstein condensates with two different hyperfine states denoted by

|1⟩ and |2⟩ Due to strong inter-component repulsion, interfaces (so called domain walls) may divide the condensate into segregated domains in the same way as in the mixture of oil and water. A classical model to describe this involves the two component Gross-Pitaevskii (GP) system derived from the following GP functional (cf. [40])

E = 1 2

∫

Ω

∑2 j=1

( ~²

2m|∇Ψj|²+ V_j|Ψj|²+1

2g_jj|Ψj|⁴ )

+ g₁₂|Ψ1|²|Ψ2|²dx .

Here ~ is Planck constant, m is the atom mass, Ω is the domain for condensate dwelling, Vj’s are trapping potentials, and Ψ_j’s are wave functions corresponding to states |j⟩’s. Besides, g_ij ∼ aij, where a_jj’s and a₁₂ are the intraspecies and interspecies scattering lengths. From the variational principle, the model of double condensates can be written as i~∂Ψj/∂t = δE/δΨ^∗_j for j = 1, 2, that is,

i~∂Ψ_j

∂t =−~²

2m△Ψj + V_jΨ_j+ g_jj|Ψj|²Ψ_j+ g₁₂|Ψ3−j|²Ψ_j, j = 1, 2 ,

called the coupled Gross-Pitaevskii (GP) equations (cf. [1] and [40]) giving conservation laws

as follows: ∫

Ω

|Ψj|² = N_j, for t > 0 , j = 1, 2 , where Nj’s are numbers of atoms.

To study phase separation of double condensates, as is explained in [45], we may switch off trapping potentials V_j’s and let V_j ≡ 0, j = 1, 2. Due to Feshbach resonance (cf. [26]), we may further set g₁₂ = _2m^~2 Λ and g_jj’s as nonnegative constants, where Λ is a large parameter tending to infinity. Then the condition g₁₂² > g₁₁g₂₂ for phase separation (cf. [3]) is fulfilled and the GP functional becomes

E =

∫

Ω

∑2 j=1

(~²

2m|∇Ψj|²+1

2g_jj|Ψj|⁴ )

+ ~²

2mΛ|Ψ1|²|Ψ2|²dx .

To find standing wave solutions of the coupled GP equations, one sets Ψ₁(x, t) = e^−iϵ1t/~u(x) and Ψ2(x, t) = e^−iϵ2t/~v(x). Here ϵj’s are chemical potentials and u, v are the corresponding condensate amplitudes (cf. [18]). Then the coupled GP equations become a class of nonlinear elliptic systems that reads as follows:















−_2m^~2∆u + g₁₁u³+_2m^~2 Λ v²u = ϵ₁u in Ω,

−_2m^~2∆v + g₂₂v³+_2m^~2 Λ u²v = ϵ₂v in Ω, u, v > 0 in Ω,

u = v = 0 on ∂Ω .

Due to conservation laws, we may regard ϵ_j’s as eigenvalues and u, v as eigenfunctions satisfying normalization conditions ∫

Ω

u² = N₁,

∫

Ω

v² = N₂.

By suitable scaling on u, v and spatial variables, the nonlinear elliptic systems with the normalization conditions above can be transformed into

− ∆u + αu³+ Λv²u = λ1u in Ω, (1.1)

− ∆v + βv³+ Λu²v = λ2v in Ω, (1.2)

u > 0, v > 0 in Ω, (1.3)

u = 0, v = 0 on ∂Ω , (1.4)

∫

Ω

u² =

∫

Ω

v² = 1 . (1.5)

Hereafter, we assume that Ω is a bounded smooth domain in R^N. Then solutions of (1.1)-(1.5) can be regarded as critical points of the GP functional

E_Λ(u, v) =

∫

Ω

(|∇u|²+|∇v|²) + α

2u⁴+β

2v⁴+ Λu²v², (1.6)

on the space (u, v) ∈ H0¹(Ω)× H0¹(Ω) with a constraint given by (1.5). The eigenvalues λ_j’s are Lagrange multipliers with respect to (1.5). Both eigenvalues λ_j = λ_j,Λ’s and eigenfunctions u = u_Λ, v = v_Λ depend on the parameter Λ. The system of equations (1.1)-(1.5) derived from the GP functional (1.6) is the type of systems we study here. Recently, several interesting results related to equations (1.1)-(1.5) have also been published in [8, 20, 21, 22, 44].

In this paper, we restrict our attention to solutions (u_Λ, v_Λ) of (1.1)-(1.5) such that the associated eigenvalues λ_j,Λ’s are uniformly bounded, that is, we assume here that

sup

Λ>0

max{λ1,Λ, λ_2,Λ} ≤ C , (1.7)

where C denotes a positive constant independent of Λ. It is obvious that (1.7) is equivalent to E_Λ(u_Λ, v_Λ)≤ C. In particular, observe that, the ground state or least energy solution satisfies this condition. Indeed by taking u and v with disjoint support, we derive an upper bound on inf E_Λ(u, v) independent of Λ. More generally, we consider here all bound state solutions that satisfy a boundedness condition on the energy.

Formally, as Λ→ +∞ (up to a subsequence), (u∞, v_∞)-the limit of (u_Λ, v_Λ) satisfies

−∆u∞+ αu³_∞= λ1,∞u_∞ in Ωu, (1.8) and

−∆v∞+ βv_∞³ = λ_2,∞v_∞ in Ω_v, (1.9) where Ω_u = {x ∈ Ω : u_∞(x) > 0} and Ωv = {x ∈ Ω : v_∞(x) > 0} are positivity domains composed of finitely disjoint domains with positive Lebesgue measure, and each λ_j,∞ is the limit of λ_j,Λ’s as Λ → ∞ (up to a subsequence). Effective numerical simulations for (1.8) and (1.9) can be found in [4], [5] and [17]. Several works deal with the convergence of (u_Λ, v_Λ). One may refer to Chang-Lin-Lin-Lin [17] for the pointwise convergence of (u_Λ, v_Λ) away from the interface γ ≡ {x ∈ Ω : u∞(x) = v_∞(x) = 0}; Wei-Weth [46] for the uniform equicontinuity of (u_Λ, v_Λ); and Noris-Tavares-Terracini-Verzini [39] for the uniform H¨older continuity of (u_Λ, v_Λ).

However, until now, the uniform Lipschitz continuity of the (u_Λ, v_Λ)’s has not yet been obtained.

One of the results here is the uniform Lipschitz continuity of the (u_Λ, v_Λ)’s when the spatial dimension is N = 1 i.e. Ω = (a, b) (see Lemma 2.4). For higher dimensions, the problem is still open.

To understand formally the connection between F_ε the Cahn-Hilliard and E_Λ the Gross- Pitaevskii functionals, we set u = 1 + ρ, v = 1− ρ and ε = 1/√

Λ a small parameter tending to zero. Then (1.6) becomes

E_Λ(u, v) = 2 ε²

[

F_ε(ρ) +

∫

Ω

ε²

4α (1 + ρ)⁴+ε²

4β (1− ρ)⁴ ]

, which is dominated by the Cahn-Hilliard energy F_ε.

One might think that near the interface, the profile of bounded solutions of (1.1)-(1.5) is quite similar to that of bounded solutions of the scalar Allen-Cahn equation. However, this is not the case. As we will see ((1.16) below), the blow up equation is a system, not a scalar equation. One of the main goals of the paper is to study this system.

Here, we completely classify the one-dimensional solution of this system (see Theorem 1.3 below). In particular, we establish the symmetry, monotonicity, uniqueness and nondegeneracy

of solutions of (1.16). This leads us to believe that there is an extended De Giorgi conjecture for this new system. When the spatial dimension is N = 2, we provide sufficient conditions that give this De Giorgi conjecture for solutions of (1.16) (see Theorem 1.8).

To derive the asymptotic behavior of (u_Λ, v_Λ)’s near the interface γ = {x ∈ Ω : u∞(x) = v_∞(x) = 0}, it is sufficient to consider the point xΛ∈ Ω such that uΛ(x_Λ) = v_Λ(x_Λ) = m_Λ→ 0 and x_Λ → x∞∈ γ ⊂ Ω as Λ → +∞ (up to a subsequence). For simplicity, the mention “up to a subsequence” will be understood in the remaining of this paper. When N = 1 and Ω = (a, b), the estimate of m_Λ’s is stated as follows:

Theorem 1.1. Assume that Ω = (a, b) ⊂ R, (uΛ, v_Λ) solves the system (1.1)-(1.5) and (1.7) holds. Let

m_Λ= u_Λ(x_Λ) = v_Λ(x_Λ)→ 0 as Λ→ +∞ , (1.10) and

x_Λ→ x_∞ ∈ Ω as Λ→ +∞ . (1.11)

Then it holds that

m⁴_ΛΛ→ C0 as Λ → +∞ , (1.12)

where C₀ is a positive constant. On the other hand, if (1.12) holds, then Λ^1/4min(|xΛ−a|, |xΛ− b|) → +∞.

In higher dimension, without loss of generality, we may assume C0 = 1. Let

˜

u_Λ(y) = 1

m_Λu_Λ(m_Λy + x_Λ) , v˜_Λ(y) = 1

m_Λv_Λ(m_Λy + x_Λ) , (1.13) for y ∈ ΩΛ≡ {y ∈ R^N : m_Λy + x_Λ∈ Ω} → R^N (in general) as Λ→ ∞. Then (˜uΛ, ˜v_Λ) satisfies

−∆˜uΛ+ m⁴_Λα˜u³_Λ+ m⁴_ΛΛ˜v_Λ²u˜_Λ= m²_Λλ₁u˜_Λ in Ω_Λ, (1.14)

−∆˜vΛ+ m⁴_Λβ ˜v_Λ³ + m⁴_ΛΛ˜u²_Λ˜v_Λ = m²_Λλ₁˜v_Λ in Ω_Λ. (1.15) In view of (1.12), we expect that in any dimension, the limit of (˜u_Λ, ˜v_Λ)–(U, V ) solves the following blow-up nonlinear elliptic system

∆U = V²U , ∆V = U²V , U, V ≥ 0 in R^N. (1.16) Here we are only able to establish this fact when the dimension is N = 1. This is the statement in the next result.

Theorem 1.2. Under the same hypotheses as in Theorem 1.1, assume xΛ → x∞ ∈ Ω as Λ→ +∞. Then there exist positive functions U(y), V (y) ∈ C^∞(R) such that, as Λ → ∞,

˜

u_Λ → U, ˜vΛ→ V in C_loc² (R) , where (U, V ) satisfies {

U^′′ = V²U in R ,

V^′′= U²V in R , (1.17)

and U (0) = V (0) = 1. Moreover,

U^′2+ V^′2− U²V² ≡ T_∞ in R , (1.18) where

T_∞=|Ω|⁻¹ lim

Λ→∞

1 2

[

λ_1,Λ+ λ_2,Λ+ 3

∫

Ω

(u^′_Λ)²+ (v^′_Λ)² ]

( up to a subsequence) (1.19) is a positive constant.

To prove the existence of (U, V ) in Theorem 1.2, we need the asymptotic behavior of m_Λ satisfying (1.12) which can be derived from the uniform Lipschitz continuity of (uΛ, vΛ). Until now, the uniform Lipschitz continuity of (u_Λ, v_Λ) holds only when the spatial dimension is one (see Lemma 2.4). This is the reason why the result of Theorem 1.2 is only one dimensional.

A more general model is obtained when trapping potentials Vj’s are turned on. The system (1.1)-(1.5) then is the form:

− u^′′+ P₁(x)u + αu³+ Λv²u = λ₁u in (a, b), (1.20)

− v^′′+ P₂(x)u + βv³+ Λu²v = λ₂v in (a, b), (1.21)

u > 0, v > 0 in (a, b), (1.22)

u(a) = u(b) = 0, v(a) = v(b) = 0 (1.23)

∫ b a

u² =

∫ b a

v² = 1, (1.24)

where Pj, j = 1, 2 are C¹([a, b]) functions. Assume P_j ≥ 0 , |P_j^′| ≤ M < 1

(b− a)³ [

π²+b− a

8 (α + β) ]

in (a, b) , j = 1, 2 , (1.25) where M is a positive constant independent of Λ. Then Theorem 1.1 and 1.2 also hold for the system (1.20)-(1.24). We refer to Section 3.1 for the details of proofs.

Next, we study the limiting system (1.16) in dimension N = 1, that is, (1.17). The existence of an nontrivial solution to (1.17) is given in Lemma 4.1. Using the method of moving planes, we are able to completely classify the one-dimensional solutions of this system (1.16).

Theorem 1.3. Let N = 1 and (U, V ) be an nonnegative solution of (1.17). Then the following properties hold.

(1) (Symmetry)There exists x₀ ∈ R such that

V (y− x0) = U (x₀− y) , for y∈ R . (2) (Asymptotic behavior) Either

{

U (−∞) = 0, U^′(−∞) = 0, U^′ > 0, U^′(∞) = √ T_∞, V (∞) = 0, V^′(∞) = 0, V^′ < 0, V^′(−∞) = −√

T_∞, or likewise with U and V interchanged, where T_∞ > 0 is defined in (1.18).

(3) (Nondegeneracy) (U, V ) is nondegenerate, that is, if (ϕ, ψ) is a bounded solution of the linearized system

ϕ^′′ = V²ϕ + 2U V ψ , ψ^′′ = U²ψ + 2U V ϕ in R , (1.26) then, it must be the case that (ϕ, ψ) = c(U^′, V^′) for some constant c.

Note that from Theorem 1.3 (2), both U and V are unbounded on R. This is one of the main difficulties in the analysis. In fact, because this is a system, even in dimension one, carrying out the moving planes procedure turns out to be somewhat involved. The question to know whether such a result holds in higher dimension is an open problem.

Remark 1.4. Without loss of generality, we may set x0 = 0 and then Theorem 1.3 (1) gives U (y) = V (−y) for y ∈ R. Whether or not the solution to (1.17) is unique up to rescaling remains open. Theorem 1.3 (3) shows local uniqueness. 

Remark 1.5. Instead of Bose-Einstein condensates, the same system (1.17) also describes a stationary membrane (representing a domain wall) in a static gravitational field of a black hole (cf. [24]). 

Using Theorem 1.1 and 1.3, we can also derive the asymptotic expansion of the minimizing energy as follows:

Theorem 1.6. Assume Ω = (−1, 1) and α = β = 0. Then the minimizing energy EΛ≡ min

{

E_Λ(u, v) : (u, v)∈ H0¹(Ω)²,

∫

Ω

u² =

∫

Ω

v² = 1 and u(x) = v(−x) , ∀x ∈ (−1, 1) }

satisfies

2π²− B1Λ⁻¹⁴ ≤ EΛ≤ 2π²− B2Λ⁻¹⁴ , (1.27) as Λ→ ∞ (up to a subsequence), where Bj, j = 1, 2 are positive constants independent of Λ.

Remark 1.7. In [23], 2π² − EΛ is assumed to satisfy 2π²− EΛ = QΛ⁻¹⁴ + o(Λ⁻¹⁴), where the constant Q can be calculated formally. Here we give a rigorous proof of (1.27) which can be regarded as a partial result of the above assumption. 

As analogue of De Giorgi’s conjecture for Allen-Cahn equation, the previous results lead us to state the following conjecture for system (1.16).

Conjecture: At least up to the dimension N = 8, under the monotonicity condition

∂U

∂yN

> 0, ∂V

∂yN

< 0, (1.28)

a solution (U, V ) of the system (1.16) is necessarily one-dimensional, i.e. there exist a ∈ R^N such that U (y) = U₀(a· y) and V (y) = V0(a· y) for y ∈ R^N, where U₀, V₀ :R → R are smooth functions.

Just note that by Theorem 1.3, the monotonicity condition (1.28) holds in the case of space dimension N = 1. When the dimension is N ≥ 2, Theorem 1.1 and 1.2 are still open. Note that the uniform Lipschitz continuity of the (u_Λ, v_Λ)’s is still open as well in the case of N ≥ 2.

We now derive further results in dimension N = 2. To this end, in addition to the previous ones, we require the following assumptions:

(H0) x_Λ→ x∞ ∈ Ω and mΛ= u_Λ(x_Λ) = v_Λ(x_Λ)→ 0 as Λ → ∞.

(H1) m⁴_ΛΛ→ C0 > 0 as Λ→ +∞ , i.e. (1.12) holds.

(H2) u˜_Λ → U , ˜vΛ → V strongly in H_loc¹ (R²) as Λ → ∞, where (˜uΛ, ˜v_Λ) is defined in (1.13) and (U, V ) solves (1.16).

(H3) For any large R, 1 R⁴

∫

B2R\BR

(U²+ V²)≤ C (independent of R) . (1.29)

Under the assumptions (H0)-(H3), we can give an affirmative answer to the above conjecture.

Theorem 1.8. Let N = 2 and assume that conditions (H0)-(H3) hold. Then the solution (U, V ) of (1.16) with (1.28) must be one-dimensional.

Remark 1.9. The remaining question then is to know under which conditions, the previous assumptions, in particular the growth condition (H3), hold. The hypothesis (H3) is satisfied if we can show the following natural growth condition

U (x) + V (x) = O(|x|) as |x| → ∞ . (1.30) Note that one dimensional solutions do satisfy such a growth condition. On the other hand, the hypothesis (H3) is equivalent to the condition of frequency function given by

N (m_ΛR) ≤ 1 + oΛ(1) , for any large R (independent of Λ) , (1.31) where N (·) defined below in (6.4) is the frequency function of (uΛ, v_Λ)’s, and o_Λ(1) is a quantity tending to zero as Λ goes to infinity (see Theorem 6.1). 

The system of equations (1.1)-(1.4) is one particular case of parameter-dependent systems of elliptic equations with k components:















−∆ui = f_i(u_i)u_i−

∑k

j=1

j̸=i

α_ijf_ij(u_j)u_i in Ω,

u₁, . . . , u_k > 0 in Ω, u₁ =· · · = uk = 0 on ∂Ω ,

(1.32)

where Ω ⊂ R^N is a smooth bounded domain, f_i, f_ij : [0,∞) → R are continuous locally Lipschitz functions and α_ij > 0 are parameters for i, j = 1, . . . , k, j ̸= i. Two special cases of (1.32) have been investigated in the literature. The case fij(t) = t for i ̸= j corresponds to a Lotka-Volterra type system modelling the interaction between biological species in population ecology. In particular, this case has been considered by Dancer and Du [19], Conti, Terracini and Verzini [16] and Caffarelli and Lin [13]. Note that in population dynamics as well, phase separation is known to occur if the repulsion of competition terms is strong enough. Another case where the right hand side f_i(u_i)u_i−∑k

j=1

j̸=iα_ijf_ij(u_j)u_i is replaced by A(x)Π^k_i=1u^a_iⁱ arises in combustion theory and has been considered recently by Caffarelli and Roquejoffre in [14].

The paper is organized as follows: In Section 2, we give some preliminaries. In Section 3, Theorem 1.1 is proved using blow-up analysis. In Section 4 and 5, we provide the proof of Theorem 1.2, 1.3 and 1.8, respectively. The frequency function and the argument to show the hypothesis (H3) are given in Section 6. Finally, we make a comparison between the Allen-Cahn equation and (1.16), and we propose several open problems in Section 7.

Notations. In this paper, C is denoted as a generic constant which may vary between lines.

U (±∞) represents limy→±∞U (y) as usual.

Acknowledgments: The research of Berestycki is partially supported by the PREFERED project of the ANR, France. The research of Lin is partially supported by NCTS, NSC and TIMS of Taiwan. The research of Wei is partially supported by an Earmarked Grant from RGC of Hong Kong. The research of Zhao is partially supported by NSFC (Project 11101155) and the Fundamental Research Funds for the Central Universities.

2 Preliminary

From now on, up to Section 5, we only consider the one dimensional problem. In this section, we will give some basic estimates and relations which are often used later.

Lemma 2.1. In the interval Ω = (a, b)⊂ R, assume that (uΛ, v_Λ) solves system (1.1)-(1.5) and that (1.7) holds. Then there is a constant C independent of Λ such that

∥uΛ∥C^1/2([a,b])≤ C, ∥vΛ∥C^1/2([a,b]) ≤ C. (2.1) Furthermore, for any 0 < γ < ¹₂,

uΛ → u∞, vΛ→ v∞ in C^γ([a, b]), (2.2) with

u_∞v_∞ ≡ 0 in (a, b). (2.3)

Proof. Testing (1.1) against u^′_Λ, we have

∫

Ω

|u^′Λ|²+ α

∫

Ω

u⁴_Λ+ Λ

∫

Ω

u²_Λv²_Λ= λ₁.

Thus∥uΛ∥H₀¹([a,b]) ≤ C. Thus, (2.1) and (2.2) are standard results of Sobolev Imbedding. Then (2.3) clearly follows from the above equality.

Remark 2.2. Actually for Ω⊂ R^N, we also know that u_Λand v_Λare always uniformly bounded (see [17, Lemma 2.1]). 

In what follows, for simplicity, we write (u, v) and (λ₁, λ₂) rather than (u_Λ, v_Λ) and (λ_1,Λ, λ_2,Λ), respectively.

Lemma 2.3. Under the same hypotheses as in Lemma 2.1, there exists a positive constant T_Λ and two positive constants C1, C2 independent of Λ such that

u^′2+ v^′2− Λu²v²− α

2u⁴−β

2v⁴+ λ₁u²+ λ₂v² = T_Λ in (a, b) , (2.4) and

0 < C₁ < T_Λ< C₂ < +∞ , (2.5)

Proof. Multiplying (1.1) on both sides by 2u^′ and (1.2) by 2v^′, we have (u^′2)^′− α

2(u⁴)^′− Λv²(u²)^′+ λ₁(u²)^′ = 0 , in (a, b) , (v^′2)^′− β

2(v⁴)^′ − Λu²(v²)^′+ λ₂(v²)^′ = 0 , in (a, b) . Adding the two equalities, we get (2.4).

Now we claim 0 < C1 < TΛ < C2 < +∞. Integrating (2.4) over (a, b), we get

∫ _b

a

u^′2+

∫ _b

a

v^′2− Λ

∫ _b

a

u²v²− α 2

∫ _b

a

u⁴−β 2

∫ _b

a

v⁴ + λ₁+ λ₂ = T_Λ(b− a). (2.6) Using (1.1), (1.2), (1.5) and integrating by parts, we obtain

∫ _b

a

u^′2+ α

∫ _b

a

u⁴+ Λ

∫ _b

a

u²v² = λ₁, (2.7)

∫ b a

v^′2+ β

∫ b a

v⁴+ Λ

∫ b a

u²v² = λ2. (2.8)

Combining (2.6)-(2.8) yields 2

∫ b a

u^′2+ 2

∫ b a

v^′2+ Λ

∫ b a

u²v²+α 2

∫ b a

u⁴+ β 2

∫ b a

v⁴ = TΛ(b− a).

Since from assumption (1.7) we know that λ1, λ2 are uniformly bounded with respect to Λ, then it is obvious that T_Λ < C₂ <∞. On the other hand, Poincar´e’s inequality shows that

∫ _b

a

u^′2 ≥ C

∫ _b

a

u² = C₁ > 0.

This gives T_Λ > C₁ > 0. Here we have used condition (1.5). Therefore, the proof of Lemma 2.3 is complete.

We now state the uniform Lipschitz continuity of u_Λ and v_Λ. Lemma 2.4. Under the same hypotheses as in Lemma 2.1, we have

∥u^′∥L^∞ ≤ C, ∥v^′∥L^∞ ≤ C , where C is a positive constant independent of Λ.

Proof. By Lemma 2.3 and (1.4), it is easy to check that u^′(a)²+ v^′(a)² ≤ C and u^′(b)²+ v^′(b)² ≤ C, which implies that

|u^′(a)| ≤ C, |u^′(b)| ≤ C, |v^′(a)| ≤ C, |v^′(b)| ≤ C. (2.9) Integrating (1.1) from a to x, we get

u^′(x)− u^′(a) = α

∫ x a

u³ + Λ

∫ x a

v²u− λ1

∫ x a

u . (2.10)

For x = b, by (1.7), (2.1), (2.9) and (2.10), this shows that Λ

∫ _b

a

v²u≤ C. (2.11)

Then by (2.1) and (2.11), it follows that |u^′(x)− u^′(a)| ≤ C. Similarly, integrating (1.1) on (x, b) yields that |u^′(b)− u^′(x)| ≤ C. Combining these two estimates shows that |u^′| + |v^′| is uniformly bounded. The proof is thereby complete.

3 Proof of Theorem 1.1 and 1.2

In this section, we provide the proof of Theorem 1.1. Recall that u_∞ and v_∞ are defined in Lemma 2.1. Owing to condition (1.5), we know that ∫_b

a u²_∞ = ∫_b

a v_∞² = 1 which implies that u_∞, v_∞̸≡ 0. Hence there exists xΛ ∈ Ω such that (1.10) and (1.11) hold.

We start with an a priori bound.

Lemma 3.1. Under the hypotheses of Theorem 1.1, it holds true that lim sup

Λ→∞ Λ m⁴_Λ <∞ .

Proof. We argue by contradiction. Suppose that along a sequence Λ_j → ∞,

m⁴_ΛΛ→ ∞ as Λ = Λ_j → ∞ . (3.1)

Then let

˜

u(y) = 1 m_Λu

( y

mΛ

√Λ + x_Λ )

, v(y) =˜ 1 m_Λv

( y

mΛ

√Λ + x_Λ )

, defined for y ∈ ˜IΛ, where ˜I_Λ = {y ∈ R : (a − xΛ)m_Λ√

Λ < y < (b− xΛ)m_Λ√

Λ}. Then (˜u, ˜v) solves the system 









˜ u^′′− α

Λu˜³ − ˜v²u +˜ λ₁

m²_ΛΛu = 0˜ in I˜_Λ,

˜ v^′′− β

Λ˜v³− ˜u²v +˜ λ₂

m²_ΛΛv = 0˜ in I˜_Λ. From (2.4) (see Lemma 2.3), we have

˜

u^′2+ ˜v^′2− ˜u²v˜² − α

2Λu˜⁴− β

2Λ˜v⁴+ λ₁

m²_ΛΛu˜²+ λ₂

m²_ΛΛ˜v² = T_Λ

m⁴_ΛΛ in I˜_Λ. (3.2) On the other hand, Lemma 2.4 gives

˜u(y) − 1

m_Λu(x_Λ)

 ≤ C|y|

m²_Λ√ Λ, 

˜v(y) − 1

m_Λv(x_Λ)

 ≤ C|y|

m²_Λ√

Λ for y∈ ˜IΛ. (3.3) Since mΛ = uΛ(xΛ) = vΛ(xΛ) → 0 and m⁴ΛΛ → ∞, then these inequalities show that ˜u and ˜v are uniformly bounded and equicontinuous on any compact subinterval of ˜I_Λ. Owing to (1.11)

and (3.1), it is obvious that ˜Ω_Λ tends to the entire real line R as Λ → ∞. Here we have used the fact that m_Λ→ 0 as Λ → ∞. Thus, by (3.1), (3.3) and the Arzela-Ascoli Theorem,

˜

u→ 1, ˜v → 1 in C_loc(R) , ˜u^′, ˜v^′ → 0 in L^∞_loc(R) as Λ→ ∞ , (3.4) Hence (3.2) and (3.4) imply

T_Λ

m⁴_ΛΛ → −1 as Λ→ ∞ .

However, this contradicts (2.5) and (3.1). Therefore, the proof of Lemma 3.1 is complete.

We can now prove Theorem 1.1. We argue by contradiction. In view of Lemma 3.1, we assume that Λ m⁴_Λ → 0 as Λ → ∞. As in (1.13), let

˜

u(y) = 1

m_Λu(m_Λy + x_Λ), v(y) =˜ 1

m_Λv(m_Λy + x_Λ) , (3.5) for y ∈ IΛ, where I_Λ =

(a−xΛ

mΛ ,^b−x_m ^Λ

Λ

)

tends to the entire real line as Λ goes to infinity since we assume x_Λ→ x∞ ∈ (a, b). As before, ˜u and ˜v satisfy





˜

u^′′− m⁴Λα˜u³− m⁴ΛΛ˜v²u + m˜ ²_Λλ₁u = 0˜ in I_Λ,

˜

v^′′− m⁴Λβ ˜v³− m⁴ΛΛ˜u²v + m˜ ²_Λλ₂v = 0˜ in I_Λ. From Lemma 2.3, we have

˜

u^′2+ ˜v^′2− m⁴ΛΛ˜u²˜v²− α

2m⁴_Λu˜⁴−β

2m⁴_Λv˜⁴+ λ₁m²_Λu˜²+ λ₂m²_Λv˜² = T_Λ in I_Λ. (3.6) Lemma 2.4 tells us that

˜

u(y) = 1

m_Λ[u(xΛ) + O(1)mΛy],

so ˜u is locally uniformly bounded and so is ˜v. By elliptic regularity, we know that ˜u and

˜

v are bounded in C_loc² (R) and thus the Arzel`a–Ascoli theorem yields that there exists U(y), V (y)∈ C²(R) such that

˜

u→ U, ˜v → V in C_loc² (R).

Passing to the limit in the associated equations, we see that U , V satisfy the following equations U^′′ = V^′′ = 0 inR.

Since U , V ≥ 0, they have to be constants. Furthermore, from (3.6), U and V satisfy U^′2+ V^′2 = T_∞.

Here T_Λ → T_∞and T_∞ > 0. Thus, this is a contradiction with the fact that U , V are constants.

Therefore, we know that Λm⁴_Λ converges to some positive constant C₀. This proves (1.12).

Next, we show that Λ^1/4min(|xλ− a|, |xλ− b|) → +∞. To this end, we define

˜

u(y) = Λ^1/4u( y

Λ^1/4 + x_Λ), ˜v(y) = Λ^1/4v( y

Λ^1/4 + x_Λ).

Then ˜u and ˜v satisfy









˜ u^′′− α

Λu˜³− ˜v²u +˜ λ1

√Λu = 0˜ in (a− xΛ, b− xΛ)√⁴ Λ,

˜ v^′′− β

Λv˜³− ˜u²v +˜ λ₂

√Λv = 0˜ in (a− xΛ, b− xΛ)√⁴ Λ.

From Lemma 2.3, it follows that

˜

u^′2+ ˜v^′2− ˜u²˜v²− α

2Λu˜⁴− β

2Λv˜⁴+ λ₁

√Λu˜² + λ₂

√Λv˜² = T_Λ.

Without loss of generality, we may assume Λ¹⁴(a− xΛ) → −C1 > −∞, where C1 is a positive constant. A similar argument as above shows that there exist U , V ∈ C^∞([−C1,∞)) such that

˜

u→ U, ˜v → V in C_loc² ([−C1,∞)) and











U^′′= V²U in [−C1,∞), V^′′ = U²V in [−C1,∞), U (0) = V (0) = C₀^1/4,

U (−C1) = V (−C1) = 0.

Fatou’s Lemma then yields

∫ _∞

−C¹

V²U ≤ lim inf

Λ→∞

∫ _(b−xΛ)^√4_Λ

(a−xΛ)√⁴ Λ

˜

v²u˜≤ Λ

∫ _b

a

v²u≤ C,

where the last inequality is due to (2.11). Thus, U (∞) = 0 or V (∞) = 0. Since U and V are convex on (C₁,∞), U(−C1) = V (−C1) = 0 and either U (∞) = 0 or V (∞) = 0, then we know that U ≡ 0 or V ≡ 0, which contradicts U(0) = V (0) = C0¹⁴.

Finally we prove Theorem 1.2. Recall that ˜u and ˜v satisfy









˜ u^′′− α

Λu˜³− ˜v²u +˜ λ₁

√Λu = 0˜ in (a− xΛ, b− xΛ)Λ^1/4,

˜ v^′′− β

Λv˜³− ˜u²v +˜ λ₂

√Λ˜v = 0 in (a− xΛ, b− xΛ)Λ^1/4.

From Lemma 2.3, we have

˜

u^′2+ ˜v^′2− ˜u²˜v²− α

2Λu˜⁴− β

2Λv˜⁴+ λ₁

√Λu˜² + λ₂

√Λv˜² = T_Λ.

By similar arguments as to the one we have already used, we pass to the limit in the above two equations, say ˜u → U and ˜v → V . The Maximum Principle yields that U > 0 and V > 0 and this completes the proof.

3.1 Special trapping potential case

We now observe that the previous arguments of Theorems 1.1 and 1.2 can be extended to the more general system (1.20)-(1.24). First, for Lemma 2.1, it is easy to check that

∥u∥L^∞, ∥v∥L^∞ ≤ c1, (3.7)

where c1 is a positive constant independent of Λ. Since∫b

a u² =∫b

a v² = 1, and u, v∈ H0¹((a, b)), it is obvious that ∫ b

a

u^′2,

∫ b a

v^′2 ≥ π²(b− a)⁻² > 0 , (3.8) and by Cauchy-Schwartz inequality,

∫ _b

a

u⁴ ≥ 1 b− a

(∫ b a

u² )²

= (b− a)⁻¹,

∫ _b

a

v⁴ ≥ 1 b− a

(∫ b a

v² )²

= (b− a)⁻¹. (3.9) Regarding the analogue of (2.4) in Lemma 2.3, we have

u^′2+ v^′2− P1u²− P2v²− Λu²v²− α

2u⁴− β

2v⁴+ λ₁u²+ λ₂v² (3.10)

= u^′2(a) + v^′2(a)−

∫ x a

P₁^′u²−

∫ x a

P₂^′v² ≡ ˜T_Λ(x) for x∈ (a, b) . Now we want to show that

C1 ≤ ˜TΛ ≤ C2 for x∈ (a, b) , (3.11) where C_j’s are positive constant independent of Λ. As before, we know that

∫

Ω

u^′2+

∫

Ω

P₁u²+ α

∫

Ω

u⁴+ Λ

∫

Ω

u²v² = λ₁, (3.12)

∫

Ω

v^′2+

∫

Ω

P2v²+ β

∫

Ω

v⁴+ Λ

∫

Ω

u²v² = λ2. (3.13)

Adding these relations, we obtain

∫ b a

2(

u^′2+ v^′2) +

∫ b a

α

2u⁴+β

2 v⁴+ Λ

∫ b a

u²v²+

∫ b a

∫ x a

P₁^′u²+ P₂^′v² = [u^′2(a) + v^′2(a)](b− a) . (3.14) Thus, by (1.25), (3.8), (3.9), (3.10) and (3.14), we get

T˜_Λ≥ 4π²(b− a)⁻³+1

2(α + β) (b− a)⁻²− 4M > 0 in (a, b) . (3.15) Here we have used (1.25) and the fact that

∫ x a

|P1^′| u² ≤ M ,

∫ x a

|P2^′| v² ≤ M ,