On a two-component Bose–Einstein condensate with steep potential wells

DOI 10.1007/s10231-017-0635-6

On a two-component Bose–Einstein condensate with steep potential wells

Yuanze Wu¹ · Tsung-fang Wu² · Wenming Zou³

Received: 26 July 2016 / Accepted: 24 January 2017 / Published online: 6 February 2017

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2017

Abstract In this paper, we study the following two-component systems of nonlinear Schrödinger equations

⎧⎨

⎩

u−(λa(x)+a₀(x))u+μ1u³+βv²u=0 inR³, v−(λb(x)+b0(x))v+μ2v³+βu²v=0 inR³, u, v∈H¹(R³),u, v >0 inR³,

whereλ, μ1, μ2 > 0 andβ < 0 are parameters;a(x),b(x) ≥ 0 are steep potentials and a₀(x),b₀(x)are sign-changing weight functions;a(x),b(x),a₀(x)andb₀(x)are not neces- sarily to be radial symmetric. By the variational method, we obtain a ground state solution and multi-bump solutions for such systems withλsufficiently large. The concentration behav- iors of solutions as bothλ → +∞andβ → −∞are also considered. In particular, the phenomenon of phase separations is observed in the whole spaceR³. In the Hartree–Fock theory, this provides a theoretical enlightenment of phase separation inR³for the 2-mixtures of Bose–Einstein condensates.

Keywords Bose–Einstein condensate·Steep potential well·Ground state solution· Multi-bump solution

B

wuyz850306@cumt.edu.cn Tsung-fang Wu

tfwu@nuk.edu.tw Wenming Zou

wzou@math.tsinghua.edu.cn

1 College of Sciences, China University of Mining and Technology, Xuzhou 221116, People’s Republic of China

2 Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan 3 Department of Mathematical Sciences, Tsinghua University, Beijing 100084,

People’s Republic of China

Mathematics Subject Classification 35B38·35B40·35J10·35J20

1 Introduction

In this paper, we consider the following two-component systems of nonlinear Schrödinger equations

⎧⎨

⎩

u−(λa(x)+a0(x))u+μ1u³+βv²u=0 inR³, v−(λb(x)+b₀(x))v+μ2v³+βu²v=0 inR³, u, v∈H¹(R³),u, v >0 inR³,

(P_λ,β) whereλ, μ1, μ2>0 andβ <0 are parameters. The potentialsa(x),b(x),a0(x)andb0(x) satisfy the following conditions:

(D₁) a(x),b(x)∈C(R³)anda(x),b(x)≥0 onR³.

(D2) There exista_∞ > 0 andb_∞ > 0 such thatDa := {x ∈ R³ | a(x) < a_∞}and Db:= {x ∈R³|b(x) <b_∞}are nonempty and have finite measures.

(D3) a = inta⁻¹(0)andb =intb⁻¹(0)are nonempty bounded sets and have smooth boundaries. Moreover,a =a⁻¹(0),b =b⁻¹(0)anda∩b = ∅.

(D₄) a₀(x),b₀(x)∈C(R³)and there existR,d_a,d_b >0 such that

a₀⁻(x)≤d_a(1+a(x)) and b⁻₀(x)≤d_b(1+b(x)) for|x| ≥R, wherea₀⁻(x)=max{−a0(x),0}andb⁻₀(x)=max{−b0(x),0}.

(D5) infσa(−+a0(x)) > 0 and infσb(−+b0(x)) > 0, whereσa(−+a0(x))is the spectrum of−+a₀(x) on H₀¹(a) andσb(−+b₀(x))is the spectrum of

−+b₀(x)onH₀¹(b).

Remark 1.1 Ifa0(x),b0(x)∈C(R³)are bounded, then the condition(D4)is trivial. How- ever, under the assumptions of(D4)−(D5),a0(x) andb0(x) may be sign-changing and unbounded.

Two-component systems of nonlinear Schrödinger equations like(Pλ,β)appear in the Hartree–Fock theory for a double condensate, that is, a binary mixture of Bose–Einstein condensates in two different hyperfine states|1 and|2 (cf. [25]), where the solutionsu andvare the corresponding condensate amplitudes,μjare the intraspecies and interspecies scattering lengths. The interaction is attractive ifβ > 0 and repulsive ifβ < 0. When the interaction is repulsive, it is expected that the phenomenon of phase separations will happen, that is, the two components of the system tend to separate in different regions as the interaction tends to infinity. This kind of systems also arises in nonlinear optics (cf. [2]). Due to the important application in physics, the following system

⎧⎨

⎩

u−λ1u+μ1u³+βv²u=0 in, v−λ2v+μ2v³+βu²v=0 in,

u, v=0 on∂, (1.1)

where⊂R²orR³, has attracted many attentions of mathematicians in the past decade.

We refer the readers to [7–9,15,17,19,20,29–31,33,34,36,37,39,40,44]. In these literatures, various existence theories of the solutions were established for the Bose–Einstein condensates inR² andR³. Recently, some mathematicians devoted their interest to the two coupled Schrödinger equations with critical Sobolev exponent in the high dimensions, and a number

of the existence results of the solutions for such systems were also established. See for example [13–16,18].

On the other hand, if the parameter λis sufficiently large, then λa(x) andλb(x) are called the steep potential wells under the conditions(D1)−(D3). The depth of the wells is controlled by the parameterλ. Such potentials were first introduced by Bartsch and Wang in [3] for the scalar Schrödinger equations. An interesting phenomenon for this kind of Schrödinger equations is that one can expect to find the solutions which are concentrated at the bottom of the wells as the depth goes to infinity. Due to this interesting property, such topic for the scalar Schrödinger equations was studied extensively in the past decade.

We refer the readers to [4,5,10,24,35,42,43,45,51] and the references therein. In particular, in [24], by assuming that the bottom of the steep potential wells consists of finitely many disjoint bounded domains, the authors obtained multi-bump solutions for scalar Schrödinger equations with steep potential wells, which are concentrated at any given disjoint bounded domains of the bottom as the depth goes to infinity.

We wonder what happens to the two-component Bose–Einstein condensate(P_λ,β)with steep potential wells? In the current paper, we shall explore this problem to find whether the solutions of such systems are concentrated at the bottom of the wells asλ→ +∞and when the phenomenon of phase separations of such systems can be observed in the whole space R³.

We remark that the phenomenon of phase separations for (1.1) was observed in [13,16, 20,21,38,49,50] for the ground state solution whenis a bounded domain. In particular, this phenomenon was also observed on the whole spacesR² andR³ by [48], where the system is radial symmetric! However, when the system is not necessarily radial symmetric, the phenomenon of phase separations for Bose–Einstein condensates on the whole space R³ has not been obtained yet. For other kinds of elliptic systems with strong competition, the phenomenon of phase separations has also been well studied; we refer the readers to [11,12,22] and references therein.

We recall some definitions in order to state the main results in the current paper. We say that(u₀, v0)∈H¹(R³)×H¹(R³)is a non-trivial solution of(P_λ,β)if(u₀, v0)is a solution of(Pλ,β)withu₀=0 andv0=0. We say(u₀, v0)∈Eis a ground state solution of(Pλ,β) if(u0, v0)is a non-trivial solution of(Pλ,β)and

J_λ,β(u₀, v0)=inf{J_λ,β(u, v)|(u, v)is a non-trivial solution of(P_λ,β)}, whereJ_λ,β(u, v)is the corresponding functional of(P_λ,β)and given by

J_λ,β(u, v)= 1 2

R³|∇u|²+(λa(x)+a₀(x))u²dx+1 2

R³|∇v|²+(λb(x)+b₀(x))v²dx

−μ1

4

R³u⁴dx−μ2

4

R³v⁴dx−β 2

R³u²v²dx. (1.2)

Remark 1.2 In Sect.2, we will give a variational setting of(P_λ,β)and show that the solutions of(P_λ,β)are equivalent to the positive critical points ofJ_λ,β(u, v)in a suitable Hilbert space E.

LetI_a(u)andI_b(v)be two functionals, respectively, defined onH₀¹(a)andH₀¹(b), which are given by

I_a(u)= 1 2

a

|∇u|²+a₀(x)u²dx−μ1

4

a

u⁴dx

and by

I_b(v)= 1 2

b

|∇v|²+b₀(x)v²dx−μ2

4

b

v⁴dx.

Then, by the condition(D5), it is well known that I_a(u) and I_b(v)have least energy nonzero critical points. We denote the least energy of nonzero critical points forI_a(u)and I_b(v)bym_a andm_b, respectively. Now, our first result can be stated as follows.

Theorem 1.1 Assume(D1)−(D5). Then there exists∗ > 0independent ofβ such that (P_λ,β)has a ground state solution(u_λ,β, v_λ,β)for allλ≥_∗andβ <0, which has the following properties:

(1)

R³\a|∇u_λ,β|²+u²_λ,βdx →0and

R³\b|∇v_λ,β|²+v_λ,β² dx →0asλ→ +∞.

(2)

a|∇u_λ,β|²+a0(x)u²_λ,βdx → 4ma and

b|∇v_λ,β|² +b0(x)v²_λ,βdx →4mb as λ→ +∞.

Furthermore, for each{λn} ⊂ [_∗,+∞)satisfiesλn → +∞as n → ∞andβ <0, there exists(u_0,β, v0,β)∈(H¹(R³)\{0})×(H¹(R³)\{0})such that

(3) (u_0,β, v0,β)∈H₀¹(a)×H₀¹(b)with u_0,β≡0outsideaandv0,β≡0outsideb. (4) (u_λn,β, vλn,β) → (u_0,β, v0,β) strongly in H¹(R³)× H¹(R³) as n → ∞ up to a

subsequence.

(5) u_0,βis a least energy nonzero critical point of I_a(u)andv0,βis a least energy nonzero critical point of I_b(v).

Remark 1.3 Roughly speaking, under the conditions(D₁)−(D₅), Theorem1.1obtain a solu- tion of the system(Pλ,β)in the following form(u₀+w¹_λ,β, v0+w²_λ,β), whereu₀andv0are, respectively, the least energy nonzero critical points ofI_a(u)andI_b(v)andw¹_λ,βandw_λ,β² are two perturbations withw_λ,β¹ →0 andw_λ,β² →0 strongly inH¹(R³)asλ→ +∞. It is worth to point out that the assumptiona∩b = ∅is not essential in proving Theorem1.1.

For instance, in the casea = b = , our method to prove Theorem 1.1(with some necessary modifications) still works to find out a solution of the system(Pλ,β)in the form (u_β+w³_λ,β, v_β+w_λ,β⁴ ), where(u_β, v_β)is the ground state solution of the following system

⎧⎪

⎪⎨

⎪⎪

⎩

u−a0(x)u+μ1u³+βv²u=0 in, v−b₀(x)v+μ2v³+βu²v=0 in,

u, v >0 in,

u=v=0 on∂,

andw_λ,β³ andw⁴_λ,β are also two perturbations withw_λ,β³ →0 andw_λ,β⁴ →0 strongly in H¹(R³)asλ→ +∞.

Next, we assume that the bottom of the steep potential wells consists of finitely many disjoint bounded domains. It is natural to ask whether the two-component Bose–Einstein condensate(Pλ,β)with such steep potential wells has multi-bump solutions which are con- centrated at any given disjoint bounded domains of the bottom as the depth goes to infinity.

Our second result is devoted to this study. Similar to [24], we need the following conditions on the potentialsa(x),b(x),a₀(x)andb₀(x).

(D₃) a = inta⁻¹(0)andb = intb⁻¹(0)satisfya = ⁿ∪^a

i_a=1a,ia andb = ⁿ∪^b

j_b=1b,jb, where{a,ia}and{b,j_b}are all nonempty bounded domains with smooth boundaries, anda,i_a ∩a,j_a = ∅foria = ja andb,i_b ∩b,j_b = ∅forib = jb. Moreover, a =a⁻¹(0)andb =b⁻¹(0)witha∩b= ∅.

(D₅) infσa,ia(−+a₀(x)) > 0 for alli_a = 1, . . . ,n_a and infσb,jb(−+b₀(x)) >0 for all j_b =1, . . . ,n_b, whereσa,ia(−+a₀(x))is the spectrum of−+a₀(x)on H₀¹(a,ia)andσb,jb(−+b₀(x))is the spectrum of−+b₀(x)onH₀¹(b,jb). Remark 1.4 Under the conditions(D₃)and(D₄), it is easy to see that the condition(D₅) is equivalent to the condition(D₅). For the sake of clarity, we use the condition(D₅)in the study of multi-bump solutions.

We defineI_a,ia(u)onH₀¹(a,i_a)for eachia =1, . . . ,naby I_a,ia(u)= 1

2

a,ia

|∇u|²+a0(x)u²dx−μ1

4

a,ia

u⁴dx andI_b,jb(v)onH₀¹(b,jb)for each j_b=1, . . . ,n_bby

I_b,jb(v)= 1 2

b,jb

|∇v|²+b₀(x)v²dx−μ2

4

b,jb

v⁴dx.

Then by the conditions(D₃)and(D₅), it is well known thatI_a,ia(u)and I_b,jb(v)have least energy nonzero critical points for everyia = 1, . . . ,na and every jb = 1, . . . ,nb, respectively. We denote the least energy of nonzero critical points forI_a,ia(u)andI_b,jb(v) bym_a,ia andm_b,jb, respectively. Now, our second result can be stated as the following.

Theorem 1.2 Assumeβ <0and the conditions(D1)−(D2),(D₃),(D4)and(D₅)hold. If the set Ja×Jb⊂ {1, . . . ,na} × {1, . . . ,nb}satisfying Ja= ∅and Jb= ∅, then there exists ∗(β) >0such that(Pλ,β)has a non-trivial solution(u_λ,β^Ja , v_λ,β^Jb )forλ≥∗(β)with the following properties:

(1)

R³\_a,0^Ja |∇u_λ,β^Ja |²+(u_λ,β^Ja )²dx →0and

R³\_b,0^Jb |∇v_λ,β^Jb |²+(v_λ,β^Jb )²dx→0asλ→ +∞, where_a,0^Ja = ∪

ia∈Ja

a,i_a and_b,0^Jb = ∪

jb∈Jb

b,j_b. (2)

a,ia|∇u_λ,β^Ja |²+a₀(x)(u_λ,β^Ja )²dx →4m_a,iaand

b,jb|∇v_λ,β^Jb |²+b₀(x)(v_λ,β^Jb )²dx → 4m_b,jbasλ→ +∞for all ia ∈Jaand jb∈Jb.

Furthermore, for eachβ <0and{λn} ⊂ [∗(β),+∞)satisfyingλn → +∞as n→ ∞, there exists(u_0,β^Ja , v_0,β^Jb )∈(H¹(R³)\{0})×(H¹(R³)\{0})such that

(3) (u_0,β^Ja , v_0,β^Jb )∈H₀¹(_a,0^Ja )×H₀¹(_b^Jb_,0)with u_0,β^Ja ≡0outside_a,0^Ja andv_0,β^Jb ≡0outside _b,0^Jb.

(4) (u_λ^Ja_n,β, v_λ^J_n^b_,β) → (u_0,β^Ja , v_0,β^Jb ) strongly in H¹(R³)× H¹(R³) as n → ∞ up to a subsequence.

(5) the restriction of u_0,β^Ja on_a,ialies in H₀¹(_a,ia)and is a least energy nonzero critical point of I_a,ia(u)for all ia∈ Ja, while the restriction ofv₀^J_,β^b onb,jblies in H₀¹(b,jb) and is a least energy nonzero critical point of I_b,jb(v)for all jb ∈Jb.

Corollary 1.1 Supposeβ <0and the conditions(D₁)−(D₂),(D₃),(D₄)and(D₅)hold.

Then,(P_λ,β)has at least(2^na−1)(2^nb−1)non-trivial solutions forλ≥∗(β).

Remark 1.5 (i) To the best of our knowledge, it seems that Theorem1.2is the first result for the existence of multi-bump solutions to system(P_λ,β).

(ii) Under the condition(D₃), we can see that I_a(u)=

na

ia=1

I_a,ia(u) and I_b(v)=

nb

jb=1

I_b,jb(v).

Letm_a,0 =min{ma,1, . . . ,m_a,na}andm_b,0 =min{mb,1, . . . ,m_b,nb}. Then, we must havema,0 =ma andmb,0 =mb. Without loss of generality, we assumema,0 =ma,1

andm_b,0 = m_b,1. Now, by Theorem 1.2, we can find a solution of(P_λ,β) with the same concentration behavior as the ground state solution obtained in Theorem1.1as λ→ +∞. However, we do not know these two solutions are the same or not.

Next we consider the phenomenon of phase separations for System(P_λ,β), i.e., the con- centration behavior of the solutions asβ→ −∞. In the following theorem, we may observe such a phenomenon on the whole spaceR³.

Theorem 1.3 Assume(D₁)−(D₅). Then, there exists∗∗≥∗such thatβ²

R³u²_λ,βv_λ,β² → 0asβ→ −∞forλ≥∗∗, where(uλ,β, vλ,β)is the ground state solution of(Pλ,β)obtained by Theorem1.1. Furthermore, for every{βn} ⊂ (−∞,0)withβn → −∞andλ ≥_∗∗, there exists(u_λ,0, v_λ,0)∈(H¹(R³)\{0})×(H¹(R³)\{0})satisfying the following properties:

(1) (u_λ,βn, v_λ,βn) → (u_λ,0, v_λ,0) strongly in H¹(R³)× H¹(R³) as n → ∞up to a subsequence.

(2) u_λ,0∈C(R³)andvλ,0 ∈C(R³).

(3) u_λ,0≥0andv_λ,0≥0inR³with{x∈R³|u_λ,0(x)>0} =R³\{x ∈R³|v_λ,0(x) >0}.

Furthermore,{x ∈ R³ | u_λ,0(x) > 0}and{x ∈ R³ |vλ,0(x) > 0}are connected domains.

(4) u_λ,0∈H₀¹({u_λ,0>0})and is a least energy solution of

−u+(λa(x)+a₀(x))u=μ1u³, u∈H₀¹({u_λ,0 >0}), (1.3) whilev_λ,0∈H₀¹({v_λ,0 >0})and is a least energy solution of

−v+(λa(x)+a0(x))v=μ1v³, v∈H₀¹({v_λ,0>0}). (1.4) Remark 1.6 (1) Theorem1.3is based on Theorem1.1, Thus, due to Remark1.3, the assump- tiona∩b= ∅is also not necessary in Theorem1.3and it still holds fora =b=. (2) It is natural to ask that whether the multi-bump solutions founded in Theorem1.2have the same phenomenon of phase separations as that of the ground state solution described in Theorem1.3. However, by checking the proof of Theorem1.2, we can see that∗(β)→ +∞asβ→ −∞. Thus, our method to prove Theorem1.2is invalid to assert that the system(P_λ,β)has multi-bump solutions forλlarge butβfixed andβdiverging. Due to this reason, we can not obtain the phenomenon of phase separations to the multi-bump solutions founded in Theorem1.2.

Before closing this section, we would like to cite other references studying the equations with steep potential wells. For example, in [46], the Kirchhoff-type elliptic equation with a steep potential well was studied. The Schrödinger–Poisson systems with a steep potential well were considered in [32,52]. Non-trivial solutions were obtained in [26–28] for quasilinear Schrödinger equations with steep potential wells, while the multi-bump solutions were also obtained in [28] for such equations.

In this paper, we will always denote the usual norms inH¹(R³)andL^p(R³)(p≥1) by

·and·p, respectively;CandCwill be indiscriminately used to denote various positive constants;o_n(1)will always denote the quantities tending toward zero asn→ ∞.

2 The variational setting

In this section, we mainly give a variational setting for(P_λ,β). Simultaneously, an important estimate is also established in this section, which is used frequently in this paper.

Let

E_a= {u∈D^1,2(R³)|

R³(a(x)+a₀⁺(x))u²dx<+∞}

and

Eb= {u∈D^1,2(R³)|

R³(b(x)+b⁺₀(x))u²dx<+∞},

wherea⁺₀(x)=max{a₀(x),0}andb₀⁺(x)=max{b₀(x),0}. Then, by the conditions(D₁) and(D4),EaandEbare Hilbert spaces equipped with the inner products

u, v a =

R³∇u∇v+(a(x)+a₀⁺(x))uvdx and u, v b =

R³∇u∇v+(b(x)+b₀⁺(x))uvdx,

respectively. The corresponding norms ofEaandEbare, respectively, given by ua=

R³|∇u|²+(a(x)+a₀⁺(x))u²dx ¹

2

and by

vb=

R³|∇v|²+(b(x)+b⁺₀(x))v²dx ¹

2.

Since the conditions(D1)−(D₂)hold, by a similar argument as that in [46], we can see that u ≤ max{1+ |Da|²³S⁻¹, 1

a_∞} ¹

2ua for allu∈Ea (2.1) and

v ≤ max{1+ |Db|²³S⁻¹, 1 b_∞}

¹

2vb for allv∈Eb, (2.2) whereSis the best Sobolev embedding constant fromD^1,2(R³)toL⁶(R³)and given by

S=inf{∇u²₂|u∈D^1,2(R³),u²₆=1}.

It follows that both Ea and Eb are embedded continuously into H¹(R³). Moreover, by applying the Hölder and Sobolev inequalities, we also have

u4≤ max{1+ |Da|²³S⁻¹, 1 a_∞}

¹

2

S⁻³⁸ua for allu∈E_a (2.3) and

v4≤ max{1+ |Db|²³S⁻¹, 1 b_∞}

¹

2

S⁻³⁸vb for allv∈E_b. (2.4) On the other hand, by the conditions(D₂)and(D₃), there exist two bounded open setsa

and_bwith smooth boundaries such thata ⊂_a ⊂Da,b⊂_b⊂Db,_a∩_b = ∅, dist(a,R³\_a) >0 and that dist(b,R³\_b) >0. Furthermore, by the condition(D₄), the

Hölder and the Sobolev inequalities, there exists0>2 max{1,d_a+^da+C_a∞^a,0,d_b+^db+C_b∞^b,0} such that

R³a⁻₀(x)u²dx≤

BR(0)C_a,0u²dx+

R³\BR(0)da(1+a(x))u²dx ≤λ

2u²_a (2.5) and

R³b⁻₀(x)v²dx≤

BR(0)C_b,0v²dx+

R³\BR(0)d_b(1+b(x))v²dx ≤λ

2v²_b (2.6) forλ ≥ 0, where BR(0) = {x ∈ R³ | |x| < R},Ca,0 = sup_BR(0)a₀⁻(x)andCb,0 = sup_B

R(0)b⁻₀(x). Combining (2.3)–(2.6) and the Hölder inequality, we can see that(P_λ,β) has a variational structure in the Hilbert space E = E_a ×E_b forλ ≥ 0, where E is endowed with the norm(u, v) =(u²_a+v²_b)¹². The corresponding functional of(Pλ,β) is given by (1.2). Furthermore, by applying (2.3)–(2.6) in a standard way, we can also see thatJ_λ,β(u, v)isC²inEand the solution of(P_λ,β)is equivalent to the positive critical point ofJ_λ,β(u, v)inEforλ≥0. In the case of(D₃), we can choose_aand_bas follows:

(I) a = ⁿ∪^a

ia=1_a,ia ⊂Da, wherea,ia ⊂ _a,ia and dist(a,ia,R³\_a,ia) > 0 for all ia =1, . . . ,naand_a,ia∩_a,ja = ∅foria= ja.

(II) _b = ⁿ∪^b

ib=1_b,ib ⊂ Db, whereb,i_b ⊂ _b,ib and dist(b,i_b,R³\_b,ib) > 0 for all jb=1, . . . ,nband_b,ib∩_b,jb = ∅forib= jb.

(III) _a∩_b = ∅.

Thus, (2.5)–(2.6) still hold for such_a and_b with λsufficiently large. Without loss of generality, we may assume that (2.5)–(2.6) still hold for such_a and_b withλ ≥0. It follows that the solution of(P_λ,β)is also equivalent to the positive critical point of theC² functionalJ_λ,β(u, v)inEforλ≥0under the conditions(D₁)−(D₂),(D₃)and(D₄).

The remaining of this section will be devoted to an important estimate, which is used frequently in this paper and essentially due to Ding and Tanaka [24].

Lemma 2.1 Assume(D1)−(D5). Then, there exist1≥0and C_a,b>0such that

u∈Einf_a\{0}

R³|∇u|²+(λa(x)+a0(x))u²dx

R³u²dx ≥C_a,b and

v∈Einfb\{0}

R³|∇v|²+(λb(x)+b0(x))v²dx

R³v²dx ≥C_a,b for allλ≥1.

Proof Since the conditions(D₁)−(D₄)hold, by a similar argument as [24, Lemma 2.1], we have

λ→+∞lim infσa,∗(−+λa(x)+a0(x))=infσa(−+a0(x)),

whereσa,∗(−+λa(x)+a0(x))is the spectrum of−+λa(x)+a0(x)onH¹(_a). Denote infσa(−+a0(x))byνa. Then, by the condition(D5), there exists₁ ≥0such that

σa,∗(−+λa(x)+a₀(x))≥νa

2 for λ≥₁. (2.7)

On the other hand, by the conditions(D₂)and(D₄), we havea⁻₀(x)≤C_a,0+d_a+d_aa_∞ forx ∈Da. LetD_a,R=Da∩B^c

R, whereB^c

R= {x ∈R| |x| ≥R}. Then, by the condition (D₂)once more,|D_a,R| →0 as R → +∞, which then implies that there exists R₀ >0 such that|D_a,R0|S⁻¹(Ca,0+da +daa_∞+1)≤ ¹₂. Thanks to the conditions(D1)−(D4), there exists1=1(R₀)≥₁such that

λa(x)+a₀(x)+(C_a,0+d_a+d_aa_∞+1)χD_a,R

0 ≥1 for allx ∈R³\aandλ≥1, whereχ_Da,R

0 is the characteristic function of the set_Da,R

0. It follows from the Hölder and the Sobolev inequalities that

R³\a

u²dx≤(1+2|D_a,R0|S⁻¹)

R³\a

|∇u|²+(λa(x)+a0(x))u²dx (2.8) for allu∈Ea\{0}andλ≥1. Combining (2.7)–(2.8) and the choice of_a, we have

R³u²dx ≤Ca

R³|∇u|²+(λa(x)+a0(x))u²dx for allu∈Ea\{0}andλ≥1, whereCa = max{_ν²

a,1+2|D_a,R0|S⁻¹}. By similar arguments as (2.7) and (2.8), we can

also have

R³v²dx ≤Cb

R³|∇v|²+(λb(x)+b0(x))v²dx

for allv∈E_b\{0}andλ≥1, whereC_b=max{_ν²_b,1+2|D_b,R0|S⁻¹},νb =infσb(−+ b₀(x))andD_b,R0 =Db∩B^c

R0. We completes the proof by takingC_a,b=(min{C_a,C_b})⁻¹. Remark 2.1 Under the conditions(D₁)−(D₂),(D₃),(D₄)and(D₅), we can see that νa = min

i_a=1,2,...,n_a

infσa,ia(−+a₀(x))

and νb = min

j_b=1,2,...,n_b

infσb,jb(−+b₀(x))

. Now, by a similar argument as (2.7), we get that

_a,ia u²dx≤ 2 νa

_a,ia|∇u|²+(λa(x)+a0(x))u²dx (2.9)

and

_b,jbv²dx ≤ 2 νb

_b,jb|∇v|²+(λb(x)+b₀(x))v²dx (2.10) for allia =1, . . . ,naand jb =1, . . . ,nbifλsufficiently large. Without loss of generality, we may assume (2.9) and (2.10) hold forλ≥1. It follows that Lemma2.1still holds under the conditions(D₁)−(D₂),(D₃),(D₄)and(D₅).

By Lemma2.1, we observe that

R³|∇u|²+(λa(x)+a₀(x))u²dxand

R³|∇v|²+(λb(x)+

b₀(x))v²dx are norms ofE_aandE_bforλ≥1, respectively. Therefore, we set u²_a,λ=

R³|∇u|²+(λa(x)+a₀(x))u²dx; v²_b,λ=

R³|∇v|²+(λb(x)+b₀(x))v²dx.