Blow-up solutions for two coupled Gross-Pitaevskii equations with attractive interactions
aa r X i v : . [ m a t h . A P ] A p r Blow-up solutions for two coupled Gross-Pitaevskiiequations with attractive interactions ∗ Yujin Guo a , Xiaoyu Zeng b and Huan-Song Zhou b † a Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences,P.O. Box 71010, Wuhan 430071, P. R. China b Department of Mathematics, Wuhan University of Technology, Wuhan 430070, China
Abstract
The paper is concerned with a system of two coupled time-independent Gross-Pitaevskii equations in R , which is used to model two-component Bose-Einsteincondensates with both attractive intraspecies and attractive interspecies interac-tions. This system is essentially an eigenvalue problem of a stationary nonlinearSchr¨odinger system in R , solutions of the problem are obtained by seeking mini-mizers of the associated variational functional with constrained mass (i.e. L − normconstaints). Under certain type of trapping potentials V i ( x ) ( i = 1 , V i ( x )) whenthe total interaction strength of intraspecies and interspecies goes to a critical value.An optimal blowing up rate for the solutions of the system is also given. Keywords:
Schr¨odinger equations; Gross-Pitaevskii equation; elliptic systems; constrainedminimization; blow up.
MSC:
In this paper, we study the following system of two coupled time-independent Gross-Pitaevskii equations: ( − ∆ u + V ( x ) u = µ u + b u + βu u in R , − ∆ u + V ( x ) u = µ u + b u + βu u in R . (1.1)The system (1.1) arises in describing two-component Bose-Einstein condensates (BEC),where ( V ( x ) , V ( x )) is a certain type of trapping potentials, ( µ , µ ) ∈ R × R is the ∗ Email: [email protected] ; [email protected] ; [email protected] † Corresponding author. b i ( i = 1 ,
2) and β are the interaction strength of cold atoms in-side each component (i.e. intraspecies) and between two components (i.e. interspecies),respectively. Here b i > β >
0) represents the intraspecies (or interspecies) in-teraction is attractive, otherwise, it is repulsive. Much attention has been paid to theexperimental studies of BECs since the BEC phenomena were successfully observed in1995 in the pioneering experiments [1, 12]. After that various BEC phenomena areobserved in BEC experiments such as the symmetry breaking, the collapse, the appear-ance of quantized vortices in rotating traps, the phase segregation, etc., which inspiredthe theoretical investigation on BECs, specially on the Gross-Pitaevskii equations–thefundamental model of describing the BEC. Multiple-component BECs can display moreinteresting phenomena absent in single-component BEC. In the past decade, under vari-ant conditions on V i ( x ), b i and β , the analogues of (1.1) have been investigated widely,see e.g. [3, 5, 9, 10, 11, 7, 18, 19, 24, 25, 29, 31, 33, 36, 40] and the references therein. Inthese mentioned papers, the authors are concerned with either the semiclassical statesof the system (i.e. replacing − ∆ by − ε ∆ in (1.1)) [25, 29, 19, 10, 31], or the problem(1.1) with V i ( x ) being constants [5, 9, 11, 27, 38], or the problem (1.1) in repulsive cases[3, 7, 18, 24, 33], etc. The results in these papers are mainly on studying the existence ofpositive solutions for small ε >
0, the location of the maximum point of the solutions andthe behavior of the solutions with respect to ε → L − normalized solutions of (1.1) with b i > b i > β > b i >
0) and the repulsive interspecies (i.e. β <
0) to the companion work [17]. Thecase of repulsive intraspecies and interspecies interactions was studied recently in [24, 33]and the references therein. When b i > β >
0, as mentioned in [33], (1.1) is cer-tainly a very different problem because the energy functional may not be bounded frombelow. In totally attractive case, we may expect from the single component BEC, seee.g., [12, 14, 16], that the collapse still happens if the particle number increases beyonda critical value. In addition to the intraspecies interaction among atoms in each compo-nent, there exist interspecies interactions among the components for multiple-componentBECs. Therefore, multiple-component BECs present more complicated characters thansingle component BEC, and the corresponding analytic investigations are more challeng-ing.It is well-known that the system (1.1) can be obtained from the associated timedependent nonlinear Schr¨odinger equations if one seeks for the following type standing-wave solutions( ψ ( x, t ) , ψ ( x, t )) = ( u ( x ) e − i µ t , u ( x ) e − i µ t ) , where i = − . (1.1) is essentially an eigenvalue problem of a system of two stationary nonlinear Schr¨odingerequations, which is also the system of Euler-Lagrange equations ( µ , µ are the Lagrangemultipliers) of the following constrained minimization problem,ˆ e ( b , b , β ) := inf { ( u ,u ) ∈M} E b ,b ,β ( u , u ) , (1.2)2here M is the so-called mass constraint M = n ( u , u ) ∈ X : Z R | u | dx = Z R | u | dx = 1 o (1.3)and X = H × H with H i = n u ∈ H ( R ) : Z R V i ( x ) | u ( x ) | dx < ∞ o , k u k H i = (cid:16) Z R h |∇ u | + V i ( x ) | u ( x ) | i dx (cid:17) , where i = 1 , . (1.4)The energy functional E b ,b ,β ( u , u ) is given by E b ,b ,β ( u , u ) = X i =1 Z R (cid:16) |∇ u i | + V i ( x ) | u i | − b i | u i | (cid:17) dx − β Z R | u | | u | dx , ( u , u ) ∈ X . (1.5)In this paper, we only interested in the solutions of (1.1) with constrained mass, thatis, the minimizers of (1.2). We mention that, different from the single component mini-mization problem (see, e.g. [15]), in the two-component case it seems not clear whethera minimizer of (1.2) is a ground state of (1.1). But we know that if ( u , u ) is a mini-mizer of (1.2), then ( u , u ) is a positive solution of (1.1) for some Lagrange multiplier( µ , µ ) ∈ R × R . In this paper, V i ( x ) are trapping potentials of the following type V i ( x ) ∈ L ∞ loc ( R ) , lim | x |→∞ V i ( x ) = ∞ and inf x ∈ R V i ( x ) = 0 , i = 1 , . (1.6)Throughout the paper, we assume that both inf x ∈ R (cid:0) V ( x )+ V ( x ) (cid:1) and inf x ∈ R V i ( x )are attained. Since problem (1.2) is invariant by adding suitable constants to V i ( x ) andwe may simply assume that inf x ∈ R V i ( x ) = 0. Moreover, in what follows we assume that b i > i = 1 ,
2) and β >
0, and denote the norm of L p ( R ) by k · k p for p ∈ (1 , ∞ ).We study problem (1.2) which is motivated by the recent works [4, 14, 15, 16], etc.,where the following single component minimization problem was investigated: e i ( a ) := inf { u ∈H i , k u k =1 } E ia ( u ) , a > , (1.7)and the energy functional E ia ( u ) satisfies E ia ( u ) := Z R (cid:0) |∇ u ( x ) | + V i ( x ) | u ( x ) | (cid:1) dx − a Z R | u ( x ) | dx, i = 1 or 2 . (1.8)Actually, it was proved in [4, 14] that (1.7) admits minimizers if and only if 0 < a − ∆ u + u − u = 0 in R , where u ∈ H ( R ) . (1.9)3n the other hand, the following Gagliardo-Nirenberg inequality was shown in [39] that Z R | u ( x ) | dx ≤ k Q k Z R |∇ u ( x ) | dx Z R | u ( x ) | dx, u ∈ H ( R ) , (1.10)where the identity is achieved at u ( x ) = Q ( | x | ). One can note from (1.9) that Q ( | x | )satisfies Z R |∇ Q | dx = Z R Q dx = 12 Z R Q dx, (1.11)see also Lemma 8.1.2 in [8]. Moreover, we have Q ( x ) , |∇ Q ( x ) | = O ( | x | − e −| x | ) as | x | → ∞ , (1.12)see [13, Proposition 4.1] for more details.We now introduce briefly the main results of the present paper. Our first result isconcerned with the following existence and nonexistence of minimizers for (1.2). Theorem 1.1.
Let Q be the unique positive radial solution of (1.9) and a ∗ = k Q k . If V i ( x ) satisfies (1.6) for i = 1 , . Then, (i). When < b < a ∗ , < b < a ∗ and β < p ( a ∗ − b )( a ∗ − b ) , problem (1.2) has atleast one minimizer. (ii). Either b > a ∗ or b > a ∗ or β > a ∗ − b + a ∗ − b , (1.2) has no minimizer. In order to prove Theorem 1.1, in Section 2 we introduce the following minimizationproblem O ( b , b , β ) := inf (cid:8) u i ∈ H ( R ) , k u i k = 1 , i =1 , (cid:9) R R ( |∇ u | + |∇ u | ) dx b R R | u | dx + b R R | u | dx + β R R | u | | u | dx . (1.13) We shall prove in Proposition 2.1 that if O ( b , b , β ) >
1, then there exists at least oneminimizer for (1.2). However, if O ( b , b , β ) <
1, then there is no minimizer for (1.2).Therefore, by Proposition 2.1, to establish Theorem 1.1 it suffices to evaluate O ( b , b , β ),depending on the range of ( b , b , β ). On the other hand, as shown in Lemma A.2 of theappendix, Theorem 1.1 can be proved alternatively by applying directly the Gagliardo-Nirenberg inequality (1.10) and some recaling techniques. We remark that some resultssimilar to Theorem 1.1 were also proved in [3, Theorem 2.6] by this kind of ideas.Theorem 1.1 gives a complete classification of the existence and nonexistence ofminimizers for (1.2), except that ( b , b , β ) satisfies0 < b ≤ a ∗ , < b ≤ a ∗ and β ∈ (cid:2)p ( a ∗ − b )( a ∗ − b ) , a ∗ − b a ∗ − b (cid:3) . (1.14)When ( b , b , β ) satisfies (1.14), in general it is difficult to evaluate O ( b , b , β ) so thatone cannot employ directly Proposition 2.1 to discuss the existence of minimizers for(1.2). Therefore, some new ideas are needed to address this case. Under some addi-tional assumptions on ( b , b , β ), in this paper we shall derive the following existenceand nonexistence of minimizers for (1.2). 4 heorem 1.2. Under condition (1.6) and let inf x ∈ R (cid:0) V ( x ) + V ( x ) (cid:1) be attained. If
Suppose V i ( x ) ( i = 1 , ) satisfies (1.6) and assume inf x ∈ R (cid:0) V ( x ) + V ( x ) (cid:1) is attained. If < b = b < a ∗ and β > satisfy (1.14). Then (i) Problem (1.2) has no minimizer if inf x ∈ R (cid:0) V ( x ) + V ( x ) (cid:1) = 0 . (ii) Problem (1.2) has at least one minimizer if ˆ e ( a ∗ − β, a ∗ − β, β ) < inf x ∈ R (cid:0) V ( x ) + V ( x ) (cid:1) . We mention that, for b = b ∈ (0 , a ∗ ), condition (1.14) implies β = a ∗ − b = a ∗ − b ,then the point ( b , b , β ) must be located on the segment ( a ∗ − β, a ∗ − β, β ) with β ∈ (0 , a ∗ ).In view of Theorems 1.1 to 1.3, the existence and nonexistence of minimizers for (1.2)still remain open for the case where 0 < b = b ≤ a ∗ and β > a ∗ − b − b frombelow. By applying Proposition 2.1, we shall derive Theorem 1.2 through establishingthe estimate O ( b , b , β ) > < b = b < a ∗ , it follows from (2.2) that O ( a ∗ − β, a ∗ − β, β ) = 1, and weshall give the proof of Theorem 1.3 (ii) by applying Ekeland’s variational principle [35,Theorem 5.1].The proof of Lemma A.2 in the appendix implies that the following relationshipalways holds 0 ≤ ˆ e ( a ∗ − β, a ∗ − β, β ) ≤ inf x ∈ R (cid:0) V ( x ) + V ( x ) (cid:1) . (1.16)In particular, the second inequality in (1.16) can be strict for certain potentials V ( x )and V ( x ). Here is an example: Example 1.1.
For any given points x and x in R satisfying | x − x | >
4, considerthe function 0 ≤ ζ i ( x ) ∈ C ( B ( x i )) satisfying k ζ i k = 1, where i = 1 ,
2, and define apositive constant C ζ by C ζ := X i =1 Z R (cid:16) |∇ ζ i | − a ∗ − β | ζ i | (cid:17) dx − β Z R | ζ | | ζ | dx < ∞ . Let 0 ≤ V i ( x ) ∈ C ( R ; R ) satisfy V i ( x ) = 0 in B ( x i ) and V i ( x ) ≥ C ζ in B c ( x i ) as wellas lim | x |→∞ V i ( x ) = ∞ , where i = 1 ,
2. One can check thatˆ e ( a ∗ − β, a ∗ − β, β ) ≤ E a ∗ − β,a ∗ − β,β ( ζ , ζ ) = C ζ < C ζ ≤ inf x ∈ R (cid:0) V ( x ) + V ( x ) (cid:1) . < β < a ∗ , thereexists at least one minimizer of (1.2) at ( b , b ) = ( a ∗ − β, a ∗ − β ) for some suitablepotentials V ( x ) and V ( x ). On the other hand, Theorem 1.3 (i) gives the non-existenceof minimizers for (1.2) at ( b , b ) = ( a ∗ − β, a ∗ − β ) and 0 < β < a ∗ for the case whereinf x ∈ R (cid:0) V ( x ) + V ( x ) (cid:1) = 0 is attained.Without loss of generality, in the follows one may restrict the minimization of (1.2)to non-negative vector functions ( u , u ), since E b ,b ,β ( u , u ) ≥ E b ,b ,β ( | u | , | u | ) forany ( u , u ) ∈ X , due to the fact that ∇| u i | ≤ |∇ u i | a.e. in R ( i = 1 , a > a ր a ∗ under some classes of trappingpotentials. We shall prove in Section 4 that a similar uniqueness result also holds for(1.2) if | ( b , b , β ) | is suitably small. Theorem 1.4. If V i ( x ) satisfies (1.6) for i = 1 and , then (1.2) admits a uniquenon-negative minimizer if | ( b , b , β ) | is suitably small. A similar result on uniqueness for the single component problem (1.7) was provedin [2, 28] by using the contracting map. However, this method seems not work forour problem. In this paper, we prove Theorem 1.4 by employing an implicit functiontheorem.For any fixed 0 < β < a ∗ , we finally focus on the limit behavior of the minimizersfor (1.2) as ( b , b ) ր ( a ∗ − β, a ∗ − β ), i.e., ( b + β, b + β ) ր ( a ∗ , a ∗ ) in the case thatthere is no minimizer for (1.2) at the threshold ( b , b ) = ( a ∗ − β, a ∗ − β ). In view ofTheorem 1.3 (i), we shall consider the special case where inf x ∈ R (cid:0) V ( x ) + V ( x ) (cid:1) = 0, i.e., the minima of V ( x ) coincide with those of V ( x ). More precisely, we assume that for V i ( x ) ( i = 1 ,
2) takes the form of V i ( x ) = h i ( x ) n i Y j =1 | x − x ij | p ij with C < h i ( x ) < /C in R , and h i ( x ) ∈ C α loc ( R ) for some α ∈ (0 , , (1.17)where n i ∈ N , p ij > x ik = x ij for k = j , and lim x → x ij h i ( x ) exists for all 1 ≤ j ≤ n i .Without loss of generality, we also assume that there exists 1 ≤ l ≤ min { n , n } suchthat x j = x j , where j = 1 , . . . , l ; x j = x j , where j i ∈ (cid:8) l + 1 , . . . n i (cid:9) and i = 1 , . (1.18)Note that (1.18) impliesΛ := (cid:8) x ∈ R : V ( x ) = V ( x ) = 0 (cid:9) = (cid:8) x , x , · · · , x l (cid:9) . (1.19)Define¯ p j := min (cid:8) p j , p j (cid:9) , j = 1 , . . . , l ; p := max ≤ j ≤ l min (cid:8) p j , p j (cid:9) = max ≤ j ≤ l ¯ p j , (1.20)6o that ¯Λ := (cid:8) x j : ¯ p j = p , j = 1 , . . . , l (cid:9) ⊂ Λ . (1.21)Let γ j ∈ (0 , ∞ ] be given by γ j := lim x → x j V ( x ) + V ( x ) | x − x j | p , ≤ j ≤ l. (1.22)Note that γ j < ∞ if and only if x j ∈ ¯Λ, where 1 ≤ j ≤ l . Finally, define γ =min (cid:8) γ , . . . , γ l (cid:9) so that the set Z := (cid:8) x j : γ j = γ, ≤ j ≤ l (cid:9) ⊂ ¯Λ (1.23)denotes the locations of the flattest global minima of V ( x ) + V ( x ). Using above nota-tions, our main results can be stated as follows. Theorem 1.5.
Assume that < β < a ∗ and V i ( x ) satisfies (1.17)-(1.18) for i = 1 , .Let ( u b , u b ) be a non-negative minimizer of (1.2) as ( b , b ) ր ( a ∗ − β, a ∗ − β ) . Then,for any sequence { ( b k , b k ) } satisfying ( b k , b k ) ր ( a ∗ − β, a ∗ − β ) as k → ∞ , thereexists a subsequence of { ( b k , b k ) } , still denoted by { ( b k , b k ) } , such that, for i = 1 , ,each u b ik has a unique global maximum point x ik k → ¯ x for some ¯ x ∈ Z and lim k →∞ | x ik − ¯ x | (cid:0) a ∗ − b k + b k +2 β (cid:1) p = 0 . (1.24) Moreover, for i = 1 , , lim k →∞ (cid:16) a ∗ − b k + b k + 2 β (cid:17) p u b ik (cid:16)(cid:0) a ∗ − b k + b k + 2 β (cid:1) p x + x ik (cid:17) = λ k Q k Q ( λx ) strongly in H ( R ) , where λ > is given by λ = (cid:16) p γ Z R | x | p Q ( x ) dx (cid:17) p (1.25) for p > and γ > defined in (1.20) and (1.23), respectively. Theorems 1.4 and 1.5 imply that the symmetry breaking occurs in the minimizers of(1.2). Actually, consider the trapping potentials V and V of the form V ( x ) = V ( x ) = l Y j =1 | x − x j | p , p > , where the points x j with j = 1 , · · · , l are arranged on the vortices of a regular polygon.Then there exist 0 < a ∗ ≤ a ∗∗ < a ∗ such that for 0 < b i + β < a ∗ ( i = 1 and 2),the functional (1.2) has a unique non-negative minimizer by Theorem 1.4, which hasthe same symmetry as that of V ( x ) = V ( x ). However, when a ∗∗ < b i + β < a ∗ ( i = 1 , l different non-negative minimizers, and both components of the minimizers concentrate at a zero point7f V ( x ) = V ( x ), which imply the symmetry breaking. We note that the symmetrybreaking bifurcation of ground states for single nonlinear Schr¨odinger/Gross-Pitaevskiiequations has been studied in detail in the literature, see, e.g., [20, 22, 23].This paper is organized as follows: in Section 2 we first derive the crucial Proposition2.1 on the auxiliary minimization problem O ( b , b , β ), based on which we then completethe proof of Theorem 1.1. In Section 3 we focus on the proof of Theorems 1.2 and 1.3.More exactly, we first use Proposition 2.1 to prove Theorem 1.2, and Theorem 1.3 is thenproved by applying Ekeland’s variational principle. Theorem 1.4 is then proved in Section4 to address the uniqueness of nonnegative minimizers for (1.2) as | ( b , b , β ) | is suitablysmall. In Section 5 we shall establish Proposition 5.1 on optimal energy estimates ofminimizers, upon which we finally complete in Section 6 the proof of Theorem 1.5. Inthe appendix, we give an alternative proof of Theorem 1.1 and prove a lemma as wellwhich is used in Section 2. In this section, we address the proof of Theorem 1.1 on the existence of minimizers. Westart with introducing the following auxiliary minimization problem O ( b , b , β ) := inf (cid:8) u i ∈ H ( R ) , k u i k = 1 , i =1 , (cid:9) R R ( |∇ u | + |∇ u | ) dx b R R | u | dx + b R R | u | dx + β R R | u | | u | dx . (2.1) By analyzing (2.1), our first aim is to derive the following proposition, which gives acriteria on the existence of minimizers for (1.2) based on the value of O ( b , b , β ). Proposition 2.1.
Suppose that (1.6) holds. Let b , b and β be positive. Then (i) (1.2) has at least one minimizer if O ( b , b , β ) > . (ii) (1.2) has no minimizer if O ( b , b , β ) < . To establish Proposition 2.1, we need the following compactness lemma.
Lemma 2.1.
Suppose V i ∈ L ∞ loc ( R ) satisfies lim | x |→∞ V i ( x ) = ∞ , where i = 1 , . Thenthe embedding X = H × H ֒ → L q ( R ) × L q ( R ) is compact for all ≤ q < ∞ . Since Lemma 2.1 can be proved in a similar way to that of [32, Theorem XIII.67] or[6, Theorem 2.1], we omit the proof.
Lemma 2.2.
Let O ( · ) be defined by (2.1), then O ( · ) is locally Lipschitz continuous in R . Proof.
We first prove that a ∗ max { b + β, b + β } ≤ O ( b , b , β ) ≤ a ∗ b + b + 2 β . (2.2)Indeed, the upper bound of (2.2) follows directly by taking ( Q ( x ) k Q k , Q ( x ) k Q k ) as a trial functionof (2.1). On the other hand, for any ( u , u ) ∈ H ( R ) × H ( R ) satisfying R R | u i | dx =8 , i = 1 , , it then follows from the Gagliardo-Nirenberg inequality (1.10) that R R ( |∇ u | + |∇ u | ) dx b R R | u | dx + b R R | u | dx + β R R | u | | u | dx ≥ a ∗ R R ( | u | + | u | ) dx b + β R R | u | dx + b + β R R | u | dx ≥ a ∗ max { b + β, b + β } , which then gives the lower bound of (2.2). Therefore, (2.2) is proved.Consider ( b , b , β ) , (˜ b , ˜ b , ˜ β ) ∈ R , and let { ( u n , u n ) } be a minimizing sequence of O ( b , b , β ). Since (2.1) is invariant under the rescaling: u ( x ) λu ( λx ) , λ >
0, one mayassume that Z R ( |∇ u n | + |∇ u n | ) dx = 1 for all n ∈ N + . (2.3)We then obtain that Z R | u in | dx ≤ a ∗ Z R |∇ u in | ≤ a ∗ , i = 1 , , and Z R | u n | | u n | dx ≤ Z R ( | u n | + | u n | ) dx ≤ a ∗ Z R (cid:0) |∇ u n | + |∇ u n | (cid:1) dx = 1 a ∗ . Furthermore, we have1 O ( b , b , β ) = lim n →∞ h ˜ b R R | u n | dx + ˜ b R R | u n | dx + ˜ β R R | u n | | u n | dx R R ( |∇ u n | + |∇ u n | ) dx + X i =1 b i − ˜ b i Z R | u in | dx + ( β − ˜ β ) Z R | u n | | u n | dx i ≤ lim n →∞ h O (˜ b , ˜ b , ˜ β ) + X i =1 | b i − ˜ b i | Z R | u in | dx + | β − ˜ β | Z R | u n | | u n | dx i ≤ O (˜ b , ˜ b , ˜ β ) + X i =1 | b i − ˜ b i | a ∗ + | β − ˜ β | a ∗ , i.e., 1 O ( b , b , β ) − O (˜ b , ˜ b , ˜ β ) ≤ a ∗ (cid:12)(cid:12) ( b , b , β ) − (˜ b , ˜ b , ˜ β ) (cid:12)(cid:12) . (2.4)Similarly, taking { (˜ u n , ˜ u n ) } as a minimizing sequence of O (˜ b , ˜ b , ˜ β ), and repeating theabove argument, we know that1 O (˜ b , ˜ b , ˜ β ) − O ( b , b , β ) ≤ a ∗ (cid:12)(cid:12) ( b , b , β ) − (˜ b , ˜ b , ˜ β ) (cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) O ( b , b , β ) − O (˜ b , ˜ b , ˜ β ) O ( b , b , β ) O (˜ b , ˜ b , ˜ β ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) O ( b , b , β ) − O (˜ b , ˜ b , ˜ β ) (cid:12)(cid:12)(cid:12) ≤ a ∗ (cid:12)(cid:12) ( b , b , β ) − (˜ b , ˜ b , ˜ β ) (cid:12)(cid:12) . By applying (2.2), we therefore conclude that (cid:12)(cid:12) O ( b , b , β ) − O (˜ b , ˜ b , ˜ β ) (cid:12)(cid:12) ≤ a ∗ ( b + b + 2 β )(˜ b + ˜ b + 2 ˜ β ) (cid:12)(cid:12) ( b , b , β ) − (˜ b , ˜ b , ˜ β ) (cid:12)(cid:12) , which implies that O ( · ) is locally Lipschitz continuous in R .With Lemma 2.2, we now prove Proposition 2.1. Proof of Proposition 2.1. i) Let { ( u n , u n ) } ⊂ X be a minimizing sequence ofproblem (1.2), i.e. , k u n k = k u n k = 1 and lim n →∞ E b ,b ,β ( u n , u n ) = ˆ e ( b , b , β ) . It then follows from (2.1) that E b ,b ,β ( u n , u n ) ≥ X i =1 Z R h(cid:0) − O ( b , b , β ) (cid:1) |∇ u in | + V i ( x ) | u in | i dx. (2.5)If O ( b , b , β ) >
1, (2.5) implies that { ( u n , u n ) } ⊂ X is bounded in n . Thus, by thecompactness of Lemma 2.1, there exist a subsequence of { ( u n , u n ) } and ( u , u ) ∈ X such that( u n , u n ) n ⇀ ( u , u ) weakly in X , ( u n , u n ) n → ( u , u ) strongly in L q ( R ) × L q ( R ) for all q ∈ [2 , ∞ ) . Therefore, k u k = k u k = 1 and E b ,b ,β ( u , u ) = ˆ e ( b , b , β ). This proves part one ofthe proposition.ii) Suppose now that O ( b , b , β ) <
1. One then may choose ( u , u ) ∈ M such thateach u i ( i = 1 ,
2) has compact support in R and satisfies R R ( |∇ u | + |∇ u | ) dx b R R | u | dx + b R R | u | dx + β R R | u | | u | dx ≤ δ := 1 + O ( b , b , β )2 < . (2.6)For λ >
0, define ¯ u i ( x ) = λu i ( λx ) , i = 1 , , (2.7)so that (¯ u , ¯ u ) ∈ M . Since u i ( x ) is compactly supported in R and V i ( x ) ∈ L ∞ loc ( R ),there exists a positive constant C , independent of λ >
0, such that for λ → ∞ , Z R V i ( x ) | ¯ u i | dx = Z R V i ( xλ ) | u i | dx ≤ C < ∞ , i = 1 , . (2.8)10n the other hand, by (2.6) and (2.7), we have X i =1 Z R (cid:16) |∇ ¯ u i | − b i | ¯ u i | (cid:17) dx − β Z R | ¯ u | | ¯ u | dx = λ X i =1 Z R (cid:16) |∇ u i | − b i | u i | (cid:17) dx − βλ Z R | u | | u | dx ≤ λ ( δ − (cid:16) b Z R | u | dx + b Z R | u | dx + β Z R | u | | u | dx (cid:17) → − ∞ as λ → ∞ . (2.9)It then follows from (2.8) and (2.9) thatˆ e ( b , b , β ) ≤ E b ,b ,β (¯ u , ¯ u ) → −∞ as λ → ∞ , which implies that ˆ e ( b , b , β ) does not admit any minimizer. Proposition 2.1 is thereforeestablished.We end this section by proving Theorem 1.1. Proof of Theorem 1.1. (i):
For any ( u , u ) ∈ H ( R ) × H ( R ) with R R | u i | dx =1 , i = 1 ,
2, it follows from (1.10) that O ( b , b , β ) ≥ inf {k u i k =1 ,i =1 , } a ∗ R R ( | u | + | u | ) dx b R R | u | dx + b R R | u | dx + β (cid:0) R R | u | dx R R | u | dx (cid:1) = inf {k u i k =1 ,i =1 , } a ∗ (cid:16) R R | u | dx (cid:14) R R | u | dx (cid:17) b + b R R | u | dx (cid:14) R R | u | dx + β (cid:0) R R | u | dx (cid:14) R R | u | dx (cid:1) . Setting t := (cid:0) Z R | u | dx (cid:14) Z R | u | dx (cid:1) ∈ (0 , ∞ ) and f b ,b ,β ( t ) := a ∗ (1 + t ) b + b t + βt , (2.10)and then O ( b , b , β ) ≥ inf t ∈ (0 , ∞ ) f b ,b ,β ( t ) . (2.11)Since 0 < b i < a ∗ ( i = 1 ,
2) and β < p ( a ∗ − b )( a ∗ − b ), standard calculations showthat f b ,b ,β ( t ) > t ∈ (0 , ∞ ), and alsolim t → + f b ,b ,β ( t ) = a ∗ b > t →∞ f b ,b ,β ( t ) = a ∗ b > . So, by the continuity of f b ,b ,β ( t ) we obtain that inf t ∈ (0 , ∞ ) f b ,b ,β ( t ) >
1. This estimateand (2.11) then imply that O ( b , b , β ) >
1, from which we conclude that (1.2) has atleast one minimizer by Proposition 2.1(i). 11 ii):
Consider a function 0 ≤ ϕ ∈ C ∞ ( R ) satisfying R R | ϕ | dx = 1, and set u λ ( x ) = λ k Q k Q ( λx ) , λ > , (2.12)where Q ( x ) is the unique radial positive solution of the scalar field equation (1.9). Itthen follows from (1.11) that Z R |∇ u λ | dx = λ k Q k Z R |∇ Q | dx = λ , as well as Z R | u λ | dx = λ k Q k Z R | Q | dx = 2 λ a ∗ . Suppose now that b > a ∗ . We then take ( u λ , ϕ ) as a trial function of O so that O ( b , b , β ) ≤ R R (cid:0) |∇ u λ | + |∇ ϕ | (cid:1) dx b R R | u λ | dx + b R R | ϕ | dx + β R R | u λ | | ϕ | dx = λ + R R |∇ ϕ | dx b a ∗ λ + b R R | ϕ | dx + βa ∗ R R | Q | | ϕ ( xλ ) | dx ≤ λ + R R |∇ ϕ | dx b a ∗ λ → a ∗ b < λ → ∞ . Thus, O ( b , b , β ) < b > a ∗ , one can also obtain the nonexistence of minimizers for(1.2).Assume finally that β > a ∗ − b + a ∗ − b . In this case, take ( u λ , u λ ) as a trial functionof O , where u λ ≥ O ( b , b , β ) ≤ R R |∇ u λ | dx (cid:0) b + b + β (cid:1) R R | u λ | dx = a ∗ b + b + β < . Hence, it follows again from Proposition 2.1(ii) that (1.2) has no minimizer. This com-pletes the proof of Theorem 1.1.
As discussed in the Introduction, our Theorem 1.1 gives a complete classification of theexistence of minimizers for (1.2), except that ( b , b , β ) satisfies0 < b ≤ a ∗ , < b ≤ a ∗ and β ∈ (cid:2)p ( a ∗ − b )( a ∗ − b ) , a ∗ − b a ∗ − b (cid:3) . The aim of this section is to prove Theorems 1.2 and 1.3, which are concerned with theexistence of minimizers for (1.2) when ( b , b , β ) lies in the above range. It turn out thatsuch an existence depends on whether 0 < b = b ≤ a ∗ or not.Firstly, we shall make full use of Proposition 2.1 to derive the existence of minimizersfor (1.2) in the case where 0 < b = b < a ∗ and β is close to p ( a ∗ − b )( a ∗ − b ) fromabove. We start with the following lemma.12 emma 3.1. Let < b = b < a ∗ and β = p ( a ∗ − b )( a ∗ − b ) such that | b − b | ≤ β. If O (cid:0) b , b , p ( a ∗ − b )( a ∗ − b ) (cid:1) possesses a radially symmetric (about the origin) min-imizing sequence { ( u n , u n ) } ⊂ H r ( R ) × H r ( R ) satisfying < C ≤ Z R |∇ u in | dx ≤ C < ∞ , and < C ≤ Z R | u in | dx ≤ C < ∞ , i = 1 , , (3.1) and C , C are independent of n , then we have O (cid:0) b , b , p ( a ∗ − b )( a ∗ − b ) (cid:1) > . Proof.
Since { u in ⊂ H r ( R ) } ( i = 1 ,
2) is radially symmetric, by (3.1) and the com-pactness lemma of Strauss [34], we deduce that there exists u i ( x ) ∈ H r ( R ) satisfying u in n ⇀ u i weakly in H ( R ) , i = 1 , u in n → u i strongly in L p ( R ) , ∀ p ∈ (2 , ∞ ) , i = 1 , . (3.2)Also, the assumption (3.1) implies that Z R | u i | dx ≥ C > u i , i = 1 , . Since 0 < b = b < a ∗ , without loss of generality, we may assume that b < b . Thenthe assumption on β can be simplified as0 < b < b < a ∗ and b ≤ β + b , where β = p ( a ∗ − b )( a ∗ − b ) . (3.3)Applying (1.10) and (3.2), we have O ( b , b , β ) ≥ lim n →∞ a ∗ R R ( | u n | + | u n | ) dx b R R | u n | dx + b R R | u n | dx + β R R | u n | | u n | dx = a ∗ R R ( | u | + | u | ) dx b R R | u | dx + b R R | u | dx + β R R | u | | u | dx ≥ f b ,b ,β ( t ) , t := (cid:0) Z R | u | dx (cid:14) Z R | u | dx (cid:1) ∈ (0 , ∞ ) , (3.4)where f b ,b ,β ( t ) is defined in (2.10), and the equality of (3.4) holds if and only if u ( x ) = κu ( x ) for some κ > . (3.5)Moreover, since β = p ( a ∗ − b )( a ∗ − b ), it holds that f b ,b ,β ( t ) ≥ , ∀ t ∈ (0 , ∞ ) , (3.6)and f b ,b ,β ( t ) = 1 ⇔ t = t := r a ∗ − b a ∗ − b . (3.7)13e thus deduce from (3.4) and (3.6) that O (cid:0) b , b , p ( a ∗ − b )( a ∗ − b ) (cid:1) ≥ O (cid:0) b , b , p ( a ∗ − b )( a ∗ − b ) (cid:1) >
1. Otherwise, if O (cid:0) b , b , p ( a ∗ − b )( a ∗ − b ) (cid:1) = 1 , (3.8)then (3.5) holds for t = t , where t and t are given by (3.4) and (3.7), respectively.Thus, we have u ( x ) = κu ( x ) , where κ = t = r a ∗ − b a ∗ − b > . (3.9)Together with (3.2), this implies that O ( b , b , β ) ≥ R R ( |∇ u | + |∇ u | ) dx b R R | u | dx + b R R | u | dx + β R R | u | | u | dx = 1 + κ b + b κ + βκ · R R |∇ u | dx R R | u | dx . (3.10)On the other hand, define˜ u i ( x ) = 1 √ λ i u i ( x ) where λ i := Z R | u i | dx ≤ lim n →∞ Z R | u in | dx = 1 , (3.11)so that R R | ˜ u i | dx = 1, where i = 1 ,
2. Note also from (3.9) that λ = κλ ≤ . (3.12)Therefore, by the definition of O ( · ), we deduce from (3.9), (3.11) and (3.12) that O ( b , b , β ) ≤ R R ( |∇ ˜ u | + |∇ ˜ u | ) dx b R R | ˜ u | dx + b R R | ˜ u | dx + β R R | ˜ u | | ˜ u | dx = 2 λ b + b + β · R R |∇ u | dx R R | u | dx ≤ κb + b + β · R R |∇ u | dx R R | u | dx . It then follows from (3.10) that 1 + κ b + b κ + βκ ≤ κb + b + β , i.e., κ + κ b + b κ + βκ ≤ b + b + β , where κ = r a ∗ − b a ∗ − b > , which however contradicts Lemma A.1 in the Appendix. Thus (3.8) cannot occur, then O (cid:0) b , b , p ( a ∗ − b )( a ∗ − b ) (cid:1) > Proof of Theorem 1.2.
Let { ( u n , u n ) } be a minimizing sequence of O ( b , b , β ). Bythe Schwarz symmetrization of { ( u n , u n ) } , one may assume that u in ( x ) = u in ( | x | ) ≥ , i = 1 , . (3.13)14oreover, since the problem (2.1) is invariant under the rescaling: u ( x ) λu ( λx ) for λ >
0, one can also assume that Z R ( |∇ u n | + |∇ u n | ) dx = 1 for all n ∈ N + . (3.14)By the Gagliardo-Nirenberg inequality (1.10), we then obtain that R R | u in | dx ( i = 1 , i.e., < C ≤ b Z R | u n | dx + b Z R | u n | dx + β Z R | u n | | u n | dx ≤ C < ∞ . (3.15)Under the assumption (1.15), we claim that O (cid:0) b , b , p ( a ∗ − b )( a ∗ − b ) (cid:1) > . (3.16)We prove (3.16) by considering separately the following two cases. Case 1. If Z R | u in | dx → n → ∞ , where i = 1 or 2 . (3.17)Without loss of generality, we assume that Z R | u n | dx → n → ∞ . It then follows from (3.15) that Z R | u n | | u n | dx n → Z R | u n | dx ≥ C > . By the assumption 0 < b < a ∗ , we then have O ( b , b , β ) = lim n →∞ R R (cid:0) |∇ u n | + |∇ u n | (cid:1) dx b R R | u n | dx + o (1) ≥ lim n →∞ R R |∇ u n | dx b R R | u n | dx + o (1) ≥ a ∗ b > , (3.18)where (1.10) is used. Thus, (3.16) follows immediately from (3.18) with β = p ( a ∗ − b )( a ∗ − b ). Case 2. If Z R | u in | dx ≥ C > i = 1 and 2 . (3.19)In this case, applying (1.10) and (3.14), we deduce that there exist positive constants C and C , independent of n , such that0
In order to prove part (i), we assume that inf x ∈ R (cid:0) V ( x ) + V ( x ) (cid:1) = 0, i.e., there exists x ∈ R such that V ( x ) = V ( x ) = 0. Since b = b = a ∗ − β , we take φ > φ, φ ) with ¯ x = x as a trial function ofˆ e ( a ∗ − β, a ∗ − β, β ). It then follows from (A.5) and (A.6) that0 ≤ ˆ e ( a ∗ − β, a ∗ − β, β ) ≤ lim τ →∞ E a ∗ ,a ∗ ( φ, φ ) = V ( x ) + V ( x ) = 0 , (3.21) i.e., ˆ e ( a ∗ − β, a ∗ − β, β ) = 0. Suppose now there exists a minimizer (ˆ u , ˆ u ) ∈ M forˆ e ( a ∗ − β, a ∗ − β, β ). As pointed out in the Introduction, we can assume (ˆ u , ˆ u ) to benon-negative. It then follows from (3.20) that ˆ u ≡ ˆ u ≥ R , and Z R |∇ ˆ u | dx = 12 Z R | ˆ u | dx and Z R V ( x ) | ˆ u | dx = 0 . u ( x ) is equal to (up to trans-lation) Q ( x ), but the second equality yields that ˆ u ( x ) has compact support. Therefore,the conclusion (i) of Theorem 1.3 is proved. (ii): In this case, we have0 ≤ ˆ e ( a ∗ − β, a ∗ − β, β ) < inf x ∈ R (cid:0) V ( x ) + V ( x ) (cid:1) . For M defined by (1.3), we introduce d ( ~u, ~v ) := k ~u − ~v k X , ~u, ~v ∈ M , where k ~u k X = (cid:0) k u k H + k u k H (cid:1) , ~u = ( u , u ) ∈ X . It is easy to check that ( M , d ) is a complete distance space. Hence, by Ekeland’s varia-tional principle [35, Theorem 5.1], there exists a minimizing sequence { ~u n = ( u n , u n ) } ⊂M of ˆ e ( a ∗ − β, a ∗ − β, β ) such thatˆ e ( a ∗ − β, a ∗ − β, β ) ≤ E a ∗ ,a ∗ ( ~u n ) ≤ ˆ e ( a ∗ − β, a ∗ − β, β ) + 1 n , (3.22) E a ∗ ,a ∗ ( ~v ) ≥ E a ∗ ,a ∗ ( ~u n ) − n k ~u n − ~v k X for ~v ∈ M . (3.23)Due to the compactness of Lemma 2.1, in order to show that there exists a minimizerfor ˆ e ( a ∗ − β, a ∗ − β, β ), it suffices to prove that { ~u n = ( u n , u n ) } is bounded in X uniformly w.r.t. n . We argue by contradiction. If { ~u n = ( u n , u n ) } is unbounded in X ,then there exists a subsequence of { ~u n } , still denoted by { ~u n } , such that k ~u n k X n −→ ∞ .By Gagliardo-Nirenberg inequality, we deduce from (3.22) that X i =1 Z R V i ( x ) | u in | dx ≤ E a ∗ ,a ∗ ( ~u n ) ≤ ˆ e ( a ∗ − β, a ∗ − β, β ) + 1 n . (3.24)Hence, Z R |∇ u n | + |∇ u n | dx n −→ ∞ . (3.25)We now claim that Z R |∇ u in | dx ∼ a ∗ Z R | u in | dx n −→ ∞ , i = 1 , , (3.26) Z R | u n | dx (cid:30) Z R | u n | dx n −→ . (3.27)Indeed, by (3.25), we may assume that R R |∇ u n | dx n −→ ∞ . Note from (3.22) that0 ≤ Z R |∇ u in | dx − a ∗ Z R | u in | dx ≤ ˆ e ( a ∗ − β, a ∗ − β, β ) + 1 n , for i = 1 , . (3.28)This implies that a ∗ Z R | u n | dx n −→ ∞ and Z R |∇ u n | dx (cid:30) a ∗ Z R | u n | dx n −→ . (3.29)17n the other hand, (3.22) also yields that β Z R (cid:0) | u n | − | u n | (cid:1) dx ≤ ˆ e ( a ∗ − β, a ∗ − β, β ) + 1 n . (3.30)Then, (cid:12)(cid:12)(cid:12)(cid:12) Z R | u a | dx − Z R | u a | dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:16) Z R (cid:0) | u a | − | u a | (cid:1) dx (cid:17) (cid:16) Z R (cid:0) | u a | + | u a | (cid:1) dx (cid:17) ≤ (cid:26) β (cid:0) ˆ e ( a ∗ − β, a ∗ − β, β ) + 1 n (cid:1)(cid:27) (cid:20)(cid:16) Z R | u a | dx (cid:17) + (cid:16) Z R | u a | dx (cid:17) (cid:21) Together with (3.29), we thus derive that Z R | u n | dx n −→ ∞ and Z R | u n | dx (cid:30) Z R | u n | dx n −→ . (3.31)This estimate and (3.28) then imply that Z R |∇ u n | dx n −→ ∞ and Z R |∇ u n | dx (cid:30) a ∗ Z R | u n | dx n −→ . (3.32)Therefore, (3.26) and (3.27) follow from (3.29), (3.31) and (3.32).Define now ǫ − n := Z R | u n | dx. Similar to Lemma 5.3 (i) in Section 5, there exists a sequence { y ǫ n } ⊂ R as well aspositive constants R and η such that w in ( x ) := ǫ n u in ( ǫ n x + ǫ n y ǫ n ) , i = 1 , , (3.33)satisfies Z B R (0) | w in | dx > η > , i = 1 , . (3.34)Recall from (3.30) that Z R (cid:0) | w n | − | w n | (cid:1) dx = ǫ n Z R (cid:0) | u n | − | u n | (cid:1) dx n −→ , which implies that w n − w n n −→ L ( R ) and w n − w n n −→ R . (3.35)Moreover, since lim | x |→∞ V i ( x ) = ∞ ( i = 1 , X i =1 Z R V i ( x ) | u in | dx = X i =1 Z R V i ( ǫ n x + ǫ n y ǫ n ) | w in | dx ≤ ˆ e ( a ∗ − β, a ∗ − β, β )+ 1 n . (3.36)We deduce from (3.34) and Fatou’s Lemma that { ǫ n y ǫ n } is bounded uniformly in R .18or any ϕ ( x ) ∈ C ∞ ( R ), define˜ ϕ ( x ) = ϕ (cid:0) x − ǫ n y ǫ n ǫ n (cid:1) , j ( τ, σ ) = 12 Z R (cid:12)(cid:12) u n + τ u n + σ ˜ ϕ (cid:12)(cid:12) dx, so that j ( τ, σ ) satisfies j (0 ,
0) = 12 , ∂j (0 , ∂τ = Z R | u n | dx = 1 and ∂j (0 , ∂σ = Z R u n ˜ ϕdx. Applying the implicit function theorem then gives that there exist a constant δ n > τ ( σ ) ∈ C (cid:0) ( − δ n , δ n ) , R (cid:1) such that τ (0) = 0 , τ ′ (0) = − Z R u n ˜ ϕdx, and j ( τ ( σ ) , σ ) = j (0 ,
0) = 12 . This implies that (cid:0) u n + τ ( σ ) u n + σ ˜ ϕ, u n (cid:1) ∈ M , where σ ∈ ( − δ n , δ n ) . We then obtain from (3.23) that E a ∗ ,a ∗ ( u n + τ ( σ ) u n + σ ˜ ϕ, u n ) − E a ∗ ,a ∗ ( u n , u n ) ≥ − n k ( τ ( σ ) u n + σ ˜ ϕ, k X . Setting σ → + and σ → − , respectively, we thus have (cid:12)(cid:12)(cid:12)(cid:10) E ′ a ∗ ,a ∗ ( u n , u n ) , ( τ ′ (0) u n + ˜ ϕ, (cid:11)(cid:12)(cid:12)(cid:12) ≤ n k τ ′ (0) u n + ˜ ϕ k H . (3.37)By the definition of (3.33), direct calculations yield that for n → ∞ ,12 (cid:10) E ′ a ∗ ,a ∗ ( u n , u n ) , ( ˜ ϕ, (cid:11) = 1 ǫ n Z R ∇ w n ∇ ϕdx + ǫ n Z R V ( ǫ n x + ǫ n y ǫ n ) w n ϕdx − a ∗ ǫ n Z R w n ϕdx + βǫ n Z R ( w n − w n ) w n ϕdx, (3.38)and τ ′ (0) = − Z R u n ˜ ϕdx = − ǫ n Z R w n ϕdx , k τ ′ (0) u n + ˜ ϕ k H < C,µ n := 12 h E ′ a ∗ ,a ∗ ( u n , u n ) , ( u n , i ∼ − a ∗ Z R | u n | dx = − a ∗ ǫ − n . (3.39)We then deduce from (3.37)-(3.39) that (cid:12)(cid:12)(cid:12)(cid:12) Z R ∇ w n ∇ ϕdx + ǫ n Z R V ( ǫ n x + ǫ n y ǫ n ) w n ϕdx − µ n ǫ n Z R w n ϕdx − a ∗ Z R w n ϕdx + β Z R ( w n − w n ) w n ϕdx (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cǫ n n . (3.40)19sing (3.34) and (3.35), we thus deduce from (3.40) that w n n ⇀ w in H ( R ), where w is a non-zero solution of − ∆ w + λ w − a ∗ w = 0 in R , (3.41)and λ := − lim n →∞ µ n ǫ n >
0. Similar to the above argument, one can also show that w n n ⇀ w in H ( R ), where w is also a non-zero solution of (3.41). Further, we derivefrom (3.35) that w ( x ) ≡ ± w ( x ) a.e. in R .We next show that k w i k = 1 , where i = 1 , . (3.42)Indeed, since k w in k ≡ w i
0, we have 0 < k w i k ≤
1. On the other hand,employing (3.41) and the Pohozaev identity ([8, Lemma 8.1.2]), we derive that Z R | w i | dx = 1 λ Z R |∇ w i | dx = a ∗ λ Z R | w i | dx, where i = 1 , . We then use the Gagliardo-Nirenberg inequality (1.10) to deduce that a ∗ ≤ R R | w i | dx R R |∇ w i | dx R R | w i | dx = a ∗ Z R | w i | dx, (3.43)which yields that k w i k ≥ i = 1 and 2, and therefore (3.42) follows.We now obtain from (3.42) that w in n −→ w i strongly in L ( R ) , i = 1 , . (3.44)Since { ǫ n y ǫ n } is bounded uniformly in R , there exists a subsequence, still denoted by { ǫ n } , of { ǫ n } such that ǫ n y ǫ n n −→ z ∈ R . Using Fatou’s lemma, we thus obtain from(3.36) and (3.44) thatˆ e ( a ∗ − β, a ∗ − β, β ) ≥ X i =1 Z R lim n →∞ V i ( ǫ n x + ǫ n y ǫ n ) | w in | dx = X i =1 Z R V i ( z ) | w i | dx = V ( z ) + V ( z ) , which however contradicts the assumption that ˆ e ( a ∗ − β, a ∗ − β, β ) < inf x ∈ R (cid:0) V ( x ) + V ( x ) (cid:1) , and the proof of Theorem 1.3(ii) is therefore done. The aim of this section is to prove Theorem 1.4 on the uniqueness of non-negativeminimizers for (1.2) by applying the implicit function theorem. Based on the contractingmap, a similar result for single GP energy functionals was also proved in [2, 28], howeverthis methods seems not applicable to our problem.Let λ i be the first eigenvalue of − ∆ + V i ( x ) in H i , i.e,λ i = inf n Z R (cid:0) |∇ u | + V i ( x ) (cid:17) u dx : u ∈ H i and Z R | u | dx = 1 o , i = 1 , . (4.1)20pplying Lemma 2.1 (see also [32]), one can deduce that λ i is simple and can be attainedby a positive normalized function φ i ∈ H i , which is called the first eigenfunction of − ∆ + V i ( x ) in H i , i = 1 ,
2. Define Z i = span { φ i } ⊥ = n u ∈ H i : Z R uφ i dx = 0 o , i = 1 , , so that H i = span { φ i } ⊕ Z i , i = 1 , . (4.2)We now recall the following properties. Proposition 4.1. [16, Lemma 4.2] If V i ( x ) satisfies (1.6) for i = 1 , , then (i) ker (cid:0) − ∆ + V i ( x ) − λ i (cid:1) = span { φ i } , (ii) φ i (cid:0) − ∆ + V i ( x ) − λ i (cid:1) Z i , (iii) Im (cid:0) − ∆ + V i ( x ) − λ i (cid:1) = (cid:0) − ∆ + V i ( x ) − λ i (cid:1) Z i is closed in H ∗ i , (iv) codim Im (cid:0) − ∆ + V i ( x ) − λ i (cid:1) =1,where H ∗ i denotes the dual space of H i for i = 1 , . Define now the C functional F i : X × R
7→ H ∗ i by F i ( u , u , µ i , b i , β ) = (cid:0) − ∆ + V i ( x ) − µ i (cid:1) u i − b i u i − βu j u i , i = 1 , , (4.3)where j = i and j = 1 ,
2. We then have the following lemma.
Lemma 4.1.
Let F i be defined by (4.3), where i = 1 , . Then there exist δ > anda unique function ( u i ( b , b , β ) , µ i ( b , b , β )) ∈ C (cid:0) B δ ( ~ B δ ( φ i , λ i ) (cid:1) , where i = 1 , ,such that µ i ( ~
0) = λ i , u i ( ~
0) = φ i , i = 1 , F i (cid:0) u ( b , b , β ) , u ( b , b , β ) , µ i ( b , b , β ) , b i , β (cid:1) = 0 , i = 1 , k u ( b , b , β ) k = k u ( b , b , β ) k = 1 . (4.4) Proof.
For i = 1 ,
2, define g i : ( Z × R ) × ( Z × R ) × R
7→ H ∗ i by g i (cid:0) ( z , µ ) , ( z , µ ) , s , s , b i , β (cid:1) := F i (cid:0) (1 + s ) φ + z , (1 + s ) φ + z , µ i , b i , β (cid:1) . Then g i ∈ C (( Z × R ) × ( Z × R ) × R ; H ∗ i ) and g i ((0 , λ ) , (0 , λ ) ,~
0) = F i ( φ , φ , λ i ,~
0) = 0 ,D s i g i ((0 , λ ) , (0 , λ ) ,~
0) = D u i F i ( φ , φ , λ i ,~ φ i = (cid:0) − ∆ + V i ( x ) − λ i (cid:1) φ i = 0 ,D s j g i ((0 , λ ) , (0 , λ ) ,~
0) = D u j F i ( φ , φ , λ i ,~ φ j = 0 , (4.5)21here j = i, i, j = 1 ,
2. Moreover, for any (ˆ z i , ˆ µ i ) ∈ Z i × R , we have (cid:10) D ( z i ,µ i ) g i ((0 , λ ) , (0 , λ ) ,~ , (ˆ z i , ˆ µ i ) (cid:11) = D u i F i ( φ , φ , λ i ,~ z i + D µ i F i ( φ , φ , λ i ,~ µ i = (cid:0) − ∆ + V ( x ) − λ i (cid:1) ˆ z i − ˆ µ i φ i ∈ H ∗ i , (4.6)and (cid:10) D ( z j ,µ j ) g i ((0 , λ ) , (0 , λ ) ,~ , (ˆ z j , ˆ µ j ) (cid:11) = D u j F i ( φ , φ , λ i ,~ z j + D µ j F i ( φ , φ , λ i ,~ µ j = 0 . (4.7)It then follows from (4.6) and Proposition 4.1 that for i = 1 , ,D ( z i ,µ i ) g i ((0 , λ ) , (0 , λ ) ,~
0) : Z i × R
7→ H ∗ i is an isomorphism. (4.8)We next define G : ( Z × R ) × ( Z × R ) × R
7→ H ∗ × H ∗ × R by G (cid:0) ( z , µ ) , ( z , µ ) , s , s , ( b , b , β ) (cid:1) := g (cid:0) ( z , µ ) , ( z , µ ) , s , s , b , β (cid:1) g (cid:0) ( z , µ ) , ( z , µ ) , s , s , b , β (cid:1) k (1 + s ) φ + z k − k (1 + s ) φ + z k − . Setting h i ( z i , s i ) = k (1 + s i ) φ i + z i k −
1, we then have D (( z ,µ ) , ( z ,µ ) ,s ,s ) G (cid:0) (0 , λ ) , (0 , λ ) , , ,~ (cid:1) = D ( z ,µ ) g D ( z ,µ ) g D s g D s g D ( z ,µ ) g D ( z ,µ ) g D s g D s g D ( z ,µ ) h D ( z ,µ ) h D s h D s h D ( z ,µ ) h D ( z ,µ ) h D s h D s h = ( − ∆ + V ( x ) − λ , − φ ) 0 0 00 ( − ∆ + V ( x ) − λ , − φ ) 0 00 0 2 00 0 0 2 . (4.9)We then derive from (4.8) that D (( z ,µ ) , ( z ,µ ) ,s ,s ) G (cid:0) (0 , λ ) , (0 , λ ) , , ,~ (cid:1) : ( Z × R ) × ( Z × R ) × R
7→ H ∗ × H ∗ × R is an isomorphism. Therefore, it follows from the implicit function theorem that there ex-ist δ > z i ( b , b , β ) , µ i ( b , b , β ) , s i ( b , b , β )) ∈ C ( B δ ( ~ B δ (0 , λ i , i = 1 ,
2, such that ( G (cid:0) ( z , µ ) , ( z , µ ) , s , s , ( b , b , β ) (cid:1) = G (cid:0) (0 , λ ) , (0 , λ ) , , ,~ (cid:1) = ~ ,z i ( ~
0) = 0 , µ i ( ~
0) = λ i , s i ( ~
0) = 0 , i = 1 , . (4.10)By setting u i ( b , b , β ) = (1 + s i ( b , b , β )) φ i + z i ( b , b , β ) , ( b , b , β ) ∈ B δ ( ~ , i = 1 , ,
22e then obtain from (4.10) that there exists a unique function ( u i ( b , b , β ) , µ i ( b , b , β )) ∈ C ( B δ ( ~ B δ ( φ i , λ i )), where i = 1 ,
2, such that u i ( ~
0) = (1 + s i ( ~ φ i + z i ( ~
0) = φ i , µ i ( ~
0) = λ i , (4.11)and F (cid:0) u ( b , b , β ) , u ( b , b , β ) , µ ( b , b , β ) , b , β (cid:1) F (cid:0) u ( b , b , β ) , u ( b , b , β ) , µ ( b , b , β ) , b , β (cid:1) k u ( b , b , β ) k − k u ( b , b , β ) k − = ~ , (4.12)and therefore (4.4) holds. This completes the proof of Lemma 4.1.In the following we use Lemma 4.1 to derive the uniqueness of nonnegative minimizersfor sufficiently small | ( b , b , β ) | . Proof of Theorem 1.4.
It follows from Theorem 1.1 that ˆ e ( b , b , β ) admits at leastone minimizer if 0 < b < a ∗ , 0 < b < a ∗ and β < p ( a ∗ − b )( a ∗ − b ). We first claimthat ˆ e ( · ) is a continuous function of ( b , b , β ) on the interval I := ( − a ∗ , a ∗ ) × ( − a ∗ , a ∗ ) × ( − a ∗ , a ∗ ). Indeed, for any ( b , b , β ) ∈ I , let ( u , u ) be any nonnegative minimizer ofˆ e ( b , b , β ). It then follows from (1.10) and Cauchy’s inequality thatˆ e ( b , b , β ) = E b ,b ,β ( u , u ) ≥ X i =1 (cid:0) a ∗ − | b i | − | β | (cid:1) Z R | u i | dx ≥ a ∗ X i =1 Z R | u i | dx, which implies thatthe L -norm of minimizers for ˆ e ( · ) is bounded uniformly on the interval I. (4.13)For any ( b i , b i , β i ) ∈ I , we now denote ( u i , u i ) to be the corresponding nonnegativeminimizer of ˆ e ( b i , b i , β i ), where i = 1 ,
2. Thenˆ e ( b , b , β ) = E b ,b ,β ( u , u )= E b ,b ,β ( u , u ) + b − b Z R | u | dx + b − b Z R | u | dx + ( β − β ) Z R | u | | u | dx ≥ ˆ e ( b , b , β ) + O ( | ( b , b , β ) − ( b , b , β ) | ) . (4.14)Similarly, we also haveˆ e ( b , b , β ) ≥ ˆ e ( b , b , β ) + O ( | ( b , b , β ) − ( b , b , β ) | ) . (4.15)We then derive from (4.14) and (4.15) thatlim ( b ,b ,β ) → ( b ,b ,β ) ˆ e ( b , b , β ) = ˆ e ( b , b , β ) , which implies that ˆ e ( · ) is continuous on the interval I , and the above claim is thereforeproved. 23et ( u b ,β , u b ,β ) be a non-negative minimizer of ˆ e ( b , b , β ) with ( b , b , β ) ∈ I . Wethen deduce that ( u b ,β , u b ,β ) satisfies the Euler-Lagrange system ( − ∆ + V ( x ) − µ b ,β ) u b ,β − b u b ,β − βu b ,β u b ,β = 0 in R , ( − ∆ + V ( x ) − µ b ,β ) u b ,β − b u b ,β − βu b ,β u b ,β = 0 in R , (4.16)i.e, F i ( u b ,β , u b ,β , µ b i ,β , b i , β ) = 0 , i = 1 , , (4.17)where ( µ b ,β , µ b ,β ) ∈ R is a Lagrange multiplier. We then derive from (4.13) and theabove claim that E , , ( u b ,β , u b ,β )= E b ,b ,β ( u b ,β , u b ,β ) + X i =1 b i Z R | u b i ,β | dx + β Z R | u b ,β | | u b ,β | dx =ˆ e ( b , b , β ) + O ( | ( b , b , β ) | ) → ˆ e (0 , ,
0) as ( b , b , β ) → . (4.18)On the other hand, one can check easily that ˆ e (0 , ,
0) = λ + λ , and ( φ , φ ) isthe unique non-negative minimizer of ˆ e (0 , , λ i , φ i ) is the first eigenpair of − ∆ + V i ( x ) in H i , i = 1 ,
2. We then deduce from Lemma 2.1 that u b i ,β → φ i in H i as ( b , b , β ) → (0 , , , i = 1 , . (4.19)Moreover, by (4.16) and (4.19) we have µ b i ,β = Z R |∇ u b i ,β | + V i ( x ) | u b i ,β | dx − b i Z R | u b i ,β | dx − β Z R | u b ,β | | u b ,β | dx → λ i as ( b , b , β ) → (0 , , , i = 1 , . (4.20)Applying (4.19) and (4.20), we then derive that there exists a small constant δ > k u b i ,β − φ i k H i < δ and | µ b i ,β − λ i | < δ if ( b , b , β ) ∈ B δ ( ~ , i = 1 , . (4.21)We thus conclude from (4.17) and Lemma 4.1 that µ b i ,β = µ i ( b , b , β ) , u b i ,β = u i ( b , b , β ) , if ( b , b , β ) ∈ B δ ( ~ , i = 1 , . This therefore implies that for sufficiently small | ( b , b , β ) | , ( u ( b , b , β ) , u ( b , b , β ))is a unique non-negative minimizer of ˆ e ( b , b , β ), and we are done. To simplify the notations and the proof, we denote a i = b i + β > < β < a ∗ , where i = 1 ,
2. It is then equivalent to rewriting the functional (1.5) as E a ,a ( u , u ) := X i =1 Z R (cid:16) |∇ u i | + V i ( x ) | u i | − a i | u i | (cid:17) dx + β Z R (cid:0) | u | − | u | (cid:1) dx , where ( u , u ) ∈ X . (5.1)24lso, the minimization problem (1.2) is then equivalent to the following one: e ( a , a ) := inf { ( u ,u ) ∈M} E a ,a ( u , u ) , (5.2)where M is defined by (1.3).To prove Theorem 1.5, we first need to establish the following crucial optimal energyestimates as ( a , a ) ր ( a ∗ , a ∗ ). Proposition 5.1.
Suppose < β < a ∗ and V i ( x ) satisfies (1.17) and (1.18), where i = 1 , . Then there exist two positive constants C and C , independent of a and a ,such that C (cid:16) a ∗ − a + a (cid:17) p p ≤ e ( a , a ) ≤ C (cid:16) a ∗ − a + a (cid:17) p p as ( a , a ) ր ( a ∗ , a ∗ ) , (5.3) where p > is defined by (1.20), and e ( a , a ) is defined by (5.2). Moreover, if ( u a , u a ) is a non-negative minimizer of e ( a , a ) , then there exist two positive constants C and C , independent of a and a , such that C (cid:16) a ∗ − a + a (cid:17) − p ≤ Z R | u a i | dx ≤ C (cid:16) a ∗ − a + a (cid:17) − p (5.4) as ( a , a ) ր ( a ∗ , a ∗ ) . We remark that even though the upper bound of (5.3) can be proved similarly tothat of Lemma 3 in [14], the arguments of [14] do not give the lower bound of (5.3).For this reason, as discussed below, we need employ a little more involved analysis toaddress the optimal lower bound of (5.3).In what follows, we focus on the proof of Proposition 5.1. For any fixed 0 < β < a ∗ ,denote ( u a , u a ) to be a non-negative minimizer of (5.2). We start with the followingenergy estimates of e ( a , a ). Lemma 5.1.
Under the assumptions of Proposition 5.1, there exists a constant
C > ,independent of a and a , such that e ( a ) + e ( a ) + β Z R (cid:0) | u a | − | u a | (cid:1) dx ≤ e ( a , a ) ≤ C (cid:16) a ∗ − a + a (cid:17) p p (5.5) as ( a , a ) ր ( a ∗ , a ∗ ) , where e i ( · ) is given by (1.7) for i = 1 , . Proof.
Since ( u a , u a ) is a non-negative minimizer of (5.2), we note from (1.8) that E a ,a ( u, v ) = E a ( u ) + E a ( v ) + β Z R (cid:0) | u | − | v | (cid:1) dx for all ( u, v ) ∈ X , where E ia i ( · ) is defined by (1.8) for i = 1 and 2. This relation then implies that e ( a , a ) ≥ e ( a ) + e ( a ) + β Z R (cid:0) | u a | − | u a | ) dx, which gives the lower bound of (5.5). 25timulated by Lemma 3 in [14], we next prove the upper bound of (5.5) as follows.Without loss of generality, we may assume p = ¯ p = min { p , p } > p ≤ p ,where p and ¯ p i are defined by (1.20). We proceed similarly to the proof of Lemma A2in the Appendix, and use the trial function ( φ, φ ) for φ satisfying (A.3) with ¯ x = x .Choose R > V i ( x ) ≤ C | x − x | p i for | x − x | ≤ R, i = 1 , . By the exponential decay of Q ( x ), we have Z R V i ( x ) φ ( x ) dx ≤ Cτ − p i Z R | x | p i Q ( x ) dx ≤ Cτ − p i as τ → ∞ , i = 1 , . This inequality and (A.5) then imply that E a ,a ( φ, φ ) ≤ a ∗ (cid:16) a ∗ − a + a (cid:17) τ + C (cid:0) τ − p + τ − p (cid:1) . Setting τ = (cid:16) a ∗ − a + a (cid:17) − p and using p = p ≤ p , we derive that e ( a , a ) ≤ E a ,a ( φ, φ ) ≤ C (cid:16) a ∗ − a + a (cid:17) p p , which therefore gives the upper bound of (5.5).By Lemma 5.1, for i = 1 ,
2, we have e i ( a i ) ≤ E ia i ( u a i ) ≤ C (cid:16) a ∗ − a + a (cid:17) p p (5.6)as ( a , a ) ր ( a ∗ , a ∗ ), where e i ( · ) is defined by (1.7). On the other hand, it is proved in[14, Lemma 3] that for p i := max { p ij , j = 1 , . . . , n i } > , i = 1 , , (5.7)there exists two positive constants m and M , independent of a and a , such that m ( a ∗ − a i ) pipi +2 ≤ e i ( a i ) ≤ M ( a ∗ − a i ) pipi +2 for 0 ≤ a i ≤ a ∗ , and i = 1 , . (5.8)Applying (5.8) and Lemma 5.1, we next derive the following L ( R ) − estimates of mini-mizers. Lemma 5.2.
Under the assumptions of Proposition 5.1, we have C (cid:16) a ∗ − a + a (cid:17) − p p pi ≤ Z R | u a i | dx ≤ C (cid:16) a ∗ − a + a (cid:17) − p as ( a , a ) ր ( a ∗ , a ∗ ) , (5.9) and lim ( a ,a ) ր ( a ∗ ,a ∗ ) R R | u a | dx R R | u a | dx = 1 , (5.10) where p i ≥ p , and p i > is given by (5.7) for i = 1 and . roof. We first prove the lower bound of (5.9). Pick any 0 < b < a i < a ∗ ( i = 1 , e i ( b ) ≤ E ia i ( u a i ) + a i − b Z R | u a i | dx, i = 1 , . It then follows from (5.6) and (5.8) that12 Z R | u a i | dx ≥ e i ( b ) − C (cid:0) a ∗ − a + a (cid:1) p p a i − b ≥ m ( a ∗ − b ) pipi +2 − C (cid:0) a ∗ − a + a (cid:1) p p a i − b . (5.11)Take b = a i − C (cid:0) a ∗ − a + a (cid:1) p pi +2) pi ( p , where C > mC p pi +2) pi ( p > C .We then derive from (5.11) that Z R | u a i | dx ≥ C (cid:0) a ∗ − a + a (cid:1) − p p pi , i = 1 , , (5.12)which therefore implies the lower bound of (5.9).We next prove the upper bound of (5.9). One can note from (5.6) that for i = 1 , E a i ( u a i ) ≤ C (cid:16) a ∗ − a + a (cid:17) p p as ( a , a ) ր ( a ∗ , a ∗ ) . (5.13)Without loss of generality, we may assume that a ≤ a ≤ a ∗ and ( a , a ) = ( a ∗ , a ∗ ).By (1.10), we then have E a ( u a ) ≥ a ∗ − a Z R | u a | dx ≥ (cid:16) a ∗ − a + a (cid:17) Z R | u a | dx. It thus follows from (5.13) that the upper bound of (5.9) holds for u a . Similarly, theupper bound of (5.9) holds also for u a if (5.10) is true, and then the proof is done.Now we come to prove (5.10). Recall from Lemma 5.1 that Z R (cid:0) | u a | − | u a | (cid:1) dx ≤ C (cid:16) a ∗ − a + a (cid:17) p p as ( a , a ) ր ( a ∗ , a ∗ ) , which implies that (cid:12)(cid:12)(cid:12)(cid:12) Z R | u a | dx − Z R | u a | dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:16) Z R (cid:0) | u a | − | u a | (cid:1) dx (cid:17) (cid:16) Z R (cid:0) | u a | + | u a | (cid:1) dx (cid:17) ≤ C (cid:16) a ∗ − a + a (cid:17) p p (cid:20)(cid:16) Z R | u a | dx (cid:17) + (cid:16) Z R | u a | dx (cid:17) (cid:21) (5.14)as ( a , a ) ր ( a ∗ , a ∗ ). Since it follows from (5.12) that R R | u a i | dx → ∞ as ( a , a ) ր ( a ∗ , a ∗ ), where i = 1 ,
2, we conclude (5.10) from the above estimate.27e next claim that the upper estimates of (5.5) and (5.9) are optimal. By Lemma5.1, we see that X i =1 Z R V i ( x ) | u a i ( x ) | dx ≤ e ( a , a ) ≤ C (cid:0) a ∗ − a + a (cid:1) p p as ( a , a ) ր ( a ∗ , a ∗ ) . (5.15)Set ǫ − ( a , a ) := Z R | u a ( x ) | dx, where ǫ ( a , a ) > . (5.16)It then yields from (5.9) that ǫ ( a , a ) ց a , a ) ր ( a ∗ , a ∗ ). Moreover, we deducefrom (5.5) and (5.10) that for i = 1 , Z R | u a i | dx, Z R |∇ u a i | dx ∼ ǫ − ( a , a ) as ( a , a ) ր ( a ∗ , a ∗ ) , (5.17)where f ∼ g means that f /g is bounded from below and above. In view of above facts,we next define the L ( R )-normalized function˜ w a i ( x ) := ǫ ( a , a ) u a i (cid:0) ǫ ( a , a ) x (cid:1) , i = 1 , . (5.18)It then follows from (5.10), (5.16) and (5.17) that for i = 1 , Z R | ˜ w a i | dx = 1 and m ≤ Z R |∇ ˜ w a i | dx ≤ m as ( a , a ) ր ( a ∗ , a ∗ ) , (5.19)where m > a and a . In the rest part of this section, for conveniencewe use ǫ > ǫ ( a , a ) so that ǫ ց a , a ) ր ( a ∗ , a ∗ ). Lemma 5.3.
Under the assumptions of Proposition 5.1, we have (i).
There exist a sequence { y ǫ } as well as positive constants R and η i such that for i = 1 , , the function w a i ( x ) = ˜ w a i ( x + y ǫ ) = ǫu a i ( ǫx + ǫy ǫ ) (5.20) satisfies lim ǫ → inf Z B R (0) | w a i | dx ≥ η i > . (5.21)(ii). The following estimate holds lim ( a ,a ) ր ( a ∗ ,a ∗ ) dist( ǫy ǫ , Λ) = 0 , (5.22) where Λ = (cid:8) x ∈ R : V ( x ) = V ( x ) = 0 (cid:9) is given by (1.19). Moreover, forany sequence { ( a k , a k ) } satisfying ( a k , a k ) ր ( a ∗ , a ∗ ) as k → ∞ , there exists aconvergent subsequence of { ( a k , a k ) } , still denoted by { ( a k , a k ) } , such that ǫ k y ǫ k k −→ x ∈ Λ and w a ik k → w strongly in H ( R ) , i = 1 , , (5.23) where w satisfies w ( x ) = λ k Q k Q ( λ | x − y | ) for some y ∈ R and λ > . (5.24)28 roof. (i). In order to establish (5.21), in view of (5.20) it suffices to prove that thereexist R > η i > ǫ → Z B R ( y ǫ ) | ˜ w a i | dx ≥ η i > , where i = 1 , . (5.25)We first show that (5.25) holds for ˜ w a . Indeed, if it is false, then for any R >
0, thereexists a subsequence { ˜ w a k } , where ( a k , a k ) ր ( a ∗ , a ∗ ) as k → ∞ , such thatlim k →∞ sup y ∈ R Z B R ( y ) | ˜ w a k | dx = 0 . By Lemma I.1 in [26], we then deduce that ˜ w a k k −→ L p ( R ) for any 2 < p < ∞ , whichcontradicts (5.19). Thus ˜ w a satisfies (5.25) for a sequence { y ǫ } , R > η > { y ǫ } , R > η > w a with some constant η >
0. On the contrary, if (5.25) is false for ˜ w a ,then there exists a subsequence { ˜ w a k } , where ( a k , a k ) ր ( a ∗ , a ∗ ) as k → ∞ , such thatlim k →∞ sup Z B R ( y ǫk ) | ˜ w a k | dx = 0 , where ǫ k := ǫ ( a k , a k ) > w a i is bounded uniformly in H ( R ) ∩ L γ ( R ) for all 2 ≤ γ < ∞ , we may choose γ > θ ∈ (0 ,
1) such that = − θγ + θ . It then follows from the H¨older inequality that Z B R ( y ǫk ) | ˜ w a k | | ˜ w a k | dx ≤ (cid:18) Z B R ( y ǫk ) | ˜ w a k | dx (cid:19) (cid:18) Z B R ( y ǫk ) | ˜ w a k | dx (cid:19) ≤ C (cid:18) Z B R ( y ǫk ) | ˜ w a k | γ dx (cid:19) − θ ) γ (cid:18) Z B R ( y ǫk ) | ˜ w a k | dx (cid:19) θ ≤ C (cid:18) Z B R ( y ǫk ) | ˜ w a k | dx (cid:19) θ → k → ∞ . By applying the estimate (5.25) for ˜ w a k , we use again the H¨older inequality to derivefrom the above that, for k large, Z B R ( y ǫk ) ( | ˜ w a k | − | ˜ w a k | ) dx ≥ Z B R ( y ǫk ) | ˜ w a k | dx ≥ πR (cid:18) Z B R ( y ǫk ) | ˜ w a k | dx (cid:19) ≥ η πR , a contradiction, since Lemma 5.1 implies that Z R (cid:0) | ˜ w a k − | ˜ w a k | (cid:1) dx = ǫ k Z R (cid:0) | u a k | − | u a k | (cid:1) dx → k → ∞ . Therefore, (5.25) holds also for ˜ w a with some constant η >
0, and part (i) is proved.29 ii).
We first prove (5.22). By (5.15) we note that X i =1 Z R V i ( x ) | u a i ( x ) | dx = X i =1 Z R V i ( ǫx + ǫy ǫ ) | w a i ( x ) | dx → a , a ) ր ( a ∗ , a ∗ ). On the contrary, suppose (5.22) is incorrect, then there exist δ > { ( a n , a n ) } , which satisfies ( a n , a n ) ր ( a ∗ , a ∗ ) as n → ∞ , such that ǫ n := ǫ ( a n , a n ) → ǫ n y ǫ n , Λ) ≥ δ > n → ∞ . This implies that there exists C = C ( δ ) > n →∞ V ( ǫ n y ǫ n ) + V ( ǫ n y ǫ n ) ≥ C ( δ ) > . Hence, by Fatou’s Lemma and (5.21) we obtain thatlim n →∞ X i =1 Z R V i ( ǫ n x + ǫ n y ǫ n ) | w a in ( x ) | dx ≥ X i =1 Z B R (0) lim inf n →∞ V i ( ǫ n x + ǫ n y ǫ n ) | w a in ( x ) | dx ≥ C ( δ ) min { η , η } , which however contradicts (5.26). Therefore, (5.22) holds.We prove now (5.23) and (5.24). Since ( u a , u a ) is a non-negative minimizer of (5.2),it satisfies the Euler-Lagrange system ( − ∆ u a + V ( x ) u a = µ a u a + a u a − β ( u a − u a ) u a in R , − ∆ u a + V ( x ) u a = µ a u a + a u a − β ( u a − u a ) u a in R , (5.27)where ( µ a , µ a ) is a suitable Lagrange multiplier, and µ a i = E ia i ( u a i ) − a i Z R | u a i | dx − β Z R ( − i ( u a − u a ) u a i dx, i = 1 , . It then follows from Lemma 5.1, (5.1), (5.10) and (5.17) that for i = 1 , µ a i ∼ − a i Z R | u a i | dx ∼ − ǫ − and µ a /µ a → a , a ) ր ( a ∗ , a ∗ ) . (5.28)Note also from (5.20) that w a i ( x ) defined in (5.20) satisfies the elliptic system ( − ∆ w a + ǫ V ( ǫx + ǫy ǫ ) w a = ǫ µ a w a + a w a − β ( w a − w a ) w a in R , − ∆ w a + ǫ V ( ǫx + ǫy ǫ ) w a = ǫ µ a w a + a w a − β ( w a − w a ) w a in R , (5.29)where the Lagrange multiplier ( µ a , µ a ) satisfies (5.28).For any given sequence { ( a k , a k ) } with ( a k , a k ) ր ( a ∗ , a ∗ ) as k → ∞ , we deducefrom (5.22) and (5.28) that there exists a subsequence of { ( a k , a k ) } , still denoted by { ( a k , a k ) } , such that ǫ k y ǫ k k → x ∈ Λ , ǫ k µ a ik k → − λ < λ > , (5.30)30nd w a ik k ⇀ w i ≥ H ( R ) for some w i ∈ H ( R ) , i = 1 , . Since (5.5) implies that k w a − w a k = ǫ k u a − u a k → a , a ) ր ( a ∗ , a ∗ ) , (5.31)we have w = w ≥ R . We thus write 0 ≤ w := w = w ∈ H ( R ). Passingto the weak limit in (5.29), we deduce from (5.30) and (5.31) that w satisfies − ∆ w ( x ) = − λ w ( x ) + a ∗ w ( x ) in R . (5.32)Furthermore, it follows from (5.21) and the strong maximum principle that w > w ( x ) = λ k Q k Q ( λ | x − y | ) for some y ∈ R . (5.33)Note that || w || = 1, by the norm preservation we further conclude that w a ik k → w strongly in L ( R ) . Moreover, this strong convergence holds also for all p ≥
2, since { w a ik } is bounded in H ( R ). Then, note that w a ik and w satisfy (5.29) and (5.32), respectively, a simpleanalysis shows that w a ik k → w strongly in H ( R ) , i = 1 , . Therefore, (5.23) and (5.24) are established.Applying above lemmas, we finally prove Proposition 5.1 on the optimal estimatesof e ( a , a ). Proof of Proposition 5.1.
For any sequence { ( a k , a k ) } satisfying ( a k , a k ) ր ( a ∗ , a ∗ )as k → ∞ , it follows from Lemma 5.3 (ii) that there exists a convergent subsequence,still denoted by { ( a k , a k ) } , such that (5.23) holds and ǫ k y ǫ k k → x ∈ Λ. Without lossof generality, we may assume x = x j for some 1 ≤ j ≤ l . We first claim thatlim sup k →∞ | ǫ k y ǫ k − x j | ǫ k < ∞ . (5.34)31ctually, by (5.23) and (5.24), we have for some R > e ( a k , a k ) = E a k ,a k ( u a k , u a k ) ≥ X i =1 n ǫ k h Z R |∇ w a ik ( x ) | dx − a ∗ Z R | w a ik ( x ) | dx i + a ∗ − a ik ǫ k Z R | w a ik ( x ) | dx + Z R V i ( ǫ k x + ǫ k y ǫ k ) | w a ik ( x ) | dx o ≥ X i =1 n a ∗ − a ik ǫ k Z R | w ( x ) | dx + Z B R (0) V i ( ǫ k x + ǫ k y ǫ k ) | w a ik ( x ) | dx o ≥ C a ∗ − a k − a k ǫ k + C X i =1 ǫ p ij k Z B R (0) (cid:12)(cid:12)(cid:12) x + ǫ k y ǫ k − x j ǫ k (cid:12)(cid:12)(cid:12) p ij | w a ik ( x ) | dx. (5.35)Suppose now that there exists a subsequence such that | ǫ k y ǫk − x j | ǫ k → ∞ as k → ∞ . Byusing Fatou’s Lemma, we then deduce from (5.35) and (5.21) that for any M > e ( a k , a k ) ≥ C a ∗ − a k − a k ǫ k + C M ǫ ¯ p j k ≥ CM pj (cid:16) a ∗ − a k + a k (cid:17) ¯ pj pj ≥ CM pj (cid:16) a ∗ − a k + a k (cid:17) p p , where p = max ≤ j ≤ l ¯ p j and ¯ p j = min { p j , p j } > { ǫ k } , still denoted by { ǫ k } , such that ǫ k y ǫ k − x j ǫ k → y as k → ∞ holds for some y ∈ R . By applying (5.35), then there exists a constant C > a k and a k , such that e ( a k , a k ) ≥ C (cid:16) a ∗ − a k + a k (cid:17) ¯ pj pj as ( a k , a k ) ր ( a ∗ , a ∗ ) . Since ¯ p j ≤ p , applying the upper bound of (5.5), we conclude from the above estimatethat ¯ p j = p and (5.3) holds for the above subsequence { ( a k , a k ) } .We next prove that (5.4) holds for the above subsequence { ( a k , a k ) } . Suppose thateither ǫ k >> (cid:16) a ∗ − a k + a k (cid:17) p or 0 < ǫ k << (cid:16) a ∗ − a k + a k (cid:17) p as k → ∞ , it then follows from (5.35) that e ( a k , a k ) >> (cid:16) a ∗ − a k + a k (cid:17) p p as k → ∞ , whichhowever contradicts (5.3). This completes the proof of (5.4).Since the above argument shows that Proposition 5.1 holds for any given subsequence { ( a k , a k ) } with ( a k , a k ) ր ( a ∗ , a ∗ ), an approach similar to that of [16] then gives thatProposition 5.1 holds essentially for the whole sequence { ( a , a ) } satisfying ( a , a ) ր ( a ∗ , a ∗ ). 32 Proof of Theorem 1.5
This section is devoted to the proof of Theorem 1.5 on the mass concentration of min-imizers. Using the notations as in (5.1)-(5.2), in order to prove Theorem 1.5 on theminimizers of (1.2) as ( b + β, b + β ) ր ( a ∗ , a ∗ ), it suffices to establish the followingtheorem on the minimizers of (5.2) as ( a , a ) ր ( a ∗ , a ∗ ). Theorem 6.1.
Assume that < β < a ∗ and V i ( x ) satisfies (1.17) and (1.18) for i = 1 and . For any sequence { ( a k , a k ) } satisfying ( a k , a k ) ր ( a ∗ , a ∗ ) as k → ∞ , and let ( u a k , u a k ) be the corresponding non-negative minimizer of (5.2). Then there exists asubsequence of { ( a k , a k ) } , still denoted by { ( a k , a k ) } , such that each u a ik ( i = 1 , )has a unique global maximum point x ik satisfying x ik −→ ¯ x ∈ Z and | x ik − ¯ x | (cid:0) a ∗ − a k + a k (cid:1) p −→ as k → ∞ . (6.1) Moreover, for i = 1 and , lim k →∞ (cid:16) a ∗ − a k + a k (cid:17) p u a ik (cid:16)(cid:0) a ∗ − a k + a k (cid:1) p x + x ik (cid:17) = λ k Q k Q ( λx ) strongly in H ( R ) , where λ > is given by λ = (cid:16) p γ Z R | x | p Q ( x ) dx (cid:17) p , (6.2) p > and γ > are defined in (1.20)-(1.23). Let ( u a , u a ) be a non-negative minimizer of (5.2), where ( a , a ) ր ( a ∗ , a ∗ ). Define ε := (cid:0) a ∗ − a + a (cid:1) p > . (6.3)It then follows from (5.4) and (5.5) that, as ( a , a ) ր ( a ∗ , a ∗ ), X i =1 Z R V i ( x ) | u a i ( x ) | dx ≤ e ( a , a ) < C (cid:0) a ∗ − a + a (cid:1) p p (6.4)and Z R |∇ u a i ( x ) | dx ∼ ε − , Z R | u a i ( x ) | dx ∼ ε − , (6.5)where i = 1 ,
2. Similar to (5.21), we know that there exist a sequence { y ε } as well aspositive constants R and η i such thatlim inf ε ց Z B R (0) | w a i | dx ≥ η i > , i = 1 , , (6.6)where w a i is the L ( R )-normalized function defined by w a i ( x ) = εu a i ( εx + εy ε ) , i = 1 , . (6.7)33ote from (6.5) that M ≤ Z R |∇ w a i | dx ≤ M , M ≤ Z R | w a i | dx ≤ M , i = 1 , , (6.8)where the positive constant M is independent of a and a . Proof of Theorem 6.1.
Let ε k := (cid:0) a ∗ − a k + a k (cid:1) p >
0, where ( a k , a k ) ր ( a ∗ , a ∗ )as k → ∞ , and denote ( u k ( x ) , u k ( x )) := ( u a k ( x ) , u a k ( x )) a non-negative minimizer of(5.2). Inspired by [14, 37], we shall complete the proof of Theorem 6.1 by the followingthree steps. Step 1: Decay estimate for ( u k ( x ) , u k ( x )) . Let w ik ( x ) := w a ik ( x ) ≥ { ε k } , still denoted by { ε k } , such that z k := ε k y ε k k −→ x for some x ∈ Λ , (6.9)where the set Λ is defined by (1.19). and w ik ( i = 1 ,
2) satisfies − ∆ w k + ε k V ( ε k x + z k ) w k = ε k µ k w k + a k w k − β ( w k − w k ) w k in R , − ∆ w k + ε k V ( ε k x + z k ) w k = ε k µ k w k + a k w k − β ( w k − w k ) w k in R , (6.10)where ( µ k , µ k ) is a suitable Lagrange multiplier satisfying µ ik ∼ − ε − k and µ k /µ k → a k , a k ) ր ( a ∗ , a ∗ ) , i = 1 , . Moreover, w ik k −→ w strongly in H ( R ) for some w > − ∆ w ( x ) = − λ w ( x ) + a ∗ w ( x ) in R , (6.11)where λ > w ( x ) = λ k Q k Q ( λ | x − y | ) for some y ∈ R . (6.12)By the exponential decay of Q , then for any α > Z | x |≥ R | w ik | α dx → R → ∞ uniformly for large k, where i = 1 , . (6.13)Recall from (6.10) that − ∆ w ik ( x ) ≤ c i ( x ) w ik ( x ) in R , where c i ( x ) = a ik w ik +( − i β ( w k − w k ) in R for i = 1 ,
2. By applying De Giorgi-Nash-Moser theory, we thus havemax B ( ξ ) w ik ≤ C (cid:16) Z B ( ξ ) | w ik | α dx (cid:17) α , i = 1 , , where ξ is an arbitrary point in R , and C > k w k k L α ( B ( ξ )) + k w k k L α ( B ( ξ )) . Hence (6.13) implies that w ik ( x ) → | x | → ∞ uniformly for k, i = 1 , . (6.14)34ince w ik ( i = 1 ,
2) satisfies (6.10), apply the comparison principle as in [21] to compare w ik with Ce − λ | x | ( λ > C >
R >
0, independent of k , such that w ik ( x ) ≤ Ce − λ | x | for | x | > R as k → ∞ , i = 1 , . (6.15) Step 2: The detailed concentration behavior.
For the convergent subsequence { w ik ( x ) } obtained in Step 1, let ¯ z ik be any local maximum point of u ik ( x ) , ( i = 1 , { k } , passing to a subsequence if necessary, such thatlim k →∞ | ¯ z k − ¯ z k | ε k = 0 . (6.16)For showing (6.16), we first prove thatlim sup k →∞ | ¯ z ik − z k | ε k < ∞ , i = 1 , , (6.17)where z k = ε k y ε k x ∈ Λ as k → ∞ . Indeed, if (6.17) is false, i.e., | ¯ z ik − z k | ε k k → ∞ holdsfor i = 1 or 2. Without loss of generality, we assume that | ¯ z k − z k | ε k k → ∞ . It follows from(6.7) and (6.15) that u ik (¯ z k ) = 1 ε k w ik (cid:0) ¯ z k − z k ε k (cid:1) = o (cid:0) ε k (cid:1) as k → ∞ , i = 1 , . This however leads to a contradiction, since (5.27) implies that a k u k (¯ z k ) − β (cid:0) u k (¯ z k ) − u k (¯ z k ) (cid:1) ≥ − µ k ≥ Cε − k . Therefore, (6.17) is true.By (6.17), there exists a sequence { k } such thatlim k →∞ ¯ z ik − z k ε k = y i for some y i ∈ R , i = 1 , . Set ¯ w ik ( x ) := w ik (cid:0) x + ¯ z ik − z k ε k (cid:1) = ε k u ik (cid:0) ε k x + ¯ z ik (cid:1) , i = 1 , . (6.18)By Step 1, w ik → w strongly in H ( R ) as k → ∞ , and w > k →∞ ¯ w ik ( x ) = w ( x + y i ) = λ k Q k Q ( λ | x + y i − y | ) strongly in H ( R ) , i = 1 , . Since the origin (0 ,
0) is a critical point of ¯ w ik for all k > i = 1 , w ( x + y i ). On the other hand, Q ( λ | x − z | ) possesses z as its unique critical(maximum) point. We therefore conclude that w ( x + y i ) = λ k Q k Q ( λ | x + y i − y | ) isspherically symmetric about the origin. Hence, y i = y , andlim k →∞ ¯ w ik ( x ) = λ k Q k Q ( λ | x | ) := ¯ w strongly in H ( R ) , i = 1 , . (6.19)35he estimate (6.16) is followed by (6.18) and (6.19).Similar to the discussion of proving (6.17), we know that each local maximum pointof ¯ w ik ( x ) ( i = 1 , u ik ( x ), must stay in a finiteball in R . Since V i ( x ) ∈ C α loc ( R ), we can deduce from (6.10) and standard ellipticregular theory that w ik ( x ) k −→ w ( x ) in C ,αloc ( R ) , i = 1 , . Moreover, by (6.18) and (6.16), we can further obtain that¯ w ik ( x ) k −→ ¯ w ( x ) in C ,αloc ( R ) , i = 1 , . Because the origin (0 ,
0) is the only non-degenerate critical point of ¯ w ( x ), all local max-imum points of ¯ w ik ( x ) must approach the origin and stay in a small ball B ǫ (0) as k → ∞ ,where ǫ > k , ¯ w ik ( x )has no critical points other than the origin. This gives the uniqueness of local maximumpoints for ¯ w ik ( x ) and u ik ( x ) ( i = 1 , Step 3: Completion of the proof.
Let γ j ( x ) = V ( x ) + V ( x ) | x − x j | ¯ p j , ≤ j ≤ l, (6.20)where ¯ p j > x → x j γ j ( x ) = γ j ( x j ) exists forall i ∈ { , · · · , l } in view of the assumptions on V and V . Moreover, one can note that γ j ( x j ) = γ j ≥ γ if x j ∈ ¯Λ, where γ j and ¯Λ are defined by (1.22) and (1.21), respectively.We hence obtain from (6.18) that e ( a k , a k ) = E a k ,a k ( u a k , u a k ) ≥ X i =1 n ε k h Z R |∇ ¯ w ik ( x ) | dx − a ∗ Z R | ¯ w ik ( x ) | dx i + a ∗ − a ik ε k Z R | ¯ w ik ( x ) | dx + Z R V i ( ε k x + ¯ z ik ) | ¯ w ik ( x ) | dx o ≥ X i =1 (cid:20) a ∗ − a ik ε k Z R | ¯ w ik ( x ) | dx + Z R V i ( ε k x + ¯ z ik ) | ¯ w ik ( x ) | dx (cid:21) , (6.21)where ¯ z ik is the unique global maximum point of u ik , and ¯ z ik k → x ∈ Λ, i = 1 ,
2. Wemay assume that x = x j for some 1 ≤ j ≤ l .We first claim that | ¯ z ik − x j | ε k is uniformly bounded as k → ∞ , where i = 1 , . (6.22)Otherwise, if (6.22) is false for i = 1 or 2. It then follows from (6.16) that both of themare unbounded, and hence there exists a subsequence of { ( a k , a k ) } , still denoted by { ( a k , a k ) } , such that lim k →∞ | ¯ z ik − x j | ε k = ∞ , i = 1 , .
36e then derive from Fatou’s Lemma that for any
M > k →∞ inf ε − ¯ p j k X i =1 Z R V i ( ε k x + ¯ z ik ) | ¯ w ik ( x ) | dx = lim k →∞ inf X i =1 Z R V i ( ε k x + ¯ z ik ) | ε k x + ¯ z ik − x j | ¯ p j (cid:12)(cid:12)(cid:12) x + ¯ z ik − x j ε k (cid:12)(cid:12)(cid:12) ¯ p j | ¯ w ik ( x ) | dx ≥ X i =1 Z R lim k →∞ inf (cid:16) V i ( ε k x + ¯ z ik ) | ε k x + ¯ z ik − x j | ¯ p j (cid:12)(cid:12)(cid:12) x + ¯ z ik − x j ε k (cid:12)(cid:12)(cid:12) ¯ p j | ¯ w ik ( x ) | (cid:17) dx ≥ M. This estimate and (6.21) imply that e ( a k , a k ) ≥ M ε ¯ p j k = M (cid:16) a ∗ − a k + a k (cid:17) ¯ p j (6.23)holds for arbitrary constant M >
0, which however contradicts Proposition 5.1, due tothe fact that ¯ p j ≤ p . Therefore, (6.22) is proved.We next show that ¯ p j = p , i.e., x j ∈ ¯Λ, where the set ¯Λ is defined by (1.21). By(6.22), we know that there exists a subsequence of { ( a k , a k ) } such thatlim k →∞ ¯ z ik − x j ε k = ¯ z for some ¯ z ∈ R , i = 1 , . (6.24)Since Q is a radially decreasing function and decays exponentially as | x | → ∞ , we thendeduce from (6.19) thatlim k →∞ inf ε − ¯ p j k X i =1 Z R V i ( ǫ k x + ¯ z ik ) | ¯ w ik ( x ) | dx = lim k →∞ inf X i =1 Z R V i ( ǫ k x + ¯ z ik ) | ε k x + ¯ z ik − x j | ¯ p j (cid:12)(cid:12)(cid:12) x + ¯ z ik − x j ε k (cid:12)(cid:12)(cid:12) ¯ p j | ¯ w ik ( x ) | dx ≥ γ j ( x j ) Z R | x + ¯ z | ¯ p j ¯ w dx ≥ γ j ( x j ) λ ¯ p j k Q k Z R | x | ¯ p j Q dx, (6.25)where the equality holds if and only if ¯ z = (0 , p j = p , otherwise we get that (6.23) holds with M being replaced by some C >
0, which however contradicts Proposition 5.1.By the fact ¯ p j = p , we now have x j ∈ ¯Λ and γ j ( x j ) = γ j . It then follows from(6.3), (6.21) and (6.25) as well as (1.11)thatlim k →∞ inf e ( a k , a k ) ε p k ≥ k ¯ w k + γ j Z R | x + ¯ z | p ¯ w dx ≥ a ∗ (cid:16) λ + γ j λ p Z R | x | p Q dx (cid:17) , (6.26)and“=” holds in the last inequality if and only if ¯ z = (0 , λ > k →∞ inf e ( a k , a k ) ε p k ≥ p + 4 p a ∗ (cid:16) p γ j R R | x | p Q dx (cid:17) p , (6.27)37here the equality is achieved at λ = λ := (cid:16) p γ j R R | x | p Q dx (cid:17) p . We finally remark that the limit of (6.27) actually exists and is equal to the righthand side of (6.27). To see this fact, we simply take u ( x ) = u ( x ) = αε k k Q k Q (cid:16) α | x − x j | ε k (cid:17) with x j ∈ Z as a trial function for E a k ,a k ( · , · ) and minimizes it over α >
0, which then leads to thelimit lim k →∞ inf e ( a k , a k ) ε p k ≤ p + 4 p a ∗ (cid:16) p γ R R | x | p Q dx (cid:17) p . (6.28)Since γ = min (cid:8) γ , . . . , γ l (cid:9) , it follows from (6.27) and (6.28) that γ = γ j , i.e., x j ∈ Z ,and further (6.27) is indeed an equality. This yields that λ is unique, which is inde-pendent of the choice of the subsequence, and minimizes (6.26), i.e., λ = λ . Moreover,when (6.27) becomes an equality, which implies that (6.26) is also an equality. Thus,¯ z = (0 , A Appendix: Some Proofs
In this appendix we shall establish the following lemma and provide a different proof ofTheorem 1.1.
Lemma A.1.
Suppose that positive constants b , b and β satisfy < b < b < a ∗ and < b ≤ β + b . Define l ( t ) = t + t b t + βt + b , t ∈ [0 , ∞ ) . Then we have l ( t ) > l (1) = 2 b + β + b , t ∈ (1 , ∞ ) . Proof.
Direct calculation shows that l ′ ( t ) = ( β − b ) t + b t + b (cid:0) b t + βt + b (cid:1) . (A.1)Let m ( t ) = ( β − b t + b t + b , t ∈ [0 , ∞ ) . If β ≥ b , then m ′ ( t ) = 2( β − b ) t + b >
0. This implies that l ′ ( t ) > t ∈ [0 , ∞ ),and we are done. 38e now suppose that β < b . In this case, we have m ( t ) > t ∈ (0 , ˆ t ); m ( t ) < t ∈ (ˆ t, ∞ ) , where ˆ t = b + p b ( b + b − β ) b − β > , in view of the assumption that b ≤ β + b . Thus, l ( t ) is strictly increasing in (1 , ˆ t ] andstrictly decreasing in (ˆ t, ∞ ). Since b ≤ β + b , we thus conclude that for any t > l ( t ) > lim t →∞ l ( t ) = 2 b ≥ l (1) = 2 b + β + b , and the proof is therefore complete.Inspired by [14], in the following we reprove Theorem 1.1 by using the Gagliardo-Nirenberg inequality (1.10) and some recaling techniques. For the reader’s convenience,we restate Theorem 1.1 as the following lemma. Lemma A.2.
Let Q be the unique positive radial solution of (1.9) and suppose V i ( x ) satisfies (1.6) for i = 1 , . Then,(i) If < b < a ∗ , < b < a ∗ and < β < p ( a ∗ − b )( a ∗ − b ) , then there exists atleast one minimizer for (1.2).(ii) If either b > a ∗ or b > a ∗ or β > a ∗ − b + a ∗ − b , then there is no minimizer for(1.2). Proof. ( i ) : We first note from the Gagliardo-Nirenberg inequality (1.10) that for any( u , u ) ∈ X satisfying k u k = k u k = 1, E b ,b ,β ( u , u ) ≥ X i =1 Z R h(cid:16) a ∗ − b i (cid:17) | u i | + V i ( x ) | u i | i dx − β Z R | u | | u | dx. (A.2)Since V i ( x ) ≥ < β < p ( a ∗ − b )( a ∗ − b ), one can deduce that ˆ e ( b , b , β ) ≥ < b i < a ∗ := k Q k ( i = 1 , { ( u n , u n ) } ⊂ M be a minimizing sequenceof problem (1.2) satisfying k u n k = k u n k = 1 and lim n →∞ E b ,b ,β ( u n , u n ) = ˆ e ( b , b , β ) . Taking δ ∈ ( βa ∗ − b , a ∗ − b β ), it then follows from (A.2) and Young’s inequality thatˆ e ( b , b , β ) + 1 ≥ X i =1 Z R (cid:16) a ∗ − b i | u in | + V i ( x ) | u in | (cid:17) dx − β Z R | u n | | u n | dx ≥ (cid:16) a ∗ − b − βδ (cid:17) Z R | u n | dx + (cid:16) a ∗ − b − β δ (cid:17) Z R | u n | dx. This implies that { ( u n , u n ) } is bounded in L ( R ) × L ( R ) uniformly w.r.t. n , and itis then easy to deduce that { ( u n , u n ) } is bounded uniformly in X . Therefore, by the39ompactness of Lemma 2.1, there exist a subsequence of { ( u n , u n ) } and ( u , u ) ∈ X such that ( u n , u n ) n ⇀ ( u , u ) weakly in X , ( u n , u n ) n → ( u , u ) strongly in L q ( R ) × L q ( R ) , where 2 ≤ q < ∞ . We therefore have k u k = k u k = 1 and E b ,b ,β ( u , u ) =ˆ e ( b , b , β ). This proves the existence of minimizers for the case where 0 < b < a ∗ ,0 < b < a ∗ and 0 < β < p ( a ∗ − b )( a ∗ − b ).( ii ) : Let ϕ ( x ) ∈ C ∞ ( R ) be a nonnegative smooth cutoff function such that ϕ ( x ) = 1if | x | ≤ ϕ ( x ) = 0 if | x | ≥
2. For any ¯ x ∈ R , τ > R >
0, set φ ( x ) = A Rτ τ k Q k ϕ (cid:0) x − ¯ x R (cid:1) Q (cid:0) τ | x − ¯ x | (cid:1) , (A.3)where A Rτ > R R φ dx = 1. By scaling, A Rτ depends only on theproduct Rτ . In fact, using the exponential decay of Q in (1.12), we have1 A Rτ = 1 k Q k Z R ϕ (cid:0) xRτ (cid:1) Q ( x ) dx = 1 + O (( Rτ ) −∞ ) as Rτ → ∞ . (A.4)Here we use the notation f ( t ) = O ( t −∞ ) for a function f satisfying lim t →∞ | f ( t ) | t s = 0for all s >
0. By the exponential decay of Q ( x ) and the equality (1.11), we have Z R |∇ φ | − b i Z R φ dx = τ k Q k Z R |∇ Q | − b i τ k Q k Z R Q dx + O (( Rτ ) −∞ )= (cid:16) − b i k Q k (cid:17) τ + O (( Rτ ) −∞ ) as Rτ → ∞ . (A.5)On the other hand, since the function x V i ( x ) ϕ ( x − ¯ x R ) is bounded and has compactsupport, we obtain that lim τ →∞ Z R V i ( x ) φ ( x ) dx = V i (¯ x ) (A.6)holds for almost every ¯ x ∈ R , where i = 1 , b > a ∗ . Choosing η ( x ) ∈ C ∞ ( R ) such that R R η ( x ) dx = 1, we thenderive that Z R φ η dx ≤ sup x ∈ R η ( x ) Z R φ dx = sup x ∈ R η ( x ) < ∞ , where φ > E b ,b ,β ( φ, η ) ≤ Z R |∇ φ | − b Z R φ dx + C ≤ a ∗ − b a ∗ τ + C, which implies that ˆ e ( b , b , β ) ≤ lim τ →∞ E b ,b ,β ( φ, η ) = −∞ . Similarly, this estimate is still true if b > a ∗ . Therefore, ˆ e ( b , b , β ) does not admit anyminimizer. 40inally, if β > a ∗ − b + a ∗ − b , we also deduce from (A.5) and (A.6) thatˆ e ( b , b , β ) ≤ lim τ →∞ E b ,b ,β ( φ, φ ) = −∞ , which also implies the non-existence of minimizers. This completes the proof of LemmaA.2. Acknowledgements:
This work was supported by National Natural Science Founda-tion of China (11322104, 11271360, 11471331, 11501555) and National Center for Math-ematics and Interdisciplinary Sciences.
Conflict of interest statement:
The authors declare that they have no conflict ofinterest.
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