On a two-component Bose-Einstein condensate with steep potential wells
aa r X i v : . [ m a t h . A P ] D ec On a two-component Bose-Einstein condensate with steeppotential wells
Yuanze Wu a , ∗ Tsung-fang Wu b , † Wenming Zou c ‡ a College of Sciences, China University of Mining and Technology, Xuzhou 221116, P.R. China b Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan c Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P.R. China
Abstract:
In this paper, we study the following two-component systems of nonlinear Schr¨odingerequations ∆ u − ( λa ( x ) + a ( x )) u + µ u + βv u = 0 in R , ∆ v − ( λb ( x ) + b ( x )) v + µ v + βu v = 0 in R ,u, v ∈ H ( R ) , u, v > R , where λ, µ , µ > β < a ( x ) , b ( x ) ≥ a ( x ) , b ( x )are sign-changing weight functions; a ( x ), b ( x ), a ( x ) and b ( x ) are not necessarily to be radialsymmetric. By the variational method, we obtain a ground state solution and multi-bump solutionsfor such systems with λ sufficiently large. The concentration behaviors of solutions as both λ → + ∞ and β → −∞ are also considered. In particular, the phenomenon of phase separations is observedin the whole space R . In the Hartree-Fock theory, this provides a theoretical enlightenment ofphase separation in R for the 2-mixtures of Bose-Einstein condensates. Keywords:
Bose-Einstein condensate; Steep potential well; Ground state solution; Multi-bumpsolution.
AMS
Subject Classification 2010: 35B38; 35B40; 35J10; 35J20.
In this paper, we consider the following two-component systems of nonlinear Schr¨odinger equations ∆ u − ( λa ( x ) + a ( x )) u + µ u + βv u = 0 in R , ∆ v − ( λb ( x ) + b ( x )) v + µ v + βu v = 0 in R ,u, v ∈ H ( R ) , u, v > R , ( P λ,β )where λ, µ , µ > β < a ( x ) , b ( x ) , a ( x ) and b ( x ) satisfythe following conditions:( D ) a ( x ) , b ( x ) ∈ C ( R ) and a ( x ) , b ( x ) ≥ R .( D ) There exist a ∞ > b ∞ > D a := { x ∈ R | a ( x ) < a ∞ } and D b := { x ∈ R | b ( x ) < b ∞ } are nonempty and have finite measures.( D ) Ω a = int a − (0) and Ω b = int b − (0) are nonempty bounded sets and have smooth boundaries.Moreover, Ω a = a − (0), Ω b = b − (0) and Ω a ∩ Ω b = ∅ . ∗ Corresponding author. E-mail address: [email protected] (Yuanze Wu) † E-mail address: [email protected] (Tsung-fang Wu) ‡ E-mail address:[email protected] (Wenming Zou) D ) a ( x ) , b ( x ) ∈ C ( R ) and there exist R, d a , d b > a − ( x ) ≤ d a (1 + a ( x )) and b − ( x ) ≤ d b (1 + b ( x )) for | x | ≥ R, where a − ( x ) = max {− a ( x ) , } and b − ( x ) = max {− b ( x ) , } .( D ) inf σ a ( − ∆ + a ( x )) > σ b ( − ∆ + b ( x )) >
0, where σ a ( − ∆ + a ( x )) is the spectrumof − ∆ + a ( x ) on H (Ω a ) and σ b ( − ∆ + b ( x )) is the spectrum of − ∆ + b ( x ) on H (Ω b ). Remark 1.1 If a ( x ) , b ( x ) ∈ C ( R ) are bounded, then the condition ( D ) is trivial. However,under the assumptions of ( D ) - ( D ) , a ( x ) and b ( x ) may be sign-changing and unbounded. Two-component systems of nonlinear Schr¨odinger equations like ( P λ,β ) appear in the Hartree-Fock theory for a double condensate, that is, a binary mixture of Bose-Einstein condensates in twodifferent hyperfine states | i and | i (cf. [25]), where the solutions u and v are the correspondingcondensate amplitudes, µ j are the intraspecies and interspecies scattering lengths. The interactionis attractive if β > β <
0. When the interaction is repulsive, it is expected thatthe phenomenon of phase separations will happen, that is, the two components of the system tendto separate in different regions as the interaction tends to infinity. This kind of systems also arisesin nonlinear optics (cf. [2]). Due to the important application in physics, the following system ∆ u − λ u + µ u + βv u = 0 in Ω , ∆ v − λ v + µ v + βu v = 0 in Ω ,u, v = 0 on ∂ Ω , (1.1)where Ω ⊂ R or R , has attracted many attentions of mathematicians in the past decade. Werefer the readers to [7, 8, 9, 15, 17, 19, 20, 29, 30, 31, 33, 34, 36, 37, 39, 43]. In these literatures,various existence theories of the solutions were established for the Bose-Einstein condensates in R and R . Recently, some mathematicians devoted their interest to the two coupled Schr¨odingerequations with critical Sobolev exponent in the high dimensions, and a number of the existenceresults of the solutions for such systems were also established. See for example [13, 14, 15, 16, 18].On the other hand, if the parameter λ is sufficiently large, then λa ( x ) and λb ( x ) are calledthe steep potential wells under the conditions ( D )-( D ). The depth of the wells is controlled bythe parameter λ . Such potentials were first introduced by Bartsch and Wang in [3] for the scalarSchr¨odinger equations. An interesting phenomenon for this kind of Schr¨odinger equations is that,one can expect to find the solutions which are concentrated at the bottom of the wells as the depthgoes to infinity. Due to this interesting property, such topic for the scalar Schr¨odinger equationswas studied extensively in the past decade. We refer the readers to [4, 5, 10, 24, 35, 41, 42, 44, 50]and the references therein. In particular, in [24], by assuming that the bottom of the steep potentialwells consists of finitely many disjoint bounded domains, the authors obtained multi-bump solutionsfor scalar Schr¨odinger equations with steep potential wells, which are concentrated at any givendisjoint 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 steeppotential wells? In the current paper, we shall explore this problem to find whether the solutionsof such systems are concentrated at the bottom of the wells as λ → + ∞ and when the phenomenonof phase separations of such systems can be observed in the whole space R .We remark that the phenomenon of phase separations for (1.1) was observed in [13, 16, 20,21, 38, 48, 49] for the ground state solution when Ω is a bounded domain. In particular, thisphenomenon was also observed on the whole spaces R and R by [47], where the system is radialsymmetric! However, when the system is not necessarily radial symmetric, the phenomenon ofphase seperations for Bose-Einstein condensates on the whole space R , has not been obtained yet.For other kinds of elliptic systems with strong competition, the phenomenon of phase separationshas also been well studied; we refer the readers to [11, 12, 22] and references therein.2e recall some definitions in order to state the main results in the current paper. We say that( u , v ) ∈ H ( R ) × H ( R ) is a non-trivial solution of ( P λ,β ) if ( u , v ) is a solution of ( P λ,β ) with u = 0 and v = 0. We say ( u , v ) ∈ H ( R ) × H ( R ) is a ground state solution of ( P λ,β ) if( u , v ) is a non-trivial solution of ( P λ,β ) and J λ,β ( u , v ) = inf { J λ,β ( u, v ) | ( u, v ) is a non-trivial solution of ( P λ,β ) } , where J λ,β ( u, v ) is the corresponding functional of ( P λ,β ) and given by J λ,β ( u, v ) = 12 Z R |∇ u | + ( λa ( x ) + a ( x )) u dx + 12 Z R |∇ v | + ( λb ( x ) + b ( x )) v dx − µ Z R u dx − µ Z R v dx − β Z R u v dx. (1.2) Remark 1.2
In section 2, we will give a variational setting of ( P λ,β ) and show that the solutionsof ( P λ,β ) are equivalent to the positive critical points of J λ,β ( u, v ) in a suitable Hilbert space E . Let I Ω a ( u ) and I Ω b ( v ) be two functionals respectively defined on H (Ω a ) and H (Ω b ), whichare given by I Ω a ( u ) = 12 Z Ω a |∇ u | + a ( x ) u dx − µ Z Ω a u dx and by I Ω b ( v ) = 12 Z Ω b |∇ v | + b ( x ) v dx − µ Z Ω b v dx. Then by the condition ( D ), it is well-known that I Ω a ( u ) and I Ω b ( v ) have least energy nonzerocritical points. We denote the least energy of nonzero critical points for I Ω a ( u ) and I Ω b ( v ) by m a and m b , respectively. Now, our first result can be stated as the following. Theorem 1.1
Assume ( D ) - ( D ) . Then there exists Λ ∗ > such that ( P λ,β ) has a ground statesolution ( u λ,β , v λ,β ) for all λ ≥ Λ ∗ and β < , which has the following properties: (1) R R \ Ω a |∇ u λ,β | + u λ,β dx → and R R \ Ω b |∇ v λ,β | + v λ,β dx → as λ → + ∞ . (2) R Ω a |∇ u λ,β | + a ( x ) u λ,β dx → m a and R Ω b |∇ v λ,β | + b ( x ) v λ,β dx → m b as λ → + ∞ .Furthermore, for each { λ n } ⊂ [Λ ∗ , + ∞ ) satisfying λ n → + ∞ as n → ∞ and β < , there exists ( u ,β , v ,β ) ∈ ( H ( R ) \{ } ) × ( H ( R ) \{ } ) such that (3) ( u ,β , v ,β ) ∈ H (Ω a ) × H (Ω b ) with u ,β ≡ outside Ω a and v ,β ≡ outside Ω b . (4) ( u λ n ,β , v λ n ,β ) → ( u ,β , v ,β ) strongly in H ( R ) × H ( R ) as n → ∞ up to a subsequence. (5) u ,β is a least energy nonzero critical point of I Ω a ( u ) and v ,β is a least energy nonzero criticalpoint of I Ω b ( v ) . Next, we assume that the bottom of the steep potential wells consists of finitely many disjointbounded domains. It is natural to ask whether the two-component Bose-Einstein condensate( P λ,β ) with such steep potential wells has multi-bump solutions which are concentrated at anygiven disjoint bounded domains of the bottom as the depth goes to infinity. Our second resultis devoted to this study. Similar to [24], we need the following conditions on the potentials a ( x ), b ( x ), a ( x ) and b ( x ). 3 D ′ ) Ω a = int a − (0) and Ω b = int b − (0) satisfy Ω a = n a ∪ i a =1 Ω a,i a and Ω b = n b ∪ j b =1 Ω b,j b , where { Ω a,i a } and { Ω b,j b } are all nonempty bounded domains with smooth boundaries, and Ω a,i a ∩ Ω a,j a = ∅ for i a = j a and Ω b,i b ∩ Ω b,j b = ∅ for i b = j b . Moreover, Ω a = a − (0) and Ω b = b − (0) withΩ a ∩ Ω b = ∅ .( D ′ ) inf σ a,i a ( − ∆ + a ( x )) > i a = 1 , · · · , n a and inf σ b,j b ( − ∆ + b ( x )) > j b =1 , · · · , n b , where σ a,i a ( − ∆+ a ( x )) is the spectrum of − ∆+ a ( x ) on H (Ω a,i a ) and σ b,j b ( − ∆+ b ( x )) is the spectrum of − ∆ + b ( x ) on H (Ω b,j b ). Remark 1.3
Under the conditions ( D ′ ) and ( D ) , it is easy to see that the condition ( D ′ ) isequivalent to the condition ( D ) . For the sake of clarity, we use the condition ( D ′ ) in the study ofmulti-bump solutions. We define I Ω a,ia ( u ) on H (Ω a,i a ) for each i a = 1 , · · · , n a by I Ω a,ia ( u ) = 12 Z Ω a,ia |∇ u | + a ( x ) u dx − µ Z Ω a,ia u dx and I Ω b,jb ( v ) on H (Ω b,j b ) for each j b = 1 , · · · , n b by I Ω b,jb ( v ) = 12 Z Ω b,jb |∇ v | + b ( x ) v dx − µ Z Ω b,jb v dx. Then by the conditions ( D ′ ) and ( D ′ ), it is well-known that I Ω a,ia ( u ) and I Ω b,jb ( v ) have leastenergy nonzero critical points for every i a = 1 , · · · , n a and every j b = 1 , · · · , n b , respectively. Wedenote the least energy of nonzero critical points for I Ω a,ia ( u ) and I Ω b,jb ( v ) by m a,i a and m b,j b ,respectively. Now, our second result can be stated as the following. Theorem 1.2
Assume β < and the conditions ( D ) - ( D ) , ( D ′ ) , ( D ) and ( D ′ ) hold. If the set J a × J b ⊂ { , · · · , n a } × { , · · · , n b } satisfying J a = ∅ and J b = ∅ , then there exists Λ ∗ ( β ) > suchthat ( P λ,β ) has a non-trivial solution ( u J a λ,β , v J b λ,β ) for λ ≥ Λ ∗ ( β ) with the following properties: (1) R R \ Ω Jaa, |∇ u J a λ,β | + ( u J a λ,β ) dx → and R R \ Ω Jbb, |∇ v J b λ,β | + ( v J b λ,β ) dx → as λ → + ∞ , where Ω J a a, = ∪ i a ∈ J a Ω a,i a and Ω J b b, = ∪ j b ∈ J b Ω b,j b . (2) R Ω a,ia |∇ u J a λ,β | + a ( x )( u J a λ,β ) dx → m a,i a and R Ω b,jb |∇ v J b λ,β | + b ( x )( v J b λ,β ) dx → m b,j b as λ → + ∞ for all i a ∈ J a and j b ∈ J b .Furthermore, for each β < and { λ n } ⊂ [Λ ∗ ( β ) , + ∞ ) satisfying λ n → + ∞ as n → ∞ , there exists ( u J a ,β , v J b ,β ) ∈ ( H ( R ) \{ } ) × ( H ( R ) \{ } ) such that (3) ( u J a ,β , v J b ,β ) ∈ H (Ω J a a, ) × H (Ω J b b, ) with u J a ,β ≡ outside Ω J a a, and v J b ,β ≡ outside Ω J b b, . (4) ( u J a λ n ,β , v J b λ n ,β ) → ( u J a ,β , v J b ,β ) strongly in H ( R ) × H ( R ) as n → ∞ up to a subsequence. (5) the restriction of u J a ,β on Ω a,i a lies in H (Ω a,i a ) and is a least energy nonzero critical pointof I Ω a,ia ( u ) for all i a ∈ J a , while the restriction of v J b ,β on Ω b,j b lies in H (Ω b,j b ) and is aleast energy nonzero critical point of I Ω b,jb ( v ) for all j b ∈ J b . Corollary 1.1
Suppose β < and the conditions ( D ) - ( D ) , ( D ′ ) , ( D ) and ( D ′ ) hold. Then ( P λ,β ) has at least (2 n a − n b − non-trivial solutions for λ ≥ Λ ∗ ( β ) . emark 1.4 ( i ) To the best of our knowledge, it seems that Theorem 1.2 is the first result forthe existence of multi-bump solutions to system ( P λ,β ) . ( ii ) Under the condition ( D ′ ) , we can see that I Ω a ( u ) = n a X i a =1 I Ω a,ia ( u ) and I Ω b ( v ) = n b X j b =1 I Ω b,jb ( v ) . Let m a, = min { m a, , · · · , m a,n a } and m b, = min { m b, , · · · , m b,n b } . Then we must have m a, = m a and m b, = m b . Without loss of generality, we assume m a, = m a, and m b, = m b, . Now, by Theorem 1.2, we can find a solution of ( P λ,β ) with the same concentrationbehavior as the ground state solution obtained in Theorem 1.1 as λ → + ∞ . However, we donot 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 concentrationbehavior of the solutions as β → −∞ . In the following theorem, we may observe such a phenomenonon the whole space R . Theorem 1.3
Assume ( D ) - ( D ) . Then there exists Λ ∗∗ ≥ Λ ∗ such that β R R u λ,β v λ,β → as β → −∞ for λ ≥ Λ ∗∗ , where ( u λ,β , v λ,β ) is the ground state solution of ( P λ,β ) obtained byTheorem 1.1. Furthermore, for every { β n } ⊂ ( −∞ , with β n → −∞ and λ ≥ Λ ∗∗ , there exists ( u λ, , v λ, ) ∈ ( H ( R ) \{ } ) × ( H ( R ) \{ } ) satisfying the following properties: (1) ( u λ,β n , v λ,β n ) → ( u λ, , v λ, ) strongly in H ( R ) × H ( R ) as n → ∞ up to a subsequence. (2) u λ, ∈ C ( R ) and v λ, ∈ C ( R ) . (3) u λ, ≥ and v λ, ≥ in R with { x ∈ R | u λ, ( x ) > } = R \{ x ∈ R | v λ, ( x ) > } .Furthermore, { x ∈ R | u λ, ( x ) > } and { x ∈ R | v λ, ( x ) > } are connected domains. (4) u λ, ∈ H ( { u λ, > } ) and is a least energy solution of − ∆ u + ( λa ( x ) + a ( x )) u = µ u , u ∈ H ( { u λ, > } ) , (1.3) while v λ, ∈ H ( { v λ, > } ) and is a least energy solution of − ∆ v + ( λa ( x ) + a ( x )) v = µ v , v ∈ H ( { v λ, > } ) . (1.4) Remark 1.5
In Theorem 1.2, the multi-bump solutions have been found for λ ≥ Λ ∗ ( β ) . Bychecking the proof of Theorem 1.2, we can see that Λ ∗ ( β ) → + ∞ as β → −∞ . Due to this fact,the multi-bump solutions obtained in Theorem 1.2 can not have the same phenomenon of phaseseparations as the ground state solution described in Theorem 1.3. Before closing this section, we would like to cite other references studying the equations withsteep potential wells. For example, in [45], the Kirchhoff type elliptic equation with a steep potentialwell was studied. The Schr¨odinger-Poisson systems with a steep potential well were considered in[32, 51]. Non-trivial solutions were obtained in [26, 27, 28] for quasilinear Schr¨odinger equationswith steep potential wells, while the multi-bump solutions were also obtained in [28] for suchequations.In this paper, we will always denote the usual norms in H ( R ) and L p ( R ) ( p ≥
1) by k · k and k · k p , respectively; C and C ′ will be indiscriminately used to denote various positive constants; o n (1) will always denote the quantities tending towards zero as n → ∞ .5 The variational setting
In this section, we mainly give a variational setting for ( P λ,β ). Simultaneously, an importantestimate is also established in this section, which is used frequently in this paper.Let E a = { u ∈ D , ( R ) | Z R ( a ( x ) + a +0 ( x )) u dx < + ∞} and E b = { u ∈ D , ( R ) | Z R ( b ( x ) + b +0 ( x )) u dx < + ∞} , where a +0 ( x ) = max { a ( x ) , } and b +0 ( x ) = max { b ( x ) , } . Then by the conditions ( D ) and ( D ), E a and E b are Hilbert spaces equipped with the inner products h u, v i a = Z R ∇ u ∇ v + ( a ( x ) + a +0 ( x )) uvdx and h u, v i b = Z R ∇ u ∇ v + ( b ( x ) + b +0 ( x )) uvdx, respectively. The corresponding norms of E a and E b are respectively given by k u k a = (cid:18) Z R |∇ u | + ( a ( x ) + a +0 ( x )) u dx (cid:19) and by k v k b = (cid:18) Z R |∇ v | + ( b ( x ) + b +0 ( x )) v dx (cid:19) . Since the conditions ( D )-( D ) hold, by a similar argument as that in [45], we can see that k u k ≤ (cid:18) max { |D a | S − , a ∞ } (cid:19) k u k a for all u ∈ E a (2.1)and k v k ≤ (cid:18) max { |D b | S − , b ∞ } (cid:19) k v k b for all v ∈ E b , (2.2)where S is the best Sobolev embedding constant from D , ( R ) to L ( R ) and given by S = inf {k∇ u k | u ∈ D , ( R ) , k u k = 1 } . It follows that both E a and E b are embedded continuously into H ( R ). Moreover, by applyingthe H¨older and Sobolev inequalities, we also have k u k ≤ (cid:18) max { |D a | S − , a ∞ } (cid:19) S − k u k a for all u ∈ E a (2.3)and k v k ≤ (cid:18) max { |D b | S − , b ∞ } (cid:19) S − k v k b for all v ∈ E b . (2.4)On the other hand, by the conditions ( D ) and ( D ), there exist two bounded open sets Ω ′ a and Ω ′ b with smooth boundaries such that Ω a ⊂ Ω ′ a ⊂ D a , Ω b ⊂ Ω ′ b ⊂ D b , Ω ′ a ∩ Ω ′ b = ∅ , dist(Ω a , R \ Ω ′ a ) > b , R \ Ω ′ b ) >
0. Furthermore, by the condition ( D ), the H¨older and the Sobolevinequalities, there exists Λ > { , d a + d a + C a, a ∞ , d b + d b + C b, b ∞ } such that Z R a − ( x ) u dx ≤ Z B R (0) C a, u dx + Z R \ B R (0) d a (1 + a ( x )) u dx ≤ λ k u k a (2.5)6nd Z R b − ( x ) v dx ≤ Z B R (0) C b, v dx + Z R \ B R (0) d b (1 + b ( x )) v dx ≤ λ k v k b (2.6)for λ ≥ Λ , where B R (0) = { x ∈ R | | x | < R } , C a, = sup B R (0) a − ( x ) and C b, = sup B R (0) b − ( x ).Combining (2.3)-(2.6) and the H¨older inequality, we can see that ( P λ,β ) has a variational structurein the Hilbert space E = E a × E b for λ ≥ Λ , where E is endowed with the norm k ( u, v ) k = k u k a + k v k b . The corresponding functional of ( P λ,β ) is given by (1.2). Furthermore, by applying(2.3)-(2.6) in a standard way, we can also see that J λ,β ( u, v ) is C in E and the solution of ( P λ,β )is equivalent to the positive critical point of J λ,β ( u, v ) in E for λ ≥ Λ . In the case of ( D ′ ), wecan choose Ω ′ a and Ω ′ b as follows:( I ) Ω ′ a = n a ∪ i a =1 Ω ′ a,i a ⊂ D a , where Ω a,i a ⊂ Ω ′ a,i a and dist(Ω a,i a , R \ Ω ′ a,i a ) > i a = 1 , · · · , n a and Ω ′ a,i a ∩ Ω ′ a,j a = ∅ for i a = j a .( II ) Ω ′ b = n b ∪ i b =1 Ω ′ b,i b ⊂ D b , where Ω b,i b ⊂ Ω ′ b,i b and dist(Ω b,i b , R \ Ω ′ b,i b ) > j b = 1 , · · · , n b and Ω ′ b,i b ∩ Ω ′ b,j b = ∅ for i b = j b .( III ) Ω ′ a ∩ Ω ′ b = ∅ .Thus, (2.5)-(2.6) still hold for such Ω ′ a and Ω ′ b with λ sufficiently large. Without loss of generality,we may assume that (2.5)-(2.6) still hold for such Ω ′ a and Ω ′ b with λ ≥ Λ . It follows that thesolution of ( P λ,β ) is also equivalent to the positive critical point of the C functional J λ,β ( u, v ) in E for λ ≥ Λ under the conditions ( D )-( D ), ( D ′ ) and ( D ).The remaining of this section will be devoted to an important estimate, which is used frequentlyin this paper and essentially due to Ding and Tanaka [24]. Lemma 2.1
Assume ( D ) - ( D ) . Then there exist Λ ≥ Λ and C a,b > such that inf u ∈ E a \{ } R R |∇ u | + ( λa ( x ) + a ( x )) u dx R R u dx ≥ C a,b and inf v ∈ E b \{ } R R |∇ v | + ( λb ( x ) + b ( x )) v dx R R v dx ≥ C a,b for all λ ≥ Λ . Proof.
Since the conditions ( D )-( D ) hold, by a similar argument as [24, Lemma 2.1], we havelim λ → + ∞ inf σ a, ∗ ( − ∆ + λa ( x ) + a ( x )) = inf σ a ( − ∆ + a ( x )) , where σ a, ∗ ( − ∆ + λa ( x ) + a ( x )) is the spectrum of − ∆ + λa ( x ) + a ( x ) on H (Ω ′ a ). Denoteinf σ a ( − ∆ + a ( x )) by ν a . Then by the condition ( D ), there exists Λ ′ ≥ Λ such that σ a, ∗ ( − ∆ + λa ( x ) + a ( x )) ≥ ν a λ ≥ Λ ′ . (2.7)On the other hand, by the conditions ( D ) and ( D ), we have a − ( x ) ≤ C a, + d a + d a a ∞ for x ∈ D a . Let D a,R = D a ∩ B cR , where B cR = { x ∈ R | | x | ≥ R } . Then by the condition( D ) once more, |D a,R | → R → + ∞ , which then implies that there exists R > |D a,R | S − ( C a, + d a + d a a ∞ + 1) ≤ . Thanks to the conditions ( D )-( D ), there existsΛ = Λ ( R ) ≥ Λ ′ such that λa ( x ) + a ( x ) + ( C a, + d a + d a a ∞ + 1) χ D a,R ≥ x ∈ R \ Ω ′ a and λ ≥ Λ , χ D a,R is the characteristic function of the set D a,R . It follows from the H¨older and theSobolev inequalities that Z R \ Ω ′ a u dx ≤ (1 + 2 |D a,R | S − ) Z R \ Ω ′ a |∇ u | + ( λa ( x ) + a ( x )) u dx (2.8)for all u ∈ E a \{ } and λ ≥ Λ . Combining (2.7)-(2.8) and the choice of Ω ′ a , we have Z R u dx ≤ C a Z R |∇ u | + ( λa ( x ) + a ( x )) u dx for all u ∈ E a \{ } and λ ≥ Λ , where C a = max { ν a , |D a,R | S − } . By similar arguments as (2.7) and (2.8), we can also have Z R v dx ≤ C b Z R |∇ v | + ( λb ( x ) + b ( x )) v dx for all v ∈ E b \{ } and λ ≥ Λ , where C b = max { ν b , |D b,R | S − } , ν b = inf σ b ( − ∆ + b ( x ))and D b,R = D b ∩ B cR . We completes the proof by taking C a,b = (min { C a , C b } ) − . Remark 2.1
Under the conditions ( D ) - ( D ) , ( D ′ ) , ( D ) and ( D ′ ) , we can see that ν a = min i a =1 , , ··· ,n a (cid:26) inf σ a,i a ( − ∆ + a ( x )) (cid:27) and ν b = min j b =1 , , ··· ,n b (cid:26) inf σ b,j b ( − ∆ + b ( x )) (cid:27) . Now, by a similar argument as (2.7) , we get that Z Ω ′ a,ia u dx ≤ ν a Z Ω ′ a,ia |∇ u | + ( λa ( x ) + a ( x )) u dx (2.9) and Z Ω ′ b,jb v dx ≤ ν b Z Ω ′ b,jb |∇ v | + ( λb ( x ) + b ( x )) v dx (2.10) for all i a = 1 , · · · , n a and j b = 1 , · · · , n b if λ sufficiently large. Without loss of generality, we mayassume (2.9) and (2.10) hold for λ ≥ Λ . It follows that Lemma 2.1 still holds under the conditions ( D ) - ( D ) , ( D ′ ) , ( D ) and ( D ′ ) . By Lemma 2.1, we observe that R R |∇ u | + ( λa ( x ) + a ( x )) u dx and R R |∇ v | + ( λb ( x ) + b ( x )) v dx are norms of E a and E b for λ ≥ Λ , respectively. Therefore, we set k u k a,λ = Z R |∇ u | + ( λa ( x ) + a ( x )) u dx ; k v k b,λ = Z R |∇ v | + ( λb ( x ) + b ( x )) v dx. Our interest in this section is to find a ground state solution to ( P λ,β ) under the conditions ( D )-( D ). For the sake of convenience, we always assume the conditions ( D )-( D ) hold in this section.Since J λ,β ( u, v ), the corresponding energy functional of ( P λ,β ), is C in E , it is well-known thatall non-trivial solutions of ( P λ,β ) lie in the Nehari manifold of J λ,β ( u, v ), which is given by N λ,β = { ( u, v ) ∈ E | u = 0 , v = 0 , h D [ J λ,β ( u, v )] , ( u, i E ∗ ,E = h D [ J λ,β ( u, v )] , (0 , v ) i E ∗ ,E = 0 } , where D [ J λ,β ( u, v )] is the Frech´et derivative of the functional J λ,β in E at ( u, v ) and E ∗ isthe dual space of E . If we can find ( u λ,β , v λ,β ) ∈ E such that J λ,β ( u λ,β , v λ,β ) = m λ,β and D [ J λ,β ( u λ,β , v λ,β )] = 0 in E ∗ , then ( u λ,β , v λ,β ) must be a ground state solution of ( P λ,β ), where m λ,β = inf N λ,β J λ,β ( u, v ). In what follows, we drive some properties of N λ,β .8et ( u, v ) ∈ ( E a \{ } ) × ( E b \{ } ) and define T λ,β,u,v : R + × R + → R by T λ,β,u,v ( t, s ) = J λ,β ( tu, sv ). These functions are called the fibering maps of J λ,β ( u, v ), which are closely linked to N λ,β . Clearly, ∂T λ,β,u,v ∂t ( t, s ) = ∂T λ,β,u,v ∂s ( t, s ) = 0 is equivalent to ( tu, sv ) ∈ N λ,β . In particular, ∂T λ,β,u,v ∂t (1 ,
1) = ∂T λ,β,u,v ∂s (1 ,
1) = 0 if and only if ( u, v ) ∈ N λ,β . Let A β = { ( u, v ) ∈ E | µ µ k u k k v k − β k u v k > } . (3.1)Then A β = ∅ for every β <
0. Now, our first observation on N λ,β can be stated as follows. Lemma 3.1
Assume λ ≥ Λ and β < . Then we have the following. (1) If ( u, v ) ∈ A β , then there exists a unique ( t λ,β ( u, v ) , s λ,β ( u, v )) ∈ R + × R + such that ( t λ,β ( u, v ) u, s λ,β ( u, v ) v ) ∈ N λ,β , where t λ,β ( u, v ) and s λ,β ( u, v ) are given by t λ,β ( u, v ) = (cid:18) µ k v k k u k a,λ − β k u v k k v k b,λ µ µ k u k k v k − β k u v k (cid:19) (3.2) and by s λ,β ( u, v ) = (cid:18) µ k u k k v k b,λ − β k u v k k u k a,λ µ µ k u k k v k − β k u v k (cid:19) . (3.3) Moreover, T λ,β,u,v ( t λ,β ( u, v ) , s λ,β ( u, v )) = max t ≥ ,s ≥ T λ,β,u,v ( t, s ) . (2) If ( u, v ) ∈ E \A β , then B u,v ∩ N λ,β = ∅ , where B u,v = { ( tu, sv ) | ( t, s ) ∈ R + × R + } . Proof. (1) The proof is similar to [20, Lemma 2.2], where N ,β with a ( x ) = a > b ( x ) = b > T λ,β,u,v and T λ,β,u,v be two functions on R + × R + defined by T λ,β,u,v ( t, s ) = k u k a,λ − µ k u k t − β k u v k s and by T λ,β,u,v ( t, s ) = k v k b,λ − µ k v k s − β k u v k t . Then it is easy to see that ∂T λ,β,u,v ∂t ( t, s ) = tT λ,β,u,v ( t, s ) and ∂T λ,β,u,v ∂s ( t, s ) = sT λ,β,u,v ( t, s ) . (3.4)Suppose ( u, v ) ∈ A β , λ ≥ Λ and β <
0. Then by Lemma 2.1, the two-component systems ofalgebraic equations, given by ( T λ,β,u,v ( t, s ) = 0 ,T λ,β,u,v ( t, s ) = 0 , (3.5)has a unique nonzero solution ( t λ,β ( u, v ) , s λ,β ( u, v )) in R + × R + , where ( t λ,β ( u, v ) , s λ,β ( u, v ))is characterized as (3.2) and (3.3). Hence, by (3.4), ( t λ,β ( u, v ) , s λ,β ( u, v )) is the unique onein R + × R + such that ∂T λ,β,u,v ∂t ( t, s ) = ∂T λ,β,u,v ∂s ( t, s ) = 0, that is, ( t λ,β ( u, v ) , s λ,β ( u, v )) is theunique one in R + × R + such that ( t λ,β ( u, v ) u, s λ,β ( u, v ) v ) ∈ N λ,β . It remains to show that T λ,β,u,v ( t λ,β ( u, v ) , s λ,β ( u, v )) = max t ≥ ,s ≥ T λ,β,u,v ( t, s ). Indeed, by a direct calculation, we have ∂ T λ,β,u,v ∂t ( t λ,β ( u, v ) , s λ,β ( u, v )) = − µ k u k [ t ( u, v )] < (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ T λ,β,u,v ∂t ( t λ,β ( u, v ) , s λ,β ( u, v )) ∂ T λ,β,u,v ∂t∂s ( t λ,β ( u, v ) , s λ,β ( u, v )) ∂ T λ,β,u,v ∂s∂t ( t λ,β ( u, v ) , s λ,β ( u, v )) ∂ T λ,β,u,v ∂s ( t λ,β ( u, v ) , s λ,β ( u, v )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) − t λ,β ( u, v )] µ k u k − t λ,β ( u, v ) s λ,β ( u, v ) β k u v k − t λ,β ( u, v ) s λ,β ( u, v ) β k u v k − s λ,β ( u, v )] µ k v k (cid:12)(cid:12)(cid:12)(cid:12) = 4[ t λ,β ( u, v )] [ s λ,β ( u, v )] ( µ µ k u k k v k − β k u v k ) > , which implies that ( t λ,β ( u, v ) , s λ,β ( u, v )) is a local maximum of T λ,β,u,v ( t, s ) in R + × R + . It followsfrom the uniqueness of ( t λ,β ( u, v ) , s λ,β ( u, v )), T λ,β,u,v ( t, s ) > | ( t, s ) | sufficiently small and T λ,β,u,v ( t, s ) → −∞ as | ( t, s ) | → + ∞ that ( t λ,β ( u, v ) , s λ,β ( u, v )) must be the global maximum of T λ,β,u,v ( t, s ) in R + × R + . Thus, we have T λ,β,u,v ( t λ,β ( u, v ) , s λ,β ( u, v )) = max t ≥ ,s ≥ T λ,β,u,v ( t, s ) . (2) Suppose ( u, v )
6∈ A β , λ ≥ Λ and β <
0. If B u,v ∩ N λ,β = ∅ , then there exists ( t, s ) ∈ R + × R + such that ∂T λ,β,u,v ∂t ( t, s ) = ∂T λ,β,u,v ∂s ( t, s ) = 0. It follows from (3.4) that ( t, s ) is a solutionof (3.5) in R + × R + . On the other hand, since ( u, v )
6∈ A β , λ ≥ Λ and β <
0, by Lemma 2.1,(3.5) has no solution in R + × R + , which is a contradiction. Hence, we must have B u,v ∩ N λ,β = ∅ if ( u, v )
6∈ A β , λ ≥ Λ and β < N λ,β ⊂ A β for λ ≥ Λ and β <
0. Moreover, m λ,β is well definedand nonnegative for λ ≥ Λ and β <
0. Let I a,λ ( u ) = 12 k u k a,λ − µ k u k and I b,λ ( v ) = 12 k v k b,λ − µ k v k . Then by (2.3)-(2.6), I a,λ ( u ) is well defined on E a and I b,λ ( v ) is well defined on E b . Moreover, bya standard argument, we can see that I a,λ ( u ) and I b,λ ( v ) are of C in E a and E b , respectively.Denote N a,λ = { u ∈ E a \{ } | I ′ a,λ ( u ) u = 0 } and N b,λ = { u ∈ E b \{ } | I ′ b,λ ( u ) u = 0 } . Clearly, N a,λ and N b,λ are nonempty, which together with Lemma 2.1, implies m a,λ = inf N a,λ I a,λ ( u )and m b,λ = inf N b,λ I b,λ ( v ) are well defined and nonnegative. Due to this fact, we have the following. Lemma 3.2
Assume λ ≥ Λ and β < . Then m λ,β ∈ [ m a,λ + m b,λ , m a + m b ] ,where m a and m b are the least energy of nonzero critical points for I Ω a ( u ) and I Ω b ( v ) , respectively. Proof.
Suppose ( u, v ) ∈ N λ,β . Then by Lemma 2.1 and ( u, v ) ∈ N λ,β ⊂ A β , we can seethat k u k a,λ > k v k b,λ > k u k > k v k >
0. It follows that there exists a unique( t ∗ ( u ) , s ∗ ( v )) ∈ R + × R + such that ( t ∗ ( u ) u, s ∗ ( v ) v ) ∈ N a,λ × N b,λ . Note that β <
0, so byLemma 3.1, we have J λ,β ( u, v ) ≥ J λ,β ( t ∗ ( u ) u, s ∗ ( v ) v ) ≥ I a,λ ( t ∗ ( u ) u ) + I b,λ ( s ∗ ( v ) v ) ≥ m a,λ + m b,λ . Since ( u, v ) ∈ N λ,β is arbitrary, we must have m λ,β ≥ m a,λ + m b,λ for λ ≥ Λ and β <
0. Itremains to show that m λ,β ≤ m a + m b for λ ≥ Λ and β <
0. In fact, let w a ∈ H (Ω a ) and w b ∈ H (Ω b ) be the least energy nonzero critical points of I Ω a ( u ) and I Ω b ( v ), respectively. Thenby the conditions ( D ) and ( D ), it is well-known that I Ω a ( w a ) = max t ≥ I Ω a ( tw a ) and I Ω b ( w b ) = max s ≥ I Ω b ( sw b ) . On the other hand, by the condition ( D ), we can extend w a and w b to R by letting w a = 0 outsideΩ a and w b = 0 outside Ω b such that w a , w b ∈ H ( R ). Thanks to the condition ( D ) again, we can10ee that ( w a , w b ) ∈ A β . It follows from Lemma 3.1 that there exists ( t λ,β ( w a , w b ) , s λ,β ( w a , w b )) ∈ R + × R + such that ( t λ,β ( w a , w b ) w a , s λ,β ( w a , w b ) w b ) ∈ N λ,β for λ ≥ Λ , which together with the condition ( D ) once more, implies m a + m b = I Ω a ( w a ) + I Ω b ( w b ) ≥ I Ω a ( t λ,β ( w a , w b ) w a ) + I Ω b ( s λ,β ( w a , w b ) w b )= J λ,β ( t λ,β ( w a , w b ) w a , s λ,β ( w a , w b ) w b ) ≥ m λ,β for λ ≥ Λ and β < m a,λ and m b,λ are nondecreasing for λ . On the other hand, since Lemma 2.1 hold,by the conditions ( D )-( D ), it is easy to show that m a,λ and m b,λ are positive for λ ≥ Λ and m a,λ → m a and m b,λ → m b as λ → + ∞ . This fact will help us to observe the following propertyof N λ,β , which is based on Lemma 3.2. Lemma 3.3
Assume λ ≥ Λ and β < . Then there exists d λ,β > such that N λ,β ⊂ A d λ,β β ,where A d λ,β β = { ( u, v ) ∈ E | u = 0 , v = 0 , µ µ k u k k v k − β k u v k > d λ,β } . Proof.
A similar result was obtained in [16]. But as we will see, some new ideas are needed dueto the fact that a ( x ) and b ( x ) are sign-changing. Suppose the contrary. Since N λ,β ⊂ A β , thereexists a sequence { ( u n , v n ) } ⊂ N λ,β such that µ µ k u n k k v n k = β k u n v n k + o n (1), where A β isgiven in (3.1). Clearly, one of the following two cases must happen:( a ) k u n k k v n k ≥ C + o n (1).( b ) k u n k k v n k = o n (1) up to a subsequence.Suppose case ( a ) happen. We claim that µ k u n k + β k u n v n k = o n (1) and µ k v n k + β k u n v n k = o n (1) up to a subsequence. If not, then up to a subsequence, we have µ k u n k + β k u n v n k ≥ C + o n (1) and µ k v n k + β k u n v n k ≥ C + o n (1)for λ ≥ Λ and β <
0, where C , C are nonnegative constants with C + C >
0. It follows from β < µ µ k u n k k v n k ≥ ( C + o n (1) + | β |k u n v n k )( C + o n (1) + | β |k u n v n k ) ≥ β k u n v n k + ( C + C + o n (1)) | β |k u n v n k + C C + o n (1) ≥ β k u n v n k + 12 ( C + C ) √ C + o n (1)for n large enough, which contradicts to µ µ k u n k k v n k = β k u n v n k + o n (1). This together with { ( u n , v n ) } ⊂ N λ,β , implies that k u n k a,λ = o n (1) and k v n k b,λ = o n (1) up to a subsequence. Notethat J λ,β ( u n , v n ) = ( k u n k a,λ + k v n k b,λ ). So m λ,β ≤ a ), which is impossible for λ ≥ Λ and β < b ). It follows that k u n k = o n (1) or k v n k = o n (1) up to a subsequence. Without loss of generality, we assume k u n k = o n (1). Since µ µ k u n k k v n k = β k u n v n k + o n (1), we also have β k u n v n k = o n (1) in this case. These togetherwith { ( u n , v n ) } ⊂ N λ,β , imply k u n k a,λ = o n (1). Therefore, J λ,β ( u n , v n ) = I b,λ ( v n ) + o n (1). Onthe other hand, since λ ≥ Λ and { ( u n , v n ) } ⊂ N λ,β , by Lemma 2.1 and N λ,β ⊂ A β , for all n ∈ N ,there exists a unique t ∗ ( u n ) > t ∗ ( u n ) u n ∈ N a,λ . It follows from Lemma 3.1 and β < J λ,β ( u n , v n ) ≥ J λ,β ( t ∗ ( u n ) u n , v n ) ≥ I a,λ ( t ∗ ( u n ) u n ) + I b,λ ( v n ) ≥ m a,λ + I b,λ ( v n )= m a,λ + J λ,β ( u n , v n ) + o n (1)11or λ ≥ Λ and β <
0, which is also impossible for n large enough. Thus, there exists d λ,β > N λ,β ⊂ A d λ,β β for λ ≥ Λ and β < Lemma 3.4
Suppose λ ≥ Λ and β < . Then N λ,β is a natural constraint. Proof.
Let ϕ λ,β ( u, v ) = h D [ J λ,β ( u, v )] , ( u, v ) i E ∗ ,E . Then by (2.3)-(2.6), ϕ λ,β ( u, v ) is C in E for λ ≥ Λ and β <
0. Since λ ≥ Λ and ( u, v ) ∈ N λ,β , we have h D [ ϕ λ,β ( u, v )] , ( u, v ) i E ∗ ,E = − µ k u k + µ k v k + 2 β k u v k ) ≤ − m λ,β . It follows from Lemma 3.2 that N λ,β is a natural constraint for λ ≥ Λ and β < P λ,β ). Proposition 3.1
There exists Λ ≥ Λ such that ( P λ,β ) has a ground state solution ( u λ,β , v λ,β ) for λ ≥ Λ and β < . Furthermore, we have lim λ → + ∞ Z R \ Ω ′ a |∇ u λ,β | + ( λa ( x ) + a ( x )) u λ,β dx = 0 (3.6) and lim λ → + ∞ Z R \ Ω ′ b |∇ v λ,β | + ( λb ( x ) + b ( x )) v λ,β dx = 0 . (3.7) Proof.
Let λ ≥ Λ and β <
0. Then for every { ( u n , v n ) } ⊂ N λ,β satisfying J λ,β ( u n , v n ) = m λ,β + o n (1), we can see from Lemma 2.1 that m λ,β + o n (1) ≥ J λ,β ( u n , v n ) − h D [ J λ,β ( u n , v n )] , ( u n , v n ) i E ∗ ,E = 14 k u n k a,λ + 14 k v n k b,λ ≥ C a,b ( k u n k + k v n k ) , (3.8)which together with the condition ( D ) and λ ≥ Λ , implies m λ,β + o n (1) ≥ k u n k a,λ + 14 k v n k b,λ ≥ k ( u n , v n ) k − C ( m λ,β + o n (1)) . (3.9)It follows that k ( u n , v n ) k ≤ C + 1)( m λ,β + o n (1)). Now, by Lemma 3.3, we can apply theimplicit function theorem and the Ekeland variational principle in a standard way (cf. [13, 36])to show that there exists { ( u n , v n ) } ⊂ N λ,β such that D [ J λ,β ( u n , v n )] = o n (1) strongly in E ∗ and J λ,β ( u n , v n ) = m λ,β + o n (1). Since m λ,β ≤ m a + m b , by similar arguments as (3.8) and (3.9), wehave k ( u n , v n ) k ≤ C + 1)( m a + m b + o n (1)) and ( u n , v n ) ⇀ ( u λ,β , v λ,β ) weakly in E as n → ∞ for some ( u λ,β , v λ,β ) ∈ E . Clearly, D [ J λ,β ( u λ,β , v λ,β )] = 0 in E ∗ . Suppose u λ,β = 0. Then by thefact that E a is embedded continuously into H ( R ), we have u n = o n (1) strongly in L ploc ( R ) for 2 ≤ p < . D ) and the H¨older and the Sobolev inequalities, we get Z R | u n | dx = Z D a | u n | dx + Z R \D a | u n | dx = Z R \D a | u n | dx + o n (1) ≤ (cid:18) a ∞ (cid:19) Z R \D a [ a ( x )] | u n | dx + o n (1) ≤ (cid:18) a ∞ S (cid:19) (cid:18) Z R \D a a ( x ) | u n | dx (cid:19) (cid:18) Z R |∇ u n | dx (cid:19) + o n (1) . (3.10)Since u n = o n (1) strongly in L ploc ( R ) for 2 ≤ p <
6, by the conditions ( D ) and ( D ), R D a ( λa ( x ) + a ( x )) u n dx = o n (1). It follows from (3.8) and (3.10) that Z R | u n | dx ≤ (cid:18) a ∞ S (cid:19) (cid:18) Z R \D a a ( x ) | u n | dx (cid:19) (cid:18) Z R |∇ u n | dx (cid:19) + o n (1) ≤ (cid:18) a ∞ S λ (cid:19) k u n k a,λ ( k u n k a,λ + o n (1)) + o n (1) ≤ (cid:18) a ∞ S λ (cid:19) k u n k a,λ + o n (1) (3.11)for λ ≥ Λ . Note that { ( u n , v n ) } ⊂ N λ,β and β <
0, from (3.8) and (3.11), we have k u n k a,λ ≤ (cid:18) a ∞ S λ (cid:19) k u n k a,λ + o n (1) ≤ m a + m b ) (cid:18) a ∞ S λ (cid:19) k u n k a,λ + o n (1) , (3.12)which then implies that there exists Λ ≥ Λ such that k u n k a,λ = o n (1) for λ ≥ Λ and β <
0. Itfollows from Lemma 2.1, the H¨older and the Sobolev inequalities and the boundedness of { ( u n , v n ) } in E that k u n k = o n (1), hence, µ µ k u n k k v n k = o n (1) for λ ≥ Λ and β <
0. However, itis impossible, since { ( u n , v n ) } ⊂ N λ,β , Λ ≥ Λ and Lemma 3.3 holds for λ ≥ Λ . Therefore,there exists Λ ≥ Λ such that u λ,β = 0 for λ ≥ Λ and β <
0. Similarly, we can also showthat v λ,β = 0 for λ ≥ Λ and β <
0. Since ( u n , v n ) ⇀ ( u λ,β , v λ,β ) weakly in E as n → ∞ , bythe fact that E is embedded continuously into H ( R ) × H ( R ), we have ( u n , v n ) → ( u λ,β , v λ,β )strongly in L ploc ( R ) × L ploc ( R ) as n → ∞ for 2 ≤ p <
6. It follows from the boundedness of { ( u n , v n ) } in E and the conditions ( D ) and ( D ) that R D a a − ( x ) u n dx = R D a a − ( x ) u λ,β dx + o n (1)and R D b b − ( x ) v n dx = R D b b − ( x ) v λ,β dx + o n (1), which together with D [ J λ,β ( u λ,β , v λ,β )] = 0 in E ∗ ,the Fatou lemma and the conditions ( D ) and ( D ), implies m λ,β ≤ J λ,β ( u λ,β , v λ,β ) − h D [ J λ,β ( u λ,β , v λ,β )] , ( u λ,β , v λ,β ) i E ∗ ,E = 14 ( k u λ,β k a,λ + k v λ,β k b,λ ) ≤ lim inf n →∞
14 ( k u n k a,λ + k v n k b,λ ) (3.13)= lim inf n →∞ ( J λ,β ( u λ,β , v λ,β ) − h D [ J λ,β ( u n , v n )] , ( u n , v n ) i E ∗ ,E )= m λ,β + o n (1) . Therefore, J λ,β ( u λ,β , v λ,β ) = m λ,β . Since J λ,β ( | u λ,β | , | v λ,β | ) = m λ,β and ( | u λ,β | , | v λ,β | ) ∈ N λ,β ,( | u λ,β | , | v λ,β | ) is a local minimizer of J λ,β ( u, v ) on N λ,β . Note that by Lemma 3.4, N λ,β isa natural constraint, we can follow the argument as used in [6, Theorem 2.3] to show that D [ J λ,β ( | u λ,β | , | v λ,β | )] = 0 in E ∗ . Thus, without loss of generality, we may assume u λ,β and v λ,β are13oth nonnegative. Now, since ( u λ,β , v λ,β ) ∈ E , by (2.1) and (2.2), we have u λ,β , v λ,β ∈ H ( R ). Itfollows from the conditions ( D ) and ( D ) and the Calderon-Zygmund inequality that u λ,β , v λ,β ∈ W , loc ( R ). By combining the Sobolev embedding theorem and the Harnack inequality, u λ,β and v λ,β are both positive. Hence, ( u λ,β , v λ,β ) is a ground state solution of ( P λ,β ) for β < λ ≥ Λ .It remains to show that (3.6) and (3.7) are true. Indeed, let Ω ′′ a be a bounded domain with smoothboundary in R satisfying Ω a ⊂ Ω ′′ a ⊂ Ω ′ a , dist (Ω ′′ a , R \ Ω ′ a ) > dist ( R \ Ω ′′ a , Ω a ) >
0. Then bya similar argument as (2.7), we can show that Z Ω ′′ a |∇ u λ,β | + ( λa ( x ) + a ( x )) u λ,β dx ≥ ν a Z Ω ′′ a u λ,β dx (3.14)for λ large enough. Without loss of generality, we assume (3.14) holds for λ ≥ Λ . Since ( u λ,β , v λ,β )is a ground state solution for λ ≥ Λ , by combining Lemma 2.1, (3.14) and similar arguments of(3.8) and (3.9), we can see that k ( u λ,β , v λ,β ) k ≤ C a, + d a ) C a,b + 1)( m a + m b ) and 8(4( C a, + d a ) C a,b + 1)( m a + m b ) ≥ λ R R \ Ω ′′ a a ( x ) u λ,β dx for λ ≥ Λ ,which together with the conditions( D )-( D ), imply R ( R \ Ω ′′ a ) ∩D a u λ,β dx → λ → + ∞ . It follows from the condition ( D ) and8(4( C a, + d a ) C a,b + 1)( m a + m b ) ≥ λ R R \ Ω ′′ a a ( x ) u λ,β dx once more thatlim λ → + ∞ Z R \ Ω ′′ a u λ,β dx = 0 . (3.15)Now, we choose Ψ ∈ C ∞ ( R , [0 , ( , x ∈ R \ Ω ′ a , , x ∈ Ω ′′ a . (3.16)Then u λ,β Ψ ∈ E a . Note D [ J λ,β ( u λ,β , v λ,β )] = 0 in E ∗ for λ ≥ Λ , we have that Z R ( |∇ u λ,β | + ( λa ( x ) + a ( x )) u λ,β )Ψ dx + Z R ( ∇ u λ,β ∇ Ψ) u λ,β dx = µ Z R u λ,β Ψ dx + β Z R v λ,β u λ,β Ψ dx. It follows from the H¨older and the Sobolev inequalities that Z R \ Ω ′ a ( |∇ u λ,β | + ( λa ( x ) + a ( x )) u λ,β ) dx ≤ µ Z R \ Ω ′′ a u λ,β dx + β Z R \ Ω ′ a v λ,β u λ,β dx + Z Ω ′ a \ Ω ′′ a |∇ u λ,β ||∇ Ψ || u λ,β | dx ≤ µ S − k u λ,β k a ( Z R \ Ω ′′ a u λ,β dx ) + max R |∇ Ψ |k u λ,β k a ( Z R \ Ω ′′ a u λ,β dx ) . (3.17)Since k ( u λ,β , v λ,β ) k ≤ C a, + d a ) C a,b + 1)( m a + m b ), we can conclude from (2.8), (3.15) and(3.17) that (3.6) holds. Similarly, we can also conclude that (3.7) is true.We close this section by Proof of Theorem 1.1:
By Proposition 3.1, we know that there exists Λ ≥ Λ such that( P λ,β ) has a ground state solution ( u λ,β , v λ,β ) for λ ≥ Λ and β <
0. In what follows, we will showthat ( u λ,β , v λ,β ) has the concentration behaviors for λ → + ∞ described as (1)-(5). We first verify(3)-(5). Let ( u λ n ,β , v λ n ,β ) be the ground state solution of ( P λ n ,β ) obtained by Proposition 3.1 with λ n → + ∞ as n → ∞ . Then by Lemma 3.2 and Proposition 3.1, { ( u λ n ,β , v λ n ,β ) } is bounded in E with lim n → + ∞ Z R \ Ω ′ a |∇ u λ n ,β | + ( λ n a ( x ) + a ( x )) u λ n ,β dx = 0 (3.18)14nd lim n → + ∞ Z R \ Ω ′ b |∇ v λ n ,β | + ( λ n b ( x ) + b ( x )) v λ n ,β dx = 0 . (3.19)Without loss of generality, we assume ( u λ n ,β , v λ n ,β ) ⇀ ( u ,β , v ,β ) weakly in E as n → ∞ for some( u ,β , v ,β ) ∈ E . By (2.1) and (2.2), ( u ,β , v ,β ) ∈ H ( R ) × H ( R ). For the sake of clarity, theverification of (3)-(5) is further performed through the following three steps. Step 1.
We prove that ( u ,β , v ,β ) ∈ H (Ω a ) × H (Ω b ) with u ,β = 0 on R \ Ω a and v ,β = 0on R \ Ω b .Indeed, since ( u λ n ,β , v λ n ,β ) is a ground state solution of ( P λ n ,β ), by Lemma 3.2 and a similarargument as (3.8), we get that 4 C a,b ( m a + m b ) ≥ ( k u λ n ,β k + k v λ n ,β k ). Now, by the condition( D ) and a similar argument as (3.8) again, we have m a + m b ≥ J λ n ,β ( u λ n ,β , v λ n ,β ) − h D [ J λ n ,β ( u λ n ,β , v λ n ,β )] , ( u λ n ,β , v λ n ,β ) i E ∗ ,E ≥ Z R ( λ n a ( x ) + a ( x )) u λ n ,β dx + 14 Z R ( λ n b ( x ) + b ( x )) v λ n ,β dx ≥ λ n Z R a ( x ) u λ n ,β + b ( x ) v λ n ,β dx − C ( k ( u λ n ,β k + k v λ n ,β ) k ) ≥ λ n Z R a ( x ) u λ n ,β + b ( x ) v λ n ,β dx − C ′ . It follows that lim n → + ∞ R R a ( x ) u λ n ,β dx = lim n → + ∞ R R b ( x ) v λ n ,β dx = 0. By the Fatou Lemmaand the conditions ( D ) and ( D ), we can see that u ,β = 0 on R \ Ω a and v ,β = 0 on R \ Ω b .Since ( u ,β , v ,β ) ∈ H ( R ), by the condition ( D ) again, we must have u ,β ∈ H (Ω a ) and v ,β ∈ H (Ω b ). Step 2.
We prove that ( u λ n ,β , v λ n ,β ) → ( u ,β , v ,β ) strongly in H ( R ) × H ( R ) as n → ∞ up to a subsequence.Indeed, by the choice of Ω ′ a and the Sobolev embedding theorem, we can see that u λ n ,β → u ,β strongly in L (Ω ′ a ) as n → ∞ up to a subsequence. Without loss of generality, we may assume u λ n ,β → u ,β strongly in L (Ω ′ a ). It follows from u ,β = 0 on R \ Ω a , (2.8) and (3.18) that u λ n ,β → u ,β strongly in L ( R ) as n → ∞ . By the H¨older and the Sobolev inequalities and theboundedness of { ( u λ n ,β , v λ n ,β ) } in E , we can see that u λ n ,β → u ,β strongly in L ( R ) as n → ∞ .On the other hand, by a similar argument as (3.13), we have Z R |∇ u ,β | + a ( x ) u ,β dx = Z R |∇ u ,β | dx + ( λ n a ( x ) + a ( x )) u ,β dx ≤ lim inf n →∞ Z R |∇ u λ n ,β | + ( λ n a ( x ) + a ( x )) u λ n ,β dx, which together with D [ J λ n ,β ( u λ n ,β , v λ n ,β )] = 0 in E ∗ and β <
0, implies Z R |∇ u ,β | + a ( x ) u ,β dx ≤ lim inf n →∞ µ Z R u λ n ,β dx = µ Z Ω a u ,β dx. Note that ( u ,β , v ,β ) ∈ H (Ω a ) × H (Ω b ) with u ,β = 0 on R \ Ω a and v ,β = 0 on R \ Ω b and D [ J λ n ,β ( u λ n ,β , v λ n ,β )] = 0 in E ∗ , by the condition ( D ), we can show that I ′ Ω a ( u ,β ) = 0in H − (Ω a ) and I ′ Ω b ( v ,β ) = 0 in H − (Ω b ). Recalling that u λ n ,β → u ,β strongly in L ( R ) as n → ∞ , { ( u λ n ,β , v λ n ,β ) } is bounded in E and the conditions ( D ) and ( D ) hold, we must have R D a a − ( x ) u λ n ,β dx = R D a a − ( x ) u ,β dx + o n (1). It follows from the conditions ( D ) and ( D ) again15nd the Fatou Lemma that Z R |∇ u ,β | + a ( x ) u ,β dx = lim inf n →∞ Z R |∇ u λ n ,β | + ( λ n a ( x ) + a ( x )) u λ n ,β dx ≥ lim inf n →∞ ( Z R |∇ u λ n ,β | + λ n a ( x ) u λ n ,β dx )+ Z R a +0 ( x ) u ,β dx + Z D a a − ( x ) u ,β dx. (3.20)By the conditions ( D )-( D ), the Fatou lemma and the fact u ,β = 0 on R \ Ω a , we can see from(3.20) that ∇ u λ n ,β → ∇ u ,β strongly in L ( R ) as n → ∞ up to a subsequence, which thenimplies lim inf n →∞ R R λ n a ( x ) u λ n ,β dx = 0 and lim inf n →∞ R R a ( x ) u λ n ,β dx = R R a ( x ) u ,β dx .These together with the conditions ( D )-( D ), u ,β = 0 on R \ Ω a and u λ n ,β → u ,β strongly in L ( R ) as n → ∞ , imply u λ n ,β → u ,β strongly in E a as n → ∞ up to a subsequence. Withoutloss of generality, we assume u λ n ,β → u ,β strongly in E a as n → ∞ . Similarly, we also have v λ n ,β → v ,β strongly in E b as n → ∞ , that is, ( u λ n ,β , v λ n ,β ) → ( u ,β , v ,β ) strongly in E as n → ∞ . Since E is embedded continuously into H ( R ) × H ( R ), ( u λ n ,β , v λ n ,β ) → ( u ,β , v ,β )strongly in H ( R ) × H ( R ) as n → ∞ . Step 3.
We prove that u ,β and v ,β are least energy nonzero critical points of I Ω a ( u ) and I Ω b ( v ), respectively.Indeed, since ( u λ n ,β , v λ n ,β ) is the ground state solution of ( P λ n ,β ), by Lemma 3.2, we can seethat k u λ n ,β k a,λ n + k v λ n ,β k b,λ n = 4( m a + m b ) + o n (1) . (3.21)By a similar argument as used in Step 2, we can show that λ n Z R a ( x ) u λ n ,β dx = λ n Z R b ( x ) v λ n ,β dx = o n (1)and Z R a ( x ) u λ n ,β dx = Z R a ( x ) u ,β dx + o n (1) and Z R b ( x ) v λ n ,β dx = Z R b ( x ) v ,β dx + o n (1) . These together with Step 2 and (3.21), imply Z R |∇ u ,β | + a ( x ) u ,β dx + Z R |∇ v ,β | + b ( x ) v ,β dx = 4( m a + m b ) . (3.22)We claim that Z R u λ n ,β dx ≥ C + o n (1) and Z R v λ n ,β dx ≥ C + o n (1) . (3.23)Indeed, suppose the contrary, we have either R R u λ n ,β dx = o n (1) or R R v λ n ,β dx = o n (1) up to asubsequence. Without loss of generality, we assume lim n →∞ R R u λ n ,β dx = 0. By the boundednessof { ( u λ n ,β , v λ n ,β ) } in E and the H¨older and Sobolev inequalities, β R R u λ n ,β v λ n ,β = o n (1), whichimplies J λ n ,β ( u λ n ,β , v λ n ,β ) = I b,λ n ( v λ n ,β ) + o n (1). By Lemma 2.1 and N λ,β ⊂ A β , for every n ,there exists a unique t ∗ n ( β ) > t ∗ n ( β ) u λ n ,β ∈ N a,λ n . It follows from Lemma 3.1 and β < J λ n ,β ( u λ n ,β , v λ n ,β ) ≥ J λ n ,β ( t ∗ n ( β ) u λ n ,β , v λ n ,β ) ≥ I a,λ n ( t ∗ n ( β ) u λ n ,β ) + I b,λ n ( v λ n ,β ) ≥ m a,λ n + I b,λ n ( v λ n ,β )= m a,λ n + J λ n ,β ( u λ n ,β , v λ n ,β ) + o n (1)= m a + J λ n ,β ( u λ n ,β , v λ n ,β ) + o n (1) . (3.24)16ince m a >
0, (3.24) is impossible for n large enough. Now, (3.23) together with Steps 1-2, implies Z Ω a u ,β dx ≥ C > Z Ω b v ,β dx ≥ C > . (3.25)Note that I ′ Ω a ( u ,β ) = 0 in H − (Ω a ) and I ′ Ω b ( v ,β ) = 0 in H − (Ω b ), by (3.25), we have Z R |∇ u ,β | + a ( x ) u ,β dx ≥ m a and Z R |∇ v ,β | + b ( x ) v ,β dx ≥ m b . It follows from (3.22) that u ,β is a least energy nonzero critical point of I Ω a ( u ) and v ,β is a leastenergy nonzero critical point of I Ω b ( v ).We close the proof of Theorem 1.1 by verifying (1) and (2). Supposing the contrary, thereexists { λ n } with λ n → + ∞ as n → ∞ such that one of the following cases must happen:( a ) R R \ Ω a |∇ u λ n ,β | + u λ n ,β dx ≥ C + o n (1);( b ) R R \ Ω b |∇ v λ n ,β | + v λ n ,β dx ≥ C + o n (1);( c ) | R Ω a |∇ u λ n ,β | + a ( x ) u λ n ,β dx − m a | ≥ C + o n (1);( d ) | R Ω b |∇ v λ,β | + b ( x ) v λ n ,β dx − m b | ≥ C + o n (1).By Steps 1-3 and the condition ( D ), it is easy to see that ( c ) and ( d ) can not hold, which thenimplies that we must have ( a ) or ( b ). Since (3.21) holds for { ( u λ n ,β , v λ n ,β ) } , by Steps 2-3 and thecondition ( D ), we can see that Z R \ Ω a |∇ u λ n ,β | + ( λ n a ( x ) + a ( x )) u λ n ,β dx + Z R \ Ω b |∇ v λ n ,β | + ( λ n b ( x ) + b ( x )) v λ n ,β dx → n → ∞ . It follows from the conditions ( D ) and ( D ) and Steps 1-2 that Z ( R \ Ω a ) ∩D a a − ( x ) u λ n ,β dx = Z ( R \ Ω b ) ∩D b b − ( x ) v λ n ,β dx = o n (1) , which then together with the conditions ( D ) and ( D ) and and Steps 1-2 once more, implies Z R \ Ω a |∇ u λ n ,β | + ( λ n a ( x ) + a ( x )) u λ n ,β dx + Z R \ Ω b |∇ v λ n ,β | + ( λ n b ( x ) + b ( x )) v λ n ,β dx ≥ Z R \ Ω a |∇ u λ n ,β | dx + Z R \ Ω a λ n a ( x ) u λ n ,β dx + Z R \ Ω b |∇ v λ n ,β | dx + Z R \ Ω b λ n b ( x ) v λ n ,β dx + o n (1)= Z R \ Ω a |∇ u λ n ,β | + u λ n ,β dx + Z R \ Ω b |∇ v λ n ,β | + v λ n ,β dx + o n (1) , Thus, R R \ Ω a |∇ u λ n ,β | + u λ n ,β dx + R R \ Ω b |∇ v λ n ,β | + v λ n ,β dx → n → ∞ and it is a contra-diction. We now complete the proof by taking Λ ∗ = Λ . The main task in this section is to find multi-bump solutions to ( P λ,β ) described as in Theorem 1.2.For the sake of convenience, in the present section, we always assume the conditions ( D )-( D ),( D ′ ), ( D ) and ( D ′ ) hold. Due to the conditions ( D ′ ) and ( D ′ ), in the present section, Ω ′ a andΩ ′ b will be chosen as ( I )-( III ) given in section 2.17 .1 The penalized functional and the ( P S ) condition Since we want to find multi-bump solutions of ( P λ,β ) described as in Theorem 1.2, we will makesome modifications on J λ,β ( u, v ). Similar technique was developed by del Pino and Felmer [23]and was also used in several other literatures, see for example [10, 24, 28, 44] and the referencestherein.Let J a × J b be a given subset of { , · · · , n a } × { , · · · , n b } with J a = ∅ and J b = ∅ . Without lossof generality, we assume J a × J b = { , · · · , k a } × { , · · · , k b } with 1 ≤ k a ≤ n a and 1 ≤ k b ≤ n b .Denote Ω J a a = k a ∪ i a =1 Ω ′ a,i a and Ω J b b = k b ∪ j b =1 Ω ′ b,j b . We also denote the characteristic functions of Ω J a a and Ω J b b by χ Ω Jaa and χ Ω Jbb , respectively. Now, let δ β = C a,b (cid:26) , µ + 2 | β | , µ + 2 | β | (cid:27) , (4.1)where C a,b is given by Lemma 2.1, and define f a ( x, t ) = χ Ω Jaa ( t + ) + (1 − χ Ω Jaa ) f ( t ), f b ( x, t ) = χ Ω Jbb ( t + ) + (1 − χ Ω Jbb ) f ( t ) and h ( x, t, s ) = ( χ Ω Jbb + χ Ω Jaa ) t + s + + (1 − χ Ω Jaa )(1 − χ Ω Jbb ) h ( t, s ), where t + = max { , t } , s + = max { , s } , f ( t ) = , t ≤ ,t , ≤ t ≤ δ β ,δ β t, t ≥ δ β , and h ( t, s ) = , min { t, s } ≤ ,ts, ≤ t, s ≤ δ β ,δ β t, ≤ t ≤ δ β ≤ s,δ β s, ≤ s ≤ δ β ≤ t,δ β , δ β ≤ t, s. Then it is easy to see that f a ( x, t ) and f b ( x, t ) are the modifications of t and h ( x, t, s ) is themodification of ts . Let us consider the following functional defined on E , J ∗ λ,β ( u, v ) = 12 k u k a,λ + 12 k v k b,λ − µ Z R F a ( x, u ) dx − µ Z R F b ( x, v ) dx − β Z R H ( x, u, v ) dx, where F a ( x, u ) = R u f a ( x, t ) dt , F b ( x, v ) = R v f b ( x, t ) dt and H ( x, u, v ) = 2 R u R v h ( x, t, s ) dsdt .Clearly, by the construction of f a ( x, t ), f b ( x, t ) and h ( x, t, s ), we can see that0 ≤ Z R \ Ω Jaa F a ( x, u ) dx ≤ δ β k u + k , ≤ Z R \ Ω Jba F b ( x, v ) dx ≤ δ β k v + k , ≤ Z R \ (Ω Jaa ∪ Ω Jba ) H ( x, u, v ) dx ≤ δ β k u + k k v + k ≤ δ β ( k u + k + k v + k ) . On the other hand,by Lemma 2.1, we have k u + k ≤ C − a,b k u k a,λ ≤ λC − a,b k u k a for λ ≥ Λ and u ∈ E a and k v + k ≤ C − a,b k v k b,λ ≤ λC − a,b k v k b for λ ≥ Λ and v ∈ E b . It follows that J ∗ λ,β ( u, v ) is well defined on E for λ ≥ Λ and β <
0. Moreover, by a standardargument, we can see that for λ ≥ Λ and β < J ∗ λ,β ( u, v ) is C on E and the critical point of J ∗ λ,β ( u, v ) is the solution of the following two-component systems: ∆ u − ( λa ( x ) + a ( x )) u + µ f a ( x, u ) + 2 β Z v h ( x, u, s ) ds = 0 in R , ∆ v − ( λb ( x ) + b ( x )) v + µ f b ( x, v ) + 2 β Z u h ( x, t, v ) dt = 0 in R ,u, v ∈ H ( R ) , u, v ≥ R . ( P ∗ λ,β )In what follows, we will make some investigations on the functional J ∗ λ,β ( u, v ).18 emma 4.1 Assume ( u, v ) ∈ E . Then Z R f a ( x, u ) u − F a ( x, u ) dx ≥ − δ β k u + k , Z R f b ( x, v ) v − F b ( x, v ) dx ≥ − δ β k v + k , ≥ Z R u Z v h ( x, u, s ) ds + v Z u h ( x, t, v ) dt − H ( x, u, v ) dx ≥ − δ β k u + v + k . Proof.
By the construction of f a ( x, t ), it is easy to see that f a ( x, t ) t − F a ( x, t ) = 0 for x ∈ Ω J a a .If x Ω J a a , then by the construction of f a ( x, t ), we have14 f a ( x, t ) t − F a ( x, t ) = , t ≤ δ β ,δ β − δ β t + ) , δ β ≤ t. It follows that for every u ∈ E a , we have Z R f a ( x, u ) u − F a ( x, u ) dx = Z { u ( x ) ≥ δ β }∩ ( R \ Ω Jaa ) δ β − δ β u + ) dx ≥ − δ β k u + k . By a similar argument, for every v ∈ E b , we have Z R f b ( x, v ) v − F b ( x, v ) dx ≥ − δ β k v + k . On the other hand, since Ω J a a ∩ Ω J b b = ∅ , by the construction of h ( x, t, s ), we can see that t R s h ( x, t, τ ) dτ + s R t h ( x, τ, s ) dτ − H ( x, t, s ) = 0 for x ∈ Ω J a a ∪ Ω J b b . If x Ω J a a ∪ Ω J b b , thenalso by the construction of h ( x, t, s ), we have t Z s h ( x, t, τ ) dτ + s Z t h ( x, τ, s ) dτ − H ( x, t, s )= , t, s ≤ δ β , ( t + ) δ β δ β − s ) , t ≤ δ β ≤ s, ( s + ) δ β δ β − t ) , s ≤ δ β ≤ t,δ β t − δ β )( δ β − s ) + ( s − δ β )( δ β − t )] , δ β ≤ t, s. It follows that for every ( u, v ) ∈ E , we have0 ≥ Z R u Z v h ( x, u, s ) ds + v Z u h ( x, t, v ) dt − H ( x, u, v ) dx ≥ − δ β k u + v + k , which completes the proof.With Lemma 4.1 in hands, we can verify that J ∗ λ,β ( u, v ) actually satisfies the ( P S ) conditionfor λ ≥ Λ and β < Lemma 4.2
Assume λ ≥ Λ and β < . Then J ∗ λ,β ( u, v ) satisfies the ( P S ) c condition for all c ∈ R , that is, every { ( u n , v n ) } ⊂ E satisfying J ∗ λ,β ( u n , v n ) = c + o n (1) and D [ J ∗ λ,β ( u n , v n )] = o n (1) strongly in E ∗ has a strongly convergent subsequence in E . Proof.
Suppose { ( u n , v n ) } ⊂ E satisfying J ∗ λ,β ( u n , v n ) = c + o n (1) and D [ J ∗ λ,β ( u n , v n )] = o n (1)strongly in E ∗ . Then by β <
0, Lemmas 2.1 and 4.1 and a similar argument of (3.8), we have c + o n (1) + o n (1) k ( u n , v n ) k ≥
14 (1 − µ δ β C − a,b ) k u n k a,λ + 14 (1 − µ δ β C − a,b ) k v n k b,λ . (4.2)19t follows from Lemma 2.1 and (4.1) that c + o n (1) + o n (1) k ( u n , v n ) k ≥ C a,b ( k u n k + k v n k ). Thistogether with Lemma 2.1 and the condition ( D ), implies c + o n (1) + o n (1) k ( u n , v n ) k ≥
14 (1 − µ δ β C − a,b ) k u n k a,λ + 14 (1 − µ δ β C − a,b ) k v n k b,λ ≥
14 (1 − µ δ β C − a,b ) k u n k a + 14 (1 − µ δ β C − a,b ) k v n k b − C ( k u n k + k v n k ) ≥ k ( u n , v n ) k − C ′ ( c + o n (1) + o n (1) k ( u n , v n ) k ) , since λ ≥ Λ . Thus, { ( u n , v n ) } is bounded in E and ( u n , v n ) ⇀ ( u , v ) weakly in E as n → ∞ forsome ( u , v ) ∈ E up to a subsequence. Without loss of generality, we assume ( u n , v n ) ⇀ ( u , v )weakly in E as n → ∞ . Since D [ J ∗ λ,β ( u n , v n )] = o n (1) strongly in E ∗ , it is easy to see that D [ J ∗ λ,β ( u , v )] = 0 in E ∗ , which implies o n (1) = h D [ J ∗ λ,β ( u n , v n )] − D [ J ∗ λ,β ( u , v )] , ( u n , v n ) − ( u , v ) i E ∗ ,E = k u n − u k a,λ + k v n − v k b,λ − µ Z R ( f a ( x, u n ) − f a ( x, u ))( u n − u ) dx − µ Z R ( f b ( x, v n ) − f b ( x, v ))( v n − v ) dx − β Z R ( Z v n h ( x, u n , s ) ds − Z v h ( x, u , s ) ds )( u n − u ) dx − β Z R ( Z u n h ( x, t, v n ) dt − Z u h ( x, t, v ) dt )( v n − v ) dx. (4.3)By the construction of f a ( x, t ), we can see that | Z R ( f a ( x, u n ) − f a ( x, u ))( u n − u ) dx |≤ Z Ω Jaa | ( f a ( x, u n ) − f a ( x, u ))( u n − u ) | dx + Z R \ Ω Jaa | ( f a ( x, u n ) − f a ( x, u ))( u n − u ) | dx ≤ Z Ω Jaa ( | u n | + | u | ) | u n − u | dx + 2 δ β Z R \ Ω Jaa | u || u n − u | dx + δ β k u n − u k . Since ( u n , v n ) ⇀ ( u , v ) weakly in E as n → ∞ , by (2.1) and the Sobolev embedding theorem,we have u n → u strongly in L ploc ( R ) as n → ∞ for p ∈ [1 , . (4.4)Thus, R Ω Jaa ( | u n | + | u | ) | u n − u | dx = o n (1) due to the choice of Ω J a a and the H¨older inequality. Onthe other hand, we also see from (4.4) and the H¨older inequality that 2 δ β R R \ Ω Jaa | u || u n − u | dx = o n (1). Hence, we have | Z R ( f a ( x, u n ) − f a ( x, u ))( u n − u ) dx | ≤ δ β k u n − u k + o n (1) . (4.5)Since (2.2) holds, we can also obtain the following estimates in a simiar way: | Z R ( f b ( x, v n ) − f b ( x, v ))( v n − v ) dx | ≤ δ β k v n − v k + o n (1) . (4.6)On the other hand, by the construction of h ( x, t, s ), we can see that | Z R ( Z v n h ( x, u n , s ) ds − Z v h ( x, u , s ) ds )( u n − u ) dx |≤ δ β Z R | u n − u || v n − v | dx + 2 δ β Z R | v || u n − u | dx + o n (1) (4.7)20nd | Z R ( Z u n h ( x, t, v n ) dt − Z u h ( x, t, v ) dt )( v n − v ) dx |≤ δ β Z R | u n − u || v n − v | dx + 2 δ β Z R | u || v n − v | dx + o n (1) . (4.8)By using similar arguments of (4.5) and (4.6), we can see from (4.7) and (4.8) that | Z R ( Z v n h ( x, u n , s ) ds − Z v h ( x, u , s ) ds )( u n − u ) dx |≤ δ β Z R | u n − u || v n − v | dx + o n (1) , (4.9)and | Z R ( Z u n h ( x, t, v n ) dt − Z u h ( x, t, v ) dt )( v n − v ) dx |≤ δ β Z R | u n − u || v n − v | dx + o n (1) . (4.10)Combining (4.3), (4.5)-(4.6) and (4.9)-(4.10), we can conclude that o n (1) ≥ k u n − u k a,λ + k v n − v k b,λ − δ β ( µ + 2 | β | ) k u n − u k − δ β ( µ + 2 | β | ) k v n − v k . (4.11)It follows from Lemma 2.1 and (4.1) that o n (1) ≥ C a,b ( k u n − u k + k v n − v k ), which implies u n → u and v n → v strongly in L ( R ) as n → ∞ . This together the condition ( D ), implies o n (1) ≥ k u n − u k a,λ + k v n − v k b,λ ≥ k u n − u k a + k v n − v k b + o n (1)for λ ≥ Λ and β <
0. Thus, ( u n , v n ) → ( u , v ) strongly in E as n → ∞ for λ ≥ Λ and β < J ∗ λ,β ( u, v ) is actually a penalized functional of J λ,β ( u, v ) in the sense that, some special critical points of J ∗ λ,β ( u, v ) are also critical points of J λ,β ( u, v ). Lemma 4.3
Assume λ ≥ Λ and β < . Let M > be a constant and ( u λ,β , v λ,β ) ∈ E satisfy J ∗ λ,β ( u λ,β , v λ,β ) ≤ M and D [ J ∗ λ,β ( u λ,β , v λ,β )] = 0 in E ∗ . Then (1) There exists M > such that k ( u λ,β , v λ,β ) k ≤ M . (2) R R \ Ω Jaa |∇ u λ,β | +( λa ( x )+ a ( x )) u λ,β dx → and R R \ Ω Jbb |∇ v λ,β | +( λb ( x )+ b ( x )) v λ,β dx → as λ → + ∞ . (3) There exists Λ ∗ ( β, M ) ≥ Λ such that | u λ,β | ≤ δ β on R \ Ω J a a and | v λ,β | ≤ δ β on R \ Ω J b b for λ ≥ Λ ∗ ( β, M ) . Proof. (1) Since λ ≥ Λ and β <
0, by a similar argument as (4.2), we can conclude that M ≥
14 (1 − µ δ β C − a,b ) k u λ,β k a,λ + 14 (1 − µ δ β C − a,b ) k v λ,β k b,λ . (4.12)It follows from Lemma 2.1 and (4.1) that 8 M C − a,b ≥ k u λ,β k + k v λ,β k . Now, applying the condition( D ), we can see that8 M + 8 M C − a,b ( C a, + d a + C b, + d b ) ≥ k ( u λ,β , v λ,β ) k . (4.13)21e complete this proof by taking M = 8 M + 8 M C − a,b ( C a, + d a + C b, + d b ).(2) Since R \ Ω J a a = ( R \ Ω ′ a ) ∪ ( n a ∪ i a = k a +1 Ω ′ a,i a ) and R \ Ω J b b = ( R \ Ω ′ b ) ∪ ( n b ∪ j b = k b +1 Ω ′ b,j b ), for thesake of clarity, we divide this proof into the following two steps. Step 1.
We prove thatlim λ → + ∞ Z R \ Ω ′ a |∇ u λ,β | + ( λa ( x ) + a ( x )) u λ,β dx = 0 (4.14)and lim λ → + ∞ Z R \ Ω ′ b |∇ v λ,β | + ( λb ( x ) + b ( x )) v λ,β dx = 0 . (4.15)Indeed, let { Ω ′′ a,i a } be a sequence of bounded domains with smooth boundaries in R and satisfy( a ) Ω a,i a ⊂ Ω ′′ a,i a ⊂ Ω ′ a,i a for all i a = 1 , · · · , n a .( b ) dist (Ω ′′ a,i a , R \ Ω ′ a,i a ) > dist ( R \ Ω ′′ a,i a , Ω a,i a ) > i a = 1 , · · · , n a .Denote Ω ′′ a = n a ∪ i a =1 Ω ′′ a,i a . Then by a similar argument as (2.7), we can show that Z Ω ′′ a |∇ u λ,β | + ( λa ( x ) + a ( x )) u λ,β dx ≥ ν a Z Ω ′′ a u λ,β dx (4.16)for λ large enough. Without loss of generality, we assume (4.16) holds for λ ≥ Λ . Since Lemma 2.1and (4.12) hold, we can obtain 8(( C a, + d a ) C − a,b + 1) M ≥ λ R R \ Ω ′′ a a ( x ) u λ,β dx for λ ≥ Λ by thecondition ( D ) and (4.16). Since the condition ( D ′ ) is contained in the condition ( D ), by a similarargument as (3.15), we can see that Z R \ Ω ′′ a u λ,β dx → λ → + ∞ . (4.17)Let Ψ ∈ C ∞ ( R ) be given by (3.16). Then u λ,β Ψ ∈ E a . Note D [ J ∗ λ,β ( u λ,β , v λ,β )] = 0 in E ∗ , weget that Z R ( |∇ u λ,β | + ( λa ( x ) + a ( x )) u λ,β )Ψ dx + Z R ( ∇ u λ,β ∇ Ψ) u λ,β dx = µ Z R f a ( x, u λ,β ) u λ,β Ψ dx + 2 β Z R ( Z v λ,β h ( x, u λ,β , s ) ds ) u λ,β Ψ dx. Since β <
0, by (2.8) and the construction of f a ( x, t ) and h ( x, t, s ), we have0 ≤ Z R \ Ω ′ a |∇ u λ,β | + ( λa ( x ) + a ( x )) u λ,β dx ≤ µ Z Ω Jaa ∩ ( R \ Ω ′′ a ) u λ,β dx + µ δ β Z ( R \ Ω Jaa ) ∩ ( R \ Ω ′′ a ) u λ,β dx + Z Ω ′ a \ Ω ′′ a |∇ u λ,β ||∇ Ψ || u λ,β | dx ≤ µ δ β Z R \ Ω ′′ a u λ,β dx + ( µ S − k u λ,β k a + max R |∇ Ψ |k u λ,β k a ) (cid:18) Z R \ Ω ′′ a u λ,β dx (cid:19) . (4.18)Thanks to (4.13) and (4.17), we know from (4.18) that (4.14) holds. By a similar argument, wecan also conclude that (4.15) is true. Step 2.
We prove thatlim λ → + ∞ Z Ω ′ a,ia |∇ u λ,β | + ( λa ( x ) + a ( x )) u λ,β dx = 0 for all i a ∈ { , · · · , n a }\ J a (4.19)22nd lim λ → + ∞ Z Ω ′ b,jb |∇ v λ,β | + ( λb ( x ) + b ( x )) v λ,β dx = 0 for all j b ∈ { , · · · , n b }\ J b . (4.20)In fact, let { Ω ′′′ a,i a } be a sequence of bounded domains with smooth boundaries in R and satisfy( i ) Ω ′ a,i a ⊂ Ω ′′′ a,i a and dist(Ω ′ a,i a , R \ Ω ′′′ a,i a ) > i a ∈ { , · · · , n a } .( ii ) Ω ′′′ a,i a ∩ Ω ′′′ a,j a = ∅ for all i a = j a .( iii ) ( n a ∪ i a =1 Ω ′′′ a,i a ) ∩ Ω ′ b = ∅ .For every i a ∈ { , · · · , n a }\ J a , we choose Ψ i a ∈ C ∞ ( R , [0 , i a = ( , x ∈ Ω ′ a,i a , , x ∈ R \ Ω ′′′ a,i a . Then by a similar argument as (4.18), the choice of Ω ′′′ a,i a and the construction of f a ( x, t ) and h ( x, t, s ), we can obtain that Z Ω ′ a,ia |∇ u λ,β | + ( λa ( x ) + a ( x )) u λ,β dx ≤ µ δ β Z Ω ′′′ a,ia u λ,β dx + Z Ω ′′′ a,ia \ Ω ′ a,ia |∇ u λ,β ||∇ Ψ i a || u λ,β | dx. Thanks to the choice of Ω ′′′ a,i a and (4.13), for i a ∈ { , · · · , n a }\ J a , we have Z Ω ′ a,ia |∇ u λ,β | + ( λa ( x ) + a ( x )) u λ,β dx ≤ µ δ β Z Ω ′ a,ia u λ,β dx + C (cid:18) Z R \ Ω ′′ a u λ,β dx (cid:19) . It follows from (2.9), (4.1) and (4.17) that (4.19) holds. A similar argument implies that (4.20)holds too. Now, the conclusion follows immediately from (4.14)-(4.15) and (4.19)-(4.20).(3) By (2.9) and (4.19), we have Z Ω ′ a,ia u λ,β dx → λ → + ∞ for i a ∈ { , · · · , n a }\ J a , which together with (4.17), implies Z R \ ∪ ia ∈ Ja Ω ′′ a,ia u λ,β dx → λ → + ∞ . (4.21)Let r = dist (Ω ′ a , Ω ′′ a ). Then for every x ∈ R \ Ω J a a , B r ( x ) ⊂ R \ ∪ i a ∈ J a Ω ′′ a,i a . We define φ L =min {| u λ,β | α − , L } u λ,β ρ , where ρ ∈ C ∞ ( B r ( x ) , [0 , ρ = 1 on B r ( x ) and |∇ ρ | < C r − r , α > L >
0. Then φ L ∈ E a . Since D [ J ∗ λ,β ( u λ,β , v λ,β )] = 0 in E ∗ and the conditions ( D ) and( D ) hold, by multiplying ( P ∗ λ,β ) with ( φ L ,
0) and letting L → + ∞ , we have12 Z R ρ | u λ,β | α − |∇ u λ,β | dx ≤ C β (cid:18) Z R ρ u α +3 λ,β dx + Z R ρ u α +1 λ,β dx (cid:19) + 4 Z R |∇ ρ | u α +1 λ,β dx, C β = µ + µ δ β + C a, + d a . By the Sobolev embedding theorem, we can see that (cid:18) Z B r ( x ) u α +1) λ,β dx (cid:19) ≤ C β ( α + 1) (cid:18) Z B r ( x ) u α +3 λ,β dx + (1 + 24 r ) Z B r ( x ) u α +1 λ,β dx (cid:19) . (4.22)Let α n = 3 α n − with α = 2 and r n = (1 + ( ) n − ) r , n ∈ N . Then (4.22) can be re-written as (cid:18) Z B r ( x ) u α +1) λ,β dx (cid:19) ≤ C β ( α + 1) (cid:18) Z B r ( x ) u α +3 λ,β dx + (1 + 4 | r − r | ) Z B r ( x ) u α +1 λ,β dx (cid:19) . (4.23)We replace α , r and r in (4.23) by α n , r n and r n +1 . Then we can obtain (cid:18) Z B rn +1 ( x ) u α n +1) λ,β dx (cid:19) αn +1) ≤ [ C β ( α n + 1) ] αn +1) (cid:18) Z B rn ( x ) u α n +3 λ,β dx (cid:19) αn +1) +[ C β ( α n + 1) ] αn +1) (1 + 4 | r n − r n +1 | ) αn +1) (cid:18) Z B rn ( x ) u α n +1 λ,β dx (cid:19) αn +1) . (4.24)Clearly, one of the following two cases must happen:(1) R B rn ( x ) u α n +1 λ,β dx ≤ R B rn ( x ) u α n +3 λ,β dx up to a subsequence.(2) R B rn ( x ) u α n +3 λ,β dx ≤ R B rn ( x ) u α n +1 λ,β dx up to a subsequence.If case (1) happen, then by (4.24), we can see that (cid:18) Z B rn +1 ( x ) u α n +1) λ,β (cid:19) αn +1) ≤ (cid:18) C β ( α n + 1) (1 + 1 | r n − r n +1 | ) (cid:19) αn +1) (cid:18) Z B rn ( x ) u α n +3 λ,β (cid:19) αn +1 . By iterating (4.24) and using the choice of r n and α n , we havelim n → + ∞ (cid:18) Z B rn +1 ( x ) u α n +1) λ,β (cid:19) αn +1) ≤ (cid:18) ∞ Y n =1 (cid:18) C β ( α n + 1) (1 + 1 | r n − r n +1 | ) (cid:19) αn +1) (cid:18) Z R \ ∪ ia ∈ Ja Ω ′′ a,ia u λ,β dx (cid:19) (cid:19) + ∞ Q n =1 αn +3 αn +1 ≤ C ′ β (cid:18) Z R \ ∪ ia ∈ Ja Ω ′′ a,ia u λ,β dx (cid:19) C , (4.25)where C ′ β is a constant independent of λ and x . If case (2) happen, then by iterating (4.24) andusing the choice of r n and α n once more, we havelim n → + ∞ (cid:18) Z B rn +1 ( x ) u α n +1) λ,β (cid:19) αn +1) ≤ ∞ Y n =1 (cid:18) C β ( α n + 1) (1 + 1 | r n − r n +1 | ) | B r n ( x ) | αn +3 (cid:19) αn +1) (cid:18) Z R \ ∪ ia ∈ Ja Ω ′′ a,ia u λ,β dx (cid:19) ≤ C ′ β (cid:18) Z R \ ∪ ia ∈ Ja Ω ′′ a,ia u λ,β dx (cid:19) . (4.26)24y the H¨older and the Sobolev inequalities and (4.13) and (4.21), we can conclude that (cid:18) Z R \ ∪ ia ∈ Ja Ω ′′ a,ia u λ,β dx (cid:19) → λ → + ∞ . It follows from (4.25) and (4.26) that k u λ,β k L ∞ ( B r ( x )) → λ → + ∞ , which then implies k u λ,β k L ∞ ( R \ Ω Jaa ) → λ → + ∞ . By similar arguments, we also have k v λ,β k L ∞ ( R \ Ω Jbb ) → λ → + ∞ . Now, we can choose Λ ∗ ( β, M ) ≥ Λ such that | u λ,β | ≤ δ β a.e. on R \ Ω J a a and | v λ,β | ≤ δ β a.e. on R \ Ω J b b for λ ≥ Λ ∗ ( β ). Note that by a similar argument as used in Theorem 1.1, we cansee that ( u λ,β , v λ,β ) ∈ C ( R ) × C ( R ). Hence, we must have | u λ,β | ≤ δ β on R \ Ω J a a and | v λ,β | ≤ δ β on R \ Ω J b b for λ ≥ Λ ∗ ( β, M ). In this section, we will construct critical values of J ∗ λ,β ( u, v ) by a minimax argument. The idea ofsuch a construction traces back to S´er´e [40] and also was applied in [10, 24, 28, 44].We first recall some well-known results, which are useful in this construction. For all i a =1 , · · · , n a and j b = 1 , · · · , n b , we define E Ω ′ a,ia on H (Ω ′ a,i a ) and E Ω ′ b,jb on H (Ω ′ b,j b ) as follows: E Ω ′ a,ia ( u ) = 12 Z Ω ′ a,ia |∇ u | + ( λa ( x ) + a ( x )) u dx − µ Z Ω ′ a,ia u dx, E Ω ′ b,jb ( v ) = 12 Z Ω ′ b,jb |∇ v | + ( λb ( x ) + b ( x )) v dx − µ Z Ω ′ b,jb v dx. By (2.9) and (2.10), E Ω ′ a,ia ( u ) and E Ω ′ b,jb ( v ) have the least energy nonzero critical point for all i a = 1 , · · · , n a and j b = 1 , · · · , n b if λ ≥ Λ . We denote the ground state level of E Ω ′ a,ia ( u )and E Ω ′ b,jb ( v ) by m a,i a ,λ and m b,j b ,λ , respectively. Since { Ω ′ a,i a } and { Ω ′ b,j b } are two sequences ofbounded domains, by the conditions ( D )-( D ), ( D ′ ), ( D ) and ( D ′ ) and (2.9)-(2.10), it is easyto show that m a,i a ,λ and m b,j b ,λ are positive for λ ≥ Λ . It follows that m a,i a ,λ = inf {E Ω ′ a,ia ( u ) | Z Ω ′ a,ia u dx = 4 m a,i a ,λ µ } for all i a = 1 , · · · , n a (4.27)and m b,j b ,λ = inf {E Ω ′ b,jb ( v ) | Z Ω ′ b,jb v dx = 4 m b,j b ,λ µ } for all j b = 1 , · · · , n b . (4.28)On the other hand, let W a,i a ∈ H (Ω a,i a ) and W b,j b ∈ H (Ω b,j b ) be the least energy nonzerocritical points of I Ω a,ia ( u ) and I Ω b,jb ( v ), respectively. Then by the conditions ( D ′ ) and ( D ′ ), it iswell-known that I Ω a,ia ( W a,i a ) = max t ≥ I Ω a,ia ( tW a,i a ) and I Ω b,jb ( W b,j b ) = max t ≥ I Ω b,jb ( tW b,j b ) . (4.29)Let γ ,a : [0 , k a → E a and γ ,b : [0 , k b → E b be γ ,a ( t , · · · , t k a ) = k a X i a =1 t i a RW a,i a (4.30)and γ ,b ( s , · · · , s k b ) = k b X j b =1 s j b RW b,j b , (4.31)25here R > I Ω a,ia ( RW a,i a ) ≤ , R Z Ω a,ia W a,i a dx ≥ m a,i a µ , (4.32) I Ω b,jb ( RW b,j b ) ≤ , R Z Ω b,jb W b,j b dx ≥ m b,j b µ . (4.33)for all i a = 1 , · · · , k a and j b = 1 , · · · , k b . By the condition ( D ′ ), we can extend W a,i a and W b,j b to the whole space R by letting W a,i a = 0 on R \ Ω a,i a and W b,j b = 0 on R \ Ω b,j b such that W a,i a ∈ H ( R ) and W b,j b ∈ H ( R ) for all i a = 1 , · · · , n a and j b = 1 , · · · , n b . Now, we can definea minimax value of J ∗ λ,β ( u, v ) for λ ≥ Λ and β < m J a ,J b ,λ,β = inf ( γ a ,γ b ) ∈ Γ sup [0 , ka × [0 , kb J ∗ λ,β ( γ a , γ b ) , where Γ = (cid:26) ( γ a , γ b ) | ( γ a , γ b ) ∈ C ([0 , k a × [0 , k b , E a × E b ) , ( γ a , γ b ) = ( γ ,a , γ ,b ) on ∂ ([0 , k a × [0 , k b ) (cid:27) .m J a ,J b ,λ,β may be a critical value of J ∗ λ,β ( u, v ). In order to show it, we need the following. Lemma 4.4
Assume ( γ a , γ b ) ∈ Γ and ( ξ , · · · , ξ k a , η , · · · , η k b ) ∈ [0 , R Z Ω a, W a, dx ] × · · · × [0 , R Z Ω a,ka W a,k a dx ] × [0 , R Z Ω b, W b, dx ] × · · · × [0 , R Z Ω b,kb W b,k b dx ] . Then there exist ( t ′ , · · · , t ′ k a ) ∈ [0 , k a and ( s ′ , · · · , s ′ k b ) ∈ [0 , k b such that Z Ω ′ a,ia [ γ a ( t ′ , · · · , t ′ k a )] ( x ) dx = ξ i a for all i a = 1 , · · · , k a and Z Ω ′ b,jb [ γ b ( s ′ , · · · , s ′ k b )] ( x ) dx = η j b for all j b = 1 , · · · , k b . Proof.
For every ( γ a , γ b ) ∈ Γ, we define a map e γ : [0 , k a × [0 , k b → R k a × R k b as follows: e γ ( t , · · · , t k a , s , · · · , s k b )= (cid:18) Z Ω ′ a, [ γ a ( t , · · · , t k a )] ( x ) dx, · · · , Z Ω ′ a,ka [ γ a ( t , · · · , t k a )] ( x ) dx, Z Ω ′ b, [ γ b ( s , · · · , s k b )] ( x ) dx, · · · , Z Ω ′ b,kb [ γ b ( s , · · · , s k b )] ( x ) dx (cid:19) . Note that for every ( γ a , γ b ) ∈ Γ, we have( γ a ( t , · · · , t k a ) , γ b ( s , · · · , s k b )) = ( γ ,a ( t , · · · , t k a ) , γ ,b ( s , · · · , s k b ))if ( t , · · · , t k a , s , · · · , s k b ) ∈ ∂ ([0 , k a × [0 , k b ). Then by the construction of { Ω ′ a,i a } and { Ω ′ b,j b } ,we can see that Z Ω ′ a,ia [ γ a ( t , · · · , t k a )] ( x ) dx = t i a R Z Ω a,ia W a,i a dx Z Ω ′ b,jb [ γ b ( s , · · · , s k b )] ( x ) dx = s j b R Z Ω b,jb W b,j b dx for all i a = 1 , · · · , k a , j b = 1 , · · · , k b and ( t , · · · , t k a , s , · · · , s k b ) ∈ ∂ ([0 , k a × [0 , k b ). It followsthat deg ( e γ, [0 , k a × [0 , k b , ( ξ , · · · , ξ k a , η , · · · , η k b )) = 1 , which completes the proof.With Lemma 4.4 in hands, we can obtain the following energy estimate, which can be viewedas a linking structure of J ∗ λ,β ( u, v ). Lemma 4.5
Assume λ ≥ Λ and β < . Then we have the following results. (1) If ( t , · · · , t k a , s , · · · , s k b ) ∈ ∂ ([0 , k a × [0 , k b ) , then J ∗ λ,β ( γ ,a ( t , · · · , t k a ) , γ ,b ( s , · · · , s k b )) ≤ k a X i a =1 m a,i a + k b X j b =1 m b,j b − min { m a, , · · · , m a,k a , m b, , · · · , m b,k b } . (2) k a P i a =1 m a,i a ,λ + k b P j b =1 m b,j b ,λ ≤ m J a ,J b ,λ,β ≤ k a P i a =1 m a,i a + k b P j b =1 m b,j b . Proof. (1) Since ( t , · · · , t k a , s , · · · , s k b ) ∈ ∂ ([0 , k a × [0 , k b ), there exists i ′ a ∈ { , · · · , k a } or j ′ b ∈ { , · · · , k b } such that t i ′ a ∈ { , } or s j ′ b ∈ { , } . Without loss of generality, we assume t = 1.It follows from (4.29)-(4.33) and the condition ( D ′ ) that J ∗ λ,β ( γ ,a ( t , · · · , t k a ) , γ ,b ( s , · · · , s k b ))= I a, ( RW a, ) + k a X i a =2 I a,i a ( t i a RW a,i a ) + k b X j b =1 I b,j b ( s j b RW b,j b ) ≤ k a X i a =2 m a,i a + k b X j b =1 m b,j b ≤ k a X i a =1 m a,i a + k b X j b =1 m b,j b − min { m a, , · · · , m a,k a , m b, , · · · , m b,k b } . (2) Since ( γ ,a , γ ,b ) ∈ Γ and
R >
2, by the condition ( D ′ ), we must have m J a ,J b ,λ,β ≤ sup [0 , ka × [0 , kb J ∗ λ,β ( k a X i a =1 t i a RW a,i a , k b X j b =1 s j b RW b,j b ) ≤ k a X i a =1 I a,i a ( W a,i a ) + k b X j b =1 I b,j b ( W b,j b )= k a X i a =1 m a,i a + k b X j b =1 m b,j b . On the other hand, since the condition ( D ′ ) holds, by the construction of Ω ′ a,i a and Ω ′ b,j b , it iseasy to show that m a,i a ,λ ≤ m a,i a and m b,j b ,λ ≤ m b,j b for all i a = 1 , · · · , n a , j b = 1 , · · · , n b and27 >
0. This together with (4.32)-(4.33) and Lemma 4.4, implies that for every ( γ a , γ b ) ∈ Γ, thereexist ( t ′ , · · · , t ′ k a ) ∈ [0 , k a and ( s ′ , · · · , s ′ k b ) ∈ [0 , k b such that Z Ω ′ a,ia [ γ a ( t ′ , · · · , t ′ k a )] ( x ) dx = 4 m a,i a ,λ µ for all i a = 1 , · · · , k a (4.34)and Z Ω ′ b,jb [ γ b ( s ′ , · · · , s ′ k b )] ( x ) dx = 4 m b,j b ,λ µ for all j b = 1 , · · · , k b . (4.35)Denote γ a ( t ′ , · · · , t ′ k a ) and γ b ( s ′ , · · · , s ′ k b ) by u ∗ and v ∗ . Then by (2.8)-(2.10), we have Z R \ Ω Jaa u ∗ dx ≤ C − a,b Z R \ Ω Jaa |∇ u ∗ | + ( λa ( x ) + a ( x )) u ∗ dx and Z R \ Ω Jbb v ∗ dx ≤ C − a,b Z R \ Ω Jbb |∇ v ∗ | + ( λb ( x ) + b ( x )) v ∗ dx for λ ≥ Λ . Note that { Ω ′ a,i a } and { Ω ′ b,j b } are two sequences of bounded domains with smoothboundaries, so the restriction of u ∗ on Ω ′ a,i a lies in H (Ω ′ a,i a ) for every i a = 1 , · · · , n a , while therestriction of v ∗ on Ω ′ b,j b lies in H (Ω ′ b,j b ) for every j b = 1 , · · · , n b . Now, by β <
0, (4.1) and theconstruction of f a ( x, t ), f b ( x, t ) and h ( x, t, s ), we have J ∗ λ,β ( u ∗ , v ∗ ) ≥ Z R |∇ u ∗ | + ( λa ( x ) + a ( x )) u ∗ dx − µ Z R F a ( x, u ∗ ) dx + 12 Z R |∇ v ∗ | + ( λb ( x ) + b ( x )) v ∗ dx − µ Z R F b ( x, v ∗ ) dx ≥
12 ( Z R \ Ω Jaa |∇ u ∗ | + ( λa ( x ) + a ( x )) u ∗ dx − δ β Z R \ Ω Jaa u ∗ dx ) + k a X i a =1 E Ω ′ a,ia ,λ ( u ∗ )+ 12 ( Z R \ Ω Jbb |∇ v ∗ | + ( λb ( x ) + b ( x )) v ∗ dx − δ β Z R \ Ω Jbb v ∗ dx ) + k b X j b =1 E Ω ′ b,jb ,λ ( v ∗ ) ≥ k a X i a =1 E Ω ′ a,ia ,λ ( u ∗ ) + k b X j b =1 E Ω ′ b,jb ,λ ( v ∗ ) . (4.36)Thanks to (4.27)-(4.28) and (4.34)-(4.35), (4.36) implies J ∗ λ,β ( u ∗ , v ∗ ) ≥ k a P i a =1 m a,i a ,λ + k b P j b =1 m b,j b ,λ for λ ≥ Λ and β <
0. Since ( γ a , γ b ) ∈ Γ is arbitrary, we must have k a P i a =1 m a,i a ,λ + k b P j b =1 m b,j b ,λ ≤ m J a ,J b ,λ,β for λ ≥ Λ and β <
0, which completes the proof.Let m a,b := n a P i a =1 m a,i a + n b P j b =1 m b,j b . Then we can obtain the following proposition. Proposition 4.1
Suppose β < . Then there exists Λ ∗ ( β ) ≥ Λ ∗ ( β, m a,b ) such that m J a ,J b ,λ,β is acritical value of J ∗ λ,β ( u, v ) for λ ≥ Λ ∗ ( β ) , that is, for all λ ≥ Λ ∗ ( β ) , there exists ( u λ,β , v λ,β ) ∈ E satisfying D [ J ∗ λ,β ( u λ,β , v λ,β )] = 0 in E ∗ and J ∗ λ,β ( u λ,β , v λ,β ) = m J a ,J b ,λ,β , where Λ ∗ ( β, m a,b ) isgiven by Lemma 4.3. Furthermore, for every { λ n } ⊂ [Λ ∗ ( β ) , + ∞ ) satisfying λ n → + ∞ as n → ∞ ,there exists ( u J a ,β , v J b ,β ) ∈ E such that (1) ( u J a ,β , v J b ,β ) ∈ H (Ω J a a, ) × H (Ω J b b, ) with u J a ,β = 0 on R \ Ω J a a, and v J b ,β = 0 on R \ Ω J b b, . (2) ( u λ n ,β , v λ n ,β ) → ( u J a ,β , v J b ,β ) strongly in H ( R ) × H ( R ) as n → ∞ up to a subsequence. The restriction of u J a ,β on Ω a,i a lies in H (Ω a,i a ) and is a critical point of I Ω a,ia ( u ) for every i a ∈ J a , while the restriction of v J b ,β on Ω b,j b lies in H (Ω b,j b ) and is a critical point of I Ω b,jb ( v ) for every j b ∈ J b . Proof.
Since the conditions ( D )-( D ), ( D ′ ), ( D ) and ( D ′ ) hold, by a similar argument as[24, Lemma 3.1], we can see that lim λ → + ∞ m a,i a ,λ = m a,i a and lim λ → + ∞ m b,j b ,λ = m b,j b for all i a = 1 , · · · , n a and j b = 1 , · · · , n b . Note that β <
0, so by Lemma 4.5, there exists Λ ∗ ( β ) ≥ Λ ∗ ( β, m a,b ) such that m J a ,J b ,λ,β > J ∗ λ,β ( γ ,a ( t , · · · , t k a ) , γ ,b ( s , · · · , s k b )) for λ ≥ Λ ∗ ( β ). Thanksto the construction of m J a ,J b ,λ,β and Lemma 4.2, we can use the linking theorem (cf. [1]) to showthat m J a ,J b ,λ,β is a critical value of J ∗ λ,β ( u, v ) for λ ≥ Λ ∗ ( β ), that is, there exists ( u λ,β , v λ,β ) ∈ E satisfying D [ J ∗ λ,β ( u λ,β , v λ,β )] = 0 in E ∗ and J ∗ λ,β ( u λ,β , v λ,β ) = m J a ,J b ,λ,β for all λ ≥ Λ ∗ ( β ). Inwhat follows, we will show that (1)-(3) hold. Suppose { λ n } ⊂ [Λ ∗ ( β ) , + ∞ ) satisfying λ n → + ∞ as n → ∞ . Then by Lemmas 4.3 and 4.5, { ( u λ n ,β , v λ n ,β ) } is bounded in E with Z R \ Ω Jaa |∇ u λ n ,β | + ( λ n a ( x ) + a ( x )) u λ n ,β dx → n → ∞ (4.37)and Z R \ Ω Jbb |∇ v λ n ,β | + ( λ n b ( x ) + b ( x )) v λ n ,β dx → n → ∞ . (4.38)Without loss of generality, we assume ( u λ n ,β , v λ n ,β ) ⇀ ( u J a ,β , v J b ,β ) weakly in E as n → ∞ for some( u J a ,β , v J b ,β ) ∈ E . For the sake of clarity, we divide the following proof into two steps. Step 1.
We prove that ( u J a ,β , v J b ,β ) ∈ H (Ω J a a, ) × H (Ω J b b, ) with u J a ,β = 0 on R \ Ω J a a, and v J b ,β = 0 on R \ Ω J b b, .Indeed, since β <
0, by Lemmas 4.1 and 4.5 and a similar argument as used in Step 1 ofthe proof for Theorem 1.1, we can conclude that u J a ,β = 0 on R \ Ω a and v J b ,β = 0 on R \ Ω b .On the other hand, combining (2.9) and (4.37), we can see that R Ω ′ a,ia u λ n ,β dx → i a ∈{ , · · · , n a }\ J a as n → ∞ , which together with the Fatou lemma, implies u J a ,β = 0 on Ω ′ a,i a for i a ∈ { , · · · , n a }\ J a . Since (4.38) holds, by a similar argument, we also have v J b ,β = 0 on Ω ′ b,j b for j b ∈ { , · · · , n b }\ J b . Note that { Ω a,i a } and { Ω b,j b } are two sequences of disjoint bounded domainswith smooth boundaries. So ( u J a ,β , v J b ,β ) ∈ H (Ω J a a, ) × H (Ω J b b, ) with u J a ,β = 0 on R \ Ω J a a, and v J b ,β = 0 on R \ Ω J b b, . Step 2.
We prove that ( u λ n ,β , v λ n ,β ) → ( u J a ,β , v J b ,β ) strongly in H ( R ) × H ( R ) as n → ∞ up to a subsequence, and the restriction of u J a ,β on Ω a,i a lies in H (Ω a,i a ) and is a critical point of I Ω a,ia ( u ) for every i a ∈ J a , while the restriction of v J b ,β on Ω b,j b lies in H (Ω b,j b ) and is a criticalpoint of I Ω b,jb ( v ) for every j b ∈ J b .Indeed, since ( u J a ,β , v J b ,β ) ∈ H (Ω J a a, ) × H (Ω J b b, ) with u J a ,β = 0 on R \ Ω J a a, and v J b ,β = 0 on R \ Ω J b b, , by D [ J ∗ λ n ,β ( u λ n ,β , v λ n ,β )] = 0 in E ∗ and the condition ( D ′ ), we can see that the restrictionof u J a ,β on Ω a,i a , denoted by u J a i a ,β , lies in H (Ω a,i a ) and I ′ Ω a,ia ( u J a i a ,β ) = 0 in H − (Ω a,i a ) for all i a ∈ J a , while the restriction of v J b ,β on Ω b,j b , denoted by v J b j b ,β , lies in H (Ω b,j b ) and I ′ Ω b,jb ( v J b j b ,β ) = 0in H − (Ω b,j b ) for all j b ∈ J b . Now, since β < D ) contains the condition ( D ′ ),by the construction of f a ( x, t ), f b ( x, t ) and h ( x, t, s ) and a similar argument as used in Step 2 of theproof for Theorem 1.1, we can conclude that ( u λ n ,β , v λ n ,β ) → ( u J a ,β , b J b ,β ) in E as n → ∞ . Since E is embedded continuously into H ( R ) × H ( R ), ( u λ n ,β , v λ n ,β ) → ( u J a ,β , v J b ,β ) in H ( R ) × H ( R )as n → ∞ .In the following part, we will use a deformation argument to obtain the solution described byTheorem 1.2. Let ε = min { √ m a, , · · · , √ m a,k a , √ m b, , · · · , √ m b,k b } . For 0 < ε < ε , we29efine D a,ε = (cid:26) u ∈ E a | (cid:18) Z R \ Ω Jaa, |∇ u | + u dx (cid:19) < ε, (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) Z Ω a,ia |∇ u | + a ( x ) u dx (cid:19) − √ m a,i a (cid:12)(cid:12)(cid:12)(cid:12) < ε, ∀ i a ∈ J a (cid:27) and D b,ε = (cid:26) v ∈ E b | (cid:18) Z R \ Ω Jbb, |∇ v | + v dx (cid:19) < ε, (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) Z Ω b,jb |∇ v | + b ( x ) v dx (cid:19) − √ m b,j b (cid:12)(cid:12)(cid:12)(cid:12) < ε, ∀ j b ∈ J b (cid:27) . Let D ε = D a,ε ∩D b,ε and J m Ja,Jb λ,β = (cid:26) ( u, v ) ∈ E | J ∗ λ,β ( u, v ) ≤ m J a ,J b (cid:27) , where m J a ,J b = k a P i a =1 m a,i a + k b P j b =1 m b,j b . Then we have the following. Proposition 4.2
Assume β < and < ε < ε . Then there exists Λ ∗ ( β, ε ) ≥ Λ ∗ ( β ) such that J ∗ λ,β ( u, v ) has a critical point in D ε ∩ J m Ja,Jb λ,β for λ ≥ Λ ∗ ( β, ε ) . Proof.
Suppose the contrary, since Lemma 4.2 holds, there exist { λ n } and { c n,β } with λ n → + ∞ as n → ∞ and c n,β > n such that k D [ J ∗ λ n ,β ( u, v )] k E ∗ ≥ c n,β for all ( u, v ) ∈ D ε ∩ J m Ja,Jb λ n ,β . (4.39)For the sake of clarity, we divide the following proof into several steps. Step 1.
We prove that there exists N ∈ N and a constant σ > k J ∗ λ n ,β ( u, v ) k E ∗ ≥ σ for every ( u, v ) ∈ ( D ε \D ε ) ∩ J m Ja,Jb λ n ,β and n ≥ N .Suppose the contrary, there exists a subsequence of { λ n } , still denoted by { λ n } , such that k J ∗ λ n ,β ( u λ n ,β , v λ n ,β ) k E ∗ → n → ∞ for some ( u λ n ,β , v λ n ,β ) ∈ ( D ε \D ε ) ∩ J m Ja,Jb λ n ,β . By a similarargument as used in Proposition 4.1, we can see that ( u λ n ,β , v λ n ,β ) → ( u J a ,β , v J b ,β ) strongly in H ( R ) × H ( R ) and E as n → ∞ for some ( u J a ,β , v J b ,β ) ∈ H (Ω J a a, ) × H (Ω J b b, ) with u J a ,β = 0on R \ Ω J a a, and v J b ,β = 0 on R \ Ω J b b, . The restriction of u J a ,β on Ω a,i a , denoted by u J a i a ,β , lies in H (Ω a,i a ) and I ′ Ω a,ia ( u J a i a ,β ) = 0 in H − (Ω a,i a ) for all i a ∈ J a , while the restriction of v J b ,β on Ω b,j b ,denoted by v J b j b ,β , lies in H (Ω b,j b ) and I ′ Ω b,jb ( v J b j b ,β ) = 0 in H − (Ω b,j b ) for all j b ∈ J b . Clearly, oneof the following two cases must occur:(1 ∗ ) R Ω a,ia ( u J a i a ,β ) dx ≥ C for all i a ∈ J a and R Ω b,jb ( v J b j b ,β ) dx ≥ C for all j b ∈ J b .(2 ∗ ) There exists i ′ a ∈ J a or j ′ b ∈ J b such that R Ω a,i ′ a ( u J a i ′ a ,β ) dx = 0 or R Ω b,j ′ b ( v J b j ′ b ,β ) dx = 0.If case (1 ∗ ) happens, then we must have I Ω a,ia ( u J a i a ,β ) ≥ m a,i a and I Ω b,jb ( v J b j b ,β ) ≥ m b,j b for all i a ∈ J a and j b ∈ J b . Since ( u λ n ,β , v λ n ,β ) ∈ J m Ja,Jb λ n ,β , by a similar argument as used in Step 3 ofthe proof for Theorem 1.1, we can show that I Ω a,ia ( u J a i a ,β ) = m a,i a and I Ω b,jb ( v J b j b ,β ) = m b,j b for all i a ∈ J a and j b ∈ J b . It follows from the condition ( D ) that Z Ω a,ia |∇ u λ n ,β | + a ( x ) u λ n ,β dx → m a,i a for all i a ∈ J a Z Ω b,jb |∇ v λ n ,β | + b ( x ) v λ n ,β dx → m b,j b for all j b ∈ J b as n → ∞ , which then implies ( u λ n ,β , v λ n ,β ) ∈ D ε for n large enough. It is impossible since( u λ n ,β , v λ n ,β ) ∈ ( D ε \D ε ) ∩ J m Ja,Jb λ n ,β for all n . Thus, we must have the case (2 ∗ ). Without loss ofgenerality, we assume R Ω a, ( u J a ,β ) dx = 0. It follows from the condition ( D ) that (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) Z Ω a, |∇ u λ n ,β | + a ( x ) u λ n ,β dx (cid:19) − √ m a, (cid:12)(cid:12)(cid:12)(cid:12) → √ m a, = 4 ε as n → ∞ , which implies ( u λ n ,β , v λ n ,β ) ∈ E \D ε for n large enough. It also contradicts to the fact that( u λ n ,β , v λ n ,β ) ∈ ( D ε \D ε ) ∩ J m Ja,Jb λ n ,β for all n . Step 2.
We construct a descending flow on J m Ja,Jb λ n ,β for every n ≥ N .Let η : E → [0 ,
1] be a local Lipschitz continuous function and satisfy η ( u, v ) = ( , ( u, v ) ∈ D ε , , ( u, v ) ∈ E \D ε . Since J ∗ λ n ,β ( u, v ) is C for every n ≥ N , there exists a pseudo-gradient vector field of J ∗ λ n ,β ( u, v ),denoted by ^ D [ J ∗ λ n ,β ( u, v )] = ( ^ D [ J ∗ λ n ,β ( u, v )] , ^ D [ J ∗ λ n ,β ( u, v )] ), which satisfies( a ∗ ) h D [ J ∗ λ n ,β ( u, v )] , ^ D [ J ∗ λ n ,β ( u, v )] i E ∗ ,E ∗ ≥ k D [ J ∗ λ n ,β ( u, v )] k E ∗ for all ( u, v ) ∈ E ;( b ∗ ) k ^ D [ J ∗ λ n ,β ( u, v )] k E ∗ ≤ k D [ J ∗ λ n ,β ( u, v )] k E ∗ for all ( u, v ) ∈ E .Let −→V n : J m Ja,Jb λ n ,β → E ∗ be a continuous map and given by −→V n ( u, v ) = ( V ,n ( u, v ) , V ,n ( u, v )) = − η ( u, v ) k ^ D [ J ∗ λ n ,β ( u, v )] k E ∗ ( ^ D [ J ∗ λ n ,β ( u, v )] , ^ D [ J ∗ λ n ,β ( u, v )] )for ( u, v ) ∈ J m Ja,Jb λ n ,β \K = { ( u, v ) ∈ J m Ja,Jb λ n ,β | D [ J ∗ λ n ,β ( u, v )] = 0 } and −→V n ( u, v ) = ( V ,n ( u, v ) , V ,n ( u, v )) = (0 , u, v ) ∈ J m Ja,Jb λ n ,β ∩ K = { ( u, v ) ∈ J m Ja,Jb λ n ,β | D [ J ∗ λ n ,β ( u, v )] = 0 } . Clearly, k−→V n ( u, v ) k E ∗ ≤ u, v ) ∈ J m Ja,Jb λ n ,β . Furthermore, by (4.39), Step 1 and the definition of η , we can see that −→V n ( u, v )is locally Lipschitz. Now, let us consider the flow −→ ρ n ( τ ) = ( ρ ,n ( τ ) , ρ ,n ( τ )) given by the followingtwo-component system of ODE dρ ,n ( τ ) dτ = V ,n ( ρ ,n ( τ ) , ρ ,n ( τ )) ,dρ ,n ( τ ) dτ = V ,n ( ρ ,n ( τ ) , ρ ,n ( τ )) , ( ρ ,n (0) , ρ ,n (0)) = ( u, v ) ∈ J m Ja,Jb λ n ,β . By ( a ∗ ), ( b ∗ ) and a direct calculation, we can see that dJ ∗ λ n ,β ( ρ ,n ( τ ) , ρ ,n ( τ )) dτ = (cid:28) ( ∂J ∗ λ n ,β ∂u ( ρ ,n , ρ ,n ) , ∂J ∗ λ n ,β ∂v ( ρ ,n , ρ ,n )) , ( V ,n ( ρ ,n , ρ ,n ) , V ,n ( ρ ,n , ρ ,n )) (cid:29) E ∗ ,E ∗ ≤ − η ( ρ ,n , ρ ,n ) k D [ J ∗ λ n ,β ( ρ ,n , ρ ,n )] k E ∗ ≤ .
31t follows that −→ ρ n ( τ ) = ( ρ ,n ( τ ) , ρ ,n ( τ )) is a descending flow on J m Ja,Jb λ n ,β . Furthermore, for every τ >
0, we have −→ ρ n ( τ ) = −→ ρ n (0) if −→ ρ n (0) = ( u, v ) ∈ E \D ε . Step 3.
For every n ≥ N , we construct a map −→ ρ n ( τ ) = ( ρ ,n ( τ ) , ρ ,n ( τ )) ∈ Γ for all τ > [0 , ka × [0 , kb J ∗ λ n ,β ( ρ ,n ( τ n ) , ρ ,n ( τ n )) < m J a ,J b − σ ∗ for some τ n > , (4.40)where σ ∗ > n ≥ N and γ ,a and γ ,b be given by (4.30) and (4.31). We consider −→ ρ n ( τ ) =( ρ ,n ( τ ) , ρ ,n ( τ )), where ( ρ ,n (0) , ρ ,n (0)) = ( γ ,a , γ ,b ). Since W a,i a and W b,j b are the least energynonzero critical points of I Ω a,ia ( u ) and I Ω b,jb ( v ) for all i a ∈ J a and j b ∈ J b , by the choice of R and ε , we can see that ( γ ,a ( t , · · · , t k a ) , γ ,b ( s , · · · , s k b )) ∈ E \D ε for every ( t , · · · , t k a , s , · · · , s k b ) ∈ ∂ ([0 , k a × [0 , k b ). It follows from the construction of( ρ ,n ( τ ) , ρ ,n ( τ )) that( ρ ,n ( τ )( t , · · · , t k a ) , ρ ,n ( τ )( s , · · · , s k b )) = ( γ ,a ( t , · · · , t k a ) , γ ,b ( s , · · · , s k b ))for every ( t , · · · , t k a , s , · · · , s k b ) ∈ ∂ ([0 , k a × [0 , k b ) and τ >
0. Thus, −→ ρ n ( τ ) ∈ Γ for all τ > t , · · · , t k a , s , · · · , s k b ) ∈ [0 , k a × [0 , k b , one of thefollowing two cases must occur:( a ∗ ) ( γ ,a ( t , · · · , t k a ) , γ ,b ( s , · · · , s k b )) ∈ E \D ε .( b ∗ ) ( γ ,a ( t , · · · , t k a ) , γ ,b ( s , · · · , s k b )) ∈ D ε .If case ( a ∗ ) happen, then by Step 2, we must have J ∗ λ n ,β ( ρ ,n ( τ )( t , · · · , t k a ) , ρ ,n ( τ )( s , · · · , s k b )) ≤ J ∗ λ n ,β ( γ ,a ( t , · · · , t k a ) , γ ,b ( s , · · · , s k b ))for all τ >
0. Moreover, by (4.29) and the choice of { W a,i a } and { W b,j b } , we can see that J ∗ λ n ,β ( γ ,a ( t , · · · , t k a ) , γ ,b ( s , · · · , s k b )) = k a X i a =1 m a,i a + k b X j b =1 m b,j b if and only if t i a = s j b = R for all i a ∈ J a and j b ∈ J b . Since ( γ ,a ( t , · · · , t k a ) , γ ,b ( s , · · · , s k b )) ∈ E \D ε in this case, there exists i ′ a ∈ J a or j ′ b ∈ J b such that t i ′ a = R or s j ′ b = R . It follows theconstruction of γ ,a and γ ,b and the condition ( D ′ ) that m ∗ a,b = sup ( u,v ) ∈ P J ∗ λ n ,β ( u, v ) = sup ( u,v ) ∈ P ( I Ω a ( u ) + I Ω b ( v )) < k a X i a =1 m a,i a + k b X j b =1 m b,j b , (4.41)where P = ( γ ,a ([0 , k a ) × γ ,b ([0 , k b )) \D ε . If case ( b ∗ ) happen, then two subcases may occur:( b ∗ ) ( ρ ,n ( τ )( t , · · · , t k a ) , ρ ,n ( τ )( s , · · · , s k b )) ∈ D ε for all τ > b ∗ ) There exists τ ∗ n > ρ ,n ( τ ∗ n )( t , · · · , t k a ) , ρ ,n ( τ ∗ n )( s , · · · , s k b )) ∈ E \D ε .In the subcase ( b ∗ ), by Step 2 and the Taylor expansion, we can calculate that J ∗ λ n ,β ( ρ ,n ( τ )( t , · · · , t k a ) , ρ ,n ( τ )( s , · · · , s k b )) ≤ J ∗ λ n ,β ( γ ,a ( t , · · · , t k a ) , γ ,b ( s , · · · , s k b )) − Z τ η ( ρ ,n , ρ ,n ) k D [ J ∗ λ n ,β ( ρ ,n , ρ ,n )] k E ∗ dν ≤ k a X i a =1 m a,i a + k b X j b =1 m b,j b − τ min { c n,β , σ } , (4.42)32here c n,β is given by (4.39) and σ is given by Step 1. It follows that J ∗ λ n ,β ( ρ ,n ( τ )( t , · · · , t k a ) , ρ ,n ( τ )( s , · · · , s k b )) < k a X i a =1 m a,i a + k b X j b =1 m b,j b − σ for τ ≥ τ n = σ min { c n,β ,σ } . In the subcase ( b ∗ ), there must exist 0 ≤ τ ∗ n,a < τ ∗ n,b ≤ τ ∗ n such that( ρ ,n ( τ ∗ n,a )( t , · · · , t k a ) , ρ ,n ( τ ∗ n,a )( s , · · · , s k b )) ∈ ∂ D ε and ( ρ ,n ( τ ∗ n,b )( t , · · · , t k a ) , ρ ,n ( τ ∗ n,b )( s , · · · , s k b )) ∈ ∂ D ε . For the sake of convenience, we respectively denote ( ρ ,n ( τ ∗ n,a )( t , · · · , t k a ) , ρ ,n ( τ ∗ n,a )( s , · · · , s k b ))and ( ρ ,n ( τ ∗ n,b )( t , · · · , t k a ) , ρ ,n ( τ ∗ n,b )( s , · · · , s k b )) by ( u n, , v n, ) and ( u n, , v n, ). Since Lemma 2.1holds for λ ≥ Λ , by a similar argument as used for (4.14) in [24], we can see that one of the followingfour cases must happen:( a ∗∗ ) R Ω a |∇ ( u n, − u n, ) | + a ( x )( u n, − u n, ) dx ≥ ε .( b ∗∗ ) R Ω b |∇ ( v n, − v n, ) | + b ( x )( v n, − v n, ) dx ≥ ε .( c ∗∗ ) R R \ Ω Jaa, |∇ ( u n, − u n, ) | + ( u n, − u n, ) dx ≥ ε .( d ∗∗ ) R R \ Ω Jbb, |∇ ( v n, − v n, ) | + ( v n, − v n, ) dx ≥ ε .By similar arguments as (2.1)-(2.2), we can see that in any case, there exists a constant C ( ε ) > k ( u n, , v n, ) − ( u n, , v n, ) k ≥ C ( ε ). On the other hand, by Step 2, we can see that( u n, , v n, ) ∈ J m Ja,Jb λ n ,β and ( u n, , v n, ) ∈ J m Ja,Jb λ n ,β . It follows from k−→V n ( u, v ) k E ∗ ≤ u, v ) ∈ J m Ja,Jb λ n ,β and the Taylor expansion that τ ∗ n,b − τ ∗ n,a ≥ C ( ε ). Now, by Step 1, we have J λ n ,β ( ρ ,n ( τ ∗ n )( t , · · · , t k a ) , ρ ,n ( τ ∗ n )( s , · · · , s k b )) ≤ J λ n ,β ( γ ,a ( t , · · · , t k a ) , γ ,b ( s , · · · , s k b )) − Z τ ∗ n,b τ ∗ n,a η ( ρ ,n , ρ ,n ) k D [ J λ n ,β ( ρ ,n , ρ ,n )] k E ∗ dν ≤ k a X i a =1 m a,i a + k b X j b =1 m b,j b − C ( ε ) σ . (4.43)Let τ n = max { τ n , τ ∗ n } and σ ∗ = min { C ( ε ) σ , σ , k a P i a =1 m a,i a + k b P j b =1 m b,j b − m ∗ a,b } . Then (4.40)follows from (4.41)-(4.43) and Step 2.Since −→ ρ n ( τ ) = ( ρ ,n ( τ ) , ρ ,n ( τ )) ∈ Γ for all τ >
0, by the definition of m J a ,J b ,λ n ,β and Step3, we can see that m J a ,J b ,λ n ,β ≤ m J a ,J b − σ ∗ , which is impossible since Lemma 4.5 holds and m a,i a ,λ n → m a,i a and m b,j b ,λ n → m b,j b as n → ∞ for all i a ∈ J a and j b ∈ J b .We close this section by Proof of Theorem 1.2:
Suppose β <
0. Then by Proposition 4.2, for every ε ∈ (0 , ε ), thereexists Λ ∗ ( β, ε ) > Λ ∗ ( β ) such that J ∗ λ,β ( u, v ) has a critical point ( u J a λ,β , v J b λ,β ) ∈ D ε ∩ J m Ja,Jb λ,β for all λ ≥ Λ ∗ ( β, ε ). Thanks to Lemma 4.3 and the choice of Λ ∗ ( β, ε ), ( u J a λ,β , v J b λ,β ) is also a critical point of J λ,β ( u, v ). Since u J a λ,β and v J b λ,β are both nonnegative by the construction of J ∗ λ,β ( u, v ), we can use asimilar argument as used in the proof of Theorem 1.1 to show that u J a λ,β and v J b λ,β are both positive,33hich implies ( u J a λ,β , v J b λ,β ) is a solution of ( P λ,β ). Clearly, ( u J a λ,β , v J b λ,β ) satisfies the concentrationbehaviors of (1) and (2), since Λ ∗ ( β, ε ) → + ∞ as ε → u J a λ,β , v J b λ,β ) ∈ D ε . Furthermore,since ( u J a λ,β , v J b λ,β ) ∈ J m Ja,Jb λ,β , by a similar argument as used in Proposition 4.1, we can see that theproperties (3) and (4) are also hold. It remains to show that the concentration behavior (5) is alsotrue. Indeed, by a similar argument as used in Proposition 4.1, the restriction of u J a ,β on Ω a,i a ,denoted by u J a i a ,β , lies in H (Ω a,i a ) and is a critical point of I Ω a,ia ( u ) for every i a ∈ J a , while therestriction of v J b ,β on Ω b,j b , denoted by v J b j b ,β , lies in H (Ω b,j b ) and is a critical point of I Ω b,jb ( v )for every j b ∈ J b . Since ( u J a λ,β , v J b λ,β ) ∈ D ε for all λ ≥ Λ ∗ ( β, ε ), it is easy to see that u J a i a ,β and v J b j b ,β are nonzero for all i a ∈ J a and j b ∈ J b . It follows from ( u J a λ,β , v J b λ,β ) ∈ J m Ja,Jb λ,β that u J a i a ,β and v J b j b ,β are the least energy nonzero critical points of I Ω a,ia ( u ) and I Ω b,jb ( v ) for every i a ∈ J a and j b ∈ J b ,respectively. The proof of this theorem can be finished by taking Λ ∗ ( β ) = Λ ∗ ( β, ε ). In this section, we study the phenomenon of phase separations to ( P λ,β ), that is, we study theconcentration behavior of the solutions for ( P λ,β ) as β → −∞ . Proof of Theorem 1.3:
Suppose λ ≥ Λ ∗ and { β n } ⊂ ( −∞ ,
0) satisfying β n → −∞ as n → ∞ .Let ( u λ,β n , v λ,β n ) be the ground state solution of ( P λ,β n ) obtained by Theorem 1.1. Then byLemma 3.2 and a similar argument of (3.9), we can see that { ( u λ,β n , v λ,β n ) } is bounded in E .Without loss of generality, we assume ( u λ,β n , v λ,β n ) ⇀ ( u λ, , v λ, ) weakly in E as n → ∞ for some( u λ, , v λ, ) ∈ E . Since E is embedded continuously into H ( R ) × H ( R ), we have ( u λ, , v λ, ) ∈ H ( R ) × H ( R ). Clearly, u λ, ≥ v λ, ≥ R . In what follows, we verify that ( u λ, , v λ, )satisfies (1)-(4). For the sake of clarity, we divide the following proof into several steps. Step 1.
We prove that there exists Λ ∗∗ ≥ Λ ∗ such that u λ, = 0 and v λ, = 0 in R for λ ≥ Λ ∗∗ in the sense of almost everywhere.Indeed, suppose u λ, = 0 a.e. in R . Since u λ,β n ⇀ u λ, weakly in E a as n → ∞ and { ( u λ,β n , v λ,β n ) } is bounded in E , by a similar argument as (3.12), we can see that k u λ,β n k a,λ ≤ Cλ − k u λ,β n k a,λ + o n (1) . It follows that there exists Λ ∗∗ ≥ Λ ∗ such that k u λ,β n k a,λ = o n (1) for λ ≥ Λ ∗∗ , which togetherLemma 2.1 and the boundedness of { ( u λ,β n , v λ,β n ) } in E , implies k u λ,β n k = o n (1). Thanks tothe fact that ( u λ,β n , v λ,β n ) is the ground state solution of ( P λ,β n ) with β n < n , we alsohave β n R R u λ,β n v λ,β n dx → n → ∞ . Hence, J λ,β n ( u λ,β n , v λ,β n ) = I b,λ ( v λ,β n ) + o n (1). On theother hand, by Lemma 2.1, for every n ∈ N , there exists t n > t n u λ,β n ∈ N a,λ with λ ≥ λ ∗∗ . It follows from Lemma 3.1 and β n < J λ,β n ( u λ,β n , v λ,β n ) ≥ J λ,β n ( t n u λ,β n , v λ,β n ) ≥ I a,λ ( t n u λ,β n ) + I b,λ ( v λ,β n ) ≥ m a,λ + J λ,β n ( u λ,β n , v λ,β n ) + o n (1) . (5.1)Since m a,λ > λ ≥ λ ∗∗ , (5.1) is impossible for n large enough. By a similar argument, we canalso show that v λ, = 0 in R for λ ≥ Λ ∗∗ in the sense of almost everywhere. Step 2.
We prove that { u λ,β n } , { v λ,β n } ⊂ C ( R ) and k u λ,β n k C ( R ) ≤ C and k v λ,β n k C ( R ) ≤ C for some C > { u λ,β n } , { v λ,β n } ⊂ C ( R ). On the other hand, since ( u λ,β n , v λ,β n ) is the ground state solution of ( P λ,β n ) obtained byTheorem 1.1 and { ( u λ,β n , v λ,β n ) } is bounded in E , we can use a similar argument as used in (3) ofLemma 4.3 to show that { u λ,β n } and { v λ,β n } are bounded in L ∞ ( R ), that is, k u λ,β n k L ∞ ( R ) ≤ C and k v λ,β n k L ∞ ( R ) ≤ C for some C >
0, which implies k u λ,β n k C ( R ) ≤ C and k v λ,β n k C ( R ) ≤ C .34 tep 3. We prove that u λ, , v λ, ∈ C ( R ) and are all local Lipschitz in R .Indeed, since the conditions ( D )-( D ) hold, by [46, Theorem 1.7] and Step 2, we can see that {∇ u λ,β n } and {∇ v λ,β n } are bounded in L ∞ ( R ). On the other hand, for every n , by a similarargument as used in (3) of Lemma 4.3, we can show that u λ,β n , v λ,β n ∈ L γ ( R ) for all γ ≥
2. Thanksto the Calderon-Zygmund inequality and conditions ( D )-( D ), we have u λ,β n , v λ,β n ∈ W ,γloc ( R )for all γ ≥
2. Together with the Sobolev embedding theorem, it implies u λ,β n , v λ,β n ∈ C ( R ).It follows that { u λ,β n } and { v λ,β n } are bounded in C ( R ). Now, by applying the Ascoli-Arzel´atheorem, we can conclude that u λ,β n → u λ, and v λ,β n → v λ, strongly in C loc ( R ) as n → ∞ with u λ, , v λ, ∈ C ( R ). This together with the boundness of { u λ,β n } and { v λ,β n } in C ( R ) again,implies u λ, and v λ, are all local Lipschitz in R . Step 4.
We prove that ( u λ,β n , v λ,β n ) → ( u λ, , v λ, ) strongly in H ( R ) × H ( R ) as n → ∞ .Furthermore, u λ, ∈ H ( { u λ, > } ) and is a least energy solution of (1.3), while v λ, ∈ H ( { v λ, > } ) and is a least energy solution of (1.4).Indeed, since u λ, ∈ C ( R ) and is local Lipschitz in R , we can conclude that ∂ { u λ, > } , theboundary of the set { u λ, > } , is local Lipschitz. It follows from u λ, ∈ H ( R ) and u λ, = 0 in R \{ u λ, > } that u λ, ∈ H ( { u λ, > } ). Similarly, we have v λ, ∈ H ( { v λ, > } ). Let I ∗ a,λ ( u )and I ∗ b,λ ( v ) respectively be the corresponding functional of (1.3) and (1.4). By a similar argumentas used in Lemma 2.1, we can show that C Z { u λ, > } u dx ≤ Z { u λ, > } |∇ u | + ( λa ( x ) + a ( x )) u dx for all u ∈ H ( { u λ, > } ) (5.2)and C Z { v λ, > } v dx ≤ Z { v λ, > } |∇ v | + ( λb ( x ) + b ( x )) v dx for all v ∈ H ( { v λ, > } ) (5.3)if λ ≥ Λ . It follows that the Nehari manifolds of I ∗ a,λ ( u ) and I ∗ b,λ ( v ) are both well defined if λ ≥ Λ . Let m ∗ a,λ = inf N ∗ a,λ I ∗ a,λ ( u ) and m ∗ b,λ = inf N ∗ b,λ I ∗ b,λ ( v ) , where N ∗ a,λ and N ∗ b,λ are respectively the Nehari manifolds of I ∗ a,λ ( u ) and I ∗ b,λ ( v ). Then m ∗ a,λ > m ∗ b,λ > λ ≥ Λ . For every ε >
0, there exist u ε ∈ N ∗ a,λ and v ε ∈ N ∗ b,λ such that I ∗ a,λ ( u ε ) < m ∗ a,λ + ε and I ∗ b,λ ( v ε ) < m ∗ b,λ + ε. (5.4)Since { ( u λ,β n , v λ,β n ) } is bounded in E , by the fact that ( u λ,β n , v λ,β n ) is the ground state solution( P λ,β n ) and β n → −∞ , we have R R u λ,β n v λ,β n dx → n → ∞ . It follows from the Fatou lemmathat R R u λ, v λ, dx = 0, which implies { u λ, > } ∩ { v λ, > } = ∅ . Hence, by u ε ∈ N ∗ a,λ and v ε ∈ N ∗ b,λ , we can see that ( u ε , v ε ) ∈ N λ,β n for all n . Now, by (5.4), we have2 ε + m ∗ a,λ + m ∗ b,λ ≥ I ∗ a,λ ( u ε ) + I ∗ b,λ ( v ε ) = J λ,β n ( u ε , v ε ) ≥ m λ,β n for all n. Since ε > n ∈ N is arbitrary, we can conclude that m ∗ a,λ + m ∗ b,λ ≥ lim sup n →∞ m λ,β n . (5.5)On the other hand, note that ( u λ,β n , v λ,β n ) is the ground state solution ( P λ,β n ) and ( u λ,β n , v λ,β n ) ⇀ ( u λ, , v λ, ) weakly in E as n → ∞ , by β n <
0, we can see that Z { u λ, > } |∇ u λ, | + ( λa ( x ) + a ( x )) u λ, dx ≤ µ Z { u λ, > } u λ, dx and Z { v λ, > } |∇ v λ, | + ( λb ( x ) + b ( x )) v λ, dx ≤ µ Z { v λ, > } v λ, dx.
35t follows from (5.2) and (5.3) that there exist 0 < t ≤ < s ≤ t u λ, ∈ N ∗ a,λ and s v λ, ∈ N ∗ b,λ . Now, since ( u λ,β n , v λ,β n ) ⇀ ( u λ, , v λ, ) weakly in E as n → ∞ , by a similarargument as (3.13), we can see thatlim inf n → + ∞ m λ,β n = lim inf n → + ∞ J λ,β n ( u λ,β n , v λ,β n )= 14 lim inf n → + ∞ ( k u λ,β n k a,λ + k v λ,β n k b,λ ) ≥
14 ( k u λ, k a,λ + k v λ, k b,λ ) ≥
14 ( k t u λ, k a,λ + k s v λ, k b,λ )= I ∗ a,λ ( t u λ, ) + I ∗ b,λ ( s v λ, ) ≥ m ∗ a,λ + m ∗ b,λ . (5.6)Hence, by combining (5.5) and (5.6), we must have the following results:( a ∗∗∗ ) lim n →∞ k u λ,β n k a,λ = k u λ, k a,λ and lim n →∞ k v λ,β n k b,λ = k v λ, k b,λ .( b ∗∗∗ ) u λ, ∈ N ∗ a,λ with I ∗ a,λ ( u λ, ) = m ∗ a,λ and v λ, ∈ N ∗ b,λ with I ∗ b,λ ( v λ, ) = m ∗ b,λ .By ( a ∗∗∗ ) and Lemma 2.1, we know that k u λ,β n − u λ, k a,λ = k v λ,β n − v λ, k b,λ = o n (1). Thanks toLemma 2.1 once more and the condition ( D ), we observe that k u λ,β n − u λ, k a = k v λ,β n − v λ, k b = o n (1) for λ ≥ Λ . Since by (5.2) and (5.3), N ∗ a,λ and N ∗ b,λ are a natural constraint in H ( { u λ, > } )and H ( { v λ, > } ), respectively, which together with ( b ∗∗∗ ), implies u λ, and v λ, are a least energysolution of (1.3) and (1.4), respectively. Step 5.
We prove that { x ∈ R | u λ, ( x ) > } and { x ∈ R | v λ, ( x ) > } are connecteddomains and { x ∈ R | u λ, ( x ) > } = R \{ x ∈ R | v λ, ( x ) > } .Indeed, since u λ, and v λ, are respectively a least energy solution of (1.3) and (1.4), we musthave { x ∈ R | u λ, ( x ) > } and { x ∈ R | v λ, ( x ) > } are connected domains. It remains toshow that { x ∈ R | u λ, ( x ) > } = R \{ x ∈ R | v λ, ( x ) > } . To the contrary, we suppose thatthere exists an open set Ω satisfying { x ∈ R | u λ, ( x ) > } $ Ω $ R \{ x ∈ R | v λ, ( x ) > } .Furthermore, it has a locally lipschitz boundary. Since Ω $ R \{ x ∈ R | v λ, ( x ) > } , by a similarargument as in Step 3, we can show that u λ, is a least energy solution of the following equation: − ∆ u + ( λa ( x ) + a ( x )) u = µ u , u ∈ H (Ω) . Since u λ, ≥ u λ, > { x ∈ R | u λ, ( x ) > } $ Ω.We complete the proof by showing that β R R u λ,β v λ,β → β → −∞ , where ( u λ,β , v λ,β )is the ground state solution of ( P λ,β ) obtained by Theorem 1.1. Indeed, if not, then there ex-ists { β n } ⊂ ( −∞ ,
0) such that β n R R u λ,β n v λ,β n ≤ − C . By (5.5) and (5.6), we must have β n R R u λ,β n v λ,β n → β n → −∞ up to a subsequence, which is a contradiction. This work was partly completed when the first author was visiting Tsinghua University, and heis grateful to the members in the Department of Mathematical Sciences at Tsinghua Universityfor their invitation and hospitality. The first author was supported by the Fundamental ResearchFunds for the Central Universities (2014QNA67), China. The second author was supported by theNational Science Council and the National Center for Theoretical Sciences (South), Taiwan. Thethird author was supported by NSFC(11025106, 11371212, 11271386) and the Both-Side TsinghuaFund.Y. Wu and W. Zou thank Prof. A. Szulkin for his suggestions and discussions when he wasvisiting Tsinghua University. They are also grateful to Dr. Zhijie Chen for careful reading themanuscript and some suggestions to improve it.36 eferences [1] A. Ambrosetti, P. Rabinowitz, Dual variational methods in critical point theory and appli-cations,
J. Funct. Anal., (1973), 349-381.[2] N. Akhmediev, A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev.Lett., (1999), 2661-2664.[3] T. Bartsch, Z.-Q. Wang, Existence and multiplicity results for superlinear elliptic problemson R n , Comm. Partial Differential Equations, (1995), 1725-1741.[4] T. Bartsch, Z.-Q. Wang, Multiple positive solutions for a nonlinear Schr¨odinger equations, Z. Angew. Math. Phys., (2000), 366-384.[5] T. Bartsch, A. Pankov, Z.-Q. Wang, Nonlinear Schr¨odinger equations with steep potentialwell, Commun. Contemp. Math., (2001), 549-569.[6] K. Brown, Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations, (2003), 481-499.[7] T. Bartsch, Z.-Q. Wang, W. Wei, Bound states for a coupled Schr¨odinger system,
J. FixedPoint Theory Appl., (2007), 353-367.[8] T. Bartsch, N. Dancer, Z.-Q. Wang, A Liouville theorem, a-priori bounds, and bifurcatingbranches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial DifferentialEquations, (2010), 345-361.[9] T. Bartsch, Bifurcation in a multicomponent system of nonlinear Schr¨odinger equations, J.Fixed Point Theory Appl., (2013), 37-50.[10] T. Bartsch, Z. Tang, Multibump solutions of nonlinear Schr¨odinger equations with steeppotential well and indefinite potential, Discrete Contin. Dyn. Syst., (2013), 7-26.[11] L. A. Caffarelli, F.-H. Lin, Singularly perturbed elliptic systems and multi-valued harmonicfunctions with free boundaries, J. Amer. Math. Soc., (2008), 847-862.[12] L. A. Caffarelli, J. M. Roquejoffre, Uniform H¨oder estimates in a class of elliptic systemsand applications to singular limits in models for diffusion flames, Arch. Ration. Mech. Anal., (2007), 457-487.[13] Z. Chen, W. Zou, Positive least energy solutions and phase separation for coupled Schr¨odingerequations with critical exponent,
Arch. Rational Mech. Anal., (2012), 515-551.[14] Z. Chen, W. Zou, Ground states for a system of Schr¨odinger equations with critical exponent,
J. Funct. Anal., (2012), 3091-3107.[15] Z. Chen, W. Zou, An optimal constant for the existence of least energy solutions of a coupledSchr¨odinger system,
Calc. Var. Partial Differential Equations, (2013), 695-711.[16] Z. Chen, W. Zou, Positive least energy solutions and phase separation for coupled Schr¨odingerequations with critical exponent: higher dimensional case, arXiv:1209.2522v2 [math.AP]; Calc. Var. Partial Differential Equations, accepted.[17] Z. Chen, W. Zou, Standing waves for coupled Schr¨odinger equations with decaying potentials,
Journal of Mathematical Physics,
Trans. Amer. Math. Soc., accepted.[19] Z. Chen, C. S. Lin, W. Zou, Multiple sign-changing and semi-nodal solutions for coupledSchr¨odinger equations,
J. Differential Equations, (2013), 4289-4311.3720] Z. Chen, C. S. Lin, W. Zou, Infinitely many sign-changing and semi-nodal solutions for anonlinear Schr¨odinger system, arXiv:1212.3773v1 [math.AP].[21] Z. Chen, C. S. Lin, W. Zou, Sign-changing solutions and phase separation for an ellipticsystem with critical exponent,
Comm. Partial Differential Equations, (2014), 1827-1859.[22] M. Conti, S. Terracini, G. Verzini, Asymptotic estimates for the spatial segregation of com-petitive systems, Adv. Math., (2005), 524-560.[23] M. del Pino, P. Felmer, Local mountain passes for semilinear elliptic problems in unboundeddomains,
Calc. Var. Partial Differential Equations, (1996), 121-137.[24] Y. Ding, K. Tanaka, Multiplicity of positive solutions of a nonlinear Schr¨odinger equation, manuscripta math., (2003), 109-135.[25] B. Esry, C. Greene, J. Burke, J. Bohn, Hartree-Fock theory for double condesates, Phys.Rev. Lett., (1997), 3594-3597.[26] M. Furtado, E. Silva, M. Xavier, Multiplicity and concentration of solutions for ellipticsystems with vanishing potentials, J. Differential Equations, (2010), 2377-2396.[27] Y. Guo, Z. Tang, Ground state solutions for quasilinear Schr¨odinger systems,
J. Math. Anal.Appl., (2012), 322-339.[28] Y. Guo, Z. Tang, Multibump bound states for quasilinear Schr¨odinger systems with criticalfrequency,
J. Fixed Point Theory Appl., (2012), 135-174.[29] N. Hirano, Multiple existence of nonradial positive solutions for a coupled nonlinearSchr¨odinger system, NoDEA Nonlinear Differential Equations Appl., (2009), 159-188.[30] N. Ikoma, Uniqueness of positive solutions for a nonlinear elliptic system, NoDEA NonlinearDifferential Equations Appl., (2009), 555-567.[31] N. Ikoma, K. Tanaka, A local mountain pass type result for a system of nonlinear Schr¨odingerequations, Calc. Var. Partial Differential Equations, (2011), 449-480.[32] Y. Jiang, H.-S. Zhou, Schr¨odinger-Poisson system with steep potential well, J. DifferentialEquations, (2011), 582-608.[33] T.C. Lin, W. Wei, Spikes in two coupled nonlinear Schr¨odinger equations,
Ann. Inst. H.Poincar´e Anal. Non Lin´eaire, (2005), 403-439.[34] T.C. Lin, W. Wei, Spikes in two-component systems of nonlinear Schr¨odinger equations withtrapping potentials, J. Differential Equations, (2006), 538-569.[35] X. Liu, Y. Huang, J. Liu, Sign-changing solutions for an asymptotically linear Schr¨odingerequations with deepening potential well,
Adv. Differential Equations, (2011), 1-30.[36] T.C. Lin, T.F. Wu, Existence and multiplicity of positive solutions for two coupled nonlinearSchr¨odinger equations, Discrete Contin. Dyn. Syst., (2013), 2911-2938.[37] L.A. Maia, E. Silva, On a class of coupled elliptic systems in R n , NoDEA Nonlinear Differ-ential Equations Appl., (2007), 303-313.[38] B. Noris, H. Tavares, S. Terracini, G. Verzini, Uniform H¨older bounds for nonlinearSchr¨odinger systems with strong competition, Comm. Pure Appl. Math., (2010), 267-302.[39] J. Royo-Letelier, Segregation and symmetry breaking of strongly coupled two-componentBose-Einstein condensates in a harmonic trap, Calc. Var. Partial Differential Equations, (2014), 103-124. 3840] E. S´er´e, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z., (1992), 27-42.[41] C. Stuart, H.-S. Zhou, Positive eigenfunctions of a Schr¨oinger operator,
J. London Math.Soc., (2005), 429-441.[42] C. Stuart, H.-S. Zhou, Global branch of solutions for nonlinear Schr¨odinger equations withdeepening potential well, Proc. London Math. Soc., (2006), 655-681.[43] B. Sirakov, Least energy solitary waves for a system of nonlinear Schr¨odinger equations in R n , Commun. Math. Phys., (2007), 199-221.[44] Y. Sato, K. Tanaka, Sign-changing multi-bump solutions nonlinear Schr¨odinger equationswith steep potential wells,
Trans. Amer. Math. Soc., (2009), 6205-6253.[45] J. Sun, T.F. Wu, Ground state solutions for an indefinite Kirchhoff type problem with steeppotential well,
J. Differential Equations, (2014), 1771-1792.[46] N. Soave, A. Zilio, Uniform bounds for strongly competing systems: the optimal Lipschitzcase, arXiv:1407.6674v1 [math.AP].[47] S. Terracini, G. Verzini, Multipulse phases in k-mixtures of Bose-Einstein condensates,
Arch.Ration. Mech. Anal., (2009), 717-741.[48] J. Wei, T. Weth, Radial solutions and phase separation in a system of two coupledSchr¨odinger equations,
Arch. Ration. Mech. Anal., (2008), 83-106.[49] J. Wei, T. Weth, Asymptotic behaviour of solutions of planar elliptic systems with strongcompetition,
Nonlinearity, (2008), 305-317.[50] Z. Wang, H.-S. Zhou, Positive solutions for nonlinear Schr¨odinger equations with deepeningpotential well, J. Eur. Math. Soc., (2009), 545-573.[51] L. Zhao, H. Liu, F. Zhao, Existence and concentration of solutions for the Schr¨odinger-Poissonequations with steep well potential, J. Differential Equations,255