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ScienceDirect J. Differential Equations ••• (••••) •••–••• www.elsevier.com/locate/jde

Standing waves of a weakly coupled Schrödinger system with distinct potential functions ✩ Jun Wang a , Junping Shi b,∗ a Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu, 212013, PR China b Department of Mathematics, College of William and Mary, Williamsburg, VA 23187-8795, USA

Received 3 December 2014; revised 13 September 2015

Abstract The standing wave solutions of a weakly coupled nonlinear Schrödinger system with distinct trapping potential functions in RN (1 ≤ N ≤ 3) are considered. This type of system arises from models in Bose– Einstein condensates theory and nonlinear optics. The existence of a positive ground state solution is shown when the coupling constant is larger than a sharp threshold value, which is explicitly defined in terms of potential functions and system parameters. It is also shown that such solutions concentrate near the minimum points of potential functions, and multiple positive concentration solutions exist when the topological structure of the set of minimum points satisfies certain condition. Variational approach is used for the existence and concentration of positive solutions. © 2015 Elsevier Inc. All rights reserved. MSC: 35J47; 35J50; 35Q55 Keywords: Coupled nonlinear Schrödinger system; Positive ground solutions; Variational methods

✩

Partially supported by National Natural Science Foundation of China (Nos. 11201186, 11571140, 11371090, 11171135), NSF of Jiangsu Province (BK2012282, BK20150478), Jiangsu University Foundation grant (11JDG117), China Postdoctoral Science Foundation funded project (2012M511199, 2013T60499). * Corresponding author. E-mail addresses: [email protected] (J. Wang), [email protected] (J. Shi). http://dx.doi.org/10.1016/j.jde.2015.09.052 0022-0396/© 2015 Elsevier Inc. All rights reserved.

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1. Introduction and main results In this paper we consider the existence, multiplicity and nonexistence of standing wave solutions of the nonlinear Schrödinger system

(Aε )

⎧ 2 3 2 ⎪ ⎨−ε u + P (x)u = μu + βv u −ε 2 v + Q(x)v = νv 3 + βu2 v ⎪ ⎩ u, v > 0, u, v ∈ H 1 (RN ),

in RN , in RN ,

where 1 ≤ N ≤ 3, μ, ν > 0, ε is a small positive parameter, P (x) and Q(x) are positive potential functions, β > 0 is a coupling constant. The nonlinear Schrödinger equation is a canonical and universal equation in physics which is of essential importance in condensed matter, nonlinear optics, continuum mechanics and plasma physics. The coupled nonlinear Schrödinger equations have been the focus of many recent theoretical studies because of recent experimental advances in multi-component Bose–Einstein condensates [1]. The two-component coupled nonlinear Schrödinger equations (also known as Gross–Pitaevskii equations) can be written in the following form: ⎧ ∂ ⎪ ⎨−i h¯ 1 + V1 (x)1 = ∂t ⎪ ⎩−i h ∂ + V (x) = ¯ 2 2 2 ∂t

h¯ 2 1 + μ|1 |2 1 + β|2 |2 1 , 2m h¯ 2 2 + ν|2 |2 2 + β|1 |2 2 , 2m

in RN , t > 0,

(1.1)

in RN , t > 0.

In the context of Bose–Einstein condensates, the complex-valued j (x, t) (j = 1, 2) are the wave functions of two interacting condensates; Vj (x) are the trapping potentials; the interaction strength parameters μ, ν and β are determined by the scattering lengths for binary collisions of like and unlike bosons. The physically realistic spatial dimensions are 1 ≤ N ≤ 3. When N = 2, problem (1.1) arises in the Hartree–Fock theory for a double condensate, i.e., a binary mixture of Bose–Einstein condensates in two different hyperfine states (see [2,3]). In the attractive case, the components of a vector solution tend to go along with each other, leading to synchronization. In the repulsive case, the components tend to segregate from each other, leading to phase separations. These phenomena have been documented in experiments as well as in numerical simulations (see [4,5] and references therein). Another recent interest on coupled nonlinear Schrödinger equations is on the propagation of soliton-like pulses in birefringent nonlinear fibers. Experiments have proved the existence of self-trapping of incoherent beam in a nonlinear medium [6,5]. Such findings are significant since optical pulses propagating in a linear medium have a natural tendency to broaden in time (dispersion) and space (diffraction). In the context of optical propagation, j in (1.1) denotes the j -th component of the beam in Kerr-like photorefractive media; the positive constants μ, ν indicate the self-focusing strength in the component of the beam; and the coupling constant β measures the interaction between the two components of the beam. The sign of β determines whether the interactions of states are repulsive or attractive. A standing wave solution of (1.1) is of the following form: 1 (x, t) = e−iEt/h¯ u(x) and 2 (x, t) = e−iEt/h¯ v(x),

(1.2)

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where (u(x), v(x)) describes the spatial profile of the wave functions. Typically the wave functions tend to zero when |x| → ∞. Substituting (1.2) into (1.1), and renaming the parameters by h¯ 2 , P (x) = V1 (x) − E, Q(x) = V2 (x) − E, 2m

ε=

we obtain the elliptic system (Aε ). Before going further, we point out that the system (Aε ) possesses a trivial solution (0, 0) and semi-trivial solutions of type (u, 0) or (0, v). A solution (u, v) of (Aε ) is nontrivial if u = 0 and v = 0. A solution (u, v) with u > 0 and v > 0 is called a positive solution. A solution is called a ground state solution (or positive ground state solution) if its energy is minimal among all the nontrivial solutions (or all the nontrivial positive solutions) of (Aε ). Here the energy functional corresponding to (Aε ) is defined by 1 L(u, v) = 2

1 (ε |∇u| + P (x)u + ε |∇v| + Q(x)v ) − 4 2

2

2

2

2

(μu4 + 2βu2 v 2 + νv 4 ),

2

RN

RN

for (u, v) ∈ E ≡ H 1 (RN ) × H 1 (RN ). There have been extensive mathematical studies in recent years for the corresponding autonomous system with ε = 1, P (x) ≡ λ1 > 0 and Q(x) ≡ λ2 > 0: ⎧ 3 2 ⎪ ⎨−u + λ1 u = μu + βv u 3 2 −v + λ2 v = νv + βu v ⎪ ⎩ u, v > 0, u, v ∈ H 1 (RN ).

in RN , in RN ,

(1.3)

The existence, multiplicity, bifurcation, concentration behavior of positive solutions of (Aε ) with ε = 1 and (1.3) (in RN or a bounded domain of RN ) have been considered in, for example, [7–25] and references therein. In particular for the autonomous case (1.3), the existence, uniqueness of positive solutions can be summarized as follows: Theorem 1.1. Suppose that λi , μ, ν, β > 0, μ = ν and 1 ≤ N ≤ 3. 1. If λ1 = λ2 =λ, then (1.3) has a positive ground state solution (u, v) when β ∈ (0, min{μ, ν}) (max{μ, ν}, ∞), which can be expressed by (u(x), v(x)) =

√ λ(β − ν) w1 ( λx), 2 β − μν

√ λ(β − μ) w1 ( λx) , β 2 − μν

(1.4)

and w1 is the unique positive (radially symmetric) solution of the problem: w − w + w 3 = 0 in RN , w ∈ H 1 (RN );

(1.5)

and (1.3) has no positive solution when β ∈ [min{μ, ν}, max{μ, ν}]. Moreover when β ∈ (max{μ, ν}, ∞), the positive solution of (1.3) is unique up to a translation.

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2. If λ1 = λ2 , then there exist β0 = β0 (μ, ν, λ1 , λ2 ) ∈ (0, min{μ, ν}] and β1 = β1 (μ, ν, λ2 /λ 1 ) ≥ max{μ, ν} such that (1.3) has a positive ground state solution when β ∈ (0, β0 ) (β1 , ∞), and (1.3) has no positive solution when β ∈ [min{μ, ν}, max{μ, ν}], where N N β1 (μ, ν, z) = max μz, νz 2 −1 , νz−1 , μz1− 2 ,

for z > 0, N = 1, 2, 3.

(1.6)

The existence of the positive ground state solution of (1.3) in Theorem 1.1 was proved in [23], and the uniqueness was proved in [26]. Partial uniqueness results for the case β ∈ (0, min{μ, ν}) and λ1 = λ2 case were also proved in [10,11,26]. The semiclassical case (Aε ) with trapping potentials P (x) and Q(x) has been studied in [27, 13,16,20,21]. Lin and Wei [16] proved the existence and asymptotic concentration behavior of a ground state solution of (Aε ) with −∞ < β < β0 for a small β0 > 0. Pomponio [21] proved a similar result for (Aε ) with μ = μ(x), ν = ν(x) and β < 0. In [20], for small β > 0, Montefusco, Pellacci and Squassina showed the existence of nonnegative ground state solutions of (Aε ) concentrating around the local minimum (possibly degenerate) points of the potentials, which are in the same region. Moreover, if β > 0 is small, then one component of the ground state solution converges to zero. Ikoma and Tanaka [13] connected the solutions of (Aε ) with the limiting system (1.3) with λ1 = P (x0 ) and λ2 = Q(x0 ) for a fixed x0 ∈ RN . Assume that there exists an open bounded set ⊂ RN such that inf m(x0 ) < inf m(x0 ),

x0 ∈

x0 ∈∂

(1.7)

where m(x0 ) is the ground state energy level of (1.3) with (λ1 , λ2 ) = (P (x0 ), Q(x0 )), they showed the existence of positive solution concentrating at an x0 ∈ which achieves the minimal energy. Recently Long and Peng [28] proved that (Aε ) (with n ≥ 2 variables) has a positive solution for β > 0 small, whose components may have spikes clustering at the same point as ε → 0. Note that these results are all for β being negative or being positive but close to zero. We also mention that in the last 20 years, various existence and concentration results for the scalar nonlinear Schrödinger equations have also been obtained in, for example, [29–37]. For the large β > 0 case, Chen and Zou [27] proved the existence of positive solutions for (Aε ) which concentrate around local minima of the potentials for the case that P , Q may vanish in R3 and large β. More precisely, they assume that there exist a bounded open subset of R3 which is similar to the one in [13] so that (1.7) holds, and β > β˜0 := max{μ, ν} · max ¯ x∈

P (x) Q(x) , , Q(x) P (x)

(1.8)

then for small ε, a ground state solution (Aε ) exists, and it concentrates near the point x0 where m(x0 ) achieves the minimal energy in (1.7). In this paper we consider the semiclassical case of (Aε ) with positive trapping potential functions P (x), Q(x) and large coupling strength β. Throughout the paper we assume that (PQ0) P , Q ∈ C(RN , R+ ), and P , Q are bounded.

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We also use the following notations. P0 = inf P (x), x∈RN

P∞ = lim inf P (x)

Q0 = inf Q(x), x∈RN

|x|→∞

Q∞ = lim inf Q(x) |x|→∞

and P ∞ = lim sup P (x), |x|→∞

and

∞

Q

= lim sup Q(x), |x|→∞

P0 = {x ∈ R : P (x) = P0 } and Q0 = {x ∈ R : Q(x) = Q0 }. N

N

Because of the assumption (PQ0), P ∞ , Q∞ and P∞ , Q∞ are all finite. By using the notation defined above, we define a critical value characterized by ν, μ, P0 , Q0 , P∞ and Q∞ to guarantee the existence of a positive ground state solution of (Aε ). Set

Q0 Q∞ ˆ , β1 , (1.9) β1 = max β1 P0 P∞ where the function β1 (z) = β1 (μ, ν, z) is defined as in (1.6) with fixed μ and ν. Our main existence and concentration of a positive ground state solution of (Aε ) is as follows. Theorem 1.2. Suppose that 1 ≤ N ≤ 3, P , Q satisfy (PQ0) and (PQ1) 0 < P0 < P∞ < ∞, 0 < Q0 < Q∞ < ∞ and V = P0 ∩ Q0 = ∅. Let βˆ1 be defined as in (1.9). Then for each β > βˆ1 and small ε > 0, (i) (Aε ) possesses at least one positive ground state solution wε = (uε , vε ) in E. (ii) Let Bε be the set of all positive ground state solutions of (Aε ). Then Bε is compact in E. If in addition, P , Q are uniformly continuous in RN , then we have that (iii) there exists a maximum point xε of uε + vε such that lim dist(xε , V ) = 0, and subject ε→0

to a subsequence, xε → y0 ∈ V and hε (x) = wε (εx + xε ) = (uε (εx + xε ), vε (εx + xε )) converges in E to a positive ground state solution of (1.3) with λ1 = P (y0 ) and λ2 = Q(y0 ) 1,σ (RN )]2 with σ ∈ (0, 1), and there exist C1 , C2 > 0 such that as ε → 0. Moreover wε ∈ [Cloc lim |wε (x)| = lim |∇wε (x)| = 0, |wε (x)| ≤ C1 e−

|x|→∞

|x|→∞

C2 ε |x−xε |

, x ∈ RN .

(1.10)

Our existence of positive ground state result in Theorem 1.2 is different from the one in [27] from several aspects. The result in [27] imposes weaker conditions on P , Q outside of open subset and considers positive ground state solutions concentrating near x0 ∈ where the ground state energy m(x0 ) for (1.3) with (λ1 , λ2 ) = (P (x0 ), Q(x0 )) achieves the minimum (see (1.7) and (1.8)). Our result considers positive ground state solutions concentrating near global minima points of P and Q (which necessarily achieves minimal m(x0 )) but with simplified conditions on P , Q as m(x0 ) in (1.7) cannot be explicitly expressed. Our result holds for 1 ≤ N ≤ 3 and the one in [27] is only for N = 3 as they used Hardy inequality for the proof. Our condition (1.9) on β reveals the dependence on the spatial dimension N , and it is almost optimal compared with earlier result for autonomous system (see Theorem 1.1 and [23]).

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Next we state the multiplicity and concentration of positive solutions of (Aε ). Here we first recall a definition from Ljusternik–Schnirelmann category theory. If Y is a closed subset of a topological space X, then the Ljusternik–Schnirelmann category catX (Y ) is the least number of closed and contractible sets in X which cover Y . In view of the condition (PQ1), the set V is compact. We also define O δ = {x ∈ RN : dist(x, O) ≤ δ} for any subset O of RN and δ > 0. Then we have the following result on the existence of multiple positive solutions. Theorem 1.3. Suppose that 1 ≤ N ≤ 3, P , Q satisfy (PQ0) and (PQ1). For each δ > 0, there exist an εδ > 0 such that for each β > βˆ1 and ε ∈ (0, εδ ), (Aε ) has at least catV δ (V ) distinct nontrivial solutions. Additionally, if P , Q are uniformly continuous functions, and assume that wε = (uε , vε ) is any nontrivial solution of (Aε ), xε is a maximum point of |uε | + |vε |, then subject to a subsequence, xε → y0 ∈ V as ε → 0, and the estimate (1.10) holds. The existence of multiple nontrivial solutions in Theorem 1.3 appears to be the first multiplicity result for solutions of (Aε ) with large β. Similar results for scalar nonlinear Schrödinger equations have been obtained in [30,38,39]. In the proof of Theorems 1.2 and 1.3, we overcome the difficulty of lack of compactness by combining the Nehari manifold methods and the condition (PQ1) to recover the local compactness. Secondly to exclude the semi-trivial solutions as the ground state of (Aε ), we carefully analyze the relationship between the ground state of scalar autonomous equation and the one for the nonautonomous equation, and prove that the ground state solutions in Theorems 1.2 and 1.3 are nontrivial when the parameter β satisfies the condition (1.9). Finally we have the following nonexistence results. Theorem 1.4. Suppose that 1 ≤ N ≤ 3, P , Q satisfy (PQ0). (i) If either (a) P (x) ≥ Q(x) for all x ∈ RN , μ < ν and μ ≤ β ≤ ν, or (b) P (x) ≤ Q(x) for all x ∈ RN , ν < μ and ν ≤ β ≤ μ, then (Aε ) does not have any nontrivial positive solution for all ε > 0. (ii) Suppose that P , Q satisfies ˜ > 0 or (PQ2) If P (x) ≥ P ∞ = P∞ > 0 and Q(x) ≥ Q∞ = Q∞ > 0 for all x ∈ RN , |P| N ∞ N ˜ ˜ ˜ |Q| > 0, where P = {x ∈ R : P (x) > P } and Q = {x ∈ R : Q(x) > Q∞ }. then there exist 0 < βˆ2 ≤ βˆ3 ≤ βˆ1 such that for β ∈ (0, βˆ2 ) ∪ (βˆ3 , ∞), (Aε ) has no positive ground state solution for all ε > 0. The nonexistence result in Theorem 1.4 completes the picture of existence/nonexistence of positive (ground state) solutions when β varies, similar to the one for autonomous system (1.3) given in Theorem 1.1. We also mention that when the two trapping potential functions P (x) and Q(x) are identical, then results in Theorems 1.2–1.4 can be simplified and strengthened with βˆ3 = βˆ1 = max{μ, ν} and βˆ2 = min{μ, ν}. Corollary 1.5. Suppose that 1 ≤ N ≤ 3, P (x) ≡ Q(x). (a) If P (x) satisfies (P1) 0 < P0 < P∞ < ∞.

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Then for β > max{μ, ν} and small ε > 0, the results in Theorems 1.2 and 1.3 hold. Moreover for β > max{μ, ν}, each positive solution (u, v) of (Aε ) satisfies v(x) =

β −μ β 2 − μν 3 u(x), ε2 u − P (x)u + u = 0, in RN , u ∈ H 1 (RN ). β −ν β −ν

(b) (Aε ) has no positive ground state solution for all ε > 0 if either (b1) μ = ν and β ∈ [min{μ, ν}, max{μ, ν}]; or (b2) β ∈ (0, min{μ, ν}) ∪ (max{μ, ν}, ∞), and (P2) P (x) ≥ P ∞ = P∞ > 0 for all x ∈ RN , and |P+ | > 0, where P+ = {x ∈ RN : P (x) > P∞ }. For the proof of our theorems, we shall consider an equivalent system to (Aε ). For this purpose, making a change of variable εy = x, we can rewrite (Aε ) as the following equivalent equation ⎧ 3 2 ⎪ ⎨−u + P (εx)u = μu + βv u 3 2 −v + Q(εx)v = νv + βu v ⎪ ⎩ u, v > 0, u, v ∈ H 1 (RN ).

(Pε )

in RN , in RN ,

Thus, our theorems for (Aε ) are equivalent to the results for (Pε ). So, in the sequel we focus on the system (Pε ). Throughout this paper, we always assume that μ, ν, β > 0. In Section 2, we provide basic variational setup of the problem and prove some preliminary estimates. We prove the existence, concentration and properties of positive ground state solutions (Theorem 1.2) in Section 3, and we prove the multiplicity result (Theorem 1.3) and nonexistence result (Theorem 1.4) in Sections 4 and 5 respectively. 2. Variational setting and preliminary results Throughout the paper, we use the following notation: • (·,·) is the inner product of H 1 (RN ) defined by (u, v) =

(∇u∇v + uv), and the correRN

1

sponding norm is u = (u, u) 2 ; • · M is an equivalent norm of

H 1 (RN )

defined by

u2M

=

(|∇u|2 + M|u|2 ), for a RN

positive function or constant M; defined by 2∗ = 6 when N = 3, and 2∗ = ∞ when N = 1, 2; • 2∗ is a critical exponential (|∇u|2 + |u|2 ) • S1 =

inf

u∈H 1 (RN )\{0}

RN

⎛ ⎜ ⎝

RN

⎞1/2 ⎟ u4 ⎠

.

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• Let w1 (x) be the unique positive solution of (1.5) with maxx∈RN w(x) = w(0). Then for √ λ λ, μ > 0, wλ,μ (x) = w1 ( λx) is the unique positive solution of μ w − λw + μw 3 = 0, in RN , w ∈ H 1 (RN ).

(2.1)

For Banach spaces X and Y with norms · X and · Y , the norm of the product space 1 X × Y is defined by (x, y)X×Y := (x2X + y2Y ) 2 ; and for Hilbert spaces X and Y with inner products (·,·)X and (·,·)Y , the inner product of X × Y is defined by ((x, y), (w, z))X×Y = (x, w)X + (y, z)Y . For any ε > 0, let Hp,ε = {u ∈ H 1 (RN ) : RN P (εx)u2 < ∞} denote the Hilbert space endowed with inner product (u, v)p,ε = (∇u∇v + P (εx)uv), for u, v ∈ Hp,ε , RN

and the induced norm denoted by u2p,ε = (u, u)p,ε . Similarly, one can define Hilbert space Hq,ε = {u ∈ H 1 (RN ) : RN Q(εx)u2 < ∞}. Clearly, since P (x) and Q(x) are positive bounded continuous functions, it follows that · p,ε , · q,ε and · are equivalent norms uniformly for ε > 0. The positive solutions of (1.3) are limits of the solutions of (Pε ) in some sense. The existence of such solutions is stated in Theorem 1.1, and here we recall some basic properties of the positive solutions of (1.3). For the proof of these results one can refer to [40,41]. Lemma 2.1. Suppose that β > 0. Let (uβ , vβ ) be a positive solution of (1.3). Then 1. (uβ , vβ ) satisfies lim |uβ (x)| = lim |vβ (x)| = 0,

|x|→∞

|x|→∞

lim |∇uβ (x)| = lim |∇vβ (x)| = 0,

|x|→∞

|x|→∞

(2.2)

1,σ and uβ , vβ ∈ Cloc (RN ) with σ ∈ (0, 1). Furthermore there exist C, c > 0 such that uβ (x) + −c|x−x β | , where x ∈ RN such that |u (x ) + v (x )| = max vβ (x) ≤ Ce β β β β β x∈RN |uβ (x) + vβ (x)|. 2. Let Bβ be the set of all ground state solutions of (1.3). Then Bβ is compact in E.

Now we give a variational framework for the solutions of (Pε ). Let Eε = Hp,ε × Hq,ε . Since P and Q are bounded positive functions, we know that for each ε > 0, Eε = E. For z ∈ Eε we define a functional 1 Lε (z) = Lε (u, v) = (|∇u|2 + P (εx)u2 + |∇v|2 + Q(εx)v 2 ) 2 RN

−

1 4

(μu4 + +2βu2 v 2 + νv 4 ).

RN

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It is routine to verify that Lε ∈ C 2 (Eε , R) and critical points of Lε are weak solutions of (Pε ) (see [16,23]). In order to find nontrivial critical points for Lε , we consider the following Nehari manifold for (Pε ): Nε = {z = (u, v) ∈ Eε : z = 0, Lε (z)z = 0}.

(2.3)

That is, z = (u, v) ∈ Nε satisfies

(|∇u| + P (εx)u + |∇v| + Q(εx)v ) = 2

2

2

2

RN

(μu4 + 2βu2 v 2 + νv 4 ).

(2.4)

RN

This implies that for (u, v) ∈ Nε , Lε |Nε (u, v) =

1 4

(|∇u|2 + P (εx)u2 + |∇v|2 + Q(εx)v 2 ) RN

1 = 4

(μu4 + 2βu2 v 2 + νv 4 ).

(2.5)

RN

Hence Lε is bounded from below away from zero on Nε . And we define Aε =

inf

z=(u,v)∈Nε

Lε (u, v),

(2.6)

then Aε > 0 because of (2.4) and (2.5). Similarly we define corresponding energy functional and Nehari manifold for the limiting equation (1.3) as follows: 1 1 Lλ1 λ2 (u, v) = (|∇u|2 + λ1 u2 + |∇v|2 + λ2 v 2 ) − (μu4 + 2βu2 v 2 + νv 4 ), (2.7) 2 4 RN

RN

and Nλ1 λ2 = {z = (u, v) ∈ E : z = 0, Lλ1 λ2 (z)z = 0}. We set Aλ1 λ2 =

inf

w∈Nλ1 λ2

Lλ1 λ2 (w).

(2.8)

A particular case is that for some y0 ∈ RN , λ1 = P (y0 ) and λ2 = Q(y0 ) in (1.3). We denote the corresponding notation by LP (y0 )Q(y0 ) , NP (y0 )Q(y0 ) and AP (y0 )Q(y0 ) respectively. The following lemma shows the relation between the ground state energy level Aε and the one for the limiting equation AP (y0 )Q(y0 ) . The proof is similar to [41, Lemma 4.1] (also see [42]), and we omit the details here.

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Lemma 2.2. Let AP (y0 )Q(y0 ) and Aε be defined as in (2.6) and (2.8). For β > 0, we have the following conclusions. 1. Suppose that for some y0 ∈ RN , AP (y0 )Q(y0 ) is attained by some (u0 , v0 ) ∈ NP (y0 )Q(y0 ) , then lim sup Aε ≤ AP (y0 )Q(y0 ) . Moreover if y0 ∈ V and Aε is attained by some (uε , v ε ) ∈ Nε for ε→0

all ε > 0 small, then lim Aε = AP (y0 )Q(y0 ) . ε→0

2. Suppose that for λ1 , λ2 > 0, Aλ1 λ2 is attained by some (u0 , v0 ) ∈ Nλ1 λ2 , P (εx) → λ1 and Q(εx) → λ2 uniformly on any bounded subset of RN as ε → 0, then lim sup Aε ≤ Aλ1 λ2 . ε→0

The next lemma states the monotonicity of the energy level Aλ1λ2 in the parameters λ1 and λ2 . Lemma 2.3. Let Aλ1 λ2 be defined as in (2.8). If λ1 ≤ λˆ 1 and λ2 ≤ λˆ 2 , then Aλ1 λ2 ≤ Aλˆ 1 λˆ 2 . Moreover, if one of the inequalities is strict and Aλˆ 1 λˆ 2 is attained by z = (u, v) ∈ Nλˆ 1 λˆ 2 (u, v = 0), then Aλ1 λ2 < Aλˆ 1 λˆ 2 . Proof. The first part of the conclusion follows from the minimax characterization 0 < Aλ1 λ2 =

inf

z∈Nλ1 λ2

Lλ1 λ2 (z) =

max Lλ1 λ2 (tw),

inf

w=(u,v)∈E,w≡0 t>0

and the inequality Lλˆ 1 λˆ 2 (tw) ≥ Lλ1 λ2 (tw), for t > 0 and w ∈ E, when λ1 ≤ λˆ 1 and λ2 ≤ λˆ 2 . Now we give the proof of second part of the conclusion. Since Aλˆ 1 λˆ 2 is attained, we can choose z1 = (u1 , v1 ) ∈ Nλˆ 1 λˆ 2 and w = (u, v) ∈ E (u, v ≡ 0) be such that Aλˆ 1 λˆ 2 = Lλˆ 1 λˆ 2 (z1 ) = max Lλˆ 1 λˆ 2 (tw). On the other hand, let z2 = (u2 , v2 ) ∈ E be such that Lλ1 λ2 (z2 ) = t>0

max Lλ1 λ2 (tw). Therefore, one sees that t>0

Aλˆ 1 λˆ 2

≥ Lλˆ 1 λˆ 2 (z2 ) = Lλ1 λ2 (z2 ) + (λˆ 1 − λ1 ) ≥ Aλ1 λ2 + (λˆ 1 − λ1 )

RN

as required.

u22

+ (λˆ 2 − λ2 )

RN

u22 + (λˆ 2 − λ2 )

v22

RN

v22 ,

RN

2

The following Mountain-Pass geometry characterization of the functional Lε can be shown by standard methods. Lemma 2.4 (Mountain-Pass geometry). For any β > 0, the functional Lε satisfies the following conditions: (i) There exist positive constants ϑ, α such that Lε (z) ≥ ϑ for z = α. (ii) There exists e ∈ Eε with e > α such that Lε (e) < 0.

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11

From Lemma 2.4, one can apply the Ambrosetti–Rabinowitz Mountain-Pass Theorem without (PS)c condition (see [43]), and it follows that for any small ε > 0, there exists a (PS)c -sequence {zn } ⊂ Eε (with c = Aε defined below) such that Lε (zn ) → Aε = inf max Lε (γ (t)) γ ∈ 0≤t≤1

and

Lε (zn ) → 0, as n → ∞,

(2.9)

where = {γ ∈ C(Eε , R) : Lε (γ (0)) = 0, Lε (γ (1)) < 0}. As in [36, Proposition 3.11] (also see [44]), we shall use the following equivalent characterization of Aε , which is more appropriate to our purpose, given by Aε =

inf

max Lε (tz) = Aε .

z∈Eε \{0} t>0

(2.10)

3. Existence and concentration of positive solution 3.1. Compactness lemma for the functional Lε In order to obtain the existence of positive solutions for (Pε ), we prove some compactness lemma for the functional Lε to analyze the Palais–Smale sequence properties for the functional Lε . Throughout this subsection we assume that (PQ0), (PQ1) and β > 0 hold. From (PQ1) we know that P0 < P∞ and Q0 < Q∞ , hence we can choose τ, σ > 0 such that P0 ≤ τ < P∞ ,

and

Q0 ≤ σ < Q∞ .

(3.1)

We assume that {zn } ⊂ Eε is a sequence satisfying Lε (zn ) → A,

Lε (zn ) → 0, as n → ∞,

(3.2)

for some A satisfying 0 < A ≤ Aτ σ < AP∞ Q∞ ,

(3.3)

where τ and σ are given in (3.1). Note that the inequalities in (3.3) follow from (3.1) and Lemma 2.3. First, we notice that the condition (3.2) implies that {zn } is bounded in E. Lemma 3.1. Assume that {zn } satisfies (3.2). Then {zn } is bounded in E. Proof. Let {zn } = {(un , vn )} ⊂ Eε such that (3.2) is satisfied. It follows from P (εx) ≥ P0 , Q(εx) ≥ Q0 , and Lε (zn ) − Lε (zn )zn = A + o(1) that 1 A + o(1) = (|∇un |2 + P (εx)u2n + |∇vn |2 + Q(εx)vn2 ) ≥ czn 2 . (3.4) 4 RN

That is, {zn } is bounded.

2

From the Lions Concentration-Compactness Principle, we have the following lemma for any fixed ε > 0.

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12

Lemma 3.2. Assume that {zn } satisfies (3.2) and zn 0 in Eε , then one of the following conclusions holds: (i) zn → 0 in Eε as n → ∞; or a sequence {yn } ⊂ RN and positive constants r, δ > 0 such that (ii) there exist |zn |2 ≥ δ > 0.

lim inf n→∞

Br (yn )

Proof. Assume that zn = (un , vn ). Suppose that (ii) does not occur, i.e., there exists r > 0 such that lim sup |zn |2 = lim sup (u2n + vn2 ) = 0. n→∞

y∈RN

n→∞

Br (y)

y∈RN

Br (y)

Then by the Lions Concentration-Compactness Principle (see [45,46,43]), we deduce that zn → 0 in Lt (RN ) × Lt (RN ) for t ∈ (2, 2∗ ). So one infers from Lε (zn ) → 0 that

(|∇un |

2

RN

+ P (εx)u2n ) +

(|∇vn |

2

+ Q(εx)vn2 ) =

RN

(μu4n + 2βu2n vn2 + νvn4 ) + o(1) → 0,

RN

as n → ∞. Since P ≥ P0 and Q ≥ Q0 , then it follows that zn → 0 in Eε as n → ∞.

2

Next we shall prove that indeed only the first alternative in Lemma 3.2 occurs if (3.3) is satisfied. Lemma 3.3. Assume that {zn = (un , vn )} satisfies (3.2), (3.3) and zn 0 in Eε . Then for fixed ε > 0, zn → 0 in Eε as n → ∞. Proof. From the proof of Lemma 2.2, for any z = (u, v) ∈ E \ {(0, 0)}, the function g(t) = LP∞ Q∞ (tz) has a unique global maximum point tz > 0 and tz z ∈ NP∞ Q∞ . Hence we can choose a positive sequence {tn } such that {tn zn } ⊂ NP∞ M∞ . We argue by contradiction. Suppose that zn = (un , vn ) 0 in Eε , we first claim that lim sup tn ≤ 1.

(3.5)

n→∞

Assume by contradiction, there exist η > 0 and a subsequence (still denoted by {tn }) such that tn ≥ 1 + η for all n ∈ N. From Lε (zn )zn = o(1) we have that

(|∇un |2 + P (εx)u2n ) + RN

(|∇vn |2 + Q(εx)vn2 ) =

RN

(μu4n + 2βu2n vn2 + νvn4 ) + o(1). (3.6)

RN

Moreover, since {tn zn } ⊂ NP∞ Q∞ , then we see that (|∇un |2 + P∞ u2n ) + (|∇vn |2 + Q∞ vn2 ) = tn2 (μu4n + 2βu2n vn2 + νvn4 ). RN

RN

RN

(3.7)

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13

We deduce from (3.6) and (3.7) that

(P∞ − P (εx))u2n +

RN

(Q∞ − Q(εx))u2n = (tn2 − 1)

RN

(μu4n + 2βu2n vn2 + νvn4 ) + o(1). (3.8)

RN

By the definition of P∞ and Q∞ , for any > 0, there exists R = R() > 0 such that P (εx) ≥ P∞ −

and Q(εx) ≥ Q∞ −

for |x| ≥ R.

(3.9)

Since {zn } is a bounded sequence from Lemma 3.1, then we have that zn → 0 in L2 (BR (0)) × L2 (BR (0)), and from (3.9), there exists C > 0 such that (tn2 − 1)

(μu4n + 2βu2n vn2 + νvn4 ) ≤ C + o(1).

(3.10)

RN

Since zn 0 in Eε , it follows from Lemma 3.2 that there exist a sequence {yn } ∈ RN and positive constants r, δ > 0 such that (u2n + vn2 ) ≥ δ.

lim inf n→∞

(3.11)

Br (yn )

If we set wn (x) = z(x + yn ) = (u(x + yn ), v(x + yn )), then there exists a function w = (u, v), up to a subsequence, such that wn w in E, wn → w in [L2loc (RN )]2 and wn (x) → w(x) a.e. in RN . Moreover, by (3.11), there exists a subset in RN with positive measure such that w = 0 a.e. in . It follows from (3.10) and Fatou’s lemma that 0 < [(1 + η)2 − 1]

(μu4 + 2βu2 v 2 + νv 4 ) ≤ C,

(3.12)

for any > 0, which yields a contradiction. Thus (3.5) holds. Next we prove that indeed lim supn→∞ tn ≤ 1 cannot happen. Then we obtain a contradiction and zn → 0 in Eε . For this purpose, we distinguish the following two cases: (1) lim sup tn = 1; n→∞

(2) lim sup tn < 1. n→∞

Case 1. lim sup tn = 1. In this case, there exists a subsequence, still denoted by {tn } such that n→∞

tn → 1 as n → ∞. Hence, from (3.2) and (3.3), o(1) + Aτ σ ≥ Lε (zn ) ≥ Lε (zn ) + AP∞ Q∞ − LP∞ Q∞ (tn zn ).

(3.13)

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14

We can estimate that Lε (un , vn ) − LP∞ Q∞ (tn un , tn vn ) 1 − tn2 1 tn = (|∇un |2 + |∇vn |2 ) + P (εx)u2n − P∞ u2n 2 2 2 RN

+

1 2

Q(εx)vn2 − RN

RN

tn2

Q∞ vn2 + (1 − tn4 )

2 RN

RN

μu4n + 2βu2n vn2 + νvn4 .

(3.14)

RN

From the boundedness of {zn }, tn → 1, zn 0 in Eε as n → ∞ and (3.9), we obtain that Lε (zn ) − LP∞ Q∞ (tn zn ) ≥ o(1) − C.

(3.15)

Taking the limit as n → ∞ in (3.13), we have Aτ σ ≥ AP∞ Q∞ . On the other hand, from (3.3), we have that Aτ σ < AP∞ Q∞ . This is a contradiction. Case 2. lim sup tn < 1. Without loss of generality, we may suppose that tn < 1 for all n ∈ N. n→∞

From (3.9), {tn zn } ⊂ NP∞ Q∞ , zn → 0 in [L2loc (RN )]2 and zn ≤ C, we see that AP∞ Q∞ ≤ LP∞ Q∞ (tn zn ) = Lε (tn zn ) +

tn2 2

(P∞ − P (εx))u2n +

tn2 2

RN

(Q∞ − Q(εx))vn2 RN

≤ Lε (zn ) + C + o(1) ≤ Aτ σ + C + o(1).

(3.16)

Let n → ∞ in (3.16), we get Aτ σ ≥ AP∞ Q∞ , which again is in contradiction with Aτ σ < AP∞ Q∞ from (3.3). 2 The next lemma states that the functional Lε satisfies (PS)A -condition. Lemma 3.4. Assume that {zn } ⊂ Eε satisfies (3.2) and (3.3). Then for fixed ε > 0, {zn } has a convergent subsequence in Eε . That is, the functional Lε satisfies the (PS)A -condition. Proof. From Lemma 3.1, {zn } is bounded in E, hence there exists z = (u, v) ∈ Eε such that zn z in Eε , and z is a critical point of Lε . Set hn = un − u and kn = vn − v. By Brezis–Lieb Lemma (see [43]), we have

|∇hn | =

RN

RN

RN

4

RN

|∇vn |2 −

|hn | =

2

RN

|∇kn |2 =

|∇u| + o(1),

2

RN

RN

|∇un | −

2

RN

|u|4 + o(1),

RN

|kn |4 = RN

|un | −

RN

|∇v|2 + o(1),

4

|vn |4 −

RN

|v|4 + o(1).

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15

Hence, as in [47], one can verify that Lε (hn , kn ) = Lε (un , vn ) − Lε (u, v) + o(1) and Lε (hn , kn ) → 0 as n → ∞. Thus, it follows from Lε (u, v) = 0 that 1 Lε (z) = Lε (u, v) = 4

(|∇u|2 + |∇v|2 + P (εx)u2 + Q(εx)v 2 ) ≥ 0.

(3.17)

RN

So from (3.17) we infer that Lε (hn , kn ) = Lε (un , vn ) − Lε (u, v) + o(1) → A − d1 as n → ∞, where d1 = Lε (z) ≥ 0. Hence {φn = (hn , kn )} is a (PS)c sequence with c = A − d1 . Moreover we have A − d1 ≤ A ≤ Aτ σ < AP∞ Q∞ thus (3.3) is satisfied, therefore Lemma 3.3 implies that hn = un − u → 0 and kn = vn − v → 0 in H 1 (RN ). 2 Finally we prove that Lε also satisfies (PS)A -condition if it is restricted to the Nehari manifold Nε . Lemma 3.5. Assume that {zn } ⊂ Nε satisfies Lε (zn )zn = 0

Lε (zn ) → A,

and Lε |Nε (zn ) → 0,

as n → ∞

(3.18)

and (3.3). Then for fixed ε > 0, {zn } has a convergent subsequence in Nε . Proof. Suppose that {zn } = {(un , vn )} ⊂ Nε satisfy (3.18) and (3.3). Then there exists a sequence {ln } ⊂ R such that o(1) = Lε |Nε (zn ) = Lε (zn ) − ln Jε (zn ),

(3.19)

where Jε (zn ) =

(|∇un | + |∇vn | 2

2

+ P (εx)u2n

RN

+ Q(εx)vn2 ) −

(μu4n + 2βu2n vn2 + νvn4 ). (3.20)

RN

So, it follows from zn ∈ Nε that Jε (zn )zn = 2

(|∇un |2 + |∇vn |2 + P (εx)u2n + Q(εx)vn2 ) − 4

RN

RN

= −2

(μu4n + 2βu2n vn2 + νvn4 )

(|∇un |2 + |∇vn |2 + P (εx)u2n + Q(εx)vn2 ) ≤ 0.

(3.21)

RN

On the other hand, from Lemma 3.1, {zn } is bounded in E, hence there exists C > 0 such that Jε (zn )zn ≥ −C. Hence subject to a subsequence we may assume that Jε (zn )zn → l ≤ 0. If l = 0, then we see that |Jε (zn )zn | ≥ 2 (|∇un |2 + |∇vn |2 + P (εx)u2n + Q(εx)vn2 ), (3.22) RN

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16

which implies that zn → 0 in E thus contradicting with Lε (zn ) → A > 0. So we must have l < 0. It follows that ln → 0 as n → ∞, and therefore Lε (zn ) = o(1). So zn is a (PS)A sequence for Lε in Eε , and the conclusion follows from Lemma 3.4. 2 3.2. Existence of positive ground state solution In this subsection we consider the existence of positive solution for large β. We assume that (PQ0) and (PQ1) are satisfied. We first recall results on the related scalar equations −u + P (εx)u = μu3 ,

u ∈ H 1 (RN ),

(3.23)

−v + Q(εx)v = νv 3 ,

v ∈ H 1 (RN ).

(3.24)

and

Following the idea of the proof of Theorem 1.1 in [41], one can prove the following results about the positive solutions of (3.23). Lemma 3.6. Suppose that (P1) is satisfied. Then for all sufficiently small ε > 0, we have that (i) (3.23) has at least one positive ground state solution uε in H 1 (RN ). (ii) Let Mε be the set of all positive solutions of (3.23). Then Mε is compact in H 1 (RN ). Moreover, if P is uniformly continuous, then the following results hold. (iii) there exists a maximum point xε of uε such that lim dist(εxε , P0 ) = 0, εxε → y0 ∈ P0 ε→0 P (y0 ) and hε (x) = uε (x + xε ) converges in H 1 (RN ) to wP (y0 ),μ = w1 ( P (y0 )x), which μ is the unique positive solution of −u + P (y0 )u = μu3 ,

u ∈ H 1 (RN ),

(3.25)

1,σ as ε → 0. Moreover uε ∈ Cloc (RN ) with σ ∈ (0, 1), and there exist constants C, c > 0 such that c

lim uε (x) = lim |∇uε (x)| = 0, |uε (x)| ≤ Ce− ε |x−xε | , x ∈ RN .

|x|→∞

|x|→∞

A parallel result as in Lemma 3.6 holds for (3.24). Let Uε and Vε denote a positive ground state solution of (3.23) and (3.24) respectively. To obtain a nontrivial solution for (Pε ), as in [23] we shall prove that for small enough ε > 0 we have Aε < min{Lε (Uε , 0), Lε (0, Vε )},

(3.26)

therefore the minimizer achieving Aε is a nontrivial solution of (Pε ). For some y0 ∈ V , we assume that AP (y0 )Q(y0 ) is attained. Then we infer from Lemma 2.2 that for ε > 0 sufficiently small,

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Aε ≤ AP (y0 )Q(y0 ) + δε ,

17

(3.27)

where δε > 0 and δε → 0 as ε → 0. Moreover, from Lemma 3.6 we conclude that μ μ μ Lε (Uε , 0) = Uε4 = Uε (x + xε )4 = wP4 (y0 ),μ + oε (1) 4 4 4 RN

RN

RN

= LP (y0 )Q(y0 ) (wP (y0 ),μ , 0) + oε (1), ν ν ν 4 Vε4 = Vε (x + x˜ε )4 = wQ(y + oε (1) Lε (0, Vε ) = 0 ),ν 4 4 4 RN

RN

RN

= LP (y0 )Q(y0 ) (0, wQ(y0 ),ν ) + oε (1),

(3.28)

where oε (1) → 0 as ε → 0, and xε , x˜ε are the maximum points of Uε and Vε respectively. So in order to prove (3.26), it is sufficient to show that AP (y0 )Q(y0 ) < min{LP (y0 )Q(y0 ) (wP (y0 ),μ , 0), LP (y0 )Q(y0 ) (0, wQ(y0 ),ν )}.

(3.29)

We shall find some conditions on P , Q, μ, ν to ensure (3.29) holds. First, as in Lemma 3.3 of [23], we know that for β ≥ 0, AP (y0 )Q(y0 ) =

inf

z=(u,v)∈E\{(0,0)}

J (z),

(3.30)

where J (z) = J (u, v) =

(u2P (y0 ) + v2Q(y0 ) )2 . 4 (μu4 + 2βu2 v 2 + νv 4 )

(3.31)

RN

For (s, t) ∈ = {(s, t) : s ≥ 0, t ≥ 0, (s, t) = (0, 0)}, we define a function √ √ g(s, t) = J ( swP (y0 ),μ , twQ(y0 ),ν ) N

=

N

(sμ−1 S12 P (y0 )2− 2 + tν −1 S12 Q(y0 )2− 2 )2 N N 4S12 (s 2 μ−1 P (y0 )2− 2 + t 2 ν −1 Q(y0 )2− 2 + 2stβμ−1 ν −1 P (y0 )Q(y0 )hˆ P (y0 )Q(y0 ) )

,

(3.32) where R hˆ P (y0 )Q(y0 ) =

N

w12 ( P (y0 )x)w12 ( Q(y0 )x)dx

. w14 (x)dx

RN

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18

We claim that min{P (y0 )− 2 , Q(y0 )− 2 } ≤ hˆ P (y0 )Q(y0 ) ≤ P (y0 )− 4 Q(y0 )− 4 . N

N

N

N

(3.33)

From Hölder inequality we have that

1 1 w12 ( P (y0 )x)w12 ( Q(y0 )x)dx ≤ ( w12 ( P (y0 )x)dx) 2 ( w12 ( Q(y0 )x)dx) 2

RN

RN − N4

= P (y0 )

− N4

RN

w14 (x)dx,

Q(y0 )

RN

which gives the upper bound for hˆ P (y0 )Q(y0 ) in (3.33). To prove the lower bound for hˆ P (y0 )Q(y0 ) , without loss of generality, we assume that Q(y0 ) ≥ P (y0 ). Since w1 (x) is radially√symmetric and strictly decreasing in |x|, it follows that for λ ≥ 1 and x ∈ RN , w1 (x) ≥ w1 ( λx). So a direct computation shows that

w12 (

P (y0 )x)w12 (

RN N

≥P (y0 )− 2

Q(y0 )x)dx = P (y0 )

w12 (x)w12 ( RN

w14 (

RN

− N2

N Q(y0 ) x)dx = Q(y0 )− 2 P (y0 )

Q(y0 ) x)dx P (y0 )

w14 (x)dx. RN

We claim that g attains its minimum over in the interior, which implies that (3.29) holds. Clearly, along the boundary of , we have that N

P (y0 )2− 2 S12 g(s, 0) = = J (wP (y0 ),μ , 0) = LP (y0 )Q(y0 ) (wP (y0 ),μ , 0), 4μ N

Q(y0 )2− 2 S12 g(0, t) = = J (0, wQ(y0 ),ν ) = LP (y0 )Q(y0 ) (0, wQ(y0 ),ν ). 4ν

(3.34)

Note that the function g is the ratio of two quadratic forms of s and t , and elementary analysis shows that the function g(s, ˜ t) =

(as + bt)2 , a, b, c, d, e > 0, cs 2 + 2dst + et 2

(3.35)

does not attain its minimum in on the boundary if and only if ad − bc > 0,

and bd − ae > 0.

(3.36)

And the minimum is achieved when t = ad − bc and s = bd − ae. For the function g(s, t), the relations in (3.36) are equivalent to

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19 N

Q(y0 )1− 2 β hˆ P (y0 )Q(y0 ) − μ > 0, P (y0 )

and

P (y0 )1− 2 β hˆ P (y0 )Q(y0 ) − ν > 0, Q(y0 )

(3.37)

or equivalently

1− N2 1− N2 ) ) Q(y P (y 0 0 β hˆ P (y0 )Q(y0 ) > max μ, ν . P (y0 ) Q(y0 )

(3.38)

Then one infers from (3.33) and (3.38) that if N N P (y0 )1− 2 Q(y0 )1− 2 β > max P (y0 ) , Q(y0 ) μ, ν · max P (y0 ) Q(y0 ) N N Q0 P0 1− 2 P0 Q0 1− 2 Q0 = max μ ,ν ,ν ,μ , := β1 P0 Q0 Q0 P0 P0

N 2

N 2

(3.39)

where β1 (z) = β1 (μ, ν, z) is defined in (1.6), then (3.29) holds. Summarizing the calculations above, we have the following lemma. Lemma 3.7. For β > β1 (Q0 /P0 ) defined as in (3.39) and sufficiently small ε > 0, any ground state solution of (Pε ) is a nontrivial one. That is, if z = (u, v) is a ground state solution of (Pε ), then u ≡ 0 and v ≡ 0. Remark 3.8. As in [23], one can also find other conditions to guarantee (3.29) holds. We omit the details and leave it for interested readers. Also we note that if P (x) ≡ Q(x), then P0 = Q0 and β1 (1) = max{μ, ν}. In order to prove the existence of a positive solution when β is large, we use a variational approach in which the compactness lemma (Lemma 3.4) is crucial. Let βˆ1 be defined as in (1.9). Then for β > βˆ1 , each of AP∞ Q∞ and AP0 Q0 is attained by a respective positive ground state solution from part 2 of Theorem 1.1 and (1.6), and from Lemma 2.3 we have that AP0 Q0 < AP∞ Q∞ . Now we are in a position to prove the existence of a positive ground state solution of (Pε ), which implies the existence part (i) of Theorem 1.2. Proposition 3.9. Suppose that P , Q satisfy (PQ0) and (PQ1). Let βˆ1 be defined as in (1.9). Then for β > βˆ1 and ε > 0 sufficiently small, Aε is attained by zε = (uε , vε ) ∈ Eε such that uε > 0 and vε > 0. Proof. First, from Lemma 2.4, the functional Lε satisfies a mountain pass geometry for β > 0. Then by using a version of the mountain pass theorem without (PS)c condition [43, Theorem 1.15], there exists a sequence {zn } ⊂ Eε satisfying Lε (zn ) → Aε ,

and Lε (zn ) → 0,

as n → ∞.

(3.40)

From Lemma 3.1, {zn } is bounded in Eε . Hence there exists z = (u, v) ∈ Eε such that zn z in Eε .

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Since β > βˆ1 , it follows that each of AP∞ Q∞ and AP0 Q0 is attained by a respective positive ground state solution from part 2 of Theorem 1.1 and (1.6). Moreover from the condition (PQ1), one can choose ξ > 1 such that P0 < ξ P0 < P∞ and Q0 < ξ Q0 < Q∞ . Define τ = ξ P0 and σ = ξ Q0 . Since β1 (ξ Q0 /ξ P0 ) = β1 (Q0 /P0 ), then Aτ σ is also attained when β > βˆ1 . Then from Lemma 2.3 we have AP0 Q0 < Aτ σ < AP∞ Q∞ . It follows from Lemma 2.2 that Aε ≤ AP0 Q0 + ηε , where ηε > 0 and ηε → 0 as ε → 0. On the other hand it follows from AP0 Q0 < Aτ σ that there exists η > 0 such that AP0 Q0 < AP0 Q0 + η ≤ Aτ σ . So for ε > 0 small enough we deduce that Aε ≤ AP0 Q0 + ηε ≤ AP0 Q0 + η ≤ Aτ σ < AP∞ Q∞ . Now from (3.40) and Lemma 3.4, we obtain that zn → z in Eε . Moreover, since β > βˆ1 ≥ β1 (Q0 /P0 ), from Lemma 3.7, we know that for z = (u, v), u ≡ 0 and v ≡ 0. Thus we prove that z ∈ Nε such that Lε (z) = Aε and Lε (z) = 0. Finally we prove that u, v > 0. In fact, since (|u|, |v|) ∈ Nε and Aε = Lε (|u|, |v|), we conclude that (|u|, |v|) is a nonnegative solution of (Pε ). Using the strong maximum principle we infer that |u|, |v| > 0. Thus Aε is attained by a positive zε = (uε , vε ), where uε = |u|, vε = |v|. 2 Remark 3.10. If P (x) ≡ Q(x), then again P0 = Q0 and P∞ = Q∞ , so for β > βˆ1 = max{μ, ν}, the conclusion of Proposition 3.9 holds. To conclude this subsection we prove part (ii) of Theorem 1.2. Lemma 3.11. Suppose that the assumptions of Theorem 1.2 are satisfied. Let Bε denote the set of all positive ground state solutions of (Pε ). Then Bε is compact in E for all small ε > 0. Proof. Let {zn } ⊂ Bε ∩ Nε be a bounded sequence satisfying Lε (zn ) = Aε and Lε (zn ) = 0. Without loss of generality we assume that zn z ∈ Eε . Then it follows from the weak continuity of Lε that Lε (z) = 0. Set wn = zn − z. As in Lemma 3.4, we can prove that wn → 0 in E. Hence z ∈ Bε . 2 3.3. Concentration of positive ground state solutions In this subsection we study the concentration phenomenon of the positive ground state solutions obtained in Subsection 3.2. We begin with the following lemma which is needed in proving the concentration of positive ground state solutions. Lemma 3.12. Assume that β > 0 and λ1 , λ2 > 0. Let {˜zn = (u˜ n , v˜n )} ⊂ Nλ1 λ2 be a sequence satisfying Lλ1 λ2 (˜zn ) → Aλ1 λ2 as n → ∞, where Lλ1 λ2 and Aλ1 λ2 are defined in (2.7) and (2.8). Then either {˜zn } has a subsequence strongly convergent in E or there exists {yn } ⊂ RN such that the sequence w˜ n (x) = z˜ n (x + yn ) converges strongly in E. In particular Aλ1 λ2 is attained by some z ∈ Nλ1 λ2 . Proof. First, it follows from the Ekeland’s variational principle (see Theorem 8.5 of [43]) on Nλ1 λ2 that there exists a minimizing sequence {(un , vn )} ⊂ Nλ1 λ2 such that (a1 )

1 Lλ1 λ2 (un , vn ) ≤ Aλ1 λ2 + , n

1 Lλ1 λ2 (u, v) ≥ Lλ1 λ2 (un , vn ) − (un − u, vn − v)E , n (a3 ) (u˜ n , v˜n ) − (un , vn )E → 0. (a2 )

∀(u, v) ∈ Nλ1 λ2 , (3.41)

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We claim that Lλ1 λ2 (un , vn ) → 0 as n → ∞.

(3.42)

In fact, it is easy to check that {(un , vn )} is bounded and (un , vn )E ≥ δ > 0. Moreover, for a given (ϕ, φ) ∈ E such that ϕ, φ ≤ 1, we define Fn (t, s) = Lλ1 λ2 (un + sun + tϕ, vn + svn + tφ),

(3.43)

where Lλ1 λ2 is defined in (2.7). Obviously, Fn (0, 0) = 0 and F ∈ C 1 (R2 , R). A direct computation shows that ∂Fn 2 2 (0, 0) = 2(un λ1 + vn λ2 ) − 4 (μu4n + 2βu2n vn2 + νvn4 ) ∂s RN

= −2(un 2λ1 + vn 2λ2 ) ≤ −cδ < 0.

(3.44)

By the implicit function theorem, there exists a C 1 function sn (t) defined on some interval (−τn , τn ) for τn > 0, such that sn (0) = 0 and Fn (t, sn (t)) = 0,

t ∈ (−τn , τn ).

(3.45)

∂Fn ∂Fn (0, 0) + (0, 0)sn (0) = 0. ∂t ∂s

(3.46)

Differentiating (3.45), we have that

Since ∂Fn ∂t (0, 0) =2 (∇un ∇ϕ + λ1 un ϕ + ∇vn ∇φ + λ2 vn φ) RN

−4

(μu3n ϕ

+ βvn2 un ϕ

RN

+ βu2n vn φ

+ νvn3 φ) ≤ c,

(3.47)

then we obtain that |sn (0)| ≤ c.

(3.48)

Let ϕn,t = un + sn (t)un + tϕ and φn,t = vn + sn (t)vn + tφ. Then it follows from (3.45) that (ϕn,t , φn,t ) ∈ Nλ1 λ2 for t ∈ (−τn , τn ). Furthermore, we deduce from (a2 ) of (3.41) that 1 Lλ1 λ2 (ϕn,t , φn,t ) − Lλ1 λ2 (un , vn ) ≥ − (sn (t)un + tϕ, sn (t)vn + tφ)E . n

(3.49)

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Note that Lλ1 λ2 (un , vn )(un , vn ) = 0, hence by using Taylor expansion we have that Lλ1 λ2 (ϕn,t , φn,t ) − Lλ1 λ2 (un , vn ) = Lλ1 λ2 (un , vn )(sn (t)un + tϕ, sn (t)vn + tφ) + R(n, t) = 2tLλ1 λ2 (un , vn )(ϕ, φ) + R(n, t), (3.50) where R(n, t) = o((sn (t)un + tϕ, sn (t)vn + tφ)) as t → 0. It follows from (3.48) that sn (t) sn (t) lim sup (3.51) un + ϕ, vn + φ ≤ c. t t n→∞ Thus, R(n, t) = o(t) as t → 0. One can deduce from (3.49)–(3.51) that c |Lλ1 λ2 (un , vn )(ϕ, φ)| ≤ , n

t → 0.

(3.52)

That is, the claim (3.42) holds. So {zn } is a (PS)Aλ1 λ2 -sequence of Lλ1 λ2 . Similar to the proof of Lemma 3.1, one can verify that {zn } is a bounded sequence in E. Therefore, choosing a subsequence if necessary, we have that zn z = (u, v) in E and Lλ1 λ2 (z) = 0. In order to prove the conclusion of this lemma, we distinguish the following two cases: Case 1. If z = 0, then it follows that z ∈ Nλ1 λ2 . Moreover, one sees that 1 1 Aλ1 λ2 ≤ Lλ1 λ2 (z) = Lλ1 λ2 (z) − (Lλ1 λ2 (z), z) = (|∇u|2 + λ1 u2 + |∇v|2 + λ2 v 2 ) 4 4 RN

≤ lim inf n→∞

(|∇un |

2

+ λ1 u2n

+ |∇vn |

2

+ λ2 vn2 ) = lim inf n→∞

1 Lλ1 λ2 (zn ) − Lλ1 λ2 (zn )zn 4

RN

= Aλ1 λ2 . So we obtain that 2 2 2 2 2 2 lim (|∇un | + λ1 un ) = (|∇u| + λ1 u ), lim (|∇vn | + λ2 vn ) = (|∇v|2 + λ2 v 2 ). n→∞ RN

n→∞ RN

RN

RN

Thus it follows from Brezis–Lieb lemma (see Lemma 1.32 of [43]) and Sobolev’s inequality that un − u and vn − v → 0 as n → ∞. Case 2. If z = 0, as in Lemma 3.2, we have that there exist {yn } ⊂ RN , r, δ > 0 such that lim inf |zn |2 ≥ δ. (3.53) n→∞

Br (yn )

Set wn (x) = zn (x +yn ) = (wn1 , wn2 ), then wn1 λ1 = un λ1 and wn2 λ2 = vn λ2 , Lλ1 λ2 (wn ) → Aλ1 λ2 and Lλ1 λ2 (wn ) → 0. Then there exists w ∈ E with w = 0 such that wn w in E. Then the conclusion follows from same arguments as in Case 1. Finally it follows from (a3 ) that the conclusions of this lemma hold. 2

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Now we are ready to prove the concentration of positive state solutions for small ε. Lemma 3.13. Suppose that P , Q satisfy (PQ0) and (PQ1), and P , Q are uniformly continuous. Assume that β > βˆ1 . Let zε = (uε , vε ) be a positive ground state solution of (Pε ). Then there is a maximum point yε of uε + vε and y0 ∈ V such that dist(εyε , V ) → 0, εyε → y0 , zε (x + yε ) converges in E to a positive ground state solution of (1.3) with λ1 = P (y0 ) = P0 and λ2 = Q(y0 ) = Q0 , as ε → 0. Proof. Since β > βˆ1 , then AP0 Q0 is attained by some positive z0 ∈ NP0 Q0 . From Proposition 3.9, for small ε > 0, Aε is attained by some positive zε ∈ Nε . Therefore from Lemma 2.2, we have lim Aε = AP0 Q0 < AP∞ Q∞ .

ε→0

(3.54)

Let {εj } be a sequence of positive numbers converging to 0 as j → ∞, and let {zj = zεj } ⊂ Nεj satisfying Lεj (zj ) = Aεj and Lεj (zj ) = 0. Then from Lemma 3.1, one sees that the sequence {zj } is bounded in E. So we can assume that zj z in E as j → ∞. To prove the main results we divide the proof into the following three steps. Step 1. {zj } is nonvanishing. Similar to the proof of Lemma 3.2, one can verify that there exist r, δ > 0 and a sequence {yj } ⊂ RN such that |zj |2 ≥ δ > 0.

lim inf j →∞

(3.55)

Br (yj )

For j ∈ N, define yj ∈ RN to be a maximum point of uj + vj such that ! " uj (yj ) + vj (yj ) = max uj (y) + vj (y) . y∈RN

(3.56)

We claim that there exists > 0 (independent of j ) such that |zj (yj )| ≥ > 0 for all j ∈ N. To the contrary, we assume that |zj (yj )| → 0 as j → ∞. We deduce from (3.55) that 0<δ≤

|zj |2 ≤ c|zj (yj )|2 → 0 as j → ∞.

Br (yj )

This is a contradiction. Furthermore, from (3.55) one deduces that there exist R > 0 and δ > 0 such that lim inf j →∞

BR (yj )

|zj |4 ≥ δ > 0.

(3.57)

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Step 2. {εj yj } is bounded. In order to prove this conclusion we set wj (x) = zj (x + yj ) = (u1j (x), vj1 (x)), ˆ εj (x) = Q(εj (x + yj )). Pˆεj (x) = P (εj (x + yj )) and Q p

p

Then from (3.57) we have wj w = (u1 , v 1 ) = 0 in E and wj → w in Lloc (RN ) × Lloc (RN ) for p ∈ (2, 2∗ ). We prove that wj → w in E. In fact, from the proof of Lemma 2.2, we can choose tj > 0 such that tj wj ∈ NP0 Q0 . Set w˜ j = tj wj = (u˜ 1j , v˜j1 ). It follows from (PQ1), zj ∈ Nεj and Lemma 2.2 that 1 1 LP0 Q0 (w˜ j ) ≤ (|∇ u˜ 1j |2 + Pˆεj (x)(u˜ 1j )2 ) + (|∇ v˜j1 |2 + Qˆ εj (x)(v˜j1 )2 ) 2 2 RN

−

1 4

RN

(μ(u˜ 1j )4 + 2β(u˜ 1j )2 (v˜j1 )2 + ν(v˜j1 )4 ) RN

= Lεj (tj zj ) ≤ Lεj (zj ) = Aεj = AP0 Q0 + o(1). On the other hand, LP0 Q0 (w˜ j ) ≥ AP0 Q0 , thus lim LP0 Q0 (w˜ j ) = AP0 Q0 . Since wj w in E, j →∞

it is easy to check that {tj } is bounded. Without loss of generality we can assume that tj → t ≥ 0 as j → ∞. If t = 0, we have that w˜ j = tj wj → 0 in view of the boundedness of wj , and hence LP0 Q0 (w˜ j ) → 0 as j → ∞, which contradicts with AP0 Q0 > 0. So we must have t > 0 and the weak limit w˜ = (u˜ 1 , v˜ 1 ) of w˜ j is different from zero. Since tn → t > 0 and wn w, we have from the uniqueness of the weak limit that w˜ = tw = 0 and w˜ ∈ NP0 Q0 . We choose λ1 = P0 and λ2 = Q0 in Lemma 3.12, then w˜ j → w˜ in E, and consequently wj → w in E. This proves the claim of wj → w in E. To further consider the limit w, notice that wj = (u1j , vj1 ) solves ⎧ 3 2 ⎪ ⎨−u + P (εj x + εj yj )u = μu + βv u −v + Q(εj x + εj yj )v = νv 3 + βu2 v ⎪ ⎩ u, v > 0, u, v ∈ H 1 (RN ),

in RN , in RN ,

(3.58)

with corresponding energy functional ˜ ε (u, v) = 1 L j 2

RN

ˆ εj v 2 ) − 1 (|∇u|2 + Pˆεj u2 + |∇v|2 + Q 4

(μu4 + 2βu2 v 2 + νv 4 ). RN

We show that {εj yj } is bounded by using an idea of [42]. Assume by contradiction that |εj yj | → ∞ as j → ∞. Without loss of generality we may assume that P (εj yj ) → P˜ ∞ and Q(εj yj ) → Q˜ ∞ as j → ∞. It follows from the assumption (PQ1) that P0 < P˜ ∞ and Q0 < Q˜ ∞ . Since P and Q are uniformly continuous functions, it follows that there exists R > 0 such that for all |x| ≤ R, as j → ∞,

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|Pˆεj (x) − P˜ ∞ | ≤ |P (εj (x + yj )) − P (εj yj )| + |P (εj yj ) − P˜ ∞ | ≤ cεj |x| + |P (εj yj ) − P˜ ∞ | → 0, ˜ ∞| ˆ εj (x) − Q |Q ˜ j yj )| + |Q(εj yj ) − Q ˜ ∞ | ≤ cεj |x| + |Q(εj yj ) − Q ˜ ∞ | → 0. ≤ |Q(εj (x + yj )) − Q(ε In addition, for any φ = (φ1 , φ2 ) ∈ C0∞ (RN ) × C0∞ (RN ), we deduce from wj → w in E that

lim L˜εj (wj )φ = lim

j →∞

j →∞ RN

[(∇u1j ∇φ1 + Pˆεj (x)u1j φ1 ) + (∇vj1 ∇φ2 + Qˆ εj (x)vj1 φ2 )]

− lim

j →∞ RN

[μ(u1j )3 φ1 + ν(vj1 )3 φ2 ] − β lim

j →∞ RN

[(vj1 )2 u1j φ1 + (u1j )2 vj1 φ2 ]

˜ ∞ v 1 φ2 )] [(∇u1 ∇φ1 + P˜∞ u1 φ1 ) + (∇v 1 ∇φ2 + Q

= RN

[μ(u ) φ1 + ν(v ) φ2 ] − β

−

1 3

[(v 1 )2 u1 φ1 + (u1 )2 v 1 φ2 ] = 0.

1 3

RN

RN

˜ ∞ , and the corresponding Thus, w = (u1 , v 1 ) is a solution of (1.3) with λ1 = P˜∞ and λ2 = Q energy functional satisfies

1 1 1 2 LP˜∞ Q˜ ∞ (w) = u P˜ ∞ + v 1 2Q˜ ∞ − 2 4

[μ(u1 )4 + 2β(u1 )2 (v 1 )2 + ν(v 1 )4 ] RN

≥ AP˜ ∞ Q˜ ∞ .

(3.59)

By using standard arguments one checks that AP˜ ∞ Q˜ ∞ is attained by z = (u, v) = 0, and z may be a semi-trivial solution of (1.3) with λ1 = P˜ ∞ and λ2 = Q˜ ∞ . Furthermore, since β > βˆ1 , ˜ ∞ , then from Lemma 2.3 we have A ˜ ∞ ˜ ∞ > AP0 Q0 . On the other hand, P0 < P˜ ∞ and Q0 < Q P Q ˜ (wj )wj = L (zj )zj = 0 that one deduces from L εj

AP0 Q0

εj

˜ ε (wj ) = lim ≥ lim Aεj = lim L j j →∞

⎡

⎢1 ≥ lim inf ⎣ j →∞ 4

j →∞

j →∞

˜ ε (wj ) − 1 L ˜ ε (wj ) (wj )wj L j 2 j ⎤

⎥ (μ(u1j )4 + ν(vj1 )4 ) + 2β(u1j )2 (vj1 )2 )⎦

RN

) * 1 ≥ μ(u1 )4 + ν(v 1 )4 + 2β(u1 )2 (v 1 )2 = LP˜∞ Q˜ ∞ (w) ≥ AP˜ ∞ Q˜ ∞ . 4 RN

(3.60)

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That is a contradiction. Therefore the sequence {εj yj } is bounded. Without loss of generality, we assume that εj yj → y0 , and consequently w = (u1 , v 1 ) is a solution of (1.3) with λ1 = P (y0 ) and λ2 = Q(y0 ). Step 3. We prove that y0 ∈ V . / V . Again it follows from Lemma 2.3 that We argue by contradiction. Assume that y0 ∈ (3.61)

AP0 Q0 < AP (y0 )Q(y0 ) .

˜ ∞ replaced by P (y0 ) By using the same arguments as in the proof of Step 2 (with P˜ ∞ and Q and Q(y0 )), we derive a contradiction by using (3.59), (3.60) and (3.61). So y0 ∈ V . Finally we prove that w = (u1 , v 1 ) is a positive ground state solution of (1.3) with λ1 = P0 = P (y0 ) and ˜ ∞ replaced by P0 and Q0 ), λ2 = Q0 = Q(y0 ). Indeed from (3.54) and (3.60) (with P˜ ∞ and Q we obtain that lim Aεj = AP0 Q0 ≥ LP0 Q0 (w) ≥ AP0 Q0 ,

j →∞

which implies that LP0 Q0 (w) = AP0 Q0 and w is a ground state solution of (1.3). Since β > βˆ1 , then w must be positive from Lemma 3.7. 2 In order to obtain exponential decay of positive solutions of (Pε ), we need the following regularity results, which can be found in, for example, [44, Proposition 2-3]. Lemma 3.14. Let z ∈ H 1 (RN ) satisfying −z + (Q(x) + H (x))z = f (x, z),

z ∈ H 1 (RN ),

N

N + N 2 where Q(x) ≥ 0 in RN , Q ∈ L∞ loc (R , R ), and H ∈ L (R ), f is a Caratheodory function such that

0 ≤ f (x, s) ≤ Cf (s + s r−1 ),

x ∈ RN , s ≥ 0,

where 2 < r < 2∗ if N = 3, 2 < r < ∞ if N = 1, 2. Then z ∈ Lp (RN ) for all 2 ≤ p < ∞. Furthermore, there is a positive constant Cp depending on p, Cf and Q such that |z|Lp (RN ) ≤ Cp zH 1 (RN ) . Moreover, the dependence on Q of Cp can be given uniformly on Cauchy seN

quences Qk in L 2 (RN ). t

Lemma 3.15. Suppose that t > N , k ∈ L 2 ( ) and z ∈ H 1 ( ) satisfies in the weak sense −z ≤ k(x), where is an open subset of RN . Then for any ball B2R (y) ⊂ , one has that sup z ≤ C(|z+ |L2 (B2R (y)) + |k|

BR (y)

where C depends on N , t and R.

t

L 2 (B2R (y))

)

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Now we are ready to prove the following exponential decay results for the positive ground state solution of (Pε ). Lemma 3.16. Under the assumptions of Theorem 1.2, if zε = (uε , vε ) is a positive ground state 1,σ solution of (Pε ), one has that when ε > 0 is small, uε , vε ∈ Cloc (RN ) for σ ∈ (0, 1), and lim uε (x) = lim vε (x) = lim |∇uε (x)| = lim |∇vε (x)| = 0,

|x|→∞

|x|→∞

|x|→∞

|x|→∞

(3.62)

and there exist C, c > 0 such that uε (x) + vε (x) ≤ Ce−c|x−yε | ,

(3.63)

where yε satisfies |uε (yε ) + vε (yε )| = max |uε (x) + vε (x)|. x∈RN

1,σ Proof. The proof of uε , vε ∈ Cloc (RN ) for σ ∈ (0, 1) and (3.62) can be proved the same way as in (i) of Lemma 2.1. In the following we prove the exponential decay of wε = uε +vε . Let εj → 0 be a positive sequence, let {zj = (uj , vj )} be a sequence of positive ground state solutions such that Lεj (zj ) = Aεj and Lεj (zj ) = 0, and let yj be the maximum point as defined in (3.56). As in the proof of Lemma 3.13, we have that (u1j (x), vj1 (x)) = zj (x + yj ) = (uj (x + yj ), vj (x + yj )) satisfies (3.58). Let wj = u1j + vj1 . Then from (3.58), wj satisfies

ˆ εj (x))wj = gj (x), −wj + (Pˆεj (x) + Q

in RN ,

(3.64)

where gj (x) = Pˆεj (x)vj1 + Qˆ εj (x)u1j + μ(u1j )3 + ν(vj1 )3 + β(u1j (vj2 )2 + vj1 (u1j )2 ).

(3.65)

So we deduce from Lemma 3.14 that wj ∈ Lt (RN ) for all t ≥ 2 and |wj |Lt (RN ) ≤ Nt wj H 1 (RN ) ,

(3.66)

for some Nt not depending on j . As in the proof of Lemma 3.13, we may assume that u1j → u1 and vj1 → v 1 in H 1 (RN ), hence for any l ∈ (2, 2∗ ], lim

R→∞ |x|≥R

[(u1j )2 + (u1j )l + (vj1 )2 + (vj1 )l ] = 0,

uniformly for j ∈ N.

(3.67)

From (3.64) it follows that −wj ≤ gj (x)

in RN ,

(3.68)

and the estimate (3.66) implies that for all t ≥ 2, there exists C > 0 such that |gj |Lt (RN ) ≤ C for j ∈ N. Thus by Lemma 3.15 we infer that for all y ∈ RN , sup wj ≤ c(|wj |L2 (B2 (y)) + |gj |Lt (B2 (y)) ).

B1 (y)

(3.69)

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This implies that |wj |∞ is uniformly bounded. Now combining the limit (3.67) with the inequality (3.69), we reach that lim wj (x) = 0 uniformly for all j ∈ N.

|x|→∞

Form this we deduce that there is ε0 > 0 such that lim wε (x) = lim [uε (x) + vε (x)] = 0 uniformly for all ε ∈ (0, ε0 ].

|x|→∞

|x|→∞

So by using the same arguments as in the proof of [41, Theorem 3.8], we can show (3.63) holds. 2 Now we complete the proof of Theorem 1.2. Proof of Theorem 1.2. From Lemmas 3.9–3.11, one sees that the conclusions (i)–(ii) of Theorem 1.2 hold. The conclusions (iii)–(iv) of Theorem 1.2 follow from Lemmas 3.13 and 3.16. The results for (Pε ) go back to (Aε ) with the change of variables: x → x/ε. 2 P ε) 4. Multiple positive solutions for (P In this section we prove the existence of multiple positive solutions of (Pε ) by using Ljusternik–Schnirelmann category theory. Our methods are inspired by the work of [38] for scalar nonlinear elliptic equations. Later, some authors use this methods for other problems, for instance, see [48,41,49] and the references therein. Since V = P0 ∩ Q0 = ∅, we shall make good use of the ground state solution of (1.3) with (λ1 , λ2 ) = (P0 , Q0 ) = (P (y0 ), Q(y0 )) for y0 ∈ P0 ∩ Q0 . Fix a δ > 0 and we define a nonincreasing cutoff function J ∈ C0∞ (R+ , [0, 1]) by ⎧ ⎪ 0 ≤ s ≤ 2δ , ⎨1, J (s) = smooth function, 2δ ≤ s ≤ δ, ⎪ ⎩ 0, s ≥ δ. Since V = P0 ∩ Q0 = ∅, then we assume that z0 = (u0 , v0 ) is a positive ground state solution of (1.3) with (λ1 , λ2 ) = (P0 , Q0 ). For any y ∈ V , we define

εx − y εx − y (Uε,y (x), Vε,y (x)) = J (|εx − y|) u0 ( ), v0 ( ) . ε ε

(4.1)

Then (Uε,y , Vε,y ) ∈ Eε , and there exists a unique t1,ε (y) > 0 such that t1,ε (Uε,y , Vε,y ) ∈ Nε . So we define a mapping ε : V → Nε by ε (y) = t1,ε (y)(Uε,y , Vε,y ). Now we study the behavior of Lε (ε (y)) as ε → 0. That is, the following results holds.

(4.2)

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Lemma 4.1. Assume that (PQ0) and (PQ1) hold. For β > 0, one has that lim Lε (ε (y)) = ε→0

AP0 Q0 uniformly for y ∈ V .

The proof of this lemma is similar to [48, Lemma 11] and we omit the details. Moreover, by using similar arguments as in Lemma 3.13 one can prove the following result. Lemma 4.2. Suppose that the assumptions of Theorem 1.3 are satisfied. Let εn → 0 and {zn } ⊂ Nεn such that Lεn (zn ) → AP0 Q0 , then there exists a sequence {yn } ⊂ RN such that zn (x + yn ) has a convergent subsequence in E, and εn yn → y ∈ V . From the condition (PQ1), V is a compact set in RN , so Vδ = {x ∈ RN : dist(x, V ) ≤ δ}. For δ > 0, let = (δ) be a positive number so that Vδ ⊂ B (0). Let γ : RN → RN be a function defined by ⎧ ⎨x, γ (x) = x ⎩ , |x|

|x| ≤ , |x| ≥ .

Then we define a barycenter type map Kε : Nε → RN by γ (εx)u2 Kε (z) = Kε (u, v) =

RN

2 RN

u2

+

γ (εx)v 2 RN

2

.

(4.3)

v2

RN

Similar to the proof of Lemma 4.1, by using Lebesgue’s Theorem it is easy to verify that lim Kε (ε (y)) = y uniformly for y ∈ V . ε→0

Let κ(ε) be any positive function such that κ(ε) → 0 as ε → 0. We define the following set: Oε = {z ∈ Nε : Lε (z) ≤ AP0 Q0 + κ(ε)}.

(4.4)

In fact, for any y ∈ V , we deduce from Lemma 4.1 that κ(ε) = |Lε (ε (y)) − AP0 Q0 | → 0 as ε → 0+ . That is, ε (y) ∈ Oε and Oε = ∅ for ε > 0. The following lemma provides a relation between the image of Oε under Kε and the neighborhood Vδ of V . Lemma 4.3. Suppose that the assumptions of Theorem 1.3 are satisfied. Then for δ > 0 satisfying Vδ ⊂ B (0), and Kε , Oε defined as in (4.3) and (4.4) respectively, we have that lim sup dist(Kε (z), Vδ ) = 0. ε→0 z∈Oε

Proof. Let {εn } be a positive sequence such that εn → 0. By definition, there exists {zn } ⊂ Oεn such that dist(Kεn (zn ), Vδ ) = sup dist(Kεn (z), Vδ ) + o(1) as n → ∞. Thus it is sufficient to find z∈Oεn

a sequence {y˜n } ⊂ Vδ satisfying |Kεn (zn ) − y˜n | = o(1) as n → ∞. From LP0 Q0 (tzn ) ≤ Lε (tzn ) for t ≥ 0 and {zn } ⊂ Oεn ⊂ Nεn , we obtain that AP0 Q0 ≤ Aεn ≤ Lεn (zn ) ≤ AP0 Q0 + κ(εn ). This

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leads to Lεn (zn ) → AP0 Q0 as n → ∞. By Lemma 4.2 one sees that there exists a sequence {yn } ⊂ RN such that y˜n = εn yn ∈ Vδ for n sufficiently large. Hence

+ y˜n ) − y˜n )u2n (ξ

(γ (εn ξ Kεn (zn ) = y˜n +

RN

+ y˜n )

2

(γ (εn ξ + y˜n ) − y˜n )vn2 (ξ + y˜n ) +

u2n (ξ + y˜n )

RN

2

RN

. vn2 (ξ + y˜n )

RN

Since εn ξ + y˜n → y ∈ V uniformly for ξ in any compact subset of RN , we have that Kεn (zn ) = y˜n + o(1) as n → ∞, which implies the desired convergence. 2 We now prove the existence of multiple positive solutions. Lemma 4.4. Suppose that the assumptions of Theorem 1.3 are satisfied. Then (Pε ) has at least catVδ (V ) distinct nontrivial solutions for ε > 0 small. Proof. For a fixed small δ > 0, by using Lemmas 4.1 and 4.3, there exists εδ > 0 small enough such that, the diagram ε

Kε

V −−→ Oε −−→ Vδ

(4.5)

is well defined for ε ∈ (0, εδ ). It is known that lim Kε (ε (y)) = y

ε→0

uniformly for y ∈ V .

(4.6)

For ε > 0 small enough, we denote Kε (ε (y)) = y + ζ (y) for y ∈ V , where |ζ (y)| < δ/2 uniformly for y ∈ V . Define η(t, y) = y + (1 − t)ζ (y). Then η : [0, 1] × V → Vδ is continuous. Obviously, η(0, y) = Kε (ε (y)), η(1, y) = y for all y ∈ V . Thus, we obtain that the composite mapping Kε ◦ ε is homotopic to the inclusion mapping id : V → Vδ . So it follows from Lemma 2.2 of [38] that catOε (Oε ) ≥ catVδ (V ).

(4.7)

Next, let us choose a function κ(ε) > 0 such that κ(ε) → 0 as ε → 0, and AP0 Q0 + κ(ε) < AP∞ Q∞ for ε > 0 sufficiently small. We deduce from Lemma 3.5 that Lε satisfies the Palais– Smale condition on Oε for small ε > 0. Hence, by the Ljusternik–Schnirelmann theory of critical points (see Theorem 2.1 of [38] or [43]), it follows that Lε has at least catOε (Oε ) distinct critical points in Oε . Furthermore, since β > βˆ1 , and all the critical values are less than AP0 Q0 + κ(ε), it follows from the proof of Lemma 3.7 that these critical points are nontrivial for ε > 0 sufficiently small. 2 Next to prove the properties of positive solutions obtained in Lemma 4.4, we state the following lemma which can be proved with similar arguments as in the proof of Lemma 3.13.

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Lemma 4.5. Suppose that the assumptions of Theorem 1.3 are satisfied. If for n ∈ N, zn = (un , vn ) is a solution of (Pε ) satisfying Lεn (zn ) = 0, Lεn (zn ) → AP0 Q0 , and there exist r, δ > 0 and a sequence {yn } ⊂ RN (as in Lemma 4.2) such that lim inf

|zn |2 ≥ δ > 0,

n→∞

Br (yn )

˜ v) ˜ in E as n → ∞ with u, ˜ v˜ = 0, then we have wn (x) = (u˜ n , v˜n ) = zn (x + yn ) → w = (u, that u˜ n , v˜n ∈ L∞ (RN ) and u˜ n L∞ (RN ) + v˜n L∞ (RN ) ≤ C for n ∈ N, and lim u˜ n (x) = |x|→∞

lim v˜n (x) = 0 uniformly for n ∈ N and |u˜ n (x)| + |v˜n (x)| ≤ ce−c|x−yn | for n ∈ N and x ∈ RN .

|x|→∞

Now we prove the following lemma which shows the concentration phenomenon and exponential decay for the positive solutions of (Pε ). Lemma 4.6. Suppose that the assumptions of Theorem 1.3 hold. If zε = (uε , vε ) is a solution of 1,σ (RN ) with σ ∈ (0, 1), (Pε ) and xε is a maximum point of uε + vε , then we have that uε , vε ∈ Cloc and lim P (εxε ) = P0 , lim Q(εxε ) = Q0 ,

ε→0

(4.8)

ε→0

lim uε (x) = lim vε (x) = 0,

|x|→∞

|x|→∞

lim |∇uε (x)| = lim |∇vε (x)| = 0.

|x|→∞

|x|→∞

(4.9)

Furthermore there exist constants C, c > 0 (independent of ε) such that |uε (x)| + |vε (x)| ≤ Ce−c|x−xε | for all x ∈ RN . Proof. Let {εn } be a positive sequence converging to 0, and let zn be a nontrivial solution of (Pεn ). Let {yn } be the sequence in RN given by Lemma 4.2 and let wn (x) = zn (x + yn ) = (u˜ n (x), v˜n (x)). Then, up to a subsequence, it follows from Lemmas 4.2 and 4.5 that wn → 1,σ w = (u, ˜ v), ˜ u, ˜ v˜ ≡ 0, and εn yn → y ∈ V . Then uε , vε ∈ Cloc (RN ) with σ ∈ (0, 1) follows from Lemma 3.16. As in [50,41], we shall prove that there exists a δ > 0 such that wn L∞ (RN ) = |u˜ n | + |v˜n |L∞ (RN ) ≥ δ > 0. Argue by contradiction, if wn L∞ (RN ) → 0 as n → ∞, one infers from zn is a solution of (Pε ) that

(|∇ u˜ n |

2

+ Pεn (yn + εn x)u˜ 2n ) +

RN

RN

≤ μu˜ n 2L∞ (RN ) as

u˜ 2n + νv˜n 2L∞ (RN )

RN

→ 0,

(|∇ v˜n |2 + Qεn (yn + εn x)v˜n2 )

RN

v˜n2 + 2βv˜n 2L∞ (RN )

u˜ 2n

RN

n → ∞.

This implies that u˜ n 2 + v˜n 2 → 0 as n → ∞. However, u˜ n → u˜ = 0 and v˜n → v˜ = 0 as n → ∞, which is a contradiction. Thus there exists a δ > 0 such that wn L∞ (RN ) ≥ δ > 0. Let x˜n be the global maximum point of |u˜ n | + |v˜n |. Then we infer from Lemma 4.5 and the claim above, that {x˜n } ⊂ BR (0) for some R > 0. Thus, the global maximum of |un | +|vn | given by xn = yn + x˜n which gives εn xn = εn yn + εn x˜n . Since {x˜n } is bounded, it follows that εn xn → y ∈ V .

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So we obtain (4.8). Finally, we infer from the above arguments, Lemma 4.4 and the boundedness of {x˜n } that |un (x)| + |vn (x)| ≤ ce−c|x−xn +x˜n | ≤ ce−c|x−xn | , and the estimates in (4.9). 2 Proof of Theorem 1.3. The results for (Aε ) can be obtained from the ones for (Pε ) via change of variable x → x/ε. Therefore the results in Theorem 1.3 follow from Lemma 4.5 and Lemma 4.6. 2 5. Nonexistence of positive solution In this section, we prove the nonexistence results in Theorem 1.4. Part (i) is quite straightforward, and it has been used in previous work as well (see for example [10]). Assume that (Pε ) has a nontrivial solution (u, v). Multiplying the equation of u in (Pε ) by v, the equation of v by u, and integrating over RN , we obtain uv[(P (εx) − Q(εx)) + (β − μ)u2 + (ν − β)v 2 ] = 0.

(5.1)

RN

Then the conclusion (i) of Theorem 1.4 follows from (5.1). So in the rest of this section we prove the conclusion (ii) of Theorem 1.4 under the assumption (PQ2). To do this, we assume that AP∞ Q∞ is attained by a nontrivial positive ground state solution. According to Theorem 1.1, we know that there exist 0 < βˆ2 < βˆ3 such that for β ∈ (0, βˆ2 ) ∪ (βˆ3 , ∞), the limiting equation (1.3) with (λ1 , λ2 ) = (P∞ , Q∞ ) has a positive ground state solution. We prove the following result. Proposition 5.1. Assume that (PQ0) and (PQ2) are satisfied, and (1.3) with (λ1 , λ2 ) = (P∞ , Q∞ ) has a nontrivial positive ground state solution for β ∈ (0, βˆ2 ) ∪ (βˆ3 , ∞), then (Pε ) has no positive ground state solution for β ∈ (0, βˆ2 ) ∪ (βˆ3 , ∞). Proof. We first claim that Aε = AP∞ Q∞ for any ε > 0. In fact, since P (x) ≥ P∞ and Q(x) ≥ Q∞ for all x ∈ RN , then Aε ≥ AP∞ Q∞ from the proof of Lemma 2.3. Next we show that AP∞ Q∞ ≥ Aε for any fixed ε > 0. Since β ∈ (0, βˆ2 ) ∪ (βˆ3 , ∞), we know that (1.3) with (λ1 , λ2 ) = (P∞ , Q∞ ) has a nontrivial positive ground state solution (u∞ , v ∞ ). Moreover, (u∞ , v ∞ ) is the unique global maximum of LP∞ Q∞ (tu∞ , tv ∞ ). Set wn = (u∞ (· − yn ), v ∞ (· − yn )), where {yn ∈ RN } is a sequence satisfying |yn | → ∞ as n → ∞. As in Lemma 2.2, it follows that there exists tn > 0 such that tn wn ∈ Nε is the unique global maximum of Lε (twn ) for each n. Then we can calculate that Aε ≤ Lε (tn wn ) t2 = LP∞ Q∞ (tn u , tn v ) + n 2 ∞

≤ AP∞ Q∞ +

tn2 2

RN

∞

[(P (εx + εyn ) − P∞ )(u∞ )2 + (Q(εx + εyn ) − Q∞ )(v ∞ )2 ]

RN

[(P (εx + εyn ) − P∞ )(u∞ )2 + (Q(εx + εyn ) − Q∞ )(v ∞ )2 ].

(5.2)

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It is clear that for any > 0, there exists R > 0 such that

(P (εx + εyn ) − P∞ )(u∞ )2 ≤ c.

(5.3)

|x|≥R

On the other hand, from Lebesgue’s dominated convergence theorem we have lim

n→∞ |x|

(P (εx + εyn ) − P∞ )(u∞ )2 =

( lim P (εx + εyn ) − P∞ )(u∞ )2 = 0. n→∞

(5.4)

|x|

Combining (5.3) and (5.4), we have lim sup n→∞

(P (εx + εyn ) − P∞ )(u∞ )2 = 0.

(5.5)

RN

(Q(εx + εyn ) − Q∞ )(v ∞ )2 . So it follows from (5.2) that

A similar estimate holds for RN

AP ∞ Q∞ = Aε for any fixed ε > 0. We complete the proof by using a contradiction argument. Assume that for some ε0 > 0 that there exists a positive zˆ such that zˆ = (u, ˆ v) ˆ ∈ Nε0 and Aε0 = Lε0 (ˆz). We know that zˆ is the unique global maximum of Lε0 (t zˆ ). Hence AP∞ Q∞ = Aε0 ≤ LP∞ Q∞ (t ∞ zˆ ) = max LP∞ Q∞ (t zˆ ). t>0

(5.6)

On the other hand, by using (PQ2), we have LP∞ Q∞ (z) ≤ Lε0 (z) for any z ∈ E. Thus combing with (5.6), we have AP∞ Q∞ ≤ LP∞ Q∞ (t ∞ zˆ ) ≤ Lε0 (t∞ zˆ ) ≤ Lε0 (ˆz) = Aε0 = AP∞ Q∞ .

(5.7)

This implies AP∞ Q∞ = LP∞ Q∞ (t ∞ zˆ ) = Lε0 (t ∞ zˆ ). Moreover, z∞ = t ∞ zˆ = (u∞ , v ∞ ) satisfies (1.3) with λ1 = P∞ and λ2 = Q∞ . So one has LP∞ Q∞ (z∞ ) = Lε0 (z∞ ) +

1 2

)

* (P∞ − P (ε0 x))(u∞ )2 + (Q∞ − Q(ε0 x))(v ∞ )2 .

(5.8)

RN

On the other hand we deduce from (PQ2) that 1 2

) * (P∞ − P (ε0 x))(u∞ )2 + (Q∞ − Q(ε0 x))(v ∞ )2 < 0.

(5.9)

RN

Equations (5.8) and (5.9) together imply that LP∞ Q∞ (z∞ ) < Lε0 (z∞ ). This is a contradiction. 2

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Acknowledgment The authors thank the anonymous referee for his or her very valuable comments, which led to an improvement of this paper. This work was done when J.W. visited Department of Mathematics, College of William and Mary during the academic year 2013–14 under the support of China Scholarship Council (201208320496), and he thanks Department of Mathematics, College of William and Mary for their support and kind hospitality. References [1] M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, E.A. Cornell, Observation of Bose–Einstein condensation in a dilute atomic vapor, Science 269 (1995) 198–201. [2] B. Esry, C. Greene, J. Burke, J. Bohn, Hartree–Fock theory for double condensates, Phys. Rev. Lett. 78 (1997) 3594–3597. [3] E. Timmermans, Phase separation of Bose–Einstein condensates, Phys. Rev. Lett. 81 (1998) 5718–5721. [4] S.-M. Chang, C.-S. Lin, T.-C. Lin, W.-W. Lin, Segregated nodal domains of two-dimensional multispecies Bose– Einstein condensates, Phys. D 196 (2004) 341–361. [5] M. Mitchell, M. Segev, Self-trapping of incoherent white light, Nature 387 (1997) 880–883. [6] M. Mitchell, Z.-G. Chen, M.-F. Shih, M. Segev, Self-trapping of partially spatially incoherent light, Phys. Rev. Lett. 77 (1996) 490–493. [7] A. Ambrosetti, E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc. (2) 75 (2007) 67–82. [8] T. Bartsch, N. Dancer, Z.-Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Differential Equations 37 (2010) 345–361. [9] T. Bartsch, Z.-Q. Wang, J.-C. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl. 2 (2007) 353–367. [10] Z.-J. Chen, W.-M. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal. 205 (2012) 515–551. [11] Z.-J. Chen, W.-M. Zou, An optimal constant for the existence of least energy solutions of a coupled Schrödinger system, Calc. Var. Partial Differential Equations 48 (2013) 695–711. [12] D.G. de Figueiredo, O. Lopes, Solitary waves for some nonlinear Schrödinger systems, Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008) 149–161. [13] N. Ikoma, K. Tanaka, A local mountain pass type result for a system of nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations 40 (2011) 449–480. [14] T.-C. Lin, J.-C. Wei, Ground state of N coupled nonlinear Schrödinger equations in Rn , n ≤ 3, Comm. Math. Phys. 255 (2005) 629–653. [15] T.-C. Lin, J.-C. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005) 403–439. [16] T.-C. Lin, J.-C. Wei, Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials, J. Differential Equations 229 (2006) 538–569. [17] Z.-L. Liu, Z.-Q. Wang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phys. 282 (2008) 721–731. [18] Z.-L. Liu, Z.-Q. Wang, Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Stud. 10 (2010) 175–193. [19] L. Maia, E. Montefusco, B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations 229 (2006) 743–767. [20] E. Montefusco, B. Pellacci, M. Squassina, Semiclassical states for weakly coupled nonlinear Schrödinger systems, J. Eur. Math. Soc. (JEMS) 10 (2008) 47–71. [21] A. Pomponio, Coupled nonlinear Schrödinger systems with potentials, J. Differential Equations 227 (2006) 258–281. [22] Y. Sato, Z.-Q. Wang, On the multiple existence of semi-positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013) 1–22. [23] B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in Rn , Comm. Math. Phys. 271 (2007) 199–221.

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