Existence and nonexistence of bound state solutions for Schrödinger systems with linear and nonlinear couplings

J. Math. Anal. Appl. 475 (2019) 350–363

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Existence and nonexistence of bound state solutions for Schrödinger systems with linear and nonlinear couplings ✩ Haijun Luo a , Zhitao Zhang b,c,∗ a

College of Mathematics and Econometrics, Hunan University, Changsha 410082, Hunan, PR China Academy of Mathematics and Systems Science, the Chinese Academy of Sciences, Beijing 100190, PR China c School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, PR China b

a r t i c l e

i n f o

Article history: Received 17 October 2018 Available online 13 February 2019 Submitted by Y. Du Keywords: Nonlinear Schrödinger equations Bound state solution Perturbation method Pohozaev–Nehari identity

a b s t r a c t We study the Schrödinger systems with linear and nonlinear coupling terms (doubly coupled nonlinear Schrödinger system for short) which arise naturally in nonlinear optics, and in the Hartree–Fock theory for Bose–Einstein condensates, among other physical problems. First, for small linear coupling constant, we get existence of a nontrivial bound state solution to the system via perturbation method, furthermore, we prove each component of the bound state solution is nonnegative by energy estimate. Second, we establish a version of Pohozaev–Nehari identity and prove a nonexistence result for the more general system when the spatial dimension N ≥ 4. © 2019 Elsevier Inc. All rights reserved.

1. Introduction 1 1 In this paper, we first study the existence of radially symmetric solutions in Hrad (RN ) × Hrad (RN ) of the Schrödinger system:

⎧ 3 2 ⎪ ⎨ −Δu + λ1 u = μ1 u + βuv − εv 3 2 −Δv + λ2 v = μ2 v + βu v − εu ⎪ ⎩ u, v ∈ H 1 (RN ),

in RN , in RN ,

(1.1)

where N = 2, 3, λ1 = λ2 = λ > 0, μ1 , μ2 > 0, 0 < β < min{μ1 , μ2 } and ε ∈ R is a small linear coupling 1 constant, Hrad (RN ) denotes the space which contains all radially symmetric functions in H 1 (RN ). ✩ H. Luo is supported by the Fundamental Research Funds for the Central Universities, No. 531107051206, and Z. Zhang is supported by National Natural Science Foundation of China, No. 11771428. * Corresponding author at: Academy of Mathematics and Systems Science, the Chinese Academy of Sciences, Beijing 100190, PR China. E-mail addresses: [email protected] (H. Luo), [email protected] (Z. Zhang).

https://doi.org/10.1016/j.jmaa.2019.02.045 0022-247X/© 2019 Elsevier Inc. All rights reserved.

H. Luo, Z. Zhang / J. Math. Anal. Appl. 475 (2019) 350–363

351

These types of systems arise when one considers standing wave solutions of time-dependent doubly coupled nonlinear Schrödinger system of the form ⎧ 1 ∂ 2 2 ⎪ ⎪ for t > 0, x ∈ Ω, ⎪ −i Φ = ΔΦ + μ1 |Φ| Φ + β |Ψ| Φ − εΨ ⎨ ∂t 2 ∂ 1 2 2 −i Ψ = ΔΨ + μ2 |Ψ| Ψ + β |Φ| Ψ − εΦ for t > 0, x ∈ Ω, ⎪ ⎪ ∂t 2 ⎪ ⎩ Φ = Φ(t, x) ∈ C, Ψ = Ψ(t, x) ∈ C, t > 0, x ∈ Ω,

(1.2)

√ where i = −1. System (1.2) models naturally many physical problems, especially in nonlinear optics. Physically, the solutions Φ and Ψ denote the first and second component of the beam in Kerr-like photorefractive media (see [1]). The positive constant μj is for self-focusing in the j-th component of the beam, j = 1, 2. The nonlinear coupling constant β is the interaction between the two components of the beam. As β > 0, the interaction is attractive, but the interaction is repulsive if β < 0. The linear coupling is generated either by a twist applied to the fiber in the case of circular polarization, or by an elliptic deformation of the fibers core in the case of circular polarizations. Problem (1.2) also arises in the Hartree–Fock theory for Bose–Einstein condensates, see [12,14,22,24,27] for more detail. When system (1.1) doesn’t admit the linear coupling term, i.e. ε = 0, there have been many interesting mathematical works on the existence and on qualitative properties of solutions after the work (see [17]) by Lin and Wei. For example, for the existence of ground state or bound state solutions, please see [2,4,20,21,25] and the references therein; for semiclassical states or singularly perturbed settings, please see [8–10,18,19, 23,30]; for the uniqueness of positive solutions, please see [28,32]. Here, we just mention some results about the existence of bound state solutions for nonlinear Schrödinger systems with nonlinear coupling constants sufficiently small. In [2], Ambrosetti and Colorado studied the following nonlinear Schrödinger systems:  ⎧ 3 ⎪ βij u2i uj ⎨ − Δuj + λj uj = μj uj + ⎪ ⎩

j = 1, 2, 3,

i=j

(1.3)

uj ∈ H (R ), 1

N

where N = 2, 3, λj > 0, μj > 0, j = 1, 2, 3. Under the assumptions βij = βji , βij = εbij ≥ 0, i = j, the authors proved that for ε sufficiently small, (1.3) has a positive radial bound state uε = (u1,ε , u2,ε , u3,ε ) such that uε → (U1 , U2 , U3 ) as ε → 0, where Uj is the unique positive radial solution of −Δuj + λj uj = μj u3j

in RN ,

uj ∈ H 1 (RN ).

Later, Colorado in [7] extended the result above to the systems whose nonlinear interaction constants are partially sufficiently small. However, when linear coupling terms and nonlinear coupling terms to system (1.1) both exist, only a few interesting results have been obtained in [5,16,26,31]. In [16,31], the authors are mainly concerned with the existence and qualitative properties of ground state solutions. For the spatial dimension N = 1, the existence of bound state has been investigated by the topological methods in [5]. Under the assumptions λ1 = λ2 and μ1 = μ2 , Tian and Zhang in [26] considered the existence of bound state and ground state solutions by bifurcation methods. In this paper, we deal with the existence of bound states to system (1.1) when N = 2, 3 and λ1 = λ2 . It is worth pointing out that we don’t need to assume μ1 = μ2 . Without loss of generality, we may assume that λ1 = λ2 = λ = 1 and μ1 ≤ μ2 , hence 0 < β < μ1 . Moreover, system (1.1) can be also written into ⎧ 3 2 ⎪ ⎨ −Δu + u = μ1 u + βuv − εv 3 2 −Δv + v = μ2 v + βu v − εu ⎪ ⎩ u, v ∈ H 1 (RN ).

in RN , in RN ,

(1.4)

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Considering the fact that we are mainly concerned with the solutions for Schrödinger system with small linear coupling constant ε, we first study the limit system to (1.4): ⎧ 3 2 ⎪ ⎨ −Δu + u = μ1 u + βuv 3 2 −Δv + v = μ2 v + βu v ⎪ ⎩ u, v ∈ H 1 (RN ).

in RN , in RN ,

(1.5)

 μ1 u4 + μ2 v 4 + 2βu2 v 2

(1.6)

The energy functional associated with system (1.5) is defined by 1 I0 (u, v) = 2



 1 |∇u|2 + u2 + |∇v|2 + v 2 − 4

RN

 RN

1 1 for every (u, v) ∈ H, where H := Hrad (RN ) × Hrad (RN ). It is easy to see that system (1.5) has two semitrivial solutions (i.e., those solutions with one component

1 μi w(x)

being zero) (U1 , 0) and (0, U2 ), where Ui (x) =

(i = 1, 2) and w is the unique positive solution of

−Δw + w = w3 in RN , w(0) = max w(x), w(x) → 0 as |x| → ∞. x∈RN

Furthermore, we know

Ui =

inf

1 (RN )\{0} u∈Hrad

u 2   1/2 , μi R N u 4

i = 1, 2.

(1.7)

By direct computations, we have I0 (U1 , 0) =

1 I0 (w, 0), μ1

I0 (0, U2 ) =

1 I0 (0, w). μ2

Since μ1 ≤ μ2 , then I0 (w, 0) ≥ I0 (0, w), whence I0 (U1 , 0) ≥ I0 (0, U2 ).

(1.8)

Besides, (1.5) has a synchronized solution of the form 

β

β



u (x), v (x) =



μ2 − β w(x), μ1 μ2 − β 2



μ1 − β w(x) μ1 μ2 − β 2

(1.9)

for β ∈ (0, μ1 ). Let us introduce the so-called Nehari manifold and its variant by N = {(u, v) ∈ H : (u, v) = 0, I0 (u, v)(u, v) = 0}, and M=

⎧ ⎨ ⎩

 (u, v) ∈ H : u = 0, v = 0,

 |∇u|2 + u2 =

RN

μ1 u4 + βu2 v 2 , RN



 |∇v|2 + v 2 = RN

μ2 v 4 + βu2 v 2 RN

⎫ ⎬ ⎭

.

H. Luo, Z. Zhang / J. Math. Anal. Appl. 475 (2019) 350–363

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Then we can consider the following minimization problems: c0 =

inf

(u,v)∈N

I0 (u, v),

A0 =

inf

(u,v)∈M

I0 (u, v).

In what follows, we put our attention to the system (1.4). First, we introduce some terms and notations. Solutions of (1.4) are the critical points (u, v) ∈ E := H 1 (RN ) × H 1 (RN ) of the corresponding energy functional defined by 1 Iε (u, v) = 2





|∇u| + u + |∇v| + v 2

2

2

2



1 − 4

RN





4

4

2 2

μ1 u + μ2 v + 2βu v

RN



 +ε

uv

RN

= I0 (u, v) + εG(u, v), where  G(u, v) =

uv. RN

Since N = 2, 3, by Sobolev embedding, Iε is well defined and of class C 1 . If (u, v) = 0, then we say that such a critical point is nonzero. Furthermore, if u = 0, v = 0, we call it nontrivial. Noticing that the kind of doubly coupled nonlinear Schrödinger systems (for example, see (1.4) and (1.10)) has no semi-trivial solution of the form (u, 0) or (0, v), we don’t distinguish the two concepts and only call nontrivial solution later. Among nontrivial solutions of (1.4), we shall distinguish between the bound states and the ground states. Definition 1.1. We say that (u, v) ∈ E is a bound state of (1.4) if (u, v) is a nontrivial critical point of Iε . A bound state (u∗ , v ∗ ) such that its energy is minimal among all the nontrivial bound states, namely Iε (u∗ , v ∗ ) = min{Iε (u, v) : (u, v) ∈ E \ {(0, 0)}, Iε (u, v) = 0}, is called a ground state of (1.4). With regard to existence and asymptotic behavior of ground state solutions to (1.4), we recall these theorems from [31, Theorem 1.2,1.4,1.5 and 1.6] as follows: Theorem A. Suppose that N = 2 or 3, λ1 = λ2 = 1, μ1 ≤ μ2 , 0 < β < μ1 . For any given ε ∈ (−1, 0), system (1.4) admits a radial ground state solution (uε , vε ) with uε > 0, vε > 0. Moreover, for any sequence εn → 0− , there exists a subsequence (still denoted by εn ) such that (uεn , vεn ) → (0, U2 )

strongly in H

when μ1 < μ2 ; while (uεn , vεn ) → (0, U2 ) or (U1 , 0)

strongly in H

when μ1 = μ2 . Note that system (1.4) is invariant under the following transformation: σ : R × R × E → R × R × E,

σ(ε, β, u, v) = (−ε, β, u, −v).

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Using the σ-invariance of system (1.4), we can see that (u, v) is a ground state solution of the system (1.4) with coupling coefficients ε and β if and only if (u, −v) is a ground state solution of the system (1.4) with coupling coefficients −ε and β. Thus, by Theorem A, we obtain the following result on the ground states to system (1.4) for ε ∈ (0, 1). Corollary 1.2. Suppose that N = 2 or 3, λ1 = λ2 = 1, μ1 ≤ μ2 , 0 < β < μ1 . For any given ε ∈ (0, 1), system (1.4) admits a radial ground state solution (uε , vε ) with uε > 0, vε < 0. Moreover, for any sequence εn → 0+ , there exists a subsequence (still denoted by εn ) such that (uεn , vεn ) → (0, −U2 )

strongly in H

when μ1 < μ2 ; while (uεn , vεn ) → (0, −U2 ) or (U1 , 0)

strongly in H

when μ1 = μ2 . In [11, Lemma 2.2 and Theorem 3.1] Dancer and Wei proved that (uβ , v β ) is non-degenerate in the space of radially symmetric functions in E, i.e., in the space H. Taking this fact into account, we shall make use of the perturbation methods to obtain bound state solutions of system (1.4) with small linear coupling constant ε. That is, Theorem 1.3. Suppose that N = 2 or 3, λ1 = λ2 = 1, μ1 ≤ μ2 . For any given β ∈ (0, μ1 ), there exists ε0 = ε0 (β) such that for |ε| < ε0 , (1.4) has a radial bound state (uε , vε ) with (uε , vε ) → (uβ , v β ) in H as ε → 0. Moreover, for any given β ∈ (0, μ1 /3), there exists ε∗ = ε∗ (β) such that for 0 < ε < ε∗ , the previous bound state (uε , vε ) satisfies uε ≥ 0, vε ≥ 0 and uε + vε > 0 in RN . We remark that the bound state solutions obtained in Theorem 1.3 are different from the ground state solutions in Theorem A and Corollary 1.2 for ε sufficiently small according to their asymptotic behavior. In the proof of Theorem 1.3, to prove that uε and vε are nonnegative is the main difficulty resulting from the mixed effect of both linear coupling and nonlinear coupling. Our idea is as follows. First, we divide (uε, vε ) + − − + − into two parts, i.e., the positive part (u+ ε , vε ) and the negative part (uε , vε ), where w = max{w, 0}, w = + + − − max{−w, 0}. Second, we make use of different characterizations on (uε , vε ) and (uε , vε ) respectively to obtain their H 1 -norm estimates. Finally, we prove the conclusion that uε and vε are nonnegative by contradiction. Finally, we study the nonexistence result of solutions to general system (1.1) in higher dimension, that is, ⎧ 3 2 ⎪ ⎨ −Δu + λ1 u = μ1 u + βuv − εv −Δv + λ2 v = μ2 v 3 + βu2 v − εu ⎪ ⎩ u, v ∈ H 1 (RN ) ∩ L4 (RN ),

in RN , in RN ,

(1.10)

√ where N ≥ 4, λ1 , λ2 > 0, μ1 , μ2 , β ∈ R and |ε| < λ1 λ2 . For the critical or supercritical cases (note that N ≥ 4), we have the following nonexistence result similar to that of the single elliptic equations. √ Theorem 1.4. Let N ≥ 4, λ1 , λ2 > 0, |ε| < λ1 λ2 and μ1 , μ2 , β ∈ R. Then there is no nontrivial solution  2 (u, v) ∈ H 1 (RN ) ∩ L4 (RN ) to system (1.10).

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Generally speaking, we usually study the system (1.10) in the lower dimension (N ≤ 3) because of its physical background. But then Theorem 1.4 can give an reason for the previous consideration from the view of mathematics. In the proof of Theorem 1.4, we first develop a kind of Pohozaev identity for system (1.10), and then establish a version of Pohozaev–Nehari identity to prove this theorem. This paper is organized as follows. In Section 2, we first prove the existence of nontrivial bound state solutions via perturbation method. And then we show that every component of the bound state solution obtained above is nonnegative, which is involved with estimating their H 1 -norm respectively. In Section 3, the nonexistence of nontrivial bound states to system (1.10) in higher dimension is given by the Pohozaev– Nehari identity. 2. Existence of bound state solutions In what follows, to prove Theorem 1.3, we give several propositions. First, as to the minimizer for A0 , by [15, Proposition 2.3 and 2.5, Remark 2.6] and [6, Theorem 4.2], we have Theorem B. For any 0 < β < μ1 , (uβ , v β ) given in (1.9) is a unique minimizer for A0 with I0 (uβ , v β ) = A0 > max{I0 (U1 , 0)), I0 (0, U2 )}. Moreover, A0 is also given by A0 =

inf

(u,v)∈M

 1

u 2 + v 2 . 4

(2.1)

Next we find out the minimizer for c0 and give its another equivalent characterization. Lemma 2.1. For any 0 < β < μ1 , then it holds that c0 = I(0, U2 ) = 14 U2 2 . Moreover, c0 can be characterized as follows:  2

u 2 + v 2 1  inf . c0 = 4 (u,v)∈H\{(0,0)} RN (μ1 u4 + μ2 v 4 + 2βu2 v 2 ) Proof. The proof is standard, we omit it, please see [7].

(2.2)

2

Now we give the existence result of bound state solutions for system (1.4). Theorem 2.2. Under the assumptions in Theorem 1.3, for any given β ∈ (0, μ1 ), there exists ε0 = ε0 (β) such that for |ε| < ε0 , (1.4) has a radial bound state (uε , vε ) with (uε , vε ) → (uβ , v β ) in H as ε → 0. Proof. First, we recall that Iε (u, v) =

1 2





1 |∇u|2 + u2 + |∇v|2 + v 2 − 4

RN





RN

= I0 (u, v) + εG(u, v), where  G(u, v) =

uv. RN

μ1 u4 + μ2 v 4 + 2βu2 v 2 + ε



RN

uv

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H. Luo, Z. Zhang / J. Math. Anal. Appl. 475 (2019) 350–363

For any given β ∈ (0, μ1 ), let us consider the critical point (uβ , v β ) of the unperturbed functional I0 . Thanks to [11, Lemma 2.2 and Theorem 3.1], we know (uβ , v β ) is a non-degenerate critical point in the space H. Thus, a straightforward application of the Local Inverse Theorem yields the existence of a critical point (uε , vε ) of Iε for any |ε| < ε0 with ε0 being a sufficiently small positive constant dependent of β; see [3] for more details. Moreover, (uε , vε ) → (uβ , v β ) in H as ε → 0. 2 Let (uε , vε ) be the radial bound state solution obtained by Lemma 2.1, we first study its positive part That is,

+ (u+ ε , vε ).

Lemma 2.3. Let β ∈ (0, μ1 ) be fixed, for ε sufficiently small, there exists a unique pair of positive constants √ √ + (tε , sε ) such that ( tε u+ ε , sε vε ) ∈ M. Furthermore, tε = 1 + o(1), sε = 1 + o(1). + β β + Proof. First, from (uε , vε ) ∈ H, it follows that (u+ ε , vε ) ∈ H. Note that (uε , vε ) → (u , v ), we get uε = + 0, vε = 0 for ε sufficiently small. √ √ + √ + √ + Next, we consider the function g(t, s) = I0 ( tu+ ε , svε ) : (0, ∞) ×(0, ∞) → R. Set ( tε uε , sε vε ) ∈ M, then we find g  (tε , sε ) = 0, that is, (tε , sε ) satisfies the following system:



  4 2 + 2 + 2 ) + sε RN β (u+ tε RN μ1 (u+ ε ε ) (vε ) = uε ,   2 2 4 + + + 2 tε RN β (u+ ε ) (vε ) + sε RN μ2 (vε ) = vε .

(2.3)

We denote   4  μ1 uβ  A0 :=  RN  β 2  β 2  Nβ u v R

 β 2  β 2   v  N β u R  β 4  ,  μ v RN 2

where | · | denotes the determinant of a matrix. Notice that β ∈ (0, μ1 ), by the Hölder inequality, then we know A0 > 0. Similarly, we define    4  + 2 + 2 ) β (u ) (v )   RN μ1 (u+ N ε ε ε R Aε :=  . 2 + 2 + 4   RN β (u+ μ (vε ) ε ) (vε ) RN 2 We apply the Hölder inequality once again to deduce that Aε > 0. Thus, (2.3) has a unique solution of the form   4 + 2 + 2 + 2 + 2 μ (u+ ε ) vε − RN β (uε ) (vε ) uε

RN 1 tε = , Aε   + 2 + 2 + 2 + 4 + 2 N β (uε ) (vε ) vε − RN μ2 (vε ) uε

. sε = R Aε + Since (uε , vε ) satisfies (1.4), multiplying the first equation in (1.4) by u+ ε and the second one by vε , and N then integrating over in R , we have     + 2  4  + 2 2 uε  = μ1 u+ + β u v − ε u+ ε ε ε ε vε , RN

 + 2 vε  =



RN

RN

 4 μ2 vε+ +



RN

RN

 2 βu2ε vε+ − ε



uε vε+ .

RN

− N Notice that u− as ε → 0, again by Lemma A.1 in [29], then we can apply the ε → 0, vε → 0 a.e. in R Lebesgue’s Dominated Convergence Theorem to deduce that

H. Luo, Z. Zhang / J. Math. Anal. Appl. 475 (2019) 350–363





2 u+ ε



 − 2 vε → 0,

RN



u− ε

2 

vε+

2

→ 0,

357

as ε → 0.

(2.4)

RN 2

2

+ Substituting the values of u+ ε and vε into tε , sε respectively, combining with (2.4), we easily see that tε → 1, sε → 1 as ε → 0. 2 − In what follows, concerned with the negative part (u− ε , vε ) of (uε , vε ) in Lemma 2.1, we have the following result.

Lemma 2.4. Let β ∈ (0, μ1 /3) be fixed. Then there exists a ε1 = ε1 (β)(< ε0 ) such that for any 0 < ε < ε1 − − 2 − 2 2 and (u− ε , vε ) = (0, 0), it holds that uε + vε ≥ δ0 U2 for some δ0 > 0 independently of ε. − Proof. For (u− ε , vε ) = (0, 0), by Lemma 2.1, we have

⎛ ⎞     − 2      2 4 4 2 2 + μ2

uε + vε− 2 ≥ U2 2 ⎝μ1 u− vε− + 2β u− vε− ⎠ . ε ε RN

RN

(2.5)

RN

− Since (uε , vε ) satisfies (1.4), multiplying the first equation in (1.4) by u− ε and the second one by vε , and N then integrating over in R , we get

 − 2  − 2 uε  + vε  =





4 μ1 u− ε

RN



+

RN

 4 μ1 u− + ε

RN



+ −ε

u− ε vε

 2  + 2 β u− vε + ε

RN





uε vε−

−ε

 4 μ2 vε− +



 2 2 β u− vε ε

+ RN



RN

− u+ ε vε − ε



RN



RN



4 μ2 vε−

 − 2 vε − ε

βu2ε

=



RN



+





RN

 2  − 2 2β u− vε ε

RN

 2  − 2 β u+ vε + 2ε ε

RN



− u− ε vε

RN

+ u− ε vε .

RN

Recall that the definition of u± and v ± , by the Young’s inequality, for ε > 0, then we obtain  − 2  − 2 uε  + vε  ≤



 4 μ1 u− + ε

RN



β



2 u− ε



RN

 4 μ1 u− + ε

≤ RN

2 vε+

 +

β



2 u− ε



RN

In what follows, we estimate the two terms

 +

 2  − 2 2β u− vε ε

 2  − 2 β u+ vε + ε ε

 4 μ2 vε− +

2 vε+



RN

RN



RN



+

 4 μ2 vε− +

RN



+





  

u− ε

2

 2  + vε−

RN

 2  − 2 2β u− vε ε

(2.6)

RN

      2  − 2 2 + vε− 2 . β u+ vε + ε u− ε ε

RN

 RN

2

2

+ β (u− ε ) (vε ) and

 RN

2

2

− β (u+ ε ) (vε ) in (2.6).

H. Luo, Z. Zhang / J. Math. Anal. Appl. 475 (2019) 350–363

358

+ β β By Theorem 2.2, we know (uε , vε ) → (uβ , v β ) in H as ε → 0, and hence, (u+ ε , vε ) → (u , v ) a.e. in β β 4 N 4 N R and (uε , vε ) → (u , v ) in L (R ) × L (R ). Therefore, by Lemma A.1 in [29], then we can apply the Lebesgue’s Dominated Convergence Theorem to obtain N



 + 4 uε →

RN





uβ

4



RN



 + 4 vε →

, RN



vβ

4

RN

as ε → 0. Then there exists ε2 = ε2 (β) > 0 such that when 0 < ε < ε2 , we have ⎛    ⎝    RN ⎛    ⎝    RN

⎞1/2  ⎛  1   + 4  β 4  uε ⎠ − ⎝ u ⎠ ≤ ⎝  2  RN RN  ⎞1/2 ⎛ ⎞1/2  ⎛   1   + 4  β 4  vε ⎠ − ⎝ v ⎠ ≤ ⎝  2  RN RN

⎞1/2

⎞1/2 ⎛  3 ⎝  β 4 ⎠ ≤ u , 2

⎞1/2

⎛





u

β 4

⎞1/2 ⎠

,

⎞1/2  β 4 v ⎠ ,

that is, ⎛ ⎝





+ 4

uε

⎠

RN

RN

⎛ ⎝





+ 4

vε

⎞1/2 ⎠

⎞1/2 ⎛  3 ⎝  β 4 ⎠ ≤ v . 2

RN

(2.7)

RN

Besides, it follows from (2.5) that ⎛ ⎝



RN

⎞1/2 2 2  − 4

u− + vε−

, uε ⎠ ≤ ε√ μ1 U2

⎛ ⎝





− 4

vε

RN

⎞1/2 ⎠

2

≤

2

−

u− ε + vε

. √ μ2 U2

(2.8)

Recall that 

and U2 (x) =

1 μ2 w(x),

β

β





u (x), v (x) =

β − μ2 w(x), 2 β − μ1 μ2



β − μ1 w(x) 2 β − μ1 μ2

where −Δw + w = w3 . By the Hölder inequality and direct computations, and

combine with (2.7), (2.8), then we get 

 2  + 2 3 β u− vε ≤ β ε 2

RN



2  − 2  μ2 μ1 − β  u−  + vε  . ε μ1 μ1 μ2 − β 2

Similarly, we have  RN

 − 2    2  − 2 μ2 − β  3 − 2 vε  .  u β v β u+ ≤ + ε ε ε 2 μ1 μ2 − β 2

Set f1 (β) =

μ1 − β , μ1 μ2 − β 2

f2 (β) =

μ2 − β , μ1 μ2 − β 2

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359

and notice that β ∈ (0, μ1 /3), then we easily see that f1 (β) ≤ f1 (0) =

1 , μ2

f2 (β) ≤ f2 (0) =

1 , μ1

which implies that 

 2  + 2 β u− vε + ε

RN



RN

2  − 2   2  − 2 3β  u−  + vε  . β u+ ≤ vε ε ε μ1

(2.9)

Take ε1 = ε1 (β) = min{ε2 , δ0 }, where δ0 = 12 (1 − 3β/μ1 ), together with (2.6) and (2.9), then we obtain  RN

 4 μ1 u− + ε



 4 μ2 vε− +

RN



      2  − 2 2 + vε− 2 vε 2β u− ≥ δ0 u− ε ε

RN

for 0 < ε < ε1 , which jointly with (2.5) gives 2 − 2 2

u− ε + vε ≥ δ0 U2 . 2

With the above preparations at hand, now we give the proof of our main theorem, i.e., Theorem 1.3. Proof of Theorem 1.3. First, the existence of bound state solutions in (1.4) follows from Theorem 2.2, that is, fix β ∈ (0, μ1 ), there exists ε0 = ε0 (β) such that for |ε| < ε0 , (1.4) has a radial bound state (uε , vε ) with (uε , vε ) → (uβ , v β ) in H as ε → 0. Next, we prove that for any given β ∈ (0, μ1 /3), there exists a ε∗ = ε∗ (β) such that for 0 < ε < ε∗ , the previous bound state (uε , vε ) satisfies uε ≥ 0, vε ≥ 0 and uε + vε > 0 in RN . We assume by contradiction that there exists a sequence of bound state solutions (uεn , vεn ) with − (uεn , vε−n ) = (0, 0) when εn → 0. Thus, for n large enough, we have εn < ε1 . By Lemma 2.4, we have 2 − 2 2

u− εn + vεn ≥ δ0 U2

(2.10)

for some δ0 > 0 independently of ε. On the other hand, since (uε , vε ) → (uβ , v β ) in H as ε → 0, we know + u+ εn = 0, vεn = 0 for n large enough. By Lemma 2.3, there exists a unique pair of positive constants (tεn , sεn ) √ √ + such that ( tεn u+ εn , sεn vεn ) ∈ M. In virtue of the characterization for A0 in Theorem B, we obtain 2 + 2 β 2 β 2 tεn u+ εn + sεn vεn ≥ u + v .

(2.11)

By Lemma 2.4, we know tεn → 1, sεn → 1 as n → ∞ and hence easily obtain a contradiction. In fact, note that (uεn , vεn ) → (uβ , v β ) in H, then we have

uεn 2 + vεn 2 → uβ 2 + v β 2

as n → ∞.

(2.12)

Besides, we also have the following decompositions: 2 − 2

uεn 2 = u+ εn + uεn ,

vεn 2 = vε+n 2 + vε−n 2 .

(2.13)

Combining with (2.10)–(2.13), for n large enough, we obtain a contradiction. Thus, there exists a ε∗ = ε∗ (β) such that for 0 < ε < ε∗ , uε and vε are nonnegative. Finally, we prove that for 0 < ε < ε∗ , it holds that uε + vε > 0 in RN . In fact, since (uε , vε ) solves (1.4) and uε ≥ 0, vε ≥ 0, then we have

H. Luo, Z. Zhang / J. Math. Anal. Appl. 475 (2019) 350–363

360

−Δ(uε + vε ) + (1 + ε)(uε + vε ) = μ1 u3ε + βuε vε2 + βu2ε vε + μ2 vε3 ≥ 0 in RN . Note that (uε , vε ) → (uβ , v β ) in H as ε → 0, we may assume that uε = 0, vε = 0 (here if necessary we can take smaller ε∗ ). Thus, the conclusion follows from the strong maximum principle. 2 3. Nonexistence result To establish the Pohozaev identity for system (1.10), we first recall the Pucci–Serrin variational identity for locally Lipschitz continuous of a general class of equations, see [13, Lemma 1]. N N 1 Lemma 3.1. Let φ ∈ L∞ loc (R ) and L(s, ξ) : R × R → R be a function of class C in s and ξ such that for N any s ∈ R, the map ξ → L(s, ξ) is strictly convex. Suppose that u : R → R is a locally Lipschitz continuous solution of

−div (Lξ (u, Du)) + Ls (u, Du) = φ

in D (RN ).

Then N  





Di hj Dξi L(u, Du)Dj u −

i,j=1 N R

(h · Du)φ

(div h)L(u, Du) =

RN

(3.1)

RN

for every h ∈ Cc1 (RN , RN ).  2 Lemma 3.2. (Pohozaev identity) Let (u, v) ∈ H 1 (RN ) ∩ L4 (RN ) be a weak solution to system (1.10), then we have the following Pohozaev type identity: 

N −2 2 N = 4



RN





 N |∇u|2 + |∇v|2 + 2



λ1 u2 + λ2 v 2 + 2εuv

RN 4

4

2 2

μ1 u + μ2 v + 2βu v



(3.2)

.

RN

 2 Proof. Since (u, v) ∈ H 1 (RN ) ∩ L4 (RN ) , by the elliptic regularity theory, we easily see that u, v ∈ C 2 (RN ). Considering the equation for u, we let L(u, Du) =

1 1 1 1 |Du|2 + u2 − μ1 u4 − βu2 v 2 , 2 2 4 2

φ = −εv and hk (x) = η(x/k)x for all x ∈ RN and k ≥ 1, where η ∈ Cc1 (RN ) satisfies η(x) = 1 if |x| ≤ 1 and η(x) = 0 if |x| ≥ 2. Then for every k ≥ 1, we know that hk ∈ Cc1 (RN , RN ) and Di hjk (x) = Di η

x x

j

x

δij , ∀i, j = 1, . . . , N, k k k x x x (div hk )(x) = Dη · + Nη , k k k +η

(3.3)

where δij is the Kronecker symbol, i.e., δij = 1 if i = j and δij = 0 if i = j. Besides, since supp Dhk ∈ B2k (RN ) \ Bk (RN ) for any k ≥ 1, then there exists a positive constant C1 such that |Dhk (x)| ≤

C1 ≤ C1 , k

for x ∈ RN , k ≥ 1,

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361

  x  which jointly with (3.3) implies that there exists C > 0 such that Di η xk kj  ≤ C for every x ∈ RN , k ≥ 1, i, j = 1, . . . , N . On the other hand, by (3.1) in Lemma 3.1, we have −ε

    N  x x   x j Dξi L(u, Du)Dj u Di η η x · Du v = k k k i,j=1

RN

RN



+

η

x k

RN



−

Dη

Dξ L(u, Du) · Du

(3.4)

 x x x · L(u, Du) − L(u, Du). Nη k k k

RN

RN

As to the equation for v, we set L(v, Dv) =

1 1 1 1 |Dv|2 + v 2 − μ2 v 4 − βu2 v 2 , 2 2 4 2

φ = −εu,

then similarly we have     N  x x   x j Dξi L(v, Dv)Dj v −ε Di η η x · Dv u = k k k i,j=1 RN



RN

η

+ RN



−

x k

Dξ L(v, Dv) · Dv

(3.5)

 x x x · L(v, Dv) − L(v, Dv). Dη Nη k k k

RN

RN

  Integrating by parts with respect to div uvη xk x , we have       x η x · Du v = k RN

 x  η x · Du v k

B2k (0)



  x  η x · Dv u − k

=− B2k (0)

 x x x uv N η + Dη · k k k

B2k (0)

      x x  x x x · Dv u − + Dη · . uv N η η =− k k k k RN

RN

Combining (3.4) with (3.5), we obtain  ε

 x x x uv N η + Dη · k k k

RN

=

N  

Di η

i,j=1 N R



+ RN

η

x x 

x  k

j

k

k

 Dξi L(u, Du)Dj u + Dξi L(v, Dv)Dj v

Dξ L(u, Du) · Du + Dξ L(v, Dv) · Dv



H. Luo, Z. Zhang / J. Math. Anal. Appl. 475 (2019) 350–363

362

 −

Dη

RN



−

Nη

x x   · L(u, Du) + L(v, Dv) k k x  k

 L(u, Du) + L(v, Dv) .

RN

  x  Since there exists C > 0 such that Di η xk kj  ≤ C for every x ∈ RN , k ≥ 1, i, j = 1, . . . , N , we can apply the Lebesgue’s Dominated Convergence Theorem to deduce  ε

  N uv =

RN

RN



−

Dξ L(u, Du) · Du + Dξ L(v, Dv) · Dv



  N L(u, Du) + L(v, Dv) .

RN

Substituting the formulae of L(u, Du) and L(v, Dv) into the previous equality, we get (3.2). 2 Remark 3.3. We define the energy functional corresponding to system (1.10) by 1 I(u, v) = 2

 RN

1 − 4



|∇u| + λ1 u + |∇v| + λ2 v



2

2

2

2



 +ε

uv

RN



4

4

2 2

μ1 u + μ2 v + 2βu v



(3.6)

.

RN

 2 Since (u, v) ∈ H 1 (RN ) ∩ L4 (RN ) , we easily see that I is well defined and of class C 1 . Then we remark that the Pohozaev identity can be reduced formally by  dI (u(x/t), v(x/t))    dt

= 0.

t=1

With the Pohozaev identity at hand, we now start to prove Theorem 1.4.  2 Proof of Theorem 1.4. Let (u, v) ∈ H 1 (RN ) ∩ L4 (RN ) be a solution of system (1.10), then we obtain the Nehari type identity I  (u, v)(u, v) = 0, that is, 



|∇u|2 + λ1 u2 + |∇v|2 + λ2 v 2 + 2ε

RN



 uv =

RN

 μ1 u4 + μ2 v 4 + 2βu2 v 2 .

RN

Combining (3.2) in Lemma 3.2 with (3.7), we obtain the following Pohozaev–Nehari identity: 

N −4 4

RN

Note that N ≥ 4 and |ε| <

√



|∇u| + |∇v| 2

2



N + 4





λ1 u2 + λ2 v 2 + 2εuv = 0.

RN

λ1 λ2 , then we infer that u ≡ 0, v ≡ 0. 2

Acknowledgments The authors thank the referees for their careful reading and helpful suggestions.

(3.7)

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References [1] N. Akhmediev, A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett. 82 (1999) 2661–2664. [2] A. Ambrosetti, E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc. (2) 75 (1) (2007) 67–82. [3] A. Ambrosetti, A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on RN , Progr. Math., vol. 240, Birkhäuser Verlag, Basel, 2006. [4] T. Bartsch, Z.Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Partial Differ. Equ. 19 (3) (2006) 200–207. [5] J. Belmonte-Beitia, V.M. Pérez-García, P.J. Torres, Solitary waves for linearly coupled nonlinear Schrödinger equations with inhomogeneous coefficients, J. Nonlinear Sci. 19 (4) (2009) 437–451. [6] 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 (2) (2012) 515–551. [7] E. Colorado, Positive solutions to some systems of coupled nonlinear Schrödinger equations, Nonlinear Anal. 110 (2014) 104–112. [8] E.N. Dancer, K.L. Wang, Z.T. Zhang, Dynamics of strongly competing systems with many species, Trans. Amer. Math. Soc. 364 (2) (2012) 961–1005. [9] E.N. Dancer, K.L. Wang, Z.T. Zhang, The limit equation for the Gross–Pitaevskii equations and S. Terracini’s conjecture, J. Funct. Anal. 262 (3) (2012) 1087–1131. [10] E.N. Dancer, K.L. Wang, Z.T. Zhang, Addendum to “The limit equation for the Gross-Pitaevskii equations and S. Terracini’s conjecture” [J. Funct. Anal. 262 (3) (2012) 1087–1131] [MR2863857], J. Funct. Anal. 264 (4) (2013) 1125–1129. [11] E.N. Dancer, J.C. Wei, Spike solutions in coupled nonlinear Schrödinger equations with attractive interaction, Trans. Amer. Math. Soc. 361 (3) (2009) 1189–1208. [12] B. Deconinck, P.G. Kevrekidis, H.E. Nistazakis, D.J. Frantzeskakis, Linearly coupled Bose–Einstein condesates: from Rabi oscillations and quasiperiodic solutions to oscillating domain walls and spiral waves, Phys. Rev. A 70 (2004) 705. [13] M. Degiovanni, A. Musesti, M. Squassina, On the regularity of solutions in the Pucci–Serrin identity, Calc. Var. Partial Differential Equations 18 (3) (2003) 317–334. [14] D.S. Hall, M.R. Matthews, J.R. Ensher, C.E. Wieman, Dynamics of component separation in a binary mixture of Bose– Einstein condensates, Phys. Rev. Lett. 81 (1998) 1539–1542. [15] N. Ikoma, K. Tanaka, A local mountain pass type result for a system of nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations 40 (3–4) (2011) 449–480. [16] K. Li, Z.T. Zhang, Existence of solutions for a Schrödinger system with linear and nonlinear couplings, J. Math. Phys. 57 (8) (2016) 081504, 17 pp. [17] T.C. Lin, J.C. Wei, Ground state of N coupled nonlinear Schrödinger equations in Rn , n ≤ 3, Comm. Math. Phys. 255 (3) (2005) 629–653. [18] T.C. Lin, J.C. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (4) (2005) 403–439. [19] T.C. Lin, J.C. Wei, Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials, J. Differential Equations 229 (2) (2006) 538–569. [20] Z.L. Liu, Z.Q. Wang, Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Stud. 10 (1) (2010) 175–193. [21] L.A. Maia, E. Montefusco, B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations 229 (2) (2006) 743–767. [22] C.J. Myatt, et al., Production of two overlapping Bose–Einstein condensates by sympathetic cooling, Phys. Rev. Lett. 78 (1997) 586–589. [23] B. Noris, H. Tavares, S. Terracini, G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure Appl. Math. 63 (3) (2010) 267–302. [24] Ch. Rüegg, et al., Bose–Einstein condensate of the triplet states in the magnetic insulator TlCuCl3, Nature 423 (2003) 62–65. [25] B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in Rn , Comm. Math. Phys. 271 (1) (2007) 199–221. [26] R.S. Tian, Z.T. Zhang, Existence and bifurcation of solutions for a double coupled system of Schrödinger equations, Sci. China Math. 58 (8) (2015) 1607–1620. [27] E. Timmermans, Phase separation of Bose–Einstein condensates, Phys. Rev. Lett. 81 (1998) 5718–5721. [28] J.C. Wei, W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Commun. Pure Appl. Anal. 11 (3) (2012) 1003–1011. [29] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. [30] Z.T. Zhang, Variational, Topological, and Partial Order Methods with Their Applications, Dev. Math., vol. 29, Springer, Heidelberg, 2013. [31] Z.T. Zhang, H.J. Luo, Symmetry and asymptotic behavior of ground state solutions for Schrödinger systems with linear interaction, Commun. Pure Appl. Anal. 17 (3) (2018) 787–806. [32] Z.T. Zhang, W. Wang, Structure of positive solutions to a Schrödinger system, J. Fixed Point Theory Appl. 19 (1) (2017) 877–887.