Coupled nonlinear Schrödinger equations with harmonic potential

The initial value problem for a coupled nonlinear Schrödinger system with unbounded potential is investigated. In the defocusing case, global well-posedness is obtained. In the focusing case, the existence and stability/instability of standing waves are established. Moreover, global well-posedness is discussed via the potential well method.


Introduction
Consider the initial value problem for a Schrödinger system with power-type nonlinearities ⎧ ⎪ ⎨ ⎪ ⎩ iu j + u j − |x| 2 u j − μ m k=1 a jk |u k | p |u j | p−2 u j = 0; where u j : R × R N → C for some N ≥ 2, j ∈ [1, m], μ = ±1 and a jk = a k j are positive real numbers. The nonlinear m-component coupled nonlinear Schrödinger system arises in many physical problems such as nonlinear optics and Bose-Einstein condensates. It models physical systems in which the field has more than one component. In nonlinear optics [2], u j denotes the jth component of the beam in Kerr-like photo-refractive media. The coupling constant a jk acts as the interaction between the jth and the kth components of the beam. This system arises also in the Hartree-Fock theory for a two component Bose-Einstein condensate. Readers are referred, for instance, to [14,30] for the derivation and applications of this system. Well-posedness issues in the energy space of (CNLS) p were recently investigated by many authors [19,25,26]. If μ = 1, the energy is always positive and (1.1) is said to be defocusing, otherwise a control of the Sobolev norm of a solution with the energy is no longer possible and a local solution may blow-up in finite time, in such a case (1.1) is focusing. Before going further let us recall some historic facts about this problem. The one component model case given by a pure power nonlinearity is of particular interest. The question of well-posedness in the energy space was widely investigated. Denote for p > 1 the Schrödinger problem (NLS) p iu + u − |x| 2 u ± u|u| p−1 = 0, u : R × R N → C.
, 2}, local well-posedness holds in the energy space [9,21]. When p < 1 + 4 N or p ≥ 1 + 4 N with a defocusing nonlinearity, the solution exists globally [6]. For p = 1 + 4 N , there exists a sharp condition [31] to the global existence; moreover, the standing waves are stable under some sufficient conditions [12]. When p > 1 + 4 N , the solution blows up in a finite time for a class of sufficiently large data and globally exists for a class of sufficiently small data [7,8,28]; moreover, the standing waves are unstable under suitable assumptions [13].
In two space dimensions, similar results about global well-posedness and instability of the Schrödinger equation with harmonic potential and exponential nonlinearity exist [23].
Intensive work has been done in the last few years about coupled Schrödinger systems [18,19,24,29]. These works have been mainly on 2-systems or with small couplings. Moreover, most works treat the focusing case by considering the stationary associated problem [3][4][5]15,27]. Despite the partial progress made so far, many difficult questions remain open and little is known about m-systems for m ≥ 3.
In this note, we combine in some meaning the two problems (NLS) p and (CNLS) p . Thus, we have to overcome two difficulties. The first one is the presence of a potential term and the second is the existence of coupled nonlinearities.
The purpose of this manuscript is twofold. First, global well-posedness of (1.1) is obtained in the defocusing case. Second, in the focusing case, the existence of ground states and the stability/instability of standing waves are obtained; moreover, using the potential well method [22], the global existence of solutions is discussed.
The rest of the paper is organized as follows. The next section contains the main results and some technical tools needed in the sequel. The third and fourth sections are devoted to prove well-posedness of (1.1). In section five, the existence of ground states is established. The sixth section contains a discussion of global existence of solutions via the potential well method. The last section is devoted to obtain stability/instability of standing waves. Finally, a proof of the Virial identity is given in the appendix. Denoting H 1 (R N ) the usual Sobolev space, define the conformal space 1 2 and the product space Denote the real numbers called, respectively, mass critical and energy critical exponents We mention that C will denote a constant which may vary from line to line and if A and B are non-negative real numbers, For simplicity, denote the usual Sobolev Space W s, p := W s, p (R N ) and H s := W s,2 . If X is an abstract space C T (X ) := C([0, T ], X ) stands for the set of continuous functions valued in X and X rd is the set of radial elements in X ; moreover, for an eventual solution to (1.1), T * > 0 denotes its lifespan.

Main results and background
In what follows, we give the main results and some estimates needed in the sequel.

Main results
First, local well-posedness of the Schrödinger problem (1.1) is claimed.
Then, there exist T * > 0 and a unique maximal solution to (1.1), Moreover, (2) u satisfies conservation of the energy and the mass;

Remark 2.2
The unnatural condition p ≥ 2 seems to be technical and yields to the restriction N ≤ 4.
In the critical case, global existence for small data holds in the energy space.
If α, β ∈ R, the following quantity is called constraint and it minimizes the problem Theorem 2.6 Take N ≥ 2, 1 < p < p * and two real numbers (α, β) ∈ G p . Then (1) m := m α,β is nonzero and independent of (α, β); (2) there is a minimizer of (2.2), which is some nontrivial solution to (2.1). Now, the global existence of a solution to the focusing problem (1.1) is discussed using the potential well method [22].
(2) if u 0 ∈ V ε ( ) and u is the solution to (1.1) given by Theorem 2.1, (3) e it is said to be orbitally stable if, for any σ > 0 there exists ε > 0 such that if u 0 ∈ V ε ( ), then T σ ( ) = ∞. Otherwise, the standing wave e it is said to be nonlinearly unstable; (4) the set In the case of coupled nonlinear Schrödinger systems, it seems that there is no result of uniqueness of ground states. So, we define a weaker stability as follows [9].

Definition 2.9
For μ > 0, define (2) G μ is said to be stable if, G μ = ∅ and for any ε > 0 there exists σ > 0 such that where u ∈ C(R, H ) is a global solution to (1.1) with data u 0 ∈ H . Theorem 2.10 Take 2 ≤ N ≤ 4, 1 < p < p * and be a ground state solution to (2.1). Then so, the standing wave e it is nonlinearly unstable.
In what follows, we collect some intermediate estimates.

Tools
First, let us recall some known results [6,10,11] about the free propagator associated with (1.1).

Proposition 2.11
There exists a family of operators U := U (t, s), Moreover, we have the following elementary properties: (1) U (t, t) = I d; (2) (t, s) → U (t, s) is continuous; Thanks to Duhamel formula, it yields A standard tool to study Schrödinger problems is the so-called Strichartz type estimate.

Proposition 2.12 If u is a solution to the inhomogeneous Schrödinger problem
In order to control an eventual solution to (1.1), the following Strichartz estimate [6] will be useful.

Proposition 2.15
Take two admissible pairs (q, r ) and (α, β). Then, for any time slab I , . Any solution to (1.1) formally enjoys the so-called Virial identity.
For the reader convenience, a proof of the Virial identity is given in the Appendix. Recall the so-called generalized Pohozaev identity [16].

Proposition 2.17 ∈ H is a solution to (2.1) if and only if S
The following Gagliardo-Nirenberg inequality [20] will be useful.

Proposition 2.19
(4) if xu ∈ L 2 and ∇u ∈ L 2 , then u ∈ L 2 and Remark 2.20 Using the previous inequality, we get u xu + ∇u .
Let us close this subsection with some absorption result. 1 1−θ , we conclude the proof by a continuity argument.

Local well-posedness
This section is devoted to prove Theorem 2.1. The proof contains three steps. First, the existence of a local solution to (1.1) is obtained using a classical fixed point method, second we show uniqueness and finally global existence in the subcritical case is established. In this section, the nonlinearity is assumed to be defocusing (μ = 1), indeed the sign of the nonlinearity has no local effect.

Local existence
Let us discuss two cases.
• Subcritical case: 2 ≤ p < p * . For T > 0 and R 2 := C H , we denote B T (R) the centered ball with radius R of the space We prove the existence of some small T, R > 0 such that φ is a contraction of the ball B T (R). Take u, v ∈ B T (R), using Strichartz estimate, we have To derive the contraction, consider the function With the mean value theorem, via the fact that p ≥ 2, it follows that Using Hölder inequality and Sobolev embedding, compute via a symmetry argument Thus, for small T > 0, we get Thanks to Hölder inequality and Sobolev embedding, we obtain Using Hölder inequality, Sobolev embedding, compute via a symmetry argument Thanks to the previous inequality and (3.2), it yields Since p < p * , by (3.1) and the previous inequality, φ is a contraction of B T (R) for some R, T > 0 small enough. The existence of a local solution to (1.1) follows with a classical fixed point Picard argument.
• Critical case: p = p * . Take the admissible couple (q, ) and the centered ball with radius R > 0 of the space Taking account of Strichartz estimate, Hölder inequality and Sobolev embedding, write for 1 Now, using Propositions 2.12-2.15, Taking account of the previous equality via Strichartz estimate, it follows that for small T > 0, Thus, for small T > 0, we get .
Thanks to Hölder inequality and Sobolev embedding, we obtain Collecting the estimates (3.4)-(3.5), it yields By ( Proof It is sufficient to write, using the previous computation via Duhamel formula,

Uniqueness
In what follows, we prove the uniqueness of a solution to the Cauchy problem (1.1). Let T > 0 be a positive time, u, v ∈ C T (H ) two solutions to (1.1) and w := u − v. Then Applying Strichartz estimate with the admissible pair (q, r ) = ( 4 p N ( p−1) , 2 p) and denoting for simplicity L q T (L r ) the norm of (L q T (L r )) (m) , we have Taking T > 0 small enough, with a continuity argument, we may assume that max j=1,...,m Using previous computation with we have Uniqueness follows for small time and then for all time with a translation argument.

Global existence in the subcritical case
The global existence is a consequence of the conservation laws and previous calculations. Let u ∈ C([0, T * ), H ) be the unique maximal solution of (1.1). By contradiction, suppose that T * < ∞. Consider for 0 < s < T * , the problem By the same arguments used in the local existence, we can find a real number τ > 0 and a solution v = H ). Using the conservation laws, we see that τ does not depend on s. Letting s be close to T * such that T * < s + τ, the solution can be extended after T * ; this contradicts the maximality of T * and finishes the proof.  Besides, the solution depends continuously on the initial data in the sense that there exists δ 0 depending on δ, such that for any δ 1 ∈ (0, δ 0 ), if − ϕ H ≤ δ 1 and v is the local solution of (1.1) with initial data ϕ, then v is defined on I and for any admissible couple (q, r ), Proof The proposition follows from a contraction mapping argument. Let the function Using Hölder inequality, Sobolev embedding and denoting the quantity , we compute via a symmetry argument Thus, Thanks to Hölder inequality and Sobolev embedding, it yields Using Hölder inequality and Sobolev embedding, it yields Moreover, taking in the previous inequality v = 0, we get for small δ > 0, With a classical Picard argument, for small a = 2δ, b > 0, there exists u ∈ X a,b a solution to (1.1) satisfying The rest of the proposition is a consequence of the fixed point properties.
Now, we are ready to prove Theorem 2.3.
Proof of Theorem 2.3 Using the previous proposition via the fact that it suffices to prove that xu + ∇u remains small on the whole interval of existence of u. Write with conservation of the energy and Sobolev's inequality So, by Lemma 2.21, if ξ( ) is sufficiently small, then xu + ∇u stays small for any time.

The stationary problem
The goal of this section is to prove that the elliptic problem (2.1) has a ground state solution. Let us start with some notations. For u := (u 1 , . . . , u m ) ∈ H and λ, α, β ∈ R, we introduce the scaling (u λ j ) α,β := e αλ u j (e −βλ .) and the differential operator We extend the previous operator as follows, if A : Finally, we introduce the quantity Now, we prove Theorem 2.6 about the existence of a ground state solution to the stationary problem (2.1).
The last point is a consequence of the equality ∂ λ H (u λ ) = £H (u λ ).
The next intermediate result is the following. Then, there exists n 0 ∈ N such that K (u n 1 , . . . , u n m ) > 0 for all n ≥ n 0 . Proof Write Thus, Let us read an auxiliary result.

Lemma 5.4 Let (α, β) ∈ G p . Then
Proof Denoting by a the right hand side of the previous equality, it is sufficient to prove that m α,β ≤ a. Take u ∈ H such that K (u) < 0. Because lim λ→−∞ K Q (u λ ) = 0, by the previous lemma, there exists some λ < 0 such that K (u λ ) > 0. With a continuity argument, there exists λ 0 ≤ 0 such that K (u λ 0 ) = 0, then since λ → H (u λ ) is increasing, we get This closes the proof.
So the following sequences are bounded Thus, for any real number a, the following sequence is also bounded Choosing a > 0 near to zero, via the fact that 2λ < p − 1, it follows that (φ n ) is bounded in H. • Second step: the limit of (φ n ) is nonzero and m > 0. Taking account of the compact injection (2.5), take Assume that φ = 0. Using Hölder inequality Similarly, we have H (φ) ≤ m. Moreover, thanks to Lemma 5.4, we can assume that K (φ) = 0 and S(φ) = H (φ) ≤ m. So that φ is a minimizer satisfying (5.1) and • Third step: the limit φ is a solution to (2.1). There is a Lagrange multiplier η ∈ R such that S (φ) = ηK (φ). Thus With a previous computation, for Therefore, £ 2 S(φ) < 0. Thus, η = 0 and S (φ) = 0. So, φ is a ground state and m is independent of (α, β).

Invariant sets and applications
This section is devoted to obtain the existence of global solutions to the system (1.1). Precisely, we prove Theorem 2.7. We start with a classical result about stable sets under the flow of ( Proof Let ∈ A + α,β and u ∈ C T * (H ) be the maximal solution to (1.1). Assume that u(t 0 ) / ∈ A + α,β for some t 0 ∈ (0, T * ). Since S(u) is conserved, we have K α,β (u(t 0 )) < 0. So, with a continuity argument, there exists a positive time t 1 ∈ (0, t 0 ) such that K α,β (u(t 1 )) = 0 and S(u(t 1 )) < m. This contradicts the definition of m. The proof is similar in the case of A − α,β . The previous stable sets are independent of the parameter (α, β). Lemma 6.2 For (α, β) ∈ G p , the sets A + α,β and A − α,β are independent of (α, β).

Orbital stability
This section is devoted to prove Theorem 2.10 about stability of standing waves. Denote K := K 1,− 2 N and I := K 1,0 . First, let us do some computations.
and ∈ H a ground state solution to (2.1) yields

Stable ground state
In this subsection, taking 1 < p < p * and m := inf u∈H {S(u), u = μ}, we establish the first point of Theorem 2.10. Let us start by proving that G μ = ∅. Take a minimizing sequence S(v n ) −→ m and v n = μ.
So, for some ε > 0 and large n, thanks to Gagliardo-Nirenberg inequality (2.4), So v ∈ G μ . This achieves the proof. Now, we prove that G μ is stable. The proof proceeds by contradiction. Suppose that there exists a sequence u n 0 ∈ H such that, when n goes to infinity for some sequence of positive real numbers (t n ) and ε 0 > 0, where u n ∈ C(R, H ) is the global solution to (1.1) with data u n 0 . Let us denote n := u n (t n ). Taking account of the definition of G μ , there exists v n ∈ H such that for a subsequence Then, arguing as in (7.1 Using the conservation laws, it follows that n → μ and S( n ) → m.
Arguing as previously, there exists ∈ G μ such that for a subsequence This contradicts (7.2) and finishes the proof.

Unstable ground state
The proof of the second part of Theorem 2.10 is based on several lemmas.
The next auxiliary result reads.
Then, there exist two real numbers ε 1 > 0, σ 1 > 0 such that for Proof With a continuity argument, there exist ε 1 > 0 and σ 1 > 0 such that Thus, with Taylor expansion By the previous lemma, It follows that The proof is finished.

Appendix
We give a proof of Proposition 2.16 about Virial identity. Let u ∈ H , a solution to (1.1). Denote the quantity Multiplying Eq. (1.1) by 2u j and examining the imaginary parts, ∂ t (|u j | 2 ) = −2 (ū j u j ).
Recall the identity Then, Moreover, On the other hand Write (∂ kū j f j,k (u)) =