On coupled nonlinear Schrodinger equations with harmonic potential

The initial value problem for some coupled nonlinear Schrodinger system with unbounded potential is investigated. In the defocusing case, global well-posedness is obtained. For the focusing sign, existence of global and non global solutions is discussed via potential well method. Moreover, existence of ground state and instability of standing waves are proved.


Introduction
Consider the initial value problem for a Schrödinger system with power-type nonlinearities a jk |u k | p |u j | p−2 u j = 0; where u j : R × R N → C for j ∈ [1, m], µ = ±1 and a jk = a kj are positive real numbers. The m-component coupled nonlinear Schrödinger system with power-type nonlinearities (CNLS) p iu j + ∆u j = ± m k=1 a jk |u k | p |u j | p−2 u j , 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 j th component of the beam in Kerr-like photorefractive media. The coupling constant a jk acts to the interaction between the j th and the k th 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 [12,29] for the derivation and applications of this system. Well-posedness issues in the energy space of (CNLS) p were recently investigated by many authors [23,24,17]. a jk |u j (t)u k (t)| p dx = E(u(0)).
If µ = 1, the energy is always positive and we say that the problem (1.1) is defocusing, otherwise it 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. We denote for p > 1 the Schrödinger problem (NLS) p iu + ∆u − |x| 2 u ± u|u| p−1 = 0, u : R × R N → C.

For 1 < p < N +2
N −2 if N ≥ 3 and 1 < p < ∞ if N ∈ {1, 2}, local well-posedness in the conformal space was established [19,9]. By [6], when p < 1 + 4 N or p ≥ 1 + 4 N with a defocusing nonlinearity, the solution to the Cauchy problem (1.1) exists globally. For p = 1 + 4 N , there exists a sharp condition [30] to the global existence for the Cauchy problem (1.1). When p > 1 + 4 N , the solution to the Cauchy problem (1.1) blows up in a finite time for a class of sufficiently large data and globally exists for a class of sufficiently small data [7,8,27].
In two space dimensions, similar results about global well-posedness and instability of the Schrödinger equation with harmonic potential and exponential nonlinearity exist [21].
Intensive work has been done in the last few years about coupled Schrödinger systems [17,28,16,22]. 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 [3,25,13,4,5]. 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.
It is the purpose of this manuscript to obtain global well-posedness of (1.1) in the defocusing sign. In the focusing case, using potential well method [20], we discuss global and non global existence of solutions, via existence of ground state. Moreover, strong instability of standing waves is proved.
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, existence of ground state is established. The sixth section contains a discussion of global and non-global existence of solutions via potential well method. The last section is devoted to obtaining strong instability of standing waves. In appendix, we give a proof of the Virial identity. Denoting H 1 (R N ) the usual Sobolev space, define the conformal space and the product space

Denote the real numbers
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, A B means that A ≤ CB. For 1 ≤ r ≤ ∞ and (s, T ) ∈ [1, ∞) × (0, ∞), we denote the Lebesgue space L r := L r (R N ) with the usual norm . r := . L r , . := . 2 and For simplicity, we 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), we denote T * > 0 it's lifespan.

Main results and background
In what follows, we give the main results and some estimates needed in the sequel. Moreover, (2) u satisfies conservation of the energy and the mass; (3) T * = ∞ in the defocusing subcritical case (µ = 1, 1 < p < p * ).
In the critical case, global existence for small data holds in the energy space. Now, we are interested on the focusing problem (1.1). For u := (u 1 , ..., u m ) ∈ H, we define the action If α, β ∈ R, we call constraint Definition 2.3. We say that Ψ := (ψ 1 , ..., ψ m ) is a ground state solution to (1.1) if and it minimizes the problem , then e it Ψ is a global solution of (1.1) said standing wave.
The last result concerns instability by blow-up for standing waves of the Schrödinger problem (1.1). Indeed, near ground state, there exist infinitely many data giving finite time blowing-up solutions to (1.1).
In what follows, we collect some intermediate estimates.
Definition 2.11. A pair (q, r) of positive real numbers is admissible if In order to control an eventual solution to (1.1), we will use the following Strichartz estimate [6].
. Any solution to (1.1) formally enjoys the so-called Virial identity, which proof is given in appendix.
Proposition 2.16. Recall some continuous and compact injections.
We close this subsection with some absorption result [26].

Local well-posedness
This section is devoted to prove Theorem 2.1. The proof contains three steps. First we prove the existence of a local solution to (1.1), second we show uniqueness and finally we establish global existence in the subcritical case. In this section, we assume that µ = 1, indeed the sign of the nonlinearity has no local effect.
3.1. Local existence. We use a standard fixed point argument. For T > 0, we denote the space Define, for u := (u 1 , .., u m ), the function where T (t)Ψ := (U(t)ψ 1 , ..., U(t)ψ m ). We prove the existence of some small T, R > 0 such that φ is a contraction on the ball B T (R) whith center zero and radius R. Take u, v ∈ E T , using Propositions 2.9-2.12 and denoting g( Thus, for small T > 0, . Then, .
To derive the contraction, consider the function With the mean value Theorem Using Hölder inequality, Sobolev embedding and denoting the quantity , we compute via a symmetry argument Let estimate the quantity Thanks to Hölder inequality and Sobolev embedding, we obtain With the same way Using Hölder inequality, Sobolev embedding and denoting the quantity , we compute via a symmetry argument Collecting the estimates (3.8)-(3.10), it follows that for T > 0 small enough, φ is a contraction satisfying Taking in the last inequality v = 0, yields Since p * < p ≤ p * if N ∈ [3, 6] and p * < p < p * if N = 2, 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.

3.2.
Uniqueness. In what follows, we prove uniqueness of 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) = ( 4p N (p−1) , 2p) 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, whith a continuity argument, we may assume that max j=1,...,m u j L ∞ T (H 1 ) ≤ 1. Using previous computation with Uniqueness follows for small time and then for all time with a translation argument.

3.3.
Global existence in the subcritical case. The global existence is a consequence of energy conservation and previous calculations. Let u ∈ C([0, T * ), H) be the unique maximal solution of (1.1). We prove that u is global. 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 = (v 1 , ..., v m ) to (P s ) on C [s, s + τ ], H). Using the conservation of energy we see that τ does not depend on s. Thus, if we let s be close to T * such that T * < s + τ, this fact contradicts the maximality of T * .

Global existence in the critical case
In this section N ∈ [3,6]. We establish global existence of a solution to (1.1) in the critical case p = p * for small data as claimed in Theorem 2.   We give an auxiliary result. 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 Ψ − 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 where a, b > 0 are sufficiently small to fix later. Using Strichartz estimate, we get .
Using Hölder inequality, Sobolev embedding and denoting the quantity , we compute via a symmetry argument Using Hölder inequality and Sobolev embedding, yields Using Hölder inequality and Sobolev embedding, 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 u (S(I)) (m) ≤ 2δ.
The rest of the Proposition is a consequence of the fixed point properties.
We are ready to prove Theorem 2.2.
Proof of Theorem 2.2. 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 2.18, if ξ(Ψ) 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 (2.2) 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 : Denote also the constraint Finally, we introduce the quantity Now, we prove Theorem 2.5 about existence of a ground state solution to the stationary problem (2.2).

Proof. With a direct computation
The last point is a consequence of the equality ∂ λ H(u λ ) = £H(u λ ).
The next intermediate result is the following. We read an auxiliary result.
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 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 m α,β ≤ H(u λ 0 ) ≤ H(u).
This closes the proof.
With a rearrangement argument via Lemma 5.4, we can assume that (φ n ) is radial decreasing and satisfies (5.11).

Invariant sets and applications
This section is devoted to obtain global and non global existence of solutions to the system (1.1). Precisely, we prove Theorem 2.6. We start with a classical result about stable sets under the flow of (1.1).
Take the real function Q(t) := m j=1 R N |x| 2 |u j (t)| 2 dx. Thanks to Virial identity (2.4), we get We infer that there exists δ > 0 such that K 1,− 2 N (u(t)) < −δ for large time. Otherwise, there exists a sequence of positive real numbers t n → +∞ such that K 1, −2 N (u(t n )) → 0. By the definition of m 1,− 2 N and Lemma 6.2, yields This absurdity finishes the proof of the claim. Thus Q ′′ < −8δ. Integrating twice, Q becomes negative for some positive time. This contradiction closes the proof.

Strong instability
This section is devoted to prove Theorem 2.7 about strong instability of standing waves. We keep notations of thr previous section, namely, K := K 1,− 2 N and I := K 1,0 . Lemma 7.1. Let v ∈ H such that I(v) ≤ 0 and K(v) ≤ 0. Then, for any λ > 1, Proof.
(1) Compute We took the real function defined on (1, ∞) by Then, the derivative satisfies when r tends to one This implies, via the fact that p > p 1 := 1+ 2 N 2 (1+ √ 1 + N 2 ), f is decreasing near to one. Since f (1) = 0, we get f < 0 near to one. The proof of the second point of the Lemma is finished.
The next intermediate result reads as follows.
Then K 1,− 2 N (u λ (t)) = 0 because Ψ is a ground state. Finally, with a continuity argument K 1,− 2 N (u λ (t)) < 0. Now, we are ready to prove the instability result. Proof of Theorem 2.7. Take u λ ∈ C T * (H) the maximal solution to (1.1) with data Ψ λ , where λ > 1 is close to one and Ψ is a ground state solution to (2.2). With the previous Lemma, we get u λ (t) ∈ A − 1,− 2 N , for any t ∈ (0, T * ).

Appendix
We give a proof of Proposition 2.13 about Virial identity. Let u ∈ H, a solution to (1.1) such that xu ∈ L 2 . Denote the quantity V (t) := m j=1 xu j (t) 2 .