#### Coupled nonlinear Schrödinger equations with harmonic potential

Coupled nonlinear Schrödinger equations with harmonic potential
H. Hezzi 0 1 2 3
M. M. Nour 0 1 2 3
T. Saanouni 0 1 2 3
0 T. Saanouni Qassim University , Buraidah, Kingdom of Saudi Arabia
1 M. M. Nour University of Zalingei, Faculty of Education , Zalingei , Sudan
2 H. Hezzi Umm Al-Qura university , Adham, Makkah, Kingdom of Saudi Arabia
3 T. Saanouni (
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. Mathematics Subject Classification 35Q55 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Main results and background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Local well-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Contents
3.1 Local existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Global existence in the subcritical case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Global existence in the critical case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 The stationary problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Invariant sets and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 Orbital stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1 Stable ground state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Unstable ground state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction
Consider the initial value problem for a Schrödinger system with power-type nonlinearities
⎧
⎪⎨ iu˙ j + u j − |x|2u j − μ
⎪⎩ u j(0, .) = ψ j,
m
k=1
a jk|uk|p |u j|p−2u j = 0;
(1.1)
where u j : R × RN → C for some N ≥ 2, j ∈ [1, m], μ = ±1 and a jk = akj are positive real numbers.
The nonlinear m-component coupled nonlinear Schrödinger system
m
k=1
(CNLS)p iu˙ j + u j = ±
a jk|uk|p |u j |p−2u j, j ∈ [1, m],
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].
A solution u := (u1, . . . , um) to (1.1) formally satisfies, respectively, conservation of the mass and the energy
M(u j) := RN |u j(x, t)|2 dx = M(ψ j);
E(u(t)) := 2 j=1 RN |∇u j (t)|2 + |xu j(t)|2 + μp m a jk|u j(t)uk(t)|p dx = E(u(0)).
1 m
k=1
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|2u ± u|u|p−1 = 0, u : R × RN → C.
For 1 < p < NN+−22 if N ≥ 3 and 1 < p < ∞ if N ∈ {1, 2}, local well-posedness holds in the energy space
[9,21]. When p < 1 + N4 or p ≥ 1 + N4 with a defocusing nonlinearity, the solution exists globally [6]. For
p = 1 + N4 , there exists a sharp condition [31] to the global existence; moreover, the standing waves are stable
under some sufficient conditions [12]. When p > 1 + N4 , 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 [3–5,15,27]. 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(RN ) the usual Sobolev space, define the conformal space
endowed with the complete norm
and the product space
Denote the real numbers called, respectively, mass critical and energy critical exponents
:=
u ∈ H 1(RN ) s. t
RN
|x |2|u(x )|2 dx < ∞
u
:=
u 2L2(RN ) +
x u 2L2(RN ) + ∇u 2L2(RN )
.
For simplicity, denote the usual Sobolev Space W s, p := W s, p(RN ) and H s := W s,2. If X is an abstract space
CT (X ) := C ([0, T ], X ) stands for the set of continuous functions valued in X and Xrd is the set of radial
elements in X ; moreover, for an eventual solution to (1.1), T ∗ > 0 denotes its lifespan.
2 Main results and background
In what follows, we give the main results and some estimates needed in the sequel.
2.1 Main results
First, local well-posedness of the Schrödinger problem (1.1) is claimed.
Theorem 2.1 Let 2 ≤ N ≤ 4 and ∈ H . Assume that 2 ≤ p ≤ p∗ if N > 2 and 2 ≤ p < p∗ if N = 2.
Then, there exist T ∗ > 0 and a unique maximal solution to (1.1),
u ∈ C ([0, T ∗), H ).
Moreover,
(
1
) u ∈
4p
L N(p−1) ([0, T ∗], W 1,2 p)
(m)
;
(
2
) u satisfies conservation of the energy and the mass;
(
3
) T ∗ = ∞ in the defocusing subcritical case (μ = 1, 2 ≤ p < p∗).
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.
Theorem 2.3 Let 3 ≤ N ≤ 4 and p = p∗. There exists 0 > 0 such that if := (ψ1, . . . , ψm ) ∈ H satisfies
mj=1 RN (|∇ψ j |2 + |x ψ j |2) dx ≤ 0; then, the system (1.1) possesses a unique global solution u ∈ C (R, H ).
Now, we are interested on the focusing problem (1.1). For u := (u1, . . . , um ) ∈ H , define the action
1 m
S(u) := 2
j=1
u j 2
If α, β ∈ R, the following quantity is called constraint
2Kα,β (u) :=
(2α + (N − 2)β) ∇u j 2 + (2α + Nβ) u j 2 + (2α + β(N + 2)) x u j 2
a jk
RN
(2 pα + Nβ)|u j uk | p dx .
m
j=1
1
− p
m
j,k=1
Definition 2.4
:= (ψ1, . . . , ψm ) is said to be a ground state solution to (1.1) if
m
k=1
ψ j − ψ j − |x |2ψ j +
a jk |ψk | p|ψ j | p−2ψ j = 0, 0 =
∈ Hrd
and it minimizes the problem
inf {S(u) s. t Kα,β (u) = 0}.
mα,β := 0=u∈H
Remark 2.5 If
∈ H is a solution to (2.1), then eit
is a global solution of (1.1) said standing wave.
Now, the existence of a ground state solution to (1.1) is claimed. Define the set G p := {(α, β) ∈ R∗+ × R+ s.
t α( p − 1) > β}.
Theorem 2.6 Take N ≥ 2, 1 < p < p∗ and two real numbers (α, β) ∈ G p. Then
(
1
) m := mα,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].
Theorem 2.7 Take 2 ≤ N ≤ 4 and 1 < p < p∗. Let := (ψ1, .., ψm ) ∈ H and u ∈ CT ∗ (H ) the maximal
solution to (1.1). If there exist (α, β) ∈ G p and t0 ∈ [0, T ∗) such that u(t0) ∈ Aα+,β := {u ∈ H s. t S(u) <
m and Kα,β (u) ≥ 0}, then u is global.
(2.1)
(2.2)
The last result concerns stability for standing waves.
Definition 2.8 For ε > 0 and
∈ H , define
(
1
) the set
Vε( ) :=
v ∈ H, s. t inf v − eit
t∈R
H < ε ;
(
2
) if u0 ∈ Vε( ) and u is the solution to (1.1) given by Theorem 2.1,
Tε(u0) := sup{T > 0, s. t u(t ) ∈ Vε( ), for any t ∈ [0, T )};
(
3
) eit is said to be orbitally stable if, for any σ > 0 there exists ε > 0 such that if u0 ∈ Vε( ), then
Tσ ( ) = ∞. Otherwise, the standing wave eit is said to be nonlinearly unstable;
(
4
) the set
ε( ) := {v ∈ V ( ), s. t E (v) < E ( ),
v ≤
and K1,− N2 (v) < 0}.
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
(
1
) the set
(
2
) Gμ is said to be stable if, Gμ = ∅ and for any ε > 0 there exists σ > 0 such that
Gμ :=
v ∈ H, s. t S(v) = ui∈nHf{S(u),
u = μ} ;
where u ∈ C (R, H ) is a global solution to (1.1) with data u0 ∈ H .
Theorem 2.10 Take 2 ≤ N ≤ 4, 1 < p < p∗ and
(
1
) if p < p∗, so Gμ is stable for any μ > 0;
(
2
) if
be a ground state solution to (2.1). Then
m
j=1
Proposition 2.11 There exists a family of operators U := U (t, s), U (t ) := U (t, 0) such that u(t, x ) :=
U (t, s)φ (x ) is solution to the linear problem
Moreover, we have the following elementary properties:
(
1
) U (t, t ) = I d;
(
2
) (t, s) → U (t, s) is continuous;
(
3
) U (t, s)∗ = U (t, s)−1;
Thanks to Duhamel formula, it yields
Proposition 2.12 If u is a solution to the inhomogeneous Schrödinger problem
(
1
) u(t ) = −i 0t U (t − s)h(s, x )ds;
(
2
) ∇u(t ) = −i 0t U (t − s)[∇h + 2x u] ds;
(
3
) x u(t ) = −i 0t U (t − s)[x h + 2∇u] ds.
(2.4)
(2.5)
Remark 2.13 Taking the derivative of the equation satisfied by u, we obtain the second point. For the last one,
it is sufficient to multiply the same equation with x .
A standard tool to study Schrödinger problems is the so-called Strichartz type estimate.
Definition 2.14 A pair (q, r ) of positive real numbers is admissible if
2 ≤ r < ∞
and N
1 1
2 − r
2
= q .
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 ,
(
1
)
(
2
)
U (t )φ Lq (I,Lr ) ≤ Cq φ , ∀φ ∈ L2;
t
0 U (t − s)h(s, x )ds Lq (I,Lr ) ≤ Cα,|I | h Lα (I,Lβ ), ∀h ∈ Lα (I, Lβ ).
Any solution to (1.1) formally enjoys the so-called Virial identity.
Proposition 2.16 Let u := (u1, . . . , um ) ∈ H , a solution to (1.1) such that x u ∈ L2. Then,
1 ⎛
8 ⎝
m
j=1
⎞
⎠
m
j=1
x u j (t ) 2
=
( ∇u j 2
− x u j 2) −
N ( p − 1)
2 p
m
j,k=1
a j,k
RN
The following Gagliardo–Nirenberg inequality [20] will be useful.
Proposition 2.18 Take 1 < p ≤ p∗. Then, for any (u1, . . . , um ) ∈ H ,
m
j,k=1
RN
|u j uk | p dx ≤ C ⎝
⎛
m
j=1
⎞
2
∇u j ⎠
(p−1)N
2 ⎛
⎝
m
j=1
⎞
2
u j ⎠
(
1
) W s, p(RN ) → Lq (RN ) whenever 1 < p < q < ∞, s > 0 and
1 1 s
p ≤ q + N ;
Proposition 2.19
(
2
) for 2 < p < 2 p∗,
Hr1d (RN ) → → L p(RN );
(
3
) for 2 ≤ p < 2 p∗,
(
4
) if xu ∈ L2 and ∇u ∈ L2, then u ∈ L2 and
Remark 2.20 Using the previous inequality, we get u
Let us close this subsection with some absorption result.
Lemma 2.21 Let T > 0 and X ∈ C([0, T ], R+) such that
where a, b > 0, θ > 1, a < (1 − θ1 ) (θb1) θ1 and X (0) ≤
1 1 . Then
(θb) θ−1
X ≤ a + b X θ on [0, T ],
θ
X ≤ θ − 1 a on [0, T ].
m
j=1
m
k=1 0
t
m
j,k=1
(RN ) → → L p(RN );
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.
3.1 Local existence
Let us discuss two cases.
• Subcritical case: 2 ≤ p < p∗. For T > 0 and R2 := C
radius R of the space
H , we denote BT (R) the centered ball with
endowed with the complete distance
Define, for u := (u1, . . . , um ), the function
ET :=
4p
u ∈ C([0, T ], H ) s. t u, ∇u, xu ∈ L N(p−1) ([0, T ], L2p)
d(u, v) :=
u j − v j
4p
L∞T(L2)∩LTN(p−1) (L2p)
:=
m
j=1
u j − v j T .
φ(u)(t) := T (t)
− i
T (t − s) a1k |uk |p|u1|p−2u1, . . . , amk |uk |p|um |p−2um ds,
where T (t) := (U (t)ψ1, . . . , U (t)ψm ). We prove the existence of some small T , R > 0 such that φ is a
contraction of the ball BT (R). Take u, v ∈ BT (R), using Strichartz estimate, we have
d(φ(u), φ(v))
|uk |p|u j |p−2u j − |vk |p|v j |p−2v j
4p 2p .
LTp(4−N)+N (L 2p−1 )
(2.6)
To derive the contraction, consider the function
f j,k :
Cm
C, (u1, . . . , um) → |uk |p|u j |
→
p−2u j .
With the mean value theorem, via the fact that p ≥ 2, it follows that
| f j,k (u) − f j,k (v)|
max{|uk |p−1|u j |
p−1
+ |uk |p|u j |p−2, |vk |p|v j |
p−2
+ |vk |p−1|v j |
p−1
}|u − v|.
Using Hölder inequality and Sobolev embedding, compute via a symmetry argument
.
Thus, for small T > 0, we get
φ(u) T + ∇(φ(u)) T + xφ(u) T ≤ C
H +
f j,k (u)
4p 2p
L p(4−N)+N (L 2p−1 )
T
+ ∇( f j,k (u))
4p 2p
L p(4−N)+N (L 2p−1 )
T
+ x f j,k (u)
4p 2p
L p(4−N)+N (L 2p−1 )
T
R
Thanks to Hölder inequality and Sobolev embedding, we obtain
Using Hölder inequality, Sobolev embedding, compute via a symmetry argument
|uk | p−1|u j | p−1 + |uk | p|u j | p−2 |x u|
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 BT (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, r ) := ( N2−N2 , N 2−2 N2N2 +4 ) and the centered ball with
radius R > 0 of the space
endowed with the complete distance
FT :=
u, ∇u, x u ∈ (LqT (Lr ) m
d(u, v) =
u − v T ,
u T :=
u j LqT (Lr ).
m
j=1
Taking account of Strichartz estimate, Hölder inequality and Sobolev embedding, write for ρ1 := r1 − N1 ,
d(φ (u), φ (v))
|uk | p∗−1|u j | p∗−1 + |uk | p∗ |u j | p∗−2 |u − v| LqT (Lr )
m
j,k=1
m
j,k=1
u − v LqT (Lr )
u − v LqT (Lr ) u 2(qp∗−1)
LT (Lρ )
u − v LqT (Lr ) u 2(qp∗−1)
LT (W 1,r )
R2( p∗−1) u − v LqT (Lr ).
Now, using Propositions 2.12–2.15,
This implies that
Taking account of the previous equality via Strichartz estimate, it follows that for small T > 0,
R
φ (u) T ≤ 6 + C f j,k (u) LqT (Lr );
R
∇(φ (u)) T ≤ 6 + C ∇( f j,k (u)) LqT (Lr ) + T x φ (u) L∞T(L2);
R
x φ (u) T ≤ 6 +
x f j,k (u) LqT (Lr ) + T ∇(φ (u)) L∞T(L2).
Thus, for small T > 0, we get
Thanks to Hölder inequality and Sobolev embedding, we obtain
|uk | p∗−1|u j | p∗−1 + |uk | p∗ |u j | p∗−2 |x u| LqT (Lr )
x u LqT (Lr ) u 2(qp∗−1)
LT (Lρ )
R2 p∗−1.
(3.4)
(3.5)
Collecting the estimates (3.4)–(3.5), it yields
Proof It is sufficient to write, using the previous computation via Duhamel formula,
u L∞T(H )
H +
u LqT (W 1,r ) +
x u LqT (Lr )
u 2L(Tp(∗W−11,)r ).
q
m
k=1
3.2 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 ∈ CT (H ) two solutions to (1.1) and w := u − v. Then
i w˙ j +
w j − |x |2w j =
a jk |uk | p|u j | p−2u j − |vk | p|v j | p−2v j ,
w j (0, .) = 0.
LqT (Lr ) the norm of (LqT (Lr ))(m), we have
4 p
Applying Strichartz estimate with the admissible pair (q, r ) = ( N ( p−1) , 2 p) and denoting for simplicity
Taking T > 0 small enough, with a continuity argument, we may assume that
Using previous computation with
m
j,k=1
f j,k (u) − f j,k (v) LqT (Lr ).
max
j=1,...,m
u j L∞T(H 1) ≤ 1.
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 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
(Ps )
⎧
⎪⎨ i v˙ j +
⎪⎩ v j (s, .) = u j (s, .).
m
j,k=1
v j − |x |2v j =
a jk |vk | p |v j | p−2v j ;
By the same arguments used in the local existence, we can find a real number τ > 0 and a solution v =
(v1, ..., vm ) to (Ps ) on C [s, s + τ ], 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.
4 Global existence in the critical case
In this section, 3 ≤ N ≤ 4. We establish Theorem 2.3 about global existence of a solution to (1.1) in the
critical case p = p∗, for small data.
Several norms have to be considered in the analysis of the critical case. Letting I ⊂ R a time slab, define
u M(I ) :=
∇u
x u
2(N+2) 2N(N+2) +
L N−2 (I,L N2+4 )
2(N+2) 2N(N+2) ;
L N−2 (I,L N2+4 )
u S(I ) :=
u
2(N+2) 2(N+2) .
L N−2 (I,L N−2 )
Let M (R) be the completion of Cc∞(R1+N ) endowed with the norm . M(R), and M (I ) be the set consisting
of the restrictions to I of functions in M (R). An important quantity closely related to the mass and the energy
is the functional defined for u ∈ H by
m
j=1
RN
ξ(u) :=
|∇u j |2 + |x u j |2 dx .
Let us give an auxiliary result.
Proposition 4.1 Let p = p∗,
for any interval I = [0, T ], if
:= (ψ1, . . . , ψm ) ∈ H and A :=
H . There exists δ := δA > 0 such that
then there exits a unique solution u ∈ C (I, H ) of (1.1) which satisfies u ∈
Moreover,
M (I ) ∩ L 2(NN+2) (I × RN ) (m).
m ⎛
j,k=1
⎝
Besides, the solution depends continuously on the initial data in the sense that there exists δ0 depending on δ,
such that for any δ1 ∈ (0, δ0), if − ϕ H ≤ δ1 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 ),
u − v (Lq (I,Lr )∩H )(m) ≤ C δ1.
Proof The proposition follows from a contraction mapping argument. Let the function
φ (u)(t ) := T (t )
− i
T (t − s) a1k |uk | N−2 |u1| N−2 u1, . . . , amk |uk | N−2 |um | 4N−−N2 um ds.
N 4−N N
Define A :=
H and the set
⎧
Xa,b := ⎨ u ∈ (M (I ))m
s. t
m
j=1
u j M(I ) ≤ a and
u j S(I ) ≤ b
m
j=1
⎫
⎬
⎭
where a, b > 0 are sufficiently small to fix later. Using Strichartz estimate, we get
φ (u) − φ (v) M(I )
∇( f j,k (u) − f j,k (v))
+ x ( f j,k (u) − f j,k (v))
2N
L2T L N+2
Using Hölder inequality, Sobolev embedding and denoting the quantity
2N
L2T L N+2 ⎠
⎞
.
(K) :=
x ( f j,k (u) − f j,k (v))
2N ,
L2T L N+2
we compute via a symmetry argument
(K)
2 2 N 4−N
|uk | N−2 |u j | N−2 + |uk | N−2 |u j | N−2 |x (u − v)| L2T L N+2
2N
x (u − v)
x (u − v)
x (u − v)
2(N+2)
LT N−2
2(N+2)
LT N−2
2(N+2)
LT N−2
2N(N+2)
L N2+4
2N(N+2)
L N2+4
2N(N+2)
L N2+4
2 2 N 4−N
|uk | N−2 |u j | N−2 + |uk | N−2 |u j | N−2
LTN2+2 L N2+2
2 2
uk SN(−I2) u j S(I ) +
N−2
N 4−N
uk SN(−I2) u j S(I )
N−2
4
u (NS−(I2))m .
Write
Thus,
∂i f j,k (u) − f j,k (v) = ∂i u∂i ( f j,k )(u) − ∂i v∂i ( f j,k )(v)
= ∂i (u − v)∂i ( f j,k )(u) + ∂i v ∂i ( f j,k )(u) − ∂i ( f j,k )(v) .
∇ f j,k (u) − f j,k (v)
Using Hölder inequality and Sobolev embedding, it yields
(I2)
∇u|(u − v)| |uk | N−2 |u j | N−2 + |uk | N−2 |u j | N−2
4−N
2
∇(u − v) 2(N+2)
L N−2
T
2N(N+2) ⎜
L N2+4 ⎝
2
N−2
uk 2(N+2)
L N−2
T
2
N−2
u j 2(N+2)
L N−2
T
2(N+2)
L N−2
N
N−2
+ uk 2(N+2)
L N−2
T
u − v (M(I ))(m)
u − v (M(I ))(m) u (NS−(I2))m .
4
2(N+2)
L N−2
4−N
N−2
u j 2(N+2)
L N−2
T
L 2(NN−+22) ⎟⎠
2 2
N−2 u j S(I ) +
N−2
uk S(I )
N 4−N
N−2 u j S(I )
N−2
uk S(I )
2N
L2T L N+2
2(N+2)
L N−2
⎞
Then
∇u
2(N+2)
L N−2
T
2N(N+2)
L N2+4
u − v (S(I ))m ⎜
⎝
N
N−2
+ uk 2(N+2)
L N−2
T
2(N+2)
L N−2
u (M(I ))(m) u − v (S(I ))m u N−2
(S(I ))m .
6−2N
N−2
u j 2(N+2)
L N−2
T
6−N
4
N
⎛
6−2N
4−N
N−2
uk 2(N+2)
L N−2
T
⎞
L 2(NN−+22) ⎟⎠
2N
L2T L N+2
2(N+2)
L N−2
2
N−2
u j 2(N+2)
L N−2
T
2(N+2)
L N−2
φ(u) − φ(v) (M(I ))(m)
a N−2 u − v (M(I ))m + ba N−2 u − v (S(I ))m
6−N
4 6−N
(a N−2 + ba N−2 ) u − v (M(I ))m .
Moreover, taking in the previous inequality v = 0, we get for small δ > 0,
φ(u) (S(I ))(m) ≤ δ + Ca N−2 ;
φ(u) (M(I ))(m) ≤ C A + Cba N−2 .
4
4
With a classical Picard argument, for small a = 2δ, b > 0, there exists u ∈ Xa,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.
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 x u + ∇u remains small on the whole interval of existence of u. Write with
conservation of the energy and Sobolev’s inequality
N − 2
N
m
j,k=1
RN
( x u
+ ∇u )2 ≤ 2E ( ) +
N N
a jk |u j (x , t )| N−2 |uk (x , t )| N−2 dx
N
≤ C ξ( ) + ξ( ) N−2
+ C
N
∇u j 2 N−2
N
≤ C ξ( ) + ξ( ) N−2
+ C ( x u
2N
+ ∇u ) N−2 .
m
j=1
So, by Lemma 2.21, if ξ( ) is sufficiently small, then x u
+ ∇u stays small for any time.
5 The stationary problem
and the differential operator
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 := (u1, . . . , um ) ∈ H and λ, α, β ∈ R, we introduce the scaling
We extend the previous operator as follows, if A : H 1(RN ) → R, then
Denote also the constraint
Kα,β (u) := ∂λ S((uλ)α,β ) |λ=0
(uλj )α,β := eαλu j (e−βλ.)
£α,β : H 1 → H 1, u j → ∂λ((uλj )α,β )|λ=0.
£α,β A(u j ) := ∂λ( A((uλj )α,β ))|λ=0.
Finally, we introduce the quantity
1
Hα,β (u) := S(u) − 2α + β(N + 2) Kα,β (u)
1
= 2α + (N + 2)β ⎣
⎡ m
j=1
+ 2 ∇u j 2) + 1p (α( p − 1) − β)
m
j,k=1
a jk
RN
⎤
|u j uk | p dx ⎦ .
Now, we prove Theorem 2.6 about the existence of a ground state solution to the stationary problem (2.1).
Remark 5.1 (i) The proof of the Theorem 2.6 is based on several lemmas;
(ii) we write, for easy notation, uλj := (uλj)α,β, K := Kα,β, KQ := KαQ,β, £ := £α,β and H := Hα,β.
Lemma 5.2 Let (α,β) ∈ Gp. Then
(
1
) min £H(u), H(u) ≥ 0 for all u ∈ H;
(
2
) λ → H(uλ) is increasing.
Proof With a direct computation
£
£H(u) = £ 1 − 2α + (N + 2)β S(u)
−1
= 2α + (N + 2)β (£ − (2α + (N + 2)β))(£ − (2α + (N − 2)β)) S(u)
Since £ − (2α + (N − 2)β) ∇uj 2 = £ − (2α + (N + 2)β) xuj 2 = 0, we have £ − (2α + (N −
2)β) £ − (2α + Nβ) ( ∇uj 2 + xuj 2) = 0 and
≥ 21p 22αα(+p −(21+) −N)2ββ (2α(p − 1) + 2β)
m
j,k=1
ajk RN |ujuk|p dx ≥ 0.
The last point is a consequence of the equality ∂λH(uλ) = £H(uλ).
The next intermediate result is the following.
Lemma 5.3 Let (α,β) ∈ R2 satisfying 2α + (N − 2)β > 0, 2α + Nβ ≥ 0, 2α + (N + 2)β ≥ 0 and
0 = (un1,...,unm) be a bounded sequence of H such that
Thus,
⎛ m
linm ⎝ j=1
⎞
KQ(unj)⎠ = 0.
Then, there exists n0 ∈ N such that K(un1,...,unm) > 0 for all n ≥ n0.
Proof Write
Using Proposition 2.18, via the fact that p∗ < p < p∗, it yields
KQ(unj) = (2α + (N − 2)β) ∇unj 2 + (2α + Nβ) unj 2 + (2α + (N + 2)β) xunj 2 → 0,
m
j,k=1
⎛ m ⎞ ⎛ m ⎞
ajk RN |unjukn|p dx = o⎝ j=1 ∇unj 2⎠ = o⎝ j=1 KQ(unj)⎠ .
K(un1,...,unm) = 2 j=1
1 m KQ(unj) − (2pα2+p Nβ) m ajk RN |unjukn|p dx
j,k=1
1 m
2 j=1
KQ(unj) ≥ 0.
Let us read an auxiliary result.
Lemma 5.4 Let (α, β) ∈ G p. Then
inf $H (u) s. t K (u) ≤ 0%.
mα,β = 0=u∈H
Proof Denoting by a the right hand side of the previous equality, it is sufficient to prove that mα,β ≤ a. Take
u ∈ H such that K (u) < 0. Because limλ→−∞ K Q(uλ) = 0, by the previous lemma, there exists some λ < 0
such that K (uλ) > 0. With a continuity argument, there exists λ0 ≤ 0 such that K (uλ0 ) = 0, then since
λ → H (uλ) is increasing, we get
mα,β ≤ H (uλ0 ) ≤ H (u).
This closes the proof.
Proof of Theorem 2.6 Let (φn) := (φ1n, . . . , φmn ) be a minimizing sequence, namely
0 = (φn) ∈ H, K (φn) = 0 and lim H (φn) = linm S(φn) = m.
n
(5.1)
With a rearrangement argument via Lemma 5.4, we can assume that (φn) is radial decreasing and satisfies
(5.1).
• First step: (φn) is bounded in H.
First subcase α = 0. Write
α ⎝
⎛ m
j=1
−
m
j,k=1
⎞ β ⎛
a jk RN |φ njφkn|p dx⎠ = 2 ⎝ 2
( ∇φ nj 2 − xφ nj 2)
m
j=1
m
j=1
− N
×
m
j=1
Denoting λ := 2βα , it yields
m
j=1
So the following sequences are bounded
m
j,k=1
m
j=1
a jk RN |φ njφkn|p dx
( ∇φ nj 2 − xφ nj 2) − N
−
φ nj 2H1 +
1 −
+
a jk RN |φ njφkn|p dx;
m
j,k=1
a jk RN |φ njφkn|p dx;
2λ
m
j=1
and
The equality K (φn) = 0 implies that
Thus, for any real number a, the following sequence is also bounded
m
j=1
m
j=1
Assume that φ = 0. Using Hölder inequality
With lower semicontinuity of the H norm,
0 = lim inf K (φn)
n
n n p n p n p
φ j φk p ≤ φ j 2p φk 2p →
Now, by Lemma 5.3, it yields K (φn) > 0 for large n. This contradiction implies that
m
j=1
m
j=1
≥
2α + (N − 2)β
2
lim inf
n
n 2
∇φ j
+
2α + (N + 2)β
2
lim inf
n
+
2α + Nβ
2
lim inf
n
−
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
1
m = H (φ) = 2α + (N + 2)β ⎣
⎡ m
j=1
β( φ j 2
+ 2 ∇φ j 2)+ 1p (α( p − 1) − β)
m
j,k=1
a jk RN |φ j φk |p d x⎦ > 0.
⎤
• 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
0 = K (φ) = £S(φ) = S (φ), £(φ) = η K (φ), £(φ) = η£K (φ) = η£2 S(φ).
With a previous computation, for ( A) := −£2 S(φ) − (2α + (N − 2)β)(2α + (N + 2)β)S(φ), we have
=
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).
6 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 (1.1). Define the sets
Aα−,β := {u ∈ H s. t S(u) < m and Kα,β (u) < 0};
Aα+,β := {u ∈ H s. t S(u) < m and Kα,β (u) ≥ 0}.
Lemma 6.1 For (α, β) ∈ G p, the sets Aα+,β and Aα−,β are invariant under the flow of (1.1).
Proof Let ∈ Aα+,β and u ∈ CT ∗ (H ) be the maximal solution to (1.1). Assume that u(t0) ∈/ Aα+,β for some
t0 ∈ (0, T ∗). Since S(u) is conserved, we have Kα,β (u(t0)) < 0. So, with a continuity argument, there exists
a positive time t1 ∈ (0, t0) such that Kα,β (u(t1)) = 0 and S(u(t1)) < m. This contradicts the definition of m.
The proof is similar in the case of Aα−,β .
The previous stable sets are independent of the parameter (α, β).
Lemma 6.2 For (α, β) ∈ G p, the sets Aα+,β and Aα−,β are independent of (α, β).
Proof Let (α, β) and (α , β ) ∈ G p. By Theorem 2.6, the reunion Aα+,β ∪ Aα−,β is independent of (α, β). So,
it is sufficient to prove that Aα+,β is independent of (α, β).The rescaling uλ := eαλu(e−βλ.) implies that a
neighborhood of zero is in Aα+,β . If S(u) < m and Kα,β (u) = 0, then u = 0. So, Aα+,β is open. Moreover, this
rescaling with λ → −∞ gives that Aα+,β is contracted to zero and so it is connected. Now, write
Aα+,β = Aα+,β ∩ ( Aα+,β ∪ Aα−,β ) = ( Aα+,β ∩ Aα+,β ) ∪ ( Aα+,β ∩ Aα−,β ).
Since by the definition, Aα−,β is open and 0 ∈ Aα+,β ∩ Aα+,β , using a connectivity argument, we have Aα+,β =
Aα+,β .
Now, we prove Theorem 2.7. With a translation argument, we assume that t0 = 0. Thus, S( ) < m and with
Lemma 6.1, u(t) ∈ A1+,1 for any t ∈ [0, T ∗). Moreover,
m ≥
1
S − 2 + N K1,1 (u)
= H1,1(u)
1
= N + 4 ⎣
2
≥ 4 + N
⎡ m
j=1
m
j=1
1
( u j 2 + 2 ∇u j 2) + p ( p − 2)
m
j,k=1
a jk RN |u j uk |p dx⎦
⎤
Then, since the L2 norm is conserved, we have
Moreover, using the energy identity and Proposition 2.18, it yields
m 1 m
j=1 RN |∇u j |2 + |xu j |2 dx = 2E( ) + p
j,k=1
m
⎞
⎝
j=1
u j 2⎠
⎞
.
Finally, T ∗ = ∞ because
7 Orbital stability
m
I := K1,0. First, let us do some computations.
N
Proposition 7.1 For v ∈ H, λ ∈ R, vλ := λ 2 v(λ.) and
This section is devoted to prove Theorem 2.10 about stability of standing waves. Denote K := K1,− N2 and
m
j=1
m
j=1
m
j=1
∂λ E(vλ) =
λ ∇v j 2 − λ−3 xv j 2 −
N
2
K (v) = ∂λ E(vλ)|λ=1;
∂λ2 E(vλ)|λ=1 =
∇v j 2 + 3 xv j 2 −
N ( p − 1)(N ( p − 1) − 1)
2 p
∂λ2 E( λ)|λ=1 =
So, for some ε > 0 and large n, thanks to Gagliardo–Nirenberg inequality (2.4),
u = μ}, we establish the first point of
1 1
m + ε ≥ 2 vn 2H − 2 p
⎛
m
j,k=1
a jk
⎛ m
j=1
|vnj vkn|p dx
n 2
∇v j ⎠
1
≥ 2 vn (2H1)m ⎜⎝ 1 − C ⎝
⎝
j=1
vnj 2⎠
⎞
N−p(2N−2) −1⎞
⎟
⎠
Since p < p∗, it follows that vn is bounded in H . Thanks to the compact Sobolev injections (2.6), this implies
that there exists v ∈ H such that
Now, with the lower semicontinuity of the H norm, it follows that
So v ∈ Gμ. This achieves the proof.
vn −→ v in Lq for any q ∈ [2, 2 p∗);
vn
Now, we prove that Gμ is stable. The proof proceeds by contradiction. Suppose that there exists a sequence
n
u0 ∈ H such that, when n goes to infinity
n
inf u0 −
∈Gμ
1
n
H <
and
for some sequence of positive real numbers (tn) and ε0 > 0, where un ∈ C(R, H ) is the global solution to
(1.1) with data u0n. Let us denote n := un(tn). Taking account of the definition of Gμ, there exists vn ∈ H
such that for a subsequence
Then, arguing as in (7.1), it follows that (vn) is bounded in H . Thus,
So, taking account of the compact Sobolev injection (2.6), there exists v ∈ H such that
This implies, via the lower semicontinuity of the H norm, that
max(sup vn H , sup u0n H )
n n
vn −→ v in Lq for any q ∈ [2, 2 p∗);
vn
So, we have the strong convergence vn −→ v in H . Thus, u0n −→ v in H . Then
Using the conservation laws, it follows that
Arguing as previously, there exists
∈ Gμ such that for a subsequence
This contradicts (7.2) and finishes the proof.
u0n → μ and S(u0n) → m.
n → μ and S( n) → m.
n −→
in H.
Thus, with Taylor expansion
By the previous lemma,
It follows that
The proof is finished.
7.2 Unstable ground state
So, ∂λ I (vλ)|λ=1,v=
proof.
The next auxiliary result reads.
The proof of the second part of Theorem 2.10 is based on several lemmas.
Lemma 7.2 Assume that ∂λ2 E( λ)|λ=1 < 0. Then, there exist two real numbers ε0 > 0, σ0 > 0 and a mapping
λ : Vε0 ( ) → (1 − σ0, 1 + σ0) such that I (vλ) = 0 for any v ∈ Vε0 ( ).
Therefore, &S ( )∂λ( λ)|λ=1, ∂λ( λ)|λ=1' ≥ 0 because
= 0. A direct application of implicit theorem concludes the
Lemma 7.3 Assume that ∂λ2 E( λ)|λ=1 < 0. Then, there exist two real numbers ε1 > 0, σ1 > 0 such that for
any v ∈ Vε1 ( ) satisfying v ≤ , we have
E( ) < E(v) + (λ − 1)K (v), for some λ ∈ (1 − σ1, 1 + σ1).
Proof With a continuity argument, there exist ε1 > 0 and σ1 > 0 such that
∂λ2 E(vλ) < 0, ∀(λ, v) ∈ (1 − σ1, 1 + σ1) × Vε1 ( ).
E(vλ) < E(v) + (λ − 1)K (v), ∀(λ, v) ∈ (1 − σ1, 1 + σ1) × Vε1 ( ).
∀v ∈ Vε1 ( ), ∃λ ∈ (1 − σ1, 1 + σ1) s. t I (vλ) = 0.
Thus, ∀v ∈ Vε1 ( ) there exists λ ∈ (1 − σ1, 1 + σ1) such that
S(vλ) ≥ S( ).
E(vλ) = S(vλ) − M(vλ)
≥ S( ) − M(vλ)
≥ S( ) − M( ) = E( ).
Lemma 7.4 Assume that ∂λ2 E( λ)|λ=1 < 0. Then, for u0 ∈ ε1 there exists a real number σ0 > 0 such that
the solution u to (1.1) given by Theorem 2.1 satisfies
K (u(t)) < −σ0, for all t ∈ [0, T (u0)).
Proof Let u0 ∈ ε1 , then E(u0) < E( ), u0 ≤ and K (u0) < 0. Put σ2 := E( ) − E(u0) > 0. With
the previous lemma, there exists λ ∈ (1 − σ1, 1 + σ1) such that
(λ − 1)K (u(t)) + E(u(t)) > E( ), ∀t ∈ [0, T (u0)).
By conservation of the energy, there exists λ ∈ (1 − σ1, 1 + σ1) such that
(λ − 1)K (u(t)) > σ2, ∀ t ∈ [0, T (u0)).
So, by a continuity argument via K (u0) < 0, we have K (u(t)) < 0 for all t ∈ [0, T (u0)). Then, λ − 1 < 0
and −σ0 := − 1σ−2λ < 0 for any t ∈ [0, T (u0)). The proof is closed.
Now, we are ready to prove the crucial result of this subsection. By Proposition 7.1, it follows that
∂2 E( λ)|λ=1 < 0 and ∂λ E( λ) = 2Nλ K ( λ). Then, 1 is a maximum for λ → E( λ) and λ1 (K ( λ)) <
λ
K ( ) = 0 as λ > 1 near to one (we denote λ = 1+). Thus, λ ∈ ε for ε = ε(λ) > 0 and λ = 1+. Take
u0 = λ, for λ = 1+, then
By the previous lemma, there exists σ0 > 0 such that
Now, if eit is orbitally stable, T (u0) = ∞ (for some positive real number ε) and K (u) < −σ0 on R+. With
virial identity, mj=1 xu j (t) 2 becomes negative for long time. This absurdity finishes the proof.
8 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,
Thus, for a(x) := |x|2, we get
m
j=1
V (t) :=
xu j (t) 2.
∂t (|u j |2) = −2 (u¯ j u j ).
V (t) = −2 j=1 RN |x|2 (u¯ j u j ) dx
RN
RN
(x.∇u j )u¯ j dx
(∂ka∂k u j )u¯ j dx.
Compute, for g the nonlinearity in (1.1),
∂t (∂k u j u¯ j ) = (∂k u˙ j u¯ j ) + (∂k u j u¯˙ j )
= (i u˙ j ∂k u¯ j ) − (i ∂k u˙ j u¯ j )
Recall the identity
Then,
(u¯ j ∂k (− u j + |x|2u j − f j,k (u)))
= (u¯ j ∂k u j − ∂k u¯ j u j ) − (u¯ j ∂k (|x|2u j ) − ∂k u¯ j |x|2u j ) + (u¯ j ∂k f j,k (u) − ∂k u¯ j f j,k (u)).
1
2 ∂k (|u j |2) − 2∂l (∂k u j ∂l u¯ j ) =
(u¯ j ∂k u j − ∂k u¯ j u j ).
RN ∂ka (u¯ j ∂k u j − ∂k u¯ j u j ) dx =
RN ∂ka
1
2 ∂k (|u j |2) − 2∂l (∂k u j ∂l u¯ j ) dx
= 2 RN ∂l ∂ka (∂k u j ∂l u¯ j ) dx
∂k a (u¯ j ∂k f j,k (u) − ∂k u¯ j f j,k (u)) dx =
∂k a (∂k [u¯ j f j,k (u)] − 2∂k u¯ j f j,k (u)) dx
au¯ j f j,k (u) − 2 (∂k a∂k u¯ j f j,k (u)) dx
a j k
RN
|uk u j | p dx − 2
RN
∂k a (∂k u¯ j f j,k (u)) dx .
m
j =1
m
j =1
RN
l=1
= p
(∂k u¯ j f j,k (u)) =
(∂k u¯ j f j,k (u)) =
a jl (∂k u¯ j |ul | p|u j | p−2u j )
p p
a jl ∂k (|u j | )|ul | .
1
p
1
m
j,l=1
m
j,l=1
Moreover,
On the other hand
RN
Write
Then
Finally
1
2
V (t ) = 4
( ∇u j
2
− x u j
2
) − 2 N
1 −
1
p
m
j,l=1
a jl
RN
|u j ul | p dx .
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