Strong instability of standing waves for a system NLS with quadratic interaction

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Strong instability of standing waves for a system NLS with quadratic interaction

van Duong Dinh

To cite this version:

van Duong Dinh. Strong instability of standing waves for a system NLS with quadratic interaction.

2018. �hal-01936100�

STRONG INSTABILITY OF STANDING WAVES FOR A SYSTEM NLS WITH QUADRATIC INTERACTION

VAN DUONG DINH

Abstract. We study the strong instability of standing waves for a system of nonlinear Schr¨odinger equations with quadratic interaction under the mass resonance condition in dimensiond= 5.

1. Introduction We consider the system NLS equations

( i∂

u +

∆u = λvu,

i∂

v +

∆v = µu

, (1.1)

where u, v : R × R

→ C , m and M are positive constants, ∆ is the Laplacian in R

and λ, µ are complex constants.

The system (1.1) is regarded as a non-relativistic limit of the system of nonlinear Klein- Gordon equations

(

2c²m

∂

u −

∆u +

u = −λvu,

1

2c²M

∂

v −

∆v +

v = −µu

, under the mass resonance condition

M = 2m. (1.2)

Indeed, the modulated wave functions (u

, v

) := (e

u, e

v) satisfy (

2c²m

∂

u

− i∂

u

−

∆u

= −e

λv

u

,

1

2c²M

∂

v

− i∂

v

−

∆v

= −e

µu

. (1.3) We see that the phase oscillations on the right hand sides vanish if and only if (1.2) holds, and the system (1.3) formally yields (1.1) as the speed of light c tends to infinity. The system (1.1) also appears in the interaction process for waves propagation in quadratic media (see e.g. [3]).

The system (1.1) has attracted a lot of interest in past several years. The scattering theory and the asymptotic behavior of solutions have been studied in [10–12, 14]. The Cauchy problem for (1.1) in L

, H

and in the weighted L

space hxi

L

= F(H

) under mass resonance condition have been studied in [13]. The space-time analytic smoothing effect has been studied in [7–9]. The sharp threshold for scattering and blow-up for (1.1) under the mass resonance condition in dimension d = 5 has been studied in [6]. The existence, stability of standing waves and the characterization of finite time blow-up solutions with minimal mass have been studied recently in [4].

2010Mathematics Subject Classification. 35Q44; 35Q55.

Key words and phrases. System NLS quadratic interaction, ground states, instability, blow-up.

1

Let us recall the local well-posedness in H

for (1.1) due to [13]. To ensure the conservation law of total charge, it is natural to consider the following condition:

∃ c ∈ R \{0} : λ = cµ. (1.4)

Proposition 1.1 (LWP in H

[13]). Let d ≤ 6 and let λ and µ satisfy (1.4). Then for any (u

, v

) ∈ H

× H

, there exists a unique paire of local solutions (u, v) ∈ Y (I) × Y (I) of (1.1) with initial data (u(0), v(0)) = (u

, v

), where

Y (I) = (C ∩ L

)(I, H

) ∩ L

(I, W

) for d = 1, Y (I ) = (C ∩ L

)(I, H

) ∩ L

(I, W

) for d = 2, where 0 <

0

= 1 −

0

< 1 with r

sufficiently large,

Y (I ) = (C ∩ L

)(I, H

) ∩ L

(I, W

) for d ≥ 3.

Moreover, the solution satisfies the conservation of mass and energy: for all t ∈ I , M (u(t), v(t)) := ku(t)k

+ ckv(t)k

= M (u

, v

),

E(u(t), v(t)) := 1

2m k∇u(t)k

+ c

4M k∇v(t)k

+ Re(λhv(t), u

(t)i) = E(u

, v

), where h·, ·i is the scalar product in L

.

We now assume that λ and µ satisfy (1.4) with c > 0 and λ 6= 0, µ 6= 0. By change of variables

u(t, x) 7→

r c

2 |µ|u t, r 1

2m x

!

, v(t, x) 7→ − λ 2 v t,

r 1 2m x

! , the system (1.1) becomes

( i∂

u + ∆u = −2vu,

i∂

v + κ∆v = −u

, (1.5)

where κ =

is the mass ratio. Note that the mass and the energy now become M (u(t), v(t)) = ku(t)k

+ 2kv(t)k

,

E(u(t), v(t)) = 1

2 (k∇u(t)k

+ κk∇v(t)k

) − Re(hv(t), u

(t)i).

The local well-posedness in H

for (1.5) reads as follows.

Proposition 1.2 (LWP in H

). Let d ≤ 6. Then for any (u

, v

) ∈ H

× H

, there exists a unique pair of local solutions (u, v) ∈ Y (I) × Y (I ) of (1.5) with initial data (u(0), v(0)) = (u

, v

). Moreover, the solution satisfies the conservation of mass and energy: for all t ∈ I ,

M(u(t), v(t)) := ku(t)k

+ 2kv(t)k

= M (u

, v

), E(u(t), v(t)) := 1

2 (k∇u(t)k

+ κk∇v(t)k

) − Re(hv(t), u

(t)i) = E(u

, v

).

The main purpose of this paper is to study the strong instability of standing waves for the system (1.5) under the mass resonance condition κ =

in dimension d = 5. Let d = 5 and consider

( i∂

u + ∆u = −2vu,

i∂

v +

∆v = −u

, (1.6)

We call a standing wave a solution to (1.6) of the form (e

φ

, e

ψ

), where ω ∈ R is a frequency and (φ

, ψ

) ∈ H

× H

is a non-trivial solution to the elliptic system

( −∆φ

+ ωφ

= 2ψ

φ

,

−

∆ψ

+ 2ωψ

= φ

. (1.7)

We are interested in showing the strong instability of ground state standing waves for (1.6).

Let us first introduce the notion of ground states related to (1.6). Denote S

(u, v) := E(u, v) + ω

2 M(u, v) = 1

2 K(u, v) + ω

2 M (u, v) − P(u, v), where

K(u, v) = k∇uk

+ 1

2 k∇vk

, M(u, v) = kuk

+ 2kvk

, P(u, v) = Re ˆ

vu

dx.

We also denote the set of non-trivial solutions of (1.7) by

A

:= {(u, v) ∈ H

× H

\{(0, 0)} : S

(u, v) = 0}.

Definition 1.3. A pair of functions (φ, ψ) ∈ H

× H

is called ground state for (1.7) if it is a minimizer of S

over the set A

. The set of ground states is denoted by G

. In particular,

G

= {(φ, ψ) ∈ A

: S

(φ, ψ) ≤ S

(u, v), ∀(u, v) ∈ A

}.

We have the following result on the existence of ground states for (1.7).

Proposition 1.4. Let d = 5, κ =

and ω > 0. Then the set G

is not empty, and it is characterized by

G

= {(u, v) ∈ H

× H

\{(0, 0)} : S

(u, v) = d(ω), K

(u, v) = 0}, where

K

(u, v) = ∂

S

(γu, γv)|

= K(u, v) + ωM(u, v) − 3P (u, v) is the Nehari functional and

d(ω) := inf {S

(u, v) : (u, v) ∈ H

× H

\{(0, 0)}, K

(u, v) = 0}. (1.8) The existence of real-valued ground states for (1.7) was proved in [13] (actually for d ≤ 5 and κ > 0). Here we proved the existence of ground states (not necessary real-valued) and proved its characterization. This characterization plays an important role in the study of strong instability of ground states standing waves for (1.6). We only state and prove Proposition 1.4 for d = 5 and κ =

. However, it is still available for d ≤ 5 and κ > 0.

We also recall the definition of the strong instability of standing waves.

Definition 1.5. We say that the standing wave (e

φ

, e

ψ

) is strongly unstable if for any ε > 0, there exists (u

, v

) ∈ H

× H

such that k(u

, v

) − (φ

, ψ

)k

< ε and the corresponding solution (u(t), v(t)) to (1.6) with initial data (u(0), v(0)) = (u

, v

) blows up in

finite time.

Our main result of this paper is the following.

Theorem 1.6. Let d = 5, κ =

, ω > 0 and (φ

, ψ

) ∈ G

. Then the ground state standing

waves (e

φ

, e

ψ

) for (1.6) is strongly unstable.

To our knowledge, this paper is the first one addresses the strong instability of standing waves for a system of nonlinear Schr¨ odinger equations with quadratic interaction. In [2], Colin-Colin-Ohta proved the instability of standing waves for a system of nonlinear Schr¨ odinger equations with three waves interaction. However, they only studied the orbital instability not strong instability by blow-up, and they only consider a special standing wave solution (0, 0, e

ϕ), where ϕ is the unique positive radial solution to the elliptic equation

−∆ϕ + 2ωϕ − |ϕ|

ϕ = 0.

This paper is organized as follows. In Section 2, we show the existence of ground states and its characterization given in Proposition 1.4. In Section 3, we give the proof of the strong instability of standing waves given in Theorem 1.6.

2. Exitence of ground states

We first show the existence of ground states given in Proposition 1.4. To do so, we need the following profile decomposition.

Proposition 2.1 (Profile decomposition). Let d = 5 and κ =

. Le (u

, v

)

be a bounded sequence in H

× H

. Then there exist a subsequence, still denoted by (u

, v

)

, a family (x

)

of sequences in R

and a sequence (U

, V

)

of H

× H

-functions such that

(1) for every j 6= k,

|x

− x

| → ∞ as n → ∞; (2.1) (2) for every l ≥ 1 and every x ∈ R

,

u

(x) =

l

X

j=1

U

(x − x

) + u

(x), v

(x) =

l

X

j=1

V

(x − x

) + v

(x), with

lim sup

n→∞

k(u

, v

)k

→ 0 as l → ∞, (2.2) for every q ∈ (2, 10/3).

Moreover, for every l ≥ 1,

M(u

, v

) =

l

X

j=1

M(U

, V

) + M(u

, v

) + o

(1), (2.3)

K(u

, v

) =

l

X

j=1

K(U

, V

) + K(u

, v

) + o

(1), (2.4)

P (u

, v

) =

l

X

j=1

P(U

, V

) + P(u

, v

) + o

(1), (2.5) where o

(1) → 0 as n → ∞.

We refer the reader to [4, Proposition 3.5] for the proof of this profile decomposition.

The proof of Proposition 1.4 is done by several lemmas. To simplify the notation, we denote for ω > 0,

H

(u, v) := K(u, v) + ωM(u, v).

It is easy to see that for ω > 0 fixed,

H

(u, v) ∼ k(u, v)k

. (2.6)

Note also that

S

(u, v) = 1

2 K

(u, v) + 1

2 P(u, v) = 1

3 K

(u, v) + 1

6 H

(u, v).

Lemma 2.2. d(ω) > 0.

Proof. Let (u, v) ∈ H

× H

\{(0, 0)} be such that K

(u, v) = 0 or H(u, v) = 3P (u, v). We have from Sobolev embedding that

P(u, v) ≤ ˆ

|v||u|

dx . kvk

kuk

. k∇vk

k∇uk

. [H

(u, v)]

. [P(u, v)]

. This implies that there exists C > 0 such that

P (u, v) ≥ r 1

C > 0.

Thus

S

(u, v) = 1

2 K(u, v) + 1

2 P (u, v) ≥ 1 2

r 1 C > 0.

Taking the infimum over all (u, v) ∈ H

× H

\{(0, 0)} satisfying K

(u, v) = 0, we get the

result.

We now denote the set of all minimizers for d(ω) by M

:=

(u, v) ∈ H

× H

\{(0, 0)} : K

(u, v) = 0, S

(u, v) = d(ω) . Lemma 2.3. The set M

is not empty.

Proof. Let (u

, v

)

be a minimizing sequence for d(ω), i.e. (u

, v

) ∈ H

× H

\{(0, 0)}, K

(u

, v

) = 0 for any n ≥ 1 and lim

S

(u

, v

) = d(ω). Since K

(u

, v

) = 0, we have that H

(u

, v

) = 3P (u

, v

) for any n ≥ 1. We also have that

S

(u

, v

) = 1

3 K

(u

, v

) + 1

6 H

(u

, v

) → d(ω) as n → ∞.

This yields that there exists C > 0 such that

H

(u

, v

) ≤ 6d(ω) + C

for all n ≥ 1. By (2.6), (u

, v

)

is a bounded sequence in H

× H

. We apply the profile decomposition given in Proposition 2.1 to get up to a subsequence,

u

(x) =

l

X

j=1

U

(x − x

) + u

(x), v

(x) =

l

X

j=1

V

(x − x

) + v

(x)

for some family of sequences (x

)

in R

and (U

, V

)

a sequence of H

× H

-functions satisfying (2.2) – (2.5). We see that

H

(u

, v

) =

l

X

j=1

H

(U

, V

) + H

(u

, v

) + o

(1).

This implies that

K

(u

, v

) = H

(u

, v

) − 3P(u

, v

)

=

l

X

j=1

H

(U

, V

) + H

(u

, v

) − 3P (u

, v

) + o

(1)

=

l

X

j=1

K

(U

, V

) + 3

l

X

j=1

P (U

, V

) − 3P(u

, v

) + H

(u

, v

) + o

(1).

Since K

(u

, v

) = 0 for any n ≥ 1, P (u

, v

) → 2d(ω) as n → ∞ and H

(u

, v

) ≥ 0 for any n ≥ 1, we infer that

l

X

j=1

K

(U

, V

) + 3

l

X

j=1

P (U

, V

) − 6d(ω) ≤ 0 or

l

X

j=1

H

(U

, V

) − 6d(ω) ≤ 0.

By H¨ older’s inequality and (2.2), it is easy to see that lim sup

P (u

, v

) = 0 as l → ∞.

Thanks to (2.5), we have that

2d(ω) = lim

n→∞

P (u

, v

) =

∞

X

j=1

P (U

, V

).

We thus obtain

∞

X

j=1

K

(U

, V

) ≤ 0 and

∞

X

j=1

H

(U

, V

) ≤ 6d(ω). (2.7) We now claim that K

(U

, V

) = 0 for all j ≥ 1. Indeed, suppose that if there exists j

≥ 1 such that K

(U

, V

) < 0, then we see that the equation K

(γU

, γV

) = γ

H

(U

, V

)−

3γ

P (U

, V

) = 0 admits a unique non-zero solution γ

:= H

(U

, V

)

3P (U

, V

) ∈ (0, 1).

By the definition of d(ω), we have d(ω) ≤ S

(γ

U

, γ

V

) = 1

6 H

(γ

U

, γ

V

) = γ

6 H(U

, V

) < 1

6 H

(U

, V

) which contradicts to the second inequality in (2.7). We next claim that there exists only one j such that (U

, V

) is non-zero. Indeed, if there are (U

, V

) and (U

, V

) non-zero, then by (2.7), both H

(U

, V

) and H

(U

, V

) are strictly smaller than 6d(ω). Moreover, since

K

(U

, V

) = 0,

d(ω) ≤ S

(U

, V

) = 1

6 H

(U

, V

) < d(ω)

which is absurd. Therefore, without loss of generality we may assume that the only one non-zero profile is (U

, V

). We will show that (U

, V

) ∈ M

. Indeed, we have P (U

, V

) = 2d(ω) > 0 which implies (U

, V

) 6= (0, 0). We also have

K

(U

, V

) = 0 and S

(U

, V

) = 1

2 P (U

, V

) = d(ω).

This shows that (U

, V

) is a minimizer for d(ω). The proof is complete.

Lemma 2.4. M

⊂ G

.

Proof. Let (φ, ψ) ∈ M

. Since K

(φ, ψ) = 0, we have H

(φ, ψ) = 3P (φ, ψ). On the other hand, since (φ, ψ) is a minimizer for d(ω), there exists a Lagrange multiplier γ ∈ R such that

S

(φ, ψ) = γK

(φ, ψ).

This implies that

0 = K

(φ, ψ) = hS

(φ, ψ), (φ, ψ)i = γhK

(φ, ψ), (φ, ψ)i.

A direct computation shows that

hK

(φ, ψ), (φ, ψ)i = 2K(φ, ψ)+2ωM (φ, ψ)−9P (φ, ψ) = 2H

(φ, ψ)−9P (φ, ψ) = −3P(φ, ψ) < 0.

Therefore, γ = 0 and S

(φ, ψ) = 0 or (φ, ψ) ∈ A

. It remains to show that S

(φ, ψ) ≤ S

(u, v) for all (u, v) ∈ A

. Let (u, v) ∈ A

. We have K

(u, v) = hS

(u, v), (u, v)i = 0. By the definition of d(ω), we get S

(φ, ψ) ≤ S

(u, v). The proof is complete.

Lemma 2.5. G

⊂ M

.

Proof. Let (φ

, ψ

) ∈ G

. Since M

is not empty, we take (φ, ψ) ∈ M

. We have from Lemma 2.4 that (φ, ψ) ∈ G

. Thus, S

(φ

, ψ

) = S

(φ, ψ) = d(ω). It remains to show that K

(φ

, ψ

) = 0. Since (φ

, ψ

) ∈ A

, S

(φ

, ψ

) = 0. This implies that

K

(φ

, ψ

) = hS

(φ

, ψ

), (φ

, ψ

)i = 0.

The proof is complete.

Proof of Proposition 1.4. The proof of Proposition 1.4 follows immediately from Lemmas 2.3,

2.4 and 2.5. 2

3. Strong instability of standing waves

We are now able to study the strong instability of standing waves for (1.6). Note that the local well-posedness in H

× H

for (1.6) in 5D is given in Proposition 1.2. Let us start with the following so-called Pohozaev’s identities.

Lemma 3.1. Let d = 5, κ =

and ω > 0. Let (φ

, ψ

) ∈ H

× H

be a solution to (1.7).

Then the following identities hold

2K(φ

, ψ

) = 5P (φ

, ψ

), 2ωM (φ

, ψ

) = P (φ

, ψ

).

Proof. We only make a formal calculation. The rigorous proof follows from a standard approximation argument. Multiplying both sides of the first equation in (1.7) with φ

, integrating over R

and taking the real part, we have

k∇φ

k

+ ωkφ

k

= 2Re ˆ

ψ

φ

dx.

Multiplying both sides of the second equation in (1.7) with ψ

, integrating over R

and taking the real part, we get

1

2 k∇ψ

k

+ 2ωkψ

k

= Re ˆ

ψ

φ

dx.

We thus obtain

K(φ

, ψ

) + 2ωM (φ

, ψ

) = 3P (φ

, ψ

). (3.1)

Multiplying both sides of the first equation in (1.7) with x · ∇φ

, integrating over R

and taking the real part, we see that

−Re ˆ

∆φ

x · ∇φ

dx + ωRe ˆ

φ

x · ∇φ

dx = 2Re ˆ

ψ

φ

x · ∇φ

dx.

A direct computation shows that Re

ˆ

∆φ

x · ∇φ

dx = 3

2 k∇φ

k

, Re

ˆ

φ

x · ∇φ

dx = − 5

2 kφ

k

, Re

ˆ

ψ

φ

x · ∇φ

dx = − 5 2 Re

ˆ

ψ

(φ

)

dx − 1 2 Re

ˆ

φ

x · ∇ψ

dx.

It follows that

− 3

2 k∇φ

k

− 5

2 ωkφ

k

= −5Re ˆ

ψ

φ

dx − Re ˆ

φ

x · ∇ψ

dx.

Similarly, multiplying both sides of the second equation in (1.7) with x · ∇ψ

, integrating over R

and taking the real part, we have

− 3

4 k∇ψ

k

− 5ωkψ

k

= Re ˆ

φ

x · ∇ψ

dx.

We thus get

3

2 K(φ

, ψ

) + 5

2 ωM (φ

, ψ

) = 5P (φ

, ψ

). (3.2)

Combining (3.1) and (3.2), we prove the result.

We also have the following exponential decay of solutions to (1.7).

Lemma 3.2. Let d = 5, κ =

and ω > 0. Let (φ

, ψ

) ∈ H

× H

be a solution to (1.7).

Then the following properties hold

• (φ

, ψ

) ∈ W

× W

for every 2 ≤ p < ∞. In particular, (φ

, ψ

) ∈ C

× C

and

|D

φ

(x)| + |D

ψ

(x)| → 0 as |x| → ∞ for all |β| ≤ 2;

• ˆ

e

(|∇φ

|

+ |φ

|

)dx < ∞, ˆ

e

(|∇ψ

|

+ 4|ψ

|

)dx < ∞.

In particular, (|x|φ

, |x|ψ

) ∈ L

× L

.

Proof. The follows the argument of [1, Theorem 8.1.1]. Let us prove the first item. We note that if (φ

, ψ

) ∈ L

× L

for some 2 ≤ p < ∞, then ψ

φ

, φ

∈ L

. It follows that (φ

, ψ

) ∈ W

× W

. By Sobolev embedding, we see that

(φ

, ψ

) ∈ L

× L

for some q ≥ p

2 satisfying 1 q ≥ 2

p − 2

5 . (3.3)

We claim that (φ

, ψ

) ∈ L

× L

for any 2 ≤ p < ∞. Since (φ

, ψ

) ∈ H

× H

, the Sobolev embedding implies that (φ

, ψ

) ∈ L

× L

for any 2 ≤ p <

. It remains to show the claim for any p sufficiently large. To see it, we define the sequence

1 q

= 2

− 1

15 + 2 5 × 2

.

We have

1 q

− 1 q

= − 1

15 × 2

< 0.

This implies that

n

is decreasing and

n

→ −∞ as n → ∞. Since q

= 3 (we take (φ

, ψ

) ∈ L

× L

to prove our claim), it follows that there exists k ≥ 0 such that

1 q

> 0 for 0 ≤ n ≤ k and 1 q

≤ 0.

We will show that (φ

, ψ

) ∈ L

× L

. If (φ

, ψ

) ∈ L

× L

for some 0 ≤ n

≤ k − 1, then by (3.3), (φ

, ψ

) ∈ L

× L

for some q ≥

satisfying

≥

n0

−

. By the choice of q

, it is easy to check that

n0

−

=

n0+1

. In particular, (φ

, ψ

) ∈ L

× L

. By induction, we prove (φ

, ψ

) ∈ L

× L

. Applying again (3.3), we have

(φ

, ψ

) ∈ L

× L

for all q ≥ q

2 such that 1 q ≥ 1

q

.

This shows that (φ

, ψ

) belongs to L

× L

for any p sufficiently large. The claim follows.

Using the claim, we have in particular ψ

φ

, φ

∈ L

for any 2 ≤ p < ∞. Hence (φ

, ψ

) ∈ W

× W

for any 2 ≤ p < ∞. By H¨ older’s inequality, we see that ∂

(ψ

φ

), ∂

(φ

) ∈ L

for any 2 ≤ p < ∞ and any 1 ≤ j ≤ 5. Thus (∂

φ

, ∂

ψ

) ∈ W

× W

for any 2 ≤ p < ∞ and any 1 ≤ j ≤ 5, or (φ

, ψ

) ∈ W

× W

for any 2 ≤ p < ∞. By Sobolev embedding, (φ

, ψ

) ∈ C

× C

for all 0 < δ < 1. In particular, |D

φ

(x)| + |D

ψ

(x)| → 0 as |x| → ∞

for all |β| ≤ 2.

To see the second item. Let ε > 0 and set χ

(x) := e

|x|

1+ε|x|

. For each ε > 0, the function χ

is bounded, Lipschitz continuous and satisfies |∇χ

| ≤ χ

a.e. Multiplying both sides of the first equation in (1.7) by χ

φ

, integrating over R

and taking the real part, we have

Re ˆ

∇φ

· ∇(χ

φ

)dx + ˆ

χ

|φ

|

dx = 2Re ˆ

χ

ψ

φ

dx.

Since ∇(χ

φ

) = χ

∇φ

+ ∇χ

φ

, the Cauchy-Schwarz inequality implies that Re

ˆ

∇φ

· ∇(χ

φ

)dx = ˆ

χ

|∇φ

|

dx + Re ˆ

∇χ

∇φ

φ

dx

≥ ˆ

χ

|∇φ

|

dx − ˆ

|∇χ

||∇φ

||φ

|dx

≥ ˆ

χ

|∇φ

|

dx − 1 2

ˆ

χ

(|∇φ

|

+ |φ

|

)dx.

We thus get

ˆ

χ

(|∇φ

|

+ |φ

|

)dx ≤ 4Re ˆ

χ

ψ

φ

dx. (3.4) Similarly, multiplying both sides of the second equation in (1.7) with χ

ψ

, integrating over R

and taking the real part, we get

ˆ

χ

(|∇ψ

|

+ 4|ψ

|

)dx ≤ 8 3 Re

ˆ

χ

ψ

φ

dx. (3.5)

By the first item, there exists R > 0 large enough such that |v(x)| ≤

for |x| ≥ R. We have that

4Re ˆ

χ

ψ

φ

dx ≤ 4 ˆ

χ

|ψ

||φ

|

dx

= 4 ˆ

|x|≤R

χ

|ψ

||φ

|

dx + ˆ

|x|≥R

χ

|ψ

||φ

|

dx

≤ 4 ˆ

|x|≤R

e

|ψ

||φ

|

dx + 1 2

ˆ

χ

|φ

|

dx.

We thus get from (3.4) that ˆ

χ

(|∇φ

|

+ |φ

|

)dx ≤ 8 ˆ

|x|≤R

e

|ψ

||φ

|

dx. (3.6) Letting ε → 0, we obtain

ˆ

e

(|∇φ

|

+ |φ

|

)dx ≤ 8 ˆ

|x|≤R

e

|ψ

||φ

|

dx < ∞.

Similarly, by (3.5) and (3.6), ˆ

χ

(|∇ψ

|

+ 4|ψ

|

)dx ≤ 2 3 4

ˆ

|x|≤R

e

|ψ

||φ

|

dx + 1 2

ˆ

χ

|φ

|

dx

!

≤ 16 3

ˆ

|x|≤R

e

|ψ

||φ

|

dx.

Letting ε → 0, we get ˆ

e

(|∇ψ

|

+ 4|ψ

|

)dx ≤ 16 3

ˆ

|x|≤R

e

|ψ

||φ

|

dx < ∞.

The proof is complete.

We also need the following virial identity related to (1.6).

Lemma 3.3. Let d = 5 and κ =

. Let (u

, v

) ∈ H

×H

be such that (|x|u

, |x|v

) ∈ L

×L

. Then the corresponding solution to (1.6) with initial data (u(0), v(0)) = (u

, v

) satisfies

d

dt

(kxu(t)k

+ 2kxv(t)k

) = 8

k∇u(t)k

+ 1

2 k∇v(t)k

− 20Re ˆ

v(t)u

(t)dx.

Proof. The above identity follows immediately from [4, Lemma 3.1] with χ(x) = |x|

. Now let us denote for (u, v) ∈ H

× H

\{(0, 0)},

Q(u, v) := K(u, v) − 5

2 P (u, v).

It is obvious that

d

dt

(kxu(t)k

+ 2kxv(t)k

) = 8Q(u(t), v(t)). (3.7) Note that if we take

u

(x) = γ

u(γx), v

(x) = γ

v(γx), (3.8)

then

S

(u

, v

) = 1

2 K(u

, v

) + ω

2 M (u

, v

) − P (u

, v

)

= γ

2 K(u, v) + ω

2 M (u, v) − γ

P (u, v).

It is easy to see that

Q(u, v) = ∂

S

(u

, v

)|

.

Lemma 3.4. Let d = 5, κ =

and ω > 0. Let (φ

, ψ

) ∈ G

. Then S

(φ

, ψ

) = inf

S

(u, v) : (u, v) ∈ H

× H

\{(0, 0)}, Q(u, v) = 0 . Proof. Denote m := inf

S

(u, v) : (u, v) ∈ H

× H

\{(0, 0)}, Q(u, v) = 0 . Since (φ

, ψ

) is a solution of (1.7), it follows from Lemma 3.1 that Q(φ

, ψ

) = K

(φ

, ψ

) = 0. Thus

S

(φ

, ψ

) ≥ m. (3.9)

Now let (u, v) ∈ H

× H

\{(0, 0)} be such that Q(u, v) = 0. If K

(u, v) = 0, then by Proposition 1.4, S

(u, v) ≥ S

(φ

, ψ

). If K

(u, v) 6= 0, we consider K

(u

, v

) = γ

K(u, v) + ωM(u, v) − γ

P (u, v), where (u

, v

) is as in (3.8). Since lim

K

(u

, v

) = ωM(u, v) > 0 and lim

K

(u

, v

) = −∞, there exists γ

> 0 such that K

(u

, v

) = 0. It again follows from Proposition 1.4, S

(u

, v

) ≥ S

(φ

, ψ

). On the other hand,

∂

S

(u

, v

) = γK (u, v) − 5

2 γ

P (u, v).

We see that the equation ∂

S

(u

, v

) = 0 admits a unique non-zero solution 2K(u, v)

5P (u, v)

= 1

since Q(u, v) = 0. This implies that ∂

S

(u

, v

) > 0 if γ ∈ (0, 1) and ∂

S

(u

, v

) < 0 if γ ∈ (1, ∞). In particular, S

(u

, v

) ≤ S

(u, v) for all γ > 0. Hence S

(u

, v

) ≤ S

(u, v).

We thus obtain S

(φ

, ψ

) ≤ S

(u, v) for any (u, v) ∈ H

× H

\{(0, 0)} satisfying Q(u, v) = 0.

Therefore,

S

(φ

, ψ

) ≤ m. (3.10)

Combining (3.9) and (3.10), we prove the result.

Let (φ

, ψ

) ∈ G

. Define B

:=

(u, v) ∈ H

× H

\{(0, 0)} : S

(u, v) < S

(φ

, ψ

), Q(u, v) < 0 .

Lemma 3.5. Let d = 5, κ =

, ω > 0 and (φ

, ψ

) ∈ G

. The set B

is invariant under the flow of (1.6).

Proof. Let (u

, v

) ∈ B

. We will show that the corresponding solution (u(t), v(t)) to (1.6) with initial data (u(0), v(0)) = (u

, v

) satisfies (u(t), v(t)) ∈ B

for any t in the existence time.

Indeed, by the conservation of mass and energy, we have

S

(u(t), v(t)) = S

(u

, v

) < S

(φ

, ψ

) (3.11) for any t in the existence time. It remains to show that Q(u(t), v(t)) < 0 for any t as long as the solution exists. Suppose that there exists t

> 0 such that Q(u(t

), v(t

)) ≥ 0. By the continuity of the function t 7→ Q(u(t), v(t)), there exists t

∈ (0, t

] such that Q(u(t

), v(t

)) = 0. It follows from Lemma 3.4 that S

(u(t

), v(t

)) ≥ S

(φ

, ψ

) which contradicts to (3.11). The

proof is complete.