Nonlinear Hartree equation in high energy-mass

Nonlinear Analysis: Real World Applications 34 (2017) 97–109

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Nonlinear Analysis: Real World Applications www.elsevier.com/locate/nonrwa

Nonlinear Hartree equation in high energy-mass Juan Huang a,∗ , Jian Zhang a,b a b

College of Mathematics and Software Science, Sichuan Normal University, Chengdu, 610066, China College of Mathematics and Information, China west Normal University, Nanchong, 637002, China

article

info

Article history: Received 9 March 2016 Received in revised form 2 August 2016 Accepted 6 August 2016

Keywords: Hartree equation Higher energy-mass Energy-mass control map

abstract This paper is concerned with the Cauchy problem of the nonlinear Hartree equation. By constructing a constrained variational problem, we get a refined Gagliardo–Nirenberg inequality and the best constant for this inequality. We thus derive two conclusions. Firstly, by establishing and analyzing the invariant manifolds, we obtain a new criteria for global existence and blowup of the solutions. Secondly, we get other sufficient condition for global existence with the discussing of the Bootstrap argument. And based on these two conclusions, we also deduce socalled energy-mass control maps, which expose the relationship between the initial data and the solutions. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Bose–Einstein condensation (BEC) in gases with weak attractive two-body interactions can be found in system of 7 Li atoms as long as the gas in the trap has a sufficiently low density [1,2]. By the heuristic discussion, we have that the dynamical evolution of the bosonic system in its mean-field regime is described by nonlinear Hartree equations. Thus, the aim of this paper is to study the following Hartree equation:  iϕt + ∆ϕ + (V (x) ∗ |ϕ|2 )ϕ = 0, (1.1) ϕ(0, x) = ϕ0 , x ∈ RN where ϕ(t, x) : R+ × RN −→ C is a complex function, ∆ is Laplacian operator on RN , V (x) = |x|1α (0 < α < min{N, 4}) and ∗ is the standard convolution in RN . The initial data ϕ0 ∈ H 1 (RN ), and  H 1 (RN ) = W 1,2 (RN ) is the standard Sobolev space with the norm ∥ϕ∥2H 1 = RN (|∇ϕ|2 + |ϕ|2 )dx. The success of experiment in atomic BEC has stimulated great interest in the properties of Cauchy problem (1.1). For instance, Cazenave [3] stated the local well-posedness of Cauchy problem (1.1) in H 1 (RN ) ∗ Corresponding author. E-mail addresses: [email protected] (J. Huang), [email protected] (J. Zhang).

http://dx.doi.org/10.1016/j.nonrwa.2016.08.002 1468-1218/© 2016 Elsevier Ltd. All rights reserved.

J. Huang, J. Zhang / Nonlinear Analysis: Real World Applications 34 (2017) 97–109

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when 0 < α < min{N, 4} Lions [4] studied the existence of the standing wave of Cauchy problem (1.1) for α = 1 (mass subcritical case). Miao [5] showed the existence of the standing waves and the blow up criterion of Cauchy problem (1.1) for L2 critical (i.e., α = 2). And for L2 supercritical (i.e., 2 < α < min{N, 4}), Wang [6] and Chen and Guo [7] discussed strong instability of the standing waves of Cauchy problem (1.1) in supercritical case with and without harmonic potential respectively. In this paper, we plan to discuss the Cauchy problem (1.1) for L2 supercritical. As we mentioned, Wang [6] and Chen and Guo [7] had discussed this issue before, however. We notice that in [6,7], they obtained the condition for blowup of Cauchy problem (1.1) by constructing a type of cross constrained variational problem and some of these functionals haven’t clear physical meanings. Can we use the energy and mass to character the criterion for global existence and blowup? Cazenave also mentioned this topic in their monographs [3]. At the meantime, a collapse processing is observed in the experiments if the number of 7 Li atoms in the trap exceeds some threshold value. So this problem is also pursued strongly in Physics (see [8] and the references therein). This is one of the motivations for discussing this problem. However, in spite of quite a number of contributions dealing with this problem (see [9–12] as well as the other relevant references), almost all of them still need to restrict the energy-mass functional(or some other functionals)to be less than its minimum in one subset. What will happen when the energy-mass is less than the minmax value of the energy-mass functional in the whole space? Obviously, this minmax value is larger than its minimum in one subset. Consequently, we also can answer the question: what will happen when the energy-mass functional is less than a number which is larger than the minimum in one subset? To fill this gap, we do more detailed discussion for the properties of the equation and energy-mass functional and extend the argument to the case that we mentioned before. Actually, we get a sharp condition for the global existence of the solution in higher energy-mass. In addition, these conditions are precisely computed. Finally, we expect to find some details of the energy-mass criteria, which is called energy-mass control map. This control map shows the relationships between the initial data and some behaviors of the solutions corresponding to these initial data. Furthermore, we find a set which indicate both global existence and blowup. To get more information of this set, we divide it into two parts, and then strip a global existence subset from one of the parts. In fact, to our knowledge no results have been already known in this direction. The plan of this paper is as follows. In Section 2, we give some concerned preliminaries and obtain a refined Gagliardo–Nirenberg inequality. And in Section 3, we give a new criterion of global existence and blowup for Cauchy problem (1.1). And the other sufficient condition for global existence is also obtained by Bootstrap argument. In the last section, we derive the so-called energy-mass control map by discussing the relations of the results which is obtained in the Section 3. 2. Preliminaries 2.1. Notations and some known lemmas In this subsection, we will give some concerned preliminaries. Throughout this paper, C denotes various positive constants. For simplicity, here and hereafter, we denote ∥ · ∥p to denote the norm of Lp (RN ) and   ·dx by ·dx unless stated otherwise. Moreover, we denote Σ := {u ∈ H 1 (RN ) : |x|u ∈ L2 (RN )} RN and  41  |ϕ(x)|2 |ϕ(y)|2 dxdy ∥ϕ∥V = . (2.1) |x − y|α RN ×RN Now, we begin by recalling the Hardy–Littlewood–Sobolev inequality.

J. Huang, J. Zhang / Nonlinear Analysis: Real World Applications 34 (2017) 97–109

Lemma 2.1 ([13,14]). Let 0 < β < N and suppose that f ∈ Ls (RN ), h ∈ Lr (RN ) with 1 < s, r < ∞, then   1s   r1  |f (x)||h(y)| s r |f (x)| dx |h(x)| dx dxdy ≤ C |x − y|β RN ×RN

1 s

+

99

1 r

+

β N

= 2 and

for some constant C > 0. Proposition 2.1 ([3]). Suppose N ≥ 3 and 2 < α < min{N, 4}. Then for any ϕ0 ∈ H 1 (RN ), there is a unique solution ϕ(t, x) of Cauchy problem (1.1) in C([0, T ); H 1 (RN )) for time T ∈ (0, 1] (maximal existence time) with the following property: either T = ∞ or else T < ∞ and limt→T − ∥ |∇ϕ(t, x)|2 ∥ = ∞, where t → T − means that t → T and t < T . Furthermore, for all t ∈ [0, T ), the solution ϕ satisfies the following conservative laws: (1) Conservation of mass: 

2



|ϕ0 |2 dx ≡ M (ϕ0 ).

|ϕ(t, x)| dx =

M (ϕ) = RN

(2.2)

RN

(2) Conservation of energy: 

|∇ϕ|2 dx −

E(ϕ) = RN

1 2

 RN ×RN

|ϕ(x)|2 |ϕ(y)|2 dxdy ≡ E(ϕ0 ). |x − y|α

Lemma 2.2 ([15] Gagliardo–Nirenberg’s inequality). Let 1 < p < symmetric solution of the nonlinear elliptic equation −∆u + u − |u|p−1 u = 0,

n+2 (n−2)+

(2.3)

and Θ be the positive and spherically

u ∈ H 1 (RN ).

Then the best constant C∗ > 0 of the Gagliardo–Nirenberg’s inequality,  |u|

p+1

 dx ≤ C∗

N (p−1)   p+1  N (p−1) 2 − 4 4 2 |u| dx |∇u| dx

2

(2.4)

is given by 2(p + 1) C∗ = N (p − 1)



4 − (p − 1)(N − 2) N (p − 1)

 N (p−1)−4 4

−(p−1)

∥Θ∥2

.

Next, by direct computations, we get the following variance identity. Lemma 2.3. Let ϕ0 ∈ Σ and ϕ(t, x) be a solution of the Cauchy problem (1.1). Then    d2 |ϕ(x)|2 |ϕ(y)|2 2 2 2 |x| |ϕ| dx = 8 |∇ϕ| dx − 2α dxdy. dt2 |x − y|α RN ×RN

(2.5)

By a similar way as in [7], we can prove this lemma. Here we omit the proof. Also, we need the following concentration-compactness principle which was given in [16,17]. Lemma 2.4. For a bounded sequence {un }n∈N ⊂ H 1 (RN ), there is a subsequence of {un }n∈N (still denoted by {un }) and a sequence {U (j) }j≥1 in H 1 (RN ) and for any j ≥ 1, a family (xjn ) ⊂ RN such that

J. Huang, J. Zhang / Nonlinear Analysis: Real World Applications 34 (2017) 97–109

100

(i) If j ̸= k, |xjn − xkn | → ∞, as n → ∞. (ii) For every l ≥ 1,

un (x) =

l 

U (j) (x − xjn ) + rnl (x).

(2.6)

j=1

Moreover, for any p ∈ (2, N2N −2 ), lim sup ∥rnl ∥p → 0

as l → ∞.

(2.7)

∥U (j) ∥22 + ∥rnl ∥22 + o(1),

(2.8)

n→∞

(iii)

∥un ∥22 =

l  j=1

∥∇un ∥22 =

l 

∥∇U (j) ∥22 + ∥∇rnl ∥22 + o(1).

(2.9)

j=1

Lemma 2.5 ([15]). Let ϕ ∈ Σ . One has   12   21  2 2 2 2 |xϕ| dx |∇ϕ| dx . |ϕ| dx ≤ N 2.2. A refined Gagliardo–Nirenberg inequality In this subsection, we obtained a refined Gagliardo–Nirenberg inequality of convolution type which is a general case of the result in [5]. Moreover, we determine the best constant of the refined Gagliardo–Nirenberg inequality. First, we begin with the following lemma: By Lemmas 2.1 and 2.2, we have 4−α ∥u∥4V ≤ Cα ∥∇u∥α . 2 ∥u∥2

(2.10)

To compute Cα , it will suffice to minimize the functional: J(u) :=

4−α ∥∇u∥α 2 ∥u∥2 ∥u∥4V

and Jα =

inf

u∈H 1 (RN )\{0}

J(u).

(2.11)

Lemma 2.6. If U is the minimizer of J(u), then U satisfies − ∆U + AU − B(V (x) ∗ |U |2 )U = 0, where A=

4 − α ∥∇U ∥22 , α ∥U ∥22

B=

4Jα . α∥U ∥4−α ∥∇U ∥α−2 2 2

(2.12)

J. Huang, J. Zhang / Nonlinear Analysis: Real World Applications 34 (2017) 97–109

101

Proposition 2.2. For 0 ≤ α < min{N, 4}, then Jα =

min

u∈H 1 (RN )

J(u).

Proof. Choose a sequence {un }n∈N ⊂ H 1 (RN ) with J(un ) → Jα . Assume, without loss of generality, that ∥un ∥2 = 1 and ∥∇un ∥2 = 1, then J(un ) =

1 → Jα . ∥un ∥4V

Obviously, {un }n∈N is bounded in H 1 (RN ). Then by Lemma 2.4, we have (2.6)–(2.9) hold. It follows from (2.8) and (2.9) that l 

∥U (j) ∥22 ≤ 1,

j=1

l 

∥∇U (j) ∥22 ≤ 1.

(2.13)

j=1

Then, by Lemma 2.1, we have ∥rnl ∥4V ≤ C∥rnl ∥4 4N . It follows from (2.7) that 2N −α

lim sup ∥rnl ∥V → 0. n→∞

Hence, by (2.6), we get  2  2     l  l (j) j   (j) j  U (x − x ) U (y − x )    n n   j=1  j=1  |x − y|α

RN ×RN

=

l   j=1

+2

RN ×RN

l−1 

|U (j) (x − xjn )|2 |U (j) (y − xjn )|2 dxdy |x − y|α

l 



j=1 k=j+1

+2

dxdy

i=1

|x − y|α

RN ×RN

 l−1  l  j=1 k=j+1

2  l   |U (j) (x − xjn )U (k) (x − xkn )|  U (i) (y − xin )

RN ×RN

dxdy

|U (j) (x − xjn )|2 |U (k) (y − xkn )|2 dxdy |x − y|α

= I + 2II + 2III.

(2.14)

By (i) of Lemmas 2.4 and 2.1, we have II → 0 and III → 0 as n → ∞. And without loss of generality, we assume that all U (j) are continuous and compactly supported. Thus, l   |U (j) (x)|2 |U (j) (y)|2 I= dxdy. |x − y|α N N j=1 R ×R It follows that  l 4 l     (j) j  U (x − xn ) → ∥U (j) ∥4V    j=1

V

j=1

Hence, we have l  j=1

∥U (j) ∥4V =

1 . Jα

as n → ∞.

102

J. Huang, J. Zhang / Nonlinear Analysis: Real World Applications 34 (2017) 97–109

Thus, by (2.11), we have j 4−α Jα ∥U (j) ∥4V ≤ ∥U (j) ∥4−α ∥∇U (j) ∥α }∥∇U (j) ∥α 2 ≤ sup{∥U ∥2 2. 2

(2.15)

j≥1

Next, we discuss the problem in two cases, 0 < α < 2 and 2 ≤ α < min{N, 4}. And it is easy to have ∥U (j0 ) ∥2 = 1. Above all, we have that there exists a unique j0 such that ∥U (j0 ) ∥2 = 1,

∥∇U (j0 ) ∥2 = 1

and ∥U (j0 ) ∥4V =

1 . Jα

It follows that − ∆U (j0 ) + The proof is complete.

4 − α (j0 ) 4 U − Jα (V (x) ∗ |U (j0 ) |2 )U (j0 ) = 0. α α

(2.16)



Remark 2.1. If U is the minimizer of J(u), then |U | is also a minimizer. Moreover, |U | ∈ H 1 (RN ) and J(|U |) ≤ J(U ). Hence we can assume that U > 0. Remark 2.2. The uniqueness of U can be obtained by means of the method in Kwong [18]. Hence, we obtain a refined Gagliardo–Nirenberg inequality: Theorem 2.1. Let 0 < α < min{N, 4}(N ≥ 3) and Q be the ground state solution of the nonlinear elliptic equation: u ∈ H 1 (RN ).

− ∆u + u − (V (x) ∗ |u|2 )u = 0,

Then the best constant Cα > 0 of the Gagliardo–Nirenberg’s inequality,   4−α   α2  2 |u(x)|2 |u(y)|2 2 2 dxdy ≤ C |u| dx |∇u| dx , α |x − y|α RN ×RN RN RN

(2.17)

(2.18)

is given by 4 Cα = 4−α



α 4−α

 α2

∥Q∥−2 2 .

(2.19)

Proposition 2.3. If Q satisfies (2.17), then I can be obtained by u(x) = µQ(λx + η) for some µ ∈ C, λ > 0 and η ∈ RN . Moreover,  α 4−α 4−α 2 Jα = ∥Q∥22 . 4 α Proof. On the one hand, multiplying (2.17) with x∇Q and integrating on RN , we get N −2 N 2N − α ∥∇Q∥22 + ∥Q∥22 = ∥Q∥4V . 2 2 4 On the other hand, multiplying (2.17) with Q and integrating on RN , we have ∥∇Q∥22 + ∥Q∥22 − ∥Q∥4V = 0. Thus, we have ∥Q∥4V =

4 ∥Q∥22 . 4−α

J. Huang, J. Zhang / Nonlinear Analysis: Real World Applications 34 (2017) 97–109

Taking µ =



α 4Jα

 21   N +2−α 4−α 4

and λ =

α

Jα = The proof is complete.

 4−α  12 α

1 ∥U (j0 ) ∥4V

=

103

. It follows that

4−α 4



4−α α

 α2

∥Q∥22 .



3. Global existence and blowup results 3.1. A new criteria for global existence and blowup In this subsection, we shall give some invariant evolution flows generated by the Cauchy problem (1.1). For convenience, if 2 < α < min{N, 4}, we define   α α (4 − α) 2 +1 (4 − α) 2 +1 1 N 2 2 2 2 G := u ∈ H (R ) : F (∥u∥H 1 ) < E(u) + M (u) < ∥Q∥2 , ∥u∥H 1 < ∥Q∥2 α α 8α 2 4α 2 and  B :=

1

N

u ∈ H (R ) :

where F (S) := S −

F (∥u∥2H 1 )

Cα 2 2 S

 α α (α − 2)(4 − α) 2 +1 (4 − α) 2 +1 2 2 2 < E(u) + M (u) < ∥Q∥2 , ∥u∥H 1 > ∥Q∥2 , α α α 2 +2 α 2 +1

for any S > 0. Then, denote G¯ := G ∪ {0}.

Proposition 3.1. Let 2 < α < min{N, 4} and N ≥ 3. Then G¯ and B are invariant under the flows generated by the Cauchy problem (1.1). More precisely, if the initial data ϕ0 ∈ G¯ (resp. B), then the solution ϕ(t, x) of the Cauchy problem (1.1) still satisfies ϕ(t, x) ∈ G¯ (resp. B) for all t ∈ [0, T ). Proof. Suppose ϕ0 ∈ G¯ and ϕ(t, x) is the unique solution of Cauchy problem (1.1) with the initial data ϕ0 . By local well-posedness of Cauchy problem (1.1), one has E(ϕ(t)) = E(ϕ0 ) and M (ϕ(t)) = M (ϕ0 )t ∈ [0, T ). α

Thus from E(ϕ0 ) + M (ϕ0 ) <

(4−α) 2 +1 ∥Q∥22 α 8α 2

=

1 2Cα ,

it follows that

E(ϕ(t)) + M (ϕ(t)) <

1 , 2Cα

t ∈ [0, T ).

(3.1)

¯ we only need to prove To check that ϕ(t) ∈ G, α

∥ϕ(t)∥2H 1 < If otherwise, we have ∥ϕ(t)∥2H 1 ≥

1 Cα .

(4 − α) 2 +1 1 ∥Q∥22 = , α Cα 4α 2

t ∈ [0, T ).

(3.2)

Then there would exist, by continuity, a t1 ∈ [0, T ) such that ∥ϕ(t)∥2H 1 =

1 . Cα

(3.3)

However, it follows from Lemma 2.2 that  E(ϕ(t1 )) + M (ϕ(t1 )) = (|∇ϕ(t1 )|2 + |ϕ(t1 )|2 )dx −  ≥ (|∇ϕ(t1 )|2 + |ϕ(t1 )|2 )dx −

1 ∥ϕ(t1 )∥4V 2 Cα ∥ϕ(t1 )∥4−α ∥∇ϕ(t1 )∥α 2 2 2  2  Cα ≥ (|∇ϕ(t1 )|2 + |ϕ(t1 )|2 )dx − (|∇ϕ(t1 )|2 + |ϕ(t1 )|2 )dx . 2

(3.4)

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J. Huang, J. Zhang / Nonlinear Analysis: Real World Applications 34 (2017) 97–109

Recall the real value function F (S) := S −

Cα 2 S 2

for any S > 0.

(3.5)

Obviously, F (S) reaches the maximum  Fmax = F

1 Cα

 =

1 . 2Cα

(3.6)

Therefore it follows from (3.4) and (3.6) that E(ϕ(t1 )) ≥

1 , 2Cα

(3.7)

which violates E(ϕ(t1 )) + M (ϕ(t1 )) = E(ϕ0 ) + M (ϕ(t0 )) < 2C1 α . Then, combining with (3.4), we obtain that (3.2) is true. Hence, G¯ is invariant under the flow generated by the Cauchy problem (1.1). By the similar argument as above, we can show that B are also invariant under the flows generated by the Cauchy problem (1.1).  We can now state the main result of this subsection. Theorem 3.1. Let 2 < α < min{N, 4}(N ≥ 3) and Q be the ground state solution of the nonlinear elliptic equation (2.15). Denote ϕ(t, x) the solution of the Cauchy problem (1.1) corresponding to the initial datum ϕ0 . Then one has that ¯ then ϕ(t, x) globally exists in H 1 (RN ); (1) if ϕ0 ∈ G, (2) if ϕ0 ∈ B ∩ Σ , then ϕ(t, x) blows up in a finite time. Proof. It is obvious that for any ϕ ∈ H 1 (RN ) \ {0}, we get E(ϕ) + M (ϕ) > F (∥ϕ∥2H 1 ). (1) Let ϕ(t, x) be the solution of the Cauchy problem (1.1) with the initial datum ϕ0 in t ∈ [0, T ). If ¯ it follows from Proposition 3.1 that ϕ ∈ G. ¯ Hence ϕ0 ∈ G,  α (4 − α) 2 +1 ∥Q∥22 (3.8) [| ▽ ϕ|2 + |ϕ|2 ]dx < α 4α 2 and we easily obtain ϕ is bounded in H 1 (RN ). Therefore, the solution ϕ of the Cauchy problem (1.1) globally exists in H 1 (RN ). (2) Suppose ϕ0 ∈ B, it follows from Proposition 3.1 that ϕ ∈ B, which reads α

E(ϕ) + M (ϕ) <

α

(α − 2)(4 − α) 2 +1 ∥Q∥22 α α 2 +2

and ∥ϕ∥2H 1 >

(4 − α) 2 +1 ∥Q∥22 . α α 2 +1

It follows that d2 dt2



2

|ϕ(x)|2 |ϕ(y)|2 dxdy |x − y|α RN ×RN  4α(E(ϕ) + M (ϕ)) − (4α − 8) |∇ϕ|2 dx − 4α|ϕ|2 dx  4α(E(ϕ) + M (ϕ)) − (4α − 8) (|∇ϕ|2 + |ϕ|2 )dx   α−2 2 4α E(ϕ0 ) + M (ϕ0 ) − ∥ϕ∥H 1 α   α (α − 2)(4 − α) 2 +1 2 4α (E(ϕ0 ) + M (ϕ0 )) − ∥Q∥ α 2 . α 2 +2 

2

|x| |ϕ| dx = 8 = < < <

2



|∇ϕ| dx − 2α

(3.9)

J. Huang, J. Zhang / Nonlinear Analysis: Real World Applications 34 (2017) 97–109

105

 2 2 d2 Therefore, dt |x| |ϕ| dx ≤ −C < 0 is got. Then using the method of Glassey [19] and Lemma 2.5, we get 2 that the solution ϕ of the Cauchy problem (1.1) blows up in a finite time. The proof is complete.  α

α

+1 2 (4−α) 2 +1 ∥Q∥22 > (α−2)(4−α) ∥Q∥22 for 2 < α < min{N, 4}(N ≥ 3). If lowering α α +2 2 2 8α α α +1 (α−2)(4−α) 2 ∥Q∥22 in case (1), Theorem 3.1 becomes a sharp condition of global existence α α 2 +2

Remark 3.1. It is clear that

the energy-mass to and blowup for Cauchy problem (1.1).

α

In fact, the set {ϕ ∈ H 1 (RN ) : E(ϕ) + M (ϕ) < α (4−α) 2 +1 α 2 4α

(α−2)(4−α) 2 +1 ∥Q∥22 , α α 2 +2

α

(4−α) 2 +1 ∥Q∥22 α α 2 +1

< ∥ϕ0 ∥2H 1 <

∥Q∥22 } equals ∅. (We refer to Proposition 4.2 in Section 4 for more details.) So we can use any α

α

+1 2 (4−α) 2 +1 ∥Q∥22 and (4−α)α2 ∥Q∥22 to divide α +1 2 α 4α α (α−2)(4−α) 2 +1 ∥Q∥22 } by norm into two parts, in which one α +2 α2

number between

the subspace {u ∈ H 1 (RN ) : E(ϕ0 ) + M (ϕ0 ) < corresponds to global existence and the other

corresponds to blowup.

3.2. Bootstrap argument Now we are at the point to do the bootstrap argument of Cauchy problem (1.1). Let us begin with the following proposition. Proposition 3.2. Let 2 < α < min{N, 4}(N ≥ 3) and Q be the ground state solution of the nonlinear elliptic α+2  (4−α) α−2 equation (2.15). If the initial data of Cauchy problem (1.1) ϕ0 satisfy |∇ϕ0 |2 dx < 2  4−α α 2 α−2 α α−2 (

4 α−2

∥Q∥2

|ϕ0 |2 dx) α−2

, then E(ϕ(t, x)) > 0.

Proof. Let ϕ ∈ H 1 (RN ) \ {0}, we have   Cα 1 4 2 ∥ϕ(t)∥4−α ∥∇ϕ(t)∥α E(ϕ(t)) = |∇ϕ(t)| dx − ∥ϕ(t)∥V ≥ |∇ϕ(t)|2 dx − 2 2 2 2   2−α/2  α/2−1   Cα 2 2 2 = |∇ϕ(t)| dx 1 − |ϕ(t)| dx |∇ϕ(t)| dx . 2 Then, by Proposition 2.1, the result is proved.

(3.10)



Now, by Theorem 2.1 and Proposition 2.1, we have  1 Cα |∇ϕ(t)|2 dx = E(ϕ(t)) + ∥ϕ(t)∥4V ≤ E(ϕ(t)) + ∥ϕ(t)∥4−α ∥∇ϕ(t)∥α 2 2 2 2 Cα = E(ϕ0 ) + M (ϕ0 )2−α/2 ∥∇ϕ(t)∥α 2. 2

(3.11)

With this estimate in hand, the result follows from the following bootstrap argument: Lemma 3.1 ([20] Bootstrap argument). Let f = f (t) be a nonnegative continuous function on [0, T ] such that, for every t ∈ [0, T ], f (t) ≤ ε1 + ε2 f (t)θ , where ε1 , ε2 > 0 and θ > 1 are constant such that   1 1 , ε1 < 1 − θ (θε2 )1/(θ−1)

f (0) ≤

1 . (θε2 )1/(θ−1)

106

J. Huang, J. Zhang / Nonlinear Analysis: Real World Applications 34 (2017) 97–109

Then, for every t ∈ [0, T ], we have f (t) ≤

θ ε1 . θ−1

 From this lemma, Propositions 2.1 and 3.2, taking f (t) = |∇ϕ(t)|2 dx, θ = ε2 = C2α M (ϕ(t))4−α , we can infer the following discussion of global existence.

α 2,

ε1 = E(ϕ(t)) and then

Now, denote   2(4 − α)α−1 2(4 − α)α−1 2 2 2 1 N 2 ∥Q∥2 , ∥u∥H 1 ≤ α +1 ∥Q∥2 . K := u ∈ H (R ) : F (∥u∥H 1 ) < E(u) + M (u) < α αα α 2 (α − 2) 2 −1 Then we give the main results of this subsection. Theorem 3.2. Let 2 < α < min{N, 4}(N ≥ 3) and Q be the ground state solution of the nonlinear elliptic equation (2.15). Denote ϕ(t, x) the solution of the Cauchy problem (1.1) corresponding to the initial datum ϕ0 . If ϕ0 ∈ K, then ϕ(t, x) globally exists in H 1 (RN ). 4. Energy-mass control map From Section 3, we have known some sufficient conditions for global existence and blowup. But, there is no results about the relation of them. So in this section we will try to explore more details of these results in the above section, and show the so called energy-mass control map of Cauchy problem (1.1). To give a concise and clear representation both in expressions and graphs, we denote H(α) :=

2(4 − α)α−1 ∥Q∥22 , αα

G(α) :=

(4 − α) 2 +1 ∥Q∥22 , α 8α 2

α

2(4 − α)α−1 ∥Q∥22 , α α (α − 2) 2 −1 α (4 − α) 2 +1 g(α) := ∥Q∥22 , α 4α 2 h(α) :=

α 2 +1

and α

α

(α − 2)(4 − α) 2 +1 (4 − α) 2 +1 2 ∥Q∥ , b(α) := ∥Q∥22 . α α 2 α 2 +2 α 2 +1 Hence, in order to show the comparison of H(α), G(α) and B(α), we give Fig. 1 for details. Furthermore, we also give the relationship among h(α), g(α) and b(α) in Fig. 2.Thus, we obtain the following propositions. B(α) :=

¯ := K ∪ {0} ⊃ G. ¯ Proposition 4.1. If 2 < α < min{N, 4}(N ≥ 3), we have K Now, consider E := K ∩ B. Thus,  2(4 − α)α−1 ∥Q∥22 , E = u ∈ H 1 (RN ) : F (∥u∥2H 1 ) < E(u) + M (u) < αα  α (4 − α) 2 +1 2(4 − α)α−1 2 2 2 ∥Q∥2 < ∥u∥H 1 ≤ α +1 ∥Q∥2 . α α α 2 +1 α 2 (α − 2) 2 −1 But from Theorems 3.1 and 3.2, we find if the initial data ϕ0 ∈ E ∩Σ , the solutions of Cauchy problem (1.1) are global existence and also blow up in a finite time, which is a contradiction. Hence, we conjecture that E ∩ Σ = ∅. But there still exists a problem: does E equal to empty set? To solve this problem, we first need the exact position relation of E and B, we give Fig. 3 for details. Here k(α) :=

2(4−α)

3α −3 α α 2 [α(4−α)2− 2 (α−2) 2 −1 −4] α +2 α−2 (α−2) α2

∥Q∥22 . Hence, we divide E into two subsets: E+ := E ∩ {u ∈

H 1 (RN ) : E(u) + M (u) ≥ 0} and E− := E ∩ {u ∈ H 1 (RN ) : E(u) + M (u) < 0}. Thus, by Proposition 3.2, we obtain the following proposition.

J. Huang, J. Zhang / Nonlinear Analysis: Real World Applications 34 (2017) 97–109

107

Fig. 1. Comparison of energy-mass.

Fig. 2. Comparison of norm.

Fig. 3. Second coordinate of the lowest point in E.

Proposition 4.2. If 2 < α < min{N, 4}, N ≥ 3, then  E− ⊂ D := u ∈ H 1 (RN ) : E(u) + M (u) < 0, ∥u∥2H 1 <

(4 − α)α−1 ∥Q∥22 (α − 2)α/2−1 αα/2

 = ∅.

Proposition 4.3. Let 2 < α < min{N, 4} and N ≥ 3. There exists a α∗ ∈ (2, 4) such that if 2 < α < α∗ , then E+ \ Σ ̸= ∅. α+2 4

∥Q∥2 Proof. For any u ∈ H 1 (RN ), put u ˆ = λ ∥u∥ u (λ > 0). Then, on the one hand, if λ ∈ ( (4−α) α+2 1 H

√

α−1 2(4−α) 2 α+2 α−2 α 4 (α−2) 4 α−1 √ 2(4−α) 2 α α2 α+2 (4−α) 4 α+2 α 4

), it follows

(0,

λ ∈ (

α (4−α) 2 +1 α +1 α2

α

∥Q∥22 < ∥ˆ u∥2H 1 ≤

2(4−α)α−1 ∥Q∥22 . α α α 2 +1 (α−2) 2 −1

), the energy-mass satisfies 0 < E(ˆ u) + M (ˆ u) < √

,

α−1 2(4−α) 2 α α2

,

4

On the other hand, if λ ∈

2(4−α)α−1 ∥Q∥22 . αα

Thus, E+ ̸= ∅ for

). From calculation by computer, we find there exists a number 3.385 such α+2 4

that if α ∈ (2, 3.385) ∩ (2, min{N, 4}), then ( (4−α) α+2 α

4

√

,

2(4−α) α α2

α−1 2

) ̸= ∅. This completes the proof.



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J. Huang, J. Zhang / Nonlinear Analysis: Real World Applications 34 (2017) 97–109

Fig. 4. Energy-mass control map.

Remark 4.1. Actually, from the proof of Proposition 4.3, we have 3.385 is not the best number of this case and the best number should be less than 3.385. And by a direct computation, we also find the best number is larger than 3. Hence, we have the main results of this section, which can be treated as the improvement of Theorems 3.1 and 3.2. Theorem 4.1. Let 2 < α < min{N, 4}, N ≥ 3 and Q be the ground state solution of the nonlinear elliptic equation (2.15). Denote ϕ(t, x) the solution of the Cauchy problem (1.1) corresponding to the initial datum ϕ0 . ¯ \ (E ∩ Σ ), then ϕ(t, x) globally exists in H 1 (RN ); (1) If the initial datum ϕ0 ∈ K (2) If the initial datum ϕ0 ∈ (B \ D) ∩ Σ , then ϕ(t, x) blows up in a finite time. To give an intuitive and visual feeling of Theorem 4.1, we draw Fig. 4. It is worth to point out that we do not know what are the blank parts of the first quadrants in the energy-mass control maps correspond to. And this is the next problem we will pursue. 1 Remark 4.2. From [21], we know that the condition ϕ0 ∈ Σ can be changed into ϕ0 ∈ Hradial . That means 1 E ∩ Hradial = ∅ and there are also have some corresponding changes in Theorem 4.1 and Fig. 4.

Acknowledgments The authors would like to thank the referees for many valuable suggestions which help them to improve the paper so much. This work was supported in part by the National Science Foundation of China (Grant Nos. 11347102, 11371267, 11401409 and 11571245) and the Scientific Research Found of Sichuan Provincial Education Department (Grant No. 15ZA0031).

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References [1] J. Froehlich, E. Lenzmann, Mean-field limit of quantum bose gases and nonlinear Hartree equation, in: Seminaire EDP, Ecole Polytechnique, 2003–2004. [2] E.P. Gross, in: E. Meeron (Ed.), Physics of Many-Particle Systems, Vol. 1, Gordon Breach, New York, 1966, pp. 231–406. [3] T. Cazenave, Semilinear Schr¨ odinger Equations, in: Courant Lecture Notes, vol. 10, American Mathematical Society, Providence, RI, 2003. [4] P.L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case Part 1, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire 1 (1984) 109–145. [5] C. Miao, G. Xu, L. Zhao, On the blow up phenomenon for the L2 -critical focusing Hartree equation in R3 , Colloq. Math. 119 (2010) 23–50. [6] Y. Wang, Strong instability of standing waves for Hartree equation with harmonic potential, Physica D 237 (2008) 998–1005. [7] J. Chen, B. Guo, Strong instability of standing waves for a nonlocal Schr¨ odinger equation, Physica D 227 (2007) 142–148. [8] E.A. Kuznetsov, J.J. Rasmussen, K. Rypdal, S.K. Turitsyn, Sharper criteria for the wave collapse, Physica D 87 (1995) 273–284. [9] G. Chen, J. Zhang, Energy criterion of global existence for supercritical nonlinear Schr¨ odinger equation with harmonic potential, J. Math. Phys. 48 (2007) 073513. [10] J. Huang, J. Zhang, X. Li, Stability of standing waves for the L2 -critical Hartree equations with harmonic potential, Appl. Anal. 92 (10) (2013) 2076–2083. [11] J. Huang, J. Zhang, Sharp conditions of global existence and scattering for a focusing energy-critical Schr¨ odinger equation, J. Math. Anal. Appl. 410 (2) (2014) 561–576. [12] J. Huang, J. Zhang, The Cauchy problem of a focusing energy -critical nonlinear Schr¨ odinger equation, Nonlinear Anal. Ser. A: Theory Methods Appl. 80 (2013) 122–134. [13] M. Reed, B. Simon, Fourier analysis, self-adjointness, in: Methods of Modern Mathematical Physics, Vol. II, Elsevier (Singapore) Pvt. Ltd., 2003. [14] E.M. Stein, Singular Integral and Differentiability Properties of Functions, in: Princeton Math. Series, vol. 30, Princeton Univ. Press, 1970. [15] M.I. Weinstein, Nonlinear Schr¨ odinger equations and sharp interpolations estimates, Comm. Math. Phys. 87 (1983) 567–576. [16] T. Hmidi, S. Keraani, Blowup theory for the critical nonlinear Schr¨ odinger equations revisited, Int. Math. Res. Not. (46) (2005) 2815–2828. [17] S. Zhu, J. Zhang, H. Yang, Limiting profile of the blow-up solutions for the fourth-order nonlinear Schr¨ odinger equation, Dyn. Partial Differ. Equ. 7 (2) (2010) 187–205. [18] M.K. Kwong, Uniqueness of positive solutions of △u−u+up = 0 in RN , Arch. Ration. Mech. Anal. 105 (3) (1989) 243–266. [19] R.T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schr¨ odinger equations, J. Math. Phys. 18 (1977) 1794–1797. [20] R. Carles, P.A. Markowich, C. Sparber, On the Gross-Pitaevskii equation for trapped dipolar quantum gases, Nonlinearity 21 (2008) 2569–2590. [21] T. Ogawa, Y. Tsutsumi, Blow-up of H 1 solutions of the nonlinear Schr¨ odinger equation, J. Differential Equations 92 (1991) 317–330.