Mathematics in Applied Sciences and Engineering https://ojs.lib.uwo.ca/mase
Volume 4, Number 1, March 2023, pp.61-78 https://doi.org/10.5206/mase/15536

ENERGY CRITERIA OF GLOBAL EXISTENCE FOR THE HARTREE

EQUATION WITH COULOMB POTENTIAL

NA TANG, CHENGLIN WANG, AND JIAN ZHANG

Abstract. This paper studies a class of Hartree equations with Coulomb potential. Combined with

the conservation of mass and energy, we analyze the variational characteristics of the corresponding

nonlinear elliptic equation. According to the range of parameters, we construct the evolution invariant

flows of the equation in different cases. Then the sharp energy thresholds for global existence and blow-

up of solutions are discussed in detail.

1. Introduction

In this paper, we study a class of Hartree equations with Coulomb potential:

iϕt + ∆ϕ + β|x|−1ϕ + (|x|−γ ∗ |ϕ|2)ϕ + |ϕ|pϕ = 0, t > 0,x ∈ Rn, (1.1)

where

n ≥ 3, 2 < γ < min{4, n}, 0 ≤ β <
(n− 2)2(γ − 2)

2(γ − 1)
, 0 < p <

4

n− 4
,

and ϕ = ϕ(t,x) is a complex value wave function of (t,x) ∈ R+ ×Rn.
Equation (1.1) is considered as the first-principle model for beam-matter interaction in X-ray free

electron lasers (XFEL)[1, 4, 9]. The parameter β denotes the strength of an electron beam interaction

with external Coulomb force. Recent developments using XFEL include the motion of atoms, measuring

the dynamics of atomic vibrations and biomolecular imaging [3, 8, 23]. Besides, in the context of BEC,

such a model equation is also known as the Gross-Pitaevskii for dipole Bose-Einstein condensation with

Coulomb potential[24].

For (1.1), the local well-posedness was established in [6, 10]. Feng and Zhao [10] obtained the global

well-posedness for (1.1) under some assumptions. In [15], authors proved the existence of ground states

and normalized solutions for (1.1) with harmonic potential. If we remove the term β|x|−1 in (1.1), this
equation may occur blow up in finite time for the whole range of p, see [25, 26]. To our knowledge,

the existence of blowup and the sharp criteria of global existence for (1.1) has not been studied in the

literature.

We recall the Hartree equation:

iϕt + ∆ϕ + (|x|−(n−2) ∗ |ϕ|α)|ϕ|α−2ϕ = 0, t > 0,x ∈ Rn. (1.2)

When α = 2, the equation (1.2) becomes Choquard-Pekar equation, which occurs in the modelling of

quantum semiconductor devices, the electron transport and the electron-electron interaction(see [17]).

There are numerous results for equation (1.2). When n ≥ 3, 2 ≤ α ≤ 1 + 4
n−2 , Genev and Venkov

[13] proved the local and global well-posedness and the existence of blow-up solutions. The dynamics

Received by the editors 23 November 2022; accepted 24 March 2023; published online 30 March 2023.

2010 Mathematics Subject Classification. Primary 54C40, 14E20; Secondary 46E25, 20C20.

Key words and phrases. Hartree equation, Coulomb potential, blow-up, global existence, sharp energy threshold.

61



62 N. TANG, C. WANG, AND J. ZHANG

of blow-up solutions was investigated in [5, 20, 22, 28, 29]. In [2, 12, 21], they showed the sharp criteria

for blow-up and scattering in H1(Rn). Huang, Zhang, Chen [16] and Tian, Yang, Zhou [25] showed the
sharp criteria of global existence for the Hartree equation with subcritical perturbations. And Leng,

Li, Zheng [18] showed the sharp criteria of global existence for the Hartree equation with supercritical

perturbations. In [26], they detected the dynamical properties of blow-up solutions. Lieb [17] showed

the uniqueness of the radial symmetric standing wave in R3.
The nonlinear Schrödinger equation with Coulomb potential is as follows:

iϕt + ∆ϕ + β|x|−1ϕ = λf(|ϕ|2)ϕ, t > 0,x ∈ Rn. (1.3)

When β > 0, it provides a quantum mechanical description of Coulomb force between two charged

particles and corresponds to having an external attractive long-range potential due to the presence of a

positively charged atomic nucleus(see [19]). When β ≤ 0 and f(|ϕ|2) = |x|−1 ∗ |ϕ|2 , Chadam, Glassey
[5] obtained the existence of the unique global solution in H1(R3). Hayashi, Ozawa [14] showed the
global existence and a decay rate of solutions when the initial data belongs to a weighted-L2 space.

For (1.1), we construct different invariant flows under different parameter ranges. Then we obtain the

sharp energy thresholds for global existence and blow-up of solutions for (1.1). We mainly consider the

following cases:

(1) 0 < p <
2

n
, 2 < γ < min{n, 4};

(2) p =
2

n
, 2 < γ < min{n, 4};

(3)
2

n
< p <

4

n
, 2 < γ < min{n, 4};

(4) p =
4

n
, 2 < γ < min{n, 4};

(5)
4

n
< p <

4

n− 2
, 2 < γ <

np

2
;

(6)
4

n
< p <

4

n− 2
,
np

2
≤ γ < min{4,n}.

This paper is organized as follows: in Section 2, we establish some basic facts including local well-

posedness, the conservation laws of mass and energy, and sharp inequalities. In Section 3, we give the

sharp energy thresholds of blow-up and global existence for (1.1).

2. Preliminaries

We impose the initial data of (1.1) as follows

ϕ(0,x) = ϕ0, x ∈ Rn. (2.1)

For the Cauchy problem (1.1) and (2.1), we define the energy space as

H1(Rn) := {v : v ∈ L2(Rn),∇v ∈ L2(Rn)}, (2.2)

and introduce the inner product

(u,v) :=

∫
∇u ·∇v + uvdx, (2.3)

whose associated norm denoted by ‖ · ‖H1 . Here and hereafter, for simplicity, we use
∫
·dx to denote∫

Rn ·dx.

Lemma 2.1. [6, 10] Assume ϕ0 ∈ H1(Rn), there exists a unique solution ϕ(t) of the Cauchy problem
(1.1) and (2.1) in C([0,T); H1(Rn)) for some T ∈ (0,∞] (maximal existence time). We have the



GLOBAL EXISTENCE AND BLOWUP OF THE HARTREE EQUATION WITH COULOMB POTENTIAL 63

alternatives T = ∞ (global existence) or else T < ∞ and lim
t→T
‖ϕ(t)‖H1 = ∞ (blow up). Moreover for

all t ∈ [0,T), the solution ϕ(t) satisfies the following:
(i) Conservation of mass: ∫

|ϕ(t)|2dx =
∫
|ϕ0|2dx. (2.4)

(ii) Conservation of energy:

E(ϕ(t)) =

∫
1

2
|∇ϕ(t)|2 −

β

2
|x|−1|ϕ(t)|2 −

1

4
(|x|−γ ∗ |ϕ(t)|2)|ϕ(t)|2 −

1

p + 2
|ϕ(t)|p+2dx = E(ϕ0). (2.5)

By a direct calculation, we have the following result.

Lemma 2.2. Let ϕ0 ∈ H1(Rn),
∫
|x|2|ϕ0|2dx < ∞ and ϕ(t,x) be a solution of the Cauchy problem

(1.1) and (2.1). Put J(t) :=
∫
|x|2|ϕ(t,x)|2dx, then one has

J
′′
(t) =

∫
8|∇ϕ|2 − 4β|x|−1|ϕ|2 − 2γ(|x|−γ ∗ |ϕ|2)|ϕ|2 −

4np

p + 2
|ϕ|p+2dx

= 8γE(ϕ0) +

∫
8γ − 4np
p + 2

|ϕ|p+2 − 4(γ − 2)|∇ϕ|2 + (4γ − 4)β|x|−1|ϕ|2dx.
(2.6)

Lemma 2.3. [27] Let ϕ0 ∈ H1(Rn) and
∫
|x|2|ϕ0|2dx < ∞. Then the following estimate holds:∫

|ϕ|2dx ≤
2

n
(

∫
|∇ϕ|2dx)

1
2 (

∫
|x|2|ϕ|2dx)

1
2 . (2.7)

Lemma 2.4. [27] For 0 < p <
4

n− 2
and v ∈ H1(Rn),

‖v‖p+2p+2 ≤
2(p + 2)

np‖∇R‖p2
‖v‖

4 − (n− 2)p
2

2 ‖∇v‖
np

2
2 , (2.8)

where R is the unique positive ground state solution of equation:

−∆R +
4 − (n− 2)p

np
R−|R|pR = 0,R ∈ H1(Rn). (2.9)

Lemma 2.5. [7, 29] For 0 < γ < min{4,n} and v ∈ H1(Rn), one has

‖(|x|−γ ∗ |v|2)|v|2‖1 ≤
4

γ‖∇W‖22
‖v‖4−γ2 ‖∇v‖

γ
2, (2.10)

where W is a positive ground state solution of equation:

−∆W +
4 −γ
γ

W − (|x|−γ ∗ |W|2)W = 0,W ∈ H1(Rn). (2.11)

Lemma 2.6. [11] Assume 1 < α < n,v ∈ W1,α(Rn), then∫
|v|α

|x|α
dx ≤ (

α

n−α
)α

∫
|∇v|αdx. (2.12)

In the end, for simplicity, we denote

c0 =
1

2
+
β

4
−

β

(n− 2)2
,

1

a0
=

1

2
−

β

(n− 2)2
.



64 N. TANG, C. WANG, AND J. ZHANG

3. Sharp energy thresholds

In this section, we state the sharp criteria for global existence and blow up of (1.1). According to

the range of parameters p and γ, we show the results in the following six cases.

Case I: 0 < p <
2

n
, 2 < γ < min{n, 4}. In this case, we have three theorems. Let

a1 =
2

2
np
2 np‖∇R‖p2

‖ϕ0‖
4−2np+2p

2
2 ,a2 =

1

2γγ‖∇W‖22
‖ϕ0‖

4−2γ
2 ,

D1 = (
2 −np
2γ − 2

)
np−2
2γ−np + (

2 −np
2γ − 2

)
2γ−2
2γ−np ,

D2 =
np

2
[
np(2 −np)
4γ(γ − 1)

]
np−2
2γ−np + γ[

np(2 −np)
4γ(γ − 1)

]
2γ−2
2γ−np ,

b1 = [
22−np(np)2γ−2γ2−np‖∇R‖2pγ−2p2 ‖∇W‖

4−2np
2

(a0D1)2γ−np
]

1
4−2np+2pγ−2p ,

b2 = [
22−np(np)2γ−2γ2−np‖∇R‖2pγ−2p2 ‖∇W‖

4−2np
2

(a0D2)2γ−np
]

1
4−2np+2pγ−2p ,

K1 =
np− 2

4γ
[

(n− 2)2(2γa1 −npa1)
2
np np

(2γ − 4)(n− 2)2 − 4β(γ − 1)
]
np

2−np .

Under the constraint : ‖ϕ0‖2 < b1, we define two invariant sets:
G1 = {ϕ ∈ H1 : E(ϕ) + c0‖ϕ‖22 < K1,‖ϕ‖2H1 < y1},

B1 = {ϕ ∈ H1 : E(ϕ) + c0‖ϕ‖22 < K1,‖ϕ‖2H1 > y1},
where y1 is the unique positive maximizer of :

f1(y) :=
1

a0
y −a1y

np
2 −a2yγ. (3.1)

Let ỹ1 > 0 be the first positive root of equation f
′
1(y) =

d

dy
f1(y) = 0.

Theorem 3.1. For 0 < p <
2

n
and 2 < γ < min{n, 4}. Assume ‖ϕ0‖2 < b1, then the following facts

are true:

(i) When ϕ0 ∈ G1 ∪{0} and f1(ỹ1) < K1, the solution ϕ(t,x) of the Cauchy problem (1.1) and
(2.1) exists globally in t ∈ (0,∞).

(ii) When ϕ0 ∈ B1 and |x|ϕ0 ∈ L2(Rn), the solution ϕ(t,x) of the Cauchy problem (1.1) and (2.1)
blows up in a finite time.

Proof. Firstly, according to (2.8) , (2.10) and (2.12), we estimate the energy functional E(ϕ), for all

t ∈ (0,T],

E(ϕ(t)) + c0‖ϕ(t)‖22 ≥
1

2
‖∇ϕ‖22 −

β

(n− 2)2
‖∇ϕ‖22 −

β

4
‖ϕ‖22 + c0‖ϕ‖

2
2

−
1

γ‖∇W‖22
‖ϕ‖4−γ2 ‖∇ϕ‖

γ
2 −

2

np‖∇R‖p2
‖ϕ‖

4−(n−2)p
2

2 ‖∇ϕ‖
np
2

2

≥ [
1

2
−

β

(n− 2)2
]‖ϕ‖2H1 −

1

2γγ‖∇W‖22
‖ϕ‖4−2γ2 ‖ϕ‖

2γ
H1

(3.2)

−
2

2
np
2 np‖∇R‖p2

‖ϕ‖
4−2np+2p

2
2 ‖ϕ‖

np
H1
.



GLOBAL EXISTENCE AND BLOWUP OF THE HARTREE EQUATION WITH COULOMB POTENTIAL 65

Let y = ‖ϕ(t)‖2
H1
≥ 0, for all t ∈ (0,T],

E(ϕ(t)) + c0‖ϕ(t)‖22 ≥ f1(‖ϕ(t)‖
2
H1 ) = f1(y), (3.3)

where f1 is defined in (3.1).

Secondly, we claim that the maximum of f1(y) on [0, +∞) is greater than 0. Let

g(y) =
1

a0
−a1y

np
2
−1 −a2yγ−1.

It follows that f1(y) = yg(y), lim
y→0+

g(y) = lim
y→+∞

g(y) = −∞ and g′(y) has only one zero point

y0 = [
(2 −np)a1
2(γ − 1)a2

]
2

2γ−np .

Thus the maximum of g(y) on [0, +∞) is g(y0). From ‖ϕ0‖2 < b1, we can obtain

a
2γ−2
1 a

2−np
2 < (a0D1)

np−2γ,

which implies

g(y0) =
1

a0
−a1y

np
2
−1

0 −a2y
γ−1
0 > 0.

Note that f1(y) → 0− as y → 0+ and f1(y) → −∞ as y → +∞. Therefore, f1(y) has the unique
positive maximizer y1 on [0,∞) and f1(y1) ≥ y0g(y0) > 0.

Thirdly, we prove the invariance of G1 and B1. When f1(ỹ1) < K1, combined with the structure

of f1(y), we can easily know that G1 is a nonempty set. If ϕ0 ∈ G1, f1(ỹ1) < K1 and ϕ(t,x) is the
corresponding solution of the Cauchy problem (1.1) and (2.1), then by Lemma 2.1, we have for all

t ∈ [0,T),
f1(‖ϕ‖2H1 ) ≤ E(ϕ) + c0‖ϕ‖

2
2 < K1. (3.4)

We only need to prove ‖ϕ‖2
H1

< y1. Otherwise, by the continuity of ϕ(t) there exists t ∈ [0,T) such
that ‖ϕ(t)‖2

H1
= y1, and then

f1(‖ϕ(t)‖2H1 ) = f1(y1) > K1,

which contradicts (3.4). Thus ‖ϕ‖2
H1

< y1, which implies the solution ϕ(t,x) of the Cauchy problem

(1.1) and (2.1) exists globally in t ∈ (0,∞). We can obtain B1 is a nonempty invariant set by the same
token.

Finally, we prove the statement (ii) of Theorem 3.1. From (2.6), we have

J′′(t) = 8γE(ϕ0) +

∫
8γ − 4np
p + 2

|ϕ|p+2 − 4(γ − 2)|∇ϕ|2 + (4γ − 4)β|x|−1|ϕ|2dx

≤ 8γE(ϕ) +
16γ − 8np

np
‖ϕ‖

4−(n−2)p
2

2 ‖∇ϕ‖
np
2

2 − 4(γ − 2)‖∇ϕ‖
2
2

+ (4γ − 4)β[
2

(n− 2)2
‖∇ϕ‖22 +

1

2
‖ϕ‖22]

≤ 8γ[E(ϕ) + c0‖ϕ‖22] + H1(y),

(3.5)

where

H1(y) = (8γ − 4np)a1y
np
2 + [

8β(γ − 1)
(n− 2)2

− 4γ + 8]y.

H′1(y) has only one zero point y
∗ on [0, +∞),

y∗ = [
(n− 2)2(2γ −np)npa1

(n− 2)2(2γ − 4) − 4β(γ − 1)
]

2
2−np .



66 N. TANG, C. WANG, AND J. ZHANG

H1(y) is increasing on(0,y
∗) and decreasing on (y∗, +∞), so the maximum of H1(y) is

H1(y
∗) = (4 − 2np)[

(n− 2)2(2γa1 −npa1)
2
np np

(n− 2)2(2γ − 4) − 4β(γ − 1)
]
np

2−np = −8γK1. (3.6)

By the invariance of B1, if ϕ0 ∈ B1 then for all t ∈ [0,T),

f1(‖ϕ‖2H1 ) ≤ E(ϕ) + c0‖ϕ‖
2
2 < K1.

Inserting the results into (3.5), we obtain

J′′(t) ≤ 8γ[E(ϕ) + c0‖ϕ‖22] + H1(y
∗) < 0.

Therefore from Lemma 2.1 and 2.3, it must be the case T < ∞, which implies that the solution ϕ(t,x)
of the Cauchy problem (1.1) and (2.1) blows up in a finite time. This completes the proof of Theorem

3.1. �

Under the constraint : b1 ≤‖ϕ0‖2 < b2, we define two invariant sets:

G2 = {ϕ ∈ H1 : E(ϕ) + c0‖ϕ‖22 < K1,‖ϕ‖2H1 < y2},

B2 = {ϕ ∈ H1 : E(ϕ) + c0‖ϕ‖22 < K1,‖ϕ‖2H1 > y2},
where y2 is the unique positive maximizer of equation (3.1).Let ỹ2 > 0 be the first positive root of the

equation f′1(y) =
d

dy
f1(y) = 0 under the constraint b1 ≤‖ϕ0‖2 < b2.

Theorem 3.2. For 0 < p <
2

n
and 2 < γ < min{n, 4}. Assume b1 ≤ ‖ϕ0‖2 < b2, then the following

facts are true:

(i) When ϕ0 ∈ G2 ∪{0} and f1(ỹ2) < K1, the solution ϕ(t,x) of the Cauchy problem (1.1) and
(2.1) exists globally in t ∈ (0,∞).

(ii) When ϕ0 ∈ B2 and |x|ϕ0 ∈ L2(Rn), the solution ϕ(t,x) of the Cauchy problem (1.1) and (2.1)
blows up in a finite time.

Proof. Firstly, we claim that f1(y) ≤ 0 and f1(y) has two extrema on [0, +∞). When b1 ≤‖ϕ0‖2 < b2,
we have

f′1(y) =
1

a0
−
npa1

2
y
np
2
−1 −γa2yγ−1,

and

f′′1 (y) =
np(2 −np)a1

4
y
np
2
−2 −γ(γ − 1)a2yγ−2.

Then f′1(y) →−∞ as y → 0+ or y → +∞, and f′′1 (y) has only one zero point ym,

ym = [
(2 −np)npa1
4(γ − 1)γa2

]
2

2γ−np ,

so the maximum of f′1(y) on [0,∞) is

f′1(ym) =
1

a0
−
npa1

2
[
(2 −np)npa1
4(γ − 1)γa2

]
np−2
2γ−np −γa2[

(2 −np)npa1
4(γ − 1)γa2

]
2γ−2
2γ−np . (3.7)

By b1 ≤‖ϕ0‖2 < b2, we can get

(a0D1)
np−2γ ≤ a2γ−21 a

2−np
2 < (a0D2)

np−2γ,

which implies that f1(y) ≤ 0 and f′1(ym) > 0. Note that f′1(y) is increasing on (0,ym) and decreasing
on (ym, +∞). Therefore f′1(y) has two zero points on [0, +∞), it follows that f1(y) has two extrema on
[0, +∞). Let y3 represent the minimal point and y2 represent the maximal point. It is not hard to find



GLOBAL EXISTENCE AND BLOWUP OF THE HARTREE EQUATION WITH COULOMB POTENTIAL 67

y3 < y2 , f1(y2) > K1.

And then, the same as the proof of Theorem 3.1, we can verify that both G2 and B2 are nonempty

invariant sets. Thus we obtain that the solution ϕ(t,x) of the Cauchy problem (1.1) and (2.1) exists

globally in t ∈ (0,∞). Besides, we can also verify

J′′(t) ≤ 8γ[E(ϕ) + c0‖ϕ‖22] + H1(y
∗) < 0.

Therefore from Lemma 2.1 and 2.3, it must be the case T < ∞, which implies the solution ϕ(t,x) of
the Cauchy problem (1.1) and (2.1) blows up in a finite time. This completes the proof of Theorem 3.2.

�

Under the constraint : ‖ϕ0‖2 ≥ b2, we define the following invariant set:
B3 = {ϕ ∈ H1 : E(ϕ) + c0‖ϕ‖22 < K1,‖ϕ‖2H1 > yk},

where yk is the unique positive solution of f1(y) = K1. Then we get a sufficient condition for blow-up

of solutions.

Theorem 3.3. Let 0 < p <
2

n
, 2 < γ < min{n, 4} and |x|ϕ0 ∈ L2(Rn). When ‖ϕ0‖2 ≥ b2 and

ϕ0 ∈ B3, the solution ϕ(t,x) of the Cauchy problem (1.1) and (2.1) blows up in a finite time.

Proof. Firstly, we claim that f1(y) ≤ 0 and f1(y) has no extrema on [0, +∞). When ‖ϕ0‖2 ≥ b2, we
have

f′1(y) =
1

a0
−
npa1

2
y
np
2
−1 −γa2yγ−1,

and

f′′1 (y) =
np(2 −np)a1

4
y
np
2
−2 −γ(γ − 1)a2yγ−2.

Then f′1(y) →−∞ as y → 0+ or y → +∞, and f′′1 (y) has only one zero point ym,

ym = [
(2 −np)npa1
4(γ − 1)γa2

]
2

2γ−np ,

so the maximum of f′1(y) on [0,∞) is

f′1(ym) =
1

a0
−
npa1

2
[
(2 −np)npa1
4(γ − 1)γa2

]
np−2
2γ−np −γa2[

(2 −np)npa1
4(γ − 1)γa2

]
2γ−2
2γ−np . (3.8)

By ‖ϕ0‖2 ≥ b2, we can get
a
2γ−2
1 a

2−np
2 ≥ (a0D2)

np−2γ,

it follows that f1(y) ≤ 0 and f′1(ym) < 0. Therefore f1(y) is decreasing on [0, +∞), which implies f1(y)
has no extrema on [0, +∞). By the monotonicity of f1(y), there exists unique yk ∈ (0, +∞) such that
f1(y) = K1 .

And then, the same as the proof of Theorem 3.1 and 3.2, we can verify that B3 is a nonempty

invariant set. Besides, we can also verify

J′′(t) ≤ 8γ[E(ϕ) + c0‖ϕ‖22] + H1(y
∗) < 0.

Therefore from Lemma 2.1 and 2.3, it must be the case T < ∞, which implies the solution ϕ(t,x) of
the Cauchy problem (1.1) and (2.1) blows up in a finite time. This completes the proof of Theorem

3.3. �

Case II: p =
2

n
, 2 < γ < min{n, 4}. Denote

y4 =
γ‖∇W‖22[((n− 2)2 − 2β)‖∇R‖

2
n
2 − (n− 2)

2‖ϕ‖
2
n
2 ]

2γ−2(n− 2)2‖∇R‖
2
n
2

,



68 N. TANG, C. WANG, AND J. ZHANG

K2 =
‖∇W‖22[((n− 2)2(γ − 2) − 2β(γ − 1))‖∇R‖

2
n
2 − (n− 2)

2(γ − 1)‖ϕ‖
2
n
2 ]

2γ−1(n− 2)4‖∇R‖
4
n
2

× [((n− 2)2 − 2β)‖∇R‖
2
n
2 − (n− 2)

2‖ϕ‖
2
n
2 ].

We define two invariant sets:

G4 = {ϕ ∈ H1 : E(ϕ) + c0‖ϕ‖22 < K2,‖ϕ‖2H1 < y4,‖ϕ‖
2
n
2 < (

γ − 2
γ − 1

−
2β

(n− 2)2
)‖∇R‖

2
n
2 },

B4 = {ϕ ∈ H1 : E(ϕ) + c0‖ϕ‖22 < K2,‖ϕ‖2H1 > y4,‖ϕ‖
2
n
2 < (

γ − 2
γ − 1

−
2β

(n− 2)2
)‖∇R‖

2
n
2 }.

Theorem 3.4. For p =
2

n
and 2 < γ < min{n, 4}, the following facts are ture:

(i) When ϕ0 ∈ G4 ∪{0}, the solution ϕ(t,x) of the Cauchy problem (1.1) and (2.1) exists globally
in t ∈ (0,∞).

(ii) When ϕ0 ∈ B4 and |x|ϕ0 ∈ L2(Rn), the solution ϕ(t,x) of the Cauchy problem (1.1) and (2.1)
blows up in a finite time.

Proof. Firstly, according to (2.8) , (2.10) and (2.12), we estimate the energy functional E(ϕ), for all

t ∈ (0,T],

E(ϕ(t)) + c0‖ϕ(t)‖22 ≥
1

2
‖∇ϕ‖22 −

β

(n− 2)2
‖∇ϕ‖22 −

β

4
‖ϕ‖22 + c0‖ϕ‖

2
2

−
1

24−γγ‖∇W‖22
‖ϕ‖4H1 −

1

2‖∇R‖
2
n
2

‖ϕ‖
2
n
2 ‖ϕ‖

2
H1

= [
1

2
−

β

(n− 2)2
]‖ϕ‖2H1 −

1

24−γγ‖∇W‖22
‖ϕ‖4H1

−
1

2‖∇R‖
2
n
2

‖ϕ‖
2
n
2 ‖ϕ‖

2
H1.

(3.9)

Let y = ‖ϕ(t)‖2
H1
≥ 0, for all t ∈ (0,T],

E(ϕ(t)) + c0‖ϕ(t)‖22 ≥ f2(‖ϕ(t)‖
2
H1 ) = f2(y), (3.10)

where

f2(y) = [
1

2
−

β

(n− 2)2
−
‖ϕ0‖

2
n
2

2‖∇R‖
2
n
2

]y −
1

24−γγ‖∇W‖22
y2,

f′2(y) = [
1

2
−

β

(n− 2)2
−
‖ϕ0‖

2
n
2

2‖∇R‖
2
n
2

] −
1

23−γγ‖∇W‖22
y.

By ‖ϕ‖
2
n
2 < (

γ − 2
γ − 1

−
2β

(n− 2)2
)‖∇R‖

2
n
2 , we know

‖ϕ‖
2
n
2 < (1 −

2β

(n− 2)2
)‖∇R‖

2
n
2 ,

so f′2(y) has only one zero point y4 on [0, +∞),

y4 =
γ‖∇W‖22[((n− 2)2 − 2β)‖∇R‖

2
n
2 − (n− 2)

2‖ϕ‖
2
n
2 ]

2γ−2(n− 2)2‖∇R‖
2
n
2

.

Then the maximum of f2(y) is



GLOBAL EXISTENCE AND BLOWUP OF THE HARTREE EQUATION WITH COULOMB POTENTIAL 69

f2(y4) =
γ‖∇W‖22[((n− 2)2 − 2β)‖∇R‖

2
n
2 − (n− 2)

2‖ϕ‖
2
n
2 ]

2

2γ(n− 2)4‖∇R‖
4
n
2

.

Secondly, we prove the invariance of G4 and B4. Combined with the structure of f2(y), we can easily

know both G4 and B4 are nonempty sets. If ϕ0 ∈ G4, by Lemma 2.1, the corresponding solution ϕ(t,x)
of Cauchy problem (1.1) and (2.1) satisfies: for all t ∈ [0,T),

f2(‖ϕ(t)‖2H1 ) ≤ E(ϕ(t)) + c0‖ϕ(t)‖
2
2 < K2, (3.11)

and

‖ϕ‖
2
n
2 < (

γ − 2
γ − 1

−
2β

(n− 2)2
)‖∇R‖

2
n
2 .

We only need to prove ‖ϕ‖2
H1

< y4. Otherwise, by the continuity of ϕ(t) there exists t ∈ [0,T) such
that ‖ϕ(t)‖2

H1
= y4, then by computation we can get

f2(‖ϕ(t)‖2H1 ) = f2(y4) > K2,

which contradicts (3.11). Thus ‖ϕ‖2
H1

< y4, which implies the solution ϕ(t,x) of the Cauchy problem

(1.1) and (2.1) exists globally in t ∈ (0,∞). We can obtain the invariance of B4 by the same token.
Finally, we prove the statement (ii) of Theorem 3.4. From (2.6), we have

J′′(t) = 8γE(ϕ0) +

∫
8γ − 4np
p + 2

|ϕ|p+2 − 4(γ − 2)|∇ϕ|2 + (4γ − 4)β|x|−1|ϕ|2dx

≤ 8γ[E(ϕ) + c0‖ϕ‖22] + H2(y),
(3.12)

where

H2(y) = [
4(γ − 1)‖ϕ0‖

2
n
2

‖∇R‖
2
n
2

− 4(γ − 2) +
8β(γ − 1)
(n− 2)2

]y.

When ‖ϕ‖
2
n
2 < (

γ − 2
γ − 1

−
2β

(n− 2)2
)‖∇R‖

2
n
2 , the maximum of H2(y) on [y4, +∞) is :

H2(y4) =
24−γγ‖∇W‖22[((n− 2)2 − 2β)‖∇R‖

2
n
2 − (n− 2)

2‖ϕ‖
2
n
2 ]

(n− 2)4‖∇R‖
4
n
2

× [(n− 2)2(γ − 1)‖ϕ‖
2
n
2 − ((n− 2)

2(γ − 2) − 2β(γ − 1))‖∇R‖
2
n
2 ]

= −8γK2.

By the invariance of B4, if ϕ0 ∈ B4, then for all t ∈ [0,T),

f2(‖ϕ‖2H1 ) ≤ E(ϕ) + c0‖ϕ‖
2
2 < K2, ‖ϕ‖

2
H1 > y4.

Inserting the results into (3.12), we obtain

J′′(t) ≤ 8γ[E(ϕ) + c0‖ϕ‖22] + H2(y4) < 0.

Therefore from Lemma 2.1 and 2.3, it must be the case T < ∞, which implies the solution ϕ(t,x) of
the Cauchy problem (1.1) and (2.1) blows up in a finite time. This completes the proof of Theorem

3.4. �



70 N. TANG, C. WANG, AND J. ZHANG

Case III:
2

n
< p <

4

n
, 2 < γ < min{n, 4}. Denote

K3 =
np− 4
4npγ

[
(n− 2)2(2γ −np)‖ϕ‖

4−(n−2)p
2

2

[(2γ − 4)(n− 2)2 − 4β(γ − 1)]
np
4 ‖∇R‖p2

]
4

4−np ,

D3 = (
4 −np
4γ − 4

)
np−4
4γ−np + (

4 −np
4γ − 4

)
4γ−4
4γ−np ,

b3 = [
‖∇R‖4pγ−4p2 ‖∇W‖

8−2np
2

2npγ−8γ+4(a0D3)4γ−np
]

1
8+4pγ−4p−2np ,

a3 =
1

2‖∇R‖P2
‖ϕ0‖

4−(n−2)p
2

2 ,a4 =
1

2γ‖∇W‖22
‖ϕ0‖

4−2γ
2 ,

f3(y) :=
1

a0
y −

2

np‖∇R‖p2
‖ϕ0‖

4−(n−2)p
2

2 y
np
4 −

1

2γγ‖∇W‖22
‖ϕ0‖

4−2γ
2

2 y
γ. (3.13)

Let ỹ3 > 0 and y5 be the first and second positive roots of the equation f
′
3(y) =

d

dy
f3(y) = 0 respectively.

Then we define two invariant sets:

G5 = {ϕ ∈ H1 : E(ϕ) + c0‖ϕ‖22 < K3,‖ϕ‖2H1 < y5,‖ϕ‖2 < b3},

B5 = {ϕ ∈ H1 : E(ϕ) + c0‖ϕ‖22 < K3,‖ϕ‖2H1 > y5,‖ϕ‖2 < b3}.

Theorem 3.5. For
2

n
< p <

4

n
and 2 < γ < min{n, 4}, the following facts are ture:

(i) When ϕ0 ∈ G5 ∪{0} and f3(ỹ3) < K3, the solution ϕ(t,x) of the Cauchy problem (1.1) and
(2.1) exists globally in t ∈ (0,∞).

(ii) When ϕ0 ∈ B5 and |x|ϕ0 ∈ L2(Rn), the solution ϕ(t,x) of the Cauchy problem (1.1) and (2.1)
blows up in a finite time.

Proof. Firstly, according to (2.8) , (2.10) and (2.12), we estimate the energy functional E(ϕ), for all

t ∈ (0,T],

E(ϕ(t)) + c0‖ϕ(t)‖22 ≥
1

2
‖∇ϕ‖22 −

β

(n− 2)2
‖∇ϕ‖22 −

β

4
‖ϕ‖22 + c0‖ϕ‖

2
2

−
1

γ‖∇W‖22
‖ϕ‖4−γ2 ‖∇ϕ‖

γ
2 −

2

np‖∇R‖p2
‖ϕ‖

4−(n−2)p
2

2 ‖∇ϕ‖
np
2

2

≥ [
1

2
−

β

(n− 2)2
]‖ϕ‖2H1 −

1

2γγ‖∇W‖22
‖ϕ‖4−2γ2 ‖ϕ‖

2γ
H1

(3.14)

−
2

np‖∇R‖p2
‖ϕ‖

4−(n−2)p
2

2 ‖ϕ‖
np
2

H1
.

Let y = ‖ϕ(t)‖2
H1
≥ 0, for all t ∈ (0,T],

E(ϕ(t)) + c0‖ϕ(t)‖22 ≥ f3(‖ϕ(t)‖
2
H1 ) = f3(y), (3.15)

where f3 is defined in (3.13). And then

f′3(y) =
1

a0
−

1

2‖∇R‖p2
‖ϕ0‖

4−(n−2)p
2

2 y
np
4
−1 −

1

2γ‖∇W‖22
‖ϕ0‖

4−2γ
2 y

γ−1

=
1

a0
−a3y

np
4
−1 −a4yγ−1,

f′′3 (y) = −
(np− 4)‖ϕ0‖

4−(n−2)p
2

2

8‖∇R‖p2
y
np
4
−2 −

(γ − 1)‖ϕ0‖
4−2γ
2

2γ‖∇W‖22
yγ−2.



GLOBAL EXISTENCE AND BLOWUP OF THE HARTREE EQUATION WITH COULOMB POTENTIAL 71

We can verify that f′′3 (y) has only one zero point

y0 = [
2γ−3(4 −np)‖∇W‖22

(γ − 1)‖ϕ0‖
2−2γ+n−2

2
p

2 ‖∇R‖
p
2

]
4

4γ−np ,

f′′3 (y) → +∞ as y → 0+ and f′′3 (y) → −∞ as y → +∞. Thus the maximum of f′3(y) on [0,∞) is
f′3(y0). By ‖ϕ0‖2 < b3, we can get

a
4γ−4
3 a

4−np
4 < (a0D3)

np−4γ,

which implies f′3(y0) > 0. Note that lim
y→+∞

f′3 = −∞, so there exists a unique y5 ∈ (y0, +∞) such that
f′3(y) = 0. Thus f3(y) is increasing on (y0,y5) and decreasing on (y5, +∞). So the maximum of f3(y)
on [0, +∞) is f3(y5).

Secondly, we prove the invariance of G5 and B5. When f3(ỹ3) < K3, combined with the structure

of f3(y), we can easily know both G5 and B5 are nonempty sets. If ϕ0 ∈ G5, by Lemma 2.1, the
corresponding solution ϕ(t,x) of Cauchy problem (1.1) and (2.1) satisfies: for all t ∈ [0,T),

f3(‖ϕ(t)‖2H1 ) ≤ E(ϕ(t)) + c0‖ϕ(t)‖
2
2 < K3, ‖ϕ‖2 < b3. (3.16)

We only need to prove ‖ϕ‖2
H1

< y5. Otherwise, by the continuity of ϕ(t) there exists t ∈ [0,T) such
that ‖ϕ(t)‖2

H1
= y5, then by computation we get

f3(‖ϕ(t)‖2H1 ) = f3(y5) > K3,

which contradicts (3.16). Thus ‖ϕ‖2
H1

< y5, which implies the solution ϕ(t,x) of the Cauchy problem

(1.1) and (2.1) exists globally in t ∈ (0,∞). We can obtain B5 is a nonempty invariant set by the same
token.

Finally, we prove the statement (ii) of Theorem 3.5. From (2.6), we have

J′′(t) = 8γE(ϕ0) +

∫
8γ − 4np
p + 2

|ϕ|p+2 − 4(γ − 2)|∇ϕ|2 + (4γ − 4)β|x|−1|ϕ|2dx

≤ 8γE(ϕ) +
16γ − 8np

np
‖ϕ‖

4−(n−2)p
2

2 ‖∇ϕ‖
np
2

2 − (4γ − 2)‖∇ϕ‖
2
2

+ (4γ − 4)β[
2

(n− 2)2
‖∇ϕ‖22 +

1

2
‖ϕ‖22]

≤ 8γ[E(ϕ) + c0‖ϕ‖22] + H3(y),

(3.17)

where

H3(y) =
16γ − 8np
np‖∇R‖p2

‖ϕ0‖
4−(n−2)p

2
2 y

np
4 + [−4(γ − 2) +

8β(γ − 1)
(n− 2)2

]y.

Then H′3 has only one zero point y
∗ on [0,∞),

y∗ = [
(n− 2)2(2γ −np)‖ϕ0‖

4−(n−2)p
2

2

[(n− 2)2(2γ − 4) − 4β(γ − 1)]‖∇R‖p2
]

4
4−np .

H3(y) is increasing on (0,y∗) and decreasing on (y∗, +∞). So the maximum of H3(y) on [0, +∞) is :

H3(y∗) =
8 − 2np
np

[
(n− 2)

np
2 (2γ −np)‖ϕ0‖

4−(n−2)p
2

2

[(n− 2)2(2γ − 4) − 4β(γ − 1)]
np
4 ‖∇R‖p2

]
4

4−np = −8γK3.

By the invariance of B5, if ϕ0 ∈ B5, then for all t ∈ [0,T),

f3(‖ϕ‖2H1 ) ≤ E(ϕ) + c0‖ϕ‖
2
2 < K3, ‖ϕ‖

2
H1 > y5.

Inserting the results into (3.17), we obtain

J′′(t) ≤ 8γ[E(ϕ) + c0‖ϕ‖22] + H3(y∗) < 0.



72 N. TANG, C. WANG, AND J. ZHANG

Therefore from Lemma 2.1 and 2.3, it must be the case T < ∞, which implies the solution ϕ(t,x) of
the Cauchy problem (1.1) and (2.1) blows up in a finite time. This completes the proof of Theorem

3.5. �

Case IV: p =
4

n
, 2 < γ < min{n, 4}. Denote

y6 =
γ‖∇W‖22[((n− 2)2 − 2β)‖∇R‖

4
n
2 − (n− 2)

2‖ϕ‖
4
n
2 ]

2γ−2(n− 2)2‖∇R‖
4
n
2

,

K4 =
‖∇W‖22[((n− 2)2(γ − 2) − 2β(γ − 1))‖∇R‖

4
n
2 − (n− 2)

2(γ − 1)‖ϕ‖
4
n
2 ]

2γ−1(n− 2)4‖∇R‖
8
n
2

× [((n− 2)2 − 2β)‖∇R‖
4
n
2 − (n− 2)

2‖ϕ‖
4
n
2 ].

We define two invariant sets:

G6 = {ϕ ∈ H1 : E(ϕ) + c0‖ϕ‖22 < K4,‖ϕ‖2H1 < y6,‖ϕ‖
4
n
2 < (

γ − 2
γ − 1

−
2β

(n− 2)2
)‖∇R‖

4
n
2 },

B6 = {ϕ ∈ H1 : E(ϕ) + c0‖ϕ‖22 < K4,‖ϕ‖2H1 > y6,‖ϕ‖
4
n
2 < (

γ − 2
γ − 1

−
2β

(n− 2)2
)‖∇R‖

4
n
2 }.

Theorem 3.6. For p =
4

n
and 2 < γ < min{n, 4}, the following facts are ture:

(i) When ϕ0 ∈ G6 ∪{0}, the solution ϕ(t,x) of the Cauchy problem (1.1) and (2.1) exists globally
in t ∈ (0,∞).

(ii) When ϕ0 ∈ B6 and |x|ϕ0 ∈ L2(Rn), the solution ϕ(t,x) of the Cauchy problem (1.1) and (2.1)
blows up in a finite time.

Proof. Firstly, according to (2.8) , (2.10) and (2.12), we estimate the energy functional E(ϕ), for all

t ∈ (0,T],

E(ϕ(t)) + c0‖ϕ(t)‖22 ≥
1

2
‖∇ϕ‖22 −

β

(n− 2)2
‖∇ϕ‖22 −

β

4
‖ϕ‖22 + c0‖ϕ‖

2
2

−
1

24−γγ‖∇W‖22
‖ϕ‖4H1 −

1

2‖∇R‖
4
n
2

‖ϕ‖
4
n
2 ‖ϕ‖

2
H1

= [
1

2
−

β

(n− 2)2
]‖ϕ‖2H1 −

1

24−γγ‖∇W‖22
‖ϕ‖4H1

−
1

2‖∇R‖
4
n
2

‖ϕ‖
4
n
2 ‖ϕ‖

2
H1.

(3.18)

Let y = ‖ϕ(t)‖2
H1
≥ 0, for all t ∈ (0,T],

E(ϕ(t)) + c0‖ϕ(t)‖22 ≥ f4(‖ϕ(t)‖
2
H1 ) = f4(y), (3.19)

where

f4(y) = [
1

2
−

β

(n− 2)2
−
‖ϕ0‖

4
n
2

2‖∇R‖
4
n
2

]y −
1

24−γγ‖∇W‖22
y2,

f′4(y) = [
1

2
−

β

(n− 2)2
−
‖ϕ0‖

4
n
2

2‖∇R‖
4
n
2

] −
1

23−γγ‖∇W‖22
y.



GLOBAL EXISTENCE AND BLOWUP OF THE HARTREE EQUATION WITH COULOMB POTENTIAL 73

By ‖ϕ‖
4
n
2 < (

γ − 2
γ − 1

−
2β

(n− 2)2
)‖∇R‖

4
n
2 , we know

‖ϕ‖
4
n
2 < (1 −

2β

(n− 2)2
)‖∇R‖

4
n
2 ,

so f′4(y) has only one zero point y6 on [0, +∞),

y6 =
γ‖∇W‖22[((n− 2)2 − 2β)‖∇R‖

4
n
2 − (n− 2)

2‖ϕ‖
4
n
2 ]

2γ−2(n− 2)2‖∇R‖
4
n
2

.

Then the maximum of f4(y) is

f4(y6) =
γ‖∇W‖22[((n− 2)2 − 2β)‖∇R‖

4
n
2 − (n− 2)

2‖ϕ‖
4
n
2 ]

2

2γ(n− 2)4‖∇R‖
8
n
2

.

Secondly, we prove the invariance of G6 and B6. Combined with the structure of f4(y), we can easily

know both G6 and B6 are nonempty sets. If ϕ0 ∈ G6, by Lemma 2.1, the corresponding solution ϕ(t,x)
of Cauchy problem (1.1) and (2.1) satisfies: for all t ∈ [0,T),

f4(‖ϕ(t)‖2H1 ) ≤ E(ϕ(t)) + c0‖ϕ(t)‖
2
2 < K4, (3.20)

‖ϕ‖
4
n
2 < (

γ − 2
γ − 1

−
2β

(n− 2)2
)‖∇R‖

4
n
2 .

We only need to prove ‖ϕ‖2
H1

< y6. Otherwise, by the continuity of ϕ(t) there exists t ∈ [0,T) such
that ‖ϕ(t)‖2

H1
= y6. Then by computation we get

f4(‖ϕ(t)‖2H1 ) = f4(y6) > K4,

which contradicts (3.20). Thus ‖ϕ‖2
H1

< y6, which implies the solution ϕ(t,x) of the Cauchy problem

(1.1) and (2.1) exists globally in t ∈ (0,∞). We can obtain the invariance of B6 by the same token.
Finally, we prove the statement (ii) of Theorem 3.6. From (2.6), we have

J′′(t) = 8γE(ϕ0) +

∫
8γ − 4np
p + 2

|ϕ|p+2 − 4(γ − 2)|∇ϕ|2 + (4γ − 4)β|x|−1|ϕ|2dx

≤ 8γ[E(ϕ) + c0‖ϕ‖22] + H4(y),
(3.21)

where

H4(y) = [
4(γ − 1)‖ϕ0‖

4
n
2

‖∇R‖
4
n
2

− 4(γ − 2) +
8β(γ − 1)
(n− 2)2

]y.

When ‖ϕ‖
4
n
2 < (

γ − 2
γ − 1

−
2β

(n− 2)2
)‖∇R‖

4
n
2 , the maximum of H4(y) on [y6, +∞) is :

H4(y6) =
24−γγ‖∇W‖22[((n− 2)2 − 2β)‖∇R‖

4
n
2 − (n− 2)

2‖ϕ‖
4
n
2 ]

(n− 2)4‖∇R‖
8
n
2

× [(n− 2)2(γ − 1)‖ϕ‖
4
n
2 − ((n− 2)

2(γ − 2) − 2β(γ − 1))‖∇R‖
4
n
2 ]

= −8γK4.

By the invariance of B6, if ϕ0 ∈ B6, then for all t ∈ [0,T),

f4(‖ϕ‖2H1 ) ≤ E(ϕ) + c0‖ϕ‖
2
2 < K4, ‖ϕ‖

2
H1 > y6.

Inserting the results into (3.21), we obtain

J′′(t) ≤ 8γ[E(ϕ) + c0‖ϕ‖22] + H4(y6) < 0.



74 N. TANG, C. WANG, AND J. ZHANG

Therefore from Lemma 2.1 and 2.3, it must be the case T < ∞, which implies the solution ϕ(t,x) of
the Cauchy problem (1.1) and (2.1) blows up in a finite time. This completes the proof of Theorem 3.6.

�

Case V:
4

n
< p <

4

n− 2
, 2 < γ <

np

2
. Denote

K5 =
(n− 2)2(γ − 2) − 2β(γ − 1)

2(n− 2)2γ
Y 2.

Then we have two invariant sets:

G7 = {ϕ ∈ H1 : E(ϕ) +
(β + 1)γ − 1

4γ
‖ϕ‖22 < K5,‖ϕ‖2 <

2

n− 2
Y,‖∇ϕ‖2 < Y},

B7 = {ϕ ∈ H1 : E(ϕ) +
(β + 1)γ − 1

4γ
‖ϕ‖22 < K5,‖ϕ‖2 <

2

n− 2
Y,‖∇ϕ‖2 > Y},

where Y is shown in the proof of the following theorem:

Theorem 3.7. For
4

n
< p <

4

n− 2
and 2 < γ <

np

2
, the following facts are ture:

(i) When ϕ0 ∈ G7 ∪{0}, the solution ϕ(t,x) of the Cauchy problem (1.1) and (2.1) exists globally
in t ∈ (0,∞).

(ii) When ϕ0 ∈ B7 and |x|ϕ0 ∈ L2(Rn), the solution ϕ(t,x) of the Cauchy problem (1.1) and (2.1)
blows up in a finite time.

Proof. Firstly, according to (2.8) , (2.10) and (2.12), we estimate the energy functional E(ϕ), for all

t ∈ (0,T],

E(ϕ(t)) +
β(γ − 1)

4γ
‖ϕ(t)‖22 ≥

1

2
‖∇ϕ‖22 −

β

(n− 2)2
‖∇ϕ‖22 −

β

4
‖ϕ‖22 +

β(γ − 1)
4γ

‖ϕ‖22

−
1

γ‖∇W‖22
‖ϕ‖4−γ2 ‖∇ϕ‖

γ
2 −

2

np‖∇R‖p2
‖ϕ‖

4−(n−2)p
2

2 ‖∇ϕ‖
np
2

2

= [
1

2
−

β

(n− 2)2
]‖∇ϕ‖22 −

β

4γ
‖ϕ‖22 −

1

γ‖∇W‖22
‖ϕ‖4−γ2 ‖∇ϕ‖

γ
2

−
2

np‖∇R‖p2
‖ϕ‖

4−(n−2)p
2

2 ‖∇ϕ‖
np
2

2 . (3.22)

Let y = ‖∇ϕ(t)‖2 ≥ 0, for all t ∈ (0,T],

E(ϕ(t)) +
β(γ − 1)

4γ
‖ϕ(t)‖22 ≥ ~1(‖∇ϕ(t)‖2) = ~1(y), (3.23)

where

~1(y) = [
1

2
−

β

(n− 2)2
]y2 −

β

4γ
‖ϕ‖22 −

1

γ‖∇W‖22
‖ϕ‖4−γ2 y

γ −
2

np‖∇R‖p2
‖ϕ‖

4−(n−2)p
2

2 y
np
2 ,

~′1(y) = [1 −
2β

(n− 2)2
−

1

‖∇W‖22
‖ϕ‖4−γ2 y

γ−2 −
1

‖∇R‖p2
‖ϕ‖

4−(n−2)p
2

2 y
np
2
−2]y = ~2(y)y.

Thus ~2(y) = 0 has only one positive solution,

~′2(y) = −(γ − 2)
1

‖∇W‖22
‖ϕ‖4−γ2 y

γ−3 −
np− 4

2‖∇R‖p2
‖ϕ‖

4−(n−2)p
2

2 y
np
2
−3 < 0,

which implies ~2 is decreasing on [0, +∞). Note that ~2(0) = 1 −
β

(n− 2)2
> 0 and



GLOBAL EXISTENCE AND BLOWUP OF THE HARTREE EQUATION WITH COULOMB POTENTIAL 75

~2[(
((n− 2)2 −β)‖∇W‖22

(n− 2)2‖ϕ‖4−γ2
)

1
γ−2 ] = −

‖ϕ‖
4−(n−2)p

2
2

‖∇R‖p2
(
((n− 2)2 −β)‖∇W‖22

(n− 2)2‖ϕ‖4−γ2
)

1
γ−2 (

np
2
−2) < 0.

Since ~2 is continuous on [0, +∞), there exists a unique positive Y,

Y ∈ [0, (
((n− 2)2 −β)‖∇W‖22

(n− 2)2‖ϕ‖4−γ2
)

1
γ−2 ],

such that ~2(Y ) = 0, thus the maximum of ~1(y) is ~1(Y ).
Secondly, we prove the invariance of G7 and B7. Combined with the structure of ~1(y), we can

easily know both G7 and B7 are nonempty sets. If ϕ0 ∈ G7, by Lemma 2.1 and ‖ϕ‖2 <
2

n− 2
Y , the

corresponding solution ϕ(t,x) of Cauchy problem (1.1) and (2.1) satisfies: for all t ∈ [0,T),

~1(‖∇ϕ(t)‖22) ≤ E(ϕ) +
β(γ − 1)

4γ
‖ϕ‖22 <

(n− 2)2(γ − 2) − 2β(γ − 1)
2(n− 2)2γ

Y 2 < ~1(Y ). (3.24)

We only need to prove ‖∇ϕ‖2 < Y . Otherwise, by the continuity of ϕ(t) there exists t ∈ [0,T) such
that ‖∇ϕ(t)‖2 = Y , then by computation we get

~1(‖∇ϕ(t)‖2) = ~1(Y ) ≤ E(ϕ) +
β(γ − 1)

4γ
‖ϕ‖22,

which contradicts (3.24). Thus ‖∇ϕ‖2 < Y , which implies the solution ϕ(t,x) of the Cauchy problem
(1.1) and(2.1) exists globally in t ∈ (0,∞). We can obtain the invariance of B7 by the same token.

Finally, we prove the statement (ii) of Theorem 3.7. From (2.6), we have

J′′(t) ≤ 8γE(ϕ0) +
∫
−4(γ − 2)|∇ϕ|2 + (4γ − 4)β[

2

(n− 2)2
|∇ϕ|2 +

1

2
|ϕ|2]dx

= 8γ[E(ϕ) +
β(γ − 1)

4γ
‖ϕ‖22] −

4(n− 2)2(γ − 2) − 8β(γ − 1)
(n− 2)2

‖∇ϕ‖22

≤ 8γ
(n− 2)2(γ − 2) − 2β(γ − 1)

2γ(n− 2)2
Y 2 −

4(n− 2)2(γ − 2) − 8β(γ − 1)
(n− 2)2

Y 2 = 0.

Therefore from Lemma 2.1 and 2.3, it must be the case T < ∞, which implies the solution ϕ(t,x) of
the Cauchy problem (1.1) and (2.1) blows up in a finite time. This completes the proof of Theorem 3.7.

�

Case VI:
4

n
< p <

4

n− 2
,
np

2
≤ γ < min{4,n}. Denote

K6 =
(n− 2)2(np− 4) −β(2np− 4)

2(n− 2)2np
Y ′2.

Then we have two invariant sets:

G8 = {ϕ ∈ H1 : E(ϕ) +
(β + 1)γ − 1

4γ
‖ϕ‖22 < K6,‖ϕ‖2 <

2

(n− 2)
Y ′,‖∇ϕ‖2 < Y ′},

B8 = {ϕ ∈ H1 : E(ϕ) +
(β + 1)γ − 1

4γ
‖ϕ‖22 < K6,‖ϕ‖2 <

2

(n− 2)
Y ′,‖∇ϕ‖2 > Y ′},

where Y ′ is shown in the proof of the following theorem:

Theorem 3.8. For
4

n
< p <

4

n− 2
and

np

2
≤ γ < min{4,n}, the following facts are ture:

(i) When ϕ0 ∈ G8 ∪{0} , the solution ϕ(t,x) of the Cauchy problem (1.1) and (2.1) exists globally
in t ∈ (0,∞).



76 N. TANG, C. WANG, AND J. ZHANG

(ii) When ϕ0 ∈ B8 and |x|ϕ0 ∈ L2(Rn), the solution ϕ(t,x) of the Cauchy problem (1.1) and (2.1)
blows up in a finite time.

Proof. Firstly, according to (2.8) , (2.10) and (2.12), we estimate the energy functional E(ϕ), for all

t ∈ (0,T],

E(ϕ(t)) +
β(np− 2)

4np
‖ϕ(t)‖22 ≥

1

2
‖∇ϕ‖22 −

β

(n− 2)2
‖∇ϕ‖22 −

β

4
‖ϕ‖22 +

β(np− 2)
4np

‖ϕ‖22

−
1

γ‖∇W‖22
‖ϕ‖4−γ2 ‖∇ϕ‖

γ
2 −

2

np‖∇R‖p2
‖ϕ‖

4−(n−2)p
2

2 ‖∇ϕ‖
np
2

2

= [
1

2
−

β

(n− 2)2
]‖∇ϕ‖22 −

β

2np
‖ϕ‖22 −

1

γ‖∇W‖22
‖ϕ‖4−γ2 ‖∇ϕ‖

γ
2

−
2

np‖∇R‖p2
‖ϕ‖

4−(n−2)p
2

2 ‖∇ϕ‖
np
2

2 . (3.25)

Let y = ‖∇ϕ(t)‖2 ≥ 0, for all t ∈ (0,T],

E(ϕ(t)) +
β(np− 2)

4np
‖ϕ(t)‖22 ≥ ~3(‖∇ϕ(t)‖2) = ~3(y), (3.26)

where

~3(y) = [
1

2
−

β

(n− 2)2
]y2 −

β

2np
‖ϕ‖22 −

1

γ‖∇W‖22
‖ϕ‖4−γ2 y

γ −
2

np‖∇R‖p2
‖ϕ‖

4−(n−2)p
2

2 y
np
2 ,

~′3(y) = [1 −
2β

(n− 2)2
−

1

‖∇W‖22
‖ϕ‖4−γ2 y

γ−2 −
1

‖∇R‖p2
‖ϕ‖

4−(n−2)p
2

2 y
np
2
−2]y = ~2(y)y.

The same as the proof of Theorem 3.7, there exists a unique positive Y ′ such that ~2(Y ′) = 0, thus the
maximum of ~3(y) is ~3(Y ′).

Secondly, we prove the invariance of G8 and B8. Combined with the structure of ~3(y), we can easily
know both G8 and B8 are nonempty sets. If ϕ0 ∈ G8, by Lemma 2.1 and ‖ϕ‖22 <

8

(n− 2)2
Y ′2, the

corresponding solution ϕ(t,x) of Cauchy problem (1.1) and (2.1) satisfies: for all t ∈ [0,T),

~3(‖∇ϕ(t)‖22) ≤ E(ϕ) +
β(np− 2)

4np
‖ϕ‖22 <

(n− 2)2(np− 4) −β(2np− 4)
2(n− 2)2np

Y ′2 < ~3(Y ′). (3.27)

We only need to prove ‖∇ϕ‖2 < Y ′. Otherwise, by the continuity of ϕ(t) there exists t ∈ [0,T) such
that ‖∇ϕ(t)‖2 = Y ′, then by computation we get

~3(‖∇ϕ(t)‖2) = ~3(Y ′) ≤ E(ϕ) +
β(np− 2)

4np
‖ϕ‖22,

which contradicts (3.27). Thus ‖∇ϕ‖2 < Y ′, which implies the solution ϕ(t,x) of the Cauchy problem
(1.1) and (2.1) exists globally in t ∈ (0,∞) . We can obtain the invariance of B8 by the same token.

Finally, we prove the statement (ii) of Theorem 3.8. From (2.6), we have

J′′(t) = 4npE(ϕ0) −
∫

(2np− 8)|∇ϕ|2 − (np− 2γ)(|x|−γ ∗ |ϕ|2)|ϕ|2 − (2np− 4)β|x|−1|ϕ|2dx

≤ 4npE(ϕ0) −
∫

(2np− 8)|∇ϕ|2 − (2np− 4)β[
2

(n− 2)2
|∇ϕ|2 +

1

2
|ϕ|2]dx

= 4np[E(ϕ) +
β(np− 2)

4np
‖ϕ‖22] −

(n− 2)2(2np− 8) −β(4np− 8)
(n− 2)2

‖∇ϕ‖22

≤ 4np
(n− 2)2(np− 4) −β(2np− 4)

2(n− 2)2np
Y ′2 −

(n− 2)2(2np− 8) −β(4np− 8)
(n− 2)2

Y ′2 = 0.

(3.28)



GLOBAL EXISTENCE AND BLOWUP OF THE HARTREE EQUATION WITH COULOMB POTENTIAL 77

Therefore from Lemma 2.1 and 2.3, it must be the case T < ∞, which implies the solution ϕ(t,x) of
the Cauchy problem (1.1) and (2.1) blows up in a finite time. This completes the proof of Theorem 3.8.

�

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N. Tang, School of Mathematical Sciences, University of Electronic Science and Technology of China,

Chengdu, Sichuan, 711731, P. R. China

Email address: tangna514@163.com

C. Wang, School of Mathematical Sciences, University of Electronic Science and Technology of China,

Chengdu, Sichuan, 711731, P. R. China

Email address: wangchenglinedu@163.com

J. Zhang, corresponding author, School of Mathematical Sciences, University of Electronic Science and

Technology of China, Chengdu, Sichuan, 711731, P. R. China

Email address: zhangjian@uestc.edu.cn