# Existence of PrescribedL

Authors: Yisheng Huang; Zeng Liu; Yuanze Wu
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

By using the standard scaling arguments, we show that the infimum of the following
minimization problem:I
ρ
2=
inf
{
(
1
/
2
)
∫
ℝ
3
|
∇
u
|
2
d
x
+
(
1
/
4
)
∬
ℝ
3
(
|
u
(
x
)
|
2
|
u
(
y
)
|
2
/
|
x
-
y
|
)
d
x
d
y −  (
1
/
p
)
∫
ℝ
3
|
u
|
p
d
x
:
u
∈
B
ρ
} can be achieved for p
∈
(
2,3
) and ρ
>
0 small, where B
ρ
:
=
{
u
∈
H
1
(
ℝ
3
)
:
∥
u
∥
2
=
ρ
}. Moreover, the properties of I
ρ
2
/
ρ
2 and the associated Lagrange multiplier
λ
ρ are also given if p
∈
(
2,8
/
3
].


---

## Body

## 1. Introduction

In this paper, we consider the nonlinear Schrödinger-Poisson type equation:(1)
-
Δ
u
+
(
|
x
|
-
1
*
|
u
|
2
)
u
-
|
u
|
p
-
2
u
=
λ
u
,
in
ℝ
3
, whereλ
∈
ℝis a parameter,p
∈
(
2,6
), and*denotes the convolution. Problems like (1) have attracted considerable attentions
recently since a pair(
u
,
λ
), solution of (1), corresponds to a solitary wave of the formψ
(
x
,
t
)
=
e
-
i
λ
t
u
(
x
)of the evolution equation: (2)
i
ψ
t
+
Δ
ψ
-
(
|
x
|
-
1
*
|
ψ
|
2
)
ψ
+
|
ψ
|
p
-
2
ψ
=
0
,
in
ℝ
3
×
ℝ
+
, which was obtained by approximation of a special case of
Hatree-Fock equation with the frequencyλdescribing a quantum mechanical system of many particles. For
more mathematical and physical background of (2), we refer to [1–4] and the references therein.In the case that the frequencyλis a fixed and assigned parameter, the critical points of the
following functional defined inH
1
(
ℝ
3
;
ℝ
): (3)
E
(
u
)
=
1
2
∫
ℝ
3
‍
|
∇
u
|
2
d
x
-
λ
2
∫
ℝ
3
‍
|
u
|
2
d
x
+
1
4
∬
ℝ
3
|
u
(
x
)
|
2
|
u
(
y
)
|
2
|
x
-
y
|
d
x
d
y
-
1
p
∫
ℝ
3
‍
|
u
|
p
d
x
, are the solutions of (1), whereE
(
u
)is obviously well defined and is aC
1functional for eachp
∈
(
2,6
)(cf. [5]). Such case has
been extensively studied by using variational methods in the past decades including the
existence, nonexistence, and multiplicity of solutions; see, for example, [5–12] and
the references therein.On the other hand, the physicists are often interested in the solutions with
prescribedL
2-norm and unknown frequencyλ, such a solution is called a “normalized
solution,” which is associated with the existence of stable standing waves.
Precisely, by a “normalized solution”(
u
ρ
,
λ
ρ
)of (1), we
mean that (4)
(
u
ρ
,
λ
ρ
)
∈
H
1
(
ℝ
3
;
ℂ
)
×
ℝ
solves
(
1
)
with
∥
u
ρ
∥
2
=
ρ
. Clearly, this kind of solutions can be obtained as the
constrained critical points of theC
1functional (5)
I
(
u
)
=
1
2
∫
ℝ
3
‍
|
∇
u
|
2
d
x
+
1
4
∬
ℝ
3
|
u
(
x
)
|
2
|
u
(
y
)
|
2
|
x
-
y
|
d
x
d
y
-
1
p
∫
ℝ
3
‍
|
u
|
p
d
x
, on the constraint (6)
B
ρ
=
{
u
∈
H
1
(
ℝ
3
;
ℂ
)
:
∥
u
∥
2
=
ρ
}
. Thus, the frequencyλ
ρ
∈
ℝcannot be fixed any longer and it will appear as a Lagrange
multiplier associated with the critical pointu
ρonB
ρ. Among all the critical points ofIconstrained onB
ρ, we are interested in the ones with minimal energy since the
corresponded standing waves are orbitally stable under the flow of (2) and can provide us some information on the
dynamics of (2). Therefore, we are reduced
to study the minimization problem (7)
I
ρ
2
=
min
u
∈
B
ρ
I
(
u
)
, forp
∈
(
2,10
/
3
). Here we note that, for eachρ
>
0,I
ρ
2
∈
(
-
∞
,
0
]ifp
∈
(
2,10
/
3
), andI
ρ
2
=
-
∞ifp
∈
(
10
/
3,6
)(cf. [13, Remark 1.1]
or (15) below).
Whenp
∈
(
10
/
3,6
)(nowI
ρ
2
=
-
∞), by using a mountain pass argument, it was proved in [14] thatIhas a critical point constrained onB
ρat a strictly positive energy level
forρ
>
0small, and this critical point is orbitally unstable.The main difficulty of considering (7) is
the lack of compactness for the (bounded) minimizing sequence{
u
n
}
⊂
B
ρ. We recall that the necessary and sufficient condition due to
Lions [15, 16] in order that any minimizing sequence for (7) is relatively compact is the strong subadditivity inequality:
(8)
I
ρ
2
<
I
μ
2
+
I
ρ
2
-
μ
2
,
∀
0
<
μ
<
ρ
.In the rangep
∈
{
8
/
3
}
∪
(
3,10
/
3
), by using the standard scaling arguments, Bellazzini and
Siciliano in [17] proved that (8) holds forρ
>
0large. In the rangep
∈
(
2,3
), Bellazzini and Siciliano also showed in [18] that (8) holds forρ
>
0small, where they developed a new abstract theorem which
guarantees the following condition(
M
D
)fors
>
0small:(MD) The functions
↦
I
s
2
/
s
2is monotone decreasing.We remark that their abstract theorem heavily relies on the behavior
ofI
ρ
2near zero; that is, to use the abstract theorem, one has to
verify some extra conditions, such as (9)
ρ
↦
I
ρ
2
is
continuous
;
lim
ρ
→
0
I
ρ
2
ρ
2
=
0
; these are unnecessary if one can show (8) by using the standard scaling arguments
like [17]. However, as mentioned in [18], the authors were not sure whether (8) can be proved or not by using the standard
scaling arguments ifp
∈
(
2,3
). Therefore, the first aim of this paper is to show that
(8) holds forρ
>
0small whenp
∈
(
2,3
)by using the standard scaling arguments. To achieve this aim,
we introduce a new subsetB
ρ
∩
𝒫ofB
ρ(see details in Section 3), then we consider the minimization problem (7) constrained onB
ρ
∩
𝒫instead ofB
ρ, and we use the standard scaling arguments to prove that
(8) holds forρ
>
0small. Moreover, we can get a specific estimate
onρthat allows us to obtain the sign and the behavior of the
Lagrange multiplierλ
ρifp
∈
(
2,8
/
3
]; these are not considered in [18].The other aim of this paper is to study the properties of the Lagrange
multiplierλ
ρand the ratioI
ρ
2
/
ρ
2corresponding to the solution(
u
ρ
,
λ
ρ
)of (1)
with∥
u
ρ
∥
2
=
ρ. It is known thatλ
ρandI
ρ
2
/
ρ
2are interpreted in physics as the frequency and the ratio
between the infimum of the energy of the standing waves with fixed charge and the charge
itself, respectively, and the relevance of the energy/charge ratio for the existence of
standing waves in field theories has been discussed under a general framework in [19].Our main results read as follows.Theorem 1.
All the minimizing sequences for (7)
are precompact inH
1
(
ℝ
3
;
ℂ
)up to translations provided that one of the following
conditions holds (1)
p
∈
(
2,8
/
3
]and0
<
ρ
<
ρ
¯
1
:
=
3
(
3
p
-
8
)
/
8
(
3
-
p
)
π
(
3
p
-
10
)
/
8
(
3
-
p
)
S
3
/
2
(
(
p
-
2
)
/
2
p
)
1
/
4
(
3
-
p
), whereSis defined by (12);
(2)
p
∈
(
8
/
3,3
)and0
<
ρ
<
ρ
¯
2for someρ
¯
2
>
0. In particular, (1) has a
solution(
u
ρ
,
λ
ρ
)
∈
H
1
(
ℝ
3
;
ℂ
)
×
ℝsuch thatI
(
u
ρ
)
=
I
ρ
2and∥
u
ρ
∥
2
=
ρ. Moreover, if the above assumption (1) holds
and(
u
ρ
,
λ
ρ
)is a solution of (1) with∥
u
ρ
∥
2
=
ρ
>
0andI
(
u
ρ
)
=
I
ρ
2, thenλ
ρ
<
0,λ
ρ
→
0andI
ρ
2
/
ρ
2
→
0asρ
→
0, respectively.Theorem 2.
Letp
∈
(
2,12
/
5
]and let ρ
>
0. If(
u
ρ
,
λ
ρ
)is a solution of (1) with∥
u
ρ
∥
2
=
ρ, then we have (i)
λ
ρ
<
0,I
(
u
ρ
)
<
0,λ
ρ
→
0asρ
→
0and there exists a positive
constantC
1, independent ofρ, such thatλ
ρ
∈
(
-
C
1
,
0
);
(ii)
there exists a positive constantC
2, independent ofρ, such thatI
(
u
ρ
)
/
ρ
2
∈
(
-
C
2
,
0
). In particular, ifI
(
u
ρ
)
=
I
ρ
2, thenI
ρ
2
/
ρ
2
∈
(
-
C
2
,
0
).Remarks. (a) We point out that parts of Theorem 1 are already contained in [18,
Theorem 4.1]. In the proof of Theorem 1,
withρ
¯
1in hand, we can obtain some additional information of the
Lagrange multiplierλ
ρand the ratioI
ρ
2
/
ρ
2whenp
∈
(
2,8
/
3
], and these are not contained in [18, Theorem 4.1]. However, we do not know
whetherρ
¯
1is optimal or not.(b) Theorem2(
i
)shows that (1) has only the zero solution ifp
∈
(
2,12
/
5
]andλ
≥
0. In the case ofp
∈
(
2,3
), it was shown in [5,
20] (see also [13, Remark 1.4]) that there existsλ
0
<
0such that (1) has only the zero solution forλ
∈
(
-
∞
,
λ
0
). The nonexistence results of nonzero solutions of (1) were also discussed in [13] forp
∈
[
3,10
/
3
].(c) As we have anticipated, the existence of minimizers forI
ρ
2is related to the existence and stability of the standing wave
solutions to (2). For the existence of
stable standing wave solutions to (2), we
refer to [4, 14, 17, 18, 20, 21] and the references therein.This paper is organized as follows. In Section2, we
give some preliminaries. Section 3 is devoted to the
proof of the main theorems, especially in the proof of Theorem 1, we first define a new subset ofB
ρ and then analyze the properties of minimizing sequences
forI
ρ
2constrained on the new subset, and finally, we prove that
(8) holds whenp
∈
(
2,8
/
3
]andp
∈
(
8
/
3,3
), respectively.


## 2. Preliminaries

Throughout this paper, all the functions, unless otherwise stated, are complex valued, but
for simplicity we will writeL
q
(
ℝ
3
),H
1
(
ℝ
3
)and𝒟
1,2
(
ℝ
3
)defined in the following: (i)
L
q
(
ℝ
3
)is the usual Lebesgue space endowed with the
norm∥
u
∥
q
:
=
(
∫
ℝ
3
‍
|
u
|
q
d
x
)
1
/
q, whereq
∈
[
1
,
∞
);
(ii)
H
1
(
ℝ
3
)is the usual Sobolev space endowed with the norm
(10)
∥
u
∥
:
=
(
∫
ℝ
3
‍
(
|
∇
u
|
2
+
|
u
|
2
)
d
x
)
1
/
2
;
(iii)
𝒟
1,2
(
ℝ
3
)is the completion ofC
0
∞
(
ℝ
3
)with respect to the norm (11)
∥
∇
u
∥
2
=
(
∫
ℝ
3
‍
|
∇
u
|
2
d
x
)
1
/
2
;
(iv)
Sis the best Sobolev imbedding constant
of𝒟
1,2
(
ℝ
3
)
↪
L
6
(
ℝ
3
)defined as (12)
S
:
=
inf
{
∥
∇
u
∥
2
:
u
∈
𝒟
1,2
(
ℝ
3
)
,
∥
u
∥
6
=
1
}
.Moreover, the letterCwill denote a suitable positive constant, whose value may
change in the same line, and the symbolo
(
1
)denotes a quantity which goes to zero. We also
useO
(
1
)to denote a bounded quantity.Letϕ
u
(
x
)
=
|
x
-
y
|
-
1
*
|
u
|
2, and then, for eachu
∈
H
1
(
ℝ
3
),ϕ
uis the unique solution of the Poisson
equation-
Δ
ϕ
=
4
π
|
u
|
2 and is usually interpreted as the Coulombian potential of the
electrostatic field generated by the charge density|
u
|
2. Evidently, see, for example [5], (13)
∫
ℝ
3
‍
|
∇
ϕ
u
|
2
d
x
=
4
π
∫
ℝ
3
‍
ϕ
u
|
u
|
2
d
x
=
4
π
∬
ℝ
3
|
u
(
x
)
|
2
|
u
(
y
)
|
2
|
x
-
y
|
d
x
d
y
,
(14)
∫
ℝ
3
‍
ϕ
u
|
u
|
2
d
x
≤
∥
ϕ
u
∥
6
∥
u
∥
12
/
5
2
≤
4
π
S
2
∥
u
∥
12
/
5
4
. For eachρ
>
0, letu
∈
B
ρandu
t
(
x
)
=
t
3
/
2
u
(
t
x
)
(
t
>
0
), and we have∥
u
t
∥
2
=
∥
u
∥
2
=
ρ, that is,u
t
∈
B
ρ. Let (15)
f
u
(
t
)
=
I
(
u
t
)
=
t
2
2
∫
ℝ
3
‍
|
∇
u
|
2
d
x
+
t
4
∫
ℝ
3
‍
ϕ
u
|
u
|
2
d
x
-
t
(
3
/
2
)
(
p
-
2
)
p
∫
ℝ
3
‍
|
u
|
p
d
x
; it is clear thatI
ρ
2
≤
0for allρ
>
0sincef
u
(
t
)
→
0ast
→
0.We now recall an abstract result on the constrained minimization problem(16)
J
ρ
2
=
inf
u
∈
B
ρ
J
(
u
)
,
(
we
agree
J
0
=
0
)
, whereρ
>
0,B
ρ
=
{
u
∈
H
1
(
ℝ
3
)
:
∥
u
∥
2
=
ρ
},J
ρ
2
>
-
∞is assumed, and (17)
J
(
u
)
=
1
2
∥
∇
u
∥
2
2
+
T
(
u
)
, for some functionalT
∈
C
1
(
H
1
(
ℝ
3
)
,
ℝ
).Lemma 3 (see [17, 18]).
LetT
∈
C
1
(
H
1
(
ℝ
3
)
,
ℝ
). Letρ
>
0and{
u
n
}
⊂
B
ρbe a minimizing sequence forJ
ρ
2weakly convergent, up to translations, to a nonzero
functionu
¯. Assume that (8) holds and that (18)
T
(
u
n
-
u
¯
)
+
T
(
u
¯
)
=
T
(
u
n
)
+
o
(
1
)
;
(19)
T
(
α
n
(
u
n
-
u
¯
)
)
-
T
(
u
n
-
u
¯
)
=
o
(
1
)
w
h
e
r
e
α
n
=
ρ
2
-
∥
u
¯
∥
2
2
∥
u
n
-
u
¯
∥
2
;
(20)
〈
T
′
(
u
n
)
,
u
n
〉
=
O
(
1
)
;
(21)
〈
T
′
(
u
n
)
-
T
′
(
u
m
)
,
u
n
-
u
m
〉
=
o
(
1
)
a
s
n
,
m
⟶
∞
. Then∥
u
n
-
u
¯
∥
→
0. In particular it follows thatu
¯
∈
B
ρandJ
(
u
¯
)
=
J
ρ
2.As pointed out in [18], Lemma 3 is a variant of the concentration-compactness
principle of Lions [15, 16]. Assumption (18)
shows thatTpossesses the Brizis-Lieb splitting property and (19) is the homogeneity
ofT. If, in addition, the condition (8) holds, then one can show that dichotomy
does not occur; that is,u
¯
∈
B
ρ. Furthermore, if (20) and (21) are also fulfilled,
then{
u
n
}strongly converges tou
¯inH
1
(
ℝ
3
). Finally we recall the following results obtained in [17, 18].Lemma 4 (see [18]).
Ifp
∈
(
2,3
), thenI
ρ
2
<
0for allρ
>
0.Lemma 5 (see [17, Lemma
3
.
1]).
Ifp
∈
(
2
,
10
/
3
), then, for everyρ
>
0, the functionalIis bounded below and coercive onB
ρ.Remark 6.
Forp
∈
(
2,3
), it follows from Lemmas 4 and 5 that each
minimizing sequence forI
ρ
2is bounded from below and above by two positive constants
in𝒟
1,2
(
ℝ
3
)andH
1
(
ℝ
3
), up to a subsequence, respectively.


## 3. Proof of the Main Theorems

Before proving our main theorems, we need some preliminary lemmas. First, we set(22)
𝒫
:
=
{
u
∈
H
1
(
ℝ
3
)
:
Q
(
u
)
=
0
,
I
(
u
)
=
min
t
>
0
I
(
u
t
)
}
, whereu
t
(
x
)
=
t
3
/
2
u
(
t
x
)witht
>
0andQ
(
u
)is a functional onH
1
(
ℝ
3
)defined as (23)
Q
(
u
)
=
∫
ℝ
3
‍
|
∇
u
|
2
d
x
+
1
4
∫
ℝ
3
‍
ϕ
u
|
u
|
2
d
x
-
3
(
p
-
2
)
2
p
∫
ℝ
3
‍
|
u
|
p
d
x
. It was shown in [13,
Lemma 2.1] that ifu
ρis a constrained critical point ofIonB
ρassociated with the Lagrange
multiplierλ
ρ
,thenQ
(
u
ρ
)
=
0, which is nothing but a linear combination
of〈
E
′
(
u
ρ
)
,
u
ρ
〉
=
0(recall thatE
(
u
)is given by (3)) and the following Pohozaev identity for (1) (cf. [5, 9]) (24)
1
2
∥
∇
u
ρ
∥
2
2
-
3
λ
ρ
2
∥
u
ρ
∥
2
2
+
5
4
∫
ℝ
3
‍
ϕ
u
ρ
|
u
ρ
|
2
d
x
-
3
p
∥
u
ρ
∥
p
p
=
0
. The following lemma shows thatB
ρ
∩
𝒫is well defined.Lemma 7.
Letp
∈
(
2,3
)and let ρ
>
0. For eachu
∈
B
ρwithI
(
u
)
<
0, there exists a uniquet
u
>
0such thatI
(
u
t
u
)
=
min
{
I
(
u
t
)
:
t
>
0
}; moreover,u
t
u
∈
B
ρ
∩
𝒫.Proof.
We divide the proof into two cases.
Case   1
(
p
∈
(
2,8
/
3
)
). Letu
∈
B
ρ, for simplicity, and we will writef
u
′
(
t
),f
u
′′
(
t
)andf
u
′′′
(
t
), the derivatives off
u
(
t
)ont, instead ofd
f
u
(
t
)
/
d
t,d
2
f
u
(
t
)
/
d
t
2andd
3
f
u
(
t
)
/
d
t
3. From (15), we have (25)
f
u
′
(
t
)
=
t
∫
ℝ
3
‍
|
∇
u
|
2
d
x
+
1
4
∫
ℝ
3
‍
ϕ
u
|
u
|
2
d
x
-
3
(
p
-
2
)
2
p
t
(
3
p
-
8
)
/
2
∫
ℝ
3
‍
|
u
|
p
d
x
. Noting that(
3
p
-
8
)
/
2
∈
(
-
1,0
)sincep
∈
(
2,8
/
3
), then, by (25),lim
t
→
0
f
u
′
(
t
)
=
-
∞andlim
t
→
∞
f
u
′
(
t
)
=
+
∞; thus there existst
u
>
0such thatf
u
′
(
t
u
)
=
0. If there exists anothers
u
>
0such thatf
u
′
(
s
u
)
=
0, without loss of generality, we may assume
thats
u
>
t
u, and then we get (26)
0
=
f
u
′
(
s
u
)
-
f
u
′
(
t
u
)
=
(
s
u
-
t
u
)
∫
ℝ
3
‍
|
∇
u
|
2
d
x
+
3
(
p
-
2
)
2
p
(
t
u
(
3
p
-
8
)
/
2
-
s
u
(
3
p
-
8
)
/
2
)
∫
ℝ
3
‍
|
u
|
p
d
x
>
0
, a contradiction. Therefore,t
uis unique and it is clear thatI
(
u
t
u
)
=
min
{
I
(
u
t
)
:
t
>
0
}. Moreover,u
t
u
∈
B
ρ
∩
𝒫because off
u
′
(
t
u
)
t
u
=
0.
Case  2
(
p
∈
[
8
/
3,3
)
). By Lemma 4,
we know that the setA
ρ
:
=
{
u
∈
B
ρ
:
I
(
u
)
<
0
}
≠
∅. Letu
∈
A
ρ, iff
u
′
(
t
)
>
0for allt
>
0; that is,f
u
(
t
)is strictly increasing, then we obtain
thatf
u
(
t
)
<
f
u
(
1
)
=
I
(
u
)
<
0for allt
∈
(
0,1
). However, it is easy to see thatlim
t
→
0
f
u
(
t
)
=
0; this is a contradiction. On the other hand, we know
thatf
u
(
t
)
→
∞ast
→
∞; hence there is at
u
>
0such thatf
u
′
(
t
u
)
=
0,u
t
u
∈
B
ρ
∩
𝒫and (27)
f
u
(
t
u
)
=
min
{
f
u
(
t
)
:
t
>
0
}
≤
f
u
(
1
)
=
I
(
u
)
<
0
. Next, we will show thatt
uis unique. Arguing by contradiction, suppose that there is
anothers
u
>
0such thatf
u
(
t
u
)
=
f
u
(
s
u
)
=
min
{
f
u
(
t
)
:
t
>
0
}, without loss of generality, we may assume
thats
u
>
t
u, and then we have (28)
f
u
′
(
t
u
)
=
f
u
′
(
s
u
)
=
0
,
f
u
′′
(
t
u
)
≥
0
,
f
u
′′
(
s
u
)
≥
0
. According to (28), there existsω
u
∈
(
t
u
,
s
u
)such thatf
u
′′
(
ω
u
)
=
0. After a simple calculation, we get (29)
f
u
′′
(
t
)
=
∥
∇
u
∥
2
2
-
3
(
p
-
2
)
(
3
p
-
8
)
4
p
t
(
3
p
-
10
)
/
2
∫
ℝ
3
‍
|
u
|
p
d
x
,
(30)
f
u
′′′
(
t
)
=
-
3
(
p
-
2
)
(
3
p
-
8
)
(
3
p
-
10
)
8
p
t
(
3
p
-
12
)
/
2
×
∫
ℝ
3
‍
|
u
|
p
d
x
. Ifp
=
8
/
3
,then, by (29),f
u
′′
(
t
)
>
0for allt
>
0, which contradictsf
u
′′
(
ω
u
)
=
0. Ifp
∈
(
8
/
3,3
), then, by (30),f
u
′′′
(
t
)
>
0for allt
>
0. Noting thatω
u
∈
(
t
u
,
s
u
), we have (31)
0
≤
f
u
′′
(
t
u
)
<
f
u
′′
(
ω
u
)
=
0
, again a contradiction. Therefore,t
u
>
0is unique.Lemma 8.
Letp
∈
(
2,3
)andρ
>
0. For each{
u
n
}
⊂
B
ρsuch thatI
(
u
n
)
→
I
ρ
2
<
0asn
→
∞andI
(
u
n
)
<
0for alln
∈
ℕ, there exists a bounded sequence{
t
n
}
⊂
ℝ
+such that{
u
n
t
n
}
⊂
B
ρ
∩
𝒫andI
(
u
n
t
n
)
→
I
ρ
2asn
→
∞withI
(
u
n
t
n
)
<
0for alln
∈
ℕ; that is,{
u
n
t
n
}is also a minimizing sequence forI
ρ
2constrained onB
ρ.Proof.
It follows from Lemma7 that, for
eachu
n, there existst
n
>
0such thatu
n
t
n
∈
B
ρ
∩
𝒫andI
(
u
n
t
n
)
=
min
{
I
(
u
n
t
)
:
t
>
0
}
≤
I
(
u
n
)
<
0; therefore, we have (32)
I
ρ
2
≤
I
(
u
n
t
n
)
≤
I
(
u
n
)
⟶
I
ρ
2
, asn
→
∞, that is,{
u
n
t
n
}is a minimizing sequence. Next, we will show
that{
t
n
}is bounded. Indeed, from Remark 6,{
u
n
}and{
u
n
t
n
}are bounded from below and above by two positive constants
in𝒟
1,2
(
ℝ
3
)andH
1
(
ℝ
3
), respectively. Noting that∫
ℝ
3
‍
|
∇
u
n
t
n
|
2
d
x
=
t
n
2
∫
ℝ
3
‍
|
∇
u
n
|
2
d
x; therefore,{
t
n
}is bounded from below and above by two positive constants.Remark 9.
Thanks to the Lemma8, we know
thatI
ρ
2
=
inf
{
I
(
u
)
:
u
∈
B
ρ
}
=
inf
{
I
(
u
)
:
u
∈
B
ρ
∩
𝒫
}, and, in the following, we will consider the minimization
problem (7) restricted
toB
ρ
∩
𝒫instead ofB
ρ. By Lemmas 4
and 8, for eachρ
>
0, if{
u
n
}
⊂
B
ρ
∩
𝒫satisfyingI
(
u
n
)
→
I
ρ
2asn
→
∞, then, up to a subsequence, we may assume
thatI
(
u
n
)
<
0. It follows from Lemma 5 that{
u
n
}is bounded inH
1
(
ℝ
3
); by the results of [17, 18], we may assume
thatu
n
⇀
u
≠
0asn
→
∞inH
1
(
ℝ
3
).
The following estimates of the elements ofB
ρ
∩
𝒫are crucial to proving the strong subadditivity inequality
(8).Lemma 10.
Letp
∈
(
2,3
)andρ
>
0. For eachu
∈
B
ρ
∩
𝒫, it holds (33)
∥
∇
u
∥
2
2
≤
(
3
(
p
-
2
)
2
p
)
4
/
(
10
-
3
p
)
S
-
6
(
p
-
2
)
/
(
10
-
3
p
)
ρ
2
(
6
-
p
)
/
(
10
-
3
p
)
,
∥
u
∥
p
p

≤

(
3
(
p

-

2
)
2
p
)
3
(
p
-
2
)
/
(
10
-
3
p
)
S
-
6
(
p
-
2
)
/
(
10
-
3
p
)
ρ
2
(
6
-
p
)
/
(
10
-
3
p
)
.Proof.
Sinceu
∈
B
ρ
∩
𝒫, (34)
∥
∇
u
∥
2
2
+
1
4
∫
ℝ
3
‍
ϕ
u
|
u
|
2
d
x
-
3
(
p
-
2
)
2
p
∥
u
∥
p
p
=
0
. Noting that∫
ℝ
3
‍
ϕ
u
|
u
|
2
d
x
≥
0(see (13)), by using the Hölder inequality, we get (35)
∥
∇
u
∥
2
2
≤
3
(
p
-
2
)
2
p
∥
u
∥
p
p
≤
3
(
p
-
2
)
2
p
∥
u
∥
2
(
6
-
p
)
/
2
∥
u
∥
6
3
(
p
-
2
)
/
2
≤
3
(
p
-
2
)
2
p
S
-
3
(
p
-
2
)
/
2
∥
u
∥
2
(
6
-
p
)
/
2
∥
∇
u
∥
2
3
(
p
-
2
)
/
2
, which implies that (36)
∥
∇
u
∥
2
2
≤
(
3
(
p
-
2
)
2
p
)
4
/
(
10
-
3
p
)
S
-
6
(
p
-
2
)
/
(
10
-
3
p
)
ρ
2
(
6
-
p
)
/
(
10
-
3
p
)
. On the other hand, we have (37)
∥
u
∥
p
p
≤
∥
u
∥
2
(
6
-
p
)
/
2
∥
u
∥
6
3
(
p
-
2
)
/
2
≤
S
-
3
(
p
-
2
)
/
2
∥
u
∥
2
(
6
-
p
)
/
2
∥
∇
u
∥
2
3
(
p
-
2
)
/
2
≤
(
3
(
p
-
2
)
2
p
)
3
(
p
-
2
)
/
(
10
-
3
p
)
×
S
-
6
(
p
-
2
)
/
(
10
-
3
p
)
ρ
2
(
6
-
p
)
/
(
10
-
3
p
)
; this concludes the proof of this lemma.Remark 11.
Letp
∈
(
3,10
/
3
). It was shown in [13, Theorem 1.1] thatI
ρ
2
<
0if and only ifρ
∈
(
ρ
¯
,
∞
), where the positive numberρ
¯is defined as (38)
ρ
¯
=
inf
{
ρ
>
0
:
I
ρ
2
<
0
}
. Therefore, after a simple calculation, we can show that both
of Lemmas 7 and 10 hold ifp
∈
(
3,10
/
3
)andρ
∈
(
ρ
¯
,
∞
).
Motivated by [17], we will use the standard
scaling arguments to prove that the strong subadditivity inequality (8) holds forp
∈
(
2,3
). First, we consider the case ofp
∈
(
2,8
/
3
].Lemma 12.
Forp
∈
(
2,8
/
3
], let (39)
ρ
¯
1
=
3
(
3
p
-
8
)
/
8
(
3
-
p
)
π
(
3
p
-
10
)
/
8
(
3
-
p
)
S
3
/
2
(
p
-
2
2
p
)
1
/
4
(
3
-
p
)
>
0
. Then (40)
I
ρ
2
<
I
μ
2
+
I
ρ
2
-
μ
2
∀
0
<
μ
<
ρ
<
ρ
¯
1
.Proof.
By Lemma8 and Remark 9, for each{
u
n
}
⊂
B
ρ
∩
𝒫satisfyingI
(
u
n
)
→
I
ρ
2
<
0asn
→
∞, we may assume that, for alln,I
(
u
n
)
≤
I
ρ
2
/
2, which implies that (41)
∥
u
n
∥
p
p
≥
-
p
2
I
ρ
2
. Noting thatt
u
n
∈
B
t
ρ(t
>
0), we have (42)
I
(
t
u
n
)
=
t
2
2
∫
ℝ
3
‍
|
∇
u
n
|
2
d
x
+
t
4
4
∫
ℝ
3
‍
ϕ
u
n
|
u
n
|
2
d
x
-
t
p
p
∫
ℝ
3
‍
|
u
n
|
p
d
x
=
t
2
(
I
(
u
n
)
+
t
2
-
1
4
∫
ℝ
3
‍
ϕ
u
n
|
u
n
|
2
d
x
d
d
d
-
t
p
-
2
-
1
p
∫
ℝ
3
‍
|
u
n
|
p
d
x
)
=
t
2
(
I
(
u
n
)
+
g
(
t
,
u
n
)
)
, where (43)
g
(
t
,
u
)
=
t
2
-
1
4
∫
ℝ
3
‍
ϕ
u
|
u
|
2
d
x
-
t
p
-
2
-
1
p
∫
ℝ
3
‍
|
u
|
p
d
x
. We calculate the derivative ofg
(
t
,
u
)ont: (44)
d
g
(
t
,
u
)
d
t
=
t
2
∫
ℝ
3
‍
ϕ
u
|
u
|
2
d
x
-
p
-
2
p
t
p
-
3
∥
u
∥
p
p
. Lettingd
g
(
t
,
u
)
/
d
t
=
0, we see from (14) that (45)
t
p
-
4
=
p
∫
ℝ
3
‍
ϕ
u
|
u
|
2
d
x
2
(
p
-
2
)
∥
u
∥
p
p
≤
2
π
p
∥
u
∥
12
/
5
4
S
2
(
p
-
2
)
∥
u
∥
p
p
. Furthermore, (46)
d
2
g
(
t
,
u
)
d
t
2
=
1
2
∫
ℝ
3
‍
ϕ
u
|
u
|
2
d
x
-
(
p
-
2
)
(
p
-
3
)
p
t
p
-
4
∥
u
∥
p
p
>
0
.
Now we divide the value ofp into two cases to discussd
g
(
t
,
u
n
)
/
d
t.
Case  1
(
p
∈
(
2,12
/
5
)
). It follows from Lemma 10, (14), and the Hölder
inequality that (47)
d
g
(
t
,
u
n
)
d
t
|
t
=
1
=
1
2
∫
ℝ
3
‍
ϕ
u
n
|
u
n
|
2
d
x
-
p
-
2
p
∫
ℝ
3
‍
|
u
n
|
p
d
x
≤
2
π
S
2
∥
u
n
∥
12
/
5
4
-
p
-
2
p
∥
u
n
∥
p
p
≤
2
π
S
2
∥
u
n
∥
p
6
p
/
(
6
-
p
)
∥
u
n
∥
6
2
(
12
-
5
p
)
/
(
6
-
p
)
-
p
-
2
p
∥
u
n
∥
p
p
=
∥
u
n
∥
p
p
(
p
-
2
p
2
π
S
12
(
p
-
3
)
/
(
6
-
p
)
∥
∇
u
n
∥
2
2
(
12
-
5
p
)
/
(
6
-
p
)
d
d
d
d
d
d
d
d
×
∥
u
n
∥
p
p
2
/
(
6
-
p
)
-
p
-
2
p
)
≤
∥
u
n
∥
p
p
(
2
π
S
12
(
p
-
3
)
/
(
10
-
3
p
)
(
3
(
p
-
2
)
2
p
)
(
8
-
3
p
)
/
(
10
-
3
p
)
d
d
d
d
d
d
d
d
×
ρ
8
(
3
-
p
)
/
(
10
-
3
p
)
-
p
-
2
p
)
.
Case  2
(
p
∈
[
12
/
5,8
/
3
]
)
. Again by Lemma 10,
(14), and the Hölder inequality,
we have (48)
d
g
(
t
,
u
n
)
d
t
|
t
=
1
=
1
2
∫
ℝ
3
‍
ϕ
u
n
|
u
n
|
2
d
x
-
p
-
2
p
∫
ℝ
3
‍
|
u
n
|
p
d
x
≤
2
π
S
2
∥
u
n
∥
12
/
5
4
-
p
-
2
p
∥
u
n
∥
p
p
≤
2
π
S
2
∥
u
n
∥
2
2
(
5
p
-
12
)
/
3
(
p
-
2
)
∥
u
n
∥
p
2
p
/
3
(
p
-
2
)
-
p
-
2
p
∥
u
n
∥
p
p
≤
∥
u
n
∥
p
p
×
(
2
π
S
2
∥
u
n
∥
2
2
(
5
p
-
12
)
/
3
(
p
-
2
)
∥
u
n
∥
p
p
(
8
-
3
p
)
/
3
(
p
-
2
)
-
p
-
2
p
)
≤
∥
u
n
∥
p
p
(
2
π
S
12
(
p
-
3
)
/
(
10
-
3
p
)
(
3
(
p
-
2
)
2
p
)
(
8
-
3
p
)
/
(
10
-
3
p
)
d
d
d
d
d
d
d
d
d
d
×
ρ
8
(
3
-
p
)
/
(
10
-
3
p
)
-
p
-
2
p
)
. Let (49)
ρ
¯
1
=
3
(
3
p
-
8
)
/
8
(
3
-
p
)
π
(
3
p
-
10
)
/
8
(
3
-
p
)
S
3
/
2
(
p
-
2
2
p
)
1
/
4
(
3
-
p
)
. Then by (47), (48), and (49), we know that, for
each0
<
ρ
<
ρ
¯
1, there holds (50)
ρ
~
1
(
ρ
)
:
=
2
π
p
p
-
2
S
12
(
p
-
3
)
/
(
10
-
3
p
)
×
(
3
(
p
-
2
)
2
p
)
(
8
-
3
p
)
/
(
10
-
3
p
)
ρ
8
(
3
-
p
)
/
(
10
-
3
p
)
<
1
,
d
g
(
t
,
u
n
)
d
t
|
t
=
1
<
0
, for alln
∈
ℕ. On the other hand, for eachε
∈
(
0,1
-
ρ
~
1
(
ρ
)
), it follows from (41), (44), (45), (46), and Lemma 10
that, for allt
∈
(
1
,
(
ρ
~
1
(
ρ
)
/
(
1
-
ε
)
)
1
/
(
p
-
4
)
)and alln
∈
ℕ, (51)
d
g
(
t
,
u
n
)
d
t
=
p
-
2
p
∥
u
n
∥
p
p
t
p
-
3
(
p
∫
ℝ
3
‍
ϕ
u
n
|
u
n
|
2
d
x
2
(
p
-
2
)
∥
u
n
∥
p
p
t
4
-
p
-
1
)
≤
ε
(
p
-
2
)
2
I
ρ
2
<
0
. This, together with the mean value theorem and (41), yields that for
allt
∈
(
1
,
(
(
1
-
ε
)
-
1
ρ
~
1
(
ρ
)
)
1
/
(
p
-
4
)
)and alln
∈
ℕ, (52)
g
(
t
,
u
n
)
=
g
(
1
,
u
n
)
+
d
g
(
t
,
u
n
)
d
t
|
t
=
θ
t
n
(
t
-
1
)
<
-
C
(
t
-
1
)
, whereθ
t
n
∈
(
1
,
t
)andC
>
0depend only onε,p
,andρ. By (42),
we have (53)
I
(
t
ρ
)
2
≤
I
(
t
u
n
)
=
t
2
(
I
(
u
n
)
+
g
(
t
,
u
n
)
)
≤
t
2
I
(
u
n
)
-
C
t
2
(
t
-
1
)
=
t
2
I
ρ
2
-
C
t
2
(
t
-
1
)
+
o
(
1
)
, then (54)
I
(
t
ρ
)
2
<
t
2
I
ρ
2
∀
t
∈
(
1
,
(
ρ
~
1
(
ρ
)
1
-
ε
)
1
/
(
p
-
4
)
)
,
ρ
∈
(
0
,
ρ
¯
1
)
. Clearly,ρ
~
1(cf. (50)) is strictly increasing onρ, and then(
ρ
~
1
/
(
1
-
ε
)
)
1
/
(
p
-
4
)is strictly decreasing onρsincep
∈
(
2,3
).
Let(55)
h
(
ρ
)
:
=
(
ρ
~
1
(
ρ
)
1
-
ε
)
1
/
(
p
-
4
)
. For eachρ
∈
(
0
,
ρ
¯
1
), letμ
∈
(
0
,
ρ
) without loss of generality, we may assume
thatμ
>
ρ
2
-
μ
2. Choosingε
∈
(
0
,
min
{
ρ
¯
1
-
ρ
,
1
-
ρ
~
1
(
ρ
)
,
μ
-
ρ
2
-
μ
2
}
), then by (50) we know thath
(
ρ
2
-
μ
2
)
>
h
(
μ
)
>
h
(
ρ
)
>
1. (a)
Ifρ
/
μ
∈
(
1
,
h
(
μ
)
), then by (54) (56)
I
ρ
2
=
I
(
ρ
2
/
μ
2
)
μ
2
<
ρ
2
μ
2
I
μ
2
=
I
μ
2
+
ρ
2
-
μ
2
μ
2
I
μ
2
=
I
μ
2
+
ρ
2
-
μ
2
μ
2
I
(
μ
2
/
(
ρ
2
-
μ
2
)
)
(
ρ
2
-
μ
2
)
.
(b)
Ifρ
/
μ
∉
(
1
,
h
(
μ
)
), then there existsk
∈
ℕsuch that(
ρ
/
μ
)
1
/
k
∈
(
1
,
h
(
ρ
)
). Therefore (57)
(
ρ
μ
)
1
/
k
∈
(
1
,
h
(
(
ρ
μ
)
(
k
-
i
)
/
k
μ
)
)
,
∀
i
=
1,2
,
…
,
k
.
It follows from (54) that
(58)
I
ρ
2
=
I
(
ρ
/
μ
)
2
/
k
(
(
ρ
/
μ
)
2
(
k
-
1
)
/
k
μ
2
)
<
(
ρ
μ
)
2
/
k
I
(
ρ
/
μ
)
2
(
k
-
1
)
/
k
μ
2
<
⋯
<
ρ
2
μ
2
I
μ
2
. Combining the above cases (a) and (b), we can show that
(59)
I
ρ
2
<
I
μ
2
+
I
ρ
2
-
μ
2
∀
0
<
μ
<
ρ
<
ρ
¯
1
. Thus the conclusion of this lemma holds.Remark 13.
For the case ofp
=
8
/
3, it has been proved in [4, 17] that the strong subadditivity
inequality (8) holds
forρ
>
0small. By using the result of [17], we can give a specific estimate of lower bound
ofρsuch that (8) holds; that is, (8) holds
for allρ
∈
(
0
,
(
8
π
)
-
3
/
4
S
3
/
2
). However, if we plugp
=
8
/
3into (49), then we haveρ
¯
1
=
(
8
π
)
-
3
/
4
S
3
/
2, which coincides with the one given in [17].Next, we will show (8)
forp
∈
(
8
/
3,3
). We point out that the case ofp
∈
(
8
/
3,3
)is quite different from the case ofp
∈
(
2,8
/
3
]since the inequality (48) does not hold anymore. Inspired by [18], we will give some estimates forI
ρ
2in Lemmas 14
and 15, and these are crucial for the proof
of (8) ifp
∈
(
8
/
3,3
).Lemma 14.
Letp
∈
(
8
/
3,3
)andρ
>
0be fixed. If there existsu
∈
B
ρ
∩
𝒫such thatI
(
u
)
≤
I
ρ
2
/
2and (60)
∥
u
∥
p
p
>
3
∥
∇
u
∥
2
2
, then there exist positive constantsC
3andC
4dependent onpandρ, such that (61)
I
μ
2
≤
-
C
3
μ
2
(
6
-
p
)
/
(
10
-
3
p
)
+
C
4
μ
2
(
18
-
5
p
)
/
(
10
-
3
p
)
∀
μ
>
0
.Proof.
From the assumptions of the lemma, we see that(62)
∥
∇
u
∥
2
2
+
1
4
∫
ℝ
3
‍
ϕ
u
|
u
|
2
d
x
-
3
(
p
-
2
)
2
p
∥
u
∥
p
p
=
0
,
(63)
1
2
∥
∇
u
∥
2
2
+
1
4
∫
ℝ
3
‍
ϕ
u
|
u
|
2
d
x
-
1
p
∥
u
∥
p
p
=
I
(
u
)
≤
I
ρ
2
2
. By (60),
(62), and (63), we deduce that (64)
I
ρ
2
2
≥
I
(
u
)
=
-
1
2
∥
∇
u
∥
2
2
+
3
p
-
8
2
p
∥
u
∥
p
p
>
4
(
p
-
3
)
p
∥
∇
u
∥
2
2
. Combining (62) and (64), and using Lemma
10, we also obtain
(65)
1
4
∫
ℝ
3
‍
ϕ
u
|
u
|
2
d
x
=
3
(
p
-
2
)
2
p
∥
u
∥
p
p
-
∥
∇
u
∥
2
2
≤
(
3
(
p
-
2
)
2
p
)
4
/
(
10
-
3
p
)
S
-
6
(
p
-
2
)
/
(
10
-
3
p
)
×
ρ
2
(
6
-
p
)
/
(
10
-
3
p
)
+
p
8
(
3
-
p
)
I
ρ
2
. For eacht
>
0, letu
t
(
x
)
=
t
4
/
(
10
-
3
p
)
u
(
t
2
(
p
-
2
)
/
(
10
-
3
p
)
x
), we have∥
u
t
∥
2
=
t
∥
u
∥
2
=
t
ρ. It follows from (60), (64), and
(65) that (66)
I
(
t
ρ
)
2
≤
I
(
u
t
)
=
1
2
t
2
(
6
-
p
)
/
(
10
-
3
p
)
∥
∇
u
∥
2
2
+
1
4
t
2
(
18
-
5
p
)
/
(
10
-
3
p
)
∫
ℝ
3
‍
ϕ
u
|
u
|
2
d
x
-
1
p
t
2
(
6
-
p
)
/
(
10
-
3
p
)
∥
u
∥
p
p
≤
(
1
2
-
3
p
)
t
2
(
6
-
p
)
/
(
10
-
3
p
)
∥
∇
u
∥
2
2
+
1
4
t
2
(
18
-
5
p
)
/
(
10
-
3
p
)
∫
ℝ
3
‍
ϕ
u
|
u
|
2
d
x
≤
6
-
p
16
(
3
-
p
)
I
ρ
2
ρ
2
(
6
-
p
)
/
(
10
-
3
p
)
(
t
ρ
)
2
(
6
-
p
)
/
(
10
-
3
p
)
+
(
(
3
(
p
-
2
)
2
p
)
4
/
(
10
-
3
p
)
S
-
6
(
p
-
2
)
/
(
10
-
3
p
)
d
d
d
×
ρ
8
(
p
-
3
)
/
(
10
-
3
p
)
d
d
d
+
p
8
(
3
-
p
)
I
ρ
2
ρ
2
(
18
-
5
p
)
/
(
10
-
3
p
)
(
3
(
p
-
2
)
2
p
)
4
/
(
10
-
3
p
)
)
×
(
t
ρ
)
2
(
18
-
5
p
)
/
(
10
-
3
p
)
. Sett
ρ
=
μ, thenμ
∈
(
0
,
∞
)sincet
∈
(
0
,
∞
)andρis a fixed positive constant. From the above inequality, we
see that (67)
I
μ
2
≤
-
C
3
μ
2
(
6
-
p
)
/
(
10
-
3
p
)
+
C
4
μ
2
(
18
-
5
p
)
/
(
10
-
3
p
)
, for some positive constantsC
3andC
4depending onpandρ.Lemma 15.
Suppose thatp
∈
(
8
/
3,3
)and{
u
k
}
⊂
B
ρ
k
∩
𝒫satisfying∥
u
k
∥
p
p
>
3
∥
∇
u
k
∥
2
2andI
(
u
k
)
≤
I
ρ
k
2
/
2for allk
∈
ℕ. Then there exists a positive
constantCdependent onp, such that (68)
I
ρ
k
2
≥
-
C
ρ
k
2
(
5
p
-
12
)
/
(
3
p
-
8
)
∀
k
∈
ℕ
.Proof.
Following the line of the proof of Lemma14, we arrive that (69)
1
4
∫
ℝ
3
‍
ϕ
u
k
|
u
k
|
2
d
x
=
3
(
p
-
2
)
2
p
∥
u
k
∥
p
p
-
∥
∇
u
k
∥
2
2
>
7
p
-
18
6
p
∥
u
k
∥
p
p
, which, together with (14) and the Höder inequality, implies that
(70)
∥
u
k
∥
p
p
≤
3
p
2
(
7
p
-
18
)
∫
ℝ
3
‍
ϕ
u
k
|
u
k
|
2
d
x
≤
6
π
p
(
7
p
-
18
)
S
2
∥
u
k
∥
12
/
5
4
≤
6
π
p
(
7
p
-
18
)
S
2
∥
u
k
∥
2
2
(
5
p
-
12
)
/
3
(
p
-
2
)
∥
u
k
∥
p
2
p
/
3
(
p
-
2
)
, and then (71)
∥
u
k
∥
p
p
≤
(
6
π
p
(
7
p
-
18
)
S
2
)
3
(
p
-
2
)
/
(
3
p
-
8
)
∥
u
k
∥
2
2
(
5
p
-
12
)
/
(
3
p
-
8
)
=
(
6
π
p
(
7
p
-
18
)
S
2
)
3
(
p
-
2
)
/
(
3
p
-
8
)
ρ
k
2
(
5
p
-
12
)
/
(
3
p
-
8
)
. Combining (62), (69), and (71), we have (72)
I
ρ
k
2
2
>
I
(
u
k
)
=
1
2
∥
∇
u
k
∥
2
2
+
1
4
∫
ℝ
3
‍
ϕ
u
k
|
u
k
|
2
d
x
-
1
p
∥
u
k
∥
p
p
=
3
p
-
10
4
p
∥
u
k
∥
p
p
+
1
8
∫
ℝ
3
‍
ϕ
u
k
|
u
k
|
2
d
x
≥
4
(
p
-
3
)
3
p
∥
u
k
∥
p
p
≥
4
(
p
-
3
)
3
p
(
6
π
p
(
7
p
-
18
)
S
2
)
3
(
p
-
2
)
/
(
3
p
-
8
)
×
ρ
k
2
(
5
p
-
12
)
/
(
3
p
-
8
)
, and this completes the proof.Lemma 16.
Ifp
∈
(
8
/
3,3
), then there exists a positive
constantρ
¯
2such that (73)
I
ρ
2
<
I
μ
2
+
I
ρ
2
-
μ
2
∀
0
<
μ
<
ρ
<
ρ
¯
2
.Proof.
Suppose thatρ
>
0and{
u
n
}
⊂
B
ρ
∩
𝒫satisfyingI
(
u
n
)
→
I
ρ
2asn
→
∞. It follows from Remark 9 that, up to a subsequence,I
(
u
n
)
≤
I
ρ
2
/
2
<
0for alln
∈
ℕ. By Lemma 5,
it is easy to see that{
u
n
}is bounded inH
1
(
ℝ
3
). Noting thatt
u
n
∈
B
t
ρ, then, by (42), we have (74)
I
(
t
u
n
)
=
t
2
(
I
(
u
n
)
+
g
(
t
,
u
n
)
)
, whereg
(
t
,
u
)is given by (43). Obviously, (75)
d
g
(
t
,
u
n
)
d
t
|
t
=
1
=
1
2
∫
ℝ
3
‍
ϕ
u
n
|
u
n
|
2
d
x
-
(
p
-
2
)
p
∥
u
n
∥
p
p
=
I
(
u
n
)
-
1
2
∥
∇
u
n
∥
2
2
+
1
4
∫
ℝ
3
‍
ϕ
u
n
|
u
n
|
2
d
x
+
(
3
-
p
)
p
∥
u
n
∥
p
p
=
I
(
u
n
)
-
3
2
∥
∇
u
n
∥
2
2
+
1
2
∥
u
n
∥
p
p
, sinceu
n
∈
B
ρ
∩
𝒫and (34)
holds. Moreover, (76)
d
2
g
(
t
,
u
n
)
d
t
2
=
1
2
∫
ℝ
3
‍
ϕ
u
n
|
u
n
|
2
d
x
-
(
p
-
2
)
(
p
-
3
)
p
t
p
-
4
∥
u
n
∥
p
p
>
0
, for allt
>
0andn
∈
ℕ.
We claim that there existsρ
¯
2
>
0such that for eachρ
∈
(
0
,
ρ
¯
2
)and for each{
u
n
}
⊂
B
ρ
∩
𝒫satisfyingI
(
u
n
)
≤
I
ρ
2
/
2
<
0andI
(
u
n
)
→
I
ρ
2asn
→
∞, we have (77)
∥
u
n
∥
p
p
≤
3
∥
∇
u
n
∥
2
2
.
Indeed, if not, we can find{
ρ
k
}and{
u
n
k
}
⊂
B
ρ
k
∩
𝒫such thatρ
k
→
0ask
→
∞and for eachk
∈
ℕ,I
(
u
n
k
)
→
I
ρ
k
2
<
0asn
→
∞, but∥
u
n
k
∥
p
p
>
3
∥
∇
u
n
k
∥
2
2. Fork
=
1, there existsn
1
>
0such thatI
(
u
n
1
1
)
<
I
ρ
1
2
/
2
<
0, and it can be deduced from Lemma 14 that (78)
I
μ
2
≤
-
C
3
μ
2
(
6
-
p
)
/
(
10
-
3
p
)
+
C
4
μ
2
(
18
-
5
p
)
/
(
10
-
3
p
)
∀
μ
>
0
, whereC
3andC
4are positive constants dependent onpandρ
1. On the other hand, we know that for
eachk
∈
ℕthere existsn
k
>
0such thatI
(
u
n
k
k
)
<
I
ρ
k
2
/
2
<
0. Then by Lemma 15, we obtain (79)
I
ρ
k
2
≥
-
C
ρ
k
2
(
5
p
-
12
)
/
(
3
p
-
8
)
∀
k
∈
ℕ
, whereCis a positive constant depending only
onp. Noting that (78) holds for allμ
>
0, by (78)
and (79), we deduce that
(80)
-
C
ρ
k
2
(
5
p
-
12
)
/
(
3
p
-
8
)
≤
I
ρ
k
2
≤
-
C
3
ρ
k
2
(
6
-
p
)
/
(
10
-
3
p
)
+
C
4
ρ
k
2
(
18
-
5
p
)
/
(
10
-
3
p
)
, which is a contradiction forklarge sincep
∈
(
8
/
3,3
)implies (81)
2
(
5
p
-
12
)
3
p
-
8
>
2
(
6
-
p
)
10
-
3
p
. Thus we have shown the claim. Now for
eachρ
∈
(
0
,
ρ
¯
2
)and for all{
u
n
}
⊂
B
ρ
∩
𝒫withI
(
u
n
)
≤
I
ρ
2
/
2andI
(
u
n
)
→
I
ρ
2asn
→
∞, using (77), we have (82)
d
g
(
t
,
u
n
)
d
t
|
t
=
1
≤
I
(
u
n
)
≤
I
ρ
2
2
<
0
. By (76),
similarly as in the proofs of (45) and
(51), we get that
(83)
t
4
-
p
=
2
(
p
-
2
)
p
∥
u
n
∥
p
p
∫
ℝ
3
‍
ϕ
u
n
|
u
n
|
2
d
x
=
2
(
p
-
2
)
p
∥
u
n
∥
p
p
(
6
(
p
-
2
)
/
p
)
∥
u
n
∥
p
p
-
4
∥
∇
u
n
∥
2
2
≥
3
(
p
-
2
)
7
p
-
18
>
1
. Now, we can chooseε
>
0so small that there exists a positive
constantCdependent onp,ρ
,andε, such that (84)
d
g
(
t
,
u
n
)
d
t
≤
-
C
<
0
∀
t
∈
(
1
,
(
3
(
1
-
ε
)
(
p
-
2
)
7
p
-
18
)
1
/
(
4
-
p
)
)
. Since, for eachn,g
(
1
,
u
n
)
=
0, it follows that, for eacht
∈
(
1
,
(
3
(
1
-
ε
)
(
p
-
2
)
/
(
7
p
-
18
)
)
1
/
(
4
-
p
)
), (85)
I
(
t
ρ
)
2
≤
I
(
t
u
n
)
=
t
2
(
I
(
u
n
)
+
g
(
t
,
u
n
)
)
≤
t
2
(
I
(
u
n
)
+
d
g
(
t
,
u
n
)
d
t
|
t
=
θ
t
n
(
t
-
1
)
)
≤
t
2
(
I
(
u
n
)
-
C
(
t
-
1
)
)
=
t
2
I
ρ
2
-
C
t
2
(
t
-
1
)
+
o
(
1
)
, whereθ
t
n
∈
(
1
,
t
), namely,I
(
t
ρ
)
2
<
t
2
I
ρ
2for allt
∈
(
1
,
(
3
(
1
-
ε
)
(
p
-
2
)
/
(
7
p
-
18
)
)
1
/
(
4
-
p
)
). Thus we complete the proof of this lemma by using the
arguments in the proof of Lemma 12.Lemma 17.
Letρ
>
0. Assume that(
u
ρ
,
λ
ρ
)is a solution of (1) with∥
u
ρ
∥
2
=
ρ. (a)
Ifp
∈
(
2,12
/
5
], thenλ
ρ
<
0.
(b)
Ifp
∈
(
12
/
5,8
/
3
]andλ
ρ
≥
0, then (86)
ρ
≥
ρ
¯
3
:
=
(
6
-
p
6
π
p
)
(
10
-
3
p
)
/
8
(
3
-
p
)
×
S
3
/
2
(
3
(
p
-
2
)
2
p
)
(
8
-
3
p
)
/
8
(
3
-
p
)
.Proof.
Since(
u
ρ
,
λ
ρ
)is a solution of (1), it follows that (87)
∥
∇
u
ρ
∥
2
2
+
∫
ℝ
3
‍
ϕ
u
ρ
|
u
ρ
|
2
d
x
-
∥
u
ρ
∥
p
p
=
λ
ρ
∥
u
ρ
∥
2
2
,
(88)
∥
∇
u
ρ
∥
2
2
+
1
4
∫
ℝ
3
‍
ϕ
u
ρ
|
u
ρ
|
2
d
x
-
3
(
p
-
2
)
2
p
∥
u
ρ
∥
p
p
=
0
. Thus, from (87) and (88), after a simple
calculation, we have (89)
λ
ρ
∥
u
ρ
∥
2
2
=
p
-
6
3
(
p
-
2
)
∥
∇
u
ρ
∥
2
2
+
5
p
-
12
6
(
p
-
2
)
∫
ℝ
3
‍
ϕ
u
ρ
|
u
ρ
|
2
d
x
, which yields that(
a
)holds. Moreover, ifp
∈
(
12
/
5,8
/
3
]andλ
ρ
≥
0, then (89) implies that (90)
5
p
-
12
6
(
p
-
2
)
∫
ℝ
3
‍
ϕ
u
ρ
|
u
ρ
|
2
d
x
≥
6
-
p
3
(
p
-
2
)
∥
∇
u
ρ
∥
2
2
. Thus we get from (88) that (91)
3
(
p
-
2
)
2
p
∥
u
ρ
∥
p
p
=
∥
∇
u
ρ
∥
2
2
+
1
4
∫
ℝ
3
‍
ϕ
u
ρ
|
u
ρ
|
2
d
x
≤
9
(
p
-
2
)
4
(
6
-
p
)
∫
ℝ
3
‍
ϕ
u
ρ
|
u
ρ
|
2
d
x
≤
9
(
p
-
2
)
π
(
6
-
p
)
S
2
∥
u
ρ
∥
12
/
5
4
. By using the Hölder inequality, it can be deduced from
(91) and Lemma 10 that (92)
6
-
p
6
π
p
S
2
∥
u
ρ
∥
p
p
≤
∥
u
ρ
∥
12
/
5
4
≤
∥
u
ρ
∥
2
2
(
5
p
-
12
)
/
3
(
p
-
2
)
∥
u
ρ
∥
p
2
p
/
3
(
p
-
2
)
≤
(
S
-
2
(
8
-
3
p
)
/
(
10
-
3
p
)
(
3
(
p
-
2
)
2
p
)
(
8
-
3
p
)
/
(
10
-
3
p
)
×
ρ
8
(
3
-
p
)
/
(
10
-
3
p
)
(
3
(
p
-
2
)
2
p
)
(
8
-
3
p
)
/
(
10
-
3
p
)
)
∥
u
ρ
∥
p
p
, and this means that (93)
ρ
≥
(
6
-
p
6
π
p
)
(
10
-
3
p
)
/
8
(
3
-
p
)
S
3
/
2
(
3
(
p
-
2
)
2
p
)
(
3
p
-
8
)
/
8
(
3
-
p
)
. Thus(
b
)holds. At this point, the lemma is proved.Proof of Theorem1.
It follows from Lemmas12 and 16 that (8) holds. LetT
(
u
)
=
∫
ℝ
3
‍
ϕ
u
|
u
|
2
d
x
/
4
-
∥
u
∥
p
p
/
p. From the results of [17, 18], we know that (18), (19), (20), and
(21) hold. Therefore, by Lemma 3, all the minimizing sequences for (7) are precompact and then (1) has a solution(
u
ρ
,
λ
ρ
). Lemma 17
shows that, forp
∈
(
2,8
/
3
],λ
ρ
<
0sinceρ
¯
1
<
ρ
¯
3, whereρ
¯
1andρ
¯
3are given by (49) and (86),
respectively.
To complete the proof of Theorem1, we
need to show thatλ
ρ
→
0andI
ρ
2
/
ρ
2
→
0asρ
→
0provided that the assumption(
1
)of Theorem 1
holds. Indeed, since(
u
ρ
,
λ
ρ
)is the solution of (1), it follows from (87), (88), and Lemma 10 that (94)
0
≤
3
4
∫
ℝ
3
‍
ϕ
u
ρ
|
u
ρ
|
2
d
x
-
λ
ρ
ρ
2
=
6
-
p
2
p
∥
u
ρ
∥
p
p
≤
C
ρ
2
(
6
-
p
)
/
(
10
-
3
p
)
, which implies that (95)
0
≤
-
λ
ρ
≤
C
ρ
4
(
p
-
2
)
/
(
10
-
3
p
)
, and that is,λ
ρ
→
0asρ
→
0. On the other hand, we have (96)
I
(
u
ρ
)
=
1
2
∥
∇
u
ρ
∥
2
2
+
1
4
∫
ℝ
3
‍
ϕ
ρ
|
u
ρ
|
2
d
x
-
1
p
∥
u
ρ
∥
p
p
, this, together with (87) and (88),
gives (97)
∥
u
ρ
∥
p
p
=
3
p
2
(
p
-
3
)
I
(
u
ρ
)
+
p
4
(
3
-
p
)
λ
ρ
ρ
2
. Therefore,I
(
u
ρ
)
<
0sinceλ
ρ
<
0andp
∈
(
2,8
/
3
). Noting thatI
(
u
ρ
)
=
I
ρ
2, by Lemma 10
and (97), we obtain
(98)
0
<
3
p
2
(
p
-
3
)
I
ρ
2
ρ
2
≤
(
3
(
p
-
2
)
2
p
)
3
(
p
-
2
)
/
(
10
-
3
p
)
S
-
6
(
p
-
2
)
/
(
10
-
3
p
)
×
ρ
4
(
p
-
2
)
/
(
10
-
3
p
)
-
p
4
(
3
-
p
)
λ
ρ
⟶
0
as
ρ
⟶
0
.Proof of Theorem2.
Suppose thatp
∈
(
2,12
/
5
]and(
u
ρ
,
λ
ρ
)is a solution of (1) with∥
u
ρ
∥
2
=
ρ. Then Lemma 17 and the above proof of Theorem 1 show thatλ
ρ
<
0,I
(
u
ρ
)
<
0andλ
ρ
→
0asρ
→
0. It was proved in [5,
20] (see also [13, Remark 1.4]) that there existsλ
0
<
0such that (1) has only the zero solution whenp
∈
(
2,3
)andλ
∈
(
-
∞
,
λ
0
). Therefore,λ
ρmust be bounded; that is,(
i
)holds. For(
ii
), it is clear that (87), (88), and
(96) hold; after a simple
calculation, we have (99)
∥
∇
u
ρ
∥
2
2
=
5
p
-
12
2
(
p
-
3
)
I
(
u
ρ
)
+
3
p
-
8
4
(
3
-
p
)
λ
ρ
ρ
2
,
∫
ℝ
3
‍
ϕ
u
ρ
|
u
ρ
|
2
d
x
=
6
-
p
p
-
3
I
(
u
ρ
)
+
10
-
3
p
2
(
3
-
p
)
λ
ρ
ρ
2
,
∥
u
ρ
∥
p
p
=
3
p
2
(
p
-
3
)
I
(
u
ρ
)
+
p
4
(
3
-
p
)
λ
ρ
ρ
2
. On the other hand, sinceϕ
u
ρis the solution of the Poisson
equation-
Δ
ϕ
=
4
π
|
u
ρ
|
2, multiplying this equation by|
u
ρ
|and integrating, we obtain (100)
4
π
∥
u
ρ
∥
3
3
=
∫
ℝ
3
‍
∇
ϕ
ρ
∇
|
u
ρ
|
d
x
≤
1
2
∥
∇
ϕ
ρ
∥
2
2
+
1
2
∥
∇
u
ρ
∥
2
2
=
2
π
∫
ℝ
3
‍
ϕ
u
ρ
|
u
ρ
|
2
d
x
+
1
2
∥
∇
u
ρ
∥
2
2
. It follows from (99) and (100) that
(101)
0
<
3
p
2
(
p
-
3
)
I
(
u
ρ
)
+
p
4
(
3
-
p
)
λ
ρ
ρ
2
=
∥
u
ρ
∥
p
p
≤
∥
u
ρ
∥
2
2
(
3
-
p
)
∥
u
ρ
∥
3
3
(
p
-
2
)
≤
∥
u
ρ
∥
2
2
(
3
-
p
)
(
1
2
∫
ℝ
3
‍
ϕ
u
ρ
|
u
ρ
|
2
d
x
+
∥
∇
u
ρ
∥
2
2
)
p
-
2
=
(
2
p
)
p
-
2
∥
u
ρ
∥
2
2
(
3
-
p
)
×
(
p
(
2
p
-
3
)
2
(
p
-
3
)
I
(
u
ρ
)
+
p
4
(
3
-
p
)
λ
ρ
ρ
2
)
p
-
2
≤
(
2
p
)
p
-
2
∥
u
ρ
∥
2
2
(
3
-
p
)
×
(
3
p
2
(
p
-
3
)
I
(
u
ρ
)
+
p
4
(
3
-
p
)
λ
ρ
ρ
2
)
p
-
2
, which implies that (102)
0
<
3
p
2
(
p
-
3
)
I
(
u
ρ
)
+
p
4
(
3
-
p
)
λ
ρ
ρ
2
≤
(
2
p
)
(
p
-
2
)
/
(
3
-
p
)
∥
u
ρ
∥
2
2
=
(
2
p
)
(
p
-
2
)
/
(
3
-
p
)
ρ
2
. Therefore we get (103)
0
<
3
p
2
(
p
-
3
)
I
(
u
ρ
)
ρ
2
+
p
4
(
3
-
p
)
λ
ρ
≤
(
2
p
)
(
p
-
2
)
/
(
3
-
p
)
, so that there existsC
>
0such thatI
(
u
ρ
)
/
ρ
2
∈
(
-
C
,
0
)since, by(
i
),λ
ρ
<
0is bounded.



---

*Source: 398164-2013-08-12.xml*