Table Of ContentDIVERGENT SOLUTIONS TO THE 5D HARTREE EQUATIONS
DAOMIN CAO AND QING GUO
1 Abstract. We consider the Cauchy problem for the focusing Hartree equation iut +
1 ∆u + ( −3 u2)u = 0 in R5 with the initial data in H1, and study the divergent
0 propert|y·o|f in∗fi|ni|te-variance and nonradial solutions. Letting Q be the ground state
2 solution of Q+∆Q+( −3 Q2)Q = 0 in R5, we prove that if u H1 satisfying
− |·| ∗| | 0 ∈
n M(u )E(u ) < M(Q)E(Q) and u u > Q Q , then the corresponding
0 0 0 2 0 2 2 2
a k∇ k k k k∇ k k k
solution u(t) either blows up in finite forward time, or exists globally for positive time
J
1 and there exists a time sequence tn →+∞ such that k∇u(tn)k2 →+∞. A similar result
holds for negative time.
1
]
P
MSC: 35Q55, 35A15, 35B30.
A
.
h Keywords: Hartree equation; Blow up; Profile decomposition; Divergence
t
a
m
[
1
v
3 1. Introduction
5
0 In this paper, we consider the following Cauchy problem for the 5D Hartree equation
2
.
1
0
1 iut +∆u+(V u 2)u = 0, (x,t) R5 R,
1 ∗| | ∈ × (1.1)
: ( u(x,0) = u0(x) H1(R5),
v ∈
i
X
r where V(x) = x −3 and denotes the convolution in R5.
a Hartree type|n|onlinear∗ity ( 2−N u 2)u in RN describes the dynamics of the mean-field
|·| ∗| |
limits of many-body quantum systems such as coherent states and condensates. The case
N = 4 gives the L2-critical Hartree equation, the solution of which, by the authors in [21],
scatters when the mass of the initial data is strictly less than that of the ground state. A
largeamount of work has been devoted to the theory of scattering for the Hartree equation,
see for example [20], [5], [6], [22], [3].
It iswell known fromGinibreandVelo [4] that, (1.1) is locallywell-posed inH1. Namely,
foru H1,thereexist0 < T andauniquesolutionu(t) C([0,T);H1)to(1.1).When
0
∈ ≤ ∞ ∈
T < , we have lim u(t) and say that solution u blows up in finite positive
t↑T 2
∞ k∇ k → ∞
time. On the other hand, when T = , the solution is called positively global. Note that
∞
the local theory gives nothing about the behavior of u(t) as t + . Solutions of
2
k∇ k ↑ ∞
1
2 DAOMINCAO AND QINGGUO
(1.1) admits the following conservation laws in energy space H1 :
L2 norm : M(u)(t) u(x,t) 2dx = M(u );
0
− ≡ | |
Z
1 1 u(x,t) 2 u(y,t) 2
Energy : E(u)(t) u(x,t) 2dx | | | | dxdy = E(u );
≡ 2 |∇ | − 4 x y 3 0
Z Z ZR5×R5 | − |
Momentum : P(u)(t) Im u(x,t) u(x,t)dx = P(u ).
0
≡ ∇
Z
In [3], it is proved that if u H1, M(u )E(u ) < M(Q)E(Q) and u u >
0 0 0 0 2 0 2
∈ k∇ k k k
Q Q , then the solution u(t) to (1.1) blows up in finite time provided xu <
2 2 0 L2
k∇ k k k k k ∞
or u is radial. Note that it is sharp in the sense that u(t) = eitQ(x) solves (1.1) and does
0
not blow-up in finite time.
In this paper, in the spirit of Holmer and Roudenko [9] dealing with the cubic 3D
Schro¨dinger equation, without assuming finite variance and radiality we obtain the follow-
ing result:
Theorem 1.1. Suppose that u H1, M(u )E(u ) < M(Q)E(Q) and u u >
0 0 0 0 2 0 2
∈ k∇ k k k
Q Q . Then either u(t) blows up in finite forward time, or u(t) is forward global and
2 2
k∇ k k k
there exists a time sequence t + such that u(t ) + . A similar statement
n n 2
→ ∞ k∇ k → ∞
holds for negative time.
Remark 1.2. Using the same argument as in the introduction of [9] (see more details in
Appendix B there), via the Galilean transformation, we will always assume in this paper
that P(u) = 0. That is, we need only show Theorem 1.1 under the condition P(u) = 0. In
fact, on the one hand, by [9], the dichotomy result of Proposition 2.1 andProposition 2.2 in
section2belowispreserved bytheGalileantransformation; Ontheotherhand, wegetfrom
therelationshipbetween u(t)withnonzero momentum anditsGalileantransformationu˜(t)
satisfying
P(u)
u˜(x,t) = eixξe−it|ξ|2u(x 2ξt,t) with ξ =
− M(u)
that
P(u)2
P(u˜) = 0, M(u˜) = M(u) = M(Q), u˜ 2 = u 2 .
k∇ kL2 k∇ kL2 − 2M(u)
Thus, Theorem 1.1 is also true by Galilean transformation.
In this paper, H1 denotes the usual Sobolev space W1,2(R5) and
1
4
u u(x) 2V(x y) u(y) 2dxdy .
LV
k k ≡ | | − | |
(cid:18)Z ZR5×R5 (cid:19)
As usual, we denote the Lp norm as and use the convention that c always stands for
p
k·k
the variant absolute constants.
The rest of this paper is organized as follows. In section 2 we recall the dichotomy
and scattering results. In section 3, we discuss blow-up of solutions based on the virial
identity and its localized versions. Section 4 is devoted to the variational characterization
DIVERGENT SOLUTIONS 3
of the ground state and can be taken as a preparation for section 5, in which we set up
the inductive argument that will be continued in section 7 and section 8. In section 6
we introduce the linear and nonlinear profile decomposition lemmas that needed in the
argument in section 7 and section 8, where we give proof of Theorem 1.1.
2. Ground state and dichotomy
As in [25], let C be the best constant in the following Hardy-Littlewood-Sobolev
HLS
inequality
u(x) 2 u(y) 2
| | | | dxdy C u u 3. (2.1)
x y 3 ≤ HLSk k2k∇ k2
Z Z | − |
Then it is attained at Q, which is the unique radial positive solution to
Q ∆Q = (V Q2)Q. (2.2)
− ∗
The uniqueness of the ground state of (2.2) can be obtain by the same method as in the
cases of dimension three and four ([17] and [16] ) by means of Newton’s theorem [18]. In
fact, it suffices to note that the convolution term in (1.1) is none other than the Newton
potential in R5.
From (2.2) we have
Q 2dx+ Q 2dx Q 4 = 0,
| | |∇ | −k kLV
Z Z
and the Pohozhaev identity
5 3 7
Q 2dx+ Q 2dx Q 4 = 0.
2 | | 2 |∇ | − 4k kLV
Z Z
These two equalities imply that
4
Q 4 = Q 2 = 4 Q 2.
k kLV 3k∇ k2 k k2
As a consequence,
Q 4 4 1
C = k kLV = , (2.3)
HLS Q Q 3 3 Q Q
k k2k∇ k2 k k2k∇ k2
and therefore
1
E(Q) = Q 2. (2.4)
6k∇ k2
Let
u u
2 2
η(t) = k∇ k k k . (2.5)
Q Q
2 2
k∇ k k k
By (2.1), (2.3) and (2.4) we have
E(u)M(u)
3η(t)2 3η(t)2 2η(t)3. (2.6)
≥ E(Q)M(Q) ≥ −
4 DAOMINCAO AND QINGGUO
Thus it is not difficult to observe that if 0 M(u)E(u)/M(Q)E(Q) < 1, then there exist
≤
two solutions 0 λ < 1 < λ of the following equation of λ
−
≤
E(u)M(u)
= 3λ2 2λ3. (2.7)
E(Q)M(Q) −
On the other hand, if E(u) < 0, there exists exactly one solution λ > 1 to (2.7).
By the H1 local theory [4] , there exist T < 0 < T + such that (T ,T ) is
− + − +
−∞ ≤ ≤ ∞
the maximal time interval of existence for u(t) solving (1.1) , and if T < + then
+
∞
u(t) + as t T ,
2 +
k∇ k → ∞ ↑
A similar conclusion holds if T > . Moreover, as a consequence of the continuity of
−
−∞
the flow u(t), we have the following dichotomy proposition :
Proposition 2.1. (Global versus blow-up dichotomy) Let u H1, and let I = (T ,T )
0 − +
∈
be the maximal time interval of existence of u(t) solving (1.1). Suppose that
M(u)E(u) < M(Q)E(Q). (2.8)
If (2.8) holds and
u u < Q Q , (2.9)
0 2 0 2 2 2
k k k∇ k k k k∇ k
then I = ( ,+ ), i.e., the solution exists globally in time, and for all time t R,
−∞ ∞ ∈
u(t) u(t) < Q Q . (2.10)
2 2 2 2
k k k∇ k k k k∇ k
If (2.8) holds and
u u > Q Q , (2.11)
0 2 0 2 2 2
k k k∇ k k k k∇ k
then for t I,
∈
u(t) u(t) > Q Q . (2.12)
2 2 2 2
k k k∇ k k k k∇ k
Proof. Multiplying theformulaofenergybyM(u)andusing theHardy-Littlewood-Sobolev
inequality (2.1), we obtain
1 1
E(u)M(u) = u 2 u 2 u 4 u 2
2k∇ kL2k kL2 − 4k kLVk kL2
1 1
u 2 u 2 C u 3 u 3.
≥ 2k∇ k2k k2 − 4 HLSk∇ k2k k2
Define f(x) = 1x2 1C x3. Then f′(x) = x(1 3CHLSx), and f′(x) = 0 when x =
2 − 4 HLS − 4 0
0 and x = Q Q = 4 1 by (2.3). Note that f(0) = 0 and f(x ) = 1 Q 2 Q 2.
1 k∇ k2k k2 3CHLS 1 6k∇ k2k k2
Thus f has two extrema: a local minimum at x and a local maximum at x . (2.8) implies
0 1
that E(u )M(u ) < f(x ), which combined with energy conservation deduces that
0 0 1
f( u u ) E(u)M(u ) = E(u)M(u) < f(x ). (2.13)
2 2 0 1
k∇ k k k ≤
If initially u u < x , i.e., (2.9) holds, then by (2.13) and the continuity
0 2 0 2 1
k∇ k k k
of u(t) in t, we have u(t) u(t) < x for all t I. In particular, the H1 norm
2 2 2 1
k∇ k k∇ k k k ∈
of the solution is bounded, which implies the global existence and (2.10) in this case.
If initially u u > x , i.e., (2.11) holds, then by (2.13) and the continuity
0 2 0 2 1
of u(t) ink∇t, wekhkavek u(t) u(t) > x for all t I, which proves (2.12). (cid:3)
2 2 2 1
k∇ k k∇ k k k ∈
DIVERGENT SOLUTIONS 5
The following is another statement of the Dichotomy Proposition in terms of λ and η(t)
defined by (2.7) and (2.5) respectively, which will be useful in the sequel.
Proposition 2.2. Let M(u)E(u) < M(Q)E(Q) and 0 λ < 1 < λ be defined as (2.7).
−
≤
Then exactly one of the following holds:
(1) The solution u(t) to(1.1) is global and
1 E(u)M(u)
η(t)2 λ2, t ( ,+ )
3E(Q)M(Q) ≤ ≤ − ∀ ∈ −∞ ∞
(2) 1 < λ η(t), t (T ,T ).
− +
≤ ∀ ∈
To easily understand, one can refer to the figure in [9] describing the relationship be-
tween M(u)E(u)/M(Q)E(Q) and η(t). Whether the solution is of the first or second type
in Proposition 2.2 is determined by the initial data. Note that the second case does not
assert finite-time blow-up.
In the remainder of this section, we will review the Strichartz estimates and some facts
aboutthescattering. Itiswell-known thatapairofexponents (q,r)isStrichartzadmissible
if
2 5 5 10
+ = , 2 q , 2 r .
q r 2 ≤ ≤ ∞ ≤ ≤ 3
Similarly for s > 0, we say that (q,r) is H˙ s(R5) admissible and denote it by (q,r) Λ if
s
∈
2 5 5 10 10
+ = s, 4 < q , r < .
q r 2 − ≤ ∞ 5 2s ≤ 3
−
Correspondingly, we denote (q′,r′) the dual H˙ s(R5) admissible by (q′,r′) Λ′ if (q,r)
∈ s ∈
Λ with (q′,r′) is the H¨older dual to (q,r). We define the following Strichartz norm
−s
kukS(H˙ 21) = sup kukLqtLrx
(q,r)∈Λ1
2
and the dual Strichartz norm
u = inf u = inf u ,
k kS′(H˙−21) (q′,r′)∈Λ′12 k kLqt′Lrx′ (q,r)∈Λ−21 k kLqt′Lrx′
where (q′,r′) is the H¨older dual to (q,r).
So we have the following Strichartz estimates
t
eit∆φ c φ and ei(t−t1)∆f( ,t1)dt1 c f .
S(L2) 2 S′(L2)
k k ≤ k k · ≤ k k
(cid:13)Z0 (cid:13)S(L2)
(cid:13) (cid:13)
Together with Sobolev embedding, we o(cid:13)btain (cid:13)
(cid:13) (cid:13)
t
keit∆φkS(H˙ 12) ≤ ckφkH˙ 21 and ei(t−t1)∆f(·,t1)dt1 1 ≤ ckD21fkS′(L2).
(cid:13)Z0 (cid:13)S(H˙ 2)
(cid:13) (cid:13)
(cid:13) (cid:13)
(cid:13) (cid:13)
6 DAOMINCAO AND QINGGUO
In fact, we also have the following Kato inhomogeneous Strichartz estimate [10]
t
ei(t−t1)∆f( ,t1)dt1 c f . (2.14)
· 1 ≤ k kS′(H˙−21)
(cid:13)Z0 (cid:13)S(H˙ 2)
(cid:13) (cid:13)
In the sequel we will w(cid:13)(cid:13)rite S(H˙ 12;I) to indi(cid:13)(cid:13)cate a restriction to a time subinterval I
⊂
( ,+ ).
−∞ ∞
For the first case of the dichotomy proposition (Proposition 2.2), we have furthermore
scattering results that will be used in the future discussion. We omit the proofs since they
are similar to those in [3].
Lemma 2.3. (Small data) Let u A, then there exists δ = δ (A) > 0 such
k 0kH˙ 21 ≤ sd sd
that eit∆u δ , then u solving (1.1) is global and
k 0kS(H˙ 12) ≤ sd
u 2 eit∆u , (2.15)
k kS(H˙ 21) ≤ k 0kS(H˙ 12)
1
kD2ukS(L2) ≤ 2cku0kH˙ 12. (2.16)
(Note that by Strichartz estimates, the hypotheses are satisfied if u cδ .)
k 0kH˙ 21 ≤ sd
Theorem 2.4. (Scattering). Suppose that 0 < M(u)E(u)/M(Q)E(Q) < 1 and the first
case of Proposition 2.2 holds, then u(t) scatters as t + or t . That is, there
→ ∞ → −∞
exist φ H1 such that
±
∈
lim u(t) eit∆φ = 0. (2.17)
± H1
t→±∞k − k
Consequently,
lim u(t) = 0 (2.18)
LV
t→±∞k k
and
1 E(u)M(u)
lim η(t)2 = . (2.19)
t→±∞ 3E(Q)M(Q)
Lemma 2.5. (Existence of wave operators) Suppose that φ+ H1 and
∈
1
φ+ 2 φ+ 2 < E(Q)M(Q). (2.20)
2k k2k∇ k2
Then there exists v H1 such that the corresponding solution v to (1.1) exists globally
0
∈
and satisfies
1
v(t) v Q Q , M(v) = φ+ 2, E(v) = φ+ 2,
k∇ k2k 0k2 ≤ k∇ k2k k2 k k2 2k∇ k2
and
lim v(t) eit∆φ+ = 0.
H1
t→+∞k − k
Moreover, if eit∆φ+ δ , then
k kS(H˙ 21) ≤ sd
kvkS(H˙ 12) ≤ 2keit∆φ+kS(H˙ 21), kD21vkS(L2) ≤ 2ckφ+kH˙ 21.
DIVERGENT SOLUTIONS 7
3. Virial Identity and Blow-Up Conditions
From now on we will focus on the second case of Proposition 2.2. Using the classical
virial identity we first derive the upper bound of the finite blow-up time under the finite
variance hypothesis.
Proposition 3.1. Suppose that xu < + . Let M(u) = M(Q), E(u) < E(Q) and
0 2
k k ∞
suppose that the second case of Proposition 2.2 holds ( λ > 1 is defined in (2.7)). Let r(t)
be the scaled variance given by
xu 2
r(t) = k k2 .
48λ2(λ 1)E(Q)
−
Then blow-up occurs in forward time before t , where t = r′(0)+ r′(0)2 +2r(0).
b b
Proof. The virial identity gives p
24E(u) 4 u 2
r′′(t) = − k∇ k2.
48λ2(λ 1)E(Q)
−
Using (2.3) we obtain
1 E(u) u 2
r′′(t) = k∇ k2 .
2λ2(λ 1) E(Q) − Q 2
− (cid:18) k∇ k2(cid:19)
By the definition of λ and η,
3λ2 2λ3 η(t)2
r′′(t) = − − .
2λ2(λ 1)
−
Since η(t) λ > 1, we have
≥
r′′(t) 1,
≤ −
which by integrating in time twice gives
1
r(t) t2 +r′(0)t+r(0).
≤ −2
Note that t is the positive root of the polynomial on the right hand side, which deduces
b
that r(t) t .
b
≤ (cid:3)
The next result is related to the local virial identity. Let ϕ C∞(RN) be radial such
∈ c
that ϕ′′ 2 and
≤
x 2, x 1,
ϕ(x) = | | | | ≤
0, x 2.
(
| | ≥
For R > 0 define
x
z (t) = R2φ( ) u(x,t) 2dx. (3.1)
R
R | |
Z
8 DAOMINCAO AND QINGGUO
Then by direct calculations we obtain the following local virial identity:
x 1 x
z′′(t) =4 ∂ ∂ ϕ ∂ u¯∂ udx ∆2ϕ u 2dx (3.2)
R j k R j k − R2 R | |
Xj,k Z (cid:16) (cid:17) Z (cid:16) (cid:17)
x y
+R ϕ ϕ V(x y) u(x) 2 u(y) 2dxdy.
∇ R −∇ R ∇ − | | | |
Z Z
(cid:16) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17)
Set
x y x y
I = 3 (2x R∂ ϕ ) (2y R∂ ϕ ) j − j u(x) 2 u(y) 2dxdy,
j j j j
− R − − R x y 5| | | |
Xj Z Z h (cid:16) (cid:17) (cid:16) (cid:17) i | − |
and by the definition of ϕ, we have
z′′(t) = 24E(u) 4 u 2 +A (u(t)),
R − k∇ k2 R
where
x x
A (u(t)) =4 ∂ ∂ ϕ ∂ u¯∂ udx+4 [∂2ϕ 2] u 2dx
R j k R j k j R − |∇ |
Xj6=k Z|x|>R (cid:16) (cid:17) Xj Z|x|≤R (cid:16) (cid:17)
1 x
∆2ϕ u 2dx+I.
− R2 R | |
Z|x|>R (cid:16) (cid:17)
ObservethatI vanishesintheregion x , y R, whileintheregion x , y R,I becomes
| | | | ≤ | | | | ≥
6 V(x y) u(x) 2 u(y) 2dydx. In other cases, since the integral is symmetric
|x|≥2R |y|≥2R − | | | |
with respect to x and y, I is bounded by
R R
x y x y
6 (2x R∂ ϕ ) (2y R∂ ϕ ) j − j u(x) 2 u(y) 2dxdy,
j j j j
− R − − R x y 5| | | |
Xj Z Z|x|≥Rh (cid:16) (cid:17) (cid:16) (cid:17) i | − |
which is bounded by c |u(x)|2|u(y)|2dxdy. Thus, for a suitable radial function ϕ such
|x|≥R |x−y|3
that ϕ′′ 2, we have the following control
R R
≤
1
A (u(t)) c u 2 + u 4 . (3.3)
R ≤ R2k kL2(|x|>R) k kLV(|x|>R)
(cid:18) (cid:19)
The local virial identity will give another version of Proposition 3.1, for which, without
the assumption of finite variance, we will assumes that the solution is suitably localized
in H1 for all times.
Proposition 3.2. Let M(u) = M(Q), E(u) < E(Q) and suppose that the second case
of Proposition 2.2 holds ( λ > 1 is defined in (2.7)). Select γ such that 0 < γ <
min λ 1,1 . Suppose that there is a radius R c such that for all t, there holds that
{ − } ≥ 6γ
q
6γE(Q)
u 4 < , (3.4)
k kLV(|x|≥R) c
DIVERGENT SOLUTIONS 9
where the absolute constant c is determined in (3.3). Let r˜(t) be the scaled local variance
given by
z (t)
R
r˜(t) = .
48λ2(λ 1 γ)E(Q)
− −
Then blow-up occurs in forward time before t , where t = r˜′(0)+ r˜′(0)2 +2r˜(0).
b b
Proof. In view of the assumptions, by the local virial identity and pthe same steps as in the
proof of Proposition 3.1
1
r˜′′(t) = 24E(u) 4 u 2 +A (u(t))
48λ2(λ 1 γ)E(Q) − k∇ k2 R
− −
1 (cid:0) A (u(t)) (cid:1)
= 3λ2 2λ3 η(t)2 + R
2λ2(λ 1 γ) − − 24E(Q)
− − (cid:18) (cid:19)
3λ2 −2λ3 −η(t)2 + Rc2kuk2L2(|x|>R) + ckuk4LV(|x|>R)
≤ 2λ2(λ 1 γ) 48E(Q)λ2(λ 1 γ) 48E(Q)λ2(λ 1 γ)
− − − − − −
1
3λ2 2λ3 η(t)2 +γη(t)2
≤ 2λ2(λ 1 γ) − −
− −
1. (cid:0) (cid:1)
≤
Finally, we complete our proof just the same as in the proof of Proposition 3.1 .
(cid:3)
Remark 3.3. Note that by Hardy-Littlewood-Sobolev inequalities, H¨older estimates and
Sobolev embedding, the assumption (3.4) is satisfied by u which is H1 bounded and H1
localized, i.e. for any ǫ > 0 there exists R > 0 large enough such that u ǫ.
H1(|x|≥R)
k k ≤
Wewillfinallygiveaquantified proofoffinite-timeblow-upforradialsolutions, forwhich
we need the following radial Sobolev embedding: Let u H1(Rd) be radially symmetric,
∈
then
k|x|d−21uk2∞ ≤ ckuk2k∇uk2. (3.5)
Proposition 3.4. Let M(u) = M(Q), E(u) < E(Q). Suppose u is radial and the second
case of Proposition 2.2 holds ( λ > 1 is defined in (2.7)). Select γ such that 0 < γ <
5
min λ 1,1 . Suppose that R max c , cE(Q) 4 , where the absolute constant c
{ − } ≥ 6γ 12γ
(cid:26) (cid:27)
is determined by the two in (3.3) and (3.5q). Le(cid:16)t r˜(t) b(cid:17)e the scaled local variance given by
z (t)
R
r˜(t) = .
48λ2(λ 1 γ)E(Q)
− −
Then blow-up occurs in forward time before t , where t = r˜′(0)+ r˜′(0)2 +2r˜(0).
b b
p
10 DAOMINCAO AND QINGGUO
Proof. Again from the local virial identity,
1
r˜′′(t) = 24E(u) 4 u 2+A (u(t))
48λ2(λ 1 γ)E(Q) − k∇ k2 R
− −
3λ2 −2λ3 −η(t)2 + (cid:0) Rc2kuk2L2(|x|>R) + cku(cid:1)k4LV(|x|>R) .
≤ 2λ2(λ 1 γ) 48E(Q)λ2(λ 1 γ) 48E(Q)λ2(λ 1 γ)
− − − − − −
The radial Sobolev embedding (3.5) implies that for any p 2,
≥
c p+2 p−2
u p u 2 u 2 .
k kLp(|x|>R) ≤ R2p−2k kL2(|x|>R)k∇ kL2(|x|>R)
This, combined with Hardy-Littlewood-Sobolev inequalities and H¨older estimates, implies
that
c 11 9 cE(Q)2
u 4 u 2 u 2 u 5 u 5 η(t)2.
k kLV(|x|>R) ≤ k kL170(R5)k kL170(|x|>R) ≤ R45k kL2(R5)k∇ kL2(R5) ≤ R45
Thus in view of the assumptions, we have
1
r˜′′(t) 3λ2 2λ3 η(t)2 +γη(t)2 1.
≤ 2λ2(λ 1 γ) − − ≤
− −
(cid:0) (cid:1)
Arguing the same as in the proof of the preceding propositions we can complete our proof.
(cid:3)
4. Variational Characterization of the Ground State
In this section we deal with the variation characterization of Q defined in section 2. It
is an important preparation for the “near boundary case” in section 5. Since the time
dependence plays no role in this section, we will write u = u(x) for now.
Proposition 4.1. There exists a function ǫ(ρ) with ǫ(ρ) 0 as ρ 0 such that the
→ →
following holds: suppose there is λ > 0 satisfying
M(u)E(u)
3λ2 2λ3 ρλ3, (4.1)
M(Q)E(Q) − − ≤
(cid:12) (cid:12)
(cid:12) (cid:0) (cid:1)(cid:12)
and (cid:12) (cid:12)
(cid:12) (cid:12)
u u λ, λ 1,
2 2
k k k∇ k λ ρ ≥ (4.2)
Q Q − ≤ λ2, λ 1.
(cid:12)k k2k∇ k2 (cid:12) ( ≤
(cid:12) (cid:12)
Then there exists θ R an(cid:12)d x R5 such th(cid:12)at
∈ (cid:12) 0 ∈ (cid:12)
u eiθλ52β−2Q λ(β−1 x0) β12ǫ(ρ) (4.3)
− ·− 2 ≤
(cid:13) (cid:13)
and (cid:13) (cid:0) (cid:1)(cid:13)
(cid:13) (cid:13)
u eiθλ25β−2Q λ(β−1 x0) λβ−21ǫ(ρ), (4.4)
∇ − ·− 2 ≤
where β = M(u). (cid:13)(cid:13) h (cid:0) (cid:1)i(cid:13)(cid:13)
M(Q) (cid:13) (cid:13)
