# Dynamiscs of radial threshold solutions for generalized energy-critical
Hartree equation

Xuemei Li  Xuemei Li
Laboratory of Mathematics and Complex Systems,
Ministry of Education,
School of Mathematical Sciences,
Beijing Normal University,
Beijing, 100875, People’s Republic of China<EMAIL_ADDRESS>,
Chenxi Liu  Chenxi Liu
Laboratory of Mathematics and Complex Systems,
Ministry of Education,
School of Mathematical Sciences,
Beijing Normal University,
Beijing, 100875, People’s Republic of China<EMAIL_ADDRESS>,
Xingdong Tang  Xingdong Tang
School of Mathematics and Statistics,
Nanjing Univeristy of Information Science and Technology,
Nanjing, 210044, People’s Republic of China<EMAIL_ADDRESS>and  Guixiang
Xu  Guixiang Xu
Laboratory of Mathematics and Complex Systems,
Ministry of Education,
School of Mathematical Sciences,
Beijing Normal University,
Beijing, 100875, People’s Republic of China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study long time dynamics of radial threshold solutions for
the focusing, generalized energy-critical Hartree equation and classify all
radial threshold solutions. The main arguments are the spectral theory of the
linearized operator, the modulational analysis and the concentration
compactness rigidity argument developed by T. Duyckaerts and F. Merle to
classify all threshold solutions for the energy critical NLS and NLW in [18,
19], later by D. Li and X. Zhang in [43, 44] in higher dimensions. The new
ingredient here is to solve the nondegeneracy of positive bubble solutions
with nonlocal structure in $\dot{H}^{1}(\mathbb{R}^{N})$ (i.e. the spectral
assumption in [51]) by the nondegeneracy result of positive bubble solution in
$L^{\infty}(\mathbb{R}^{N})$ in [45] and the Moser iteration method in [16],
which is related to the spectral analysis of the linearized operator with
nonlocal structure, and plays a key role in the construction of the special
threshold solutions, and the classification of all threshold solutions.

###### Key words and phrases:

Concentration compactness rigidity argument; Hartree equation; Modulation
analysis; Nondegeneracy of ground state; Spectral theory; Uniqueness.

###### 2010 Mathematics Subject Classification:

Primary: 35Q41, Secondary: 35Q55

##  1\. Introduction

In this paper, we consider the focusing, generalized energy-critical Hartree
equation

$i\partial_{t}u+\Delta
u+\left(I_{\lambda}*|u|^{p}\right)|u|^{p-2}u=0,\quad(t,x)\in\mathbb{R}\times\mathbb{R}^{N},$
(1.1)

where $N\geq 3$, $0<\lambda<N$, $\lambda\leq 4$, $p=\frac{2N-\lambda}{N-2}\geq
2$ and the Riesz-potential stands for

$I_{\lambda}(x):=\frac{\Gamma(\frac{\lambda}{2})}{\pi^{\frac{N}{2}}\Gamma(\frac{N-\lambda}{2})2^{N-\lambda}|x|^{\lambda}}.$

The generalized Hartree equation can be considered as a classical limit of a
field equation and it describes a quantum mechanical non-relativistic many-
boson system interacting through a two body potential
$V(x)=\frac{1}{|x|^{\lambda}}$ , see [28]. As for the mean-field limit of
many-body quantum systems, we can refer to the work by Hepp in [32], see also
[9, 10, 26, 28, 61]. Lieb and Yau use it to develop the theory for stellar
collapse in [49] .

The equation (1.1) enjoys several symmetries: If $u(t,x)$ is a solution, then

  1. (a)

so is $\delta^{\frac{N-2}{2}}u(\delta^{2}t,\delta x)$, $\delta>0$ by scaling
invariance;

  2. (b)

so is $u(t+t_{0},x)$ for $t_{0}\in\mathbb{R}$ by time translation invariance;

  3. (c)

so is $u(t,x+x_{0})$ for $x_{0}\in\mathbb{R}^{N}$ by spatial translation
invariance;

  4. (d)

so is $e^{i\theta_{0}}u(t,x)$, $\theta_{0}\in\mathbb{R}$ by phase rotation
invariance;

  5. (e)

so is $\overline{u(-t,x)}$ by time reversal.

The Cauchy problem for (1.1) was developed by A. K. Arora and S. Roudenko in
[1, 3, 5, 13]. Namely, if $u_{0}\in\dot{H}^{1}(\mathbb{R}^{N})$, there exists
a unique solution defined on a maximal interval
$I=\left(-T_{-}(u),T_{+}(u)\right)$ and the energy

$\displaystyle E(u(t)):=\frac{1}{2}\int_{\mathbb{R}^{N}}\big{|}\nabla
u(t,x)\big{|}^{2}dx-\frac{1}{2p}\int_{\mathbb{R}^{N}}\left(I_{\lambda}*|u|^{p}\right)\big{|}u(t,x)\big{|}^{p}\
dx$ (1.2)

is conserved by time translation invariance of equation (1.1). The exponent
$p=\frac{2N-\lambda}{N-2}$ is energy-critical exponent since the scaling
transform

$u(t,x)\rightarrow\delta^{\frac{N-2}{2}}u\left(\delta^{2}t,\delta
x\right),\;\delta>0$

makes the equation (1.1) and the energy (1.2) invariant.

For long time behavior of solutions for (1.1), the ground state plays an
important role, which is the solution of the elliptic equation with nonlocal
potential

$\displaystyle-\Delta W(x)=\left(I_{\lambda}*W^{p}\right)\;W^{p-1},\quad
x\in\mathbb{R}^{N}.$ (1.3)

According to [17], we have the variational characterization of $W$, which can
be proved by combining sharp Sobolev inequality [7, 48, 64] with sharp Hardy-
Littlewood-Sobolev inequality [47, 48]. Recently, the authors give an
alternative proof of the existence of the extremizer of this sharp Hardy-
Littlewood-Sobolev inequality in $\mathbb{R}^{N}$ by using the stereographic
projection and sharp Hardy-Littlewood-Sobolev inequality on the sphere in
[45]. That is,

######  Proposition 1.1 ([17, 45]).

Let $N\geq 3$, $0<\lambda<N$, $\lambda\leq 4$, and
$p=\frac{2N-\lambda}{N-2}\geq 2$. Then for any
$f\in\dot{H}^{1}(\mathbb{R}^{N}),$ we have

$\displaystyle\left(\int_{\mathbb{R}^{N}}\left(I_{\lambda}*|f|^{p}\right)\;|f(x)|^{p}\right)^{\frac{1}{2p}}\leq$
$\displaystyle\;C(N,\lambda)\big{\|}\nabla f\big{\|}_{2},$

where
$C(N,\lambda)=\left[(\frac{1}{2\sqrt{\pi}})^{N-\lambda}\frac{\Gamma(\frac{\lambda}{2})}{\Gamma(N-\frac{\lambda}{2})}\big{(}\frac{\Gamma(N)}{\Gamma(\frac{N}{2})}\big{)}^{\frac{N-\lambda}{N}}\right]^{\frac{1}{2p}}\cdot\left[\frac{N(N-2)}{4}2^{2/N}\pi^{\frac{N+1}{N}}\Gamma(\frac{N+1}{2})^{-\frac{2}{N}}\right]^{-\frac{1}{2}}$
and if

$\displaystyle\left(\int_{\mathbb{R}^{N}}\left(I_{\lambda}*|f|^{p}\right)\;|f(x)|^{p}\right)^{\frac{1}{2p}}=$
$\displaystyle\;C(N,\lambda)\big{\|}\nabla f\big{\|}_{2},$

then there exist $\lambda_{0}>0$, $x_{0}\in\mathbb{R}^{N}$,
$\theta_{0}\in[0,2\pi)$, such that

$\displaystyle
f(x)=e^{i\theta_{0}}\lambda^{-\frac{N-2}{2}}_{0}W\left(\lambda^{-1}_{0}(x+x_{0})\right),$

where

$W=\left[N(N-2)\right]^{\frac{N-2}{4}}\big{(}\frac{1}{1+|x|^{2}}\big{)}^{\frac{N-2}{2}}.$
(1.4)

As for the solution of (1.1) with energy below the threshold $E(W)$, C. Guzmán
and C. Xu make use of the concentration compactness rigidity argument, which
is originally developed by C. Kenig and F. Merle in [35, 36], (see also [54])
to show that

######  Theorem 1.2 ([31]).

Let $N=3$, $0<\lambda<N$, $p=\frac{2N-\lambda}{N-2}\geq 2$, $W$ be the ground
state solution of (1.3) and $u$ be a radial solution of (1.1) in
$\dot{H}^{1}(\mathbb{R}^{N})$ such that

$E(u_{0})<E(W)\;\text{and}\;\;\|u_{0}\|_{\dot{H}^{1}(\mathbb{R}^{N})}<\|W\|_{\dot{H}^{1}(\mathbb{R}^{N})}.$

Then the solution $u$ is global in time and scatters both forward and backward
in time, i.e. there exists $u_{\pm}\in\dot{H}^{1}(\mathbb{R}^{N})$, such that

$\lim_{t\rightarrow\pm\infty}\big{\|}u(t)-e^{it\Delta}u_{\pm}\big{\|}_{\dot{H}^{1}(\mathbb{R}^{N})}=0.$

######  Remark 1.3.

Using the Virial identity, we can also show the blow-up result: if
$E(u_{0})<E(W)$, $\big{\|}\nabla u_{0}\big{\|}_{L^{2}}>\big{\|}\nabla
W\big{\|}_{L^{2}}$ and $x\cdot u_{0}\in L^{2}(\mathbb{R}^{N})$, then the
solution blows up at finite time $T_{\pm}$, which can be shown from [54]. So
far, there have been many other dynamics results of the Hartree equation. In
the mass and energy critical cases, please refer to [38, 41, 42, 54, 55, 57,
58, 68], and we can refer to [2, 4, 12, 29, 39, 52, 53, 56, 69] for other
cases.

In this paper, we follow the argument by T. Duyckaerts and F. Merle in [18,
19], D. Li and X. Zhang in [43, 44] and consider the dynamics of radial
solutions of (1.1) with threshold energy $E(W)$. Our goal is to classify the
threshold solutions with

$\displaystyle u_{0}\in\dot{H}^{1}(\mathbb{R}^{N})\,\text{is radial,
and}\;\;E(u_{0})=E(W).$

For the case $p=2$ and $\lambda=4<N$, C. Miao, Y. Wu and the fourth author
study the dynamics of the radial threshold solution and classify the
corresponding radial solutions in the energy space
$\dot{H}^{1}(\mathbb{R}^{N})$ under the spectral assumption (i. e.
nondegeneracy of the ground state ) in [51]. Based on the results by authors
in [45, 46], the motivation in this paper is to remove the spectral assumption
in [51] and solve the nondegeneracy of the positive bubble solutions of (1.1)
in the energy space $\dot{H}^{1}(\mathbb{R}^{N})$. That is,

######  Proposition 1.4.

Let $N\geq 3$, $0<\lambda<N$, $\lambda\leq 4$ and
$p=\frac{2N-\lambda}{N-2}\geq 2$. Then the nontrivial solution
$U_{c,\delta,x_{0}}(x):=\frac{c}{\left(1+\delta^{2}|x-x_{0}|^{2}\right)^{\frac{N-2}{2}}}$
of the equation

$-\Delta u-\left(I_{\lambda}*|u|^{p}\right)|u|^{p-2}u=0,\quad
x\in\mathbb{R}^{N}$

with constant sign is non-degenerate. More precisely, any energy solution to
the linearized equation

$-\Delta\varphi=p\left[I_{\lambda}*(|u|^{p-1}\varphi)\right]u^{p-1}+(p-1)\left(I_{\lambda}*|u|^{p}\right)u^{p-2}\varphi$
(1.5)

must be the linear combination of the functions $\varphi_{1}$, $\cdots$,
$\varphi_{N}$ and $\varphi_{N+1}$ defined by

$\varphi_{j}(x):=\frac{\partial U_{1,1,0}}{\partial x_{j}}(x),1\leq j\leq
N,\text{~{}and ~{} }\varphi_{N+1}(x):=\frac{N-2}{2}U_{1,1,0}(x)+x\cdot\nabla
U_{1,1,0}(x).$

######  Remark 1.5.

The uniqueness and nondegeneracy of the ground state itself are very
interesting and important problems, we can refer to [21, 22, 50, 65] and
references therein, and they also play a key role in long-time dynamics of
nonlinear dispersive equations (for example, NLS, NLW, Hartree, Boson star,
etc.), especially in spectral analysis of the linearized operator, the
construction of the threshold solutions, and the classification of the
threshold solutions. Now we give some remarks about Proposition 1.4 as
following.

  1. (a)

In [46], The first, third and fourth authors show the nondegeneracy of the
ground state in the Newtonian potential case $\lambda=4,N=6$ by using the
spherical harmonic expansion of the Newtonian potential, which is related to
Gegenbauer identity in [6, 40].

  2. (b)

Motivated by R. Frank and E. Lieb in [23, 24], the authors make use of the
spherical harmonic decomposition and the Funk-Hecke formula of the spherical
harmonic functions (see also [6, 15]) to obtain the nondegeneracy of the
ground state in $L^{\infty}(\mathbb{R}^{N})$ in [45] for the general case .

  3. (c)

Based on the nondegeneracy of positive bubble solution in
$L^{\infty}(\mathbb{R}^{N})$ in [45], we only need show
$L^{\infty}(\mathbb{R}^{N})$-regularity of $\dot{H}^{1}(\mathbb{R}^{N})$
solution of equation (1.5) to complete the proof of Proposition 1.4, which
will be shown by the Moser iteration method (please refer to [16], and
references therein) in Appendix A.

After showing the nondegeneracy result in Proposition 1.4, we can follow the
standard argument in [18, 19, 43, 44, 51] to study the dynamics of threshold
solution and classify all radial threshold solutions. The classification of
threshold solutions is more complicate than that in Theorem 1.2. In fact, the
ground state $W$ is a new solution which doesn’t satisfy the conclusion in
Theorem 1.2. Besides $W$, there also exist two other special radial threshold
solutions $W^{\pm}$.

######  Theorem 1.6.

There exist two radial solutions $W^{\pm}$ of (1.1) on the maximal lifespan
interval $(T_{-}(W^{\pm}),T_{+}(W^{\pm}))$ with initial data
$W^{\pm}_{0}\in\dot{H}^{1}(\mathbb{R}^{N})$ such that

  1. (a)

$E(W^{\pm})=E(W)$, $T_{+}(W^{\pm})=\infty$ and

$\displaystyle\lim_{t\rightarrow+\infty}W^{\pm}(t)=W\;\text{in}\;\dot{H}^{1}.$

  2. (b)

$\big{\|}\nabla W^{-}_{0}\big{\|}_{2}<\big{\|}\nabla W\big{\|}_{2}$,
$T_{-}(W^{-})=\infty$ and $W^{-}$ scatters for the negative time.

  3. (c)

$\big{\|}\nabla W^{+}_{0}\big{\|}_{2}>\big{\|}\nabla W\big{\|}_{2}$. Moreover,
if $N\geq 5$, $T_{-}(W^{+})<\infty$.

######  Remark 1.7.

There is no $L^{2}$-regularity for $W^{+}$ and $W$ when $N=3,4$ from the
construction in Section 3 and there is additional $L^{2}$-regularity for
threshold solution in Proposition 5.1 in Section 5. We also expect that
$T_{-}(W^{+})<\infty$ for the case $N=3,4$.

Next, we classify all radial threshold solutions as follows.

######  Theorem 1.8.

Suppose $u_{0}\in\dot{H}^{1}\big{(}\mathbb{R}^{N}\big{)}$ is radial, and such
that $E(u_{0})=E(W)$. Let $u$ be a solution of (1.1) with initial data $u_{0}$
and $I$ be the maximal interval of existence. Then the following holds:

  1. (a)

If $\big{\|}\nabla u_{0}\big{\|}_{2}<\big{\|}\nabla W\big{\|}_{2}$, then
$I=\mathbb{R}$. Furthermore, either $u=W^{-}$ up to the symmetries of the
equation, or
$\big{\|}u\big{\|}_{L^{2p}_{t}L^{\frac{2Np}{Np-2p-2}}_{x}}<\infty$.

  2. (b)

If $\big{\|}\nabla u_{0}\big{\|}_{2}=\big{\|}\nabla W\big{\|}_{2}$, then $u=W$
up to the symmetries of the equation.

  3. (c)

If $\big{\|}\nabla u_{0}\big{\|}_{2}>\big{\|}\nabla W\big{\|}_{2}$, and
$u_{0}\in L^{2}(\mathbb{R}^{N})$, then either $u=W^{+}$ up to the symmetries
of the equation, or $I$ is finite.

######  Remark 1.9.

When $N=3,\,4$ in the supercritical case (c), the above result shows that any
$L^{2}(\mathbb{R}^{N})$-solution blows up in both time directions from the
fact that $W^{+}\not\in L^{2}(\mathbb{R}^{N})$.

######  Remark 1.10.

Because of radial assumption, the symmetries of the equation here refer to the
symmetries under scaling, time translation, phase rotation and time reverse.
The above result is a generalization of the classification of radial threshold
solutions by Miao, Wu and the fourth author in [51], which consider the case
$\lambda=4<N$, and $p=2$. For $\dot{H}^{1}(\mathbb{R}^{N})$ critical NLS, Q.
Su and Z. Zhao classify nonradial, subcritical threshold solution in [63]. In
addition, for subcritical NLS, we can refer to [20] by T. Duyckaerts and S.
Roudenko, [11] by L. Campos, L. G. Farah and S. Roudenko. and T. Zhou also
classifies threshold solutions of subcritical Hartree equation under spectral
assumption in [69].

Last, the paper is organized as follows. In Section 2, we recall the Cauchy
problem and the properties of the ground state. We also state the spectral
properties of the linearized operator $\mathcal{L}$ around $W$, which rely on
the nondegeneracy of positive bubble solutions in Proposition 1.4. In Section
3, we prove Theorem 1.6 except for the negative time behavior, in particular,
we construct two special solutions $W^{\pm}$ of (1.1) except for the negative
time behavior. In Section 4, we discuss the modulational stability around $W$.
In Section 5 and Section 6, we study the solutions with initial data
satisfying Theorem 1.8 part (a) and (c), and also obtain the proof of Theorem
1.6 for the negative time behavior. In Section 7, we analyze the property of
the exponentially decay solution of the linearized equation to establish the
uniqueness of the special solutions and this will imply the proof of Theorem
1.8. In appendix A, we show $L^{\infty}(\mathbb{R}^{N})$-regularity of
$\dot{H}^{1}(\mathbb{R}^{N})$ solution of equation (1.5) by the Moser
iteration method. In appendix B and appendix C, we prove the positivity and
spectral properties of the linearized operator in Proposition 2.13 and
Proposition 2.14.∎

### Acknowledgements.

G. Xu was partly supported by National Key Research and Development Program of
China (No. 2020YFA0712900) and by NSFC (No. 12371240). X. Tang were partly
supported by NSFC (No. 12001284).

##  2\. Preliminaries

In this section, we give some preliminary results for the context.

###  2.1. The linear estimates and the Cauchy problem.

In this subsection, we firstly recall some results on the Strichartz estimate
and Cauchy problem of (1.1). Let $I$ be an interval, and denote

$\displaystyle
Z(I):=L^{2p}\left(I;L^{\frac{2Np}{Np-2p-2}}({\mathbb{R}^{N}})\right),\;S(I):=L^{2p}\left(I;L^{\frac{2Np}{Np-2}}({\mathbb{R}^{N}})\right),$
$\displaystyle
N(I):=L^{\frac{2p}{2p-1}}\left(I;L^{\frac{2Np}{Np+2}}({\mathbb{R}^{N}})\right),\;l(I):=Z(I)\cap
L^{2p}\left(I;\dot{W}^{\frac{2Np}{Np-2}}({\mathbb{R}^{N}})\right),$
$\displaystyle\big{\|}u\big{\|}_{l(I)}:=\big{\|}u\big{\|}_{Z(I)}+\big{\|}\nabla
u\big{\|}_{S(I)}.$

Here we write $N(I)$ to represent the inhomogeneous space-time norm on
$I\times\mathbb{R}^{N}$ and the single $N$ to represent spatial dimension in
the context without confusion.

A solution of (1.1) on an interval $I$ with $0\in I$ is a function $u\in
C^{0}(I,\dot{H}^{1}(\mathbb{R}^{N}))$ such that $u\in Z(J)$ for all interval
$J\Subset I$ and

$\displaystyle
u(t)=e^{it\Delta}u_{0}+i\int^{t}_{0}e^{i(t-s)\Delta}\left(I_{\lambda}*|u(s,\cdot)|^{p}\right)(x)\;|u|^{p-2}u(s,x)\;ds.$

We have

######  Lemma 2.1 ([13, 66]).

Consider

$\left\\{\begin{aligned} i\partial_{t}u+\Delta u=&\;f,\quad
x\in\mathbb{R}^{N},\;t\in[0,T),\\\
u(0)=&\;u_{0}\in\dot{H}^{1}(\mathbb{R}^{N}),\end{aligned}\right.$ (2.1)

where $\nabla f\in N(0,T)$, then for some constant $C>0$, we have

$\displaystyle\sup_{t\in[0,T)}\big{\|}u\big{\|}_{\dot{H}^{1}}+\big{\|}u\big{\|}_{l(0,T)}\leq
C\left(\big{\|}u_{0}\big{\|}_{\dot{H}^{1}}+\big{\|}\nabla
f\big{\|}_{N(0,T)}\right).$

######  Lemma 2.2 ([30, 62]).

For $\alpha\in(0,N)$, then there exists a constant $C(N,\alpha)$ such that for
any $r\in(\frac{N}{N-\alpha},\infty)$,

$\left\|\int_{\mathbb{R}^{N}}\frac{f(y)}{|x-y|^{N-\alpha}}\;dy\right\|_{L^{r}(\mathbb{R}^{N})}\leq
C(N,\alpha)\big{\|}f\big{\|}_{L^{\frac{Nr}{N+\alpha r}}(\mathbb{R}^{N})}.$

######  Lemma 2.3 ([8, 27, 37]).

For every $f\in\dot{H}^{1}(\mathbb{R}^{N})$, there is a constant $C$ such that

$\big{\|}f\big{\|}_{L^{\frac{2N}{N-2}}(\mathbb{R}^{N})}\leq
C\big{\|}f\big{\|}^{\frac{N-2}{N}}_{\dot{H}^{1}(\mathbb{R}^{N})}\big{\|}f\big{\|}^{\frac{2}{N}}_{\dot{B}^{1}_{2,\infty}(\mathbb{R}^{N})}.$

######  Theorem 2.4 ([1, 3, 5]).

For any $u_{0}\in\dot{H}^{1}(\mathbb{R}^{N})$ and $t_{0}\in\mathbb{R}$, there
exists a unique maximal-lifespan solution
$u:I\times\mathbb{R}^{N}\rightarrow\mathbb{C}$ to (1.1) with $u(t_{0})=u_{0}$.
This solution also has the following properties:

  1. (a)

$I$ is an open neighborhood of $t_{0}$.

  2. (b)

The energy of the solution $u$ are conserved, that is, for all $t\in I$, we
have

$\displaystyle E(u(t))=E(u_{0}).$

  3. (c)

If $u^{(n)}_{0}$ is a sequence converging to $u_{0}$ in $\dot{H}^{1}$ and
$u^{(n)}:I^{(n)}\times\mathbb{R}^{N}\rightarrow\mathbb{C}$ are the associated
maximal-lifespan solutions, then $u^{(n)}$ converges locally uniformly to $u$.

  4. (d)

There exists $\eta_{0}$, such that if
$\big{\|}u_{0}\big{\|}_{\dot{H}^{1}}<\eta_{0}$, then $u$ is a global solution.
Indeed, the solution also scatters to $0$ in $\dot{H}^{1}(\mathbb{R}^{N})$.

###  2.2. Properties of ground state.

From (1.3) and Proposition 1.1, we have

$\displaystyle\big{\|}\nabla
W\big{\|}^{2}_{2}=C(N,\lambda)^{-\frac{p}{p-1}},\quad
E(W)=\frac{p-1}{2p}C(N,\lambda)^{-\frac{p}{p-1}}.$ (2.2)

Using the characterization of $W$ in Proposition 1.1, the refined Sobolev
inequality in Lemma 2.3, the analogue concentration compactness principle
(profile decomposition in $\dot{H}^{1}_{rad}$) to the proof of Proposition
$3.1$ in [57], we have

######  Proposition 2.5.

Let $u\in\dot{H}^{1}(\mathbb{R}^{N})$ be radial and $E(u)=E(W)$. Then there
exists a function $\varepsilon=\varepsilon(\rho)$, such that

$\displaystyle\inf_{\theta\in\mathbb{R},\;\mu>0}\big{\|}u_{\theta,\mu}-W\big{\|}_{\dot{H}^{1}}\leq\varepsilon(\delta(u)),\quad\lim_{\rho\rightarrow
0}\varepsilon(\rho)=0,$

where $u_{\theta,\mu}(x)=e^{i\theta}\mu^{-\frac{N-2}{2}}u(\mu^{-1}x)$,
$\displaystyle\delta(u)=\Big{|}\int_{\mathbb{R}^{N}}\left(\big{|}\nabla
u\big{|}^{2}-\big{|}\nabla W\big{|}^{2}\right)\;dx\Big{|}$.

###### Proof.

We follow the argument in Proposition $2.7$ in [51] and argue by
contradiction. Let $u_{n}\in\dot{H}^{1}$ be any radial sequence and satisfy

$E(u_{n})=E(W),\;\text{and}\;\Big{|}\int_{\mathbb{R}^{N}}\left(\big{|}\nabla
u_{n}\big{|}^{2}-\big{|}\nabla W\big{|}^{2}\right)\;dx\Big{|}\longrightarrow
0.$ (2.3)

Hence, from the profile decomposition in $\dot{H}^{1}_{rad}(\mathbb{R}^{N})$,
there exists a subsequence of $\\{u_{n}\\}^{\infty}_{n=1}$ and a sequence
$\\{U^{(j)}\\}_{j\geq 1}$ in $\dot{H}^{1}_{rad}$ and for any $j\geq 1$, a
family $\\{\lambda^{j}_{n}\\}$ such that

  1. (1)

If $j\not=k$, then

$\displaystyle\frac{\lambda^{j}_{n}}{\lambda^{k}_{n}}+\frac{\lambda^{k}_{n}}{\lambda^{j}_{n}}\longrightarrow+\infty,\text{}\;n\rightarrow+\infty$

  2. (2)

For every $l\geq 1$, we have

$\displaystyle
u_{n}(x)=\sum^{l}_{j=1}\frac{1}{\left(\lambda^{j}_{n}\right)^{(N-2)/2}}U^{(j)}\left(\frac{x}{\lambda^{j}_{n}}\right)+r^{l}_{n}(x),\;\text{with}\;\lim_{l\rightarrow+\infty}\limsup_{n\rightarrow+\infty}\big{\|}r^{l}_{n}\big{\|}_{\dot{B}^{1}_{2,\infty}}\rightarrow
0.$

  3. (3)

We have

$\displaystyle\big{\|}\nabla
u_{n}\big{\|}^{2}_{L^{2}}=\sum^{l}_{j=1}\big{\|}\nabla
U^{(j)}\big{\|}^{2}_{L^{2}}+\big{\|}\nabla
r^{l}_{n}\big{\|}^{2}_{L^{2}}+o_{n}(1),$

By Lemma 2.2, Lemma 2.3, the orthogonality between $\lambda^{j}_{n}$ and
$\lambda^{k}_{n}$ for $j\not=k$, we have

$\displaystyle\lim_{l\rightarrow+\infty}\limsup_{n\rightarrow+\infty}\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\lambda}(x-y)\left|r^{l}_{n}(x)\right|^{p}\left|r^{l}_{n}(y)\right|^{p}\;dxdy=0,$
$\displaystyle\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\lambda}(x-y)|u_{n}(x)|^{p}|u_{n}(y)|^{p}\;dxdy=\lim_{l\rightarrow+\infty}\sum^{l}_{j=1}\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\lambda}(x-y)\left|U^{(j)}(x)\right|^{p}\left|U^{(j)}(y)\right|^{p}\;dxdy.$

By Proposition 1.1, we have

$\displaystyle
C(N,\lambda)^{-2p}\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\lambda}(x-y)\left|U^{(j)}(x)\right|^{p}\left|U^{(j)}(y)\right|^{p}\;dxdy\leq\big{\|}\nabla
U^{(j)}\big{\|}^{2p}_{L^{2}},$

thus,

$\displaystyle
C(N,\lambda)^{-2p}\sum^{l}_{j=1}\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\lambda}(x-y)\left|U^{(j)}(x)\right|^{p}\left|U^{(j)}(y)\right|^{p}\;dxdy\leq\sum^{l}_{j=1}\big{\|}\nabla
U^{(j)}\big{\|}^{2p}_{L^{2}}.$

This yields that

$\displaystyle C(N,\lambda)^{-2p}\leq$
$\displaystyle\lim_{n\rightarrow+\infty}\limsup_{l\rightarrow+\infty}\frac{\displaystyle\sum^{l}_{j=1}\big{\|}\nabla
U^{(j)}\big{\|}^{2p}_{L^{2}}}{\displaystyle\sum^{l}_{j=1}\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\lambda}(x-y)\left|U^{(j)}(x)\right|^{p}\left|U^{(j)}(y)\right|^{p}\;dxdy}$
$\displaystyle\leq$
$\displaystyle\lim_{n\rightarrow+\infty}\limsup_{l\rightarrow+\infty}\frac{\displaystyle\left(\sum^{l}_{j=1}\big{\|}\nabla
U^{(j)}\big{\|}^{p}_{L^{2}}\right)^{2}}{\displaystyle\sum^{l}_{j=1}\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\lambda}(x-y)\left|U^{(j)}(x)\right|^{p}\left|U^{(j)}(y)\right|^{p}\;dxdy}$
$\displaystyle\leq$
$\displaystyle\lim_{n\rightarrow+\infty}\frac{\displaystyle\big{\|}\nabla
u_{n}\big{\|}^{2p}_{L^{2}}}{\displaystyle\sum^{l}_{j=1}\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\lambda}(x-y)\left|u_{n}(x)\right|^{p}\left|u_{n}(y)\right|^{p}\;dxdy}$
$\displaystyle=$ $\displaystyle C(N,\lambda)^{-2p},$

where in the last step we use (2.3). Therefore, we conclude that only one term
$U^{(j)}$ is nonzero, and

$\displaystyle C(N,\lambda)^{-2p}=\frac{\displaystyle\big{\|}\nabla
U^{(j_{0})}\big{\|}^{2p}_{L^{2}}}{\displaystyle\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\lambda}(x-y)\left|U^{(j_{0})}(x)\right|^{p}\left|U^{(j_{0})}(y)\right|^{p}\;dxdy}.$

By Proposition 1.1 for the radial case, there exist $\theta_{0}\in[0,2\pi)$,
$\lambda_{0}>0$ such that

$U^{(j_{0})}(x)=e^{i\theta_{0}}\lambda^{-\frac{N-2}{2}}_{0}W(\lambda^{-1}_{0}x).$

This contradicts with the assumption and completes the proof. ∎

###  2.3. The coercive property.

By the convex analysis in [54], we have

$\displaystyle\forall u\in\dot{H}^{1},\;\;E(u)\leq E(W),\;\;\big{\|}\nabla
u\big{\|}_{L^{2}}\leq\big{\|}\nabla
W\big{\|}_{L^{2}}\Longrightarrow\frac{\big{\|}\nabla
u\big{\|}^{2}_{L^{2}}}{\big{\|}\nabla
W\big{\|}^{2}_{L^{2}}}\leq\frac{E(u)}{E(W)},$ (2.4) $\displaystyle\forall
u\in\dot{H}^{1},\;\;E(u)\leq E(W),\;\;\big{\|}\nabla
u\big{\|}_{L^{2}}\geq\big{\|}\nabla
W\big{\|}_{L^{2}}\Longrightarrow\frac{\big{\|}\nabla
u\big{\|}^{2}_{L^{2}}}{\big{\|}\nabla
W\big{\|}^{2}_{L^{2}}}\geq\frac{E(u)}{E(W)}.$

This, together with the energy conservation, the variational characterization
of $W$ and the continuity argument, implies that

######  Lemma 2.6.

Let $u\in C(I,\dot{H}^{1}(\mathbb{R}^{N}))$ be a radial solution of (1.1) with
initial data $u_{0}$, and $I=(-T_{-},T_{+})$ its maximal interval of
existence. Assume that $E(u_{0})=E(W),$ then

  1. (a)

if $\big{\|}\nabla u_{0}\big{\|}_{2}<\big{\|}\nabla W\big{\|}_{2}$, then
$\big{\|}\nabla u(t)\big{\|}_{2}<\big{\|}\nabla W\big{\|}_{2}$ for $t\in I$.

  2. (b)

if $\big{\|}\nabla u_{0}\big{\|}_{2}=\big{\|}\nabla W\big{\|}_{2}$, then $u=W$
up to the symmetry of the equation.

  3. (c)

if $\big{\|}\nabla u_{0}\big{\|}_{2}>\big{\|}\nabla W\big{\|}_{2}$, then
$\big{\|}\nabla u(t)\big{\|}_{2}>\big{\|}\nabla W\big{\|}_{2}$ for $t\in I$.

###### Proof.

The proof is analogue to Lemma $2.8$ in [51] and Proposition 3.1 in [54]. ∎

###  2.4. Monotonicity formula

We define the localized Virial term

$\displaystyle
V_{R}(t)=\int_{\mathbb{R}^{N}}\phi_{R}(x)\big{|}u(t,x)\big{|}^{2}dx,$ (2.5)

where $\phi(x)$ be a positive smooth radial function such that
$\phi(x)=\frac{|x|^{2}}{2}$ for $|x|\leq 1$, $\phi(x)=0$ for $|x|\geq 2$,
$\phi^{\prime\prime}(r)\leq 1,\;r\geq 0$ and
$\phi_{R}(x)=R^{2}\phi\left(\frac{x}{R}\right),R>0$, which is useful to
analyze the gradient variant from W in Section 4. By simple calculations as in
[51, 54, 60], we have

######  Lemma 2.7.

Let $u(t,x)$ be a radial solution to (1.1), $V_{R}(t)$ be defined by (2.5),
then

$\displaystyle\partial_{t}V_{R}(t)=$
$\displaystyle\;2\mathrm{Im}\int_{\mathbb{R}^{N}}\overline{u}\;\nabla
u\cdot\nabla\phi_{R}\;dx,$ $\displaystyle\partial^{2}_{t}V_{R}(t)=$
$\displaystyle\;4\int_{\mathbb{R}^{N}}\Big{|}\nabla
u(t,x)\Big{|}^{2}dx-4\int_{\mathbb{R}^{N}}\left(\
I_{\lambda}*|u|^{p}\right)\;|u(x)|^{p}\;dx+A_{R}\big{(}u(t)\big{)},$

where

$\displaystyle A_{R}\big{(}u(t)\big{)}=$
$\displaystyle\;\int_{\mathbb{R}^{N}}\left(4\phi^{\prime\prime}\Big{(}\frac{|x|}{R}\Big{)}-4\right)\big{|}\nabla
u(t,x)\big{|}^{2}dx+\int_{\mathbb{R}^{N}}\big{(}-\Delta\Delta\phi_{R}(x)\big{)}\big{|}u(t,x)\big{|}^{2}dx$
$\displaystyle+O\left(\int_{|x|\geq
R}\left(I_{\lambda}*|u|^{p}\right)\;|u(x)|^{p}\;dx\right).$

###  2.5. Basic properties of the linearized operator.

We consider a radial solution $u$ of (1.1) close to $W$ and write $u$ as

$\displaystyle u(t,x)=W(x)+h(t,x),$

then $h$ satisfies the difference equation

$\displaystyle i\partial_{t}h+\Delta h=Vh+iR(h),$

where the linear operator $V$ and the remainder $R(h)$ are defined by

$\displaystyle Vh:=$
$\displaystyle\;-(p-1)(I_{\lambda}*W^{p})W^{p-2}h_{1}-p[I_{\lambda}*(W^{p-1}h_{1})]W^{p-1}-i(I_{\lambda}*W^{p})W^{p-2}h_{2},$
(2.6) $\displaystyle R(h):=$ $\displaystyle
i\Big{(}(I_{\lambda}*|W+h|^{p})|W+h|^{p-2}(W+h)-(I_{\lambda}*W^{p})W^{p-2}W-(p-1)(I_{\lambda}*W^{p})W^{p-2}h_{1}$
$\displaystyle-p[I_{\lambda}*(W^{p-1}h_{1})]W^{p-1}-i(I_{\lambda}*W^{p})W^{p-2}h_{2}\Big{)}.$
(2.7)

Let $h_{1}=\mathrm{Re}h$ and $h_{2}=\mathrm{Im}h$ (In the following context,
we will always denote the complex value function
$h=h_{1}+ih_{2}=(h_{1},h_{2})$ without confusion). Then $h$ is a solution of
the equation

$\displaystyle\partial_{t}h+\mathcal{L}h=-iR(h),\quad\mathcal{L}:=\begin{pmatrix}0&-L_{-}\\\
L_{+}&0\end{pmatrix},$ (2.8)

where the self-adjoint operators $L_{\pm}$ are defined by

$\displaystyle L_{+}h_{1}:=$ $\displaystyle-\Delta
h_{1}-(p-1)(I_{\lambda}*W^{p})W^{p-2}h_{1}-p[I_{\lambda}*(W^{p-1}h_{1})]W^{p-1},$
(2.9) $\displaystyle L_{-}h_{2}:=$ $\displaystyle-\Delta
h_{2}-(I_{\lambda}*W^{p})W^{p-2}h_{2}.$ (2.10)

Now we first give some basic estimates about the linearized equation (2.8).
Firstly, we give nonlinear estimates as follows.

######  Lemma 2.8.

Let $V$, $R$ be defined by (2.6) and (2.7), respectively, $I$ be a interval
with $|I|\leq 1$, and $g,h\in l(I)$, $u,v\in
L^{\frac{2N}{N-2}}(\mathbb{R}^{N})$, then

$\displaystyle\big{\|}\nabla\big{(}Vh\big{)}\big{\|}_{N(I)}\lesssim$
$\displaystyle\;|I|^{\frac{p-1}{p}}\big{\|}h\big{\|}_{l(I)},$ (2.11)
$\displaystyle\big{\|}\nabla\big{(}R(g)-R(h)\big{)}\big{\|}_{N(I)}\lesssim$
$\displaystyle\;\big{\|}g-h\big{\|}_{l(I)}\Big{(}|I|^{\frac{2p-3}{2p}}\big{(}\big{\|}g\big{\|}_{l(I)}+\big{\|}h\big{\|}_{l(I)}\big{)}+\big{\|}g\big{\|}^{2(p-1)}_{l(I)}+\big{\|}h\big{\|}^{2(p-1)}_{l(I)}\Big{)},$
(2.12)
$\displaystyle\big{\|}R(u)-R(v)\big{\|}_{L^{\frac{2N}{N+2}}(\mathbb{R}^{N})}\lesssim$
$\displaystyle\;\Big{(}\big{\|}u\big{\|}_{L^{\frac{2N}{N-2}}}+\big{\|}v\big{\|}_{L^{\frac{2N}{N-2}}}+\big{\|}u\big{\|}^{2(p-1)}_{L^{\frac{2N}{N-2}}}+\big{\|}v\big{\|}^{2(p-1)}_{L^{\frac{2N}{N-2}}}\Big{)}\big{\|}u-v\big{\|}_{L^{\frac{2N}{N-2}}}.$
(2.13)

######  Remark 2.9.

Let $f(z)=|z|^{p-2}z$. The complex derivative of $f$ is that

$f_{z}(z)=\frac{p}{2}|z|^{p-2},\quad
f_{\bar{z}}(z)=\frac{p-2}{2}|z|^{p-4}z^{2}.$

For $z_{1},z_{2}\in\mathbb{C}$, we have

$f(z_{1})-f(z_{2})=\int_{0}^{1}[f_{z_{1}}(z_{2}+\theta(z_{1}-z_{2}))(z_{1}-z_{2})+f_{\bar{z_{1}}}(z_{2}+\theta(z_{1}-z_{2}))\overline{(z_{1}-z_{2})}]d\theta.$

Therefore,

$f(z_{1})-f(z_{2})\lesssim(|z_{1}|^{p-2}+|z_{2}|^{p-2})|z_{1}-z_{2}|.$

Also, by [14], we get

$\big{|}|z_{1}|^{\alpha}-|z_{2}|^{\alpha}\big{|}\lesssim\left\\{\begin{aligned}
&(|z_{1}|^{{\alpha}-1}+|z_{2}|^{{\alpha}-1})|z_{1}-z_{2}|&\alpha\geq 1;\\\
&|z_{1}-z_{2}|^{\alpha}&0<\alpha\leq 1.\end{aligned}\right.$

###### Proof of Lemma 2.8.

We follow the argument in Lemma $2.10$ in [51]. We firstly prove (2.11).
Recall that

$Vh:=\;-(p-1)(I_{\lambda}*W^{p})W^{p-2}h_{1}-p[I_{\lambda}*(W^{p-1}h_{1})]W^{p-1}-i(I_{\lambda}*W^{p})W^{p-2}h_{2}.$

Hence, we get

$\displaystyle\big{|}\nabla\big{(}Vh\big{)}\big{|}\lesssim$
$\displaystyle\;|(I_{\lambda}*W^{p})W^{p-3}\nabla
Wh|+|(I_{\lambda}*W^{p})W^{p-2}\nabla h|+|(I_{\lambda}*(W^{p-1}\nabla
W))W^{p-2}h|$ $\displaystyle\;+|(I_{\lambda}*(W^{p-1}h_{1}))W^{p-2}\nabla
W|+|I_{\lambda}*(W^{p-2}\nabla Wh_{1})W^{p-1}|$
$\displaystyle\;+|(I_{\lambda}*(W^{p-1}\nabla h_{1})W^{p-1}|.$ (2.14)

As for the first term on the right of (2.5), by Lemma 2.2 , Remark 2.9 and
Hölder’s inequality, we have

$\displaystyle\big{\|}(I_{\lambda}*W^{p})W^{p-3}\nabla
Wh\big{\|}_{N(I)}\lesssim$ $\displaystyle\big{\|}\nabla
W\big{\|}_{S(I)}\big{\|}W\big{\|}^{p-3}_{Z(I)}\big{\|}h\big{\|}_{Z(I)}\big{\|}I_{\lambda}*W^{p}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2Np}{2Np+2p^{2}-Np^{2}-2p}}}$
$\displaystyle\lesssim$ $\displaystyle\big{\|}\nabla
W\big{\|}_{S(I)}\big{\|}W\big{\|}^{p-3}_{Z(I)}\big{\|}h\big{\|}_{Z(I)}\big{\|}W^{p}\big{\|}_{L_{t}^{2}L_{x}^{\frac{2Np}{(Np-2p-2)p}}}$
$\displaystyle=$ $\displaystyle\big{\|}W\big{\|}^{2p-3}_{Z(I)}\big{\|}\nabla
W\big{\|}_{S(I)}\big{\|}h\big{\|}_{Z(I)}.$

As for the rest term on the right of (2.5), by simple calculations, we have

$\displaystyle\big{\|}\nabla\big{(}Vh\big{)}\big{\|}_{N(I)}\lesssim\big{\|}W\big{\|}^{2(p-1)}_{Z(I)}\big{\|}\nabla
h\big{\|}_{S(I)}+\big{\|}W\big{\|}^{2p-3}_{Z(I)}\big{\|}\nabla
W\big{\|}_{S(I)}\big{\|}h\big{\|}_{Z(I)}.$

Since $W\in L^{\frac{2Np}{Np-2p-2}}_{x}\cap\dot{W}^{1,\frac{2Np}{Np-2}}_{x},$
we have that

$\displaystyle\big{\|}W\big{\|}_{Z(I)}\lesssim|I|^{\frac{1}{2p}},\;\;\big{\|}\nabla
W\big{\|}_{S(I)}\lesssim|I|^{\frac{1}{2p}},$

which implies (2.11).

As for (2.12) and (2.13), recall that

$\displaystyle R(h):=$
$\displaystyle\;i\Big{(}(I_{\lambda}*|W+h|^{p})|W+h|^{p-2}(W+h)-(I_{\lambda}*W^{p})W^{p-1}-(p-1)(I_{\lambda}*W^{p})W^{p-2}h_{1}$
$\displaystyle-p[I_{\lambda}*(W^{p-1}h_{1})]W^{p-1}-i(I_{\lambda}*W^{p})W^{p-2}h_{2}\Big{)},$

so, we have

$\displaystyle R(u)-R(v)$
$\displaystyle=\;i\Big{(}(I_{\lambda}*|W+u|^{p})|W+u|^{p-2}(W+u)-(I_{\lambda}*|W+v|^{p})|W+v|^{p-2}(W+v)$
$\displaystyle\qquad-(p-1)(I_{\lambda}*W^{p})W^{p-2}(u_{1}-v_{1})-p\left(I_{\lambda}*(W^{p-1}(u_{1}-v_{1}))\right)W^{p-1}$
$\displaystyle\qquad-i(I_{\lambda}*W^{p})W^{p-2}(u_{2}-v_{2})\Big{)}.$

As for the term
$(I_{\lambda}*|u|^{p})|u|^{p-2}u-(I_{\lambda}*|v|^{p})|v|^{p-2}v$, by Lemma
2.2 , Remark 2.9 and Hölder’s inequality, we estimate when $p>3$ that

$\displaystyle\big{\|}(I_{\lambda}*|u|^{p})|u|^{p-2}u-(I_{\lambda}*|v|^{p})|v|^{p-2}v\big{\|}_{L^{\frac{2N}{N+2}}}$
$\displaystyle=$
$\displaystyle\big{\|}(I_{\lambda}*|u|^{p})|u|^{p-2}(u-v)\big{\|}_{L^{\frac{2N}{N+2}}}+\big{\|}(I_{\lambda}*(|u|^{p}-|v|^{p}))|u|^{p-2}v\big{\|}_{L^{\frac{2N}{N+2}}}$
$\displaystyle+\big{\|}(I_{\lambda}*|v|^{p})(|u|^{p-2}-|v|^{p-2})v\big{\|}_{L^{\frac{2N}{N+2}}}$
$\displaystyle\lesssim$
$\displaystyle\big{\|}u\big{\|}^{p-2}_{L^{\frac{2N}{N-2}}}\big{\|}u-v\big{\|}_{L^{\frac{2N}{N-2}}}\big{\|}I_{\lambda}*|u|^{p}\big{\|}_{L^{\frac{2N}{2N+2p-Np}}}+\big{\|}u\big{\|}^{p-2}_{L^{\frac{2N}{N-2}}}\big{\|}v\big{\|}_{L^{\frac{2N}{N-2}}}\big{\|}I_{\lambda}*(|u|^{p}-|v|^{p})\big{\|}_{L^{\frac{2N}{2N+2p-Np}}}$
$\displaystyle+\big{\|}|u|^{p-2}-|v|^{p-2}\big{\|}_{L^{\frac{2N}{(N-2)(p-2)}}}\big{\|}v\big{\|}_{L^{\frac{2N}{N-2}}}\big{\|}I_{\lambda}*|u|^{p}\big{\|}_{L^{\frac{2N}{2N+2p-Np}}}$
$\displaystyle\lesssim$
$\displaystyle\big{\|}u\big{\|}^{p-2}_{L^{\frac{2N}{N-2}}}\big{\|}u-v\big{\|}_{L^{\frac{2N}{N-2}}}\big{\|}u^{p}\big{\|}_{L^{\frac{2N}{(N-2)p}}}+\big{\|}u\big{\|}^{p-2}_{L^{\frac{2N}{N-2}}}\big{\|}v\big{\|}_{L^{\frac{2N}{N-2}}}\big{\|}|u|^{p}-|v|^{p}\big{\|}_{L^{\frac{2N}{(N-2)p}}}$
$\displaystyle+\big{\|}|u|^{p-2}-|v|^{p-2}\big{\|}_{L^{\frac{2N}{(N-2)(p-2)}}}\big{\|}v\big{\|}_{L^{\frac{2N}{N-2}}}\big{\|}|u|^{p}\big{\|}_{L^{\frac{2N}{(N-2)p}}}$
$\displaystyle\lesssim$
$\displaystyle\big{\|}u\big{\|}^{2p-2}_{L^{\frac{2N}{N-2}}}\big{\|}u-v\big{\|}_{L^{\frac{2N}{N-2}}}+\big{\|}u\big{\|}^{p-2}_{L^{\frac{2N}{N-2}}}\big{\|}v\big{\|}_{L^{\frac{2N}{N-2}}}\big{\|}(|u|^{p-1}+|v|^{p-1})|u-v|\big{\|}_{L^{\frac{2N}{(N-2)p}}}$
$\displaystyle+\big{\|}(|u|^{p-3}-|v|^{p-3})|u-v|\big{\|}_{L^{\frac{2N}{(N-2)(p-2)}}}\big{\|}v\big{\|}_{L^{\frac{2N}{N-2}}}\big{\|}|u|^{p}\big{\|}_{L^{\frac{2N}{(N-2)p}}}$
$\displaystyle\lesssim$
$\displaystyle\big{\|}u\big{\|}^{2p-2}_{L^{\frac{2N}{N-2}}}\big{\|}u-v\big{\|}_{L^{\frac{2N}{N-2}}}+\big{\|}u\big{\|}^{2p-3}_{L^{\frac{2N}{N-2}}}\big{\|}v\big{\|}_{L^{\frac{2N}{N-2}}}\big{\|}u-v\big{\|}_{L^{\frac{2N}{N-2}}}+\big{\|}u\big{\|}^{p-2}_{L^{\frac{2N}{N-2}}}\big{\|}u-v\big{\|}_{L^{\frac{2N}{N-2}}}\big{\|}v\big{\|}^{p}_{L^{\frac{2N}{N-2}}}$
$\displaystyle+\big{\|}u-v\big{\|}_{L^{\frac{2N}{N-2}}}\big{(}\big{\|}u\big{\|}^{2p-3}_{L^{\frac{2N}{N-2}}}\big{\|}v\big{\|}_{L^{\frac{2N}{N-2}}}+\big{\|}u\big{\|}^{p}_{L^{\frac{2N}{N-2}}}\big{\|}v\big{\|}^{p-2}_{L^{\frac{2N}{N-2}}}\big{)}$
$\displaystyle\lesssim$
$\displaystyle\big{\|}u-v\big{\|}_{L^{\frac{2N}{N-2}}}\big{(}\big{\|}u\big{\|}_{L^{\frac{2N}{N-2}}}+\big{\|}v\big{\|}_{L^{\frac{2N}{N-2}}}+\big{\|}u\big{\|}^{2(p-1)}_{L^{\frac{2N}{N-2}}}+\big{\|}v\big{\|}^{2(p-1)}_{L^{\frac{2N}{N-2}}}\big{)}.$

When $2\leq p\leq 3$ and the other terms can also be estimated similarly.
Above all, we can obtain (2.13).

Last, we can show (2.12) from the expression of $R(u)-R(v)$ above, and
completes the proof. ∎

Because of the linear operator $V$ in the linearized equation (2.8), we will
use the following integral summation argument.

######  Lemma 2.10 ([18]).

Let $t_{0}>0$, $p\in[1,+\infty[$, $a_{0}\neq 0$, $E$ be a normed vector space,
and $f\in L^{p}_{\rm loc}(t_{0},+\infty;E)$ such that

$\exists\;\tau_{0}>0,\;\exists\;C_{0}>0,\;\forall\;t\geq
t_{0},\quad\|f\|_{L^{p}(t,t+\tau_{0},E)}\leq C_{0}e^{a_{0}t}.$ (2.15)

Then for $t\geq t_{0}$,

$\displaystyle\|f\|_{L^{p}(t,+\infty,E)}\leq$
$\displaystyle\;\frac{C_{0}e^{a_{0}t}}{1-e^{a_{0}\tau_{0}}},\;\text{if
}\;a_{0}<0;$ (2.16) $\displaystyle\|f\|_{L^{p}(t_{0},t,E)}\leq$
$\displaystyle\;\frac{C_{0}e^{a_{0}t}}{1-e^{-a_{0}\tau_{0}}},\;\text{if
}\;a_{0}>0.$ (2.17)

By the Strichartz estimate, Lemma 2.8 and Lemma 2.10, we have

######  Lemma 2.11.

Let $C$, $c>0$, and $v$ be a solution of (2.8) with

$\big{\|}v(t)\big{\|}_{\dot{H}^{1}}\leq Ce^{-ct},$

then for any $(q,r)$ with $\frac{2}{q}=N(\frac{1}{2}-\frac{1}{r})$,
$q\in[2,+\infty]$, we have for large $t$

$\displaystyle\big{\|}v\big{\|}_{l(t,+\infty)}+\big{\|}\nabla
v\big{\|}_{L^{q}(t,+\infty;L^{r})}\leq Ce^{-ct}.$

###### Proof.

We follow the argument in Lemma $2.12$ in [51]. For small $\tau_{0}$, by the
Strichartz estimate and Lemma 2.8, we have on $I=[t,t+\tau_{0}]$

$\displaystyle\big{\|}v\big{\|}_{l(I)}+\big{\|}\nabla
v\big{\|}_{L^{q}(I;L^{r})}\leq$ $\displaystyle
Ce^{-ct}+C\big{\|}\nabla(Vv)\big{\|}_{N(I)}+C\big{\|}\nabla
R(v)\big{\|}_{N(I)}$ $\displaystyle\leq$ $\displaystyle
Ce^{-ct}+C|I|^{\frac{p-1}{p}}\big{\|}v\big{\|}_{l(I)}+\big{\|}v\big{\|}^{2}_{l(I)}+\big{\|}v\big{\|}^{2p-1}_{l(I)}.$

By choosing sufficiently small $\tau_{0}$, the continuous argument gives that

$\big{\|}v\big{\|}_{l(I)}+\big{\|}\nabla v\big{\|}_{L^{q}(I;L^{r})}\leq
Ce^{-ct}.$

Thus this implies the desired result by Lemma 2.10. ∎

###  2.6. Spectral properties of the linearized operator.

Because of the phase rotation invariance and the scaling invariance of (1.1),
we know that $iW\in\dot{H}^{1}(\mathbb{R}^{N})$ and
$\widetilde{W}=\dfrac{N-2}{2}W+x\cdot\nabla W\in\dot{H}^{1}(\mathbb{R}^{N})$
are elements of the null-space of $\mathcal{L}$ in $\dot{H}^{1}_{rad}$. In
fact, they are the only elements of the null-space of $\mathcal{L}$ in
$\dot{H}^{1}_{rad}$. From Proposition 1.4, we have

######  Lemma 2.12.

Let $\mathcal{L}$ be defined by (2.8). Then

$\displaystyle\big{\\{}u\in\dot{H}^{1}_{rad}(\mathbb{R}^{N}),\mathcal{L}u=0\big{\\}}=\text{span}\big{\\{}iW,\widetilde{W}\big{\\}}.$

Namely,

$\big{\\{}u\in\dot{H}^{1}_{rad}(\mathbb{R}^{N}),L_{-}u=0\big{\\}}=\text{span}\big{\\{}W\big{\\}};\quad\big{\\{}u\in\dot{H}^{1}_{rad}(\mathbb{R}^{N}),L_{+}u=0\big{\\}}=\text{span}\big{\\{}\widetilde{W}\big{\\}}.$

Now we define the linearized energy $\Phi$ by

$\displaystyle\Phi(h):=$
$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}(L_{+}h_{1})h_{1}\;dx+\frac{1}{2}\int_{\mathbb{R}^{N}}(L_{-}h_{2})h_{2}\;dx$
(2.18) $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}\big{|}\nabla
h_{1}\big{|}^{2}-\frac{p-1}{2}\int_{\mathbb{R}^{N}}(I_{\lambda}*W^{p})W^{p-2}h_{1}^{2}\;dx-\frac{p}{2}\int_{\mathbb{R}^{N}}(I_{\lambda}*W^{p-1}h_{1})W^{p-1}h_{1}\;dx$
$\displaystyle+\frac{1}{2}\int_{\mathbb{R}^{N}}\big{|}\nabla
h_{2}\big{|}^{2}-\frac{1}{2}\int_{\mathbb{R}^{N}}(I_{\lambda}*W^{p})W^{p-2}h_{1}^{2}\;dx,$

and denote by $B(g,h)$ the bilinear symmetric form associated to $\Phi$,

$\displaystyle
B(g,h):=\frac{1}{2}\int_{\mathbb{R}^{N}}(L_{+}g_{1})h_{1}\;dx+\frac{1}{2}\int_{\mathbb{R}^{N}}(L_{-}g_{2})h_{2}\;dx,\;\;\forall\;g,h\in\dot{H}^{1}(\mathbb{R}^{N}).$
(2.19)

Let us specify the important coercivity properties of $\Phi$. Consider the
three orthogonal directions $W,iW,$ and $\widetilde{W}$ in the Hilbert space
$\dot{H}^{1}(\mathbb{R}^{N},\mathbb{C})$. Let
$H:=\text{span}\\{W,iW,\widetilde{W}\\},$ and $H^{\bot}$ be its orthogonal
subspace in $\dot{H}^{1}_{\text{rad}}(\mathbb{R}^{N})$, that is,

$\displaystyle
H^{\bot}:=\left\\{v\in\dot{H}^{1}_{\text{rad}}(\mathbb{R}^{N}),(iW,v)_{\dot{H}^{1}}=(\widetilde{W},v)_{\dot{H}^{1}}=(W,v)_{\dot{H}^{1}}=0\right\\}.$

Following the arguments in Proposition $2.13$ and Proposition $2.14$ in [51],
we have the following coercivity properties of the linearized energy $\Phi$ on
$H^{\bot}$ and $G_{\bot}$.

######  Proposition 2.13.

There exists a constant $c>0$ such that for all radial function $h\in
H^{\bot}$, we have

$\displaystyle\Phi(h)\geq c\big{\|}h\big{\|}^{2}_{\dot{H}^{1}}.$ (2.20)

For the sake of completeness, we will show it in Appendix B.

######  Proposition 2.14.

Let $\sigma(\mathcal{L})$ be the spectrum of the operator $\mathcal{L}$,
defined on $L^{2}(\mathbb{R}^{N})$ with domain $H^{2}(\mathbb{R}^{N})$ and let
$\sigma_{ess}(\mathcal{L})$ be its essential spectrum. Then we have

  1. (a)

The operator $\mathcal{L}$ admits two radial eigenfunctions
$\mathcal{Y}_{\pm}\in\mathcal{S}(\mathbb{R}^{N})$ with real eigenvalues $\pm
e_{0}$, $e_{0}>0$, that is, $\mathcal{L}\mathcal{Y}_{\pm}=\pm
e_{0}\mathcal{Y}_{\pm},\;\mathcal{Y}_{+}=\overline{\mathcal{Y}}_{-},\;e_{0}>0.$

  2. (b)

There exists a constant $c>0$ such that for all radial function $h\in
G_{\bot}$, we have

$\displaystyle\Phi(h)\geq c\big{\|}h\big{\|}^{2}_{\dot{H}^{1}},$

where

$G_{\bot}=\left\\{v\in\dot{H}^{1},(iW,v)_{\dot{H}^{1}}=(\widetilde{W},v)_{\dot{H}^{1}}=B(\mathcal{Y}_{\pm},v)=0\right\\}.$

  3. (c)

$\sigma_{ess}(\mathcal{L})=\\{i\xi:\xi\in\mathbb{R},\\},\quad\sigma(\mathcal{L})\cap\mathbb{R}=\\{-e_{0},0,e_{0}\\}.$

######  Remark 2.15.

The proofs of (b) and (c) of Proposition 2.14 are analogue to that of
Proposition $2.14$ in [51], but there is a little difference in the proof of
(a). Here, we consider the case $N\geq 3$ rather than the case $N>4=\lambda$
in [51]. However, when $N=3,4,\;W\notin L^{2}(\mathbb{R}^{N})$, we will give
the proof of (a) for the completeness in Appendix C.

######  Remark 2.16.

As a consequence of the definition of $\Phi$ and $B$, we have that for any
$f,g\in\dot{H}^{1}(\mathbb{R}^{N}),\mathcal{L}f,\mathcal{L}g\in\dot{H}^{1}(\mathbb{R}^{N})$,
and $h\in H^{\bot}$

$\displaystyle\Phi(iW)=$
$\displaystyle\Phi(\widetilde{W})=\Phi(\mathcal{Y}_{\pm})=0,\;\;\Phi(W)=-\big{\|}\nabla
W\big{\|}^{2}_{2}<0,$ (2.21) $\displaystyle B(f,g)=$ $\displaystyle
B(g,f),\;\;B(\mathcal{L}f,g)=-B(f,\mathcal{L}g),$ (2.22) $\displaystyle
B(iW,f)=$ $\displaystyle B(\widetilde{W},f)=0,\;\;B(W,h)=0.$ (2.23)

######  Corollary 2.17.

As a consequence of Proposition 2.13, we have

$\displaystyle B(\mathcal{Y}_{+},\mathcal{Y}_{-})\not=0.$ (2.24)

###### Proof.

If $B(\mathcal{Y}_{+},\mathcal{Y}_{-})=0$, then for any
$h\in\text{span}\\{iW,\widetilde{W},\mathcal{Y}_{\pm}\\}$, which is of
dimension 4, we have

$\Phi(h)=0.$

But by Proposition 2.13, we know that $\Phi$ is positive on
$H^{\bot}\cap\dot{H}^{1}_{\text{rad}}$, which is of codimension 3. It is a
contradiction. ∎

######  Corollary 2.18.

As a consequence of Proposition 2.14, we have

$\displaystyle\int\nabla
W\cdot\nabla\mathcal{Y}_{1}\;dx\not=0,\quad\text{where}\;\mathcal{Y}_{1}=\mathrm{Re}\mathcal{Y}_{+}.$
(2.25)

###### Proof.

Note that

$(W,iW)_{\dot{H}^{1}}=(W,\widetilde{W})_{\dot{H}^{1}}=0.$

By (1.1), we have $L_{+}(W)=2(p-1)\Delta W.$ If
$(W,\mathcal{Y}_{1})_{\dot{H}^{1}}=0,$ then by (2.22), we obtain

$\displaystyle\pm e_{0}B(\mathcal{Y}_{\pm},W)=$ $\displaystyle
B(\mathcal{L}\mathcal{Y}_{\pm},W)=-B(\mathcal{Y}_{\pm},\mathcal{L}W)$
$\displaystyle=$ $\displaystyle\mp\frac{1}{2}\int L_{-}\mathcal{Y}_{2}\cdot
L_{+}(W)=\pm\frac{1}{2}\int e_{0}\mathcal{Y}_{1}\cdot L_{+}(W)=\pm(p-1)\int
e_{0}\mathcal{Y}_{1}\cdot\Delta W=0,$

which means that $W\in G_{\bot}$. By Proposition 2.14, we have
$\Phi(W)\gtrsim\big{\|}W\big{\|}^{2}_{\dot{H}^{1}}$, which contradicts with
(2.21). ∎

##  3\. Existence of special threshold solutions $W^{\pm}$

In this section, we follow the arguments in [18, 19, 43, 44] and [51] to show
the existence of special threshold solutions $U^{a}$ and $W^{\pm}$ in the
energy space, we can also refer to [59] for the ideas from the existence of
minimal mass blow-up solutions for mass-critical NLS. We firstly construct a
family of high-order approximate solutions to (1.1) by use of the spectral
property of the linearized operator $\mathcal{L}$ and a series of the elliptic
iterations, and then obtain the special solution $U^{a}$ by the compactness
argument. The proof here is similar to those in Section 3 in [51], we only
give the sketch of proof.

###  3.1. A family of approximate solutions converging to $W$ as
$t\rightarrow+\infty$.

######  Proposition 3.1.

Let $a\in\mathbb{R}$. There exists a sequence of functions
$(\mathcal{Z}^{a}_{j})_{j\geq 1}$ in $\mathcal{S}(\mathbb{R}^{N})$ such that
$\mathcal{Z}^{a}_{1}=a\mathcal{Y}_{+}$ and if

$\displaystyle
h^{a}_{k}(t,x):=\sum^{k}_{j=1}e^{-je_{0}t}\mathcal{Z}^{a}_{j},\;k\in{\mathbb{Z}}^{+},$
(3.1)

then we have as $t\rightarrow+\infty$

$\displaystyle\epsilon_{k}:=$
$\displaystyle\partial_{t}h^{a}_{k}+\mathcal{L}h^{a}_{k}-R(h^{a}_{k})=O(e^{-(k+1)e_{0}t})\quad\text{in
}\;\mathcal{S}(\mathbb{R}^{N}).$ (3.2)

######  Remark 3.2.

Let

$\displaystyle U^{a}_{k}(t,x):=W(x)+h^{a}_{k}(t,x),$ (3.3)

then as $t\rightarrow+\infty$, we have the high-order error estimate

$\displaystyle\epsilon_{k}:=$ $\displaystyle i\partial_{t}U^{a}_{k}+\Delta
U^{a}_{k}+[I_{\lambda}*|U^{a}_{k}|^{p}]|U^{a}_{k}|^{p-2}U^{a}_{k}=O(e^{-(k+1)e_{0}t})\quad\text{in
}\;\mathcal{S}(\mathbb{R}^{N}).$

###### Sketch of proof.

Using the eigenfunctions of the real eigenvalues of the linearized operator,
we can define $h^{a}_{1}:=e^{-e_{0}t}\mathcal{Z}^{a}_{1}$. By the definition
of (2.7), we know that $R$ has at least the quadratic term, that is,
$R\left(e^{-e_{0}t}\mathcal{Z}^{a}_{1}\right)=O(e^{-2e_{0}t})$, which yields
(3.2) for $k=1$. we can use a series of the elliptic iterations to cancel low-
order error term, which together with the induction argument implies the
existence of $\mathcal{Z}^{a}_{j}$, $j=2,\ldots$. ∎

###  3.2. Construction of special solutions near an approximate solution.

Now we show the existence of special radial threshold solution which can be
proved by the fixed point argument.

######  Proposition 3.3.

Let $a\in\mathbb{R}$. There exist $k_{0}>0$ and $t_{0}\geq 0$ such that for
any $k\geq k_{0}$, there exists a radial solution $U^{a}$ of (1.1) such that
for $t\geq t_{0}$

$\displaystyle\left\|U^{a}-U^{a}_{k}\right\|_{l(t,+\infty)}\leq
e^{-(k+\frac{1}{2})e_{0}t}.$ (3.4)

Furthermore $U^{a}$ is the unique solution of (1.1) satisfying (3.4) for large
$t$, and $U^{a}\in L^{2}(\mathbb{R}^{N})$ if $N\geq 5$. Finally, $U^{a}$ is
independent of $k$ and satisfies for $t\geq t_{0}$,

$\displaystyle\big{\|}U^{a}(t)-W-ae^{-e_{0}t}\mathcal{Y}_{+}\big{\|}_{\dot{H}^{1}(\mathbb{R}^{N})}\leq
e^{-\frac{3}{2}e_{0}t}.$ (3.5)

###### Sketch of proof.

The proof depends on the fixed point argument and Strichartz estimate in Lemma
2.1. Let $e:=U^{a}-U^{a}_{k}=h^{a}-h^{a}_{k}$ and it satisfies

$\partial_{t}e+\mathcal{L}e=R(h^{a}_{k}+e)-R(h^{a}_{k})-\varepsilon_{k},$

where $h^{a}:=U^{a}-W,\;h^{a}_{k}:=U^{a}_{k}-W,\;\varepsilon_{k}$ is defined
in Remark 3.2. This equation may be rewritten as

$\displaystyle i\partial_{t}e+\Delta e=$ $\displaystyle
Ve+iR(h^{a}_{k}+e)-iR(h^{a}_{k})-i\varepsilon_{k}.$

Let us define the mapping

$\displaystyle\mathcal{M}_{k}(e)(t):=-\int_{t}^{+\infty}e^{i(t-s)\Delta}\Big{(}iVe(s)-R\big{(}h^{a}_{k}(s)+e(s)\big{)}+R(h^{a}_{k}(s))+\varepsilon_{k}(s)\Big{)}ds.$

Thus the existence of a solution $U^{a}$ of (1.1) satisfying (3.4) for $t\geq
t_{0}$ is equivalent to the following fixed point problem

$\displaystyle\forall\;t\geq t_{0},\quad e(t)=\mathcal{M}_{k}(e)(t)\text{ and
}\|e\|_{l(t,+\infty)}\leq e^{-\left(k+\frac{1}{2}\right)e_{0}t}.$ (3.6)

Let us fix $k$ and $t_{0}$. Denote

$\displaystyle B_{l}^{k}$ $\displaystyle:=\big{\\{}e\in
Z(t_{0},+\infty),\;\nabla e\in S(t_{0},+\infty);\;\sup_{t\geq
t_{0}}e^{\left(k+\frac{1}{2}\right)e_{0}t}\|e\|_{l(t,+\infty)}\leq
1\big{\\}}.$

By using the Strichartz estimate and Lemmar 2.8, we can show that if $t_{0}$
and $k$ are large enough, the mapping $\mathcal{M}_{k}$ is a contraction map
on $B_{l}^{k}$.

Finally, by the finite propagation of finite energy solutions, we can obtain
that the analogue estimate $U^{a}\in L^{2}(\mathbb{R}^{N})$ if $N\geq 5$ from
the derivative estimate of the localized mass to that in [18]. For this
purpose, define a positive radial function $\psi$ on $\mathbb{R}^{N}$,such
that $\psi=1$ if $|x|\leq 1$ and $\psi=0$ if $|x|\geq 2$. We also define that
for $R>0$ and large $t$,

$F_{R}(t):=\int_{\mathbb{R}^{N}}|U^{a}(t,x)|^{2}\psi(\frac{x}{R})dx.$

Then, we have

$\displaystyle F_{R}^{{}^{\prime}}(t)$ $\displaystyle=\frac{2}{R}Im\int
U^{a}\nabla\bar{U}^{a}\cdot(\nabla\psi)(\frac{x}{R})dx=\frac{2}{R}Im\int
W\nabla(\bar{U}^{a}-W)\cdot(\nabla\psi)(\frac{x}{R})dx$
$\displaystyle+\frac{2}{R}Im\int(U^{a}-W)\nabla
W\cdot(\nabla\psi)(\frac{x}{R})dx+\frac{2}{R}Im\int(U^{a}-W)\nabla(\bar{U}^{a}-\bar{W})\cdot(\nabla\psi)(\frac{x}{R})dx$

Using(3.5), $\|U^{a}(t)-W\|_{\dot{H}^{1}}\leq Ce^{-e_{0}t}$ and the Hardy
inequality, we get

$|F_{R}^{{}^{\prime}}(t)|\leq
C\|U^{a}(t)-W\|_{\dot{H}^{1}}(\|U^{a}(t)\|_{\dot{H}^{1}}+\|W\|_{\dot{H}^{1}})\leq
Ce^{-e_{0}t},$

with a constant C independent od R, and thus, integrating between sufficiently
large t and $+\infty$,

$\left|F_{R}(t)-\int_{\mathbb{R}^{N}}|W(x)|^{2}\psi(\frac{x}{R})dx\right|\leq
Ce^{-e_{0}t}.$

Letting R goes to $+\infty$, we get $\|U^{a}(t)\|_{L^{2}}=\|W\|_{L^{2}}$ and
$U^{a}(t)\in L^{2}(\mathbb{R}^{N})$ when $N\geq 5$, which completes the proof
from mass conservation law. ∎

###  3.3. Construction of $W^{\pm}$.

Note that by (3.5), and the energy conservation, we have

$E(U^{a})=E(W).$

In addition, by (3.5), we have

$\displaystyle\big{\|}\nabla U^{a}(t)\big{\|}^{2}_{2}=\big{\|}\nabla
W\big{\|}^{2}_{2}+2ae^{-e_{0}t}\int_{\mathbb{R}^{N}}\nabla
W\cdot\nabla\mathcal{Y}_{1}\;dx+O\left(e^{-\frac{3}{2}e_{0}t}\right)\quad\text{as}\;t\rightarrow+\infty.$

By (2.25), replacing $\mathcal{Y}_{+}$ by $-\mathcal{Y}_{+}$ if necessary, we
may assume that

$\displaystyle\int\nabla W\cdot\nabla\mathcal{Y}_{1}\;dx>0,$

which implies that $\big{\|}\nabla U^{a}(t)\big{\|}^{2}_{2}-\big{\|}\nabla
W\big{\|}^{2}_{2}$ has the sign of $a$ for large positive time. Thus, by Lemma
2.6, $\big{\|}\nabla U^{a}(t_{0})\big{\|}^{2}_{2}-\big{\|}\nabla
W\big{\|}^{2}_{2}$ has the sign of $a$.

Let

$\displaystyle W^{+}(t,x)=U^{+1}(t+t_{0},x),\quad
W^{-}(t,x)=U^{-1}(t+t_{0},x),$ (3.7)

which yields two radial solutions $W^{\pm}(t,x)$ of (1.1) for $t\geq 0$, and
satisfies

$\displaystyle E(W^{\pm}(t))=E(W),\;\big{\|}\nabla
W^{-}(0)\big{\|}_{2}<\big{\|}\nabla W\big{\|}_{2},\;\big{\|}\nabla
W^{+}(0)\big{\|}_{2}>\big{\|}\nabla W\big{\|}_{2},$

and

$\displaystyle\big{\|}W^{\pm}(t)-W\big{\|}_{\dot{H}^{1}}\leq
Ce^{-e_{0}t},\;\;t\geq 0.$

Moreover, we have $W^{\pm}\in L^{2}$ when $N\geq 5$. To conclude the proof of
Theorem 1.6, it remains to show the dynamics of $W^{\pm}$ for the negative
time, we will leave the proof in Section 5.3 and Section 6.5. ∎

##  4\. Modulation Analysis

For the radial function $u\in\dot{H}^{1}(\mathbb{R}^{N})$, we define the
gradient variant from $W$ as

$\displaystyle\delta(u)=\left|\int_{\mathbb{R}^{N}}\Big{(}|\nabla
u(x)|^{2}-|\nabla W(x)|^{2}\Big{)}dx\right|.$

By Proposition 2.5, we know that if

$\displaystyle E(u)=E(W),$ (4.1)

and $\delta(u)$ is small enough, then there exist $\widetilde{\theta}$ and
$\widetilde{\mu}$ such that

$\displaystyle
u_{\widetilde{\theta},\widetilde{\mu}}=W+\widetilde{u},\;\;\text{with}\;\big{\|}\widetilde{u}\big{\|}_{\dot{H}^{1}}\leq\varepsilon(\delta(u)),$

where $u_{\widetilde{\theta},\widetilde{\mu}}$ is defined as Proposition 2.5,
and $\varepsilon(\delta)\rightarrow 0$ as $\delta\rightarrow 0$. The goal in
this section is that for radial threshold solution of (1.1), we choose
parameters $\widetilde{\theta}$ and $\widetilde{\mu}$ to obtain the dynamics
of these parameters and their derivatives in terms of the gradient variant
$\delta(u)$.

First, by the implicit theorem and the variational characterization of $W$ in
Proposition 2.5, we have the following orthogonal decomposition, which is
analogue to Lemma $4.1$ in [51].

######  Lemma 4.1.

There exist $\delta_{0}>0$ and a positive function $\epsilon(\delta)$ defined
for $0<\delta\leq\delta_{0}$, which tends to $0$ when $\delta$ tends to $0$,
such that for all radial $u$ in $\dot{H}^{1}(\mathbb{R}^{N})$ satisfying
(4.1), there exists a couple $(\theta,\mu)\in\mathbb{R}\times(0,+\infty)$ such
that $v=u_{\theta,\mu}$ satisfies

$\displaystyle v\bot iW,\quad v\bot\widetilde{W}.$ (4.2)

The parameters $(\theta,\mu)$ are unique in
$\mathbb{R}/2\pi{\mathbb{Z}}\times\mathbb{R},$ and the mapping
$u\mapsto(\theta,\mu)$ is $C^{1}$.

###### Proof.

First by Proposition 2.5, there exist a function $\epsilon$ and
$\theta_{1},\mu_{1}>0$ such that $u_{\theta_{1},\mu_{1}}=W+g$ and

$\big{\|}u_{\theta_{1},\mu_{1}}-W\big{\|}_{\dot{H}^{1}}\leq\epsilon\left(\delta(u)\right).$
(4.3)

Now consider the functional

$\displaystyle J(\theta,\mu,f)=$
$\displaystyle\left(J_{0}(\theta,\mu,f),J_{1}(\theta,\mu,f)\right)=\left((f_{\theta,\mu},iW)_{\dot{H}^{1}},(f_{\theta,\mu},\widetilde{W})_{\dot{H}^{1}}\right)$
$\displaystyle=$
$\displaystyle\left((e^{i\theta}\mu^{-\frac{N-2}{2}}f(x/\mu),iW)_{\dot{H}^{1}},(e^{i\theta}\mu^{-\frac{N-2}{2}}f(x/\mu),\widetilde{W})_{\dot{H}^{1}}\right).$

Hence, by simple computations, we have

$\displaystyle J(0,1,W)=0,\quad\left|\frac{\partial
J}{\partial(\theta,\mu)}(0,1,W)\right|=\left|\begin{array}[]{cc}\big{\|}\nabla
W\big{\|}^{2}_{L^{2}}&0\\\
0&-\big{\|}\nabla\widetilde{W}\big{\|}^{2}_{L^{2}}\end{array}\right|\not=0.$

Thus by the implicit theorem, there exist $\epsilon_{0},\eta_{0}>0$, we have
that for any $h\in\dot{H}^{1}(\mathbb{R}^{N})$ with
$\big{\|}h-W\big{\|}_{\dot{H}^{1}}<\epsilon_{0}$, there exists a unique
$\left(\widetilde{\theta}(h),\widetilde{\mu}(h)\right)\in C^{1}$ such that
$|\widetilde{\theta}|+|\widetilde{\mu}-1|<\eta_{0}$ and

$\displaystyle J(\widetilde{\theta},\widetilde{\mu},h)=0.$

By (4.3), this implies that there exists a unique
$\left(\widetilde{\theta}_{1}(u),\widetilde{\mu}_{1}(u)\right)$ such that

$\displaystyle
J\left(\widetilde{\theta}_{1}+\theta_{1},\widetilde{\mu}_{1}\mu_{1},u\right)=J\left(\widetilde{\theta}_{1},\widetilde{\mu}_{1},u_{\theta_{1},\mu_{1}}\right)=0.$

This completes the proof by taking $\theta=\widetilde{\theta}_{1}+\theta_{1}$
and $\mu=\widetilde{\mu}_{1}\mu_{1}$. ∎

Let $u(t)$ be a radial solution of (1.1) satisfying (4.1) and write

$\displaystyle\delta(t):=\delta(u(t))=\left|\int_{\mathbb{R}^{N}}\Big{(}|\nabla
u(t,x)|^{2}-|\nabla W(x)|^{2}\Big{)}dx\right|.$ (4.4)

Let $D_{\delta_{0}}$ be the open set of all time $t$ in the domain of
existence of $u$ such that $\delta(t)<\delta_{0}$. On $D_{\delta_{0}}$, by
Lemma 4.1, there exist $\theta(t)$ and $\mu(t)$, which are $C^{1}$ functions
of $t$ such that we have the following decomposition

$\displaystyle u_{\theta(t),\mu(t)}(t,$ $\displaystyle
x)=\Big{(}1+\alpha(t)\Big{)}W(x)+h(t,x),$ (4.5) $\displaystyle 1+\alpha(t)=$
$\displaystyle\frac{1}{\big{\|}W\big{\|}^{2}_{\dot{H}^{1}}}\left(u_{\theta(t),\mu(t)},W\right)_{\dot{H}^{1}},\quad
h\in H^{\bot}\cap\dot{H}^{1}_{\text{rad}}.$

We will make use of additional conditions to show that $u$ converges
exponentially to $W$ in the positive time in $\dot{H}^{1}(\mathbb{R}^{N})$ up
to the constant modulation parameters in Section 5 and Section 6. The reason
why we make this decomposition is that the linearized energy $\Phi$ is
coercive on $H^{\bot}\cap\dot{H}^{1}_{\text{rad}}$.

######  Lemma 4.2.

Let $u$ be a radial solution of (1.1) satisfying (4.1). Then taking a smaller
$\delta_{0}$ if necessary, the following estimates hold for $t\in
D_{\delta_{0}}$:

$\displaystyle\big{|}\alpha(t)\big{|}\approx\big{\|}\alpha(t)W(\cdot)+h(t,\cdot)\big{\|}_{\dot{H}^{1}}\approx$
$\displaystyle\big{\|}h(t,\cdot)\big{\|}_{\dot{H}^{1}}\approx\delta(t),$ (4.6)
$\displaystyle\big{|}\alpha^{\prime}(t)\big{|}+\big{|}\theta^{\prime}(t)\big{|}+\left|\frac{\mu^{\prime}(t)}{\mu(t)}\right|\leq$
$\displaystyle\;C\mu^{2}(t)\delta(t).$ (4.7)

###### Sketch of the proof.

The proof mainly uses the coercivity of the linearized energy $\Phi$ on
$H^{\bot}\cap\dot{H}^{1}_{rad}$ in Proposition 2.13 and the orthogonal
decomposition in Lemma 4.1, we can refer to Lemma 4.2 in [51] for more
details. ∎

##  5\. Convergence to $W$ for the supercritical threshold solution

In this section, we follows the argument in Section $5$ in [51] to show the
following result, which implies the dynamics of special threshold solution
$W^{+}$ of Theorem 1.6 in the negative time and is also the first step in the
proof of case (c) of Theorem 1.8.

######  Proposition 5.1.

Consider a radial solution $u\in H^{1}(\mathbb{R}^{N})$ of (1.1) such that

$\displaystyle E(u)=E(W),\quad\big{\|}\nabla
u_{0}\big{\|}_{L^{2}}>\big{\|}\nabla W\big{\|}_{L^{2}},$ (5.1)

which is globally defined for the positive time. Then there exist
$\theta_{0}\in\mathbb{R}/(2\pi{\mathbb{Z}}),\mu_{0}\in(0,+\infty),c,C>0$ such
that

$\displaystyle\forall\;t\geq
0,\quad\big{\|}u-W_{\theta_{0},\mu_{0}}\big{\|}_{\dot{H}^{1}}\leq Ce^{-ct},$
(5.2)

and the negative time of existence of $u$ is finite.

###  5.1. Exponential convergence of the gradient variant

######  Lemma 5.2 ([51]).

Under the assumptions of Proposition 5.1, then there exists $C$ such that for
any $R>0$, and all $t\geq 0$,

$\displaystyle\big{|}\partial_{t}V_{R}(t)\big{|}\leq CR^{2}\;\delta(t),$ (5.3)

where $V_{R}(t)$ is defined as in (2.5).

###### Proof.

By Lemma 2.5, Lemma 2.6 and Lemma 4.2, we can obtain the result for both large
$\delta(t)$ and small $\delta(t)$, which is similar to Lemma $5.2$ in [51]. We
omit the details here. ∎

######  Lemma 5.3.

Under the assumptions of Proposition 5.1, there exist $C>0$ and $R_{1}\geq 1$
such that for $R\geq R_{1}$, and all $t\geq 0$,

$\displaystyle\partial^{2}_{t}V_{R}(t)\leq$ $\displaystyle-2(p-1)\delta(t),$
(5.4) $\displaystyle\partial_{t}V_{R}(t)>$ $\displaystyle\;0.$ (5.5)

###### Proof.

By Lemma 2.7, we have

$\displaystyle\partial^{2}_{t}V_{R}(t)=$
$\displaystyle\;4\int_{\mathbb{R}^{N}}\Big{|}\nabla
u(t,x)\Big{|}^{2}dx-4\int_{\mathbb{R}^{N}}\left(\
I_{\lambda}*|u|^{p}\right)\;|u(x)|^{p}\;dx+A_{R}\big{(}u(t)\big{)},$

where $A_{R}\big{(}u(t)\big{)}$ is defined in Lemma 2.7.

By $E(u)=E(W)=\frac{p-1}{2p}\big{\|}\nabla W\big{\|}^{2}_{L^{2}}$, we have

$\displaystyle 4\int_{\mathbb{R}^{N}}\Big{|}\nabla
u(t,x)\Big{|}^{2}dx-4\int_{\mathbb{R}^{N}}\left(\
I_{\lambda}*|u|^{p}\right)\;|u(x)|^{p}\;dx=-4(p-1)\delta(t).$

To prove (5.4), it suffices to show that for $R\geq R_{1}$

$\displaystyle A_{R}\big{(}u(t)\big{)}\leq 2(p-1)\delta(t).$

We divide the proof into three steps:

Step 1: General bound on $A_{R}(u(t))$. We claim that there exists $C>0$ such
that for any $R>0$, $t\geq 0$, we have

$\displaystyle
A_{R}\big{(}u(t)\big{)}\leq\frac{C}{R^{2}}+\frac{C}{R^{\frac{p(N-1)}{N}}}\big{\|}u(t)\big{\|}^{\frac{p(N+1)}{N}}_{\dot{H}^{1}}.$
(5.6)

Indeed, on one hand, by the facts that $\phi^{\prime\prime}(r)\leq 1,\;r\geq
0$, $\big{|}-\Delta\Delta\phi_{R}\big{|}\lesssim R^{-2},$ we have

$\displaystyle A_{R}\big{(}u(t)\big{)}\leq
C\left(\frac{\big{\|}u(t)\big{\|}^{2}_{L^{2}}}{R^{2}}+\int_{|x|>R}(I_{\lambda}*|u|^{p})|u|^{p}\;dx\right).$
(5.7)

On the other hand, by the Hardy-Littlewood-Sobolev inequality, mass
conservation and the Sobolev inequality, we have

$\displaystyle\int_{|x|>R}(I_{\lambda}*|u|^{p})|u|^{p}\;dx$
$\displaystyle\lesssim\big{\|}u(t)\big{\|}^{p}_{L^{\frac{2N}{N-2}}(|x|>R)}\big{\|}u(t)\big{\|}^{p}_{L^{\frac{2N}{N-2}}}$
$\displaystyle\lesssim\big{\|}u(t)\big{\|}^{p}_{L^{\frac{2N}{N-2}}}\big{\|}u(t)\big{\|}^{\frac{2p}{N}}_{L^{\infty}(|x|>R)}\big{\|}u(t)\big{\|}^{\frac{pN-2p}{N}}_{L^{2}}$
$\displaystyle\lesssim\big{\|}\nabla
u(t)\big{\|}^{p}_{L^{2}}\big{\|}u(t)\big{\|}^{\frac{2p}{N}}_{L^{\infty}(|x|>R)}$
$\displaystyle\lesssim\big{\|}\nabla
u(t)\big{\|}^{p}_{L^{2}}\left(\frac{1}{R^{N-1}}\big{\|}\nabla
u(t)\big{\|}_{L^{2}}\big{\|}u(t)\big{\|}_{L^{2}}\right)^{\frac{p}{N}}$
$\displaystyle\lesssim\frac{1}{R^{\frac{p(N-1)}{N}}}\big{\|}\nabla
u(t)\big{\|}_{L^{2}}^{\frac{p(N+1)}{N}}.$

which together with (5.7) gives (5.6).

Step 2: Refined bound on $A_{R}(u(t))$ when $\delta(t)$ small. We claim that
there exist $\delta_{1}$, $C>0$ such that for any $R>1$, and $t\geq 0$, and
$\delta(t)\leq\delta_{1}$, we have

$\displaystyle\Big{|}A_{R}(u(t))\Big{|}\leq
C\left(\frac{1}{R^{\frac{N-2}{2}}}\delta(t)+\delta(t)^{2}\right).$ (5.8)

To do so, we first show that for small $\delta_{1}$,

$\displaystyle\mu_{-}:=\inf\Big{\\{}\mu(t),t\geq
0,\delta(t)\leq\delta_{1}\Big{\\}}>0.$ (5.9)

Indeed, by (4.5) and Lemma 4.2, we have

$\displaystyle
u_{\theta(t),\mu(t)}(t)=W+V,\;\text{with}\;\;\big{\|}V(t)\big{\|}_{\dot{H}^{1}}\approx\delta(t).$

By the mass conservation, we have

$\displaystyle\big{\|}u_{0}\big{\|}^{2}_{L^{2}}\geq$
$\displaystyle\int_{|x|\leq\frac{1}{\mu(t)}}\big{|}u(t,x)\big{|}^{2}dx=\frac{1}{\mu(t)^{2}}\int_{|x|\leq
1}\big{|}u_{\theta(t),\mu(t)}\big{|}^{2}dx$ $\displaystyle\gtrsim$
$\displaystyle\frac{1}{\mu(t)^{2}}\left(\int_{|x|\leq 1}W^{2}dx-\int_{|x|\leq
1}\big{|}V(t)\big{|}^{2}dx\right).$

While by the Sobolev inequality, we have

$\displaystyle\big{\|}V(t)\big{\|}_{L^{2}(|x|\leq
1)}\lesssim\big{\|}V(t)\big{\|}_{L^{\frac{2N}{N-2}}(|x|\leq
1)}\lesssim\big{\|}V(t)\big{\|}_{\dot{H}^{1}}\lesssim\delta(t),$

which implies that

$\displaystyle\big{\|}u_{0}\big{\|}^{2}_{L^{2}}\geq\frac{1}{\mu(t)^{2}}\left(\int_{|x|\leq
1}W^{2}dx-C\delta(t)^{2}\right).$

We obtain (5.9) if choosing sufficiently small $\delta_{1}$. Now by (4.5) and
the change of variable, we have

$\displaystyle\Big{|}A_{R}(u(t))\Big{|}=\Big{|}A_{R}\big{(}(W+V)_{-\theta(t),1/\mu(t)}\big{)}\Big{|}=\Big{|}A_{R\mu(t)}\big{(}W+V\big{)}\Big{|}.$

Note that

$\displaystyle A_{R\mu(t)}(W)=0,\;\text{and}\;\;\big{\|}\nabla
W\big{\|}_{L^{2}(|x|\geq\rho)}\approx\big{\|}W\big{\|}_{L^{\frac{2N}{N-2}}(|x|\geq\rho)}\approx\rho^{-\frac{N-2}{2}}\;\text{for}\;\rho\geq
1,$

we have

$\displaystyle\Big{|}A_{R}(u(t))\Big{|}=$
$\displaystyle\Big{|}A_{R\mu(t)}(W+V)-A_{R\mu(t)}(W)\Big{|}$
$\displaystyle\lesssim$ $\displaystyle\;\int_{|x|\geq R\mu(t)}\big{|}\nabla
W\cdot\nabla V(t)\big{|}+\big{|}\nabla V(t)\big{|}^{2}dx$
$\displaystyle+\int_{R\mu(t)\leq|x|\leq
2R\mu(t)}\frac{1}{\big{(}R\mu(t)\big{)}^{2}}\left(\big{|}W\cdot
V(t)\big{|}+\big{|}V(t)\big{|}^{2}\right)dx$ $\displaystyle+\iint_{|x|\geq
R\mu(t)}I_{\lambda}(x-y)\big{|}W(x)+V(t,x)\big{|}^{p}\big{|}W(y)+V(t,y)\big{|}^{p}$
$\displaystyle\qquad\quad\qquad\quad-I_{\lambda}(x-y)W(x)^{p}W(y)^{p}\;dxdy$
$\displaystyle\lesssim$
$\displaystyle\frac{1}{\big{(}R\mu(t)\big{)}^{\frac{N-2}{2}}}\big{\|}V(t)\big{\|}_{\dot{H}^{1}}+\big{\|}V(t)\big{\|}^{2}_{\dot{H}^{1}}+\big{\|}V(t)\big{\|}^{3}_{\dot{H}^{1}}+\big{\|}V(t)\big{\|}^{2p}_{\dot{H}^{1}},$

which together with (5.9) implies that (5.8) if $\delta_{1}$ is small enough.

Step 3: Conclusion. From (5.8), there exist $\delta_{2}>0$ and $R_{2}\geq 1$
such that if $R\geq R_{2}$, $\delta(t)\leq\delta_{2}$, then we have

$\displaystyle\Big{|}A_{R}(u(t))\Big{|}\leq
C\left(R^{-\frac{N-2}{2}}\delta(t)+\delta(t)^{2}\right)\leq 2(p-1)\delta(t).$

Now Let

$\displaystyle
f_{R_{3}}(\delta):=\frac{C}{R_{3}^{2}}+\frac{C}{R_{3}^{\frac{p(N-1)}{N}}}\big{(}\delta+\big{\|}W\big{\|}^{2}_{\dot{H}^{1}}\big{)}^{\frac{p(N+1)}{2N}}-2(p-1)\delta,$

where $C$ is given by (5.6). Choosing $R_{3}$ large enough such that
$f_{R_{3}}(\delta_{3})\leq 0$, and $f^{\prime}_{R_{3}}(\delta_{3})\leq 0$, we
have for any $R\geq R_{3}$, $\delta\geq\delta_{3}$ that

$\displaystyle f_{R}(\delta)\leq f_{R_{3}}(\delta)\leq 0,$

which implies that $A_{R}(u(t))\leq 2(p-1)\delta(t)$, so we conclude the proof
of (5.4) with $R_{1}=\max(R_{2},R_{3})$.

Finally, using the positivity of $V_{R}(t)$, $\partial^{2}_{t}V_{R}(t)<0$ for
$t>0$ and $u$ is defined on $[0,+\infty)$, we have $\partial_{t}V_{R}(t)>0$ by
the convexity analysis. ∎

######  Lemma 5.4.

Under the assumptions of Proposition 5.1, there exist $c>0$, $C>0$ such that
for $R\geq R_{1}$ ( which is given in Lemma 5.3), and all $t\geq 0$,

$\displaystyle\int^{+\infty}_{t}\delta(s)ds\leq$ $\displaystyle\;Ce^{-ct}.$
(5.10)

###### Proof.

Fix $R\geq R_{1}$. By (5.3), (5.4) and (5.5), we have

$\displaystyle
2(p-1)\int^{T}_{t}\delta(s)ds\leq-\int^{T}_{t}\partial^{2}_{s}V_{R}(s)ds=\partial_{t}V_{R}(t)-\partial_{t}V_{R}(T)\leq
CR^{2}\delta(t).$

Let $T\rightarrow+\infty$, we have

$\displaystyle\int^{+\infty}_{t}\delta(s)ds\leq C\delta(t).$

By the Gronwall inequality, we have (5.10). ∎

###  5.2. Convergence of the modulation parameters.

By the variational characterization of W in Proposition 2.5, the relation
between modulation parameters and the gradient variant which is shown in Lemma
4.2, and Lemma 5.4, we can follow the argument as that in Section $5.2$ in
[51] to obtain the convergence of the modulation parameters. Thus, we have

$\displaystyle\lim_{t\rightarrow+\infty}\delta(t)=0.\;\;$
$\displaystyle\lim_{t\rightarrow+\infty}\theta(t)=\theta_{0},\;\;\lim_{t\rightarrow+\infty}\mu(t)=\mu_{0}$
(5.11) $\displaystyle\delta(t)\approx\big{|}\alpha(t)\big{|}=$
$\displaystyle\big{|}\alpha(t)-\alpha(+\infty)\big{|}$ $\displaystyle\leq$
$\displaystyle C\int^{+\infty}_{t}\big{|}\alpha^{\prime}(s)\big{|}ds\leq
C\int^{+\infty}_{t}\mu^{2}(s)\delta(s)ds\leq Ce^{-ct},$ (5.12)
$\displaystyle\big{\|}u-W_{\theta(t),\mu(t)}\big{\|}_{\dot{H}^{1}}+\big{|}\alpha^{\prime}(t)\big{|}+\big{|}\theta^{\prime}(t)\big{|}\leq
C\delta(t)\leq Ce^{-ct},$

which yields (5.2). ∎

###  5.3. Blowup for the negative times

It is a consequence of the positivity of $\partial_{t}V_{R}(t)$ in (5.3) and
the time reversal symmetry. Suppose that $u$ is also global for the negative
time. Applying Lemma 5.2, Lemma 5.3 and Lemma 5.4 to $\overline{u}(-t)$, we
know that they also hold for the negative time. Hence by (5.11), we know that

$\lim_{t\rightarrow\pm\infty}\delta(t)=0.$

By Lemma 5.2 and Lemma 5.3, we know that $\partial_{t}V_{R}(t)>0$ and
$\partial_{t}V_{R}(t)\longrightarrow 0,\;\text{as}\;t\rightarrow\pm\infty.$ By
Lemma 5.3, we have $\partial^{2}_{t}V_{R}(t)<0.$ This implies that
$\partial_{t}V_{R}(t)\equiv 0$. It is a contradiction and completes the proof.
∎

##  6\. Convergence to $W$ for the subcritical threshold solution

In this section, we consider the subcritical threshold solution $u$ of (1.1).
Similar to that in Section 5, the following result will give the dynamics of
special threshold solution $W^{-}$ of Theorem 1.6 in the negative time and is
also the first step in the proof of case (a) of Theorem 1.8.

######  Proposition 6.1.

Let $u\in\dot{H}^{1}(\mathbb{R}^{N})$ be a radial solution of (1.1), and
$I=(T_{-},T_{+})$ be its maximal interval of existence. Assume that

$\displaystyle E(u_{0})=E(W),\quad\big{\|}\nabla
u_{0}\big{\|}_{L^{2}}<\big{\|}\nabla W\big{\|}_{L^{2}}.$ (6.1)

Then

$I=\mathbb{R}.$

Furthermore, if $u$ does not scatter for the positive time, that is,

$\displaystyle\big{\|}u\big{\|}_{Z(0,+\infty)}=\infty,$ (6.2)

then there exist $\theta_{0}\in\mathbb{R},\mu_{0}>0,c,C>0$ such that

$\displaystyle\forall\;t\geq
0,\quad\big{\|}u-W_{\theta_{0},\mu_{0}}\big{\|}_{\dot{H}^{1}}\leq Ce^{-ct},$

and

$\displaystyle\big{\|}u\big{\|}_{Z(-\infty,0)}<\infty.$

An analogous assertion holds on $(-\infty,0]$.

###  6.1. Compactness properties.

We recall some key results of Theorem 1.2 and Remark 1.3. Note that the fact
that the solution with

$\displaystyle E(u)<E(W),\quad\big{\|}\nabla
u_{0}\big{\|}_{L^{2}}<\big{\|}\nabla W\big{\|}_{L^{2}},$

is global well-posed and scatters in both time directions. By Lemma 2.6, the
concentration compactness principle and the stability theory as the proof of
Proposition $4.2$ in [54] and Lemma $6.2$ in [51], we can show the following
pre-compact property of minimal energy blow-up solution.

######  Lemma 6.2 ([51, 54]).

Let $u$ be a radial solution of (1.1) satisfying

$\displaystyle E(u_{0})=E(W),\;\;\big{\|}\nabla
u_{0}\big{\|}_{L^{2}}<\big{\|}\nabla
W\big{\|}_{L^{2}},\;\;\big{\|}u\big{\|}_{Z(0,T_{+})}=+\infty.$

Then there exists a continuous functions $\lambda(t)$ such that the set

$\displaystyle K:=\big{\\{}(u(t))_{\lambda(t)},t\in[0,T_{+})\big{\\}}$ (6.3)

is pre-compact in $\dot{H}^{1}_{\text{rad}}(\mathbb{R}^{N})$, where
$(u(t))_{\lambda(t)}:=\lambda(t)^{-\frac{N-2}{2}}u(t,\frac{x}{\lambda(t)})$.

Let $u$ be a solution of (1.1), and $\lambda(t)$ be as in Lemma 6.2. Consider
$\delta_{0}$ as in Section 4. The parameters $\theta(t)$, $\mu(t)$ and
$\alpha(t)$ are defined for $t\in
D_{\delta_{0}}=\\{t:\delta(t)<\delta_{0}\\}$. By (4.5) and Lemma 4.2, there
exists a constant $C_{0}>0$ such that for any $t\in D_{\delta_{0}},$, we have

$\int_{\frac{1}{\mu(t)}\leq|x|\leq\frac{2}{\mu(t)}}\big{|}\nabla
u(t,x)\big{|}^{2}dx=\int_{1\leq|x|\leq
2}\frac{1}{\mu(t)^{N}}\left|e^{i\theta(t)}\nabla
u\left(t,\frac{x}{\mu(t)}\right)\right|^{2}dx\geq\int_{1\leq|x|\leq
2}\big{|}\nabla W\big{|}^{2}-C_{0}\delta(t).$

Taking a smaller $\delta_{0}$ if necessary, we can assume that the right hand
side of the above inequality is bounded from below by a positive constant
$\varepsilon_{0}$ on $D_{\delta_{0}}$. Thus, we have

$\int_{\frac{\lambda(t)}{\mu(t)}\leq|x|\leq\frac{2\lambda(t)}{\mu(t)}}\frac{1}{\lambda(t)^{N}}\left|\nabla
u\left(t,\frac{x}{\lambda(t)}\right)\right|^{2}dx\geq\int_{1\leq|x|\leq
2}\big{|}\nabla W\big{|}^{2}-C_{0}\delta(t),\;\text{for any}\;t\in
D_{\delta_{0}}.$

By the compactness of $\overline{K}$, it follows that for any $t\in
D_{\delta_{0}}$, we have $\big{|}\mu(t)\big{|}\sim\big{|}\lambda(t)\big{|}$.
As a consequence, we may modify $\lambda(t)$ such that $K$ defined by (6.3)
remains pre-compact in $\dot{H}^{1}$ and for any $t\in D_{\delta_{0}}$, we
have

$\lambda(t)=\mu(t).$ (6.4)

As a consequence of Lemma 6.2, we have

######  Corollary 6.3.

Let $u$ be a radial solution of (1.1) satisfying (6.1) and not scatter for the
positive time. Then

  1. (a)

$T_{+}=+\infty.$

  2. (b)

$\displaystyle\lim_{t\rightarrow\infty}\sqrt{t}\lambda(t)=\infty$;

As a direct consequence of (a) in Corollary 6.3, we have

######  Corollary 6.4.

Let $u$ be a radial solution of (1.1) with the maximal lifespan interval $I$
and satisfy

$\displaystyle E(u_{0})\leq E(W),\quad\big{\|}\nabla
u_{0}\big{\|}_{2}\leq\big{\|}\nabla W\big{\|}_{2},$

then $I=\mathbb{R}.$

The proofs of these results are same as the proofs of Corollary $6.3$ and
Corollary $6.4$ in [51], which is the consequence of the compactness of
$\overline{K}$ in $\dot{H}^{1}(\mathbb{R}^{N})$, so we omit the details.

###  6.2. Convergence in the ergodic mean.

######  Lemma 6.5.

Let $u$ be a radial solution of (1.1) satisfying (6.1) and (6.2). Then

$\displaystyle\lim_{T\rightarrow+\infty}\frac{1}{T}\int^{T}_{0}\delta(t)dt=0,$

where $\delta(t)$ is defined by (4.4).

###### Proof.

By Lemma 2.7 and the Hardy inequality, we have

$\displaystyle\big{|}\partial_{t}V_{R}(t)\big{|}\leq$ $\displaystyle CR^{2}.$

In addition, by $E(u)=E(W)=\frac{p-1}{2p}\big{\|}\nabla
W\big{\|}^{2}_{L^{2}}$, we have

$\displaystyle\partial^{2}_{t}V_{R}(t)=$ $\displaystyle
4(p-1)\delta(t)+A_{R}(t),$

where $A_{R}\big{(}u(t)\big{)}$ is defined in Lemma 2.7. By the compactness of
$\overline{K}$ in $\dot{H}^{1}$, for any $\epsilon>0$, there exists
$\rho_{\epsilon}>0$ such that

$\displaystyle\int_{|x|\geq\frac{\rho_{\epsilon}}{\lambda(t)}}\big{|}\nabla
u(t,x)\big{|}^{2}dx\leq\epsilon,$

which together with Lemma 5.3 implies for $\displaystyle
R\geq\rho_{\epsilon}/\lambda(t)$, that

$\displaystyle\forall\;t\geq 0,\;\big{|}A_{R}(t)\big{|}\leq\epsilon.$ (6.5)

Fix $\epsilon$, choosing $\epsilon_{0}$ and $M_{0}$ such that

$\displaystyle
2C\epsilon^{2}_{0}=\epsilon,\;\;M_{0}\epsilon_{0}\geq\rho_{\epsilon}.$

By Corollary 6.3 (b), there exists $t_{0}\geq 0$ such that

$\displaystyle\forall\;t\geq t_{0},\;\;\lambda(t)\geq\frac{M_{0}}{\sqrt{t}}.$

For $T\geq t_{0}$, let

$\displaystyle R:=\epsilon_{0}\sqrt{T}.$

For $t\in[t_{0},T]$, we have

$\displaystyle
R=\epsilon_{0}\sqrt{T}\;\frac{M_{0}}{\sqrt{t}\lambda(t)}=\frac{\sqrt{T}}{\sqrt{t}}\;\frac{M_{0}\epsilon_{0}}{\lambda(t)}\geq\frac{\rho_{\epsilon}}{\lambda(t)},$

which yields that

$\displaystyle 4(p-1)\int^{T}_{t}\delta(s)ds\leq$
$\displaystyle\int^{T}_{t}\partial^{2}_{s}V_{R}(s)ds+\big{|}A_{R}(s)\big{|}\big{(}T-t_{0}\big{)}\leq
2CR^{2}+\epsilon T=2\epsilon T.$

Let $T\rightarrow+\infty$, we have

$\displaystyle\lim_{T\rightarrow+\infty}\frac{1}{T}\int^{T}_{0}\delta(s)ds\leq\frac{\epsilon}{2},$

which concludes the proof. ∎

######  Corollary 6.6.

Let $u$ be a radial solution of (1.1) satisfying (6.1) and (6.2). Then there
exists a sequence $t_{n}$ such that $t_{n}\rightarrow+\infty$ and

$\displaystyle\lim_{n\rightarrow+\infty}\delta(t_{n})=0.$

###  6.3. Exponential convergence.

As the same with Section 5, we will give the exponential convergence of the
gradient variant, which will prove the Proposition 6.1. The two ingredients of
the proof of Proposition 6.1 are the local virial argument (lemma 6.7) and a
precise control of the variations of the parameter $\lambda(t)$ (Lemma 6.8).

######  Lemma 6.7.

Let $u$ be a radial solution of (1.1) satisfying (6.1), (6.2) and (6.4), and
$\lambda(t)$ be as in Lemma 6.2. Then there exists a constant $C$ such that if
$\;0\leq\sigma<\tau$, we have

$\displaystyle\int^{\tau}_{\sigma}\delta(t)dt\leq C\left(\sup_{\sigma\leq
t\leq\tau}\frac{1}{\lambda(t)^{2}}\right)\Big{(}\delta(\sigma)+\delta(\tau)\Big{)}.$

The proof is analogue to that of Lemma 6.7 in [51]. For the sake of
completeness, We give the proof.

###### Proof.

For $R>0$, Let us consider the function $V_{R}(t)$ defined as in (2.5).

By the same estimate as that in Lemma 5.2, there is a constant $C_{0}$
independent of $t\geq 0$ such that

$\displaystyle\Big{|}\partial_{t}V_{R}(t)\Big{|}\leq C_{0}R^{2}\delta(t).$

Now we show that if $\epsilon>0$, there exists $R_{\epsilon}$ such that for
any $R\geq R_{\epsilon}/\lambda(t)$, then

$\displaystyle\partial^{2}_{t}V_{R}(t)\geq(4(p-1)-\epsilon)\delta(t).$ (6.6)

Indeed, by Lemma 2.7 and $E(u)=E(W)=\frac{p-1}{2p}\big{\|}\nabla
W\big{\|}^{2}_{L^{2}}$, we have

$\displaystyle\partial^{2}_{t}V_{R}(t)=$ $\displaystyle
4(p-1)\delta(t)+A_{R}(t),$

where $A_{R}\big{(}u(t)\big{)}$ is defined in Lemma 2.7.

To prove (6.6), it suffices to show that if $\epsilon>0$, there exists
$R_{\epsilon}$ such that for any $R\geq R_{\epsilon}/\lambda(t)$,

$\displaystyle\Big{|}A_{R}(t)\Big{|}\lesssim\epsilon\delta(t).$

For the case when $\delta(t)\geq\delta_{0}$. As the estimate (6.5), we can use
the compactness of $\overline{K}$ in $\dot{H}^{1}$ to show that for any $t\geq
0$, $\epsilon>0$, there exists $\rho_{\epsilon}>0$, such that for any
$R\geq\rho_{\epsilon}/\lambda(t)$, we have

$\displaystyle\big{|}A_{R}(t)\big{|}\leq\epsilon\lesssim\epsilon\delta(t).$

For the case when $\delta(t)<\delta_{0}$. As the proof of Step 2 in Lemma 5.3,
we can show that for any $t\geq 0$, $\rho>1$, there exists $C>0$ such that for
any $R\geq\rho/\lambda(t)$, we have

$\displaystyle\big{|}A_{R}(t)\big{|}\leq
C\left(\rho^{-\frac{N-2}{2}}\delta(t)+\delta(t)^{2}\right),$

which implies (6.6) if choosing $\rho$ large enough.

By (6.6), there exists $R_{2}$ such that for any $R\geq R_{2}/\lambda(t)$,

$\displaystyle\partial^{2}_{t}V_{R}(t)\geq 2(p-1)\delta(t).$

Finally, if taking $R=R_{2}\displaystyle\sup_{\sigma\leq
t\leq\tau}\left(\frac{1}{\lambda(t)}\right),$ and integrating between $\sigma$
and $\tau$, we have

$\displaystyle
2(p-1)\int^{\tau}_{\sigma}\delta(t)dt\leq\int^{\tau}_{\sigma}\partial^{2}_{t}V_{R}(t)dt=\partial_{t}V_{R}(\tau)-\partial_{t}V_{R}(\sigma)\leq
CR^{2}\Big{(}\delta(\tau)+\delta(\sigma)\Big{)},$

which implies the result. ∎

In order to get the exponential convergence of the gradient variant, we need
to control the variations of $\frac{1}{\lambda(t)^{2}}$.

######  Lemma 6.8 ([51]).

Let $u$ be a radial solution of (1.1) satisfying (6.1), (6.2) and (6.4), and
$\lambda(t)$ be as in Lemma 6.2. Then there is a constant $C_{0}>0$ such that
for any $\sigma,\tau>0$ with $\sigma+\frac{1}{\lambda(\sigma)^{2}}\leq\tau$,
we have

$\displaystyle\left|\frac{1}{\lambda(\tau)^{2}}-\frac{1}{\lambda(\sigma)^{2}}\right|\leq
C_{0}\int^{\tau}_{\sigma}\delta(t)dt.$

Thus, we have

######  Lemma 6.9 ([51]).

Let $u$ be a radial solution of (1.1) satisfying (6.1), (6.2) and (6.4), and
$\lambda(t)$ be as in Lemma 6.2. Then there exists $C$ such that for all
$t\geq 0$,

$\displaystyle\int^{+\infty}_{t}\delta(s)ds\leq$ $\displaystyle\;Ce^{-ct}.$
(6.7)

###  6.4. Convergence of the modulation parameters

Now by Lemma 6.7, Lemma 6.8 and (6.7), we can get that there exists
$\lambda_{0}\in(0,+\infty)$ such that

$\lim_{t\rightarrow+\infty}\lambda(t)=\lambda_{0}.$ (6.8)

The argument is standard, and we can refer to Section 6.4 in [51] for details.

By (6.7), there exists $t_{n}\rightarrow+\infty$ such that

$\delta(t_{n})\rightarrow 0.$

Fix such $\\{t_{n}\\}_{n\in\mathbb{N}}$. The similar proof as that of (5.11),
we can show that

$\displaystyle\lim_{t\rightarrow+\infty}\delta(t)=0.$ (6.9)

Now for large $t$, $\alpha(t)$ is well defined by Lemma 4.2, (6.4), (6.8), and
(6.9). Furthermore, we have

$\displaystyle\delta(t)\approx\big{|}\alpha(t)\big{|}=$
$\displaystyle\big{|}\alpha(t)-\alpha(+\infty)\big{|}$ $\displaystyle\leq$
$\displaystyle C\int^{+\infty}_{t}\big{|}\alpha^{\prime}(s)\big{|}ds\leq
C\int^{+\infty}_{t}\mu^{2}(s)\delta(s)ds\leq Ce^{-ct}.$

Finally, by Lemma 4.2, (6.4), and (6.8), there exists $\theta_{\infty}$ such
that for large $t$, we have

$\displaystyle\Big{|}\theta(t)-\theta_{\infty}\Big{|}\leq
C\int^{\infty}_{t}\big{|}\theta^{\prime}(s)\big{|}ds\leq
C\int^{+\infty}_{t}\mu^{2}(s)\delta(s)ds\leq Ce^{-ct}.$

###  6.5. Scattering for the negative times.

By Corollary 6.4, we know that $u$ is globally well-defined. Assume that $u$
does not scatter for the negative time. Then by the analogue estimates as
those from Subsection 6.1 to Subsection 6.3, we have

  1. (a)

there exists $\lambda(t)$, defined for $t\in\mathbb{R}$, such that the set

$\displaystyle
K:=\Big{\\{}\big{(}u(t)\big{)}_{\lambda(t)},\;t\in\mathbb{R}\Big{\\}}$

is pre-compact in $\dot{H}^{1}_{\text{rad}}(\mathbb{R}^{N})$.

  2. (b)

there exists an decreasing sequence $t^{\prime}_{n}\rightarrow-\infty$ as
$n\rightarrow+\infty$, such that

$\displaystyle\lim_{n\rightarrow+\infty}\delta(t^{\prime}_{n})=0.$

  3. (c)

there is a constant $C>0$ such that if $\;-\infty<\sigma<\tau<\infty$,

$\displaystyle\int^{\tau}_{\sigma}\delta(t)dt\leq C\left(\sup_{\sigma\leq
t\leq\tau}\frac{1}{\lambda(t)^{2}}\right)\big{(}\delta(\sigma)+\delta(\tau)\big{)}.$

  4. (d)

there is a constant $C>0$ such that for any $\sigma,\tau$ with
$\sigma+\frac{1}{\lambda(\sigma)^{2}}\leq\tau$, we have

$\displaystyle\left|\frac{1}{\lambda(\tau)^{2}}-\frac{1}{\lambda(\sigma)^{2}}\right|\leq
C\int^{\tau}_{\sigma}\delta(t)dt.$

From (b)-(d), we know that

$\displaystyle\lim_{t\rightarrow\pm\infty}\delta(t)=0$

and $\frac{1}{\lambda(t)^{2}}$ is bounded for $t\in\mathbb{R}$. Hence we have

$\displaystyle\int^{\tau}_{\sigma}\delta(s)\;ds\leq
C\Big{(}\delta(\sigma)+\delta(\tau)\Big{)}.$

Let $\sigma\rightarrow-\infty$ and $\tau\rightarrow+\infty$, we have

$\delta(t)=0,\;\forall\;t\in\mathbb{R}.$

Thus $u=W$ up to the symmetry of (1.1), which contradicts with
$\big{\|}u_{0}\big{\|}_{\dot{H}^{1}}<\big{\|}W\big{\|}_{\dot{H}^{1}}$. ∎

##  7\. Uniqueness and the classification result

In this section, we follows the arguments in [18, 19] and [51].

###  7.1. Exponentially small solutions of the linearized equation.

Consider the radial functions $v(t)$ and $g(t)$

$\displaystyle v\in C^{0}([t_{0},+\infty),\dot{H}^{1}),\;\;g\in
C^{0}([t_{0},+\infty),L^{\frac{2N}{N+2}})\;\;\text{and}\;\;\nabla g\in
N(t_{0},+\infty)$

such that

$\displaystyle\partial_{t}v+\mathcal{L}v=g,$
$\displaystyle\;\;(x,t)\in\mathbb{R}^{N}\times[t_{0},+\infty),$
$\displaystyle\big{\|}v(t)\big{\|}_{\dot{H}^{1}}\leq Ce^{-c_{1}t},$
$\displaystyle\;\;\;\;\big{\|}g(t)\big{\|}_{L^{\frac{2N}{N+2}}(\mathbb{R}^{N})}+\big{\|}\nabla
g(t)\big{\|}_{N(t,+\infty)}\leq Ce^{-c_{2}t},$

where $0<c_{1}<c_{2}$. For any $c>0$, we denote by $c^{-}$ a positive number
arbitrary close to $c$ and such that $0<c^{-}<c$.

By the Strichartz estimate and Lemma 2.10, we have

######  Lemma 7.1 ([18, 19, 51]).

Under the above assumptions, then for any $(q,r)$ with
$\frac{2}{q}=N(\frac{1}{2}-\frac{1}{r})$, $q\in[2,+\infty]$, we have

$\displaystyle\big{\|}v\big{\|}_{l(t,+\infty)}+\big{\|}\nabla
v\big{\|}_{L^{q}(t,+\infty;L^{r})}\leq Ce^{-c_{1}t}.$

Now we show that $v$ must decay almost as fast as $g$, except in the direction
$\mathcal{Y}_{+}$ where the decay is of order $e^{-e_{0}t}$.

######  Proposition 7.2 ([18, 19, 51]).

The following self-improving estimates hold.

  1. (a)

if $c_{2}\leq e_{0}$ or $e_{0}<c_{1}$, then

$\displaystyle\big{\|}v(t)\big{\|}_{\dot{H}^{1}}\leq Ce^{-c_{2}^{-}t},$

  2. (b)

If $c_{2}>e_{0}$, then there exists $a\in\mathbb{R}$ such that

$\displaystyle\big{\|}v(t)-ae^{-e_{0}t}\mathcal{Y}_{+}\big{\|}_{\dot{H}^{1}}\leq
Ce^{-c_{2}^{-}t}.$

###  7.2. Uniqueness.

######  Proposition 7.3.

Let $u$ be a radial solution of (1.1), defined on $[t_{0},+\infty)$, such that
$E(u)=E(W)$ and

$\displaystyle\exists\;c,C>0,\;\big{\|}u-W\big{\|}_{\dot{H}^{1}}\leq
Ce^{-ct},\;\forall\;t\geq t_{0}.$ (7.1)

Then there exists a unique $a\in\mathbb{R}$ such that $u=U^{a}$ is the
solution of (1.1) defined in Proposition 3.3.

For the sake of completeness, We give the details.

###### Proof.

Let $u=W+v$ be a radial solution of (1.1) for $t\geq t_{0}$ satisfying (7.1).
Then $v$ satisfies the linearized equation (2.8), that is,

$\displaystyle\partial_{t}v+\mathcal{L}v=R(v),$

Step 1:  We will show that there exists $a\in\mathbb{R}$ such that

$\displaystyle\forall\;\eta>0,\;\big{\|}v(t)-ae^{-e_{0}t}\mathcal{Y}_{+}\big{\|}_{\dot{H}^{1}}+\big{\|}v(t)-ae^{-e_{0}t}\mathcal{Y}_{+}\big{\|}_{l(t,+\infty)}\leq
C_{\eta}e^{-2^{-}e_{0}t}.$ (7.2)

Indeed, we only need to show

$\displaystyle\big{\|}v(t)\big{\|}_{\dot{H}^{1}}\leq Ce^{-e_{0}t},$
$\displaystyle\;\;\big{\|}R(v(t))\big{\|}_{L^{\frac{2N}{N+2}}(\mathbb{R}^{N})}+\big{\|}\nabla
R(v(t))\big{\|}_{N(t,+\infty)}\leq Ce^{-2e_{0}t},$ (7.3)

which together with Proposition 7.2 gives (7.2). By Lemma 2.8 and Lemma 2.11
it suffices to show the first estimate.

By (7.1), Lemma 2.8 and Lemma 2.11, we have

$\displaystyle\big{\|}R(v(t))\big{\|}_{L^{\frac{2N}{N+2}}(\mathbb{R}^{N})}+\big{\|}\nabla
R(v(t))\big{\|}_{N(t,+\infty)}\leq Ce^{-2ct}.$

Then Proposition 7.2 gives that

$\displaystyle\big{\|}v(t)\big{\|}_{\dot{H}^{1}}\leq
C\Big{(}e^{-e_{0}t}+e^{-\frac{3}{2}ct}\Big{)}.$

If $\dfrac{3}{2}\;c\geq e_{0}$, we obtain the first inequality in (7.3). If
not, an iteration argument gives the first inequality in (7.3).

Step 2:  We will use the induction argument to show that

$\displaystyle\forall\;m>0,\;\big{\|}u(t)-U^{a}(t)\big{\|}_{\dot{H}^{1}}+\big{\|}u(t)-U^{a}(t)\big{\|}_{l(t,+\infty)}\leq
e^{-mt}.$ (7.4)

By Proposition 3.3 and Step 1, (7.4) holds with $m=\frac{3}{2}e_{0}$. Let us
assume that (7.4) holds with $m=m_{1}>e_{0}$, we will show that it also holds
for $m=m_{1}+\frac{1}{2}e_{0}$, which yields (7.4) by iteration.

If writing $u^{a}(t)=U^{a}-W$, then we have

$\displaystyle\partial_{t}\big{(}v-u^{a}\big{)}+\mathcal{L}\big{(}v-u^{a}\big{)}=R(v)-R(u^{a}).$

By assumption, we have

$\displaystyle\big{\|}v(t)-u^{a}(t)\big{\|}_{\dot{H}^{1}}+\big{\|}v(t)-u^{a}(t)\big{\|}_{l(t,+\infty)}\leq
Ce^{-m_{1}t}.$

By Lemma 2.8 and Lemma 2.10, we have

$\displaystyle\big{\|}R(v(t))-R(u^{a}(t))\big{\|}_{L^{\frac{2N}{N+2}}(\mathbb{R}^{N})}+\big{\|}\nabla\big{(}R(v(t))-R(u^{a}(t))\big{)}\big{\|}_{N(t,+\infty)}\leq
Ce^{-(m_{1}+e_{0})t}.$

By Proposition 7.2 and Lemma 7.1, we have

$\displaystyle\forall\;m>0,\;\big{\|}v(t)-u^{a}(t)\big{\|}_{\dot{H}^{1}}+\big{\|}v(t)-u^{a}(t)\big{\|}_{l(t,+\infty)}\leq
Ce^{-(m_{1}+\frac{3}{4}e_{0})t}\leq e^{-(m_{1}+\frac{1}{2}e_{0})t}.$

Step 3: End of the proof.  Using (7.4) with $m=(k_{0}+1)e_{0}$, (where $k_{0}$
is given by Proposition 3.3), we get that for large $t>0$,

$\displaystyle\big{\|}u(t)-U^{a}_{k_{0}}(t)\big{\|}_{l(t,+\infty)}\leq
e^{-(k_{0}+\frac{1}{2})e_{0}t}.$

By the uniqueness in Proposition 3.3, we get that $u=U^{a}$, which concludes
the proof. ∎

######  Corollary 7.4.

For any $a\not=0$, there exists $T_{a}\in\mathbb{R}$ such that

$\left\\{\begin{array}[]{rl}U^{a}(t)=W^{+}(t-T_{a}),&\text{if}\;\;a>0,\\\
U^{a}(t)=W^{-}(t-T_{a}),&\text{if}\;\;a<0,\end{array}\right.$

###### Proof.

Let $a\not=0$ and choose $T_{a}$ such that $|a|e^{-e_{0}T_{a}}=1$. By (3.5),
we have

$\displaystyle\big{\|}U^{a}(t+T_{a})-$ $\displaystyle
W-\text{sign}(a)e^{-e_{0}t}\mathcal{Y}_{+}\big{\|}_{\dot{H}^{1}}$
$\displaystyle=$
$\displaystyle\big{\|}U^{a}(t+T_{a})-W-ae^{-e_{0}(t+T_{a})}\mathcal{Y}_{+}\big{\|}_{\dot{H}^{1}}\leq
e^{-\frac{3}{2}e_{0}(t+T_{a})}\leq Ce^{-\frac{3}{2}e_{0}t}.$ (7.5)

In addition, $U^{a}(t+T_{a})$ satisfies the assumption of Proposition 7.3,
this implies that there exists $\widetilde{a}$ such that

$\displaystyle U^{a}(t+T_{a})=U^{\widetilde{a}}(t).$

By (7.2) and (3.5), we know that $\widetilde{a}=1$ if $a>0$ and
$\widetilde{a}=-1$ if $a=-1$, this completes the proof.∎

######  Remark 7.5.

This shows that $U^{a}(t)=W^{\pm}(t)$ up to time translation.

###  7.3. Proof of Theorem 1.8.

Let us first show (a). Let $u$ be a radial solution to (1.1) on
$[-T_{-},T_{+}]$ such that, replacing if necessary $u(t)$ by
$\overline{u}(-t)$,

$E(u_{0})=E(W),\big{\|}\nabla u_{0}\big{\|}_{L^{2}}<\big{\|}\nabla
W\big{\|}_{L^{2}},\;\text{and}\;\big{\|}u\big{\|}_{Z(0,T_{+})}=+\infty.$

Then by Proposition 6.1, we know that $T_{-}=T_{+}=\infty$, scatters in
negative time and there exist $\theta_{0}\in\mathbb{R},\mu_{0}>0$ and $c,C>0$
such that

$\displaystyle\big{\|}u(t)-W_{\theta_{0},\mu_{0}}\big{\|}_{\dot{H}^{1}}\leq
Ce^{-ct}\;\text{for any}\;t\geq 0.$

This shows that $u_{-\theta_{0},\frac{1}{\mu_{0}}}$ fulfills the assumptions
of Proposition 7.3. By
$\big{\|}u\big{\|}_{\dot{H}^{1}}<\big{\|}W\big{\|}_{\dot{H}^{1}}$ and
Corollary 7.4, we know that there exist $a<0$ and $T_{a}$ such that

$\displaystyle u_{-\theta_{0},\frac{1}{\mu_{0}}}(t)=U^{a}(t)=W^{-}(t-T_{a}),$

which shows (a).

(b) is a consequence of the variational characterization of $W$.

The proof of (c) is similar to that of (a). Let $u$ be a radial solution of
(1.1) defined on $[0,+\infty)$ (Replacing if necessary $u(t)$ by
$\overline{u}(-t)$) such that

$\displaystyle E(u)=E(W),\;\big{\|}\nabla u_{0}\big{\|}_{L^{2}}>\big{\|}\nabla
W\big{\|}_{L^{2}},\;\text{and}\;u_{0}\in L^{2}.$

then by Proposition 5.1, there exist $\theta_{0}\in\mathbb{R},\mu_{0}>0$ and
$c,C>0$ such that

$\displaystyle\big{\|}u(t)-W_{\theta_{0},\mu_{0}}\big{\|}_{\dot{H}^{1}}\leq
Ce^{-ct}.$

Similar to the proof of (a), we know that there exist $a>0$ and $T_{a}$ such
that

$\displaystyle u_{-\theta_{0},\frac{1}{\mu_{0}}}(t)=U^{a}(t)=W^{+}(t-T_{a}).$

This shows (c). The proof of Theorem 1.8 is completed. ∎

##  Appendix A Nondegeneracy of energy solution of equation (1.5)

From Remark 1.5 (b) and (c) , it suffices to show $L^{\infty}$-regularity of
$\dot{H}^{1}(\mathbb{R}^{N})$ solution of equation (1.5) to prove Proposition
1.4. Now we use the Moser iteration method in [16] to show this result in this
appendix.

Recalling equation (1.5), energy solution $u\in\dot{H}^{1}(\mathbb{R}^{N})$
satisfies

$-\Delta
u=(p-1)\left(I_{\lambda}*W^{p}\right)W^{p-2}u+p\left[I_{\lambda}*(W^{p-1}u)\right]W^{p-1}.$
(A.1)

Let $\beta\geq 1,\;T>0$, we define the cutoff function

$\varphi(t)=\begin{cases}\beta T^{\beta-1}(t-T)+T^{\beta},&if\;t\geq T;\\\
|t|^{\beta},&if\;-T\leq t\leq T;\\\ \beta
T^{\beta-1}(T-t)+T^{\beta},&if\;t\leq-T.\end{cases}$ (A.2)

Since $\varphi$ is convex and Lipschitz, we have

$(-\Delta)^{s}\varphi(u)\leq\varphi^{\prime}(u)(-\Delta)^{s}u,\;s\in[0,1]$
(A.3)

in the weak sense. By the equation (A.1), we get

$\displaystyle\int_{\mathbb{R}^{N}}(-\Delta
u)\varphi(u)\varphi^{\prime}(u)\;dx=$
$\displaystyle\,(p-1)\int_{\mathbb{R}^{N}}\left(I_{\lambda}*W^{p}\right)W^{p-2}u\varphi(u)\varphi^{\prime}(u)\;dx$
$\displaystyle\,\,+p\int_{\mathbb{R}^{N}}\left[I_{\lambda}*(W^{p-1}u)\right]W^{p-1}\varphi(u)\varphi^{\prime}(u)\;dx.$

Using (A.3) and integration by parts, we get

$\int_{\mathbb{R}^{N}}(-\Delta
u)\varphi(u)\varphi^{\prime}(u)\;dx=\int_{\mathbb{R}^{N}}|\nabla
u|^{2}(\varphi^{\prime}(u))^{2}\;dx+\int_{\mathbb{R}^{N}}|\nabla
u|^{2}\varphi(u)\varphi^{\prime\prime}(u)\;dx.$

By (A.2), we know that the last integral is nonnegative, therefore, we have

$\displaystyle\int_{\mathbb{R}^{N}}|\nabla
u|^{2}(\varphi^{\prime}(u))^{2}\;dx\leq$
$\displaystyle\,(p-1)\int_{\mathbb{R}^{N}}\left(I_{\lambda}*W^{p}\right)W^{p-2}|u||\varphi(u)\varphi^{\prime}(u)|\;dx$
$\displaystyle\,\,+p\int_{\mathbb{R}^{N}}\left[I_{\lambda}*(W^{p-1}|u|)\right]W^{p-1}\varphi(u)|\varphi^{\prime}(u)|\;dx.$

Applying the weak-Young inequality and noting that
$W=c\left(\frac{t}{t^{2}+|x|^{2}}\right)^{\frac{N-2}{2}}\in
L^{p}(\mathbb{R}^{N}),\;p>(N-2)/N$, we have

$I_{\lambda}*W^{p}\in L^{\infty}(\mathbb{R}^{N}).$

Using that $\varphi(u)\varphi^{\prime}(u)\leq\beta u^{2\beta-1}$ and the
Hölder inequality, we have

$\int_{\mathbb{R}^{N}}\left(I_{\lambda}*W^{p}\right)W^{p-2}|u||\varphi(u)\varphi^{\prime}(u)|\;dx\leq
C\beta\int_{\mathbb{R}^{N}}|u|^{2\beta}.$

Similarly, we can get

$\int_{\mathbb{R}^{N}}\left[I_{\lambda}*(W^{p-1}|u|)\right]W^{p-1}\varphi(u)|\varphi^{\prime}(u)|\;dx\leq
C\beta\int_{\mathbb{R}^{N}}|u|^{2\beta}.$

Thus, by the Sobolev inequality, we have

$\|\varphi(u)\|_{L^{2^{*}}}^{2}\leq\|\varphi(u)\|_{\dot{H}^{1}}^{2}=\int_{\mathbb{R}^{N}}|\nabla
u|^{2}(\varphi^{\prime}(u))^{2}\;dx\leq
C\beta\int_{\mathbb{R}^{N}}|u|^{2\beta},$

where $2^{*}=\frac{2N}{N-2}$. Let $T\rightarrow\infty$, we obtain

$\left(\int_{\mathbb{R}^{N}}|u|^{2^{*}\beta}\right)^{\frac{2}{2^{*}}}\leq
C\beta\int_{\mathbb{R}^{N}}|u|^{2\beta},$

where C changing from line to line, but independent of $\beta$. Therefore,

$\left(\int_{\mathbb{R}^{N}}|u|^{2^{*}\beta}\right)^{\frac{1}{2^{*}\beta}}\leq(C\beta)^{\frac{1}{2\beta}}\left(\int_{\mathbb{R}^{N}}|u|^{2\beta}\right)^{\frac{1}{2\beta}}.$
(A.4)

Because $u\in\dot{H}^{1}(\mathbb{R}^{N})$, we know $u\in L^{\frac{2N}{N-2}}$
by the Sobolev inequality. Therefore, we can take $\beta_{1}=\frac{N}{N-2}\geq
1$. From now on, we can follow exactly the iteration argument. That is, we
define $\beta_{m+1},m\geq 1$, so that

$2\beta_{m+1}=2^{*}\beta_{m}.$

Thus,

$\beta_{m+1}=\left(\frac{2^{*}}{2}\right)^{m}\beta_{1},$

and replacing in equation (A.4), we know that

$\left(\int_{\mathbb{R}^{N}}|u|^{2\beta_{m+1}}\right)^{\frac{1}{2\beta_{m+1}}}=\left(\int_{\mathbb{R}^{N}}|u|^{2^{*}\beta_{m}}\right)^{\frac{1}{2^{*}\beta_{m}}}\leq\left(C\beta_{m}\int_{\mathbb{R}^{N}}|u|^{2\beta_{m}}\right)^{\frac{1}{2\beta_{m}}}.$

Defining $C_{m+1}:=C\beta_{m}$ and

$A_{m}:=\big{(}\int_{\mathbb{R}^{N}}|u|^{2\beta_{m+1}}\big{)}^{\frac{1}{2\beta_{m+1}}},$

we conclude that there exists a constant $C_{0}>0$ independent of $m$, such
that

$A_{m+1}\leq\prod_{k=1}^{m}C_{k}^{\frac{1}{2\beta_{k}}}A_{1}\leq C_{0}A_{1}.$

Thus, we obtain

$\|u\|_{L^{\infty}(\mathbb{R}^{N})}\leq C_{0}A_{1}<+\infty.$

This together with the result in [45] completes the proof of Proposition 1.4.
∎

##  Appendix B Coercivity of $\Phi$ on $H^{\bot}\cap\dot{H}^{1}_{rad}$

In this Appendix, we follow the argument in [51] and prove Proposition 2.13.
We divide the proof into two steps.

Step 1: Nonnegative. We claim that for any function $h\in\dot{H}^{1}$,
$\big{(}h,W\big{)}_{\dot{H}^{1}}=0$, there exists

$\displaystyle\Phi(h)\geq 0.$

Indeed, let

$\displaystyle I(u)=\frac{\big{\|}\nabla u\big{\|}^{2p}_{2}}{\big{\|}\nabla
W\big{\|}^{2p}_{2}}-\frac{\displaystyle\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\lambda}(x-y)|u(x)|^{p}|u(y)|^{p})\;dxdy}{\displaystyle\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\lambda}(x-y)W(x)^{p}W(y)^{p}\;dxdy}.$

By Proposition 1.1, we have

$\displaystyle\forall\;u\in\dot{H}^{1},\quad I(u)\geq 0.$

Take $\displaystyle h\in\dot{H}^{1}$ with
$(W,h)_{\dot{H}^{1}}=0,\;\alpha\in\mathbb{R}$, we consider the expansion of
$I(W+\alpha h)$ in $\alpha$ of order $2$. Note that

$\displaystyle\big{\|}\nabla(W+\alpha h)\big{\|}^{2p}_{2}=\big{\|}\nabla
W\big{\|}^{2p}_{2}\left(1+p\alpha^{2}\frac{\big{\|}h\big{\|}^{2}_{\dot{H}^{1}}}{\big{\|}W\big{\|}^{2}_{\dot{H}^{1}}}+O(\alpha^{4})\right),$

and

$\displaystyle\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\lambda}(x-y)\big{|}W(x)+\alpha
h(x)\big{|}^{p}\big{|}W(y)+\alpha h(y)\big{|}^{p}\;dxdy$ $\displaystyle=$
$\displaystyle\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\lambda}(x-y)W(x)^{p}W(y)^{p}\;dxdy$
$\displaystyle+p\alpha\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\lambda}(x-y)\left(W(x)^{p}W(y)^{p-1}h_{1}(y)+W(y)^{p}W(x)^{p-1}h_{1}(x)\right)\;dxdy$
$\displaystyle+\;\alpha^{2}\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\lambda}(x-y)\Big{[}\left(\frac{p(p-1)}{2}W(x)^{p-2}h_{1}(x)^{2}+\frac{p}{2}W(x)^{p-2}h_{2}(x)^{2}\right)W(y)^{p}$
$\displaystyle\qquad\qquad+\left(\frac{p(p-1)}{2}W(y)^{p-2}h_{1}(y)^{2}+\frac{p}{2}W(y)^{p-2}h_{2}(y)^{2}\right)W(x)^{p}$
$\displaystyle\qquad\qquad+p^{2}W(x)^{p-1}h_{1}(x)W(y)^{p-1}h_{1}(y)\Big{]}\;dxdy$
$\displaystyle+O(\alpha^{3})$ $\displaystyle=$
$\displaystyle\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\lambda}(x-y)W(x)^{p}W(y)^{p}\;dxdy$
$\displaystyle\times\Big{(}1+\frac{\alpha^{2}}{\big{\|}W\big{\|}^{2}_{\dot{H}^{1}}}\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\lambda}(x-y)\big{[}p(p-1)W(x)^{p-2}h_{1}(x)^{2}W(y)^{p}$
$\displaystyle+pW(x)^{p-2}h_{2}(x)^{2}W(y)^{p}+p^{2}W(x)^{p-1}h_{1}(x)W(y)^{p-1}h_{1}(y)\big{]}\;dxdy+O(\alpha^{3})\Big{)},$

where we use the facts that

$\displaystyle\int_{\mathbb{R}^{N}}(I_{\lambda}*W^{p})W(y)^{p-1}h_{1}(y)\;dy=$
$\displaystyle\int_{\mathbb{R}^{N}}-\Delta W(y)\cdot h_{1}(y)dy=0,$
$\displaystyle\int_{\mathbb{R}^{N}}(I_{\lambda}*W^{p})W(x)^{p-1}h_{1}(x)\;dx=$
$\displaystyle\int_{\mathbb{R}^{N}}-\Delta W(x)\cdot h_{1}(x)dy=0,$
$\displaystyle\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\lambda}(x-y)W(x)^{p-2}h_{1}(x)^{2}dx=$
$\displaystyle\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\lambda}(x-y)W(y)^{p-2}h_{1}(y)^{2}dx,$
$\displaystyle\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\lambda}(x-y)W(x)^{p}W(y)^{p}\;dxdy=$
$\displaystyle\int_{\mathbb{R}^{N}}\big{|}\nabla W\big{|}^{2}dx.$

Therefore we have

$\displaystyle I(W+\alpha h)=$
$\displaystyle\frac{p\alpha^{2}}{\big{\|}W\big{\|}^{2}_{\dot{H}^{1}}}\Big{(}\big{\|}h\big{\|}^{2}_{\dot{H}^{1}}-\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\lambda}(x-y)\big{[}(p-1)W(x)^{p-2}h_{1}(x)^{2}W(y)^{p}$
$\displaystyle\qquad\qquad+W(x)^{p-2}h_{2}(x)^{2}W(y)^{p}+W(x)^{p-1}h_{1}(x)W(y)^{p-1}h_{1}(y)\big{]}\
dxdy\Big{)}$ $\displaystyle+O(\alpha^{3}).$

Since $I(W+\alpha h)\geq 0$ for all $\alpha\in\mathbb{R}$, we have

$\displaystyle\begin{aligned} 2\Phi(h)&=\int_{\mathbb{R}^{N}}\big{|}\nabla
h\big{|}^{2}dx-\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\lambda}(x-y)\big{[}(p-1)W(x)^{p-2}h_{1}(x)^{2}W(y)^{p}\\\
&\qquad+W(x)^{p-2}h_{2}(x)^{2}W(y)^{p}+W(x)^{p-1}h_{1}(x)W(y)^{p-1}h_{1}(y)\big{]}\
dxdy\\\ &\geq 0.\end{aligned}$

Step 2: Coercivity. We show that there exists a constant $c_{*}>0$ such that
for any radial function $h\in H^{\bot}$

$\displaystyle\Phi(h)\geq c_{*}\big{\|}h\big{\|}^{2}_{\dot{H}^{1}}.$

Note that $\Phi(h)=\Phi_{1}(h_{1})+\Phi_{2}(h_{2})$, where

$\displaystyle\Phi_{1}(h_{1}):=$
$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}(L_{+}h_{1})h_{1}\;dx$
$\displaystyle=$ $\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}\big{|}\nabla
h_{1}\big{|}^{2}\;dx-\frac{p-1}{2}\int_{\mathbb{R}^{N}}(I_{\lambda}*W^{p})W(x)^{p-2}h_{1}(x)^{2}\;dx$
$\displaystyle\qquad\qquad\qquad\quad-\frac{p}{2}\int_{\mathbb{R}^{N}}(I_{\lambda}*W^{p-1}h_{1})W(x)h_{1}(x)\;dx,$

and

$\displaystyle\Phi_{2}(h_{2}):=$
$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}(L_{-}h_{2})h_{2}\;dx$
$\displaystyle=$ $\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}\big{|}\nabla
h_{2}\big{|}^{2}\;dx-\frac{1}{2}\
\int_{\mathbb{R}^{N}}(I_{\lambda}*W^{p})W(x)^{p-2}h_{2}(x)^{2}\;dx.$

By Step 1, $L_{+}$ is nonnegative on $\\{W\\}^{\bot}$ (in the sense of
$\dot{H}^{1}$) and $L_{-}$ is nonnegative. We will deduce the coercivity by
the compactness argument in [51, 67].

We first show that there exists a constant $c$ such that for any radial real-
valued $\dot{H}^{1}$-function $h_{1}\in\\{W,\widetilde{W}\\}^{\bot}$ (in the
sense of $\dot{H}^{1}$),

$\displaystyle\Phi_{1}(h_{1})\geq c\big{\|}h_{1}\big{\|}^{2}_{\dot{H}^{1}}.$

Assume that the above inequality does not hold. Then there exists a sequence
of real-valued radial $\dot{H}^{1}$-functions $\\{f_{n}\\}_{n}$ such that

$\displaystyle f_{n}\in
H^{\bot},\quad\lim_{n\rightarrow+\infty}\Phi_{1}(f_{n})=0,\quad\big{\|}f_{n}\big{\|}_{\dot{H}^{1}}=1.$
(B.1)

Extracting a subsequence from $\\{f_{n}\\}$, we may assume that

$\displaystyle f_{n}\rightharpoonup f_{*}\;\text{in}\;\dot{H}^{1}.$

The weak convergence of $f_{n}\in H^{\bot}$ to $f_{*}$ implies that $f_{*}\in
H^{\bot}$. In addition, by compactness, we have

$\displaystyle\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\lambda}(x-y)W(y)^{p}W(x)^{p-2}|f_{n}(x)|^{2}\;dxdy$
$\displaystyle\longrightarrow$
$\displaystyle\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\lambda}(x-y)W(y)^{p}W(x)^{p-2}|f_{*}(x)|^{2}\;dxdy,$
$\displaystyle\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\lambda}(x-y)W(x)^{p-1}|f_{n}(x)|W(y)^{p-1}|f_{n}(y)|\;dxdy$
$\displaystyle\longrightarrow$
$\displaystyle\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\lambda}(x-y)W(x)^{p-1}|f_{*}(x)|W(y)^{p-1}|f_{*}(y)|\;dxdy.$

Thus by the Fatou lemma, (B.1) and Step 1, we have

$\displaystyle
0\leq\Phi_{1}(f_{*})\leq\liminf_{n\rightarrow+\infty}\Phi_{1}(f_{n})=0.$

$f_{*}$ is the solution to the following minimizing problem

$\displaystyle
0=\min_{f\in\Omega\backslash\\{0\\}}\frac{\displaystyle\int_{\mathbb{R}^{N}}L_{+}f\cdot
f\;dx}{\big{\|}f\big{\|}^{2}_{\dot{H}^{1}}},\quad\Omega=\left\\{f\in\dot{H}^{1}_{rad},(f,W)_{\dot{H}^{1}}=(f,\widetilde{W})_{\dot{H}^{1}}=0\right\\}.$

Thus for some Lagrange multipliers $\lambda_{0},\lambda_{1}$, we have

$\displaystyle L_{+}f_{*}=\lambda_{0}\Delta W+\lambda_{1}\Delta\widetilde{W}.$

Note that $(W,\widetilde{W})_{\dot{H}^{1}}=0$ and $L_{+}(\widetilde{W})=0$, we
have

$\displaystyle
0=\int_{\mathbb{R}^{N}}f_{*}L_{+}(\widetilde{W})=\int_{\mathbb{R}^{N}}(L_{+}f_{*})\widetilde{W}=\lambda_{1}\big{\|}\widetilde{W}\big{\|}^{2}_{\dot{H}^{1}}\Longrightarrow\lambda_{1}=0.$

Thus

$\displaystyle L_{+}f_{*}=\lambda_{0}\Delta
W=-\lambda_{0}\big{(}|x|^{-4}*W^{2}\big{)}W=\frac{\lambda_{0}}{2}L_{+}W.$

By $\text{Null}(L_{+})=\text{span}\\{\widetilde{W}\\}$ in Lemma 2.12, there
exists $\mu_{1}$ such that

$\displaystyle f_{*}=\frac{\lambda_{0}}{2}W+\mu_{1}\widetilde{W}.$

Using $f_{*}\in H^{\bot}$, we get $\mu_{1}=0$ and $\lambda_{0}=0$, therefore,
we have

$\displaystyle f_{*}=0,\;\;\text{and}\;\;f_{n}\rightharpoonup
0\;\text{in}\;\dot{H}^{1}.$

Now, by compactness, we have

$\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\lambda}(x-y)W(y)^{p}W(x)^{p-2}|f_{n}(x)|^{2}\;dxdy\longrightarrow
0,$
$\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}I_{\lambda}(x-y)W(x)^{p-1}|f_{n}(x)|W(y)^{p-1}|f_{n}(y)|\;dxdy\longrightarrow
0.$

By $\Phi_{1}(f_{n})\rightarrow 0$ in (B.1), we get that

$\displaystyle\big{\|}\nabla f_{n}\big{\|}_{2}\rightarrow 0,$

which contradicts $\big{\|}f_{n}\big{\|}_{\dot{H}^{1}}=1$ in (B.1).

Using the same argument, we can show that there exists a constant $c$, such
that for any real-valued radial $\dot{H}^{1}$-function
$h_{2}\in\\{W\\}^{\bot}$, we have

$\Phi_{2}(h_{2})\geq c\big{\|}h_{2}\big{\|}^{2}_{\dot{H}^{1}}.$

This completes the proof. ∎

##  Appendix C Spectral properties of the linearized operator

In this Appendix, we follow the argument in [18, 19, 51] to give the proof of
Proposition 2.14.

###  C.1. Existence, symmetry of the eigenfunctions

Note that

$\overline{\mathcal{L}v}=-\mathcal{L}\overline{v},$

so that if $e_{0}>0$ is an eigenvalue of $\mathcal{L}$ with the eigenfunction
$\mathcal{Y}_{+}$, $-e_{0}$ is also an eigenvalue with eigenfunction
$\mathcal{Y}_{-}=\overline{\mathcal{Y}}_{+}$.

Now we show the existence of $\mathcal{Y}_{+}$. Let
$\mathcal{Y}_{1}=\mathrm{Re}\mathcal{Y}_{+},\mathcal{Y}_{2}=\mathrm{Im}\mathcal{Y}_{+}$,
it suffices to show that

$-L_{-}\mathcal{Y}_{2}=\;e_{0}\mathcal{Y}_{1},\quad
L_{+}\mathcal{Y}_{1}=\;e_{0}\mathcal{Y}_{2}.$ (C.1)

From the proof of the coercivity property of $\Phi$ on
$H^{\bot}\cap\dot{H}^{1}_{rad}$ in Proposition 2.13, we know that $L_{-}$ on
$L^{2}$ with domain $H^{2}$ is self-adjoint and nonnegative, By Theorem 3.35
in [34, Page 281], it has a unique square root $(L_{-})^{1/2}$ with domain
$H^{1}$.

Assume that there exists a function $f\in H^{4}_{rad}$ such that

$\displaystyle\mathcal{P}f=-e^{2}_{0}f_{1},\quad\mathcal{P}:=\big{(}L_{-}\big{)}^{1/2}\big{(}L_{+}\big{)}\big{(}L_{-}\big{)}^{1/2}.$

Then taking

$\displaystyle\mathcal{Y}_{1}:=\big{(}L_{-}\big{)}^{1/2}f,\quad\mathcal{Y}_{2}:=\frac{1}{e_{0}}\big{(}L_{+}\big{)}\big{(}L_{-}\big{)}^{1/2}f$

would yield a solution of (C.1), which implies the existence of the radial
$\mathcal{Y}_{\pm}$ by the rotation invariance of the operator $\mathcal{L}$.

It suffices to show that the operator $\mathcal{P}$ on $L^{2}$ with domain
$H^{4}_{rad}$ has a strictly negative eigenvalue. Since $\mathcal{P}$ is a
relatively compact, self-adjoint, perturbation of $\big{(}-\Delta\big{)}^{2}$,
then by the Weyl theorem [33, 34], we know that

$\displaystyle\sigma_{ess}(\mathcal{P})=[0,+\infty).$

Thus we only need to show that $\mathcal{P}$ has at least one negative
eigenvalue $-e^{2}_{0}$.

######  Lemma C.1.

$\displaystyle\sigma_{-}(\mathcal{P}):=\inf\\{\big{(}\mathcal{P}f,f\big{)}_{L^{2}},f\in
H^{4}_{rad},\big{\|}f\big{\|}_{L^{2}}=1\\}<0.$

###### Proof.

Note that

$\displaystyle\big{(}\mathcal{P}f,f\big{)}_{L^{2}}=\big{(}L_{+}F,F\big{)}_{L^{2}},\quad
F:=\big{(}L_{-}\big{)}^{1/2}f.$

Thus it suffices to find $F$ such that there exists $g\in H^{4}_{rad}$,
$F=(\Delta+V)g$ and

$\displaystyle\big{(}L_{+}F,F\big{)}_{L^{2}}<0,$ (C.2)

where $V:=(I_{\lambda}*W^{p})W^{p-2}$.

We divide into two cases.

First assume that N=3, 4, in these cases $W\notin L^{2}$, we will use the
localization method. Let $W_{a}:=\chi(x/a)W(x)$, where $\chi$ is a smooth,
radial positive function such that $\chi(r)=1$ for $r\leq 1$ and $\chi(r)=0$
for $r\geq 2$. We first claim

$\exists a>0,\;E_{a}:=\int L_{+}W_{a}W_{a}<0$ (C.3)

Recall that $-\Delta W=(I_{\lambda}*W^{p})W^{p-1}$. Thus

$\displaystyle L_{+}W_{a}=$
$\displaystyle-(p-2)(I_{\lambda}*W^{p})\chi(x/a)W^{p-1}-p\big{[}I_{\lambda}*(\chi(x/a)W^{p})\big{]}W^{p-1}$
$\displaystyle-\frac{2}{a}\nabla\chi(x/a)\nabla
W-\frac{1}{a^{2}}\Delta\chi(x/a)W.$

Hence

$\displaystyle\int L_{+}W_{a}W_{a}=$
$\displaystyle-(p-2)\int(I_{\lambda}*W^{p})\chi^{2}(x/a)W^{p}-p\int\big{[}I_{\lambda}*(\chi(x/a)W^{p})\big{]}W^{p}$
$\displaystyle-\frac{2}{a}\int\nabla\chi(x/a)\nabla
WW-\frac{1}{a^{2}}\int\Delta\chi(x/a)W^{2}.$

Let $A:=\frac{2}{a}\int\nabla\chi(x/a)\nabla
WW,\;B:=\frac{1}{a^{2}}\int\Delta\chi(x/a)W^{2}$. According to the explicit
expression (1.4) of W, $W\leq C|x|^{-(N-2)}$ and $\nabla W\leq C|x|^{-(N-1)}$
at infinity, which gives $|(A)|+|(B)|\leq\frac{C}{a}$ if $N=3$,
$|(A)|+|(B)|\leq\frac{C}{a^{2}}$ if $N=4$. Hence (C.3).

Let us fix $a$ such that (C.3) holds. Recall that W is not in $L^{2}$. Thus
$\Delta+V$ is a self-adjoint operator on $L^{2}$, with domain $H^{2}$, and
without eigenfunction. In particular the orthogonal of its range $R(\Delta+V)$
is $0$, and thus $R(\Delta+V)$ is dense in $L^{2}$. Let $\epsilon>0$, and
consider $G_{\epsilon}\in H^{2}$ such that

$\big{\|}(\Delta+V)G_{\epsilon}-(\Delta+V-1)W_{a}\big{\|}_{L^{2}}\leq\epsilon.$

Taking

$F_{\epsilon}:=\big{(}L_{-}+1\big{)}^{-1}L_{-}G_{\epsilon},$

we obtain
$\big{\|}(\Delta+V-1)(F_{\epsilon}-W_{a})\big{\|}_{L^{2}}\leq\epsilon$ which
implies
$\big{\|}F_{\epsilon}-W_{a}\big{\|}_{H^{2}}\leq\epsilon\big{\|}(\Delta+V-1)^{-1}\big{\|}_{L^{2}\to
L^{2}}$. Hence for some constant $C_{0}$, we have

$\displaystyle\left|\int_{\mathbb{R}^{N}}L_{+}F_{\epsilon}\cdot
F_{\epsilon}-\int_{\mathbb{R}^{N}}L_{+}W_{a}\cdot W_{a}\right|\leq
C_{0}\epsilon.$

As a consequence of (C.3), we get (C.2) for
$F=F_{\epsilon},\epsilon=\frac{-E_{a}}{2C_{0}}$, which shows the claim in the
case $N=3,4$.

Assume now that $N\geq 5$, so that $W\in L^{2}$ and more generally in all
space $H^{s}(\mathbb{R}^{N}).$ In this case

$\displaystyle\text{Ran}(L_{-})^{\bot}=\text{Null}(L_{-})=\text{span}\\{W\\},$

Thus

$\displaystyle\text{Ran}(L_{-})=\\{f\in L^{2},(f,W)_{L^{2}}=0\\}.$ (C.4)

Note that $L_{+}$ is a self-adjoint compact perturbation of $-\Delta$ and

$\displaystyle\big{(}L_{+}W,W\big{)}_{L^{2}}=-(2p-2)\int|\nabla W|^{2}\;dx<0,$

which implies that $L_{+}$ has a negative eigenvalue. Let $Z$ be the
eigenfunction for this eigenvalue (it is radial by the minimax principle).
Note that $L_{+}\widetilde{W}=0$, then for any real number $\alpha$, we have

$\displaystyle
E_{0}:=\int_{\mathbb{R}^{d}}L_{+}(Z+\alpha\widetilde{W})\cdot(Z+\alpha\widetilde{W})=\int_{\mathbb{R}^{d}}L_{+}Z\cdot
Z<0.$ (C.5)

Note that

$\big{(}\widetilde{W},W\big{)}_{L^{2}}\not=0,$

we can choose the real number $\alpha_{1}$ such that

$\displaystyle\big{(}Z+\alpha_{1}\widetilde{W},W\big{)}_{L^{2}}=0,$

which means that

$\displaystyle\left((L_{-}+1)(Z+\alpha_{1}\widetilde{W}),W\right)_{L^{2}}=$
$\displaystyle\left(Z+\alpha_{1}\widetilde{W},(L_{-}+1)W\right)_{L^{2}}$
$\displaystyle=$
$\displaystyle\big{(}Z+\alpha_{1}\widetilde{W},W\big{)}_{L^{2}}=0.$

By (C.4), for any $\epsilon>0$, there exists a function $G_{\epsilon}\in
H^{2}_{rad}$ such that

$\displaystyle\big{\|}L_{-}G_{\epsilon}-\big{(}L_{-}+1\big{)}(Z+\alpha_{1}\widetilde{W})\big{\|}_{L^{2}}<\epsilon.$

Taking

$F_{\epsilon}:=\big{(}L_{-}+1\big{)}^{-1}L_{-}G_{\epsilon},$

we obtain that

$\displaystyle\big{\|}\big{(}L_{-}+1\big{)}\big{(}F_{\epsilon}-(Z+\alpha_{1}\widetilde{W})\big{)}\big{\|}_{L^{2}}\leq\epsilon,$

which implies that

$\displaystyle\big{\|}F_{\epsilon}-(Z+\alpha_{1}\widetilde{W})\big{\|}_{H^{2}}\leq\epsilon\big{\|}\big{(}L_{-}+1\big{)}^{-1}\big{\|}_{L^{2}\rightarrow
L^{2}}.$

Hence for some constant $C_{0}$, we have

$\displaystyle\left|\int_{\mathbb{R}^{d}}L_{+}F_{\epsilon}\cdot
F_{\epsilon}-\int_{\mathbb{R}^{d}}L_{+}(Z+\alpha_{1}\widetilde{W})\cdot(Z+\alpha_{1}\widetilde{W})\right|\leq
C_{0}\epsilon.$

By (C.5), we have (C.2) for $F=F_{\epsilon},\epsilon=-\frac{E_{0}}{2C_{0}}$. ∎

###  C.2. Decay of the eigenfunctions at infinity

By the bootstrap argument, we know that $\mathcal{Y}_{\pm}\in C^{\infty}\cap
H^{\infty}$. In fact, we have $\mathcal{Y}_{\pm}\in\mathcal{S}$. By (C.1), it
suffices to show that $\mathcal{Y}_{1}\in\mathcal{S}$. Note that
$\mathcal{Y}_{1}$ satisfies that

$\displaystyle\left(e^{2}_{0}+\Delta^{2}\right)\mathcal{Y}_{1}=$
$\displaystyle-p\Delta\left(I_{\lambda}*(W^{p-1}\mathcal{Y}_{1})W^{p-1}\right)-p\left(I_{\lambda}*W^{p}\right)\left(I_{\lambda}*(W^{p-1}\mathcal{Y}_{1})\right)W^{2p-3}$
$\displaystyle-(p-1)\Delta\left((I_{\lambda}*W^{p})W^{p-2}\mathcal{Y}_{1}\right)-(p-1)\left((I_{\lambda}*W^{p})^{2}W^{2p-4}\mathcal{Y}_{1}\right)$
$\displaystyle-(I_{\lambda}*W^{p})W^{p-2}\Delta\mathcal{Y}_{1}.$

Thus,

$\displaystyle\left(e_{0}-\Delta\right)^{2}\mathcal{Y}_{1}=$
$\displaystyle-(p-1)\Delta\left((I_{\lambda}*W^{p})W^{p-2}\mathcal{Y}_{1}\right)-(p-1)\left((I_{\lambda}*W^{p})^{2}W^{2p-4}\mathcal{Y}_{1}\right)$
$\displaystyle-p\Delta\left(I_{\lambda}*(W^{p-1}\mathcal{Y}_{1})W^{p-1}\right)-p\left(I_{\lambda}*W^{p}\right)\left(I_{\lambda}*(W^{p-1}\mathcal{Y}_{1})\right)W^{2p-3}$
$\displaystyle-(I_{\lambda}*W^{p})W^{p-2}\Delta\mathcal{Y}_{1}-2e_{0}\Delta\mathcal{Y}_{1}.$

Because of the existence of the nonlocal interaction on the right hand side,
the decay estimate in [18] does not work. From the Bessel potential theory in
[62], we know that the integral kernel $G$ of the operator
$\left(e_{0}-\Delta\right)^{-2}$ is

$\displaystyle
G(x)=\frac{1}{(4\pi)^{2}}\int^{\infty}_{0}e^{-\frac{e_{0}}{4\pi}\delta}e^{-\frac{\pi}{\delta}|x|^{2}}\delta^{\frac{-N+4}{2}}\frac{d\delta}{\delta}.$

Hence we have

  1. (1)

$G(x)=\frac{|x|^{-N+4}}{\gamma(4)}+o(|x|^{-N+4}),\;|x|\rightarrow 0$;

  2. (2)

there exists $c>0$ such that

$G(x)=o(e^{-c|x|}),\;|x|\rightarrow+\infty.$

Then the conclusion follows by the analogue estimates in [25, 39].∎

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