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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 211, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/211\hfil Cross-constrained problems]
{Cross-constrained problems for nonlinear Schr\"odinger
equation with harmonic potential}

\author[R. Xu, C. Xu \hfil EJDE-2012/211\hfilneg]
{Runzhang Xu, Chuang Xu}  % in alphabetical order

\address{Runzhang Xu \newline
Department of Applied Mathematics,
Harbin Engineering University, 150001, China}
\email{xurunzh@yahoo.com.cn}

\address{Chuang Xu \newline
Department of Mathematiccal and Statistical Sciences,
University of Alberta, Edmonton T6G 2G1, Alberta, Canada \newline
Department of Mathematics, Harbin Institute of Technology, 150001, China}
\email{xuchuang6305@163.com}

\thanks{Submitted August 3, 2012. Published November 27, 2012.}
\subjclass[2000]{78A60, 35Q55}
\keywords{Cross-constrained problem; blow up; global existence;
\hfill\break\indent  invariant manifold;  harmonic potential}

\begin{abstract}
 This article studies a nonlinear Sch\"odinger equation with harmonic
 potential by constructing different cross-constrained problems. By
 comparing the different cross-constrained problems, we derive
 different sharp criterion and  different invariant manifolds that
 separate the global solutions and blowup solutions. Moreover, we
 conclude that some manifolds are empty due to the essence of the
 cross-constrained problems. Besides, we compare the three
 cross-constrained problems and the three depths of the potential
 wells. In this way, we explain the gaps in [J. Shu and J.
 Zhang,  Nonlinear Shr\"odinger equation with harmonic
 potential, Journal of Mathematical Physics, 47, 063503 (2006)],
 which was pointed out in  [R.  Xu and Y. Liu,  Remarks on nonlinear
 Schr\"odinger equation with harmonic potential, Journal of
 Mathematical Physics, 49, 043512 (2008)].
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks


\section{Introduction}\label{sec1}

In this paper, we study the following initial-value problem for
the  nonlinear Sch\"odinger equation with harmonic potential:
\begin{equation}\label{2}
\begin{gathered}
 i\varphi_t+\Delta \varphi-|x|^2\varphi+|\varphi|^{p-1}\varphi=0,\quad
 t>0,\; x\in\mathbb{R}^N, \\
 \varphi(0,x)=\varphi_0(x).
\end{gathered}
\end{equation}
Hereafter we will use the following notation:
$\varphi(x,t):\mathbb{R}^N\times[0,T_a)\to \mathbb{C}$ is a
complex valued wavefunction;
$0<T_a\leqslant+\infty$ is the maximal existence time;
 $N$ is the space dimension;
 $i=\sqrt{-1}$;  $\Delta $ is the Laplace operator on $\mathbb{R}^N$;
 $p$ is the exponent of the nonlinear function,
$\frac{4}{N}+1<p\leqslant\frac{N+2}{N-2}$;
$\| \cdot \|_{H^1}$ is the norm of $H^1(\mathbb{R}^N)$;
$\| \cdot \|_{L^p}$ is the norm of $L^p(\mathbb{R}^N)$;
$\int \cdot dx= \int_{\mathbb{R}^N} \cdot dx$;
$C$ is a positive constant that varies from expression to expression.

Note a more general form of \eqref{2} is
\begin{equation}\label{3}
\begin{gathered}
 i\varphi_t+\Delta \varphi-V(x)\varphi+|\varphi|^{p}\varphi=0,\quad
t>0,\; x\in\mathbb{R}^N, \\
\varphi(0,x)=\varphi_0(x).
\end{gathered}
\end{equation}
It is well-known that
\begin{equation}\label{1}
\begin{gathered}
i\varphi_t+\Delta \varphi+|\varphi|^{p-1}\varphi=0,\quad  t>0,\;
 x\in\mathbb{R}^N, \\
 \varphi(0,x)=\varphi_0(x),
\end{gathered}
\end{equation}
is one of the basic evolution models for nonlinear waves in various
branches of physics.
Many papers have studied equation \eqref{1}. In \cite{JG},  Ginibre
and  Velo established the local existence of the Cauchy problems in the
energy space $H^1(\mathbb{R}^N)$.  Glassey \cite{RT},  Tsutsumi
\cite{MT}, Ogawa and  Tsutsumi \cite{TY,TY2} proved that
for some initial data, especially for a class of sufficiently large
data, the solutions of the Cauchy problem for \eqref{1} blow up in
finite time. Strauss and  Cazenave also mentioned this topic in their
monographs \cite{W} and \cite{T} respectively. There are also
 many mathematicians who addressed these problems with
harmonic potential.  It is found that for sufficiently small
initial data, the solutions of the Cauchy problem for \eqref{1}
globally exist (cf. \cite{JG2,NY,CGL,JT,NKM}, etc).
Zhang \cite{J1} studied the global existence of \eqref{1} and the
relationship
between the Schr\"odinger equation and its ground state.
 For \eqref{3},  Fujiwara \cite{F}
proved the smoothness of Schr\"odinger kernel for potentials of
quadratic growth. It is shown that quadratic
potentials are the highest order potentials for local well-posedness
of the equation  \cite{O}.  Yajima \cite{Y} showed that for super-quadratic
potentials, the Schr\"odinger kernel is nowhere $C^1$.

When $p>1+4/N$,  Cazenave \cite{T},  Tsurumi and  Wadati \cite{TT3} and
Carles \cite{RC1, RC2} showed that the solutions of the
Cauchy problem of \eqref{2} blow up in  finite time for some initial
data, especially for a class of sufficiently large initial data; while
the solutions of the Cauchy problem of \eqref{2} globally exist for
other initial data, especially for a class of sufficiently small
initial data, see \cite{RC1}, \cite{RC2} and \cite{TT3}. When
$1<p<1+4/N$,  Zhang \cite{J2} proved that global solutions of the
Cauchy problem of \eqref{2} exist for any initial data in the energy
space. When $p=1+4/N$,  Zhang \cite{J3} showed that there exists a
sharp condition of the global existence. In \cite{GJ},  Chen
and  Zhang derived  a global existence condition for the
supercritical case for \eqref{2}. Moreover,  Shu and  Zhang
\cite{JJ} also studied \eqref{2} for its global existence and
blowup.

 Shu and  Zhang \cite{JJ} studied \eqref{2} by
constructing a cross-constrained problem, which  originated from
\cite{J4}. The main idea of the cross-constrained
variational method introduced in \cite{J4} can be described as
follows. In the energy functional, there are more than two terms,
like $\int|\nabla\varphi|^2 dx$, $\int |x|^2|\varphi|^2 dx$ and
$\int|\varphi|^{p+1} dx$ for problem \eqref{2}.
It is well-known that the ``nonlinear
source'' is controlled by the ``potential energy'',
 using the variational method. If the
``potential energy'' is not as simple as being composed of just one
term, then one can give some various combinations of the terms and  consider
different cases of these combinations. For instance, for problem
\eqref{2}, we can  use various combinations of the terms
$\int|\nabla\varphi|^2 dx$, $\int |x|^2|\varphi|^2 dx$ and
$\int|\varphi|^2 dx$ to control the nonlinear term
$\int|\varphi|^{p+1} dx$. Then we define the corresponding Nehari
functional and potential energy functional to construct the variational
problem, which is the so-called cross-constrained potential well
method. This approach seems to work in the sense of finding the
relationships between these different terms or functionals.
But sometimes it may also arouse some
confusion because of the complex structure.
It may explain the occurrence of some self-contradiction criteria in
\cite{JJ}. Although
Xu and  Liu  pointed out the self-contradiction in \cite{X1},
they still never make a clear statement about the relationships
among these different so-called cross-constrained problems. In other
words,   Xu and  Liu just found the problem, but they
did not clarify the essence behind it.  So in
this paper we mainly aim at a comprehensive study of the
so-called cross-constrained problems, and finding the relations between
the cross-constrained functionals and cross-constrained manifolds.
Then occurrence of all of the gaps and problems mentioned above can
be well explained. In the end, we simply illustrate the spatial structure by three concentric spheres.

In this paper, we study the Cauchy problem of \eqref{2} by
constructing different cross-constrained problems and therefore
derive different sharp criteria for both  global existence and
blowup. Moreover, we compare three different invariant manifolds
defined in order to separate the global solutions and the blowup
solutions of the Cauchy problem \eqref{2}. We also dig the reason
that some invariant manifolds are empty, which was previously pointed out in
\cite{X1}.

The organization of this paper is as follows. In Section 2,
we give some concerned preliminaries. In Section 3, we construct
three variational problems and invariant manifolds. In Section 4, we
derive sharp criteria for both global existence and blowup. In
Section 5, we compute the potential depth of one of the potential
wells. In Section 6, we compare the three variational problems,
point out some of the invariant manifolds are empty and explain why
such phenomenon happens. Further we reveal the relationships
among these different cross-constrained variational problems and the
different manifolds.

\section{Preliminaries}

In this section, we like to  introduce some functionals and a Hilbert
space, which will be used to construct
different cross-constrained  problems. For \eqref{2}, we first equip
the following space
\begin{equation}\label{4}
H=\big\{\psi \in H^1(\mathbb{R}^N): \int |x|^2|\psi|^2
dx<\infty\big\}
\end{equation} with the inner product


\begin{equation}\label{5}
\langle\psi, \phi\rangle :=\int \nabla \psi \nabla
\overline{\phi}+\psi\overline{\phi}+|x|^2\psi\overline{\phi} dx,
\end{equation}
whose associated norm is $\|\cdot\|_H$.

Further we  define the energy functional
\begin{equation}\label{6}E(\varphi)=\int \frac{1}{2}|\nabla
\varphi|^2+\frac{1}{2}|x|^2|\varphi|^2-\frac{1}{p+1}|\varphi|^{p+1}
dx,\end{equation}
and the following four auxiliary functionals
\begin{gather}\label{7}
P(\varphi)=\int  \frac{1}{2}|\nabla
\varphi|^2+\frac{1}{2}|\varphi|^2+\frac{1}{2}|x|^2|\varphi|^2
-\frac{1}{p+1}|\varphi|^{p+1} dx,
\\
\label{8}
I_1(\varphi)=\int |\nabla
\varphi|^2+|\varphi|^2+|x|^2|\varphi|^2-\frac{N(p-1)}{2(p+1)}|\varphi|^{p+1}
dx,
\\
\label{9}
I_2(\varphi)=\int |\nabla
\varphi|^2+|\varphi|^2-\frac{N(p-1)}{2(p+1)}|\varphi|^{p+1}dx,
\\
\label{10} I_3(\varphi)=\int |\nabla
\varphi|^2+|x|^2|\varphi|^2-\frac{N(p-1)}{2(p+1)}|\varphi|^{p+1}
dx.
\end{gather}
In the above four functionals, $P(\varphi)$ is
composed of both energy and mass. And $I_i(\varphi)$ $(i=1,2,3)$ can
be considered as Nehari functionals. Throughout this paper, we assume
\begin{equation} \label{eH}
\begin{gathered}
 1+\frac{4}{N}<p<\frac{N+2}{N-2},\quad \text{for } N\geqslant3;\\
 1+\frac{4}{N}<p<+\infty,\quad \text{for } N=1,2.
\end{gathered}
\end{equation}
We now state the local well-posedness.

\begin{lemma}[\cite{TK}]\label{le1}
Let $\varphi_0\in H$. Then there exists a unique solution $\varphi$
of the Cauchy problem \eqref{2} in $C([0,T]; H)$ for some 
$T\in (0,\infty]$ (maximal existence time), and either $T=\infty$ (global
existence) or else $T<\infty$ and
 $$
\lim_{t\to T}\|\varphi\|_H=\infty\quad \text{(blowup)}.
$$
\end{lemma}

Now we have the conservation laws for both energy and mass.

\begin{lemma}[\cite{TC,RT,YJ}] \label{le2}
Let $\varphi_0\in H$ and $\varphi$ be a solution of the Cauchy
problem \eqref{2} in $C([0,T]; H)$.  Then one has
\begin{gather}\label{11}
\int |\varphi|^2 dx=\int |\varphi_0|^2 dx,\\ \label{12} 
E(\varphi)\equiv E(\varphi_0),\\ \label{13} 
P(\varphi)\equiv P(\varphi_0).
\end{gather}
\end{lemma}

We introduce the following lemma, which will be used for proving the
blowup phenomenon in Section 4.

\begin{lemma}\label{le3}
Let $\varphi_0\in H$ and $\varphi$ be a
solution of the Cauchy problem \eqref{2} in $C([0,T]; H)$, Set
$J(t)=\int |x|^2|\varphi|^2 dx$. Then one has
\begin{equation}\label{14}
J''(t)=8\int\Big(|\nabla
\varphi|^2-|x|^2|\varphi|^2-\frac{N(p-1)}{2(p+1)}|\varphi|^{p+1}\Big)dx.
\end{equation}
\end{lemma}

\section{Three variational problems and invariant manifolds}

First we define the following  three Nehari manifolds,
\begin{gather*}
M_1 :=\{\psi\in H\backslash\{0\}: I_1(\psi)=0 \},\\
M_2 :=\{\psi\in H\backslash\{0\}: I_2(\psi)=0 \},\\
M_3 :=\{\psi\in H\backslash\{0\}: I_3(\psi)=0 \}.
\end{gather*}
Now we consider the following cross-constrained problems
 \begin{gather}\label{15}
d_1=\inf_{\psi\in M_1} P(\psi),\\ \label{16}
d_2=\inf_{\psi\in M_2} P(\psi),\\ \label{17}
d_3=\inf_{\psi\in M_3} P(\psi),
\end{gather}
respectively.
First we have the following lemma.

\begin{lemma}\label{le4} 
$d_i>0$ for $i=1,2,3$.
\end{lemma}

\begin{proof}
(i) For any $\varphi\in M_1\cup M_2$, we have
$$\int|\nabla\varphi|^2+|\varphi|^2 dx\leqslant\frac{N(p-1)}{2(p+1)}\int|\varphi|^{p+1}dx.$$
By Sobolev embedding inequality, this implies
$$
\int|\nabla\varphi|^2+|\varphi|^2 dx\geqslant C.
$$
Note by assumption \eqref{eH},
$$
\frac{1}{2}-\frac{1}{p+1}\cdot\frac{2(p+1)}{N(p-1)}>0.
$$ 
Hence
$P(\varphi)\geqslant C>0$, which verifies $d_i>0$ ($i=1,2$).

(ii) For $\varphi\in M_3$, we have 
$$
\|\nabla\varphi\|_2^2\leqslant\frac{N(p-1)}{2(p+1)}\int |\varphi|^pdx,
$$
which implies from Gagliardo-Nirenberg inequality and Cauchy-Schwartz 
inequality that there exists a constant
$C(p,N)>0$ such that 
\begin{align*}
  C(p,N)&\leqslant \|\nabla\varphi\|_2^{\frac{Np-(N+4)}{2}}
 \cdot\|\varphi\|_2^{\frac{(N+2)-(N-2)p}{2}}\\ 
&\leqslant \frac{1}{2}\Big(\|\nabla\varphi\|_2^{Np-(N+4)}
 +\|\varphi\|_2^{(N+2)-(N-2)p}\Big).
\end{align*}
This yields
$\|\nabla\varphi\|_2\geqslant C>0$ or 
$\|\varphi\|_2\geqslant C>0$.
Thus 
$$P(\varphi)=\frac{1}{2}\|\varphi\|_2^2+\frac{Np-(N+4)}{2N(p-1)}
\Big[\|\nabla\varphi\|_2+\int|x|^2|\varphi|^2dx\Big]\geqslant C>0,
$$
which proves $d_3>0$.
\end{proof}

Next we give the invariance of some manifolds.

\begin{theorem}\label{th1} For $i=1,2,3$, define
\begin{equation}\label{29} 
\mathcal{G}_i :=\{\psi\in H: P(\psi)<d_i,\,
I_i(\psi)>0\}\cup\{0\}
\end{equation}
Then $\mathcal{G}_i$ is an invariant manifold
of \eqref{2}; that is, if $\varphi_0\in \mathcal{G}_i$, then the
solution $\varphi(x, t)$ of the Cauchy problem \eqref{2} also
satisfies $\varphi(x, t)\in \mathcal{G}_i$ for any $t\in [0, T)$.
\end{theorem}

\begin{proof}
If $\varphi_0=0$, from the mass conservation law; i.e., \eqref{11},
we can find that $\varphi=0$ for $t\in[0, T)$; i.e.,
 $\varphi(x, t)\in \mathcal {G}_i$. 
If $\varphi_0\neq 0$,  we have $\varphi_0\in
\mathcal {G}_i\backslash\{0\}$; i.e., $P(\varphi_0)<d_i$ and
$I_i(\varphi_0)>0$. 
By Lemma \ref{le1}, there exists a unique
$\varphi(x, t)\in C([0, T); H)$ with $0<T\leqslant\infty$ such that
$\varphi(x, t)$ is a solution of problem \eqref{2}. Now we shall
show that $\varphi(x, t)\in \mathcal {G}_i$ for any $t\in [0, T)$.
By \eqref{13}, we have
\begin{equation}\label{1-12}
 P(\varphi(x, t))=P(\varphi_0)\geqslant d_i.
\end{equation}

Next we show $I_i(\varphi)>0$ for $t\in[0, T)$. Note that
$I_i(\varphi_0)>0$. Arguing by contradiction, by the
continuity of $I_i(\varphi)$, suppose that  there were a $t_2\in[0, T)$
such that $I_i(\varphi(x ,t_2 ))=0$. If $\varphi(x,t_2)=0$, then by
\eqref{11}, we have $0=\int|\varphi(x,t_2)|^2 dx=\int|\varphi_0|^2
dx$, which indicates $\varphi_0=0$. Contradiction.
So $\varphi(x,t_2)\neq0$, by the definition of
$d_i$, we have $P(\varphi(x,t_2))\geqslant d_i$, which contradicts
\eqref{1-12}. Therefore $I_i(\varphi)>0$ for all $t\in[0, T)$.

Combining all of the analysis above, we arrive at  
$\varphi(x,t)\in \mathcal {G}_i$ for any $t\in [0, T)$. The proof is complete.
\end{proof}

By a similar argument, we can obtain the following result.

\begin{theorem}\label{th2}
For $i=1,2,3$, define 
\begin{equation*}%\label{32} 
\mathcal{B}_i :=\{\psi\in H: P(\psi)<d_i,\, I_i(\psi)<0\}
\end{equation*} 
Then $\mathcal{B}_i$ is an invariant manifold of \eqref{2}.
\end{theorem}

\section{Sharp conditions for global existence}

\begin{theorem}\label{th3-0}
If $\varphi_0\in \mathcal{G}_i$ ($i=1,2,3$), then the solution
$\varphi(x, t)$ of the Cauchy problem \eqref{2} globally exists on
$t\in [0, \infty)$.
\end{theorem}

\begin{proof}
Here we  prove only the case $\varphi_0\neq0$, for $\varphi_0=0$ 
is a trivial case. For any nontrivial $\varphi_0\in\mathcal{G}_i$ ($i=1,2,3$),
let $\varphi_i(x, t)$ be the solution of the Cauchy problem
\eqref{2} with initial condition $\varphi_i(x,0)=\varphi_0$, 
and $0<T\leqslant\infty$ be the maximal existence time. 
It follows from Theorem \ref{th1} that $\varphi_i(x,t)\in\mathcal{G}_i$ 
($i=1,2,3$) for all $t\in[0,T)$. Fix $t\in[0,T)$, and simply 
denote $\varphi_i(x,t)$ by $\varphi_i$, then the definition 
of $\mathcal{G}_i$ implies that 
$$
d_i>P(\varphi_i),\ I_i(\varphi_i)>0\ (i=1,2,3).
$$
For $i=1,2,3$, it always follows from $I_i(\varphi_i)>0$ that 
$$
\frac{1}{p+1}|\varphi_i|^{p+1} dx<\frac{2}{N(p-1)}\int |\nabla
\varphi_i|^2+|\varphi_i|^2+|x|^2|\varphi_i|^2 dx
$$
Thus we obtain 
\begin{equation}\label{33}
\begin{split}
d_i&>P(\varphi_i)\\
&=\int \frac{1}{2}|\nabla
\varphi_i|^2+\frac{1}{2}|\varphi_i|^2+\frac{1}{2}|x|^2|\varphi_i|^2
-\frac{1}{p+1}|\varphi_i|^{p+1} dx\\ 
&> \Big(\frac{1}{2}-\frac{2}{N(p-1)}\Big)\int |\nabla
\varphi_i|^2+|\varphi_i|^2+|x|^2|\varphi_i|^2 dx,
\end{split}
\end{equation} 
which yields
$$
\int |\nabla \varphi_i|^2+|\varphi_i|^2+|x|^2|\varphi_i|^2
dx<\frac{2N(p-1)d_i}{N(p-1)-4}.
$$ 
Therefore, it follows from Lemma \ref{le1}
that $\varphi$  globally exists on $t\in [0, \infty)$.
 At this point, we proved this theorem.
\end{proof}

\begin{theorem}\label{th3}
If $\varphi_0\in \mathcal {B}_i\ (i=1,2,3)$, then the solution
$\varphi(x, t)$ of the Cauchy problem \eqref{1} blows up in  finite
time.
\end{theorem}

\begin{proof}
We prove this theorem case by case.

\noindent\textbf{Case I}: 
$\varphi_0\in \mathcal {B}_1\cup\mathcal {B}_2$.
In this case, Theorem \ref{th1} implies that the solution
$\varphi(x, t)$ of the Cauchy problem \eqref{1} satisfies that
$\varphi(x, t)\in \mathcal {B}_1\cup\mathcal {B}_2$ for $t\in [0, T)$.
 For $J(t)=\int
|x|^2|\varphi|^2 dx$, the definitions of $P(\varphi)$ and 
$I_i(\varphi)$ $(i=1,2)$ imply that
\begin{equation}\label{39}
J''(t)<-8\int|\varphi|^2 dx.
\end{equation}
Then 
$$
J'(t)<J'(0)-8\Big(\int|\varphi|^2 dx\Big)\ t.
$$
 Further we have
$$
J(t)<J(0)+J'(0)t-4\Big(\int|\varphi|^2 dx\Big) t^2.
$$
Note that $I_1(\varphi)<0$ yields
$\int|\varphi|^2 dx>0$. Therefore there exists a $T_1\in (0,\infty)$
such that $J(t)>0$ for $t\in[0,T_1)$ and $J(T_1)=0$. By the
inequality (see \cite{YJ})
$$
\|\varphi\|^2\leqslant\frac{2}{N}\|\nabla\varphi\|\cdot\|x\varphi\|
$$
we obtain $\lim_{t\to T_1}\|\nabla\varphi\|=\infty$, which indicates
$$
\underset{t\to T_1}{\lim}\|\varphi\|_H=\infty.
$$

\noindent\textbf{Case ii}: $\varphi_0\in \mathcal {B}_3$.
In this case,  Theorem \ref{th1} implies that the solution
$\varphi(x, t)$ of the Cauchy problem \eqref{1} satisfies that
$\varphi(x, t)\in \mathcal {B}_3$ for $t\in [0, T)$. 
 For $J(t)=\int |x|^2|\varphi|^2 dx$, \eqref{7} and \eqref{10} imply that
\begin{equation}\label{40}
J''(t)<-16\int|x|^2|\varphi|^2 dx.
\end{equation}

Now we show that there exists a $T_1\in (0,\infty)$ such that
$J(t)>0$ for $t\in[0,T_1)$ and $J(T_1)=0$. Arguing by contradiction,
suppose $\forall t\in[0,\infty)$, $J(t)>0$. Set
$$
g(t)=\frac{J'(t)}{J(t)}.
$$ 
It is easy to show that
\begin{equation}\label{21}
g'(t)=\frac{J''(t)}{J(t)}-\Big(\frac{J'(t)}{J(t)}\Big)^2<-16-g^2(t).
\end{equation}
Next we like to show $g(t)\neq0$ for any $t\in[0,\infty)$. Arguing
by contradiction again, suppose there is a $t_0$ such that
$g(t_0)=0$. By \eqref{21}, we have $g(t)<0$ for $t\in(t_0,\infty)$.
For  any fixed $t_1>t_0$, dividing \eqref{21} by $g^2(t)$, we have
$$
\frac{g'(t)}{g^2(t)}<-\frac{16}{g^2(t)}-1<-1.
$$ 
Further we derive
$$
\int_{t_1}^{t}\frac{g'(\tau)}{g^2(\tau)} d\tau<\int_{t_1}^{t}-1
d\tau,
$$ 
namely, 
\begin{equation}\label{111}
\frac{1}{g(t)}>\frac{1}{g(t_1)}+(t-t_1),
\end{equation} 
which indicates that there exists a $t_2>t_1$ such that
\begin{equation}\label{41}
g(t)>0 \quad \text{for any } t\in(t_2,\infty)
\end{equation} 
This contradicts $g(t)<0$ for $t\in(t_0,\infty)$. 
Hence we have $g(t)\neq0$ for any
$t\in[0,\infty)$. By \eqref{111}, for $t\in(0,\infty)$ , we have
$$
\frac{1}{g(t)}>\frac{1}{g(0)}+t.
$$  
Hence, $J'(t)>0$ for
$t\in(|\frac{1}{g(0)}|,\infty)$. Therefore $J(t)$ is increasing in
$(|\frac{1}{g(0)}|,\infty)$. Let $t_0=|\frac{1}{g(0)}|$. 
By 
$$
J''(t)<-16J(t)<0,
$$  
we have for $t>t_0$,
$$
J'(t)<J'(t_0)+16J(t_0)t_0-16J(t_0)t.
$$
Further
$$
J(t)-J(0)<(J'(t_0)+16J(t_0)t_0)t-8J(t_0)t^2;
$$ 
i.e.,
$$
J(t)<J(0)+(J'(t_0)+16J(t_0)t_0)t-8J(t_0)t^2,
$$
which implies that there exists $0<T_1\leqslant t_0$ such that
 $J(t)>0$ for $t\in[0,T_1)$ and
$J(T_1)=0$. Again by the inequality (see \cite{YJ})
$$
\|\varphi\|^2\leqslant\frac{2}{N}\|\nabla\varphi\|\cdot\|x\varphi\|
$$
we obtain 
$$\lim_{t\to T_1} \|\nabla\varphi\|=\infty.
$$

So far we have shown that for the initial data
 $\varphi\in\mathcal {B}_i $, the solution of the Cauchy problem \eqref{2} 
blows up in finite time. This completes the proof of the theorem.
\end{proof}

\begin{remark} \rm
It is clear that 
$$
\{\psi\in H, P(\psi)<d_i\}=\mathcal {G}_i\cup \mathcal {B}_i,
$$ 
which indicates Theorem \ref{th3-0} and Theorem \ref{th3} are
sharp.
\end{remark}


By Theorem \ref{th2}, we obtain another condition for global existence of
the solution of \eqref{2}.

\begin{corollary}
If $\varphi_0$ satisfies $\|\varphi_0\|_H^2<2d_i$, then the solution
$\varphi$ of the Cauchy problem \eqref{2} globally exists on
$t\in[0, \infty)$.
\end{corollary}

\begin{proof}
We  consider only the nontrivial case. Suppose $\varphi_0\neq0$, from
$\|\varphi_0\|_H^2<2d_i$, we have $P(\varphi_0)<d_i$. Moreover, we
claim that $I_i(\varphi_0)>0$. Otherwise, there is a
$0<\mu\leqslant1$ such that $I_i(\mu\varphi_0)=0$. Thus
$P(\mu\varphi_0)\geqslant d_i$. On the other hand,
$$
\|\mu\varphi_0\|^2_H=\mu^2\|\varphi_0\|^2_H<2\mu^2d_i<2d_i.
$$ 
It follows that $P(\mu\varphi_0)<d_i$. This is a contradiction.
Therefore we have $\varphi_0\in \mathcal {G}_i$. Thus Theorem
\ref{th2} implies this corollary.
\end{proof}

\section{Computing  $d_1$}

In this section, we compute $d_1$ using the method definded by 
Payne and  Sattinger \cite{L}.
Since $M_1$ is a closed nonempty set, there exists an $\omega_1\in M_1$ 
such that 
$$
P(\omega_1)=\underset{\psi\in M_1}{\inf}P(\psi)=d_1,
$$ 
where $\omega_1$ is a solution of the following  Euler equation
\begin{equation*}
\Delta\omega-|x|^2\omega+|\omega|^{p-1}\omega-\omega=0.
\end{equation*}
We define
\begin{align*}
C_{p,1}&=\frac{\big(\int|\nabla\omega_1|^2+|\omega_1|^2+|x|^2|\omega_1|^2
dx\big)^{1/2}}{\big(\int|\omega_1|^{p+1}dx\big)^{\frac{1}{p+1}}}\\
&=\inf_{\psi\in M_1}\frac{\big(\int |\nabla\psi|^2+|\psi|^2+|x|^2|\psi|^2
dx\big)^{1/2}}
{\big(\int|\psi|^{p+1}dx\big)^{\frac{1}{p+1}}}
\end{align*}
be the Sobolev constant from 
$H$ to $L^{p+1}(\mathbb{R}^N)$. By $I_1(\omega_1)=0$, we
derive
\begin{equation}\label{42}
d_1=P(\omega_1)=\frac{N(p-1)-4}{4(p+1)}\int |\omega_1|^{p+1}dx.
\end{equation}
Using the definition of $C_{p,1}$, we obtain
\begin{equation}\label{43}
\Big(\int|\omega_1|^{p+1} dx\Big)^{\frac{1}{p+1}}
=\Big(\frac{2(p+1)C_{p,1}^2}{N(p-1)}\Big)^{\frac{1}{p-1}}.
\end{equation}
Combining \eqref{42} and \eqref{43}, we have
\begin{equation}\label{44}
d_1=\frac{N(p-1)-4}{4(p+1)}\Big(\frac{2(p+1)C_{p,1}^2}{N(p-1)}\Big)
^{\frac{p+1}{p-1}}.
\end{equation}

\section{Cross-constrained problems and some trivial manifolds}

In this section, we like to address the relations of  some
cross-constrained various manifolds for problem
\eqref{2}. 

Next we define three manifolds as follows.
First, we define some new potential well depths.
 For $i<j$, $i,j=1,2,3$,
\begin{equation}\label{45}
d_{i,j}=\min\{d_i,d_j\}
\end{equation}
and
\begin{equation}\label{47}
d_{1,2,3}=\min\{d_1,d_2,d_3\}.
\end{equation}
Second, we define the following manifolds. Note similar as the proof of
Theorem \ref{th1}, it is trivial to show that these manifolds are 
also invariant. For $i<j$, $i,j=1,2,3$,
\begin{gather}\label{48}
\mathcal {G}_{i,j}: =\{\psi\in H: P(\psi)<d_{i,j}, I_i(\psi)>0,
I_j(\psi)>0 \}\cup\{0\}, \\ \label{49}
\mathcal {V}_{+i,-j}: =\{\psi\in H: P(\psi)<d_{i,j},
I_i(\psi)>0, I_j(\psi)<0 \}, \\ \label{50}
\mathcal {V}_{-i,+j}: =\{\psi\in H: P(\psi)<d_{i,j},
I_i(\psi)<0, I_j(\psi)>0 \}, \\ \label{51}
\mathcal {B}_{i,j}: =\{\psi\in H: P(\psi)<d_{i,j}, I_i(\psi)<0,
I_j(\psi)<0 \}.
\end{gather}
We have the following theorems to clarify the relations
among all the invariant manifolds.

\begin{theorem}\label{th5}
$\mathcal {G}_{i,j}=\mathcal {G}_i\cap\mathcal {G}_j$; 
$\mathcal {V}_{+i,-j}=\emptyset$; $\mathcal {V}_{-i,+j}=\emptyset $; 
$\mathcal {B}_{i,j}=\mathcal {B}_i\cap\mathcal {B}_j$, where $i<j$ and
$i,j=1,2,3$.
\end{theorem}

\begin{theorem}\label{th6} 
For $i<j$ and $i,j=1,2,3$, define
\begin{gather*}
\mathcal {G}_{ij}: =\{\psi\in H: P(\psi)<d_{i,j},I_j(\psi)>0\},\\
\mathcal {B}_{ij}: =\{\psi\in H: P(\psi)<d_{i,j},I_j(\psi)<0\}.
\end{gather*}
Then 
$$
\mathcal {G}_{1i}\subset\mathcal {G}_{i1},\quad
\mathcal {B}_{i1}\subset\mathcal {B}_{1i}.
$$
\end{theorem}


\begin{remark} \rm
Theorem \ref{th5} can well explain the gaps in \cite{JJ}, which was
pointed out in \cite{X1}.
\end{remark}

\begin{remark} \rm
It is easy to see that the reason for $\mathcal {V}_{-1,+i}=\emptyset$
($i=2,3$) is due to the fact $I_1(\varphi)>I_i$ ($i=2,3$) for
$\varphi\neq0$. Now we like to analyze why $\mathcal
{V}_{+2,-3}=\emptyset$, $\mathcal {V}_{-2,+3}=\emptyset$, $\mathcal
{V}_{+1,-i}=\emptyset$  for $i=2,3$. In fact, we know that $\mathcal
{V}_{+1,-2}$ is a subset of both $\mathcal {G}_1$ and $\mathcal
{B}_2$. We have proved that $\mathcal{G}_1$ is a manifold of all the
global solutions of the Cauchy problem \eqref{2} for
$P(\varphi)<d_1$ while $\mathcal{B}_2$ is a manifold of all the
blowup solutions of the Cauchy problem \eqref{2} for
$P(\varphi)<d_2$. Hence by Lemma \ref{le1}, there should be no
intersection of the two manifolds, which indicates $\mathcal
{V}_{+1,-2}=\emptyset$.  Hence it is natural to deduce that the two
surfaces
\[
S_{12,1}: =\{\psi\in H:P(\psi)<d_{1,2}, I_1(\psi)=0 \}
\]
and
\[
S_{12,2}: =\{\psi\in H:P(\psi)<d_{1,2}, I_2(\psi)=0 \}
\]
coincide.
Similarly,
\[
S_{13,1}: =\{\psi\in H:  P(\psi)<d_{1,3}, I_1(\psi)=0\}
\]
and
\[
S_{13,3}: =\{\psi\in H: P(\psi)<d_{1,3}, I_3(\psi)=0\}
\]
coincide. Further we define the following three Nehari
manifolds
$$
S_i: =\{\psi\in H: P(\psi)<d_{1,2,3}, I_i(\psi)=0 \},
$$
for $i=1,2,3$.
It is easy to see that $S_1=S_2=S_3$.
\end{remark}

To have an intuitive feeling of the relations among some of the
above manifolds, we draw the Figures \ref{fig1} and \ref{fig2}.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=.8\textwidth]{fig1.jpg}
\end{center}
\caption{The relations among $I_1(\varphi)$, $I_2(\varphi)$ and
$I_3(\varphi)$, where $d_1>d_2>d_3$; The intersections of the three
spheres and red surface represent the three manifolds
$I_i(\varphi)=0$ ($i=1,2,3$), respectively.}
\label{fig1}
\end{figure}


\begin{figure}[ht]
\begin{center}
\begin{overpic}[scale=0.42,tics=20]{fig2}
\put(110,106){\makebox(0,0)[cc]{\small$I_3(\varphi)=0$}}
\put(88.62,120){\makebox(0,0)[cc]{\small$I_2(\varphi)=0$}}
\put(48.72,106){\makebox(0,0)[cc]{\small$I_1(\varphi)=0$}}
\put(145.74,110){\makebox(0,0)[cc]{\small$O$}}
\put(140.5,94.92){\makebox(0,0)[cc]{\small$I_3(\varphi)<0$}}
\put(140.28,67.2){\makebox(0,0)[cc]{\small$I_2(\varphi)<0$}}
\put(140.98,32.34){\makebox(0,0)[cc]{\small$I_1(\varphi)<0$}}
\put(140,120.26){\makebox(0,0)[cc]{\small$I_3(\varphi)>0$}}
\put(140,148.26){\makebox(0,0)[cc]{\small$I_2(\varphi)>0$}}
\put(140,184.8){\makebox(0,0)[cc]{\small$I_1(\varphi)>0$}}
\end{overpic}
\end{center}
\caption{Cross section for Figure 1}
\label{fig2}
\end{figure}


\subsection*{Acknowledgements}

This work is supported by grants:
11101102,  11101104,  11031002
from the National Natural Science Foundation of China; 
20102304120022 from the Ph. D. Programs Foundation of Ministry of Education 
of China;
1252G020  from the Support Plan for
the Young College Academic Backbone of Heilongjiang Province;
A201014 from the Natural Science Foundation of Heilongjiang Province;
12521401 from the Science and Technology Research Project of Department 
of Education  of Heilongjiang Province;
and from Foundational Science Foundation of Harbin Engineering University
 and Fundamental  Research Funds for the Central Universities.

The second author was supported by China Scholarship Council
 and University of Alberta Doctoral Recruitment Scholarship.

The authors want to thank the anonymous referee for his/her valuable 
suggestions which greatly improve the presentation of this article.
The authors also thank  Yufeng Wang and Dr. Yanyou Chai for their 
kind help.

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