Blow-up Solutions to Nonlinear Schrödinger Equation with a Potential

Abstract

This is a sequel to the paper “Characterization of the ground state to the intercritical NLS with a linear potential by the virial functional” by the same authors. We continue to study the Cauchy problem for a nonlinear Schrödinger equation with a potential. In the previous chapter, we investigated some minimization problems and showed global existence of solutions to the equation with initial data, whose action is less than the value of minimization problems and positive virial functional. In particular, we saw that such solutions are bounded. In this chapter, we deal with solutions to the equation with initial data, whose virial functional is negative contrary to the previous paper and show that such solutions are unbounded.

1. Introduction

In this chapter, we consider the Cauchy problem of the following nonlinear Schrödinger equation with a linear potential:

$$i{\partial }_{t}u+{\mathrm{\Delta }}_{V}u=-{\left|u\right|}^{p-1}u,\left(t,x\right)\in \mathbb{R}\times {\mathbb{R}}^d,$$E1

where $d\ge 1$, $1<p<2^{\ast }-1$,

$$2^{\ast }≔\left\{\begin{array}{ll}\infty & \text{if}d\in \left\{1,2\right\},\\ \frac{2d}{d-2}& \text{if}d\ge 3,\end{array}\right.$$E2

and ${\mathrm{\Delta }}_V≔\mathrm{\Delta }-V={\sum }_{j=1}^d\frac{{\partial }^2}{\partial x_j^2}-V$. In particular, we consider the Cauchy problem of Eq. (1) with initial condition

$$u\left(0,\cdot \right)=u_0\in H^1\left({\mathbb{R}}^d\right).$$E3

Eq. (1) with $V\in L^{\infty }\left({\mathbb{R}}^d\right)$ is a model proposed to describe the local dynamics at a nucleation site (see [1]).

Eq. (1) is locally well-posed in the energy space $H^1\left({\mathbb{R}}^d\right)$ under some assumptions, where Eq. (1) is called local well-posedness in $H^1\left({\mathbb{R}}^d\right)$ if Eq. (1) satisfies all of the following conditions:

• There is uniqueness in $H^1\left({\mathbb{R}}^d\right)$ for a solution to Eq. (1).

• For each $u_0\in H^1\left({\mathbb{R}}^d\right)$, there exists a solution to Eq. (1) with Eq. (3) defined on a maximal existence interval $\left(T_{\min },T_{\max }\right)$, where $T_{\max }=T_{\max }\left(u_0\right)\in \left(0,\infty \right]$ and $T_{\min }=T_{\min }\left(u_0\right)\in \left[-\infty ,0\right)$.

• There is the blow-up alternative. That is, if $T_{\max }<\infty$ (resp. $T_{\min }>-\infty$), then we have

  $$\underset{t\uparrow T_{\max }}{lim}\parallel u\left(t\right){\parallel }_{H_x^1}=\infty \left(\text{resp}.\underset{t\downarrow T_{\min }}{lim}\parallel u\left(t\right){\parallel }_{H_x^1}=\infty \right).$$E4

• The solution depends on continuously on the initial condition. That is, if $u_{0,n}\to u_0$ in $H^1\left({\mathbb{R}}^d\right)$, then for any closed interval $I\subset \left(T_{\min },T_{\max }\right)$, there exists $n_0\in \mathbb{N}$ such that for any $n\ge n_0$, the solution $u_n$ to Eq. (1) with $u_n\left(0,x\right)=u_{0,n}\left(x\right)$ is defined on $C_t\left(I;H^1\left({\mathbb{R}}^d\right)\right)$ and satisfies $u_n\to u$ in $C_t\left(I;H^1\left({\mathbb{R}}^d\right)\right)$ as $n\to \infty$, where $u$ is the solution to Eq. (1) with $u\left(0,x\right)=u_0\left(x\right)$.

To state a local well-posedness result, we define the space

$${\mathcal{K}}_0\left({\mathbb{R}}^d\right)≔{\overline{\left\{f\in L^{\infty }\left({\mathbb{R}}^d\right):suppf\text{is\hspace{0.17em}compact}.\right\}}}^{\parallel \cdot {\parallel }_{\mathcal{K}}},$$E5

where

$$\parallel f{\parallel }_{\mathcal{K}}≔\underset{x\in {\mathbb{R}}^d}{sup}{\int }_{{\mathbb{R}}^d}\frac{\mid f\left(y\right)\mid }{{\left|x-y\right|}^{d-2}}\mathit{dy}.$$E6

We note that

$$L^{\frac{d}{2}-\epsilon }\left({\mathbb{R}}^d\right)\cap L^{\frac{d}{2}+\epsilon }\left({\mathbb{R}}^d\right)\hookrightarrow L^{\frac{d}{2},1}\left({\mathbb{R}}^d\right)\hookrightarrow \mathcal{K}\left({\mathbb{R}}^d\right)≔\left\{f:\parallel f{\parallel }_{\mathcal{K}}<\infty \right\}$$E7

for some $\epsilon >0$, where the space $L^{p,q}\left({\mathbb{R}}^d\right)$ denotes the usual Lorentz space.

Theorem 1 (Local well-posedness, [2, 3, 4]) Let $d\ge 1$ and $1<p<2^{\ast }-1$. If $V$ satisfies one of the following, then Eq. (1) is locally well-posed in $H^1\left({\mathbb{R}}^d\right)$.

• $V\in L^{\eta }\left({\mathbb{R}}^d\right)+L^{\infty }\left({\mathbb{R}}^d\right)$ for $\eta \ge 1$ if $d=1$ and $\eta >\frac{d}{2}$ if $d\ge 2$,

• $\parallel V_{-}{\parallel }_{\mathcal{K}}<4\pi$ and $V\in L^{\frac{3}{2}}\left({\mathbb{R}}^3\right)\cap {\mathcal{K}}_0\left({\mathbb{R}}^3\right)$, where $V_{-}≔min\left\{V\left(x\right),0\right\}$.

Moreover, the solution $u$ to Eq. (1) conserves its mass and energy with respect to time $t$, where they are defined as

$$\begin{array}{c}\left(\text{Mass}\right)M\left[u\left(t\right)\right]≔\parallel u\left(t\right){\parallel }_{L_x^2}^2,\\ \left(\text{Energy}\right)E_V\left[u\left(t\right)\right]≔\frac{1}{2}\parallel u\left(t\right){\parallel }_{{\dot{H}}_x^1}^2+\frac{1}{2}{\int }_{{\mathbb{R}}^d}V\left(x\right){\left|u(t,x)\right|}^2-\frac{1}{p+1}\parallel u\left(t\right){\parallel }_{L_x^{p+1}}^{p+1}.\end{array}$$E8

We turn to time behaviors of the solution to Eq. (1). A solution to Eq. (1) has various kinds of time behaviors by the choice of initial data. For example, we can consider the following time behaviors.

• (Scattering) We say that the solution $u$ to Eq. (1) scatters in positive time (resp. negative time) if $T_{\max }=\infty$ (resp. $T_{\min }=-\infty$) and there exists ${\psi }_{+}\in H^1\left({\mathbb{R}}^d\right)$ (resp. ${\psi }_{-}\in H^1\left({\mathbb{R}}^d\right)$) such that

$$\underset{t\to +\infty }{lim}\parallel u\left(t\right)-e^{\mathit{it}{\mathrm{\Delta }}_V}{\psi }_{+}{\parallel }_{H_x^1}=0\left(\text{resp}.\underset{t\to -\infty }{lim}\parallel u\left(t\right)-e^{\mathit{it}{\mathrm{\Delta }}_V}{\psi }_{-}{\parallel }_{H_x^1}=0\right),$$E9

where $e^{\mathit{it}{\mathrm{\Delta }}_V}f$ is a solution to the corresponding linear equation with Eq. (1)

$$i{\partial }_{t}u\left(t,x\right)+{\mathrm{\Delta }}_{V}u\left(t,x\right)=0,u\left(0,x\right)=f\left(x\right).$$E10

We say that $u$ scatters when $u$ scatters in positive and negative time.

• (Blow-up) We say that the solution $u$ to Eq. (1) blows up in positive time (resp. negative time) if $T_{\max }<\infty$ (resp. $T_{\min }>-\infty$). We say that $u$ blows up when $u$ blows up in positive and negative time.

• (Grow-up) We say that the solution $u$ to Eq. (1) grows up in positive time (resp. negative time) if $T_{\max }=\infty$ (resp. $T_{\min }=-\infty$) and

$$\underset{t\to \infty }{limsup}\parallel u\left(t\right){\parallel }_{H_x^1}=\infty ,\left(\text{resp}.\underset{t\to -\infty }{limsup}\parallel u\left(t\right){\parallel }_{H_x^1}=\infty \right).$$E11

We say that $u$ grows up when $u$ grows up in positive and negative time.

• (Standing wave) We say that the solution $u$ to Eq. (1) is a standing wave if $u=e^{\mathit{i\omega t}}{Q}_{\omega ,V}$ for some $\omega \in \mathbb{R}$, where $Q_{\omega ,V}$ satisfies the elliptic equation

$$-\omega Q_{\omega ,V}+{\mathrm{\Delta }}_V{Q}_{\omega ,V}=-{\left|Q_{\omega ,V}\right|}^{p-1}{Q}_{\omega ,V}.$$E12

In particular, $Q_{\omega ,V}$ is ground state to Eq. (12) if

$$Q_{\omega ,V}\in \left\{\varphi \in {\mathcal{A}}_{\omega ,V}:S_{\omega ,V}\left(\varphi \right)\le S_{\omega ,V}\left(\psi \right)\text{for}\mathrm{any}\psi \in {\mathcal{A}}_{\omega ,V}\right\}≕{\mathcal{G}}_{\omega ,V},$$E13

where $S_{\omega ,V}\left(f\right)≔\frac{\omega }{2}M\left[f\right]+E_V\left[f\right]$ (and)

$${\mathcal{A}}_{\omega ,V}≔\left\{\psi \in H^1\left({\mathbb{R}}^d\right)\backslash \left\{0\right\}:S_{\omega ,V}^{\prime }\left(\psi \right)=0\right\}.$$E14

We know the following results (Theorems 2 and 3) for time behaviors of the solutions to Eq. (1). For related results, we also list [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38].

Theorem 2 (Hong, [3]) Let $d=p=3$, $u_0\in H^1\left({\mathbb{R}}^3\right)$, and $Q_{1,0}\in {\mathcal{G}}_{1,0}$. Suppose that $V$ satisfies $V\in L^{\frac{3}{2}}\left({\mathbb{R}}^3\right)\cap {\mathcal{K}}_0\left({\mathbb{R}}^3\right)$, $V\ge 0$, $x\cdot \nabla V\in L^{\frac{3}{2}}\left({\mathbb{R}}^3\right)$, and $x\cdot \nabla V\le 0$. We also assume that

$$M\left[u_0\right]E_V\left[u_0\right]<M\left[Q_{1,0}\right]E_0\left[Q_{1,0}\right]\text{and}\parallel u_0{\parallel }_{L^2}\parallel u_0{\parallel }_{{\dot{H}}_V^1}<\parallel Q_{1,0}{\parallel }_{L^2}\parallel Q_{1,0}{\parallel }_{{\dot{H}}^1}.$$E15

Then, the solution $u$ to Eq. (1) with Eq. (3) scatters.

Theorem 3 (Hamano–Ikeda, [4]) Let $d=3$, $\frac{7}{3}<p<5$, $u_0\in H^1\left({\mathbb{R}}^3\right)$, and $Q_{1,0}\in {\mathcal{G}}_{1,0}$. Suppose that $V$ satisfies $V\ge 0$ and $x\cdot \nabla V\in L^{\frac{3}{2}}\left({\mathbb{R}}^3\right)$. We also assume that

$$M{\left[u_0\right]}^{\frac{1-s_c}{s_c}}{E}_V\left[u_0\right]<M{\left[Q_{1,0}\right]}^{\frac{1-s_c}{s_c}}{E}_0\left[Q_{1,0}\right],$$E16

where $s_c≔\frac{d}{2}-\frac{2}{p-1}$.

1. (Scattering)

   If $V\in L^{\frac{3}{2}}\left({\mathbb{R}}^3\right)\cap {\mathcal{K}}_0\left({\mathbb{R}}^3\right)$, $x\cdot \nabla V\le 0$, and

   $$\parallel u_0{\parallel }_{L^2}^{\frac{1-s_c}{s_c}}\parallel u_0{\parallel }_{{\dot{H}}^1}<\parallel Q_{1,0}{\parallel }_{L^2}^{\frac{1-s_c}{s_c}}\parallel Q_{1,0}{\parallel }_{{\dot{H}}^1},$$E17

   then $\left(T_{\min },T_{\max }\right)=\mathbb{R}$, that is, exists globally in time. Moreover, if $u_0$ and $V$ are radially symmetric, then $u$ scatters.

2. (Blow-up or grow-up)

   If “$V\in L^{\frac{3}{2}}\left({\mathbb{R}}^3\right)\cap {\mathcal{K}}_0\left({\mathbb{R}}^3\right)$ or $V\in L^{\sigma }\left({\mathbb{R}}^3\right)$ for some $\frac{3}{2}<\sigma \le \infty$,” $2V+x\cdot \nabla V\ge 0$, and

   $$\parallel u_0{\parallel }_{L^2}^{\frac{1-s_c}{s_c}}\parallel u_0{\parallel }_{{\dot{H}}_V^1}>\parallel Q_{1,0}{\parallel }_{L^2}^{\frac{1-s_c}{s_c}}\parallel Q_{1,0}{\parallel }_{{\dot{H}}^1},$$E18

   then $u$ blows up or grows up. Furthermore, if one of the following holds:

   • “$u_0$ and $V$ are radially symmetric,” $x\cdot \nabla V\ge 0$, and $V\in L^{\infty }\left({\mathbb{R}}^3\right)$,

   • ${\mathit{xu}}_0\in L^2\left({\mathbb{R}}^3\right)$,

then $u$ blows up.

Remark 1 Mizutani [39] proved that for any $\psi \in H^1$, there exists ${\varphi }_{\pm }\in H^1\left({\mathbb{R}}^3\right)$ such that

$$\underset{t\to \pm \infty }{lim}\parallel e^{\mathit{it}{\mathrm{\Delta }}_V}\psi -e^{\mathit{it}\mathrm{\Delta }}{\varphi }_{\pm }{\parallel }_{H_x^1}=0$$E19

under the assumptions $V\in L^{\frac{3}{2}}\left({\mathbb{R}}^3\right)$ and $V\ge 0$, where the double-sign corresponds. Combining this limit and scattering part in Theorem 3 (or Theorem 2), we can see that the nonlinear solution $u$ to Eq. (1) approaches to a free solution $e^{\mathit{it}\mathrm{\Delta }}{\varphi }_{\pm }$ as $t\to \pm \infty$ for some ${\varphi }_{\pm }\in H^1\left({\mathbb{R}}^3\right)$.

We realize that there is no potential, which satisfies scattering and blow-up or grow-up parts in Theorem 1 at the same time. Indeed, if $V$ satisfies $x\cdot \nabla V\le 0$ and $2V+x\cdot \nabla V\ge 0$, then $V\notin L^{\frac{3}{2}}\left({\mathbb{R}}^3\right)$. Then, we consider a minimization problem

$$n_{\omega ,V}≔inf\left\{S_{\omega ,V}\left(f\right):f\in H^1\left({\mathbb{R}}^d\right)\backslash \left\{0\right\},K_V\left(f\right)=0\right\}$$E20

to get a potential $V$, which deduces scattering and blow-up or grow-up at the same time. It proved in [40] that the condition Eq. (16) can be rewritten as the following by using $n_{\omega ,V}$.

Proposition 1 Let $d\ge 3$, $1+\frac{4}{d}<p<2^{\ast }-1$, $f\in H^1\left({\mathbb{R}}^d\right)$, and $Q_{1,0}\in {\mathcal{G}}_{1,0}$. Assume that $V$ satisfies (A2) with $\mid \mathfrak{a}\mid \le 1$ and (A6) below. Then, the following two conditions are equivalent.

1. $M{\left[f\right]}^{\frac{1-s_c}{s_c}}{E}_V\left[f\right]<M{\left[Q_{1,0}\right]}^{\frac{1-s_c}{s_c}}{E}_0\left[Q_{1,0}\right]$,

2. There exists $\omega >0$ such that $S_{\omega ,V}\left(f\right)<n_{\omega ,V}$.

Using $n_{\omega ,V}$, we expect that if $S_{\omega ,V}\left(u_0\right)<n_{\omega ,V}$ and $K_V\left(u_0\right)\ge 0$, then the solution $u$ scatters and if $S_{\omega ,V}\left(u_0\right)<n_{\omega ,V}$ and $K_V\left(u_0\right)<0$, then the solution $u$ blows up or grows up, where $K_V$ is called virial functional and is defined as

$$\begin{array}{c}{K}_V\left(f\right)≔{\left.\frac{d}{\mathit{d\lambda }}\right|}_{\lambda =0}{S}_{\omega ,V}\left(e^{\mathit{d\lambda }}f\left(e^{2\lambda }\cdot \right)\right)\\ =2\parallel f{\parallel }_{{\dot{H}}^1}^2-{\int }_{{\mathbb{R}}^d}\left(x\cdot \nabla V\right){\left|f\left(x\right)\right|}^2\mathit{dx}-\frac{\left(p-1\right)d}{p+1}\parallel f{\parallel }_{L^{p+1}}^{p+1}.\end{array}$$E21

It is well known that $K_V\left(u\left(t\right)\right)$ denotes variance of the solution and if ${\mathit{xu}}_0\in L^2\left({\mathbb{R}}^d\right)$ then

$$K_V\left(u\left(t\right)\right)=\frac{1}{4}\cdot \frac{d^2}{{\mathit{dt}}^2}\parallel \mathit{xu}\left(t\right){\parallel }_{L_x^2}^2$$E22

for each $t\in \left(T_{\min },T_{\max }\right)$. We also consider a minimization problem $r_{\omega ,V}$, which restricts $n_{\omega ,V}$ to radial functions, that is,

$$r_{\omega ,V}≔inf\left\{S_{\omega ,V}\left(f\right):f\in H_{\mathrm{rad}}^1\left({\mathbb{R}}^d\right)\backslash \left\{0\right\},K_V\left(f\right)=0\right\}$$E23

and expect for radial initial data $u_0$ and radial potential $V$ that if $S_{\omega ,V}\left(u_0\right)<r_{\omega ,V}$ and $K_V\left(u_0\right)\ge 0$, then the solution $u$ scatters and if $S_{\omega ,V}\left(u_0\right)<r_{\omega ,V}$ and $K_V\left(u_0\right)<0$, then the solution $u$ blows up. For more general minimization problems

$$\begin{array}{l}{n}_{\omega ,V}^{\alpha ,\beta }≔inf\left\{S_{\omega ,V}\left(f\right):f\in H^1\left({\mathbb{R}}^d\right)\backslash \left\{0\right\},K_{\omega ,V}^{\alpha ,\beta }\left(f\right)=0\right\},\\ r_{\omega ,V}^{\alpha ,\beta }≔inf\left\{S_{\omega ,V}\left(f\right):f\in H_{\mathrm{rad}}^1\left({\mathbb{R}}^d\right)\backslash \left\{0\right\},K_{\omega ,V}^{\alpha ,\beta }\left(f\right)=0\right\}\end{array}$$E24

with

$$\alpha >0,\beta \ge 0,2\alpha -\mathit{d\beta }\ge 0,$$E25

the authors showed in [40, 41] the following results (Theorems 4 and 5) Eq. (27), where the functional $K_{\omega ,V}^{\alpha ,\beta }$ is given as

$$K_{\omega ,V}^{\alpha ,\beta }\left(f\right)≔{\left.\frac{d}{\mathit{d\lambda }}\right|}_{\lambda =0}{S}_{\omega ,V}\left(e^{\mathit{\alpha \lambda }}f\left(e^{\mathit{\beta \lambda }}\cdot \right)\right).$$E26

Here, we realize $n_{\omega ,V}=n_{\omega ,V}^{d,2}$, $r_{\omega ,V}=r_{\omega ,V}^{d,2}$, and $K_V=K_{\omega ,V}^{d,2}$.

To state the results, we give the assumptions of the potential $V$: Let $\mathfrak{a}\in {\left(\mathbb{N}\cup \left\{0\right\}\right)}^d$.

1. A1. $V\in L^{\frac{3}{2}}\left({\mathbb{R}}^3\right)\cap {\mathcal{K}}_0\left({\mathbb{R}}^3\right)$

2. A2. $x^{\mathfrak{a}}{\partial }^{\mathfrak{a}}V\in L^{\frac{d}{2}}\left({\mathbb{R}}^d\right)+L^{\sigma }\left({\mathbb{R}}^d\right)$ for some $\frac{d}{2}\le \sigma <\infty$

3. A3. $x^{\mathfrak{a}}{\partial }^{\mathfrak{a}}V\in L^{\frac{d}{2}}\left({\mathbb{R}}^d\right)+L^{\infty }\left({\mathbb{R}}^d\right)$

4. A4. $x^{\mathfrak{a}}{\partial }^{\mathfrak{a}}V\in L^{\eta }\left({\mathbb{R}}^d\right)+L^{\sigma }\left({\mathbb{R}}^d\right)$ for some $\frac{d}{2}<\eta \le \sigma <\infty$

5. A5. $x^{\mathfrak{a}}{\partial }^{\mathfrak{a}}V\in L^{\eta }\left({\mathbb{R}}^d\right)+L^{\infty }\left({\mathbb{R}}^d\right)$ for some $\frac{d}{2}<\eta <\infty$

6. A6. $V\ge 0$, $x\cdot \nabla V\le 0$, $2V+x\cdot \nabla V\ge 0$

7. A7. $V\ge 0$, $x\cdot \nabla V\le 0$, $\omega \ge {\omega }_0$ for

$${\omega }_0≔-\frac{1}{2}\mathrm{ess}{inf}_{x\in {\mathbb{R}}^d}\left(2V+x\cdot \nabla V\right).$$E27

We note that the third inequality implies $2V+x\cdot \nabla V+2\omega \ge 0$ a.e. $x\in {\mathbb{R}}^d$.

Theorem 4 Let $d\ge 3$ and $1+\frac{4}{d}<p<2^{\ast }-1$.

• (Non-radial case) Let $V$ satisfy (A2) with $\mid \mathfrak{a}\mid \le 1$ and (A6). Then, for each $\left(\alpha ,\beta \right)$ with Eq. (25) and $\omega >0$, $n_{\omega ,V}^{\alpha ,\beta }=n_{\omega ,0}^{\alpha ,\beta }$ holds. Moreover, if $x\cdot \nabla V<0$, then $n_{\omega ,V}^{\alpha ,\beta }$ is never attained.

• (Radial case) Let $V$ satisfy (A3) with $\mid \mathfrak{a}\mid \le 1$ and (A7). Let $V$ be radially symmetric. Then, $r_{\omega ,V}^{\alpha ,\beta }$ is attained for each $\left(\alpha ,\beta \right)$ with Eq. (25). Moreover, if $V$ satisfies (A3) with $\mid \mathfrak{a}\mid \le 2$ and $3x\cdot \nabla V+x{\nabla }^2{\mathit{Vx}}^T\le 0$, then ${\mathcal{M}}_{\omega ,V,\mathrm{rad}}^{\alpha ,\beta }={\mathcal{G}}_{\omega ,V,\mathrm{rad}}$ holds, where ${\nabla }^{2}V$ denotes the Hessian matrix of $V$,

$$\begin{array}{l}{\mathcal{M}}_{\omega ,V,\mathrm{rad}}^{\alpha ,\beta }≔\left\{\varphi \in H_{\mathrm{rad}}^1\left({\mathbb{R}}^d\right):S_{\omega ,V}\left(\varphi \right)=r_{\omega ,V}^{\alpha ,\beta },K_{\omega ,V}^{\alpha ,\beta }\left(\varphi \right)=0\right\},\\ {\mathcal{G}}_{\omega ,V,\mathrm{rad}}≔\left\{\varphi \in {\mathcal{A}}_{\omega ,V,\mathrm{rad}}:S_{\omega ,V}\left(\varphi \right)\le S_{\omega ,V}\left(\psi \right)\text{for}\mathrm{any}\psi \in {\mathcal{A}}_{\omega ,V,\mathrm{rad}}\right\},\\ {\mathcal{A}}_{\omega ,V,\mathrm{rad}}≔\left\{\psi \in H_{\mathrm{rad}}^1\left({\mathbb{R}}^d\right)\backslash \left\{0\right\}:S_{\omega ,V}^{\prime }\left(\psi \right)=0\right\}.\end{array}$$E28

The inequality $n_{\omega ,V}^{\alpha ,\beta }\le r_{\omega ,V}^{\alpha ,\beta }$ holds by their definitions and the attainability of $n_{\omega ,V}^{\alpha ,\beta }$ and $r_{\omega ,V}^{\alpha ,\beta }$ deduces the following corollary.

Corollary 1 Under the all assumptions of (Non-radial case) in Theorem 4, we have

$$n_{\omega ,V}^{\alpha ,\beta }<r_{\omega ,V}^{\alpha ,\beta }.$$E29

Remark 2 In the case of $V=0$, it is well known that $n_{\omega ,0}^{\alpha ,\beta }$ and $r_{\omega ,0}^{\alpha ,\beta }$ are attained by $Q_{\omega ,0}\in {\mathcal{G}}_{\omega ,0}$. That is, $n_{\omega ,0}^{\alpha ,\beta }=r_{\omega ,0}^{\alpha ,\beta }=S_{\omega ,0}\left(Q_{\omega ,0}\right)$ holds.

Then, we investigate global existence of a solution to time-dependent Eq. (1).

Theorem 5 (Global well-posedness in $H^1$) Let $d\ge 3$ and $1+\frac{4}{d}<p<2^{\ast }-1$.

• (Non-radial case) Let $u_0\in H^1\left({\mathbb{R}}^d\right)$ and $Q_{\omega ,0}\in {\mathcal{G}}_{\omega ,0}$. Suppose that $V$ satisfies “(A1) or (A4) with $\mid \mathfrak{a}\mid =0$,” (A2) with $\mid \mathfrak{a}\mid =1$, and (A6). We also assume that there exist $\left(\alpha ,\beta \right)$ satisfying Eq. (25) and $\omega >0$ such that

  $$S_{\omega ,V}\left(u_0\right)<S_{\omega ,0}\left(Q_{\omega ,0}\right)\left(=n_{\omega ,V}^{\alpha ,\beta }\right),K_{\omega ,V}^{\alpha ,\beta }\left(u_0\right)\ge 0.$$E30

  Then, the solution $u$ to Eq. (1) with Eq. (3) exists globally in time. In particular, it follows that

  $$\underset{t\in \mathbb{R}}{sup}\parallel u\left(t\right){\parallel }_{H_x^1}<\infty .$$E31

• (Radial case) Let $u_0\in H_{\mathrm{rad}}^1\left({\mathbb{R}}^d\right)$ and $Q_{\omega ,V}\in {\mathcal{G}}_{\omega ,V,\mathrm{rad}}$. Suppose that $V$ is radially symmetric and satisfies “(A1) or (A5) with $\mid \mathfrak{a}\mid =0$,” (A3) with $\mid \mathfrak{a}\mid =1,2$, (A7), and $3x\cdot \nabla V+x{\nabla }^2{\mathit{Vx}}^T\le 0$. If there exist $\left(\alpha ,\beta \right)$ with Eq. (25) and $\omega >0$ satisfying $\omega \ge {\omega }_0$ such that

$$S_{\omega ,V}\left(u_0\right)<S_{\omega ,V}\left(Q_{\omega ,V}\right)\left(=r_{\omega ,V}^{\alpha ,\beta }\right),K_{\omega ,V}^{\alpha ,\beta }\left(u_0\right)\ge 0,$$E32

then the solution $u$ to Eq. (1) with Eq. (3) exists globally in time.

1.1 Main theorem

In the previous paper, the authors handled the solution $u$ to Eq. (1) with initial data $u_0$ satisfying $S_{\omega ,V}\left(u_0\right)<m_{\omega ,V}$ and $K_V\left(u_0\right)\ge 0$, where $m_{\omega ,V}$ denotes $n_{\omega ,V}$ or $r_{\omega ,V}$. We note that $m_{\omega ,V}$ is $m_{\omega ,V}^{\alpha ,\beta }$ with $\left(\alpha ,\beta \right)=\left(d,2\right)$ and $m_{\omega ,V}^{\alpha ,\beta }$ is independent of $\left(\alpha ,\beta \right)$. In this chapter, we are interested in the solutions to Eq. (1) with initial data satisfying $S_{\omega ,V}\left(u_0\right)<m_{\omega ,V}$ and $K_V\left(u_0\right)<0$. Our main theorem is the following:

Theorem 6 Let $d\ge 3$ and $1+\frac{4}{d}<p<1+\frac{4}{d-2}$.

• (Non-radial case) Let $u_0\in H^1\left({\mathbb{R}}^d\right)$ and $Q_{\omega ,0}\in {\mathcal{G}}_{\omega ,0}$. Suppose that $V$ satisfy “(A1) or (A4) with $\mid \mathfrak{a}\mid =0$,” (A2) with $\mid \mathfrak{a}\mid =1$, and (A6). We also assume that there exists $\omega >0$ such that

  $$S_{\omega ,V}\left(u_0\right)<S_{\omega ,0}\left(Q_{\omega ,0}\right)\left(=n_{\omega ,V}\right),K_V\left(u_0\right)<0.$$E33

  Then, the solution $u$ to Eq. (1) with Eq. (3) blows up or grows up. Moreover, $u$ blows up under the additional assumption ${\mathit{xu}}_0\in L^2\left({\mathbb{R}}^d\right)$.

• (Radial case) Let $u_0\in H_{\mathrm{rad}}^1\left({\mathbb{R}}^d\right)$ and $Q_{\omega ,V}\in {\mathcal{G}}_{\omega ,V,\mathrm{rad}}$. Suppose that $V$ is radially symmetric and satisfies “(A1) or (A5) with $\mid \mathfrak{a}\mid =0$,” (A3) with $\mid \mathfrak{a}\mid =1,2$, (A7), and $3x\cdot \nabla V+x{\nabla }^2{\mathit{Vx}}^T\le 0$. We also assume that there exists $\omega >0$ satisfying $\omega \ge {\omega }_0$ such that

  $$S_{\omega ,V}\left(u_0\right)<S_{\omega ,V}\left(Q_{\omega ,V}\right)\left(=r_{\omega ,V}\right),K_V\left(u_0\right)<0.$$E34

Then, the solution $u$ to Eq. (1) with Eq. (3) blows up.

Remark 3 Let $V$ be a potential in Theorem 6. Combining Theorems 5 and 6, we complete bounded and unbounded dichotomy of $\left\{u_0\in H^1\left({\mathbb{R}}^d\right):S_{\omega ,V}\left(u_0\right)<S_{\omega ,0}\left(Q_{\omega ,0}\right)\right\}$ and global existence and blow-up dichotomy of $\left\{u_0\in H_{\mathrm{rad}}^1\left({\mathbb{R}}^d\right):S_{\omega ,V}\left(u_0\right)<S_{\omega ,V}\left(Q_{\omega ,V}\right)\right\}$ by using sign of the virial functional of initial data.

Remark 4 The following potential satisfies all of conditions in Theorem 6:

$$V\left(x\right)=\frac{\gamma {\left\{log\left(1+|x|\right)\right\}}^{\theta }}{{\left|x\right|}^{\mu }},\left(\gamma >0,0\le \theta \le \mu <2,\mu >0\right).$$E35

Theorem 6 with the potential Eq. (35) having $\theta =0$ was considered in the previous paper [19] by the authors. As the other example, we put

$$V\left(x\right)≔\frac{\gamma }{{⟨x⟩}^{\mu }},\left(\gamma >0,0<\mu <2\right),$$E36

where $⟨\cdot ⟩$ is called the Japanese bracket and is defined as ${\left(1+{\left|\cdot \right|}^2\right)}^{\frac{1}{2}}$.

1.2 Organization of the paper

The organization of the rest of this chapter is as follows. In Section 2, we collect some notations and tools used throughout this chapter. In Section 3, we prove non-radial case in Theorem 6 by using an argument in [13]. In Section 4, we show radial case in Theorem 6 by using an argument in [33].

2. Preliminaries

In this section, we define some notations and collect some tools, which are used throughout this chapter.

2.1 Notation and definition

For $1\le p\le \infty$, $L^p=L^p\left({\mathbb{R}}^d\right)$ denotes the usual Lebesgue space. For a Banach space $X$, we use $L^q\left(I;X\right)$ to denote the Banach space of functions $f:I\times {\mathbb{R}}^d\to \mathbb{C}$ whose norm is

$$\parallel f{\parallel }_{L^q\left(I;X\right)}≔\parallel \parallel f\left(t\right){\parallel }_X{\parallel }_{L^q\left(I\right)}<\infty .$$E37

We extend our notation as follows: If a time interval is not specified, then the $t$-norm is evaluated over $\left(-\infty ,\infty \right)$. To indicate a restriction to a time subinterval $I\subset \left(-\infty ,\infty \right)$, we will write as $L^q\left(I\right)$. $H^s\left({\mathbb{R}}^d\right)$ and ${\dot{H}}^s\left({\mathbb{R}}^d\right)$ are the usual Sobolev spaces, whose norms $\parallel f{\parallel }_{H^s}≔\parallel {\left(1-\mathrm{\Delta }\right)}^{\frac{s}{2}}f{\parallel }_{L^2}$ and $\parallel f{\parallel }_{{\dot{H}}^s}≔\parallel {\left(-\mathrm{\Delta }\right)}^{\frac{s}{2}}f{\parallel }_{L^2}$ respectively. We also define the Sobolev spaces $H_V^s\left({\mathbb{R}}^d\right)$ and ${\dot{H}}_V^s\left({\mathbb{R}}^d\right)$ with the potential $V$ via norms $\parallel f{\parallel }_{H_V^s}≔\parallel {\left(1-{\mathrm{\Delta }}_V\right)}^{\frac{s}{2}}f{\parallel }_{L^2}$ and $\parallel f{\parallel }_{{\dot{H}}_V^s}≔\parallel {\left(-{\mathrm{\Delta }}_V\right)}^{\frac{s}{2}}f{\parallel }_{L^2}$ respectively.

2.2 Some tools

Proposition 2 Let $p\ge 1$. For $f\in H_{\mathrm{rad}}^1\left({\mathbb{R}}^d\right)$, we have

$$\parallel f{\parallel }_{L^{p+1}\left(R\le |x|\right)}^{p+1}\le \frac{C}{R^{\frac{\left(d-1\right)\left(p-1\right)}{2}}}\parallel f{\parallel }_{L^2\left(R\le |x|\right)}^{\frac{p+3}{2}}\parallel f{\parallel }_{{\dot{H}}^1\left(R\le |x|\right)}^{\frac{p-1}{2}}$$E38

for any $R>0$, where the implicit constant $C$ is independent of $R$ and $f$.

To state the next proposition, we define two functions:

$${\mathcal{X}}_R≔R^2\mathcal{X}\left(\frac{\mid x\mid }{R}\right),$$E39

where $\mathcal{X}:\left[0,\infty \right)\to \left[0,\infty \right)$ (forms)

$$\mathcal{X}\left(r\right)≔\left\{\begin{array}{ll}{r}^2& \left(0\le r\le 1\right),\\ \mathit{\text{smooth}}& \left(1\le r\le 3\right),\\ 0& \left(3\le r\right)\end{array}\right.$$E40

and satisfies ${\mathcal{X}}^{″}\left(r\right)\le 2$.

$${\mathcal{Y}}_R\left(x\right)≔\mathcal{Y}\left(\frac{\mid x\mid }{R}\right),$$E41

where $\mathcal{Y}:\left[0,\infty \right)\to \left[0,\infty \right)$ (forms)

$$\mathcal{Y}\left(r\right)≔\left\{\begin{array}{ll}0& \left(0\le r\le \frac{1}{2}\right),\\ \mathit{\text{smooth}}& \left(\frac{1}{2}\le r\le 1\right),\\ 1& \left(1\le r\right)\end{array}\right.$$E42

and satisfies $0\le {\mathcal{Y}}^{\prime }\left(r\right)\le 3$.

Proposition 3 (Localized virial identity, [3]) Let $w$ be ${\mathcal{X}}_R$ or ${\mathcal{Y}}_R$ defined as Eqs. (39) and (41) respectively. For the solution $u$ to Eq. (1), we define

$$I_w\left(t\right)≔{\int }_{{\mathbb{R}}^d}w\left(x\right){\left|u(t,x)\right|}^2\mathit{dx}.$$E43

Then, we have

$$\begin{array}{l}{I}_{w^{\prime }}\left(t\right)=2Im{\int }_{{\mathbb{R}}^d}\frac{x\cdot \nabla u}{\mid x\mid }\overline{u}{w}^{\prime }\mathit{dx},\\ I_{{w^{\prime }}^{\prime }}\left(t\right)={\int }_{{\mathbb{R}}^d}{F}_1{\left|x\cdot \nabla u\right|}^2\mathit{dx}+4{\int }_{{\mathbb{R}}^d}\frac{w^{\prime }}{\mid x\mid }{\left|\nabla u\right|}^2\mathit{dx}-{\int }_{{\mathbb{R}}^d}{F}_2{\left|u\right|}^{p+1}\mathit{dx}\\ -{\int }_{{\mathbb{R}}^d}{F}_3{\left|u\right|}^2\mathit{dx}-2{\int }_{{\mathbb{R}}^d}\frac{w^{\prime }}{\mid x\mid }\left(x\cdot \nabla V\right){\left|u\right|}^2\mathit{dx}.\end{array}$$E44

where

$$\begin{array}{l}{F}_1\left(w,|x|\right)≔4\left(\frac{{w^{\prime }}^{\prime }}{{\left|x\right|}^2}-\frac{w^{\prime }}{{\left|x\right|}^3}\right),F_2\left(w,|x|\right)≔\frac{2\left(p-1\right)}{p+1}\left({w^{\prime }}^{\prime }+\frac{d-1}{\mid x\mid }{w}^{\prime }\right),\\ F_3\left(w,|x|\right)≔w^{\left(4\right)}+\frac{2\left(d-1\right)}{\mid x\mid }{w}^{\left(3\right)}+\frac{\left(d-1\right)\left(d-3\right)}{{\left|x\right|}^2}{w^{\prime }}^{\prime }+\frac{\left(d-1\right)\left(3-d\right)}{{\left|x\right|}^3}{w}^{\prime }.\end{array}$$E45

3. Non-radial case of main theorem

In this section, we prove (Non-radial case) for Theorem 6. First, we recall rewriting of $n_{\omega ,V}$, which is given in [40].

Lemma 1 Let $d\ge 3$, $1+\frac{4}{d}<p<1+\frac{4}{d-2}$, and $Q_{\omega ,0}\in {\mathcal{G}}_{\omega ,0}$. Assume that $V$ satisfies (A2) with $\mid \mathfrak{a}\mid \le 1$ and (A6). Then,

$$S_{\omega ,0}\left(Q_{\omega ,0}\right)=n_{\omega ,V}=inf\left\{T_{\omega ,V}\left(f\right):f\in H^1\left({\mathbb{R}}^d\right)\backslash \left\{0\right\},K_V\left(f\right)\le 0\right\}$$E46

holds, where the functional $T_{\omega ,V}$ is defined as

$$T_{\omega ,V}\left(f\right)≔S_{\omega ,V}\left(f\right)-\frac{1}{4}{K}_V\left(f\right).$$E47

Next, we give uniform estimate of the virial functional $K_V$.

Lemma 2 Under the all assumptions of (Non-radial) in Theorem 6, there exists $\delta >0$ such that

$$\underset{t\in \left(T_{\min },T_{\max }\right)}{sup}{K}_V\left(u\left(t\right)\right)\le -\delta <0.$$E48

Proof: Let $\delta ≔4\left\{S_{\omega ,V}\left(Q_{\omega ,V}\right)-S_{\omega ,V}\left(u_0\right)\right\}>0$. Applying Lemma 1, we have

$$\begin{array}{c}{S}_{\omega ,V}\left(Q_{\omega ,V}\right)\le T_{\omega ,V}\left(u\left(t\right)\right)=S_{\omega ,V}\left(u_0\right)-\frac{1}{4}{K}_V\left(u\left(t\right)\right)\\ =S_{\omega ,V}\left(Q_{\omega ,V}\right)-\frac{1}{4}\delta -\frac{1}{4}{K}_V\left(u\left(t\right)\right),\end{array}$$E49

which implies the desired result.

The blow-up result with ${\mathit{xu}}_0\in L^2\left({\mathbb{R}}^d\right)$ of (Non-radial case) in Theorem 1.1 follows immediately from Lemma 2.

Proof of blow-up part in (Non-radial case) for Theorem 6: We assume that the solution $u$ exists globally in time for contradiction. When ${\mathit{xu}}_0\in L^2\left({\mathbb{R}}^d\right)$, we have Eq. (22). Combining Eq. (22) and Lemma 2, there exists $\delta >0$ such that

$$\frac{d^2}{{\mathit{dt}}^2}\parallel \mathit{xu}\left(t\right){\parallel }_{L^2}^2=4K_V\left(u\left(t\right)\right)<-4\delta <0$$E50

for any $t\in \mathbb{R}$. Therefore, we obtain $\parallel \mathit{xu}\left(t\right){\parallel }_{L^2}^2<0$ if $\mid t\mid$ is sufficiently large. However, this is contradiction.

We consider Lemmas 3 and 4 to prove blow-up or grow-up part in (Non-radial case) for Theorem 6.

Lemma 3 Let $d\ge 3$ and $1+\frac{4}{d}<p<1+\frac{4}{d-2}$. We assume that $u\in C\left(\right[0,\infty );H^1)$ be a solution to Eq. (1) satisfying $C_0≔{sup}_{t\in \left[0,\infty \right)}\parallel u\left(t\right){\parallel }_{{\dot{H}}_x^1}<\infty$. Then, it follows that

$$\parallel u\left(t\right){\parallel }_{L^2\left(|x|\ge R\right)}^2\le o_R\left(1\right)+\eta$$E51

for any $\eta >0$, $R>0$, and $t\in \left[0,\frac{\mathit{\eta R}}{6C_0\parallel u{\parallel }_{L_x^2}}\right]$, where $o_R\left(1\right)$ goes to zero as $R\to \infty$ and is independent of $t$.

Proof: We consider $I_{{\mathcal{Y}}_R}$ given in Eq. (43). Using Proposition 3,

$$\begin{array}{ll}I\left(t\right)& =I\left(0\right)+{\int }_0^t{I}^{\prime }\left(s\right)\mathit{ds}\le I\left(0\right)+{\int }_0^t\mid I^{\prime }\left(s\right)\mid \mathit{ds}\\ & \le I\left(0\right)+\frac{2t}{R}\parallel {\mathcal{Y}}^{\prime }{\parallel }_{L^{\infty }}\underset{t\in \left[0,\infty \right)}{sup}\parallel u\left(t\right){\parallel }_{{\dot{H}}_x^1}\parallel u{\parallel }_{L_x^2}\le I\left(0\right)+\frac{6C_0\parallel u{\parallel }_{L_x^2}t}{R}\end{array}$$E52

for any $t\in \left[0,\infty \right)$. By the definition of ${\mathcal{Y}}_R$, we have

$$I\left(0\right)={\int }_{{\mathbb{R}}^d}{\mathcal{Y}}_R\left(x\right){\left|u_0\left(x\right)\right|}^2\mathit{dx}\le \parallel u_0{\parallel }_{L^2\left(|x|\ge \frac{R}{2}\right)}^2=o_R\left(1\right)$$E53

and hence, we obtain

$$\parallel u\left(t\right){\parallel }_{L^2\left(|x|\ge R\right)}^2\le I\left(t\right)\le o_R\left(1\right)+\eta .$$E54

Lemma 4 Let $d\ge 3$ and $1+\frac{4}{d}<p<1+\frac{4}{d-2}$. Let $u\in C\left(\right[0,\infty );H^1\left({\mathbb{R}}^d\right))$ be a solution to Eq. (1). Then, for $q\in \left(p+1,2^{\ast }\right)$, there exist constants $C=C\left(q,\parallel u_0{\parallel }_{L^2},C_0\right)>0$ and ${\theta }_q>0$ such that the estimate

$$I_{{\mathcal{X}}_R}{’}^{\prime }\left(t\right)\le 4K_V\left(u\left(t\right)\right)+C\parallel u\left(t\right){\parallel }_{L^2\left(R\le |x|\right)}^{\left(p+1\right){\theta }_q}+\frac{C}{R^2}$$E55

holds for any $R>0$ and $t\in \left[0,\infty \right)$, where ${\theta }_q≔\frac{2\left\{q-\left(p+1\right)\right\}}{\left(p+1\right)\left(q-2\right)}\in \left(0,\frac{2}{p+1}\right)$, $C_0$ is given in Lemma 3, and $I_{{\mathcal{X}}_R}$ is defined as Eq. (43).

Proof: Using Proposition 3, we have

$$I_{{\mathcal{X}}_R}^{″}\left(t\right)=4K_V\left(u\left(t\right)\right)+{\mathcal{R}}_1+{\mathcal{R}}_2+{\mathcal{R}}_3+{\mathcal{R}}_4,$$E56

where ${\mathcal{R}}_k={\mathcal{R}}_k\left(t\right)$$\left(k=\mathrm{1,2,3,4}\right)$ are defined as

$$\begin{array}{l}{\mathcal{R}}_1≔4{\int }_{{\mathbb{R}}^d}\left\{\frac{1}{{\left|x\right|}^2}{\mathcal{X}}^{″}\left(\frac{r}{R}\right)-\frac{R}{{\left|x\right|}^3}{\mathcal{X}}^{\prime }\left(\frac{\mid x\mid }{R}\right)\right\}{\left|x\cdot \nabla u\right|}^2\mathit{dx}\\ +4{\int }_{{\mathbb{R}}^d}\left\{\frac{R}{\mid x\mid }{\mathcal{X}}^{\prime }\left(\frac{\mid x\mid }{R}\right)-2\right\}{\left|\nabla u\left(t,x\right)\right|}^2\mathit{dx},\end{array}$$E57

$${\mathcal{R}}_2≔-\frac{2\left(p-1\right)}{p+1}{\int }_{{\mathbb{R}}^d}\left\{{\mathcal{X}}^{″}\left(\frac{\mid x\mid }{R}\right)+\frac{\left(d-1\right)R}{\mid x\mid }{\mathcal{X}}^{’}\left(\frac{\mid x\mid }{R}\right)-2d\right\}{\left|u\left(t,x\right)\right|}^{p+1}\mathit{dx},$$E58

$$\begin{array}{l}{\mathcal{R}}_3≔-{\int }_{{\mathbb{R}}^d}\left\{\frac{1}{R^2}{\mathcal{X}}^{\left(4\right)}\left(\frac{\mid x\mid }{R}\right)+\frac{2\left(d-1\right)}{R\mid x\mid }{\mathcal{X}}^{\left(3\right)}\left(\frac{\mid x\mid }{R}\right)+\frac{\left(d-1\right)\left(d-3\right)}{{\left|x\right|}^2}\mathcal{X}{’}^{\prime }\left(\frac{\mid x\mid }{R}\right)\right.\\ \left.+\frac{\left(d-1\right)\left(3-d\right)R}{{\left|x\right|}^3}{\mathcal{X}}^{\prime }\left(\frac{\mid x\mid }{R}\right)\right\}{\left|u\left(t,x\right)\right|}^2\mathit{dx},\end{array}$$E59

$${\mathcal{R}}_4≔2{\int }_{R\le \mid x\mid }\left\{2-\frac{R}{\mid x\mid }{\mathcal{X}}^{’}\left(\frac{\mid x\mid }{R}\right)\right\}\left(x\cdot \nabla V\right){\left|u(t,x)\right|}^2\mathit{dx}.$$E60

We set

$$\mathrm{\Omega }≔\left\{x\in {\mathbb{R}}^d:\frac{1}{{\left|x\right|}^2}{\mathcal{X}}^{″}\left(\frac{\mid x\mid }{R}\right)-\frac{R}{{\left|x\right|}^3}{\mathcal{X}}^{\prime }\left(\frac{\mid x\mid }{R}\right)\le 0\right\}.$$E61

By the inequality ${\mathcal{X}}^{\prime }\left(\frac{\mid x\mid }{R}\right)\le \frac{2\mid x\mid }{R}$, we have

$${\mathcal{R}}_1\le 4{\int }_{{\mathrm{\Omega }}^c}\left\{{\mathcal{X}}^{″}\left(\frac{r}{R}\right)-2\right\}{\left|\nabla u\left(t,x\right)\right|}^2\mathit{dx}\le 0,$$E62

where ${\mathrm{\Omega }}^c$ denotes a complement of $\mathrm{\Omega }$.

Next, we estimate ${\mathcal{R}}_2$. Applying Hölder’s inequality and Sobolev’s embedding, we have

$$\begin{array}{l}{\mathcal{R}}_2\le C\parallel u\left(t\right){\parallel }_{L^{p+1}\left(R\le |x|\right)}^{p+1}\le C\parallel u\left(t\right){\parallel }_{L^q\left(R\le |x|\right)}^{\left(p+1\right)\left(1-{\theta }_q\right)}\parallel u\left(t\right){\parallel }_{L^2\left(R\le |x|\right)}^{\left(p+1\right){\theta }_q}\\ \le C\parallel u\left(t\right){\parallel }_{H^1}^{\left(p+1\right)\left(1-{\theta }_q\right)}\parallel u\left(t\right){\parallel }_{L^2\left(R\le |x|\right)}^{\left(p+1\right){\theta }_q}\le C\parallel u\left(t\right){\parallel }_{L^2\left(R\le |x|\right)}^{\left(p+1\right){\theta }_q}.\end{array}$$E63

Next, we estimate ${\mathcal{R}}_3$.

$${\mathcal{R}}_3\le \frac{C}{R^2}\parallel u\left(t\right){\parallel }_{L^2\left(R\le |x|\right)}^2\le \frac{C}{R^2}.$$E64

Finally, ${\mathcal{R}}_4$ is estimated as ${\mathcal{R}}_4\le 0$ by ${\mathcal{X}}^{\prime }\left(\frac{\mid x\mid }{R}\right)\le \frac{2\mid x\mid }{R}$ and $x\cdot \nabla V\le 0$, which completes the proof of the lemma.

Proof of blow-up or grow-up part in (Non-radial case) for Theorem 6. We assume that

$$T_{\max }=\infty \text{and}\underset{t\in \left[0,\infty \right)}{sup}\parallel u\left(t\right){\parallel }_{{\dot{H}}_x^1}<\infty$$E65

for contradiction. By Lemmas 2, 3, and 4, there exists $\delta >0$ such that

$$I_{{\mathcal{X}}_R}^{″}\left(s\right)\le -4\delta +C\parallel u\left(s\right){\parallel }_{L_x^2\left(R\le |x|\right)}^{\left(p+1\right){\theta }_q}+\frac{C}{R^2}\le -4\delta +C{\eta }^{\frac{\left(p+1\right){\theta }_q}{2}}+o_R\left(1\right)$$E66

for any $\eta >0$, $R>0$, and $s\in \left[0,\frac{\mathit{\eta R}}{6C_0\parallel u_0{\parallel }_{L^2}}\right]$. We take $\eta ={\eta }_0>0$ sufficiently small such as

$$C{\eta }_0^{\frac{\left(p+1\right){\theta }_q}{2}}\le 2\delta .$$E67

and set

$$T=T\left(R\right)≔{\alpha }_{0}R≔\frac{{\eta }_{0}R}{6C_0\parallel u_0{\parallel }_{L^2}}.$$E68

Applying Eq. (67), integrating Eq. (66) over $s\in \left[0,t\right]$, and integrating over $t\in \left[0,T\right]$, we have

$$\begin{array}{ll}{I}_{{\mathcal{X}}_R}\left(T\right)& \le I_{{\mathcal{X}}_R}\left(0\right)+I_{{\mathcal{X}}_R}^{\prime }\left(0\right)T+\frac{1}{2}\left(-2\delta +o_R\left(1\right)\right)T^2\\ & =I_{{\mathcal{X}}_R}\left(0\right)+I_{{\mathcal{X}}_R}^{\prime }\left(0\right){\alpha }_{0}R+\frac{1}{2}\left(-2\delta +o_R\left(1\right)\right){\alpha }_0^2{R}^2.\end{array}$$E69

Here, we can see

$$I_{{\mathcal{X}}_R}\left(0\right)=o_R\left(1\right)R^2\text{and}{I}_{{\mathcal{X}}_R}^{\prime }\left(0\right)=o_R\left(1\right)R.$$E70

Indeed, we get

$$I_{{\mathcal{X}}_R}\left(0\right)\le R\parallel u_0{\parallel }_{L^2\left(|x|\le \sqrt{R}\right)}^2+{\mathit{cR}}^2\parallel u_0{\parallel }_{L^2\left(\sqrt{R}\le |x|\right)}=o_R\left(1\right)R^2,$$E71

and

$$I_{{\mathcal{X}}_R}^{\prime }\left(0\right)\le 4\sqrt{R}\parallel u_0{\parallel }_{{\dot{H}}^1}\parallel u_0{\parallel }_{L^2\left(|x|\le \sqrt{R}\right)}+\mathit{cR}\parallel u_0{\parallel }_{{\dot{H}}^1}\parallel u_0{\parallel }_{L^2\left(\sqrt{R}\le |x|\right)}=o_R\left(1\right)R.$$E72

Combining Eqs. (69) and (70), we get

$$I_{{\mathcal{X}}_R}\left(T\right)\le \left(o_R\left(1\right)-{\mathit{\delta \alpha }}_0^2\right)R^2.$$E73

We take $R>0$ such as $o_R\left(1\right)-{\mathit{\delta \alpha }}_0^2<0$. However, this contradicts $I_{{\mathcal{X}}_R}\left(T\right)\ge 0$.

4. Radial case of main theorem

In this section, we prove (Radial case) for Theorem 6. First, we introduce another characterization of $r_{\omega ,V}$.

Lemma 5 Let $d\ge 3$, $1+\frac{4}{d}<p<1+\frac{4}{d-2}$, and $Q_{\omega ,V}\in {\mathcal{G}}_{\omega ,V,\mathrm{rad}}$. Assume that $V$ is radially symmetric and satisfies (A3) with $\mid \mathfrak{a}\mid \le 2$, (A7), and $3x\cdot \nabla V+x{\nabla }^2{\mathit{Vx}}^T\le 0$. Then,

$$S_{\omega ,V}\left(Q_{\omega ,V}\right)=r_{\omega ,V}=inf\left\{U_{\omega ,V}\left(f\right):f\in H_{\mathrm{rad}}^1\left({\mathbb{R}}^d\right)\backslash \left\{0\right\},K_V\left(f\right)\le 0\right\}$$E74

holds, where the functional $U_{\omega ,V}$ is defined as

$$U_{\omega ,V}\left(f\right)≔S_{\omega ,V}\left(f\right)-\frac{1}{d\left(p-1\right)}{K}_V\left(f\right).$$E75

Proof: The lemma follows from proof of Lemma 1 (see [40], Lemma 4.3) combined $2\omega +2V+x\cdot \nabla V\ge 0$.

Proof of (Radial case) for Theorem 6. Assume that the solution $u$ to Eq. (1) exists globally in time for contradiction. We consider $I_{{\mathcal{X}}_R}$ again and recall

$$I_{{\mathcal{X}}_R}^{″}\left(t\right)=4K_V\left(u\left(t\right)\right)+{\mathcal{R}}_1+{\mathcal{R}}_2+{\mathcal{R}}_3+{\mathcal{R}}_4,$$E76

where ${\mathcal{R}}_k$$\left(1\le k\le 4\right)$ are defined as Eqs. (57) $\sim$ (60). We use same estimates with proof of blow-up or grow-up for ${\mathcal{R}}_1$, ${\mathcal{R}}_3$, and ${\mathcal{R}}_4$. Applying Proposition 2 and the Young’s inequality, we have

$$\begin{array}{ll}{\mathcal{R}}_2& \le \frac{C}{R^{\frac{\left(d-1\right)\left(p-1\right)}{2}}}\parallel u\left(t\right){\parallel }_{L^2\left(R\le |x|\right)}^{\frac{p+3}{2}}\parallel u\left(t\right){\parallel }_{{\dot{H}}^1\left(R\le |x|\right)}^{\frac{p-1}{2}}\\ & \le \frac{C}{R^{\frac{2\left(d-1\right)\left(p-1\right)}{5-p}}{\epsilon }^{\frac{4}{5-p}}}\parallel u{\parallel }_{L^2}^{\frac{2\left(p+3\right)}{5-p}}+2\left\{d\left(p-1\right)-4\right\}\epsilon \parallel u{\parallel }_{{\dot{H}}^1}^2\\ & \le \frac{C}{R^{\frac{2\left(d-1\right)\left(p-1\right)}{5-p}}{\epsilon }^{\frac{4}{5-p}}}\parallel u{\parallel }_{L^2}^{\frac{2\left(p+3\right)}{5-p}}+4d\left(p-1\right)\epsilon U_{\omega ,V}\left(u\right)\end{array}$$E77

for each positive $\epsilon >0$, which is chosen later. Collecting these estimates, we have

$$\begin{array}{l}{I}_{{\mathcal{X}}_R}^{″}\left(t\right)\\ \le 4K_V\left(u\right)+4d\left(p-1\right)\epsilon U_{\omega ,V}\left(u\right)+\frac{C}{R^{\frac{2\left(d-1\right)\left(p-1\right)}{5-p}}{\epsilon }^{\frac{4}{5-p}}}+\frac{C}{R^2}\\ =4d\left(p-1\right)\left\{S_{\omega ,V}\left(u\right)-U_{\omega ,V}\left(u\right)\right\}+4d\left(p-1\right)\epsilon U_{\omega ,V}\left(u\right)+\frac{C}{R^{\frac{2\left(d-1\right)\left(p-1\right)}{5-p}}{\epsilon }^{\frac{4}{5-p}}}+\frac{C}{R^2}\\ <4d\left(p-1\right)\left(1-\delta \right)S_{\omega ,V}\left(Q_{\omega ,V}\right)+4d\left(p-1\right)\left(\epsilon -1\right)U_{\omega ,V}\left(u\right)+\frac{C}{R^{\frac{2\left(d-1\right)\left(p-1\right)}{5-p}}{\epsilon }^{\frac{4}{5-p}}}+\frac{C}{R^2}\\ \le 4d\left(p-1\right)\left(\epsilon -\delta \right)S_{\omega ,V}\left(Q_{\omega ,V}\right)+\frac{C}{R^{\frac{2\left(d-1\right)\left(p-1\right)}{5-p}}{\epsilon }^{\frac{4}{5-p}}}+\frac{C}{R^2},\end{array}$$E78

where the second inequality is used $S_{\omega ,V}\left(u\right)<\left(1-\delta \right)S_{\omega ,V}\left(Q_{\omega ,V}\right)$ for some $\delta \in \left(0,1\right)$ and the third inequality is used $S_{\omega ,V}\left(Q_{\omega ,V}\right)\le U_{\omega ,V}$ (see Lemma 5). Taking $\epsilon \in \left(0,\delta \right)$ and sufficiently large $R>0$, there exists $\eta >0$ such that $I_{{\mathcal{X}}_R}^{″}\left(t\right)<-\eta <0$ for each $t\in \mathbb{R}$. However, this inequality implies that if $\mid t\mid$ is sufficiently large, then . This is contradiction and hence, we complete the proof.

5. Conclusions

In this chapter, our main result is Theorem 6. Combining the main result and a previous result (Theorem 5), we can classify time behavior of solutions to Eq. (1) with initial data below the ground state in the sense of their action $S_{\omega ,V}$ by using sign of the virial functional for the initial data. More precisely, for the solution $u\left(t\right)$ with $S_{\omega ,V}\left(u_0\right)<S_{\omega ,0}\left(Q_{\omega ,0}\right)$, if $K_V\left(u_0\right)\ge 0$ then $u$ is bounded in $H^1\left({\mathbb{R}}^d\right)$ and if $K_V\left(u_0\right)<0$ then $u$ is unbounded in $H^1\left({\mathbb{R}}^d\right)$. In addition, for the radial solution $u\left(t\right)$ with $S_{\omega ,V}\left(u_0\right)<S_{\omega ,V}\left(Q_{\omega ,V}\right)$, if $K_V\left(u_0\right)\ge 0$, then $u$ exists globally in time and if $K_V\left(u_0\right)<0$ then $u$ blows up.

Acknowledgments

M.H. is supported by JSPS KAKENHI Grant Number JP22J00787. M.I. is supported by JSPS KAKENHI Grant Number JP19K14581 and JST CREST Grant Number JPMJCR1913.

Conflict of interest

The authors declare no conflict of interest.