On Blow-Up Solutions for the Fourth-Order Nonlinear Schrödinger Equation with Mixed Dispersions

Abstract

In this paper, we consider blow-up solutions for the fourth-order nonlinear Schrödinger equation with mixed dispersions. We study the dynamical properties of blow-up solutions for this equation, including the ${\dot{H}}^{{\gamma }_c}$-concentration and limiting profiles, which extend and improve the existing results in the literature.

1. Introduction

In this paper, we study the nonlinear fourth-order Schrödinger equation with mixed dispersions

$$\left\{\begin{array}{c}i{\psi }_t-{\Delta }^2\psi +\mu \Delta \psi +{\left|\psi \right|}^p\psi =0,(t,x)\in [0,T^{\ast })\times {\mathbb{R}}^N,\hfill \\ \psi (0,x)={\psi }_0\left(x\right),\hfill \end{array}\right.$$

(1)

where $\mu \in \mathbb{R}$, $\psi :[0,T^{\ast })\times {\mathbb{R}}^N\to \mathbb{C}$ is a complex valued function, $0<T^{\ast }\le \infty$, $0<p<4^{\ast }$ (where $4^{\ast }=+\infty$ if $N=1,2,3,4$ and $4^{\ast }=\frac{8}{N-4}$ if $N\ge 5$). Karpman in [1] first introduced the fourth-order Schrödinger Equation (1) to stabilize soliton instabilities. Karpman and Shagalov in [2] also proposed a small fourth-order dispersion term to describe the propagation of intense laser beams in a bulk medium with Kerr nonlinearity. In recent years, there has been a great deal of interest in using higher-order operators to model physical phenomena (see [3,4,5,6,7,8]).

When $\mu =0$, Equation (1) entails the scaling invariance

$${\psi }_{\lambda }(t,x)={\lambda }^{\frac{4}{p}}\psi ({\lambda }^{4}t,\lambda x),\lambda >0.$$

This implies that if $\psi$ solves (1) with $\mu =0$, then ${\psi }_{\lambda }$ solves the same equation with the initial data ${\psi }_{\lambda }(0,x)={\lambda }^{\frac{4}{p}}{\psi }_0\left(\lambda x\right)$. A direct computation shows

$$\parallel {\psi }_{\lambda }{\left(0\right)\parallel }_{{\dot{H}}^{\gamma }}={\lambda }^{\gamma +\frac{4}{p}-\frac{N}{2}}{\parallel {\psi }_0\parallel }_{{\dot{H}}^{\gamma }}.$$

This implies that the Sobolev ${\dot{H}}^{{\gamma }_c}$-norm and Lebesgue $L^{p_c}$-norm are invariant under the scaling $\psi \mapsto {\psi }_{\lambda }$, where

$${\gamma }_c:=\frac{N}{2}-\frac{4}{p}\mathrm{and}{p}_c:=\frac{2N}{N-2{\gamma }_c}=\frac{Np}{4}.$$

Although there is not any scaling invariance for Equation (1) with $\mu \ne 0$, ${\gamma }_c$ and $p_c$ are referred to as the critical Sobolev and Lebesgue exponents of (1), respectively. When $0\le {\gamma }_c\le 2$, i.e., $\frac{8}{N}\le p\le 4^{\ast }$, Equation (1) is referred to as ${\dot{H}}^{{\gamma }_c}$-critical. In particular, when ${\gamma }_c=0$ and ${\gamma }_c=2$, Equation (1) is referred to as $L^2$-critical (or mass-critical) and ${\dot{H}}^2$-critical (or energy-critical), respectively.

If the initial data ${\psi }_0\in H^2$, then Equation (1) reflects the mass and energy conservation laws:

$$\begin{array}{c}\hfill {\parallel \psi \left(t\right)\parallel }_{L^2}={\parallel {\psi }_0\parallel }_{L^2},E\left(\psi \left(t\right)\right)=E\left({\psi }_0\right),\end{array}$$

where the energy E is defined by

$$E\left(\psi \left(t\right)\right)=\frac{1}{2}{\parallel \Delta \psi \left(t\right)\parallel }_{L^2}^2+\frac{\mu }{2}{\parallel \nabla \psi \left(t\right)\parallel }_{L^2}^2-\frac{1}{p+2}{\parallel \psi \left(t\right)\parallel }_{L^{p+2}}^{p+2}.$$

(2)

If the initial data ${\psi }_0\in {\dot{H}}^{{\gamma }_c}\cap {\dot{H}}^2$ with ${\gamma }_c\le 1$, then the equation only assumes energy conservation. The conservation of mass is no longer available in this setting.

Recently, Equation (1) was investigated extensively in [9,10,11,12,13,14,15,16,17,18]. The local well-posedness in $H^2$ was studied in [9,13,15]. The global well-posedness for (1) in $H^2$ was studied by Fibich, Ilan, and Papanicolaou in [19]. The global properties, including the sharp threshold of scattering and blow-up, asymptotical behavior, and scattering were investigated in [12,15,16,17,18,20]. When $0<p<\frac{8}{N}$, it follows that all the solutions of (1) exist globally using the mass conservation. Boulenger and Lenzmann in [21] proved the existence of radial blow-up solutions for (1) with $\frac{8}{N}\le p\le 4^{\ast }$. When $\mu =0$, the dynamical properties of the blow-up solutions of (1) were investigated in [22,23,24,25,26,27,28]. However, when $\mu \ne 0$, the dynamical properties of the blow-up solutions of (1) have not yet been discussed.

The aim of this paper is to consider the dynamical properties of the blow-up solutions of (1) with $\mu \ne 0$. However, compared with the case $\mu =0$ considered in [25,26,28], there are two major difficulties in the analysis of the blow-up solutions of (1). One is the loss of mass conservation due to the initial data ${\psi }_0\in {\dot{H}}^{{\gamma }_c}\cap {\dot{H}}^2$; the other is the loss of scaling invariance to (1) with $\mu \ne 0$. Since there is no scaling invariance for $\mu \ne 0$, we choose the ground states of the equations

$${\Delta }^{2}Q+{(-\Delta )}^{{\gamma }_c}Q-{\left|Q\right|}^{p}Q=0,$$

(3)

and

$${\Delta }^{2}R+{\left|R\right|}^{p_c-2}R-{\left|R\right|}^{p}R=0,$$

(4)

to describe some of the concentration properties and limiting profiles of the blow-up solutions to (1), respectively, where (3) and (4) arise in the study of the optimal constants of inequalities (12) and (14) (see [25]).

The structure of this paper is as follows: In Section 2, we provide some preliminary information, including the local well-posedness of (1), the profile decomposition of the bounded sequences in ${\dot{H}}^{{\gamma }_c}\cap {\dot{H}}^2$, and the localized virial to (1). In Section 3, we investigate the dynamical properties of the blow-up solutions of (1) with $\mu \ne 0$ in the $L^2$-critical and $L^2$-supercritical cases, including the concentration properties and limiting profiles.

2. Preliminaries

First, we recall the local well-posedness for the Cauchy problem (1).

Lemma 1 

([14]). Let $0<p<4^{\ast }$ and ${\psi }_0\in H^2$. Then, there exists $T=T(\parallel {\psi }_0{\parallel }_{H^2})$, such that (1) admits a unique solution $\psi \in C([0,T),H^2)$. If $T^{\ast }<\infty$, then ${\parallel \Delta \psi \left(t\right)\parallel }_{L^2}\to \infty$ as $t\uparrow T^{\ast }$, where $T^{\ast }$ is the maximal existence time of solution $\psi \left(t\right)$. Moreover, for all $[0,T^{\ast })$, the following mass and energy conservation laws follow:

$$M\left(\psi \left(t\right)\right)={\int }_{{\mathbb{R}}^N}{\left|\psi (t,x)\right|}^{2}dx=M\left({\psi }_0\right),$$

(5)

$$E\left(\psi \left(t\right)\right)=E\left({\psi }_0\right),$$

(6)

where $E\left(\psi \right(t\left)\right)$ is defined by (2).

Next, in order to study the existence of the blow-up solutions, we recall the localized virial to (1) established in [21]. Let $\phi :{\mathbb{R}}^N\to \mathbb{R}$ be a radial function which satisfies ${\nabla }^j\phi \in L^{\infty }$, for $1\le j\le 6$,

$$\phi \left(r\right):=\left\{\begin{array}{c}\frac{r^2}{2}forr\le 1\hfill \\ const.forr\ge 10,\hfill \end{array}\right.and{\phi }^{″}\left(r\right)\le 1,forr\ge 0.$$

For $R>0$, we define ${\phi }_R\left(r\right):=R^2\phi \left(\frac{r}{R}\right)$. When $\psi \in C([0,T^{\ast });H^2)$, we define the localized virial of $\psi \left(t\right)$ by

$$M_{{\phi }_R}\left(\psi \left(t\right)\right):=2Im{\int }_{{\mathbb{R}}^N}\overline{\psi }(t,x)\nabla {\phi }_R\left(x\right)\nabla \psi (t,x)dx.$$

(7)

Boulenger and Lenzmann in [21] obtained the following time evolution of $M_{{\phi }_R}\left(\psi \left(t\right)\right)$.

Lemma 2 

([21], Lemma 3.1). Let $0<p<4^{\ast }$ and $R>0$. Let $\psi \in C([0,T^{\ast });H^2)$ be a radial solution to (1), then,

$$\begin{array}{cc}\hfill \frac{d}{dt}{M}_{{\phi }_R}\left(\psi \left(t\right)\right)\le & 2NpE\left(\psi \left(t\right)\right)-{(Np-8)\parallel \Delta \psi \left(t\right)\parallel }_{L^2}^2-(Np-4)\mu {\parallel \nabla \psi \left(t\right)\parallel }_{L^2}^2+X_{\mu }\left[\psi \left(t\right)\right]\hfill \\ \hfill & +\mathcal{O}\left(\frac{1}{R^4}+\frac{{\parallel \nabla \psi \left(t\right)\parallel }_{L^2}^2}{R^2}+\frac{{\parallel \nabla \psi \left(t\right)\parallel }_{L^2}^{p/2}}{R^{\frac{(N-1)p}{2}}}+\frac{\left|\mu \right|}{R^2}\right)\hfill \\ \hfill =& 4Q\left(\psi \left(t\right)\right)+X_{\mu }\left[\psi \left(t\right)\right]+\mathcal{O}\left(\frac{1}{R^4}+\frac{{\parallel \nabla \psi \left(t\right)\parallel }_{L^2}^2}{R^2}+\frac{{\parallel \nabla \psi \left(t\right)\parallel }_{L^2}^{p/2}}{R^{\frac{(N-1)p}{2}}}+\frac{\left|\mu \right|}{R^2}\right),\hfill \end{array}$$

for any $t\in [0,T^{\ast })$, where

$$X_{\mu }\left[\psi \left(t\right)\right]\le \left\{\begin{array}{c}0for\mu \le 0,\hfill \\ A_0{\left|\mu \right|\parallel \nabla \psi \left(t\right)\parallel }_{L^2}^{2}for\mu <0,\hfill \end{array}\right.$$

with some constant $A_0>0$.

Lemma 3 

([29], Proposition 1.32). Let $s_0\le s\le s_1$. Then, ${\dot{H}}^{s_0}\cap {\dot{H}}^{s_1}$ is included in ${\dot{H}}^s$, and

$${\parallel v\parallel }_{{\dot{H}}^s}\le {\parallel v\parallel }_{{\dot{H}}^{s_0}}^{1-\theta }{\parallel v\parallel }_{{\dot{H}}^{s_1}}^{\theta },forallv\in {\dot{H}}^{s_0}\cap {\dot{H}}^{s_1},$$

(8)

where $s=(1-\theta )s_0+\theta s_1$.

Lemma 4 

([26], Theorem 1.1). If $0<p<4^{\ast }$, then

$${\parallel v\parallel }_{L^{p+2}}^{p+2}\le \frac{4(p+2)}{4(p+2)-Np}{\left(\frac{4(p+2)-Np}{Np}\right)}^{\frac{Np}{8}}\frac{1}{{\parallel R\parallel }_{L^2}^p}{\parallel v\parallel }_{L^2}^{\frac{4(p+2)-Np}{4}}{\parallel \Delta v\parallel }_{L^2}^{\frac{Np}{4}},$$

(9)

for all $v\in H^2$, where $R\in H^2$ is a ground state of the equation

$${\Delta }^{2}R+R-{\left|R\right|}^{p}R=0.$$

(10)

Moreover, the following Pohozaev’s identities follow.

$$\begin{array}{c}\hfill {\parallel \Delta R\parallel }_{L^2}^2=\frac{Np}{4(p+2)}{\parallel R\parallel }_{L^{p+2}}^{p+2}=\frac{Np}{8-p(N-4)}{\parallel R\parallel }_{L^2}^2.\end{array}$$

(11)

Lemma 5 

([25], Proposition 3.2). Let $\frac{8}{N}<p<4^{\ast }$. Then, for all $v\in {\dot{H}}^{{\gamma }_c}\cap {\dot{H}}^2$

$${\parallel v\parallel }_{L^{p+2}}^{p+2}\le \frac{p+2}{2}\frac{1}{{\parallel Q\parallel }_{{\dot{H}}^{{\gamma }_c}}^p}{\parallel v\parallel }_{{\dot{H}}^{{\gamma }_c}}^p{\parallel \Delta v\parallel }_{L^2}^2,$$

(12)

where $Q\in {\dot{H}}^{{\gamma }_c}\cap {\dot{H}}^2$ is a ground state of (3). Moreover, the following Pohozaev’s identities follow.

$$\begin{array}{c}\hfill {\parallel Q\parallel }_{{\dot{H}}^2}^2=\frac{2}{p+2}{\parallel Q\parallel }_{L^{p+2}}^{p+2}=\frac{2}{p}{\parallel Q\parallel }_{{\dot{H}}^{{\gamma }_c}}^2.\end{array}$$

(13)

Lemma 6 

([25], Proposition 3.2). Let $\frac{8}{N}<p<4^{\ast }$. Then, for all $v\in L^{p_c}\cap {\dot{H}}^2$

$${\parallel v\parallel }_{L^{p+2}}^{p+2}\le \frac{p+2}{2}\frac{1}{{\parallel R\parallel }_{L^{p_c}}^p}{\parallel v\parallel }_{L^{p_c}}^p{\parallel \Delta v\parallel }_{L^2}^2,$$

(14)

where $R\in L^{p_c}\cap {\dot{H}}^2$ is a ground state solution of the elliptic Equation (4). Moreover, the following Pohozaev’s identities hold true:

$$\begin{array}{c}\hfill {\parallel R\parallel }_{{\dot{H}}^2}^2=\frac{2}{p+2}{\parallel R\parallel }_{L^{p+2}}^{p+2}=\frac{2}{p}{\parallel R\parallel }_{L^{p_c}}^2.\end{array}$$

(15)

Since the uniqueness of the ground state solutions to (10), (3) and (4) is still unknown, to study the dynamical properties of blow-up solutions, we introduce the notions of Sobolev and Lebesgue ground states. Denote

$$\begin{array}{cc}\hfill G_0\left(u\right)& :={\parallel u\parallel }_{L^{\frac{8}{N}+2}}^{\frac{8}{N}+2}÷\left[{\parallel u\parallel }_{L^2}^{\frac{8}{N}}{\parallel \Delta u\parallel }_{L^2}^2\right],u\in H^2,\hfill \\ \hfill G\left(u\right)& :={\parallel u\parallel }_{L^{p+2}}^{p+2}÷\left[{\parallel u\parallel }_{{\dot{H}}^{{\gamma }_{\mathrm{c}}}}^p{\parallel \Delta u\parallel }_{L^2}^2\right],u\in {\dot{H}}^{{\gamma }_{\mathrm{c}}}\cap {\dot{H}}^2,\hfill \\ \hfill K\left(u\right)& :={\parallel u\parallel }_{L^{p+2}}^{p+2}÷\left[{\parallel u\parallel }_{L^{p_{\mathrm{c}}}}^p{\parallel \Delta u\parallel }_{L^2}^2\right],u\in L^{p_{\mathrm{c}}}\cap {\dot{H}}^2.\hfill \end{array}$$

Definition 1 

(Ground states).

1. 

We call the  Sobolev ground states  the maximizers of $G_0$ and G, which are solutions to (10) and (3), respectively. We denote the set of Sobolev ground states of $G_0$ and G by ${\mathcal{G}}_0$ and $\mathcal{G}$, respectively.

2. 

We call the  Lebesgue ground states  the maximizers of K, which are solutions to (4). We denote the set of Lebesgue ground states by $\mathcal{K}$.

It follows from the optimal constants in (9), (12), and (14) that all the Sobolev ground states have the same ${\dot{H}}^{{\gamma }_{\mathrm{c}}}$-norm and all the Lebesgue ground states have the same $L^{p_{\mathrm{c}}}$-norm. We thus denote

$$\begin{array}{c}\hfill G_0:={\parallel Q\parallel }_{L^2},\forall Q\in {\mathcal{G}}_0,G_1:={\parallel Q\parallel }_{{\dot{H}}^{{\gamma }_{\mathrm{c}}}},\forall Q\in \mathcal{G},G_2:={\parallel R\parallel }_{L^{p_{\mathrm{c}}}},\forall R\in \mathcal{K}.\end{array}$$

(16)

Finally, we recall the following two compactness lemmas:

Lemma 7 

([28], Compactness lemma I). Suppose that ${\left\{u_n\right\}}_{n=1}^{\infty }$ is a bounded sequence in $H^2$ and satisfies

$$\underset{n\to \infty }{lim\; sup}\parallel \Delta u_n{\parallel }_{L^2}\le M,\underset{n\to \infty }{lim\; sup}{\parallel u_n\parallel }_{L^{8/N+2}}\ge m>0.$$

Then, there exist ${\left\{x_n\right\}}_{n=1}^{\infty }\subset {\mathbb{R}}^N$ and $V\in H^2$, such that, up to a subsequence,

$$u_n(·+x_n)\rightharpoonup Vweaklyin{H}^2$$

with

$${\parallel V\parallel }_{L^2}^p\ge \frac{2}{p+2}\frac{m^{p+2}}{M^2}{G}_0^p.$$

Lemma 8 

([25], Compactness lemma II). Let $\frac{8}{N}<p<4^{\ast }$. Let ${\left\{v_n\right\}}_{n=1}^{\infty }$ be a bounded sequence in ${\dot{H}}^{{\gamma }_c}\cap {\dot{H}}^2$, such that

$$\underset{n\to \infty }{lim\; sup}\parallel v_n{\parallel }_{{\dot{H}}^2}\le M,\underset{n\to \infty }{lim\; sup}{\parallel v_n\parallel }_{L^{p+2}}\ge m.$$

• Then, there exist $V\in {\dot{H}}^{{\gamma }_c}\cap {\dot{H}}^2$ and a sequence ${\left\{y_n\right\}}_{n=1}^{\infty }$ in ${\mathbb{R}}^N$, such that up to a subsequence,

  $$v_n(·+y_n)\rightharpoonup V\mathrm{weakly}\mathrm{in}{\dot{H}}^{{\gamma }_c}\cap {\dot{H}}^2,$$
  with

  $$\begin{array}{c}\hfill {\parallel V\parallel }_{{\dot{H}}^{{\gamma }_c}}^p\ge \frac{2}{p+2}\frac{m^{p+2}}{M^2}{G}_1^p.\end{array}$$

  (17)
• Then, there exist $W\in L^{p_c}\cap {\dot{H}}^2$ and a sequence ${\left\{z_n\right\}}_{n=1}^{\infty }$ in ${\mathbb{R}}^N$, such that up to a subsequence,

  $$v_n(·+z_n)\rightharpoonup W\mathrm{weakly}\mathrm{in}{L}^{p_c}\cap {\dot{H}}^2,$$
  with

  $$\begin{array}{c}\hfill {\parallel W\parallel }_{L^{p_c}}^p\ge \frac{2}{p+2}\frac{m^{p+2}}{M^2}{G}_2^p.\end{array}$$

  (18)

Remark 1. 

The lower bounds (17) and (18) are optimal. Indeed, taking $v_n=Q$ in the first case and $v_n=R$ in the second case where $Q\in \mathcal{G}$ and $R\in \mathcal{K}$, we obtain the equalities.

3. Dynamic of Blow-Up Solutions in the ${\mathit{L}}^{\mathbf{2}}$-Critical and ${\mathit{L}}^{\mathbf{2}}$-Supercritical Cases

In this section, we study the dynamical properties of the blow-up solutions for (1) in the $L^2$-critical and $L^2$-supercritical cases.

3.1. The Sharp Threshold Mass of Blow-Up and Global Existence

It easily follows from the local well-posedness that the solution of (1) with small initial data exists globally, and the solution may blow up in finite time for some large initial data. Therefore, whether there is a sharp threshold of global existence and blow-up for (1) is of particular interest. Next, we obtain the sharp threshold mass of global existence and blow-up for (1) by using the scaling argument and the inequality (9).

Theorem 1. 

Let ${\psi }_0\in H^2$, $\mu >0$, $p=\frac{8}{N}$. Then, we obtain the following sharp threshold mass of the global existence and blow-up:

(i) If $\parallel {\psi }_0{\parallel }_{L^2}\le G_0$, then all solutions of (1) exist globally.

(ii) For any $\rho >G_0$, there exist initial data ${\psi }_0$, such that $\parallel {\psi }_0{\parallel }_{L^2}=\rho$ and the corresponding solution $\psi \left(t\right)$ of (1) blows up in finite time.

Remark 2. 

When $\mu =0$, Fibich, Ilan, and Papanicolaou in [19] proved that all solutions of (1) with initial data $\parallel {\psi }_0{\parallel }_{L^2}<G_0$ exist globally. When $\mu >0$, we prove that all solutions of (1) with initial data $\parallel {\psi }_0{\parallel }_{L^2}\le G_0$ exist globally. This suggest that the defocusing second-order dispersion term may prevent the occurrence blow-up.

Proof. 

(i) When $\parallel {\psi }_0{\parallel }_{L^2}<G_0$, we deduce from (2) and (9) that

$$\begin{array}{cc}\hfill E\left({\psi }_0\right)& =E\left(\psi \left(t\right)\right)=\frac{1}{2}{\parallel \Delta \psi \left(t\right)\parallel }_{L^2}^2+\frac{\mu }{2}{\parallel \nabla \psi \left(t\right)\parallel }_{L^2}^2-\frac{1}{p+2}{\parallel \psi \left(t\right)\parallel }_{L^{p+2}}^{p+2}\hfill \\ & \ge \left(\frac{1}{2}-\frac{\parallel {\psi }_0{\parallel }_{L^2}^p}{2G_0^p}\right){\parallel \Delta \psi \left(t\right)\parallel }_{L^2}^2.\hfill \end{array}$$

Due to $\parallel {\psi }_0{\parallel }_{L^2}<G_0$, we have that ${\parallel \Delta \psi \left(t\right)\parallel }_{L^2}$ is uniformly bounded for all times t. Therefore, (i) follows from the conservation of mass and Lemma 1.

When $\parallel {\psi }_0{\parallel }_{L^2}=G_0$, we prove this result by contradiction. If the solution $\psi \left(t\right)$ of (1) blows up in finite time, then there exists $T^{\ast }>0$, such that ${lim}_{t\to T^{\ast }}{\parallel \Delta \psi \left(t\right)\parallel }_{L^2}=\infty$. Set

$${\rho }^2\left(t\right)={\parallel \Delta R\parallel }_{L^2}/{\parallel \Delta \psi \left(t\right)\parallel }_{L^2}andv(t,x)=\rho {\left(t\right)}^{N/2}\psi (t,\rho \left(t\right)x).$$

Let ${\left\{t_n\right\}}_{n=1}^{\infty }$ be any time sequence, such that $t_n\to T^{\ast }$, ${\rho }_n:=\rho \left(t_n\right)$ and $v_n\left(x\right):=v(t_n,x)$. Then, the sequence $\left\{v_n\right\}$ satisfies

$$\parallel v_n{\parallel }_{L^2}=\parallel \psi \left(t_n\right){\parallel }_{L^2}=\parallel {\psi }_0{\parallel }_{L^2}=G_0,\parallel \Delta v_n{\parallel }_{L^2}={\rho }_n^2\parallel \Delta \psi \left(t_n\right){\parallel }_{L^2}={\parallel \Delta R\parallel }_{L^2}.$$

(19)

Observe that

$$\begin{array}{cc}\hfill 0\le \frac{1}{2}\parallel \Delta v_n{\parallel }_{L^2}^2-\frac{1}{p+2}{\parallel v_n\parallel }_{L^{p+2}}^{p+2}=& {\rho }_n^4\left(\frac{1}{2}\parallel \Delta \psi \left(t_n\right){\parallel }_{L^2}^2-\frac{1}{p+2}{\parallel \psi \left(t_n\right)\parallel }_{L^{p+2}}^{p+2}\right)\hfill \\ \hfill =& {\rho }_n^4\left(E\left({\psi }_0\right)-\frac{\mu }{2}{\parallel \nabla \psi \left(t_n\right)\parallel }_{L^2}^2\right)\hfill \\ \hfill \le & {\rho }_n^{4}E\left({\psi }_0\right)\to 0,asn\to \infty .\hfill \end{array}$$

(20)

This implies that

$$\underset{n\to \infty }{lim}\parallel v_n{\parallel }_{L^{p+2}}^{p+2}=\frac{p+2}{2}{\parallel \Delta R\parallel }_{L^2}^2.$$

Thus, we deduce from (19) that there exist subsequences, still denoted by $\left\{v_n\right\}$ and $u\in H^2\setminus \left\{0\right\}$, such that

$$u_n:={\tau }_{x_n}{v}_n\rightharpoonup u\ne 0weaklyin{H}^2,$$

for some $\left\{x_n\right\}\subseteq {\mathbb{R}}^N$. This implies that there exists $C_0>0$, such that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\parallel \nabla v_n{\parallel }_{L^2}^2=\underset{n\to \infty }{lim}{\parallel \nabla u_n\parallel }_{L^2}^2\ge C_0>0.\end{array}$$

(21)

On the other hand, we deduce from (9) and ${\parallel \psi \left(t\right)\parallel }_{L^2}=\parallel {\psi }_0{\parallel }_{L^2}={\parallel R\parallel }_{L^2}$ that

$$\begin{array}{c}\hfill \frac{1}{2}{\parallel \Delta \psi \left(t\right)\parallel }_{L^2}^2-\frac{1}{p+2}{\parallel \psi \left(t\right)\parallel }_{L^{p+2}}^{p+2}\ge 0,\end{array}$$

for all $t\in [0,T^{\ast })$. This implies that

$$\begin{array}{c}\hfill \frac{\mu }{2}{\parallel \nabla \psi \left(t\right)\parallel }_{L^2}^2\le E\left({\psi }_0\right),\end{array}$$

for all $t\in [0,T^{\ast })$. We consequently obtain that

$$\begin{array}{c}\hfill \parallel \nabla v_n{\parallel }_{L^2}^2={\rho }_n^2{\parallel \nabla \psi \left(t_n\right)\parallel }_{L^2}^2\le \frac{2{\rho }_n^2}{\mu }E\left({\psi }_0\right)\to 0,asn\to \infty ,\end{array}$$

which is a contradiction with (21). Thus, the solution $\psi \left(t\right)$ of (1) exists globally.

(ii) Let $R\in {\mathcal{G}}_0$ be radial. We define the initial data ${\psi }_0\left(x\right)=c{\lambda }^{\frac{N}{2}}R\left(\lambda x\right)$ with $c=\frac{\rho }{G_0}$ and some $\lambda >1$. Then, $\parallel {\psi }_0{\parallel }_{L^2}=\rho$. Applying the Poho$\breve{\mathrm{z}}$aev identity for the following equation:

$${\Delta }^{2}R+R-{\left|R\right|}^{p}R=0,$$

(22)

i.e., $\frac{1}{2}{\parallel \Delta R\parallel }_{L^2}^2=\frac{1}{p+2}{\parallel R\parallel }_{L^{p+2}}^{p+2}$, we deduce that

$$\begin{array}{cc}\hfill E\left({\psi }_0\right)& =\frac{{\left|c\right|}^2{\lambda }^4}{2}{\parallel \Delta R\parallel }_{L^2}^2+\frac{{\mu \left|c\right|}^2{\lambda }^2}{2}{\parallel \nabla R\parallel }_{L^2}^2-\frac{{\left|c\right|}^{p+2}{\lambda }^{\frac{Np}{2}}}{p+2}{\parallel R\parallel }_{L^{p+2}}^{p+2}\hfill \\ \hfill & =-\frac{{\left|c\right|}^2{\lambda }^4}{2}{\left(\right|c|}^p-{1)\parallel \Delta R\parallel }_{L^2}^2+\frac{{\mu \left|c\right|}^2{\lambda }^2}{2}{\parallel \nabla R\parallel }_{L^2}^2.\hfill \end{array}$$

(23)

Now, taking $\lambda$, such that

$$\frac{{\mu \parallel \nabla R\parallel }_{L^2}^2}{{\left(\right|c|}^p-{1)\parallel \Delta R\parallel }_{L^2}^2}<{\lambda }^2.$$

This implies $E\left({\psi }_0\right)<0$. Thus, the solution $\psi$ of (1) with initial data ${\psi }_0$ blows up by applying the same method as that of Theorem 3 in [21]. □

3.2. The $L^2$-Critical Case

In this subsection, we investigate some dynamical properties of the blow-up solutions for (1) with $\mu \ne 0$ in the $L^2$-critical case.

Theorem 2. 

($L^2$-concentration) Let ${\psi }_0\in H^2$, $\mu \ne 0$, $p=\frac{8}{N}$. If the solution $\psi \left(t\right)$ of (1) blows up in finite time $T^{\ast }>0$. Let $a\left(t\right)$ be a real-valued non-negative function defined on $[0,T^{\ast })$ satisfying ${a\left(t\right)\parallel \Delta \psi \left(t\right)\parallel }_{L^2}^{\frac{1}{2}}\to \infty$ as $t\to T^{\ast }$. Then, there exists $x\left(t\right)\in {\mathbb{R}}^N$, such that

$$\underset{t\to T^{\ast }}{lim\; inf}{\int }_{|x-x(t\left)\right|\le a\left(t\right)}{\left|\psi (t,x)\right|}^{2}dx\ge G_0^2,$$

(24)

where $G_0$ is defined by (16).

Remark 3. 

By a similar analysis as that in Remark 2, this theorem gives the $L^2$-concentration and rate of $L^2$-concentration of the blow-up solutions of (1).

Proof. 

Let $R\in {\mathcal{G}}_0$; we set

$${\rho }^2\left(t\right)={\parallel \Delta R\parallel }_{L^2}/{\parallel \Delta \psi \left(t\right)\parallel }_{L^2}andv(t,x)={\rho }^{\frac{N}{2}}\left(t\right)\psi (t,\rho \left(t\right)x).$$

Let ${\left\{t_n\right\}}_{n=1}^{\infty }$ be any time sequence, such that $t_n\to T^{\ast }$, ${\rho }_n:=\rho \left(t_n\right)$ and $v_n\left(x\right):=v(t_n,x)$. Then, the sequence $\left\{v_n\right\}$ satisfies

$$\parallel v_n{\parallel }_{L^2}=\parallel \psi \left(t_n\right){\parallel }_{L^2}=\parallel {\psi }_0{\parallel }_{L^2},\parallel \Delta v_n{\parallel }_{L^2}={\rho }_n^2\parallel \Delta \psi \left(t_n\right){\parallel }_{L^2}={\parallel \Delta R\parallel }_{L^2}.$$

(25)

Observe that

$$\begin{array}{cc}\hfill |E_0\left(v_n\right)|=& \left|\frac{1}{2}{\int }_{{\mathbb{R}}^N}|\Delta v_n{\left(x\right)|}^{2}dx-\frac{1}{p+2}{\int }_{{\mathbb{R}}^N}{\left|v_n\left(x\right)\right|}^{p+2}dx\right|\hfill \\ \hfill =& {\rho }_n^4\left|\frac{1}{2}{\int }_{{\mathbb{R}}^N}|\Delta \psi (t_n,x){|}^{2}dx-\frac{1}{p+2}{\int }_{{\mathbb{R}}^N}{\left|\psi (t_n,x)\right|}^{p+2}dx\right|\hfill \\ \hfill \le & {\rho }_n^4\left|E\left({\psi }_0\right)+\frac{\left|\mu \right|}{2}{\int }_{{\mathbb{R}}^N}{|\nabla \psi (t_n,x)|}^{2}dx\right|.\hfill \end{array}$$

(26)

Thus, applying the inequality (8), we deduce that $E_0\left(v_n\right)\to 0$, as $n\to \infty$. This implies ${\int }_{{\mathbb{R}}^N}|v_n{\left(x\right)|}^{p+2}dx\to \frac{p+2}{2}{\parallel \Delta R\parallel }_{L^2}^2$.

Set $m^{p+2}=\frac{p+2}{2}{\parallel \Delta R\parallel }_{L^2}^2$ and $M={\parallel \Delta R\parallel }_{L^2}$. Then, it follows from Lemma 7 that there exist $V\in H^2$ and ${\left\{x_n\right\}}_{n=1}^{\infty }\subset {\mathbb{R}}^N$, such that, up to a subsequence,

$$v_n(·+x_n)={\rho }_n^{N/2}\psi (t_n,{\rho }_n(·+x_n))\rightharpoonup Vweaklyin{H}^2$$

(27)

with

$${\parallel V\parallel }_{L^2}\ge G_0.$$

(28)

Note that

$$\frac{a\left(t_n\right)}{{\rho }_n}=\frac{a\left(t_n\right){\parallel \Delta \psi \left(t_n\right)\parallel }_{L^2}^{1/2}}{{\parallel \Delta R\parallel }_{L^2}^{1/2}}\to \infty ,asn\to \infty .$$

Then, for every $r>0$, there exists $n_0>0$, such that for every $n>n_0$, $r{\rho }_n<a\left(t_n\right)$. Therefore, using (27), we obtain

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim\; inf}\underset{y\in {\mathbb{R}}^N}{sup}{\int }_{|x-y|\le a\left(t_n\right)}{\left|\psi (t_n,x)\right|}^{2}dx& \ge \underset{n\to \infty }{lim\; inf}\underset{y\in {\mathbb{R}}^N}{sup}{\int }_{|x-y|\le r{\rho }_n}{\left|\psi (t_n,x)\right|}^{2}dx\hfill \\ \hfill & \ge \underset{n\to \infty }{lim\; inf}{\int }_{|x-x_n|\le r{\rho }_n}{\left|\psi (t_n,x)\right|}^{2}dx\hfill \\ \hfill & =\underset{n\to \infty }{lim\; inf}{\int }_{\left|x\right|\le r}{\rho }_n^N{\left|\psi (t_n,{\rho }_n(x+x_n))\right|}^{2}dx\hfill \\ \hfill & =\underset{n\to \infty }{lim\; inf}{\int }_{\left|x\right|\le r}{\left|v(t_n,x+x_n)\right|}^{2}dx\hfill \\ \hfill & \ge \underset{n\to \infty }{lim\; inf}{\int }_{\left|x\right|\le r}{\left|V\left(x\right)\right|}^{2}dx,foreveryr>0,\hfill \end{array}$$

which means that

$$\underset{n\to \infty }{lim\; inf}\underset{y\in {\mathbb{R}}^N}{sup}{\int }_{|x-y|\le a\left(t_n\right)}|\psi (t_n,x){|}^{2}dx\ge {\int }_{{\mathbb{R}}^N}{\left|V\left(x\right)\right|}^{2}dx.$$

Since the sequence ${\left\{t_n\right\}}_{n=1}^{\infty }$ is arbitrary, we obtain

$$\underset{t\to T^{\ast }}{lim\; inf}\underset{y\in {\mathbb{R}}^N}{sup}{\int }_{|x-y|\le a\left(t\right)}{\left|\psi (t,x)\right|}^{2}dx\ge {\int }_{{\mathbb{R}}^N}{\left|R\left(x\right)\right|}^{2}dx.$$

(29)

Observe that for every $t\in [0,T^{\ast })$, the function $g\left(y\right):={\int }_{|x-y|\le a\left(t\right)}{\left|\psi (t,x)\right|}^{2}dx$ is continuous on $y\in {\mathbb{R}}^N$ and $g\left(y\right)\to 0$ as $\left|y\right|\to \infty$. So, there exists a function $x\left(t\right)\in {\mathbb{R}}^N$, such that for every $t\in [0,T^{\ast })$

$$\underset{y\in {\mathbb{R}}^N}{sup}{\int }_{|x-y|\le a\left(t\right)}{\left|\psi (t,x)\right|}^{2}dx={\int }_{|x-x(t\left)\right|\le a\left(t\right)}{\left|\psi (t,x)\right|}^{2}dx.$$

This and (29) yield (24). □

Next, we study the limiting profile of the blow-up $H^2$-solutions with critical norms. To do so, we recall the following characterization of the ground states:

Lemma 9 

(Characterization of ground states [28]). Let $p=\frac{8}{N}$. If $u\in H^2$ is such that ${\parallel u\parallel }_{L^2}=G_0$ and

$$E_0\left(u\right):=\frac{1}{2}{\parallel u\parallel }_{{\dot{H}}^2}^2-\frac{1}{p+2}{\parallel u\parallel }_{L^{p+2}}^{p+2}=0,$$

then there exists $R\in {\mathcal{G}}_0$, such that u is of the form

$$u\left(x\right)=e^{i\theta }{\lambda }^{\frac{N}{2}}R(\lambda x+x_0),$$

for $\theta \in {\mathbb{R}}^N,\lambda >0$ and $x_0\in {\mathbb{R}}^N$.

Theorem 3. 

Let ${\psi }_0\in H^2$, $\mu <0$, $p=\frac{8}{N}$. Assume $\parallel {\psi }_0{\parallel }_{L^2}=G_0$ and the corresponding solution ψ of (1) blows up in finite time $T^{\ast }>0$, then there exist $R_1\in {\mathcal{G}}_0$, $\rho \left(t\right)>0$, $x\left(t\right)\in {\mathbb{R}}^N$ and $\theta \left(t\right)\in [0,2\pi )$, such that

$${\rho }^{N/2}\left(t\right)\psi (t,\rho \left(t\right)(·+x\left(t\right)))e^{i\theta \left(t\right)}\to R_{1}stronglyin{H}^2,ast\to T^{\ast }.$$

(30)

Proof. 

We use the notations in the proof of Theorem 2. Assume that $\parallel {\psi }_0{\parallel }_{L^2}=G_0$. Recall that we have verified that ${\parallel V\parallel }_{L^2}\ge G_0$ in the proof of Theorem 2. Whence

$$G_0\le {\parallel V\parallel }_{L^2}\le \underset{n\to \infty }{lim\; inf}\parallel v_n{\parallel }_{L^2}=\underset{n\to \infty }{lim\; inf}\parallel \psi \left(t_n\right){\parallel }_{L^2}={\parallel {\psi }_0\parallel }_{L^2}=G_0,$$

and then,

$$\underset{n\to \infty }{lim}\parallel v_n{\parallel }_{L^2}={\parallel V\parallel }_{L^2}=G_0,$$

(31)

which implies

$$v_n(·+x_n)\to Vstronglyin{L}^{2}asn\to \infty .$$

We infer from the inequality (8) that

$$\parallel \nabla (v_n(·+x_n)-V){\parallel }_{L^2}^2\le C\parallel v_n(·+x_n){-V\parallel }_{L^2}{\parallel \Delta (v_n(·+x_n)-V)\parallel }_{L^2}.$$

From $\parallel \Delta v_n(·+x_n){\parallel }_{L^2}\le C$, we obtain

$$\nabla v_n(·+x_n)\to \nabla Vin{L}^{2}asn\to \infty .$$

Next, we will prove that $v_n(·+x_n)$ converges to V strongly in $H^2$. For this purpose, we estimate as follows:

$$\begin{array}{cc}\hfill 0=\underset{n\to \infty }{lim}\left|E_0\left(v_n\right)\right|=& \left|\frac{1}{2}{\int }_{{\mathbb{R}}^N}{|\Delta R\left(x\right)|}^{2}dx-\frac{1}{p+2}\underset{n\to \infty }{lim}{\int }_{{\mathbb{R}}^N}{\left|v_n\left(x\right)\right|}^{p+2}dx\right|\hfill \\ \hfill =& \left|\frac{1}{2}{\int }_{{\mathbb{R}}^N}{|\Delta R\left(x\right)|}^{2}dx-\frac{1}{p+2}{\int }_{{\mathbb{R}}^N}{\left|V\left(x\right)\right|}^{p+2}dx\right|.\hfill \end{array}$$

(32)

Thus, we infer from the inequality (9) that

$$\frac{1}{2}{\int }_{{\mathbb{R}}^N}{|\Delta R\left(x\right)|}^{2}dx=\frac{1}{p+2}{\int }_{{\mathbb{R}}^N}{\left|V\left(x\right)\right|}^{p+2}dx\le \frac{1}{2}\frac{{\parallel V\parallel }_{L^2}^p}{G_0^p}{\parallel \Delta V\parallel }_{L^2}^2=\frac{1}{2}{\parallel \Delta V\parallel }_{L^2}^2.$$

(33)

On the other hand, we deduce from (25) that ${\parallel \Delta V\parallel }_{L^2}\le {lim\; inf}_{n\to \infty }\parallel \Delta v_n(·+x_n){\parallel }_{L^2}={\parallel \Delta R\parallel }_{L^2}$. Hence, we have ${\parallel Q\parallel }_{H^2}={\parallel V\parallel }_{H^2}$ and

$$v_n(·+x_n)\to Vstronglyin{H}^{2}asn\to \infty .$$

(34)

This and (33) imply that

$$E_0\left(V\right)=\frac{1}{2}{\int }_{{\mathbb{R}}^N}{|\Delta V\left(x\right)|}^{2}dx-\frac{1}{p+2}{\int }_{{\mathbb{R}}^N}{\left|V\left(x\right)\right|}^{p+2}dx=0.$$

Up to now, we have verified that

$${\parallel V\parallel }_{L^2}=G_{0}and{E}_0\left(V\right)=0.$$

Applying Lemma 9, there exists $R_1\in {\mathcal{G}}_0$, such that

$$V\left(x\right)=e^{i\theta }{R}_1(x+x_0)forsome\theta \in [0,2\pi ),x_0\in {\mathbb{R}}^N$$

and

$${\rho }_n^{N/2}\psi (t_n,{\rho }_n(·+x_0))\to e^{i\theta }{R}_1(·+x_0)stronglyin{H}^{2}asn\to \infty .$$

Since the sequence ${\left\{t_n\right\}}_{n=1}^{\infty }$ is arbitrary, we infer that there are two functions $x\left(t\right)\in {\mathbb{R}}^N$ and $\theta \left(t\right)\in [0,2\pi )$, such that

$${\rho }^{N/2}\left(t\right)e^{i\theta \left(t\right)}\psi (t,\rho \left(t\right)(x+x\left(t\right)))\to R_{1}stronglyin{H}^{2}ast\to T^{\ast }.$$

□

3.3. The $L^2$-Supercritical Case

In this subsection, we investigate some dynamical properties of the blow-up solutions for (1) with ${\psi }_0\in {\dot{H}}^{{\gamma }_c}\cap {\dot{H}}^2$ in the $L^2$-supercritical case. The main difficulty in this consideration is the lack of conservation of mass.

Theorem 4. 

Let $\mu \in \mathbb{R}$, $\frac{8}{N}<p<4^{\ast }$, ${\psi }_0\in {\dot{H}}^{\gamma }\cap {\dot{H}}^2$ with $\gamma =min\{{\gamma }_c,1\}$. If the solution $\psi \left(t\right)$ of (1) blows up in finite time $T^{\ast }>0$ and satisfies

$$\underset{t\in [0,T^{\ast })}{sup}{\parallel \psi \left(t\right)\parallel }_{{\dot{H}}^{{\gamma }_c}}<\infty if{\gamma }_c\le 1,\underset{t\in [0,T^{\ast })}{sup}{\parallel \psi \left(t\right)\parallel }_{{\dot{H}}^{{\gamma }_c}\cap {\dot{H}}^1}<\infty if1<{\gamma }_c<2.$$

(35)

Assume that $a\left(t\right)>0$, such that

$${a\left(t\right)\parallel \Delta \psi \left(t\right)\parallel }_{L^2}^{\frac{1}{2-{\gamma }_c}}\to \infty ,$$

(36)

as $t\to T^{\ast }$. Then, there exist $x_1\left(t\right),x_2\left(t\right)\in {\mathbb{R}}^N$, such that

$$\underset{t\to T^{\ast }}{lim\; inf}{\int }_{|x-x_1\left(t\right)|\le a\left(t\right)}{\left|{(-▵)}^{\frac{{\gamma }_c}{2}}\psi (t,x)\right|}^{2}dx\ge G_1^2,$$

(37)

and

$$\underset{t\to T^{\ast }}{lim\; inf}{\int }_{|x-x_2\left(t\right)|\le a\left(t\right)}{\left|\psi (t,x)\right|}^{p_c}dx\ge G_2^{p_c}.$$

(38)

Remark 4. 

The assumption ${\psi }_0\in {\dot{H}}^{\gamma }\cap {\dot{H}}^2$ with $\gamma =min\{{\gamma }_c,1\}$ guarantees that the energy $E\left(\psi \right)$ is well-defined.

Proof. 

Let $Q\in \mathcal{G}$; we set

$$\rho \left(t\right)={\parallel \Delta Q\parallel }_{L^2}^{\frac{1}{2-{\gamma }_c}}/{\parallel \Delta \psi \left(t\right)\parallel }_{L^2}^{\frac{1}{2-{\gamma }_c}}andv(t,x)={\rho }^{\frac{4}{p}}\left(t\right)\psi (t,\rho \left(t\right)x).$$

Let ${\left\{t_n\right\}}_{n=1}^{\infty }$ be an any time sequence, such that $t_n\to T^{\ast }$, ${\rho }_n=\rho \left(t_n\right)$ and $v_n\left(x\right)=v(t_n,x)$. Then, it follows from assumption (35) that $v_n$ satisfies $\parallel v_n{\parallel }_{{\dot{H}}^{{\gamma }_c}}={\parallel \psi \left(t_n\right)\parallel }_{{\dot{H}}^{{\gamma }_c}}<\infty$ uniformly in n. Moreover, by some direct computations, we obtain

$$\parallel \Delta v_n{\parallel }_{L^2}={\rho }_n^{2-{\gamma }_c}\parallel \Delta \psi \left(t_n\right){\parallel }_{L^2}={\parallel \Delta Q\parallel }_{L^2},$$

and

$$\begin{array}{cc}\hfill |E_0\left(v_n\right)|=& \left|\frac{1}{2}{\int }_{{\mathbb{R}}^N}|\Delta v_n{\left(x\right)|}^{2}dx-\frac{1}{p+2}{\int }_{{\mathbb{R}}^N}{\left|v_n\left(x\right)\right|}^{p+2}dx\right|\hfill \\ \hfill =& {\rho }_n^{2(2-{\gamma }_c)}\left|\frac{1}{2}{\int }_{{\mathbb{R}}^N}|\Delta \psi (t_n,x){|}^{2}dx-\frac{1}{p+2}{\int }_{{\mathbb{R}}^N}{\left|\psi (t_n,x)\right|}^{p+2}dx\right|\hfill \\ \hfill =& {\rho }_n^{2(2-{\gamma }_c)}\left|E\left(\psi \left(t_n\right)\right)-\frac{\mu }{2}{\int }_{{\mathbb{R}}^N}{|\nabla \psi (t_n,x)|}^{2}dx\right|\hfill \\ \hfill =& \frac{{\parallel \Delta Q\parallel }_{L^2}^2}{\parallel \Delta \psi \left(t_n\right){\parallel }_{L^2}^2}\left|E\left({\psi }_0\right)-\frac{\mu }{2}{\int }_{{\mathbb{R}}^N}{|\nabla \psi (t_n,x)|}^{2}dx\right|.\hfill \end{array}$$

(39)

When $0<{\gamma }_c<1$, applying the inequality (8), that is

$$\parallel \nabla \psi \left(t_n\right){\parallel }_{L^2}^2\le \parallel \Delta \psi \left(t_n\right){\parallel }_{L^2}^{\frac{2(1-{\gamma }_c)}{2-{\gamma }_c}}{\parallel \psi \left(t_n\right)\parallel }_{{\dot{H}}^{{\gamma }_c}}^{\frac{2}{2-{\gamma }_c}},$$

(40)

we have $E_0\left(v_n\right)\to 0$ as $n\to \infty$. When $1\le {\gamma }_c<2$, it follows from (35) that $E_0\left(v_n\right)\to 0$ as $n\to \infty$. These imply that $\parallel v_n{\parallel }_{L^{p+2}}^{p+2}\to \frac{p+2}{2}{\parallel \Delta Q\parallel }_{L^2}^2$ as $n\to \infty$.

Set $m^{p+2}=\frac{p+2}{2}{\parallel \Delta Q\parallel }_{L^2}^2$ and $M={\parallel \Delta Q\parallel }_{L^2}$. Then, it follows from Lemma 8 that there exist $V\in {\dot{H}}^{{\gamma }_c}\cap {\dot{H}}^2$ and ${\left\{x_n\right\}}_{n=1}^{\infty }\subset {\mathbb{R}}^N$, such that up to a subsequence,

$$v_n(·+x_n)={\rho }_n^{\frac{4}{p}}\psi (t_n,{\rho }_n·+x_n)\rightharpoonup Vweaklyin{\dot{H}}^{{\gamma }_c}\cap {\dot{H}}^2$$

with

$${\parallel V\parallel }_{{\dot{H}}^{{\gamma }_c}}\ge G_1.$$

(41)

By the definition of ${\dot{H}}^{{\gamma }_c}$, we have

$${(-\Delta )}^{\frac{{\gamma }_c}{2}}{\rho }_n^{\frac{4}{p}}\psi (t_n,{\rho }_n·+x_n)\rightharpoonup {(-\Delta )}^{\frac{{\gamma }_c}{2}}Vweaklyin{L}^2.$$

Thus, for any $R>0$,

$${\int }_{\left|x\right|\le R}{|(-\Delta )}^{\frac{{\gamma }_c}{2}}{V\left(x\right)|}^{2}dx\le \underset{n\to \infty }{lim\; inf}{\int }_{|x-x_n|\le {\rho }_{n}R}{\left|{(-\Delta )}^{\frac{{\gamma }_c}{2}}\psi (t_n,x)\right|}^{2}dx.$$

In view of the assumption $a\left(t_n\right)/{\rho }_n\to \infty$, this implies immediately

$${\int }_{\left|x\right|\le R}{|(-\Delta )}^{\frac{{\gamma }_c}{2}}{V|}^{2}dx\le \underset{n\to \infty }{lim\; inf}\underset{y\in {\mathbb{R}}^N}{sup}{\int }_{|x-y|\le a\left(t_n\right)}{\left|{(-\Delta )}^{\frac{{\gamma }_c}{2}}\psi (t_n,x)\right|}^{2}dx.$$

Then, we can prove this theorem by a similar argument as that in Theorem 3. The proof of (38) is similar, so we omit it. This completes the proof. □

Let us now study the limiting profile of the blow-up ${\dot{H}}^{{\gamma }_c}\cap {\dot{H}}^2$ solutions with critical norms. To do so, we recall the following characterization of the ground states.

Lemma 10 

(Characterization of ground states [25]). Let $\frac{8}{N}<p<4^{\ast }$.

1. 

If $u\in {\dot{H}}^{{\gamma }_c}\cap {\dot{H}}^2$ is such that ${\parallel u\parallel }_{{\dot{H}}^{{\gamma }_c}}=G_1$ and

$$E_0\left(u\right):=\frac{1}{2}{\parallel u\parallel }_{{\dot{H}}^2}^2-\frac{1}{p+2}{\parallel u\parallel }_{L^{p+2}}^{p+2}=0,$$

then, u is of the form

$$u\left(x\right)=e^{i\theta }{\lambda }^{\frac{4}{p}}Q(\lambda x+x_0),$$

for some $Q\in \mathcal{G}$, $\theta \in {\mathbb{R}}^N,\lambda >0$ and $x_0\in {\mathbb{R}}^N$.

2. 

If $u\in L^{p_c}\cap {\dot{H}}^2$ is such that ${\parallel u\parallel }_{L^{p_c}}=G_2$ and

$$H\left(u\right):=\frac{1}{2}{\parallel u\parallel }_{{\dot{H}}^2}^2-\frac{1}{p+2}{\parallel u\parallel }_{L^{p+2}}^{p+2}=0,$$

then, u is of the form

$$u\left(y\right)=e^{i\vartheta }{\rho }^{\frac{4}{p}}R(\rho y+y_0),$$

for some $R\in \mathcal{K}$, $\vartheta \in {\mathbb{R}}^N,\rho >0$ and $y_0\in {\mathbb{R}}^N$.

Proposition 1 

(Limiting profile with critical norms). Let $\mu \in \mathbb{R}$, $\frac{8}{N}<p<4^{\ast }$, ${\psi }_0\in {\dot{H}}^{\gamma }\cap {\dot{H}}^2$ with $\gamma =min\{{\gamma }_c,1\}$, and the corresponding solution $\psi \left(t\right)$ of (1) blows up in the finite time $T^{\ast }>0$.

1. 

Assume that

$$\begin{array}{c}\hfill \underset{t\in [0,T^{\ast })}{sup}{\parallel \psi \left(t\right)\parallel }_{{\dot{H}}^{{\gamma }_c}}=G_1.\end{array}$$

(42)

If $1<{\gamma }_c<2$, assume further that ${sup}_{t\in [0,T^{\ast })}{\parallel \psi \left(t\right)\parallel }_{{\dot{H}}^1}<\infty$. Then, there exists $Q_1\in \mathcal{G}$, $\theta \left(t\right)\in \mathbb{R},\lambda \left(t\right)>0$ and $y\left(t\right)\in {\mathbb{R}}^N$, such that

$$e^{i\theta \left(t\right)}{\lambda }^{\frac{4}{p}}\left(t\right)\psi (t,\lambda \left(t\right)·+y\left(t\right))\to Q_1\mathrm{strongly}\mathrm{in}{\dot{H}}^{{\gamma }_c}\cap {\dot{H}}^2,$$

as $t\uparrow T^{\ast }$.

2. 

Assume that

$$\begin{array}{c}\hfill \underset{t\in [0,T^{\ast })}{sup}{\parallel \psi \left(t\right)\parallel }_{{\dot{H}}^{{\gamma }_c}}<\infty ,\underset{t\in [0,T^{\ast })}{sup}{\parallel \psi \left(t\right)\parallel }_{L^{p_c}}=G_2.\end{array}$$

(43)

If $1<{\gamma }_c<2$, assume further that ${sup}_{t\in [0,T^{\ast })}{\parallel \psi \left(t\right)\parallel }_{{\dot{H}}^1}<\infty$. Then, there exist $Q_2\in \mathcal{K},\vartheta \left(t\right)\in \mathbb{R},\rho \left(t\right)>0$ and $z\left(t\right)\in {\mathbb{R}}^N$, such that

$$e^{i\vartheta \left(t\right)}{\rho }^{\frac{4}{p}}\left(t\right)\psi (t,\rho \left(t\right)·+z\left(t\right))\to Q_2\mathit{strongly}\mathit{in}{L}^{p_c}\cap {\dot{H}}^2,$$

as $t\uparrow T^{\ast }$.

Proof. 

We only treat the first term, the second one is similar. It is enough to show that for any ${\left(t_n\right)}_{n\ge 1}$ satisfying $t_n\uparrow T^{\ast }$, there exists a subsequence still denoted by ${\left(t_n\right)}_{n\ge 1}$, $Q_1\in \mathcal{G}$, sequences ${\theta }_n\in \mathbb{R},{\lambda }_n>0$ and $y_n\in {\mathbb{R}}^N$, such that

$$\begin{array}{c}\hfill e^{it{\theta }_n}{\lambda }_n^{\frac{4}{p}}\psi (t_n,{\lambda }_n·+y_n)\to Q_1\mathrm{strongly}\mathrm{in}{\dot{H}}^{{\gamma }_c}\cap {\dot{H}}^2,\end{array}$$

(44)

as $n\to \infty$. Using the notation given in the proof of Theorem 4, we have

$$v_n(·+y_n)={\lambda }_n^{\frac{4}{p}}\psi (t_n,{\lambda }_n·+y_n)\rightharpoonup V\mathrm{weakly}\mathrm{in}{\dot{H}}^{{\gamma }_c}\cap {\dot{H}}^2,$$

as $n\to \infty$ with ${\parallel V\parallel }_{{\dot{H}}^{{\gamma }_c}}\ge G_1$. By the semi-continuity of weak convergence, (41) and (42), we have

$$G_1\le {\parallel V\parallel }_{{\dot{H}}^{{\gamma }_c}}\le \underset{n\to \infty }{lim\; inf}\parallel v_n{\parallel }_{{\dot{H}}^{{\gamma }_c}}=\underset{n\to \infty }{lim\; inf}{\parallel \psi \left(t_n\right)\parallel }_{{\dot{H}}^{{\gamma }_c}}\le G_1.$$

We thus obtain

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\parallel v_n{\parallel }_{{\dot{H}}^{{\gamma }_c}}={\parallel V\parallel }_{{\dot{H}}^{{\gamma }_c}}=G_1.\end{array}$$

(45)

This shows that $v_n(·+y_n)\to V$ strongly in ${\dot{H}}^{{\gamma }_c}$ as $n\to \infty$. Using the sharp Gagliardo–Nirenberg inequality (12), we have

$$v_n(·+y_n)\to V\mathrm{strongly}\mathrm{in}{L}^{p+2},$$

as $n\to \infty$. Using (39) and (45), the sharp Gagliardo–Nirenberg inequality (12) yields

$${\parallel Q\parallel }_{{\dot{H}}^2}^2=\frac{2}{p+2}\underset{n\to \infty }{lim}\parallel v_n{\parallel }_{L^{p+2}}^{p+2}=\frac{2}{p+2}{\parallel V\parallel }_{L^{p+2}}^{p+2}\le {\left(\frac{{\parallel V\parallel }_{{\dot{H}}^{{\gamma }_c}}}{G_1}\right)}^p{\parallel V\parallel }_{{\dot{H}}^2}^2={\parallel V\parallel }_{{\dot{H}}^2}^2.$$

This combined with

$${\parallel V\parallel }_{{\dot{H}}^2}\le \underset{n\to \infty }{lim\; inf}\parallel v_n{\parallel }_{{\dot{H}}^2}={\parallel Q\parallel }_{{\dot{H}}^2}$$

shows that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\parallel v_n{\parallel }_{{\dot{H}}^2}={\parallel V\parallel }_{{\dot{H}}^2}={\parallel Q\parallel }_{{\dot{H}}^2}.\end{array}$$

(46)

Combining (45), (46) and the fact $v(·+y_n)\rightharpoonup V$ weakly in ${\dot{H}}^{{\gamma }_c}\cap {\dot{H}}^2$, we conclude that

$$v_n(·+y_n)\to V\mathrm{strongly}\mathrm{in}{\dot{H}}^{{\gamma }_c}\cap {\dot{H}}^2,$$

as $n\to \infty$. In particular, we have

$$E_0\left(V\right)=\underset{n\to \infty }{lim}{E}_0\left(v_n\right)=0.$$

Therefore, we have proved that $V\in {\dot{H}}^{{\gamma }_c}\cap {\dot{H}}^2$ and satisfies

$${\parallel V\parallel }_{{\dot{H}}^{{\gamma }_c}}=G_1,E_0\left(V\right)=0.$$

Applying Lemma 10, there exists $Q_1\in \mathcal{G}$, such that $V\left(y\right)=e^{i\theta }{\lambda }^{\frac{4}{p}}{Q}_1(\lambda y+y_0)$ for some $\theta \in \mathbb{R},\lambda >0$ and $y_0\in {\mathbb{R}}^N$. We thus obtain

$$v_n(·+y_n)={\lambda }_n^{\frac{4}{p}}\psi (t_n,{\lambda }_n·+y_n)\to V=e^{i\theta }{\lambda }^{\frac{4}{p}}{Q}_1(\lambda ·+y_0)\mathrm{strongly}\mathrm{in}{\dot{H}}^{{\gamma }_c}\cap {\dot{H}}^2,$$

as $n\to \infty$. Redefining variables, we prove (44). The proof is complete. □