Sharp thresholds of blow-up and global existence for the Schrödinger equation with combined power-type and Choquard-type nonlinearities

Abstract

In this paper, we consider the sharp thresholds of blow-up and global existence for the nonlinear Schrödinger–Choquard equation

$$ i\psi _{t}+\Delta \psi =\lambda _{1} \vert \psi \vert ^{p_{1}}\psi +\lambda _{2}\bigl(I _{\alpha } \ast \vert \psi \vert ^{p_{2}}\bigr) \vert \psi \vert ^{p_{2}-2}\psi . $$

We derive some finite time blow-up results. Due to the failure of this equation to be scale invariant, we obtain some sharp thresholds of blow-up and global existence by constructing some new estimates. In particular, we prove the global existence for this equation with critical mass in the $L^{2}$-critical case. Our obtained results extend and improve some recent results.

1 Introduction

In this paper, we study the sharp threshold of blow-up and global existence for the nonlinear Schrödinger–Choquard equation

$$ \left \{ \textstyle\begin{array}{l} i\psi _{t}+\Delta \psi =\lambda _{1} \vert \psi \vert ^{p_{1}}\psi +\lambda _{2}(I _{\alpha }\ast \vert \psi \vert ^{p_{2}}) \vert \psi \vert ^{p_{2}-2}\psi , \\ \psi (0,x) = \psi _{0} (x), \end{array}\displaystyle \right . $$

(1.1)

where $\psi (t,x):[0,T^{*})\times \mathbb{R}^{N} \rightarrow \mathbb{C}$ and $0< T^{*}\leq \infty$, $N\geq 3$, $\psi _{0} \in H^{1}$, $\lambda _{1},\lambda _{2} \in \mathbb{R}$, $0< p_{1}< \frac{4}{N-2}$, $1+\frac{\alpha }{N}< p_{2}<1+\frac{2+\alpha }{N-2}$, $I_{\alpha }: \mathbb{R}^{N}\rightarrow \mathbb{R}$ is the Riesz potential defined by

$$ I_{\alpha }(x)=\frac{\varGamma (\frac{N-\alpha }{2})}{\varGamma (\frac{ \alpha }{2})\pi ^{N/2}2^{\alpha } \vert x \vert ^{N-\alpha }}, $$

where Γ is the Gamma function and $\max \{0,N-4\}<\alpha <N$.

When $\lambda _{2}=0$, Eq. (1.1) is the classical Schrödinger equation which appears in various areas of physics, such as nonlinear plasmas and nonlinear optics; see [2, 18]. This class of equations received a great deal of attention from mathematicians see [2, 18]. Particularly, from scaling invariance of (1.1) with $\lambda _{2}=0$, Weinstein [19] and Zhang [21] obtained the sharp threshold of blow-up and global existence for the $L^{2}$-critical nonlinearity and $L^{2}$-supercritical nonlinearity, respectively.

When $\lambda _{1}=0$, $0<\alpha <N$ and $1+\frac{\alpha }{N}< p_{2}<\frac{N+ \alpha }{N-2}$, under the assumption that the local well-posedness holds for (1.1), Chen and Guo [3] derived the existence of blow-up solutions and the instability of standing waves. When $0<\alpha <N$ and $1+\frac{\alpha }{N}< p_{2}<1+\frac{2+\alpha }{N}$, Squassina et al. in [1] studied the soliton dynamics of (1.1) under the assumption that the solution ψ of (1.1) is in $C([0,\infty ),H^{2})\cap C^{1}((0,\infty ),L^{2})$. The dynamical properties of blow-up solutions have been investigated in [11]. In [8], Feng and Yuan systematically studied the Cauchy problem (1.1) for general $\max \{0,N-4\}< \alpha <N$ and $2\leq p_{2}<\frac{N+\alpha }{N-2}$. More precisely, they studied the local well-posedness, global existence, the existence of blow-up solutions and the dynamics of blow-up solutions. The sharp threshold of global existence and blow-up, the instability of standing wave of (1.1) with $\lambda _{1}=0$ and a harmonic potential have been investigated in [5].

From the local well-posedness of (1.1) with $\lambda _{1}=0$ or $\lambda _{2}=0$, for small initial data $\psi _{0}$, the solution $\psi (t)$ to (1.1) exists globally, and the solution $\psi (t)$ may blow up for some large initial data. Hence, whether there are some sharp thresholds of global existence and blow-up for (1.1) is a very interesting problem. In particular, the sharp thresholds of global existence and blow-up for nonlinear Schrödinger equations are pursued strongly in [2, 4, 6, 7, 9, 12–24]. However, in these papers, the scale invariance plays an important role in the study of the sharp threshold of blow-up and global existence. When $\lambda _{1}\neq 0$ and $\lambda _{2}\neq 0$, there is no any scaling invariance for Eq. (1.1). Therefore, the study of the sharp threshold of blow-up and global existence for (1.1) with $\lambda _{1}\neq 0$ and $\lambda _{2}\neq 0$ is of particular interest.

To study this problem, we mainly use the idea of Zhang and Zhu [22], where they studied sharp criteria for the Davey–Stewartson system

$$ i\psi _{t}+\Delta \psi =\lambda _{1} \vert \psi \vert ^{p}\psi +\lambda _{2}E\bigl( \vert \psi \vert ^{2}\bigr)\psi . $$

(1.2)

Due to the failure of (1.1) to be scale invariant, motivated by the idea in [22], we must construct some new estimates to establish some sharp thresholds of blow-up and global existence for (1.1). We will derive sharp thresholds of blow-up and global existence for (1.1) in the following three cases: (i) $\lambda _{1}<0$ and $\lambda _{2}<0$; (ii) $\lambda _{1}>0$ and $\lambda _{2}<0$; (iii) $\lambda _{1}<0$ and $\lambda _{2}>0$. However, the authors in [22] only studied sharp criteria for (1.2) with $\lambda _{1}<0$ and $\lambda _{2}<0$. Therefore, we extend and improve these sharp thresholds for the Davey–Stewartson system to the Schrödinger–Choquard equation. In particular, we can prove the global existence for this equation with critical mass in the $L^{2}$-critical case.

This paper is organized as follows: in Sect. 2, we recall some preliminaries. In Sect. 3, we will derive some sufficient conditions on existence of blow-up solutions. In Sect. 4, we will derive some sharp thresholds of blow-up and global existence for (1.1) by constructing some new estimates. Section 5 is a concluding section.

2 Preliminaries

In order to study the sharp threshold of blow-up and global existence for (1.1), we first make the following assumption about the local well-posedness of (1.1).

Assumption 1

Let $\psi _{0} \in H^{1}$, $0< p_{1}< \frac{4}{N-2}$ and $1+\frac{\alpha }{N}< p_{2}<1+\frac{2+\alpha }{N-2}$ with $N\geq 3$. Then, there exist $T^{*}>0$ and a unique maximal solution $u\in C([0,T^{*}),H^{1})$. In addition, if $T^{\ast }< \infty$, then $\| \psi (t)\| _{H^{1}}\rightarrow \infty$ as $t\uparrow T^{\ast }$. Moreover, the solution $\psi (t)$ satisfies

$$\begin{aligned}& \bigl\Vert \psi (t) \bigr\Vert _{L^{2}}= \Vert \psi _{0} \Vert _{L^{2}}, \end{aligned}$$

(2.1)

$$\begin{aligned}& E\bigl(\psi (t)\bigr)=E(\psi _{0} ), \end{aligned}$$

(2.2)

for all $0\leq t< T^{*}$, where $E(\psi (t))$ is defined by

$$\begin{aligned} E\bigl(\psi (t)\bigr):={} &\frac{1}{2} \int _{\mathbb{R}^{N}} \bigl\vert \nabla \psi (t,x) \bigr\vert ^{2} \,dx+\frac{\lambda _{1}}{p_{1}+2} \int _{\mathbb{R}^{N}} \bigl\vert \psi (t,x) \bigr\vert ^{p _{1}+2}\,dx \\ &{}+\frac{\lambda _{2}}{2p_{2}} \int _{\mathbb{R}^{N}} \bigl(I_{\alpha }\ast \vert \psi \vert ^{p_{2}}\bigr) (t,x) \bigl\vert \psi (t,x) \bigr\vert ^{p_{2}}\,dx. \end{aligned}$$

(2.3)

Remark

When $0< p_{1}<\frac{4}{N-2}$ and $2\leq p_{2}<1+\frac{2+ \alpha }{N-2}$, this assumption can be easily proved by Strichartz’s estimates and a fixed point argument; see [2, 8]. When $1+\frac{\alpha }{N}< p_{2}<2$, we deduce from the Hardy–Littlewood–Sobolev inequality that $\int _{\mathbb{R}^{N}} (I _{\alpha }\ast |\psi |^{p_{2}})|\psi |^{p_{2}}\,dx$ is well-defined for $\psi \in H^{1}$. Thus, we assume that the local well-posedness of (1.1) holds for $\frac{N+\alpha }{N}< p_{2}<2$. However, we cannot prove this result since the nonlinearity $(I_{\alpha }\ast |\psi |^{p _{2}})|\psi |^{p_{2}-2}\psi$ is singular when $\frac{N+\alpha }{N}< p _{2}<2$. Consequently, the case of $\frac{N+\alpha }{N}< p_{2}<2$ will be the object of a future investigation.

By the same argument as that in [2], we can easily derive the following lemma.

Lemma 2.1

Let$\psi _{0} \in \varSigma :=\{u\in H^{1}, xu\in L^{2}\}$, and the solution$\psi (t)$to (1.1) exists on the interval$[0,T^{*})$. Then, $\psi (t) \in \varSigma$for all$t\in [0,T^{*})$. Moreover, let$F(t)=\int _{\mathbb{R}^{N}} |x\psi (t,x)|^{2}\,dx$, then

$$ F'(t)=-4\operatorname{Im} \int _{\mathbb{R}^{N}}\psi (t,x)x\cdot \nabla \bar{\psi }(t,x)\,dx:=-4h(t), $$

(2.4)

and

$$\begin{aligned} F''(t)={}&{-}4h'(t) \\ ={}& 8 \int _{\mathbb{R}^{N}} \bigl\vert \nabla \psi (t,x) \bigr\vert ^{2}\,dx+\frac{4N \lambda _{1}p_{1}}{p_{1}+2} \int _{\mathbb{R}^{N}} \bigl\vert \psi (t,x) \bigr\vert ^{p_{1}+2}\,dx \\ &{}+\lambda _{2}\frac{4p_{2}N-4N-4\alpha }{p_{2}} \int _{\mathbb{R}^{N}} \bigl(I _{\alpha }\ast \vert \psi \vert ^{p_{2}}\bigr) (t,x) \bigl\vert \psi (t,x) \bigr\vert ^{p_{2}}\,dx. \end{aligned}$$

(2.5)

Finally, we recall two important Gagliardo–Nirenberg type inequalities; see [8, 19].

Lemma 2.2

([19])

LetQbe the ground state solution of the following elliptic equation:

$$ -\Delta Q+Q- \vert Q \vert ^{p+2}Q=0 \quad \textit{in } \mathbb{R}^{N}. $$

(2.6)

Then, the optimal constant in the Gagliardo–Nirenberg inequality,

$$ \Vert \psi \Vert _{L^{p+2}}^{p+2} \leq C_{*} \Vert \psi \Vert _{L^{2}}^{p+2- \frac{Np}{2}} \Vert \nabla \psi \Vert _{L^{2}}^{\frac{Np}{2}}, $$

(2.7)

is

$$ C_{*}=\frac{2(p+2)(2(p+2)-Np)^{\frac{Np-4}{4}}}{(Np)^{\frac{Np}{4}} \Vert Q \Vert _{L^{2}}^{p}}. $$

(2.8)

In particular, in the$L^{2}$-critical case, i.e., $p=\frac{4}{N}$, $C_{*}=\frac{p+2}{2\|Q\|_{L^{2}}^{p}}$.

Lemma 2.3

([8])

LetRbe the ground state solution of the following elliptic equation:

$$ -\Delta R+R-\bigl(I_{\alpha }\ast \vert R \vert ^{p}\bigr) \vert R \vert ^{p-2}R=0\quad \textit{in } \mathbb{R}^{N}. $$

(2.9)

The best constant in the Gagliardo–Nirenberg type inequality

$$ \int _{\mathbb{R}^{N}} \bigl(I_{\alpha }\ast \vert \psi \vert ^{p}\bigr) \vert \psi \vert ^{p}\,dx \leq C^{*} \Vert \nabla \psi \Vert _{L^{2}}^{Np-N-\alpha } \Vert \psi \Vert _{L^{2}} ^{N+\alpha -Np+2p} $$

(2.10)

is

$$ C^{*}=\frac{2p}{2p-Np+N+\alpha } \biggl( \frac{2p-Np+N+\alpha }{Np-N- \alpha } \biggr)^{\frac{Np-N-\alpha }{2}} \Vert R \Vert _{L^{2}}^{2-2p}. $$

(2.11)

In particular, in the$L^{2}$-critical case, i.e., $p=1+\frac{2+ \alpha }{N}$, $C^{*}=p\|R\|_{L^{2}}^{2-2p}$.

This inequality has been extended to the fractional case; see [10].

Finally, we recall the following compactness lemma is vital in the proof of global existence; see [7].

Lemma 2.4

Let$N\geq 2$, $0< p<\frac{4}{N-2}$. Let$\{u_{n}\}$be a bounded sequence in$H^{1}$such that

$$ \limsup_{n\rightarrow \infty } \Vert u_{n} \Vert _{\dot{H}^{1}} \leq M, \qquad \limsup_{n\rightarrow \infty } \Vert u_{n} \Vert _{L^{p+2}} \geq m. $$

Then there exist a sequence$(x_{n})_{n\geq 1}$in$\mathbb{R}^{N}$and$U \in H^{1}\setminus \{0\}$such that up to a subsequence,

$$ u_{n}(\cdot +x_{n}) \rightharpoonup U\quad \textit{weakly in } H^{1}. $$

3 The existence of blow-up solutions

In this section, we will derive the sufficient conditions about existence of blow-up solutions.

Theorem 3.1

Let$\psi _{0}\in \varSigma$, $\lambda _{1}<0$, $h_{0}:=\operatorname{Im} \int _{\mathbb{R}^{N}} \bar{\psi }_{0}x\nabla \psi _{0}\,dx>0$and$\frac{4}{N}< p_{1}< \frac{4}{N-2}$with$N\geq 3$. Then, the solution$\psi (t)$of (1.1) blows up in each of the following three cases:

• (1) $\lambda _{2}>0$, $1+\frac{\alpha }{N}< p_{2}<1+\frac{Np_{1}+2\alpha }{2N}$, and$E(\psi _{0})<0$;

• (2) $\lambda _{2}<0$, $1+\frac{2+\alpha }{N}< p_{2}<1+\frac{Np_{1}+2 \alpha }{2N}$, and$E(\psi _{0})<0$;

• (3) $\lambda _{2}<0$, $1+\frac{\alpha }{N}< p_{2}\leq 1+ \frac{2+\alpha }{N}$, and$E(\psi _{0})+C\|\psi _{0}\|_{L^{2}}^{\frac{2Np _{1}+2p_{1}\alpha -4Np_{2}+4N+4\alpha }{Np_{1}-2Np_{2}+2N+2\alpha }}<0$for some constantC.

More precisely, there is$T^{*}\in (0, C\frac{\|x\psi _{0}\|_{L ^{2}}^{2}}{y_{0}} ]$such that

$$ \lim_{t\rightarrow T^{*}} \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}=\infty . $$

Proof

In the following, we will prove $F'(t)<0$ and $F''(t)<0$ for all $t\in [0,T^{*})$. More precisely, we will prove that

$$ h'(t)\geq c \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}^{2}>0 $$

(3.1)

for some constant $c>0$, where $h(t)$ is defined by (2.4). Thus, it follows from (2.5) that $F''(t)<0$ for all $t\in [0,T^{*})$. This shows that $F(t)$ is concave and the solution $\psi (t)$ of (1.1) blows up. Indeed, it follows from $y(0)=y_{0}>0$ that $h(t)>h(0)>0$ for all $t>0$. On the other hand, we deduce from Hölder’s inequality that

$$ h(t)\leq \bigl\Vert x \psi (t) \bigr\Vert _{L^{2}} \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}} $$

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

$$ \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}\geq \frac{h(t)}{ \Vert x \psi _{0} \Vert _{L^{2}}}. $$

(3.2)

We deduce from (3.1) and (3.2) that

$$ \textstyle\begin{cases} h'(t)\geq c\frac{h^{2}(t)}{ \Vert x \psi _{0} \Vert _{L^{2}}^{2}}, \\ h(0) = h_{0}>0. \end{cases} $$

(3.3)

This shows that there is $T^{*}\in (0, \frac{\|x \psi _{0}\|_{L ^{2}}^{2}}{cy_{0}} ]$ such that $\|\nabla \psi (t)\|_{L^{2}} \rightarrow \infty$ as $t\rightarrow T^{*}$.

Case (i): $\lambda _{2}>0$, $Np_{1}>2Np_{2}-2N-2\alpha$, and $E(\psi _{0})<0$. We deduce from (2.5), (2.2), and our assumptions that

$$\begin{aligned} h'(t)= {}&{-}2 \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}^{2}-\frac{N\lambda _{1}p_{1}}{p _{1}+2} \bigl\Vert \psi (t) \bigr\Vert _{L^{p_{1}+2}}^{p_{1}+2} \\ &{}-\lambda _{2} \frac{p_{2}N-N- \alpha }{p_{2}} \int _{\mathbb{R}^{N}} \bigl(I_{\alpha }\ast \vert \psi \vert ^{p_{2}}\bigr) (t) \bigl\vert \psi (t) \bigr\vert ^{p_{2}}\,dx \\ = {}& {-}2 \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}^{2} \\ &{}+Np_{1} \biggl(\frac{1}{2} \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}^{2}+\frac{\lambda _{2}}{2p_{2}} \int _{\mathbb{R}^{N}} \bigl(I_{\alpha }\ast \bigl\vert \psi (t) \bigr\vert ^{p_{2}}\bigr) \bigl\vert \psi (t) \bigr\vert ^{p _{2}}\,dx-E(\psi _{0}) \biggr) \\ &{} -\lambda _{2}\frac{p_{2}N-N-\alpha }{p_{2}} \int _{\mathbb{R}^{N}} \bigl(I _{\alpha }\ast \bigl\vert \psi (t) \bigr\vert ^{p_{2}}\bigr) \bigl\vert \psi (t) \bigr\vert ^{p_{2}}\,dx \\ ={} & \frac{Np_{1}-4}{2} \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}^{2}-Np_{1}E(\psi _{0}) \\ &{}+ \frac{\lambda _{2}}{2p_{2}}(Np_{1}-2Np_{2}+2N+2\alpha ) \int _{\mathbb{R}^{N}} \bigl(I_{\alpha }\ast \bigl\vert \psi (t) \bigr\vert ^{p_{2}}\bigr) \bigl\vert \psi (t) \bigr\vert ^{p _{2}}\,dx \\ \geq {}& \frac{Np_{1}-4}{2} \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}^{2}. \end{aligned}$$

This implies that (3.1) holds.

Case (ii): $\lambda _{2}<0$, $Np_{1}+2N+2\alpha >2Np_{2}$, $p_{2}>1+\frac{ \alpha +2}{N}$ and $E(\psi _{0})<0$. We deduce from (2.5), (2.2), and our assumptions that

$$\begin{aligned} h'(t)={} &{-}2 \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}^{2}-\frac{N\lambda _{1}p_{1}}{p _{1}+2} \bigl\Vert \psi (t) \bigr\Vert _{L^{p_{1}+2}}^{p_{1}+2} \\ &{}-(p_{2}N-N-\alpha ) \biggl(2E(\psi _{0})- \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}^{2}- \frac{2 \lambda _{1}}{p_{1}+2} \bigl\Vert \psi (t) \bigr\Vert _{L^{p_{1}+2}}^{p_{1}+2} \biggr) \\ ={} & (p_{2}N-N-\alpha -2) \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}^{2}-2(p_{2}N-N- \alpha )E(\psi _{0}) \\ &{}-\frac{\lambda _{1}}{p_{1}+2}(Np_{1}-2Np_{2}+2N+2\alpha ) \bigl\Vert \psi (t) \bigr\Vert _{L^{p_{1}+2}}^{p_{1}+2} \\ \geq{} & (p_{2}N-N-\alpha -2) \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}^{2}. \end{aligned}$$

This implies that (3.1) holds.

Case (iii): $\lambda _{2}<0$, $1+\frac{\alpha }{N}< p_{2}\leq 1+\frac{2+ \alpha }{N}$, and $E(\psi _{0})+C\|\psi _{0}\|_{L^{2}}^{\frac{2Np_{1}+2p _{1}\alpha -4Np_{2}+4N+4\alpha }{Np_{1}-2Np_{2}+2N+2\alpha }}<0$ for some constant C.

We deduce from $p_{1}>\frac{4}{N}$ that there is a constant ε such that $p_{1}>\frac{2(2+\varepsilon )}{N}$. Let $\theta :=\frac{2(2+\varepsilon )}{p_{1}N}<1$. Therefore, it follows from (2.2) and our assumptions that

$$\begin{aligned} h'(t)= {}&{-}2 \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}^{2}-\frac{N\lambda _{1}p_{1} \theta }{p_{1}+2} \bigl\Vert \psi (t) \bigr\Vert _{L^{p_{1}+2}}^{p_{1}+2}-\frac{N\lambda _{1}p_{1}(1-\theta )}{p_{1}+2} \bigl\Vert \psi (t) \bigr\Vert _{L^{p_{1}+2}}^{p_{1}+2} \\ &{}-\lambda _{2}\frac{p_{2}N-N-\alpha }{p_{2}} \int _{\mathbb{R}^{N}} \bigl(I _{\alpha }\ast \bigl\vert \psi (t) \bigr\vert ^{p_{2}}\bigr) \bigl\vert \psi (t) \bigr\vert ^{p_{2}}\,dx \\ \geq{} & {-}2 \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}^{2} \\ &{}+Np_{1} \theta \biggl(\frac{1}{2} \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}^{2}-E(\psi _{0})+ \frac{\lambda _{2}}{2p_{2}} \int _{\mathbb{R}^{N}} \bigl(I_{\alpha }\ast \vert \psi \vert ^{p_{2}}\bigr) (t) \bigl\vert \psi (t) \bigr\vert ^{p_{2}}\,dx \biggr) \\ &{}-\frac{N\lambda _{1}p_{1}(1-\theta )}{p_{1}+2} \bigl\Vert \psi (t) \bigr\Vert _{L^{p_{1}+2}} ^{p_{1}+2}-\lambda _{2}\theta \frac{p_{2}N-N-\alpha }{p_{2}} \int _{\mathbb{R}^{N}} \bigl(I_{\alpha }\ast \bigl\vert \psi (t) \bigr\vert ^{p_{2}}\bigr) \bigl\vert \psi (t) \bigr\vert ^{p _{2}}\,dx \\ \geq {}& \biggl(-2+\frac{Np_{1}\theta }{2} \biggr) \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}^{2}-Np_{1}\theta E(\psi _{0})-\frac{N\lambda _{1}p_{1}(1- \theta )}{p_{1}+2} \bigl\Vert \psi (t) \bigr\Vert _{L^{p_{1}+2}}^{p_{1}+2} \\ &{}+\frac{\lambda _{2}\theta (Np_{1}-2p_{2}N+2N+2\alpha )}{2p_{2}} \int _{\mathbb{R}^{N}} \bigl(I_{\alpha }\ast \bigl\vert \psi (t) \bigr\vert ^{p_{2}}\bigr) \bigl\vert \psi (t) \bigr\vert ^{p _{2}}\,dx. \end{aligned}$$

Applying Young’s inequality, we have

$$\begin{aligned} \int _{\mathbb{R}^{N}} I_{\alpha }\ast \bigl\vert \psi (t) \bigr\vert ^{p_{2}} \bigl\vert \psi (t) \bigr\vert ^{p _{2}}\,dx &\leq \bigl\Vert \psi (t) \bigr\Vert _{L^{\frac{2Np_{2}}{N+\alpha }}}^{2p_{2}} \\ &\leq \bigl\Vert \psi (t) \bigr\Vert _{L^{2}}^{2p_{2}- \frac{2(p_{1}+2)(Np_{2}-N-\alpha )}{Np_{1}}} \bigl\Vert \psi (t) \bigr\Vert _{L^{p_{1}+2}}^{ \frac{2(p_{1}+2)(Np_{2}-N- \alpha )}{Np_{1}}} \\ &\leq C(\delta ) \bigl\Vert \psi (t) \bigr\Vert _{L^{2}}^{\frac{2Np_{1}+2p_{1}\alpha -4Np _{2}+4N+4\alpha }{Np_{1}-2Np_{2}+2N+2\alpha }}+ \delta \bigl\Vert \psi (t) \bigr\Vert _{L^{p_{1}+2}}^{p_{1}+2}. \end{aligned}$$

Therefore, we can choose $\delta >0$ enough small such that

$$ \delta \frac{ \vert \lambda _{2} \vert \theta (Np_{1}-2p_{2}N+2N+2\alpha )}{2p_{2}}< \frac{N \vert \lambda _{1} \vert p_{1}(1-\theta )}{p_{1}+2}, $$

which implies

$$\begin{aligned} y'(t) \geq& \varepsilon \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}^{2}-Np_{1}\theta E \\ &{}-C( \delta ) \frac{ \vert \lambda _{2} \vert \theta (Np_{1}-2p_{2}N+2N+2\alpha )}{2p _{2}} \Vert \psi _{0} \Vert _{L^{2}}^{\frac{2Np_{1}+2p_{1}\alpha -4Np_{2}+4N+4 \alpha }{Np_{1}-2Np_{2}+2N+2\alpha }}. \end{aligned}$$

Therefore, if $Np_{1}\theta E+C(\delta )\frac{|\lambda _{2}|\theta (Np _{1}-2p_{2}N+2N+2\alpha )}{2p_{2}}\|\psi _{0}\|_{L^{2}}^{\frac{2Np_{1}+2p _{1}\alpha -4Np_{2}+4N+4\alpha }{Np_{1}-2Np_{2}+2N+2\alpha }}<0$, then

$$ y'(t)\geq \varepsilon \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}^{2}. $$

This implies that (3.1) holds. □

4 Sharp conditions of blow-up and global existence

From the local well-posedness of the nonlinear Schrödinger–Choquard equation, for small initial data $\psi _{0}$, the solution $\psi (t)$ to (1.1) exists globally, and the solution $\psi (t)$ may blow up for some large initial data. Therefore, whether there are some sharp thresholds of global existence and blow-up for (1.1) is a very interesting problem. For Eq. (1.1), there are two nonlinearities and there is no scaling invariance, which are the main difficulties. We obtain the following sharp conditions of blow-up and global existence for (1.1) by constructing some new estimates.

4.1 $L^{2}$-Critical case

Theorem 4.1

Let$\psi _{0}\in H^{1}$, $\lambda _{1}=-1$, $\lambda _{2}=1$, $p_{1}=\frac{4}{N}$and$1+\frac{\alpha }{N}< p_{2}<1+ \frac{2+\alpha }{N}$. Assume thatQis the ground state solution of (2.6). Then, we have the following sharp threshold mass of blow-up and global existence.

• (i) If$\|\psi _{0}\|_{L^{2}}\leq \|Q\|_{L^{2}}$, then the solution of (1.1) exists globally.

• (ii) If the initial data$\psi _{0}=c\rho ^{\frac{N}{2}} Q(\rho x)$satisfies$|x|\psi _{0}\in L^{2}$, where the complex numbercsatisfying$|c|> 1$, and the real number$\rho >0$, then the solutionψof (1.1) with initial data$\psi _{0}$blows up in finite time.

Proof

(i) We firstly consider the case $\|\psi _{0}\|_{L^{2}}<\|Q\|_{L^{2}}$. It follows from (2.3) and (2.7) that

$$\begin{aligned} E(\psi _{0})={}&E\bigl(\psi (t)\bigr) \\ ={} &\frac{1}{2} \int _{\mathbb{R}^{N}} \bigl\vert \nabla \psi (t,x) \bigr\vert ^{2} \,dx-\frac{1}{p _{1}+2} \int _{\mathbb{R}^{N}} \bigl\vert \psi (t,x) \bigr\vert ^{p_{1}+2}\,dx \\ &{}+ \frac{1}{2p_{2}} \int _{\mathbb{R}^{N}} \bigl(I_{\alpha }\ast \vert \psi \vert ^{p_{2}}\bigr) (t,x) \bigl\vert \psi (t,x) \bigr\vert ^{p_{2}}\,dx \\ \geq {}& \biggl(\frac{1}{2}-\frac{ \Vert \psi _{0} \Vert _{L^{2}}^{p_{1}}}{2 \Vert Q \Vert _{L^{2}}^{p_{1}}} \biggr) \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}^{2}. \end{aligned}$$

Due to $\|\psi _{0}\|_{L^{2}}<\|Q\|_{L^{2}}$, we find that $\|\nabla \psi (t)\|_{L^{2}}$ is uniformly bounded for all time t. Therefore, (i) follows from the conservation of mass and Proposition 2.1.

When $\|\psi _{0}\|_{L^{2}}=\|Q\|_{L^{2}}$, if the solution $\psi (t)$ of (1.1) blows up in finite time, then there exists $T^{*}>0$ such that $\lim_{t\rightarrow T^{*}}\|\nabla \psi (t)\|_{L^{2}}=\infty$. Set

$$ \rho (t)= \Vert \nabla Q \Vert _{L^{2}}/ \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}} \quad \mbox{and}\quad v(t,x)=\rho ^{\frac{N}{2}}(t) \psi \bigl(t,\rho (t) x\bigr). $$

Let $\{t_{n}\}_{n=1}^{\infty }$ be an any time sequence such that $t_{n}\rightarrow T^{*}$, $\rho _{n}:=\rho (t_{n})$ and $v_{n}(x):=v(t _{n},x)$. Then, the sequence $\{v_{n}\}$ satisfies

$$ \Vert v_{n} \Vert _{L^{2}}= \bigl\Vert \psi (t_{n}) \bigr\Vert _{L^{2}}= \Vert \psi _{0} \Vert _{L^{2}}= \Vert Q \Vert _{L^{2}},\qquad \Vert \nabla v_{n} \Vert _{L^{2}}=\rho _{n} \bigl\Vert \nabla \psi (t_{n}) \bigr\Vert _{L^{2}}= \Vert \nabla Q \Vert _{L^{2}}. $$

(4.1)

Observe that

$$\begin{aligned} H(v_{n}) := &\frac{1}{2} \int _{\mathbb{R}^{N}} \bigl\vert \nabla v_{n}(x) \bigr\vert ^{2}\,dx-\frac{1}{p _{1}+2} \int _{\mathbb{R}^{N}} \bigl\vert v_{n}(x) \bigr\vert ^{p_{1}+2}\,dx \\ = &\rho _{n}^{2} \biggl(\frac{1}{2} \int _{\mathbb{R}^{N}} \bigl\vert \nabla \psi (t _{n},x) \bigr\vert ^{2}\,dx-\frac{1}{p_{1}+2} \int _{\mathbb{R}^{N}} \bigl\vert \psi (t_{n},x) \bigr\vert ^{p _{1}+2}\,dx \biggr) \\ = &\rho _{n}^{2} \biggl(E(\psi _{0})- \frac{1}{2p_{2}} \int _{\mathbb{R} ^{N}} \bigl(I_{\alpha }\ast \vert \psi \vert ^{p_{2}}\bigr) (t,x) \bigl\vert \psi (t,x) \bigr\vert ^{p_{2}}\,dx \biggr). \end{aligned}$$

(4.2)

Thus, we deduce from the Gagliardo–Nirenberg inequality (2.10) and $1+\frac{\alpha }{N}< p_{2}<1+\frac{2+\alpha }{N}$ that

$$ \rho _{n}^{2} \biggl(E(\psi _{0})- \frac{1}{2p_{2}} \int _{\mathbb{R}^{N}} \bigl(I _{\alpha }\ast \vert \psi \vert ^{p_{2}}\bigr) (t,x) \bigl\vert \psi (t,x) \bigr\vert ^{p_{2}}\,dx \biggr) \rightarrow 0, \quad \mbox{as }n\rightarrow \infty . $$

This, together with (4.2) implies that $\int _{\mathbb{R}^{N}} |v _{n}(x)|^{p_{1}+2}\,dx\rightarrow (2/N+1)\|\nabla Q\|_{L^{2}}^{2}$. Thus, we deduce from (4.1) that there exist a subsequence, still denoted by $\{v_{n}\}$, and $u\in H^{1}\backslash \{0\}$ such that

$$ u_{n}:=\tau _{x_{n}}v_{n}\rightharpoonup u \neq 0 \quad \mbox{weakly in }H^{1}, $$

for some $\{x_{n}\}\subseteq \mathbb{R}^{N}$. This implies that there exists $C_{0}>0$ such that

$$\begin{aligned} &\liminf_{n\rightarrow \infty } \int _{\mathbb{R}^{N}} \bigl(I_{\alpha } \ast \vert u_{n} \vert ^{p_{2}}\bigr) (x) \bigl\vert u_{n}(x) \bigr\vert ^{p_{2}}\,dx \\ &\quad = \liminf _{n\rightarrow \infty } \int _{\mathbb{R}^{N}} \bigl(I_{\alpha } \ast \vert v_{n} \vert ^{p_{2}}\bigr) (x) \bigl\vert v_{n}(x) \bigr\vert ^{p_{2}}\,dx\geq C_{0}>0. \end{aligned}$$

(4.3)

On the other hand, we deduce from (2.7) and $\|\psi (t)\|_{L^{2}}= \|\psi _{0}\|_{L^{2}}=\|Q\|_{L^{2}}$ that

$$\begin{aligned} \frac{1}{2} \int _{\mathbb{R}^{N}} \bigl\vert \nabla \psi (t,x) \bigr\vert ^{2} \,dx-\frac{1}{p _{1}+2} \int _{\mathbb{R}^{N}} \bigl\vert \psi (t,x) \bigr\vert ^{p_{1}+2}\,dx\geq 0, \end{aligned}$$

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

$$\begin{aligned} \frac{1}{2p_{2}} \int _{\mathbb{R}^{N}} \bigl(I_{\alpha }\ast \vert \psi \vert ^{p_{2}}\bigr) (t,x) \bigl\vert \psi (t,x) \bigr\vert ^{p_{2}}\,dx\leq E(\psi _{0}), \end{aligned}$$

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

$$\begin{aligned} \int _{\mathbb{R}^{N}} \bigl(I_{\alpha }\ast \vert v_{n} \vert ^{p_{2}}\bigr) (x) \bigl\vert v_{n}(x) \bigr\vert ^{p _{2}}\,dx &=\rho _{n}^{Np_{2}-N-\alpha } \int _{\mathbb{R}^{N}} \bigl(I_{\alpha }\ast \vert \psi \vert ^{p_{2}}\bigr) (t_{n},x) \bigl\vert \psi (t_{n},x) \bigr\vert ^{p_{2}}\,dx \\ &\leq \rho _{n}^{Np_{2}-N-\alpha }E(\psi _{0}) \rightarrow 0,\quad \mbox{as }n\rightarrow \infty , \end{aligned}$$

which is a contradiction with (4.3). Thus, the solution $\psi (t)$ of (1.1) exists globally.

(ii) Since $|x|\psi _{0}\in L^{2}$, $J(t)=\int _{\mathbb{R}^{N}} |x \psi (t,x)|^{2}\,dx$ is well-defined, and it follows from Lemma 2.1 that

$$ J''(t)=16E(\psi _{0})+\frac{4Np_{2}-4N-4\alpha -8}{p_{2}} \int _{\mathbb{R}^{N}} \bigl(I_{\alpha }\ast \vert \psi \vert ^{p_{2}}\bigr) (t,x) \bigl\vert \psi (t,x) \bigr\vert ^{p _{2}}\,dx. $$

(4.4)

By the definition of initial data $\psi _{0}(x)=c\rho ^{\frac{N}{2}} Q( \rho x)$ and the Pohoz̆aev identity for Eq. (2.6), i.e., $\frac{1}{2}\|\nabla Q\|_{L^{2}}^{2}=\frac{1}{p_{1}+2}\|Q\|^{p_{1}+2} _{L^{p_{1}+2}}$, we deduce that

$$\begin{aligned} E(\psi _{0}) &=\frac{ \vert c \vert ^{2}\rho ^{2}}{2} \int _{\mathbb{R}^{N}} \bigl\vert \nabla Q(x) \bigr\vert ^{2} \,dx-\frac{ \vert c \vert ^{p_{1}+2}\rho ^{2}}{p_{1}+2} \int _{\mathbb{R}^{N}} \bigl\vert Q(x) \bigr\vert ^{p_{1}+2}\,dx \\ &\quad {}+\frac{ \vert c \vert ^{2p_{2}}\rho ^{Np_{2}-N-\alpha }}{2p_{2}} \int _{\mathbb{R} ^{N}} \bigl(I_{\alpha }\ast \vert Q \vert ^{p_{2}}\bigr) (x) \bigl\vert Q(x) \bigr\vert ^{p_{2}}\,dx \\ &=-\frac{ \vert c \vert ^{2}\rho ^{2}}{2}\bigl( \vert c \vert ^{p_{1}}-1\bigr) \Vert \nabla Q \Vert _{L^{2}}^{2} \\ &\quad {}+\frac{ \vert c \vert ^{2p _{2}}\rho ^{Np_{2}-N-\alpha }}{2p_{2}} \int _{\mathbb{R}^{N}} \bigl(I_{\alpha }\ast \vert Q \vert ^{p_{2}}\bigr) (x) \bigl\vert Q(x) \bigr\vert ^{p_{2}}\,dx. \end{aligned}$$

(4.5)

Thanks to $Np_{2}-N-\alpha <2$, we can take ρ large enough such that

$$ E(\psi _{0})< 0. $$

It follows from (4.4) that $F''(t)<16 E(\psi _{0}) < 0$. By the standard concave argument, the solution ψ of (1.1) with the initial data $\psi _{0}$ blows up in finite time. □

4.2 $L^{2}$-Supercritical case

Theorem 4.2

Let$\lambda _{1}=\lambda _{2}=-1$, $p_{1}>\frac{4}{N}$, and$\psi \in C([0,T^{*}),H^{1})$be a solution of (1.1). Then we have the following sharp criteria of blow-up and global existence for (1.1).

• (1) $\|\psi _{0}\|_{L^{2}}<\|R\|_{L^{2}}$, $p_{2}=1+\frac{2+\alpha }{N}$, and$E(\psi _{0})< \frac{Np_{1}-4}{2Np_{1}} (1 -\frac{\| \psi _{0} \|^{2p_{2}-2}_{L^{2}}}{\|R\|^{2p_{2}-2}_{L^{2}}} )y_{0}^{2}$. If$\|\nabla \psi _{0}\|_{L^{2}}< y_{0}$, then the solution$\psi (t)$of (1.1) exists globally; If$\|\nabla u_{0}\|_{L^{2}}>y_{0}$, then the solution$\psi (t)$of (1.1) blows up, whereRis the ground state solution of (2.9) with$p=1+\frac{2+\alpha }{N}$, $y_{0}$is defined by (4.8).

• (2) $1+\frac{\alpha +2}{N}< p_{2}<1+\frac{Np_{1}+2\alpha }{2N}$and$E(\psi _{0})<\frac{Np_{2}-N-\alpha -2}{2(Np_{2}-N-\alpha )}y_{1}^{2}$. If$\|\nabla \psi _{0}\|_{L^{2}}< y_{1}$, then the solution$\psi (t)$of (1.1) exists globally; If$\|\nabla \psi _{0}\|_{L^{2}}>y_{1}$, then the solution$\psi (t)$of (1.1) blows up, where$y_{1}$is the unique positive solution of the equation$f(y)=0$and$f(y)$is defined in (4.13) with$1+\frac{\alpha +2}{N}< p_{2}<1+\frac{Np_{1}+2 \alpha }{2N}$.

• (3) $1+\frac{Np_{1}+2\alpha }{2N}< p_{2}<1+\frac{2+\alpha }{N-2}$and$E(\psi _{0})<\frac{Np_{1}-4}{2Np_{1}}y_{2}^{2}$. If$\|\nabla \psi _{0}\|_{L^{2}}< y_{2}$, then the solution$\psi (t)$of (1.1) exists globally; If$\|\nabla \psi _{0}\|_{L^{2}}>y_{2}$, then the solution$\psi (t)$of (1.1) blows up, where$y_{2}$is the unique positive solution of the equation$f(y)=0$and$f(y)$is defined in (4.13) with$1+\frac{Np_{1}+2\alpha }{2N}< p_{2}<1+\frac{2+\alpha }{N-2}$.

Proof

Case (1): $p_{2}=1+\frac{2+\alpha }{N}$. First, we deduce from (2.7) and (2.10) that

$$\begin{aligned} E\bigl(\psi (t)\bigr)\geq{} & \frac{1}{2} \bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{L^{2}}-\frac{C_{*}}{p _{1}+2} \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}^{\frac{Np_{1}}{2}} \bigl\Vert \psi (t) \bigr\Vert ^{p_{1}+2-\frac{Np_{1}}{2}}_{L^{2}} \\ & {}-\frac{C^{*}}{2p_{2}} \bigl\Vert \nabla \psi (t) \bigr\Vert ^{2}_{L^{2}} \bigl\Vert \psi (t) \bigr\Vert ^{2p_{2}-2}_{L^{2}} \\ \geq{} &h\bigl( \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}\bigr), \end{aligned}$$

(4.6)

where $C_{*}$ and $C^{*}$ are defined by (2.8) and (2.11), respectively, $h(y)$ is defined by

$$ h(y)=\frac{1}{2}y^{2}-\frac{C_{*}}{p_{1}+2} \Vert \psi _{0} \Vert ^{p_{1}+2-\frac{Np _{1}}{2}}_{L^{2}}y^{\frac{Np_{1}}{2}} - \frac{C^{*}}{2p_{2}} \Vert \psi _{0} \Vert ^{2p_{2}-2}_{L^{2}}y^{2},\quad y\in [0,\infty ). $$

By a simple computation, we find that $h(y)$ is continuous on $[0,\infty )$ and

$$ h'(y)= \biggl(1-\frac{C^{*}}{p_{2}} \Vert \psi _{0} \Vert ^{2p_{2}-2}_{L^{2}} \biggr)y- \frac{C _{*}}{p_{1}+2}\frac{Np_{1}}{2} \Vert \psi _{0} \Vert ^{p_{1}+2- \frac{Np_{1}}{2}}_{L^{2}}y^{\frac{Np_{1}}{2}-1}. $$

(4.7)

By the assumption $\|\psi _{0}\|_{L^{2}}<\|R\|_{L^{2}}$, $1-\frac{C ^{*}}{p_{2}}\| \psi _{0}\|^{2p_{2}-2}_{L^{2}}>0$. Thus, the equation $h'(y)=0$ has a unique positive root:

$$ y_{0}= \biggl(\frac{1-\frac{C^{*}}{p_{2}} \Vert \psi _{0} \Vert ^{2p_{2}-2}_{L ^{2}}}{\frac{C_{*}}{p_{1}+2}\frac{Np_{1}}{2} \Vert \psi _{0} \Vert ^{p_{1}+2-\frac{Np _{1}}{2}}_{L^{2}}} \biggr)^{\frac{2}{Np_{1}-4}}. $$

(4.8)

This implies that $h(y)$ is increasing on the interval $[0,y_{0})$, decreasing on the interval $[y_{0},\infty )$ and

$$ h_{\max }=h(y_{0})=\frac{Np_{1}-4}{2Np_{1}} \biggl(1 -\frac{ \Vert \psi _{0} \Vert ^{2p_{2}-2}_{L^{2}}}{ \Vert R \Vert ^{2p_{2}-2}_{L^{2}}} \biggr)y_{0}^{2}. $$

(4.9)

By (2.2) and $E(\psi _{0})< h(y_{0})$, we have

$$ h\bigl( \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}\bigr)\leq E\bigl(\psi (t)\bigr)=E(\psi _{0})< h(y_{0}), \quad \mbox{for all }t\in \bigl[0,T^{*}\bigr). $$

(4.10)

Now, we claim that if $\|\nabla \psi _{0}\|_{L^{2}}< y_{0}$, then $\|\nabla \psi (t)\|_{L^{2}}< y_{0}$, for all $t\in [0,T^{*})$. This implies the solution $\psi (t)$ of (1.1) exists globally. Let us prove this result by contradiction. If not, by the continuity of $\|\nabla \psi (t)\|_{L^{2}}$, there exists $t_{0}\in [0,T^{*})$ such that $\|\nabla \psi (t_{0})\|_{L^{2}}=y_{0}$. Thus, $h(\|\nabla \psi (t_{0})\|_{L^{2}})=h(y_{0})=h_{\max }$. Moreover, taking $t=t_{0}$ in (4.10), it follows that

$$ h\bigl( \bigl\Vert \nabla \psi (t_{0}) \bigr\Vert _{L^{2}}\bigr)=h(y_{0})=h_{\max }\leq E\bigl(\psi (t)\bigr)=E( \psi _{0})< h_{\max }, $$

which is a contradiction. Thus, the solution $\psi (t)$ of (1.1) exists globally.

On the other hand, if $\|\nabla \psi _{0}\|_{L^{2}}>y_{0}$, by the same argument, it follows that $\|\nabla \psi (t)\|_{L^{2}}>y_{0}$ for all $t\in [0,T^{*})$. Thus, by (2.2), (2.5), (2.7), and the assumption $E(\psi _{0})< \frac{Np_{1}-4}{2Np_{1}} (1 -\frac{ \| \psi _{0}\|^{2p_{2}-2}_{L^{2}}}{\|R\|^{2p_{2}-2}_{L^{2}}} )y _{0}^{2}$, we deduce that

$$\begin{aligned} F''(t)={} &8 \bigl\Vert \nabla \psi (t) \bigr\Vert ^{2}_{L^{2}}-\frac{4Np_{1}}{p_{1}+2} \bigl\Vert \psi (t) \bigr\Vert _{L^{p_{1}+2}}^{p_{1}+2}- \frac{8}{p_{2}} \int _{\mathbb{R} ^{N}} \bigl(I_{\alpha }\ast \vert \psi \vert ^{p_{2}}\bigr) (t,x) \bigl\vert \psi (t,x) \bigr\vert ^{p_{2}}\,dx \\ ={} & 4Np_{1}E(\psi _{0})-2(Np_{1}-4) \bigl\Vert \nabla \psi (t) \bigr\Vert ^{2}_{L^{2}} \\ &{}+ \frac{2(Np _{1}-4)}{p_{2}} \int _{\mathbb{R}^{N}} \bigl(I_{\alpha }\ast \vert \psi \vert ^{p_{2}}\bigr) (t,x) \bigl\vert u(t,x) \bigr\vert ^{p _{2}}\,dx \\ \leq{} & 4Np_{1}E(\psi _{0})-2(Np_{1}-4) \bigl\Vert \nabla \psi (t) \bigr\Vert ^{2}_{L^{2}}+2(Np _{1}-4)\frac{ \Vert \psi _{0} \Vert ^{2p_{2}-2}_{L^{2}}}{ \Vert R \Vert ^{2p_{2}-2}_{L ^{2}}} \bigl\Vert \nabla \psi (t) \bigr\Vert ^{2}_{L^{2}} \\ \leq {}&2(Np_{1}-4) \biggl(1 -\frac{ \Vert \psi _{0} \Vert ^{2p_{2}-2}_{L^{2}}}{ \Vert R \Vert ^{2p_{2}-2}_{L^{2}}} \biggr)y_{0}^{2} \\ &{}-2(Np_{1}-4) \biggl(1 - \frac{ \Vert \psi _{0} \Vert ^{2p_{2}-2}_{L^{2}}}{ \Vert R \Vert ^{2p_{2}-2}_{L^{2}}} \biggr) \bigl\Vert \nabla \psi (t) \bigr\Vert ^{2}_{L^{2}}< 0. \end{aligned}$$

(4.11)

Therefore, by the classical argument for Schrödinger equations, the solution $\psi (t)$ of (1.1) blows up.

Case (2): $1+\frac{\alpha +2}{N}< p_{2}<1+\frac{Np_{1}+2\alpha }{2N}$ and $E(\psi _{0})<\frac{Np_{2}-N-\alpha -2}{2(Np_{2}-N-\alpha )}y_{1}^{2}$. Similarly, we define a function $g(y)$ on $[0,\infty )$ by

$$ g(y)=\frac{1}{2}y^{2}-\frac{C_{*}}{p_{1}+2} \Vert \psi _{0} \Vert ^{(p_{1}+2)-\frac{Np _{1}}{2}}_{L^{2}}y^{\frac{ Np_{1}}{2}}- \frac{C^{*}}{2p_{2}} \Vert \psi _{0} \Vert ^{N+\alpha -Np_{2}+2p_{2}}_{L^{2}}y^{Np_{2}-N-\alpha },\quad y\in [0,\infty ). $$

Thus, it follows that $E(\psi (t))\geq g(\|\nabla \psi (t)\|_{L^{2}})$, $g(y)$ is continuous on $[0,\infty )$ and

$$\begin{aligned} g'(y) &=y-\frac{C_{*}}{p_{1}+2} \frac{Np_{1}}{2} \Vert \psi _{0} \Vert ^{(p_{1}+2)-\frac{Np _{1}}{2}}_{L^{2}} y^{\frac{ Np_{1}}{2}-1} \\ &\quad {}-\frac{C^{*}}{2p_{2}}(Np_{2}-N-\alpha ) \Vert \psi _{0} \Vert ^{N+\alpha -Np _{2}+2p_{2}}_{L^{2}}y^{Np_{2}-N-\alpha -1}. \end{aligned}$$

(4.12)

Next, we define a function $f(y)$ by

$$\begin{aligned} f(y)={} &1-\frac{C_{*}}{p_{1}+2}\frac{Np_{1}}{2} \Vert \psi _{0} \Vert ^{(p_{1}+2)-\frac{Np _{1}}{2}}_{L^{2}} y^{\frac{ Np_{1}}{2}-2} \\ &{}-\frac{C^{*}}{2p_{2}}(Np_{2}-N-\alpha ) \Vert \psi _{0} \Vert ^{N+\alpha -Np _{2}+2p_{2}}_{L^{2}}y^{Np_{2}-N-\alpha -2}. \end{aligned}$$

(4.13)

For the equation $f(y)=0$, there is a unique positive solution $y_{1}$. Indeed, by the assumption $1+\frac{\alpha +2}{N}< p_{2}<1+\frac{Np _{1}+2\alpha }{2N}$, for $y>0$, we have

$$\begin{aligned} f'(y) = &-\frac{C_{*}}{p_{1}+2} \frac{Np_{1}}{2}\biggl(\frac{ Np_{1}}{2}-2\biggr) \Vert \psi _{0} \Vert ^{(p_{1}+2)-\frac{Np_{1}}{2}}_{L^{2}} y^{\frac{ Np_{1}}{2}-3} \\ &{}-\frac{C^{*}}{2p_{2}}(Np_{2}-N-\alpha ) (Np_{2}-N- \alpha -2) \Vert \psi _{0} \Vert ^{N+\alpha -Np_{2}+2p_{2}}_{L^{2}}y^{Np_{2}-N-\alpha -3}< 0, \end{aligned}$$

(4.14)

which implies that $f(y)$ is decreasing on $[0,\infty )$. Due to $f(0)=1$, there exists a unique $y_{1}>0$ such that $f(y_{1})=0$. Therefore, we have

$$ f(y)>0 \quad \mbox{for all }y\in [0,y_{1}) \quad \mbox{and} \quad f(y)< 0\quad \mbox{for all }y\in (y_{1},+\infty ). $$

This implies that $g(y)$ is increasing on $[0,y_{1})$, decreasing on $(y_{1},+\infty )$ and

$$\begin{aligned} g_{\max }&=g(y_{1}) \\ &= \biggl( \frac{1}{2}-\frac{1}{Np_{2}-N-\alpha } \biggr)y _{1}^{2} \\ &\quad {} +\frac{C_{*}}{p_{1}+2}\frac{Np_{1}-2(Np_{2}-N-\alpha )}{2(Np_{2}-N- \alpha )} \Vert \psi _{0} \Vert ^{p_{1}+2-\frac{Np_{1}}{2}}_{L^{2}}y^{\frac{Np _{1}}{2}}. \end{aligned}$$

(4.15)

On the other hand, we deduce from (2.2) and the assumption $E(u_{0})<\frac{Np_{2}-N-\alpha -2}{2(Np_{2}-N-\alpha )}y_{1}^{2}$ that

$$\begin{aligned} g\bigl( \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}\bigr)&\leq E\bigl(\psi (t)\bigr)=E(\psi _{0}) \\ &< \biggl(\frac{1}{2}-\frac{1}{Np_{2}-N-\alpha } \biggr)y_{1}^{2} \\ &\quad {} +\frac{C_{*}}{p_{1}+2}\frac{Np_{1}-2(Np_{2}-N-\alpha )}{2(Np_{2}-N- \alpha )} \Vert \psi _{0} \Vert ^{p_{1}+2-\frac{Np_{1}}{2}}_{L^{2}}y^{\frac{Np _{1}}{2}}=g(y_{1}). \end{aligned}$$

(4.16)

By the same argument as Case (1), we find that if $\|\nabla \psi _{0}\| _{L^{2}}< y_{1}$, then, for all $t\in [0,T^{*})$, $\|\nabla \psi (t)\| _{L^{2}}< y_{1}$, which implies the solution $\psi (t)$ of (1.1) exists globally.

And if $\|\nabla \psi _{0}\|_{L^{2}}>y_{1}$, by the same way, it follows that $\|\nabla \psi (t)\|_{L^{2}}>y_{1}$ for all $t\in [0,T^{*})$. Thus, it follows from (2.2) and (2.5) that

$$\begin{aligned} F''(t)={} &8 \bigl\Vert \nabla \psi (t) \bigr\Vert ^{2}_{L^{2}}-\frac{4Np_{1}}{p_{1}+2} \bigl\Vert \psi (t) \bigr\Vert _{L^{p_{1}+2}}^{p_{1}+2} \\ &{}-\frac{4p_{2}N-4N-4\alpha }{p_{2}} \int _{\mathbb{R}^{N}} \bigl(I_{\alpha } \ast \vert \psi \vert ^{p_{2}}\bigr) (t,x) \bigl\vert \psi (t,x) \bigr\vert ^{p_{2}}\,dx \\ = {}& 8(Np_{2}-N-\alpha )E(\psi _{0})-4(Np_{2}-N- \alpha -2) \bigl\Vert \nabla \psi (t) \bigr\Vert ^{2}_{L^{2}} \\ &{}-\frac{4(Np_{1}-2(Np_{2}-N-\alpha ))}{p_{1}+2} \bigl\Vert \psi (t) \bigr\Vert _{L^{p _{1}+2}}^{p_{1}+2} \\ < {} &4(Np_{2}-N-\alpha -2)y_{1}^{2}-4(Np_{2}-N- \alpha -2)y_{1}^{2}=0. \end{aligned}$$

(4.17)

This implies that the solution of (1.1) blows up.

Case (3): $1+\frac{Np_{1}+2\alpha }{2N}< p_{2}<1+\frac{2+\alpha }{N-2}$ and $E(\psi _{0})<\frac{Np_{1}-4}{2Np_{1}}y_{2}^{2}$. By the same argument as Case (2), we have

$$\begin{aligned} g_{\max }&=g(y_{2}) \\ &= \biggl( \frac{1}{2}-\frac{2}{Np_{1}} \biggr)y_{2} ^{2} \\ &\quad {} +\frac{C^{*}}{2p_{2}}\frac{2(Np_{2}-N-\alpha )-Np_{1}}{Np_{1}} \Vert \psi _{0} \Vert ^{N+\alpha -Np_{2}+2p_{2}}_{L^{2}}y_{2}^{Np_{2}-N-\alpha }, \end{aligned}$$

(4.18)

where $y_{2}$ is the unique positive solution of (4.13). Thus, we deduce from (2.2) and the assumption $E(\psi _{0})<\frac{Np _{1}-4}{2Np_{1}}y_{2}^{2}$ that

$$\begin{aligned} g\bigl( \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}\bigr)&\leq E \bigl(\psi (t)\bigr)=E(\psi _{0}) \\ &< \biggl(\frac{1}{2}-\frac{2}{Np_{1}} \biggr)y_{2}^{2} +\frac{C^{*}}{2p _{2}}\frac{2(Np_{2}-N-\alpha )-Np_{1}}{Np_{1}} \Vert \psi _{0} \Vert ^{N+ \alpha -Np_{2}+2p_{2}}_{L^{2}}y_{2}^{Np_{2}-N-\alpha } \\ &=g(y_{2}). \end{aligned}$$

By the same argument as Case (1), we find that if $\|\nabla \psi _{0}\| _{L^{2}}< y_{1}$, then, for all $t\in [0,T^{*})$, $\|\nabla \psi (t)\| _{L^{2}}< y_{1}$, which implies the solution $\psi (t)$ of (1.1) exists globally.

And if $\|\nabla \psi _{0}\|_{L^{2}}>y_{2}$, in the same way, it follows that $\|\nabla \psi (t)\|_{L^{2}}>y_{2}$ for all $t\in [0,T^{*})$. Thus, it follows from (2.2) and (2.5) that

$$\begin{aligned} F''(t)= {}&8 \bigl\Vert \nabla \psi (t) \bigr\Vert ^{2}_{L^{2}}-\frac{4Np_{1}}{p_{1}+2} \bigl\Vert \psi (t) \bigr\Vert _{L^{p_{1}+2}}^{p_{1}+2} \\ &{}-\frac{4p_{2}N-4N-4\alpha }{p_{2}} \int _{\mathbb{R}^{N}} \bigl(I_{\alpha } \ast \vert \psi \vert ^{p_{2}}\bigr) (t,x) \bigl\vert \psi (t,x) \bigr\vert ^{p_{2}}\,dx \\ = {}& 4Np_{1}E(\psi _{0})-2(Np_{1}-4) \bigl\Vert \nabla \psi (t) \bigr\Vert ^{2}_{L^{2}} \\ &{}-\frac{4(Np_{2}-N-\alpha )-2Np_{1}}{p_{2}} \bigl\Vert \psi (t) \bigr\Vert _{L^{p_{1}+2}} ^{p_{1}+2} \\ < {}&2(Np_{1}-4)y_{2}^{2}-2(Np_{1}-4)y_{2}^{2}=0. \end{aligned}$$

(4.19)

This implies that the solution $\psi (t)$ of (1.1) blows up. □

Theorem 4.3

Let$\lambda _{1}=1$, $\lambda _{2}=-1$, $1+\frac{Np_{1}+2\alpha }{2N}< p _{2}<1+\frac{2+\alpha }{N-2}$, and$E(\psi _{0})< \frac{Np_{2}-N- \alpha -2}{2(Np_{2}-N-\alpha )}x_{0}^{2}$, and$\psi \in C([0,T^{*}),H ^{1})$be a solution of (1.1). If$\|\nabla \psi _{0}\|< x_{0}$, then the solution$\psi (t)$of (1.1) exists globally; If$\|\nabla \psi _{0}\|>x_{0}$, then the solution$\psi (t)$of (1.1) blows up, where$x_{0}$is defined by (4.21).

Proof

Applying (2.10), it follows that

$$\begin{aligned} E\bigl(\psi (t)\bigr)&=\frac{1}{2} \bigl\Vert \nabla \psi (t) \bigr\Vert ^{2}_{L^{2}}+ \frac{1}{p _{1}+2} \bigl\Vert \psi (t) \bigr\Vert ^{p_{1}+2}_{L^{p_{1}+2}} -\frac{1}{2p_{2}} \int _{\mathbb{R}^{N}} \bigl(I_{\alpha }\ast \vert \psi \vert ^{p_{2}}\bigr) (t,x) \bigl\vert \psi (t,x) \bigr\vert ^{p _{2}}\,dx \\ &\geq \frac{1}{2} \bigl\Vert \nabla \psi (t) \bigr\Vert ^{2}_{L^{2}}- \frac{C^{*}}{2p_{2}} \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}^{Np_{2}-N-\alpha } \bigl\Vert \psi (t) \bigr\Vert _{L^{2}}^{N+\alpha -Np_{2}+2p_{2}} \\ &=f\bigl( \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}\bigr), \end{aligned}$$

(4.20)

where the $C^{*}$ are defined by (2.11) and

$$ f(x):=\frac{1}{2}x^{2}-\frac{C^{*}}{2p_{2}} \Vert \psi _{0} \Vert _{L^{2}}^{N+ \alpha -Np_{2}+2p_{2}}x^{Np_{2}-N-\alpha }. $$

By a simple computation, we find that the unique positive solution $x_{0}$ of $f'(x)=0$ is given by

$$ x_{0}= \biggl(\frac{2p_{2}}{C^{*}(Np_{2}-N-\alpha ) \Vert \psi _{0} \Vert _{L^{2}} ^{N+\alpha -Np_{2}+2p_{2}}} \biggr)^{\frac{1}{Np_{2}-N-\alpha -2}}. $$

(4.21)

This implies that f is increasing on $(0,x_{0})$ and decreasing on $(x_{0},\infty )$. By a simple computation, it follows that

$$ f(x_{0})=\frac{Np_{2}-N-\alpha -2}{2(Np_{2}-N-\alpha )}x_{0}^{2}. $$

By (2.2) and the assumption $E(\psi _{0})< f(x_{0})$, it follows that

$$ f\bigl( \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}\bigr)\leq E( \psi _{0})< f(x_{0}), \quad \forall t\in \bigl[0,T^{*}\bigr). $$

If $\|\nabla \psi _{0}\|_{L^{2}}< x_{0}$, it follows from the continuity argument that $\|\nabla \psi (t)\|_{L^{2}}< x_{0}$ for all $t\in [0,T ^{*})$. Therefore, the solution $\psi (t)$ of (1.1) exists globally.

If $\|\nabla \psi _{0}\|_{L^{2}}>x_{0}$, we deduce from the continuity argument that $\|\nabla \psi (t)\|_{L^{2}}>x_{0}$ for all $t\in [0,T ^{*})$. We choose $\delta >0$ small enough so that

$$ E(\psi _{0})\leq (1-\delta )f(x_{0}). $$

This implies that

$$\begin{aligned} 8(Np_{2}-N-\alpha )E(\psi _{0})&\leq 8(Np_{2}-N-\alpha ) (1-\delta )f(x _{0}) \\ &=4(Np_{2}-N- \alpha -2) (1-\delta )x_{0}^{2}. \end{aligned}$$

(4.22)

Thus, we deduce from (2.2), (2.5) and (4.22) that

$$\begin{aligned} F''(t)={} &8 \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}^{2}+\frac{4Np_{1}}{p_{1}+2} \bigl\Vert \psi (t) \bigr\Vert _{L^{p_{1}+2}}^{p_{1}+2} \\ &{}-\frac{4p_{2}N-4N-4\alpha }{p_{2}} \int _{\mathbb{R}^{N}} \bigl(I_{\alpha } \ast \vert \psi \vert ^{p_{2}}\bigr) (t,x) \bigl\vert \psi (t,x) \bigr\vert ^{p_{2}}\,dx \\ ={} &4(2-Np_{2}+N+\alpha ) \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}^{2}+\frac{4Np_{1}-8(Np _{2}-N-\alpha )}{p_{1}+2} \bigl\Vert \psi (t) \bigr\Vert _{L^{p_{1}+2}}^{p_{1}+2} \\ &{}+8(Np_{2}-N-\alpha )E(\psi _{0}) \\ \leq{} & -4(Np_{2}-N-\alpha -2)\delta x_{0}^{2}< 0. \end{aligned}$$

(4.23)

Therefore, by the classical argument for Schrödinger equations, the solution $\psi (t)$ of (1.1) blows up. □

Theorem 4.4

Let$\lambda _{1}=-1$, $\lambda _{2}=1$, $1+\frac{\alpha +2}{N}< p_{2}<1+\frac{Np _{1}+2\alpha }{2N}$, and$E(\psi _{0})< \frac{Np_{2}-N-\alpha -2}{2(Np _{2}-N-\alpha )}x_{0}^{2}$, and$\psi \in C([0,T^{*}),H^{1})$be a solution of (1.1). If$\|\nabla \psi _{0}\|< x_{1}$, then the solution$\psi (t)$of (1.1) exists globally; If$\|\nabla \psi _{0}\|>x _{1}$, then the solution$\psi (t)$of (1.1) blows up, where$x_{1}$is defined by (4.25).

Proof

Applying (2.7), it follows that

$$\begin{aligned} E\bigl(\psi (t)\bigr)&=\frac{1}{2} \bigl\Vert \nabla \psi (t) \bigr\Vert ^{2}_{L^{2}}- \frac{1}{p _{1}+2} \bigl\Vert \psi (t) \bigr\Vert ^{p_{1}+2}_{L^{p_{1}+2}} +\frac{1}{2p_{2}} \int _{\mathbb{R}^{N}} \bigl(I_{\alpha }\ast \vert \psi \vert ^{p_{2}}\bigr) (t,x) \bigl\vert \psi (t,x) \bigr\vert ^{p _{2}}\,dx \\ &\geq \frac{1}{2} \bigl\Vert \nabla \psi (t) \bigr\Vert ^{2}_{L^{2}}- \frac{C_{*}}{p_{1}+2} \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}^{\frac{Np_{1}}{2}} \bigl\Vert \psi (t) \bigr\Vert ^{p_{1}+2-\frac{Np_{1}}{2}}_{L^{2}} \\ &=f\bigl( \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}\bigr), \end{aligned}$$

(4.24)

where the $C_{*}$ are defined by (2.8) and

$$ f(x):=\frac{1}{2}x^{2}-\frac{C_{*}}{p_{1}+2} \Vert \psi _{0} \Vert ^{p_{1}+2-\frac{Np _{1}}{2}}_{L^{2}}x^{\frac{Np_{1}}{2}}. $$

By a simple computation, we find that the unique positive solution $x_{1}$ of $f'(x)=0$ is given by

$$ x_{1}= \biggl(\frac{2(p_{1}+2)}{C_{*}Np_{1} \Vert \psi _{0} \Vert _{L^{2}}^{p_{1}+2-\frac{Np _{1}}{2}}} \biggr)^{\frac{2}{Np_{1}-4}}. $$

(4.25)

This implies that f is increasing on $(0,x_{1})$ and decreasing on $(x_{1},\infty )$. By a simple computation, it follows that

$$ f(x_{1})=\frac{Np_{1}-4}{2Np_{1}}x_{1}^{2}. $$

By (2.2) and the assumption $E(\psi _{0})< f(x_{1})$, it follows that

$$ f\bigl( \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}\bigr)\leq E( \psi _{0})< f(x_{1}),\quad \forall t\in \bigl[0,T^{*}\bigr). $$

If $\|\nabla \psi _{0}\|_{L^{2}}< x_{1}$, it follows from the continuity argument that $\|\nabla \psi (t)\|_{L^{2}}< x_{1}$ for all $t\in [0,T ^{*})$. Therefore, the solution $\psi (t)$ of (1.1) exists globally.

If $\|\nabla \psi _{0}\|_{L^{2}}>x_{1}$, we deduce from the continuity argument that $\|\nabla \psi (t)\|_{L^{2}}>x_{1}$ for all $t\in [0,T ^{*})$. We can choose $\delta >0$ small enough so that

$$ E(\psi _{0})\leq (1-\delta )f(x_{1}). $$

This implies that

$$ 4Np_{1}E(\psi _{0})\leq 4Np_{1}(1-\delta )f(x_{1})=2(Np_{1}-4) (1- \delta )x_{1}^{2}. $$

(4.26)

Thus, we deduce from (2.2), (2.5) and (4.26) that

$$\begin{aligned} F''(t)={} &8 \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}^{2}-\frac{4Np_{1}}{p_{1}+2} \bigl\Vert \psi (t) \bigr\Vert _{L^{p_{1}+2}}^{p_{1}+2} \\ &{}+\frac{4p_{2}N-4N-4\alpha }{p_{2}} \int _{\mathbb{R}^{N}} \bigl(I_{\alpha } \ast \vert \psi \vert ^{p_{2}}\bigr) (t,x) \bigl\vert \psi (t,x) \bigr\vert ^{p_{2}}\,dx \\ ={} &(8-2Np_{1}) \bigl\Vert \nabla \psi (t) \bigr\Vert _{L^{2}}^{2} \\ &{}+\frac{4Np_{2}-4N-4\alpha -2Np_{1}}{p_{2}} \int _{\mathbb{R}^{N}} \bigl(I_{\alpha }\ast \vert \psi \vert ^{p_{2}}\bigr) (t,x) \bigl\vert \psi (t,x) \bigr\vert ^{p_{2}}\,dx \\ &{}+4Np_{1}E(\psi _{0}) \\ \leq{} & -2(Np_{1}-4)\delta x_{1}^{2}< 0. \end{aligned}$$

(4.27)

Therefore, by the classical argument for Schrödinger equations, the solution $\psi (t)$ of (1.1) blows up. □

5 Conclusions

In this paper, we obtain some sharp thresholds of blow-up and global existence for the nonlinear Schrödinger–Choquard equation. We firstly obtain some sufficient conditions about existence of blow-up solutions. Due to the loss of scaling invariance for this equation, we derive some sharp thresholds of blow-up and global existence by constructing some new estimates. In particular, we prove the global existence for this equation with critical mass in the $L^{2}$-critical case.