Global existence, blow-up and stability of standing waves for the Schrödinger-Choquard equation with harmonic potential

1.   Introduction

In this article, we conduct a comprehensive investigation into the Cauchy problem of the following Schrödinger-Choquard equation with harmonic potential:

where $N\geq3$, $V(x) = a^2|x|^{2}$ ($a\neq0$) and $\varphi_{0}\in\Sigma$, where $\Sigma$ will be defined in the next section; also, $0 < T \leq\infty$, $\varphi: [0, T) \times \mathbb{R}^{N} \rightarrow \mathbb{C}$ is a complex valued function, $0 < p_{1} < \frac{4}{N-2}$, $1+\frac{\alpha}{N} < p_{2} < 1+\frac{2+\alpha}{N-2}$, $a, \lambda_{1}, \lambda_{2}\in\mathbb{R}$, $I_\alpha:\mathbb{R}^{N} \rightarrow \mathbb{R}$ is the Riesz potential given by

with $\alpha\in(0, N)$, and $\Gamma$ denotes the gamma function.

The model (1.1) is a kind of nonlinear Schrödinger equation (NLS) with combined nonlinearities, and it has a wide range of physical applications. For example, it can be used to describe a quantum system with an infinite number of particles; see, e.g., ^[1,26,27]. Additionally, it can also describe the phenomenon of Bose-Einstein condensation in gases found in systems of Rb or Na atoms with very weak two-body interactions; see, e.g., ^[7,8,35]. From the mathematical point of view, the presence of combined nonlinearities in Eq (1.1) leads to a loss of scaling invariance and significant changes in the variational structure of the corresponding energy functional. As a result, this kind of model has attracted a great deal of interest; moreover, more and more attention is being paid to the orbital stability of standing waves and the dynamics of blow-up solutions to Eq (1.1). In the current paper, we are particularly interested in the cases that $\lambda_{2} = -1$, $p_2 = 1+\frac{2+\alpha}{N}$, $\lambda_{1}\in\mathbb{R}$ and $0 < p_1 < \frac{4}{N-2}$.

When $a = 0$, Eq (1.1) can be rewritten as below:

For $\lambda_{1} = 0$, it means that Eq (1.2) only involves the Choquard-type nonlinearity whose Cauchy problem has been extensively studied; see ^{[2,3,4,6,10,14,16,17,23]} for examples. Especially, Chen and Guo ^[10] studied the existence of blow-up solutions and the instability of standing waves for $N-3 < \alpha < N-2$ and $1+\frac{2+\alpha}{N}\leq p_{2} < \frac{N+\alpha}{N-2}$. Feng and Yuan ^[14] surveyed the local and global well-posedness of Eq (1.2) for $max\{0, N-4\} < \alpha < N$ and $2\leq p_{2} < 1+\frac{2+\alpha}{N-2}$ by employing Cazenave's method based on a compactness argument (see ^[9]). Furthermore, the limiting behaviors of blow-up solutions at finite-time are investigated in the $L^2$-critical case of $p_2 = 1+\frac{2+\alpha}{N}$. For $\lambda_{2} = 0$, Eq (1.2) only involves the power-type nonlinearity. Zhang ^[36] established the sharp criterion for the global existence and blow-up of the solutions by constructing the corresponding cross-invariant manifolds. Then, the author showed the strong instability of standing waves based on the property of the cross-constrained variational problem.

For $a = 1$, Eq (1.1) corresponds to the following form:

For $\lambda_1 = 0$ in Eq (1.3), the model has been discussed in ^{[12,18,32,34]}. It is particularly worth mentioning that Feng ^[12] obtained the sharp thresholds for global existence and blow-up in both the $L^2$-critical and $L^2$-subcritical cases. Moreover, the author showed the stability of standing waves in the $L^2$-critical case and the instability of standing waves in the $L^2$-supercritical case. For the special case of $p_2 = 2$, the finite-time blow-up solutions and instability of standing waves have been studied in ^[32]. For $\lambda_{2} = 0$, Eq (1.3) degenerates into the well-known Gross-Pitaevskii equation which has been widely investigated. In particular, there has been a large amount of results on the Cauchy problem for the Gross-Pitaevskii equation; see ^[25,29,37] for examples. For the case that the nonlocal term $(I_{\alpha}*|\varphi|^{p_{2}})|\varphi|^{p_{2}-{2}}\varphi$ is replaced by $|\varphi|^{p_{2}}\varphi$ in Eq (1.3), Zhang and Zhang ^[38] studied the stability and instability of standing waves by applying a compact embedding (see ^[35]).

Now, we return to the nonlinear Schrödinger-Choquard equation without harmonic potential, that is, Eq (1.2). For this situation, there exist a few studies concerning the issues of global existence and finite-time blow-up; see ^{[11,13,21,22,28]} for examples. For the case that $0 < p_{1} < \frac{4}{N}$ and $p_{2} = 2$, Tian et al. ^[30] constructed some invariant flows to obtain the exact energy threshold of blow-up and global existence for Eq (1.2) with the focusing $L^{2}$-subcritical term $|\varphi|^{p_{1}}\varphi$. Moreover, Tian and Zhu ^[31] investigated the sharp energy criterion for the blow-up solutions and global existence for Eq (1.2) with a focusing or defocusing perturbation, and they utilized the Strichartz estimate to obtain the lower bound of the blow-up rate. Recently, Shi and Liu ^[28] derived the sharp threshold for global existence and blow-up of the solutions for $0 < p_{1} < \frac{4}{N}$ and $p_{2} = 1+\frac{2+\alpha}{N}$ with $\lambda_1 = 1$, $\lambda_2 = -1$; they obtained the $L^{2}$-concentration and blow-up rate of the explosive solutions for Eq (1.2). Regarding the stability issues of standing waves, it should be mentioned that Liu and Shi ^[22] showed the orbital stability of standing waves in the setting of $0 < p_{1} < \frac{4}{N}$ and $p_{2} = 1+\frac{2+\alpha}{N}$ with $\lambda_1 = \lambda_2 = -1$ by using the profile decomposition theory and variational methods. In addition, the profile decomposition technique has also been applied to explore the existence of orbitally stable standing waves for fractional Schrödinger equations including combined Hartree-type and power-type nonlinear terms. See ^[5,15] for examples.

To our knowledge, the Cauchy problem of the NLS including harmonic potential and combined Choquard and power-type nonlinearities has not been investigated for $\lambda_{2} < 0$, $p_2 = 1+\frac{2+\alpha}{N}$, $\lambda_{1}\neq 0$ and $0 < p_1 < \frac{4}{N-2}$. Motivated by the works mentioned above, in the presence of harmonic potential, we shall research the existence of a global solution and the limiting dynamics of blow-up solutions as well as the stability of standing waves with prescribed mass to Eq (1.1) in the case that $\lambda_{2} = -1$, $p_2 = 1+\frac{2+\alpha}{N}$, $\lambda_{1} = \pm1$ and $0 < p_1 < \frac{4}{N-2}$.

In the present work, by taking advantage of the ground state solutions to the $L^{2}$-critical elliptic Eqs (2.11) and (2.12) and the scaling techniques, we first obtain some criteria for the global existence and blow-up of the solutions in light of ^{[12,22,31,38]}. More precisely, we derive the sharp threshold for global existence and blow-up with defocusing $L^2$-subcritical perturbation by using the sharp Gagliardo-Nirenberg inequality and scaling techniques together with the ground state for Eq (2.11). In the case of focusing $L^2$-subcritical perturbation, the major obstacle to guaranteeing the collapse of solutions to Eq (1.1) lies in that it is difficult to choose $E(\varphi_{0})$ to ensure the second-order derivative $J^{{\prime}{\prime}}(t) < -C < 0$ of $J(t) = \int|x|^2|\varphi(t, x)|^2dx$ (see (2.5)). In order to get around this difficulty, we argue by contradiction, together with scaling arguments. Then a sharp condition for blow-up is also derived. For the focusing $L^{2}$-critical perturbation case, we make use of the ground-state solutions to the $L^{2}$-critical elliptic equations given by Eqs (2.11) and (2.12) and scaling techniques, as well as the sharp Gagliardo-Nirenberg inequality, to get the criterion for global existence and blow-up. In the situation involving defocusing $L^{2}$-supercritical perturbation, with the aid of the Hardy-Littlewood-Sobolev inequality, the interpolation inequality and Young's inequality, we verify the existence of a global solution to Eq (1.1).

Second, in terms of the refined compactness argument, scaling techniques and the variational characterization of the ground state solution to the Choquard equation given by Eq (2.11) with $p_{2} = 1+\frac{2+\alpha}{N}$, we explore the limiting properties of blow-up solutions for the $L^{2}$-critical Choquard equation with $L^{2}$-subcritical perturbation, including the $L^{2}$-mass concentration and blow-up rate. With regard to the dynamical properties of blow-up solutions to Eq (1.1), the main obstacle is the loss of scaling invariance, caused by the combined nonlinearities. To overcome this obstacle, following the clues of ^[12,14,28], we apply the ground state solution of the $L^{2}$-critical Choquard equation without harmonic potential to describe the limiting behaviors of blow-up solutions at blow-up time.

Finally, the orbital stability of standing waves is investigated in the presence of $L^{2}$-subcritical perturbation, focusing $L^{2}$-critical perturbation and defocusing $L^{2}$-supercritical perturbation by using variational methods and some compactness arguments in light of ^[12,22,38]. Our results supplement the conclusions of some known works ^{[12,22,28,31]}.

The structure of this paper is as below. In Section 2, some notations and preliminaries are given. In Section 3, the criteria for the global existence and finite time blow-up of Eq (1.1) are considered. In Section 4, we focus on the limiting dynamics of blow-up solutions in the case of ${L^{2}}$-subcritical perturbation, including the mass concentration phenomenon and the dynamical properties of blow-up solutions with minimal mass. In Section 5, the stability of standing waves is covered. In the last section, the conclusion of this paper is given.

2.   Notations and preliminaries

Throughout this manuscript, for convenience, we abbreviate $\int _{\mathbb{R}^{N}} \cdot dx$ by $\int \cdot dx$, use $||\cdot||_{H^{1}}$ to denote $||\cdot||_{H^{1}(\mathbb{R}^{N})}$ and replace $||\cdot||_{L^{r}(\mathbb{R}^{N})}$ by $||\cdot||_{r}$. Meanwhile, we may as well assume that $a = 1$ and utilize $C$ to represent a positive constant that may be different from line to line.

The energy space associated with the Cauchy problem described by Eq (1.1) is given by

equipped with the norm

In addition, the energy functional defined on $\Sigma$ is denoted by

To investigate the stability of standing waves and blow-up solutions of Eq (1.1), we first give the following local well-posedness result to Eq (1.1).

Proposition 2.1. Assume that $\varphi_{0}\in \Sigma$, $N\geq3$, $0 < p_{1} < \frac{4}{N-2}$ and $2\leq p_{2} < 1+\frac{2+\alpha}{N-2}$. Then there exists $T = T(||\varphi_{0}||_\Sigma)$ such that Eq (1.1) admits a unique solution $\varphi(t)\in C([0, T), \Sigma)$. Let $[0, T)$ be the maximal interval on which the solution $\varphi(t)$ is well-defined: if $T < \infty$, then $||\varphi||_\Sigma\rightarrow \infty$ as $t \rightarrow T$. Moreover, for arbitrary $t\in[0, T)$, the solution $\varphi(t)$ obeys the conservation laws of mass and energy as below:

When $0 < p_{1} < \frac{4}{N-2}$ and $2\leq p_{2} < 1+\frac{2+\alpha}{N-2}$, this proposition can be easily proved by applying Strichartz estimates and a fixed-point argument (see ^[9,12,14]).

Proposition 2.2. Assume that $\lambda_{2} = -1$, $p_2 = 1+\frac{2+\alpha}{N}$, $\lambda_{1}\in\mathbb{R}$ and $0 < p_1 < \frac{4}{N-2}$. Let $\varphi_{0}\in \Sigma$ and $\varphi(t)$ be a solution of Eq (1.1) in $C([0, T);\Sigma)$. Moreover, let $J(t) = \int|x|^2|\varphi(t, x)|^2dx$; then,

and

Proof. Based on the work of Cazenave ^[9], by a formal computation, it is easy to obtain that

and

where

and

Combining (2.3) and (2.6)–(2.8), one can conclude that (2.5) holds true. □

Lemma 2.3. (^[33]) Let $\varphi \in H^{1}(\mathbb{R}^{N})$; then, one has that

Lemma 2.4. (^[14]) The best constant in the Gagliardo-Nirenberg-type inequality

is

where Q is the ground-state solution of the elliptic equation

In particular, in the $L^{2}$-critical case in which $p_{2} = 1+\frac{2+\alpha}{N}, \; C_{\alpha, p_{2}} = \frac{N+2+\alpha}{N}||Q||_{2}^{-\frac{4+2\alpha}{N}}$.

Lemma 2.5. (^[33]) Let $0 < p_{1} < \frac{4}{N-2}$, $N\geq3$. Then, for any $\varphi\in H^{1}(\mathbb{R}^{N})$, we have the sharp Gagliardo-Nirenberg inequality

where $C_{p_{1}, N} = \frac{p_1+2}{2||W||_2^{p_1}}$ and $W$ is the ground state solution of the elliptic equation

In particular, in the $L^{2}$-critical case in which $p_{1} = \frac{4}{N}, \; C_{p_{1}, N} = \frac{N+2}{N}||W||_{2}^{-\frac{4}{N}}$.

3.   Global existence and blow-up for Eq (1.1)

In this section, we are devoted to showing the blow-up and global existence of solutions to Eq (1.1) with $\lambda_{1} = \pm1$, $\lambda_{2} = -1$ and $0 < p_{1}\leq\frac{4}{N-2}$, $p_{2} = 1+\frac{2+\alpha}{N}$. Before stating the results, we first review some arguments about the ground-state solution of Eq (2.11) or Eq (2.12), which are of great significance in the study of the criteria of global existence or finite-time blow-up to Eq (1.1). From ^[19,20,24], the existence of ground-state solution $Q$ (or $W$) to Eq (2.11) (or Eq (2.12)) is given, respectively. The ground-state solution of the $L^{2}$-critical elliptic equation given by Eq (2.11) will be a powerful ingredient to characterize the limiting properties for the blow-up solutions at finite time in the subsequent section.

Through simple calculations, one could immediately get the Poho$\breve{z}$aev identities related to Eq (2.11) with $p_{2} = 1+\frac{2+\alpha}{N}$ and Eq (2.12) with $p_{1} = \frac{4}{N}$ as follows:

3.1.   Sharp threshold for global existence and blow-up in the case of ${L^{2}}$-subcritical perturbation

Applying the local well-posedness theory to the NLS, the solution of Eq (1.1) exists globally when the initial data size becomes sufficiently small, and, for some cases with large initial data, the global solution may not exist. Therefore, in this subsection, we focus on whether the sharp threshold of global and blow-up solutions exists for Eq (1.1) with focusing or defocusing ${L^{2}}$-subcritical perturbation.

Case 1. $\lambda_{1} = 1$.

Theorem 3.1. Assume that $\varphi_{0}\in\Sigma$, $\lambda_{1} = 1, \lambda_{2} = -1$, $0 < p_{1} < \frac{4}{N}$ and $p_{2} = 1+\frac{2+\alpha}{N}$. Let $Q$ be the ground-state solution of $L^{2}$-critical elliptic equation (2.11). Then the following assertions hold:

(i) Global existence: If $\| \varphi_{0} \|_{2} < \| Q\|_{2}$, then the solution $\varphi(t, x)$ of Eq (1.1) exists globally in $t \in [0, +\infty)$.

(ii) Blow-up: Let $\varphi_{0} = c\rho^{\frac{N}{2}}Q(\rho x)$, where $|c| > 1$, $\rho > 0$ and the following equation is satisfied

Then the corresponding solution $\varphi(t, x)$ of problem (1.1) blows up in finite time.

Proof. (i) First, from the mass conservation described by Eq (2.2) and Lemma 2.4, we have

Furthermore, we obtain from Eqs (2.1), (2.3) and (3.4) that

Combining this with $||\varphi_{0}||_{2} < ||Q||_{2}$, we conclude that $||\nabla\varphi||_{2}^{2}$ and $||x\varphi||_{2}^{2}$ are uniformly bounded for $t\in [0, +\infty)$. Therefore, the solution $\varphi(t, x)$ exists globally.

(ii) Now we prove the second part of Theorem 3.1. We first obtain from Proposition 2.2 that

and, from Eq (3.3), we get

Since $\varphi_{0} = c\rho^{\frac{N}{2}}Q(\rho x)$, by utilizing Poho$\breve{z}$aev identities (3.1), (3.6) and (3.7), one has that

At the same time, using the exponential decay of ground state solution $Q(x)$ (see ^[24]), i.e.,

we conclude that $\varphi_{0} = c\rho^{\frac{N}{2}}Q(\rho x) \in H^{1}(\mathbb{R}^{N})$ and $x\varphi_{0}\in L^{2}(\mathbb{R}^{N})$, i.e. $\varphi_{0}\in\Sigma$. In addition, one has that $||\varphi_{0}||_{2} = |c|||Q||_{2} > ||Q||_{2}$. Furthermore, we infer from Eq (3.5) that $J^{{\prime}{\prime}}(t) < 16E(\varphi_{0}) < 0$. Therefore, the solution $\varphi(t, x)$ to Eq (1.1) blows up in finite time. □

As a conclusion of the above theorem, one has the following

Corollary 3.2. Assume that $\varphi_{0}\in\Sigma$, $\lambda_{1} = 1, \lambda_{2} = -1$, $0 < p_{1} < \frac{4}{N}$ and $p_{2} = 1+\frac{2+\alpha}{N}$. If the solution $\varphi(t)$ to Eq (1.1) blows up at finite time T, then there exists $C > 0$ such that for all $t\in[0, T)$,

Proof. From Eq (2.5) and Theorem 3.1, one has

Then, we can easily obtain that

Therefore,

□

Case 2. $\lambda_{1} = -1$.

This situation is different from Case I. The major obstacle to prove the collapse of solutions is that the second-order derivative $J^{{\prime}{\prime}}(t)$ of $J(t) = \int|x|^2|\varphi(t, x)|^2dx$ has the following form:

Due to the third term $\frac{16-4Np_{1}}{p_{1}+2}\int|\varphi|^{p_{1}+2}dx > 0$, it is difficult to choose $E(\varphi_{0})$ to guarantee the blow-up of solutions. Therefore, it is of particular interest to investigate whether there exists a sharp criterion for global existence. In the theorem below, by taking advantage of the scaling argument and reduction to absurdity, we show that the blow-up solutions exist and derive the sharp threshold mass of blow-up versus global existence for Eq (1.1).

Theorem 3.3. Assume that $\varphi_{0}\in\Sigma$, $\lambda_{1} = \lambda_{2} = -1, \; 0 < p_{1} < \frac{4}{N}$ and $p_{2} = 1+\frac{2+\alpha}{N}$. Let $Q$ be the ground-state solution to $L^{2}$-critical elliptic equation (2.11). Then, we have the following conclusions:

(i) Global existence: If $\| \varphi_{0} \|_{2} < \| Q\|_{2},$ then the solution $\varphi(t, x)$ of Eq (1.1) exists globally in $t \in [0, +\infty)$.

(ii) Blow-up: Let $\varphi_{0} = c\rho^{\frac{N}{2}}Q(\rho x)$ where $|c|\geq1, \rho > 0$ and it satisfies the following equation:

Then blow-up of the corresponding solution $\varphi(t, x)$ to Eq (1.1) occurs in finite time.

Proof. (i) First, from Eqs (2.1) and (2.3) we get

Since $0 < p_{1} < \frac{4}{N}$, then $\frac{Np_{1}}{2} < 2$. Hence we infer from Lemma 2.5 and Young's inequality that, for any $0 < \varepsilon < \frac{1}{2}$, there exists a constant $C(\varepsilon, ||\varphi_{0}||_{2})$ such that

Thus, from Eqs (3.4) and (3.9), we have the following estimate:

Thus

Let $\varepsilon$ be small enough and ultilizing the fact that $||\varphi_{0}||_{2} < ||Q||_{2}$, then we conclude that $||\nabla\varphi||_{2}^{2}$ and $||x\varphi||_{2}^{2}$ are uniformly bounded for all $t\in [0, +\infty)$. Therefore, the conclusion holds.

(ii) Assume by contradiction that the corresponding solution $\varphi(t, x)$ exists globally with $T = +\infty$ and there exists $C > 0$ such that

In the case that $\lambda_{1} = \lambda_{2} = -1$, we derive from Eq (2.5) and the Poho$\breve{z}$aev identity given by Eq (3.1) that

Then utilizing the conservation of mass and interpolating between $L^{2}(\mathbb{R}^{N})$ and $L^{\frac{2N}{N-2}}(\mathbb{R}^{N})$, together with the Sobolev embedding $H^{1}(\mathbb{R}^{N})\hookrightarrow L^{\frac{2N}{N-2}}(\mathbb{R}^{N})$, we have

From Eq (3.10) and $||\varphi_{0}||_{2} = |c|||Q||_{2}$, we get

If we choose $|c| > 1$, it follows from Eqs (3.8), (3.11) and (3.12) that

Moreover, taking

then

for all $t\in [0, +\infty)$ with some constant $C > 0$. Therefore, there must exist $\widetilde{T} < +\infty$ such that

Thus by Lemma 2.4, one obtains that $\lim_{t\rightarrow \widetilde{T}}||\varphi(t)||_{H^{1}} = +\infty$, which gives a contradiction to Eq (3.10). Thus, we conclude that blow-up of the solution $\varphi(t, x)$ to Eq (1.1) occurs in some time $0 < T < \infty$. The proof is done. □

Remark 3.4. (i) In the case that $a = 0$, $\lambda_{1} = 1, \lambda_{2} = -1$, $0 < p_{1} < \frac{4}{N}$, $p_{2} = 1+\frac{2+\alpha}{N}$ and $\| \varphi_{0} \|_{2} < \| Q\|_{2}$, it is proved in ^[28] that the corresponding solution $\varphi(t, x)$ is global. In addition, the authors showed that there exists $\varphi_{0}$ such that $\| \varphi_{0} \|_{2} \geq \| Q\|_{2}$, and that the solution $\varphi(t, x)$ with the initial data $\varphi_{0}$ blows up in finite time. The above two theorems (Theorems 3.1 and 3.3) reveal that the sharp threshold for global existence of Eq (1.1) with $a\in\mathbb{R}, \lambda_{1} = \pm1$ and $\lambda_{2} = -1$ is the same as that for Theorem 3.1 of ^[28] when considered for $a = 0, \lambda_{1} = 1, \lambda_{2} = -1$. Hence, Theorem 3.3 can be seen as a supplement to Theorem 3.1 of ^[28].

(ii) Theorems 3.1 and 3.3 can also be seen as complements to Theorem 3.2 of ^[12], where the case that $a\neq0$, $\lambda_{1} = 0$, $\lambda_{2} < 0$ and $p_{2} = 1+\frac{2+\alpha}{N}$ is studied.

3.2.   Global existence and blow-up in the case of focusing ${L^{2}}$-critical perturbation

Now we consider the case with double $L^{2}$-critical nonlinear terms, i.e., $p_{1} = \frac{4}{N}$ and $p_{2} = 1+\frac{2+\alpha}{N}$. In what follows, by applying the sharp Gagliardo-Nirenberg inequality, Poho$\breve{z}$aev identity, virial-type identity and scaling approach, we obtain the criteria of global existence and blow-up for Eq (1.1).

Theorem 3.5. Assume that $\lambda_{1} = \lambda_{2} = -1$, $p_{1} = \frac{4}{N}$ and $p_{2} = 1+\frac{2+\alpha}{N}$. Let $Q$ (or $W$) be the ground state solution of $L^{2}$-critical elliptic equation (2.11) (or (2.12)). Then then the following results hold true.

(i) Global existence: If $\varphi_{0}\in\Sigma$ and $\| \varphi_{0} \|_{2}$ satisfies

then the solution $\varphi(t, x)$ of Eq (1.1) exists globally for $t \in [0, +\infty)$.

(ii) Blow-up: For any $\nu > 0$, there exists $\varphi_{0}\in\Sigma$ satisfying

such that the solution $\varphi(t, x)$ to Eq (1.1) blows up in finite time.

Proof. (i) First, from Eqs (2.2) and (2.3) and Lemmas 2.4 and 2.5, one has that

Thus, we conclude from Eqs (3.13) and (3.15) that $||\nabla\varphi||_{2}^{2}$ and $||x\varphi||_{2}^{2}$ are uniformly bounded for all $t\in [0, +\infty)$. Therefore, the solution $\varphi(t, x)$ to Eq (1.1) becomes global.

(ii) On one hand, if $||Q||_{2}\leq||W||_{2}$, it follows from Eq (3.14) that $||\varphi_{0}||_{2}^{2} = ||Q||_{2}^{2}+\nu$. Now, take $\varphi_{0}^{\tau, \rho} = \tau\rho^{\frac{N}{2}}Q(\rho x)$, where $\tau = \frac{||Q||_{2}^{2}+\nu}{||Q||_{2}^{2}} > 1$ and $\rho$ satisfies

similar to Theorem 3.1, we have that $\varphi_{0}^{\tau, \rho}\in\Sigma$. Injecting $\varphi_{0}^{\tau, \rho}$ into the energy functional, then we get

where, in the last step, we use the Poho$\breve{z}$aev identity given by Eq (3.1). Furthermore, combining Eqs (3.16) and (3.17), we deduce that

Thus, from virial-type identity (2.5) and Eq (3.18), we find that

which implies that the collapse of solution $\varphi(t, x)$ to Eq (1.1) must happen at finite time $T$.

On the other hand, if $||Q||_{2} > ||W||_{2}$, then, from Eq (3.14) we derive $||\varphi_{0}||_{2}^{2} = ||W||_{2}^{2}+\nu$. Take the initial data $\varphi_{0}^{\mu, \lambda} = \mu\lambda^{\frac{N}{2}}W(\lambda x)$, where $\mu = \frac{||W||_{2}^{2}+\nu}{||W||_{2}^{2}} > 1$ and $\lambda$ satisfies

By the exponential decay of ground state $W(x)$ (see ^[9]), one can derive $\varphi_{0}^{\mu, \lambda}\in\Sigma$. Using the same argument as that for $||Q||_{2}\leq||W||_{2}$, one has

Hence, we infer that there exists $0 < T < +\infty$ such that the solution $\varphi(t, x)$ to Eq (1.1) blows up at time $T$. Thus, the proof of Theorem 3.5 is completed. □

Remark 3.6. The result of Theorem 3.5 is a generalization to Theorem 3.6 and Remark 3.7 in ^[31], where the case that $a = 0$, $\lambda_{1} = \lambda_{2} = -1$, $p_{1} = \frac{4}{N}$, $p_{2} = 2$ and $N-\alpha = 2$ is considered.

3.3.   Global existence in the case of defocusing ${L^{2}}$-supercritical perturbation

Theorem 3.7. Assume that $\varphi_{0}\in\Sigma$, $\lambda_{1} = 1$, $\lambda_{2} = -1$, $\frac{4}{N} < p_{1} < \frac{4}{N-2}$ and $p_{2} = 1+\frac{2+\alpha}{N}$. Then, the solution $\varphi(t, x)$ of Eq (1.1) exists globally.

Proof. Since $p_{1} > \frac{4}{N}$ and $p_{2} = 1+\frac{2+\alpha}{N}$, according to the Hardy-Littlewood-Sobolev inequality and interpolation inequality, for $\varphi\in\Sigma$, one can find that $0 < \theta = \frac{2(p_{1}+2)}{Np_{1}p_{2}} < 1$ such that $\frac{1}{\frac{2Np_{2}}{N+\alpha}} = \frac{\theta}{p_{1}+2}+\frac{1-\theta}{2}$ and

Combining Young's inequality, mass conservation and Eq (3.19), for $0 < \varepsilon < \frac{2p_{2}}{p_{1}+2}$, there exists a constant $C(\varepsilon, p_{1}, p_{2}, ||\varphi_{0}||_{2}) > 0$ such that

which yields that

Thus we obtain the boundedness of $||\nabla\varphi||_{2}^{2}$ and $||x\varphi||_{2}^{2}$ for arbitrary $t\in [0, +\infty)$, which implies that the solution $\varphi(t, x)$ for Eq (1.1) exists globally. This completes the proof of Theorem 3.7. □

4.   Limiting dynamics of blow-up solutions in the case of ${L^{2}}$-subcritical perturbation

This section is concerned with the limiting dynamics of blow-up solutions to Eq (1.1) with $\lambda_{1}\in \mathbb{R}$, $\lambda_{2} = -1$, $0 < p_{1} < \frac{4}{N}$ and $p_{2} = 1+\frac{2+\alpha}{N}$, including the mass concentration phenomenon and the dynamical properties of blow-up solutions with minimal mass. Without loss of generality, we deal with the cases in which $\lambda_{1} = \pm1$. To achieve this goal, let us first review a refined compactness lemma established in ^[14].

Lemma 4.1. Let $p_{2} = 1+\frac{2+\alpha}{N}$ and $\{v_n\}_{n = 1}^{\infty}$ be a bounded sequence in $H^{1}(\mathbb{R}^{N})$ satisfying

Then, there exists $\{x_n\}_{n = 1}^{\infty}\subset \mathbb{R}^{N}$ such that, up to a subsequence,

with $\|V\|_{2}\geq(\frac{m}{p_{2}M})^{\frac{1}{2p_{2}-2}}\|Q(x)\|_{2}$, where $Q(x)$ is the ground state solution of Eq (2.11).

Using the refined compactness argument, we are able to establish the following concentration property of blow-up solutions to Eq (1.1).

Theorem 4.2. ($L^2$-concentration) Assume that $\varphi_{0}\in\Sigma, \lambda_{1} = \pm1, \lambda_{2} = -1$, $0 < p_{1} < \frac{4}{N}$ and $p_{2} = 1+\frac{2+\alpha}{N}$. Let $\varphi(t, x)$ be a corresponding solution of Eq (1.1) blowing up in finite time $T$, and suppose that $g(t):[0, T)\mapsto \mathbb{R}$ is a real-valued nonnegative function satisfying that $g(t)||\nabla \varphi(t)||_{2}\rightarrow +\infty$ as $t \rightarrow T$. Then, there exists a function $x(t)\in \mathbb{R}^{N}$ for $t < T$ satisfying

where $Q(x)$ is the ground state solution of Eq (2.11) with $p_{2} = 1+\frac{2+\alpha}{N}$.

Proof. Take

where $\{t_n\}_{n = 1}^{\infty}\subseteq [0, T)$ is an arbitrary time sequence satisfying that $t_n \rightarrow T$ as $n \rightarrow \infty$. Then, the following equations hold:

We now introduce the functional $G(\phi) = \frac{1}{2}\int|\nabla \phi|^{2}dx-\frac{1}{2p_{2}}\int(I_{\alpha}*|\phi|^{p_{2}})|\phi|^{p_{2}}dx$; then,

From Lemma 2.5, we discover that

Hence, by applying $||\nabla \varphi(t_{n})||_{2}\rightarrow +\infty$ as $n \rightarrow +\infty$ and $0 < p_{1} < \frac{4}{N}$, we deduce that $|G(v_{n})|\rightarrow 0$ as $n \rightarrow +\infty$, which yields

Take $M = ||\nabla Q||_{2}^{2}$ and $m = p_{2}||\nabla Q||_{2}^{2}$; then,

Due to Lemma 4.1, there exist $V\in H^{1}(\mathbb{R}^{N})$ and $\{x_n\}_{n = 1}^{\infty}\subset \mathbb{R}^{N}$ such that, up to a subsequence,

and

Therefore, using the fact that the $L^{2}$-norm is weakly lower semi-continuous, one has the following inequality:

Under the hypothesis on $g(t)$, one derives

For sufficiently large $n$, one can get that $R\rho_n < g(t_n)$. Combining Eqs (4.5) and (4.7), one obtains

This and Eq (4.6) infer that

Furthermore, owing to the arbitrariness of the sequence $\{t_n\}_{n = 1}^{\infty}$, one has that

For every $t\in[0, T)$, it is easy to see that the function $k(y): = \int_{|x-y|\leq g(t)}|\varphi(t, x)|^{2}dx$ is continuous and $\lim_{|y|\rightarrow \infty}k(y) = 0$. Thus, there is a function $x(t)\in\mathbb{R}^{N}$ satisfying that

Thus, the above identity together with Eq (4.8) leads to Eq (4.1). □

Theorem 4.3. (Location of $L^2$-concentration point) Assume that $\varphi_{0}\in\Sigma$, $\lambda_{1} = 1$, $\lambda_{2} = -1$, $0 < p_{1} < \frac{4}{N}$ and $p_{2} = 1+\frac{2+\alpha}{N}$. Let $\varphi(t, x)$ be a corresponding solution of Eq (1.1) that blows up in finite time $T$ and $||\varphi_{0}||_{2} = ||Q||_{2}$. Then, there exists $x_{0}\in \mathbb{R}^{N}$ such that

where $Q$ is the ground-state solution to $L^{2}$-critical Choquard equation (2.11).

Proof. By Theorem 4.2, we get

which, together with mass conservation, as described by Eq (2.1), implies that

This means that

Therefore, we obtain

From Eq (4.4), applying Lemma 2.4, one knows that, for any $\varepsilon > 0$ and any real-valued function $\theta(x)$

which indicates that

Let $\theta_{j}(x) = x_{j}$ for every $j = 1, 2, \cdots, N$ in Eq (4.12); then, combining this Eqs (2.2) and (2.3), we deduce from Eq (1.1) and $G(\varphi)\leq E(\varphi)$ that

This implies that

where $\{t_n\}_{n = 1}^{\infty}$, $\{t_m\}_{m = 1}^{\infty}\subset[0, T)$ and $\lim\limits_{n\rightarrow \infty}t_n = \lim\limits_{m\rightarrow \infty}t_m = T$. It yields that

That is,

Now, let $x_{0} = \frac{\lim\limits_{t\rightarrow T}\int|\varphi(t, x)|^{2}xdx}{||Q||_{2}^{2}}$; then, $x_{0}\in \mathbb{R}^{N}$ and

On the other hand, we deduce from Corollary 3.2 and Eq (4.11) that

which implies that

and

Hence, for any $r_{0} > 0$, we obtain

Thus, for any $\varepsilon > 0$, there exists sufficiently large $r_{0} > 0$ such that

Owing to Eqs (4.11) and (4.16), one discovers that

It follows from Eqs (4.13) and (4.17) that

which infers that

Therefore, the conclusion given by Eq (4.9) holds. □

In what follows, we research the blow-up rate of blow-up solutions for Eq (1.1) with $\|\varphi_{0}\|_{2} = \|Q\|_{2}$.

Theorem 4.4. (Blow-up rate) Assume that $\varphi_{0}\in\Sigma$, $\lambda_{1} = 1$, $\lambda_{2} = -1$, $0 < p_{1} < \frac{4}{N}$, $p_{2} = 1+\frac{2+\alpha}{N}$ and $\|\varphi_{0}\|_{2} = \|Q\|_{2}$, where $Q$ is the ground state solution to the $L^{2}$-critical Choquard equation given by Eq (2.11). Suppose that the solution $\varphi(t, x)$ of Eq (1.1) blows up in finite time $T$; then, there exists a constant $C > 0$ such that

Proof. Let $g\in C_{0}^{\infty}(\mathbb{R}^{N})$ be a nonnegative radial function such that

For $M > 0$, we define $g_{M}(x) = M^{2}g(\frac{x}{M})$ and $f_{M}(t) = \int g_{M}(x-x_{0})|\varphi(t, x)|^{2}dx$, where $x_{0} = \frac{\lim\limits_{t\rightarrow T}\int|\varphi(t, x)|^{2}xdx}{||Q||_{2}^{2}}$. Then, we deduce from Eq (4.12) that, for every $t\in[0, T)$

which indicates that

As a matter of fact, by integrating on both sides from $t$ to $T$, we obtain

In addition, we deduce from Eq (4.9) that

Thus, from Eqs (4.19) and (4.20), we get that $f_M(t)\leq C(T-t)^{2}$. Now, fix $t\in[0, T)$; thus, one has that

Finally, based on the uncertainty principle, we have

Thus, the whole proof is completed. □

Remark 4.5. (i) In the case that $a = 0$, $\lambda_{1} = 1$ and $\lambda_{2} = -1$, ^[28] has demonstrated the $L^2$-concentration property. The conclusion of Theorem 4.2, considering the situation in which $a\neq0$, $\lambda_{1} = \pm1$ and $\lambda_{2} = -1$, supplements the result of Theorem 4.2 in ^[28].

(ii) For $a = 0$, $\lambda_{1} = 1$ and $\lambda_{2} = -1$, the authors of ^[28] obtained the location of the $L^2$-concentration point and blow-up rate of the blow-up solutions with a minimal mass (see Theorems 4.3 and 4.4). Our conclusions in Theorems 4.3 and 4.4 can be seen as complements to the corresponding ones in ^[28].

5.   Orbital stability of standing waves

In this part, we initiate a study on the orbital stability of standing waves for the $L^{2}$-critical Schrödinger-Choquard equation, i.e., Eq (1.1), in the presence of focusing and defocusing $L^{2}$-subcritical perturbation, focusing $L^{2}$-critical perturbation and defocusing $L^{2}$-supercritical perturbation. Here, the standing waves are solutions to Eq (1.1) possessing the form of $\varphi(t, x) = e^{i\gamma t}u(x)$, where $\gamma\in\mathbb{R}$ represents a frequency and $u\in\Sigma$ is a nonzero solution to the stationary equation

To study the stability of standing waves, for $M > 0$, we deal with the variational problem as follows:

where

The main result of this section is as follows:

Theorem 5.1. Assume that $\varphi_{0}\in\Sigma$, $\lambda_{2} = -1$ and $p_{2} = 1+\frac{2+\alpha}{N}$. Let $Q$ be the ground state solution of $L^{2}$-critical Choquard equation (2.11). Then the standing waves for Eq (1.1) are orbitally stable in the following cases:

(i) $\lambda_{1} = \pm1$, $0 < p_{1} < \frac{4}{N}$ and $0 < M < \| Q\|_{2}^{2}$;

(ii) $\lambda_{1} = -1$, $p_{1} = \frac{4}{N}$ and $M$ satisfies that $1-\big(\frac{M}{||Q||_{2}}\big)^{\frac{4+2\alpha}{N}}- \big(\frac{M}{||W||_{2}}\big)^{\frac{4}{N}} > 0$, where $W$ is the ground state solution to Eq (2.12) with $p_{1} = \frac{4}{N}$;

(iii) $\lambda_{1} = 1$, $p_{1} > \frac{4}{N}$ and any $M > 0$.

To show Theorem 5.1, we need the vital lemma below.

Lemma 5.2. Assume that $\varphi_{0}\in\Sigma$, $\lambda_{2} = -1$ and $p_{2} = 1+\frac{2+\alpha}{N}$. Let $Q$ be the ground state solution of $L^{2}$-critical Choquard equation (2.11). If one of the following conditions holds:

(i) $\lambda_{1} = \pm1$, $0 < p_{1} < \frac{4}{N}$ and $0 < M < \| Q\|_{2}^{2}$;

(ii) $\lambda_{1} = -1$, $p_{1} = \frac{4}{N}$ and $M$ satisfies that $1-\big(\frac{M}{||Q||_{2}}\big)^{\frac{4+2\alpha}{N}}- \big(\frac{M}{||W||_{2}}\big)^{\frac{4}{N}} > 0$, where $W$ is the ground state solution to Eq (2.12) with $p_{1} = \frac{4}{N}$;

(iii) $\lambda_{1} = 1$, $p_{1} > \frac{4}{N}$ and any $M > 0$,

then there exists $u\in\Sigma$ such that $E(u) = d_{M}$ and $||u||_{2}^{2} = M$.

Proof. Suppose that $\{u_{n}\}_{n = 1}^{\infty}$ is a minimizing sequence of Eq (5.2) satisfying

Then, in Case (i), we have

When $\lambda_{1} = 1$, using Eq (2.5), it is easy to get

On the other hand, for $\lambda_{1} = -1$, by applying Young's inequality, for any $0 < \varepsilon < \frac{1}{2}$, we can derive

In Case (ii), we could easily obtain that

In Case (iii), similarly, one can discover that

By Eq (5.3), taking $n$ large enough such that $E(u_{n}) < d_{M}+1$, then, from Eqs (5.4)–(5.7), we know that $||\nabla u_{n} ||_{2}^{2}$ and $||xu_{n}||_{2}^{2}$ are both bounded, which yields that $\{u_{n}\}_{n = 1}^{\infty}$ is bounded in $\Sigma$. Therefore, there exists a subsequence, still denoted by $\{u_{n}\}$, and $u\in\Sigma$ satisfying

and

Furthermore, we deduce from the compact embedding $\Sigma \hookrightarrow L^{2}(\mathbb{R}^{N})$ that

From Eqs (5.8) and (5.9), it follows that

Moreover, using the Gagliardo-Nirenberg inequality given by Eq (2.4), we infer from the Brezis-Lieb lemma (see Lemma 2.4 in ^[24]) that

Therefore, from Eqs (5.9)–(5.12), one obtains

By the definition of variational problem (5.2), it is obvious that $d_{M}\leq E(u)$. Thus, we conclude that $d_{M} = E(u)$. Therefore, the lemma is proved completely. □

In what follows, define

Then, for any $u(x)\in S_{M}$, by the Euler-Lagrange theorem, there exists $\gamma > 0$ such that $u(x)$ is a solution to Eq (5.1); also, we generally refer to $e^{i\gamma t}u(x)$ as the orbit of $u(x)$. On the other hand, if $u\in S_{M}$, that is, $u$ is a minimizer of $d_{M}$, then $e^{i\gamma t}u\in S_{M}$. In other words, $e^{i\gamma t}u$ is also a minimizer of $d_{M}$. Subsequently, we state the definition of the orbital stability of the set $S_{M}$ as follows.

Definition 5.3. The set $S_{M}\subset\Sigma$ is called orbitally stable if for arbitrary $\varepsilon > 0$, there exists $\delta > 0$ such that, for any initial value $\varphi_{0}$ satisfying

then the corresponding solution $\varphi(t)$ of Eq (1.1) satisfies

Proof of Theorem 5.1. First, by contradiction, assume that there exist $\varepsilon_{0}$ and a sequence $\{\varphi_{0, n}\}_{n = 1}^{\infty}$ such that

and that there exists a sequence $\{t_{n}\}_{n = 1}^{\infty}$ such that the corresponding sequence $\{\varphi_{n}(t_{n})\}_{n = 1}^{\infty}$ of solutions to Eq (1.1) satisfies

Owing to Eq (5.13) and Lemma 5.2, we discover that

and

It follows from Eqs (5.15), (5.16), (2.2) and (2.3) that $\{\varphi_{n}(t_{n})\}_{n = 1}^{\infty}$ is still a minimizing sequence for the variational problem given by Eq (5.2). Therefore, combining the arguments of Lemma 5.2, there exists $u_{0}\in S_{M}$ such that

which contradicts Eq (5.14). Thus, we arrive at the conclusion of Theorem 5.1. □

Remark 5.4. In the case in which $a = 0$, $\lambda_{1} = \lambda_{2} = -1$, $0 < p_{1} < \frac{4}{N}$, $p_{2} = 1+\frac{2+\alpha}{N}$ and $M < \| Q\|_{2}$, it is shown in ^[22] that the standing waves are orbitally stable. In addition, for the cases that $a\neq0$, $\lambda_{1} = 0$, $\lambda_{2} < 0$, $p_{2} = 1+\frac{2+\alpha}{N}$ and $M < \| Q\|_{2}$, Feng, in ^[12] proved the existence of orbitally stable standing waves. The conclusion of Case (i) in Theorem 5.1 is a generalization and complement to Theorem 3.1 of ^[22] and Theorem 4.2 of ^[12].

The conclusions of Cases (ii) and (iii) in Theorem 5.1 are new and interesting in the literature.

6.   Conclusions

In this work, we study in details the global existence, blow-up and stability of standing waves for the $L^{2}$-critical Schrödinger-Choquard equation with harmonic potential. More precisely, by using the ground state solutions and scaling techniques, some criteria for the global existence and blow-up of the solutions for Eq (1.1) are obtained. Then we apply the refined compactness argument, scaling techniques and the variational characterization of the ground state solution to research the limiting dynamics of blow-up solutions for the $L^{2}$-critical Choquard equation with $L^{2}$-subcritical perturbation. In addition, we employ the variational methods to prove the orbital stability of standing waves in the presence of $L^{2}$-subcritical perturbation, focusing $L^{2}$-critical perturbation and defocusing $L^{2}$-supercritical perturbation.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was partially supported by the Jiangxi Provincial Natural Science Foundation (Grant Nos. 20212BAB211006, 20224BAB201005 and 20232BAB201009) and National Natural Science Foundation of China (Grant No. 11761032).

Conflict of interest

The authors declare that no conflicts of competing interests exists.