Instability of standing waves for the inhomogeneous Gross-Pitaevskii equation

1.   Introduction

In this paper, we consider the following inhomogeneous Gross-Pitaevskii equation

where $\psi:[0, T^*)\times \mathbb{R}^{N}\rightarrow \mathbb{C}$ is a complex valued function, $\psi_0\in \Sigma$, $T^*\in (0, \infty]$, $b\in (0, \min\{2, N\})$, $p\in (0, \frac{4-2b}{N-2})$. Thus, the Cauchy problem (1.1) is local well-posedness in the energy space $\Sigma$, where $\Sigma$ is defined by

with the norm

When $\omega\in (-N, \infty)$, we notice that

there exist positive constants $C_1(\omega)$ and $C_2(\omega)$ such that

for all $u\in \Sigma$.

The inhomogeneous Gross-Pitaevskii equation (1.1) arises naturally in nonlinear optics for the propagation of laser beams. (1.1) also appears in Bose-Einstein condensation, where the harmonic potential $|x|^2$ may model a confining magnetic potential. The nonlinearity $|x|^{-b}|\psi|^{p}\psi$ describes the propagation of waves in the inhomogeneous medium, see ^[1,2,3] for the related physical backgrounds.

Recently, this type of equations has been studied extensively in ^{[4,5,6,7,8,9,10,11,12,13,14]}. Eq (1.1) enjoys a class of special solutions, which are called standing waves, namely solutions of the form $e^{i\omega t}u_\omega(x)$, where $\omega \in \mathbb{R}$ is a frequency and $u_\omega\in \Sigma$ is a nontrivial solution to the elliptic equation

Note that (1.3) can be written as $S'_\omega(u_\omega) = 0$, where

is the action functional. We also define the following functionals

where $\alpha = \frac{Np}{2}+b$ and $u^\lambda(x): = \lambda^{\frac{N}{2}}u(\lambda x)$. The ground state for (1.3) is defined by

where

is the set of all nontrivial solutions for (1.3).

We firstly recall the definitions of stability and instability of standing waves, see ^[15].

Definition 1.1. Let $\psi(t, x) = e^{i\omega t}u_\omega(x)$ be a standing wave solution of (1.1).

1. The solution $\psi(t, x) = e^{i\omega t}u_\omega(x)$ is said to be orbitally stable if, for any $\varepsilon > 0$, there exists $\delta > 0$ such that for any initial data $\psi_0$ satisfying $\|\psi_0-u_\omega\|_{\Sigma} < \delta$, then the corresponding solution $\psi(t)$ of (1.1) exists globally in time and satisfies

2. The solution $\psi(t, x) = e^{i\omega t}u_\omega(x)$ is said to be unstable if $\psi(t, x) = e^{i\omega t}u_\omega(x)$ is not stable.

3. The solution $\psi(t, x) = e^{i\omega t}u_\omega(x)$ is said to strongly unstable if for any $\varepsilon > 0$, there exists $\psi_0\in \Sigma$ such that $\|\psi_0-u_\omega\|_{\Sigma} < \varepsilon$, and the corresponding solution $\psi(t)$ of (1.1) blows up in finite time.

Next, we recall some known instability results for nonlinear Schrödinger equations. The strong instability was first studied by Berestycki and Cazenave, see ^[15]. Later, Le Coz in ^[16] gave an alternative, simple proof of the classical result of Berestycki and Cazenave. The key point is to establish the finite time blow-up by using the variational characterization of ground states as minimizers of the action functional and the virial identity. More precisely, based on the variational characterization of ground states on the Pohozaev manifold $\mathcal{N}: = \{v\in H^1, \; \; Q(v) = 0\}$ or the Nehari manifold, ones can obtain the key estimate $Q(\psi(t))\leq 2(S_\omega(\psi_0)-S_\omega(u_\omega))$, where $u_\omega$ is the ground state solution. Then, it follows from the virial identity and the choice of initial data $\psi_0$ that

This implies that the solution $\psi(t)$ blows up in a finite time. Thus, ones can prove the strong instability of ground state standing waves, see, e.g., ^{[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33]}.

For the nonlinear Schrödinger equation with a harmonic potential, i.e., $b = 0$ in (1.1), by constructing the cross-invariant manifolds of the evolution flow and defining the cross-invariant variational problems, Zhang in ^[32] studied the sharp threshold of global existence and blow-up, and proved the strong instability of standing waves. Recently, by using the idea of Zhang in ^[32], Ardila and Dinh in ^[34] studied the strong instability of standing waves of (1.1). More precisely, when $\frac{4-2b}{N} < p < \frac{4-2b}{N-2}$, defining the following variational problems

and

Ardila and Dinh in ^[34] obtained some sharp thresholds of global existence and blow-up. Under the assumption $d(\omega)\leq m(\omega)$, they proved that the standing wave $e^{i\omega t}u_\omega(x)$ is strongly unstable. However, this assumption is still vague. It is hard to determine for which $\omega$, $d(\omega)\leq m(\omega)$ is true, see also Remark 5.1 in ^[32].

Motivated by this work, we will further study the strong instability of standing waves of (1.1) from a different perspective. Let $u\in \Sigma$, we define

It is obvious that $f(\lambda)\rightarrow +\infty$, as $\lambda \rightarrow 0^+$. Thus, there is no maximum point of $f(\lambda)$ on $(0, \infty)$. It is hard to establish the variational characterization of ground states in the Pohozaev manifold $\mathcal{N}$. On the other hand, we define

It is obvious that equation $f_1'(\lambda) = 0$ has unique solution $\lambda_0 > 0$, and $f_1(\lambda)$ has the unique maximum point on $(0, \infty)$. Based on this fact, ones can easily obtain the variational characterization of ground states in the Nehari manifold by using the compact embedding $\Sigma \hookrightarrow L^q$ with $q\in [2, 2^*)$. But it is difficult to obtain the key estimate $Q(\psi(t))\leq 2(S_\omega(\psi_0)-S_\omega(u_\omega))$.

Since there is no any maximum of function $f(\lambda)$ on $(0, \infty)$, we assume that $\lambda = 1$ is the local maximum point of $f(\lambda)$, i.e., $\partial_\lambda^2S_\omega(u_\omega^\lambda)|_{\lambda = 1}\leq 0$. In this assumption, we will study the strong instability of standing waves of (1.1). Moreover, we obtain that there exists $\omega_* > 0$ such that for all $\omega > \omega_*$, the standing wave $\psi(t, x) = e^{i\omega t}u_\omega(x)$ is unstable.

Firstly, we establish the variational characterization of the ground states of (1.3). To this aim, we recall the existence of minimizing problem (1.8) established by Ardila and Dinh in ^[34].

Lemma 1.2. Let $a > 0$, $N\geq 1$, $0 < b < \min\{2, N\}$, $0 < p < \frac{4-2b}{N-2}$, $\omega > -aN$. Then $d(\omega) > 0$ and $d(\omega)$ is attained by a function which is a solution to the elliptic equation (1.3). Moreover, every minimizer is the form $e^{i\theta}u$, where $u\in \Sigma$ is a real-valued, positive and spherically symmetric function.

Since the embedding $\Sigma \hookrightarrow L^q$ with $q\in [2, 2^*)$ is compact, this result can be easily proved by (1.2). Based on this existence result, we can easily obtain the following variational characterization of the ground states to (1.3). So we omit the proof.

Theorem 1.3. Let $N\geq 1$, $0 < b < \min\{2, N\}$, $\omega > -aN$, $0 < p < \frac{4-2b}{N-2}$. Then $u_\omega\in\mathcal{G}_{\omega}$ if and only if $u_\omega$ solves the minimizing problem (1.8).

Based on this variational characterization of the ground states, we can obtain the key estimate $Q(\psi(t))\leq 2(S_\omega(\psi_0)-S_\omega(u_\omega))$ under the assumption $\partial_\lambda^2S_\omega(u_\omega^\lambda)|_{\lambda = 1}\leq 0$. Therefore, we can obtain the following instability and strong instability of standing waves of (1.1).

Theorem 1.4. Let $N\geq 1$, $0 < b < \min\{2, N\}$, $\frac{4-2b}{N} < p < \frac{4-2b}{N-2}$. Assume that $u_\omega$ is the ground state related to (1.3). Then, there exists $\omega_* > 0$ such that for all $\omega > \omega_*$, the standing wave $\psi(t, x) = e^{i\omega t}u_\omega(x)$ is unstable.

Theorem 1.5. Let $N\geq 1$, $0 < b < \min\{2, N\}$, $\omega > 0$, $\frac{4-2b}{N} < p < \frac{4-2b}{N-2}$. Assume that $u_\omega$ is the ground state related to (1.3) and $\partial_\lambda^2S_\omega(u_\omega^\lambda)|_{\lambda = 1}\leq 0$. Then the standing wave $\psi(t, x) = e^{i\omega t}u_\omega(x)$ is strongly unstable by blow-up.

Combining this theorem and Lemma 3.5, we can easily obtain the following corollary.

Corollary 1.6. Let $N\geq 1$, $0 < b < \min\{2, N\}$, $\omega > 0$, $\frac{4-2b}{N} < p < \frac{4-2b}{N-2}$. Assume that $u_\omega$ is the ground state related to (1.3). Then, there exists $\omega_* > 0$ such that for any $\omega > \omega_*$, the standing wave $\psi(t, x) = e^{i\omega t}u_\omega(x)$ is strongly unstable by blow-up.

Remark. Ardila and Dinh in ^[34] proved the standing wave $e^{i\omega t}u_\omega(x)$ is strongly unstable under the assumption $d(\omega)\leq m(\omega)$. However, it is hard to determine for which $\omega$, $d(\omega)\leq m(\omega)$ is true, see also Remark 5.1 in ^[32]. In this corollary, when $\omega$ is large, we prove that the standing wave $\psi(t, x) = e^{i\omega t}u_\omega(x)$ is strongly unstable by blow-up.

This paper is organized as follows: in Section 2, we will collect some preliminaries such as the local well-posedness of (1.1), the virial identity to (1.1), Pohozaev's identities related to (1.3). In section 3, we will prove the instability of the standing wave $e^{i\omega t}u_\omega(x)$. In section 4, we will prove the strong instability of the standing wave $e^{i\omega t}u_\omega(x)$.

2.   Preliminaries

In this section, we recall some useful results. Firstly, we recall the following local well-posedness result for (1.1). By a similar argument as that in [15,Theorem 9.2.6], we can estabilish the following local well-posedness result for (1.1).

Lemma 2.1. Let $N\geq 1$, $0 < b < \min\{2, N\}$, $0 < p < \frac{4-2b}{N-2}$, and $\psi_0 \in \Sigma$. Then, there exists $T = T(\|\psi_0\|_{\Sigma})$ such that (1.1) admits a unique solution $\psi \in C([0, T), \Sigma)$. Assume that $[0, T^{\ast })$ is the maximal time interval of solution $\psi(t)$. If $T^{\ast } < \infty$, then $\| \psi(t)\| _{\Sigma}\rightarrow \infty$ as $t\uparrow T^{\ast }$. Moreover, the solution $\psi(t)$ depends continuously on initial data $\psi_0$ and satisfies the following mass and energy conservation laws

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

In order to study the strong instability of standing waves, we need to prove the existence of blow-up solutions. In order to study the strong instability of standing waves, we need to prove the existence of blow-up solutions. Following the classical convexity method of Glassey, by some formal computations which are made rigorously in ^[15]), we can obtain the following lemma.

Lemma 2.2. Let $N\geq 1$, $0 < b < \min\{2, N\}$, $0 < p < \frac{4-2b}{N-2}$. Assume that $\psi_0 \in \Sigma: = \{v\in H^1\; and\; |x|v \in L^2\}$ and $\psi \in C([0, T^*), \Sigma)$ is the corresponding solution of (1.1). Then, $\psi(t)\in \Sigma$ for all $t\in [0, T^*)$ and the function $J(t)$ belongs to $C^2[0, T^*)$, where $J(t) = \int_{\mathbb{R}^N} |x|^2 |\psi(t, x)|^2dx$. Furthermore, we have

for all $t\in [0, T^*)$, where $Q(u)$ is defined by (1.6).

Next, we recall the following Pohozaev's identities related to (1.3), see [34,Lemma 5.1].

Lemma 2.3. [34,Lemma 5.1] If $u\in \Sigma$ and satisfies equation (1.3), then the following properties hold:

and

Lemma 2.4. [35,Theorem 1.2] Let $0 < p < \frac{4-2b}{N-2}$ and $0 < b < \min\{2, N\}$. Then, for all $u\in H^1$,

where the best constant $C_{opt}$ is given by

where $Q$ is the ground state of the elliptic eqation

Moreover, the following Pohozaev's identities hold true:

3.   Non-linear instability

In this section, we will prove Theorem 1.4. Based on the variational characterization of the ground states, we can obtain the following result.

Lemma 3.1. Let $N\geq 1$, $0 < b < \min\{2, N\}$, $\omega > -aN$, $\frac{4-2b}{N} < p < \frac{4-2b}{N-2}$, $u_\omega$ be the ground state related to (1.3). Assume that $v\in \Sigma$, and $\int_{\mathbb{R}^N}|x|^{-b} |v(x)|^{p+2}dx = \int_{\mathbb{R}^N}|x|^{-b}|u_\omega(x)|^{p+2}dx$, then $S_\omega(u_\omega)\leq S_\omega(v)$.

Proof. Firstly, we notice that $S_\omega(v)$ can be written as

By the variational characterization of ground state $u_\omega$, we have

This implies that $d_1 = S(u_\omega) = \frac{p}{2(p+2)}\int_{\mathbb{R}^N}|x|^{-b} |u_\omega(x)|^{p+2}dx$. We set

Since it is clear that $d_2\leq d_1$, we show $d_1\leq d_2$. For any $v\in \Sigma\backslash \{0\}$ satisfying $K_\omega(v) < 0$, there exists $\mu_0\in (0, 1)$ such that $K_\omega(\mu_0v) = 0$. Thus, we have

Hence, we have $d_1\leq d_2$.

Finally, we define

Since $d_3\leq S(u_\omega)$, it suffices to prove $d_3\geq S(u_\omega)$. By $d_1 = d_2$, for any $v\in \Sigma\backslash \{0\}$ satisfying $\int_{\mathbb{R}^N}|x|^{-b} |v(x)|^{p+2}dx = \int_{\mathbb{R}^N}|x|^{-b} |u_\omega(x)|^{p+2}dx$, we have $K_\omega(v)\geq 0$. Thus, we have

Therefore, we obtain $d_3\geq S(u_\omega)$. This complete the proof.

In order to study the instability, for any $u_\omega\in \Sigma$ and $\varepsilon > 0$, we define

Lemma 3.2. Let $N\geq 1$, $0 < b < \min\{2, N\}$, $\omega > -aN$, $\frac{4-2b}{N} < p < \frac{4-2b}{N-2}$. Assume that $u_\omega \in \mathcal{G}_\omega$ and $\partial^2_\lambda S_\omega (u_\omega^{\lambda})|_{\lambda = 1} < 0$. Then there exist $\varepsilon, \delta > 0$, and a mapping

such that $K_\omega(v^{\lambda(v)}) = 0$ for any $v\in U_{\varepsilon}(u_\omega)$.

Proof. Let

Since $u_\omega$ is a minimizer of $S_\omega(v)$ constrained on the manifold $\mathcal{N}: = \{v\in \Sigma\backslash \{0\}, \; \; K_\omega(v) = 0\}$, then

Next, since

then

Combining (3.1) and (3.2), we have $\langle\partial_\lambda u_\omega^\lambda|_{\lambda = 1}, u_\omega\rangle\neq 0$ and so

Thanks to $\partial_\lambda F(u_\omega, 1) = K_\omega(u_\omega) = 0$, applying the implicit function theorem, there exist $\varepsilon, \delta > 0$, and a mapping

such that $K_\omega(v^{\lambda(v)}) = 0$ for all $v\in U_{\varepsilon}(u_\omega)$. This completes the proof.

Lemma 3.3. Let $N\geq 1$, $0 < b < \min\{2, N\}$, $\omega > 0$, $\frac{4-2b}{N} < p < \frac{4-2b}{N-2}$. Assume that $u_\omega \in \mathcal{G}_\omega$ and $\partial^2_\lambda S_\omega (u_\omega^{\lambda})|_{\lambda = 1} < 0$. Then there exist $\varepsilon_0, \delta_0 > 0$ such that, for any $v\in U_{\varepsilon_0}(u_\omega)$

for some $\lambda(v)\in (1-\delta_0, 1+\delta_0)$.

Proof. Since $\partial^2_\lambda S_\omega (u_\omega^{\lambda})|_{\lambda = 1} < 0$ and $\partial^2_\lambda S_\omega (v^{\lambda})$ is continuous in $\lambda$ and $v$, we know that there exists $\varepsilon_0, \delta_0 > 0$ such that $\partial^2_\lambda S_\omega (v^{\lambda}) < 0$ for any $\lambda\in (1-\delta_0, 1+\delta_0)$ and $v\in U_{\varepsilon_0}(u_\omega)$. Noticing that $\partial_\lambda S_\omega (v^{\lambda})|_{\lambda = 1} = Q(v)$, applying the Taylor expansion for the function $S_\omega (v^{\lambda})$ at $\lambda = 1$, we have

By Lemma 3.2, we choose $\varepsilon_0 < \varepsilon$ and $\delta_0 < \delta$, then there exists $\lambda(v)\in (1-\delta_0, 1+\delta_0)$ such that $K_\omega (v^{\lambda(v)}) = 0$ for all $v\in U_{\varepsilon_0}(u_\omega)$. Therefore, we have $S_\omega (v^{\lambda(v)})\geq S_\omega (u_\omega)$. This, together with (3.3) implies that

for some $\lambda(v)\in (1-\delta_0, 1+\delta_0)$.

Let $u_\omega \in \mathcal{G}_\omega$, we define

and

where $\psi(t)$ is a solution of (1.1) with initial data $\psi_0$. Then, we have the following lemma.

Lemma 3.4. Let $N\geq 1$, $0 < b < \min\{2, N\}$, $\omega > -aN$, $\frac{4-2b}{N} < p < \frac{4-2b}{N-2}$. Assume that $u_\omega \in \mathcal{G}_\omega$ and $\partial^2_\lambda S_\omega (u_\omega^{\lambda})|_{\lambda = 1} < 0$. Then, for any $\psi_0\in \mathcal{C}_\omega$, there exists $\delta_2 = \delta_2(\psi_0) > 0$ such that $Q(\psi(t))\leq -\delta_2$ for all $t\in [0, T(\psi_0))$.

Proof. Let $\psi_0\in \mathcal{C}_\omega$, and $\delta_1 = S_\omega (u_\omega)-S_\omega (\psi_0) > 0$. We deduce from Lemma 3.3 that

which implies

Thus, $Q(\psi(t))\neq 0$. Due to $\psi_0\in \mathcal{C}_\omega$, then $Q(\psi_0) < 0$. It follows from the continuity of $Q(\psi(t))$ that

Thus, $\lambda(\psi(t))\in (1-\delta_0, 1)$. Combining (3.4), we have

Thus, $Q(\psi(t))\leq \delta_2$ with $\delta_2 = -\frac{\delta_1}{\delta_0}$, for $0\leq t < T(\psi_0)$.

Lemma 3.5. Let $N\geq 1$, $0 < b < \min\{2, N\}$, $\omega > -aN$, $\frac{4-2b}{N} < p < \frac{4-2b}{N-2}$. Assume that $u_\omega \in \mathcal{G}_\omega$. Then, there exists $\omega_* > 0$ such that $\partial^2_\lambda S_\omega (u_\omega^{\lambda})|_{\lambda = 1}\leq 0$ for all $\omega > \omega_*$.

Proof. Let $u_\omega \in \mathcal{G}_\omega$, $f(\lambda)$ be defined by (1.9), it follows from Lemma 2.3 that $Q(u_\omega) = 0$, i.e., $f'(1) = 0$. We consequently obtain

Thus, $\partial^2_\lambda S_\omega (u_\omega^{\lambda})|_{\lambda = 1}\leq 0$ if and only if

So it is sufficient to prove that

Let $u_\omega(x) = \omega^{\frac{2- b}{2p}}\tilde{u}_\omega(\sqrt{\omega}x)$, then $\tilde{u}_\omega$ satisfies

Since

it is sufficient to prove that

Let $V\in H^1\setminus \{0\}$ be a ground state solution to the elliptic problem

then

where

and

Then, by a similar argument as that in Lemma 3.1, we have

and

where

In addition, we infer from $\tilde{K}_0(V) = 0$ that for $\lambda > 1$

Then, for any $\lambda > 1$, there exists $\omega(\lambda)$ such that $\tilde{K}_\omega(\lambda V) < 0$. This and (3.7) imply that

On the other hand, we deduce from $\tilde{K}_\omega(\tilde{u}_\omega) = 0$ that

Then, for any $\lambda > 1$, $\tilde{K}_0(\lambda\tilde{u}_\omega) < 0$. We consequently deduce from (3.6) and (3.8) that for any $\omega > \omega(\lambda)$,

This implies

Notice that

there exists $\lambda(\omega) > 0$ such that $\tilde{K}_0(\lambda(\omega) \tilde{u}_\omega) = 0$. This and (3.6) yield that

This implies that $\liminf_{\omega \rightarrow \infty}\lambda(\omega)\geq 1$ and

On the other hand, we deduce from $\tilde{K}_\omega(\tilde{u}_\omega) = 0$ that $\tilde{K}_0(\tilde{u}_\omega) < 0$. This implies that

Combining this and (3.10), it follows that

This implies that $\lim_{\omega \rightarrow \infty}\tilde{K}_0(\tilde{u}_\omega) = 0$. We consequently obtain that

as $\omega \rightarrow \infty$. Thus, we see from (3.9) that

This completes the proof.

Proof of Theorem 1.4. Let $f(\lambda)$ be defined by (1.9), then

for all $\lambda\geq 1$. This, together with Lemma 2.3 implies that

and

for all $\lambda > 1$. On the other hand, it follows from Brezis-Lieb's lemma that $u_\omega^\lambda\rightarrow u_\omega$ as $\lambda \rightarrow 1$. Thus, for any $\varepsilon > 0$, there exists $\lambda_0 > 1$ such that $\|u_\omega^{\lambda_0}- u_\omega\|_{\Sigma} < \varepsilon$.

Next, let $\psi_0 = u_\omega^{\lambda_0}$, then $\psi_0\in U_\varepsilon(u_\omega)$, $S_\omega(\psi_0) < S_\omega(u_\omega)$ and $Q(\psi_0) < 0$. Thus, $\psi_0\in \mathcal{C}_\omega$ and there exists $\delta_2 = \delta_2(\psi_0) > 0$ such that $Q(\psi(t))\leq -\delta_2$ for all $t\in [0, T(\psi_0))$. Then, we deduce from Lemma 2.2 that

for all $t\in [0, T(\psi_0))$. If $e^{i\omega t}u_\omega$ is orbitally stable, then $T(\psi_0) = +\infty$ and $Q(\psi(t))\leq -\delta_2$ for all $t\in [0, \infty)$. This implies that $J(t)$ becomes negative for long time. This is an contradiction. Moreover, applying Lemma 3.5, there exists $\omega_* > 0$ such that $\partial^2_\lambda S_\omega (u_\omega^{\lambda})|_{\lambda = 1} < 0$ for all $\omega > \omega_*$. Thus, when $\omega > \omega_*$, the standing wave $e^{i\omega t}u_\omega$ is unstable.

4.   Strong instability

In this section, we will prove Theorem 1.5. To this end, we firstly establish the following key estimate.

Lemma 4.1. Let $N\geq 1$, $0 < b < \min\{2, N\}$, $\omega > 0$, $\frac{4-2b}{N} < p < \frac{4-2b}{N-2}$. Assume that $u_\omega \in \mathcal{G}_\omega$ and $\partial^2_\lambda S_\omega (u_\omega^{\lambda})|_{\lambda = 1}\leq 0$. Suppose further that $v\in \Sigma\backslash \{0\}$ such that

Then it holds that

Remark. It is easy to see from Lemma 4.1 that for $u_\omega\in \mathcal{G}_\omega$ satisfying $\partial^2_\lambda S_\omega(u_\omega^{\lambda})|_{\lambda = 1}\leq 0$,

Indeed, if there exists $v\in \Sigma\backslash \{0\}$ satisfying $\|v\|_{L^2} = \|u_\omega\|_{L^2}$, $S_\omega(v) < S_\omega(u_\omega)$, $K_\omega(v) < 0$ and $Q(v) = 0$, then by this lemma,

which is a contradiction.

Proof. If $K_\omega(v) = 0$, we infer from Theorem 1.3 and $Q(v) < 0$ that

which is the desired estimate (4.1).

When $K_\omega(v) < 0$, we notice that

Since $\lim_{\lambda\rightarrow 0}K_\omega(v^\lambda) = \omega\|v\|_{L^2}^2 > 0$ and $K_\omega(v) < 0$, there exists $\lambda_0\in (0, 1)$ such that $K_\omega(v^{\lambda_0}) = 0$. Applying Theorem 1.3, it follows that

When $\int_{\mathbb{R}^N}|x|^2|v(x)|^2dx\geq \int_{\mathbb{R}^N}|x|^2|u_\omega(x)|^2dx$, it follows from $Q(u_\omega) = 0$ that

which is the desired estimate (4.1).

When $\int_{\mathbb{R}^N}|x|^2|v(x)|^2dx < \int_{\mathbb{R}^N}|x|^2|u_\omega(x)|^2dx$, we define

If $f_1(\lambda_0)\leq f_1(1)$, then we deduce from Theorem 1.3 and $Q(v)\leq0$ that

which is the desired estimate (4.1).

In what follows, we will prove $f_1(\lambda_0)\leq f_1(1)$, which is equivalent to

In views of (1.9), the condition $\partial^2_\lambda S_\omega(u^{\lambda})|_{\lambda = 1}\leq 0$ is equivalent to

Combining (4.4) and $Q(u_\omega) = 0$, we can obtain that

This, together with (4.3), it suffices to show that

Let $\alpha = 2\beta$, then (4.5) is equivalent to

Let

From the Taylor expansion of $\lambda^\beta$ at $\lambda = 1$, there exists $\xi\in (\lambda_0^2, 1)$ such that

Due to $\beta > 1$ and $\xi\in (\lambda_0^2, 1)$, it follows that

Thus, we have $h(\lambda_0) > 0$. This implies that (4.5) holds. This completes the proof.

Based on this lemma, we define an invariant set $\mathcal{B}_\omega$ under the flow of (1.1).

Lemma 4.2. Let $N\geq 1$, $0 < b < \min\{2, N\}$, $\omega > 0$, $\frac{4-2b}{N} < p < \frac{4-2b}{N-2}$. Assume that $u_\omega \in \mathcal{G}_\omega$ and $\partial^2_\lambda S_\omega (u_\omega^{\lambda})|_{\lambda = 1}\leq 0$. Then, the set $\mathcal{B}_\omega$ is invariant under the flow of (1.1), that is, if $\psi_0\in \mathcal{B}_\omega$, then the solution $\psi(t)$ to (1.1) with initial data $\psi_0$ belongs to $\mathcal{B}_\omega$.

Proof. Let $\psi_0\in \mathcal{B}_\omega$, it follows from Lemma 2.1 that there exists a unique solution $\psi\in C([0, T^*), \Sigma)$. By the conservations of mass and energy, we have $S_\omega(\psi(t)) = S_\omega(\psi_0) < S_\omega(u_\omega)$ and $\|\psi(t)\|_{L^2} = \|\psi_0\|_{L^2} = \|u_\omega\|_{L^2}$ for any $t\in [0, T^*)$. In addition, by the continuity of the function $t\mapsto K_\omega(\psi(t))$ and Theorem 1.3, if there exists $t_0\in[0, T^*)$ such that $K_\omega(\psi(t_0)) = 0$, then $S_\omega(u_\omega)\leq S_\omega(\psi(t_0))$, which contradicts with $S_\omega(u_\omega) > S_\omega(\psi(t))$ for all $t\in [0, T^*)$. Therefore, the solution $\psi(t)$ satisfies $K_\omega(\psi(t)) < 0$ for all $t\in [0, T^*)$.

Finally, we prove that if $Q(\psi_0) < 0$, then $Q(\psi(t)) < 0$ for all $t\in[0, T^*)$. Let us prove this by contradiction. If not, there exists $t_0\in[0, T^*)$ such that $Q(\psi(t_0)) = 0$. Applying Lemma 4.1, we have

which is a contradiction with $S_\omega(u_\omega) > S_\omega(\psi(t))$ for all $t\in [0, T^*)$. This ends the proof.

Lemma 4.3. Let $N\geq 1$, $0 < b < \min\{2, N\}$, $\omega > 0$, $\frac{4-2b}{N} < p < \frac{4-2b}{N-2}$. Assume that $u_\omega \in \mathcal{G}_\omega$ and $\partial^2_\lambda S_\omega (u_\omega^{\lambda})|_{\lambda = 1}\leq 0$. Then, $u_\omega^\lambda \in \mathcal{B}_\omega$ for any $\lambda > 1$.

Proof. Firstly, it easily follows that

Next, we define

Thus, it follows from the assumption

that

for any $\lambda\geq 1$. This, together with Pohozaev identity related to (1.3), implies that

We consequently obtain that $K_\omega(u_\omega^\lambda) < 0$ for any $\lambda\geq 1$.

Let $f(\lambda)$ be defined by (1.9), then

for all $\lambda\geq 1$. This, together with Pohozaev identity related to (1.3), implies that

and

for any $\lambda > 1$. Thus, $u^{\lambda} \in \mathcal{B}_\omega$ for any $\lambda > 1$. This finishes the proof.

Proof of Theorem 1.5. Let $\omega > 0$ and $u_\omega$ be the ground state related to (1.3). By Lemma 4.3, we have $u_\omega^{\lambda}\in \mathcal{B}_\omega$. Let $\psi^\lambda \in C([0, T^*), \Sigma)$ be the solution of (1.1) with the initial data $u_\omega^{\lambda}$, then $\psi^\lambda(t)\in \mathcal{B}_\omega$ for all $t\in[0, T^*)$. Thus, by a classical argument, it follows that $\psi^\lambda\in \Sigma$ and

for all $t\in[0, T^*)$. This implies that the solution $\psi^\lambda$ of (1.1) with the initial data $u_\omega^{\lambda}$ blows up in finite time. Hence, the result follows, since $u_\omega^\lambda\rightarrow u_\omega$ as $\lambda \downarrow 1$.

5.   Conclusions

In this paper, we consider the instability of standing waves for an inhomogeneous Gross-Pitaevskii equation (1.1). We firstly proved that there exists $\omega_* > 0$ such that for all $\omega > \omega_*$, the ground state standing wave $\psi(t, x) = e^{i\omega t}u_\omega(x)$ is unstable. Then, we deduce that if $\partial_\lambda^2S_\omega(u_\omega^\lambda)|_{\lambda = 1}\leq 0$, the ground state standing wave $e^{i\omega t}u_\omega(x)$ is strongly unstable by blow-up. This result is a complement to the partial result of Ardila and Dinh in ^[34].

Acknowledgments

The authors would like to express their sincere thanks to the referees for the valuable comments and suggestions which helped to improve the original paper. This work is supported by the NWNU-LKQN2019-7.

Conflict of interest

All authors declare no conflicts of interest in this paper.