A comprehensive characterization of the robust isolated calmness of Ky Fan $k$-norm regularized convex matrix optimization problems

1.   Introduction and main results

In this paper, we consider the following Cauchy problem for the nonlinear Schrödinger equation with an attractive inverse-power potential

where $N\geq3$, $\psi:[0, T^*)\times\mathbb{R}^N\rightarrow \mathbb{C}$ is an unknown complex valued function with $0 < T^*\leq\infty$, $\psi_0\in H^1(\mathbb{R}^N)$, $\gamma\in(0, +\infty)$, $\sigma\in(0, 2)$ and $\frac{4}{N} < \alpha < \frac{4}{N-2}$.

In the case $\sigma = 1$, i.e., the operator $\Delta+\frac{\gamma}{|x|}$ with Coulomb potential, (1.1) describes the situation where the wave function of an electron, satisfying the Schrödinger evolution equation, is influenced by m nuclei, see ^[19] for a broader introduction. In the case $0 < \sigma < 2$, i.e., the operator $\Delta+\frac{\gamma}{|x|^\sigma}$ with slowly decaying potentials, we refer the readers to ^[15] and references therein. It also attracted a great deal of attention from mathematicians in recent years, see, e.g. ^[4,13,14,23].

Recently, this type of equations has been studied widely in ^{[1,4,5,6,7,8,9,10,11,13,17,20,22,23,24,25,26]}. In particular, Eq (1.1) enjoys a class of special solutions, which are called standing waves, namely solutions of the form $\psi(t, x) = e^{i\omega t}u(x)$, where $\omega\in \mathbb{R}$ is a frequency and $u\in H^1(\mathbb{R}^N)$ satisfies the elliptic equation

The Eq (1.2) is variational and its action functional is defined by

where the corresponding energy functional $E_\gamma(u)$ is defined by

For the evolutional type Eq (1.1), one of the most interesting problems is to consider the stability of standing waves, which is defined as follows:

Definition 1.1. Assume $u$ is a solution of (1.2). The standing wave $e^{i\omega t}u(x)$ is called orbitally stable in $H^1(\mathbb{R}^N)$, if for every $\epsilon > 0$, there exists $\delta > 0$ such that if $\psi_0\in H^1(\mathbb{R}^N)$ satisfies

then the solution $\psi(t)$ to (1.1) with $\psi|_{t = 0} = \psi_0$ satisfies

There are two main methods to study the stability of standing waves. One is the stability or instability criterion in ^[16] proposed by Grillakis, Shatah and Strauss. It says that the standing wave $e^{i\omega t}u_\omega$ is stable if $\frac{\partial}{\partial\omega}\| u_\omega\|^{2}_{L^{2}} > 0$ and unstable if not, see ^[12] for more details. However, it is hard to estimate the sign of $\frac{\partial}{\partial\omega}\| u_\omega\|^{2}_{L^{2}}$ for nonlinear Schrödinger equations without scaling invariance, e.g. (1.1). The other is the constrained minimization approach introduced by Cazenave and Lions in ^[3]. In this paper, we thus take into account the orbital stability of the set of minimizers by using the method from Cazenave and Lions in ^[3].

Now we recall the following definition of the orbital stability of the set $\mathcal{M}$.

Definition 1.2. The set $\mathcal{M}\subset H^1(\mathbb{R}^N)$ is called orbitally stable if for every $\epsilon > 0$, there exists $\delta > 0$ such that for any initial data $\psi_0\in H^1(\mathbb{R}^N)$ satisfying

the corresponding solution $u$ to (1.1) satisfies

Based on the above definition, in order to study the stability, we require that the solution of (1.1) exists globally, at least for initial data $\psi_0$ enough close to $\mathcal{M}$. In the $L^2$-subcritical case, all solutions for (1.1) exist globally. Therefore, the stability of standing waves has been studied widely in this case, see, e.g. ^[3,4,14,23]. However, in the $L^2$-supercritical case, we know that the solution of (1.1) with small initial data exists globally, but for some large initial data, the solution of (1.1) may blow up in finite time by the local well-posedness theory of NLS, see ^[2] for further inference.

For Eq (1.1), in the mass subcritical and critical cases, i.e., $0 < \alpha\leq \frac{4}{N}$, Dinh in ^[4] and Li-Zhao in ^[23] studied the stability of set of minimizers by using the concentration compactness principle. In the mass supercritical case, i.e., $\frac{4}{N} < \alpha < \frac{4}{N-2}$, Fukaya-Ohta in ^[14] proved the strong instability of standing wave $e^{i\omega t}u$ under the assumption $\partial_\lambda^2E_\gamma(u_\lambda)|_{\lambda = 1}\leq 0$ with $u_\lambda(x): = \lambda^{\frac{N}{2}}u(\lambda x)$. Therefore, whether there are stable standing waves is an interesting problem in the mass supercritical case. In this paper, we will solve this problem by considering the following minimization problem

where

In the $L^2$-supercritical case, the energy $E_\gamma(u)$ is unbounded from below on $S(a)$. Actually, when $\frac{4}{N} < \alpha < \frac{4}{N-2}$, taking $u\in S(a)$ and setting $u_\lambda(x): = \lambda^{\frac{N}{2}}u(\lambda x)$, then $\| u\|^{2}_{L^{2}} = \| u_\lambda\|^{2}_{L^{2}} = a$, and we will find that

as $\lambda\rightarrow \infty$, where $2 < \frac{N\alpha}{2} < \frac{2N}{N-2}$. Thus, we can not discuss the global minimization problem (1.5) to study the existence and stability of standing waves for (1.1). However, inspired by the thought in ^[18], we consider the further constrained minimization problem:

The sets $V(a)$ and $\partial V(a)$ are given by

for an appropriate $r_0 > 0$, depending only on $a_0$ but not on $a\in (0, a_0)$. However, compared with the case (i.e., $\gamma = 0$) considered in ^[18], the energy functional of (1.2) is not invariant under the scaling transform due to the inhomogeneous nonlinearity $\gamma|x|^{-\sigma} u$. Furthermore, we cannot prove the preconceived limiting problem of the energy functional under translation sequences. In fact, we can solve the minimization problem (1.7) by proving the boundedness of the translation sequences. We now denote all the energy minimizers of (1.7) by

Our main results are as follows:

Theorem 1.3. Let $N\geq3$, $0 < \sigma < 2$ and $\frac{4}{N} < \alpha < \frac{4}{N-2}$. Then there exists a $\gamma_0 > 0$ sufficiently small such that $0 < \gamma < \gamma_0$ and for any $a\in(0, a_0)$, the following properties hold:

(i) $\emptyset\neq\mathcal{M}(a)\subset V(a)$.

(ii) $\mathcal{M}(a)$ is orbitally stable.

This paper is organized as follows: In Section 2, we firstly give some variational problems and then prove that the solution $\psi(t)$ of (1.1) with the initial data $\psi_0$ exists globally. In Section 3, we will show that $\mathcal{M}(a)$ is orbitally stable.

2.   The variational problem

In this section, we first establish the following classical inequalities: If $N\geq2$ and $\alpha\in[2, \frac{2N}{N-2})$, then the following Gagliardo-Nirenberg inequality holds that

where $C(\alpha)$ is the sharp constant. Let $1\leq p < \infty$, if $\sigma < N$ is such that $0\leq\sigma\leq p$, then $\frac{|u(\cdot)|^p}{|\cdot|^\sigma}\in L^1(\mathbb{R}^N)$ for any $u\in H^1(\mathbb{R}^N)$. Moreover,

where $C_1 = \left(\frac{p}{N-\sigma}\right)^\sigma$.

Secondly, by (2.1) and (2.2), we have

Next, setting

we now consider the function $f(a, r)$ defined on $(0, \infty)\times(0, \infty)$ by

and, for each $a\in(0, \infty)$, its restriction $g_a(r)$ defined on $(0, \infty)$ by $r\mapsto g_a(r) = f(a, r)$. For further reference, note that for any $N\geq3$, $\eta_0\in(0, 2)$, $\eta_1\in(-2, 0)$, $\eta_2\in(0, \frac{4}{N})$ and $\eta_3\in(0, \frac{4}{N-2})$.

Lemma 2.1. [2,Theorem 9.2.6] Let $N\geq3$, $0 < \sigma < 2$ and $\frac{4}{N} < \alpha < \frac{4}{N-2}$. Then for any initial data $\psi_0\in H^1(\mathbb{R}^N)$, there exists $T = T(\|\psi_0\|_{H^1})$ such that (1.1) admits a unique solution with $\psi(0) = \psi_0$. Let $[0, T^*)$ be the maximal time interval on which the solution $\psi$ is well-defined, if $T^* < \infty$, then $\|\psi(t)\|_{H^1}\rightarrow \infty$ as $t\uparrow T^*$.Moreover, for all $0 < t < T^*$, the solution $\psi(t)$ satisfies the following conservations of mass and energy

where the energy $E_\gamma(u)$ is defined by (1.4).

Lemma 2.2. For every $a > 0$, the function $g_a(r) = f(a, r)$ has a unique global maximum and the maximum value satisfies

where

with

Proof. According to the definition of $g_a(r)$, we have

Hence, the equation $g'_a(r) = 0$ has a unique solution given by

Noticing that $g_a(r)\rightarrow -\infty$ as $r\rightarrow0$ and $g_a(r)\rightarrow -\infty$ as $r\rightarrow \infty$, we obtain that $r_a$ is the unique global maximum point of $g_a(r)$. Actually, the maximum value is

According to the definition of $a_0$, we have $\max\limits_{r > 0}g_{a_0}(r) = 0$. Hence, the lemma follows.

Now let $a_0 > 0$ be given by (2.5) and $r_0: = r_{a_0} > 0$ being determined by (2.7). Note that by the proof of Lemma 2.2, we have $f(a_0, r_0) = 0$ and $f(a, r_0)\geq 0$ for all $a\in (0, a_0)$. We denote

We now consider the following local minimization problem:

And if $(a_1, r_1)\in(0, \infty)\times(0, \infty)$ be such that $f(a_1, r_1)\geq0$, then for any $a_2\in(0, a_1]$, by Lemma 2.2 and direct calculations, we have

Theorem 2.3. Let $N\geq3$, $0 < \sigma < 2$, $\frac{4}{N} < \alpha < \frac{4}{N-2}$ and $a\in(0, a_0)$, there exists a $\gamma_0 > 0$ sufficiently small such that $0 < \gamma < \gamma_0$. Then, there exists $u\in H^1(\mathbb{R}^N)$ such that $E_\gamma(u) = m_\gamma(a, r_0)$.

Lemma 2.4. For any $a\in (0, a_0)$, the following property holds,

Proof. For any $u\in \partial V(a)$, we have $\|\nabla u\|_{L^2}^2 = r_0$. Thus, by (2.3) and Lemma 2.2, we have

Now let $u\in S(a)$ be arbitrary but fixed. We denote $u_\lambda(x): = \lambda^{\frac{N}{2}}u(\lambda x)$ for $\lambda\in(0, \infty)$. It is obvious that $u_\lambda\in S(a)$ for any $\lambda\in(0, \infty)$. We set the map on $(0, \infty)$ by

Noticing that $0 < \sigma < 2, \; \; 2 < \frac{N\alpha}{2} < \frac{2N}{N-2}$, thus $E_\gamma(u_\lambda) < 0$ and $\|\nabla u_\lambda\|^2_{L^2} = \lambda^2\|\nabla u\|^2_{L^2} < r_0$ for sufficiently small $\lambda > 0$. The proof follows.

Lemma 2.5. Let $\frac{4}{N} < \alpha < \frac{4}{N-2}$, $r_0 > 0$ be determined as Lemma 2.2 and $0 < a < a_0$ be as in Lemma 2.4. Then, there exists $\delta > 0$ such that for any initial data $\psi_0\in H^1(\mathbb{R}^N)$ and $\inf\limits_{u\in\mathcal{M}(a)}\|\psi_0-u\|_{H^1} < \delta$, the solution $\psi(t)$ of (1.1) with the initial data $\psi_0$ exists globally.

Proof. Firstly, we denote the right hand of (2.10) by $A$. According to the continuity of energy functional $E_\gamma(u)$ with respect to $u\in \mathcal{M}(a)$, we deduce $E_\gamma(u) = m_\gamma(a, r_0) < A$ and $\|\nabla u\|_{L^2}^2 < r_0$. Moreover, there exists $\delta > 0$ such that for any $\psi_0\in H^1(\mathbb{R}^N)$ and $\|\psi_0-u\|_{H^1} < \delta$, we have

Secondly, let us prove this result by contradiction. If not, there exists $\psi_0\in H^1(\mathbb{R}^N)$ such that $\|\psi_0-u\|_{H^1} < \delta$ and the solution $\psi(t)$ with an initial value of $\psi_0$ blows up in finite time. By continuity, there exists $T_1 > 0$ such that $\|\nabla \psi(T_1)\|_{L^2}^2 > r_0$. We now assume the initial value $\tilde{\psi_0} = \frac{\sqrt{a}}{\| \psi_0\|_{L^2}}\psi_0$. When $\delta > 0$ is sufficiently small, we have

When $\sqrt{a} < \| \psi_0\|_{L^2}$, $\|\nabla \tilde{\psi_0}\|_{L^2}^2 < \|\nabla \psi_0\|_{L^2}^2 < r_0$. When $\sqrt{a} > \| \psi_0\|_{L^2}$, considering $0 < a < a_0$, we have $\|\nabla \tilde{\psi_0}\|_{L^2}^2 < r_0$. This implies that $\tilde{\psi}_0\in V(a)$. Since the solution of (1.1) is continuously dependent on the initial data and $\|\nabla \psi(T_1)\|_{L^2}^2 > r_0$, there exists $T_2 > 0$ such that $\|\nabla\tilde{\psi}(T_2)\|_{L^2}^2 > r_0$, where $\tilde{\psi}(t)$ is the solution of (1.1) that satisfies $\tilde{\psi}(t)|_{t = 0} = \tilde{\psi}_0$. We consequently deduce from the continuity that there exists $T_3 > 0$ such that $\|\nabla\tilde{\psi}(T_3)\|_{L^2}^2 = r_0$. This indicates that $\tilde{\psi}(T_3)\in \partial V(a)$. Then we infer from Lemma 2.4 that $A > E_\gamma(\tilde{\psi}_0) > E_\gamma(\tilde{\psi}(T_3))\geq \inf\limits_{u\in \partial V(a)}E_\gamma(u)$, which is a contradiction. The proof follows.

Lemma 2.6. It holds that

(i) $a\in (0, a_0)$, $a\mapsto m_\gamma(a, r_0)$ is a continuous mapping.

(ii) Let$a\in (0, a_0)$. For all $\mu\in (0, a)$, we have

Proof. (ⅰ) Let $a\in (0, a_0)$ be arbitrary and $\{a_n\}_{n\geq1}\subset (0, a_0)$ be such that $a_n\rightarrow a$. According to the definition of $m_\gamma(a_n, r_0)$, we know that there exists $u_n\in V(a)$ such that

We set $v_n: = \sqrt{\frac{a}{a_n}}u_n$ and hence $v_n\in S(a)$. We find that $v_n\in V(a)$. Indeed, if $a_n\geq a$, then

If $a_n < a$, by Lemma 2.2, (2.4) and (2.9) we have $f(a_n, r)\geq f(a_n, r_0)\geq f(a_0, r_0) = 0$ for any $r\in [\frac{a_n}{a}r_0, r_0]$. Indeed, since $f(a, r)$ is a non-increasing function, then we have

And then by direct calculations we have

Hence, we deduce from (2.3) and (2.14) that $f(a_n, \|\nabla u_n\|^2_{L^2}) < 0$, thus $\|\nabla u_n\|^2_{L^2} < \frac{a_n}{a}r_0$ and

Since $v_n\in V(a)$ we can write

where

Since $\|\nabla u_n\|^2_{L^2} < r_0$, also $\|u_n\|^{\alpha+2}_{L^{\alpha+2}}$ and $\int_{\mathbb{R}^N}\frac{|u_n|^2}{|x|^\sigma}dx$ are uniformly bounded. Thus, we have

Combining (2.14) and (2.17), we get

Now, let $u\in V(a)$ be such that

Set $u_n: = \sqrt{\frac{a_n}{a}}u$ and hence $u_n\in S(a_n)$. Clearly, $\|\nabla u\|_{L^2}^2 < r_0$ and $a_n\rightarrow a$ imply $\|\nabla u_n\|_{L^2}^2 < r_0$ for $n$ large enough, so that $u_n\in V(a)$. Also, $E_\gamma(u_n)\rightarrow E_\gamma(u)$. We thus have

Therefore, since $\varepsilon > 0$ is arbitrary, we deduce that $m_\gamma(a_n, r_0)\rightarrow m_\gamma(a, r_0)$. The point (ⅰ) follows.

(ⅱ) Note that, fixed $\mu\in (0, a)$, it is sufficient to prove that the following holds

Indeed, if (2.19) holds, then we have

Next, we prove that (2.19) holds. According to the definition of $m_\gamma(\mu, r_0)$, we have that there exists $u\in V(\mu)$ such that

By (2.9), $f(a, r)\geq0$ for any $r\in[\frac{\mu}{a}r_0, r_0]$. Hence, we can deduce from (2.3) and (2.20) that

Now we set $v(x): = u(\theta^{-\frac{1}{N}}x)$. On the one hand, we note that $\|v\|_{L^2}^2 = \theta\|u\|_{L^2}^2 = \theta \mu$ and also, because of (2.21), $\|\nabla v\|_{L^2}^2 = \theta\|\nabla u\|_{L^2}^2 < r_0$. Thus $v\in V(\theta\mu)$. On the other hand, we obtain from (2.2) that

and it follows easily that

for $0 < \gamma < \gamma_0$ and $\gamma_0$ small sufficiently. We can obtain that

Lemma 2.7. Let $\{v_n\}_{n\geq1}\subset B(r_0)$ be such that $\int_{\mathbb{R}^N}\frac{|v_n|^2}{|x|^\sigma}dx\rightarrow 0$. Then, there exist a $\beta_0 > 0$ such that

Proof. Indeed, using the Sobolev inequality, we obtain that

Now, since $f(a_0, r_0) = 0$, we have

Lemma 2.8. For any $a\in (0, a_0)$, let $\{u_n\}_{n\geq1}\subset B(r_0)$ be such that $\|u_n\|_{L^2}^2\rightarrow a$ and $E_\gamma(u_n)\rightarrow m_\gamma(a, r_0)$. Then, there exists a $\beta_1 > 0$ and a sequence $\{y_n\}_{n\geq1}\subset \mathbb{R}^N$ such that

Proof. We assume by contradiction that (2.22) does not hold. Since $\{u_n\}_{n\geq1}\subset B(r_0)$ and $\|u_n\|_{L^2}^2\rightarrow a$, the sequence $\{u_n\}_{n\geq1}$ is bounded in $H^1(\mathbb{R}^N)$. From Lemma $I.1$ in ^[21] and since $\frac{4}{N}+2 < \alpha+2 < \frac{2N}{N-2}$, we deduce that $\|u_n\|_{L^{\alpha+2}}\rightarrow 0$ as $n\rightarrow \infty$. At this point, Lemma 2.7 implies $E_\gamma(u_n)\geq o_n(1)$. This contradicts the fact that $m_\gamma(a, r_0) < 0$ and the lemma follows.

Theorem 2.9. For any $a\in (0, a_0)$, if $\{u_n\}_{n\geq1}\subset B(r_0)$ is such that $\|u_n\|^2_{L^2} \rightarrow a$ and $E_\gamma(u_n)\rightarrow m_\gamma(a, r_0)$ then, up to translation, $u_n\rightarrow u$ in $H^1(\mathbb{R}^N)$.

Proof. Let $\{u_n\}_{n\geq1}\subset B(r_0)$ be a minimizing sequence of the energy functional $E_\gamma(u)$, that is,

By similar proof as ^[23], we known that $\{u_n\}_{n\geq1}$ is bounded in $H^1(\mathbb{R}^N)$. By Lemma 2.6, Lemma 2.8 and Rellich compactness theorem, there exist sequences $\{u_{n_k}\}_{k\geq1}\subset H^1(\mathbb{R}^N)$ and $\{y_{n_k}\}_{k\geq1}\subset \mathbb{R}^N$ such that for any $\epsilon > 0$, there exists $R(\epsilon) > 0$ such that for all $k\geq1$,

Denote $\tilde{u}_{n_k}(\cdot) = u_{n_k}(\cdot+y_{n_k})$, then there exists $\tilde{u}$ such that $\tilde{u}_{n_k}\rightharpoonup\tilde{u}$ weakly in $H^1(\mathbb{R}^N)$, $\tilde{u}_{n_k}\rightarrow \tilde{u}$ strongly in $L^r_{loc}(\mathbb{R}^N)$ with $r\in[2, \frac{2N}{N-2})$, combining with (2.23) we have

Thus $\int_{\mathbb{R}^N}|\tilde{u}|^2dx = \beta_2$. Indeed, we assume by contradiction that $\int_{\mathbb{R}^N}|\tilde{u}|^2dx = \bar{\beta_2} < \beta_2$. According to Brézis-Lieb Lemma and $\tilde{u}_{n_k} = (\tilde{u}_{n_k}-\tilde{u})+\tilde{u}$, we have

Then,

Hence,

which contradicts Lemma 2.6. Thus $\int_{\mathbb{R}^N}|\tilde{u}|^2dx = \beta_2$, i.e., $\tilde{u}_{n_k}\rightarrow \tilde{u}$ strongly in $L^2(\mathbb{R}^N)$. By the Gagliardo-Nirenberg inequality, $\tilde{u}_{n_k}\rightarrow \tilde{u}$ strongly in $L^s(\mathbb{R}^N)$, where $s\in[2, \frac{2N}{N-2})$. Next, our aim is to show that $\{y_{n_k}\}_{k\geq1}\subset {\mathbb{R}^N}$ is bounded. If it was not the case, we deduce that

Hence $\lim\limits_{n\rightarrow \infty}E_\gamma(u_{n_k}) = \lim\limits_{n\rightarrow \infty}E_\gamma^\infty(u_{n_k})$, where $E_\gamma^\infty(u_{n_k})$ is the corresponding energy functional of the sequence $\{u_{n_k}\}_{k\geq1}\subset H^1(\mathbb{R}^N)$ via translation transformation. On the other hand, we know that $\lim\limits_{n\rightarrow \infty}E_\gamma^\infty(u_n)$ is attained by a nontrivial function $u_{r_0}\in B(r_0)$. Hence,

According to the definition of $E_\gamma^\infty(u_{n_k})$, we see that $\tilde{u}$ is a minimizer of $\lim\limits_{k\rightarrow \infty}E_\gamma^\infty(u_{n_k})$. Consequently,

which contradicts $\lim\limits_{k\rightarrow \infty}E_\gamma(u_{n_k})\geq\lim\limits_{k\rightarrow \infty}E_\gamma^\infty(u_{n_k})$. Thus, $\{y_{n_k}\}_{k\geq1}\subset{\mathbb{R}^N}$ is bounded. We can deduce, up to subsequences, $\lim\limits_{k\rightarrow \infty}y_{n_k} = y_0$ for some $y_0\in\mathbb{R}^N$. Consequently,

as $k\rightarrow \infty$ for any $s\in[2, \frac{2N}{N-2})$, that is, $u_{n_k}$ converges strongly to $\tilde{u}(x-y_0)$ in $L^s(\mathbb{R}^N)$ for $s\in[2, \frac{2N}{N-2})$. We denote $u(x) = \tilde{u}(x-y_0)$. Hence,

We see that $E_\gamma(u) = m_\gamma(a, r_0)$ and hence $u_{n_k}\rightarrow u$ in $H^1(\mathbb{R}^N)$.

Proof of Theorem 2.3. The existence of $\gamma_0$ and a minimizer for $E_\gamma(u)$ on $V(a)$ were proved in Lemma 2.6, Lemma 2.7 and Theorem 2.9.

3.   Orbital stability

Proof of Theorem 1.3. (ⅰ) The property that $\mathcal{M}(a)\subset V(a)$ is non-empty follows from Theorem 2.3.

(ⅱ) We will show that $\mathcal{M}(a)$ is orbitally stable. Firstly, we note that the solution $\psi$ of (1.1) exists globally by Lemma 2.5. Now we suppose by contradiction that there exist $\epsilon_0 > 0$, a sequence of initial data $\{\psi_{0, n}\}_{n\geq1}\subset H^1(\mathbb{R}^N)$ and a sequence $\{t_n\}_{n\geq1}\subset\mathbb{R}$ such that for all $n\geq1$,

where $\psi_n(t)$ is the solution to (1.1) with initial data $\psi_{0, n}$. Next, we claim that there exists $v\in\mathcal{M}(a)$ such that

Indeed, by (3.1), we see that for each $n\geq1$, there exists $v_n\in V(a)$ such that

Since $\{v_n\}_{n\geq1}\subset\mathcal{M}(a)$ is a minimizing sequence of (1.7) and by the same argument as in Lemma 2.3, there exists $v\in\mathcal{M}(a)$ such that

By (3.2) and (3.3), the claim follows. And then, due to $v\in V(a)$, we have $\tilde{\psi_n} = \frac{\sqrt{a}\psi_{n}(t_n)}{\|\psi_{n}(t_n)\|_{L^2}}\in V(a)$ and

which implies that $\{\tilde{\psi_n}\}_{n\geq1}$ is a minimizing sequence for (1.7). Thanks to the compactness of all minimizing sequence of (1.7), there is a $\tilde{u}\in V(a)$ satisfies

Moreover, by the definition of $\tilde{\psi_n}$, it follows easily that

Consequently, we have

which contradicts (3.1). The proof is now complete.

4.   Conclusions

In this work, we study the stability of the set of energy minimizers in the mass supercritical case. In the mass supercritical case, the energy functional is unbounded from below on $S(a)$. Thus, we consider the further constrained minimization problem to study the existence and stability of standing waves for (1.1). And, the energy functional is not invariant under the scaling transform due to inverse-power potentials $\gamma|x|^{-\sigma} u$. It is intrinsically difficult for us to prove the compactness of minimizing sequences. By a rather delicate analysis, we can overcome this difficulty by proving the boundedness of any translation sequence.

Acknowledgments

This work is supported by the Outstanding Youth Science Fund of Gansu Province (No. 20JR10RA111).

Conflict of interest

The author declares no conflicts of interest.