Strong instability of standing waves for nonlinear Schr\"{o}dinger equations with a partial confinement

TL;DR

This paper proves that ground state solutions of certain nonlinear Schrödinger equations with partial confinement are strongly unstable and blow up under specific critical or supercritical conditions.

Contribution

It establishes the strong instability of ground states for NLS with a harmonic potential in higher dimensions when the nonlinearity is critical or supercritical.

Findings

Ground states are strongly unstable by blowup under specified conditions.

Instability occurs for nonlinearities that are L^2-critical or supercritical in N-1 dimensions.

Results apply to N ≥ 2 with a one-dimensional harmonic potential.

Abstract

We study the instability of standing wave solutions for nonlinear Schr\"{o}dinger equations with a one-dimensional harmonic potential in dimension $N\ge 2$. We prove that if the nonlinearity is $L^2$-critical or supercritical in dimension $N-1$, then any ground states are strongly unstable by blowup.

Equations

$$i{\partial }_{t}u=-\Delta u+x_N^{2}u-|u{|}^{p-1}u,(t,x)\in \mathbb{R}\times {\mathbb{R}}^N,$$

$$X=\{v\in H^1({\mathbb{R}}^N):x_{N}v\in L^2({\mathbb{R}}^N)\}$$

$$\Vert v{\Vert }_X={\left(\Vert \nabla v{\Vert }_{L^2}^2+\Vert v{\Vert }_{L^2}^2+\Vert x_{N}v{\Vert }_{L^2}^2\right)}^{1/2}.$$

$$\Vert u(t){\Vert }_{L^2}^2=\Vert u_0{\Vert }_{L^2}^2,E(u(t))=E(u_0)$$

$$E(v)=\frac{1}{2}\Vert \nabla v{\Vert }_{L^2}^2+\frac{1}{2}\Vert x_{N}v{\Vert }_{L^2}^2-\frac{1}{p+1}\Vert v{\Vert }_{L^{p+1}}^{p+1}.$$

$$-\Delta \varphi +x_N^2\varphi +\omega \varphi -|\varphi {|}^{p-1}\varphi =0,x\in {\mathbb{R}}^N,$$

$$S_{\omega }(v)$$

$$=\frac{1}{2}\Vert \nabla v{\Vert }_{L^2}^2+\frac{1}{2}\Vert x_{N}v{\Vert }_{L^2}^2+\frac{\omega }{2}\Vert v{\Vert }_{L^2}^2-\frac{1}{p+1}\Vert v{\Vert }_{L^{p+1}}^{p+1}$$

$${\mathcal{G}}_{\omega }=\{\varphi \in {\mathcal{A}}_{\omega }:S_{\omega }(\varphi )\le S_{\omega }(v)\text{mbox}forallv\in {\mathcal{A}}_{\omega }\},$$

$${\mathcal{A}}_{\omega }=\{v\in X:S_{\omega }^{\prime }(v)=0,v\ne 0\}$$

$${\mathcal{G}}_{\omega }=\{v\in X:S_{\omega }(v)=d(\omega ),K_{\omega }(v)=0,v\ne 0\},$$

$$K_{\omega }(v)={\partial }_{\lambda }{S}_{\omega }(\lambda v){|}_{\lambda =1}=\Vert \nabla v{\Vert }_{L^2}^2+\Vert x_{N}v{\Vert }_{L^2}^2+\omega \Vert v{\Vert }_{L^2}^2-\Vert v{\Vert }_{L^{p+1}}^{p+1}$$

$$d(\omega )=\inf \{S_{\omega }(v):v\in X,K_{\omega }(v)=0,v\ne 0\}.$$

$$\Vert v{\Vert }_{L^2}^2\le 2\Vert {\partial }_{N}v{\Vert }_{L^2}\Vert x_{N}v{\Vert }_{L^2},$$

$$C_1(\omega )\Vert v{\Vert }_X^2\le \Vert \nabla v{\Vert }_{L^2}^2+\Vert x_{N}v{\Vert }_{L^2}^2+\omega \Vert v{\Vert }_{L^2}^2\le C_2(\omega )\Vert v{\Vert }_X^2$$

$$i{\partial }_{t}u=-\Delta u-|u{|}^{p-1}u,(t,x)\in \mathbb{R}\times {\mathbb{R}}^N,$$

$$-\Delta \varphi +\omega \varphi -|\varphi {|}^{p-1}\varphi =0,x\in {\mathbb{R}}^N$$

$$i{\partial }_{t}u=-\Delta u+|x{|}^{2}u-|u{|}^{p-1}u,(t,x)\in \mathbb{R}\times {\mathbb{R}}^N,$$

$$-\Delta \varphi +|x{|}^2\varphi +\omega \varphi -|\varphi {|}^{p-1}\varphi =0,x\in {\mathbb{R}}^N$$

$$i{\partial }_{t}u=-\Delta u+(x_1^2+\cdots +x_d^2)u-|u{|}^{p-1}u,(t,x)\in \mathbb{R}\times {\mathbb{R}}^N,$$

$$1+4/N<p<\min \{1+4/(N-d),1+4/(N-2)\}$$

$$\Sigma =\{v\in H^1({\mathbb{R}}^N):|x|v\in L^2({\mathbb{R}}^N)\}.$$

$$t\mapsto F(t)=\sum _{j=1}^{N-1}{\int }_{{\mathbb{R}}^N}{x}_j^2|u(t,x){|}^{2}dx$$

$$F^{\prime \prime }(t)=16P(u(t))$$

$$P(v)=\frac{1}{2}\sum _{j=1}^{N-1}\Vert {\partial }_{j}v{\Vert }_{L^2}^2-\frac{\alpha }{2(p+1)}\Vert v{\Vert }_{L^{p+1}}^{p+1},\alpha =\frac{(N-1)(p-1)}{2}.$$

$$F^{\prime }(t)=2\sum _{j=1}^{N-1}\mathrm{Im}{\int }_{{\mathbb{R}}^N}{x}_j^2\overline{u}\left(-\Delta u+x_N^{2}u-|u{|}^{p-1}u\right)dx$$

$$=-2\sum _{j=1}^{N-1}\mathrm{Im}{\int }_{{\mathbb{R}}^N}{x}_j^2\overline{u}\Delta udx=4\sum _{j=1}^{N-1}\mathrm{Im}{\int }_{{\mathbb{R}}^N}\overline{u}{x}_j{\partial }_{j}udx.$$

$$F^{\prime \prime }(t)=-4\sum _{j=1}^{N-1}\mathrm{Im}{\int }_{{\mathbb{R}}^N}{\partial }_{t}u\left(2x_j{\partial }_j\overline{u}+\overline{u}\right)dx.$$

$$v^{\lambda }(x)={\lambda }^{(N-1)/2}v(\lambda x_1,...,\lambda x_{N-1},x_N)$$

$${\partial }_{\lambda }{v}^{\lambda }(x){|}_{\lambda =1}=\sum _{j=1}^{N-1}{x}_j{\partial }_{j}v(x)+\frac{N-1}{2}v(x),$$

$$E(v^{\lambda })=\frac{{\lambda }^2}{2}\sum _{j=1}^{N-1}\Vert {\partial }_{j}v{\Vert }_{L^2}^2-\frac{{\lambda }^{\alpha }}{p+1}\Vert v{\Vert }_{L^{p+1}}^{p+1}+\frac{1}{2}\Vert {\partial }_{N}v{\Vert }_{L^2}^2+\frac{1}{2}\Vert x_{N}v{\Vert }_{L^2}^2,$$

$$P(v)=\frac{1}{2}{\partial }_{\lambda }E(v^{\lambda }){|}_{\lambda =1}.$$

Full text

1 Introduction

In this paper, we study the instability of standing wave solutions $e^{i\omega t}\phi_{\omega}(x)$ for the nonlinear Schrödinger equation with a one-dimensional harmonic potential

[TABLE]

where $N\geq 2$, $x_{N}$ is the $N$-th component of $x=(x_{1},...,x_{N})\in\mathbb{R}^{N}$, $\Delta$ is the Laplacian in $x$, and $1<p<1+4/(N-2)$. Here, $1+4/(N-2)$ stands for $\infty$ if $N=2$.

The Cauchy problem for (1.1) is locally well-posed in the energy space $X$ (see [6, Theorem 9.2.6]). Here, the energy space $X$ for (1.1) is defined by

[TABLE]

with the norm

[TABLE]

Proposition 1**.**

Let $1<p<1+4/(N-2)$. For any $u_{0}\in X$ there exist $T_{\max}=T_{\max}(u_{0})\in(0,\infty]$ and a unique maximal solution $u\in C([0,T_{\max}),X)\cap C^{1}([0,T_{\max}),X^{*})$ of (1.1) with initial condition $u(0)=u_{0}$. The solution $u(t)$ is maximal in the sense that if $T_{\max}<\infty$, then $\displaystyle{\|u(t)\|_{X}\to\infty}$ as $t\nearrow T_{\max}$.

Moreover, the solution $u(t)$ satisfies the conservation laws

[TABLE]

for all $t\in[0,T_{\max})$, where the energy $E$ is defined by

[TABLE]

Next, we consider the stationary problem

[TABLE]

where $\omega\in\mathbb{R}$. Note that if $\phi(x)$ solves (1.3), then $e^{i\omega t}\phi(x)$ is a solution of (1.1). Moreover, (1.3) can be written as $S_{\omega}^{\prime}(\phi)=0$, where

[TABLE]

is the action. The set of all ground states for (1.3) is defined by

[TABLE]

where

[TABLE]

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

Then, we have the following result on the existence of ground states for (1.3).

Proposition 2**.**

Let $1<p<1+4/(N-2)$ and $\omega\in(-1,\infty)$. Then, the set $\mathcal{G}_{\omega}$ is not empty, and it is characterized by

[TABLE]

where

[TABLE]

is the Nehari functional, and

[TABLE]

Although Proposition 2 can be proved by the standard concentration compactness argument, for the sake of completeness, we give the proof of Proposition 2 in Section 3.

Here, we remark that by Heisenberg’s inequality

[TABLE]

for any $\omega\in(-1,\infty)$ there exist positive constants $C_{1}(\omega)$ and $C_{2}(\omega)$ such that

[TABLE]

for all $v\in X$.

Now we state our main result in this paper.

Theorem 1**.**

Assume that $N\geq 2$, $1+4/(N-1)\leq p<1+4/(N-2)$, and let $\phi_{\omega}\in\mathcal{G}_{\omega}$ for $\omega\in(-1,\infty)$. Then, for any $\omega\in(-1,\infty)$, the standing wave solution $e^{i\omega t}\phi_{\omega}$ of (1.1) is strongly unstable in the following sense. For any $\varepsilon>0$ there exists $u_{0}\in X$ such that $\|u_{0}-\phi_{\omega}\|_{X}<\varepsilon$ and the solution $u(t)$ of (1.1) with $u(0)=u_{0}$ blows up in finite time.

Notice that Theorem 1 covers the physically relevant case $N=3$ and $p=3$ as a borderline case.

Here, we recall some known results related to Theorem 1. First, we consider the nonlinear Schrödinger equations without potential

[TABLE]

where $1<p<1+4/(N-2)$. For any $\omega\in(0,\infty)$, there exists a unique positive radial solution $\phi_{\omega}(x)$ of the stationary problem

[TABLE]

(see [13] for the uniqueness). When $1<p<1+4/N$, the standing wave solution $e^{i\omega t}\phi_{\omega}$ of (1.9) is orbitally stable for all $\omega>0$ (see [7]). While, if $1+4/N\leq p<1+4/(N-2)$, then the standing wave solution $e^{i\omega t}\phi_{\omega}$ of (1.9) is strongly unstable for all $\omega>0$ (see [3] and also [6, Theorem 8.2.2]).

Next, we consider the nonlinear Schrödinger equations with a harmonic potential

[TABLE]

where $1<p<1+4/(N-2)$. For any $\omega\in(-N,\infty)$, there exists a unique positive radial solution $\phi_{\omega}(x)$ of the stationary problem

[TABLE]

(see [11, 12] for the uniqueness).

When $\omega$ is sufficiently close to $-N$, the standing wave solution $e^{i\omega t}\phi_{\omega}$ of (1.10) is orbitally stable for any $1<p<1+4/(N-2)$ (see [9]). We remark that $N$ is the first eigenvalue of $-\Delta+|x|^{2}$.

On the other hand, when $\omega$ is sufficiently large, the standing wave solution $e^{i\omega t}\phi_{\omega}$ of (1.10) is orbitally stable for the case $1<p\leq 1+4/N$ (see [8, 9]), and it is strongly unstable for the case $1+4/N<p<1+4/(N-2)$ (see [17] and also [10] for an earlier result on the orbital instability).

Finally, we consider the nonlinear Schrödinger equations with a partial confinement of the form

[TABLE]

where $N\geq 2$, $1\leq d\leq N-1$, $x=(x_{1},...,x_{d},x_{d+1},...,x_{N})$. The typical case is that $N=3$ and $d=2$. Recently, Bellazzini, Boussaïd, Jeanjean and Visciglia [2] constructed orbitally stable standing wave solutions of (1.11) for the case

[TABLE]

(see Theorem 1 and Remark 1.9 of [2]). It should be remarked that the bottom of the spectrum of $-\Delta+(x_{1}^{2}+\cdots+x_{d}^{2})$ is not an eigenvalue, so that unlike (1.10) with a complete confinement, the existence of stable standing wave solutions for (1.11) is highly nontrivial in the $L^{2}$-supercritical case $p>1+4/N$.

We also remark that for the case $d\geq 2$, the assumption (1.12) becomes $1+4/N<p<1+4/(N-2)$. On the other hand, for the case $d=1$, the assumption (1.12) becomes $1+4/N<p<1+4/(N-1)$, and there is a chance to consider the case $1+4/(N-1)\leq p<1+4/(N-2)$. This is our main motivation for Theorem 1 in the present paper (see also [1, 5, 20] for related results).

Although it is not clear whether the standing wave solutions constructed by [2] are ground states in the sense of (1.4) (see Definition 1.1 and Remark 1.10 of [2]), it would be safe to conclude from our Theorem 1 that the upper bound on $p$ in (1.12) is optimal for the existence of stable standing wave solutions of (1.11).

The rest of the paper is organized as follows. In Section 2, we give the proof of Theorem 1. The proof is based on a virial type identity (2.1) associated with the scaling (2.2), the characterization of ground states (1.5) by the minimization problem on the Nehari manifold, and Lemma 1 below. We remark that the classical method by Berestycki and Cazenave [3] is not applicable to (1.1) directly. Instead, we use and modify the ideas of Zhang [21] and Le Coz [14], which give an alternative approach to the strong instability (see also [17, 18, 19] for recent developments).

In Section 3, we give the proof of Proposition 2. The proof is based on the standard concentration compactness argument.

2 Proof of Theorem 1

We define

[TABLE]

First, we derive a virial type identity.

Proposition 3**.**

Let $1<p<1+4/(N-2)$. If $u_{0}\in\Sigma$, then the solution $u(t)$ of (1.1) with $u(0)=u_{0}$ satisfies $u\in C([0,T_{\max}),\Sigma)$. Moreover, the function

[TABLE]

is in $C^{2}[0,T_{\max})$, and satisfies

[TABLE]

for all $t\in[0,T_{\max})$, where

[TABLE]

Proof.

We state formal calculations for the identity (2.1) only. These formal calculations can be justified by the classical regularization argument as in [6, Proposition 6.5.1] (see also [16]).

Let $u(t,x)$ be a smooth solution of (1.1). Then, we have

[TABLE]

Moreover, we have

[TABLE]

Here, we consider the scaling

[TABLE]

for $\lambda>0$ and $x=(x_{1},...,x_{N-1},x_{N})\in\mathbb{R}^{N}$. Then, we have

[TABLE]

and

[TABLE]

Thus, we have

[TABLE]

As stated above, these formal calculations can be justified by the regularization argument. ∎

Notice that

[TABLE]

for the case $1+4/(N-1)\leq p<1+4/(N-2)$.

The following lemma is a modification of the ideas of Zhang [21] and Le Coz [14] (see also [17, 18, 19]).

Lemma 1**.**

Assume that $1+4/(N-1)\leq p<1+4/(N-2)$ and $\omega\in(-1,\infty)$. If $v\in X$ satisfies $P(v)\leq 0$ and $v\neq 0$, then $d(\omega)\leq S_{\omega}(v)-P(v)$.

Proof.

Since $\omega>-1$ and $v\neq 0$, by Heisenberg’s inequality (1.7), we have

[TABLE]

Then, it follows from $P(v)\leq 0$ that

[TABLE]

for $\lambda>0$. Since $\alpha\geq 2$ and $v\neq 0$, there exists $\lambda_{0}\in(0,\infty)$ such that $K_{\omega}(v^{\lambda_{0}})=0$. Here, we remark that

[TABLE]

for the case $\alpha=2$.

Then, by the definition (1.6) of $d(\omega)$, we have $d(\omega)\leq S_{\omega}(v^{\lambda_{0}})$.

Moreover, since $\alpha\geq 2$, the function

[TABLE]

attains its maximum at $\lambda=1$.

Thus, since $P(v)\leq 0$ again, we have

[TABLE]

This completes the proof. ∎

Once we have obtained Lemma 1, the rest of the proof is the same as in the classical argument of Berestycki and Cazenave [3].

Lemma 2**.**

Assume that $1+4/(N-1)\leq p<1+4/(N-2)$ and $\omega\in(-1,\infty)$. The set

[TABLE]

is invariant under the flow of (1.1). That is, if $u_{0}\in\mathcal{B}_{\omega}$, then the solution $u(t)$ of (1.1) with $u(0)=u_{0}$ satisfies $u(t)\in\mathcal{B}_{\omega}$ for all $t\in[0,T_{\max})$.

Proof.

This follows from the conservation laws (1.2), Lemma 1, and the continuity of the function $t\mapsto P(u(t))$. ∎

Theorem 2**.**

Assume that $1+4/(N-1)\leq p<1+4/(N-2)$ and $\omega\in(-1,\infty)$. If $u_{0}\in\mathcal{B}_{\omega}\cap\Sigma$, then the solution $u(t)$ of (1.1) with $u(0)=u_{0}$ blows up in finite time.

Proof.

Let $u_{0}\in\mathcal{B}_{\omega}\cap\Sigma$ and let $u(t)$ be the solution of (1.1) with $u(0)=u_{0}$. Then, it follows from Lemma 2 and Proposition 3 that $u(t)\in\mathcal{B}_{\omega}\cap\Sigma$ for all $t\in[0,T_{\max})$.

Moreover, by the virial identity (2.1), the conservation laws (1.2) and Lemma 1, we have

[TABLE]

for all $t\in[0,T_{\max})$. This implies $T_{\max}<\infty$. ∎

Finally, we give the proof of Theorem 1.

Proof of Theorem 1. First, by the elliptic regularity theory, we see that $\phi_{\omega}\in\Sigma$ (see, e.g., [6, Theorem 8.1.1]).

Next, since $S_{\omega}^{\prime}(\phi_{\omega})=0$, the function

[TABLE]

attains its maximum at $\lambda=1$. Thus, we have

[TABLE]

for all $\lambda>1$. Moreover, since $P(\phi_{\omega})=0$, we have

[TABLE]

for all $\lambda>1$.

Therefore, we see that $\lambda\phi_{\omega}\in\mathcal{B}_{\omega}\cap\Sigma$ for all $\lambda>1$, and it follows from Theorem 2 that the solution $u(t)$ of (1.1) with $u(0)=\lambda\phi_{\omega}$ blows up in finite time. Hence, the result follows, since $\lambda\phi_{\omega}\to\phi_{\omega}$ in $X$ as $\lambda\to 1$. ∎

3 Proof of Proposition 2

In this section, we prove Proposition 2 by using the standard concentration compactness argument. Throughout this section, we assume that $1<p<1+4/(N-2)$ and $\omega\in(-1,\infty)$.

We define

[TABLE]

Note that by (1.8), there exists a positive constant $C_{0}$ depending only on $\omega$ and $p$ such that

[TABLE]

We also remark that by (3.1) and (1.6), we have

[TABLE]

Lemma 3**.**

$d(\omega)>0$.

Proof.

Let $v\in X$ satisfy $K_{\omega}(v)=0$ and $v\neq 0$.

Then, by $K_{\omega}(v)=0$, the Sobolev inequality and (3.2), there exist positive constants $C_{1}$ and $C_{2}$ depending only on $N$, $p$ and $\omega$ such that

[TABLE]

Since $v\neq 0$, we have $J_{\omega}(v)>0$ and $J_{\omega}(v)^{(p-1)/2}\geq 1/C_{2}$.

Thus, by (3.3), we have

[TABLE]

This completes the proof. ∎

Lemma 4**.**

If $v\in X$ satisfies $K_{\omega}(v)<0$, then $d(\omega)<J_{\omega}(v)$.

Proof.

Since $K_{\omega}(v)<0$ and

[TABLE]

for $\lambda>0$, there exists $\lambda_{0}\in(0,1)$ such that $K_{\omega}(\lambda_{0}v)=0$.

Thus, by (3.3) and (3.1), we have

[TABLE]

This completes the proof. ∎

The following lemma is a variant of the classical result of Lieb [15] (see also [2, Lemma 3.4]).

Lemma 5**.**

Assume that a sequence $(u_{n})_{n\in\mathbb{N}}$ is bounded in $X$, and satisfies

[TABLE]

Then, there exist a sequence $(y^{n})_{n\in\mathbb{N}}$ in $\mathbb{R}^{N-1}$ and $u\in X\setminus\{0\}$ such that $(\tau_{y^{n}}u_{n})_{n\in\mathbb{N}}$ has a subsequence which converges to $u$ weakly in $X$.

Here we define

[TABLE]

for $x=(x_{1},...,x_{N-1},x_{N})\in\mathbb{R}^{N}$ and $y=(y_{1},...,y_{N-1})\in\mathbb{R}^{N-1}$.

Proof.

Without loss of generality, we may assume that

[TABLE]

Moreover, we put

[TABLE]

and for $y=(y_{1},...,y_{N-1})\in\mathbb{Z}^{N-1}$, we define

[TABLE]

Then, by the definition of $C_{3}$, we see that for any $n\in\mathbb{N}$, there exists $y^{n}\in\mathbb{Z}^{N-1}$ such that

[TABLE]

where we put

[TABLE]

Here, we define $v_{n}=\tau_{-y^{n}}u_{n}$. Then, we have

[TABLE]

for all $n\in\mathbb{N}$. In particular, $\|v_{n}\|_{L^{p+1}(Q_{0})}^{p+1}>0$ for all $n\in\mathbb{N}$.

Moreover, by the Sobolev inequality, we have

[TABLE]

for all $n\in\mathbb{N}$, where $C_{4}$ is a positive constant depending only on $N$ and $p$.

Thus, we have

[TABLE]

Since $(v_{n})_{n\in\mathbb{N}}$ is bounded in $X$, there exist a subsequence $(v_{n^{\prime}})$ of $(v_{n})$ and $u\in X$ such that $(v_{n^{\prime}})$ converges to $u$ weakly in $X$.

Finally, since the embedding $X(Q_{0})\hookrightarrow L^{p+1}(Q_{0})$ is compact, it follows from (3.5) that

[TABLE]

which implies $u\neq 0$. This completes the proof. ∎

We define the set of all minimizers for (1.6) by

[TABLE]

Lemma 6**.**

The set $\mathcal{M}_{\omega}$ is not empty.

Proof.

Let $(u_{n})$ be a sequence in $X$ such that $K_{\omega}(u_{n})=0$, $u_{n}\neq 0$ for all $n\in\mathbb{N}$, and $S_{\omega}(u_{n})\to d(\omega)$.

Then, by (3.2) and $J_{\omega}(u_{n})=S_{\omega}(u_{n})\to d(\omega)$, we see that the sequence $(u_{n})_{n\in\mathbb{N}}$ is bounded in $X$.

Moreover, it follows from $K_{\omega}(u_{n})=0$ and Lemma 3 that

[TABLE]

Thus, by Lemma 5, there exist a sequence $(y^{n})$ in $\mathbb{R}^{N-1}$, a subsequence of $(\tau_{y^{n}}u_{n})$, which is denoted by $(v_{n})$, and $v\in X\setminus\{0\}$ such that $(v_{n})$ converges to $v$ weakly in $X$. By the weakly lower semicontinuity of $J_{\omega}$, we have

[TABLE]

Moreover, by the Brezis-Lieb Lemma (see [4]), we have

[TABLE]

which implies $K_{\omega}(v)\leq 0$.

Indeed, suppose that $K_{\omega}(v)>0$. Since $K_{\omega}(v_{n})=0$, we have $K_{\omega}(v_{n}-v)<0$ for large $n$. Then, by Lemma 4, we have $d(\omega)<J_{\omega}(v_{n}-v)$, and

[TABLE]

On the other hand, by $v\neq 0$ and (3.2), we have $J_{\omega}(v)>0$. This is a contradiction. Thus, we obtain $K_{\omega}(v)\leq 0$.

Furthermore, by Lemma 4 and (3.6), we have $K_{\omega}(v)=0$. Since $v\neq 0$ again, it follows from (1.6) and (3.6) that

[TABLE]

Hence, we have $S_{\omega}(v)=d(\omega)$ and $v\in\mathcal{M}_{\omega}$.

This completes the proof. ∎

Lemma 7**.**

$\mathcal{M}_{\omega}\subset\mathcal{G}_{\omega}$.

Proof.

Let $\phi\in\mathcal{M}_{\omega}$. Then, there exists a Lagrange multiplier $\mu\in\mathbb{R}$ such that $S_{\omega}^{\prime}(\phi)=\mu K_{\omega}^{\prime}(\phi)$. Thus, we have

[TABLE]

Here, by (3.4), $K_{\omega}(\phi)=0$ and $\phi\neq 0$, we have

[TABLE]

Thus, we have $\mu=0$ and $S_{\omega}^{\prime}(\phi)=0$, which shows that $\phi\in\mathcal{A}_{\omega}$.

Moreover, for any $v\in\mathcal{A}_{\omega}$, we have $K_{\omega}(v)=\langle S_{\omega}^{\prime}(v),v\rangle=0$ and $v\neq 0$, so it follows from the definition (1.6) of $d(\omega)$ that $S_{\omega}(\phi)=d(\omega)\leq S_{\omega}(v)$.

Therefore, we have $\phi\in\mathcal{G}_{\omega}$, and we conclude that $\mathcal{M}_{\omega}\subset\mathcal{G}_{\omega}$. ∎

Finally, we give the proof of Proposition 2.

Proof of Proposition 2. By Lemma 7, it is enough to show that $\mathcal{G}_{\omega}\subset\mathcal{M}_{\omega}$.

Let $\phi\in\mathcal{G}_{\omega}$. By Lemma 7, we can take an element $v\in\mathcal{M}_{\omega}$. Then, since $v\in\mathcal{A}_{\omega}$ and $\phi\in\mathcal{G}_{\omega}$, by the definition (1.4) of $\mathcal{G}_{\omega}$, we have

[TABLE]

On the other hand, since $\phi$ satisfies $K_{\omega}(\phi)=0$ and $\phi\neq 0$, by the definition (1.6) of $d(\omega)$, we have $d(\omega)\leq S_{\omega}(\phi)$.

Hence, we have $S_{\omega}(\phi)=d(\omega)$ and $\phi\in\mathcal{M}_{\omega}$.

This completes the proof. ∎

Acknowledgments. The author thanks Louis Jeanjean and Noriyoshi Fukaya for useful discussions and suggestions. This work was supported by JSPS KAKENHI Grant Number 15K04968.