# Conditional Latent ODEs for Motion Prediction in Autonomous Driving

Truong G. Khang
KAIST CS
<EMAIL_ADDRESS>
&Yongjae Kim
KAIST ME
<EMAIL_ADDRESS>
&Andrea Finazzi
The Robotics Program - KAIST EE
<EMAIL_ADDRESS>

###### Abstract

This paper addresses imitation learning for motion prediction problem in
autonomous driving, especially in multi-agent setting. Different from previous
methods based on GAN, we present the conditional latent ordinary differential
equation (cLODE) to leverage both the generative strength of conditional VAE
and the continuous representation of neural ODE. Our network architecture is
inspired from the Latent-ODE model. The experiment shows that our method
outperform the baseline methods in the simulation of multi-agent driving and
is very efficient in term of GPU memory consumption. Our code and docker image
are publicly available: `https://github.com/TruongKhang/cLODE`;
`https://hub.docker.com/r/kim4375731/clode`.

##  1 Introduction

In the field of autonomous driving, predicting the motion of the surrounding
agents is a crucial and challenging task to solve complex navigation problems
in dynamic environments. Imitation Learning (IL) is a set of well-known
approaches that try to model the behaviour of drivers from real-world data.
Behavioural Cloning (BC), Inverse Reinforcement Learning (IRL), and Generative
Adversarial IL (GAIL) are three of the most popular approaches. They formulate
the imitation task with a supervised learning approach, as a policy
optimization problem, or via a GAN-inspired generative procedure,
respectively.

However, a main drawback of the above-mentioned methods is that the generated
trajectories cannot capture time-wise correlation. In particular, relying on
RNNs, these models learn a transformation between subsequent observations,
failing at representing consistency of purpose in the generated driving
behaviour.

Motivated by this analysis, we propose to control the generated trajectories
by using a Neural Ordinary Differential Equation (NODE) (Chen u. a., 2018). To
build a generative model for imitation learning, we use a Variational Auto-
Encoder (VAE) which is easier to train with ODE compared to GAN. We achieve
these two goals by using Latent-ODE (Rubanova u. a., 2019) which can
explicitly learn a representation of the trajectory space. The generated
trajectories sampled from this space mimic the driving style of the expert
demonstrations. Our method takes the history data, including the actions and
the observations in the past, to estimate a posterior for generating the
action trajectory. Therefore, we regard our method as a conditional variation
of Latent ODEs (cLODE).

The contribution of this work is twofold:

  * •

We propose a conditional Latent-ODE model for imitation learning.

  * •

We train and evaluate our method by using the US-101 dataset (Colyar und
Halkias, 2007) in a multi-agent setting. The experiments demonstrate that our
method outperforms the baseline AGen (Si u. a., 2019).

##  2 Related works

##### Imitation learning

Imitation learning (Hussein u. a., 2017) is a set of techniques aiming at
mimicking the behaviour of an expert agent by learning from data or, as this
approach is often referred to, learning from demonstration. Among the most
popular algorithms, adopted in many works also in the context of autonomous
driving to solve diverse tasks, are Behavioral Cloning (BC), Inverse
Reinforcement Learning (IRL), and Generative Adversarial Imitation Learning
(GAIL).

BC (Bain und Sammut, 1995), formulated as a supervised learning approach, is a
data-efficient approach that has the advantage of being relatively easy to
implement (Kuefler u. a., 2017). However, as it relies on the assumption that
the data is i.i.d., BC is particularly sensible to the issue known as
covariate shift (Kuefler u. a., 2017).

To cope with this limitation, IRL (Ho u. a., 2016; Finn u. a., 2016; Ho und
Ermon, 2016) solves the imitation learning problem by learning the unknown
reward function which the expert is supposed to be acting upon. Although this
method well generalizes to out-of-distribution situations, its dual-step
nature makes it less efficient than direct learning approaches, such as GAIL.

Generative adversarial reinforcement learning, deriving from Generative
Adversarial Networks (GANs), trains a generator model to fool a critic to the
point that the latter is unable to distinguish between generated and expert
trajectories (Ho und Ermon, 2016; Kuefler u. a., 2017). Song u. a. (2018);
Bhattacharyya u. a. (2018) introduce multi-agent GAIL, which has been extended
by Si u. a. (2019) with recursive online adaptation.

Also, several works combine one or more of these solutions to overcome well-
known limitations (Jena u. a., 2021; Sasaki u. a., 2020; Wang u. a., 2019;
Torabi u. a., 2018).

##### Latent ordinary differential equations

Recurrent and residual models, as well as normalizing flows, perform the
generative process by learning a transformation that is applied in an
iterative manner to the latent variable.

Chen u. a. (2018) define the evolution of the hidden variable in the time
limit, parameterizing its dynamics with neural network-defined ordinary
differential equations. The resulting representation describes the nature of
the latent space in the continuous domain, offering several advantages in
terms of memory and parameter efficiency, scalability and flexibility with
respect to time-wise constraints.

As an evolution of this work, Rubanova u. a. (2019) define a continuous-time
state dynamics model for RNNs, combining the power of recurrent networks for
structured data generation with the highly information-efficient ODE-based
latent dynamics introduced by Chen u. a. (2018). The models introduced by the
authors (ODE-RNN and Latent ODE) reduce data loss and are better suited for
irregularly-sampled data, compared to other works aiming at improving RNN time
flexibility (Che u. a., 2018; Lipton u. a., 2016).

##  3 Method

###  3.1 Conditional Latent ODEs for imitation learning

Let $\tau$ be an expert trajectory which consists of $T$ observations and
actions $\tau=(a_{1},o_{1},…,a_{T},o_{T})$. We denote
$h_{t-1}=(a_{1},o_{1},…,a_{t-1},o_{t-1})$ as the history of actions and
observations before the time step $t$. The goal of imitation learning is to
learn a policy $\pi_{\theta}=p(a_{t}|h_{t-1};\theta)$, parameterized by
$\theta$, which can imitate the expert behaviour $\tau$ and predict an action
at any time step $t$ from the history of actions and observations $h_{t-1}$.

We estimate $\theta$ by maximizing the likelihood with respect to the
trajectory $\tau$, $p(\tau|\theta)$. Because each observation $o_{t}$ can be
estimated from the action $a_{t}$ in a deterministic way via a kinematic
model, the likelihood can be regarded as the likelihood of the action
variables only. Therefore, the likelihood $p(\tau|\theta)$ is represented via
the action distribution as follows

$p(\tau|\theta)=p(a_{1}|\theta)\prod_{t=1}^{T-1}p(a_{t+1}|h_{t},\theta)$ (1)

The action distribution conditioned to the history data is the policy that we
aim to learn. To compute $p(a_{t+1}|h_{t},\theta)$, we use a latent-variable
model instead of directly using a neural network with a GAN-based loss as in
previous works (Ho und Ermon, 2016; Bhattacharyya u. a., 2018). As a result,
the log-probability of the action is approximated by the Evidence Lower Bound
(ELBO) using a posterior distribution $q_{\phi}(z|a_{t+1},h_{t})$
parameterized by $\phi$

$ELBO(\theta,\phi)=E_{q}\log
p(a_{t+1}|z,h_{t})-KL(q(z|h_{t},a_{t+1})||p(z|h_{t}))$ (2)

This is similar to the loss form of conditional Variational Auto-Encoder
(cVAE) (Sohn u. a., 2015). Note that we can output a new observation $o_{t+1}$
when the action $a_{t+1}$ is known. This leads to
$q(z|h_{t},a_{t+1})=q(z|h_{t+1})$. Combined with Eq. 1, the formula of $ELBO$
over all time steps is

$ELBO_{\tau}(\theta,\phi)=\sum_{t=0}^{T-1}E_{z\sim q(z)}\log
p(a_{t+1}|z,h_{t})-\sum_{t=0}^{T-1}KL(q(z|h_{t+1})||p(z|h_{t}))$ (3)

where $p(z|h_{0})=p(z)=\mathcal{N}(0,I)$ is the prior distribution of the
latent variable $z$. We notice that using a neural network for encoding each
$h_{t}$ to compute KL-divergence in Eq. 3 is expensive. When we minimize the
KL-divergence term, we aim to encode all actions and observations of the
trajectory $\tau$ into the same distribution of $z$. This means that we model
the entire trajectory $\tau$ by a latent variable $z$. To achieve this goal,
inspired by the Latent-ODE model (Rubanova u. a., 2019), we assume that the
trajectory $\tau$ is generated from an Ordinary Differential Equation (ODE) in
the latent space. We use an ODE-RNN network which runs backward in time to
estimate the posterior distribution of $z$ given the entire history data
$h_{T}$. Therefore, we can replace the encoder term, $KL$, in Eq. 3 simply by

$\sum_{t=0}^{T-1}KL(q(z|h_{t},a_{t+1})||p(z|h_{t}))\rightarrow
KL(q(z|h_{T})||p(z))$ (4)

where $q(z|h_{T})=\mathcal{N}(\mu_{z},\sigma_{z})$, and
$(\mu_{z},\sigma_{z})=\textit{ODE-RNN}(h_{T})$.

After estimating the latent distribution, we sample $z_{1}$ as a starting
point from this distribution to generate the trajectory of actions. Given
$z_{1}$, we use a neural ODE to represent the trajectory of actions in the
latent space. We denote $z_{t}$ as the hidden state of the action $a_{t}$ at
each time step $t$. $z_{t}$ can be estimated by solving the ODE with the
starting point $z_{1}$ as follows

$\frac{dz}{dt}=f_{\lambda}(z,t);\quad(z_{1},z_{2},...,z_{T+k})=ODESolve(f_{\lambda},z_{1},(1,...,T+k))$
(5)

Finally, each state $z_{t}$ at time step $t$ is used to predict the
distribution of actions $p(a_{t}|z,h_{t-1})$ in Eq. 3.

$p(a_{t}|z,h_{t-1})=p(a_{t}|z)=\mathcal{N}(\mu_{z_{t}},\sigma_{z_{t}});\quad(\mu_{z_{t}},\sigma_{z_{t}})=h(z_{t})$
(6)

where $h(.)$ is a Multi-Layer Perceptron (MLP).

Figure 1: Illustration of our proposed architecture for the conditional Latent
ODEs (cLODE).

###  3.2 Network architecture

Fig. 1 illustrates the overview of our proposed architecture cLODE. Similar to
the VAE-based generative models (Kingma und Welling, 2013; Rezende u. a.,
2014), the architecture includes two parts: an encoder and a decoder. The
encoder is used to embed the history data into the distribution of the latent
variable $z$. The decoder then generates the action based on the sampled
hidden state. We describe the detail of each part in Appendix.

##  4 Results

###  4.1 Experimental Setup

We validate the performance of cLODE on the US 101 human driving data from the
Next Generation SIMulation (NGSIM) dataset (Colyar und Halkias, 2007). As a
baseline, we use an imitative algorithm for autonomous driving, Adaptable
Generative Prediction Networks for Autonomous Driving (AGen) (Si u. a., 2019).
We can make a direct comparison between the two algorithms because both are
based on the NGSIM dataset, which is a widely used benchmark dataset for many
autonomous driving problems.

The dimension of one observation pair is on $\mathbf{R}^{66}$, where we
consider multiple concatenations of the observation history for the encoder
network of cLODE. The number of vehicle agents throughout the whole experiment
is fixed to 22 to incorporate multi-agent interactions into the driving data,
although NGSIM also supports single-agent simulations. For training the 22
cLODE agents on the NGSIM data, we set batch size $B$, learning rate $l_{r}$,
simulation update rate $dt$ as $B=50$, $l_{r}=1e-3$, $dt=0.1$, respectively.

###  4.2 Evaluation Metrics

For fair comparison, we adopt the same metric as AGen, which is root mean
squared error (RMSE) of predicted vehicle roll-out trajectory with repect to
the dataset Ground-truth. Specifically, if we want to evaluate RMSE of a value
$v$ at time $t$ over $m$ samples,

${\rm
RMSE}=\sqrt{\frac{1}{m}\sum^{m}_{i=1}\left(v^{(i)}_{t}-\hat{v}^{(i)}_{t}\right)^{2}}$

(7)

where $v^{(i)}_{t}$ is the Ground-truth value of $v$ and $\hat{v}^{(i)}_{t}$
is the predicted value. Note that $i$ is the sample index. Also, if we define
the vehicle positional error at time $t$ of the top-view trajectory as
$p_{t}$, with corresponding longitudinal error $x_{t}$ and lateral error
$y_{t}$: $p_{t}=\sqrt{x_{t}^{2}+y_{t}^{2}}$. Note that both metrics are
calculated along the base coordinate which is fixed at the start point of the
vehicle trajectory. As such, we can calculate the total positional
RMSE($p_{t}$) as well as the longitudinal RMSE($x_{t}$) and the lateral
RMSE($y_{t}$) if trajectory data are given.

###  4.3 Prediction Results

####  4.3.1 Computation time

All experiments are conducted with an ordinary desktop computer equipped with
3.6GHz Intel Core i7 CPU, 64GB RAM and NVIDIA GeForce 1080 GPU. The memory
consumption of cLODE at test time on GPU is up to 598MB, which is way lower
than that of AGen since it takes 6,547MB memory on the same GPU. This
indicates that cLODE provides more memory-efficient inference than the
baseline. However, cLODE needs longer time to roll-out the predicted vehicle
trajectories, as it requires 71.18(s) to predict 200 time steps where it took
57.79(s) for AGen. This result is equivalent to 0.36(s) per time step for
cLODE versus 0.29(s) per time step for AGen, meaning that cLODE makes 1.24
times slower prediction. We conjecture this is mainly because cLODE pays
constant memory cost along the neural ODE computation. Nevertheless, such a
downside is deemed to be a marginal sacrifice considering about 10-fold gain
on the memory consumption efficiency.

####  4.3.2 RMSE

Table 1: Positional RMSEs at 2.5(s) with 22 vehicle agents.

|  Longitudinal (m) |  Lateral (m) |  Total (m)
---|---|---|---
AGen |  1.64 |  1.89 |  2.51
cLODE with 5 obs. |  1.44 |  1.64 |  2.18
cLODE with 10 obs. |  1.47 |  1.70 |  2.25
cLODE with 20 obs. |  1.45 |  1.70 |  2.24
cLODE with 50 obs. |  1.53 |  1.79 |  2.36
cLODE with 100 obs. |  1.50 |  1.74 |  2.29

Figure 2: Positional RMSE comparison between AGen and cLODE with standard
deviation.

Figure 3: Positional RMSE comparison between AGen and cLODE up to 200 time
stpes. Figure 4: Ablation study of observation history length on RMSE
performance.

The positional RMSE results at 2.5(s) on the vehicle trajectory are shown in
Table 1. We focus on the record at this specific time because as time step
increases the vehicle position predictions also tend to increasingly deviate
from the Ground-truth data in general. As indicated in the table, even though
AGen is an domain-adapted model boosted by a generative predictor trained with
PS-GAIL, cLODE outperformed over all RMSE metrics. The comparison between the
RMSE results of the two models up to 2.5(s) is delineated in Fig. 2. The
figure is with standard deviation of the data over 330 trajectory samples and
22 vehicle agents, which shows that the difference in variance is not
significant. The RMSE throughout 0$\sim$200 time steps is also depicted in
Fig. 3, which suggests that the RMSE is consistently smaller with cLODE model.

####  4.3.3 Ablation Study on the Length of Observation History

As the transition model of real-world vehicle dynamics is not a Markov
Process, observation length of the vehicle state when we train the cLODE model
may relate to prediction performance. With the data shown in Table 1 and Fig.
4 we noted that even with just 5 observations, the cLODE model successfully
predicts the Ground-truth vehicle trajectories, outperforming the AGen
results. Surprisingly, stacking more observations in the history do not help
increasing the performance. After augmenting to more than 20 observations, the
trajectory prediction performance gradually deteriorates. We conjecture this
is because, during training, the redundant information fed to the model
negatively affects the learning process. This phenomenon is commonly known as
causal confusion in imitation learning (de Haan u. a., 2019).

##  5 Discussion and conclusions

In this report, we proposed a generative imitation learning model for
autonomous driving, the conditional latent ordinary differential equation
(cLODE). cLODE combines VAE with neural ODE for a more accurate prediction of
multi-agent vehicle trajectories. Utilizing a Latent-ODE-inspired
architecture, the model successfully predicts the vehicle state transitions in
the latent space, outperforming the adaptive generative prediction networks
AGen on RMSE metrics on myriads of NGSIM experiments. Furthermore, we found
that the appropriate length of the observation history helps increase the
prediction performance from the ablation studies.

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##  Appendix A Appendix

###  A.1 The Details of Network Architecture

##### Encoder

For each time step $t$, we first concatenate the observation $o_{t}$ and the
action $a_{t}$, and then use a MLP with 6 fully connected layers to output an
embedding vector $e_{t}$. All embedding vectors at all time steps are
propagated backward into the first time step $t=1$ via an ODE-RNN model to
estimate a representation of the observed trajectory $\tau$. ODE-RNN estimates
an intermediate embedding $\hat{e}_{t-1}$ at the step $t-1$ based on
$(e_{t},...,e_{T})$ by solving the ODE, and then updates the current embedding
$e_{t}$ with $\hat{e}_{t}$ via a GRU.

##### Decoder

The neural ODE used for decoding trajectories has 3 layers with 256 units for
each. After estimating the hidden variable $z_{t}$ by solving the ODE, we use
an MLP with 3 fully connected layers to predict the action $\hat{a}_{t}$.

##### Training and Prediction

We train cLODE by optimizing the ELBO loss computed from Eq. 3,4, and 6. After
the model is trained, it can predict an action at any time given an arbitrary
length of the history data.


0pt section[2em]2em0pt subsection[5em]3em5pt

# On global minimizers for a mass constrained problem

Louis Jeanjean
Laboratoire de Mathématiques (CNRS UMR 6623),
Université de Bourgogne Franche-Comté,
Besançon 25030, France
e-mail<EMAIL_ADDRESS>

Sheng-Sen Lu
LMAM and School of Mathematical Sciences,
Peking University,
Beijing 100871, PR China
e-mail<EMAIL_ADDRESS>
Abstract. In any dimension $N\geq 1$, for given mass $m>0$ and for the $C^{1}$
energy functional

$I(u):=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx-\int_{\mathbb{R}^{N}}F(u)dx,$

we revisit the classical problem of finding conditions on $F\in
C^{1}(\mathbb{R},\mathbb{R})$ insuring that $I$ admits global minimizers on
the mass constraint

$S_{m}:=\left\\{u\in
H^{1}(\mathbb{R}^{N})~{}|~{}\|u\|^{2}_{L^{2}(\mathbb{R}^{N})}=m\right\\}.$

Under assumptions that we believe to be nearly optimal, in particular without
assuming that $F$ is even, any such global minimizer, called energy ground
state, proves to have constant sign and to be radially symmetric monotone with
respect to some point in $\mathbb{R}^{N}$. Moreover, we manage to show that
any energy ground state is a least action solution of the associated free
functional. This last result settles, under general assumptions, a long
standing open problem.

2010 MSC: 35J60, 35Q55

Key words: Nonlinear scalar field equations, prescribed $L^{2}$-norm
solutions, constrained minimization problems, energy ground states versus
least action solutions.

##  1 Introduction

Let $N\geq 1$ and $I:H^{1}(\mathbb{R}^{N})\to\mathbb{R}$ be a $C^{1}$ energy
functional defined by

$I(u):=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx-\int_{\mathbb{R}^{N}}F(u)dx$

where $F(t):=\int^{t}_{0}f(\tau)d\tau$ for some function $f\in
C(\mathbb{R},\mathbb{R})$.

In this paper we focus on the minimization problem

$E_{m}:=\inf_{u\in S_{m}}I(u),$ ($\text{Inf}_{m}$)

where $m>0$ is prescribed and

$S_{m}:=\left\\{u\in
H^{1}(\mathbb{R}^{N})~{}|~{}\|u\|^{2}_{L^{2}(\mathbb{R}^{N})}=m\right\\}.$

By a direct application of Lagrange multiplier’s rule, if $u\in S_{m}$ solves
($\text{Inf}_{m}$) then there exists $\mu=\mu(u)\in\mathbb{R}$ such that

$-\Delta u=f(u)-\mu u\qquad\mbox{in}~{}H^{1}(\mathbb{R}^{N}).$ (1.1)

A minimizer of ($\text{Inf}_{m}$) is often called an energy ground state and
$E_{m}$ the ground state energy.

The study of problem ($\text{Inf}_{m}$) naturally arises in the search of
standing waves for nonlinear scalar field equations the form

$i\psi_{t}+\Delta\psi+f(\psi)=0,\qquad\psi:\mathbb{R}\times\mathbb{R}^{N}\to\mathbb{C}.$
(1.2)

By standing waves, we mean solutions to (1.2) of the special form
$\psi(t,x)=e^{i\mu t}u(x)$ with $\mu\in\mathbb{R}$ and $u\in
H^{1}(\mathbb{R}^{N})$. Clearly $\psi(t,x)$ satisfies (1.2) if $u(x)$
satisfies (1.1) for the corresponding $\mu\in\mathbb{R}$.

The study of such type of equations, which already saw major contributions
forty years ago, [1, 7, 22, 23, 27], now lies at the root of several models
linked with current physical applications (such as nonlinear optics, the
theory of water waves, …). For these equations, finding solutions with a
prescribed $L^{2}$-norm is particularly relevant since this quantity is
preserved along the time evolution. In addition, if the solutions correspond
to energy ground states, then, in most situations, it is possible to prove
that the associated standing waves are orbitally stable. This likely explains
why the study of problem ($\text{Inf}_{m}$) is still the object of an intense
activity. Among many others possible choices, we refer to [4, 5, 6, 10, 12,
13, 15, 21, 25, 26] and to the references therein.

Our first main result concerns the solvability of ($\text{Inf}_{m}$). It can
be viewed as an extension of the one of [25] already obtained in a very
general setting. The following assumptions on $f\in C(\mathbb{R},\mathbb{R})$
will be required.

  * $(f1)$

$\lim_{t\to 0}f(t)/t=0$.

  * $(f2)$

When $N\geq 3$,

$\limsup_{t\to\infty}\frac{|f(t)|}{|t|^{\frac{N+2}{N-2}}}<\infty;$

when $N=2$,

$\lim_{t\to\infty}\frac{f(t)}{e^{\alpha t^{2}}}=0\quad\forall\alpha>0;$

and also for any $N\geq 1$,

$\limsup_{t\to\infty}\frac{f(t)t}{|t|^{2+\frac{4}{N}}}\leq 0.$

  * $(f3)$

There exists $\zeta\neq 0$ such that $F(\zeta)>0$.

###### Theorem 1.1

Assume that $N\geq 1$, $f\in C(\mathbb{R},\mathbb{R})$ satisfies
$(\hyperlink{f1}{f1})-(\hyperlink{f3}{f3})$. Then

$E_{m}:=\inf_{u\in S_{m}}I(u)>-\infty$

and the map $m\mapsto E_{m}$ is nonincreasing and continuous. Moreover,

  * $(i)$

there exists a number $m^{*}\in[0,\infty)$ such that

$E_{m}=0\quad\text{if}~{}0<m\leq m^{*},\qquad
E_{m}<0\quad\text{when}~{}m>m^{*};$

  * $(ii)$

when $m>m^{*}$, the global infimum $E_{m}$ is achieved and thus
($\text{Inf}_{m}$) has an energy ground state $v\in S_{m}$ with
$I(v)=E_{m}<0$;

  * $(iii)$

when $0<m<m^{*}$, $E_{m}=0$ is not achieved;

  * $(iv)$

$m^{*}=0$ if in addition

$\lim_{t\to 0}\frac{F(t)}{|t|^{2+\frac{4}{N}}}=+\infty,$ (A.1)

and $m^{*}>0$ if in addition

$\limsup_{t\to 0}\frac{F(t)}{|t|^{2+\frac{4}{N}}}<+\infty.$ (A.2)

###### Remark 1.2

  * $(i)$

As it will be clear from the proof of Theorem 1.1 $(ii)$, see also Remark 2.4,
when $m>m^{*}$ we also show that any minimizing sequence for
($\text{Inf}_{m}$) is, up to a subsequence and up to translations in
$\mathbb{R}^{N}$, strongly convergent.

  * $(ii)$

When $0<m<m^{*}$, it is proved in Theorem 1.1 $(iii)$ that the global infimum
$E_{m}=0$ is not achieved, but this does not mean that the constrained
functional $I_{|S_{m}}$ may not admit critical points with positive energies,
see the companion work [17].

  * $(iii)$

In the case $m^{*}>0$, studying existence and nonexistence of global
minimizers with respect to $E_{m^{*}}=0$ seems to be a delicate issue. Since
it exceeds our scope of the present paper, we shall not explore further
general conditions on $f$ that ensure the existence or nonexistence but refer
the interested reader to [17] and [25, Theorem 1.4] for some existence
results.

  * $(iv)$

For convenience of statement, we introduce the notation

$m\succeq_{f}m^{*}$

with the understanding that $m\geq m^{*}$ if $m^{*}>0$ and $E_{m^{*}}=0$ is
achieved, and $m>m^{*}$ if otherwise. As one may observe, when
$m\succeq_{f}m^{*}$ and for any minimizer $v\in S_{m}$ of ($\text{Inf}_{m}$),
the associated Lagrange multiplier $\mu=\mu(v)$ is positive. Indeed, from the
Pohozaev identity corresponding to (1.1), see [1, Proposition 1],

$P(v):=\frac{N-2}{2N}\int_{\mathbb{R}^{N}}|\nabla
v|^{2}dx+\frac{1}{2}\mu\int_{\mathbb{R}^{N}}|v|^{2}dx-\int_{\mathbb{R}^{N}}F(v)dx=0$

and the fact that $I(v)=E_{m}\leq 0$, we have

$0\geq I(v)=I(v)-P(v)=\frac{1}{N}\int_{\mathbb{R}^{N}}|\nabla
v|^{2}dx-\frac{1}{2}\mu m$

and hence $\mu>0$.

###### Remark 1.3

Let us give some examples of nonlinearities satisfying
$(\hyperlink{f1}{f1})-(\hyperlink{f3}{f3})$.

  * $(i)$

$f(t)=|t|^{p-2}t+A|t|^{q-2}t$ with

$A\in\mathbb{R}\qquad\text{and}\qquad 2<q<p<2+\frac{4}{N}.$

In particular, (A.1) and (A.2) hold when $A\geq 0$ and when $A<0$
respectively.

  * $(ii)$

$f(t)=|t|^{p-2}t-|t|^{q-2}t$ with

$\left\\{\begin{aligned} &2<p<q<\infty,&\qquad&\text{if}~{}N=1,2,\\\
&2<p<q\leq\frac{2N}{N-2},&\qquad&\text{if}~{}N\geq 3.\end{aligned}\right.$

In particular, (A.1) and (A.2) hold if $p<2+\frac{4}{N}$ and if $p\geq
2+\frac{4}{N}$ respectively, and when $N=1,2,3$ we cover the so-called cubic-
quintic nonlinearity

$f(t)=|t|^{2}t-|t|^{4}t$

which attracts much attention due to its physical relevance, see for example
[4, 5, 20, 21, 26].

These examples are only some special cases, our Theorem 1.1 and the subsequent
Theorems 1.4 and 1.6 apply to more general nonlinearities, in particular to
those which are not a sum of powers.

Our next result shows that any energy ground state has constant sign and
enjoys symmetry and monotonicity properties.

###### Theorem 1.4

Assume that $N\geq 1$, $f\in C(\mathbb{R},\mathbb{R})$ satisfies
$(\hyperlink{f1}{f1})-(\hyperlink{f3}{f3})$, and in addition $f$ is locally
Lipschitz continuous when $N=1$. Let $m\succeq_{f}m^{*}$, where $m^{*}\geq 0$
is the number given by Theorem 1.1. Then any minimizer $v\in S_{m}$ of
($\text{Inf}_{m}$) satisfies the following properties:

  * $(i)$

$v$ has constant sign,

  * $(ii)$

$v$ is radially symmetric up to a translation in $\mathbb{R}^{N}$,

  * $(iii)$

$v$ is monotone with respect to the radial variable.

Our last theorem is the heart of the present paper, it answers for nonlinear
scalar field equations a long standing open problem. To explain what is at
stake, we need the following definition.

###### Definition 1.5

For given $\mu>0$, a nontrivial solution $w\in H^{1}(\mathbb{R}^{N})$ of the
free problem

$\left\\{\begin{aligned} -\Delta u&=f(u)-\mu
u\quad\text{in}~{}\mathbb{R}^{N},\\\ u&\in
H^{1}(\mathbb{R}^{N}),\end{aligned}\right.$ ($Q_{\mu}$)

is said to be a least action solution if it achieves the infimum of the
$C^{1}$ action functional

$J_{\mu}(u):=I(u)+\frac{1}{2}\mu\int_{\mathbb{R}^{N}}|u|^{2}dx$

among all the nontrivial solutions, namely

$J_{\mu}(w)=A_{\mu}:=\inf\\{J_{\mu}(u)~{}|~{}u\in
H^{1}(\mathbb{R}^{N})\setminus\\{0\\},J^{\prime}_{\mu}(u)=0\\}.$

For future reference, the value $A_{\mu}$ is called the least action of
($Q_{\mu}$).

The open problem can be now formulated as follows.

Open Problem: We know that a ground state energy minimizer $v\in S_{m}$ is a
nontrivial solution to ($Q_{\mu}$), where $\mu=\mu(v)>0$ is the associated
Lagrange multiplier. Is the minimizer $v\in S_{m}$ a least action solution to
($Q_{\mu}$)? In other words, does an energy ground state is necessarily a
least action solution?

For some odd $f\in C(\mathbb{R},\mathbb{R})$ satisfying
$(\hyperlink{f1}{f1})-(\hyperlink{f3}{f3})$ it is known that there exists a
unique positive solution to ($Q_{\mu}$) and that it is a least action
solution, see for example [5, 20, 21]. Thus, in such situations, Theorem 1.4
implies that any energy ground state is a least action solution. However,
apart in some particular cases of this type where some uniqueness property was
used, the Open Problem remained unsolved until recently. In 2020 a positive
answer was given in [11] for a related problem of biharmonic type with a power
nonlinearity, see [11, Proposition 3.9 and Theorem 1.3]. Also, very recently
in [10], the authors answered positively the Open Problem for nonlinearities
modeled on $f(t)=|t|^{p-2}t$, $p\in(2,2^{*})$ which allow to benefit from a
Nehari manifold in the sense given in [28], see [10, Theorem 1.3] for more
details. Note that the results of [10] also hold when the analog of problem
($Q_{\mu}$) is set on an arbitrary domain $\Omega\subset\mathbb{R}^{N}$.
Finally, we mention [14] in which the Open Problem was claimed to be solved
for a nonlinearity which is a sum of two powers.

Our result in that direction covers all the previous particular cases, at
least when the associated equations are set on all the space $\mathbb{R}^{N}$.

###### Theorem 1.6

Assume that $N\geq 1$, $f\in C(\mathbb{R},\mathbb{R})$ satisfies
$(\hyperlink{f1}{f1})-(\hyperlink{f3}{f3})$, and in addition $f$ is locally
Lipschitz continuous when $N=1$. Let $m\succeq_{f}m^{*}$ and denote by
$\mu(v)$ the Lagrange multiplier corresponding to an arbitrary minimizer $v\in
S_{m}$ of ($\text{Inf}_{m}$), where $m^{*}\geq 0$ is the number given by
Theorem 1.1. Then the following statements hold.

  * $(i)$

Any minimizer $v\in S_{m}$ of ($\text{Inf}_{m}$) is a least action solution of
($Q_{\mu}$) with $\mu=\mu(v)>0$. In particular,

$A_{\mu}=E_{m}+\frac{1}{2}\mu m.$

  * $(ii)$

For given $\mu\in\\{\mu(v)~{}|~{}v\in S_{m}~{}\text{is a minimizer
of}~{}\eqref{eq:Inf_m}\\}$, any least action solution $w\in
H^{1}(\mathbb{R}^{N})$ of ($Q_{\mu}$) is a minimizer of ($\text{Inf}_{m}$),
namely

$\|w\|^{2}_{L^{2}(\mathbb{R}^{N})}=m\qquad\text{and}\qquad I(w)=E_{m}.$

###### Remark 1.7

  * $(i)$

The conclusions of Theorem 1.6 (ii) were also observed in [10, 11, 14] in the
corresponding frames.

  * $(ii)$

For alternative variational characterizations of the energy ground states, in
related problems, we refer to [8, 9, 13]. Note that in [8, 9], a variational
characterization of the associated Lagrange multiplier is proposed, see also
[10, Theorem 1.2] in that direction.

  * $(iii)$

It is known, see for example [4, 17, 21], that under the assumptions
$(\hyperlink{f1}{f1})-(\hyperlink{f3}{f3})$ there may exist least action
solutions which are not energy ground states.

The paper is organized as follows. In Section 2 we prove Theorem 1.1. Section
3 is devoted to the proof of Theorem 1.4. Finally, in Section 4 we prove
Theorem 1.6.

Notations. Throughout this paper, for any $p\in[1,\infty)$,
$L^{p}(\mathbb{R}^{N})$ is the usual Lebesgue space endowed with the norm

$\|u\|_{L^{p}(\mathbb{R}^{N})}:=\left(\int_{\mathbb{R}^{N}}|u|^{p}dx\right)^{1/p},$

and $H^{1}(\mathbb{R}^{N})$ the usual Sobolev space endowed with the norm

$\|u\|_{H^{1}(\mathbb{R}^{N})}:=\left(\|\nabla
u\|^{2}_{L^{2}(\mathbb{R}^{N})}+\|u\|^{2}_{L^{2}(\mathbb{R}^{N})}\right)^{1/2}.$

Moreover, for given $u\in H^{1}(\mathbb{R}^{N})$ and any $s\in\mathbb{R}$, we
define the scaling function

$s\diamond u:=e^{Ns/2}u(e^{s}\cdot),$

which remains in $H^{1}(\mathbb{R}^{N})$ and preserves the $L^{2}$ norm when
$s\in\mathbb{R}$ varies.

##  2 Existence and nonexistence

This section aims to prove Theorem 1.1 and in particular we shall show the
existence and nonexistence of minimizers of ($\text{Inf}_{m}$) for suitable
range of the mass $m>0$. As a necessary preparation, we have the following
lemma the proof of which is standard.

###### Lemma 2.1

Assume that $N\geq 1$ and $f\in C(\mathbb{R},\mathbb{R})$ satisfies
$(\hyperlink{f1}{f1})-(\hyperlink{f2}{f2})$. Then the following statements
hold.

  * $(i)$

For any bounded sequence $\\{u_{n}\\}$ in $H^{1}(\mathbb{R}^{N})$,

$\lim_{n\to\infty}\int_{\mathbb{R}^{N}}F(u_{n})dx=0$

if $\lim_{n\to\infty}\|u_{n}\|_{L^{\infty}(\mathbb{R}^{N})}=0$, and

$\limsup_{n\to\infty}\int_{\mathbb{R}^{N}}F(u_{n})dx\leq 0$

if $\lim_{n\to\infty}\|u_{n}\|_{L^{2+4/N}(\mathbb{R}^{N})}=0$.

  * $(ii)$

There exists $C=C(f,N,m)>0$ such that

$I(u)\geq\frac{1}{4}\|\nabla u\|^{2}_{L^{2}(\mathbb{R}^{N})}-C(f,N,m)$

for any $u\in H^{1}(\mathbb{R}^{N})$ satisfying
$\|u\|^{2}_{L^{2}(\mathbb{R}^{N})}\leq m$. In particular, $I$ is coercive on
$S_{m}$.

To proceed further, we recall the global infimum

$E_{m}:=\inf_{u\in S_{m}}I(u)$

and make below a detailed study of its basic properties.

###### Lemma 2.2

Assume that $N\geq 1$ and $f\in C(\mathbb{R},\mathbb{R})$ satisfies
$(\hyperlink{f1}{f1})-(\hyperlink{f3}{f3})$. Then the following statements
hold.

  * $(i)$

$-\infty<E_{m}\leq 0$ for all $m>0$.

  * $(ii)$

There exists $m_{0}>0$ such that $E_{m}<0$ for any $m>m_{0}$.

  * $(iii)$

$E_{m}<0$ for all $m>0$ if (A.1) holds, and $E_{m}=0$ for small $m>0$ if (A.2)
holds.

  * $(iv)$

For any $m>m^{\prime}>0$ one has

$E_{m}\leq\frac{m}{m^{\prime}}E_{m^{\prime}}.$ (2.1)

If $E_{m^{\prime}}$ is achieved then the inequality is strict.

  * $(v)$

The function $m\mapsto E_{m}$ is nonincreasing and continuous.

Proof. $(i)$ By Lemma 2.1 $(ii)$, $I$ is bounded from below on $S_{m}$ and
thus $E_{m}>-\infty$. For a fixed $u\in S_{m}\cap L^{\infty}(\mathbb{R}^{N})$,
we have $\|\nabla(s\diamond u)\|_{L^{2}(\mathbb{R}^{N})}\to 0$ and
$\|s\diamond u\|_{L^{\infty}(\mathbb{R}^{N})}\to 0$ as $s\to-\infty$. In view
of Lemma 2.1 $(i)$, $E_{m}\leq\lim_{s\to-\infty}I(s\diamond u)=0$.

$(ii)$ From $(\hyperlink{f3}{f3})$ and Step 1 of [1, Proof of Theorem 2],
there exists $u\in H^{1}(\mathbb{R}^{N})$ such that
$\int_{\mathbb{R}^{N}}F(u)dx>0$. For any $m>0$, set
$u_{m}:=u(m^{-1/N}\cdot\|u\|^{2/N}_{L^{2}(\mathbb{R}^{N})}\cdot x)\in S_{m}$.
Since

$\begin{split}I(u_{m})&=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u_{m}|^{2}dx-\int_{\mathbb{R}^{N}}F(u_{m})dx\\\
&=\frac{m^{\frac{N-2}{N}}}{2\|u\|^{2(N-2)/N}_{L^{2}(\mathbb{R}^{N})}}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx-\frac{m}{\|u\|^{2}_{L^{2}(\mathbb{R}^{N})}}\int_{\mathbb{R}^{N}}F(u)dx\\\
&=:Am^{\frac{N-2}{N}}-Bm=:g(m),\end{split}$

it follows that $E_{m}\leq I(u_{m})=g(m)<0$ for any sufficiently large $m>0$.

$(iii)$ When (A.1) holds, we choose $u\in S_{m}\cap
L^{\infty}(\mathbb{R}^{N})$. For

$D:=\int_{\mathbb{R}}|\nabla
u|^{2}dx\bigg{/}\int_{\mathbb{R}^{N}}|u|^{2+4/N}dx>0,$

by (A.1), there exists $\delta>0$ such that $F(t)\geq D|t|^{2+4/N}$ for all
$|t|\leq\delta$. Since $\|s\diamond
u\|_{L^{\infty}(\mathbb{R}^{N})}\leq\delta$ for some $s<0$, it is clear that

$\begin{split}E_{m}\leq I(s\diamond
u)&\leq\frac{1}{2}\int_{\mathbb{R}}|\nabla(s\diamond
u)|^{2}dx-D\int_{\mathbb{R}^{N}}|s\diamond u|^{2+4/N}dx\\\
&=\frac{1}{2}e^{2s}\int_{\mathbb{R}}|\nabla u|^{2}dx-
De^{2s}\int_{\mathbb{R}^{N}}|u|^{2+4/N}dx\\\
&=-\frac{1}{2}e^{2s}\int_{\mathbb{R}}|\nabla u|^{2}dx<0.\end{split}$

When (A.2) is satisfied, there exists $C_{f}>0$ such that $F(t)\leq
C_{f}|t|^{2+4/N}$ for any $t\in\mathbb{R}$. By the Gagliardo-Nirenberg
inequality,

$\int_{\mathbb{R}^{N}}F(u)dx\leq C_{f}C_{N}m^{2/N}\|\nabla
u\|^{2}_{L^{2}(\mathbb{R}^{N})}\qquad\text{for all}~{}u\in S_{m}.$

For any $m>0$ small enough such that $C_{f}C_{N}m^{2/N}\leq 1/4$, we have

$I(u)\geq\frac{1}{4}\|\nabla u\|^{2}_{L^{2}(\mathbb{R}^{N})}>0,$

and thus $E_{m}\geq 0$. From Item $(i)$, it follows that $E_{m}=0$ for $m>0$
small.

$(iv)$ Let $t:=m/m^{\prime}>1$. For any $\varepsilon>0$ there exists $u\in
S_{m^{\prime}}$ such that $I(u)\leq E_{m^{\prime}}+\varepsilon$. Clearly,
$v:=u(t^{-1/N}\cdot)\in S_{m}$ and then

$\begin{split}E_{m}\leq
I(v)&=tI(u)+\frac{1}{2}t^{\frac{N-2}{N}}\left(1-t^{\frac{2}{N}}\right)\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx\\\ &<tI(u)\\\
&\leq\frac{m}{m^{\prime}}(E_{m^{\prime}}+\varepsilon).\end{split}$ (2.2)

Since $\varepsilon>0$ is arbitrary, we see that the inequality (2.1) holds. If
$E_{m^{\prime}}$ is achieved, for example, at some $u\in S_{m^{\prime}}$, then
we can let $\varepsilon=0$ in (2.2) and thus the strict inequality follows.

$(v)$ By Item $(i)$ and (2.1), it is clear that $E_{m}$ is nonincreasing. Our
remaining task is to prove the continuity and this is equivalent to show that
for given $m>0$ and any positive sequence $\\{m_{k}\\}$ such that $m_{k}\to m$
as $k\to\infty$, one has $\lim_{k\to\infty}E_{m_{k}}=E_{m}$. We first claim
that

$\limsup_{k\to\infty}E_{m_{k}}\leq E_{m}.$ (2.3)

Indeed, for any $u\in S_{m}$ and each $k\in\mathbb{N}^{+}$, set
$u_{k}:=\sqrt{m_{k}/m}\cdot u\in S_{m_{k}}$. Since $u_{k}\to u$ strongly in
$H^{1}(\mathbb{R}^{N})$, it is clear that $\lim_{k\to\infty}I(u_{k})=I(u)$ and
thus

$\limsup_{k\to\infty}E_{m_{k}}\leq\limsup_{k\to\infty}I(u_{k})=I(u).$

By the arbitrariness of $u\in S_{m}$, we conclude that (2.3) holds. To
complete the proof of the continuity, we only need to show

$\liminf_{k\to\infty}E_{m_{k}}\geq E_{m}.$ (2.4)

For each $k\in\mathbb{N}^{+}$, there exists $v_{k}\in S_{m_{k}}$ such that

$I(v_{k})\leq E_{m_{k}}+\frac{1}{k}.$ (2.5)

Setting $t_{k}:=(m/m_{k})^{1/N}$ and $\tilde{v}_{k}:=v_{k}(\cdot/t_{k})\in
S_{m}$, we have

$\begin{split}E_{m}\leq I(\tilde{v}_{k})&\leq
I(v_{k})+\big{|}I(\tilde{v}_{k})-I(v_{k})\big{|}\\\ &\leq
E_{m_{k}}+\frac{1}{k}+\big{|}I(\tilde{v}_{k})-I(v_{k})\big{|}=:E_{m_{k}}+\frac{1}{k}+C(k),\end{split}$

where

$\begin{split}C(k)&\leq\frac{1}{2}|t^{N-2}_{k}-1|\cdot\int_{\mathbb{R}^{N}}|\nabla
v_{k}|^{2}dx+|t^{N}_{k}-1|\cdot\int_{\mathbb{R}^{N}}|F(v_{k})|dx\\\
&=:\frac{1}{2}|t^{N-2}_{k}-1|\cdot A(k)+|t^{N}_{k}-1|\cdot B(k).\end{split}$

Since $t_{k}\to 1$, the proof of (2.4) can be reduced to showing that $A(k)$
and $B(k)$ are bounded. To justify the boundedness, by (2.5) and (2.3), we
have $\limsup_{k\to\infty}I(v_{k})\leq E_{m}$. Noting that $v_{k}\in
S_{m_{k}}$ and $m_{k}\to m$, it follows from Lemma 2.1 $(ii)$ that
$\\{v_{k}\\}$ is bounded in $H^{1}(\mathbb{R}^{N})$. Since $f\in
C(\mathbb{R},\mathbb{R})$ satisfies $(\hyperlink{f1}{f1})$ and
$(\hyperlink{f2}{f2})$, it is clear that $A(k)$ and $B(k)$ are both bounded
and thus the continuity is proved. $\square$

###### Remark 2.3

When $N\geq 2$ the function $m\mapsto E_{m}$ is concave. To see this we define
for given $u\in S_{1}$ and any $m>0$ the real function

$\Phi_{u}(m):=I(u(m^{-\frac{1}{N}}\cdot))=m^{1-\frac{2}{N}}\|\nabla
u\|^{2}_{L^{2}(\mathbb{R}^{N})}-m\int_{\mathbb{R}^{N}}F(u)dx.$

It is clear that

$E_{m}=\inf_{u\in S_{1}}\Phi_{u}(m).$

Each function $m\mapsto\Phi_{u}(m)$ being concave since

$\Phi^{\prime\prime}_{u}(m)=-\frac{2}{N}\Big{(}1-\frac{2}{N}\Big{)}m^{-\frac{2}{N}-1}\|\nabla
u\|^{2}_{L^{2}(\mathbb{R}^{N})}\leq 0,$

we deduce that $m\mapsto E_{m}$ is a concave function. In particular, it is
continuous.

Proof of Theorem 1.1. We define

$m^{*}:=\inf\\{m>0~{}|~{}E_{m}<0\\}.$

It is easily seen from Lemma 2.2 that $m^{*}\in[0,\infty)$,

$E_{m}=0\quad\text{if}~{}0<m\leq m^{*},\qquad\qquad
E_{m}<0\quad\text{when}~{}m>m^{*};$ (2.6)

in particular, $m^{*}=0$ if (A.1) holds, and $m^{*}>0$ if (A.2) holds. Let us
first show that if $0<m<m^{*}$ then $E_{m}=0$ is not achieved. Indeed,
assuming by contradiction that $E_{m}=0$ is achieved for some $m\in(0,m^{*})$,
we infer from Lemma 2.2 $(iv)$ that

$E_{m^{*}}<\frac{m^{*}}{m}E_{m}=0$

and this leads a contradiction since $E_{m^{*}}=0$ by (2.6). The rest is to
prove that the global infimum $E_{m}$ is achieved when $m>m^{*}$.

Fix $m>m^{*}$ and let $\\{u_{n}\\}\subset S_{m}$ be any minimizing sequence
with respect to $E_{m}$. It is clear that $\\{u_{n}\\}$ is bounded in
$H^{1}(\mathbb{R}^{N})$ by Lemma 2.1 $(ii)$ and then one may assume that up to
a subsequence $\lim_{n\to\infty}\int_{\mathbb{R}^{N}}|\nabla u_{n}|^{2}dx$ and
$\lim_{n\to\infty}\int_{\mathbb{R}^{N}}F(u_{n})dx$ exist. Since $E_{m}<0$ by
(2.6), we deduce that $\\{u_{n}\\}$ is non-vanishing, namely

$\lim_{n\to\infty}\left(\sup_{y\in\mathbb{R}^{N}}\int_{B(y,1)}|u_{n}|^{2}dx\right)>0.$
(2.7)

Indeed, if (2.7) were not true, then $u_{n}\to 0$ in
$L^{2+4/N}(\mathbb{R}^{N})$ by Lions Lemma [23, Lemma I.1] and thus

$\lim_{n\to\infty}\int_{\mathbb{R}^{N}}F(u_{n})dx\leq 0$

via Lemma 2.1 $(i)$; noting that
$I(u_{n})\geq-\int_{\mathbb{R}^{N}}F(u_{n})dx$, we obtain a contradiction:

$0>E_{m}=\lim_{n\to\infty}I(u_{n})\geq-\lim_{n\to\infty}\int_{\mathbb{R}^{N}}F(u_{n})dx\geq
0.$

Since $\\{u_{n}\\}$ is non-vanishing, there exists a sequence
$\\{y_{n}\\}\subset\mathbb{R}^{N}$ and a nontrivial element $v\in
H^{1}(\mathbb{R}^{N})$ such that up to a subsequence
$u_{n}(\cdot+y_{n})\rightharpoonup v$ in $H^{1}(\mathbb{R}^{N})$ and
$u_{n}(\cdot+y_{n})\to v$ almost everywhere on $\mathbb{R}^{N}$. Set
$m^{\prime}:=\|v\|^{2}_{L^{2}(\mathbb{R}^{N})}\in(0,m]$ and
$w_{n}:=u_{n}(\cdot+y_{n})-v$. It is clear that

$\lim_{n\to\infty}\|w_{n}\|^{2}_{L^{2}(\mathbb{R}^{N})}=m-m^{\prime}$ (2.8)

and, using the splitting result [16, Lemma 2.6],

$E_{m}=\lim_{n\to\infty}I(u_{n})=\lim_{n\to\infty}I(v+w_{n})=I(v)+\lim_{n\to\infty}I(w_{n}).$
(2.9)

We shall prove below a claim and then conclude the whole proof.

Claim. $\lim_{n\to\infty}\|w_{n}\|_{L^{2}(\mathbb{R}^{N})}=0$. In particular,
$m^{\prime}=m$ by (2.8).

Let $t_{n}:=\|w_{n}\|^{2}_{L^{2}(\mathbb{R}^{N})}$ for each
$n\in\mathbb{N}^{+}$. If $\lim_{n\to\infty}t_{n}>0$, then (2.8) gives that
$m^{\prime}\in(0,m)$. In view of the definition of $E_{t_{n}}$ and Lemma 2.2
$(v)$, we obtain

$\lim_{n\to\infty}I(w_{n})\geq\lim_{n\to\infty}E_{t_{n}}=E_{m-m^{\prime}}.$

From (2.9) and (2.1), it follows

$E_{m}\geq I(v)+E_{m-m^{\prime}}\geq
E_{m^{\prime}}+E_{m-m^{\prime}}\geq\frac{m^{\prime}}{m}E_{m}+\frac{m-m^{\prime}}{m}E_{m}=E_{m}.$

Thus necessarily $I(v)=E_{m^{\prime}}$ and this shows that $E_{m^{\prime}}$ is
achieved at $v\in S_{m^{\prime}}$. But then still using (2.9) and (2.1), we
obtain a contradiction:

$E_{m}\geq
E_{m^{\prime}}+E_{m-m^{\prime}}>\frac{m^{\prime}}{m}E_{m}+\frac{m-m^{\prime}}{m}E_{m}=E_{m},$

and so the claim is proved.

Conclusion. Clearly, $v\in S_{m}$ by the above claim and thus $I(v)\geq
E_{m}$. Since the claim and Lemma 2.1 $(i)$ imply that

$\lim_{n\to\infty}\int_{\mathbb{R}^{N}}F(w_{n})dx\leq 0,$ (2.10)

we also have $\lim_{n\to\infty}I(w_{n})\geq 0$. Therefore, by (2.9) we get
$E_{m}\geq I(v)$ and hence $E_{m}<0$ is achieved at $v\in S_{m}$. $\square$

###### Remark 2.4

One can deduce further that $u_{n}(\cdot+y_{n})\to v$ in
$H^{1}(\mathbb{R}^{N})$. Indeed, from (2.9), (2.10) and the fact that
$I(v)=E_{m}$, it follows

$\|\nabla w_{n}\|_{L^{2}(\mathbb{R}^{N})}\to 0\qquad\text{as}~{}n\to\infty.$

Since $\lim_{n\to\infty}\|w_{n}\|_{L^{2}(\mathbb{R}^{N})}=0$, we obtain the
strong convergence.

##  3 Sign, symmetry and monotonicity

This section is devoted to the proof of Theorem 1.4. Unless otherwise noted,
for given $\mu>0$ we use the notations

$g_{\mu}(t):=-\mu t+f(t)\qquad\text{and}\qquad G_{\mu}(t):=-\frac{1}{2}\mu
t^{2}+F(t).$

To deal with the case $N=1$, a special treatment is required. To be more
precise, we shall make use of the following classification result which is
deduced by means of simple methods of ordinary differential equations.

###### Lemma 3.1

Assume that $N=1$, $f$ is a locally Lipschitz continuous function on
$\mathbb{R}$ satisfying $(\hyperlink{f1}{f1})$, and $w\in H^{1}(\mathbb{R})$
is a nontrivial critical point of $J_{\mu}$ for some $\mu>0$. Then $w$ have a
sign. More precisely we have

  * $(a)$

If $w$ is negative somewhere, then
$\zeta_{-}:=\sup\\{t<0~{}|~{}G_{\mu}(t)=0\\}\in(-\infty,0)$,

$g_{\mu}(\zeta_{-})<0,$

and after a suitable translation of the origin $w$ satisfies

    * $(a1)$

$w(x)=w(-x)$ for any $x\in\mathbb{R}$,

    * $(a2)$

$w(x)<0$ for any $x\in\mathbb{R}$,

    * $(a3)$

$w(0)=\zeta_{-}$,

    * $(a4)$

$w^{\prime}(x)>0$ for any $x>0$.

  * $(b)$

If $w$ is positive somewhere, then
$\zeta_{+}:=\inf\\{t>0~{}|~{}G_{\mu}(t)=0\\}\in(0,\infty)$,

$g_{\mu}(\zeta_{+})>0,$

and after a suitable translation of the origin $w$ satisfies

    * $(b1)$

$w(x)=w(-x)$ for any $x\in\mathbb{R}$,

    * $(b2)$

$w(x)>0$ for any $x\in\mathbb{R}$,

    * $(b3)$

$w(0)=\zeta_{+}$,

    * $(b4)$

$w^{\prime}(x)<0$ for any $x>0$.

In particular, $w$ is a translation of the unique solution to the initial
value problem $-u^{\prime\prime}=g_{\mu}(u)$ with $u(0)=\zeta_{-}$ (or
$u(0)=\zeta_{+}$) and $u^{\prime}(0)=0$.

Proof. By regularity $w\in C^{2}(\mathbb{R},\mathbb{R})$ and thus

$-w^{\prime\prime}=g_{\mu}(w)\qquad\text{in}~{}\mathbb{R}.$ (3.1)

Since $|w(x)|$ and $|w^{\prime}(x)|$ decay to zero exponentially as
$|x|\to\infty$, we have

$\frac{1}{2}|w^{\prime}(x)|^{2}+G_{\mu}(w(x))=0\qquad\text{for}~{}x\in\mathbb{R}.$
(3.2)

Without loss of generality, we only consider the case when $w$ is negative
somewhere. By translating the point where $w$ achieves its negative minimum to
the origin, one may assume that $w^{\prime}(0)=0$. In view of (3.2),

$G_{\mu}(w(0))=0\qquad\text{and}\qquad\zeta_{-}>-\infty.$

Since $(\hyperlink{f1}{f1})$ gives that $G_{\mu}(t)<0$ for any $t<0$ close
enough to the origin, we also have $\zeta_{-}<0$. Now assume by contradiction
that $g_{\mu}(\zeta_{-})\geq 0$. Since $w(0)\leq\zeta_{-}$, there exists
$x^{*}\in\mathbb{R}$ such that $w(x^{*})=\zeta_{-}$. Then

$w^{\prime}(x^{*})=0\qquad\text{and}\qquad
w^{\prime\prime}(x^{*})=-g_{\mu}(\zeta_{-})\leq 0$

via (3.2) and (3.1) respectively. If $g_{\mu}(\zeta_{-})>0$, then since
whenever $w(x)=\zeta_{-}$ one also has $w^{\prime}(x)=0$ and
$w^{\prime\prime}(x)<0$, $w$ can never go above $\zeta_{-}<0$, which is
impossible. On the other hand, if $g_{\mu}(\zeta_{-})=0$, then by uniqueness
the conditions

$w(x^{*})=\zeta_{-}\qquad\text{and}\qquad
w^{\prime}(x^{*})=w^{\prime\prime}(x^{*})=0$

imply $w\equiv\zeta_{-}$, which is also impossible. With the desired
conclusion $g_{\mu}(\zeta_{-})<0$ at hand, there exists $\varepsilon>0$ such
that $G_{\mu}(t)>0$ for any $t\in(\zeta_{-}-\varepsilon,\zeta_{-})$. If
$w(0)<\zeta_{-}$, then $w(x_{*})\in(\zeta_{-}-\varepsilon,\zeta_{-})$ for some
$x_{*}\in\mathbb{R}$ and so

$\frac{1}{2}|w^{\prime}(x_{*})|^{2}+G_{\mu}(w(x_{*}))>0.$

This contradicts (3.2), and therefore $w(0)=\zeta_{-}$. Since $w$ is the
global solution of (3.1) with the initial conditions $w(0)=\zeta_{-}$ and
$w^{\prime}(0)=0$, the rest follows from a standard adaptation of some
arguments in [1, Proof of Theorem 5]. $\square$

###### Remark 3.2

Even though the nonlinearity $f$ in Lemma 3.1 is locally Lipschitz continuous,
the nontrivial critical points of $J_{\mu}$ (if exist) are not necessarily
unique up to a translation in $\mathbb{R}$ and up to a sign since we allow $f$
to be not odd.

In the higher dimensional case $N\geq 2$, the radial symmetry of minimizers
will be obtained as a direct consequence of a general symmetry result in [24],
and the proof of the monotonicity relies on Lemma 3.3 below. We remark that
the first part of Lemma 3.3 is well known and the second part is a simple
corollary of [2, Lemma 3.2].

###### Lemma 3.3

Let $v$ be a nonnegative measurable function defined on $\mathbb{R}^{N}$ such
that for any $\alpha>0$ the function $(v-\alpha)^{+}$ belongs to
$H^{1}(\mathbb{R}^{N})$ and has compact support, and denote by $v^{*}$ the
Schwarz rearrangement of $v$. Then

$\int_{\mathbb{R}^{N}}|\nabla v^{*}|^{2}dx\leq\int_{\mathbb{R}^{N}}|\nabla
v|^{2}dx.$ (3.3)

Moreover, if the equality in (3.3) holds then the level set

$\chi_{\alpha}:=\\{x\in\mathbb{R}^{N}~{}|~{}v(x)>\alpha\\}$

is equivalent to a ball for any $\alpha\in(0,\text{ess sup}(v))$.

Proof of Theorem 1.4 By Lemma 3.1, the case $N=1$ is proved. We treat below
the case $N\geq 2$. For given minimizer $v\in S_{m}$ of ($\text{Inf}_{m}$), we
set

$v^{+}:=\max\\{0,v\\}\qquad\text{and}\qquad v^{-}:=\min\\{0,v\\}.$

If $m^{\pm}:=\|v^{\pm}\|^{2}_{L^{2}(\mathbb{R}^{N})}\neq 0$, then Lemma 2.2
$(iv)$ gives that

$E_{m}=I(v)=I(v^{+})+I(v^{-})\geq
E_{m^{+}}+E_{m^{-}}\geq\frac{m^{+}}{m}E_{m}+\frac{m^{-}}{m}E_{m}=E_{m},$

and thus $E_{m^{\pm}}$ is achieved at $v^{\pm}\in S_{m^{\pm}}$. Using Lemma
2.2 $(iv)$ again, we obtain a contradiction:

$E_{m}\geq
E_{m^{+}}+E_{m^{-}}>\frac{m^{+}}{m}E_{m}+\frac{m^{-}}{m}E_{m}=E_{m}.$

Hence $v$ has constant sign. Since any minimizer of ($\text{Inf}_{m}$) is a
solution of ($Q_{\mu}$) for some $\mu>0$ and then by regularity must be of
class $C^{1}$, we also deduce from [24, Theorem 2] that $v$ is radially
symmetric up to a translation in $\mathbb{R}^{N}$. To proceed further, without
loss of generality, we may assume that $v\geq 0$ and $v(x)=\overline{v}(|x|)$
for some one variable function $\overline{v}:[0,\infty)\to[0,\infty)$. By the
fact that $v(x)\to 0$ as $|x|\to\infty$, it can be seen that $v$ is bounded
and for any $\alpha>0$ the function $(v-\alpha)^{+}$ belongs to
$H^{1}(\mathbb{R}^{N})$ and has compact support. Since the Schwarz
rearrangement $v^{*}$ satisfies

$v^{*}\in
S_{m}\qquad\text{and}\qquad\int_{\mathbb{R}^{N}}F(v^{*})dx=\int_{\mathbb{R}^{N}}F(v)dx,$

it follows from Lemma 3.3 that $E_{m}\leq I(v^{*})\leq I(v)=E_{m}$ and thus

$\int_{\mathbb{R}^{N}}|\nabla v^{*}|^{2}dx=\int_{\mathbb{R}^{N}}|\nabla
v|^{2}dx.$

By Lemma 3.3 again, for any $\alpha\in(0,\max(v))$, the level set
$\chi_{\alpha}$ is equivalent to a ball. We now assume by contradiction that
$\overline{v}$ is not nonincreasing. Then

$\overline{v}(r_{2})>\overline{v}(r_{1})>0\qquad\text{for
some}~{}r_{2}>r_{1}\geq 0.$

Since $\overline{v}(r)\to 0$ as $r\to\infty$, there exists $r_{3}>r_{2}$ such
that $\overline{v}(r_{3})=\overline{v}(r_{1})$. Denoting
$a:=\overline{v}(r_{1})$ and $b:=\overline{v}(r_{2})$, one may see that for
any $\alpha\in(a,b)$ the level set $\chi_{\alpha}$ is nonempty but not
equivalent to a ball. This gives a contradiction and thus $v$ is nonincreasing
with respect to the radial variable. $\square$

##  4 Least action characterization

In this section we show the least action characterization for any minimizer
$v\in S_{m}$ of ($\text{Inf}_{m}$) by using a mountain pass characterization
of nontrivial solutions of ($Q_{\mu}$) with $\mu=\mu(v)>0$. This mountain pass
characterization, see Lemma 4.1 below, is the core of the proof of Theorem
1.6. It highlights the role of the $L^{2}$ mass and seems to have not been
formulated before. Some of our arguments are motivated by [3, 18, 19].

###### Lemma 4.1

Assume that $N\geq 1$, $\mu>0$ and $f\in C(\mathbb{R},\mathbb{R})$ satisfies
the conditions $(\hyperlink{f1}{f1})$ and

  * $(f2)^{\prime}$

when $N\geq 3$,
$\limsup_{t\to\infty}\frac{|f(t)|}{|t|^{\frac{N+2}{N-2}}}<\infty$,
when $N=2$, for any $\alpha>0$

$\lim_{t\to\infty}\frac{f(t)}{e^{\alpha t^{2}}}=0,$

and suppose in addition that $f$ is locally Lipschitz continuous when $N=1$.
Then for any nontrivial critical point $w\in H^{1}(\mathbb{R}^{N})$ of
$J_{\mu}$, any $\delta>0$ and any $M>0$, there exist a constant
$T=T(w,\delta,M)>0$ and a continuous path $\gamma:[0,T]\to
H^{1}(\mathbb{R}^{N})$ satisfying

  * $(i)$

$\gamma(0)=0$, $J_{\mu}(\gamma(T))<-1$,
$\max_{t\in[0,T]}J_{\mu}(\gamma(t))=J_{\mu}(w)$;

  * $(ii)$

$\gamma(\tau)=w$ for some $\tau\in(0,T)$, and

$J_{\mu}(\gamma(t))<J_{\mu}(w)$

for any $t\in[0,T]$ such that
$\|\gamma(t)-w\|_{L^{2}(\mathbb{R}^{N})}\geq\delta$;

  * $(iii)$

$m(t):=\|\gamma(t)\|^{2}_{L^{2}(\mathbb{R}^{N})}$ is a strictly increasing
continuous function with $m(T)>M$.

Proof. When $N\geq 3$ and for the given $w\in H^{1}(\mathbb{R}^{N})$, we set

$\gamma(t):=\left\\{\begin{aligned}
&w\left(\frac{\cdot}{t}\right),&&\qquad\text{for}~{}t>0,\\\
&~{}~{}~{}0,&&\qquad\text{for}~{}t=0.\end{aligned}\right.$

Note that
$m(t):=\|\gamma(t)\|^{2}_{L^{2}(\mathbb{R}^{N})}=t^{N}\|w\|^{2}_{L^{2}(\mathbb{R}^{N})}$
and by the Pohozaev identity

$\begin{split}J_{\mu}(\gamma(t))&=\frac{1}{2}t^{N-2}\int_{\mathbb{R}^{N}}|\nabla
w|^{2}dx-t^{N}\int_{\mathbb{R}^{N}}G_{\mu}(w)dx\\\
&=\frac{1}{2}\Big{(}t^{N-2}-\frac{N-2}{N}t^{N}\Big{)}\int_{\mathbb{R}^{N}}|\nabla
w|^{2}dx.\end{split}$

For any $\delta>0$ and $M>0$, we can thus choose a large constant $T=T(w,M)>0$
such that the continuous path $\gamma:[0,T]\to H^{1}(\mathbb{R}^{N})$
satisfies Items $(i)-(iii)$ of Lemma 4.1.

In the case of $N=1$, without loss of generality, we only consider the
situation when the given $w\in H^{1}(\mathbb{R})$ is negative somewhere. Then
the statement $(a)$ of Lemma 3.1 holds and we can define a negative continuous
function $W:\mathbb{R}\to\mathbb{R}$ by

$W(x)=\left\\{\begin{aligned} &~{}~{}w(x),&&\qquad\text{for}~{}x\geq 0,\\\
&\zeta_{-}-x^{4},&&\qquad\text{for}~{}x\in[-\varepsilon,0),\\\
&\zeta_{-}-\varepsilon^{4},&&\qquad\text{for}~{}x<-\varepsilon.\end{aligned}\right.$

Here $\varepsilon>0$ is a chosen small constant such that

$\frac{1}{2}|W^{\prime}(x)|^{2}-G_{\mu}(W(x))=8x^{6}-G_{\mu}(\zeta_{-}-x^{4})<0\qquad\text{for}~{}x\in[-\varepsilon,0),$
(4.1)

and it follows from $G_{\mu}(\zeta_{-})=0$ and $g_{\mu}(\zeta_{-})<0$. Setting

$\gamma(t):=\left\\{\begin{aligned} &W(|\cdot|-\ln
t),&&\qquad\text{for}~{}t>0,\\\
&~{}~{}~{}~{}~{}~{}~{}~{}0,&&\qquad\text{for}~{}t=0,\end{aligned}\right.$

one may see that the path $\gamma:[0,\infty)\to H^{1}(\mathbb{R})$ is
continuous,

$m(t):=\|\gamma(t)\|^{2}_{L^{2}(\mathbb{R})}=\left\\{\begin{aligned}
&\|w\|^{2}_{L^{2}(\mathbb{R})}-\int^{-\ln t}_{\ln
t}|w(x)|^{2}dx,&&\qquad\text{for}~{}t\in(0,1),\\\
&\|w\|^{2}_{L^{2}(\mathbb{R})},&&\qquad\text{for}~{}t=1,\\\
&\|w\|^{2}_{L^{2}(\mathbb{R})}+2\int^{0}_{-\ln
t}|W(x)|^{2}dx,&&\qquad\text{for}~{}t>1,\end{aligned}\right.$

and

$J_{\mu}(\gamma(t))=\left\\{\begin{aligned} &J_{\mu}(w)-\int^{-\ln t}_{\ln
t}\Big{(}\frac{1}{2}|w^{\prime}(x)|^{2}-G_{\mu}(w(x))\Big{)}dx,&&\qquad\text{for}~{}t\in(0,1),\\\
&J_{\mu}(w),&&\qquad\text{for}~{}t=1,\\\ &J_{\mu}(w)+2\int^{0}_{-\ln
t}\Big{(}\frac{1}{2}|W^{\prime}(x)|^{2}-G_{\mu}(W(x))\Big{)}dx,&&\qquad\text{for}~{}t>1.\end{aligned}\right.$

By the fact that $G_{\mu}(w(x))<0$ for $x\neq 0$ and (4.1), we have

$J_{\mu}(\gamma(t))<J_{\mu}(w)\qquad\text{for}~{}t\neq 1$

and

$J_{\mu}(\gamma(t))<J_{\mu}(w)-2G_{\mu}(\zeta_{-}-\varepsilon^{4})\cdot(\ln
t-\varepsilon)\to-\infty\qquad\text{as}~{}t\to\infty.$

Noting also that $m(t)$ is strictly increasing and $m(t)\to\infty$ as
$t\to\infty$, for any $\delta>0$ and $M>0$ there exists a large constant
$T=T(w,M)>0$ such that $\gamma:[0,T]\to H^{1}(\mathbb{R})$ is a desired
continuous path of Lemma 4.1 when $N=1$.

To cope with the remaining case $N=2$, we adapt some arguments from [3,
Proposition 2]. For the given $w\in H^{1}(\mathbb{R}^{2})$, we define
$\Psi:[0,\infty)\times(0,\infty)\to\mathbb{R}$ by

$\Psi(\theta,s):=J_{\mu}\left(\theta
w\left(\frac{\cdot}{s}\right)\right)=\frac{1}{2}\theta^{2}\int_{\mathbb{R}^{2}}|\nabla
w|^{2}dx-s^{2}\int_{\mathbb{R}^{2}}G_{\mu}(\theta w)dx.$

It can be easily seen that

$\begin{split}\Psi_{\theta}(\theta,s)&=\theta\int_{\mathbb{R}^{2}}|\nabla
w|^{2}dx-s^{2}\int_{\mathbb{R}^{2}}g_{\mu}(\theta w)wdx,\\\
\Psi_{s}(\theta,s)&=-2s\int_{\mathbb{R}^{2}}G_{\mu}(\theta w)dx,\\\
\frac{d}{d\theta}\int_{\mathbb{R}^{2}}G_{\mu}(\theta
w)dx&=\int_{\mathbb{R}^{2}}g_{\mu}(\theta w)wdx.\end{split}$

Since the Nehari and Pohozaev identities give respectively

$\int_{\mathbb{R}^{2}}g_{\mu}(w)wdx=\int_{\mathbb{R}^{2}}|\nabla
w|^{2}dx\qquad\text{and}\qquad\int_{\mathbb{R}^{2}}G_{\mu}(w)dx=0,$

there exist two positive constants $\theta_{1}<1<\theta_{2}$ such that

$\frac{d}{d\theta}\int_{\mathbb{R}^{2}}G_{\mu}(\theta
w)dx>0\qquad\text{for}~{}\theta\in[\theta_{1},\theta_{2}],$

and thus

$\int_{\mathbb{R}^{2}}G_{\mu}(\theta w)dx\left\\{\begin{aligned}
&<0,&&\qquad\text{for}~{}\theta\in[\theta_{1},1),\\\
&=0,&&\qquad\text{for}~{}\theta=1,\\\
&>0,&&\qquad\text{for}~{}\theta\in(1,\theta_{2}].\end{aligned}\right.$ (4.2)

As a direct consequence,

$\Psi_{s}(\theta,s)\left\\{\begin{aligned}
&>0,&&\qquad\text{for}~{}(\theta,s)\in[\theta_{1},1)\times(0,\infty),\\\
&=0,&&\qquad\text{for}~{}(\theta,s)\in\\{1\\}\times(0,\infty),\\\
&<0,&&\qquad\text{for}~{}(\theta,s)\in(1,\theta_{2}]\times(0,\infty).\end{aligned}\right.$
(4.3)

On the other hand, noting that

$\Psi_{\theta}(1,s)=\int_{\mathbb{R}^{2}}|\nabla
w|^{2}dx-s^{2}\int_{\mathbb{R}^{2}}g_{\mu}(w)wdx=(1-s^{2})\int_{\mathbb{R}^{2}}|\nabla
w|^{2}dx,$

for any $s\neq 1$ there exists $\vartheta_{s}\in(0,1)$ such that

$\Psi_{\theta}(\theta,s)\left\\{\begin{aligned}
&>0,&&\qquad\text{for}~{}(\theta,s)\in[1-\vartheta_{s},1+\vartheta_{s}]\times(0,1),\\\
&<0,&&\qquad\text{for}~{}(\theta,s)\in[1-\vartheta_{s},1+\vartheta_{s}]\times(1,\infty).\end{aligned}\right.$
(4.4)

Also, with at hand the continuous function

$h(t):=\left\\{\begin{aligned}
&\frac{g_{\mu}(t)}{t},&&\qquad\text{for}~{}t\neq 0,\\\
&-\mu,&&\qquad\text{for}~{}t=0,\end{aligned}\right.$

one may find a small constant $s^{*}\in(0,1)$ such that

$\Psi_{\theta}(\theta,s)=\theta\left(\int_{\mathbb{R}^{2}}|\nabla
w|^{2}dx-s^{2}\int_{\mathbb{R}^{2}}h(\theta
w)w^{2}dx\right)>0\qquad\text{for}~{}(\theta,s)\in(0,1]\times(0,s^{*}].$ (4.5)

Now for any $\delta>0$ we fix a small constant
$\varepsilon=\varepsilon(\delta)>0$ such that $1-\varepsilon>s^{*}$ and

$\left\|w\left(\frac{\cdot}{s}\right)-w\right\|_{L^{2}(\mathbb{R}^{2})}<\delta\qquad\text{for}~{}s\in[1-\varepsilon,1+\varepsilon],$

and denote by $\eta(t)=(\theta(t),s(t)):[0,\infty)\to\mathbb{R}^{2}$ the
piecewise linear curve joining

$\begin{split}(0,s^{*})\to(1-\theta^{*},s^{*})\to(1-\theta^{*},1-\varepsilon)&\to(1,1-\varepsilon)\\\
&\to(1,1)\\\
&\to(1,1+\varepsilon)\to(1+\theta^{*},1+\varepsilon)\to(1+\theta^{*},\infty).\end{split}$

Here $\theta^{*}=\theta^{*}(w,\delta)\in(0,1)$ is a chosen constant satisfying

$\theta^{*}\leq\min\\{1-\theta_{1},\theta_{2}-1,\vartheta_{1-\varepsilon},\vartheta_{1+\varepsilon}\\},$

and each segment is horizontal or vertical. Let
$0=:t_{0}<t_{1}<\cdots<t_{6}<t_{7}:=\infty$ be such that for each
$k=0,1,\cdots,7$, the element $\eta(t_{k})\in\mathbb{R}^{2}$ is an end point
of a linear segment of the piecewise linear curve $\eta$. We define

$\gamma(t):=\theta(t)w\left(\frac{\cdot}{s(t)}\right),\qquad t\geq 0.$

Then the function $J_{\mu}(\gamma(t))=\Psi(\eta(t))$ is strictly increasing on
$(t_{0},t_{1})$, $(t_{1},t_{2})$ and $(t_{2},t_{3})$ by (4.5), (4.3) and (4.4)
respectively. One may also see that $J_{\mu}(\gamma(t))$ is constant on
$(t_{3},t_{4})$, $(t_{4},t_{5})$ by (4.3), and strictly decreasing on
$(t_{5},t_{6})$ and $(t_{6},t_{7})$ via (4.4) and (4.3) respectively.
Moreover, using (4.2),

$\begin{split}J_{\mu}(\gamma(t))&=\frac{1}{2}(1+\theta^{*})^{2}\int_{\mathbb{R}^{2}}|\nabla
w|^{2}dx-s^{2}(t)\int_{\mathbb{R}^{2}}G_{\mu}((1+\theta^{*})w)dx\\\
&\to-\infty\qquad\text{as}~{}t\to\infty.\end{split}$ (4.6)

Finally we observe that the mass function
$m(t):=\|\gamma(t)\|^{2}_{L^{2}(\mathbb{R}^{2})}=\theta^{2}(t)s^{2}(t)\|w\|^{2}_{L^{2}(\mathbb{R}^{2})}$
is strictly increasing and

$m(t)=(1+\theta^{*})^{2}s^{2}(t)\|w\|^{2}_{L^{2}(\mathbb{R}^{2})}\to\infty\qquad\text{as}~{}t\to\infty.$
(4.7)

Since for any $M>0$ we can deduce from (4.6) and (4.7) the existence of a
large constant $T=T(w,\delta,M)>0$ such that

$J_{\mu}(\gamma(T))<-1\qquad\text{and}\qquad m(T)>M,$

the continuous path $\gamma:[0,T]\to H^{1}(\mathbb{R}^{2})$ is a desired one
and this completes the proof of Lemma 4.1. $\square$

Proof of Theorem 1.6. To prove Item $(i)$, denoting by $w\in
H^{1}(\mathbb{R}^{N})$ an arbitrary nontrivial critical point of $J_{\mu}$, we
only need to show that

$J_{\mu}(w)\geq J_{\mu}(v)=E_{m}+\frac{1}{2}\mu m.$

For a fixed $\delta>0$ and $M:=m>0$, let $\gamma:[0,T]\to
H^{1}(\mathbb{R}^{N})$ be the continuous path given by Lemma 4.1. In view of
Lemma 4.1 $(i)$ and $(iii)$, there exists $t_{0}\in(0,T)$ such that

$\|\gamma(t_{0})\|^{2}_{L^{2}(\mathbb{R}^{N})}=m$

and thus

$\begin{split}J_{\mu}(w)=\max_{t\in[0,T]}J_{\mu}(\gamma(t))&\geq
J_{\mu}(\gamma(t_{0}))\\\
&=I(\gamma(t_{0}))+\frac{1}{2}\mu\int_{\mathbb{R}^{N}}|\gamma(t_{0})|^{2}dx\\\
&\geq E_{m}+\frac{1}{2}\mu m.\end{split}$

We now prove Item $(ii)$. In view of Item $(i)$, an arbitrary least action
solution $w\in H^{1}(\mathbb{R}^{N})$ of ($Q_{\mu}$) satisfies

$J_{\mu}(w)=A_{\mu}=E_{m}+\frac{1}{2}\mu m.$ (4.8)

Assume by contradiction that $\|w\|^{2}_{L^{2}(\mathbb{R}^{N})}\neq m$. Then,
for

$\delta:=\left|\sqrt{m}-\|w\|_{L^{2}(\mathbb{R}^{N})}\right|>0\qquad\text{and}\qquad
M:=m>0,$

we have the continuous path $\gamma:[0,T]\to H^{1}(\mathbb{R}^{N})$ given by
Lemma 4.1. Noting that by Lemma 4.1 $(iii)$ there exists $t_{0}\in(0,T)$ such
that

$\|\gamma(t_{0})\|^{2}_{L^{2}(\mathbb{R}^{N})}=m\qquad\text{and}\qquad\|\gamma(t_{0})-w\|_{L^{2}(\mathbb{R}^{N})}\geq\delta,$

it follows from Lemma 4.1 $(ii)$ a contradiction:

$\begin{split}J_{\mu}(w)&>J_{\mu}(\gamma(t_{0}))\\\
&=I(\gamma(t_{0}))+\frac{1}{2}\mu\int_{\mathbb{R}^{N}}|\gamma(t_{0})|^{2}dx\\\
&\geq E_{m}+\frac{1}{2}\mu m.\end{split}$

Since we have proved $\|w\|^{2}_{L^{2}(\mathbb{R}^{N})}=m$, it is easy to see
further that $I(w)=E_{m}$ by (4.8). $\square$

## Acknowledgements

The authors thank Prof. Thierry Cazenave for useful remarks on a preliminary
version of this work. Sheng-Sen Lu acknowledges the support of the China
Postdoctoral Science Foundation (No. 2020M680174) and the National Natural
Science Foundation of China (Nos. 11771324 and 11831009).

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