# Normalized solutions to Schrödinger equations in the strongly sublinear
regime

Jarosław Mederski  and  Jacopo Schino
Institute of Mathematics,
Polish Academy of Sciences,
ul. Śniadeckich 8, 00-656 Warsaw, Poland<EMAIL_ADDRESS>
Department of Mathematics
North Carolina State University
2311 Stinson Drive, 27607 Raleigh, NC, USA
and
Faculty of Mathematics, Informatics and Mechanics
University of Warsaw
ul. Banacha 2, 02-097, Warsaw, Poland<EMAIL_ADDRESS>
###### Abstract.

We look for solutions to the Schrödinger equation

$-\Delta u+\lambda u=g(u)\quad\text{in }\mathbb{R}^{N}$

coupled with the mass constraint $\int_{\mathbb{R}^{N}}|u|^{2}\,dx=\rho^{2}$,
with $N\geq 2$. The behaviour of $g$ at the origin is allowed to be strongly
sublinear, i.e., $\lim_{s\to 0}g(s)/s=-\infty$, which includes the case

$g(s)=\alpha s\ln s^{2}+\mu|s|^{p-2}s$

with $\alpha>0$ and $\mu\in\mathbb{R}$, $2<p\leq 2^{*}$ properly chosen. We
consider a family of approximating problems that can be set in
$H^{1}(\mathbb{R}^{N})$ and the corresponding least-energy solutions, then we
prove that such a family of solutions converges to a least-energy one to the
original problem. Additionally, under certain assumptions about $g$ that allow
us to work in a suitable subspace of $H^{1}(\mathbb{R}^{N})$, we prove the
existence of infinitely many solutions.

MSC 2010: 35J10, 35J20, 35J61, 58E05

## Introduction

We look for solutions $(u,\lambda)\in H^{1}(\mathbb{R}^{N})\times\mathbb{R}$
to the following equation

(1.1) $-\Delta u+\lambda u=g(u)\quad\mathrm{in}\ \mathbb{R}^{N},\ N\geq 2$

paired with the constraint

(1.2) $\int_{\mathbb{R}^{N}}|u|^{2}\,dx=\rho^{2},$

where $\rho>0$ is fixed and $g$ satisfies suitable assumptions (see (g0)–(g4)
below). Equations like (1.1) arise when one looks for standing-wave solutions
to the evolution equation

(1.3) $\mathrm{i}\partial_{t}\Psi+\Delta\Psi+g(\Psi)=0,$

i.e., $\Psi(t,x)=e^{\mathrm{i}\lambda t}u(x)$. The condition (1.2), instead,
is motivated by the property that, formally, the quantity

$\int_{\mathbb{R}^{N}}|\Psi(t,x)|^{2}\,dx$

is conserved in time. Moreover, the $L^{2}$-norm squared of a solution to
(1.3) has a precise physical meaning in the contexts which this equation is
derived from: it represents, e.g., the power supply in nonlinear optics and
the total number of atoms in the Bose–Einstein condensation. The model for $g$
that inspires this paper is

(1.4) $g(s)=s\ln s^{2},$

which was introduced in [4] to obtain the separability of non-interacting
subsystems; observe that, in this case,

$\lim_{s\to 0}\frac{g(s)}{s}=-\infty,$

which is known in the literature as the strongly sublinear (or infinite-mass)
regime. After that, both (1.1) – possibly with an external potential – and
(1.3), in both cases with $g$ as in (1.4), raised great interest: we mention
[13, 31, 33, 34, 35] for the former context and [8, 9, 11, 14] (see also [10,
Section 9.3]) for the latter one. The common feature of most of the articles
above is that the authors can work in the subspace of $H^{1}(\mathbb{R}^{N})$

${\mathcal{H}}:=\left\\{u\in
H^{1}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}u^{2}|\ln
u^{2}|\,dx<\infty\right\\},$

where the energy functional (see (1.7) below) is of class ${\mathcal{C}}^{1}$
(we mention that, instead, the approach of [13] is based on critical point
theory for lower semi-continuous functionals, while that of [14] on a
truncation argument). Of course, this strategy is impossible with generic
nonlinearities as in this paper; speaking of which, generic nonlinear terms
were recently considered in [15, 24].

Concerning normalized solutions, i.e., in the presence of (1.2), to the best
of our knowledge, they are only treated in two very recent pieces of work: in
[30], with

(1.5) $g(s)=\alpha s\ln s^{2}+\mu|s|^{p-2}s,$

where $\alpha,\mu\in\mathbb{R}$ and $2<p\leq 2^{*}$, and in [37], which mainly
concerns systems of two equations. It is worth mentioning that, when $g$ is as
in (1.4), if $(v,0)$ solves (1.1) and $v\neq 0$, then $(u,\lambda)$ given by

$u=\frac{\rho}{|v|_{2}}v,\quad\lambda=2\ln\left(\frac{\rho}{|v|_{2}}\right)$

is a solution to (1.1)–(1.2). Nonetheless, the approach of [30, 37] still
makes use of ${\mathcal{H}}$. In particular, to recover compactness, the
authors consider the subspace of radially symmetric functions as in [9, 11],
which embeds compactly into $L^{q}(\mathbb{R}^{N})$ for every $q\in[2,2^{*})$;
this is different from the classical result concerning the subspace of radial
functions in $H^{1}(\mathbb{R}^{N})$, which embeds compactly into
$L^{q}(\mathbb{R}^{N})$ only for $q\in(2,2^{*})$. The additional compact
embedding into $L^{2}(\mathbb{R}^{N})$ proves to be extremely useful owing to
the constraint (1.2); in particular, this is what allows the authors to work
with every $\rho>0$ when $2<p<2+4/N$, where $p$ is the same as in (1.5). Once
again, here we cannot utilize the compact embedding into
$L^{2}(\mathbb{R}^{N})$ because $g$ need not be given by (1.5).

The function $g$ in (1.1) satisfies the following assumptions. Define
$G(s)=\int_{0}^{s}g(t)\,dt$ and

(1.6)
$G_{+}(s)=\begin{cases}\displaystyle\int_{0}^{s}\max\\{g(t),0\\}\,dt&\text{if
}s\geq 0,\\\ \displaystyle\int_{s}^{0}\max\\{-g(t),0\\}\,dt&\text{if
}s<0.\end{cases}$

  * (g0)

$g\colon\mathbb{R}\to\mathbb{R}$ is continuous and $g(0)=0$.

  * (g1)

$\lim_{s\to 0}G_{+}(s)/|s|^{2}=0$ and $\limsup_{s\to 0}g_{+}(s)/s<\infty$,
where $g_{+}=G_{+}^{\prime}$.

  * (g2)

If $N\geq 3$, then $\limsup_{|s|\to\infty}|g(s)|/|s|^{2^{*}-1}<\infty$, where
$2^{*}=\frac{2N}{N-2}$.
If $N=2$, then $\lim_{|s|\to\infty}g(s)/e^{\alpha s^{2}}=0$ for all
$\alpha>4\pi$.

  * (g3)

$\lim_{|s|\to\infty}G_{+}(s)/|s|^{2+4/N}=0$.

  * (g4)

There exists $\xi_{0}\neq 0$ such that $G(\xi_{0})>0$.

The assumptions (g0), (g2), and (g4) are classical in the context of elliptic
problems (cf. [2, 3]), while (g1) is a weaker version of what is normally
assumed in the presence of $L^{2}$-constraints, i.e., $\lim_{s\to 0}g(s)/s=0$;
(g3) indicates that the growth of $G_{+}$ is mass-subcritical at infinity.
With these assumptions, we can deal with nonlinearities that are much more
general than (1.5); for example, we can consider functions $g$ that behave
like $-|s|^{\omega-1}s$ or $|s|^{\omega-1}s\ln s^{2}$ for $|s|$ small, with
$0<\omega<1$. We can also cover the case when $g$ goes to $0$ more slowly than
any power, e.g., when $g$ behaves like $1/\ln s^{2}$ for $|s|$ small.

Let us define

${\mathcal{D}}:=\left\\{u\in
H^{1}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}|u|^{2}\,dx\leq\rho^{2}\right\\}\quad\text{and}\quad{\mathcal{S}}:=\left\\{u\in
H^{1}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}|u|^{2}\,dx=\rho^{2}\right\\}.$

Sometimes, we will write explicitly ${\mathcal{D}}(\rho)$ and
${\mathcal{S}}(\rho)$. Let also $J\colon
H^{1}(\mathbb{R}^{N})\to\mathbb{R}\cup\\{\infty\\}$ be the energy functional
associated with (1.1)–(1.2), i.e.,

(1.7) $J(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}\,dx-\int_{\mathbb{R}^{N}}G(u)\,dx.$

Our first result is the existence of a solution to (1.1)–(1.2), with
additional properties. We point out that by _solution_ to (1.1) we mean a pair
$(u,\lambda)\in H^{1}(\mathbb{R}^{N})\times\mathbb{R}$ such that
$J^{\prime}(u)v=-\lambda uv$ for every
$v\in{\mathcal{C}}^{\infty}_{0}(\mathbb{R}^{N})$.

######  Theorem 1.1.

If $g$ satisfies (g0)–(g4), then there exists $\bar{\rho}>0$ such that for
every $\rho>\bar{\rho}$ there exist $\lambda>0$ and $u\in{\mathcal{S}}$ such
that $J(u)=\min_{{\mathcal{D}}}J<0$ and $(u,\lambda)$ is a solution to
(1.1)–(1.2). Moreover, $u$ has constant sign and is, up to a translation,
radial and radially monotonic.

######  Remark 1.2.

Of course, (g0)–(g4) include the classical case $\lim_{s\to 0}g(s)/s=0$, which
implies that $J$ is of class ${\mathcal{C}}^{1}$ in $H^{1}(\mathbb{R}^{N})$.
In this case, with the notations of Theorem 1.1, $u$ is actually a critical
point of $J|_{\mathcal{S}}$. In addition, if $\lim_{s\to
0}G(s)/|s|^{2+4/N}=\infty$ (i.e., in the mass-subcritical regime), then
$\bar{\rho}=0$ in Theorem 1.1 (see, e.g., [28] for more details). An important
example in this context is given by nonlinear media with _saturation_ [18],
where $g(s)=s^{3}/(1+s^{2})$.

######  Remark 1.3.

Under the assumptions (g0)–(g4), if there exists a solution $(u,\lambda)\in
H^{1}(\mathbb{R}^{N})\times\mathbb{R}$ to (1.1)–(1.2) with $J(u)\leq 0$, then
$\lambda>0$. As a matter of fact, $J(u)\leq 0$ implies that
$\int_{\mathbb{R}^{N}}G(u)\,dx$ is finite, thus from [24, Proposition 3.1]
$(u,\lambda)$ satisfies the Pohožaev identity
$(N-2)\int_{\mathbb{R}^{N}}|\nabla
u|^{2}\,dx=2N\int_{\mathbb{R}^{N}}G(u)-\frac{1}{2}\lambda|u|^{2}\,dx$ and,
consequently, $0\geq J(u)=\frac{1}{N}|\nabla
u|_{2}^{2}-\frac{\lambda}{2}\rho^{2}$.

Our next result concerns the limit of the ground state energy map
$\rho\mapsto\inf_{{\mathcal{D}}(\rho)}J$ as $\rho\to\infty$.

######  Proposition 1.4.

If $g$ satisfies (g0)–(g4), then

$\lim_{\rho\to\infty}\inf_{{\mathcal{D}}(\rho)}J=-\infty.$

Moreover, we obtain the relative compactness in $H^{1}(\mathbb{R}^{N})$ of
minimizing sequences for $\inf_{{\mathcal{D}}(\rho)}J$ up to translations.
Here and in the sequel, with a small abuse of notations, if $(u_{n})_{n}$ is a
sequence in $X$, we write $u_{n}\in X$ instead of $(u_{n})_{n}\subset X$.

######  Theorem 1.5.

If (g0)–(g4) hold, then there exists $\bar{\rho}>0$ such that for every
$\rho>\bar{\rho}$, if $0<\rho_{n}\to\rho$ and
$u_{n}\in{\mathcal{D}}(\rho_{n})$ is such that
$J(u_{n})\to\inf_{{\mathcal{D}}(\rho)}J$, then $u_{n}$ is relatively compact
in $H^{1}(\mathbb{R}^{N})$ up to translations.

It is well known that such a relative compactness is an important step toward
the orbital stability of solutions to (1.1)–(1.2). Unfortunately, we currently
do not have the necessary tools to investigate the dynamics of (1.3) in this
general setting, hence this issue is postponed to future work. Moreover,
unlike the classical approach when $J$ is of class ${\mathcal{C}}^{1}$, where
the existence of a constrained minimizer for $J$ and, consequently, a solution
to (1.1)–(1.2) is a by-product of the aforementioned relative compactness,
here we prove first the existence of a solution to (1.1)–(1.2) and then use
this fact to prove the compactness result.

When $g$ is as in (1.5), we additionally have the following non-existence
result.

######  Theorem 1.6.

Let $g$ be given by (1.5) with $\alpha>0$, $\mu\in\mathbb{R}$, and $2<p\leq
2^{*}$ ($p>2$ if $N=2$).

(i) If $\rho\gg 1$ and either $-\frac{\alpha p}{p-2}e^{-p/2}<\mu\leq 0$, or
$\mu>0$ and $2<p<2+4/N$, then there exists a solution $(u,\lambda)\in
H^{1}(\mathbb{R}^{N})\times(0,\infty)$ to (1.1)–(1.2) such that $G(u)\in
L^{1}(\mathbb{R}^{N})$ and $J(u)=\min_{{\mathcal{D}}}J<0$.

(ii) If $\mu\leq-\frac{\alpha p}{p-2}e^{-p/2}$, then there exist no solutions
$(u,\lambda)\in H^{1}(\mathbb{R}^{N})\times[0,\infty)$ to (1.1)–(1.2) such
that $G(u)\in L^{1}(\mathbb{R}^{N})$.

######  Remark 1.7.

In Theorem 1.1, we look for (ground state) solutions with sufficiently large
$\rho$ because this implies that the energy is negative, which is used, in
turn, to ensure such an existence. Then, as we pointed out in Remark 1.3, a
consequence is that $\lambda>0$. On the other hand, in the context of Theorem
1.6 (i) and $\mu>0$, ground state solutions still exist for all $\rho>0$ as in
Corollary 1.9 below, but with no information about the Lagrange multiplier or
the energy.

We follow the approach of [24] and consider a family of approximating
problems. Note that the approximations are different from those in [15], where
the nonlinearity is modified in a neighbourhood of infinity, and are more
similar to those in [14], although not the same as the nonlinearity therein is
of the form (1.4).

Let $g_{-}(s):=g_{+}(s)-g(s)$ and $G_{-}(s):=G_{+}(s)-G(s)\geq 0$ for
$s\in\mathbb{R}$. In view of (g1) and (g3), $G_{+}(u)\in
L^{1}(\mathbb{R}^{N})$ for $u\in H^{1}(\mathbb{R}^{N})$; however, $G_{-}(u)$
may not be integrable unless $G_{-}(s)\lesssim|s|^{2}$ for small $|s|$.
Moreover, for any $u\in H^{1}(\mathbb{R}^{N})$, $J$ is differentiable at $u$
along functions in ${\mathcal{C}}_{0}^{\infty}(\mathbb{R}^{N})$, but it is not
of class ${\mathcal{C}}^{1}$ in general. In order to overcome this problem,
for every $\varepsilon\in(0,1)$ let us take an even function
$\varphi_{\varepsilon}\colon\mathbb{R}\to[0,1]$ such that
$\varphi_{\varepsilon}(s)=|s|/\varepsilon$ for $|s|\leq\varepsilon$,
$\varphi_{\varepsilon}(s)=1$ for $|s|\geq\varepsilon$. We introduce a new
functional,

(1.8) $J_{\varepsilon}(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}\,dx+\int_{\mathbb{R}^{N}}G_{-}^{\varepsilon}(u)\,dx-\int_{\mathbb{R}^{N}}G_{+}(u)\,dx,$

such that
$G_{-}^{\varepsilon}(s)=\int_{0}^{s}\varphi_{\varepsilon}(t)g_{-}(t)\,dt$,
$s\in\mathbb{R}$, and now observe that $G_{-}^{\varepsilon}(s)\leq
c_{\varepsilon}|s|^{2}$ for every $|s|\leq 1$ and some constant
$c_{\varepsilon}>0$ depending only on $\varepsilon>0$. Hence, for
$\varepsilon\in(0,1)$, $J_{\varepsilon}$ is of class ${\mathcal{C}}^{1}$ on
$H^{1}(\mathbb{R}^{N})$.

The idea of considering the $L^{2}$-disc ${\mathcal{D}}$ instead of the
$L^{2}$-sphere ${\mathcal{S}}$ was introduced in [5] in a context where the
nonlinear term is mass-supercritical and Sobolev-subcritical at infinity
(i.e., $\lim_{|s|\to\infty}G(s)/|s|^{2+4/N}=\infty$,
$\lim_{|s|\to\infty}G(s)/|s|^{2^{*}}=0$) and mass-critical or -supercritical
at the origin (i.e., $\limsup_{s\to 0}G(s)/|s|^{2+4/N}<\infty$); later on, it
was exploited again in [25], in a similar context that allows systems with
Sobolev-critical nonlinearities, in [28], for equations and systems of
equations in the mass-critical or -subcritical setting, and in [6], where
polyharmonic operators and Hardy-type potentials are considered. One of the
reasons for this choice was that the embedding
$H^{1}(\mathbb{R}^{N})\hookrightarrow L^{2}(\mathbb{R}^{N})$ is not compact,
even if considering radial functions or with symmetries as in [20], which, in
turn, causes that the limit point of a weakly convergent sequence in
${\mathcal{S}}$ need not belong to ${\mathcal{S}}$; on the contrary, limit
points of weakly convergent sequences in ${\mathcal{D}}$ stay in
${\mathcal{D}}$, and the additional information that such a weak limit point
belongs to the set one is working with proves to be useful. In this paper,
this strategy is fundamental: it allows the critical point found in Theorem
2.1 below – the existence result for the perturbed problem – to be a minimizer
of the corresponding energy functional over the whole set ${\mathcal{D}}$, and
this fact plays an important role in the proof of Theorem 1.1 because the
“candidate minimizer” of $J|_{\mathcal{D}}$ might not, initially, belong to
${\mathcal{S}}$.

Now, we turn to the problem of finding multiple solutions (in fact, infinitely
many) to (1.1)–(1.2). In this case, we need to change both the assumptions
about $g$ and our approach. The idea is to build a subspace of
$H^{1}(\mathbb{R}^{N})$ in the spirit of ${\mathcal{H}}$, but with a more
general setting. The issues with using the previous approach to obtain an
infinite sequence of solutions are mainly two. First, even at the level of the
perturbed functional, one can obtain an arbitrary large number of solutions if
$\rho$ is sufficiently large, but it is unclear whether infinitely many
solutions exist; this is caused by the lack of compact embedding into
$L^{2}(\mathbb{R}^{N})$, which in turn raises the need for each of the usual
minimax values (cf. Theorem 4.9 below) to be negative, for which we need
$\rho$ to be large (and how large depends differently on each minimax level).
Second, fixed a family (depending on $\varepsilon$) of critical values of the
perturbed functional obtained in the way we have just briefly mentioned, and
considering the corresponding family of critical points, it is not clear
whether this family converges to a critical point of the unperturbed
functional, unlike what happens in the Proof of Theorem 1.1, where we can
exploit the information that the critical levels are minima.

We assume that the right-hand side in (1.1) is given by

(1.9) $g(s)=f(s)-a(s)$

together with the following assumptions, where $F(s)=\int_{0}^{s}f(t)\,dt$,
$A(s)=\int_{0}^{s}a(t)\,dt$, $F_{+}$ is defined via (1.6) replacing $g$ with
$f$, and $f_{+}=F_{+}^{\prime}$.

  * (A)

$A\in{\mathcal{C}}^{1}(\mathbb{R})$ is an N-function that satisfies the
$\Delta_{2}$ and $\nabla_{2}$ conditions globally and such that $\lim_{s\to
0}A(s)/s^{2}=\infty$ and $s\mapsto a(s)s$ is convex.

  * (f0)

$f\colon\mathbb{R}\to\mathbb{R}$ is continuous and odd.

  * (f1)

$\lim_{s\to 0}f_{+}(s)/s=0$.

  * (f2)

If $N\geq 3$, then $|f(s)|\lesssim a(s)+|s|+|s|^{2^{*}-1}$;
If $N=2$, then for all $q\geq 2$ and $\alpha>4\pi$ there holds $|f(s)|\lesssim
a(s)+|s|+|s|^{q-1}\left(e^{\alpha s^{2}}-1\right)$.

  * (f3)

$\limsup_{|s|\to\infty}f_{+}(s)/|s|^{1+4/N}<\infty$.

Observe that (f3) implies

(1.10) $\eta:=\limsup_{|s|\to\infty}\frac{F_{+}(s)}{|s|^{2+4/N}}<\infty.$

Let us define

$W:=\left\\{u\in H^{1}(\mathbb{R}^{N}):A(u)\in L^{1}(\mathbb{R}^{N})\right\\}$

and, for a subgroup ${\mathcal{O}}\subset{\mathcal{O}}(N)$,

$W_{{\mathcal{O}}}:=\left\\{u\in W:u(g\cdot)=u\text{ for all
}g\in{\mathcal{O}}\right\\}.$

If $N=4$ or $N\geq 6$, with the purpose of finding non-radial solutions (cf.
[1, 23]), fix $2\leq M\leq N/2$ such that $N-2M\neq 1$ and let
$\tau(x)=(x_{2},x_{1},x_{3})$ for
$x=(x_{1},x_{2},x_{3})\in\mathbb{R}^{M}\times\mathbb{R}^{M}\times\mathbb{R}^{N-2M}$
($x_{3}$ is removed form the definition of $\tau$ if $N=2M$). Then, let

${\mathcal{X}}:=\left\\{u\in
W_{{\mathcal{O}}(M)\times{\mathcal{O}}(M)\times{\mathcal{O}}(N-2M)}:u\circ\tau=-u\right\\}$

(${\mathcal{O}}(N-2M)$ is removed from the definition of ${\mathcal{X}}$ if
$N=2M$) and observe that ${\mathcal{X}}\cap W_{{\mathcal{O}}(N)}=\\{0\\}$.

For $2<p<2^{*}$, let us denote by $C_{N,p}$ the best constant in the
Gagliardo–Nirenberg inequality:

(1.11) $|u|_{p}\leq C_{N,p}|\nabla
u|_{2}^{N(1/2-1/p)}|u|_{2}^{1-N(1/2-1/p)}\quad\forall\,u\in
H^{1}(\mathbb{R}^{N}).$

Our multiplicity result reads as follows.

######  Theorem 1.8.

Let $g$ be given by (1.9) such that (A), (f0)–(f3), and

(1.12) $2\eta C_{N,2+4/N}^{2+4/N}\rho^{4/N}<1$

hold, where $\eta$ is defined in (1.10). Then there exist infinitely many
solutions $(u,\lambda)\in W_{{\mathcal{O}}(N)}\times\mathbb{R}$ to
(1.1)–(1.2), one of which – say, $(\bar{u},\bar{\lambda})$ – having the
property that $J(\bar{u})=\min_{{\mathcal{S}}\cap W}J$. If, in addition, $N=4$
or $N\geq 6$, then there exist infinitely many solutions
$(v,\mu)\in{\mathcal{X}}\times\mathbb{R}$, one of which – say,
$(\bar{v},\bar{\mu})$ – having the property that
$J(\bar{v})=\min_{{\mathcal{S}}\cap{\mathcal{X}}}J$.

By setting

$A(s):=\begin{cases}-\alpha s^{2}\ln s^{2}&\text{if }0\leq|s|<e^{-3},\\\
3\alpha s^{2}+4\alpha e^{-3}|s|-\alpha e^{-6}&\text{if }|s|\geq
e^{-3},\end{cases}\qquad F(s)=\alpha s^{2}\ln
s^{2}+A(s)+\frac{\mu}{p}|s|^{p},$

with $\alpha>0$, and $\mu,p$ in the right range, we check that (A) and
(f0)–(f3) hold and we recover the case (1.5).

######  Corollary 1.9.

Let $g$ be given by (1.5) with $\alpha>0$, $\mu\geq 0$, and $2<p\leq 2+4/N$.
If (1.12) holds, then there exists a solution $(u,\lambda)\in
W\times\mathbb{R}$ to (1.1)–(1.2) such that $J(u)=\min_{{\mathcal{S}}\cap
W}J$.

In particular, we get a solution to (1.1)–(1.2) with (1.5) for any $\rho>0$
provided that $2<p<2+4/N$. Moreover, (f3) allows us to handle terms $g$ that
grow “too fast” to be included in the usual $H^{1}(\mathbb{R}^{N})$ setting.
For example, letting

$A(s):=\begin{cases}\displaystyle-\alpha s^{2}\ln s^{2}&\text{if
}0\leq|s|<e^{-3},\\\
\displaystyle\frac{10\alpha}{p}e^{3p-6}|s|^{p}+2\alpha\left(3-\frac{5}{p}\right)e^{-6}&\text{if
}|s|\geq e^{-3},\end{cases}$

with $\alpha>0$, we can allow $f_{-}(s)\approx|s|^{p-2}s$ for $|s|\gg 1$ for
every $p>2$, even when $N\geq 3$ and $p>2^{*}$. At the same time, we can treat
nonlinearities that are “more singular” at the origin, for example, letting

$A(s):=\frac{1}{q}|s|^{q},\quad 1<q<2,$

and taking any $f$ that satisfies (f0)–(f3). The price to pay for this
different approach is that we need (1.9) and (A) to construct a proper
functional setting (i.e., an Orlicz one); this prevents us from dealing, e.g.,
with terms that behave like $1/\ln s^{2}$ near the origin, which instead are
allowed by the use of the approximating problems to find least-energy
solutions. On the other hand, we can weaken the assumptions about $f$, the
“remaining” part of the nonlinearity: it can be controlled by $a$ instead of
the usual conditions with an $H^{1}(\mathbb{R}^{N})$-setting (in particular,
we can deal with Sobolev-supercritical nonlinearities), we do not need $F>A$
anywhere, $f_{+}$ can have a mass-critical growth at infinity (provided $\rho$
is small), and there are no restrictions on $\rho$ if $f_{+}$ has a mass-
subcritical growth at infinity; none of these scenarios can be handled with
the previous approach. Except for [30, Theorems 1.2 and 1.3], where the
nonlinear term has the very specific form (1.5), Theorem 1.8 seems to be the
first multiplicity result when normalized solutions are sought in the strongly
sublinear regime, and also the first time at all that more than two solutions
are obtained.

The paper is structured al follows: in Section 2, we obtain least-energy
solutions to the perturbed problem; in Section 3, we prove Theorems 1.1, 1.5,
1.6, and Proposition 1.4; in Section 4, we obtain infinitely many solutions to
(1.1)–(1.2).

##  2\. Solutions to the perturbed problem

Recall from (1.8) the definition of $J_{\varepsilon}\colon
H^{1}(\mathbb{R}^{N})\to\mathbb{R}$. The main purpose of this section is to
prove the following result, where we denote
$c_{\varepsilon}(\rho):=\inf_{{\mathcal{D}}(\rho)}J_{\varepsilon}$.

######  Theorem 2.1.

Let $g$ satisfy (g0)–(g4). Then for every $\beta>0$ there exists
$\bar{\rho}>0$ such that for every $\rho>\bar{\rho}$ and $\varepsilon>0$ there
exist $\lambda_{\varepsilon}>0$ and $u_{\varepsilon}\in H^{1}(\mathbb{R}^{N})$
such that $J_{\varepsilon}(u_{\varepsilon})=c_{\varepsilon}(\rho)<-\beta$ and

(2.1) $\begin{cases}-\Delta
u_{\varepsilon}+\lambda_{\varepsilon}u_{\varepsilon}=g_{+}(u_{\varepsilon})-\varphi_{\varepsilon}(u_{\varepsilon})g_{-}(u_{\varepsilon})\quad\text{in
}\mathbb{R}^{N}\\\
\int_{\mathbb{R}^{N}}|u_{\varepsilon}|^{2}\,dx=\rho^{2}.\end{cases}$

Moreover, every such $u_{\varepsilon}$ has constant sign and, up to a
translation, is radial and radially monotonic.

We begin by proving a series of lemmas.

######  Lemma 2.2.

If $g$ satisfies (g0)–(g3), then $J_{\varepsilon}|_{\mathcal{D}}$ is coercive
and bounded below.

###### Proof.

From (g1) and (g3), for every $\delta>0$ there exists $C_{\delta}>0$ such that
for every $s\in\mathbb{R}$

$G_{+}(s)\leq C_{\delta}|s|^{2}+\delta|s|^{2+4/N}.$

Consequently, from (1.11), for every $u\in{\mathcal{D}}$ there holds

$\begin{split}J_{\varepsilon}(u)&\geq\int_{\mathbb{R}^{N}}\frac{1}{2}|\nabla
u|^{2}-G_{+}(u)\,dx\geq\frac{1}{2}|\nabla
u|_{2}^{2}-C_{\delta}|u|_{2}^{2}-\delta|u|_{2+4/N}^{2+4/N}\\\
&\geq\frac{1}{2}|\nabla u|_{2}^{2}-C_{\delta}\rho^{2}-\delta
C_{N,2+4/N}^{2+4/N}\rho^{4/N}|\nabla u|_{2}^{2}.\end{split}$

Taking $\delta<\left(2C_{N,2+4/N}^{2+4/N}\rho^{4/N}\right)^{-1}$ we conclude.
∎

######  Lemma 2.3.

If $g$ satisfies (g0)–(g2) and (g4), then for every $\beta>0$ there exists
$\bar{\rho}>0$ such that

$c_{\varepsilon}(\rho)<-\beta$

for every $\rho>\bar{\rho}$ and every $\varepsilon\in(0,1)$.

###### Proof.

Arguing as in [2, Proof of Theorem 2] and observing that
$G^{\varepsilon}:=G_{+}-G_{-}^{\varepsilon}\geq G$, there exists $u\in
H^{1}(\mathbb{R}^{N})$ such that
$\int_{\mathbb{R}^{N}}G^{\varepsilon}(u)\,dx\geq 1$ for every
$\varepsilon\in(0,1)$. Define
$u_{\rho}:=u(|u|_{2}^{2/N}\rho^{-2/N}\cdot)\in{\mathcal{S}}(\rho)$. There
holds

$\begin{split}J_{\varepsilon}(u_{\rho})&=\int_{\mathbb{R}^{N}}\frac{1}{2}|\nabla
u_{\rho}|^{2}-G^{\varepsilon}(u_{\rho})\,dx=\frac{|\nabla
u|_{2}^{2}}{2|u|_{2}^{2(N-2)}}\rho^{2-4/N}-\frac{1}{|u|_{2}^{2}}\int_{\mathbb{R}^{N}}G^{\varepsilon}(u)\,dx\rho^{2}\\\
&\leq\frac{|\nabla
u|_{2}^{2}}{2|u|_{2}^{2(N-2)}}\rho^{2-4/N}-\frac{1}{|u|_{2}^{2}}\rho^{2}\to-\infty\end{split}$

as $\rho\to\infty$, whence the statement. ∎

######  Lemma 2.4.

If (g0)–(g3) hold, then for every $\rho_{1},\rho_{2}>0$ there holds
$c_{\varepsilon}(\sqrt{\rho_{1}^{2}+\rho_{2}^{2}})\leq
c_{\varepsilon}(\rho_{1})+c_{\varepsilon}(\rho_{2})$.

###### Proof.

Fix $\delta>0$. From the definition of $c_{\varepsilon}$, which is finite due
to Lemma 2.2, there exist
$u_{1},u_{2}\in{\mathcal{C}}_{0}^{\infty}(\mathbb{R}^{N})$ such that
$|u_{i}|_{2}\leq\rho_{i}$ and $J_{\varepsilon}(u_{i})\leq
c_{\varepsilon}(\rho_{i})+\delta$, $i=1,2$. In virtue of the translation
invariance, we can assume that $u_{1}$ and $u_{2}$ have disjoint supports.
Then
$|u_{1}+u_{2}|_{2}^{2}=|u_{1}|_{2}^{2}+|u_{2}|_{2}^{2}\leq\rho_{1}^{2}+\rho_{2}^{2}$
and so

$c_{\varepsilon}\left(\sqrt{\rho_{1}^{2}+\rho_{2}^{2}}\right)\leq
J_{\varepsilon}(u_{1}+u_{2})\leq
c_{\varepsilon}(\rho_{1})+c_{\varepsilon}(\rho_{2})+2\delta.\qed$

The following lemma is inspired from [29, Lemma 3.2 (ii)].

######  Lemma 2.5.

Assume (g0)–(g3) hold and let $\rho_{1},\rho_{2}>0$. If there exist
$u_{i}\in{\mathcal{D}}(\rho_{i})$ such that
$J_{\varepsilon}(u_{i})=c_{\varepsilon}(\rho_{i})$, $i=1,2$, and
$(u_{1},u_{2})\neq(0,0)$, then
$c_{\varepsilon}(\sqrt{\rho_{1}^{2}+\rho_{2}^{2}})<c_{\varepsilon}(\rho_{1})+c_{\varepsilon}(\rho_{2})$.

###### Proof.

For every $s\geq 1$ and $i=1,2$ there holds

$c_{\varepsilon}(\sqrt{s}\rho_{i})\leq
J_{\varepsilon}\bigl{(}u_{i}(s^{-1/N}\cdot)\bigr{)}=s\left(\frac{1}{2s^{2/N}}\int_{\mathbb{R}^{N}}|\nabla
u_{i}|^{2}\,dx-\int_{\mathbb{R}^{N}}G^{\varepsilon}(u_{i})\,dx\right)\leq
sJ_{\varepsilon}(u_{i})=sc_{\varepsilon}(\rho_{i}).$

Moreover, if $s>1$ and $u_{i}\neq 0$, then
$c_{\varepsilon}(\sqrt{s}\rho_{i})<sc_{\varepsilon}(\rho_{i})$. Let us assume,
without loss of generality, that $u_{1}\neq 0$. If $\rho_{1}\geq\rho_{2}$,
then

$c_{\varepsilon}\left(\sqrt{\rho_{1}^{2}+\rho_{2}^{2}}\right)<\frac{\rho_{1}^{2}+\rho_{2}^{2}}{\rho_{1}^{2}}c_{\varepsilon}(\rho_{1})=c_{\varepsilon}(\rho_{1})+\frac{\rho_{2}^{2}}{\rho_{1}^{2}}c_{\varepsilon}(\rho_{1})\leq
c_{\varepsilon}(\rho_{1})+c_{\varepsilon}(\rho_{2}).$

If $\rho_{1}<\rho_{2}$, then

$c_{\varepsilon}\left(\sqrt{\rho_{1}^{2}+\rho_{2}^{2}}\right)\leq\frac{\rho_{1}^{2}+\rho_{2}^{2}}{\rho_{2}^{2}}c_{\varepsilon}(\rho_{2})=\frac{\rho_{1}^{2}}{\rho_{2}^{2}}c_{\varepsilon}(\rho_{2})+c_{\varepsilon}(\rho_{2})<c_{\varepsilon}(\rho_{1})+c_{\varepsilon}(\rho_{2}).\qed$

######  Remark 2.6.

(a) Lemmas 2.4 and 2.5 still hold if $J_{\varepsilon}$ is replaced with $J$.
(b) As proved in [28, Lemma 2.5 (ii)], Lemma 2.5 holds under the weaker
assumption that $J_{\varepsilon}(u_{1})=c_{\varepsilon}(\rho_{1})$ or
$J_{\varepsilon}(u_{2})=c_{\varepsilon}(\rho_{2})$.

######  Lemma 2.7.

Let $g$ satisfy (g0)–(g4). Then, for every $\beta>0$, there exists
$\bar{\rho}>0$ such that for every $\rho>\bar{\rho}$ and $\varepsilon>0$ we
have $c_{\varepsilon}(\rho)<-\beta$ and, if $0<\rho_{n}\to\rho$ and
$u_{n}\in{\mathcal{D}}(\rho_{n})$ is such that $J_{\varepsilon}(u_{n})\to
c_{\varepsilon}(\rho)$, then there exists
$u\in{\mathcal{D}}(\rho)\setminus\\{0\\}$ such that, up to translations,
$u_{n}\to u$ in $L^{p}(\mathbb{R}^{N})$, $2<p<2^{*}$.

###### Proof.

Let $\bar{\rho}>0$ be determined by Lemma 2.3 and consider $u_{n}$ and
$\rho_{n}$ as in the statement. Then $u_{n}$ is bounded in view of Lemma 2.2.
If

$\lim_{n}\max_{y\in\mathbb{R}^{N}}\int_{B_{1}(y)}|u_{n}|^{2}\,dx=0,$

then $\lim_{n}|u_{n}|_{p}=0$ for every $2<p<2^{*}$ due to Lions’ lemma [21,
Lemma I.1 in Part 2], hence, taking into account (g1) and (g3),
$\lim_{n}\int_{\mathbb{R}^{N}}G_{+}(u_{n})\,dx=0$ and, consequently,
$c_{\varepsilon}(\rho)=\lim_{n}J_{\varepsilon}(u_{n})\geq 0$, a contradiction
with Lemma 2.3. Therefore, there exist $y_{n}\in\mathbb{R}^{N}$ and
$u\in{\mathcal{D}}(\rho)\setminus\\{0\\}$ such that, up to a subsequence,
$u_{n}(\cdot-y_{n})\rightharpoonup u$ in $H^{1}(\mathbb{R}^{N})$ and
$u_{n}(\cdot-y_{n})\to u$ a.e. in $\mathbb{R}^{N}$. Let us denote
$v_{n}(x):=u_{n}(x-y_{n})-u(x)$. In virtue of the Brezis–Lieb lemma [7,
Theorem 2],

$\lim_{n}J_{\varepsilon}(u_{n})-J_{\varepsilon}(v_{n})=J_{\varepsilon}(u).$

We want to prove that

$\lim_{n}\max_{y\in\mathbb{R}^{N}}\int_{B_{1}(y)}|v_{n}|^{2}\,dx=0,$

which, as before, will yield the statement of the Lemma. If this is not the
case, then, as before, there exist $z_{n}\in\mathbb{R}^{N}$ and $v\in
H^{1}(\mathbb{R}^{N})\setminus\\{0\\}$ such that, denoting
$w_{n}(x):=v_{n}(x-z_{n})-v(x)$, we have $w_{n}\rightharpoonup 0$ in
$H^{1}(\mathbb{R}^{N})$, $w_{n}\to 0$ a.e. in $\mathbb{R}^{N}$, and

$\lim_{n}J_{\varepsilon}(v_{n})-J_{\varepsilon}(w_{n})=J_{\varepsilon}(v).$

Observe that, again owing to the Brezis–Lieb lemma,

$\lim_{n}|u_{n}|_{2}^{2}-|w_{n}|_{2}^{2}=\lim_{n}|u_{n}|_{2}^{2}-|v_{n}|_{2}^{2}+|v_{n}|_{2}^{2}-|w_{n}|_{2}^{2}=|u|_{2}^{2}+|v|_{2}^{2},$

whence, denoting $\beta:=|u|_{2}>0$ and $\gamma:=|v|_{2}>0$, there holds

$\rho^{2}-\beta^{2}-\gamma^{2}\geq\limsup_{n}|u_{n}|_{2}^{2}-\beta^{2}-\gamma^{2}=\limsup_{n}|w_{n}|_{2}^{2}=:\delta^{2}\geq
0.$

If $\delta>0$, then let us set
$\tilde{w}_{n}:=\frac{\delta}{|w_{n}|_{2}}w_{n}\in{\mathcal{S}}(\delta)$. Via
explicit computations (cf., e.g., [29, Lemma 2.4]), we have
$\lim_{n}J_{\varepsilon}(w_{n})-J_{\varepsilon}(\tilde{w}_{n})=0$, hence,
together with Lemma 2.4 and since $c_{\varepsilon}$ is non-increasing,

(2.2)
$\begin{split}c_{\varepsilon}(\rho)&=\lim_{n}J_{\varepsilon}(u_{n})=J_{\varepsilon}(u)+J_{\varepsilon}(v)+\lim_{n}J_{\varepsilon}(w_{n})=J_{\varepsilon}(u)+J_{\varepsilon}(v)+\lim_{n}J_{\varepsilon}(\tilde{w}_{n})\\\
&\geq
c_{\varepsilon}(\beta)+c_{\varepsilon}(\gamma)+c_{\varepsilon}(\delta)\geq
c_{\varepsilon}(\sqrt{\beta^{2}+\gamma^{2}+\delta^{2}})\geq
c_{\varepsilon}(\rho).\end{split}$

Thus all the inequalities in (2.2) are in fact equalities and, in particular,
$J_{\varepsilon}(u)=c_{\varepsilon}(\beta)$ and
$J_{\varepsilon}(v)=c_{\varepsilon}(\gamma)$. Therefore Lemma 2.5 yields
$c_{\varepsilon}(\beta)+c_{\varepsilon}(\gamma)+c_{\varepsilon}(\delta)>c_{\varepsilon}(\sqrt{\beta^{2}+\gamma^{2}+\delta^{2}})$,
which contradicts (2.2). If $\delta=0$, then $w_{n}\to 0$ in
$L^{2}(\mathbb{R}^{N})$, which, together with (1.11), implies
$\lim_{n}\int_{\mathbb{R}^{N}}G_{+}(w_{n})\,dx=0$, whence
$\liminf_{n}J_{\varepsilon}(w_{n})\geq 0$. Then (2.2) becomes

$c_{\varepsilon}(\rho)=\lim_{n}J_{\varepsilon}(u_{n})=J_{\varepsilon}(u)+J_{\varepsilon}(v)+\lim_{n}J_{\varepsilon}(w_{n})\geq
c_{\varepsilon}(\beta)+c_{\varepsilon}(\gamma)\geq
c_{\varepsilon}(\sqrt{\beta^{2}+\gamma^{2}})=c_{\varepsilon}(\rho)$

and we reach a contradiction as before. ∎

Let us recall that every solution $(u_{\varepsilon},\lambda_{\varepsilon})$ to
the differential equation in (2.1) satisfies the Pohožaev identity

(2.3) $\int_{\mathbb{R}^{N}}(N-2)|\nabla
u_{\varepsilon}|^{2}+N\lambda_{\varepsilon}|u_{\varepsilon}|^{2}\,dx=2N\int_{\mathbb{R}^{N}}G_{+}(u_{\varepsilon})-G_{-}^{\varepsilon}(u_{\varepsilon})\,dx.$

###### Proof of Theorem 2.1.

Let $\bar{\rho}>0$ be determined by Lemma 2.3 and consider a minimizing
sequence $u_{n}\in{\mathcal{D}}(\rho)$ for $c_{\varepsilon}(\rho)$. Then, in
virtue of Lemmas 2.2 and 2.7, there exists
$u_{\varepsilon}\in{\mathcal{D}}(\rho)\setminus\\{0\\}$ such that, up to
subsequences and translations, $u_{n}\rightharpoonup u_{\varepsilon}$ in
$H^{1}(\mathbb{R}^{N})$ and $u_{n}\to u_{\varepsilon}$ in
$L^{p}(\mathbb{R}^{N})$ for every $2<p<2^{*}$ and a.e. in $\mathbb{R}^{N}$.
From (g1) and (g3) we have
$\lim_{n}\int_{\mathbb{R}^{N}}G_{+}(u_{n})\,dx=\int_{\mathbb{R}^{N}}G_{+}(u_{\varepsilon})\,dx$,
therefore, using Fatou’s Lemma, $c_{\varepsilon}(\rho)\leq
J_{\varepsilon}(u_{\varepsilon})\leq\lim_{n}J_{\varepsilon}(u_{n})=c_{\varepsilon}(\rho)<-\beta$.
In particular, there exists $\lambda_{\varepsilon}\in\mathbb{R}$ such that

$-\Delta
u_{\varepsilon}+\lambda_{\varepsilon}u_{\varepsilon}=g^{\varepsilon}(u_{\varepsilon})\quad\text{in
}\mathbb{R}^{N}.$

Note that $\lambda_{\varepsilon}=0$ if $u_{\varepsilon}$ is an interior point
of ${\mathcal{D}}(\rho)$, i.e., $|u_{\varepsilon}|_{2}<\rho$. If
$\lambda_{\varepsilon}\leq 0$, then (2.3) yields
$0>J_{\varepsilon}(u_{\varepsilon})\geq|\nabla u_{\varepsilon}|_{2}^{2}/N\geq
0$, a contradiction, therefore $\lambda_{\varepsilon}>0$ and
$u_{\varepsilon}\in{\mathcal{S}}(\rho)$. The last statement follows from [16,
Theorem 1.4] (see also [22]). ∎

######  Remark 2.8.

The proof of Theorem 2.1 shows that every minimizing sequence for
$c_{\varepsilon}(\rho)$ is relatively compact in $H^{1}(\mathbb{R}^{N})$ up to
a translation. As a matter of fact, from
$\lim_{n}\int_{\mathbb{R}^{N}}G_{+}(u_{n})\,dx=\int_{\mathbb{R}^{N}}G_{+}(u_{\varepsilon})\,dx$
and $\lim_{n}J_{\varepsilon}(u_{n})=J_{\varepsilon}(u_{\varepsilon})$ we get
$\lim_{n}|\nabla u_{n}|_{2}=|\nabla u_{\varepsilon}|_{2}$. Moreover,

$\rho=|u_{\varepsilon}|_{2}\leq\liminf_{n}|u_{n}|_{2}\leq\limsup_{n}|u_{n}|_{2}\leq\rho,$

i.e., $\lim_{n}|u_{n}|_{2}=|u_{\varepsilon}|_{2}$. Recalling that
$u_{n}\rightharpoonup u_{\varepsilon}$ in $H^{1}(\mathbb{R}^{N})$, we obtain
$u_{n}\to u_{\varepsilon}$ in $H^{1}(\mathbb{R}^{N})$.

##  3\. Solutions to (1.1) and related properties

Throughout this section, it is assumed that (g0)–(g4) hold. We consider the
family of functions and positive numbers
$(u_{\varepsilon},\lambda_{\varepsilon})_{\varepsilon\in(0,1)}$ given by
Theorem 2.1.

######  Lemma 3.1.

The quantities $|\nabla u_{\varepsilon}|_{2}$ and $\lambda_{\varepsilon}$ are
bounded.

###### Proof.

From (g0), (g1), and (g3), for every $\delta>0$ there exists $C_{\delta}>0$
such that for every $s\in\mathbb{R}$

$G_{+}(s)\leq C_{\delta}|s|^{2}+\delta|s|^{2+4/N}.$

Let us first consider the case $N\geq 3$. Since $\lambda_{\varepsilon}>0$,
from (2.3) we obtain

$\frac{1}{2^{*}}|\nabla
u_{\varepsilon}|_{2}^{2}<\int_{\mathbb{R}^{N}}G_{+}(u_{\varepsilon})\,dx\leq
C_{\delta}|u_{\varepsilon}|_{2}^{2}+\delta|u_{\varepsilon}|_{2+4/N}^{2+4/N}\leq
C_{\delta}\rho^{2}+\delta C_{N,2+4/N}^{2+4/N}\rho^{4/N}|\nabla
u_{\varepsilon}|_{2}^{2}.$

Taking $\delta<\left(2^{*}C_{N,2+4/N}^{2+4/N}\rho^{4/N}\right)^{-1}$, we
obtain the boundedness of $|\nabla u_{\varepsilon}|_{2}$. In addition, this
and again (2.3) yield

(3.1)
$0<\frac{2^{*}}{2}\lambda_{\varepsilon}\rho^{2}<\int_{\mathbb{R}^{N}}|\nabla
u_{\varepsilon}|^{2}+\frac{2^{*}}{2}\lambda_{\varepsilon}|u_{\varepsilon}|^{2}+2^{*}G_{-}^{\varepsilon}(u_{\varepsilon})\,dx=2^{*}\int_{\mathbb{R}^{N}}G_{+}(u_{\varepsilon})\,dx\leq
C<\infty$

and $\lambda_{\varepsilon}$ is bounded as well. Now, let us consider the case
$N=2$. From (2.3) we get

(3.2)
$\rho^{2}\lambda_{\varepsilon}\leq\int_{\mathbb{R}^{N}}\lambda_{\varepsilon}|u_{\varepsilon}|^{2}+2G_{-}^{\varepsilon}(u_{\varepsilon})\,dx=2\int_{\mathbb{R}^{N}}G_{+}(u_{\varepsilon})\,dx\leq
C_{\delta}\rho^{2}+\delta C_{N,2+4/N}^{2+4/N}\rho^{4/N}|\nabla
u_{\varepsilon}|_{2}^{2}.$

Moreover, again using (2.3) we obtain

$-\beta>J_{\varepsilon}(u_{\varepsilon})=\frac{1}{2}\left(|\nabla
u_{\varepsilon}|_{2}^{2}-\rho^{2}\lambda_{\varepsilon}\right),$

which, together with (3.2), implies

$\frac{1}{2}|\nabla u_{\varepsilon}|_{2}^{2}\leq
C_{\delta}\rho^{2}-\beta+\delta C_{N,2+4/N}^{2+4/N}\rho^{4/N}|\nabla
u_{\varepsilon}|_{2}^{2}.$

Taking $\delta<\left(2C_{N,2+4/N}^{2+4/N}\rho^{4/N}\right)^{-1}$, we obtain
the boundedness of $|\nabla u_{\varepsilon}|_{2}$, hence that of
$\lambda_{\varepsilon}$ follows from (3.2). ∎

Note that for every $\varepsilon\in(0,1)$ and every $u\in{\mathcal{D}}$ there
holds $J_{\varepsilon}(u_{\varepsilon})\leq J_{\varepsilon}(u)\leq J(u)$,
whence
$J_{\varepsilon}(u_{\varepsilon})\leq\inf_{{\mathcal{D}}(\rho)}J=:c(\rho)$.

###### Proof of Theorem 1.1.

From Lemma 3.1 there exist $u\in{\mathcal{D}}$ and $\lambda\geq 0$ such that,
up to a subsequence, $u_{\varepsilon}\rightharpoonup u$ in
$H^{1}(\mathbb{R}^{N})$ and $\lambda_{\varepsilon}\to\lambda$ as
$\varepsilon\to 0^{+}$. Arguing as in [24, Proof of Theorem 1.1] we obtain
that $\varphi_{\varepsilon}(u_{\varepsilon})g_{-}(u_{\varepsilon})v\to
g_{-}(u)v$ a.e. for every $v\in{\mathcal{C}}_{0}^{\infty}(\mathbb{R}^{N})$ and
that $g_{+}(u_{\varepsilon})v$ and
$\varphi_{\varepsilon}(u_{\varepsilon})g_{-}(u_{\varepsilon})v$ are uniformly
integrable (and tight). We deduce that for every
$v\in{\mathcal{C}}_{0}^{\infty}(\mathbb{R}^{N})$

$-\lambda\int_{\mathbb{R}^{N}}uv\,dx\leftarrow
J_{\varepsilon}^{\prime}(u_{\varepsilon})v\to J^{\prime}(u)v,$

i.e.,

$-\Delta u+\lambda u=g(u)\quad\text{in }\mathbb{R}^{N}.$

Moreover, (3.1) (when $N\geq 3$) or (3.2) (when $N=2$) yields

$\int_{\mathbb{R}^{N}}G_{-}^{\varepsilon}(u_{\varepsilon})\,dx\leq\int_{\mathbb{R}^{N}}G_{+}(u_{\varepsilon})\,dx\leq
C$

hence, in view of Fatou’s lemma, $G_{-}(u)\in L^{1}(\mathbb{R}^{N})$. In
particular, from [24, Proposition 3.1], $(u,\lambda)$ satisfies the Pohožaev
identity

(3.3) $(N-2)\int_{\mathbb{R}^{N}}|\nabla
u|^{2}\,dx=2N\int_{\mathbb{R}^{N}}G(u)-\frac{1}{2}\lambda|u|^{2}\,dx.$

We can assume that $|\nabla u_{\varepsilon}|_{2}$ and
$\int_{\mathbb{R}^{N}}G_{-}^{\varepsilon}(u_{\varepsilon})\,dx$ are
convergent. Since
$\int_{\mathbb{R}^{N}}G_{+}(u_{\varepsilon})\,dx\to\int_{\mathbb{R}^{N}}G_{+}(u)$
(remember that each $u_{\varepsilon}$ is radially symmetric), we have
$J(u)\leq\lim_{\varepsilon}J_{\varepsilon}(u_{\varepsilon})<0$ from Lemma 2.3;
in particular, $u\neq 0$. If $\lambda=0$, then from (3.3) we have
$J(u)=|\nabla u|_{2}^{2}/N>0$, therefore $\lambda>0$. We prove that
$u_{\varepsilon}\to u$ in $H^{1}(\mathbb{R}^{N})$. Since $\lambda>0$, from
(3.3) there follows

$\begin{split}\int_{\mathbb{R}^{N}}(N-2)|\nabla
u|^{2}+N\lambda|u|^{2}+2NG_{-}(u)\,dx&\leq\lim_{\varepsilon}\int_{\mathbb{R}^{N}}(N-2)|\nabla
u_{\varepsilon}|^{2}+N\lambda|u_{\varepsilon}|^{2}+2NG^{\varepsilon}_{-}(u_{\varepsilon})\,dx\\\
&=2N\lim_{\varepsilon}\int_{\mathbb{R}^{N}}G_{+}(u_{\varepsilon})\,dx=2N\int_{\mathbb{R}^{N}}G_{+}(u)\,dx\\\
&=\int_{\mathbb{R}^{N}}(N-2)|\nabla
u|^{2}+N\lambda|u|^{2}+2NG_{-}(u)\,dx,\end{split}$

which yields $|u_{\varepsilon}|_{2}\to|u|_{2}$; in particular,
$u\in{\mathcal{S}}$. Furthermore, in a similar fashion we obtain
$J(u)\leq\lim_{\varepsilon}J_{\varepsilon}(u_{\varepsilon})\leq c(\rho)\leq
J(u)$, which shows that $|\nabla u_{\varepsilon}|_{2}\to|\nabla u|_{2}$ and
$J(u)=c(\rho)$. Since (2.1) is translation-invariant, we can assume that every
$u_{\varepsilon}$ is radial, hence so is $u$. Moreover, because $u\neq 0$,
there exists $x\in\mathbb{R}^{N}$ such that $u(x)\neq 0$ and so
$u_{\varepsilon}(x)$ has the same sign as $u(x)$ for every $\varepsilon$
sufficiently small; since every $u_{\varepsilon}$ has constant sign, it has
everywhere the same sign as $u(x)$, hence $u$ has constant sign too. Finally,
if there exist $x,y,z\in\mathbb{R}^{N}$ such that $|x|<|y|<|z|$ and either
$u(y)<\min\\{u(x),u(z)\\}$ or $u(y)>\max\\{u(x),u(z)\\}$, then arguing as
before we obtain a contradiction. ∎

######  Remark 3.2.

(i) The Proof of Theorem 1.1 contains the relevant result that
$c_{\varepsilon}(\rho)\to c(\rho)$ as $\varepsilon\to 0^{+}$.
(ii) Unlike the proof of Theorem 2.1, we cannot use the information
$\lambda>0$ to deduce $u\in{\mathcal{S}}$ because we do not know whether $u$
is a critical point of $J|_{\mathcal{D}}$.

###### Proof of Proposition 1.4.

It follows from Lemma 2.3 and Remark 3.2 (i). ∎

###### Proof of Theorem 1.5.

Let $u_{n}$ and $\rho_{n}$ as in the statement. We prove preliminarily that
$u_{n}$ is relatively compact up to translations in $L^{p}(\mathbb{R}^{N})$,
$2<p<2^{*}$. Fix $\varepsilon>0$. Arguing (by contradiction) as in the proof
of Lemma 2.7, we find $\beta,\gamma>0$, $\delta\geq 0$,
$u\in{\mathcal{D}}(\beta)\setminus\\{0\\}$,
$v\in{\mathcal{D}}(\gamma)\setminus\\{0\\}$, $y_{n},z_{n}\in\mathbb{R}^{N}$,
and $v_{n},w_{n}\in H^{1}(\mathbb{R}^{N})$ bounded, none of which depending on
$\varepsilon$, such that $\rho^{2}-\beta^{2}-\gamma^{2}\geq\delta^{2}$ and
$\lim_{n}J_{\varepsilon}(u_{n})-J_{\varepsilon}(w_{n})=J_{\varepsilon}(u)+J_{\varepsilon}(v)$,
where $v_{n}(x)=u_{n}(x-y_{n})-u(x)$ and $w_{n}(x)=v_{n}(x-z_{n})-v(x)$. If
$\delta>0$, then
$\tilde{w}_{n}:=\frac{\delta}{|w_{n}|_{2}}w_{n}\in{\mathcal{S}}(\delta)$ and,
again from [29, Lemma 2.4],

$\begin{split}c_{\varepsilon}(\rho)&=\lim_{n}J_{\varepsilon}(u_{n})=J_{\varepsilon}(u)+J_{\varepsilon}(v)+\lim_{n}J_{\varepsilon}(\tilde{w}_{n})\geq
c_{\varepsilon}(\beta)+c_{\varepsilon}(\gamma)+c_{\varepsilon}(\delta)\\\
&\geq c_{\varepsilon}(\sqrt{\beta^{2}+\gamma^{2}+\delta^{2}})\geq
c_{\varepsilon}(\rho).\end{split}$

Letting $\varepsilon\to 0^{+}$, observing that $J_{\varepsilon}(u)\to J(u)$
and $J_{\varepsilon}(v)\to J(v)$ from the monotone convergence theorem, and
using Remark 3.2 (i), we obtain

$\begin{split}c(\rho)&\geq J(u)+J(v)+\lim_{\varepsilon\to
0^{+}}\lim_{n}J_{\varepsilon}(\tilde{w}_{n})\geq\lim_{\varepsilon\to
0^{+}}c_{\varepsilon}(\beta)+\lim_{\varepsilon\to
0^{+}}c_{\varepsilon}(\gamma)+\lim_{\varepsilon\to
0^{+}}c_{\varepsilon}(\delta)\\\ &\geq\lim_{\varepsilon\to
0^{+}}c_{\varepsilon}(\sqrt{\beta^{2}+\gamma^{2}+\delta^{2}})\geq
c(\rho).\end{split}$

In particular, $c_{\varepsilon}(\beta)\to J(u)$ and
$c_{\varepsilon}(\gamma)\to J(v)$ as $\varepsilon\to 0^{+}$, therefore

$c(\beta)\leq J(u)=\lim_{\varepsilon\to 0^{+}}c_{\varepsilon}(\beta)\leq
c(\beta),\quad c(\gamma)\leq J(v)=\lim_{\varepsilon\to
0^{+}}c_{\varepsilon}(\gamma)\leq c(\gamma),$

and so, from Lemma 2.4 and Remark 2.6 (a),

$\begin{split}\lim_{\varepsilon\to
0^{+}}c_{\varepsilon}(\sqrt{\beta^{2}+\gamma^{2}+\delta^{2}})&\leq\lim_{\varepsilon\to
0^{+}}c_{\varepsilon}(\sqrt{\beta^{2}+\gamma^{2}})+\lim_{\varepsilon\to
0^{+}}c_{\varepsilon}(\delta)\leq
c(\sqrt{\beta^{2}+\gamma^{2}})+\lim_{\varepsilon\to
0^{+}}c_{\varepsilon}(\delta)\\\ &<c(\beta)+c(\gamma)+\lim_{\varepsilon\to
0^{+}}c_{\varepsilon}(\delta)=\lim_{\varepsilon\to
0^{+}}c_{\varepsilon}(\sqrt{\beta^{2}+\gamma^{2}+\delta^{2}}),\end{split}$

a contradiction. If $\delta=0$, then, again as in the proof of Lemma 2.7,
$\liminf_{n}J_{\varepsilon}(w_{n})\geq 0$ and we get

$c_{\varepsilon}(\rho)=\lim_{n}J_{\varepsilon}(u_{n})=J_{\varepsilon}(u)+J_{\varepsilon}(v)+\lim_{n}J_{\varepsilon}(w_{n})\geq
c_{\varepsilon}(\beta)+c_{\varepsilon}(\gamma)\geq
c_{\varepsilon}(\sqrt{\beta^{2}+\gamma^{2}})\geq c_{\varepsilon}(\rho),$

thus we obtain a contradiction arguing in a similar (and simpler) way as
before. Having proved that $u_{n}$ is relatively compact up to translations in
$L^{p}(\mathbb{R}^{N})$, $2<p<2^{*}$, now we prove that the same convergence
holds in $H^{1}(\mathbb{R}^{N})$. Recalling that $u_{n}$ is bounded, there
exists $u\in{\mathcal{D}}(\rho)$ such that, up to translations and
subsequences, $u_{n}\rightharpoonup u$ in $H^{1}(\mathbb{R}^{N})$ and
$u_{n}\to u$ in $L^{p}(\mathbb{R}^{N})$, $2<p<2^{*}$, whence

$c(\rho)\leq J(u)\leq\lim_{n}J(u_{n})=c(\rho)$

and so $|\nabla u_{n}|_{2}^{2}\to|\nabla u|_{2}^{2}$. Moreover,
$J(u)=c(\rho)<0$, which implies, in particular, that $G(u)\in
L^{1}(\mathbb{R}^{N})$. Assume by contradiction that $|u|_{2}<\rho$; then, for
every $v\in{\mathcal{C}}_{0}^{\infty}(\mathbb{R}^{N})$ and every
$t\in\mathbb{R}$ sufficiently small, $|u+tv|_{2}<\rho$. Next, we check that
$J$ is differentiable at $u$ along every
$v\in{\mathcal{C}}_{0}^{\infty}(\mathbb{R}^{N})$. Of course, this only needs
to be checked for the functional $\int_{\mathbb{R}^{N}}G(\cdot)\,dx$. Let
$t\neq 0$: from (g0) and (g2),
$\frac{1}{t}[G(u+tv)-G(u)]=\int_{0}^{1}g(u+stv)\,ds\,v$ and the family
$\\{\int_{0}^{1}g(u+stv)\,ds\,v\\}_{t}$ is uniformly integrable (and tight),
thus the claim holds true. There follows that $J^{\prime}(u)[v]=0$ for every
$v\in{\mathcal{C}}_{0}^{\infty}(\mathbb{R}^{N})$, i.e., $-\Delta u=g(u)$ in
$\mathbb{R}^{N}$. Then, from [24, Proposition 3.1], $u$ satisfies (3.3) with
$\lambda=0$, whence $J(u)=|\nabla u|_{2}^{2}/N>0$, which is impossible. ∎

###### Proof of Theorem 1.6.

Since $G(s)=\alpha/2(\ln s^{2}-1)s^{2}+\mu/p|s|^{p}$, it is clear that
(g0)–(g3) hold. We want to prove that (g4) is satisfied if and only if
$\mu>-\frac{\alpha p}{p-2}e^{-p/2}$. If $\mu\geq 0$, then clearly (g4) holds,
so let us consider the case $\mu<0$. Let $s>0$. Of course, $G(s)>0$ if and
only if

$\widetilde{G}(s):=\frac{\alpha}{2}\left(\ln
s^{2}-1\right)+\frac{\mu}{p}s^{p-2}>0.$

Since

$\widetilde{G}^{\prime}(s)=\frac{\alpha}{s}+\mu\frac{p-2}{p}s^{p-3},$

we have

$\max\widetilde{G}=\frac{\alpha}{2}\left(\ln\left(\frac{\alpha
p}{\mu(2-p)}\right)^{2/(p-2)}-1\right)-\frac{\alpha}{p-2}$

and the claim holds because (g4) is equivalent to $\max\widetilde{G}>0$. Point
(i) is then a consequence of Theorem 1.1. Concerning point (ii), observe that
$\max G=0$ if and only if $\mu=-\frac{\alpha p}{p-2}e^{-p/2}$. Since, from
[24, Proposition 3.1], every solution to (1.1) satisfies the Pohožaev identity
(3.3), the statement easily holds true if $\lambda>0$ or $N\geq 3$ or
$\mu<-\frac{\alpha p}{p-2}e^{-p/2}$. Now let us consider the case $\lambda=0$,
$N=2$, and $\mu=-\frac{\alpha p}{p-2}e^{-p/2}$ and assume by contradiction
that such a $u$ exists. Since (3.3) reads $\int_{\mathbb{R}^{N}}G(u)\,dx=0$,
$G\leq 0$, and $u$ is continuous (se, e.g., [32, p. 271]), necessarily
$u\equiv 0$ or $u\equiv\pm\bar{s}$, where $\bar{s}$ is the unique $s>0$ such
that $G(s)=0$, which contradicts $|u|_{2}=\rho\in(0,\infty)$. ∎

##  4\. Multiple solutions

###  4.1. Functional setting

Let us recall from [27] that a function $A\colon\mathbb{R}\to\mathbb{R}$ is
called an _N-function_ if and only if it is nonnegative, even, convex, and
satisfies

$\lim_{s\to 0}\frac{A(s)}{s}=\lim_{s\to\infty}\frac{A(s)}{s}=\infty.$

It is said to satisfy the _$\Delta_{2}$ condition_ globally if and only if
there exists $K>0$ such that

$A(2s)\leq KA(s)\quad\text{for all }s\in\mathbb{R}.$

It is said to satisfy the _$\nabla_{2}$ condition_ globally if and only if
there exist $\ell>1$ such that

$2\ell A(s)\leq A(\ell s)\quad\text{for all }s\in\mathbb{R}.$

Equivalently, $A$ satisfies the $\Delta_{2}$ (respectively, $\nabla_{2}$)
condition globally if and only if there exists $C>1$ such that

(4.1) $CA(s)\geq sA^{\prime}(s)\;\text{(respectively, $sA^{\prime}(s)\geq
CA(s)$)}\quad\text{for all }s\in\mathbb{R}.$

From now on, $A$ shall denote the same function as in (A). We can define the
Orlicz space associated with $A$ as

$V:=\left\\{u\in L^{1}_{\textup{loc}}(\mathbb{R}^{N}):A(u)\in
L^{1}(\mathbb{R}^{N})\right\\}.$

If we define the norm

$\|u\|_{V}:=\inf\left\\{\kappa>0:\int_{\mathbb{R}^{N}}A(u/\kappa)\,dx\leq
1\right\\},$

then $(V,\|\cdot\|_{V})$ is a reflexive Banach space (cf. [27, Theorem
IV.I.10]). Moreover, the following holds true.

######  Lemma 4.1.

(i) Let $u_{n},u\in V$. Then $\lim_{n}\|u_{n}-u\|_{V}=0$ if and only if
$\lim_{n}\int_{\mathbb{R}^{N}}A(u_{n}-u)\,dx=0$.
(ii) Let $X\subset V$. Then $X$ is bounded if and only if

$\left\\{\int_{\mathbb{R}^{N}}A(u)\,dx:u\in X\right\\}$

is bounded.
(iii) Let $u_{n},u\in V$. If $u_{n}\to u$ a.e. and
$\int_{\mathbb{R}^{N}}A(u_{n})\,dx\to\int_{\mathbb{R}^{N}}A(u)\,dx$, then
$\|u_{n}-u\|_{V}\to 0$.

###### Proof.

The first two points follow from [27, Theorem III.IV.12 and Corollary
III.IV.15] respectively, while the third one is a consequence of the previous
two and [7, Theorem 2, Example (b)]. ∎

Note that $W=H^{1}(\mathbb{R}^{N})\cap V$. Since $(V,\|\cdot\|_{V})$ is a
reflexive Banach space, so is $(W,\|\cdot\|_{W})$, with

$\|u\|_{W}^{2}:=|\nabla u|_{2}^{2}+|u|_{2}^{2}+\|u\|_{V}^{2}.$

We obtain the following variant of Lions’ lemma.

######  Lemma 4.2.

Suppose that $u_{n}\in W$ is bounded and for some $r>0$

(4.2)
$\lim_{n\to\infty}\sup_{y\in\mathbb{R}^{N}}\int_{B(y,r)}|u_{n}|^{2}\,dx=0.$

Then $u_{n}\to 0$ in $L^{p}(\mathbb{R}^{N})$ for every $p\in[2,2^{*})$.

###### Proof.

In view of Lions’ lemma [21, Lemma I.1 in Part 2], $u_{n}\to 0$ in
$L^{p}(\mathbb{R}^{N})$ for every $p\in(2,2^{*})$. We prove the convergence in
$L^{2}(\mathbb{R}^{N})$. Take any $p\in(2,2^{*})$ and $\varepsilon>0$. We find
$\delta>0$ such that

$\begin{split}|s|^{2}&\leq\varepsilon A(s)\quad\hbox{ if }|s|\in[0,\delta],\\\
|s|^{2}&\leq\delta^{2-p}|s|^{p}\quad\hbox{ if }|s|>\delta,\end{split}$

Hence, we get

$\limsup_{n\to\infty}\int_{\mathbb{R}^{N}}|u_{n}|^{2}\,dx\leq\varepsilon\limsup_{n\to\infty}\int_{\mathbb{R}^{N}}A(u_{n})\,dx+\delta^{2-p}\limsup_{n\to\infty}\int_{\mathbb{R}^{N}}|u_{n}|^{p}\,dx=\varepsilon\limsup_{n\to\infty}\int_{\mathbb{R}^{N}}A(u_{n})\,dx.$

In view of Lemma 4.1 (ii), we conclude by letting $\varepsilon\to 0$. ∎

Let ${\mathcal{O}}$ be any subgroup ${\mathcal{O}}(N)$ such that
$\mathbb{R}^{N}$ is compatible with ${\mathcal{O}}$ (cf. [20]), i.e.,
$\lim_{|y|\to\infty}m(y,r)=\infty$ for some $r>0$, where, for
$y\in\mathbb{R}^{N}$,

$m(y,r):=\sup\big{\\{}n\in\mathbb{N}:\hbox{there exist
}g_{1},\dots,g_{n}\in{\mathcal{O}}\hbox{ such that }B(g_{i}y,r)\cap
B(g_{j}y,r)=\emptyset\hbox{ for }i\neq j\big{\\}}.$

In view of [20], $H^{1}_{{\mathcal{O}}}(\mathbb{R}^{N})$ embeds compactly into
$L^{p}(\mathbb{R}^{N})$ for $2<p<2^{*}$. As for the subspace
$W_{\mathcal{O}}:=W\cap H^{1}_{{\mathcal{O}}}(\mathbb{R}^{N})$, we obtain the
compact embedding also in $L^{2}(\mathbb{R}^{N})$ .

######  Corollary 4.3.

Suppose that $u_{n}\in W_{{\mathcal{O}}}$ is bounded and $u_{n}\to 0$ in
$L^{2}_{\textup{loc}}(\mathbb{R}^{N})$. Then $u_{n}\to 0$ in
$L^{p}(\mathbb{R}^{N})$ for every $p\in[2,2^{*})$.

###### Proof.

Suppose that

(4.3) $\int_{B(y_{n},1)}|u_{n}|^{2}\,dx\geq c>0$

for some sequence $y_{n}\in\mathbb{R}^{N}$ and a constant $c$. Observe that in
the family $\\{B(hy_{n},1)\\}_{h\in{\mathcal{O}}}$ we find an increasing
number of disjoint balls provided that $|y_{n}|\to\infty$. Since $u_{n}$ is
bounded in $L^{2}(\mathbb{R}^{N})$ and invariant with respect to
${\mathcal{O}}$, by (4.3) $(y_{n})$ must be bounded. Then for sufficiently
large $r$ one obtains

$\int_{B(0,r)}|u_{n}|^{2}\,dx\geq\int_{B(y_{n},1)}|u_{n}|^{2}\,dx\geq c>0,$

and we get a contradiction with the convergence of $u_{n}$ in
$L^{2}_{\textup{loc}}(\mathbb{R}^{N})$. Therefore by Lemma 4.2 we conclude. ∎

Finally, we have the following result.

######  Proposition 4.4.

If (A) and (f0)–(f3) hold, then the functional $J|_{W}\colon W\to\mathbb{R}$
is of class ${\mathcal{C}}^{1}$ and, for every $u\in W\cap{\mathcal{S}}$,
$J|_{W\cap{\mathcal{S}}}^{\prime}(u)=0$ if and only if there exists
$\lambda\in\mathbb{R}$ such that $(u,\lambda)$ is a solution to (1.1)–(1.2).

###### Proof.

Let $B\colon\mathbb{R}\to\mathbb{R}$ be the complementary N-function of $A$.
From [27, Theorem II.V.8], it satisfies the $\Delta_{2}$ and $\nabla_{2}$
conditions globally because $A$ does. Let us recall (cf. [27, Definition
III.IV.2, Corollary III.IV.5, and Corollary IV.II.9]) that $V^{\prime}$, the
dual space of $V$, is isomorphic to the Orlicz space associated with $B$. From
[27, Theorem I.III.3],

$B\bigl{(}a(s)\bigr{)}=sa(s)-A(s)\leq(C-1)A(s),$

where $C>1$ is the constant given in the characterization of the $\Delta_{2}$
condition – i.e., (4.1). Then, for every $u,v\in V$, we have

$\left|\int_{\mathbb{R}^{N}}a(u)v\,dx\right|\leq\int_{\mathbb{R}^{N}}|a(u)||v|\,dx\lesssim\|a(u)\|_{V^{\prime}}\|v\|_{V}.$

Now we can use the same argument as in [12, Lemma 2.1] to obtain that the
functionals $u\mapsto\int_{\mathbb{R}^{N}}A(u)\,dx$ and
$u\mapsto\int_{\mathbb{R}^{N}}F(u)\,dx$ belong to ${\mathcal{C}}^{1}(W)$. The
remaining part is obvious. ∎

###  4.2. Proof of Theorem 1.8

We recall the definitions $F_{-}:=F_{+}-F$ and $f_{-}:=F_{-}^{\prime}$.

######  Lemma 4.5.

If (A), (f0)–(f3), and (1.12) hold, then $J|_{W\cap{\mathcal{S}}}$ is coercive
and bounded below.

###### Proof.

For every $\delta>0$, there exists $C_{\delta}>0$ such that for every
$s\in\mathbb{R}$

$F_{+}(s)\leq C_{\delta}|s|^{2}+(\eta+\delta)|s|^{2+4/N}.$

For every $u\in W\cap{\mathcal{S}}$ there holds

$\begin{split}J(u)&\geq\int_{\mathbb{R}^{N}}\frac{1}{2}|\nabla
u|^{2}+A(u)-F_{+}(u)\,dx\\\ &\geq\int_{\mathbb{R}^{N}}\frac{1}{2}|\nabla
u|^{2}+A(u)-C_{\delta}|u|^{2}-(\eta+\delta)|u|^{2+4/N}\,dx\\\
&\geq\frac{1}{2}|\nabla u|_{2}^{2}+\int_{\mathbb{R}^{N}}A(u)\,dx-
C_{\delta}\rho^{2}-(\eta+\delta)C_{N,2+4/N}^{2+4/N}\rho^{4/N}|\nabla
u|_{2}^{2}.\end{split}$

We conclude by Lemma 4.1 (ii) and taking $\delta$ so small that
$2(\eta+\delta)C_{N,2+4/N}^{2+4/N}\rho^{4/N}<1$. ∎

######  Remark 4.6.

Setting $f_{\textup{p}}:=\max\\{f,0\\}$ and $f_{\textup{n}}:=\max\\{-f,0\\}$,
we have that $f_{-}(s)=f_{\textup{n}}(s)$ if $s\geq 0$ and
$f_{-}(s)=-f_{\textup{p}}(s)$ if $s<0$. In particular, $f_{-}(s)s\geq 0$ for
every $s\in\mathbb{R}$.

######  Lemma 4.7.

If (A) and (f0)–(f3) hold, then $J|_{W_{{\mathcal{O}}}}\cap{\mathcal{S}}$
satisfies the Palais–Smale condition.

###### Proof.

Let $u_{n}\in W_{{\mathcal{O}}}\cap{\mathcal{S}}$ such that $J(u_{n})$ is
bounded and $J|_{W_{{\mathcal{O}}}\cap{\mathcal{S}}}^{\prime}(u_{n})\to 0$.
From Lemma 4.5, $u_{n}$ is bounded in $W$, therefore there exists $u\in
W_{{\mathcal{O}}}$ such that $u_{n}\rightharpoonup u$ in $W_{\mathcal{O}}$ up
to a subsequence. Then, from Corollary 4.3, $u_{n}\to u$ in
$L^{p}(\mathbb{R}^{N})$ for every $p\in[2,2^{*})$; in particular,
$u\in{\mathcal{S}}$. Up to a second subsequence, we can assume that $u_{n}\to
u$ a.e. in $\mathbb{R}^{N}$. Additionally, from [3, Lemma 3], there exist
$\lambda_{n}\in\mathbb{R}$ such that

(4.4) $-\Delta u_{n}+\lambda_{n}u_{n}-g(u_{n})u_{n}\to 0\quad\text{in
}W_{\mathcal{O}}^{\prime},$

where $W_{\mathcal{O}}^{\prime}$ is the dual space of $W_{\mathcal{O}}$.
Testing (4.4) with $u_{n}$, we obtain that $\lambda_{n}$ is bounded as well,
hence there exists $\lambda\in\mathbb{R}$ such that $\lambda_{n}\to\lambda$ up
to a subsequence, and $(u,\lambda)$ is a solution to (1.1). Finally, from the
Nehari identity and the fact that $\lambda_{n}\to\lambda$, we obtain

$\begin{split}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}+a(u)u+f_{-}(u)u\,dx&=\int_{\mathbb{R}^{N}}f_{+}(u)u-\lambda|u|^{2}\,dx=\lim_{n}\int_{\mathbb{R}^{N}}f_{+}(u_{n})u_{n}-\lambda|u_{n}|^{2}\,dx\\\
&=\lim_{n}\int_{\mathbb{R}^{N}}|\nabla
u_{n}|^{2}+a(u_{n})u_{n}+f_{-}(u_{n})u_{n}\,dx,\end{split}$

which, together with Remark 4.6, implies that $|\nabla
u_{n}|_{2}^{2}\to|\nabla u|_{2}^{2}$ and
$\int_{\mathbb{R}^{N}}a(u_{n})u_{n}\,dx\to\int_{\mathbb{R}^{N}}a(u)u\,dx$. It
remains to prove that $\|u_{n}-u\|_{V}\to 0$. In virtue of Lemma 4.1(iii), it
suffices to prove that
$\int_{\mathbb{R}^{N}}A(u_{n})\,dx\to\int_{\mathbb{R}^{N}}A(u)\,dx$.
Additionally, since $A$ satisfies the $\nabla_{2}$ condition globally, this
occurs if the sequence $a(u_{n})u_{n}$ is bounded above by an integrable
function, which holds true from (A) and [7, Example (b)]. ∎

######  Lemma 4.8.

For every $k\geq 1$ there exist $\pi_{k}\colon\mathbb{S}^{k-1}\to
W_{{\mathcal{O}}(N)}\cap{\mathcal{S}}$ and
$\tilde{\pi}_{k}\colon\mathbb{S}^{k-1}\to{\mathcal{X}}\cap{\mathcal{S}}$ odd
and continuous.

###### Proof.

The existence of $\pi_{k}$ was proved in [3, Theorem 10, Lemma 8] (with the
notations therein, note that we can take any number different from $0$ instead
of $\zeta$ because we do not need $V(u)\geq 1$). Then, following [17, Proof of
Lemma 3.4] and [23, Remark 4.2], we take
$\phi\in{\mathcal{C}}^{\infty}(\mathbb{R})$ odd such that $0\leq\phi\leq 1$
and $\phi(t)=1$ for all $t\geq 1$, and define for $\sigma\in\mathbb{S}^{k-1}$

$\tilde{\pi}_{k}(\sigma)(x):=\pi_{k}(\sigma)(x)\phi(|x_{1}|-|x_{2}|)\quad\text{for
all
}x=(x_{1},x_{2},x_{3})\in\mathbb{R}^{M}\times\mathbb{R}^{M}\times\mathbb{R}^{N-2M}.\qed$

We make use of the following abstract theorem, where ${\mathcal{G}}$ stands
for the Krasnoselsky genus [32, Chapter 5].

######  Theorem 4.9.

Let $E$ be a Banach space, $H\supset E$ a Hilbert space with scalar product
$(\cdot|\cdot)$, $R>0$, ${\mathcal{M}}:=\left\\{u\in E:(u|u)=R\right\\}$, and
$I\in{\mathcal{C}}^{1}(E)$ even such that $I|_{\mathcal{M}}$ is bounded below.
For every $k\geq 1$, define

$c_{k}:=\inf_{A\in\Gamma_{k}}\sup_{u\in
A}I(u),\quad\Gamma_{k}:=\left\\{A\subset{\mathcal{M}}:A=-A=\overline{A}\text{
and }{\mathcal{G}}(A)\geq k\right\\}.$

Then, for every $k\geq 1$, $-\infty<c_{1}\leq\dots\leq c_{k}\leq
c_{k+1}\leq\dots$ and the following holds: if there exist $k\geq 1$ and $h\geq
0$ such that $c_{k}=\dots=c_{k+h}<\infty$ and $I|_{\mathcal{M}}$ satisfies the
Palais–Smale condition at the level $c_{k}$, then

${\mathcal{G}}\left(\left\\{u\in{\mathcal{M}}:I(u)=c_{k}\text{ and
}I|_{\mathcal{M}}^{\prime}(u)=0\right\\}\right)\geq h+1$

(in particular, taking $h=0$, every $c_{k}$ is a critical value of
$I|_{\mathcal{M}}$).

###### Proof.

This theorem is basically (part of) [17, Theorem 2.1] (see also [26]), so we
omit the proof. The only difference is that the values $c_{k}$ are critical
regardless of their sign, which is a consequence of $I|_{\mathcal{M}}$
satisfying the Palais–Smale condition at any level. ∎

###### Proof of Theorem 1.8.

Let us set $E=W_{{\mathcal{O}}(N)}$ (respectively, $E={\mathcal{X}}$),
$H=L^{2}(\mathbb{R}^{N})$, $R=\rho^{2}$, ${\mathcal{M}}={\mathcal{S}}\cap E$,
and $I=J|_{E}$. From Lemmas 4.5 and 4.7,
$I|_{\mathcal{M}}=J_{{\mathcal{S}}\cap E}$ is bounded below and satisfies the
Palais–Smale condition; moreover, from Lemma 4.8,
$\pi_{k}(\mathbb{S}^{k-1})\in\Gamma_{k}$ (respectively,
$\tilde{\pi}_{k}(\mathbb{S}^{k-1})\in\Gamma_{k}$) for every $k$, so the
numbers $c_{k}$ are finite. Applying Theorem 4.9, we conclude the part about
the existence of infinitely many solutions. Concerning the existence of a
least-energy solution, we consider a sequence $u_{n}\in{\mathcal{S}}\cap E$
such that $\lim_{n}J(u_{n})=\inf_{{\mathcal{S}}\cap E}J$. From Ekeland’s
variational principle, we can assume that $u_{n}$ is a Palais–Smale sequence
for $J|_{{\mathcal{S}}\cap E}$, hence argue as above to obtain a solution
$(\bar{u},\bar{\lambda})\in\mathbb{R}\times({\mathcal{S}}\cap E)$ to
(1.1)–(1.2) such that $J(\bar{u})=\min_{{\mathcal{S}}\cap E}J$. The fact that
$\min_{{\mathcal{S}}\cap W_{{\mathcal{O}}(N)}}J=\min_{{\mathcal{S}}\cap W}J$
follows from the properties of the Schwartz rearrangement [19, Chapter 3]. ∎

Data availability statement. Data sharing not applicable to this article as no
datasets were generated or analysed during the current study.

Acknowledgements. The authors would like to thank the referee for valuable
comments concerning the improvement of Theorem 1.6 and for indicating the very
recent preprint [36], where properties of the ground state energy have been
studied in the context of strongly sublinear nonlinearities. J.M. was partly
supported by the National Science Centre, Poland (Grant No.
2017/26/E/ST1/00817). J.S. was partly supported by the National Science
Centre, Poland (Grant No. 2020/37/N/ST1/00795).
J.S. is a member of GNAMPA.

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# Universal Dynamic Scaling and Contact Dynamics in Quenched Quantum Gases

Jia-nan Cui  School of Science, Beijing University of Posts and
Telecommunications, Beijing 100876, China  Zhengqiang Zhou  School of Science,
Beijing University of Posts and Telecommunications, Beijing 100876, China
Mingyuan Sun<EMAIL_ADDRESS>School of Science, Beijing University of
Posts and Telecommunications, Beijing 100876, China  State Key Lab of
Information Photonics and Optical Communications, Beijing University of Posts
and Telecommunications, Beijing 100876, China

###### Abstract

Recently universal dynamic scaling is observed in several systems, which
exhibit a spatiotemporal self-similar scaling behavior, analogous to the
spatial scaling near phase transition. The latter one arises from the emergent
continuous scaling symmetry. Motivated by this, we investigate the possible
relation between the scaling dynamics and continuous scaling symmetry in this
paper. We derive a theorem that the scaling invariance of the quenched
Hamiltonian and the initial density matrix can lead to the universal dynamic
scaling. It is further demonstrated both in a two-body system analytically and
in a many-body system numerically. For the latter one, we calculate the
dynamics of quantum gases quenched from the zero interaction to a finite
interaction via the non-equilibrium high-temperature virial expansion. A
dynamic scaling of the momentum distribution appears in certain momentum-time
windows at unitarity as well as in the weak interacting limit. Remarkably,
this universal scaling dynamics persists approximately with smaller scaling
exponents even if the scaling symmetry is fairly broken. Our findings may
offer a new perspective to interpret the related experiments. We also study
the Contact dynamics in the BEC-BCS crossover. Surprisingly, the half-way time
displays a maximum near unitarity while some damping oscillations occur on the
BEC side due to the dimer state, which can be used to detect possible two-body
bound states in experiments.

##  I Introduction

Quench dynamics refers to the evolution of a closed system after a sudden
change of either the initial state or the Hamiltonian. Given its simple
operation in experiments, it becomes one of the most studied non-equilibrium
dynamics, especially in ultracold atomic gases [1, 2, 3, 4, 5, 6, 7, 8, 9].
The advance in experiments further prompts extensive theoretical studies [10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33]. For instance, the Cambridge group observed several universal
phenomena in a three-dimensional (3D) Bose gas quenched from noninteracting to
unitarity, such as the exponential form of the momentum distribution at the
final steady state [3]. Later, the low temperature condensed part and the
thermal gas part of the data are respectively explained by using the time-
dependent Bogoliubov theory [23, 24] and the non-equilibrium high-temperature
virial expansion [27]. On the other hand, some characteristics of the quenched
Hamiltonian can lead to novel dynamical phenomena [34, 35, 36, 37, 38]. For
instance, the Chern number of a topological insulator can be revealed by the
linking number in quench dynamics [35, 36, 37]. The discrete scaling symmetry
of the quenched Hamiltonian may give rise to a dynamical fractal [38].

Recently, universal scaling dynamics is observed in quenched Bose gases [4, 5,
6, 7, 9]. The scaling property of spin correlations is investigated both in a
quasi-1D [4] and 2D [9] spinor Bose gas, while other works demonstrate the
dynamic scaling behavior of the momentum distribution in 1D [5], 2D [7] and 3D
[6] Bose gases respectively. Specifically, the evolution of the momentum
distribution exhibits a spatiotemporal self-similar scaling feature in certain
momentum-time windows, which is analogous to the spatial scaling around the
critical point of phase transition [39, 40]. The latter arises from the
continuous scaling invariance of the system emerged at the critical point,
where the correlation length is divergent. The scaling dynamics has been
studied in various systems by a number of theoretical works [41, 42, 43, 44,
45, 46, 47, 48, 49, 50, 51, 52, 53], mainly in the framework of non-thermal
fixed points. However, there are some disagreements between the theory and the
experiment, for example, the scaling exponent [5, 6]. On the other hand, the
possible relation between the scaling symmetry and the dynamic scaling has not
been discussed yet. If the Hamiltonian $\hat{H}(\mathcal{P})$ has a continuous
scaling symmetry, i.e,
$\hat{H}(\zeta\mathcal{P})=\zeta^{\beta}\hat{H}(\mathcal{P})$ with a scaling
exponent $\beta$ ($\zeta$ is a positive constant), it can play an important
role in quench dynamics through the time-evolution operator $e^{-it\hat{H}}$.

In this letter we investigate this problem by deriving a theorem that connects
the continuous scaling symmetry of the quenched Hamiltonian and the initial
density matrix to the dynamic scaling of the momentum distribution. We further
demonstrate its validity both in a two-body system and in a many-body system.
Surprisingly, even if the scaling symmetry required by the theorem is fairly
broken, an approximate scaling dynamics can still occur with modified scaling
exponents in certain momentum-time windows, which offers a new perspective to
interpret the experimental observations. At last, we calculate the Contact
dynamics to investigate how the short-range correlations are built up in
quenched quantum gases [11, 14, 18, 25], since the Contact is a universal
quantity to characterize the property of short-range interactions [54, 55].
Recently, the Contact dynamics also receives the experimental interest [1, 56,
57, 58].

##  II Universal dynamic scaling

Theorem. If the initial density matrix $\rho_{0}=\int d\kappa
w_{\kappa}|\psi_{\kappa}\rangle\langle\psi_{\kappa}|$ and the quenched
Hamiltonian $\hat{H}({\bf p_{1}},{\bf p_{2}},...)$ both have a continuous
scaling symmetry, i.e.,

$\begin{cases}w_{\zeta\kappa}=\zeta^{\delta}w_{\kappa}\\\ \hat{H}(\zeta{\bf
p_{1}},\zeta{\bf p_{2}},...)=\zeta^{\gamma}\hat{H}({\bf p_{1}},{\bf
p_{2}},...).\end{cases}$

where, $\delta$ and $\gamma$ denote the scaling exponents. The momentum
distribution $n({\bf k},t)$ will exhibit a dynamic scaling behavior in the
momentum-time domain, which can be expressed as

$n({\bf k},t)=\tilde{t}^{\alpha}n(\tilde{t}^{\beta}{\bf k},t_{0})$ (1)

with $\beta=1/\gamma$ and $\alpha=2d/\gamma$. Here, $\kappa$ represents the
set of the quantum numbers for the eigenstate $|\psi_{\kappa}\rangle$. For
example, for non-interacting quantum gases, $\kappa=\\{\bf
p_{1},p_{2},...\\}$, and for an interacting two-body system with no bound
states, $\kappa=\\{\bf P,q\\}$ with ${\bf P}$ and ${\bf q}$ being the total
and relative momentum respectively. $\zeta$ is a positive number (i.e., the
scaling factor). $\tilde{t}=t/t_{0}$ and $d$ denotes the dimensionality of the
system. The proof is displayed in the following.

For a given initial density matrix $\rho_{0}=\int d\kappa
w_{\kappa}|\psi_{\kappa}\rangle\langle\psi_{\kappa}|$ at time $t=0$, the
dynamics of the momentum distribution $n({\bf k},t)$ governed by the quenched
Hamiltonian $\hat{H}$ at time $t>0$ can be expressed as

$n({\bf k},t)=\int d\kappa
w_{\kappa}\langle\psi_{\kappa}|e^{it\hat{H}}\hat{n}_{\bf
k}e^{-it\hat{H}}|\psi_{\kappa}\rangle$ (2)

For simplicity, we define a scaling operator $\hat{S}_{\zeta}$ as $\langle{\bf
p_{1}},{\bf p_{2}},...|\hat{S}_{\zeta}|\phi_{\kappa}\rangle=\langle{\bf
p_{1}}/\zeta,{\bf
p_{2}}/\zeta,...|\phi_{\kappa}\rangle\equiv\zeta^{Nd}\langle{\bf p_{1}},{\bf
p_{2}},...|\phi_{\zeta\kappa}\rangle$, with $N$ being the total particle
number (see the appendix). By inserting the basis of the eigenstates of the
quenched Hamiltonian $\hat{H}$, one can connect the momentum distribution
$n({\bf k},t)$ with $n(\zeta{\bf k},t\zeta^{-\gamma})$, which is displayed as
follows (see the appendix).

$n({\bf k},t)=\zeta^{2d}n(\zeta{\bf
k},t\zeta^{-\gamma})\equiv\tilde{t}^{\alpha}n(\tilde{t}^{\beta}{\bf k},t_{0})$
(3)

Note that $\delta=-Nd$ from the constraint $\int d\kappa w_{\kappa}=1$. Here,
we have used the property that $|\phi_{\zeta\kappa}\rangle$ is also the
eigenfunction of the Hamiltonian with the eigenvalue
$\zeta^{\gamma}E_{\kappa}$ [59]. In the following, we demonstrate this theorem
by directly calculating the quench dynamics of the momentum distribution in a
two-body system as well as in a many-body system via the framework of the
high-temperature virial expansion.

Two-body system. We first consider a two-body system with equal mass ($m$) and
an s-wave interaction between them. Essentially, it can be transformed into a
one-body problem and the Hamiltonian can be written as
$\hat{H}=-\nabla^{2}/m+\hat{V}({\bf r})$ in the relative coordinate. We take
the reduced Planck constant and Boltzmann constant equal to unity for
simplicity in this paper. If the system is quenched to a finite scattering
length $a_{s}$ at $t=0$ with an initial state $|{\bf k}={\bf 0}\rangle$, the
momentum distribution $n({\bf k},t)$ for finite momenta (${\bf k}>0$) after
the quench can be expressed as

$n({\bf k},t)=|\frac{1}{2\pi
i}\int_{-\infty}^{+\infty}d\omega\frac{t_{2}(s)e^{-i\omega
t}}{s(s-\varepsilon_{\bf k})}|^{2}$ (4)

with the two-body scattering matrix

$t_{2}(s)=\frac{4\pi}{m}\frac{1}{a_{s}^{-1}-\sqrt{-ms}}$ (5)

Here, $s=\omega+i0^{+}$ and $\varepsilon_{\bf k}={\bf k}^{2}/m$. For any
finite scattering length, the dynamics of the momentum distribution does not
follow Eq. (1), due to the breakdown of the scale invariance. However, in the
unitary limit ($a_{s}\rightarrow\infty$), the scale invariance recovers and
Eq. (4) can be simplified to

$n({\bf k},t)=\frac{64\pi t}{mk^{4}}|F(\frac{1-i}{\sqrt{2m}}k\sqrt{t})|^{2}$
(6)

where, the function $F(z)=1-\frac{\sqrt{\pi}}{2}\frac{{\rm
erf}(z)}{z}e^{z^{2}}$ with ${\rm erf}(z)$ being the error function. One can
see that it indeed satisfies Eq. (1) and exhibits the dynamic scaling with the
scaling exponents $\beta=1/2$ and $\alpha=3$ as the theorem says.

If one considers the weak interacting limit, $a_{s}\rightarrow 0$, to the
lowest order, the momentum distribution can be written as

$n({\bf k},t)=\frac{32\pi^{2}a_{s}^{2}}{k^{4}}[1-{\rm cos}(\frac{k^{2}t}{m})]$
(7)

Remarkably, it also displays the dynamic scaling behavior but with the scaling
exponents $\beta=1/2$ and $\alpha=2$. Because the scattering length is much
smaller than other relevant length scales, the system remains scale-invariant
approximately and our theorem is still valid in such cases. But the scaling
exponents can depend on the specific approximation, which is also indicated in
Eq. (3).

Many-body system. For a many-body system, the particle density (or the
distance between nearest particles) introduces a new length scale, which
generally prevents the system from owning the continuous scaling symmetry,
such as the non-interacting or unitary Fermi gas. However, in the low-density
limit or the high-energy regime, the influence of this length scale can be
negligible and the many-body system may also exhibit scale invariant, similar
to the two-body system. In the following, we employ the high-temperature
virial expansion to demonstrate the dynamic scaling, where our theorem is
applicable.

Figure 1: The momentum distribution $\delta n_{\bf k}$ at the large momentum
region for various times $t/t_{\lambda}$ = 0.02(black), 0.04(red), 0.06(blue),
0.08(pink), 0.10(green), 0.12(navy) in the quench dynamics of quantum gases.
Here, $\lambda=\sqrt{2\pi/(mT)}$ is the thermal wavelength, while
$t_{\lambda}=1/T$. $\tilde{t}=t/t_{0}$ with $t_{0}/t_{\lambda}=0.02$. The
results in the BEC-BCS crossover are displayed with the scattering length
labelled in the first row. In the second row, the corresponding scalings of
both the horizontal and vertical axes are taken via Eq. (1), to demonstrate
whether there exists a dynamic scaling behavior. The scaling exponents
$\alpha$ and $\beta$ are shown in the corresponding graphs.

If one considers that a quantum gas is quenched from a non-interacting
Hamiltonian $\hat{H}_{0}$ with an equilibrium state to an interacting
Hamiltonian $\hat{H}$ at time $t=0$, the evolution of the momentum
distribution $n({\bf k},t)$ for $t>0$ can be expressed as

$n({\bf
k},t)=\frac{\text{Tr}[e^{-\beta(\hat{H}_{0}-\mu\hat{N})}e^{it\hat{H}}\hat{n}_{\bf
k}e^{-it\hat{H}}]}{\text{Tr}[e^{-\beta(\hat{H}_{0}-\mu\hat{N})}]}$ (8)

Here $\beta=1/(k_{B}T)$ with $T$ being the temperature. $\mu$ and $N$ are the
chemical potential and the total number of particles, respectively. At high
temperature, it can be expanded in powers of the fugacity $z=e^{\beta\mu}\ll
1$ in the framework of virial expansion [27]. In the following, we will
consider the change of the momentum distribution $\delta n({\bf k},t)=n({\bf
k},t)-n({\bf k},0)$ for simplicity, since it represents the property of the
quenched Hamiltonian $\hat{H}$. To the second order of $z$, it can be
expressed as

$\displaystyle\delta n({\bf k},t)=$ $\displaystyle 2z^{2}\sum_{\bf
k_{1},k_{2}}f({\bf k_{1}})f({\bf k_{2}})$ $\displaystyle\\{2{\rm Re}[g({\bf
k_{1}},{\bf k_{2}},t)]\delta_{{\bf k_{1}},{\bf k}}+|g({\bf k_{1}},{\bf
k_{2}},t)|^{2}\\}$ (9)

with

$g({\bf k_{1}},{\bf k_{2}},t)=\frac{1}{2\pi
i}\int_{-\infty}^{+\infty}d\omega\frac{t_{2}(s)e^{-it(\omega-\varepsilon_{\bf
q_{1}})}}{(s-\varepsilon_{\bf q_{1}})(s-\varepsilon_{\bf q_{2}})}$ (10)

Here, $s=\omega+i0^{+}$, ${\bf q}_{1}=({\bf k}_{1}-{\bf k}_{2})/2$, ${\bf
q}_{2}={\bf k}-({\bf k}_{1}+{\bf k}_{2})/2$, $\varepsilon_{\bf q}={\bf
q}^{2}/m$ and $f({\bf k_{1}})=e^{-2\beta\varepsilon_{\bf k_{1}}}$. One can
calculate Eq. (9) numerically to obtain the dynamics of the momentum
distribution.

The initial distribution function $f({\bf k_{1}})=e^{-2\beta\varepsilon_{\bf
k_{1}}}$ does not have a scaling symmetry. However, if one considers the large
momentum region $k\lambda\gg 1$ and $t/t_{\lambda}\ll 1$($t_{\lambda}=1/T$),
this Gaussian function can be regarded as a Dirac delta function approximately
and the scaling symmetry recovers. Therefore, according to our theorem, a
scaling dynamics can occur at unitarity [60] and indeed this is ture as shown
in Fig. 1. Moreover, we calculated the results in the whole BEC-BCS crossover.
The finite scattering length breaks the scaling symmetry of the Hamiltonian
and the dynamic scaling disappears. Nevertheless, in the weak interacting
limit, i.e, $a_{s}\rightarrow 0$, the scattering length is much smaller than
all other relevant length scales in the system and the scaling symmetry
recovers approximately, which further leads to the recurring scaling dynamics.
Note that to the lowest order (i.e., the order of $a_{s}^{2}$), $\beta=1/2$
and $\alpha=2$ due to the prefactor $a_{s}^{2}$ in this approximation, which
does not follow the theorem quantitatively. All the numerical results agree
well with the above analysis based on the theorem. Our results imply that the
scaling dynamics can occur in a specific momentum-time window, as long as the
initial state and the Hamiltonian are scale-invariant approximately in this
domain.

Actually, if one takes the initial distribution function as $f({\bf
k_{1}})=\delta({\bf k_{1}})$ mathematically, Eq. (9) can be simplified to

$\delta n({\bf k},t)=2z^{2}|g({\bf 0},{\bf 0},t)|^{2}$ (11)

which is the same with the result of the two-body system (see Eq. (4)), except
the prefactor $2z^{2}$. Thus, it is straightforward to show that $\delta
n({\bf k},t)=\tilde{t}^{3}\delta n(\tilde{t}^{1/2}{\bf k},t_{0})$ for all
finite momenta at unitarity and $\delta n({\bf k},t)=\tilde{t}^{2}\delta
n(\tilde{t}^{1/2}{\bf k},t_{0})$ as $a_{s}\rightarrow 0$ (to the order of
$a_{s}^{2}$), which are in excellent agreement with our numerical results (see
Fig. 1).

Figure 2: The quench dynamics of the momentum distribution $|\delta n_{\bf
k}|$ at finite momentum region for various times $t/t_{\lambda}$= =
0.1(black), 0.2(red), 0.3(blue), 0.4(pink), 0.5(green). The results for the
unitarity are plotted with unscaled axes (a) and scaled axes (b), while they
are only demonstrated in scaled axes for $\lambda/a_{s}=-100$ (c) and
$\lambda/a_{s}=100$ (d). According to the theorem, they are not expected to
display any strict dynamic scaling behavior, due to the breakdown of the scale
invariance of the initial density matrix. However, an approximate scaling
dynamics can still be seen in all three cases with smaller scaling exponents
$\alpha$ and $\beta$ displayed in the corresponding graphs.

Approximate dynamic scaling. Surprisingly, even if the scale-invariant
condition is not strictly satisfied, an approximate scaling dynamics can still
occur in certain momentum-time windows as shown in Fig. 2. If one considers
the momentum region $k\lambda\in[4,10]$, the strict dynamic scaling is not
expected given the missing scaling invariance of the initial density matrix.
However, there exists an approximate dynamic scaling with modified scaling
exponents, which may arise from the approximate scaling behaviors of the
initial density matrix (or the main contributing part) with different scaling
exponents in certain momentum regions.

On the one hand, it demonstrates that the dynamic scaling can be robust
against the proper breakdown of the continuous scaling symmetry for either the
initial density matrix or the Hamiltonian. On the other hand, it could be
closely related to the recent experimental observations [5, 6], since the
experimental condition does not strictly satisfy the scaling symmetry either
and the scaling exponents vary significantly in different systems. Here, we
would like to emphasize that there are two key features, on which both the
experiment and our theory agree well with each other qualitatively. One is
that both the scaling exponents $\beta$ and $\alpha$ are smaller than that of
the strict dynamic scaling in the theorem. The other is the scaling exponents
depend on the chosen momentum-time windows. Therefore, our results provide a
new perspective to interpret the experimental observations.

##  III Contact dynamics

In the following, we further study universal features of the Contact dynamics
in quenched quantum gases, which correspond to the large-momentum limit. In
the framework of high-temperature virial expansion, it can be directly
obtained from the momentum distribution, through the relation
$C(t)=\lim_{k\rightarrow\infty}k^{4}\delta n({\bf k},t)$. To the second order
of $z$, the Contact dynamics in the BEC-BCS crossover are displayed in Fig. 3.
Owing to the initial non-interacting equilibrium state, they all start from
zero and gradually approach to a steady state at long evolution times. At
unitarity, one can derive an analytic form, expressed as

$C(a_{s}\rightarrow\infty)=\frac{64z^{2}}{\lambda^{4}}{\rm
arctan}(\frac{t}{t_{\lambda}})$ (12)

Figure 3: The Contact dynamics after the quench for
$\lambda/a_{s}=$-3(black),-1(red),0(blue),1(pink),3(green). They all start
from zero and approach to a steady value at long times. On the BEC side, it
exhibits damping oscillations.

In the whole BEC-BCS crossover, as $t\rightarrow 0$, the Contact can be
expanded in powers of $t$. To the lowest order, it can be written as

$C(t\rightarrow 0)=\frac{64z^{2}}{\lambda^{4}t_{\lambda}}t$ (13)

which is independent of the scattering length and consistent with Ref. [61]’s
result. Actually in this limit, one can obtain the momentum distribution
$\delta n({\bf k},t)\approx(64z^{2}/\lambda^{4}t_{\lambda})t/k^{4}$ for large
momenta. As expected, it satisfies the dynamic scaling relation (see Eq. (1)),
because the scattering length is much larger than other relevant length scales
in this regime and the system can be regarded as being scale-invariant. On the
BEC side, there exist some damping oscillations, which are associated with the
dimer state. Roughly speaking, they result from the coupling between the dimer
state and the scattering states after the quench. The decoherence of the
scattering states further leads to the damping, due to different scattering
energies.

For $t\rightarrow\infty$, one can obtain the Contact of the final steady
state, which can be written analytically as

$\displaystyle C(\infty)=$
$\displaystyle\frac{z^{2}}{\lambda^{4}}[(32\pi-\frac{64\lambda^{2}}{a_{s}^{2}}\Theta(a_{s}))$
$\displaystyle+\frac{16\sqrt{2}\lambda}{a_{s}}(\pi+\frac{2\lambda^{2}}{a_{s}^{2}}\Theta(a_{s}))e^{\frac{\lambda^{2}}{2\pi
a_{s}^{2}}}{\rm erfc}(\frac{\lambda}{\sqrt{2\pi}|a_{s}|})]$

Here, $\Theta(x)$ and erfc($x$) are respectively the unit step function and
the complementary error function. It exhibits a maximum around the unitarity,
as displayed in Fig. 4(a). We further define a half-way time $\tau$ as
$C(t=\tau)=C(\infty)/2$ to quantify how fast the Contact evolves [3, 27]. The
results for the BEC-BCS crossover are displayed in Fig. 4(b). Surprisingly, a
maximum occurs around the unitarity, indicating the evolving time is
determined mainly by the difference between the final state and the initial
state (see Fig. 4(a)), instead of the interaction strength.

Figure 4: The Contact dynamics in the BEC-BCS crossover. (a) The Contact at
infinite time ($t\rightarrow\infty$) in the whole BEC-BCS crossover, obtained
from Eq. (LABEL:Contf). (b)The half-way time $\tau$ as a function of the
interaction. (c)The relative value at the peak $\Delta
C_{p}=C_{p}-C(t\rightarrow\infty)$ for
$\lambda/a_{s}=$3(black),4(red),5(blue), which obeys a power-law decay with
the same power approximately. (d) The frequencies of oscillations at various
interactions on the BEC side, with a linear fitting (red line).

We quantitatively investigate the damping oscillations on the BEC side to
reveal the relation between oscillations and the dimer state. The relative
values at the peak $\Delta C_{p}=C_{p}-C(\infty)$ obey a universal power-law
decay with the same power approximately for various interactions (see Fig.
4(c)), implying the intial state plays an important role on the damping rate
instead of the interaction. However, the frequencies are proportional to
$a_{s}^{-2}$ approximately (see Fig. 4(d)), which is associated to the energy
difference between the dimer state and the scattering state. It will be
determined by the dimer’s binding energy when the energy of the scattering
state can be negligible. Therefore, these oscillations of the Contact dynamics
can be used as a tool to detect possible dimer states in experiments.

##  IV Conclusion

In summary, we find a theorem showing that the continuous scale invariance of
the Hamiltonian and the initial density matrix can give rise to the universal
scaling dynamics. We first demonstrate it in the two-body system analytically.
Then, for quantum gases quenched from the non-interacting equilibrium state,
the dynamic scaling appears at large momentum region for both the unitary
limit and the weak interacting limit. Remarkably, an approximate dynamic
scaling with smaller scaling exponents is also found at finite momentum
region, which may offer a new interpretation to the experimental observations.
Our results can also guide the future exploration of the scaling dynamics in
various systems. The calculated Contact dynamics exhibits several novel
features such as a maximum of the half-way time around the unitary limit and
the damping oscillations on the BEC side, which can be explored by the
experiment in near future [3, 6, 56, 57].

## Acknowledgments

We thank Zheyu Shi, Pengfei Zhang, Ran Qi, Zhiyuan Yao, Lei Pan, Ren Zhang,
and Hui Zhai for inspiring discussion. The project was supported by NSFC Grant
No. 12004049, Fund of State Key Laboratory of IPOC (BUPT) (No. 600119525,
505019124).

## Appendix: Proof of the theorem

In the following, we perform the derivation of the theorem (i.e., Eq. (3) in
the main text) step by step. We first display the scaling behavior of each
term separately and then put them together to give the final result. Given the
quenched Hamiltonian is scale-invariant, if the wavefunction
$|\phi_{\eta}\rangle$ is an eigenfunction of the Hamiltonian with the
eigenvalue $E_{\eta}$( $\eta$ denotes $Nd$ quantum numbers with $N$ and $d$
being the total particle number and the dimensionality respectively), a scaled
wavefunction $|\phi_{\zeta\eta}\rangle$ ($\zeta$ is a positive number) will
also be an eigenfunction with the eigenvalue
$E_{\zeta\eta}=\zeta^{\gamma}E_{\eta}$ [59]. Therefore, if assuming
$\phi_{\zeta\eta}(\zeta{\bf k_{1}},\cdots,\zeta{\bf k_{N}})=A\phi_{\eta}({\bf
k_{1}},\cdots,{\bf k_{N}})$ (A is a real number), one can have

$\displaystyle\langle\phi_{\zeta\eta_{1}}|\phi_{\zeta\eta_{2}}\rangle$
$\displaystyle=\int d{\bf k_{1}}\cdots d{\bf
k_{N}}\phi_{\zeta\eta_{1}}^{*}({\bf k_{1}},\cdots,{\bf
k_{N}})\phi_{\zeta\eta_{2}}({\bf k_{1}},\cdots,{\bf k_{N}})$
$\displaystyle=\zeta^{Nd}\int d{\bf k_{1}}\cdots d{\bf
k_{N}}\phi_{\zeta\eta_{1}}^{*}(\zeta{\bf k_{1}},\cdots,\zeta{\bf
k_{N}})\phi_{\zeta\eta_{2}}(\zeta{\bf k_{1}},\cdots,\zeta{\bf k_{N}})$
$\displaystyle=\zeta^{Nd}\int d{\bf k_{1}}\cdots d{\bf
k_{N}}A\phi_{\eta_{1}}^{*}({\bf k_{1}},\cdots,{\bf
k_{N}})A\phi_{\eta_{2}}({\bf k_{1}},\cdots,{\bf k_{N}})$
$\displaystyle=A^{2}\zeta^{Nd}\langle\phi_{\eta_{1}}|\phi_{\eta_{2}}\rangle$

According to the condition
$\langle\phi_{\zeta\eta_{1}}|\phi_{\zeta\eta_{2}}\rangle=\delta(\zeta\eta_{1}-\zeta\eta_{2})=\zeta^{-Nd}\delta(\eta_{1}-\eta_{2})$,
one can obtain $A=\zeta^{-Nd}$ and
$\langle\phi_{\zeta\eta_{1}}|\phi_{\zeta\eta_{2}}\rangle=\zeta^{-Nd}\langle\phi_{\eta_{1}}|\phi_{\eta_{2}}\rangle$.
Based on this, it is straightforward to derive that

$\langle\psi_{\kappa}|e^{it\hat{H}}|\phi_{\eta}\rangle=\zeta^{Nd}\langle\psi_{\zeta\kappa}|e^{it\zeta^{-\gamma}\hat{H}}|\phi_{\zeta\eta}\rangle.$
(16)

For the term involved with the momentum distribution, it can be transformed as
follows.

$\displaystyle\langle\phi_{\eta_{1}}|\hat{n}_{\bf k}|\phi_{\eta_{2}}\rangle$
$\displaystyle=\int d{\bf k_{1}}\cdots d{\bf k_{N}}\phi_{\eta_{1}}^{*}({\bf
k_{1}},\cdots,{\bf k_{N}})$ $\displaystyle\hskip
14.22636pt\cdot\sum_{m}\delta({\bf k_{m}}-{\bf k})\phi_{\eta_{2}}({\bf
k_{1}},\cdots,{\bf k_{N}})$ $\displaystyle=\sum_{m}\int d{\bf
k_{m}}f_{\eta_{1},\eta_{2}}({\bf k_{m}})\delta({\bf k_{m}}-{\bf k})$
$\displaystyle=\sum_{m}\int d{\bf
k_{m}}\zeta^{3d}f_{\zeta\eta_{1},\zeta\eta_{2}}(\zeta{\bf
k_{m}})\delta(\zeta{\bf k_{m}}-\zeta{\bf k})$
$\displaystyle=\zeta^{2d}\sum_{m}\int d{\bf
k_{m}}f_{\zeta\eta_{1},\zeta\eta_{2}}({\bf k_{m}})\delta({\bf k_{m}}-\zeta{\bf
k})$ $\displaystyle=\zeta^{2d}\langle\phi_{\zeta\eta_{1}}|\hat{n}_{\zeta{\bf
k}}|\phi_{\zeta\eta_{2}}\rangle$

Here, we have first integrated over all other momenta (except ${\bf k_{m}}$)
and then regularized the scaling divergent term (see Eq. (LABEL:tm1)), to
obtain $f_{\eta_{1},\eta_{2}}({\bf k_{m}})=\int d{\bf k_{1}}\cdots d{\bf
k_{m-1}}d{\bf k_{m+1}}\cdots d{\bf k_{N}}\phi_{\eta_{1}}^{*}({\bf
k_{1}},\cdots,{\bf k_{N}})\phi_{\eta_{2}}({\bf k_{1}},\cdots,{\bf k_{N}})$ in
one-particle space, which satisfies $f_{\eta_{1},\eta_{2}}({\bf
k_{m}})=\zeta^{2d}f_{\zeta\eta_{1},\zeta\eta_{2}}(\zeta{\bf k_{m}})$.

With Eq. (16) and (LABEL:tm3), one can derive the dynamic scaling of the
momentum distribution as follows.

$\displaystyle n({\bf k},t)$ $\displaystyle=\int d\kappa
w_{\kappa}\langle\psi_{\kappa}|e^{it\hat{H}}\hat{n}_{\bf
k}e^{-it\hat{H}}|\psi_{\kappa}\rangle$ $\displaystyle=\int d\kappa\int
d\eta_{1}\int
d\eta_{2}w_{\kappa}\langle\psi_{\kappa}|e^{it\hat{H}}|\phi_{\eta_{1}}\rangle\langle\phi_{\eta_{1}}|\hat{n}_{\bf
k}|\phi_{\eta_{2}}\rangle$ $\displaystyle\hskip
14.22636pt\cdot\langle\phi_{\eta_{2}}|e^{-it\hat{H}}|\psi_{\kappa}\rangle$
$\displaystyle=\zeta^{2d}\int d\kappa
w_{\kappa}\langle\psi_{\kappa}|e^{it\zeta^{-\gamma}\hat{H}}\hat{n}_{\zeta{\bf
k}}e^{-it\zeta^{-\gamma}\hat{H}}|\psi_{\kappa}\rangle$
$\displaystyle=\zeta^{2d}n(\zeta{\bf k},t\zeta^{-\gamma})$
$\displaystyle\equiv\tilde{t}^{\alpha}n(\tilde{t}^{\beta}{\bf k},t_{0})$

Here, $w_{\kappa}=\zeta^{-\delta}w_{\zeta\kappa}$ with $\delta=-Nd$, which can
be obtained from the relation $\int d\kappa w_{\kappa}=\zeta^{Nd}\int d\kappa
w_{\zeta\kappa}=\zeta^{Nd+\delta}\int d\kappa w_{\kappa}$. We set
$\tilde{t}=t/t_{0}$, the scaling exponent $\beta=1/\gamma$ and
$\alpha=2d/\gamma=2d\beta$. Thus, the theorem is proved.

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