# Infinitely many normalized solutions for a quasilinear Schrödinger equation

Xianyong Yang1 Fukun Zhao2
1\. School of Mathematics and Computer Science, Yunnan Minzu University,
Kunming 650500 P.R. China
2\. School of Mathematics, Yunnan Normal University, Kunming 650500 P.R. China
Corresponding author. E-mail<EMAIL_ADDRESS>
###### Abstract

In this paper, we are concerned with the following quasilinear Schrödinger
equation

$\begin{cases}-\Delta u+\mu
u-\Delta(u^{2})u=g(u)~{}~{}\hbox{in}~{}~{}\mathbb{R}^{N}\\\
\int_{\mathbb{R}^{N}}|u|^{2}dx=m,\\\ u\in H^{1}(\mathbb{R}^{N}),\end{cases}$

where $N\geq 2$, $m>0$ is a given constant, $\mu\in\mathbb{R}$ is a Lagrange
multiplier, and $g$ satisfies the well-known Berestycki–Lions condition. The
existence of infinitely many normalized solutions is obtained via a minimax
argument.

_Keywords: Quasilinear Schrödinger equation; Normalized solutions;
Berestycki–Lions nonlinearity._

Mathematics Subject Classification: 35A15, 35B38, 35J10, 35J62.

##  1 Introduction and main results

Assume $N\geq 2$ and consider the following quasilinear Schrödinger equation

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

which is related to the standing waves $\psi(x,t)=u(x)e^{-iEt}$ of the
following Schrödinger equation

$-i\psi_{t}-\Delta\psi+\mu\psi-h(|\psi|^{2})\psi-\Delta(|\psi|^{2})\psi=0,$
(1.2)

where $i$ denotes the imaginary unit, $\lambda=\mu-E$, $g(t)=h(|t|^{2})t$. The
quasilinear Schrödinger equation (1.2) is derived as a model of several
physical phenomena. For example, it was used for the superfluid film equation
in plasma physics by Kurihara [20]. It also appears in the theory of
Heisenberg ferromagnetism and magnons (see [19] and [36]), in dissipative
quantum mechanics and the condensed matter theory [33]. In literature the
study of standing waves of (1.1) has been pursed in two main directions, which
opened two different challenging research fields.

The first topic regards the search for solutions of (1.1) with a prescribed
frequency $\mu$ and free mass, which is called unconstrained problem. To this
aim, we consider the associated energy functional to (1.1)

$I(u)={1\over 2}\int_{\mathbb{R}^{N}}\big{(}1+2u^{2}\big{)}|\nabla
u|^{2}dx+\lambda\int_{\mathbb{R}^{N}}u^{2}dx-\int_{\mathbb{R}^{N}}G(u)dx$

defined on the natural space

$X=\bigg{\\{}u\in H^{1}(\mathbb{R}^{N})|\int_{\mathbb{R}^{N}}u^{2}|\nabla
u|^{2}dx<\infty\bigg{\\}},$

where $G(u)=\int_{0}^{u}g(s)ds.$ A function $u\in X$ is called a weak solution
of (1.1) if

$\lim_{t\rightarrow 0}\frac{I(u+t\varphi)-I(u)}{t}=0$

for all $\varphi\in C_{0}^{\infty}(\mathbb{R}^{N}).$ Recall that
$22^{\ast}=\frac{4N}{N-2}$ corresponds to a critical exponent of (1.1) when
$g(u)=|u|^{r-2}u$ and $N\geq 3$, see [26]. Different from the semilinear case
that $\kappa=0$, there is a non-convex term $\Delta(u^{2})u$ appears in the
quasilinear problem (1.1), and hence the search of solutions of such an
equation must face a challenge: the functional

$W(u)=\int_{\mathbb{R}^{N}}u^{2}|\nabla u|^{2}dx$

associated with the quasilinear term is non differentiable in the space $X$
when $N\geq 2$. Consequently, the standard critical point theory can not be
applied directly. In order to overcome this difficulty, several approaches
have been successfully developed in the last decade, such as the constraint
minimization ([29]), the Nehari manifold method ([27]), the perturbation
method ([32]) and the nonsmooth critical point theory ([6, 22]). It seems that
the first contribution to the equation (1.1) via variational methods dues to
the work [11]. After this work, a series of subsequent studies have been done
concerning with the existence, multiplicity and qualitative properties of
solutions to (1.1), see for example [1, 12, 18, 25, 30, 28, 31, 35].

The second topic of investigation of the problem (1.1) consists of prescribing
the mass $m>0$, thus conserved by $\psi$ in time

$\int_{\mathbb{R}^{N}}\psi(x,t)dx=m,\forall t\in[0,+\infty),$

and letting the frequency $\mu$ to be free. Such a problem is usually called
constrained one. The constrained problem has a significant relevance in
Physics, not only for the quantum probability normalization, but also because
the mass may also have specific meanings, such as the power supply in
nonlinear optics, or the total number of atoms in Bose-Einstein condensation.
Moreover, the investigation of constrained problems can give a better insight
of the dynamical properties, as the orbital stability of solutions of (1.2),
see for example [9]. For the case $\kappa=0$, (1.1) is reduced to the
semilinear Schrödinger equation whose normalized solutions are studied widely
in recent years. We can refer the interested readers to [7, 9, 13, 15, 39] and
the references therein.

In recent years, there has been an increasing interest in studying of
normalized solutions to the quasilinear Schrödinger equation. To obtain
normalized solutions, a natural method is to look for the minimizer of the
following constrained problem:

$e(m):=\inf_{u\in S(m)}E(u),$

where

$S(m)=\bigg{\\{}u\in H^{1}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}u^{2}|\nabla
u|^{2}dx<\infty~{}\text{and}~{}\int_{\mathbb{R}^{N}}|u|^{2}dx=m\bigg{\\}},$

and

$E(u)={1\over 2}\int_{\mathbb{R}^{N}}\big{(}1+2u^{2}\big{)}|\nabla
u|^{2}dx-\int_{\mathbb{R}^{N}}G(u)dx.$

It is proved in[10, Theorem 4.6] that each minimizer of $e(m)$, corresponds to
a Lagrange multiplier $\lambda<0$ such that $(u,\lambda)$ solves weakly
$\eqref{eqA1}$ when $g(u)=|u|^{r-2}u$. Moreover, it is easy to see that
$e(m)>-\infty$ for $2<r<4+\frac{4}{N}$ and $e(m)=-\infty$ for
$4+\frac{4}{N}<r<\frac{4N}{N-2}$. From this point of view, $r=4+\frac{4}{N}$
plays a critical exponent for the constrained problem. By this constrained
argument, Colin, Jeanjean and Squassina [10] first prove the following result
for $L_{2}$ -constrained problem (see also [16]).

###### Theorem 1.1

Assume that $g(u)=|u|^{r-2}u$, where $r\in\big{(}2,\frac{4N}{N-2}\big{)}$ if
$N\geq 3$ and $r\in(2,\infty)$ if $N=1,2$. Then

  * $(1)$

For all $m>0$ and $r\in(2,2+\frac{4}{N})$, $e(m)\in(-\infty,0)$ and $e(m)$ has
a minimizer.

  * $(2)$

For $r\in(2+\frac{4}{N},4+\frac{4}{N})$, there exists $m(r,N)>0$ such that

(i)If $m\in\big{(}0,m(r,N)\big{)},e(m)=0$ and $e(m)$ has no minimizer;

(ii)If $m=m(r,N),m(c)=0$ and $e(m)$ has a minimizer;

(iii) If $m\in\big{(}c(r,N),+\infty\big{)},e(m)<0$ and $e(m)$ has a minimizer;

  * $(3)$

For all $m>0$, $r\in\big{(}4+\frac{4}{N},\frac{4N}{N-2}\big{)}$ if $N\geq 3$
and $r\in\big{(}4+\frac{4}{N},\infty)$ if $N=1,2$, there holds $e(m)=-\infty$.

  * $(4)$

For $r=4+\frac{4}{N}$, there exists $m_{N}>0$ such that

$e(m)=\begin{cases}0,~{}~{}~{}~{}~{}~{}\hbox{if}~{}m\in(0,m_{N}),\\\
-\infty,~{}~{}\hbox{if}~{}m\in(m_{N},\infty).\end{cases}$

Moreover, $e(m)$ has no minimizer for all $m>0$.

  * $(5)$

The standing waves obtained as minimizer of $e(m)$ are orbitally stable.

Interestingly, by a combination of analytical and numerical arguments in
dimension $N=3$, Caliari and Squassina [5] studied the explicit bounds on
$m(r,N)$ and $m_{N}$ given in Theorem 1.1. In particular, for
$r=\frac{13}{3}$, they find that there exists
$\underline{m},~{}\overline{m}>0$ such that

$e(m)=\begin{cases}0,~{}~{}~{}~{}~{}~{}\hbox{if}~{}m\in(0,\underline{m}),\\\
-\infty,~{}~{}\hbox{if}~{}m\in(\overline{m},\infty).\end{cases}$

Moreover, $\underline{m}\approx 19.73$ and $\overline{m}\approx 85.09$.
However, the constrained minimization argument does not use much of smoothness
of the variational functional, so it is not suitable for dealing with the
existence of multiple solutions. Subsequently, Jeanjean, Luo and Wang [17]
prove the existence of two normalized solutions for $L_{2}$ -subcritical case
via the perturbation approach. More precisely, they proved the following
result.

###### Theorem 1.2

Assume that $N\geq 1$ and $g(u)=|u|^{r-2}u$ with
$r\in(2+\frac{4}{N},4+\frac{4}{N})$.

  * $(1)$

There exists a $m_{0}\in(0,m(r,N))$ such that for any $m\in(0,m(r,N))$ the
functional $E$ admits a critical point $v_{m}$ on $S(m)$ which is a local
minimum of $E$ when $m\in(m_{0},m(r,N))$ and a global minimum of $E$ when
$m\in(m(r,N),+\infty)$. In particular,

$E(v_{m})\begin{cases}>0,~{}~{}\hbox{if}~{}m\in(m_{0},m(r,N));\\\
=0,~{}~{}\hbox{if}~{}m=m(r,N);\\\
<0,~{}~{}\hbox{if}~{}m=(m(r,N),+\infty).\end{cases}$

  * $(2)$

Assuming in addition that $r\in(2+\frac{4}{N},2+\frac{4}{N-2})$ if $N\geq 5$
there exists a second critical point $u_{m}$ on $S(m)$ which satisfies

(i) $E(u_{m})>0$ for all $m\in(m_{0},+\infty)$ and is a mountain pass level;

(ii)$E(u_{m})>E(v_{m})$ for any $m\in(m_{0},+\infty)$.

Very recently, Ye and Yu [47] study the $L_{2}$-critical problem to (1.1) if
$g(u)=|u|^{r-2}r$ for $r=2+\frac{4}{N}$ and $r=4+\frac{4}{N}$. They prove the
following result.

###### Theorem 1.3

Assume that $g(u)=|u|^{r-2}u$. Then

  * $(1)$

For $r=2+\frac{4}{N}$, there exists $m_{N}>0$ such that

(i)If $m\in\big{(}0,m_{N}\big{]},e(m)=0$ and $E(u)$ has no critical point on
the constraint $S(m)$;

(ii) If $m\in\big{(}m_{N},+\infty\big{)},e(m)<0$ and $e(m)$ has a minimizer;

  * $(2)$

For $r=4+\frac{4}{N}$, there exists $m_{N}>0$ such that

(i) $E(u)$ has no critical point on the constraint $S(m)$ for all
$m\in\big{(}0,m_{N}\big{]}$;

(ii) $E(u)$ has a critical point on the constraint $S(m)$ for all
$m\in\big{(}m_{N},+\infty\big{)}$ when $N\leq 3$.

We also point out that some works focus on the existence and asymptotic
behavior of normalized solutions related to some quasilinear elliptic
equations with potential function, that is, $\mu$ is replaced by $V(x)$
satisfying different assumptions in (1.1), we can refer the interested readers
to [2, 21, 50, 48, 49] and the references therein for this case.

However, to the best of our knowledge, the existing results in references of
the constrained problem (1.1) mainly focus on the case where $g$ is a power
function. It seems that there is no work considering the existence of
normalized solutions for (1.1) when $g$ is a general nonlinearity. A natural
question is whether the nonlinearity $g(u)=|u|^{r-2}u$ can be relaxed to a
general nonlinearity $g$ or not? On the other hand, it seems that the
multiplicity of normalized solutions for (1.1) only can be found in Jeanjean,
Luo and Wang [17], where the authors partly obtained the existence of two
normalized solutions for (1.1) when $g(u)=|u|^{r-2}u$. Since such a power
nonlinearity is odd, another natural question is whether (1.1) has infinitely
many normalized solutions or not? Moreover, we note from Theorem 1.2 that
$r\in\big{(}2+\frac{4}{N},4+\frac{4}{N}\big{)}$ for $N\leq 4$ or
$r\in\big{(}2+\frac{4}{N},2+\frac{4}{N-2})$ for $r\geq 5$ are required.
However, it is well-known that the existence of solutions of (1.1) can be
obtained for the unconstrained problem when
$r\in\big{(}2,\frac{4N}{N-2}\big{)}$ and $N\geq 1$. So the last natural
questions is whether the scope of $r$ and the dimension of the space can be
relaxed to obtain multiple normalized solutions of (1.1)?

Motivated by the above results especially by [16] and [17], in the present
paper, we are concerned with the existence and multiplicity of normalized
solutions of the following quasilinear Schrödinger equation

$\begin{cases}-\Delta u+\mu
u-\Delta(u^{2})u=g(u)~{}~{}\hbox{in}~{}~{}\mathbb{R}^{N},\\\
\int_{\mathbb{R}^{N}}|u|^{2}dx=m,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(P_{\mu,m})\\\
u\in H^{1}(\mathbb{R}^{N}),\end{cases}$

where $g$ satisfies the well-known Berestycki–Lions condition, which is
regarded as an almost optimal assumption of the existence of solutions for
nonlinear Schrödinger equation (see [3, 4]). To state our main results, we
need the following assumptions.

  * $(g_{1})$

$g\in C(\mathbb{R},\mathbb{R})$.

  * $(g_{2})$

$\lim\limits_{s\rightarrow 0}\frac{g(s)}{s}=0$.

  * $(g_{3})$

For $p=4+\frac{4}{N}$,
$\lim\limits_{s\rightarrow\infty}\frac{|g(s)|}{|s|^{p-1}}=0$.

  * $(g_{4})$

There exists some $s_{0}>0$ such that $G(s_{0})>0$.

  * $(g_{5})$

$g(-s)=-g(s)$ for all $s\in\mathbb{R}$.

  * $(g_{6})$

For $q=2+\frac{4}{N}$, $\liminf\limits_{s\rightarrow
0}\frac{|g(s)|}{|s|^{q-2}s}=\infty$.

  * $(g_{7})$

$g(s)s\geq 0$ for all $s\in\mathbb{R}$.

###### Definition 1.1

For any $m>0$, a solution $u\in H^{1}(\mathbb{R}^{N})$ to (1.1) is called a
least energy normalized solution, if $u\in S(m)$ and

$E(u)=\min\\{E(u):u\in
S(m),\mu\in\mathbb{R}~{}\text{and}~{}(u,\mu)~{}\text{solves}~{}(P_{\mu,m})\\}.$

Now, our main results of this paper can be stated as follows.

###### Theorem 1.4

Assume that $(g_{1})-(g_{4})$ hold and $N\geq 2$. Then

  * $(1)$

There exists $m_{1}\geq 0$ such that $(P_{\mu,m})$ has at least a ground state
normalized solution for $m>m_{1}$.

  * $(2)$

In addition that $(g_{6})$, $(P_{\mu,m})$ has at least a ground state
normalized solution for any $m>0$.

###### Theorem 1.5

Assume that $(g_{1})-(g_{5})$ hold and $N\geq 2$. Then

  * $(1)$

For any $k\in\mathbb{N}$, there exists $m_{k}\geq 0$ such that $(P_{\mu,m})$
has at least $k$ radial solutions for $m>m_{k}$.

  * $(2)$

In addition that $(g_{6})$, $(P_{\mu,m})$ has infinitely many radial solutions
for any $m>0$.

We immediately conclude the following result from Theorem 1.5:

###### Corollary 1.1

Assume that $g(u)=|u|^{r-2}u$ hold and $N\geq 2$.

  * $(1)$

$(P_{\mu,m})$ has infinitely many radial solutions for any $2<r<2+\frac{4}{N}$
and $m>0$.

  * $(2)$

For any $k\in\mathbb{N}$, there exists $m_{k}>0$ such that $(P_{\mu,m})$ has
at least $k$ radial solutions for $2+\frac{4}{N}\leq r<4+\frac{4}{N}$ when
$m>m_{k}$.

###### Remark 1.1

If $e(m)<0$ for $r\in\big{(}2,4+\frac{4}{N})$, by Theorem 1.1 and Theorem 1.2,
we have $e(m)=b_{1}^{m}$ and $m_{1}=m(r,N)$, where $b_{1}^{m}$ is given in
definition 3.1 below. In other words, the minimizer of $E|_{S(m)}$ can be
characterized by mountain pass level. Clearly, Corollary 1.1 improves Theorems
1.1–1.2.

###### Theorem 1.6

Assume that $(g_{1})-(g_{5})$, $(g_{7})$ hold and $N\geq 2.$ Then

  * $(1)$

$e(m)$ has a minimizer for $m>m_{1}$, where $m_{1}$ is given in Theorem 1.4.

  * $(2)$

In addition that $(g_{6})$, $e(m)$ has a minimizer for any $m>0$.

###### Remark 1.2

It is a natural question that if $e(m)$ has a minimizer for $m\in(0,m_{1}]$.
For semilinear case, by proving a strict subadditivity inequality as follows

$e_{m_{1}+m_{2}}<e_{m_{1}}+e_{m_{2}},$

Shibata [40] gives a negative answer. For quasilinear case, however, it is not
clear this subadditivity inequality holds, because the functional $E$ is non
differentiable in the space $X$ when $N\geq 2$. So this problem still remains
open.

To prove Theorems 1.4–1.5, we employ a new approach introduced by Hirata and
Tanaka [14]. One advantage of this method is that it can be suitably adapted
to derive multiplicity results of normalized solutions by using minimax and
deformation arguments. Firstly, we consider a augmented functional
$I^{m}(\lambda,u)$, which is not well defined in the normal Sobolev space
$\mathbb{R}\times H^{1}(\mathbb{R}^{N})$. Consequently, one of the
difficulties to deal with (1.1) stems from the fact that there is no suitable
working space where the energy functional enjoys both the smoothness and
compactness properties. In order to overcome this difficulty, inspired by
[26], we make a change of variables and transform the quasilinear problem into
a semilinear one.

Since $g$ only satisfies the Berestycki–Lions condition, it impossible for us
to verify the boundedness of $(PS)$ or $(C_{c})$ sequence because of the
absence of Ambrosetti-Rabinowize condition and monotonicity condition.
Inspired by [14], we use a new deformation argument under a new version of the
Palias–Samle condition related to a Pohozaev functional. Compared with the
unconstrained case, we find that the corresponding functional seems to
satisfies the compactness condition only at the negative energy level but not
at positive energy level (see Lemma 2.4). Consequently, we have to construct a
family of negative minimax values $b^{k}_{m}$ to prove Theorems 1.4–1.5. To
this aim, a key problem is prove that none of $b^{k}_{m}$ is equal to
$-\infty$ and hence $b^{k}_{m}$ is well defined. In [14], Hirata and Tanaka
accomplished it by comparing $b^{k}_{m}$ and the least energy of the following
classic problem

$-\Delta u+u=|u|^{r-2}u,~{}~{}x\in\mathbb{R}^{N},$ (1.3)

where $2<r<2^{\ast}.$ This argument heavily relies on scaling properties of
functional. However, it seems to fail for the quasilinear case because of the
nonhomogeneous properties of change of variables and nonlinearity. In this
paper, we will borrow from a Pohozaev Mountain developed in [9] to overcome
this difficulty. We point out that a new property of change of variables play
a important role in proof, see Lemma 2.1–(14).

To look for a minimizer of $e(m)$, our work uses in particular some arguments
of Steiner rearrangement (see [23, 24]) and a new scaling of [10]. We will see
that a key point is show that $e(m)<0$. Indeed, for the homogeneous case that
$g(u)=|u|^{r-2}u$, one can achieve this by using a scaling argument. However,
for the nonhomogeneous case, it is not easy to see whether $e(m)<0$ holds or
not. In this paper, by using minimax method, we first find a least energy
normalized solution with negative energy to $(P_{\mu,m})$, which concludes
that $e(m)<0$. From this perspective, we gives a strategy for finding a
minimizer for constraint problems.

The paper is organized as follows. In Section 2, we will focus on the
variational frame and deformation lemma. In Section 3, we will construct the
negative minimax level and and analyze their behavior. In Section 4, we will
devote to the proofs of main results.

Notation. For the sake of notational simplicity, we omit integral symbols $dx$
without causing confusion. Throughout this paper, $\rightarrow$ and
$\rightharpoonup$ denote the strong convergence and the weak convergence,
respectively. $\|\cdot\|$ denotes the norm in $H^{1}(\mathbb{R}^{N})$.
$\|(\lambda,u)\|:=\sqrt{\lambda^{2}+\|u\|^{2}}$ denotes the norm of
$(\lambda,u)$ in $\mathbb{R}\times H^{1}(\mathbb{R}^{N})$. For
$A\subset\mathbb{R}\times H^{1}(\mathbb{R}^{N})$ and $\rho>0$,
$N_{\rho}(A)=\\{(\lambda,u):\hbox{dist}((\lambda,u),A)<\rho\\}$.
$\|\cdot\|_{p}$ denotes the norm in $L^{p}(\mathbb{R}^{N})$ for $1\leq
p\leq\infty$. $\partial
D_{k}=\\{x\in\mathbb{R}^{k}:|x|=1\\},~{}~{}D_{k}=\\{x\in\mathbb{R}^{k}:|x|\leq
1\\}.$ $\Sigma_{\rho}=\big{\\{}v\in
H^{1}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}\big{(}|\nabla
v|^{2}+f^{2}(v)\big{)}\leq\rho\big{\\}}$ and
$\partial\Sigma_{\rho}=\big{\\{}v\in
H^{1}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}\big{(}|\nabla
v|^{2}+f^{2}(v)\big{)}=\rho\big{\\}}.$ $C_{0}$, $C$, $C_{i}$ denote various
positive constants whose value may change from line to line but are not
essential to the analysis of the proof.

##  2 Palais-Smale-Pohozaev condition and deformation theory

###  2.1 Palais-Smale-Pohozaev condition

For the sake of notational simplicity, we denote $\mu=e^{\lambda}$ . Since we
are looking for normalized solutions to equation $(P_{\mu,m})$, we consider
the following energy functional:

$I^{m}(\lambda,u)={1\over 2}\int_{\mathbb{R}^{N}}\big{(}1+2u^{2}\big{)}|\nabla
u|^{2}+{e^{\lambda}\over
2}\bigg{(}\int_{\mathbb{R}^{N}}u^{2}-m\bigg{)}-\int_{\mathbb{R}^{N}}G(u),$

which is not well defined in the normal Sobolev space $\mathbb{R}\times
H^{1}(\mathbb{R}^{N})$. In order to overcome this difficulty, inspired by
[26], we make a change of variables $v=f^{-1}(u)$, where $f$ is defined by

$\begin{cases}f(0)=0,\\\
f^{\prime}(t)=\big{(}1+2|f(t)|^{2}\big{)}^{-\frac{1}{2}},~{}~{}~{}\forall~{}t>0,\\\
f(t)=-f(-t),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\forall~{}t<0.\end{cases}$

###### Lemma 2.1

The function $f$ satisfies the following properties:

  * $(1)$

$f\in C^{\infty}(\mathbb{R},\mathbb{R})$ is uniquely defined function and
invertible.

  * $(2)$

$0<f^{\prime}(t)\leq 1$ for all $t\in\mathbb{R}$.

  * $(3)$

$|f(t)|\leq|t|$ for all $t\in\mathbb{R}$.

  * $(4)$

$\lim\limits_{t\rightarrow 0}\frac{f(t)}{t}=1.$

  * $(5)$

$\lim\limits_{t\rightarrow\infty}\frac{|f(t)|^{2}}{|t|}=\sqrt{2}.$

  * $(6)$

$\frac{1}{2}f(t)\leq tf^{\prime}(t)\leq f(t)$ for all $t\geq 0$.

  * $(7)$

$|f(t)|\leq 2^{\frac{1}{4}}|t|^{\frac{1}{2}}$ for all $t\in\mathbb{R}$.

  * $(8)$

There exists a positive constant $C>0$ such that

$f(t)\geq\begin{cases}C|t|,~{}~{}|t|\leq 1,\\\
C|t|^{\frac{1}{2}},|t|>1.\end{cases}$

  * $(9)$

$|f(t)|f^{\prime}(t)\leq\frac{1}{\sqrt{2}}$ for all $t\in\mathbb{R}$.

  * $(10)$

For all $\theta>0$, there is a constant $C(\theta)>0$ such that $|f(\theta
t)|^{2}\leq C(\theta)|f(t)|^{2}$ for all $t\in\mathbb{R}$.

  * $(11)$

The function $f(t)f^{\prime}(t)t^{-1}$ is decreasing for $t>0$.

  * $(12)$

The function $f^{r}(t)f^{\prime}(t)t^{-1}$ is increasing for $r\geq 3$ and
$t>0$.

  * $(13)$

There exists $C>0$ such that $|t|^{r}\leq C|f(t)|^{2}+C|f(t)|^{2r}$ for all
$r\geq 2$ and $t\in\mathbb{R}$.

  * $(14)$

There exists a positive constant $C>0$ such that $f^{2}(rt)\geq Crf^{2}(t)$
for all $t\in\mathbb{R}$ and $r\geq 1$; $f^{2}(rt)\geq Cr^{2}f^{2}(t)$ for all
$t\in\mathbb{R}$ and $r\leq 1$.

  * $(15)$

Assume $(g_{1})$–$(g_{3})$ hold. For any $\varepsilon>0$, there exists
$C_{\varepsilon}>0$ such that
$|G[f(t)]|\leq\varepsilon|f(t)|^{2}+C_{\varepsilon}|t|^{\frac{p}{2}}$ for all
$t\in\mathbb{R}$.

  * $(16)$

Assume $(g_{1})$, $(g_{3})$ and $(g_{6})$ hold. Let $r\in(q,+\infty)$ if
$N=2$, and $r\in(q,2^{\ast})$ if $N\geq 3$, then for any $L>0$ there exists
$C_{L}>0$ such that $G[f(t)]\geq
L|t|^{q}-C_{L}|t|^{r}~{}~{}\hbox{for}~{}~{}\hbox{all}~{}~{}t\in\mathbb{R}.$

Proof: $(1)$-$(13)$ can be found in [43]. Here we only give the proof for
$(16)$ because $(14)$-$(15)$ is simple. For any $L>0$, by $(g_{6})$ and $(8)$,
there exists $0<\delta=\delta(L)<1$ such that

$G[f(t)]>\frac{L}{C}\bigg{|}f(t)\bigg{|}^{q}\geq L|t|^{q}$

for any $0<|t|\leq\delta$. From $(g_{3})$ , $(5)$ and $r>q>\frac{p}{2}$, we
have

$\lim_{t\rightarrow\infty}\frac{G[f(t)]}{-L|t|^{q}+|t|^{r}}\geq\lim_{t\rightarrow\infty}\frac{-|f(t)|^{p}}{-L|t|^{q}+|t|^{r}}=\lim_{t\rightarrow\infty}\frac{-|t|^{\frac{p}{2}}}{-L|t|^{q}+|t|^{r}}=0.$

Then there exists $M=M(L)>0$ such that

$G[f(t)]>L|t|^{q}-|t|^{r}$

for any $|t|>M$. For $\delta\leq|t|\leq M$, by $(g_{1})$, there exists
$C(L)>0$ such that

$G[f(t)]\geq-C(L)\geq-C(L)\delta^{-r}|t|^{r}.$

Consequently, for $t\in\mathbb{R}$,

$G[f(t)]\geq
L|t|^{q}-\big{(}1+C(L)\delta^{-r}\big{)}|t|^{r}:=L|t|^{q}-C_{L}|t|^{r}.$

After making the change of variables, we consider the functional

$J^{m}(\lambda,v)={1\over 2}\|\nabla v\|_{2}^{2}+{e^{\lambda}\over
2}\big{(}\|f(v)\|^{2}_{2}-m\big{)}-\int_{\mathbb{R}^{N}}G[f(v)]$

and the corresponding semilinear problem

$-\Delta v+e^{\lambda}\big{(}f(v)f^{\prime}(v)-m\big{)}=g[f(v)]f^{\prime}(v).$

Let us recall the principle of symmetric criticality (see [42]) as follows.

###### Lemma 2.2

Assume that the action of the topological group $G$ on the Hilbert space $X$
is isometric. If $\varphi\in C^{1}(X,\mathbb{R})$ is invariant and if $u$ is a
critical point of $\varphi$ restricted to $Fix(G)$, then $u$ is a critical
point of $\varphi$.

###### Lemma 2.3

Assume that $(g_{1})-(g_{4})$ hold. Then $J^{m}\in C^{1}(\mathbb{R}\times
H^{1}(\mathbb{R}^{N}),\mathbb{R})$. Moreover, if $(\lambda,v)$ is a critical
point for $J^{m}|_{\mathbb{R}\times H_{r}^{1}(\mathbb{R}^{N})}$, then
$(e^{\lambda},f(v))$ solves equation $(P_{\mu,m})$.

Proof: $J^{m}\in C^{1}(\mathbb{R}\times H^{1}(\mathbb{R}^{N}),\mathbb{R})$
directly follows from $(g_{1})-(g_{4})$ and Lemma 2.1. Suppose that
$(\lambda,v)$ is a critical point for $J^{m}|_{\mathbb{R}\times
H_{r}^{1}(\mathbb{R}^{N})}$, by Lemma 2.2, $(\lambda,v)$ is also a critical
point for $J^{m}$. Consequently,

$\displaystyle 0=\langle\partial_{v}J^{m}(\lambda,v),\varphi\rangle$
$\displaystyle=\int_{\mathbb{R}^{N}}\nabla
v\nabla\varphi+e^{\lambda}\int_{\mathbb{R}^{N}}f(v)f^{\prime}(v)\varphi-\int_{\mathbb{R}^{N}}g[f(v)]f^{\prime}(v)\varphi,~{}~{}~{}~{}~{}~{}~{}~{}\forall\varphi\in
C_{0}^{\infty}(\mathbb{R}^{N})$

and

$\displaystyle 0=\partial_{\lambda}J^{m}(\lambda,v)={e^{\lambda}\over
2}\big{(}\|f(v)\|_{2}^{2}-m\big{)}.$

Let $u=f(v)$ and $\varphi=\frac{\psi}{f^{\prime}(v)}$. Then

$\displaystyle\int_{\mathbb{R}^{N}}(1+2u^{2})\nabla
u\nabla\psi+\int_{\mathbb{R}^{N}}2|\nabla
u|^{2}u\psi+e^{\lambda}\int_{\mathbb{R}^{N}}u\psi-\int_{\mathbb{R}^{N}}g(u)\psi=0,~{}~{}~{}~{}~{}~{}~{}~{}\forall\psi\in
C_{0}^{\infty}(\mathbb{R}^{N})$

and

$\displaystyle{e^{\lambda}\over 2}\bigg{(}\|u\|_{2}^{2}-m\bigg{)}=0.$

Consequently, $(\mu,u)$ solves equation $(P_{\mu,m})$ with $\mu=e^{\lambda}$.

Now, we introduce the Pohozaev functional as follows:

$P(\lambda,v)={N-2\over 2}\|\nabla v\|_{2}^{2}+{Ne^{\lambda}\over
2}\|f(v)\|_{2}^{2}-N\int_{\mathbb{R}^{N}}G[f(v)].$

For $c\in\mathbb{R}$, let

$K^{m}_{c}=\\{(\lambda,v)\in\mathbb{R}\times
H_{r}^{1}(\mathbb{R}^{N}):J^{m}(\lambda,v)=c,~{}\partial_{\lambda}J^{m}(\lambda,v)=0,~{}\partial_{v}J^{m}(\lambda,v)=0,~{}P(\lambda,v)=0\\}.$

###### Definition 2.1

For $c\in\mathbb{R}$, we call $J^{m}(\lambda,v)$ satisfies $(PSP)_{c}$
condition, if any sequence $\\{(\lambda_{n},v_{n})\\}\subset\mathbb{R}\times
H_{r}^{1}(\mathbb{R}^{N})$ with

$J^{m}(\lambda_{n},v_{n})\rightarrow c,$ (2.1)
$\partial_{\lambda}J^{m}(\lambda_{n},v_{n})\rightarrow 0,$ (2.2)
$\partial_{v}J^{m}(\lambda_{n},v_{n})\rightarrow
0~{}~{}\hbox{in}~{}~{}H_{r}^{1}(\mathbb{R}^{N})^{\ast},$ (2.3)
$P(\lambda_{n},v_{n})\rightarrow 0$ (2.4)

as $n\rightarrow\infty$, then $\\{(\lambda_{n},v_{n})\\}$ has a strongly
convergent subsequence in $\mathbb{R}\times H_{r}^{1}(\mathbb{R}^{N})$.

###### Lemma 2.4

Assume that $(g_{1})-(g_{4})$ hold. Then $J^{m}(\lambda,v)$ satisfies
$(PSP)_{c}$ condition for all $c<0$.

Proof: If $\\{(\lambda_{n},v_{n})\\}\subset\mathbb{R}\times
H_{r}^{1}(\mathbb{R}^{N})$ satisfying (2.1)–(2.4), then

$\frac{N}{2}me^{\lambda_{n}}=P(\lambda_{n},v_{n})-NJ^{m}(\lambda_{n},v_{n})+\|\nabla
v_{n}\|_{2}^{2}\geq P(\lambda_{n},v_{n})-NJ^{m}(\lambda_{n},v_{n}),$

implies that $\\{\lambda_{n}\\}$ is bounded from below. Since

$\partial_{\lambda}J^{m}(\lambda_{n},v_{n})={e^{\lambda_{n}}\over
2}\big{(}\|f(v_{n})\|_{2}^{2}-m\big{)}\rightarrow 0,$

one has $\|f(v_{n})\|_{2}^{2}\rightarrow m$. Next, we prove that $\\{\nabla
v_{n}\\}$ is bounded in $L^{2}(\mathbb{R}^{N})$. We argue by contradiction
that

$\|\nabla v_{n}\|_{2}^{2}\rightarrow\infty.$ (2.5)

For small $\varepsilon>0,$ by Lemma 2.1 and the Gagliardo–Nirenberg
inequality, there exists $C_{\varepsilon}>0$ such that

$\displaystyle\int_{\mathbb{R}^{N}}\big{|}g[f(v_{n})]f^{\prime}(v_{n})v_{n}\big{|}\leq$
$\displaystyle
C_{\varepsilon}\|f(v_{n})\|_{2}^{2}+\varepsilon\|f(v_{n})\|_{p}^{p}$ (2.6)
$\displaystyle\leq
C_{\varepsilon}\|f(v_{n})\|_{2}^{2}+\varepsilon\|f(v_{n})\|_{2}^{\theta
p}\|f(v_{n})\|_{22^{\ast}}^{(1-\theta)p}$ $\displaystyle\leq
C_{\varepsilon}\|f(v_{n})\|_{2}^{2}+\varepsilon C\|f(v_{n})\|_{2}^{\theta
p}\|v_{n}\|_{2^{\ast}}^{\frac{(1-\theta)p}{2}}$ $\displaystyle\leq
C_{\varepsilon}\|f(v_{n})\|_{2}^{2}+\varepsilon C_{N}\|f(v_{n})\|_{2}^{\theta
p}\|\nabla v_{n}\|_{2}^{\frac{(1-\theta)p}{2}}$
$\displaystyle=C_{\varepsilon}\|f(v_{n})\|_{2}^{2}+\varepsilon
C_{N}\|f(v_{n})\|_{2}^{\frac{4}{N}}\|\nabla v_{n}\|_{2}^{2},$

where $\theta=\frac{1}{N+1}$. Similarly, one has

$\|v_{n}\|_{2}^{2}\leq C\|f(v_{n})\|_{2}^{2}+C\|f(v_{n})\|_{p}^{p}\leq
C\|f(v_{n})\|_{2}^{2}+C_{N}\|f(v_{n})\|_{2}^{\frac{4}{N}}\|\nabla
v_{n}\|_{2}^{2}.$ (2.7)

For large $n$, from (2.6)-(2.7) and Lemma 2.1, we have

$\displaystyle 2C_{N}\|\nabla v_{n}\|_{2}$ (2.8)
$\displaystyle\geq\sqrt{\|\nabla
v_{n}\|_{2}^{2}+C\|f(v_{n})\|_{2}^{2}+C_{N}\|f(v_{n})\|_{2}^{\frac{4}{N}}\|\nabla
v_{n}\|_{2}^{2}}$
$\displaystyle\geq\|\partial_{v}J^{m}(\lambda_{n},v_{n})\|\sqrt{\|\nabla
v_{n}\|_{2}^{2}+\|v_{n}\|_{2}^{2}}$
$\displaystyle\geq\langle\partial_{v}J^{m}(\lambda_{n},v_{n}),v_{n}\rangle$
$\displaystyle=\|\nabla
v_{n}\|_{2}^{2}+e^{\lambda_{n}}\int_{\mathbb{R}^{N}}f(v_{n})f^{\prime}(v_{n})v_{n}-\int_{\mathbb{R}^{N}}g[f(v_{n})]f^{\prime}(v_{n})v_{n}$
$\displaystyle\geq\|\nabla
v_{n}\|_{2}^{2}+\frac{1}{2}e^{\lambda_{n}}\|f(v_{n})\|_{2}^{2}-C_{\varepsilon}\|f(v_{n})\|_{2}^{2}-\varepsilon
C_{N}\|f(v_{n})\|_{2}^{\frac{4}{N}}\|\nabla v_{n}\|_{2}^{2}$
$\displaystyle\geq\big{(}1-\varepsilon C_{N}(2m)^{\frac{2}{N}}\big{)}\|\nabla
v_{n}\|_{2}^{2}-2C_{\varepsilon}m,$

which contradicts (2.5) if we choose sufficiently small $\varepsilon$ .
Moreover, the proof of (LABEL:eqB16) implies that $\\{\lambda_{n}\\}$ is
bounded from above. By claim $2$ of Lemma 2.2 in [45], up to a subsequence, we
may assume that $\lambda_{n}\rightarrow\lambda_{0}$, and

$\begin{cases}v_{n}\rightharpoonup
v_{0}~{}~{}\hbox{in}~{}~{}H_{r}^{1}(\mathbb{R}^{N}),\\\ v_{n}\rightarrow
v_{0}~{}~{}\hbox{in}~{}~{}L^{r}(\mathbb{R}^{N})~{}~{}\hbox{for}~{}~{}2<r<2^{\ast},\\\
v_{n}\rightarrow
v_{0}~{}~{}\hbox{in}~{}~{}L_{loc}^{s}(\mathbb{R}^{N})~{}~{}\hbox{for}~{}~{}2\leq
s<2^{\ast},\\\ v_{n}\rightarrow
v_{0}~{}~{}a.e.~{}~{}\hbox{in}~{}~{}\mathbb{R}^{N}.\end{cases}$

For small $\varepsilon>0$, by Lemma 2.1, there exists $C_{\varepsilon}>0$ such
that

$\displaystyle\bigg{|}\int_{\mathbb{R}^{N}}\big{(}g[f(v_{n})]f^{\prime}(v_{n})-g[f(v_{0})]f^{\prime}(v_{0})\big{)}\big{(}v_{n}-v_{0}\big{)}\bigg{|}$
(2.9)
$\displaystyle\leq\int_{\mathbb{R}^{N}}\big{(}\varepsilon|f(v_{n})||f^{\prime}(v_{n})|+C_{\varepsilon}|f(v_{n})|^{p-1}|f^{\prime}(v_{n})|+\varepsilon|f(v_{0})||f^{\prime}(v_{0})|+C_{\varepsilon}|f(v_{0})|^{p-1}|f^{\prime}(v_{0})|\big{)}\big{|}v_{n}-v_{0}\big{|}$
$\displaystyle\leq\int_{\mathbb{R}^{N}}\big{(}\varepsilon|v_{n}|+C_{\varepsilon}|v_{n}|^{\frac{p-2}{2}}+\varepsilon|v_{0}|+C_{\varepsilon}|v_{0}|^{\frac{p-2}{2}}\big{)}\big{|}v_{n}-v_{0}\big{|}$
$\displaystyle\leq\varepsilon
C+C_{\varepsilon}\bigg{(}\big{\|}v_{n}\big{\|}_{\frac{p}{2}}^{\frac{p-2}{2}}+\big{\|}v_{0}\big{\|}_{\frac{p}{2}}^{\frac{p-2}{2}}\bigg{)}\big{\|}v_{n}-v_{0}\big{\|}_{\frac{p}{2}}$
$\displaystyle=\varepsilon C+o_{n}(1).$

Now, by view of Lemma 2.6 in [46], we have

$\displaystyle o_{n}(1)$
$\displaystyle=\big{\langle}\partial_{v}J^{m}(\lambda_{n},v_{n})-\partial_{v}J^{m}(\lambda_{n},v_{0}),v_{n}-v_{0}\big{\rangle}$
(2.10) $\displaystyle=\|\nabla v_{n}-\nabla
v_{0}\|_{2}^{2}+e^{\lambda_{n}}\int_{\mathbb{R}^{N}}\big{(}f(v_{n})f^{\prime}(v_{n})-f(v_{0})f^{\prime}(v_{0})\big{)}\big{(}v_{n}-v_{0}\big{)}$
$\displaystyle~{}~{}~{}-\int_{\mathbb{R}^{N}}\big{(}g[f(v_{n})]f^{\prime}(v_{n})-g[f(v_{0})]f^{\prime}(v_{0})\big{)}\big{(}v_{n}-v_{0}\big{)}$
$\displaystyle\geq C\|v_{n}-v_{0}\|^{2}+o_{n}(1),$

which implies that $v_{n}\rightarrow v$ in $H_{r}^{1}(\mathbb{R}^{N})$, and
the proof is completed.

###  2.2  Deformation Lemma

Following [14], we introduce the augmented functional to construct a
deformation flow as follows:

$\mathcal{F}^{m}(\theta,\lambda,v)={e^{(N-2)\theta}\over 2}\|\nabla
v\|^{2}+{e^{\lambda}\over
2}\bigg{(}e^{N\theta}\|f(v)\|^{2}_{2}-m\bigg{)}-e^{N\theta}\int_{\mathbb{R}^{N}}G[f(v)].$

Without causing symbol confusion, in this subsection we denote $J^{m}$ and
$\mathcal{F}^{m}$ by $J$ and $\mathcal{F}$, respectively. By a direct
calculation, we obtain the following Lemma.

###### Lemma 2.5

For any $(\theta,\lambda,v)\in\mathbb{R}^{2}\times H_{r}^{1}(\mathbb{R}^{N})$,
$\varphi\in H_{r}^{1}(\mathbb{R}^{N})$ and $\beta>0$,

$\mathcal{F}(\theta,\lambda,v)=J\big{(}\lambda,v(e^{-\theta}x)\big{)},$
$~{}~{}~{}~{}\partial_{\theta}\mathcal{F}(\theta,\lambda,v)=P\big{(}\lambda,v(e^{-\theta}x)\big{)},$
$~{}~{}~{}~{}\partial_{\lambda}\mathcal{F}(\theta,\lambda,v)=\partial_{\lambda}J\big{(}\lambda,v(e^{-\theta}x)\big{)},$
$~{}~{}~{}~{}\big{\langle}\partial_{v}\mathcal{F}(\theta,\lambda,v),\varphi(x)\big{\rangle}=\big{\langle}\partial_{v}J\big{(}\lambda,v(e^{-\theta}x)\big{)},\varphi\big{(}e^{-\theta}x\big{)}\big{\rangle},$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathcal{F}(\theta+\beta,\lambda,v(e^{\beta}x))=\mathcal{F}(\theta,\lambda,v(x)).$

We introduce a metric on $M:=\mathbb{R}^{2}\times H_{r}^{1}(\mathbb{R}^{N})$
by

$\|(\alpha,\nu,h)\|^{2}_{(\theta,\lambda,v)}=\big{|}(\alpha,\nu,\|h(e^{-\theta}\cdot)\|)\big{|}^{2}$

for any $(\alpha,\nu,h)\in T_{(\theta,\lambda,v)}M$. Then $M$ be a Hilbert
manifold. We also denote the dual norm on $T^{\ast}_{(\theta,\lambda,v)}M$ by
$\|\cdot\|_{(\theta,\lambda,v),\ast}$.

Let

$\mathcal{D}\mathcal{F}=(\mathcal{D}_{\theta}\mathcal{F},\mathcal{D}_{\lambda}\mathcal{F},\mathcal{D}_{v}\mathcal{F}).$

By a direct computation, we have

$\|\mathcal{D}\mathcal{F}(\theta,\lambda,v)\|_{(\theta,\lambda,v),\ast}=\big{|}P(\lambda,v(e^{-\theta}\cdot))\big{|}^{2}+\big{|}\partial_{\lambda}J\big{(}\lambda,v(e^{-\theta}\cdot)\big{)}\big{|}^{2}+\big{\|}\partial_{v}J\big{(}\lambda,v(e^{-\theta}\cdot)\big{)}\big{\|}^{2}.$

We introduce a natural distance in Sobolev space $\mathbb{R}^{2}\times
H_{r}^{1}(\mathbb{R}^{N})$ as follows:

$\displaystyle\hbox{dist}\big{(}(\theta_{1},\lambda_{1},v_{1}),(\theta_{2},\lambda_{2},v_{2})\big{)}$
$\displaystyle=\inf\bigg{\\{}\int_{0}^{1}||\dot{\sigma}(t)||dt:\sigma(t)\in
C^{1}([0,1],\mathbb{R}^{2}\times
H_{r}^{1}(\mathbb{R}^{N})),\sigma(0)=(\theta_{1},\lambda_{1},v_{1}),\sigma(1)=(\theta_{2},\lambda_{2},v_{2})\bigg{\\}}.$

For $c\in\mathbb{R}$, we denote

$\widetilde{K^{m}_{c}}=\\{(\theta,\lambda,v)\in\mathbb{R}^{2}\times
H_{r}^{1}(\mathbb{R}^{N}):\mathcal{F}(\theta,\lambda,v)=c,~{}\mathcal{D}\mathcal{F}(\theta,\lambda,v)=(0,0,0)\\}.$

By Lemma 2.5, we have

$\widetilde{K^{m}_{c}}=\\{(\theta,\lambda,v(e^{\theta}x)):\theta\in\mathbb{R},~{}(\lambda,v)\in
K^{m}_{c}\\}.$

For $\rho>0$, $\widetilde{A}\in\mathbb{R}^{2}\times
H_{r}^{1}(\mathbb{R}^{N})$, $c\in\mathbb{R}$, let us denote

$\widetilde{N}_{\rho}(\widetilde{A})=\big{\\{}(\theta_{1},\lambda_{1},v_{1})\in\mathbb{R}^{2}\times
H_{r}^{1}(\mathbb{R}^{N}):\inf_{(\theta_{2},\lambda_{2},v_{2})\in\widetilde{A}}\hbox{dist}\big{(}(\theta_{1},\lambda_{1},v_{1}),(\theta_{2},\lambda_{2},v_{2})\big{)}<\rho\big{\\}},$
$\mathcal{F}^{c}=\\{(\theta,\lambda,v)\in\mathbb{R}^{2}\times
H_{r}^{1}(\mathbb{R}^{N}):\mathcal{F}(\theta,\lambda,v)\leq c\\}.$

The following Lemma can be founded in ([14].

###### Lemma 2.6

If $c<0$, then for any $\overline{\varepsilon}>0$ and $\rho>0$ there exist
$\varepsilon\in(0,\overline{\varepsilon})$ and a continuous map
$\widetilde{\eta}(t,\theta,\lambda,v):[0,1]\rightarrow\mathbb{R}^{2}\times
H_{r}^{1}(\mathbb{R}^{N})$ such that

  * $(1)$

$\widetilde{\eta}(0,\theta,\lambda,v)=(\theta,\lambda,v)$ for all
$(\theta,\lambda,v)\in\mathbb{R}^{2}\times H_{r}^{1}(\mathbb{R}^{N}).$

  * $(2)$

$\widetilde{\eta}(0,\theta,\lambda,v)=(\theta,\lambda,v)$ for all
$(\lambda,v)\in\mathcal{F}^{c-\overline{\varepsilon}}$ and $t\in[0,1].$

  * $(3)$

$\mathcal{F}(\widetilde{\eta}(t,\theta,\lambda,v))\leq\mathcal{F}(\theta,\lambda,v)$
for all $(t,\theta,\lambda,v)\in[0,1]\times\mathbb{R}^{2}\times
H_{r}^{1}(\mathbb{R}^{N}).$

  * $(4)$

$\widetilde{\eta}(1,\mathcal{F}^{c+{\varepsilon}}\setminus\widetilde{N}_{\rho}(\widetilde{K}_{c}))\subset\mathcal{F}^{c-{\varepsilon}},$
$\widetilde{\eta}(1,\mathcal{F}^{c+{\varepsilon}})\subset\mathcal{F}^{c-{\varepsilon}}\cup\widetilde{N}_{\rho}(\widetilde{K}_{c}).$

  * $(5)$

If $\widetilde{K}_{c}=\emptyset$, then
$\widetilde{\eta}(1,\mathcal{F}^{c+{\varepsilon}})\subset\mathcal{F}^{c-{\varepsilon}}$.

  * $(6)$

Let
$\widetilde{\eta}(t,\theta,\lambda,v)=(\widetilde{\eta_{1}}(t,\theta,\lambda,v),\widetilde{\eta_{2}}(t,\theta,\lambda,v),\widetilde{\eta_{3}}(t,\theta,\lambda,v))$,
we have

$\widetilde{\eta_{1}}(t,\theta,\lambda,-v)=\widetilde{\eta_{1}}(t,\theta,\lambda,v),$
$\widetilde{\eta_{2}}(t,\theta,\lambda,-v)=\widetilde{\eta_{2}}(t,\theta,\lambda,v),$
$\widetilde{\eta_{3}}(t,\theta,\lambda,-v)=-\widetilde{\eta_{3}}(t,\theta,\lambda,v)$

for all $(t,\theta,\lambda,v)\in[0,1]\times\mathbb{R}^{2}\times
H_{r}^{1}(\mathbb{R}^{N}).$

###### Lemma 2.7

( Deformation Lemma) If $c<0$, then

  * (i)

$K_{c}^{m}$ is compact in $\mathbb{R}\times H_{r}^{1}(\mathbb{R}^{N})$ and
$K_{c}^{m}\cap\\{\mathbb{R}\times\\{0\\}\\}=\emptyset$.

  * $({ii})$

For any open neighborhood $N$ of $K_{c}^{m}$ and $\overline{\varepsilon}>0$,
there exist $\varepsilon\in(0,\overline{\varepsilon})$ and a continuous map
$\eta(t,\lambda,v):[0,1]\rightarrow\mathbb{R}\times H_{r}^{1}(\mathbb{R}^{N})$
such that

  * $(1)$

$\eta(0,\lambda,v)=(\lambda,v)$ for all $(\lambda,v)\in\mathbb{R}\times
H_{r}^{1}(\mathbb{R}^{N})$.

  * $(2)$

$\eta(t,\lambda,v)=(\lambda,v)$ for all $(\lambda,v)\in
J^{c-\overline{\varepsilon}}$ and $t\in[0,1].$

  * $(3)$

$J(\eta(t,\lambda,v))\leq J(\lambda,v)$ for all
$(t,\lambda,v)\in[0,1]\times\mathbb{R}\times H_{r}^{1}(\mathbb{R}^{N}).$

  * $(4)$

$\eta(1,J^{c+{\varepsilon}}\setminus N)\subset J^{c-{\varepsilon}},$
$\eta(1,J^{c+{\varepsilon}})\subset J^{c-{\varepsilon}}\cup N.$

  * $(5)$

If $K_{c}^{m}=\emptyset$, then $\eta(1,J^{c+{\varepsilon}})\subset
J^{c-{\varepsilon}}$.

  * $(6)$

Let $\eta(t,\lambda,v)=(\eta_{1}(t,\lambda,v),\eta_{2}(t,\lambda,v))$, we have

$\eta_{1}(t,\lambda,-v)=\eta_{1}(t,\lambda,v),$
$\eta_{2}(t,\lambda,-v)=-\eta_{2}(t,\lambda,v),$

for all $(t,\lambda,v)\in[0,1]\times\mathbb{R}\times
H_{r}^{1}(\mathbb{R}^{N}).$

Proof: Consider the following maps:

$\pi:\mathbb{R}^{2}\times H_{r}^{1}(\mathbb{R}^{N})\rightarrow\mathbb{R}\times
H_{r}^{1}(\mathbb{R}^{N}),~{}~{}(\theta,\lambda,v(x))\mapsto\big{(}\lambda,v(e^{-\theta}x)\big{)},$
$l:\mathbb{R}\times H_{r}^{1}(\mathbb{R}^{N})\rightarrow\mathbb{R}^{2}\times
H_{r}^{1}(\mathbb{R}^{N}),~{}~{}(\lambda,v(x))\mapsto(0,\lambda,v(x)).$

By Lemma 2.5, we have

$\pi(l(\lambda,v))=(\lambda,v)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hbox{for}~{}~{}\hbox{all}~{}~{}(\lambda,v)\in\mathbb{R}\times
H_{r}^{1}(\mathbb{R}^{N}),$
$l(\pi(\theta,\lambda,v))=\big{(}0,\lambda,v(e^{-\theta}x)\big{)}~{}~{}~{}~{}~{}~{}~{}~{}\hbox{for}~{}~{}\hbox{all}~{}~{}(\theta,\lambda,v)\in\mathbb{R}^{2}\times
H_{r}^{1}(\mathbb{R}^{N}),$
$\mathcal{F}(\theta,\lambda,v)=J(\pi(\theta,\lambda,v))~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hbox{for}~{}~{}\hbox{all}~{}~{}(\theta,\lambda,v)\in\mathbb{R}^{2}\times
H_{r}^{1}(\mathbb{R}^{N}).$

$\pi(\widetilde{K_{c}})=K_{c}.$ From those results above and Lemma 2.6,
deformation Lemma follows. We can refer the readers to Lemma 3.5 in [44] for
details.

##  3 Minimax methods

###  3.1 Construction of multidimensional odd path

In what follows, we set

$J_{\lambda}(v)={1\over 2}\|\nabla v\|_{2}^{2}+{e^{\lambda}\over
2}\|f(v)\|_{2}^{2}-\int_{\mathbb{R}^{N}}G[f(v)],$

and

$\lambda_{0}=\begin{cases}\ln\bigg{(}2\sup\limits_{s\neq
0}\frac{G(s)}{s^{2}}\bigg{)}~{}~{}~{}~{}\hbox{if}~{}~{}\sup\limits_{s\neq
0}\frac{G(s)}{s^{2}}<\infty,\\\
\infty~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hbox{if}~{}~{}\sup\limits_{s\neq
0}\frac{G(s)}{s^{2}}=\infty.\end{cases}$

###### Definition 3.1

For $k=1$ and $\lambda\in(-\infty,\lambda_{0})$, let

$a_{1}(\lambda)=\inf_{\gamma\in{\Gamma}_{1}(\lambda)}\max_{\xi\in[0,1]}{J}_{\lambda}(\gamma(\xi)),$

where

$\Gamma_{1}(\lambda)=\\{\gamma\in
C([0,1],H_{r}^{1}(\mathbb{R}^{N})):\gamma(0)=0,{J}_{\lambda}(\gamma(1))<0\\};$

For $k\geq 2$ and $\lambda\in(-\infty,\lambda_{0})$, let

$a_{k}(\lambda)=\inf_{\gamma\in{\Gamma}_{k}(\lambda)}\max_{\xi\in
D_{k}}{J}_{\lambda}(\gamma(\xi)),$

where

$\Gamma_{k}(\lambda)=\\{\gamma\in
C(D_{k},H_{r}^{1}(\mathbb{R}^{N})):\gamma(-\xi)=-\gamma(\xi)~{}\hbox{for}~{}\xi\in
D_{k},{J}_{\lambda}(\gamma(\xi))<0~{}\hbox{for}~{}\xi\in\partial D_{k}\\}.$

###### Lemma 3.1

Assume that $\lambda\in(-\infty,\lambda_{0})$ and $(g_{1})-(g_{4})$ hold,
$\Gamma_{1}(\lambda)\neq\emptyset.$ In addition $(g_{5})$ holds, then
$\Gamma_{k}(\lambda)\neq\emptyset$ for any $k\geq 1.$

Proof: Now, we prove the case for $k\geq 2$. Let
$H(s):=-\frac{e^{\lambda}}{2}f^{2}(s)+G[f(s)]$, for any
$\lambda\in(-\infty,\lambda_{0})$, by the definition of $\lambda_{0},$ there
exists $s_{\lambda}>0$ such that $H(s_{\lambda})>0.$ If $(g_{1})-(g_{5})$
hold, there exists $\delta>0$ such that

$H(s)>0,~{}\forall
s\in(s_{\lambda}-\delta,s_{\lambda}+\delta)\cup(-s_{\lambda}-\delta,-s_{\lambda}+\delta).$

For $R\gg 1$ and $h\in[0,1]$, let

$A(R,h)=\\{x\in\mathbb{R}^{N}:R-h\leq|x|\leq R+h\\}$

and

$\chi(R,h;x)=\begin{cases}1,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hbox{if}~{}~{}x\in
A(R,h),\\\
0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hbox{otherwise}.\end{cases}$

Consider the polyhedron

$\Sigma:=\\{t=(t_{1},...,t_{k})\in\mathbb{R}^{k}:\max\limits_{1\leq i\leq
k}|t_{i}|=1\\}$

and define the odd map $\gamma:\Sigma\rightarrow H_{r}^{1}(\mathbb{R}^{N})$ by

$\gamma(t)(\cdot)=\sum_{i=1}^{k}\hbox{sgn}(t_{i})\big{(}s_{\lambda}+(|t_{i}|-1)\delta\big{)}\chi(2ki,|t_{i}|;\cdot).$

For all $t=(t_{1},...,t_{k})\in\sum$, we have
$\chi(2ki,|t_{i}|;x)\cap\chi(2kj,|t_{j}|;\cdot)=0$ if $i\neq j$ and

$\displaystyle\int_{\mathbb{R}^{N}}H(\gamma(t)(x))$
$\displaystyle=\sum_{i=1}^{k}H\bigg{(}\hbox{sgn}(t_{i})\big{(}s_{\lambda}+(|t_{i}|-1)\delta\big{)}\bigg{)}\int_{\mathbb{R}^{N}}\chi(2ki,|t_{i}|;x)$
(3.1) $\displaystyle\geq
H(s_{\lambda})\int_{\mathbb{R}^{N}}\chi(2k{i_{0}},1;\cdot)\geq C_{0}.$

Let

$\chi_{\varepsilon}(R,h;x)=\begin{cases}1,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hbox{if}~{}~{}R-h\leq|x|\leq
R+h,\\\ 1-\frac{1}{\varepsilon
h}\big{(}R-h-|x|\big{)},~{}~{}\hbox{if}~{}R-h-\varepsilon h\leq|x|\leq R-h,\\\
1-\frac{1}{\varepsilon
h}\big{(}|x|-R-h\big{)},~{}~{}\hbox{if}~{}~{}R+h\leq|x|\leq R+h+\varepsilon
h,\\\
0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hbox{otherwise}.\end{cases}$

It is easy to see that

  * (i)

$\chi_{\varepsilon}(R,h;\cdot)\in H_{r}^{1}(\mathbb{R}^{N});$

  * (ii)

For fixed $R>0$ and small $\varepsilon>0$,
$\chi_{\varepsilon}(R,h;\cdot)~{}~{}\hbox{is}~{}~{}\hbox{continuous}~{}~{}\hbox{in}~{}~{}H_{r}^{1}(\mathbb{R}^{N})~{}~{}\hbox{for}~{}~{}h\in[0,1];$

  * (iii)

$\chi_{\varepsilon}(R,h;\cdot)\rightarrow\chi(R,h;\cdot)~{}~{}\hbox{in}~{}~{}L^{r}(\mathbb{R}^{N})~{}~{}\hbox{as}~{}~{}\varepsilon\rightarrow
0~{}~{}\hbox{uniformly}~{}~{}\hbox{for}~{}~{}h\in[0,1]~{}~{}\hbox{and}~{}~{}R\leq
C;$

  * (iv)

$\chi_{\varepsilon}(R,h;\cdot)\rightarrow
0~{}~{}\hbox{in}~{}~{}L^{r}(\mathbb{R}^{N})~{}~{}\hbox{as}~{}~{}h\rightarrow
0~{}~{}\hbox{uniformly}~{}~{}\hbox{for}~{}~{}\hbox{small}~{}~{}\varepsilon>0~{}~{}\hbox{and}~{}~{}R\leq
C.$

  * (v)

For each $t=(t_{1},...,t_{k})\in\sum$,
$\hbox{supp}(\chi_{\varepsilon}(2ki,|t_{i}|;\cdot))\cap\hbox{supp}(\chi_{\varepsilon}(2kj,|t_{j}|;\cdot))=\emptyset~{}~{}\hbox{if}~{}~{}i\neq
j$.

Let $\gamma_{\varepsilon}:\sum\rightarrow H_{r}^{1}(\mathbb{R}^{N})$ defined
by

$\gamma_{\varepsilon}(t)(\cdot)=\sum_{i=1}^{k}\hbox{sgn}(t_{i})\big{(}s_{\lambda}+(|t_{i}|-1)\delta\big{)}\chi_{\varepsilon}(2ki,|t_{i}|;\cdot).$

From (i)–(ii), (iv), $\gamma_{\varepsilon}$ is odd and continuous. We observe
that $\int_{\mathbb{R}^{N}}H(v)$ is continuous in $L^{r}(\mathbb{R}^{N})$ for
$r\in[2,2^{\ast}]$. By using (3.1), (iii) and (v), there holds

$\displaystyle\int_{\mathbb{R}^{N}}H(\gamma_{\varepsilon}(t)(x))$
$\displaystyle=\sum_{i=1}^{k}H\bigg{(}\hbox{sgn}(t_{i})\big{(}s_{\lambda}+(|t_{i}|-1)\delta\big{)}\bigg{)}\int_{\mathbb{R}^{N}}\chi_{\varepsilon}(2ki,|t_{i}|;\cdot)$
(3.2) $\displaystyle\geq
H(s_{\lambda})\int_{\mathbb{R}^{N}}\chi_{\varepsilon}(2k{i_{0}},1;\cdot)$
$\displaystyle\rightarrow
H(s_{\lambda})\int_{\mathbb{R}^{N}}\chi(2k{i_{0}},1;\cdot)$ $\displaystyle\geq
C_{0}.$

For small $\varepsilon$, since $\Sigma$ is homeomorphic to $\partial D_{k}$,
we may assume that

$\int_{\mathbb{R}^{N}}H(\gamma_{\varepsilon}(t))>C~{}~{}\hbox{for}~{}~{}\hbox{all}~{}~{}t\in\partial
D_{k}.$

Note that

$\displaystyle J_{\lambda}\big{(}\gamma_{\varepsilon}(t)(L^{-1}\cdot)\big{)}$
$\displaystyle={1\over
2}L^{N-2}\|\nabla\gamma_{\varepsilon}(t)\|_{2}^{2}-L^{N}\int_{\mathbb{R}^{N}}H(\gamma_{\varepsilon}(t)(\cdot)),$

there exists some large $L>0$ such that

$J_{\lambda}\big{(}\gamma_{\varepsilon}(t)(L^{-1}\cdot)\big{)}<0~{}~{}\hbox{for}~{}~{}\hbox{all}~{}~{}t\in\partial
D_{k}.$

Let

$\eta(s,t)(\cdot)=s\gamma_{\varepsilon}(t)(L^{-1}\cdot).$

Then $D_{k}=\\{st|s\in[0,1],t\in\sum\\}$, and hence
$\eta\in\Gamma_{k}(\lambda).$

If $(g_{1})-(g_{4})$ hold, there exist $s_{\lambda}>0$ and $\delta>0$ such
that

$H(s)>0,~{}\forall s\in(s_{\lambda}-\delta,s_{\lambda}+\delta).$

For $t\in[0,1]$, we set
$\gamma_{\varepsilon}(t)(\cdot)=\big{(}s_{\lambda}+(t-1)\delta\big{)}\chi_{\varepsilon}(R,t;\cdot).$
Then

$\int_{\mathbb{R}^{N}}H(\gamma_{\varepsilon}(1))=H(s_{\lambda})\int_{\mathbb{R}^{N}}\chi_{\varepsilon}(R,1;\cdot)\geq
C.$

Take large $L>0$ and let
$\eta(t)(\cdot)=\gamma_{\varepsilon}(t)(L^{-1}\cdot)$, we have
$\eta\in\Gamma_{1}(\lambda).$

For any $\varepsilon>0$, by Lemma 2.1 and the Sobolev theorem, there exists
$C_{\varepsilon}>0$ such that

$\bigg{|}\int_{\mathbb{R}^{N}}G[f(v)]\bigg{|}\leq\varepsilon\big{(}\|\nabla
v\|_{2}^{2}+\|f(v)\|_{2}^{2}\big{)}+C_{\varepsilon}\big{(}\|\nabla
v\|_{2}^{2}+\|f(v)\|_{2}^{2}\big{)}^{\frac{2^{*}}{2}}.$

For all $v\in\partial\Sigma_{\rho}$ and small $\varepsilon>0$, we have

$\displaystyle J_{\lambda}(v)$ $\displaystyle={1\over 2}\|\nabla
v\|_{2}^{2}+{e^{\lambda}\over 2}\|f(v)\|_{2}^{2}-\int_{\mathbb{R}^{N}}G[f(v)]$
(3.3) $\displaystyle\geq\big{(}C_{\lambda}-\varepsilon\big{)}\big{(}\|\nabla
v\|_{2}^{2}+\|f(v)\|_{2}^{2}\big{)}-C_{\varepsilon}\big{(}\|\nabla
v\|_{2}^{2}+\|f(v)\|_{2}^{2}\big{)}^{\frac{2^{*}}{2}}$
$\displaystyle\geq{C_{\lambda}\over 2}\rho-
C_{\lambda}^{\prime}\rho^{\frac{2^{*}}{2}}.$

Then for any small $\rho>0$, there exists $\alpha_{\rho}>0$ such that
$J_{\lambda}(v)\geq\alpha_{\rho}$ for $v\in\partial\Sigma_{\rho}$. By the
definition of $a_{k}(\lambda)$, we get that $0\leq m_{k}\leq
m_{k+1},~{}\forall~{}k\in\mathbb{N}^{\ast},$ where

$m_{k}:=2\inf\limits_{\lambda\in(-\infty,\lambda_{0})}\frac{a_{k}(\lambda)}{e^{\lambda}}.$

Furthermore, we have the following result:

###### Lemma 3.2

Assume that $\lambda\in(-\infty,\lambda_{0})$ and $(g_{1})-(g_{4})$ hold. For
any $k\in\mathbb{N}$,

  * $(1)$

$m_{k}=0$, if $(g_{6})$ holds.

  * $(2)$

$m_{k}>0$, if $\limsup\limits_{s\rightarrow
0}\frac{|g(s)|}{|s|^{q-2}s}<\infty.$

Proof: Denote $E_{k}=\hbox{span}\\{e_{1},e_{2},...,e_{k}\\}$, where
$\\{e_{i}\\}$ is a sequence of orthonormal basis in
$H_{r}^{1}(\mathbb{R}^{N})$. Since $E_{k}$ is a finite dimensional space, for
any $2\leq s\leq 2^{\ast}$ there exists $c_{s}>0$ such that

$c_{s}\|v\|_{s}\leq\|v\|\leq c_{s}^{-1}\|v\|_{s},~{}~{}~{}~{}\forall v\in
E_{k}.$ (3.4)

Thus, we may choose some $R>0$ such that

$\frac{1}{2}\|v\|^{2}-\frac{1}{s}\|v\|^{s}_{s}<0,~{}~{}~{}~{}\forall v\in
E_{k}\setminus B_{R}(0).$

Let $\eta:\mathbb{R}^{k}\rightarrow E_{k}$ defined by

$\eta(\xi)=R\sum_{i=1}^{k}\xi_{i}e_{i},$

where $\xi=(\xi_{1},...,\xi_{k})\in\mathbb{R}^{k}$. For any $\xi\in\partial
D_{k}$, we have

$\frac{1}{2}\|\eta(\xi)\|^{2}-\frac{1}{q}\|\eta(\xi)\|^{q}_{q}<0.$ (3.5)

Let
$\gamma(\xi)(\cdot)=e^{\frac{\lambda}{q-2}}\eta(\xi)(e^{\frac{\lambda}{2}}\cdot)$.
For any $\xi\in\partial D_{k}$ with $L>\frac{1}{q}$, it follow (3.4) and Lemma
2.1-$(16)$ that

$\displaystyle J_{\lambda}(\gamma(\xi))$ $\displaystyle\leq{1\over
2}\|\nabla\gamma(\xi)\|_{2}^{2}+{e^{\lambda}\over
2}\|\gamma(\xi)\|_{2}^{2}-L\|\gamma(\xi)\|_{q}^{q}+C_{L}\|\gamma(\xi)\|_{r}^{r}$
$\displaystyle=e^{\lambda}\big{(}{1\over
2}\|\eta(\xi)\|^{2}-L\|\eta(\xi)\|_{q}^{q}+C_{L}e^{\frac{r-q}{q-2}\lambda}\|\eta(\xi)\|_{r}^{r}\big{)}$
$\displaystyle\leq e^{\lambda}\bigg{(}{1\over
2}\|\eta(\xi)\|^{2}-\frac{1}{q}\|\eta(\xi)\|^{q}_{q}+\big{(}\frac{1}{q}-L\big{)}\|\eta(\xi)\|^{q}_{q}+C_{L}e^{\frac{r-q}{q-2}\lambda}\|\gamma_{k}(\xi)\|^{r}_{r}\bigg{)}$
$\displaystyle\leq e^{\lambda}\bigg{(}{1\over
2}\|\eta(\xi)\|^{2}-\frac{1}{q}\|\eta(\xi)\|^{q}+\big{(}\frac{1}{q}-L\big{)}(Rc_{q})^{q}+C_{L}e^{\frac{r-q}{q-2}\lambda}(Rc_{r}^{-1})^{r}\bigg{)},$

which implies that $\gamma\in\Gamma_{k}(\lambda)$ as
$\lambda\rightarrow-\infty$. Consequently,

$\displaystyle\limsup_{\lambda\rightarrow-\infty}\frac{a_{k}(\lambda)}{e^{\lambda}}$
$\displaystyle\leq\max_{\xi\in D_{k}}\bigg{(}{1\over
2}\|\eta(\xi)\|^{2}-L\|\eta(\xi)\|^{q}_{q}\bigg{)}$
$\displaystyle\leq\max_{\xi\in D_{k}}\bigg{(}{1\over
2}\|\eta(\xi)\|^{2}-L(c_{q})^{q}\|\eta(\xi)\|^{q}\bigg{)}$
$\displaystyle\leq\max_{s\geq 0}\bigg{(}{1\over
2}|s|^{2}-L(c_{q})^{q}|s|^{q}\bigg{)}$ $\displaystyle=\bigg{(}{1\over
2}|s|^{2}-L(c_{q})^{q}|s|^{q+1}\bigg{)}\bigg{|}_{s=\big{(}Lq(c_{q})^{q}\big{)}^{-\frac{1}{q-2}}}$
$\displaystyle\rightarrow 0$

as $L\rightarrow+\infty$, and hence $(1)$ holds. If
$\limsup\limits_{s\rightarrow 0}\frac{|g(s)|}{|s|^{q-2}s}<\infty,$ it follows
from $(g_{1})$ and $(g_{3})$ that

$G[f(s)]\leq C|s|^{q}~{}~{}\hbox{for}~{}~{}\hbox{all}~{}~{}s\in\mathbb{R}.$

For any $\lambda\rightarrow-\infty$ and
$\gamma_{\lambda}\in{\Gamma}_{k}(\lambda)$, let
$\eta_{\lambda}(\xi)(\cdot)=e^{-\frac{\lambda}{q-2}}\gamma_{\lambda}(\xi)(e^{-\frac{\lambda}{2}}\cdot)$.
Since $e^{\frac{\lambda}{q-2}}\leq 1$, by Lemma 2.1 and Sobolev theorem, one
has

$\displaystyle\max_{\xi\in D_{k}}J_{\lambda}(\lambda,\gamma_{\lambda}(\xi))$
$\displaystyle=\max_{\xi\in D_{k}}\bigg{(}{1\over
2}\|\nabla\gamma_{\lambda}(\xi)\|_{2}^{2}+{e^{\lambda}\over
2}\|f[\gamma_{\lambda}(\xi)]\|_{2}^{2}-\int_{\mathbb{R}^{N}}G[f(\gamma_{\lambda}(\xi))]\bigg{)}$
$\displaystyle\geq\max_{\xi\in D_{k}}\bigg{(}{1\over
2}\|\nabla\gamma_{\lambda}(\xi)\|_{2}^{2}+{e^{\lambda}\over
2}\|f[\gamma_{\lambda}(\xi)]\|_{2}^{2}-C\|\gamma_{\lambda}(\xi)\|_{q}^{q}\bigg{)}$
$\displaystyle\geq e^{\lambda}\max_{\xi\in D_{k}}\bigg{(}{1\over
2}\big{(}\|\nabla\eta_{\lambda}(\xi)\|_{2}^{2}+{1\over
2}\|f[\eta_{\lambda}(\xi)]\|_{2}^{2}-C\|\eta_{\lambda}(\xi)\|_{q}^{q}\bigg{)}$
$\displaystyle\geq e^{\lambda}\max_{\xi\in D_{k}}\bigg{(}{1\over
2}\big{(}\|\nabla\eta_{\lambda}(\xi)\|_{2}^{2}+{1\over
2}\|f[\eta_{\lambda}(\xi)]\|_{2}^{2}-\varepsilon\|f[\eta_{\lambda}(\xi)]\|_{2}^{2}-C_{\varepsilon}\|f\eta_{\lambda}(\xi)]\|_{2^{*}}^{2^{*}}\bigg{)}$
$\displaystyle\geq e^{\lambda}\max_{\xi\in D_{k}}\bigg{(}{1\over
4}\big{(}\|\nabla\eta_{\lambda}(\xi)\|_{2}^{2}+\|f[\eta_{\lambda}(\xi)]\|_{2}^{2}\big{)}-C\big{(}\|\nabla\eta_{\lambda}(\xi)\|_{2}^{2}+\|f[\eta_{\lambda}(\xi)]\|_{2}^{2}\big{)}^{\frac{2^{*}}{2}}\bigg{)}$
$\displaystyle=e^{\lambda}\max_{\rho\geq
0}\bigg{(}\frac{1}{4}\rho-C\rho^{\frac{2^{\ast}}{2}}\bigg{)}$
$\displaystyle=e^{\lambda}\frac{1}{2N}\bigg{(}\frac{N-2}{4NC}\bigg{)}^{\frac{N-2}{2}}$

Since $\gamma_{\lambda}$ is arbitrary, we have
$m_{k}\geq\frac{1}{2N}\big{(}\frac{N-2}{4NC}\big{)}^{\frac{N-2}{2}}$, and the
proof is completed.

###### Lemma 3.3

Assume that $\lambda_{0}=\infty$ and $(g_{1})-(g_{4})$ hold. Then
$\lim\limits_{\lambda\rightarrow+\infty}\frac{a_{k}(\lambda)}{e^{\lambda}}=+\infty.$

Proof: We denote $e^{\lambda}$ by $\mu$ for simplicity. Since
$0<a_{1}(\mu)\leq a_{k}(\mu)$ for each $k\in\mathbb{N}^{\ast}$, it is
sufficient to show the conclusion for $k=1$. Let $w_{r,s}(\cdot)=rv(s\cdot)$
for any $v\in H_{r}^{1}(\mathbb{R}^{N})$, by Lemma 2.1, we have

$\displaystyle J_{\mu}(w_{r,s})$ $\displaystyle={1\over 2}\|\nabla
w_{r,s}\|_{2}^{2}+{\mu\over
2}\|f(w_{r,s})\|_{2}^{2}-\int_{\mathbb{R}^{N}}G[f(w_{r,s})]$ (3.6)
$\displaystyle={1\over 2}r^{2}s^{2-N}\|\nabla v\|_{2}^{2}+{\mu\over
2}s^{-N}\|f(rv)\|_{2}^{2}-s^{-N}\int_{\mathbb{R}^{N}}G[f(rv)]$
$\displaystyle\geq{1\over 2}r^{2}s^{2-N}\|\nabla v\|_{2}^{2}+\big{(}{\mu\over
2}-\varepsilon\big{)}s^{-N}\|f(rv)\|_{2}^{2}-C_{\varepsilon}s^{-N}r^{\frac{p}{2}}\|v\|_{\frac{p}{2}}^{\frac{p}{2}}$
$\displaystyle\geq{1\over 2}r^{2}s^{2-N}\|\nabla
v\|_{2}^{2}+Cr\big{(}{\mu\over
2}-C_{\varepsilon}\big{)}s^{-N}\|f(v)\|_{2}^{2}-\varepsilon
s^{-N}r^{\frac{p}{2}}\|v\|_{\frac{p}{2}}^{\frac{p}{2}}$
$\displaystyle\geq{1\over 2}r^{2}s^{2-N}\|\nabla
v\|_{2}^{2}+\bigg{[}Cr\big{(}{\mu\over
2}-C_{\varepsilon}\big{)}s^{-N}-\varepsilon
s^{-N}r^{\frac{p}{2}}\bigg{]}\|f(v)\|_{2}^{2}-C\varepsilon
s^{-N}r^{\frac{p}{2}}\|f(v)\|_{p}^{p}.$

Letting
$w_{\mu}(\cdot)=\mu^{\frac{2}{p-2}}v\big{(}\mu^{\frac{p-4}{2p-4}}\cdot\big{)}$
and taking $r=\mu^{\frac{2}{p-2}}$ with $s=\mu^{\frac{p-4}{2p-4}}$, by (3.6),
there exists $\kappa>0$ such that

$\frac{J_{\mu}(w_{\mu})}{\mu}\geq{1\over 2}\|\nabla v\|_{2}^{2}+\big{(}\kappa-
C_{\varepsilon}\mu^{-1}\big{)}\|f(v)\|_{2}^{2}-\varepsilon\|f(v)\|_{p}^{p}.$
(3.7)

For $A>0$, let

$J_{A,\varepsilon}(v)={1\over 2}\|\nabla
v\|_{2}^{2}+A\|f(v)\|_{2}^{2}-\varepsilon\|f(v)\|_{p}^{p}.$

As in the proof of Theorem 1.1 of [45], there exists $v_{A,\varepsilon}\in
H_{r}^{1}(\mathbb{R}^{N})$ such that
$J_{A,\varepsilon}(v_{A,\varepsilon})=c(A,\varepsilon)>0$ and
$J^{\prime}_{A,\varepsilon}(v_{A,\varepsilon})=0$, where the minimax level is
defined by

$c(A,\varepsilon):=\inf\limits_{\gamma\in\Gamma_{A,\varepsilon}}\max\limits_{t\in[0,1]}J_{A,\varepsilon}(\gamma(t)),$

and

$\Gamma_{A,\varepsilon}=\\{\gamma\in
C\big{(}[0,1],H_{r}^{1}(\mathbb{R}^{N})\big{)}:\gamma(0)=0,~{}J_{A,\varepsilon}(\gamma(1))<0\\}.$

Claim 1: $\lim_{\varepsilon\rightarrow 0}c(A,\varepsilon)=+\infty.$

Since

$\displaystyle\|\nabla
v_{A,\varepsilon}\|_{2}^{2}+A\|f(v_{A,\varepsilon})\|_{2}^{2}$
$\displaystyle\leq\|\nabla
v_{A,\varepsilon}\|_{2}^{2}+2A\int_{\mathbb{R}^{N}}f(v_{A,\varepsilon})f^{\prime}(v_{A,\varepsilon})v_{A,\varepsilon}$
(3.8) $\displaystyle=\varepsilon
p\int_{\mathbb{R}^{N}}\big{|}f(v_{A,\varepsilon})\big{|}^{p-2}f(v_{A,\varepsilon})f^{\prime}(v_{A,\varepsilon})v_{A,\varepsilon}$
$\displaystyle\leq
C\varepsilon\|f(v_{A,\varepsilon}\|_{2}^{2}+C\varepsilon\|f(v_{A,\varepsilon})\|_{22^{\ast}}^{22^{\ast}}$
$\displaystyle\leq C\varepsilon\big{(}\|\nabla
v_{A,\varepsilon}\|_{2}^{2}+\|f(v_{A,\varepsilon})\|_{2}^{2}\big{)}+C\varepsilon\bigg{(}\|\nabla
v_{A,\varepsilon}\|_{2}^{2}+\|f(v_{A,\varepsilon})\|_{2}^{2}\bigg{)}^{\frac{2^{*}}{2}},$

one has $\|\nabla
v_{A,\varepsilon}\|_{2}^{2}+\|f(v_{A,\varepsilon})\|_{2}^{2}\geq C$ for samll
$\varepsilon$. Moreover, (3.8) leads to

$\|\nabla
v_{A,\varepsilon}\|_{2}^{2}+\|f(v_{A,\varepsilon})\|_{2}^{2}\rightarrow+\infty~{}~{}\hbox{as}~{}~{}\varepsilon\rightarrow
0.$

Consequently,

$\displaystyle c(A,\varepsilon)$
$\displaystyle=J_{A,\varepsilon}(v_{A,\varepsilon})-\frac{2}{p}\langle
J^{\prime}_{A,\varepsilon}(v_{A,\varepsilon}),v_{A,\varepsilon}\rangle$ (3.9)
$\displaystyle\geq\bigg{(}1-\frac{2}{p}\bigg{)}\big{(}\|\nabla
v_{A,\varepsilon}\|_{2}^{2}+A\|f(v_{A,\varepsilon})\|_{2}^{2}\big{)}\rightarrow+\infty~{}~{}\hbox{as}~{}~{}\varepsilon\rightarrow
0.$

Claim 2: $c(A,\varepsilon)$ is continuous with respect to $A$.

Without loss of generality, we may assume $\varepsilon=1$. For fixed $A$, we
denote $J_{\delta}:=J_{A+\delta,1}$, so do $c_{\delta}$ and $\Gamma_{\delta}$.
Clearly, $c_{\delta}\leq c_{0}$ if $\delta<0$. Suppose that
$\delta_{k}\rightarrow 0^{-}$, we claim that
$\lim_{k\rightarrow\infty}c_{\delta_{k}}=c_{0}$. Otherwise, up to a
subsequence, $\lim_{k\rightarrow\infty}c_{\delta_{k}}:=\underline{c}<c_{0}$,
because $c(A,1)\leq c(B,1)$ if $A<B$. Take $\\{v_{k}\\}\subset
H_{r}^{1}(\mathbb{R}^{N})$ with $\|v_{k}\|$=1 such that

$\max\limits_{t\geq 0}J_{\delta_{k}}(tv_{k})\leq
c_{\delta_{k}}+\frac{1}{k^{2}}.$

Using the proof of Lemma 2.5 in [45] and Proposition 3.11 in [38], there
exists $\gamma_{k}\in\Gamma_{k}$ with $\gamma_{k}(t)=\psi_{k}(t)v_{k}$ such
that

$\max\limits_{t\in[0,1]}J_{\delta_{k}}(\gamma_{k}(t))=\max\limits_{t\geq
0}J_{\delta_{k}}(tv_{k})\leq c_{\delta_{k}}+\frac{1}{k^{2}}.$

According to Theorem $4.3$ of [34], there exist $\\{w_{k}\\}\subset
H_{r}^{1}(\mathbb{R}^{N})$ and $\\{t_{k}\\}\subset[0,1]$ such that

$\displaystyle\|w_{k}-\gamma_{k}(t_{k})\|\leq\frac{1}{k},$ (3.10)
$\displaystyle
J_{\delta_{k}}(w_{k})\in\big{(}c_{\delta_{k}}-\frac{1}{k},c_{\delta_{k}}+\frac{1}{k}\big{)},$
$\displaystyle\|J^{\prime}_{\delta_{k}}(w_{k})\|\leq\frac{1}{k}.$

Let $\max\limits_{t\geq 0}J_{0}(tv_{k}):=J_{0}(\varphi(v_{k})v_{k})$. Then

$\displaystyle c_{0}$ $\displaystyle\leq\max\limits_{t\geq 0}J_{0}(tv_{k})$
(3.11) $\displaystyle=J_{0}(\varphi(v_{k})v_{k})$
$\displaystyle=J_{\delta_{k}}(\varphi(v_{k})v_{k})-\delta_{k}\|f[\varphi(v_{k})v_{k}]\|_{2}^{2}$
$\displaystyle\leq\max\limits_{t\in[0,1]}J_{\delta_{k}}(tv_{k})-\delta_{k}\|f[\varphi(v_{k})v_{k}]\|_{2}^{2}$
$\displaystyle\leq
c_{\delta_{k}}+\frac{1}{k^{2}}-\delta_{k}\|f[\varphi(v_{k})v_{k}]\|_{2}^{2}.$

Up to a subsequence, if ${\varphi(v_{k})}$ is bounded above, by

$\|f[\varphi(v_{k})v_{k}]\|_{2}^{2}\leq\varphi^{2}(v_{k})$

and (3.12), a contradiction will be obtained. Thus we may assume that
$\varphi(v_{k})\geq 1$ for all $k$, by Lemma 2.1, we have

$\displaystyle C_{1}\varphi^{2}(v_{k})$
$\displaystyle\geq\varphi^{2}(v_{k})\|\nabla
v_{k}\|_{2}^{2}+2A\int_{\mathbb{R}^{N}}f[\varphi(v_{k})v_{k}]f^{\prime}[\varphi(v_{k})v_{k}]\varphi(v_{k})v_{k}$
$\displaystyle=p\int_{\mathbb{R}^{N}}\big{|}f[\varphi(v_{k})v_{k}]\big{|}^{p-2}f[\varphi(v_{k})v_{k}]f^{\prime}[\varphi(v_{k})v_{k}]\varphi(v_{k})v_{k}$
$\displaystyle\geq
C_{2}(\varphi(v_{k}))^{\frac{p}{2}}\|v_{k}\|_{\frac{p}{2}}^{\frac{p}{2}},$

which leads to

$\varphi(v_{k})\leq\|v_{k}\|_{\frac{p}{2}}^{\frac{p}{8-2p}}.$

If, along a subsequence, $\|v_{k}\|_{\frac{p}{2}}\geq C>0$, then we get upper
bound for ${\varphi(v_{k})}$. Otherwise,

$v_{k}\rightarrow 0~{}~{}\hbox{in}~{}~{}L^{\frac{p}{2}}(\mathbb{R}^{N}).$

We claim that there exist $R,\delta>0$ and $\\{y_{k}\\}\subset\mathbb{R}^{N}$
such that

$\liminf_{n\rightarrow\infty}\int_{B_{R}(y_{k})}|w_{k}|^{2}\geq\delta.$

If not, for any $r>0$

$\lim_{k\rightarrow\infty}\sup_{y\in\mathbb{R}^{N}}\int_{B_{r}(y)}|w_{k}|^{2}=0.$

By (LABEL:eqBB150), $\\{w_{k}\\}$ is bounded in $H_{r}^{1}(\mathbb{R}^{N})$.
It follows from Lemma 1.21 of [42] and its proof that $v_{k}\rightarrow 0$ in
$L^{s}(\mathbb{R}^{N})$, where $2\leq s<2^{*}$. Consequently,

$\displaystyle 0<c_{0}\leq c_{\delta_{k}}$
$\displaystyle=\lim_{k\rightarrow\infty}\bigg{(}J_{\delta_{k}}(w_{k})-\langle
J^{\prime}_{\delta_{k}}(w_{k}),w_{k}\rangle\bigg{)}$ $\displaystyle\leq
p\int_{\mathbb{R}^{N}}|f(w_{k})|^{p-2}f(w_{k})f^{\prime}(w_{k})w_{k}+\|f(w_{k})\|_{p}^{p}$
$\displaystyle\leq C\|w_{k}\|_{\frac{p}{2}}^{\frac{p}{2}}\rightarrow 0,$

a contradiction. In view of

$\psi_{k}(t)=\|\psi_{k}(t)v_{k}\|\leq\|w_{k}-\psi_{k}(t)v_{k}\|+\|w_{k}\|\leq
C,$

we have for large $k$,

$\displaystyle\|v_{k}\|_{2}$ $\displaystyle\geq\|v_{k}\|_{2,B_{R}(y_{k})}$
$\displaystyle\geq{\psi_{k}(t)}^{-1}\big{(}\|w_{k}\|_{2,B_{R}(y_{k})}-\|w_{k}-\psi_{k}(t)v_{k}\|_{2,B_{R}(y_{k})}\big{)}$
$\displaystyle\geq C^{-1}\big{(}\delta-\frac{1}{k}\big{)}$
$\displaystyle\geq\frac{\delta}{2C},$

a contradiction.

Suppose that $\delta_{k}\rightarrow 0^{+}$. If, up to a subsequence,
$\lim_{k\rightarrow\infty}c_{\delta_{k}}:=\overline{c}>c_{0}$. Take
$\\{v_{k}\\}\subset H_{r}^{1}(\mathbb{R}^{N})$ with $\|v_{k}\|$=1 such that

$\max\limits_{t\geq 0}J_{0}(tv_{k})\leq c_{0}+\frac{1}{k^{2}}.$

Observe that

$\displaystyle c_{\delta_{k}}$ $\displaystyle\leq\max\limits_{t\geq
0}J_{\delta_{k}}(tv_{k})$ (3.12)
$\displaystyle=J_{\delta_{k}}(\varphi_{\delta_{k}}(v_{k})v_{k})$
$\displaystyle=J_{0}(\varphi_{\delta_{k}}(v_{k})v_{k})+\delta_{k}\|f[\varphi_{\delta_{k}}(v_{k})v_{k}]\|_{2}^{2}$
$\displaystyle\leq\max\limits_{t\in[0,1]}J_{0}(tv_{k})+\delta_{k}\|f[\varphi_{\delta_{k}}(v_{k})v_{k}]\|_{2}^{2}$
$\displaystyle\leq
c_{0}+\frac{1}{k^{2}}+\delta_{k}\|f[\varphi_{\delta_{k}}(v_{k})v_{k}]\|_{2}^{2}$
$\displaystyle\leq
c_{0}+\frac{1}{k^{2}}+\delta_{k}\varphi_{\delta_{k}}(v_{k})^{2},$

where $\max\limits_{t\geq
0}J_{\delta_{k}}(tv_{k}):=J_{\delta_{k}}(\varphi_{\delta_{k}}(v_{k})v_{k})$.
Similarly, we can obtain a contradiction by proving that
$\\{\varphi_{\delta_{k}}(v_{k})\\}$ is bounded above.

For any $\gamma\in\Gamma_{1}(\mu)$ and $\xi\in[0,1]$, let

$\eta(\xi)(\cdot)=\mu^{-\frac{2}{p-2}}\gamma(\xi)\big{(}\mu^{-\frac{p-4}{2p-4}}\cdot\big{)}.$

In view of (3.7), we get $\eta\in\Gamma_{\kappa-
C_{\varepsilon}\mu^{-1},\varepsilon}$ and

$\displaystyle\max_{\xi\in[0,1]}J_{\mu}(\gamma(\xi))=\max_{\xi\in[0,1]}J_{\mu}({\eta(\xi)}_{\mu})\geq\mu\max_{\xi\in[0,1]}J_{\kappa-
C_{\varepsilon}\mu^{-1},\varepsilon}(\eta(\xi))\geq\mu c\big{(}\kappa-
C_{\varepsilon}\mu^{-1},\varepsilon\big{)},$

which implies that

$\frac{a_{1}(\mu)}{\mu}\geq c\big{(}\kappa-
C_{\varepsilon}\mu^{-1},\varepsilon\big{)}.$

It follows claim $2$ that

$\lim\limits_{\mu\rightarrow+\infty}\frac{a_{1}(\mu)}{\mu}\geq\lim\limits_{\mu\rightarrow+\infty}c\big{(}\kappa-
C_{\varepsilon}\mu^{-1},\varepsilon\big{)}=c\big{(}\kappa,\varepsilon\big{)}.$

Since $\varepsilon>0$ is arbitrary, by claim $1$, we have

$\lim\limits_{\mu\rightarrow+\infty}\frac{a_{1}(\mu)}{\mu}=+\infty.$

###  3.2 Construction of Pohozaev Mountain and Minimax level

Following [9], we introduce the Pohozaev level set

$\Omega:=\\{(\lambda,v)\in\mathbb{R}\times
H_{r}^{1}(\mathbb{R}^{N}):P(\lambda,v)>0\\}\cup\\{(\lambda,0):\lambda\in\mathbb{R}\\}.$

As in the proof of (3.3), we have $P(\lambda,v)>0$ for
$v\in\partial\Sigma_{\rho}$ if $\rho$ is small. Hence
$\\{(\lambda,0):\lambda\in\mathbb{R}\\}\subset\hbox{int}(\Omega)$, and hence

$\partial\Omega=\\{(\lambda,v)\in\mathbb{R}\times
H_{r}^{1}(\mathbb{R}^{N}):P(\lambda,v)=0,v\neq 0\\}$

Combining Theorem $1.1$ in [18] and a Pohozaev identity given in Lemma $3.1$
in [10], we have $\partial\Omega\neq\emptyset$. We also remark that
nonlinearity odd is required and thus the existence of infinitely many
solutions can be obtain in [18]. However, we can still prove the existence of
solutions if we remove the odd hypothesis.

###### Lemma 3.4

The following statements hold:

  * $(1)$

$J_{\lambda}(v)\geq 0$ for all $(\lambda,v)\in\Omega$.

  * $(2)$

$J_{\lambda}(v)\geq a_{1}(\lambda)>0$ for all $(\lambda,v)\in\partial\Omega.$

  * $(3)$

$E_{m}\geq B_{m}>-\infty$, where

$E_{m}:=\inf_{(\lambda,v)\in\partial\Omega}J^{m}(\lambda,v)~{}~{}\hbox{and}~{}~{}B_{m}:=\inf_{\lambda\in(-\infty,\lambda_{0})}\bigg{(}a_{1}(\lambda)-{e^{\lambda}\over
2}m\bigg{)}.$

Proof: Note that

$J_{\lambda}(v)\geq J_{\lambda}(v)-\frac{1}{N}P(\lambda,v)=\frac{1}{N}\|\nabla
v\|_{2}^{2}\geq 0$

and $(1)$ follows.

For any $(\lambda,v)\in\partial\Omega$, we claim that $\lambda<\lambda_{0}.$
Otherwise, by the definition of $\lambda_{0}$, we have

$G(s)-{Ne^{\lambda}\over 2}s^{2}\leq
0~{}~{}\hbox{for}~{}~{}\hbox{all}~{}~{}s\in\mathbb{R}.$

This leads to

$\displaystyle 0=P(\lambda,v)={N-2\over 2}\|\nabla
v\|_{2}^{2}-N\bigg{(}\int_{\mathbb{R}^{N}}G[f(v)]-{e^{\lambda}\over
2}\|f(v)\|^{2}\bigg{)}\geq{N-2\over 2}\|\nabla v\|_{2}^{2}>0,$

a contradiction. For any $(\lambda,v)\in\partial\Omega$, we have

$\displaystyle 0<{N-2\over 2}\|\nabla
v\|_{2}^{2}=N\bigg{(}\int_{\mathbb{R}^{N}}G[f(v)]-{e^{\lambda}\over
2}\|f(v)\|_{2}^{2}\bigg{)}.$

Set $h(t)=\frac{d}{dt}\big{(}J_{\lambda}(v(\cdot/t))\big{)}$. It follows from

$h^{\prime}(t)=\frac{N-2}{2}t^{(N-2)}\|\nabla v\|_{2}^{2}(1-t^{2})$

that

$\max_{t\geq 0}J_{\lambda}(v(\cdot/t))=J_{\lambda}(v).$

Since $J_{\lambda}(v(\cdot/t)\rightarrow-\infty$ as $t\rightarrow+\infty$,
there exists $t_{0}>0$ such that $J_{\lambda}(v(\cdot/t_{0}))<0$. Let

$\gamma(t)=\begin{cases}v(\cdot/tt_{0}),~{}~{}~{}\hbox{if}~{}~{}0<t\leq 1,\\\
0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hbox{if}~{}~{}t=0.\end{cases}$

Clearly, $\gamma\in{\Gamma}_{1}(\lambda)$ and

${J_{\lambda}}(v)=\max_{\xi\in[0,1]}J_{\lambda}(\gamma(t))\geq
a_{1}(\lambda).$

Passing to the infimum over ${\Gamma}_{1}(\lambda)$, $(2)$ follows. Hence
$E_{m}\geq B_{m}$, while the fact that $B_{m}>-\infty$ follows from Lemma 3.3.

Next, let us define a family of minimax values as follows.

###### Definition 3.2

For any $k=1$, set

$b^{m}_{1}=\inf_{\Upsilon\in\Gamma^{m}_{1}}\max_{\xi\in[0,1]}J^{m}(\gamma(\xi)),$

where

$\Gamma^{m}_{1}=\\{\Upsilon=(\Upsilon_{1},\Upsilon_{2})\in
C([0,1],\mathbb{R}\times
H_{r}^{1}(\mathbb{R}^{N})):\Upsilon~{}\hbox{satisfies}~{}(\rm{{i}})-(\rm{{ii}})~{}~{}\hbox{below}\\}.$

  * (i)

$J^{m}(\Upsilon(0))\leq B_{m}-1$.

  * (ii)

$\Upsilon(1)\notin\Omega$ and $J^{m}(\Upsilon(1))\leq B_{m}-1$.

For any $k\geq 2$, set

$b^{m}_{k}=\inf_{\Upsilon\in\Gamma^{m}_{k}}\max_{\xi\in
D_{k}}J^{m}(\gamma(\xi)),$

where

$\Gamma^{m}_{k}=\\{\Upsilon=(\Upsilon_{1},\Upsilon_{2})\in
C(D_{k},\mathbb{R}\times
H_{r}^{1}(\mathbb{R}^{N})):\Upsilon~{}\hbox{satisfies}~{}(\rm{{i}})-(\rm{{iii}})~{}~{}\hbox{below}\\}.$

  * (i)

$\Upsilon\in\Gamma^{m}_{k}$ to be $\mathbb{Z}_{2}$-equivariant, that is
$\Upsilon_{1}(-\xi)=\Upsilon_{1}(\xi)$,
$\Upsilon_{2}(-\xi)=-\Upsilon_{2}(\xi)$ for all $\xi\in D_{k}$.

  * (ii)

$J^{m}(\Upsilon(0))\leq B_{m}-1$.

  * (iii)

$\Upsilon|_{\partial D_{k}}\cap\Omega=\emptyset$ and
$J^{m}(\Upsilon|_{\partial D_{k}})\leq B_{m}-1$.

###### Lemma 3.5

For any $k\in\mathbb{N}$, $\Gamma^{m}_{k}\neq\emptyset$. Moreover, $B_{m}\leq
b^{m}_{k}<0$ if $m>m_{k}$.

Proof: Here we only prove the case for $k\geq 2$, since the case $k=1$ is
similar. In order to prove $B_{m}\leq b^{m}_{k}$, it suffices to prove that
any path in $\Gamma^{m}_{k}$ passes through $\partial\Omega$. In fact, for all
$\Upsilon\in\Gamma^{m}_{k}$, there exist $\xi_{1}\in\hbox{int}~{}(D_{k})$ and
$\xi_{2}\in\hbox{int}~{}(D_{k})$ such that $P(\Upsilon(\xi_{1}))>0$ and
$P(\Upsilon(\xi_{2}))<0$. Let

$\alpha:=\sup\\{t\in[0,1]:\chi(t)>0\\},~{}~{}\beta:=\inf\\{t\in[0,1]:\chi(t)<0\\},$

where $\chi(t)=P(\Upsilon(t\xi_{1}+(1-t)\xi_{2})).$ Clearly,
$0<\alpha\leq\beta<1$ and $\chi(\alpha)=\chi(\beta)=0$. We divide the proof
into two cases as follows.

Case 1: If $\alpha=\beta$, for small $\delta>0$, by the definition of $\alpha$
and $\beta$, there exist $t_{1},t_{2}\in(\alpha-\delta,\alpha+\delta)$ such
that $\chi(t_{1})>0$ and $\chi(t_{2})<0$. By the fact that
$\\{(\lambda,0)|\lambda\in\mathbb{R}\\}\subset\hbox{int}(\Omega)$, we have
$\Upsilon_{2}(\alpha\xi_{1}+(1-\alpha)\xi_{2}))\neq 0$, and hence
$\Upsilon(\alpha\xi_{1}+(1-\alpha)\xi_{2}))\in\partial\Omega$.

Case 2: If $\alpha<\beta$, for small $\delta>0$, there exist
$t_{1}\in(\alpha-\delta,\alpha)$ such that $\chi(t_{1})>0$ and $\chi(t)\equiv
0$ for $t\in[\alpha,\beta]$. Thus
$\Upsilon(\alpha\xi_{1}+(1-\alpha)\xi_{2}))\in\partial\Omega$.

Fix $\lambda\in(-\infty,\lambda_{0})$, for any $\gamma\in\Gamma_{k}(\lambda)$
and $\xi\in\partial D_{k}$, we have

$\displaystyle J_{\lambda}\big{(}\gamma(\xi)(L^{-1}\cdot)\big{)}$
$\displaystyle={1\over
2}L^{N-2}\|\nabla\gamma(\xi)\|_{2}^{2}+{e^{\lambda}\over
2}L^{N}\|f(\gamma(\xi))\|_{2}^{2}-L^{N}\int_{\mathbb{R}^{N}}G[f(\gamma(\xi))]$
$\displaystyle\leq{1\over
2}L^{N}\|\nabla\gamma(\xi)\|_{2}^{2}+{e^{\lambda}\over
2}L^{N}\|f(\gamma(\xi))\|_{2}^{2}-L^{N}\int_{\mathbb{R}^{N}}G[f(\gamma(\xi))]$
$\displaystyle\leq
L^{N}M\rightarrow-\infty,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hbox{as}~{}~{}L\rightarrow\infty,$

where $M=\max\limits_{\xi\in\partial D_{k}}J_{\lambda}(\gamma(\xi))<0$.
Consequently,
$\lim\limits_{L\rightarrow\infty}J_{\lambda}\big{(}\gamma(\xi)(L^{-1}\cdot)\big{)}=-\infty$
uniformly for $\xi\in\partial D_{k}$. Taking large $L>0$ and setting

$\gamma_{0}(\xi)(\cdot)=\begin{cases}\gamma(2\xi)(\cdot)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hbox{if}~{}~{}|\xi|\in[0,\frac{1}{2}],\\\
\gamma\big{(}\frac{\xi}{|\xi|}\big{)}\big{(}((2|\xi|-1)L+1)^{-1}\cdot\big{)}~{}~{}\hbox{if}~{}~{}|\xi|\in(\frac{1}{2},1],\end{cases}$

then $\gamma_{0}\in\Gamma_{k}(\lambda)$. Moreover,

$\max_{\xi\in D_{k}}J_{\lambda}(\gamma_{0}(\xi))=\max_{\xi\in
D_{k}}J_{\lambda}(\gamma(\xi)),~{}~{}\max_{\xi\in\partial
D_{k}}J_{\lambda}(\gamma_{0}(\xi))\leq B_{m}-1.$ (3.13)

Now, let us define $\Upsilon=(\Upsilon_{1},\Upsilon_{2})$ as follows:

$\Upsilon_{1}(\xi)=\begin{cases}\lambda+R(1-2|\xi|)~{}~{}~{}\hbox{if}~{}~{}|\xi|\in[0,\frac{1}{2}],\\\
\lambda~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hbox{if}~{}~{}|\xi|\in(\frac{1}{2},1],\end{cases}$
$\Upsilon_{2}(\xi)=\begin{cases}0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hbox{if}~{}~{}|\xi|\in[0,\frac{1}{2}],\\\
\gamma_{0}\big{(}\frac{\xi}{|\xi|}(2|\xi|-1)\big{)}~{}~{}\hbox{if}~{}~{}|\xi|\in(\frac{1}{2},1].\end{cases}$

It is easy to see that

$J^{m}(\Upsilon(\xi))=-\frac{e^{\lambda+R(1-2|\xi|)}}{2}m~{}~{}\hbox{for}~{}~{}|\xi|\in[0,\frac{1}{2}],$
$J^{m}(\Upsilon(\xi))=J^{m}(\lambda,\Upsilon_{2}(\xi))=J_{\lambda}(\Upsilon_{2}(\xi))-\frac{e^{\lambda}}{2}m~{}~{}\hbox{for}~{}~{}|\xi|\in(\frac{1}{2},1].$

By choosing a large $R>0$, we have $J^{m}(\Upsilon(0))\leq B_{m}-1$. If
$\xi\in\partial D_{k}$, by (3.13), we have for large $L$

$J^{m}(\Upsilon(\xi))=J_{\lambda}(\gamma_{0}(\xi))-\frac{e^{\lambda}}{2}m\leq
B_{m}-1$

and

$P(\Upsilon(\xi))=P(\lambda,\gamma_{0}(\xi))=P\big{(}\lambda,\gamma_{0}(\xi)(L+1)^{-1}\cdot)\big{)}<0.$

Consequently, $\Upsilon\in\Gamma_{k}^{m}$. By (3.13), we have

$\displaystyle b^{m}_{k}$ $\displaystyle\leq\max_{\xi\in
D_{j}}J^{m}(\Upsilon(\xi))$
$\displaystyle=\max_{|\xi|\in(\frac{1}{2},1]}J^{m}(\Upsilon(\xi))$
$\displaystyle=\max_{\xi\in
D_{k}}J_{\lambda}(\gamma_{0}(\xi))-\frac{e^{\lambda}}{2}m$
$\displaystyle=\max_{\xi\in
D_{k}}J_{\lambda}(\gamma(\xi))-\frac{e^{\lambda}}{2}m.$

Since $\gamma$ is arbitrary, for any $\lambda\in(-\infty,\lambda_{0}),$

$b^{m}_{k}\leq a_{k}(\lambda)-\frac{e^{\lambda}}{2}m.$ (3.14)

Hence

$\inf_{\lambda(-\infty,\lambda_{0})}\frac{b^{m}_{k}}{e^{\lambda}}\leq\inf_{\lambda\in(-\infty,\lambda_{0})}\frac{a_{k}(\lambda)}{e^{\lambda}}-\frac{m}{2}=\frac{m_{k}}{2}-\frac{m}{2}<0,$

and hence $b^{m}_{k}<0$. This completes the proof of Lemma 3.5.

In order to study the multiplicity of radial for equation $(P_{\mu,m})$, let
us define another family of minimax values as follows:

###### Definition 3.3

For $k\geq 2$, we set

$c_{k}^{m}=\inf_{A\in\Lambda^{m}_{k}}\max_{(\lambda,v)\in A}J^{m}(\lambda,v),$

where

$\Lambda_{k}^{m}=\\{\Upsilon(\overline{D_{k+j}\setminus
Y}):j\in\mathbb{N},\Upsilon\in\Gamma^{m}_{k+j},Y\subset
D_{k+j}\setminus\\{0\\}~{}~{}\hbox{is}~{}~{}\hbox{closed},~{}-Y=Y,~{}\hbox{and}~{}~{}\hbox{genus}(Y)\leq
j\\}.$

Let $P_{2}:\mathbb{R}\times H_{r}^{1}(\mathbb{R}^{N})\rightarrow
H_{r}^{1}(\mathbb{R}^{N})$ be a projection defined by

$P_{2}(\lambda,v)=v~{}~{}~{}\hbox{for}~{}~{}(\lambda,v)\in\mathbb{R}\times
H_{r}^{1}(\mathbb{R}^{N}).$

###### Lemma 3.6

We have the following results:

  * (i)

$\Lambda^{m}_{k}\neq\emptyset.$

  * (ii)

$\Lambda^{m}_{k}\subset\Lambda^{m}_{k+1}$, and thus $c^{m}_{k}\leq
c^{m}_{k+1}$.

  * (iii)

$c^{m}_{k}\leq b^{m}_{k}.$

  * (iv)

Let $A\in\Lambda^{m}_{k}$ and $Z\subset\mathbb{R}\times
H_{r}^{1}(\mathbb{R}^{N})$ be $\mathbb{Z}_{2}$-invariant, closed, and such
that $0\notin\overline{P_{2}(Z)}$ and $\hbox{genus}\overline{P_{2}(Z)}\leq
i<n$. Then $\overline{A\setminus Z}\in\Lambda^{m}_{k-i}.$

Proof: The proof is given in [14] and [8].

###### Lemma 3.7

The following statements hold:

  * (i)

$b_{1}^{m}$ and $c_{k}^{m}$ are critical values of $J^{m}(\lambda,v)$ for all
$k\geq 2$.

  * $({ii})$

If $c_{k}=c_{k+1}=\cdot\cdot\cdot=c_{k+j}:=c$, then
$\hbox{genus}(P_{2}(K^{m}_{c}))\geq j+1$.

Proof: Since the $(PSP)_{c}$ condition holds for $c<0$ by Lemma 2.4, we can
develop deformation theory given in Lemma 2.7. We can also observe that the
negative minimax values $b_{1}^{m}$ and $c_{k}^{m}$ are stable under the
deformation. Thus (i) follows from Lemma 2.7, see [44] for details.
Essentially, the proof of (ii) is similar as Proposition 3.3 in [14]. Here we
give some details for completeness. By Lemma 2.7–(i), $P_{2}(K_{c}^{m})$ is
compact, symmetric with respect to $0$ and $0\notin P_{2}(K_{c}^{m})$. By the
fundamental properties of genus [37], we have

  * (1)

$\hbox{genus}(P_{2}(K_{c}^{m}))\leq\infty$,

  * (2)

there exists $\delta>0$ small such that
$\hbox{genus}(\overline{P_{2}(N_{\delta}(K_{c}^{m}))})=\hbox{genus}(P_{2}(K_{c}^{m}))$.

Let $\varepsilon\in(0,1)$ and $A\in\Lambda^{m}_{k+j}$ such that
$A\subset{(J^{m})}^{c-\varepsilon}$. By Lemma 2.7–(4), one has

$\eta(1,\overline{A\setminus
P_{2}(N_{\delta}(K_{c}))})\subset{(J^{m})}^{c-\varepsilon}.$

We argue by contradiction that $\hbox{genus}(P_{2}(K_{c}^{m}))\leq j$. By
Lemma 2.4–(iv) in [14], one can check that

$\overline{A\setminus
P_{2}(N_{\delta}(K_{c}^{m}))}\in\Lambda^{m}_{k+j-\hbox{genus}(\overline{P_{2}(N_{\delta}(K_{c}^{m}))})}\subset\Lambda_{k}^{m}.$

Through a similar argument to (i), we have $\eta(1,\overline{A\setminus
P_{2}(N_{\delta}(K_{c}^{m}))})\in\Lambda^{m}_{k}$, and hence

$J^{m}(\eta(1,\overline{A\setminus P_{2}(N_{\delta}(K_{c}^{m}))}))\geq c.$

This leads to $c\leq c-\varepsilon$, a contradiction.

##  4 The proofs of main results

The proof of Theorems 1.4–1.5

Proof: By Lemma 2.3 and Lemma 3.7, it suffices to prove that $b_{1}^{m}$ be
least energy on

$\\{u:u\in
S(m),\mu\in\mathbb{R}~{}\hbox{and}~{}(u,\mu)~{}\hbox{solves}~{}(P_{\mu,m})\\}.$

We argue by contradiction that there exists $(\mu_{\ast},u_{\ast})$ solves
$(P_{\mu,m})$ with $E(u_{\ast})<b_{1}^{m}<0$, that is, $(\mu_{\ast},v_{\ast})$
satisfies

$-\Delta v+\mu f(v)f^{\prime}(v)=g[f(v)]f^{\prime}(v)$

and $\|f(v_{\ast})\|_{2}^{2}=m$, where $v_{\ast}:=f^{-1}(u_{\ast})$. By
corollary B.4 in [42], we have $v_{\ast}$ satisfies Pohozaev identity

$\frac{N-2}{2}\|\nabla
v_{\ast}\|_{2}^{2}+{N\mu_{\ast}}\|f(v_{\ast})\|_{2}^{2}-N\int_{\mathbb{R}^{N}}G[f(v_{\ast})]=0.$

Since

$NE(u\ast)-\|\nabla v\ast\|^{2}+N\mu_{\ast}m=\frac{N-2}{2}\|\nabla
v_{\ast}\|_{2}^{2}+{N\mu_{\ast}}\|f(v_{\ast})\|_{2}^{2}-N\int_{\mathbb{R}^{N}}G[f(v_{\ast})]=0,$

$\mu_{\ast}>0$. Next we claim that $v\mapsto J_{\log\mu_{\ast}}(v)$ satisfies
mountain pass geometry, that is, $\log\mu_{\ast}<\lambda_{0}$. To this aim, it
is suffice to show $J_{\log\mu_{\ast}}(v)<0$ for some $v\in
H^{1}_{r}(\mathbb{R}^{N})$. Recall that
$H(s):=-\frac{\mu_{\ast}}{2}f^{2}(s)+G[f(s)]$. If $N\geq 3$, by
$P(\lambda_{\ast},v_{\ast})=0$, we have

$\displaystyle\int_{\mathbb{R}^{N}}H(v_{\ast})={N-2\over 2N}\|\nabla
v_{\ast}\|_{2}^{2}>0.$

Thus, we take $v=v_{\ast}$ for $N\geq 3$. If $N=2$, by
$P(\lambda_{\ast},v_{\ast})=0$, we have

$\int_{\mathbb{R}^{N}}H(v_{\ast})=0$

and

$\frac{d}{ds}\int_{\mathbb{R}^{N}}H(sv_{\ast})\big{|}_{s=1}=\int_{\mathbb{R}^{N}}-\frac{\mu_{\ast}}{2}f(v_{\ast})f^{\prime}(v_{\ast})v_{\ast}+g[f(v_{\ast})]f^{\prime}(v_{\ast})v_{\ast}=\int_{\mathbb{R}^{N}}|\nabla
v_{\ast}|^{2}>0.$

So we can choose some $t_{\ast}\rightarrow 1^{+}$ such that
$\int_{\mathbb{R}^{N}}H(t^{\ast}v_{\ast})>0.$ Thus, we take $v=v_{\ast}$ for
$N=2$. Consequently, as $\theta\rightarrow\infty$,

$\displaystyle J_{\log\mu_{\ast}}(v(\cdot/\theta))={1\over
2}\theta^{N-2}\|\nabla
v\|_{2}^{2}-\theta^{N}\int_{\mathbb{R}^{N}}H(v)\rightarrow-\infty.$

Fix large $\theta>0$ and let

$\gamma(t)=\begin{cases}v(\cdot/t\theta),~{}~{}~{}\hbox{if}~{}~{}0<t\leq 1,\\\
0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hbox{if}~{}~{}t=0.\end{cases}$

Clearly, $\gamma\in{\Gamma}_{1}(\log\mu_{\ast})$. It follows (3.14) that

$E(u_{\ast})<b_{1}^{m}\leq
a_{1}(\log\mu_{\ast})-\frac{\mu_{\ast}}{2}m\leq\max_{t\in[0,1]}J_{\log\mu_{\ast}}(\gamma(t))-\frac{\mu_{\ast}}{2}m=J_{\log\mu_{\ast}}(v_{\ast})-\frac{\mu_{\ast}}{2}m=E(u_{\ast}),$

a contradiction.

The proof of Theorems 1.6

Proof: If $m>m_{1}$, by Lemma 3.7, there exists $(\lambda_{m},v_{m})\in
K_{b^{1}_{m}}^{m}$. Let $u_{m}:=f(v_{m})$. Then
$J^{m}(\lambda_{m},v_{m})=b^{1}_{m}$ and

$0=\partial_{\lambda}J^{m}(\lambda_{m},v_{m})={e^{\lambda_{m}}\over
2}\bigg{(}\|u_{m}\|_{2}^{2}-m\bigg{)},$

which implies that $\|u_{m}\|_{2}^{2}=m$ and

$\displaystyle 0>b^{1}_{m}=J^{m}(\lambda_{m},v_{m})$ $\displaystyle={1\over
2}\|\nabla v_{m}\|_{2}^{2}+{e^{\lambda_{m}}\over
2}\big{(}\|f(v_{m})\|^{2}_{2}-m\big{)}-\int_{\mathbb{R}^{N}}G[f(v_{m})]$
$\displaystyle={1\over
2}\int_{\mathbb{R}^{N}}\big{(}1+2u_{m}^{2}\big{)}|\nabla
u_{m}|^{2}-\int_{\mathbb{R}^{N}}G(u_{m}).$

Consequently, $e(m)\leq b^{1}_{m}<0$. Let $\\{u_{n}\\}\subset X$ be a
minimizing sequence for $e(m)$, that is, $\\{u_{n}\\}\subset S(m)$ with
$E(u_{n})\rightarrow e_{m}$. Let $u_{n}^{\ast}$ be the Steiner rearrangement
of $u_{n}$. By $(g_{5})$ and $(g_{7})$, and (ii) of Proposition 2.1 in [41],
we have

$\int_{\mathbb{R}^{N}}G(u_{n})=\int_{\mathbb{R}^{N}}G(u_{n}^{\ast}).$

From standard rearrangement inequalities, we have
$\|u_{n}\|^{2}_{2}=\|u_{n}^{\ast}\|^{2}_{2}$ and

$\int_{\mathbb{R}^{N}}\big{(}1+2(u_{n}^{\ast})^{2}\big{)}|\nabla
u_{n}^{\ast}|^{2}\leq\int_{\mathbb{R}^{N}}\big{(}1+2u_{n}^{2}\big{)}|\nabla
u_{n}|^{2}.$

Thus, $\\{u_{n}\\}$ can be replaced by $\\{u_{n}^{\ast}\\}$ as minimizing
sequence. We still denote $\\{u_{n}^{\ast}\\}$ by $\\{u_{n}\\}$ for
simplicity. By the Gagliardo-Nirenberg inequality, for any $2<r<22^{\ast}$,
there exists $c(r,N)>0$ such that

$\displaystyle\|u_{n}\|_{r}^{r}$ $\displaystyle\leq
c(r,N)\big{(}\|u_{n}\|_{2}^{2}\big{)}^{\frac{3N+2-(N-2)(r-1)}{2N+4}}\big{(}\|\nabla
u_{n}\|_{2}^{2}\big{)}^{\frac{N(r-2)}{2N+4}}$ (4.1) $\displaystyle\leq
c(r,N)m^{\frac{3N+2-(N-2)(r-1)}{2N+4}}\bigg{(}\int_{\mathbb{R}^{N}}\big{(}1+2u_{n}^{2}\big{)}|\nabla
u_{n}|^{2}\bigg{)}^{\frac{N(r-2)}{2N+4}}.$

For any $2<r<p$ and $\varepsilon>0$, by $(g_{1})-(g_{3})$, there exists
$C_{\varepsilon}>0$ such that

$|G(t)|<\varepsilon|t|^{2}+\varepsilon|t|^{p}+C_{\varepsilon}|t|^{r},$

Consequently, for small $\varepsilon$,

$\displaystyle E(u_{n})$ $\displaystyle={1\over
2}\int_{\mathbb{R}^{N}}\big{(}1+2u_{n}^{2}\big{)}|\nabla
u_{n}|^{2}-\int_{\mathbb{R}^{N}}G(u_{n})$ $\displaystyle\geq{1\over
2}\int_{\mathbb{R}^{N}}\big{(}1+2u^{2}\big{)}|\nabla
u|^{2}-\varepsilon\|u_{n}\|_{2}^{2}-C_{\varepsilon}\|u_{n}\|_{r}^{r}-\varepsilon\|u_{n}\|_{p}^{p}$
$\displaystyle\geq\int_{\mathbb{R}^{N}}\big{(}1+2u_{n}^{2}\big{)}|\nabla
u_{n}|^{2}-c(r,N)C_{\varepsilon}m^{\frac{3N+2-(N-2)(r-1)}{2N+4}}\bigg{(}\int_{\mathbb{R}^{N}}\big{(}1+2u_{n}^{2}\big{)}|\nabla
u_{n}|^{2}\bigg{)}^{\frac{N(r-2)}{2N+4}}$
$\displaystyle~{}~{}~{}~{}-c(p,N)\varepsilon
m^{\frac{3N+2-(N-2)(p-1)}{2N+4}}\int_{\mathbb{R}^{N}}\big{(}1+2u_{n}^{2}\big{)}|\nabla
u_{n}|^{2}-\varepsilon m$
$\displaystyle\geq\frac{1}{4}\int_{\mathbb{R}^{N}}\big{(}1+2u_{n}^{2}\big{)}|\nabla
u_{n}|^{2}-m-Cc(r,N)m^{\frac{3N+2-(N-2)(r-1)}{2N+4}}\bigg{(}\int_{\mathbb{R}^{N}}\big{(}1+2u_{n}^{2}\big{)}|\nabla
u_{n}|^{2}\bigg{)}^{\frac{N(r-2)}{2N+4}}$

Since $2<r<p=4+\frac{4}{N}$ and $e(m)<0$, $\\{u_{n}\\}$ and $\\{\nabla
u_{n}^{2}\\}$ are bounded in $H^{1}(\mathbb{R}^{N})$ and
$L^{2}(\mathbb{R}^{N})$, respectively. By the interpolation inequality, we may
assume that

$\begin{cases}u_{n}\rightharpoonup
u~{}~{}\hbox{in}~{}~{}H_{r}^{1}(\mathbb{R}^{N}),\\\ u_{n}\rightarrow
u~{}~{}\hbox{in}~{}~{}L^{r}(\mathbb{R}^{N})~{}~{}\hbox{for}~{}~{}2<r<22^{\ast},\\\
u_{n}\rightarrow u~{}~{}a.e.~{}~{}\hbox{in}~{}~{}\mathbb{R}^{N}.\end{cases}$

A standard argument implies that

$\int_{\mathbb{R}^{N}}G(u_{n})\rightarrow\int_{\mathbb{R}^{N}}G(u_{n})$

and

$\liminf_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\big{(}1+2u_{n}^{2}\big{)}|\nabla
u_{n}|^{2}\leq\int_{\mathbb{R}^{N}}\big{(}1+2u^{2}\big{)}|\nabla u|^{2}.$

Then

$E(u)\leq\liminf_{n\rightarrow\infty}E(u_{n})=e(m)<0.$

From this, we have $u\neq 0$. To prove Theorems 1.6, it suffices to prove that
$\|u\|_{2}^{2}=m$. We argue by contradiction that $0<\|u\|_{2}^{2}=c<m$,
because $0<\|u\|_{2}^{2}\leq m$. Consider the scaling
$v(\cdot)=u\big{(}\sigma^{-\frac{1}{N}}\cdot\big{)}$, where
$\sigma=\frac{m}{\rho}$. Then $\|v\|_{2}^{2}=m$ and

$\displaystyle E(v)$
$\displaystyle=\bigg{(}\frac{m}{\rho}\bigg{)}^{1-\frac{2}{N}}\bigg{(}{1\over
2}\int_{\mathbb{R}^{N}}\big{(}1+2u^{2}\big{)}|\nabla
u|^{2}\bigg{)}-\frac{m}{\rho}\int_{\mathbb{R}^{N}}G(u)$
$\displaystyle\leq\frac{m}{\rho}\bigg{(}{1\over
2}\int_{\mathbb{R}^{N}}\big{(}1+2u^{2}\big{)}|\nabla
u|^{2}\bigg{)}-\frac{m}{\rho}\int_{\mathbb{R}^{N}}G(u)$ $\displaystyle<E(u),$

which gives a contradiction, and the proof is completed.

Acknowledgments

This work is supported partially by NSFC (No. 12161091).

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