# Normalized solutions to a class of $(2,q)$-Laplacian equations
in the strongly sublinear regime

Rui Ding
School of Mathematics
East China University of Science and Technology
Shanghai 200237, PR China<EMAIL_ADDRESS>,  Chao Ji
School of Mathematics
East China University of Science and Technology
Shanghai 200237, PR China<EMAIL_ADDRESS>and  Patrizia Pucci
Dipartimento di Matematica e Informatica
Università degli Studi di Perugia
06123, Italy<EMAIL_ADDRESS>
###### Abstract.

In this paper, we consider the existence and multiplicity of normalized
solutions for the following $(2,q)$-Laplacian equation

($\mathscr{E}_{\lambda}$) $\left\\{\begin{aligned} &-\Delta
u-\Delta_{q}u+\lambda u=g(u),\quad x\in\mathbb{R}^{N},\\\
&\|u\|_{2}^{2}=c^{2},\\\ \end{aligned}\right.$

where $1<q<N$, $\Delta_{q}=\operatorname{div}\left(|\nabla u|^{q-2}\nabla
u\right)$ is the $q$-Laplacian operator, $\lambda$ is a Lagrange multiplier
and $c>0$ is a constant. The nonlinearity $g:\mathbb{R}\rightarrow\mathbb{R}$
is continuous and the behaviour of $g$ at the origin is allowed to be strongly
sublinear, i.e., $\lim\limits_{s\rightarrow 0}g(s)/s=-\infty$, which includes
the logarithmic nonlinearity

$g(s)=s\log s^{2}.$

We consider a family of approximating problems that can be set in
$H^{1}\left(\mathbb{R}^{N}\right)\cap D^{1,q}\left(\mathbb{R}^{N}\right)$ and
the corresponding least-energy solutions. Then, we prove that such a family of
solutions converges to a least-energy solution to the original problem.
Additionally, under certain assumptions about $g$ that allow us to work in a
suitable subspace of $H^{1}\left(\mathbb{R}^{N}\right)\cap
D^{1,q}\left(\mathbb{R}^{N}\right)$, we prove the existence of infinitely many
solutions of the above $(2,q)$-Laplacian equation.

##  1\. Introduction

In this paper, we are concerned with the existence and multiplicity of
normalized solutions to the following $(2,q)$-Laplacian equation

($\mathscr{E}_{\lambda}$) $\left\\{\begin{array}[]{l}-\Delta
u-\Delta_{q}u+\lambda u=g(u),{\quad x\in\mathbb{R}^{N},}\\\
\int_{\mathbb{R}^{N}}u^{2}dx=c^{2},\\\ \end{array}\right.$

where $\Delta_{q}u=\operatorname{div}\left(|\nabla u|^{q-2}\nabla u\right)$ is
the $q$-Laplacian of $u$, $u\in X$, with
$X:=H^{1}\left(\mathbb{R}^{N}\right)\cap D^{1,q}\left(\mathbb{R}^{N}\right)$,
$c>0$, $\frac{2N}{N+2}<q<2$, $N\geq 2$ or $2<q<N$, $N\geq 3$.

In recent years, the $(p,q)$-Laplacian equation has received considerable
attention. The $(p,q)$-Laplacian equation comes from the general reaction-
diffusion equation

(1.1)
$\partial_{t}u-\Delta_{p}u-\Delta_{q}u=f(x,u),\quad(t,x)\in\mathbb{R}\times\mathbb{R}^{N},$

This equation has a wide range of applications in physics and related sciences
such as plasma physics, biophysics, and chemical reaction design. In such
applications, the function $u$ describes a concentration;
$\Delta_{p}u+\Delta_{q}u$ corresponds to the diffusion and $f(x,u)$ is the
reaction related to source and loss processes. Taking the stationary version
of (1.1), with $p=2$, we obtain the $(2,q)$-Laplacian equation
($\mathscr{E}_{\lambda}$).

An important way to find the nontrivial solutions for
($\mathscr{E}_{\lambda}$) is to search for solutions with prescribed mass, and
in this case $\lambda\in\mathbb{R}$ is part of the unknown. This approach
seems to be particularly meaningful from the physical point of view, because
in nonlinear optics and the theory of Bose-Einstein condensates, there is a
conservation of mass, see [12, 19]. Hence, motivated by the above facts, we
look for solutions of ($\mathscr{E}_{\lambda}$) in $X$ with a prescribed
$L^{2}$-norm.

Due to our scope, we would like to mention [6], where Baldelli and Yang
studied the existence of normalized solutions of the following
$(2,q)$-Laplacian equation in all possible cases according to the value of
$p$,

(1.2) $\left\\{\begin{array}[]{l}-\Delta u-\Delta_{q}u=\lambda
u+|u|^{p-2}u,\quad x\in\mathbb{R}^{N},\\\
\int_{\mathbb{R}^{N}}|u|^{2}dx=c^{2}.\end{array}\right.$

In the $L^{2}$-subcritical case, they studied a global minimization problem
and obtained a ground state solution for problem (1.2). While in the
$L^{2}$-critical case, they proved several nonexistence results, also extended
in the $L^{q}$-critical case. Finally, for the $L^{2}$-supercritical case,
they derived a ground state and infinitely many radial solutions.

In a recent work [9], Cai and Rădulescu studied the following
$(p,q)$-Laplacian equation with $L^{p}$-constraint

(1.3)
$\left\\{\begin{array}[]{l}-\Delta_{p}u-\Delta_{q}u+\lambda|u|^{p-2}u=f(u),\quad
x\in\mathbb{R}^{N},\\\ \int_{\mathbb{R}^{N}}|u|^{p}dx=c^{p},\\\ u\in
W^{1,p}\left(\mathbb{R}^{N}\right)\cap
W^{1,q}\left(\mathbb{R}^{N}\right).\end{array}\right.$

For problem (1.3), they established the existence of ground states, and
revealed some basic behaviors of the ground state energy $E_{c}$ as $c>0$
varies. The analysis developed in [9] allows us to provide the general growth
assumptions imposed to the reaction $f$.

On the other hand, in the past decade, the following logarithmic Schrödinger
equation has received considerable attention

$None$ $-{\varepsilon}^{2}\Delta u+V(x)u=u\log u^{2},\quad
x\in\mathbb{R}^{N},$

where $\varepsilon>0$ and $V:\mathbb{R}^{N}\rightarrow\mathbb{R}$. The class
of equations $(E_{\varepsilon})$ has some physical applications, such as
quantum mechanics, quantum optics, nuclear physics, transport and diffusion
phenomena. For further details we refer to [34]. The class of equations
$(E_{\varepsilon})$ also raises many difficult mathematical questions. For
example, there exists $u\in H^{1}(\mathbb{R}^{N})$ such that
$\int_{\mathbb{R}^{N}}u^{2}\log u^{2}\,dx=-\infty$. Thus, the underlying
energy functional associated to $(E_{\varepsilon})$ is no longer of class
$C^{1}$. In order to overcome this technical difficulty some authors have used
different techniques to establish the existence, multiplicity and
concentration of the solutions under some assumptions on the potential
$V=V(x)$, which can be seen in [2, 3, 16, 26, 27, 31] and the references
therein. See also [4] for the normalized solutions.

In particular, in the paper [23], Mederski and Schino investigated the
normalized solutions of the following Schrödinger equation

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

coupled with the mass constraint $\int_{\mathbb{R}^{N}}|u|^{2}dx=\rho^{2}$,
when $N\geq 2$. In [23] the behaviour of $g$ at the origin is allowed to be
strongly sublinear, i.e., $\lim\limits_{s\rightarrow 0}g(s)/s=-\infty$, which
includes the logarithmic nonlinearity represented as the special case

(1.4) $g(s)=s\log s^{2}.$

Under some assumptions about $g$, Mederski and Schino in [23] proved the
existence of infinitely many normalized solutions of the above problem.

Motivated by the results mentioned above, this paper aims to investigate the
existence and multiplicity of normalized solutions to the $(2,q)$-Laplacian
equation ($\mathscr{E}_{\lambda}$), where $g$ is allowed to be strongly
sublinear at the origin. Before stating the main results of this paper, we
present the conditions required on $g$. Define $G(s)=\int_{0}^{s}g(t)dt$ and
$\bar{q}:=\left(1+\frac{2}{N}\right)\min\\{2,q\\}$,
${q^{\prime}}:=\max\left\\{2^{*},q^{*}\right\\}$ and $s^{*}:=\frac{sN}{N-s}$.
Let

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

$(g_{0})$ $g:\mathbb{R}\rightarrow\mathbb{R}$ is continuous and $g(0)=0$.
$(g_{1})$ $\lim\limits_{s\rightarrow 0}G_{+}(s)/|s|^{2}=0$.
$(g_{2})$ If $N\geq 3$, then
$\lim\sup\limits_{|s|\rightarrow\infty}|g(s)|/|s|^{q^{\prime}-1}=0$.
If $N=2$, then when $q<2$, $\lim\limits_{|s|\rightarrow\infty}g(s)/e^{\alpha
s^{2}}=0$ $\forall\alpha>0$, when $q>2$,
$\lim\sup\limits_{|s|\rightarrow\infty}|g(s)|/|s|^{q^{*}}=0$.
$(g_{3})$ $\lim\limits_{|s|\rightarrow\infty}G_{+}(s)/|s|^{\bar{q}}=0$.
$(g_{4})$ There exists $\xi_{0}\neq 0$ such that $G\left(\xi_{0}\right)>0$.
Let us define

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

Solutions of ($\mathscr{E}_{\lambda}$) are critical points of the functional
$J:X\rightarrow\mathbb{R}\cup\\{\infty\\}$,

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

on the constraint $\mathcal{S}(c)$, with $\lambda\in\mathbb{R}$ appearing as a
Lagrange multiplier The energy functional $J$ is not well defined in $X$.
Indeed, when $g(u)=u\log u^{2}$, there exists $u\in X$ such that
$\int_{\mathbb{R}^{N}}u^{2}\log u^{2}dx=-\infty$, as shown in details in the
Appendix. Furthermore, the reason of the restriction $\frac{2N}{N+2}<q$ when
$N\geq 2$ appears in Remark 6.1 of the Appendix.

Our first result is the existence of the normalized solutions to
($\mathscr{E}_{\lambda}$), with additional properties. Specifically, we define
the normalized solution to ($\mathscr{E}_{\lambda}$) as a pair $(u,\lambda)\in
X\times\mathbb{R}$ such that $J^{\prime}(u)v=-\lambda uv$ for every
$v\in\mathcal{C}_{0}^{\infty}\left(\mathbb{R}^{N}\right)$.

######  Theorem 1.1.

If $g$ satisfies $(g_{0})-(g_{4})$, then there exists $\bar{c}>0$ such that
for every $c>\bar{c}$, there exist $\lambda>0$ and $u\in\mathcal{S}(c)$ such
that $J(u)=\min_{\mathcal{D}}J<0$ and $(u,\lambda)$ is a solution to
($\mathscr{E}_{\lambda}$). Moreover, $u$ has constant sign and is, up to a
translation, radial and radially monotonic.

Equation ($\mathscr{E}_{\lambda}$) presents several challenges. On one hand,
for the double-phase operator, due to the appearance of $q$-Laplacian term,
the solution space $X$ is not an Hilbertian space. This makes some estimates
more difficult and complicated. For example, it is challenging to apply the
Brézis-Lieb lemma to bounded Palais-Smale sequences. This is clear from the
proof of the Lemma 3.8 in Section 3. On the other hand, the nonlinearity $g$
is strongly sublinear near zero. This in particular gives that the
corresponding functional $J$ is not well-defined in the solution space $X$.
This difficulty is overcome with the arguments adopted in the proof of Lemma
6.2. To handle this, we consider a series of perturbed functionals as in (1.7)
below and in Lemma 3.2 we are able to prove that the solutions to these
perturbed problems are bounded in $X$ with respect to $\varepsilon$. Besides,
their weak limits are minimizers of the original functional (1.6) in the set
$\mathcal{D}(c)$. Importantly, the solutions to the perturbed problems belong
to the set $\mathcal{S}(c)$, which is crucial when proving that the solutions
of the original problem also belong to $\mathcal{S}(c)$.

Given functions $f_{1},\,f_{2}:\mathbb{R}\rightarrow\mathbb{R}$. We introduce
the following notation: $f_{1}\lesssim f_{2}$ provided that $f_{1}(s)\leq
Cf_{2}(s)$ for all $s\in\mathbb{R}$, where $C$ is a positive constant
independent of $s$.

Let $g_{+}(s)=G_{+}^{\prime}(s)$, $g_{-}(s):=g_{+}(s)-g(s)$, and
$G_{-}(s):=G_{+}(s)-G(s)\geq 0$ for $s\in\mathbb{R}$. Thus, $G_{+}(u)\in
L^{1}\left(\mathbb{R}^{N}\right)$ for all $u\in X$ in view of $(g_{1})$ and
$(g_{3})$. However, $G_{-}(u)$ may not be integrable unless
$G_{-}(s)\lesssim|s|^{2}$ for small $|s|$. In order to overcome this
difficulty, for every $\varepsilon\in(0,1)$, let us take an even function
$\varphi_{\varepsilon}:\mathbb{R}\rightarrow[0,1]$ such that
$\varphi_{\varepsilon}(s)=|s|/\varepsilon$ for
$|s|\leq\varepsilon,\varphi_{\varepsilon}(s)=1$ for $|s|\geq\varepsilon$. Now
we introduce a new perturbed functional

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

where
$G_{-}^{\varepsilon}(s)=\int_{0}^{s}\varphi_{\varepsilon}(t)g_{-}(t)dt,s\in\mathbb{R}$.
Since $G_{-}^{\varepsilon}(s)\leq c_{\varepsilon}|s|^{2}$ for every $|s|\leq
1$ and some constant $c_{\varepsilon}>0$ depending only on $\varepsilon>0$,
clearly $J_{\varepsilon}$ is of class $\mathcal{C}^{1}(X)$ for any
$\varepsilon\in(0,1)$.

Now, we turn back to the main question of finding multiple normalized
solutions (in fact, infinitely many) for ($\mathscr{E}_{\lambda}$). To this
aim, we need to modify both the assumptions on $g$ and our approach.

We assume that the right-hand side in ($\mathscr{E}_{\lambda}$) is given in
the form

(1.8) $g(s)=f(s)-a(s),$

together with the following assumptions on $a$ and $f$, where
$F(s)=\int_{0}^{s}f(t)dt$, $A(s)=\int_{0}^{s}a(t)dt$, $F_{+}$ is defined via
(1.5) 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
$\lim\limits_{s\rightarrow 0}A(s)/s^{2}=\infty$ and $s\mapsto a(s)s$ is convex
in $\mathbb{R}$.
$(f_{0})$ $f:\mathbb{R}\rightarrow\mathbb{R}$ is continuous and odd.
$(f_{1})$ $\lim\limits_{s\rightarrow 0}f_{+}(s)/s=0$.
$(f_{2})$ If $N\geq 3$, then $|f(s)|\lesssim a(s)+|s|+|s|^{q^{\prime}-1}$,
again with ${q^{\prime}}:=\max\left\\{2^{*},q^{*}\right\\}$.
If $N=2$, then when $q<2$ for all $m\geq 2$ and $\alpha>4\pi$ there holds
$|f(u)|\lesssim a(s)+|s|+|s|^{m-1}\left(e^{\alpha s^{2}}-1\right)$, when
$q>2$, $|f(s)|\lesssim a(s)+|s|+|s|^{q^{*}-1}$.
$(f_{3})$
$\lim\sup\limits_{|s|\rightarrow\infty}f_{+}(s)/|s|^{\bar{q}-1}<\infty$ and

(1.9)
$\eta:=\limsup_{|s|\rightarrow\infty}\frac{F_{+}(s)}{|s|^{\bar{q}}}<\infty.$

Let us define

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

To present our results, we introduce some notations. For a subgroup
$\mathcal{O}\subset\mathcal{O}(N)$,

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

If $N=4$ or $N\geq 6$, in order to find non-radial solutions of
($\mathscr{E}_{\lambda}$), following [21, 23], we fix $M$, with $2\leq M\leq
N/2$ such that $N-2M\neq 1$ and we put
$\tau(x)=\left(x_{2},x_{1},x_{3}\right)$ for
$x=\left(x_{1},x_{2},x_{3}\right)\in\mathbb{R}^{M}\times\mathbb{R}^{M}\times\mathbb{R}^{N-2M}$.
Then, let

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

here we agree that the components corresponding to $N-2M$ do not exist when
$N=2M$ and observe that $\mathcal{X}\cap W_{\mathcal{O}(N)}=\\{0\\}$. For
$2<p<2^{*}$, the number $C_{N,p}$ denotes the best constant in the Gagliardo-
Nirenberg inequality of Lemma 2.1.

Our multiplicity result reads as follows.

######  Theorem 1.2.

Let $g$ be as in (1.8). Assume that (A), $(f_{0})-(f_{3})$, and

(1.12)
$2(\eta+\delta)C_{N,\bar{q}}^{\bar{q}}c^{\bar{q}\left(1-\delta_{\bar{q}}\right)}<1$

hold, where $\eta$ is defined in (1.9). Then there exist infinitely many
solutions $(u,\lambda)\in W_{\mathcal{O}(N)}\times\mathbb{R}$ to
($\mathscr{E}_{\lambda}$), one of which say, $(\bar{u},\bar{\lambda})$ having
the property that $J(\bar{u})=\min_{\mathcal{S}(c)\cap W}{J(u)}$. 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
$J(\bar{v})=\min_{\mathcal{S}(c)\cap\mathcal{X}}J$.

Let us now present the main difficulties in proving Theorem 1.2. To prove the
multiplicity results stated in Theorem 1.2, we work in the subspace of
$W_{\mathcal{O}}$ where the functional $J$ is well-defined. From Corollary
5.4, we have the compact embedding $W_{\mathcal{O}}\hookrightarrow
L^{m}\left(\mathbb{R}^{N}\right)$ for all $2\leq m<q^{\prime}$. This result,
which is due to Lemma 5.3, allows us to show that
$\left.J\right|_{{W_{\mathcal{O}}}\cap\mathcal{S}(c)}$ satisfies the Palais-
Smale condition, see Lemma 5.9. Then, to obtain Palais-Smale sequences we make
use of classical minimax arguments, see Theorem 5.11 proved in [14].
Furthermore, the proof of Theorem 5.11 is based on the construction of
suitable mappings from $\mathbb{S}^{k-1}$ to
$W_{\mathcal{O}}\cap\mathcal{S}(c)$, given in Lemma 5.10, which relies on the
existence of some special mappings introduced in [7].

The paper is organized as follows. In Section 2 we introduce the functional
setting and we give some preliminaries. We study the perturbed problem
($\mathscr{P}_{\varepsilon}$) in Section 3. Then, in Section 4, we prove
Theorem 1.1. In the last section, we focus on the multiplicity of normalized
solutions of ($\mathscr{E}_{\lambda}$) and prove Theorem 1.2.

##  2\. Preliminaries

In this section, we introduce some preliminary properties useful for the
proofs of our main results. As already defined in the Introduction,
$X=H^{1}\left(\mathbb{R}^{N}\right)\cap D^{1,q}\left(\mathbb{R}^{N}\right)$ is
endowed with the norm

$\|u\|_{X}=\|\nabla u\|_{2}+\|u\|_{2}+\|\nabla u\|_{q},$

where $\|\cdot\|_{q}$ denotes the usual $L^{q}$-norm in $\mathbb{R}^{N}$.

In the following, $B_{R}(x)$ denotes the open ball of $\mathbb{R}^{N}$ centred
at $x$ with radius $R>0$. Moreover, $C$ and $C_{i}$ denote any positive
constant, whose value is not relevant.

######  Lemma 2.1 (The Gagliardo-Nirenberg inequality – Corollary 2.1 of
[32]).

When $N\geq 3$, let $2<p<2^{*}$ and $\delta_{p}=\frac{N(p-2)}{2p}$, then there
exists a constant
$C_{N,p}=\left(\frac{p}{2\left\|W_{p}\right\|_{2}^{p-2}}\right)^{\frac{1}{p}}>0$
such that

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

where $W_{p}$ is the unique positive radial solution of $-\Delta
W+\left(\frac{1}{\delta_{p}}-1\right)W=\frac{2}{p\delta_{p}}|W|^{p-2}W$ in
$\mathbb{R}^{N}$.

######  Lemma 2.2 (The Trudinger-Moser inequality – Lemma 2.1 of [10]).

If $\alpha>0$ and $u\in H^{1}\left(\mathbb{R}^{2}\right)$, then

$\int_{\mathbb{R}^{2}}\left(e^{\alpha u^{2}}-1\right)\mathrm{d}x<\infty.$

Moreover, if $\|\nabla u\|_{2}\leq 1,\|u\|_{2}\leq M<\infty$ with $M>0$ and
$0<\alpha<4\pi$, then there exists a constant $C>0$, which depends only on $M$
and $\alpha$, such that

$\int_{\mathbb{R}^{2}}\left(e^{\alpha u^{2}}-1\right)\mathrm{d}x\leq
C\left(M,\alpha\right).$

######  Lemma 2.3 (The Sobolev embedding Theorem – Lemma 1.2 of [9] and [5]).

The space $X$ is embedded continuously into $L^{m}\left(\mathbb{R}^{N}\right)$
for $m\in\left[2,q^{\prime}\right]$ and compactly into $L_{\text{loc
}}^{m}\left(\mathbb{R}^{N}\right)$ for $m\in\left[1,q^{\prime}\right)$.

The space $X_{r}:=\\{u\in X:u\text{ is radially symmetric}\\}$ is compactly
embedded into $L^{m}\left(\mathbb{R}^{N}\right)$ for all
$m\in\left(2,q^{\prime}\right)$.

For the next lemma, we can take a similar argument as that of the classical
concentration-compactness Lemma. See, for instance,[33, Lemma 1.21].

######  Lemma 2.4.

Let $r>0$. If $(u_{n})_{n}$ is a bounded sequence in $X$ which satisfies

$\sup_{x\in\mathbb{R}^{N}}\int_{B_{r}(x)}\left|u_{n}\right|^{2}dx\rightarrow
0,\quad\text{ as }n\rightarrow\infty,$

then,

$\left\|u_{n}\right\|_{m}\rightarrow 0\quad\text{ as }n\rightarrow\infty$

holds for any $m\in\left(2,q^{\prime}\right)$, where
${q^{\prime}}=\max\left\\{2^{*},q^{*}\right\\}$.

######  Lemma 2.5 (Lemma 2.7, [17]).

Assume $s>1$. Let $\Omega$ be an open set in $\mathbb{R}^{N},\alpha,\beta$
positive numbers and $a(x,\xi)$ in
$C\left(\Omega\times\mathbb{R}^{N},\mathbb{R}^{N}\right)$ such that
(1) $\alpha|\xi|^{s}\leq a(x,\xi)\xi$ for all
$(x,\xi)\in\Omega\times\mathbb{R}^{N}$,
(2) $|a(x,\left.\xi)|\leq\beta|\xi|^{s-1}\right.$ for all
$(x,\xi)\in\Omega\times\mathbb{R}^{N}$,
(3) $(a(x,\xi)-a(x,\eta))(\xi-\eta)>0$ for all
$(x,\xi)\in\Omega\times\mathbb{R}^{N}$ with $\xi\neq\eta$,
(4) $a(x,\gamma\xi)=\gamma|\gamma|^{p-2}a(x,\xi)$ for all
$(x,\xi)\in\Omega\times\mathbb{R}^{N}$ and
$\gamma\in\mathbb{R}\backslash\\{0\\}$.
Consider $(u_{n})_{n}$, $u\in W^{1,s}(\Omega)$, then $\nabla
u_{n}\rightarrow\nabla u$ in $L^{s}(\Omega)$ if and only if

$\lim_{n\rightarrow\infty}\int_{\Omega}\Big{(}a\left(x,\nabla
u_{n}(x)\right)-a(x,\nabla u(x))\Big{)}\left(\nabla u_{n}(x)-\nabla
u(x)\right)dx=0.$

######  Lemma 2.6 (The Vitali convergence theorem).

Let $(v_{n})_{n}$ be a sequence in $L^{m}(\Omega)$ with $1\leq m<\infty$, and
let $\mu$ denote the Lebesgue measure on $\mathbb{R}^{N}$. Assume that
(i) $\forall\eta>0,\exists\sigma>0$ and $N>0$ such that
$\int_{A}\left|v_{n}\right|^{m}dx<\eta,\forall n>N$ and $\forall
A\subset\Omega$ measurable with $\mu(A)<\sigma$,
(ii) $v_{n}\rightarrow v$ for a.e. $x\in\mathbb{R}^{N}$.
Then $v\in{L^{m}}(\Omega)$ and that $v_{n}\rightarrow v$ in ${L^{m}}(\Omega)$.

######  Remark 2.7.

Let us end the section by recalling the next elementary inequality. For any
number $s>1$ there exists a constant $C(s)>0$ such that for all
$x,y\in\mathbb{R}^{N}$ with $|x|+|y|\neq 0$,

$\left\langle|x|^{s-2}x-|y|^{s-2}y,x-y\right\rangle\geq
C(s)\begin{cases}\dfrac{|x-y|^{2}}{(|x|+|y|)^{2-s}},&\text{ if }1\leq s<2\\\
|x-y|^{s},&\text{ if }s\geq 2.\end{cases}$

##  3\. The perturbed problem

The key functional $J_{\varepsilon}:X\rightarrow\mathbb{R}$ is defined in
(1.7). Put $m_{\varepsilon}(c)=\inf_{\mathcal{D}(c)}J_{\varepsilon}$. The main
purpose of this section is to prove the following result.

######  Theorem 3.1.

Let $g$ satisfy $(g_{0})-\left(g_{4}\right)$, then for every $\beta>0$, there
exists $\bar{c}>0$ such that for every $c>\bar{c}$ and $\varepsilon>0$, there
exist $\lambda_{\varepsilon}>0$ and $u_{\varepsilon}\in X$ such that
$J_{\varepsilon}\left(u_{\varepsilon}\right)=m_{\varepsilon}(c)<-\beta$ and

($\mathscr{P}_{\varepsilon}$) $\left\\{\begin{array}[]{l}-\Delta
u_{\varepsilon}-\Delta_{q}u_{\varepsilon}+\lambda_{\varepsilon}u_{\varepsilon}=g_{+}\left(u_{\varepsilon}\right)-\varphi_{\varepsilon}\left(u_{\varepsilon}\right)g_{-}\left(u_{\varepsilon}\right),\quad
x\in\mathbb{R}^{N},\\\
\int_{\mathbb{R}^{N}}\left|u_{\varepsilon}\right|^{2}dx=c^{2}.\end{array}\right.$

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 3.2.

If $g$ satisfies $(g_{0})-(g_{3})$, then
$\left.J_{\varepsilon}\right|_{\mathcal{D}(c)}$ is bounded from below.

###### Proof.

Observing that

$\left\\{\begin{array}[]{l}\bar{q}\delta_{\bar{q}}=q\left(1+\delta_{q}\right)<2,\,\,\text{
if }\frac{2N}{N+2}<q<2,\\\
\bar{q}\delta_{\bar{q}}=2<q\left(1+\delta_{q}\right),\,\,\text{ if
}2<q<N,\end{array}\right.$

thus

$\bar{q}\delta_{\bar{q}}=\min\left\\{2,q\left(1+\delta_{q}\right)\right\\},\,\,\text{for}\,\,\frac{2N}{N+2}<q<2\,\,\text{and}\,\,2<q<N.$

From $(g_{1})$ and $(g_{3})$, for any $\delta>0$, there exists $C_{\delta}>0$
such that for every $s\in\mathbb{R}$

$G_{+}(s)\leq C_{\delta}|s|^{2}+\delta|s|^{\bar{q}}.$

Consequently, from Lemma 2.1, for every $u\in\mathcal{D}(c)$, we have

$\displaystyle J_{\varepsilon}(u)$ $\displaystyle\geq\frac{1}{2}\|\nabla
u\|^{2}_{2}-\int_{\mathbb{R}^{N}}G_{+}(u)dx$
$\displaystyle\geq\frac{1}{2}\|\nabla
u\|_{2}^{2}-C_{\delta}\|u\|_{2}^{2}-\delta\|u\|_{\bar{q}}^{\bar{q}}$
$\displaystyle\geq\frac{1}{2}\|\nabla
u\|_{2}^{2}-C_{\delta}\|u\|_{2}^{2}-\delta
C_{N,\bar{q}}^{\bar{q}}c^{\bar{q}\left(1-\delta_{\bar{q}}\right)}\|\nabla
u\|_{2}^{\bar{q}\delta_{\bar{q}}}$ $\displaystyle\geq\|\nabla
u\|_{2}^{2}\left(\frac{1}{2}-\delta
C_{N,\bar{q}}^{\bar{q}}c^{\bar{q}\left(1-\delta_{\bar{q}}\right)}\|\nabla
u\|_{2}^{\bar{q}\delta_{\bar{q}}-2}\right)-C_{\delta}c^{2}.$

If $\frac{2N}{N+2}<q<2$, then
$\bar{q}\delta_{\bar{q}}=q\left(1+\delta_{q}\right)<2$, so that
$\left.J_{\varepsilon}\right|_{\mathcal{D}(c)}$ is coercive and bounded from
below.
If $2<q<N$, then $\bar{q}\delta_{\bar{q}}=2$. Thus, taking
$\delta<\left(2C_{N,\bar{q}}^{\bar{q}}c^{\bar{q}\left(1-\delta_{\bar{q}}\right)}\right)^{-1}$,
we easily get that $\left.J_{\varepsilon}\right|_{\mathcal{D}(c)}$ is bounded
from below. ∎

######  Lemma 3.3.

If $g$ satisfies $(g_{0})-(g_{2})$ and $(g_{4})$, then for every $\beta>0$,
there exists $\bar{c}>0$ such that

$m_{\varepsilon}(c)<-\beta$

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

###### Proof.

Take $R>0$ and set

$u_{R}(x)=\begin{cases}\xi_{0}&\text{ if }|x|\leq R\\\ \xi_{0}(R+1-|x|)&\text{
if }R<|x|\leq R+1\\\ 0&\text{ if }|x|>R+1\end{cases}$

where $\xi_{0}$ is the constant determined in $(g_{4})$. It is easy to see
that $u_{R}\in X$.

Arguing as in [30, Lemma 2.3], for $R$ large enough,
$\int_{\mathbb{R}^{N}}G(u_{R})dx\geq 1$. Putting
$G^{\varepsilon}:=G_{+}-G_{-}^{\varepsilon}\geq G$, we see that
$\int_{\mathbb{R}^{N}}G^{\varepsilon}(u_{R})dx\geq 1$ for any
$\varepsilon\in(0,1)$. Now,
$u_{c}:=u_{R}\left(\|u_{R}\|_{2}^{2/N}c^{-2/N}x\right)\in\mathcal{S}(c)$, and

$\displaystyle J_{\varepsilon}\left(u_{c}\right)$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u_{c}|^{2}dx+\frac{1}{q}\int_{\mathbb{R}^{N}}|\nabla
u_{c}|^{q}dx-\int_{\mathbb{R}^{N}}G^{\varepsilon}\left(u_{c}\right)dx$
$\displaystyle={c^{\frac{2(N-2)}{N}}}\frac{\|\nabla
u_{R}\|_{2}^{2}}{2\|u_{R}\|_{2}^{\frac{2(N-2)}{N}}}+{c^{\frac{2(N-q)}{N}}}\frac{\|\nabla
u_{R}\|_{q}^{q}}{q\|u_{R}\|_{2}^{\frac{2(N-q)}{N}}}-c^{2}\frac{\int_{\mathbb{R}^{N}}G^{\varepsilon}(u_{R})dx}{\|u_{R}\|_{2}^{2}}$
$\displaystyle\leq{c^{\frac{2(N-2)}{N}}}\frac{\|\nabla
u_{R}\|_{2}^{2}}{2\|u_{R}\|_{2}^{\frac{2(N-2)}{N}}}+{c^{\frac{2(N-q)}{N}}}\frac{\|\nabla
u_{R}\|_{q}^{q}}{q\|u_{R}\|_{2}^{\frac{2(N-q)}{N}}}-c^{2}\frac{1}{\|u_{R}\|_{2}^{2}}\rightarrow-\infty$

as $c\rightarrow\infty$,since $\frac{2(N-2)}{N}<2$ and $\frac{2(N-q)}{N}<2$.
This concludes the proof. ∎

######  Lemma 3.4.

If $(g_{0})-(g_{3})$ are satisfied, then for any $c_{1},\,c_{2}>0$, there
holds

$m_{\varepsilon}\left(\sqrt{c_{1}^{2}+c_{2}^{2}}\right)\leq
m_{\varepsilon}\left(c_{1}\right)+m_{\varepsilon}\left(c_{2}\right).$

###### Proof.

We know that $m_{\varepsilon}(c)=\inf_{\mathcal{D}(c)}J_{\varepsilon}$. From
Lemma 3.2, it is easy to see that $m_{\varepsilon}(c)$ is finite for any
$c>0$. Fix $c_{1},\,c_{2}>0$ and $\delta>0$. Then there exist $u_{1},u_{2}\in$
$\mathcal{C}_{0}^{\infty}\left(\mathbb{R}^{N}\right)$ such that
$\|u_{i}\|_{2}\leq c_{i}$ and $J_{\varepsilon}\left(u_{i}\right)\leq
m_{\varepsilon}\left(c_{i}\right)+\delta$, where $c_{i}>0$ and $i=1,2$. In
virtue of the translation invariance, we can assume that $u_{1}$ and $u_{2}$
have disjoint supports. Then
$\left\|u_{1}+u_{2}\right\|_{2}^{2}=\left\|u_{1}\right\|_{2}^{2}+\left\|u_{2}\right\|_{2}^{2}\leq
c_{1}^{2}+c_{2}^{2}$ and so

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

Letting $\delta\rightarrow 0$, we get
$m_{\varepsilon}\left(\sqrt{c_{1}^{2}+c_{2}^{2}}\right)\leq
m_{\varepsilon}\left(c_{1}\right)+m_{\varepsilon}\left(c_{2}\right)$, as
stated. ∎

######  Lemma 3.5.

Let $(g_{0})-(g_{3})$ hold, and let $c_{1},c_{2}>0$. If there exist
$u_{i}\in\mathcal{D}\left(c_{i}\right)$ such that
$J_{\varepsilon}\left(u_{i}\right)=m_{\varepsilon}\left(c_{i}\right),i=1,2$,
and $\left(u_{1},u_{2}\right)\neq(0,0)$, then
$m_{\varepsilon}\left(\sqrt{c_{1}^{2}+c_{2}^{2}}\right)<m_{\varepsilon}\left(c_{1}\right)+m_{\varepsilon}\left(c_{2}\right)$.

###### Proof.

First of all, for any $s>0$

$\int_{\mathbb{R}^{N}}\left|\left(u(s^{-1/N}x\right)\right|^{2}dx=s\int_{\mathbb{R}^{N}}|u(x)|^{2}dx.$

For every $s\geq 1$ and $i=1,2$, we have

$\displaystyle m_{\varepsilon}\left(\sqrt{s}c_{i}\right)$ $\displaystyle\leq
J_{\varepsilon}\left(u_{i}\left(s^{-1/N}x\right)\right)$
$\displaystyle=s\left(\frac{1}{2s^{2/N}}\int_{\mathbb{R}^{N}}\left|\nabla
u_{i}\right|^{2}dx+\frac{1}{qs^{q/N}}\int_{\mathbb{R}^{N}}\left|\nabla
u_{i}\right|^{q}dx-\int_{\mathbb{R}^{N}}G^{\varepsilon}\left(u_{i}\right)dx\right)$
$\displaystyle\leq
sJ_{\varepsilon}\left(u_{i}\right)=sm_{\varepsilon}\left(c_{i}\right).$

Moreover, if $s>1$ and $u_{i}\neq 0$, then
$m_{\varepsilon}\left(\sqrt{s}c_{i}\right)<sm_{\varepsilon}\left(c_{i}\right)$
for $i=1,2$. Without loss of generality, assume that $u_{1}\neq 0$. If
$c_{1}\geq c_{2}$, then

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

If $c_{1}<c_{2}$, then

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

The proof is completed. ∎

######  Remark 3.6.

(a) Lemmas 3.4 and 3.5 still hold if $J_{\varepsilon}$ is replaced with $J$.
(b) Under the weaker assumption that $J_{\varepsilon}\left(u_{1}\right)=$
$m_{\varepsilon}\left(c_{1}\right)$ or
$J_{\varepsilon}\left(u_{2}\right)=m_{\varepsilon}\left(c_{2}\right)$, Lemma
3.5 still holds (see [25]).

######  Lemma 3.7.

Assume that $c>0$ and $(g_{0})-(g_{3})$ hold. Let
${(\tilde{u}_{n})_{n}}\subset\mathcal{D}(c)$ be a minimizing sequence for
$m_{\varepsilon}(c)$, then $m_{\varepsilon}(c)$ possesses another minimizing
sequence ${(u_{n})_{n}}\subset\mathcal{D}(c)$ and
$\lambda_{\varepsilon}\in\mathbb{R}$ such that for all $\varphi\in X^{*}$

$\left\|u_{n}-\tilde{u}_{n}\right\|_{X}\rightarrow 0,\quad
J_{\varepsilon}^{\prime}\left(u_{n}\right)\varphi+\lambda_{\varepsilon}\int_{\mathbb{R}^{N}}u_{n}\varphi
dx\rightarrow 0\text{ as }n\rightarrow\infty.$

In addition, if $\lim\limits_{n\rightarrow\infty}\left\|u_{n}\right\|_{2}<c$,
then $\lambda_{\varepsilon}=0$.

###### Proof.

Let $(\tilde{u}_{n})_{n}$ be a minimizing sequence of $m_{\varepsilon}(c)$,
which is bounded in $X$ by Lemma 3.2. Thus, by Lemma 2.3, there exists
$\tilde{u}_{\varepsilon}{\in X}$ such that, up to a subsequence, if necessary,
$\tilde{u}_{n}\rightharpoonup\tilde{u}_{\varepsilon}$ in $X$ and
$\tilde{u}_{n}\rightarrow\tilde{u}_{\varepsilon}$ in
$L_{loc}^{m}\left(\mathbb{R}^{N}\right)$ for every $m$, with $2<m<q^{\prime}$
and $\tilde{u}_{n}\rightarrow\tilde{u}_{\varepsilon}$ for a.e.
$x\in\mathbb{R}^{N}$. Therefore, from Fatou’s Lemma, we have that
$\tilde{u}_{\varepsilon}\in\mathcal{D}(c)$.

Using Ekeland’s variational principle (see [33, Theorem 2.4]), we derive a new
minimizing sequence ${(u_{n})_{n}}\subset\mathcal{D}(c)$, that is also a
Palais-Smale sequence for $J_{\varepsilon}$ on $\mathcal{D}(c)$. Therefore,
there exist ${(\lambda_{n})_{n}}\subset\mathbb{R}$ , such that for all
$\varphi\in X^{*}$

$\left\|u_{n}-\tilde{u}_{n}\right\|_{X}\rightarrow 0,\quad
J^{\prime}\left(u_{n}\right)\varphi+\lambda_{n}\int_{\mathbb{R}^{N}}u_{n}\varphi
dx\rightarrow 0\text{ as }n\rightarrow\infty.$

Let $\varphi=u_{n}$, it is easy to show that $(\lambda_{n})_{n}$ is bounded in
$\mathbb{R}$. Thus, we may assume
$\lambda_{n}\rightarrow\lambda_{\varepsilon}\text{ as }n\rightarrow\infty$, up
to a subsequence if necessary.

If $\tilde{u}_{\varepsilon}\in\mathcal{S}(c)$, i.e.,
$\left\|\tilde{u}_{\varepsilon}\right\|_{2}=c$, then
$\lim\limits_{n\rightarrow\infty}\left\|\tilde{u}_{n}\right\|_{2}=c$. Using
Lemma 2.3,

$\lim_{n\rightarrow\infty}\Big{|}\left\|u_{n}\right\|_{2}-\left\|\tilde{u}_{n}\right\|_{2}\Big{|}\leq\lim_{n\rightarrow\infty}\left\|u_{n}-\tilde{u}_{n}\right\|_{2}\leq\lim_{n\rightarrow\infty}C\left\|u_{n}-\tilde{u}_{n}\right\|_{X}=0,$

which shows that $\lim\limits_{n\rightarrow\infty}\left\|u_{n}\right\|_{2}=c$.

If $\tilde{u}_{\varepsilon}\in\mathcal{D}(c)\backslash\mathcal{S}(c)$, then
$\tilde{u}_{\varepsilon}$ is an interior point of $\mathcal{D}(c)$, i.e.,
$\left\|\tilde{u}_{\varepsilon}\right\|_{2}<c$. Therefore,
$\widetilde{u}_{\varepsilon}$ is a local minimizer of $J$ on $X$. Hence

$J^{\prime}\left(\tilde{u}_{\varepsilon}\right)\varphi=0\quad\mbox{for all
}\varphi\in{X^{*}},$

and $\lambda_{\varepsilon}=0$.

Moreover, by Lemma 2.3,

$\displaystyle\lim\limits_{n\rightarrow\infty}\left\|u_{n}\right\|_{2}$
$\displaystyle\leq\lim_{n\rightarrow\infty}\left\|\tilde{u}_{n}\right\|_{2}+\lim_{n\rightarrow\infty}\left\|u_{n}-\tilde{u}_{n}\right\|_{2}$
$\displaystyle\leq\lim_{n\rightarrow\infty}\left\|\tilde{u}_{n}\right\|_{2}+C\lim_{n\rightarrow\infty}\left\|u_{n}-\tilde{u}_{n}\right\|_{X}$
$\displaystyle\leq c,$

where $C$ is a positive constant.

If $\lim\limits_{n\rightarrow\infty}\left\|u_{n}\right\|_{2}<c$, then
$\lim\limits_{n\rightarrow\infty}\left\|\tilde{u}_{n}\right\|_{2}<c$,
otherwise

$0<\lim_{n\rightarrow\infty}\Big{|}\left\|u_{n}\right\|_{2}-\left\|\tilde{u}_{n}\right\|_{2}\Big{|}\leq\lim_{n\rightarrow\infty}C\left\|u_{n}-\tilde{u}_{n}\right\|_{X}.$

The Fatou Lemma shows that
$\tilde{u}_{\varepsilon}\in\mathcal{D}(c)\backslash\mathcal{S}(c)$. Thus, if
$\lim\limits_{n\rightarrow\infty}\left\|u_{n}\right\|_{2}<c$, then
$\lambda_{\varepsilon}=0$. This ends the proof. ∎

######  Lemma 3.8.

If $g$ satisfy $(g_{0})-(g_{4})$, then, for every $\beta>0$, there exists
$\bar{c}>0$ such that $m_{\varepsilon}(c)<-\beta$ for every $c>\bar{c}$ and
$\varepsilon>0$. If $0<c_{n}\rightarrow c$ and
$\tilde{u}_{n}\in\mathcal{D}\left(c_{n}\right)$ is such that
$J_{\varepsilon}\left(\tilde{u}_{n}\right)\rightarrow m_{\varepsilon}(c)$,
then there exists another minimizing sequence
${(u_{n})_{n}\subset}\mathcal{D}(c)$ such that
$\left\|u_{n}-\tilde{u}_{n}\right\|_{X}\rightarrow 0$ and up to translations,
$u_{n}\rightarrow u$ in $L^{m}\left(\mathbb{R}^{N}\right)$ for all $m$, with
$2<m<q^{\prime}$.

###### Proof.

Let $\bar{c}$ be the number determined by Lemma 3.3 and
${(\tilde{u}_{n})_{n}}\subset\mathcal{D}(c)$ be a minimizing sequence of
$m_{\varepsilon}(c)$. Hence, an application of Lemma 3.7 gives that
$m_{\varepsilon}(c)$ possesses another minimizing sequence
${(u_{n})_{n}}\subset\mathcal{D}(c)$ and $\lambda_{\varepsilon}\in\mathbb{R}$
such that for all $\varphi\in X^{*}$

$\left\|u_{n}-\tilde{u}_{n}\right\|_{X}\rightarrow 0,\quad
J_{\varepsilon}^{\prime}\left(u_{n}\right)\varphi+\lambda_{\varepsilon}\int_{\mathbb{R}^{N}}u_{n}\varphi
dx\rightarrow 0\text{ as }n\rightarrow\infty,$

and $(u_{n})_{n}$ is also a Palais-Smale sequence for $J_{\varepsilon}$ on
$\mathcal{D}(c)$. Similarly to the proof of Lemma 3.2, we have that
$(u_{n})_{n}$ is bounded in $X$. If

$\lim_{n\rightarrow\infty}\sup_{y\in\mathbb{R}^{N}}\int_{B_{R}(y)}\left|u_{n}(x)\right|^{2}dx=0$

for any $R>0$, due to Lemma 2.4, $\left\|u_{n}\right\|_{m}\rightarrow 0$ for
any $m$, with $2<m<q^{\prime}$. From $(g_{1})$ and $(g_{3})$, for any
$\delta>0$, there exists $C_{\delta}>0$ such that for every $s\in\mathbb{R}$,

$G_{+}(s)\leq\delta|s|^{2}+C_{\delta}|s|^{\bar{q}}.$

Thus,
$\lim\limits_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}G_{+}\left(u_{n}\right)dx=0$
and
$m_{\varepsilon}(c)=\lim\limits_{n\rightarrow\infty}J_{\varepsilon}\left(u_{n}\right)\geq
0$, which contradicts Lemma 3.3. Hence, there exist $\varepsilon_{0}>0$ and a
sequence $(y_{n})_{n}\subset\mathbb{R}^{N}$ such that

$\int_{B_{R}\left(y_{n}\right)}\left|u_{n}(x)\right|^{2}dx\geq\varepsilon_{0}>0,$

for some $R>0$. Moreover, we have $u_{n}\left(x+y_{n}\right)\rightharpoonup
u_{c}\not\equiv 0$ in $X$ for some $u_{c}\in X$.

Put $v_{n}(x):=u_{n}\left(x+y_{n}\right)-u_{c}(x)$. Then,
$v_{n}\rightharpoonup 0$ in $X$, and $u_{n}\left(x+y_{n}\right)\rightarrow
u_{c}$ for a.e. $x\in\mathbb{R}^{N}$ by Lemma 2.3. Therefore, we obtain as
$n\to\infty$

(3.1) $\displaystyle\left\|\nabla u_{n}\right\|_{2}^{2}=\left\|\nabla
u_{n}\left(\cdot+y_{n}\right)\right\|_{2}^{2}=\left\|\nabla
v_{n}\right\|_{2}^{2}+\left\|\nabla u_{c}\right\|_{2}^{2}+o_{n}(1).$

Moreover, from the Brézis-Lieb Lemma (see [14, Lemma 3.2]), we have as
$n\to\infty$

(3.2)
$\displaystyle\left\|u_{n}\right\|_{2}^{2}=\left\|u_{n}\left(\cdot+y_{n}\right)\right\|_{2}^{2}=\left\|v_{n}\right\|_{2}^{2}+\left\|u_{c}\right\|_{2}^{2}+o_{n}(1),$
$\displaystyle\int_{\mathbb{R}^{N}}G_{\varepsilon}\left(u_{n}\right)dx=$
$\displaystyle\int_{\mathbb{R}^{N}}G_{\varepsilon}\left(u_{n}\left(\cdot+y_{n}\right)\right)dx=\int_{\mathbb{R}^{N}}G_{\varepsilon}\left(v_{n}\right)dx+\int_{\mathbb{R}^{N}}G_{\varepsilon}(u_{c})dx+o_{n}(1).$

Now, we prove that as $n\to\infty$

(3.3) $\left\|\nabla u_{n}\right\|_{q}^{q}=\left\|\nabla
u_{n}\left(\cdot+y_{n}\right)\right\|_{q}^{q}=\left\|\nabla
v_{n}\right\|_{q}^{q}+\left\|\nabla u_{c}\right\|_{q}^{q}+o_{n}(1).$

To apply the Brézis-Lieb Lemma, we only need to show that, up to a
subsequence,

$\nabla u_{n}(x+y_{n})\rightarrow\nabla u_{c}\enspace\text{for a.e.}\enspace
x\in\mathbb{R}^{N}.$

Since $J_{\varepsilon}^{\prime}$ and $J_{\varepsilon}$ possess translation
invariance, $(u_{n}(\cdot+y_{n}))_{n}$ is also a Palais-Smale sequence for
$J_{\varepsilon}$ on $\mathcal{D}(c)$. Hence, for the remainder of the proof,
we can substitute $(u_{n}(\cdot+y_{n}))_{n}$ with $(u_{n})_{n}$.

Let $\psi\in C_{0}^{\infty}\left(\mathbb{R}^{N}\right)$ such that
$0\leq\psi\leq 1$ in $\mathbb{R}^{N}$, $\psi(x)=1$ for $x\in B_{1/2}(0)$ and
$\psi(x)=0$ for $x\in\mathbb{R}^{N}\backslash B_{1}(0)$. Now, take $R>0$ and
define $\psi_{R}(x)=\psi(x/R)$ for $x\in\mathbb{R}^{N}$. Recall from An
application Lemma 3.7 gives

$\lim_{n\rightarrow\infty}\left(J_{\varepsilon}^{\prime}\left(u_{n}\right)\left(u_{n}-u_{c}\right)\psi_{R}+\lambda_{\varepsilon}\int_{\mathbb{R}^{N}}u_{n}\left(u_{n}-u_{c}\right)\psi_{R}{dx}\right)=0.$

Lemma 2.3 implies , up to a possible subsequencethat, that $u_{n}\rightarrow
u_{c}$ in $L_{\text{loc }}^{2}\left(\mathbb{R}^{N}\right)$. Thus

$\displaystyle\lim_{n\rightarrow\infty}$
$\displaystyle\int_{\mathbb{R}^{N}}u_{n}\left(u_{n}-u_{c}\right)\psi_{R}dx$
$\displaystyle\leq$
$\displaystyle\lim_{n\rightarrow\infty}\left[\left(\int_{\mathbb{R}^{N}}u_{n}^{2}dx\right)^{\frac{1}{2}}\left(\int_{\mathbb{R}^{N}}\left(u_{n}-u_{c}\right)^{2}\psi_{R}^{2}dx\right)^{\frac{1}{2}}\right]$
$\displaystyle\leq$
$\displaystyle\lim_{n\rightarrow\infty}\left[\left(\int_{\mathbb{R}^{N}}u_{n}^{2}dx\right)^{\frac{1}{2}}\left(\int_{B_{R}(0)}\left(u_{n}-u_{c}\right)^{2}dx\right)^{\frac{1}{2}}\right]$
$\displaystyle=$ $\displaystyle 0.$

Therefore,
$\lim\limits_{n\rightarrow\infty}J_{\varepsilon}^{\prime}\left(u_{n}\right)\left(u_{n}-u_{c}\right)\psi_{R}=0$.
Put $g^{\varepsilon}=(G^{\varepsilon})^{\prime}$, where
$G^{\varepsilon}:=G_{+}-G_{-}^{\varepsilon}\geq G$. The definition (1.7) of
$J_{\varepsilon}^{\prime}\left(u_{n}\right)\left(u_{n}-u_{c}\right)\psi_{R}$,
gives

(3.4) $\displaystyle\int_{\mathbb{R}^{N}}$ $\displaystyle{\left[\nabla
u_{n}-\nabla u_{c}+\left|\nabla u_{n}\right|^{q-2}\nabla u_{n}-\left|\nabla
u_{c}\right|^{q-2}\nabla u_{c}\right]\left(\nabla u_{n}-\nabla
u_{c}\right)\psi_{R}dx}$ $\displaystyle=$ $\displaystyle
J_{\varepsilon}^{\prime}\left(u_{n}\right)\left(\left(u_{n}-u_{c}\right)\psi_{R}\right)-\int_{\mathbb{R}^{N}}\nabla
u_{n}u_{n}\nabla\psi_{R}dx-\int_{\mathbb{R}^{N}}\left|\nabla
u_{n}\right|^{q-2}\nabla u_{n}u_{n}\nabla\psi_{R}dx$ $\displaystyle\quad+$
$\displaystyle\int_{\mathbb{R}^{N}}g^{\varepsilon}(u_{n})u_{n}\psi_{R}dx+\int_{\mathbb{R}^{N}}\nabla
u_{n}u_{c}\nabla\psi_{R}+\int_{\mathbb{R}^{N}}\left|\nabla
u_{n}\right|^{q-2}\nabla u_{n}u_{c}\nabla\psi_{R}dx$ $\displaystyle\quad-$
$\displaystyle\int_{\mathbb{R}^{N}}g^{\varepsilon}(u_{n})u_{c}\psi_{R}dx-\int_{\mathbb{R}^{N}}\nabla
u_{n}\nabla u_{c}\psi_{R}dx-\int_{\mathbb{R}^{N}}\left|\nabla
u_{c}\right|^{q-2}\nabla u_{c}\nabla u_{n}\psi_{R}dx$ $\displaystyle\quad+$
$\displaystyle\int_{\mathbb{R}^{N}}\left|\nabla
u_{c}\right|^{q}\psi_{R}dx+\int_{\mathbb{R}^{N}}\left|\nabla
u_{c}\right|^{2}\psi_{R}dx,$

since
$\lim\limits_{n\rightarrow\infty}J_{\varepsilon}^{\prime}\left(u_{n}\right)\left(u_{n}-u_{c}\right)\psi_{R}=0$.
The boundedness of $(u_{n})_{n}$ in $X$ and the fact that $u_{c}\in X$ imply
that as $n\rightarrow\infty$

$\begin{gathered}\int_{\mathbb{R}^{N}}\nabla
u_{n}u_{n}\nabla\psi_{R}dx\rightarrow\int_{\mathbb{R}^{N}}\nabla
u_{n}u_{c}\nabla\psi_{R}{dx},\\\ \int_{\mathbb{R}^{N}}\left|\nabla
u_{n}\right|^{q-2}\nabla
u_{n}u_{n}\nabla\psi_{R}dx\rightarrow\int_{\mathbb{R}^{N}}\left|\nabla
u_{n}\right|^{q-2}\nabla u_{n}u_{c}\nabla\psi_{R}dx.\end{gathered}$

Since $u_{n}\rightharpoonup u_{c}$ in $X$, following [24], we are able to show
that as $n\rightarrow\infty$

$\begin{gathered}\int_{\mathbb{R}^{N}}\left|\nabla u_{c}\right|^{q-2}\nabla
u_{c}\nabla u_{n}\psi_{R}dx\rightarrow\int_{\mathbb{R}^{N}}\left|\nabla
u_{c}\right|^{q}\psi_{R}dx,\\\ \int_{\mathbb{R}^{N}}\nabla u_{n}\nabla
u_{c}\psi_{R}dx\rightarrow\int_{\mathbb{R}^{N}}\left|\nabla
u_{c}\right|^{2}\psi_{R}dx.\end{gathered}$

Now we prove that as $n\rightarrow\infty$

$\begin{gathered}\int_{\mathbb{R}^{N}}g^{\varepsilon}(u_{n})u_{n}\psi_{R}dx,\;\int_{\mathbb{R}^{N}}g^{\varepsilon}(u_{n})u_{c}\psi_{R}dx\rightarrow\int_{\mathbb{R}^{N}}g^{\varepsilon}(u_{c})u_{c}\psi_{R}dx.\end{gathered}$

When $N\geq 3$ or $N=2$ and $q>2$, by $(g_{1})$ and $(g_{2})$, and since
$G_{-}^{\varepsilon}(s)\leq c_{\varepsilon}|s|^{2}$ for every $|s|\leq 1$, for
any $\delta>0$ there exists $C_{\delta}>0$ such that for every
$s\in\mathbb{R}$

$g^{\varepsilon}(s)\leq C_{\delta}|s|+\delta|s|^{{q^{\prime}-1}}.$

This, we have

$\displaystyle\int_{\mathbb{R}^{N}}$ $\displaystyle
g^{\varepsilon}(u_{n}-u_{c})(u_{n}-u_{c})\psi_{R}dx$ $\displaystyle\leq$
$\displaystyle
C_{\delta}\int_{\mathbb{R}^{N}}|u_{n}-u_{c}|^{2}\psi_{R}dx+\delta\int_{\mathbb{R}^{N}}|u_{n}-u_{c}|^{{q^{\prime}}}\psi_{R}dx$
$\displaystyle\leq$ $\displaystyle
C_{\delta}\int_{B_{R}(0)}|u_{n}-u_{c}|^{2}dx+\delta\int_{B_{R}(0)}|u_{n}-u_{c}|^{{q^{\prime}}}dx$

By Lemma 2.3, up to a subsequence
$\lim\limits_{n\rightarrow\infty}\int_{B_{R}(0)}|u_{n}-u_{c}|^{2}dx=0$ and
$\int_{B_{R}(0)}|u_{n}-u_{c}|^{{q^{\prime}}}dx<C$. Thus, for any $\eta>0$, we
can choose $0<\delta<\frac{\eta}{2C}$, then for all measurable set $A\subset
B_{R}(0)$,

(3.5)
$\delta\int_{A}|u_{n}-u_{c}|^{{q^{\prime}}}dx\leq\delta\int_{B_{R}(0)}|u_{n}-u_{c}|^{{q^{\prime}}}dx<\frac{\eta}{2}.$

And since
$\lim\limits_{n\rightarrow\infty}\int_{B_{R}(0)}|u_{n}-u_{c}|^{2}dx=0$, there
exists $N\in\mathbb{N}$ such that

(3.6)
$C_{\delta}\int_{B_{R}(0)}|u_{n}-u_{c}|^{2}dx<\frac{\eta}{2}\quad\text{for all
}n>N.$

Hence, by (3.5) and (3.6), for $\forall\eta>0,\exists\,\sigma>0$ and
$N\in\mathbb{N}$ such that

$\int_{\mathbb{R}^{N}}g^{\varepsilon}(u_{n}-u_{c})(u_{n}-u_{c})\psi_{R}dx<\eta,$

$\forall n>N$ and $\forall A\subset\Omega$ measurable with $\mu(A)<\sigma$. So
$g^{\varepsilon}(u_{n}-u_{c})(u_{n}-u_{c})\psi_{R}$ satisfies Lemma 2.6(i).
Since $g^{\varepsilon}(u_{n}-u_{c})(u_{n}-u_{c})\psi_{R}\rightarrow 0$ for
a.e. $x\in\mathbb{R}^{N}$, it also satisfies Lemma 2.6(ii). Applying Lemma
2.6, we obtain

$\int_{\mathbb{R}^{N}}g^{\varepsilon}(u_{n}-u_{c})(u_{n}-u_{c})\psi_{R}dx\rightarrow
0\text{ as }n\rightarrow\infty.$

From the Brézis-Lieb Lemma [8],

$\int_{\mathbb{R}^{N}}g^{\varepsilon}(u_{n}-u_{c})(u_{n}-u_{c})\psi_{R}dx=\int_{\mathbb{R}^{N}}g^{\varepsilon}(u_{n})u_{n}\psi_{R}dx-\int_{\mathbb{R}^{N}}g^{\varepsilon}(u_{c})u_{c}\psi_{R}dx+o_{n}(1).$

Therefore,

$\int_{\mathbb{R}^{N}}g^{\varepsilon}(u_{n})u_{n}\psi_{R}dx\rightarrow\int_{\mathbb{R}^{N}}g^{\varepsilon}(u_{c})u_{c}\psi_{R}dx,\quad\text{
as }n\rightarrow\infty.$

When $N=2$ and $q<2$, for every $\delta>0$, $\alpha>0$, $s\in\mathbb{R}$ and
$m>2$ there exists $C_{\delta}>0$ such that

$g^{\varepsilon}(s)\leq C_{\delta}|s|+\delta|s|^{{m-1}}\left(e^{\alpha
s^{2}}-1\right).$

Then, by $\left(e^{s}-1\right)^{r}\leq e^{sr}-1$ for $r>1$ and $s\geq 0$, and
the Hölder inequality, we have

$\displaystyle\int_{\mathbb{R}^{N}}$ $\displaystyle
g^{\varepsilon}(u_{n}-u_{c})(u_{n}-u_{c})\psi_{R}dx$ $\displaystyle\leq$
$\displaystyle
C_{\delta}\int_{\mathbb{R}^{2}}|u_{n}-u_{c}|^{2}dx+\delta\int_{\mathbb{R}^{2}}|u_{n}-u_{c}|^{{m}}(e^{\alpha|u_{n}-u_{c}|^{2}}-1)\psi_{R}dx$
$\displaystyle\leq$ $\displaystyle
C_{\delta}\int_{B_{R}(0)}|u_{n}-u_{c}|^{2}dx+\delta\int_{B_{R}(0)}|u_{n}-u_{c}|^{{m}}(e^{\alpha|u_{n}-u_{c}|^{2}}-1)\psi_{R}dx$
$\displaystyle\leq$ $\displaystyle
C_{\delta}\int_{B_{R}(0)}|u_{n}-u_{c}|^{2}dx+\delta\int_{B_{R}(0)}|u_{n}-u_{c}|^{{2m}}dx\int_{B_{R}(0)}(e^{2\alpha|u_{n}-u_{c}|^{2}}-1)dx.$

Define
$\bar{u}_{n}(x):=\sqrt{\frac{\alpha}{\pi}}\left(u\left(x\right)-u_{n}\left(x\right)\right)$.
Then

$\int_{\mathbb{R}^{2}}|\nabla
v|^{2}dx=\frac{\alpha}{\pi}\int_{\mathbb{R}^{2}}|\nabla u|^{2}dx.$

We can choose $\alpha>0$ small enough such that $\|\nabla v\|_{2}^{2}<1$.
Moreover, by Lemma 2.2 we have

$\int_{B_{R}(0)}(e^{2\alpha|u_{n}-u_{c}|^{2}}-1)dx=\int_{B_{R}(0)}(e^{2\pi|v_{n}|^{2}}-1)dx\leq\int_{\mathbb{R}^{2}}(e^{2\pi|v_{n}|^{2}}-1)dx\leq
C_{1},$

for some $C_{1}>0$. Hence

$\displaystyle\int_{\mathbb{R}^{N}}$ $\displaystyle
g^{\varepsilon}(u_{n}-u_{c})(u_{n}-u_{c})\psi_{R}dx$ $\displaystyle\leq$
$\displaystyle
C_{\delta}\int_{B_{R}(0)}|u_{n}-u_{c}|^{2}dx+C_{1}\delta\int_{B_{R}(0)}|u_{n}-u_{c}|^{{2m}}dx.$

Applying a variant of the Lebesgue Dominated Convergence Theorem,

$\int_{\mathbb{R}^{N}}g^{\varepsilon}(u_{n}-u_{c})(u_{n}-u_{c})\psi_{R}dx\rightarrow
0\text{ as }n\rightarrow\infty.$

Employing the similar arguments as before, we obtain

$\int_{\mathbb{R}^{2}}g^{\varepsilon}(u_{n})u_{n}\psi_{R}dx\rightarrow\int_{\mathbb{R}^{2}}g^{\varepsilon}(u_{c})u_{c}\psi_{R}dx,\quad\text{
as }n\rightarrow\infty.$

Similarly, we can prove

$\int_{\mathbb{R}^{N}}g^{\varepsilon}(u_{n})u_{c}\psi_{R}dx\rightarrow\int_{\mathbb{R}^{N}}g^{\varepsilon}(u_{c})u_{c}\psi_{R}dx,\quad\text{
as }n\rightarrow\infty.$

Thus, letting $n\rightarrow\infty$ in (3.4), we get

$\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\left[\nabla u_{n}-\nabla
u_{c}+\left|\nabla u_{n}\right|^{q-2}\nabla u_{n}-\left|\nabla
u_{c}\right|^{q-2}\nabla u_{c}\right]\left(\nabla u_{n}-\nabla
u_{c}\right)\psi_{R}dx=0.$

Now, $\int_{\mathbb{R}^{N}}\left(\left|\nabla u_{n}\right|^{q-2}\nabla
u_{n}-\left|\nabla u_{c}\right|^{q-2}\nabla u_{c}\right)\left(\nabla
u_{n}-\nabla u_{c}\right)\psi_{R}dx\geq 0$ by Remark 2.7. Therefore,

$\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\left(\left|\nabla
u_{n}\right|^{q-2}\nabla u_{n}-\left|\nabla u_{c}\right|^{q-2}\nabla
u_{c}\right)\left(\nabla u_{n}-\nabla u_{c}\right)\psi_{R}dx=0,$

and

$\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\left(\nabla u_{n}-\nabla
u_{c}\right)\left(\nabla u_{n}-\nabla u_{c}\right)\psi_{R}dx=0.$

Applying Lemma 2.5 to $a(x,\xi)=|\xi|^{s-2}\xi$ with $s=2$ and $s=q$, and
following the steps given in [24], we are able to prove that $\nabla
u_{n}\rightarrow\nabla u_{c}$ for a.e. $x\in\mathbb{R}^{N}$ up to a
subsequence.

We next claim that

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

which, by Lemma 2.4, will yield the statement of Lemma 2.5. If this is not
true, then, as before, there exist $z_{n}\in\mathbb{R}^{N}$ and $v_{c}\in
X\backslash\\{0\\}$ such that, denoting
$w_{n}(x):=v_{n}\left(x-z_{n}\right)-v_{c}(x)$, we have $w_{n}\rightharpoonup
0$ in $X,w_{n}\rightarrow 0$ for a.e. $x\in\mathbb{R}^{N}$, and

$\lim_{n\rightarrow\infty}\big{[}J_{\varepsilon}\left(v_{n}\right)-J_{\varepsilon}\left(w_{n}\right)\big{]}=J_{\varepsilon}(v).$

Note that, once more due to the Brézis-Lieb Lemma,

$\lim_{n\rightarrow\infty}\left(\left\|u_{n}\right\|_{2}^{2}-\left\|w_{n}\right\|_{2}^{2}\right)=\lim_{n\rightarrow\infty}\left(\left\|u_{n}\right\|_{2}^{2}-\left\|v_{n}\right\|_{2}^{2}+\left\|v_{n}\right\|_{2}^{2}-\left\|w_{n}\right\|_{2}^{2}\right)=\|u_{c}\|_{2}^{2}+\|v_{c}\|_{2}^{2},$

whence, denoting $\beta:=\|u_{c}\|_{2}>0$ and $\gamma:=\|v_{c}\|_{2}>0$, there
holds

$c^{2}-\beta^{2}-\gamma^{2}\geq\liminf_{n}\left\|u_{n}\right\|_{2}^{2}-\beta^{2}-\gamma^{2}=\liminf_{n}\left\|w_{n}\right\|_{2}^{2}=:\delta^{2}\geq
0.$

If $\delta>0$, then let us set
$\tilde{w}_{n}:=\frac{\delta}{\left\|w_{n}\right\|_{2}}w_{n}\in\mathcal{S}(\delta)$.
Via explicit computations we have
$\lim\limits_{n\rightarrow\infty}\big{[}J_{\varepsilon}\left(w_{n}\right)-J_{\varepsilon}\left(\tilde{w}_{n}\right)\big{]}=0$,
hence, together with Lemma 3.4 and since $m_{\varepsilon}(c)$ is non-
increasing with respect to $c>0$,

(3.7) $\displaystyle m_{\varepsilon}(c)$
$\displaystyle=\lim\limits_{n\rightarrow\infty}J_{\varepsilon}\left(u_{n}\right)$
$\displaystyle=J_{\varepsilon}(u_{c})+J_{\varepsilon}(v_{c})+\lim_{n\rightarrow\infty}J_{\varepsilon}\left(w_{n}\right)$
$\displaystyle=J_{\varepsilon}(u_{c})+J_{\varepsilon}(v_{c})+\lim_{n\rightarrow\infty}J_{\varepsilon}\left(\tilde{w}_{n}\right)$
$\displaystyle\geq
m_{\varepsilon}(\beta)+m_{\varepsilon}(\gamma)+m_{\varepsilon}(\delta)$
$\displaystyle\geq
m_{\varepsilon}\left(\sqrt{\beta^{2}+\gamma^{2}+\delta^{2}}\right)$
$\displaystyle\geq m_{\varepsilon}(c).$

Thus all the inequalities in (3.7) are in fact equalities and, in particular,
$J_{\varepsilon}(u_{c})=m_{\varepsilon}(\beta)$ and $J_{\varepsilon}(v_{c})=$
$m_{\varepsilon}(\gamma)$. Therefore Lemma 3.5 yields that
$m_{\varepsilon}(\beta)+m_{\varepsilon}(\gamma)+m_{\varepsilon}(\delta)>m_{\varepsilon}\left(\sqrt{\beta^{2}+\gamma^{2}+\delta^{2}}\right)$,
which contradicts (3.7).

If $\delta=0$, then $w_{n}\rightarrow 0$ in
$L^{2}\left(\mathbb{R}^{N}\right)$, which, together with Lemma 2.1, implies
that
$\lim\limits_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}G_{+}\left(w_{n}\right)dx=0$,
whence $\lim\limits\inf_{n}J_{\varepsilon}\left(w_{n}\right)\geq 0$. Then
(3.7) becomes

$m_{\varepsilon}(c)=\lim_{n\rightarrow\infty}J_{\varepsilon}\left(u_{n}\right)=J_{\varepsilon}(u_{c})+J_{\varepsilon}(v_{c})+\lim_{n\rightarrow\infty}J_{\varepsilon}\left(w_{n}\right)\geq
m_{\varepsilon}(\beta)+m_{\varepsilon}(\gamma)\geq
m_{\varepsilon}\left(\sqrt{\beta^{2}+\gamma^{2}}\right)=m_{\varepsilon}(c)$

and we get a contradiction as before. ∎

Let us recall that every solution
$\left(u_{\varepsilon},\lambda_{\varepsilon}\right)$ of equation
($\mathscr{P}_{\varepsilon}$) satisfies the following Pohozaev identity

(3.8) $\frac{N-2}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx+\frac{N-q}{q}\int_{\mathbb{R}^{N}}|\nabla u|^{q}dx+\frac{\lambda
N}{2}\int_{\mathbb{R}^{N}}|u|^{2}dx=N\int_{\mathbb{R}^{N}}G^{\varepsilon}(u)dx.$

Proof of Theorem 3.1. Let $\bar{c}>0$ be determined by Lemma 3.3 and let
$(u_{n})_{n}\subset\mathcal{D}(c)$ be a minimizing sequence for
$m_{\varepsilon}(c)$. Then, in virtue of Lemmas 3.2 and 3.8, there exists
$u_{\varepsilon}\in\mathcal{D}(c)\backslash 0$ such that, up to subsequences
and translations, $u_{n}\rightharpoonup u_{\varepsilon}$ in $X$ and
$u_{n}\rightarrow u_{\varepsilon}$ in $L^{m}\left(\mathbb{R}^{N}\right)$ for
every $m$, with $2<m<q^{\prime}$ and $u_{n}\rightarrow u_{\varepsilon}$ for
a.e. $x\in\mathbb{R}^{N}$. From $(g_{1})$ and $(g_{3})$ we have
${\lim\limits_{n\rightarrow\infty}}\int_{\mathbb{R}^{N}}G_{+}\left(u_{n}\right)dx=\int_{\mathbb{R}^{N}}G_{+}\left(u_{\varepsilon}\right)dx$,
therefore, using Fatou’s Lemma, $m_{\varepsilon}(c)\leq
J_{\varepsilon}\left(u_{\varepsilon}\right)\leq\lim\limits_{n\rightarrow\infty}J_{\varepsilon}\left(u_{n}\right)=m_{\varepsilon}(c)<-\beta$.
In particular, there exists $\lambda_{\varepsilon}\in\mathbb{R}$ such that

$-\Delta
u_{\varepsilon}-\Delta_{q}u_{\varepsilon}+\lambda_{\varepsilon}u_{\varepsilon}=g^{\varepsilon}\left(u_{\varepsilon}\right),\quad
x\in\mathbb{R}^{N}.$

By Lemma 3.7, note that $\lambda_{\varepsilon}=0$ if
$\lim\limits_{n\rightarrow\infty}\left\|u_{n}\right\|_{2}<c$. If
$\lambda_{\varepsilon}\leq 0$, then (3.8) yields
$0>J_{\varepsilon}\left(u_{\varepsilon}\right)\geq\left(\left\|\nabla
u_{\varepsilon}\right\|_{2}^{2}+\left\|\nabla
u_{\varepsilon}\right\|_{q}^{q}\right)/N\geq 0$, which is a contradiction.
Therefore $\lambda_{\varepsilon}>0$ and
$\lim\limits_{n\rightarrow\infty}\left\|u_{n}\right\|_{2}=c$.

If $\|u_{\varepsilon}\|_{2}=c_{0}<c$, then arguing as in Lemma 3.8 we derive
that $u_{n}\rightarrow u_{\varepsilon}$ a.e. for $x\in\mathbb{R}^{N}$ and
$\nabla u_{n}\rightarrow\nabla u_{\varepsilon}$ for a.e. $x\in\mathbb{R}^{N}$.
Thus,

$m_{\varepsilon}(c)=\lim_{n\rightarrow\infty}J_{\varepsilon}\left(u_{n}\right)=J_{\varepsilon}(u_{c})+\lim_{n\rightarrow\infty}J_{\varepsilon}(u_{n}-u_{c})\geq
m_{\varepsilon}(c_{0})+m_{\varepsilon}(c-c_{0})\geq m_{\varepsilon}(c).$

and we reach a contradiction as before. Therefore
$\lim\limits_{n\rightarrow\infty}\left\|u_{n}\right\|_{2}=\left\|u_{\varepsilon}\right\|_{2}$.
Finally, Theorem 1.4 of [15], see also [20], implies that such
$u_{\varepsilon}$ has constant sign and, up to translations, $u_{\varepsilon}$
is radial and radially monotone. $\Box$

######  Remark 3.9.

The proof of Theorem 3.1 shows that every minimizing sequence for
$m_{\varepsilon}(c)$ is relatively compact in $X$ up to a translation.
Specifically, from
$\lim\limits_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}G_{+}\left(u_{n}\right)dx=$
$\int_{\mathbb{R}^{N}}G_{+}\left(u_{\varepsilon}\right)dx$ and
$\lim\limits_{n\rightarrow\infty}J_{\varepsilon}\left(u_{n}\right)=J_{\varepsilon}\left(u_{\varepsilon}\right)$
we get $\lim\limits_{n\rightarrow\infty}\left\|\nabla
u_{n}\right\|_{2}=\left\|\nabla u_{\varepsilon}\right\|_{2}$ and
$\lim\limits_{n\rightarrow\infty}\left\|\nabla u_{n}\right\|_{q}=\left\|\nabla
u_{\varepsilon}\right\|_{q}$. Moreover, since

$\lim\limits_{n\rightarrow\infty}\left\|u_{n}\right\|_{2}=\left\|u_{\varepsilon}\right\|_{2},$

we obtain that $u_{n}\rightarrow u_{\varepsilon}$ in $X$.

##  4\. Proof of Theorem 1.1

Throughout this section, we assume that $(g_{0})-(g_{4})$ hold. The family
$\\{u_{\varepsilon}\\}_{\varepsilon}$ and the related Lagrange multipliers
$\\{\lambda_{\varepsilon}\\}_{\varepsilon}$ are given by Theorem 3.1, where
$\varepsilon\in(0,1)$.

######  Lemma 4.1.

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

###### Proof.

By $(g_{0}),(g_{1})$, and $(g_{3})$, for every $\delta>0$ there exists
$C_{\delta}>0$ such that for every $s\in\mathbb{R}$

$G_{+}(s)\leq C_{\delta}|s|^{2}+\delta|s|^{\bar{q}}.$

Let us first consider the case $N\geq 3$. Since $\lambda_{\varepsilon}>0$,
from (3.8) we obtain

$\frac{1}{2^{*}}\left\|\nabla
u_{\varepsilon}\right\|_{2}^{2}+\frac{1}{q^{*}}\left\|\nabla
u_{\varepsilon}\right\|_{q}^{q}<\int_{\mathbb{R}^{N}}G_{+}\left(u_{\varepsilon}\right)dx\leq
C_{\delta}\left\|u_{\varepsilon}\right\|_{2}^{2}+\delta
C_{N,\bar{q}}^{\bar{q}}c^{\bar{q}\left(1-\delta_{\bar{q}}\right)}\|\nabla
u\|_{2}^{\bar{q}\delta_{\bar{q}}}.$

Taking
$\delta<\left(C_{N,\bar{q}}^{\bar{q}}c^{\bar{q}\left(1-\delta_{\bar{q}}\right)}\right)^{-1}$,
we obtain the boundedness of $\left\|\nabla u_{\varepsilon}\right\|_{2}$ and
$\left\|\nabla u_{\varepsilon}\right\|_{q}$. In addition, this and again (3.8)
yield

(4.1) $\displaystyle 0$ $\displaystyle<\frac{1}{2}\lambda_{\varepsilon}c^{2}$
$\displaystyle<\int_{\mathbb{R}^{N}}\left(\frac{1}{2^{*}}\left|\nabla
u_{\varepsilon}\right|^{2}+\frac{1}{q^{*}}\left|\nabla
u_{\varepsilon}\right|^{q}+\frac{1}{2}\lambda_{\varepsilon}\left|u_{\varepsilon}\right|^{2}+G_{-}^{\varepsilon}\left(u_{\varepsilon}\right)\right)dx$
$\displaystyle=\int_{\mathbb{R}^{N}}G_{+}\left(u_{\varepsilon}\right)dx\leq
C<\infty,$

and $\lambda_{\varepsilon}$ is bounded as well. Now, let us consider the case
$N=2$. From (3.8) we get

(4.2) $\displaystyle c^{2}\lambda_{\varepsilon}$
$\displaystyle\leq\int_{\mathbb{R}^{N}}\left(\lambda_{\varepsilon}\left|u_{\varepsilon}\right|^{2}+2G_{-}^{\varepsilon}\left(u_{\varepsilon}\right)\right)dx$
$\displaystyle\leq\int_{\mathbb{R}^{N}}2G_{+}\left(u_{\varepsilon}\right)dx$
$\displaystyle\leq C_{\delta}\left\|u_{\varepsilon}\right\|_{2}^{2}+\delta
C_{N,\bar{q}}^{\bar{q}}c^{\bar{q}\left(1-\delta_{\bar{q}}\right)}\|\nabla
u\|_{2}^{\bar{q}\delta_{\bar{q}}}.$

Moreover, again using (3.8) we obtain

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

which, together with (4.2), implies

$\frac{1}{2}\left\|\nabla
u_{\varepsilon}\right\|_{2}^{2}+\frac{1}{2}\left\|\nabla
u_{\varepsilon}\right\|_{q}^{q}\leq C_{\delta}c^{2}-\beta+\delta
C_{N,\bar{q}}^{\bar{q}}c^{\bar{q}\left(1-\delta_{\bar{q}}\right)}\|\nabla
u\|_{2}^{\bar{q}\delta_{\bar{q}}}.$

Taking
$\delta<\left(2C_{N,\bar{q}}^{\bar{q}}c^{\bar{q}\left(1-\delta_{\bar{q}}\right)}\right)^{-1}$,
we obtain the boundedness of $\left\|\nabla u_{\varepsilon}\right\|_{2}^{2}$
and $\left\|\nabla u_{\varepsilon}\right\|_{q}^{q}$. Hence, the boundedness of
the family $\lambda_{\varepsilon}$ follows from (4.2). ∎

Note that for every $\varepsilon\in(0,1)$ and every $u\in\mathcal{D}$ there
holds $J_{\varepsilon}\left(u_{\varepsilon}\right)\leq J_{\varepsilon}(u)\leq
J(u)$, whence
$J_{\varepsilon}\left(u_{\varepsilon}\right)\leq\inf_{\mathcal{D}(c)}J=:m(c)$.
Now we are ready to complete the proof of Theorem 1.1.

Proof of Theorem 1.1. From Lemma 4.1 there exist $u\in\mathcal{D}(c)$ and
$\lambda\geq 0$ such that, up to a subsequence,
$u_{\varepsilon}\rightharpoonup u$ in $X$ and
$\lambda_{\varepsilon}\rightarrow\lambda$ as $\varepsilon\rightarrow 0^{+}$.
Arguing as in [22, Theorem 1.1], we obtain that
$\varphi_{\varepsilon}\left(u_{\varepsilon}\right)g_{-}\left(u_{\varepsilon}\right)v\rightarrow
g_{-}(u)v$ for a.e. $x\in\mathbb{R}^{N}$ for every
$v\in\mathcal{C}_{0}^{\infty}\left(\mathbb{R}^{N}\right)$ and that
$g_{+}\left(u_{\varepsilon}\right)v$ and
$\varphi_{\varepsilon}\left(u_{\varepsilon}\right)g_{-}\left(u_{\varepsilon}\right)v$
are uniformly integrable (and tight). We deduce that for every
$v\in\mathcal{C}_{0}^{\infty}\left(\mathbb{R}^{N}\right)$

$-\lambda\int_{\mathbb{R}^{N}}uvdx\leftarrow
J_{\varepsilon}^{\prime}\left(u_{\varepsilon}\right)v\rightarrow
J^{\prime}(u)v,\quad{\mbox{as $\varepsilon\rightarrow 0^{+}$}},$

i.e.,

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

Moreover, (4.1) (when $N\geq 3$) or (4.2) (when $N=2$ ) yields

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

Hence, in view of Fatou’s Lemma, $G_{-}(u)\in
L^{1}\left(\mathbb{R}^{N}\right)$. In particular, as shown in [9], the couple
$(u,\lambda)$ satisfies the Pohožaev identity

(4.3) $\frac{N-2}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx+\frac{N-q}{q}\int_{\mathbb{R}^{N}}|\nabla u|^{q}dx+\frac{\lambda
N}{2}\int_{\mathbb{R}^{N}}|u|^{2}dx=N\int_{\mathbb{R}^{N}}G(u)dx.$

We can assume that $\left\|\nabla u_{\varepsilon}\right\|_{2}$, $\left\|\nabla
u_{\varepsilon}\right\|_{q}$ and
$\int_{\mathbb{R}^{N}}G_{-}^{\varepsilon}\left(u_{\varepsilon}\right)dx$ are
convergent. Since
$\int_{\mathbb{R}^{N}}G_{+}\left(u_{\varepsilon}\right)dx\rightarrow\int_{\mathbb{R}^{N}}G_{+}(u)dx$
(remember that each $u_{\varepsilon}$ is radially symmetric), we have
$J(u)\leq\lim\limits_{\varepsilon}J_{\varepsilon}\left(u_{\varepsilon}\right)<0$
from Lemma 3.3; in particular, $u\neq 0$.

If $\lambda=0$, then from (4.3) we have $J(u)=(\|\nabla u\|_{2}^{2}+\|\nabla
u\|_{q}^{q})/N>0$, therefore $\lambda>0$. We prove that
$u_{\varepsilon}\rightarrow u$ in $X$. Since $\lambda>0$, from (4.3) there
follows

$\displaystyle\int_{\mathbb{R}^{N}}$ $\displaystyle\left(\frac{N-2}{2}|\nabla
u|^{2}+\frac{N-q}{q}|\nabla
u|^{2}+\frac{N\lambda}{2}|u|^{2}+NG_{-}(u)\right)dx$ $\displaystyle\leq$
$\displaystyle\lim_{\varepsilon\rightarrow
0}\int_{\mathbb{R}^{N}}\left(\frac{N-2}{2}|\nabla
u_{\varepsilon}|^{2}+\frac{N-q}{q}|\nabla
u_{\varepsilon}|^{2}+\frac{N\lambda}{2}\left|u_{\varepsilon}\right|^{2}+NG_{-}^{\varepsilon}\left(u_{\varepsilon}\right)\right)dx$
$\displaystyle=$ $\displaystyle\lim_{\varepsilon\rightarrow
0}N\int_{\mathbb{R}^{N}}G_{+}\left(u_{\varepsilon}\right)dx$ $\displaystyle=$
$\displaystyle N\int_{\mathbb{R}^{N}}G_{+}(u)dx$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}\left(\frac{N-2}{2}|\nabla
u|^{2}+\frac{N-q}{q}|\nabla
u|^{2}+\frac{N\lambda}{2}|u|^{2}+NG_{-}(u)\right)dx$

This yields that $\left\|u_{\varepsilon}\right\|_{2}\rightarrow\|u\|_{2}$; in
particular, $u\in\mathcal{S}(c)$. Furthermore, in a similar way we obtain
$J(u)\leq$
$\lim\limits_{\varepsilon}J_{\varepsilon}\left(u_{\varepsilon}\right)\leq
m(c)\leq J(u)$, which shows that $\left\|\nabla
u_{\varepsilon}\right\|_{2}\rightarrow\|\nabla u\|_{2}$ and $J(u)=m(c)$.

Since ($\mathscr{P}_{\varepsilon}$) 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. $\Box$

##  5\. Proof of Theorem 1.2

In this section, we first present some fundamental concepts and properties
concerning the Orlicz spaces. For more details we refer to [1, 13, 29]

######  Definition 5.1.

An $N$-function $A$ is a nonnegative continuous function
$A:\mathbb{R}\rightarrow[0,\infty)$ that satisfies the following conditions:
$\phantom{i}(i)$ $A$ is convex and even.
$(ii)$ $\lim\limits_{t\rightarrow 0}\frac{A(t)}{t}=0$ and
$\lim\limits_{t\rightarrow\infty}\frac{A(t)}{t}=\infty$.

An N-function $A$ is said to satisfy the $\Delta_{2}$ condition globally if
and only if there exists $K>0$ such that

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

An N-function $A$ is said to satisfy the $\nabla_{2}$ condition globally if
and only if there exists $\ell>1$ such that

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

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

(5.1) $CA(s)\geq sa(s)\text{ (respectively, }sa(s)\geq CA(s)\text{ ) for all
}s\in\mathbb{R}.$

From now on, $A$ denotes the same function given in assumtion $(A)$.

The Orlicz space $V$ associated with $A$ in $(A)$ is given by

$V:=\left\\{u\in L_{\text{loc }}^{1}\left(\mathbb{R}^{N}\right):A(u)\in
L^{1}\left(\mathbb{R}^{N}\right)\right\\},$

endowed with the norm

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

Since $A$ is an $N$-function that satisfies the $\Delta_{2}$ and $\nabla_{2}$
conditions globally, $\left(V,\|\cdot\|_{V}\right)$ is a reflexive Banach
space. Moreover, we shall use the following lemma.

######  Lemma 5.2.

(i) Let $u_{n},u\in V$. Then $\lim\limits_{n}\left\|u_{n}-u\right\|_{V}=0$ if
and only if
$\lim\limits_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}A\left(u_{n}-u\right)dx=0$.
(ii) Let $\bar{V}\subset V$. Then $\bar{V}$ is bounded in $V$ if and only if

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

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

###### Proof.

(i) and (ii) follow from [29] respectively, while (iii) is a consequence of
(i), (ii) and [8]. ∎

The space $W$ introduced in (1.10) is exactly given by $W=X\cap V$. Since
$X=(X,\|\cdot\|_{X})$ and $\left(V,\|\cdot\|_{V}\right)$ are reflexive Banach
spaces, so is $\left(W,\|\cdot\|_{W}\right)$, with

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

The next result is a variant of Lions’ lemma.

######  Lemma 5.3.

Suppose that $(u_{n})_{n}$ is a bounded sequence in $W$ and that for some
$r>0$

$\lim_{n\rightarrow\infty}\sup_{y\in\mathbb{R}^{N}}\int_{B_{r}(y)}\left|u_{n}\right|^{2}dx=0.$

Then $u_{n}\rightarrow 0$ in $L^{p}\left(\mathbb{R}^{N}\right)$ for every
$p\in\left[2,q^{\prime}\right)$,
$q^{\prime}=\max\left\\{2^{*},q^{*}\right\\}$.

###### Proof.

Lions’ Lemma 2.4 implies that $u_{n}\rightarrow 0$ in
$L^{m}\left(\mathbb{R}^{N}\right)$ for every $m\in\left(2,q^{\prime}\right)$.
We claim that the strong convergence occurs also in
$L^{2}\left(\mathbb{R}^{N}\right)$. Take any $m\in\left(2,q^{\prime}\right)$
and $\varepsilon>0$. Then, there exists $\delta>0$ such that

$\displaystyle|s|^{2}\leq\varepsilon A(s)\quad\text{ if }|s|\in[0,\delta],$
$\displaystyle|s|^{2}\leq\delta^{2-m}|s|^{m}\quad\text{ if }|s|>\delta.$

Hence, we get

$\displaystyle\limsup_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\left|u_{n}\right|^{2}dx\leq\varepsilon\limsup_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}A\left(u_{n}\right)dx+\delta^{2-m}\limsup_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\left|u_{n}\right|^{m}dx=\varepsilon\limsup_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}A\left(u_{n}\right)dx.$

Letting $\varepsilon\rightarrow 0^{+}$, we conclude the proof thanks to Lemma
5.2 (ii). ∎

Let $\mathcal{O}$ be any subgroup $\mathcal{O}(N)$ such that $\mathbb{R}^{N}$
is compatible with $\mathcal{O}$ (see [18]), i.e.,
$\lim\limits_{|y|\rightarrow\infty}m(y,r)=$ $\infty$ for some $r>0$, where,
for all $y\in\mathbb{R}^{N}$

$m(y,r):=\sup\left\\{n\in\mathbb{N}\text{ : there exist
}g_{1},\ldots,g_{n}\in\mathcal{O}\text{ such that
}{B_{r}}\left(g_{i}y\right)\cap B_{r}\left(g_{j}y\right)=\emptyset\text{ for
}i\neq j\right\\}\text{. }$

In view of [18], the space $H_{\mathcal{O}}^{1}\left(\mathbb{R}^{N}\right)$
embeds compactly into $L^{m}\left(\mathbb{R}^{N}\right)$ for all $m$, with
$2<m<2^{*}$. Using the same argument, we obtain that $W_{\mathcal{O}}$,
defined in (1.11), embeds compactly into $L^{m}\left(\mathbb{R}^{N}\right)$
for all $m$, with $2\leq m<q^{\prime}$, as proved in the next result.

######  Corollary 5.4.

Suppose that $(u_{n})_{n}$ is a bounded sequence in $W_{\mathcal{O}}$ and that
$u_{n}\rightarrow 0$ in $L_{\text{loc }}^{2}\left(\mathbb{R}^{N}\right)$. Then
$u_{n}\rightarrow 0$ in $L^{m}\left(\mathbb{R}^{N}\right)$ for every
$m\in\left[2,q^{\prime}\right)$.

###### Proof.

Suppose that

(5.2) $\int_{B_{1}\left(y_{n}\right)}\left|u_{n}\right|^{2}dx\geq C>0,$

for some sequence $(y_{n})_{n}\subset\mathbb{R}^{N}$ and a suitable constant
$C$. Observe that in the family
$\left\\{B_{1}\left(hy_{n}\right)\right\\}_{h\in\mathcal{O}}$, we find an
increasing number of disjoint balls provided that
$\left|y_{n}\right|\rightarrow\infty$. Since $(u_{n})_{n}$ is bounded in
$L^{2}\left(\mathbb{R}^{N}\right)$ and invariant with respect to
$\mathcal{O}$, by (5.2) the sequence $(y_{n})_{n}$ must be bounded. Then, for
sufficiently large $r$ we obtain

$\int_{B_{r}(0)}\left|u_{n}\right|^{2}dx\geq\int_{B_{r}\left(y_{n}\right)}\left|u_{n}\right|^{2}dx\geq
c>0.$

This contradicts (5.2). Therefore, the conclusion holds thanks to Lemma 5.3. ∎

Then, we have the following result.

######  Proposition 5.5.

If $(A)$ and $(f_{0})-(f_{3})$ hold, then the functional
$\left.J\right|_{W}:W\rightarrow\mathbb{R}$ is of class $\mathcal{C}^{1}$.
Moreover, $\left.J\right|_{W\cap\mathcal{S}(c)}^{\prime}(u)=0$, with $u\in
W\cap\mathcal{S}(c)$, if and only if there exists $\lambda\in\mathbb{R}$ such
that $(u,\lambda)$ is a solution of ($\mathscr{E}_{\lambda}$).

###### Proof.

Let $B:\mathbb{R}\rightarrow\mathbb{R}$ be the complementary of the
$N$-function of $A$. From [29, Theorem II.V.8], the function $B$ satisfies the
$\Delta_{2}$ and $\nabla_{2}$ conditions globally as $A$ also satisfies these
conditions. Let us recall (see [29, Definition III.IV.2, Corollary III.IV.5,
and Corollary IV.II.9]) that $V^{*}$, the dual space of $V$, is isomorphic to
the Orlicz space associated with $B$. From [29, Theorem I.III.3],

$B(a(s))=sa(s)-A(s)\leq(C-1)A(s),$

where $C>1$ is the constant given in the characterization of the $\Delta_{2}$
condition (5.1). Then, for every $u,v\in V$, we have

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

Now we proceeding as in the proof of Lemma 2.1 in [11], we obtain that, under
the assumptions $(f_{1})-(f_{3})$, both functionals
$u\mapsto\int_{\mathbb{R}^{N}}A(u)dx$ and
$u\mapsto\int_{\mathbb{R}^{N}}F(u)dx$ are of class
$\mathcal{C}^{1}(V)\subset\mathcal{C}^{1}(W)$. The remaining part is obvious.
∎

To prove Theorem 1.2, we need some important preliminary results. As usually,
we adopt the notations $F_{-}:=F_{+}-F$ and $f_{-}:=F_{-}^{\prime}$.

######  Lemma 5.6.

If $(A),(f_{0})-(f_{3})$, and (1.12) hold, then
$\left.J\right|_{W\cap\mathcal{S}(c)}$ is bounded below.

###### Proof.

From (1.9), for every $\delta>0$, there exists $C_{\delta}>0$ such that for
every $s\in\mathbb{R}$

$F_{+}(s)\leq C_{\delta}|s|^{2}+(\eta+\delta)|s|^{\bar{q}}.$

Hence, for every $u\in W\cap\mathcal{S}(c)$ there holds

$\displaystyle J(u)$
$\displaystyle\geq\int_{\mathbb{R}^{N}}\left(\frac{1}{2}|\nabla
u|^{2}+\frac{1}{q}|\nabla u|^{q}+A(u)-F_{+}(u)\right)dx$
$\displaystyle\geq\int_{\mathbb{R}^{N}}\left(\frac{1}{2}|\nabla
u|^{2}+\frac{1}{q}|\nabla
u|^{q}+A(u)-C_{\delta}|u|^{2}-(\eta+\delta)|u|^{\bar{q}}\right)dx$
$\displaystyle\geq\frac{1}{2}\|\nabla u\|_{2}^{2}+\frac{1}{q}\|\nabla
u\|_{q}^{q}+\int_{\mathbb{R}^{N}}A(u)dx-(\eta+\delta)C_{N,\bar{q}}^{\bar{q}}c^{\bar{q}\left(1-\delta_{\bar{q}}\right)}\|\nabla
u\|_{2}^{\bar{q}\delta_{\bar{q}}}.$

Taking $\delta$ so small that
$2(\eta+\delta)C_{N,\bar{q}}^{\bar{q}}c^{\bar{q}\left(1-\delta_{\bar{q}}\right)}<1$,
we conclude that $J(u)$ bounded from below on $W\cap\mathcal{S}(c)$, since
$A(s)$ is nonnegative. ∎

######  Remark 5.7.

Setting $f_{\mathrm{p}}:=\max\\{f,0\\}$ and $f_{\mathrm{n}}:=\max\\{-f,0\\}$,
we see that

$F_{+}(s)=\begin{cases}\int_{0}^{s}\max\\{f(t),0\\}dt\quad\text{ if }s\geq
0,\\\ \int_{s}^{0}\max\\{-f(t),0\\}dt\quad\text{ if }s<0.\end{cases}$

Hence, being $f_{+}(s)=F_{+}^{\prime}(s)$, $f_{-}(s):=f_{+}(s)-f(s)$, and
$F_{-}(s):=F_{+}(s)-F(s)\geq 0$ for $s\in\mathbb{R}$, it holds
$f_{-}(s)=\max\\{-f(s),0\\}=f_{\mathrm{n}}(s)$ if $s\geq 0$ and
$f_{-}(s)=-\max\\{f(s),0\\}=-f_{\mathrm{p}}(s)$ if $s<0$. Therefore, we
conclude that, $f_{-}(s)s\geq 0$ for every $s\in\mathbb{R}$.

######  Lemma 5.8.

If $(A)$ holds, then $s\mapsto a(s)s$ is nonnegative for all $s\in\mathbb{R}$.

###### Proof.

Since $A$ is an even function, $s\mapsto a(s)s$ also is an even function.
]tcrThus, we only need to prove that $s\mapsto a(s)s$ is nonnegative on
$\left(0,\infty\right)$.

Suppose, by contradiction, that there exists $s_{1}>0$, such that
$a(s_{1})s_{1}<0$, then there exists $s_{2}\in\left(0,s_{1}\right)$ such that
$a(s_{2})s_{2}>0$. Otherwise $a(s)\leq 0$ on $(0,s_{1})$, and thus
$A(s_{1})<0$, which is a contradiction. Since $s\mapsto a(s)s$ is convex in
$\mathbb{R}$ by$(A)$, the following inequality holds

$\frac{a(0)0-a\left(s_{1}\right)s_{1}}{0-s_{1}}\geq\frac{a(0)0-a\left(s_{2}\right)s_{2}}{0-s_{2}}.$

Hence, $a(s_{1})\geq a(s_{2})$, which is the required contradiction. ∎

######  Lemma 5.9.

If $(A)$ and $(f_{0})-(f_{3})$ hold, then
$\left.J\right|_{{W_{\mathcal{O}}}\cap\mathcal{S}(c)}$ satisfies the Palais-
Smale condition.

###### Proof.

Let $(u_{n})_{n}$ be a sequence in $W_{{\mathcal{O}}}\cap\mathcal{S}(c)$ such
that $(J\left(u_{n}\right))_{n}$ is bounded in $\mathbb{R}$ and
$\left.J\right|_{W_{\mathcal{O}}\cap\mathcal{S}(c)}^{\prime}\left(u_{n}\right)\rightarrow
0$ as $n\to\infty$. From Lemma 5.6 and Lemma 5.2 (ii), the sequence
$(u_{n})_{n}$ 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, $u_{n}\rightarrow u$ in
$L^{m}\left(\mathbb{R}^{N}\right)$ for every $m\in\left[2,q^{\prime}\right)$
by Corollary 5.4. In particular, $u\in\mathcal{S}(c)$. Up to a subsequence
again, we can assume that $u_{n}\rightarrow u$ for a.e. in $\mathbb{R}^{N}$.
Additionally, from [7, Lemma 3], there exists a sequence $(\lambda_{n})_{n}$
in $\mathbb{R}$ such that

(5.3) $-\Delta u_{n}+\lambda_{n}u_{n}-g\left(u_{n}\right)\rightarrow
0\quad\text{ in }W_{\mathcal{O}}^{*},$

where $W_{\mathcal{O}}^{*}$ is the dual space of $W_{\mathcal{O}}$. Testing
(5.3) with $(u_{n})_{n}$, we obtain that $(\lambda_{n})_{n}$ is bounded as
well. Hence, there exist $\lambda\in\mathbb{R}$ such that
$\lambda_{n}\rightarrow\lambda$ up to a subsequence, and $(u,\lambda)$ is a
solution of ($\mathscr{E}_{\lambda}$). Finally, from the Nehari identity and
the fact that $\lambda_{n}\rightarrow\lambda$, we obtain that

$\displaystyle\int_{\mathbb{R}^{N}}\left(|\nabla u|^{2}+|\nabla
u|^{q}+a(u)u+f_{-}(u)u\right)dx$
$\displaystyle=\int_{\mathbb{R}^{N}}\left(f_{+}(u)u-\lambda|u|^{2}\right)dx$
$\displaystyle=\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\left(f_{+}\left(u_{n}\right)u_{n}-\lambda\left|u_{n}\right|^{2}\right)dx$
$\displaystyle=\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\left(\left|\nabla
u_{n}\right|^{2}+|\nabla
u|^{q}+a\left(u_{n}\right)u_{n}+f_{-}\left(u_{n}\right)u_{n}\right)dx,$

Now, $f_{-}(s)s\geq 0$ for every $s\in\mathbb{R}$ by Remark 5.7, so that by
Fatou’s Lemma

$\limsup_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}f\left(u_{n}\right)u_{n}dx\geq\int_{\mathbb{R}^{N}}f(u)udx.$

Similarly, $a(s)s\geq 0$ for every $s\in\mathbb{R}$ by $(A)$ and by Lemma 5.8,
so that

$\limsup_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}a\left(u_{n}\right)u_{n}dx\geq\int_{\mathbb{R}^{N}}a(u)udx.$

Additionally, since $u_{n}\rightharpoonup u$ in $W_{\mathcal{O}}$,

$\liminf_{n\rightarrow\infty}\left\|\nabla u_{n}\right\|_{2}^{2}\geq\|\nabla
u\|_{2}^{2}\quad\text{and}\quad\liminf_{n\rightarrow\infty}\left\|\nabla
u_{n}\right\|_{q}^{q}\geq\|\nabla u\|_{q}^{q}.$

The above properties imply that, up to subsequences, as $n\rightarrow\infty$

$\displaystyle\left\|\nabla u_{n}\right\|_{2}^{2}\rightarrow\|\nabla
u\|_{2}^{2},\quad\left\|\nabla u_{n}\right\|_{q}^{q}\rightarrow\|\nabla
u\|_{q}^{q},$
$\displaystyle\int_{\mathbb{R}^{N}}f\left(u_{n}\right)u_{n}dx\rightarrow\int_{\mathbb{R}^{N}}f(u)udx\quad\mbox{and}\quad\int_{\mathbb{R}^{N}}a\left(u_{n}\right)u_{n}dx\rightarrow\int_{\mathbb{R}^{N}}a(u)udx.$

It remains to prove that $\left\|u_{n}-u\right\|_{V}\rightarrow 0$. In virtue
of Lemma 5.2 (iii), it suffices to show that
$\int_{\mathbb{R}^{N}}A\left(u_{n}\right)dx\rightarrow\int_{\mathbb{R}^{N}}A(u)dx$.
Additionally, since $A$ satisfies the $\nabla_{2}$ condition globally, we have
that $sa(s)\geq CA(s)$. Since
$\int_{\mathbb{R}^{N}}a\left(u_{n}\right)u_{n}dx\rightarrow\int_{\mathbb{R}^{N}}a(u)udx$,
by a variant of the Lebesgue Dominated Convergence theorem, we have that
$\int_{\mathbb{R}^{N}}A\left(u_{n}\right)dx\rightarrow\int_{\mathbb{R}^{N}}A(u)dx$.
∎

######  Lemma 5.10.

For every $k\geq 1$ there exist the functions
$\gamma_{k}:\mathbb{S}^{k-1}\rightarrow W_{\mathcal{O}(N)}\cap\mathcal{S}(c)$
and
$\tilde{\gamma}_{k}:\mathbb{S}^{k-1}\rightarrow\mathcal{X}\cap\mathcal{S}(c)$
which are odd and continuous.

###### Proof.

Fix $k\in\mathbb{N}$. In view of [14, Lemma 3.4], there exist constants
$R(k)>2(k+1)$ and $c_{k}>0$ such that, for any $R\geqslant R(k)$, there exists
an odd continuous mapping $\tau_{k,R}:\mathbb{S}^{k-1}\rightarrow
H^{1}\left(\mathbb{R}^{N}\right)$ having the properties that
$\tau_{k,R}[\sigma]$ is a radial function,
$\operatorname{supp}\left(\tau_{k,R}[\sigma]\right)\subset\bar{B}_{R}(0)$ for
any $\sigma\in\mathbb{S}^{k-1}$. Therefore, $\tau_{k,R}$ is also an odd
continuous mapping from $\mathbb{S}^{k-1}$ to $W_{\mathcal{O}(N)}$.

Fix $c>0$ and $k\in\mathbb{N}$, we can define an odd continuous mapping
$\gamma_{k}:\mathbb{S}^{k-1}\rightarrow W_{\mathcal{O}(N)}\cap\mathcal{S}(c)$
as follows:

${\gamma_{k}}[\sigma](x):=\tau_{k}[\sigma]\left(c^{-1/N}\cdot\left\|\tau_{k}[\sigma]\right\|_{L^{2}\left(\mathbb{R}^{N}\right)}^{2/N}\cdot
x\right),\quad x\in\mathbb{R}^{N}\text{ and }\sigma\in\mathbb{S}^{k-1}.$

Let $\chi:\mathbb{R}\rightarrow[0,1]$ be an odd smooth function such that
$\chi(t)=1$ for any $t\geqslant 1$. Let us introduce

$\pi_{k,R}[\sigma](x):=\tau_{k,R}[\sigma](x)\cdot\chi\left(\left|x_{1}\right|-\left|x_{2}\right|\right),$

where $\sigma\in\mathbb{S}^{k-1}$ and
$x=\left(x_{1},x_{2},x_{3}\right)\in\mathbb{R}^{M}\times\mathbb{R}^{M}\times\mathbb{R}^{N-2M}$.
Clearly, $\pi_{k,R}$ is an odd continuous mapping from $\mathbb{S}^{k-1}$ to
$\mathcal{X}$.

Fix $c>0$ and $k\in\mathbb{N}$, we can define an odd continuous mapping
$\gamma_{k}:\mathbb{S}^{k-1}\rightarrow\mathcal{X}\cap\mathcal{S}(c)$ as
follows:

$\bar{\gamma_{k}}[\sigma](x):=\pi_{k}[\sigma]\left(c^{-1/N}\cdot\left\|\pi_{k}[\sigma]\right\|_{L^{2}\left(\mathbb{R}^{N}\right)}^{2/N}\cdot
x\right),\quad x\in\mathbb{R}^{N}\text{ and }\sigma\in\mathbb{S}^{k-1},$

as required. ∎

In what follows, we make use of the next abstract theorem, where $\mathcal{G}$
stands for the Krasnoselsky genus [28, Chapter 5].

######  Theorem 5.11 ([14, Theorem 2.1]).

Let $E$ be a Banach space, let $H\subset E$ be a Hilbert space with scalar
product $(\cdot\mid\cdot)$. Fix $R>0$ and put $\mathcal{M}:=\\{u\in
E\,:\,(u\mid u)=R\\}$. Asumme that $I\in\mathcal{C}^{1}(E)$ is an even
functional and that $\left.I\right|_{\mathcal{M}}$ is bounded below. For every
$k\geq 1$, set

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

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

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

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

Hence we can conclude in the following way.

###### Proof of Theorem 1.2.

Take $E=W_{\mathcal{O}(N)}$ (respectively, $E=\mathcal{X}$),
$H=L^{2}\left(\mathbb{R}^{N}\right)$, $R=c^{2},\mathcal{M}=$
$\mathcal{S}(c)\cap E$, and $I=\left.J\right|_{E}$. From Lemmas 5.6 and 5.9
the functional $\left.I\right|_{\mathcal{M}}=J_{\mathcal{S}(c)\cap E}$ is
bounded below and satisfies the Palais-Smale condition. Moreover, Lemma 5.10
implies that $\gamma_{k}\left(\mathbb{S}^{k-1}\right)\in\Gamma_{k}$
(respectively, $\tilde{\gamma}_{k}\left(\mathbb{S}^{k-1}\right)\in\Gamma_{k}$)
for every $k$. Thus, the numbers $c_{k}$ are finite. Applying Theorem 5.11, we
conclude there exist infinitely many solutions of ($\mathscr{E}_{\lambda}$).

Concerning the existence of a least-energy solution, we consider a minimizing
sequence $(u_{n})_{n}$ in $\mathcal{S}(c)\cap E$ such that
$\lim\limits_{n\rightarrow\infty}{J}\left(u_{n}\right)=\inf_{\mathcal{S}(c)\cap
E}J$. From Ekeland’s variational principle, we can assume that $(u_{n})_{n}$
is a Palais-Smale sequence for $\left.J\right|_{\mathcal{S}(c)\cap E}$. Hence,
arguing as above, we obtain a solution
$(\bar{u},\bar{\lambda})\in\mathbb{R}\times(\mathcal{S}(c)\cap E)$ of
($\mathscr{E}_{\lambda}$) such that $J(\bar{u})=\min_{\mathcal{S}(c)\cap E}J$.
The fact that $\min_{\mathcal{S}(c)\cap
W_{\mathcal{O}(N)}}J=\min_{\mathcal{S}(c)\cap W}J$ follows from the properties
of the Schwartz rearrangement. ∎

### Acknowledgements

C. Ji was partially supported by National Natural Science Foundation of China
(No. 12171152).

P. Pucci is a member of the Gruppo Nazionale per l’Analisi Matematica, la
Probabilità e le loro Applicazioni (GNAMPA) of the Instituto Nazionale di Alta
Matematica (INdAM) and this paper was written under the auspices of
GNAMPA–INdAM.

##  6\. Appendix

In this section we show that the energy functional $J$ is not well defined in
$X$.

######  Remark 6.1.

Since we consider the case where

(6.1) $2<\bar{q}=\left(1+\frac{2}{N}\right)\min\\{2,q\\},$

we have either $\frac{2N}{N+2}<q<2,\quad N\geq 2$ or $2<q<N,\quad N\geq 3$.

######  Lemma 6.2.

For $\frac{2N}{N+2}<q<2$, $N\geq 2$ or $2<q<N$, $N\geq 3$, there exists $u\in
X$ such that $\int_{\mathbb{R}_{N}}u^{2}\log u^{2}dx=-\infty$.

###### Proof.

It is enough to consider a smooth function that satisfies

$u(x)=\begin{cases}\left(|x|^{N/2}\log(|x|)\right)^{-1},&|x|\geq 3,\\\
0,&|x|\leq 2.\end{cases}$

From, we know that $u\in H^{1}(\mathbb{R}^{N})$ and
$\int_{\mathbb{R}_{N}}u^{2}\log u^{2}dx=-\infty$ for $N\geq 2$, so we only
need to prove that $u\in D^{1,q}(\mathbb{R}^{N})$, that is
$\int_{\mathbb{R}^{N}}|\nabla u|^{q}dx<\infty$.

If $2<q<N$, $N\geq 3$, choose $R>0$ large enough such that $u(x)<1$ on
$\mathbb{R}^{N}\backslash B_{R}(0)$. Hence,

$\int_{\mathbb{R}^{N}\backslash B_{R}(0)}|\nabla
u|^{q}dx<\int_{\mathbb{R}^{N}\backslash B_{R}(0)}|\nabla u|^{2}dx<\infty.$

Since ${u}\in C^{\infty}(\mathbb{R}^{N})$, we deduce that

$\int_{B_{R}(0)}|\nabla u|^{q}dx<\infty,$

and so

$\int_{\mathbb{R}^{N}}|\nabla u|^{q}dx{=}\int_{\mathbb{R}^{N}\backslash
B_{R}(0)}|\nabla u|^{q}dx+\int_{B_{R}(0)}|\nabla u|^{q}dx<\infty.$

If $\frac{2N}{N+2}<q<2$, $N\geq 2$, we compute the following integral directly

$\displaystyle\int_{B_{1}(0)}|\nabla u|^{q}dx$
$\displaystyle=\int_{B_{1}(0)}\left[\sum_{i=1}^{N}\left|\frac{-\frac{N}{2}|x|^{\frac{N}{2}-1}\log(|x|)+|x|^{\frac{N}{2}-1}}{|x|^{N}|\log(|x|)|^{2}}\cdot\frac{x_{i}}{|x|}\right|^{2}\right]^{\frac{q}{2}}dx$
$\displaystyle=\int_{B_{1}(0)}\left|\frac{|x|^{\frac{N}{2}-1}-\frac{N}{2}|x|^{\frac{N}{2}-1}\log(|x|)}{|x|^{N}|\log(|x|)|^{2}}\right|^{q}dx$
$\displaystyle\leq
C_{1}\left(\int_{B_{1}(0)}\frac{|x|^{-q-\frac{N}{2}q}}{|\log(|x|)|^{2q}}dx+\frac{|x|^{-q-\frac{N}{2}q}}{|\log(|x|)|^{q}}\right)dx$
$\displaystyle=C_{2}\int_{1}^{\infty}\left(\frac{r^{-1+N-q-\frac{N}{2}q}}{|\log
r|^{2q}}dr+\frac{r^{-1+N-q-\frac{N}{2}q}}{|\log r|^{q}}\right)dr<\infty,$

Where $C_{1}$ and $C_{2}$ are positive constants. Since
$\int_{1}^{\infty}\frac{r^{\alpha}}{|{\ln r}|^{\beta}}dr<\infty$ for all
$\alpha<-1$, $\beta>0$ and since $-1+N-q-\frac{N}{2}q<-1$, we have

$\int_{1}^{\infty}\left(\frac{r^{-1+N-q-\frac{N}{2}q}}{|\log
r|^{2q}}\right)dr<\infty,\qquad\int_{1}^{\infty}\left(\frac{r^{-1+N-q-\frac{N}{2}q}}{|\log
r|^{q}}\right)dr<\infty.$

Hence, $\int_{\mathbb{R}^{N}}|\nabla u|^{q}dx<\infty$ holds. ∎

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