\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{mathrsfs}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 49, pp. 1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/49\hfil 
 Characterization of a homogeneous Orlicz space]
{Characterization of a homogeneous Orlicz space}

\author[W. Arriagada, J. Huentutripay \hfil EJDE-2017/49\hfilneg]
{Waldo Arriagada, Jorge Huentutripay}

\address{Waldo Arriagada \newline
Department of Applied Mathematics and Sciences, Khalifa University,
Al Zafranah, P.O. Box 127788, Abu Dhabi, United Arab Emirates}
\email{waldo.arriagada@kustar.ac.ae}

\address{Jorge Huentutripay \newline
Instituto de Ciencias F\'{i}sicas y Matem\'aticas,
Universidad Austral de Chile, Casilla 567, Valdivia, Chile}
\email{jorge.huentutripay@uach.cl}

\dedicatory{Communicated by Vicentiu Radulescu}

\thanks{Submitted August 31, 2016. Published February 16, 2017.}
\subjclass[2010]{46E30, 46T30, 35J20, 35J50}
\keywords{Homogeneous space; Orlicz space; eigenvalue problem;  $\phi$-Laplacian}

\begin{abstract}
 In this article we define and characterize the homogeneous Orlicz space
 $\mathscr{D}^{1,\Phi}_{\rm o}(\mathbb{R}^{N})$ where $\Phi:\mathbb{R}\to [0,+\infty)$
 is the $N$-function generated by an odd, increasing and not-necessarily
 differentiable homeomorphism $\phi:\mathbb{R}\to\mathbb{R}$.
 The properties of $\mathscr{D}^{1,\Phi}_{\rm o}(\mathbb{R}^{N})$ are treated
 in connection with the $\phi$-Laplacian eigenvalue problem
 $$
 -\operatorname{div}\Big(\phi(|\nabla u|)\frac{\nabla u}{|\nabla u|}\Big)
 =\lambda\,g(\cdot)\phi(u)\quad\text{in }\mathbb{R}^N
 $$
 where $\lambda\in\mathbb{R}$ and $g:\mathbb{R}^N\to\mathbb{R}$ is measurable.
 We use a classic Lagrange rule to prove that solutions of the $\phi$-Laplace
 operator exist and are non-negative.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}\label{Int}

Let $N\geq 2$ be an integer. A broad subclass of maximization problems in
an open domain $\Omega\subset\mathbb{R}^N$ involves critical Sobolev exponents.
Several articles are motivated by the ideas and methods in the seminal paper by
 Brezis and  Nirenberg \cite{BN}, mainly when $\Omega$ is bounded.
The case $\Omega$ unbounded is treated in  \cite{bence,panx}.
The reference \cite{BTW} contains significant results on semilinear problems
also in the unbounded case, which are largely treated via concentration-compactness
 methods. In that reference the authors introduce the space
\begin{equation}
\mathscr{D}^{1,p}(\Omega)=\{u\in L^{p^*}(\Omega): |\nabla u| \in L^p(\Omega)\}
\label{Dnor}
\end{equation}
where $1<p<N$ and $p^*=pN/(N-p)$ is the conjugate exponent. This space
is equipped with the norm
$\|u\|_{1,p}=\|u\|_{p^*}+\||\nabla u|\|_{p}$
where $\|\cdot\|_{p}$ is the norm in $L^p(\Omega)$. On the other hand,
the completion of the space $\mathcal D(\Omega)$ of $C^{\infty}$-functions
 with compact support in $\Omega$ with respect to the norm $\|\cdot\|_{1,p}$
is denoted by $\mathscr D^{1,p}_{\rm o}(\Omega)$.
Equivalently,
$$
\mathscr D^{1,p}_{\rm o}(\Omega)=\operatorname{cl}_{{\mathscr{D}^{1,p}}(\Omega)}
\mathcal D(\Omega)
$$
 where $\operatorname{cl}_{X}(Y)$ is the closure operator of $Y$ in $X$.
This space is endowed with the gradient seminorm
$\|u\|_{\rm o,\,p}=\||\nabla u|\|_{p}$. It can be easily proved that this
is actually a norm on $\mathscr D^{1,p}_{\rm o}(\Omega)$ which is equivalent
to $\|u\|_{1,p}$. It is moreover known that the two spaces thus defined are
reflexive and Banach for the respective norms. Somewhat surprisingly,
a fundamental characterization (see \cite[Lemma 1.2]{BTW}) in the (unbounded)
case $\Omega=\mathbb{R}^N$ asserts that
$\mathscr D^{1,p}_{\rm o}(\mathbb{R}^N)=\mathscr{D}^{1,p}(\mathbb{R}^N)$.
This equivalence motivates the problem whether this space is still meaningful
in a larger context or not and raises the issue about the use and place
of this \emph{extended} space in analysis, particularly in optimization and
differential equations. In this paper we answer positively the former question
and provide an application which well suits the latter via a fundamental
formulation in Orlicz spaces, see below. An exhaustive treatment on the
theory of these function spaces can be found in the classic textbook by
  Krasnosel'skii and  Rutic'kii \cite{kr} and, more recently, in
references \cite{Hu-Ma,ku-Jo-Fu,RR}. The papers and monographs by
 Gossez \cite{Go2,Go,Go-Ma} are particularly detailed and have played a
paramount role in the subject as well.

Orlicz spaces constitute a natural \emph{extension} of the notion of an $L^p$ space:
the function $t\mapsto |t|^{p}$ entering the definition of $L^p$ is replaced
by a more general $N$-function $\Phi:\mathbb{R}\to [0,+\infty)$
(sometimes called a Young function). The typical approach in the references
mentioned above is mostly developed in $\mathbb{R}^N$ with the Lebesgue measure.
One is naturally led to the question whether the properties and structure of
classic Orlicz spaces are preserved in a much more general measure space
$(\Omega,\Sigma,\mu)$. The monograph by J. Musielak \cite{Mu} studies the
properties associated with the generalized Orlicz space
$L^{\Phi}(\Omega,\Sigma,\mu)$ (such as embeddings of and compactness in
generalized Orlicz classes) in the setting of modular and parameter-dependent
families of Orlicz spaces.

An interesting source of research is given by the case of exponents $p(x)$,
 where $p:\Omega\to(1,+\infty)$ is a bounded function.
The article \cite{Ra} and excellent book \cite{RaRe} are representatives
in the case of nonhomogeneous differential operators containing one or
 more power-type nonlinearities with variable exponents. The theory there
is developed in great generality including many possible pathologies
of the Young function. As a yet another significant contribution,
the paper by Fu and Shan \cite{FuSh} gives sufficient conditions for
removability of isolated singular points of elliptic equations in the
Sobolev space $W^{1,p(x)}$, which was first studied by Kov\'a\v{c}ik
and R\'akosn\'ik.

In this manuscript we consider the homogeneous Orlicz space
$\mathscr{D}^{1,\Phi}_{\rm o}(\mathbb{R}^{N})$. It corresponds to the completion
of $\mathcal{D}(\mathbb{R}^N)$ with respect to a suitable norm, see
Section \ref{hosect}. If additional hypotheses are fulfilled this space
constitutes a natural source of solutions of minimization problems with
constraints for a wide class of energy functionals in the generalized-Laplacian
form. For example, in the article \cite{fin} the following quasilinear
elliptic problem is considered,
\begin{equation}
-\operatorname{div}\left(\varphi(|\nabla u|)\nabla u\right)
=b(|u|)u+\lambda f(x,u)\quad\text{in }\mathbb{R}^N\label{ante}
\end{equation}
where the function $\varphi(t)t$ is non-homogeneous.
The term $b(|u|)u$ denotes a critical Sobolev growth coefficient, $f(x,u)$
is a subcritical term and $\lambda>0$ is a parameter.
The authors prove that any non-negative solution of this problem
can be regarded as a critical point of the variational formulation
\begin{align*}
&\text{minimize}\quad \int_{\mathbb{R}^N}\big(\Phi(|\nabla u|)-B(u)
 -\lambda F(x,u)\big)dx\\
&\text{such that}\quad u\in \mathscr D^{1,\Phi}_{\rm o}(\mathbb{R}^N) %\label{ES}
\end{align*}
where $B(t)$ and $F(x,t)$ are the primitives of $b(t)t$ and $f(x,t)$,
respectively, and $\Phi(t)=\int_0^s\varphi(t)tdt$. Due to some topological
restrictions on $\mathscr D^{1,\Phi}_{\rm o}(\mathbb{R}^N)$ standard methods
to prove convergence of minimizing sequences for this problem are useless.
The techniques employed in \cite{fin} consist of a modification of the
concentration-compactness principle for Mountain-pass problems.

In this article we assume that $\phi:\mathbb{R}\to\mathbb{R}$ is an increasing,
odd and not-necessarily differentiable homeomorphism and define the associated
$N$-function
\begin{equation}\label{one}
\Phi(t)=\int_0^t \phi(s)\,ds.
\end{equation}
Motivated by the ideas discussed above, we provide a characterization of the
 homogeneous Orlicz space $\mathscr{D}^{1,\Phi}_{\rm o}(\mathbb{R}^{N})$
generated by $\Phi$. This characterization asserts that the latter space is
an \emph{extension} of \eqref{Dnor} in a precise sense and naturally leads
to the following application. Let $g:\mathbb{R}^N\to\mathbb{R}$ be a
measurable function and $\lambda$ be a real number. Under additional global
restrictions on $\Phi$ and $g$, existence of nontrivial solutions of the
 $\phi$-Laplacian equation
\begin{equation}\label{equa3}
-\operatorname{div}\Big(\phi(|\nabla u|)\frac{\nabla u}{|\nabla u|}\Big)
=\lambda g(\cdot)\phi(u)\quad\text{in }\mathbb{R}^N
\end{equation}
can be proved. We address this question and solve the associated optimization
problem by implementing a version of Lagrange multipliers rule \cite{FBr}
on the source space $\mathscr{D}^{1,\Phi}_{\rm o}(\mathbb{R}^{N})$.
 We prove that solutions of the $\phi$-Laplace operator exist and are non-negative.

\section{$N$-functions}\label{Orl}

This is a brief overview on Orlicz spaces. Fundamental definitions and properties
 can be found in several monographs, articles and books. For further details
we refer the reader to \cite{kr,ku-Jo-Fu,Mu}.

A convex, even and continuous function $\Phi:\mathbb{R}\to[0,+\infty)$ satisfying
$\Phi(t)=0$ if and only if $t=0$ and such that
$$
\frac{\Phi(t)}{t}\to 0\text{ as } t\to 0\quad\text{and}\quad
\frac{\Phi(t)}{t}\to +\infty\text{ as } t\to +\infty
$$
is called an $N$-function. Equivalently \cite{Go}, $\Phi$ can be represented
in the integral form \eqref{one}, where $\phi:\mathbb{R}\to\mathbb{R}$ is a
non-decreasing, odd function which is right-continuous for $t\geq 0$ and which
satisfies $\phi(t)=0$ if and only if $t=0$ and $\phi(t)\to +\infty$ as
$t\to +\infty$. The $N$-function $\Phi$ satisfies a global $\Delta_2$-condition
(see \cite[pp. 266]{Ad}) if there exists $\mathcal{C}>0$ such that
$$
\Phi(2t)\leq \mathcal{C}\Phi(t)
$$
for all $t\geq 0$.

\begin{lemma}[\cite{Ad}] \label{deldos}
The $N$-function $\Phi$ satisfies a global $\Delta_2$-condition if and only
if
\begin{equation}
q_\Phi:=\sup_{s>0} \frac{s\phi(s)}{\Phi(s)}<+\infty.\label{Delta3}
\end{equation}
\end{lemma}

\subsection{Conjugates}
The reciprocal function $\psi(s)$ of $\phi$ is defined for $s\geq 0$ by
$$
\psi(s)=\sup\left\{t:\phi(t)\leq s\right\}.
$$
Both functions $\phi$ and $\psi$ have the same properties. Hence the integral
$$
\overline{\Phi}(t)=\int_0^t \psi(s)\,ds
$$
is an $N$-function, called the conjugate (or complementary) $N$-function of $\Phi$.
The pair $\Phi,\overline{\Phi}$ is called a pair of complementary $N$-functions.
If $\phi$ is continuous and increases monotonically then the reciprocal $\psi$
is the ordinary inverse of $\phi$.

\begin{lemma}[{\cite[Lemma 2.5]{fin}}] \label{lem2.2}
The complementary $N$-function $\overline{\Phi}$ satisfies a global
 $\Delta_2$-condition if and only if
\begin{equation}
p_\Phi:=\inf_{s>0} \frac{s\phi(s)}{\Phi(s)}>1. \label{Delta4}
\end{equation}
\end{lemma}

The Sobolev conjugate $N$-function $\Phi_{*}$ of $\Phi$
is defined as
\[
\Phi_{*}^{-1}(t)=\int_0^t \frac{{\Phi}^{-1}(s)}{s^{1+\frac{1}{N}}}\,ds
\]
where $\Phi^{-1}$ denotes the inverse function of
$\Phi|_{[0,+\infty)}$. It is known \cite{RR} that the Sobolev conjugate
exists if and only if
\begin{equation}\label{def100}
\int_0^1\frac{{\Phi}^{-1}(s)}{s^{1+\frac{1}{N}}}\,ds<
+\infty\quad\text{and}\quad
\lim_{t\to+\infty}\int_0^t \frac{{\Phi}^{-1}(s)}{s^{1+\frac{1}{N}}}\,ds=+\infty.
\end{equation}
Moreover, it is known \cite{glms} that if conditions \eqref{def100} are
fulfilled then
\begin{equation}
\lim_{t\to+\infty}\frac{\Phi(t)}{\Phi_*(kt)}=0\label{crelen}
\end{equation}
for all $k>0$.

\begin{proposition}[\cite{fin}] \label{pelo}
If conditions \eqref{def100} are met and $q_{\Phi}<N$ then the following
estimates hold:
\begin{itemize} 
\item[(a)] 
$\min\{\rho^{p_\Phi},\rho^{q_\Phi}\}\Phi(t)
\leq\Phi(\rho t) \leq\max\{\rho^{p_\Phi},\rho^{q_\Phi}\}\Phi(t)$;

\item[(b)] $\min\{r^{p_\Phi^*},r^{q_\Phi^*}\}\Phi_*(t)\leq\Phi_*(r t)
\leq\max\{r^{p_\Phi^*},r^{q_\Phi^*}\}\Phi_*(t)$;

\item[(c)] $\min\{r^{p_\Phi^*/(p_\Phi^*-1)},r^{q_\Phi^*/(q_\Phi^*-1)}\}
\overline{\Phi_*}(t)\leq\overline{\Phi_*} (rt)$\\
$\leq\max\{r^{p_\Phi^*/(p_\Phi^*-1)},r^{q_\Phi^*/(q_\Phi^*-1)}\} 
\overline{\Phi_*}(t)$
\end{itemize}
for $r,t\geq 0$ and where $p_{\Phi}^*=p_{\Phi}\,N/(N-p_{\Phi})$ and
$q_{\Phi}^*=q_{\Phi}\,N/(N-q_{\Phi})$ are the conjugate exponents.
\end{proposition}

Note that Proposition \ref{pelo} ensures that both the Sobolev conjugate
$N$-function $\Phi_{*}$ and its complementary $\overline{\Phi_{*}}$ satisfy a
 global $\Delta_2$-condition provided $q_\Phi<N$.

\begin{lemma}\label{lamos}
Let $1<r<N$ be such that
\begin{equation}
0<A=\liminf_{s\to 0^+} \frac{\phi(s)}{s^{r-1}}\le \mathsf{B}
=\limsup_{s\to 0^+} \frac{\phi(s)}{s^{r-1}}<+\infty. \label{hypop}
\end{equation}
Then for $\varepsilon>0$ sufficiently small there exists $s_0=s_0(\varepsilon)>0$
such that for all $0<s<s_0$,
\begin{itemize}
\item[(a)] $\frac{(\mathsf{A}-\varepsilon)}{r}s^{r}\leq\Phi(s)
\leq \frac{(\mathsf{B}+\varepsilon)}{r}s^{r}$,

\item[(b)] $\Big(\frac{s\,r^{*}}{\overline{\mathsf{A}}}\Big)^{1/r^{*}}
\leq \Phi_{*}(s)
\leq \Big(\frac{s\,r^{*}}{\overline{\mathsf{B}}}\Big)^{1/r^{*}}$
\end{itemize}
where $\overline{\mathsf{B}}=r^{1/r}/(\mathsf{B}+\varepsilon)^{1/r}$,
$\overline{\mathsf{A}}=r^{1/r}/(\mathsf{A}-\varepsilon)^{1/r}$ and
$r^{*}=(N-r)/Nr$ is the Sobolev conjugate exponent.
\end{lemma}

\begin{proof}
If $\varepsilon>0$ is small then there exists $s_0=s_0(\varepsilon)>0$ such
that if $0<s<s_0$ then by definition
$$
\mathsf{A}-\varepsilon\leq\frac{\phi(s)}{s^{r-1}}\leq \mathsf{B}+\varepsilon.
$$
Denote  $t=\Phi(s)$ and $t_0=\Phi(s_0)$. The monotonicity of $\Phi$ and simple
integration yield
\[
\frac{(\mathsf{A}-\varepsilon)}{r}{(\Phi^{-1}(t))}^{r}
\leq t\leq \frac{(\mathsf{B}+\varepsilon)}{r}{(\Phi^{-1}(t))}^{r}
\]
provided $0<t<t_0$. Hence
$\overline{\mathsf{B}}\,t^{1/r}\leq \Phi^{-1}(t)\leq \overline{\mathsf{A}}\,t^{1/r}$
for all $0<t<t_0$. If $s<t<t_0$ we integrate (from $s$ to $t$) the latter
inequalities with respect to a new variable. This gives
$$
\frac{\overline{\mathsf{B}}}{r^{*}}(t^{r^{*}}-s^{r^{*}})
\leq \Phi_{*}^{-1}(t)-\Phi_{*}^{-1}(s)
\leq \frac{\overline{\mathsf{A}}}{r^{*}}(t^{r^{*}}-s^{r^{*}}).
$$
Letting $s\to 0^{+}$ we get
$$
\frac{\overline{\mathsf{B}}}{r^{*}}t^{r^{*}}\leq \Phi_{*}^{-1}(t)
\leq \frac{\overline{\mathsf{A}}}{r^{*}}t^{r^{*}}
$$
provided $0<t<t_0$. Finally, the change of variables $s=\Phi_{*}^{-1}(t)$ and
$s_0=\Phi_{*}^{-1}(t_0)$ and the inequality above yield the estimate in (b)
provided $0<s<s_0$.
\end{proof}

\section{Function spaces}
\subsection{Orlicz classes}
Let $\Phi,\overline{\Phi}$ be a pair of complementary $N$-functions and let
$\Omega$ denote an open domain in $\mathbb{R}^N$. The Orlicz class
${\mathcal{L}}_{\Phi}(\Omega)$ is the set of (equivalence classes of)
real-valued measurable functions $u$ such that $\Phi(u)\in L^1(\Omega)$.
In general, ${\mathcal{L}}_{\Phi}(\Omega)$ is not a vector space \cite{Go}.
However, the linear hull $L_{\Phi}(\Omega)$ of ${\mathcal{L}}_{\Phi}(\Omega)$
equipped with the Luxemburg norm
$$
\|u\|_{\Phi,\Omega}=\inf\big\{k>0:\int_{\Omega}\Phi\Big(\frac{u}{k}\Big)\leq 1
\big\}
$$
is a normed linear space, called the Orlicz space generated by the $N$-function
$\Phi$. It is known \cite{kr} that the vector space thus defined is complete.

The closure in $L_\Phi(\Omega)$ of the space of bounded measurable functions
with compact support in $\overline{\Omega}$ is denoted by $E_{\Phi}(\Omega)$.
This space is separable and Banach with the inherited norm.
The following lemma gives a useful characterization of a particular type of
sequences in $E_{\Phi}$ in the unbounded case $\Omega=\mathbb{R}^N$.

\begin{lemma}\label{facil}
Let $z\in E_{\Phi}(\mathbb{R}^N)$ and fix an integer $k>1$. Define the function
\[
z_k(x)=  \begin{cases}
z(x) & \text{if }   |x|>k \\
0 & \text{if }  |x|\leq k.
\end{cases}
\]
Then $\|z_k\|_{\Phi,\mathbb{R}^N}\to 0$ as $k\to +\infty$.
\end{lemma}

\begin{proof}
If $\varepsilon >0$ is sufficiently small then
$z/\varepsilon\in E_{\Phi}(\mathbb{R}^N)
\subseteq{\mathcal{L}}_{\Phi}(\mathbb{R}^N)$.
 The latter implies $\Phi(z/\varepsilon)\in L^1(\mathbb{R}^N)$ and then there
exists a positive integer $k_0$ such that if $k\geq k_0$ then
$$
\int_{\mathbb{R}^N}\Phi\big(\frac{z_k}{\varepsilon} \big)\,dx
=\int_{\mathbb{R}^N\backslash B_k(0)}\Phi\big(\frac{z}{\varepsilon} \big)\,dx
\leq 1
$$
where $B_k(0)$ denotes the ball of radius $k$ and center at zero in $\mathbb{R}^N$.
The definition of the Luxemburg norm hence yields
$\|z_k\|_{\Phi,\mathbb{R}^N}\leq\varepsilon$ provided $k\geq k_0$.
\end{proof}

In general,
$E_{\Phi}(\Omega)\subseteq{\mathcal{L}}_{\Phi}(\Omega)\subseteq L_\Phi(\Omega)$
but if $\Phi$ satisfies a global $\Delta_2$-condition then
$E_{\Phi}(\Omega)=L_\Phi(\Omega)$ and \emph{vice-versa}. In this case,
a known result \cite[Theorem 8.20]{Ad} ensures that $L_{\Phi}(\Omega)$ and
$L_{\overline{\Phi}}(\Omega)$ are reflexive and separable provided
$\overline{\Phi}$ satisfies a global $\Delta_2$-condition as well.
Since this result remains valid after replacing $\Phi$ by its Sobolev
conjugate $\Phi_*$ (provided the latter exists), Proposition \ref{pelo} guarantees
the validity of the following result.

\begin{corollary}\label{nosel}
If \eqref{def100} are satisfied and $q_{\Phi}<N$ then the Orlicz space
$L_{\Phi_*}(\Omega)$ is reflexive.
\end{corollary}

It is well known \cite{Ad,Go} that one can identify the dual space of
$E_\Phi(\Omega)$ with $L_{\overline\Phi}(\Omega)$ and the dual space of
$E_{\overline\Phi}(\Omega)$ with $L_\Phi(\Omega)$. Moreover, if
$u\in L_{\Phi}(\Omega)$ and $v\in L_{\overline{\Phi}}(\Omega)$ then the inequality
\begin{eqnarray}\label{hol}
\int_{\Omega}|uv|\,dx\leq 2\|u\|_{\Phi,\Omega}\,\|v\|_{\overline{\Phi},\Omega}
\end{eqnarray}
holds. This estimate is an extension of H\"{o}lder's inequality to Orlicz spaces.
\medskip

\noindent{\emph{An Orlicz-Sobolev space.}} The Orlicz-Sobolev space
$W^1L_{\Phi}(\Omega)$ is the vector space of functions in
$L_{\Phi}(\Omega)$ with first distributional derivatives in $L_{\Phi}(\Omega)$.
This space is Banach with the norm
\begin{equation}
|\!|\!| u|\!|\!|_{\Omega}=\|u\|_{\Phi,\Omega}+\sum_{i=1}^{N}
\|\partial_{x_i} u\|_{\Phi,\Omega}
\label{orso}
\end{equation}
where $\partial_{x_i}$ denotes the partial derivative $\partial /\partial x_i$.
Usually, $W^1L_\Phi(\Omega)$ is identified with a subspace of the product
$L_{\Phi}(\Omega)^{N+1}=\Pi L_{\Phi}(\Omega)$. The space $W^1L_\Phi(\Omega)$
is not separable in general.

\subsection{Approximation properties}\label{molli}
In what follows we  consider $\Omega=\mathbb{R}^N$
in which case further characterizations are possible.
The Luxemburg norm $\|\cdot\|_{\Phi,\mathbb{R}^N}$ will be simply denoted by
$\|\cdot\|_{\Phi}$. The symbol $\mathcal{D}(\mathbb{R}^N)$ denotes the space of
$C^{\infty}$-functions with compact support in $\mathbb{R}^N$.
We choose a mollifier $\rho\in \mathcal{D}(\mathbb{R}^N)$; i.e. $\rho$ is a
real-valued function such that
\begin{itemize}
\item[(a)] $\rho(x)\geq 0$, if $x\in\mathbb{R}^N$;

\item[(b)] $\rho(x)=0$, if $ |x|\geq 1$;

\item[(c)] $\int_{\mathbb{R}^N}\rho(x)\,dx=1$.
\end{itemize}
If $\varepsilon$ is positive, it is clear that the function
$\rho_\varepsilon(x)=\varepsilon^{-N}\rho(x/\varepsilon)$ is non-negative,
belongs to $\mathcal{D}(\mathbb{R}^N)$ and satisfies $\rho_{\varepsilon}(x)=0$
provided $|x|\geq \varepsilon$. In addition,
\begin{equation}
 \int_{\mathbb{R}^N}\rho_\varepsilon(x)\,dx=1.\label{int1}
\end{equation}

If $u\in L_{\Phi}(\mathbb{R}^N)$ we define the regularized function
$u_\varepsilon$ of $u$ by the convolution
\[
u_\varepsilon(x)=(\rho_\varepsilon * u)(x)
=\int_{\mathbb{R}^N}u(x-y)\rho_\varepsilon(y)\,dy.
\]
It is easy to see that if $u$ has compact support in $\mathbb{R}^N$ then
$u_\varepsilon$ belongs to $\mathcal{D}(\mathbb{R}^N)$.

\begin{proposition}\label{25a3}
If $u\in L_{\Phi}(\mathbb{R}^N)$ then $u_\varepsilon\in L_{\Phi}(\mathbb{R}^N)$ and
$\|u_\varepsilon\|_{\Phi}\leq\|u\|_{\Phi}$.
\end{proposition}

\begin{proof}
Let $\lambda=\|u\|_{\Phi}$. Jensen's inequality \cite[pp. 18]{Go} yields
\begin{equation}
\int_{\mathbb{R}^N}\Phi\Big(\frac{u_\varepsilon(x)}{\lambda}\Big)\,dx\leq
\int_{\mathbb{R}^N}\Big(\int_{\mathbb{R}^N}\Phi\Big(\frac{u(x-y)}{\lambda}\Big)
\rho_\varepsilon (y)\,dy \Big)dx. \label{not1}
\end{equation}
Define the function $F(x,y)=\Phi(u(x-y)/\lambda)\rho_\varepsilon (y)$.
It is clear from the definition of $\lambda$ that
\begin{equation}
\int_{\mathbb{R}^N}F(x,y)\,dx=\rho_\varepsilon(y)
\int_{\mathbb{R}^N}\Phi\Big(\frac{u(x-y)}{\lambda}\Big)dx
\leq \rho_\varepsilon(y).
\end{equation}
Integration of this inequality with respect to $y$ and condition \eqref{int1}
imply $F\in L^1(\mathbb{R}^N\times\mathbb{R}^N)$. Hence Fubini's theorem and
\eqref{not1} yield
$$
\int_{\mathbb{R}^N}\Phi\Big(\frac{u_\varepsilon(x)}{\lambda}\Big)\,dx
\leq\int_{\mathbb{R}^N}\Big(\int_{\mathbb{R}^N}\Phi\Big(\frac{u(x-y)}{\lambda}\Big)
\,dx\Big)\rho_\varepsilon(y)\,dy\leq 1
$$
and then $u_{\varepsilon}\in L_{\Phi}(\mathbb{R}^N)$.
By definition of the Luxemburg norm,
$\|u_\varepsilon\|_{\Phi}\leq\lambda=\|u\|_{\Phi}$.
\end{proof}

\begin{lemma}[\cite{Go3}] \label{181819}
If $u\in E_{\Phi}(\mathbb{R}^N)$ then $\| u_\varepsilon -u\|_{\Phi}\to 0$ as
$\varepsilon\to 0$.
\end{lemma}

\section{The homogeneous Orlicz space $\mathscr D^{1,\Phi}_{\rm o}(\mathbb{R}^{N})$}
\label{hosect}

In what follows we assume that $\phi:\mathbb{R}\to\mathbb{R}$
is an odd, non-decreasing and not-necessarily differentiable homeomorphism which
generates the $N$-function \eqref{one}. We suppose that condition \eqref{Delta3}
is fulfilled; i.e. $\Phi$ satisfies a global $\Delta_2$-condition.
 We will assume that \eqref{def100} are met as well, so that the Sobolev
conjugate $\Phi_*$ is defined. The set $B_R(x_0)\subseteq \mathbb{R}^N$ will denote
the ball of radius $R$ with center at $x_0\in\mathbb{R}^N$. As mentioned previously,
the operator $\partial_{x_i}$ will denote the partial derivative
$\partial /\partial x_i$, $i=1,\dots,N$. We start out by defining the space
$$
\mathscr{D}^{1,\Phi}(\mathbb{R}^N)=\left\{u\in L_{\Phi_{\ast}}(\mathbb{R}^N):
|\nabla u| \in L_{\Phi}(\mathbb{R}^N)\right\}.
$$

\begin{proposition}
The space $\mathscr{D}^{1,\Phi}(\mathbb{R}^N)$ equipped with the norm
\begin{equation}
\|u\|_{1,\Phi}=\|u\|_{\Phi_*}+\||\nabla u|\|_{\Phi}. \label{otnor}
\end{equation}
is complete.
\end{proposition}

\begin{proof}
Let $\{u_n\}$ be a Cauchy sequence in $\mathscr{D}^{1,\Phi}(\mathbb{R}^N)$; that is,
\begin{equation}
\|u_n-u_m\|_{\Phi_*}\to 0\quad\text{and}\quad\||\nabla u_n-\nabla u_m|\|_{\Phi}\to 0
\label{cause}
\end{equation}
as $n,m\to +\infty$. Since $L_{\Phi_{\ast}}(\mathbb{R}^N)$ is a Banach space we
can find $u\in L_{\Phi_{\ast}}(\mathbb{R}^N)$ such that $u_n\to u$ in
$L_{\Phi_{\ast}}(\mathbb{R}^N)$. The second condition in \eqref{cause} implies that
$\left\{\partial_{x_i} u_n\right\}$ is a Cauchy sequence in $L_{\Phi}(\mathbb{R}^N)$.
Then for each index $i=1,\dots,N$ there exists $\omega_i\in L_{\Phi}(\mathbb{R}^N)$
such that $\partial_{x_i} u_n\to \omega_i$ in $L_{\Phi}(\mathbb{R}^N)$.
Since $\partial_{x_i} u_n$ is the weak derivative of $u_n$ we have
$\partial_{x_i} u_n\in L_{\Phi}(\mathbb{R}^N)$. Then
$$
-\int_{\mathbb{R}^N} u_n\,\partial_{x_i} \psi\,dx=\int_{\mathbb{R}^N}
\partial_{x_i} u_n\,\psi\,dx
$$
for all $\psi\in\mathcal{D}(\mathbb{R}^N)$. H\"{o}lder's inequality \eqref{hol}
and uniqueness of limits yield
$$
-\int_{\mathbb{R}^N} u\,\partial_{x_i} \psi\,dx
=\int_{\mathbb{R}^N} \omega_i\,\psi\,dx.
$$
Thus, we get $\partial_{x_i} u=\omega_i\in L_{\Phi}(\mathbb{R}^N)$ and
$\|u_n-u\|_{1,\Phi}\to 0$ as $n\to +\infty$.
\end{proof}

\begin{definition} \label{def4.1} \rm
The homogeneous Orlicz space $\mathscr{D}^{1,\Phi}_{\rm o}(\mathbb{R}^N)$
is the completion of $\mathcal{D}(\mathbb{R}^N)$ with respect to the
norm \eqref{otnor}. Equivalently,
$$
\mathscr{D}^{1,\Phi}_{\rm o}(\mathbb{R}^N)
=\operatorname{cl}_{{\mathscr{D}^{1,\Phi}}(\mathbb{R}^N)}\mathcal D(\mathbb{R}^N)
$$
where $\operatorname{cl}_{{\mathscr{D}^{1,\Phi}}(\mathbb{R}^N)}$ denotes the closure
operator.
\end{definition}

The space $\mathscr{D}^{1,\Phi}_{\rm o}(\mathbb{R}^N)$ is endowed with the
 seminorm
\begin{equation}
\|u\|_{{\rm o},\Phi}=\||\nabla u|\|_{\Phi}.\label{nrara}
\end{equation}

\begin{lemma} \label{lem4.1}
On $\mathscr{D}^{1,\Phi}_{\rm o}(\mathbb{R}^N)$ the seminorm \eqref{nrara}
 defines a norm which is equivalent to \eqref{otnor}.
\end{lemma}

\begin{proof}
By \cite[Theorem 3.4]{dt}, if $u\in\mathcal{D}(\mathbb{R}^{N})$ then
\begin{equation}
\|u\|_{\Phi_*}\leq\mathscr C(N)\,\||\nabla u|\|_{\Phi}
=\mathscr C(N)\,\|u\|_{{\rm o},\Phi}
\label{comis}
\end{equation}
where  $\mathscr C(N)$ is a positive constant. This inequality extends to all of
 $\mathscr D^{1,\Phi}_{\rm o}(\mathbb{R}^{N})$ by density.
\end{proof}

We remark that since
$\mathcal{D}(\mathbb{R}^N)\subseteq \mathscr{D}^{1,\Phi}_{\rm o}(\mathbb{R}^N)$,
the inclusions
\begin{equation}\label{inc}
W^1L_\Phi(\mathbb{R}^N)\subseteq\mathscr{D}^{1,\Phi}_{\rm o}
(\mathbb{R}^N)\subseteq\mathscr{D}^{1,\Phi}(\mathbb{R}^N)
\end{equation}
hold. Example \ref{ejo1} below proves that there exist $N$-functions $\Phi$ for
which the inclusion 
$W^1 L_\Phi (\mathbb{R}^N) \subseteq \mathscr{D}^{1,\Phi}(\mathbb{R}^N)$
is strict.

The following theorem is the main result in this article.

\begin{theorem}\label{li}
Assume that there exists $1<r<N$ such that estimates \eqref{hypop} are fulfilled.
If $q_\Phi<N$ then the reversed inclusion
$\mathscr{D}^{1,\Phi}(\mathbb{R}^N)\subseteq\mathscr D^{1,\Phi}_{\rm o}
(\mathbb{R}^{N})$  holds as well. That is,
$$
\mathscr{D}^{1,\Phi}_{\rm o}(\mathbb{R}^N)
=\{u\in L_{\Phi_{\ast}}(\mathbb{R}^N): |\nabla u| \in L_{\Phi}(\mathbb{R}^N)\}.
$$
\end{theorem}

\begin{proof}
Take $u\in \mathscr D^{1,\Phi}(\mathbb{R}^{N})$ and define 
$\omega\in\mathcal{D}(\mathbb{R}^{N})$ by
\[
\omega(x)= \begin{cases}
0 & \text{if } |x|\geq 2, \\
1 & \text{if } |x|\leq 1.
\end{cases}
\]
Next, form the functions
$$
\omega_{k}(x)=\omega\left(\frac{x}{k}\right)\quad\text{and}\quad
u_k(x)=u(x)\,\omega_k(x),\quad k\in\mathbb{N}.
$$
For each fixed $k\in\mathbb{N}$ we consider the sequence of regularized functions
$v_{n}^k=\rho_{1/n}\,*\,u_k$, $n\in\mathbb{N}$, where
$\rho_{1/n}(x)=(1/n)^{-N}\rho(nx)$ and $\rho$ is the mollifier satisfying
(a), (b) and (c) in \S \ref{molli}.
Note that as $u_k$ has compact support the convolution
$v_{n}^k\in \mathcal D(\mathbb{R}^N)$. Moreover, since
$\partial_{x_i} v_{n}^k=\rho_{1/n}\,*\,\partial_{x_i} u_k\in E_{\Phi}(\mathbb{R}^N)$,
Lemma \ref{181819} implies
$$
\|\partial_{x_i} v_{n}^k-\partial_{x_i} u_k\|_\Phi \to 0\quad\text{as }
n\to +\infty.
$$
Then, for $k\in\mathbb{N}$, we have
$$
\lim_{n\to +\infty}\||\nabla v_{n}^k-\nabla u_k|\|_{\Phi}=0.
$$
For every natural number $k$, Cantor's diagonalization method produces an
integer $n_k\in\mathbb{N}$ (which depends only on $k$) such that if we set
 $v_k=v_{n_k}^k=\rho_{1/n_k}\,*\,u_k$, then
$$
\|| \nabla v_{k}-\nabla u_k|\| _{\Phi}\leq \frac{1}{k},\quad k\in\mathbb{N}.
$$
The triangle inequality thus implies
$$
\| | \nabla v_{k}-\nabla u|\|_{\Phi}\leq \frac{1}{k}
+\| | \nabla u_k-\nabla u|\|_{\Phi}.
$$
We must prove that
\begin{equation}\label{2122}
\lim_{k\to +\infty}\| | \nabla u_{k}-\nabla u|\|_{\Phi}=0.
\end{equation}

We note that the product rule yields
 $\partial_{x_i} u_k=u\,\partial_{x_i} \omega_k +\omega_k\,\partial_{x_i} u$
and hence
$$
\| | \nabla u_k-\nabla u |\|_{\Phi}\leq \|(1-\omega_k ) |\nabla
u|\|_{\Phi} +\| |u\,\nabla \omega_k | \|_{\Phi}.
$$
Since $\Phi$ is increasing,
$$
\int_{\mathbb{R}^{N}} \Phi \Big((1-\omega_k )\frac{|\nabla u|}{\lambda}
\Big)\,dx
\leq \int_{\mathbb{R}^{N}\backslash \overline{{B}_k(0)}}
\Phi\Big(\frac{|\nabla u|}{\lambda} \Big)\,dx
$$
where the parameter $\lambda>0$ is arbitrary. Note that
$L_{\Phi}(\mathbb{R}^N)=\mathcal L_{\Phi}(\mathbb{R}^N)$ since $\Phi$ satisfies a
$\Delta_2$-condition. Therefore $\Phi(|\nabla u|/\lambda)\in L^1(\mathbb{R}^N)$.
The definition of the Luxemburg norm thus implies
$$
\|(1-\omega_k)|\nabla u|\|_{\Phi}\to 0\quad\text{as}\quad k\to+\infty.
$$
To prove \eqref{2122} we need $\| |u\nabla \omega_k | \|_{\Phi}\to 0$ as
$k\to +\infty$. This is the case. Indeed, if $\varepsilon>0$ is sufficiently small,
there exists $s_0=s_0(\varepsilon)$ such that items (a) and (b) from
Lemma \ref{lamos} will be satisfied for all $0<s<s_0$.
Also, note that \eqref{crelen} implies
\begin{equation}
\mathsf{C}:=\sup_{s\geq s_0}\,\frac{\Phi(s)}{\Phi_*(s)}<+\infty.
\label{huye}
\end{equation}
We define the sets $\Omega_1=\{x\in\mathbb{R}^N:|u(x)|< s_0\}$ and
$\Omega_2 = \{x\in\mathbb{R}^N:|u(x)|\geq s_0\}$ and take the closed annulus
$A_k=\overline{B_{2k}(0)}\backslash B_k(0)\subseteq\mathbb{R}^N$.
Choose $\lambda$ positive and denote by $M=\sup_{\mathbb{R}^N}\partial_{x_i}\omega$.
We take $k$ sufficiently large such that $k>M/\lambda$. The monotonicity of $\Phi$
and \eqref{huye} yield
\begin{equation}
\begin{aligned}
&\int_{A_k}\Phi\Big(\frac{1}{\lambda}|\partial_{x_i} w_k| |u| \Big)\,dx\\
&= \int_{A_k\cap \Omega_1}\Phi\Big(\frac{1}{\lambda k} |\partial_{x_i} w| |u|
\Big)\,dx
+\int_{A_k\cap \Omega_2}\Phi\Big(\frac{1}{\lambda k} |\partial_{x_i} w|\, |u| \Big)
\,dx \\
&\leq \int_{A_k\cap \Omega_1}\Phi\left(\frac{M}{\lambda k} |u|
\right)\,dx +\int_{A_k\cap \Omega_2}\Phi(|u|)\,dx \\
&\leq \int_{A_k\cap \Omega_1}\Phi\Big(\frac{M}{\lambda k} |u|
\Big)\,dx
+\mathsf{C}\int_{A_k}\Phi_*(|u|)\,dx.
\end{aligned}\label{altiro}
\end{equation}
Since $u\in L_{\Phi_*}(\mathbb{R}^N)$ it is evident that
$\int_{A_k}\Phi_*(|u|)\,dx\to 0$ as $k\to+\infty$.

Note that the choice of $k$ above implies that $M|u|/\lambda k<s_0$ on $\Omega_1$.
Item (a) in Lemma \ref{lamos} yields the following estimate for the integral on
the right-hand side in \eqref{altiro},
\begin{equation}
\int_{A_k\cap \Omega_1}\Phi\Big(\frac{M}{\lambda k} |u|\Big)\,dx
\leq (\mathsf{B}+\varepsilon)\frac{M^r}{r\lambda^r k^r}
\int_{A_k\cap \Omega_1}|u|^r\,dx.
\label{fract}
\end{equation}
Since $\Phi_*$ satisfies a global $\Delta_2$-condition,
 $\Phi_*(|u|)\in L_1(A_k\cap\Omega_1)$. Item (b) in Lemma \ref{lamos} yields
$$
\mathscr{A}(r,\varepsilon) |u|^{\frac{Nr}{N-r}}\leq\Phi_*(|u|)
$$
where $\mathscr{A}(r,\varepsilon)$ is positive. Therefore
$|u|^r\in L^{\frac{N}{N-r}}(A_k\cap \Omega_1)$ and then H\"{o}lder's inequality,
with $p=N/(N-r)$ and $q=N/r$, implies
\begin{align*}
\int_{A_k\cap \Omega_1}|u|^r\,dx
&\leq \big(\operatorname{meas}(A_k\cap\Omega_1)\big)^{r/N}
\Big(\int_{A_k\cap\Omega_1}|u|^{\frac{Nr}{N-r}}\,dx\Big)^{\frac{N-r}{N}}\\
&\leq  \big(\operatorname{meas}(\overline{B_{2k}(0)})\big)^{r/N}
\Big(\int_{A_k\cap\Omega_1}|u|^{\frac{Nr}{N-r}}\,dx\Big)^{\frac{N-r}{N}}
\end{align*}
where $\operatorname{meas}(\overline{B_{2k}(0)})=\pi^{N/2}(2k)^N/\Gamma(N/2+1)$
is the volume of the closed ball $\overline{B_{2k}(0)}$ and $\Gamma$ is Euler's
gamma function. Thus, we obtain
$$
\int_{A_k\cap \Omega_1}|u|^r\,dx
\leq \mathscr B k^r\Big(\int_{A_k\cap
\Omega_1}|u|^{\frac{Nr}{N-r}}\,dx\Big)^{\frac{N-r}{N}}
$$
where $\mathscr B=\mathscr B(r,N)$ is a positive constant. Therefore,
estimate \eqref{fract} yields
$$
\int_{A_k\cap \Omega_1}\Phi\Big(\frac{M}{\lambda k} |u|\Big)dx
\leq \mathscr B\cdot(\mathsf{B}+\varepsilon)\frac{M^r}{r\lambda^r}
\Big(\int_{A_k\cap\Omega_1}|u|^{\frac{Nr}{N-r}}\,dx\Big)^{\frac{N-r}{N}}.
$$
Since the integral on the right tends to $0$ as $k\to+\infty$, from \eqref{altiro}
we obtain
 $$
\int_{A_k}\Phi\Big(\frac{1}{\lambda} |\partial_{x_i} w_k| |u| \Big)\,dx\to 0
\quad\text{as } k\to +\infty.
$$
The definition of the Luxemburg norm thus ensures
$\| |u\nabla \omega_k | \|_{\Phi}\to 0$ as $k\to +\infty$ and hence
\eqref{2122} holds.

To conclude the proof we must show that $\|v_k -u\|_{\Phi_*}\to 0$ as
$k\to+\infty$. Notice that
 $v_k-u\in \mathscr D^{1,\Phi}(\mathbb{R}^{N})\cap L^1(\mathbb{R}^N)$
and hence inequality \eqref{comis} does not apply in this case.
 We proceed as follows, instead. The triangle inequality and Proposition \ref{25a3}
yield
\begin{align*}
\|v_k -u\|_{\Phi_*}
&= \|\rho_{1/n_k}*u_k -u\|_{\Phi_*} \\
&\leq \|\rho_{1/n_k}*(\omega_k\,u-u)\|_{\Phi_*}+\|\rho_{1/n_k}*u-u\|_{\Phi_*} \\
&\leq \|\omega_k\,u-u\|_{\Phi_*}+\|\rho_{1/n_k}*u-u\|_{\Phi_*}.
\end{align*}
Since $\Phi_*$ satisfies a global $\Delta_2$-condition we have
$\omega_k u-u\in \mathscr D^{1,\Phi}(\mathbb{R}^N)
\subseteq L_{\Phi_{*}}(\mathbb{R}^N)
=E_{\Phi_{*}}(\mathbb{R}^N)$. Lemma \ref{facil} (with $z_k=\omega_k u-u$) produces
$\|\omega_k\,u-u\|_{\Phi_*}\to 0$ as $k\to+\infty$. Lemma \ref{181819}
in turn implies that $\|\rho_{1/n_k}*u-u\|_{\Phi_*}\to 0$ as $k\to+\infty$
and hence the inequality above ensures that $v_k \to u$ in
 $L_{\Phi_*}(\mathbb{R}^N)$. Along with \eqref{2122}, the latter implies
 $\|v_k-u\|_{1,\Phi}\to 0$ as $k\to+\infty$. The proof of the theorem is complete.
\end{proof}

\begin{example} \label{examp4.1} \rm
 We define 
\[
\phi_1(s)= \frac{|s|^{p-2}s}{\log(1+|s|)},
\]
 where $p>2$. In this case, 
\[
\Phi_1(s)=\int_0^s\phi_1(t)\,dt
=\frac{|s|^{p}}{p\log(1+|s|)}+\frac{1}{p} \int_0^{|s|}
\frac{t^{p}}{(1+t)(\ln (1+t))^2}\,dt.
\]
If we take $\alpha=p-1$ and $\beta=1$ in \cite[Example III]{clore}, 
then we obtain 
$$
p_{\Phi_1}=\inf_{s>0} \frac{s\phi_1(s)}{\Phi_1(s)}
= p-1\quad{\text{and}}\quad q_{\Phi_1}=\sup_{s>0} \frac{s\phi_1(s)}{\Phi_1(s)}= p.
$$ 
By Lemma \ref{deldos}, $\Phi_1$ satisfies a $\Delta_2$-condition. 
Since $p>2$ estimate \eqref{Delta4} is also fulfilled (i.e. the complementary 
$N$-function $\overline{\Phi_1}$ satisfies a $\Delta_2$-condition). 
On the other hand, the choice $r=p-1$ and L'H\^{o}pital's rule yield 
$$
\liminf_{s\to 0^+} \frac{\phi_1(s)}{s^{r-1}}
=\limsup_{s\to 0^+} \frac{\phi_1(s)}{s^{r-1}}
=\lim_{s\to 0^+} \frac{\phi_1(s)}{s^{r-1}}
=\lim_{s\to 0^+} \frac{s}{\log(1+s)}=1.
$$ 
Conditions \eqref{hypop} are met in this case and hence Theorem \ref{li} 
implies $\mathscr{D}^{1,\Phi_1}(\mathbb{R}^N)=\mathscr D^{1,\Phi_1}_{\rm o}
(\mathbb{R}^{N})$.
\end{example}

\begin{example} \label{examp4.2} \rm
Consider the function $\phi_2(s)=|s|^{p-2}s\,\log(1+\mu+|s|)$ where 
$p>1$ and $\mu> 0$ is a parameter. A simple calculation shows that
$$
\Phi_2(s)=\int_0^s\phi_2(t)\,dt=\frac{|s|^{p}}{p}\log(1+\mu+|s|)
-\frac{1}{p} \int_0^{|s|}\frac{t^{p}}{1+\mu+t}\,dt.
$$ 
For values $s>0$ we consider the differentiable function 
$$
g_{\mu}(s)=\frac{\int_0^s\frac{t^p}{1+\mu+t}dt}{s^p\log(1+\mu+s)}.
$$ 
A simple application of L'H\^{o}pital's rule proves that $g_{\mu}(s)\to 0$ as 
$s\to 0$ and also $g_{\mu}(s)\to 0$ as $s\to +\infty$. Since 
$$
s^p\log(1+\mu+s)= p\int_0^s t^{p-1}\log(1+\mu+t)dt+\int_0^s\frac{t^p}{1+\mu+t}dt
$$ 
it is evident that $0<g_{\mu}(s)< 1$ if $s>0$. It follows that 
$$
\frac{s\phi_2(s)}{\Phi_2(s)}=\frac{p}{1-g_{\mu}(s)}\geq \lim_{s\to 0^+} 
\frac{s\phi_2(s)}{\Phi_2(s)}=p
$$ 
for all $s>0$. Therefore
\begin{equation}
p_{\Phi_2}=\inf_{s>0} \frac{s\phi_2(s)}{\Phi_2(s)}
=\lim_{s\to 0^+} \frac{s\phi_2(s)}{\Phi_2(s)}=p.
\label{minp}
\end{equation}
On the other hand, the implicit function theorem allows to determine a 
local maximum of $g_{\mu}$ at $s=s^*>0$ from the equation 
$$
s^{p+1}\log(1+\mu+s)=\Big(\int_0^s \frac{t^p}{1+\mu+t}dt\Big)
\Big(p(1+\mu+s)\log(1+\mu+s)+s\Big).
$$ 
The condition $g_{\mu}(s)\to 0$ as $s\to +\infty$ ensures that 
$s^*$ is also global. Therefore, 
$$
q_{\Phi_2}=\sup_{s>0} \frac{s\phi_2(s)}{\Phi_2(s)}
=\max_{s>0} \frac{s\phi_2(s)}{\Phi_2(s)}=\frac{p}{1-g_{\mu}(s^*)}<+\infty.
$$ 
By Lemma \ref{deldos}, $\Phi_2$ satisfies a $\Delta_2$-condition. 
Bound \eqref{minp} implies that estimate \eqref{Delta4} is also fulfilled 
in this case (i.e. $\overline{\Phi_2}$ satisfies a $\Delta_2$-condition). 
Furthermore, if we choose $r=p$ then 
$$
0<\liminf_{s\to 0^+} \frac{\phi_2(s)}{s^{r-1}}
=\limsup_{s\to 0^+} \frac{\phi_2(s)}{s^{r-1}}=\lim_{s\to 0^+} 
\frac{\phi_2(s)}{s^{r-1}}=\log(1+\mu)<+\infty.
$$ 
Hence conditions \eqref{hypop} are fulfilled. Theorem \ref{li} yields 
$\mathscr{D}^{1,\Phi_2}(\mathbb{R}^N)=\mathscr D^{1,\Phi_2}_{\rm o}(\mathbb{R}^{N})$.
\end{example}

\begin{example}\label{ejo1}\rm
This example proves that there exists an $N$-function $\Phi$ for which the
 corresponding Orlicz-Sobolev space $W^1L_{\Phi}(\mathbb{R}^N)$ is in general 
a proper subset of $\mathscr D^{1,\Phi}(\mathbb{R}^{N})$. Consider $p>1$ 
and set the real homeomorphism $\phi(t)=|t|^{p-2}t$. Let us define a 
function $$u(x)=(1+\|x\|^2)^{-s}$$ where $\|x\|$ is the Euclidean norm of 
$x\in\mathbb{R}^N$ and $s$ is a positive quantity to be fixed later. 
It is easy to see that
\[
|\nabla u(x)|= \frac{2s\|x\|\quad}{(1+\|x\|^2)^{s+1}}.
\]
We take spherical coordinates 
$\mathbf{F}:(x_1,\dots,x_N)\to(\rho,\varphi_1,\ldots,\varphi_{N-1})$ in 
$\mathbb{R}^N$ defined by
\begin{gather*}
x_1 = \rho\cos\varphi_1\\
x_i = \rho\sin\varphi_1\sin\varphi_2\dots\sin\varphi_{i-1}\cos\varphi_{i},\quad 
i=2,\dots,N-1\\
x_N = \rho\sin\varphi_1\sin\varphi_2\dots\sin\varphi_{N-2}\sin\varphi_{N-1}
\end{gather*}
where $\rho=(x_1^2+\ldots+x_N^2)^{1/2}$ and $\varphi_i\in[0,\pi]$ for 
$i=1,\dots, N-2$ and $\varphi_{N-1}\in[0,2\pi]$. 
A simple computation yields the Jacobian:
\begin{align*}
\mathbf{J}_{\mathbf{F}}(\rho,\varphi_1,\ldots,\varphi_{N-1})
&=\frac{\partial(x_1,x_2,\ldots,x_N)}{\partial(\rho,\varphi_1,\ldots,
\varphi_{N-1})} \\
&=\rho^{N-1}(\sin\varphi_1)^{N-2}\,(\sin\varphi_2)^{N-3}
\dots(\sin\varphi_{N-3})^{2}\,\sin\varphi_{N-2}.
\end{align*}
Let us define the integral 
$$
I:= \int_{\mathbb{R}^N\backslash
B_1(0)}\frac{dx}{(1+\|x\|^2)^{sr}}
$$ 
where $1<r<N$. (Obviously, $u^r\in L^1(\mathbb{R}^N)$ if and only if $I$ is finite). 
Change to spherical coordinates and further integration yields
\begin{align*}
I &= \int_{1}^{+\infty}\int_0^{2\pi}\int_0^{\pi}\dots
 \int_0^{\pi}\,\frac{\mathbf{J}_{\mathbf{F}}(\rho,\varphi_1,
 \ldots,\varphi_{N-1})}{(1+\rho^2)^{sr}}\,d\varphi_{1}\ldots
  d\varphi_{N-2}\,d\varphi_{N-1}\,d\rho \\
&=\mathscr{C}\int_{1}^{+\infty}\frac{\rho^{N-1}}{(1+\rho^2)^{sr}}\,d\rho
\end{align*}
where $\mathscr{C}$ depends on 
$\int_0^{\pi}\sin^{k}\varphi_{N-k-1}\,d\varphi_{N-k-1}$, for all index 
$k=1,\dots, N-2$. The limit comparison test for improper integrals yields
$$
\int_{1}^{+\infty}\frac{\rho^{N-1}}{(1+\rho^2)^{sr}}\,d\rho<+\infty
$$
if and only if $N<2sr$. If we set $r=p$ in the latter inequality, we obtain 
that convergence of the integral is equivalent to the condition $s>N/2p$. 
Thus if $s\leq N/2p$ we get $u\not\in L^p(\mathbb{R}^N)$. Likewise, in the 
particular case $r=p^*=Np/(N-p)$, convergence of the integral means $s>(N-p)/2p$. 
Therefore, 
$$
u\not\in L^p(\mathbb{R}^N)\text{  and   }u\in L^{p^*}(\mathbb{R}^N)
\text{  if and only if  } s\in \Big(\frac{N-p}{2p},\frac{N}{2p}\Big].
$$

The same argument we employed above proves that
$$
J:= \int_{\mathbb{R}^N\backslash B_1(0)}{|\nabla u|}^p\,dx
={(2s)}^{p}\mathscr{C}\int_{1}^{+\infty}
 \frac{\rho^{N+p-1}}{(1+\rho^2)^{(s+1)p}}\,d\rho.
$$ 
Hence, the integral $J$ is finite if and only if $N+p-2sp-2p<0$. 
That is, 
$$
|\nabla u|\in L^p(\mathbb{R}^N)\text{  if and only if  }
 s\in \Big(\frac{N-p}{2p},+\infty\Big).
$$ 
We conclude that $u\in\mathscr{D}^{1,\Phi}(\mathbb{R}^{N})$ and 
$u\not\in W^1L_{\Phi}(\mathbb{R}^{N})$ (with $\Phi(t)={|t|}^p/p$) 
provided the parameter $s\in \big((N-p)/2p,N/2p\big]$.
\end{example}

\section{Application}

In this section the number $p_\Phi$ defined in \eqref{Delta4} plays a paramount role. 
We prove existence of nontrivial and non-negative solutions of equation 
\eqref{equa3} under the assumptions made at the beginning of Section \ref{hosect}. 
Additionally we will require the following hypotheses:
\begin{itemize}
\item[(H0)] Condition \eqref{Delta4} is fulfilled 
(i.e. $\overline{\Phi}$ satisfies a $\Delta_2$-condition);

\item[(H1)] $q_\Phi< N$ and $q_\Phi<p_\Phi^*=p_\Phi\, N/(N-p_\Phi)$ 
(the conjugate exponent);

\item[(H2)] $g\in L^{q_\Phi^*/(q_\Phi^*-p_\Phi)}(\mathbb{R}^N)\cap
L^\infty(\mathbb{R}^N)$ and the positive part $g^+\not\equiv 0$.
\end{itemize}

We define functionals 
$$
I(u)=\int_{\mathbb{R}^N}\Phi(|\nabla u|)dx\quad\text{and}\quad 
G(u)=\int_{\mathbb{R}^{N}}g(x)\Phi(u)dx.
$$ 
Since $\Phi$ satisfies a global $\Delta_2$-condition, the functional $I$ 
is well-defined on $\mathscr{D}^{1,\Phi}_{\rm o}(\mathbb{R}^{N})$ and 
real-valued there. Further, \cite[Lemma A.3]{fin} ensures that $I$ 
is of class $C^1$ with Fr\'echet derivative 
$$
{I}'(u)(v)= \int_{\mathbb{R}^N}\phi(|\nabla u|) 
\frac{\nabla u}{|\nabla u|}\cdot\nabla vdx.
$$ 
Application of the same lemma (with the term $f(x,t)=g(x)\phi(t)$ 
in \eqref{ante}) shows that $G$ is real-valued on 
$\mathscr{D}^{1,\Phi}_{\rm o}(\mathbb{R}^{N})$ and that 
$G:\mathscr D^{1,\Phi}_{\rm o}(\mathbb{R}^N)\to\mathbb{R}$ is of class $C^1$ 
as well with Fr\'echet derivative 
$$
{G}'(u)(v)=\int_{\mathbb{R}^N}g(x)\phi(u)vdx
$$ 
where $u,v\in \mathscr{D}^{1,\Phi}_{\rm o}(\mathbb{R}^{N})$.

\begin{proposition}\label{88}
Let $\{u_n\}$ be a sequence in $\mathscr{D}^{1,\Phi}_{\rm o}(\mathbb{R}^N)$ such 
that $u_n \rightharpoonup u$ (weak convergence). Then there exists a 
subsequence denoted again by $\{u_n\}$ such that $G(u_{n})\to G(u)$.
\end{proposition}

\begin{proof}
By definition there exists $d'>0$ such that 
$\|u_n\|_{\Phi_*}\leq\mathscr C(N)\, \|u_n\|_{{\rm o},\Phi}\leq d'$ 
for all $n\in\mathbb{N}$, where $\mathscr C(N)$ is the constant in \eqref{comis}. 
Choose $R>0$ and let $B_R$ be a ball of radius $R$ centered at $0$. 
For each natural number $n$ we have $G(u_{n})-G(u)=I_{n}^R+J_{n}^R$, where
$$
I_{n}^R=\int_{B_R}g(x)\,(\Phi(u_{n})-\Phi(u))\,dx,\quad
J_{n}^R=\int_{\mathbb{R}^N\backslash B_R}g(x)\,(\Phi(u_{n})-\Phi(u))\,dx.
$$
Let us define $A_{R,n}=\{x\in\mathbb{R}^N\backslash B_R : 0\leq u_n(x)\leq 1\}$ 
and $C_{R,n}=\{x\in\mathbb{R}^N\backslash B_R : u_n(x)\geq 1\}$. 
Let $\sigma=q_\Phi^*/(q_\Phi^*-p_\Phi)$. Items (a) and (b) in 
Proposition \ref{pelo} applied with $\rho=u_n\in A_{R,n}$ and $t=1$ yield 
$$
|\Phi(u_n)|^{q^*_\Phi/p_\Phi}
\leq |u_n^{p_\Phi}\Phi(1)|^{q^*_\Phi/p_\Phi}
=|u_n|^{q^*_\Phi}(\Phi(1))^{q^*_\Phi/p_\Phi}
\leq\frac{(\Phi(1))^{q^*_\Phi/p_\Phi}}{\Phi_*(1)}\Phi_*(u_n).
$$ 
Hence Holder's inequality produces
\begin{align*}
\int_{A_{R,n}}|g\Phi(u_n)|\,dx 
&\leq  \Phi(1)\Big(\int_{A_{R,n}}|g|^\sigma\,dx\Big)^{1/\sigma}
\Big(\int_{A_{R,n}}|u_n|^{q_\Phi^*}\,dx\Big)^{p_\Phi/q_\Phi^*} \\
&\leq C_1\Big(\int_{\mathbb{R}^N\backslash B_R}|g|^\sigma\,dx\Big)^{1/\sigma}
\Big(\int_{\mathbb{R}^N}\Phi_*(u_n)\,dx\Big)^{p_\Phi/q_\Phi^*}
\end{align*}
where $C_1=\Phi(1)/(\Phi_*(1))^{p_\Phi/q^*_\Phi}$. Since 
$\sigma\leq p_\Phi^*/(p_\Phi^*-q_\Phi)$ by interpolation we have 
$g\in L^{p_\Phi^*/(p_\Phi^*-q_\Phi)}(\mathbb{R}^N)$ as well. 
If $u\in C_{R,n}$ then analogue arguments as the ones used above yield
\[
\int_{C_{R,n}}|g\Phi(u_n)|\,dx 
\leq C_2 \Big(\int_{\mathbb{R}^N\backslash B_R}|g|^{\sigma^*}\,dx\Big)^{1/\sigma^*}
\Big(\int_{\mathbb{R}^N}\Phi_*(u_n)\,dx\Big)^{q_\Phi/p_\Phi^*}
\]
where $\sigma^*=p_\Phi^*/(p_\Phi^*-q_\Phi)$ and $C_2>0$. 
Since $\|u_n\|_{\Phi_*}\leq d'$ the integral $\int_{\mathbb{R}^N}\Phi_*(u_n)\,dx$ 
is bounded and then the two inequalities above imply 
$$
\int_{\mathbb{R}^N}|g\Phi(u_n)|\,dx<+\infty.
$$ 
Thus, given $\varepsilon>0$, there exists $R_0=R_0(\varepsilon)>0$ such that
$$
\int_{\mathbb{R}^N\backslash B_{R_0}}|g\Phi(u_n)|\,dx<\varepsilon/4.
$$
One can similarly prove that
$$
\int_{\mathbb{R}^N\backslash B_{R_1}}|g\Phi(u)|\,dx<\varepsilon/4
$$
for $R_1$ large enough. Thus, if $R_2=\max\{R_0,R_1\}$ then we have 
$|J_n^{R_2}|<\varepsilon/2$ for $n\in\mathbb{N}$.

Let us study now $I^{R_2}_n$. Since the injection 
$L_{\Phi^*}(B_{R_2})\hookrightarrow L_{\Phi}(B_{R_2})$ is continuous 
(see \cite[Theorem 8.16]{Ad}) the inclusions \eqref{inc} yield 
$u_n,u\in W^1L_\Phi(B_{R_2})$ and hence there exist $d,\tilde d>0$ such that
$$
|\!|\!| u_n|\!|\!|_{B_{R_2}}\leq d \|u_n\|_{{\rm o},\Phi}\leq \tilde d
$$
for all $n\in\mathbb{N}$ where $|\!|\!|\cdot|\!|\!|_{B_{R_2}}$ is the norm 
\eqref{orso} on the ball $B_{R_2}$. Since the imbedding 
$W^1L_\Phi(B_{R_2})\hookrightarrow L_{\Phi}(B_{R_2})$ is compact 
(see \cite[Theorem 2.2]{glms}) we have $u_{n}\to u$ in
$L_{\Phi}(B_{R_2})$. Thus, passing to a subsequence (denoted by 
$\{u_n\}$ again) we can further assume that $u_{n}\to u$, a.e. in 
$B_{R_2}$ and that there exists $w\in L_{\Phi}(B_{R_2})$ such that 
$ |u_{n} |\leq w$, a.e. in $B_{R_2}$, for all $n\in\mathbb{N}$.
By Lebesgue's dominated convergence on $B_{R_2}$,
$$
\lim_{n\to +\infty}\int_{B_{R_2}}|\Phi(u_{n})-\Phi(u)|\,dx=0.
$$
Thus, for $n$ sufficiently large, 
$|I_{n}^{R_2}|\leq \|g\|_{\infty} \|\Phi(u_{n})-\Phi(u)\|_{L^1 (B_{R_2})}
\leq\varepsilon/2$. Since 
$|G(u_{n})-G(u)|\leq |I_{n}^{R_2}|+|J_{n}^{R_2}|$ the result is proved.
\end{proof}

\begin{lemma}[Lagrange multipliers rule \rm{\cite{FBr}}] \label{exis}
Let $v_0\in\mathscr{D}^{1,\Phi}_{\rm o}(\mathbb{R}^N)$ be such that \\
$G'(v_0)\neq 0$. 
If $I$ has a local minimum at $v_0$ with respect to the set 
$\{v:G(v)=G(v_0)\}$ then there exists $\lambda\in\mathbb{R}$ such that 
$I'(v_0)=\lambda G'(v_0)$.
\end{lemma}

Lagrange multipliers rule motivates the following definition. 
A pair $(\lambda,u)\in\mathbb{R}\times\mathscr{D}^{1,\Phi}_{\rm o}(\mathbb{R}^{N})$
is a solution of \eqref{equa3} if 
$\phi(|\nabla u|)\in L_{\overline{\Phi}}(\mathbb{R}^N)$ and
\[
\int_{\mathbb{R}^N}\phi(|\nabla u|)\frac{\nabla u}{|\nabla u|}\cdot\nabla\theta\,dx
=\lambda\int_{\mathbb{R}^N}g(x)\phi(u)\,\theta\,dx
\]
for all $\theta\in\mathscr{D}^{1,\Phi}_{\rm o}(\mathbb{R}^{N})$. 
If $(\lambda,u)$ is a solution of \eqref{equa3} and $u\not\equiv 0$ 
we call $\lambda$ an eigenvalue of \eqref{equa3} with corresponding eigenfunction 
$u$. That is, $\lambda$ is the eigenvalue associated to the eigenfunction $u$.
 Note that the inclusion on the right in \eqref{inc} ensures that any solution 
$u$ belongs to $L_{\Phi_*}(\mathbb{R}^N)$ and
 $ |\nabla u| \in L_{\Phi}(\mathbb{R}^N)$.

\begin{theorem}\label{mainth}
The optimization problem
\[
\inf_{{G}(u)=\mu>0}{I}(u)
\]
has a nontrivial solution $u_{\mu}\in\mathscr{D}^{1,\Phi}_{\rm o}(\mathbb{R}^{N})$. 
Define the nonzero number
\begin{equation}
\lambda_\mu
=\frac{\int_{\mathbb{R}^N}\phi(|\nabla u_\mu|)|\nabla
u_\mu|dx}{\int_{\mathbb{R}^N}g(x)\phi(u_\mu) u_\mu\,dx}. \label{forlammu}
\end{equation}
Then $u_\mu$ is a non-negative eigenfunction of equation \eqref{equa3} 
with associated eigenvalue $\lambda=\lambda_\mu$.
\end{theorem}

\begin{proof}
The first part is motivated by the ideas in the proof of \cite[Theorem 3.1]{Hu-Ma}. 
Compare also with the proof of \cite[Theorem 2.2]{MiRaRe}. 
We prove that for any $\mu>0$, the set 
$\mathcal{M}_{\mu}=\{u\in \mathscr{D}^{1,\Phi}_{\rm o}(\mathbb{R}^N):{G}(u)=\mu\}$ 
is not empty. Since $G(0)=0$, by continuity of $G$, it will be sufficient to 
find $\overline u\in \mathcal{D}(\mathbb{R}^N)$ such that $G(\overline u)\ge\mu$.

Since $g^+\not\equiv 0$ in $\mathbb{R}^N$ there exists a compact subset $K$ of 
$\mathbb{R}^N$, with $\operatorname{meas}(K)>0$, such that $g>0$ on $K$. 
If $r\in\mathbb{R}$ we define $u_r(x)=r\chi_{K}(x)$ where
 $\chi_K:\mathbb{R}^N\to\mathbb{R}$ is the characteristic function
$$
\chi_K(x)=\begin{cases}
1 &\text{if } x\in K,\\
0 &\text{if } x\in K^c.
\end{cases}
$$
We choose $r_0>0$ such that the number 
$\mu_0={G}(u_{r_0})-\mu=\Phi(r_0)\int_K g\,dx-\mu$ be strictly positive. 
Let $\Omega\subset\mathbb{R}^N$ be a bounded domain such that $K\subset\Omega$. 
Since the function 
$$
u\in L_{\Phi}(\Omega)\mapsto \Phi(u)\in L^1(\Omega)
$$ 
is continuous, we have that $\Phi(u_\varepsilon)$ converges to 
$\Phi(r_0\chi_{K})$ in $L^1(\Omega)$, as $\varepsilon\to 0^+$ where 
$u_\varepsilon\in\mathcal{D}(\Omega)$ is the regularized function of 
$r_0\chi_{K}$ and $\mathcal D(\Omega)$ denotes the space of $C^{\infty}$-functions 
with compact support in $\Omega$. H\"{o}lder's inequality yields 
$G(u_\varepsilon)\to\mu+\mu_0$ and hence we can choose $\varepsilon_0$ 
sufficiently small such that $G(\overline u)=G(u_{\varepsilon_0})\ge\mu$.

Denote by $\beta=\inf_{\mathcal M_{\mu}}{I}$ and let $\{u_{n}\}$ be a 
sequence in $\mathcal{M}_{\mu }$ such that
$$
\lim_{n\to +\infty}{I}(u_{n})=\beta.
$$
Hence, there exists $\mathcal{C}>1$ such that for each $n\in\mathbb{N}$,
$$
I(u_n)=\int_{\mathbb{R}^N}\Phi(|\nabla u_n|)dx\leq \mathcal{C}.
$$ 
Since $\Phi(u/t)\leq\Phi(u)/t$ for $t\geq 1$ (convexity), we get 
$$
\int_{\mathbb{R}^N}\Phi\Big(\frac{|\nabla u_n|}{\mathcal{C}}\Big)dx
\leq \int_{\mathbb{R}^N}\frac{\Phi(|\nabla u_n|)}{\mathcal{C}}dx\leq 1
$$ 
and by definition of the Luxemburg norm, 
$\|u_n\|_{{\rm o},\Phi}=\||\nabla u_n|\|_{\Phi}\leq\mathcal{C}$. 
That is, the minimizing sequence $\{u_n\}$ is bounded in 
$\mathscr{D}^{1,\Phi}_{\rm o}(\mathbb{R}^N)$. Inclusions \eqref{inc} 
imply that this space is a closed subspace of $L_{\Phi_*}(\mathbb{R}^N)$. 
Corollary \ref{nosel} proves that $\mathscr{D}^{1,\Phi}_{\rm o}(\mathbb{R}^N)$ 
is itself reflexive. Then there exists 
$u_\mu\in\mathscr{D}^{1,\Phi}_{\rm o}(\mathbb{R}^N)$ and a subsequence in 
$\mathcal{M}_{\mu}$, denoted again by $\{u_{n}\}$, such that 
$u_n \rightharpoonup u_\mu$ in the weak topology. As the function $G$ 
is sequentially continuous with respect to this weak topology, 
Proposition \ref{88} yields 
$$
G(u_\mu)= \lim_{n\to +\infty}{G}(u_{n})=\mu
$$ 
and hence $u_\mu\in\mathcal M_\mu$. Since the convex functional $I$ 
is continuously Fr\'echet-differentiable on 
$\mathscr{D}^{1,\Phi}_{\rm o}(\mathbb{R}^N)$ we obtain by 
\cite[Corollary III.8]{Br},
$$
\beta\le {I}(u_\mu)\leq\liminf_{n\to +\infty }{I}(u_{n})=\beta
$$
which is what we wanted to prove.

On the other hand, as $|g\Phi(u_\mu)|\leq |g\phi(u_\mu)u_\mu|$ and 
$\mu\neq 0$, we obtain both $g\phi(u_\mu)u_\mu\not\equiv 0$ and 
$g\phi(u_\mu)\not\equiv 0$ in $\mathbb{R}^N$. The latter implies that there 
exists $K'\subseteq \mathbb{R}^N$, with $\operatorname{meas}(K')>0$, 
such that $g\phi(u_\mu)\neq 0$ on $K'$ and the sign of $g\phi(u_\mu)$ on $K'$ 
is constant. Thus, for a suitable $r\in \mathbb{R}$,
$$
\int_{\mathbb{R}^N}g(x)\phi(u_\mu)r\chi_{K'}\,dx
 >\int_{\mathbb{R}^N}g(x)\phi(u_\mu)u_\mu\,dx
$$
where $\chi_{K'}$ is the characteristic function on $K'$. 
Since $g\phi(u_\mu)\in L_{\overline{\Phi}_*}(\mathbb{R}^N)$ and as the
regularized function $(r\,\chi_{K'})_{\varepsilon}\in\mathcal D(\mathbb{R}^N)$ 
converges to $r\chi_{K'}$ in $L_{\Phi_{*}}(\mathbb{R}^N)$,
$$
G'(u_\mu)(u_1)=\int_{\mathbb{R}^N}g(x)\phi(u_\mu)u_1\,dx
>\int_{\mathbb{R}^N}g(x)\phi(u_\mu)u_\mu\,dx=G'(u_\mu)(u_\mu)
$$
where $u_1=(r\,\chi_{K'})_\varepsilon$ for $\varepsilon>0$ sufficiently small.
 Notice that $G'(u_\mu)\not\equiv 0$ (otherwise, $0>G'(u_\mu)(u_\mu)=0$ in the 
above strict inequality). By Lemma \ref{exis} there exists
 $\lambda_u\in\mathbb{R}$ such that
\begin{equation}\label{112233}
 \int_{\mathbb{R}^N}\phi(|\nabla u_\mu|
) \frac{\nabla u_\mu}{|\nabla
u_\mu|}\cdot\nabla u\,dx=\lambda_\mu \int_{\mathbb{R}^N
}g(x)\phi(u_\mu)\,u\,dx
\end{equation}
for all $u\in\mathscr{D}^{1,\Phi}_{\rm o}(\mathbb{R}^N)$. Thus, $u_\mu$ 
is a weak solution of \eqref{equa3}. We then set $u=u_\mu$ in 
\eqref{112233} and we obtain the value of the eigenvalue in \eqref{forlammu}.

Since $\Phi$ is even it is clear that $G(|u_\mu|)=G(u_\mu)$. Moreover, 
the chain rule implies $|\nabla|u_\mu||=|\nabla u_\mu|$ and hence the 
equivalence $I(|u_\mu|)=I(u_\mu)$ follows as well. Therefore, we can take 
$u_\mu(x)\ge 0$ for a.e. $x\in \mathbb{R}^N$. The proof of the theorem is complete.
\end{proof}

\subsection*{Acknowledgements}
We are grateful to the anonymous referee who made several remarks and improved 
the list of references.

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