# Existence of positive solution for a class of quasilinear Schrödinger
equations with potential vanishing at infinity on nonreflexive Orlicz-Sobolev
spaces

Lucas da Silva∗ Unidade Acadêmica de Matemática
Universidade Federal de Campina Grande,
58429-970, Campina Grande - PB - Brasil<EMAIL_ADDRESS>and  Marco
Souto  Unidade Acadêmica de Matemática
Universidade Federal de Campina Grande,
58429-970, Campina Grande - PB - Brasil<EMAIL_ADDRESS>
###### Abstract.

In this paper we investigate the existence of positive solution for a class of
quasilinear problem on an Orlicz-Sobolev space that can be nonreflexive

$-\Delta_{\Phi}u+V(x)\phi(|u|)u=K(x)f(u)\mbox{ in }\mathbb{R}^{N},$

where $N\geq 2$, $V,K$ are nonnegative continuous functions and $f$ is a
continuous function with a quasicritical growth. Here we extend the Hardy-type
inequalities presented in [4] to nonreflexive Orlicz spaces. Through
inequalities together with a variational method for non-differentiable
functionals we will obtain a ground state solution. We analyze also the
problem with $V=0$.

###### Key words and phrases:

Orlicz-Sobolev spaces; Variational methods; Quasilinear Elliptic Problems;
$\Delta_{2}-$condition

###### 2020 Mathematics Subject Classification:

Primary: 35J15, 35J20, 35J62; Secondary: 46E30.

M. Souto was partially supported by CNPq/Brazil 309.692/2020-2

∗Corresponding author

##  1\. Introduction

This paper concerns the existence of a positive solution for a class of
quasilinear elliptic problem of the type

$None$ $\left\\{\;\begin{aligned}
-\Delta_{\Phi}u+V(x)\phi(|u|)u=K(x)f(u),\text{ in }\;\mathbb{R}^{N}\\\ u\in
D^{1,\Phi}(\mathbb{R}^{N}),\;u\geq 0,\text{ in
}\;\mathbb{R}^{N}\;\;\;\;\;\;\;\end{aligned}\right.$

for $N\geq 2$ and assuming that $V,K:\mathbb{R}^{N}\longrightarrow\mathbb{R}$
and $f:\mathbb{R}\longrightarrow\mathbb{R}$ are continuous functions with $V$,
$K$ being nonnegative functions and $f$ having a quasicritical growth. It is
important to recall that

$\Delta_{\Phi}u=\text{div}(\phi(|\nabla u|)\nabla u),$

where $\Phi:\mathbb{R}\longrightarrow\mathbb{R}$ is a $N$-function of the form

(1.1) $\Phi(t)=\int_{0}^{|t|}s\phi(s)ds,$

and $\phi:(0,\infty)\longrightarrow(0,\infty)$ is a $C^{1}$ function verifying
some technical assumptions. This type of problem driven by a $N$-function
$\Phi$ appears in a lot of physical applications, such as Nonlinear
Elasticity, Plasticity, Generalized Newtonian Fluid, Non-Newtonian Fluid and
Plasma Physics. The reader can find more details about this subject in [8],
[12], [25] and their references.

Recently, motivated by above applications, these types of problems have been
studied frequently, we would like to cite Bonanno, Bisci and Radulescu [10,
11], Cerny [29], Clément, Garcia-Huidobro and Manásevich [28], Donaldson [30],
Fuchs and Li [17], Fuchs and Osmolovski [18], Fukagai, Ito and Narukawa [9],
Gossez [16], Le and Schmitt [31], Mihailescu and Radulescu [19, 20],
Mihailescu and Repovs [21], Mihailescu, Radulescu and Repovs [22], Mustonen
and Tienari [32], Orlicz [33] and their references. In all of these works, the
so-called $\Delta_{2}$-condition was assumed on $\Phi$ and $\tilde{\Phi}$
(Complementary function of $\Phi$), which ensures that the Orlicz-Sobolev
space $W^{1,\Phi}(\Omega)$ and $D^{1,\Phi}(\Omega)$ are reflexive Banach
spaces. This assertion is used several times in order to get a nontrivial
solution for elliptic problems taking into account the weak topology and the
classical variational methods to $C^{1}$ functionals.

In recent years, problems without the $\Delta_{2}$-condition of the function
$\tilde{\Phi}$ are being studied. This type of problem brings us many
difficulties when we intend to apply variational methods. For example, in our
article, the energy functional associated with the problem might not be
$C^{1}$, so classical minimax type results cannot be used here. To get around
this difficulty, some recent articles suggest the use of the minimax theory
developed by Szulkin [3], we cite [5]. In this paper, Alves and Carvalho study
a class of problem

$\left\\{\;\begin{aligned} -\Delta_{\Phi}u+V(x)\phi(u)u&=f(u),\;\text{ in
}\mathbb{R}^{N}&\\\ u\in W^{1,\Phi}(\mathbb{R}^{N})&\text{ with }N\geq
1&\end{aligned}\right.$

when $V$ is $\mathbb{Z}^{N}$-periodic and $f$ is a continuous function
satisfying some technical conditions. The $\Delta_{2}$-condition of
$N$-function $\tilde{\Phi}$ has not been required.

Another work in which the $\Delta_{2}$-condition of $N$-function
$\tilde{\Phi}$ can be relaxed, is found in [7]. Silva, Carvalho, Silva and
Gonçalves study a class of problem

$\left\\{\;\begin{aligned} -\Delta_{\Phi}u&=g(x,u),\;\text{ in }\Omega&\\\
u&=0,\text{ on }\partial\Omega&\end{aligned}\right.$

where $\Omega\subset\mathbb{R}^{N}$, $N\geq 2$, is a bounded domain with
smooth boundary.

In this article, to overcome the lack of differentiability of the functional
$J$ we present a new approach to work with problems where the
$\Delta_{2}$-condition of $N$-function $\tilde{\Phi}$ is relaxed. Here, we
will use a weaker version of the mountain pass theorem for functionals that
are Gateaux differentiable. Another important point that comes up in this
paper is that we cannot use the standard analysis because
$D^{1,\Phi}(\mathbb{R}^{N})$ might not be reflexive. This difficulty brings us
many problems in order to apply variational methods. In order to overcome
these difficulties, we consider the weak∗ topology recovering some compactness
required in variational methods and one of the results involving the weak∗
topology is obtained in Lemma 3.4.

Based on the papers above, we assume that
$\phi:(0,\infty)\longrightarrow(0,\infty)$ is $C^{1}$ and satisfies the
following hypotheses:

$None$ $t\longmapsto t\phi(t)\;\text{ is increasing for }\;t>0.$ $None$
$\displaystyle\lim_{t\rightarrow 0^{+}}t\phi(t)=0\;\;\text{ and
}\;\;\displaystyle\lim_{t\rightarrow+\infty}t\phi(t)=+\infty.$ $None$
$1\leq\ell=\inf_{t>0}{\dfrac{\phi(t)t^{2}}{\Phi(t)}}\leq\sup_{t>0}{\dfrac{\phi(t)t^{2}}{\Phi(t)}}=m<N,\;\;\;m<\ell^{*}\text{
and }m\neq 1.$ $None$ $t\longmapsto\dfrac{\phi(t)}{t^{m-2}}\text{ is
nonincreasing for }t>0.$

Hereafter, we will say that $\Phi\in\mathcal{C}_{m}$, if there is a constant
$C>0$ satisfying

$None$ $\Phi(t)\geq C|t|^{m},\;\text{ for all }\;t\in\mathbb{R}.$

Next, we show some examples of functions $\Phi$ that can be considered in the
present paper. If $\ell>1$, we can consider

  1. $i)$

$\Phi(t)=|t|^{p}/p,$ for $p>1$,

  2. $ii)$

$\Phi(t)=|t|^{p}/p+|t|^{q}/q,$ where $1<p<q<N$ with $q\in(p,p^{*})$,

  3. $iii)$

$\Phi(t)=(1+t)^{\alpha}-1,$ $\alpha\in(1,\frac{N}{N-2})$

  4. $iv)$

$\Phi(t)=t^{p}\ln(1+t),$ $1<\frac{-1+\sqrt{1+4N}}{2}<p<N-1$, $N\geq 3$.

To the best of our knowledge, there seems to be no legitimate example of a
$N$-function that satisfies the conditions $(\phi_{1})-(\phi_{3})$ with
$\ell=1$ described in the literature. Lacking an illustrative example, we
present the function

$\displaystyle\Phi_{\alpha}(t)=|t|\ln(|t|^{\alpha}+1)\;\text{ for
}0<\alpha<\frac{N}{N-1}-1.$

as an example of $N$-function that satisfies $(\phi_{1})-(\phi_{3})$ for the
case $\ell=1$.

Here we would like to point out that if $1<\ell\leq m<+\infty$, then function
$\Phi$ and its complementary function $\tilde{\Phi}$ given by

$\tilde{\Phi}_{\alpha}(s)=\max_{t\geq
0}\\{st-\Phi_{\alpha}(t)\\},\leavevmode\nobreak\ \leavevmode\nobreak\ \text{
for }\;t\geq 0.$

satisfy the $\Delta_{2}$-condition. It is well known in the literature that
$D^{1,\Phi}(\mathbb{R}^{N})$ is reflexive when $\Phi$ and $\tilde{\Phi}$
satisfy the $\Delta_{2}$-condition. Thus, in our paper the space
$D^{1,\Phi}(\mathbb{R}^{N})$ can be nonreflexive, because we are also
considering the case $\ell=1$

The particular case of the problem $(P)$ with $\Phi(t)=|t|^{2}/2$ (The
Laplacian Case) is a study by Alves and Souto [4], where the authors show the
existence of ground state solutions to obtain a positive solution for the
following Schrödinger equation

$\left\\{\;\begin{aligned} -\Delta u+V(x)&u=K(x)f(u),&\;\mathbb{R}^{N}\\\
&u\in D^{1,2}(\mathbb{R}^{N})&\end{aligned}\right.$

Also in this paper, the authors impose some conditions under the potentials
$V$ and $K$ so that it would be possible to show that the space

$\displaystyle E=\Big{\\{}u\in
D^{1,2}(\mathbb{R}^{N})/\int_{\mathbb{R}^{N}}V(x)|u|^{2}dx<+\infty\Big{\\}}$

with norm

$\displaystyle\lVert u\lVert_{E}^{2}=\int_{\mathbb{R}^{N}}(|\nabla
u|^{2}+V(x)|u|^{2})dx$

is compactly embedded in the weighted Lebesgue space
$L^{p}_{K}(\mathbb{R}^{N})$ for some $p\in(2,2^{*})$, where

$\displaystyle
L^{p}_{K}(\mathbb{R}^{N}):=\left\\{u:\mathbb{R}^{N}\longrightarrow\mathbb{R}:u\text{
is measurable and}\int_{\mathbb{R}^{N}}K(x)|u|^{p}dx<+\infty\right\\}$

The purpose of this new paper is to extend the results presented by [4] to
Orlicz-Sobolev space where the conditions can be relaxed. For that, we
introduce the following assumptions on the potential $V$ and the coefficient
$K$:
$(K_{0})$ $V>0$, $K\in L^{\infty}(\mathbb{R}^{N})$ and $K$ is positive almost
everywhere.
(I) If $\\{A_{n}\\}\subset\mathbb{R}^{N}$ is a sequence of Borelian sets such
that $\displaystyle\sup_{n}|A_{n}|<+\infty$, then

$None$ $\lim_{r\rightarrow+\infty}\int_{A_{n}\cap
B_{r}^{c}(0)}K(x)dx=0,\;\text{ uniformly in }n\in\mathbb{N}.$

(II) One of the below conditions occurs:

$None$ $\dfrac{K}{V}\in L^{\infty}(\mathbb{R}^{N})$

or there are $a_{1},a_{2}\in(m,\ell^{*})$ and a $N$-function
$A(t)=\int_{0}^{|t|}sa(s)ds$ verifying the following properties:

(1.2) $\displaystyle a_{1}\leq\dfrac{a(t)t^{2}}{A(t)}\leq a_{2}$

and

$None$
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\dfrac{K(x)}{H(x)}\longrightarrow
0\text{ when }|x|\rightarrow+\infty}$

where $\displaystyle
H(x)=\min_{s>0}\left\\{V(x)\dfrac{\Phi(s)}{A(s)}+\dfrac{\Phi_{*}(s)}{A(s)}\right\\}$.
Hereafter, we say that $(V,K)\in\mathcal{K}_{1}$ if $(K_{0}),(K_{1})$ and
$(K_{2})$ hold. When $(K_{0}),(K_{1})$ and $(K_{3})$ hold, then we say that
$(V,K)\in\mathcal{K}_{2}$.

At this point, it is very important to note that $(K_{1})$ is weaker than any
one of the below conditions:
${(a)}$ There are $r\geq 1$ and $\rho\geq 0$ such that $K\in
L^{r}(\mathbb{R}^{N}\setminus B_{\rho}(0))$;
${(b)}$ $K(x)\rightarrow 0$ as $|x|\rightarrow\infty$;
${(c)}$ $K=H_{1}+H_{2}$, with $H_{1}$ and $H_{2}$ verifying $(a)$ and $(b)$
respectively.

Now, for every $n\in\mathbb{N}$, fix $z_{n}=(n,0,\cdots,0)$. Consider
$\left\\{B_{\frac{1}{2^{n}}}(z_{n})\right\\}_{n\in\mathbb{N}}$ the disjoint
sequence of open balls in $\mathbb{R}^{N}$ and the nonnegative function
$H_{1}:\mathbb{R}^{N}\longrightarrow\mathbb{R}$ given by

$\displaystyle 0\leq H_{1}(x)\leq 1,\;\forall
x\in\mathbb{R}^{N},\;\;H_{1}(z_{n})=1,\;\forall
n\in\mathbb{N},\;\;\;H_{1}\equiv 0\;\text{ in
}\;\mathbb{R}^{N}\setminus\Big{\\{}\bigcup_{n\in\mathbb{N}}B_{\frac{1}{2^{n}}}(z_{n})\Big{\\}}$

and

$\displaystyle\int_{B_{\frac{1}{2^{n}}}(z_{n})}H_{1}(x)dx\leq\dfrac{1}{2^{n}},\;\;\;\forall
n\in\mathbb{N}.$

Without any difficulties, we can see that the functions

(1.3) $\displaystyle V(x)=K(x)=H_{1}(x)+\dfrac{1}{\ln(2+|x|)}$

satisfies the item $(c)$ and consequently verifies the condition $(K_{1})$. In
addition, clearly $V$ and $K$ satisfy the conditions $(K_{0})$ and $(K_{2})$.
However, these functions do not verify the condition $(K_{3})$.

Now consider the functions

(1.4) ${K(x)=H_{1}(x)+\dfrac{1}{ln(2+|x|)}}$

and

(1.5)
$\displaystyle\begin{split}V(x)=\left(|x|H_{1}(x)\right)^{\frac{m^{*}-m}{m^{*}-a_{1}}}+\left(\dfrac{1}{ln(2+|x|)}\right)^{\frac{m^{*}-m}{m^{*}-a_{2}}}+\left(|x|H_{1}(x)\right)^{\frac{\ell^{*}-\ell}{\ell^{*}-a_{2}}}+\left(\dfrac{1}{ln(2+|x|)}\right)^{\frac{\ell^{*}-\ell}{\ell^{*}-a_{1}}}\end{split}$

As mentioned before, these functions verifies $(K_{0})$ and $(K_{1})$.
Furthermore, the condition $(K_{3})$ is satisfied. However, these functions do
not verifies the condition $(K_{2})$.

This paper is organized as follows: in Section $2$, we review some properties
of Orlicz and Orlicz-Sobolev spaces that will be used throughout this paper.

In Section $3$, we present the space and show some important properties
involving the energy functional related to the problem $(P)$.

In Section 4, we will study the case $(V,K)\in\mathcal{K}_{1}$. In our first
main result, by means of some conditions imposed on $\Phi$ and $f$, we will
show that the problem $(P)$ has a $C^{1,\alpha}_{loc}(\mathbb{R}^{N})$
positive ground state solution. More specifically, we will assume that
$\Phi\in\mathcal{C}_{m}$ and $f:\mathbb{R}\longrightarrow\mathbb{R}$ satisfies
the following conditions

$None$ $\displaystyle\lim_{t\to 0}\dfrac{f(t)}{t\phi(t)}=0\;\;\;\text{ and
}\;\;\;\displaystyle\limsup_{t\to\infty}\dfrac{f(t)}{t\phi_{*}(t)}=0.$

$(f_{2})$ $s^{1-m}f(s)$ is an increasing function in $(0,+\infty)$.
$(f_{3})$ $F(t)=\int_{0}^{t}f(s)ds$ is $m$-superlinear at infinity, that is,

$\lim_{|t|\rightarrow+\infty}\dfrac{F(t)}{|t|^{m}}=+\infty.$

Under these conditions, our first main result can be stated as follows.

######  Theorem 1.1.

Assume that $(V,K)\in\mathcal{K}_{1}$ and $\Phi\in\mathcal{C}_{m}$. Suppose
that $(\phi_{1})-(\phi_{4})$ and $(f_{1})-(f_{3})$ hold. Then problem $(P)$
possesses a nonnegative solutions that are locally bounded.

To study the regularity of the solutions provided by Theorem 1.1, we add the
following assumptions:

$(\phi_{5})$ There are $0<\delta<1$, $C_{1},C_{2}>0$ and $1<\beta\leq\ell^{*}$
such that

$C_{1}t^{\beta-1}\leq t\phi(t)\leq C_{2}t^{\beta-1},\;\text{ for
}\;t\in[0,\delta].$

$(\phi_{6})$ There are constants $\delta_{0}>0$ and $\delta_{1}>0$ such that

$\delta_{0}\leq\dfrac{(\phi(t)t)^{\prime}}{\phi(t)}\leq\delta_{1}\;\text{
for}\;t>0.$

We are in position to state the following regularity result:

######  Theorem 1.2.

Supoose that $\Phi$ satisfies $(\phi_{5})-(\phi_{6})$. Under the assumptions
of Theorem 1.1, the problem $(P)$ possesses a
$C^{1,\alpha}_{loc}(\mathbb{R}^{N})$ positive ground state solution.

For the second main result, we consider the problem $(P)$ without condition
$\mathcal{C}_{m}$. The removal of this condition on the $N$-function $\Phi$,
forced us to present a more restricted growth condition that $(f_{1})$, this
constraint under the nonlinearity $f$, will be necessary to show that the
nonnegative solutions of $(P)$, are positive. In this way, we will consider
$B:\mathbb{R}\rightarrow[0,\infty)$ being a $N$-function given by
$B(t)=\int_{0}^{|t|}b(s)sds$, where $b:(0,\infty)\rightarrow(0,\infty)$ is a
function satisfying the following conditions:

$None$ $t\longmapsto tb(t)\;\text{ is increasing for }\;t>0,$ $None$
$\displaystyle\lim_{t\rightarrow 0^{+}}tb(t)=0\;\;\text{ and
}\;\;\displaystyle\lim_{t\rightarrow+\infty}tb(t)=+\infty,$

and there exist $b_{1}\in[m,\ell^{*}]$ such that

$None$ $b_{1}=\inf_{t>0}\dfrac{b(t)t^{2}}{B(t)}\;\;\text{ and
}\;\;\ell^{*}\geq\sup_{t>0}\dfrac{b(t)t^{2}}{B(t)},\;\;\forall
t>0.\vspace*{-0.2cm}$

For this case, we assume that $f:\mathbb{R}\longrightarrow\mathbb{R}$
satisfies the conditions $(f_{2})$ and $(f_{3})$. Moreover, we will consider
the following growth condition

$None$ $\displaystyle\lim_{t\to 0}\dfrac{f(t)}{t\phi(t)}=0\;\;\;\text{ and
}\;\;\;\displaystyle\limsup_{t\to\infty}\dfrac{|f(t)|}{tb(t)}=0.$

Our second main result can be written in the following form.

######  Theorem 1.3.

Assume that $(V,K)\in\mathcal{K}_{1}$. Suppose that $(\phi_{1})-(\phi_{4})$,
$(f_{2})$, $(f_{3})$ and $(f_{4})$ hold. Then problem $(P)$ possesses a
nonnegative solutions that are locally bounded. If $\Phi$ also satisfies
$(\phi_{5})$ and $(\phi_{6})$, the solutions for the problem $(P)$ are
$C^{1,\alpha}_{loc}(\mathbb{R}^{N})$ positive ground state solution.

In Section 5, we will study the case that $(V,K)\in\mathcal{K}_{2}$. As in
section 4, we initially analyzed the problem $(P)$ with
$\Phi\in\mathcal{C}_{m}$. In this sense, we assume that
$f:\mathbb{R}\longrightarrow\mathbb{R}$ satisfies $(f_{2})$ and $(f_{3})$.
Moreover, we will consider the following condition

$None$ $\displaystyle\limsup_{t\to 0}\dfrac{|f(t)|}{ta(t)}<\infty\;\;\;\text{
and }\;\;\;\displaystyle\lim_{t\to\infty}\dfrac{f(t)}{t\phi_{*}(t)}=0.$

Our first main result of this fifth section can be stated as follows.

######  Theorem 1.4.

Assume that $(V,K)\in\mathcal{K}_{2}$ and $\Phi\in\mathcal{C}_{m}$. Suppose
that $(\phi_{1})-(\phi_{4})$, $(f_{2})$, $(f_{3})$ and $(f_{5})$ hold. Then
problem $(P)$ possesses a nonnegative solutions that are locally bounded. If
$\Phi$ also satisfies $(\phi_{5})$ and $(\phi_{6})$, the solutions for the
problem $(P)$ are $C^{1,\alpha}_{loc}(\mathbb{R}^{N})$ positive ground state
solution.

In a second moment, as in Theorem 1.3, we relax the condition
$\Phi\in\mathcal{C}_{m}$ and present a more restricted growth condition that
$(f_{5})$. More precisely, $f:\mathbb{R}\longrightarrow\mathbb{R}$ satisfies

$None$ $\displaystyle\limsup_{t\to 0}\dfrac{|f(t)|}{ta(t)}<\infty\;\;\;\text{
and }\;\;\;\displaystyle\lim_{t\to\infty}\dfrac{f(t)}{tb(t)}=0.$

Furthermore, we will assume that $f$ satisfies $(f_{2})$ and $(f_{3})$.

Our second main result of this fifth section can be written in the following
form.

######  Theorem 1.5.

Assume that $(V,K)\in\mathcal{K}_{2}$. Suppose that $(\phi_{1})-(\phi_{4})$,
$(f_{2})$, $(f_{3})$ and $(f_{6})$ hold. Then problem $(P)$ possesses a
nonnegative solutions that are locally bounded. If $\Phi$ also satisfies
$(\phi_{5})$ and $(\phi_{6})$, the solutions for the problem $(P)$ are
$C^{1,\alpha}_{loc}(\mathbb{R}^{N})$ positive ground state solution.

In Section 6, we will study the zero mass case, this is, $V=0$. In what
follows, we assume that $f$ satisfies $(f_{2})$ and $(f_{3})$. Moreover, we
will consider the following growth condition

$None$ $\displaystyle\lim_{t\to 0}\dfrac{f(t)}{t\phi_{*}(t)}=0\;\;\;\text{ and
}\;\;\;\displaystyle\lim_{t\to\infty}\dfrac{f(t)}{t\phi_{*}(t)}=0.$

Our main result stated in this section is described next.

######  Theorem 1.6.

Suppose that $(\phi_{1})-(\phi_{4})$ and $(K_{0})-(K_{1})$ hold. Assume that
$f$ satisfies $(f_{2})$, $(f_{3})$, $(f_{7})$. If $V\equiv 0$, then the
problem $(P)$ has a nonnegative ground state solution.

##  2\. Basics On Orlicz-Sobolev Spaces

In this section we recall some properties of Orlicz and Orlicz-Sobolev spaces,
which can be found in [1,9,23,24]. First of all, we recall that a continuous
function $\Phi:\mathbb{R}\rightarrow[0,+\infty)$ is a $N$-function if:

  * (i)

$\Phi$ is convex;

  * (ii)

$\Phi(t)=0\Leftrightarrow t=0$;

  * (iii)

$\Phi$ is even;

  * (iv)

$\displaystyle\lim_{t\rightarrow 0}\dfrac{\Phi(t)}{t}=0\text{ and
}\lim_{t\rightarrow+\infty}\dfrac{\Phi(t)}{t}=+\infty$.

We say that a $N$-function $\Phi$ verifies the $\Delta_{2}$-condition, and we
denote by $\Phi\in(\Delta_{2})$, if there are constants $K>0,\;t_{0}>0$ such
that

$\Phi(2t)\leq K\Phi(t),\leavevmode\nobreak\ \leavevmode\nobreak\ \forall t\geq
t_{0}.$

In the case of $|\Omega|=+\infty$, we will consider that $\Phi\in(\Delta_{2})$
if $t_{0}=0$. For instance, it can be shown that $\Phi(t)=|t|^{p}/p$ for $p>1$
satisfies the $\Delta_{2}$-condition, while $\Phi(t)=(e^{t^{2}}-1)/2$.

If $\Omega$ is an open set of $\mathbb{R}^{N}$, where $N$ can be a natural
number such that $N\geq 1$, and $\Phi$ a $N$-function then define the Orlicz
space associated with $\Phi$ as

$L^{\Phi}(\Omega)=\left\\{u\in L^{1}_{\text{loc}}(\Omega):\leavevmode\nobreak\
\int_{\Omega}\Phi\left(\frac{|u|}{\lambda}\right)dx<+\infty\leavevmode\nobreak\
\text{for some}\leavevmode\nobreak\ \lambda>0\right\\}.$

The space $L^{\Phi}(\Omega)$ is a Banach space endowed with the Luxemburg norm
given by

$\|u\|_{L^{\Phi}(\Omega)}=\inf\left\\{\lambda>0:\int_{\Omega}\Phi\left(\frac{|u|}{\lambda}\right)dx\leq
1\right\\}.$

In the case that $\Phi$ verifies $\Delta_{2}$-condition we have

$L^{\Phi}(\Omega)=\left\\{u\in L^{1}_{\text{loc}}(\Omega):\leavevmode\nobreak\
\int_{\Omega}\Phi(|u|)dx<+\infty\right\\}.$

The complementary function $\tilde{\Phi}$ associated with $\Phi$ is given by
the Legendre transformation, that is,

$\tilde{\Phi}(s)=\max_{t\geq 0}\\{st-\Phi(t)\\},\leavevmode\nobreak\
\leavevmode\nobreak\ \forall\;t\geq 0.$

The functions $\Phi$ and $\tilde{\Phi}$ are complementary each other and
satisfy the inequality below

$\tilde{\Phi}(\Phi^{\prime}(t))\leq\Phi(2t),\;\;\forall\;t>0.$

Moreover, we also have a Young type inequality given by

$st\leq\Phi(t)+\tilde{\Phi}(s),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \forall s,t\geq 0.$

Using the above inequality, it is possible to establish the following Holder
type inequality:

$\left|\int_{\Omega}uvdx\right|\leq
2\|u\|_{L^{\Phi}(\Omega)}\|v\|_{{L^{\tilde{\Phi}}(\Omega)}},\leavevmode\nobreak\
\leavevmode\nobreak\ \text{for all}\leavevmode\nobreak\ \leavevmode\nobreak\
u\in L^{\Phi}(\Omega)\leavevmode\nobreak\ \leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ \leavevmode\nobreak\ v\in
L^{\tilde{\Phi}}(\Omega).$

If $|\Omega|<\infty$, the space $E^{\Phi}(\Omega)$ denotes the closing of
$L^{\infty}(\Omega)$ in $L^{\Phi}(\Omega)$ with respect to the norm
$\lVert\cdot\lVert_{\Phi}.$ When $|\Omega|=\infty$, the space
$E^{\Phi}(\Omega)$ denotes the closure of $C^{\infty}_{0}(\Omega)$ in
$L^{\Phi}(\Omega)$ with respect to norm $\lVert\cdot\lVert_{\Phi}.$ In any of
these cases, $L^{\Phi}(\Omega)$ is the dual space of
$E^{\tilde{\Phi}}(\Omega)$, while $L^{\tilde{\Phi}}(\Omega)$ is the dual space
of $E^{{\Phi}}(\Omega)$. Moreover, $E^{{\Phi}}(\Omega)$ and
$E^{\tilde{\Phi}}(\Omega)$ are separable and all continuous functional
$M:E^{{\Phi}}(\Omega)\longrightarrow\mathbb{R}$ is of the form

$\displaystyle M(v)=\int_{\Omega}v(x)g(x)dx,\;\;\;\;\text{for some
function}\;\;g\in L^{\tilde{\Phi}}(\Omega).$

Another important function related to function $\Phi$, it is the Sobolev
conjugate function ${\Phi}_{*}$ of $\Phi$ defined by

$\displaystyle\Phi_{*}^{-1}(t)=\int_{0}^{t}\dfrac{\Phi^{-1}(s)}{s^{(N+1)/N}}ds\;\;\text{for}\;\;t>0\;\;\text{
when }\;\;\int_{1}^{+\infty}\dfrac{\Phi^{-1}(s)}{s^{(N+1)/N}}ds=+\infty.$

######  Lemma 2.1.

Consider $\Phi$ a $N$-function of the form (1.1) and satisfying $(\phi_{1})$
and $(\phi_{2})$. Set

$\xi_{0}(t)=\min\\{t^{\ell},t^{m}\\}\leavevmode\nobreak\ \leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ \leavevmode\nobreak\
\xi_{1}(t)=\max\\{t^{\ell},t^{m}\\},\leavevmode\nobreak\ \leavevmode\nobreak\
\forall t\geq 0.$

Then $\Phi$ satisfies

$\xi_{0}(t)\Phi(\rho)\leq\Phi(\rho
t)\leq\xi_{1}(t)\Phi(\rho),\leavevmode\nobreak\ \leavevmode\nobreak\
\forall\rho,t\geq 0$

and

$\xi_{0}(\|u\|_{\Phi})\leq\int_{\Omega}\Phi(u)dx\leq\xi_{1}(\|u\|_{\Phi}),\;\;\;\forall
u\in L^{\Phi}(\Omega).$

######  Lemma 2.2.

If $\Phi$ is an N-function of the form (1.1) satisfying $(\phi_{1})$,
$(\phi_{2})$ and $(\phi_{3})$, then

(2.1) $\xi_{2}(t)\Phi_{*}(\rho)\leq\Phi_{*}(\rho
t)\leq\xi_{3}(t)\Phi_{*}(\rho),\;\;\;\forall\rho,t>0$

and

$\displaystyle\xi_{2}(\Arrowvert
u\Arrowvert_{\Phi_{*}})\leq\int_{\Omega}\Phi_{*}(u)dx\leq\xi_{3}(\Arrowvert
u\Arrowvert_{\Phi_{*}}),\;\;\;\forall u\in L^{\Phi_{*}}(\Omega),$

where

$\xi_{2}(t)=\min\\{t^{\ell^{*}},t^{m^{*}}\\}\hskip 2.84544pt\text{ and }\hskip
2.84544pt\xi_{3}(t)=\max\\{t^{\ell^{*}},t^{m^{*}}\\},\;t\geq 0.$

######  Lemma 2.3.

(2.1) is equivalent to

${\ell^{*}}\leq\dfrac{\Phi^{\prime}_{*}(t)t^{2}}{\Phi_{*}(t)}\leq{m^{*}},\;\;\;\forall
t>0$

######  Lemma 2.4.

If $\Phi$ is an N-function of the form (1.1) satisfying $(\phi_{1})$,
$(\phi_{2})$ and $(\phi_{3})$, then

$\tilde{\Phi}(\rho t)\leq t^{\frac{m}{m-1}}\tilde{\Phi}(\rho),\;\text{ for all
}\;\rho>0\;\text{ and }\;0\leq t<1.$

######  Lemma 2.5.

If $\Phi$ is $N$-function and $(\int_{\Omega}\Phi(|u_{n}|)dx)$ is a bounded
sequence, then $(u_{n})$ is a bounded sequence in $L^{\Phi}(\Omega)$. When
$\Phi\in(\Delta_{2})$, the equivalence is valid.

######  Lemma 2.6.

If $\Phi$ is an N-function of the form (1.1) satisfying $(\phi_{1})$,
$(\phi_{2})$ and $(\phi_{3})$ with $\ell=1$, then
$\tilde{\Phi}\notin(\Delta_{2})$.

######  Lemma 2.7.

Let $\Omega\subset\mathbb{R}^{N}$ be a domain, if $\Phi\in(\Delta_{2})$ and
$(u_{n})$ a sequence in $L^{\Phi}(\Omega)$ with $u_{n}\longrightarrow u$ in
$L^{\Phi}(\Omega)$, there is $H\in L^{\Phi}(\Omega)$ and a subsequence
$(u_{n_{j}})$ such that
$i)\;\;\;|u_{n_{j}}(x)|\leq H(x)$ a.e. in $\Omega$.
$ii)\;\;\;u_{n_{j}}(x)\longrightarrow u(x)$ a.e. in $\Omega$ and all
$j\in\mathbb{N}$.

For a $N$-function $\Phi$, the corresponding Orlicz-Sobolev space is defined
as the Banach space

$W^{1,\Phi}(\Omega)=\left\\{u\in L^{\Phi}(\Omega):\dfrac{\partial u}{\partial
x_{i}}\in L^{\Phi}(\Omega),i=1,...,N\right\\},$

with the norm

(2.2) $\displaystyle\lVert u\Arrowvert_{1,\Phi}=\Arrowvert\nabla
u\Arrowvert_{\Phi}+\Arrowvert u\Arrowvert_{\Phi}.$

If $\Phi\in(\Delta_{2})$, the space $D^{1,\Phi}(\mathbb{R}^{N})$ is defined to
be the complement of the space $C^{\infty}_{0}(\mathbb{R}^{N})$ with respect
to the standard

(2.3) $\displaystyle|u|_{D^{1,\Phi}(\mathbb{R}^{N})}=\lVert
u\lVert_{\Phi_{*}}+\lVert\nabla u\lVert_{\Phi}.$

Since the Orlicz-Sobolev inequality

(2.4) $\displaystyle\lVert u\lVert_{\Phi_{*}}\leq S_{N}\lVert\nabla
u\lVert_{\Phi},$

holds for $u\in D^{1,\Phi}(\mathbb{R}^{N})$ with a constant $S_{N}>0$, the
norm (2.3) is equivalent to the norm

(2.5) $\displaystyle\|u\|_{D^{1,\Phi}(\mathbb{R}^{N})}=\lVert\nabla
u\lVert_{\Phi},$

on $D^{1,\Phi}(\mathbb{R}^{N})$. In this paper, we will use (2.5) as the norm
of $D^{1,\Phi}(\mathbb{R}^{N})$. Clearly

$D^{1,\Phi}(\mathbb{R}^{N})\xhookrightarrow[cont\;]{}L^{\Phi_{*}}(\mathbb{R}^{N}).$

The space $L^{\Phi}(\mathbb{R}^{N})$ is separable and reflexive when the
$N$-functions $\Phi$ and $\tilde{\Phi}$ satisfy the $\Delta_{2}$-condition.
Knowing that $D^{1,\Phi}(\mathbb{R}^{N})$ can be seen as a closed subspace of
the space $L^{\Phi_{*}}(\mathbb{R}^{N})\times(L^{\Phi}(\mathbb{R}^{N}))^{N}$,
then $D^{1,\Phi}(\mathbb{R}^{N})$ is reflexive when the $N$-functions $\Phi$,
$\tilde{\Phi}$, $\Phi_{*}$ and $\tilde{\Phi}_{*}$ satisfy the
$\Delta_{2}$-condition .

The following lemma is an immediate consequence of the Banach-Alaoglu-Bourbaki
theorem [14].

######  Lemma 2.8.

Assume that $\Phi$ is an N-function of the form (1.1) satisfying $(\phi_{1})$,
$(\phi_{2})$ and $(\phi_{3})$. If $(u_{n})\subset D^{1,\Phi}(\mathbb{R}^{N})$
is a bounded sequence, then there exists a subsequence of $(u_{n})$, which we
will still denote by $(u_{n})$, and $u\in D^{1,\Phi}(\mathbb{R}^{N})$ such
that

(2.6) $\displaystyle u_{n}\xrightharpoonup[\quad]{\ast}u\;\;\;\text{ in
}\;L^{\Phi_{*}}(\mathbb{R}^{N})\;\;\;\;\text{ and }\;\;\;\;\dfrac{\partial
u_{n}}{\partial x_{i}}\xrightharpoonup[\quad]{\ast}\dfrac{\partial u}{\partial
x_{i}}\;\;\;\text{ in }\;L^{\Phi}(\mathbb{R}^{N})$

or equivalently,

$\displaystyle\int_{\mathbb{R}^{N}}u_{n}vdx\longrightarrow\int_{\mathbb{R}^{N}}uvdx,\;\;\forall
v\in E^{\tilde{\Phi}_{*}}(\mathbb{R}^{N})$

and

$\displaystyle\int_{\mathbb{R}^{N}}\dfrac{\partial u_{n}}{\partial
x_{i}}wdx\longrightarrow\int_{\mathbb{R}^{N}}\dfrac{\partial u}{\partial
x_{i}}wdx,\;\;\forall w\in E^{\tilde{\Phi}}(\mathbb{R}^{N}).$

From now on, we denote the limit (2.6) by
$u_{n}\xrightharpoonup[\quad]{\ast}u$ in $D^{1,\Phi}(\mathbb{R}^{N})$. As an
immediate consequence of the last lemma, we have the following corollary.

######  Corollary 2.9.

If $(u_{n})\subset D^{1,\Phi}(\mathbb{R}^{N})$ is a bounded sequence with
$u_{n}\longrightarrow u$ in $L^{\Phi}_{loc}(\mathbb{R}^{N})$, then $u\in
D^{1,\Phi}(\mathbb{R}^{N})$.

The last lemma is crucial when the space $D^{1,\Phi}(\mathbb{R}^{N})$ can be
nonreflexive. And, this happens, for example, when
$\Phi_{\alpha}(t)=|t|\ln(|t|^{\alpha}+1)$, for $0<\alpha<\frac{N}{N-1}-1$,
because $\tilde{\Phi}_{\alpha}$ does not verify the
$\linebreak\Delta_{2}$-condition. Here we emphasize that the condition
$(\phi_{3})$ guarantees that $\Phi$ and $\tilde{\Phi}$ verifies the
$\Delta_{2}$-condition when $\ell>1$, for more details see Fukagai and
Narukawa [9].

##  3\. Preliminary results

This section focuses on preparing some preliminaries for proving Theorems 1.1,
1.3, 1.4 and 1.5. Since the potential V may vanish at infinity, we cannot
study equation $(P)$ on the Sobolev space $D^{1,\Phi}(\mathbb{R}^{N})$ by
variational methods. As in [4], we work in the space $\linebreak
E=\big{\\{}u\in
D^{1,\Phi}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}V(x)\Phi(|u|)dx<+\infty\big{\\}}$
with norm

$\displaystyle\lVert u\lVert_{E}=\lVert
u\lVert_{D^{1,\Phi}(\mathbb{R}^{N})}+\lVert u\lVert_{V,\Phi},$

where

$\displaystyle\lVert
u\lVert_{V,\Phi}=\inf\Big{\\{}\alpha>0:\int_{\mathbb{R}^{N}}V(x)\Phi\left({|u|}/{\alpha}\right)dx\leq
1\Big{\\}}$

is the norm of Banach Space

$\displaystyle
L^{\Phi}_{V}(\mathbb{R}^{N})=\Big{\\{}u:\mathbb{R}^{N}\longrightarrow\mathbb{R}\;\text{measurable
}:\int_{\mathbb{R}^{N}}V(x)\Phi(|u|)dx<+\infty\Big{\\}}.$

It is immediate that $E$ is continuously embedded in the spaces
$D^{1,\Phi}(\mathbb{R}^{N})$ and $L^{\Phi}_{V}(\mathbb{R}^{N})$.

Now let us list some properties involving the space $E$.

######  Lemma 3.1.

$(E,\lVert\cdot\lVert_{E})$ is a Banach space.

######  Lemma 3.2.

$E=\overline{C^{\infty}_{0}(\mathbb{R}^{N})}^{\lVert\cdot\lVert_{E}}$

Proof: It follows the same ideas as Theorem 8.21, which can be found in [1].
For that reason, we will omit your proof.

∎

######  Lemma 3.3.

$E$ is compactly embedded in $L^{\Phi}_{loc}(\mathbb{R}^{N})$.

######  Lemma 3.4.

Suppose $(u_{n})\subset E$ is a bounded sequence, then there is $u\in E$ such
that $u_{n}\xrightharpoonup[\quad]{\ast}u$ in $D^{1,\Phi}(\mathbb{R}^{N})$.

Proof: Since $(u_{n})$ is a bounded sequence in $E$, then $(u_{n})$ is a
bounded sequence in $D^{1,\Phi}(\mathbb{R}^{N})$ and by the Lemma 2.8, there
is $u\in D^{1,\Phi}(\mathbb{R}^{N})$ such that
$u_{n}\xrightharpoonup[\quad]{\ast}u$ in $D^{1,\Phi}(\mathbb{R}^{N})$. Let us
show that $u\in E$, because by Lemma 3.3 and Corollary 2.9, we can conclude
that less than one subsequence

$\displaystyle u_{n}(x)\longrightarrow
u(x),\;\;\;\;a.e.\;\;\;\text{in}\;\;\;\mathbb{R}^{N}.$

By Fatou’s Lemma

$\displaystyle\int_{\mathbb{R}^{N}}V(x)\Phi(|u(x)|)dx\leq\liminf_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}V(x)\Phi(|u_{n}(x)|)dx.$

Since $(u_{n})$ is bounded in $E$, then $(u_{n})$ is bounded in
$L^{\Phi}_{V}(\mathbb{R}^{N})$. As $\Phi\in(\Delta_{2})$, there is $C>0$ such
that

$\displaystyle\int_{\mathbb{R}^{N}}V(x)\Phi(|u_{n}(x)|)dx\leq
C,\;\;\;\;\forall n\in\mathbb{N}$

Therefore,

$\displaystyle\int_{\mathbb{R}^{N}}V(x)\Phi(|u(x)|)dx<+\infty,$

showing that $u\in E$, and the proof is complete.

∎

Now, we consider the functional $Q:E\longrightarrow\mathbb{R}$ which is given
by

$Q(u)=\int_{\mathbb{R}^{N}}\Phi(|\nabla
u|)dx+\int_{\mathbb{R}^{N}}V(x)\Phi(|u|)dx.$

It is well known in the literature that $Q\in C^{1}(E,\mathbb{R})$ when $\Phi$
and $\tilde{\Phi}$ satisfy the condition $(\Delta_{2})$ and this occurs when
we have the condition satisfied to $\ell>1$. When $\ell=1$, we know that
$\tilde{\Phi}\notin(\Delta_{2})$ and therefore cannot guarantee the
differentiability of functional $Q$. However, we will show that the functional
$Q$ is continuous and Gateaux-differentiable with derivative
$Q^{\prime}:E\longrightarrow E^{*}$ defined by

$\displaystyle Q^{\prime}(u)v=\int_{\mathbb{R}^{N}}\phi(|\nabla u|)\nabla
u\nabla vdx+\int_{\mathbb{R}^{N}}V(x)\phi(|u|)uvdx,\;\;\;\forall u,v\in E$

is continuous from the norm topology of $E$ to the weak∗-topology of $E^{*}$.

######  Lemma 3.5.

The functional $Q$ is Gateaux differentiable, that is, $Q^{\prime}(u)v$ exists
for every $u,v\in E$ with

$\displaystyle Q^{\prime}(u)v=\int_{\mathbb{R}^{N}}\phi(|\nabla u|)\nabla
u\nabla vdx+\int_{\mathbb{R}^{N}}V(x)\phi(|u|)uvdx.$

Proof: See Lemma 4.1 in [5].

∎

######  Lemma 3.6.

Let $\Phi$ an N-function of the form (1.1) satisfying $(\phi_{1})$,
$(\phi_{2})$ and $(\phi_{3})$. If $u_{n}\longrightarrow u$ in
$L^{\Phi}_{V}(\mathbb{R}^{N})$, then there exists $H_{1}\in
L^{\Phi}_{V}(\mathbb{R}^{N})$ and a subsequence $\\{u_{n_{j}}\\}$ such that
$i)\;\;|u_{n_{j}}(x)|\leq H_{1}(x)$ for every $x\in\mathbb{R}^{N}$ and every
$j\in\mathbb{N}$
$ii)\;\;u_{n_{j}}(x)\longrightarrow u(x)$ $a.e.$ in $\mathbb{R}^{N}$ and every
$j\in\mathbb{N}$.

Proof: It is enough to repeat the same argument explored in [24, Lemma $2.5$].

∎

As an immediate consequence of the Lemma 3.6, we have the following result.

######  Lemma 3.7.

The functional $Q:E\longrightarrow\mathbb{R}$ is continuous in the norm
topology.

######  Lemma 3.8.

The Gateaux derivative $Q^{\prime}:E\longrightarrow E^{*}$ is continuous from
the norm topology of $E$ to the weak∗ topology of $E^{*}$.

Proof: By the Proposition $3.2$ of [14] is sufficient prove that, any sequence
$(u_{n})\subset E$ such that $u_{n}\longrightarrow u$ in $E$, implies

$\langle Q^{\prime}(u_{n}),v\rangle\longrightarrow\langle
Q^{\prime}(u),v\rangle,\;\;\;\forall v\in E.$

Consider $(u_{n})\subset E$ such that $u_{n}\longrightarrow u$ in $E$, then

$|\nabla u_{n}|\longrightarrow|\nabla u|\;\;\text{ in
}\;\;L^{\Phi}(\mathbb{R}^{N})\;\;\text{ and }\;\;u_{n}\longrightarrow
u\;\text{ in }\;L^{\Phi}_{V}(\mathbb{R}^{N}).$

By Lemma 3.6, there are $H_{1}\in L^{\Phi}_{V}(\mathbb{R}^{N})$, $H_{2}\in
L^{\Phi}_{V}(\mathbb{R}^{N})$ and a subsequence
$\\{u_{n_{j}}\\}\subset\\{u_{n}\\}$ such that
i) $|u_{n_{j}}(x)|\leq H_{1}(x),\;$ for $x\in\mathbb{R}^{N}$ and
$j\in\mathbb{N}$
ii) $|\nabla u_{n_{j}}(x)|\leq H_{2}(x),\;$ for $x\in\mathbb{R}^{N}$ and
$j\in\mathbb{N}$
iii) $u_{n_{j}}(x)\rightarrow u(x),\;$ $a.e.$ in $\mathbb{R}^{N}$ and
$j\in\mathbb{N}$
iv) $|\nabla u_{n_{j}}(x)|\longrightarrow|\nabla u(x)|,\;$ $a.e.$ in
$\mathbb{R}^{N}$ and $j\in\mathbb{N}$.

Set $v\in E$ arbitrary. By the continuity of the function $\phi$, it follows
that

$\phi(|\nabla u_{n_{j}}(x)|)\nabla u_{n_{j}}(x)\nabla
v(x)\longrightarrow\phi(|\nabla u(x)|)\nabla u(x)\nabla v(x),\;a.e.\text{ in
}\mathbb{R}^{N}.$

Also, by $(\phi_{1})$ the function $\phi(t)t$ is increasing for every $t>0$,
thus

$\displaystyle\phi(|\nabla u_{n_{j}}(x)|)\nabla u_{n_{j}}(x)\nabla
v(x)|\leq\phi(|\nabla u_{n_{j}}(x)|)|\nabla u_{n_{j}}(x)||\nabla
v(x)|\leq\phi(|H(x)|)|H(x)|\nabla v(x)|.$

Hence by Lebesgue dominated convergence theorem

$\displaystyle\int_{\mathbb{R}^{N}}\phi(|\nabla u_{n_{j}}|)\nabla
u_{n_{j}}\nabla vdx\longrightarrow\int_{\mathbb{R}^{N}}\phi(|\nabla u|)\nabla
u\nabla vdx,$

with this, we ensure that

$\displaystyle\int_{\mathbb{R}^{N}}\phi(|\nabla u_{n}|)\nabla u_{n}\nabla
vdx\longrightarrow\int_{\mathbb{R}^{N}}\phi(|\nabla u|)\nabla u\nabla vdx.$

Similarly, we have

$\displaystyle\int_{\mathbb{R}^{N}}\phi(|u_{n}|)u_{n}vdx\longrightarrow\int_{\mathbb{R}^{N}}\phi(|u|)uvdx.$

Therefore,

$\displaystyle\langle Q^{\prime}(u_{n}),v\rangle\longrightarrow\langle
Q^{\prime}(u),v\rangle.$

By the arbitrariness of $v\in E$, we conclude the results.

∎

##  4\. Proof of Theorems 1.1 and 1.3

Initially, note that the condition $(f_{4})$ implies that
$\displaystyle\lim_{t\rightarrow+\infty}\frac{f(t)}{\phi_{*}(t)t}=0$. Then, by
the conditions $(f_{1})$ or $(f_{4})$, given $\varepsilon>0$ there exists
$\delta_{0}>0$, $\delta_{1}>0$ and $C_{\varepsilon}>0$ such that

(4.1) $\displaystyle K(x)|f(t)|\leq\varepsilon
C_{1}\big{(}V(x)t\phi(t)+t\phi_{*}(t)\big{)}+C_{\varepsilon}K(x)t\phi_{*}(t)\chi_{[\delta_{0},\delta_{1}]}(t),$

for every $t\geq 0$ and $x\in\mathbb{R}^{N}$, where
$C_{1}=\max\big{\\{}\lVert{K}\lVert_{\infty},\big{\lVert}\frac{K}{V}\big{\lVert}_{\infty}\big{\\}}$.
This inequality yields that the functional
$\mathcal{F}:E\longrightarrow\mathbb{R}$ given by

(4.2) $\mathcal{F}(u)=\int_{\mathbb{R}^{N}}K(x)F(u)dx$

is well defined and belongs to $C^{1}(E,\mathbb{R})$ with derivative

$\mathcal{F}^{\prime}(u)v=\int_{\mathbb{R}^{N}}K(x)f(u)vdx,\quad\forall u,v\in
E.$

From the results presented in the previous section, we can conclude that the
energy functional $J:E\longrightarrow\mathbb{R}$ associated with the problem
$(P)$, which is given by

$J(u)=\int_{\mathbb{R}^{N}}\Phi(|\nabla
u|)dx+\int_{\mathbb{R}^{N}}V(x)\Phi(|u|)dx-\int_{\mathbb{R}^{N}}K(x)F(u)dx.$

is a continuous and Gateaux-differentiable functional such that
$J^{\prime}:E\longrightarrow E^{*}$ given by

$\displaystyle J^{\prime}(u)v=\int_{\mathbb{R}^{N}}\phi(|\nabla u|)\nabla
u\nabla
vdx+\int_{\mathbb{R}^{N}}V(x)\phi(|u|)uvdx-\int_{\mathbb{R}^{N}}K(x)f(u)vdx$

is continuous from the norm topology of $E$ to the weak∗-topology of $E^{*}$.

Once that we intend to find nonnegative solutions for the problem $(P)$, we
will assume that

(4.3) $\displaystyle f(s)=0,\;\;\;\forall s\in(-\infty,0].$

The convexity of the functional $Q$ together with the Gateaux-
differentiability of the functional $J$ allows us to present a definition of a
critical point for $J$. In this sense, we will say that $u\in E$ is a critical
point for the functional $J$ if

(4.4) $\displaystyle
Q(v)-Q(u)\geq\int_{\mathbb{R}^{N}}{K(x)f(u)(v-u)}dx,\;\;\;\;\forall v\in E.$

Our next lemma establishes that a critical point $u$ in the sense (4.4) is a
weak solution for $(P)$.

######  Proposition 4.1.

If $u\in E$ is a critical point of $J$ in $E$, then u is a weak solution to
$(P)$.

Proof: See Lemma 4.1 in [5].

∎

Now, let us check that $J$ also satisfies the mountain pass geometry

######  Lemma 4.2.

There are $\rho,\eta>0$ such that $J(u)\geq\eta$ if $\lVert u\lVert_{E}=\rho$.

Proof: Consider $0<\varepsilon<\frac{1}{2C_{1}}$ with
$C_{1}=\big{\lVert}\frac{K}{V}\big{\lVert}_{\infty}$. By (4.1), there is
$C_{\varepsilon}>0$, such that

(4.5) $\displaystyle
K(x)|F(t)|\leq\frac{1}{2}V(x)\Phi(t)+C_{\varepsilon}\Phi_{*}(t),\;\;\forall
t\geq 0\;\text{ and }\;x\in\mathbb{R}^{N}.$

Thus,

$\displaystyle J(u)$ $\displaystyle\geq\int_{\mathbb{R}^{N}}\Phi(|\nabla
u|)dx+\frac{1}{2}\int_{\mathbb{R}^{N}}V(x)\Phi(|u|)dx-
C_{\varepsilon}\int_{\mathbb{R}^{N}}\Phi_{*}(|u|)dx$ $\displaystyle\geq
C_{2}\big{(}\xi_{0}(\lVert\nabla u\lVert_{\Phi})+\xi_{0}(\lVert
u\lVert_{V,\Phi})\big{)}-C_{2}\xi_{3}(\lVert u\lVert_{\Phi_{*}}),$

for some $C_{2}>0$, where $\xi_{0}(t)=\min\\{t^{\ell},\,t^{m}\\}$ and
$\xi_{3}(t)=\max\\{t^{\ell^{*}},\,t^{m^{*}}\\}.$ Choose $\rho>0$ such that

$\displaystyle\lVert u\lVert_{E}=\lVert
u\lVert_{D^{1,\Phi}(\mathbb{R}^{N})}+\lVert u\lVert_{V,\Phi}=\rho<1.$

As $E$ is continuously embedded in $L^{\Phi_{*}}(\mathbb{R}^{N})$, we get
$\lVert u\lVert_{\Phi^{*}}\leq 1$. Furthermore,

$\displaystyle J(u)\geq C_{2}\big{(}\lVert\nabla u\lVert_{\Phi}^{m}+\lVert
u\lVert_{V,\Phi}^{m}\big{)}-C_{2}\lVert u\lVert_{\Phi_{*}}^{\ell^{*}}.$

Using classical inequality

$\displaystyle(x+y)^{\alpha}\leq
2^{\alpha-1}(x^{\alpha}+y^{\alpha}),\;\;x,y\geq 0\;\;\text{ with
}\;\;\;\alpha>1,$

we concluded that

$\displaystyle J(u)\geq C_{3}\lVert u\lVert_{E}^{m}-C_{3}\lVert
u\lVert_{E}^{\ell^{*}},$

for some positive constant $C_{3}$. Since $0<m<\ell^{*}$, there is $\eta>0$
such that

$\displaystyle J(u)\geq\eta\;\;\;\text{for all}\;\;\lVert u\lVert_{E}=\rho.$

∎

######  Lemma 4.3.

There is $e\in E$ with $\lVert e\lVert_{E}>\rho$ and $J(e)<0$.

Proof: Consider $\psi\in C^{\infty}_{0}(\mathbb{R}^{N})\setminus\\{0\\}$ and
$C_{1}\in\mathbb{R}$ such that

(4.6) $\displaystyle
C_{1}>\xi_{1}(\lVert\psi\lVert_{D^{1,\Phi}(\mathbb{R}^{N})})+\xi_{1}(\lVert\psi\lVert_{V,\Phi}).$

By $(f_{3})$, there exists $C_{2}>0$ satisfying

$\displaystyle F(t)\geq C_{1}|t|^{m}-C_{2},\;\;\;\;\forall t\in\mathbb{R}.$

Thus

$\displaystyle K(x)F(t)\geq C_{1}K(x)|t|^{m}-C_{2}K(x),\;\;\;\;\forall
t\in\mathbb{R}\;\;\text{ and }\;\;x\in\mathbb{R}^{N}.$

That said, considering $t>0$, we have

$\displaystyle J(t\psi)\leq$
$\displaystyle\int_{\mathbb{R}^{N}}\Phi(t|\nabla\psi|)dx+\int_{\mathbb{R}^{N}}V(x)\Phi(t|\psi|)dx-
C_{1}t^{m}\int_{\mathbb{R}^{N}}K(x)|\psi|^{m}dx+C_{3}|supp(\psi)|,$

before that, it follows from the Lemma 2.1 that

$\displaystyle J(t\psi)\leq$
$\displaystyle\xi_{1}(t)\big{(}\xi_{1}(\lVert\psi\lVert_{D^{1,\Phi}(\mathbb{R}^{N})})+\xi_{1}(\lVert\psi\lVert_{V,\Phi})\big{)}-C_{1}t^{m}\int_{\mathbb{R}^{N}}K(x)|\psi|^{m}dx+C_{3}|supp(\psi)|,$

therefore, for $t>1$,

(4.7) $\displaystyle\begin{split}J(t\psi)\leq
t^{m}\big{(}\xi_{1}(\lVert\psi\lVert_{D^{1,\Phi}(\mathbb{R}^{N})})+\xi_{1}(\lVert\psi\lVert_{V,\Phi})\big{)}-C_{1}t^{m}\int_{\mathbb{R}^{N}}K(x)|\psi|^{m}dx+C_{3}|supp(\psi)|.\end{split}$

By (4.6) and (4.7), we can conclude that

$\displaystyle J(t\psi)\longrightarrow-\infty\;\;\text{ as
}\;\;t\rightarrow+\infty.$

The last limit guarantees the existence of $t>0$ large enough that the result
is verified with $e=t\psi$.

∎

In what follows, let us denote by $c>0$ the mountain pass level associated
with $J$, that is,

$\displaystyle c=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}J(\gamma(t))$

where

$\displaystyle\Gamma=\\{\gamma\in C([0,1],X):\;\gamma(0)=0\;\text{ and
}\;\gamma(1)=e\\}.$

Associated with $c$, we have a Cerami sequence $(u_{n})\subset E$, that is,

(4.8) $\displaystyle J(u_{n})\longrightarrow c\;\;\;\;\text{ and
}\;\;\;\;(1+\lVert u_{n}\lVert)\lVert
J^{\prime}(u_{n})\lVert_{*}\longrightarrow 0.$

The above sequence is obtained from the Ghoussoub-Preiss theorem, see [6,
Theorem $5.46$].

To show that the sequences obtained in (4.8), let us prove a Hardy Type
Inequality.

######  Proposition 4.4 (Hardy Type Inequality).

Suppose that $(V,K)\in\mathcal{K}_{1}$, then $E$ is compactly embedded in
$L^{Z}_{K}(\mathbb{R}^{N})$, where $Z(t)=\int_{0}^{|t|}sz(s)ds$ is a
$N$-function satisfying

(4.9) $0<z_{1}\leq\dfrac{t^{2}z(t)}{Z(t)}\leq z_{2},\;\;\;\;\forall t\geq 0,$

where $m<z_{1}\leq z_{2}<\ell^{*}$.

######  Remark 1.

The inequality (4.9) implies the following inequalities

$\xi_{0,Z}(t)Z(\rho)\leq Z(\rho t)\leq\xi_{1,Z}(t)Z(\rho),\leavevmode\nobreak\
\leavevmode\nobreak\ \forall\rho,t\geq 0$

when

$\xi_{0,Z}(t)=\min\\{t^{z_{1}},t^{z_{2}}\\}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ \leavevmode\nobreak\
\xi_{1,Z}(t)=\max\\{t^{z_{1}},t^{z_{2}}\\},\leavevmode\nobreak\
\leavevmode\nobreak\ \forall t\geq 0.$

In the same way of remark 3, we have

$\displaystyle\lim_{t\rightarrow 0}\dfrac{Z(|t|)}{\Phi(|t|)}=0\;\;\;\;\text{ e
}\;\;\;\;\lim_{t\rightarrow\infty}\dfrac{Z(|t|)}{\Phi_{*}(|t|)}=0.$

Proof of Proposition 4.4: We will assume that $(K_{2})$ is true. In this case,
by Remark 1, given $\varepsilon>0$, there are $0<s_{0}<s_{1}$ and $C>0$, such
that

(4.10) $\displaystyle K(x)Z(|s|)\leq\varepsilon
C(V(x)\Phi(|s|)+\Phi_{*}(|s|))+CK(x)\chi_{[s_{0},s_{1}]}(|s|)\Phi_{*}(|s|),$

for all $s\in\mathbb{R}$ and $x\in\mathbb{R}^{N}$. Thus, for $r>0$ large
enough,

(4.11) $\displaystyle\int_{B_{r}(0)^{c}}K(x)|(|u|)dx\leq\varepsilon
CQ(u)+C\Phi_{*}(s_{1})\int_{A_{u}\cap B_{r}(0)^{c}}K(x)dx,\;\;\;\;\forall u\in
E,$

where

$\displaystyle
Q(u)=\int_{\mathbb{R}^{N}}V(x)\Phi(|u|)dx+\int_{\mathbb{R}^{N}}\Phi_{*}(|u|)dx$

and

$\displaystyle A_{u}=\\{x\in\mathbb{R}^{N}:\;s_{0}\leq|u(x)|\leq s_{1}\\}.$

Consider $(v_{n})$ a bounded sequence in $E$. To see that the operator
$i:E\longrightarrow L^{B}_{K}(\mathbb{R}^{N})$ is compact just prove that
$(v_{n})$ has a convergent subsequence on $L^{B}_{K}(\mathbb{R}^{N})$. By
Lemma 3.4, there is $v\in E$ such that $v_{n}\xrightharpoonup[\quad]{\ast}v$
in $D^{1,\Phi}(\mathbb{R}^{N})$, or equivalently
$w_{n}\xrightharpoonup[\quad]{\ast}0$ in $D^{1,\Phi}(\mathbb{R}^{N})$, where
$w_{n}=v_{n}-v$. Still because of the boundedness $M_{1}>0$ verifying

$\displaystyle\int_{\mathbb{R}^{N}}V(x)\Phi(|w_{n}|)dx\leq M_{1}\;\;\;\;\text{
and }\;\;\;\;\int_{\mathbb{R}^{N}}\Phi_{*}(|w_{n}|)dx\leq
M_{1},\;\;\;\;\forall n\in\mathbb{N},$

implying that $(Q(w_{n}))$ is bounded.

On the other hand, defining

$\displaystyle A_{n}=\\{x\in\mathbb{R}^{N}:\;s_{0}\leq|w_{n}(x)|\leq s_{1}\\}$

the last inequality implies that

$\displaystyle\Phi_{*}(s_{0})|A_{n}|\leq\int_{A_{n}}\Phi_{*}(|w_{n}|)dx\leq
M_{1},\;\;\;\;\forall n\in\mathbb{N}.$

With this, we can guarantee that $\sup_{n\in\mathbb{N}}\,|A_{n}|<+\infty$.
Therefore, by $K_{1})$

(4.12) $\displaystyle\int_{A_{n}\cap
B_{r}(0)^{c}}K(x)dx<\dfrac{\varepsilon}{\Phi_{*}(s_{1})},\;\;\;\;\forall
n\in\mathbb{N}.$

From (4.11) and (4.12),

$\displaystyle\int_{B_{r}(0)^{c}}K(x)Z(|w_{n}|)dx$
$\displaystyle\leq\varepsilon CM_{1}+\Phi_{*}(s_{1})\int_{A_{n}\cap
B_{r}(0)^{c}}K(x)dx$ $\displaystyle\leq\varepsilon(CM_{1}+1),\;\;\;\;\forall
n\in\mathbb{N},$

and hence,

(4.13)
$\displaystyle\limsup_{n\to\infty}\int_{B_{r}(0)^{c}}K(x)Z(|w_{n}|)dx\leq\varepsilon(CM_{1}+1).$

Since $E$ is compactly embedded in $L^{\Phi}_{loc}(\mathbb{R}^{N})$, so by the
Corollary 2.9, there is a subsequence of $(v_{n})$, still denoted by itself,
that such

$\displaystyle v_{n}\longrightarrow v\;\;\;\;\text{ in
}\;\;\;\;L^{\Phi}(B_{r}(0)).$

Thus, there is a subsequence of $(v_{n})$, still denoted by itself, verifying

$\displaystyle v_{n}(x)\longrightarrow v(x)\;\;\;\;a.e.\;\text{
in}\;B_{r}(0),$

that is,

$\displaystyle w_{n}(x)\longrightarrow 0\;\;\;\;a.e.\;\text{ in }\;B_{r}(0).$

Consider the functions $P:\mathbb{R}\longrightarrow\mathbb{R}$ and
$Q:\mathbb{R}\longrightarrow\mathbb{R}$ given by

$\displaystyle P(t)=Z(|t|)\;\;\;\;\text{ and }\;\;\;\;Q(t)=\Phi_{*}(|t|).$

Of course $P$ and $Q$ are continuous in addition

$\displaystyle\lim_{|t|\rightarrow+\infty}\dfrac{P(t)}{Q(t)}=0.$

Finally, it follows from the boundedness of $(v_{n})$ in $E$ and from the fact
that $\Phi_{*}\in(\Delta_{2})$ the existence of a constant $C_{1}>0$, so that

$\displaystyle\int_{\mathbb{R}^{N}}Q(w_{n})dx\leq\int_{\mathbb{R}^{N}}\Phi_{*}(|w_{n}|)dx<C_{1},\;\;\;\;\forall
n\in\mathbb{N}.$

Invoking the Lemma Strauss

$\displaystyle\lim_{n\rightarrow\infty}\int_{B_{r}(0)}P(w_{n})dx=0.$

Therefore,

(4.14)
$\displaystyle\limsup_{n\rightarrow\infty}\int_{B_{r}(0)}K(x)Z(|w_{n}|)dx=0.$

According to (4.13) and (4.14), we get

$\displaystyle\limsup_{n\to\infty}\int_{\mathbb{R}^{N}}K(x)Z(|w_{n}|)dx\leq$
$\displaystyle\limsup_{n\rightarrow\infty}\int_{B_{r}(0)}K(x)Z(|w_{n}|)dx$
$\displaystyle+\limsup_{n\to\infty}\int_{B_{r}(0)^{c}}K(x)Z(|w_{n}|)dx$
$\displaystyle\leq$ $\displaystyle\varepsilon(CM_{1}+1).$

By the arbitrariness of $\varepsilon>0$, it follows that

$\displaystyle\limsup_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}K(x)Z(|w_{n}|)dx=0.$

As $Z$ verifies $\Delta_{2}$-condition, we have that

$\displaystyle w_{n}\longrightarrow 0\;\;\;\;\text{ in
}\;\;\;\;L^{Z}_{K}(\mathbb{R}^{N}),$

in other words

$\displaystyle v_{n}\longrightarrow v\;\;\;\;\text{ in
}\;\;\;\;L^{Z}_{K}(\mathbb{R}^{N}),$

showing the result for the case $(K_{2})$.

∎

Next lemma is an important step to prove that the Cerami sequence obtained in
(4.8) is bounded.

######  Lemma 4.5.

Let $(v_{n})$ be a bounded sequence in $E$ such that
$v_{n}\xrightharpoonup[\quad]{\ast}v$ in $D^{1,\Phi}(\mathbb{R}^{N})$ .
Suppose that $f$ satisfies $(f_{1})$ or $(f_{4})$, then

(4.15)
$\displaystyle\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}K(x)F(v_{n})dx=\int_{\mathbb{R}^{N}}K(x)F(v)dx,$
(4.16)
$\displaystyle\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}K(x)f(v_{n})v_{n}dx=\int_{\mathbb{R}^{N}}K(x)f(v)vdx$

and

(4.17)
$\displaystyle\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}K(x)f(v_{n})\psi
dx=\int_{\mathbb{R}^{N}}K(x)f(v)\psi dx,\;\;\;\forall\psi\in
C^{\infty}_{0}(\mathbb{R}^{N}).$

Proof: As in (4.1), given $\varepsilon>0$, there exists $\delta_{0}>0$,
$\delta_{1}>0$ and $C_{\varepsilon}>0$ such that

(4.18) $\displaystyle K(x)f(t)\leq\varepsilon
C_{1}\big{(}V(x)t\phi(t)+t\phi_{*}(t)\big{)}+C_{\varepsilon}K(x)tb(t),\;\;\;\forall
t\geq 0\text{ and }x\in\mathbb{R}^{N}$

where
$C_{1}=\max\\{\lVert{K}\lVert_{\infty},\big{\lVert}\frac{K}{V}\big{\lVert}_{\infty}\\}$.
Hence,

(4.19) $\displaystyle K(x)F(t)\leq\varepsilon
C_{1}(V(x)\Phi(t)+\Phi_{*}(t))+C_{\varepsilon}B(t),\;\;\;\forall t\geq
0\;\text{ and }\;x\in\mathbb{R}^{N}.$

From Proposition (4.4),

(4.20)
$\displaystyle\int_{\mathbb{R}^{N}}K(x)B(v_{n})dx\rightarrow\int_{\mathbb{R}^{N}}K(x)B(v)dx$

then there is $r_{0}>0$, so that

(4.21)
$\displaystyle\int_{B_{{r_{0}}}^{c}(0)}K(x)B(v_{n})dx<\frac{\varepsilon}{C_{\varepsilon}},\;\;\;\forall
n\in\mathbb{N}.$

Moreover, as $(v_{n})$ is bounded in $E$, there is a constant $M_{1}>0$
satisfying

$\displaystyle\int_{\mathbb{R}^{N}}V(x)\Phi(|v_{n}|)dx\leq M_{1}\;\;\;\;\text{
and }\;\;\;\;\int_{\mathbb{R}^{N}}\Phi_{*}(|v_{n}|)dx\leq
M_{1},\;\;\;\;\forall n\in\mathbb{N}.$

Combining the last inequalities with (4.19) and (4.21),

$\displaystyle\left|\int_{B_{r_{0}}^{c}(0)}K(x)F(v_{n})dx\right|\leq\varepsilon(C_{1}M_{1}+1),$

for all $n\in\mathbb{N}$. Therefore

(4.22)
$\displaystyle\limsup_{n\rightarrow+\infty}\int_{B_{r_{0}}^{c}(0)}K(x)F(v_{n})dx\leq\varepsilon(C_{1}M_{1}+1).$

On the other hand, using $(f_{2})$ and the compactness lemma of Strauss [13,
Theorem A.I, p. 338], it follows that

(4.23)
$\displaystyle\lim_{n\rightarrow+\infty}\int_{B_{r_{0}}(0)}K(x)F(v_{n})dx=\int_{B_{r_{0}}(0)}K(x)F(v)dx.$

In light of this, we can conclude that

$\displaystyle\lim_{n\rightarrow+\infty}\int_{\mathbb{R}^{N}}K(x)F(v_{n})dx=\int_{\mathbb{R}^{n}}K(x)F(v)dx.$

To show (4.16), consider $r_{0}>0$ given in (4.21). By (4.18),

$\displaystyle\int_{B_{r_{0}}^{c}(0)}K(x)f(v_{n})v_{n}dx\leq$
$\displaystyle\varepsilon
C_{1}\left(m\int_{B_{r_{0}}^{c}(0)}V(x)\Phi(|v_{n}|)dx+m^{*}\int_{B_{r_{0}}^{c}(0)}\Phi_{*}(|v_{n}|)dx\right)$
$\displaystyle+C_{\varepsilon}z_{2}\int_{B_{r_{0}}^{c}(0)}K(x)Z(v_{n})dx$
$\displaystyle\leq$ $\displaystyle\varepsilon((m^{*}+m)C_{1}M+z_{2}),$

for all $n\in\mathbb{N}$. Therefore

(4.24)
$\displaystyle\limsup_{n\rightarrow+\infty}\int_{B_{r_{0}}^{c}(0)}K(x)f(u_{n})u_{n}dx\leq\varepsilon((m^{*}+m)C_{1}M+z_{2}),.$

On the other hand, using $(f_{1})$ or $(f_{4})$ and the compactness lemma of
Strauss [13, Theorem A.I, p. 338], we can conclude that

(4.25)
$\displaystyle\lim_{n\rightarrow+\infty}\int_{B_{r_{0}}(0)}K(x)f(u_{n})u_{n}dx=0.$

Thus, the limit (4.15) is obtained from (4.24) and (4.25). Related the limit
(4.17), it follows directly from the condition $(f_{1})$ or $(f_{4})$ together
with a version of the compactness lemma of Strauss for non-autonomous problem.

∎

Now, we can prove that the Cerami sequence $(u_{n})$ obtained is bounded.

######  Lemma 4.6.

Let $(u_{n})$ the Cerami sequence given in (4.8). There is a constant $M>0$
such that $J(tu_{n})\leq M$ for every $t\in[0,1]$ and $n\in\mathbb{N}$.

Proof: Let $t_{n}\in[0,1]$ be such that
$J(t_{n}u_{n})=\displaystyle\max_{t\in[0,1]}J(tu_{n})$. If $t_{n}=0$ and
$t_{n}=1$, we are done. Thereby, we can assume $t_{n}\in(0,1)$ ,and so
$J^{\prime}(t_{n}u_{n})u_{n}=0$. From this,

$\displaystyle mJ(t_{n}u_{n})=$ $\displaystyle
mJ(t_{n}u_{n})-J^{\prime}(t_{n}u_{n})(t_{n}u_{n})$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}\big{(}m\Phi(|\nabla(t_{n}u_{n})|)-\phi(|\nabla(t_{n}u_{n})|)|\nabla(t_{n}u_{n})|^{2}\big{)}dx$
$\displaystyle+\int_{\mathbb{R}^{N}}V(x)$
$\displaystyle\big{(}m\Phi(|t_{n}u_{n}|)-\phi(|t_{n}u_{n}|)|t_{n}u_{n}|^{2}\big{)}dx+\int_{\mathbb{R}^{N}}K(x)\mathcal{H}(t_{n}u_{n})dx,$

where $\mathcal{H}(s)=sf(s)-mF(s)$. By $(f_{2})$ the a function
$\mathcal{H}(s)$ is increasing for all $s>0$ and decreasing for $s<0,$ and by
$(\phi_{4})$ the function $s\longmapsto m\Phi(s)-\phi(s)s^{2}$ is increasing
for $s>0$. Thus,

$\displaystyle mJ(tu_{n})\leq mJ(t_{n}u_{n})\leq
mJ(u_{n})-J^{\prime}(u_{n})u_{n}=mJ(u_{n})-o_{n}(1),\;\;\forall t\in[0,1].$

Since $(J(u_{n}))$ is bounded, there is $M>0$ such that

$\displaystyle J(tu_{n})\leq M,\;\;\forall t\in[0,1]\;\text{ and
}\;n\in\mathbb{N}.$

∎

######  Proposition 4.7.

The Cerami sequence $(u_{n})$ given in (4.8) is bounded.

Proof: Suppose for contradiction that, up to a subsequence, $\lVert
u_{n}\lVert_{E}\longrightarrow\infty$ when $n\rightarrow+\infty$, then we have
the following cases:
$i)\;\;\lVert\nabla u_{n}\lVert_{\Phi}\longrightarrow+\infty$ and $(\lVert
u_{n}\lVert_{V,\Phi})$ is bounded
$ii)\;\;\lVert u_{n}\lVert_{V,\Phi}\longrightarrow\infty$ and $(\lVert\nabla
u_{n}\lVert_{\Phi})$ is bounded
$iii)\;\;\lVert\nabla u_{n}\lVert_{\Phi}\longrightarrow+\infty$ and $\lVert
u_{n}\lVert_{V,\Phi}\longrightarrow+\infty$.

In the case $iii)$, consider

$\displaystyle w_{n}=\dfrac{u_{n}}{\lVert u_{n}\lVert_{E}},\;\;\;\;\forall
n\in\mathbb{N}.$

Since $\lVert w_{n}\lVert_{E}=1$, by Lemma 3.4, there exists $w\in E$ such
that $w_{n}\xrightharpoonup[\quad]{\ast}w$ in $D^{1,\Phi}(\mathbb{R}^{N})$.
Now, let us show that $w=0$. Before that, as $J(u_{n})\longrightarrow c$, we
have $J(u_{n})\geq 0$, for every $n$ large enough. Thus, there is
$n_{0}\in\mathbb{N}$ such that

(4.26) $\displaystyle\int_{\mathbb{R}^{N}}\Phi(|\nabla
u_{n}|)dx+\int_{\mathbb{R}^{N}}V(x)\Phi(|u_{n}|)dx\geq\int_{\mathbb{R}^{N}}K(x)F(u_{n})dx,\;\forall
n\geq n_{0}.$

As $\lVert\nabla u_{n}\lVert_{\Phi}\geq 1$ and $\lVert
u_{n}\lVert_{V,\Phi}\geq 1$ for every $n\geq n_{1}$, we have

$\displaystyle\int_{\mathbb{R}^{N}}\Phi(|\nabla u_{n}|)dx\leq\lVert\nabla
u_{n}\lVert_{\Phi}^{m}\;\text{ and
}\;\int_{\mathbb{R}^{N}}V(x)\Phi(|u_{n}|)dx\leq\lVert
u_{n}\lVert_{V,\Phi}^{m},\;\forall n\geq n_{2}.$

So, by (4.26)

$\displaystyle\lVert\nabla u_{n}\lVert_{\Phi}^{m}+\lVert
u_{n}\lVert_{V,\Phi}^{m}\geq\int_{\mathbb{R}^{N}}K(x)F(u_{n})dx,\;\;\;\;\forall
n\geq n_{2}.$

Therefore, there is a constant $C>0$ such that

$\displaystyle C(\lVert\nabla u_{n}\lVert_{\Phi}+\lVert
u_{n}\lVert_{V,\Phi})^{m}\geq\int_{\mathbb{R}^{N}}K(x)F(u_{n})dx,\;\;\;\;\forall
n\geq n_{2},$

or equivalently

$\displaystyle C(\lVert
u_{n}\lVert_{E})^{m}\geq\int_{\mathbb{R}^{N}}K(x)F(u_{n})dx,\;\;\;\;\forall
n\geq n_{2}$

where $n_{2}=\max\\{n_{0},n_{1}\\}$. Thus,

$\displaystyle C\geq\int_{\mathbb{R}^{N}}K(x)\dfrac{F(u_{n})}{\lVert
u_{n}\lVert_{E}^{m}}dx\geq\int_{\mathbb{R}^{N}}K(x)\dfrac{F(u_{n})}{\lvert
u_{n}\lvert^{m}}|w_{n}|^{m}dx.$

The condition $(f_{4})$ implies that for every $\tau>0$, there is $\xi>0$
sufficiently large such that

$\displaystyle\dfrac{F(s)}{|s|^{m}}\geq\tau,\;\;\;\;\forall\;|s|\geq\xi.$

So

$\displaystyle
1+C\geq\int_{\Omega\cap\\{|u_{n}|\geq\xi\\}}K(x)\dfrac{F(u_{n})}{\lvert
u_{n}\lvert^{m}}|w_{n}|^{m}dx\geq\tau\int_{\Omega\cap\\{|u_{n}|\geq\xi\\}}K(x)|w_{n}|^{m}dx,$

where $\Omega=\\{x\in\mathbb{R}^{N}:\;w(x)\neq 0\\}$. By Fatou’s Lemma,

$\displaystyle
1+C\geq\tau\int_{\Omega}K(x)|w(x)|^{m}dx,\;\;\;\;\forall\;\tau>0.$

Therefore,

$\displaystyle\int_{\Omega}K(x)|w(x)|^{m}dx=0.$

As $K(x)>0$ for almost everywhere in $\mathbb{R}^{N}$, we have $w=0$.

Note that for every $M>1$, there is $n_{0}\in\mathbb{N}$ such that,
$\dfrac{M}{\lVert\nabla u_{n}\lVert_{\Phi}}\in[0,1]$, for all $n\geq n_{0}.$
Given this, we get

$\displaystyle J(t_{n}u_{n})\geq$ $\displaystyle J(\dfrac{M}{\lVert\nabla
u_{n}\lVert_{\Phi}}u_{n})=J(\dfrac{M}{\lVert\nabla
u_{n}\lVert_{\Phi}}|u_{n}|)=J(Mw_{n})$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}\Phi(M|\nabla
w_{n}|)dx+\int_{\mathbb{R}^{N}}V(x)\Phi(M|w_{n}|)dx-\int_{\mathbb{R}^{N}}K(x)F(Mw_{n})dx$
$\displaystyle\geq$ $\displaystyle
MQ(u_{n})-\int_{\mathbb{R}^{N}}K(x)F(Mw_{n})dx.$

By definition of the sequence $(w_{n})$, we have $\lVert\nabla
w_{n}\lVert_{\Phi}\leq 1$ and $\lVert w_{n}\lVert_{V,\Phi}\leq 1,$ for all
$n\in\mathbb{N}.$ Then,

$\displaystyle\int_{\mathbb{R}^{N}}\Phi(|\nabla w_{n}|)dx\geq\lVert\nabla
w_{n}\lVert_{\Phi}^{m},\text{ and
}\int_{\mathbb{R}^{N}}V(x)\Phi(|w_{n}|)dx\geq\lVert
w_{n}\lVert_{V,\Phi}^{m},\;\forall\;n\in\mathbb{N}.$

So there is $C>0$ such that

$\displaystyle Q(w_{n})\geq\lVert\nabla w_{n}\lVert_{\Phi}^{m}+\lVert
w_{n}\lVert_{V,\Phi}^{m}\geq C(\lVert\nabla w_{n}\lVert_{\Phi}+\lVert
w_{n}\lVert_{V,\Phi})^{m},\;\;\;\;\forall\;n\in\mathbb{N}.$

Thus

$\displaystyle J(t_{n}u_{n})$ $\displaystyle\geq MC(\lVert
w_{n}\lVert_{E})^{m}-\int_{\mathbb{R}^{N}}K(x)F(Mw_{n})dx$
$\displaystyle=MC-\int_{\mathbb{R}^{N}}K(x)F(Mw_{n})dx.$

By Lemma 4.5,

$\displaystyle\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}K(x)F(Mw_{n})dx=0,$

therefore,

$\displaystyle\liminf_{n\rightarrow\infty}J(t_{n}u_{n})\geq
M,\;\;\;\;\forall\;M\geq 1.$

which constitutes a contradiction with Lemma 4.6, once that $(J(t_{n}u_{n}))$
is bounded from above. Therefore $(u_{n})$ is bounded.

The cases $i)$ and $ii)$ are analogous to the case $iii)$.

∎

Since that the Cerami sequence $(u_{n})$ given in (6.5) is bounded in $E$, by
Lemma 2.8, we can assume that for some subsequence, there is $u\in E$ such
that

(4.27) $\displaystyle u_{n}\xrightharpoonup[\quad]{\ast}u\;\;\text{ in
}D^{1,\Phi}(\mathbb{R}^{N}),\;\;\;\text{ and }\;\;\;u_{n}(x)\longrightarrow
u(x)\;\;a.e.\;\;\mathbb{R}^{N}.$

######  Lemma 4.8.

The Cerami sequence $(u_{n})$ given in (6.5) satisfies

$\int_{\mathbb{R}^{N}}\Phi(|\nabla
u|)dx\leq\displaystyle\liminf_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\Phi(|\nabla
u_{n}|)dx.$

Proof: Consider $\varphi\in L^{\infty}(\mathbb{R}^{N})$ arbitrary. For every
$R>1$, define the function

$\displaystyle\omega_{R}(t)=\left\\{\begin{array}[]{lr}1\;,&\text{se }\;x\in
B_{R}(0)\\\ 0\;,&\text{ se }x\in B_{R}^{c}(0)\end{array}\right..$

It is clear that $\omega_{R}\in E^{\tilde{\Phi}}(\mathbb{R}^{N})$, because
$\omega_{R}\in L^{\infty}(\mathbb{R}^{N})$ and
$supp(\omega_{R})\subset\subset\mathbb{R}^{N}$. As a consequence, we have
$\varphi\omega_{R}\in E^{\tilde{\Phi}}(\mathbb{R}^{N})$. We know that
$L^{\Phi}(\mathbb{R}^{N})\xhookrightarrow[cont\;]{}L^{1}_{loc}(\mathbb{R}^{N})$,
such as $u_{n},u\in D^{1,\Phi}(\mathbb{R}^{N}),$ then

$\displaystyle\dfrac{\partial u_{n}}{\partial x_{i}},\;\dfrac{\partial
u}{\partial x_{i}}\in L^{1}_{loc}(\mathbb{R}^{N}),\;\;\;\;\forall
i=1,2,\cdots,N$

and hence

$\displaystyle\dfrac{\partial u_{n}}{\partial
x_{i}}\omega_{R},\;\dfrac{\partial u}{\partial x_{i}}\omega_{R}\in
L^{1}_{loc}(\mathbb{R}^{N}),\;\;\;\;\forall i=1,2,\cdots,N.$

By (4.27), $u_{n}\xrightharpoonup[\quad]{\ast}u$ in
$D^{1,\Phi}(\mathbb{R}^{N})$, thus

$\displaystyle\int_{\mathbb{R}^{N}}\left(\dfrac{\partial u_{n}}{\partial
x_{i}}\omega_{R}\right)\varphi dx=\int_{\mathbb{R}^{N}}\dfrac{\partial
u_{n}}{\partial
x_{i}}(\omega_{R}\varphi)dx\rightarrow\int_{\mathbb{R}^{N}}\dfrac{\partial
u}{\partial
x_{i}}(\omega_{R}\varphi)dx=\int_{\mathbb{R}^{N}}\left(\dfrac{\partial
u}{\partial x_{i}}\omega_{R}\right)\varphi dx.$

By the arbitrariness of $\varphi$ in $L^{\infty}(\mathbb{R}^{N})$,

$\displaystyle\dfrac{\partial u_{n}}{\partial
x_{i}}\omega_{R}\xrightharpoonup[\quad]{}\dfrac{\partial u}{\partial
x_{i}}\omega_{R},\;\;\text{ in }\;\;L^{1}(\mathbb{R}^{N})$

because $(L^{1}(\mathbb{R}^{N}))^{*}=L^{\infty}(\mathbb{R}^{N})$. Therefore,
applying [15, Theorem $2.1$], we can conclude that

$\displaystyle\int_{B_{R}(0)}\Phi(|\nabla
u|)dx\leq\liminf_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\Phi(|\nabla
u_{n}|\omega_{R})dx\leq\liminf_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\Phi(|\nabla
u_{n}|)dx.$

Passing the limit at $R\rightarrow+\infty$, we get

$\displaystyle\int_{\mathbb{R}^{N}}\Phi(|\nabla
u|)dx\leq\liminf_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\Phi(|\nabla
u_{n}|)dx,$

which completes the proof.

∎

Fix $v\in C^{\infty}_{0}(\mathbb{R}^{N})$. By boundedness of Cerami sequence
$(u_{n})$, we have $J^{\prime}(u_{n})(v-u_{n})=o_{n}(1)$, hence, since $\Phi$
is a convex function, it is possible to show that

(4.28)
$\displaystyle\begin{split}Q(v)-Q(u_{n})\geq\int_{\mathbb{R}^{N}}K(x)f(u_{n})(v-u_{n})dx+o_{n}(1).\end{split}$

For the Lemma 4.8, we have to

$\displaystyle\liminf_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\Phi(|\nabla
u_{n}|)dx\geq\int_{\mathbb{R}^{N}}\Phi(|\nabla u|)dx.$

Now, due to the Fatou’s Lemma, we have

$\displaystyle\liminf_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}V(x)\Phi(|u_{n}|)dx\geq\int_{\mathbb{R}^{N}}V(x)\Phi(|u|)dx.$

Therefore,

(4.29) $\displaystyle\liminf_{n\rightarrow\infty}Q(u_{n})\geq Q(u).$

From (4.28) and (4.29) together with the Lemma 4.5, we get

$\displaystyle
Q(v)-Q(u)\geq\int_{\mathbb{R}^{N}}K(x)f(u)(v-u)dx,\;\;\;\;\forall\;v\in
C^{\infty}_{0}(\mathbb{R}^{N}).$

As $E=\overline{C^{\infty}_{0}(\mathbb{R}^{N})}^{\lVert\cdot\lVert_{E}}$ and
$\Phi\in(\Delta_{2})$, we conclude that

(4.30) $\displaystyle
Q(v)-Q(u)\geq\int_{\mathbb{R}^{N}}K(x)f(u)(v-u)dx,\;\;\;\;\forall\;v\in E.$

In other words, $u$ is a critical point of the $J$ functional. And from
Proposition 4.1, we can conclude that $u$ is a weak solution for $(P)$. Now,
we substitute $v=u^{+}:=\max\\{0,u(x)\\}$ in (4.30) and use (4.3) to get

$\displaystyle-\int_{\mathbb{R}^{N}}\Phi(|\nabla
u^{-}|)dx-\int_{\mathbb{R}^{N}}V(x)\Phi(u^{-})dx\geq\int_{\mathbb{R}^{N}}K(x)f(u)u^{-}dx=0,$

which leads to

$\displaystyle\int_{\mathbb{R}^{N}}\Phi(|\nabla u^{-}|)dx=0\;\;\text{ and
}\;\;\int_{\mathbb{R}^{N}}V(x)\Phi(u^{-})dx=0$

whence it is readily inferred that $u^{-}=0$, therefore, $u$ is a weak
nonnegative solution.

Note that $u$ is nontrivial. In the sense, consider a sequence
$(\varphi_{k})\subset C^{\infty}_{0}(\mathbb{R}^{N})$ such that
$\varphi_{k}\rightarrow u$ in $D^{1,\Phi}(\mathbb{R}^{N})$. Since $(u_{n})$
bounded, we get
$J^{\prime}(u_{n})(\varphi_{k}-u_{n})=o_{n}(1)\lVert\varphi_{k}\lVert-
o_{n}(1)$. As $\Phi$ is convex, we can show that

(4.31) $\displaystyle
Q(\varphi_{k})-Q(u_{n})\geq\int_{\mathbb{R}^{N}}K(x)f(u_{n})(\varphi_{k}-u_{n})dx+o_{n}(1)\lVert\varphi_{k}\lVert-
o_{n}(1).$

Since $(\lVert\varphi_{k}\lVert)_{k\in\mathbb{N}}$ is a bounded sequence, it
follows from (4.31) and from Lemma (4.5) that

$\displaystyle
Q(\varphi_{k})\geq\limsup_{n\to\infty}Q(u_{n})+\int_{\mathbb{R}^{N}}K(x)f(u)(\varphi_{k}-u)dx.$

Now, note that being $\Phi\in(\Delta_{2})$ and $\varphi_{k}\rightarrow u$ in
$E$, we conclude from the inequality above that

(4.32) $\displaystyle Q(u)\geq\limsup_{n\rightarrow\infty}Q(u_{n}).$

From (4.29) and (4.32),

(4.33) $\displaystyle\lim_{n\rightarrow\infty}Q(u_{n})=Q(u).$

By Lemma 4.5, we have

(4.34)
$\displaystyle\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}K(x)F(u_{n})dx=\int_{\mathbb{R}^{N}}K(x)F(u)dx,$

Therefore, by (4.8), (4.33) and (4.34), we conclude

$\displaystyle
0<c=\lim_{n\rightarrow\infty}J(u_{n})=\int_{\mathbb{R}^{N}}\Phi(|\nabla
u|)dx-\int_{\mathbb{R}^{N}}K(x)F(u)dx=J(u),$

that is, $u\neq 0$.

###  4.1. Proof of Theorem 1.1

Assuming the assumptions of Theorem 1.1, the above argument guarantees the
existence of a nontrivial weak solution for problem $(P)$. This subsection
makes heavy use of fact that $\Phi\in\mathcal{C}_{m}$ to study the regularity
of nonnegative solutions of the problem $(P)$, thereby showing Theorem 1.1.

We begin by presenting a technical result, which can be found in [2].

######  Lemma 4.9.

Let $u\in E$ be a nonnegative solution of $(P)$, $x_{0}\in\mathbb{R}^{N}$ and
$R_{0}>0$. Then

$\displaystyle\int_{\mathcal{A}_{k,t}}|\nabla u|^{m}dx\leq
C\left(\int_{\mathcal{A}_{k,s}}\left|\dfrac{u-k}{s-t}\right|^{m^{*}}dx+(k^{m^{*}}+1)|\mathcal{A}_{k,s}|\right)$

where $0<t<s<R_{0}$, $k>1$, $\mathcal{A}_{k,\rho}=\\{x\in
B_{\rho}(x_{0}):u(x)>k\\}$ and $C>0$ is a constant that does not depend on
$k$.

Proof: Let $u\in E$ be a weak solution nonnegative of $(P)$ and
$x_{0}\in\mathbb{R}^{N}$. Moreover, fix $0<t<s<R_{0}$ and $\zeta\in
C^{\infty}_{0}(\mathbb{R}^{N})$ verifying

$\displaystyle 0\leq\zeta\leq 1,\quad supp(\zeta)\subset
B_{s}(x_{0}),\quad\zeta\equiv 1\;\text{ on }\;B_{t}(x_{0})\;\;\text{ and
}\;\;|\nabla\zeta|\leq\dfrac{2}{s-t}.$

For $k>1$, set $\varphi=\zeta^{m}(u-k)^{+}$ and

$\displaystyle J=\int_{\mathcal{A}_{k,s}}\Phi(|\nabla u|)\zeta^{m}dx,$

Using $\varphi$ as a test function and $\ell\Phi(t)\leq\phi(t)t^{2}$, we find

$\displaystyle\ell J\leq
m\int_{\mathcal{A}_{k,s}}\zeta^{m-1}(u-k)^{+}\phi(|\nabla u|)|\nabla
u||\nabla\zeta|dx-\int_{\mathcal{A}_{k,s}}V(x)\phi(u)u\zeta^{m}(u-k)^{+}dx$
$\displaystyle+\int_{\mathcal{A}_{k,s}}K(x)f(u)\zeta^{m}(u-k)^{+}dx.$

As in (4.1), given $\eta>0$, there exists $C_{\varepsilon}>0$ such that

$\displaystyle K(x)f(t)\leq\eta
C_{1}V(x)t\phi(t)+C_{\varepsilon}t\phi_{*}(t),\;\;\;\forall t\geq 0\text{ and
}x\in\mathbb{R}^{N}.$

where $C_{1}=\lVert{K}\lVert_{\infty}$. Thus, setting $\eta=\frac{1}{C_{1}}$,
there exists a constant $C_{2}>0$ such that

(4.35) $\displaystyle\ell J\leq
m\int_{\mathcal{A}_{k,s}}\zeta^{m-1}(u-k)^{+}\phi(|\nabla u|)|\nabla
u||\nabla\zeta|dx+C_{2}\int_{\mathcal{A}_{k,s}}\phi_{*}(|u|)u\zeta^{m}(u-k)^{+}dx.$

For each $\tau\in(0,1)$, the Young’s inequalities gives

(4.36) $\displaystyle\begin{split}\phi(|\nabla u|)|\nabla
u||\nabla\zeta|\zeta^{m-1}(u-k)^{+}\leq\tilde{\Phi}(\phi(|\nabla u|)|\nabla
u|\zeta^{m-1}\tau)+C_{3}\Phi\Big{(}\Big{|}\dfrac{u-k}{s-t}\Big{|}\Big{)}.\end{split}$

It follows from Lemma 2.4,

(4.37) $\displaystyle\tilde{\Phi}(\phi(|\nabla u|)|\nabla
u|\zeta^{m-1}\tau)\leq C_{4}(\tau\zeta^{m-1})^{\frac{m}{m-1}}{\Phi}(|\nabla
u|).$

From (4.35), (4.36) and (4.37),

$\displaystyle\ell J\leq
mC_{4}\tau^{\frac{m}{m-1}}\int_{\mathcal{A}_{k,s}}\Phi(|\nabla
u|)\zeta^{m}+mC_{3}\int_{\mathcal{A}_{k,s}}\Phi\Big{(}\Big{|}\dfrac{u-k}{s-t}\Big{|}\Big{)}dx+C_{2}\int_{\mathcal{A}_{k,s}}u\phi_{*}(u)\zeta^{m}(u-k)^{+}dx.$

Choosing $\tau\in(0,1)$ such that $0<mC_{4}\tau^{\frac{m}{m-1}}<\ell$, we
derive

(4.38) $\displaystyle J\leq
C_{5}\int_{\mathcal{A}_{k,s}}\Phi\Big{(}\Big{|}\dfrac{u-k}{s-t}\Big{|}\Big{)}dx+C_{5}\int_{\mathcal{A}_{k,s}}u\phi_{*}(u)\zeta^{m}(u-k)^{+}dx.$

By Young’s inequalities,

(4.39) $\displaystyle u\phi_{*}(u)\zeta^{m}(u-k)^{+}\leq
C_{6}\Phi_{*}\left(\Big{|}\dfrac{u-k}{s-t}\Big{|}\right)+C_{6}\Phi_{*}(k).$

Therefore, a combination of (4.38) and (4.39), yields

(4.40) $\displaystyle J\leq
C_{7}\int_{\mathcal{A}_{k,s}}\Phi\Big{(}\Big{|}\dfrac{u-k}{s-t}\Big{|}\Big{)}dx+C_{7}\int_{\mathcal{A}_{k,s}}\Phi_{*}\Big{(}\Big{|}\dfrac{u-k}{s-t}\Big{|}\Big{)}dx+C_{7}\int_{\mathcal{A}_{k,s}}\Phi_{*}(k)dx.$

Now, using that $\ell\leq m<m^{*}$ and applying the Lemmas 2.1 and 2.2 for
functions $\Phi$ and $\Phi_{*},$ respectivamente, we get

$\displaystyle\Phi\Big{(}\Big{|}\dfrac{u-k}{s-t}\Big{|}\Big{)}\leq\Phi(1)\Big{(}\Big{|}\dfrac{u-k}{s-t}\Big{|}^{m^{*}}+1\Big{)},\;\Phi_{*}\Big{(}\Big{|}\dfrac{u-k}{s-t}\Big{|}\Big{)}\leq\Phi_{*}(1)\Big{(}\Big{|}\dfrac{u-k}{s-t}\Big{|}^{m^{*}}+1\Big{)}\text{
and }\Phi_{*}(k)\leq(k^{m^{*}}+1).$

From (4.40) and the inequality above,

$\displaystyle J\leq
C_{8}\left(\int_{\mathcal{A}_{k,s}}\Big{|}\dfrac{u-k}{s-t}\Big{|}^{m^{*}}dx+(k^{m^{*}}+1)|\mathcal{A}_{k,s}|\right)$

The result follows from the fact that $\Phi\in\mathcal{C}_{m}$.

∎

######  Lemma 4.10.

Let $u\in E$ be a nonnegative solution of $(P)$. Then, $u\in
L^{\infty}_{loc}(\mathbb{R}^{N})$.

Proof: To begin with, consider $\Lambda$ a compact subset on $\mathbb{R}^{N}$.
Fix $R_{1}\in(0,1)$ and $x_{0}\in\Lambda$. Given $K>1$, define the sequences

$\displaystyle\sigma_{n}=\dfrac{R_{1}}{2}+\dfrac{R_{1}}{2^{n+1}},\;\;\overline{\sigma}_{n}=\dfrac{\sigma_{n}+\sigma_{n+1}}{2}\;\;\text{
and }\;\;K_{n}=\dfrac{K}{2}\left(1-\dfrac{1}{2^{n+1}}\right).$

For every $n\in\mathbb{N}$, we consider

$\displaystyle
J_{n}=\int_{\mathcal{A}_{K_{n},\sigma_{n}}}\big{(}(u-K_{n})^{+}\big{)}^{m^{*}}dx\;\;\text{
and
}\;\;\xi_{n}=\xi\left(\dfrac{2^{n+1}}{R_{1}}\Big{(}|x-x_{0}|-\dfrac{R_{1}}{2}\Big{)}\right),\;\;x\in\mathbb{R}^{N}$

with $\xi\in C^{1}(\mathbb{R})$ satisfying

$\displaystyle 0\leq\xi\leq 1,\quad\xi(t)=1\;\text{ for
}\;t\leq\dfrac{1}{2}\quad\text{ and }\quad\xi(t)=0\;\text{ for
}\;t\geq\dfrac{3}{4}..$

Repeating the same arguments used in the proof of Lemma 3.6 in [2], there
exists a constant $C=C(N,m,R_{1})>0$ such that

(4.41) $\displaystyle\begin{split}J_{n+1}\leq
CD^{n}J_{n}^{1+\omega},\;\;n=0,1,2,\cdots\end{split}$

where $D=2^{(m+m^{*}){\frac{m^{*}}{m}}}$ and $\omega=\frac{m^{*}}{m}-1$.

Note that

(4.42)
$\displaystyle\begin{split}J_{0}&=\int_{\mathcal{A}_{K_{0},{\sigma_{0}}}}\big{(}(u-K_{0})^{+}\big{)}^{m^{*}}dx\leq\int_{B_{R_{1}}(x_{0})}\big{(}(u-K_{0})^{+}\big{)}^{m^{*}}dx\end{split}$

Then, by the Lebesgue’s Theorem,
$\displaystyle\lim_{K\rightarrow\infty}J_{0}=0$, from where it follows that

$\displaystyle J_{0}\leq
C^{-\frac{1}{\omega}}D^{-\frac{1}{\omega^{2}}},\;\;\text{ for all }\;K\geq
K^{*}$

for some $K^{*}\geq 1$ that depends on $x_{0}$. Fix $K=K^{*}$. Thus, by [27,
Lemma 4.7], we deduce that

$\displaystyle J_{n}\longrightarrow 0\;\;\text{ as}\;\;n\rightarrow\infty.$

On the other hand,

$\displaystyle\lim_{n\rightarrow\infty}J_{n}=\lim_{n\rightarrow\infty}\int_{\mathcal{A}_{K_{n},\sigma_{n}}}\big{(}(u-K_{0})^{+}\big{)}^{m^{*}}dx=\int_{\mathcal{A}_{\frac{K^{*}}{2},\frac{R_{1}}{2}}}\big{(}(u-\frac{K^{*}}{2})^{+}\big{)}^{m^{*}}dx$

Hence,

$\displaystyle\int_{\mathcal{A}_{\frac{K^{*}}{2},\frac{R_{1}}{2}}}\big{(}(u-\frac{K^{*}}{2})^{+}\big{)}^{m^{*}}dx=0,$

leading to

$\displaystyle u(x)\leq\frac{K^{*}}{2},\;\;\text{ a.e. }\text{in
}B_{\frac{R_{1}}{2}}(x_{0}).$

Since $x_{0}$ is arbitrary and $\Lambda$ is a compact subset, the last
inequality ensures that

$\displaystyle u(x)\leq\frac{M}{2},\;\;\text{ a.e. }\text{in }\Lambda$

for some constant $M>0$. By the arbitrariness of $\Lambda$, we conclude that
$u\in L^{\infty}_{loc}(\mathbb{R}^{N})$.

∎

This Lemmas guarantees that the Theorem 1.1 is valid. The proof of Theorem 1.2
will be divided into the following Lemmas:

######  Lemma 4.11.

$u\in C^{1,\alpha}_{loc}(\mathbb{R}^{N})$.

###### Proof.

It is enough to apply the regularity theorem due to Lieberman [27, Theorem
1.7]. And this is possible due to the condition $(\phi_{6})$. ∎

######  Corollary 4.12.

Let $u\in E$ be a nonnegative solution of $(P)$. Then, $u$ is positive
solution.

Proof: If $\Omega\subset\mathbb{R}^{N}$ is a bounded domain, the Lemma 4.10
implies that $u\in C^{1}(\overline{\Omega})$. Using this fact, in the sequel,
we fix $M_{1}>\max\big{\\{}\lVert\nabla
u\lVert_{L^{\infty}(\overline{\Omega})},1\big{\\}}$ and

$\displaystyle\varphi(t)=\left\\{\begin{array}[]{lr}\;\;\;\;\phi(t)\;\;\;,\;\;\;\;\;\;\text{
for }0<t\leq M_{1}\vspace*{0.2cm}\\\
\dfrac{\phi(M_{1})}{M_{1}^{\beta-2}}t^{\beta-2}\;,\text{ for }t\geq
M_{1}\end{array}\right.,$

where $\beta$ is given in the hypothesis $(\phi_{5})$. Still by the condition
$(\phi_{5})$, there are $\alpha_{1},\alpha_{2}>0$ satisfying

(4.43)
$\displaystyle\varphi(|y|)|y|^{2}=\phi(|y|)|y|^{2}\geq\alpha_{1}|y|^{\beta}\;\;\;\text{
and }\;\;\;|\varphi(|y|)y|\leq\alpha_{2}|y|^{\beta-1},\;\;\;\forall
y\in\mathbb{R}^{N}.$

Now, consider the vector measurable functions
$G:\Omega\times\mathbb{R}\times\mathbb{R}^{N}\longrightarrow\mathbb{R}^{N}$
given by $\linebreak G(x,t,p)=\frac{1}{\alpha_{1}}\varphi(|p|)p$. From (4.43),

(4.44)
$\displaystyle|G(x,t,p)|\leq\frac{\alpha_{2}}{\alpha_{1}}|p|^{\beta-1}\;\;\;\text{
and }\;\;\;pG(x,t,p)\geq|p|^{\beta-1},\;\;\text{ for all
}\;(x,t,p)\in\Omega\times\mathbb{R}\times\mathbb{R}^{N}.$

We next will consider the scalar measurable function
$B:\Omega\times\mathbb{R}\times\mathbb{R}^{N}\longrightarrow\mathbb{R}$ given
by $G(x,t,p)=\frac{1}{\alpha_{1}}\big{(}V(x)\phi(|t|)t-K(x)f(t)\big{)}$.
Remember that from the inequality (4.1), there will be a constant $C_{1}>0$
satisfying

(4.45) $\displaystyle K(x)|f(t)|\leq
C_{1}V(x)\phi(|t|)|t|+C_{1}\phi_{*}(|t|)|t|,\;\;\;\forall
t\in\mathbb{R}^{N}\;\text{ and }\;x\in\mathbb{R}^{N}.$

Fix $M\in(0,\infty)$. Through the condition $(\phi_{5})$ and by a simple
computation yields there exists $C_{2}=C_{2}(M)>0$ verifying

$\displaystyle|B(x,t,p)|\leq C_{2}|t|^{\beta-1},\;\text{ for every
}\;(x,t,p)\in\Omega\times(-M,M)\times\mathbb{R}^{N}.$

By the arbitrariness of $M$, we can conclude that functions $G$ and $B$
fulfill the structure required by Trudinger [26]. Also, as $u$ is a weak
solution of $(P)$, we infer that $u$ is a quasilinear problem solution

$\displaystyle-div\,G(x,u,\nabla u(x))+B(x,u,\nabla u(x))=0\;\text{ in
}\Omega.$

By [26, Theorem 1.1], we deduce that $u>0$ in $\Omega$. As $\Omega$ arbitrary,
we conclude that $u>0$ in $\mathbb{R}^{N}$.

∎

###  4.2. Proof of Theorem 1.3

Assuming the assumptions of Theorem 1.3, the arguments at the beginning of
this section 4, guarantees the existence of a nontrivial weak solution for
problem $(P)$. Then, to prove the Theorem 1.3, it is enough to show the
regularity of nonnegative solutions of the problem $(P)$. Following the same
arguments as the proof of Lemma 4.9, we state the following result.

######  Lemma 4.13.

Let $u\in E$ be a nonnegative solution of $(P)$, $x_{0}\in\mathbb{R}^{N}$ and
$R_{0}>0$. Then

$\displaystyle\int_{\mathcal{A}_{k,t}}|\nabla u|^{\ell}dx\leq
C\left(\int_{\mathcal{A}_{k,s}}\left|\dfrac{u-k}{s-t}\right|^{\ell^{*}}dx+(k^{\ell^{*}}+1)|\mathcal{A}_{k,s}|\right)$

where $0<t<s<R_{0}$, $k>1$, $\mathcal{A}_{k,\rho}=\\{x\in
B_{\rho}(x_{0}):u(x)>k\\}$ and $C>0$ is a constant that does not depend on
$k$.

The following result is proved in an entirely analogous way to Lemma 4.10, it
is enough to change $m$ for $\ell$ in every proof.

######  Lemma 4.14.

Let $u\in E$ be a nonnegative solution of $(P)$. Then, $u\in
L^{\infty}_{loc}(\mathbb{R}^{N})$.

Arguing as in the proof of Corollary 4.12, the nonnegative solution of $(P)$
is positive solution. Showing the Theorem 1.3.

######  Remark 2.

In the Theorems 1.1 and 1.3, assuming that
$\displaystyle\inf_{x\in\mathbb{R}^{N}}V(x)>0$, the nonnegatives solution for
$(P)$ and they go to zero at infinity, that is,
$\displaystyle\lim_{|x|\rightarrow\infty}u(x)=0$.

##  5\. Proof of Theorems 1.4 and 1.5

In the beginning, we are going to present the ardy Type Inequality for case
$(K,V)\in\mathcal{K}_{2}$.

######  Remark 3.

The inequality (1.2) implies the following inequalities

$\xi_{0,A}(t)A(\rho)\leq A(\rho t)\leq\xi_{1,A}(t)A(\rho),\leavevmode\nobreak\
\leavevmode\nobreak\ \forall\rho,t\geq 0$

when

$\xi_{0,A}(t)=\min\\{t^{a_{1}},t^{a_{2}}\\}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{and }\leavevmode\nobreak\ \leavevmode\nobreak\
\xi_{1,A}(t)=\max\\{t^{a_{1}},t^{a_{2}}\\},\leavevmode\nobreak\
\leavevmode\nobreak\ \forall t\geq 0.$

Besides by Lemma 2.1 and Lemma 2.2, we have

$\displaystyle\lim_{t\rightarrow 0}\dfrac{A(t)}{\Phi(t)}=0\;\;\text{ and
}\;\;\lim_{|t|\rightarrow\infty}\dfrac{A(t)}{\Phi_{*}(t)}=0$

######  Proposition 5.1 (Hardy Type Inequality).

If $(V,K)\in\mathcal{K}_{2}$, then $E$ is compactly embedded
in$L^{A}_{K}(\mathbb{R}^{N})$ where $A$ is given in $K_{3}$.

Proof: As $E$ is continuously embedded in $L^{\Phi_{*}}(\mathbb{R}^{N})$,
there exists $C_{1}>0$ such that

(5.1) $\displaystyle\lVert u\lVert_{\Phi_{*}}\leq C_{1}\lVert
u\lVert_{E},\;\;\forall u\in E.$

By $(K_{3})$ given $\varepsilon>0$ there is $r>0$ large enough such that

(5.2) $\displaystyle K(x)A(s)\leq\varepsilon(V(x)\Phi(|s|)+\Phi_{*}(|s|)),$

for all $s\in\mathbb{R}$ and $|x|\geq r$. On the other hand, by the Remark 3,
there is a constant $C_{2}>0$ such that

$\displaystyle A(t)\leq C_{2}\Phi(t)+C_{2}\Phi_{*}(t),\;\;\;\;\forall t>0.$

Hence, for each $x\in B_{r}(0)$,

(5.3) $\displaystyle\begin{split}K(x)A(t)\leq
C_{2}\left\lVert\frac{K}{V}\right\lVert_{L^{\infty}(B_{r}(0))}V(x)\Phi(t)+C_{2}\lVert
K\lVert_{\infty}\Phi_{*}(t),\;\;\;\;\forall t>0.\end{split}$

Combining (5.2) and (5.3),

(5.4)
$\displaystyle\begin{split}K(x)A(t)\leq&C_{3}V(x)\Phi(t)+C_{3}\Phi_{*}(t),\;\;\;\;\forall
t>0\;\text{ and }\;x\in\mathbb{R}^{N}\end{split}$

with $C_{3}=\max\\{1,C_{2}\lVert
K\lVert_{\infty},C_{2}\left\lVert\frac{K}{V}\right\lVert_{L^{\infty}(B_{r}(0))}\\}$.
By the inequalities (5.1) and (5.4), we get

$\displaystyle\int_{\mathbb{R}^{N}}K(x)A\left(\dfrac{|u|}{C_{3}\lVert
u\lVert_{E}+C_{1}\lVert u\lVert_{E}}\right)dx\leq
C_{3}\int_{\mathbb{R}^{N}}V(x)\Phi\left(\dfrac{|u|}{\lVert
u\lVert_{V,\Phi}}\right)dx+C_{3}\int_{\mathbb{R}^{N}}\Phi_{*}\left(\dfrac{|u|}{\lVert
u\lVert_{\Phi_{*}}}\right)dx\leq C_{4}$

where $C_{4}$ is a positive constant that does not depend on $u$. So we can
conclude that $E\subset L^{A}_{K}(\mathbb{R}^{N})$.

Now, consider $(v_{n})$ a bounded sequence in $E$. To see that the operator
$i:E\longrightarrow L^{A}_{K}(\mathbb{R}^{N})$ is compact just prove that
$(v_{n})$ has a convergent subsequence on $L^{A}_{K}(\mathbb{R}^{N})$. Since
$(v_{n})$ is bounded in $E$, we have that $(v_{n})$ is bounded in
$D^{1,\Phi}(\mathbb{R}^{N})$, so there is $u\in E$ such that
$v_{n}\xrightharpoonup[\quad]{\ast}v$ in $D^{1,\Phi}(\mathbb{R}^{N})$, or
equivalently $w_{n}\xrightharpoonup[\quad]{\ast}0$ in
$D^{1,\Phi}(\mathbb{R}^{N})$, where $w_{n}=v_{n}-v$. By the limitation of
$(v_{n})$ in $E$ and $\Phi,\Phi_{*}\in(\Delta_{2})$, there is $M_{1}>0$ such
that

(5.5) $\displaystyle\int_{\mathbb{R}^{N}}V(x)\Phi(|w_{n}|)dx\leq
M_{1},\;\text{ and }\;\int_{\mathbb{R}^{N}}\Phi_{*}(|w_{n}|)dx\leq
M_{1},\;\forall n\in\mathbb{N}.$

Thus, by (5.2) and (5.5), we obtain

(5.6) $\displaystyle\int_{B_{r}(0)^{c}}K(x)A(|w_{n}|)dx\leq\varepsilon
M_{1},\;\;\;\;\forall n\in\mathbb{N}.$

Again, we use the Corollary 2.9 and the fact that $E$ is compactly embedded in
$L^{\Phi}_{loc}(\mathbb{R}^{N})$ to ensure the existence of a subsequence of
$(v_{n})$, still denoted by itself, such that

$\displaystyle v_{n}\longrightarrow v\;\;\;\;\text{ in
}\;\;\;\;L^{\Phi}(B_{r}(0)).$

Thus, there is a subsequence of $(v_{n})$, still denoted by itself, that such

$\displaystyle v_{n}(x)\longrightarrow v(x)\;\;\;\;a.e.\;\text{ in
}\;B_{r}(0),$

that is,

$\displaystyle w_{n}(x)\longrightarrow 0\;\;\;\;a.e.\;\text{ in }\;B_{r}(0).$

Consider the functions $P:\mathbb{R}\longrightarrow\mathbb{R}$ and
$Q:\mathbb{R}\longrightarrow\mathbb{R}$ given by

$\displaystyle P(t)=A(|t|)\;\;\;\;\text{ e }\;\;\;\;Q(t)=\Phi_{*}(|t|).$

Clearly $P$ and $Q$ are continuous, moreover

$\displaystyle\lim_{|t|\rightarrow+\infty}\dfrac{P(t)}{Q(t)}=0.$

Finally, it follows from the limitation of $(v_{n})$ in $E$ that there is
$C_{1}>0$, such that

$\displaystyle\int_{\mathbb{R}^{N}}Q(w_{n})dx\leq\int_{\mathbb{R}^{N}}\Phi_{*}(|w_{n}|)dx<C_{1}.$

Then, by compactness Lemma of Strauss [13, Theorem A.I, p. $338$],

$\displaystyle\int_{B_{r}(0)}P(w_{n})dx\longrightarrow 0.$

Therefore,

(5.7)
$\displaystyle\lim_{n\rightarrow\infty}\int_{B_{r}(0)}K(x)A(|w_{n}|)dx=0.$

By (5.6) e (5.7), we have

$\displaystyle\limsup_{n\to\infty}\int_{\mathbb{R}^{N}}K(x)A(|w_{n}|)dx\leq$
$\displaystyle\limsup_{n\rightarrow\infty}\int_{B_{r}(0)}A(|w_{n}|)dx$
$\displaystyle+\limsup_{n\to\infty}\int_{B_{r}(0)^{c}}K(x)A(|w_{n}|)dx$
$\displaystyle\leq$ $\displaystyle\varepsilon(CM_{1}+1).$

By the arbitrariness of $\varepsilon>0$, it follows that

$\displaystyle\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}K(x)A(|w_{n}|)dx=0.$

As $A\in(\Delta_{2})$,

$\displaystyle w_{n}\longrightarrow 0\;\;\;\;\text{ in
}\;\;\;\;L^{A}_{K}(\mathbb{R}^{N}),$

in other words

$\displaystyle v_{n}\longrightarrow v\;\;\;\;\text{ in
}\;\;\;\;L^{A}_{K}(\mathbb{R}^{N}),$

which completes the proof.

∎

Note that the condition $(f_{6})$ implies that
$\displaystyle\lim_{t\rightarrow+\infty}\frac{f(t)}{\phi_{*}(t)t}=0$. Then, by
the conditions $(f_{5})$ or $(f_{6})$, given $\varepsilon>0$ there exists
$\delta_{0}>0$, $\delta_{1}>0$ and $C_{\varepsilon}>0$ such that

(5.8) $\displaystyle K(x)|f(t)|\leq\varepsilon
K(x)a(t)t+\varepsilon\lVert{K}\lVert_{\infty}\phi_{*}(t)t+C_{\varepsilon}K(x)\phi_{*}(t)t\chi_{[\delta_{0},\delta_{1}]}(t),$

for all $t\geq 0$ and $x\in\mathbb{R}^{N}$. This inequality together with
Lemma 5.1 yields that the functional $\mathcal{F}:E\longrightarrow\mathbb{R}$,
given by

(5.9) $\mathcal{F}(u)=\int_{\mathbb{R}^{N}}K(x)F(u)dx$

is well defined and belongs to $C^{1}(E,\mathbb{R})$ with derivative

$\mathcal{F}^{\prime}(u)v=\int_{\mathbb{R}^{N}}K(x)f(u)vdx,\quad\forall u,v\in
E.$

Therefore, we can conclude that the energy functional
$J:E\longrightarrow\mathbb{R}$ associated to problem $(P)$, which is given by

$J(u)=\int_{\mathbb{R}^{N}}\Phi(|\nabla
u|)dx+\int_{\mathbb{R}^{N}}V(x)\Phi(|u|)dx-\int_{\mathbb{R}^{N}}K(x)F(u)dx.$

is a continuous and Gateaux-differentiable functional such that
$J^{\prime}:E\longrightarrow E^{*}$ given by

$\displaystyle J^{\prime}(u)v=\int_{\mathbb{R}^{N}}\phi(|\nabla u|)\nabla
u\nabla
vdx+\int_{\mathbb{R}^{N}}V(x)\phi(|u|)uvdx-\int_{\mathbb{R}^{N}}K(x)f(u)vdx$

is continuous from the norm topology of $E$ to the weak∗-topology of $E^{*}$.
Once that we intend to find nonnegative solutions, we will assume that

(5.10) $\displaystyle f(s)=0,\;\;\;\forall s\in(-\infty,0].$

From (5.8) and $(f_{3})$, it follows that $J$ satisfies the geometry of the
mountain pass. Hence, there is a Cerami sequence $(u_{n})\subset E$, such
that,

(5.11) $\displaystyle J(u_{n})\longrightarrow c\;\;\;\;\text{ and
}\;\;\;\;(1+\lVert u_{n}\lVert)\lVert
J^{\prime}(u_{n})\lVert_{*}\longrightarrow 0$

where $c$ is the mountain pass level given by

$\displaystyle c=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}J(\gamma(t))$

with

$\displaystyle\Gamma=\\{\gamma\in C([0,1],X):\;\gamma(0)=0\;\text{ and
}\;J(\gamma(1))\leq 0\\}.$

As in the previous section, the above sequence is obtained from the Ghoussoub-
Preiss theorem, see [6, Theorem $5.46$].

In order to show that the Cerami sequence obtained in (5.11) is bounded, we
present the following result.

######  Lemma 5.2.

Let $(v_{n})$ be a bounded sequence in $E$ such that
$v_{n}\xrightharpoonup[\quad]{\ast}v$ in $D^{1,\Phi}(\mathbb{R}^{N})$ .
Suppose that $f$ satisfies $(f_{5})$ or $(f_{6})$, then

(5.12)
$\displaystyle\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}K(x)F(v_{n})dx=\int_{\mathbb{R}^{N}}K(x)F(v)dx,$
(5.13)
$\displaystyle\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}K(x)f(v_{n})v_{n}dx=\int_{\mathbb{R}^{N}}K(x)f(v)vdx$

and

(5.14)
$\displaystyle\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}K(x)f(v_{n})\psi
dx=\int_{\mathbb{R}^{N}}K(x)f(v)\psi dx,\;\;\;\forall\psi\in
C^{\infty}_{0}(\mathbb{R}^{N}).$

Proof: As in (5.8), given $\varepsilon>0$, there exists $\delta_{0}>0$,
$\delta_{1}>0$ $C_{1}>0$ and $C_{\varepsilon}>0$ such that

(5.15) $\displaystyle K(x)|f(t)|\leq
C_{1}K(x)a(t)t+\varepsilon\lVert{K}\lVert_{\infty}\phi_{*}(t)t+C_{\varepsilon}K(x)\phi_{*}(t)t\chi_{[\delta_{0},\delta_{1}]}(t),$

for all $t\geq 0$ and $x\in\mathbb{R}^{N}$. By the condition $(K_{3})$, there
is $r_{0}>0$ sufficiently large satisfying

$\displaystyle
K(x)A(t)\leq\varepsilon\left(V(x){\Phi(t)}+{\Phi_{*}(t)}\right),\;\;\;\;\forall
t>0\;\text{ and }\;|x|\geq r_{0}.$

From the above inequalities, we have

(5.16) $\displaystyle K(x)F(t)\leq\varepsilon C_{1}V(x)\Phi(t)+\varepsilon
C_{2}\Phi_{*}(t)+C_{\varepsilon}K(x)\Phi_{*}(\delta_{1})\chi_{[\delta_{0},\delta_{1}]}(t),\;\forall
t>0\text{ and }|x|\geq r_{0}.$

Repeating the same arguments used in the proof of Proposition 2.1, it follows
that

(5.17)
$\displaystyle\limsup_{n\rightarrow+\infty}\int_{B_{r_{0}}^{c}(0)}K(x)F(v_{n})dx\leq\varepsilon
C_{3},$

for some constant $C_{3}>0$ that does not depend on $n$ and $\varepsilon$. On
the other hand, the compactness lemma of Strauss [13, Theorem A.I, p. 338],
guarantees that

(5.18)
$\displaystyle\lim_{n\rightarrow+\infty}\int_{B_{r_{0}}(0)}K(x)F(v_{n})dx=\int_{B_{r_{0}}(0)}K(x)F(v)dx.$

In light of this, we can conclude that

$\displaystyle\lim_{n\rightarrow+\infty}\int_{\mathbb{R}^{N}}K(x)F(v_{n})dx=\int_{\mathbb{R}^{n}}K(x)F(v)dx.$

In the same way, we can get the limit (5.13). Related the limit (5.14), it
follows directly from the condition $(f_{5})$ or $(f_{6})$ together with a
version of the compactness lemma of Strauss for non-autonomous problem.

∎

Repeating the same arguments used in the proof of Lemma 4.6 and of Proposition
4.7, it follows that the Cerami sequence $(u_{n})$ given in (5.11) is bounded,
up to some subsequence, we can assume that there is $u\in E$ such that

(5.19) $\displaystyle u_{n}\xrightharpoonup[\quad]{\ast}u\;\;\text{ in
}D^{1,\Phi}(\mathbb{R}^{N}),\;\;\;\text{ and }\;\;\;u_{n}(x)\longrightarrow
u(x)\;\;a.e.\;\;\mathbb{R}^{N}.$

As in the previous section, we can conclude that $u$ is a nonnegative solution
and $u\neq 0$. It remains to study the sign of the nonnegative solution for
the problem $(P)$.

###  5.1. Proof of Theorem 1.4

Assuming the assumptions of Theorem 1.4, the existence of a nontrivial weak
solution for problem $(P)$ is posed by the arguments presented above. In this
subsection, as in the subsection 4.1, we heavily use the fact that
$\Phi\in\mathcal{C}_{m}$ to study the regularity and positiveness of
nonnegative solutions of the problem $(P)$, therefore showing Theorem 1.4.

We begin by presenting a technical result, but crucial for the study that
follows.

######  Lemma 5.3.

Let $u\in E$ be a nonnegative solution of $(P)$, $x_{0}\in\mathbb{R}^{N}$ and
$R_{0}>0$. Then

$\displaystyle\int_{\mathcal{A}_{k,t}}|\nabla u|^{m}dx\leq
C\left(\int_{\mathcal{A}_{k,s}}\left|\dfrac{u-k}{s-t}\right|^{m^{*}}dx+(k^{m^{*}}+1)|\mathcal{A}_{k,s}|\right)$

where $0<t<s<R_{0}$, $k>1$, $\mathcal{A}_{k,\rho}=\\{x\in
B_{\rho}(x_{0}):u(x)>k\\}$ and $C>0$ is a constant that does not depend on
$k$.

Proof: Let $u\in E$ be a weak solution nonnegative of $(P)$ and
$x_{0}\in\mathbb{R}^{N}$. Moreover, fix $0<t<s<R_{0}$. By the condition
$(f_{5})$ together with the Remark 3, given $\varepsilon>0$ there exists
$C_{\varepsilon}>0$ such that

(5.20) $\displaystyle K(x)|f(t)|\leq\varepsilon
C_{1}V(x)\phi(t)t+C_{\varepsilon}\phi_{*}(t)t,\;\;\;\forall x\in
B_{R_{0}}(x_{0})$

where
$C_{1}=\Big{\lVert}\frac{K}{V}\Big{\lVert}_{L^{\infty}({B_{R_{0}}(x_{0})})}$.
Now, we can repeat the same arguments used in the proof of Lemma 4.9.

∎

We consider $u\in E$ a nonnegative solution of $(P)$, repeating the same
arguments used in the proof of Lemma 4.10, it follows that $u\in
C^{1,\alpha}_{loc}(\mathbb{R}^{N})$. Also, $u>0$ in $\mathbb{R}^{N}$.Since the
proof of this fact is done in an analogous way to Corollary (4.12), we will
omit their respective proof here.

###  5.2. Proof of Theorem 1.5

Here, we are considering the assumptions of Theorem 1.3, the arguments at the
beginning of this section 5, guarantees the existence of a nontrivial weak
solution for problem $(P)$. Then, to prove the Theorem 1.3, it is enough to
show the regularity and positiveness of nonnegative solutions of the problem
$(P)$. In this case, by the condition $(f_{6})$ together with the Remark 3,
given $\varepsilon>0$, $x_{0}\in\mathbb{R}^{N}$ and $R_{0}>0$, there exists
$C_{\varepsilon}>0$ such that

(5.21) $\displaystyle K(x)|f(t)|\leq\varepsilon
C_{1}V(x)\phi(t)t+C_{\varepsilon}b(t)t,\;\;\;\forall x\in B_{R_{0}}(x_{0})$

where
$C_{1}=\Big{\lVert}\frac{K}{V}\Big{\lVert}_{L^{\infty}({B_{R_{0}}(x_{0})})}$.
Therefore, without great difficulties, we can show the following result:

######  Lemma 5.4.

Let $u\in E$ be a nonnegative solution of $(P)$, $x_{0}\in\mathbb{R}^{N}$ and
$R_{0}>0$. Then

$\displaystyle\int_{\mathcal{A}_{k,t}}|\nabla u|^{\ell}dx\leq
C\left(\int_{\mathcal{A}_{k,s}}\left|\dfrac{u-k}{s-t}\right|^{\ell^{*}}dx+(k^{\ell^{*}}+1)|\mathcal{A}_{k,s}|\right)$

where $0<t<s<R_{0}$, $k>1$, $\mathcal{A}_{k,\rho}=\\{x\in
B_{\rho}(x_{0}):u(x)>k\\}$ and $C>0$ is a constant that does not depend on
$k$.

We consider $u\in E$ a nonnegative solution of $(P)$, the preceding lemma
guarantees that $u\in C^{1,\alpha}_{loc}(\mathbb{R}^{N})$. Also, $u>0$ in
$\mathbb{R}^{N}$. Since the proof of these facts are made in an analogous way
to Lemmas 4.10 and Corollary (4.12), we will omit their respective proof here.

######  Remark 4.

In the Theorems 1.4 and 1.5, assuming that
$\displaystyle\inf_{x\in\mathbb{R}^{N}}V(x)>0$, the nonnegatives solution for
$(P)$ and they go to zero at infinity, that is,
$\displaystyle\lim_{|x|\rightarrow\infty}u(x)=0$.

##  6\. Zero mass case

In this section, we consider the zero mass case, and give the proof of Theorem
1.6. Here, the coefficient $K$ only satisfies the conditions $(K_{0})$ and
$(K_{1})$, furthermore, we are considering $V=0$. For $(f_{7})$, there is
$C>0$ such that

(6.1) $\displaystyle K(x)|f(t)|\leq C\phi_{*}(|t|)|t|,\quad
x\in\mathbb{R}^{N}\;\text{ and }\;t\in\mathbb{R}.$

The inequality above yields that
$\mathcal{F}:D^{1,\Phi}(\mathbb{R}^{N})\longrightarrow\mathbb{R}$ defined by

(6.2) $\mathcal{F}(u)=\int_{\mathbb{R}^{N}}K(x)F(u)dx$

is well defined, $\mathcal{F}\in C^{1}(D^{1,\Phi}(\mathbb{R}^{N}),\mathbb{R})$
with derivative

$\mathcal{F}^{\prime}(u)v=\int_{\mathbb{R}^{N}}K(x)f(u)vdx,\quad\forall u,v\in
D^{1,\Phi}(\mathbb{R}^{N}).$

As in the previous sections, the energy functional
$J:D^{1,\Phi}(\mathbb{R}^{N})\longrightarrow\mathbb{R}$ associated with the
problem $(P)$, which is given by

$\displaystyle J(u)=\int_{\mathbb{R}^{N}}\Phi(|\nabla
u|)dx-\int_{\mathbb{R}^{N}}K(x)F(u)dx,$

is continuous and Gateaux-differentiable with derivative
$J^{\prime}:D^{1,\Phi}(\mathbb{R}^{N})\longrightarrow(D^{1,\Phi}(\mathbb{R}^{N}))^{*}$
defined by

$\displaystyle J^{\prime}(u)v=\int_{\mathbb{R}^{N}}\phi(|\nabla u|)\nabla
u\nabla vdx-\int_{\mathbb{R}^{N}}K(x)f(u)vdx$

continuous from the norm topology of $D^{1,\Phi}(\mathbb{R}^{N})$ to the
weak∗-topology of $(D^{1,\Phi}(\mathbb{R}^{N}))^{*}$.

Once that we intend to find nonnegative solutions for the problem $(P)$, we
will assume that

(6.3) $\displaystyle f(s)=0,\;\;\;\forall s\in(-\infty,0].$

We will say that $u\in D^{1,\Phi}(\mathbb{R}^{N})$ is a critical point for the
functional $J:D^{1,\Phi}(\mathbb{R}^{N})\longrightarrow\mathbb{R}$ if

(6.4) $\displaystyle
Q(v)-Q(u)\geq\int_{\mathbb{R}^{N}}{K(x)f(u)(v-u)}dx,\;\;\;\;\forall v\in
D^{1,\Phi}(\mathbb{R}^{N}).$

Similar to the proof of [5, Lemma 4.1], we give the following result.

######  Proposition 6.1.

If $u\in D^{1,\Phi}(\mathbb{R}^{N})$ is a critical point of $J$ in
$D^{1,\Phi}(\mathbb{R}^{N})$, then u is a weak solution to $(P)$.

Also due to the inequality (6.1), it follows that $J$ satisfies the geometry
of the mountain pass. Therefore, by [6, Theorem $5.46$], there is a sequence
$(u_{n})\subset D^{1,\Phi}(\mathbb{R}^{N})$ such that

(6.5) $\displaystyle J(u_{n})\longrightarrow c\;\;\;\;\text{ and
}\;\;\;\;(1+\lVert u_{n}\lVert)\lVert J^{\prime}(u_{n})\lVert\longrightarrow
0.$

where $c$ is the mountain pass level given by

$\displaystyle c=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}J(\gamma(t))$

with

$\displaystyle\Gamma=\\{\gamma\in C([0,1],X):\;\gamma(0)=0\;\text{ and
}\;\gamma(1)=e\\}.$

Now, similar to the proof of Lemma 4.6, we give the following Lema.

######  Lemma 6.2.

Let $(u_{n})$ the Cerami sequence given in (6.5). There is a constant $M>0$
such that $J(tu_{n})\leq M$ for every $t\in[0,1]$ and $n\in\mathbb{N}$.

######  Proposition 6.3.

The Cerami sequence $(u_{n})$ given in (6.5) is bounded in
$D^{1,\Phi}(\mathbb{R}^{N})$.

Proof: Suppose for contradiction that, up to a subsequence, $\lVert
u_{n}\lVert_{D^{1,\Phi}(\mathbb{R}^{N})}\longrightarrow\infty$ and set
$w_{n}=\frac{u_{n}}{\lVert\nabla u_{n}\lVert_{\Phi}}$. Since $\lVert\nabla
w_{n}\lVert_{\Phi}=1$, there is $w\in D^{1,\Phi}(\mathbb{R}^{N})$ such that
$w_{n}\xrightharpoonup[\quad]{\ast}w$ in $D^{1,\Phi}(\mathbb{R}^{N})$ and
$w_{n}(x)\rightarrow w(x)$ almost everywhere in $\mathbb{R}^{N}$. As in the
proof of Proposition 4.7, for each $\tau>0$, there is $\xi>0$ sufficiently
large such that

$\displaystyle
1\geq\tau\int_{\Omega\cap\\{|u_{n}|\geq\xi\\}}K(x)|w_{n}|^{m}dx,\;\;\;\forall
n\geq n_{0}$

for some $n_{0}\in\mathbb{N}$ where $\Omega=\\{x\in\mathbb{R}^{N}:\;w(x)\neq
0\\}$. By Fatou’s Lemma

$\displaystyle 1\geq\tau\int_{\Omega}K(x)|w|^{m}dx,\;\;\text{ for all
}\tau>0.$

Therefore,

$\displaystyle\int_{\Omega}K(x)|w(x)|^{m}dx=0.$

By $(K_{0})$, we have $K(x)>0$ almost everywhere in $\mathbb{R}^{N}$,
therefore $|\Omega|=0$, showing that $w=0$.

Note that for every $\sigma>1$, there is $n_{0}\in\mathbb{N}$ such that
$\dfrac{\sigma}{\lVert\nabla u_{n}\lVert_{\Phi}}\in[0,1]$, for all $n\geq
n_{0}.$ Given this, we get

$\displaystyle J(t_{n}u_{n})\geq\sigma\int_{\mathbb{R}^{N}}\Phi(|\nabla
w_{n}|)dx-\int_{\mathbb{R}^{N}}K(x)F(\sigma
w_{n})dx=\sigma-\int_{\mathbb{R}^{N}}K(x)F(\sigma w_{n})dx,\;\;\;\forall n\geq
n_{0}.$

We claim that

(6.6) $\displaystyle\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}K(x)F(\sigma
w_{n})dx=0,$

We postpone for a moment the verification of (6.6). But if it were true, one
would get

$\displaystyle\liminf_{n\rightarrow\infty}J(t_{n}u_{n})\geq\sigma,\;\text{ for
every }\sigma\geq 1,$

which constitutes a contradiction with Lemma 6.2, once that $(J(t_{n}u_{n}))$
is bounded from above. Therefore $(u_{n})$ is bounded.

Now, it remains to prove the limit (6.6). From $(f_{7})$, given
$\varepsilon>0$, there exists $\delta_{0}>0$, $\delta_{1}>0$ and $C_{1}>0$
such that

(6.7)
$\displaystyle|f(t)|\leq\varepsilon\phi_{*}(|t|)|t|+C_{1}\chi_{[\delta_{0},\delta_{1}]}(t),\;\;\text{
for }t\in\mathbb{R}.$

Note that $F_{n}=\\{x\in\mathbb{R}^{N}:|\sigma w_{n}(x)|\geq\delta_{0}\\}$ is
such that

$\displaystyle\Phi_{*}(\delta_{0})|F_{n}|\leq\int_{F_{n}}\Phi_{*}(|\sigma
w_{n}(x)|)dx\leq\int_{\mathbb{R}^{N}}\Phi_{*}(|\sigma w_{n}(x)|)dx\leq
C_{2}\int_{\mathbb{R}^{N}}\Phi_{*}(|w_{n}(x)|)dx,$

for some constant $C_{2}>0$ that does not depend on $n$. Since $(w_{n})$ is
bounded in $D^{1,\Phi}(\mathbb{R}^{N})$, there is $C_{3}>0$ satisfying
$\int_{\mathbb{R}^{N}}\Phi_{*}(|w_{n}|)dx<C_{3}$, for all $n\in\mathbb{N}$.
Thus, $\displaystyle\sup_{n\in\mathbb{N}}|F_{n}|<+\infty$. From $(K_{1})$, we
have

$\displaystyle\lim_{r\rightarrow+\infty}\int_{F_{n}\cap
B_{r}^{c}(0)}K(x)dx=0,\;\text{ uniformly in }n\in\mathbb{N},$

thus, there is $r_{0}>0$, so that

$\displaystyle\int_{F_{n}\cap
B_{{r_{0}}}^{c}(0)}K(x)dx<\dfrac{\varepsilon}{C_{1}},\;\;\;\forall
n\in\mathbb{N}.$

By (6.7), it follows that

$\displaystyle\int_{B_{r_{0}}^{c}(0)}K(x)F(\sigma w_{n})dx$
$\displaystyle\leq\varepsilon\lVert
K\lVert_{\infty}\int_{B_{r_{0}}^{c}(0)}\Phi_{*}(|\sigma
w_{n}(x)|)dx+C_{1}\int_{F_{n}\cap B_{r_{0}}^{c}(0)}K(x)dx$
$\displaystyle\leq\varepsilon(\lVert K\lVert_{\infty}+1),$

for all $n\in\mathbb{N}$. Therefore

(6.8)
$\displaystyle\limsup_{n\rightarrow+\infty}\int_{B_{r_{0}}^{c}(0)}K(x)F(\sigma
w_{n})dx\leq\varepsilon(\lVert K\lVert_{\infty}+1).$

On the other hand, using $(f_{2})$ and the compactness lemma of Strauss [13,
Theorem A.I, p. 338], we can conclude that

(6.9) $\displaystyle\lim_{n\rightarrow+\infty}\int_{B_{r_{0}}(0)}K(x)F(\sigma
w_{n})dx=0.$

Thus, the limit (6.6) is obtained from (6.8) and (6.9).

∎

Since that the Cerami sequence $(u_{n})$ given in (6.5) is bounded in
$D^{1,\Phi}(\mathbb{R}^{N})$, by Lemma 2.8, we can assume that for some
subsequence, there is $u\in D^{1,\Phi}(\mathbb{R}^{N})$ such that

(6.10) $\displaystyle u_{n}\xrightharpoonup[\quad]{\ast}u\;\;\text{ in
}D^{1,\Phi}(\mathbb{R}^{N}),\;\;\;\text{ and }\;\;\;u_{n}(x)\longrightarrow
u(x)\;\;a.e.\;\;\mathbb{R}^{N}.$

Given these limits, we can obtain, as in Lemma 4.8, the inequality

(6.11) $\displaystyle\int_{\mathbb{R}^{N}}\Phi(|\nabla
u|)dx\leq\displaystyle\liminf_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\Phi(|\nabla
u_{n}|)dx.$

To complete the proof of Theorem 1.6, we need to show the following limits:

(6.12)
$\displaystyle\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}K(x)f(u_{n})u_{n}dx=\int_{\mathbb{R}^{N}}K(x)f(u)udx$

and

(6.13)
$\displaystyle\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}K(x)f(u_{n})\psi
dx=\int_{\mathbb{R}^{N}}K(x)f(u)\psi dx,\;\;\;\forall\psi\in
C^{\infty}_{0}(\mathbb{R}^{N}).$

Consider $\varepsilon,\delta_{0},\delta_{1},C_{1}>0$ as in (6.7). To verify
(6.12), consider $E_{n}=\\{x\in\mathbb{R}^{N}:|u_{n}(x)|\geq\delta_{0}\\}$
which satisfies $\sup_{n}|E_{n}|<\infty$. From $(K_{1})$, there is $r_{0}>0$,
so that

$\displaystyle\int_{E_{n}\cap
B_{{r_{0}}}^{c}(0)}K(x)dx<\dfrac{\varepsilon}{\delta_{1}C_{1}},\;\;\;\forall
n\in\mathbb{N}.$

By (6.7), it follows that

$\displaystyle\int_{B_{r_{0}}^{c}(0)}K(x)f(u_{n})u_{n}dx$
$\displaystyle\leq\varepsilon m^{*}\lVert
K\lVert_{\infty}\int_{B_{r_{0}}^{c}(0)}\Phi_{*}(|u_{n}|)dx+C_{1}\delta_{1}\int_{F_{n}\cap
B_{r_{0}}^{c}(0)}K(x)dx$ $\displaystyle\leq\varepsilon(m^{*}\lVert
K\lVert_{\infty}+1),$

for all $n\in\mathbb{N}$. Therefore

(6.14)
$\displaystyle\limsup_{n\rightarrow+\infty}\int_{B_{r_{0}}^{c}(0)}K(x)f(u_{n})u_{n}dx\leq\varepsilon(m^{*}\lVert
K\lVert_{\infty}+1).$

On the other hand, using $(f_{7})$ and the compactness lemma of Strauss [13,
Theorem A.I, p. 338], we can conclude that

(6.15)
$\displaystyle\lim_{n\rightarrow+\infty}\int_{B_{r_{0}}(0)}K(x)f(u_{n})u_{n}dx=0.$

Thus, the limit (6.12) is obtained from (6.14) and (6.15). Related the limit
(6.13), it follows directly from the condition $(f_{7})$ together with a
version of the compactness lemma of Strauss for non-autonomous problem.

Fix $v\in C^{\infty}_{0}(\mathbb{R}^{N})$. By boundedness of Cerami sequence
$(u_{n})$, we have $J^{\prime}(u_{n})(v-u_{n})=o_{n}(1)$, hence, since $\Phi$
is a convex function, it is possible to show that

(6.16) $\displaystyle\begin{split}\int_{\mathbb{R}^{N}}\Phi(|\nabla
v|)dx-\int_{\mathbb{R}^{N}}\Phi(|\nabla
u_{n}|)dx\geq\int_{\mathbb{R}^{N}}K(x)f(u_{n})(v-u_{n})dx+o_{n}(1).\end{split}$

From (6.11), (6.16) and (6.13), we obtain

$\displaystyle\int_{\mathbb{R}^{N}}\Phi(|\nabla
v|)dx-\int_{\mathbb{R}^{N}}\Phi(|\nabla
u|)dx\geq\int_{\mathbb{R}^{N}}K(x)f(u)(v-u)dx.$

By the arbitrariness of $v\in C^{\infty}_{0}(\mathbb{R}^{N})$, it follows that

$\displaystyle\int_{\mathbb{R}^{N}}\Phi(|\nabla
v|)dx-\int_{\mathbb{R}^{N}}\Phi(|\nabla
u|)dx\geq\int_{\mathbb{R}^{N}}K(x)f(u)(v-u)dx,\;\;\;\;\forall\;v\in
C^{\infty}_{0}(\mathbb{R}^{N}).$

Since
$D^{1,\Phi}(\mathbb{R}^{N})=\overline{C^{\infty}_{0}(\mathbb{R}^{N})}^{\lVert\cdot\lVert_{D^{1,\Phi}(\mathbb{R}^{N})}}$
and $\Phi\in(\Delta_{2})$, we conclude that

(6.17) $\displaystyle\int_{\mathbb{R}^{N}}\Phi(|\nabla
v|)dx-\int_{\mathbb{R}^{N}}\Phi(|\nabla
u|)dx\geq\int_{\mathbb{R}^{N}}K(x)f(u)(v-u)dx,\;\;\;\;\forall\;v\in
D^{1,\Phi}(\mathbb{R}^{N}).$

In other words, $u$ is a critical point of the $J$ functional. And from
Proposition 6.1, we can conclude that $u$ is a weak solution for $(P)$. Now,
we substitute $v=u^{+}:=\max\\{0,u(x)\\}$ in (6.17) and use (6.3) to get

$\displaystyle-\int_{\mathbb{R}^{N}}\Phi(|\nabla
u^{-}|)dx\geq\int_{\mathbb{R}^{N}}K(x)f(u)u^{-}dx=0,$

which leads to

$\displaystyle\int_{\mathbb{R}^{N}}\Phi(|\nabla u^{-}|)dx=0,$

which yields $\lVert\nabla u^{-}\lVert_{\Phi}=0$. Hence, given the inequality
(2.4) we can conclude that $u^{-}=0$, and therefore, $u$ is a weak nonnegative
solution.

It remains to show that $u$ is nontrivial. For this, consider a sequence
$(\varphi_{k})\subset C^{\infty}_{0}(\mathbb{R}^{N})$ such that
$\varphi_{k}\rightarrow u$ in $D^{1,\Phi}(\mathbb{R}^{N})$. Since $(u_{n})$ is
a limited Cerami sequence, it follows that
$J^{\prime}(u_{n})(\varphi_{k}-u_{n})=o_{n}(1)\lVert\varphi_{k}\lVert-
o_{n}(1)$. As $\Phi$ is convex, it is possible to show that

(6.18)
$\displaystyle\int_{\mathbb{R}^{N}}\Phi(|\nabla\varphi_{k}|)dx-\int_{\mathbb{R}^{N}}\Phi(|\nabla
u_{n}|)dx\geq\int_{\mathbb{R}^{N}}K(x)f(u_{n})(\varphi_{k}-u_{n})dx+o_{n}(1)\lVert\varphi_{k}\lVert-
o_{n}(1).$

Because $(\lVert\varphi_{k}\lVert)_{k\in\mathbb{N}}$ is a bounded sequence, it
follows from (6.12), (6.13) and (6.18) that

$\displaystyle\int_{\mathbb{R}^{N}}\Phi(|\nabla\varphi_{k}|)dx\geq\limsup_{n\to\infty}\int_{\mathbb{R}^{N}}\Phi(|\nabla
u_{n}|)dx+\int_{\mathbb{R}^{N}}K(x)f(u)(\varphi_{k}-u)dx.$

Knowing that $\Phi\in(\Delta_{2})$ and $\varphi_{k}\rightarrow u$ in
$D^{1,\Phi}(\mathbb{R}^{N})$, the last inequality ensures that

(6.19) $\displaystyle\int_{\mathbb{R}^{N}}\Phi(|\nabla
u|)dx\geq\limsup_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\Phi(|\nabla
u_{n}|)dx.$

From (6.11) and (6.19),

(6.20)
$\displaystyle\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\Phi(|\nabla
u_{n}|)dx=\int_{\mathbb{R}^{N}}\Phi(|\nabla u|)dx.$

Repeating the same arguments used in the proof of limit (6.6), that

(6.21)
$\displaystyle\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}K(x)F(u_{n})dx=\int_{\mathbb{R}^{N}}K(x)F(u)dx,$

Therefore, by (6.20) and (6.21), we conclude

$\displaystyle
0<c=\lim_{n\rightarrow\infty}J(u_{n})=\int_{\mathbb{R}^{N}}\Phi(|\nabla
u|)dx-\int_{\mathbb{R}^{N}}K(x)F(u)dx=J(u),$

that is, $u\neq 0$.

##  7\. Acknowledgments

The authors are grateful to the Paraíba State Research Foundation (FAPESQ),
Brazil, and the Conselho Nacional de Desenvolvimento Científico e Tecnológico
(CNPq), Brazil, whose funds partially supported this paper.

## Declarations

  * •

Ethical Approval . Not applicable.

  * •

Competing interests. On behalf of all authors, the corresponding author states
that there is no Competing interests.

  * •

Authors Contributions. Lucas da Silva wrote the main manuscript text under
supervision of Marco Souto. All authors reviwed the manuscript.

  * •

Funding. This study was financed in part by the Paraíba State Research
Foundation (FAPESQ), Brazil. Marco Souto was partially supported by the
Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Brazil,
309.692/2020-2.

  * •

Availability of data and materials . Data sharing and materials not applicable
to this article as no datasets and materials were generated or analysed during
the current study.

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