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\title{Standing wave solutions of a quasilinear degenerate Schr\"odinger equation with unbounded potential}
%Please write the title of your paper here. Please use capital letters only for the first letter of the title and for proper names

%Please write the full names of all authors here, using the order first name - middle names (optionally initials only) - last (family) name.
\author[1]{\textbf{Nejmeddine Chorfi}}
\author[2]{\textbf{Vicen\c tiu D. R\u adulescu}\footnote{Corresponding author. Email: vicentiu.radulescu@math.cnrs.fr}}%Please write the email address of the corresponding author here
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\affil[1]{Department of Mathematics, College of Sciences, King Saud University, Riyadh, Saudi Arabia}
\affil[2]{Institute of Mathematics ``Simion Stoilow" of the Romanian Academy, P.O. Box 1-764,
          014700 Bucharest, Romania \& Department of Mathematics, University of Craiova, Street A.I. Cuza No. 13,
          200585 Craiova, Romania}
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\newcommand\shorttitle{Standing waves of a quasilinear degenerate Schr\"odinger equation}


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\begin{document}

\maketitle


\pagestyle{plain}



\begin{center}
\noindent
\begin{minipage}{0.85\textwidth}

\medskip
\begin{center}
\noindent Received XX XXXXXX 2016, appeared XX XXXXXX 2016 % Please do not change this line, it will be edited by the technical editors.
\smallskip

\noindent Communicated by Patrizia Pucci
\end{center}
\bigskip

{\small{
\noindent {\bf Abstract.} We are concerned with the existence of entire distributional nontrivial solutions for a new class of nonlinear partial differential equations. The differential operator was introduced by A.~Azzolini {\it et al.} \cite{azo1,azo2} and it is described by a potential with different growth near zero and at infinity. The main result generalizes a property established by P.~Rabinowitz in relationship with the existence of nontrivial standing waves of the Schr\"odinger equation with lack of compactness. The proof combines arguments based on the mountain pass and energy estimates.}
\smallskip

% Please enter at most 6 keywords here with lowercase letters separated by commas.
\noindent {\bf{Keywords:}} nonlinear Schr\"odinger equation; nonhomogeneous differential operator; mountain pass.
\smallskip

% Please enter at most 5 Mathematics Subject Classification codes here. Please use 2010 classification codes, which can be found on the following link: http://www.ams.org/msc//msc2010.html.
\noindent{\bf{2010 Mathematics Subject Classification:}} Primary: 35J60. Secondary: 58E05.
}
\end{minipage}
\end{center}




\section{Introduction}
The Schr\"odinger equation has a basic role in quantum theory and it plays the role of Newton's conservation laws  of energy in classical
mechanics, that is, it predicts the future behaviour of a dynamical system. The {\it linear}  Schr\"odinger equation
 provides a thorough description of particles in a non-relativistic setting.
The structure of the
{\it nonlinear} form of the Schr\"odinger equation is much more complicated.
The most common applications
 of this equation vary from Bose-Einstein condensates and nonlinear optics,
 stability of Stokes waves in water, propagation of the  electric field in optical fibers
to the self-focusing and
collapse of Langmuir waves in plasma physics
and the behaviour of deep water waves and
freak waves (or rogue waves)
in the ocean.
The nonlinear Schr\"odinger equation also describes phenomena arising in
the theory of Heisenberg ferromagnets and magnons,
self-channelling of a high-power
ultra-short laser in matter,  condensed matter theory,
 dissipative quantum mechanics,  electromagnetic fields, plasma physics
(e.g., the Kurihara superfluid film equation). We refer to
M.J.~Ablowitz, B.~Prinari and A.D.~Trubatch \cite{abl}, T.~Cazenave \cite{cazenave},
C.~Sulem and P.L.~Sulem \cite{sulem} for a modern overview and relevant applications.

In a seminal paper, P. Rabinowitz \cite{rabi} showed how variational arguments based on the mountain pass theorem can be applied to obtain existence results for nonlinear Schr\"odinger-type equations  with lack of compactness. P.~Rabinowitz \cite{rabi} studied the nonlinear Schr\"odinger equation
\bb\label{eq1}
-\Delta u+a(x)u=f(x,u)\quad\mbox{in}\ \RR^N\ (N\geq 3),
\bbb
where $a$ is a positive potential and $f$ has a subcritical growth. The existence of nontrivial standing waves of problem \eqref{eq1} strongly
relies on the mountain pass theorem.

We point out that the mountain pass theorem was established by A.~Ambrosetti and P.~Rabinowitz \cite{ambrab} and it is a basic tool in nonlinear analysis. The limiting version of this result, which corresponds to {\it mountains of zero altitude} is due to P.~Pucci and J. Serrin \cite{pucser1,pucser2,pucser3}.
We also refer to H.~Brezis and L.~Nirenberg \cite{brenir} who proved a version of the mountain pass theorem that includes the limiting case corresponding to mountains of zero altitude. Their proof combines
a pseudo-gradient lemma, an original perturbation argument and the Ekeland variational
principle \cite{ekeland}.
For further related results and applications of the mountain pass theorem, we refer to  Y.~Jabri \cite{jabri}, I.~Peral \cite{peral},
 P.~Pucci and V.~R\u adulescu \cite{pucrad}, D. Repov\v{s} \cite{repaa}, and M. Ro\c siu \cite{rosiu}.

Problems like \eqref{eq1} are obtained by substituting the ansatz
$$\psi (x,t)=\exp (-iEt/\hbar )u(x)$$
into the Schr\"odinger equation
$$i\hbar\psi_t=-\frac{\hbar^2}{2m}\,\Delta\psi+V(x)\psi -\gamma |\psi|^{p-1}\psi\quad\mbox{in}\ \RR^N\times (0,\infty),$$
where $1<p<(N+2)/(N-2)$. Here, $\hbar$ is the Plank constant divided by $2\pi$, $\psi$ is the wave function, $m$ is the magnetic quantum number, $V$ is the potential energy, and $\gamma$ is a constant that depends on the number of particles.

Under natural hypotheses, P. Rabinowitz \cite{rabi} proved that problem \eqref{eq1} has a nontrivial distributional solution. This result was generalized by F.~Gazzola and V.~R\u adulescu \cite{gazrad} in two nonsmooth settings (Degiovanni and Clarke theories) and by M.~Mih\u ailescu and V.~R\u adulescu \cite{mrad} in the framework of singular potentials of Hardy or Caffarelli-Kohn-Nirenberg type.

In some recent papers, A.~Azzollini {\it et al.} \cite{azo1,azo2} introduced a new class of differential operators with a variational structure. They considered nonhomogeneous operators of the type $$\divv [\phi'(|\nabla u|^2)\nabla u],$$
where $\phi\in C^1(\RR_+,\RR_+)$ has a different growth near zero and at infinity.
Such a behaviour occurs if $\phi(t)=2[\sqrt{1+t}-1]$, which corresponds to the prescribed mean curvature operator (capillary surface operator), which is defined by
$$\divv\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right).$$
More generally, $\phi(t)$ behaves like $t^{q/2}$ for small $t$ and $t^{p/2}$ for large $t$, where $1<p<q<N$. Such a behaviour is fulfilled if
$$\phi(t)=\frac 2p\,[(1+t^{q/2})^{p/q}-1],$$
which generates the differential operator
$$\divv \left[(1+|\nabla u|^q)^{(p-q)/q}|\nabla u|^{q-2}\nabla u \right]\,.$$

The main purpose of this paper is to study problem \eqref{eq1} in the new abstract setting introduced by A.~Azzollini {\it et al.} \cite{azo1,azo2}.
In the next section, we introduce the main hypotheses and we state the basic result of this paper. The proof and related comments are developed in the final section of this paper.

\section{The main result}
We study the following quasilinear Schr\"odinger equation
\bb\label{eq2}
-\divv [\phi'(|\nabla u|^2)\nabla u]+a(x)|u|^{\alpha-2}u=f(x,u)\quad\mbox{in}\ \RR^N\ (N\geq 3).
\bbb

Throughout this paper we assume that $\alpha$, $p$, $q$, and $s$ are real numbers satisfying the following properties:
\bb\label{pq}
\left\{\begin{array}{ll}
&\di 1<p<q<N\\
&\di 1<\alpha<\frac{p^*q'}{p'}\\
&\di \max\{\alpha ,q\}<s<p^*:=\frac{pN}{N-p},
\end{array}\right.
\bbb
where $p'$ denotes the conjugate exponent of $p$, that is, $p'=p/(p-1)$.

We assume that the potential $a$ in \eqref{eq2} is singular and that it satisfies the following hypotheses:

\smallskip
\noindent ($a_1$) $a\in L^\infty_{loc}(\RR^N\setminus\{0\})$ and  $\essinf_{\RR^N}a>0$;

\smallskip\noindent
($a_2$) $\lim_{x\ri 0}a(x)=\lim_{|x|\ri\infty}a(x)=+\infty$.

\smallskip A potential satisfying these conditions is $a(x)=\exp(|x|)/|x|$, for $x\in\RR^N\setminus\{0\}$.

We assume that the nonlinearity $f:\RR^N\times\RR\ri\RR$ is a Carath\'eodory function characterized by the following conditions:

\smallskip
\noindent ($f_1$) $f(x,u)=o(u^{\alpha-1})$ as $u\ri 0^+$, uniformly for a.e. $x\in\RR^N$;

\smallskip
\noindent ($f_2$) $f(x,u)=O(u^{s-1})$ as $u\ri \infty$, uniformly for a.e. $x\in\RR^N$;

\smallskip
\noindent ($f_3$) there exists $\theta>\alpha$ such that $0<\theta F(x,u)\leq uf(x,u)$ for all $u>0$, a.e. $x\in\RR^N$, where $F(x,u)=\int_0^uf(x,t)dt$;

\smallskip
\noindent ($f_4$) if $\alpha<q$ then $\lim_{u\ri +\infty}F(x,u)/u^q=+\infty$ uniformly for a.e. $x\in\RR^N$.

Next, we assume that the differential operator in problem \eqref{eq2} is generated by the function $\phi\in C^1(\RR_+,\RR_+)$  having the following properties:

\smallskip
\noindent ($\phi_1$) $\phi (0)=0;$

\smallskip
\noindent ($\phi_2$) there exists $c_1>0$ such that $\phi(t)\geq c_1t^{p/2}$ if $t\geq 1$ and   $\phi(t)\geq c_1t^{q/2}$ if $0\leq t\leq 1$;

\smallskip
\noindent ($\phi_3$) there exists $c_2>0$ such that $\phi(t)\leq c_2t^{p/2}$ if $t\geq 1$ and   $\phi(t)\leq c_2t^{q/2}$ if $0\leq t\leq 1$;

\smallskip
\noindent ($\phi_4$) there exists $0<\mu<1/s$ such that $2t\phi'(t)\leq s\mu\phi (t)$ for all $t\geq 0$;

\smallskip
\noindent ($\phi_5$) the mapping $t\mapsto \phi (t^2)$ is strictly convex.

\smallskip Since our hypotheses allow that $\phi'$ approaches 0, problem \eqref{eq2} is {\it degenerate} and no ellipticity condition is assumed.

In order to state the main abstract result of this paper, we need to describe the functional setting corresponding to problem \eqref{eq2}.

In what follows, we denote by $\|\,\cdot\,\|_r$ the Lebesgue norm for all $1\leq r\leq\infty$ and by $C^\infty_c(\RR^N)$ the  space of all $C^\infty$ functions with a compact support.

\begin{definition}\label{d1}
We define the function space $L^p(\RR^N)+L^q(\RR^N)$ as the completion of $C^\infty_c(\RR^N)$ in the norm
$$\|u\|_{L^p+L^q}:=\inf\{\|v\|_p+\|w\|_q;\ v\in L^p(\Omega),\ w\in L^q(\Omega),\ u=v+w\}.$$
\end{definition}

For more properties of the Orlicz space $L^p(\RR^N)+L^q(\RR^N)$ we refer to M.~Badiale, L.~Pisani, and S. Rolando \cite[Sect. 2]{badiale}.

Throughout this paper we denote $$\|u\|_{p,q}=\|u\|_{L^p+L^q}.$$

A key role in our arguments is played by the function space
$${\mathcal X}:=\overline{C^\infty_c(\RR^N)}^{\|\,\cdot\,\|},$$
where $$\|u\|:=\|\nabla u\|_{p,q}+\left(\int_{\RR^N}a(x)|u|^\alpha dx\right)^{1/\alpha}.$$

We notice that $\calX$ is continuously embedded in the reflexive Banach space $\calW$ defined in \cite[p. 202]{azo2}, where $\calW$ is the completion
of $C^\infty_c(\RR^N)$ in the norm $\|u\|=\|\nabla u\|_{p,q}+\|u\|_\alpha $.

\begin{definition}\label{d2}
A weak solution of problem \eqref{eq2} is a function $u\in \calX\setminus\{0\}$ such that for all $v\in\calX$
$$\int_{\RR^N}\left[\phi'(|\nabla u|^2)\nabla u\nabla v+a(x)|u|^{\alpha-2}uv-f(x,u)v\right]dx=0.$$
\end{definition}

The energy functional associated to problem \eqref{eq2} is $\calE:\calX\ri\RR$ defined by
$$\calE (u):=\frac 12\int_{\RR^N}\phi (|\nabla u|^2)dx+\frac 1\alpha\int_{\RR^N}a(x)|u|^\alpha dx-\int_{\RR^N}F(x,u)dx.$$

Our assumptions imply that
$$\phi (|\nabla u|^2)\simeq\left\{
\begin{array}{lll}
&\di |\nabla u|^p,&\quad\mbox{if}\ |\nabla u|\gg 1;\\
&\di |\nabla u|^q,&\quad\mbox{if}\ |\nabla u|\ll 1.
\end{array}\right.
$$

 We also observe that $\calE$ is well-defined on $\calX$, of class $C^1$ (see also A.~Azzollini \cite[Theorem 2.5]{azo1}). Moreover,  for all $u$, $v\in\calX$ its G\^ateaux directional derivative is given by
$$\calE'(u)(v)=\int_{\RR^N}\left[\phi'(|\nabla u|^2)\nabla u\nabla v+a(x)|u|^{\alpha-2}uv-f(x,u)v \right]dx.$$

The main result of this paper establishes the following existence property.

\begin{theorem}\label{t1}
Assume that hypotheses \eqref{pq}, ($a_1$), ($a_2$), ($f_1$)-($f_4$), and ($\phi_1$)-($\phi_5$) are fulfilled. Then problem \eqref{eq2} admits at least one weak solution.
\end{theorem}

We point out that a related existence property was established by A.~Azzollini, P.~d'Avenia, and A.~Pomponio \cite[Theorem 1.3]{azo2} but under the assumption that the potential $a$ reduces to a positive constant. Our setting is different and it corresponds to variable potentials which blow-up both at the origin and at infinity. The lack of compactness due to the unboundedness of the domain is handled in \cite{azo2} by restricting the study to the case of {\it radially symmetric} weak solutions. In such a setting, a key role in the arguments developed in \cite{azo2} is played by the {\it compact embedding} of a related function space with radial symmetry into a certain class of Lebesgue spaces.

The approach developed in this paper is general and cannot be reduced to radially symmetric solutions, due to the presence of the general potential $a$. A central role in the arguments developed in \cite{azo2} is played by the fact that the space $\calW$ is continuously embedded in $L^{p^*}(\RR^N)$, provided that $1<p<\min\{q,N\}$, $1<p^*q'/p'$ and $\alpha \in (1,p^*q'/p')$. By interpolation, the same continuous embedding holds in every Lebesgue space $L^r(\RR^N)$ for every $r\in [\alpha, p^*]$.

\section{Proof of the main result}
We give the proof of Theorem \ref{t1} by using the following version of the mountain pass lemma of A.~Ambrosetti and P.~Rabinowitz \cite{ambrab} (see also H.~Brezis and L.~Nirenberg \cite{brenir}).

\begin{theorem}\label{munte}
Let $\calX$ be a real Banach space and assume that $\calE:\calX\ri\RR$ is a $C^1$-functional that satisfies the following geometric hypotheses:

\noindent (i) $\calE(0)=0$ and there exist positive numbers $a$ and $r$ such that $\calE(u)\geq a$ for all $u\in\calX$ with $\|u\|=r$;

\noindent (ii) there exists $e\in\calX$ with $\|e\|>r$ such that $\calE(e)<0$.

Set
$${\mathcal P}:=\{ p\in C([0,1];\calX);\ p(0)=0,\ p(1)=e\}$$
and
$$c:=\inf_{p\in{\mathcal P}}\sup_{t\in [0,1]}\calE (p(t)).$$

Then there exists a sequence $(u_n)\subset\calX$ such that
$$\lim_{n\ri\infty}\calE(u_n)=c\quad\mbox{and}\quad\lim_{n\ri\infty}\|\calE'(u_n)\|_{\calX^*}=0.$$

Moreover, if $\calE$ satisfies the Palais-Smale condition at the level $c$, then $c$ is a critical value of $\calE$.
\end{theorem}

We split the proof into several steps.
\subsection{Existence of a mountain and of a valley}
Fix $r\in (0,1)$ and let $u\in\calX$ with $\|u\|=r$. Using hypotheses ($\phi_1$) and  ($\phi_2$) we have
\bb\label{due}
\calE(u)\di\geq\frac{c_1}{2}\int_{[|\nabla u|\leq 1]}|\nabla u|^qdx+\frac{c_2}{2}\int_{[|\nabla u|> 1]}|\nabla u|^pdx+\frac 1\alpha \int_{\RR^N}a(x)|u|^\alpha dx-\int_{\RR^N}F(x,u)dx.
\bbb

Fix $\ep>0$. Using hypotheses ($f_1$) and ($f_2$), there exists $C_\ep>0$ such that
\bb\label{tre}
|F(x,u)|\leq\ep|u|^\alpha +C_\ep |u|^{p^*}\quad\mbox{for all $u\in\RR$, a.e. $x\in\RR^N$}.
\bbb
Setting $c:=\min\{c_1/2,c_2/2\}$, relation \eqref{tre} and hypothesis ($a_1$) yield
\bb\label{quatro}
\begin{array}{ll}
\calE(u)&\di\geq c\max\left\{ \int_{[|\nabla u|\leq 1]}|\nabla u|^qdx,\int_{[|\nabla u|> 1]}|\nabla u|^pdx\right\}+\frac 1\alpha \int_{\RR^N}a(x)|u|^\alpha dx\\
&\di -\ep\int_{\RR^N}|u|^\alpha dx-C_\ep\int_{\RR^N}|u|^{p^*}dx\\
&\di\geq c\,\|\nabla u\|_{p,q}^q+\frac 1\alpha\int_{\RR^N}a(x)|u|^\alpha dx-\frac{\ep}{a_0}\int_{\RR^N}a(x)|u|^\alpha dx-C_\ep\int_{\RR^N}|u|^{p^*}dx.
\end{array}\bbb

We have already observed that $\calX$ is continuously embedded into the function space $\calW$ defined in \cite[p. 584]{azo1}. Using now \cite[Corollary 2.4]{azo1}, it follows that $\calX$ is continuously embedded in $L^{p^*}(\RR^N)$, hence there exists $C>0$ such that
$$\|u\|_{p^*}\leq C\,\|u\|\quad\mbox{for all}\ u\in\calX.$$
Returning to \eqref{quatro}, we deduce that
\bb\label{cinque}\calE(u)\geq c\,\|\nabla u\|_{p,q}^q+\left(\frac 1\alpha-\frac{\ep}{a_0} \right)\int_{\RR^N}a(x)|u|^\alpha dx-C\,\|u\|^{p^*}_{p^*}.\bbb
Recall that $\max\{\alpha ,q\}<p^*$, see hypothesis \eqref{pq}. Taking $r\in(0,1)$ small enough, relation \eqref{cinque} yields that there is $a>0$ such that
\bb\label{aa} \calE(u)\geq a\quad\mbox{for all $u\in\calX$ with $\|u\|=r$}.\bbb
This shows the existence of a ``mountain" around the origin.

\smallskip
Next, we prove the existence of a ``valley" over the chain of mountains. This is essentially due to the relationship between the exponents $p$, $q$ and $\alpha$. For this purpose, we fix $w\in C^\infty_c(\RR^N)\setminus\{0\}$ and $t>0$. Using hypothesis ($\phi_3$) we have
\bb\label{vale}\begin{array}{ll}
\calE(tw)&\di\leq \frac{c_2}{2}\int_{[|\nabla (tw)|\leq 1]}|\nabla (tw)|^qdx+\frac{c_2}{2}\int_{[|\nabla (tw)|> 1]}|\nabla (tw)|^pdx\\ &\di +\frac 1\alpha\int_{\RR^N}a(x)|tw|^\alpha dx-
\int_{\RR^N}F(x,tw)dx\\
&\di\leq \frac{c_2}{2}\left(t^q\int_{\RR^N}|\nabla w|^qdx+t^p\int_{\RR^N}|\nabla w|^pdx\right)+\frac{t^\alpha}{\alpha} \int_{\RR^N}a(x)|w|^\alpha dx\\
&\di - \ep t^\alpha \int_{\RR^N}|w|^\alpha dx-C_\ep t^{p^*}\int_{\RR^N}|w|^{p^*}dx.
\end{array}
\bbb
Since $w$ is fixed, relation \eqref{vale} and hypothesis \eqref{pq} imply that $\lim_{t\ri +\infty}\calE(tw)=-\infty$. Thus, there exists $t_0>0$ such that $\calE(t_0w)<0$.

\smallskip We have checked until now the geometric hypotheses of the mountain pass lemma. We argue in what follows that the corresponding setting is non-degenerate, that is, the associated min-max value given by Theorem \ref{munte} is positive.

Set
$$c:=\inf_{p\in{\mathcal P}}\max_{t\in [0,1]}\calE (p(t)),$$
where
$${\mathcal P}:=\{ p\in C([0,1];\calX);\ p(0)=0,\ p(1)=t_0w\}.$$

We observe that for all $p\in{\mathcal P}$
$$c\geq \calE (p(0))=\calE(0)=0.$$
In fact, we claim that
\bb\label{claim}
c>0.\bbb
Arguing by contradiction, we assume that $c=0$. In particular, this means that for all $\ep>0$ there exists $q\in{\mathcal P}$ such that
$$0\leq\max_{t\in [0,1]}\calE (q(t))<\ep\,.$$
Fix $\ep<a$, where $a$ is given by \eqref{aa}. Then $q(0)=0$ and $q(1)=t_0w$, hence
$$\|q(0)\|=0\quad\mbox{and}\quad \|q(1)\|>r.$$
Using the continuity of $q$, there exists $t_1\in (0,1)$ such that $\|q(t_1)\|=r$, hence
$$\|\calE(q(t_1))\|=a>\ep,$$
which is a contradiction. This shows that our claim \eqref{claim} is true.

Applying Theorem \ref{munte}, we find a Palais-Smale sequence for the level $c>0$, that is, a sequence $(u_n)\subset\calX$ such that
\bb\label{pasma}\lim_{n\ri\infty}\calE(u_n)=c\quad\mbox{and}\quad\lim_{n\ri\infty}\|\calE'(u_n)\|_{\calX^*}=0.\bbb

\subsection{The boundedness of the Palais-Smale sequence}
We prove in what follows that the sequence $(u_n)$ described in \eqref{pasma} is bounded in $\calX$. Indeed, using both information in relation \eqref{pasma}, we have
\bb\label{pas1}
\begin{array}{ll}
c+O(1)+o(\|u_n\|)&\di =\calE(u_n)-\frac 1\theta \calE'(u_n)u_n\\

&\di =\int_{\RR^N}\left[\frac 12\phi(|\nabla u_n|^2)-\frac 1\theta\phi'(|\nabla u_n|^2)|\nabla u_n|^2 \right]dx\\

&\di +\left(\frac 1\alpha -\frac 1\theta \right)\int_{\RR^N}a(x)|u_n|^\alpha dx\\

&\di+\int_{\RR^N}\left[\frac 1\theta f(x,u_n)u_n-F(x,u_n) \right]dx.
\end{array}
\bbb

Using hypothesis ($f_3$), relation \eqref{pas1} yields
\bb\label{pas11}
\begin{array}{ll}
c+O(1)+o(\|u_n\|)&\di =\calE(u_n)-\frac 1\theta \calE'(u_n)u_n\\

&\di \geq\int_{\RR^N}\left[\frac 12\phi(|\nabla u_n|^2)-\frac 1\theta\phi'(|\nabla u_n|^2)|\nabla u_n|^2 \right]dx\\

&\di +\left(\frac 1\alpha -\frac 1\theta \right)\int_{\RR^N}a(x)|u_n|^\alpha dx.
\end{array}
\bbb

Using hypothesis ($\phi_4$) we have for all $t\geq 0$
$$\frac 12\phi(t)-\frac 1\theta \phi'(t)t\geq\frac{1-\mu s}{2}\phi (t),$$where $\mu s\in (0,1)$. Hypothesis ($f_3$) yields that $\theta>\alpha$.
Thus, returning to \eqref{pas11} we deduce that there exists $c_0>0$ such that for all $n\in\NN$
\bb\label{pas2}
\calE(u_n)-\frac 1\theta \calE'(u_n)u_n\geq c_0\left[\min\{\|\nabla u_n\|^q_{p,q},\|\nabla u_n\|^p_{p,q} \}+ \int_{\RR^N}a(x)|u_n|^\alpha dx\right].\bbb
Combining relations \eqref{pas11} and \eqref{pas2}, we deduce that the sequence $(u_n)\subset\calX$ is bounded.

Since $\calX$ is a closed subset of $\calW$, using Proposition 2.5 in \cite{azo2} we deduce that the sequence $(u_n)$ converges weakly (up to a subsequence) in $\calX$ and strongly in $L^s_{loc}(\RR^N)$ to some $u_0$. We show in what follows that $u_0$ is a solution of problem \eqref{eq2}.

Fix $\zeta\in C^\infty_c(\RR^N)$ and set $\Omega:={\rm supp}\, (\zeta)$. Define
$$A(u)=\frac 12\int_\Omega \phi (|\nabla u|^2)dx+\frac 1\alpha\int_\Omega a(x)|u|^\alpha dx$$
and
$$B(u)=\int_\Omega F(x,u)dx.$$
Using \eqref{pasma} we have
\bb\label{un}A'(u_n)(\zeta)-B'(u_n)(\zeta)\ri 0\quad\mbox{as}\ n\ri\infty .\bbb
Since $u_n\ri u_0$ in $L^s(\Omega)$ and the mapping $u\mapsto F(x,u)$ is compact from $\calX$ into $L^1$, it follows that
\bb\label{deux}
B(u_n)\ri B(u_0)\ \mbox{and}\ B'(u_n)(\zeta)\ri B'(u_0)(\zeta)\quad\mbox{as}\ n\ri\infty .\bbb
Combining relations \eqref{un} and \eqref{deux} we deduce that
\bb\label{trois}
A'(u_n)(\zeta)\ri B'(u_0)(\zeta)\quad\mbox{as}\ n\ri\infty .\bbb

Using hypothesis ($\phi_5$), we obtain that the nonlinear mapping $A$ is convex. Therefore
\bb\label{quatre}
A(u_n)\leq A(u_0)+A'(u_n)(u_n-u_0)\quad\mbox{for all}\ n\in\NN.\bbb
Using \eqref{trois} in combination with $u_n\rightharpoonup u_0$ in $\calX$, relation \eqref{quatre} yields
$$\limsup_{n\ri\infty} A(u_n)\leq A(u_0).$$
But $A$ is lower semicontinuous, since it is convex and continuous. It follows that
$$A(u_0)\leq\liminf_{n\ri\infty}A(u_n).$$
We conclude that
$$A(u_n)\ri A(u_0)\quad\mbox{as}\ n\ri\infty .$$

From now on, with the same arguments as in \cite[p. 210]{azo2} (see also \cite[p. 59]{gazrad}), we deduce that
$$\nabla u_n\ri\nabla u_0\quad\mbox{as $n\ri\infty$ in}\ L^p(\RR^N)+L^q(\RR^N)$$
and
$$\int_{\RR^N}a(x)|u_n|^\alpha dx\ri \int_{\RR^N}a(x)|u_0|^\alpha dx\quad\mbox{as}\ n\ri\infty\,.$$
Therefore
$$\int_\Omega \phi'(|\nabla u_0|^2)\nabla u_0\nabla\zeta dx+\int_\Omega a(x)|u_0|^{\alpha-2}u_0\zeta dx-\int_\Omega f(x,u_0)\zeta dx=0.$$
By density, we obtain that this identity holds for all $\zeta\in\calX$, hence $u_0$ is a solution of problem \eqref{eq2}.

\subsection{Proof of Theorem \ref{t1} completed}
It remains to argue that the solution $u_0$ is  nontrivial. For this purpose we use some ideas developed in \cite{gazrad,mrad}.

Using the fact that $(u_n)$ is a Palais-Smale sequence, relation \eqref{pasma} implies that if $n$ is a positive integer sufficiently large then
\bb\label{11}
\begin{array}{ll}
\di\frac c2&\di\leq E(u_n)-\frac 12E'(u_n)u_n\\
&\di =\frac 12\int_{\RR^N}\left[\phi (|\nabla u_n|^2)-\phi'(|\nabla u_n|^2)|\nabla u_n|^2 \right]dx\\
&\di +\left(\frac 1\alpha -\frac 12\right)\int_{\RR^N}a(x)|u_n|^\alpha dx+\int_{\RR^N}\left[\frac 12f(x,u_n)u_n-F(x,u_n) \right]dx.
\end{array}\bbb
Using hypothesis ($\phi_5$) that concerns the convexity of the map $t\mapsto\phi (t^2)$, we deduce that
$$\phi(t^2)-\phi(0)\leq\phi'(t^2)t^2.$$
Using now ($\phi_1$) we obtain
$$\phi(t^2)\leq\phi'(t^2)t^2,$$
hence
\bb\label{12}
\phi(|\nabla u_n|^2)\leq\phi'(|\nabla u_n|^2)|\nabla u_n|^2.\bbb

We first assume that $\alpha\geq 2$. Thus, relations \eqref{11} and \eqref{12} combined with hypothesis ($f_3$) imply that for all $n$ large enough we have
\bb\label{13}
\begin{array}{ll}
\di \frac c2&\di\leq \int_{\RR^N}\left[\frac 12f(x,u_n)u_n-F(x,u_n) \right]dx\\
&\di \leq \frac 12\int_{\RR^N}f(x,u_n)u_ndx.
\end{array}\bbb

Hypotheses ($f_1$) and ($f_2$) show that for all $\ep>0$ there exists $C_\ep>0$ such that
$$|f(x,u)|\leq\ep |u|^{p^*-1}+C_\ep |u|^{\alpha-1},\quad\mbox{for all $u\in\RR$, a.e. $x\in\RR^N$}.$$
Returning to \eqref{13} we obtain for all $n$ large enough
$$\frac c2\leq \frac\ep 2\,\|u_n\|_{p^*}^{p^*}+C_\ep\,\|u_n\|^\alpha_\alpha\,.$$
Since $(u_n)$ is bounded in $L^{p^*}(\RR^N)$, we fix $\ep>0$ small enough such that
$$\frac\ep 2\sup_n\|u_n\|^{p^*}_{p^*}\leq\frac c4\,.$$
It follows that for all $n\geq n_0$
$$\frac c4\leq C_0\,\|u_n\|^\alpha_\alpha\,,$$
where $C_0$ is a positive constant.

In order to show that $u_0\not=0$ we argue by contradiction. Assume that $u_0=0$. In particular, this implies that
\bb\label{14}
u_n\ri 0\quad\mbox{in}\ L^\alpha_{loc}(\RR^N).\bbb

Let $k$ be a positive integer and set
\bb \label{omega}\omega:=\{x\in\RR^N;\ 1/k<|x|<k\}.\bbb
Using \eqref{14}, it follows that if $k$ is large enough then
$$C_0\int_\omega |u_n|^\alpha dx\leq\frac c8\quad\mbox{for all $n\geq n_0$}\,.$$
Therefore
\bb\label{15}
\begin{array}{ll}
\di\frac c8&\di\leq C_0\int_{\RR^N\setminus\omega}|u_n|^\alpha dx\\
&\di\leq\frac{C_0}{\inf_{|x|\leq 1/k}a(x)}\int_{|x|\leq 1/k}a(x)|u_n|^\alpha dx+
\frac{C_0}{\inf_{|x|\geq k}a(x)}\int_{|x|\geq k}a(x)|u_n|^\alpha dx\\
&\di\leq C_0M\left[\frac{1}{\inf_{|x|\leq 1/k}a(x)}+\frac{1}{\inf_{|x|\geq k}a(x)} \right],
\end{array}\bbb
where $M=\sup_n\int_{\RR^N}a(x)|u_n|^\alpha dx$.

Choosing $k$ large enough and using hypothesis ($a_2$), relation \eqref{15} implies that $c=0$, a contradiction.

It remains to study the case $1<\alpha<2$. Relations \eqref{11} and \eqref{12} imply that for all $n$ large enough we have
\bb\label{16}
\begin{array}{ll}
\di\frac c2&\di\leq E(u_n)-\frac 12E'(u_n)u_n\\
&\di \leq\left(\frac 1\alpha -\frac 12\right)\int_{\RR^N}a(x)|u_n|^\alpha dx+\frac 12\int_{\RR^N}f(x,u_n)u_ndx.
\end{array}\bbb

We argue again by contradiction and assume that $u_0=0$. With the same choice of $\omega$ as in \eqref{omega} and with similar estimates in \eqref{16} as above, we obtain a contradiction.

Summarizing, we have obtained that $u_0$ is a nontrivial solution of problem \eqref{eq2}. $\quad\square$

\subsection{Final remarks} (i) The study of Orlicz spaces $L^p+L^q$ has been initiated by  M.~Badiale, L.~Pisani, and S.~Rolando \cite{badiale}.

(ii) We point out that with a similar analysis we can treat the case of potentials satisfying $\liminf_{|x|\ri\infty}a(x)=0$, which is a particular {\it critical frequency case}, see for details J. Byeon and Z.Q. Wang \cite{byeon}.

(iii) The existence of solutions of problem \eqref{eq2} in the case of a null potential $a$ was established by H.~Berestycki and P.L.~Lions \cite{bere}, where the authors used a {\it double-power} growth hypothesis on the nonlinearity, that is, $f(x,\cdot)$ has a subcritical behaviour at infinity and a supercritical growth near the origin.

(iv) We expect that new and interesting results can be established if the nonhomogeneous  operator in problem \eqref{eq2} is a replaced by a differential operator with {\it two} competing potentials $\phi_1$ and $\phi_2$. We refer to operators of the type
$${\rm div}\,\left((\phi_1'(|\nabla u|^2)+\phi_2'(|\nabla u|^2))|\nabla u|^2 \right),$$
where $\phi_1$ and $\phi_2$ have different growth decay. This new abstract framework is inspired by the analysis developed in Chapter 3.3 of the recent monograph by V.~R\u adulescu and D.~Repov\v{s} \cite{radrep} in the framework of nonlinear problems with {\it variable exponents}.

(v) A new research direction in strong relationship with several relevant applications is the study of problems described by the nonlocal term
$$M\left[\int_{\RR^N}\phi (|\nabla u|^2)|\nabla u|^2 \right].$$
We refer here to the pioneering papers by P.~Pucci {\it at all.} \cite{autuori,pucibero, pucana} related to Kirchhoff problems involving nonlocal operators associated to the standard Laplace, $p$-Laplace or $p(x)$-Laplace operators.



% For the acknowledgements section, please do not number the section, please begin it with \section*{Acknowledgements}
\section*{Acknowledgements}
The  first  author would like to extend his sincere appreciation to the Deanship of Scientific Research at King Saud University
for funding this Research group No (RG-1435-026).  The second author thanks the
Visiting Professor Programming at King Saud University for funding this work.

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\end{document}




