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%\title[On nonlocal type equations via linking methods]{On nonlocal type equations via linking methods}
\title[Sublinear Schr\"odinger equations with indefinite potentials]{Infinitely many solutions for a class of sublinear Schr\"odinger equations with indefinite potentials}
\author{Anouar Bahrouni, Hichem Ounaies and Vicen\c{t}iu D. R\u{a}dulescu}
\address[A. Bahrouni]{Mathematics Department, Faculty of Sciences, University of Monastir, 5019 Monastir, Tunisia} \email{bahrounianouar@yahoo.fr}
\address[H. Ounaies]{Mathematics Department, Faculty of Sciences, University of Monastir, 5019 Monastir, Tunisia} \email{hichem.ounaies@fsm.rnu.tn}
\address[V. R\u{a}dulescu]{Department of Mathematics, Faculty of Science, King Abdulaziz University,
Jeddah, Saudi Arabia} \email{vicentiu.radulescu@math.cnrs.fr}
%
\thanks{{\it 2010 Mathematics Subject Classification.} 35J20, 35J20, 35B38.}
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\keywords{Nonlinear Schr\"odinger equation, sign-changing potential, sublinear equation, concentration-compactness principle, variational method, infinitely many solutions.}
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%\thanks{Work performed under the auspicies of G.N.A.M.P.A., of I.N.D.A.M.}
%21/03/11
%\thanks{V. R\u adulescu acknowledges the support through Grant CNCS PCE-47/2011.}
\begin{document}

\begin{abstract}
In this paper we are concerned with qualitative properties of entire solutions to a Schr\"odinger equation with sublinear nonlinearity and sign-changing potentials.
Our analysis considers three distinct cases and we establish sufficient conditions for the existence of infinitely many solutions.
\end{abstract}
\maketitle

\section{Historical perspectives of the Schr\"odinger equation}
The Schr\"odinger equation plays the role of Newton's laws and conservation of energy in classical %%@
mechanics, that is, it predicts the future behaviour of a dynamic system. The linear
Schr\"odinger equation is a central tool of quantum mechanics, which provides a thorough description of a particle in a non-relativistic setting. Schr\"odinger's linear equation is
$$\Delta\psi +\frac{8\pi^2m}{\hbar^2}\,\left(E-V(x)\right)\psi =0\,,$$
where $\psi$ is the Schr\"odinger wave function, $m$ is the mass, $\hbar$ denotes Planck's %%@
constant, $E$ is the energy, and $V$ stands for the potential energy.

The structure of the %%@
nonlinear Schr\"odinger equation is much more complicated. This  equation
is a prototypical
dispersive nonlinear
partial differential equation that has been central for
almost four decades now to a variety of areas in mathematical physics.
The relevant fields of
application  vary from Bose-Einstein condensates and nonlinear optics (see Byeon and Wang \cite{byeon}), propagation of the  electric field in optical fibers (see
Hasegawa and Kodama \cite{hase}, Malomed \cite{malo})
to the self-focusing and
collapse of Langmuir waves in plasma physics (see Zakharov \cite{zakh})
and the behaviour of deep water waves and
freak waves (the so-called rogue waves)
in the ocean (see  Onorato, Osborne,
Serio and Bertone \cite{onor}).
The nonlinear Schr\"odinger equation also describes various 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 (see  Avron, Herbst and Simon \cite{avron1}), plasma physics
(e.g., the Kurihara superfluid film equation). We refer to
Ablowitz, Prinari and Trubatch \cite{abl},
Sulem \cite{sulem} for a modern overview, including applications.

Schr\"odinger also established the  classical derivation of his equation, based upon the analogy %%@
between mechanics and optics, and closer to de Broglie's ideas. He  developed a perturbation
method, inspired by the work of Lord Rayleigh in acoustics, proved the equivalence between his
wave mechanics and Heisenberg's matrix, and introduced the time dependent Schr\"odinger's equation
\begin{equation}\label{oheq}
i\hbar\psi_t=-\frac{\hbar^2}{2m}\,\Delta\psi+V(x)\psi-\gamma |\psi|^{p-1}\psi\qquad
\mbox{in $\R^N$ ($N\geq 2$)},\end{equation}
where $p<2N/(N-2)$ if $N\geq 3$ and $p<+\infty$ if $N=2$.
In physical problems,
a cubic nonlinearity corresponding to $p = 3$ is common; in this case problem \eqref{oheq} is called the
Gross-Pitaevskii equation.
In the study of Eq.~\eqref{oheq}, Floer and Weinstein \cite{floer} and Oh \cite{oh} supposed that the potential $V$ is bounded
and possesses a non-degenerate critical point at $x=0$. More precisely, it is assumed that
$V$ belongs to the class ($V_a$) (for some real number $a$) introduced in Kato \cite{kato}.
Taking $\gamma>0$ and $\hbar >0$
sufficiently small and using a Lyapunov-Schmidt type reduction, Oh \cite{oh} proved the existence
of
 standing wave solutions of problem \eqref{oheq}, that is, a solution of the form
\begin{equation}\label{oheq1}
\psi (x,t)=e^{-iEt/\hbar}u(x)\,.\end{equation}
Using the ansatz \eqref{oheq1}, we reduce the nonlinear Schr\"odinger equation \eqref{oheq}  to
the semilinear elliptic equation
$$-\frac{\hbar^2}{2m}\,\Delta u+\left(V(x)-E\right)u=|u|^{p-1}u\,.$$
The  change of variable $y=\hbar^{-1}x$ (and replacing $y$ by $x$) yields
\begin{equation}\label{oheqoheq}
-\Delta u+2m\left(V_\hbar (x)-E\right)u=|u|^{p-1}u\qquad\mbox{in $\R^N$}\,,\end{equation}
where $V_\hbar (x)=V(\hbar x)$.

If for some $\xi\in\R^N\setminus\{0\}$, $V(x + s\xi) = V (x)$ for all $s \in \R$, equation
\eqref{oheq} is invariant
under the Galilean transformation
$$\psi(x,t)\longmapsto \psi(x-\xi t,t)\exp\left(i\xi\cdot x/\hbar -\frac %%@
12i|\xi|^2t/\hbar\right)\psi(x-\xi t, t)\,.$$
Thus, in this case, standing waves reproduce solitary waves traveling in the direction
of $\xi$.
In other words, Schr\"odinger  discovered that the standing waves are scalar waves rather than
vector electromagnetic waves. This is an important difference, vector electromagnetic waves are
mathematical waves which describe a direction (vector) of force, whereas the wave motions of space
are scalar waves which are simply described by their wave-amplitude. The importance of this
discovery was pointed out by Albert Einstein, who wrote:
``The Schr\"odinger method, which has in a certain sense the character of a field theory, does
indeed deduce the existence of only discrete states, in surprising agreement with empirical facts.
It does so on the basis of differential equations applying a kind of resonance argument". ({\it On
Quantum Physics}, 1954).

In a celebrated paper, Rabinowitz \cite{R1}
proved that problem \eqref{oheqoheq} has a ground-state solution (mountain-pass solution) for $\hbar
>0$ small, under the assumption that $\inf_{x\in\R^N}V(x)>E$.
After making
a standing wave ansatz, Rabinowitz reduces the problem to that of studying the
semilinear elliptic equation
\begin{equation}\label{phreqq}
-\Delta u +V(x)u=f(x,u)\qquad\mbox{in }\R^N,
\end{equation}
under suitable conditions on $V$ and assuming that $f$ is smooth, superlinear
and has a subcritical growth.


\section{Introduction and main results}
In the present paper we are concerned with the existence of infinitely many solutions of the semilinear Schr\"odinger equation
\begin{equation}\label{main}
-\Delta u+V(x)u=a(x) g(u) \qquad x\in{\mathbb{R}^{N}} \  (N\geq 3),
\end{equation}
where $V$, $a$ are functions changing sign  and the nonlinearity $g$ has a sublinear growth.
 Such problems in $\mathbb{R}^{N}$ arise naturally in various branches of physics and present challeging mathematical difficulties.

 If problem \eqref{main} is considered in a bounded domain $\Omega$, with the Dirichlet boundary condition, then there is a large literature on existence and a multiplicity of solutions (see \cite{A2,B5,K1,K2,T1, tehrani,W2}).
 In particular,  Kajikiya \cite{K1} has considered such sublinear case with sign-changing nonlinearity and has proved the existence of infinitely many solutions.

 If $\Omega$ is an unbounded domain and especially if $\Omega=\mathbb{R}^{N}$, the existence and multiplicity of nontrivial solutions for problem \eqref{main} have been extensively investigated in the literature over the past several decades, both for  sublinear and superlinear nonlinearities.

 In the superlinear case, we can cite the references \cite{ A1, A3,B1,B4,C3,costa, D2,D3,F,R1, W1}. In particular,   Costa and Tehrani \cite{C3} have considered the following problem
 \begin{equation}\label{costa}
 -\Delta u-\lambda h(x) u=a(x)g(u), \ \ u>0 \ \ \mbox{in} \ \ \mathbb{R}^{N},
 \end{equation}
  where $\lambda >0, h$ is a positive function, $a$ changes the sign in $\mathbb{R}^{N}$, $N\geq 3$, and $g$ is a superlinear function. With furthermore assumptions on $h$, $a$ and  $g$, they proved the existence of $\lambda_{1}(h)>0$ such that problem \eqref{costa} admits one positive solution for $0<\lambda<\lambda_{1}(h) $ and two positive solutions for $\lambda_{1}(h)<\lambda<\lambda_{1}(h)+\epsilon $, for some  $\epsilon>0$.

   In recent years, many authors have studied the question of existence and multiplicity of solutions for problem \eqref{main} with sublinear nonlinearity, see  \cite{B2, MBH, MBN, B6, C1,C2, K3,T2}.
   In most of the problems studied in the papers cited in the references above, $V$ and $a$ are considered to be positive. In particular, Brezis and Kamin \cite{B6} gave  a sufficient and necessary condition for the existence of bounded positive solutions of problem \eqref{main} with $V=0$ and $a>0$.

     Balabane, Dolbeault and Ounaies \cite{B2} proved that for each integer $k$, the equation \eqref{main} has a radially compactly supported solution that has $k$ zeros in its support,  provided that $V=a=-1$ and $g(u)=\left|u \right|^{-2 \theta} u$,
 where $\theta \in ]0,\frac{1}{2} [$.

  Zhang and Wang \cite{Z1} proved the existence of infinitely many solutions for equation \eqref{main} with $g(u)=\left|u\right|^{p-1} u$, $0<p<1$ and the potentials $V>0, a>0$ satisfy the following assumptions:

  \smallskip
  \noindent
  $\left(S_{1}\right) \ \ V\in C\left(\mathbb{R}^{N}, \mathbb{R}\right)$ \ \ and there exists $r>0$ such that
  $$m\left\{x\in{B\left(y,r\right)}; \ \ V(x)\leq M\right\}\rightarrow 0  \ \ \mbox{as} \ \left|y\right|\rightarrow +\infty, \ \ \forall  \ M>0,$$
  where $m$ is the Lebesgue measure in $\mathbb{R}^{N}.$\\
  $\left(S_{2}\right) \ \ a:\mathbb{R}^{N} \rightarrow \mathbb{R}$ \ is a  continuous function and $a\in L^{\frac{2}{1-p}}\left(\mathbb{R}^N{}\right) \ \ 0<p<1$.

\smallskip
  If $V$, $a$ both change sign on $\mathbb{R}^{N}$, various difficulties arise. To authors' knowledge, few results are known in this case. On this subject,  Costa and Tehrani \cite{C2} have proved the existence of at least one non trivial solution for the following equation:
  $$-\Delta u+V(x)u=\lambda u+g(x,u),$$
  under the following conditions:

  \smallskip\noindent
  $\left(VC_{1}\right)$ \ \ $V\in C^{\beta}\left(\mathbb{R}^{N}\right) \ \ \left(0<\beta<1 \right)$ and $\displaystyle \lim_{\left|x\right|\rightarrow +\infty}V(x)=0$; \\
  $\left(VC_{2}\right)$ \ \ $ \displaystyle \int_{\mathbb{R}^{N}} \left(\left|\nabla \varphi\right|^{2}+V(x)\varphi^{2} \right)dx<0$ for some $\varphi \in C^1_0\left(\mathbb{R}^{N}\right)$; \\
  $\left(GC_{1}\right)$ \ \ $\left|g(x,s)\right|\leq b_{1}(x)\left|s \right|^{\alpha}+b_{2}(x)$ \ \ for some $0<\alpha<1$ and a class of integrable functions $b_{1}$ and $b_{2}$; \\
  $\left(GC_{2}\right)$ \ \ $\lambda<0$ is an eigenvalue of the Schr\"odinger operator $L_{V}=-\Delta+V(x)$ in $\mathbb{R}^{N}$. \\
  $\left(GC_{2}\right)$ $\displaystyle \lim_{\stackrel{\left\|u_{0}\right\|\rightarrow +\infty}{u_{0}\in {\rm Ker}\left(-\Delta+V-\lambda\right)}} \frac{1}{\left\|u_{0}\right\|^{2\alpha}} \displaystyle \int_{\mathbb{R}^{N}} G\left(x,u_{0}(x)\right) dx=\pm \infty$.

  \smallskip
  Tehrani \cite{T2} studied the following perturbed equation:
  \begin{equation}\label{perturbed}
  -\Delta u+V(x)u=a(x) g(u)+f,
  \end{equation}
  where $a$, $V$ change sign on $\mathbb{R}^{N}$, $f\in L^{2}\left(\mathbb{R}^{N}\right)$, and $g$ is a sublinear function. With further assumptions on $a$, $V$, $f$, and $g$, he proved the existence of at least one non trivial solution.

  Costa and Chabrowski \cite{C1} considered the following $p$-Laplacian equation:
  \begin{equation}\label{chabr}
  -\Delta_{p}u-\lambda V(x)\left|u\right|^{p-2} u=a(x) \, \left|u\right|^{q-2} u, \ \ x\in \mathbb{R}^{N},
  \end{equation}
  where $\lambda \in \mathbb{R}$ is a parameter, $1<q<p<p^{\ast}=\frac{Np}{N-p}, \ \ V\in L^{\frac{N}{p}}\left(\mathbb{R}^{N}\right)\cap L^{\infty}\left(\mathbb{R}^{N}\right),  a \in L^{\infty}\left(\mathbb{R}^{N}\right)$ and \ \ $\displaystyle \lim_{\left|x\right|\rightarrow +\infty} a(x)=a_{\infty}<0$. With further assumptions on $a$ and $V$, they proved the existence of $\lambda_{1}>0$ and $\lambda_{-1}<0$ such that problem \eqref{chabr} admits at least one positive solution for $\lambda_{-1}<\lambda<\lambda_{1}$ and two positive solutions for $\lambda>\lambda_{1}$ and $\lambda <\lambda_{-1}$.

   Benrhouma \cite{MB} has proved the existence of at least three solutions for equation \eqref{perturbed} with $g(u)=\left|u\right|^p {\rm sgn}\,(u)$, $0<p<1$, $V$ changes the sign, and $a<0$.

  In all works cited above, where $a$ and $V$ change sign, the authors proved the existence of at most three solutions.
In this paper, we prove the existence of infinitely many solutions of problem \eqref{main} with $a$ and $V$ changing sign, under various assumptions on these potential functions.

Denote by $s$ the best  Sobolev constant,
$$s=\inf\left\{\left\|\nabla u\right\|_{2}^{2}, \ \ u\in W^{1,2}\left(\mathbb{R}^{N}\right), \ \ \int_{\mathbb{R}^{N}} \vert u(x) \vert^{\frac{2N}{N-2} }dx =1\right\},\qquad N\geq 3.$$

We suppose the following hypotheses on $g$:\\
$\left(G_{1}\right) \ \ g\in C \left(\mathbb{R}, \mathbb{R}\right)$, $g $ is odd and   there exist $c>0 $ ,  $q \in ] 0,1[$  such that
$$\left|g(x)\right|\leq c \, \left|x\right|^{q}, \ \ \mbox{for  \  all} \  x  \in{\mathbb{R}};$$
$\left(G_{2}\right) \ \ \lim_{x\rightarrow 0} \frac{G(x)}{\left|x\right|^{2}}=+\infty$, where $G\left(x\right)= \displaystyle \int_{0}^{x}g(t)dt, \ \ \forall\, x\in \mathbb{R};$\\
$\left(G_{3}\right)$ $G$ is positive on $\mathbb{R}\setminus\{0\}$.

\smallskip
We give three theorems on the existence of infinitely many solutions to the nonlinear problem \eqref{main}.

\begin{thm}\label{troi thm}
Assume that $g(x)=\left|x\right|^{q-1} x$, $0<q<1$, and $V$ satisfies:\\
 $\left(V_{1}\right)$  $\ \ V\in L^{\infty}\left(\mathbb{R}^{N}\right), \ \ \displaystyle \lim_{\left|x\right|\rightarrow +\infty} V(x)=v_{\infty}>0$ \ \ and
$$\left\|V^{-}\right\|_{\frac{N}{2}}<s,$$
where $u^{\mp}(x)=\max\{\mp u(x),0\}$, for all $x\in \mathbb{R}^{N}$ and for all $u\in E$
 and $a$ satisfies:
\\ $\left(A_{1}\right) \ \ a\in L^{\infty}\left(\mathbb{R}^{N}\right), \displaystyle \lim_{\left|x\right|\rightarrow +\infty}a(x)=a_{\infty}<0$ and there  exist $y=\left(y_{1},\cdots, y_{N}\right) \in \mathbb{R}^{N}$, $R_{0}>0$ such that
 $$ a(x)>0, \ \ for \ all \ \ x\in B\left(y, R_{0}\right).$$

Then problem \eqref{main}   possesses a sequence of nontrivial solutions converging to $0$.
\end{thm}

In the next two theorems, we change the assumption of boundedness of $a$ by the integrability condition. The last assumption was supported to make the energy functional associated to problem \eqref{main} well defined and to guarantee that the functional $F(u)=\int_{\mathbb{R}^{N}}a(x)G(u(x)) \, dx$ has a compact gradient. This compactness property in turn was used to prove the required Palais-Smale condition, which is essential in the application of the critical point theory. Then we have the following two multiplicity properties.


\begin{thm}\label{prem thm}
Suppose that $g$ satisfies $\left(G_{1}\right), \left(G_{2}\right), \left(G_{3}\right)$, and the potentials $V$ and $a$ satisfy the following hypotheses:\\
$\left(V_{2}\right) \ \ V\in L^{\frac{N}{2}}\left(\mathbb{R}^{N}\right)$ \ \ and
$$\left\|V^{-}\right\|_{\frac{N}{2}}< s;$$
$\left(A_{2}\right) \ \ a\in L^{\frac{2^{\ast}}{2^{\ast}-\left(q+1\right)}}\left(\mathbb{R}^{N}\right)$  and there exist $y \in \mathbb{R}^{N}$ and  $R_{0}>0$  such that
$$a(x)>0, \ \ \forall \, x\in B\left(y,R_{0}\right).$$

Then problem \eqref{main} possesses a bounded sequence of nontrivial solutions.
\end{thm}

\begin{thm}\label{deux thm}
Assume that $g$ satisfies $\left(G_{1}\right),\left(G_{2}\right), \left(G_{3}\right)$, $V$ satisfies $\left(V_{1}\right)$ and $a$ satisfies  \\
$\left(A_{3}\right) \ \ a\in L^{\frac{2}{1-q}}\left(\mathbb{R}^{N}\right)$,  and there exist $y=\left(y_{1},\ldots, y_{N}\right)\in \mathbb{R}^{N}$ and  $R_{0}>0$  such that
$$a(x)>0, \ \ \forall \, x\in B\left(y,R_{0}\right).$$

Then problem \eqref{main} possesses a bounded sequence of nontrivial solutions.
\end{thm}


This paper is organized as follows. In section $2$, we give some notations, we present the variational framework and we recall some definitions, and standard results. Next, the sections $3$, $4$ and $5$ are dedicated to the proof of Theorems \ref{troi thm}, \ref{prem thm} and \ref{deux thm}.

\section{Notations and preliminary results}
     In this section we present some notations and preliminaries that will be useful in the sequel. We denote:\\
 $\left( \ast\right) \ \ \left\|u\right\|_{m}=\left(\displaystyle \int_{\mathbb{R}^{N}}\left|u(x)\right|^{m} dx\right)^{\frac{1}{m}}, \ \ 1\leq m < +\infty$;\\
 $\left(\ast\right)$\ \ $2^{\ast}=\frac{2N}{N-2}$ if $N\geq 3$ and $2^\ast =+\infty$ if $n\in\{1,2\}$;\\
 $\left(\ast\right)\ \  B_{R}$ denotes the ball centered at the origin of radius $R>0$ in $\mathbb{R}^{N}$ and $B_{R}^{c}=\mathbb{R}^{N}\backslash B_{R}$; \\
 $ \left( \ast\right) \ \ F'(u):$ the Fr\'echet derivative of $F$ at $u$.

 Let $F_{1}$, $F_{2}$ be Banach spaces and $T: F_{1}\rightarrow F_{2}$. $T$ is said to be a sequentially compact operator if given any bounded sequence $\left(x_{n}\right)$ in $F_{1}$ then $\left(T\left(x_{n}\right)\right)$ has a convergent subsequence in $F_{2}$.

 Let $E=H^{1}\left(\mathbb{R}^{N}\right)\cap L^{q+1}\left(\mathbb{R}^{N}\right)$ ($0<q<1$) be the reflexive Banach space endowed  with the norm
 $$\left\|u\right\|=\left\|\nabla u\right\|_{2}+\left\|u\right\|_{q+1}\,.$$


 Let $X=D^{1,2}\left(\mathbb{R}^{N}\right)=\left\{u\in L^{2^{\ast}}\left(\mathbb{R}^{N}\right); \  \nabla u\in \left(L^{2}\left(\mathbb{R}^{N}\right)\right)^{N}\right\}$, endowed with the norm
 $$\left\|u\right\|_{X}=\left(\int_{\mathbb{R}^{N}}\left|\nabla u(x)\right|^{2} dx\right)^{1/2}$$
   is a Hilbert space. Moreover,  the embedding $X\subset  L^{2^{\ast}}\left(\mathbb{R}^{N}\right)$ is continuous, which implies that
 $$S:=\inf\left\{\int_{\mathbb{R}^{N}}\left|\nabla u(x)\right|^{2} dx;\ u\in X,\ \int_{\mathbb{R}^{N}}\left| u(x)\right|^{2^\ast} dx=1 \right\}>0.$$
 We refer to Willem \cite[pp. 8-9]{willembook} for more details.

 Let
 $$Y=\left\{u\in H^{1}\left(\mathbb{R}^{N}\right); \ \ \int_{\mathbb{R}^{N}}V^{+}(x) u^{2}(x) dx < +\infty \right\},$$
under the hypotheses $V\in L^{\infty}\left(\mathbb{R}^{N}\right)$ and ${\rm esslim}_{x\rightarrow +\infty} V(x)>0$. We endow $Y$ with the inner product
 $$\left\langle u,v\right\rangle=\displaystyle \int_{\mathbb{R}^{N}} \nabla u \nabla v+\displaystyle \int_{\mathbb{R}^{N}} V^{+}(x) u v dx$$
 and the associated norm $\left\| \ \ \right\|_{Y}$, which is equivalent to the usual norm
 $$\left\|u\right\|_{H^{1}}=\left\|\nabla u\right\|_{2}+\left\|u\right\|_{2}. $$

  Consider the following functionals
 $$\qquad \, \qquad \, \qquad I(u)=\frac{1}{2} \int_{\mathbb{R}^{N}}\left(\left|\nabla u\right|^{2}(x)+V(x) u^{2}(x)\right) dx -\int_{\mathbb{R}^{N}}a(x) G\left(u(x)\right) dx,$$
 $$\phi \left(u\right)=\frac{1}{2}\int_{\mathbb{R}^{N}}V(x)u^{2}(x) dx-\int_{\mathbb{R}^{N}}a(x)G\left(u(x)\right) dx,$$
 $$\quad \psi (u)=-\frac{1}{2} \int_{\mathbb{R}^{N}}V^{-}(x) u^{2}(x) dx-\int_{\mathbb{R}^{N}} a(x) G\left(u(x)\right) dx,$$
Under suitable assumptions  on $a$, $G$, $V$ (to be fixed later), $I$,  $\phi$  and $\psi$ are well defined and of class $C^{1}$ on $X$,  $Y$ or $E$. A critical point of $I$ is a weak solution of problem \eqref{main}.

Next, let us recall that a  Palais-Smale sequence for the functional $I$, for short we write $\left(PS\right)$, is a sequence $\left(u_{n}\right)$ such that
 $$I(u_{n}) \ \ \mbox{is bounded} \ \ \mbox{and} \ \ \left\|I{'}(u_{n})\right\|\rightarrow 0.$$
 The functional $I$ is said to satisfy the Palais-Smale condition if any $\left(PS\right)$-sequence possesses a convergent subsequence.

 A first main difficulty that appears in the study of problem \eqref{main} is the loss of compactness. In order to overcome this difficulty, we use the Lions compactness principle \cite{L2}. A second main difficulty is to satisfy the geometric conditions required by the Ambrosetti-Rabinowitz theorem \cite{A2}. We use a geometrical construction  of subset to overcome this difficulty. Let us give a based definition and recall the mountain pass theorem of Ambrosetti and Rabinowitz.

 \begin{definition}
 Let $E$ be a Banach space. A subset $A$ of $E$ is said to be symmetric if $u\in E$ implies $-u\in E$. For a closed symmetric set $A$ which does not contain the origin, we define the genus $\gamma \left(A\right)$ of $A$ by the smallest integer $k$ such that there exists an odd continuous mapping from $A$ to $\mathbb{R}^{k} \backslash  \left\{0\right\}$. If there does not exist such a $k$, we define $\gamma \left(A\right)=+\infty$. We set $\gamma \left( \emptyset \right)=0.$ Let $\Gamma_{k}$ denote the family of closed symmetric subsets $A$ of $E$ such that $0\notin A$ and $\gamma\left(A\right)\geq k$.
 \end{definition}

 \begin{thm}[Theorem of Ambrosetti-Rabinowitz \cite{A2}] \label{zer thm}
 Let $E$ be an infinite dimensional Banach space and $I\in C^{1}\left(E, \mathbb{R}\right)$ satisfy:\\
$\left(1\right) \ \ I$ is even, bounded from below, $I\left(0\right)=0$ and $I$ satisfies the Palais-Smale condition.\\
$\left(2\right)$ For each $k\in \mathbb{N}$, there exists  $A_{k}\in \Gamma_{k}$ such that
$$ \displaystyle \sup_{u\in A_{k} }  I\left(u\right)< 0.$$
Under assumptions $\left(1\right)-\left(2\right)$, we define $c_{k}$ by
$$c_{k}=\displaystyle \inf_{A \in \Gamma_{k}}\displaystyle \sup_{u\in A}I\left(u\right).$$
Then each $c_{k}$ is a critical value of $I$, $c_{k}\leq c_{k+1}<0$ for $k\in \mathbb{N}$ and $\left(c_{k}\right)$ converges to zero. Moreover, if $c_{k}=c_{k+1}=\cdots =c_{k+p}=c,$ then $\gamma\left(K_{c}\right)\geq p+1$. The critical set $K_{c}$ is defined by
$$K_{c}=\left\{u\in E; \  I'\left(u\right)=0, \  I\left(u\right)=c\right\}\,.$$
\end{thm}
%Our approach is based on applying the above Ambrosetti-Rabinowitz theorem.

\section {Proof of Theorem \ref{troi thm}}
 In this section we consider the case where $a$ is bounded and we define $I$ on the function space $E=H^{1}\left(\mathbb{R}^{N}\right)\cap L^{q+1}\left(\mathbb{R}^{N}\right)$.

\begin{lemma}\label{spalais}
Assume that $\left(A_{1}\right)$ and $\left(V_{1}\right)$  hold. Then  any $\left(PS\right)$-sequence of $I$ is bounded in $E$.
\end{lemma}
\begin{proof}
By standard arguments, $I$ is well defined and of class $C^{1}$ on $E$.

Let $\left(u_{n}\right)$ be a $\left(PS\right)$-sequence of $I$. Then there exists $\alpha>0$ such that $I(u_{n})\leq \alpha$. Applying H\"older inequality, conditions $ \left(A_{1}\right)$ and $\left(V_{1}\right)$, we have
$$\begin{array}{ll}
\alpha &\di\geq I\left(u_{n}\right)=\frac{1}{2} \displaystyle \int_{\mathbb{R}^{N}} \left(\left|\nabla u_{n}\right|^{2}(x)+ V(x) u_{n}(x)^{2}\right) dx -\frac{1}{q+1} \int_{\mathbb{R}^{N}}a(x)\left|u_{n}(x)\right|^{q+1} dx \\
&\di\geq \frac{1}{2} \displaystyle \int_{\mathbb{R}^{N}}\left|\nabla u_{n}(x)\right|^{2} dx-\frac{1}{2} \displaystyle \int_{\mathbb{R}^{N}}V^{-}(x)u_{n}(x)^{2} dx-\frac{1}{q+1}\displaystyle \int_{\mathbb{R}^{N}}a^{+}(x)\left|u_{n}\right|^{q+1}(x) dx \\
&+ \di\frac{1}{q+1}\displaystyle \int_{\mathbb{R}^{N}} a^{-}(x)\left|u_{n}\right|^{q+1}(x)dx\\
&\geq \di\frac{1}{2}\displaystyle \int_{\mathbb{R}^{N}} \left|\nabla u_{n}\right|^{2}(x) dx-\frac{\left\|V^{-}\right\|_{\frac{N}{2}}}{2s} \left\|\nabla u_{n}\right\|_{2}^{2}-\frac{1}{q+1} \displaystyle \int_{\mathbb{R}^{N}} a^{+}(x)\left|u_{n}\right|^{q+1}(x) dx.
\end{array}$$
By $\left(A_{1}\right)$, there exists $R>0$ such that
\begin{equation}\label{spre maj}
-\left\|a\right\|_{\infty} \leq a(x) \leq \frac{a_{\infty}}{2} <0,\ \ \forall  \left|x\right|\geq R \ \ and \ \ a^{+}\in L^{m}\left(\mathbb{R}^{N}\right), \ \ \forall \, 1\leq m \leq + \infty.
\end{equation}
Combining $\left(3.8\right)$  and \eqref{spre maj}, we infer that
\begin{align*}
\alpha & \geq I\left(u_{n}\right)\geq \frac{1}{2} \left\| \nabla u_{n}\right\|^{2}_{2}- \frac{\left\|V^{-}\right\|_{\frac{N}{2}}}{2s} \left\| \nabla u_{n}\right\|_{2}^{2}
-s^{\frac{-q-1}{2}}\left\|a^{+}\right\|_{\frac{2^{\ast}}{2^{\ast-\left(q+1\right)}}} \left\|\nabla u_{n}\right\|_{2}^{q+1}\\
&\geq \left(\frac{1}{2}-\frac{\left\|V^{-}\right\|_{\frac{N}{2}}}{2s}\right) \left\| \nabla u_{n}\right\|_{2}^{2}-s^{\frac{-q-1}{2}}\left\|a^{+}\right\|_{\frac{2^{\ast}}{2^{\ast}-(q+1)}} \left\|\nabla u_{n}\right\|_{2}^{q+1},
\end{align*}
hence there exists $\beta>0$ such that
\begin{equation}\label{sdeux maj}
\left\|\nabla u_{n}\right\|_{2}\leq \beta \qquad \forall \, n \in \mathbb{N}.
\end{equation}
On the other hand, there exists $c>0$ such that
\begin{align*}
c+\frac{\left\|u_{n}\right\|}{2} &\geq -\frac{1}{2} \left\langle I'\left(u_{n}\right),u_{n}\right\rangle+I\left( u_{n}\right)=\left(\frac{1}{2}-\frac{1}{q+1}\right) \displaystyle \int_{\mathbb{R}^{N}}a(x)\left|u_{n}\right|^{q+1}(x)dx \\
& =\left(\frac{1}{q+1}-\frac{1}{2}\right)\displaystyle \int_{\mathbb{R}^{N}}a^{-}(x) \left|u_{n}\right|^{q+1}( dx-\left(\frac{1}{q+1}-\frac{1}{2}\right)\displaystyle \int_{\mathbb{R}^{N}}a^{+}(x)\left|u_{n}\right|^{q+1} dx\\
&=\left(\frac{1}{q+1}-\frac{1}{2}\right)\displaystyle \int_{\mathbb{R}^{N}}\left(a^{-}(x)+\chi_{B_{R}}(x)\right) \left|u_{n}\right|^{q+1} dx\\ &-\left(\frac{1}{q+1}-\frac{1}{2}\right)\displaystyle \int_{\mathbb{R}^{N}}\left(a^{+}(x)+\chi_{B_{R}}(x)\right) \left|u_{n}\right|^{q+1} dx\\
&\geq \left(\frac{1}{q+1}-\frac{1}{2}\right) \min\left\{\frac{-a_{\infty}}{2}, 1\right\} \displaystyle \int_{\mathbb{R}^{N}}\left|u_{n}\right|^{q+1}(x) dx\\
&-s^{\frac{-q-1}{2}}\left(\frac{1}{q+1}-\frac{1}{2}\right)\left\|a^{+}+\chi_{B_{R}}\right\|_{\frac{2^{\ast}}{2^{\ast}-(q+1)}} \left\|\nabla u_{n}\right\|_{2}^{q+1}.
\end{align*}
Thus, there is a constant $c>0$ such that
\begin{equation*}
\displaystyle \int_{\mathbb{R}^{N}}\left|u_{n}\right|^{q+1} dx\leq c \left(\left\|\nabla u_{n}\right\|_{2}+ \left\|u_{n}\right\|_{q+1}+\left\|a^{+}\right\|_{\frac{2^{\ast}}{2^{\ast}+ (q+1)}} \left\|\nabla u_{n}\right\|_{2}^{q+1}\right).
\end{equation*}
Relation \eqref{sdeux maj} yields
\begin{equation}\label{stroi maj}
\left\|u_{n}\right\|_{q+1}^{q+1}\leq c+ c\left\|u_{n}\right\|_{q+1} \ \ \mbox{for  \  all} \ \ n\in \mathbb{N}.
\end{equation}
Combining \eqref{sdeux maj} and \eqref{stroi maj}, we get
$$\left\|u_{n}\right\|\leq c \ \ \forall \, n\in \mathbb{N}.$$
The proof is complete.
\end{proof}

We need the following lemma to prove that the Palais-Smale condition is satisfied for $I$ on $E$.

\begin{lemma}\label{apprendix}
 There exists a constant $c>0$ such that for all real numbers $x$, $y$,
\begin{equation}\label{xy}
\left|\left|x+y\right|^{q+1}-\left|x\right|^{q+1}-\left|y\right|^{q+1}\right|\leq c \left|x\right|^{q}\,|y|.
\end{equation}
\end{lemma}
\begin{proof}
If $x=0$, the inequality \eqref{xy} is trivial.

Suppose that $x\neq 0$. We consider the continous function $f$ defined on $\mathbb{R}\backslash \left\{0\right\}$ by
$$f(t)=\frac{\left|1+t\right|^{q+1}-\left|t\right|^{q+1}-1}{\left|t\right|}.$$
Then $\lim_{\left|t\right|\rightarrow +\infty}f(t)=0$ and $\lim_{t\rightarrow  0\pm}f(t)=\pm \left(q+1\right)$. Thus, there exists a constant $c>0$ such that $\left|f(t)\right|\leq c, \ \ \forall\, t\in \mathbb{R}\backslash \left\{0\right\}$. In particular $\left|f\left(\frac{y}{x}\right)\right|\leq c$, so
$$\left|\left|1+\frac{y}{x}\right|^{q+1}-\left|\frac{y}{x}\right|^{q+1}-1\right|\leq c \left|\frac{y}{x}\right|.$$
Multiplying by $\left|x\right|^{q+1}$, we obtain the desired result.
\end{proof}

\begin{lemma}\label{Palais-Smalee}
Assume $\left(A_{1}\right)$ and $\left(V_{1}\right)$ hold. Then $I$ satisfies the Palais-Smale condition on $E$.
\end{lemma}
\begin{proof}
Let $\left(u_{n}\right)$ be a $\left(PS\right)$-sequence. By Lemma \ref{spalais}, $\left(u_{n}\right)$ is bounded in $E$. Then there exists a subsequence $u_{n}\rightharpoonup u$ in $E$, $u_{n}\rightarrow u$ in $L^{p}_{loc}\left(\mathbb{R}^{N}\right)$ for all  $1\leq p\leq 2^{\ast}$ and $u_{n}\rightarrow u$ a.e. in $\mathbb{R}^{N}$.

Fix $\phi \in D\left(\mathbb{R}^{N}\right)$. By the weak convergence of $\left(u_{n}\right)$ to $u$, we get
\begin{equation}\label{squat maj}
\int_{\mathbb{R}^{N}}\nabla u_{n} \nabla \phi(x) +V(x) u_{n} \phi(x) dx\rightarrow \int_{\mathbb{R}^{N}} \nabla u \nabla \phi+ V(x) u \phi (x) dx.
\end{equation}
By compactness Sobolev embedding,
 $u_{n}\rightarrow u$ in $L^{q+1}\left({\rm supp}\,\left( \phi \right)\right)$, hence there exists a function $h \in L^{q+1}\left(\mathbb{R}^{N}\right)$ such that
$$a(x) \left|u_{n}\right|^{q-1}u_{n} \phi \rightarrow a(x) \left| u \right|^{q-1} u \phi \ \ \mbox{a.e. \ in}  \ \mathbb{R}^{N}$$
and
$$\left|a\right| \left|u_{n}\right|^{q} \left| \phi \right|\leq \left\| a \right\|_{\infty} \left|h\right| \left|\phi\right| \ \ \mbox{in} \ \ \mathbb{R}^{N}.$$
Using the Lebesgue dominated convergence theorem, we deduce that
\begin{equation}\label{scinq maj}
\displaystyle \int_{\mathbb{R}^{N}}a(x) \left|u_{n}\right|^{q-1} u_{n} \phi(x) dx \rightarrow \int_{\mathbb{R}^{N}} a(x)\left|u\right|^{q-1} u \phi(x) dx.
\end {equation}
Combining relations \eqref{squat maj} and \eqref{scinq maj}, we obtain
$$0=\displaystyle \lim_{n\rightarrow +\infty}\left\langle I'\left(u_{n}\right), \phi \right\rangle= \left\langle I'\left(u\right), \phi\right\rangle, \ \ \forall \, \phi \in D\left(\mathbb{R}^{N}\right).$$
 Then
 \begin{equation}\label{Id}
 \left\langle I'(u),u\right\rangle=0.
 \end{equation}
 Since $u_{n}\rightharpoonup u$ in $E$, we have $\left\|u \right\|\leq \displaystyle \liminf_{n\rightarrow + \infty}\left\|u_{n}\right\|= \displaystyle \lim_{n\rightarrow +\infty}\left\|u_{n}\right\|$. We distinguish two cases: \\
\textbf{A}) \ \ \textbf{Compactness}: \ \ $\left\|u\right\|=\displaystyle \lim_{n\rightarrow +\infty} \left\|u_{n}\right\|$, then
$$
\limsup_{n\rightarrow +\infty} \left\|u_{n}\right\|_{q+1}
%= \left\|u \right\|_{q+1}+\left\|\nabla u\right\|_{2}-\liminf_{n \rightarrow +\infty} \left\|\nabla %u_{n}\right\|_{2}\\
\leq \left\|u \right\|_{q+1}+\left\|\nabla u\right\|_{2}-\liminf_{n\rightarrow +\infty } \left\|\nabla u_{n}\right\|_{2}.
$$
Since
 $$\left\|\nabla u\right\|_{2}\leq \liminf_{n\rightarrow +\infty}\left\|\nabla u_{n}\right\|_{2},\ \ \left\|u \right\|_{q+1}\leq \liminf_{n\rightarrow +\infty} \left\|u_{n}\right\|_{q+1},$$
we get
$$\left\|u \right\|_{q+1}\leq \liminf_{n\rightarrow +\infty}\left\|u_{n}\right\|_{q+1}\leq \limsup_{n\rightarrow +\infty} \left\|u_{n}\right\|_{q+1}\leq \left\|u\right\|_{q+1},$$
thus
$$\left\{\begin{array}{cll}
u_{n}&\rightarrow u \ \ \mbox{a.e.  \ in} \ \ \mathbb{R}^{N}\\
\left\|u_{n}\right\|_{q+1}&\rightarrow \left\|u \right\|_{q+1}.
\end{array}
\right.$$
By Brezis-Lieb lemma \cite{B7}, we infer that
\begin{equation}\label{ssix maj}
u_{n}\rightarrow u \ \ \mbox{in} \ \ L^{q+1}\left(\mathbb{R}^{N}\right).
\end{equation}
 Therefore $\left\|\nabla u_{n}\right\|_{2}\rightarrow \left\|\nabla u\right\|_{2}$. On the other hand
$$\int_{\mathbb{R}^{N}} \left|\nabla u_{n}-\nabla u\right|^{2} dx=\int_{\mathbb{R}^{N}}\left|\nabla u_{n}\right|^{2} dx+\int_{\mathbb{R}^{N}}\left|\nabla u\right|^{2}dx - 2 \int_{\mathbb{R}^{N}} \nabla u_{n} \nabla u \, dx,$$
hence
$$\int_{\mathbb{R}^{N}}\nabla u_{n} \nabla u dx\rightarrow \int_{\mathbb{R}^{N}}\left|\nabla u\right|^{2} \, dx .$$
Therefore
\begin{equation}\label{sep maj}
\left\| \nabla u_{n}-\nabla u\right\|_{2}\rightarrow 0.
\end{equation}
Combining relations \eqref{ssix maj} and \eqref{sep maj}, we deduce that $u_{n}\rightarrow u$ in $E$ and the (PS) condition for $I$ is satisfied.

\textbf{B}) \ \ \textbf{Dichotomy}: $\left\|u \right\|< \displaystyle \lim_{n\rightarrow +\infty}\left\|u_{n}\right\|$. We prove that this case cannot occur. Set $v_{n}=u_{n}-u$.

\textbf{\textsl{Step}} $1$: \ \ there exists $\left(y_{n}\right)\subset \mathbb{R}^{N}$ such that $v_{n}\left(.+y_{n}\right)\rightharpoonup v\neq 0$ in $E$.
If not,  for all $\left(y_{n}\right)\subset \mathbb{R}^{N}$, $v_{n}\left(.+y_{n}\right) \rightharpoonup 0$ in $E$. Then
$$\forall R>0 \ \ \ \sup_{y\in{\mathbb{R}^{N}}}\int_{B\left(y,R\right)} \left|v_{n}\right|^{q+1}(x) dx \rightarrow 0.$$
By \cite[Lemma I.1, p. 231]{L2},
\begin{equation}\label{hu maj}
v_{n}\rightarrow 0 \ \ in \ \ L^{p}\left(\mathbb{R}^{N}\right), \ \ \forall \, q+1<p<2^{\ast}.
\end{equation}\label{maj}
On the other hand,
\begin{equation}\label{I}\begin{array}{ll} {\small
\di\left\langle I'\left(u_{n}\right), u_{n}\right\rangle } &{\small =\di\int_{\mathbb{R}^{N}}\left(\left|\nabla u_{n}\right|^{2}+ V(x)u_{n}^{2}\right) dx-\int_{\mathbb{R}^{N}}a(x)\left|u_{n}\right|^{q+1} dx}\\
&{\small =\di\int_{\mathbb{R}^{N}}\left(\left|\nabla v_{n}\right|^{2}+ V(x) v_{n}^{2}\right) dx+ \int_{\mathbb{R}^{N}}\left|\nabla u\right|^{2} dx+\int_{\mathbb{R}^{N}}\left(V(x) u^{2}+2 \nabla v_{n} \nabla u\right) dx }\\
&{\small +\di\int_{\mathbb{R}^{N}}2 V(x)v_{n} u dx
-\int_{\mathbb{R}^{N}} a(x)\left(\left|u_{n}\right|^{q+1}-\left|u\right|^{q+1}\right) dx-\int_{\mathbb{R}^{N}} a(x)\left| u\right|^{q+1} dx.  }
\end{array}\end{equation}
By \eqref{xy} in Lemma \ref{apprendix}, we obtain
\begin{align*}
\left|a(x)\right| \left|\left|u_{n}\right|^{q+1}-\left|u\right|^{q+1}-\left|v_{n}\right|^{q+1}\right|&=\left|a(x)\right| \left|\left|v_{n}+u\right|^{q+1}-\left|u\right|^{q+1}-\left|v_{n}\right|^{q+1}\right|\\
&\leq c  \left|a(x)\right| \left|u\right|^{q} v_{n}.
\end{align*}
Since $v_{n}\rightharpoonup 0$ in $E$, we deduce that
\begin{equation}\label{vn}
\displaystyle \lim_{n\rightarrow +\infty} \int_{\mathbb{R}^{N}} a(x)\left(\left|u_{n}\right|^{q+1}-\left|u\right|^{q+1}\right) dx=\displaystyle \lim_{n\rightarrow +\infty} \int_{\mathbb{R}^{N}} a(x) \left|v_{n}\right|^{q+1}(x)dx.
\end{equation}
Using H\"older inequality in combination with relations \eqref{spre maj} and \eqref{hu maj}, we obtain
\begin{equation}\label{hv}
\displaystyle \int_{\mathbb{R}^{N}} \left(a^{+}(x)+\chi_{B_{R}}(x)\right) \left|v_{n}\right|^{q+1} dx \leq \left\|a^{+}+\chi_{B_{R}}\right\|_{\frac{2}{1-q}} \left\|v_{n}\right\|_{L^{2}\left(B\left(0,R\right)\right)}^{q+1}\rightarrow 0,
\end{equation}
Passing to the limit in \eqref{I} and using \eqref{spre maj}, \eqref{hu maj}, \eqref{vn} and \eqref{hv}, we get
\begin{align*}
0=\lim_{n\rightarrow +\infty}\left\langle I'\left(u_{n}\right),u_{n}\right\rangle &= \left\langle I{'}\left(u\right), u\right\rangle+\lim_{n\rightarrow +\infty}\left( \int_{\mathbb{R}^{N}}\left(\left|\nabla v_{n}\right|^{2}-a(x)\left|v_{n}\right|^{q+1}\right) dx\right)\\
&=\left\langle I'\left(u\right), u\right\rangle+\lim_{n\rightarrow +\infty} \left(\int_{\mathbb{R}^{N}}\left|\nabla v_{n}\right|^{2}+\left(a^{-}(x)+\chi_{B_{R}}\right)\left|v_{n}\right|^{q+1}\right)\\
&-\displaystyle \lim_{n\rightarrow +\infty} \int_{\mathbb{R}^{N}}\left(a^{+}(x)+\chi_{B_{R}}\right) \left|v_{n}\right|^{q+1} dx \\
&=\lim_{n\rightarrow +\infty} \int_{\mathbb{R}^{N}}\left| \nabla v_{n}\right|^{2}+\left(a^{-}(x)+\chi_{B_{R}}\right) \left|v_{n}\right|^{q+1} dx.\\
&\geq \lim_{n\rightarrow +\infty}\int_{\mathbb{R}^{N}}\left( \left|\nabla v_{n}\right|^{2}+ \min\left(\frac{-a_{\infty}}{2}, 1\right) \left|v_{n}\right|^{q+1}\right) dx \\
&\geq \lim_{n\rightarrow +\infty} \min\left(1, \min\left(\frac{-a_{\infty}}{2}, 1\right)\right) \int_{\mathbb{R}^{N}} \left( \left|\nabla v_{n}\right|^{2}+ \left|v_{n}\right|^{q+1}\right) dx.
\end{align*}
Then $v_{n}\rightarrow 0$ in $E$, which yields a contradiction.

\textbf{Step $2$}: \ \ $\left(y_{n}\right)$ is not bounded. Indeed, suppose that $\left(y_{n}\right)$ is bounded, there exists a subsequence of $\left(y_{n}\right)$, also denoted by $\left(y_{n}\right)$, such that $y_{n}\rightarrow y_{0}$.
Then for all $\phi \in D\left(\mathbb{R}^{N}\right)$
$$0=\displaystyle \lim_{n\rightarrow +\infty}\int_{\mathbb{R}^{N}}\phi \left(x-y_{n}\right) v_{n} dx=\displaystyle \lim_{n\rightarrow +\infty}\int_{\mathbb{R}^{N}}\phi\left(x\right) v_{n}\left(x+y_{n}\right) dx=\int_{\mathbb{R}^{N}} \phi (x) v(x) dx.$$
Hence  $v=0$ a.e. \ in  $\mathbb{R}^{N}$, a contradiction.

\textbf{Step} $3$: We show that $v$ is a solution of the following problem:
$$\left(P_{\infty}\right)\left\{ \begin{array}{lll}
 -\Delta u+v_{\infty} u=a_{\infty}\left|u\right|^{q-1} u \ \ \ \ \mbox{in}\ \mathbb{R}^{N} \\
u\in{E}
\end{array}
\right.
$$
We first prove that problem $\left(P_{\infty}\right)$  admits only the trivial solution. Thus, since $v$ solves $\left(P_{\infty}\right)$, we will obtain a contradiction.

Since $\left(y_{n}\right)$ is not bounded, then $u_{n}\left(.+y_{n}\right)\rightharpoonup v$ in $E$. In fact, $u\left(.+y_{n}\right)\rightharpoonup \psi \in E$, hence
$$0=\lim_{n\rightarrow +\infty}\int_{\mathbb{{R}^{N}}} u\left(x+y_{n}\right) \phi (x) dx=\int_{\mathbb{{R}^{N}}}\psi(x) \phi(x) dx, \ \ \ \forall \, \phi \in D\left(\mathbb{R}^{N}\right).$$
It follows that
  $\psi=0$ a.e. Therefore
\begin{equation}\label{neuf maj}
 u_{n}\left(.+y_{n}\right)\rightharpoonup v \ \ \mbox{in} \ \ E.
\end{equation}
Let $\phi\in D\left(\mathbb{R}^{N}\right)$. We have
\begin{align*}
\left\langle I'\left(u_{n}\right), \phi \left(.-y_{n}\right)\right\rangle &=\int_{\mathbb{R}^{N}}\left(\nabla u_{n} \nabla \phi \left(x-y_{n}\right)+ V(x)u_{n}\phi(x-y_{n})\right) dx\\ &-\int_{\mathbb{R}^{N}}a(x)\left|u_{n}\right|^{q-1} u_{n} \phi \left(x-y_{n}\right) dx\\
&= \int_{\mathbb{R}^{N}}\nabla u_{n}\left(x+y_{n}\right) \nabla \phi(x)+ V(x+y_{n}) u_{n}(x+y_{n}) \phi (x) dx\\
&-\int_{\mathbb{R}{N}}a(x+y_{n}) \left|u_{n}\right|^{q-1}(x+y_{n})u_{n}(x+y_{n}) \phi(x) dx.
\end{align*}
Relation \eqref{neuf maj} yields
\begin{equation}\label{dix maj}
\int_{\mathbb{R}^{N}}\nabla u_{n}\left(x+y_{n}\right) \nabla \phi(x) dx\rightarrow \int_{\mathbb{R}^{N}}\nabla v(x) \nabla \phi(x) dx.
\end{equation}
 Since $\left(u_{n}\left(.+y_{n}\right)\right)$ is bounded in $E$, $u_{n}\left(.+y_{n}\right)\rightarrow v$ in $L^{p}_{loc}\left(\mathbb{R}^{N}\right)$, for all $1\leq p\leq 2^{\ast}$ (up a subsequence), $u_{n}\left(x+y_{n}\right)\rightarrow v$ a.e. in $\mathbb{R}^{N}$ and there exists $K\in L^{p}\left(\mathbb{R}^{N}\right)$ such that $\phi \, \left|u_{n}\left(.+y_{n}\right)\right|\leq \left| K\right|$ in $\mathbb{R}^{N}$, $1\leq p \leq 2^{\ast}$. Then, by $\left(V_{1}\right)$, we obtain
$$\left\{\begin{array}{lll}
V\left(x+y_{n}\right) u_{n}\left(x+y_{n}\right) \phi &\rightarrow v_{\infty} v \phi \ \ \mbox{a.e. \  in} \ \ \mathbb{R}^{N}\\
\left|V\left(x+y_{n}\right) u_{n}\left(x+y_{n}\right) \phi\right| & \leq \left\|V\right\|_{\infty} \left|K\right| \left|\phi \right| \in L^{1}\left(\mathbb{R}^{N}\right).
\end{array}
\right. $$
 Applying Lebesgue's dominated convergence theorem, we deduce that
\begin{equation}\label{onze maj}
\int_{\mathbb{R}^{N}} V\left(x+y_{n}\right) u_{n}\left(x+y_{n}\right) \phi(x) dx \rightarrow v_{\infty} \int_{\mathbb{R}^{N}} v(x) \phi(x) dx.
\end{equation}
From hypothesis $\left(A_{1}\right)$, we find
$$\left\{ \begin{array}{lll}
a(x+y_{n}) \left|u_{n}\left(x+y_{n}\right)\right|^{q-1} u_{n}(x+y_{n}) \phi & \rightarrow a_{\infty} \left|v\right|^{q-1} v \phi \ \ \mbox{a.e.\ in}\ \R^N\\
\left|a(x+y_{n})\right|  \left|u_{n}(x+y_{n})\right|^{q} \left| \phi \right| & \leq \left\|a \right\|_{\infty} \left|K\right|^{q} \left| \phi \right| \, \in L^{1}\left(\mathbb{R}^{N}\right).
\end{array}
\right.
$$
Next, by Lebesgue's dominated convergence theorem, we obtain
\begin{equation}\label{a}
\displaystyle \lim_{n\rightarrow +\infty}\int_{\mathbb{R}^{N}} a(x+y_{n})\left|u_{n}\right|^{q-1}(x+y_{n})u_{n}(x+y_{n}) dx=a_{\infty} \int_{\mathbb{R}^{N}} \left|v\right|^{q-1} v \phi(x)dx.
\end{equation}
Combining relations \eqref{dix maj}, \eqref{onze maj} and \eqref{a}, we deduce that for all
$\phi \in D\left(\mathbb{R}^{N}\right)$,
$$\begin{array}{ll} 0&\di=\displaystyle \lim_{n\rightarrow +\infty}\left\langle I'\left(u_{n}\right), \phi\left(.-y_{n}\right)\right\rangle\\ &\di= \int_{\mathbb{R}^{N}}\left(\nabla v(x)\nabla \phi(x) +v_{\infty} v \phi \right)dx-a_{\infty}\int_{\mathbb{R}^{N}}\left|v\right|^{q-1}v \phi(x) dx\,.\end{array}$$
 Thus, $v$ is a weak solution of problem $\left(P_{\infty}\right)$, hence $v=0$, which yields a contradiction. From steps 1, 2, and 3, we conclude that the dichotomy does not occur. The proof is complete.
\end{proof}

\begin{lemma}\label{cubee}
Assume $\left( A_{1}\right)$ and $\left(V_{1}\right)$ are fulfilled. Then for each $k\in \mathbb{N}$, there exists  $A_{k} \in \Gamma_{k}$ such that
$ \sup_{u\in A_{k}} I\left(u\right)<0$.
\end{lemma}
\begin{proof}
We use some ideas developed  in \cite{K1}.

Let $R_{0}$ and $y_{0}$ fixed by assumption $\left(A_{1}\right)$ and consider the cube
$$D\left(R_{0}\right)=\left\{\left(x_{1},\cdots,x_{N}\right)\in{\mathbb{R}^{N}}: \ \ \left|x_{i}-y_{i}\right|<R_{0}, \ \ 1\leq i\leq N\right\}.$$
Fix $k\in \mathbb{N}$ arbitrarily. Let $n \in\mathbb{N}$ be the smallest integer such that $n^{N}\geq k$. We divide $D\left(R_{0}\right)$ equally into $n^{N}$ small cubes, denote them by $D_{i}$ with $1\leq i \leq n^{N}$, by planes parallel to each face of $D\left(R_{0}\right)$. The edge of $D_{i}$ has the length of $a=\frac{R_{0}}{n}$. We construct  new cubes $E_{i}$ in $D_{i}$ such that $E_{i}$ has the same center as that of $D_{i}$. The faces of $E_{i}$ and $D_{i}$ are parallel and the edge of $E_{i}$ has the length of $\frac{a}{2}$. Thus, we can construct a function $\psi_{i}, \ 1\leq i \leq k$, such that
$${\rm supp}\, \left(\psi_{i}\right) \subset D_{i}, \ \ {\rm supp}\,\left(\psi_{i}\right)\cap {\rm supp}\,\left(\psi_{j}\right)=\emptyset \ \ \left(i\neq j\right),$$
$$\psi_{i}(x)=1 \ \ \mbox{for} \ \ x\in E_{i}, \ \ 0\leq \psi_{i}(x)\leq 1, \ \ \forall\, x\in \mathbb{R}^{N}.$$
We denote
\begin{equation} \label{l'en V}
S^{k-1}=\left\{\left(t_{1},\cdots, t_{k}\right)\in \mathbb{R}^{k}: \ \ \displaystyle \max_{1\leq i \leq k}\left|t_{i}\right|=1\right\}
\end{equation}
\begin{equation}\label{l'en w}
W_{k}=\left\{\sum_{i=1}^{k}t_{i}\psi_{i}(x): \ \ \left(t_{1},\cdots, t_{k}\right)\in S^{k-1}\right\}\subset E.
\end{equation}
Since the mapping $\left(t_{1},\cdots, t_{k}\right)\rightarrow \sum_{i=1}^{k} t_{i} \psi_{i}$ from $S^{k-1}$ to $W_{k}$ is odd and homeomorphic, we have $\gamma\left(W_{k}\right)=\gamma \left(S^{k-1}\right)=k$. But $W_{k}$ is compact in $E$, thus there is a constant $\alpha_{k}>0$ such that
$$\left\|u\right\|^{2} \leq \alpha_{k} \ \ \mbox{for  \ all} \ \ u\in W_{k}.$$
We recall the following inequality:
\begin{equation}\label{interpolation}
\left\| u\right\|_{2}\leq c \left\|\nabla u\right\|_{2}^{r} \left\|u\right\|_{q+1}^{1-r}\leq c \left\|u\right\|\,,
\end{equation}
with $r=\frac{2^{\ast} \left(q-1\right)}{2\left(2^{\ast}-q-1\right)}$. Then there is a constant $c_{k}>0$ such that
$$\left\|u \right\|_{2}^{2} \leq c_{k} \ \ \ \mbox{for \  all} \ \ u\in W_{k}.$$
Let $z>0$ and $u=\sum_{i=1}^{k}t_{i} \psi_{i}(x) \in W_{k}$. We have
\begin{equation} \label{pre maj}
I \left(z u\right) \leq \frac{z^{2}}{2}\alpha_{k}+z^{2} \frac{\left\|V\right\|_{\infty}}{2} c_{k}-\frac{1}{q+1}\sum_{i=1}^{k}\displaystyle \int_{D_{i}} a(x) \left|z t_{i} \psi_{i}\right|^{q+1}  dx.
\end{equation}
By \eqref{l'en V}, there exists  $j \in \left[1,k\right]$ such that $\left|t_{j}\right|=1$ and $\left|t_{i}\right|\leq 1$ for $i\neq j$. Then
\begin{equation} \label{deux maj}\begin{array}{ll}
&\di \sum_{i=1}^{k}\displaystyle \int_{D_{i}} a(x) \left|z t_{i} \psi_{i}\right|^{q+1}  dx =\displaystyle \int_{E_{j}}a(x) \left|z t_{j} \psi_{j}\right|^{q+1}  dx
+ \\ &\displaystyle \int_{D_{j}\backslash E_{j}}a(x) \left|z t_{j} \psi_{j}(x)\right|^{q+1} dx
+\sum_{i\neq j}\int_{D_{i}}a(x) \left|z t_{i} \psi_{i}\right|^{q+1} dx.
\end{array}\end{equation}
Since $\psi_{j}(x)=1$ for $x\in E_{j}$ and $\left|t_{j}\right|=1$, we have
\begin{equation} \label{troi maj}
\displaystyle \int_{E_{j}}a(x) \left|z t_{j} \psi_{j}\right|^{q+1}  dx= \left|z\right|^{q+1} \displaystyle \int_{E_{j}}a(x)dx.
\end{equation}
On the other hand by  $\left(A_{1}\right)$,
\begin{equation}\label{quat maj}
\displaystyle \int_{D_{j}\backslash E_{j}}a(x) \left|z t_{j} \psi_{j}\right|^{q+1}  dx+ \displaystyle \sum_{i\neq j} \displaystyle \int_{D_{i}}a(x) \left|z t_{i} \psi_{i}\right|^{q+1}  dx\geq 0.
\end{equation}
Relations \eqref{pre maj}, \eqref{deux maj}, \eqref{troi maj} and \eqref{quat maj} yield
\begin{equation}\label{cinq maj}
\frac{I\left(z u\right)}{z^{2}}\leq \frac{\alpha_{k}}{2}+ \frac{\left\|V\right\|_{\infty}}{2} c_{k} - \frac{\left|z\right|^{q+1}}{z^{2}}\displaystyle \inf_{1\leq i \leq k}\left(\displaystyle \int_{E_{i}}a(x) dx\right),
\end{equation}
By  \eqref{cinq maj}, we conclude that
 $$\displaystyle \lim_{z\rightarrow 0} \sup_{u\in W_{k}} \frac{I\left(z u\right)}{z^{2}}=-\infty.$$

 We fix $z$ so small such that
$$\sup\left\{I(u), u\in A_{k}\right\} < 0, \ \ \mbox{where} \ A_{k}=z W_{k} \in \ \Gamma_{k}.$$
 This concludes the proof.
\end{proof}

\begin{lemma}\label{boundedd}
Assume $\left(A_{1}\right)$ and $\left(V_{1}\right)$ hold. Then $I$ is bounded from below.
\end{lemma}
\begin{proof}
By $\left(A_{1}\right)$, we get
\begin{equation}
a^{+} \in L^{p} \left(\mathbb{R}^{N}\right)\ \ \mbox{for  \ all} \ \  1 \leq p \leq +\infty.
\end{equation}
Then
\begin{align*}
I\left(u\right)&=\frac{1}{2}\int_{\mathbb{R}^{N}} \left( \left| \nabla u\right|^{2}+V(x) u^{2}\right) dx-\frac{1}{q+1}\int_{\mathbb{R}^{N}} a(x) \left|u \right|^{q+1} dx \\
&\geq \frac{1}{2} \int_{\mathbb{R}^{N}} \left(\left|\nabla u\right|^{2}-V^{-}(x)u^{2}\right) dx-\frac{1}{q+1} \int_{\mathbb{R}^{N}} a^{+}(x) \left| u \right|^{q+1} dx\\
&\geq \left(\frac{1}{2}-\frac{\left\| V^{-}\right\|_{\frac{N}{2}}}{2s} \right) \left\| \nabla u\right\|_{2}^{2}- \frac {\left\| a^{+}\right\|_{\frac{2^{\ast}}{2^{\ast}-q-1}}}{s^{\frac{q+1}{2}}} \left\| \nabla u\right\|_{2}^{q+1}.
\end{align*}
In view of $\left(V_{1}\right)$ we conclude the proof.
\end{proof}

{\bf Proof of Theorem \ref{troi thm} concluded}.
We have $I(0)=0$ and $I$ is  even. Combining Lemmas \ref{Palais-Smalee}, \ref{cubee} and \ref{boundedd}, we deduce that the conditions $(1)$ and $(2)$ of Theorem \ref{zer thm} are satisfied. Thus, there exists a sequence  $\left(u_{n}\right)\subset E$ such that $I(u_{n})<0$, $I'\left(u_{n}\right)=0$ and $I(u_{n})\rightarrow 0$, for all $n\geq 0$, hence $u_{n}$ is a weak solution of problem \eqref{main}.

By $\left(V_{1}\right)$, we deduce that
\begin{align*}
\frac{1}{q+1} \left\langle I'(u_{n}),u_{n}\right\rangle- I(u_{n}) &= \left(\frac{1}{q+1}-\frac{1}{2}\right)\int_{\mathbb{R}^{N}}\left(\left|\nabla u_{n}\right|^{2}+V(x) u_{n}^{2}\right)dx\\
&\geq \left(\frac{1}{q+1}-\frac{1}{2}\right)\left(\frac{1}{2}-\frac{1}{2s} \left\|V^{-}\right\|_{\frac{N}{2}}\right) \left\|\nabla u_{n}\right\|_{2}^{2}.
\end{align*}
It follows that
\begin{equation}\label{grad}
\displaystyle \lim_{n\rightarrow +\infty} \int_{\mathbb{R}^{N}} \left|\nabla u_{n}\right|^{2} dx=0.
\end{equation}
On the other hand, by H\"older inequality, \eqref{spre maj} and \eqref{grad}, we have
\begin{align*}
\displaystyle 0&=\lim_{n\rightarrow +\infty} \left(I(u_{n})-\frac{1}{2}\left\langle I'(u_{n}), u_{n}\right\rangle\right) = \displaystyle \lim_{n\rightarrow +\infty}\left(\frac{1}{2}-\frac{1}{q+1}\right) \int_{\mathbb{R}^{N}} a(x) \left|u_{n}\right|^{q+1}\\
&= \left(\frac{1}{q+1}-\frac{1}{2}\right) \displaystyle \lim _{n\rightarrow +\infty} \left(\int_{\mathbb{R}^{N}} \left(a^{-}(x)+\chi_{B_{R}}\right) \left|u_{n}\right|^{q+1} dx
- \int_{\mathbb{R}^{N}} \left(a^{+}(x)+\chi_{B_{R}}\right) \left|u_{n}\right|^{q+1}\right)\\
&\geq \left(\frac{1}{q+1}-\frac{1}{2}\right) \displaystyle \lim _{n\rightarrow +\infty} \left(\int_{\mathbb{R}^{N}} \left(a^{-}(x)+\chi_{B_{R}}\right) \left|u_{n}\right|^{q+1} dx-\frac{\left\|a^{+}+\chi_{B_{R}}\right\|^{\frac{2^{\ast}}{2^{\ast}-q-1}}}{s^{\frac{q+1}{2}}} \left\|\nabla u_{n}\right\|_{2}^{q+1}\right)\\
& =\left(\frac{1}{q+1}-\frac{1}{2}\right)\displaystyle \lim_{n\rightarrow +\infty} \int_{\mathbb{R}^{N}} \left(a^{-}(x)+\chi_{B_{R}}\right) \left|u_{n}\right|^{q+1} dx \\ &\di\geq \left(\frac{1}{q+1}-\frac{1}{2}\right) \min\left(\frac{-a_{\infty}}{2}, 1\right) \lim_{n\rightarrow +\infty}\left\|u_{n}\right\|_{q+1}^{q+1}.
\end{align*}
This shows that
\begin{equation}\label{norm}
\displaystyle \lim _{n\rightarrow +\infty} \int_{\mathbb{R}^{N}} \left|u_{n}\right|^{q+1} dx=0,
\end{equation}
hence $ \lim_{n\rightarrow +\infty}u_{n}=0$ in $E$. This concludes the proof.\qed

\section{Proof of Theorem \ref{prem thm}}
In this section, we define $I$ and $\phi$ on $X$. We use  standard arguments based on the fact that $I'$ is a sequentially compact operator in order to prove that $I$ satisfies the Palais-Smale condition. Then we deduce that problem \eqref{main}  admits infinitely many nontrivial solutions in $X$.

To prove  Theorem \ref{prem thm}, we  need the following auxiliary results.

\begin{lemma}\label{compact}
Assume $\left(A_{2}\right)$, $\left(V_{2}\right)$ and $\left(G_{1}\right)$ are satisfied. Then  $\phi'$ is a sequentially compact operator on $X$.
\end{lemma}
\begin{proof}
By  standard arguments, the functionals $I$ and $\phi$ are well defined and of class $C^{1}$ on $X$.

Let $\left(u_{n}\right) \subset X$ be a bounded sequence. Then for all $h\in X$,
$$\left\langle \phi'\left(u_{n}\right)-\phi'\left(u\right),h\right\rangle=\displaystyle \int_{\mathbb{R}^{N}}\left[V(x)\left(u_{n}-u\right)-a(x)\left(g(u_{n})-g(u)\right)\right]h(x) dx.$$
Let $R>0$ and $h\in X$ be such that $\left\|h\right\|=1$. We have
$$\left\langle \phi'\left(u_{n}\right)-\phi'\left(u\right), h\right\rangle=J_{1}\left(n,h,R\right)+J_{2}\left(n,h,R\right),$$
where
$$J_{1}\left(n,h,R\right)= \displaystyle \int_{B_{R}}\left[V(x)\left(u_{n}-u\right)-a(x) \left(g(u_{n})-g(u)\right)\right]h(x)dx$$
$$J_{2}\left(n,h,R\right)= \displaystyle \int_{B_{R}^{c}}\left[ V(x)\left(u_{n}-u\right)-a(x)\left(g(u_{n})-g(u)\right)\right] h(x) dx .$$
  By H\"older's inequality, $\left(V_{2}\right), \left(A_{2}\right)$ and $\left(G_{1}\right)$ we obtain
\begin{align*}
&\di\left|J_{2}\left(n,h,R\right)\right| \leq \displaystyle \int_{B_{R}^{c}}\left|V(x)\left(u_{n}-u\right)h(x)-a(x)\left(g\left(u_{n}\right)-g\left(u\right)\right) h(x)\right| dx\leq\\
&\di  \left( \displaystyle \int_{B_{R}^{c}}\left|V(x)\right|^{\frac{N}{2}} dx\right)^{\frac{2}{N}} \left(\displaystyle \int_{B_{R}^{c}}\left|u_{n}-u\right|^{2^{\ast}} dx\right)^{\frac{1}{2^{\ast}}} \left(\displaystyle \int_{B_{R}^{c}}\left|h(x)\right|^{2^{\ast}} dx\right)^{\frac{1}{2^{\ast}}}+\\
& c \left(\displaystyle \int_{B_{R}^{c}}\left|a(x)\right|^{\frac{2^{\ast}}{2^{\ast}-(q+1)}}dx\right)^{\frac{2^{\ast}-(q+1)}{2^{\ast}}} \left(\displaystyle \int_{B_{R}^{c}}\left(\left|u_{n}(x)+u(x)\right|\right)^{2^{\ast}}dx\right)^{\frac{q}{2^{\ast}}} \left(\displaystyle \int_{B_{R}^{c}}\left|h(x)\right|^{2^{\ast}}dx\right)^{\frac{1}{2^{\ast}}}\leq \\
&\di \left[\left(\displaystyle \int_{B_{R}^{c}}\left|V(x)\right|^{\frac{N}{2}}dx\right)^{\frac{2}{N}}+\left(\int_{B_{R}^{c}}\left|a(x)\right|^{\frac{2^{\ast}}{2^{\ast}-(q+1)}}dx\right)^{\frac{2^{\ast}-(q+1)}{2^{\ast}}}\right].
\end{align*}

 The last expression can be made arbitrarily small by taking $R>0$ large enough.

  For $J_{1}$, since $V\in L^{\frac{N}{2}}\left(\mathbb{R}^{N}\right)$ and $a\in L^{\frac{2^{\ast}}{2^{\ast-(q+1)}}}\left(\mathbb{R}^{N}\right)$, we deduce that for all $\epsilon >0$, there exists $ \eta>0$ such that
$$ \left(\displaystyle \int_{K}\left|a(x)\right|^{\frac{2^{\ast}}{2^{\ast}-(q+1)}}dx\right)^{\frac{2^{\ast}-(q+1)}{2^{\ast}}}+\left(\displaystyle \int_{K}\left|V(x)\right|^{\frac{N}{2}}dx\right)^{\frac{2}{N}}< \epsilon ,
 $$
for all $K\subset B_{R}$ with $m\left(K\right)< \eta$ (\mbox{see  Dunford-Pettis} \cite{del}). Moreover,
\begin{align*}
\displaystyle &\di\int_{K}\left|V(x)\left(u_{n}-u\right)-a(x)\left(g\left(u_{n}\right)-g\left(u\right)\right)\right| \left|h(x)\right| dx
\leq\\ &\di  c    \left(\displaystyle \int_{K}\left|V(x)\right|^{\frac{N}{2}}dx\right)^{\frac{2}{N}}
+c \left(\displaystyle \int_{K}\left|a(x)\right|^{\frac{2^{\ast}}{2^{\ast}-(q+1)}}dx\right)^{\frac{2^{\ast}-(q+1)}{2^{\ast}}}
 \leq c\epsilon,
 \end{align*}
where $c$ is independent of $n$ and $h$. By using the Vitali convergence theorem, we deduce that $J_{1}\left(n,h,R\right)\rightarrow 0$ as $n\rightarrow +\infty$ uniformly for $\left\|h\right\|=1$. We conclude that $\phi'\left(u_{n}\right)\rightarrow \phi'\left(u\right)$ strongly in $X'$. The proof is complete.
\end{proof}

\begin{lemma}\label{bounded}
Assume that $\left(V_{2}\right)$, $\left(A_{2}\right)$ and $\left(G_{1}\right)$ are satisfied. Then any $\left(PS\right)$-sequence of $I$ is bounded in $X$.
\end{lemma}
\begin{proof}
Let $\left(u_{n}\right)\subset X$ be a $\left(PS\right)$-sequence. Then there exists $\alpha>0$ such that $I\left(u_{n}\right)\leq \alpha$. By H\"older inequality and conditions $\left(A_{2}\right), \left(V_{2}\right), \left(G_{1}\right)$, we have
\begin{align*}
\alpha\geq I\left(u_{n}\right)&\geq \frac{1}{2} \int_{\mathbb{R}^{N}}\left(\left|\nabla u_{n}\right|^{2}(x)-V^{-}(x)u_{n}(x)^{2}\right) dx-\int_{\mathbb{R}^{N}}a(x)G(u_{n}(x))dx\\
&\geq \left(\frac{1}{2}-\frac{1}{2s}\left\|V^{-}\right\|_{\frac{N}{2}}\right) \left\|u_{n}\right\|_{X}^{2}- s^{\frac{-q-1}{2}}\left\|a\right\|_{\frac{2^{\ast}}{2^{\ast}-(q+1)}}\left\|u_{n}\right\|_{X}^{q+1}.
\end{align*}
 Since $0<q<1$, the last inequality shows that $\left(u_{n}\right)$ is bounded in $X$. The proof is complete.
\end{proof}

As a consequence, we obtain the following result.

\begin{lemma}\label{Palais-Smale}
Assume that $\left(V_{2}\right)$, $\left(A_{2}\right)$  and $ \left(G_{1}\right)$ are satisfied. Then $I$ satisfies the $\left(PS\right)$ condition in $X$.
\end{lemma}
\begin{proof}
Set
\[ \begin{array}{lll}
F:&D^{1,2}\left(\mathbb{R}^{N}\right)& \rightarrow \left(D^{1,2}\left(\mathbb{R}^{N}\right)\right)'\\
  & u     &\longmapsto  F(u), \ \ \left\langle F(u), v\right\rangle= \displaystyle \int_{\mathbb{R}^{N}} \nabla u \nabla v \ \ dx, \ \ \forall v \in D^{1,2}\left(\mathbb{R}^{N}\right).
\end{array}
  \]
Then  $F$ is an isomorphism.
  Let $\left(u_{n}\right)$ be a (PS) sequence of $I$, hence
  \begin{equation}\label{n}
  u_{n}=F^{-1}( \phi'(u_{n}))+o(1).
 \end{equation}
 By Lemma \ref{bounded}, $\left(u_{n}\right)$ is bounded in $X$. Sine $\phi'$ is a compact operator and using \eqref{n}, we deduce that $\left(u_{n}\right)$ is strongly convergent in $X$ (up a subsequence).
\end{proof}

\begin{lemma}\label{bounded operator}
Assume $\left(G_{1}\right), \left(V_{2}\right)$ and $\left(A_{2}\right)$ are satisfied. Then $I$ is bounded from below.
\end{lemma}
\begin{proof}
By $\left(G_{1}\right), \left(V_{2}\right)$ and $\left(A_{2}\right)$, we have
\begin{align*}
I(u)&=\frac{1}{2}\int_{\mathbb{R}^{N}}\left(\left|\nabla u(x)\right|^{2}+V(x)u^{2}(x)\right)dx-\int_{\mathbb{R}^{N}}a(x)G\left(u(x)\right) dx\\
&\geq \frac{1}{2}\int_{\mathbb{R}^{N}}\left|\nabla u(x)\right|^{2}dx- \frac{1}{2}\int_{\mathbb{R}^{N}}{V^{-}(x)u^{2}(x) dx}-\int_{\mathbb{R}^{N}}{a(x)G\left(u(x)\right) dx}\\
&\geq \frac{1}{2}\int_{\mathbb{R}^{N}}{\left|\nabla u(x)\right|^{2} dx}-\frac{1}{2s} \left\|V^{-}\right\|_{\frac{N}{2}} \left\|u\right\|_{X}^{2} -s^{\frac{-q-1}{2}}\left\|a\right\|_{\frac{2^{\ast}}{2^{\ast}-(q+1})} \left\|u\right\|_{X}^{q+1}\\
&\geq \left(\frac{1}{2}-\frac{1}{2s}\left\|V^{-}\right\|_{\frac{N}{2}}\right) \left\|u\right\|_{X}^{2}- s^{\frac{-q-1}{2}} \left\|a\right\|_{\frac{2^{\ast}}{2^{\ast}-(q+1)}}\left\|u\right\|_{X}^{q+1}.
\end{align*}
Since $1<q+1<2$, we deduce that $I$ is bounded from below. The proof is complete.
\end{proof}

Next, we prove the geometric condition required by Theorem \ref{zer thm}.

\begin{lemma}\label{cube}
Assume $\left(A_{2}\right)$, $\left(V_{2}\right)$, $\left(G_{1}\right)$, $\left(G_{2}\right), \  and \ \left(G_{3}\right)$ are satisfied. Then for each $k\in\mathbb{N}$, there exists an $A_{k}\in \Gamma_{k}$ such that
$ \sup_{u\in{A_{k}}} I\left(u\right)<0$.
\end{lemma}

\begin{proof}
By using conditions $\left(G_{2}\right)$ and $\left(G_{3}\right)$, the proof is similar to that Lemma \ref{cubee}.
\end{proof}

{\bf Proof of theorem \ref{prem thm} concluded.}
The energy functional
 $I$ is even and $I\left(0\right)=0$. By Lemmas \ref{Palais-Smale} and \ref{bounded operator}, condition $(1)$ of Theorem \ref{zer thm} is satisfied.
In view of Lemma \ref{cube}, condition $(2)$ of theorem \ref{zer thm} is also satisfied.
Thus, there exists a sequence $\left(u_{k}\right)$  such that  $c_{k}=I(u_{k})$ is a critical value of $I$, $c_{k}<0$, $c_{k}\rightarrow 0$, for all $k\geq 0$. This means that $ \left(u_{k}\right)$ are weak solutions of problem \eqref{main} and $\left(u_{k}\right)$ is a $(PS)$ sequence of $I$. Then, by Lemma \ref{bounded}, $\left(u_{k}\right)$ is bounded.\qed

\begin{remark}
If $g(x)=\left|x\right|^{q-1}x, \ \ 0<q<1$, then $u_{n}\rightarrow 0$ in $X$. In fact, by $\left( V_{2}\right)$, we have
\begin{align*}
0=\frac{1}{q+1}\left\langle I'\left(u_{n}\right), u_{n}\right\rangle-I\left(u_{n}\right)&=\left(\frac{1}{q+1}-\frac{1}{2}\right) \displaystyle \int_{\mathbb{R}^{N}}\left(\left|\nabla u_{n}\right|^{2}+ V(x) \left|u_{n}\right|^{2}\right) dx\\
& \geq \left(\frac{1}{q+1}-\frac{1}{2}\right) \left(1-\frac{\left\|V^{-}\right\|_{\frac{N}{2}}}{s}\right) \displaystyle \int_{\mathbb{R}^{N}} \left|\nabla u_{n}\right|^{2}(x) dx.
\end{align*}
Since $I'\left(u_{n}\right)=0$ and $\displaystyle \lim_{n\rightarrow +\infty} I\left(u_{n}\right)=0$, we deduce that
$u_{n}\rightarrow 0 $ in $ X.$
\end{remark}

\section{Proof of Theorem \ref{deux thm}}
In this section we change the condition $\left(V_{2}\right)$ by $\left(V_{1}\right)$ and we suppose that $a$ satisfies $(A_{3})$. Under the last conditions, the functional $I$ is not well defined both on $X$ and on $E$, then we define it on the space $Y$. We first establish that $\left(Y, \left\langle \ \ \right\rangle\right)$ is a Hilbert space and it is embedded into $L^{p}\left(\mathbb{R}^{N}\right)$ for $2\leq p\leq 2^{\ast}$. The proof of the following result relies on standard arguments and we will omit it.

\begin{lemma}\label{Y space}
Assume that $\left(V_{1}\right)$ holds. Then
$$u\rightarrow \left(\int_{\mathbb{R}^{N}}\left(\left|\nabla u(x)\right|^{2}+V^{+}(x) u^{2}(x)\right) dx\right)^{\frac{1}{2}},$$
defines a norm on $Y$, which is equivalent to the usual norm in $H^{1}\left(\mathbb{R}^{N}\right)$
$$\left\|u\right\|_{H^{1}}=\left\|\nabla u\right\|_{2}+\left\|u\right\|_{2}.$$
\end{lemma}



  By using Lemma \ref{Y space},  the proof of Theorem \ref{deux thm}, with slight modification,  is similar to that of Theorem \ref{prem thm}.

\begin{remark}
If $g(x)=\left|x \right|^{q-1}x, \ \ 0<q<1$, then $u_{n}\rightarrow 0$ in $Y$.
\end{remark}
\begin{remark}
In Theorems \ref{troi thm}, \ref{prem thm}, and \ref{deux thm},  we can suppose that $u_{0}$ is a nonnegative solution of \eqref{main}, since
$$I(u_{0})=I\left(\left|u_{0}\right|\right)=c_{0}.$$
In such a case, $u_{0}$ is called a ground state for $I$.
\end{remark}

\medskip
{\bf Acknowledgements.} The authors are grateful to the anonymous referee and to the handling Editor for the careful reading of the paper and for their constructive remarks.

\begin{thebibliography}{99}
\bibitem{abl} Ablowitz, M.J.,  Prinari, B., Trubatch, A.D.,
{\it Discrete and Continuous Nonlinear Schr\"odinger Systems},
Cambridge Univ. Press, Cambridge, 2004.

\bibitem{A1}  Adachi, S.,  Tanaka, K., Four positive solutions for the equation
$-\Delta u+u=a(x)u^{p}+f(x)$  in $\mathbb{R}^{N}$, {\it Calc. Var. Partial Differential Equations} {\bf 11} (2000), 63-95.

\bibitem{A0} Adams, R.A., {\it Sobolev Spaces}, Pure and Applied Mathematics, Vol. 65, Academic Press, New York--London, 1975.

\bibitem{A2} Ambrosetti, A., Rabinowitz, P.H., Dual variational methods in critical point theory and applications, {\it J. Funct. Anal.} {\bf 14} (1973),  349-381.

\bibitem{A3}  Ambrosetti, A., Badiale, M., The dual variational principle and elliptic problems with discontinuous nonlinearities, {\it J. Math. Anal. Appl.} {\bf 140} (1989),  363-373.

\bibitem{avron1}  Avron, J.,  Herbst, I.,  Simon, B.,  Schr\"odinger operators with electromagnetic fields. III.
Atoms in homogeneous magnetic field, {\it Commun. Math. Phys.} {\bf 79} (1981), 529-572.

\bibitem{B1} Bahri, A., Lions, P.-L., On the existence of a positive solution of semilinear elliptic equations in unbounded domains, {\it Ann. Inst. H. Poincar\'e, Anal. Non Lin\'eaire} {\bf 14} (1997), 365-413.

\bibitem{B2} Balabane, M., Dolbeault, J., Ounaies, H., Nodal solutions for a sublinear elliptic equation, {\it  Nonlinear Anal.} {\bf 52} (2003),  219-237.

\bibitem{B4} Bartsch, T., Liu, Z, Weth, T., Sign changing solutions for superlinear Schr\"odinger equations, {\it Comm. Partial Differential Equations} {\bf 29} (2004), 25-42.

\bibitem{MB} Benrhouma, M., Study of multiplicity and uniqueness of solutions for a class of nonhomogeneous sublinear elliptic equations, {\it Nonlinear Anal.} {\bf 74} (2011),  2682-2694.

\bibitem{MBH} Benrhouma, M., Ounaies, H., Existence and uniqueness of positive solution for nonhomogeneous sublinear elliptic equation, {\it J. Math. Anal. Appl.} {\bf 358} (2009), 307-319.

\bibitem{MBN}  Benrhouma, M., Ounaies, H., Existence of solutions for a perturbation sublinear elliptic equation in $\mathbb{R}^{N}$, {\it Nonlinear Differential Equations and Applications (NoDEA)} {\bf 5} (2010), 647-662.

\bibitem{B6} Brezis, H., Kamin, S., Sublinear elliptic equations in $\mathbb{R}^{n}$, {\it Manuscr. Math.} {\bf 74} (1992),  87-106.

\bibitem{B7} Brezis, H., Lieb, E.H., A relation between pointwise convergence of functions and convergence of functionals, {\it Proc. Amer. Math. Soc.} {\bf 88} (1983), 486-490.

\bibitem{B5} Brezis, H., Oswald, L., Remarks on sublinear elliptic equations, {\it Nonlinear Anal.} {\bf 10} (1986),  55-64.

\bibitem{byeon} Byeon, J., Wang, Z.Q., Standing waves with a critical frequency for nonlinear Schr\"odinger equations, {\it Arch. Ration. Mech. Analysis} {\bf 165} (2002), 295-316.

\bibitem{C1} Chabrowski, J., Costa, D.G.,  On a class of Schr\"odinger-type equations with indefinite weight functions, {\it Comm. Partial Differential Equations} {\bf 33} (2008),  1368-1394.

\bibitem{C3}  Costa, D.G., Tehrani, H., Existence of positive solutions for a class of indefinite elliptic problems in $\mathbb{R}^{N}$, {\it Calc. Var. Partial Differential Equations} {\bf 13} (2001), 159-189.

\bibitem{C2} Costa, D.G., Tehrani, H., Unbounded perturbations of resonant Schr\"odinger equations, in {\it Contemp. Math. Amer. Math. Soc.}, Providence, RI, {\bf 357} (2004),  101-110.

\bibitem{costa} Costa, D.G., Tehrani, H., Ramos, M., Non-zero solutions for a Schr\"odinger
equation with indefinite linear and nonlinear terms, {\it Proc. Roy. Soc. Edinburgh Sect. A} {\bf 134} (2004), 249-258.

\bibitem{del} Dellacherie, C., Meyer, P.A., {\it Probabilit\'es et Potentiel}, Hermann, Paris, 1983.

\bibitem{D2} Dong, W., Mei, L., Multiple solutions for an indefinite superlinear elliptic problem on $\mathbb{R}^{N}$, {\it Nonlinear Anal.} {\bf 73} (2010),  2056-2070.

\bibitem{D3} Du, Y., Multiplicity of positive solutions for an indefinite superlinear elliptic problem on $\mathbb{R}^{N}$, {\it Ann. Inst. H. Poincar\'e, Anal. Non Lin\'eaire} {\bf 21} (2004), 657-672.

\bibitem{floer} Floer, A., Weinstein, A., Nonspreading wave packets for the cubic Schr\"odinger equation with a bounded potential, {\it J.~Funct. Anal.} {\bf 69} (1986), 397-408.

\bibitem{F} Furtado, M.F., Maia, L.A.,  Medeiros, E.S., Positive and nodal solutions for a nonlinear Schr\"odinger equation with indefinite potential, {\it Advanced Nonlinear Studies} {\bf 8} (2008), 353-373.

\bibitem{hase}
Hasegawa, A., Kodama, Y., {\it Solitons in Optical Communications}, Academic Press, San Diego,
1995.

%\bibitem{H2}  Hirano, N., Existence of entire positive solutions for nonhomogenous elliptic equations, %{\it Nonlinear Anal.} {\bf 29} (1997),  889-901.

%\bibitem{H1} Hirano, N., Existence of infinitely many solutions for sublinear elliptic problems, {\it J. %Math. Anal. Appl.} {\bf 278} (2003),  83-92.

\bibitem{K1} Kajikiya, R., A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations, {\it J. Funct. Anal.} {\bf 225} (2005),  352-370.

\bibitem{K2} Kajikiya, R., Multiple solutions of sublinear Lane-Emden elliptic equations, {\it Calc. Var. Partial Differential Equations} {\bf 26} (2006), 29-48.

    \bibitem{kato} Kato, T., Remarks on holomorphic families of Schr\"odinger and Dirac operators,
in {\it Differential equations}, (Knowles I., Lewis R., Eds.), North-Holland Math. Stud., Vol. 92, North-Holland,
Amsterdam, 1984, pp. 341-352.

%\bibitem{kristaly} Kristaly, A., A double eigenvalue problem for Schr\"odinger equations involving
%sublinear nonlinearities at infinity, {\it Electronic Journal of Differential Equations}, {\bf 2007} %(2007), 1-11.

\bibitem{K3}  Kristaly, A., Multiple solutions of a sublinear Schr\"odinger equation, {\it Nonlinear Differential Equations and Applications (NoDEA)} {\bf 14} (2007), 291-302.

\bibitem{L2} Lions, P.-L., The concentration-compactness principle in the calculus of variations. The locally compact case, part 2, {\it Ann. Inst. H. Poincar\'e, Anal. Non Lin\'eaire} {\bf 2} (1984),  223-283.

\bibitem{malo}
  Malomed, B.A., Variational methods in nonlinear fiber optics and related fields, {\it Progress in Optics} {\bf 43}
(2002), 69-191.

\bibitem{oh} Oh, Y.G., Existence of semi-classical bound states of nonlinear Schr\"odinger equations with potentials
of the class ($V_a$), {\it Communications in Partial Differential Equations} {\bf 13} (1988), 1499-1519.

\bibitem{onor}  Onorato, M., Osborne, A.R., Serio, M., Bertone, S., Freak waves in random oceanic sea states, {\it
Phys. Rev. Lett.} {\bf 86} (2001), 5831-5834.

\bibitem{R1} Rabinowitz, P.H., On a class of nonlinear Schr\"odinger equations, {\it Z. Angew. Math. Phys.} {\bf 43} (1992), 270-291.

    \bibitem{sulem} Sulem, C., Sulem, P.-L., {\it The Nonlinear Schr\"odinger Equation. Self-focusing and Wave Collapse},
Applied Mathematical Sciences, Vol. 139, Springer-Verlag, New York, 1999.

\bibitem{T1}   Taira, K., Umezu, K., Positive solutions of sublinear elliptic boundary value problems, {\it Nonlinear Anal.} {\bf 29} (1997),  711-761.

\bibitem{T2} Tehrani, H., Existence results for an indefinite unbounded perturbation of a resonant Schr\"odinger equation, {\it J. Differential Equations} {\bf 236} (2007), 1-28.

\bibitem{tehrani} Tehrani, H., Infinitely many solutions for indefinite semilinear elliptic equations
without symmetry, {\it Communications in Partial Differential Equations} {\bf 21} (1996), 541-557.

\bibitem{willembook} Willem, M., {\it Minimax Theorems}, Progress in Nonlinear Differential Equations and their Applications, vol. 24, Birkh\"auser, Boston,  1996.

\bibitem{W2} Wu,  T.F., On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function, {\it J. Math. Anal. Appl.} {\bf 318} (2006),  253-276.

\bibitem{W1} Wu, T.F., Multiple positive solutions for a class of concave-convex elliptic problems in $\mathbb{R}^{N}$ involving sign-changing weight, {\it J. Funct. Anal.} {\bf 258} (2010),  99-131.

\bibitem{zakh}  Zakharov, V.E., {\it Collapse and Self-focusing of Langmuir Waves},
Handbook of Plasma Physics, (M. N. Rosenbluth and R. Z. Sagdeev, eds.), vol. 2 (A. A. Galeev and R. N. Sudan, eds.)
81-121, Elsevier (1984).

\bibitem{Z1} Zhang, Q., Wang, Q., Multiple solutions for a class of sublinear Schr\"odinger equations, {\it J. Math. Anal. Appl.} {\bf 389} (2012), 511-518.

\end{thebibliography}
\end{document}

\begin{proof}
By $\left(V_{1}\right)$, there exists $R>0$ such that
$$\frac{v_{\infty}}{2}\leq V^{+}(x)\leq \left\|V\right\|_{\infty} \ \ \mbox{for \  all} \ \ x  \in  B_{R}^{c}.$$
Then
\begin{align*}
&\displaystyle \int_{\mathbb{R}^{N}}\left(\left|\nabla u(x)\right|^{2}+V^{+}(x)u^{2}(x)\right) dx =\\ &\displaystyle \int_{\mathbb{R}^{N}}\left|\nabla u(x)\right|^{2} dx+ \displaystyle \int_{B_{R}} V^{+}(x)u^{2}(x) dx +\displaystyle \int_{B_{R}^{c}}V^{+}(x)u^{2}(x)\leq\\
& \displaystyle \int_{\mathbb{R}^{N}}\left|\nabla u(x)\right|^{2} dx+\left\|V^{+}\right\|_{L^{\frac{N}{2}}\left(B_{R}\right)} \left\|u\right\|_{2^{\ast}}^{2}+\left\|V\right\|_{\infty} \int_{B_{R}^{c}}u^{2}(x) dx\leq\\
& \int_{\mathbb{R}^{N}}\left|\nabla u(x)\right|^{2} dx+s^{-1}\left\|V^{+}\right\|_{L^{\frac{N}{2}}\left(B_{R}\right)} \left\|\nabla u\right\|_{2}^{2}+\left\|V\right\|_{\infty} \displaystyle \int_{B_{R}^{c}}u^{2}(x) dx \leq\\
& \left(1+c_{1}\right) \int_{\mathbb{R^{N}}}\left|\nabla u(x)\right|^{2} dx+\left\|V\right\|_{\infty} \displaystyle \int_{B_{R}^{c}} u^{2}(x) dx\leq \\
& \di\max\left((1+c_{1}), \left\|V\right\|_{\infty}\right) \displaystyle \int_{\mathbb{R}^{N}}\left(\left|\nabla u(x)\right|^{2}+u^{2}(x)\right) dx,
\end{align*}
with $r_{1}=s^{-1}\left\|V^{+}\right\|_{L^{\frac{N}{2}}\left(B_{R}\right)}$.

On the other hand, we have
$$\displaystyle \int_{B_{R}}\left|u(x)\right|^{2} dx\leq \left(m\left(B_{R}\right)\right)^{\frac{2}{N}} \left(\int_{B_{R}}\left|u(x)\right|^{2^{\ast}}dx \right)^{\frac{2}{2^{\ast}}}\leq \frac{\left(m\left(B_{R}\right)\right)^{\frac{2}{N}}}{s} \displaystyle \int_{\mathbb{R}^{N}}\left|\nabla u(x)\right|^{2} dx.$$
Then
\begin{align*}
&\displaystyle \int_{\mathbb{R}^{N}}\left(\left|\nabla u(x)\right|^{2}+\left|u(x)\right|^{2} \right) dx =\\ &\displaystyle \int_{\mathbb{R}^{N}}\left|\nabla u(x)\right|^{2} dx+\displaystyle \int_{B_{R}}\left|u(x)\right|^{2} dx+\int_{B_{R}^{c}} \left|u(x)\right|^{2} dx\leq\\
& \left(1+ \frac{\left(m\left(B_{R}\right)\right)^{\frac{2}{N}}}{s}\right) \displaystyle \int_{\mathbb{R}^{N}}\left|\nabla u(x)\right|^{2} dx+\frac{2}{v_{\infty}}\int_{B_{R}^{c}} V^{+}(x)\left|u(x)\right|^{2} dx\leq\\
& \max\left(\frac{2}{v_{\infty}}, \left(1+ \frac{\left(m\left(B_{R}\right)\right)^{\frac{2}{N}}}{s}\right)\right) \displaystyle \int_{\mathbb{R}^{N}}\left(\left|\nabla u(x)\right|^{2}+V^{+}(x) \left|u(x)\right|^{2} \right) dx .
\end{align*}
 It follows that $\left\| \ \  \right\|_{Y}$ is equivalent to $\left\| \ \ \right\|_{H^{1}}$. We conclude that $\left(Y, \ \ \left\| \ \ \right\|_{Y} \right)$ is a Hilbert space. By the Sobolev embedding theorem, $Y\hookrightarrow L^{p}$, $ 2 \leq p \leq2^{\ast}$, see \cite{A0}.
\end{proof} 