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%French
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\begin{document}

\begin{frontmatter}


\selectlanguage{english}
\title{Standing waves for the fractional Schr\"odinger equation}

\vspace{-2.6cm}

\selectlanguage{french}
\title{Ondes stationnaires de l'\'equation de Schr\"odinger fractionnaire }

\selectlanguage{english}
\author[GM]{Giovanni Molica Bisci}
\ead{gmolica@unirc.it}
\author[VR1,VR2]{Vicen\c tiu D. R\u{a}dulescu}
%\thanks[label3]{Correspondence address: Vicen\c tiu R\u{a}dulescu, University of Craiova,
%Department of Mathematics,
%200585 Craiova, Romania. E-mail: {\tt vicentiu.radulescu@imar.ro}}
\ead{vicentiu.radulescu@math.cnrs.fr}
%\ead[url]{http://www.math.ucv.ro/\~{}radulescu}
\address[GM]{Dipartimento PAU,
          Universit\`a `Mediterranea' di Reggio Calabria,
          Via Graziella, Feo di Vito, 89124 Reggio Calabria, Italy}
\address[VR1]{Department of Mathematics, Faculty of Sciences, King Abdulaziz University,
P.O. Box 80203, Jeddah 21589, Saudi Arabia}  
\address[VR2]{Institute of Mathematics ``Simion Stoilow" of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania}


\begin{abstract}
% Text of abstract in English
We investigate the existence of multiple standing waves for a class of fractional Schr\"odinger equations. The nonlinear term is assumed to have a superlinear behaviour at the origin and a sublinear decay at infinity. The proof combines variational methods and energy estimates in a suitable fractional Sobolev space.
\vskip 0.5\baselineskip

\selectlanguage{francais}
% Text of abstract in French
\noindent{\bf R\'esum\'e} \vskip 0.5\baselineskip \noindent
Nous \'etudions l'existence de plusieurs ondes stationnaires pour une classe
d'\'equations de Schr\"odinger fractionnaires. Le terme non lin\'eaire a un comportement sur-lin\'eaire autour de l'origine et une croissance sous-lin\'eaire \`a l'infini. La preuve combine des m\'ethodes variationnelles avec des estimations de l'\'energie dans un espace de Sobolev fractionnaire.
\end{abstract}
\end{frontmatter}

% now the Version fran\c{c}aise abr\'eg\'ee, if it exists
\selectlanguage{francais}
\section*{Version fran\c{c}aise abr\'eg\'ee}
Soit $(-\Delta)^s$ l'op\'erateur de Laplace fractionnaire d'ordre $s\in (0,1)$. Nous rappelons que $(-\Delta)^su=\mathfrak{F}^{-1}(|\xi|^{2s}(\mathfrak{F}u)(\xi)),
$ o\`u $\mathfrak{F}u:\RR^n\to\RR$ est la transform\'ee de Fourier de $u$.

Dans cette Note nous \'etudions
le probl\`eme non lin\'eaire
$$
(-\Delta)^s u +V(x)u= \lambda(f(x,u)+\mu g(x,u)),\quad x\in
\mathbb{R}^n\ (n\geq 3). \eqno(P)
$$

On suppose que le potential $V:\RR^n\rightarrow \RR$ satisfait les hypoth\`eses suivantes:
\begin{itemize}
\item[$(\rm p_1)$] $V\in C(\mathbb{R}^n)$ \textit{et} $\displaystyle\inf_{x\in\mathbb{R}^n}V(x)>0$;
\item[$(\rm p_2)$] \textit{pour tout } $M>0$ {\it il existe}\ $r_0>0$\ {\it tel que }\
$\lim_{|y|\to+\infty}|\{x\in B(y,r_0):V(x)\leq M\}|=0.$
\end{itemize}

Les fonctions continues $f,g:\mathbb{R}^n\times \mathbb{R}\to\mathbb{R}$ ont un comportement sur-lin\'eaire autour de l'origine et une croissance sous-lin\'eaire \`a l'infini. Plus pr\'ecis\'ement, on suppose que $f$ et $g$ v\'erifient les conditions suivantes:
\begin{itemize}
 \item[$(\rm h_1)$] \textit{il existe $W\in L^1(\mathbb{R}^n)\cap L^\infty(\mathbb{R}^n)\setminus\{0\}$
et $q\in (0,1)$  tels que}
 $$
\max\{|f(x,t)|,|g(x,t)|\}\leq W(x)|t|^{q} \quad\mbox{\it pour tout }
(x,t)\in \mathbb{R}^n\times \mathbb{R}\,;
$$

\item[$(\rm h_2)$] $\displaystyle\lim_{t\to 0}\frac{f(x,t)}{t}=
\lim_{t\to 0}\frac{g(x,t)}{t}=0$ \textit{uniform\'ement pour} $x\in
\mathbb{R}^n$\,;

\item[$(\rm h_3)$] \textit{il existe $s_0\in \mathbb{R}$ tel que}
$\displaystyle\sup_{\sigma>0}\left(\min_{|x|\leq \sigma}\int_0^{s_0}f(x,\tau)\,d\tau\right)>0\,.$
\end{itemize}

\smallskip

\noindent Consid\'erons l'espace de Sobolev fractionnaire avec poids $E_s^n(V)$ d\'efinit par
$$
E_s^n(V):=\left\{u\in H^{s}(\mathbb{R}^{n});\  \int_{\mathbb{R}^{n}}|\xi|^{2s}|\mathfrak{F}u(\xi)|^2\, d\xi+\int_{\mathbb{R}^{n}} V(x)|u(x)|^{2}dx<\infty\right\}.
$$

On dit que $u\in E_s^n(V)$ est une solution  du probl\`eme $(P)$ si pour tout $v\in E_s^n(V)$
$$
 \int_{\RR^n}
|\xi|^{2s}\mathfrak{F}u(\xi)\mathfrak{F}v(\xi)\, d\xi+\int_{\RR^n}V(x)u(x)v(x)\, dx=
 \lambda \int_{\RR^n} f(x,u(x))v(x)\, dx+\lambda\mu \int_{\RR^n} g(x,u(x))v(x)\, dx.
$$

Le r\'esultat principal de cette Note est le suivant.


\smallskip
\begin{theoreme} \label{th-1fr}
Supposons que les hypoth\`eses {$(\rm h_1)$--$(\rm h_3)$} et {$(\rm p_1)$--$(\rm p_2)$} sont satisfaites. Alors les propri\'et\'es suivantes sont vraies.

$(i)$ Il existe $\mu_0>0$ tel que pour chaque
 $\mu\in [-\mu_0,\mu_0]$ il y a un intervalle ouvert
 $\emptyset\not= \Sigma_\mu\subset(0,\infty)$ et $\kappa_\mu>0$ avec la propri\'et\'e que le probl\`eme
$(P)$ admet au moins deux solutions non triviales
  $v_{\lambda\mu}$ et $w_{\lambda\mu}$ v\'erifiant
$$
\max\{ \|v_{\lambda\mu}\|_{E_s^n(V)}, \|w_{\lambda\mu}\|_{E_s^n(V)}\}\leq
\kappa_\mu,\qquad \mbox{pour tout  $\lambda\in \Sigma_\mu$.}
$$


$(ii)$ Il existe
$\lambda_0>0$ tel que pour chaque
 $\lambda\in (\lambda_0,\infty)$ il y a un intervalle ouvert
 $\emptyset\not= \Lambda_\lambda\subset \mathbb{R}$  et $\kappa_\lambda>0$ avec la propri\'et\'e que le probl\`eme
$(P)$ admet au moins deux solutions non trivialles $v_{\lambda\mu}$ et $w_{\lambda\mu}$  v\'erifiant
$\mathcal J_{\lambda,\mu}(v_{\lambda\mu})<0
<\mathcal J_{\lambda,\mu}(w_{\lambda\mu})$ et
$\max\{\|v_{\lambda\mu}\|_{E_s^n(V)},\|w_{\lambda\mu}\|_{E_s^n(V)}\}\leq \kappa_\lambda$, pour tout
  $\mu\in \Lambda_\lambda$.\end{theoreme}

\smallskip
La preuve combine le principe variationnel de Ricceri \cite{R1,R2} avec le th\'eor\`eme du col de Ambrosetti et Rabinowitz \cite{ambrab}.


\vspace{0.3cm}
$$\line(1,0){300}$$
\selectlanguage{english}

\vspace{1cm}
Let $\mathcal{S}$ be the Schwartz space of rapidly decaying $C^{\infty}(\erre^n)$ functions.
For every $u\in \mathcal{S}$,  the fractional Laplace operator acting on $u$ is defined by
$(-\Delta)^su=\mathfrak{F}^{-1}(|\xi|^{2s}(\mathfrak{F}u)(\xi)),
$ where $s\in (0,1)$ and $\mathfrak{F}u:\RR^n\to\RR$ denotes the Fourier transform of $u$.

 In this Note, we study the combined effects of two nonlinear terms and two real parameters
 for the
 existence of multiple solutions to the following fractional parametric Schr\"odinger equation
\begin{equation} \label{Plm}
(-\Delta)^s u +V(x)u= \lambda(f(x,u)+\mu g(x,u)),\quad x\in
\mathbb{R}^n,\ n>2.
\end{equation}



We  assume that the potential $V$ satisfies the following hypotheses:
\begin{itemize}
\item[$(\rm p_1)$] $V\in C(\mathbb{R}^n)$ \textit{with} $\displaystyle\inf_{x\in\mathbb{R}^n}V(x)>0$\,;
\end{itemize}
\begin{itemize}
\item[$(\rm p_2)$] \textit{for any $M>0$ there exists $r_0>0$ such that }
$\lim_{|y|\to+\infty}|\{x\in B(y,r_0):V(x)\leq M\}|=0,
$
\textit{where $B(y,r)$ is the open ball in $\mathbb{R}^n$ with center $y$
and radius $r>0$ and $|\cdot|$ denotes the Lebesgue measure in $\mathbb{R}^n$.}
\end{itemize}

 We assume that the continuous functions $f,\,g:\mathbb{R}^n\times \mathbb{R}\to\mathbb{R}$
  have a superlinear behaviour at the origin and a sublinear decay at infinity.
 More precisely, we suppose that $f$ and $g$ satisfy the following conditions:

\begin{itemize}
 \item[$(\rm h_1)$] \textit{there exist $W\in L^1(\mathbb{R}^n)\cap L^\infty(\mathbb{R}^n)\setminus\{0\}$
 and $q\in (0,1)$  such that}
 $$
\max\{|f(x,t)|,|g(x,t)|\}\leq W(x)|t|^{q} \quad\mbox{for all }
(x,t)\in \mathbb{R}^n\times \mathbb{R}\,;
$$

\item[$(\rm h_2)$] $\displaystyle\lim_{s\to 0}\frac{f(x,t)}{t}=
\lim_{t\to 0}\frac{g(x,t)}{t}=0$ \textit{uniformly for each} $x\in
\mathbb{R}^n$\,;

\item[$(\rm h_3)$] \textit{there exist $s_0\in \mathbb{R}$ such that}
$$\displaystyle\sup_{\sigma>0}\left(\min_{|x|\leq \sigma}\int_0^{s_0}f(x,\tau)\,d\tau\right)>0\,.$$
\end{itemize}

Let $H^s(\RR^n)$ denote the fractional Sobolev space of all functions $u \in L^2(\RR^n)$ such that
$$\mbox{the map}\,\,\,
(x,y)\longmapsto \frac{u(x)-u(y)}{|x-y|^{(n+2s)/2}}\,\,\, \mbox{is in}\,\,\,
L^2(\RR^{2n}, dxdy\big).$$
On $H^s(\RR^n)$ we consider the norm
\begin{equation}\label{normoggi}
\|u\|_{H^{s}(\RR^n)}:=\left(\int_{\RR^n} |u(x)|^2\,dx+\int_{\RR^{2n}}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dxdy\right)^{1/2}.
\end{equation}

Consider the weighted fractional Sobolev space
\begin{equation}\label{X0}
E_s^n(V):=\left\{u\in H^{s}(\mathbb{R}^{n});\  \int_{\mathbb{R}^{n}}|\xi|^{2s}|\mathfrak{F}u(\xi)|^2\, d\xi+\int_{\mathbb{R}^{n}} V(x)|u(x)|^{2}dx<\infty\right\}
\end{equation}
endowed with the norm
$$
\|u\|_{E_s^n(V)}:=\left(\int_{\mathbb{R}^{n}}|\xi|^{2s}|\mathfrak{F}u(\xi)|^2\, d\xi+\int_{\mathbb{R}^{n}} V(x)|u(x)|^{2}dx\right)^{1/2}\,.
$$

We say that $u\in E_s^n(V)$ is a solution of problem (\ref{Plm}) if for all $v\in E_s^n(V)$
$$
 \int_{\RR^n}
|\xi|^{2s}\mathfrak{F}u(\xi)\mathfrak{F}v(\xi)\, d\xi+\int_{\RR^n}V(x)u(x)v(x)\, dx
  \displaystyle =\lambda \int_{\RR^n} f(x,u(x))v(x)\, dx+\lambda\mu \int_{\RR^n} g(x,u(x))v(x)\, dx.
$$

For all  $(x,t)\in \RR^n\times \RR$ we set $ F(x,t):=\int_0^tf(x,\tau)d\tau$ and $G(x,t):=\int_0^tg(x,\tau)\,d\tau.$

\noindent The energy functional $\mathcal J_{\lambda,\mu}:E_s^n(V)\to \RR$ associated to problem (\ref{Plm})  is defined by
$$
%\begin{equation}\label{JKlambda}
\mathcal J_{\lambda,\mu}(u) :=\frac 1 2 \left(\int_{\mathbb{R}^{n}}|\xi|^{2s}|\mathfrak{F}u(\xi)|^2\, d\xi+\int_{\mathbb{R}^{n}} V(x)|u(x)|^{2}dx\right)
-\lambda \int_{\mathbb{R}^{n}} F(x,u(x))\,dx-\lambda\mu \int_{\mathbb{R}^{n}} G(x,u(x))\,dx.
$$
%\end{equation}

A straightforward computation shows that $\mathcal J_{\lambda,\mu}$ is sequentially weakly lower semicontinuous, coercive, bounded from below, and satisfies the Palais-Smale condition for all $\lambda,\,\mu\in \mathbb{R}$, see \cite{molrad} for detailed proofs.

The main result of this Note is the following.

\begin{theorem}\label{th1}
Assume that  $f$, $g$ and $ V$  satisfy hypotheses {$(\rm h_1)$--$(\rm h_3)$} and {$(\rm p_1)$--$(\rm p_2)$}. Then the following properties hold.

$(i)$ There exists $\mu_0>0$ such that for all
 $\mu\in [-\mu_0,\mu_0]$ there are an open interval
 $\emptyset\not=\Sigma_\mu\subset(0,\infty)$ and  $\kappa_\mu>0$ with the property that problem
$($\ref{Plm}$)$ has at least two distinct nontrivial
solutions  $v_{\lambda\mu}$ and $w_{\lambda\mu}$ satisfying
$$
\max\{ \|v_{\lambda\mu}\|_{E_s^n(V)}, \|w_{\lambda\mu}\|_{E_s^n(V)}\}\leq
\kappa_\mu\,,
$$
provided that $\lambda\in \Sigma_\mu$.


$(ii)$ There exists
$\lambda_0>0$ such that for all
 $\lambda\in (\lambda_0,\infty)$ there are an open interval
 $\emptyset\not=\Lambda_\lambda\subset \mathbb{R}$  and  $\kappa_\lambda>0$ with the property that problem
$($\ref{Plm}$)$ has at least two distinct  nontrivial
solutions $v_{\lambda\mu}$ and $w_{\lambda\mu}$  satisfying
$\mathcal J_{\lambda,\mu}(v_{\lambda\mu})<0
<\mathcal J_{\lambda,\mu}(w_{\lambda\mu})$ and
$\max\{\|v_{\lambda\mu}\|_{E_s^n(V)},\|w_{\lambda\mu}\|_{E_s^n(V)}\}\leq \kappa_\lambda$, provided that
  $\mu\in \Lambda_\lambda$.
\end{theorem}
\medskip

\indent We point out that this result is no longer valid for all
$\lambda\in \mathbb{R}$. For
instance, consider the functions
$$
f(x,t):=\frac{\sin^2
t}{(1+|x|^\alpha)^{\beta}}\quad \mbox{and}\quad g(x,t):=\frac{\arctan^2
t}{(1+|x|^\alpha)^{\beta}}\qquad \forall\, (x,t)\in \RR^n\times \RR\,,
$$
with
$\alpha,\beta>0$ such that $\alpha\beta>n\geq 3$.
In such a case, the  nonlocal fractional equation
$$
(-\Delta)^s u +V(x)u=\frac{ \lambda}{(1+|x|^\alpha)^{\beta}}(\sin^2
u+\mu \arctan^2
u),\quad x\in
\mathbb{R}^n
$$
\noindent has only the trivial solution,
whenever $0<\lambda<(1+|\mu|\pi)^{-1}S_2^{-2}$ and $\mu$ is arbitrary. Here, $S_2$ denotes the best Sobolev embedding constant of the embedding $E^n_s(V)\hookrightarrow L^2(\RR^n)$.
More generally, the following non-existence result holds.

\begin{proposition}\label{fotetel-1}
Assume that  $f,g$ and $ V$  satisfy hypotheses {$(\rm h_1)$--$(\rm h_3)$} and {$(\rm p_1)$--$(\rm p_2)$}. Assume that $f(x,t):=W(x)h(t)$ and
$g(x,s):=W(x)k(t)$
for all $(x,t)\in \RR^n\times\RR$,
where $h$ and $k$ are Lipschitz continuous functions with
Lipschitz constants $L_h$ and $L_k$, and $W$ satisfying hypothesis $(\rm h_1)$.

Then the fractional Schr\"odinger equation
$$
(-\Delta)^s u +V(x)u=\lambda W(x)(h(u)+\mu k(u)),\quad x\in
\mathbb{R}^n
$$
 admits only the trivial solution, provided that
 $\mu\in \RR$ and
$$0<\lambda<\frac{1}{\|W\|_{L^\infty(\RR^n)}(L_h+|\mu|L_k)S_2^2}\,.$$
\end{proposition}

\section{Proof of Theorem \ref{th1}.}
Fix $\sigma>0$ and $\eta\in \erre$. For every $\varepsilon>0$, define $u_\varepsilon^{\eta}\in E_s^n(V)$ as follows
\begin{equation}\label{Test}
u_\varepsilon^{\eta}(x):= \left\{
\begin{array}{ll}
0 & \mbox{ if $x \in \RR^n \setminus B(0,\sigma)$} \\
\displaystyle\frac{\eta}{\varepsilon} \left(\sigma- |x| \right)
& \mbox{ if $x \in B(0,\sigma) \setminus B(0,\sigma-\varepsilon)$} \\
\eta & \mbox{ if $x \in B(0,\sigma-\varepsilon).$}
\end{array}
\right.
\end{equation}
Note that $u_\varepsilon^{\eta}\in E_s^n(V)$ and $\|u_{\varepsilon}^{\eta}\|_\infty:=\displaystyle\max_{x\in \RR^n}|u_{\varepsilon}^{\eta}(x)|\leq |\eta|.
$

\begin{lemma}\label{JDE-F(u-0)0} There exists
$u_0\in E_s^n(V)$ such that
$\displaystyle\int_{\mathbb{R}^n}
F(x,u_0(x))\,dx>0.
$
\end{lemma}

{\it Proof.} By hypothesis $(\rm h_3)$, there exist $\sigma_0>0$ and $s_0\in \mathbb{R}$ such that
$$\displaystyle\min_{|x|\leq \sigma_0}\int_0^{s_0}f(x,\tau)\,d\tau>0\,.$$
 Fix $\varepsilon\in (0,\sigma_0/2)$ and denote
$\omega_0:=\min_{|x|\leq \sigma_0}F(x,s_0)>0$. Let $u_\varepsilon^{s_0}\in E_s^n(V)$ be the function obtained in (\ref{Test}) by replacing $\sigma$ with $\sigma_0$, as well as $\eta$ with $s_0$. We have
$$\begin{array}{ll}
\displaystyle\int_{\mathbb{R}^n}
F(x,u_\varepsilon^{s_0}(x))\,dx
&=\displaystyle \int_{|x|\leq \sigma_0-\varepsilon}
F(x,u_\varepsilon^{s_0}(x))\,dx+\int_{|x|\geq \sigma_0}
F(x,u_\varepsilon^{s_0}(x))\,dx +\int_{\sigma_0-\varepsilon<|x|< \sigma_0}
F(x,u_\varepsilon^{s_0}(x))\,dx\\
&\geq\displaystyle \omega_0|B_{\sigma_0/2}|-\int_{\sigma_0-\varepsilon<|x|< \sigma_0}
|F(x,u_\varepsilon^{s_0}(x))|\, dx \\
&\geq\displaystyle \omega_0|B_{\sigma_0/2}|-
 \max_{ |x|\in[\sigma_0/2, \sigma_0]\,, |t|\leq |s_0|}
 |F(x,t)|(|B_{\sigma_0}|
-|B_{\sigma_0-\varepsilon}|).
\end{array}$$
Since
$$
\max_{ |x|\in[\sigma_0/2, \sigma_0]\,, |t|\leq |s_0|}
 |F(x,t)|(|B_{\sigma_0}|
-|B_{\sigma_0-\varepsilon} |)\rightarrow 0\quad\mbox{as $\varepsilon\to 0^+$},
$$
 there exists $\varepsilon_0>0$ such that
$$
\omega_0|B_{\sigma_0/2}|>
 \max_{ |x|\in[\sigma_0/2, \sigma_0]\,, |t|\leq |s_0|}
 |F(x,t)|(|B_{\sigma_0}|
-|B_{\sigma_0-\varepsilon_0}|).
$$
 Thus
$u_0:=u_{\varepsilon_0}^{s_0}\in E_s^n(V)$ is the required function.
\qed

\smallskip
For every $\mu\in \RR$, we define
$L_\mu:E_s^n(V)\rightarrow \RR$ by
 $$L_\mu(u):=\int_{\mathbb{R}^n} F(x,u(x))\,dx+\mu \int_{\mathbb{R}^n} G(x,u(x))\,dx.$$

 For all $\rho\in \mathbb{R}$, we set
$$
\chi(\varrho):=\frac{\sup\left\{L_\mu(u):
 \|u\|_{E^n_s(V)}< \sqrt{2\varrho}\right\}}{\varrho}\,.
$$

\begin{lemma}\label{F-hat-zero}
We have
$\displaystyle\lim_{\varrho\to 0^+}\chi(\varrho)=0.
$
\end{lemma}

{\it Proof.} Fix arbitrarily $\varepsilon>0$ and $p\in
(2,2^*_s)$, where $2^*_s:=2n/(n-2s)$ is the Sobolev fractional critical exponent. Taking into account the growth rates of $f$ and $g$, we obtain
for all $u\in E_s^n(V)$
$$
L_\mu(u)\leq(1+|\mu|)(\varepsilon
S_2^2\|u\|_{E^n_s(V)}^2+c(\varepsilon)S_p^p\|u\|_{E^n_s(V)}^p).
$$
Thus for every $\varrho>0$ we have
$$
0\leq \chi(\varrho)\leq (1+|\mu|)(2\varepsilon
S_2^2+c(\varepsilon)2^{\frac{p}{2}}S_p^p\varrho^{\frac{p}{2}-1}).
$$
Taking
$\varrho\to 0^+$, the right-hand side of the above inequality tends
to zero due to the arbitrariness of $\varepsilon>0$, which concludes the
proof.\qed

\smallskip
{\it Proof of Theorem \ref{th1}}.
$(i)$ Let $u_0\in E^n_s(V)$ be the element from Lemma \ref{JDE-F(u-0)0} and set
$$
\mu_0:=\frac{\int_{\mathbb{R}^n} F(x,u_0(x))\,dx}{\left|\int_{\mathbb{R}^n} G(x,u_0(x))\,dx\right|+1}\,.
$$
Let
$\Phi(u):=\displaystyle\frac{1}{2}\|u\|_{E^n_s(V)}^2$ and
$\Psi:=L_\mu$ for every fixed
$\mu\in [-\mu_0,\mu_0] $.
Then $\mathcal J_{\lambda,\mu}=\Phi-\lambda J_\mu.$

By straightforward computation we deduce that for every $\mu\in [-\mu_0,\mu_0]$
we have
\begin{equation}\label{nulla-u-0}
L_\mu(u_0)=\displaystyle\int_{\mathbb{R}^n} F(x,u_0(x))\,dx+\mu \int_{\mathbb{R}^n} G(x,u_0(x))\,dx\geq \mu_0>0.
\end{equation}
\indent Using relation (\ref{nulla-u-0}) and Lemma \ref{F-hat-zero}, there exists $\varrho_\mu>0$   such
that
$$
\varrho_\mu<\min\left\{1,\frac{\|u_0\|_{E^n_s(V)}^2}{2}\right\}$$
and
$$
\chi(\varrho_\mu):=\frac{\sup\{L_\mu(u):\ \|u\|_{E^n_s(V)}<
\sqrt{2\varrho_\mu}\}}{\varrho_\mu}<\frac{L_\mu (u_0)}{\|u_0\|_{E^n_s(V)}^2}.
$$
Now
all the assumptions of Ricceri's variational principle \cite{R1,R2} are fulfilled.
Thus there are an open interval $\emptyset\not=\Sigma_\mu\subset [0,\overline
a_\mu]$ and  $\kappa_\mu>0$ such that for all  $\lambda\in
\Sigma_\mu$, the functional $\mathcal J_{\lambda,\mu}=\Phi-\lambda L_\mu$
admits at least three distinct
critical points $u_{\lambda,\mu}^i\in E^n_s(V)$ ($i\in \{1,2,3\}$),
having norm less than $\kappa_\mu$.

$(ii)$ Define
$$
c_{0}:=\int_{\mathbb{R}^n}
|G(x,u_0(x))|dx \quad\mbox{and}\quad
\lambda_0:=\frac{\|u_0\|_{E_s^n(V)}^2}{2\int_{\mathbb{R}^{n}} F(x,u_0(x))\,dx}\,.
$$
For all
$\lambda>\lambda_0$ we set
$$
\mu_\lambda^*:=\frac{1}{1+c_0}
\left(1-\frac{\lambda_0}{\lambda}\right)\displaystyle\int_{\mathbb{R}^{n}} F(x,u_0(x))\,dx.
$$

Fix $\lambda>\lambda_0$ and
$\mu\in(-\mu_\lambda^*,\mu_\lambda^*)\equiv \Lambda_\lambda$. Since $J_{\lambda,\mu}$
is coercive and bounded from below,  there  exists  $v_{\lambda\mu}\in E^n_s(V)$ such that
$\mathcal J_{\lambda,\mu}(v_{\lambda\mu})
 =\inf_{u\in E^n_s(V)}\mathcal J_{\lambda,\mu}(u).$
 By straightforward computation we deduce that
$\inf_{u\in {E_s^n(V)}}\mathcal J_{\lambda,\mu}(u)<0$ for all $\lambda>\lambda_0$.
Therefore
$\mathcal J_{\lambda,\mu}(v_{\lambda\mu})<0$.

On the other hand, since $\mathcal J_{\lambda,\mu}$ satisfies the Palais-Smale condition and has the mountain pass geometry for all
$\lambda>\lambda_0$ and $\mu\in (-\mu_\lambda^*,\mu_\lambda^*)$,
we find $w_{\lambda\mu}\in E^n_s(V)$ such that
$\mathcal J_{\lambda,\mu}'(w_{\lambda\mu})=0$ and
$$
\mathcal J_{\lambda,\mu}(w_{\lambda\mu})\geq \Big(\frac{1}{4}-\lambda(1+|\mu|)
c(\lambda,\mu)S_p^p\varrho^{p-2}\Big)\varrho^2>0.
$$
 The mountain pass
level $\mathcal J_{\lambda,\mu}(w_{\lambda\mu})$ is characterized by
\begin{equation}\label{mountain-pass-char}
\mathcal J_{\lambda,\mu}(w_{\lambda\mu})=\inf_{g \in
\Gamma}\max_{t\in [0,1]}\mathcal J_{\lambda,\mu} (g(t)),
\end{equation}
 where $\Gamma:=\left\{g\in C([0,1];E^n_s(V)):g(0)=0,\ g(1)= u_0\right\}.$

 Define the function $g_0:[0,1]\to E^n_s(V)$  by
$g_0(t):=tu_0$. Since $g_0\in \Gamma$,  using relation
(\ref{mountain-pass-char}) we obtain for all $\mu\in \Lambda_\lambda$
$$
\mathcal J_{\lambda,\mu}(w_{\lambda\mu})
\leq \max_{t\in [0,1]}\mathcal J_{\lambda,\mu} (tu_0)
\leq \frac{1}{2}\|u_0\|^2_{E^n_s(V)}+\lambda \max_{t\in [0,1]}\left\{
\left|\displaystyle\int_{\mathbb{R}^{n}} F(x,tu_0(x))\,dx\right|+\mu_\lambda^*\left|\displaystyle\int_{\mathbb{R}^{n}} G(x,tu_0(x))\,dx\right|\right\}\,.$$
Thus for all $\mu\in \Lambda_\lambda$
  we have
$$\begin{array}{ll}
\displaystyle \|w_{\lambda\mu}\|_{E^n_s(V)}^2
&\displaystyle\leq 2\lambda(1+\mu_\lambda^*)
\|W\|_{L^{\frac{2}{1-q}}(\RR^n)}S_2^{q+1}\|w_{\lambda\mu}\|_{E^n_s(V)}^{q+1}\\
&\displaystyle + \|u_0\|^2_{E^n_s(V)}+2\lambda \max_{t\in [0,1]}\left\{
\left|\displaystyle\int_{\mathbb{R}^{n}} F(x,tu_0(x))\,dx\right|+\mu_\lambda^*\left|\displaystyle\int_{\mathbb{R}^{n}} G(x,tu_0(x))\,dx\right|\right\}.
\end{array}$$

 \indent Since $q+1<2$, there exists
 $\kappa_\lambda^1>0$ such that  $\|w_{\lambda\mu}\|_{E^n_s(V)}\leq
 \kappa_\lambda^1$ for every $\mu\in \Lambda_\lambda$.
 Moreover, since $\mathcal J_{\lambda,\mu}(v_{\lambda\mu})<0$
for every $\mu\in \Lambda_\lambda$,
 a similar argument as above shows
the existence of $\kappa_\lambda^2>0$ such that
$\|v_{\lambda\mu}\|_{E^n_s(V)}\leq  \kappa_\lambda^2,$ for every $\mu\in
\Lambda_\lambda$. Thus letting
$\kappa_\lambda:=\max\{\kappa_\lambda^1,\kappa_\lambda^2\}$, the
proof of Theorem \ref{th1} is completed.\qed

\smallskip
Using the definition of $\mu_\lambda^*$, we observe that
 for every $\lambda>\lambda_0$ we have
$$
\mu^*_\lambda<\frac{\int_{\mathbb{R}^{n}} F(x,u_0(x))\,dx}{1+c_0}\,.
$$
Since the right-hand side does not depend on $\lambda$,
 we have a uniform estimation of $\Lambda_\lambda$, namely
 $$
\Lambda_\lambda\subset \left[-\,{(1+c_0)^{-1}}{\int_{\mathbb{R}^{n}} F(x,u_0(x))\,dx},
\,{(1+c_0)^{-1}}{\int_{\mathbb{R}^{n}} F(x,u_0(x))\,dx}\right],
$$
for all $\lambda>\lambda_0$.\par

\smallskip
We refer to Brezis \cite{brezis} and Ciarlet \cite{ciarlet} as basic references
for the mathematical
methods employed in this paper.

\bigskip
\textbf{Acknowledgements.}
This work has been completed while V. R\u adulescu was visiting the  City University of Hong Kong in February 2015. He would like to express his gratitude to Professor Philippe G. Ciarlet for  this invitation and for enlightened discussions.


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\end{document}
