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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 151, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/151\hfil 
 Multiple solutions for a fractional $p$-Laplacian equation]
{Multiple solutions for a fractional $p$-Laplacian equation with sign-changing
potential}

\author[V. Ambrosio \hfil EJDE-2016/151\hfilneg]
{Vincenzo Ambrosio}

\address{Vincenzo Ambrosio \newline
Dipartimento di Matematica e Applicazioni ``R. Caccioppoli'',
Universit\`a degli Studi di Napoli Federico II,
via Cinthia, 80126 Napoli, Italy}
\email{vincenzo.ambrosio2@unina.it}

\thanks{Submitted May 10, 2016. Published June 20, 2016.}
\subjclass[2010]{35A15, 35R11, 35J92}
\keywords{Fractional $p$-Laplacian; sign-changing potential; fountain theorem}

\begin{abstract}
 We use a variant of the fountain Theorem to prove the existence of infinitely
 many weak solutions for the fractional $p$-Laplace equation
 \[
 (-\Delta)_{p}^s u + V(x) |u|^{p-2}u = f(x, u) \quad \text{in }  \mathbb{R}^N,
 \]
 where $s\in (0,1)$, $p\geq 2$, $N\geq 2$, $(- \Delta)_{p}^s$ is the
 fractional $p$-Laplace operator, the nonlinearity $f$ is $p$-superlinear
 at infinity and the potential $V(x)$ is allowed to be sign-changing.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

In this article we are interested in the study of the nonlinear fractional
$p$-Laplacian equation
\begin{equation}\label{P}
(- \Delta)_{p}^s u + V(x) |u|^{p-2}u = f(x, u) \quad \text{in }  \mathbb{R}^N,
\end{equation}
where $s\in (0,1)$, $p\geq 2$ and $N\geq 2$. 
Here $(- \Delta)_{p}^s$ is the fractional $p$-Laplace operator defined, for $u$ smooth enough, by setting
$$
(- \Delta)_{p}^su(x)=2\lim_{\epsilon \to 0}
\int_{\mathbb{R}^N\setminus B_{\epsilon}(x)}
\frac{|u(x)-u(y)|^{p-2} (u(x)-u(y))}{|x-y|^{N+sp}} dy, \quad x\in \mathbb{R}^N,
$$
up to some normalization constant depending upon $N$ and $s$.

When $p=2$,  equation \eqref{P} arises in the study of the nonlinear Fractional
Schr\"odinger equation
\[
\imath \frac{\partial \psi}{\partial t}+(- \Delta)^s \psi=H(x,\psi) \quad\text{in }
\mathbb{R}^N\times \mathbb{R}
\]
when looking for standing wave functions
${\psi(x,t)= u(x) e^{-\imath c t}}$, where $c$ is a constant.
This equation was introduced by Laskin \cite{Laskin1, Laskin2} and comes
from an extension of the Feynman path integral from the Brownian-like
to the Levy-like quantum mechanical paths.

Nowadays there are many articles related to the nonlinear fractional
Schr\"odinger equation: see for instance
\cite{A, Cheng, DDPW, FQT, FLS, MBR, Secchi1} and references therein.
More recently, a new nonlocal and nonlinear operator was considered, namely
the fractional $p$-Laplacian.
In the works of Franzina and Palatucci \cite{FP} and of Lindgren and
Linqvist \cite{LL}, the eigenvalue problem associated with $(-\Delta)^s_{p}$
is studied, in particular some properties of the first eigenvalue and of the
higher order eigenvalues are obtained.
Torres \cite{Torres} established an existence result for the problem \eqref{P}
when $f$ is $p$-superlinear and subcritical.
Iannizzotto et al. \cite{ILPS} investigated existence and multiplicity of
solutions for a class of quasi-linear nonlocal problems involving the
$p$-Laplacian operator.

Motivated by the above papers, we aim to study the multiplicity of nontrivial
 weak solutions to \eqref{P},
when $f$ is $p$-superlinear and $V(x)$ can change sign.
More precisely, we require that the potential $V(x)$ satisfies the following
assumptions:
\begin{itemize}
\item[(A1)] $V \in C(\mathbb{R}^N)$ is bounded from below;

\item[(A2)] There exists $r>0$ such that
$$
\lim_{|y|\to \infty} |\{x\in \mathbb{R}^N: |x-y|\leq r, V(x)\leq M\}|=0
$$
for any $M>0$,
\end{itemize}
while the nonlinearity $f:\mathbb{R}^N\times \mathbb{R}\to \mathbb{R}$ and its primitive
${F(x, t) = \int_{0}^{t} f(x, z) \, dz}$ are such that
\begin{itemize}
\item[(A3)] $f\in C(\mathbb{R}^N\times \mathbb{R})$,  and there exist
$c_1>0$ and $p<\nu<p_{s}^{*}$ such that
$$
|f(x,t)|\leq c_1(|t|^{p-1}+|t|^{\nu-1}) \quad \forall (x,t)\in \mathbb{R}^N\times \mathbb{R},
$$
where $p^{*}_{s}= \frac{Np}{N-sp}$ if $sp<N$ and
$p^{*}_{s}=\infty$ for $sp\geq N$.

\item[(A4)] $F(x,0)\equiv 0, F(x,t)\geq 0$ for all
$(x,t)\in \mathbb{R}^N\times \mathbb{R}$ and
$$
\lim_{|t| \to \infty} \frac{F(x,t)}{|t|^{p}}=+\infty
\text{ uniformly in } x\in \mathbb{R}^N.
$$

\item[(A5)] There exists $\theta \geq 1$ such that
$$
\theta \mathcal{F}(x,t)\geq \mathcal{F}(x,\tau t) \quad
\forall (x,t)\in \mathbb{R}^N\times \mathbb{R} \text{ and } \tau \in [0,1]
$$
where ${\mathcal{F}(x,t)=tf(x,t)-pF(x,t)}$.

\item[(A6)] $f(x,-t)=-f(x,t)$ for all $(x,t)\in \mathbb{R}^N\times \mathbb{R}$.
\end{itemize}
We recall that the conditions (A1) and (A2) on the potential $V$
and (A3)--(A6) with $p=2$ and $s=1$, have been used in \cite{ZX}
to extend the well-known multiplicity result due to Bartsch and Wang \cite{BW}.
Examples of $V$ and $f$ satisfying the above assumptions are
\[
V(x)=
\begin{cases}
2n|x|-2n(n-1)+c_{0} & \text{if } n-1\leq |x| \leq (2n-1)/2\\
-2n|x|+2n^{2}+c_{0} & \text{if } (2n-1)/2\leq |x|\leq n.
\end{cases},
\]
for $n\in \mathbb{N}$  and $c_{0}\in \mathbb{R}$;
and
$$
f(x,t)=a(x) |t|^{p-2}t \ln(1+|t|) \quad \forall (x,t)\in \mathbb{R}^N \times \mathbb{R},
$$
where $a(x)$ is a continuous bounded function with positive lower bound.
Our main result can be stated as follows.

\begin{theorem}\label{thm1}
Assume that {\rm (A1)--(A6)} are satisfied.
Then the problem \eqref{P} has  infinitely many nontrivial weak solutions.
\end{theorem}

To prove Theorem \ref{thm1}, we will consider the  family of functionals
\[
\mathcal{J}_{\lambda}(u)= \frac{1}{p} \Big[\iint_{\mathbb{R}^{2N}}
\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}} \,dx\,dy
+\int_{\mathbb{R}^N} V(x) |u(x)|^{p}\,dx \Big]
- \lambda \int_{\mathbb{R}^N} F(x,u) \,dx,
\]
with $\lambda \in [1,2]$ and $u\in E$,
where $E$ is the completion of $C^{\infty}_{0}(\mathbb{R}^N)$ with
respect to the norm
$$
\|u\|_{E}^{p}=\iint_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}} \,dx\,dy
+ \int_{\mathbb{R}^N} V(x)|u(x)|^{p}\,dx,
$$
and we will show that $\mathcal{J}_{\lambda}$ satisfies the assumptions of the
 following variant
of fountain Theorem due to Zou \cite{Zou}.

\begin{theorem}[\cite{Zou}]\label{FT}
Let $(E, \|\cdot\|)$ be a Banach space, $E=\overline{\oplus _{j\in \mathbb{N}} X_{j}}$,
with $\dim X_{j}<\infty$ for any $j\in \mathbb{N}$.
Set $Y_{k}=\oplus_{j=1}^{k} X_{j}$ and
$Z_{k}=\overline{\oplus_{j=k}^{\infty} X_{j}}$.
Let $\mathcal{J}_{\lambda}: E \to \mathbb{R}$ a family of $C^{1}(E, \mathbb{R})$ functionals
defined by
$$
\mathcal{J}_{\lambda}(u)=A(u)-\lambda B(u), \quad \lambda \in [1, 2].
$$
Assume that $\mathcal{J}_{\lambda}$ satisfies the following assumptions:
\begin{itemize}
\item[(i)] $\mathcal{J}_{\lambda}$ maps bounded sets to bounded sets uniformly for
$\lambda \in [1, 2]$, $\mathcal{J}_{\lambda}(-u)=\mathcal{J}_{\lambda}(u)$
for all $(\lambda, u)\in [1, 2] \times E$.

\item[(ii)] $B(u) \geq 0$ for all $u\in E$, and $A(u)\to \infty$ or
$B(u)\to \infty$ as $\|u\|\to \infty$.

\item[(iii)] There exists $r_{k}>\rho_{k}$ such that
$$
\beta_{k}(\lambda)=\max_{u\in Y_{k}, \|u\|=r_{k}} \mathcal{J}_{\lambda}(u)<
\alpha_{k}(\lambda)=\inf_{u\in Z_{k}, \|u\|=\rho_{k}} \mathcal{J}_{\lambda}(u),
\quad \forall  \lambda \in [1, 2].
$$
\end{itemize}
Then
$$
\alpha_{k}(\lambda)\leq \xi_{k}(\lambda)
=\inf_{\gamma \in \Gamma_{k}} \max_{u\in B_{k}} \mathcal{J}_{\lambda}(\gamma(u)) \quad
\forall \lambda \in [1, 2],
$$
where
$$
B_{k}=\{u\in Y_{k}: \|u\|\leq r_{k} \}
\text{ and }
\Gamma_{k}=\{\gamma \in C(B_{k}, X): \gamma \text{ is odd },
 \gamma=Id \text{ on } \partial B_{k} \}.
$$
Moreover, for a.e. $\lambda \in [1,2]$, there exists a sequence
$\{u^{k}_{m}(\lambda)\}_{m\in \mathbb{N}}\subset E$ such that
\begin{align*}
\sup_{m\in \mathbb{N}} \|u^{k}_{m}(\lambda)\|<\infty, \mathcal{J}'_{\lambda}(u^{k}_{m}(\lambda))\to 0,
\quad
 \mathcal{J}_{\lambda}(u^{k}_{m}(\lambda))\to \xi_{k}(\lambda) \quad\text{as } m \to \infty.
\end{align*}
\end{theorem}


\begin{remark} \label{rmk1} \rm
By using (A1) we know that there exists $V_{0}>0$ such that
$V_1(x)=V(x)+V_{0}\geq 1$ for any $x\in \mathbb{R}^N$.
Let $f_1(x,t)=f(x,t)+V_{0}|t|^{p-2}t$ for all $(x,t) \in \mathbb{R}^N\times \mathbb{R}$.
Then it is easy to verify that the study of \eqref{P} is equivalent to
investigate the  problem
\[
(- \Delta)_{p}^s u + V_1(x) |u|^{p-2}u = f_1(x, u) \quad \text{in }
 \mathbb{R}^N.
\]
Hence, from now on, we assume that $V(x)\geq 1$ for any
$x\in \mathbb{R}^N$ in (A1).
\end{remark}

\section{Preliminaries and functional setting}

In this preliminary Section, for the reader's convenience, we recall
some basic results related to the fractional Sobolev spaces.
For more details about this topic we refer to \cite{DPV}.

Let  $u:\mathbb{R}^N\to \mathbb{R}$ be a measurable function. We say that
$u$ belongs to the space $W^{s,p}(\mathbb{R}^N)$ if
$u\in L^{p}(\mathbb{R}^N)$ and
$$
[u]_{W^{s,p}(\mathbb{R}^N)}^{p}
:=\iint_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}} \,dx\,dy<\infty.
$$
Then $W^{s,p}(\mathbb{R}^N)$ is a Banach space with respect to the norm
$$
\|u\|_{W^{s,p}(\mathbb{R}^N)}
=\big[[u]^{p}_{W^{s,p}(\mathbb{R}^N)} + |u|^{p}_{L^{p}(\mathbb{R}^N)}\big]^{1/p}.
$$
We recall the main embeddings results for this class of fractional Sobolev spaces:

\begin{theorem}[\cite{DPV}] \label{ce}
Let $s\in (0,1)$ and $p\in [1,\infty)$ be such that $s p<N$.
Then there exists $C=C(N, p, s)>0$ such that
\[
|u|_{L^{p^{*}_{s}}(\mathbb{R}^N)} \leq C \|u\|_{W^{s,p}(\mathbb{R}^N)}.
\]
for any $u\in W^{s,p}(\mathbb{R}^N)$.
Moreover the embedding $W^{s,p}(\mathbb{R}^N)\subset L^{q}(\mathbb{R}^N)$
is locally compact whenever $q<p^{*}_{s}$.
\begin{itemize}
\item If $sp=N$ then $W^{s,p}(\mathbb{R}^N)\subset L^{q}(\mathbb{R}^N)$
 for any $q\in [p, \infty)$.
\item If $sp>N$ then $W^{s,p}(\mathbb{R}^N)\subset
 C^{0,\frac{sp-N}{p}}_{\rm loc}(\mathbb{R}^N)$.
\end{itemize}
\end{theorem}

Now we give the definition of weak solution for the problem \eqref{P}.
Taking into account the presence of the potential $V(x)$, we denote
by $E$ the closure of $C^{\infty}_{0}(\mathbb{R}^N)$  with respect to the norm
$$
\|u\|:=\Big([u]^{p}_{W^{s,p}(\mathbb{R}^N)}+|u|_{V}^{p} \Big)^{1/p}, \quad
|u|^{p}_{V}=\int_{\mathbb{R}^N} V(x) |u(x)|^{p}\,dx.
$$
Equivalently
$$
E=\big\{u\in L^{p^{*}_{s}}(\mathbb{R}^N):
[u]_{W^{s,p}(\mathbb{R}^N)},|u|_{V}<\infty \big\}.
$$
Let us denote by $(E^{*}, \|\cdot\|_{*})$ the dual space of $(E, \|\cdot\|)$.
We define the nonlinear operator $A: E \to E^{*}$ by setting
\begin{align*}
\langle A(u),v \rangle
&=\iint_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+sp}}(v(x)-v(y))
\,dx dy\\
&\quad  + \int_{\mathbb{R}^N} V(x)|u|^{p-2}uv \,dx,
\end{align*}
for $u,v\in E$.
Here $\langle \cdot, \cdot \rangle$ denotes the duality pairing between
$E$ and $E^{*}$.
Let
$$
B(u)=\int_{\mathbb{R}^N} F(x,u)\,dx
$$
for $u\in E$,
and we set $\mathcal{J}(u)=\frac{1}{p} \langle A(u), u \rangle -B(u)$ for $u\in E$.

\begin{definition} \label{def1} \rm
We say that $u\in E$ is a weak solution to \eqref{P} if $u$ satisfies
$$
\langle A(u),v \rangle=\langle B'(u),v \rangle
$$
for all $v\in E$.
\end{definition}

Now we show a compactness result.

\begin{lemma}\label{T1}
Under the assumption {\rm (A1)} and {\rm (A2)}, the embedding
$E\subset L^{q}(\mathbb{R}^N)$ is compact for any $q\in [p, p^{*}_{s})$.
\end{lemma}

\begin{proof}
Let $\{u_n\}\subset E$ such that $u_n\rightharpoonup 0$ in $E$. We have
to show that $u_n\to 0$ in $L^{q}(\mathbb{R}^N)$ for $q\in [p, p^{*}_{s})$.
 By the interpolation inequality we only need to consider $q=p$.
 By using the Theorem \ref{ce} we know that $u_n \to 0$ in
$L^{p}_{\rm loc}(\mathbb{R}^N)$. Thus it suffices to show that,
for any $\epsilon>0$, there exists $R>0$ such that
$$
\int_{B_{R}^{c}(0)} |u_n|^{p} \,dx < \epsilon;
$$
here $B_{R}^{c}(0) = \mathbb{R}^N \setminus B_{R}(0)$.
Let $\{y_{i}\}_{i\in \mathbb{N}}$ be a sequence of points in $\mathbb{R}^N$
satisfying $\mathbb{R}^N\subset \bigcup_{i\in \mathbb{N}} B_{r}(y_{i})$ and such
that each point $x$ is contained in at most $2^N$ such balls $B_{r}(y_{i})$.
We recall that we are assuming $V(x)\geq 1$ for any $x\in \mathbb{R}^N$.
Let
\begin{align*}
A_{R,M} = \{x\in B_{R}^{c} : V(x) \geq M\}, \quad
B_{R,M} = \{x\in B_{R}^{c} : V(x) < M\}.
\end{align*}
Then
$$
\int_{A_{R,M}} |u_n|^{p} \,dx
\leq \frac{1}{M} \int_{\mathbb{R}^N} V(x) |u_n|^{p} \,dx
$$
and this can be made arbitrarily small by choosing $M$ large.

Take $\gamma>1$ such that $p\gamma \leq p^{*}_{s}$ and let
$\gamma'= \frac{\gamma}{\gamma-1}$ be the dual exponent of $\gamma$.
Then for fixed $M>0$ we have
\begin{align*}
\int_{B_{R,M}} |u_n|^{p} \,dx
& \leq \sum_{i\in \mathbb{N}} \int_{B_{R,M}\cap B_{r}(y_{i})} |u_n|^{p} \,dx \\
&\leq \sum_{i\in \mathbb{N}} \Bigl(\int_{B_{R,M}\cap B_{r}(y_{i})} |u_n|^{p\gamma} \,dx
	\Bigr)^{1/\gamma} |B_{R,M}\cap B_{r}(y_{i})|^{1/\gamma'}\\
&\leq \epsilon_{R} \sum_{i\in \mathbb{N}} \Bigl(\int_{B_{R,M}\cap B_{r}(y_{i})}
|u_n|^{p\gamma} \,dx	\Bigr)^{1/\gamma}\\
&\leq C \epsilon_{R} \sum_{i\in \mathbb{N}} \|u_n\|^{p}_{W^{s,p}(B_{r}(y_{i}))} \\
&\leq C \epsilon_{R} 2^N \|u_n\|^{p}_{W^{s,p}(\mathbb{R}^N)}
\end{align*}
where $\epsilon_{R}=\sup_{y_{i}}  |B_{R,M}\cap B_{r}(y_{i})|^{1/\gamma'}$.
By assumption (A1) we can infer that $\epsilon_{R} \to 0$ as $R \to \infty$.
 Then we may make this term small by choosing $R$ large.
\end{proof}

Next  we prove the following result which will be fundamental later.

\begin{lemma}\label{Stype}
If $u_n \rightharpoonup u$ in $E$ and $\langle A(u_n),u_n-u \rangle\to 0$
then $u_n \to u$ in $E$.
\end{lemma}

\begin{proof}
Firstly, let us observe that for any $u, v\in E$
$$
|\langle A(u), v \rangle |\leq [u]_{W^{s,p}(\mathbb{R}^N)}^{p-1}[v]_{W^{s,p}(\mathbb{R}^N)}+|v|_{V}^{p-1}|v|_{V}\leq \|u\|^{p-1}\|v\|.
$$
Then, elementary calculations yield
\begin{equation}  \label{TV}
\begin{aligned}
0&\leq (\|u_n\|^{p-1}-\|u\|^{p-1})(\|u_n\|-\|u\|)  \\
&\leq \|u_n\|^{p}-\langle A(u_n), u \rangle-\langle A(u), u_n \rangle+ \|u\|^{p}  \\
&= \langle A(u_n), u_n \rangle-\langle A(u_n), u \rangle-\langle A(u), u_n \rangle
 +\langle A(u), u \rangle  \\
&=\langle A(u_n), u_n-u \rangle-\langle A(u), u_n-u \rangle=:I_n+II_n.
\end{aligned}
\end{equation}
By the hypotheses of the lemma, it follows that $I_n , II_n\to 0$ as $n \to \infty$
so, in view of \eqref{TV}, we have $\|u_n\|\to \|u\|$ as $n \to \infty$.
Since it is well known that the weak convergence and the norm convergence
in a uniformly convex space imply the strong convergence, to conclude the
proof it will be sufficient to prove that $E$ is uniformly convex.

Fix $\varepsilon\in (0, 2)$ and let $u, v\in E$ such that $\|u\|, \|v\|\leq 1$
and $\|u-v\|> \varepsilon$.
By using  that
$$
|\frac{a+b}{2}|^{p}+|\frac{a-b}{2}|^{p}
\leq \frac{1}{2}(|a|^{p}+|b|^{p}) \quad \text{ for any } a, b\in \mathbb{R},
$$
it follows that
\begin{align*}
&\|\frac{u+v}{2}\|^{p}+\|\frac{u-v}{2}\|^{p}\\
&=\Big\{\big[\frac{u+v}{2}\big]^{p}_{W^{s,p}(\mathbb{R}^N)}
 +\big[\frac{u-v}{2}\big]^{p}_{W^{s,p}(\mathbb{R}^N)}
 +|\frac{u+v}{2}|^{p}_{V}+|\frac{u-v}{2}|^{p}_{V}\Big\}\\
&\leq \frac{1}{2} \Big([u]^{p}_{W^{s,p}(\mathbb{R}^N)}+[v]^{p}_{W^{s,p}
 (\mathbb{R}^N)}+|u|^{p}_{V}+|v|^{p}_{V}\Big) \\
&=\frac{1}{2}\big(\|u\|^{p}+\|v\|^{p}\big)=1,
\end{align*}
which gives  ${\|\frac{u+v}{2}\|^{p}< 1-\frac{\varepsilon^{p}}{2^{p}}}$.
Choosing
$$
\delta=1-\bigl[1-\Bigl(\frac{\varepsilon}{2}\Bigr)^{p} \bigr]^{1/p}>0,
$$
we can infer that $\|\frac{u+v}{2}\|< 1-\delta$. Then $E$ is uniformly convex.
\end{proof}


Let us introduce a family of functionals $\mathcal{J}_{\lambda}: E \to \mathbb{R}$ defined by
$$
\mathcal{J}_{\lambda}(u)=\frac{1}{p} \langle A(u),u \rangle-\lambda B(u), \quad
\lambda \in [1, 2].
$$
After integrating, from (A3), we obtain  that for any
$(x,t)\in \mathbb{R}^N\times \mathbb{R}$,
\begin{equation}\label{F}
|F(x,t)|\leq \frac{c_1}{p}|t|^{p}+\frac{c_1}{\nu}|t|^{\nu}
\leq c_1(|t|^{p}+|t|^{\nu}).
\end{equation}
By using (A1), (A2), \eqref{F} and Lemma \ref{T1} follows that
$\mathcal{J}_{\lambda}$ is well defined on $E$.
Moreover $\mathcal{J}_{\lambda}\in C^{1}(E, \mathbb{R})$, and
\begin{equation}\label{derivative}
\mathcal{J}_{\lambda}'(u)v= \langle A(u), v \rangle-\lambda  \langle B'(u),v \rangle
\end{equation}
where
$$
\langle B'(u),v \rangle=\int_{\mathbb{R}^N} f(x,u) v\,dx.
$$
Then the critical points of $\mathcal{J}_1=\mathcal{J}$ are weak solutions to \eqref{P}.

To apply the Theorem \ref{FT}, we can observe that $E$ is a separable
($C_{0}^{\infty}(\mathbb{R}^N)$ is separable and dense in
$W^{s,p}(\mathbb{R}^N)$) and reflexive Banach space, so there exist
 $(\phi_n)\subset E$ and   $(\phi^{*}_n)\subset E^{*}$ such that
$E=\overline{\operatorname{span}\{\phi_n: n\in \mathbb{N}\}}$,
$E^{*}=\overline{\operatorname{span}\{\phi^{*}_n: n\in \mathbb{N}\}}$
and $\langle \phi^{*}_n, \phi_{m} \rangle=1$ if $n=m$ and zero otherwise.
Then, for any $n\in \mathbb{N}$, we set $X_n=\operatorname{span}\{\phi_n\}$,
 $Y_n=\oplus_{j=1}^{n} X_{j}$ and $Z_n=\overline{\oplus_{j=n}^{\infty} X_{j}}$.


\section{Proof of Theorem \ref{thm1}}

In this section we give the proof of the main result of this paper.
Firstly we prove the following Lemmas:

\begin{lemma}\label{lem1}
Assume that  {\rm (A1)--(A3)} are satisfied.
Then there exists $k_1\in \mathbb{N}$ and a sequence $\rho_{k}\to \infty$ as
$k \to \infty$ such that
$$
\alpha_{k}(\lambda)=\inf_{u\in Z_{k}, \|u\|=\rho_{k}} \mathcal{J}_{\lambda}(u), \quad
\forall k\geq k_1.
$$
\end{lemma}

\begin{proof}
Let us define
$$
b_{p}(k)=\sup_{u\in Z_{k}, \|u\|=1} |u|_{L^{p}(\mathbb{R}^N)},\quad
b_{\nu}(k)=\sup_{u\in Z_{k}, \|u\|=1} |u|_{L^{\nu}(\mathbb{R}^N)}.
$$
We aim to prove that
\begin{equation}\label{bp}
b_{p}(k)\to 0,   b_{\nu}(k)\to 0 \quad \text{as } k\to \infty.
\end{equation}
It is clear that $b_{p}(k)$ and $b_{\nu}(k)$ are decreasing with respect
to $k$ so there exist $b_{p}, b_{\nu} \geq 0$ such that
$b_{p}(k) \to b_{p}$ and  $b_{\nu}(k) \to b_{\nu}$ as $k\to \infty$.
For any $k\geq 0$, there exists $u_{k}\in Z_{k}$ such that $\|u_{k}\|=1$
and $|u_{k}|_{p}\geq \frac{b_{p}(k)}{2}$.

Taking into account that $E$ is reflexive, we can assume that
$u_{k} \rightharpoonup u$ in $E$.
Now, for any $\phi^{*}_n\in \{\phi^{*}_{j}\}_{j\in \mathbb{N}}$, we can see that
$\langle \phi^{*}_n, u_{k} \rangle=0$ for $k>n$, so
$\langle \phi^{*}_n, u\rangle=\lim_{k \to \infty}
\langle \phi^{*}_n, u_{k} \rangle= 0$.
Then $\langle \phi^{*}_n, u\rangle=0$ for any
$\phi^{*}_n\in \{\phi^{*}_{j}\}_{j\in \mathbb{N}}$, which gives $u=0$.
Since $E$ is compactly embedded in $L^{p}(\mathbb{R}^N)$
 by Lemma \ref{T1}, we have
$u_{k}\to 0$ in $L^{p}(\mathbb{R}^N)$, which implies that $b_{p}=0$.
Similarly we can prove  $b_{\nu}=0$.
Then, for any $u\in Z_{k}$ and $\lambda \in [1, 2]$, we can see that
\begin{align*}
\mathcal{J}_{\lambda}(u)
&=\frac{1}{p} \langle A(u), u \rangle -\lambda B(u)\\
&\geq \frac{\|u\|^{p}}{p}-2\int_{\mathbb{R}^N} F(x,u)\,dx \\
&\geq \frac{\|u\|^{p}}{p}-2c_1(|u|^{p}_{L^{p}(\mathbb{R}^N)}+|u|_{L^{\nu}(\mathbb{R}^N)}^{\nu}) \\
&\geq \frac{\|u\|^{p}}{p}-2c_1(b_{p}^{p}(k)\|u\|^{p}+b_{\nu}^{\nu}(k)\|u\|^{\nu}).
\end{align*}
By using \eqref{bp}, we can find $k_1\in \mathbb{N}$ such that
$$
2c_1b_{p}^{p}(k)\leq \frac{1}{2p} \quad \forall k\geq k_1.
$$
For each $k\geq k_1$, we choose
$$
\rho_{k}:=(8pc_1b_{\nu}^{\nu}(k))^{\frac{1}{p-\nu}}.
$$
Let us note that
\begin{equation}\label{defrho}
\rho_{k}\to \infty \quad \text{as } k\to \infty,
\end{equation}
since $\nu>p$.
Then we deduce that
$$
\alpha_{k}(\lambda):=\inf_{u\in Z_{k}, \|u\|
=\rho_{k}} \mathcal{J}_{\lambda}(u)\geq \frac{1}{4p}\rho_{k}^{p}>0
$$
for any $k\geq k_1$.
\end{proof}

\begin{lemma}\label{lem2}
Assume that {\rm (A1)--(A4)} hold. Then for the positive integer $k_1$
and the sequence $\rho_{k}$ obtained in Lemma \ref{lem1},
there exists $r_{k}>\rho_{k}$ for any $k\geq k_1$ such that
$$
\beta_{k}(\lambda)=\max_{u\in Y_{k}, \|u\|=r_{k}} \mathcal{J}_{\lambda}(u)<0.
$$
\end{lemma}

\begin{proof}
Firstly we prove that for any finite dimensional subspace
$F \subset E$ there exists a constant $\delta>0$ such that
\begin{equation}\label{finiteineq}
|\{x\in \mathbb{R}^N: |u(x)|\geq \delta \|u\|\}| \geq \delta, \quad
\forall u\in F\setminus\{0\}.
\end{equation}
We argue by contradiction and we suppose that for any $n\in \mathbb{N}$ there
exists $0\neq u_n\in F$ such that
$$
\bigl|\bigl\{x\in \mathbb{R}^N: |u_n(x)|\geq \frac{1}{n} \|u\|\bigr\}\bigr|
<\frac{1}{n}, \quad \forall n\in \mathbb{N}.
$$
Let $v_n:=\frac{u_n}{\|u_n\|}\in F$ for all $n\in \mathbb{N}$.
Then $\|v_n\|=1$ for all $n\in \mathbb{N}$ and
\begin{equation}\label{2.20}
\bigl|\bigl\{x\in \mathbb{R}^N: |v_n(x)|
\geq \frac{1}{n} \bigr\}\bigr|<\frac{1}{n}, \quad \forall n\in \mathbb{N}.
\end{equation}
Up to a subsequence, we may assume that $v_n \to v$ in $E$ for some
$v\in F$ since $F$ is a finite dimensional space.
Clearly $\|v\|=1$. By using Lemma \ref{T1} and the fact that all
norms are equivalent on $F$, we deduce that
\begin{equation}\label{2.21}
|v_n-v|_{L^{p}(\mathbb{R}^N)}\to 0 \quad \text{as } n \to \infty.
\end{equation}
Since $v\neq 0$, there exists $\delta_{0}>0$ such that
\begin{equation}\label{2.22}
|\{x\in \mathbb{R}^N: |v(x)|\geq \delta_{0} \}|\geq \delta_{0}.
\end{equation}
Set
$$
\Lambda_{0}:=\{x\in \mathbb{R}^N: |v(x)|\geq \delta_{0} \}
$$
and for all  $n\in \mathbb{N}$,
$$
\Lambda_n:=\bigl\{x\in \mathbb{R}^N: |v_n(x)|\geq \frac{1}{n} \bigr\},
\quad \Lambda^{c}_n:=\mathbb{R}^N\setminus \Lambda_n.
$$
Taking into account \eqref{2.20} and \eqref{2.22}, we obtain
$$
|\Lambda_n\cap \Lambda_{0}|\geq |\Lambda_{0}|-|\Lambda^{c}_n|
\geq \delta_{0}-\frac{1}{n}\geq \frac{\delta_{0}}{2}.
$$
for $n$ large enough.
Therefore,
\begin{align*}
\int_{\mathbb{R}^N} |v_n-v|^{p}\,dx
&\geq \int_{\Lambda_n\cap \Lambda_{0}} |v_n-v|^{p}\,dx\\
&\geq \int_{\Lambda_n\cap \Lambda_{0}} (|v|^{p}-|v_n|^{p})\,dx\\
&\geq \Bigl(\delta_{0}-\frac{1}{n}\Bigr)^{p}|\Lambda_n\cap \Lambda_{0}| \\
& \geq \Bigl(\frac{\delta_{0}}{2}\Bigr)^{p+1}>0
\end{align*}
which contradicts \eqref{2.21}.

Now, by using  that $Y_{k}$ is finite dimensional and \eqref{finiteineq},
we can find $\delta_{k}>0$ such that
\begin{equation}\label{ter}
|\{x\in \mathbb{R}^N: |u(x)|\geq \delta_{k} \|u\|\}|
\geq \delta_{k}, \quad \forall u\in Y_{k}\setminus\{0\}.
\end{equation}
By (A4), for any $k\in \mathbb{N}$ there exists a constant $R_{k}>0$ such that
$$
F(x,u)\geq \frac{|u|^{p}}{\delta^{p+1}_{k}} \quad \forall x\in \mathbb{R}^N
\text{ and } |u|\geq R_{k}.
$$
Set
$$
A_{u}^{k}=\{x\in \mathbb{R}^N: |u(x)|\geq \delta_{k}\|u\| \}
$$
and let us observe that, by \eqref{ter}, $|A_{u}^{k}|\geq \delta_{k}$
for any $u\in Y_{k}\setminus\{0\}$.
Then for any $u\in Y_{k}$ such that $\|u\|\geq \frac{R_{k}}{\delta_{k}}$, we have
\begin{align*}
\mathcal{J}_{\lambda}(u)&\leq \frac{1}{p}\|u\|^{p}-\int_{\mathbb{R}^N} F(x,u)\,dx \\
&\leq \frac{1}{p}\|u\|^{p}-\int_{A_{u}^{k}} \frac{|u|^{p}}{\delta^{p+1}_{k}}\,dx \\
&\leq  \frac{1}{p}\|u\|^{p}-\|u\|^{p}
=-\bigl(\frac{p-1}{p}\bigr)\|u\|^{p}.
\end{align*}
Choosing $r_{k}>\max\{\rho_{k}, \frac{R_{k}}{\delta_{k}}\}$ for all
$k\geq k_1$, follows that
$$
\beta_{k}(\lambda)=\max_{u\in Y_{k}, \|u\|=r_{k}} \mathcal{J}_{\lambda}(u)
\leq -\Bigl(\frac{p-1}{p}\Bigr)r_{k}^{p}<0, \quad \forall k\geq k_1.
$$
\end{proof}

By using \eqref{F} and Lemma \ref{T1} we can see that $\mathcal{J}_{\lambda}$
maps bounded sets to bounded sets uniformly for $\lambda \in [1,2]$.
Moreover, by (A6), $\mathcal{J}_{\lambda}$ is even. Then  condition (i)
in Theorem \ref{FT} is satisfied.
Condition (ii) is clearly true, while (iii) follows by Lemma \ref{lem1}
and Lemma \ref{lem2}.
Then, by Theorem \ref{FT},  for any $k\geq k_1$ and $\lambda \in [1,2]$
there exists a sequence $\{u^{k}_{m}(\lambda)\}\subset E$ such that
\[
\sup_{m\in \mathbb{N}} \|u^{k}_{m}(\lambda)\|<\infty, \quad
\mathcal{J}'_{\lambda}(u^{k}_{m}(\lambda))\to 0, \quad
\mathcal{J}_{\lambda}(u^{k}_{m}(\lambda))\to \xi_{k}(\lambda) \quad
\text{as } m \to \infty
\]
where
$$
\xi_{k}(\lambda)=\inf_{\gamma \in \Gamma_{k}}
\max_{u\in B_{k}} \mathcal{J}_{\lambda}(\gamma(u))
$$
with
$$
B_{k}=\{u\in Y_{k}: \|u\|\leq r_{k} \},\quad
\Gamma_{k}=\{\gamma \in C(B_{k}, X): \gamma \text{ is odd },
 \gamma=Id \text{ on } \partial B_{k} \}.
$$
In particular, from the proof of Lemma \ref{lem1}, we deduce that for
any $k \geq k_1$ and $\lambda \in [1, 2]$
\begin{equation}\label{2.27}
\frac{1}{4p}\rho_{k}^{p}=:c_{k}\leq\xi_{k}(\lambda)
\leq d_{k}:=\max_{u\in B_{k}} \mathcal{J}_1(u),
\end{equation}
and $c_{k}\to \infty$ as $k \to \infty$ by \eqref{defrho}.
As a consequence, for any $k \geq k_1$, we can choose $\lambda_n \to 1$
(depending on $k$) and get the corresponding sequences satisfying
\begin{equation}\label{2.28}
\sup_{m\in \mathbb{N}} \|u^{k}_{m}(\lambda_n)\| <\infty, \quad
\mathcal{J}'_{\lambda_n}(u^{k}_{m}(\lambda_n))\to 0 \quad \text{as } m \to \infty.
\end{equation}
Now, we prove that for any $k\geq k_1$, $\{u^{k}_{m}(\lambda_n)\}_{m\in \mathbb{N}}$
admits a strongly convergent subsequence $\{u^{k}_n\}_{n\in \mathbb{N}}$,
 and that such subsequence is bounded.

\begin{lemma}\label{claim1}
For each $\lambda_n$ given above, the sequence
$\{u^{k}_{m}(\lambda_n)\}_{m\in \mathbb{N}}$ has a strong convergent subsequence.
\end{lemma}

\begin{proof}
By \eqref{2.28} we may assume, without loss of generality, that as
$m \to \infty$,
$$
u^{k}_{m}(\lambda_n)\rightharpoonup u^{k}_n \text{ in } E
$$
for some $u^{k}_n \in E$.
By Lemma \ref{T1} we have
\begin{equation}\label{conv}
u^{k}_{m}(\lambda_n)\to u^{k}_n \quad \text{in }
L^{p}(\mathbb{R}^N) \cap L^{\nu}(\mathbb{R}^N).
\end{equation}
By (A3)  and H\"older inequality it follows that
\begin{align*}
&\Bigl|\int_{\mathbb{R}^N} f(x, u^{k}_{m}(\lambda_n))
 (u^{k}_{m}(\lambda_n)- u^{k}_n)\,dx \Bigr| \\
&\leq c_1|u^{k}_{m}(\lambda_n)|_{p}^{p-1} |u^{k}_{m}(\lambda_n)- u^{k}_n|_{p}+c_1
|u^{k}_{m}(\lambda_n)|_{\nu}^{\nu-1} |u^{k}_{m}(\lambda_n)- u^{k}_n|_{\nu}
\end{align*}
so, by using \eqref{conv}, we obtain
\[
\lim_{m \to \infty} \int_{\mathbb{R}^N} f(x, u^{k}_{m}(\lambda_n))
(u^{k}_{m}(\lambda_n)- u^{k}_n)\,dx=0.
\]
Since
$ \mathcal{J}'_{\lambda_n}(u^{k}_{m}(\lambda_n)) \to 0$  as
$m \to \infty$, and
\[
\langle \mathcal{J}_{\lambda}'(u),v \rangle=\langle A(u),v \rangle
-\lambda \langle B'(u),v \rangle,
\]
 we deduce that
$$
\langle A(u^{k}_{m}(\lambda_n)), u^{k}_{m}(\lambda_n)- u^{k}_n \rangle \to 0
\quad  \text{as } m\to \infty.
$$
Then, by using Lemma \ref{Stype}, we infer that
$$
u^{k}_{m}(\lambda_n)\to u^{k}_n \quad \text{in } E \text{ as } m\to \infty.
$$
\end{proof}

Therefore, without loss of generality, we may assume that
$$
\lim_{m \to \infty} u^{k}_{m}(\lambda_n)=u^{k}_n, \quad \forall n\in \mathbb{N}, k\geq k_1.
$$
As a consequence, we obtain
\begin{equation}\label{2.32}
\mathcal{J}'_{\lambda_n}(u^{k}_n)= 0,  \mathcal{J}_{\lambda_n}(u^{k}_n)\in [c_{k}, d_{k}],
\quad \forall n\in \mathbb{N}, k \geq k_1.
\end{equation}

\begin{lemma}\label{claim2}
For any $k\geq k_1$, the sequence $\{u^{k}_n\}_{n\in \mathbb{N}}$ is bounded.
\end{lemma}

\begin{proof}
For simplicity we set $u_n=u^{k}_n$.
We suppose by contradiction that, up to a subsequence,
\begin{equation}\label{2.33}
\|u_n\|\to \infty  \quad \text{as } n\to \infty.
\end{equation}
Let $w_n=u_n/\|u_n\|$ for any $n\in \mathbb{N}$.
Then, up to subsequence, we may assume that
\begin{equation} \label{2.34}
 \begin{gathered}
w_n\rightharpoonup w \quad\text{ in } E   \\
w_n\to w \quad \text{in } L^{p}(\mathbb{R}^N)\cap L^{\nu}(\mathbb{R}^N)\\
w_n\to w \quad \text{a.e. in } \mathbb{R}^N.
\end{gathered}
\end{equation}
Now we distinguish two cases.
\smallskip

\noindent\textbf{Case $w=0$.}
As in \cite{J}, we can say that for any $n\in \mathbb{N}$ there exists
$t_n \in [0,1]$ such that
\begin{equation}\label{2.36}
\mathcal{J}_{\lambda_n}(t_nu_n)=\max_{t\in [0,1]} \mathcal{J}_{\lambda_n}(t u_n).
\end{equation}
Since \eqref{2.33} holds, for any $j\in \mathbb{N}$, we can choose 
$r_{j}=(2jp)^{1/p} w_n$ such that
\begin{equation}\label{2.361}
r_{j}\|u_n\|^{-1}\in (0,1)
\end{equation}
provided $n$ is large enough.
By \eqref{2.34}, $F(\cdot,0)=0$ and the continuity of $F$, we can see that
\begin{equation}\label{2.362}
F(x,r_{j}w_n)\to F(x,r_{j}w)=0 \quad \text{a.e. } x\in \mathbb{R}^N
\end{equation}
as $n\to \infty$ for any $j\in \mathbb{N}$.
Then, taking into account \eqref{F}, \eqref{2.34}, \eqref{2.362}, $(A4)$ 
and by using the Dominated Convergence Theorem we deduce that
\begin{equation}\label{2.37}
F(x,r_{j}w_n)\to 0 \quad \text{in } L^{1}(\mathbb{R}^N)
\end{equation}
as $n\to \infty$ for any $j\in \mathbb{N}$.
Then \eqref{2.36}, \eqref{2.361} and \eqref{2.37} yield
$$
\mathcal{J}_{\lambda_n}(t_n u_n)\geq \mathcal{J}_{\lambda_n}(r_{j}w_n)
\geq 2j-\lambda_n \int_{\mathbb{R}^N} F(x,r_{j}w_n)\,dx\geq j
$$
provided $n$ is large enough and for any $j\in \mathbb{N}$.
As a consequence
\begin{equation}\label{3.31}
\mathcal{J}_{\lambda_n}(t_nu_n)\to \infty \quad \text{as } n\to \infty.
\end{equation}
Since $\mathcal{J}_{\lambda_n}(0)=0$ and 
$\mathcal{J}_{\lambda_n}(u_n)\in [c_{k}, d_{k}]$, we deduce that 
$t_n\in (0,1)$ for $n$ large enough. Thus, by \eqref{2.36} we have
\begin{equation}\label{2.38}
\langle{\mathcal{J}'_{\lambda_n}(t_nu_n),t_nu_n}\rangle
=t_n \frac{d}{dt}\Bigr|_{t=t_n} \mathcal{J}_{\lambda_n}(t u_n)=0.
\end{equation}
Taking into account (A5), \eqref{2.38} and \eqref{derivative} we obtain
\begin{align*}
\frac{1}{\theta}\mathcal{J}_{\lambda_n}(t_nu_n)
&=\frac{1}{\theta}
 \Bigl(\mathcal{J}_{\lambda_n}(t_nu_n)- \frac{1}{p}\langle{\mathcal{J}'_{\lambda_n}
 (t_nu_n),t_nu_n}\rangle \Bigr)\\
&= \frac{\lambda_n}{\theta p} \int_{\mathbb{R}^N} \mathcal{F}(x,t_nu_n)\,dx \\
&\leq \frac{\lambda_n}{p} \int_{\mathbb{R}^N} \mathcal{F}(x, u_n)\,dx \\
&= \mathcal{J}_{\lambda_n}(u_n)- \frac{1}{p}\langle{\mathcal{J}'_{\lambda_n}(u_n), 
u_n}\rangle=\mathcal{J}_{\lambda_n}(u_n)
\end{align*}
which contradicts \eqref{2.32} and \eqref{3.31}.
\smallskip

\noindent\textbf{Case $w\not\equiv 0$.} %\label{zdiv0}
Thus the set $\Omega:=\{x\in \mathbb{R}^N: w(x)\neq 0\}$ has 
positive Lebesgue measure. By using \eqref{2.33} and that $w\not\equiv 0$,
we have
\begin{equation}\label{3.34}
|u_n(x)|\to \infty \quad \text{a.e. } x\in \Omega \text{ as } n\to \infty.
\end{equation}
Putting together \eqref{2.34}, \eqref{3.34}, and $(A4)$, and by applying 
Fatou's Lemma, we can easily deduce that
\begin{align*}
\frac{1}{p}-\frac{\mathcal{J}_{\lambda_n}(u_n)}{\|u_n\|^{p}}
&=\lambda_n \int_{\mathbb{R}^N}\frac{F(x,u_n(x))}{\|u_n\|^{p}}\,dx \\
&\geq \lambda_n \int_{\Omega} |w_n|^{p} \frac{F(x,u_n(x))}{|u_n|^{p}}\,dx 
 \to \infty \text{ as }  n\to \infty
\end{align*}
which gives a contradiction because of \eqref{2.32}.

Then, we have proved that the sequence $\{u_n\}$ is bounded in $E$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1}]
Taking into account Lemma \ref{claim2} and \eqref{2.32}, 
for each $k\geq k_1$, we can use similar arguments to those in the proof 
of Lemma \ref{claim1}, to show that the sequence $\{u_n^{k} \}$ admits 
a strong convergent subsequence with the limit $u^{k}$ being just a critical 
point of $\mathcal{J}_1=\mathcal{J}$. Clearly, $\mathcal{J}(u^{k})\in [c_{k}, d_{k}]$ for all $k\geq k_1$. 
Since $c_{k}\to \infty$ as $k\to \infty$ in \eqref{2.27}, we deduce the 
existence of infinitely many nontrivial critical points of $\mathcal{J}$. 
As a consequence, we have that \eqref{P} possesses infinitely many nontrivial
 weak solutions.
\end{proof}

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\end{document}