\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 230, pp. 1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}
\begin{document}
\title[\hfilneg EJDE-2017/230\hfil Kirchhoff-type Schr\"odinger-Choquard equations]
{Existence of infinitely many solutions for degenerate
Kirchhoff-type Schr\"odinger-Choquard equations}
\author[S. Ling, V. R\u{a}dulescu \hfil EJDE-2017/230\hfilneg]
{Sihua Liang, Vicen\c{t}iu D. R\u{a}dulescu}
\address{Sihua Liang \newline
College of Mathematics,
Changchun Normal University,
Changchun 130032, Jilin, China}
\email{liangsihua@126.com}
\address{Vicen\c{t}iu R\u{a}dulescu \newline
Department of Mathematics, Faculty of Sciences,
King Abdulaziz University, P.O. Box 80203,
Jeddah 21589, Saudi Arabia. \newline
Department of Mathematics, University of Craiova,
200585 Craiova, Romania}
\email{vicentiu.radulescu@imar.ro}
\dedicatory{Communicated by Giovanni Molica Bisci}
\thanks{Submitted July 10, 2017. Published September 22, 2017.}
\subjclass[2010]{35R11, 35A15, 47G20}
\keywords{Kirchhoff-type problems; Schr\"odinger-Choquard equations;
\hfill\break\indent fractional $p$-Laplacian; critical exponent;
variational methods}
\begin{abstract}
In this article we study a class of Kirchhoff-type
Schr\"odinger-Choquard equations involving the
fractional $p$-Laplacian. By means of Kajikiya's new version of
the symmetric mountain pass lemma, we obtain the existence of
infinitely many solutions which tend to zero under a suitable value
of $\lambda$. The main feature and difficulty of our equations arise
in the fact that the Kirchhoff term $M$ could vanish at zero, that is,
the problem is {\em degenerate}. To our best knowledge, our result
is new even in the framework of Schr\"odinger-Choquard problems.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks
\section{Introduction and statement of main result}
In this article, we consider a class of Kirchhoff-type
Schr\"odinger-Choquard equations involving the
fractional $p$-Laplacian of the form
\begin{equation} \label{e1.1}
M(\|u\|_s^p)[(-\Delta)^s_pu + V(x)|u|^{p-2}u]
= \lambda f(x,u) + (\mathcal{K}_\mu*|u|^{p_{\mu,s}^*})|u|^{p_{\mu,s}^*-2}u,\\
\end{equation}
in $\mathbb{R}^N$,
where hereafter $\mathcal{K}_\mu(x)=|x|^{-\mu}$,
\begin{equation}\label{e1.2}
\begin{aligned}
\|u\|_{s}&=\Big([u]_{s}^p
+\int_{\mathbb{R}^N}V(x)|u|^pdx\Big)^{1/p}[u]_{s}\\
&=\Big(\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\,dx\,dy
\Big)^{1/p},
\end{aligned}
\end{equation}
$M:\mathbb{R}^+_0\to\mathbb{R}^+_0$ is a Kirchhoff
function, $V:\mathbb{R}^N\to\mathbb{R}^+$ is a scalar
potential, $p_{\mu,s}^*=(pN-p\mu/2)/(N-ps)$ is the critical exponent
in the sense of Hardy-Littlewood-Sobolev inequality,
$f:\mathbb{R}^N\times\mathbb{R}\to\mathbb{R}$ is a
Carath\'{e}odory function, and $(-\Delta )^s_p$ is the associated
fractional operator which, up to a normalization constant, is
defined as
\begin{equation*}
(-\Delta)_p^s\varphi(x)=2 \lim_{\varepsilon\to
0^+}\int_{\mathbb{R}^N\setminus
B_\varepsilon(x)}\frac{|\varphi(x)-\varphi(y)|^{p-2}
(\varphi(x)-\varphi(y))}{|x-y|^{N+ps}}\,dy,\quad
x\in\mathbb{R}^{N},
\end{equation*}
along functions $\varphi\in C_0^\infty(\mathbb{R}^N)$. Henceforward
$B_\varepsilon(x)$ denotes the ball of $\mathbb{R}^N$ centered at
$x\in\mathbb{R}^N$ and radius $\varepsilon>0$.
Nonlocal operators can be seen as the infinitesimal generators of
L\'{e}vy stable diffusion processes \cite{r8}. Moreover, they allow
us to develop a generalization of quantum mechanics and also to
describe the motion of a chain or an array of particles that are
connected by elastic springs as well as unusual diffusion processes
in turbulent fluid motions and material transports in fractured
media (for more details see for example \cite{r8,r3,r7} and the
references therein). Indeed, the literature on nonlocal fractional
operators and on their applications is quite large, see for example
the recent monograph \cite{MBRS}, the extensive paper \cite{DMV} and
the references cited there. The literature on non-local operators
and their applications is quite large, here we just quote a few, see
\cite{AFP,cencelj,dAS, moliradu, molrad1, molrad2, PP,PS,PXZ,PXZ2,xiang1, xiang2}.
This paper is motivated by some works appeared in recent years. On the
one hand, the following Choquard or nonlinear
Schr\"odinger-Newton equation
\begin{align}\label{eq1.01}
-\Delta u+V(x)u=(\mathcal{K}_\mu*u^2)u+\lambda f(x,u)\quad\text{in
}\mathbb{R}^N,
\end{align}
was elaborated by Pekar \cite{Pekar} in the framework of
quantum mechanics. Subsequently, it was adopted as an approximation
of the Hartree-Fock theorey, see \cite{Bongers}. Recently, {\em
Penrose} \cite{Penrose} settled it as a model of self-gravitational
collapse of a quantum mechanical wave function. The first
investigations for existence and symmetry of the solutions to
\eqref{eq1.01} go back to the works of Lieb \cite{Lieb} and
Lions \cite{Lions2}. Equations of type \eqref{eq1.01} have
been extensively studied, see e.g. \cite{AFY1,MS1,MS2,Ye}. For
the critical case in the sense of Hardy-Littlewood-Sobolev
inequality, we refer the interested reader to \cite{GY2} for
recent existence results in a smooth bounded domain of
$\mathbb{R}^{N}$. In the setting of the fractional Laplacian,
Wu \cite{DW} investigated existence and stability of solutions
for the equations
\begin{align}\label{eq1.02}
(-\Delta)^s u+\omega u=(\mathcal{K}_\mu*|u|^{q})|u|^{q-2}u+\lambda
f(x,u)\quad\text{in }\mathbb{R}^N,
\end{align}
where $q=2$, $\lambda=0$ and $\mu \in (N-2s, N)$. Subsequently,
D'Avenia and Squassina in \cite{PDSS} studied some properties
of solutions for \eqref{eq1.02} with $\lambda=0$, such as
regularity, existence, multiplicity, nonexistence, symmetry as well
as decays properties. In particular, they claimed the nonexistence
of solutions as $q\in (2-\mu/N, 2_{\mu,s}^*)$.
In the critical case that corresponds to
$q=2_{\mu,s}^*$, Mukherjee and Sreenadh~\cite{MK} obtained
existence and multiplicity results for
solutions of \eqref{eq1.02} as $\omega=0$ and $f(x,u)=u$ in a smooth
bounded domain of~$\mathbb{R}^{N} (N\geq3)$.
On the other hand, L\"{u} \cite{Lu} studied the following
Kirchhoff-type equation
\begin{equation} \label{eq11}
-\Big(a+b\int_{\mathbb{R}^3}|\nabla u|^2dx\Big)\Delta u
+V_{\lambda}(x)u=(\mathcal{K}_\mu*{u^{q}})|u|^{q-2}u \quad\text{in
}\mathbb{R}^3,
\end{equation}
where $a\in \mathbb{R}^+$, $b\in \mathbb{R}^+_0$ are given numbers,
$V_{\lambda}(x)=1+\lambda g(x)$, $\lambda\in \mathbb{R}^+$ is a
parameter and $g(x)$ is a continuous potential function on
$\mathbb{R}^3$, $q\in (2, 6-\mu)$. By using the Nehari manifold and
the concentration compactness principle, the author obtained the
existence of ground state solutions for~\eqref{eq11} if the
parameter $\lambda$ is large enough. Indeed, the study of
Kirchhoff-type problems, which arise in various models of physical
and biological systems, have received more and more attention in
recent years. More precisely, Kirchhoff established a model given
by the equation
\begin{equation} \label{kirchhoff}
\rho\frac{\partial ^2u}{\partial
t^2}-\Big(\frac{p_0}{h}+\frac{E}{2L} \int_0^L\left|\frac{\partial
u}{\partial x}\right|^2dx\Big) \frac{\partial ^2u}{\partial
x^2}=0,
\end{equation}
where $\rho$, $p_0$, $h$, $E$, $L$ are constants which represent
some physical meanings respectively. Equation \eqref{kirchhoff} extends
the classical D'Alembert wave equation by considering the effects of
the changes in the length of the strings during the vibrations.
Recently, Fiscella and Valdinoci \cite{FV} proposed a
steady-state Kirchhoff model involving the fractional Laplacian by
taking into account the nonlocal aspect of the tension arising from
nonlocal measurements of the fractional length of the string, see
\cite[Appendix A]{FV} for more details. Very recently, by using the mountain
pass theorem and the Ekeland variational principle,
Pucci {\em et al.} \cite{PXZ3} studied the existence of solutions
for problem \eqref{e1.1} under some appropriate assumptions.
Motivated by the above works, we are interested in the
existence of infinitely many solutions for problem \eqref{e1.1} in a
possibly degenerate Kirchhoff setting. It is worth mentioning that the method
exploited in this paper, is Kajikiya's new version of
the symmetric mountain pass lemma, which was first used to study Kirchhoff-type
fractional $p$-Laplacian problems in \cite{BFL, WZ}.
As far as we know, there is no result to investigate multiplicity of solutions for
\eqref{e1.1} in the literature.
Throughout the paper, without explicit mention, we assume
\begin{itemize}
\item[(A1)] $V:\mathbb{R}^N\to\mathbb{R}^+$
is a continuous function and there exists $V_0>0$ such that
$\inf_{\mathbb{R}^N} V \geq V_0$.
\end{itemize}
$M:\mathbb{R}^+_0\to
\mathbb{R}^+_0$ is assumed to be continuous and to satisfy
\begin{itemize}
\item[(A2)] For any $\tau>0$ there exists $m=m(\tau)>0$
such that $M(t)\geq m$ for all $t\geq\tau$.
\item[(A3)] There exists $\theta\in[1,p_s^*/p)$ if $p\geq 2$
and $\theta\in [1,\min\{p_s^*/p,p_{\mu,s}^*\})$ if $1
0$ such that
$M(t)\geq m_0 t^{\theta-1}$ for all $t\in [0,1]$.
\end{itemize}
A prototype for $M$, due to Kirchhoff, is given by
\begin{equation}\label{prot}
M(t)=a +b\theta t^{\theta-1}\quad \text{for }t\in \mathbb{R}^+_0,\;
a,\,b\ge0,\;a+b>0.
\end{equation}
When $M(t)\ge c>0$ for all $t\in\mathbb R^+_0$, Kirchhoff equations
like~\eqref{e1.1} are said to be \textit{non-degenerate} and this
happens for example if $a>0$ in the model case \eqref{prot}. While,
if $M(0)=0$ but $M(t)>0$ for all $t\in\mathbb R^+$, Kirchhoff
equations as~\eqref{e1.1} are called \textit{degenerate}. Of course,
for \eqref{prot} this occurs when $m_0=0$.
Concerning the function
$f:\mathbb{R}^N\times\mathbb{R}\to\mathbb{R}$, we suppose
that
\begin{itemize}
\item[(A5)] There exist $q \in (1, \theta p)$ and a
nonnegative function $w\in L^{\vartheta}(\mathbb{R}^N)$ such that
$|f(x,t)|\leq w(x)t^{q-1}$ for all
$(x,t)\in\mathbb{R}^N\times\mathbb{R}^+$, where $\vartheta =
p_s^*/(p_s^*-q)$.
\item[(A6)] There exist $\xi\in (1, p)$, $\delta>0$, $a_0>0$ and
a nonempty open subset $\Omega$ of $\mathbb{R}^N$ such that
\[
F(x,t)\geq a_0t^{\xi}\quad\text{for all } (x,t)\in\Omega\times(0,\delta).
\]
\end{itemize}
A simple example of $f$, verifying (A5)-(A6), is
$f(x,t)=(1+|x|^2)^{(l-2)/2}(t^+)^{l-1}$ with $10$ such that for any
$\lambda\in(0,\overline{\lambda})$ equation \eqref{e1.1} admits a
sequence of solutions $\{u_n\}_n$ in $E$ with $\mathcal
J_\lambda(u_n) < 0$, $\mathcal J_\lambda(u_n) \to 0$ and
$\{u_n\}_n$ converges to zero as $n \to\infty$.
\end{theorem}
\begin{remark}\rm
With the help of the Ekeland variational principle, Pucci {\em et al.}
\cite{PXZ3} just obtained the existence of a nontrivial nonnegative solution
for problem \eqref{e1.1} under the same hypotheses as Theorem \ref{thm1.1},
see \cite[Theorem 1.3]{PXZ3} for more details.
Therefore, our result and approach are completely different from those developed
by Pucci {\em et al.} in \cite{PXZ3}.
\end{remark}
The proof of Theorem \ref{thm1.1} is mainly based on the application
of the symmetric mountain pass lemma, introduced by Kajikiya in
\cite{k2}. For this, we need a truncation argument which allow us to
control from below functional $\mathcal J_\lambda$. Furthermore, as
usual in elliptic problems involving critical nonlinearities, we
must pay attention to the lack of compactness at critical level
corresponding to the space
$L^{p^*_s}(\mathbb R^N)$. To overcome this difficulty, we fix
parameter $\lambda$ under a suitable threshold strongly depending on
assumptions (A3) and (A4).
The paper is organized as follows. In Section~\ref{sec preliminaries}
we discuss the variational formulation of the
equation \eqref{e1.1} and introduce some topological notions. In
Section~\ref{sec ps} we prove the Palais-Smale condition for the
functional $\mathcal J_\lambda$. In Section~\ref{sec truncation} we
introduce a truncation argument for our functional. In
Section~\ref{sec T1.1} we prove Theorem~\ref{thm1.1}.
\section{Preliminaries}\label{sec preliminaries}
We first provide some basic functional setting that will be used in
the next sections. The critical exponent $p^*_s$ is defined as
$Np/(N-ps).$ Let $L^p(\mathbb{R}^N,V)$ denote the Lebesgue space of
real valued functions, with $V(x)|u|^p\in L^1({\mathbb{R}}^N),$
equipped with norm
$$
\|u\|_{p,V}=\Big(\int_{\mathbb{R}^N}V(x)|u|^p
\,dx\Big)^{1/p}\quad \text{for all }u\in L^p(\mathbb{R}^N,V).
$$
The embedding $W^{s,p}_V(\mathbb{R}^N)\hookrightarrow
L^{\nu}(\mathbb{R}^N)$ is continuous for any $\nu\in [p,p_s^*]$ by
\cite[Theorem 6.7]{r28}, namely there exists a positive constant
$C_\nu$ such that
\begin{align*}
\|u\|_{L^{\nu}(\mathbb{R}^N)}\leq C_\nu \|u\|_s\quad\text{for all }
u\in W^{s,p}_V(\mathbb{R}^N).
\end{align*}
Next, we recall the Hardy-Littlewood-Sobolev
inequality, see \cite[Theorem 4.3]{LL}. Hereafter we denote by
$\|\cdot\|_q$ the norm of Lebesgue space $L^q(\mathbb{R}^N)$.
\begin{theorem}
Assume that $10$ such that
\begin{align*}
\iint_{\mathbb R^{2N}}\frac{|u(x)|\cdot|v(y)|}{|x-y|^\mu}\,dx\,dy\leq
C(N,\mu,r,t)\|u\|_r\|v\|_t
\end{align*}
for all $u\in L^r(\mathbb{R}^N)$ and $v\in L^t(\mathbb{R}^N)$.
\end{theorem}
Note that, by the Hardy-Littlewood-Sobolev inequality, the
integral
\begin{align*}
\iint_{\mathbb{R}^{2N}}\frac{|u(x)|^q|u(y)|^q}{|x-y|^\mu}\,dx\,dy
\end{align*}
is finite, whenever $|u|^q\in L^t(\mathbb{R}^N)$ for some $t>1$
satisfying
\[
\frac{2}{t}+\frac{\mu}{N}=2, \quad\text{that is }t=\frac{2N}{2N-\mu}.
\]
Hence, by the fractional Sobolev embedding theorem, if
$u\in W_{V}^{s,p}(\mathbb{R}^N)$ this occurs provided that $tq \in[p,
p_s^*]$. Thus, $q$ has to satisfy
\begin{align*}
\tilde p_{\mu,s}=\frac{(N-\mu/2)p}{N}\leq q\leq
\frac{(N-\mu/2)p}{N-sp}=p_{\mu,s}^*.
\end{align*}
Hence, $\tilde p_{\mu,s}$ is called the lower critical exponent and
$p_{\mu,s}^*$ is said to be the upper critical exponent in the sense
of the Hardy-Littlewood-Sobolev inequality.
By the Hardy-Littlewood-Sobolev inequality, there exists
$\widetilde{C}(N,\mu)>0$ such that
\begin{align*}
\iint_{\mathbb{R}^{2N}}
\frac{|u(x)|^{p^*_{\mu,s}}|u(y)|^{p^*_{\mu,s}}}{|x-y|^{\mu}}\,dx\,dy
\leq \widetilde{C}(N,\mu)\|u\|_{p_s^*}^{2p_{\mu,s}^*},
\end{align*}
for all $u\in W_V^{s,p}(\mathbb{R}^N)$. In other words, there
exists $C(N,\mu)=\widetilde{C}(N,\mu)C_{p_s^*}^{2p_{\mu,s}^*}>0$
such that
\begin{equation}\label{em2}
\begin{aligned}
\int_{\mathbb{R}^N}(\mathcal{K}_\mu*|u|^{p_{\mu,s}^*})
|u|^{p_{\mu,s}^*}dx
&=\iint_{\mathbb{R}^{2N}}
\frac{|u(x)|^{p^*_{\mu,s}}|u(y)|^{p^*_{\mu,s}}}{|x-y|^{\mu}}\,dx\,dy\\
&\leq C(N,\mu)\|u\|_{s}^{2p_{\mu,s}^*},
\end{aligned}
\end{equation}
for all $u\in W_V^{s,p}(\mathbb{R}^N)$. Let us define
\begin{align}\label{em1}
S^*=\inf_{u\in W_V^{s,p}(\mathbb{R}^N)\setminus\{0\}}
\frac{\|u\|_s^p}{(\int_{\mathbb{R}^N}(\mathcal{K}_\mu*|u|^{p_{\mu,s}^*})
|u|^{p_{\mu,s}^*}dx)^{p/2p_{\mu,s}^*}}.
\end{align}
Clearly, $S^*>0$.
\begin{theorem}[{see \cite[Theorem 2.3]{PXZ3}}] \label{thm2.1} % lem2.1
Let $\{u_n\}_n$ be a bounded sequence in $L^{p_s^*}(\mathbb{R}^N)$
such that $u_n\to u$ a.e. in $\mathbb{R}^N$ as
$n\to\infty$. Then
\begin{align*}
&\iint_{\mathbb{R}^{2N}}
\frac{|u_n(x)|^{p_{\mu,s}^*}|u_n(y)|^{p_{\mu,s}^*}}{|x-y|^\mu}\,dx\,dy\\
&-\iint_{\mathbb{R}^{2N}}
\frac{|u_n(x)-u(x)|^{p_{\mu,s}^*}|u_n(y)-u(y)|^{p_{\mu,s}^*}}{|x-y|^\mu}\,dx\,dy\\
&\to\iint_{\mathbb{R}^{2N}}
\frac{|u(x)|^{p_{\mu,s}^*}|u(y)|^{p_{\mu,s}^*}}{|x-y|^\mu}\,dx\,dy
\quad \text{as }n\to\infty.
\end{align*}
\end{theorem}
To handle the degenerate Kirchhoff coefficient we need
appropriate lower and upper bounds for $M$, given by (A2) and
(A3). Indeed, condition (A3) implies that $M(t)>0$ for any
$t>0$ and consequently by (A2) for all $t\in(0,1]$ we have
$M(t)/\mathscr M(t)\le\theta/t$. Thus, integrating on
$[t,1]$, with $00$ there exists
$\delta_\varepsilon=\mathscr M(\varepsilon)/\varepsilon^\theta >0$
such that
\begin{equation}\label{e2.4}
\mathscr M(t)\leq \delta_\varepsilon t^\theta\quad\text{for any
}t\geq\varepsilon.
\end{equation}
To prove the multiplicity result stated in Theorem \ref{thm1.1}, we
will use some topological results introduced by Krasnoselskii in
\cite{K}. For the sake of completeness and for reader's convenience,
we recall here some basic notions on the Krasnoselskii's genus. Let
$X$ be a Banach space and let us denote by $\Sigma$ the class of all
closed subsets $A\subset X\setminus\left\{0\right\}$ that are
symmetric with respect to the origin, that is, $u\in A$ implies
$-u\in A$.
Let $A\in\Sigma$. The Krasnoselskii's genus $\gamma(A)$ of $A$ is
defined as being the least positive integer $n$ such that there is
an odd mapping $\phi\in C(A,\mathbb R^n)$ such that $\phi(x)\neq0$
for any $x\in A$. If $n$ does not exist, we set $\gamma(A)=\infty$.
Furthermore, we set $\gamma(\emptyset)=0$.
In the sequel we recall only the properties of the genus that
will be used throughout this work. More information on this subject
may be found in the references \cite{k2, K, r1}.
\begin{proposition}\label{prop3.1}
Let $A$ and $B$ be closed symmetric subsets of $X$ which do not
contain the origin. Then the following hold.
\begin{itemize}
\item[(1)] If there exists an odd continuous mapping from $A$ to $B$,
then $\gamma(A) \leq \gamma(B)$.
\item[(2)] If there is an odd homeomorphism from $A$ to $B$, then $\gamma(A)
= \gamma(B)$.
\item[(3)] If $\gamma(B) < \infty$, then $\gamma\overline{(A\setminus B)}
\geq \gamma(A) - \gamma(B)$.
\item[(4)] Then $n$-dimensional sphere $S^n$ has a genus of
$n+1$ by the Borsuk-Ulam theorem.
\item[(5)] If $A$ is compact, then $\gamma (A) < +\infty$ and there exists
$ \delta> 0$ such that
$N_{\delta} (A)\subset\Sigma $ and $\gamma (N_{\delta} (A)) =\gamma (A)$,
with $N_\delta(A)=\left\{x\in X:
\operatorname{dist}(x,A)\leq\delta\right\}$.
\end{itemize}
\end{proposition}
We conclude this section by recalling the symmetric mountain-pass lemma
introduced by Kajikiya in \cite{k2}. The proof of Theorem
\ref{thm1.1} is based on the application of the following result.
\begin{lemma}\label{lem3.1} Let $E$ be an infinite-dimensional
space and $J \in C^1(E, \mathbb{R})$ and suppose that the following
conditions hold.
\begin{itemize}
\item[(1)] $J(u)$ is even, bounded from below, $J(0) = 0$ and $J(u)$
satisfies the local Palais-Smale condition,
i.e. for some $\bar{c} > 0$, in the case when every sequence
$\{u_n\}_n$ in $E$ satisfying
$\lim_{n\to\infty}J(u_n) = c < \bar{c}$ and
$\lim_{n\to\infty}\|J'(u_n)\|_{E'} = 0$
has a convergent subsequence;
\item[(2)] For each $n \in \mathbb{N}$, there exists $A_n \in \Sigma_n$
such that $\sup_{u\in A_n}J(u) < 0$.
\end{itemize}
Then either (i) or (ii) below holds.
\begin{itemize}
\item[(i)] There exists a sequence $\{u_n\}_n$ such that $J'(u_n) = 0$, $J(u_n) < 0$
and $\{u_n\}_n$ converges to zero.
\item[(ii)] There exist two sequences $\{u_n\}_n$ and $\{v_n\}_n$ such
that $J'(u_n) = 0$, $J(u_n) =0$, $u_n \neq 0$,
$\lim_{n \to \infty}u_n = 0$; $J'(v_n) = 0$,
$J(v_n) < 0,$ $\lim_{n \to \infty}J(v_n) = 0$,
and $\{v_n\}_n$ converges to a non-zero limit.
\end{itemize}
\end{lemma}
\section{The Palais-Smale condition}\label{sec ps}
Throughout this paper, we consider $N>ps$ with $s\in(0,1)$
and $p\in(1,\infty)$, $M(0)=0$ and we assume $M$ and $V$ satisfy
(A1)--(A4), without further mentioning.
To apply Lemma \ref{lem3.1}, we discuss now the
compactness property for the functional $\mathcal J_\lambda$, given
by the Palais-Smale condition. We recall that $\{u_n\}_n\subset
W^{s,p}_V(\mathbb{R}^N)$ is a Palais-Smale sequence for $\mathcal
J_\lambda$ at level $c\in\mathbb R$ if
\begin{equation}\label{e3.1}
\mathcal J_\lambda(u_n)\to c\quad\text{and}\quad \mathcal
J'_\lambda(u_n)\to 0\quad\text{in $(W^{s,p}_V(\mathbb{R}^N))'$ as
}n\to\infty.
\end{equation}
We say that $\mathcal J_\lambda$ satisfies the Palais-Smale
condition at level $c$ if any Palais-Smale sequence $\{u_n\}_n$ at
level $c$ admits a convergent subsequence in
$W^{s,p}_V(\mathbb{R}^N)$.
Before going to prove Theorem~\ref{thm1.1}, we first give some
auxiliary lemmas.
\begin{lemma}[{see \cite[Lemma 4.1]{PXZ3}}] \label{lem4.1}
If {\rm (A5)} holds, then there exist $\rho \in (0,1]$ and
$\lambda_0 = \lambda_0(\rho)>0$,
$\ell= \ell(\rho)$, such that $\mathcal J_\lambda(u)\geq \ell >0$
for any $u\in W^{s,p}_V(\mathbb{R}^N)$, with $\|u\|_s = \rho$, and
for all $\lambda\le\lambda_0$.
\end{lemma}
Set
$$
c_\lambda=\inf\{\mathcal J_\lambda(u): u\in
\overline{B_{\rho}}\},
$$
where $B_{\rho} =\{u \in
W_V^{s,p}(\mathbb{R}^N): \|u\|_s <\rho \}$ and $\rho \in(0,1]$
is the number determined in Lemma~\ref{lem4.1}.
\begin{lemma}[{see \cite[Lemma 4.2]{PXZ3}}] \label{lem4.2}
If {\rm (A5)} and {\rm (A6)} hold, then $c_\lambda<0$
for each $\lambda\in(0,\lambda_0]$.
\end{lemma}
\begin{lemma}\label{lem4.3}
If {\rm (A5)} and {\rm (A6)} hold, then
there exists $\lambda^*>0$ such that, up to a subsequence, $\{u_n\}_n$
strongly converges to some $u_\lambda$ in $W_V^{s,p}(\mathbb{R}^N)$
for all $\lambda\in (0,\lambda^*]$ .
\end{lemma}
\begin{proof}
Because of the degenerate nature of \eqref{e1.1}, two situations must be
considered: either \text{$\inf_{n\in\mathbb{N}}
\|u_n\|_s = d_\lambda >0$} or
$\inf_{n\in\mathbb{N}}\|u_n\|_s = 0$. For
this, we divide the proof in two cases.\smallskip
\noindent (i)\ {\em Case} $\inf_{n\in\mathbb N}\|u_n\|_s = d_\lambda > 0$.
By Lemmas \ref{lem4.1} and \ref{lem4.2} and the Ekeland variational principle,
applied in $\overline{B}_{\rho}$, there exists a sequence $(u_n)_n \subset
B_{\rho}$ such that
\begin{align}\label{e3.3}
c_\lambda\leq \mathcal J_\lambda(u_n)\leq c_\lambda+1/n \quad
\text{and} \quad \mathcal J_\lambda(v)\geq \mathcal J_\lambda(u_n)
-\|u_n-v\|_s\big/n
\end{align}
for all $n \in \mathbb{N}$ and for any $v\in \overline{B}_{\rho}$.
Fixed $n \in \mathbb{N}$, for all $\upsilon \in S_V$, where
$S_V = \{u \in W_V^{s,p}(\mathbb{R}^N): \|u\|_s = 1\}$, and for all
$\sigma > 0$ so small that $u_n + \sigma\upsilon \in \overline{B}_{\rho}$,
we have
\begin{align*}
\mathcal J_\lambda(u_n + \sigma\upsilon) - \mathcal J_\lambda(u_n)
\geq - \frac{\sigma}{n}
\end{align*}
by \eqref{e3.3}. Since $\mathcal J_\lambda$ is
G$\hat{\text{a}}$teaux differentiable in $W_V^{s,p}(\mathbb{R}^N)$,
we get
\begin{align*}
\langle \mathcal J'_\lambda(u_n),\upsilon \rangle = \lim_{\sigma
\to 0} \frac{\mathcal J_\lambda(u_n + \sigma\upsilon) -
\mathcal J_\lambda(u_n)}{\sigma} \geq - \frac{1}{n}
\end{align*}
for all $\upsilon \in S_V$. Hence
\begin{align*}
|\langle \mathcal J'_\lambda(u_n),\upsilon \rangle | \leq
\frac{1}{n},
\end{align*}
since $\upsilon \in S_V$ is arbitrary. Consequently,
$\mathcal J'_\lambda(u_n) \to 0$ in $(W_V^{s,p}(\mathbb{R}^N))'$ as $n
\to \infty$ and clearly, up to a subsequence, the bounded
sequence $(u_n)_n$ weakly converges to some $u_\lambda \in
\overline{B}_{\rho}$ and has the following properties
\begin{equation}\label{eq3.4}
\begin{gathered}
u_n\rightharpoonup u_{\lambda}\ \ {\rm weakly\ in}\ W_V^{s,p}(\mathbb{R}^N),
\quad \|u_n\|_{s}\to \alpha_\lambda,\\
\int_{\mathbb{R}^N}(\mathcal{K}_\mu*|u_n-u_\lambda|^{p_{\mu,s}^*})
|u_n-u_\lambda|^{p_{\mu,s}^*}dx\to \kappa_\lambda,\\
u_n\to u_{\lambda}\text{ a.e. in } \mathbb{R}^N,\quad
|u_n|^{p_{\mu,s}^*}\rightharpoonup|u_\lambda|^{p_{\mu,s}^*}
\text{ weakly in } L^{\frac{p_s^*}{p_{\mu,s}^*}}(\mathbb{R}^N),\\
|u_n|^{p_{\mu,s}^*-2}u_n\rightharpoonup
|u_\lambda|^{p_{\mu,s}^*-2}u_\lambda\quad \text{weakly in }
L^{\frac{p_s^*}{p_{\mu,s}^*-1}}(\mathbb{R}^N),
\end{gathered}
\end{equation}
as $n\to\infty$, by \cite[Lemma 2.1]{capu}. Clearly
$\alpha_\lambda > 0$ since we are in the case in which $d_\lambda > 0$.
Therefore $M(\|u_n\|_s^p) \to M(\alpha_\lambda^p) > 0$ as $n \to \infty$, by
continuity of $M$ and the fact that 0 is the unique zero of $M$.
By the Hardy-Littlewood-Sobolev inequality and that
${p_s^*}/p_{\mu,s}^*={2N}/(2N-\mu)$, the Riesz potential defines a
linear continuous map $\mathcal{K}_\mu*(\,\cdot\,):
L^{\frac{p_s^*}{p_{\mu,s}^*}}(\mathbb{R}^N)\to $
$L^{\frac{2N}{\mu}}(\mathbb{R}^N)$. Since
$|u_n|^{p_{\mu,s}^*}\rightharpoonup|u_\lambda|^{p_{\mu,s}^*}$ weakly
in $L^{\frac{p_s^*}{p_{\mu,s}^*}}(\mathbb{R}^N)$, then as
$n\to\infty$
\begin{align}\label{c3.5}
\mathcal{K}_\mu*|u_n|^{p_{\mu,s}^*}\rightharpoonup
\mathcal{K}_\mu*|u_\lambda|^{p_{\mu,s}^*}\quad \text{weakly in }
L^{\frac{2N}{\mu}}(\mathbb{R}^N).
\end{align}
Note that for any subset $U\subset \mathbb{R}^N$, it holds
\begin{align*}
\int_{U}||u_n|^{p_{\mu,s}^*-2}u_nu_\lambda|^{\frac{2N}{2N-\mu}}dx
&\leq\int_U|u_n|^{(p_{\mu,s}^*-1)\frac{p_s^*}{p_{\mu,s}^*}}
|u_\lambda|^{\frac{p_s^*}{p_{\mu,s}^*}}dx\\
&\leq\|u_n\|_{p_s^*}^{\frac{p_{\mu,s}^*-1}{p_{\mu,s}^*}}(\int_U|u_\lambda|^{p_s^*}dx
)^{\frac{1}{p_{\mu,s}^*}}\\
&\leq C(\int_U|u_\lambda|^{p_s^*}dx)^{\frac{1}{p_{\mu,s}^*}}.
\end{align*}
This and $u_\lambda\in L^{p_s^*}(\mathbb{R}^N)$ imply
that the sequence
$\{||u_n|^{p_{\mu,s}^*-2}u_nu_\lambda|^{\frac{2N}{2N-\mu}}\}_n$ is
equi-integrable in $L^1(\mathbb{R})$. Moreover,
$|u_n|^{p_{\mu,s}^*-2}u_nu_\lambda\to
|u_\lambda|^{p_{\mu,s}^*}$ a.e. in $\mathbb{R}^N$ as
$n\to\infty$. Hence the Vitali convergence theorem yields
that $|u_n|^{p_{\mu,s}^*-2}u_nu_\lambda \to
|u_\lambda|^{p_{\mu,s}^*}$ strongly in
$L^{\frac{2N}{2N-\mu}}(\mathbb{R}^N)$. Thus,
\begin{align}\label{c3.6}
\lim_{n\to\infty}\int_{\mathbb{R}^N}(\mathcal{K}_\mu*|u_n|^{p_{\mu,s}^*})|u_n|^{p_{\mu,s}^*-2}u_nu_\lambda
\,dx=\int_{\mathbb{R}^N}(\mathcal{K}_\mu*|u_\lambda|^{p_{\mu,s}^*})
|u_\lambda|^{p_{\mu,s}^*}dx
\end{align}
by \eqref{c3.5}. Similarly,
\begin{align}\label{c3.7}
\lim_{n\to\infty}\int_{\mathbb{R}^N}(\mathcal{K}_\mu*|u_\lambda|^{p_{\mu,s}^*})|u_\lambda|^{p_{\mu,s}^*-2}u_\lambda
u_n
\,dx=\int_{\mathbb{R}^N}(\mathcal{K}_\mu*|u_\lambda|^{p_{\mu,s}^*})
|u_\lambda|^{p_{\mu,s}^*}dx.
\end{align}
For any subset $U\subset \mathbb{R}^N$, by (A5) we have
\begin{align*}
\int_{U}|f(x,u_n)(u_n-u_\lambda)|dx\leq
\|w\|_{L^{\frac{p_s^*}{p_s^*-q}}(U)}\|u_n-u_\lambda\|_{p_s^*}\leq
C\|w\|_{L^{\frac{p_s^*}{p_s^*-q}}(U)}.
\end{align*}
It follows from $w\in L^{\frac{p_s^*}{p_s^*-q}}(\mathbb{R}^N)$ that
sequence $\{f(x,u_n)(u_n-u_\lambda)\}_n$ is equi-integrable in
$L^1(\mathbb{R}^N)$. Clearly, $f(x,u_n)(u_n-u_\lambda)\to 0$
a.e. in $\mathbb{R}^N$ as $n\to\infty$. Hence the Vitali
convergence theorem implies that
\begin{align}\label{e3.8}
\lim_{n\to\infty}\int_{\mathbb{R}^N}f(x,u_n)(u_n-u_\lambda)dx=0.
\end{align}
A similar argument shows that
\begin{gather}\label{e3.9}
\lim_{n\to\infty}\int_{\mathbb{R}^N}f(x,u_\lambda)(u_n-u_\lambda)dx=0, \\
\label{e3.10}
\lim_{n\to\infty}\int_{\mathbb{R}^N}f(x,u_n)u_\lambda\,dx=
\lim_{n\to\infty}\int_{\mathbb{R}^N}f(x,u_\lambda)u_\lambda \,dx.
\end{gather}
Furthermore, by (A6) for equi-integrability, we get
\begin{align}\label{e3.11}
\lim_{n\to\infty}\int_{\mathbb{R}^N}F(x,u_n)dx=
\lim_{n\to\infty}\int_{\mathbb{R}^N}F(x,u_\lambda)dx.
\end{align}
Let us now introduce, for simplicity, for all $v\in
W_V^{s,p}(\mathbb{R}^N)$ the linear functional $L(v)$ on
$W_V^{s,p}(\mathbb{R}^N)$ defined by
\begin{align*}
\langle L(v),w\rangle
&=\iint_{\mathbb{R}^{2N}}\frac{|v(x)-v(y)|^{p-2}(v(x)-v(y))
(w(x)-w(y))}{|x-y|^{N+2s}}\,dx\,dy
\\ &\quad +\int_{\mathbb{R}^N}V(x)|v|^{p-2}v wdx\\
&=\langle v,w\rangle_{s,p}+\int_{\mathbb{R}^N}V(x)|v|^{p-2}v wdx
\end{align*}
for all $w\in W_V^{s,p}(\mathbb{R}^N)$. The H\"older inequality
gives
\[
\left|\langle L(v),w\rangle\right|\leq
[v]_s^{p-1}[w]_s+\|v\|_{p,V}^{p-1}\|w\|_{p,V} \leq
\|v\|_s^{p-1}\|w\|_s.
\]
Thus, for each $v\in W_V^{s,p}(\mathbb{R}^N)$, the linear functional
$L(v)$ is continuous on $W_V^{s,p}(\mathbb{R}^N)$. Hence, the weak
convergence of $\{u_n\}_n$ in $W_V^{s,p}(\mathbb{R}^N)$ gives that
\begin{align}\label{e3.12}
\lim_{n\to\infty}\langle
L(u_{\lambda}),u_n-u_{\lambda}\rangle=0.
\end{align}
Since $\{u_n\}_n$ in bounded in $W^{s,p}_V(\mathbb{R}^N)$, then
$\{L(u_n)\}_n$ is bounded in $(W^{s,p}_V(\mathbb{R}^N))'$. Hence
there exist some functional $\xi\in(W^{s,p}_V(\mathbb{R}^N))'$
and a subsequence of $\{u_n\}_n$, still denoted by $\{u_n\}_n$, such
that
\begin{equation}\label{e3.13}
\lim_{n\to\infty}\langle L(u_n),v\rangle=\langle\xi,v\rangle
\end{equation}
for any $v\in W^{s,p}_V(\mathbb{R}^N)$. Then, $\langle \mathcal
J_\lambda'(u_n),u_\lambda\rangle\to 0$, \eqref{e3.12}
and \eqref{e3.13} give
\begin{equation} \label{e3.14}
M(\alpha_\lambda^p)\langle\xi,u_\lambda\rangle=\lambda\int_{\mathbb{R}^N}
f(x,u_\lambda)u_\lambda\,dx
+\int_{\mathbb{R}^N}(\mathcal{K}*|u_\lambda|^{p_{\mu,s}^*})|u_\lambda|^{p_{\mu,s}^*}dx.
\end{equation}
By \eqref{e3.14} and (A6), we have
$\langle\xi,u_\lambda\rangle\geq 0$.
Since $\{u_n\}_n$ is a $(PS)$ sequence, we deduce from
Theorem~\ref{thm2.1}, \eqref{c3.5}-\eqref{e3.14} that
\begin{align}\label{e3.15}
o(1)&=\langle \mathcal J_\lambda'(u_n)-\mathcal
J_\lambda'(u_{\lambda}),u_n-u_{\lambda}\rangle
\nonumber\\
&=M(\|u_n\|_s^p)\|u_n\|_{s}^p-M(\|u_n\|_s^p)\langle L(u_n),u_{\lambda}\rangle
-M(\|u\|_s^p)\langle L(u_{\lambda}), u_n-u_\lambda\rangle\nonumber\\
&\quad -\lambda\int_{\mathbb{R}^N}[f(x,u_n)-f(x,u_{\lambda})](u_n-u_{\lambda})dx
\nonumber\\
&\quad -\int_{\mathbb{R}^N}\Big[(\mathcal{K}_\mu*|u_n|^{p_{\mu,s}^*})
|u_n|^{p_{\mu,s}^*-2}u_n \nonumber \\
&\quad -(\mathcal{K}_\mu*|u_\lambda|^{p_{\mu,s}^*})|u_{\lambda}|^{p_{\mu,s}^*-2}
u_{\lambda}\Big]
(u_n-u_{\lambda})dx \nonumber \\
&=M(\alpha_\lambda^p)[\alpha_{\lambda}^p-\langle\xi,u_{\lambda}\rangle]-
\int_{\mathbb{R}^N}(\mathcal{K}_\mu*|u_n-u_\lambda|^{p_{\mu,s}^*})|u_n-u_\lambda|^{p_{\mu,s}^*}dx
+o(1) \nonumber \\
&=M(\|u_n\|_s^p)\langle L(u_n)-L(u_{\lambda}), u_n-u_\lambda\rangle \nonumber \\
&\quad -\int_{\mathbb{R}^N}(\mathcal{K}_\mu*|u_n-u_\lambda|^{p_{\mu,s}^*})
|u_n-u_\lambda|^{p_{\mu,s}^*}dx+o(1).
\end{align}
Hence the above inequality yields
\begin{equation} \label{e3.16}
\begin{aligned}
&\lim_{n\to\infty}M(\|u_n\|_s^p)\langle
L(u_n)-L(u_{\lambda}), u_n-u_{\lambda}\rangle \\
&=\lim_{n\to\infty}\int_{\mathbb{R}^N}(\mathcal{K}_\mu*|u_n-u_\lambda|^{p_{\mu,s}^*})
|u_n-u_\lambda|^{p_{\mu,s}^*}dx.
\end{aligned}
\end{equation}
Now, since $\{u_n\}_n$ is a minimizing $(PS)_{c_\lambda}$ sequence,
by \eqref{e3.16} we also get as $n\to\infty$
\begin{align} \label{e3.17}
c_\lambda&=\frac{1}{p}\mathscr{M}(\|u_n\|_{s}^p)
-\lambda\int_{\mathbb{R}^N} F(x,u_n)dx
-\frac{1}{2p_{\mu,s}^*}\int_{\mathbb{R}^N}(\mathcal{K}_\mu*
|u_n|^{p_{\mu,s}^*})|u_n|^{p_{\mu,s}^*}dx+o(1) \nonumber \\
&\geq\frac{1}{\theta p}M(\|u_n\|_s^p)\|u_n\|_s^p
-\lambda\int_{\mathbb{R}^N} F(x,u_n)dx \nonumber \\
&\quad -\frac{1}{2p_{\mu,s}^*}
\int_{\mathbb{R}^N}(\mathcal{K}_\mu*|u_n|^{p_{\mu,s}^*})
|u_n|^{p_{\mu,s}^*}dx+o(1)\\
&=\frac{1}{\theta p}M(\|u_n\|_s^p)\langle
L(u_n)-L(u_\lambda),u_n-u_\lambda\rangle+\frac{1}{\theta
p}M(\|u_n\|_s^p)\langle L(u_n),u_\lambda\rangle
\nonumber \\
&\quad-\lambda\int_{\mathbb{R}^N} F(x,u_n)dx-\frac{1}{2p_{\mu,s}^*}
\int_{\mathbb{R}^N}(\mathcal{K}_\mu*|u_n|^{p_{\mu,s}^*})
|u_n|^{p_{\mu,s}^*}dx+o(1) \nonumber \\
&=\frac{1}{\theta p}\int_{\mathbb{R}^N}
(\mathcal{K}_\mu*|u_n-u_\lambda|^{p_{\mu,s}^*})|u_n-u_\lambda|^{p_{\mu,s}^*}dx
+\frac{1}{\theta p}M(\|u_n\|_s^p)\langle L(u_n),u_\lambda\rangle
\nonumber \\
&\quad -\lambda\int_{\mathbb{R}^N} F(x,u_n)dx-\frac{1}{2p_{\mu,s}^*}
\int_{\mathbb{R}^N}(\mathcal{K}_\mu*|u_n|^{p_{\mu,s}^*})
|u_n|^{p_{\mu,s}^*}dx+o(1). \nonumber
\end{align}
Note that $M(\|u_n\|_s^p)\langle
L(u_n),u_\lambda\rangle=M(\alpha_\lambda^p)\langle\xi,u_\lambda\rangle+o(1)$.
Then by \eqref{e3.17} we get
\begin{align*}
M(\|u_n\|_s^p)\langle L(u_n),u_\lambda\rangle=\lambda
\int_{\mathbb{R}^N}f(x,u_\lambda)u_\lambda\,dx+\int_{\mathbb{R}^N}
(\mathcal{K}_\mu*|u_\lambda|^{p_{\mu,s}^*})|u_\lambda|^{p_{\mu,s}^*}dx+o(1).
\end{align*}
Inserting this equality in \eqref{e3.17} and using
Theorem~\ref{thm2.1}, we deduce
\begin{align*}
c_\lambda&\geq\frac{1}{\theta p}\int_{\mathbb{R}^N}
(\mathcal{K}_\mu*|u_n-u_\lambda|^{p_{\mu,s}^*})|u_n-u_\lambda|^{p_{\mu,s}^*}dx\\
&\quad -\frac{1}{2p_{\mu,s}^*}
\int_{\mathbb{R}^N}(\mathcal{K}_\mu*|u_n|^{p_{\mu,s}^*})
|u_n|^{p_{\mu,s}^*}dx\\
&\quad+\frac{1}{\theta p}\int_{\mathbb{R}^N}
(\mathcal{K}_\mu*|u_\lambda|^{p_{\mu,s}^*})|u_\lambda|^{p_{\mu,s}^*}dx
+\frac{\lambda}{\theta p}\int_{\mathbb{R}^N}f(x,u_\lambda)u_\lambda
\,dx \\
&\quad -\lambda\int_{\mathbb{R}^N} F(x,u_n)dx+o(1)\\
&\geq \frac{1}{\theta p}\int_{\mathbb{R}^N}
(\mathcal{K}_\mu*|u_n-u_\lambda|^{p_{\mu,s}^*})|u_n-u_\lambda|^{p_{\mu,s}^*}dx\\
&\quad-\frac{1}{2p_{\mu,s}^*}\Big(\int_{\mathbb{R}^N}(\mathcal{K}_\mu*|u_n|^{p_{\mu,s}^*})
|u_n|^{p_{\mu,s}^*}dx
-\int_{\mathbb{R}^N}(\mathcal{K}_\mu*|u_\lambda|^{p_{\mu,s}^*})
|u_\lambda|^{p_{\mu,s}^*}dx\Big)\\
&\quad+\frac{\lambda}{\theta
p}\int_{\mathbb{R}^N}f(x,u_\lambda)u_\lambda\,dx
-\lambda\int_{\mathbb{R}^N} F(x,u_n)dx+o(1)\\
&\geq \big(\frac{1}{\theta p}-\frac{1}{2p_{\mu,s}^*}\big)
\int_{\mathbb{R}^N}(\mathcal{K}_\mu*|u_n-u_\lambda|^{p_{\mu,s}^*})
|u_n-u_\lambda|^{p_{\mu,s}^*}dx\\
&\quad+\frac{\lambda}{\theta
p}\int_{\mathbb{R}^N}f(x,u_\lambda)u_\lambda\,dx
-\lambda\int_{\mathbb{R}^N} F(x,u_n)dx+o(1),
\end{align*}
thanks to $\theta p<2p_{\mu,s}^*$. Hence
\begin{equation}\label{e3.18}
\begin{aligned}
c_\lambda&\geq\big(\frac{1}{\theta p}-\frac{1}{2p_{\mu,s}^*}\big)\int_{\mathbb{R}^N}
(\mathcal{K}_\mu*|u_n-u_\lambda|^{p_{\mu,s}^*})|u_n-u_\lambda|^{p_{\mu,s}^*}dx\\
&\quad +\frac{\lambda}{\theta p}\int_{\mathbb{R}^N}f(x,u_\lambda)
u_\lambda\,dx-\lambda\int_{\mathbb{R}^N} F(x,u_\lambda)dx+o(1).
\end{aligned}
\end{equation}
Clearly, $|F(x,u_{\lambda})|\leq w(x)|u_{\lambda}|^q$ for all
$x\in\mathbb{R}^N$ and for all $\lambda\in(0,\lambda_0]$
thanks to (A5). In view of the choice of $\rho$ in Lemma \ref{lem4.1},
we know that $\rho$ is independent of $\lambda$.
Thus, $\{u_\lambda\}_{\lambda\in(0,\lambda_0]}$ is uniformly
bounded in $W_V^{s,p}(\mathbb{R}^N)$. Furthermore, there exists $C>0$,
which does not depend on $\lambda$, such that
$\int_{\mathbb{R}^N} F(x,u_\lambda)dx\le C$ and
$\big|\int_{\mathbb{R}^N}f(x,u_\lambda)u_\lambda\,dx \big| \le C$.
Hence by \eqref{e3.18}, we deduce
\begin{align*}
(\frac{1}{\theta p}-\frac{1}{2p_{\mu,s}^*})\int_{\mathbb{R}^N}
(\mathcal{K}_\mu*|u_n-u_\lambda|^{p_{\mu,s}^*})|u_n-u_\lambda|^{p_{\mu,s}^*}dx\leq
c_\lambda +2C\lambda+o(1).
\end{align*}
This and Lemma~\ref{lem4.2} imply that
\begin{equation} \label{e3.19}
\lim_{\lambda\to 0}\kappa_\lambda=\lim_{\lambda\to
0}\lim_{n\to\infty}\int_{\mathbb{R}^N}
(\mathcal{K}_\mu*|u_n-u_\lambda|^{p_{\mu,s}^*})|u_n-u_\lambda|^{p_{\mu,s}^*}dx=0.
\end{equation}
Let us now recall the well-known Simon inequalities.
We refer to \cite[Formula 2.2]{simon} (see also \cite[p. 713]{fili}). There
exist positive numbers $C_p$ and $\widetilde{C}_p$, depending only
on $p$, such that
\begin{equation} \label{simon}
|\xi-\eta|^{p}
\leq \begin{cases}
C_p (|\xi|^{p-2}\xi-|\eta|^{p-2}\eta)(\xi-\eta)
& \text{for } p\geq2\\[3pt]
\widetilde{C}_p\big[(|\xi|^{p-2}\xi-|\eta|^{p-2}\eta)\\
\times (\xi-\eta) \big]^{p/2}(|\xi|^p+|\eta|^p)^{(2-p)/2}
& \text{for }10\},
&\text{if } \kappa_\lambda\not\equiv 0, \\
\lambda_0,&\text{if }\kappa_\lambda\equiv 0.
\end{cases}
\]
If $\kappa_\lambda\not\equiv 0$, then $\lambda^{*}=
\inf\{\lambda\in(0,\lambda_0]:\kappa_{\lambda}>0\}>0$.
Otherwise, there exists a sequence $(\lambda_k)_k$, with
$\kappa_{\lambda_k}>0$, such that $\lambda_k\to 0$ as
$k\to\infty$. Thus, \eqref{c23} implies that
\begin{align*}
\kappa_{\lambda_k}^{1-\frac{p}{2p_{\mu,s}^*}}\geq
C^{-\frac{p}{2p_{\mu,s}^*}}(N,\mu)C_p^{-1}M(\alpha_\lambda^p),
\end{align*}
which means that $\alpha_{\lambda_k}\to0$ as
$k\to\infty$. Without loss of generality, we assume that
$\alpha_{\lambda_k}\in(0,1]$ for all $k\geq 1$. Then
\begin{align*}
M(\alpha_{\lambda_k}^p)\alpha_{\lambda_k}^p
&\geq
M(\alpha_{\lambda_k}^p)[\alpha_{\lambda_k}^p-\langle\xi,u_{\lambda_k}\rangle]\\
&=\delta_{\lambda_k}
\geq\left[ C^{-\frac{p}{2p_{\mu,s}^*}}(N,\mu)C_p^{-1}M(\alpha_{\lambda_k}^p)
\right]^{1/(1-{p}/{2p_{\mu,s}^*})}.
\end{align*}
Hence, (A4) gives
\[
\alpha_{\lambda_k}^{p-\frac{p^2}{2p_{\mu,s}^*}}\geq
C^{-\frac{p}{2p_{\mu,s}^*}}(N,\mu)C_p^{-1}
m_0^{\frac{p}{2p_{\mu,s}^*}}\alpha_{\lambda_k}^{\frac{p^2(\theta-1)}{2p_{\mu,s}^*}},
\]
that is,
\[
\alpha_{\lambda_k}^{p-\frac{\theta p}{2p_{\mu,s}^*}}\geq
C^{-\frac{p}{2p_{\mu,s}^*}}(N,\mu)C_p^{-1}
m_0^{\frac{p}{2p_{\mu,s}^*}}.
\]
This is impossible, since $\theta0$ such that
$u_n\to u_{\lambda}$ strongly in $W_V^{s,p}(\mathbb{R}^N)$ as $n\to\infty$
for all $\lambda\in(0,\lambda^{*}]$.
\smallskip
\noindent (ii)\ {\em Case} $\inf_{n\in\mathbb{N}}\|u_n\|_s =0$.
If $0$ is an isolated point for the real sequence $\{\|u_n\|_s\}_n$,
then there is a subsequence $\{u_{n_k}\}_k$ such that
$$\inf_{k\in\mathbb N}\|u_{n_k}\|_s = d>0,$$
and we can proceed as before. Otherwise, $0$ is an accumulation
point of the sequence $\{\|u_n\|_s\}_n$ and so there exists a
subsequence $\{u_{n_k}\}_k$ of $\{u_{n}\}_n$ such that
$u_{n_k}\to 0$ strongly in $W_V^{s,p}(\mathbb{R}^N)$ as $n\to\infty$.
\smallskip
In conclusion, $\mathcal J_\lambda$ satisfies the $(PS)$ condition
in $W_V^{s,p}(\mathbb{R}^N)$ at the level $c_\lambda$ in all the
possible cases.
\end{proof}
\section{A truncation argument}\label{sec truncation}
We note that our functional $\mathcal J_\lambda$ is not bounded from
below in $W_V^{s,p}(\mathbb{R}^N)$. Indeed, if $\varepsilon =
1$, it follows from \eqref{e2.4} that
\begin{equation} \label{M2}
\mathscr{M}(t)\leq \mathscr{M}(1)t^\theta\quad\text{for all } t\geq
1.
\end{equation}
Furthermore, $F(x,t)\geq0$ for all $(x,t)\in\mathbb{R}^N\times
\mathbb{R}$ by (A5) and (A6). Let $u\in
C_0^\infty(\mathbb{R}^N)$, with $u\geq 0$ a.e. in $\mathbb{R}^N$
such that $\|u\|_s=1$. Then for all $t\geq 1$, we have
\begin{equation}\label{dopo}
\begin{aligned}
\mathcal J_\lambda(tu)
&\leq \frac{1}{p}\mathscr{M}(1)t^{\theta
p}\|u\|_{s}^{\theta p}
-\frac{t^{2p_{\mu,s}^*}}{2p_{\mu,s}^*}\int_{\mathbb{R}^N}
(\mathcal{K}_\mu*u^{p_{\mu,s}^*})
u^{p_{\mu,s}^*}dx \\
&\leq \frac{1}{p}\mathscr{M}(1)t^{\theta p}\|u\|_{s}^{\theta p}
-\frac{t^{2p_{\mu,s}^*}}{2p_{\mu,s}^*}(S^*)^{-p^*_{\mu,s}}
\|u\|_s^{2p_{\mu,s}^*}\\
&=\frac{1}{p}\mathscr{M}(1)t^{\theta p}
-\frac{1}{2p_{\mu,s}^*}(S^*)^{-2p_{\mu,s}^*}t^{2p_{\mu,s}^*}.
\end{aligned}
\end{equation}
Hence, $\mathcal J_\lambda(tu)\to-\infty$ as
$t\to\infty$, since $\theta
\frac{C(N,\mu)}{2p_{\mu,s}^*}R_1^{2p_{\mu,s}^*},
\end{equation}
since $p\leq p\theta< p_{\mu,s}^*$, and we define
\begin{equation}\label{stella}
\lambda_*=\frac{1}{2\,C^q_w\, R^q_1}\Big(\frac{\mathscr
M(1)}{p}R^{p\theta}_1-\frac{C(N,\mu)}{2p_{\mu,s}^*}R_1^{2p_{\mu,s}^*}\Big),
\end{equation}
so that $\mathcal G_{\lambda_*}(R_1)>0$. From this, we consider
\[
R_0=\max\left\{t\in(0,R_1):\,\,\mathcal G_{\lambda_*}(t)\leq0\right\}.
\]
Since by $q0$, it easily
follows that $\mathcal G_{\lambda_*}(R_0)=0$.
We can choose $\psi\in C^\infty_0([0,\infty),[0,1])$ such that
$\psi(t)=1$ if $t\in[0,R_0]$ and $\psi(t)=0$ if $t\in[R_1,\infty)$.
Thus, we consider the truncated functional
$$
\mathcal I_\lambda(u)
=\frac{1}{p}\mathscr{M}(\|u\|_s^p)-\lambda\int_{\mathbb{R}^N} F(x,u)dx
-\psi(\|u\|_s)\frac{1}{2p_{\mu,s}^*}
\int_{\mathbb{R}^N}(\mathcal{K}_\mu*|u|^{p_{\mu,s}^*})|u|^{p_{\mu,s}^*}dx.
$$
It immediately follows that $\mathcal I_\lambda(u)\to\infty$
as $\|u\|_s \to\infty$, by (A2) and (A3). Hence,
$\mathcal I_\lambda$ is coercive and bounded from below.
Now, we prove a local Palais-Smale and a topological result for the
truncated functional $\mathcal I_\lambda$.
\begin{lemma}\label{localpalais}
There exists $\overline{\lambda}>0$ such that, for any
$\lambda\in(0,\overline{\lambda})$
\begin{enumerate}
\item[(i)] if $\mathcal I_\lambda(u)\leq0$ then $\|u\|_s < R_0$
and $\mathcal J_\lambda(v)=\mathcal I_\lambda(v)$ for any $v$
in a sufficiently small neighborhood of $u$;
\item[(ii)]
$\mathcal I_\lambda$ satisfies the $(PS)_{c_\lambda}$ condition on
$W_V^{s,p}(\mathbb{R}^N)$.
\end{enumerate}
\end{lemma}
\begin{proof}
Considering $\lambda_0$, $\lambda^*$ and $\lambda_*$ given
respectively by Lemma \ref{lem4.1}, \ref{lem4.3} and \eqref{stella}, we choose
$\overline{\lambda}$ sufficiently small such that
$\overline{\lambda}\leq\min\left\{\lambda_0, \lambda^*,
\lambda_*\right\}$. Let $\lambda<\overline{\lambda}$.
For proving (i) we assume that $\mathcal I_\lambda(u)\leq0$. When
$\|u\|_s \geq 1$, by using (A2), (A3) with $\tau=1$ and
$\lambda < \lambda_*$, we see that
\[
\mathcal I_\lambda(u)\geq\frac{m}{p\theta}\|u\|_s^p
-\frac{\lambda_*}{q}C_w^q\|u\|_s^q>0,
\]
where the last inequality follows by $q
0$ and \eqref{R1} we have
\[
\frac{m}{p\theta}R^p_1-\frac{\lambda_*}{q}C_w^q R^q_1>0.
\]
Thus, we get the contradiction $0\geq\mathcal I_\lambda(u)>0$.
Similarly, when $R_1\leq \|u\|_s <1$, by using \eqref{e2.3},
\eqref{R1} and $\lambda<\lambda_*$, we obtain
\[
\mathcal I_\lambda(u)\geq\frac{\mathscr
M(1)}{p}\|u\|_s^{p\theta}-\frac{\lambda_*}{q}C_w^q\|u\|_s^q>0,
\]
where the last inequality follows by $q
0$ we have
\[
\frac{\mathscr M(1)}{p}R^{p\theta}_1-\frac{\lambda_*}{q}C_w^q R^q_1>0.
\]
We get again the contradiction $0\geq\mathcal I_\lambda(u)>0$. When
$\|u\|_s < R_1$, since $\phi(t)\leq 1$ for any $t\in[0,\infty)$ and
$\lambda<\lambda_*$, we have
\[
0\geq\mathcal I_\lambda(u)\geq\mathcal G_\lambda(\|u\|_s)
\geq\mathcal G_{\lambda_*}(\|u\|_s),
\]
and this yields $\|u\|_s\leq R_0$, by definition of $R_0$.
Furthermore, for any $u\in B(0,R_0/2)$ we have $\mathcal
I_\lambda(u)=\mathcal J_\lambda(u)$.
Arguing exactly as Lemma \ref{lem4.3} we know that $\mathcal I_\lambda$
satisfies the $(PS)_{c_\lambda}$ condition on
$W_V^{s,p}(\mathbb{R}^N)$ for $\lambda<\lambda^*$. This completes
the proof of Lemma \ref{localpalais}.
\end{proof}
\begin{lemma}\label{genus}
For any $\lambda>0$ and $n\in\mathbb N$, there exists
$\varepsilon=\varepsilon(\lambda, n)>0$ such that
\[
\gamma(\mathcal I_\lambda^{-\varepsilon})\geq n,
\]
where $\mathcal I_\lambda^{-\varepsilon}=\left\{u\in
W_V^{s,p}(\mathbb{R}^N):\,\,\mathcal
I_\lambda(u)\leq-\varepsilon\right\}$.
\end{lemma}
\begin{proof}
Fix $\lambda>0$, $n\in\mathbb N$. Let $Y_n$ be a $n$-dimensional
subspace of $W_V^{s,p}(\mathbb{R}^N)$. For any $u \in Y_n$, $u \neq
0$ write $u = r_n\phi$ with $\phi \in Y_n$, $\|\phi\|_s = 1$ and
$\bar\phi=\int_{\Omega}|\phi|^\xi\,dx>0$. Then, by (A6) and
continuity of $M$, for all $r_n$, with $00$ and $n\in\mathbb N$, the number $c_n$ is
negative.
\end{lemma}
\begin{proof}
Let $\lambda>0$ and $n\in\mathbb N$. By Lemma \ref{genus}, there
exists $\varepsilon>0$ such that $\gamma(\mathcal
I_\lambda^{-\varepsilon})\geq n$. Since also $\mathcal I_\lambda$ is
continuous and even, $\mathcal I_\lambda^{-\varepsilon}\in\Sigma_n$.
From $\mathcal I_\lambda(0)=0$ we have $0\not\in\mathcal
I_\lambda^{-\varepsilon}$. Furthermore $\sup_{u\in\mathcal
I_\lambda^{-\varepsilon}}\mathcal I_\lambda(u)\leq-\varepsilon$. In
conclusion, remembering also that $\mathcal I_\lambda$ is bounded
from below, we get
\[
-\infty