\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{mathrsfs} \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 $10$ 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 $q0$ 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 $q0$ 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