\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small Two nonlinear days in Urbino 2017,\newline \emph{Electronic Journal of Differential Equations}, Conference 25 (2018), pp. 149--165.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2018 Texas State University.} \vspace{8mm}} \begin{document} \setcounter{page}{149} \title[\hfilneg EJDE-2018/conf/25\hfil Fractional Schr\"odinger equations] {Nonhomogeneous sublinear fractional Schr\"odinger equations} \author[Teresa Isernia \hfil EJDE-2018/conf/25\hfilneg] {Teresa Isernia} \address{Teresa Isernia \newline Dipartimento di Ingegneria Industriale e Scienze Matematiche, Universit\`a Politecnica delle Marche, Via Brecce Bianche, 12, 60131 Ancona, Italy} \email{teresa.isernia@unina.it} \thanks{Published September 15, 2018} \subjclass[2010]{35A15, 35J60, 35R11} \keywords{Fractional Laplacian; sublinear growth; variational methods} \begin{abstract} We study the existence, uniqueness and multiplicity for the sublinear fractional problem \[ (-\Delta)^{s}u + V(x) u + a(x) |u|^{p} \operatorname{sgn}(u) = f \quad\text{in } \mathbb{R}^N, \] where $s\in (0, 1)$, $N>2s$, $(-\Delta)^{s}$ is the fractional Laplacian, $p\in (0, 1)$, $f\in L^2(\mathbb{R}^N) \cap L^{\frac{p+1}{p}}(\mathbb{R}^N)$, $V:\mathbb{R}^N\to \mathbb{R}$ and $a:\mathbb{R}^N\to \mathbb{R}$ are positive bounded functions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} In this article we consider the nonlinear fractional Schr\"odinger equation \begin{equation}\label{P} \begin{gathered} (-\Delta)^{s}u + V(x) u + a(x) |u|^{p} \operatorname{sgn}(u) = f \quad\text{in } \mathbb{R}^N\\ u\in H^s(\mathbb{R}^N)\cap L^{p+1}(\mathbb{R}^N) \end{gathered} \end{equation} where $N>2s$, $s\in (0, 1)$, $p\in (0, 1)$, $f\in L^2(\mathbb{R}^N) \cap L^{(p+1)/p}(\mathbb{R}^N)$, $V:\mathbb{R}^N\to \mathbb{R}$ and $a:\mathbb{R}^N\to \mathbb{R}$ are positive bounded functions. The nonlocal operator $(-\Delta)^{s}$ is the fractional Laplacian which is defined as \[ (-\Delta)^{s} u(x) = C_{N, s} \operatorname{P.V.} \int_{\mathbb{R}^N} \frac{u(x)- u(y)}{|x-y|^{N+2s}} dy, \] for any $u:\mathbb{R}^N\to \mathbb{R}$ sufficiently smooth. The symbol P.V. stands for the Cauchy principal value and $C_{N, s}$ is a dimensional constant depending on $N$ and $s$; see \cite{CS}. In the previous decade a great attention has been devoted to the study of fractional and nonlocal operators of elliptic type since these operators arise in a quite natural way in many different contexts such as phase transition phenomena, minimal surface, game theory, continuum mechanics, crystal dislocation, optimization, water waves and so on. For more details the interested reader may consult \cite{DNPV, MBRS} and references therein. A basic motivation for the study of problem \eqref{P} is related to the search of standing wave solutions of the type $\psi(x, t)= u(x) e^{-\imath c t}$ for the time dependent fractional Schr\"odinger equation \begin{equation}\label{FSE} \imath \frac{\partial \psi}{\partial t} = (-\Delta)^{s} \psi + (V(x)+c) \psi - g(|\psi|) \end{equation} where $V: \mathbb{R}^N\to \mathbb{R}$ is an external potential and $g$ is a suitable nonlinearity. The fractional Schr\"odinger equation \eqref{FSE} was introduced by Laskin \cite{L} and it is a fundamental equation of fractional quantum mechanic in the study of particles on stochastic fields modeled by Levy processes. Very recently, the study of problems of fractional Schr\"odinger equations has attracted the attention of many mathematicians. Indeed several existence and multiplicity results have been established, under different assumptions on the potential $V$ and nonlinear term, by using suitable variational methods; see \cite{A2, A3, AD, AF, AH, AI, DPV, FQT, FS, FLS}. In particular way, a special attention has been devoted to the study of fractional Schr\"odinger equations involving superlinear nonlinearities; see for instance \cite{AM, A1, Secchi}. On the contrary, to our knowledge, only few results deal with fractional problems with sublinear terms; see \cite{FP, I, YL}. The aim of this article is to consider equation \eqref{P} under the following assumptions: \begin{itemize} \item[(H1)] $f\in L^2(\mathbb{R}^N) \cap L^{\frac{p+1}{p}}(\mathbb{R}^N)$, $f\geq 0$ ($f\not \equiv 0$); \item[(H2)] $V\in L^{\infty}(\mathbb{R}^N)$ and $\lim_{|x|\to \infty} V(x)= v_{\infty}\geq 0$; \item[(H3)] $a\in L^{\infty}(\mathbb{R}^N)$, $\lim_{|x|\to \infty} a(x)= a_{\infty}>0$ and there exists $\alpha>0$ such that $a(x)>\alpha$ a.e.\ in $\mathbb{R}^N$. \end{itemize} Our main result is the following theorem. \begin{theorem}\label{thmmain} Assume that {\rm (H1)--(H3)} hold. Then, there exists a positive constant $c$, such that for every $f>0$ a.e.\ in $\mathbb{R}^N$, $\|f\|_{L^2(\mathbb{R}^N)}2s$. Then there exists a sharp constant $S_{*}= S(N,s)>0$ such that for any $u\in \mathcal{D}^{s,2}(\mathbb{R}^N)$, \[ \|u\|^2_{ L^{2^{*}_{s}}(\mathbb{R}^N)}\leq S_{*} \iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}} \,dx\,dy. \] Moreover, $H^s(\mathbb{R}^N)$ is continuously embedded in $L^{q}(\mathbb{R}^N)$ for any $q\in [2, 2^{*}_{s}]$ and compactly in $L^{q}_{\rm loc}(\mathbb{R}^N)$ for any $q\in [1, 2^{*}_{s})$. \end{theorem} Let $\mathbb{X}:=H^s(\mathbb{R}^N)\cap L^{p+1}(\mathbb{R}^N)$ equipped with the norm \[ \|u\|^2:= \iint_{\mathbb{R}^{2N}} \frac{|u(x)- u(y)|^2}{|x-y|^{N+2s}} \,dx\,dy +\int_{\mathbb{R}^N} |u|^{p+1} dx. \] Since we are interested in weak solutions to \eqref{P}, we look for critical points of the functional $\mathcal{I}:\mathbb{X}\to \mathbb{R}$ defined by \begin{align*} \mathcal{I}(u) &= \frac{1}{2}\iint_{\mathbb{R}^{2N}} \frac{|u(x)- u(y)|^2}{|x-y|^{N+2s}} \,dx\,dy + \frac{1}{2}\int_{\mathbb{R}^N} V(x)|u|^2 dx + \frac{1}{p+1}\int_{\mathbb{R}^N} a(x) |u|^{p+1} dx\\ &\quad - \int_{\mathbb{R}^N} fu \, dx. \end{align*} It is standard to check that $\mathcal{I}$ is well-defined in $\mathbb{X}$, $\mathcal{I} \in C^{1}(\mathbb{X}, \mathbb{R})$ and its differential is given by \begin{align*} \langle \mathcal{I}'(u), \varphi \rangle &= \iint_{\mathbb{R}^{2N}} \frac{(u(x)- u(y))(\varphi(x)- \varphi(y))}{|x-y|^{N+2s}} \,dx\,dy + \int_{\mathbb{R}^N} V(x)u \varphi dx\\ &\quad + \int_{\mathbb{R}^N} a(x) |u|^{p}\varphi dx - \int_{\mathbb{R}^N} f\varphi \, dx \end{align*} for any $u, \varphi \in \mathbb{X}$. We begin by proving the following Lemma to obtain the existence of a local minimum for $\mathcal{I}$. \begin{lemma}\label{lem1} Assume that {\rm (H1)--(H3)} hold. Then, there exist positive constants $\kappa, \rho$ and $L$ such that if $\|f\|_{L^2(\mathbb{R}^N)}< L$ then $\mathcal{I}(u) \geq \kappa$ whenever $\|u\|=\rho$. \end{lemma} \begin{proof} Let $u\in \mathbb{X}$. By using (H2), (H3), H\"older inequality and Young inequality we obtain \begin{equation} \label{t1} \begin{aligned} \mathcal{I}(u) &\geq \frac{1}{2}\iint_{\mathbb{R}^{2N}} \frac{|u(x)- u(y)|^2}{|x-y|^{N+2s}} \,dx\,dy - \frac{\|V\|_{L^{\infty}(\mathbb{R}^N)}}{2} \int_{\mathbb{R}^N} |u|^2 dx\\ &\quad + \frac{\alpha}{p+1} \int_{\mathbb{R}^N} |u|^{p+1} dx - \Big( \int_{\mathbb{R}^N} |f|^2 dx \Big)^{1/2} \Big( \int_{\mathbb{R}^N} |u|^2 dx \Big)^{1/2} \\ &\geq \frac{1}{2}\iint_{\mathbb{R}^{2N}} \frac{|u(x)- u(y)|^2}{|x-y|^{N+2s}} \,dx\,dy - \frac{\|V\|_{L^{\infty}(\mathbb{R}^N)}}{2} \int_{\mathbb{R}^N} |u|^2 dx\\ &\quad + \frac{\alpha}{p+1} \int_{\mathbb{R}^N} |u|^{p+1} dx -\frac{1}{2} \int_{\mathbb{R}^N} |f|^2 dx - \frac{1}{2} \int_{\mathbb{R}^N} |u|^2 dx \\ &= \frac{1}{2}\iint_{\mathbb{R}^{2N}} \frac{|u(x)- u(y)|^2}{|x-y|^{N+2s}} \,dx\,dy - \frac{1}{2} \left(\|V\|_{L^{\infty}(\mathbb{R}^N)}+1\right) \int_{\mathbb{R}^N} |u|^2 dx \\ &\quad + \frac{\alpha}{p+1} \int_{\mathbb{R}^N} |u|^{p+1} dx - \frac{1}{2}\int_{\mathbb{R}^N} |f|^2 dx. \end{aligned} \end{equation} Now, by interpolation, there exists $r=\frac{( 2_s^*-2)(p+1)}{2( 2_s^* - (p+1))} \in (0, 1)$ such that \begin{equation} \label{t2} \begin{aligned} \int_{\mathbb{R}^N} |u|^2 dx &\leq \Big( \int_{\mathbb{R}^N} |u|^{p+1} dx \Big)^{\frac{2r}{p+1}} \Big( \int_{\mathbb{R}^N} |u|^{ 2_s^*}dx \Big)^{\frac{2(1-r)}{ 2_s^*}} \\ &\leq S_{*}^{1-r} \Big( \int_{\mathbb{R}^N} |u|^{p+1} dx \Big)^{\frac{2r}{p+1}} \Big( \iint_{\mathbb{R}^{2N}} \frac{|u(x)- u(y)|^2}{|x-y|^{N+2s}} \,dx\,dy \Big)^{1-r} \\ &\leq \frac{\alpha}{(p+1) (\|V\|_{L^{\infty}(\mathbb{R}^N)}+1)} \int_{\mathbb{R}^N} |u|^{p+1} dx\\ &\quad + C_1 \Big( \iint_{\mathbb{R}^{2N}} \frac{|u(x)- u(y)|^2}{|x-y|^{N+2s}} \,dx\,dy \Big)^{\frac{(1-r)(p+1)}{p+1-2r}} \end{aligned} \end{equation} where we used Theorem \ref{Sobolev}, H\"older inequality and Young inequality. Therefore, putting together \eqref{t1} and \eqref{t2}, we obtain \begin{align*} \mathcal{I}(u) &\geq \frac{1}{2}\iint_{\mathbb{R}^{2N}} \frac{|u(x)- u(y)|^2}{|x-y|^{N+2s}} \,dx\,dy - \frac{1}{2} \frac{\alpha}{p+1}\int_{\mathbb{R}^N} |u|^{p+1} dx \\ &\quad - C_1 \Big( \iint_{\mathbb{R}^{2N}} \frac{|u(x)- u(y)|^2}{|x-y|^{N+2s}} \,dx\,dy \Big)^{\frac{(1-r)(p+1)}{p+1-2r}} + \frac{\alpha}{p+1} \int_{\mathbb{R}^N} |u|^{p+1} dx \\ &- \frac{1}{2}\int_{\mathbb{R}^N} |f|^2 dx\\ &= \frac{1}{2}\iint_{\mathbb{R}^{2N}} \frac{|u(x)- u(y)|^2}{|x-y|^{N+2s}} \,dx\,dy + \frac{1}{2} \frac{\alpha}{p+1}\int_{\mathbb{R}^N} |u|^{p+1} dx \\ &\quad - C_1 \left( \iint_{\mathbb{R}^{2N}} \frac{|u(x)- u(y)|^2}{|x-y|^{N+2s}} \,dx\,dy \right)^{\frac{(1-r)(p+1)}{p+1-2r}} - \frac{1}{2}\int_{\mathbb{R}^N} |f|^2 dx. \end{align*} Now, since $\frac{2(1-r)(p+1)}{p+1-2r}>2$, we can see that \begin{align*} &\frac{1}{2}\iint_{\mathbb{R}^{2N}} \frac{|u(x)- u(y)|^2}{|x-y|^{N+2s}} \,dx\,dy - C_1 \Big( \iint_{\mathbb{R}^{2N}} \frac{|u(x)- u(y)|^2}{|x-y|^{N+2s}} \,dx\,dy \Big)^{\frac{(1-r)(p+1)}{p+1-2r}} \\ & \geq \frac{1}{4} \iint_{\mathbb{R}^{2N}} \frac{|u(x)- u(y)|^2}{|x-y|^{N+2s}} \,dx\,dy \end{align*} for $\iint_{\mathbb{R}^{2N}} \frac{|u(x)- u(y)|^2}{|x-y|^{N+2s}} \,dx\,dy \leq \beta$, with $\beta$ sufficiently small. Hence, for ${\iint_{\mathbb{R}^{2N}} \frac{|u(x)- u(y)|^2}{|x-y|^{N+2s}} \,dx\,dy \leq \beta}$ and $\|u\|_{L^{p+1}(\mathbb{R}^N)}\leq 1$, by using the fact that for any $x\geq 0$ and $y\in [0, 1]$ it holds $(x+y)^2 \leq 2 x^2 + 2y^{p+1}$, we obtain \begin{equation}\label{eq1} \begin{aligned} \mathcal{I}(u) &\geq \frac{1}{4} \iint_{\mathbb{R}^{2N}} \frac{|u(x)- u(y)|^2}{|x-y|^{N+2s}} \,dx\,dy + \frac{\alpha}{2(p+1)} \int_{\mathbb{R}^N} |u|^{p+1} dx\\ &\quad - \frac{1}{2} \int_{\mathbb{R}^N} |f|^2 dx \\ &\geq \min\big\{\frac{1}{4}, \frac{\alpha}{2(p+1)} \big\} \Big(\iint_{\mathbb{R}^{2N}} \frac{|u(x)- u(y)|^2}{|x-y|^{N+2s}} \,dx\,dy - \int_{\mathbb{R}^N} |u|^{p+1} dx\Big) \\ &\quad - \frac{1}{2} \int_{\mathbb{R}^N} |f|^2 dx \\ &\geq C_{2} \|u\|^2- \frac{1}{2} \|f\|_{L^2(\mathbb{R}^N)}^2. \end{aligned} \end{equation} Taking $\rho= \min\{\beta, 1\}$ and $\|f\|_{L^2(\mathbb{R}^N)}^2= C_{2} \rho^2$, then for $\|u\|=\rho$ we obtain $\mathcal{I}(u)\geq \kappa:=\frac{1}{2}C_{2}\rho^2$. \end{proof} Let us define \begin{equation}\label{eq2} m:=\inf_{u\in B_{\rho}} \mathcal{I}(u), \end{equation} where $B_{\rho}=\{u\in \mathbb{X} : \|u\|<\rho\}$ and $\rho$ is defined in Lemma \ref{lem1}. \begin{lemma}\label{lem2} Under assumptions {\rm (H1)--(H3)} it holds $-\infty0$. Then we have \begin{align*} \mathcal{I}(t\psi) &= \frac{t^2}{2}\iint_{\mathbb{R}^{2N}} \frac{|\psi(x) - \psi(y)|^2}{|x-y|^{N+2s}} \,dx\,dy + \frac{t^2}{2}\int_{\mathbb{R}^N} V(x)|\psi|^2 dx \\ &\quad + \frac{t^{p+1}}{p+1}\int_{\mathbb{R}^N} a(x) |\psi|^{p+1} dx - \int_{\mathbb{R}^N} f\psi \, dx<0 \end{align*} for $t$ sufficiently small. As a consequence $m<0$. It is clear that $m>-\infty$ in view of \eqref{eq1}. \end{proof} \section{Existence of a first solution} \label{Sect3} \noindent In this section we study of the existence and uniqueness of solutions to \eqref{P}. \begin{theorem} Assume that {\rm (H1)--(H3)} hold. Then there exists $u_1\in \mathbb{X}$ which is a nonnegative solution to \eqref{P}. Moreover $u_1$ converges to zero in $\mathbb{X}$ as $\|f\|_{L^2(\mathbb{R}^N)}\to 0$. \end{theorem} \begin{proof} Let $\{u_n\}\subset \mathbb{X}$ be a minimizing sequence of \eqref{eq2}. Since $\{u_n\}$ is bounded in $\mathbb{X}$, we may assume, up to a subsequence, that \begin{gather*} u_n\rightharpoonup u_1 \quad\text{ in } \mathbb{X}, \\ u_n \to u_1 \quad \text{in } L^{q}_{\rm loc}(\mathbb{R}^N), \forall q\in [1, 2_s^*), \\ u_n \to u_1 \quad \text{a.e. in } \mathbb{R}^N. \end{gather*} Our aim is to prove that $u_n\to u_1$ in $\mathbb{X}$. Set $v_n= u_n-u_1$. Let us compute $\mathcal{I}(u_n)$: \begin{equation} \label{eq3} \begin{aligned} \mathcal{I}(u_n) &=\mathcal{I}(v_n+ u_1) \\ &= \frac{1}{2}\iint_{\mathbb{R}^{2N}} \frac{|v_n(x)- v_n(y)|^2}{|x-y|^{N+2s}} \,dx\,dy + \frac{1}{2}\iint_{\mathbb{R}^{2N}} \frac{|u_1(x)- u_1(y)|^2}{|x-y|^{n+2s}} \,dx\,dy \\ &\quad + \iint_{\mathbb{R}^{2N}} \frac{(v_n(x)- v_n(y))(u_1(x)- u_1(y))} {|x-y|^{N+2s}} \,dx\,dy + \frac{1}{2}\int_{\mathbb{R}^N} V(x) |v_n|^2 dx \\ &\quad + \frac{1}{2}\int_{\mathbb{R}^N} V(x) |u_1|^2 dx + \int_{\mathbb{R}^N} V(x) v_nu_1 dx - \int_{\mathbb{R}^N} fv_n dx - \int_{\mathbb{R}^N} fu_1 \, dx \\ &\quad + \frac{1}{p+1} \int_{\mathbb{R}^N} a(x) (|u_n|^{p+1} - (|u_1|^{p+1}+|v_n|^{p+1})) dx \\ &\quad + \frac{1}{p+1}\int_{\mathbb{R}^N} a(x) |u_1|^{p+1}dx + \frac{1}{p+1} \int_{\mathbb{R}^N} a(x) |v_n|^{p+1} dx \\ &= \mathcal{I}(u_1) + \frac{1}{2}\iint_{\mathbb{R}^{2N}} \frac{|v_n(x)- v_n(y)|^2}{|x-y|^{N+2s}} \,dx\,dy + \frac{1}{2}\int_{\mathbb{R}^N} V(x) |v_n|^2 dx \\ &\quad + \frac{1}{p+1} \int_{\mathbb{R}^N} a(x) |v_n|^{p+1} dx - \int_{\mathbb{R}^N} fv_n dx \\ &\quad + \iint_{\mathbb{R}^{2N}} \frac{(v_n(x)- v_n(y))(u_1(x)- u_1(y))} {|x-y|^{N+2s}} \,dx\,dy + \int_{\mathbb{R}^N} V(x) v_nu_1 dx \\ &\quad +\frac{1}{p+1} \int_{\mathbb{R}^N} a(x) (|u_n|^{p+1} - (|u_1|^{p+1}+|v_n|^{p+1})) dx. \end{aligned} \end{equation} From the fact that $v_n\rightharpoonup 0$ in $\mathbb{X}$ we infer that \begin{equation} \label{eq4} \begin{gathered} \iint_{\mathbb{R}^{2N}} \frac{(v_n(x)- v_n(y))(u_1(x)- u_1(y))}{|x-y|^{N+2s}} \,dx\,dy \to 0 \\ \int_{\mathbb{R}^N} fv_n dx \to 0 \quad\text{and}\quad \int_{\mathbb{R}^N} V(x) v_nu_1 dx\to 0. \end{gathered} \end{equation} Moreover, taking into account that \[ -c|x|^{p} |y| \leq |x+y|^{p+1}-(|x|^{p+1}+ |y|^{p+1}) \leq c |x|^{p}|y| \] holds for any $x, y\in \mathbb{R}$ with $c>0$ (independent of $x$ and $y$), we obtain \begin{equation} \label{eq5} \begin{aligned} &\left|\frac{1}{p+1} \int_{\mathbb{R}^N} a(x) (|u_n|^{p+1}- (|u_1|^{p+1}+|v_n|^{p+1})) dx \right| \\ &\leq \frac{c}{p+1} \int_{\mathbb{R}^N} a(x) |v_n||u_1|^{p+1} dx\to 0. \end{aligned} \end{equation} Let us note that \begin{equation} \label{t3} \begin{aligned} \rho^2&>\iint_{\mathbb{R}^{2N}} \frac{|u_n(x)- u_n(y)|^2}{|x-y|^{N+2s}} \,dx\,dy \\ &= \iint_{\mathbb{R}^{2N}} \frac{|v_n(x)- v_n(y)|^2}{|x-y|^{N+2s}} \,dx\,dy + \iint_{\mathbb{R}^{2N}} \frac{|u_1(x)- u_1(y)|^2}{|x-y|^{N+2s}} \,dx\,dy \\ &\quad + 2 \iint_{\mathbb{R}^{2N}} \frac{(v_n(x)- v_n(y))(u_1(x)- u_1(y))} {|x-y|^{N+2s}} \,dx\,dy\,. \end{aligned} \end{equation} Using \eqref{eq4} we can see that \[ \iint_{\mathbb{R}^{2N}} \frac{|v_n(x)- v_n(y)|^2}{|x-y|^{n+2s}} \,dx\,dy \leq \rho^2 \] for $n$ large enough. Then, taking the limit in \eqref{eq3} and by using \eqref{eq4} and \eqref{eq5}, we obtain \begin{equation} \label{t4} \begin{aligned} m&= \mathcal{I}(u_1)+ \lim_{n\to \infty} \Bigl\{ \frac{1}{2}\iint_{\mathbb{R}^{2N}} \frac{|v_n(x)- v_n(y)|^2}{|x-y|^{N+2s}} \,dx\,dy + \frac{1}{2}\int_{\mathbb{R}^N} V(x) |v_n|^2 dx\\ &\quad + \frac{1}{p+1} \int_{\mathbb{R}^N} a(x) |v_n|^{p+1} dx\Bigr\} \\ &\geq \mathcal{I}(u_1) + \lim_{n\to \infty}\Big\{ \frac{1}{4} \iint_{\mathbb{R}^{2N}} \frac{|v_n(x)- v_n(y)|^2}{|x-y|^{N+2s}} \,dx\,dy \\ &\quad + \frac{\alpha}{2(p+1)} \int_{\mathbb{R}^N} a(x) |v_n|^{p+1} dx\Big\} \geq m \end{aligned} \end{equation} where in the last inequality we have estimated as in Lemma \ref{lem1}. Then, \eqref{t3} and \eqref{t4} yield $v_n\to 0$ in $\mathbb{X}$, $0>m=\mathcal{I}(u_1)$, $u_1\in B_{\rho}$ and $\mathcal{I}'(u_1)=0$. Thus $u_1$ is a weak solution to \eqref{P}. To prove that $u_1$ is a nonnegative solution to \eqref{P}, note that $m\leq \mathcal{I}(|u_n|)\leq \mathcal{I}(u_n)$, thus $\mathcal{I}(|u_n|)\to m$, and we can choose $u_n\geq 0$. As before, up to a subsequence $u_n\to u_1$ a.e. in $\mathbb{R}^N$, $u_n\to u_1$ in $\mathbb{X}$ and $u_1$ is a nonnegative solution to \eqref{P}. Let $\{f_n\}\subset L^2(\mathbb{R}^N)$ be such that $\|f_n\|_{L^2(\mathbb{R}^N)}\to 0$ and let $u_{f_n}$ be a solution to \eqref{P}. Now we aim to prove that $u_{f_n}\to 0$ in $\mathbb{X}$. Since $u_{f_n}$ is a solution to \eqref{P}, we obtain \begin{align*} \langle \mathcal{I}'(u_{f_n}), u_{f_n} \rangle =0, \end{align*} that is, \begin{equation} \label{eq6} \begin{aligned} &\iint_{\mathbb{R}^{2N}} \frac{|u_{f_n}(x)-u_{f_n}(y)|^2}{|x-y|^{N+2s}} \,dx\,dy + \int_{\mathbb{R}^N} V(x)|u_{f_n}|^2 dx + \int_{\mathbb{R}^N} a(x) |u_{f_n}|^{p+1} dx \\ &=\int_{\mathbb{R}^N} f_nu_{f_n} dx. \end{aligned} \end{equation} Recalling that $\mathcal{I}(u_{f_n})<0$, by using \eqref{eq6} and H\"older inequality we deduce \begin{align*} 0&>\mathcal{I}(u_{f_n})\\ &= \frac{1}{2} \Big( - \int_{\mathbb{R}^N} a(x) |u_{f_n}|^{p+1} dx + \int_{\mathbb{R}^N} f_nu_{f_n} dx \Big) + \frac{1}{p+1} \int_{\mathbb{R}^N} a(x) |u_{f_n}|^{p+1} dx \\ &\quad- \int_{\mathbb{R}^N} f_nu_{f_n} dx \\ &= \frac{1-p}{2(p+1)} \int_{\mathbb{R}^N} a(x) |u_{f_n}|^{p+1} dx - \frac{1}{2}\int_{\mathbb{R}^N} f_nu_{f_n} dx \\ &\geq \frac{1-p}{2(p+1)} \int_{\mathbb{R}^N} a(x) |u_{f_n}|^{p+1} dx - \frac{1}{2}\int_{\mathbb{R}^N} f_nu_{f_n} dx\\ &\geq \frac{(1-p)\alpha}{2(p+1)} \|u_{f_n}\|_{L^{p+1}(\mathbb{R}^N)}^{p+1} - \frac{1}{2} \|f_n\|_{L^2(\mathbb{R}^N)} \|u_{f_n}\|_{L^2(\mathbb{R}^N)}. \end{align*} Observing that $p\in (0, 1)$ and $\alpha>0$, we obtain \begin{align*} 0&<\frac{(1-p)\alpha}{2(p+1)} \|u_{f_n}\|_{L^{p+1}(\mathbb{R}^N)}^{p+1} \\ &\leq \frac{1}{2} \|f_n\|_{L^2(\mathbb{R}^N)} \|u_{f_n}\|_{L^2(\mathbb{R}^N)} \\ &\leq c \|f_n\|_{L^2(\mathbb{R}^N)}\to 0, \end{align*} thus $u_{f_n}\to 0$ in $L^{p+1}(\mathbb{R}^N)$. Now, by using \eqref{eq6}, H\"older inequality and Sobolev inequality, there exists $r\in (0, 1)$ (defined in Lemma \ref{lem1}) such that \begin{align*} &\iint_{\mathbb{R}^{2N}} \frac{|u_{f_n}(x)-u_{f_n}(y)|^2}{|x-y|^{N+2s}} \,dx\,dy \\ &=- \int_{\mathbb{R}^N} V(x)|u_{f_n}|^2 dx - \int_{\mathbb{R}^N} a(x) |u_{f_n}|^{p+1} dx +\int_{\mathbb{R}^N} f_nu_{f_n} dx \\ &\leq \|V\|_{L^{\infty}(\mathbb{R}^N)} \|u_{f_n}\|_{L^2(\mathbb{R}^N)}^2 + \|a\|_{L^{\infty}(\mathbb{R}^N)} \|u_{f_n}\|_{L^{p+1}(\mathbb{R}^N)}^{p+1} + \|f_n\|_{L^2(\mathbb{R}^N)} \|u_{f_n}\|_{L^2(\mathbb{R}^N)}\\ &\leq \|V\|_{L^{\infty}(\mathbb{R}^N)} \Big( \|u_{f_n}\|_{L^{p+1}(\mathbb{R}^N)}^{r} \|u_{f_n}\|_{L^{ 2_s^*}(\mathbb{R}^N)}^{1-r} \Big) + \|a\|_{L^{\infty}(\mathbb{R}^N)} \|u_{f_n}\|_{L^{p+1}(\mathbb{R}^N)}^{p+1} \\ &\quad + \|f_n\|_{L^2(\mathbb{R}^N)} \|u_{f_n}\|_{L^2(\mathbb{R}^N)}\\ &\leq C \|V\|_{L^{\infty}(\mathbb{R}^N)} \|u_{f_n}\|_{L^{p+1}(\mathbb{R}^N)}^{r} + \|a\|_{L^{\infty}(\mathbb{R}^N)} \|u_{f_n}\|_{L^{p+1}(\mathbb{R}^N)}^{p+1} \\ &\quad + \|f_n\|_{L^2(\mathbb{R}^N)} \|u_{f_n}\|_{L^2(\mathbb{R}^N)}\to 0. \end{align*} Combining this with $u_{f_n}\to 0$ in $L^{p+1}(\mathbb{R}^N)$, we deduce the thesis. \end{proof} Now we recall the notion of supersolution and subsolution to \eqref{P}: \begin{definition} \label{def3.2}\rm We say that $\overline{u}\in \mathbb{X}$ is a weak supersolution to \eqref{P} if \begin{equation} \label{supersol} \begin{aligned} &\iint_{\mathbb{R}^{2N}} \frac{(\overline{u}(x) - \overline{u}(y)) (\varphi(x) - \varphi(y))}{|x-y|^{N+2s}} \,dx\,dy + \int_{\mathbb{R}^N} V(x) \overline{u} \varphi \, dx\\ & + \int_{\mathbb{R}^N} a(x) |\overline{u}|^{p} \operatorname{sgn}(\overline{u}) \varphi \, dx \\ &\leq \int_{\mathbb{R}^N} f\varphi\, dx \end{aligned} \end{equation} for any $\varphi\in \mathbb{X}$ such that $\varphi \geq 0$ a.e. in $\mathbb{R}^N$. \end{definition} \begin{definition} \label{def3.3}\rm We say that $\underline{u}\in \mathbb{X}$ is a weak subsolution to \eqref{P} if \begin{equation} \label{subsol} \begin{aligned} &\iint_{\mathbb{R}^{2N}} \frac{(\underline{u}(x) - \underline{u}(y)) (\varphi(x) - \varphi(y))}{|x-y|^{N+2s}} \,dx\,dy + \int_{\mathbb{R}^N} V(x) \underline{u} \varphi \, dx \\ &+ \int_{\mathbb{R}^N} a(x) |\underline{u}|^{p} \operatorname{sgn}(\underline{u}) \varphi \, dx \\ &\leq \int_{\mathbb{R}^N} f\varphi\, dx \end{aligned} \end{equation} for any $\varphi\in \mathbb{X}$ such that $\varphi \geq 0$ a.e. in $\mathbb{R}^N$. \end{definition} \begin{theorem}\label{thmsupersub} Assume that {\rm (H1)--(H3)} and \eqref{h4} hold. Let $\underline{u}\in \mathbb{X}$ be a subsolution to \eqref{P} and $\overline{u}\in \mathbb{X}$ be a supersolution to \eqref{P}. Then $\underline{u}\leq \overline{u}$ a.e. in $\mathbb{R}^N$. \end{theorem} \begin{proof} From \eqref{supersol} and \eqref{subsol} it follows that \begin{equation} \label{eq7} \begin{aligned} &\iint_{\mathbb{R}^{2N}} \frac{((\underline{u}- \overline{u})(x) - (\underline{u}- \overline{u})(y)) (\varphi(x)- \varphi(y))}{|x-y|^{N+2s}} \,dx\,dy + \int_{\mathbb{R}^N} V(x) (\underline{u}- \overline{u})\varphi \, dx \\ &+ \int_{\mathbb{R}^N} a(x) (|\underline{u}|^{p} \operatorname{sgn}(\underline{u}) - |\overline{u}|^{p} \operatorname{sgn}(\overline{u}))\varphi \, dx \leq 0. \end{aligned} \end{equation} Assume $\underline{u}\not\leq \overline{u}$ and let $\varphi:= (\underline{u}- \overline{u})^{+}$ in \eqref{eq7}, then we have \begin{align*} 0&\leq \iint_{\mathbb{R}^{2N}} \frac{|((\underline{u}- \overline{u})(x) - (\underline{u}- \overline{u})(y))^{+}|^2}{|x- y|^{N+2s}} \,dx\,dy + \int_{\mathbb{R}^N} V(x)|(\underline{u}- \overline{u})^{+}|^2 dx \\ &\leq - \int_{\mathbb{R}^N} a(x) (|\underline{u}|^{p} \operatorname{sgn}(\underline{u}) - |\overline{u}|^{p} \operatorname{sgn}(\overline{u})) (\underline{u}- \overline{u})^{+}<0, \end{align*} and this gives a contradiction. Thus we have $\underline{u} \leq \overline{u}$ a.e. in $\mathbb{R}^N$. \end{proof} At this point we are ready to prove that the problem \eqref{P} admits a unique weak solution. \begin{theorem} \label{thm3.5} Under assumptions {\rm (H1)--(H3)} and \eqref{h4}, problem \eqref{P} admits a unique solution. \end{theorem} \begin{proof} Let $u_1$ and $u_2$ be two solutions to \eqref{P}. Then by Theorem \ref{thmsupersub} follows that $u_1\leq u_2$ and $u_2\leq u_1$, that is $u_1= u_2$ a.e. in $\mathbb{R}^N$. \end{proof} \section{Existence of a second solution} In this section we show the existence of a second solution to \eqref{P} under assumption \eqref{h5}. \begin{lemma}\label{lem41} Under assumption \eqref{h5}, there exists $\varphi_{0}\in \mathbb{X}\setminus B_{\rho}$ such that $\mathcal{I}(\varphi_{0})<0$. \end{lemma} \begin{proof} Let $\varphi\in C^{\infty}_{c}(\mathbb{R}^N)$ satisfying \eqref{h5}. Then, as $t\to +\infty$ \begin{align*} \mathcal{I}(t\varphi) &= \frac{t^2}{2}\iint_{\mathbb{R}^{2N}} \frac{|\varphi(x) - \varphi(y)|^2}{|x-y|^{N+2s}} \,dx\,dy + \frac{t^2}{2}\int_{\mathbb{R}^N} V(x)|\varphi|^2 dx\\ &\quad + \frac{t^{p+1}}{p+1}\int_{\mathbb{R}^N} a(x) |\varphi|^{p+1} dx - \int_{\mathbb{R}^N} f\varphi \, dx \to -\infty. \end{align*} Thus, choosing $t_{0}$ sufficiently large such that $\|t_{0}\varphi\|>\rho$ and $\mathcal{I}(t_{0}\varphi)<0$, we can take $\varphi_{0}= t_{0}\varphi$ to complete the proof. \end{proof} Let us consider the problem \begin{equation}\label{eq10} M=\inf_{\gamma \in \Gamma} \max_{t\in [0, 1]} \mathcal{I}(\gamma(t)) \end{equation} where \[ \Gamma= \{\gamma \in C([0, 1], \mathbb{X}) : \gamma(0)=0, \, \gamma(1)=\varphi_{0}\} \] with $\varphi_{0}$ as in Lemma \ref{lem41}. \begin{lemma}\label{lem42} Under assumption {\rm (H1)--(H3)} and \eqref{h5} it results $M>0$. \end{lemma} \begin{proof} Let $\gamma \in \Gamma$, then $\gamma(0)=0\in B_{\rho}$ and $\gamma(1)=\varphi_{0}\in \mathbb{X}\setminus B_{\rho}$. Then, there exists $\tau\in (0, 1)$ such that $\|\gamma(\tau)\|=\rho$, and applying Lemma \ref{lem1} we have $\mathcal{I}(\gamma(\tau))\geq \kappa>0$, and thus $M>0$. \end{proof} By applying Ekeland's variational principle to \eqref{eq10} there exists $\{u_n\}\subset \mathbb{X}$ such that $\mathcal{I}(u_n)\to M$ and $\mathcal{I}'(u_n)\to 0$ in $\mathbb{X}'$. In this case $\{u_n\}$ is called a (PS) sequence of the functional $\mathcal{I}$ at level $M$. \begin{theorem}\label{thm43} Under assumptions {\rm (H1)--(H3)} and \eqref{h5}, $\{u_n\}$ is bounded in $\mathbb{X}$. \end{theorem} \begin{proof} Firstly note that it is possible to find a positive constant $b$ such that, for $n$ large, $$ |\langle \mathcal{I}'(u_n), u_n \rangle |\leq \|u_n\| \quad \text{and}\quad |\mathcal{I}(u_n)|\mathcal{I}(u_n) \\ &\geq \frac{1}{2} \iint_{\mathbb{R}^{2N}} \frac{|u_n(x)- u_n(y)|^2}{|x-y|^{N+2s}} \,dx\,dy - \frac{\|V\|_{L^{\infty}(\mathbb{R}^N)}}{2} \int_{\mathbb{R}^N} |u_n|^2 dx \\ &\quad + \frac{\alpha}{p+1} \int_{\mathbb{R}^N} |u_n|^{p+1} dx - \Big( \int_{\mathbb{R}^N} |f|^2 dx \Big)^{1/2} \Big( \int_{\mathbb{R}^N} |u_n|^2 dx \Big)^{1/2} \\ &\geq \frac{1}{2} \iint_{\mathbb{R}^{2N}} \frac{|u_n(x)- u_n(y)|^2}{|x-y|^{N+2s}} \,dx\,dy - \frac{\|V\|_{L^{\infty}(\mathbb{R}^N)} +1}{2} \int_{\mathbb{R}^N} |u_n|^2 dx \\ &\quad+ \frac{\alpha}{p+1} \int_{\mathbb{R}^N} |u_n|^{p+1} dx - \frac{1}{2} \int_{\mathbb{R}^N} |f|^2 dx \\ &\geq \frac{1}{2} \iint_{\mathbb{R}^{2N}} \frac{|u_n(x)- u_n(y)|^2}{|x-y|^{N+2s}} \,dx\,dy + \frac{\alpha}{p+1} \int_{\mathbb{R}^N} |u_n|^{p+1} dx - \frac{1}{2} \int_{\mathbb{R}^N} |f|^2 dx\\ &\quad- C_{3} \Big( \iint_{\mathbb{R}^{2N}} \frac{|u_n(x) - u_n(y)|^2}{|x-y|^{N+2s}} \,dx\,dy\Big)^{\frac{2(1-r)}{\beta}+r} \\ &\quad -C_{4} \Big( \iint_{\mathbb{R}^{2N}} \frac{|u_n(x)- u_n(y)|^2}{|x-y|^{N+2s}} \,dx\,dy\Big)^{r} \\ &\geq \frac{1}{2} \iint_{\mathbb{R}^{2N}} \frac{|u_n(x) - u_n(y)|^2}{|x-y|^{N+2s}} \,dx\,dy + \frac{\alpha}{p+1} \int_{\mathbb{R}^N} |u_n|^{p+1} dx - \frac{1}{2} \int_{\mathbb{R}^N} |f|^2 dx\\ &\quad - C_{3} \Big( \iint_{\mathbb{R}^{2N}} \frac{|u_n(x) - u_n(y)|^2}{|x-y|^{N+2s}} \,dx\,dy\Big)^{\frac{2(1-r)}{\beta}+r}\\ &\quad -C_{4} \Big( \iint_{\mathbb{R}^{2N}} \frac{|u_n(x)- u_n(y)|^2}{|x-y|^{N+2s}} \,dx\,dy\Big)^{r}. \end{align*} Note that since $r\in (0, 1)$, $\beta \in (1, p+1)$ and $p\in (0, 1)$ we can infer that \begin{align*} 0 < \frac{2(1-r)}{\beta}+r <2, \end{align*} from which it follows that $\{u_n\}$ is bounded in $H^s(\mathbb{R}^N)$, in contrast with \eqref{eq11}. Thus, $\{u_n\}$ is bounded in $L^{p+1}(\mathbb{R}^N)$. From this we deduce that \begin{equation} \label{eq12} \begin{aligned} \int_{\mathbb{R}^N} |u_n|^2 dx &\leq \Big( \int_{\mathbb{R}^N} |u_n|^{p+1} dx \Big)^{\frac{2(1-r)}{p+1}} \Big( \int_{\mathbb{R}^N} |u_n|^{ 2_s^*} dx \Big)^{\frac{2r}{ 2_s^*}} \\ &\leq S_{*}^{r} \Big( \int_{\mathbb{R}^N} |u_n|^{p+1} dx \Big)^{\frac{2(1-r)}{p+1}} \Big( \iint_{\mathbb{R}^{2N}} \frac{|u_n(x)- u_n(y)|^2}{|x-y|^{N+2s}} \,dx\,dy\Big)^{r} \\ &\leq C \Big( \iint_{\mathbb{R}^{2N}} \frac{|u_n(x)- u_n(y)|^2}{|x-y|^{N+2s}} \,dx\,dy\Big)^{r}. \end{aligned} \end{equation} Therefore, by \eqref{eq12} we obtain \begin{align*} b&>\mathcal{I}(u_n) \\ &\geq \frac{1}{2} \iint_{\mathbb{R}^{2N}} \frac{|u_n(x)- u_n(y)|^2}{|x-y|^{N+2s}} \,dx\,dy - C \int_{\mathbb{R}^N} |u_n|^2 dx + \frac{\alpha}{p+1} \int_{\mathbb{R}^N} |u_n|^{p+1} dx \\ &\quad - \frac{1}{2} \int_{\mathbb{R}^N} |f|^2 dx \\ &\geq \frac{1}{2} \iint_{\mathbb{R}^{2N}} \frac{|u_n(x)- u_n(y)|^2}{|x-y|^{N+2s}} \,dx\,dy - C \Big( \iint_{\mathbb{R}^{2N}} \frac{|u_n(x)- u_n(y)|^2}{|x-y|^{N+2s}} \,dx\,dy\Big)^{r}\\ &\quad - \frac{1}{2} \int_{\mathbb{R}^N} |f|^2 dx. \end{align*} This implies that $\{u_n\}$ is bounded in $H^s(\mathbb{R}^N)$. \end{proof} \begin{theorem}\label{thm44} Under assumptions {\rm (H1)--(H3)} and \eqref{h5}, problem \eqref{P} admits a second solution $u_2\in \mathbb{X}$. \end{theorem} \begin{proof} By Theorem \ref{thm43} $\{u_n\}\subset \mathbb{X}$ is bounded, thus \begin{equation} \label{conv} \begin{aligned} & u_n\rightharpoonup u_2 \quad \text{in } \mathbb{X}, \\ & u_n\to u_2 \quad \text{in } L^{q}_{\rm loc}(\mathbb{R}^N), \quad \forall q\in [1, 2^{*}_{s}), \\ & u_n\to u_2 \quad \text{ a. e. in } \mathbb{R}^N. \end{aligned} \end{equation} Let $\varphi \in C^{\infty}_{c}(\mathbb{R}^N)$, then we can infer that \begin{align*} &\lim_{n\to \infty} \Big\{\iint_{\mathbb{R}^{2N}} \frac{(u_n(x)- u_n(y))(\varphi(x)- \varphi(y))}{|x-y|^{N+2s}} \,dx\,dy + \int_{\mathbb{R}^N} V(x) u_n\varphi \, dx \Big\}\\ &= \iint_{\mathbb{R}^{2N}} \frac{(u_2(x)- u_2(y))(\varphi(x)- \varphi(y))} {|x-y|^{N+2s}} \,dx\,dy + \int_{\mathbb{R}^N} V(x) u_2\varphi \, dx. \end{align*} From $u_n\to u_2$ in $L^{p+1}(\operatorname{supp} \varphi)$, it follows that there exist a subsequence still denoted by $\{u_n\}$ and a function $w\in L^{p+1}(\operatorname{supp}\varphi)$ such that $|u_n|\leq |w|$ and \begin{gather*} |a| |u_n|^{p} |\varphi| \leq \|a\|_{L^{\infty}(\mathbb{R}^N)} |w|^{p} |\varphi| \in L^{1}(\mathbb{R}^N),\\ a |u_n|^{p} \operatorname{sgn}(u_n) \varphi \to a |u_2|^{p} \operatorname{sgn}(u_2) \varphi \quad \text{ a.e. in } \mathbb{R}^N. \end{gather*} By using the Dominated convergence theorem we obtain \[ \lim_{n\to \infty} \int_{\mathbb{R}^N} a(x) |u_n|^{p} \operatorname{sgn}(u_n) \varphi \, dx = \int_{\mathbb{R}^N} a(x) |u_2|^{p} \operatorname{sgn}(u_2)\varphi \, dx. \] Therefore, for every $\varphi \in C^{\infty}_{c}(\mathbb{R}^N)$ \begin{equation}\label{eq13} \langle \mathcal{I}'(u_n), \varphi \rangle \to \langle \mathcal{I}'(u_2), \varphi \rangle. \end{equation} Since $\{u_n\}$ is a $(PS)$ sequence for $\mathcal{I}$ on $\mathbb{X}$, we have $\langle \mathcal{I}'(u_n), \varphi \rangle \to 0$, that combined with \eqref{eq13} gives $\langle \mathcal{I}'(u_2), \varphi \rangle$=0, hence $u_2$ is a weak solution to \eqref{P}. Now we prove that $u_2\neq u_1$, where $u_1$ is the first solution to \eqref{P}. Since $u_n\rightharpoonup u_2$ in $\mathbb{X}$, then up to a subsequence $\|u_2\|\leq \lim_{n\to \infty} \|u_n\|$. We distinguish two cases: \smallskip \noindent\textbf{Case 1: compactness.} We show that $u_n\to u_2$ in $\mathbb{X}$. \\ By Theorem \eqref{thm43} $\{u_n\}$ is bounded in $\mathbb{X}$, so up to a subsequence we can say that \begin{equation} \label{D1} \begin{aligned} &\lim_{n\to \infty} \Big[\iint_{\mathbb{R}^{2N}} \frac{|u_n(x)- u_n(y)|^2}{|x-y|^{N+2s}} \,dx\,dy + \int_{\mathbb{R}^N} |u_n|^{p+1} dx \Big]\\ &= \iint_{\mathbb{R}^{2N}} \frac{|u_2(x)- u_2(y)|^2}{|x-y|^{N+2s}} \,dx\,dy + \int_{\mathbb{R}^N} |u_2|^{p+1} dx \end{aligned} \end{equation} from which it follows that \begin{align*} \limsup_{n\to \infty} \int_{\mathbb{R}^N} |u_n|^{p+1} dx &= \iint_{\mathbb{R}^{2N}} \frac{|u_2(x)- u_2(y)|^2}{|x-y|^{N+2s}} \,dx\,dy + \int_{\mathbb{R}^N} |u_2|^{p+1} dx \\ &\quad - \limsup_{n\to \infty} \iint_{\mathbb{R}^{2N}} \frac{|u_n(x)- u_n(y)|^2}{|x-y|^{N+2s}} \,dx\,dy\\ &\leq \iint_{\mathbb{R}^{2N}} \frac{|u_2(x)- u_2(y)|^2}{|x-y|^{N+2s}} \,dx\,dy + \int_{\mathbb{R}^N} |u_2|^{p+1} dx \\ &\quad- \liminf_{n\to \infty} \iint_{\mathbb{R}^{2N}} \frac{|u_n(x)- u_n(y)|^2}{|x-y|^{N+2s}} \,dx\,dy. \end{align*} We also know that \begin{gather*} \iint_{\mathbb{R}^{2N}} \frac{|u_2(x)- u_2(y)|^2}{|x-y|^{N+2s}} \,dx\,dy \leq \liminf_{n\to \infty} \iint_{\mathbb{R}^{2N}} \frac{|u_n(x) - u_n(y)|^2}{|x-y|^{N+2s}} \,dx\,dy, \\ \int_{\mathbb{R}^N} |u_2|^{p+1} dx \leq \liminf_{n\to \infty} \int_{\mathbb{R}^N} |u_n|^{p+1} dx; \end{gather*} therefore \[ \int_{\mathbb{R}^N} |u_2|^{p+1} dx \leq \liminf_{n\to \infty} \int_{\mathbb{R}^N} |u_n|^{p+1} dx \leq \limsup_{n\to \infty} \int_{\mathbb{R}^N} |u_n|^{p+1} dx \leq \int_{\mathbb{R}^N} |u_2|^{p+1} dx \] so we deduce that \begin{equation}\label{D2} u_n \to u_2 \quad \text{ in } L^{p+1}(\mathbb{R}^N). \end{equation} Putting together \eqref{D1} and \eqref{D2} we obtain \[ \lim_{n\to \infty} \iint_{\mathbb{R}^{2N}} \frac{|u_n(x)- u_n(y)|^2}{|x-y|^{N+2s}} \,dx\,dy = \iint_{\mathbb{R}^{2N}} \frac{|u_2(x)- u_2(y)|^2}{|x-y|^{N+2s}} \,dx\,dy. \] Now, \begin{equation} \begin{split}\label{D3} &\iint_{\mathbb{R}^{2N}} \frac{|(u_n-u_2)(x) - (u_n-u_2)(y)|^2}{|x-y|^{N+2s}} \,dx\,dy \\ &= \iint_{\mathbb{R}^{2N}} \frac{|[u_n(x) - u_n(y)]-[u_2(x)-u_2(y)]|^2}{|x-y|^{N+2s}}\,dx\,dy\\ &=\iint_{\mathbb{R}^{2N}} \frac{|u_n(x) - u_n(y)|^2}{|x-y|^{N+2s}}\,dx\,dy + \iint_{\mathbb{R}^{2N}} \frac{|u_2(x)-u_2(y)|^2}{|x-y|^{N+2s}}\,dx\,dy\\ &\quad - 2\iint_{\mathbb{R}^{2N}} \frac{[u_n(x) - u_n(y)][u_2(x)-u_2(y)]}{|x-y|^{N+2s}}\,dx\,dy \end{split} \end{equation} and by using \eqref{conv} we obtain \begin{equation}\label{D4} \begin{aligned} &\lim_{n\to \infty} \iint_{\mathbb{R}^{2N}} \frac{[u_n(x) - u_n(y)][u_2(x)-u_2(y)]}{|x-y|^{N+2s}}\,dx\,dy \\ &=\iint_{\mathbb{R}^{2N}} \frac{|u_2(x)-u_2(y)|^2}{|x-y|^{N+2s}} \,dx\,dy. \end{aligned} \end{equation} Combining \eqref{D3} and \eqref{D4} we have \begin{equation}\label{D5} \lim_{n\to \infty} \iint_{\mathbb{R}^{2N}} \frac{|(u_n-u_2)(x) - (u_n-u_2)(y)|^2}{|x-y|^{N+2s}} \,dx\,dy =0. \end{equation} By \eqref{D2} and \eqref{D5} follows that $u_n\to u_2$ in $\mathbb{X}$. \smallskip \noindent\textbf{Case 2: dichotomy.} Assume that $\|u_2\|< \lim_{n\to \infty} \|u_n\|$. Let $v_n(x)= u_n(x)- u_2(x)$ be such that $v_n\rightharpoonup 0$ in $\mathbb{X}$. \smallskip \noindent \textbf{Step 1:} We show that there exists a sequence $\{y_n\}\subset \mathbb{R}^N$ such that \begin{equation}\label{B} v_n(\cdot + y_n)\rightharpoonup v_1\neq 0 \quad \text{in } \mathbb{X}. \end{equation} Assume by contradiction that for any $R>0$, \[ \lim_{n\to \infty} \sup_{y\in \mathbb{R}^N} \int_{B_{R}(y)} |v_n|^{p+1} dx =0. \] By using a variant of the Lions compactness principle (see \cite{FQT, Secchi}) we can infer \begin{equation}\label{lions1} v_n\to 0 \quad \text{in } L^{q}(\mathbb{R}^N) \quad \text{for all } q\in [p+1, 2_s^*). \end{equation} Taking into account that $u_n(x)= v_n(x)+ u_2(x)$, we obtain \begin{align*} &\langle \mathcal{I}'(u_n), u_n \rangle \\ &= \iint_{\mathbb{R}^{2N}} \frac{|v_n(x)- v_n(y)|^2}{|x-y|^{N+2s}} \,dx\,dy + \iint_{\mathbb{R}^{2N}} \frac{|u_2(x)- u_2(y)|^2}{|x-y|^{N+2s}} \,dx\,dy \\ &\quad + 2 \iint_{\mathbb{R}^{2N}} \frac{(v_n(x)- v_n(y))(u_2(x)-u_2(y))}{|x-y|^{N+2s}} \,dx\,dy + \int_{\mathbb{R}^N} V(x)|v_n|^2 dx \\ &\quad + \int_{\mathbb{R}^N} V(x)|u_2|^2 dx + 2\int_{\mathbb{R}^N} V(x)v_n u_2 dx + \int_{\mathbb{R}^N} a(x) ( |u_n|^{p+1}- |u_2|^{p+1}) \, dx \\ &\quad + \int_{\mathbb{R}^N} a(x) |u_2|^{p+1} dx -\int_{\mathbb{R}^N} fv_n dx - \int_{\mathbb{R}^N} f_2 dx\\ &= \langle \mathcal{I}'(u_2), u_2\rangle + \iint_{\mathbb{R}^{2N}} \frac{|v_n(x)- v_n(y)|^2}{|x-y|^{N+2s}} \,dx\,dy + \int_{\mathbb{R}^N} V(x)|v_n|^2 dx \\ &\quad + \int_{\mathbb{R}^N} a(x) \big( |u_n|^{p+1}- (|u_2|^{p+1} + |v_n|^{p+1})\big) \, dx + \int_{\mathbb{R}^N} a(x)|v_n|^{p+1} dx \\ &\quad -\int_{\mathbb{R}^N} fv_n dx + 2 \iint_{\mathbb{R}^{2N}} \frac{(v_n(x)- v_n(y))(u_2(x)-u_2(y))}{|x-y|^{N+2s}} \,dx\,dy \\ &\quad + 2\int_{\mathbb{R}^N} V(x)v_n u_2 dx. \end{align*} Putting together this and \eqref{lions1} we infer \begin{align*} 0&= \lim_{n\to \infty} \langle \mathcal{I}'(u_n), u_n \rangle \\ &= \langle \mathcal{I}'(u_2), u_2\rangle + \lim_{n\to \infty} \iint_{\mathbb{R}^{2N}} \frac{|v_n(x)- v_n(y)|^2}{|x-y|^{N+2s}} \,dx\,dy + \int_{\mathbb{R}^N} a(x)|v_n|^{p+1} dx. \end{align*} Hence $v_n\to 0$ in $\mathbb{X}$, and this gives a contradiction. \smallskip \noindent\textbf{Step 2:} Now we prove that $\{y_n\}$ is not a bounded sequence. Assume by contradiction that $\{y_n\}$ is bounded. Then, up to a subsequence $y_n\to y$. Let $\varphi \in C^{\infty}_{c}(\mathbb{R}^N)$. From the facts that $y_n\to y$ and $v_n\rightharpoonup 0$ in $\mathbb{X}$ it follows that \begin{equation}\label{F} \lim_{n\to \infty} \int_{\mathbb{R}^N} \varphi(x- y_n) v_n(x) \, dx =0. \end{equation} By using \eqref{B} we can infer \begin{equation}\label{G} \begin{aligned} \lim_{n\to \infty} \int_{\mathbb{R}^N} \varphi(x-y_n) v_n(x) \, dx &= \lim_{n\to \infty} \int_{\mathbb{R}^N} \varphi(x) v_n(x+y_n)\, dx \\ &= \int_{\mathbb{R}^N} \varphi(x) v_1(x) \, dx. \end{aligned} \end{equation} Putting together \eqref{F} and \eqref{G} we deduce that \[ \int_{\mathbb{R}^N} \varphi(x) v_1(x) \, dx =0 \quad \forall \varphi \in C^{\infty}_{c}(\mathbb{R}^N), \] that implies that $v_1(x)=0$ a.e. in $\mathbb{R}^N$, and this is a contradiction because of \eqref{B}. Thus $\{y_n\}$ is not bounded. \smallskip \noindent \textbf{Step 3:} $v_1$ is a solution to \eqref{P1}. Since $\{y_n\}$ is not bounded, \begin{equation}\label{A3} u_n(x+y_n)\rightharpoonup v_1 \quad \text{in } \mathbb{X}. \end{equation} Now, let $\varphi \in C^{\infty}_{c}(\mathbb{R}^N)$. Since $\{u_n\}$ is a (PS) sequence for $\mathcal{I}$, we have $\langle \mathcal{I}'(u_n), \varphi(\cdot- y_n)\rangle \to 0$. On the other hand we have \begin{equation} \label{C1} \begin{aligned} &\langle \mathcal{I}'(u_n), \varphi(\cdot -y_n)\rangle \\ &= \iint_{\mathbb{R}^{2N}} \frac{(u_n(x) - u_n(y))(\varphi(x-y_n) - \varphi(y-y_n))}{|x-y|^{N+2s}} \,dx\,dy \\ &\quad + \int_{\mathbb{R}^N} V(x) u_n(x) \varphi(x-y_n)\, dx + \int_{\mathbb{R}^N}a(x) |u_n(x)|^{p} \operatorname{sgn}(u_n(x))\varphi(x-y_n) \, dx \\ &\quad - \int_{\mathbb{R}^N} f(x)\varphi(x-y_n)\, dx\\ &=\iint_{\mathbb{R}^{2N}} \frac{(u_n(x+y_n) - u_n(y+y_n)) (\varphi(x)- \varphi(y))}{|x-y|^{N+2s}}\,dx\,dy \\ &\quad + \int_{\mathbb{R}^N} V(x+y_n)u_n(x+y_n) \varphi(x)\, dx \\ &\quad + \int_{\mathbb{R}^N} a(x+y_n)|u_n(x+y_n)|^{p} \operatorname{sgn}(u_n(x+y_n))\varphi(x) \, dx \\ &\quad - \int_{\mathbb{R}^N} f(x)\varphi(x-y_n)\, dx. \end{aligned} \end{equation} Since $|y_n|\to +\infty$, $f\in L^2(\mathbb{R}^N)$ and $\varphi \in C^{\infty}_{c}(\mathbb{R}^N)$, we have \begin{equation}\label{C2} \int_{\mathbb{R}^N} f(x)\varphi(x-y_n)\, dx \to 0. \end{equation} Moreover, by \eqref{A3} we have \begin{equation} \label{C3} \begin{aligned} &\lim_{n\to \infty} \iint_{\mathbb{R}^{2N}} \frac{(u_n(x+y_n) - u_n(y+y_n)) (\varphi(x)- \varphi(y))}{|x-y|^{N+2s}}\,dx\,dy \\ &=\iint_{\mathbb{R}^{2N}} \frac{(v_1(x) - v_1(y)) (\varphi(x)- \varphi(y))}{|x-y|^{N+2s}}\,dx\,dy. \end{aligned} \end{equation} Since $u_n(\cdot+y_n)\to v_1$ in $L^2(\operatorname{supp} \varphi)$, there exists a subsequence denoted again by $u_n(\cdot +y_n)$ and a function $h\in L^2(\operatorname{supp} \varphi)$ such that $|u_n(\cdot+y_n)|\leq |h|$. Then, by (H2) it follows that \begin{gather*} V(x+y_n) u_n(x+y_n) \varphi \to v_{\infty} v_1 \varphi \quad \text{a.e. in } \mathbb{R}^N\\ |V(x+y_n)u_n(x+y_n) \varphi|\leq \|V\|_{L^{\infty}(\mathbb{R}^N)} |h| |\varphi| \in L^{1}(\mathbb{R}^N). \end{gather*} Thus, by the Dominated Convergence Theorem we obtain \begin{equation}\label{C4} \int_{\mathbb{R}^N} V(x+y_n) u_n(x+y_n) \varphi(x)\, dx \to v_{\infty} \int_{\mathbb{R}^N} v_1 \varphi\, dx. \end{equation} Similarly, since $u_n(\cdot+y_n)\to v_1$ in $L^{p+1}(\operatorname{supp} \varphi)$, there exist a subsequence denoted again by $u_n(\cdot +y_n)$ and a function $\tilde{h}\in L^{p+1}(\operatorname{supp} \varphi)$ such that $|u_n(\cdot+y_n)|\leq |\tilde{h}|$. Then, by (H3) we infer that \begin{gather*} a(x+y_n)|u_n(x+y_n)|^{p} \operatorname{sgn}(u_n(x +y_n)) \varphi \to a_{\infty}|v_1|^{p} \operatorname{sgn}(v_1) \varphi \quad \text{a.e. in } \mathbb{R}^N, \\ |a(x+y_n)|u_n(x +y_n)|^{p}\varphi|\leq \|a\|_{L^{\infty} (\mathbb{R}^N)} |\tilde{h}|^{p}|\varphi|\in {\rm L^{1}(\mathbb{R}^N)}. \end{gather*} Thus, by the Dominated Convergence Theorem we obtain \begin{equation} \label{C5} \begin{aligned} &\lim_{n\to \infty} \int_{\mathbb{R}^N} a(x+y_n)|u_n(x+y_n)|^{p} \operatorname{sgn}(u_n(x+y_n))\varphi(x) \, dx\\ &= a_{\infty} \int_{\mathbb{R}^N} |v_1|^{p} \operatorname{sgn}(v_1)\varphi \, dx. \end{aligned} \end{equation} Putting together \eqref{C1}, \eqref{C2}, \eqref{C3}, \eqref{C4} and \eqref{C5} we obtain \begin{align*} &\iint_{\mathbb{R}^{2N}} \frac{(v_1(x) - v_1(y)) (\varphi(x)- \varphi(y))}{|x-y|^{N+2s}}\,dx\,dy + v_{\infty} \int_{\mathbb{R}^N} v_1 \varphi\, dx \\ & +a_{\infty}\int_{\mathbb{R}^N} |v_1|^{p} (v_1)\varphi \, dx=0, \end{align*} that is $v_1$ is a weak solution to \begin{equation}\label{P1} \begin{gathered} (-\Delta)^{s}u + v_{\infty}u+a_{\infty}|u|^{p}\operatorname{sgn}(u)= 0 \quad\text{in } \mathbb{R}^N\\ u\in H^s(\mathbb{R}^N)\cap L^{p+1}(\mathbb{R}^N) \end{gathered} \end{equation} But this problem only possesses the trivial solution, thus $v_1=0$ and this is an absurd in view of \eqref{B}. From Steps 1, 2 and 3 we conclude that the dichotomy does not occur. Then $\mathcal{I}(u_n)\to M= \mathcal{I}(u_2)>0(>\mathcal{I}(u_1))$ and $u_2\neq u_1$. \end{proof} \begin{thebibliography}{00} \bibitem{AM} C. O. Alves, O. H. Miyagaki; \emph{Existence and concentration of solution for a class of fractional elliptic equation in $\mathbb{R}^N$ via penalization method}, Calc. Var. Partial Differential Equations, \textbf{55} (2016), no. 3, Art. 47, 19 pp. \bibitem{A1} V. Ambrosio; \emph{Ground states for superlinear fractional Schr\"odinger equations in $\mathbb{R}^N$}, Ann. Acad. Sci. Fenn. Math., \textbf{41} (2016), 745--756. \bibitem{A2} V. Ambrosio; \emph{Ground states for a fractional scalar field problem with critical growth}, Differential Integral Equations, \textbf{30} (2017), no. 1-2, 115--132. \bibitem{A3} V. Ambrosio; \emph{Multiplicity of positive solutions for a class of fractional Schr\"odinger equations via penalization method}, Ann. Mat. Pura Appl., (4) \textbf{196} (2017), no. 6, 2043--2062. \bibitem{AD} V. Ambrosio, P. d'Avenia; \emph{Nonlinear fractional magnetic Schr\"odinger equation: Existence and multiplicity}, J. Differential Equations, \textbf{264} (2018), no. 5, 3336--3368. \bibitem{AF} V. Ambrosio, G. M. Figueiredo; \emph{Ground state solutions for a fractional Schr\"odinger equation with critical growth}, Asymptotic Analysis, \textbf{105} (2017), no. 3-4, pp. 159--191. \bibitem{AH} V. Ambrosio \& H. Hajaiej; \emph{Multiple solutions for a class of nonhomogeneous fractional Schr\"odinger equations in $\mathbb{R}^N$}, J. Dyn. Diff. Equat. (in press) DOI: 10.1007/s10884-017-9590-6. \bibitem{AI} V Ambrosio \& T. Isernia; \emph{Concentration phenomena for a fractional Schr\"odinger-Kirchhoff type equation}, Math. Methods Appl. Sci., \textbf{41} (2018), no. 2, 615--645. \bibitem{B} M. Benrhouma; \emph{Study of multiplicity and uniqueness of solutions for a class of nonhomogeneous sublinear elliptic equations}, Nonlinear Anal. \textbf{74} (2011), no. 7, 2682--2694. \bibitem{BO} M. Benrhouma, H. Ounaies; \emph{Existence and uniqueness of positive solution for nonhomogeneous sublinear elliptic equations}, J. Math. Anal. Appl. \textbf{358} (2009), no. 2, 307--319. \bibitem{CS} L. Caffarelli, L. Silvestre; \emph{An extension problem related to the fractional Laplacian}, Comm. Partial Differential Equations \textbf{32} (2007), no. 7-9, 1245--1260. \bibitem{CT} D. Costa, H. Tehrani; \emph{Unbounded perturbations of resonant Schr\"odinger equations}, Contemp. Math. \textbf{357}, Amer. Math. Soc., Providence, RI, 2004, 101--110. \bibitem{DNPV} E. Di Nezza, G. Palatucci, E. Valdinoci; \emph{Hitchhiker's guide to the fractional Sobolev spaces}, Bull. Sci. Math., \textbf{136} (2012), 521--573. \bibitem{DPV} S. Dipierro, G. Palatucci, E. Valdinoci; \emph{Existence and symmetry results for a Schr\"odinger type problem involving the fractional Laplacian}, Matematiche (Catania), \textbf{68} (2013), no. 1, 201--216. \bibitem{FQT} P. Felmer, A. Quaas, J. Tan; \emph{Positive solutions of the nonlinear Schr\"odinger equation with the fractional Laplacian}, Proc. Roy. Soc. Edinburgh Sect. A \textbf{142} (2012), 1237--1262. \bibitem{FS} G. M. Figueiredo, G. Siciliano; \emph{A multiplicity result via Ljusternick-Schnirelmann category and Morse theory for a fractional Schr\"odinger equation in $\mathbb{R}^N$}, NoDEA Nonlinear Differential Equations Appl., \textbf{23} (2016), no. 2, Art. 12, 22 pp. \bibitem{FLS} R. L. Frank, E. Lenzmann, L. Silvestre; \emph{Uniqueness of radial solutions for the fractional Laplacian}, Commun. Pur. Appl. Math., \textbf{69} (2016), 1671--1726. \bibitem{FP} Y. Fu, P. Pucci; \emph{Multiplicity existence for sublinear fractional Laplacian problems}, Appl. Anal., \textbf{96} (2017), no. 9, 1497--1508. \bibitem{I} T. Isernia; \emph{Positive solution for nonhomogeneous sublinear fractional equations in $\mathbb{R}^N$}, Complex Var. Elliptic Equ., \textbf{63} (2018), no. 5, 689--714. \bibitem{L} N. Laskin; \emph{Fractional quantum mechanics and L\`evy path integrals}, Phys. Lett. A, \textbf{268} (2000), no. 4-6, 298--305. 81S40. \bibitem{Lions} P.-L. Lions; \emph{The concentration-compactness principle in the calculus of variations. The locally compact case. Part 2.}, Annales Inst. H. Poincar\'e Analyse Non Lin\'eaire, \textbf{2} (1984), 223--283. \bibitem{MBRS} G. Molica Bisci, V. R\u{a}dulescu, R. Servadei; \emph{Variational Methods for Nonlocal Fractional Problems. With a Foreword by Jean Mawhin}, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 162 Cambridge, 2016. \bibitem{Secchi} S. Secchi; \emph{Ground state solutions for nonlinear fractional Schr\"odinger equations in $\mathbb{R}^N$}, J. Math. Phys. \textbf{54} (2013), 031501. \bibitem{T} H. Tehrani; \emph{Existence results for an indefinite unbounded perturbation of a resonant Schr\"odinger equation}, J. Differential Equations, \textbf{236} (2007), no. 1, 1--28. \bibitem{YL} L. Yang, Z. Liu; \emph{Multiplicity and concentration of solutions for fractional Schr\"odinger equation with sublinear perturbation and steep potential well}, Comput. Math. Appl., \textbf{72} (2016), no. 6, 1629--1640. \end{thebibliography} \end{document}