\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 260, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/260\hfil Non-local fractional Laplacian equations] {Multiple solutions for perturbed non-local fractional Laplacian equations} \author[M. Ferrara, L. Guerrini, B. Zhang \hfil EJDE-2013/260\hfilneg] {Massimiliano Ferrara, Luca Guerrini, Binlin Zhang} % in alphabetical order \address{Massimiliano Ferrara \newline University of Reggio Calabria and CRIOS University Bocconi of Milan, Via dei Bianchi presso Palazzo Zani, 89127 Reggio Calabria, Italy} \email{massimiliano.ferrara@unirc.it} \address{Luca Guerrini \newline Department of Management, University Polytecnic of Marche, Pizza Martelli 8, 60121 Ancona, Italy} \email{luca.guerrini@univpm.it} \address{Binlin Zhang \newline MEDAlics Research Center, University Dante Alighieri, Via del Torrione, 89125 Reggio Calabria, Italy \newline Department of Mathematics, Heilongjiang Institute of Technology, 150050 Harbin, China} \email{zhbinlin@gmail.com} \thanks{Submitted October 21, 2013. Published November 26, 2013.} \subjclass[2000]{49J35, 35A15, 35S15, 47G20, 45G05} \keywords{Variational methods; integrodifferential operators; \hfill\break\indent fractional Laplacian} \begin{abstract} In article we consider problems modeled by the non-local fractional Laplacian equation \begin{gather*} (-\Delta)^s u=\lambda f(x,u)+\mu g(x,u) \quad\text{in } \Omega\\ u=0 \quad\text{in } \mathbb{R}^n\setminus \Omega, \end{gather*} where $s\in (0,1)$ is fixed, $(-\Delta )^s$ is the fractional Laplace operator, $\lambda,\mu$ are real parameters, $\Omega$ is an open bounded subset of $\mathbb{R}^n$ ($n>2s$) with Lipschitz boundary $\partial \Omega$ and $f,g:\Omega\times\mathbb{R}\to\mathbb{R}$ are two suitable Carath\'{e}odory functions. By using variational methods in an appropriate abstract framework developed by Servadei and Valdinoci \cite{svmountain} we prove the existence of at least three weak solutions for certain values of the parameters. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction}\label{sec:introduzione} There are a lot interesting problems in the standard framework of the Laplacian (and, more generally, of uniformly elliptic operators), widely studied in the literature. A natural question is whether or not the existence results got in this classical context can be extended to the non-local framework of the fractional Laplacian type operators. In this spirit, we study the existence of weak solutions for the following general non-local equation depending on two real parameters $\lambda$ and $\mu$ given by \begin{equation}\label{op} \begin{gathered} -\mathcal{ L}_K u=\lambda f(x, u)+\mu g(x,u) \quad\text{in } \Omega\\ u=0 \quad\text{in } \mathbb{R}^n\setminus \Omega\,. \end{gathered} \end{equation} More precisely, in our setting $\Omega$ is an open bounded subset of $\mathbb{R}^n$, $n>2s$, (where $s\in (0,1)$) with smooth boundary $\partial\Omega$, while $\mathcal L_K$ is the integrodifferential operator defined as \begin{equation}\label{lk} \mathcal L_Ku(x):= \int_{\mathbb{R}^n}\Big(u(x+y)+u(x-y)-2u(x)\Big)K(y)\,dy, \quad x\in \mathbb{R}^n, \end{equation} with the kernel $K:\mathbb{R}^n\setminus\{0\}\to(0,+\infty)$ such that \begin{gather}\label{kernel} m K\in L^1(\mathbb{R}^n), \quad\text{where }m(x)=\min\{|x|^2,1\};\\ \label{kernelfrac} \text{there exists $\theta>0$ such that $K(x)\geq \theta |x|^{-(n+2s)}$ for all $x\in \mathbb{R}^n \setminus\{0\}$};\\ \label{evenkernel} K(x)=K(-x)\quad \text{for all } x\in \mathbb{R}^n \setminus\{0\}. \end{gather} A model for $K$ is given by the singular kernel $K(x):=|x|^{-(n+2s)}$ which gives rise to the fractional Laplace operator $-(-\Delta)^s$, which, up to normalization factors, may be defined as \begin{equation} \label{2} -(-\Delta)^s u(x):= \int_{\mathbb{R}^n}\frac{u(x+y)+u(x-y)-2u(x)}{|y|^{n+2s}}\,dy, \quad x\in \mathbb{R}^n. \end{equation} The homogeneous Dirichlet datum in~\eqref{op} is given in $\mathbb{R}^n\setminus \Omega$ and not simply on the boundary $\partial \Omega$, as it happens in the classical case of the Laplacian, consistently with the non-local nature of the operator~$\mathcal L_K$. Moreover, we will assume that $f,g:\Omega\times\mathbb{R}\to\mathbb{R}$ are two suitable Carath\'{e}odory functions with subcritical growth. Recently, a great attention has been focused on the study of fractional and nonlocal operators of elliptic type, both for the pure mathematical research and in view of concrete real-world applications. This type of operators arises in a quite natural way in many different contexts, such as, among the others, the thin obstacle problem, optimization, finance, phase transitions, stratified materials, anomalous diffusion, crystal dislocation, soft thin films, semipermeable membranes, flame propagation, conservation laws, ultra-relativistic limits of quantum mechanics, quasi-geostrophic flows, multiple scattering, minimal surfaces, materials science and water waves. In this work, motivated by this large interest, we prove the existence of non-trivial weak solutions of problem~\eqref{op} using variational and topological methods. Our variational approach is realizable checking that the associated energy functional verifies the assumptions requested by a general critical point theorem obtained by Ricceri in \cite[Theorem 1.1]{Ricceri2009} (see Theorem \ref{application} below) and thanks to a suitable variational setting developed by Servadei and Valdinoci in \cite{svmountain}. Indeed, the nonlocal analysis that we perform here in order to use Theorem is quite general and successfully exploited for other goals in several recent contributions; see \cite{svlinking,servadeivaldinociBN} and \cite{valpal} for an elementary introduction to this topic and for a list of related references. By a weak solutions of \eqref{op} we mean a solution of the problem \begin{equation}\label{problemaK0f} \begin{aligned} &\int_{\mathbb{R}^n\times\mathbb{R}^n } (u(x)-u(y))(\varphi(x)-\varphi(y))K(x-y) dx\,dy\\ &= \lambda \int_\Omega f(x,u(x))\varphi(x)dx +\mu \int_\Omega g(x,u(x))\varphi(x)dx, \quad \forall \varphi \in X_0,\;\forall u\in X_0. \end{aligned} \end{equation} Hence, Problem \eqref{problemaK0f} represents the weak formulation of~\eqref{op}. Note that, to write such a weak formulation, we need to assume condition \eqref{evenkernel}. The space $X_0$ in which we set problem~\eqref{problemaK0f} is a functional space, inspired by, but not equivalent to, the usual fractional Sobolev space. This new space was introduced in \cite{sv}; see also \cite{svmountain}. The choice of this space is motivated by the fact that it allows us to correctly encode the Dirichlet boundary datum in the weak formulation. We will recall its definition in Section~\ref{sec:preliminary}, to make the present paper self-contained. In the current literature \cite{colorado, cabretan, capella, fiscella, Molica1, Molica2, sBNRES, svmountain, svlinking, servadeivaldinociBN,servadeivaldinociCFP, tan} the authors studied non-local fractional Laplacian equations with superlinear and subcritical or critical nonlinearities, while in \cite{fsv, fsvBS} the asymptotically linear case was exploited. In this paper we are interested in equations depending on two real parameters. In many mathematical problems deriving from applications the presence of one (or more) parameter is a relevant feature, and the study of how solutions depend on parameters is an important topic. Here, exploiting an abstract critical point theorem recalled in Theorem \ref{thm:ricceri} and by using Proposition \ref{propo}, we prove in the main result (see Theorem \ref{application}) the existence of an open interval $\Lambda\subseteq(0,+\infty)$ and a positive real number $\rho$ with the following property: for each $\lambda\in\Lambda$ and each subcritical Carath\'{e}odory function $g$ there exists $\delta>0$ such that, for each $\mu\in[0,\delta]$, problem \eqref{op} has at least three weak solutions whose norms in $X_0$ are less than $\rho$. We would like to stress that we do not assume any condition (only the natural subcritical growth condition) on the perturbation term $g$. As an application of Theorem~\ref{application}, we consider the model problem \begin{gather*} (-\Delta)^s u=\lambda f(u)+\mu (1+|u|^{\beta}) \quad\text{in } \Omega\\ u=0 \quad\text{in } \mathbb{R}^n\setminus \Omega, \end{gather*} where $\beta\in (0,2^*-1)$. Here the exponent $2^*:=2n/(n-2s)$ is the fractional critical Sobolev exponent. Notice that when~$s=1$ the above exponent reduces to the classical critical Sobolev exponent~$2_*:=2n/(n-2)$. In this framework, the cited Theorem~\ref{application} reduces to the following result. \begin{theorem}\label{application2} Let $s\in (0,1)$, $n>2s$ and $\Omega$ be an open bounded set of $\mathbb{R}^n$ with Lipschitz boundary. Further, let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function such that \[ \sup_{s\in\mathbb{R}}\frac{|f(s)|}{1+|s|^{\sigma-1}}<+\infty, \] for some $1<\sigma<2^*$. Assume that there exist three positive real constants $c, d$ and $1<\gamma<2$, such that \begin{itemize} \item [(I1')] $F(s):=\int_0^s f(t)dt> 0,$ for every $s\in ]0,c]$; \item [(I2')] There exists $2<\alpha<2^*$, such that \[ \limsup_{s\to 0}\frac{ F(s)}{|s|^{\alpha}}<+\infty; \] \item [(I3')] $|F(s)|\leq d(1+|s|^{\gamma})$ for every $s\in\mathbb{R}$. \end{itemize} Then, there exist an open interval $\Lambda\subseteq(0,+\infty)$ and a positive real number $\rho$ with the following property: For each $\lambda\in\Lambda$ there exists $\delta>0$ such that, for each $\mu\in[0,\delta]$, the problem \begin{equation}\label{2s} \begin{gathered} (-\Delta)^s u=\lambda f(u)+\mu (1+|u|^{\beta}) \quad \text{in } \Omega\\ u=0 \quad\text{in } \mathbb{R}^n\setminus \Omega, \end{gathered} \end{equation} where $\beta\in (0,2^*-1)$, has at least three weak solutions $u_{i}\in H^s(\mathbb{R}^n)$ (with $i=1,2,3$) such that $u_i=0$ a.e. in $\mathbb{R}^n\setminus \Omega$ and whose norms are less than $\rho$. \end{theorem} The article is organized as follows. In Section~\ref{sec:preliminary} we recall the definition of the functional space we work in and we give some notations. In Section~\ref{sec:premain3} we prove our main result (Theorem~\ref{application}) and its consequence (Theorem \ref{application2}) for fractional Laplacian equations; see Remark \ref{rema}. A direct application of our result, studying a non-local equation driven by the fractional Laplacian, is then presented; see Example \ref{exa}. \section{Preliminaries}\label{sec:preliminary} This section we introduce the notation used, and give some preliminary results which will be useful in the sequel. \subsection{The functional space $X_0$}\label{subsec:X0} We briefly recall the definition of the functional space~$X_0$, firstly introduced in \cite{sv}. The reader familiar with this topic may skip this section and go directly to the next one. The functional space $X$ denotes the linear space of Lebesgue measurable functions from $\mathbb{R}^n$ to $\mathbb{R}$ such that the restriction to $\Omega$ of any function $g$ in $X$ belongs to $L^2(\Omega)$ and the map $(x,y)\mapsto (g(x)-g(y))\sqrt{K(x-y)}$ is in $L^2\big((\mathbb{R}^n\times\mathbb{R}^n) \setminus ({\mathcal C}\Omega\times {\mathcal C}\Omega), dxdy\big)$ (here ${\mathcal C}\Omega:=\mathbb{R}^n \setminus\Omega$). Also, we denote by $X_0$ the linear subspace of $X$, $$ X_0:=\big\{g\in X : g=0\text{ a.e. in } \mathbb{R}^n\setminus \Omega\big\}. $$ We remark that $X$ and $X_0$ are non-empty, since $C^2_0 (\Omega)\subseteq X_0$ by \cite[Lemma~11]{sv}. Moreover, the space $X$ is endowed with the norm defined as \begin{equation}\label{norma} \|g\|_X:=\|g\|_{L^2(\Omega)}+\Big(\int_Q |g(x)-g(y)|^2K(x-y)dx\,dy\Big)^{1/2}\,, \end{equation} where $Q=(\mathbb{R}^n\times\mathbb{R}^n)\setminus \mathcal O$ and ${\mathcal{O}}=({\mathcal{C}}\Omega)\times({\mathcal{C}}\Omega) \subset\mathbb{R}^n\times\mathbb{R}^n$. It is easily seen that $\|\cdot\|_X$ is a norm on $X$ (see, for instance, \cite{svmountain} for a proof). By \cite[Lemmas~6 and 7]{svmountain} in the sequel we can take the function \begin{equation}\label{normaX0} X_0\ni v\mapsto \|v\|_{X_0}=\Big(\int_Q|v(x)-v(y)|^2K(x-y)\,dx\,dy\Big)^{1/2} \end{equation} as norm on $X_0$. Also $(X_0, \|\cdot\|_{X_0})$ is a Hilbert space (for this see \cite[Lemmas~7]{svmountain}), with scalar product \begin{equation}\label{scalarproduct} \langle u,v\rangle_{X_0}:=\int_Q \big( u(x)-u(y)\big) \big( v(x)-v(y)\big)\,K(x-y)\,dx\,dy. \end{equation} Note that in \eqref{normaX0} (and in the related scalar product) the integral can be extended to all $\mathbb{R}^n\times\mathbb{R}^n$, since $v\in X_0$ (and so $v=0$ a.e. in $\mathbb{R}^n\setminus \Omega$). While for a general kernel~$K$ satisfying conditions~\eqref{kernel}--\eqref{evenkernel} we have that $X_0\subset H^s(\mathbb{R}^n)$, in the model case $K(x):=|x|^{-(n+2s)}$ the space $X_0$ consists of all the functions of the usual fractional Sobolev space $H^s(\mathbb{R}^n)$ which vanish almost everywhere outside $\Omega$ (see \cite[Lemma~7]{servadeivaldinociBN}). Here $H^s(\mathbb{R}^n)$ denotes the usual fractional Sobolev space endowed with the norm (the so-called \emph{Gagliardo norm}) \begin{equation}\label{gagliardonorm} \|g\|_{H^s(\mathbb{R}^n)}=\|g\|_{L^2(\mathbb{R}^n)}+ \Big(\int_{\mathbb{R}^n\times\mathbb{R}^n} \frac{|g(x)-g(y)|^2}{|x-y|^{n+2s}}\,dx\,dy\Big)^{1/2}\,. \end{equation} Before concluding this subsection, we recall the embedding properties of~$X_0$ into the usual Lebesgue spaces (see \cite[Lemma~8]{svmountain}). The embedding $j:X_0\hookrightarrow L^{\nu}(\mathbb{R}^n)$ is continuous for any $\nu\in [1,2^*]$, while it is compact whenever $\nu\in [1,2^*)$. Hence, for any $\nu\in [1,2^*]$ there exists a positive constant $c_\nu$ such that \begin{equation}\label{Poincare1} \|v\|_{L^{\nu}(\mathbb{R}^n)}\leq c_{\nu} \|v\|_{X_0}, \end{equation} for any $v\in X_0$. For further details on the fractional Sobolev spaces we refer the reader to \cite{valpal} and to the references therein, while for other details on $X$ and $X_0$ we refer to \cite{sv}, where these functional spaces were introduced and various properties of these spaces were proved. \section{Main Results}\label{sec:premain3} \subsection{Multiple critical points}\label{subsec:Ricceri} For the reader's convenience, we recall the revised form of Ricceri's three critical points theorem (see \cite[Theorem 1]{Ricceri2009}) and \cite[Proposition 3.1]{Ricceri2000a}. \begin{theorem}[{\cite[Theorem 1]{Ricceri2009}}] \label{thm:ricceri} Let $E$ be a reflexive real Banach space, $ \Phi\colon E \to \mathbb{R} $ be a continuously G\^{a}teaux differentiable and sequentially weakly lower semicontinuous functional whose G\^{a}teaux derivative admits a continuous inverse on $E^*$ and $\Phi$ is bounded on each bounded subset of $E$; $\Psi\colon E \to \mathbb{R} $ is a continuously G\^{a}teaux differentiable functional whose G\^{a}teaux derivative is compact; $ I \subseteq \mathbb{R} $ an interval. Assume that \begin{equation}\label{qiangzhi} \lim_{\|u\|_E \to +\infty } (\Phi(u)+\lambda \Psi (u))=+\infty \end{equation} for all $ \lambda \in I $, and that there exists $h\in \mathbb{R}$ such that \begin{equation}\label{t2} \sup_{\lambda \in I} \inf_{u \in E} (\Phi (u)+ \lambda (\Psi (u)+h)) < \inf_{u \in E} \sup_{\lambda \in I} (\Phi (u)+ \lambda (\Psi (u)+ h)). \end{equation} Then, there exists an open interval $ \Lambda \subseteq I $ and a positive real number $ \rho $ with the following property: For every $ \lambda \in \Lambda $ and every $ C^1 $ functional $ J\colon E \mapsto \mathbb{R} $ with compact derivative, there exists $ \delta > 0 $ such that, for each $ \mu \in [0,\delta] $ the equation \[ \Phi '(u)+\lambda \Psi '(u)+\mu J'(u)=0, \] has at least three solutions in $E$ whose norms are less than $ \rho $. \end{theorem} \begin{proposition}[{\cite[Proposition 3.1]{Ricceri2000a}}] \label{propo} Let $E$ be a non-empty set and $\Phi, \Psi$ two real functions on $E$. Assume that there are $r > 0$ and $ u_0, u_1 \in E$ such that \[ \Phi(u_0)=-\Psi(u_0)=0, \quad \Phi(u_1)>r, \quad \sup_{ u \in \Phi^{-1} ( ]-\infty ,r ] ) } (-\Psi(u)) < r \frac{-\Psi(u_1)}{\Phi(u_1)}. \] Then, for each $ h $ satisfying \[ \sup_{ u \in \Phi^{-1} ( ]-\infty ,r ] ) } (-\Psi(u)) < h < r \frac{-\Psi(u_1)}{\Phi(u_1)}, \] one has \[ \sup_{\lambda \geq 0} \inf_{u \in E}(\Phi(u)+\lambda(h +\Psi(u))) < \inf_{u\in E} \sup_{\lambda \geq 0} (\Phi(u)+\lambda(h +\Psi(u))). \] \end{proposition} For several related topics and a careful analysis of the abstract framework we refer the reader to the recent monograph~\cite{kristaly}. \subsection{Three weak solutions}\label{teoremanostro} Let us denote $$ F(x,s):=\int_0^s f(x,t)dt, $$ for a.e. $x\in\Omega$ and every $s\in\mathbb{R}$. For the rest of this article, the nonlinearity $f$ in \eqref{op} is a Carath\'eodory function $f:\Omega\times \mathbb{R}\to \mathbb{R}$ satisfying the following growth conditions: There exist $a_1, a_2\geq 0$ and $\sigma\in (1, 2^*)$ such that \begin{equation}\label{crescita} |f(x,t)|\le a_1+a_2|t|^{\sigma-1}\quad \text{a.e. } x\in \Omega,\, t\in \mathbb{R}. \end{equation} This assumption says that the function $f$ has a subcritical growth. Our main result is the following multiplicity theorem. \begin{theorem}\label{application} Let $s\in (0,1)$, $n>2s$, $\Omega$ be an open bounded set of $\mathbb{R}^n$ with Lipschitz boundary and let $K:\mathbb{R}^n\setminus\{0\}\to(0,+\infty)$ be a function satisfying conditions \eqref{kernel}--\eqref{evenkernel}. Further, let $f:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Carath\'{e}odory such that condition~\eqref{crescita} holds. In addition, assume that there exist three positive real constants $c, d$ and $1<\gamma<2$, such that \begin{itemize} \item [(I1)] $F(x,s)> 0$ for a.e. $x\in\Omega$ and every $s\in ]0,c]$; \item [(I2)] There exists $2<\alpha<2^*$, such that \[ \limsup_{s\to 0}\frac{\sup_{x\in\Omega} F(x,s)}{|s|^{\alpha}}<+\infty; \] \item [(I3)] $|F(x,s)|\leq d(1+|s|^{\gamma})$ for a.e. $x\in\Omega$ and every $s\in\mathbb{R}$. \end{itemize} Then, there exist an open interval $\Lambda\subseteq(0,+\infty)$ and a positive real number $\rho$ with the following property: For each $\lambda\in\Lambda$ and each Carath\'{e}odory function $g:\Omega\times\mathbb{R}\to\mathbb{R}$ satisfying \[ \sup_{(x,t)\in\Omega\times\mathbb{R}}\frac{|g(x,t)|}{1+|t|^{\beta-1}}<+\infty, \] for some $1<\beta<2^*$, there exists $\delta>0$ such that, for each $\mu\in[0,\delta]$, problem \eqref{op} has at least three weak solutions whose norms in $X_0$ are less than $\rho$. \end{theorem} \begin{proof} The idea of the proof consists in applying Theorem \ref{thm:ricceri} and Proposition \ref{propo}. The energy functional corresponding to problem \eqref{op} is defined on $E:=X_0$ as \begin{equation}\label{erfuc} H(u):=\Phi(u)+\lambda\Psi(u)+\mu J(u), \end{equation} where \begin{gather}\label{phi} \Phi(u):=\frac 1 2 \int_{Q}|u(x)-u(y)|^2 K(x-y)\,dx\,dy, \\ \label{psi} \Psi(u):=-\int_{\Omega}F(x,u(x))dx, \\ \label{:J} J(u):=-\int_{\Omega}G(x,u(x))dx, \end{gather} where $$ G(x,s):=\int_0^s g(x,t)dt, $$ for a.e. $x\in\Omega$ and every $s\in\mathbb{R}$. Of course, $\Phi$ is a continuously G\^{a}teaux differentiable and sequentially weakly lower semicontinuous functional whose G\^{a}teaux derivative admits a continuous inverse on $E^{*}$, moreover, $\Psi$ and $J$ are continuously G\^{a}teaux differentiable functionals whose G\^{a}teaux derivative is compact. Obviously, $\Phi$ is bounded on each bounded subset of $E$ under our assumptions. Taking into account \eqref{phi}, \eqref{psi} and \eqref{:J}, for each $u$ and $v\in E$, one has \begin{gather*} \langle\Phi '(u),v\rangle = \int_{Q} (u(x)-u(y))(v(x)-v(y))K(x-y)\, dx\,dy,\\ \langle\Psi '(u),v\rangle =-\int_{\Omega}f(x,u(x))v(x)\,dx, \\ \langle J'(u),v\rangle =-\int_{\Omega}g(x,u(x))v(x)\,dx. \end{gather*} Now, from (I3) and \eqref{Poincare1} it follows that \begin{align*} \lambda\Psi(u) & =-\lambda\int_{\Omega}F(x,u(x))dx\\ & \geq -\lambda d\int_{\Omega}(1+|u(x)|^{\gamma})dx\\ & \geq -\lambda d(\operatorname{meas}(\Omega) +\|u\|^{\gamma}_{L^{\gamma}(\Omega)})\\ & \geq -C_\lambda(1+\|u\|^{\gamma}_{X_0}). \end{align*} for a suitable positive constant $C_\lambda$. Thus, we obtain \[ \Phi(u)+\lambda\Psi(u)\geq \frac{1}{2}\|u\|_{X_0}^{2} -C_\lambda(1+\|u\|_{X_0}^{\gamma}), \] Since $\gamma<2$, it follows that \[ \lim_{\|u\|_{X_0}\to +\infty } (\Phi(u)+\lambda \Psi (u)) =+\infty\quad\forall u\in E,\;\lambda\in (0,+\infty). \] Then, assumption \eqref{qiangzhi} of Theorem \ref{thm:ricceri} is satisfied. Next, we prove that $$ \sup_{\lambda \in I} \inf_{u \in E} (\Phi (u)+ \lambda (\Psi (u)+h)) < \inf_{u \in E} \sup_{\lambda \in I} (\Phi (u)+ \lambda (\Psi (u)+ h)), $$ for some $h\in \mathbb{R}$. This means that assumption \eqref{t2} is also satisfied. For our goal it suffices to verify the conditions of Proposition \ref{propo}. Hence, let us take $u_0\equiv 0$ and observe that \[ \Phi(u_0)=-\Psi(u_0)=0. \] At this point, we claim that there exist $r> 0$ and $u_1\in X$ such that $\Phi(u_1)>r$ and \eqref{t2} is satisfied. By (I2) there exist $\eta\in[0,1]$ and $C_1>0$, such that \[ F(x,s)0$, by the Sobolev embedding theorem, for a suitable positive constant $C_2$ we have \[ -\Psi(u)=\int_{\Omega}F(x,u(x))dx 2$, it follows that \begin{equation}\label{eq:lim} \lim_{r\to 0^+}\frac{\sup_{\|u\|^{2}_{X_0}\leq 2r}(-\Psi(u))}{r}=0. \end{equation} Let $u_1 \in C^2_0(\Omega)$ be a function positive in $\Omega$, with $u|_{\partial \Omega} = 0$ and $$ \max_{x\in\overline{\Omega}} u(x) \leq c. $$ Note that $C^2_0 (\Omega)\subseteq X_0$; see \cite[Lemma~11]{sv}. Then, of course, $u_1 \in E$ and in addition $\Phi(u_1) > 0$. In view of (I1) we also have $$ -\Psi(u_1) = \int_{\Omega} F(x,u_1(x)) dx > 0. $$ Therefore, from \eqref{eq:lim}, we can find $r \in \left( 0,\Phi(u_1)\right)$ such that \[ \sup_{ u \in \Phi^{-1} ( ]-\infty ,r ] ) } (-\Psi(u)) = \sup_{\|u\|^{2}_{X_0} \leq 2r} (-\Psi(u)) < r\frac{-\Psi(u_1)}{\Phi(u_1)}. \] The proof is complete. \end{proof} The following is special case of Theorem \ref{application}. \begin{corollary}\label{application555} Let $s\in (0,1)$, $n>2s$, $\Omega$ be an open bounded set of $\mathbb{R}^n$ with Lipschitz boundary and let $K:\mathbb{R}^n\setminus\{0\}\to(0,+\infty)$ be a function satisfying conditions \eqref{kernel}--\eqref{evenkernel}. Further, let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function such that \[ \sup_{s\in\mathbb{R}}\frac{|f(s)|}{1+|s|^{\sigma-1}}<+\infty, \] for some $1<\sigma<2^*$. In addition, assume that there exist three positive real constants $c, d$ and $1<\gamma<2$, such that conditions {\rm (I1')--(I3')} hold. Then, there are an open interval $\Lambda\subseteq(0,+\infty)$ and a positive real number $\rho$ such that, for each $\lambda\in\Lambda$ the problem \begin{gather*} -\mathcal L_K u=\lambda f(u) \quad\text{in } \Omega\\ u=0 \quad \text{in } \mathbb{R}^n\setminus \Omega, \end{gather*} has at least three weak solutions whose norms in $X_0$ are less than $\rho$. \end{corollary} \begin{remark}\label{rema}\rm In the model case in which $K(x):=|x|^{-(n+2s)}$ the space $X_0$ can be characterized as follows $$ X_0=\big\{v\in H^s(\mathbb{R}^n) : v=0\text{ a.e. in } \mathbb{R}^n\setminus \Omega\big\}; $$ for details see \cite[Lemma 7-b]{servadeivaldinociBN}. Then, is immediate to observe that Theorem \ref{application2} in the Introduction is a direct consequence of Theorem \ref{application}. \end{remark} \begin{example}\label{exa}\rm Let $\Omega$ be an open bounded set of $\mathbb{R}^3$ with Lipschitz boundary. A simple example of application of Theorem \ref{application2} can be produced considering the real function \[ f(t):= \begin{cases} 0 & \text{if } t<0\\ t^{\alpha-1} & \text{if } 0\leq t\leq 1 \\ t^{\gamma-1} & \text{if } t> 1, \end{cases} \] where $2<\alpha<3$ and $1<\gamma<2$. In such a case there exist an open interval $\Lambda\subseteq(0,+\infty)$ and a positive real number $\rho$ with the following property: For each $\lambda\in\Lambda$ there exists $\delta>0$ such that, for each $\mu\in[0,\delta]$, the Dirichlet problem \begin{gather*} (-\Delta)^{1/2} u=\lambda f(u)+\mu (1+|u|^{\beta}) \quad\text{in } \Omega\\ u=0 \quad\text{in } \mathbb{R}^n\setminus \Omega, \end{gather*} where $\beta\in (0,2)$, has at least three non-trivial weak solutions $u_{i}\in H^{1/2}(\mathbb{R}^3)$ (with $i=1,2,3$) such that $u_i=0$ a.e. in $\mathbb{R}^n\setminus \Omega$ and whose norms are less than $\rho$. \end{example} \begin{remark}\label{discontinuo}\rm We point out that, recently, Teng in \cite{teng}, by using a non-smooth critical point theorem due to Arcoya and Carmona \cite{AC}, studied the existence of two non-trivial weak solutions for a parametric non-local hemivariational inequalities with the Dirichlet boundary condition. More precisely, they proved, for suitable value of the real parameters $\lambda$ and $\mu$, the existence of two non-trivial weak solutions for the Dirichlet problem \begin{gather*} -\mathcal L_K u\in \lambda(\partial j(x,u)+\mu \partial k(x,u)) \quad\text{in } \Omega\\ u=0 \quad\text{in } \mathbb{R}^{n}\setminus\Omega, \end{gather*} where $j,k:\Omega\times \mathbb{R}\to \mathbb{R}$ are suitable measurable functions such that, for almost every $x\in\Omega$, one has that $j(x,\cdot)$ and $k(x,\cdot)$ are locally Lipschitz continuous functions. Here, the expressions $\partial j(x,\cdot)$ and $\partial k(x,\cdot)$ denote the generalized subdifferential in the sense of Clarke. We just observe that Theorem \ref{application} and the smooth version of the above mentioned result are independent. \end{remark} \subsection*{Acknowlegements} B. Zhang was supported by the project ``Variational, topological and algebraic methods for nonlinear elliptic problems" when he was visiting the Medalics Research Centre in Reggio Calabria (Italy). B. Zhang was also supported by the Natural Science Foundation of Heilongjiang Province of China (A201306). \begin{thebibliography}{99} \bibitem{AC} D. Arcoya, J. Carmona; \emph{A nondifferentiable extension of a theorem of Pucci and Serrin and applications}, J. Differential Equations \textbf{235} (2007), 683--700. \bibitem{colorado} B. Barrios, E. Colorado, A. De Pablo, U. Sanchez; \emph{On some critical problems for the fractional Laplacian operator}, to appear in J. Differential Equations, available online at http://arxiv.org/abs/1106.6081. \bibitem{cabretan} X. Cabr\'e, J. Tan; \emph{Positive solutions of nonlinear problems involving the square root of the Laplacian}, Adv. Math. \textbf{224} (2010), 2052--2093. \bibitem{capella} A. Capella; \emph{Solutions of a pure critical exponent problem involving the half-Laplacian in annular-shaped domains}, Commun. Pure Appl. Anal. \textbf{10} (2011), 1645--1662. \bibitem{valpal} 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{fiscella} A. Fiscella; \emph{Saddle point solutions for non-local elliptic operators}, preprint (2012). \bibitem{fsv} A. Fiscella, R. Servadei, E. Valdinoci; \emph{A resonance problem for non-local elliptic operators}, Z. Anal. Anwendungen \textbf{32} (2013) 411--431. \bibitem{fsvBS} A. Fiscella, R. Servadei, E. Valdinoci; \emph{Asymptotically linear problems driven by the fractional Laplacian operator}, in preparation. \bibitem{kristaly} A. Krist\'aly, V. R\u{a}dulescu, Cs. Varga; \emph{Variational Principles in Mathematical Physics, Geometry, and Economics}. Qualitative Analysis of Nonlinear Equations and Unilateral Problems. With a Foreword by Jean Mawhin, Encyclopedia of Mathematics and its Applications, 136, Cambridge University Press, Cambridge (2010). \bibitem{Molica1} G. Molica Bisci, R. Servadei; \emph{A bifurcation result for non-local fractional equations}, to appear in Analysis and Applications. \bibitem{Molica2} G. Molica Bisci; \emph{Fractional equations with bounded primitive}, {Appl. Math. Lett.} \textbf{27} (2014), 53--58. \bibitem{Ricceri2000a} B. Ricceri; \emph{Existence of three solutions for a class of elliptic eigenvalue problems}, {Math. Comput. Modelling} \textbf{32} (2000), 1485--1494. \bibitem{Ricceri2009} B. Ricceri; \emph{A three critical points theorem revisited}, {Nonlinear Anal.} \textbf{70} (2009), 3084--3089. \bibitem{sY} R. Servadei; \emph{The Yamabe equation in a non-local setting}, Adv. Nonlinear Anal., \textbf{2} (2013) 235-270. \bibitem{sBNRES} R. Servadei; \emph{A critical fractional Laplace equation in the resonant case}, to appear in Topol. Methods Nonlinear Anal. \bibitem{sv} R. Servadei, E. Valdinoci; \emph{Lewy-Stampacchia type estimates for variational inequalities driven by nonlocal operators}, Adv. Nonlinear Stud. \textbf{12} (2012) 597-619. \bibitem{svmountain} R. Servadei, E. Valdinoci; \emph{Mountain Pass solutions for non-local elliptic operators}, J. Math. Anal. Appl. \textbf{389} (2012), 887--898. \bibitem{svlinking} R. Servadei, E. Valdinoci; \emph{Variational methods for non-local operators of elliptic type}, Discrete Contin. Dyn. Syst., \textbf{33} (2013) 2105--2137. \bibitem{servadeivaldinociBN} R. Servadei, E. Valdinoci; \emph{The Br\'{e}zis-Nirenberg result for the fractional Laplacian}, to appear in Trans. AMS. \bibitem{servadeivaldinociCFP} R. Servadei, E. Valdinoci; \emph{Fractional Laplacian equations with critical Sobolev exponent}, preprint (2012), available at http://www.math.utexas.edu/mp\_arc-bin/mpa?yn=12-58. \bibitem{tan} J. Tan; \emph{The Brezis-Nirenberg type problem involving the square root of the Laplacian}, Calc. Var. Partial Differential Equations \textbf{36} (2011), 21--41. \bibitem{teng} K. Teng; \emph{Two nontrivial solutions for hemivariational inequalities driven by nonlocal elliptic operators}, Nonlinear Analysis: Real World Applications \textbf{14} (2013), 867--874. \end{thebibliography} \end{document}