Content-Type: multipart/mixed; boundary="-------------1205180645177" This is a multi-part message in MIME format. ---------------1205180645177 Content-Type: text/plain; name="12-61.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="12-61.keywords" integrodifferential operators, fractional Laplacian, variational techniques, Saddle Point Theorem, Palais--Smale condition. ---------------1205180645177 Content-Type: application/x-tex; name="fiscella-servadei-valdinoci.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="fiscella-servadei-valdinoci.tex" % version of May 17, 2012 \documentclass[11pt]{amsart} \usepackage{graphicx, color} \usepackage{amscd} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{mathrsfs} \textwidth=6in \textheight=9.5in \topmargin=-0.5cm \oddsidemargin=0.5cm \evensidemargin=0.5cm %\usepackage[notref,notcite]{showkeys} \newtheorem{theorem}{Theorem} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{claim}{Claim} \newtheorem{step}{Step} \newenvironment{proofthmmain}{\mbox{\emph{Proof of Theorem~\ref{lapfra0f}:}}} {\hspace*{\fill}\mbox{$\Box$}} \numberwithin{equation}{section} \newcommand{\RR}{\mathbb R} \newcommand{\NN}{\mathbb N} \newcommand{\PP}{\mathbb P} \renewcommand{\le}{\leqslant} \renewcommand{\leq}{\leqslant} \renewcommand{\ge}{\geqslant} \renewcommand{\geq}{\geqslant} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \baselineskip=16pt plus 1pt minus 1pt \begin{document} %\hfill\today\bigskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title[A resonance problem for non-local elliptic operators] {A resonance problem \\for non-local elliptic operators} \thanks{The second author was supported by the MIUR National Research Project {\it Variational and Topological Methods in the Study of Nonlinear Phenomena}, while the third one by the MIUR National Research Project {\it Nonlinear Elliptic Problems in the Study of Vortices and Related Topics} and the FIRB project A\&B ({\it Analysis and Beyond}). All the authors were supported by the ERC grant $\epsilon$ ({\it Elliptic Pde's and Symmetry of Interfaces and Layers for Odd Nonlinearities}).} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \author[A. Fiscella]{Alessio Fiscella} \address{Dipartimento di Matematica, Universit\`a di Milano\\ Via Cesare Saldini 50, 20133 Milano, Italy} \email{\tt alessio.fiscella@unimi.it} \author[R. Servadei]{Raffaella Servadei} \address{Dipartimento di Matematica, Universit\`a della Calabria, Ponte Pietro Bucci 31 B, 87036 Arcavacata di Rende (Cosenza), Italy} \email{\tt servadei@mat.unical.it} \author[E. Valdinoci]{Enrico Valdinoci} \address{Dipartimento di Matematica, Universit\`a di Milano, Via Cesare Saldini 50, 20133 Milano, Italy} \email{\tt valdinoci@mat.uniroma2.it} \keywords{integrodifferential operators, fractional Laplacian, variational techniques, Saddle Point Theorem, Palais--Smale condition.\\ \phantom{aa} 2010 AMS Subject Classification: Primary: 49J35, 35A15, 35S15; Secondary: 47G20, 45G05.} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{abstract} In this paper we consider a resonance problem driven by a non-local integrodifferential operator $\mathcal L_K$ with homogeneous Dirichlet boundary conditions. This problem has a variational structure and we find a solution for it using the Saddle Point Theorem. We prove this result for a general integrodifferential operator of fractional type and from this, as a particular case, we derive an existence theorem for the following fractional Laplacian equation $$ \left\{ \begin{array}{ll} (-\Delta)^s u=\lambda a(x)u+f(x,u) & {\mbox{ in }} \Omega\\ u=0 & {\mbox{ in }} \mathbb{R}^n\setminus \Omega\,, \end{array} \right.$$ when $\lambda$ is an eigenvalue of the related non-homogenous linear problem with homogeneous Dirichlet boundary data. Here the parameter $s\in (0,1)$ is fixed, $\Omega$ is an open bounded set of $\RR^n$, $n>2s$, with Lipschitz boundary, $a$ is a Lipschitz continuous function, while $f$ is a sufficiently smooth function. This existence theorem extends to the non-local setting some results, already known in the literature in the case of the Laplace operator $-\Delta$\,. \end{abstract} \maketitle \tableofcontents \section{Introduction}\label{sec:intro} Nonlinear elliptic problems modeled by \begin{equation}\label{1} \left\{ \begin{array}{ll} -\Delta u=\lambda u+f(x,u) & {\mbox{ in }} \Omega\\ u=0 & {\mbox{ on }} \partial \Omega\,, \end{array} \right. \end{equation} where $\Omega\subset \RR^n$, $n>2$, is an open bounded set, $\lambda$ is a positive\footnote{Throughout this paper, by ``positive'', we mean ``strictly positive''.} parameter and the perturbation $f$ is a function satisfying different growth conditions (asymptotically linear, superlinear, subcritical or critical, for instance), were widely studied in the literature (see, for instance, \cite{am, struwe, willem} and references therein). In some recent papers these problems were treated in a non-local setting: in this framework see, for instance, \cite{fiscella} for the asymptotically linear case, \cite{cabretan, svmountain, svlinking} for subcritical nonlinearities and \cite{capella, sY, servadeivaldinociBN, servadeivaldinociBNLOW, servadeivaldinociCFP, tan} for the critical case. Aim of this paper is to consider the non-local version of problem~\eqref{1} in the case when the perturbation~$f:\overline{\Omega}\times \mathbb{R}\to \mathbb{R}$ is a function such that \begin{equation}\label{f1} f\in C(\overline{\Omega}\times\mathbb{R},\mathbb{R}); \end{equation} \begin{equation}\label{f2} \mbox{there exists a constant}\,\, M> 0\,\, \mbox{such that}\,\,\left|f(x,t)\right|\leq M\,\,\mbox{for any}\,\,(x,t)\in\Omega\times\mathbb{R}; \end{equation} \begin{equation}\label{f3} {\displaystyle F(x,t)=\int^{t}_{0}f(x,s)ds\rightarrow +\infty}\,\,\mbox{as}\,\,\left|t\right|\rightarrow +\infty\,\, \mbox{uniformly for}\,\, x\in \Omega. \end{equation} To be precise, in this paper we deal with the following problem \begin{equation}\label{op} \left\{\begin{array}{ll} -\mathcal L_K u=\lambda a(x)u+f(x,u) & \mbox{in } \Omega\\ u=0 & \mbox{in } \mathbb{R}^{n}\setminus \Omega\,, \end{array} \right. \end{equation} where $s\in(0,1)$ is fixed, $n> 2s$\,, $\Omega\subset\mathbb{R}^{n}$ is an open bounded set with Lipschitz boundary and $a:\overline{\Omega}\rightarrow\mathbb{R}$ is such that \begin{equation}\label{conda} a\,\,\,\, \mbox{is a positive Lipschitz continuous function in}\,\, \overline \Omega\,. \end{equation} Finally $\mathcal L_K$ is the non-local operator defined as follows \begin{equation}\label{lk} \mathcal L_Ku(x)= \int_{\RR^n}\Big(u(x+y)+u(x-y)-2u(x)\Big)K(y)\,dy\,, \,\,\,\,\, x\in \RR^n\,, \end{equation} where $K:\mathbb{R}^n\setminus\{0\}\rightarrow(0,+\infty)$ is a function with the properties that \begin{equation}\label{kernel} {\mbox{$m K\in L^1(\RR^n)$, where $m(x)=\min \{|x|^2, 1\}$\,;}} \end{equation} \begin{equation}\label{kernelfrac} \mbox{there exists}\,\, \theta>0\,\, \mbox{such that}\,\, K(x)\geq \theta |x|^{-(n+2s)}\,\, \mbox{for any}\,\, x\in \RR^n \setminus\{0\}\,;\\ \end{equation} \begin{equation}\label{evenkernel} K(x)=K(-x)\,\, \mbox{for any}\,\, x\in \RR^n \setminus\{0\}\,. \end{equation} A typical example for $K$ is given by $K(x)=\left|x\right|^{-(n+2s)}$. In this case problem \eqref{op} becomes \begin{equation}\label{p} \left\{\begin{array}{ll} (-\Delta)^{s} u=\lambda a(x)u+f(x,u) & \mbox{in } \Omega\\ u=0 & \mbox{in } \mathbb{R}^{n}\setminus \Omega\,, \end{array}\right. \end{equation} where $(-\Delta)^{s}$ is the fractional Laplace operator which (up to normalization factors) may be defined as \begin{equation} -(-\Delta)^s u(x)= \int_{\mathbb{R}^{n}}\frac{u(x+y)+u(x-y)-2u(x)}{|y|^{n+2s}}\,dy \end{equation} for $x\in\mathbb{R}^{n}$. We refer to \cite{valpal} and references therein for further details on the fractional Laplacian. One of the motivations for studying \eqref{p} (and, more generally, \eqref{op}) is trying to extend some important results, which are well known for the classical case of the Laplacian $-\Delta$ (see, e.g., \cite[Chapter~4 and Theorem~4.12]{rabinowitz}), to a non-local setting. The conditions we consider on $a$ and $f$ are classical in the nonlinear analysis (see, e.g., conditions $(p1)$, $(p2)$ and $(p7)$ in \cite[Theorem~4.12]{rabinowitz}) and, roughly speaking, they state that problem \eqref{op} is a suitable perturbation from the following non-homogenous eigenvalue problem \begin{equation}\label{problema autovalori} \left\{\begin{array}{ll} -\mathcal L_K u=\lambda a(x) u & \mbox{in } \Omega\\ u=0 & \mbox{in } \mathbb{R}^{n}\setminus\Omega\,. \end{array} \right. \end{equation} We recall that there exists a non-decreasing sequence of positive eigenvalues $\lambda_k$ for which \eqref{problema autovalori} admits non-trivial solutions. We will study problem \eqref{problema autovalori} in Subsection~\ref{subsec:eigenvalue}. Finally, note that, thanks to \eqref{f3}, the nonlinearity $f$ cannot be the trivial function. As a model for $f$ we can take the functions $$f(x,t)=M>0 \,\,\,\,\, \mbox{or}\,\,\,\,\, f(x,t)=b(x)\arctan t\,,$$ with $b\in Lip(\overline \Omega)$ and $b>0$ in $\Omega$\,. In the first case $u\equiv 0$ does not solve \eqref{op}, while in the second one the trivial function is a solution of \eqref{op}\,. In general, the function $u\equiv 0$ in $\RR^n$ is a solution of problem~\eqref{op} if and only if $f(\cdot, 0)=0$\,. This is an important difference with respect to the other works in the subject, such as~\cite{sY, svmountain, svlinking, servadeivaldinociBN, servadeivaldinociBNLOW, servadeivaldinociCFP}, where the trivial function is always a solution. The aim of this paper is to find solutions for \eqref{op} via variational methods. For this, firstly we need the weak formulation of \eqref{op}, which is given by the following problem (for this, it is worth to assume \eqref{evenkernel}) \begin{equation}\label{wf} \left\{\begin{array}{lll} \displaystyle\int_{\mathbb{R}^{2n}} (u(x)-u(y))(\varphi(x)-\varphi(y))K(x-y) dx\,dy= \displaystyle\lambda \int_{\Omega} a(x)u(x)\varphi(x)\,dx\\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad+\displaystyle\int_\Omega f(x, u(x))\varphi(x)\,dx \,\,\,\,\,\,\,\forall \,\varphi \in X_0\\ u\in X_0\,. \end{array} \right. \end{equation} Here 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 \begin{equation} \mbox{the map}\,\,\, (x,y)\mapsto (g(x)-g(y))\sqrt{K(x-y)}\,\,\, \mbox{is in}\,\,\, L^2\big(\mathbb{R}^{2n} \setminus ({\mathcal C}\Omega\times {\mathcal C}\Omega), dxdy\big)\,,\label{5.1} \end{equation} where ${\mathcal C}\Omega:=\mathbb{R}^n \setminus\Omega$. Moreover, \begin{equation} X_0=\{g\in X : g=0\,\, \mbox{a.e. in}\,\, \mathbb{R}^n\setminus \Omega\}\,.\label{5.2} \end{equation} We remark that $X$ and $X_0$ are non-empty since $C^2_0 (\Omega)\subseteq X_0$, by \cite[Lemma~11]{sv} and \eqref{kernel}\,. Working in $X_0$ allows us to encode the Dirichlet datum $u=0$ in $\RR^n\setminus \Omega$ in the weak formulation. The main result of the present paper can be stated as follows: \begin{theorem} \label{Thsaddle} Let $s\in(0,1)$, $n> 2s$ and $\Omega$ be an open bounded subset of $\mathbb{R}^{n}$ with Lipschitz boundary. Let $K:\mathbb{R}^n\setminus\{0\}\rightarrow(0,+\infty)$ be a function satisfying \eqref{kernel}--\eqref{evenkernel} and let $f:\overline{\Omega}\times \mathbb{R}\to \mathbb{R}$ and $a:\overline{\Omega}\rightarrow\mathbb{R}$ be two functions verifying \eqref{f1}--\eqref{f3} and \eqref{conda}, respectively. Moreover, assume that $\lambda$ is an eigenvalue of the non-homogeneous linear problem in~\eqref{problema autovalori}. Then, problem \eqref{op} admits a solution $u\in X_{0}$. \end{theorem} In the classical case of the Laplacian~$-\Delta$ the counterpart of Theorem~\ref{Thsaddle} is given in \cite[Theorem~4.12]{rabinowitz}: in this sense Theorem~\ref{Thsaddle} may be seen as the natural extension of classical results to the non-local fractional setting. The strategy for proving Theorem~\ref{Thsaddle} is based on the fact that problem~\eqref{wf} can be seen as the Euler--Lagrange equation of a suitable functional (see \eqref{J}). Hence, the solutions of \eqref{wf} can be found as critical points of this functional: at this purpose, along the paper, we will exploit the Saddle Point Theorem by Rabinowitz (see \cite{rabinowitz78, rabinowitz}). This paper is organized as follows. In Section~\ref{sec:preliminary} we will give some notations and we will recall some basic facts on the spectral theory for the operator $-\mathcal L_K$\,, while in Section~\ref{sec:lemmi} we will state and prove some technical lemmas useful along the paper. Finally, in Section~\ref{sec saddle} we will prove Theorem~\ref{Thsaddle} by making use of the classical Saddle Point Theorem. \section{Some preliminary facts}\label{sec:preliminary} \subsection{Notations}\label{subsec:notations} In the sequel the spaces $X$ and $X_0$ (whose definitions were recalled in the Introduction) will be endowed, respectively, with the norms 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} and \begin{equation}\label{normaX0} \|g\|_{X_0}=\left(\int_Q|g(x)-g(y)|^2K(x-y)\,dx\,dy\right)^{1/2}\,. \end{equation} Here $Q=\mathbb{R}^{2n}\setminus \mathcal O$\,, with ${\mathcal{O}}= ({\mathcal{C}}\Omega)\times({\mathcal{C}}\Omega)\subset\mathbb{R}^{2n}$and $\mathcal C\Omega=\mathbb{R}^n\setminus \Omega$\,. Note that, since $g\in X_0$ is such that $g=0$ a.e. in $\RR^n\setminus\Omega$, then in \eqref{normaX0} the integral on $Q$ can be extended to all $\RR^{2n}$\,. Moreover, the norm on $X_0$ given in \eqref{normaX0} is equivalent to the usual one defined in \eqref{norma}, by \cite[Lemmas~6 and 7]{svmountain}. With the norm given in \eqref{normaX0}, $X_0$ is a Hilbert space with scalar product defined as \begin{equation}\label{prodottoscalare} \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} For this see \cite[Lemma~7]{svmountain}. For further details on $X$ and $X_0$ and also for their properties we refer to \cite{sY, svmountain, svlinking, servadeivaldinociBN, servadeivaldinociBNLOW, servadeivaldinociCFP}\,. Note that, since $a\in L^\infty(\Omega)$ by \eqref{conda}, all the embeddings properties of $X_0$ into the usual Lebesgue space $L^2(\Omega)$ still hold true in $L^2(\Omega,\, \mu)$, with $\mu(\cdot)=a(\cdot)dx$\,, defined as \begin{eqnarray*} &&L^2(\Omega,\, \mu):=\Big\{g:\Omega \to \mathbb R\,\, \mbox{s.t.}\,\, g\,\, \mbox{is measurable in $\Omega$ and} \\ &&\qquad \,\, \int_\Omega a(x)|g(x)|^2\,dx=\int_\Omega |g|^2\,d\mu<+\infty\Big\}\,. \end{eqnarray*} In the following we will denote by $H^s(\Omega)$ the usual fractional Sobolev space endowed with the norm (the so-called \emph{Gagliardo norm}) \begin{equation}\label{gagliardonorm} \|g\|_{H^s(\Omega)}=\|g\|_{L^2(\Omega)}+ \Big(\int_{\Omega\times \Omega}\frac{\,\,\,|g(x)-g(y)|^2}{|x-y|^{n+2s}}\,dx\,dy\Big)^{1/2}\,. \end{equation} We remark that, even in the model case in which $K(x)=|x|^{-(n+2s)}$, the norms in \eqref{norma} and \eqref{gagliardonorm} are not the same, because $\Omega\times\Omega$ is strictly contained in $Q$. For further details on the fractional Sobolev spaces we refer to~\cite{valpal} and to the references therein. \subsection{An eigenvalue problem}\label{subsec:eigenvalue} This subsection is devoted to the study of the non-homogeneous eigenvalue problem \eqref{problema autovalori}. More precisely, we consider the weak formulation of \eqref{problema autovalori}, which consists in the following eigenvalue problem \begin{equation}\label{autovalore debole} \left\{\begin{array}{l} \displaystyle\int_{\mathbb{R}^{2n}} (u(x)-u(y))(\varphi(x)-\varphi(y))K(x-y) dx\,dy= \displaystyle\lambda\int_{\Omega} a(x)u(x)\varphi(x)dx \\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \forall\,\varphi \in X_0\\ u\in X_0. \end{array} \right. \end{equation} We recall that $\lambda\in\mathbb{R}$ is an eigenvalue of problem~\eqref{autovalore debole} provided there exists a non-trivial solution $u\in X_0$ of problem \eqref{autovalore debole} and, in this case, any solution will be called an eigenfunction corresponding to the eigenvalue $\lambda$. For the proof of the next result we refer to \cite[Proposition~9 and Appendix A]{svlinking}\,, where the problem~\eqref{autovalore debole} with $a\equiv 1$ was considered (the case of $a\not\equiv 1$ can be proved similarly, just replacing the classical space~$L^2(\Omega)$ with~$L^2(\Omega,\,\mu)$)\,. \begin{proposition} \label{autovalori} Let $s\in(0,1)$, $n>2s$, $\Omega$ be an open bounded subset of $\mathbb{R}^{n}$ and let $K:\mathbb{R}^{n}\setminus \left\{0\right\}\rightarrow(0,+\infty)$ be a function satisfying assumptions \eqref{kernel}--\eqref{evenkernel}. Moreover, let $a:\overline{\Omega}\rightarrow\mathbb{R}$ be a function verifying \eqref{conda}. Then, \renewcommand{\labelenumi}{(\roman{enumi})} \begin{enumerate} \item problem \eqref{autovalore debole} admits an eigenvalue $\lambda_1$ which is positive and that can be characterized as follows \begin{equation*} \lambda_1=\min_{{u\in X_0} \atop{\|u\|_{L^2(\Omega,\, \mu)}=1}}\int_{\mathbb{R}^{2n}}\left|u(x)-u(y)\right|^{2}K(x-y)dx\,dy, \end{equation*} or, equivalently, \begin{equation} \lambda_1=\min_{u\in X_0\setminus{\left\{0\right\}}} \frac{\displaystyle\int_{\mathbb{R}^{2n}}\left|u(x)-u(y)\right|^{2}K(x-y)dx\,dy} {\displaystyle \int_{\Omega}a(x)\left|u(x)\right|^{2}dx}\,,\label{lambda1} \end{equation} where $\|\cdot\|_{L^2(\Omega,\,\mu)}$ denotes the $L^2$--norm with respect to the measure $\mu(x)=a(x)dx$; \item there exists a non-negative function $e_1\in X_0$, which is an eigenfunction corresponding to $\lambda_1$, attaining the minimum in \eqref{lambda1}, that is $\|e_1\|_{L^2(\Omega,\,\mu)}=1$ and \begin{equation*} \lambda_1=\int_{\mathbb{R}^{2n}}\left|e_1(x)-e_1(y)\right|^{2}K(x-y)dx\,dy; \end{equation*} \item $\lambda_1$ is simple, that is if $u\in X_0$ is a solution of the following equation \begin{equation*} \int_{\mathbb{R}^{2n}}(u(x)-u(y))(\varphi(x)-\varphi(y))K(x-y)dx\,dy=\lambda_1\int_{\Omega}a(x)u(x)\varphi(x)dx\quad \forall\varphi\in X_0, \end{equation*} then $u=\zeta e_1$, with $\zeta\in\mathbb{R}$; \item the set of the eigenvalues of problem \eqref{autovalore debole} consists of a sequence $\left\{\lambda_k\right\}_{k\in\mathbb{N}}$ with\footnote{As usual, here we call $\lambda_1$ the \emph{first eigenvalue} of the operator $-\mathcal L_K$\,. This notation is justified by~\eqref{ordine lambdak}. Notice also that some of the eigenvalues in the sequence $\big\{ \lambda_k\big\}_{{k\in\NN}}$ may repeat, i.e. the inequalities in~\eqref{ordine lambdak} may be not always strict.} \begin{equation} 0<\lambda_1<\lambda_2\leq\ldots\leq\lambda_k\leq\lambda_{k+1}\leq\ldots\label{ordine lambdak} \end{equation} and \begin{equation*} \lambda_k\rightarrow +\infty\,\,\mbox{as }\,\,k\rightarrow +\infty. \end{equation*} Moreover, for any $k\in\mathbb{N}$ the eigenvalues can be characterized as follows: \begin{equation*} \lambda_{k+1}=\min_{{u\in \PP_{k+1}}\atop{\|u\|_{L^2(\Omega,\, \mu)}=1} } \int_{\mathbb{R}^{2n}}\left|u(x)-u(y)\right|^{2}K(x-y)dx\,dy, \end{equation*} or, equivalently, \begin{equation} \lambda_{k+1}=\min_{u\in \mathbb P_{k+1}\setminus{\left\{0\right\}}}\frac{\displaystyle \int_{\mathbb{R}^{2n}}\left|u(x)-u(y)\right|^{2}K(x-y)dx\,dy} {\displaystyle\int_{\Omega}a(x)\left|u(x)\right|^{2}dx},\label{lambdak+1} \end{equation} where \begin{equation}\label{pk+1} \mathbb P_{k+1}:=\left\{u\in X_0:\,\left\langle u,e_j\right\rangle_{X_0}=0\quad\forall j=1,\ldots,k\right\}; \end{equation} \item for any $k\in\mathbb{N}$ there exists a function $e_{k+1}\in\mathbb P_{k+1}$, which is an eigenfunction corresponding to $\lambda_{k+1}$, attaining the minimum in \eqref{lambdak+1}, that is $\|e_{k+1}\|_{L^2(\Omega,\,\mu)}=1$ and \begin{equation} \lambda_{k+1}=\int_{\mathbb{R}^{2n}}\left|e_{k+1}(x)-e_{k+1}(y)\right|^{2}K(x-y)dx\,dy;\label{ek} \end{equation} \item the sequence $\left\{e_k\right\}_{k\in\mathbb{N}}$ of eigenfunctions corresponding to $\lambda_k$ is an orthonormal basis of $L^{2}(\Omega, \mu)$ and an orthogonal basis of $X_0$; \item each eigenvalue $\lambda_k$ has finite multiplicity; more precisely, if $\lambda_k$ is such that \begin{equation*} \lambda_{k-1}<\lambda_k=\ldots=\lambda_{k+h}<\lambda_{k+h+1} \end{equation*} for some $h\in\mathbb{N}_0$, then the set of all the eigenfunctions corresponding to $\lambda_k$ agrees with \begin{equation*} span\left\{e_k,\ldots,e_{k+h}\right\}. \end{equation*} \end{enumerate} \end{proposition} In particular, Proposition~\ref{autovalori} gives a variational characterization of the eigenvalues $\lambda_k$ of $-\mathcal L_K$ (see formulas~\eqref{lambda1} and \eqref{lambdak+1})\,. Another interesting characterization of the eigenvalues is given in the next result. For the proof we refer to \cite[Proposition~5]{sY}\,, where the case $a\equiv 1$ was treated (again, the case of $a\not\equiv 1$ can be proved likewise)\,. \begin{proposition}\label{propcaratterizzazione} Let $\big\{ \lambda_k\big\}_{{k\in\NN}}$ be the sequence of the eigenvalues given in Proposition~\ref{autovalori} and let $\big\{e_k\big\}_{{k\in\NN}}$ be the corresponding sequence of eigenfunctions\,. Then, for any $k\in \NN$ the eigenvalues can be characterized as follows: $$\lambda_k=\max_{u\in \mbox{span}\{ e_1, \dots, e_k\}\setminus\{0\}}\frac{{\displaystyle \int_{\RR^{2n}}|u(x)-u(y)|^2 K(x-y)dx\,dy}}{{\displaystyle \int_\Omega a(x)|u(x)|^2\,dx}}\,.$$ \end{proposition} %% Both Propositions~\ref{autovalori} and~\ref{propcaratterizzazione} %% will be useful in the sequel. We conclude this subsection with some notation. In what follows, without loss of generality, we will fix $\lambda=\lambda_k$ with $k\in \NN$ such that $\lambda_k<\lambda_{k+1}$ and we will denote by $\mathbb H_k$ the linear subspace of $X_0$ generated by the first $k$ eigenfunctions of $-\mathcal L_{K}$\,, i.e. $$\mathbb H_k:=\mbox{span}\left\{e_{1},\ldots,e_{k}\right\}\,,$$ while $\mathbb P_{k+1}$ will be the space defined in \eqref{pk+1}\,. Here $e_j$ and $\lambda_j$\,, $j\in \NN$\,, are the eigenfunctions and the eigenvalues of $-\mathcal L_{K}$, as defined in Proposition~\ref{autovalori}\,. It is immediate to observe that $\mathbb P_{k+1}=\mathbb H^{\bot}_k$ with respect to the scalar product in $X_0$ defined as in formula~\eqref{prodottoscalare}. Thus, since $X_0$ is a Hilbert space (see \cite[Lemma~7]{svmountain} and \eqref{prodottoscalare}), we can write it as a direct sum as follows $$X_0=\mathbb H_k\oplus\mathbb P_{k+1}\,.$$ Moreover, since $\left\{e_1\,, \dots\,, e_k\,, \dots \right\}$ is an orthogonal basis of $X_0$\,, it follows that $$\mathbb P_{k+1}=\overline{\mbox{span}\left\{e_j:\,\,j\geq k+1\right\}}\,.$$ Also we will set \begin{equation}\label{E0k} \mathbb E^0_k:=\mbox{span}\left\{e_j:\,\,\lambda_j=\lambda_k\right\} \quad\mbox{and}\quad \mathbb E^-_k:=\mbox{span}\left\{e_j:\,\,\lambda_j<\lambda_k\right\}\,.\end{equation} Note that with this notation, if $u\in \mathbb H_k$, then we can write it as $$u=u^0 +u^-\,,\,\,\, \mbox{with}\,\,\, u^0 \in \mathbb E^0_k\,\,\, \mbox{and}\,\,\, u^- \in \mathbb E^-_k\,.$$ \section{Some technical lemmas}\label{sec:lemmi} In this section we prove some technical lemmas, which will be useful in order to apply the Saddle Point Theorem to problem~\eqref{wf}\,. \begin{lemma}\label{lemma1} Let $K:\mathbb{R}^{n}\setminus \left\{0\right\}\rightarrow(0,+\infty)$ satisfy assumptions \eqref{kernel}--\eqref{evenkernel} and let $a:\overline \Omega \to \RR$ verify \eqref{conda}\,. Then, for any $u\in\mathbb P_{k+1}$ \[\int_{\RR^{2n}}|u(x)-u(y)|^2 K(x-y)\,dx\,dy-\lambda_k \int_\Omega a(x)|u(x)|^2\,dx\geq \left(1-\frac{\lambda_k}{\lambda_{k+1}}\right)\|u\|^{2}_{X_0}\,. \] \end{lemma} \begin{proof} If $u\equiv 0$, then the assertion is trivial. Now, let $u\in\mathbb P_{k+1}\setminus\{0\}$\,. By the variational characterization of $\lambda_{k+1}$ given in \eqref{lambdak+1} we get that $$\|u\|_{L^2(\Omega,\, \mu)}^2\leq \frac{1}{\lambda_{k+1}}\|u\|_{X_0}^2\,.$$ As a consequence of this and taking into account that $\lambda_k$ is positive (since $\lambda_k\geq \lambda_1>0$), we obtain $$\begin{aligned} \int_{\RR^{2n}}|u(x)-u(y)|^2 K(x-y)\,dx\,dy-\lambda_k \int_\Omega a(x)|u(x)|^2\,dx & \geq \|u\|_{X_0}^2-\frac{\lambda_k}{\lambda_{k+1}}\|u\|_{X_0}^2\\ & = \left(1-\frac{\lambda_k}{\lambda_{k+1}}\right)\|u\|^{2}_{X_0}, \end{aligned}$$ concluding the proof. \end{proof} Note that, if $\lambda_k=\lambda_{k+1}$, then Lemma~\ref{lemma1} is trivial. The interesting case is when $\lambda_k<\lambda_{k+1}$\,. \begin{lemma}\label{lemma2} Let $K:\mathbb{R}^{n}\setminus \left\{0\right\}\rightarrow(0,+\infty)$ satisfy assumptions \eqref{kernel}--\eqref{evenkernel} and let $a:\overline \Omega \to \RR$ verify \eqref{conda}\,. Then, there exists a positive constant $M^*$, depending on $k$, such that \[\int_{\RR^{2n}}|u(x)-u(y)|^2 K(x-y)\,dx\,dy-\lambda_k \int_\Omega a(x)|u(x)|^2\,dx\leq -M^*\|u^-\|^{2}_{X_0}\] for all $u\in \mathbb H_k$, where $u=u^-+u^0$, $u^-\in E^-_k$ and $u^0\in E^0_k$\,. \end{lemma} \begin{proof} Of course, if $u\equiv 0$, then the assertion is trivial. Hence, assume that $u\in \mathbb H_k\setminus\{0\}$\,. Let $h\in \NN$ be the multiplicity of $\lambda_k$ ($h$ is finite thanks to Proposition~\ref{autovalori}-$(vii)$), that is suppose that \begin{equation}\label{h} \lambda_{k-h-1}<\lambda_{k-h}= \dots = \lambda_k<\lambda_{k+1}\,. \end{equation} With this notation, $u$ can be written as follows $$u=u^-+u^0\,,$$ with $$u^-\in \mathbb E^-_k=\mbox{span}\left\{e_{1},\ldots,e_{k-h-1}\right\}\,\,\,\,\, \mbox{and}\,\,\,\,\, u^0\in \mathbb E^0_k=\mbox{span}\left\{e_{k-h},\ldots,e_{k}\right\}\,.$$ Notice that~$u^0$ is a linear combination of eigenfunctions corresponding to the same eigenvalue~$\lambda_{k-h}=\dots=\lambda_k$, hence it is also an eigenfunction corresponding to~$\lambda_k$. Hence, by~\eqref{autovalore debole}, $$ \|u^0\|^2_{X_0}=\lambda_k\|u^0\|^2_{L^2(\Omega,\mu)}.$$ Also,~$u^-$ and~$u^0$ are orthogonal both in~$X_0$ and in~$L^2(\Omega,\,\mu)$, therefore \begin{equation}\label{ortou} \begin{aligned} \|u\|_{X_0}^2-\lambda_k\|u\|_{L^2(\Omega,\, \mu)}^2 & =\|u^-\|_{X_0}^2+\|u^0\|_{X_0}^2-\lambda_k\left(\|u^-\|_{L^2(\Omega,\, \mu)}^2 +\|u^0\|_{L^2(\Omega,\, \mu)}^2\right) \\ & = \|u^-\|_{X_0}^2-\lambda_k\|u^-\|_{L^2(\Omega,\, \mu)}^2\,. \end{aligned} \end{equation} Now, note that $u^-\in \mathbb E^-_k= \mbox{span}\left\{e_{1},\ldots,e_{k-h-1}\right\}$\,. Hence, by this and Proposition~\ref{propcaratterizzazione} we get \begin{equation}\label{caratterizzazioneu-} \|u^-\|_{X_0}^2\leq \lambda_{k-h-1}\|u^-\|_{L^2(\Omega,\, \mu)}^2\,. \end{equation} Finally, \eqref{ortou} and \eqref{caratterizzazioneu-} yield $$\begin{aligned} \|u\|_{X_0}^2-\lambda_k\|u\|_{L^2(\Omega,\, \mu)}^2 & = \|u^-\|_{X_0}^2-\lambda_k\|u^-\|_{L^2(\Omega,\, \mu)}^2\\ & \leq \|u^-\|_{X_0}^2-\frac{\lambda_k}{\lambda_{k-h-1}}\|u^-\|_{X_0}^2\\ & = \left(1-\frac{\lambda_k}{\lambda_{k-h-1}}\right)\|u^-\|_{X_0}^2\,, \end{aligned}$$ which gives the desired assertion with $$M^*:=\frac{\lambda_k}{\lambda_{k-h-1}}-1\,.$$ Note that $M^*>0$, thanks to \eqref{h}\,. \end{proof} Finally, in the next two results we discuss some properties of the function~$F$ defined as in \eqref{f3}\,. \begin{lemma}\label{lemma3} Let $f:\overline{\Omega}\times\mathbb{R}\rightarrow\mathbb{R}$ satisfy \eqref{f1}--\eqref{f3}\,. Then, there exists a positive constant $\widetilde M$, depending on $\Omega$, such that \[\left|\int_\Omega F(x,u(x))\,dx\right|\leq \widetilde M\|u\|_{X_0} \] for all $u\in X_0$\,. \end{lemma} \begin{proof} Using the definition of $F$ and \eqref{f2}, it is easy to see that $$\left|\int_\Omega F(x,u(x))\,dx\right|=\left|\int_\Omega \int_0^{u(x)}f(x,t)\,dt\,dx\right|\leq M\int_\Omega |u(x)|\,dx\,,$$ so that, by H\H older inequality and \cite[Lemma~8]{svmountain} we get \begin{equation} \left|\int_\Omega F(x,u(x))\,dx\right|\leq M\left|\Omega\right|^{1/2}\|u\|_{L^2(\Omega)}\leq \widetilde M\|u\|_{X_0} \end{equation} for all $u\in X_0$, where $\widetilde M$ is a positive constant depending on $\Omega$. Hence, the assertion is proved. \end{proof} \begin{lemma}\label{lemma4} Let $f:\overline{\Omega}\times\mathbb{R}\rightarrow\mathbb{R}$ satisfy \eqref{f1}--\eqref{f3}\,. Then, \[\lim_{u\in\, \mathbb E^0_k \atop{\|u\|_{X_0}\rightarrow+\infty}}\int_\Omega F(x,u(x))\,dx=+\infty\,.\] \end{lemma} \begin{proof} \noindent We argue by contradiction and suppose that there exists a positive constant $C$ and a sequence $u_j\in E^0_k$ such that \begin{equation}\label{tj} t_j:=\|u_j\|_{X_0}\rightarrow+\infty \end{equation} and \begin{equation} \int_\Omega F(x,u_j(x))dx\leq C\,.\label{assurdo} \end{equation} Let $v_j:=u_j/\|u_j\|_{X_0}$\,. Of course, $v_j$ is bounded in $X_0$. Hence, since $\mathbb E^0_k$ is finite dimensional, there exists $v\in \mathbb E^0_k$ such that $v_j$ converges to $v$ strongly in $X_0$. Note also that $v\not \equiv 0$, since $$\|v\|=\lim_{j\to +\infty}\|v_j\|=1\,.$$ Furthermore, recalling~\cite[Lemma~8]{svmountain}, \begin{equation}\label{convergenzavj1} v_j\rightarrow v \quad\mbox{in } L^{q}(\mathbb{R}^{n})\,\,\,\mbox{for any}\,\,\, q\in [1,2^{*}) \end{equation} and, by applying \cite[Theorem~IV.9]{brezis}, up to a subsequence (still denoted by $v_j$) \begin{equation}\label{convergenzavj2} v_j\rightarrow v \quad\mbox{a.e. in } \mathbb{R}^{n} \end{equation} as $j\to +\infty$\,. Now, we define $i(r):=\displaystyle\inf_{x\in\overline{\Omega},\,\left|t\right|\geq r} F(x,t)$ for $r>0$. By \eqref{f3} it follows that \begin{equation} \lim_{r\rightarrow+\infty}i(r)=+\infty.\label{infinito} \end{equation} Note that \begin{equation}\label{inf} \displaystyle\inf_{x\in\overline{\Omega},\,t \in\mathbb{R}}F(x,t)\,\,\, \mbox{is finite.} \end{equation} Indeed, by \eqref{f3} it follows that for any $H>0$ there exists $R>0$ such that \begin{equation} F(x,t)> H\,\,\,\mbox{for any} \left|t\right|> R\,\,\mbox{and any}\,\, x\in\Omega.\label{maggiore} \end{equation} Moreover, if $\left|t\right|\leq R$, by \eqref{f2} we have \begin{equation} \left|F(x,t)\right|\leq M\left|t\right|\leq MR=:C_R,\label{minore} \end{equation} for any $x\in\Omega$\,. Hence, by \eqref{maggiore} and \eqref{minore} we can conclude that $$F(x,t)\geq -C_R\,\,\,\, \mbox{for any}\,\,\, (x,t)\in\Omega\times\mathbb{R}\,,$$ which implies \eqref{inf}\,. As a consequence of~\eqref{inf}, we may define $$ \omega^*:= -\min\left\{ -1,\; \displaystyle\inf_{x\in\overline{\Omega},\,t \in\mathbb{R}}F(x,t) \right\}.$$ Notice that~$\omega^*\ge0$ and~$F(x,t)\ge-\omega^*$ for any~$ x\in\overline{\Omega}$ and any~$t \in\mathbb{R}$. Now, we fix $h>0$ and set $\Omega_{j,\,h}=\left\{x\in\Omega:\,\, \left|t_j v_j(x)\right|\geq h\right\}$. Thus, we get \begin{equation}\label{disuguaglianza} \begin{aligned} \int_\Omega F(x,t_j v_j(x))dx & = \int_{\Omega_{j,\,h}} F(x,t_j v_j(x))dx+ \int_{\Omega\setminus\Omega_{j,\,h}} F(x,t_j v_j(x))dx\\ & \geq \left|\Omega_{j,\,h}\right| i(h)-\omega^*\, \left|\Omega\right|\,. \end{aligned} \end{equation} Since $v\not\equiv 0$, there exists a set $\Omega^\sharp$ with $\left|\Omega^\sharp\right|>0$ and a constant $\delta>0$ such that $\left|v(x)\right|\geq \delta$ a.e. $x\in \Omega^\sharp$. Then, by \eqref{convergenzavj2} and Egorov Theorem, there exists a measurable set $\Omega^*\subseteq\Omega^\sharp$ such that~$|\Omega^*|\ge|\Omega^\sharp|/2>0$ and the limit in~\eqref{convergenzavj2} is uniform in~$\Omega^*$. In particular, if~$j$ is large enough, $$ \sup_{x\in\Omega^*}|v_j(x)-v(x)|\le\frac\delta4$$ and therefore~$ |v_j(x)|\ge3\delta/4$ a.e.~$x\in\Omega^*$\,. So, by \eqref{tj}, for $h$ fixed above there exists $j_h$ such that $\left|t_j v_j(x)\right|\geq h$ for any $j\geq j_h$ and a.e. $x\in\Omega^*$. As a consequence of this, we have that $\Omega^*\subseteq\Omega_{j,\,h}$ for $j\geq j_h$. Finally, by \eqref{assurdo} and \eqref{disuguaglianza}, we have \begin{equation*} C\geq\int_\Omega F(x,t_j v_j(x))dx\geq \left|\Omega^*\right|i(h) -\omega^*\,\left|\Omega\right| \end{equation*} for $j\geq j_h$. Passing to the limit as $h\to +\infty$ and taking into account \eqref{infinito}, we get a contradiction. This proves the assertion. \end{proof} \section{Main result of the paper}\label{sec saddle} This section is devoted to the proof of Theorem~\ref{Thsaddle}, which is the main result of the present paper. At this purpose, first of all we observe that problem~\eqref{wf} has a variational structure, indeed it is the Euler--Lagrange equation of the functional $\mathcal J:X_0\to \mathbb{R}$ defined as follows \begin{equation}\label{J} \mathcal J(u)=\frac 1 2 \int_{\RR^{2n}}|u(x)-u(y)|^2 K(x-y)\,dx\,dy-\frac \lambda 2 \int_\Omega a(x)|u(x)|^2\,dx-\int_\Omega F(x, u(x)dx\,, \end{equation} where $F$ was introduced in \eqref{f3}. Note that the functional $\mathcal J$ is Fr\'echet differentiable in $u\in X_0$ and for any $\varphi\in X_0$ $$\begin{aligned} \langle \mathcal J'(u), \varphi\rangle & = \int_{\RR^{2n}} \big(u(x)-u(y)\big)\big(\varphi(x)-\varphi(y)\big)K(x-y)\,dx\,dy\\ & \qquad \qquad \qquad \qquad \qquad \qquad -\lambda \int_\Omega a(x)u(x)\varphi(x)\,dx-\int_\Omega f(x, u(x))\varphi(x)\,dx\,. \end{aligned}$$ Thus, critical points of $\mathcal J$ are weak solutions to problem~\eqref{op}. In order to find these critical points, in the sequel we will apply the Saddle Point Theorem by Rabinowitz (see \cite{rabinowitz78, rabinowitz}). For this, as usual for minimax theorems, we have to check that the functional $\mathcal J$ has a particular geometric structure (as stated, in our case, in conditions~($I_{3}$) and ($I_{4}$) of \cite[Theorem~4.6]{rabinowitz}) and that it satisfies the Palais--Smale compactness condition (see, for instance, \cite[p.~3]{rabinowitz}). \subsection{Geometry of the functional~$\mathcal J$}\label{subsec:geometry} In this subsection we will prove that the functional $\mathcal J$ has the geometric features required by the Saddle Point Theorem. \begin{proposition}\label{prop Hk ortogonale} Let $K:\mathbb{R}^{n}\setminus \left\{0\right\}\rightarrow(0,+\infty)$ satisfy assumptions \eqref{kernel}--\eqref{evenkernel}. Moreover, let $\lambda=\lambda_k<\lambda_{k+1}$ for some $k\in \NN$ and let $f$ and $a$ be two functions satisfying \eqref{f1}--\eqref{f3} and \eqref{conda}, respectively. Then \begin{equation} \liminf_{u\in\mathbb P_{k+1} \atop{\|u\|_{X_{0}}\rightarrow +\infty}}\frac{\mathcal J(u)}{\|u\|^{2}_{X_{0}}}>0. \label{coercivit? Hk ortogonale} \end{equation} \end{proposition} \begin{proof} Since $u\in\mathbb P_{k+1}$, by Lemmas~\ref{lemma1} and \ref{lemma3} we have \[\mathcal J(u)\geq\frac{1}{2}\left(1-\frac{\lambda_k}{\lambda_{k+1}}\right)\|u\|^{2}_{X_0}-\widetilde M\|u\|_{X_0}\,. \] Hence, dividing both the sides of this expression by $\|u\|^{2}_{X_0}$ and passing to the limit as $\|u\|_{X_0}\rightarrow +\infty$, we get \eqref{coercivit? Hk ortogonale}, since $\lambda_k<\lambda_{k+1} $ by assumption. \end{proof} \begin{proposition}\label{prop Hk} Let $K:\mathbb{R}^{n}\setminus \left\{0\right\}\rightarrow(0,+\infty)$ satisfy assumptions \eqref{kernel}--\eqref{evenkernel}. Moreover, let $\lambda=\lambda_k<\lambda_{k+1}$ for some $k\in \NN$ and let $f$ and $a$ be two functions satisfying \eqref{f1}--\eqref{f3} and \eqref{conda}, respectively. Then $$\lim_{u\in\mathbb H_k \atop{\|u\|_{X_{0}}\rightarrow +\infty}}\mathcal J(u)= -\infty\,.$$ \end{proposition} \begin{proof} Since $u\in \mathbb H_k$, we can write $u=u^- +u^0$, with $u^-\in \mathbb E^-_k$ and $u^0\in \mathbb E^0_k$\,. Also, $\mathcal J(u)$ can be written as follows \begin{equation} \begin{alignedat}2 &\mathcal J(u)=\frac{1}{2}\int_{\RR^{2n}}|u(x)-u(y)|^2 K(x-y)\,dx\,dy-\frac{\lambda_k}{2}\int_\Omega a(x)|u(x)|^2dx\\ &\qquad\qquad\qquad-\int_\Omega \Big(F(x,u^0(x)+u^-(x))-F(x,u^0(x))\Big)\,dx-\int_\Omega F(x,u^0(x))\,dx.\label{4.15} \end{alignedat} \end{equation} First of all, note that, by \eqref{f2}, H\H older inequality and \cite[Lemma~8]{svmountain}, it follows that \begin{equation}\label{4.16b} \begin{aligned} \Big|\int_\Omega \Big(F(x,u^0(x)+u^-(x))-& F(x,u^0(x))\Big)\,dx\Big| =\left|\int_\Omega\int^{u^0(x) +u^-(x)}_{u^0(x)}f(x,t)dt\,dx\right|\\ &\leq M\int_\Omega\left|u^-(x)\right|dx\leq M\left|\Omega\right|^{1/2}\|u^-\|_{L^2(\Omega)}\\ & \leq \overline M\|u^-\|_{X_0}\,, \end{aligned} \end{equation} where $\overline M$ denotes a positive constant depending on $\Omega$\,. Thus, by \eqref{4.15}, \eqref{4.16b} and Lemma~\ref{lemma2}, we get \begin{equation} \mathcal J(u)\leq -M^*\|u^-\|^{2}_{X_0}+\overline M\|u^-\|_{X_0}-\int_\Omega F(x,u^0(x))\,dx.\label{4.16c} \end{equation} Beware that the first norm in the right hand side of~\eqref{4.16c} is squared, while the second one is not. Moreover, by orthogonality we have $$\|u\|^{2}_{X_0}=\|u^0\|^{2}_{X_0}+\|u^-\|^{2}_{X_0}\,.$$ Then, as $\|u\|_{X_0}\rightarrow +\infty$, we have that at least one of the two norms, either $\|u^0\|_{X_0}$ or $\|u^-\|_{X_0}$, goes to infinity. Suppose that $\|u^0\|_{X_0}\rightarrow+\infty$ (in this case $\|u^-\|_{X_0}$ can be finite or not). Then, \eqref{4.16c}, the fact that $u^0\in \mathbb E^0_k$ and Lemma~\ref{lemma4} show that $\mathcal J(u)\rightarrow -\infty$ as $\|u\|_{X_0}\rightarrow +\infty$ and so, Proposition~\ref{prop Hk} follows. Otherwise, assume that $\|u^0\|_{X_0}$ is finite. In this setting, of course, \begin{equation}\label{u-infty} \|u^-\|_{X_0}\rightarrow+\infty\,. \end{equation} and, by Lemma~\ref{lemma3}, $\displaystyle\int_\Omega F(x,u^0(x))\,dx$ is also finite. Moreover, by \eqref{4.16c} and \eqref{u-infty}, we have that $\mathcal J(u)\rightarrow -\infty$ as $\|u\|_{X_0}\rightarrow +\infty$\,. This completes the proof of Proposition~\ref{prop Hk}\,. \end{proof} \subsection{The Palais--Smale condition}\label{subsec:PS} In this subsection we discuss a compactness property for the functional~$\mathcal J$, given by the Palais--Smale condition. First of all, as usual when using variational methods, we prove the boundedness of a Palais--Smale sequence for~$\mathcal J$. We say that $u_j$ is a Palais--Smale sequence for $\mathcal J$ at level $c\in \RR$ if \begin{equation}\label{palais smale1} |\mathcal J(u_{j})|\leq c, \end{equation} and \begin{equation}\label{palais smale2} \sup\Big\{|\langle \mathcal J^{'}(u_{j}),\varphi\rangle|:\,\varphi\in X_0,\,\|\varphi\|_{X_{0}}=1\Big\}\rightarrow 0\,\,\,\, \mbox{as $j\rightarrow +\infty$} \end{equation} hold true. \begin{proposition}\label{prop palais} Let $K:\mathbb{R}^{n}\setminus \left\{0\right\}\rightarrow(0,+\infty)$ satisfy assumptions \eqref{kernel}--\eqref{evenkernel}. Moreover, assume that $\lambda=\lambda_k<\lambda_{k+1}$ for some $k\in\mathbb{N}$ and let $f$ and $a$ be two functions satisfying \eqref{f1}--\eqref{f3} and \eqref{conda}, respectively. Finally, let $c\in\mathbb{R}$ and let $u_j$ be a sequence in $X_{0}$ verifying \eqref{palais smale1} and \eqref{palais smale2}\,. Then, the sequence $u_j$ is bounded in $X_{0}$. \end{proposition} \begin{proof} Let $u_{j}=u^{0}_{j}+u^{-}_{j}+u^{+}_{j}$, where $u^{0}_{j}\in \mathbb E^0_k$, $u^{-}_{j}\in \mathbb E^-_k$ and $u^{+}_{j}\in\mathbb P_{k+1}$. In order to prove Proposition~\ref{prop palais}, we will show that the sequences $u^{0}_{j}$, $u^{-}_{j}$ and $u^{+}_{j}$ are bounded in $X_0$\,. First of all, by \eqref{palais smale2}, for large $j$, we get \begin{equation} \begin{aligned} &\|u^{\pm}_{j}\|_{X_0}\ge\left|\langle \mathcal J^{'}(u_j), u^{\pm}_{j} \rangle\right|\\ & \qquad=\left|\int_{\RR^{2n}} \Big(u_j(x)-u_j(y)\Big)\Big(u^{\pm}_{j}(x)-u^{\pm}_{j}(y)\Big)K(x-y)\,dx\,dy\right.\\ & \left.\qquad \qquad \qquad -\lambda_k\int_\Omega a(x)|u_j^\pm(x)|^2\,dx-\int_\Omega f(x, u_j(x))u^{\pm}_{j}(x)\,dx\right|\,.\label{4.17} \end{aligned} \end{equation} While, by \eqref{f2}, the H\H older inequality and \cite[Lemma~8]{svmountain} \begin{equation} \left|\int_\Omega f(x,u_j(x))u^\pm_{j}(x)\,dx\right|\leq \tilde M\|u^\pm_{j}\|_{X_0},\label{4.17a} \end{equation} with $\tilde M$ positive constant. Finally, taking into account that $\big\{e_1, \dots, e_k \dots\big\}$ is a orthogonal basis of $X_0$ and of $L^2(\Omega, d\mu)$\,, $d\mu=a(\cdot)dx$\,, we get that \begin{equation}\label{j'u+} \begin{aligned} \langle \mathcal J^{'}(u_j), u^{\pm}_{j}\rangle & =\int_{\RR^{2n}} |u^\pm_{j}(x)-u^\pm_{j}(y)|^2K(x-y)\,dx\,dy-\lambda_k\int_\Omega a(x)|u_j^\pm(x)|^2\,dx\\ & \qquad \qquad \qquad \qquad \qquad \qquad -\int_\Omega f(x, u_j(x))u^\pm_{j}(x)\,dx\,. \end{aligned} \end{equation} Now, by Lemma~\ref{lemma1} (applied with $u=u^{+}_{j}\in \mathbb P_{k+1}$) and \eqref{4.17}--\eqref{j'u+} we get \[\left(1-\frac{\lambda_k}{\lambda_{k+1}}\right) \|u^{+}_{j}\|^{2}_{X_0}-\tilde M \|u^{+}_{j}\|_{X_0}\leq \|u^{+}_{j}\|_{X_0}, \] which shows that the sequence $u^{+}_j$ is bounded in $X_0$. Moreover, again by \eqref{4.17}--\eqref{j'u+} and Lemma~\ref{lemma2} (applied to $u^{-}_{j}\in \mathbb E^-_k\subset \mathbb H_k$), it follows that \[\|u^{-}_{j}\|_{X_0}\geq -\langle \mathcal J'(u_j), u_j^-\rangle \geq M^*\|u^{-}_{j}\|^{2}_{X_0}-\tilde M \|u^{-}_{j}\|_{X_0}, \] and so also $u^{-}_j$ is bounded in $X_0$. It remains to show that the sequence $u^{0}_j$ is bounded in $X_0$\,. At this purpose, we point out that~$u^0_j\in\mathbb E^0_k$ and so, by~\eqref{E0k}, $u^0_j$ is an eigenfunctions corresponding to~$\lambda_k$. Accordingly, by~\eqref{autovalore debole}, \begin{equation*} \frac12\int_{\RR^{2n}}|u^0_j(x)-u^0_j(y)|^2 K(x-y)\,dx\,dy=\frac{\lambda_k}2 \int_\Omega a(x)|u_j^0(x)|^2\,dx\,.\end{equation*} Therefore, by \eqref{palais smale1} and orthogonality, we see that \begin{equation}\label{4.19} \begin{aligned} c & \geq\left|\mathcal J(u_j)\right|\\ & = \left|\frac{1}{2}\int_{\RR^{2n}} \Big(|u^{+}_{j}(x)-u^{+}_{j}(y)|^2+|u^{-}_{j}(x)-u^{-}_{j}(y)|^2\Big) K(x-y)\,dx\,dy\right.\\ &\quad\quad\left.-\frac{\lambda_k}{2}\int_\Omega a(x)\Big(|u^{+}_{j}(x)|^2+|u^{-}_{j}(x)|^2\Big)\,dx-\int_\Omega\Big(F(x, u_j(x))-F(x,u^{0}_{j}(x))\Big)dx\right.\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad\left.-\int_\Omega F(x,u^{0}_{j}(x))\,dx\right|. \end{aligned} \end{equation} By \cite[Lemma~8]{svmountain} and the H\H older inequality we get that there exists a positive constant $C$, possibly depending on $\Omega$, such that \begin{equation}\label{4.19a} \left|\lambda_k\int_\Omega a(x)\left(|u^{+}_{j}(x)|^2 +|u^{-}_{j}(x)|^2\right)dx\right|\leq \lambda_k \|a\|_{L^\infty(\Omega)}\left( \|u^{+}_{j}\|^{2}_{X_0}+\|u^{-}_{j}\|^{2}_{X_0}\right)\leq 2C\,, \end{equation} and \begin{equation}\label{4.19b} \begin{aligned} \left|\int_\Omega\Big(F(x, u_j(x))-F(x,u^{0}_{j}(x))\Big)\,dx\right|& \leq\int_\Omega \left|\int^{u^{0}_{j}(x)+u^{-}_{j}(x)+u^{+}_{j}(x)}_{u^{0}_{j}(x)}f(x,t)dt\right|dx\\ &\leq M\int_\Omega\Big(|u^{-}_{j}(x)|+|u^{+}_{j}(x)|\Big)dx\\ & \leq M_*\left(\|u^{-}_{j}\|_{X_0}+\|u^{+}_{j}\|_{X_0}\right)\leq C\,, \end{aligned} \end{equation} since the sequences $u^{-}_j$ and $u^{+}_j$ are bounded in $X_0$ and \eqref{f2} holds true. Here $M_*$ is a positive constant. Hence, by \eqref{4.19}--\eqref{4.19b} it is easy to see that $$\begin{aligned} \left|\int_\Omega F(x,u^{0}_{j}(x))\,dx\right|& \leq |\mathcal J(u_j)|+\left|\frac{1}{2}\int_{\RR^{2n}} \Big(|u^{+}_{j}(x)-u^{+}_{j}(y)|^2+|u^{-}_{j}(x)-u^{-}_{j}(y)|^2\Big) K(x-y)\,dx\,dy\right.\\ &\qquad \qquad\left.-\frac{\lambda_k}{2}\int_\Omega a(x)\Big(|u^{+}_{j}(x)|^2+|u^{-}_{j}(x)|^2\Big)\,dx\right.\\ & \qquad \qquad \left.-\int_\Omega\Big(F(x, u_j(x))-F(x,u^{0}_{j}(x))\Big)dx\right|\\ & \leq c+\frac 1 2 \Big(\|u^+\|_{X_0}^2+ \|u^-\|_{X_0}^2\Big)+2C\leq \tilde C \end{aligned}$$ where $\tilde C$ is a positive constant independent of $j$. Here we have used again the fact that the sequences $u^{-}_j$ and $u^{+}_j$ are bounded in $X_0$\,. Hence, the integral $\displaystyle\int_\Omega F(x,u^{0}_{j}(x))\,dx$ is bounded. As a consequence, being $u^0\in \mathbb E^0_k$, by Lemma~\ref{lemma4} it follows that also the sequence $u^{0}_j$ is bounded in $X_0$, concluding the proof of Proposition~\ref{prop palais}\,. \end{proof} Now it remains to check the validity of the Palais--Smale condition, that is we have to show that every Palais--Smale sequence $u_j$ for $\mathcal J$ at level $c\in \RR$ strongly converges in $X_0$, up to a subsequence. This will be done in the next result. \begin{proposition}\label{prop:PScond} Let $K:\mathbb{R}^{n}\setminus \left\{0\right\}\rightarrow(0,+\infty)$ satisfy assumptions \eqref{kernel}--\eqref{evenkernel}. Moreover, assume that $\lambda=\lambda_k<\lambda_{k+1}$ for some $k\in\mathbb{N}$ and let $f$ and $a$ be two functions satisfying \eqref{f1}--\eqref{f3} and \eqref{conda}, respectively. Let $u_j$ be a sequence in $X_{0}$ satisfying \eqref{palais smale1} and \eqref{palais smale2}. Then, there exists $u_\infty\in X_0$ such that $u_j$ strongly converges to some $u_\infty$ in $X_0$\,. \end{proposition} \begin{proof} Since, by Proposition~\ref{prop palais}, $u_j$ is bounded in~$X_0$ and $X_{0}$ is a reflexive space (being a Hilbert space, by \cite[Lemma~7]{svmountain}), up to a subsequence, there exists $u_\infty\in X_{0}$ such that $u_{j}$ converges to $u_\infty$ weakly in $X_{0}$, that is \begin{equation} \begin{alignedat}2 \int_{\mathbb{R}^{2n}} (u_j(x)-u_j(y))&(\varphi(x)-\varphi(y)) K(x-y)\,dx\,dy\to \\ & \int_{\mathbb{R}^{2n}} (u_\infty(x)-u_\infty(y))(\varphi(x)-\varphi(y))K(x-y)\,dx\,dy \end{alignedat}\label{convergenza un} \end{equation} for any $\varphi\in X_0$\,, as $j\rightarrow +\infty$. Moreover, by applying \cite[Lemma~8]{svmountain} and \cite[Theorem~IV.9]{brezis}, up to a subsequence \begin{equation} \begin{alignedat}2 &u_j\rightarrow u_\infty && \quad\mbox{in } L^{q}(\mathbb{R}^{n})\,\,\,\mbox{for any}\,\, q\in [1,2^{*}) \\ &u_j\rightarrow u_\infty && \quad\mbox{a.e. in } \mathbb{R}^{n}\label{convergenza classica} \end{alignedat} \end{equation} as $j\rightarrow +\infty$. By \eqref{palais smale2} we have \begin{equation}\label{smale modificata} \begin{aligned} 0\leftarrow \langle \mathcal J'(u_j),u_j-u_\infty \rangle& =\int_{\RR^{2n}} \left|u_j(x)-u_j(y)\right|^{2}K(x-y)\,dx\,dy\\ &\qquad -\int_{\RR^{2n}} \big(u_j(x)-u_j(y)\big)\big(u_\infty(x)-u_\infty(y)\big)K(x-y)\,dx\,dy\\ &\qquad -\lambda_k\int_\Omega a(x)u_j(x)(u_j(x)-u_\infty(x))dx\\ & \qquad -\int_{\Omega}f(x,u_j(x))(u_j(x)-u_\infty(x))dx \end{aligned} \end{equation} as $j\to +\infty$\,. Now, by using the H\H older inequality, \eqref{f2} and \eqref{convergenza classica}, we get \begin{equation}\label{conv l2} \begin{aligned} \left|\lambda_k\int_\Omega a(x)u_j(x)\right.& \left.(u_j(x)-u_\infty(x))dx+\int_{\Omega}f(x,u_j(x))(u_j(x)-u_\infty(x))dx\right|\\ &\leq \left(\lambda_k \|a\|_{L^\infty (\Omega)} \|u_j\|_{L^2(\Omega)}+M\left|\Omega\right|^{1/2}\right)\|u_j-u_\infty\|_{L^{2}(\Omega)}\rightarrow 0 \end{aligned} \end{equation} as $j\rightarrow+\infty$. Hence, passing to the limit in \eqref{smale modificata} and taking into account \eqref{convergenza un} and \eqref{conv l2}, it follows that \[\int_{\RR^{2n}} \left|u_j(x)-u_j(y)\right|^{2}K(x-y)\,dx\,dy\rightarrow\int_{\RR^{2n}} \left|u_\infty(x)-u_\infty(y)\right|^{2}K(x-y)\,dx\,dy\,, \] that is \begin{equation} \|u_j\|_{X_{0}}\rightarrow \|u_\infty\|_{X_{0}}\label{convergenza norma} \end{equation} as $j\rightarrow +\infty$. Finally, we have that $$\begin{aligned} \|u_j-u_\infty\|^{2}_{X_{0}}& =\|u_j\|^{2}_{X_{0}}+\|u_\infty\|^{2}_{X_{0}}\\ & \qquad -2\int_{\mathbb{R}^{2n}} \big(u_j(x)-u_j(y)\big)\big(u_\infty(x)-u_\infty(y)\big)K(x-y)\,dx\,dy\\ & \rightarrow 2\|u_\infty\|^{2}_{X_{0}}-2\|u_\infty\|^{2}_{X_{0}}=0 \end{aligned}$$ as $j\rightarrow +\infty$, thanks to \eqref{convergenza un} and \eqref{convergenza norma}. Hence, $u_j\to u_\infty$ strongly in $X_0$ as $j\to +\infty$ and this completes the proof of Proposition~\ref{prop:PScond}\,. \end{proof} \subsection{Proof of Theorem~\ref{Thsaddle}} In this section we will prove Theorem~\ref{Thsaddle}, as an application of the Saddle Point Theorem \cite[Theorem~4.6]{rabinowitz}\,. At first, we prove that $\mathcal J$ satisfies the geometric structure required by the Saddle Point Theorem. For this note that by Proposition~\ref{prop Hk ortogonale} for any $H>0$ there exists $R>0$ such that, if $u\in\mathbb P_{k+1}$ and $\|u\|_{X_{0}}\geq R$\,, then \begin{equation}\label{H} \mathcal J(u)\geq H\,. \end{equation} While, if $u\in\mathbb P_{k+1}$ with $\|u\|_{X_{0}}\leq R$, by applying $\eqref{f2}$, the H\H older inequality and \cite[Lemma~8]{svmountain} we have \begin{equation}\label{finedim} \begin{aligned} \mathcal J(u) & \geq -\frac{\lambda_k}{2}\int_\Omega a(x)|u(x)|^2dx-\int_{\Omega}F(x,u(x))dx\\ & \geq -\frac{\lambda_k}{2}\|a\|_{L^\infty(\Omega)}\|u\|^{2}_{L^2(\Omega)}-M\int_\Omega |u(x)|\,dx\\ & \geq -\frac{\lambda_k}{2}\|a\|_{L^\infty(\Omega)}\|u\|^{2}_{X_{0}}-M_*\|u\|_{X_0}\\ & \geq -\frac{\lambda_k}{2}\|a\|_{L^\infty(\Omega)}R^2-M_*R=:-C_R\,. \end{aligned} \end{equation} Here $M_*$ is a positive constant. Hence, by \eqref{H} and \eqref{finedim} we get \begin{equation}\label{cr} \mathcal J(u)\geq -C_{R}\qquad \mbox{for any}\,\, u\in\mathbb P_{k+1}\,. \end{equation} Moreover, by Proposition~\ref{prop Hk}, there exists $T>0$ such that, for any $u\in \mathbb H_k$ with $\|u\|_{X_{0}}=T$, we have \begin{equation}\label{cr1} \mathcal J(u)<-C_R\,. \end{equation} Thus, by \eqref{cr} and \eqref{cr1} it easily follows that $$\sup_{u\in \mathbb H_k,\atop{\|u\|_{X_{0}}=T}}\mathcal J(u)<-C_R\leq\inf_{u\in\mathbb P_{k+1}}\mathcal J(u)\,,$$ so that the functional~$\mathcal J$ has the geometric structure of the Saddle Point Theorem (see assumptions~$(I_3)$ and $(I_4)$ of \cite[Theorem~4.6]{rabinowitz}). Since $\mathcal J$ satisfies also the Palais--Smale condition by Proposition~\ref{prop:PScond}, the Saddle Point Theorem provides the existence of a critical point $u\in X_0$ for the functional~$\mathcal J$\,. This concludes the proof of Theorem~\ref{Thsaddle}\,. \begin{thebibliography}{99} \bibitem{am} {\sc A. Ambrosetti and A. Malchiodi}, Nonlinear analysis and semilinear elliptic problems, {\em Cambridge Studies in Advanced Mathematics}, 104 {\em Cambridge University Press}, Cambridge (2007). \bibitem{colorado} {\sc B. Barrios, E. Colorado, A. De Pablo and U. Sanchez}, {\em On some critical problems for the fractional Laplacian operator}, to appear in J. Differential Equations, available online at {\tt http://arxiv.org/abs/1106.6081}. \bibitem{brezis} {\sc H. Brezis}, Analyse fonctionelle. Th\'{e}orie et applications, {\em Masson}, Paris (1983). \bibitem{cabretan} {\sc X. Cabr\'e and J. Tan}, {\em Positive solutions of nonlinear problems involving the square root of the Laplacian}, Adv. Math. 224, no. 5, 2052--2093 (2010). \bibitem{capella} {\sc A. Capella}, {\em Solutions of a pure critical exponent problem involving the half-Laplacian in annular-shaped domains}, Commun. Pure Appl. Anal. 10, no. 6, 1645--1662 (2011). \bibitem{valpal} {\sc E. Di Nezza, G. Palatucci and E. Valdinoci}, {\em Hitchhiker's guide to the fractional Sobolev spaces}, preprint, available at {\tt http://arxiv.org/abs/1104.4345}\,. \bibitem{fiscella}{\sc A. Fiscella}, {\em Saddle point solutions for non-local elliptic operators}, preprint (2012). \bibitem{rabinowitz78} {\sc P.H. Rabinowitz}, Some minimax theorems and applications to nolinear partial differential equations, {\em Nonlinear Analysis: a collection of papers in honor of Erich R\H othe}, {\em Academic Press}, New York, (1978), 161--177. \bibitem{rabinowitz} {\sc P.H. Rabinowitz}, Minimax methods in critical point theory with applications to differential equations, {\em CBMS Reg. Conf. Ser. Math.}, 65, {\em American Mathematical Society}, Providence, RI (1986). \bibitem{sY}{\sc R. Servadei}, {\em The Yamabe equation in a non-local setting}, preprint, available at {\tt http://www.ma.utexas.edu/mp$\_$arc-bin/mpa?yn=12-40}\,. \bibitem{sv}{\sc R. Servadei and E. Valdinoci}, {\em Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators}, preprint, available at {\tt http://www.ma.utexas.edu/mp$\_$arc-bin/mpa?yn=11-103}\,. \bibitem{svmountain}{\sc R. Servadei and E. Valdinoci}, {\em Mountain Pass solutions for non-local elliptic operators}, J. Math. Anal. Appl., 389, 887--898 (2012). \bibitem{svlinking}{\sc R. Servadei and E. Valdinoci}, {\em Variational methods for non-local operators of elliptic type}, preprint, available at {\tt http://www.math.utexas.edu/mp$\_$arc-bin/mpa?yn=11-131}. \bibitem{servadeivaldinociBN}{\sc R. Servadei and E. Valdinoci}, {\em The Brezis-Nirenberg result for the fractional Laplacian}, preprint, available at {\tt http://www.ma.utexas.edu/mp$\_$arc-bin/mpa?yn=11-196}. \bibitem{servadeivaldinociBNLOW}{\sc R. Servadei and E. Valdinoci}, {\em A Brezis-Nirenberg result for non-local critical equations in low dimension}, preprint, available at {\tt http://www.ma.utexas.edu/mp$\_$arc-bin/mpa?yn=12-41}\,. \bibitem{servadeivaldinociCFP}{\sc R. Servadei and E. Valdinoci}, {\em Fractional Laplacian equations with critical Sobolev exponent}, preprint (2012), available at {\tt http://www.math.utexas.edu/mp$\_$arc-bin/mpa?yn=12-58}\,. \bibitem{struwe} {\sc M. Struwe}, Variational methods, Applications to nonlinear partial differential equations and Hamiltonian systems, {\em Ergebnisse der Mathematik und ihrer Grenzgebiete}, 3, {\em Springer Verlag}, Berlin--Heidelberg (1990). \bibitem{tan} {\sc J. Tan}, {\em The Brezis-Nirenberg type problem involving the square root of the Laplacian}, Calc. Var. Partial Differential Equations, 36, no.~1-2, 21--41 (2011). \bibitem{willem} {\sc M. Willem}, Minimax theorems, {\em Progress in Nonlinear Differential Equations and their Applications}, 24, {\em Birkh\"{a}user}, Boston (1996). \end{thebibliography} \end{document} ---------------1205180645177--