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integrodifferential operators, fractional Laplacian, variational 
techniques, Saddle Point Theorem, Palais--Smale condition.
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\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}

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