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Yamabe problem, nonlinear analysis, Mountain Pass Theorem, Linking Theorem, critical nonlinearities, best fractional critical Sobolev constant, Palais--Smale condition, variational 
techniques, integrodifferential 
operators, fractional Laplacian.
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\begin{document}
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\title[The Yamabe equation in a non-local setting]{The Yamabe equation in a non-local setting}

\thanks{The author was supported by
the MIUR National Research Project {\it Variational and Topological Methods
in the Study of Nonlinear Phenomena} and by the
ERC grant $\epsilon$ ({\it Elliptic Pde's and Symmetry of Interfaces and Layers
for Odd Nonlinearities}).}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}


\keywords{Mountain Pass Theorem, Linking Theorem, critical nonlinearities, best fractional critical Sobolev constant, Palais--Smale condition, variational
techniques, integrodifferential
operators, fractional Laplacian.\\
\phantom{aa} 2010 AMS Subject Classification: Primary:
49J35, 35A15, 35S15;
Secondary: 47G20, 45G05.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{abstract}
Aim of this paper is to study the following elliptic equation driven by a general non-local integrodifferential operator~$\mathcal L_K$
$$\left\{
\begin{array}{ll}
\mathcal L_K u+\lambda u+|u|^{2^*-2}u=0 &  \mbox{in } \Omega\\
u=0 & \mbox{in } \RR^n\setminus \Omega\,,
\end{array}\right.$$
where $s\in (0,1)$, $\Omega$ is an open bounded set of $\RR^n$, $n>2s$, with Lipschitz boundary, $\lambda$ is a positive real parameter, $2^*=2n/(n-2s)$ is a fractional critical Sobolev exponent, while $\mathcal L_K$ is the non-local integrodifferential operator
$$\mathcal L_Ku(x)=
\frac12
\int_{\RR^n}\Big(u(x+y)+u(x-y)-2u(x)\Big)K(y)\,dy\,,
\,\,\,\,\, x\in \RR^n\,.$$

As a concrete example, we consider the case when $K(x)=|x|^{-(n+2s)}$\,, which gives rise to the fractional Laplace operator $-(-\Delta)^s$\,. In this framework, in the existence result proved along the paper, we show that our problem admits a non-trivial solution for any $\lambda>0$, provided $n\geq 4s$ and $\lambda$ is different from the eigenvalues of $(-\Delta)^s$\,. This result may be read as the non-local fractional counterpart of the one obtained by Capozzi, Fortunato and Palmieri in \cite{capfortpalm} (see also \cite{as})  and by Gazzola and Ruf in \cite{gazzolaruf} for the classical Laplace equation with critical nonlinearities.

In this sense the present work may be seen as the extension of some classical results for the Laplacian to the case of non-local fractional operators.

\end{abstract}

\maketitle

\tableofcontents

\section{Introduction}\label{sec:introduzione}
\subsection{Critical non-local integrodifferential equations}\label{subsec:intro1}
In the literature there are many papers related to the study of the critical elliptic equation
\begin{equation}\label{1} \left\{
\begin{array}{ll}
-\Delta u-\lambda u=|u|^{2_*-2}u & {\mbox{ in }} \Omega\\
u=0 & {\mbox{ on }} \partial \Omega\,, \quad 2_*=2n/(n-2)\,, n>2\,,
\end{array} \right.
\end{equation}
where $\Omega\subset \RR^n$, $n>2$, is an open, bounded set and $\lambda$ is a positive parameter
(see, for instance, \cite{am, struwe, willem} and references therein).

Equation~\eqref{1} is particularly relevant for its relations with problems arising in differential geometry and in physics, where a lack of compactness occurs.
One of the most important problem which gives rise to equation~\eqref{1} is the Yamabe problem: given a $n$ dimensional compact Riemannian manifold $(\mathcal M, g)$, $n>2$, with scalar curvature $k=k(x)$, find a metric $\tilde g$ conformal to $g$ with constant scalar curvature $\tilde k$\,. If we put
$$\tilde g=u^{4/(n-2)}g\,,$$
where $u>0$ is the conformal factor, then the Yamabe problem can be formulated as follows: find $u>0$ satisfying the equation
$$-4\frac{n-1}{n-2}\Delta_{\mathcal M}u=\tilde k u^{2^*-1}-k(x)u \quad \mbox{in}\,\, \mathcal M\,.$$
Here $\Delta_{\mathcal M}$ is the Laplace-Beltrami operator on $\mathcal M$ with respect to the metric $g$\,.

Motivated by the interest shown in the literature for non-local operators of elliptic type,
in this paper we study the non-local counterpart of problem~\eqref{1}, that is we consider critical problems modeled by
\begin{equation}\label{1s}
\left\{
\begin{array}{ll}
(-\Delta)^s u-\lambda u=|u|^{2^*-2}u & {\mbox{ in }} \Omega\\
u=0 & {\mbox{ in }} \RR^n\setminus \Omega\,,
\end{array} \right.
\end{equation}
where $s\in (0,1)$ is fixed and $(-\Delta)^s$ is the fractional Laplace operator defined, up to normalization factors, as
\begin{equation} \label{2}
-(-\Delta)^s u(x)=
\frac12
\int_{\RR^n}\frac{u(x+y)+u(x-y)-2u(x)}{|y|^{n+2s}}\,dy\,,
\,\,\,\,\, x\in \RR^n\end{equation}
(see \cite{valpal} and references therein for further details on the
fractional Laplacian), while $\Omega\subset \RR^n$, $n>2s$, is open, bounded and with
Lipschitz boundary, $\lambda>0$ and
\begin{equation}\label{2star}
2^*=2n/(n-2s)
\end{equation}
is the fractional critical Sobolev exponent.

In \cite{svmountain, svlinking} we studied problems of the type
\begin{equation}\label{1sq}
\left\{
\begin{array}{ll}
(-\Delta)^s u-\lambda u=|u|^{q-2}u & {\mbox{ in }} \Omega\\
u=0 & {\mbox{ in }} \RR^n\setminus \Omega\,,
\end{array} \right.
\end{equation}
where $q\in (2, 2^*)$\,, that is we considered non-local equations with subcritical growth, and in this setting we extended the validity of some existence results known in the classical subcritical case of the Laplacian to the non-local framework. Also, in \cite{servadeivaldinociBN} we considered the critical equation~\eqref{1s} in the case when the parameter $\lambda\in (0, \lambda_{1,\,s})$, where $\lambda_{1,\,s}$ denotes the first eigenvalue of the fractional Laplace operator~$(-\Delta)^s$ with homogeneous Dirichlet boundary conditions. In this framework we proved a non-local Brezis-Nirenberg type result, i.e. we showed that problem~\eqref{1s} admits a non-trivial solution for any $\lambda\in (0, \lambda_{1,\,s})$, provided $n\geq 4s$ (see \cite{bn, servadeivaldinociBN})\,. The case when $n<4s$ was treated in the paper \cite{servadeivaldinociBNLOW}, where we proved the existence of non-trivial weak solution for the equation~\eqref{1s} for any value of the parameter $\lambda>0$ different from the eigenvalues of $(-\Delta)^s$ \,.

Aim of this paper is to complete the study of problem~\eqref{1s}, by showing that the existence result obtained in \cite[Theorem~4]{servadeivaldinociBN} holds true for any $\lambda>0$ different from the eigenvalues of $(-\Delta)^s$\,.

More precisely, along this paper we consider the following general non-local equation
\begin{equation}\label{op}
\left\{
\begin{array}{ll}
\mathcal L_K u+\lambda u+|u|^{2^*-2}u=0 &  \mbox{in } \Omega\\
u=0 & \mbox{in } \RR^n\setminus \Omega\,,
\end{array}\right.
\end{equation}
where $\mathcal L_K$ is the non-local operator defined as
\begin{equation}\label{lk}
\mathcal L_Ku(x)=
\frac12
\int_{\RR^n}\Big(u(x+y)+u(x-y)-2u(x)\Big)K(y)\,dy\,,
\,\,\,\,\, x\in \RR^n\,,
\end{equation}
with the kernel $K:\RR^n\setminus\{0\}\rightarrow(0,+\infty)$ such 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}
As a model for $K$ we can take  the singular kernel $K(x)=|x|^{-(n+2s)}$ which gives rise to the fractional Laplace operator $-(-\Delta)^s$
defined in \eqref{2}.

Note that in problem~\eqref{op} the Dirichlet datum is given
in $\RR^n\setminus \Omega$ and not simply on $\partial \Omega$,
consistently with the non-local character of the operator~$\mathcal L_K$\,.

Along the paper we will be interested in the weak formulation of~\eqref{op} given by the following problem (for this, it is convenient to assume \eqref{evenkernel})
\begin{equation}\label{problemaK0f}
\left\{\begin{array}{l} {\displaystyle \int_{\RR^{2n} }
(u(x)-u(y))(\varphi(x)-\varphi(y))K(x-y) dx\,dy-\lambda \int_\Omega
u(x)
\varphi(x)\,dx}\\
\\
\qquad \qquad \qquad \qquad \qquad \qquad
{\displaystyle = \int_\Omega |u(x)|^{2^*-2}u(x)\varphi(x)dx}\,\,\,\,\,\,\, \forall\,\, \varphi \in X_0\\
\\
u\in X_0\,.
\end{array}\right.
\end{equation}

In order to study problem~\eqref{1s} (and, in general, \eqref{op}) the usual fractional Sobolev spaces are not adequate. Hence, to overcome this difficulty in \cite{svmountain} (see also \cite{sv}) we introduced a new functional space $X_0$, which, in our opinion, is the suitable space in which working in. The definition of $X_0$ will be given in Subsection~\ref{SN}.
In the model case $\mathcal L_K=-(-\Delta )^s$ the space $X_0$ consists of all the functions of the usual fractional Sobolev space $H^s(\RR^n)$ which vanish a.e. outside $\Omega$ (see \cite[Lemma~7]{servadeivaldinociBN})\,.


\subsection{Main theorems of the paper}\label{susec:mainresults}
It is easily seen that problem~\eqref{op} admits the trivial function $u\equiv 0$ as a solution. The aim of this paper is to find non-trivial weak solutions for \eqref{op}\,, that is non-trivial solutions of \eqref{problemaK0f}\,.
For this, we will use a variational approach adapting the techniques of \cite{bn, capfortpalm} (see also \cite{am, struwe, willem} and references therein) to our non-local setting.

Problem~\eqref{problemaK0f} is the Euler--Lagrange equation of the functional $\mathcal J_{K,\,\lambda}:X_0\to \RR$ defined as follows
\begin{equation}\label{JKlambda}
\mathcal J_{K,\,\lambda}(u) =\frac 1 2 \int_{\RR^{2n}}|u(x)-u(y)|^2 K(x-y)\,dx\,dy-\frac \lambda 2 \int_\Omega |u(x)|^2\,dx
-\frac{1}{2^*}\int_\Omega |u(x)|^{2^*}\,dx\,.
\end{equation}

Thus, our goal will be finding critical points of~$\mathcal J_{K,\,\lambda}$ and, for this, we will use some variants of the classical Mountain Pass and Linking Theorems due to Ambrosetti and Rabinowitz (see \cite{ar, rabinowitz}), which take into account that, since the embedding $X_0\hookrightarrow L^{2^*}(\RR^n)$ is not compact (see Lemma~\ref{lemmaembeddings}), the functional~$\mathcal J_{K,\,\lambda}$ does not verify the Palais--Smale condition globally, but only in a suitable range related to the best fractional critical Sobolev constant defined as
\begin{equation}\label{SK}
{\displaystyle S_K:=\inf_{v\in X_0\setminus\{0\}}S_K(v)\,,}
\end{equation}
where
\begin{equation}\label{SKv}
{\displaystyle  X_0\setminus\{0\} \ni v\mapsto S_K(v):=\frac{{\displaystyle\int_{\RR^{2n}}|v(x)-v(y)|^2 K(x-y)\,dx\,dy}}{{\displaystyle\Big(\int_\Omega |v(x)|^{2^*}dx\Big)^{2/2^*}}}\,.}
\end{equation}
The constant $S_K$ is well defined, as it can be seen by Lemma~\ref{lemmaembeddings} below, and it is strictly positive.

Along the paper we also need the following function
\begin{equation}\label{SKlambdav}
\begin{aligned}
{\displaystyle X_0\setminus\{0\} \ni v\mapsto S_{K,\,\lambda}(v)}& {\displaystyle :=\frac{{\displaystyle\int_{\RR^{2n}}|v(x)-v(y)|^2 K(x-y)\,dx\,dy-\lambda\int_\Omega |v(x)|^2\,dx}}{{\displaystyle\Big(\int_\Omega |v(x)|^{2^*}dx\Big)^{2/2^*}}}\,.}
\end{aligned}
\end{equation}
Note that, since $v\in X_0$, the integrals over $\Omega$ in \eqref{SKv} and in \eqref{SKlambdav} can be extended to all $\RR^n$, that is
$${\displaystyle S_K(v)=\frac{{\displaystyle\int_{\RR^{2n}}|v(x)-v(y)|^2 K(x-y)\,dx\,dy}}{{\displaystyle\Big(\int_{\RR^n} |v(x)|^{2^*}dx\Big)^{2/2^*}}}}$$
and
$${\displaystyle S_{K,\,\lambda}(v)=\frac{{\displaystyle\int_{\RR^{2n}}|v(x)-v(y)|^2 K(x-y)\,dx\,dy-\lambda\int_{\RR^n} |v(x)|^2\,dx}}{{\displaystyle\Big(\int_{\RR^n} |v(x)|^{2^*}dx\Big)^{2/2^*}}}\,.}$$
Hence, $S_K(\cdot)$ and $S_{K,\,\lambda}(\cdot)$ do not depend on the domain $\Omega$, while $S_K$ does, since $X_0$ depends on $\Omega$\,.

In the usual fractional Sobolev space $H^s(\RR^n)$ the counterparts of $S_K$ and $S_{K,\,\lambda}(\cdot)$ are given, respectively, by the constant $S_s$ defined as follows (see also Lemma~\ref{lemmaSs} in the sequel)
\begin{equation}\label{Ss}\begin{split}&
{\displaystyle S_s:=\inf_{v\in H^s(\RR^n)\setminus\{0\}}S_s(v)}\,,
\\ &\qquad
\, \,\, {\displaystyle  H^s(\RR^n)\setminus\{0\}\ni v\mapsto S_s(v):=\frac{ {\displaystyle\int_{\RR^{2n}}\frac{|v(x)-v(y)|^2}{|x-y|^{n+2s}} \,dx\,dy} }{ {\displaystyle\Big(\int_{\RR^n} |v(x)|^{2^*}dx\Big)^{2/2^*}}}\,,}
\end{split}\end{equation}
and by the function
\begin{equation}\label{Sslambdav}
{\displaystyle H^s(\RR^n)\setminus\{0\}\ni v\mapsto S_{s,\,\lambda}(v):=\frac{{\displaystyle\int_{\RR^{2n}}\frac{|v(x)-v(y)|^2}{|x-y|^{n+2s}} \,dx\,dy-\lambda \int_{\RR^n} |v(x)|^2\,dx}}{{\displaystyle\Big(\int_{\RR^n} |v(x)|^{2^*}dx\Big)^{2/2^*}}}\,.}
\end{equation}


Note that for any $\lambda>0$
$${\displaystyle S_{K,\,\lambda}:=\inf_{v\in X_0\setminus\{0\}}S_{K,\,\lambda}(v)\leq S_K}$$ and $$S_{s,\,\lambda}:=\inf_{v\in H^s(\RR^n)\setminus\{0\}}S_{s,\,\lambda}(v)\leq S_s\,.$$ The interesting case is when the strict inequality holds true. Indeed, in this setting a suitable compactness propriety holds true for the functional~$\mathcal J_{K,\,\lambda}$\,. This fact allows us to prove our main existence result, by suitably using critical points theorems.

In the sequel $\big\{ \lambda_k\big\}_{{k\in\NN}}$ will denote the sequence of the
eigenvalues of the operator~$-\mathcal L_K$ with homogeneous Dirichlet boundary data with
$$0<\lambda_1<\lambda_2\le \dots \le \lambda_k\le \lambda_{k+1}\le \dots$$
and
$${\mbox{$\lambda_k\to +\infty$ as $ k\to +\infty\,.$}}$$
Moreover, $\big\{e_k\big\}_{{k\in\NN}}$ will be the sequence
of eigenfunctions corresponding to $\lambda_k$\,. We recall that this sequence is an orthonormal basis of $L^2(\Omega)$ and an orthogonal basis of $X_0$\,. For a complete study of the spectrum of the integrodifferential operator~$-\mathcal L_K$ we refer to \cite[Proposition~9 and Appendix~A]{svlinking}\,.

The main result of the present paper is the following existence theorem:
\begin{theorem}\label{generalkernel0f}
Let $s\in (0,1)$, $n>2s$ and $\Omega$ be an open bounded set of
$\RR^n$ with Lipschitz boundary and let
$K:\RR^n\setminus\{0\}\rightarrow(0,+\infty)$ be a function satisfying conditions~\eqref{kernel}--\eqref{evenkernel}\,. Furthermore, assume that
\begin{equation}\label{SKV}
\begin{aligned}
& \mbox{there exists $u_0\in X_0\setminus\{0\}$ with $u_0\geq 0$ a.e. in $\RR^n$ such that}\\
& \qquad \qquad \quad \mbox{$S_{K,\, \lambda}(u)<S_K$ for any $u\in \mathcal U$,}\\
& \mbox{where}\,\, \mathcal U=\begin{cases}
\mbox{span}\{u_0\} & \mbox{if}\,\,\, \lambda\in (0,\, \lambda_1)\\
\mbox{span}\{e_1, \dots, e_k, u_0\} & \mbox{if}\,\,\, \lambda\in [\lambda_k,\, \lambda_{k+1})\,, \,\, k\in \NN\,.
\end{cases}
\end{aligned}
\end{equation}
Then, for any $\lambda>0$ problem~\eqref{problemaK0f} admits a
solution $u\in X_0$, which is not identically zero.
\end{theorem}

Note that when $\lambda\in (0,\, \lambda_1)$ condition~\eqref{SKV} is exactly
\begin{equation}\label{SKlambdaSK}
S_{K,\,\lambda}<S_K\,,
\end{equation}
so that Theorem~\ref{generalkernel0f} reduces to \cite[Theorem~1]{servadeivaldinociBN}\,.
Also, a concrete case in which condition~\eqref{SKV} is satisfied is presented in the forthcoming Theorem~\ref{lapfra0}\,.

As a matter of fact, when dealing with the model kernel $K(x)=|x|^{-(n+2s)}$, problem~\eqref{op} reduces to~\eqref{1s}\,,
whose weak formulation is given by
\begin{equation}\label{problemalapfrac0}
\left\{\begin{array}{l}
{\displaystyle \int_{\RR^{2n}}}
{\displaystyle \frac{(u(x)-u(y))(\varphi(x)-\varphi(y))}{|x-y|^{n+2s}}\,dx\,dy -\lambda
\int_\Omega u(x) \varphi(x)\,dx}\\
\\
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
{\displaystyle =\int_\Omega |u(x)|^{2^*-2}u(x)\varphi(x)dx}\\
\\
\qquad \qquad \qquad \qquad \qquad \qquad \qquad\,\,\,\,\,\,\, \forall\,\, \varphi \in H^s(\RR^n)\,\,\mbox{with}\,\,\, \varphi=0\,\,\, \mbox{a.e. in}\,\,\, \RR^n\setminus\Omega\\
\\
u\in H^s(\RR^n) \,\,\mbox{with}\,\,\, u=0\,\,\, \mbox{a.e. in}\,\,\, \RR^n\setminus\Omega\,.
\end{array}\right.
\end{equation}


In the fractional Laplace setting the counterpart of Theorem~\ref{generalkernel0f} is given by the following result:
\begin{theorem}\label{lapfra0}
Let $s\in (0,1)$, $n\geq 4s$\,, $\Omega$ be an open bounded set of $\RR^n$
with Lipschitz boundary and $\lambda$ be a positive parameter.

Then, problem~\eqref{problemalapfrac0} admits a
solution $u\in H^s(\RR^n)$, which is not identically zero and such that $u=0$ a.e. in $\RR^n\setminus \Omega$\,,  provided $\lambda$ is not an eigenvalue of $(-\Delta)^s$ with homogeneous Dirichlet boundary data.
\end{theorem}

This existence theorem may be seen as the natural extension to the non-local framework of a well-know result obtained by Capozzi, Fortunato and Palmieri in \cite{capfortpalm}, where the classical critical Laplace equation~\eqref{1} was studied (for more general critical nonlinearities see \cite{gazzolaruf}). Indeed, when $s=1$, Theorem~\ref{lapfra0} reads as \cite[Theorem~0.1]{capfortpalm} (see also \cite[Corollary~1]{gazzolaruf} and \cite[Theorem~2.24]{willem}).

The proof of Theorem~\ref{lapfra0} is based on the fact that, in the fractional Laplace setting, condition~\eqref{SKV} is related to the strict inequality $S_{s,\,\lambda}<S_s$, which holds true in any dimension $n$ provided $n\geq 4s$\,. For more details, see Section~\ref{sec:prooffraclap}.

In general the main existence results proved in the present paper are obtained through variational methods (precisely applying classical minimax theorems to $\mathcal J_{K,\,\lambda}$).  The main difficulties are related to the encoding
the Dirichlet boundary datum in the variational formulation. Working in the space~$X_0$ allows us to overcome this difficulty. Moreover, since equations~\eqref{1s} and \eqref{op} have a critical growth, another problem is related to the lack of compactness in the embedding $X_0\hookrightarrow L^{2^*}(\RR^n)$ (or, in the case of the fractional Laplacian, $H^s(\RR^n)\hookrightarrow L^{2^*}(\RR^n)$). In order to overcome these difficulties, first we will show that the functional $\mathcal J_{K,\,\lambda}$ satisfies the Palais--Smale condition at any level smaller than a certain threshold related to the best fractional critical Sobolev constant. Then, after showing that $\mathcal J_{K,\,\lambda}$ has the right geometric structure, we will prove that the minimax critical level of $\mathcal J_{K,\,\lambda}$ lies below such a threshold.

When dealing with a non-local setting, in the pure power case all the results presented here extend the existence theorems obtained in \cite{svmountain, svlinking, servadeivaldinociBN} to the critical case. Furthermore, the present paper extends some classical results for the Laplacian to the case of non-local fractional operators (see, for instance, \cite{as, bn, capfortpalm, gazzolaruf}).

We also would like to recall the recent papers \cite{colorado, tan}, where the authors studied a non-local version of the Brezis-Nirenberg equation for an operator conceptually different from the one treated in the present work.

The paper is organized as follows. Section~\ref{sec:preliminary}
is devoted to some preliminary results: since the functional analytic space we will work is not a standard space, first we introduce it and then we recall some of its embeddings into the usual Lebesgue spaces. In Section~\ref{sec:strategy} we illustrate the strategy, based on a variational approach, we will use in order to prove the main result of the paper. In Section~\ref{sec:compactness} we show that the Euler--Lagrange functional associated with problem~\eqref{problemaK0f} satisfies a suitable compactness condition, while Section~\ref{sec:geometry} is devoted to the study of the geometric structure of the same functional. In Section~\ref{sec:dimtheorem1} we prove
Theorem~\ref{generalkernel0f} via variational techniques, while in Section~\ref{sec:prooffraclap} we discuss the case of the fractional Laplacian operator and we prove Theorem~\ref{lapfra0}.

\section{Some preliminary results}\label{sec:preliminary}
In this section we recall some preliminary results which will be useful along the paper.

As we said in the Introduction, in our setting the fractional Sobolev space is not enough in order to study problems~\eqref{1s} and \eqref{op}\,. For this reason in \cite{svmountain} (see also \cite{sv}) we introduced a new functional analytic space~$X_0$ inspired by (but not equivalent to) the fractional Sobolev spaces which seems to be the appropriate space in which work in. Since this space is not a standard one, here we recall its definition and some of its properties which we will use along the paper.
The reader familiar with this topic may skip it.

\subsection{The functional setting}\label{SN}
In our framework the functional space $X$ denotes the linear space of Lebesgue
measurable functions from $\RR^n$ to $\RR$ such that the restriction
to $\Omega$ of any function $g$ in $X$ belongs to $L^2(\Omega)$ and
$$\mbox{the map}\,\,\,
(x,y)\mapsto (g(x)-g(y))\sqrt{K(x-y)}\,\,\, \mbox{is in}\,\,\,
L^2\big(\RR^{2n} \setminus ({\mathcal C}\Omega\times
{\mathcal C}\Omega), dxdy\big)\,,$$
where ${\mathcal C}\Omega:=\RR^n \setminus\Omega$\,, while
$$X_0=\{g\in X : g=0\,\, \mbox{a.e. in}\,\,
\RR^n\setminus
\Omega\}\,.$$
We note that $X$ and $X_0$ are non-empty, since $C^2_0 (\Omega)\subseteq X_0$ by \cite[Lemma~11]{sv} (for this we need condition~\eqref{kernel}).

The space $X$ is endowed with the norm defined as
\begin{equation}\label{norma}
\|g\|_X=\|g\|_{L^2(\Omega)}+\Big(\int_Q |g(x)-g(y)|^2K(x-y)dx\,dy\Big)^{1/2}\,,
\end{equation}
where $Q=\RR^{2n}\setminus \mathcal
O$ and $
{\mathcal{O}}=({\mathcal{C}}\Omega)\times({\mathcal{C}}\Omega)\subset\RR^{2n}$\,.
It is easily seen that $\|\cdot\|_X$ is a norm on $X$ (see, for instance, \cite{svmountain} for a proof).

In the following we 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 $Q$
(this makes the classical fractional Sobolev space approach not
sufficient for studying the problem).

For further details on the fractional Sobolev spaces we refer to~\cite{valpal}
and to the references therein.

By \cite[Lemmas~6 and 7]{svmountain} in the sequel we can take the function
\begin{equation}\label{normaX0}
X_0\ni v\mapsto \|v\|_{X_0}=\left(\int_Q|v(x)-v(y)|^2K(x-y)\,dx\,dy\right)^{1/2}
\end{equation}
as norm on $X_0$. Also $\left(X_0, \|\cdot\|_{X_0}\right)$ is a Hilbert space,
with scalar product
\begin{equation}\label{scalarproduct}
\langle u,v\rangle_{X_0}=\int_Q
\big( u(x)-u(y)\big) \big( v(x)-v(y)\big)\,K(x-y)\,dx\,dy\,.
\end{equation}

Note that in \eqref{normaX0} (and in the related scalar product) the integral can be extended to all $\RR^{2n}$, since $v\in X_0$ (and so $v=0$ a.e. in $\RR^n\setminus \Omega$).

Along the paper we need some embeddings of the spaces $X_0$ and $H^s(\RR^n)$ into the usual Lebesgue spaces. With respect to these embeddings, the space $X_0$ behaves like $H^1_0(\Omega)$, while the fractional Sobolev space $H^s(\RR^n)$ as the usual Sobolev space $H^1(\RR^n)$: this is due to the fact that the functions $v\in X_0$ are such that $v=0$ a.e. in $\RR^n\setminus \Omega$ and so $X_0$ may be seen as a space of functions defined in the bounded set $\Omega$\,. Indeed, the following results, proved in \cite[Lemma~8]{svmountain} and in \cite[Lemma~9]{servadeivaldinociBN}, hold true:
\begin{lemma}\label{lemmaembeddings}
Let $K:\RR^n\setminus\{0\}\rightarrow (0,+\infty)$
satisfy assumptions~\eqref{kernel}--\eqref{evenkernel}\,.
Then, the following assertions hold true:
\begin{itemize}
\item [$a)$] if $\Omega$ has a Lipschitz boundary, then the embedding $X_0\hookrightarrow L^\nu(\RR^n)$ is compact for any $\nu\in [1, 2^*)$;
\item[$b)$] the embedding $X_0\hookrightarrow L^{2^*}(\RR^n)$ is continuous.
\end{itemize}
\end{lemma}

The counterpart of Lemma~\ref{lemmaembeddings} in the usual fractional Sobolev spaces is given by the following one proved in \cite[Theorem~6.5]{valpal}:
\begin{lemma}\label{lemmaSs}
The embedding $H^s(\RR^n)\hookrightarrow L^\nu(\RR^n)$ is continuous for any $\nu\in [2, 2^*]$\,.
\end{lemma}

\subsection{A variational characterization of the eigenvalues of $-\mathcal L_K$}\label{subsec:autovalori}
In this subsection we will give a variational characterization of the eigenvalues of the integrodifferential operator~$-\mathcal L_K$, which will be useful in the sequel.
A complete study of the spectrum of the $-\mathcal L_K$ can be found in \cite[Proposition~9 and Appendix~A]{svlinking}\,.

\begin{proposition}\label{propcaratterizzazione}
Let $\big\{ \lambda_k\big\}_{{k\in\NN}}$ be the sequence of the
eigenvalues of the operator~$-\mathcal L_K$ with homogeneous Dirichlet boundary data and
$$0<\lambda_1<\lambda_2\le \dots \le \lambda_k\le \lambda_{k+1}\le \dots$$
and let $\big\{e_k\big\}_{{k\in\NN}}$  be the sequence
of eigenfunctions corresponding to $\lambda_k$\,.

Then, for any $k\in \NN$ the eigenvalues can be characterized as follows:
$$\lambda_k=\max_{u\in \mbox{span}\{ e_1, \dots, e_k\}\setminus\{0\}}\frac{{\displaystyle \int_{\RR^{2n}}|u(x)-u(y)|^2 K(x-y)dx\,dy}}{{\displaystyle \int_\Omega |u(x)|^2\,dx}}\,.$$
\end{proposition}
\begin{proof}
Let $k\in \NN$\,. Since $\lambda_k$ is the eigenvalue corresponding to the eigenfunction $e_k$, we have that
\begin{equation}\label{minore}
\begin{aligned}
\lambda_k & =\frac{{\displaystyle \int_{\RR^{2n}}|e_k(x)-e_k(y)|^2 K(x-y)dx\,dy}}{{\displaystyle \int_\Omega |e_k(x)|^2\,dx}}\\
& \leq \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 |u(x)|^2\,dx}}\,.
\end{aligned}
\end{equation}
Furthermore, let $u\in \mbox{span}\{ e_1, \dots, e_k\}\setminus\{0\}$\,. Then
$$u=\sum_{i=1}^k u_ie_i\,\,\,\, \mbox{with}\,\,\,\, \sum_{i=1}^k u_i^2\not=0$$
and
$$\frac{{\displaystyle \int_{\RR^{2n}}|u(x)-u(y)|^2 K(x-y)dx\,dy}}{{\displaystyle \int_\Omega |u(x)|^2\,dx}} = \frac{{\displaystyle \sum_{i=1}^k u_i^2\|e_i\|_{X_0}}}{{\displaystyle \sum_{i=1}^k u_i^2}}\\ = \frac{{\displaystyle \sum_{i=1}^k u_i^2\lambda_i}}{{\displaystyle \sum_{i=1}^k u_i^2}}\\ \leq \lambda_k\,,$$
since $\big\{e_1, \dots, e_k\big\}$ are orthonormal
in $L^2(\Omega)$ and  orthogonal in $X_0$ (see \cite[Proposition~9--$f)$]{svlinking}) and $\lambda_k\geq \lambda_i$ for any $i=1, \dots, k$\,. As a consequence, passing to the maximum over $u\in \mbox{span}\{ e_1, \dots, e_k\}\setminus\{0\}$ we get
$$\max_{u\in\, \mbox{span}\{ e_1, \dots, e_k\}\setminus\{0\}}\frac{{\displaystyle \int_{\RR^{2n}}|u(x)-u(y)|^2 K(x-y)dx\,dy}}{{\displaystyle \int_\Omega |u(x)|^2\,dx}}\leq \lambda_k\,,$$
which together with \eqref{minore} gives the assertion.
\end{proof}

In the sequel it will be useful also the following regularity result for the eigenvalues of $-\mathcal L_K$\,, whose proof can be found in \cite[Proposition~4]{servadeivaldinociBNLOW} in the setting of the fractional Laplacian (in the general case we can proceed in the same way):
\begin{proposition}\label{proplinfty}
Let~$e\in X_0$
and~$\lambda>0$ be such that
\begin{equation}\label{e+}
\langle e,\varphi\rangle_{X_0}=\lambda\int_\Omega e(x)\varphi(x)\,dx
\end{equation}
for any~$\varphi\in X_0$. Then~$e\in L^\infty(\Omega)$\,.
\end{proposition}

\section{Strategy for proving Theorem~\ref{generalkernel0f}}\label{sec:strategy}

This section is devoted to the study of problem~\eqref{problemaK0f} which is the
Euler--Lagrange equation of the functional $\mathcal J_{K,\,\lambda}$ defined in \eqref{JKlambda}\,.
Notice that this functional is well defined thanks
to the definition of $X_0$ and Lemma~\ref{lemmaembeddings}-$b)$.
Moreover, $\mathcal J_{K,\,\lambda}$ is Fr\'echet
differentiable in $u\in X_0$ and for any $\varphi\in X_0$
$$\begin{aligned}
\langle \mathcal J_{K,\,\lambda}'(u), \varphi\rangle & = \int_{\RR^{2n}}
\big(u(x)-u(y)\big)\big(\varphi(x)-\varphi(y)\big)K(x-y)\,dx\,dy -\lambda \int_\Omega u(x) \varphi(x)\,dx \\
& \qquad \qquad \qquad \qquad\qquad \qquad \qquad \qquad \qquad
 -\int_\Omega |u(x)|^{2^*-2}(x)u(x) \varphi(x)\,dx\,.
\end{aligned}$$

In order to prove the existence of solutions for problem~\eqref{problemaK0f},
we will apply some variants of the Mountain Pass Theorem and of the Linking Theorem (see, for instance, \cite{ar, bnc, r78, rabinowitz}) which take into account the fact that in the critical setting there is a lack of compactness in the embedding $X_0\hookrightarrow L^{2^*}(\RR^n)$ (see Lemma~\ref{lemmaembeddings}-$b)$). As a consequence of this, as it happens in classical case, also in the non-local critical setting the functional $\mathcal J_{K,\,\lambda}$ does not verify the Palais--Smale condition globally, but only in an energy range determined by the best fractional critical Sobolev constant $S_K$ given in formula~\eqref{SK}\,.

In what follows, $\lambda_1$ will be the first eigenvalue of $-\mathcal L_K$ with homogeneous Dirichlet boundary data. In the case when $\lambda\in (0, \lambda_1)$, problem~\eqref{problemaK0f} was studied in \cite{servadeivaldinociBN} in a setting more general than the power function. Hence, here we consider only the case when $\lambda\geq \lambda_1$ and, in order to get our goal, we will use appropriately the Linking Theorem of Rabinowitz (see \cite[Theorem~5.3]{rabinowitz}).

Since $\lambda\geq \lambda_1$, we can suppose that
$$\lambda\in
[\lambda_k, \lambda_{k+1})\,\, \mbox{ for some}\,\, k\in \NN\,,$$
where $\lambda_k$ is the $k$-th eigenvalue of the operator
$-\mathcal L_K$ with homogeneous Dirichlet boundary conditions.
We recall that, in what follows, $e_k$ will be the $k$--th eigenfunction corresponding to the eigenvalue $\lambda_k$ of $-\mathcal L_K$, and
\begin{equation}\label{Pk+1}
\PP_{k+1}:= \big\{ u\in X_0 {\mbox{ s.t. }}
\langle u,e_j\rangle_{X_0}=0 \quad\forall j=1,\dots,k
\big\}\,,
\end{equation}
while
${\mbox{span}} \{e_1, \dots, e_k\}$ will denote the linear subspace
generated by the first $k$ eigenfunctions of $-\mathcal L_K$
for any $k\in \NN$\,. Finally, we recall that $\mathcal J_{K,\,\lambda}$ satisfies the Palais--Smale condition at level $c\in \RR$ ($(PS)_c$-condition for short) if any sequence $u_j$ in $X_0$ such that
$$\mathcal J_{K,\,\lambda}(u_j)\to c$$
and
$$\sup\Big\{ \big|\langle\,\mathcal J_{K,\,\lambda}'(u_j),\varphi\,\rangle
\big|\,: \; \varphi\in X_0\,, \|\varphi\|_{X_0}=1\Big\}\to 0$$
as $j\to +\infty$ admits a subsequence strongly convergent in $X_0$\,.

In order to prove Theorem~\ref{generalkernel0f}, our strategy consists in adapting the techniques used in the classical case of the Laplacian in the critical setting (see \cite{am, struwe, willem} and references therein) to the non-local critical framework. Roughly, in order to apply \cite[Theorem~5.3]{rabinowitz}
we will prove these three facts:
\begin{itemize}
\item compactness conditions for $\mathcal J_{K,\,\lambda}$: the functional $\mathcal J_{K,\,\lambda}$ satisfies the $(PS)_c$-condition at any level $c$ smaller than a certain threshold related to the best fractional critical Sobolev constant, i.e., to be precise, for any $c$ such that $c<\frac s n S_K^{n/(2s)}$\,;
\item geometric structure of $\mathcal J_{K,\,\lambda}$: the functional~$\mathcal J_{K,\,\lambda}$ has the geometry required by the Linking Theorem;
\item estimate of the critical level of $\mathcal J_{K,\,\lambda}$: the Linking critical level of $\mathcal J_{K,\,\lambda}$ lies below the threshold where the $(PS)_c$-condition holds true. For this we will use assumption~\eqref{SKV}\,.
\end{itemize}


\section{A local Palais--Smale condition for the functional~$\mathcal J_{K,\,\lambda}$}\label{sec:compactness}
We start by proving that the functional~$\mathcal J_{K,\,\lambda}$ satisfies the Palais--Smale condition in a suitable energy range involving the best fractional critical Sobolev constant~$S_K$ given in \eqref{SK}.
\begin{proposition}\label{propPS}
Let $\lambda\in[\lambda_k, \lambda_{k+1})$ for some $k\in \NN$ and let $c\in \RR$ be such that
\begin{equation}\label{cS}
c<\frac s n S_K^{n/(2s)}\,.
\end{equation}
Let $u_j$ be a
sequence in $X_0$ such that
\begin{equation}\label{Jc0}
\mathcal J_{K,\,\lambda}(u_j)\to c
\end{equation}
and
\begin{equation}\label{J'00}
\sup\Big\{ \big|\langle\,\mathcal J_{K,\,\lambda}'(u_j),\varphi\,\rangle
\big|\,: \; \varphi\in X_0\,, \|\varphi\|_{X_0}=1\Big\}\to 0
\end{equation}
as $j\to +\infty$.

Then, there exists $u_\infty\in X_0$ such that, up to a subsequence,
$\|u_j-u_\infty\|_{X_0}\to 0$ as $j\to +\infty$\,.
\end{proposition}
\begin{proof}

We proceed by steps.

\medskip

\begin{step}\label{claim2}
The sequence $u_j$ is bounded in $X_0$\,.
\end{step}
\begin{proof}
The proof of Step~\ref{claim2} is quite standard (see, for instance, \cite{servadeivaldinociBN})\,, but we prefer to repeat it for reader's convenience.
For any $j\in \NN$ by \eqref{Jc0} and \eqref{J'00} it easily follows
that there exists $\kappa>0$ such that
\begin{equation}\label{jlimitato0}
|\mathcal J_{K,\,\lambda}(u_j)|\leq \kappa\,,
\end{equation}
and
\begin{equation}\label{j'limitato0}
\Big|\langle \mathcal J_{K,\,\lambda}'(u_j),
\frac{u_j}{\|u_j\|_{X_0}}\rangle\Big| \leq \kappa\,.
\end{equation}
As a consequence of \eqref{jlimitato0} and \eqref{j'limitato0} we have
\begin{equation}\label{2k}
\mathcal J_{K,\,\lambda}(u_j)-\frac 1 2 \langle \mathcal J_{K,\,\lambda}'(u_j),
u_j\rangle\leq \kappa \left(1+ \|u_j\|_{X_0}\right)\,.
\end{equation}

Moreover, it is easy to see that
$$\mathcal J_{K,\,\lambda}(u_j)-\frac 1 2 \langle \mathcal J_{K,\,\lambda}'(u_j), u_j\rangle  =
-\left(\frac{1}{\,2^*} -\frac 1 2\right)\|u_j\|_{L^{2^*}(\Omega)}^{2^*}= \frac s n\, \|u_j\|_{L^{2^*}(\Omega)}^{2^*}\,,$$
so that, thanks to \eqref{2k}, we get that for any $j\in \NN$
\begin{equation}\label{norma2star}
\|u_j\|_{L^{2^*}(\Omega)}^{2^*} \leq \kappa_*\left(1+\|u_j\|_{X_0}\right)
\end{equation}
for a suitable positive constant $\kappa_*$\,.

Consequently, recalling that~$2^*>2$, the use of the H\"older inequality yields
\begin{equation*}
\|u_j\|_{L^{2}(\Omega)}^{2}\le |\Omega|^{2s/n}\|u_j\|_{L^{2^*}(\Omega)}^{2}
\le \kappa_*^{2/2^*} |\Omega|^{2s/n} \left(1+\|u_j\|_{X_0}\right)^{2/2^*}\,.
\end{equation*}
That is, using that~$2/2^*<1$,
\begin{equation}\label{norma2}
\|u_j\|_{L^{2}(\Omega)}^{2}\le
\widetilde\kappa \left(1+\|u_j\|_{X_0}\right) \,,
\end{equation}
for a suitable~$\widetilde\kappa>0$
that does not depend on~$j$.
By~\eqref{jlimitato0}, \eqref{norma2star} and~\eqref{norma2}, we conclude that
\begin{eqnarray*}&& \kappa\ge\mathcal J_{K,\,\lambda}(u_j) =\frac12\|u_j\|_{X_0}^2-\frac\lambda2
\|u_j\|_{L^{2}(\Omega)}^{2}-\frac1{2^*}\|u_j\|_{L^{2^*}(\Omega)}^{2^*}\\
&&\qquad\ge \frac12\|u_j\|_{X_0}^{2}-\underline\kappa \left(1+\|u_j\|_{X_0}\right) \,,  \end{eqnarray*}
with~$\underline\kappa>0$ independent of~$j$\,.
Hence, the proof of Step~\ref{claim2} is complete\,.
\end{proof}

\begin{step}\label{claim3}
Problem~\eqref{problemaK0f} admits a solution $u_\infty\in X_0$\,.
\end{step}
\begin{proof}
Since $u_j$ 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, still denoted by $u_j$, there exists $u_\infty \in X_0$
such that~$u_j\to u_\infty$ weakly in~$X_0$, that is
\begin{equation}\label{convergenze0}
\begin{aligned}
 & \int_{\RR^{2n}}\big(u_j(x)-u_j(y)\big)\big(\varphi(x)-\varphi(y)\big) K(x-y)\,dx\,dy \to \\
& \qquad \qquad
\int_{\RR^{2n}}\big(u_\infty(x)-u_\infty(y)\big)\big(\varphi(x)-\varphi(y)\big)
K(x-y)\,dx\,dy  \quad \mbox{for any}\,\, \varphi\in X_0
\end{aligned}
\end{equation}
as $j\to +\infty$\,. Moreover, by Step~\ref{claim2}, \eqref{norma2star}, Lemma~\ref{lemmaembeddings}-$b)$ and the fact that $L^{2^*}(\RR^n)$ is a reflexive space we have that, up to a subsequence
\begin{equation}\label{convergenzaweak}
u_j \to u_\infty \quad \mbox{weakly in}\,\, L^{2^*}(\RR^n)
\end{equation}
as $j\to +\infty$, while by Lemma~\ref{lemmaembeddings}-$a)$, up to a subsequence,
\begin{equation}\label{convergenze0bis}
u_j \to u_\infty \quad \mbox{in}\,\, L^\nu(\RR^n)
\end{equation}
\begin{equation}\label{convergenze0ter}
u_j \to u_\infty \quad \mbox{a.e. in}\,\, \RR^n
\end{equation}
as $j\to +\infty$ for any $\nu\in [1, 2^*)$\,.

As a consequence of \eqref{norma2star} it is easy to see that $|u_j|^{2^*-2}u_j$ is bounded in $L^{2^*/(2^*-1)}(\Omega)$ uniformly in $j\in \NN$. Hence, by this and \eqref{convergenzaweak} we get that
\begin{equation}\label{convergenzaweak1}
|u_j|^{2^*-2}u_j \to |u_\infty|^{2^*-2}u_\infty \quad \mbox{weakly in}\,\, L^{2^*/(2^*-1)}(\Omega)
\end{equation}
as $j\to +\infty$\,.

Since $\big(2^*/(2^*-1)\big)'=2^*$, by \eqref{convergenzaweak1} it is easily seen that
$$\int_\Omega |u_j(x)|^{2^*-2}u_j(x)\varphi(x)\,dx \to  \int_\Omega |u_\infty(x)|^{2^*-2}u_\infty(x)\varphi(x)\,dx \quad \mbox{for any} \quad \varphi\in L^{2^*}(\Omega)\,,$$
and so, in particular,
\begin{equation}\label{convffi1}
\int_\Omega |u_j(x)|^{2^*-2}u_j(x)\varphi(x)\,dx \to  \int_\Omega |u_\infty(x)|^{2^*-2}u_\infty(x)\varphi(x)\,dx \quad \mbox{for any} \quad \varphi\in X_0
\end{equation}
as $j\to +\infty$ (here we use Lemma~\ref{lemmaembeddings}-$b)$).

By \eqref{J'00}, for any $\varphi \in X_0$
$$\begin{aligned}
0\leftarrow \langle \mathcal J_{K,\,\lambda}'(u_j), \varphi\rangle & =
\int_{\RR^{2n}}\big(u_j(x)-u_j(y)\big)\big(\varphi(x)-\varphi(y)\big) K(x-y)\,dx\,dy-\lambda \int_\Omega u_j(x)\varphi(x)\,dx\\
& \qquad \qquad \qquad \quad \quad  - \int_\Omega |u_j(x)|^{2^*-2}u_j(x)\varphi(x)\,dx\,,
\end{aligned}$$
so that, passing to the limit in this expression as $j\to +\infty$ and taking into account \eqref{convergenze0}, \eqref{convergenze0bis} and \eqref{convffi1} we get
$$\begin{aligned}
\int_{\RR^{2n}}\big(u_\infty(x)-u_\infty(y)\big)& \big(\varphi(x)-\varphi(y)\big) K(x-y)\,dx\,dy-\lambda \int_\Omega u_\infty(x)\varphi(x)\,dx\\
& \qquad \qquad \qquad \qquad \qquad \qquad  - \int_\Omega |u_\infty(x)|^{2^*-2}u_\infty(x)\varphi(x)\,dx=0
\end{aligned}$$
for any $\varphi \in X_0$, that is $u_\infty$ is a solution of problem~\eqref{problemaK0f} and Step~\ref{claim3} follows.
\end{proof}

\begin{step}\label{step3}
The following relation holds true:
$$\mathcal J_{K,\,\lambda}(u_\infty) =\frac s n\, \int_\Omega|u_\infty(x)|^{2^*}dx\geq 0\,.$$
\end{step}
\begin{proof}
By Step~\ref{claim3}, taking $\varphi=u_\infty\in X_0$ as a test function in \eqref{problemaK0f}\,, we get
$${\displaystyle \int_{\RR^{2n}}
|u_\infty(x)-u_\infty(y)|^2 K(x-y) dx\,dy -\lambda \int_\Omega
|u_\infty(x)|^2\,dx= \int_\Omega |u_\infty(x)|^{2^*}dx}\,,$$
so that
$$\mathcal J_{K,\,\lambda}(u_\infty) =\left(\frac 1 2-\frac{1}{2^*}\right) \int_\Omega|u_\infty(x)|^{2^*}dx=\frac s n\int_\Omega|u_\infty(x)|^{2^*}dx\geq 0\,.$$
Hence, Step~\ref{step3} is proved.
\end{proof}

\begin{step}\label{step4}
The following equality holds true:
$$\begin{aligned}
\mathcal J_{K,\,\lambda}(u_j)  =\mathcal J_{K,\,\lambda}(u_\infty) + \frac 1 2 \int_{\RR^{2n}}|u_j(x)-&u_\infty(x)-u_j(y)+u_\infty(y)|^2 K(x-y)dx\,dy\\
& -\frac{1}{2^*}\int_\Omega |u_j(x)-u_\infty(x)|^{2^*}dx+\textit{o}(1)
\end{aligned}$$
as $j\to +\infty$\,.
\end{step}

\begin{proof}
First of all, we observe that by Step~\ref{claim2} and Lemma~\ref{lemmaembeddings}-$b)$, the sequence $u_j$ is bounded in $X_0$ and in $L^{2^*}(\Omega)$. Hence, since \eqref{convergenze0ter} holds true, by the Brezis-Lieb Lemma (see \cite[Theorem~1]{bl}), we get
\begin{equation}\label{bl1}
\begin{aligned}
\int_{\RR^{2n}}|u_j(x)-u_j(y)|^2 & K(x-y)\,dx\,dy \\ & =\int_{\RR^{2n}}|u_j(x)-u_\infty(x)-u_j(y)+u_\infty(y)|^2 K(x-y)\,dx\,dy\\
 & \qquad \qquad + \int_{\RR^{2n}}|u_\infty(x)-u_\infty(y)|^2 K(x-y)\,dx\,dy + \textit{o}(1)
\end{aligned}
\end{equation}
and
\begin{equation}\label{bl2}
\int_\Omega|u_j(x)|^{2^*}\,dx= \int_\Omega |u_j(x)-u_\infty(x)|^{2^*}\,dx+ \int_\Omega|u_\infty(x)|^{2^*}\,dx+ \textit{o}(1)
\end{equation}
as $j\to +\infty$\,.

Therefore, using also the definition of $\mathcal J_{K,\,\lambda}$, by \eqref{convergenze0bis},  \eqref{bl1} and \eqref{bl2} we deduce that
$$\begin{aligned}
\mathcal J_{K,\,\lambda}(u_j) & =\frac 1 2 \int_{\RR^{2n}}|u_j(x)-u_\infty(x)-u_j(y)+u_\infty(y)|^2 K(x-y)\,dx\,dy\\
 & \qquad \qquad + \frac 1 2\int_{\RR^{2n}}|u_\infty(x)-u_\infty(y)|^2 K(x-y)\,dx\,dy -\frac \lambda 2 \int_\Omega |u_\infty(x)|^2 dx\\
 & \qquad \qquad -\frac{1}{2^*}\int_\Omega |u_j(x)-u_\infty(x)|^{2^*}\,dx-\frac{1}{2^*} \int_\Omega|u_\infty(x)|^{2^*}\,dx+\textit{o}(1)\\
& =\mathcal J_{K,\,\lambda}(u_\infty)+\frac 1 2 \int_{\RR^{2n}}|u_j(x)-u_\infty(x)-u_j(y)+u_\infty(y)|^2 K(x-y)\,dx\,dy\\
& \qquad \qquad -\frac{1}{2^*}\int_\Omega |u_j(x)-u_\infty(x)|^{2^*}\,dx+\textit{o}(1)
\end{aligned}$$
as $j\to +\infty$\,, which gives the desired assertion.
\end{proof}

\begin{step}\label{step5}
The following equality holds true:
$$\begin{aligned}
\int_{\RR^{2n}}|u_j(x)-u_\infty(x)-u_j(y)+u_\infty(y)|^2 & K(x-y)dx\,dy = \int_\Omega |u_j(x)-u_\infty(x)|^{2^*}dx+\textit{o}(1)
\end{aligned}$$
as $j\to +\infty$\,.
\end{step}

\begin{proof}
First of all, note that, as a consequence of \eqref{convergenzaweak}, \eqref{convergenzaweak1} and \eqref{bl2}, we get
\begin{equation}\label{ujuinfty}
\begin{aligned}
\int_\Omega\left(|u_j(x)|^{2^*-2}u_j(x)\right. & \left.-|u_\infty(x)|^{2^*-2}u_\infty(x)\right)\Big(u_j(x)-u_\infty(x)\Big)\,dx\\
& = \int_\Omega |u_j(x)|^{2^*}\,dx -\int_\Omega |u_\infty(x)|^{2^*-1}u_\infty(x)u_j(x)\,dx\\
& \qquad -\int_\Omega |u_j(x)|^{2^*-2}u_j(x) u_\infty(x)\,dx+ \int_\Omega  |u_\infty(x)|^{2^*}\,dx\\
& = \int_\Omega |u_j(x)|^{2^*}\,dx- \int_\Omega  |u_\infty(x)|^{2^*}\,dx + \textit{o}(1)\\
& = \int_\Omega |u_j(x)-u_\infty(x)|^{2^*}\,dx+ \textit{o}(1)
\end{aligned}
\end{equation}
as $j\to +\infty$\,.


By \eqref{J'00} and Steps~\ref{claim2} and \ref{claim3}, it is easily seen that
\begin{equation}\label{conto1}
\textit{o}(1)=\langle\mathcal J'_{K,\,\lambda}(u_j), u_j-u_\infty\rangle=\langle\mathcal J'_{K,\,\lambda}(u_j)-\mathcal J'_{K,\,\lambda}(u_\infty), u_j-u_\infty\rangle
\end{equation}
as $j\to +\infty$\,.
Moreover, by \eqref{convergenze0bis} and \eqref{ujuinfty}
\begin{equation}\label{conto2}
\begin{aligned}
\langle\mathcal J'_{K,\,\lambda}(u_j) -& \mathcal J'_{K,\,\lambda} (u_\infty), u_j -u_\infty\rangle  \\=
& \int_{\RR^{2n}}|u_j(x)-u_\infty(x)-u_j(y)+u_\infty(y)|^2 K(x-y)dx\,dy \\
&  \qquad -\lambda \int_\Omega |u_j(x)-u_\infty(x)|^2\,dx\\
&  \qquad -
\int_\Omega\left(|u_j(x)|^{2^*-2}u_j(x)-|u_\infty(x)|^{2^*-2}u_\infty(x)\right)(u_j(x)-u_\infty(x))\,dx\\
& = \int_{\RR^{2n}}|u_j(x)-u_\infty(x)-u_j(y)+u_\infty(y)|^2 K(x-y)dx\,dy \\
& \qquad - \int_\Omega |u_j(x)-u_\infty(x)|^{2^*}\,dx+ \textit{o}(1)
\end{aligned}
\end{equation}
as $j\to +\infty$\,.
Hence, the assertion of Step~\ref{step5} comes from \eqref{conto1} and \eqref{conto2}\,.
\end{proof}

Now, we can conclude the proof of Proposition~\ref{propPS}\,.

\medskip

By Step~\ref{step5} it is easy to see that
$$\begin{aligned}
\frac 1 2 \int_{\RR^{2n}}|u_j(x)- & u_\infty(x)- u_j(y)+u_\infty(y)|^2 K(x-y)dx\,dy -\frac{1}{2^*}\int_\Omega |u_j(x)-u_\infty(x)|^{2^*}dx\\
&= \left(\frac 1 2 -\frac{1}{2^*}\right)\int_{\RR^{2n}}|u_j(x)-u_\infty(x)-u_j(y)+u_\infty(y)|^2 K(x-y)dx\,dy +\textit{o}(1)\\
&= \frac s n\int_{\RR^{2n}}|u_j(x)-u_\infty(x)-u_j(y)+u_\infty(y)|^2 K(x-y)dx\,dy +\textit{o}(1)\,,
\end{aligned}$$
and so, as a consequence of this and Step~\ref{step4},
\begin{equation}\label{bo}
\begin{aligned}
\mathcal J_{K,\,\lambda}(u_\infty) & +\frac s n\int_{\RR^{2n}}|u_j(x)-u_\infty(x)-u_j(y)+u_\infty(y)|^2 K(x-y)dx\,dy\\
& =\mathcal J_{K,\,\lambda}(u_j) +\textit{o}(1)\\
& =c+\textit{o}(1)\,,
\end{aligned}
\end{equation}
as $j\to +\infty$\,, thanks to \eqref{Jc0}\,.
Now, by Step~\ref{claim2} the sequence $\|u_j\|_{X_0}$ is bounded in $\RR$. Hence, up to a subsequence, if necessary, we can assume that
\begin{equation}\label{L1}
\|u_j-u_\infty\|_{X_0}^2=\int_{\RR^{2n}}|u_j(x)-u_\infty(x)-u_j(y)+u_\infty(y)|^2 K(x-y)\,dx\,dy\to L
\end{equation}
and so, again as a consequence of Step~\ref{step5},
$$\int_\Omega |u_j(x)-u_\infty(x)|^{2^*}\,dx\to L$$
as $j\to +\infty$\,.
Of course $L\in [0, +\infty)$ and, by definition of $S_K$, it holds true
$$L\geq L^{2/2^*}S_K\,,$$
so that
$$L=0 \quad \mbox{or} \quad L\geq S_K^{n/(2s)}\,.$$
The case $L\geq S_K^{n/(2s)}$ can not occur. Otherwise, by \eqref{bo}, \eqref{L1} and Step~\ref{step3}, we would get
$$c=\mathcal J_{K,\,\lambda}(u_\infty)+\frac s n\,L \geq \frac s n\,L \geq \frac s n\, S_K^{n/(2s)}\,,$$
which contradicts \eqref{cS}\,. Thus $L=0$ and so, by \eqref{L1}, we obtain that
$$\|u_j-u_\infty\|_{X_0}\to 0$$
as $j\to +\infty$\,.
This ends the proof of Proposition~\ref{propPS}\,.
\end{proof}

\section{The geometry of the functional $\mathcal J_{K,\,\lambda}$}\label{sec:geometry}
Along this section we show that the functional $\mathcal J_{K,\,\lambda}$ has the
geometric structure required by \cite[Theorem~5.3]{rabinowitz}. In a non-local setting for nonlinearities with subcritical growth a proof of this is given in \cite{svlinking}. Essentially, here we adapt the proof given in \cite[Subsection~4.2]{svlinking} to the critical case. With respect to the classical case of the Laplacian some technical differences arise.

\begin{proposition}\label{lemmaJpositivoL}
Let $\lambda\in [\lambda_k, \lambda_{k+1})$  and let $\PP_{k+1}$ be as in \eqref{Pk+1} for some $k\in \NN$\,. Then,
there exist $\rho>0$ and $\beta>0$ such that for any $u\in \PP_{k+1}$ with
$\|u\|_{X_0}=\rho$ it results that $\mathcal J_{K,\,\lambda}(u)\geq
\beta$\,.
\end{proposition}
\begin{proof}
Let $u$ be a function in $\PP_{k+1}$. Thanks to the choice of $\lambda$ and to the variational characterization of $\lambda_{k+1}$ (see \cite[Proposition~9-$d)$]{svlinking}) we get
\begin{equation}\label{infJfrac0}
\begin{aligned}
\mathcal J_{K,\,\lambda}(u)& = \frac 1 2\int_{\RR^{2n}}\!\!|u(x)-u(y)|^2 K(x-y)dx\,dy-\frac \lambda 2 \int_\Omega\!\! |u(x)|^2\,dx-\frac{1}{2^*}\int_\Omega\!\! |u(x)|^{2^*}dx\\
& \geq \frac{1}{2}\left(1-\frac{\lambda}{\lambda_{k+1}}\right)\int_{\RR^{2n}} |u(x)-u(y)|^2
K(x-y)dx\,dy-\frac{1}{2^*}\int_\Omega |u(x)|^{2^*}dx\,.
\end{aligned}
\end{equation}

Using \cite[Lemma~6]{svmountain} and \eqref{kernelfrac}, from \eqref{infJfrac0} we
deduce that
\begin{equation}\label{infJfrac20}
\begin{aligned}
\mathcal J_{K,\,\lambda}(u) & \geq \frac{1}{2}\left(1-\frac{\lambda}{\lambda_{k+1}}\right)\int_{\RR^{2n}} |u(x)-u(y)|^2 K(x-y)\,dx\,dy \\
& \qquad -\frac{c^{2^*/2}}{2^*}\left( \int_{\RR^{2n}}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\, dx\,dy\right)^{2^*/2}\\
& \ge \frac{1}{2}\left(1-\frac{\lambda}{\lambda_{k+1}}\right) \int_{\RR^{2n}} |u(x)-u(y)|^2 K(x-y)\,dx\,dy\\
& \qquad
-\frac{1}{2^*}
\Big(\frac{c}{\theta}\Big)^{2^*/2}
\left( \int_{\RR^{2n}} |u(x)-u(y)|^2 K(x-y)\, dx\,dy\right)^{2^*/2}\\
& =\frac{1}{2}\left(1-\frac{\lambda}{\lambda_{k+1}}\right) \|u\|_{X_0}^2-\frac{1}{2^*}
\Big(\frac{c}{\theta}\Big)^{2^*/2}
\|u\|_{X_0}^{2^*}\,.
\end{aligned}
\end{equation}
Hence, for a suitable positive constant $\kappa$ we have
$$\mathcal J_{K,\,\lambda}(u)\geq \frac{1}{2}\left(1-\frac{\lambda}{\lambda_{k+1}}\right) \|u\|_{X_0}^2\left(1-\kappa
\|u\|_{X_0}^{2^*-2}\right)\,.$$

Now, let $u\in \PP_{k+1}$ be such that $\|u\|_{X_0}=\rho>0$. Since $2^*>2$, choosing $\rho$ sufficiently small (i.e. $\rho$ such that $1-\kappa\rho^{2^*-2}>0$), we get
$$\inf_{
{u\in \PP_{k+1}}\atop{\|u\|_{X_0}=\rho}
} \mathcal J_{K,\,\lambda}(u)\geq \frac{1}{2}\left(1-\frac{\lambda}{\lambda_{k+1}}\right) \rho^2(1-\kappa \rho^{2^*-2})=:\beta>0\,,$$
and this ends the proof of Proposition~\ref{lemmaJpositivoL}\,.
\end{proof}

\begin{proposition}\label{lemmaJnegativoL}
Let $\lambda\in [\lambda_k, \lambda_{k+1})$ for some $k\in \NN$\,. Then, $\mathcal J_{K,\,\lambda}(u)\leq 0$ for any $u\in \mbox{span}\{ e_1, \dots, e_k\}$\,.
\end{proposition}
\begin{proof}
Let $u\in \mbox{span}\{ e_1, \dots, e_k\}$. Then
$$u(x)=\sum_{i=1}^{k}u_i e_i(x)\,,$$ with $u_i\in \RR$, $i=1, \dots,
k$\,. Since $\big\{e_1, \dots, e_k, \dots\big\}$ is an orthonormal
basis of $L^2(\Omega)$ and an orthogonal one of $X_0$ by \cite[Proposition~9--$f)$]{svlinking}, we get
\begin{equation}\label{ortoL2}
\int_\Omega |u(x)|^2\,dx=\sum_{i=1}^k u_i^2
\end{equation}
and
\begin{equation}\label{ortoX0}
\begin{aligned}
\int_{\RR^{2n}} |u(x)-u(y)|^2 K(x-y)\,dx\,dy
=\sum_{i=1}^k u_i^2 \|e_i\|_{X_0}^2\,.
\end{aligned}
\end{equation}
Then, by \eqref{ortoL2} and \eqref{ortoX0} and using \cite[Proposition~9-$b)$ and $e)$]{svlinking} (here we also use the fact that $e_i\in X_0$ for $i=1, \dots, k$, see \cite[Proposition~9--$f)$]{svlinking}) we get
$$\begin{aligned}
\mathcal J_{K,\,\lambda}(u) & = \frac 1 2\sum_{i=1}^k u_i^2
\Big(\|e_i\|_{X_0}^2-\lambda\Big)-\frac{1}{2^*}\|u\|_{L^{2^*}(\Omega)}^{2^*}\\
& \leq \frac 1 2\sum_{i=1}^k u_i^2 \big(\lambda_i-\lambda\big)\leq 0\,,
\end{aligned}$$
since $\lambda_i\leq \lambda_k\leq \lambda$ for
any $i=1, \dots, k$\,. Hence, Proposition~\ref{lemmaJnegativoL} follows.
\end{proof}

\begin{proposition}\label{lemmaJnegativo1LF}
Let $\lambda\geq 0$ and
let $\mathbb F$ be a finite dimensional subspace of $X_0$. Then, there exists $R>\rho$ such that $\mathcal J_{K,\,\lambda}(u)\leq 0$ for any $u\in \mathbb F$ with $\|u\|_{X_0}\geq R$\,, where $\rho$ is given in Proposition~\ref{lemmaJpositivoL}.
\end{proposition}
\begin{proof}
Let $u\in \mathbb F$. Then, the non-negativity of $\lambda$ gives
$$\begin{aligned}
\mathcal J_{K,\,\lambda} (u)& = \frac 1 2 \|u\|_{X_0}^2-\frac \lambda 2\|u\|_{L^2(\Omega)}^2-\frac{1}{2^*}
\|u\|_{L^{2^*}(\Omega)}^{2^*}\\
& \leq \frac 1 2 \|u\|_{X_0}^2-\frac{1}{2^*}
\|u\|_{L^{2^*}(\Omega)}^{2^*}\\
& \leq \frac 1 2 \|u\|_{X_0}^2-\frac{\kappa}{2^*}
\|u\|_{X_0}^{2^*}\,,
\end{aligned}$$
for some positive constant $\kappa$, thanks to the fact that in any finite dimensional space all the norms are equivalent.

Hence, if $\|u\|_{X_0}\to +\infty$, then
$$\mathcal J_{K,\,\lambda} (u)\to -\infty\,,$$
since $2^*>2$ by assumption, and so the assertion of Proposition~\ref{lemmaJnegativo1LF} follows.
\end{proof}

\section{Proof of Theorem~\ref{generalkernel0f}}\label{sec:dimtheorem1}
As we said at the beginning of Section~\ref{sec:strategy}, in the case when $\lambda\in (0, \lambda_1)$ our thesis follows by \cite[Theorem~2]{servadeivaldinociBN}, where more general critical nonlinearities than the power (critical) function were studied.

Now, let us consider the case when $\lambda\in [\lambda_k, \lambda_{k+1})$ for some $k\in \NN$\,.
Let $u_0$ be the function given in condition~\eqref{SKV}\,.
Note that we can choose $u_0\geq 0$ a.e. in $\RR^n$, since if $v\in X_0$, then its positive part $v^+$ and its negative part $v^-$ belong to $X_0$ by \cite[Lemma~12]{sv}\,.

Starting from $u_0$ we construct the functions $z, \tilde z\in \PP_{k+1}$ as follows
\begin{equation}\label{z}
z=u_0-\sum_{i=1}^k \Big(\int_\Omega u_0(x) e_i(x)\,dx\Big)e_i\,, \quad \tilde z=z/\|z\|_{X_0}\,.
\end{equation}
Moreover, let $V=\mbox{span}\{e_1, \dots, e_k\}$ and $W=\PP_{k+1}$\,. It is easily seen that $V$ is finite dimensional and $V\oplus W=X_0$\,.

By Propositions~\ref{lemmaJpositivoL}--\ref{lemmaJnegativo1LF} we get that
$\mathcal J_{K, \,\lambda}$ has the geometric structure required by the Linking Theorem (see assumptions~$(I'_1)$ and $(I_5)$ of \cite[Theorem~5.3]{rabinowitz} and also \cite[Remark~5.5-$(iii)$]{rabinowitz}), that is
$$\inf_{
{u\in W}\atop{\|u\|_{X_0}=\rho}
} \mathcal J_{K,\,\lambda}(u)\geq\beta>0\,,$$
$$\sup_{u\in V}\mathcal J_{K,\,\lambda}(u)\leq 0$$
and
$$\sup_{
{u\in \mathbb F}\atop{\|u\|_{X_0}\geq R}
} \mathcal J_{K,\,\lambda}(u)\leq 0\,, \quad \mathbb F=V\oplus \mbox{span}\{\tilde z\}\,,$$
where $\beta$ and $R$ are as in Propositions~\ref{lemmaJpositivoL} and \ref{lemmaJnegativo1LF}, respectively, and $\tilde z$ is as in \eqref{z}\,.

Now, let $c_L$ be the Linking critical level of $\mathcal J_{K,\,\lambda}$, i.e.
$$c_L=\inf_{h\in \Gamma}\max_{u\in Q}\mathcal J_{K,\,\lambda}(h(u))\,,$$
where
$$\Gamma=\big\{h\in C(\overline Q; X_0): h=id\,\, \mbox{on}\,\, \partial Q \big\}$$
and
$$Q=\Big(\overline B_R\cap V\Big)\oplus \big\{r\tilde z : r\in (0, R)\big\}\,.$$

Our goal is to show that $c_L$ is smaller than $\frac s n\, S_K^{n/(2s)}$\,, that is proving that the Linking critical level of $\mathcal J_{K,\,\lambda}$ lies below the threshold where the $(PS)_c$--condition holds true.  For this, first of all we note that for any $u\in X_0\setminus\{0\}$
\begin{equation}\label{max}
\max_{\zeta\geq 0}\mathcal J_{K,\,\lambda}(\zeta u)=\frac s n\,S_{K,\,\lambda}^{n/(2s)}(u)
\end{equation}
(see, for instance, \cite[Proposition~20]{servadeivaldinociBN}).

By definition of $c_L$ we get that for any $h\in \Gamma$
$$c_L\leq \max_{u\in Q}\mathcal J_{K,\,\lambda}(h(u))$$
and so, in particular, taking $h=id$ on $\overline Q$
\begin{equation}\label{c1}
c_L\leq \max_{u\in Q}\mathcal J_{K,\,\lambda}(u)\leq \max_{u\in \mathbb F}\mathcal J_{K,\,\lambda}(u)\,,
\end{equation}
being $Q\subset \mathbb F$\,.
Since $\mathbb F$ is a linear space,
\begin{equation}\label{maxt}
\begin{aligned}
\max_{u\in \mathbb F}\mathcal J_{K,\,\lambda}(u) & =\max_{u\in \mathbb F \atop \zeta\not =0}\mathcal J_{K,\,\lambda}(|\zeta|\cdot \frac{u}{|\zeta|})\\
& = \max_{u\in \mathbb F \atop \zeta> 0}\mathcal J_{K,\,\lambda}(\zeta u)\\
& \leq \max_{u\in \mathbb F \atop \zeta\geq 0}\mathcal J_{K,\,\lambda}(\zeta u)\,.
\end{aligned}
\end{equation}
Hence, \eqref{max}--\eqref{maxt} and assumption~\eqref{SKV} yield
\begin{equation}\label{cL}
c_L\leq \max_{u\in \mathbb F\atop \zeta\geq 0}\mathcal J_{K,\,\lambda}(\zeta u)=\frac s n\,\max_{u \in \mathbb F} S_{K,\,\lambda}^{n/(2s)}(u)<\frac s n\, S_K^{n/(2s)}\,,
\end{equation}
since $\mathbb F=V\oplus \mbox{span}\{\tilde z\}=\mbox{span}\{e_1, \dots, e_k, u_0\}$ by construction of $\tilde z$\,.


Therefore, the Linking Theorem \cite[Theorem~5.3]{rabinowitz} yields that problem~\eqref{problemaK0f} admits a
solution $u\in X_0$ with critical value $\mathcal J_{K,\,\lambda}(u)=c_L\geq \beta$\,. Since $\beta>0= \mathcal J_{K,\,\lambda}(0)$, we deduce that $u$ is not identically zero and this concludes the proof of  Theorem~\ref{generalkernel0f}.


\section{The fractional Laplace case: proof of Theorem~\ref{lapfra0}}\label{sec:prooffraclap}
In this section, as a concrete application, we consider the model fractional kernel
$$K(x)=|x|^{-(n+2s)},$$ which gives rise to the fractional Laplace operator $-(-\Delta )^s$ defined is \eqref{2}\,.
Hence, here we study problem~\eqref{1s} or, to be more precise,
its weak formulation, that is problem \eqref{problemalapfrac0}\,.

In order to prove Theorem~\ref{lapfra0} we will apply Theorem~\ref{generalkernel0f}. For doing this, it is enough to show that condition~\eqref{SKV} is satisfied in the fractional Laplace setting.
First of all, we have to construct the function $u_0$ and the linear space~$\mathcal U$ given in~\eqref{SKV} in a suitable way (i.e. in such a way that \eqref{SKV} is verified).

\subsection{Construction of the function~$u_0$ and of the linear space~$\mathcal U$}\label{subsec:setting}
In order to construct such $u_0$, we can proceed as in \cite[Section~4]{servadeivaldinociBN}\,. To illustrate the procedure we need the following result proved in \cite[Theorem~1.1]{cotsiolis}:

\begin{theorem}\label{claimubar}
The infimum in formula~\eqref{Ss} is attained, that is
$$S_s=S_s(\tilde u)\,,$$
where
\begin{equation}\label{utilde}
\tilde u(x)=\kappa{\Big(\mu^2+|x-x_0|^2\Big)^{-(n-2s)/2}}\,, \quad x\in \RR^n\end{equation}
with $\kappa\in \RR\setminus\{0\}$, $\mu>0$ and $x_0\in \RR^n$ fixed constants.

Equivalently, the function $\bar u$ defined as
\begin{equation}\label{baru}
\bar u(x)=\frac{\tilde u(x)}{\|\tilde u\|_{L^{2^*}(\RR^n)}}\end{equation}
is such that
\begin{equation}\label{utildeSs}
{\displaystyle S_s=\inf_{{v\in H^s(\RR^n)} \atop{\|v\|_{L^{2^*}(\RR^n)}=1}}{\displaystyle\int_{\RR^{2n}}\frac{|v(x)-v(y)|^2}{|x-y|^{n+2s}} \,dx\,dy}=\int_{\RR^{2n}}\frac{|\bar u(x)-\bar u(y)|^2}{|x-y|^{n+2s}} \,dx\,dy\,.}
\end{equation}
\end{theorem}

In what follows we suppose that, up to a translation, $x_0=0$ in \eqref{utilde}\,.
As in the classical case of the Laplacian, following \cite[Section~4]{servadeivaldinociBN}, we can construct an explicit solution of the limiting
critical problem
\begin{equation}\label{limitingproblem}
(-\Delta )^s u=|u|^{2^*-2}u \quad {\mbox{ in }} \RR^n
\end{equation}
as follows:
\begin{equation}\label{ustar}
{\displaystyle u^*(x)=\bar u\left(\frac{x}{S_s^{1/(2s)}}\right)}\,, \qquad x\in \RR^n\,.
\end{equation}
Note that the function~$u^*\in H^s(\RR^n)$ is a solution of problem~\eqref{limitingproblem} satisfying the property
\begin{equation}\label{normaSs}
\|u^*\|_{L^{2^*}(\RR^n)}^{2^*}=S_s^{n/(2s)}\,.
\end{equation}

Now, starting from $u^*$ we can define the family of functions $U_\varepsilon$ as
\begin{equation}\label{Uepsilon}
U_\varepsilon(x)=\varepsilon^{-(n-2s)/2}\,u^*(x/\varepsilon)\,, \qquad x\in \RR^n
\end{equation}
for any $\varepsilon>0$\,.

Finally, we need to put appropriately $U_\varepsilon$  to zero outside $\Omega$\,. For this let us fix $\delta>0$ such that
$$B_{4\delta}\subset \Omega$$
and let us consider the cut-off function $\eta \in C^\infty(\RR^n)$ such that $0\leq \eta \leq 1$ in
$\RR^n$\,, $\eta\equiv 1$ in $B_\delta$ and $\eta\equiv
0$ in $\mathcal C B_{2\delta}$, where $B_{\delta}=B(0, \delta)$ and $\mathcal C B_\delta=\RR^n\setminus B_\delta$\,. For
every $\varepsilon>0$ we denote by $u_\varepsilon$ the following family of
functions
\begin{equation}\label{uepsilon}
u_\varepsilon(x)=\eta(x)U_\varepsilon(x)\,, \quad x\in \RR^n\,,
\end{equation}
where $U_\varepsilon$ is given in \eqref{Uepsilon}\,.

Of course $u_\varepsilon\in H^s(\RR^n)$, $u_\varepsilon \geq 0$ a.e. in $\RR^n$ and $u_\varepsilon=0$ a.e. in $\RR^n\setminus\Omega$\,. Hence, by \cite[Lemma~7]{servadeivaldinociBN} it is easily seen that $u_\varepsilon \in X_0$\,.
Moreover, the function $u_\varepsilon$ satisfies the following crucial estimates:

\begin{proposition}\label{stimaUepsilon}
Let $s\in (0,1)$ and $n>2s$. Then, the  following estimates hold true
$${\displaystyle \int_{\RR^{2n}} \frac{|u_\varepsilon(x)-u_\varepsilon(y)|^2}{|x-y|^{n+2s}}\,dx\,dy
\le S_s^{n/(2s)}+\mathcal O(\varepsilon^{n-2s})}\,,$$
$$\int_{\RR^n} |u_\varepsilon(x)|^2\,dx\geq \begin{cases}
C_s\varepsilon^{2s}+ \mathcal O(\varepsilon^{n-2s}) & \mbox{if} \quad n>4s\\
C_s\varepsilon^{2s}\big|\log\varepsilon\big|+ \mathcal O(\varepsilon^{2s})  & \mbox{if} \quad n=4s\\
C_s\varepsilon^{n-2s}+ \mathcal O(\varepsilon^{2s}) & \mbox{if} \quad n<4s\,,
\end{cases}$$
$$\int_{\RR^n} |u_\varepsilon(x)|^{2^*}\,dx= S_s^{n/(2s)}
+ \mathcal O(\varepsilon^n\,)\,,$$
$$\int_{\RR^n} |u_\varepsilon(x)|^{2^*-1}\,dx= \mathcal O(\varepsilon^{(n-2s)/2}\,)\,,$$
and
$$\int_{\RR^n} |u_\varepsilon(x)|\,dx= \mathcal O(\varepsilon^{(n-2s)/2}\,)$$
as $\varepsilon \to
0$\,, for some positive constant $C_s$ depending on $s$\,.

As a consequence, if $\varepsilon>0$ is sufficiently small, then for any $\lambda>0$
\begin{equation}\label{SslambdaSs}
S_{s,\,\lambda}(u_\varepsilon)<S_s\,,
\end{equation}
provided $n\geq 4s$\,.
\end{proposition}

\begin{proof}
The first three estimates were proved in \cite[Propositions~21 and 22 and Subsection~4.2.1]{servadeivaldinociBN}\,. It remains to show the last two ones. Let us start by proving the $L^{2^*-1}$-estimate.
By \eqref{Uepsilon} and \eqref{uepsilon} we have that
$$\begin{aligned}
\int_{\RR^n} |u_\varepsilon(x)|^{2^*-1}\,dx & \leq \int_{B_{2\delta}} |U_\varepsilon(x)|^{2^*-1}\,dx\\
 & = \kappa^*\Big(\varepsilon^{-(n-2s)/2}\Big)^{2^*-1}\int_{B_{2\delta}}
 \left[{\Big(\mu^2+|x/\varepsilon|^2\Big)^{-(n-2s)/2}}\right]^{2^*-1}\,dx\\
 & = \kappa^*\varepsilon^{(n-2s)/2}\int_0^{2\delta/\varepsilon}
 \frac{r^{n-1}}{\Big(\mu^2+r^2\Big)^{(n+2s)/2}}\,\,dr\\
 & \cong \tilde \kappa \varepsilon^{(n-2s)/2}\left(1+\int_1^{2\delta/\varepsilon}r^{-2s-1}\,dr\right)\\
 & = \mathcal O(\varepsilon^{(n-2s)/2})
\end{aligned}$$
as $\varepsilon\to 0$.

In a similar way we get
$$\begin{aligned}
\int_{\RR^n} |u_\varepsilon(x)|\,dx & \leq \int_{B_{2\delta}} |U_\varepsilon(x)|\,dx\\
 & = \kappa^*\varepsilon^{-(n-2s)/2}\int_{B_{2\delta}}
 {\Big(\mu^2+|x/\varepsilon|^2\Big)^{-(n-2s)/2}}\,dx\\
 & = \kappa^*\varepsilon^{(n+2s)/2}\int_0^{2\delta/\varepsilon}
 \frac{r^{n-1}}{\Big(\mu^2+r^2\Big)^{(n-2s)/2}}\,\,dr\\
 & \cong \tilde \kappa \varepsilon^{(n+2s)/2}\left(1+\int_1^{2\delta/\varepsilon}r^{2s-1}\right)\\
 & = \mathcal O(\varepsilon^{(n-2s)/2})
\end{aligned}$$
as $\varepsilon\to 0$. In both cases, here $\kappa^*$ and $\tilde \kappa$ are positive constants.

Finally, let us show \eqref{SslambdaSs}\,. If $n>4s$, by definition of $S_{s,\,\lambda}(\cdot)$ and the previous estimates we get
$$\begin{aligned}
{\displaystyle S_{s,\,\lambda}(u_\varepsilon)}& {\displaystyle =\frac{{\displaystyle\int_{\RR^{2n}}\frac{|u_\varepsilon(x)-u_\varepsilon(y)|^2}{|x-y|^{n+2s}} \,dx\,dy-\lambda\int_\Omega |u_\varepsilon(x)|^2\,dx}}{{\displaystyle\Big(\int_\Omega |u_\varepsilon(x)|^{2^*}dx\Big)^{2/2^*}}}}\\
& {\displaystyle \leq\frac{S_s^{n/(2s)}+\mathcal O(\varepsilon^{n-2s})-\lambda C_s\varepsilon^{2s}}{\Big(S_s^{n/(2s)}+\mathcal O(\varepsilon^{n})\Big)^{2/2^*}}}\\
& {\displaystyle \leq S_s +\varepsilon^{2s}\Big(\mathcal O(\varepsilon^{n-4s})-\lambda C_s \Big)<S_s\,,}
\end{aligned}$$
for any $\lambda>0$, provided $\varepsilon>0$ is sufficiently small.

If $n=4s$, similarly we get
$$\begin{aligned}
{\displaystyle S_{s,\,\lambda}(u_\varepsilon)}
& {\displaystyle \leq\frac{S_s^{n/(2s)}+\mathcal O(\varepsilon^{n-2s})-\lambda C_s\varepsilon^{2s}|\log \varepsilon|+ \mathcal O(\varepsilon^{2s})}{\Big(S_s^{n/(2s)}+\mathcal O(\varepsilon^{n})\Big)^{2/2^*}}}\\
& {\displaystyle \leq S_s +\mathcal O(\varepsilon^{2s})-\lambda C_s\varepsilon^{2s}|\log \varepsilon|}\\
& {\displaystyle \leq S_s +\varepsilon^{2s}|\log \varepsilon|\Big(|\log \varepsilon|^{-1}-\lambda C_s \Big)<S_s\,,}
\end{aligned}$$
for any $\lambda>0$, provided $\varepsilon>0$ is sufficiently small. This concludes the proof of Proposition~\ref{stimaUepsilon}\,.
\end{proof}

Now, we construct the function~$u_0$ given in \eqref{SKV} as follows:
\begin{equation}\label{u0}
u_0=u_\varepsilon\,\,\, \mbox{for $\varepsilon$ sufficiently small\,,}
\end{equation}
and the linear space $\mathcal U$ as
\begin{equation}\label{F}
\mathcal U=\mathcal U_\varepsilon:=\begin{cases}
\mbox{span}\{u_\varepsilon\} & \mbox{if}\,\,\, \lambda\in (0,\, \lambda_{1,\,s})\\
\mbox{span}\{e_{1,\,s}, \dots, e_{k,\,s}, u_\varepsilon\} & \mbox{if}\,\,\, \lambda\in [\lambda_{k,\,s}\, \lambda_{k+1,\,s})\,, \,\, k\in \NN\,,
\end{cases}
\end{equation}
where the functions~$e_{k,\,s}$,
are the eigenfunctions of $(-\Delta)^s$ corresponding to~$\lambda_{k,\,s}$ for any $k\in \NN$ (see \cite[Proposition~9]{svlinking} for a detailed spectral theory for the fractional Laplacian)\,.

In the next subsection we will show that, with this choice of $u_0$ and $\mathcal U$, condition~\eqref{SKV} is satisfied.

\subsection{A non-degeneracy estimate}\label{subsec:estimate} Note that, if $\lambda\in (0,\, \lambda_{1,\,s})$, then condition~\eqref{SKV} reduces to
\begin{equation}\label{condBN}
S_{s,\,\lambda}(u_0)=S_{s,\,\lambda}(u_\varepsilon)<S_s\,,
\end{equation}
which is always satisfied, provided $\varepsilon$ is sufficiently small, thanks to \eqref{SslambdaSs}\,.

Now, it remains to consider the case when $\lambda\in [\lambda_{k,\,s}\, \lambda_{k+1,\,s})$ for some $k\in \NN$\,. For this we need the following result:
\begin{proposition}\label{propMepsilon}
Let $s\in (0,1)$ and $n>2s$\,. Let $\lambda\in [\lambda_{k,\,s}, \lambda_{k+1,\,s})$ for some $k\in \NN$ and let $\mathcal U_\varepsilon$ be the linear space defined in \eqref{F}\,. Finally, let
\begin{equation}\label{mepsilon}
{\displaystyle M_\varepsilon:=
\max_{{u\in \mathcal U_\varepsilon}\atop{\|u\|_{L^{2^*}(\Omega)}=1}}S_{s,\,\lambda}(u)\,,}
\end{equation}
where the function~$H^s(\RR^n)\setminus\{0\}\ni v\mapsto S_{s,\,\lambda}(v)$ is defined as in \eqref{Sslambdav}\,.
Then
\begin{itemize}
\item [$a)$] $M_\varepsilon$ is achieved in $u_M\in \mathcal U_\varepsilon$ and $u_M$ can be written as follows\footnote{Beware that $u_M$, $v$ and $t$
(and also $\tilde v$ in \eqref{2an} below) depend on $\varepsilon$\,. For simplicity we omit this dependence in the notation.}
\begin{equation}\label{uM}
u_M=v+tu_\varepsilon\,,\,\,\, \mbox{with}\,\,\, v\in \mbox{span}\{e_{1,\,s}, \dots, e_{k,\,s}\}\,\, \mbox{and}\,\,\, t\geq 0\,;
\end{equation}
\item [$b)$] the following estimate holds true
$$M_\varepsilon\leq \begin{cases}
\big(\lambda_{k,\,s}-\lambda\big)\|v\|_{L^2(\Omega)}^2 & \mbox{if}\,\,\,\, t=0\\
 \begin{aligned}
 \big(\lambda_{k,\,s}-\lambda\big)& \|v\|_{L^2(\Omega)}^2+S_{s,\,\lambda}(u_\varepsilon)\Big(1+\mathcal O(\varepsilon^{(n-2s)/2)})\|v\|_{L^2(\Omega)}\Big)\\
  & \qquad \quad \qquad \qquad \qquad \qquad \qquad
+\mathcal O(\varepsilon^{(n-2s)/2)})\,\|v\|_{L^2(\Omega)}\end{aligned} & \mbox{if}\,\,\,\, t>0
 \end{cases}$$
as $\varepsilon \to 0$\,, where $v$ is given in \eqref{uM}\,.
\end{itemize}
\end{proposition}

\begin{proof}
First of all, note that $u_\varepsilon\in X_0$ and $e_{i,\,s}\in X_0$ for any $i=1, \dots, k$, thanks to \cite[Proposition~9-$b)$ and $e)$]{svlinking} and to \cite[Lemma~7]{servadeivaldinociBN}\,. As a consequence of this $\mathcal U_\varepsilon\subset X_0$ and, by definition of $S_{s,\,\lambda}$\,,
$${\displaystyle M_\varepsilon:=\max_{{u\in \mathcal U_\varepsilon}\atop{\|u\|_{L^{2^*}(\Omega)}=1}}\left(\int_{\RR^{2n}}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}} \,dx\,dy-\lambda \int_\Omega |u(x)|^2\,dx\right)}\,.$$

Let us start by proving part~$a)$. The maximum in formula~\eqref{mepsilon} is achieved thanks to the Weierstrass Theorem
(applied here in a finite dimensional space). Thus, let $u_M\in \mathcal U_\varepsilon$ be the function such that
$${\displaystyle M_\varepsilon=\int_{\RR^{2n}}\frac{|u_M(x)-u_M(y)|^2}{|x-y|^{n+2s}} \,dx\,dy-\lambda \int_\Omega |u_M(x)|^2\,dx} \,\,\,\,\,\, \mbox{and}\,\,\,\,\,\, \|u_M\|_{L^{2^*}(\Omega)}=1\,.$$
Note that $u_M\not \equiv 0$\,. Moreover, since $u_M\in \mathcal U_\varepsilon$\,, by definition of $\mathcal U_\varepsilon$ we have that
$$u_M=v+t u_\varepsilon\,,$$
for some $v\in \mbox{span}\{e_{1,\,s}, \dots, e_{k,\,s}\}$ and $t\in \RR$\,.
We can suppose that $t\geq 0$\, (otherwise, if $t\leq 0$ we can replace~$u_M$
with~$-u_M$). This concludes the proof of assertion~$a)$\,.

Now, let us show $b)$\,. First of all, note that, if $t=0$ in formula~\eqref{uM}, then $u_M=v\in \mbox{span}\{e_{1,\,s}, \dots, e_{k,\,s}\} $ and
$$\begin{aligned}
M_\varepsilon & {\displaystyle =\int_{\RR^{2n}}\frac{|u_M(x)-u_M(y)|^2}{|x-y|^{n+2s}} \,dx\,dy-\lambda \int_\Omega |u_M(x)|^2\,dx}\\
& {\displaystyle =\int_{\RR^{2n}}\frac{|v(x)-v(y)|^2}{|x-y|^{n+2s}} \,dx\,dy-\lambda \int_\Omega |v(x)|^2\,dx}\\
& \leq (\lambda_{k,\,s}-\lambda)\|v\|_{L^2(\Omega)}^2\,,
\end{aligned}$$
by Proposition~\ref{propcaratterizzazione}. Thus, assertion~$b)$ holds true when $t=0$\,.

Now, let us consider the case when $t>0$ in~\eqref{uM}\,.
By the
H\H older inequality and the properties of $u_M$,
we can bound $u_M$ as follows
\begin{equation}\label{1an}
\|u_M\|^2_{L^2(\Omega)}
\leq |\Omega|^{n/(2s)} \|u_M\|^2_{L^{2^*}(\Omega)}= |\Omega|^{n/(2s)}.
\end{equation}
Also,
since $u_M\in \mathcal U_\varepsilon$\,, by definition of $\mathcal U_\varepsilon$ we have that
\begin{equation}\label{2an}
u_M=\tilde v+t z_\varepsilon\,,
\end{equation}
where
$${\displaystyle \tilde v=
v+t\sum_{i=1}^k \Big(\int_\Omega u_\varepsilon(x) e_{i,\,s}(x)\,dx\Big)
e_{i,\,s}\in \mbox{span}\{e_{1,\,s}, \dots, e_{k,\,s}\}}$$
and
\begin{equation}\label{3an}
{\displaystyle z_\varepsilon=u_\varepsilon-\sum_{i=1}^k \Big(\int_\Omega u_\varepsilon(x) e_{i,\,s}(x)\,dx\Big)e_{i,\,s}\,,}
\end{equation}
so that $\tilde v$ and $z_\varepsilon$ are orthogonal in $L^2(\Omega)$ (see \cite[Proposition~9--$f)$]{svlinking})\,.
As a consequence of this, we get that
\begin{equation}\label{4an}
\|u_M\|^2_{L^2(\Omega)}=\|\tilde v\|^2_{L^2(\Omega)}+t^2
\|z_\varepsilon\|^2_{L^2(\Omega)}\ge \|\tilde v\|^2_{L^2(\Omega)}.
\end{equation}
By \eqref{1an} and \eqref{4an}, we have obtained that~$\|u_M\|_{L^2(\Omega)}$,
and~$\|\tilde v\|_{L^2(\Omega)}$
are all bounded uniformly in~$\eps$ by a suitable~$\tilde c>0$.

Now, we claim that, up to renaming such a constant,
\begin{equation}\label{tlimitato}
t\leq \tilde c\,.
\end{equation}
For this note that by Proposition~\ref{proplinfty} we have that $e_{i,\,s}\in L^\infty(\Omega)$ for any $i\in \NN$\,, so that also
$\tilde v\in \mbox{span}\{e_{1,\,s}, \dots,
e_{k,\,s}\}$ does. Hence, $\tilde v\in L^{2^*}(\Omega)$, since $\Omega$ is bounded. Moreover, by the equivalence of the norms in a finite dimensional space,
we also have
\begin{equation}\label{plus}
\|\tilde v\|_{L^{2^*}(\Omega)}\le\tilde c\,.
\end{equation}
By Propositions~\ref{proplinfty} and \ref{stimaUepsilon} we have
$$ \left|\int_\Omega u_\varepsilon(x)e_{i,s}(x)\,dx\right|\le \|u_\varepsilon\|_{L^1(\Omega)}\|e_{i,s}\|_{L^\infty(\Omega)}
={\mathcal{O}}(\varepsilon^{(n-2s)/2})$$
as $\varepsilon\to 0$\,.
As a consequence, using \eqref{3an} and again Proposition~\ref{stimaUepsilon},
\begin{eqnarray*}
\|z_\varepsilon\|_{L^{2^*}(\Omega)}
&\ge&\|u_\varepsilon\|_{L^{2^*}(\Omega)}-
\sum_{i=1}^k \left|\int_\Omega u_\varepsilon(x)e_{i,s}(x)\,dx\right| \|e_{i,s}\|_{L^{2^*}(\Omega)}\\
& = & S_s^{(n-2s)/(4s)}+\mathcal O(\varepsilon^{(n-2s)/2})\\
&\ge& \frac{S^{(n-2s)/(4s)}_s}{2}
\end{eqnarray*}
for $\varepsilon$ sufficiently small.
So, by~\eqref{2an}, the fact that $t\geq 0$ and \eqref{plus} we have
$$ \frac{S^{(n-2s)/(4s)}_s\,t}{2}
\le
t\|z_\varepsilon\|_{L^{2^*}(\Omega)}\le\|u_M\|_{L^{2^*}(\Omega)}
+\|\tilde v\|_{L^{2^*}(\Omega)}
\le 1+\tilde c\,,$$
thanks to the properties of $u_M$\,. Therefore the claim in~\eqref{plus} follows.

Note that by Proposition~\ref{proplinfty} also
$v\in \mbox{span}\{e_{1,\,s}, \dots,
e_{k,\,s}\}$ belongs to $L^\infty(\Omega)$\,.

Finally, the convexity\footnote{If $f$ is a differentiable convex function, then $f(y)\geq f(x)+f'(x)(y-x)$. Here we take $f(s)=s^{2^*}$\,, $x=tu_\varepsilon$ and $y=u_M=v+tu_\varepsilon$\,.}, the monotonicity properties of the integrals, \eqref{tlimitato} and the fact that in $\mbox{span}\{e_{1,\,s}, \dots, e_{k,\,s}\}$ all the norms are equivalent yield
\begin{equation}\label{convexity0}
\begin{aligned}
1=\|u_M\|_{L^{2^*}(\Omega)}^{2^*} & =\int_\Omega |u_M(x)|^{2^*}\,dx\\
& \geq \int_\Omega |tu_\varepsilon(x)|^{2^*}\,dx + 2^*\int_\Omega \big(tu_\varepsilon(x)\big)^{2^*-1} v(x)\,dx\\
& \geq \|tu_\varepsilon\|_{L^{2^*}(\Omega)}^{2^*}-2^*\tilde c\,^{2^*-1}\|u_\varepsilon\|_{L^{2^*-1}(\Omega)}^{2^*-1}\|v\|_{L^\infty(\Omega)}\\
& \geq \|tu_\varepsilon\|_{L^{2^*}(\Omega)}^{2^*}-\hat c\,\|u_\varepsilon\|_{L^{2^*-1}(\Omega)}^{2^*-1}\|v\|_{L^2(\Omega)}\,,
\end{aligned}
\end{equation}
for some positive constant $\hat c$\,, so that
\begin{equation}\label{convexity}
\|tu_\varepsilon\|_{L^{2^*}(\Omega)}^{2^*}\leq 1+\hat c\,\|u_\varepsilon\|_{L^{2^*-1}(\Omega)}^{2^*-1}\|v\|_{L^2(\Omega)}\,.
\end{equation}
Now, we are ready to prove the inequality in part~$b)$\,. By
\eqref{uM} we get
\begin{equation}\label{Mepsilon1}
\begin{aligned}
M_\varepsilon & ={\displaystyle\int_{\RR^{2n}}\frac{|u_M(x)-u_M(y)|^2}{|x-y|^{n+2s}} \,dx\,dy-\lambda \int_\Omega |u_M(x)|^2\,dx}\\
& = {\displaystyle\int_{\RR^{2n}}\frac{|v(x)+tu_\varepsilon(x)-v(y)-tu_\varepsilon(y)|^2}{|x-y|^{n+2s}} \,dx\,dy-\lambda \int_\Omega |v(x)+tu_\varepsilon(x)|^2\,dx}\\
& ={\displaystyle\int_{\RR^{2n}}\frac{|v(x)-v(y)|^2}{|x-y|^{n+2s}} \,dx\,dy}+t^2{\displaystyle\int_{\RR^{2n}}\frac{|u_\varepsilon(x)-u_\varepsilon(y)|^2}{|x-y|^{n+2s}} \,dx\,dy}\\
& \qquad \qquad +2t\int_{\RR^{2n}} \frac{\big(u_\varepsilon(x)-u_\varepsilon(y)\big)\big(v(x)-v(y)\big)}{|x-y|^{n+2s}}\,dxdy\\
& \qquad -
\lambda  {\displaystyle\int_\Omega |v(x)|^2\,dx}-\lambda t^2 {\displaystyle\int_\Omega |u_\varepsilon(x)|^2\,dx}-2\lambda t\int_\Omega u_\varepsilon(x)v(x)\,dx\\
& \leq \big(\lambda_{k,\,s}-\lambda\big)\|v\|_{L^2(\Omega)}^2+
S_{s,\,\lambda}(u_\varepsilon)\|t u_\varepsilon\|_{L^{2^*}(\Omega)}^2\\
& \qquad +2t\int_{\RR^{2n}} \frac{\big(u_\varepsilon(x)-u_\varepsilon(y)\big)\big(v(x)-v(y)\big)}{|x-y|^{n+2s}}
-2\lambda t\int_\Omega u_\varepsilon(x)v(x)\,dx\,,
\end{aligned}
\end{equation}
thanks to Proposition~\ref{propcaratterizzazione} and to the definition of $S_{s,\,\lambda}(\cdot)$\,.

Now we write
$$ v=\sum_{i=1}^k v_i e_{i,s}$$
for some~$v_i\in\RR$\,, so that
$$ \|v\|_{L^2(\Omega)}^2=\sum_{i=1}^k v_i^2\,.$$
Also (see \eqref{scalarproduct}),
$$ \langle u_\eps,v\rangle_{X_0}=\sum_{i=1}^k
v_i \langle u_\eps,e_{i,s}\rangle_{X_0}=
\sum_{i=1}^k\lambda_{i,s}v_i\int_{\Omega} u_\eps(x) e_{i,s}(x)\,dx.$$
So, by the H\H older inequality,
$$\begin{aligned}
|\langle u_\eps,v\rangle_{X_0}| &
\le \sum_{i=1}^k \lambda_{i,s}|v_i|\,
\| u_\eps\|_{L^1(\Omega)} \|e_{i,s}\|_{L^\infty(\Omega)}\\
& \le  \lambda_{k+1,\,s}\| u_\eps\|_{L^1(\Omega)}
\|v\|_{L^\infty(\Omega)}\\
& \leq \bar\kappa\| u_\eps\|_{L^1(\Omega)}\| v\|_{L^2(\Omega)}\,,
\end{aligned}$$
for a suitable~$\bar\kappa>0$ possibly depending on~$k$.
More explicitly,
\begin{equation}\label{Mepsilon2}
\left |\int_{\RR^{2n}} \frac{\big(u_\varepsilon(x)-u_\varepsilon(y)\big) \big(v(x)-v(y)\big)}{|x-y|^{n+2s}}\,dxdy\right|\\
\leq \bar \kappa\|v\|_{L^2(\Omega)}\|u_\varepsilon\|_{L^1(\Omega)}.
\end{equation}
Gathering the results in \eqref{Mepsilon1} and \eqref{Mepsilon2}, and using again
the H\H older inequality we get
$$\begin{aligned}
M_\varepsilon & \leq \big(\lambda_{k,\,s}-\lambda\big)\|v\|_{L^2(\Omega)}^2+
S_{s,\,\lambda}(u_\varepsilon)\|t u_\varepsilon\|_{L^{2^*}(\Omega)}^2+2t\bar \kappa\|v\|_{L^2(\Omega)}\|u_\varepsilon\|_{L^1(\Omega)}\\
& \qquad \qquad \qquad \qquad \qquad \qquad
+2\lambda t\|u_\varepsilon\|_{L^1(\Omega)}\|v\|_{L^\infty(\Omega)}\\
& \leq \big(\lambda_{k,\,s}-\lambda\big)\|v\|_{L^2(\Omega)}^2+
S_{s,\,\lambda}(u_\varepsilon)\|t u_\varepsilon\|_{L^{2^*}(\Omega)}^2+\kappa\|u_\varepsilon\|_{L^1(\Omega)}\|v\|_{L^2(\Omega)}\,,
\end{aligned}$$
for some positive $\kappa$\,, thanks to \eqref{tlimitato} and
the equivalence of the norms in the
finite dimensional space~$\mbox{span}\{e_{1,\,s}, \dots, e_{k,\,s}\}$\,.

This and \eqref{convexity} yield
$$\begin{aligned}
M_\varepsilon & \leq \big(\lambda_{k,\,s}-\lambda\big)\|v\|_{L^2(\Omega)}^2+
S_{s,\,\lambda}(u_\varepsilon)\Big(1+\hat c\,\|u_\varepsilon\|_{L^{2^*-1}(\Omega)}^{2^*-1}\|v\|_{L^2(\Omega)}\Big)^{2/2^*}\!\!+\kappa\|u_\varepsilon\|_{L^1(\Omega)}\|v\|_{L^2(\Omega)}\,,
\end{aligned}$$
so that, by Proposition~\ref{stimaUepsilon}
$$\begin{aligned}
M_\varepsilon & \leq \big(\lambda_{k,\,s}-\lambda\big)\|v\|_{L^2(\Omega)}^2+
S_{s,\,\lambda}(u_\varepsilon)\Big(1+\mathcal O(\varepsilon^{(n-2s)/2})\|v\|_{L^2(\Omega)}\Big)^{2/2^*}+\mathcal O(\varepsilon^{(n-2s)/2})\|v\|_{L^2(\Omega)}\\
& \leq \big(\lambda_{k,\,s}-\lambda\big)\|v\|_{L^2(\Omega)}^2+
S_{s,\,\lambda}(u_\varepsilon)\Big(1+\mathcal O(\varepsilon^{(n-2s)/2})\|v\|_{L^2(\Omega)}\Big)+\mathcal O(\varepsilon^{(n-2s)/2})\|v\|_{L^2(\Omega)}\,,
\end{aligned}$$
being $2<2^*$ (since $s>0$)\,. Then, assertion~$b)$ is proved.

This ends the proof of Proposition~\ref{propMepsilon}\,.
\end{proof}

\medskip

\rm{Now, we are ready to prove Theorem~\ref{lapfra0}\,.}

\subsection{End of the proof of Theorem~\ref{lapfra0}}\label{subsec:prooftheorem4}
When $\lambda\in (0, \lambda_{1,\,s})$ condition~\eqref{SKV} reduces to
\eqref{condBN}. Hence, Theorem~\ref{lapfra0} comes from \cite[Theorem~4]{servadeivaldinociBN}.

Let us consider the case when $\lambda> \lambda_{1,\,s}$.
As we said at the beginning of Section~\ref{sec:prooffraclap}, in order to prove Theorem~\ref{lapfra0}, the idea consists of applying Theorem~\ref{generalkernel0f} (for this we have to show that condition~\eqref{SKV} is satisfied). If we assume that $\lambda\in (\lambda_{k,\,s}\, \lambda_{k+1,\,s})$ for some $k\in \NN$, in order to prove condition~\eqref{SKV} it is enough to show that
\begin{equation}\label{stimaMepsilon}
{\displaystyle M_\varepsilon:=
\max_{{u\in \mathcal U_\varepsilon}\atop{\|u\|_{L^{2^*}(\Omega)}=1}}S_{s,\,\lambda}(u)<S_s\,,}
\end{equation}
taking into account the scale invariance of the function $S_{s,\,\lambda}(\cdot)$\,.

By Proposition~\ref{propMepsilon}-$a)$
$${\displaystyle M_\varepsilon=\int_{\RR^{2n}}\frac{|u_M(x)-u_M(y)|^2}{|x-y|^{n+2s}} \,dx\,dy-\lambda \int_\Omega |u_M(x)|^2\,dx}\,,$$
with
$$u_M=v+t u_\varepsilon\,,$$
for some $v\in \mbox{span}\{e_{1,\,s}, \dots, e_{k,\,s}\}$ and $t\geq 0$\,.

If $t=0$, by Proposition~\ref{propMepsilon}-$b)$ and the choice of $\lambda> \lambda_{k,\,s}$, we get that
$$M_\varepsilon\leq
\big(\lambda_{k,\,s}-\lambda\big)\|v\|_{L^2(\Omega)}^2< 0<S_s\,,$$
and so \eqref{stimaMepsilon} is proved.

Now, suppose that $t>0$\,. Again by Proposition~\ref{propMepsilon}-$b)$ we deduce that for $\varepsilon$ sufficiently small
\begin{equation}\label{R}
\begin{aligned}
M_\varepsilon & \leq \big(\lambda_{k,\,s}-\lambda\big)\|v\|_{L^2(\Omega)}^2+S_{s,\,\lambda}(u_\varepsilon)\Big(1+\mathcal O(\varepsilon^{(n-2s)/2})\|v\|_{L^2(\Omega)}\Big)\\
 & \qquad \qquad \qquad \qquad \qquad \qquad +\mathcal O(\varepsilon^{(n-2s)/2)}\|v\|_{L^2(\Omega)}\\
& =\mathcal O(\varepsilon^{n-2s})+S_{s,\,\lambda}(u_\varepsilon)\Big(1+\mathcal O(\varepsilon^{(n-2s)/2})\|v\|_{L^2(\Omega)}\Big)\,,
\end{aligned}
\end{equation}
since the parabola $\big(\lambda_{k,\,s}-\lambda\big)\|v\|_{L^2(\Omega)}^2+\mathcal O(\varepsilon^{(n-2s)/2})\|v\|_{L^2(\Omega)}$ stays always below its vertex,
that is
$$\big(\lambda_{k,\,s}-\lambda\big)\|v\|_{L^2(\Omega)}^2+\mathcal O(\varepsilon^{(n-2s)/2})\|v\|_{L^2(\Omega)}\leq \frac{1}{4(\lambda-\lambda_{k,\,s})}\mathcal O(\varepsilon^{n-2s})=\mathcal O(\varepsilon^{n-2s})\,.$$

We have to distinguish the cases $n>4s$ and $n=4s$\,.
Let us start with the first one, i.e. $n>4s$\,.
Thus, by \eqref{R} and Proposition~\ref{stimaUepsilon}
$$\begin{aligned}
M_\varepsilon & \leq \mathcal O(\varepsilon^{n-2s})+S_{s,\,\lambda}(u_\varepsilon)\Big(1+\mathcal O(\varepsilon^{(n-2s)/2})\|v\|_{L^2(\Omega)}\Big)\\
& \leq \mathcal O(\varepsilon^{n-2s})+\Big(S_s+\mathcal O(\varepsilon^{n-2s})-\lambda C_s\varepsilon^{2s}\Big)\Big(1+\mathcal O(\varepsilon^{(n-2s)/2})\|v\|_{L^2(\Omega)}\Big)\\
& = S_s+\mathcal O(\varepsilon^{n-2s})-\lambda C_s\varepsilon^{2s}\\
& = S_s+\varepsilon^{2s}\Big(\mathcal O(\varepsilon^{n-4s})-\lambda C_s\Big)\\
& < S_s
\end{aligned}$$
as $\varepsilon \to 0$\,. Here we use also the fact that $\lambda$ is positive (being $\lambda>\lambda_{k,\,s}>0$). This ends the proof of \eqref{stimaMepsilon} when $n>4s$\,.

Now, let $n=4s$\,. Arguing as above, by \eqref{R} and Proposition~\ref{stimaUepsilon} we get
$$\begin{aligned}
M_\varepsilon & \leq \mathcal O(\varepsilon^{2s})+S_{s,\,\lambda}(u_\varepsilon)\Big(1+\mathcal O(\varepsilon^s)\|v\|_{L^2(\Omega)}\Big)\\
& \leq S_s+\mathcal O(\varepsilon^{2s})-\lambda C_s\varepsilon^{2s}|\log \varepsilon|\\
& = S_s+\varepsilon^{2s}|\log \varepsilon|\Big(|\log \varepsilon|^{-1}-\lambda C_s\Big)\\
& < S_s
\end{aligned}$$
as $\varepsilon \to 0$\,. This conclude the proof in the case when $n=4$\,.

Hence, condition~\eqref{SKV} is satisfied when $n\geq 4s$ for any $\lambda>0$ different from the eigenvalues of $(-\Delta)^s$\,. Then, Theorem~\ref{generalkernel0f} provides a solution $u\in X_0\setminus\{0\}$ of problem~\eqref{problemalapfrac0}. Since $X_0\subseteq H^s(\RR^n)$ by \cite[Lemma~5-$b)$]{svmountain} (see also \cite[Lemma~7]{servadeivaldinociBN}), the proof
of Theorem~\ref{lapfra0} is complete.

\medskip

Note that, of course we can prove Theorem~\ref{lapfra0} directly, by applying the Mountain Pass and the Linking Theorem to the Euler--Lagrange functional
$$\mathcal J_{s,\,\lambda}(u) =
\frac 1 2 \int_{\RR^{2n}}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy-
\frac \lambda 2 \int_\Omega |u(x)|^2\,dx -\frac{1}{2^*}\int_\Omega |u(x)|^{2^*}dx$$
associated with problem~\eqref{problemalapfrac0}.
In this framework condition~\eqref{SKV} (or, to be precise, \eqref{condBN} when $\lambda\in (0,\, \lambda_{1,\,s})$ and \eqref{stimaMepsilon} when $\lambda\in (\lambda_{k,\,s}\, \lambda_{k+1,\,s})$\,, $k\in \NN$) says that the Mountain Pass or the Linking critical level of $\mathcal J_{s,\,\lambda}$ stays below the threshold where the $(PS)$--condition holds true.


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