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Critical nonlinearities, best critical Sobolev constant, variational 
techniques, Mountain Pass Theorem, Linking Theorem, integrodifferential 
operators, fractional Laplacian.
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\title[A Brezis-Nirenberg result for non-local critical equations in low dimension]{
A Brezis-Nirenberg result\\ for non-local critical equations\\ in low dimension}

\thanks{The first author was supported by
the MIUR National Research Project {\it Variational and Topological Methods
in the Study of Nonlinear Phenomena}, while the second 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}). Both the authors were supported 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}

\author[E. Valdinoci]{Enrico Valdinoci}
\address{Dipartimento di Matematica,
          Universit\`a di Milano,
          Via Cesare Saldini 50, 20133 Milano, Italy}
\email{\tt enrico.valdinoci@unimi.it}

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

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


\begin{abstract}
The present paper is devoted to the study of the following non-local fractional equation involving critical nonlinearities
$$\left\{
\begin{array}{ll}
(-\Delta)^s u-\lambda u=|u|^{2^*-2}u & {\mbox{ in }} \Omega\\
u=0 & {\mbox{ in }} \RR^n\setminus \Omega\,,
\end{array} \right.$$
where $s\in (0,1)$ is fixed, $(-\Delta )^s$ is the fractional Laplace operator, $\lambda$ is a positive parameter, $2^*$ is the fractional critical Sobolev exponent and $\Omega$ is an open bounded subset of $\RR^n$, $n>2s$\,, with Lipschitz boundary. In the recent papers \cite{sY, servadeivaldinociBN, servadeivaldinociCFP} we investigated the existence of non-trivial solutions for this problem when $\Omega$ is an open bounded subset of $\RR^n$ with $n\geq 4s$ and, in this framework, we prove some existence results.

Aim of this paper is to complete the investigation carried on in \cite{sY, servadeivaldinociBN, servadeivaldinociCFP}, by considering the case when $2s<n<4s$\,. In this context, we prove an existence theorem for our problem, which may be seen as a Brezis-Nirenberg type result in low dimension.
In particular when $s=1$ (and consequently $n=3$) our result is the classical result obtained by Brezis and Nirenberg in the famous paper~\cite{bn}\,.
In this sense the present work may be considered 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}

The quest of nontrivial solutions of the equation
$$ {\mathcal{L}} u=\Psi(u)$$
with Dirichlet boundary data is one of the classical
topic of research of the modern nonlinear analysis.
Typically, ${\mathcal{L}}$ is an elliptic operator (in most of the cases, the
Laplacian) and~$\Psi:\RR\rightarrow\RR$ is a reasonably smooth
function (called ``the nonlinearity'' in jargon).
When~$\Psi(0)=0$, we have that~$u\equiv0$ is a trivial solution of the problem,
and the task is to find another (hence non-trivial) solution -- i.e., a solution that does
not vanish identically.

Such a solution is usually sought by variational and topological methods:
namely, the problem is reduced to the search of (non-trivial) critical points
of an energy functional. The change of topology of the sublevel sets of
this functional is the geometric indication of the presence of a critical point,
in combination with suitable compactness conditions that ensure that the critical point
``does not escape towards infinity''.

In the celebrated paper~\cite{bn},
the case of~${\mathcal{L}}:=\Delta$ and~$\Psi(u):=-\lambda u-|u|^{2_*-2}u$ was taken
into account. In this setting, $\lambda$ is a positive parameter and
\begin{equation}\label{SOBOLEV} 2_*:=\frac{2n}{n-2}\end{equation} is
the Sobolev critical exponent (to make sense of which, it is necessary to require~$n\ge3$).
The technical problem posed by such an exponent is that the Sobolev embedding is
not compact. Nevertheless, in~\cite{bn}, a fine variational analysis showed
that non-trivial solutions exist when~$n\ge4$ and $\lambda$ lies in a suitable range
of parameters. This result was extended in~\cite{servadeivaldinociBN} for the case of
the fractional Laplacian~$-(-\Delta)^s$, with~$s\in(0,1)$: in this case, the dimensional
restriction becomes~$n\ge4s$ (which is consistent with the one in~\cite{bn} when~$s=1$).

In the case of the Laplacian, the case~$n=3$ was treated in detail in~\cite{druet}.
The purpose of the present paper is to extend this analysis to the case of
the fractional Laplacian under the dimensional restriction~$n\in (2s,4s)$. Notice once more
that this condition reduces to the classical one~$n=3$ when~$s=1$, but for different
values of~$s\in(0,1)$ it may cover different dimensions (for instance, it boils down to~$n=1$
when~$s\in(1/4, 1/2)$, but it becomes~$n=2$, $3$ for~$s=7/8$).

We recall that
fractional and non-local
operators of elliptic type were widely studied in the literature, both
for their interesting theoretical
structure and in view of concrete applications in many fields
such as, among the others,
optimization,
finance, phase transitions,
stratified materials,
anomalous diffusion, crystal dislocation,
soft thin films,
semipermeable membranes, flame propagation,
conservation laws,
ultra-relativistic
limits of quantum mechanics,
quasi-geostrophic flows,
multiple scattering,
minimal surfaces,
materials science and
water waves (we refer, e.g., to \cite{valpal} for an introduction to these topics and their applications).

Combining
the interest of the mathematical community both for non-local operators and for pde's with nonlinearities with critical growth (in the sense of Sobolev), in the recent papers \cite{sY, servadeivaldinociBN, servadeivaldinociCFP} we studied critical non-local equations using variational techniques (see also \cite{sv, svmountain, svlinking} for the subcritical case).

In these works our model problem is given by the following critical non-local fractional equation
\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,
which (up to normalization factors) may be defined 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}
$\Omega\subset \RR^n$, $n>2s$, is open, bounded and with
Lipschitz boundary, $\lambda>0$ and $$2^*=\frac{2n}{n-2s}$$
is the fractional critical Sobolev exponent (notice that when~$s=1$ the above exponent
reduces to the classical Sobolev exponent~$2_*$ in~\eqref{SOBOLEV}).
The homogeneous Dirichlet datum in~\eqref{1s} is given
in $\RR^n\setminus \Omega$ and not simply on $\partial \Omega$, as it
happens in the classical case of the Laplacian,
consistently with the non-local nature of the operator~$(-\Delta)^s$\,.

In this context in \cite{sY, servadeivaldinociBN} we have shown that
if $n\geq 4s$, then for any $\lambda>0$ different from the eigenvalue of the fractional Laplacian
with homogeneous Dirichlet datum,
problem~\eqref{1s} admits a non-trivial solution.

We thought that it was interesting to check what happens in lower dimension, that is when $2s<n<4s$\,.
Aim of this paper is to cover this case, in order to treat completely the critical setting.
The result we obtain in this case is the following:

\begin{theorem}\label{lapfra0}
Let $s\in (0,1)$, $2s<n<4s$ and $\Omega$ be an open bounded set of $\RR^n$
with Lipschitz boundary.

Then, there exists $\lambda_s>0$ such that for any
$\lambda>\lambda_s$ different from the eigenvalues of $(-\Delta)^s$ problem~\eqref{1s} admits a weak
solution $u\in H^s(\RR^n)$, which is not identically zero,
and such that $u=0$ a.e. in $\RR^n\setminus \Omega$\,.
\end{theorem}

We will prove Theorem~\ref{lapfra0} by using the Mountain Pass and the Linking Theorem in a combined way. The main difficulty to be
overcame is the lack of compactness. In order to solve this problem we first will show that the Palais--Smale condition holds true in
a suitable range related to the best fractional Sobolev constant $S_s$ (see Section~\ref{sec:PS} for more details) and then we will
prove that the critical Mountain Pass and Linking levels are below this threshold.

It is interesting to notice that when $s=1$ Theorem~\ref{lapfra0} reads as the results obtained for the Laplacian in
\cite[Corollary~1.1]{bn} for a bounded domain of $\RR^3$ and in \cite[Theorem~1]{comte} (see also \cite[Theorem~1.2]{bn}) in the case
of the ball in $\RR^3$\,. Hence, we think that Theorem~\ref{lapfra0} may be seen as the natural extension of some classical results
well know for the Laplacian (see \cite{bn, capfortpalm, comte, gazzolaruf} and also \cite{struwe, willem}) to the non-local fractional
framework. With this respect, we recall the recent papers~\cite{colorado, tan}, where a non-local version of the Brezis-Nirenberg result was also
considered for an fractional operator different than the one treated here.
See also~\cite{fiscella}
for another nonlinear analysis problem for fractional operators.

We remark that
problem~\eqref{1s} has a variational structure and we are interested in the notion
of solution given by the following weak formulation:
\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}
This formulation represents the Euler-Lagrange equation associated with the functional $\mathcal J_{s,\,\lambda}: X_0\to \RR$ defined as
\begin{equation}\label{jsl}
\begin{aligned}
\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\,.
\end{aligned}
\end{equation}
Here the functional space $X_0$ is given by
$$X_0=\big\{u\in H^s(\RR^n) : u=0\,\, \mbox{a.e. in}\,\, \RR^n\setminus \Omega\}\,,$$
while $H^s(\RR^n)$ is the usual fractional Sobolev space (we refer to \cite{sY, sv, svmountain, svlinking, servadeivaldinociBN, servadeivaldinociCFP} for a general definition of $X_0$ and its properties and, for instance, to \cite{valpal} for $H^s(\RR^n)$)\,.

In order to get non-trivial weak solutions for \eqref{1s} (i.e non-trivial solution of~\eqref{problemalapfrac0}), as in the classical case of the Laplacian, we will use variational techniques. More precisely, we will apply suitable variants of the Mountain Pass Theorem and of the Linking Theorem (see e.g. \cite{ar, r78, rabinowitz}) to the functional $\mathcal J_{s,\,\lambda}$\,.

As usual in the critical case, the problem related to the variational formulation of \eqref{1s} is the lack of compactness, due to the fact that $H^s(\RR^n)$ is not compactly embedded in $L^{2^*}(\RR^n)$\,. To overcome this difficulty we will show that, even if the functional $\mathcal J_{s,\,\lambda}$ does not
verify globally the Palais--Smale condition, it satisfies such a condition in a suitable range  related to the best fractional critical Sobolev constant $S_s$ for the embedding of $H^s(\RR^n)$ into $L^{2^*}(\RR^n)$\, which is defined as
\begin{equation}\label{Ss}
{\displaystyle S_s:=\inf_{v\in H^s(\RR^n)\setminus\{0\}}S_s(v)\,,}
\end{equation}
where for any $v\in H^s(\RR^n)\setminus\{0\}$
\begin{equation}\label{Ssv}
{\displaystyle 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{equation}
Along the paper we also consider the constant $S_{s,\,\lambda}$ defined as follows
\begin{equation}\label{Sslambda}
{\displaystyle S_{s,\,\lambda}:=\inf_{v\in H^s(\RR^n)\setminus\{0\}}S_{s,\,\lambda}(v)\,,}
\end{equation}
where for any $v\in H^s(\RR^n)\setminus\{0\}$
\begin{equation}\label{Sslambdav}
{\displaystyle 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}
The consistency of the definitions of $S_s$ and $S_{s,\,\lambda}$ and the fact that they are strictly positive
follows from~\cite[Lemma~14]{servadeivaldinociBN} and~\cite[Theorem~6.5]{valpal}.
It is easily seen that~when $\lambda=0$, then $S_{s,\,\lambda}=S_s$. On the other hand,~$S_{s,\,\lambda}(v)<S_{s}(v)$ for any~$v\in
H^s(\RR^n)\setminus\{0\}$
and any~$\lambda>0$, so~$S_{s,\,\lambda}\le S_{s}$\,. The interesting case is when the {\em strict} inequality
holds true, namely when~$S_{s,\,\lambda}< S_{s}$\,,
since such a non-degeneracy condition will allow us to recover the necessary compactness
and to obtain the existence of a non-trivial solution for our problem (the fact that the inequality
is strict will be proved in Corollary~\ref{corollary} here below).

The paper is organized as follows. In Section~\ref{sec:preliminary}
we discuss the setting of the problem. In Section~\ref{sec:regularity} we give an $L^\infty$--regularity result for the eigenvalues of the fractional Laplacian.
In Section~\ref{sec:PS} we prove a compactness property for the functional~$\mathcal J_{s,\,\lambda}$, while Section~\ref{sec:estimate} is devoted to an estimate involving the fractional critical Sobolev constants. In Section~\ref{sec:geometry} we study the geometric structure of~$\mathcal J_{s,\,\lambda}$ and in Section~\ref{sec:estimate2} we prove a crucial estimate which allow us to apply classical critical points theorems to our functional. Finally, in Section~\ref{sec:proof} we give a proof of Theorem~\ref{lapfra0} by combining different topological critical point theories.

\section{Brief summary of the functional setting}\label{sec:preliminary}
In this section we briefly recall some preliminary results on the fractional Sobolev space $H^s(\RR^n)$ and on the functional space~$X_0$, which was first introduced in \cite{sv} (see also \cite{sY, svmountain, svlinking, servadeivaldinociBN, servadeivaldinociCFP}, where some properties of $X_0$ were proved)\,.
The reader familiar with this topic may skip this section and may go directly to the next one.

In the following we denote by $H^s(\RR^n)$ the usual fractional
Sobolev space endowed with the norm (the so-called \emph{Gagliardo norm})
\begin{equation}\label{gagliardonorm}
\|g\|_{H^s(\RR^n)}=\|g\|_{L^2(\RR^n)}+
\Big(\int_{\RR^{2n}}\frac{\,\,\,|g(x)-g(y)|^2}{|x-y|^{n+2s}}\,dx\,dy\Big)^{1/2}\,,
\end{equation}
while $X_0$ is the functional space defined as
$$X_0=\big\{u\in H^s(\RR^n) : u=0\,\, \mbox{a.e. in}\,\, \RR^n\setminus \Omega\}\,.$$
For this see, for instance, \cite[Lemma~7-$b)$]{servadeivaldinociBN}.

By \cite[Theorem~6.5]{valpal} (used here with $p=2$) we can take the function
\begin{equation}\label{normaX0}
X_0\ni v\mapsto \|v\|_{X_0}=\left(\int_{\RR^{2n}}\frac{\,\,\,|v(x)-v(y)|^2}{|x-y|^{n+2s}}\,dx\,dy\right)^{1/2}
\end{equation}
as norm on $X_0$. The following result, proved in \cite[Lemma~7]{svmountain}, holds true:

\begin{lemma}\label{lemmanormaequiv}
$\left(X_0, \|\cdot\|_{X_0}\right)$ is a Hilbert space,
with scalar product
$$\langle u,v\rangle_{X_0}=\int_Q
\big( u(x)-u(y)\big) \big( v(x)-v(y)\big)\,K(x-y)\,dx\,dy\,.$$
\end{lemma}

Now we discuss some results related to the embeddings of the spaces $X_0$ and $H^s(\RR^n)$ into the usual Lebesgue spaces.
In \cite[Lemma~8]{svmountain} and~\cite[Lemma~9]{servadeivaldinociBN}
some embeddings results were proved in a general setting. In the case of the fractional Laplacian these results read as follows:
\begin{lemma}\label{lemmaembeddings}
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}

In what follows, we denote by $\lambda_{1,\,s}$ the first eigenvalue of the operator~$(-\Delta)^s$ with homogeneous Dirichlet boundary conditions. For the existence and the basic properties
of the eigenvalues and the eigenfunctions of the fractional Laplacian~$(-\Delta)^s$ we refer to \cite[Proposition~9 and Appendix~A]{svlinking}, where a spectral theory for general integrodifferential non-local operators was developed.



\section{A regularity result for the eigenvalues of~$(-\Delta)^s$}\label{sec:regularity}
In this section we prove a regularity result for the eigenvalues of the fractional Laplace operator~$(-\Delta)^s$\,. This result can be easily extended to general non-local integrodifferential operators as the ones treated in \cite{sY, sv, svmountain, svlinking, servadeivaldinociBN, servadeivaldinociCFP} (see the
definition of $\mathcal L_K$ in all these papers).

\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~$e\in L^\infty(\Omega)$
and there exists~$C>0$, possibly depending on~$n$, $s$ and~$\lambda$,
such that
\begin{equation}\label{9:0}
\|e\|_{L^\infty(\Omega)}\le C\|e\|_{L^2(\Omega)}\,.\end{equation}
\end{proposition}

\begin{proof} We may assume that $e$ does not vanish identically
(otherwise there is nothing to prove). Let~$\delta>0$,
to be taken appropriately small
in what follows (the choice of~$\delta$
will be done on~\eqref{DELTA} below).
Up to multiplying~$e$ by a small constant, we may and do assume that
\begin{equation}\label{9:0:1}
\|e\|_{L^2(\Omega)}=\delta\,.
\end{equation}

We perform a simple observation on the positive
part of a function~$v\in X_0$, that is~$v^+:=\max\{ v,0\}$\,.
We claim that
\begin{equation}\label{v+}
\big(v(x)-v(y)\big)
\big(v^+(x)-v^+(y)\big)\ge \big|v^+(x)-v^+(y)\big|^2
\end{equation}
for any~$x$, $y\in\RR^n$. To check~\eqref{v+}, since the role of~$x$
and~$y$ is symmetric, we can always suppose~$v(x)\ge v(y)$.
Also~\eqref{v+}
is clearly an identity when~$x$, $y\in \{v\ge 0\}$
and when~$x$, $y\in \{v< 0\}$. So it only remains to check~\eqref{v+}
when~$x\in\{v\ge0\}$ and~$y\in \{v< 0\}$. In this case
$$v^+(x)-v^+(y)=v(x)<v(x)+|v(y)|=v(x)-v(y) \,,$$
and so, if we multiply by~$v^+(x)-v^+(y)\ge0$ both sides,
we obtain~\eqref{v+}.

Now, for any~$k\in\NN$,
we let~$C_k:=1-2^{-k}$, $v_k:=e-C_k$, $w_k:= v_k^+$
and~$U_k:= \|w_k\|_{L^2(\Omega)}^2$.
Notice that $w_k\in X$, since $X$ is a linear space, and a.e. $x\in \RR^n\setminus \Omega$ we have that
$$e(x)-C_k=-C_k\leq 0\,.$$
Hence $w_k(x)=(-C_k)^+=0$
in~$\RR^n\setminus \Omega$\,, so that $w_k\in X_0$\,. Moreover, we get that
$$0\le w_k\le |e|+|C_k|\le |e|+1\in L^2(\Omega)\,,$$
being $\Omega$ bounded, and
$$ \lim_{k\rightarrow+\infty} w_k=(e-1)^+\,.$$
Therefore, by the Dominated Convergence Theorem,
\begin{equation}\label{8:0}
\lim_{k\rightarrow+\infty}
U_k=\int_\Omega \Big[ (e(x)-1)^+\Big]^2\,dx
\,.\end{equation}
Moreover, for any $k\in \NN$ we have $C_{k+1}>C_k$ and so $v_{k+1}<v_k$ a.e. in $\RR^n$\,, from which we deduce that
\begin{equation}\label{11:0}
w_{k+1}\le w_k\,\,\, \mbox{a.e. in}\,\, \RR^n\,.
\end{equation}
Let also~$A_k:=C_{k+1}/(C_{k+1}-C_k)=2^{k+1}-1$ for any $k\in \NN$\,. We claim that for any $k\in \NN$
\begin{equation}\label{99:0}
{\mbox{$e< A_k w_k$
on $\{w_{k+1}>0\}$\,.}}
\end{equation}
To check this, let~$x\in \{w_{k+1}>0\}$\,. Then~$e(x)-C_{k+1}>0$ and so by the properties of $C_k$ we have $e(x)>C_{k+1}>C_k$\,. Hence, $w_k(x)=v_k(x)=e(x)-C_k$ and
\begin{eqnarray*}
&& A_k w_k(x)= A_k(e(x)-C_k)=\frac{C_{k+1}}{C_{k+1}-C_k}e(x)-
\frac{C_k C_{k+1}}{C_{k+1}-C_k}\\
&&\qquad =e(x)+\frac{C_{k}}{C_{k+1}-C_k} \left(e(x)-
 C_{k+1}\right)> e(x).
\end{eqnarray*}
This proves~\eqref{99:0}.

Notice also that
$$ v_{k+1}(x)-v_{k+1}(y)=e(x)-e(y)$$
for any~$x$, $y\in\RR^n$.
Using this,~\eqref{e+}, \eqref{v+} and the fact that $v_{k+1}^+ =w_{k+1}\in X_0$, we obtain
\begin{equation*}
\begin{aligned}
\int_{\RR^{2n}}
\frac{\big|w_{k+1}(x)-w_{k+1}(y)\big|^2}{|x-y|^{n+2s}}\,dx\,dy &
=
\int_{\RR^{2n}}
\frac{\big| v_{k+1}^+(x)-v_{k+1}^+(y)\big|^2}{|x-y|^{n+2s}}\,dx\,dy
\\ & \le
\int_{\RR^{2n}}
\frac{\big(v_{k+1}(x)-v_{k+1}(y)\big)\big( v_{k+1}^+(x)-v_{k+1}^+(y)\big)}{|x-y|^{n+2s}}\,dx\,dy\\
& = \int_{\RR^{2n}}
\frac{\big(e(x)-e(y)\big)\big( v_{k+1}^+(x)-v_{k+1}^+(y)\big)}{|x-y|^{n+2s}}\,dx\,dy\\
& =\lambda\int_\Omega e (x)w_{k+1}(x)\,dx\\
& =\lambda\int_{\{w_{k+1}>0\}}e (x)w_{k+1}(x)\,dx\,.
\end{aligned}
\end{equation*}
Therefore, as a consequence of this, recalling~\eqref{11:0} and~\eqref{99:0} and using the fact that $\lambda$ is positive, we deduce
\begin{equation}\label{2:5}
\begin{aligned}
\int_{\RR^{2n}}
\frac{\big|w_{k+1}(x)-w_{k+1}(y)\big|^2}{|x-y|^{n+2s}}\,dx\,dy &
=\lambda\int_{\{w_{k+1}>0\}}e (x)w_{k+1}(x)\,dx\\
&\le
\lambda A_k\int_{\{w_{k+1}>0\}}w_k (x)w_{k+1}(x)\,dx\\
& \le
\lambda A_k\int_{\{w_{k+1}>0\}}w_{k}^2(x)\,dx
\le 2^{k+1} \lambda U_k\,.
\end{aligned}
\end{equation}
Now we claim that
\begin{equation}\label{set}
\{ w_{k+1}>0\}\subseteq \{ w_k>2^{-(k+1)}\}.
\end{equation}
To establish this, we
observe that if~$x\in \{w_{k+1}>0\}$ then
$$ 0<v_{k+1}^+(x)=\max\{e(x)-C_{k+1},0\}\,.$$
Hence~$e(x)-C_{k+1}>0$. Accordingly
$$ v_k(x)=e(x)-C_k>C_{k+1}-C_k=2^{-(k+1)}\,,$$
so that, as a consequence, $w_k(x)=v_k(x)>2^{-(k+1)}$\,,
which proves~\eqref{set}.

As a consequence of~\eqref{set}, we obtain
\begin{equation}\label{5:2}
\begin{aligned}
U_k & =\|w_k\|_{L^2(\Omega)}^2\\
& \ge\int_{\{ w_k>2^{-(k+1)}\}} w_k^2(x)\,dx\\
& \ge 2^{-2(k+1)}\big| \{ w_k\ge 2^{-(k+1)}\} \big|\\
&\ge 2^{-2(k+1)} \big| \{ w_{k+1}>0\} \big|\,.
\end{aligned}
\end{equation}

Now we use the H\H older inequality (with exponents~$2^*/2$ and~$n/(2s)$)
and the fractional Sobolev inequality (see, e.g., Theorem~6.5
in~\cite{valpal}) to get that
\begin{eqnarray*}
U_{k+1} &\le&\left( \int_\Omega |w_{k+1}(x)|^{2^*}\,dx\right)^{2/2^*}
\big|  \{ w_{k+1}>0\} \big|^{2s/n}
\\ &\le&
\tilde c \int_{\RR^{2n}}
\frac{|w_{k+1}(x)-w_{k+1}(y)|^2}{|x-y|^{n+2s}}\,dx\,dy
\;\big|  \{ w_{k+1}>0\} \big|^{2s/n}\,,
\end{eqnarray*}
for some positive constant $\tilde c$ depending only on $n$ and $s$\,.
Consequently, by~\eqref{2:5} and~\eqref{5:2}, we see that
\begin{equation}\label{8:8}\begin{split}
& U_{k+1}\le(\tilde c \,2^{k+1}
\lambda U_k)\,(2^{2(k+1)} U_k)^{2s/n}
=2^{1+(4s/n)}\tilde c\lambda \cdot \big(2^{1+(4s/n)}\big)^k U_k^{1+(2s/n)}
\\ &\qquad\le
\Big( (1+2^{1+(4s/n)}\tilde c\lambda )\cdot 2^{1+(4s/n)}\Big)^k
U_k^{1+(2s/n)}
= C^k U_k^{\beta}
\,,\end{split}\end{equation}
where~$\beta:=1+(2s/n)>1$ and~$C>1$ only depends on~$n$, $s$ and~$\lambda$.
Now we are ready to perform our choice of~$\delta$: namely we assume that~$\delta>0$
is so small that
\begin{equation}\label{DELTA}
\delta^{\beta-1}<\frac{1}{C^{1/(\beta-1)}}
\end{equation}
We also fix
$$ \eta\in \left(\delta^{\beta-1},\frac{1}{C^{1/(\beta-1)}}\right).$$
Notice that, since $C>1$ and $\beta>1$
\begin{equation}\label{00:8}
\eta\in(0,1)\,.
\end{equation}
Moreover
\begin{equation}\label{00:8:1}
\delta^{\beta-1}\le \eta\;{\mbox{ and }}\;C \eta^{\beta-1}\le 1
\,.\end{equation}
We claim that
\begin{equation}\label{00:9}
U_k \le\delta \eta^k\,.
\end{equation}
The proof is by induction. First of all
$$ U_0=\|e^+\|_{L^2(\Omega)}\le\|e\|_{L^2(\Omega)}=\delta\,,$$
which is~\eqref{00:9} when~$k=0$. Let us now suppose that~\eqref{00:9}
holds true for~$k$ and let us prove it for~$k+1$. For this, we use~\eqref{8:8}
and~\eqref{00:8:1}:
\begin{eqnarray*}
U_{k+1}\le C^k U_k^{\beta}\le C^k (\delta \eta^k)^{\beta}=\delta(C\eta^{\beta-1})^k\delta^{\beta-1}\eta^k
\le \delta\eta^{k+1}\,.
\end{eqnarray*}
This proves~\eqref{00:9}. Then, by~\eqref{00:8} and~\eqref{00:9}, we conclude that
$$ \lim_{k\rightarrow+\infty}U_k=0\,.$$
Hence, by~\eqref{8:0},~$(e-1)^+=0$ a.e. in~$\Omega$,
that is~$e\le 1$ a.e. in~$\Omega$.
By replacing~$e$ with~$-e$, we obtain~$\|e\|_{L^\infty(\Omega)}\le 1$.
Then,~\eqref{9:0} follows by recalling the scaling in~\eqref{9:0:1}.
This completes the proof of Proposition~\ref{proplinfty}\,.
\end{proof}

\section{A compactness condition for the functional~$\mathcal J_{s,\,\lambda}$}\label{sec:PS}
Now we are ready for studying problem~\ref{problemalapfrac0}\,.
In this section we prove a  compactness property for the functional~$\mathcal J_{s,\,\lambda}$\,.

First of all, we note that this functional is well defined thanks
to Lemma~\ref{lemmaembeddings}-$b)$.
Moreover, $\mathcal J_{s,\,\lambda}$ is Fr\'echet
differentiable in $u\in X_0$ and for any $\varphi\in X_0$
$$\begin{aligned}
\langle \mathcal J_{s,\,\lambda}'(u), \varphi\rangle & = \int_{\RR^{2n}}\frac{(u(x)-u(y))(\varphi(x)-\varphi(y))}{|x-y|^{n+2s}}\,dx\,dy -\lambda \int_\Omega u(x) \varphi(x)\,dx \\
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad
 -\int_\Omega |u(x)|^{2^*-2}u(x) \varphi(x)\,dx\,.
\end{aligned}$$
Thus, critical points of $\mathcal J_{s,\,\lambda}$ are solutions to
problem~\eqref{problemalapfrac0}. In order to find these critical points,
we will apply some variants of the Mountain
Pass Theorem and of the Linking Theorem  which take into account the lack of compactness due to the fact that the embedding $X_0\hookrightarrow L^{2^*}(\RR^n)$ is only continuous.

In the sequel we prove that, for suitable values of $c$, the functional $\mathcal J_{s,\,\lambda}$ satisfies the following Palais--Smale condition at level $c\in \RR$ (or $(PS)_c$--condition for short):
\begin{center}
every Palais--Smale sequence at level $c$ (or~$(PS)_c$--sequence for short) $u_j$ in $X_0$ admits a subsequence strongly convergent in $X_0$\,.
\end{center}
We say that $u_j$ in $X_0$ is a $(PS)_c$--sequence for~$\mathcal J_{s,\,\lambda}$ if
\begin{center}
$\mathcal J_{s,\,\lambda}(u_j)\to c$\,\, and \,\,
$\sup\Big\{ \big|\langle\,\mathcal J_{s,\,\lambda}'(u_j),\varphi\,\rangle
\big|\,: \; \varphi\in X_0\,, \|\varphi\|_{X_0}=1\Big\}\to 0$\,\,
as\,\, $j\to +\infty$\,.
\end{center}

\begin{proposition}\label{propPS}
Let $\lambda>0$ and let $c\in \RR$ be such that
\begin{equation}\label{cS}
c<\frac s n S_s^{n/(2s)}\,.
\end{equation}
Then, the functional~$\mathcal J_{s,\,\lambda}$ satisfies the $(PS)_c$--condition.
\end{proposition}
\begin{proof}
Let $u_j\in X_0$ be a $(PS)_c$--sequence for ~$\mathcal J_{s,\,\lambda}$, that is let $u_j$ be such that
\begin{equation}\label{Jc0}
\mathcal J_{s,\,\lambda}(u_j)\to c
\end{equation}
and
\begin{equation}\label{J'00}
\sup\Big\{ \big|\langle\,\mathcal J_{s,\,\lambda}'(u_j),\varphi\,\rangle
\big|\,: \; \varphi\in X_0\,, \|\varphi\|_{X_0}=1\Big\}\to 0
\end{equation}
as $j\to +\infty$.
In order to prove Proposition~\ref{propPS}, 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}\,, where more general non-local operators~$\mathcal L_K$ than the fractional Laplacian were considered (but only in the case when $\lambda>0$ is less than the first eigenvalue of $-\mathcal L_K$)\,. Here, for the sake of clarity and also for reader's convenience, we prefer to repeat the proof of Step~\ref{claim2}.

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_{s,\,\lambda}(u_j)|\leq \kappa\,,
\end{equation}
and
\begin{equation}\label{j'limitato0}
\Big|\langle \mathcal J_{s,\,\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_{s,\,\lambda}(u_j)-\frac 1 2 \langle \mathcal J_{s,\,\lambda}'(u_j),
u_j\rangle\leq \kappa \left(1+ \|u_j\|_{X_0}\right)\,.
\end{equation}
Moreover
$$\mathcal J_{s,\,\lambda}(u_j)-\frac 1 2 \langle \mathcal J_{s,\,\lambda}'(u_j), u_j\rangle  =
-\left(\frac{1}{\,2^*} -\frac 1 2\right)\|u_j\|_{L^{2^*}(\Omega)}^{2^*}= \frac s n\, \|u_j\|_{L^{2^*}(\Omega)}^{2^*}\,,$$
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_{s,\,\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~$j$,
and so Step~\ref{claim2} is proved\,.
\end{proof}

\begin{step}\label{claim3}
Problem~\eqref{problemalapfrac0} admits a solution $u_\infty\in X_0$\,.
\end{step}
\begin{proof}
Since $u_j$ is bounded in $X_0$\,, which is a reflexive space by Lemma~\ref{lemmanormaequiv}, then, 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}
\int_{\RR^{2n}}\!\!\!\frac{(u_j(x)-u_j(y))(\varphi(x)-\varphi(y))}{|x-y|^{n+2s}}\,dx\,dy \to
\int_{\RR^{2n}}\!\!\!\frac{(u_\infty(x)-u_\infty(y))(\varphi(x)-\varphi(y))}{|x-y|^{n+2s}}\,dx\,dy
\end{equation}
for any $\varphi\in X_0$\,, as $j\to +\infty$\,. Moreover, by Step~\ref{claim2}, \eqref{norma2star}, Lemma~\ref{lemmaembeddings} 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}
\begin{equation}\label{convergenze0bis}
u_j \to u_\infty \quad \mbox{in}\,\, L^\nu(\RR^n)\,\, \mbox{for any}\,\, \nu\in [1, 2^*)
\end{equation}
and
\begin{equation}\label{convergenze0ter}
u_j \to u_\infty \quad \mbox{a.e. in}\,\, \RR^n
\end{equation}
as $j\to +\infty$\,.

As a consequence of \eqref{norma2star} and Step~\ref{claim2}, we have that~$\| u_j\|_{L^{2^*}(\Omega)}$
is bounded uniformly in~$j$, hence~$|u_j|^{2^*-2}u_j$
is bounded in~$L^{2^*/(2^*-1)}(\Omega)$, uniformly in~$j$\,. Accordingly, by~\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 the conjugate exponent of~$2^*/(2^*-1)$ is~$2^*$, it follows from~\eqref{convergenzaweak1} that
\begin{equation}\label{3.13bis}
\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)\,,\end{equation}
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 $j\to +\infty$\,, thanks to Lemma~\ref{lemmaembeddings}.

By \eqref{J'00}, for any $\varphi \in X_0$ as $j\to +\infty$
$$\begin{aligned}
0\leftarrow \langle \mathcal J_{s,\,\lambda}'(u_j), \varphi\rangle & =
\int_{\RR^{2n}}\frac{(u_j(x)-u_j(y))(\varphi(x)-\varphi(y))}{|x-y|^{n+2s}}\,dx\,dy-\lambda \int_\Omega u_j(x)\varphi(x)\,dx\\
& \qquad \qquad \qquad \qquad \qquad \qquad  - \int_\Omega |u_j(x)|^{2^*-2}u_j(x)\varphi(x)\,dx\,,
\end{aligned}$$
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
$$\begin{aligned}
\int_{\RR^{2n}}&\frac{(u_\infty(x)-u_\infty(y))(\varphi(x)-\varphi(y))}{|x-y|^{n+2s}}\,dx\,dy-\lambda \int_\Omega u_\infty(x)\varphi(x)\,dx\\
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad  - \int_\Omega |u_\infty(x)|^{2^*-2}u_\infty(x)\varphi(x)\,dx=0
\end{aligned}$$
for any $\varphi \in X_0$, that is $u_\infty$ is a solution of problem~\eqref{problemalapfrac0} and Step~\ref{claim3} follows.
\end{proof}

\begin{step}\label{step3}
The following relations hold true:
\begin{itemize}
\item[$a)$] ${\displaystyle \mathcal J_{s,\,\lambda}(u_\infty) =\frac s n\, \int_\Omega|u_\infty(x)|^{2^*}dx\geq 0\,;}$\\
    \\
\item[$b)$] $\begin{aligned}
{\displaystyle \mathcal J_{s,\,\lambda}(u_j)  = \mathcal J_{s,\,\lambda}(u_\infty) + \frac 1 2 \int_{\RR^{2n}}} & {\displaystyle \frac{|u_j(x)-u_\infty(x)-u_j(y)+u_\infty(y)|^2}{|x-y|^{n+2s}}dx\,dy}\\
& {\displaystyle \qquad \qquad \quad -\frac{1}{2^*}\int_\Omega |u_j(x)-u_\infty(x)|^{2^*}dx+\textit{o}(1)}
\end{aligned}$\\
as $j\to +\infty$\,;\\
\\
\item[$c)$] ${\displaystyle \int_{\RR^{2n}}\frac{|u_j(x)-u_\infty(x)-u_j(y)+u_\infty(y)|^2}{|x-y|^{n+2s}}dx\,dy = \int_\Omega |u_j(x)-u_\infty(x)|^{2^*}dx+\textit{o}(1)}$\\
    \\
as $j\to +\infty$\,.
\end{itemize}
\end{step}

\begin{proof}
For the proof of this step we refer to Steps~3--5 of \cite[Proposition~7]{sY}, where a general non-local integrodifferential operator with kernel $K$ was considered.
\end{proof}

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

\medskip

By Step~\ref{step3}-$c)$ it easily follows that
$$\begin{aligned}
\frac 1 2 \int_{\RR^{2n}}& \frac{|u_j(x)- u_\infty(x)- u_j(y)+u_\infty(y)|^2}{|x-y|^{n+2s}}dx\,dy -\frac{1}{2^*}\int_\Omega |u_j(x)-u_\infty(x)|^{2^*}dx\\
&= \left(\frac 1 2 -\frac{1}{2^*}\right)\int_{\RR^{2n}}\frac{|u_j(x)-u_\infty(x)-u_j(y)+u_\infty(y)|^2 }{|x-y|^{n+2s}}dx\,dy +\textit{o}(1)\\
&= \frac s n\int_{\RR^{2n}}\frac{|u_j(x)-u_\infty(x)-u_j(y)+u_\infty(y)|^2}{|x-y|^{n+2s}}dx\,dy +\textit{o}(1)\,,
\end{aligned}$$
and so, recalling Step~\ref{step3}-$b)$,
\begin{equation}\label{bo}
\begin{aligned}
\mathcal J_{s,\,\lambda}(u_\infty) & +\frac s n\int_{\RR^{2n}}\frac{|u_j(x)-u_\infty(x)-u_j(y)+u_\infty(y)|^2}{|x-y|^{n+2s}}dx\,dy\\
& =\mathcal J_{s,\,\lambda}(u_j) +\textit{o}(1)\\
& =c+\textit{o}(1)\,,
\end{aligned}
\end{equation}
as $j\to +\infty$\,, thanks to \eqref{Jc0}\,.
Since the sequence $\|u_j\|_{X_0}$ is bounded in $\RR$ by Step~\ref{claim2}, then we can assume that up to a subsequence, if necessary,
\begin{equation}\label{L1}
\|u_j-u_\infty\|_{X_0}^2=\int_{\RR^{2n}}\frac{|u_j(x)-u_\infty(x)-u_j(y)+u_\infty(y)|^2}{|x-y|^{n+2s}}\,dx\,dy\to L
\end{equation}
and so, again as a consequence of Step~\ref{step3}-$c)$,
$$\int_\Omega |u_j(x)-u_\infty(x)|^{2^*}\,dx\to L$$
as $j\to +\infty$\,.
Of course $L\in [0, +\infty)$ and, by definition of $S_s$ and $S_s(\cdot)$ (see~\eqref{Ss} and~\eqref{Ssv}), we have that
$$ S_s\le S_s(u_j-u_\infty)\rightarrow\frac{L}{L^{2/2^*}}$$
as~$j\rightarrow+\infty$. As a consequence,
$$L\geq L^{2/2^*}S_s\,.$$
Thus
$${\mbox{either}}\quad L=0 \quad \mbox{or} \quad L\geq S_s^{n/(2s)}\,.$$
The case $L\geq S_s^{n/(2s)}$ can not occur. Otherwise, by \eqref{bo}, \eqref{L1} and Step~\ref{step3}-$a)$, we would get
$$c=\mathcal J_{s,\,\lambda}(u_\infty)+\frac s n\,L \geq \frac s n\,L \geq \frac s n\, S_s^{n/(2s)}\,,$$
which contradicts \eqref{cS}\,. Thus $L=0$ and so, by \eqref{L1}, we obtain that
$$\|u_j-u_\infty\|_{X_0}\to 0$$
as $j\to +\infty$\,.
Hence, the sequence~$u_j$ has a (strongly) convergent subsequence and this concludes the proof of Proposition~\ref{propPS}\,.
\end{proof}


\section{A non-degeneracy estimate}\label{sec:estimate}
As we said in the Introduction, to get our goal we have to show that
$$S_{s,\,\lambda}<S_s$$
for suitable values of $\lambda$. For this it is enough to prove that there exists a non-trivial function $\tilde v\in H^s(\RR^n)$ such that
$$S_{s,\,\lambda}(\tilde v)<S_s\,,$$
for some $\lambda$ (recall~\eqref{Sslambda})\,.
This section is devoted to the construction of such $\tilde v$\,. To this aim we will proceed as in \cite{servadeivaldinociBN}. First of all, we recall some results appeared in \cite{cotsiolis, servadeivaldinociBN}\,.

\begin{proposition}\label{claimubar}(\,\cite[Theorem~1.1]{cotsiolis}\,)
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{proposition}

In what follows we suppose that, up to a translation, $x_0=0$ in \eqref{utilde}\,.
Following \cite[Section~4]{servadeivaldinociBN}, we consider the function
\begin{equation}\label{ustar}
{\displaystyle u^*(x)=\bar u\left(\frac{x}{S_s^{1/(2s)}}\right)}\,, \qquad x\in \RR^n\,,
\end{equation}
which is an explicit solution of the limiting problem
\begin{equation}\label{limitingproblem}
(-\Delta )^s u=|u|^{2^*-2}u \quad {\mbox{ in }} \RR^n
\end{equation}
such that
\begin{equation}\label{normaSs}
\|u^*\|_{L^{2^*}(\RR^n)}^{2^*}=S_s^{n/(2s)}\,.
\end{equation}

Moreover, we construct the families of functions $U_\varepsilon$ and $u_\varepsilon$ as follows
\begin{equation}\label{Uepsilon}
U_\varepsilon(x)=\varepsilon^{-(n-2s)/2}\,u^*(x/\varepsilon)\,, \qquad x\in \RR^n
\end{equation}
and
\begin{equation}\label{uepsilon}
u_\varepsilon(x)=\eta(x)U_\varepsilon(x)\,, \quad x\in \RR^n\,,
\end{equation}
for any $\varepsilon>0$\,.
Here $\eta \in C^\infty(\RR^n)$ is a function 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 $\mathcal C B_\delta=\RR^n\setminus B_\delta$\,, with $\delta>0$ fixed so that
$$B_{4\delta}\subset \Omega\,.$$

Note that $u_\varepsilon\in X_0$ and $u_\varepsilon=0$ a.e. in $\RR^n\setminus\Omega$\,.
Moreover, the function $u_\varepsilon$ satisfies the following crucial estimates, whose proofs can be found in \cite[Propositions~21 and 22 and Subsection~4.2.1]{servadeivaldinociBN} and in \cite[Proposition~12]{sY}.

\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 $s$\,.
\end{proposition}

As a consequence of Proposition~\ref{stimaUepsilon} we have

\begin{corollary}\label{corollary}
Let $s\in (0,1)$\,, $2s<n<4s$ and let $u_\varepsilon$ be as in \eqref{uepsilon}\,.
Then, there exists $\lambda_s>0$ such that for any $\lambda>\lambda_s$
\begin{equation}\label{SslambdaSs}
S_{s,\,\lambda}(u_\varepsilon)<S_s\,,
\end{equation}
provided $\varepsilon>0$ is sufficiently small.
\end{corollary}
\begin{proof} We observe that
$$ \frac{n}{2s}\left( 1-\frac{2}{2^*}\right)=1\,.$$
Hence,
by~\eqref{Sslambdav} and Proposition~\ref{stimaUepsilon} we get
\begin{equation}\label{agg0}\begin{aligned}
{\displaystyle S_{s,\,\lambda}(u_\varepsilon)}& {\displaystyle \leq\frac{S_s^{n/(2s)}+\mathcal O(\varepsilon^{n-2s})-\lambda C_s\varepsilon^{n-2s}+ \mathcal O(\varepsilon^{2s})}{\Big(S_s^{n/(2s)}+\mathcal O(\varepsilon^{n})\Big)^{2/2^*}}}\\
& {\displaystyle \leq S_s +\varepsilon^{n-2s}\Big(\mathcal O(1)-\lambda \widetilde C_s\Big)+\mathcal O(\varepsilon^{2s})<S_s\,,}
\end{aligned}\end{equation}
if $\lambda$ is large enough, say $\lambda>\lambda_s>0$, provided $\varepsilon>0$ is sufficiently small.
Thus, the assertion of Corollary~\ref{corollary} is proved.
\end{proof}

\section{The geometric structure of the functional~$\mathcal J_{s,\,\lambda}$}\label{sec:geometry}

In this section we analyze the geometry of the functional~$\mathcal J_{s,\,\lambda}$\,.
Precisely, we will show that $\mathcal J_{s,\,\lambda}$ has the geometric structure of the Mountain Pass Theorem when $\lambda<\lambda_{1,\,s}$ (see \cite{ar, struwe, willem}), while it has the one required by the Linking Theorem when $\lambda\geq \lambda_{1,\,s}$
(see, for instance, \cite{r78, rabinowitz, struwe, willem})\,.

\subsection{The Mountain Pass geometry}\label{subsec:mountain}
In this subsection we prove that the functional~$\mathcal J_{s,\,\lambda}$ has the
geometric features required by the Mountain Pass Theorem (see, e.g., \cite{ar} and also \cite[Theorem~2.2]{bn}).

\begin{proposition}\label{propgeometriaMP}
Let $\lambda\in (0, \lambda_{1,\,s})$\,. Then,
there exist $\rho>0$\,, $\beta>0$ and $e\in X_0$ such that
\begin{itemize}
\item[$a)$] for any $u\in X_0$ with
$\|u\|_{X_0}=\rho$ it results that $\mathcal J_{s,\,\lambda}(u)\geq
\beta$\,;
\\
\item[$b)$] $e\ge 0$ a.e. in $\RR^n$,
$\|e\|_{X_0}>\rho$ and $\mathcal J_{s,\,\lambda}(e)<\beta$\,.
\end{itemize}
In particular, we can construct $e$ as follows
\begin{equation}\label{e}
e=e_\varepsilon:=\zeta_\varepsilon u_\varepsilon
\end{equation}
with $\zeta_\varepsilon$ large enough and $u_\varepsilon$ as in Corollary~\ref{corollary}, with $\varepsilon$ sufficiently small.
\end{proposition}
\begin{proof}
For part~$a)$ we can argue, for instance, as in \cite[Proposition~18]{servadeivaldinociBN}, where an operator more general than the fractional Laplacian was considered together with nonlinearities more general than the power function. Note that here we need the assumption $\lambda<\lambda_{1,\,s}$\,.

For part~$b)$ we essentially proceed as in \cite[Proposition~19]{servadeivaldinociBN}\,. We prefer to repeat it not only for the reader's convenience, but also in order to clarify
the choice of $e$ given in formula~\eqref{e}\,.

Let $u\in X_0$ be such that $\|u\|_{X_0}=1$ and $u\ge 0$ a.e. in $\RR^n$:
this choice is possible, thanks to \cite[Lemma~12]{sv}. Also, let $\zeta>0$\,. By the fact that $\lambda$ is positive, we have
$$\begin{aligned}
\mathcal J_{s,\,\lambda}(\zeta u) & = \frac{\zeta^2}{2}\int_{\RR^{2n}}\frac{|u(x)- u(y)|^2}{|x-y|^{n+2s}}dx\,dy-\frac \lambda 2\, \zeta^2 \int_\Omega |u(x)|^2\,dx-\frac{\zeta^{2^*}}{\,2^*}\int_\Omega
|u(x)|^{2^*}\,dx\\
& \leq \frac{\zeta^2}{2}\|u\|_{X_0}^2
-\frac{\zeta^{2^*}}{\,2^*}\int_\Omega
|u(x)|^{2^*}\,dx\\
& = \frac{\zeta^2}{2}
-\frac{\zeta^{2^*}}{\,2^*}\int_\Omega
|u(x)|^{2^*}\,dx\,.
\end{aligned}$$
Since $2^*>2>1$, passing to the limit as $\zeta\to +\infty$ (for~$u$ fixed), we get that
$\mathcal J_{s,\,\lambda}(\zeta u)\to -\infty$, so that the assertion follows
taking $e=\zeta u$, with $\zeta$ sufficiently large.

In particular, we can take
$u=u_\varepsilon/\|u_\varepsilon\|_{X_0}\in X_0$ (note that
$u_\varepsilon\not \equiv 0$ by assumption) so that we can choose $e=e_\varepsilon=\zeta_\varepsilon u_\varepsilon$ with $\zeta_\varepsilon$ large enough\,.
\end{proof}

\subsection{The Linking geometry}\label{subsec:linking}
This subsection is devoted to the case when $\lambda\geq \lambda_{1,\,s}$. We can suppose that
$\lambda\in [\lambda_{k,\,s}\,, \lambda_{k+1,\,s})$ for some $k\in \NN$\,, where $\lambda_{k,\,s}$ is the $k$-th eigenvalue of $(-\Delta)^s$ with homogeneous Dirichlet boundary data.

Arguing as in \cite{sY, servadeivaldinociCFP}, we prove that the functional~$\mathcal J_{s,\,\lambda}$ has the geometry of the Linking Theorem (see \cite{r78, rabinowitz}).
In what follows we denote by
$$\PP_{k+1,\,s}:= \big\{ u\in X_0 {\mbox{ s.t. }}
\langle u,e_{j,\,s}\rangle_{X_0}=0 \quad\forall j=1,\dots,k
\big\}\,,$$
where the function~$e_{k,\,s}$
is an eigenfunction of $(-\Delta)^s$ corresponding to~$\lambda_{k,\,s}$ for any $k\in \NN$ (see \cite[Proposition~9]{svlinking}
for a detailed spectral theory of the fractional Laplacian)\,.

\begin{proposition}\label{propgeometriaL}
Let $\lambda\in [\lambda_{k,\,s}, \lambda_{k+1,\,s})$ for some $k\in \NN$\,. Then,
there exist $\rho>0$ and $\beta>0$ such that
\begin{itemize}
\item [$a)$] for any $u\in \PP_{k+1}$ with
$\|u\|_{X_0}=\rho$ it results that $\mathcal J_{s,\,\lambda}(u)\geq
\beta$\,;
\\
\item[$b)$] $\mathcal J_{s,\,\lambda}(u)\leq 0$ for any $u\in \mbox{span}\{e_{1,\,s}, \dots, e_{k,\,s}\}$\,;
\\
\item[$c)$] for any finite dimensional subspace~$\mathbb F$ of $X_0$\,, there exists $R>\rho$ such that $$\sup_{{u\in \mathbb F}\atop{\|u\|_{X_0}\geq R}
} \mathcal J_{s,\,\lambda}(u)\leq 0\,.$$\
\end{itemize}
In particular we can construct $\mathbb F$ as follows
\begin{equation}\label{F}
\mathbb F=\mathbb F_\varepsilon:=\mbox{span}\{e_{1,\,s}, \dots, e_{k,\,s}\}\oplus \mbox{span}\{\tilde z_\varepsilon\}\,,
\end{equation}
with
$\tilde z_\varepsilon=z_\varepsilon/\|z_\varepsilon\|_{X_0}$\,,
${\displaystyle z_\varepsilon=u_\varepsilon-\sum_{i=1}^k \Big(\int_\Omega u_\varepsilon(x) e_{i,\,s}(x)\,dx\Big)e_{i,\,s}}$ and $u_\varepsilon$ as in Corollary~\ref{corollary}, with $\varepsilon$ small enough.
\end{proposition}
\begin{proof}
The proof can be done as in \cite[Propositions~8--10]{sY}\,. The particular choice of the finite dimensional subspace $\mathbb F$ given in formula~\eqref{F} will be justified in the sequel (see Proposition~\ref{propMepsilon} and \eqref{BRS}).
\end{proof}

\section{A crucial estimate}\label{sec:estimate2}
In this section we prove a result which will be crucial in the sequel in order to estimate the critical level of the functional~$\mathcal J_{s,\,\lambda}$\,.

For this, we need the following characterization of the eigenvalues of the fractional Laplacian~$(-\Delta)^s$, given in \cite[Proposition~5]{sY} for a general non-local fractional operator. For a detailed spectral theory of the fractional Laplacian and, in general, of non-local integrodifferential operators, we refer to \cite[Proposition~9]{svlinking}\,.

\begin{proposition}\label{propcaratterizzazioneautovalori}
Let $k\in \NN$ and let $\lambda_{k,\,s}$ be the $k$-th eigenvalue of the operator~$(-\Delta)^s$ with homogeneous Dirichlet boundary data and let $e_{i,\,s}$ be an eigenfunction of $(-\Delta)^s$ corresponding to~$\lambda_{i,\,s}$ for any $i=1,\dots, k$\,.

Then,
\begin{equation}\label{carlambdak}
{\displaystyle\lambda_{k,\,s}=\max_{v\in \mbox{{\footnotesize span}}\{e_{1,\,s}, \dots, e_{k,\,s}\}\setminus\{0\}} \frac{\|v\|_{X_0}^2}{\|v\|_{L^2(\Omega)}^2}\,.}
\end{equation}
\end{proposition}

The main result of this section is the following one:
\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 $\mathbb F_\varepsilon$ be the linear space defined in \eqref{F}\,. Finally, let
\begin{equation}\label{mepsilon}
{\displaystyle M_\varepsilon:=
\max_{{u\in \mathbb F_\varepsilon}\atop{\|u\|_{L^{2^*}(\Omega)}=1}}S_{s,\,\lambda}(u)\,.}
\end{equation}
Then,
\begin{itemize}
\item [$a)$] $M_\varepsilon$ is achieved in $u_M\in \mathbb F_\varepsilon$ and $u_M$ can be written as follows\footnote{Beware that $u_M$, $v$ and $t$ 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 $v$ and $t$ are given in \eqref{uM}\,.
\end{itemize}
Here the function~$X_0\setminus\{0\}\ni v\mapsto S_{s,\,\lambda}(v)$ is defined as in \eqref{Sslambdav}\,.
\end{proposition}

\begin{proof}
Let us start by proving part~$a)$. First of all note that, by definition of $S_{s,\,\lambda}$
$${\displaystyle M_\varepsilon:=\max_{{u\in \mathbb F_\varepsilon}\atop{\|u\|_{L^{2^*}(\Omega)}=1}}\Big(\|u\|_{X_0}^2-\lambda\|u\|_{L^2(\Omega)}^2\Big)}\,.$$
Moreover, the maximum in formula~\eqref{mepsilon} is achieved thanks to the Weierstrass Theorem. Now, let $u_M\in \mathbb F_\varepsilon$ be the function such that
$$\|u_M\|_{X_0}^2-\lambda\|u_M\|_{L^2(\Omega)}^2=M_\varepsilon \,\,\, \mbox{and}\,\,\, \|u_M\|_{L^{2^*}(\Omega)}=1\,.$$
Note that $u_M\not \equiv 0$\,. Moreover, since $u_M\in \mathbb F_\varepsilon$\,, by definition of $\mathbb F_\varepsilon$ we have that
$$u_M=\tilde v+t z_\varepsilon$$
for some $\tilde v\in \mbox{span}\{e_{1,\,s}, \dots, e_{k,\,s}\}$ and $t\in \RR$\,.
As a consequence of this and thanks to the definition of $z_\varepsilon$ (see Proposition~\ref{propgeometriaL}), we have that
$$u_M=v+t u_\varepsilon\,,$$
where
$${\displaystyle v=\tilde v-t\sum_{i=1}^k \Big(\int_\Omega u_\varepsilon(x) e_{i,\,s}(x)\,dx\Big)e_{i,\,s}\in \mbox{span}\{e_{1,\,s}, \dots, e_{k,\,s}\}}\,.$$
Moreover, we can suppose that $t\geq 0$\,(otherwise
we change the sign of~$u_M$)\,.
This concludes the proof of assertion~$a)$\,.

The proof of part~$b)$ was done in \cite[Proposition~13--$b)$]{sY}. For this we used Propositions~\ref{proplinfty} and \ref{stimaUepsilon}\,.
\end{proof}

\section{Proof of Theorem~\ref{lapfra0}}\label{sec:proof}
Here we will prove our main theorem, using classical critical points results.

Roughly speaking, Propositions~\ref{propgeometriaMP} and \ref{propgeometriaL} say that the functional $\mathcal J_{s,\,\lambda}$ has the geometry required by the Mountain Pass and the Linking Theorems, respectively when $\lambda< \lambda_{1,\,s}$ and $\lambda\geq \lambda_{1,\,s}$\,.

Since, by Proposition~\ref{propPS}, the functional~$\mathcal J_{s,\,\lambda}$ verifies the $(PS)_c$--condition only in an energy range bounded above by the constant $(s/n)\,S_s^{n/(2s)}$\,, in order to apply the classical critical points theorems cited above to $\mathcal J_{s,\,\lambda}$\,, it is enough to show that the Mountain Pass critical level and the Linking critical level of $\mathcal J_{s,\,\lambda}$ stay below this threshold. This is what we will do in the next subsection.

\subsection{Estimates of the critical level of $\mathcal J_{s,\,\lambda}$}\label{subsec:estimate}
Let $\lambda_s>0$ be as in Corollary~\ref{corollary}\,. Of course, only the two cases $\lambda_s< \lambda_{1,\,s}$ and $\lambda_s\geq \lambda_{1,\,s}$ can occur. Let us consider them separately.

\textbf{Case 1: $\lambda_s< \lambda_{1,\,s}$\,.}
This case will be dealt with by using either the
Mountain Pass Theorem when~$\lambda\in (\lambda_s,\lambda_{1,s})$
or the Linking Theorem when~$\lambda\ge\lambda_{1,s}$\,.

First of all, let $\lambda\in (\lambda_s,\lambda_{1,s})$\,. Hence, by the Mountain Pass Theorem, the critical level of $\mathcal J_{s,\,\lambda}$
is given by
\begin{equation}\label{cdef}
c_{{\rm MP},\,\lambda}=\inf_{P\in\mathcal P} \max_{v\in P([0,1])} \mathcal J_{s,\,\lambda}(v)\,,
\end{equation}
where
$$\mathcal P=\Big\{P\in C([0,1]; X_0) : P(0)=0\,, \,\, P(1)=e\Big\}$$
with $e=e_\varepsilon:=\zeta_\varepsilon u_\varepsilon$ given in Proposition~\ref{propgeometriaMP}\,.
We claim that for any $\lambda\in (\lambda_s,\lambda_{1,s})$ it holds true
\begin{equation}\label{AR1}
\beta\leq c_{{\rm MP},\,\lambda}<\frac s n\, S_s^{n/(2s)}\,,
\end{equation}
where $\beta$ is given in Proposition~\ref{propgeometriaMP} and $S_s$ is defined in formula~\eqref{Ss}\,.

For this we first observe that, for any $P\in \mathcal P$, the function $t\mapsto \|P(t)\|_{X_0}$ is continuous in $[0,1]$, that $\|P(0)\|_{X_0}=\|0\|_{X_0}=0<\rho$, and that $\|P(1)\|_{X_0}=\|e\|_{X_0}>\rho$, with $\rho$ as in Proposition~\ref{propgeometriaMP}\,.
Accordingly, there exists $\overline t\in (0,1)$ such that $\|P(\overline t)\|_{X_0}=\rho$\,. Thus,
$$\max_{v\in P([0,1])} \mathcal J_{s,\,\lambda}(v)\geq \mathcal J_{s,\,\lambda}(P(\overline t))\geq \inf_{
{v\in X_0}\atop{\|v\|_{X_0}=\rho}
} \mathcal J_{s,\,\lambda}(v)\,.$$
As a consequence \begin{equation}\label{AR2}
c_{{\rm MP},\,\lambda}\geq \beta\,,\end{equation} where $\beta$ is given in Proposition~\ref{propgeometriaMP}\,.

Now, recalling that $e=e_\varepsilon=\zeta_\varepsilon u_\varepsilon$ in Proposition~\ref{propgeometriaMP}, we have that the map
$[0,1]\ni t\mapsto t\, \zeta_\varepsilon u_\varepsilon$ belongs
to $\mathcal P$. By \cite[Proposition~20]{servadeivaldinociBN}, we have that
$$\inf_{P\in {\mathcal P}}\max_{v\in P([0,1])} \mathcal J_{s,\,\lambda}(v)\leq \max_{\zeta\geq 0} \mathcal J_{s,\,\lambda}(\zeta u_\varepsilon)=\frac s n\, S_{s,\,\lambda}^{n/(2s)}(u_\varepsilon)\,,$$
while, by Corollary~\ref{corollary} we get that for any $\lambda>\lambda_s$
$$\frac s n\, S_{s,\,\lambda}^{n/(2s)}(u_\varepsilon)<\frac s n\, S_s^{n/(2s)}$$
for $\varepsilon$ sufficiently small.
Hence, putting together these two inequalities we conclude that
\begin{equation}\label{AR3}
c_{{\rm MP},\,\lambda}<\frac s n\, S_s^{n/(2s)}\,,
\end{equation}
provided $\lambda\in (\lambda_s,\lambda_{1,s})$\,. Thus, \eqref{AR1} follows from~\eqref{AR2} and~\eqref{AR3}.

\medskip

Now, let~$\lambda\ge\lambda_{1,s}$\,. We can suppose that $\lambda\in [\lambda_{k,\,s}, \lambda_{k+1,\,s})$ for some $k\in \NN$\,. In fact we suppose that $\lambda\in (\lambda_{k,\,s}, \lambda_{k+1,\,s})$ since in the setting of Theorem~\ref{lapfra0} the parameter $\lambda$ can not be an eigenvalue of the fractional Laplace operator $(-\Delta)^s$\,.

Here we have to estimate the Linking critical level of $\mathcal J_{s,\,\lambda}$
given by
$$c_{{\rm L},\,\lambda}=\inf_{h\in \Gamma}\max_{u\in Q}\mathcal J_{s,\,\lambda}(h(u))\,,$$
where
\begin{equation}\label{BRS}\begin{split}
&\Gamma=\big\{h\in C(\overline Q; X_0): h=id\,\, \mbox{on}\,\, \partial Q \big\}\,,\\
&Q=\Big(\overline B_R\cap \mbox{span}\{e_{1,\,s}, \dots, e_{k,\,s}\}\Big)\oplus \big\{r\tilde z_\varepsilon : r\in (0, R)\big\}\,,\end{split}\end{equation}
and $R$ and $\tilde z_\varepsilon$ are as in Proposition~\ref{propgeometriaL}\,. Here we use the particular
choice of~$\mathbb F=\mathbb F_\varepsilon$
performed in~\eqref{F}.
We claim that for any $\lambda\geq \lambda_{1,\,s}$
\begin{equation}\label{AR4}
c_{{\rm L},\,\lambda}<\frac s n\, S_s^{n/(2s)}\,,
\end{equation}
where $S_s$ is given in formula~\eqref{Ss}\,.
For this, note that
by definition of $c_{{\rm L},\,\lambda}$ for any $h\in \Gamma$
\begin{equation}\label{h1}
c_{{\rm L},\,\lambda}\leq \max_{u\in Q}\mathcal J_{s,\,\lambda}(h(u))
\end{equation}
and so, in particular, taking $h=id$ and using the fact that $Q\subset \mathbb F_\varepsilon$, we have
\begin{equation}\label{h2}
c_{{\rm L},\,\lambda}\leq \max_{u\in Q}\mathcal J_{s,\,\lambda}(u) \leq \max_{u\in \mathbb F_\varepsilon}\mathcal J_{s,\,\lambda}(u)\,,
\end{equation}
where $\mathbb F_\varepsilon$ is given in \eqref{F}\,.

Since $\mathbb F_\varepsilon$ is a linear space,
\begin{equation}\label{maxt}
\begin{aligned}
\max_{u\in \mathbb F_\varepsilon}\mathcal J_{s,\,\lambda}(u) & =\max_{u\in \mathbb F_\varepsilon \atop \zeta\not =0}\mathcal J_{s,\,\lambda}(|\zeta|\cdot \frac{u}{|\zeta|})\\
& = \max_{u\in \mathbb F_\varepsilon \atop \zeta> 0}\mathcal J_{s,\,\lambda}(\zeta u)\\
& \leq \max_{u\in \mathbb F_\varepsilon \atop \zeta\geq 0}\mathcal J_{s,\,\lambda}(\zeta u)\,.
\end{aligned}
\end{equation}
Hence, putting together \eqref{h1}--\eqref{maxt}, we have that
\begin{equation}\label{cL}
c_{{\rm L},\,\lambda}\leq \max_{u\in \mathbb F_\varepsilon \atop \zeta\geq 0}\mathcal J_{s,\,\lambda}(\zeta u)\,.
\end{equation}
On the other hand, by \cite[Proposition~20]{servadeivaldinociBN}, we have that for any $u\in X_0\setminus\{0\}$
\begin{equation}\label{max}
\max_{\zeta\geq 0}\mathcal J_{s,\,\lambda}(\zeta u)=\frac s n\,S_{s,\,\lambda}^{n/(2s)}(u)=\frac s n\,\left(\frac{\|u\|_{X_0}-\lambda\|u\|_{L^2(\Omega)}^2}{\|u\|_{L^{2^*}(\Omega)}^{2^*}}\right)^{n/(2s)}\,,
\end{equation}
also thanks to \eqref{Sslambdav}\,.
Thus, in order to prove \eqref{AR4}, by \eqref{cL} it is enough to show that
$$\max_{u\in \mathbb F_\varepsilon \atop \zeta\geq 0}\mathcal J_{s,\,\lambda}(\zeta u)< \frac s n\,S_s^{n/(2s)}\,,$$
that is, thanks to \eqref{max} and the scale invariance of the right-hand side in \eqref{max}, we have to prove that
\begin{equation}\label{stimaMepsilon}
M_\varepsilon<S_s
\end{equation}
for some $\varepsilon>0$\,, where $M_\varepsilon$ is defined in \eqref{mepsilon}\,.

By Proposition~\ref{propMepsilon}-$a)$
$$M_\varepsilon=\|u_M\|_{X_0}^2-\lambda\|u_M\|_{L^2(\Omega)}^2$$
with
$$u_M=v+tu_\varepsilon\,,\,\,\, v\in \mbox{span}\{e_{1,\,s}, \dots, e_{k,\,s}\}\,,\,\, t\geq 0\,, \,\,\, \|u_M\|_{L^{2^*}(\Omega)}=1\,.$$

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 $t>0$\,.
By Propositions~\ref{propMepsilon}-$b)$ and~\ref{stimaUepsilon} with $2s<n<4s$ we deduce that
$$\begin{aligned}
M_\varepsilon & \leq \big(\lambda_{k,\,s}-\lambda\big)\|v\|_{L^2(\Omega)}^2+S_{s,\,\lambda}(u_\varepsilon)\Big(1+\mathcal O(\varepsilon^{(n-2s)/2})\|v\|_{L^2(\Omega)}\Big) +\mathcal O(\varepsilon^{(n-2s)/2)}\|v\|_{L^2(\Omega)}\\
& \leq \big(\lambda_{k,\,s}-\lambda\big)\|v\|_{L^2(\Omega)}^2+\Big(S_s+\mathcal O(\varepsilon^{n-2s})-\lambda C_s\varepsilon^{n-2s}+ \mathcal O(\varepsilon^{2s})\Big)\Big(1+\mathcal O(\varepsilon^{(n-2s)/2})\|v\|_{L^2(\Omega)}\Big)\\
& \qquad \qquad \qquad +\mathcal O(\varepsilon^{(n-2s)/2})\|v\|_{L^2(\Omega)}\\
& \leq \big(\lambda_{k,\,s}-\lambda\big)\|v\|_{L^2(\Omega)}^2+S_s+\mathcal O(\varepsilon^{n-2s})-\lambda C_s\varepsilon^{n-2s}+ \mathcal O(\varepsilon^{2s})+\mathcal O(\varepsilon^{(n-2s)/2})\|v\|_{L^2(\Omega)}\\
& \leq S_s+\mathcal O(\varepsilon^{n-2s})-\lambda C_s\varepsilon^{n-2s}+ \mathcal O(\varepsilon^{2s})\,,\\
\end{aligned}$$
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
$$\big(\lambda_{k,\,s}-\lambda\big)\|v\|_{L^2(\Omega)}^2+\mathcal O(\varepsilon^{(n-2s)/2})\|v\|_{L^2(\Omega)}\leq \frac{1}{4(\lambda-\lambda_{k,\,s})}\mathcal O(\varepsilon^{n-2s})=\mathcal O(\varepsilon^{n-2s})\,.$$
Thus,
$$\begin{aligned}
M_\varepsilon & \leq S_s+\mathcal O(\varepsilon^{n-2s})-\lambda C_s\varepsilon^{n-2s}+ \mathcal O(\varepsilon^{2s})\\
& =S_s+\varepsilon^{n-2s}\Big(\mathcal O(1)-\lambda C_s+ \mathcal O(\varepsilon^{4s-n})\Big)\\
& <S_s
\end{aligned}$$
as $\varepsilon \to 0$\,, thanks to the fact that $\lambda>\lambda_s$\,. This ends the proof of \eqref{stimaMepsilon}\,.

Ultimately, we have shown that the critical level of $\mathcal J_{s,\,\lambda}$ stays below $(s/n)\,S_s^{n/(2s)}$, provided $\lambda>\lambda_s$ is not an eigenvalue of $(-\Delta)^s$\,.

\smallskip

\textbf{Case 2: $\lambda_s\geq \lambda_{1,\,s}$\,.} In this setting the only critical level of $\mathcal J_{s,\,\lambda}$ to be estimated is the one given by the Linking Theorem. For doing this, we can argue as in the last part of Case~1 (where we estimate the Linking critical level). In this way we get that for any $\lambda>\lambda_s$ different from an eigenvalue of $(-\Delta)^s$
$$c_{{\rm L},\,\lambda}<\frac s n\, S_s^{n/(2s)}\,.$$

Now, we can conclude the proof of Theorem~\ref{lapfra0}\,.

\subsection{End of the proof of Theorem~\ref{lapfra0}}
In Subsection~\ref{subsec:estimate} we have shown that there exists $\lambda_s>0$ such that for any $\lambda>\lambda_s$, with $\lambda$ not an eigenvalue of $(-\Delta)^s$\,, the critical level of $\mathcal J_{s,\,\lambda}$ stays below $(s/n)\,S_s^{n/(2s)}$ and therefore
the $(PS)_c$--condition is satisfied (see, in particular, \eqref{AR1},
\eqref{AR4} and Proposition~\ref{propPS})\,.

Let us consider two cases: either~$0<\lambda_s<\lambda_{1,s}$ or~$\lambda_s\ge\lambda_{1,s}$. Considering separately this two situations we have
\begin{itemize}
\item $0<\lambda_s<\lambda_{1,\,s}$\,: in this case, for any $\lambda\in (\lambda_s,\, \lambda_{1,\,s})$ problem~\eqref{problemalapfrac0} admits a Mountain Pass solution $u\in X_0$ at level $\mathcal J_{s,\,\lambda}(u)=c_{\rm{MP},\lambda}$. Moreover, $u$ is non-trivial since $\mathcal J_{s,\,\lambda}(u)=c_{\rm{MP},\lambda}\geq \beta>0=\mathcal J_{s,\,\lambda}(0)$\,. While for any $\lambda> \lambda_{1,\,s}>\lambda_s$, with $\lambda$ different from an eigenvalue of $(-\Delta)^s$, problem~\eqref{problemalapfrac0} admits a non-trivial Linking solution $u\in X_0$ at level $\mathcal J_{s,\,\lambda}(u)=c_{\rm{L},\lambda}\geq \beta$\,;

\item $\lambda_s\geq\lambda_{1,\,s}$\,: in this case problem~\eqref{problemalapfrac0} admits a non-trivial Linking solution $u$ at level $\mathcal J_{s,\,\lambda}(u)=c_{\rm{L},\lambda}\geq \beta$\,, provided $\lambda>\lambda_s$ is not an eigenvalue of the fractional Laplacian.
\end{itemize}

Ultimately, for any value of $\lambda>\lambda_s$ which is not an eigenvalue of $(-\Delta)^s$ we get a non-trivial weak solution for problem~\eqref{problemalapfrac0}\,. This concludes the proof of Theorem~\ref{lapfra0}\,.


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