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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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\title[Fractional Laplacian equations with critical Sobolev exponent]{
Fractional Laplacian equations\\ with critical Sobolev exponent}

\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 valdinoci@mat.uniroma2.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}
In this paper we complete the study of the following elliptic equation driven by a general non-local integrodifferential operator~$\mathcal L_K$
$$\left\{
\begin{array}{ll}
\mathcal L_K u+\lambda u+|u|^{2^*-2}u+f(x, u)=0 &  \mbox{in } \Omega\\
u=0 & \mbox{in } \RR^n\setminus \Omega\,,
\end{array}\right.$$
started in \cite{servadeivaldinociBN}. Here $s\in (0,1)$, $\Omega$ 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 and $f$ is a lower order perturbation of the critical power $|u|^{2^*-2}u$\,, while $\mathcal L_K$ is the integrodifferential operator defined as
$$\mathcal L_Ku(x)=
\int_{\RR^n}\Big(u(x+y)+u(x-y)-2u(x)\Big)K(y)\,dy\,,
\,\,\,\,\, x\in \RR^n\,.$$

Under suitable growth condition on $f$, we show that this problem admits non-trivial solutions for any positive parameter~$\lambda$\,. This existence theorem extends some results obtained in \cite{sY, servadeivaldinociBN, servadeivaldinociBNLOW}\,.

In the model case, that is when $K(x)=|x|^{-(n+2s)}$ (this gives rise to the fractional Laplace operator $-(-\Delta)^s$), the existence result proved along the paper may be read as the non-local fractional counterpart of the one obtained in \cite{gazzolaruf} (see also \cite{capfortpalm}) in the framework of the classical Laplace equation with critical nonlinearities.

\end{abstract}

\maketitle

\tableofcontents

\section{Introduction}\label{sec:introduzione}
\subsection{Non-local integrodifferential equations with critical nonlinearities}\label{subsec:intro1}
Elliptic problems with critical nonlinearities are widely studied in the literature for their interest in many fields, such as differential geometry and physics, to name but a few.

On the other hand, a deep interest was shown also for non-local operators, thanks to their applications in a wide range of contexts, such as
the thin obstacle problem, optimization, finance, phase transitions, stratified materials, anomalous diffusion, crystal dislocation, soft thin films, semipermeable membranes, flame propagation, conservation laws, ultra-relativistic limits of quantum mechanics, quasi-geostrophic flows,
multiple scattering, minimal surfaces, materials science and water waves.

For all these reasons in \cite{sY, servadeivaldinociBN, servadeivaldinociBNLOW} we studied non-local equations with critical growth. To be precise, \cite{sY, servadeivaldinociBNLOW} were devoted to the study of the critical 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}
while in \cite{servadeivaldinociBN} we considered a lower order perturbation of the critical nonlinearity $|u|^{2^*-2}u$ in non-local ellitic equations, that is we studied problems modeled by
\begin{equation}\label{1sf}
\left\{
\begin{array}{ll}
(-\Delta)^s u-\lambda u=|u|^{2^*-2}u+f(x,u) & {\mbox{ in }} \Omega\\
u=0 & {\mbox{ in }} \RR^n\setminus \Omega\,,
\end{array} \right.
\end{equation}
where $f$ has subcritical growth at infinity (for instance, $f$ behaves like $|u|^{q-2}u$, with $q\in (2, 2^*)$). In both these problems $s\in (0,1)$ is fixed, $\Omega$ is an open bounded subset of $\RR^n$, $n>2s$, with Lipschitz boundary, $\lambda>0$\,,
\begin{equation}\label{2star}
2^*=2n/(n-2s)
\end{equation}
is the fractional critical Sobolev exponent,
and $(-\Delta)^s$ is the fractional Laplace operator defined, up to normalization factors, as
\begin{equation} \label{2}
-(-\Delta)^s u(x)=
\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).

Along the present 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+f(x, u)=0 &  \mbox{in } \Omega\\
u=0 & \mbox{in } \RR^n\setminus \Omega\,,
\end{array}\right.
\end{equation}
where $\mathcal L_K$ is the integrodifferential operator defined as follows
\begin{equation}\label{lk}
\mathcal L_Ku(x)=
\int_{\RR^n}\Big(u(x+y)+u(x-y)-2u(x)\Big)K(y)\,dy\,,
\,\,\,\,\, x\in \RR^n\,,
\end{equation}
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}
A model for $K$ is given by the singular kernel $K(x)=|x|^{-(n+2s)}$ which gives rise to the fractional Laplace operator $-(-\Delta)^s$
defined in \eqref{2}.

Finally, the perturbation in
equation~\eqref{op} is a Carath\'eodory function
$f:\Omega\times \RR\to \RR$ verifying the following conditions:
\begin{equation}\label{supf}
\sup\Big\{|f(x,t)| : \mbox{a.e.} \,\, x\in \Omega\,,\,\, t\in [0, M]\Big\}<+\infty\,\,\, \mbox{for any}\,\, M>0\,;
\end{equation}
\begin{equation}\label{cond0}
{\displaystyle \lim_{t\to 0}\frac{f(x,t)}{t}=0}\,\, \mbox{uniformly in}\,\, x\in \Omega\,;
\end{equation}
\begin{equation}\label{condinfinito}
{\displaystyle \lim_{|t|\to +\infty}\frac{f(x,t)}{|t|^{2^*-1}}=0}\,\, \mbox{uniformly in}\,\, x\in \Omega\,;
\end{equation}
\begin{equation}\label{Fpositiva}
{\displaystyle F(x,t)\geq 0\,\, \mbox{a.e.}\,\, x\in \Omega\,,\,\, t\in \RR\,;}
\end{equation}
\begin{equation}\label{condFf}
{\displaystyle F(x,t)\leq \frac 1 2\, f(x,t)\,t+\frac s n\,|t|^{2^*}\,\, \mbox{a.e.}\,\, x\in \Omega\,,\,\, t\in \RR\,.}
\end{equation}
Here $F$ is the primitive of $f$ with respect to the second variable, i.e.
\begin{equation}\label{F}
{\displaystyle F(x,t)=\int_0^t f(x,\tau)d\tau}\,.
\end{equation}
As a model for $f$ we can take~$f(x,t)=a(x)|t|^{q-2}t$, with $a\in
L^\infty(\Omega)$\,, $a(x)\geq 0$ a.e. $x\in \Omega$ and $q\in (2, 2^*)$\,.

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~$\mathcal L_K$\,.

Aim of this paper is to study the existence of non-trivial weak solutions of~\eqref{op}, that is non-trivial solutions of the following problem
\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
{\displaystyle = \int_\Omega |u(x)|^{2^*-2}u(x)\varphi(x)dx+\int_\Omega f(x,u(x))\varphi(x)dx}\,\,\,\,\,\,\, \forall\,\, \varphi \in X_0\\
\\
u\in X_0\,,
\end{array}\right.
\end{equation}
which represents the weak formulation of~\eqref{op}. Note that, in order to write such a weak formulation, it is convenient to assume \eqref{evenkernel}.

The space $X_0$ in which we set the weak problem is a functional space, inspired by, but not equivalent to, the usual fractional Sobolev space. The choice of this space is motivated by the fact that it allows us to correctly encode the Dirichlet boundary datum in the weak formulation. This space was introduced in \cite{sv} (see also \cite{svmountain}). We will recall its definition and properties in Appendix~\ref{appendix}, in order to make the present paper self-contained.
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})\,.

Before stating our main results, we need to introduce some notations.

\subsection{Notations}\label{subsec:notations}
Along the present paper $\big\{ \lambda_k\big\}_{{k\in\NN}}$ 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 choose
a normalization for which
this sequence of eigenfunctions provides 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~5]{sY}, \cite[Proposition~9 and Appendix~A]{svlinking} and \cite[Proposition~4]{servadeivaldinociBNLOW}\,.


In the sequel we will also refer to the best fractional critical Sobolev constants $S_K$ and $S_{K,\,\lambda}$ defined as follows, respectively
\begin{equation}\label{SK}
{\displaystyle S_K:=\inf_{v\in X_0\setminus\{0\}}S_K(v)}
\end{equation}
and
\begin{equation}\label{SKlambda}
{\displaystyle S_{K,\,\lambda}:=\inf_{v\in X_0\setminus\{0\}}S_{K,\,\lambda}(v)\,,}
\end{equation}
where for any $v\in X_0\setminus\{0\}$
\begin{equation}\label{SKv}
{\displaystyle  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^*}}}=\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^*}}}\,.}
\end{equation}
and
\begin{equation}\label{SKlambdav}
\begin{aligned}
{\displaystyle  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^*}}}}\\
& {\displaystyle =\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^*}}}\,.}
\end{aligned}
\end{equation}
In the model case of the fractional Laplacian~$-(-\Delta)^s$ the counterparts of $S_K$ and $S_{K,\,\lambda}$ are the constants $S_s$ and $S_{s,\,\lambda}$ defined as
\begin{equation}\label{Ss}
{\displaystyle S_s:=\inf_{v\in H^s(\RR^n)\setminus\{0\}}S_s(v)\,,}
\end{equation}
and
\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{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}
and
\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}
All these constants are well-defined thanks to \cite[Theorem~6.5]{valpal}, \cite[Lemma~8]{svmountain}, \cite[Lemma~10]{svlinking} and \cite[Lemmas~9 and 14]{servadeivaldinociBN}. We refer, in particular, to \cite{sY, servadeivaldinociBN, servadeivaldinociBNLOW} for further comments on these constants.

\subsection{Main results of the paper}\label{subsec:mainresults}
As we said before, our aim is to show that problem~\eqref{op} has non-trivial solutions. Of course, the trivial function (i.e., $u$ identically zero)
is a solution of the problem, since $f(x,0)=0$, by assumption~\eqref{cond0}.
In order to get our goal, we will use variational methods and we will look for non-trivial critical points of the functional $\mathcal J_{K,\,\lambda}:X_0\to \RR$ given by
\begin{equation}\label{JKlambda}
\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
\\
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad
-\frac{1}{2^*}\int_\Omega |u(x)|^{2^*}\,dx
-\int_\Omega F(x, u(x))dx\,,
\end{aligned}
\end{equation}
where $F$ is defined as in \eqref{F}\,, since \eqref{problemaK0f} represents the Euler--Lagrange equation of the functional~$\mathcal J_{K,\,\lambda}$\,.

To this purpose we will use some variants of the classical Mountain Pass and Linking Theorems due to Ambrosetti and Rabinowitz (see \cite{ar, bnc, r78, rabinowitz}), which take into account the lack of compactness which occurs when dealing with critical nonlinearities. As in the classical case of the Laplacian, here the main difficulty is due to the fact that 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~$S_K$ introduced in \eqref{SK}\,.

The main result of the present paper is the following:

\begin{theorem}\label{generalkernel0f}
Let $s\in (0,1)$, $n>2s$ and $\Omega$ be an open bounded set of
$\RR^n$ with Lipschitz boundary \,. Let
$K:\RR^n\setminus\{0\}\rightarrow(0,+\infty)$ be a function satisfying conditions~\eqref{kernel}--\eqref{evenkernel} and let $f:\Omega\times \RR \to \RR$ be a Carath\'eodory function verifying \eqref{supf}--\eqref{condFf}\,. Finally, assume that
\begin{equation}\label{u0S}
\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}

When dealing with the model problem~\eqref{1sf}, the weak formulation is given by
\begin{equation}\label{problemalapfrac0f}
\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
{\displaystyle =\int_\Omega |u(x)|^{2^*-2}u(x)\varphi(x)dx+\int_\Omega
f(x,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)\,,
\end{array}\right.
\end{equation}
which represents the Euler--Lagrange equation of the functional $\mathcal J_{s,\,\lambda}: H^s(\RR^n)\to \RR$ defined as
$$\mathcal J_{s,\,\lambda}(u)=\frac 1 2 \int_{\RR^{2n}}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy-\frac \lambda 2 \int_\Omega |u(x)|^2\,dx
-\frac{1}{2^*}\int_\Omega |u(x)|^{2^*}dx-\int_\Omega F(x, u(x))\,dx\,,$$
with $F$ given in \eqref{F}\,.

In this framework Theorem~\ref{generalkernel0f} reads as follows:

\begin{theorem}\label{lapfra0f}
Let $s\in (0,1)$, $n>2s$ and $\Omega$ be an open bounded set of $\RR^n$
with
Lipschitz boundary.
Let $f:\Omega\times \RR \to \RR$ be a Carath\'eodory function
verifying \eqref{supf}--\eqref{condFf} and assume that
\begin{equation}\label{u0Sfrac}
\begin{aligned}
& \mbox{there exists $u_0\in H^s(\RR^n)\setminus\{0\}$ with $u_0\geq 0$ a.e. in $\Omega$}\\
& \qquad \qquad \mbox{and $u_0=0$ a.e. in $\RR^n\setminus \Omega$\,, such that}\\
& \qquad \qquad \quad \mbox{$S_{s,\, \lambda}(u)<S_s$ 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{problemalapfrac0f} admits a
solution $u\in H^s(\RR^n)$, which is not identically zero, such that $u=0$ a.e. in $\RR^n\setminus \Omega$\,.
\end{theorem}

The main results proved in the present paper are obtained by
adapting the variational and topological techniques used in the classical critical case (see \cite{am, as, bn, capfortpalm, gazzolaruf, struwe, willem}) to the non-local fractional setting.  The main difficulties are related to the
encoding of the Dirichlet boundary datum in the variational formulation and to the lack of compactness.
To overcome the first difficulty we work in the functional space~$X_0$, while for the second one we apply suitable variants of some classical critical points theorems which take into account that the Palais--Smale condition holds true in a suitable energy range, related to the best fractional critical Sobolev constant~$S_K$ (or $S_s$ for the fractional Laplacian).
Of course, differently from the classical case of the Laplacian,
the fact that the leading operator is non-local provides
technical and conceptual complications, and some estimates
become more delicate and involved.

\subsection{Some comments on the main theorems}\label{subsec:remarks}

The existence theorems proved along this paper may be seen as the natural extension to the non-local framework of well-know results obtained in \cite{as, bn, capfortpalm, gazzolaruf} (see also \cite{am, struwe, willem}), where the classical critical Laplace equation was considered.

Also aim of this paper is to complete the study of problem~\eqref{op}, started in \cite{servadeivaldinociBN}, where only the case when the positive parameter~$\lambda$ is less than the first eigenvalue of $-\mathcal L_K$ was considered.
Note that when $\lambda\in (0, \lambda_1)$, the assertion of Theorem~\ref{generalkernel0f} holds true under weaker assumptions on the data of the problem. To be precise, Theorem~\ref{generalkernel0f} is still valid if we do not assume condition~\eqref{Fpositiva} and if we replace \eqref{u0S} with the following one
\begin{equation}\label{u0SBN}
\begin{aligned}
& \mbox{there exists $u_0\in X_0\setminus\{0\}$ with $u_0\geq 0$ a.e. in $\RR^n$\,, such that}\\
& \quad \qquad \qquad \sup_{\zeta\geq 0}\mathcal J_{K,\,\lambda}(\zeta u_0)<\frac s n\, S_K^{n/(2s)}\,,
\end{aligned}
\end{equation}
which is weaker than \eqref{u0S} and it is the assumption considered in \cite[Theorem~1]{servadeivaldinociBN}\,.
In this sense Theorem~\ref{generalkernel0f} may be seen as an extension of \cite[Theorem~1]{servadeivaldinociBN}\,.

When there is no lower order perturbation, i.e. when $f\equiv 0$, in \eqref{op}, this non-local equation was studied in \cite{sY}: in this setting \cite[Theorem~1]{sY} is a particular case of Theorem~\ref{generalkernel0f}\,.
Also in \cite{sY} there are applications to the fractional Laplacian in dimension $n\geq 4$, while for $2s<n<4s$ we refer to \cite{servadeivaldinociBNLOW}. Both \cite[Theorem~2]{sY} and \cite[Theorem~1]{servadeivaldinociBNLOW} represent particular cases of Theorem~\ref{lapfra0f}\,.

In the context of non-local problems, we would like to recall also the recent paper~\cite{colorado}, where the authors studied the effect of a lower order perturbation in a critical elliptic problem driven by a non-local operator defined starting from the power of the eigenvalues of $-\Delta$\,. In this framework we refer also to \cite{tan}, where a non-local version of the Brezis-Nirenberg result was given in the case of a regional
square root of the Laplacian (which is an operator related, but different, from
the one considered here).

As we said before, problem~\eqref{op} will be treated via variational and topological tools. Hence, we will need to prove that the functional~$\mathcal J_{K,\,\lambda}$ has suitable geometric features, verifies a compactness condition and, due to the critical character of problem, that the minimax critical level of $\mathcal J_{K,\,\lambda}$ satisfies adequate estimates.

For the geometry, what are important are the growth conditions~\eqref{cond0} and \eqref{condinfinito} and, in the case when $\lambda\geq \lambda_1$ also \eqref{Fpositiva}\,. This last assumption is natural and it is
assumed also in the classical case of the Laplacian, when the parameter $\lambda$ is greater or equal to the first eigenvalue of $-\Delta$ with homogeneous Dirichlet boundary data (see, for instance, \cite{gazzolaruf}).

Condition~\eqref{condFf} will be crucial in the proof of the Palais--Smale condition (see Step~\ref{step3} in the proof of Proposition~\ref{propPS}). This is only the generalization of the assumption required in the classical case of the Laplacian (see, for instance, \cite[formula~$(2.34)$]{bn} and \cite[Section~3]{gazzolaruf}). Of course, in the model case $f(x,t)=a(x)|t|^{q-2}t$, with $a\in
L^\infty(\Omega)$\,, $a(x)\geq 0$ a.e. $x\in \Omega$ and $q\in (2, 2^*)$\,, condition~\eqref{condFf} is trivially satisfied.

The paper is organized as follows. Section~\ref{sec:proofgeneral} will be devoted to the proof of
Theorem~\ref{generalkernel0f}, while in Section~\ref{sec:prooffraclap} we will discuss the case of the fractional Laplacian operator and we will prove Theorem~\ref{lapfra0f}. Finally, since the space~$X_0$ in which we will work in is not a standard one, in Appendix~\ref{appendix} we will recall its definition and properties, in order to make this paper self-contained.

\section{The general case: proof of Theorem~\ref{generalkernel0f}}\label{sec:proofgeneral}

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 $X_0$, to \cite[Lemmas~5, 6 and 9]{servadeivaldinociBN} and to the fact that $\Omega$ is bounded.
Moreover, $\mathcal J_{K,\,\lambda}$ is Fr\'echet
differentiable in $u\in X_0$ and for any $\varphi\in X_0$
$$\begin{aligned}
\langle \mathcal J_{K,\,\lambda}'(u), \varphi\rangle & = \int_{\RR^{2n}}
\big(u(x)-u(y)\big)\big(\varphi(x)-\varphi(y)\big)K(x-y)\,dx\,dy -\lambda \int_\Omega u(x) \varphi(x)\,dx \\
& \qquad \qquad \qquad \qquad
 -\int_\Omega |u(x)|^{2^*-2}(x) u(x)\varphi(x)\,dx-\int_\Omega f(x, u(x))\varphi(x)\,dx\,.
\end{aligned}$$

Hence, critical points of $\mathcal J_{K,\,\lambda}$ are solutions to
problem~\eqref{problemaK0f}. Goal of this paper is finding these critical points. For this
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, as usual, a lack of compactness in the embedding $X_0\hookrightarrow L^{2^*}(\RR^n)$ (see, for instance, \cite[Lemma~9]{servadeivaldinociBN}). As a consequence of this, as 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}\,.

This is the reason why we can not exploit directly the Mountain Pass or the Linking Theorem. In order to overcome the difficult related to the Palais--Smale condition, we have to estimate the minimax critical level of the functional~$\mathcal J_{K,\,\lambda}$ in order to show that it is below the threshold where the compactness condition is satisfied. For this the key-point is condition~\eqref{u0S}\,.

In the case when $\lambda\in (0, \lambda_1)$, problem~\eqref{problemaK0f} was studied in \cite[Theorem~1]{servadeivaldinociBN} using the Mountain Pass Theorem in the version given in \cite[Theorem~2.2]{bn}\,. In this framework note that we do not need assumptions~\eqref{Fpositiva} and \eqref{condFf} and, moreover, that under hypothesis~\eqref{Fpositiva}, condition~\eqref{u0S} implies \eqref{u0SBN}\,. Then, Theorem~\ref{generalkernel0f} follows directly by \cite[Theorem~1]{servadeivaldinociBN}\,.

Hence, in the present paper we focus our attention to the case when $\lambda\geq \lambda_1$ and, in order to get our goal, we will use appropriately \cite[Theorem~5.3]{rabinowitz}\,.

Since $\lambda\geq \lambda_1$, we can suppose that
$$\lambda\in
[\lambda_k, \lambda_{k+1})\,\, \mbox{ for some}\,\, k\in \NN\,.$$

The idea consists in proving that the functional~$\mathcal J_{K,\,\lambda}$ satisfies the Palais--Smale condition at any level $c\in \RR$, provided $c$ belongs to a suitable range which depends on the best fractional critical constant~$S_K$\,. Then, after proving that the geometry of $\mathcal J_{K,\,\lambda}$ fits with the requirements of the Linking Theorem~\cite[Theorem~5.3]{rabinowitz}, we will show that the related Linking critical value of~$\mathcal J_{K,\,\lambda}$ stays below the energy level in which the compactness condition holds true. Assumption~\eqref{u0S} will be crucial in the proof of this last part.

\subsection{The Palais--Smale condition for the functional~$\mathcal J_{K,\,\lambda}$}\label{sub:compactness}
We start by proving that the functional~$\mathcal J_{K,\,\lambda}$ satisfies the Palais--Smale condition at any level $c$\,, whenever $c$ stays below a suitable threshold depending on $S_K$\,.
\begin{proposition}\label{propPS}
Let $\lambda\in[\lambda_k,\,\lambda_{k+1})$ for some $k\in \NN$ and
let $f$ be a Carath\'eodory function satisfying conditions
\eqref{supf}--\eqref{condFf}\,. Let $c\in \RR$ be such that
\begin{equation}\label{cS}
c<\frac s n S_K^{n/(2s)}
\end{equation}
and 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}
For this proof we adapt the arguments used in the classical elliptic setting
to the non-local fractional framework.
When $f\equiv 0$ the Palais--Smale condition for $\mathcal J_{K,\,\lambda}$ was proved in \cite[Proposition~7]{sY}\,, while in the case when $\lambda<\lambda_1$ a proof can be found in \cite[Subsection~3.1.1]{servadeivaldinociBN}\,.
Here some non-trivial differences arise with respect to these cases.
Now, 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, but we prefer to repeat it for reader's convenience, since some steps of this proof will be used in the sequel.

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}
so that
\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, by H\"older inequality, the fact that $\Omega$ is bounded, \cite[Lemma~6]{svmountain} and \eqref{kernelfrac}
\begin{equation}\label{H}
\int_\Omega |u_j(x)|\,dx\le
|\Omega|^{(2^*-1)/2^*}\left(\int_\Omega
|u_j(x)|^{2^*}\,dx\right)^{1/2^*}\le \bar \kappa
|\Omega|^{(2^*-1)/2^*}\|u_j\|_{X_0}\,,
\end{equation}
for some positive constant~$\bar \kappa$\,.

Hence, by \cite[Lemma~5]{servadeivaldinociBN} and \eqref{H} we obtain
\begin{equation}\label{jj'0}
\begin{aligned}
\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^*} -\int_\Omega F(x, u_j(x))\,dx\\
& \qquad \qquad \qquad \qquad  +\frac 1 2 \int_\Omega f(x, u_j(x)) \,u_j(x)\,dx\\
& \geq \frac s n\, \|u_j\|_{L^{2^*}(\Omega)}^{2^*}-\varepsilon\Big(1+\frac{2^*}{2}\Big)\|u_j\|_{L^{2^*}(\Omega)}^{2^*}\\
& \qquad \qquad \qquad \qquad  -\frac 3 2 M(\varepsilon)\int_\Omega |u_j(x)|\,dx\\
& \geq \frac s n\, \|u_j\|_{L^{2^*}(\Omega)}^{2^*}-\varepsilon\Big(1+\frac{2^*}{2}\Big)\|u_j\|_{L^{2^*}(\Omega)}^{2^*}
-C_\varepsilon\|u_j\|_{X_0}\,,
\end{aligned}
\end{equation}
where $C_\varepsilon$ is a positive constant, possibly depending on $|\Omega|$\,.

Choosing $\varepsilon>0$ small enough (i.e. $\varepsilon$ such that $\frac s n>\varepsilon \Big(1+\frac{2^*}{2}\Big)$), \eqref{2k} and~\eqref{jj'0} imply 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 $\kappa_*$\,.
As a consequence of this and using again the H\"older inequality we get
\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*}
so that, since $2<2^*$,
\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~$\widetilde\kappa>0$ not depending on~$j$.

Hence, by \eqref{jlimitato0} and \cite[Lemma~5]{servadeivaldinociBN} (used here with $\varepsilon=1$), we get
$$\kappa\geq \mathcal J_{K,\,\lambda}(u_j) \geq \frac 1 2 \|u_j\|_{X_0}^2-\frac \lambda 2 \|u_j\|_{L^2(\Omega)}^2 -\Big(\frac{1}{2^*}+1\Big)\|u_j\|_{L^{2^*}(\Omega)}^{2^*} -M(1)\|u_j\|_{L^1(\Omega)}\,,$$
so that, by \eqref{H}, \eqref{norma2star} and \eqref{norma2} we get that for any $j\in \NN$
$$\|u_j\|_{X_0}^2 \leq \kappa_{**}\left(1+\|u_j\|_{X_0}\right)$$
for some positive constant $\kappa_{**}$ independent of $j$\,. Hence, the proof of Step~\ref{claim2} is complete\,.
\end{proof}

\begin{step}\label{claim3}
Problem~\eqref{problemaK0f} admits\footnote{As a matter of fact,
the solution $u_\infty$
constructed in Step~\ref{claim3} will turn out to
be the limit of $u_j$ in the topology of~$X_0$, but this
will be achieved only at the end of the proof, see~\eqref{END} below.
Notice that, at this stage, the existence of
the solution given by Step~\ref{claim3}
does not end the proof of Theorem~\ref{generalkernel0f} because
\eqref{Jc0} and \eqref{J'00} are in use.}
a solution $u_\infty\in X_0$\,.
\end{step}
\begin{proof}
This proof can be found in \cite[Claim~3 of Subsection~3.1.1]{servadeivaldinociBN}\,. Note that for this part we do not need conditions~\eqref{Fpositiva} and \eqref{condFf}\,.
\end{proof}

\begin{step}\label{step3}
The following equality holds true:
$$\begin{aligned}
\mathcal J_{K,\,\lambda}(u_\infty) & =\frac s n\, \int_\Omega|u_\infty(x)|^{2^*}dx +\frac 1 2\, \int_\Omega f(x, u_\infty(x))u_\infty(x)dx-\int_\Omega F(x, u_\infty(x))dx\geq 0\,.
\end{aligned}$$
\end{step}
\begin{proof}
By Step~\ref{claim3}, taking $\varphi=u_\infty$ as a test function in \eqref{problemaK0f}\,, we get
$$\begin{aligned}
{\displaystyle \int_{\RR^{2n}}
|u_\infty(x)-u_\infty(y)|^2 K(x-y) dx\,dy} & {\displaystyle -\lambda \int_\Omega
|u_\infty(x)|^2\,dx= \int_\Omega |u_\infty(x)|^{2^*}dx}\\
&{\displaystyle  \qquad \qquad +\int_\Omega f(x,u_\infty(x))u_\infty(x)dx\,,}
\end{aligned}$$
so that
$$\begin{aligned}
\mathcal J_{K,\,\lambda}(u_\infty) & =\left(\frac 1 2-\frac{1}{2^*}\right) \int_\Omega|u_\infty(x)|^{2^*}dx +\frac 1 2\, \int_\Omega f(x, u_\infty(x))u_\infty(x)dx-\int_\Omega F(x, u_\infty(x))dx\\
&=\frac s n\, \int_\Omega|u_\infty(x)|^{2^*}dx +\frac 1 2\, \int_\Omega f(x, u_\infty(x))u_\infty(x)dx -\int_\Omega F(x, u_\infty(x))dx\geq 0\,.
\end{aligned}$$
The last inequality follows from assumption~\eqref{condFf}\,. 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~\cite[Lemma~9]{servadeivaldinociBN}, the sequence $u_j$ is bounded in $X_0$ and in $L^{2^*}(\RR^n)$.
Since $L^{2^*}(\RR^n)$ 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 \cite[Lemma~8]{svmountain}, 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 $\nu\in[1, 2^*)$ (see, for instance \cite[Theorem~IV.9]{brezis}).

Moreover, 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 $j\to +\infty$\,.

Finally, we claim that
\begin{equation}\label{convergenzaF}
\int_\Omega F(x, u_j(x))dx \to \int_\Omega F(x, u_\infty(x))dx
\end{equation}
as $j\to +\infty$\,.
At this purpose, note that, since $u_j$ is bounded in $L^{2^*}(\Omega)$, there exists $\kappa>0$ such that
\begin{equation}\label{k2star}
\|u_j\|_{L^{2^*}(\Omega)}\leq \kappa\,\,\, \mbox{for any}\,\, j\in \NN\,.
\end{equation}
Now, let us fix $\delta>0$. By~\cite[formula~(2.2) in Lemma~5]{servadeivaldinociBN} (used here with $\varepsilon=\delta/(2\kappa)$, we have that
\begin{equation}\label{Fdelta}
|F(x,u_j(x))|\leq \frac{\delta}{2\kappa}\,|u_j(x)|^{2^*}+M(\delta/(2\kappa))|u_j(x)|\,\, \mbox{a.e. in}\,\, \Omega\,.
\end{equation}
Then, thanks to the Sobolev embeddings of $X_0$, we have that $F(\cdot, u_j(\cdot)) \in L^1(\Omega)$ and, putting $\eta(\delta)=\left(\frac{\delta}{2\kappa M(\delta/(2\kappa))}\right)^{2^*/(2^*-1)}$, for any measurable subset $\Omega'$ of $\Omega$ such that
\begin{equation}\label{omega'}
|\Omega'|\leq \eta(\delta)
\end{equation}
we get
\begin{equation}\label{assolutacont}
\begin{aligned}
\int_{\Omega'} |F(x,u_j(x))|\,dx& \leq \frac{\delta}{2\kappa}\int_{\Omega'} |u_j(x)|^{2^*}\,dx+M(\delta/(2\kappa))\int_{\Omega'}|u_j(x)|\,dx\\
& \leq \frac{\delta}{2\kappa}\,\|u_j\|_{L^{2^*}(\Omega)}+M(\delta/(2\kappa))\|u_j\|_{L^1(\Omega')}\\
& \leq \frac{\delta}{2}+M(\delta/(2\kappa))|\Omega'|^{(2^*-1)/2^*}\|u_j\|_{L^{2^*}(\Omega)}\\
& \leq \frac{\delta}{2}+M(\delta/(2\kappa))\eta(\delta)^{(2^*-1)/2^*}\kappa\\
& = \delta\,,
\end{aligned}
\end{equation}
thanks to \eqref{k2star}--\eqref{omega'} and H\H older inequality. Hence, $F(\cdot, u_j(\cdot))$ is uniformly integrable\footnote{Or, according to
the different terminologies,
absolutely continuous in $\Omega$, uniformly with respect to $j\in \NN$\,.}
in~$\Omega$\,.
Moreover, by \eqref{convergenze0ter} and the
fact that the map $t\mapsto F(\cdot, t)$ is continuous in $t\in \RR$, we get
\begin{equation}\label{convF0}
 F(\cdot, u_j(\cdot)) \to  F(\cdot, u_\infty(\cdot)) \quad \mbox{a.e. in} \quad \Omega
\end{equation}
as $j\to +\infty$\,. Thus, by \eqref{assolutacont} and \eqref{convF0} and the Vitali Convergence Theorem, \eqref{convergenzaF} holds true.

Therefore, using also the definition of $\mathcal J_{K,\,\lambda}$, by \eqref{convergenze0bis} and \eqref{bl1}--\eqref{convergenzaF} we deduce that
$$\begin{aligned}
\mathcal J_{K,\,\lambda}(u_j) & =\frac 1 2 \int_{\RR^{2n}}|u_j(x)-u_\infty(x)-u_j(y)+u_\infty(y)|^2 K(x-y)\,dx\,dy\\
 & \qquad \qquad + \frac 1 2\int_{\RR^{2n}}|u_\infty(x)-u_\infty(y)|^2 K(x-y)\,dx\,dy -\frac \lambda 2 \int_\Omega |u_\infty(x)|^2 dx\\
 & \qquad \qquad -\frac{1}{2^*}\int_\Omega |u_j(x)-u_\infty(x)|^{2^*}\,dx-\frac{1}{2^*} \int_\Omega|u_\infty(x)|^{2^*}\,dx\\
 & \qquad \qquad - \int_\Omega F(x, u_\infty(x))dx+\textit{o}(1)\\
& =\mathcal J_{K,\,\lambda}(u_\infty)+\frac 1 2 \int_{\RR^{2n}}|u_j(x)-u_\infty(x)-u_j(y)+u_\infty(y)|^2 K(x-y)\,dx\,dy\\
& \qquad \qquad -\frac{1}{2^*}\int_\Omega |u_j(x)-u_\infty(x)|^{2^*}\,dx+\textit{o}(1)
\end{aligned}$$
as $j\to +\infty$\,, so that the assertion of Step~\ref{step4} follows.
\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, we note that, as a consequence of \eqref{convergenzaweak} and \eqref{bl2}, it holds true that
\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)(u_j(x)-u_\infty(x))\,dx\\
& = \int_\Omega |u_j(x)|^{2^*}\,dx -\int_\Omega |u_\infty(x)|^{2^*-2}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 $j\to +\infty$\,.

Furthermore, arguing exactly as in the proof of \eqref{convergenzaF}
(that is, using~\cite[formula~(2.1) in Lemma~5]{servadeivaldinociBN}
and the Vitali Convergence Theorem), we deduce that
\begin{equation}\label{convergenzaf}
\int_\Omega f(x, u_j(x))u_j(x)\,dx \to \int_\Omega f(x, u_\infty(x))u_\infty(x)\,dx
\end{equation}
as $j\to +\infty$\,.

Also, by \eqref{convergenze0ter} and the
fact that the map $t\mapsto f(\cdot, t)$ is continuous in $t\in \RR$, it is easily seen that
\begin{equation}\label{convfae}
 f(\cdot, u_j(\cdot)) \to  f(\cdot, u_\infty(\cdot)) \quad \mbox{a.e. in} \quad \Omega
\end{equation}
as $j\to +\infty$.
Now, by~\cite[Lemma~5]{servadeivaldinociBN}, the fact that $\Omega$ is bounded and \eqref{k2star}, it follows that $f(\cdot, u_j(\cdot))$ is bounded in $L^{2^*/(2^*-1)}(\Omega)$, so that, also thanks to \eqref{convfae}, we get
\begin{equation}\label{convf0}
 f(\cdot, u_j(\cdot)) \to  f(\cdot, u_\infty(\cdot)) \quad \mbox{weakly in} \quad L^{2^*/(2^*-1)}(\Omega)\,,
\end{equation}
which yields
\begin{equation}\label{convergenzaf1}
\int_\Omega f(x, u_j(x))u_\infty(x)\,dx \to \int_\Omega f(x, u_\infty(x))u_\infty(x)\,dx
\end{equation}
as $j\to +\infty$\,, being $u_\infty\in L^{2^*}(\Omega)$\,.

Again by \cite[Lemma~5]{servadeivaldinociBN} we get that $f(\cdot, u_\infty(\cdot))$ is bounded in $L^{2^*/(2^*-1)}(\Omega)$, so that, by \eqref{convergenzaweak}, we deduce that
\begin{equation}\label{convergenzaf2}
\int_\Omega f(x, u_\infty(x))u_j(x)\,dx \to \int_\Omega f(x, u_\infty(x))u_\infty(x)\,dx
\end{equation}
as $j\to +\infty$\,.

Now, 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,
\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\\
& \qquad -\int_\Omega \left(f(x, u_j(x))-f(x, 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 $j\to +\infty$\,, thanks to \eqref{convergenze0bis}, \eqref{ujuinfty}, \eqref{convergenzaf},  \eqref{convergenzaf1} and  \eqref{convergenzaf2}\,.
Putting together \eqref{conto1} and \eqref{conto2} we get the assertion of Step~\ref{step5}\,.
\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, Step~\ref{step4} and \eqref{Jc0}, we obtain
\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 $j\to +\infty$\,.
Note that, 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 $L\in [0, +\infty)$. Moreover, by definition of $S_K$ we have that
$$L\geq L^{2/2^*}S_K\,,$$
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 \eqref{L1} yields
\begin{equation}\label{END}
\|u_j-u_\infty\|_{X_0}\to 0\end{equation}
as $j\to +\infty$\,.
This ends the proof of Proposition~\ref{propPS}\,.
\end{proof}

\subsection{The geometry of the functional $\mathcal J_{K,\,\lambda}$}\label{sub:geometry}
This subsection is devoted to the study of the geometric structure of the functional $\mathcal J_{K,\,\lambda}$\,. Precisely, we will show that it fits with the geometric assumptions of the Linking Theorem given in \cite[Theorem~5.3]{rabinowitz}. Remember that we consider only the case when $\lambda\geq \lambda_1$\,, since the other one was treated in \cite{servadeivaldinociBN}\,.

Here we argue essentially as in the classical case of the Laplacian (see \cite{am, struwe, willem} and references therein), but some nontrivial technical differences arise.

In the pure non-local critical case, i.e. when $f\equiv 0$\,, the proof of the geometric features of $\mathcal J_{K,\,\lambda}$ was done in \cite[Propositions~8--10]{sY} (see also \cite{servadeivaldinociBNLOW} for the case $\mathcal L_K=-(-\Delta)^s$)\,. Here we generalize this result to the perturbed setting.

\begin{proposition}\label{lemmaJpositivoL}
Let $\lambda\in [\lambda_k, \lambda_{k+1})$ for some $k\in \NN$ and let $f$ be a Carath\'eodory function
satisfying conditions \eqref{supf}--\eqref{condinfinito}. 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 $\PP_{k+1}$. By \cite[Lemma~6]{servadeivaldinociBN} we get that for any $\varepsilon>0$
\begin{equation}\label{infJfrac0}
\begin{aligned}
\mathcal J_{K,\,\lambda}(u)& \geq \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\\
& \qquad \qquad \qquad -\frac{1}{2^*}\int_\Omega |u(x)|^{2^*}dx-\varepsilon\int_\Omega |u(x)|^2dx - \delta(\varepsilon)\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-\varepsilon\|u\|_{L^2(\Omega)}^2\\
& \qquad \qquad \qquad
 -\Big(\frac{1}{2^*}+\delta(\varepsilon)\Big)\|u\|_{L^{2^*}(\Omega)}^{2^*} \\
& \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 \qquad \qquad -\varepsilon|\Omega|^{(2^*-2)/2^*}
\|u\|_{L^{2^*}(\Omega)}^2 -\Big(\frac{1}{2^*}+\delta(\varepsilon)\Big)\|u\|_{L^{2^*}(\Omega)}^{2^*}\,,\\
\end{aligned}
\end{equation}
thanks to \cite[Lemma~16]{svlinking} and to the fact that $L^{2^*}(\Omega)\hookrightarrow
L^2(\Omega)$ continuously (since $\Omega$ is bounded and $2<2^*$,
so that one can use the H\"older
inequality).

By \eqref{kernelfrac} and \cite[Lemma~6]{svmountain}, from \eqref{infJfrac0} we
deduce that for any $\varepsilon>0$
\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 -\varepsilon c|\Omega|^{(2^*-2)/2^*}
 \int_{\RR^{2n}}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\, dx\,dy \\
& \qquad -\Big(\frac{1}{2^*}+\delta(\varepsilon)\Big)c^{2^*/2}\left( \int_{\RR^{2n}}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\, dx\,dy\right)^{2^*/2}\\
& \ge \left[\frac{1}{2}\left(1-\frac{\lambda}{\lambda_{k+1}}\right) -\frac{\varepsilon
c|\Omega|^{(2^*-2)/2^*}}{\theta}\right]\int_{\RR^{2n}} |u(x)-u(y)|^2 K(x-y)\,dx\,dy\\
& \qquad
-\Big(\frac{1}{2^*}+\delta(\varepsilon)\Big)
\Big(\frac{c}{\theta}\Big)^{2^*/2}
\left( \int_{\RR^{2n}} |u(x)-u(y)|^2 K(x-y)\, dx\,dy\right)^{2^*/2}\\
& =\left[\frac{1}{2}\left(1-\frac{\lambda}{\lambda_{k+1}}\right) -\frac{\varepsilon
c|\Omega|^{(2^*-2)/2^*}}{\theta}\right]\|u\|_{X_0}^2-\Big(\frac{1}{2^*}+\delta(\varepsilon)\Big)
\Big(\frac{c}{\theta}\Big)^{2^*/2}
\|u\|_{X_0}^{2^*}\,.
\end{aligned}
\end{equation}
We recall that $\theta$ was introduced in \eqref{kernelfrac}.
Now, we can choose $\varepsilon>0$ such that $2\varepsilon c
|\Omega|^{(2^*-2)/2^*}<\theta\big(1-\lambda/\lambda_{k+1}\big)$, so that, by \eqref{infJfrac20}
it is easily seen that
$$\mathcal J_{K,\,\lambda}(u)\geq \alpha \|u\|_{X_0}^2\left(1-\kappa
\|u\|_{X_0}^{2^*-2}\right)\,,$$
for suitable positive constants $\alpha$ and $\kappa$\,.

Finally, 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
$$\inf_{
{u\in \PP_{k+1}}\atop{\|u\|_{X_0}=\rho}
} \mathcal J_{K,\,\lambda}(u)\geq \alpha \rho^2(1-\kappa \rho^{2^*-2})=:\beta>0\,,$$
so that the proof of Proposition~\ref{lemmaJpositivoL} is complete.
\end{proof}

\begin{proposition}\label{lemmaJnegativoL}
Let $\lambda\in [\lambda_k, \lambda_{k+1})$ for some $k\in \NN$ and
let $f$ be a Carath\'eodory function
satisfying conditions \eqref{supf}--\eqref{Fpositiva}\,. 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}, the fact that $e_i\in X_0$ for any $i\in \NN$ and \eqref{Fpositiva}, we get
$$\begin{aligned}
\mathcal J_{K,\,\lambda}(u) & = \frac 1 2\sum_{i=1}^k u_i^2
\Big(\|e_i\|_{X_0}^2-\lambda\Big)-\frac{1}{2^*}\int_\Omega |u(x)|^{2^*}dx-\int_\Omega F(x, u(x))\,dx\\
& \leq \frac 1 2\sum_{i=1}^k u_i^2
\Big(\|e_i\|_{X_0}^2-\lambda\Big)\\
& = \frac 1 2\sum_{i=1}^k u_i^2 \big(\lambda_i-\lambda\big)\leq 0\,,
\end{aligned}$$
since $\lambda_i\leq \lambda_k\leq \lambda$ for
any $i=1, \dots, k$\,. This concludes the proof of Proposition~\ref{lemmaJnegativoL}\,.
\end{proof}

\begin{proposition}\label{lemmaJnegativo1LF}
Let $\lambda\geq 0$\,, let $f$ be a Carath\'eodory function
satisfying conditions \eqref{supf}--\eqref{Fpositiva} and
let $\mathbb F$ be a finite dimensional subspace of $X_0$. Then, there exist $R>\rho$ 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 $u\in \mathbb F$. Since $\lambda\geq 0$ and \eqref{Fpositiva} holds true, we obtain that
$$\begin{aligned}
\mathcal J_{K,\,\lambda} (u)& \leq \frac 1 2 \|u\|_{X_0}^2-\frac{1}{2^*}
\|u\|_{L^{2^*}(\Omega)}^{2^*}\\
& \leq \frac 1 2 \|u\|_{X_0}^2-\frac{\kappa}{2^*}
\|u\|_{X_0}^{2^*}\,,
\end{aligned}$$
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. This ends the proof of Proposition~\ref{lemmaJnegativo1LF}\,.
\end{proof}

By Propositions~\ref{lemmaJpositivoL}--\ref{lemmaJnegativo1LF} we deduce that the functional~$\mathcal J_{K,\,\lambda}$ has the geometric features required by the Linking Theorem~\cite[Theorem~5.3]{rabinowitz}\,. Now we have to choose carefully the finite dimensional space~$\mathbb F$ in which
we apply such theorem in order to have the suitable estimate on the critical level (i.e. an estimate which fits with the range where the Palais--Smale condition holds true).

To this purpose, let $u_0$ be the function given in assumption~\eqref{u0S} and $u_0^{(k+1)}$ its projection on $\PP_{k+1}$ and $\tilde u_0^{(k+1)}$ the normalization of $u_0^{(k+1)}$ in $X_0$, i.e.
\begin{equation}\label{tildeu0}
u_0^{(k+1)}=u_0-\sum_{i=1}^k \Big(\int_\Omega u_0(x) e_i(x)\,dx\Big)e_i\,, \quad \tilde u_0^{(k+1)}= u_0^{(k+1)}/\|u_0^{(k+1)}\|_{X_0}\,.
\end{equation}

Here we take $$\mathbb F=V\oplus\mbox{span}\left\{\tilde u_0^{(k+1)}\right\}\,,$$
where $V=\mbox{span}\{e_1, \dots, e_k\}$\,. Note that $V$ is finite dimensional and
$V\oplus W=X_0$, with $W=\PP_{k+1}$\,, thanks to \cite[Proposition~9--$f)$]{svlinking}\,.
Moreover, by Propositions~\ref{lemmaJpositivoL}--\ref{lemmaJnegativo1LF} we get that there exist $\rho>0$, $\beta>0$ and $R>\rho$ such that
$$\mathcal J_{K, \,\lambda}(u)\geq \beta>0 \quad \mbox{for any}\,\, u\in W \,\, \mbox{with}\,\,
\|u\|_{X_0}=\rho>0\,,$$
$$\mathcal J_{K, \,\lambda}(u)\leq 0 \quad \mbox{for any} \,\, u\in V$$
and
$$\mathcal J_{K, \,\lambda}(u)\leq 0 \quad \mbox{for any} \,\, u\in \mathbb F\,\, \mbox{with}\,\,
\|u\|_{X_0}\geq R\,.$$

Hence, the geometric structure of $\mathcal J_{K, \,\lambda}$ fits with the one 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}).

The Linking critical level $c_L$ of $\mathcal J_{K,\,\lambda}$ is given by
$$c_L=\inf_{h\in \Gamma}\max_{u\in Q}\mathcal J_{K,\,\lambda}(h(u))\,,$$
where
$$\Gamma=\big\{h\in C(\overline Q; X_0): h=id\,\, \mbox{on}\,\, \partial Q \big\}$$
and
$$Q=\Big(\overline B_R\cap V\Big)\oplus \left\{r\,\tilde u_0^{(k+1)} : r\in (0, R)\right\}\,.$$

Now we are ready to estimate the critical level~$c_L$\,. This will be done in the next subsection.

\subsection{Estimates of the critical level of $\mathcal J_{K,\,\lambda}$}\label{subsec:dimtheorem1}
Due to the critical nature of our problem, we have to estimate the Linking critical level~$c_L$ in order to show that it stays below the threshold under which the Palais--Smale condition is satisfied by $\mathcal J_{K,\,\lambda}$\,.

For this, first of all note that, by definition of $c_L$\,, for any $h\in \Gamma$
$$c_L\leq \max_{u\in Q}\mathcal J_{K,\,\lambda}(h(u))$$
from which, taking $h=id$ in $\overline Q$, we deduce that
\begin{equation}\label{cll}
c_L\leq \max_{u\in Q}\mathcal J_{K,\,\lambda}(u)\,.
\end{equation}

Furthermore, recall that $\mathbb F$ is a linear space and so, as a consequence,
\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, by \eqref{cll}, \eqref{maxt} and the fact that $Q\subset \mathbb F$, we obtain that
\begin{equation}\label{cL}
c_L\leq \max_{u\in Q}\mathcal J_{K,\,\lambda}(u)\leq \max_{u\in \mathbb F}\mathcal J_{K,\,\lambda}(u) \leq \max_{u\in \mathbb F\atop \zeta\geq 0}\mathcal J_{K,\,\lambda}(\zeta u)\,.
\end{equation}

Thanks to \eqref{Fpositiva} and \cite[Proposition~20]{servadeivaldinociBN} we also have
\begin{equation}\label{max}
\begin{aligned}
\max_{\zeta\geq 0}\mathcal J_{K,\,\lambda}(\zeta u) & =
\max_{\zeta\geq 0}\left(\frac{\zeta^2}{2} \int_{\RR^{2n}}|u(x)-u(y)|^2 K(x-y)\,dx\,dy-\frac {\zeta^2\lambda}{2} \int_\Omega |u(x)|^2\,dx\right. \\
 & \qquad \qquad \qquad \qquad \left.-\frac{\zeta^{2^*}}{2^*}\int_\Omega |u(x)|^{2^*}\,dx-\int_\Omega F(x, \zeta u(x))\,dx\right) \\
& \leq \max_{\zeta\geq 0} \left(\frac{\zeta^2}{2} \int_{\RR^{2n}}|u(x)-u(y)|^2 K(x-y)\,dx\,dy -\frac {\zeta^2\lambda}{2} \int_\Omega |u(x)|^2\,dx\right.
\\
& \qquad \qquad \qquad \qquad \left.-\frac{\zeta^{2^*}}{2^*}\int_\Omega |u(x)|^{2^*}\,dx
\right)\\
& =\frac s n\, S_{K,\,\lambda}^{n/(2s)}(u)
\end{aligned}
\end{equation}
for any $u\in X_0\setminus\{0\}$\,.

Finally, taking into account that $\mathbb F=V\oplus \mbox{span}\{\tilde u_0^{(k+1)}\}=\mbox{span}\{e_1, \dots, e_k, u_0\}$ (here we use the construction of $\tilde u_0^{(k+1)}$), by \eqref{u0S}, \eqref{cL} and \eqref{max} we deduce that
\begin{equation}\label{cL1}
c_L\leq \max_{u\in \mathbb F\atop \zeta\geq 0}\mathcal J_{K,\,\lambda}(\zeta u)\leq\frac s n\,\max_{u \in \mathbb F} S_{K,\,\lambda}^{n/(2s)}(u)<\frac s n\, S_K^{n/(2s)}\,.
\end{equation}

Ultimately, we have shown that the Linking critical level of $\mathcal J_{K,\,\lambda}$ stays below the threshold where the Palais--Smale condition is verified.

\medskip

Therefore, the conclusion of Theorem~\ref{generalkernel0f} follows from the  Linking Theorem \cite[Theorem~5.3]{rabinowitz}\,.

\section{Applications to critical fractional Laplace equations}\label{sec:prooffraclap}
In this section we briefly discuss the case when $K$ is the model kernel
$$K(x)=|x|^{-(n+2s)},$$
which gives rise to the fractional Laplace operator~$-(-\Delta)^s$
and we prove Theorem~\ref{lapfra0f}\,. For this
it is enough to note that, in this setting, condition~\eqref{u0Sfrac} implies \eqref{u0S}, thanks to \cite[Lemma~7]{servadeivaldinociBN} and to the fact that
$$\inf_{v\in H^s(\RR^n)\setminus\{0\}}S_s(v)\leq \inf_{v\in X_0\setminus\{0\}}S_s(v)\,,$$
since $X_0\subseteq H^s(\RR^n)$ by \cite[Lemma~5-$b)$]{svmountain}\,.

Hence, Theorem~\ref{generalkernel0f} provides a non-trivial solution $u\in X_0\setminus\{0\}$ of problem~\eqref{problemalapfrac0f} for any $\lambda>0$\,. Now, Theorem~\ref{lapfra0f} follows from the fact that $X_0\subseteq H^s(\RR^n)$ by \cite[Lemma~5-$b)$]{svmountain} (alternatively, we can use \cite[Lemma~7]{servadeivaldinociBN}, since the solution $u$ provided by Theorem~\ref{generalkernel0f} is such that $u=0$ a.e. in $\RR^n\setminus\Omega$)\,.

\medskip

%%% As a final remark, we would like to note that we can prove
%%% Theorem~\ref{lapfra0f} directly applying the Linking Theorem to the
%%% functional~$\mathcal J_{s,\,\lambda}$\,.

\appendix

\section{The functional setting}\label{appendix}
Here we recall the definition of the functional analytic spaces $X$ and $X_0$ we work in.

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
$$X_0=\{g\in X : g=0\,\, \mbox{a.e. in}\,\,
\RR^n\setminus
\Omega\}\,.$$
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).

Along the paper we also 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.

Some relations between the space $X_0$ and the usual fractional Sobolev space $H^s(\RR^n)$ were proved in
\cite[Lemma~5-$b)$]{svmountain} and in \cite[Lemma~7]{servadeivaldinociBN}, while, as for the embeddings of $X_0$ and $H^s(\RR^n)$ into the classical Lebesgue spaces, many properties were proved in \cite{valpal, svmountain, servadeivaldinociBN}. We would like to note that, with respect to such embeddings, the fractional Sobolev space $H^s(\RR^n)$ behaves like the usual Sobolev space $H^1(\RR^n)$, while $X_0$ as $H^1_0(\Omega)$\,. 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 $\Omega$\,.


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