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\title[Multiplicity results for elliptic subcritical fractional equations]{Multiplicity results for elliptic fractional\\ equations with subcritical term}

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\author[G. Molica Bisci]{Giovanni Molica Bisci}
\address{Dipartimento PAU,
          Universit\`a `Mediterranea' di Reggio Calabria,
          Via Graziella, Feo di Vito, 89124 Reggio Calabria, Italy}
\email{\tt gmolica@unirc.it}

\author[V. D. R\u{a}dulescu]{Vicen\c{t}iu D. R\u{a}dulescu}
\address{Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia}\email{vicentiu.radulescu@math.cnrs.fr}
%

\keywords{Fractional Laplacian, variational methods, multiple solutions.\\
\phantom{aa} 2010 AMS Subject Classification: Primary:
49J35, 35A15, 35S15;
Secondary: 47G20, 45G05.}

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\begin{abstract}
In the present paper, by using  variational methods, we study the existence of multiple nontrivial weak solutions for parametric nonlocal equations,
driven by the fractional Laplace operator $(-\Delta)^{s}$, in which the nonlinear term has a sublinear growth at infinity. More precisely,
a critical point result for differentiable functionals is exploited, in order to prove
the existence of an open interval of positive eigenvalues for which the treated problem
admits at least two nontrivial weak solutions in a suitable fractional Sobolev space.
\end{abstract}

\maketitle

\tableofcontents

\section{Introduction}\label{sec:introduzione}

Nonlocal fractional equations appear in many fields and a lot of interest has been devoted to this kind of problems and to their concrete applications; see, for instance the seminal papers \cite{CSCPDE, Css, CSapp} and \cite{AP, colorado, cabretan, capella, DiPi, fsv, molicaservadei, servadeiKavian, tan}, as well as, the references therein.\par
In this note, motivated by this large interest in the current literature, exploiting variational methods, we study the existence of multiple weak solutions for the following fractional problem
\begin{equation}\label{op}
\left\{
\begin{array}{ll}
(-\Delta)^s u=\lambda f(u) &  \mbox{in } \Omega\\
u=0 & \mbox{in } \RR^n\setminus \Omega,
\end{array}\right.
\end{equation}
where
$\Omega$ is an open bounded subset of $\RR^n$ ($n>2s$ and $s\in ]0,1[$) with Lipschitz boundary $\partial \Omega$, $\lambda$ is real parameter, $(-\Delta)^{s}$ denotes the fractional Laplace operator
which, up to normalization factor, may be defined as
\begin{equation}\label{FraZ}
(-\Delta)^s u(x):=
-\int_{\mathbb{R}^{n}}\frac{u(x+y)+u(x-y)-2u(x)}{|y|^{n+2s}}\,dy, \quad\, (\forall\, x\in \RR^n)
\end{equation}
 while $f$ is a subcritical term.\par
 Precisely, we assume that $f:\RR\to \RR$ is a continuous
function that is \textit{superlinear at zero}, that is,
\begin{equation}\label{supf}
{\displaystyle \lim_{t\to 0}\frac{f(t)}{t}=0},
\end{equation}
and \textit{sublinear at infinity}, that is,
\begin{equation}\label{condinfinito}
{\displaystyle \lim_{|t|\to +\infty}\frac{f(t)}{t}=0}.
\end{equation}
\indent Further,
we require that
\begin{equation}\label{Segno}
\sup_{t\in \RR}F(t)>0,
\end{equation}
where
\begin{equation*}
{\displaystyle F(t):=\int_0^t f(z)dz},
\end{equation*}
for any $t\in \RR$.\par
\indent Note that conditions~\eqref{supf} and \eqref{condinfinito} are standard assumptions to be satisfied in presence of subcritical terms. Now, let us denote
\begin{equation}\label{X0}
X_0:=\big\{v\in H^s(\RR^n) : v=0\,\, \mbox{a.e. in}\,\,
\RR^n\setminus
\Omega\big\},
\end{equation}
our ambient framework endowed by the norm
$$
\|v\|_{X_0}:=\left(\int_{\RR^n\times \RR^n}\frac{(v(x)-v(y))^2}{|x-y|^{n+2s}}\,dxdy\right)^{1/2},
$$
in which the functional space $H^s(\RR^n)$ denotes the fractional Sobolev space of the functions $v \in L^2(\RR^n)$ such that
$$\mbox{the map}\,\,\,
(x,y)\mapsto \frac{v(x)-v(y)}{|x-y|^{\frac{n+2s}{2}}}\,\,\, \mbox{is in}\,\,\,
L^2(\RR^n\times \RR^n, dxdy\big).$$

\indent This analytical context is inspired by (but not equivalent
to) the fractional Sobolev spaces, in order to correctly encode
the Dirichlet boundary datum in the variational formulation.\par Indeed, the nonlocal analysis that we perform here
in order to use our abstract approach is quite general and successfully exploited for other goals
in several recent contributions; see \cite{Molica, MolicaBisci, MP, sY,sv,svmountain,svlinking,servadeivaldinociBN} and \cite{valpal}
for an elementary introduction to this topic and for a list of related references.\par
\indent By a \textit{weak solution} of~\eqref{op} we mean a function $u\in X_0$ such that
\begin{equation}\label{problemaK0f}
\left\{\begin{array}{l} {\displaystyle \int_{\RR^n\times\RR^n }
\frac{(u(x)-u(y))(\varphi(x)-\varphi(y))}{|x-y|^{n+2s}}\, dxdy}\\
\\
\qquad \qquad \qquad \qquad {\displaystyle =\lambda \int_\Omega f(u(x))\varphi(x)dx},\,\,\,\,\,\,\, \forall\,\, \varphi\in X_0.
\end{array}\right.
\end{equation}
\indent As direct computations prove, Problem~\eqref{problemaK0f} represents the Euler--Lagrange equation of the $C^1$-functional $\mathcal J_{\lambda}:X_0\to \RR$ defined as
\begin{equation}\label{JKlambda}
\begin{aligned}
\mathcal J_{\lambda}(u)& :=\frac 1 2 \int_{\RR^n\times\RR^n}\frac{(u(x)-u(y))^2}{|x-y|^{n+2s}}\,dxdy\\
& \qquad \qquad \qquad \qquad
-\lambda \int_\Omega F(u(x))dx,
\end{aligned}
\end{equation}
for every $u\in X_0$.\par
The main multiplicity result of the present paper can be stated as follows:

\begin{theorem}\label{generalkernel0f}
Let $s\in ]0,1[$, $n>2s$, $\Omega$ be an open bounded set of
$\RR^n$ with Lipschitz boundary. Let $f:\RR \to \RR$ be a continuous function verifying conditions from \eqref{supf} to \eqref{Segno}.
Then, there exists an open interval $\Lambda\subset ]0,+\infty[$ and a real positive number $\kappa$ such that
for every $\lambda\in \Lambda$ problem~\eqref{op} has at least two distinct,
nontrivial weak solutions in $X_0$ whose $X_0$-norms are less than $\kappa$.
\end{theorem}

The above theorem will be proved by using variational techniques, in particular performing
a direct consequence of some general results given by Ricceri in~\cite{R1,R2}, which assures the existence of multiple critical points for a functional, under suitable regularity assumptions on it (see Theorem~\ref{abstract} below).\par
We note that, in Theorem \ref{generalkernel0f}, the superlinearity of the function $f$ at the origin (that is, hypothesis \eqref{supf}) is an
indispensable fact. Indeed, for example, if $$f (t) := (\arctan t)^2,\,\,\,\,\,(\forall\, t\in \RR)$$ an easy computation shows that \eqref{op} possesses only the trivial solution,
whenever $$\displaystyle|\lambda|<\frac{1}{c_2^2\pi},$$ where $c_2>0$ is the best Sobolev constant of the continuous embedding $X_0\hookrightarrow
L^{2}(\RR^n)$. For detailed comments and for a more general non-existence result, see Proposition \ref{fotetel-1} in Subsection \ref{subsec:comments}.\par
 Furthermore, we will give additional information as far as the
localization of the interval $\Lambda$ is concerned. More precisely, by using the notations adopted along the paper, we show that
$$
\Lambda \subset\left[\frac{\lambda_{1,s}}{c_f},\frac{2}{(1-\sigma_0)^2}\frac{(1-\sigma_0^n)\kappa_1\kappa_2}{\left(F(t_0)\sigma^n_0-(1-\sigma^n_0)\displaystyle\max_{|t|\leq |t_0|}|F(t)|\right)}\frac{t_0^2}{\tau^2}\right],
$$
see Subsection \ref{subsec:comments} for a detailed proof.\par
Finally, we point out that the results contained here can be proved, in a similar way, for the nonlocal fractional operator defined by
$$
\mathcal L_K u(x):= -\int_{\RR^n}(u(x+y)+u(x-y)-2u(x))K(y)dy, \quad x\in \RR^n,
$$
where the kernel function $K:\RR^n\setminus\{0\}\rightarrow \mathopen{]}0,+\infty[$ satisfies the following hypothesis:
\begin{itemize}
\item[] \textit{there exists} $\gamma\in\mathopen{]}0,1]$ \textit{such that}
$$
\gamma |x|^{-(n+2s)} \leq K(x) \leq \gamma^{-1} |x|^{-(n+2s)},
$$
\mbox{\it for any } $x\in \RR^n \setminus\{0\}$.
\end{itemize}
Clearly, the paradigm of the above $K$ is given by the singular kernel $K(x):=|x|^{-(n+2s)}$; in this case $\mathcal L_K$ reduces to the fractional Laplace operator given in \eqref{FraZ}.\par

\indent The present paper is organized as follows. In Section~\ref{sec:preliminary}
we give some notations and we recall some properties of the functional space we work in. We also give some tools which will be useful along the manuscript. In Section~\ref{sec:main} we study problem~\eqref{op} and we prove
Theorem~\ref{generalkernel0f}.


\section{Some preliminaries}\label{sec:preliminary}
This section is devoted to the notations used along the paper. We
also give some preliminary results which will be useful in the sequel.

\subsection{Notations and definitions}\label{subsec:notations}
In this subsection we briefly recall some properties of the functional space~$X_0$, firstly introduced in \cite{sv}, and we give some notations.
The reader familiar with this topic may skip this section and go directly to the next one.

The space~$X_0$ is defined as in \eqref{X0}, where $H^s(\RR^n)$
denotes 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^n\times\RR^n}\frac{\,\,\,(g(x)-g(y))^2}{|x-y|^{n+2s}}\,dxdy\Big)^{1/2}\,.
\end{equation}

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

Of course, the space~$X_0$ is non-empty, since $C^2_0 (\Omega)\subseteq X_0$ by \cite[Lemma~5.1]{sv} and it depends on the set $\Omega$.
Moreover, by \cite[Lemma~6]{svmountain} and the fact that any function $v\in X_0$ is such that $v=0$ a.e. in $\RR^n\setminus \Omega$, in the sequel we can take
\begin{equation}\label{normaX0}
X_0\ni v\mapsto \|v\|_{X_0}=\left(\int_{\RR^n\times \RR^n}\frac{(v(x)-v(y))^2}{|x-y|^{n+2s}}\,dxdy\right)^{1/2}
\end{equation}
as norm on $X_0$.\par
 Also $\left(X_0, \|\cdot\|_{X_0}\right)$ is a Hilbert space (for this see \cite[Lemma~7]{svmountain}), with scalar product
\begin{equation}\label{scalarproduct}
\langle u,v\rangle_{X_0}:=\int_{\RR^n\times \RR^n}
\frac{\big( u(x)-u(y)\big) \big( v(x)-v(y)\big)}{|x-y|^{n+2s}}\,dxdy.
\end{equation}

Finally, we recall that in \cite[Lemma~8]{svmountain} and in \cite[Lemma~9]{servadeivaldinociBN} the authors proved that the embedding $j:X_0\hookrightarrow
L^{\nu}(\RR^n)$ is continuous for any $\nu\in [1,2^*]$, while it is
compact whenever $\nu\in [1,2^*[$. In the sequel for any $\nu\in [1, 2^*[$, we will denote by $c_\nu$ the positive constant such that
\begin{equation}\label{Poincare1}
\|v\|_{L^{\nu}(\RR^n)}\leq c_{\nu} \|v\|_{X_0},
\end{equation}
for any $v\in
X_0$.
\subsection{Some useful tools}\label{subsec:ricceri}
The main tool used along this paper in order to prove our multiplicity result stated in Theorem~\ref{generalkernel0f} is given by a direct consequence of some general theorems due to Ricceri \cite{R1,R2} that we recall here below.

\begin{theorem}\label{abstract}
Let $(E, \|\cdot\|)$ be a separable and reflexive real Banach
space, and let $\Phi, \Psi : E\to \RR$ be two continuously G\^{a}teaux differentiable functionals.
Assume that there exists $z_0\in E$ such that $\Phi(z_0) = \Psi(z_0) = 0$ and $\displaystyle \inf_{z\in E}\Phi(z) \geq 0$
and that there exist $z_1\in E$, $\varrho > 0$ such that
\begin{itemize}
\item[$\rm (i)$] $\varrho<\Phi(z_1)$$;$
\item[$\rm (ii)$] $\displaystyle\sup_{\Phi(z)<\varrho}\Psi(z)<\varrho \displaystyle\frac{\Psi(z_1)}{\Phi(z_1)}$.
\end{itemize}

\noindent Further, put
$$
\bar a:=\frac{\zeta\varrho}{\varrho \displaystyle\frac{\Psi(z_1)}{\Phi(z_1)}-\displaystyle\sup_{\Phi(z)<\varrho}\Psi(z)},
$$
with $\zeta>1$, assume that the functional $$J_\lambda(z):=\Phi(z)-\lambda \Psi(z),\,\,\,\,\,(\forall\, z\in E)$$ is sequentially weakly lower semicontinuous, satisfies the $(\rm PS)$ condition, and
\begin{itemize}
\item[$\rm (iii)$] $\displaystyle\lim_{\|z\|\rightarrow +\infty}J_\lambda(z)=+\infty$,
\end{itemize}
for every $\lambda\in [0,\bar a]$.\par
Then there is an open interval $\Lambda \subset [0, \bar a]$ and a number $\kappa>0$ such that for each
$\lambda\in \Lambda$, the equation $J'_\lambda(z)=0$ admits at least three solutions in $E$ having
norm less than $\kappa$.
\end{theorem}

Some details and related topics on the above result can be found in the recent monograph~\cite{kristaly}.\par
\smallskip
For the sake of completeness, we also recall that the $C^1$-functional $J_\lambda:E\to\RR$ satisfies the Palais-Smale condition at level $\mu\in\RR$ when
\begin{itemize}
\item[$\textrm{(PS)}_{\mu}$] {\it Every sequence $\{z_{j}\}_{j\in \NN} \subset E$  such that
$$
J_\lambda(z_j)\to \mu, \qquad{\it and}\qquad \|J'_\lambda(z_j)\|_{E^*}\to0,
$$
as $j\rightarrow \infty$, possesses a convergent subsequence in $E$.}
\end{itemize}

\indent Here $E^*$ denotes the topological dual of $E$. Finally, we say that $J_\lambda$ satisfies the Palais-Smale condition (in short $(\rm PS)$) if $\textrm{(PS)}_{\mu}$ holds for every $\mu\in \RR$.

\smallskip

\indent Now, fix an element $x_0\in \Omega$ and choose $\tau>0$ in such a way that
 \begin{equation}\label{tiau}
 \bar B(x_0,\tau):=\{x\in\RR^n:|x-x_0|\leq \tau\}\subseteq \Omega,
 \end{equation}
 where $|\cdot|$ denotes the usual Euclidean norm in $\RR^n$.\par
\indent At this point, let $\sigma\in ]0,1[$, $t_0\in \RR$ and define $u_{\sigma}^{t_0}\in X_0$ as follows
\[
u_\sigma^{t_0}(x):= \left\{
\begin{array}{ll}
0 & \mbox{ if $x \in \RR^n \setminus B(x_0,\tau)$} \\\\
\displaystyle\frac{t_0}{(1-\sigma)\tau} \left(\tau- |x-x_0| \right)
& \mbox{ if $x \in B(x_0,\tau) \setminus B(x_0,\sigma\tau)$} \\\\
t_0 & \mbox{ if $x \in B(x_0,\sigma{\tau}),$}
\end{array}
\right.
\]
where $B(x_0,r)$ denotes the $n$-dimensional open ball with center $x_0\in \Omega$ and radius $r>0$.
\indent This function will be useful in the sequel in the proof of our theorem as well as the next preparatory Lemma. Let $n>2s$ and set
\begin{equation}\label{gio0}
\nu^{(n)}:=(2\pi)^n\left(1+\frac{1}{{\lambda_1}}\right),
\end{equation}
where
\begin{equation}\label{gio}
\lambda_1:=\inf_{u\in W^{1,2}_0(\Omega)\setminus\{0\}}\frac{\|\nabla u\|_{L^2(\Omega)}^2}{\|u\|_{L^2(\Omega)}^2}.
\end{equation}
\indent The following result holds.
\begin{lemma}\label{Stime}
Let $\sigma, s\in ]0,1[$ and $\tau$ defined as before, $
S_{n-2}
$
be the Lebesgue measure of the unit sphere in $\RR^{n-1}$ and
$$
\Gamma(t):=\int_0^{+\infty}z^{t-1}e^{-z}dz,\,\,\,\, (\forall\; t>0)
$$
be the usual Gamma function.
If $t_0\in \RR$, then $u_\sigma^{t_0}\in X_0$ and one has
\begin{equation}\label{dx}
 \|u_\sigma^{t_0}\|_{X_0}\leq \frac{|t_0|}{(1-\sigma)}\sqrt{\displaystyle \frac{\pi^{\frac{n}{2}}\tau^{n-2}(1-\sigma^n)}{\displaystyle \Gamma\left(1+\frac{n}{2}\right)}\kappa_1\kappa_2},
 \end{equation}
 where
 $$
 \kappa_1:= \left\{
\begin{array}{ll}
\displaystyle 2\nu^{(1)} &  \mbox{\rm if } n=1\\
\displaystyle \left(\pi+\frac{4}{1+2s}\right)\nu^{(2)} & \mbox{\rm if } n=2\\
\displaystyle S_{n-2}\left(\frac{\pi}{2}+\frac{2}{1+2s}\right)\nu^{(n)} & \mbox{\rm if } n\geq 3
\end{array}\right.\,\,\,\,\,\,\,{\rm and}\,\,\,\,\,\,\,
 \kappa_2:=\left(\frac{1}{2(1-s)}+\frac{2}{s}\right).
 $$
\end{lemma}
\begin{proof}
Computing the standard seminorm of the function $u_\sigma^{t_0}$ in $H^1(\RR^n)$, we easily have
\begin{eqnarray}\label{Nor}
 [u_{\sigma}^{t_0}]^2_{H^1(\RR^n)} &=& \int_{\RR^n}|\nabla u_{\sigma}^{t_0}(x)|^2\,dx=\int_{B(x_0,\tau)\setminus B(x_0,\sigma\tau)}\displaystyle\frac{t_0^2}{(1-\sigma)^2\tau^2}\,dx\nonumber\\
&=& \displaystyle\frac{t_0^2}{(1-\sigma)^2\tau^2}\left[\meas(B(x_0,\tau))-\meas(B(x_0,\sigma\tau))\right]\\
               &=&\displaystyle\frac{t_0^2}{(1-\sigma)^2}\frac{\pi^{\frac{n}{2}}\tau^{n-2}(1-\sigma^n)}{\displaystyle \Gamma\left(1+\frac{n}{2}\right)},\nonumber
\end{eqnarray}
where $``\meas(B(x_0,\tau))"$ and $``\meas(B(x_0,\sigma\tau))"$ denote respectively the Lebesgue measure of $B(x_0,\tau)$ and $B(x_0,\sigma\tau)$.\par

Since $u_{\sigma}^{t_0}\in H^{1}_0(\Omega)$, by \cite[Proposition 2.2 ]{valpal} it follows that $u_{\sigma}^{t_0}\in H^{s,2}(\Omega)$.
Moreover, the boundary $\partial \Omega$ is Lipschitz,
$\bar B(x_0,\tau)\subset \Omega$, and $u_{\sigma}^{t_0}=0$ in $\Omega\setminus \bar B(x_0,\tau)$. By \cite[Lemma 5.1]{valpal} one has that $u_{\sigma}^{t_0}\in H^{s}(\RR^n)$.

\indent Hence, since $s\in ]0,1[$, Proposition 3.4 of \cite{valpal}, yields
\begin{eqnarray}\label{obelix}
 \|u_{\sigma}^{t_0}\|^2_{X_0} &\leq& 2\left(\int_{\RR^n}\frac{1-\cos x_1}{|x|^{n+2s}}\, dx\right)\int_{\RR^n}|\xi|^{2s}|\mathfrak{F}u_{\sigma}^{t_0}(\xi)|^2\,d\xi\\
&\leq& 2\left(\int_{\RR^n}\frac{1-\cos x_1}{|x|^{n+2s}}\, dx\right)\int_{\RR^n}(1+|\xi|^{2})|\mathfrak{F}u_{\sigma}^{t_0}(\xi)|^2\,d\xi,\nonumber
\end{eqnarray}
where
where
$$
\mathfrak{F}u_{\sigma}^{t_0}(\xi):=\frac{1}{(2\pi)^{n/2}}\int_{\RR^n}e^{-i\xi\cdot x}u_{\sigma}^{t_0}(x)dx,
$$
stands for the classical Fourier transform of $u_{\sigma}^{t_0}$.\par
\indent Now, by standard arguments on the Fourier transform and Poincar\'{e} inequality, one has
\begin{eqnarray}\label{obelix2}
\int_{\RR^n}(1+|\xi|^{2})|\mathfrak{F}u_{\sigma}^{t_0}(\xi)|^2\,d\xi\leq \nu^{(n)}[u_{\sigma}^{t_0}]^2_{H^1(\RR^n)}.
\end{eqnarray}
\indent Indeed
$$
u_{\sigma}^{t_0}\in L^2(\RR^n)\,\,\,\,\,\,\,\mbox{if and only if}\,\,\,\,\,\,\,\mathfrak{F}u_{\sigma}^{t_0}\in L^2(\RR^n)
$$
and
\begin{eqnarray}\label{alf}
\|u_{\sigma}^{t_0}\|^2_{L^2(\RR^n)}=(2\pi)^{-n}\|\mathfrak{F}u_{\sigma}^{t_0}\|^2_{L^2(\RR^n)}.
\end{eqnarray}
\indent Further
$$
|\nabla u_{\sigma}^{t_0}|\in L^2(\RR^n)\,\,\,\,\,\,\,\mbox{if and only if}\,\,\,\,\,\,\,|\xi|\mathfrak{F}u_{\sigma}^{t_0}\in L^2(\RR^n)
$$
and
\begin{eqnarray}\label{alf2}
\|\nabla u_{\sigma}^{t_0}\|^2_{L^2(\RR^n)}=(2\pi)^{-n}\||\xi|\mathfrak{F}u_{\sigma}^{t_0}\|^2_{L^2(\RR^n)}.
\end{eqnarray}
\indent Relations \eqref{alf} and \eqref{alf2} give
\begin{eqnarray}\label{obelix23}
\int_{\RR^n}(1+|\xi|^{2})|\mathfrak{F}u_{\sigma}^{t_0}(\xi)|^2\,d\xi=(2\pi)^{n}\left(\|u_{\sigma}^{t_0}\|^2_{L^2(\RR^n)}+\|\nabla u_{\sigma}^{t_0}\|^2_{L^2(\RR^n)}\right).
\end{eqnarray}
\indent Hence, by \eqref{gio0} and \eqref{gio} one has that inequality \eqref{obelix2} is a direct consequence of \eqref{obelix23} taking into account that
$u_{\sigma}^{t_0}\in W^{1,2}_0(\Omega)$.\par
\indent Then, by \eqref{obelix} and \eqref{obelix2}, il follows that
\begin{eqnarray}\label{obelix3}
 \|u_{\sigma}^{t_0}\|^2_{X_0} &\leq& 2\nu^{(n)}\left(\int_{\RR^n}\frac{1-\cos x_1}{|x|^{n+2s}}\, dx\right)[u_{\sigma}^{t_0}]^2_{H^1(\RR^n)}.\nonumber
\end{eqnarray}
\noindent Thus, for $n\geq 3$, we have
\begin{eqnarray}\label{e3.2a}
 \|u_{\sigma}^{t_0}\|^2_{X_0} &\leq& 2\nu^{(n)}\left(\int_{\RR^n}\frac{1-\cos x_1}{|x|^{n+2s}}\, dx\right)[u_{\sigma}^{t_0}]^2_{H^1(\RR^n)}\nonumber\\
&=& 2\nu^{(n)}\left(\int_{\RR^{n-1}}\frac{1}{(1+|x|^2)^{\frac{n+2s}{2}}}\, dx\right)\times\nonumber\\
               &&\times \left(\int_{\RR}\frac{1-\cos t}{|t|^{1+2s}}\, dt\right)[u_{\sigma}^{t_0}]^2_{H^1(\RR^n)}.\nonumber
\end{eqnarray}

\indent The conclusion follows by \eqref{Nor} taking into account that, by direct computations it follows that, one has
$$
\int_{\RR^{n-1}}\frac{1}{(1+|x|^2)^{\frac{n+2s}{2}}}\, dx<S_{n-2}\left(\frac{\pi}{4}+\frac{1}{1+2s}\right),
$$
\noindent and
$$
\int_{\RR}\frac{1-\cos t}{|t|^{1+2s}}\, dt<\frac{1}{2(1-s)}+\frac{2}{s}.
$$
\indent Indeed

\begin{eqnarray}\label{e3.2a2}
 \int_{\RR^{n-1}}\frac{1}{(1+|x|^2)^{\frac{n+2s}{2}}}\, dx &=& S_{n-2}\int_0^{+\infty}\frac{\rho^{n-2}}{(1+\rho^2)^{\frac{n+2s}{2}}}\, d\rho\nonumber\\
&=& S_{n-2}\Bigg[\int_0^{1}\frac{\rho^{n-2}}{(1+\rho^2)^{\frac{n+2s}{2}}}\, d\rho+ \int_1^{+\infty}\frac{\rho^{n-2}}{(1+\rho^2)^{\frac{n+2s}{2}}}\, d\rho\Bigg]\nonumber\\
               &<&S_{n-2}\Bigg[\int_0^{1}\frac{(1+\rho^2)^{\frac{n-2}{2}}}{(1+\rho^2)^{\frac{n+2s}{2}}}\, d\rho+ \int_1^{+\infty}\frac{\rho^{n-2}}{(\rho^2)^{\frac{n+2s}{2}}}\, d\rho\Bigg]\nonumber\\
               &<&S_{n-2}\Bigg[\int_0^{1}\frac{d\rho}{1+\rho^2}+\int_1^{+\infty}\frac{d\rho}{\rho^{2(1+s)}}\Bigg]\nonumber\\
               &=&S_{n-2}\left(\frac{\pi}{4}+\frac{1}{1+2s}\right),\nonumber
\end{eqnarray}
and
\begin{eqnarray}\label{e3.2a3}
 \int_{\RR}\frac{1-\cos t}{|t|^{1+2s}}\, dt &=& 2\int_0^{+\infty}\frac{1-\cos t}{t^{1+2s}}\, dt\nonumber\\
&=& 2\Bigg[\int_0^{1}\frac{1-\cos t}{t^{1+2s}}\, dt+ \int_1^{+\infty}\frac{1-\cos t}{t^{1+2s}}\, dt\Bigg]\nonumber\\
               &\leq&\Bigg[\frac{1}{2}\int_0^{1}\frac{t^2}{t^{1+2s}}\, dt+2\int_1^{+\infty}\frac{1}{t^{(1+2s)}}\, dt\Bigg]\nonumber\\
               &=&\frac{1}{2(1-s)}+\frac{2}{s}.\nonumber
\end{eqnarray}
\noindent On the other hand, we also have
$$
\int_{\RR^n}\frac{1-\cos x_1}{|x|^{n+2s}}\, dx\leq\left\{
\begin{array}{ll}
\displaystyle\frac{1}{2(1-s)}+\frac{2}{s} &  \mbox{if } n=1\\
\displaystyle \left(\frac{\pi}{2}+\frac{2}{1+2s}\right)\left(\frac{1}{2(1-s)}+\frac{2}{s}\right) & \mbox{if } n=2.
\end{array}\right.
$$
\indent Clearly $u_\sigma^{t_0}:\RR^n\rightarrow\RR$ is a continuous continuous, $u_\sigma^{t_0}\in L^2(\Omega)$ and by the above computations it follows that $u_\sigma^{t_0}\in X$. Finally $u_\sigma^{t_0}=0$ in $\RR^n\setminus\Omega$. Thus $u_\sigma^{t_0}\in X_0$ and \eqref{dx} holds.
\end{proof}

\section{Proof of the Main Result}\label{sec:main}
The validity of the following Lemmas will be crucial in order to prove the main result.
\begin{lemma}\label{regularity}
Let $f:\RR \to \RR$ be a continuous function verifying condition \eqref{condinfinito}. Then,
 for every $\lambda\in \RR$, the functional $\mathcal J_{\lambda}$ is weakly lower semicontinuous on $X_0$.
\end{lemma}
\begin{proof}
The application $$u\mapsto\int_{\Omega}F(u(x))dx$$ is
continuous in the weak topology of $X_0$.\par
 Indeed, if
$\{u_{j}\}_{j\in \NN}$ is a sequence in $X_{0}$ such that
$u_{j}\rightharpoonup u$ weakly in $X_{0}$, then, by
\eqref{Poincare1} and \cite[Theorem~IV.9]{brezis}, up to a
subsequence, $u_{j}$ converges to $u$ strongly in $L^{\nu}(\Omega)$
and a.e. in $\Omega$ as $j\to +\infty$, and it is dominated by some
function $h_\nu\in L^{\nu}(\Omega)$\,, that is,
\begin{equation}\label{dominata}
|u_j(x)|\leq h_\nu(x)\quad \mbox{a.e.}\,\, x\in \Omega\,\,\,\, \mbox{for any}\, j\in \NN
\end{equation}
for any $\nu\in [1,2^{*}[$\,.\par

Due to \eqref{condinfinito}, there exists $c>0$ such that
\begin{equation}\label{crescita}
|f(t)|\le c(1+|t|),\,\,\,\,\,(\forall\,t\in \RR).
\end{equation}

Then, by the continuity of $F$ and \eqref{crescita} it follows that
$$F(u_{j}(x)) \to F(u(x))\,\,\, \mbox{a.e.}\,\, x\in \Omega$$
as $j\to \infty$ and
$$\left|F(u_{j}(x))\right|\leq c\left( |u_{j}(x)|+\frac{1}{2}u_j(x)^2\right)\leq c\left( h_1(x)+\frac{1}{2} h_2(x)^2\right)\in L^1(\Omega)$$
a.e. $x\in \Omega$ and for any $j\in \NN$\,.\par
 Hence, by applying the
Lebesgue Dominated Convergence Theorem in $L^1(\Omega)$, we have
that
$$\int_\Omega F(u_{j}(x))\,dx \to \int_\Omega F(u(x))\,dx$$
as $j\to \infty$, that is
the map  $$u\mapsto\int_{\Omega}F(u(x))dx$$
is continuous from $X_0$ with the weak topology to $\RR$. Thus, the functional~$\Psi$ is continuous from $X_0$ with the weak topology to $\RR$.\par
 On the other hand, the map
$$u\mapsto \left\|u\right\|^{2}_{X_0}$$ is lower
semicontinuous in the weak topology of $X_{0}$\,, so that the
functional $\Phi$ is lower semicontinuous in the weak
topology of $X_0$\,.
\end{proof}
\begin{lemma}\label{cps}
Let $f:\RR \to \RR$ be a continuous function verifying condition \eqref{condinfinito}. Then,
 for every $\lambda\in \RR$, the functional $\mathcal J_{\lambda}$ is coercive and satisfies the compactness $(\rm PS)$ condition.
\end{lemma}
\begin{proof}
\noindent Let us fix $\lambda\in \RR$. By \eqref{condinfinito}, there exists a positive $\delta_{\lambda}$ such that
$$
|f(t)|\leq \frac{|t|}{2c_2^2(1+|\lambda|)},
$$
\noindent for every $|t|\geq \delta_{\lambda}$.\par
 Hence, we get
$$
|F(t)|\leq \frac{t^2}{2c_2^2(1+|\lambda|)}+\max_{|t|\leq \delta_{\lambda}}|f(t)||t|,
$$
\noindent for every $t\in\RR$.\par
\indent Thus
\begin{eqnarray}\label{e3.2a}
\mathcal J_{\lambda}(u) &\geq& \Phi(u)-|\lambda| \left|\int_\Omega F(u(x))dx\right|\nonumber\\
&\geq& \frac{1}{2}\|u\|^2_{X_0}-|\lambda|\int_\Omega |F(u(x))|dx\nonumber\\
               &\geq& \frac{1}{2(1+|\lambda|)}\|u\|^2_{X_0}-c_1|\lambda| \max_{|t|\leq \delta_\lambda}|f(t)|\|u\|_{X_0},\nonumber
\end{eqnarray}
for every $u\in X_0$.\par
\indent Then the functional $\mathcal J_{\lambda}$ is bounded from below and
$\mathcal J_{\lambda}(u)\rightarrow +\infty$ whenever $\|u\|_{X_0}\rightarrow +\infty$. Hence $\mathcal J_{\lambda}$ is coercive.

Now, let us prove that $\mathcal J_{\lambda}$ satisfies the condition $\textrm{(PS)}_{\mu}$ for $\mu\in\RR$. For our goal, let $\{u_{j}\}_{j\in \NN}\subset X$ be a Palais-Smale
sequence, that is,
$$
\mathcal J_{\lambda}(u_j)\to \mu, \qquad{\rm and}\qquad \|\mathcal J'_{\lambda}(u_j)\|_{X^*_0}\to0,
$$
as $j\rightarrow +\infty$.\par
\indent Taking into account the coercivity of $\mathcal J_{\lambda}$, the sequence $\{u_{j}\}_{j\in \NN}$ is necessarily bounded in $X_0$. Since $X_0$ is reflexive, we may extract a subsequence,
which for simplicity we call again $\{u_{j}\}_{j\in \NN}$, such that
$u_{j}\rightharpoonup u$ in $X_0$.
This means that
\begin{equation}\label{convergenze0}
\begin{aligned}
 & \int_{\RR^n\times\RR^n}
\frac{\big( u_j(x)-u_j(y)\big) \big(\varphi(x)-\varphi(y)\big)}{|x-y|^{n+2s}}\,dxdy \to \\
& \qquad \qquad \qquad
\int_{\RR^n\times\RR^n}
\frac{\big( u(x)-u(y)\big) \big(\varphi(x)-\varphi(y)\big)}{|x-y|^{n+2s}}\,dxdy ,
\end{aligned}
\end{equation}
for any $\varphi\in X_0$, as $j\to \infty$.\par
We will prove that $\{u_{j}\}_{j\in \NN}$ strongly converges to $u\in X_0$. Exploiting the derivative $\mathcal J_{\lambda}(u_j)(u_j-u)$, we obtain
\begin{eqnarray}\label{jj}
\langle \Phi'(u_{j}),u_{j}-u\rangle &=& \langle \mathcal J_{\lambda}'(u_j),u_j-u\rangle\\
               &+ & \lambda\int_{\Omega}f(u_j(x))(u_j-u)(x)dx,\nonumber
\end{eqnarray}
where
\begin{eqnarray*}
\langle \Phi'(u_{j}),u_{j}-u\rangle  &=& \int_{\RR^n\times\RR^n}\frac{(u_j(x)-u_j(y))^2}{|x-y|^{n+2s}}dxdy\\\nonumber
               &-& \int_{\RR^n\times\RR^n}
\frac{\big( u_j(x)-u_j(y)\big) \big(u(x)-u(y)\big)}{|x-y|^{n+2s}}\,dxdy\nonumber
\end{eqnarray*}
\indent Since $\|\mathcal J_{\lambda}'(u_n)\|_{X^*_0}\to0$ and the sequence $\{u_j-u\}_{j\in \NN}$ is bounded in
$X$, taking into account that $|\langle
\mathcal J_{\lambda}'(u_j),u_j-u\rangle|\leq\|\mathcal J_{\lambda}'(u_j)\|_{X^*_0}\|u_j-u\|_{X_0}$, one has
\begin{eqnarray}\label{j2}
\langle \mathcal J_{\lambda}'(u_j),u_j-u\rangle\to0.
\end{eqnarray}
 \indent Further, by the growth condition \eqref{crescita}, one has

 \begin{eqnarray*}
\int_{\Omega}|f(u_j(x))||u_j(x)-u(x)|dx &\leq & c\int_{\Omega}|u_j(x)-u(x)|dx
+ c\int_{\Omega}|u_j(x)||u_j(x)-u(x)|dx\\
               &\leq& c((\meas(\Omega))^{1/2}+\|u_j\|_{L^{2}(\Omega)}) \|u_j-u\|_{L^2(\Omega)}.
\end{eqnarray*}

\indent Now, the embedding
$X_0\hookrightarrow L^2(\Omega)$ is compact, hence $u_j\to u$ strongly in $L^2(\Omega)$.
So we obtain
\indent \begin{eqnarray}\label{j3}
\int_{\Omega}|f(u_j(x))||u_j(x)-u(x)|dx\to0,
\end{eqnarray}
as $j\rightarrow \infty$.\par
\indent By \eqref{jj} relations \eqref{j2} and \eqref{j3} yield
\begin{eqnarray}\label{fin}
\langle \Phi'(u_{j}),u_{j}-u\rangle
\rightarrow 0,
\end{eqnarray}
as $j\rightarrow \infty$.\par
Hence by \eqref{fin} we can write
\begin{eqnarray}\label{fin2}
&& \int_{\RR^n\times\RR^n}\frac{(u_j(x)-u_j(y))^2}{|x-y|^{n+2s}}dxdy-\\\nonumber
               && \quad\quad\int_{\RR^n\times\RR^n}
\frac{\big( u_j(x)-u_j(y)\big) \big(u(x)-u(y)\big)}{|x-y|^{n+2s}}\,dxdy\rightarrow 0,\nonumber
\end{eqnarray}
as $j\rightarrow \infty$.\par
Thus, by \eqref{fin2} and \eqref{convergenze0} it follows that
\begin{eqnarray*}
&& \lim_{j\rightarrow \infty}\int_{\RR^n\times\RR^n}\frac{(u_j(x)-u_j(y))^2}{|x-y|^{n+2s}}\,dxdy\\\nonumber
               && \quad\quad\quad\quad\quad\quad=\int_{\RR^n\times\RR^n}\frac{\left(u(x)-u(y)\right)^{2}}{|x-y|^{n+2s}}\,dxdy.\nonumber
\end{eqnarray*}
\indent In conclusion, thanks to \cite[Proposition III.30]{brezis}, $u_j\rightarrow u$ in $X_0$. The proof is complete.
\end{proof}

\begin{lemma}\label{kappa}
Let $f:\RR \to \RR$ be a continuous function verifying condition \eqref{supf} and \eqref{condinfinito}. Then, the following property holds
$$
\lim_{\varrho\rightarrow 0^{+}}\displaystyle\frac{\displaystyle\sup_{u \in \Phi^{-1}(]-\infty,\varrho[)}\int_\Omega F(u(x))\,dx}{\varrho}=0.
$$
\end{lemma}
\begin{proof}
Due to \eqref{supf}, for an arbitrary small $\varepsilon>0$ there exists $\delta_\varepsilon>0$ such that
$$
|f(t)|<\frac{\varepsilon}{4c_2^2} |t|,
$$
for every $|t|<\delta_\varepsilon$.\par
 On the other hand, combining \eqref{crescita} with the above inequality, one has
\begin{eqnarray}\label{ora}
|F(t)|\leq \varepsilon \frac{t^2}{4c_2^2}+c(1+\delta_\varepsilon)\delta^{1-q}|t|^q,
\end{eqnarray}
\noindent for a fixed $q\in ]2,2^*[$ and for all $t\in\RR$.\par
Now, fix $\varrho>0$ and define the sets
 $$S_\varrho:=\Phi^{-1}(]-\infty,\varrho[),\,\,\,\,\,\,{\rm and}\,\,\,\,\,\,S_{2\varrho}:=\Phi^{-1}(]-\infty,2\varrho[).$$
 Of course $S_\varrho\subseteq S_{2\varrho}$. Moreover, by using \eqref{ora}, we have
\begin{eqnarray*}
 \int_\Omega F(u(x))\,dx&\leq&
\frac{\varepsilon}{2}\varrho+c(1+\delta_{\varepsilon})\delta_{\varepsilon}^{1-q}c_q^q2^{q/2}\varrho^{q/2}\\ &\equiv&
\frac{\varepsilon}{2}\varrho+\widehat{c}(\varepsilon)\varrho^{q/2},
\end{eqnarray*}

\indent where we set
$$
\widehat{c}(\varepsilon):=c(1+\delta_{\varepsilon})\delta_{\varepsilon}^{1-q}c_q^q2^{q/2}.
$$
\noindent
Thus, there exists $\varrho(\varepsilon)>0$ such that, for every $0<\varrho<\varrho(\varepsilon)$, we have
\begin{eqnarray*}
0\leq \displaystyle\frac{\displaystyle\sup_{u\in S_\varrho}\int_\Omega F(u(x))\,dx}{\varrho} &\leq&
\displaystyle\frac{\displaystyle\sup_{u\in S_{2\varrho}}\int_\Omega F(u(x))\,dx}{\varrho}\\ &\leq&
\frac{\varepsilon}{2}+\widehat{c}(\varepsilon)\varrho^{\frac{q-2}{2}}<\varepsilon,
\end{eqnarray*}
which completes the proof.
\end{proof}




\subsection{Proof of Theorem~\ref{generalkernel0f}}
The idea of the proof consists in applying Theorem \ref{abstract} to
the functional~$\mathcal J_{\lambda}$\,.\par
 To this purpose, we write the functional~$\mathcal J_{\lambda}$ as follows:
$$\mathcal J_{\lambda}(u)= \Phi(u)-\lambda \Psi(u),\,\,\,\,\,\,(\forall\, u \in  X_0)$$
\noindent where
$$
\Phi(u):=\frac 1 2 \int_{\RR^n\times\RR^n}\frac{(u(x)-u(y))^2}{|x-y|^{n+2s}}\,dxdy,
$$
while
$$\Psi(u):= \int_{\Omega} F(u(x))\,dx,$$
for every $u\in X_0$.\par
First of all, note that $X_0$ is a separable Hilbert space (see
\cite[Lemma~7]{svmountain}) and the
functionals~$\Phi$ and $\Psi$ are continuously G\^{a}teaux differentiable. Moreover, the functional
$\Phi$ is coercive in $X_0$,
$\displaystyle\inf_{u\in X_0}\Phi(u)=0$ and, by choosing $z_0\equiv 0$, of course
$$
\Phi(z_0)=\Psi(z_0)=0.
$$
\indent Now, by hypothesis \eqref{Segno}, there exists $t_0\in \RR$ such that $F(t_0)>0$. Further, let $\sigma_0\in]0,1[$ be such that
$$
F(t_0)\sigma^n_0-(1-\sigma^n_0)\displaystyle\max_{|t|\leq |t_0|}|F(t)|>0.
$$
\indent Hence, set
\[
u_{\sigma_0}^{t_0}(x):= \left\{
\begin{array}{ll}
0 & \mbox{ if $x \in \RR^n \setminus B(x_0,\tau)$} \\\\
\displaystyle\frac{t_0}{(1-\sigma_0)\tau} \left(\tau- |x-x_0| \right)
& \mbox{ if $x \in B(x_0,\tau) \setminus B(x_0,\sigma_0\tau)$} \\\\
t_0 & \mbox{ if $x \in B(x_0,\sigma_0{\tau}),$}
\end{array}
\right.
\]
where $\tau$ is as in \eqref{tiau}.\par
 One has
$$
\Psi(u_{\sigma_0}^{t_0})\geq \left[F(t_0)\sigma^n_0-(1-\sigma^n_0)\displaystyle\max_{|t|\leq |t_0|}|F(t)|\right]\omega_n\tau^n,
$$

\noindent where $\omega_n$ denotes the volume of the unit ball in $\RR^n$.\par

Indeed, since $$\|u_{\sigma_0}^{t_0}\|_\infty:=\displaystyle\max_{x\in \bar \Omega}|u_{\sigma_0}^{t_0}(x)|\leq |t_0|,$$ it follows that
\begin{equation}\label{e3.2a25551}
\int_{B(x_0,\tau) \setminus B(x_0,\sigma_0\tau)} F(u_{\sigma_0}^{t_0}(x))\,dx\geq -(1-\sigma^n_0)\displaystyle\max_{|t|\leq |t_0|}|F(t)|\tau^n\omega_n.
\end{equation}
\indent Consequently, relation \eqref{e3.2a25551} yields
\begin{eqnarray}\label{e3.2a2555}
 \Psi(u_{\sigma_0}^{t_0}) &:=& \int_{\Omega} F(u_{\sigma_0}^{t_0}(x))\,dx\nonumber\\
&=& \int_{B(x_0,\sigma_0\tau)} F(u_{\sigma_0}^{t_0}(x))\,dx+\int_{B(x_0,\tau) \setminus B(x_0,\sigma_0\tau)} F(u_{\sigma_0}^{t_0}(x))\,dx\nonumber\\
&\geq& F(t_0)\sigma^n_0\tau^n\omega_n+\int_{B(x_0,\tau) \setminus B(x_0,\sigma_0\tau)} F(u_{\sigma_0}^{t_0}(x))\,dx\nonumber\\
&\geq& \left[F(t_0)\sigma^n_0-(1-\sigma^n_0)\displaystyle\max_{|t|\leq |t_0|}|F(t)|\right]\omega_n\tau^n.\nonumber
\end{eqnarray}

Further, by Lemma \ref{Stime} we have
\begin{equation}\label{dx2}
\Phi(u_{\sigma_0}^{t_0})\leq \beta,
 \end{equation}
 where
 $$
 \beta:= \frac{t_0^2}{2(1-{\sigma_0})^2}{\displaystyle \frac{\pi^{\frac{n}{2}}\tau^{n-2}(1-\sigma_0^n)}{\displaystyle \Gamma\left(1+\frac{n}{2}\right)}\kappa_1\kappa_2}.
 $$
\indent Due to Lemma \ref{kappa}, we may take $\varrho_0>0$ such that the function $u_{\sigma_0}^{t_0}\in X_0$
verifies the following conditions:
\begin{equation}\label{Giov1}
u_{\sigma_0}^{t_0}\in \RR\setminus \bar S_{\varrho_0},
\end{equation}
and
\begin{equation}\label{Giov2}
\displaystyle\frac{\displaystyle\sup_{u \in S_{\varrho_0}}\Psi(u)}{\varrho_0}<\frac{\left[F(t_0)\sigma^n_0-(1-\sigma^n_0)\displaystyle\max_{|t|\leq |t_0|}|F(t)|\right]\omega_n\tau^n}{\beta}.
\end{equation}
\indent Relation \eqref{Giov1} clearly means
$$
\varrho_0<\Phi(u_{\sigma_0}^{t_0}).
$$
 Moreover, since \eqref{dx2} holds, it follows that
\begin{equation}\label{Giov3}
\frac{\left[F(t_0)\sigma^n_0-(1-\sigma^n_0)\displaystyle\max_{|t|\leq |t_0|}|F(t)|\right]\omega_n\tau^n}{\beta}\leq  \displaystyle\frac{\Psi(u_{\sigma_0}^{t_0})}{\Phi(u_{\sigma_0}^{t_0})}.
\end{equation}
\indent Hence, inequality \eqref{Giov2} together to \eqref{Giov3} give
\begin{equation}\label{au}
\displaystyle\sup_{\Phi(u)<\varrho_0}\Psi(u)<\varrho_0 \displaystyle\frac{\Psi(u_{\sigma_0}^{t_0})}{\Phi(u_{\sigma_0}^{t_0})}.
\end{equation}
By choosing $z_1\equiv u_\sigma^{t_0}$, hypotheses $(\rm i)$ and $(\rm ii)$ of Theorem \ref{abstract} are verified.\par
Set
$$
\bar a:=\frac{1+\varrho_0}{\displaystyle\frac{\Psi(u_{\sigma_0}^{t_0})}{\Phi(u_{\sigma_0}^{t_0})}-\frac{\displaystyle\sup_{\Phi(u)<\varrho_0}\Psi(u)}{\varrho_0}}.
$$
Note that, in such a case $\zeta:=1+\varrho_0$ and, by \eqref{au}, $\bar a$ is strictly positive.\par
Bearing in mind Lemma \ref{regularity} the functional $\mathcal J_{\lambda}$ is sequentially lower continuous in the weak topology on $X_0$. Further,
 Lemma \ref{cps} guarantees the coercivity property.

 Hence, all the assumptions of Theorem \ref{abstract} are fulfilled. Thus there is an open interval $\Lambda \subseteq [0, \bar a]$ and a number $\kappa>0$ such that for each
$\lambda\in \Lambda$, the equation $J'_\lambda(u)=0$ admits at least three solutions in $X_0$ having
norm less than $\kappa$.\par
Since one of them may be the trivial one (note that by \eqref{op} one has $f(0)=0$), we still have at least two distinct, nontrivial weak solutions of the problem \eqref{op}.
\subsection{Final comments}\label{subsec:comments}

In spite of the fact that hypotheses of Theorem \ref{generalkernel0f} are verified, the
conclusions do not hold in general for every parameter $\lambda>0$, as we mentioned in
the first section. For instance, let $f$ be a Lipschitz continuous function of constant $L$ and
take
\begin{equation}\label{au344567}
\displaystyle0\leq\lambda<\frac{1}{c_2^2L}.
\end{equation}
\indent Further, let us assume that there exists a weak solution $u_0\in X_0\setminus\{0\}$ of the problem \eqref{op}, that is,
\begin{equation}\label{problemaK0f45}
\left\{\begin{array}{l} {\displaystyle \int_{\RR^n\times\RR^n }
\frac{(u_0(x)-u_0(y))(\varphi(x)-\varphi(y))}{|x-y|^{n+2s}}\, dxdy}\\
\\
\qquad \qquad \qquad \qquad {\displaystyle =\lambda \int_\Omega f(u_0(x))\varphi(x)dx},\,\,\,\,\,\,\, \forall\,\, \varphi\in X_0.
\end{array}\right.
\end{equation}
\indent In particular, testing \eqref{problemaK0f45} with $\varphi:=u_0$, we have
\begin{equation}\label{au34}
\|u_0\|^2_{X_0}=\lambda\int_\Omega f(u_0(x))u_0(x)\,dx.
\end{equation}
On the other hand, bearing in mind that $f$ is Lipschitz continuous and $f(0)=0$, by using \eqref{au344567}, it follows that

\begin{eqnarray}\label{au345}
 \int_\Omega f(u_0(x))u_0(x)\,dx&\leq& \int_\Omega |f(u_0(x))||u_0(x)|\,dx\nonumber\\
 &=& \int_\Omega |f(u_0(x))-f(0)||u_0(x)|\,dx\\
 &\leq& L\|u_0\|_{L^2(\Omega)}^2\leq c_2^2L\|u_0\|_{X_0}^2.\nonumber
\end{eqnarray}
By \eqref{au34} and \eqref{au345} we get
\begin{equation*}
\|u_0\|^2_{X_0}\leq \lambda c_2^2 L \|u_0\|^2_{X_0}<\|u_0\|^2_{X_0},
\end{equation*}
which is a contradiction.\par
\indent More generally, according to hypotheses \eqref{supf}-\eqref{Segno} and defining the positive number
\begin{equation}\label{cf}
    c_f:=\max_{s\neq 0}\frac{|f(s)|}{|s|},
\end{equation}
 a non-existence result for our problem can be proved
whenever the parameter $\lambda$ is small enough.\par
 Namely, arguing substantially as before, and by \cite[Proposition~9]{svlinking} we have

\begin{proposition}\label{fotetel-1} Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous
function which satisfies \eqref{supf}--\eqref{Segno} and denote by $\lambda_{1,s}$ be the first positive eigenvalue of the linear problem
\begin{equation}\label{linearp}
\left\{
\begin{array}{ll}
(-\Delta)^s u=\lambda u &  \mbox{\rm in } \Omega\\
u=0 & \mbox{\rm in } \RR^n\setminus \Omega.
\end{array}\right.
\end{equation}
Then for every parameter
$$
0\leq \lambda<\frac{\lambda_{1,s}}{c_f},
$$
where $c_f$ is the constant defined in \eqref{cf},
problem~\eqref{op} admits only the trivial solution.
\end{proposition}
 \indent Finally, a natural question arises about the interval $\Lambda$ obtained in Theorem \ref{generalkernel0f}: \textit{can we estimate it}?\par

 To give a positive answer to this question, preserving the above notations, one has
$$
\frac{1+\varrho_0}{\displaystyle\frac{\alpha}{\beta}-\frac{\displaystyle\sup_{\Phi(u)<\varrho_0}\Psi(u)}{\varrho_0}}<\frac{2}{\displaystyle\frac{\alpha}{\beta}-\frac{\displaystyle\sup_{\Phi(u)<\varrho_0}\Psi(u)}{\varrho_0}},
$$
for every $\varrho_0\in ]0,1[$ and
where we set
$$
\alpha:=\left(F(t_0)\sigma^n_0-(1-\sigma^n_0)\displaystyle\max_{|t|\leq |t_0|}|F(t)|\right)\omega_n\tau^n.
$$
\indent Further, based on Lemma \ref{kappa}, one can take $\varrho_0\in ]0,1[$ and such that
$$
\frac{1}{\displaystyle\frac{\alpha}{\beta}-\frac{\displaystyle\sup_{\Phi(u)<\varrho_0}\Psi(u)}{\varrho_0}}<\frac{2\beta}{\alpha}.
$$
Thus
\begin{eqnarray*}
\bar a:=\frac{1+\varrho_0}{\displaystyle\frac{\Psi(u_{\sigma_0}^{t_0})}{\Phi(u_{\sigma_0}^{t_0})}-\frac{\displaystyle\sup_{\Phi(u)<\varrho_0}\Psi(u)}{\varrho_0}}&\leq&
\frac{1+\varrho_0}{\displaystyle\frac{\alpha}{\beta}-\frac{\displaystyle\sup_{\Phi(u)<\varrho_0}\Psi(u)}{\varrho_0}}\\ &<&
4\frac{\beta}{\alpha}.
\end{eqnarray*}
\indent In conclusion, thanks to the above computations and bearing in mind Proposition \ref{fotetel-1}, we have the following concrete localization
$$
\Lambda \subset\left[\frac{\lambda_{1,s}}{c_f}, \bar a\right]\subset\left[\frac{\lambda_{1,s}}{c_f},\frac{2}{(1-\sigma_0)^2}\frac{(1-\sigma_0^n)\kappa_1\kappa_2}{\left(F(t_0)\sigma^n_0-(1-\sigma^n_0)\displaystyle\max_{|t|\leq |t_0|}|F(t)|\right)}\frac{t_0^2}{\tau^2}\right],
$$
as desired.
\smallskip

\begin{remark}\rm{
We point out that, encoding the exact normalization constant
$$C(n,s):=\left(\int_{\RR^n}\frac{1-\cos x_1}{|x|^{n+2s}}\, dx\right)^{-1},$$ defining the fractional Laplacian operator
$$(-\Delta)^s u(x)=
-\frac{C(n,s)}{2}\int_{\mathbb{R}^{n}}\frac{u(x+y)+u(x-y)-2u(x)}{|y|^{n+2s}}\,dy, \quad\, (\forall\, x\in \RR^n)$$
our approach ensures the existence of an open interval $$\Lambda\subset \left[\frac{\lambda_{1,s}}{c_f},\frac{(1-\sigma_0^n)\nu^{(n)}}{(1-\sigma_0)^2\left(F(t_0)\sigma^n_0-(1-\sigma^n_0)\displaystyle\max_{|t|\leq |t_0|}|F(t)|\right)}\frac{t_0^2}{\tau^2}\right],$$ and a real positive number $\kappa$ such that
for every $\lambda\in \Lambda$ problem~\eqref{op} has at least two distinct,
nontrivial weak solutions in $X_0$ whose $X_0$-norms are less than $\kappa$.
}
\end{remark}

\noindent {\bf Acknowledgements.} The paper is realized with the auspices of the INdAM - GNAMPA Project 2014 titled: {\it Propriet\`{a} geometriche ed analitiche per problemi non--locali}.
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\end{document}