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Fractional Laplacian, Nonlinear Analysis
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\title[Infinitely many solutions for fractional Laplace equations]{
Infinitely many solutions for fractional Laplace equations with subcritical nonlinearity}

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


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\author[Raffaella 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}

\dedicatory{Dedicated to Patrizia Pucci\\ on the occasion of her sixtieth birthday,\\ with all my affection, esteem and gratitude.}

\keywords{Fractional Laplacian, integrodifferential
operators, variational and topological techniques, Palais--Smale condition, subcritical nonlinearities.\\
\phantom{aa} 2010 AMS Subject Classification: Primary:
49J35, 35A15, 35S15;
Secondary: 47G20, 45G05.}

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


\begin{abstract}
In this paper we discuss the existence of infinitely many solutions for a nonlocal, nonlinear equation with homogeneous Dirichlet boundary data.
Our model problem is the following one
$$ \left\{
\begin{array}{ll}
(-\Delta)^s u-\lambda u=|u|^{q-2}u+h & {\mbox{ in }} \Omega\\
u=0 & {\mbox{ in }} \RR^n\setminus \Omega\,,
\end{array} \right.
$$
where $s\in (0,1)$ is a fixed parameter, $(-\Delta)^s$ is the fractional Laplace operator,
which (up to normalization factors) may be defined as
$$-(-\Delta)^s u(x)=
\int_{\RR^n}\frac{u(x+y)+u(x-y)-2u(x)}{|y|^{n+2s}}\,dy\,,
\,\,\,\,\, x\in \RR^n\,,$$
while $\lambda$ is a real parameter, the exponent~$q\in (2, 2^*)$, with $2^*=2n/(n-2s)$, $n>2s$, the function~$h$ belongs to the space $L^2(\Omega)$ and, finally,
the set $\Omega$ is an open, bounded subset of $\RR^n$ with
Lipschitz boundary.  Here the solution is sought to satisfy $u = 0$ on $\RR^n\setminus \Omega$ and not simply on $\partial \Omega$,
consistently with the non-local character of the fractional Laplace operator.

Adapting the classical variational techniques used in order to study the standard Laplace equation with subcritical growth nonlinearities to the nonlocal framework, along the present paper we prove that this problem admits infinitely many weak solutions~$u_k$, with the property that their Sobolev norm goes to infinity as $k\to +\infty$\,, provided the exponent $q<2^*-2s/(n-2s)$\,.
In this sense, the results presented here may be seen as the extension
of some classical nonlinear analysis theorems to the case of fractional
operators.
\end{abstract}

\maketitle

\tableofcontents

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

The starting point of the present paper is represented by the following standard Laplace equation
\begin{equation}\label{1} \left\{
\begin{array}{ll}
-\Delta u-\lambda u=|u|^{q-2}u+h & {\mbox{ in }} \Omega\\
u=0 & {\mbox{ on }} \partial \Omega\,,
\end{array} \right.
\end{equation}
where $\lambda$ is a real parameter, the exponent $q\in (2,2_*)$, $2_*$ is the classical critical Sobolev exponent given by $2_*=2n/(n-2)$, $n>2$, $h\in L^2(\Omega)$ and $\Omega$ is an open bounded subset of $\RR^n$ with smooth boundary.

This problem was widely studied and in the literature there are many existence, non-existence and multiplicity results about it (see, for instance, \cite{kavian, struwe, willem} and the references therein).

Since, recently, a great attention has been focused on the study of fractional Laplacian equations, the aim of the present paper is to consider the nonlocal counterpart of equation~\eqref{1}, namely the following problem
\begin{equation}\label{1.11bis}
\left\{
\begin{array}{ll}
(-\Delta)^s u-\lambda u=|u|^{q-2}u+h & {\mbox{ in }} \Omega\\
u=0 & {\mbox{ in }} \RR^n\setminus \Omega\,.
\end{array} \right.
\end{equation}
Here, the parameter~$s\in (0,1)$ is fixed
and $(-\Delta)^s$ is the fractional Laplace operator,
which (up to normalization factors) may be defined as
\begin{equation} \label{2}
-(-\Delta)^s u(x)=
\int_{\RR^n}\frac{u(x+y)+u(x-y)-2u(x)}{|y|^{n+2s}}\,dy\,,
\,\,\,\,\, x\in \RR^n\end{equation}
(see \cite{valpal} and references therein for further details on the
fractional Laplacian), while $q\in (2, 2^*)$, $2^*=2n/(n-2s)$ is the fractional critical Sobolev exponent, $n>2s$, $h\in L^2(\Omega)$ and $\Omega$ is an open, bounded subset of $\RR^n$ with
Lipschitz boundary.  Here, the standard Dirichlet condition $u=0$ in $\partial \Omega$ is replaced with the condition that the function~$u$ vanishes outside $\Omega$\,,
consistently with the non-local character of the operator~$(-\Delta)^s$.

In \cite{svmountain, svlinking} the case $h\equiv 0$ in \eqref{1.11bis} was considered and the existence of a nontrivial weak solution was obtained using classical minimax theorems, namely the Mountain Pass and the Linking Theorems (see \cite{ar, rabinowitz}), according to different values of the parameter $\lambda$\,.

A natural question is whether or not classical existence results for equation~\eqref{1} still hold in the nonlocal framework of \eqref{1.11bis}. The aim of this paper is to answer this question, with respect to the existence of infinitely many solutions for the subcritical equation~\eqref{1.11bis}.

In \cite{bahriber, bahrilions, struwe1} (see also \cite{kavian, struwe}) problem~\eqref{1} was studied using different methods and the existence of infinitely many weak solutions for \eqref{1} (with the property that the $L^2$--norm of their gradient goes to infinity) was proved.
Along this paper we will show that this result still holds true in the nonlocal setting.

Precisely, our existence result with respect to problem~\eqref{1.11bis} can be stated as follows:
\begin{theorem}\label{lapfra0}
Let $s\in (0,1)$, $n>2s$ and $\Omega$ be an open bounded subset of $\RR^n$
with Lipschitz boundary. Assume also that $h\in L^2(\Omega)$ and $q\in (2, 2^*-2s/(n-2s))$\,, with $2^*=2n/(n-2s)$\,.

Then, for any $\lambda\in \RR$ problem~\eqref{1.11bis} admits infinitely many
weak solutions $u_k\in H^s(\RR^n)$ such
that $u_k=0$ a.e. in $\RR^n\setminus\Omega$ and
$$\int_{\RR^n\times \RR^n}\frac{|u_k(x)-u_k(y)|^2}{|x-y|^{n+2s}}\, dx\,dy\to +\infty$$
as $k\to +\infty$\,.
\end{theorem}

For a weak solution of problem~\eqref{1.11bis} we mean a function $u$ such that
\begin{equation}\label{lapfraw}
\left\{\begin{array}{l}
{\displaystyle \int_{\RR^n\times \RR^n }
\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
{\displaystyle = \int_\Omega |u(x)|^{q-2}u(x)\varphi(x)\,dx+\int_\Omega h(x)\varphi(x)\,dx}\\
\\
\qquad \qquad \qquad \qquad \forall\,\, \varphi \in H^s(\RR^n)\,\,\mbox{with}\,\,\, \varphi=0\,\,\, \mbox{a.e. in}\,\,\, \RR^n\setminus\Omega\\
\\
u\in H^s(\RR^n)\,\,\mbox{with}\,\,\, u=0\,\,\, \mbox{a.e. in}\,\,\, \RR^n\setminus\Omega\,.
\end{array}\right.
\end{equation}

Equation~\eqref{1.11bis} represents only a model for the general problem studied along this work. Indeed, in the present paper we consider the following equation
\begin{equation}\label{op}
\left\{
\begin{array}{ll}
\mathcal L_K u+\lambda u+f(x, u)+h(x)=0 &  \mbox{in } \Omega\\
u=0 & \mbox{in } \RR^n\setminus \Omega\,,
\end{array}\right.
\end{equation}
where the set $\Omega\subset \RR^n$, $n>2s$, is open, bounded and with Lipschitz boundary and $s\in (0,1)$ is fixed, while $\mathcal L_K$ is the non-local operator defined as follows:
\begin{equation}\label{lk}
\mathcal L_Ku(x)=
\int_{\RR^n}\Big(u(x+y)+u(x-y)-2u(x)\Big)K(y)\,dy\,,
\,\,\,\,\, x\in \RR^n\,.
\end{equation}
Here the kernel $K:\RR^n\setminus\{0\}\rightarrow(0,+\infty)$ is a function 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 typical model for $K$ is given by $K(x)=|x|^{-(n+2s)}$. In this case $\mathcal L_K$ is the fractional Laplace operator $-(-\Delta)^s$
defined in \eqref{2}.

Moreover, the nonlinear term in equation~\eqref{op} is a function
$f:\overline\Omega\times \RR\to \RR$ verifying the following conditions:
\begin{equation}\label{fcontinua}
f\,\,\mbox{is a continuous function in}\,\, \overline\Omega \times \RR\,;
\end{equation}
\begin{equation}\label{fc1}
f(x, \cdot)\,\,\mbox{is a $C^1(\RR)$--continuous function for any}\,\, x\in\overline\Omega\,;
\end{equation}
\begin{equation}\label{crescita}
\begin{aligned}
&\mbox{there exist}\,\, a_1, a_2>0\,\,\mbox{and}\,\, q\in (2, 2^*), 2^*=2n/(n-2s)\,,\,\, \mbox{such that}\\
&\qquad \qquad |f(x,t)|\le a_1+a_2|t|^{q-1}\,\, \mbox{for any}\,\, x\in \overline\Omega,\, t\in \RR\,;
\end{aligned}
\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{condf'}
\begin{aligned}
&\mbox{there exist}\,\, \beta\in (0,1)\,\, \mbox{and}\,\,r>0\,\, \mbox{such that for any}\,\, x\in \overline\Omega,\, t\in \RR,\, |t|\geq r\\
& \qquad \qquad \qquad \qquad 0<\frac{f(x,t)}{t}\le \beta f_t(x,t)\,,
\end{aligned}
\end{equation}
where $f_t$ denotes the derivative of $f$ with respect to the second variable, i.e.
$$f_t(x,t)=\frac{\partial f}{\partial t}(x,t)\,.$$

Note that these assumptions are the standard ones, when dealing with partial differential equations driven by the Laplace operator (or, more generally, by uniformly elliptic
operators) with homogeneous Dirichlet boundary conditions. For this we refer to \cite[Theorem~6.1]{bahriber}, \cite[p.233]{kavian} and \cite[Chapter~II,~Theorem~7.2]{struwe} (see also \cite{bahrilions, struwe1})\,.

As a model for $f$ we can take the odd
nonlinearity~$f(x,t)=a(x)|t|^{q-2}t$, with $a\in
C(\overline\Omega)$, $a>0$ in $\overline\Omega$, and $q\in (2, 2^*)$\,.
Finally, the perturbation~$h:\Omega \to \RR$ is a function such that
\begin{equation}\label{g}
h\in L^2(\Omega)\,.
\end{equation}

In order to give the weak formulation of problem~\eqref{op}, we need to work in a special functional space. Indeed, one of the difficulty in treating problem~\eqref{op} is related to the encoding the Dirichlet boundary condition in the variational formulation. With this respect the standard fractional Sobolev spaces are not enough in order to study the problem. We overcome this difficulty working in a new functional space, whose definition will be recalled here below.

In the sequel 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
$$\mbox{the map}\,\,\,
(x,y)\mapsto (g(x)-g(y))\sqrt{K(x-y)}\,\,\, \mbox{is in}\,\,\,
L^2\big((\RR^n\times \RR^n) \setminus ({\mathcal C}\Omega\times
{\mathcal C}\Omega), dxdy\big)\,,$$
where ${\mathcal C}\Omega:=\RR^n \setminus\Omega$.

Moreover, $$X_0=\{g\in X : g=0\,\, \mbox{a.e. in}\,\,
\RR^n\setminus
\Omega\}\,.$$
Both $X$ and $X_0$ are non-empty, since they contain the space~$C^2_0 (\Omega)$ (see, e.g., \cite[Lemma~11]{sv}, for this we need condition~\eqref{kernel}).
The spaces $X$ and $X_0$ were introduced in \cite{sv} (see also \cite{sY, svmountain, svlinking, servadeivaldinociBN} for the definitions and the properties of these spaces).

The weak formulation of~\eqref{op} is given by the following problem:
\begin{equation}\label{problemaK0}
\left\{\begin{array}{l}
u\in X_0,\\
\\
{\displaystyle \int_{\RR^n\times\RR^n }
(u(x)-u(y))(\varphi(x)-\varphi(y))K(x-y) dx\,dy}\\
\\
\,\,\, {\displaystyle -\lambda \int_\Omega
u(x)
\varphi(x)\,dx= \int_\Omega f(x,u(x))\varphi(x)dx+\int_\Omega h(x)\varphi(x)\,dx}\\
\\
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad  \forall\,\, \varphi \in X_0\,.
\end{array}\right.
\end{equation}
In order to write problem~\eqref{problemaK0} we need to assume \eqref{evenkernel}\,.

The main result of the present paper is given by the following one:

\begin{theorem}\label{generalkernel}
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}, let $f:\overline\Omega\times \RR\to \RR$ verify \eqref{fcontinua}--\eqref{condf'} with $q\in (2, 2^*-2s/(n-2s))$\,, $2^*=2n/(n-2s)$, and let $h:\Omega\to \RR$ satisfy \eqref{g}\,.

Then, for any $\lambda \in \RR$ problem~\eqref{op} admits infinitely many
weak solutions $u_k\in X_0$, such that
$$\int_{\RR^n\times \RR^n}|u_k(x)-u_k(y)|^2K(x-y)\, dx\,dy\to +\infty$$
as $k\to +\infty$\,.
\end{theorem}

Of course, when $K$ and $f$ are exactly as in the model, that is $K(x)=|x|^{-(n+2s)}$ and $f(x,t)=|t|^{q-2}t$\,, $t\in (2, 2^*)$\,, then Theorem~\ref{generalkernel} reduces to Theorem~\ref{lapfra0}\,.

We would like to note that Theorem~\ref{lapfra0} establishes the existence of infinitely many weak solutions for problem~\eqref{op}, provided $q<2^*-2s/(n-2s)$\,.
When $s=1$, equation \eqref{1.11bis}
reduces to the standard semilinear Laplace partial differential equation~\eqref{1}. In this framework the bound from above on the exponent $q$ becomes $q<2(n-1)/(n-2)$, which is the usual one, when dealing with the Laplace equation (see \cite[ Chapter~5, Theorem~4.6 and Remark~4.7]{kavian}).
For all these reasons Theorems~\ref{lapfra0} and \ref{generalkernel} may be seen as the fractional version
of the classical existence results given in \cite[ Chapter~5, Theorem~4.6]{kavian} in the model case $f(x,t)=|t|^{q-2}t$ and in \cite[Theorem~6.1]{bahriber} for more general nonlinearities
(see also~\cite[Chapter~II,~Theorem~7.2]{struwe} and \cite{bahrilions, struwe1}). Also, when $h\equiv 0$ (i.e. when there is no perturbation) Theorem~\ref{lapfra0} represents the nonlocal counterpart of the classical results obtained, for instance, in \cite{ar} (see also \cite{struwe} and references therein)\,.

The classical results stated in \cite[ Chapter~5, Theorem~4.6]{kavian} and in \cite[Theorem~6.1]{bahriber} are proved by means of variational and topological methods. In the present paper we will adapt these techniques to the nonlocal framework
in order to get Theorem~\ref{generalkernel}.
The main difficulty, as in the standard case of the Laplacian, will be to prove the Palais--Smale condition for the energy functional associated with the problem.

The paper is organized as follows. In Section~\ref{sec:notations}
we will give some notations, while Section~\ref{sec:plan} we will be devoted to the strategy used along the paper in order to get the main results. In Section~\ref{sec:pscondizione} we will show that the energy functional satisfies the Palais-Smale condition and in Section~\ref{sec:esistenza} we will prove the main results of the paper.

\section{Some notations}\label{sec:notations}
This section is devoted to the notations used along the present paper.
In the sequel we set $Q=(\RR^n\times\RR^n)\setminus (\mathcal C \Omega\times \mathcal C \Omega)$\,, with $\mathcal C \Omega=\RR^n\setminus \Omega$.

We recall that the spaces $X$ and $X_0$ are endowed, respectively, with the norms defined by
\begin{equation}\label{norma}
\|g\|_X =\|g\|_{L^2(\Omega)}+\Big(\int_Q |g(x)-g(y)|^2K(x-y)dx\,dy\Big)^{1/2}\,,
\end{equation}
and
\begin{equation}\label{normaX0}
\|g\|_{X_0}=\left(\int_Q|g(x)-g(y)|^2K(x-y)\,dx\,dy\right)^{1/2}\,.
\end{equation}

With the norm given in formula~\eqref{normaX0}, $X_0$ is a Hilbert space with scalar product defined as
\begin{equation}\label{prodottoscalare}
\langle u,v\rangle_{X_0}=\int_Q
\big( u(x)-u(y)\big) \big( v(x)-v(y)\big)\,K(x-y)\,dx\,dy\,.
\end{equation}
For this see \cite[Lemma~7]{svmountain}.

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 spaces $X$ and $X_0$ and for their properties we refer to \cite{sY, sv, svmountain, svlinking, servadeivaldinociBN}, while for the fractional Sobolev spaces we refer to~\cite{valpal} and to the references therein.

Finally, along the paper we denote by
$$\lambda_1<\lambda_2\leq \dots \leq \lambda_k \leq \dots$$
the divergent sequence of eigenvalues of~$-\mathcal L_{K}$  and by $e_k$ the $k$-th eigenfunction corresponding to the eigenvalue $\lambda_k$\,, namely
$$\left\{\begin{array}{ll}
-\mathcal L_K e_k=\lambda_k e_k & \mbox{in } \Omega
\\
e_k=0 & \mbox{in } \RR^{n}\setminus\Omega\,.
\end{array}
\right.$$


For the properties of the eigenvalues and the eigenfunctions of the non-local integrodifferential operator $-\mathcal L_K$ (and, in particular, of the fractional Laplacian~$(-\Delta)^s$) we refer to \cite[Proposition~9 and Appendix~A]{svlinking} (see also \cite{sY, servadeivaldinociBNLOW, servadeivaldinociREGO}).


\section{Plan for proving the main results}\label{sec:plan}
In this section, for reader's convenience, we illustrate our strategy in order to prove Theorem~\ref{generalkernel}\,.

First of all, we observe
that problem~\eqref{problemaK0} has a variational structure, indeed
it is the Euler-Lagrange equation of the functional $\mathcal
J_{K,\,\lambda}:X_0\to \RR$ defined as follows
$$\begin{aligned}
\mathcal J_{K,\,\lambda}(u) & =\frac 1 2 \int_{\RR^n\times\RR^n}|u(x)-u(y)|^2 K(x-y)\,dx\,dy-\frac \lambda 2 \int_\Omega |u(x)|^2\,dx
\\
& \qquad \qquad \qquad \qquad -\int_\Omega F(x, u(x))\,dx  -\int_\Omega h(x)u(x)\,dx\,.
\end{aligned}$$
Here the function $F$ is
the primitive of $f$ with respect to its second variable, that is
\begin{equation}\label{F}
{\displaystyle F(x,t)=\int_0^t f(x,\tau)d\tau}\,.
\end{equation}

Notice that the functional $\mathcal J_{K,\,\lambda}$ is well defined thanks
to assumptions \eqref{fcontinua} and \eqref{crescita}, to
\cite[Lemma~9-$a)$]{servadeivaldinociBN} and also thanks to condition~\eqref{g}\,. Moreover, $\mathcal J_{K,\,\lambda}$ is Fr\'echet
differentiable at $u\in X_0$ and for any $\varphi\in X_0$
$$\begin{aligned}
\langle \mathcal J_{K,\,\lambda}'(u), \varphi\rangle & = \int_{\RR^n\times \RR^n}
\big(u(x)-u(y)\big)\big(\varphi(x)-\varphi(y)\big)K(x-y)\,dx\,dy\\
& \quad
 -\lambda \int_\Omega u(x) \varphi(x)\,dx -\int_\Omega f(x, u(x))\varphi(x)\,dx-\int_\Omega h(x)\varphi(x)\,dx\,.
\end{aligned}$$

Hence, in order to prove Theorem~\ref{generalkernel}, we will look for critical points of the functional~$\mathcal J_{K,\,\lambda}$\,. To this purpose, let us introduce the functional $\mathcal I_{K,\,\lambda}:X_0\to \RR$ defined as follows
$$\mathcal I_{K,\,\lambda}(u)=\max_{\tau>0} \mathcal J_{K,\,\lambda}(\tau u)\,.$$

As we will show below, the critical points of functionals $\mathcal J_{K,\,\lambda}$ and $\mathcal I_{K,\,\lambda}$ are strictly related the ones to the others, indeed the following result holds true:

\begin{proposition}\label{prop:IJ}
Let $\lambda \in \RR$\,, $s\in (0,1)$, $n>2s$ 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}, let $f:\overline\Omega\times \RR\to \RR$ verify \eqref{fcontinua}--\eqref{condf'} and let $h:\Omega\to \RR$ satisfy \eqref{g}\,.

Then, there exists a positive constant~$\alpha$ such that
\begin{itemize}
\item[$a)$] the functional $\mathcal I_{K,\,\lambda}$ is of class~$C^1$ in $\{\mathcal I_{K,\,\lambda}>\alpha\}$\,;
\item[$b)$] for any $v\in \{\mathcal I_{K,\,\lambda}>\alpha\}$ there exists a unique $\sigma(v)>0$ such that $$\mathcal I_{K,\,\lambda}(v)=\mathcal J_{K,\,\lambda}(\sigma(v)v)\,.$$
As a consequence, for any $v\in \{\mathcal I_{K,\,\lambda}>\alpha\}$
$$\mathcal I'_{K,\,\lambda}(v)=\sigma(v)\mathcal J'_{K,\,\lambda}(\sigma(v)v)\,.$$
\end{itemize}
\end{proposition}
\begin{proof}
Let $v\in X_0$\,. First of all, note that, by definition of~$\mathcal I_{K,\,\lambda}$ we have
$$\mathcal I_{K,\,\lambda}(tv)=\mathcal I_{K,\,\lambda}(v)\,\,\, \mbox{for any}\,\,\, t>0\,.$$
Hence, without loss of generality, we can assume that $\|v\|_{L^q(\Omega)}=1$\,.

From now on, thanks in particular to assumptions~\eqref{fc1} and \eqref{condf'}, we can argue exactly as in \cite[Propositions~6.2 and 6.3]{bahriber}, where the classical case of the Laplacian (in general of second order elliptic operators) was considered: we have just to replace the $L^2$--norm of the gradient with the $X_0$--norm.
\end{proof}

As a consequence of Proposition~\ref{prop:IJ} we have the next result, which relates the critical points of $\mathcal I_{K,\,\lambda}$ to the ones of $\mathcal J_{K,\,\lambda}$\,.
In the following $M$ will denote the set
$$M:=\big\{u\in X_0 : \|u\|_{L^q(\Omega)}=1\big\}\,,$$
where $q\in (2, 2^*)$\,. Note that $M$ is well defined, since the space $X_0$ is compactly embedded into $L^q(\Omega)$, by \cite[Lemma~9]{servadeivaldinociBN}.

\begin{corollary}\label{cor:IJ}
Let all the assumptions of Proposition~\ref{prop:IJ} be satisfied and let $\alpha$ and $\sigma(v)$ be as in Proposition~\ref{prop:IJ}\,. Then, the following assertions hold true:
\begin{itemize}
\item[$a)$] if $v\in \{\mathcal I_{K,\,\lambda}>\alpha\}$ is a critical point of $\mathcal I_{K,\,\lambda}$\,, then $\sigma(v)v$ is a critical point of $\mathcal J_{K,\,\lambda}$\,;
\item[$b)$] if $v\in \{\mathcal I_{K,\,\lambda}>\alpha\}$ is a critical point of $\mathcal I_{K,\,\lambda}$ on $M$, then $v$ is a critical point of $\mathcal I_{K,\,\lambda}$ on $X_0$\,.
\end{itemize}
\end{corollary}
\begin{proof}
Assertion~$a)$ is a consequence of the fact that $\sigma(v)>0$ and of Proposition~\ref{prop:IJ}-$b)$\,.
For part~$b)$ note that if $v\in \{\mathcal I_{K,\,\lambda}>\alpha\}$ is a critical point of $\mathcal I_{K,\,\lambda}$ on $M$, then there exists $\varrho\in \RR$ such that
\begin{equation}\label{ro}
\langle \mathcal I'_{K,\,\lambda}(v), \varphi\rangle=\varrho\int_\Omega |v(x)|^{q-2}v(x)\varphi(x)\,dx
\end{equation}
for any $\varphi\in X_0$\,.
In particular, taking $\varphi=v$, since $v\in M$ we get
$$\varrho=\langle \mathcal I'_{K,\,\lambda}(v), v\rangle\,.$$
Hence, by Proposition~\ref{prop:IJ}-$b)$ and by part~$a)$ we have
$$\varrho=\langle \mathcal I'_{K,\,\lambda}(v), v\rangle=\sigma(v)\langle \mathcal J'_{K,\,\lambda}(\sigma(v)v), v\rangle=0\,.$$
As a consequence of this and of \eqref{ro} we deduce that $\langle \mathcal I'_{K,\,\lambda}(v), \varphi\rangle=0$ for any $\varphi\in X_0$, which proves assertion~$b)$\,.
\end{proof}

\begin{remark}
{\rm We would like to note that in the model case $f(x,t)=|t|^{q-2}t$, $q\in (2, 2^*)$, in the assertions of Proposition~\ref{prop:IJ} and of Corollary~\ref{cor:IJ} the open set $\{\mathcal I_{K,\,\lambda}>\alpha\}$ can be replaced with $\{\mathcal I_{K,\,\lambda}>0\}$\,.}
\end{remark}

Thanks to Corollary~\ref{cor:IJ}, in order to find critical points of $\mathcal J_{K,\,\lambda}$ it is enough to look for critical points $u\in \{\mathcal I_{K,\,\lambda}>\alpha\}$ of the functional~$\mathcal I_{K,\,\lambda}$ on $M$\,.
Hence, in order to prove Theorem~\ref{generalkernel}, this will be our strategy, namely we will look for critical points $u\in \{\mathcal I_{K,\,\lambda}>\alpha\}$ of~$\mathcal I_{K,\,\lambda}$ on $M$\,.

To this purpose we will apply \cite[ Chapter~5, Theorem~2.2]{kavian} to the functional~$\mathcal I_{K,\,\lambda}$. For this we have to prove that
\begin{itemize}
\item[$i)$] the functional~$\mathcal I_{K,\,\lambda}$ satisfies the {\em Palais--Smale compactness condition} on $M$\,;
\item[$ii)$] for $k\in \NN$ large enough it hold true that $a_k\in \RR$ and $a_k<b_k$\,,
where $a_k$ and $b_k$ are defined as follows
\begin{equation}\label{ak}
a_k:=\inf_{A\in \mathcal A_k}\max_{v\in A}\mathcal I_{K,\,\lambda}(v)\,,
\end{equation}
\begin{equation}\label{bk}
b_k:=\inf_{B\in \mathcal B_k}\max_{v\in B}\mathcal I_{K,\,\lambda}(v)
\end{equation}
with $\mathcal A_k$ and $\mathcal B_k$ given by
$$\mathcal A_k:=\Big\{g(S^{k-1})\,\,\, \mbox{such that}\,\,\, g:S^{k-1}\to M\,\,\, \mbox{is continuous and odd}\Big\}\,,$$
$$\mathcal B_k:=\Big\{g(S^k_+)\,\,\, \mbox{such that}\,\,\, g:S^k_+\to M\,\,\, \mbox{is continuous and $g_{|S^k_+}$ is odd}\Big\}\,.$$
\end{itemize}
Here $S^k$ is the unit ball in $\RR^{k+1}$\,, while $S^k_+$ is the northern hemisphere of $S^k$ that is the set
$$S^k_+=\Big\{(u_1, \dots, u_{k+1})\in S^k\,\,\, \mbox{such that}\,\,\, u_{k+1}\geq 0\Big\}\,.$$

In order to prove the properties stated above with respect to the functional~$\mathcal I_{K,\,\lambda}$ we need some preliminary results.

\subsection{Some preliminary lemmas}\label{subsec:preliminary}
In this subsection we prove some auxiliary results on $\mathcal I_{K,\,\lambda}$ which will be useful in the sequel. Here assumptions~\eqref{fc1} and \eqref{condf'} will be crucial.

First we need some notations.
In what follows $\mathcal I^0_{K,\,\lambda}$ will be the even functional~$\mathcal I_{K,\,\lambda}$ obtained
in the particular case when $h\equiv 0$, that is
$$\mathcal I^0_{K,\,\lambda}(u)=\max_{\tau>0} \mathcal J^0_{K,\,\lambda}(\tau u)\,,$$
where
$$\begin{aligned}
\mathcal J^0_{K,\,\lambda}(u) & =\frac 1 2 \int_{\RR^n\times \RR^n}|u(x)-u(y)|^2 K(x-y)\,dx\,dy-\frac \lambda 2 \int_\Omega |u(x)|^2\,dx\\
& \qquad \qquad \qquad \qquad \qquad \qquad -\int_\Omega F(x, u(x))\,dx\,.
\end{aligned}$$

Moreover, in this case we also will denote by $\sigma_0(v)$ the element given in Proposition~\ref{prop:IJ}-$b)$, i.e. $\sigma_0(v)$ will be such that
$$\mathcal I^0_{K,\,\lambda}(v)=\mathcal J^0_{K,\,\lambda}(\sigma_0(v)v)\,.$$

\smallskip

According to these notations we have the following result:
\begin{lemma}\label{lemma1}
Let $\lambda\in \RR$\,, $s\in (0,1)$, $n>2s$ 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}, let $f:\overline\Omega\times \RR\to \RR$ verify \eqref{fcontinua}--\eqref{condf'} and let $h:\Omega\to \RR$ satisfy \eqref{g}\,. Finally, let $\alpha$ be as in Proposition~\ref{prop:IJ}\,.

Then, there exist two positive constants $\bar R$ and $\bar c$\,, depending only on $\alpha$, $\lambda$, $q$, $|\Omega|$ and $\|h\|_{L^2(\Omega)}$\,, such that for any $v\in M$ with $\|v\|_{X_0}\geq \bar R$\,, the following assertions hold true:
\begin{itemize}
\item[$a)$] $\mathcal I_{K,\,\lambda}(v)>\alpha$\,;
\item[$b)$] $|\mathcal I_{K,\,\lambda}(v)-\mathcal I^0_{K,\,\lambda}(v)|\leq \bar c\left(1+\mathcal I_{K,\,\lambda}(v)^{1/q}\right)$\,;
\item[$c)$] $|\mathcal I_{K,\,\lambda}(v)-\mathcal I^0_{K,\,\lambda}(v)|\leq \bar c\left(1+\mathcal I^0_{K,\,\lambda}(v)^{1/q}\right)$\,.
\end{itemize}
\end{lemma}
\begin{proof}
Let us start by proving assertion~$a)$\,. Thanks to the definition of~$\mathcal I_{K,\,\lambda}$, in order to get our goal it is enough to show that
\begin{equation}\label{alphaI}
\mathcal J_{K,\,\lambda}(v)>\alpha\,,
\end{equation}
when $v\in M$ is such that $\|v\|_{X_0}$ is large enough.

To this purpose, note that \eqref{crescita}, \eqref{g} and the H\"older inequality yield
$$\begin{aligned}
\mathcal J_{K,\,\lambda}(v)& = \frac 1 2\|v\|_{X_0}^2-\frac \lambda 2\,\|v\|_{L^2(\Omega)}^2-\int_\Omega F(x, u_j(x))\,dx-\int_\Omega h(x)v(x)\,dx\\
& \geq \frac 1 2\|v\|_{X_0}^2-\frac \lambda 2\,\|v\|_{L^2(\Omega)}^2-a_1\,\|v\|_{L^1(\Omega)}-\frac{ a_2}{q}\,\|v\|_{L^q(\Omega)}^q-\|h\|_{L^2(\Omega)}\,\|v\|_{L^2(\Omega)}\\
& \geq \frac 1 2\|v\|_{X_0}^2-\kappa
-a_1\,|\Omega|^{(q-1)/q}\,\|v\|_{L^q(\Omega)}-\frac{ a_2}{q}\,\|v\|_{L^q(\Omega)}^q\\
& = \frac 1 2\|v\|_{X_0}^2-\kappa
-a_1\,|\Omega|^{(q-1)/q}-\frac{ a_2}{q}\,,
\end{aligned}$$
when $v\in M$\,. Here $\kappa$ is a positive constant given by $$\kappa=\max\big\{\big(\lambda\,|\Omega|^{(q-2)/q}\big)/2+|\Omega|^{(q-2)/(2q)}\,\|h\|_{L^2(\Omega)}\,, |\Omega|^{(q-2)/(2q)}\,\|h\|_{L^2(\Omega)}\big\}\,.$$
Hence, if $\|v\|_{X_0}$ is sufficiently large, we get \eqref{alphaI} and this ends the proof of~$a)$\,.

Taking into account that part~$a)$ holds true, assertions~$b)$ and $c)$ can be proved as in \cite[Lemma~6.8]{bahriber}\,.
\end{proof}

Also, in the sequel we denote by $a_k^0$ and $b_k^0$ the quantity
\begin{equation}\label{ak0}
a_k^0:=\inf_{A\in \mathcal A_k}\max_{v\in A}\mathcal I^0_{K,\,\lambda}(v)\,,
\end{equation}
\begin{equation}\label{bk0}
b_k^0:=\inf_{B\in \mathcal B_k}\max_{v\in B}\mathcal I^0_{K,\,\lambda}(v)
\end{equation}
with $\mathcal A_k$ and $\mathcal B_k$ given as above, for any $k\in \NN$.

About the sequence $a_k$ defined in \eqref{ak} we have the following result:
\begin{lemma}\label{lemmaak}
Let $\lambda\in \RR$\,, $s\in (0,1)$, $n>2s$ 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}, let $f:\overline\Omega\times \RR\to \RR$ verify \eqref{fcontinua}--\eqref{condf'} and let $h:\Omega\to \RR$ satisfy \eqref{g}\,.

Then, there exist $\tilde c>0$ and $\tilde \kappa\in \NN$ such that for any $k\geq \tilde \kappa$
$$a_k\geq \tilde c\,k^{2sq/(n(q-2))}\,.$$

In particular, $a_k\to +\infty$ as $k\to +\infty$\,.
\end{lemma}
\begin{proof}
Here we can argue exactly as in \cite[Proposition~6.10]{bahriber}. To this purpose we can use also the properties of the eigenvalues of $(-\Delta)^s$ (see \cite{servadeivaldinociNOTE}).
\end{proof}

In the sequel we also need the following result, which is a sort of Gronwall Lemma in a discrete framework: its proof can be found in \cite[ Chapter~5, Lemma~4.5]{kavian}.

\begin{lemma}\label{lemmakavian}
Let $p>1$ and $\alpha_k$ be a sequence such that $\alpha_k>0$ for any $k\in \NN$\,.
Assume that there exist a positive constant~$c_o$ and $\kappa_o\in \NN$ such that
$$\alpha_{k+1}\leq \alpha_k+c_o(1+\alpha_k^{1/(p+1)})\,\,\, \mbox{for any}\,\,\, k\geq \kappa_o\,.$$
Then, there exist a positive constant~$c^*$ and $\kappa^*\in \NN$ such that
\begin{equation}\label{akdis1}
\alpha_k\leq c^*k^{(p+1)/p}\,\,\,\,\,\mbox{for any}\,\,\, k\geq \kappa^*\,.
\end{equation}
\end{lemma}

Now, we are ready to prove the properties stated above in $i)$ and $ii)$\,: this will be done in the forthcoming sections.

\section{Some compactness conditions}\label{sec:pscondizione}
In this section we prove that the functional~$\mathcal I_{K,\,\lambda}$ satisfies the property~$i)$, that is the Palais--Smale compactness condition on $M$\,. To this purpose, first of all we need to show that
the energy functional~$\mathcal J_{K,\,\lambda}$ associated with problem~\eqref{lk} satisfies the Palais--Smale
condition in $X_0$ at any level $c\in \RR$. This will be accomplished in the forthcoming
Subsection~\ref{subsec:PS}\,.

\subsection{The Palais--Smale condition for~$\mathcal J_{K,\,\lambda}$}\label{subsec:PS}
This subsection is devoted to the proof of the Palais--Smale condition for the functional~$\mathcal J_{K,\,\lambda}$\,.
The proof can be performed as in \cite[Propositions~13, 14 and 20]{svlinking}. Due to the presence of the extra term $\int_\Omega h(x)u(x)\,dx$ in the functional $\mathcal J_{K,\,\lambda}$, we prefer to repeat the calculation, for reader's convenience and also for the sake of clarity.

In order to perform our proof we need some estimates on the nonlinearity $f$ and its primitive $F$, given in the following result whose proof can be found in \cite[Lemma~3]{svmountain}:
\begin{lemma}\label{fF}
Assume $f:\overline\Omega\times \RR\to \RR$ is a function satisfying conditions~\eqref{fcontinua}, \eqref{crescita} and \eqref{cond0}. Then, for any $\varepsilon>0$ there exists $\delta=\delta(\varepsilon)$ such that
for any $x\in \overline\Omega$ and $t\in \RR$
$$|f(x,t)|\leq 2\varepsilon |t|+q\delta(\varepsilon) |t|^{q-1}\,$$
and so, as a consequence,
$$|F(x,t)|\leq \varepsilon\,|t|^2+\delta(\varepsilon)\, |t|^{q}\,,$$
where $F$ is defined as in \eqref{F}\,.
\end{lemma}

With the next result we show that the function~$F$ (i.e. the primitive of~$f$, see \eqref{F}) satisfies a superquadratic growth condition and, moreover, $F$ and $f$ verify a sort of Ambrosetti--Rabinowitz condition (see \cite{ar, rabinowitz}).

\begin{lemma}\label{lemmamu}
Assume $f:\overline\Omega\times \RR\to \RR$ is a function satisfying conditions~\eqref{fcontinua}, \eqref{fc1} and \eqref{condf'}. Let $F$ be as in \eqref{F} and $\mu=(\beta+1)/\beta>2$\,.
Then,
\begin{itemize}
\item[$a)$] there exist two positive constant $a_3$ and $a_4$ such that for any $x\in \overline \Omega$ and $t\in \RR$
$$|F(x,t)|\geq a_3|t|^\mu-a_4\,;$$
\item[$b)$] there exists a positive constant $a_5$ such that for any $x\in \overline \Omega$ and $t\in \RR$\,, $|t|\geq r$
$$0<\mu F(x,t)\leq tf(x,t)+a_5\,.$$
\end{itemize}
\end{lemma}
\begin{proof} First of all, let us prove assertion~$a)$\,. Let $r>0$ be as in \eqref{condf'}: then, for any $x\in \overline \Omega$ and $t\in \RR$ with $|t|\geq r>0$
$$\frac{f_t(x,t)}{f(x,t)}\geq \frac{1}{\beta t}\,.$$
Here we use also the fact that $tf(x,t)>0$ for $|t|\geq r$\,.

Now, suppose that $t>r$. Integrating both terms in $[r,t]$
we obtain
$$
|f(x,t)|=f(x,t)\geq \frac{f(x,r)}{r^{1/\beta}}\,t^{\/\beta}\,.
$$
With the same arguments and using again the fact that $tf(x,t)>0$ when $|t|\geq r$ it is easy to prove that if $x\in \overline \Omega$ and $t<-r$ then it
holds
$$
|f(x,t)|\geq \frac{|f(x,-r)|}{r^{1/\beta}}\,|t|^{1/\beta}\,,
$$
so that for any $x\in \overline \Omega$ and $t\in \RR$ with $|t|\geq r$ we get
\begin{equation}\label{Flog}
|f(x,t)|\geq \tilde m(x)\,|t|^{1/\beta}\,,
\end{equation}
where $\tilde m(x)=r^{-{1/\beta}}\min\{f(x,r),\, |f(x, -r)|\}$\,.

Note that $\tilde m$ is a continuous functions, since $x\mapsto f(x,
\cdot)$ does. Moreover, $\tilde m$ is positive, being $f(x,t)>0$
for any $x\in \overline \Omega$ and $t\in \RR$ such that $|t|\geq r$ (see
\eqref{condf'})\,. Hence, for any $x\in \overline \Omega$
$$\tilde m(x)\geq \min_{x\in \overline \Omega}\tilde m(x)=:\tilde m>0\,.$$
This and \eqref{Flog} give
\begin{equation}\label{Flog1}
|f(x,t)|\geq \tilde m\,|t|^{1/\beta}
\end{equation}
for any $x\in \overline\Omega$ and $t\in \RR$ such that $|t|\geq r$\,.

Now, let $t>r$. Integrating both terms in~\eqref{Flog1} in $[r,t]$ we get
\begin{equation}\label{Flog2}
F(x,t)\geq a_3\,|t|^{\mu}+F(x,r)-a_3 r^{\mu}\,,
\end{equation}
where $a_3$ is a positive constant and $\mu=(\beta+1)/\beta$\,. Note that $\mu>2$, since $\beta\in (0,1)$ by assumption.

Arguing in the same way it is easily seen that if $t<-r$ then
$$
F(x,t)\geq a_3\,|t|^{\mu}+F(x,-r)-a_3 r^{\mu}\,,
$$
so that, by this and \eqref{Flog2} we obtain that for any $x\in \overline \Omega$ and $t\in \RR$ with $|t|\geq r$
\begin{equation}\label{Flog4}
F(x,t)\geq a_3\,|t|^{\mu}+\hat m -a_3 r^{\mu}\geq a_3\,|t|^{\mu}-a_3 r^{\mu}\,,
\end{equation}
where $\hat m(x)=\min\{F(x,r),\, F(x, -r)\}>0$ (note that $F(x,r)>0$ since $tf(x,t)>0$ for any $x\in \overline \Omega$ and $t\in \RR$, with $|t|\geq r$).

Also, the function $F$ is continuous for any $x\in \overline \Omega$ and $t\in \RR$ such that $|t|\leq r$, so that, by
the Weierstrass Theorem, it is bounded, say
\begin{equation}\label{maxM1}
|F(x,t)|\leq \widetilde M \quad \mbox{in}\,\,\, \overline \Omega\times\{|t|\leq r\}\,,
\end{equation}
where $\widetilde M$ is a positive constant\,.

Taking $a_4:=\widetilde M+a_3r^{\mu}>0$
from \eqref{Flog4} and \eqref{maxM1} it follows that
\begin{equation}\label{fMm}
F(x,t)\geq a_3\,|t|^{\mu}-a_4
\end{equation}
for any $x\in \overline\Omega$ and $t\in \RR$\,. This concludes the proof of part~$a)$\,.

Now, let us show assertion~$b)$. For this note that, since $tf(x,t)>0$ when $x\in \overline \Omega$ and $t\in \RR$ with $|t|\geq r$, then $F(x,t)>0$ in the same set.

Let $r>0$ be as in \eqref{condf'}: then, for any $x\in \overline \Omega$ and $t\in \RR$ with $|t|\geq r>0$
$$f(x,t)\leq \beta tf_t(x,t)\,.$$
Suppose $t>r$. Integrating both terms in $[r,t]$
we obtain
$$
0<\mu F(x,t)\leq tf(x,t)+\mu F(x,r)-rf(x,r)\,,
$$
so that, taking into account that $rf(x,r)>0$, we have
\begin{equation}\label{Flog001}
0<\mu F(x,t)\leq tf(x,t)+\mu F(x,r)
\end{equation}
for any $x\in \overline \Omega$ and $t\in \RR$, $|t|\geq r$\,.

With the same arguments it is easy to prove that if $t<-r$ then it
holds
$$
0<\mu F(x,t)\leq tf(x,t)+\mu F(x,-r)+rf(x,-r)\,,
$$
and so
$$
0<\mu F(x,t)\leq tf(x,t)+\mu F(x,-r)\,,
$$
since $rf(x,-r)<0$\,.

Hence, from this and \eqref{Flog001} for any $x\in \overline \Omega$ and $t\in \RR$ with $|t|\geq r$ we get
\begin{equation}\label{Flog00}
0<\mu F(x,t)\leq tf(x,t)+M(x)\,,
\end{equation}
where $M(x)=\mu\,\max\{F(x,r), F(x,-r)\}>0$, being $F(x,t)>0$\,.

Taking ${\displaystyle a_5:=\max_{x\in \overline \Omega}M(x)}$, we get assertion~$b)$\,. Note that $a_5$ exists by assumption~\eqref{fcontinua} and the Weierstrass Theorem.
\end{proof}

Assertion~$b)$ in Lemma~\ref{lemmamu} is classical and it is a sort of Ambrosetti--Rabinowitz condition, which, usually, it is assumed in order to prove that the energy functional satisfies the Palais--Smale compactness property. Of course, our condition~\eqref{condf'} is stronger than assertion~$b)$, but, in a sense, they have the same nature.

\medskip

Now we are ready to prove the main result of the present subsection.

\begin{proposition}\label{lemmasucclimitata}
Let $\lambda\in \RR$\,, $s\in (0,1)$, $n>2s$ 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}, let $f:\overline\Omega\times \RR\to \RR$ verify \eqref{fcontinua}--\eqref{condf'} and let $h:\Omega\to \RR$ satisfy \eqref{g}\,. Let $c\in \RR$
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, the sequence~$u_j$ is bounded in $X_0$\,.
\end{proposition}
\begin{proof}
For any $j\in \NN$ by \eqref{Jc0} and \eqref{J'00} it easily follows
that there exists $\kappa>0$ such that
\begin{equation}\label{jlimitato0}
|\mathcal J_{K,\,\lambda}(u_j)|\leq \kappa\,,
\end{equation}
and
\begin{equation}\label{j'limitato0}
\Big|\langle \mathcal J_{K,\,\lambda}'(u_j),
\frac{u_j}{\|u_j\|_{X_0}}\rangle\Big| \leq \kappa\,.
\end{equation}

As a consequence of \eqref{jlimitato0} and \eqref{j'limitato0} we also have
\begin{equation}\label{aggiunto}
\mathcal J_{K,\,\lambda}(u_j)-\frac 1 \mu \langle \mathcal J_{K,\,\lambda}'(u_j),
u_j\rangle\leq \kappa \left(1+ \|u_j\|_{X_0}\right)\,.
\end{equation}

Moreover, by Lemma~\ref{fF} applied with $\varepsilon=1$ we have
that
\begin{equation}\label{uj<r}
\begin{aligned} & \Big|\int_{\Omega \cap\{|u_j|\leq
\,r\}}\Big(F(x, u_j(x))-\frac 1 \mu\, f(x,
u_j(x))\, u_j(x)\,\Big)\,dx\Big|\\
& \qquad \qquad \qquad \leq \left(r^2+\delta(1)r^q+\frac 2 \mu\,
r+\frac q \mu\, \delta(1) r^{q-1}\right)|\Omega|=:\tilde \kappa\,.
\end{aligned}
\end{equation}

Now, we have to distinguish two different cases, according to the fact that $\lambda$ is less or greater than the first eigenvalue $\lambda_1$ of $-\mathcal L_K$ (see Section~\ref{sec:notations} for the notation), namely $\lambda<\lambda_1$ or $\lambda\geq \lambda_1$\,.

Let us start with the case when $\lambda<\lambda_1$\,.
Then, by the variational characterization of $\lambda_1$ (see \cite[Proposition~9-$a)$]{svlinking}), Lemma~\ref{lemmamu}-$b)$\,, \eqref{g}\,, \eqref{uj<r} and H\" older inequality we get
\begin{equation}\label{jj'0}
\begin{aligned}
\mathcal J_{K,\,\lambda}&(u_j)-\frac 1 \mu  \langle \mathcal J_{K,\,\lambda}'(u_j), u_j\rangle =
\left(\frac 1 2 -\frac 1 \mu\right)\left(\|u_j\|_{X_0}^2-\lambda \|u\|_{L^2(\Omega)}^2\right)\\
& \qquad \qquad  -\frac 1 \mu \int_\Omega \Big(\mu F(x, u_j(x))-f(x, u_j(x)) \,u_j(x)\Big)\,dx\\
& \qquad \qquad - \Big(1-\frac 1 \mu\Big)\int_\Omega h(x) u_j(x)\,dx\\
& \geq \left(\frac 1 2 -\frac 1 \mu\right)m_1^\lambda\|u_j\|_{X_0}^2\\
& \qquad \qquad -\int_{\Omega \cap\{|u_j|\leq r\}}\Big(F(x, u_j(x))-\frac 1 \mu\, f(x, u_j(x))\,u_j(x)\Big)\,dx\\
& \qquad \qquad -a_5- \Big(1-\frac 1 \mu\Big)\|h\|_{L^2(\Omega)}\|u_j\|_{L^2(\Omega)}\\
& \geq \left(\frac 1 2 -\frac 1
\mu\right)m_1^\lambda\|u_j\|_{X_0}^2-\bar \kappa-\hat \kappa \|h\|_{L^2(\Omega)}\|u_j\|_{X_0}
\end{aligned}
\end{equation}
for some $\bar \kappa$ and $\hat\kappa$ positive. Here $m_1^\lambda:=\min\big\{1, 1-\lambda/\lambda_1\big\}$\,.

By \eqref{aggiunto} and~\eqref{jj'0}, for any $j\in \NN$
$$\|u_j\|_{X_0}^2 \leq \kappa_*\left(1+\|u_j\|_{X_0}\right)$$
for a suitable positive constant $\kappa_*$\,. Hence, the assertion
of Proposition~\ref{lemmasucclimitata} is proved, when $\lambda<\lambda_1$\,.

Now, let us consider the case when $\lambda\geq \lambda_1$\,. In this context the arguments are the same as when $\lambda<\lambda_1$\,.
Nevertheless, in this situation we can not use the variational characterization of the first eigenvalues of $-\mathcal L_K$\,. For this reason some technical difficulties arise and, in order to overcome these problems, we need to introduce an additional parameter~$\gamma$.

Note that, as we said in Section~\ref{sec:notations}, the sequence~$\lambda_k$ of the eigenvalues of $-\mathcal L_K$ diverges as $k\to +\infty$\,. Hence, since $\lambda\geq \lambda_1$, we can suppose that
$$\lambda\in
[\lambda_k, \lambda_{k+1})\,\, \mbox{ for some}\,\, k\in \NN\,.$$

Now, let us fix $\gamma\in (2, \mu)$, where $\mu>2$ is given in Lemma~\ref{lemmamu}\,. As in \eqref{uj<r}, by Lemma~\ref{fF} applied with $\varepsilon=1$ we have
that
\begin{equation}\label{uj<rL}
\begin{aligned} & \Big|\int_{\Omega \cap\{|u_j|\leq
\,r\}}\Big(F(x, u_j(x))-\frac 1 \gamma\, f(x,
u_j(x))\, u_j(x)\,\Big)\,dx\Big|\\
& \qquad \qquad \qquad \leq \left(r^2+\delta(1)r^q+\frac 2 \gamma\,
r+\frac q \gamma\, \delta(1) r^{q-1}\right)|\Omega|=:\tilde \kappa\,,
\end{aligned}
\end{equation}
so that, using also Lemma~\ref{lemmamu}-$b)$, H\" older inequality, \eqref{g} and Lemma~\ref{fF},
\begin{equation}\label{jj'0L}
\begin{aligned}
\mathcal J_{K,\,\lambda}(u_j)-&\frac 1 \gamma \langle \mathcal J_{K,\,\lambda}'(u_j), u_j\rangle  = \left(\frac 1 2 -\frac 1 \gamma\right)\Big(\|u_j\|_{X_0}^2-\lambda \|u_j\|_{L^2(\Omega)}^2\Big)\\
& \quad  -\int_\Omega \Big(F(x, u_j(x))-\frac 1 \gamma\, f(x, u_j(x)) \,u_j(x)\Big)\,dx\\
& \quad - \Big(1-\frac 1 \gamma\Big)\int_\Omega h(x) u_j(x)\,dx\\
& \geq \left(\frac 1 2 -\frac 1 \gamma\right)\Big(\|u_j\|_{X_0}^2-\lambda \|u_j\|_{L^2(\Omega)}^2\Big)\\
& \quad +\left(\frac \mu \gamma-1\right)\int_{\Omega \cap\{|u_j|\geq\, r\}} F(x, u_j(x))\,dx-a_5\\
& \quad -\int_{\Omega \cap\{|u_j|\leq\, r\}}\Big(F(x, u_j(x))-\frac 1 \gamma\, f(x, u_j(x))\,u_j(x)\Big)\,dx\\
& \quad -\kappa^* \|h\|_{L^2(\Omega)}\|u_j\|_{X_0}\\
& \geq \left(\frac 1 2 -\frac 1 \gamma\right)\Big(\|u_j\|_{X_0}^2-\lambda \|u_j\|_{L^2(\Omega)}^2\Big)\\
& \quad +\left(\frac \mu \gamma-1\right)\int_{\Omega \cap\{|u_j|\geq\, r\}} F(x, u_j(x))\,dx-\bar
\kappa\\
& \quad -\kappa^* \|h\|_{L^2(\Omega)}\|u_j\|_{X_0}\\
& \geq \left(\frac 1 2 -\frac 1 \gamma\right)\Big(\|u_j\|_{X_0}^2-\lambda \|u_j\|_{L^2(\Omega)}^2\Big)\\
& \quad +a_3\left(\frac \mu \gamma-1\right)\|u_j\|_{L^\mu(\Omega)}^\mu-a_4\left(1 - \frac \mu \gamma\right)\,|\Omega|-\bar\kappa\\
& \quad -\kappa^* \|h\|_{L^2(\Omega)}\|u_j\|_{X_0}\,.
\end{aligned}
\end{equation}

Here $\bar \kappa$ and $\kappa^*$ are positive constants.
Moreover, for any $\varepsilon>0$ the Young inequality (applied here with $\mu/2$ and its conjugate $\mu/(\mu-2)$\,, $\mu>2$ by assumption, see Lemma~\ref{lemmamu}) gives
\begin{equation}\label{youngPS}
\|u_j\|_{L^2(\Omega)}^2\leq \frac{2\varepsilon}{\mu}\,\|u_j\|_{L^\mu(\Omega)}^\mu+\frac{\mu-2}{\mu}\,\varepsilon^{-2/(\mu-2)}\,|\Omega|\,.
\end{equation}
Hence, by \eqref{jj'0L} and \eqref{youngPS} we deduce that
\begin{equation}\label{young2}
\begin{aligned}
\mathcal J_{K,\,\lambda}(u_j)-\frac 1 \gamma \langle & \mathcal J_{K,\,\lambda}'(u_j), u_j\rangle
\geq \left(\frac 1 2 -\frac 1 \gamma\right)\|u_j\|_{X_0}^2\\
& \qquad -\lambda\left(\frac 1 2 -\frac 1 \gamma\right) \frac{2\varepsilon}{\mu}\,\|u_j\|_{L^\mu(\Omega)}^\mu\\
& \qquad -\lambda\left(\frac 1 2 -\frac 1 \gamma\right) \frac{\mu-2}{\mu}\,\varepsilon^{-2/(\mu-2)}\,|\Omega|\\
& \qquad +a_3\left(\frac \mu \gamma-1\right)\|u_j\|_{L^\mu(\Omega)}^\mu\\
& \qquad -a_4\left(1 - \frac \mu \gamma\right)\,|\Omega|-\bar\kappa\\
& \qquad -\kappa^* \|h\|_{L^2(\Omega)}\|u_j\|_{X_0}\\
& = \left(\frac 1 2 -\frac 1 \gamma\right)\|u_j\|_{X_0}^2\\
& \qquad +\Big[a_3\left(\frac \mu \gamma-1\right)-\lambda\left(\frac 1 2 -\frac 1 \gamma\right)\frac{2\varepsilon}{\mu}\Big]\|u_j\|_{L^\mu(\Omega)}^\mu\\
& \qquad -C_\varepsilon-\kappa^* \|h\|_{L^2(\Omega)}\|u_j\|_{X_0}\,,
\end{aligned}
\end{equation}
where $C_\varepsilon$ is a constant such that $C_\varepsilon\to +\infty$ as $\varepsilon\to 0$, being $\mu>\gamma>2$\,.

Now, choosing $\varepsilon$ so small that
$$a_3\left(\frac \mu \gamma-1\right)-\lambda\left(\frac 1 2 -\frac 1 \gamma\right)\frac{2\varepsilon}{\mu}>0\,,$$
by \eqref{young2} we get
\begin{equation}\label{jj'20}
\begin{aligned}
\mathcal J_{K,\,\lambda}(u_j)-\frac 1 \gamma \langle \mathcal J_{K,\,\lambda}'(u_j), u_j\rangle & \geq \left(\frac 1 2 -\frac 1 \gamma\right)\|u_j\|_{X_0}^2-C_\varepsilon\\
& \qquad -\kappa^* \|h\|_{L^2(\Omega)}\|u_j\|_{X_0}\,.
\end{aligned}
\end{equation}

By \eqref{aggiunto} and \eqref{jj'20} for any $j\in \NN$
$$\|u_j\|_{X_0}^2 \leq \kappa_*\left(1+\|u_j\|_{X_0}\right)$$
for a suitable positive constant $\kappa_*$\,. Hence, the assertion
of Proposition~\ref{lemmasucclimitata} is proved also in the case when $\lambda\geq \lambda_1$\,. This ends the proof of Proposition~\ref{lemmasucclimitata}\,.
\end{proof}

Now, we are ready to prove the validity of the Palais--Smale condition on $X_0$ for the functional $\mathcal J_{K,\,\lambda}$ for any value of the parameter $\lambda\in \RR$\,.

\begin{proposition}\label{prop:PS}
Let $\lambda\in \RR$\,, $s\in (0,1)$, $n>2s$ 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}\,, let $f:\overline\Omega\times \RR\to \RR$ be a function satisfying conditions~\eqref{fcontinua}--\eqref{condf'} and let $h:\Omega\to \RR$ verify \eqref{g}\,. Let $u_j$ be a sequence in $X_0$ such
that $u_j$ is bounded in $X_0$ and \eqref{J'00} holds true. 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}
Since $u_j$ is bounded in $X_0$ and $X_0$ is a reflexive space
(being a Hilbert space, by \cite[Lemma~7]{svmountain}), up to a
subsequence, still denoted by $u_j$, there exists $u_\infty \in X_0$
such that~$u_j\weak u_\infty$ weakly in~$X_0$, that is
\begin{equation}\label{convergenze0}
\begin{aligned}
 & \int_{\RR^n\times \RR^n}\big(u_j(x)-u_j(y)\big)\big(\varphi(x)-\varphi(y)\big) K(x-y)\,dx\,dy \to \\
& \qquad \qquad
\int_{\RR^n\times \RR^n}\big(u_\infty(x)-u_\infty(y)\big)\big(\varphi(x)-\varphi(y)\big)
K(x-y)\,dx\,dy
\end{aligned}
\end{equation}
for any $\varphi\in X_0$ as $j\to +\infty$\,. Moreover, by
\cite[Lemma~8]{svmountain}, up to a subsequence,
\begin{equation}\label{convergenze0bis}
\begin{aligned}
& u_j \to u_\infty \quad \mbox{in}\,\, L^2(\RR^n) \\
& u_j \to u_\infty \quad \mbox{in}\,\, L^q(\RR^n) \\
& u_j \to u_\infty \quad \mbox{a.e. in}\,\, \RR^n
\end{aligned}
\end{equation}
as $j\to +\infty$ and there exists $\ell\in L^q(\RR^n)$ such that
\begin{equation}\label{dominata20}
|u_j(x)|\leq \ell(x) \quad \mbox{a.e. in}\,\, \RR^n\,\quad \mbox{for any}\,\,j\in \NN
\end{equation}
(see, for instance \cite[Theorem~IV.9]{brezis}).

By \eqref{crescita}, \eqref{convergenze0}--\eqref{dominata20}, the
fact that the map $t\mapsto f(\cdot, t)$ is continuous in $t\in \RR$ (see assumption~\eqref{fcontinua})
and the Dominated Convergence Theorem we get
\begin{equation}\label{convf0}
\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}
and
\begin{equation}\label{convfu0}
\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 $j\to +\infty$. Moreover, by \eqref{J'00} and the boundedness of the sequence~$u_j$ in $X_0$ we have that $\langle \mathcal J_{K,\,\lambda}'(u_j), u_j\rangle  \to 0$\,, that is
$$\begin{aligned}
\int_{\RR^n\times \RR^n}|u_j(x)-u_j(y)|^2 & K(x-y)\,dx\,dy  -\lambda \int_\Omega |u_j(x)|^2\,dx\\
 & - \int_\Omega f(x,
u_j(x))u_j(x)\,dx -\int_\Omega h(x)u_j(x)\,dx\to 0
\end{aligned}$$
as $j\to +\infty$\,.
Consequently, recalling also
\eqref{convergenze0bis} and \eqref{convf0}, we deduce
that
\begin{equation}\label{convnorma10}
\begin{aligned}
\int_{\RR^n\times \RR^n}|u_j(x)-& u_j(y)|^2  K(x-y) \,dx\,dy\to  \lambda \int_\Omega
|u_\infty(x)|^2\,dx \\
& +\int_\Omega f(x, u_\infty(x))u_\infty(x)\,dx+\int_\Omega h(x)u_\infty(x)\,dx
\end{aligned}
\end{equation}
as $j\to +\infty$.

Furthermore, using again~\eqref{J'00}, we have $\langle \mathcal J_{K,\,\lambda}'(u_j), u_\infty\rangle \to 0$\,, that is
\begin{equation}\label{convj'0}
\begin{aligned}
\int_{\RR^n\times \RR^n}\big(u_j(x) & -u_j(y)\big)\big(u_\infty(x)-u_\infty(y)\big)K(x-y)\,dx\,dy\\
 & \qquad -\lambda \int_\Omega u_j(x)u_\infty(x)\,dx - \int_\Omega f(x, u_j(x))u_\infty(x)\,dx\\
& \qquad -\int_\Omega h(x)u_\infty(x)\,dx \to 0
\end{aligned}
\end{equation}
as $j\to +\infty$. By \eqref{convergenze0}, \eqref{convergenze0bis},
\eqref{convfu0} and \eqref{convj'0} we obtain
\begin{equation}\label{convnorma20}
\begin{aligned}
\int_{\RR^n\times \RR^n}|u_\infty(x)-& u_\infty(y)|^2  K(x-y)\,dx\,dy  = \lambda
\int_\Omega |u_\infty(x)|^2\,dx \\
& \quad + \int_\Omega f(x,
u_\infty(x))u_\infty(x)\,dx +\int_\Omega h(x)u_\infty(x)\,dx\,.
\end{aligned}
\end{equation}
Thus, \eqref{convnorma10} and \eqref{convnorma20} give that
$$\int_{\RR^n\times \RR^n}|u_j(x)-u_j(y)|^2 K(x-y)\,dx\,dy\to \int_{\RR^n\times \RR^n}|u_\infty(x)-u_\infty(y)|^2 K(x-y)\,dx\,dy\,,$$
namely
$$\|u_j\|_{X_0}\to \|u_\infty\|_{X_0}$$
as $j\to +\infty$. Since $u_j \weak u_{\infty}$ weakly
in the Hilbert space $X_{0}$ and $\|u_{j}\|_{X_{0}} \to \|u_{\infty}\|_{X_{0}}$,
we conclude that $u_{j} \to u_{\infty}$ strongly in $X_{0}$ as $j\to +\infty$\,.
\end{proof}

As a consequence of Proposition~\ref{prop:PS} we can prove that the functional~$\mathcal I_{K,\,\lambda}$ satisfies the Palais--Smale condition on $M$ for any value of the parameter~$\lambda\in \RR$\,. This will be done in the next subsection.

\subsection{The Palais--Smale condition on $M$ for~$\mathcal I_{K,\,\lambda}$}\label{subsec:PSM}
Along this subsection we show that the functional~$\mathcal I_{K,\,\lambda}$ satisfies the Palais--Smale condition on $M$, namely the following result holds true:

\begin{proposition}\label{propPSM}
Let $\lambda\in \RR$\,, $s\in (0,1)$, $n>2s$ 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}, let $f:\overline\Omega\times \RR\to \RR$ verify \eqref{fcontinua}--\eqref{condf'} and let $h:\Omega\to \RR$ satisfy \eqref{g}\,. Finally, let $\bar R>0$ be the constant given in Lemma~\ref{lemma1}. Let $c\in \RR$ and let $(u_j, \rho_j)$ be a sequence in $M\times \RR$ such that $\|u_j\|_{X_0}\geq \bar R$ for $j\in \NN$ sufficiently large\,,
\begin{equation}\label{Jc01}
\mathcal I_{K,\,\lambda}(u_j)\to c
\end{equation}
and
\begin{equation}\label{J'001}
\sup\Big\{ \big|\langle\,\mathcal I_{K,\,\lambda}'(u_j)+\rho_j|u_j|^{q-2}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,
$$\|u_j-u_\infty\|_{X_0}\to 0$$
and
$$\rho_j\to 0$$
as $j\to +\infty$\,.
\end{proposition}
\begin{proof}
First of all, we want to show that the sequence~$u_j$ is bounded in $X_0$\,.
By \eqref{Jc01} there exists $\kappa>0$ such that for any $j\in \NN$
\begin{equation}\label{Jlim}
|\mathcal I_{K,\,\lambda}(u_j)|\leq \kappa\,.
\end{equation}
Also, by definition of $\mathcal I_{K,\,\lambda}$, it is easily seen that
$$\mathcal J_{K,\,\lambda}(u_j)\leq \mathcal I_{K,\,\lambda}(u_j)\,,$$
so that, by this and \eqref{Jlim}
$$\mathcal J_{K,\,\lambda}(u_j)\leq \kappa$$
for any $j\in \NN$\,.

Hence,
$$\begin{aligned}
\|u_j\|_{X_0}^2 & = 2\mathcal J_{K,\,\lambda}(u_j)+\lambda\,\|u_j\|_{L^2(\Omega)}^2+2\int_\Omega F(x, u_j(x))\,dx+2\int_\Omega h(x)u_j(x)\,dx\\
& \leq 2\kappa+\lambda\,\|u_j\|_{L^2(\Omega)}^2+2a_1\,\|u_j\|_{L^1(\Omega)}+\frac{ 2a_2}{q}\,\|u_j\|_{L^q(\Omega)}^q+2\|h\|_{L^2(\Omega)}\,\|u_j\|_{L^2(\Omega)}\\
& \leq 2\kappa+\kappa^*+2a_1\,|\Omega|^{(q-1)/q}\,\|u_j\|_{L^q(\Omega)}+\frac{ 2a_2}{q}\,\|u_j\|_{L^q(\Omega)}^q\\
& = 2\kappa+\kappa^*+2a_1\,|\Omega|^{(q-1)/q}+\frac{2a_2}{q}\,,
\end{aligned}$$
thanks to condition~\eqref{crescita}, H\"older inequality (here used with $q/2>1$ and its conjugate and also with $q$ and its conjugate) and since $u_j\in M$\,. Here $\kappa^*=\max\big\{\big(\lambda\,|\Omega|^{(q-2)/q}\big)+2|\Omega|^{(q-2)/2q}\,\|h\|_{L^2(\Omega)}\,, 2|\Omega|^{(q-2)/2q}\,\|h\|_{L^2(\Omega)}\big\}>0$. Hence,
\begin{equation}\label{ujbounded}
\mbox{the sequence }\, u_j \,\mbox{ is bounded in }\,\, X_0\,.
\end{equation}

We would like to note that, since $\|u_j\|_{X_0}\geq \bar R$ for $j$ sufficiently large, say $j\geq j^*$, with $j^*\in \NN$, then $u_j\not\equiv 0$ for any $j\geq j^*$\,.
Moreover, by Lemma~\ref{lemma1}
\begin{equation}\label{Ialpha}
\mathcal I_{K,\,\lambda}(u_j)>\alpha\,\,\, \mbox{for any}\,\,\, j\geq j^*\,.
\end{equation}
Hence, the functional~$\mathcal I_{K,\,\lambda}$ is differentiable in $u_j$ for $j\geq j^*$, by Proposition~\ref{prop:IJ}\,.

Now, we claim that
\begin{equation}\label{kavianh}
\langle \mathcal I'_{K,\,\lambda}(u_j), u_{j}\rangle = 0\quad \mbox{for}\,\,\, j\geq j^*\,.
\end{equation}
To this purpose, fix $j\geq j^*$ and let $g(t):=\mathcal I_{K,\,\lambda}(tu_j)$ for any $t>0$\,. Then, by definition of $\mathcal I_{K,\,\lambda}$, the function~$g$ turns out to be constant, and so
$$0=g'(t)=\langle \mathcal I'_{K,\,\lambda}(t u_j), u_j\rangle$$
for any $t>0$\,. Taking $t=1$ we get \eqref{kavianh}\,.

By \eqref{J'001}--\eqref{kavianh} and the fact that $u_j\in M$ it is easily seen that
$$\begin{aligned}
|\rho_j|& = \rho_j\|u_j\|_{L^q(\Omega)}^q\\
& = |\langle \mathcal I'_{K,\,\lambda}(u_j)+\rho_j|u_j|^{q-2}u_j, u_j\rangle|\\
& \leq \sup\Big\{ \big|\langle\,\mathcal I_{K,\,\lambda}'(u_j)+\rho_j|u_j|^{q-2}u_j,\varphi\,\rangle
\big|\,: \; \varphi\in X_0\,, \|\varphi\|_{X_0}=1\Big\}\,\|u_j\|_{X_0}\to 0
\end{aligned}$$
as $j\to +\infty$\,. Thus, one of the assertion of Proposition~\ref{propPSM} is proved.

Now, it remains to show that, up to a subsequence, $u_j$ converges strongly to some $u_\infty$ in $X_0$\,. To this purpose, for any $j\geq j^*$ let us denote by $v_j$ the element
\begin{equation}\label{vj}
v_j:=\sigma(u_j)u_j\,,
\end{equation}
where $\sigma(u_j)>0$ is the unique constant (see Proposition~\ref{prop:IJ}-$b)$) such that
\begin{equation}\label{vjproprieta}
\mathcal I_{K,\,\lambda}(u_j)=\mathcal J_{K,\,\lambda}(v_j)\,.
\end{equation}
Note that we can apply Proposition~\ref{prop:IJ}, since \eqref{Ialpha} holds true.

By \eqref{Jc01} and \eqref{vjproprieta} we get
\begin{equation}\label{vjproprieta2}
\mathcal J_{K,\,\lambda}(v_j)\to c\,\,\,\, \mbox{as}\,\,\, j\to +\infty\,.
\end{equation}
Moreover, again by Proposition~\ref{prop:IJ}-$b)$, \eqref{J'001}, \eqref{ujbounded} and the fact that $\rho_{j} \to 0$\,, we have that for any $\varphi\in X_0$
\begin{equation}\label{vjproprieta3}
\sigma(u_j)\langle \mathcal J'_{K,\,\lambda}(v_j), \varphi\rangle = \langle \mathcal I'_{K,\,\lambda}(u_j), \varphi\rangle\to 0
\end{equation}
as $j\to +\infty$\,.

We claim that
\begin{equation}\label{sigmaj}
\mbox{there exists $\delta>0$ such that $\sigma(u_j)\geq \delta$ for $j$ large enough.}
\end{equation}
For this, we argue by contradiction and we suppose that
\begin{equation}\label{sigmajto0}
\sigma(u_j)\to 0
\end{equation}
as $j\to+\infty$\,. Then, by this, \eqref{ujbounded} and \eqref{vj} we deduce that $v_j\weak 0$ weakly in~$X_0$\,, so that, by
\cite[Lemma~8]{svmountain}, up to a subsequence,
\begin{equation}\label{convergenzevj0bis}
\begin{aligned}
& v_j \to 0 \quad \mbox{in}\,\, L^\nu(\RR^n) \\
& v_j \to 0 \quad \mbox{a.e. in}\,\, \RR^n
\end{aligned}
\end{equation}
as $j\to +\infty$ and by \cite[Theorem~IV.9]{brezis} there exists $\ell_\nu\in L^\nu(\RR^n)$ such that
\begin{equation}\label{dominata2020}
|u_j(x)|\leq \ell_\nu(x) \quad \mbox{a.e. in}\,\, \RR^n\,\quad \mbox{for any}\,\,j\in \NN
\end{equation}
for any $\nu\in [1, 2^*)$\,.

By Lemma~\ref{fF}, \eqref{convergenzevj0bis} and the Dominated Convergence Theorem we have that
\begin{equation}\label{convF0}
\int_\Omega F(x, v_j(x))\,dx \to \int_\Omega F(x, 0)\,dx=0
\end{equation}
as $j\to +\infty$, since $F(\cdot, 0)=0$\,.
Thus, by \eqref{vjproprieta2}, \eqref{convergenzevj0bis} and \eqref{convF0} we get
$$\begin{aligned}
c & =\lim_{j\to +\infty}\mathcal J_{K,\,\lambda}(v_j)\\
& = \lim_{j\to +\infty} \left(\frac 1 2 \int_{\RR^n\times\RR^n}|v_j(x)-v_j(y)|^2 K(x-y)\,dx\,dy\right.
\\
& \qquad \qquad \left. -\frac \lambda 2 \int_\Omega |v_j(x)|^2\,dx-\int_\Omega F(x, v_j(x))\,dx  -\int_\Omega h(x)v_j(x)\,dx\right)\\
& =\frac 1 2 \lim_{j\to +\infty} \|v_j\|_{X_0}^2\,,
\end{aligned}$$
namely
\begin{equation}\label{normavj}
\|v_j\|_{X_0}^2 \to 2c\geq 2\alpha>0
\end{equation}
as $j\to +\infty$\,. For the last inequality we use the fact that ${\displaystyle c=\lim_{j\to +\infty}\mathcal I_{K,\,\lambda}(u_j)\geq \alpha}$, thanks to \eqref{Jc01} and \eqref{Ialpha}\,.

By \eqref{vj} and the fact that $\sigma(u_j)>0$ by construction, we deduce that
$$\|u_j\|_{X_0}=\big(\sigma(u_j)\big)^{-1}\|v_j\|_{X_0}\,,$$
so that, by \eqref{sigmajto0} and \eqref{normavj} we conclude that
$$\|u_j\|_{X_0}\to +\infty\,,$$
as $j\to +\infty$\,. This contradicts \eqref{ujbounded} and so assertion~\eqref{sigmaj} is proved.

As a consequence of \eqref{vjproprieta3} and \eqref{sigmaj} we deduce that for any $\varphi\in X_0$
\begin{equation}\label{vjproprieta33}
\langle \mathcal J'_{K,\,\lambda}(v_j), \varphi\rangle\to 0
\end{equation}
as $j\to +\infty$\,.

Since \eqref{vjproprieta2} and \eqref{vjproprieta33} hold true, by Proposition~\ref{lemmasucclimitata} we get that the sequence~$v_j$ is bounded in $X_0$\,. Hence, the sequence~$v_j$ is bounded in $L^q(\Omega)$ by the embedding properties of $X_0$ into the classical Lebesgue spaces (see \cite[Lemma~8]{svmountain}). As a consequence of this, of \eqref{vj} and of the fact that $u_j\in M$, we get
$$\sigma(u_j)=\sigma(u_j)\|u_j\|_{L^q(\Omega)}=\|v_j\|_{L^q(\Omega)}<\kappa$$
for any $j\geq j^*$, for a suitable positive constant $\kappa$\,, that is
the sequence
\begin{equation}\label{sigmajlimitata}
\sigma(u_j)\,\,\,\, \mbox{is bounded in }\,\,[\delta,+\infty)\,.
\end{equation}

Now, we can apply Proposition~\ref{prop:PS} to the sequence~$v_j$ (remember that that the sequence $v_j$ is bounded in $X_0$ and again \eqref{vjproprieta33} holds true). Then, this and \eqref{sigmajlimitata} yield that there exist $v_\infty\in X_0$ and $\sigma^*\in \RR$ such that, up to subsequences (still denoted by $v_j$ and $\sigma(u_j)$),
\begin{equation}\label{convX0}
v_j\to v_\infty\,\,\,\,\,\mbox{strongly in }\,\,\, X_0
\end{equation}
and
\begin{equation}\label{convR}
\sigma(u_j)\to \sigma^*\,\,\,\,\,\mbox{in }\,\,\, [\delta,+\infty)
\end{equation}
as $j\to +\infty$\,.

Note that, by \eqref{sigmaj}
\begin{equation}\label{sigmastar}
\sigma^*\geq \delta>0\,.
\end{equation}

Finally, as a consequence of \eqref{vj} (remember that $\sigma(u_j)>0$ for any $j\geq j^*$ by Proposition~\ref{prop:IJ}-$b)$), \eqref{convX0}--\eqref{sigmastar}, we have
$$u_j=\big(\sigma(u_j)\big)^{-1}\,v_j\to \big(\sigma^*\big)^{-1}\,v_\infty=:u_\infty\,\,\,\,\, \mbox{strongly in}\,\,\, X_0$$
as $j\to +\infty$\,. This concludes the proof of Proposition~\ref{propPSM}\,.
\end{proof}

\begin{remark}
{\rm Note that the function~$u_\infty$ given in Proposition~\ref{propPSM} is not the trivial one, i.e. $u_\infty\not\equiv 0$ in $X_0$\,. Indeed, by \eqref{convX0} and \cite[Lemma~8]{svmountain}, up to a subsequence,
\begin{equation}\label{convergenze0bisvj}
\begin{aligned}
& v_j \to v_\infty \quad \mbox{in}\,\, L^\nu(\RR^n) \\
& v_j \to v_\infty \quad \mbox{a.e. in}\,\, \RR^n
\end{aligned}
\end{equation}
as $j\to +\infty$ and, by \cite[Theorem~IV.9]{brezis}, there exists $\ell_\nu\in L^\nu(\RR^n)$ such that
$$|v_j(x)|\leq \ell_\nu(x) \quad \mbox{a.e. in}\,\, \RR^n\,\quad \mbox{for any}\,\,j\in \NN$$
for any $\nu\in [1, 2^*)$\,.
Then, by the Dominated Convergence Theorem applied in $L^\nu(\Omega)$ (here with $\nu=1$ and $\nu=q$) and thanks to assumption~\eqref{crescita}, we get that
$$\int_\Omega F(x, v_j(x))\,dx\to \int_\Omega F(x, v_\infty(x))\,dx$$
as $j\to +\infty$\,, so that
$$\mathcal J_{K,\,\lambda}(v_j)\to \mathcal J_{K,\,\lambda}(v_\infty)\,,$$
as $j\to +\infty$, by \eqref{convX0}, \eqref{convergenze0bisvj} and \eqref{g}\,.
That is, by \eqref{vjproprieta2}
$$\mathcal J_{K,\,\lambda}(v_\infty)=c\,.$$
From this it is easy to see that $v_\infty\not \equiv 0$, since $\mathcal J_{K,\,\lambda}(0)=0<\alpha\leq c=\mathcal J_{K,\,\lambda}(v_\infty)$, thanks to the fact that $F(\cdot, 0)=0$\,.

Finally, \eqref{sigmastar} and $v_\infty\not \equiv 0$ give that $u_\infty:=\big(\sigma^*\big)^{-1}\,v_\infty$ is not the trivial function in $X_0$.}
\end{remark}


\section{Existence of infinitely many solutions}\label{sec:esistenza}
This section is devoted to the proof of Theorems~\ref{lapfra0} and~\ref{generalkernel}. The arguments we will use here are variational ones.
In order to perform our proof, we will use the preliminary results stated in Subsection~\ref{subsec:preliminary} and we will apply \cite[ Chapter~5, Theorem~2.2]{kavian}.
For this, as we already explained, it is enough to verify that conditions~$i)$ and $ii)$, stated in Section~\ref{sec:plan}, are satisfied.

In the proof of Theorem~\ref{generalkernel} the assumptions on $q$ will be crucial for our argument. In particular, we will use the fact that
\begin{equation}\label{stimaq}
q<2^*-\frac{2s}{n-2s}\,.
\end{equation}

We would like to note that, until now, we never used this condition and all the results proved in the previous sections are valid for any $q\in (2, 2^*)$\,. Assumption~\eqref{stimaq} is fundamental for the proof of the validity of $ii)$\,.

If we consider the classical setting of equation~\eqref{1}, the existence of infinitely many solutions for this problem was proved under some restriction on $q$, namely for any $q>2$ such that
$$q<\frac{2(n-1)}{n-2}\,.$$
For more details on this we refer to \cite[ Chapter~5, Remark~4.7]{kavian}\,.
Note that this bound from above on $q$ corresponds to~\eqref{stimaq} when $s=1$ (which gives the classical Laplacian case).


\subsection{Proof of Theorem~\ref{generalkernel}}\label{subsec:proof}
As we said in Section~\ref{sec:plan}, since problem~\eqref{op} has a variational nature, the proof of Theorem~\ref{generalkernel} reduces to find critical points of the functional~$\mathcal J_{K,\,\lambda}$\,. To this purpose, by Corollary~\ref{cor:IJ}, it is enough to look for critical points $u\in \{\mathcal I_{K,\,\lambda}>\alpha\}$ of the functional~$\mathcal I_{K,\,\lambda}$ on $M$ (here $\alpha>0$ is the constant given in Proposition~\ref{prop:IJ}). For this we will apply \cite[ Chapter~5, Theorem~2.2]{kavian}.

First of all, note that the functional~$\mathcal I_{K,\,\lambda}\in C^1(\{\mathcal I_{K,\,\lambda}>\alpha\})$ by Proposition~\ref{prop:IJ}-$a)$ and it verifies the Palais--Smale condition on $M$, thanks to Proposition~\ref{propPSM}\,.
Hence, by \cite[ Chapter~5, Theorem~2.2]{kavian}, we get that for any $k\in \NN$
$$a_k\leq b_k\,,$$
where $a_k$ and $b_k$ are as in \eqref{ak} and \eqref{bk}, respectively.

Now, in order to get our goal, we need to prove that $a_k\in \RR$ and that $a_k<b_k$ for $k\in \NN$ sufficiently large.

To this purpose, first of all note that, by Lemma~\ref{lemmaak} and the choice of $q>2$\,, we have that $a_k\to +\infty$ as $k\to +\infty$, so that $a_k>\alpha$ for $k\geq \tilde \kappa$, with $\tilde \kappa\in \NN$\,. This means that $a_k\in \RR$ for any $k\geq \tilde \kappa$\,.

Now, it remains to show that $a_k<b_k$ for any $k\in \NN$ large enough. To this purpose, first of all note that for any $k\in \NN$ and any $A\in \mathcal A_k$ there exists $u_A\in A\cap \mathbb P_k$ (for this see \cite[Chapter~2, Theorem~4.4 and the proof of Chapter~5, Lemma~3.3]{kavian}), where
$$\mathbb P_k:= \big\{ u\in X_0 {\mbox{ s.t. }}
\langle u,e_j\rangle_{X_0}=0 \quad\forall j=1,\dots,k-1
\big\}\,.$$
Here $\langle \cdot, \cdot \rangle$ denotes the scalar product in $X_0$ (see formula~\eqref{prodottoscalare}), while $e_j$ are the eigenfunctions of $-\mathcal L_K$ (see Section~\ref{sec:notations}).

Let us fix $k\in \NN$. By the variational characterization of $\lambda_k$ (see \cite[Proposition~9]{svlinking}), we have that
\begin{equation}\label{lambdak}
\|u_A\|_{X_0}\geq \sqrt{\lambda_k}\,\|u_A\|_{L^2(\Omega)}\,.
\end{equation}
Also, by H\"older inequality applied in $L^2(\Omega)$ (for this we need that $2<q<2^*-2s/(n-2s)$, which is our assumption) and the embedding properties of the space~$X_0$ (see \cite[Lemma~8]{svmountain}) we get
$$\begin{aligned}
\|u_A\|_{L^q(\Omega)}^q & =\int_\Omega |u_A(x)|^{q-1}|u_A(x)|\,dx\\
& \leq \|u_A\|_{L^2(\Omega)}\,\|u_A\|_{L^{2(q-1)}(\Omega)}^{q-1}\\
& \leq C \|u_A\|_{L^2(\Omega)}\,\|u_A\|_{X_0}^{q-1}
\end{aligned}$$
for a suitable $C>0$\,.
As a consequence of this and taking into account that $u_A\in A\subseteq M$ we get
$$\|u_A\|_{L^2(\Omega)}\geq C^{-1}\|u_A\|_{L^q(\Omega)}^q\,\|u_A\|_{X_0}^{1-q}=C^{-1}\|u_A\|_{X_0}^{1-q}\,,$$
so that this and \eqref{lambdak} yield
$$\|u_A\|_{X_0}\geq C^{-1}\sqrt{\lambda_k}\,\|u_A\|_{X_0}^{1-q}\,,$$
namely
$$\|u_A\|_{X_0}\geq C^{-1/q}\lambda_k^{1/(2q)}\,.$$
Since $\lambda_k\to +\infty$ as $k\to +\infty$ (see \cite[Proposition~9]{svlinking}), for $k$ large enough we get that
$$\|u_A\|_{X_0}\geq \bar R\,,$$
where $\bar R$ is the positive constant given in Lemma~\ref{lemma1}\,.
This proves that for $k\in \NN$ sufficiently large, say $k\geq \breve{\kappa}$, with $\breve{\kappa}\in \NN$\,, and for any $A\in \mathcal A_k$
$$\mbox{the set}\,\,\, A\cap \big\{\|v\|_{X_0}\geq \bar R\big\}\,\,\, \mbox{is not empty.}$$

With this assertion and using the fact that $\mathcal I_{K,\,\lambda}(v)=\mathcal I_{K,\,\lambda}(tv)$ for any $t>0$ it is easy to see that for $k\geq \breve{\kappa}$
\begin{equation}\label{ak1}
a_k=\inf_{A\in \mathcal A_k}\max_{v\in A\\ \atop \|v\|_{X_0}\geq \bar R}\mathcal I_{K,\,\lambda}(v)
\end{equation}
and
\begin{equation}\label{bk1}
b_k=\inf_{B\in \mathcal B_k}\max_{v\in B\\ \atop \|v\|_{X_0}\geq \bar R}\mathcal I_{K,\,\lambda}(v)\,.
\end{equation}
Also the same holds true for $a_k^0$ and $b_k^0$ (see formulas \eqref{ak0} and \eqref{bk0}).

Finally, we can show that
\begin{equation}\label{claimak}
a_k<b_k\,\,\, \mbox{for any $k\in \NN$ sufficiently large.}
\end{equation}
For this we argue by contradiction and we suppose that there exists $\bar \kappa\in \NN$ such that $a_k=b_k$ for any $k\geq \bar\kappa$ (note that we know that $a_k\leq b_k\,,$ for any $k\in \NN$). Let $\hat \kappa=\max\{\bar \kappa, \tilde \kappa, \breve{\kappa}\}$\,.

By \eqref{ak1} and \eqref{bk1} and the analogous formulas for $a_k^0$ and $b_k^0$ and Lemma~\ref{lemma1} we get that for any $k\in \NN$ with $k\geq \breve{\kappa}$
$$a_{k+1}^0 = b_k^0\leq b_k+\bar c\big(1+b_k^{1/q}\big)$$
and
$$a_{k+1}\leq a_{k+1}^0+\bar c\big(1+(a_{k+1}^0)^{1/q}\big)\,,$$
so that
$$a_{k+1}\leq b_k+\bar c\big(1+b_k^{1/q}\big)+\bar c\big[1+\big(b_k+\bar c\big(1+b_k^{1/q}\big)\big)^{1/q}\big]\,.$$

As a consequence of this and using the {assumption} that $a_k=b_k$ for any $k\geq \hat\kappa$, we deduce that
$$a_{k+1}\leq a_k+\bar c\big(1+a_k^{1/q}\big)+\bar c\big[1+\big(a_k+\bar c\big(1+a_k^{1/q}\big)\big)^{1/q}\big]\,,$$
which gives
\begin{equation}\label{akdis}
a_{k+1}\leq a_k+\hat c\big(1+a_k^{1/q}\big)
\end{equation}
for a suitable positive constant~$\hat c$ and for any $k\geq \hat\kappa$\,. Here we use also the fact that $a_k$ is large enough (since $a_k\to +\infty$ as $k\to +\infty$)\,.

By \eqref{akdis} and Lemma~\ref{lemmakavian} (applied here with $p=q-1>1$) there exist $c^*>0$ and $\kappa^*\in \NN$ such that
\begin{equation}\label{akdis1}
a_k\leq c^*k^{q/(q-1)}\,\,\,\,\,\mbox{for}\,\,\, k\geq \kappa^*\,.
\end{equation}

Moreover, combining \eqref{akdis1} and Lemma~\ref{lemmaak} we deduce that, for $k$ sufficiently large
$$\tilde c\,k^{2sq/n(q-2)}\leq a_k\leq c^*k^{q/(q-1)}\,,$$
which implies that
$$\frac{2sq}{n(q-2)}\leq \frac{q}{q-1}\,,$$
that is
$$q\geq 1+n/(n-2s)=2^*-2s/(n-2s)\,.$$
Clearly this contradicts the assumption on $q$ (see \eqref{stimaq}). Thus, \eqref{claimak} is proved.

Hence, by \cite[ Chapter~5, Theorem~2.2]{kavian}, the functional~$\mathcal I_{K,\,\lambda}$ admits a sequence of critical points $v_k$ on $M$ with critical value $c_k=\mathcal I_{K,\,\lambda}(v_k)\geq b_k$\,. Since $\mathcal I_{K,\,\lambda}(v_k)\geq b_k>a_k>\alpha$ for $k$ large enough (as $a_k\to +\infty$ as $k\to +\infty$), then by Corollary~\ref{cor:IJ}-$a)$ the sequence~$u_k$ defined as
$$u_k:=\sigma(v_k)v_k\,,$$
where $\sigma(v_k)$ is given as in Proposition~\ref{prop:IJ}-$b)$, is a sequence of critical points of~$\mathcal J_{K,\,\lambda}$ on $X_0$\,. In addition, again by Proposition~\ref{prop:IJ}-$b)$\,,
\begin{equation}\label{Jinfty}
\mathcal J_{K,\,\lambda}(u_k)=\mathcal J_{K,\,\lambda}(\sigma(v_k)v_k)=\mathcal I_{K,\,\lambda}(v_k)\geq b_k>a_k\to +\infty
\end{equation}
as $k\to+\infty$\,.

Finally, we have to show that $\|u_k\|_{X_0}\to +\infty$ as $k\to +\infty$\,. For this, we can argue again by contradiction. Indeed, if the sequence~$u_k$ were bounded in $X_0$, then, by the embeddings properties of $X_0$ into the usual Lebesgue spaces (see \cite[Lemma~8]{svmountain}) and assumption~\eqref{crescita}, we would get that the sequence~$\mathcal J_{K,\,\lambda}(u_k)$ is bounded in $\RR$, but this contradicts \eqref{Jinfty}. Thus, $\|u_k\|_{X_0}\to +\infty$ as $k\to +\infty$ and this concludes the proof of Theorem~\ref{generalkernel}\,.

\subsection{Proof of Theorem~\ref{lapfra0}}\label{subsec:prooflapfra}
This is a consequence of Theorem~\ref{generalkernel}, since, in the model case $K(x)=|x|^{-(n+2s)}$, we have that  $X_0=\{v\in H^s(\RR^n) : v=0 \quad \mbox{a.e. in}\quad \RR^N\setminus \Omega\}$\,, see \cite[Lemma~7]{servadeivaldinociBN}\,.

\bigskip

\textbf{Acknowledgements.} The author would like to thank Otared Kavian for
his time and his useful and interesting advices during the preparation of this paper.


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\end{document}

\begin{equation}\label{mu0}
\begin{aligned}
&\mbox{there exist}\,\, \mu>2\,\, \mbox{and}\,\,r>0\,\, \mbox{such that a.e.}\,\, x\in \Omega,\, t\in \RR,\, |t|\geq r\\
& \qquad \qquad \qquad \qquad 0<\mu F(x,t)\le tf(x,t)\,,
\end{aligned}
\end{equation}
where the function $F$ is
the primitive of $f$ with respect to its second variable, that is
\begin{equation}\label{F}
{\displaystyle F(x,t)=\int_0^t f(x,\tau)d\tau}\,.
\end{equation}
---------------1210040455348--
