\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
Two nonlinear days in Urbino 2017,\newline
\emph{Electronic Journal of Differential Equations},
Conference 25 (2018), pp. 197--212.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document} \setcounter{page}{197}
\title[\hfilneg EJDE-2018/conf/25\hfil
 Nonlocal fractional problems and $\nabla$-theorems]
{Nonlocal fractional problems and $\nabla$-theorems}

\author[D. Mugnai \hfil EJDE-2018/conf/25\hfilneg]
{Dimitri Mugnai}

\address{Dimitri Mugnai \newline
Dipartimento di Scienze Ecologiche e Biologiche (DEB),
Universit\`a della Tuscia,
Largo dell'Universit\`a - 01100 Viterbo, Italy}
\email{dimitri.mugnai@unitus.it}

\dedicatory{In memory of Anna Aloe}

\thanks{Published September 15, 2018}
\subjclass[2010]{35J20, 35S15, 47G20, 45G05}
\keywords{Nonlocal operators; fractional Laplacian; variational methods;
\hfill\break\indent  $\nabla$-theorems, $\nabla$-condition; 
 superlinear and subcritical nonlinearities}

\begin{abstract}
 We prove the multiplicity result in \cite{mbms} under more general assumptions.
 More precisely, we prove the existence of three nontrivial solutions for
 a nonlocal problem when a parameter approaches one of the eigenvalues of
 the leading operator, without assuming the Ambrosetti-Rabinowitz condition.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

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

In this article we prove the existence of three nontrivial solutions
for a class of nonlocal problems when a parameter approaches one of the
eigenvalues of the leading operator and when the nonlinear terms has
superlinear and subcritical behaviour. The result is in the spirit
of \cite{mbms}, but here the result is proved under more general assumptions.

Going into details, we consider a class of problems near resonance whose
 prototype is
\begin{equation}\label{problema}
\begin{gathered}
(-\Delta)^s u=\lambda u+f(x,u) \quad \text{in } \Omega\\
u=0 \quad \text{in } \mathbb R^n\setminus \Omega.
\end{gathered}
\end{equation}
Here $\Omega\subset \mathbb R^n$ is a bounded domain with Lipschitz continuous
boundary, $\lambda\in \mathbb R$, $f$ is a Carath\'eodory function which
is superlinear and subcritical in the sense of the fractional Sobolev exponent.
Moreover, $s\in (0,1)$ and $(-\Delta)^s$ is the fractional Laplace operator,
which (up to normalization factors) may be defined as
\begin{equation} \label{2}
-(-\Delta)^s u =
\int_{\mathbb R^n}\frac{u(x+y)+u(x-y)-2u }{|y|^{n+2s}}\,dy\,,
\quad  x\in \mathbb R^n.
\end{equation}
Actually, we shall consider more general nonlocal operators, in place of
$(-\Delta)^s$, and thus we will focus on problems of the form
\begin{equation}\label{problemaK}
\begin{gathered}
-\mathcal L_K u=\lambda u+f(x,u) \quad \text{in } \Omega\\
u=0 \quad \text{in } \mathbb R^n\setminus \Omega,
\end{gathered}
\end{equation}
where the nonlocal operator $\mathcal L_K$ is defined as
\begin{equation}\label{lk}
\mathcal L_Ku(x) = \int_{\mathbb R^n}\Big(u(x+y)+u(x-y)-2u(x) \Big)K(y)\,dy\,,
\quad  x\in \mathbb R^n,
\end{equation}
and $K:\mathbb R^n\setminus\{0\}\to (0,+\infty)$ is such that
\begin{gather}\label{kernel0}
K(-x)=K(x) \quad \text{for any $x\in \mathbb R^n \setminus\{0\}$}, \\
\label{kernel}
m K\in L^1(\mathbb R^n),\quad \text{where $m =\min \{|x|^2, 1\}$,}\\
\label{kernelfrac}
\text{there exists $\theta>0$ such that $K \geq \theta |x|^{-(n+2s)}$ for every }
x\in \mathbb R^n \setminus\{0\}.
\end{gather}

We notice that, similarly to \cite[Lemma 3.5]{valpal}, an equivalent formulation
for $\mathcal L_K$ is given, as usual up to some positive constant, by
\begin{equation}\label{LKPV}
\begin{aligned}
\mathcal L_Ku(x) &=\operatorname{P.V.}
\int_{\mathbb R^n}\big(u(x)-u(y) \big)K(x-y)\,dy \\
&=\lim_{\varepsilon\to0}\int_{|x|\geq \varepsilon}\big(u(x)-u(y) \big)K(x-y)\,dy
\end{aligned}
\end{equation}
for every $x\in \mathbb R^n$, $P.V.$ standing for the ``Cauchy principal value''.

Before stating our result, we recall that the ``boundary condition'' $u=0$ in
$\mathbb R^n\setminus \Omega$ leads to settle the problem in a particular
functional setting, namely, in view of \eqref{LKPV}, a
weak solution of \eqref{problemaK} is a function $u\in X_0$ such that
\begin{equation}\label{problemaKweak}
 \int_{\mathbb R^n \times \mathbb R^n} (u(x) -u(y))
(\varphi -\varphi(y))K(x-y) \,dxdy=\lambda \int_\Omega u \varphi \,dx
 + \int_\Omega f(x,u )\varphi dx
\end{equation}
for every $ \varphi \in X_0$. Here $X_0$ is defined as follows:
first, $X$ is the linear space
\begin{align*}
X=&\Big\{ u\in {\mathcal M}(\mathbb R^n): u_{|\Omega}\in L^2(\Omega)
 \text{ and the map }\\
& (x,y)\mapsto (g(x) -g(y))\sqrt{K(x-y)}\in L^2\big(\mathbb R^n \times
 \mathbb R^n \setminus ({\mathcal C}\Omega\times
{\mathcal C}\Omega), dx dy\big)\Big\},
\end{align*}
where ${\mathcal C}\Omega:=\mathbb R^n \setminus\Omega$. Finally,
$$
X_0=\{g\in X : g=0 \text{ a.e. in } \mathbb R^n\setminus\Omega\}.
$$
We recall that
\[
\langle u,v\rangle_{X_{0}}:= \int_{\mathbb R^n\times \mathbb R^n} \big(u(x)
-u(y)\big)\big(\varphi -\varphi(y)\big)K(x-y)\,dx\,dy
\]
makes $X_0$ a Hilbert space, see \cite[Lemma~7]{svmountain}.

Moreover, we also need to recall that $-\mathcal L_K$ admits a sequence
$\big\{ \lambda_k\big\}_{{k\in\mathbb N}}$ of eigenvalues having finite
 multiplicity and with the property that
\begin{equation}\label{lambdacrescente}
\begin{gathered}
0<\lambda_1<\lambda_2\le \dots \le \lambda_k\le \lambda_{k+1}\le \dots, \\
\lambda_k\to +\infty \quad\text{as } k\to +\infty.
\end{gathered}
\end{equation}
In addition, if $e_k$ is the eigenfunction corresponding to $\lambda_k$
normalized in $L^2(\Omega)$, then $\big\{e_k\big\}_{{k\in\mathbb N}}$
is an orthonormal basis of $L^2(\Omega)$ and an orthogonal basis of $X_0$,
see \cite{sY,svlinking}.

Finally, we say that eigenvalue $\lambda_k$, $k\geq 2$, has multiplicity
 $m\in \mathbb N$ if
$$
\lambda_{k-1}<\lambda_k=\dots =\lambda_{k+m-1}<\lambda_{k+m},
$$
and in such a case the set of all eigenfunctions associated to
$\lambda_k$ coincides with
$\operatorname{span}\{e_k,\ldots,e_{k+m-1}\}$.

In this article, for any $k\in \mathbb N$ we set
\begin{gather*}
H_k=\operatorname{span}\{e_{1},\ldots,e_k\}, \\
H^\perp_k= \big\{u\in X_{0}: \langle u,e_{j}\rangle_{X_{0}}=0\text{ for any }
 j=1,\ldots,k \big\},
\end{gather*}
so that $H_k$ has precisely dimension $k$.

In this way, the variational characterization of the eigenvalues
(see \cite{svlinking}) gives
\begin{equation}\label{poincarevincolata}
\int_{\mathbb R^n\times \mathbb R^n}|u (x)-u(y)|^2 K(x-y)\,dx\,dy
\geq \lambda_{k+1} \int_\Omega |u |^2\,dx  \text{ for all $u\in H^\perp_k$}.
\end{equation}
On the other hand, by the orthogonality properties of the eigenvalues,
a standard Fourier decomposition gives
\begin{equation}\label{antipoincare}
\int_{\mathbb R^n\times \mathbb R^n}|u(x) -u(y)|^2 K(x-y)\,dx\,dy
\leq \lambda_k \int_\Omega |u |^2\,dx  \text{ for all $u\in H_k$}.
\end{equation}

The aim of this paper is to exploit a critical point theorem \emph{of mixed type},
one of the so-called \emph{$\nabla$-theorems}, introduced by
 Marino and Saccon \cite{marsac:sns} (see also \cite{MMu, marsac, mug5}),
 which permit to provide multiplicity results in a very elegant way.
These theorems have been successfully employed in several contexts,
see, for instance, \cite{mms, Mug, mugnaiNODEA,mug4, mugpag, OL, fwang, fwang2, wzz}.
In particular, one theorem of this type was used in \cite{mbms} for showing
a multiplicity result for a problem like \eqref{problemaK}, assuming that
$f$ satisfies a growth condition of the Ambrosetti-Rabinowitz type.
Here we want to obtain the same result in a more general setting.
Indeed, we assume:

\begin{itemize}
\item[(A1)] $f:\Omega\times \mathbb R\to\mathbb R$ is a Carath\'eodory
function satisfying the following conditions:
there exist $a_1, a_2>0$ and $q\in (2, 2^*)$, $2^*=2n/(n-2s)$ such that
\begin{gather}\label{f1}
|f(x,t)|\le a_1+a_2|t|^{q-1}\quad \text{a.e. } x\in \Omega, t\in \mathbb R\,;\\
\label{f2}
 \lim_{|t|\to 0}\frac{f(x,t)}{|t|}=0 \quad  \text{uniformly in } x\in \Omega\,; \\
\label{f31F2}
 f(x,t)t - 2F(x,t)>0\quad \text{for a.e. $x\in \Omega$ and all $t\in \mathbb R$,
$t\neq 0$}, \\
\label{F1}
\lim_{|t|\to \infty}\frac{F(x,t)}{t^2}=+\infty \quad\text{uniformly in } x\in \Omega,
\end{gather}
there exist positive constants
$p>\max\big\{\frac{2n}{n+2s}(q-1),q-1\big\}$, $a_3>0$ and $R>0$ such that
\begin{gather}\label{f32}
f(x,t)t- 2F(x,t)\geq a_3|t|^p \quad \text{for a.e. $ x\in \Omega$ and every
$| t|\geq R$}; \\
\label{F2}
F(x,t)\geq 0 \text{ for a.e. $x\in \Omega$ and all $t\in \mathbb R$}.
\end{gather}
\end{itemize}
Here
\begin{equation}\label{F}
 F(x,t):=\int_0^t f(x,\tau)d\tau \quad \text{for a.e. $x\in \Omega$ and all
$t\in \mathbb R$}.
\end{equation}

As an example for $f$ we can take $f(x,t)=a(x)|t|^{q-2}t$, with
$a\in L^\infty(\Omega)$, $\inf_\Omega a>0$ (see \cite{dmnodea2}) and
$q\in (2, 2^*)$.

\begin{remark} \label{rmk1.1} \rm
The common Ambrosetti-Rabinowitz condition, i.e. there exists $\mu>2$ and
$R\geq0$ such that
\[
\leqno{(AR)}\quad \quad 0<F(x,t)\leq f(x,t)t ,
\]
for a.e. $x\in \Omega$ and all $|t|>R$, is not sufficient to ensure that
$F(x,\cdot)$ can be estimated from below by a superquadratic power,
 while it would be if $(AR)$ holds for every
$(x,t)\in \bar \Omega \times \mathbb R$ (see \cite{dmnodea2}).
 For this reason, it seems natural, in this general context, to assume
{\it a priori} some kind of control from below, as we do in \eqref{f32},
though we {\it do not} require $p>2$. Indeed,
\[
\frac{2n}{n+2s}(q-1)\geq 2
\]
if and only if
\[
q\geq 4\frac{n+s}{n+2s}>2^*,
\]
which is not an admissible occurrence.

On the other hand, by \eqref{f1} and \eqref{F} it is clear that $p\leq q$.
\end{remark}

Very close assumptions on $f$ were assumed in \cite{LAO} for studying a
fourth order problem in bounded domains through the same approach via
$\nabla-$theorems. Inspired by \cite{LAO}, our main result reads as follows.

\begin{theorem}\label{thmain}
Let $s\in (0,1)$, $n>2s$ and $\Omega$ be an open bounded subset of
$\mathbb R^n$ with continuous boundary. Let
$K:\mathbb R^n\setminus\{0\}\to (0,+\infty)$
satisfy \eqref{kernel0}--\eqref{kernelfrac} and let $f$ satisfy {\rm(A1)}.
Then, for every eigenvalue $\lambda_k$ of $-\mathcal L_K$, $k\geq 2$,
there exists a left neighborhood $\mathcal O_k$ of $\lambda_k$
such that problem~\eqref{problemaK} admits at least three nontrivial
weak solutions for all $\lambda\in \mathcal O_k$.
\end{theorem}

\begin{corollary}\label{thlapfrac}
Under the assumptions of Theorem \ref{thmain}, for every $k\geq 2$ there
exists a left neighborhood $O_k$ of the $k-$th eigenvalue $\lambda_k$ of
$(-\Delta)^s$, such that, if $\lambda\in O_k$, then \eqref{problema} admits
at least three nontrivial weak solutions.
\end{corollary}


This article is organized in the following way:
in Section~\ref{sec:preliminary} we recall some notions and notations
which will be used throughout the paper. In Section~\ref{sec:geometry} we
prove that the energy functional associated to the problem enjoys some good
geometric structures.
In Section~\ref{sec:nablacondition} we prove the $\nabla$-condition, the
main ingredient of the critical point tool that we shall use, which is
Theorem~\ref{thmarinosaccon}.
Finally, in Section~\ref{sec:proofthmain} we prove the main multiplicity
result of this paper, i.e. Theorem~\ref{thmain}, by coupling the result of
the $\nabla$-theorem due to Marino and Saccon in \cite{marsac:sns}
with a classical Linking theorem (see \cite[Theorem~5.3]{rabinowitz}),
obtaining the existence of three nontrivial solutions for problem
\eqref{problemaK}. We remark that while the existence of two nontrivial
solutions near resonance is free due to bifurcation theory, the fine
estimates on the critical value provided by Theorem \ref{thmarinosaccon}
permit to compare the critical value obtained with a Linking Theorem
and find a third nontrivial solution, being the energy of the last solution
higher than that of the former two ones.

A last comment on the notation: we will use several times the symbol
 $c$ or $C$ to denote absolute constants, which, however,
may be different from previous ones denoted in the same way.

\section{Preliminaries}\label{sec:preliminary}

First of all, we need some notation. In the sequel we endow the space $X_0$
with the norm defined as (see \cite[Lemma~6]{svmountain})
\begin{equation}\label{norma}
\|g \|=\Big(\int_{\mathbb R^n\times \mathbb R^n} |g(x) -g(y)|^2K(x-y)dx\,dy
\Big)^{1/2},
\end{equation}
which is obviously related to 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}
of the usual fractional
Sobolev space $H^s(\Omega)$.
For further details on the fractional Sobolev spaces we refer
to \cite{adams, valpal, molicalibro}
and to the references therein. We only recall the following embeddings,
which will be repeatedly used and for whose proofs we refer to \cite{svmountain}:
\begin{equation}\label{embedding}
\begin{gathered}
X_0 \hookrightarrow L^\nu(\Omega) \quad\text{for every }\nu\in[1,2^*],\\
X_0 \hookrightarrow \hookrightarrow L^\nu(\Omega) \quad\text{for every }
\nu\in[1,2^*).
\end{gathered}
\end{equation}

Problem \eqref{problemaKweak} has a variational structure: indeed, it is the
 Euler-Lagrange equation of the functional $\mathcal J_\lambda:X_0\to \mathbb R$
defined as
$$
\mathcal J_\lambda(u)=\frac 1 2 \int_{\mathbb R^n\times \mathbb R^n}|u(x)
-u(y)|^2 K(x-y)\,dx\,dy-\frac \lambda 2 \int_\Omega u^2\,dx
-\int_\Omega F(x, u )\,dx.
$$
Note that when functional $\mathcal J_\lambda$ is Fr\'echet differentiable at
 $u\in X_0$, we have that for any $\varphi\in X_0$
\begin{align*}
\langle \mathcal J'_\lambda(u), \varphi\rangle
& = \int_{\mathbb R^n\times \mathbb R^n} \big(u(x) -u(y)\big)
 \big(\varphi -\varphi(y)\big)K(x-y)\,dx\,dy\\
&\quad -\lambda \int_\Omega u \varphi \,dx
   -\int_\Omega f(x, u )\varphi \,dx,
\end{align*}
where we have denoted by $\langle \cdot,\cdot\rangle$ the duality between
$X_0'$ and $X_0$.
Thus, critical points of $\mathcal J_\lambda$ are solutions to problem
\eqref{problemaKweak}. We remark that (A1) ensures that $\mathcal J_\lambda$
is actually of class $C^1$, and so we can find solutions to \eqref{problemaKweak}
by looking for critical points to $\mathcal J_\lambda$.
This is what we shall do using the $\nabla$-theorem in the form of
Theorem~\ref{thmarinosaccon} (see Section~\ref{sec:proofthmain}) and
the classical Linking Theorem.

We conclude this section recalling that problems of the form \eqref{problemaK}
 have been widely investigated in latest years, under different assumptions
on $\lambda$ and $f$. The literature in this context is huge, and we only
refer to some recent papers and the references therein, quoting,
in addition to the already cited ones,
\cite{barrioscoloradoservadei,cm,dsv,FMV,152,molicaservadei}.

\section{Geometry of the $\nabla$-theorem}\label{sec:geometry}

In this section we show that if $k$ and $m$ in $\mathbb N$ are such that
\begin{equation}\label{hpauto}
\lambda_{k-1}<\lambda<\lambda_k=\dots = \lambda_{k+m-1}<\lambda_{k+m},
\end{equation}
then $\mathcal J_\lambda$ satisfies the geometric setting of
Theorem~\ref{thmarinosaccon} with
\[
 X_1:=H_{k-1},
 X_2:=\operatorname{span}\left\{e_k,\ldots,e_{k+m-1}\right\}\quad
 X_3:=H^\perp_{k+m-1}\,.
\]

\begin{proposition}\label{propgeometria}
Let $k$ and $m$ in $\mathbb N$ be such that \eqref{hpauto} holds and let $f$
satisfy {\rm (A1)}.
Then, there exist $\rho, R$, with $R>\rho>0$, such that
$$
 \sup_{\{u\in X_1, \|u \|\leq R\} \cup \{u\in X_1\oplus X_2 :\|u\|=R\}}
\mathcal J_\lambda(u)
<\inf_{\{u\in X_2\oplus X_3 : \|u \|=\rho\}}\mathcal J_\lambda(u)\,.
$$
\end{proposition}

\begin{proof}
Take $u\in X_1$. Then by \eqref{F2} it is straightforward to see that
\begin{equation}\label{primdis}
\mathcal J_\lambda(u)\leq \frac{\lambda_{k-1}-\lambda}{2}\int_\Omega u^2dx\leq 0,
\end{equation}
since $\lambda_{k-1}< \lambda$. Moreover, by \eqref{F1} there exists $M>0$ such that
\[
F(x,t)\geq (\lambda_k-\lambda)t^2-M
\]
for a.e. $x\in \Omega$ and all $t\in \mathbb R$. Thus, if
$u\in X_1\oplus X_2$, we obtain
\[
\mathcal J_\lambda(u)\leq \frac{\lambda_k-\lambda}{2}\int_\Omega u^2dx
-(\lambda_k-\lambda)\int_\Omega u^2dx+M|\Omega|
=-\frac{\lambda_k-\lambda}{2}\int_\Omega u^2dx+M|\Omega|,
\]
and, being all norms equivalent in $X_1\oplus X_2$, we obtain that
\begin{equation}\label{seconddis}
\lim_{u\in X_1\oplus X_2 ,\, \|u\|\to \infty} \mathcal J_\lambda(u)=-\infty.
\end{equation}

Now, by \eqref{f1} and \eqref{f2} we obtain that, fixed $\varepsilon>0$,
there exists $M_\varepsilon>0$ such that
\[
F(x,t)<\frac{\varepsilon}{2}t^2+M_\varepsilon|t|^q \quad
\text{for a.e. $x\in \Omega$ and all $t\in \mathbb R$}.
\]
Then, if $u\in X_2\oplus X_3$, by \eqref{poincarevincolata} and
\eqref{embedding} we obtain
\[
{\mathcal J_\lambda}(u)\geq \frac{1}{2}
\Big(1-\frac{\lambda+\varepsilon}{\lambda_k}\Big)\|u\|^2
-M_\varepsilon\int_\Omega|u|^qdx
\geq \frac{1}{2}\Big(1-\frac{\lambda+\varepsilon}{\lambda_k}\Big)\|u\|^2
-\tilde{M_\varepsilon}\|u\|^q
\]
for some $\tilde{M_\varepsilon}>0$. Choosing $\varepsilon<\lambda_k-\lambda$,
we can find $\rho>0$ so small that
\begin{equation}\label{infpositivo}
\inf_{\{u\in X_2\oplus X_3 : \|u \|=\rho\}}\mathcal J_\lambda(u)>0.
\end{equation}
By \eqref{primdis}, \eqref{seconddis} and \eqref{infpositivo}, the claim follows.
\end{proof}

\section{$\nabla$-condition}\label{sec:nablacondition}

To prove the $\nabla-$condition, we denote by
$P_C:X_0\to C$ the orthogonal projection of $X_0$ onto a closed subspace $C$,
and we recall the following concept.

\begin{definition}\label{def4.1} \rm
Let $C$ be a closed subspace of $X_0$ and $a,b\in \mathbb R\cup\{-\infty, +\infty\}$.
We say that $\mathcal J_\lambda$ verifies $(\nabla)(\mathcal J_\lambda, C, a, b)$
if there exists $\gamma>0$ such that
$$
\inf\big\{\|P_C \nabla \mathcal J_\lambda (u) \| :
a\leq \mathcal J_\lambda(u)\leq b,\operatorname{dist}(u, C)\leq \gamma\big\}>0\,.
$$
\end{definition}

Roughly speaking, the condition $(\nabla)(\mathcal J_\lambda, C, a, b)$
requires that $\mathcal J_\lambda$ has no critical points
$u\in C$ such that $a\leq \mathcal J_\lambda(u)\leq b$, with some uniformity.
The main purpose of this section is to prove the following result.

\begin{proposition}\label{propnabla}
Let $k$ and $m$ in $\mathbb N$ be such that \eqref{hpauto} holds and let $f$
satisfy {\rm (A1)}. Then, for any $\sigma>0$ with
$\sigma<\min\{\lambda_{k+m}-\lambda_k,\lambda_k-\lambda_{k-1}\}$
there exists $\varepsilon_\sigma>0$ such that for any
$\lambda\in [\lambda_{k-1}+\sigma, \lambda_{k+m}-\sigma]$ and for any
$\varepsilon', \varepsilon''\in (0, \varepsilon_\sigma)$, with
 $\varepsilon'<\varepsilon''$, functional~$\mathcal J_\lambda$ satisfies
 $(\nabla)(\mathcal J_\lambda, H_{k-1}\oplus H^\perp_{k+m-1}, \varepsilon',
\varepsilon'')$.
\end{proposition}

Of course, in our case $C= H_{k-1}\oplus H^\perp_{k+m-1}$, and without
mentioning any longer, we assume \eqref{hpauto} and (A1).
We start by proving the following result.

\begin{lemma}\label{lemmanabla1}
For any $\sigma$ such that
$0<\delta<\min\{\lambda_{k+m}-\lambda_k,\lambda_k-\lambda_{k-1} \}$
there exists $\varepsilon_\sigma>0$ such that for any
$\lambda\in [\lambda_{k-1}+\sigma, \lambda_{k+m}-\sigma]$
the unique critical point $u$ of $\mathcal J_\lambda$ constrained on
$H_{k-1}\oplus H^\perp_{k+m-1}$ with
$\mathcal J_\lambda(u)\in [-\varepsilon_\sigma, \varepsilon_\sigma]$,
is the trivial one.
\end{lemma}

\begin{proof}
We argue by contradiction and we suppose that there exists $\bar \sigma>0$,
a sequence $\{\mu_j\}_{j\in \mathbb N}$ in $\mathbb R$ with
\begin{equation}\label{lambdaj}
\mu_j\in [\lambda_{k-1}+\bar \sigma, \lambda_{k+m}-\bar \sigma]
\end{equation}
and a sequence $\{u_j\}_{j\in \mathbb N}\subset H_{k-1}\oplus
 H^\perp_{k+m-1}\setminus\{0\}$ such that
\begin{gather}\label{ujvincolato}
\langle\mathcal J'_{\mu_j}(u_j),\varphi\rangle =0\quad \text{for any }
 \varphi\in H_{k-1}\oplus H^\perp_{k+m-1} \text{ and any $j\in \mathbb N$}, \\
\label{ujvalorecritico}
\begin{aligned}
\mathcal J_{\mu_j}(u_j)
& =\frac 1 2 \int_{\mathbb R^n\times \mathbb R^n}|u_j -u_j(y)|^2 K(x-y)\,dx\,dy \\
&\quad -\frac{\mu_j}{2} \int_\Omega |u_j |^2\,dx
 - \int_\Omega F(x, u_j )\,dx\to 0\quad\text{as } j\to +\infty\,.
\end{aligned}
\end{gather}
By \eqref{f32} and \eqref{f1} we obtain the existence of $a_4>0$ such that
\begin{equation}\label{sopra}
f (x, t)t - 2F (x, t) \geq a_3 |t|^p -a_4 \quad
 \text{for a.e. $x\in \Omega$ and all $t\in \mathbb R$}.
\end{equation}
Taking $\varphi=u_j$ in \eqref{ujvincolato} and using \eqref{sopra},
we obtain that for any $j\in \mathbb N$,
\begin{align*}
 2\mathcal J_{\mu_j}(u_j)- \langle\mathcal J'_{\mu_j}(u_j), u_j\rangle
& =\int_{\Omega}\left(f(x,u_j)u_j-F(x,u_j)\right)dx\\
& \geq a_3\int_\Omega |u_j|^pdx-a_5
\end{align*}
for some positive constant $a_5$. Hence, by \eqref{ujvincolato} and
\eqref{ujvalorecritico}, we immediately get that
\begin{equation}\label{ujbounded}
(u_j)_{j\in \mathbb N} \quad \text{is bounded in $L^p(\Omega)$}.
\end{equation}

Now, let $v_j\in H_{k-1}$ and $w_j\in H^\perp_{k+m-1}$ be such that
 $u_j=v_j+w_j$ for any $j\in\mathbb N$. Choosing $\varphi=v_j-w_j$
in \eqref{ujvincolato} and taking into account the orthogonality properties
of $v_j$ and $w_j$, we have that for any $j\in \mathbb N$
\begin{equation}\label{split}
\begin{aligned}
0 & =\langle\mathcal J'_{\mu_j}(u_j), v_j-w_j\rangle\\
& = \int_{\mathbb R^n\times \mathbb R^n}|v_j(x) -v_j(y)|^2 K(x-y)\,dx\,dy \\
&\quad - \int_{\mathbb R^n\times \mathbb R^n}|w_j(x) -w_j(y)|^2 K(x-y)\,dx\,dy\\
 &\quad -\mu_j \int_\Omega |v_j |^2\,dx + \mu_j \int_\Omega |w_j |^2\,dx  
 - \int_\Omega f(x, u_j )(v_j -w_j )\,dx\,.
\end{aligned}
\end{equation}
By \eqref{antipoincare} and \eqref{poincarevincolata}, equation \eqref{split}
implies that for any $j\in \mathbb N$,
\begin{equation}\label{carvar2}
\begin{aligned}
\int_\Omega f(x, u_j )(v_j -w_j )\,dx 
& = \int_{\mathbb R^n\times \mathbb R^n}|v_j(x) -v_j(y)|^2 K(x-y)\,dx\,dy\\
&\quad - \int_{\mathbb R^n\times \mathbb R^n}|w_j(x) -w_j(y)|^2 K(x-y)\,dx\,dy\\
&\quad -\mu_j \int_\Omega |v_j |^2\,dx + \mu_j \int_\Omega |w_j |^2\,dx\\
& \leq \frac{\lambda_{k-1}-\mu_j}{\lambda_{k-1}}\|v_j \|^2
 + \frac{\mu_j-\lambda_{k+m}}{\lambda_{k+m}}\|w_j \|^2\\
& \leq -\frac{\bar \sigma}{\lambda_{k-1}}\|v_j \|^2
 - \frac{\bar \sigma}{\lambda_{k+m}}\|w_j \|^2\\
&\leq -\frac{\bar \sigma}{\lambda_{k+m}}\|u_j \|^2.
\end{aligned}
\end{equation}
Hence,
 there exists $C>0$ such that
 \begin{equation}\label{2.8}
 \|u_j\|^2\leq C \int_\Omega f(x, u_j )(v_j -w_j )\,dx \quad
 \text{for all }j\in \mathbb N.
 \end{equation}
Now, since
\[
\frac{2n}{n+2s}(q-1)\leq p<2^*,
\]
we immediately get that
\[
1<\frac{p}{p-q+1}\leq 2^*,
\]
and so, by \eqref{embedding}, there exists $C>0$ such that
\begin{equation}\label{2.11}
\|u\|_{\frac{p}{p-q+1}}\leq C\|u\| \quad \text{for every }u\in X_0.
\end{equation}
As a consequence, by the H\"older inequality and \eqref{ujbounded}, we obtain
\begin{equation}\label{2.9}
\int_\Omega |u_j|^{q-1}|v_j-w_j|\,dx
\leq \|u_j\|_p^{q-1}\|v_j-w_j\|_{p-q+1}
\leq c \|v_j-w_j\|_{L^{2^*}(\Omega)}
\end{equation}
for some $c>0$.
Moreover, by \eqref{f1} we obtain
\begin{equation}\label{primostep}
\begin{aligned}
\big|\int_\Omega f(x, u_j )(v_j -w_j )\,dx\big|
& \leq a_1\int_\Omega |v_j-w_j|\,dx+a_2\int_\Omega|u_j|^{q-1}|v_j-w_j|\,dx.
\end{aligned}
\end{equation}
Hence, by \eqref{primostep}, \eqref{embedding}, \eqref{2.8} and \eqref{2.9},
we obtain the existence of two constants $C_1,\,C_2>0$ such that for all
 $j\in \mathbb N$
\[
\|u_j\|^2\leq C_1\|v_j-w_j\|+C_2\|v_j-w_j\|=C_3\|v_j+w_j\|=C_3\|u_j\|,
\]
and so $(u_j)_{j\in\mathbb N}$ is bounded in $X_0$. Then, we can assume that
there exists $u_\infty\in H_{k-1}\oplus H^\perp_{k+m-1}$ such that,
by \eqref{ujvincolato},
\begin{gather}\label{convergenze0nabla}
\begin{aligned}
& \int_{\mathbb R^n\times \mathbb R^n}\big(u_j(x) -u_j(y)\big)
 \big(\varphi(x) -\varphi(y)\big) K(x-y)\,dx\,dy  \\
&\to \int_{\mathbb R^n\times \mathbb R^n}
 \big(u_\infty(x) -u_\infty(y)\big)\big(\varphi -\varphi(y)\big)
K(x-y)\,dx\,dy \quad \text{for any } \varphi\in X_0\,,
\end{aligned} \\
\label{convergenze0bisnabla}
\begin{gathered}
 u_j \to u_\infty \quad \text{in } L^q(\mathbb R^n)\\
 u_j \to u_\infty \quad \text{a.e. in } \mathbb R^n
\end{gathered}
\end{gather}
as $j\to +\infty$. Now, taking $\varphi=u_j$ in \eqref{ujvincolato}
 and using \eqref{sopra}, we obtain that for any $j\in \mathbb N$
$$
0= 2\mathcal J_{\mu_j}(u_j)- \langle\mathcal J'_{\mu_j}(u_j), u_j\rangle
=\int_{\Omega}\big(f(x,u_j)u_j-F(x,u_j)\big)dx.
$$
Passing to the limit in the equation above, by \eqref{f1} and
\eqref{convergenze0bisnabla}, we obtain
\[
0=\int_{\Omega}\left(f(x,u_\infty)u_\infty-F(x,u_\infty)\right)dx,
\]
and so \eqref{f31F2} implies  $u_\infty\equiv 0$.

From \eqref{2.8} we also obtain
\begin{align*}
\|u_j\|^2&\leq C \Big(\int_\Omega |f(x,u_j)|^{\frac{q}{q-1}}dx\Big)
\|v_j-w_j\|_{L^q(\Omega)} \\
&\leq \tilde C\Big(\int_\Omega |f(x,u_j)|^{\frac{q}{q-1}}dx\Big)\|v_j-w_j\|\\
&=\tilde C \Big(\int_\Omega |f(x,u_j)|^{\frac{q}{q-1}}dx\Big)\|u_j\|
\end{align*}
for some $\tilde C>0$ and all $j\in \mathbb N$. Since $u_j\neq 0$, we obtain
\begin{equation}\label{2.13}
\|u_j\|\leq C\Big(\int_\Omega |f(x,u_j)|^{\frac{q}{q-1}}dx\Big)
\end{equation}
for some $C>0$ and all $j\in \mathbb N$. Now, if $u_j\to 0$ in $X_0$,
from \eqref{2.13} and \eqref{f2} we would get
\[
1\leq \lim_{j\to \infty}C\frac{\big(\int_\Omega |f(x,u_j)|^{\frac{q}{q-1}}dx\big)}
{\|u_j\|}=0,
\]
which is absurd. Hence, we can assume that there is $A>0$ such that
$\|u_j\|\geq C$ for all $j\in \mathbb N$. Hence, \eqref{2.13} and the fact
that $u_j\to 0$ in $L^q(\Omega)$ would give
\[
A\leq \lim_{j\to \infty}C \Big(\int_\Omega |f(x,u_j)|^{\frac{q}{q-1}}dx\Big)=0,
\]
again a contradiction. The proof is complete.
\end{proof}

Before going on, we recall that, as showed in \cite{mbms}, we have
\begin{equation}\label{grad}
\nabla \mathcal J_\lambda(u)=u-\mathcal L_K^{-1}(\lambda u+f(x,u))
\end{equation}
for all $u\in X_0$, where
\begin{equation}\label{gradcomp}
\mathcal L_K^{-1}:L^\nu(\Omega) \to X_0
\text{ is a compact operator for all }\nu\in [1,2^*).
\end{equation}
Moreover,
\begin{equation}\label{scarica}
\langle u, \mathcal L_K^{-1}v\rangle_{X_0}=\int_\Omega u v \,dx
\end{equation}
for every $u,v\in X_0$.
The second lemma we need in order to prove the $\nabla$-condition
is the following one.

\begin{lemma}\label{lemmanabla2}
Let $\{u_j\}_{j\in \mathbb N}$ be a sequence in $X_0$ such that
\begin{gather}\label{condlemma1}
\{\mathcal J_\lambda(u_j)\}_{j\in \mathbb N} \text{ is bounded in } \mathbb R\,,\\
\label{condlemma2}
P_{{\rm{span}}\{e_k,\ldots,\,e_{k+m-1}\}} u_j\to 0\quad \text{in } X_0, \\
\label{condlemma3}
P_{H_{k-1}\oplus H^\perp_{k+m-1}}\nabla \mathcal J_\lambda(u_j)\to 0
\quad  \text{in } X_0 \text{ as $j\to +\infty$.}
\end{gather}
Then, $\{u_j\}_{j\in \mathbb N}$ is bounded in $X_0$.
\end{lemma}

\begin{proof}
Assume by contradiction that $\{u_j\}_{j\in \mathbb N}$ is unbounded in $X_0$;
 hence, we can assume that
\begin{equation}\label{ujtoinfty}
\|u_j \|\to +\infty
\end{equation}
as $j\to +\infty$ and that there exists $u_\infty\in X_0$ such that
\begin{equation}\label{convnorm}
\begin{gathered}
 \frac{u_j}{\|u_j \|}\rightharpoonup u_\infty\quad \text{in }X_0\\
 \frac{u_j}{\|u_j \|}\to u_\infty\quad
\text{in $L^\nu(\Omega)$ for any $\nu\in[1,2^*)$}
\end{gathered}
\end{equation}
as $j\to +\infty$.

Now, for shortness, set
$P_{{\operatorname{span}}\left\{e_k,\ldots,\,e_{k+m-1}\right\}}=:P$,
$P_{H_{k-1}\oplus H^\perp_{k+m-1}}=:Q$, and write
$$
u_j=Pu_j+Qu_j\,,$$ where $Pu_j\to 0$ as $j\to\infty$ by \eqref{condlemma2}.

First, by \eqref{f1} and H\"older's inequality, since $p>q-1$, we have that
there exists $c_1>0$ such that for a.e. $x\in \Omega$ and all $j\in \mathbb N$
$$
|f(x,u_j )Pu_j | \leq c_1\|Pu_j\|_\infty\big(1+\|u_j\|^{q-1}_p\big).
$$
Recalling \eqref{grad}, we have
\begin{equation}\label{add11}
\begin{aligned}
\langle Q\nabla \mathcal J_\lambda(u_j),u_j\rangle_{X_0}&= \langle \nabla
\mathcal J_\lambda(u_j),u_j\rangle_{X_0} - \langle P\nabla
\mathcal J_\lambda(u_j),u_j\rangle_{X_0}\\
&=\|u_j \|^2-\lambda \int_\Omega |u_j |^2\,dx
-\int_\Omega f(x,u_j )u_j \, dx \\
&\quad -\langle P\big(u_j-\mathcal L_K^{-1}(\lambda
 u_j+f(x,u_j))\big), u_j\rangle_{X_0}.
\end{aligned}
\end{equation}
Since $\langle Pu,v\rangle_{X_0}=\langle u,Pv\rangle_{X_0}$ for any $u,v\in X_0$,
by \eqref{scarica} \eqref{add11} reads
\begin{equation}\label{sec}
\begin{aligned}
\langle Q\nabla \mathcal J_\lambda(u_j),u_j\rangle_{X_0}
& =2\mathcal J_\lambda(u_j)+
2\int_\Omega F(x,u_j ) dx -\int_\Omega f(x,u_j )u_j dx\\
&\quad -\|Pu_j \|^2 +\lambda \int_\Omega |Pu_j |^2\, dx+\int_\Omega f(x,u_j )Pu_j dx.
\end{aligned}
\end{equation}
As a consequence, by \eqref{f32} there exists $c_2>0$ such that
\begin{align*}
2\mathcal J_\lambda(u_j)-\langle Q\nabla \mathcal J_\lambda(u_j),u_j\rangle_{X_0}
&=\int_\Omega \left(f(x,u_j )u_j dx-2F(x,u_j )\right)dx\\
&\quad +\|Pu_j \|^2 -\lambda \int_\Omega |Pu_j |^2\, dx
 -\int_\Omega f(x,u_j )Pu_j \,dx\\
&\geq a_3\int_\Omega |u_j|^pdx-c_2+\|Pu_j \|^2
 -\lambda \int_\Omega |Pu_j |^2\, dx\\
&\quad -c_1\|Pu_j\|_\infty\big(1+\|u_j\|^{q-1}_p\big).
\end{align*}
Recalling \eqref{condlemma1}-\eqref{condlemma3} and that $p>q-1$, we easily obtain
\begin{equation}\label{pq-1}
\lim_{j\to \infty} \frac{\|u_j\|^{q-1}_p}{\|u_j\|}=0,
\end{equation}
since $X_2$ has finite dimension, and so all norms are equivalent.
As a consequence of \eqref{pq-1} we also get
\begin{equation}\label{uinfty0}
u_\infty=0.
\end{equation}

Now, by \eqref{condlemma1}, \eqref{ujtoinfty} and \eqref{uinfty0} we obtain
$$
\frac{\mathcal J_\lambda(u_j)}{\|u_j \|^2}=\frac 1 2-\frac \lambda 2\frac{ \int_\Omega |u_j |^2\,
dx}{\|u_j \|^2}-\frac{ \int_\Omega F(x,u_j )\,
dx}{\|u_j \|^2}\to 0,
$$
which implies that
\begin{equation}\label{Funmezzo}
\frac{ \int_\Omega F(x,u_j )\, dx}{\|u_j \|^2}\to \frac 1 2
\end{equation}
as $j\to +\infty$. But, by \eqref{f1}, proceeding as for \eqref{2.11},
\[
\big|\int_\Omega F(x,u_j)\,dx\big|
\leq a_1\int_\Omega |u_j|\,dx+\frac{a_2}{q}\int_\Omega |u_j|^qdx
\leq \tilde{a_1}\|u_j\|+\tilde{a_2}\|u_j\|_p^{q-1}\|u_j\|,
\]
and by \eqref{pq-1} we obtain a contradiction with \eqref{Funmezzo}.
\end{proof}

As a consequence of Lemmas \ref{lemmanabla1} and \ref{lemmanabla2},
 we are  able to prove Proposition \ref{propnabla}.

\begin{proof}[Proof of Proposition \ref{propnabla}]
Assume by contradiction that there exists $\sigma>0$ such that for every
$\varepsilon_0>0$ there exist
$\bar \lambda\in [\lambda_{k-1}+\sigma,\lambda_{k+m}-\sigma]$
and $\varepsilon'<\varepsilon''$ in $(0,\varepsilon_0)$ such that
\begin{equation}\label{notnabla}
(\nabla)(\mathcal J_{\bar\lambda}, H_{k-1}\oplus H^\perp_{k+m-1}, \varepsilon',
 \varepsilon'')\quad \text{does not hold}.
\end{equation}
Take $\varepsilon>0$ associated to $\sigma$ according to Lemma~\ref{lemmanabla1}.

By \eqref{notnabla} we can find a sequence $\{u_j\}_{j\in \mathbb N}$ in $X_0$
such that
\begin{equation}\label{ujadd}
\begin{gathered}
 \mathcal J_{\bar\lambda}(u_j)\in [\varepsilon',\varepsilon'']\quad
  \text{for all } j\in \mathbb N\,,\\
 \operatorname{dist}(u_j,H_{k-1}\oplus H^\perp_{k+m-1})\to 0\\
 P_{H_{k-1}\oplus H^\perp_{k+m-1}}\nabla \mathcal J_{\bar\lambda}(u_j)\to 0\quad
 \text{in } X_0
\end{gathered}
\end{equation}
as $j\to +\infty$.

By Lemma~\ref{lemmanabla2} we obtain that $\{u_j\}_{j\in \mathbb N}$
is bounded in $X_0$, and so there exists $u_\infty\in X_0$ such that,
up to a subsequence,
\begin{equation}\label{convujadd}
\begin{gathered}
 u_j\rightharpoonup u_\infty\quad  \text{in } X_0\\
 u_j\to u_\infty\quad \text{in } L^\nu(\Omega) \text{ for any } \nu\in[1, 2^*)\\
 u_j\to u_\infty\quad \text{a.e. in } \Omega
\end{gathered}
\end{equation}
as $j\to +\infty$.

Now, by \eqref{grad} we have
\begin{equation}\label{add55}
\begin{aligned}
P_{H_{k-1}\oplus H^\perp_{k+m-1}} \nabla \mathcal J_{\bar\lambda}(u_j)
& =u_j -P_{{\operatorname{span}}\left\{e_k,\ldots,\,e_{k+m-1}\right\}}u_j\\
&\quad -P_{H_{k-1}\oplus H^\perp_{k+m-1}}\mathcal L_K^{-1}(\bar\lambda u_j+f(x,u_j)).
\end{aligned}
\end{equation}
Hence, recalling that $\mathcal L_K^{-1}:L^{q'}(\Omega)\to X_0$ is a
compact operator, see \eqref{gradcomp}, and that $f(x,u_j)\to f(x,u_\infty)$ in
$L^{q'}(\Omega)$ by Krasnoselskii's Theorem, see \cite[Theorem 2.75]{MMP},
we obtain that
$$
P_{H_{k-1}\oplus H^\perp_{k+m-1}}\mathcal L_K^{-1}(\bar \lambda
u_j+f(x,u_j))\to P_{H_{k-1}\oplus H^\perp_{k+m-1}}\mathcal L_K^{-1}(\bar \lambda
u_\infty+f(x,u_\infty))
$$
as $j\to +\infty$ and so, taking into account \eqref{ujadd}, \eqref{convujadd}
and \eqref{add55}, we deduce that
\begin{equation}\label{ujforte}
u_j\to P_{H_{k-1}\oplus H^\perp_{k+m-1}}\mathcal L_K^{-1}(\bar \lambda
u_\infty+f(x,u_\infty))=u_\infty\quad \text{in } X_0
\end{equation}
as $j\to +\infty$.

Moreover, again by \eqref{ujadd}, we obtain that $u_\infty$
is a critical point of $\mathcal J_{\bar\lambda}$ constrained on
$H_{k-1}\oplus H^\perp_{k+m-1}$.
Hence, Lemma~\ref{lemmanabla1} yields that $u_\infty\equiv 0$.
However, $0<\varepsilon'\leq \mathcal J_{\bar\lambda}(u_j)$ for every
$j\in \mathbb N$, so that, by continuity of $\mathcal J_{\bar\lambda}$,
we find $\mathcal J_{\bar\lambda}(u_\infty)>0$, which is absurd.
\end{proof}

\section{Proof of Theorem\ref{thmain}} \label{sec:proofthmain}

The proof of Theorem \ref{thmain} relies on the combination of
Theorem \ref{thmarinosaccon} below with a classical Linking Theorem,
see \cite[Theorem~5.3]{rabinowitz}.

\begin{theorem}[{\cite[Theorem 2.10]{marsac:sns}}] \label{thmarinosaccon}
Let $H$ be a Hilbert space and $X_1, X_2, X_3$ be three subspaces of $H$
such that $H=X_1\oplus X_2 \oplus X_3$ with $0<\text{dim}\,X_i<\infty$ for $i=1,2$.
Let $\mathcal I:H\to \mathbb R$ be a $C^{1,1}$ functional.
Let $\rho, \rho', \rho'', \rho_1$ be such that $0<\rho_1$,
$0\leq \rho'<\rho<\rho''$ and
$$
\Delta=\{u\in X_1\oplus X_2 : \rho'\leq \|P_2u\|\leq \rho'',
\|P_1u\|\leq \rho_1\}, \quad
T=\partial_{X_1\oplus X_2}\Delta\,,
$$
where $P_i:H\to X_i$ is the orthogonal projection of $H$ onto $X_i$\,, $i=1,2$,
and
$$
S_{23}(\rho)=\{u\in X_2\oplus X_3 : \|u\|=\rho\},\quad
B_{23}(\rho)=\{u\in X_2\oplus X_3 : \|u\|<\rho\}\,.
$$
Assume that
$$
a'=\sup \mathcal I(T)<\inf \mathcal I(S_{23}(\rho))=a''\,.
$$
Let $a,b$ be such that $a'<a<a''$, $b>\sup \mathcal I(\Delta)$ and
the assumption
$(\nabla)(\mathcal I, X_1\oplus X_3, a, b)$ holds;
the Palais-Smale condition holds at any level $c\in [a,b]$.
Then $\mathcal I$ has at least two critical points in $\mathcal I^{-1}([a,b])$.

If, furthermore,
$$
-\infty<\inf \mathcal I(B_{23}(\rho)), \quad\text{and}\quad
 a_1< \inf \mathcal I(B_{23}(\rho)),
$$
and the Palais-Smale condition holds at every $c\in [a_1, b]$,
then $\mathcal I$ has another critical level between $a_1$ and $a'$.
\end{theorem}

Hence, let us start showing that $\mathcal J_\lambda$ satisfies the
\emph{Palais-Smale condition} at any level, i.e. for all $c\in \mathbb R$
every sequence $\{u_j\}_{j\in \mathbb N}\subset X_0$ such that
\begin{gather}\label{Jc0}
\mathcal J_\lambda(u_j)\to c, \\
\label{J'00}
\mathcal J'_\lambda(u_j)\to 0\quad \text{in }X_0'
\end{gather}
as $j\to +\infty$, admits a strongly convergent subsequence in $X_0$.

\begin{proposition}\label{lemmaPS}
Let $\lambda>0$ and let $f$ satisfy {\rm (A1)}. Then,
$\mathcal J_\lambda$ satisfies the Palais-Smale condition at any
level~$c\in \mathbb R$.
\end{proposition}

\begin{proof}
Let $c\in \mathbb R$ and let $\{u_j\}_{j\in \mathbb N}$ be a sequence
satisfying \eqref{Jc0} and \eqref{J'00}. Assume by contradiction that
$\{u_j\}_{j\in \mathbb N}$ is not bounded, and so assume that
$\|u_j\|\to \infty$ as $j\to \infty$. We claim that
\begin{equation}\label{dadimostrar}
\frac{Pu_j}{\|u_j\|}\to 0 \quad \text{as }j\to \infty,
\end{equation}
where $P$ is the same projection of Lemma \ref{lemmanabla2}.
Indeed, by \eqref{f32} we can find $A,B>0$ such that
\[
f(x,t)t- 2F(x,t)\geq A|t|-B \quad \text{for a.e. $x\in \mathbb R$ and all $t\in \mathbb R$}.
\]
Then
\begin{equation}\label{quasidimo}
\begin{aligned}
2\mathcal J_\lambda(u_j)-\mathcal J_\lambda'(u_j)(u_j)
&=\int_\Omega \left(f(x,u_j )u_j dx-2F(x,u_j )\right)dx\\
&\geq A\int_\Omega|u_j|\,dx-B|\Omega|\\
&\geq \|Pu_j \|_1 -A \|Qu_j \|_1-B|\Omega|,
\end{aligned}
\end{equation}
where $Q$ is as in Lemma \ref{lemmanabla2}.

Now, write $Qu_j=v_j+w_j$, where $v_j\in X_1$ and $w_j\in X_3$ for every
$j\in \mathbb N$. As in \eqref{2.11}, we obtain
\begin{gather}\label{2.15}
\int_\Omega|u_j|^{q-1}|v_j|dx\leq C\|u_j\|^{q-1}_p\|v_j\|, \\
\label{2.16}
\int_\Omega|u_j|^{q-1}|w_j|dx\leq C\|u_j\|^{q-1}_p\|w_j\|
\end{gather}
for some $C>0$ and all $j\in \mathbb N$.
Hence, by \eqref{J'00} and \eqref{f32} there exists $c_2>0$ such that
\begin{align*}
2\mathcal J_\lambda(u_j)-\langle \nabla \mathcal J_\lambda(u_j),u_j\rangle_{X_0}
&=\int_\Omega \left(f(x,u_j )u_j dx-2F(x,u_j )\right)dx\\
&\geq a_3\int_\Omega |u_j|^pdx-c_2,
\end{align*}
so that
\begin{equation}\label{pq}
\lim_{j\to \infty} \frac{ \int_\Omega |u_j|^pdx}{\|u_j\|}=0,
\end{equation}
and so also \eqref{pq-1} holds again.

Now, by \eqref{J'00}, \eqref{f1}, \eqref{antipoincare} and \eqref{2.15} we obtain
\begin{align*}
\|v_j\|o(1)
&=\langle \nabla {\mathcal J_\lambda}(u_j),-v_j\rangle_{X_0} \\
&=-\|v_j\|^2+\lambda\int_\Omega v_j^2dx+\int_\Omega f(x,u_j)v_j\,dx\\
&\geq \Big(-1+\frac{\lambda}{\lambda_{k-1}}\Big)\|v_j\|^2
 -a_1\|v_j\|_1-a_2\int_\Omega|u_j|^{q-1}|v_j|\,dx\\
&\geq \frac{\lambda-\lambda_{k-1}}{\lambda_{k-1}}\|v_j\|^2
 -c\|v_j\|-d\|u_j\|^{q-1}_p\|v_j\|\\
&=\frac{\lambda-\lambda_{k-1}}{\lambda_{k-1}}\|v_j\|^2-\|v_j\|
 \big(c+d\|u_j\|^{q-1}_p\|v_j\|\big)
\end{align*}
for some constants $c,d>0$ and where $o(1)\to 0 $ as $j\to \infty$.
By \eqref{pq-1}, the previous inequality implies that
\begin{equation}\label{2.21}
\lim_{j\to \infty}\frac{\|v_j\|}{\|u_j\|}=0.
\end{equation}

Similarly, by using \eqref{poincarevincolata}, we find
\begin{align*}
\|w_j\|o(1)
&=\langle \nabla {\mathcal J_\lambda}(u_j),w_j\rangle_{X_0}\\
&=\|w_j\|^2-\lambda\int_\Omega w_j^2dx-\int_\Omega f(x,u_j)w_j\,dx\\
&\geq \Big(1-\frac{\lambda}{\lambda_{k+m}}\Big)\|v_j\|^2-a_1\|w_j\|_1
 -a_2\int_\Omega|u_j|^{q-1}|w_j|\,dx\\
&\geq \frac{\lambda_{k+m}-\lambda}{\lambda_{k-1}}\|w_j\|^2
 -c\|w_j\|-d\|u_j\|^{q-1}_p\|w_j\|\\
&=\frac{\lambda_{k+m}-\lambda}{\lambda_{k+m}}\|w_j\|^2-\|w_j\|
 (c+d\|u_j\|^{q-1}_p\|w_j\|),
\end{align*}
and by \eqref{pq-1} we find that
\begin{equation}\label{2.22}
\lim_{j\to \infty}\frac{\|w_j\|}{\|u_j\|}=0.
\end{equation}
By \eqref{2.21} and \eqref{2.22}, recalling that $Qu_j=v_j+w_j$, we finally get
\begin{equation}\label{qu_j}
\lim_{j\to \infty}\frac{\|Qu_j\|}{\|u_j\|}=0.
\end{equation}

Since by \eqref{embedding} there exists $c>0$ such that
\[
\|Qu_j\|_1\leq c\|Qu_j\|,
\]
using \eqref{qu_j} in \eqref{quasidimo}, being $X_2$ finite-dimensional,
\eqref{dadimostrar} holds.

Now, proceeding as in the proof of Lemma \ref{lemmanabla2} we finally
find that $ \{u_j\}_{j\in \mathbb N}$ is bounded in $X_0$.
By \eqref{f1} it is standard to prove that $ \{u_j\}_{j\in \mathbb N} $
is pre-compact, and so the Palais-Smale condition holds at every level.
\end{proof}

\begin{lemma}\label{limite}
Assume \eqref{hpauto} and {\rm (A1)}. Then
\[
\lim_{\lambda \to \lambda_k}\sup_{u\in H_{k+m-1}} \mathcal J_\lambda(u)=0.
\]
\end{lemma}

\begin{proof}
First of all, note that $\mathcal J_\lambda$ attains a maximum in $H_{k+m-1}$
by \eqref{F1}.

Now, assume by contradiction that there exist $\{\mu_j\}_{j\in \mathbb N}$,
such that
\begin{equation}\label{muj}
\mu_j\to \lambda_k
\end{equation}
as $j\to +\infty$, $\{u_j\}_{j\in \mathbb N}$
in $H_{k+m-1}$ and $\varepsilon>0$ such that for any $j\in \mathbb N$
\begin{equation}\label{sup}
\mathcal J_{\mu_j}(u_j)=\max_{u\in H_{k+m-1}}
\mathcal J_{\mu_j} (u)\geq \varepsilon.
\end{equation}
If $\{u_j\}_{j\in \mathbb N}$ were bounded, we could assume that
$u_j\to u_\infty$ in $H_{k+m-1}$. Then, by \eqref{muj} we would get
$$
\mathcal J_{\mu_j}(u_j)\to \mathcal J_{\lambda_k}(u_\infty)
$$
as $j\to +\infty$. By \eqref{sup}, \eqref{antipoincare} and \eqref{F2} we
would find that
\begin{align*}
\varepsilon
&\leq \mathcal J_{\lambda_k}(u_\infty)
 =\frac{1}{2}\|u_\infty \|^2-\frac{\lambda_k}{2}\int_\Omega
| u_\infty |^2\, dx -\int_\Omega F(x,u_\infty )\, dx\\
& \leq \frac 1 2 (\lambda_{k+m-1}-\lambda_k)\int_\Omega
|u_\infty |^2\, dx -\int_\Omega F(x,u_\infty )\, dx\leq 0,
\end{align*}
which is absurd.

Otherwise, if $\{u_j\}_{j\in\mathbb N}$ were unbounded in $X_0$,
we could assume that
$\|u_j \|\to +\infty$ as $j\to +\infty$. Therefore, \eqref{sup} and
\eqref{F1} would imply
\begin{equation}\label{ciribi}
0<\varepsilon\leq \mathcal J_{\mu_j}(u_j)= \frac{1}{2}\|u_j \|^2-
\frac{\mu_j}{2}\int_\Omega |u_j |^2\, dx-\int_\Omega F(x,u_j)\, dx.
\end{equation}
Notice that \eqref{F2} and Fatou's Lemma imply, since all norms are equivalent
in $H_{k+m-1}$, that
\[
\lim_{j\to \infty}\int_\Omega \frac{F(x,u_j)}{\|u_j\|^2}dx=+\infty,
\]
and so from \eqref{ciribi} we would get
\[
0<\varepsilon\leq \mathcal J_{\mu_j}(u_j)
= \|u_j \|^2\Big(\frac{1}{2}-
\frac{\mu_j}{2}\int_\Omega \frac{|u_j |^2}{\|u_j\|^2}\, dx
-\int_\Omega \frac{F(x,u_j)}{\|u_j\|^2}dx\Big)=-\infty
\]
another contradiction, and so the lemma holds.
\end{proof}

Applying Theorem \ref{thmarinosaccon} to $\mathcal J_\lambda$ we have
a preliminary result.

\begin{proposition}\label{2soluzioni}
Assume \eqref{hpauto} and {\rm (A1)}. Then, there exists a left neighborhood
 $\mathcal O_k$ of $\lambda_k$ such that for all $\lambda \in \mathcal O_k$,
problem~\eqref{problemaK} has two nontrivial solutions $u_i$ such that
$$
0<\mathcal J_\lambda(u_i)\leq \sup_{u\in H_{k+m-1}} \mathcal J_\lambda(u)
$$
for $i=1,2$.
\end{proposition}

\begin{proof}
To apply Theorem~\ref{thmarinosaccon} to $\mathcal J_\lambda$, fix $\sigma>0$
and find $\varepsilon_\sigma$ as in Proposition~\ref{propnabla}.
Then, for all $\lambda\in [\lambda_{k-1}+\sigma, \lambda_{k+m}-\sigma]$
and for every $\varepsilon',\varepsilon''\in(0,\varepsilon_\sigma)$,
functional $\mathcal J_\lambda$ satisfies
$(\nabla)(\mathcal J_\lambda, H_{k-1}\oplus H^\perp_{k+m-1}, \varepsilon',
\varepsilon'')$.

By Lemma~\ref{limite} there exists $\sigma_1\leq\sigma$ such that, if
$\lambda \in (\lambda_k-\sigma_1,\lambda_k)$, then
\begin{equation}\label{epsilonsup}
\sup_{u\in H_{k+m-1}} \mathcal J_\lambda(u)=\varepsilon''.
\end{equation}
Moreover, since $\lambda<\lambda_k$, Proposition~\ref{propgeometria} holds
 and $\mathcal J_\lambda$ satisfies the Palais-Smale condition at any level
by Proposition~\ref{lemmaPS}.

Then, we can apply Theorem~\ref{thmarinosaccon} and find two critical points
$u_1,u_2$ of $\mathcal J_\lambda$ with
\begin{equation}\label{epsilon}
\mathcal J_\lambda(u_i)\in [\varepsilon',\varepsilon''],\quad i=1,2,
\end{equation}
i.e. $u_1$ and $u_2$ are nontrivial solutions of
\eqref{problemaK} such that
$$
0<\mathcal J_\lambda(u_i)\leq \varepsilon'',\quad  i=1,2.
$$
\end{proof}

We are now ready to conclude with the following result.

\begin{proof}[Proof of Theorem \ref{thmain}]
Mimicking the proof of Proposition \ref{propgeometria} we see that for every
$u\in H_{k+m-1}^\perp$,
\[
 {\mathcal J_\lambda}(u)\geq \frac{1}{2}
\Big(1-\frac{\lambda+\varepsilon}{\lambda_{k+m}}\Big)\|u\|^2
 -\tilde{M_\varepsilon}\|u\|^q,
\]
so that, for $\varepsilon$ small, there exists $\rho>0$ such that
\[
\inf_{u\in H_{k+m-1}^\perp,\,  \|u\|=\rho}{\mathcal J_\lambda}(u)
\geq \frac{1}{2}\Big(1-\frac{\lambda+\varepsilon}{\lambda_{k+m}}\Big)\rho^2
 -\tilde{M_\varepsilon}\rho^q:=\alpha_\rho>0.
\]
By Lemma~\ref{limite}, we can choose $\lambda$ so close to $\lambda_k$ that
\begin{equation}\label{confr}
\sup_{u\in H_{k+m-1}} \mathcal J_\lambda (u)< \alpha_\rho.
\end{equation}
Hence, the classical Linking Theorem ensures the existence of a solution
$u_3$ of problem~\eqref{problemaK} with
\begin{equation}\label{stimasup}
\mathcal J_\lambda(u_3)
\geq \inf_{u\in H^\perp_{k+m-1},\, \|u\|=\varrho}
\mathcal J_\lambda(u)\geq \alpha_\rho.
\end{equation}
Choosing $\sigma_1$ such that in \eqref{epsilonsup} $\varepsilon''<\alpha_\rho$,
we obtain
$$
\mathcal J_\lambda(u_i)\leq \sup_{u\in H_{k+m-1}} \mathcal J_\lambda(u)
<\mathcal J_\lambda(u_3)
$$
and so $u_3\neq u_i,$ $i=1,2$. The proof of Theorem~\ref{thmain} is complete.
\end{proof}

\subsection*{Acknowledgments}
The author is a member of GNAMPA and is supported by the INdAM-GNAMPA
Project 2017 {\it Nonlinear differential equations},
by the MIUR National Research Project {\it Variational methods, with applications
to problems in mathematical physics and geometry}
(2015KB9WPT\_009) and by the FFABR
``Fondo per il finanziamento delle attivit\`a base di ricerca'' 2017.

\begin{thebibliography}{99}

\bibitem{adams}  R. A. Adams; \emph{Sobolev Spaces},
Academic Press, New York, 1975.

\bibitem{barrioscoloradoservadei}  B. Barrios, E. Colorado, R. Servadei, F. Soria;
 \emph{A critical fractional equation with concave-convex power nonlinearities},
to appear in Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire.

\bibitem{cm}  G. Carboni, D. Mugnai;
 \emph{On some fractional equations with convex-concave and logistic-type
nonlinearities}, J. Differential Equations \textbf{262} (2017), 2393--2413.

\bibitem{valpal} E. Di Nezza, G. Palatucci, E. Valdinoci;
\emph{Hitchhiker's guide to the fractional Sobolev spaces},
 Bull. Sci. Math., \textbf{136} (2012), 521--573.

\bibitem{dsv}  S. Dipierro, N. Soave, E. Valdinoci;
\emph{On fractional elliptic equations in Lipschitz sets and epigraphs:
regularity, monotonicity and rigidity results}, Math. Ann. \textbf{369} (2017),
1283--1326.

\bibitem{LAO}  C. Li, R. P. Agarwal, Z.-Q. Ou;
 \emph{Existence of three nontrivial solutions for a class of fourth-order
elliptic equations}, Topol. Methods Nonlinear Anal., to appear.

\bibitem{FMV}  M. M. Fall, F. Mahmoudi, E. Valdinoci;
\emph{Ground states and concentration phenomena for the fractional Schr\"odinger
equation}, Nonlinearity \textbf{28} (2015), 1937--1961.

\bibitem{mms} P. Magrone, D. Mugnai, R. Servadei;
\emph{Multiplicity of solutions for semilinear variational inequalities
via linking and $\nabla-$theorems}, J. Differential Equations \textbf{228} (2006),
191--225.

\bibitem{MMu}  A. Marino, D. Mugnai;
\emph{Asymptotical multiplicity and some reversed variational inequalities},
Top. Meth. Nonlin. Anal., \textbf{ 20} (2002), 43--62.

\bibitem{marsac:sns}  A. Marino, C. Saccon;
\emph{Some variational theorems of mixed type and elliptic problems with
jumping nonlinearities}, Ann. Scuola Norm. Sup. Pisa Cl. Sci.,
(4) \textbf{25} (1997), 631--665.

\bibitem{marsac}  A. Marino, C. Saccon;
\emph{Nabla theorems and multiple solutions for some noncooperative
elliptic systems}, Topol. Methods Nonlinear Anal., \textbf{ 17} (2001), 213--237.

\bibitem{mbms}   G. Molica Bisci, D. Mugnai, R. Servadei;
\emph{On multiple solutions for nonlocal fractional problems via
$\nabla-$theorems}, Differential Integral Equations ,\textbf{30} (2017), 641--666.

\bibitem{152}  G. Molica Bisci, B. A. Pansera;
\emph{Three weak solutions for nonlocal fractional equations},
 Adv. Nonlinear Stud., \textbf{ 14} (2014), 591--601.

\bibitem{molicalibro}  G. Molica Bisci, V. Radulescu, R. Servadei;
\emph{Variational Methods for Nonlocal Fractional Problems}, Cambridge
 University Press, Encyclopedia Math. Appl., 2016.

\bibitem{molicaservadei}  G. Molica Bisci, R. Servadei;
\emph{A bifurcation result for non-local fractional equations},
Anal. Appl., \textbf{13} (2015), no. 4, 371--394.

\bibitem{MMP}  D. Motreanu, V. V. Motreanu, N. S. Papageorgiou;
\emph{Topological and Variational Methods with Applications to Nonlinear
Boundary Value Problems}, Springer, New York (2014).

\bibitem{mug5}  D. Mugnai;
\emph{On a five critical points theorem}, Lect. Notes Semin. Interdiscip.
Mat. \textbf{13} (2016), 59--65.

\bibitem{Mug}  D. Mugnai;
\emph{On a reversed variational inequality}, Top.
Meth. Nonlin. Anal., \textbf{17}(2001), 321--358.

\bibitem{mugnaiNODEA}  D. Mugnai;
\emph{Multiplicity of critical points in presence of a
linking: application to a superlinear boundary value problem},
NoDEA Nonlinear Differential Equations Appl., \textbf{11} (2004), 379--391.

\bibitem{mug4}  D. Mugnai;
\emph{Four nontrivial solutions for subcritical exponential equations},
 Calc. Var. Partial Differential Equations, \textbf{32} (2008), 481--497.

\bibitem{dmnodea2}  D. Mugnai;
\emph{Addendum to: Multiplicity of critical points in presence of a linking:
application to a superlinear boundary value problem, NoDEA.
Nonlinear Differential Equations Appl. 11 (2004), no. 3, 379--391,
and a comment on the generalized Ambrosetti-Rabinowitz condition},
Nonlinear Differ. Equ. Appl., \textbf{19} (2012), 299--301.

\bibitem{mugpag}  D. Mugnai, D. Pagliardini;
\emph{Existence and multiplicity results for the fractional Laplacian in
bounded domains}, Adv. Calc. Var., to appear, DOI: 10.1515/acv-2015-0032.

\bibitem{OL}  Z.-Q. Ou, C. Li;
 \emph{Existence of three nontrivial solutions for a class of superlinear
elliptic equations}, J. Math. Anal. Appl., \textbf{390} (2012), 418--426.

\bibitem{rabinowitz}  P. H. Rabinowitz;
\emph{Minimax methods in critical point theory with applications to differential
equations},  CBMS Reg. Conf. Ser. Math. \textbf{65}
American Mathematical Society, Providence, RI (1986).

\bibitem{sY} R. Servadei;
\emph{The Yamabe equation in a non-local setting},
Adv. Nonlinear Anal., \textbf{2} (2013), 235--270.

\bibitem{svmountain} R. Servadei and E. Valdinoci;
\emph{Mountain Pass solutions for non-local elliptic operators},
J. Math. Anal. Appl., \textbf{389} (2012), 887--898.

\bibitem{svlinking} R. Servadei, E. Valdinoci;
\emph{Variational methods for non-local operators of elliptic type},
 Discrete Contin. Dyn. Syst., \textbf{33} (2013), 2105--2137.

\bibitem{fwang}  F. Wang;
\emph{Multiple solutions for some nonlinear Schr\"odinger equations with
indefinite linear part}, J. Math. Anal. Appl., \textbf{ 331} (2007), 1001--1022.

\bibitem{fwang2}  F. Wang;
\emph{Multiple solutions for some Schr\"odinger equations with convex and critical
nonlinearities in $\mathbb R^N$}, J. Math. Anal. Appl., \textbf{342} (2008), 255--276.

\bibitem{wzz}  W. Wang, A. Zang, P. Zhao;
\emph{Multiplicity of solutions for a class of fourth elliptic equations},
Nonlinear Anal., \textbf{70} (2009), 4377--4385.

\end{thebibliography}

\end{document}