\documentclass[reqno]{amsart}
\usepackage[notref,notcite]{showkeys}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 86, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/86\hfil Existence of solutions]
{Existence of solutions to nonlocal Kirchhoff equations
of elliptic type via genus theory}

\author[N. Nyamoradi, N. T. Chung \hfil EJDE-2014/86 \hfilneg]
{Nemat Nyamoradi, Nguyen Thanh Chung}  

\address{Nemat Nyamoradi \newline
Department of Mathematics, Faculty of Sciences,
Razi University, 67149 Kermanshah, Iran}
\email{nyamoradi@razi.ac.ir, neamat80@yahoo.com}

\address{Nguyen Thanh Chung  \newline
Department of Mathematics, Quang Binh University,
312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Vietnam}
\email{ntchung82@yahoo.com}


\thanks{Submitted December 15, 2013. Published April 2, 2014.}
\subjclass[2000]{34B27, 35J60, 35B05}
\keywords{Kirchhoff nonlocal operators; fractional differential equations;
 \hfill\break\indent genus properties; critical point theory}

\begin{abstract}
 In this article, we study the existence and multiplicity of
 solutions to the nonlocal Kirchhoff fractional equation
 \begin{gather*}
 \Big(a + b\int_{\mathbb{R}^{2N}} |u (x) - u (y)|^2 K (x - y)\,dx\,dy\Big)
 (- \Delta)^s u - \lambda u  = f (x, u (x)) \quad \text{in }   \Omega,\\
 u = 0 \quad \text{in }   \mathbb{R}^N \setminus \Omega,
 \end{gather*}
 where $a, b > 0$ are constants, $(- \Delta)^s$ is the fractional
 Laplace operator, $s \in (0, 1)$ is a fixed real
 number, $\lambda$ is a real parameter and $\Omega$ is an open bounded subset
 of $\mathbb{R}^N$, $N > 2 s$, with Lipschitz boundary,
 $f: \Omega \times \mathbb{R} \to \mathbb{R}$ is a continuous function.
 The proofs rely essentially on the genus properties in critical point theory.
\end{abstract}

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


\section{Introduction}

 Recently, a great attention has been focused on the study of fractional
and non-local operators of elliptic type, both for the pure mathematical
research and in view of concrete real-world applications.
This type of operators arises in a quite natural way in many different contexts,
such as, among the others, the thin obstacle problem, optimization, finance,
phase transitions, stratified materials, conservation laws.
 The literature on non-local operators and on their applications is,
therefore, very interesting and, up to now, quite large, we refer the
interested readers to
\cite{AMAES,JiZh,KiSrTr,LaVa,MiRo,Nyamoradi1,Nyamoradi2,SeVa1,SeVa2,Teng}.

In this article, we are concerned with a class of nonlocal Kirchhoff 
fractional equations of the type
\begin{equation}\label{e1.1}
      \begin{gathered}
   -\Big (a + b\int_{\mathbb{R}^{2N}} |u (x) - u (y)|^2 K (x - y)\,dx\,dy \Big)
\mathcal{L}_K u - \lambda u = f (x, u (x)) \quad \text{in }   \Omega,\\
         u = 0 \quad \text{in }   \mathbb{R}^N \setminus \Omega,
             \end{gathered}
\end{equation}
where $\Omega$ is an open bounded subset of $\mathbb{R}^N$ with
Lipschitz boundary, $N > 2s$ with $s\in (0,1)$, $a, b > 0$ are
constants, $f: \Omega \times \mathbb{R} \to \mathbb{R}$ is a
continuous function, $\lambda$ is a parameter and
\begin{equation}\label{e1.2}
\mathcal{L}_K u (x) : = \int_{\mathbb{R}^N} \Big (u (x + y) + u
(x- y) - 2 u (x) \Big) K (y)\,d y, \quad x \in \mathbb{R}^N,
\end{equation}
where $K : \mathbb{R}^N \setminus \{0 \} \to (0, +\infty)$
is a kernel function satisfying the following properties:
\begin{itemize}
\item[(K1)]  $m K \in L^1 (\mathbb{R}^N)$, where $m (x) = \min
\{|x|^2,1\}$;

\item[(K2)] there exists $\theta > 0$ such that $K
(x) \geq \theta |x|^{-(N + 2 s)}$ for any $x \in \mathbb{R}^N
\setminus \{0 \}$;

\item[(K3)] $K (x) = K (-x)$ for any $x \in
\mathbb{R}^N \setminus \{0\}$.
\end{itemize}
The homogeneous Dirichlet datum in \eqref{e1.1} is given in
$\mathbb{R}^N \setminus \Omega$ and not simply on the boundary
$\partial \Omega$, consistent with the nonlocal character of the
kernel operator $\mathcal{L}_K$.

A typical model for $K$ is given by the singular kernel
$K(x)=|x|^{- (N + 2 s)}$ which gives rise to the fractional Laplace
operator $- (- \Delta)^s$ where $s \in (0, 1)$ $(N > 2 s)$ is
fixed, which, up to normalization factors, may be defined as
\begin{equation}\label{e1.3}
- (- \Delta)^s u (x) : =  \int_{\mathbb{R}^N}\frac{u (x + y) + u(x
- y) - 2 u (x) }{|y|^{N + 2 s}}\,d y, \quad  x \in \mathbb{R}^N.
\end{equation}
The problem \eqref{e1.1} in the model case $\mathcal{L}_K = -
(-\Delta)^s$ becomes
\begin{equation}\label{e1.4}
       \begin{gathered}
   \Big(a+b\int_{\mathbb{R}^N\times \mathbb{R}^N} |u (x) - u (y)|^2
 |x - y|^{-(N+2s)}\,
d x \,d y \Big) (-\Delta)^s u - \lambda u = f(x,u(x)) ,\\
         u = 0 \quad \text{in } \mathbb{R}^N \setminus \Omega,
       \end{gathered}
\end{equation}
which is related to Kirchhoff type problems. These problems model
several physical and biological systems, where $u$ describes a
process which depends on the average of itself, such as the
population density, see \cite{ChLo,Kirchhoff}. Problem
\eqref{e1.4} with the $p$-Laplacian operator $-\Delta_pu$ has been
studied in many papers, see
\cite{AlCoMa,ChTa,Chung1,Chung2,DaHa,TFMa,Ricceri,SuTa}. Motivated
by \cite{ChTa,Nyamoradi2,SeVa1,SeVa2,SeVa3}, in this paper, we
study the existence and multiplicity of solutions for Kirchhoff
type problem \eqref{e1.1} driven by the nonlocal operator
$\mathcal L_K$.

Before proving the main results, some preliminary material on
function spaces and norms is needed. In the following, we briefly
recall the definition of the functional space $X_0$, firstly
introduce in \cite{SeVa1}, and we give some notations. We denote
$\mathrm{Q} = \mathbb{R}^{2N} \setminus \mathcal{O}$, where
$\mathcal{O} = \mathbb{R}^N \setminus \Omega \times \mathbb{R}^N
\setminus \Omega$. We denote the set $X$ by
$$
X = \big\{u : \mathbb{R}^N \to \mathbb{R}:  u|_\Omega
\in L^2 (\Omega), \; (u (x) - u (y)) \sqrt{K (x - y)}\in L^2
(\mathbb{R}^{2N} \setminus \mathcal{O}) \big\},
$$
where $u|_\Omega$ represents the restriction to $\Omega$ of
function $u (x)$. Also, we denote by $X_0$ the following linear
subspace of $X$
$$
X_0 = \{g \in X : \; g = 0 \; \textrm{a.e. in }
\mathbb{R}^N \setminus \Omega \}.
$$

We know that $X$ and $X_0$ are nonempty, since
$C_0^2 (\Omega) \subseteq X_0$ by Lemma 11 of \cite{SeVa1}.
Moreover, the linear space $X$ is endowed with the norm defined as
$$
\|u\|_X : = \|u\|_{L^2 (\Omega)} +  \Big(\int_{\mathrm{Q}} |u
(x)- u (y)|^2 K (x - y) \,dx\,dy  \Big)^{1/2}.
$$
It is easy seen that $\|\cdot\|_X$ is a norm on $X$ (see, for
instance, \cite{SeVa2} for a proof). By Lemmas 6 and 7 of
\cite{SeVa2}, in the sequel we can take the function
\begin{equation}\label{e1.5}
X_0 \ni v \mapsto \|v\|_{X_0}
= \Big(\int_{\mathrm{Q}} |v (x) -
v(y)|^2 K (x - y)\, dx\,dy  \Big)^{1/2}
\end{equation}
as norm on $X_0$. Also $(X_0, \|\cdot\|_{X_0})$ is a Hilbert
space, with scalar product
\begin{equation}\label{e1.6}
\langle u, v \rangle_{X_0} : = \int_{\mathrm{Q}} (u (x) - u (y))(v
(x) - v(y)) K(x - y) \, dx dy.
\end{equation}
Note that in \eqref{e1.5} the integral can be extended to all
$\mathbb{R}^N \times \mathbb{R}^N$, since $v \in X_0$ and so
$v = 0$ a.e. in $\mathbb{R}^N \setminus \Omega$.

In what follows, we denote by $\lambda_1$ the first eigenvalue of
the operator $\mathcal{L}_K$ with homogeneous Dirichlet boundary
data, namely the first eigenvalue of the problem
\begin{gather*}
   \mathcal{L}_K  u = \lambda u,   \quad \text{in }   \Omega,\\
         u = 0, \quad \text{in }   \mathbb{R}^N \setminus \Omega.
\end{gather*}
We refer to \cite[Proposition 9 and Appendix A]{SeVa3}, for the
existence and the basic properties of this eigenvalue, where a
spectral theory for general integro-differential nonlocal
operators was developed.

When $\lambda < \lambda_1$ we can take as a norm on $X_0$ the
function
\begin{equation}\label{e1.7}
X_0 \ni v \mapsto \|v\|_{X_0,\lambda}
= \Big(\int_{\mathrm{Q}}
|v(x) - v (y)|^2 K (x - y)\,dx\,dy  - \lambda \int_\Omega |v
(x)|^2\, dx \Big)^{1/2},
\end{equation}
since for any $v \in X_0$ it holds true (for this see \cite[Lemma 10]{SeVa3})
\begin{equation}\label{e1.8}
m_\lambda \|v\|_{X_0} \leq \|v\|_{X_0,\lambda} \leq
M_\lambda\|v\|_{X_0},
\end{equation}
where
$$
m_\lambda : = \min \Big\{\sqrt{\frac{\lambda_1
-\lambda}{\lambda_1}}, 1 \Big\},\quad  
M_\lambda : = \max \Big\{\sqrt{\frac{\lambda_1 - \lambda}{\lambda_1}}, 1 \Big\}.
$$

Let $H^s(\mathbb{R}^N)$ be the usual fractional Sobolev space
endowed with the norm (the so-called Gagliardo norm)
\begin{equation}\label{e1.9}
\|u\|_{H^s(\mathbb{R}^N)} = \|u\|_{L^2(\mathbb{R}^N)}  +
\Big(\int_{\mathbb{R}^N\times \mathbb{R}^N} \frac{|u (x) -
u(y)|^2}{|x - y|^{N + 2 s}}\, dx\,dy  \Big)^{1/2}.
\end{equation}
Also, we recall the embedding properties of $X_0$ into the usual
Lebesgue spaces (see \cite[Lemma 8]{SeVa2}). The embedding
$j : X_0 \hookrightarrow L^v(\mathbb{R}^N)$ is continuous for any
$v \in [1, 2^\ast]$  $(2^\ast = \frac{2 N}{N - 2 s})$, while it is
compact whenever $v \in [1, 2^\ast)$. Hence, for any $v \in [1,
2^\ast]$ there exists a positive constant $c_v$ such that
\begin{equation}\label{e1.10}
\|v\|_{L^v (\mathbb{R}^N)} \leq c_v \|v\|_{X_0} \leq c_v
m_{\lambda}^{-1} \|v\|_{X_0,\lambda},
\end{equation}
for any $v \in X_0$.

We are now in the position to state the notation of solution and to
state the main results of this article.

\begin{definition}\label{def1.1} \rm
We say that $u \in X_0$ is a weak solution of problem
\eqref{e1.1}, if it satisfies
\begin{align*}
 &\Big(a+b\int_Q |u (x) - u (y)|^2 K (x - y)\,dx\,dy \Big)\\
&\int_Q (u (x) - u (y)) (v(x) - v(y)) K (x - y)\, d x\, d y
- \lambda \int_\Omega u (x) v(x) \, d x\\
&- \int_\Omega f(x,u(x))v(x)\,dx = 0,\quad  \forall  v \in X_0.
\end{align*}
\end{definition}

\begin{theorem}\label{the1.2}
Assume that $f$ satisfies the following conditions:
\begin{itemize}
\item[(F1)] $f \in C(\Omega\times \mathbb{R},\mathbb{R})$ and
there exist constants $1 < \gamma_1<\gamma_2<\dots<\gamma_m <2$
and functions $a_i \in
L^\frac{2}{2-\gamma_i}(\Omega,[0,+\infty))$, $i=1,2,\dots, m$
such that
$$
|f(x,z)| \leq \sum_{i=1}^ma_i(x)|z|^{\gamma_i - 1}, \quad \forall
(x,z) \in \Omega\times \mathbb{R}.
$$
\item[(F2)] There exist and open set $\Omega_0 \subset \Omega$ and
three constants $\delta>0$, $\gamma_0 \in (1,2)$ and $\eta>0$ such
that
$$
F(x,z) \geq \eta|z|^{\gamma_0}, \quad \forall (x,z) \in
\Omega_0\times[-\delta,\delta],
$$
where $F(x,z): = \int_0^zf(x,s)\,ds$, $x \in \Omega$, $z \in
\mathbb{R}$.
\end{itemize}
Then for any $\lambda< \lambda_1.\min\{a,1\}$, problem
\eqref{e1.1} has at least one nontrivial solutions.
\end{theorem}

\begin{theorem}\label{the1.3}
Assume that $f$ and $F$ satisfy the conditions {\rm (F1), (F2)} and
\begin{itemize}
\item[(F3)] $F(x,-z)=F(x,z)$ for all $(x,z) \in \Omega \times
\mathbb{R}$.
\end{itemize}
Then for any $\lambda< \lambda_1.\min\{a,1\}$, problem
\eqref{e1.1} has infinitely many nontrivial solutions.
\end{theorem}

\section{Proofs of main results}

Our idea is to obtain the existence and multiplicity of solutions
for problem \eqref{e1.1} by using critical point theory. Consider
the functional $J: X_0 \to \mathbb{R}$ defined by
\begin{align}\label{e2.1}
\begin{split}
J(u)& = \frac{a}{2}\int_Q |u (x) - u (y)|^2 K (x - y)\,dx\,dy  +
\frac{b}{4}\Big(\int_Q |u (x) - u (y)|^2 K (x - y)\,dx\,dy \Big)^2 \\
& \quad -\frac{\lambda}{2}\int_\Omega |u(x)|^2\,dx -
\int_\Omega F(x,u(x))\,dx
\end{split}
\end{align}
and set
$$
\Psi(u) = \int_\Omega F(x,u(x))\,dx.
$$
Let us recall the following definitions and results which are used
to prove our main results, see for instance
\cite{MaWi,Rabinowitz}.

\begin{definition}\label{def2.1}
 We say that $J$ satisfies the Palais-Smale (PS) condition if any sequence
$(u_n) \in X$ for which $J(u_n)$ is bounded and $J'(u_n) \to 0$ as $n \to
\infty$ possesses a convergent subsequence.
\end{definition}

\begin{lemma}[\cite{MaWi}] \label{lem2.2}
Let $X$ be a real Banach space and $J \in C^1(X,\mathbb{R})$
satisfy the $(PS)$ condition. If $J$ is bounded from below, then
$c = \inf_X J$ is a critical value of $J$.
\end{lemma}

Let $\mathcal{X}$ be a Banach space, $g \in C^1 (\mathcal{X}, \mathbb{R})$ 
and $c \in \mathbb{R}$. We set
\begin{gather*}
\Sigma  =  \{A \subset \mathcal{X} \setminus \{0\} : \;
\textrm{$A$ is closed in $X$ and symmetric with respect to 0)} \},\\
K_c  =  \{x \in \mathcal{X} : g (x) = c, \; g' (x) = 0 \},\\
g^c  =  \{x \in \mathcal{X} :  g (x) \leq c \}.
\end{gather*}

\begin{definition}[\cite{MaWi}] \label{def2.3}
For $A \in \Sigma$, we say genus of $A$ is $j$ (denoted by
$\gamma(A) = j$) if there is an odd map $\psi \in C(A,
\mathbb{R}^j\setminus \{0\})$, and $j$ is the smallest integer
with this property.
\end{definition}

\begin{lemma}[\cite{Rabinowitz}]\label{lem2.4}
 Let $g$ be an even $C^1$ functional on $\mathcal{X}$ which satisfies the
Palais-Smale condition. If $j \in \mathbb{N}$, $j > 0$, let
\[
\Sigma_j  =  \{A \in \Sigma : \; \gamma (A) \geq j \},
 c_j = \inf_{A \in  \Sigma_j} \sup_{u \in A} g (u).
\]
\begin{itemize}
\item[(i)] If $\Sigma_j \ne \emptyset$ and $c_j \in \mathbb{R}$, then
$c_j$ is a critical value of $g$.

\item[(ii)] If there exists $r \in \mathbb{N}$ such that $c_j = c_{j + 1}
= \cdots = c_{j + r} = c \in \mathbb{R}$ and $c \ne g (0)$ , then
$\gamma (K_c) \geq r + 1$.
\end{itemize}
\end{lemma}

\begin{remark} \rm
 From \cite[Remark 7.3]{Rabinowitz}, we know that
if $K_c \subset \Sigma$ and
$\gamma (K_c) > 1$, then $K_c$ contains infinitely many distinct
points, i.e., $J$ has infinitely many distinct critical points in
$\mathcal{X}$.
\end{remark}

\begin{lemma}\label{lem2.5}
Assume that {\rm (F1)} and {\rm (F2)} hold. Then the functional $J: X_0 \to
\mathbb{R}$ is well-defined and is of class $C^1(X_0,\mathbb{R})$ and
\begin{equation}
\begin{aligned}
J'(u)(v) & = \Big(a+b\int_Q |u (x) - u (y)|^2 K (x - y)\,dx\,dy \Big)\\
&\quad\times \int_Q (u (x) - u (y)) (v(x) - v(y)) K (x - y)\, dx\,dy  \\
  & \quad - \lambda \int_\Omega u (x) v(x) \, d x - \Psi'(u)(v),
\quad  \text{ for all } v \in X_0,
\end{aligned} \label{e2.2}
\end{equation}
where $\Psi'(u)(v)=\int_\Omega f(x,u(x))v(x)\,dx$. Moreover, the
critical points of $J$ are the solutions of problem \eqref{e1.1}.
\end{lemma}

\begin{proof}
For any $u \in X_0$, by (F1) and the H\"{o}lder inequality, one
have
\begin{equation} \label{e2.3}
\begin{split}
\int_\Omega |F(x,u)|\,dx
& \leq \sum_{i=1}^m\frac{1}{\gamma_i}
\int_\Omega a_i(x)|u|^{\gamma_i}\,dx\\
& \leq \sum_{i=1}^m\frac{1}{\gamma_i}\Big(\int_\Omega |a_i(x)
|^\frac{2}{2-\gamma_i}\,dx\Big)^\frac{2-\gamma_i}{2}
\Big(\int_\Omega |u|^2\,dx\Big)^\frac{\gamma_i}{2}\\
& \leq
C_1\sum_{i=1}^m\frac{1}{\gamma_i}\|a_i\|_\frac{2-\gamma_i}{2}
\|u\|^{\gamma_i}_{X_0},
\end{split}
\end{equation}
and so $J$ is defined by \eqref{e2.1} is well-defined on $X_0$ by
(F1).

Next, we prove that \eqref{e2.2} holds. For any $u,v \in X_0$, any
function $\theta: \Omega \to (0,1)$ and any number $h \in (0,1)$,
by (F1) and the H\"{o}lder inequality, we have
\begin{equation} \label{e2.4}
\begin{split}
& \int_\Omega \max_{h \in (0,1)}|f(x,u(x)+\theta(x)hv(x))v(x)|\,dx \\
& \leq \int_\Omega \max_{h \in (0,1)}|f(x,u(x)+\theta(x)hv(x))\|v(x)|\,dx \\
& \leq \sum_{i=1}^m\int_\Omega a_i(x)|u(x)+\theta(x)v(x)|^{\gamma_i-1}|v(x)|\,dx \\
& \leq \sum_{i=1}^m\int_\Omega a_i(x)(|u(x)|^{\gamma_i-1}+|v(x)|^{\gamma_i-1})|v(x)|\,dx \\
& \leq C_2\sum_{i=1}^m\|a_i\|_\frac{2-\gamma_i}{2}(\|u\|^{\gamma_i
-1}_{X_0}+\|v\|_{X_0}^{\gamma_i-1})\|v\|_{X_0} <+\infty.
\end{split}
\end{equation}
Then by \eqref{e2.4} and the Lebesgue dominated convergence
theorem, we have
\begin{equation} \label{e2.5}
\begin{split}
\Psi'(u)(v) & = \lim_{h \to 0^+}\frac{\Psi(u+hv)-\Psi(u)}{h} \\
& = \lim_{h\to 0^+} \frac{1}{h} \int_\Omega[F(x,u(x)+hv(x))-F(x,u(x))]\,dx \\
& = \lim_{h\to 0^+}\int_\Omega f(x,u(x)+\theta(x)v(x))v(x)\,dx \\
& = \int_\Omega f(x,u(x))v(x)\,dx.
\end{split}
\end{equation}
By \eqref{e2.5}, relation \eqref{e2.2} holds. Furthermore, by a
standard argument, it is easy to show that the critical points of
the functional $J$ in $X_0$ are the solutions of problem
\eqref{e1.1}.

Let us prove now that $J'$ is continuous. It is sufficient to
verify that $\Psi'$ is continuous. Let $u_n \to u$ in $X_0$, then
$u_n\to u$ in $L^2(\Omega)$ and
\begin{equation}\label{e2.6}
      \begin{gathered}
  u_n \to u,  \quad \text{strongly in }  L^2(\Omega),\\
   u_n \to u, \quad \text{a.e. in } \Omega.
  \end{gathered}
\end{equation}
Then there exists $h \in L^2(\Omega)$ such that
$|u_n(x)| \leq h(x)$ a.e. $x \in \Omega$ and for any $n \in \mathbb N$.

By (F1), we have
\begin{equation} \label{e2.9}
\begin{split}
& |f(x,u_n(x))-f(x,u(x))|^2 \\
& \leq 2(|f(x,u_n(x))|^2+|f(x,u(x))|^2) \\
& \leq C_2\sum_{i=1}^m|a_i(x)|^2\left(|u_n(x)|^{2(\gamma_i-1)}+|u(x)|^{2(\gamma_i-1)}\right) \\
& \leq C_2\sum_{i=1}^m|a_i(x)|^2\left(|h(x)|^{2(\gamma_i-1)}+|u(x)|^{2(\gamma_i-1)}\right)\\
& :=g(x), \quad \forall n \in \mathbb N, \quad x \in \Omega
\end{split}
\end{equation}
and
\begin{equation} \label{e2.10}
\begin{split}
\int_\Omega g(x)\,dx & = C_2\sum_{i=1}^m\int_\Omega|a_i(x)|^2\left(|h(x)|^{2(\gamma_i-1)}
+|u(x)|^{2(\gamma_i-1)}\right)\,dx\\
& \leq C_2\sum_{i=1}^m\|a_i\|^2_\frac{2-\gamma_i}{2}\left(\|h\|^{2(\gamma_i
-1)}_{L^2}+\|u\|_{L^2}^{2(\gamma_i-1}\right)
<+\infty.
\end{split}
\end{equation}
By \eqref{e2.6}, \eqref{e2.9}, \eqref{e2.10}, and the Lebesgue dominated
convergence theorem, we have
\begin{equation}\label{e2.7}
\lim_{n\to \infty}\int_\Omega |f(x,u_n(x))-f(x,u(x))|^2\,dx = 0.
\end{equation}

From  \eqref{e1.10}, \eqref{e2.2}, (F1) and the H\"{o}lder
inequality, we have
\begin{align*}
|(\Psi'(u_n)-\Psi'(u),v)| & = \left|\int_\Omega [f(x,u_n(x))-f(x,u(x))]v(x)\,dx\right|\\
& \leq \int_\Omega |f(x,u_n(x))-f(x,u(x))\|v(x)|\,dx\\
& \leq \Big(\int_\Omega|f(x,u_n(x))-f(x,u(x))|^2\,dx\Big)^{1/2}\|v\|_{L^2}\\
& \leq C_3\Big(\int_\Omega|f(x,u_n(x))-f(x,u(x))|^2\,dx\Big)^{1/2}\|v\|_{X_0},
\end{align*}
which converges to $0$ as $n \to \infty$. This implies that
$\Psi'$ is continuous and the proof of Lemma \ref{lem2.5} is
complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{the1.2}]
In view of Lemma \ref{lem2.5}, $J \in C^1(X_0,\mathbb{R})$. In
what follows, we first show that $J$ is bounded from below. Since
$\lambda< \lambda_1.\min\{a,1\}$ we have $a-1+m_\lambda^2>0$,
where $m_\lambda$ is defined by \eqref{e1.8}. By (F1),
\eqref{e1.5}, \eqref{e1.7}, \eqref{e1.8} and the H\"{o}lder
inequality, we have
\begin{equation} \label{e2.11}
\begin{split}
&J(u)\\
&=\frac{a}{2}\int_Q |u (x) - u (y)|^2 K (x - y)\,dx\,dy
+ \frac{b}{4}\Big(\int_Q |u (x) - u (y)|^2 K (x - y)\,dx\,dy \Big)^2 \\
& \quad -\frac{\lambda}{2}\int_\Omega |u(x)|^2\,dx
 - \int_\Omega F(x,u(x))\,dx\\
& \geq \frac{1}{2}(a-1+m_\lambda^2)\|u\|^2_{X_0}
 -\sum_{i=1}^m\frac{1}{\gamma_i}\int_\Omega a_i(x)|u|^{\gamma_i}\,dx\\
& \geq \frac{1}{2}(a-1+m_\lambda^2)\|u\|^2_{X_0}-
C_1\sum_{i=1}^m
\frac{1}{\gamma_i}\|a_i\|_\frac{2-\gamma_i}{2}\|u\|^{\gamma_i}_{X_0}.
\end{split}
\end{equation}
As $\gamma_i \in (1,2)$, $i=1,2 ,\dots , m$, it follows from
\eqref{e2.11} that $J(u) \to +\infty$ as $\|u\|_{X_0} \to +\infty$
and $J$ is bounded from below.

Next, we prove that $J$ satisfies the (PS)-condition. Assume that
$\{u_n\}\subset X_0$ is a sequence such that $\{J(u_n)\}$ is
bounded and $J'(u_n) \to 0$ as $n \to \infty$. Since $\{u_n\}$ is
a (PS)-sequence and using the definition of $J$, there exists a
constant $C_4>0$ such that
\begin{equation}\label{e2.12}
\|u_n\|_{X_0} \leq C_4, \quad \forall n \in \mathbb N.
\end{equation}
So passing to a subsequence it necessary, it can be assumed that
$\{u_n\}$ converges weakly to $u_0$ in $X_0$ and thus $\{u_n\}$
converges strongly to $u_0$ in $L^2(\Omega)$. By \eqref{e2.12} and
(F1), we have
\begin{equation} \label{e2.13}
\begin{split}
& \big|\int_\Omega (f(x,u_n(x))-f(x,u(x)))(u_n(x)-u_0(x))\,dx\big| \\
& \leq \int_\Omega |f(x,u_n(x))-f(x,u(x))\|u_n(x)-u_0(x)|\,dx \\
& \leq \Big(\int_\Omega |f(x,u_n(x))-f(x,u_0(x))|^2\,dx\Big)^{1/2}
 \Big(\int_\Omega |u_n(x)-u_0(x)|^2\,dx\Big)^{1/2}\\
& \leq \Big(\int_\Omega 2(|f(x,u_n(x))|^2+|f(x,u_0(x))|^2)\,dx\Big)^{1/2}
 \Big(\int_\Omega |u_n(x)-u_0(x)|^2\,dx\Big)^{1/2}\\
& \leq C_5\Big(\sum_{i=1}^m\|a_i\|^2_\frac{2}{2-\gamma_i}
 (\|u_n\|^{2(\gamma_i-1)}_{X_0}+\|u_0\|^{2(\gamma_i-1)}_{X_0})\,dx\Big)^{1/2}
 \|u_n-u_0\|_{L^2(\Omega)},
\end{split}
\end{equation}
which approaches $0$ as $n\to \infty$.

Since $\lambda< \lambda_1\min\{a,1\}$,  by \eqref{e1.7}
and \eqref{e1.8}, we have
\begin{align*}
&(J'(u_n)-J'(u_0))(u_n-u_0)\\
& = \Big(a + b\int_Q |u_n(x) - u_n(y)|^2 K (x - y)\,dx\,dy\Big)\\
&\quad \times\int_Q(u_n(x) - u_n(y))\Big((u_n(x)-u_0(x))-(u_n(y)-u_0(y))\Big) 
 K (x - y)\,dx\,dy\\
& \quad -\Big(a + b\int_Q |u_0(x) - u_0(y)|^2 K (x - y)\,dx\,dy\Big)\\
&\quad \times\int_Q(u_0(x) - u_0(y))\Big((u_n(x)-u_0(x))-(u_n(y)-u_0(y))\Big)
  K (x - y)\,dx\,dy\\
&\quad -\lambda \int_\Omega |u_n(x)-u_0(x)|^2\,dx
 -\int_\Omega[f(x,u_n(x))-f(x,u_0(x))](u_n-u_0)\,dx\\
& = \Big(a + b\int_Q |u_n(x) - u_n(y)|^2 K (x - y)\,dx\,dy\Big)\\
&\quad\times \int_Q |(u_n(x)-u_0(x))-(u_n(y)-u_0(y)|^2 K (x - y)\,dx\,dy\\
&\quad -b\Big(\int_Q|u_0(x)-u_0(y)|^2K (x - y)\,dx\,dy
 -\int_Q|u_n(x)-u_n(y)|^2K (x - y)\,dx\,dy\Big)\\
&\quad \times\int_Q(u_0(x) - u_0(y))\Big((u_n(x)-u_0(x))-(u_n(y)-u_0(y))\Big)
  K (x - y)\,dx\,dy\\
&\quad -\lambda \int_\Omega |u_n(x)-u_0(x)|^2\,dx
 -\int_\Omega[f(x,u_n(x))-f(x,u_0(x))](u_n-u_0)\,dx\\
& \geq (a-1+m_\lambda^2)\|u_n-u_0\|_{X_0}^2
 -\int_\Omega[f(x,u_n(x))-f(x,u_0(x))](u_n-u_0)\,dx\\
&\quad -b\Big(\|u_0\|_{X_0}^2-\|u_n\|_{X_0}^2\Big)\int_Q(u_0(x)
- u_0(y))\\
&\quad\times \Big((u_n(x)-u_0(x))-(u_n(y)-u_0(y))\Big) K (x - y)\,dx\,dy\,.
\end{align*}
Then
\begin{equation} \label{e2.14}
\begin{split}
&(a-1+m_\lambda^2)\|u_n-u_0\|_{X_0}^2\\
&\leq (J'(u_n)-J'(u_0))(u_n-u_0)+\int_\Omega[f(x,u_n(x))-f(x,u_0(x))](u_n-u_0)\,dx\\
&\quad +b\left(\|u_0\|_{X_0}^2-\|u_n\|_{X_0}^2\right)\\
&\quad\times \int_Q(u_0(x)- u_0(y))\Big((u_n(x)-u_0(x))-(u_n(y)-u_0(y))\Big)
  K (x - y)\,dx\,dy.
\end{split}
\end{equation}
As $\{u_n\}$ converges weakly $u_0$ in $X_0$, $\{\|u_n\|_{X_0}\}$
is bounded and we have
\begin{equation}\label{e2.15}
\begin{split}
&\lim_{n\to\infty}b\left(\|u_0\|_{X_0}^2-\|u_n\|_{X_0}^2\right)
 \int_Q(u_0(x) - u_0(y))\\
&\times  \Big((u_n(x)-u_0(x))-(u_n(y)-u_0(y))\Big)
 K (x - y)\,dx\,dy=0.
\end{split}
\end{equation}
It follows from \eqref{e2.13}, \eqref{e2.14} and \eqref{e2.15}
that $\{u_n\}$ converges strongly to $u_0$ in $X_0$ and the
functional $J$ satisfies the $(PS)$ condition.

Then $d=\inf_{X_0}J(u)$ is a critical value of $J$, that is, there
exists a critical point $u^\ast\in X_0$ such that $J(u^\ast) = d$.

Finally, we show that $u^\ast\ne 0$. Let $u_0\in X_0\cap
C_0^\infty(\Omega_0)$ and $\|u_0\|_\infty \leq 1$, where
$\Omega_0$ is given by (F2). By (F2), for $t \in (0,\delta)$,  we have
\begin{equation} \label{e2.16}
\begin{split}
J(tu_0)
&=\frac{at^2}{2}\int_Q |u_0(x) - u_0(y)|^2 K (x - y)\,dx\,dy \\
&\quad  + \frac{bt^4}{4}\Big(\int_Q |u_0(x) - u_0(y)|^2 K (x - y)\,dx\,dy \Big)^2
 \\
& \quad -\frac{\lambda t^2}{2}\int_\Omega |u_0(x)|^2\,dx
 - \int_\Omega F(x,tu_0(x))\,dx\\
& \leq \frac{t^2}{2}(a-1+M_\lambda^2)\|u_0\|_{X_0}^2+\frac{bt^4}{4}\|u_0\|_{X_0}^4
 - \int_{\Omega_0} F(x,tu_0(x))\,dx\\
& \leq
\frac{t^2}{2}(a-1+M_\lambda^2)\|u_0\|_{X_0}^2+\frac{bt^4}{4}\|u_0\|_{X_0}^4-\eta
t^{\gamma_0}\int_{\Omega_0}|u_0(x)|^{\gamma_0}\,dx\,.
\end{split}
\end{equation}
As $\gamma_0\in (1,2)$, it follows from \eqref{e2.16} that
$J(tu_0)<0$ for $t>0$ small enough. Hence, $J(u^\ast) =d<0$ and
therefore, $u^\ast$ is a nontrivial critical point of $J$, and so
$u^\ast$ is a nontrivial solution of problem \eqref{e1.1}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{the1.3}]
In view of Lemma \ref{lem2.5}, $J \in C^1(X_0,\mathbb{R})$ is
bounded from below and satisfies the $(PS)$ condition. It follows
from (F3) that $J$ is even and $J(0)=0$. In order to apply Lemma
\ref{lem2.4}, we prove now that
\begin{equation}\label{e2.17}
\text{ for any } n\in \mathbb N, \text{ there exists } \epsilon>0
\text{ such that } \gamma(J^{-\epsilon}) \geq n.
\end{equation}
For any $n\in \mathbb N$, we take $n$ disjoint open sets $K_i$
such that
$$
\cup_{i=1}^nK_i \subset \Omega_0.
$$
For $i=1, 2, \dots , n$, let $u_i \in \big(X_0\cap
C_0^\infty(K_i)\big)\backslash \{0\}$ and $\|u_i\|_{X_0}=1$, and
$$
E_n = \textrm{span}\{u_1,u_2,\dots,u_n\}, \quad S_n =\{u \in
E_n: \|u\|_{X_0}=1\}.
$$
For each $u \in E_n$, there exist $\mu_i\in \mathbb{R}$, $i = 1,
2,\dots, n$ such that
\begin{equation}\label{e2.18}
u(x) = \sum_{i=1}^n\mu_i u_i(x) \quad\text{for } x\in \Omega.
\end{equation}
Then
\begin{equation}\label{e2.19}
\|u\|_{\gamma_0} = \Big(\int_\Omega
|u(x)|^{\gamma_0}\,dx\Big)^{1/\gamma_0}
=\Big(\sum_{i=1}^n |\mu_i|^{\gamma_0}\int_{K_i}|u_i(x)|^{\gamma_0}\,dx
\Big)^{1/\gamma_0}
\end{equation}
and
\begin{equation} \label{san}
\begin{split}
\|u\|_{X_0}^2
&=  \int_{\mathrm{Q}} |u (x) - u(y)|^2 K (x - y)\,dx\,dy  \\
&=  \sum_{i=1}^n\mu_i^2 \int_{\mathrm{Q}} |u_i (x) - u_i(y)|^2 K
(x - y)\, dx\,dy  \\
&=  \sum_{i=1}^n\mu_i^2 \|u_i\|_{X_0}^2 = \sum_{i=1}^n\mu_i^2.
\end{split}
\end{equation}

As all norms of a finite dimensional normed space are equivalent,
there is a constant $C_6>0$ such that
\begin{equation}\label{e2.20}
C_6\|u\|_{X_0} \leq \|u\|_{\gamma_0} \quad \text{for all } u \in E_n.
\end{equation}
By \eqref{e2.18}, \eqref{e2.19}, \eqref{e2.20}, we have
\begin{equation} \label{e2.21}
\begin{split}
J(tu)
&=\frac{at^2}{2}\int_Q |u(x) - u(y)|^2 K (x - y)\,dx\,dy  \\
&\quad + \frac{bt^4}{4}\Big(\int_Q |u(x) - u(y)|^2 K (x - y)\,dx\,dy \Big)^2 \\
& \quad -\frac{\lambda t^2}{2}\int_\Omega |u(x)|^2\,dx
 - \int_\Omega F(x,tu(x))\,dx\\
& \leq \frac{t^2}{2}(a-1+M_\lambda^2)\|u\|_{X_0}^2+\frac{bt^4}{4}\|u\|_{X_0}^4
 -\sum_{i=1}^n\int_{K_i} F(x,t\mu_iu_i(x))\,dx\\
& \leq \frac{t^2}{2}(a-1+M_\lambda^2)\|u\|_{X_0}^2
 +\frac{bt^4}{4}\|u\|_{X_0}^4-\eta t^{\gamma_0}\sum_{i=1}^n|\mu_i|^{\gamma_0}
 \int_{K_i} |u_i(x)|^{\gamma_0}\,dx\\
& \leq \frac{t^2}{2}(a-1+M_\lambda^2)\|u\|_{X_0}^2+\frac{bt^4}{4}\|u\|_{X_0}^4
 -\eta t^{\gamma_0}\|u\|_{\gamma_0}^{\gamma_0}\\
& \leq \frac{t^2}{2}(a-1+M_\lambda^2)\|u\|_{X_0}^2+\frac{bt^4}{4}\|u\|_{X_0}^4
 -\eta (C_6t)^{\gamma_0}\|u\|_{X_0}^{\gamma_0}\\
& \leq \frac{t^2}{2}(a-1+M_\lambda^2)\|u\|_{X_0}^2+\frac{bt^4}{4}\|u\|_{X_0}^4-\eta
(C_6t)^{\gamma_0}
\\
& = \frac{t^2}{2}(a-1+M_\lambda^2) +\frac{bt^4}{4}-\eta
(C_6t)^{\gamma_0}
\end{split}
\end{equation}
for all $u \in S_n$ and and sufficient small $t > 0$. In this case
(F2) is applicable, since $u$ is continuous on
$\overline{\Omega}_0$ and so $|t \mu_i u_i(x)| \leq \delta$,
�$\forall  \; x \in \Omega_0$, $i = 1, 2, �\dots, n$ can be true
for sufficiently small $t$. Then, there exist $\epsilon>0$ and
$\sigma>0$ such that
\begin{equation}\label{e2.22}
J(\sigma u) < -\epsilon \quad \text{for } u \in S_n.
\end{equation}
Let
$$
S_n^\sigma = \{\sigma u:~ u \in S_n\}, \quad  \Lambda =
\big\{(\mu_1,\mu_2,\dots,\mu_n) \in \mathbb{R}^n: ~\sum_{i=1}^n
\mu_i^2<\sigma^2\big\}.
$$
Then it follows from \eqref{e2.22} that
$$
J(u) < -\epsilon \quad \text{for all } u \in S_n^\sigma,
$$
which, together with the fact that $J \in C^1(X_0,\mathbb{R})$ and
is even, implies that
\begin{equation}\label{e2.23}
S_n^\sigma \subset J^{-\epsilon} \in \Sigma.
\end{equation}
On the other hand, it follows from \eqref{e2.18} and \eqref{san},
that
$$
S_n^\sigma = \big\{\sum_{i=1}^n\mu_i u_i:~ \sum_{i=1}^n \mu_i^2
= \sigma^2 \big\}.
$$
So, we define a map $\psi : S_n^\sigma \to
\partial \Lambda$ as follows:
$$
\psi (u) = (\mu_1, \mu_2, \dots, \mu_n), \quad \; \forall \; u
\in S_n^\sigma.
$$
It is easy to verify that
$\psi : S_n^\sigma \to \partial \Lambda$ is an odd
homeomorphic map. By Proposition 7.7 in \cite{Rabinowitz}, we get
$\gamma(S_n^\sigma) = n$ and so by some properties of the genus
(see 3$^\circ$ of \cite[Proposition 7.5]{Rabinowitz}), we have
\begin{equation}\label{e2.24}
\gamma(J^{-\epsilon}) \geq \gamma(S_n^\sigma) = n,
\end{equation}
so the proof of \eqref{e2.17} follows. Set
$$
c_n = \inf_{A \in \Sigma_n}\sup_{u \in A}J(u).
$$
It follows from \eqref{e2.24} and the fact that $J$ is bounded
from below on $X_0$ that $-\infty< c_n \leq - \epsilon < 0$, that
is, for any $n \in \mathbb N$, $c_n$ is a real negative number. By
Lemma \ref{lem2.4}, the functional $J$ has infinitely many
nontrivial critical points, and so problem \eqref{e1.1} possesses
infinitely many nontrivial solutions.
\end{proof}

\subsection*{Acknowledgments}
The authors would like to thank the anonymous referees for their suggestions
 and helpful comments which improved the presentation of the original manuscript.


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\end{document}