\documentclass[reqno]{amsart} \usepackage[notref,notcite]{showkeys} \usepackage{hyperref} \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. \begin{thebibliography}{99} \bibitem{AlCoMa} C. O. Alves, F. J. S. A. 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