\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 115, pp. 1--13.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2017 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2017/115\hfil Degenerate Kirchhoff type problems] {Existence of solutions for degenerate Kirchhoff type problems with fractional $p$-Laplacian} \author[N. Nyamoradi, L. I. Zaidan \hfil EJDE-2017/115\hfilneg] {Nemat Nyamoradi, Lahib Ibrahim Zaidan} \address{Nemat Nyamoradi (corresponding author)\newline Department of Mathematics, Faculty of Sciences, Razi University, 67149 Kermanshah, Iran} \email{nyamoradi@razi.ac.ir, neamat80@yahoo.com} \address{Lahib Ibrahim Zaidan\newline Department of Mathematics, Faculty of Sciences, Razi University, 67149 Kermanshah, Iran. \newline Education College, University Of Babylon, Babil, Iraq} \email{lahibzaidan75@yahoo.com; lahibzaidan@uobabylon.edu.iq} \dedicatory{Communicated by Paul H. Rabinowitz} \thanks{Submitted February 23, 2017. Published April 27, 2017.} \subjclass[2010]{34B27, 35J60, 35B05} \keywords{Kirchhoff nonlocal operators; fractional differential equations; \hfill\break\indent fountain theorem; mountain pass theorem; critical point theory} \begin{abstract} In this article, by using the Fountain theorem and Mountain pass theorem in critical point theory without Palais-Smale (PS) condition, we show the existence and multiplicity of solutions to the degenerate Kirchhoff type problem with the fractional $p$-Laplacian \begin{gather*} \Big(a + b\int\int_{\mathbb{R}^{2N}} \frac{|u (x) - u (y)|^p}{|x - y|^{N + ps}} \, dx \, dy\Big) (- \Delta)_p^s u = f (x, u ) \quad \text{in } \Omega,\\ u = 0 \quad \text{in } \mathbb{R}^N \setminus \Omega, \end{gather*} where $(- \Delta)_p^s$ is the fractional $p$-Laplace operator with $0 < s < 1 < p < \infty$, $\Omega$ is a smooth bounded domain of $\mathbb{R}^N$, $N > 2 s$, $a, b > 0$ are constants and $f: \Omega \times \mathbb{R} \to \mathbb{R}$ is a continuous function. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction and statement of main results} The aim of this article is to establish the existence of solutions to the Kirchhoff nonlocal problem \begin{equation}\label{e1.4} \begin{gathered} \Big(a + b \int_{\mathbb{R}^{2N}} \frac{|u (x) - u (y)|^p}{|x - y|^{N + ps}} \,dx \,dy \Big) (-\Delta)_p^s u= f(x,u) \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 and $(- \Delta)_p^s$ is the fractional $p$-Laplacian operator which, up to normalization factors, may be defined as \[ (- \Delta)_p^s u (x) = 2 \lim_{\varepsilon \to 0^+} \int_ {\mathbb{R}^{N} \setminus B_\varepsilon (x)} \frac{|u (x) - u (y)|^{p - 2} (u (x) - u (y)) }{|x-y|^{N + p s}}\, dy \] for $x \in \mathbb{R}^{N}$, where $B_\varepsilon (x) := \{y \in \mathbb{R}^{N} : |x - y| < \varepsilon \}$. As for some recent results on the fractional $p$-Laplacian, we refer to for example \cite{Iannizzotto,IannizzottoSeq,Lindgren} and the references therein. When $a \equiv1$, $b \equiv 0$ and $p = 2$, problem \eqref{e1.4} becomes the fractional Laplacian problem \begin{equation}\label{s1} \begin{gathered} (-\Delta)^s u= f(x,u) \quad \text{in } \Omega,\\ u = 0 \quad \text{in } \mathbb{R}^N \setminus \Omega. \end{gathered} \end{equation} For the basic properties of fractional Sobolev spaces and the functional framework that takes into account the problem \eqref{s1}, we refer the readers to \cite{SeVa1}. In \cite{SeVa2, SeVa3}, Servadei and Valdinoci considered the existence of nontrivial weak solutions of the problem \eqref{s1} by using variational methods. For other recent results in \eqref{s1}, the reader is referred, for example, to \cite{CaSi,SeVaS2,SeVaS3}. Fractional and nonlocal operators and on their applications is very interesting, we refer the readers to \cite{CP1,Dipierro,Felmer, FP2,FP1,MMTZ, MR1,MRS,MR, MV,Pucci1,Pucci,Secchi,ZBX} and the references therein. For the basic properties of fractional Sobolev spaces, we refer the readers to \cite{DPV,MRS}. In \cite{MV}, Molica Bisci and Vilasi studied a class of Kirchhoff nonlocal fractional equation in a bounded domain $\Omega$ and obtained three solutions by using three critical point theorem. Pucci and Saldi \cite{Pucci1} established the existence and multiplicity of nontrivial solutions for a Kirchhoff type eigenvalue problem in $\mathbb{R}^N$ involving a critical nonlinearity and the nonlocal fractional Laplacian. We refer also to \cite{FMS,FV0,MRS,MT1} for related problems. Notice that when $a \equiv 1$ and $b \equiv 0$, as $s \to 1^-$, problem \eqref{e1.4} reduces to the problem \begin{equation}\label{s2} - \Delta_p u = f (x, u ) \quad \text{in } \Omega, \end{equation} where $\Omega \subset \mathbb{R}^N$ is a smooth domain. For the case of a bounded domain, there are several articles considering the system \begin{equation*} - \Big(a + b \int_\Omega |\nabla u|^p \Big) \Delta_p u = g (x, u ) \quad \text{in } \Omega \end{equation*} where $\Omega \subset \mathbb{R}^N$ is a smooth domain, which is related to the stationary analogue of the Kirchhoff equation \begin{equation*} u_{tt} - \Big(a + b \int_\Omega |\nabla u|^p\Big) \Delta_p u = g(x,u), \end{equation*} which was proposed by Kirchhoff \cite{Kirchhoff} as an extension of the classical D'Alembert's wave equation for free vibrations of elastic string. In recent years, many authors are interesting in Kirchhoff type problems, see for example \cite{AMC1, CLBG, Bitao,Chung1,Chung2,DaHa,Pucci, Vilasi, WHLiu, Xiang1,Xiang,XWTS} and references therein. Motivated by the above works and \cite{ChTa,Nyamoradi2,SeVa1,SeVa2,SeVa3,Teng}, we study the existence and multiplicity of solutions for Kirchhoff type problem \eqref{e1.4}. Before proving our main results, some preliminary material on function spaces and norms is needed. In the following, we briefly recall the definition of the functional space $X_0$, introduced in \cite{SeVa1}, and we give some notation. 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 $$ X = \big\{u : \mathbb{R}^N \to \mathbb{R}: u|_\Omega \in L^p (\Omega), \; \int \int_{Q} \frac{|u (x) - u (y)|^p}{|x - y|^{N + ps}} \,dx \, dy < \infty \big\}, $$ where $ u|_\Omega $ represents the restriction to $ \Omega $ of function $ u(x) $. Also, we define the following linear subspace of $ X$, $$ X_0 = \big\{g \in X : g = 0 \; \text{a.e. } \text{in } \mathbb{R}^N \setminus \Omega \big\}. $$ The linear space $ X $ is endowed with the norm $$ \|u\|_X : = \|u\|_{L^2 (\Omega)} + \Big( {\int \int_{Q}} \frac{|u (x) - u (y)|^p}{|x - y|^{N + ps}} \,dx \,dy \Big)^{1/p}. $$ It is easily seen that $ \|\cdot\|_X $ is a norm on $X$ and $ C_0^\infty (\Omega) \subseteq X_0 $ (see \cite[Lemma 2.1]{Xiang}). Also, we know that $X_0$, endowed with the norm \begin{equation}\label{e1.5} \|v\|_{X_0} = \Big( \int \int_{Q} \frac{|v (x) - v (y)|^p}{|x - y|^{N + ps}} \, dx \, dy \Big)^{1/p} \quad \text{for all } v \in X_0, \end{equation} is a uniformly convex Banach space and a reflexive Banach space \cite[Remark 2.1 and Lemma 2.4]{Xiang}. We consider the nonlinear eigenvalue problem \begin{equation}\label{egn} \begin{gathered} \|u\|_{X_0}^p (- \Delta)_p^s u = \lambda |u|^{2 p - 2} u \quad \text{in } \Omega,\\ u = 0 \quad \text{in } \mathbb{R}^N \setminus \Omega, \end{gathered} \end{equation} whose eigenvalues are the critical values of the functional \begin{equation}\label{e14} J_p (u) = \| u \|_{X_0}^{2p}, \quad u \in \mathcal{M} = \Big\{u \in X_0 : \int_\Omega |u|^{2 p} \,dx = 1 \Big\}. \end{equation} We know that the first eigenvalue $ \lambda_1 : = \inf_{u \in \mathcal{M}} J_p (u) > 0 $. The first eigenfunction is denoted by $ \varphi_1 $ (see \cite{Xiang1} for the case $ \theta = 2 $). We denote the usual $L^p(\Omega)$-norm by $\|\cdot\|_p$. Since $\Omega$ is a bounded domain, it is well known that $ X_0 \hookrightarrow L^p (\Omega) $ continuously for $ p \in [1, p^*_s] $, (see \cite[Lemma 2.3]{Xiang}) and compactly for $q \in [1, p^*_s) $, where $ p^*_s : = \frac{N p}{N - s p} $. Moreover there exists $ C_q> 0 $ such that \begin{equation}\label{e21} \|u\|_{q} \leq C_q \|u\|_{X_0}, \quad u \in X_0. \end{equation} We consider the functional $ J: X_0 \to \mathbb{R} $ defined by \begin{align}\label{e2.1} J(u) = \frac{a}{p}\|u\|_{X_0}^{p} + \frac{b}{ 2 p}\|u\|_{X_0}^{2p} - \int_\Omega F(x,u(x))\,dx \end{align} and set $$ \Psi(u) = \int_\Omega F(x,u(x))\,dx, $$ where $F(x,u)=\int_0^u f(x,s)\,dx$. Obviously, the functional $ J $ is well-defined, it is of class $ C^1(X_0,\mathbb{R}) $ and \begin{equation} \begin{aligned} &\langle J'(u), v \rangle \\ & = (a+b \|u\|_{X_0}^{p}) \int\int_Q \frac{|u (x) - u (y)|^{p - 2}\big (u (x) - u (y)\big) }{|y|^{N + p s}} (v (x) - v (y)) \,dx \,dy \\ &\quad - \int_\Omega f(x,u(x))v(x)\,dx, \quad \text{for all } u, v \in X_0, \end{aligned} \label{e2.2} \end{equation} Moreover, the critical points of $ J $ are the solutions of problem \eqref{e1.4}. Let $$ E_j := \oplus_{i \leq j} \ker ((- \Delta)_p^s - \mu_i), $$ where $ 0 < \mu_1 \leq \mu_2 \leq \dots, \mu_i \leq \dots$, are the eigenvalue of $ ((- \Delta)_p^s, X_0) $ (see \cite{Iannizzotto,Lindgren,Franzina}). \begin{definition}\label{def2.1} \rm 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} Also, we need the following definition, which is a weak version of the (PS) condition, due by Cerami \cite{Cerami}. \begin{definition}\label{def1.1} \rm Let $ J \in C^1 (X, \mathbb{R}) $, we say that $ J $ satisfies the Cerami condition at the level $ c \in \mathbb{R} $ ((Ce)$_{\rm c}$ for short), if any sequence $ (u_n) \in X $ with $$ J(u_n) \to c, \quad (1 + \|u_n\|) J'(u_n) \to 0 \quad \text{as } n \to \infty, $$ possesses a convergent subsequence in $ X $; $ J $ satisfies the (Ce) condition if $ J $ satisfies the (Ce)$_{\rm c}$ for all $ c \in \mathbb{R} $. \end{definition} The assumptions on the function $f$ are stated as follows: \begin{itemize} \item[(A1)] There exists a positive constant $ C > 0 $ such that $ |f(x,t))| \leq C (|t|^{r - 1} + 1) $, for some $ 2 p < r < p^*_s $, $ x \in \Omega $ and all $ t \in \mathbb{R} $; \item[(A2)] $ \lim_{|t| \to \infty} \big(\frac1{2 p} f (x, t) t - F (x, t) + \frac{a \mu_1}{p} t^p\big) = +\infty $ uniformly in $ x \in \Omega $; \item[(A3)] there exists $ \mu > \mu_1 $ such that $ F (x, t) \geq \frac{a \mu}{p} t^p $ for $ |t| $ small; \item[(A4)] $ \lim_{|t| \to \infty} \big(\frac{a \mu_1}{p} t^p + \frac{b \mu_1}{2 p} t^{2p} - F (x, t) \big) = +\infty $ uniformly in $ x \in \Omega $. \end{itemize} Now we state our main results. \begin{theorem}\label{themain1} Assume that $f \in C (\Omega \times \mathbb{R}, \mathbb{R})$, {\rm (A1)--(A4)} hold. Then \eqref{e1.4} has at least one nontrivial solution. \end{theorem} In the next theorem we use the assumptions: \begin{itemize} \item[(A5)] $\lim_{|t| \to \infty} \frac{F (x, t)}{|t|^{2 p}} \to \infty$ uniformly in $x \in \Omega$, and there exists $L_1 \geq 0$ such that $F (x, t) \geq 0$ for all $(x, t) \in \Omega \times \mathbb{R}$ and $|t| \geq L_1$; \item[(A6)] there exists $\theta_0 > 0$ such that $$ F (x, t) \leq \frac1{2p} f (x, t) t + \theta_0 |t|^p, \quad \forall (x, t) \in \Omega \times \mathbb{R}; $$ \item[(A7)] $f (x, - t) = - f (x, t)$ for all $(x, t) \in \Omega \times \mathbb{R}$; \end{itemize} \begin{theorem}\label{thenew} Assume that {\rm (A1), (A5)--(A7)} are satisfied. Then problem \eqref{e1.4} possesses infinitely many nontrivial solutions $ \{u_k\} $ such that $ J (u_k) \to + \infty $. \end{theorem} Now, we study the existence of infinitely many solutions of the following problem, which it is a special case of problem \eqref{e1.4}, \begin{equation}\label{e.5} \begin{gathered} \begin{aligned} &\Big(a + b\int_{\mathbb{R}^{2N}} \frac{|u (x) - u (y)|^p}{|x - y|^{N + ps}}\,d xd y \Big) (-\Delta)_p^s u\\ &= g(x,u(x)) + H (x) |u|^{r_0 - 2} u \quad \text{in } \Omega, \end{aligned}\\ u = 0 \quad \text{in } \mathbb{R}^N \setminus \Omega. \end{gathered} \end{equation} with the following conditions: \begin{itemize} \item[(A8)] There exists a positive constant $ C_{G} $ such that $ | G(x,t) | \leq C_{G} ( | t |^{r-1} + 1) $ for some $ 2p < r < p^*$, $x \in \Omega $ and all $ t\in \mathbb{R}$, where $G(x,t)=\int_0^t f(x,s)\,ds$; \item[(A9)] $\lim_{|t| \to \infty} \big(\frac1{r_0} g (x, t) t - G (x, t) + a \varrho |t|^p + m |t|^{q}\big) = +\infty$ uniformly in $x \in \Omega$ where $\varrho < (\frac1p - \frac1{r_0}) \mu_1$, $1 < q< p < p^*$, $r_0 > 2 p$ and $m$ is a arbitrary positive constant; \item[(A10)] $G(x,t)\geq 0$, for all $x \in \Omega$, $t\in \mathbb{R}$; \item[(A11)] the function $H$ is a nonnegative and satisfies $0 < m \leq H \leq M$; \item[(A12)] $\lim_{| t | \to 0} \frac{g(x,t)}{| t |^{p-1}}=0$, uniformly in $x \in \Omega$; \item[(A13)] $G (x,0) = 0$ for all $x \in \Omega$ and $G(x,-t) = G(x,t)$, for all $x \in \Omega$, $t\in \mathbb{R}$. \end{itemize} \begin{theorem}\label{themain2} Assume that $ g \in C (\Omega \times \mathbb{R}, \mathbb{R}) $, $ H \in C (\Omega, \mathbb{R}) $, and {\rm (A8)--(A13)} hold. Then problem \eqref{e.5} has a sequence of solutions $ \{u_k\} $ such that $ I (u_k) \to + \infty $. \end{theorem} The proofs of the our main results are fully based on some theorems that we recalled here for the reader's convenience. \begin{theorem}[Mountain Pass Theorem \cite{AmRa,Ekeland}] \label{Mountain Pass Theorem} Let $X$ be a real Banach space and $J \in C^1(X,\mathbb{R})$ satisfying the {\rm (Ce)} condition. Suppose $J(0) = 0$, \begin{itemize} \item[(i)] there are constants $\rho, \beta > 0$ such that $J|_{\partial B_\rho} \geq \beta$ where $$ B_\rho = \{u \in X: \; \|u\| \leq \rho\}; $$ \item[(ii)] there is $u_1 \in X$ and $\|u_1\| > \rho$ such that $J(u_1) < 0$. \end{itemize} Then $J$ possesses a critical value $c \geq \beta$. Moreover c can be characterized as $$ c= \inf_{g \in \Gamma} \max_{u \in g ([0, 1])} J (u), \quad \Gamma = \{g \in C ([0, 1]): \; g (0) = 0,\; g(1) = u_1\}. $$ \end{theorem} \begin{theorem}[\cite{Rabinowitz}] \label{newthe} Let $X$ be an infinite dimensional Banach space, $X = Y \oplus Z$, where $Y$ is finite dimensional. If $J \in C^1(X, \mathbb{R})$ satisfies (Ce)$_{\rm c}$-condition for all $c > 0$, and \begin{itemize} \item[(i)] $J(0) = 0$, $J(- u) = J(u)$ for all $u \in X$; \item[(ii)] there exist constants $\rho, \alpha > 0$ such that $J|_{\partial B_\rho} \geq \alpha$; \item[(iii)] for any finite dimensional subspace $\widetilde{X} \subset X$, there is $R = R (\widetilde{X}) > 0$ such that $J (u) \leq 0$ on $\widetilde{X} \setminus B_\rho$; \end{itemize} then $J$ possesses an unbounded sequence of critical values. \end{theorem} \begin{theorem}[Fountain theorem] \label{the3} Let $ X_0 $ be a Banach space with the norm $ \|\cdot\| $ let $ X_i $ be a sequence of subspace of $ X $ with $ dim X_i < \infty $ for each $ i\in N $. Further, set \begin{equation*} X=\overline{\oplus_{\infty}^{i=1} X_i}, \quad Y_{k}=\oplus_{i=1}^{k} X_i, \quad Z_{k}=\overline{\oplus^{\infty}_{i=k} X_i} \end{equation*} Consider an even functional $ \Phi \in C^{1}(X,\mathbb{R})$. Assume that for each $ k \in \mathbb{N} $, there exists $ \rho_{k} > \gamma_{k}> 0 $ such that \begin{itemize} \item[(1)] $a_{k}:= \max_{u\in Y_{k},\| u \| = \rho_{k}} \Phi(u)\leq 0$, \item[(2)] $b_{k}:=\inf_{u\in Z_{k}, \| u \| =\gamma_{k}} \Phi(u) \to +\infty$, $k \to +\infty$, \item[(3)] $\Phi$ satisfies the $(\text{PS})_{\text{c}}$ condition for every $c > 0$. \end{itemize} Then $ \phi $ has an unbounded sequence of critical values. \end{theorem} Now, we need the following lemma about the (Ce) condition which will play an important role in the proof of our main results. \begin{lemma}\label{lemma2.5} Assume that {\rm (A1)} and {\rm (A2)} hold. Then the functional $ J: X_0 \to \mathbb{R} $ satisfies the {\rm (Ce)} condition. \end{lemma} \begin{proof} Let $ \{u_n\} $ be a (Ce)$_{\rm c}$ sequence for $ c \in \mathbb{R} $, \begin{equation}\label{e24} J(u_n) \to c, \quad (1 + \|u_n\|_{X_0}) J'(u_n) \to 0 \quad \text{as } n \to \infty. \end{equation} We first show that $\{u_n\}$ is a bounded sequence. In view of \eqref{e2.1}, \eqref{e2.2} and \eqref{e24}, one has \begin{equation}\label{e25} \begin{aligned} 1 + c & \geq J(u_n) - \frac1{2p} J'(u_n) u_n \\ & = \frac{a}{2p} \|u_n\|_{X_0}^p + \int_\Omega \Big(\frac1{2p} f(x,u_n(x))u_n(x) - F(x,u_n(x)) \Big) \,dx. \end{aligned} \end{equation} From (A2), there exists $ \theta > 0 $ such that \begin{equation}\label{e26} - \theta \leq \frac1{2p} f(x,t)t - F(x,t) + \frac{a \mu_1}{2 p} |t|^p, \quad \forall x \in \Omega, \; t \in \mathbb{R}. \end{equation} Now, We define $ u_n = \varphi_n + v_n $, where $ \varphi_n \in E_1 $ and $ v_n \in E_1^{\bot} $. By \eqref{e25} and \eqref{e26}, we have \begin{equation}\label{e27} \begin{aligned} 1 + c & \geq \frac{a}{2p} \|u_n\|_{X_0}^p - \frac{a \mu_1}{2 p} \|u_n\|_{L^p}^p \\ &\quad + \int_\Omega \Big(\frac1{2p} f(x,u_n(x))u_n(x) - F(x,u_n(x) + \frac{a \mu_1}{2 p} |u_n (x)|^p) \Big) \,dx \\ & \geq \frac{a}{2p} \big(1 - \frac{\mu_1}{\mu_2}\big) \|v_n\|_{X_0}^p - \theta |\Omega|, \end{aligned} \end{equation} which implies that $ \|v_n\|_{X_0} $ is bounded. Now, we assume that $ \{u_n\} $ is unbounded sequence, so there is a subsequence $ \{u_n\} $ (to simplify the notation) of $ \{u_n\} $ satisfying $ \|u_n\|_{X_0} \to + \infty $ as $ n \to + \infty $. Hence we have $ \frac{v_n}{\|u_n\|_{X_0}} \to 0 \in X_0 $. Since $\frac{\varphi_n}{\|u_n\|_{X_0}} $ is bounded in finite dimensional $ E_1 $, one can get $ \frac{\varphi_n}{\|u_n\|_{X_0}} \to w $ in $ E_1 $. Using \[ w_n : = \frac{u_n}{\|u_n\|_{X_0}} = \frac{\varphi_n + v_n}{\|u_n\|_{X_0}} = \frac{\varphi_n}{\|u_n\|_{X_0}} + \frac{v_n}{\|u_n\|_{X_0}} \to w, \] in $ E_1 $, yields \begin{equation}\label{e28} \frac{u_n(x)}{\|u_n\|_{X_0}} \to w (x) \quad \text{a.e. } \text{in } \Omega. \end{equation} So, by this fact $ \|w\|_{X_0} = 1 $ (we know that $ \|w_n\|_{X_0} = 1 $), $ w \in E_1 $ and \eqref{e28}, we have \begin{equation}\label{e29} |u_n(x)| \to + \infty \quad \text{as } n \to + \infty. \end{equation} In view of (A2), \eqref{e27}, \eqref{e29} and Fatou's lemma, one has \begin{equation}\label{e30} \begin{aligned} 1 + c & \geq J(u_n) - \frac1{2p} J'(u_n) u_n \\ & = \frac{a}{2p} \|u_n\|_{X_0}^p + \int_\Omega \Big(\frac1{2p} f(x,u_n(x))u_n(x) - F(x,u_n(x)) \Big) \,dx \\ & \geq \int_\Omega \Big(\frac1{2p} f(x,u_n(x))u_n(x) - F(x,u_n(x) + \frac{a \mu_1}{2 p} |u_n (x)|^p) \Big) \,dx \\ & \to + \infty \quad \text{as } n \to + \infty, \end{aligned} \end{equation} which is a contradiction. Then we get that $ \{u_n\} $ is bounded in $ X_0 $. By (A1), we can easily obtain that $ \{u_n\} $ has a convergence subsequence. Therefore, the functional $ J $ satisfies the (Ce) condition. \end{proof} \begin{proof}[Proof of Theorem \ref{themain1}] By Lemma \ref{lemma2.5}, we know that the functional $ J: X_0 \to \mathbb{R} $ satisfies the (Ce) condition. Hence, it is sufficient to show that $ J $ satisfies (i) and (ii) of Theorem \ref{themain1}. First, we claim that there are constant $\beta, \rho > 0$ such that $J (u) \geq \beta$ for all $\|u\|_{X_0} = \rho$. By (A1) and (A4), we can get \begin{equation}\label{29} F (x, t) \leq \frac{a \mu_1}{p} |t|^p + \frac{b (\lambda_1 - \varepsilon)}{2 p} |t|^{2p} + C |t|^r, \end{equation} for all $\varepsilon$ small enough, $t \in \mathbb{R}$ and $x \in \Omega$. Then, from \eqref{e14}-\eqref{e2.1} and \eqref{29}, we have \begin{align*} J(u)& = \frac{a}{p}\|u\|_{X_0}^{p} + \frac{b}{ 2 p}\|u\|_{X_0}^{2p} - \int_\Omega F(x,u(x))\,dx\\ & \geq \frac{a}{p}\|u\|_{X_0}^{p} + \frac{b}{ 2 p}\|u\|_{X_0}^{2p} - \frac{a \mu_1}{p} \|u\|_{L^p}^p + \frac{b (\lambda_1 - \varepsilon)}{2 p} \|u\|_{L^{2p}}^{2p} - C \int_\Omega |u(x)|^r\,dx \\ & \geq \frac{b}{ 2 p} \Big(1 - \frac{\lambda_1 - \varepsilon}{\lambda_1} \Big)\|u\|_{X_0}^{2p} - C C_r \|u\|_{X_0}^{r}. \end{align*} Since $ 2 p < r < p^* $ then for $ \varepsilon $ small enough, there exists $ \beta > 0 $ such that $ J (u) \geq \beta $ for all $ \|u\|_{X_0} = \rho $, where $ \rho > 0 $ small enough. Next, we will show that there exists $ u_1 \in X_0 $ and $ \|u_1\|_{X_0} > \rho $ such that $ J (u_1) < 0 $. By the definition of $ \lambda_1 $, for small enough $ \varepsilon > 0 $, we can choose $ u \in \mathcal{M} $ satisfying \begin{equation}\label{e31} \lambda_1 + \frac{\varepsilon}{p} \geq \|u\|_{X_0}^{2 p}. \end{equation} Also, in view of (A1) and (A3) that \begin{equation}\label{e32} F (x, t) > \frac{b (\lambda_1 + \varepsilon)}{2p} t^{2 p} - C. \end{equation} So, From \eqref{e31} and \eqref{e32}, one can get \begin{align*} J(t u)& = \frac{a}{p} t^p \|u\|_{X_0}^{p} + t^{2p} \frac{b}{ 2 p}\|u\|_{X_0}^{2p} - \int_\Omega F(x, t u(x))\,dx\\ & \leq \frac{a}{p} t^p \|u\|_{X_0}^{p} + \frac{b}{ 2 p} t^{2p} \|u\|_{X_0}^{2p} - \frac{b }{2p} t^{2p} (\lambda_1 + \varepsilon) + C |\Omega| \\ & \leq \frac{a}{p} t^p \|u\|_{X_0}^{p} + \frac{b}{ 2 p} t^{2p} \lambda_1 + \frac{b \varepsilon}{2 p^2} t^{2 p}- \frac{b }{2p} t^{2p} (\lambda_1 + \varepsilon) + C |\Omega| \\ & = \frac{a}{p} t^p \|u\|_{X_0}^{p} - \frac{b \varepsilon}{2 p^2} t^{2 p} + C |\Omega|. \end{align*} Then, $J(t u) \to - \infty$ as $t \to \infty$. Therefore, there exists $u_1 \in X_0$ and $\|u_1\|_{X_0} > \rho$ such that $J (u_1)< 0$. \end{proof} To prove of Theorem \ref{thenew}, we need the following lemmas. \begin{lemma}\label{lem2} Assume that {\rm (A1), (A6) and (A7)} hold. Then the functional $ J: X_0 \to \mathbb{R} $ satisfies the {\rm (Ce)} condition. \end{lemma} \begin{proof} Let $ \{u_n\} \subset X_0$ is a (Ce)$_{\rm c}$ sequence for $ c \in \mathbb{R} $, \begin{equation}\label{e240} J(u_n) \to c, \quad (1 + \|u_n\|_{X_0}) J'(u_n) \to 0 \quad \text{as } n \to \infty. \end{equation} We first claim that $\{u_n\}$ is a bounded sequence. Suppose to the contrary that $\|u_n\|_{X_0} \to \infty$. We consider $w_n : = \frac{u_n}{\|u_n\|_{X_0}}$, then $\|w_n\|_{X_0} = 1$. Going if necessary to a subsequence, we may assume that \begin{equation}\label{co1} \begin{gathered} w_n \rightharpoonup w, \quad \text{weakly in } X_0,\\ w_n \to w, \quad \text{strongly in } L^q (\Omega) \; 1 \leq q < p^*_s)\\ w_n \to w, \quad \text{a.e. } x \in \Omega. \end{gathered} \end{equation} There are only two cases need to be consider: $w = 0$ or $w \ne 0$. We firs consider the case $w = 0$. By (A6) and \eqref{e240}, one obtains \begin{align*} &\frac{1}{\|u_n\|_{X_0}^ p} \Big(J(u_n) - \frac1{2p} J'(u_n) u_n \Big) \\ & \geq \frac{a}{2p} + \frac{1}{\|u_n\|_{X_0}^p} \int_\Omega \Big(\frac1{2p} f(x,u_n(x))u_n(x) - F(x,u_n(x)) \Big) dx\\ & \geq \frac{a}{2p} - \theta_0 \int_\Omega |w_n|^p d x, \end{align*} which implies $0 \geq a/(2p)$. This is a contradiction. If $w \ne 0$, setting $\Omega_1 := \{x \in \Omega : w (x) \ne 0 \}$, obviously $|\Omega_1| > 0$ where $|\Omega_1|$ is Lebesgue measure of $\Omega_1$. For $x \in \Omega_1$, we have $|u_n(x)| \to \infty$ as $n \to \infty$. In view of (A5), one has $$ \lim_{n \to \infty} \frac{F (x, u_n(x))}{|u_n(x)|^{2 p}} |w_n(x)|^{2 p} \to \infty. $$ So, using Fatou's Lemma, we can get \begin{equation}\label{e21n} \lim_{n \to \infty} \int_\Omega \frac{F (x, u_n(x))}{|u_n(x)|^{2 p}} |w_n(x)|^{2 p} d x \to \infty. \end{equation} From (A1), it follows that $$ |F(x , t)| \leq M |t|, \quad \forall x \in \Omega, \; |t| \leq L_1. $$ Combining this with (A5), we obtain \[ F(x , t)\geq - M |t|, \quad \forall (x, t) \in \Omega \times \mathbb{R}. \] So, by \eqref{e21}, we obtain \[ \int_{\Omega \setminus \Omega_1} \frac{F(x, u_n)}{\|u_n(x)\|_{X_0}^{2 p}}\,dx \geq - \frac{M \int_{\Omega \setminus \Omega_1} |u_n| d x}{\|u_n(x)\|_{X_0}^{2 p}} \geq - \frac{M \|u_n\|_1}{\|u_n(x)\|_{X_0}^{2 p}} \geq - \frac{M C_1}{\|u_n(x)\|_{X_0}^{2 p - 1}}, \] which implies \begin{equation}\label{2100} \liminf_{n \to \infty} \int_{\Omega \setminus \Omega_1} \frac{F(x, u_n)}{\|u_n(x)\|_{X_0}^{2 p}}\,dx \geq 0, \end{equation} Using \eqref{e240}, \eqref{e21n} and \eqref{2100}, we obtain \begin{equation}\label{210} \begin{aligned} 0 &= \lim_{n \to \infty} \frac{c + o (1)}{\|u_n(x)\|_{X_0}^{2 p}} = \lim_{n \to \infty} \frac{J(u_n)}{\|u_n(x)\|_{X_0}^{2 p}} \\ &= \lim_{n \to \infty} \frac{1}{\|u_n(x)\|_{X_0}^{2 p}} \Big(\frac{a}{p} \|u_n\|_{X_0}^{p} + \frac{b}{ 2 p}\|u_n\|_{X_0}^{2p} - \int_\Omega F(x, u_n(x))\,dx \Big) \\ &= \lim_{n \to \infty} \frac{1}{\|u_n(x)\|_{X_0}^{2 p}} \Big(\frac{a}{p} \|u_n\|_{X_0}^{p} + \frac{b}{ 2 p}\|u_n\|_{X_0}^{2p} - \int_{\Omega_1} F(x, u_n(x))\,dx \\ &\quad - \int_{\Omega \setminus \Omega_1} F(x, u_n(x))\,dx \Big) \\ &\leq \frac{b}{ 2p} + \lim_{n \to \infty} \frac{a}{p\|u_n(x)\|_{X_0}^{p}} - \lim_{n \to \infty} \int_{\Omega_1} \frac{F(x, u_n)}{\|u_n(x)\|_{X_0}^{2 p}}\,dx \\ &\quad - \liminf_{n \to \infty} \int_{\Omega \setminus \Omega_1} \frac{F(x, u_n)}{\|u_n(x)\|_{X_0}^{2 p}}\,dx \\ &\leq \frac{b}{ 2p} - \lim_{n \to \infty} \int_{\Omega_1} \frac{F(x, u_n)}{\|u_n(x)\|_{X_0}^{2 p}}\,dx = - \infty, \end{aligned} \end{equation} which is a contradiction. Then we $ \{u_n\} $ is bounded in $ X_0 $. By (A1), we can easily obtain that $ \{u_n\} $ has a convergence subsequence. Therefore, the functional $ J $ satisfies the (Ce) condition. \end{proof} \begin{proof}[Proof of Theorem \ref{thenew}] Let $\{e_j\}$ is an orthonormal basis of $X_0$ and define $X_j = \mathbb{R} e_j$, $$ Y_k = \oplus_{j=1}^k X_j, \quad Z_k =\oplus_{j=k + 1}^{\infty} X_j, \quad k \in \mathbb{Z} $$ and $Y_k$ is finite-dimensional. Set $X = X_0$, $Y = Y_k$ and $Z = Z_k$. Clearly, $J (0) = 0$ and (A7) implies $J$ is even and from Lemma \ref{lem2}, $J$ satisfies the (Ce) condition. conditions (i) of Theorem \ref{newthe} is satisfied. So, we only need to verify (ii) and (ii) of Theorem \ref{newthe}. Set \begin{equation}\label{31} \beta_k (r) := \sup_{u \in Z_{k}, \| u \|_{X_0=1}} \| u \|_{r}. \end{equation} By a direct calculation, we have $\beta_k \to 0$ as $k \to \infty$ for all $1 \leq r < p^*_s$. choose \begin{equation*} \rho : = \min \Big\{ \Big( \frac{a }{4 p C \beta_{k}(1)} \Big)^{\frac{1}{1-p}} , \Big( \frac{a r}{4 p C \beta_{k}^{r} (r)} \Big)^{\frac{1}{r-p}} \Big\} \end{equation*} Then, by (A1) and \eqref{31}, for $u \in Z_k$ and $\| u \|_{X_0} = \rho $, we have \begin{align*} J (u) &= \frac{a}{p} \|u\|_{X_0}^{p} + \frac{b}{ 2p}\|u\|_{X_0}^{2p} - \int_\Omega F(x, u(x))\,dx \\ &\geq \frac{a}{p} \|u\|_{X_0}^{p} - C \|u\|_1 - \frac{C}{r} \|u\|^r_r \\ &\geq \frac{a}{p} \|u\|_{X_0}^{p} - C \beta_k (1) \|u\|_{X_0} - \frac{C}{r} \beta_k^k (r) \|u\|_{X_0}^r \\ &\geq \frac{a}{2 p} \rho^p: = \alpha > 0. \end{align*} Thus condition (ii) of Theorem \ref{newthe} is satisfied. Since all norms are equivalent in a finite dimensional space, there is a constant $\Upsilon > 0$ such that \begin{equation}\label{32} \|u\|_{2 p} \geq \Upsilon \|u\|_{X_0}, \quad \forall u \in Y. \end{equation} In view of (A5), for any $M_1 > \frac{b}{2 p \Upsilon^{2 p}}$, there is a constant $\Gamma_0 > 0$ such that $$ F (x , t) \geq M_1 t^{2 p}, \quad \forall x \in \Omega, \; |t| \geq \Gamma_0. $$ By (A1), we have $$ |F (x , t)| \leq C(1 + \Gamma_0^{r - 1}) |t|, \quad \forall x \in \Omega, \; |t| \leq \Gamma_0, $$ which implies \begin{equation}\label{30} F (x , t) \geq M_1 t^{2 p} - C' |t|, \quad \forall (x, t) \in \Omega \times \mathbb{R}, \end{equation} where $C'$ is a positive constant. Hence from \eqref{e21}, \eqref{32} and \eqref{30}, one can get \begin{align*} J (u) &\leq \frac{a}{p} \|u\|_{X_0}^{p} + \frac{b}{ 2 p}\|u\|_{X_0}^{2p} - M_1 \|u\|_{2 p}^{2 p} + C' \|u\|_1 \\ &\leq \frac{a}{p} \|u\|_{X_0}^{p} - \big(M_1 \Upsilon^{2 p} - \frac{b}{ 2 p} \big) \|u\|_{X_0}^{2 p} + C'C \|u\|_{X_0}, \quad \forall u \in Y. \end{align*} Consequently, there is a large $R = R (\widetilde{X}) > 0$ such that $J (u) \leq 0$ on $Y \setminus B_\rho$. Thus the condition (iii) of Theorem \ref{newthe} is satisfied. Then all conditions of Theorem \ref{newthe} are satisfied. Therefore, problem \eqref{e1.4} possesses infinitely many nontrivial solutions. \end{proof} To proof Theorem \ref{themain2}, wee need the following lemmas. \begin{lemma}\label{lem2.5} Assume that {\rm (A8)-(A10)} hold. Then the functional $ J: X_0 \to \mathbb{R} $ satisfies the ${\rm (PS)_c}$. condition. \end{lemma} \begin{proof} Assume that $ \{u_n\} \subset X_0 $ such that $$ J (u_n) \to c \quad \text{and } J' (u_n) \to 0 \quad \text{in } X_0^*. $$ So we first prove that $ \{u_n\} $ is bounded in $ X_0 $. By (A9), there exists $ \theta_0 $ such that \begin{equation}\label{tet} \frac{1}{r_0} g(x,t)t -G(x,t) + a\varrho |t|^{p} +m|t|^{q} > - \theta_0. \end{equation} So, by \eqref{e2.1}, \eqref{e2.2} and \eqref{tet}, we have \begin{align*} C+1 &\geq J(u_{n})-\frac{1}{n} \langle J'(u_{n}),u_{n} \rangle \\ &= a \big(\frac{1}{p}-\frac{1}{r_0} \big)\| u_{n}\|^{p}_{X_0} +b \big(\frac{1}{2p}-\frac{1}{r_0}\big)\| u_{n}\|^{2p}_{X_0} +\int_{\Omega}\big[\dfrac{1}{r_0}g(x,u_{n})-G(x,u_{n})\big] \, dx\\ &\geq a\big(\frac{1}{p}-\frac{1}{r_0} \big)\| u_{n}\|^{p}_{X_0} +b\big(\frac{1}{2p}-\frac{1}{r_0}\big)\| u_{n}\|^{2p}_{X_0} -a\varrho \int_{\Omega}| u_{n}|^{p} \, dx \\ &\quad -m\int_{\Omega}| u_{n}|^{q} \, dx -\theta_0 |\Omega|\\ &= a\big(\frac{1}{p}-\frac{1}{r_0} \big) \| u_{n}\|^{p}_{X_0} +b\big(\frac{1}{2p}-\frac{1}{r_0}\big) \| u_{n}\|^{2p}_{X_0} -a \varrho \| u_{n}\|_{p}^{p} - m\| u_{n}\|_{q}^{q} -\theta | \Omega|\\ &\geq a \big(\frac{1}{p}-\frac{1}{r_0}-\frac{\varrho}{\mu_{1}}\big) \| u_{n} \|_{X_0}^{p} +b\big(\frac{1}{2p}-\frac{1}{r_0}\big)\| u_{n}\|_{X_0}^{2p} -mC_{q}^{q}\| u_{n} \|_{X_0}^{q} - \theta_0 | \Omega |\\ &\geq a\big(\frac{1}{p}-\frac{1}{r_0}-\frac{\varrho}{\mu_{1}}\big) \| u_{n} \|_{X_0}^{p} -mC_{q}^{q}\| u_{n} \|_{X_0}^{q} - \theta_0 |\Omega|. \end{align*} This implies \begin{equation*} a \big(\frac{1}{p}-\frac{1}{r_0}-\frac{\varrho}{\mu_{1}} \big) \| u_{n} \|_{X_0}^{p} \leq C+1+ mC_{q}^{q}\| u_{n} \|_{X_0}^{q} + \theta_0 |\Omega|. \end{equation*} Since $ 1 2p $ and all norms are equivalent on a finite dimensional space, there exists large $ \rho_{k} > 0 $ such that \begin{equation*} a_{k} := \max_{a\in Y_{k}, \| u \|_{X_0=\rho_{k}}} J(u) < 0. \end{equation*} Then, condition (1) of Theorem \ref{the3} is satisfied. Set $$ \beta_k := \max \Big\{\sup_{u \in Z_{k}, \| u \|_{X_0=1}} \| u \|_{r}, \sup_{u \in Z_{k}, \| u \|_{X_0=1}} \| u \|_{r_0} \Big\}. $$ In view of $Z_{k + 1} \subset Z_k$, one has $0 < \beta_{k + 1}\leq \beta_k$ and by a direct calculation, we have $\beta_k \to 0$ as $k \to \infty$. By (A8) and (A12), for any $ \epsilon > 0 $ there exists $ \delta=\delta(\epsilon) > 0 $ such that a.e. $ x \in \Omega$ and for any $ t \in \mathbb{R} $ \begin{equation*} | G(x,t) | \leq \epsilon | t |^{p} + r \delta(\epsilon)| u |^{r}. \end{equation*} Then, by (A11), \begin{align*} J(u) &= \frac{a}{p} \| u \|_{X_0}^{p} + \frac{b}{2p} \| u \|_{X_0}^{2p} - \int_{\Omega} G(x,u) \, dx - \frac{1}{r_0} \int_{\Omega} H | u |^{r_0} \, dx\\ &\geq \frac{a}{p} \| u \| ^{p}_{X_0} - \epsilon \| u \|_{p}^{p} - r\delta(\epsilon)\| u \|_{r}^{r} -\frac{M}{r_0} \| u \|_{r_0}^{r_0}\\ &\geq \frac{a}{p} \| u \|_{X_0}^{p} + \frac{b}{2p} \| u \|_{X_0}^{2p} - \frac{M}{r_0} \| u \|_{r_0}^{r_0}\\ &\geq \frac{a}{p} \| u \| ^{p}_{X_0} - \frac{\epsilon}{\mu_{1}} \| u \|_{X_0}^{p} -r \delta(\epsilon) \| u \|_{r}^{r}- \frac{M}{r_0} \| u \|^{r_0}_{r_0}\\ &\geq (\frac{a}{p}- \frac{\epsilon}{\mu_{1}}) \| u \|^{p}_{X_0} - r \delta(\epsilon) \beta_{k}^{r} \| u \|^{r}_{X_0} - \frac{M}{r_0} \beta_{k}^{r_0} \| u \|_{X_0}^{r_0}. \end{align*} For every $ \epsilon $ with $ 0 < \epsilon < \frac{a \mu_{1}}{p} $, choose \begin{equation*} \| u \|_{X_0} = \gamma_{k} = \min \Big\{ \Big( \frac{a \mu_{1}-\epsilon p}{3r \delta(\epsilon) p \mu_{1} \beta_{k}^{r}} \Big)^{\frac{1}{r-p}} , \Big( \frac{(a \mu_{1}- \epsilon p) r_0}{3p\mu_{1} M \beta_{k}^{r_0}} \Big)^{\frac{1}{r_0-p}} \Big\}. \end{equation*} Since $ \beta_{k} \to 0 $ as $ k \to \infty $, we have $ \| u \| = \gamma_{k} \to + \infty $ as $ k \to \infty $. Hence \begin{align*} b_{k} &:= \inf_{ u\in Z_{k}, \| u \|_{X_0}=\gamma_{k}} J(u) \\ &\geq \Big( \frac{a}{p} - \frac{\epsilon}{\mu_{1}}\Big) \gamma_{k}^{p} - r\delta(\epsilon) \Big(\frac{a \mu_{1} - \epsilon p}{3r \delta(\epsilon) p \mu_{1}} \Big) \gamma_{k}^{p} - \frac{M}{r_0} \Big( \frac{(a \mu_{1} - \epsilon p )r_0}{3 p \mu_{1} M}\Big) \gamma_{k}^{p}\\ &= \frac{1}{3} \Big( \frac{a \mu_{1}- \epsilon p}{p \mu_{1}}\Big) \gamma_{k}^{p} \to + \infty, \quad \text{as } k \to \infty. \end{align*} Then, condition (2) of Theorem \ref{the3} is satisfied. So, its follows that the conditions of Theorem \ref{the3} was satisfied and we have unbounded sequence which yields that $ I(u_{k} ) \to + \infty $ then the proof is complete. \end{proof} \subsection*{Acknowledgements} The authors would like to thank the anonymous referees for their valuable suggestions and comments. \begin{thebibliography}{99} \bibitem{AmRa} A. Ambrosetti, P. H. 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