\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 148, pp. 1--13.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2013/148\hfil Existence and multiplicity of solutions] {Existence and multiplicity of solutions for a degenerate nonlocal elliptic differential equation} \author[N. T. Chung, H. Q. Toan\hfil EJDE-2013/148\hfilneg] {Nguyen Thanh Chung, Hoang Quoc Toan} % in alphabetical order \address{Nguyen Thanh Chung \newline Dept. Science Management and International Cooperation, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Vietnam} \email{ntchung82@yahoo.com} \address{Hoang Quoc Toan \newline Department of Mathematics, Hanoi University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam} \email{hq\_toan@yahoo.com} \thanks{Submitted October 23, 2012. Published June 27, 2013.} \subjclass[2000]{35J60, 35B38, 35J25} \keywords{Degenerate nonlocal problems; existence o solutions; multiplicity; \hfill\break\indent variational methods} \begin{abstract} Using variational arguments, we study the existence and multiplicity of solutions for the degenerate nonlocal differential equation \begin{gather*} - M\Big(\int_\Omega |x|^{-ap}|\nabla u|^p\,dx\Big)\operatorname{div} \Big(|x|^{-ap}|\nabla u|^{p-2}\nabla u\Big) = |x|^{-p(a+1)+c} f(x,u) \quad \text{in } \Omega,\\ u = 0 \quad \text{on } \partial\Omega, \end{gather*} where $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) and the function $M$ may be zero at zero. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} In this article, we study the boundary-value problem \begin{equation}\label{e1.1} \begin{gathered} - M\Big(\int_{\Omega}|x|^{-ap}|\nabla u|^p\,dx\Big)\operatorname{div}\Big( |x|^{-ap}|\nabla u|^{p-2}\nabla u\Big) = |x|^{-p(a+1)+c} f(x,u) \quad \text{ in } \Omega,\\ u = 0 \quad \text{on } \partial\Omega, \end{gathered} \end{equation} where $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) is a smooth bounded domain, $0 \in \Omega$, $0 \leq a < \frac{N-p}{p}$, $1 < p < N$, $00 \quad \text{for all } t \in \mathbb{R}^+. \end{equation} We refer the readers to \cite{AutColPuc,ColaPucc} where the authors studied the existence of weak solutions for elliptic equations involving $p$-polyharmonic Kirchhoff operators. Motivated by the ideas introduced in \cite{Chung,ColaPucc,XLFan,DLiu,BJXuan}, the goal of this paper is to study the existence and multiplicity of solutions for \eqref{e1.1} without condition \eqref{e1.4}. The approach is based on variational arguments. Our results complement the previous ones in the non-degenerate case. Moreover, we consider problem \eqref{e1.1} in the general case $0 \leq a < \frac{N-p}{p}$, $1 < p < N$, $00 $$ for all $t\in [0,+\infty)$ and $x \in \Omega$. We start by recalling some useful results in \cite{CafKohNir, CatWang, BJXuan}. We have known that for all $u \in C^\infty_0(\mathbb{R}^N)$, there exists a constant $C_{a,b} > 0$ such that \begin{equation}\label{e1.5} \Big(\int_{\mathbb{R}^N}|x|^{-bq}|u|^q\,dx\Big)^{p/q} \leq C_{a,b}\int_{\mathbb{R}^N} |x|^{-ap}|\nabla u|^p\,dx, \end{equation} where $$ - \infty < a < \frac{N-p}{p}, \quad a \leq b \leq a+1, \quad q = p^\ast(a,b) =\frac{Np}{N-dp}, \quad d = 1+a-b. $$ Let $W^{1,p}_0(\Omega,|x|^{-ap})$ be the completion of $C^\infty_0 (\Omega)$ with respect to the norm $$ \|u\|_{a,p} = \Big(\int_\Omega |x|^{-ap}|\nabla u|^p\,dx\Big)^{1/p}. $$ Then $W^{1,p}_0(\Omega,|x|^{-ap})$ is reflexive and separable Banach space. From the boundedness of $\Omega$ and the standard approximation argument, it is easy to see that \eqref{e1.5} holds for any $u \in W^{1,p}_0(\Omega,|x|^{-ap})$ in the sense that \begin{equation}\label{e1.6} \Big(\int_{\mathbb{R}^N}|x|^{-\alpha}|u|^l\,dx\Big)^{p/l} \leq C_{a,b}\int_{\mathbb{R}^N} |x|^{-ap}|\nabla u|^p\,dx, \end{equation} for $1 \leq l \leq p^\ast = \frac{Np}{N-p}$, $\alpha \leq (1+a)l + N\Big(1- \frac{l}{p}\Big)$; that is, the embedding $W^{1,p}_0(\Omega,|x|^{-ap}) \hookrightarrow L^l(\Omega,|x|^{-\alpha})$ is continuous, where $L^l (\Omega,|x|^{-\alpha})$ is the weighted $L^l(\Omega)$ space with the norm $$ |u|_{l,\alpha} : = |u|_{L^l(\Omega, |x|^{-\alpha})} = \Big(\int_\Omega|x|^{-\alpha}|u|^l\,dx\Big)^{1/l}. $$ In fact, we have the following compact embedding result which is an extension of the classical Rellich-Kondrachov compactness theorem. \begin{lemma}[Compactness embedding theorem \cite{BJXuan}] \label{lem1.1} Suppose that $\Omega\subset \mathbb{R}^N$ is an open bounded domain with $C^1$ boundary and that $0 \in \Omega$, where $1 < p < N$, $-\infty < a < \frac{N-p}{p}$, $1 \leq l < \frac{Np}{N-p}$ and $\alpha < (1+a)l+N\big(1- \frac{l}{p}\big)$. Then the embedding $W^{1,p}_0(\Omega,|x|^{-ap}) \hookrightarrow L^l(\Omega,|x|^{-\alpha})$ is compact. \end{lemma} \section{Main results} In this section, will we discuss the existence of weak solutions for problem \eqref{e1.1}. For simplicity, we denote $X=W^{1,p}_0(\Omega,|x|^{-ap})$. In the following, when there is no misunderstanding, we always use $c_i, C_i$ to denote positive constants. \begin{definition}\label{def2.1}\rm We say that $u \in X$ is a weak solution of problem \eqref{e1.1} if \begin{align*} &M\Big(\int_\Omega|x|^{-ap}|\nabla u|^p\,dx\Big)\int_\Omega |x|^{-ap}|\nabla u|^{p-2}\nabla u\cdot \nabla \varphi \,dx \\ &- \int_\Omega |x|^{-p(a+1)+c}f(x,u) \varphi \,dx = 0 \end{align*} for all $\varphi \in C^\infty_0(\Omega)$. \end{definition} Define \begin{equation}\label{e2.1} \Phi(u) = \frac{1}{p}\widehat{M}\Big(\int_\Omega|x|^{-ap}|\nabla u|^p\,dx\Big), \quad \Psi(u) = \int_\Omega |x|^{-p(a+1)+c}F(x,u)\,dx, \end{equation} where \[ \widehat{M}(t) = \int_0^tM(s)\,ds, \quad F(x,t) = \int_0^tf(x,s)\,ds. \] By the condition (F0) (see Theorem \ref{the2.2} below), Lemma \ref{lem1.1} implies that the energy functional $J(u) = \Phi(u)-\Psi(u): X \to \mathbb{R}$ associated with problem \eqref{e1.1} is well defined. Then it is easy to see that $J \in C^1(X,\mathbb{R})$ and $u \in X$ is a weak solution of \eqref{e1.1} if and only if $u$ is a critical point of $J$. Moreover, we have \begin{align*} J'(u)(\varphi) &= M\Big(\int_\Omega|x|^{-ap}|\nabla u|^p\,dx\Big)\int_\Omega |x|^{-ap}|\nabla u|^{p-2}\nabla u\cdot \nabla \varphi \,dx \\ &\quad - \int_\Omega |x|^{-p (a+1)+c}f(x,u)\varphi \,dx \\ & = \Phi'(u)(\varphi)-\Psi'(u)(\varphi) \end{align*} for all $\varphi \in X$. For the next theorem, we use the following assumptions: \begin{itemize} \item[(M0)] $M: \mathbb{R}^+ \to \mathbb{R}^+$ is a continuous function and satisfies $$ m_0t^{\alpha-1} \leq M(t) \quad \text{for all } t \in \mathbb{R}^+, $$ where $m_0>0$ and $\alpha>1$; \item[(F0)] $f: \Omega\times \mathbb{R} \to \mathbb{R}$ is a Carath\'{e}odory function such that $$ |f(x,t)| \leq C_1(1+|t|^{q-1}) \quad\text{for all } x \in \Omega \text{ and } t \in \mathbb{R}, $$ where $C_1>0$ and $1 q$. \end{itemize} \begin{theorem}\label{the2.2} Under assumptions {\rm (M0), (F0), (E0)}, problem \eqref{e1.1} has at least one weak solution. \end{theorem} \begin{proof} Let $\{u_m\}$ be a sequence that converges weakly to $u$ in $X$. Then, by the weak lower semicontinuity of the norm, we have $$ \liminf_{m\to \infty}\int_\Omega|x|^{-ap}|\nabla u_m|^p\,dx \geq \int_\Omega |x|^{-ap}|\nabla u|^p\,dx. $$ Combining this with the continuity and monotonicity of the function $\psi: \mathbb{R}^+ \to \mathbb{R}$, $t \mapsto \psi(t)=\frac{1}{p}\widehat{M}(t)$, we obtain \begin{equation} \label{e2.2} \begin{aligned} \liminf_{m\to \infty}\Phi(u_m) & = \liminf_{m\to \infty}\frac{1}{p}\widehat{M} \Big(\int_\Omega |x|^{-ap}|\nabla u_m|^p\,dx\Big) \\ & = \liminf_{m\to \infty}\psi\Big(\int_\Omega |x|^{-ap}|\nabla u_m|^p\,dx\Big) \\ & \geq \psi\Big(\liminf_{m\to \infty}\int_\Omega |x|^{-ap}|\nabla u_m|^p\,dx \Big) \\ & \geq \psi\Big(\int_\Omega |x|^{-ap}|\nabla u|^p\,dx\Big) \\ & = \frac{1}{p}\widehat{M}\Big(\int_\Omega |x|^{-ap}|\nabla u|^p\,dx\Big) = \Phi(u). \end{aligned} \end{equation} Using (F0), H\"older's inequality, and Lemma \ref{lem1.1}, it follows that \begin{equation} \label{e2.3} \begin{aligned} & \big|\int_\Omega |x|^{-p(a+1)+c}[F(x,u_m)-F(x,u)]\,dx\big| \\ & \leq \int_\Omega |x|^{-p(a+1)+c}|f(x,u+\theta_{m}(u_m-u))| |u_m-u|\,dx \\ & \leq C_1\int_\Omega|x|^{-p(a+1)+c}\left(1+|u+\theta_{m}(u_m-u)|^{q-1}\right) |u_m-u|\,dx \\ & \leq C_1\Big(\int_\Omega |x|^{-p(a+1)+c}\,dx\Big)^\frac{q-1}{q}\|u_m-u\|_{L^q (\Omega,|x|^{-p(a+1)+c})} \\ & \quad +C_1\|u+\theta_{m}(u_m-u)\|_{L^q(\Omega,|x|^{-p(a+1)+c}) }^{q-1}\|u_m-u\|_{L^q(\Omega,|x|^{-p(a+1)+c})}, \end{aligned} \end{equation} which tends to $0$ as $m\to \infty$, where $0 \leq \theta_m(x) \leq 1$ for all $x \in \Omega$. From \eqref{e2.2} and \eqref{e2.3}, the functional $J$ is weakly lower semi-continuous in $X$. On the other hand, by assumptions (M0) and (F0), we have \begin{equation} \label{e2.4} \begin{aligned} J(u) & = \frac{1}{p}\widehat{M}\Big(\int_\Omega|x|^{-ap}|\nabla u|^p\,dx \Big)-\int_\Omega |x|^{-p(a+1)+c}F(x,u)\,dx \\ & \geq \frac{m_0}{p}\int_0^{\|u\|^p_{a,p}}t^{\alpha-1}\,dt-c_1\int_\Omega |x|^{-p(a+1)+c}\big(1+|u|^q\big)\,dx \\ & \geq \frac{m_0}{\alpha p}\|u\|^{\alpha p}_{a,p}-c_2\|u\|^q_{a,p}-c_3. \end{aligned} \end{equation} Since $10$ and $1< \alpha_1 \leq\alpha_2$; \item[(M2)] $M$ satisfies $$ \widehat{M}(t) \geq M(t)t \text{ for all } t \in \mathbb{R}^+; $$ \item[(F1)] $f(x,t)=o\big(|t|^{\alpha_1 p-1}\big)$, $t\to 0$ uniformly for $x \in \Omega$; \item[(F2)] There exists a positive constant $\mu > \alpha_2p$ such that $$ 0< \mu F(x,t) := \int_0^t f(x,s)ds \leq f(x,t)t $$ for all $x\in \Omega$ and $|t| \geq T>0$; \item[(E1)] $\alpha_1 p < q$. \end{itemize} \begin{theorem}\label{the2.3} Under assumptions {\rm (F0)--(F2), (M1)--(M2)}, problem \eqref{e1.1} has at least one nontrivial weak solution. \end{theorem} To prove the above theorem, we need to verify the following lemmas. \begin{lemma}\label{lem2.4} Assume that {\rm (M1), (M2), (F0), (F2)} are satisfied. Then the functional $J$ satisfies the (PS) condition. \end{lemma} \begin{proof} Let $\{u_m\}\subset X$ be a sequence such that \begin{equation}\label{e2.5} J(u_m) \to \overline c<\infty, \quad J'(u_m) \to 0 \quad \text{in } X^\ast \text{ as } m\to \infty, \end{equation} where $X^\ast$ is the dual space of $X$. First, we will show that the sequence $\{u_m\}$ is bounded in $X$. Indeed, from \eqref{e2.5}, (M1), (M2) and (F2), we obtain that for all $m$ large enough, \begin{equation} \label{e2.6} \begin{aligned} &1+\overline c +\|u_m\|_{a,p} \\ & \geq J(u_m)-\frac{1}{\mu}J'(u_m)(u_m) \\ & = \frac{1}{p}\widehat{M}\Big(\int_\Omega|x|^{-ap}|\nabla u_m|^p\,dx\Big)- \frac{1}{\mu}M\Big(\int_\Omega|x|^{-ap}|\nabla u_m|^p\,dx\Big)\int_\Omega |x|^{-ap}|\nabla u_m|^p\,dx\\ & \quad -\int_\Omega |x|^{-p(a+1)+c}F(x,u_m)\,dx +\frac{1}{\mu} \int_\Omega |x|^{-p(a+1)+c}f(x,u_m)u_m\,dx \\ & \geq \Big(\frac{1}{p}-\frac{1}{\mu}\Big)M\Big(\int_\Omega|x|^{-ap}|\nabla u_m|^p\,dx\Big)\int_\Omega|x|^{-ap}|\nabla u_m|^p\,dx \\ & \quad -\int_\Omega |x|^{-p(a+1)+c}\Big(\frac{1}{\mu}f(x,u_m)u_m- F(x,u_m)\Big)\,dx \\ & \geq m_1\big(\frac{1}{p}-\frac{1}{\mu}\big)\|u_m\|^{\alpha_1p}_{a,p}-c_4. \end{aligned} \end{equation} Since $\alpha_1p>1$, it follows from \eqref{e2.6} that $\{u_m\}$ is bounded. Passing to a subsequence if necessary, there exists $u \in X$, such that $\{u_m\}$ converges weakly to $u$ in $X$. By \eqref{e2.5}, we obtain \begin{equation}\label{e2.7} \lim_{m\to \infty}J'(u_m)(u_m-u) = 0. \end{equation} By (F0) and Lemma \ref{lem1.1}, we have \begin{equation} \label{e2.8} \begin{aligned} &\big| \int_\Omega |x|^{-p(a+1)+c}f(x,u_m)(u_m-u)\,dx\big| \\ &\leq \int_\Omega |x|^{-p(a+1)+c}|f(x,u_m)||u_m-u|\,dx \\ & \leq C_1\int_\Omega |x|^{-p(a+1)+c}(1+|u_m|^{q-1})|u_m-u|\,dx \\ & \leq C_1\Big(\int_\Omega |x|^{-p(a+1)+c}\,dx\Big)^\frac{q-1}{q}\|u_m-u \|_{L^q(\Omega,|x|^{-p(a+1)+c})} \\ & \quad + C_1\|u_m\|_{L^q(\Omega,|x|^{-p(a+1)+c})}^{q-1}\|u_m- u\|_{L^q(\Omega,|x|^{-p(a+1)+c})}, \end{aligned} \end{equation} which tends to $0$ as $m\to \infty$. By \eqref{e2.7}, \eqref{e2.8} and the definition of the functional $J$, it follows that \begin{equation}\label{e2.9} \lim_{m\to\infty}M\Big(\int_\Omega|x|^{-ap}|\nabla u_m|^p\,dx\Big) \int_{\Omega}|x|^{-ap}|\nabla u_m|^{p-2}\nabla u_m\cdot (\nabla u_m-\nabla u) \,dx=0. \end{equation} Since $\{u_m\}$ is bounded in $X$, passing to a subsequence, if necessary, we may assume that $$ \int_\Omega|x|^{-ap}|\nabla u_m|^p\,dx \to t_0 \geq 0 \quad \text{as } m\to \infty. $$ If $t_0 = 0$ then $\{u_m\}$ converges strongly to $u = 0$ in $X$ and the proof is finished. If $t_0 > 0$ then by (M1) and the continuity of $M$, we obtain $$ M\Big(\int_\Omega|x|^{-ap}|\nabla u_m|^p\,dx \Big) \to M(t_0)>0 \quad \text{as } m\to\infty. $$ Thus, for $m$ sufficiently large, we have \begin{equation}\label{e2.10} 0 < c_5 \leq M\Big(\int_\Omega|x|^{-ap}|\nabla u_m|^p\,dx\Big)\leq c_6. \end{equation} From \eqref{e2.9} and \eqref{e2.10} and the condition (M1), we have \begin{equation}\label{e2.11} \lim_{m\to\infty}\int_\Omega |x|^{-ap}|\nabla u_m|^{p-2}\nabla u_m\cdot (\nabla u_m-\nabla u)\,dx=0. \end{equation} On the other hand, since $\{u_m\}$ converges weakly to $u$ in $X$, we have \begin{equation}\label{e2.12} \lim_{m\to\infty}\int_\Omega |x|^{-ap}|\nabla u|^{p-2}\nabla u\cdot (\nabla u_m-\nabla u)\,dx=0. \end{equation} By \eqref{e2.11} and \eqref{e2.12}, $$ \lim_{m\to\infty}\int_\Omega |x|^{-ap}\left(|\nabla u_m|^{p-2}\nabla u_m- |\nabla u|^{p-2}\nabla u\right)\cdot (\nabla u_m-\nabla u)\,dx=0. $$ or \begin{equation}\label{e2.13} \lim_{m\to\infty}\int_\Omega \left(|\nabla v_m|^{p-2}\nabla v_m-|\nabla v|^{p-2}\nabla v\right)\cdot (\nabla v_m-\nabla v)\,dx=0, \end{equation} where $\nabla v_m=|x|^{-a}\nabla u_m$, $\nabla v=|x|^{-a}\nabla u \in L^p(\Omega)$. We recall that the following inequalities hold \begin{equation} \label{e2.14} \begin{gathered} \langle {|\xi|^{p-2}\xi-|\eta|^{p-2}\eta,\xi-\eta} \rangle \geq c_7 \Big(|\xi|+|\eta|\Big)^{p-2}|\xi-\eta|^{2} \quad \text{ if }10$ for all $u \in X$ with $\|u\|_{a,p}=\rho$; \item[(ii)] There exists $\widehat{u} \in X$ such that $\|\widehat{u}\|_{ a,p}>\rho$ and $J(u) < 0$. \end{itemize} \end{lemma} \begin{proof} (i) By (M1), we have \begin{equation} \label{e2.15} \begin{aligned} J(u) & = \frac{1}{p}\widehat{M}\Big(\int_\Omega |x|^{-ap}|\nabla u|^p\,dx \Big)-\int_\Omega|x|^{-p(a+1)+c}F(x,u)\,dx \\ & \geq \frac{m_1}{\alpha_1p}\|u\|_{a,p}^{\alpha_1p}-\int_{\Omega}|x|^{ -p(a+1)+c}F(x,u)\,dx. \end{aligned} \end{equation} Since $\alpha_1 p < q< \min \{ p^\ast, \frac{p(N-(a+1)p+c)}{N-(a+1)p}\}$, the embeddings \[ X \hookrightarrow L^{\alpha_1p}(\Omega, |x|^{-p(a+1)+c}),\quad X\hookrightarrow L^{q}(\Omega,|x|^{-p(a+1)+c}) \] are compact. Then there are constants $c_{12}, c_{13}>0$ such that \begin{gather}\label{e2.16} \|u\|_{L^{\alpha_1p}(\Omega,|x|^{-p(a+1)+c})} \leq c_{12}\|u\|_{a,p}, \\ \label{e2.17} \|u\|_{L^q(\Omega,|x|^{-p(a+1)+c})} \leq c_{13}\|u\|_{a,p}. \end{gather} Let $\epsilon > 0$ be small enough such that $\epsilon < \frac{m_1}{\alpha_1pc^{\alpha_1p}_{12}}$. By (F0) and (F1), we obtain \begin{equation}\label{e2.18} |F(x,t)|\leq \epsilon|t|^{\alpha_1p} + c_\epsilon |t|^q \text{ for all } x\in \Omega \text{ and } t \in \mathbb{R}. \end{equation} Therefore, by \eqref{e2.15}-\eqref{e2.18}, we have \begin{align*} J(u) & \geq \frac{m_1}{\alpha_1p}\|u\|_{a,p}^{\alpha_1p}-\int_{\Omega} |x|^{-p(a+1)+c}F(x,u)\,dx \\ & \geq \frac{m_1}{\alpha_1p}\|u\|_{a,p}^{\alpha_1p}-\epsilon\int_\Omega |x|^{-p(a+1)+c}|u|^{\alpha_1p}\,dx-c_\epsilon\int_\Omega|x|^{-p(a+1)+c} |u|^q\,dx \\ & \geq \Big(\frac{m_1}{\alpha_1p}-\epsilon c^{\alpha_1p}_{12}\Big)\|u \|^{\alpha_1p}_{a,p}-c_\epsilon c_{13}^q\|u\|^q. \end{align*} Since $\alpha_1p < q$, there exist real numbers $\rho, R>0$ such that $J(u) \geq R$ for all $u \in X$ with $\|u\|_{a,p}=\rho$. (ii) By (F2), there exists $c_{14}>0$ such that \begin{equation}\label{e2.19} F(x,t) \geq c_{14}|t|^\mu \text{ for all } x \in \Omega \text{ and } |t| \geq T. \end{equation} For $w \in X \backslash\{0\}$ and $t>0$, it follows from \eqref{e2.19} that \begin{equation} \label{e2.20} \begin{aligned} J(tw) & = \frac{1}{p}\widehat{M}\Big(\int_\Omega |x|^{-ap}|\nabla tw|^p\,dx \Big)-\int_{\Omega}|x|^{-p(a+1)+c}F(x,tw)\,dx \\ & \leq \frac{m_2t^{\alpha_2p}}{\alpha_2p}\|w\|_{a,p}^{\alpha_2p}-c_{14} t^\mu\int_{\Omega}|x|^{-p(a+1)+c}|w|^\mu \,dx - c_{15}, \end{aligned} \end{equation} which tends to $-\infty$ as $t\to +\infty$ since $\alpha_2p < \mu$. Then, there exists $t_0>0$ such that $J(t_0w)<0$ and $\|t_0w\|_{a,p}>\rho$. We set $\widehat{u} = t_0w$, then Lemma \ref{lem2.5} is proved. \end{proof} \begin{proof}[Proof of Theorem \ref{the2.3}] By Lemmas \ref{lem2.4} and \ref{lem2.5}, all assumptions of the mountain pass theorem in \cite{AmbRab} are satisfied. Then the functional $J$ has a nontrivial critical point in $X$ and thus problem \eqref{e1.1} has a nontrivial weak solution. \end{proof} Next, we will use the Fountain theorem and the Dual fountain theorem in order to study the existence of infinitely many solution for \eqref{e1.1}. More exactly, we will prove the following theorems. \begin{theorem}\label{the2.6} Assume that {\rm (M1), (M2), (F0), (F2), (E1)} are satisfied. Moreover, we assume that \begin{itemize} \item[(F3)] $f(x,-t) = -f(x,t)$ for all $x \in \Omega$ and $t \in \mathbb{R}$. \end{itemize} Then problem \eqref{e1.1} has a sequence of weak solutions $\{\pm u_k\}_{k=1}^\infty$ such that $J(\pm u_k) \to +\infty$ as $k \to +\infty$. \end{theorem} \begin{theorem}\label{the2.7} Assume that {\rm (M1), (M2), (F0)--(F2)} are satisfied. Moreover, we assume that \begin{itemize} \item[(F4)] $f(x,t) \geq C_2|t|^{r-1}$, $t \to 0$, where $\alpha_2pr_k>0$ such that \begin{itemize} \item[(A1)] $\inf_{\{u \in Z_k: \|u\|=r_k\}}J(u) \to +\infty$ as $k \to \infty$; \item[(A2)] $\max_{\{u \in Y_k: \|u\|=\rho_k\}}J(u) \leq 0$. \end{itemize} Then $J$ has a sequence of critical values which tends to $+\infty$. \end{lemma} \begin{definition}\label{def2.10}\rm We say that $J$ satisfies the (PS)$^\ast_c$ condition (with respect to $(Y_n)$) if any sequence $\{u_{n_j}\}\subset X$ such that $u_{n_j} \in Y_{n_j}$, $J(u_{n_j})\to c$ and $(J|_{Y_{n_j}})'(u_{n_j})\to 0$ as $n_j \to +\infty$, contains a subsequence converging to a critical point of $J$. \end{definition} \begin{lemma}[Dual fountain theorem \cite{Willem}]\label{lem2.11} Assume that $(X,\|\cdot\|)$ is a separable Banach space, $J \in C^1(X,\mathbb{R})$ is an even functional satisfying the (PS)$^\ast_c$ condition. Moreover, for each $k=1, 2, \dots $, there exist ${\rho}_k >r_k>0$ such that \begin{itemize} \item[(B1)] $\inf_{\{u \in Z_k: \|u\|=\rho_k\}}J(u) \geq 0$; \item[(B2)] $b_k: =\max_{\{u \in Y_k: \|u\|=r_k\}}J(u) < 0$; \item[(B3)] $d_k :=\inf_{\{u \in Z_k: \|u\|=\rho_k\}}J(u) \to 0$ as $k \to \infty$. \end{itemize} Then $J$ has a sequence of negative critical values which tends to $0$. \end{lemma} \begin{proof}[Proof of Theorem \ref{the2.6}] According to (F3) and Lemma \ref{lem2.4}, $J$ is an even functional and satisfies the (PS) condition. We will prove that if $k$ is large enough, then there exist $\rho_k>r_k>0$ such that (A1) and (A2) hold. Thus, the assertion of conclusion can be obtained from the Fountain theorem. (A1): From (F0), there exists $c_{16}>0$ such that $$ |F(x,t)| \leq c_{16}(|t|+|t|^q) \quad\text{for all } x \in \Omega \text{ and all } t \in \mathbb{R}. $$ Then, using (M1) and Lemma \ref{lem1.1}, for any $u \in Z_k$, \begin{equation} \label{e2.21} \begin{split} J(u) & = \frac{1}{p}\widehat{M}\Big(\int_\Omega |x|^{-ap}|\nabla u|^p\,dx \Big)-\int_\Omega|x|^{-p(a+1)+c}F(x,u)\,dx \\ & \geq \frac{m_1}{p\alpha_1}\Big(\int_\Omega |x|^{-ap}|\nabla u|^p\,dx \Big)^{\alpha_1}-c_{16}\int_\Omega|x|^{-p(a+1)+c}(|u|+|u|^q)\,dx \\ & \geq \frac{m_1}{p\alpha_1}\|u\|_{a,p}^{\alpha_1p}-c_{17}\beta_k^q \|u\|_{a,p}^q-c_{17}\|u\|_{a,p}, \end{split} \end{equation} where \begin{equation}\label{e2.22} \beta_k = \sup\big\{\|u\|_{L^q(\Omega,|x|^{-p(a+1)+c})}: \|u\|_{a,p}=1, u \in Z_k\big\}. \end{equation} Now, we deduce from \eqref{e2.21} that for any $u \in Z_k$, $\|u\|_{a,p} = r_k = \Big(\frac{c_{17}q\beta_k^q}{m_1}\Big)^\frac{1}{\alpha_1p-q}$, \begin{equation} \label{e2.23} \begin{split} J(u) & \geq \frac{m_1}{p\alpha_1}\|u\|_{a,p}^{\alpha_1p}-c_{17}\beta_k^q \|u\|_{a,p}^q-c_{17}\|u\|_{a,p}\\ & = \frac{m_1}{p\alpha_1}\Big(\frac{c_{17}q\beta_k^q}{m_1}\Big)^{ \frac{\alpha_1p}{\alpha_1p-q}} - c_{17}\beta_k^q\Big(\frac{c_{17}q{\beta }_k^q}{m_1}\Big)^{\frac{q}{\alpha_1p-q}} -c_{17}\Big(\frac{c_{17}q{\beta}_k^q}{m_1}\Big)^\frac{1}{\alpha_1p-q} \\ & = m_1\Big(\frac{1}{\alpha_1p}-\frac{1}{q}\Big)\Big(\frac{c_{17}q\beta_k^q }{m_1}\Big)^\frac{\alpha_1p}{\alpha_1p-q}-c_{17}\Big(\frac{c_{17}q{\beta }_k^q}{m_1}\Big)^\frac{1}{\alpha_1p-q}, \end{split} \end{equation} which tends to $+\infty$ as $k \to +\infty$, because $\alpha_1p0$ such that $$ F(x,t) \geq c_{18}|t|^\mu - c_{18} \quad \text{for all } x \in \Omega \text{ and } t \in \mathbb{R}. $$ Therefore, using (M1), for any $w \in Y_k$ with $\|w\|_{a,p} = 1$ and $1 < t < \rho_k$, we have \begin{equation} \label{e2.24} \begin{split} J(tw) & = \frac{1}{p}\widehat{M} \Big(\int_\Omega |x|^{-ap}|\nabla tw|^p\,dx \Big) -\int_\Omega|x|^{-p(a+1)+c}F(x,tw)\,dx \\ & \leq \frac{m_2}{\alpha_2p}\Big(\int_\Omega |x|^{-ap}|\nabla tw|^p\,dx \Big)^{\alpha_2}-c_{18}\int_\Omega |x|^{-p(a+1)+c}|tw|^\mu \,dx-c_{19} \\ & =\frac{m_2t^{\alpha_2p}}{\alpha_2p}\|w\|_{a,p}^{\alpha_2p}-c_{18}t^\mu {\int}_\Omega |x|^{-p(a+1)+c}|w|^\mu \,dx-c_{19}. \end{split} \end{equation} Since $\mu > \alpha_2p$ and dim$(Y_k)=k$, it is easy to see that $J(u) \to -\infty$ as $\|u\|_{a,p} \to +\infty$ for $u \in Y_k$. \end{proof} To prove Theorem \ref{the2.7}, we need to verify the following lemma. \begin{lemma}\label{lem2.12} Assume that {\rm (M1), (M2), (F0), (F2)} are satisfied. Then the functional $J$ satisfies the $(PS)^\ast_c$ condition. \end{lemma} \begin{proof} Let $\{u_{n_j}\}\subset X$ be such that $u_{n_j} \in Y_{n_j}$ and $J(u_{ n_j})\to 0$ and $(J|_{Y_{n_j}})'(u_{n_j}) \to 0$ as $n_j\to \infty$. Similar to the process of verifying the (PS) condition in the proof of Lemma \ref{lem2.4}, we can get the boundedness of $\{\|u_{n_j}\|_{a,p}\}$. Going, if necessary, to a subsequence, we can assume that $\{u_{n_j }\}$ converges weakly to $u$ in $X$. As $X = \overline{\cup_{n_j}Y_{ n_j}}$, we can choose $v_{n_j} \in Y_{n_j}$ such that $v_{n_j}\to u$. Hence, \begin{equation} \label{e2.25} \begin{split} \lim_{n_j\to \infty} J'(u_{n_j})(u_{n_j}-u) & = \lim_{n_j\to \infty} J'(u_{n_j}) (u_{n_j}-v_{n_j})+\lim_{n_j\to \infty} J'(u_{n_j})(v_{n_j}-u) \\ & = \lim_{n_j\to \infty} (J|_{Y_{n_j}})'(u_{n_j})(u_{n_j}-v_{n_j}) = 0. \end{split} \end{equation} From the proof of Lemma \ref{lem2.4}, $J'$ is of $(S_+)$ type, so we can conclude that $u_{n_j} \to u$ as $n_j\to \infty$, furthermore we have $J'(u_{n_j})\to J'(u)$. Let us prove $J'(u) = 0$, i.e., $u$ is a critical point of $J$. Indeed, taking arbitrarily $w_k \in Y_k$, notice that when $n_j \geq k$ we have \begin{equation} \label{e2.26} \begin{split} J'(u)(w_k) & = (J'(u)-J'(u_{n_j}))(w_k)+J'(u_{n_j})(w_k) \\ & =(J'(u)-J'(u_{n_j}))(w_k)+(J|_{Y_{n_j}})'(u_{n_j})(w_k). \end{split} \end{equation} Going to limit in the right hand-side of \eqref{e2.26} reaches $J'(u)(w_k)= 0$ for all $w_k \in Y_k$. Thus, $J'(u)=0$ and the functional $J$ satisfies the (PS)$^\ast_c$ condition for every $c\in \mathbb{R}$. \end{proof} \begin{proof}[Proof of Theorem \ref{the2.7}] From (F0), (F2), (F3) and Lemma \ref{lem2.12}, we know that $J$ is an even functional and satisfies the (PS)$^\ast_c$ condition, the assertion of conclusion can be obtained from Dual fountain theorem. (B1): For any $v \in Z_k$, $\|v\|_{a,p}=1$ and $0\alpha_1p$, taking $\rho_k = t$ small enough and sufficiently large $k$, for $v \in Z_k$ with $\|v\|_{a,p} = 1$, we have $J(tv)\geq 0$. So for sufficiently large $k$, $$ \inf_{\{u \in Z_k: \|u\|_{a,p} = \rho_k\}} J(u) \geq 0; $$ i.e., (B1) is satisifed. (B2): For $v \in Y_k$, $\|v\|_{a,p}=1$ and $0 < t < \rho_k < 1$, we have \begin{equation} \label{e2.28} \begin{split} J(tv) & = \frac{1}{p}\widehat{M}\Big(\int_\Omega |x|^{-ap}|\nabla tv|^p \,dx\Big)-\int_\Omega|x|^{-p(a+1)+c}F(x,tv)\,dx \\ & \leq \frac{m_2}{\alpha_2p}\Big(\int_\Omega |x|^{-ap}|\nabla tv|^p\,dx \Big)^{\alpha_2} - C_2\int_\Omega |x|^{-p(a+1)+c}|tv|^{r}\,dx \\ & = \frac{m_2}{\alpha_2p}t^{\alpha_2p}\|v\|_{a,p}^{\alpha_2p}-C_2t^r \int_{\Omega} |x|^{-p(a+1)+c}|v|^{r}\,dx. \end{split} \end{equation} Condition $\alpha_2p 0$ and $\alpha> 1$ such that $$ m_1t^{\alpha-1} \leq M(t) \leq m_2t^{\alpha-1}, \quad \forall t \in \mathbb{R}^+ $$ (The original (M1) implies $\alpha_1=\alpha_2$, so we rename constant $\alpha$.); \item[(F2)] There exists a positive constant $\mu > \frac{m_2}{m_1}\alpha p$ such that $$ 0< \mu F(x,t) = \mu\int_0^t f(x,s)\,ds \leq f(x,t)t $$ for all $x\in \Omega$ and $|t| \geq T>0$ (The constant $\mu$ has been redefined); \end{itemize} \subsection*{New Lemma 2.4} \textit{Assume that} (M1), (F0), (F2) \textit{are satisfied. Then the functional $J$ satisfies the Palais-Smale condition in the space $X$.} \begin{proof} Let $\{u_m\}\subset X$ be a sequence such that \begin{equation}\label{e3n} J(u_m) \to \overline c<\infty, \quad J'(u_m) \to 0 \quad\text{in }X^\ast\text{ as } m\to \infty, \end{equation} where $X^\ast$ is the dual space of $X$. We shall show that the sequence $\{u_m\}$ is bounded in $X$. Indeed, from (\ref{e3n}), (M1) and (F2), for all $m$ large enough, we have \begin{align}\label{e4n} \begin{split} &1+\overline c +\|u_m\|_{a,p} \\ & \geq J(u_m)-\frac{1}{\mu}J'(u_m)(u_m) \\ & = \frac{1}{p}\widehat{M}\Big(\int_\Omega|x|^{-ap}|\nabla u_m|^p\,dx\Big) - \frac{1}{\mu} M\Big(\int_\Omega|x|^{-ap}|\nabla u_m|^p\,dx\Big) \int_\Omega |x|^{-ap}|\nabla u_m|^p\,dx\\ & \quad -\int_\Omega |x|^{-p(a+1)+c}F(x,u_m)\,dx +\frac{1}{\mu} \int_\Omega |x|^{-p(a+1)+c}f(x,u_m)u_m\,dx \\ & \geq \frac{m_1}{\alpha p} \Big(\int_\Omega|x|^{-ap}|\nabla u_m|^p\,dx\Big)^\alpha - \frac{m_2}{\mu}\Big(\int_\Omega|x|^{-ap}|\nabla u_m|^p\,dx\Big)^\alpha \\ & \quad -\int_\Omega |x|^{-p(a+1)+c}\big(\frac{1}{\mu}f(x,u_m)u_m-F(x,u_m)\big)\,dx \\ & \geq \big(\frac{m_1}{\alpha p}-\frac{m_2}{\mu}\big)\|u_m\|^{\alpha p}_{a,p}-c_4. \end{split} \end{align} Since $\alpha p>1$ and $\mu > \frac{m_2}{m_1}\alpha p$, from \eqref{e4n} it follows that $\{u_m\}$ is bounded. Then with similar arguments as in the proof of the original Lemma 2.4 we can show that $J$ satisfies the Palais-Smale condition. \end{proof} Theorem 2.2 remains unchanged. However, Theorems 2.3, 2.6, 2.7 and Lemma 2.12 need to be stated without assumption (M2). Their proofs are similar to the original proofs, but using the new Lemma 2.4, and replacing $\alpha_1$ and $\alpha_2$ by $\alpha$. \smallskip The authors would like to thank anonymous reader and the editor for allowing us to correct our mistake. \end{document}