\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 145, pp. 1--24.\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{9mm}} \begin{document} \title[\hfilneg EJDE-2013/145\hfil Existence of multiple solutions] {Existence of multiple solutions to elliptic equations satisfying a global eigenvalue-crossing condition} \author[D. M. Duc, L. H. Nguyen, L. Nguyen \hfil EJDE-2013/145\hfilneg] {Duong Minh Duc, Loc Hoang Nguyen, Luc Nguyen} % in alphabetical order \address{Duong Minh Duc \newline University of Sciences - Hochiminh City, Vietnam} \email{dmduc@hcmuns.edu.vn} \address{Loc Hoang Nguyen \newline Department of Mathematics, Ecole Normale Suprieure, Paris, France} \email{lnguyen@dma.ens.fr} \address{lu Nguyen \newline Department of Mathematics, Princeton University, Princeton, New Jersey 08544, USA} \email{llnguyen@princeton.edu} \thanks{Submitted April 10, 2013. Published June 25, 2013.} \subjclass[2000]{47H11, 55M25, 35J20} \keywords{Index of critical points; mountain pass type; \hfill\break\indent nonlinear elliptic equations; multiplicity of solutions} \begin{abstract} We study the multiplicity of solutions to the elliptic equation $\Delta u+ f(x,u)=0$, under the assumption that $f(x,u)/u$ crosses globally but not pointwise any eigenvalue for every $x$ in a part of the domain, when $u$ varies from $-\infty$ to $\infty$. Also we relax the conditions on uniform convergence of $f(x,s)/s$, which are essential in many results on multiplicity for asymptotically linear problems. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} Let $\Omega$ be a bounded connected open subset with smooth boundary in $\mathbb{R}^N$ $(N\geq 3)$ and $H$ be the usual Sobolev space $W^{1,2}_0(\Omega)$ with the inner product and norm \begin{gather*} \langle u,v\rangle = \int_\Omega \nabla u.\nabla v \,dx \quad \forall u,v \in W^{1,2}_0(\Omega),\\ \|u\| = \Big(\int_\Omega |\nabla u|^{2} dx\Big)^{1/2} \quad \forall u\in W^{1,2}_0(\Omega). \end{gather*} Let $f$ be a real-valued Caratheodory function on $\Omega\times \mathbb{R}$ such that the first order partial derivative in second variable $\frac{\partial f}{\partial t}(x,t)$ exists and is continuous at any $t$ in $\mathbb{R}$ for every $x$ in $\mathbb{R}$, and $f(x,0)=0$ for all $x$ in $\Omega$. Assume that there exist measurable functions $V_1$, $V_2$, $V_3$ and $V_4$ on $\Omega$ such that \begin{gather} \liminf_{|t|\to 0}\frac{f(x,t)}{t} = V_1(x)\quad \forall (x,t)\in \Omega\times\mathbb{R}.\label{c33} \\ | f(x,t) - f(x,s)| \leq V_2(x)|t-s|\quad \forall x\in \Omega,\; s,t\in \mathbb{R},\label{c1} \\ \frac{ f(x,t)-f(x,s)}{t-s} \leq V_3(x)\quad \forall x\in \Omega,\; s,t\in \mathbb{R},s\neq t,\label{c2} \\ \liminf_{|t|\to \infty}\frac{f(x,t)}{t} = V_4(x)\quad \forall (x,t)\in \Omega\times\mathbb{R}.\label{c3} \end{gather} We consider the problem (P), \begin{equation} \begin{gathered} \Delta u+ f(x,u)=0 \quad\text{in }\Omega, \\ u=0\quad \text{on }\partial \Omega. \end{gathered}\label{eqn01} \end{equation} This equation has been studied when $f$ depends only on $u$ in \cite{AZ,CC,CDN,CHV,LP,SU}. The uniform convergence in \eqref{c33} and \eqref{c3} is essential in the articles \cite{CPV,CWL,DZ,FT,LM,LL,LW,LWZ,MMP,MT,NT,PA,PRS,PS,QL,ZZD,ZB,ZL}. If $f(x,u)/u$ does not cross uniformly any $\lambda_{i}$, problem \eqref{eqn01} may not have any solution (see \cite{NT}). In this paper we relax conditions on uniform convergence of $f(x,s)/s$. In the real world we can not estimate $f(x,u)/u$ pointwise, we have only its average values by integration. On the other hand we can neglect the behavior of $f(x,u)/u$ at every $x$ in small parts of $\Omega$. With these motivations, we introduce the concept of global eigenvalue-crossing defined by \eqref{c5} and \eqref{c6}, below. Using this concept, we study problem \eqref{eqn01}, and illustrate our method by improving the results in \cite{CC}; see Theorem \ref{thm1} below. It is interesting that the conditions in Theorem \ref{thm1} are similar to the Landesman-Lazer conditions in \cite{AO,LL}. Let $\lambda _1<\lambda _2\leq \ldots $ be the eigenvalues and $\varphi _1, \varphi _2,\ldots $ be their corresponding eigenfunction of the Laplacian operator $-\Delta$ in $H$. Our result on multiplicity of solutions is stated in the following theorem. \begin{theorem} \label{thm1} Let $Y$ be the subspace of $H$ spanned by $\{\varphi _1,\varphi _2,\ldots,\varphi _k\}$, and let $Z$ be the orthogonal complement of $Y$ in $H$. Let $W$ = $|V_1| + V_2+|V_3|+|V_4|$ and $r$ be in the interval $(\frac{N}{2},\infty)$. Suppose $W\in L^{r}(\Omega)$ and \begin{gather} \int _{\Omega} \big(|\nabla u|^2-V_1 u^2 \big)dx \geq C_0 \|u\|^2 \quad \forall u \in W^{1,2}_0(\Omega ),\label{c4} \\ \int _{\Omega}\big(|\nabla z|^2-V_3z^2 \big)dx \geq C_1 \|z\|^2 \quad \forall z\in Z,\label{c5} \\ \int _{\Omega}\big(|\nabla y|^2-V_4y^2 \big)dx \leq -C_2\|y\|^2 \quad \forall y\in Y,\label{c6} \end{gather} Then (i) Problem \eqref{eqn01} has at least five solutions. (ii) Moreover, one of the following cases occurs: \begin{itemize} \item[(a)] $k$ is even and \eqref{eqn01} has two solutions that change sign. \item[(b)] $k$ is even and \eqref{eqn01} has six solutions, three of which are of the same sign. \item[(c)] $k$ is odd and \eqref{eqn01} has two solutions that change sign. \item[(d)] $k$ is odd and \eqref{eqn01} has three solutions of the same sign. \end{itemize} \end{theorem} These results have been proved in \cite{CC} under the following conditions: $f$ is a differentiable function from $\mathbb{R}$ to $\mathbb{R}$, such that $f(0) = 0$, $f'(0) < \lambda_1$, $\lim_{|t|\to\infty}\frac{f(t)}{t} \in(\lambda_{k},\lambda_{k+1})$, and $f'(t) <\gamma<\lambda_{k+1}$ for all $t$ in $\mathbb{R}$. If $f'$ is continuous on $\mathbb{R}$ and $\sup\{|f'(t)| : t \in \mathbb{R}\}= M< \infty$, we can apply Theorem \ref{thm1} to consider this case with $V_1(x) = f'(0)$, $V_2(x) =M$, $V_3(x)= \gamma$ and $V_4(x) = \frac{1}{2}(\lambda_{k+1}+\lim_{|t|\to\infty}\frac{f(t)}{t})$ for any $x$ in $\Omega$. Let $\mu$ and $\nu$ be real numbers such that $\mu < \lambda_{k} < \nu$. We have to cross $\lambda_{k}$ in order to go from $\mu$ to $\nu$. Arguing as in \cite[p. 26]{Ra}, we have \begin{gather} \int _{\Omega} (|\nabla z|^2-\mu z^2 )dx \ge (1-\frac{\mu}{\lambda_{k+1}}) \|z\|^{2}\quad \forall z \in Z.\label{0a} \\ \int _{\Omega}(|\nabla y|^2-\nu y^2 )dx \le - (\frac{\nu}{\lambda_{k}}-1)\|y\|^2 \quad \forall y\in Y.\label{0b} \end{gather} These inequalities motivated us to introduce the global conditions \eqref{c5} and \eqref{c6}. \begin{example} \label{example1.4} \rm Let $\Omega$ be the unit sphere in $\mathbb{R}^{N}$, $\gamma$ be in the interval $(\lambda_k, \lambda_{k+1})$, $\varepsilon$ be a positive real number, and $f$ be a real $C^2$-function on $\Omega\times\mathbb{R}$ such that \[ f(x,t) = \begin{cases} 0\quad & \text{if } |t|\leq \frac{1}{2}, \\ (\gamma - \varepsilon (1-|x|^{2})^{-1/N})t & \text{if }|t|\geq 1, \end{cases} \] and \[ |\frac{\partial f(x,t)}{\partial t}| \leq 4\gamma - 4\varepsilon (1-|x|^{2})^{-1/N}\quad \text{if }0\le |t| \le 1. \] Since $(1-|x|^{2})^{-1/N}$ is in $L^{\frac{N}{2}}(\Omega)$, by inequalities of Sobolev and Poincar\'{e} there is a constant $c_0$ such that for any $u$ in $W^{1,2}_0(\Omega)$, \begin{align*} \int_\Omega (1-|x|^{2})^{-1/N}u^{2}dx &\le \Big\{\int_\Omega (1-|x|^{2})^{-\frac{1}{2}}dx\Big\}^{2/N} \Big\{\int_\Omega u^{\frac{2N}{N-2}}dx\Big\}^{\frac{N-2}{N}}\\ & \le c_0\int_\Omega |\nabla u|^{2}dx. \end{align*} Let $\varepsilon$ be in the interval $(0,c_0^{-1}(\frac{\gamma}{\lambda_{k}} -1 ))$. We see that $c_2\equiv \frac{\gamma}{\lambda_{k}} -1- c_0\varepsilon$ is positive. Put $V_1(x) =0$, $V_2(x)=4\gamma - 4\varepsilon(1-|x|^{2})^{-\frac{1}{N}}$, $V_3(x=V_4(x)=gamma - \varepsilon(1-|x|^{2})^{-\frac{1}{N}}$ for any $x$ in $\Omega$. Then $f$ satisfies \eqref{c33}, \eqref{c1}, \eqref{c2}, \eqref{c3}, \eqref{c4}, \eqref{c5} and \eqref{c6}. Indeed, arguing as in \cite[p. 26]{Ra}, we have \begin{gather*} \int_\Omega [|\nabla z|^{2} - V_3z^{2}]dx \ge \int_\Omega [|\nabla z|^{2} - \gamma z^{2}]dx \ge (1-\frac{\gamma}{\lambda_{k+1}})\|z\|^{2}\quad \forall z\in Z, \\ \begin{aligned} \int_\Omega [|\nabla y|^{2} - V_4y^{2}]dx &\le \int_\Omega [(1+ c_0\varepsilon)|\nabla y|^{2} - \gamma y^{2}]dx \\ &= \sum_{j=1}^{k}[(1+ c_0\varepsilon)\lambda_j -\gamma]\int_\Omega \alpha_j^{2}\varphi _j^{2}dx\\ & =- \sum_{j=1}^{k}[\frac{\gamma}{\lambda_j} - 1- c_0\varepsilon ]\lambda_j\int_\Omega \alpha_j^{2}\varphi _j^{2}dx\\ &= -[\frac{\gamma}{\lambda_{k}} - 1- c_0\varepsilon ] \int_\Omega |\nabla y|^{2}dx\quad \forall y = \sum_{j=1}^{k}\alpha_j\varphi _j \in Y. \end{aligned} \end{gather*} \end{example} \begin{remark} \label{rmk1.5} \rm Note that the set $E = \{x \in \Omega : \gamma - \varepsilon (1-|x|^{2})^{-\frac{1}{N}} < \lambda_1\}$ is a nonempty open subset of $\Omega$. Then the Lebesgue measure of $E$ is positive, and $\frac{f(x,t)}{t}< \lambda_1$ for any $x$ in $E$ and $|t|\ge 1$. Thus $\frac{f(x,t)}{t}$ does not cross any $\lambda_{i}$ at any $x$ in $E$. \end{remark} \section{Proof of main results} For any $(x,\xi)$ in $\Omega\times \mathbb{R}$ and any $u$ in $H$ we define \begin{gather} \xi^+=\max \{\xi,0\},\quad \xi^-=\min\{\xi,0\}, \label{f2} \\ f_{\pm}(x,\xi )=f(x,\xi^{\pm})\mp V_1(x)\xi ^{\mp}, \label{f3} \\ F(x,\xi)=\int _0^{1} f(x,s\xi)\xi ds, \quad F_{\pm}(x,\xi)=\int _0^{1}f(x, s\xi^{\pm})\xi^{\pm}ds +\frac{1}{2}V_1(x)|\xi ^{\mp}|^{2}, \label{f4} \\ J(u)=\int _{\Omega}[\frac{1}{2}| \nabla u| ^2-F(x,u(x))]dx,\quad J_\pm(u)=\int _{\Omega}[\frac{1}{2}| \nabla u(x)| ^2 - F_\pm(x,u(x))]dx.\label{f5} \end{gather} Some operators in this sections may not be compact vector fields but of class $(S)_+$, which has been introduced by Browder (see \cite{BRA,BRB}). We have the definitions and properties of the class $(S)_+$ as follows. \begin{definition} \label{S} \rm Let $X$ be a subset of $H$ and $h$ be a mapping of $X$ into $H$. We say: \begin{itemize} \item[(i)] $h$ is demicontinuous if the sequence $\{h(x_m)\}$ converges weakly to $h(x)$ in $H$ for any sequence $\{x_m\}$ converging strongly to $x$ in $H$. \item[(ii)] $h$ is of class $(S)_{+}$ if $h$ is demicontinuous and has the following property : let $\{x_{m}\}$ be a sequence in $X$ such that $\{x_{m}\}$ converges weakly to $x$ in $H$. Then $\{x_{m}\}$ converges strongly to $x$ in $H$ if $\limsup_{n\to\infty}\langle h(x_{m}), x_{m} -x \rangle \le 0$. \end{itemize} \end{definition} Denote by $B_{s}(x_0)$ the open ball of radius $s$ centered at $x_0$ for any $x_0$ in $H$. Let $U$ be a bounded open subset of $H$ and $\partial U$ and $\overline{U}$ be the boundary and the closure of $U$ in $H$ respectively. Let $f$ be a mapping of class $(S)_{+}$ on $\overline{U}$ and let $p$ be in $H\setminus f(\partial U)$. By \cite[Theorems 4 and 5]{BRA}, the topological degree of $f$ on $U$ at $p$ is defined as a family of integers and is denoted by $\deg(f,U,p)$. In \cite{S1} Skrypnik showed that this topological degree is single-valued (see also \cite{BRB}). The following result was proved in \cite{BRB}. \begin{proposition} \label{Browder} Let $f$ be a mapping of class $(S)_+$ from $\overline{U}$ into $H$, and let $y$ be in $H \setminus f(\partial U)$. Then we can define the degree $\deg (f,U,y)$ as an integer satisfying the following three conditions: \begin{itemize} \item[(a)] (Normalization) If $\deg (f,U,y)\neq 0$ then there exists $x\in U$ such that $f(x)=y$. If $y\in U$ then $\deg (Id,U,y)=1$ where $Id$ is the identity mapping. \item[(b)] (Additivity) If $U_1$ and $U_2$ are two disjoint open subsets of $U$ and $y$ does not belong to $f(\overline{U}\backslash (U_1\cup U_2) ) $ then $\deg (f,U,y)=\deg (f,U_1,y)+\deg (f,U_2,y)$. \item[(c)] (Invariance under homotopy). If $\{g_t:0\leq t\leq 1\}$ is a homotopy of class $(S)_+$ and $\{y_t:t\in [0,1]\}$ is a continuous curve in $H$ such that $y_t\notin g_t(\partial U)$ for all $t\in [0,1]$, then $\deg (g_t,U,y_t)$ is constant in $t$ on $[0,1]$. \end{itemize} \end{proposition} \begin{definition} \label{iiii} \rm If $u_0$ is an isolated zero of a map $f$ of class $(S)_+$, then, by the additivity property of the degree, we can define $$ i(f,u_0)=\lim _{s \to 0}\deg (f,B_{s}(u_0),0), $$ which is called the index of $f$ at $u_0$. \end{definition} \begin{definition} \rm Let $j$ be a real-valued $C^1$-function on $H$. We say that $j$ satisfies the Palais-Smale condition if for any sequence $\{x_m\}$ in $H$ such that $\{j(x_m)\}$ is bounded and $\{\|Dj(x_m)\|\}$ converges to $0$, there is a convergent subsequence $\{x_{m_k}\}$ of $\{x_m\}$. \end{definition} \begin{definition} \label{MP} \rm Let $j$ be a real $C^1$-function defined on an open subset $U$ in $H$, and $x$ be a critical point of $j$. Then $x$ is said to be a critical point of mountain-pass type if there exists a neighborhood $V$ of $x$ contained in $U$ such that $W\cap j^{-1}(-\infty,j(x))$ is nonempty and not path-connected whenever $W$ is an open neighborhood of $x$ contained in $V$. \end{definition} \begin{definition} \rm Let $j$ be a real-valued $C^{2}$-function on $H$ and $x_0\in H$. We say $j$ satisfies the condition $(\Phi )$ at $x_0$ if: $Dj(x_0)=0$ and $0$ is a simple eigenvalue whenever it is the smallest eigenvalue of $D^{2}j(x_0)$. \end{definition} We shall extend the results in \cite{AM,Hof} to operators of class $(S)_+$ in the appendix and use them in the present and next sections. The proof of Theorem \ref{thm1} needs following lemmas. \begin{lemma} \label{ff} Let $W$ be as in Theorem \ref{thm1} and $u$ be in $H$. We have \begin{itemize} \item[(F1)] $Wu$ is in $L^{1}(\Omega)$ for any $u$ in $H$. \item[(F2)] There is a positive constant $K$ such that \begin{equation} \int _{\Omega}Wu^2\,dx \leq K\| u\| ^2\quad \forall u\in H. \label{eqn003} \end{equation} \item[(F3)] For any sequence $\{v_{m}\}$ converging weakly to $v$ in $H$, there are a measurable function $g$ on $\Omega$ and a subsequence $\{v_{m_{k}}\}$ of $\{v_{m}\}$ having the following properties: $|v_{m_{k}}| \le g$ a.e. on $\Omega$, and for any $k$, \begin{equation} \int _{\Omega}W|g|^{2}\,dx< \infty.\label{eqn004} \end{equation} \end{itemize} \end{lemma} \begin{proof} Let $q$ be in $[1,\frac{N}{N-2})$ such that $\frac{1}{r} + \frac{1}{q} = 1$. By H\"{o}lder's and Sobolev's inequalities, there is constant $c$ such that for any $u \in H$ and $s\in\{1,2\}$ \begin{equation} \int_{\Omega} W|u|^{s}dx \le \Big(\int_{\Omega} W^{r}dx\Big)^{1/r} \Big[\Big(\int_{\Omega} |u|^{sq}dx\Big)^{\frac{s}{q}}\Big]^{s} \le c^{s} \Big(\int_{\Omega} W^{r}dx\Big)^{1/r}\|u\|^{s}.\label{f12} \end{equation} Therefore, $(F1)$ and $(F2)$ are satisfied. Let $\{v_{m}\}$ be a sequence converging weakly to $v$ in $H$. By \cite[Theorem 4.9]{BR} and Rellich-Kondrachov's theorem, there exist $g$ and $v$ in $L^{2q}(\Omega)$, and a subsequence $\{v_{m_{k}}\}$ of $\{v_{m}\}$ such that $\{v_{m_{k}}\}$ converges to $v$ in $L^{2q}(\Omega)$, $\{v_{m_{k}}\}$ converges $v$ a.e on $\Omega$, and $|v_{m_{k}}|\leq g$ a.e on $\Omega$. Since $g^{2}$ is in $L^{q}(\Omega)$, $Wg^{2}$ is integrable on $\Omega$. \end{proof} \begin{lemma} \label{po} Let $v$ and $w$ be in $H$, such that $w$ is nonnegative and not equal to $0$ and \[ \int_{\Omega} [\nabla w\nabla \varphi - \frac{\partial f}{\partial t}(x,v(x))\varphi]dx = 0 \quad \forall \varphi\in H. \] Then $w >0$ a.e. on $\Omega$. \end{lemma} \begin{proof} Let $W$ and $r$ be in Theorem \ref{thm1}, $B(x,t)$ be a ball in $\mathbb{R}^{N}$ with center $x$ and radius $t$, and $p$ be in $[2,\frac{N}{N-2})$ such that $\frac{1}{p}+\frac{1}{r} = 1$. Put $U(y)=\frac{\partial f}{\partial t}(y,v(y))$ for any $y$ in $\Omega$. By \eqref{c1} and H\"{o}lder's inequality, there are positive contants $M_1$ and $M_2$ independing from $x$ and $t$ such that \begin{align*} \int_{B(x,t)}\frac{|U(y)|}{|x-y|^{N-2}}\chi_{\Omega}(y)dy &\le \int_{B(x,t)}\frac{W(y)}{|x-y|^{N-2}}\chi_{\Omega}(y)dy \\ &\le(\int_{\Omega}W^{r}dy)^{1/r}(\int_{B(x,t)}|x-y|^{p(2-N)}dy)^{1/p}\\ &\le M_1\int_0^{t}s^{p(2-N) + N-1}ds = M_2t^{\theta}, \end{align*} where $\theta = p(2-N) +N > \frac{N}{N-2}(2-N) + N = 0$. Thus $U$ is of Kato's class (see \cite{SI}). Let $x_0$ be in $\Omega$ and $\Omega '$ be an open set such that $w(x_0) >0$, $x_0 \in \Omega '$ and $\overline{\Omega '} \subset \Omega$. By Harnack's inequality \cite[Theorem 5.5]{SI}), $w(x) >0$ for any $x$ in $\Omega '$. Since $\Omega$ is connected in $\mathbb{R}^{N}$, $w(z) >0$ for any $z$ in $\Omega$. \end{proof} \begin{lemma} \label{smoothness} (i) The functional $J$ is of class $C^2$, the functionals $J_{+}$ and $J_-$ are of class $C^1$. For any $u$ and $v$ in $H$ we have \begin{gather} \langle DJ(u),v\rangle = \int _{\Omega}[\nabla u\nabla v-f(x,u)v]dx, \label{eqn05}\\ \langle DJ_{\pm}(u),v\rangle = \int _{\Omega}[\nabla u\nabla v-f_{\pm}(x,u)v]dx. \label{eqn06} \end{gather} (ii) \begin{equation} \limsup_{y\in Y,\| y\|\to\infty}\frac{J(y)}{\| y\| ^2}<0. \label{eqn07} \end{equation} \end{lemma} \begin{proof} (i) It is sufficient to prove that $J$ is of class $C^2$. The prove for $J_\pm$ are similar. Let $u,v$ be in $H$ and $x$ be in $\Omega$. By \eqref{c1}$, \eqref{f4}$ and the mean value theorem, there is $s_{x}$ in $[0,1]$ such that \begin{align*} &|F(x,u(x)+v(x)) - F(x,u(x))- f(x,u)v|\\ & =|f(x,u(x) + s_{x}v(x))v- f(x,u(x))v(x)| \\ &\le V(x)v^{2}(x). \end{align*} Hence, by \eqref{f5} and \eqref{f12}, \begin{align*} & |J(u+v) -J(u) - \int_{\Omega}[\nabla u\nabla v - f(x,u)v]dx|\\ & = \||v\|^{2} + \int_{\Omega}[F(x,u(x)+v(x)) - F(x,u(x))- f(x,u)v]dx |\\ & \le \|v\|^{2} + |\int_{\Omega}V_2v^{2}dx| \\ & \le \Big[1+c^{2}\Big(\int_{\Omega}W^{\frac{N}{2}}dx\Big)^{2/N}\Big] \|v\|^{2} \end{align*} Therefore, $J$ is Fr\'{e}chet-differentiable on $H$ and \[ \langle DJ(u),v\rangle = \int _{\Omega}[\nabla u\nabla v -f(x,u)v]dx \quad \forall u,v \in H. \] By \eqref{eqn003}$ and \eqref{c1}$, we have that for any $u$, $w$ and $v$ in $H$, \begin{equation} \begin{aligned} |\langle J(u)-J(w),v\rangle | &=\int_\Omega |\nabla(u-w)\nabla v - (f(x,u)-f(x,w))v|dx \\ &\le \|u-w\|\|v| + \int_\Omega |V_2(u-w)v|dx \\ &\le\|u-w\|\|v| + \Big\{\int_\Omega V_2(u-w)^{2}dx\Big\}^{1/2}\Big\{ \int_\Omega V_2v^{2}dx\Big\}^{1/2} \\ &\le (1+K)\|u-w\|\|v\|. \end{aligned} \label{li} \end{equation} Thus $J$ is of class $C^1$. Similarly we see that $DJ$ is Fr\'{e}chet-differentiable and \[ D^2 J(u)(v,w)= \int _{\Omega}[\nabla v\nabla w -\frac{\partial f}{\partial t}(x,u) v w]dx \quad \forall u, v, w \in H. \] Let $v$ and $w$ be in $H$ and $\{u_m\}$ be a sequence converging to $u$ in $H$. We see that $\{u_m\}$ converges to $u$ in $L^{2}(\Omega)$. Since $V_2$ is in $L^{N/2}(\Omega)$, by a result on page 30 in \cite{KR} and \eqref{c1}, we have \begin{equation} \lim_{m\to\infty} \Big\{\int_{\Omega} \big|\frac{\partial f}{\partial t}(x,u_{m}(x)) - \frac{\partial f}{\partial t}(x,u(x))\big|^{N/2} dx\Big\}^{N/2} =0.\label{jc1} \end{equation} As in \eqref{f12}, we have \begin{equation} \begin{aligned} &|[D^2 J(u_{m}) - D^2 J(u)](v,w)|\\ &= | \int _{\Omega}[\frac{\partial f}{\partial t}(x,u_{m}) -\frac{\partial f}{\partial t}(x,u)]vwdx|\\ &\le \Big\{\int _{\Omega}\big|\frac{\partial f}{\partial t}(x,u_{m}) -\frac{\partial f}{\partial t}(x,u)|v^{2}dx\big|\Big\}^{1/2}\\ &\quad\times \Big\{\int _{\Omega}|\frac{\partial f}{\partial t}(x,u_{m}) -\frac{\partial f}{\partial t}(x,u)|w^{2}dx|\Big\}^{N/2}\\ &\le c^{2}\{\int _{\Omega}|\frac{\partial f}{\partial t}(x,u_{m}(x)) -\frac{\partial f}{\partial t}(x,u(x))|^{N/2}dx\}^{N/2}\|v\|\|w\|. \end{aligned}\label{jc2} \end{equation} Combining \eqref{jc1} and \eqref{jc2}, we obtain the continuity of $D^{2}J$. (ii) Let $\{y_m\}$ be a sequence in $Y$ with $a_m=\| y_m\|\to\infty$. We shall prove that \[ \limsup_{m\to\infty}\frac{J(y_m)}{a_m^2}<0. \] Put $w_m = \frac{y_m}{a_m}$. Since $\| w_m\| =1$ and $Y$ is of finite dimension we may assume that $\{w_m\}$ converges to $w$ in $Y$ with $\|w\|=1$. When $m$ goes to $\infty$, by \eqref{c1}$, \eqref{f5}$, \eqref{eqn003} and the mean value theorem, we have \begin{equation} \begin{aligned} &|\frac{J(a_{m}w_{m})-J(a_{m}w)}{a_m^2}|\\ & \le |\frac{\int_\Omega [\frac{1}{2}(|\nabla a_{m}w_{m}|^{2} - |a_{m}\nabla w|^{2}) - F(x,a_{m}w_{m}(x))+ F(x,a_{m}w(x))]dx }{a_m^2}|\\ &\le \|w_{m}+w\|\|w_{m}-w\| + \int_\Omega V_2|w_m-w|^{2}dx\\ &\le (\|w_{m}+w\|+K\|w_{m}-w\|)\|w_{m}-w\|\to 0. \end{aligned}\label{m1} \end{equation} Thus \[ \limsup_{m \to \infty}\frac{J(a_{m}w_{m})}{a_m^2} = \limsup_{m \to \infty}\frac{J(a_{m}w)}{a_m^2}. \] Let $s$ be in $(0,1]$ and $x$ be in $\Omega$ such that $w(x) \neq 0$. Then $\lim_{m\to\infty}|sa_{m}w(x)| = \infty$ and by \eqref{c3}, \begin{equation} \liminf_{m\to\infty}\frac{f(x,s a_{m} w(x))}{s a_m w(x)}= V_4(x)\label{V2} \end{equation} Put $D=\{x\in \Omega: w(x)\neq 0\}$. By \eqref{c2}, $\frac{f(x,t)}{t}+V_2(x)\geq 0$ for any $(x,t)$ in $\Omega\times\mathbb{R}$. Therefore, by a general version of Fatou's lemma, \eqref{c1} and \eqref{c3}, we have \begin{equation} \begin{aligned} \liminf_{m \to \infty} a_{m}^{-2}\int_\Omega F(x,a_m w(x))dx &= \liminf_{m \to \infty}a_{m}^{-2}\int_\Omega \int_0^1 f(x,s a_m w(x)) a_{m}w(x)\,ds\,dx\\ &= \liminf_{m \to \infty}\int_{D}\int_0^1 \frac{f(x,s a_{m} w(x))}{s a_{m} w(x)}sw^{2}(x)\,ds\,dx \\ &\ge \int_{D}[\int_0^1 \liminf_{m \to \infty} \frac{f(x,s a_{m} w(x))}{s a_{m} w(x)}sw^{2}(x)\,ds\,dx \\ &= \int_{D}\int_0^1 V_4(x)sw^{2}(x)\,ds\,dx \\ &= \frac{1}{2}\int_{D}V_4(x)w^{2}(x)dx \end{aligned} \label{m2} \end{equation} Combining \eqref{m1}, \eqref{m2} and \eqref{c6}, we obtain \begin{align*} \limsup_{m \to \infty}\frac{J(a_{m}w_m)}{a_m^2} &= \limsup_{m \to \infty}\Big[\int_\Omega |\nabla w|^{2}dx -a_{m}^{-2}\int_\Omega F(x,s a_m w(x))dx\Big]\\ &=\frac{1}{2}\int_\Omega |\nabla w|^{2}dx -\liminf_{m \to \infty}a_{m}^{-2}\int_\Omega F(x,s a_m w(x))dx\\ &\le \frac{1}{2}\int_{\Omega}[|\nabla w|^{2}dx -V_4(x)w^{2}(x)]dx \\ &\le -\frac{1}{2}C_2\|w\|^{2} < 0, \end{align*} which completes the proof. \end{proof} \begin{lemma} \label{Lemma5.1} (i) For every $y\in Y$ and $z_1,z\in Z$, \begin{equation} \langle DJ(y+z_1)-DJ(y+z),z_1 - z\rangle \geq C_1\| z_1-z\| ^2. \label{eqn09} \end{equation} Moreover, if $\{u_m\}$ converges weakly to $u_0$ in $H$ then: (ii) \begin{equation} \limsup_{m \to \infty}\langle DJ(u_m),u_m - u_0\rangle \geq C_1\limsup_{m \to \infty}\| u_m-u_0\| ^2,\label{eqn10} \end{equation} (iii) \begin{equation} \limsup_{m \to \infty}\langle DJ_{+}(u_m),u_m - u_0\rangle \geq C_1\limsup_{m \to \infty}\| u_m-u_0\| ^2, \label{eqn11} \end{equation} (iv) \begin{equation} \limsup_{m \to \infty}\langle DJ_{-}(u_m),u_m - u_0\rangle \geq C_1\limsup_{m \to \infty}\| u_m-u_0\| ^2. \label{eqn12} \end{equation} \end{lemma} \begin{proof} (i) By \eqref{eqn05}, \eqref{c2}, \eqref{c5} and the orthogonality between $Y$ and $Z$ in $H$, we have for all $y\in Y$ and $z_1,z\in Z$, \begin{align*} &\langle DJ(y+z_1)-DJ(y+z), z_1-z\rangle\\ &=\int _{\Omega}| \nabla (z_1-z)| ^2dx -\int _{\Omega}(f(x,y+z_1)-f(x,y+z))(z_1-z)dx \\ &\geq\int _{\Omega}| \nabla (z_1-z)| ^2dx -\int _{\Omega}V_3(x)(z_1-z)^2dx \geq C_1\| z_1-z\| ^2. \end{align*} (ii) Write $u_m=y_m+z_m$ and $u_0=y_0+z_0$, where $y_m,y_0\in Y$ and $z_m,z_0\in Z$. Using the orthogonality between $Y$ and $Z$ in $H$, we obtain $y_m\rightharpoonup y_0$ and $z_m\rightharpoonup z_0$ in $H$. Since $Y$ is finite dimensional, $\{y_m\}$ converges strongly to $y_0$ in $H$. By \eqref{li}, $DJ$ is Lipschitz continuous on $H$. Thus $DJ(A)$ is bounded for any bounded subset $A$ of $H$. (i) implies that \begin{align*} &\limsup_{m \to \infty}\langle DJ(u_m),u_m - u_0\rangle \\ &= \limsup_{m \to \infty}\langle DJ(u_m),y_m - y_0 + z_m - z_0\rangle\\ &= \limsup_{m \to \infty}\langle DJ(y_m +z_m ),z_m - z_0\rangle\\ & = \limsup_{m \to \infty}\langle DJ(y_m +z_m )-DJ(y_m +z_0 ) ,z_m - z_0\rangle \\ &\quad + \lim_{m \to \infty}\langle DJ(y_m +z_0 )-DJ(y_0 +z_0 ),z_m - z_0\rangle + \lim_{m \to \infty}\langle DJ(y_0+z_0 ),z_m - z_0\rangle\\ &= \limsup_{m \to \infty}\langle DJ(y_m +z_m )-DJ(y_m +z_0 ),z_m - z_0\rangle\\ & \ge C_1\limsup_{m \to \infty}\| z_{m}-z_0\| ^2 \\ &= C_1\limsup_{m \to \infty}\| u_{m}-u_0\| ^2. \end{align*} (iii) Arguing as in Lemma \ref{smoothness}, by \eqref{f3} and \eqref{f5}, we have \begin{equation} \begin{aligned} &\langle DJ_{+}(u_m),u_m-u_0\rangle\\ &=\int _{\Omega}\nabla u_m \nabla (u_m-u_0)dx -\int _{\Omega}[f(x,u_m^+(x))- V_1 (x)u_m^-(x)](u_m(x)-u_0(x))dx\\ &= \langle DJ(u_m),u_m-u_0\rangle +\int _{\Omega}[f(x,u_{m}^{-}(x)) + V_1(x)u_m^-(x)](u_m)-u_0)dx. \end{aligned} \label{eqn13} \end{equation} Let $q$ be in $[1,\frac{N}{N-2})$ such that $\frac{1}{r} + \frac{1}{2q} = 1$. By Rellich-Kondrachov's theorem, H\"{o}lder's theorem, \eqref{c1} and \eqref{f12}, $\{u_{m}\}$ converges strongly to $u_0$ in $L^{2q}(\Omega)$ and \begin{equation} \begin{aligned} &\big|\int _{\Omega}[f(x,u_{m}^{-}(x)+ V_1 (x)u_m^-(x)](u_m(x)-u_0(x))dx\big| \\ &\le 2c^{2}\{\int _{\Omega} W^{r}dx\}^{1/r} \|u_{m}^{-}\|\|u_{m}-u_0\| \to 0 \quad\text{as }m \to 0. \end{aligned}\label{eqn13bb} \end{equation} Thus by (ii), \eqref{eqn13} and \eqref{eqn13bb}, we obtain \[ \limsup_{m\to\infty} \langle DJ_{+}(u_m),u_m-u_0\rangle =\limsup_{m\to\infty}\langle DJ(u_m),u_m-u_0\rangle \ge C_1\|u_{m}-u_0\|^{2}. \] (iv) The proof of (iv) is similar to the proof of \eqref{eqn12}, and is omitted. \end{proof} \begin{lemma} \label{S+operator} The operators $DJ$ and $DJ_{\pm}$ are of class $(S)_+$. \end{lemma} The proof of the above lemma follows from Lemma \ref{Lemma5.1} and Definition \ref{S}. \begin{lemma} \label{Castro} Let $P$ be the orthogonal projection of $H$ onto $Y$. Let $N(u)(x)=f(x,u(x))$ for all $u$ in $H$ and $x$ in $\Omega$. Then: (i) For any $y$ in $Y$, there exists a unique $\psi(y)\in Z$ such that $\psi(y)|_{\partial\Omega} = 0$, $J(y + \psi(y)) = \min_{z \in Z}J(y + z)$ and \begin{equation} (I-P)DJ(y + \psi(y)) = -\Delta \psi(y) - (I-P)N(y + \psi(y)) = 0.\label{py} \end{equation} (ii) The mapping $\psi$ is continuous on $Y$. (iii) The reduction mapping $\tilde J:Y\to\mathbb{R}$ determined by $\tilde J(y)=J(y + \psi(y))$ is of class $C^1$, and \[ D\tilde J(y)=PDJ(y + \psi(y)). \] Moreover, $y$ is a critical point of $\tilde J$ if and only if $y + \psi(y)$ is a critical point of $J$. (iv) If $u_0=y_0+\psi (y_0)$ is an isolated critical point of mountain-pass type of $J$ then $y_0$ is a critical point of mountain-pass type of $\tilde J$. (v) If $y_0 \in Y$ such that $y_0+\psi (y_0)$ is an isolated critical point of $J$, then \begin{equation} i(D\tilde J,y_0)=i (DJ,y_0+\psi (y_0)).\label{eqn08} \end{equation} \end{lemma} \begin{proof} The proofs of (i), (ii), (iii), (iv) are based on \eqref{eqn09} and can be found in \cite[Lemma 1]{Cas} and \cite[Lemma 2.1]{CC}. (v) Put $u_0=y_0 + \psi(y_0)$. Because $DJ$ and $\psi$ are continuous and $u_0$ is an isolated critical point of $J$, we can choose $M>0$ and $r >0$ such that $u_0$ is the unique critical point of $J$ in $\overline{B_{r}(u_0)}$ and \[ \| DJ(y+t\psi (y)+(1-t)z)\|\leq M\quad \forall u=y+z\in \overline{B_{r}(u_0)},\; t\in [0,1]. \] We put \begin{equation} \begin{aligned} h_1(t,u)&=PDJ(y+t\psi(y)+(1-t)z)+(1-t)(I-P)DJ(y+z)\\ &\quad +t(z-\psi (y)) \quad \forall t\in [0,1], u = y + z \in Y\oplus Z. \end{aligned} \label{eqn15} \end{equation} First we show that $u_0$ is the unique zero of $h_1(t,\cdot)$ in $\overline{B_{r}(u_0)}$ for all $t\in [0,1]$. Indeed, let $(t,u) \in [0,1]\times\overline{B_{r}(u_0)}$ such that $u=y + z$ in $Y\oplus Z$ and $h_1(t,u)=0$. By $(i)$ \begin{equation} \langle DJ(y+\psi (y)),w\rangle = 0 \quad \forall w \in Z.\label{y1} \end{equation} Thus by \eqref{eqn09}, we have \begin{align*} 0 &=\langle h_1(t,u),z-\psi(y)\rangle \\ &= (1-t)\langle DJ(y+z),z-\psi(y)>+t\| z-\psi (y)\| ^2 \\ &=(1-t)\langle DJ(y+z)-DJ(y+\psi (y)),z-\psi(y)\rangle +t\| z-\psi (y)\| ^2 \\ &\geq [(1-t)C_1+t]\| z-\psi (y)\| ^{2}, \end{align*} which implies $z=\psi(y)$. Therefore, by (i) and \eqref{eqn15}, \begin{align*} 0 &= h_1(t,u) = h_1(t, y +\psi(y)) \\ &= PDJ(y+\psi(y))+(1-t)(I-P)DJ(y+\psi(y)) \\ &= DJ(y+\psi(y)) - t(I-P)DJ(y+\psi(y))\\ &= DJ(y+\psi(y)) = DJ(u). \end{align*} By the choice of $r$, we obtain $u=u_0$. We will prove that $h_1$ is a homotopy of class $(S)_{+}$ on $\overline{B_{r}(u_0)}$. Let $\{(t_{m},u_{m})\}$ be a sequence in $[0,1]\times \overline{B_{r}(u_0)}$ such that $\{t_{m}\}$ converges to $t$ in $[0,1]$ and $\{u_{m}\}$ converges weakly to $u$ in $\overline{B_{r}(u_0)}$ and \begin{equation} \limsup_{m\to\infty}\langle h_1(t_{m},u_{m}),u_{m} - u\rangle \leq 0. \label{eqn16} \end{equation} We will show that $\{u_{m}\}$ converges strongly to $u$ in $H$. We write $u_{m}=y_{m} + z_{m}$ and $u=y + z$, where $(y,z)$ and $(y_{m},z_{m})$ are in $Y\times Z$ for any integer $n$. Since $Y$ is finite-dimensional, $\{y_m\}$ converges strongly to $y$. Using the continuity of $\psi$ and the boundedness of $\{DJ(y_m+t_m\psi(y_m)+(1-t_m)z_m)\}$ and $\{u_m\}$, we can assume that $\{PDJ(y_m+t_m\psi (y_m)+(1-t_m)z_m)\}$ and $\{PDJ(u_m)\}$ converge strongly in $H$. By \eqref{eqn15} we have \begin{align*} &\langle h_1(t_m,u_m),u_m-u\rangle\\ &=\langle PDJ(y_m+t_m\psi (y_m)+(1-t_m)z_m),u_m-u\rangle \\ &\quad -(1-t_m)\langle PDJ(u_m),u_m-u\rangle +(1-t_m)\langle DJ(u_m)- DJ(u),u_m-u\rangle \\ &\quad + (1-t_m)\langle DJ(u),u_m-u\rangle +t_m \rangle z_m,y_m-y\rangle\\ &\quad + t_m\langle z_m-z,z_m-z\rangle +t_m\langle z,z_m-z\rangle -t_m\langle \psi(y_m),u_m-u\rangle. \end{align*} Hence, \begin{align*} &\limsup_{m\to\infty}(1-t_m)\langle DJ(u_m) - DJ(u),u_m-u\rangle \\ &=\limsup_{m\to\infty}(1-t_m)\langle DJ(y+z_m) - DJ(y+z),z_m-z\rangle . \end{align*} Thus, by \eqref{eqn09}, \eqref{eqn16} is equivalent to \begin{align*} 0&\geq \limsup_{m\to\infty}\{(1-t_m)\langle DJ(y+z_m) - DJ(y+z),z_m-z\rangle +t_m\|z_m-z\|^2\}\\ &\geq \limsup_{m\to\infty}\{(1-t_m)C_1\|z_m-z\|^2+t_m\|z_m-z\|^2\}\\ &\ge \min\{C_1,1\}\lim_{m\to\infty}\|z_m-z\|^2, \end{align*} which gives the strong convergence of $\{u_m\}$ to $u$. Hence, $h_1$ is a homotopy of class $(S)_+$ on $\overline{B_r(u_0)}$. Since $h_1(0,u)=DJ(u)$, by Proposition \ref{Browder} we have \begin{equation} i(DJ,y_0+\psi(y_0))=i(h_1(1,\cdot),y_0+\psi (y_0)). \label{eqn17} \end{equation} For $(t,u)$ in $[0,1]\times \overline{B_r(u_0)}$, put \begin{equation} \begin{aligned} h_2(t,u)&=u+t[PDJ(Pu+\psi(Pu))-Pu-\psi (Pu)] \\ &\quad +(1-t)[PDJ(Pu+\psi(Pu))-Pu-\psi (y_0)] \end{aligned} \label{eqn42} \end{equation} We write $u=y+z$ with $y = P(u)$ and $z= u- P(u)$, we have $u=P(u)+ u- P(u)$ and \begin{gather*} h_2(t,u)=PDJ(y+\psi (y))+t(z-\psi(y))+(1-t)(z-\psi(y_0)),\\ h_2(0,u)= PDJ(y+\psi (y))+z-\psi (y_0),\\ h_2(1,u)=PDJ(y+\psi (y))+z-\psi (y). \end{gather*} If $h_2(t,u)=0$ for some $(t,u)$ in $[0,1]\times \overline{B_{r}(u_0)}$, then it is implied that \[ PDJ(y+\psi(y)) = Ph_2(t,u) = 0. \] Thus by \eqref{py}, we see that $y + \psi(y)$ is a critical point of $J$. Since $u_0$ is the unique critical point of $J$ in $\overline{B_r (u_0)}$, we have $y=y_0$. Hence \[ 0 = (I-P)h_2(t,u) = z-t\psi(y)-(1-t)\psi(y_0) = z - \psi(y_0). \] Thus $z=\psi(y_0)$ and $u=u_0$. Therefore $h_2(t,.)$ has a unique zero $u=u_0$ in $\overline{B_r (u_0)}$ for all $t\in [0,1]$, and $h_2$ is a homotopy on $[0,1]\times\overline{B_{r}(u_0)}$ of the compact vector fields $ h_2(0,.)$ and $ h_2(1,.)$. By the product formula and the homotopy invariance of topological degree for compact vector fields, we have \begin{equation} i(D\tilde J,y_0) = i(h_2(0,.),y_0+\psi(y_0))= i(h_2(1,.),y_0+\psi (y_0)). \label{eqn18} \end{equation} Combining \eqref{eqn17}, \eqref{eqn18} and the fact that $h_1(1,\cdot)=h_2(1,\cdot)$ we obtain (v). \end{proof} \begin{lemma} \label{localmin} There exist positive real numbers $r$ and $C$ such that \[ C\|u\|^{2} \leq \min\{J(u),J_{+}(u), J_{-}(u)\} \forall u \in B_r(0). \] \end{lemma} \begin{proof} By \eqref{f3}, \eqref{f4}, \eqref{f5} and \eqref{c4}, we have \begin{gather*} J_{+}(u) = J(u^+) + \frac{1}{2}\int_\Omega[|\nabla u^-|^2 - V_1 |u^-|^2]dx \geq J(u^+) + C_0\|u^-\|^2,\label{J+}\\ J_{-}(u) = J(u^-) + \frac{1}{2}\int_\Omega[|\nabla u^+|^2 - V_1 |u^+|^2]dx \geq J(u^-) + C_0\|u^+\|^2.\label{J-} \end{gather*} Thus, since $\|u\|^2=\|u^+\|^2+\|u^-\|^2$, it suffices to show that there exist positive constants $C$ and $r$ such that $J(u)\ge C\|u\|^2$ for all $u$ in $B_r(0)$. Assume by contradiction that there exist sequences $\{u_m\}\subset H$ and $\{s_m\}\subset \mathbb{R}$ such that $00, \] This contradiction completes the proof. \end{proof} \begin{lemma} \label{Phi} Let $u_0$ be a critical point of $J$ in $H$. Then $J$ satisfies condition $(\Phi )$ at $u_0$. \end{lemma} \begin{proof} This proof is based on the ideas by Manes and Micheletti in \cite{MM} and the regularity and strong Harnack's inequality for Schrodinger operators in \cite{SI} (see also \cite{ST}). Let $B$ be the second derivative of $J$ at $u_0$. We have \begin{equation} \langle Bw,\varphi\rangle = \int_\Omega [\nabla w \nabla \varphi - \frac{\partial f}{\partial t}(x,u_0(x))w \varphi]dx \quad \forall w,\varphi \in H.\label{b1} \end{equation} Suppose that $0$ is the smallest eigenvalue of $B$, we must show that it is simple. Let $E_1$ be its corresponding eigenspace. The proof consists of four steps. \noindent\textbf{Step 1.} Firstly, we show that $B$ is of class $(S)_+$. Let $\{w_{m}\}$ be a sequence converging weakly to $w_0$ in $H$ and \begin{equation} \limsup_{m\to \infty}\leq 0. \label{b2} \end{equation} By (F3), we can suppose that $\{w_{m}\}$ convergent pointwise to $w_0$ and there is a measurable function $g$ having the following properties: $|w_{m}|\leq g$ for every integer $m$ and $|\frac{\partial f}{\partial t}(x,u_0(x))|g^{2}$ is integrable on $\Omega$. Thus by Lebesgue's Dominated convergence theorem, we have \begin{equation} \lim_{m\to\infty} \int_\Omega \frac{\partial f}{\partial t}(x,u_0(x))(w_{m}-w_0)^{2}dx = 0. \label{ob3} \end{equation} By \eqref{b1}, we have \begin{equation} \begin{aligned} &\langle B(w_{m}) - B(w_0), w_{m}-w_0\rangle\\ &= \int_\Omega [|\nabla(w_{m}-w_0)|^{2} - \frac{\partial f}{\partial t}(x,u_0(x))(w_{m}-w_0)^{2}]dx\\ &= \|w_{m}-w_0\|^{2}- \int_\Omega \frac{\partial f}{\partial t} (x,u_0(x))(w_{m}-w_0)^{2}dx. \end{aligned} \label{b3} \end{equation} Since $\{w_{m}\}$ converges weakly to $w_0$, By \eqref{b1} and \eqref{ob3}, we obtain \begin{equation} \begin{aligned} &\lim_{m\to\infty}|\langle B(w_0), w_{m}-w_0\rangle | \\ &\le \lim_{m\to\infty}\|\int_\Omega [\nabla w_0\nabla(w_{m}-w_0)| + \lim_{m\to\infty}|\frac{\partial f}{\partial t}(x,u_0(x))w_0(w_{m}-w_0) = 0 \end{aligned}\label{b4} \end{equation} Combining \eqref{b2}, \eqref{ob3}, \eqref{b3} and \eqref{b4}, we see that $\{w_{m}\}$ converges strongly to $w_0$. Hence $B$ is of class $(S)_+$. Since $B$ is of class $(S)_+$ and $0$ is the smallest eigenvalue of $B$, by Lemma \ref{Decomposition}, we have \begin{equation} \langle Bw,w \rangle \geq 0 \quad \forall w\in H,\label{eqn39x} \end{equation} and the equality holds if and only if $Bw=0$. \noindent\textbf{Step 2.} We show that $w_0^+$ and $w_0^-$ are in $E_1$ for any $w_0$ in $E_1$. Indeed, let $w_0$ be in $E_1$. By \eqref{b1}, we have \[ 0 = \langle Bw_0,w_0\rangle = \langle Bw_0^+,w_0^+\rangle +\langle Bw_0^-,w_0^-\rangle. \] Hence, \eqref{eqn39x} implies \begin{equation} \langle Bw_0^+,w_0^+\rangle = \langle Bw_0^-,w_0^-\rangle =0. \label{eqn40x} \end{equation} By \eqref{eqn39x} and \eqref{eqn40x} we obtain \begin{equation} Bw_0^+=Bw_0^-=0. \label{eqn41} \end{equation} \noindent\textbf{Step 3.} We show that if $w_0\in E_1$ then $w_0$ is continuous and either positive, negative or identically vanishing in $\Omega$. As in Lemma \ref{po}, $ \frac{\partial f}{\partial t}(x,u_0(x))$ is of Kato's class. Using \eqref{eqn41} and results of \cite[ sections 2 and 3]{SI}, we see that $w_0^+$ and $w_0^-$ are continuous functions. If $w_0$ vanishes in $\Omega$ then this step is done. Now suppose that $w_0(x_1)\neq 0$ for some $x_1\in \Omega$. If $w_0(x_1)>0$, by Lemma \ref{po}, we see that $w_0^+$ is positive. Thus $w_0(x)=w_0^{+}(x)>0$ for all $x\in \Omega$. If $w_0(x_1)<0$, a similar argument shows that $w_0(x)<0$ for all $x$ in $\Omega$. \noindent\textbf{Step 4.} Finally, we show that $0$ is a simple eigenvalue of $B$. Indeed, let $w_1$ and $w_2$ be two distinct elements of $E$ such that $w_2(x_0)\neq 0$ for some $x_0\in \Omega$. Put $\lambda =\frac{w_1(x_0)}{w_2(x_0)}$ and $w_3=w_1-\lambda w_2$. Then $Bw_3=0$ and $w_3(x_0)=0$. Thus $w_3\equiv 0$ and therefore $w_1 = \lambda w_2$ or $0$ is simple. \end{proof} \begin{lemma} \label{Palais} $J_{+}$ and $J_{-}$ satisfy the Palais-Smale condition. \end{lemma} \begin{proof} We only give the proof for $J_{+}$, because the case of $J_{-}$ is similar. Let $\{u_m\}$ be a sequence in $H$ such that $\|DJ_{+}(u_m)\|\leq \frac{1}{m}$ for any $n$. We prove that $\{u_m\}$ has a converging subsequence. Because $DJ_{+}$ is of class $(S)_+$ it suffices to show that $\{u_m\}$ is bounded in $H$. For any $m$ in $\mathbb{N}$ and any $ \varphi$ in $H$ \begin{equation} |\langle DJ_{+}(u_m),\varphi\rangle | =\big|\int _{\Omega} [\nabla u_m\nabla \varphi-(f(x,u_{m}^{+}) +V_1(x)u_{m}^{-})\varphi]dx\big| \leq \frac{1}{m}\| \varphi\|. \label{eqn25} \end{equation} Using $\varphi= u_{m}^{-}$ in \eqref{eqn25}, by \eqref{c4}, we obtain \[ C_0\|u_{m}^{-}\|^{2}\le \int _{\Omega} [|\nabla u_{m}^{-}|^{2} -V_1(x)|u_{m}^{-}|^{2}]dx\leq \frac{1}{m}\| u_{m}^{-}\|. \] Thus $\lim_{m\to\infty}u_{m}^{-} = 0$ in $H$. Thus there is a sequence of positive real numbers $\{\varepsilon_{m}\}$ converging to $0$ such that \begin{equation} |\int _{\Omega}[\nabla u_{m}^{+}\nabla \varphi-f(x,u_{m}^{+})\varphi]dx| \leq \varepsilon_{m}\| \varphi\|\quad \forall \varphi \in H. \label{eqn26} \end{equation} Let $v_{m}$ and $w_{m}$ be in $Y$ and $Z$ respectively such that $u_{m}^{+}=v_{m}+w_{m}$ for any integer $m$. Put $a_{m}=\|u_{m}^{+}\|$ for every positive integer $n$. Suppose by contradiction that \[ \lim_{m\to \infty}\|u_{m}^{+}\|^{2}\equiv \lim_{m\to \infty}(\|v_{m}\|^{2}+ \|w_{m}\|^{2}) =\infty. \] Replacing $\{u_{m}^{+}\}$ by its subsequence, by (F3), we can assume that $\{\frac{u_{m}^{+}}{a_{m}}\}$, $\{\frac{v_{m}}{a_{m}}\}$ and $\{\frac{w_{m}}{a_{m}}\}$ converge almost everywhere on $\Omega$, and there is a measurable function $g$ such that $\frac{|u_{m}^{+}|}{a_{m}}+\frac{|v_{m}|}{a_{m}}+|\frac{w_{m}|}{a_{m}}\le g$ and $V_2g^{2}$, $V_3g^{2}$ and $V_4g^{2}$ are integrable on $\Omega$. Put $D=\{x\in \Omega : \sup_{m}u_{m}^{+}(x) < \infty\}$. We have \begin{gather} \lim_{m\to\infty} \frac{u_{m}^{+}}{a_{m}}(x) = 0 \quad \forall x\in D, \\ \lim_{m\to\infty} \frac{|v_{m}^{2}- w_{m}^{2}|}{a_{m}^{2}}(x) = \lim_{m\to\infty} \frac{u_{m}^{+}}{a_{m}}\frac{|v_{m}- w_{m}|}{a_{m}}(x) \le \lim_{m\to\infty} 2g(x)\frac{u_{m}^{+}}{a_{m}}(x) = 0\quad \forall x\in D. \label{umb} \\ \frac{|v_{m}^{2}- w_{m}^{2}|}{a_{m}^{2}}(x)\le 2g^{2}(x)\quad \forall x\in D. \label{umbb} \end{gather} Using $\varphi= w_{m}$ and $v_{m}$ in \eqref{eqn26}, by \eqref{c2}, \eqref{c5} and \eqref{c6}, we have \begin{equation} \begin{aligned} &\varepsilon_{m}\| w_{m}\| \\ &\ge \int _{\Omega} [|\nabla w_{m}|^{2} - f(x,u_{m}^{+})w_{m}]dx \\ &= \|w_{m}\|^{2} - \int _{\Omega} \frac{f(x,u_{m}^{+}(x))}{u_{m}^{+}(x)} u_{m}^{+}w_{m}dx\\ &= [\|w_{m}\|^{2} -\int _{\Omega} V_3w_{m}^{2}dx] + \int _{\Omega} V_3w_{m}^{2}dx - \int _{\Omega} \frac{f(x,u_{m}^{+}(x))}{u_{m}^{+}(x)} u_{m}^{+}w_{m}dx \\ & \ge C_1\|w_{m}\|^{2} +\int_{\Omega} [V_3- \frac{f(x,u_{m}^{+}(x))}{u_{m}^{+}(x)}]w_{m}^{2}dx - \int_{\Omega}\frac{f(x,u_{m}^{+}(x))}{u_{m}^{+}(x)}v_{m}w_{m}dx, \end{aligned} \label{eqn27} \end{equation} and \begin{equation} \begin{aligned} &\varepsilon_{m}\| v_{m}\| \\ &\ge - \int _{\Omega} [|\nabla v_{m}|^{2} - f(x,u_{m}^{+})v_{m}]dx \\ &= -[\|v_{m}\|^{2}-\int_{\Omega}V_4v_{m}^{2}dx] +\int_{\Omega}V_4v_{m}^{2}dx - \frac{f(x,u_{m}^{+}(x))}{u_{m}^{+}(x)}u_{m}^{+}v_{m}dx\\ &\ge C_2\|v_{m}\|^{2} +\int_{\Omega} [\frac{f(x,u_{m}^{+}(x))}{u_{m}^{+}(x)}-V_4]v_{m}^{2}dx + \int_{\Omega}\frac{f(x,u_{m}^{+}(x))}{u_{m}^{+}(x)}w_{m}v_{m}dx. \end{aligned}\label{eqn28} \end{equation} Put $\gamma=\min\{C_1,C_2\} >0$. Using the orthogonality of $v_{m}$ and $w_{m}$ in $H$, by \eqref{c3}, \eqref{eqn27} and \eqref{eqn28}, we obtain \begin{equation} \begin{aligned} &\frac{2\varepsilon_{m}}{a_{m}}- \gamma\\ &\ge \int_{\Omega}V_3\frac{w_{m}^{2}}{a_{m}^{2}}dx -\int_{\Omega}\frac{f(x,u_{m}^{+}(x))}{u_{m}^{+}(x)}\frac{w_{m}^{2}}{a_{m}^{2}}dx - \int_{\Omega}V_4\frac{v_{m}^{2}}{a_{m}^{2}}dx + \int_{\Omega}\frac{f(x,u_{m}^{+}(x))}{u_{m}^{+}(x)} \frac{v_{m}^{2}}{a_{m}^{2}}dx \\ &= \int_{D}[V_3-V_4]\frac{w_{m}^{2}}{a_{m}^{2}}dx +\int_{D}V_4[\frac{w_{m}^{2}}{a_{m}^{2}} - \frac{v_{m}^{2}}{a_{m}^{2}}]dx -\int_{D}\frac{f(x,u_{m}^{+}(x))}{u_{m}^{+}(x)}[\frac{w_{m}^{2}}{a_{m}^{2}} - \frac{v_{m}^{2}}{a_{m}^{2}}]dx \\ &\quad + \int_{\Omega\setminus D}[V_3-\frac{f(x,u_{m}^{+}(x))}{u_{m}^{+}(x)}] \frac{w_{m}^{2}}{a_{m}^{2}}dx - \int_{\Omega\setminus D}[V_4-\frac{f(x,u_{m}^{+}(x))}{u_{m}^{+}(x)}] \frac{v_{m}^{2}}{a_{m}^{2}}dx \\ &\ge \int_{D}[V_4- \frac{f(x,u_{m}^{+}(x))}{u_{m}^{+}(x)}] [\frac{w_{m}^{2}}{a_{m}^{2}}- \frac{v_{m}^{2}}{a_{m}^{2}}]dx \\ &\quad +\int_{\Omega\setminus D}[\frac{f(x,u_{m}^{+}(x))}{u_{m}^{+}(x)} -V_4]\frac{v_{m}^{2}}{a_{m}^{2}}dx \end{aligned} \label{eqn28b} \end{equation} By \eqref{umb}, \eqref{umbb}, \eqref{c3}, Lebesgue's dominated convergence theorem and Fatou's Lemma, as in \eqref{m2}, we have \begin{gather} \lim_{m\to\infty }\int_{D}[V_4- \frac{f(x,u_{m}^{+}(x)}{u_{m}^{+}(x)}] [\frac{w_{m}^{2}}{a_{m}^{2}}- \frac{v_{m}^{2}}{a_{m}^{2}}]dx =0,\label{nnn1} \\ \begin{aligned} &\liminf_{m\to\infty}\int_{\Omega\setminus D} [\frac{f(x,u_{m}^{+}(x))}{u_{m}^{+}(x)}-V_4] \frac{v_{m}^{2}}{a_{m}^{2}}dx \\ &\ge \int_{\Omega\setminus D}\liminf_{m\to\infty} [\frac{f(x,u_{m}^{+}(x))}{u_{m}^{+}(x)}-V_4]\frac{v_{m}^{2}}{a_{m}^{2}}dx \ge 0. \end{aligned}\label{nnn2} \end{gather} Combining \eqref{eqn28b}, \eqref{nnn1} and \eqref{nnn2}, we obtain a contradiction, which implies the boundedness of $\{u_{m}^{+}\}$. Therefore, $\{u_{m}\}$ is bounded in $H$ and we have the conclusion. \end{proof} \begin{lemma} \label{degJ0} (i) If $R$ is sufficiently large then $DJ_{+}$ and $DJ_-$ have no solution outside $B_R(0)$, and (ii) $ \deg (DJ_{+},B_R(0),0)=\deg (DJ_-,B_R(0),0)=0$. \end{lemma} \begin{proof} Using Lemma \ref{Palais} we obtain (i). It suffices to prove (ii) for $J_{+}$. Let $\eta_1$ and $\eta_2$ be two positive numbers such that $\eta _1 <\lambda _1 <\lambda_{k}<\eta_2<\lambda_{k+1}$. For any $u$ in $H$, by the Riesz representation theorem, there is a unique $\pi(u)$ in $H$ such that \[ \langle \pi (u),\varphi\rangle = \int_{\Omega }[\nabla u\nabla\varphi - \eta _2 u^+\varphi + \eta _1 u^-\varphi]dx\quad \forall \varphi \in H. \] It is easy to prove that $\pi$ is a compact-vector field on $H$. Arguing as in \cite[Lemma 3.1]{CC} we see that $0$ is the unique zero of $\pi$ and $\deg (\pi,B,0)=0$ if $B$ is a ball in $H$ containing zero. Put \[ h(s,u) = sDJ_{+}(u) + (1-s)\pi(u)\quad \forall (s,u) \in [0,1]\times H. \] By the homotopy invariance of topological degree of $S_{+}$ operators, it is sufficient to show that there exists a sufficiently large $R$ such that $h(s,u) \neq 0$ for all $sin[0,1]$ and $u\in H\setminus B_R(0)$. Suppose by contradiction that there exist a sequence $\{u_{m}\}$ in $H$ and a sequence $\{s_{m}\}$ in $[0,1]$ such that $\{s_{m}\}$ converges to $s$ in $[0,1]$, $\|u_{m}\| \geq n$ and \[ s_{m}DJ_{+}(u_{m}) + (1-s_{m})\pi(u_{m})=0 \] or \[ \int _{\Omega}\nabla u_m\nabla \varphi dx -\int _{\Omega}\{[ s_{m}f(x,u_{m}^{+})+(1-s_{m}) \eta _2 u_{m}^{+}] -[ s_{m}V_1(x)+(1-s_{m}) \eta _1 ]u_{m}^{-}\}\varphi dx= 0 \] for any $m\in \mathbb{N}$ and $\varphi \in H$. Arguing as in the proof of Lemma \ref{Palais} with $[ s_{m}f(x,u_{m}^{+})+(1-s_{m}) \eta _2 u^{+}]$ and $[ s_{m}V_1(x)+(1-s_{m}) \eta _1 ]$ instead of $f(x,u_{m}^{+})$ and $V_1(x)$ respectively , we obtain a contradiction. \end{proof} \begin{lemma} \label{degequal} (i) $u_0$ is a critical point of $J_{+}$ (respectively $J_-$) if and only if $u_0$ is a nonnegative (respectively non-positive) critical point of $J$. (ii) Moreover if $u_0$ is a common isolated critical point of both $J$ and $J_{+}$ (respectively $J_{-}$), then $i(DJ,u_0)=i(DJ_{+},u_0)$ (respectively $=i(DJ_-,u_0)$). \end{lemma} \begin{proof} (i) Suppose that $u_0$ is a critical point of $J_{+}$; i.e., \[ \int_\Omega [\nabla u_0 \nabla\varphi - f(x,u_0^+)\varphi - V_1(x)u_0^- \varphi]dx = 0 \forall \varphi \in H. \] Choosing $\varphi=u_0^-$, we have \[ \int_\Omega [|\nabla u_0^-|^2 - V_1|u_0^-|^2]dx = 0. \] By \eqref{c4}, we have $u_0^-=0$ and thus $u_0\geq 0$. (ii) Let $u_0$ be a common isolated critical point of $J$ and $J_{+}$. Choose $r>0$ such that $J$ and $J_{+}$ have no any other critical point inside $B_r(u_0)$. By the homotopy invariance property of topological degree for operators of class $(S)_+$, it is sufficient to show that: there exists $r_10$ and $v_m=a_m^{-1}u_m^-$. We can assume that $v_m\rightharpoonup v_0$ in $H$ and then $s_m\to s$ in $[0,1]$. By \eqref{c4} and \eqref{eqn33} we have \begin{equation} \begin{aligned} 0 &= \int_\Omega \{|\nabla v_{m}|^{2} - s_{m}\frac{f(x,a_{m}v_{m})}{a_{m}v_{m}}v_{m}^{2} - (1-s_{m})V_1 v_{m}^{2}\}dx\\ &= \int_\Omega (|\nabla v_{m}|^{2}-V_1 v_{m}^{2})dx - s_{m}\int_\Omega (\frac{f(x,a_{m}v_{m})}{a_{m}v_{m}}- V_1)v_{m}^{2}dx\\ &\ge C_0 + s_{m}\int_{D_{\varepsilon}}(\frac{f(x,u_{m}^{-})}{u_{m}^{-}} - V_1 )v_{m}^{2}dx - s_{m}\int_{\Omega\setminus D_{\varepsilon}} (\frac{f(x,a_{m}v_{m})}{a_{m}v_{m}}- V_1)v_{m}^{2}dx. \end{aligned}\label{vv1} \end{equation} Since $\{u_{m}^{-}\}$ converges uniformly to $0$ on $D_{\varepsilon}$, we see that $\{\frac{f(x,u_{m}^{-})}{u_{m}^{-}}- V_1\}$ converges uniformly to $0$ on $D_{\varepsilon}$ and \begin{equation} \lim_{m\to\infty} \int_{D_{\varepsilon}} (\frac{f(x,u_{m}^{-})}{u_{m}^{-}}- V_1 )v_{m}^{2}dx = 0 . \label{vv2} \end{equation} Since $r > \frac{N}{2}$, there is $q$ in the interval $(1,\infty)$ such that $\frac{1}{q}+ \frac{1}{r} + \frac{N-2}{N} =1$. By \eqref{c1} and H\"{o}lder's inequality, we have \begin{equation} \begin{aligned} &|\int_{\Omega\setminus D_{\varepsilon}}(\frac{f(x,a_{m}v_{m})}{a_{m}v_{m}} - V_1)v_{m}^{2}dx| \\ &\le 2\int_{\Omega\setminus D_{\varepsilon}}Wv_{m}^{2}dx\\ &\le 2(m(\Omega\setminus D_{\varepsilon}))^{1/q} \Big\{\int_{\Omega\setminus D_{\varepsilon}}W^{r}dx\Big\}^{1/r} \Big\{\int_{\Omega\setminus D_{\varepsilon}}|v_{m}|^{\frac{2N}{N-2}}dx \Big\}^{\frac{N-2}{N}}\to 0\quad \text{as }\varepsilon \to 0. \end{aligned}\label{vv3} \end{equation} Combining \eqref{vv1}, \eqref{vv2} and \eqref{vv3}, we find a contradiction, which completes the proof of the lemma. \end{proof} \begin{lemma} \label{Mountain} Let $u_0$ be an isolated critical point of mountain-pass type of $J_{+}$ (respectively $J_-$). Then it is also an isolated critical point of mountain-pass type of $J$. \end{lemma} \begin{proof} It suffices to prove the lemma for the case $u_0$ is an isolated critical point of mountain-pass type of $J_{+}$. We shall find a neighborhood $U$ of $u_0$ such that for all open neighborhood $V_2 \subset U$ of $u_0$, the set $V \cap J^{-1}(-\infty, J(u_0))$ is nonempty and not path-connected. By calculations, we have \begin{equation} J_{+}(u) = J(u^+)+\frac{1}{2}\int_{\Omega}[| \nabla u^-(x)|^2-V_1(x) | u^-(x)|^2]dx\quad \forall u\in H. \label{eqn35} \end{equation} Since $u_0$ is a critical point of $J_{+}$, $u_0$ is nonnegative by Lemma \ref{degequal}. Therefore, $u_0 =u_0^+$ and \[ J(u_0)= J(u_0^+) = J_{+}(u_0). \] Put $\mu=J(u_0)=J_{+}(u_0)$. By definition, there exists a neighborhood $E_0$ of $u_0$ such that the set $E \cap {J_{+}}^{-1}(-\infty, \mu)$ is nonempty and not path-connected for all open neighborhood $E \subset E_0$ of $u_0$. By Lemma \ref{localmin}, we can choose an $r$ such that $J(u^-)\geq 0$ if $\|u^-\|\leq r$. Since \[ \|u - u_0\| = \|(u^+ - u_0) + u^-\| \leq \|u^+ - u_0\| + \|u^-\|, \] there exist $\delta\in (0,r)$ such that \begin{equation} U=\{u:\| u^+-u_0\| <\delta, \| u^-\| <\delta\}\subset E_0.\label{eqn36} \end{equation} We see that $U$ is an open neighborhood of $u_0$ and \begin{gather} u^+ + tu^- \in U \forall t \in [0,1], \label{eqn37}\\ J(u) = J(u^+) + J(u^-) \ge J(u^+) = J_{+}(u^+)\quad \forall u\in U. \label{eqn38} \end{gather} Let $V$ be an open neighborhood of $u_0$ in $U$. As in \eqref{eqn36}, there is a $\delta_1 \in (0,\delta)$ such that \begin{equation} U_1=\{u:\| u^+-u_0\| <\delta_1, \| u^-\| <\delta_1\}\subset V.\label{eqn39} \end{equation} We accomplish the proof by following steps \noindent\textbf{Step 1.} We show that $V \cap J^{-1}(-\infty, \mu)$ is nonempty. If $s\geq t\geq 0$, by \eqref{eqn35} and \eqref{c4} we have \begin{equation} J_{+}(u^++su^-) - J_+(u^++tu^-) =(s^2-t^2)\int_{\Omega}[| \nabla u^-(x)|^2-V_1(x)| u^-(x)|^2]dx \ge 0. \label{eqn40} \end{equation} Since $u_0\in U_1 \subset E_0$ and $U_1$ is open, the set $U_1\cap {J_+}^{-1}(-\infty,\mu)$ is nonempty. Pick an element $v$ in this set. By \eqref{eqn40}, we have $J(v^+)={J_+}(v^+)\leq J^+(v^+ + v^-)={J_+}(v) < \mu$, and hence $v^+ \in J^{-1}(-\infty,\mu)$. Furthermore, by \eqref{eqn39}, $v^+ \in U_1$. It follows that $U_1 \cap J^{-1}(-\infty, \mu)$ is nonempty, then $V \cap J^{-1}(-\infty, \mu)$ is nonempty. \noindent\textbf{Step 2.} We show that $S\equiv V \cap J^{-1}(-\infty, \mu)$ is not path-connected. Assume by contradiction that it is path-connected. Put \begin{gather*} W_1 = \{u^++tu^-:u\in U, u^+\in V\cap J^{-1}(-\infty,\mu), t\in [0,1]\},\\ W_2 = \{u^++tu^-:u\in V\cap J^{-1}(-\infty,\mu), t\in (0,1]\},\\ W_3 = \{u:\| u^+-u_0\| <\delta_1, \| u^-\| <\delta_1\},\\ W_0 = W _1\cup W _2 \cup W _3. \end{gather*} It is clear that $W_1$, $W_2$, $W_3$ and $W_0$ are open sets in $E_0$, $W_3 = U_1$ and $u_0\in W_3\subseteq W_0$. We will show that $G=W_0\cap {J_+}^{-1}(-\infty,\mu)$ is path-connected, which yields a contradiction. For any $v$ and $w$ in $G$, we say $v\sim w$ if and only if there exists a continuous mapping $\varphi$ from $[1,2]$ into $G$ such that $\varphi (1)=v$ and $\varphi (2)=w$. Let $w_1$ and $w_2$ be in $G$. If $w_1$ and $w_2$ are in $W_1$, then by definition we see that $w_1\sim w_1^+$, $w_2\sim w_2^+$, and $w_1^+$ and $w_2^+$ are in $S$. Since $S$ is path-connected, there exists a continuous mapping $\phi$ from $[1,2]$ into $G$ such that $\phi (1)=w_1^+$ and $\phi (2)=w_2^+$. By definition $\varphi (t,\epsilon)=(\phi (t))^++\epsilon (\phi (t))^- \in W_2$ for all $(t,\epsilon)$ in $[1,2]\times(0,1]$. By \eqref{eqn38}, we obtain $(\phi (t))^+\in {J_+}^{-1}(-\infty,\mu)$. Hence, by the continuity of $J^+$ and the compactness of $\phi([1,2])$, $\varphi (t,\epsilon)$ is in $G$ if $\epsilon$ is sufficiently small for any $t$ in $[1,2]$. Note that $\varphi (1,\epsilon)=w_1^+$ and $\varphi (2,\epsilon)=w _2^+$. Thus, $w_1^+\sim w_2^+$ which gives $w_1\sim w _2$. If $w_1$ and $w_2$ belong to $W_3$ then $w_1\sim w_1^+$, $w_2\sim w_2^+$, and $w_1^+$ and $w_2^+$ are in $S$. Arguing as above we have $w_1\sim w _2$. If $w_1$ and $w_2$ are in $W _2$ then $w_1=u_1^++t_1u_1^-$ and $w_2=u_2^++t_2u_2^-$ where $t_1,t_2>0$, and $u_1$ and $u_2$ in $S$. As in the first case, there is a positive real number $\varepsilon_0$ such that $(u_1^{+} + \varepsilon u_1^{-})\sim (u_2^{+} + \varepsilon u_2^{-})$ for any $\varepsilon$ in $[0,\varepsilon_0]$. On the other hand we have $w_1\sim (u_1^{+} + \varepsilon u_1^{-})$ and $w_2\sim (u_2^{+} + \varepsilon u_2^{-})$ for any $\varepsilon$ in $(0,\min\{\varepsilon_0,t_1,t_2\}]$. Therefore $w_1\sim w_2$. Similarly $w_1\sim w_2$ for other cases. Thus we have shown that $W_0\cap {J_+}^{-1}(-\infty,\mu)$ is path-connected, contradicting the way we choose $U^+$. Thus $V \cap J^{-1}(-\infty,\mu)$ is not path-connected. This completes the proof. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1}] We use the following steps \noindent\textbf{Step 1.} Note that a weak solution $u\in H$ of \eqref{eqn01} is a critical point of $J$ and vice-versa. Moreover, it suffices to consider the case in which the set of solutions of \eqref{eqn01} is finite. In this case, all critical points of $J$ are isolated. Since $f(x,0)=0$, by Lemma $\ref{smoothness}$, $u_1\equiv0$ is a solution of \eqref{eqn01}. On the other hand, since $\varphi_1$ is positive in $\Omega$, by (ii) of Lemma \ref{smoothness}, we have \[ \lim _{m\to\infty}J^+(n\varphi _1 ) =\lim _{m\to\infty}J(n\varphi _1 )=-\infty. \] Thus by \cite[Theorem 1]{Hof} and Lemmas \ref{localmin}, \ref{Palais}, \ref{degequal} and \ref{Mountain}, $J$ has a nonnegative critical point of mountain-pass type $u_2$. Similarly $J$ has a non-positive critical point of mountain-pass type $u_3$. Let $\widetilde{J}$ be the reduction function of $J$ as in Lemma \ref{Castro}. Then it follows from (ii) of Lemma \ref{smoothness} that $\lim _{y\in Y, \| y\|\to\infty}\widetilde{J}(y)=-\infty$. Hence $\widetilde{J}$ has a global maximum at $y_0$ and $u_4=y_0+\psi (y_0)$ is a critical point of $J$. By (iv) of Lemma \ref{Castro} and Definition \ref{MP}, we see that $u_4 \not\in \{u_2, u_3\}$. By Lemma \ref{localmin}, $J$ has a strictly local minimum at $0$, which shows that $\widetilde{J}$ has a strictly local minimum at $0$. Hence $u_4$ is not equal to $0$. Thus we have found four distinct solutions $u_1,u_2,u_3$ and $u_4$ of \eqref{eqn01}. \noindent\textbf{Step 2.} Choose a sufficiently large real number $R$ such that all critical points of $J, J_{+}, J_{-}$ and $\widetilde {J}$ lie in $B_{R}(0)$. We can find four open subsets $U_1$, $U_2$, $U_3$ and $U_4$ in $B_R(0)$ such that $U_1\cap DJ^{-1}(0)=\{u_1\}$, $U_2\cap DJ^{-1}(0)=\{ u \geq 0, u \not\equiv 0\}$, $U_3\cap DJ^{-1}(0)=\{ u \leq 0, u \not\equiv 0\}$, and $U_4\cap DJ^{-1}(0)=\{u_4\}$. Moreover, we can assume that $U_1$, $U_2$ and $U_3$ are disjoint. Now we consider the case in which $k$ is even. By Corollaries 2.1 and 2.2 in \cite{Duc-Luc-Nam-Tuyen}, Lemma \ref{localmin}, Lemma \ref{Castro} and Lemma \ref{degJ0}, we have \begin{gather*} \deg(DJ, U_1,0) = 1 ,\\ \deg(DJ, B_{R}(0),0) = \deg(\nabla \widetilde{J}, B_{R}(0),0)= (-1)^{k}= 1 ,\\ \deg(DJ_{+}, B_{R}(0),0) = \deg(DJ_{-}, B_{R}(0),0)= 0 . \end{gather*} Thus by Lemma \ref{degequal} and the excision property of the degree, \begin{gather*} \deg(DJ, U_2,0) = \deg(DJ^+, U_2,0) =\deg(DJ_{+}, B_{R}(0),0) - \deg(DJ_{+}, U_1,0) = - 1,\\ \deg(DJ, U_3,0) = \deg(DJ_-, U_3,0) =\deg(DJ_{-}, B_{R}(0),0) - \deg(DJ_{-}, U_2,0) = - 1. \end{gather*} Since $y_0$ is a global maximum point of $\widetilde{J}$ and hence a global minimum point of $-\widetilde{J},$ it follows from \cite[Corollary 2.2]{Duc-Luc-Nam-Tuyen} that $i(\nabla \widetilde{J},y_0)=(-1)^k = 1$. Thus, by Lemma \ref{Castro}(v), \[ \deg(DJ, U_4,0) =1. \] If $u_4$ is not in $U_2\cup U_3$, we may assume $U_4 \cap(\cup_{i=1}^{3}U_{i}) = \emptyset$. Thus, by Proposition \ref{Browder}, \[ \deg(DJ, B_{R}(0)\setminus \overline{\cup_{i=1}^{4}U_{i}},0) = \deg(DJ, B_{R}(0),0)-\sum_{i=1}^{4} deg(DJ, U_{i},0) = 1, \] which implies that $J$ has a sign-changing critical point $u_5 \not\in \{u_1, u_2, u_3, u_4\}$. Therefore \eqref{eqn01} has two sign-changing solutions $u_4$ and $u_5$. If $u_4$ is in $U_2 \cup\, U_3$, we can assume that $u_4\in U_2$. By Lemma \ref{Phi}, $J$ satisfies $(\Phi)$ at $u_2$, and by Proposition \ref{Hofer}, we have $i(DJ, u_2)=-1$. Let $U_5$ be an open neighborhood of $u_2$ in $H$ containing no other critical point of $J$ then $deg(DJ, U_5,0)=i(DJ, u_2)=-1$. Thus, by the additivity property of the degree, \[ \deg(DJ, U_2\setminus \overline{U_4\cup U_5},0) =\deg(DJ, U_2,0)-\deg(DJ, U_4,0)-\deg(DJ, U_5,0) =-1. \] By the normalization property of the degree, there is a solution $u_5$ of \eqref{eqn01} in $U_2\setminus \overline{U_4\cup U_5}$. Hence \eqref{eqn01} has three solutions $u_2,u_4,u_5$ of the same sign. Moreover, by Proposition \ref{Browder} and the excision property of the degree, we have \[ \deg(DJ, B_{R}(0)\setminus \overline{\cup_{i=1}^{3}U_{i}},0) = \deg(DJ, B_{R}(0),0)-\sum_{i=1}^{3} deg(DJ, U_{i},0)=2, \] implying that \eqref{eqn01} has a sign-changing solution $u_6 \not\in \{u_1, u_2, u_3, u_4, u_5\}$. Next, suppose that $k$ is odd. If $u_4\not\in U_2\cup U_3$ then the proof is similar to that of the case $k$ is even. It remains to consider the case $u_4\in U_2\cup U_3$. We can assume $u_4\in U_2$. Let $U_5$ be as above. Arguing as above, we have \begin{gather*} \deg(DJ, U_4,0)=i(\nabla\widetilde{J},y_0)=(-1)^{k}=-1,\\ \deg(DJ, U_5,0)=i(DJ,u_2)=-1,\\ \deg(DJ, U_2\setminus \overline{U_4\cup U_5},0) =\deg(DJ, U_2,0)-\deg(DJ, U_4,0)-\deg(DJ, U_5,0)=1. \end{gather*} Thus, by the normalization property of the degree, there exists $u_5\in U_2\setminus \overline{U_4\cup U_5}$ with $DJ(u_5)=0$. Thus, \eqref{eqn01} has five solutions $u_1,u_2, u_3, u_4$ and $ u_5$, where $u_2$, $u_4$ and $u_5$ are of the same sign. The proof is complete. \end{proof} \section{Appendix} In this section, we extend the results of Hofer \cite{Hof} on the index at a critical point of mountain-pass type of a functional whose gradient is a compact vector field to the case where the gradient is an operator of class $(S)_+$. Throughout this section, the dual space $H^{\ast}$ is identified with $H$. Our main result of the appendix is the following theorem. \begin{theorem} \label{Hofer} Let $x_0$ be in $H$, and let $j$ be a $C^2$-real function on $H$ such that $Dj$ is of class $(S)_+$ on $H$ and $x_0$ is an isolated critical point of mountain-pass type of $j$. Assume that $D^{2}j(x_0)$ is of class $(S)_+$ and $j$ satisfies $(\Phi )$ at $x_0$. Then \[ i (Dj,x_0)=-1. \] \end{theorem} To prove the above theorem, we need the following lemmas. \begin{lemma} \label{Decomposition} Let $A$ be a bounded self-adjoint linear operator of class $(S)_{+}$ on $H$ and $Y$ be $A^{-1}(\{0\})$. Then $Y$ is finite dimensional and there exist a positive number $C$ and vector subspaces $X$ and $Z$ of $H$ such that $X$ is finite dimensional and \begin{itemize} \item[(i)] $X\oplus Y \oplus Z$ is an orthogonal decomposition of $H$, \item[(ii)] $X$, $Y$ and $Z$ are invariant under $A$, \item[(iii)] the restriction of $A$ on $X\oplus Z$ is a one-to-one mapping from $X\oplus Z$ onto itself, \item[(iv)] $\langle Ax,x\rangle \leq -C\| x\| ^2$ for all $x\in X$, \item[(v)] $\langle Az,z\rangle \geq C\| z\| ^2$ for all $z\in Z$. \end{itemize} \end{lemma} \begin{proof} Let $\{y_{m}\}$ is a sequence in $Y$ and weakly converges to $y$ in $H$. Since $\limsup_{n\to\infty}< A(y_{m}), y_{m} -y > = 0$ and $A$ is of class $(S)_{+}$, $\{y_{m}\}$ converges strongly to $y$. Thus $Y$ is locally compact. It implies $Y$ is finite dimensional. Put $E = Y^{\perp}$. We see that $\langle Au,v\rangle = \langle u,Av\rangle = 0$ for all $u \in E$, $v \in Y$. Therefore, $A(E) \subset E$. Denote by $B$ the restriction of $A$ on $E$. We see that $B$ is a bounded self-adjoint linear operator on $E$. It is clear that $B$ is one-to-one.\\\quad We shall prove that $B(E)$ is a closed subspace of $E$. Let $\{x_{m}\}$ be a sequence in $E$ such that $\{B(x_{m})\}$ converges to $y$ in $E$, we will prove that $y\in B(E)$. First, we show that $\{x_m\}$ is bounded. Suppose by contradiction that $\{\|x_{m}\|\}$ tends to $\infty$. Put $v_{m}=(\|x_{m}\| +1)^{-1}x_{m}$ for any integer $n$, then $\{\|v_{m}\|\}$ converges to $1$ and $\{B(v_{m})\}$ converges to $0$. Without loss of generality, we can (and shall) suppose that $\{v_{m}\}$ converges weakly to a vector $v_0$ in $E$. Since $A$ is of class $(S)_{+}$, and $$ \limsup_{m\to \infty}\langle A(v_{m}),v_{m}-v_0\rangle = \limsup_{m\to \infty}\langle B(v_{m}),v_{m}-v_0\rangle = 0, $$ the sequence $\{v_{m}\}$ converges to $v_0$. Thus, $\|v_0\| = 1$ and $A(v_0) = 0$, which is a contradiction. Therefore $\{x_{m}\}$ is bounded and we can suppose that it converges weakly to a vector $x_0$ in $E$. Since $\{A(x_{m})\}$ converges to $y$, by the definition of class $(S)_{+}$, the sequence $\{x_{m}\}$ converges to $x_0$ in $E$. Therefore $A(x_0) = y$ and $B(E)$ is closed. Next we show that $B(E) = E$. Otherwise, there is a vector $x$ in $E\setminus\{0\}$ such that \[ \langle B(z),x\rangle = 0\quad \text{or} \quad \langle z,A(x)\rangle = 0\quad \forall z\in E. \] Thus, $A(x)$ is in $Y$. It implies that $A(A(x))$ is also in $Y$ and \[ \langle A(x),A(x)\rangle = \langle x,A(A(x))\rangle = 0. \] It follows that $x$ is in $Y\cap E$, then $x=0$, which is impossible. This contradiction shows that $B(E) = E$. We have proved that $B$ is an one-to-one mapping from $E$ onto $E$. Thus, by the open mapping theorem, $B$ is an invertible self-adjoint bounded operator on $E$. By a result on self-adjoint operator (see \cite[ p. 172]{Lang}), there exist a positive real number $C$ and an orthogonal decomposition $X\oplus Z$ of $E$ such that $X$ and $Z$ are $A$-invariant closed subspaces of $E$ and \begin{gather} \langle A(x),x\rangle \leq -C\|x\|^{2}\quad \forall x\in X,\label{ac1}\\ \langle A(x),x\rangle \geq C\|x\|^{2}\quad \forall x\in Z.\label{ac2} \end{gather} Finally we prove that $X$ is finite dimensional. It is sufficient to show that $X$ is locally compact. Let $\{x_{m}\}$ be a sequence weakly converging to $x$ in $X$. We see that \begin{equation} \lim_{m\to \infty} = 0.\label{ac3} \end{equation} On the other hand, by \eqref{ac1}, \begin{equation} \langle A(x_{m} - x),x_{m}-x\rangle \leq 0\quad \forall n \in \mathbb{N}.\label{ac4} \end{equation} Combining \eqref{ac3} and \eqref{ac4}, we have $\limsup_{m\to \infty}\langle A(x_{m}),x_{m}-x\rangle \leq 0$. Since $A$ is of class $(S)_{+}$, $\{x_{m}\}$ converges to $x$. Therefore $X$ is locally compact and finite dimensional. The proof is complete. \end{proof} \begin{lemma} \label{Hofer1} Let $U$ be an open subset of a Hilbert space $H$, and let $j$ be a $C^{2}$-real function on $U$. Suppose that $j(0)=0$, $0$ is an isolated critical of $j$, $D j$ and $D^{2}j(0)$ are of class $(S)_+$ on $H$. Let $X\oplus Y\oplus Z$ be the decomposition of $H$ for $A=D^{2}j(0)$ as in Lemma $\ref{Decomposition}$. Then there exist a homeomorphism $G$ defined on a neighborhood of $0$ in $H$ into $H$ and a $C^1$-map $\beta$ defined on a neighborhood $V$ of $0$ in $Y$ into $X\oplus Z$ with $G(0)= \beta(0)=0$ such that \begin{itemize} \item[(a)] for all $u=x+y+z \in X\oplus Y\oplus Z$ with $\|u\|$ sufficiently small, \begin{equation} j(G(u))=-\frac{1}{2}\| x\| ^2+\frac{1}{2}\| z\| ^2 +j(y+\beta (y)), \label{eqn02} \end{equation} \item[(b)] and for all $y \in Y$, \begin{equation} P(Dj(y+\beta(y)))=0, \label{eqn03} \end{equation} where $P$ is the orthogonal projection of $H$ onto $X \oplus Z$. \end{itemize} Moreover, \begin{equation} i (Dj,0)=(-1)^{\dim (X)}i(D\psi,0),\label{eqn04} \end{equation} where $\psi (y)=j(y+\beta (y))$. \end{lemma} \begin{proof} Using lemma \ref{Decomposition} and arguing as in the proof of \cite[Theorem 3]{Hof} we obtain the existence of functions $G$ and $\beta$ satisfying \eqref{eqn02} and \eqref{eqn03}. As in the proof of the cited Theorem, we obtain \eqref{eqn04} by using the following homotopy in sense of class $(S)_+$, \[ h(t,u)= \begin{cases} P(Dj(u))+ Q(Dj(t\beta(y) +(1-t)(x+z)+y) & t\in [0,1]\\ P(Dj(x+z+(2-t)y))+Q(Dj(y+\beta(y))) & t\in [1,2] \\ (3-t)(P(Dj(x+z))+(t-2)(-x+z)+ Q(Dj(y+\beta (y))) & t\in [2,3], \end{cases} \] where $u = x+y+z\in X\oplus Y \oplus Z$, and $Q$ is the projection of $H$ onto $Y$. \end{proof} \begin{proof}[Proof of Theorem \ref{Hofer}] Using Proposition \ref{Hofer1} and arguing as in the proofs of \cite[Theorems 2 and 3]{Hof}, we obtain the theorem. \end{proof} \subsection*{Acknowledgments} We would like to thank the anonymous referees for their very helpful comments. The work is partially supported by the grant 101.01.2010.11 from the National Foundation for Science and Technology Development of Vietnam. \begin{thebibliography}{00} \bibitem{AM} H. Amman; \emph{A note on degree theory for gradient mappings}, Proc. Amer. Math. 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