\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small 2005-Oujda International Conference on Nonlinear Analysis. \newline {\em Electronic Journal of Differential Equations}, Conference 14, 2006, pp. 125--133.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2006 Texas State University - San Marcos.} \vspace{9mm}} \setcounter{page}{125} \begin{document} \title[\hfilneg EJDE/Conf/14\hfil Solutions for nonlinear elliptic problems] {Nontrivial solutions for nonlinear elliptic problems via Morse theory} \author [A. Ayoujil, A. R. El Amrouss \hfil EJDE/Conf/14\hfilneg] {Abdesslem Ayoujil, Abdel R. El Amrouss} \address{Abdesslem Ayoujil \newline University Mohamed 1er, Faculty of Sciences, Department of Mathematics, Oujda, Morocco} \email{abayoujil@yahoo.fr} \address{Abdel R. El Amrouss \newline University Mohamed 1er, Faculty of Sciences, Department of Mathematics, Oujda, Morocco} \email{amrouss@sciences.univ-oujda.ac.ma} \date{} \thanks{Published September 20, 2006.} \subjclass[2000]{58E05, 35J65, 49B27} \keywords{Minimax method; p-Laplacian; resonance elliptic equation; \hfill\break\indent critical group; Morse theory} \begin{abstract} We prove the existence of nontrivial solutions for perturbations of p-Laplacian. Our approach combine minimax arguments and Morse Theory, under the conditions on the behaviors of the perturbed function $f(x,t)$ or its primitive $F(x,t)$ near infinity and near zero. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \section{Introduction } Let $\Omega\subset\mathbb{R}^{N}$ be a bounded domain with smooth boundary $\partial\Omega$, and let $f:\Omega\times\mathbb{R}\rightarrow{\mathbb{R}}$ be a Carath\'eodory function, with some appropriate growth condition to be specified later. We consider the Dirichlet problem $$\label{e1.1} \begin{gathered} -\Delta_{p} u =f(x,u) \quad\mbox{in } \Omega, \\ \quad u=0 \quad\mbox{on } \partial\Omega, \end{gathered}$$ where $\Delta_{p}u: = \mathop{\rm div}(|{\nabla u}|^{p-2}\nabla u)$, $10$ that is simple and has an associated normalized eigenfunction $\varphi_1$ which is positive in $\Omega$. It is also known, (see \cite{ANT96}), that there exists a second eigenvalue $\lambda_2$ such that $\sigma(-\Delta_p) \cap ]\lambda_1,\lambda_2[ = \emptyset$. Here, $\sigma(-\Delta_p)$ is the spectrum of $-\Delta_p$ on $W_0^{1,p}(\Omega)$, which contains at least an increasing eigenvalue sequence obtained by the Lusternik-Schnirelaman theory. The existence of nontrivial solutions for \eqref{e1.1} has been widely treated by many authors, under various assumptions on nonlinearity $f$ and its primitive $F$, see \cite{Cos95, ELA00, LS01} and the references therein. Throughout this paper, we assume that $f$ satisfies the subcritical growth \begin{itemize} \item[(F0)] for some $q\in(1,p^{*})$, there exists a constant $c>0,$ such that $$|f(x,t)|\leq c(1+|t|^{q-1}), \quad\forall t\in\mathbb{R}, \quad\text{a.e } x\in \Omega,$$ where $p^{*}=\frac{Np}{N-p}$ if $10$ such that \begin{gather}\label{e1.3} F(x,t)\geq C_\tau|t|^\mu , \quad \text{ for a.e. } x\in\Omega, \quad 0<|t|\leq \tau,\\ \label{e1.4} \liminf_{|t|\to 0}\frac{\mu F(x,t)-tf(x,t)}{|t|^{q}}\geq \alpha, \quad\text{uniformly for a.e.} x\in\Omega, \end{gather} for some $q\in(p,p^{*})$ and $\alpha$ be a constant non positive. \end{itemize} Now, we may state the main result. \begin{theorem}\label{thm1.1} Assume (F0), (F3)--(F5) and (F1) or (F2). Then the problem \eqref{e1.1} has at least one nontrivial solution. \end{theorem} For finding critical points of $\Phi$, by applying minimax methods, we will use the following compactness condition, introduced by Cerami \cite{Cer78}, which is a generalization of the classical Plais-Smale type (PS). \begin{definition}\label{def1.1} \rm Given $c\in \mathbb{R}$, we say that $\Phi\in C^{1}(X,\mathbb{R})$ satisfies condition $(C_c)$, if \begin{itemize} \item[(i)] Every bounded sequence $(u_{n})\subset X$ such that $\Phi(u_{n})\to c$ and $\Phi'(u_{n})\to 0$ has a convergent subsequence, \item[(ii)] There is constants $\delta, R,\alpha>0$ such that $$\|\Phi'(u)\|_{X'}\|u\|_{X}\geq \alpha , \quad\forall u\in\Phi^{-1}([c-\delta,c+\delta]) \quad\text{with }\|u\|_{X}\geq R.$$ \end{itemize} If $\Phi$ satisfies condition $(C_c)$ for every $c\in\mathbb{R}$, we simply say that $\Phi$ satisfies $(C)$. \end{definition} The present paper is organized as follows. In section 2, we will compute the critical groups at zero and at a mountain pass point. In section 3, we give the proof of theorem \ref{thm1.1}. \section{Critical groups } In this section, we investigate the critical groups at zero and at a mountain pass type. To proceed, some concepts are needed. Let X be a Banach space, given a $\Phi\in C^{1}(X,\mathbb{R})$. For $\beta\in\mathbb{R}$ and $c\in\mathbb{R}$, we set \begin{gather*} \Phi^{\beta}=\{u\in X: \Phi(u)\leq \beta\}, \\ K=\{u\in X: \Phi^{'}(u)=0 \}, \\ K_{c}=\{u\in X: \Phi(u)=c , \Phi^{'}(u)=0 \}. \end{gather*} Denote by $H_{q}(A,B)$ the $q$-th homology group of the topological pair $(A,B)$ with integer coefficient. The critical groups of $\Phi$ at an isolated critical point $u\in K_c$ are defined by $$C_{q}(\Phi,u)=H_{q}(\Phi^{c}\cap U,\Phi^{c}\cap U\setminus\{u\}),~~q\in \mathbb{Z},$$ where $U$ is a neighborhood of $u$. Moreover, it is known that $C_{q}(\Phi,u)$ is independent of the choice of $U$ due to the excision property of homology. We refer the readers to \cite{Ch93,MW89} for more information. Let denote by $B_{\rho}$ the closed ball in $W^{1,p}_{0}(\Omega)$ of radius $\rho>0$ which is to be chosen later , with the center at the origin. We will show that the critical groups of $\Phi$ at zero are trivial. \begin{theorem}\label{thm2.1} Assume $(F0)$ and $(F5)$. Then, $$C_{q}(\Phi,0)\cong 0,~~\forall q\in\mathbb{Z}.$$ \end{theorem} This result will be proved by constructing a retraction of $B_{\rho}\setminus\{0\}$ to $B_{\rho}\cap\Phi^{0}\setminus\{0\}$ and by proving that $B_{\rho}\cap\Phi^{0}$ is contractible in itself. For this purpose, some technical lemmas must be proved. Note that the following lemma has been proved in case $p=2$ \cite [Lemma 1.1]{Mo97}). \begin{lemma}\label{lem2.1} Under $(F0)$ and $(F5)$, zero is a local maximum for the functional $\Phi(su)$, $s\in\mathbb{R},$ for $u\neq 0$. \end{lemma} \begin{proof} Using $(F0)$ and the hypothesis \eqref{e1.3}, we get $$\label{(e2.1)} F(x,t)\geq C_\tau|t|^{\mu}-C_{1}|t|^{q}, \quad x\in\Omega, \quad t\in\mathbb{R},$$ for some $q\in(p,p^{*})$ and $C_{1}>0$. For $u\in W_{0}^{1,p}(\Omega),u\neq 0$ and $s>0$, we have \label{e2.2} \begin{aligned} \Phi(su)&=\frac{1}{p}s^{p}\int_{\Omega}|\nabla u|^{p}dx-\int_{\Omega}F(x,su)dx\\ &\leq\frac{s^{p}}{p}\|u\|^{p}-\int_{\Omega}(C_\tau|su|^{\mu}-C_{1}|su|^{q})dx \\ &\leq\frac{s^{p}}{p}\|u\|^{p}-C_\tau s^{\mu}\|u\|^{\mu}_{\mu}+C_{1}s^{q}\|u\|^{q}_{q}. \end{aligned} Since $\mu0$ such that \label{e2.3} \Phi(su)<0, \quad\text{for all } 00$such that for all$u\in W_{0}^{1,p}(\Omega)$with$\Phi(u)=0$and$0<\|u\|\leq\rho$, we have $$\label{e2.4} \frac{d}{ds}\Phi(su)|_{s=1}>0.$$ \end{lemma} \begin{proof} For$u\in W_{0}^{1,p}(\Omega)$be such that$\Phi(u)=0$. From$(F0)$and (\eqref{e1.4}), we have $$\mu F(x,u)-f(x,u)u\geq -c|u|^{q}, \quad \text{a.e. } x\in\Omega,$$ for some$q\in(p,p^{*})$and$c>0$. Denote by$\langle.,.\rangle$the duality pairing between$W^{1,p}_{0}(\Omega)$and$W^{-1,p'}(\Omega)$. Then, since$\Phi(u)=0, we have \begin{align*} \langle\Phi^{'}(su),u\rangle|_{s=1} &=\int_{\Omega}|\nabla u|^{p}dx-\int_{\Omega}f(x,u)udx,\\ &=(1-\frac{\mu}{p})\int_{\Omega}|\nabla u|^{p}dx +\int_{\Omega}(\mu F(x,u)-f(x,u)u)dx . \end{align*} By the above inequality and the Poincar\'e's inequality, we write \begin{align*} \frac{d}{ds}\Phi(su)|_{s=1}&=\langle\Phi^{'}(su),u\rangle|_{s=1},\\ &\geq (1-\frac{\mu}{p})\|u\|^{p} -c\int_{\Omega}|u|^{q}dx,\\ &\geq (1-\frac{\mu}{p})\|u\|^{p}-C\|u\|^{q}, \end{align*} for someC> 0$. Since$\mu < p < q$, the inequality \eqref{e2.4} is verified. \end{proof} \begin{lemma}\label{lem2.3} For all$u\in W_{0}^{1,p}(\Omega)$with$\Phi(u)\leq 0$and$\|u\|\leq\rho$, we have $$\label{e2.5} \Phi(su)\leq 0, \quad\text{ for all } s\in(0,1).$$ \end{lemma} \begin{proof} Let$\|u\|\leq\rho$with$\Phi(u)\leq 0$and assume by contradiction that there exists some$s_{0}\in(0,1]$such that$\Phi(s_{0}u)>0$. Thus, by the continuity of$\Phi$, there exists an$s_{1}\in(s_{0},1]$such that$\Phi(s_{1}u)=0$. Choose$s_2 \in (s_0, 1]$such that$s_2 = \min \{s \in [s_0, 1] : \Phi(su) = 0\}$. It is easy to see that$\Phi(su) \geq 0$for each$s \in [s_0, s_2]$. Taking$u_{1}=s_{2}u$, one deduces $$\Phi(su)-\Phi(s_{2}u) \geq 0 \quad\text{implies that}\quad \frac{d}{ds}\Phi(su)|_{s=s_{2}}=\frac{d}{ds}\Phi(su_{1})|_{s=1}\leq 0.$$ However, by \eqref{e2.4} $$\frac{d}{ds}\Phi(su_{1})|_{s=1}>0.$$ This contradiction shows that \eqref{e2.5} holds. \end{proof} \begin{proof}[Proof of theorem \ref{thm2.1}] Let us fix$\rho>0$such that zero is the unique critical point of$\Phi$in$B_{\rho}$. First, by taking the mapping$h:[0,1]\times(B_{\rho}\cap\Phi^{0})\to B_{\rho}\cap\Phi^{0}$as $$h(s,u)=(1-s)u,$$$B_{\rho}\cap\Phi^{0}$is contractible in itself. Now, we prove that$(B_{\rho}\cap\Phi^{0})\setminus\{0\}$is contractible in itself too. For this purpose, define a mapping$T:B_{\rho}\setminus\{0\}\to (0,1]$by \begin{gather*} T(u)=1, \text{ for } u\in (B_{\rho}\cap\Phi^{0})\setminus\{0\},\\ T(u)=s, \text{ for } u\in B_{\rho}\setminus\Phi^{0}\quad\text{with }\Phi(su)=0, s<1. \end{gather*} >From the relations \eqref{e2.3}, \eqref{e2.4} and \eqref{e2.5}, the mapping$T$is well defined and if$\Phi(u)>0$then there exists an unique$T(u)\in(0,1)$such that $$\label{e2.6} \begin{gathered} \Phi(su)<0,\quad\forall s\in(0,T(u)), \\ \Phi(T(u)u)=0, \\ \Phi(su)>0, \quad\forall s\in(T(u),1). \end{gathered}$$ Thus, using \eqref{e2.4}, \eqref{e2.6} and the Implicit Function Theorem to get that the mapping T is continuous. Next, we define a mapping$\eta: B_{\rho}\setminus\{0\}\to (B_{\rho}\cap\Phi^{0})\setminus\{0\}$by $$\label{e2.7} \begin{gathered} \eta(u)=T(u)u, u\in B_{\rho}\setminus\{0\}\quad\text{with } \Phi(u)\geq 0,\\ \eta(u)=u, u\in B_{\rho}\setminus\{0\}\quad\text{with } \Phi(u)< 0. \end{gathered}$$ Since$T(u)=1$as$\Phi(u)=0$, the continuity of$\eta$follows from the continuity of T. Obviously,$\eta(u)=u$for$u\in(B_{\rho}\cap\Phi^{0})\setminus\{0\}$. Thus,$\eta$is retraction of$B_{\rho}\setminus\{0\}$to$(B_{\rho}\cap\Phi^{0})\setminus\{0\}$. Since$W_{0}^{1,p}(\Omega)$is infinite-dimensional,$B_{\rho}\setminus\{0\}$is contractible in itself. By the fact that retracts of contractible space are also contractible,$(B_{\rho}\cap\Phi^{0})\setminus\{0\}$is contractible in itself. From the homology exact sequence, one deduces $$H_{q}(B_{\rho}\cap\Phi^{0},(B_{\rho}\cap\Phi^{0})\setminus\{0\})=0, \quad \forall q\in\mathbb{Z}.$$ Hence $$C_{q}(\Phi,0)=H_{q}(B_{\rho}\cap\Phi^{0},(B_{\rho}\cap\Phi^{0})\setminus\{0\})= 0,\quad \forall q\in\mathbb{Z}.$$ The proof of theorem \ref{thm2.1} is completed. \end{proof} Recall that we have the following Morse relation between the critical groups and homological characterization of sublevel sets. For details of the proof, we refer readers to \cite{ELA98,Si96} for example. \begin{theorem}\label{thm2} Suppose$\Phi\in C^{1}(X,\mathbb{R})$and satisfies$(C)$condition. If$c\in\mathbb{R}$is an isolated critical value of$\Phi$, with$K_{c}=\{u_{j}\}^{n}_{j=1}$, then, for every$\varepsilon>0$sufficiently small, we have $$H_{q}(\Phi^{c+\epsilon},\Phi^{c-\epsilon})=\oplus_{1\leq j\leq n} C_{q}(\Phi,u_{j}).$$ \end{theorem} \begin{remark}\label{rmk2.1} \rm >From theorem \ref{thm2} follows that if$H_{q}(\Phi^{c+\epsilon},\Phi^{c-\epsilon})$is nontrivial for some q, then there exists a critical point$u\in K_{c}$with$C_{q}(\Phi,u)\ncong 0.$Furthermore, when$C_{q}(\Phi,0)\cong 0$for all$q$, we get that$u\neq 0.$\end{remark} We will use the following theorem, which is proved with (PS) condition see for example \cite{MW89}. \begin{theorem}\label{thm3} Assume that$\Phi\in C^{1}(X,\mathbb{R})$, there exists$u_{1}\in X, u_{2}\in X$and a bounded open neighborhood$\Omega$of$u_{0}$such that$u_{1}\in X\backslash\overline{\Omega}$and $$\inf_{\partial\Omega} \Phi>\max(\Phi(u_{0}),\Phi(u_{1})).$$ Let$\Gamma=\{g\in C([0,1],X): g(0)=u_{0},g(1)=u_{1}\}$and $$c=\inf_{g\in\Gamma}\max_{s\in[0,1]}\Phi(g(s)).$$ If$\Phi$satisfies the$(C)$condition over X and if each critical point of$\Phi$in$K_{c}$is isolated in X, then there exists$u\in K_{c}$such that$\dim C_{1}(\Phi,u)\geq 1.$\end{theorem} \begin{proof} Let$\varepsilon>0$be such that$c-\varepsilon>\max(\Phi(u_{0}),\Phi(u_{1}))$and c is the only critical value of$\Phi$in$[c-\varepsilon,c+\varepsilon]$. Consider the exact sequence $$\dots\to H_{1}(\Phi^{c+\epsilon},\Phi^{c-\epsilon})\overset{\partial}{\to} H_{0}(\Phi^{c-\epsilon},\emptyset)\overset{i_*}{\to} H_{0}(\Phi^{c+\epsilon},\emptyset)\to \dots$$ where$\partial$is the boundary homomorphism and$i_{*}$is induced by the inclusion mapping$i:(\Phi^{c-\epsilon},\emptyset)\to (\Phi^{c+\epsilon},\emptyset).$The definition of$c$implies that$u_{0}$and$u_{1}$are path connected in$\Phi^{c+\varepsilon}$but not in$\Phi^{c-\epsilon}$. Thus,$\ker i_{*}\neq \{0\}$\cite{Ch93,MW89} and, by exactness,$H_{1}(\Phi^{c+\epsilon},\Phi^{c-\epsilon})\neq\{0\}$. It follows from theorem \ref{thm2} that$\dim C_{1}(\Phi,u)\geq 1.$\end{proof} \section{Proof of main result} The proof is based on the following minimax theorem due to the second author \cite[Theorem 3.5]{ELA05} ), with Cerami condition. For this, we recall the Krasnoselskii genus. Define the class of closed symmetric subsets of X as $$\Sigma=\{A\subset X: A\text{ is closed and } A=-A\}.$$ \begin{definition}\label{def3.1} For a non empty set$A$in$\Sigma$, following Coffman \cite{Coff69}, we define the Krasnoselskii genus as $\gamma(A)=\begin{cases} \inf\{m:\exists h\in C(A,\mathbb{R}^{m}\backslash\{0\});h(-x)=-h(x)\}, \\ \infty, \text{ if \{\dots\} is empty, in particular if } 0\in A. \end{cases}$ For$A$empty we define$\gamma(A)=0.$\end{definition} Note that$A_{k}=\{C\in\Sigma: C \text{ is compact }, \gamma(C)\geq k\}$. \begin{theorem}\label{el} Let$\Phi$be a$C^{1}$functional on$X$satisfying$(C)$, let$Q$be a closed connected subset of$X$such that$\partial Q\cap\partial (-Q)\neq \emptyset$and$\beta\in\mathbb{R}$. Assume that \begin{enumerate} \item for every$K\in\mathcal{A}_{2}$, there exists$v_{K}$such that$\Phi(v_{K})\geq \beta$and$\Phi(-v_{K})\geq \beta$, \item$a=\displaystyle\sup_{\partial Q} \Phi< \beta$, \item$\displaystyle\sup_{\partial Q} \Phi< \infty$. \end{enumerate} Then$\Phi$has a critical value$c\geq \beta$given by $$c=\inf_{h\in \Gamma}\sup_{x\in Q}\Phi(h(x)),$$ where$\Gamma=\{h\in C(X,X): h(x)=x \text {for every } x\in \partial Q\}$. \end{theorem} We will establish the compactness condition under the conditions$(F0)$,$(F3)$and$(F1)$. The proof is similar for$(F0)$,$(F3)$and$(F2)$. \begin{lemma}\label{lem3.1} Assume$(F0)$,$(F3)$and$(F1)$. Then$\Phi$satisfies the condition(C). \end{lemma} \begin{proof}$(i)$First, we verify that the Palais- Small condition is satisfied on the bounded subsets of$W^{1,p}_{0}(\Omega)$. Let$(u_{n})\subset W^{1,p}_{0}(\Omega)$be bounded such that $$\label{e3.1} \Phi'(u_{n})\to 0 \quad\text{and}\quad \Phi(u_{n})\to c, \quad c\in\mathbb{R}.$$ Passing if necessary to a subsequence, we may assume that $$\begin{gathered}\label{e3.2} u_{n}\rightharpoonup u \quad\text{ weakly in } W^{1,p}_{0}(\Omega), \\ u_{n}\to u \quad\text{ strongly in } L^{p}(\Omega), \\ u_{n}(x)\to u(x) \quad\text{ a.e.in } \Omega. \end{gathered}$$ From \eqref{e3.1} and \eqref{e3.2}, we have$\langle\Phi'(u_{n}),u_{n}-u\rangle\to 0,$or equivalently $$\label{e3.3} \int_{\Omega}|\nabla u|^{p}\nabla u_{n}\nabla (u_{n}-u)dx -\int_{\Omega}f(x,u_{n})(u_{n}-u)dx\to 0.$$ Applying the H\"{o}lder inequality, we deduce that $$\label{e3.4} \int_{\Omega}f(x,u_{n})(u_{n}-u)dx\to 0.$$ Thus, it follows from \eqref{e3.3} and \eqref{e3.4} that$\langle-\Delta_{p}u_{n},u_{n}-u\rangle\to 0$. Since,$-\Delta_{p}$is of type$S^{+}$(see \cite{BM88}), we conclude that $$u_{n}\to u \quad\text{ strongly in } W^{1,p}_{0}(\Omega).$$ Now, by contradiction, we will show that$(ii)$is satisfied for every$c\in\mathbb{R}$. Let$c\in\mathbb{R}$and$(u_{n})\subset W^{1,p}_{0}(\Omega)$such that $$\label{e3.5} \Phi(u_{n})\to c, \quad \langle\Phi'(u_{n}),u_{n}\rangle \to 0 \quad \text{and } \|u_{n}\|\to +\infty.$$ Therefore, $$\label{e3.6} \lim_{n}\int_{\Omega}g(x,u_{n})dx=pc,$$ where$g(x,u_{n})=u_{n}f(x,u_{n})-pF(x,u_{n})$. Taking$v_{n}=\frac{u_{n}}{\|u_{n}\|}$, clearly$v_{n}$is bounded in$W^{1,p}_{0}(\Omega)$. So, there is a function$v\in W^{1,p}_{0}(\Omega)$and a subsequence still denote by$(u_n)$such that $$\begin{gathered}\label{e3.7} v_{n}\rightharpoonup v \quad \text{weakly in } W^{1,p}_{0}(\Omega), \\ v_{n}\to v \quad \text{ strongly in } L^{p}(\Omega), \\ v_{n}(x)\to v(x) \quad \text{ a.e.in } \Omega. \end{gathered}$$ On the other hand, in view$(F0)$and$(F3)$, it follows that $$\label{e3.8} F(x,s)\leq \frac{\lambda_2}{p}|s|^{p}+b,\quad \forall s\in\mathbb{R},\quad b\in L^{p}(\Omega).$$ Combining relations \eqref{e3.5} and \eqref{e3.8}, we obtain $$\frac{1}{p}\|u_{n}\|^{p}-\frac{\lambda_2}{p}\|u_{n}\|^{p}_{L^{p}}-b\leq C, \quad C\in \mathbb{R}.$$ Dividing by$\|u_{n}\|$and passing to the limit, we conclude $$\frac{1}{p}-\frac{\lambda_2}{p}\|v\|^{p}_{L^{p}}\leq 0,$$ and consequently$v\neq 0.$Let$\Omega_{0}=\{x\in\Omega : v(x)\neq 0\}$, via the result above we have$|\Omega_{0}|>0$and $$\label{e3.9} |u_{n}(x)|\to +\infty, \quad \text{a.e. } x\in\Omega_{0}.$$ Furthermore,$(F0)$and$(F1)$implies that there exist$M>0$and$d\in L^{1}(\Omega)$such that $$sf(x,s)-pF(x,s)\geq -M+ d(x), \quad\forall s\in\mathbb{R},\quad\text{a.e. } x\in\Omega.$$ Hence, $$\int_{\Omega}g(x,u_{n})dx\geq \int_{\Omega_{0}} g(x,u_{n})dx-M|\Omega\setminus\Omega_0|-\|d\|_{L^{1}}.$$ Using \eqref{e3.9} and Fatou's lemma, one deduces $$\lim_{n}\int_{\Omega}g(x,u_{n})d=+\infty.$$ This contradicts \eqref{e3.6}. \end{proof} Now, we will prove the geometric conditions of Theorem \ref{el}. Let denote$E(\lambda_{1})$the eigenspace associated to the eigenvalue$\lambda_{1}$. \begin{lemma}\label{lem3.2} Under the hypothesis$(F0)$,$(F3)$and$(F4)$, we have: \begin{itemize} \item[(i)]$\Phi$is anticoercive on$E(\lambda_{1})$. \item[(ii)] For all$K\in \mathcal{A}_{2}$, there exists$v_{K}\in K$and$\beta \in \mathbb{R}$such that$\Phi(v_{K})\geq \beta$and$\Phi(-v_{K})\geq \beta $. \end{itemize} \end{lemma} \begin{proof} (i) For each$v\in E(\lambda_{1})$, there exist$t\in\mathbb{R}$such that$v=t\varphi_1$. Therefore, using$(F4), we write \begin{align*} \Phi(v)&=\frac{|t|^p}{p}\int_{\Omega}|\nabla \varphi_1|^pdx-\int_{\Omega}F(x,t\varphi_1)dx\\ &=-[\int_{\Omega}F(x,t\varphi_1)dx-\frac{|t|^p}{p}]\to -\infty, \quad \text{ as }|t|\to\infty. \end{align*} ii) By the Lusternik-Schnirelaman theory, we write $$\lambda_{2}=\inf_{K\in\mathcal{A}_{2}}\sup\{\int_{\Omega}|\nabla u|^{p}dx, \int_{\Omega}|u|^{p}dx=1 \text{ and }u\in K\}.$$ Then, for allK\in\mathcal{A}_{2}$, and all$\varepsilon>0$, there exists$v_{K}\in K$such that $$\label{e3.10} (\lambda_{2}-\varepsilon)\int_{\Omega}|v_{K}|^{p}dx\leq \int_{\Omega}|\nabla v_{K}|^{p}dx.$$ Indeed, if$0 \in K$, we take$v_{K}=0$.\\ Otherwise, we consider the odd mapping $$g:K\to K', v\mapsto \frac{v}{\|v\|_{L^{p}}}.$$ By the genus properties, we have$\gamma(g(K))\geq 2$, and by the definition of$\lambda_{2},$there exist$w_{K}\in K'$such that $$\int|w_{K}|^{p}dx=1 \quad \text{ and } \quad (\lambda_{2}-\varepsilon)\leq \int_{\Omega}|\nabla w_{K}|^{p}dx.$$ Thus \eqref{e3.10} is satisfied by setting$v_{K}=g^{-1}(w_{K})$. On the other hand, the two assumptions$(F0)$and$(F4)$implies $$\label{e3.11} F(x,s)\leq (\frac{\lambda_{2}-2\varepsilon}{p})|s|^{p}+C, \quad \forall s\in\mathbb{R},$$ for some constant$C>0. Consequently, one deduces from \eqref{e3.10} and \eqref{e3.11} that \label{e3.12} \begin{aligned} \Phi(w_{K})&\geq \frac{1}{p}\int_{\Omega}|\nabla w_{K}|^{p}dx-(\frac{\lambda_{2}-2\varepsilon}{p})\int_{\Omega}|w_{K}|^{p}dx- C|\Omega|\\ &\geq \frac{1}{p} (1-\frac{\lambda_{2}-2\varepsilon}{\lambda_{2}-\varepsilon})\int_{\Omega}|\nabla w_{K}|^{p}dx- C|\Omega|. \end{aligned} The argument is similar for $$\label{e3.13} \Phi(-w_{K})\geq \frac{1}{p} (1-\frac{\lambda_{2}-2\varepsilon}{\lambda_{2}-\varepsilon})\int_{\Omega}|\nabla w_{K}|^{p}dx- C|\Omega|.$$ Finally, for everyK\in\mathcal{A}_{2}$, we have$\Phi(\pm w_{K})\geq \beta:= -C|\Omega|$, which completes the proof. \end{proof} \begin{proof}[Proof of theorem \ref{thm1.1}] Putting$Q=\{t\varphi_{1}: |t|\leq R\}$for$R>0$, clearly,$Q$is closed and compact. In view of lemma \ref{lem3.1}, we can find$t_{0}>0$such that$\Phi(\pm t_{0}\varphi_{1})<\beta$. In return for lemma \ref{lem3.2}, we may apply Theorem \ref{el} to get that$\Phi$has a critical value given by $$c=\inf_{h\in \Gamma}\sup_{x\in Q}\Phi(h(x))\geq\beta,$$ where$\Gamma=\{h\in C([0,1],W^{1,p}_{0}(\Omega)): h(0)=-t_{0}\varphi_{1},h(1)=t_{0}\varphi_{1}\}$. Therefore, there exists at least one critical point$u^{*}$of$\Phi$. More precisely,$u^{*}$is a Mountain Pass type. However, by theorem \ref{thm3}, we have$C_{1}(\Phi,u^{*})\ncong 0$. Using theorem \ref{thm2.1}, one deduces$u^{*}\neq 0$. \end{proof} \begin{thebibliography}{00} \bibitem{ANT96} A.Anane and N.Tsouli, \emph{On the second eigenvalue of the p-Laplacien}, {Nonlinear Partiel Differential Equations}, Pitmen research Notes 343(1996), 1-9. \bibitem{BM88} J.Berkovits and V.Mustonen, \emph{Nonlinear mappings of monotone type(classification and degree theory)}, Math. Univer. 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