\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 237, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/237\hfil Multiple positive solutions] {Multiple positive solutions for quasilinear elliptic equations of $p(x)$-Laplacian type with sign-changing nonlinearity} \author[K. Ho, C.-G. Kim, I. Sim \hfil EJDE-2014/237\hfilneg] {Ky Ho, Chan-Gyun Kim, Inbo Sim} % in alphabetical order \address{Ky Ho \newline Department of Mathematics, University of Ulsan, Ulsan 680-749, Korea} \email{hnky81@gmail.com} \address{Chan-Gyun Kim \newline Department of Mathematics Education, Pusan National University, Busan 609-735, Korea} \email{cgkim75@gmail.com} \address{Inbo Sim \newline Department of Mathematics, University of Ulsan, Ulsan 680-749, Korea} \email{ibsim@ulsan.ac.kr} \thanks{Submitted April 15, 2014. Published November 13, 2014.} \subjclass[2000]{35J20, 35J60, 35J70, 47J10, 46E35} \keywords{$p(x)$-Laplacian; variable exponent; sign-changing nonlinearity; \hfill\break\indent positive solutions; multiplicity} \begin{abstract} We establish sufficient conditions for the existence of multiple positive solutions to nonautonomous quasilinear elliptic equations with $p(x)$-Laplacian and sign-changing nonlinearity. For solving the Dirichlet boundary-value problem we use variational and topological methods. The nonexistence of positive solutions is also studied. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} We are concerned with the existence of multiple positive solutions for the problem \begin{equation}\label{0} \begin{gathered} -\Delta_{p(x)} u =\lambda f(x,u),\quad x \in \Omega, \\ u(x)=0,\quad x \in \partial \Omega, \end{gathered} \end{equation} where $\Delta_{p(x)} u:=\operatorname{div} (|\nabla u|^{p(x)-2}\nabla u)$ (is called $p(x)$-Laplacian), $\Omega \subset \mathbb{R}^N$ a bounded domain with smooth boundary $\partial \Omega$ for $N\ge 1$, $p \in C^1(\overline \Omega)$ with $p(x)>1$ for all $x \in \overline \Omega$, $f \in C(\overline \Omega \times \mathbb{R}, \mathbb{R})$, and $\lambda$ is a positive parameter. The problems related to the $p(x)$-Laplacian have been intensively studied. We refer the reader to \cite{Ruzicka} for motivations from electrorheological fluids, and to \cite{Edmunds1, Edmunds2, Fan1, Fan2, Fan3, Fan4, Fan5, Kovacik} for basic definitions, properties, and standard results associated with the $p(x)$-Laplacian and the variable exponent Lebesgue-Sobolev space. As far as the authors know, most studies are related to the positive nonlinearity $f(x,u)$, and very few are related to the existence of positive solutions for the sign-changing nonlinearity. Throughout this article, unless otherwise stated, we assume that for $k, l, m \in \mathbb{N}$ and $m \geq 2$. We use the following assumptions: \begin{itemize} \item[(F1)] $f(x,0)\ge 0$ for all $x \in \overline \Omega$; \item[(F2)] there exist $a_k,b_l \in C(\overline \Omega)$ and positive constants $c_l$, where $1\le k\le m,~1\le l\le m-1$ such that $$ 0 \le a_1(x) < c_1 \le b_1(x)0$; \item[(F4k)] $k \in \{2,\dots,m\}$, $a_{k} \in C^1(\overline \Omega)$, $\int_{\Omega}\alpha_k(x)\,dx>0$, where $$ \alpha_k(x):=F(x,a_{k}(x))-\max \{F(x,s): 0\leq s \leq a_{k-1}(x),\; x\in \overline{\Omega}\}, $$ where $F(x,s):=\int_0^s f(x,\tau)d\tau$ for $(x,s) \in \Omega \times \mathbb{R}$. \end{itemize} In spite of the fact that (F3) implies (F1), the reason we assumed (F1) is to compare the conditions which the researchers mentioned below used. Indeed let us briefly review the previous conditions and results which are related to \eqref{0}. When $p(x) \equiv 2$, that is, for the Laplacian case, Hess \cite{Hess} initiated the study about sufficient conditions for sign-changing nonlinearity to get at least $2m-1$ positive solutions for sufficiently large $\lambda$. Actually, his conditions was $f(x,u) = f(u)$ and $f \in C^1([0,\infty),\mathbb{R})$ with $f(0)>0$ and (F2) and (F4k) with $a_k,b_l$ constants. It is worth noting that if $f\in C^1([0,\infty),\mathbb{R})$ and $f(0)>0$ then (F3) holds automatically. The $p$-Laplacian version was established by Loc-Schmitt \cite{Loc} with $f(0) \geq 0$ (not $f(0)>0$), Hess' assumptions, and some different condition from (F4k). They only showed the existence of at least $m-1$ non-negative solutions but also discussed the necessary conditions. We emphasize that non-negativity of solutions comes from $f(0) \geq 0$ (see, Proposition~\ref{maximum} and Remark~\ref{rem3.100}). Let us note that in the above two papers the nonlinearity was autonomous. For the nonautonomous case, when $p(x) \equiv p, m=2$, Kim-Shi \cite{Kim} showed that \eqref{0} has at least two positive solutions for sufficiently large $\lambda$, under the assumptions $f(x,a_1(x))=0$, (F2), (F3) and a condition weaker than (F4k), with $k=2$, \begin{itemize} \item[(F5)] there exists an open ball $B_1$ of $\Omega$ such that $a_2\in C^1(\overline B_1)$ and $$ F(x,a_2(x))>0,\quad x \in B_1. $$ \end{itemize} They also showed the nonexistence of positive solutions of \eqref{0} for sufficiently small $\lambda$. Motivated by the above results, we shall consider the case of $p(x)$-Laplacian, $m \geq 2$ and sign-changing nonautonomous nonlinearity which are weaker than conditions of Hess, Loc-Schmitt and Kim-Shi and obtain some results which contain their results as a special case in a unified way. \section{Preliminaries} In this section we establish a basic setup and some preliminary results concerning the $p(x)$-Laplacian problems. Let $C_+(\overline \Omega):= \{ h \in C(\overline \Omega) : h(x)>1$ for all $x \in \overline \Omega\}$, and for $h \in C_+(\overline \Omega)$, we denote $h^+=\max_{\overline \Omega} h(x)$ and $h^-=\min_{\overline \Omega} h(x)$. For any $p \in C_+(\overline \Omega)$, we define the variable exponent Lebesgue space by $L^{p(x)}(\Omega):=\{u : u$ is a measurable real valued function, $\int_\Omega |u(x)|^{p(x)}\,dx <\infty\}$ with the norm $$ \|u\|_{p(x)}=\inf \big\{ \lambda>0 : \int_{\Omega} |\frac{u(x)}{\lambda}|^{p(x)}\,dx \le 1 \big\}. $$ The space $(L^{p(x)}(\Omega),\|\cdot\|_{p(x)})$ is a separable, uniformly convex Banach space, and its conjugate space is $L^{q(x)}(\Omega)$, where $1/p(x)+1/q(x)=1$ for all $x \in \overline \Omega$. The variable exponent Sobolev space $W^{1,p(x)}(\Omega)$ is defined by $$ W^{1,p(x)}(\Omega) = \{ u \in L^{p(x)}(\Omega) : |\nabla u| \in L^{p(x)}(\Omega)\} $$ with the norm $$ \|u\|_1 = \|u\|_{p(x)}+\||\nabla u|\|_{p(x)}. $$ We denote by $W_0^{1,p(x)}(\Omega)$ the closure of $C_0^\infty(\Omega)$ in $W^{1,p(x)}(\Omega)$. Then $W^{1,p(x)}(\Omega)$ and $W_0^{1,p(x)}(\Omega)$ are separable reflexive Banach spaces. Moreover, we have the compact imbedding $W^{1,p(x)}(\Omega) \hookrightarrow\hookrightarrow L^{q(x)}(\Omega)$ if $q \in C_+(\overline \Omega)$ with $q(x) < p^*(x)$ for all $x \in \overline \Omega$, where $$ p^*(x)= \begin{cases} \frac{Np(x)}{N-p(x)}, &p(x)0$ in $\Omega$, and when $u\in C^1(\overline{\Omega}),\partial u/ \partial \nu <0$ on $\partial \Omega$ where $\nu$ is the outward unit normal on $\partial \Omega$. \end{proposition} The following lemma gives estimates for a solution of $p(x)$-Laplacian which has a cut-off type nonlinear term. \begin{lemma}\label{Lem1} Let $g:\Omega \times \mathbb{R} \to \mathbb{R}$ be a continuous function such that there exists $\bar s>0$ such that $g(x,s) \ge 0$ if $(x,s) \in \Omega \times (-\infty,0]$ and $g(x,s) \le 0$ if $(x,s) \in \Omega \times [\bar s, \infty)$. If $u$ is a weak solution to problem %\label{150} \begin{gather*} -\Delta_{p(x)} u=g(x,u),\quad x \in \Omega ,\\ u(x)=0,\quad x \in \partial \Omega, \end{gather*} then $0 \le u(x) \le \bar s$ for almost all $x \in \overline \Omega$. \end{lemma} \begin{proof} Putting $\phi =(u-\bar s)^+=\max\{u-\bar s,0\}\in W_0^{1,p(x)}(\Omega)$, we have $$ \int_\Omega |\nabla u|^{p(x)-2} \nabla u \cdot \nabla \phi \,dx =\int_{\{u(x)>\bar s\}} g(x,u(x))\phi \,dx \le 0. $$ Since $$ \int_\Omega |\nabla u|^{p(x)-2} \nabla u \cdot \nabla \phi \,dx =\int_\Omega |\nabla (u-\bar s)^+|^{p(x)} \,dx\ge0, $$ $\nabla(u-\bar s)^+=0$ a.e. in $\Omega$, and thus $u\le \bar s$. In a similar manner, taking $\phi=\max\{-u,0\}\in W_0^{1,p(x)}(\Omega)$, we have $u\ge 0$ almost all $x \in \overline \Omega$. The proof is complete. \end{proof} \section{Main results} In this section, we state the main theorems and compare the conditions and results in \cite{Hess, Loc, Kim}. First, for any $\lambda\ge0$, we define the functional $I(\lambda,\cdot) : W_0^{1,p(x)}(\Omega) \to \mathbb{R}$ by $$ I(\lambda,u):=\int_\Omega \frac{1}{p(x)}|\nabla u(x)|^{p(x)} \,dx -\lambda\int_\Omega F(x,u(x))\,dx,\quad u \in W_0^{1,p(x)}(\Omega). $$ \begin{theorem}\label{main1} Assume that {\rm (F2), (F3), (F4k)} (with $k=2,\dots,m$) hold. Then, for sufficiently large $\lambda>0$, \eqref{0} has at least $m$ solutions $u_1(\lambda), \dots,u_m(\lambda)$ in which $u_1(\lambda)$ is a non-negative solution and $u_2(\lambda),\dots, u_m(\lambda)$ are positive solutions such that $0\leq \|u_1(\lambda)\|_\infty \leq c_1<\|u_2(\lambda)\|_\infty \leq c_2<\dots0$, equation \eqref{0} has other $m-1$ positive solutions $\hat u_2(\lambda),\dots, \hat u_{m}(\lambda)$ such that $\|\hat u_k(\lambda)\|_\infty \in (c_{k-1},c_{k})$ and $\hat u_k(\lambda)\neq u_k(\lambda)$ for $k=2,\dots,m$. \end{theorem} \begin{remark}\label{rem3.3} \rm Since the existence of $L$ in (F6) is guaranteed, when $f \in C^1$, Theorem \ref{main3} is just Hess' conclusion. \end{remark} We have a similar result even in the case that we replace (F4k), with $k=2$, by the weaker condition (F5). \begin{theorem}\label{main5} Assume that {\rm (F2), (F3), (F5)} for $m=2$, or {\rm (F2), (F3), (F5), (F4k)} (with $k=3,\dots,m$), for $m \ge 3$ hold. Then, for sufficiently large $\lambda>0$, \eqref{0} has at least $m-1$ positive solutions $u_2(\lambda), \dots,u_m(\lambda)$ such that $\|u_k(\lambda)\|_\infty \in (c_{k-1},c_k]$ and $I(\lambda,u_m(\lambda))<\dots0$ for sufficiently large $\lambda>0$. \end{theorem} \begin{remark}\label{rem3.3b} \rm This theorem extends Kim-Shi's result of $p$-Laplacian into the case of $p(x)$-Laplacian with more humps (for this terminology, see \cite{Hess}). \end{remark} For the nonexistence result we need only a simple assumption. \begin{theorem}\label{main8} Assume that there exists positive constants $C_1$ and $C_2$ such that $f(x,s) \le 0$ for all $(x,s) \in \Omega \times ((0,C_1) \cup (C_2,\infty))$. Then \eqref{0} has no positive solutions for small $\lambda>0$. \end{theorem} \begin{remark}\label{rem3.4} \rm The property of the first eigenvalue of $p$-Laplacian problem and Picone's identity were used in \cite{Kim}, but both are not expected in $p(x)$-Laplacian problem. \end{remark} By Theorems \ref{main5}, \ref{main6} and \ref{main8}, we have the following corollary. \begin{corollary}\label{co} Assume that {\rm (F2), (F3), (F5), (F7)} for $m=2$, or {\rm (F2), (F3), (F5), (F4k)} (with $k=3,\dots,m)$, {\rm (F7)} for $m\ge3$ hold. If $f(x,s)$ satisfies $f(x,s) \le 0$ for $(x,s) \in \Omega \times [c_m,\infty)$, then problem \eqref{0} has at least $m$ positive solutions for sufficiently large $\lambda$, and it has no positive solutions for small $\lambda>0$. Moreover, if we also assume that $(F6)$ holds, then problem \eqref{0} has at least $2m-2$ positive solutions for sufficiently large $\lambda$. \end{corollary} \section{Lemmas} For each $k=1,2,\dots,m$, let us consider the truncation of the nonlinearity $f(x,s)$ as follows; $$ f_k(x,s):= \begin{cases} f(x,0), &(x,s) \in \overline \Omega \times (-\infty,0],\\ f(x,s), &(x,s) \in \overline \Omega \times (0,c_{k}],\\ f(x,c_{k}), &(x,s) \in \overline \Omega \times (c_{k},\infty). \end{cases} $$ Then $f_k(x,s)\ge0$ for $(x,s) \in \overline \Omega \times (-\infty,0]$ and $f_k(x,s)\le0$ for $(x,s) \in \overline \Omega \times [c_{k},\infty)$. For any $\lambda\ge0$, we define the functional $I_k(\lambda,\cdot) : W_0^{1,p(x)}(\Omega) \to \mathbb{R}$ by $$ I_k(\lambda,u):=\int_\Omega \frac{1}{p(x)}|\nabla u(x)|^{p(x)} \,dx -\lambda\int_\Omega F_k(x,u(x))\,dx,\quad u \in W_0^{1,p(x)}(\Omega), $$ where $F_k(x,s):= \int_0^s f_k(x,\tau)d\tau$ for $(x,s) \in \Omega \times \mathbb{R}$. \begin{lemma}\label{PS} Assume that $f \in C(\overline \Omega \times \mathbb{R}, \mathbb{R})$. Then $I_k(\lambda,\cdot)$ is continuously Fr\'echet differentiable on $W_0^{1,p(x)}(\Omega)$, and $I'_k(\lambda,\cdot)$ is of $(S_+)$ type operator. Moreover $I_k(\lambda,\cdot)$ is sequentially weakly lower-semicontinuous, coercive on $W_0^{1,p(x)}(\Omega)$ and satisfies the Palais-Smale condition. \end{lemma} \begin{proof} Let $I_k(\lambda,\cdot)=J-\lambda J_k$, where $J(u)=\int_\Omega \frac{1}{p(x)}|\nabla u(x)|^{p(x)} \,dx$ and $J_k(u)=\int_\Omega F_k(x,u(x))\,dx$. Since $f_k(x,s)$ is bounded, it is well known that $I_k(\lambda,\cdot)$ is continuously Fr\'echet differentiable, sequentially weakly lower-semicontinuous and coercive on $W_0^{1,p(x)}(\Omega)$ (see, e.g., \cite{Fan2}). The $(S_+)$-property of $I'_k(\lambda,\cdot)$ comes from $(S_+)$-property of $J'$ (see \cite{Fan2}) and the sequentially weak continuity of $J'_k$. Since $I'_k(\lambda,\cdot)$ is of $(S_+)$ type operator, to show that $I_k(\lambda,\cdot)$ satisfies $(PS)$ condition it is enough to show every $(PS)$ sequence is bounded. Let $\{u_n\}_{n=1}^\infty$ be any $(PS)$ sequence of $I_k(\lambda,\cdot)$ in $W_0^{1,p(x)}(\Omega)$; i.e., there exists a constant $M>0$ such that $|I_k(\lambda,u_n)|\leq M$, for all $n$ and $I_k'(\lambda,u_n) \to 0$ as $n \to \infty$. It follows from the boundedness of $f_k$ and the relation between modular and norm (see \cite[Theorem 1.3]{Fan1}) that for $n$ large, we have \begin{align*} M+\|u_n\| &\geq I_k(\lambda,u_n)-\frac{1}{2p^+}I'_k(\lambda,u_n)u_n \\ &\geq \frac{1}{2p^{+}}\big(\|u_n \|^{p^{-}}-1\big)-C\int_{\Omega}|u_n|\,dx\\ &\geq \frac{1}{2p^{+}}\|u_n \|^{p^{-}}-CC_1\|u_n\|-\frac{1}{2p^{+}}, \end{align*} where $C$ is some positive constant and $C_1$ is the imbedding constant for $\|u_n\|_{L^1(\Omega)}\leq C_1\|u_n\|$. Thus $\{u_n\}_{n=1}^\infty$ is bounded in $W_0^{1,p(x)}(\Omega)$ since $p^->1$. \end{proof} \begin{lemma}\label{regularity} Assume that {\rm (F1), (F2)} hold. Let $u$ be any critical point of $I_k(\lambda,\cdot)$ for some $k \in \{ 1,2,\dots,m\}$. Then $u \in C_0^{1,\alpha}(\overline \Omega)$ for some $\alpha \in (0,1)$ and $0 \le u(x) \le c_{k}$ for all $x \in \Omega$. Assume in addition that {\rm (F3)} holds, then $u>0$ in $\Omega$ and $\partial u/ \partial \nu <0$ on $\partial \Omega$ if $u \not \equiv 0$ in $\Omega$, where $\nu$ is the outward unit normal on $\partial \Omega$. \end{lemma} \begin{proof} Let $u$ be any critical point of $I_k(\lambda,\cdot)$. By Lemma \ref{Lem1}, $0 \le u(x) \le c_{k}$ for a.e $x \in \Omega$. Since $u$ is a nonnegative bounded solution of \eqref{0}, $u \in C_0^{1,\alpha}(\overline \Omega)$ for some $\alpha \in (0,1)$ in view of $C^{1,\alpha}$-regularity result in the Proposition~\ref{reg}. Hence, $0 \le u(x) \le c_{k}$ for all $x \in \Omega$. Assume in addition that (F3) is satisfied, it follows from Proposition~\ref{maximum} that $u>0$ in $\Omega$ and $\partial u/ \partial \nu <0$ on $\partial \Omega$ if $u \not \equiv 0$ in $\Omega$. \end{proof} Fix $k \in \{1,\dots, m\}$ and denote by $\mathcal{C}_k(\lambda)$ the set of critical points of $I_k(\lambda,\cdot)$. Note that $u \in \mathcal{C}_k(\lambda)$ if and only if $u$ is a solution of \begin{equation}\label{10} \begin{gathered} -\Delta_{p(x)}u = \lambda f_k(x,u),\quad x \in \Omega, \\ u=0, \quad x \in \partial \Omega. \end{gathered} \end{equation} Since $I_k(\lambda,\cdot)$ is sequentially weakly lower-semicontinuous and coercive on the space $W_0^{1,p(x)}(\Omega)$, it follows that $I_k(\lambda,\cdot)$ has a global minimizer $u_k(\lambda) \in \mathcal{C}_k(\lambda)$ for any $\lambda>0$. \begin{lemma}\label{lem-124} Assume that {\rm (F1), (F2), (F5)} hold. Then there exists $\lambda_2>0$ such that for all $\lambda>\lambda_2$, $$ I(\lambda,u_2(\lambda))<0. $$ \end{lemma} \begin{proof} We shall show that, for large $\lambda$, there exists $v \in W_0^{1,p(x)}(\Omega)$ such that $0\le v(x) \le a_2(x)$ for all $x \in \Omega$ and $I(\lambda,v) <0=I(\lambda,0)$, which implies that $I(\lambda,u_2(\lambda))<0$. Let us define $v_\epsilon(x)$ for small $\epsilon>0$ and $B_1$ in $(F5)$ as follows: $$ v_\epsilon(x):= \begin{cases} 0, & x \in \Omega \setminus B_1^\epsilon\\ a_2^\epsilon(x), & x \in B_1^\epsilon \setminus \overline B_1\\ a_2(x), & x \in B_1, \end{cases} $$ where $B_1^\epsilon:=\{x \in \Omega : \operatorname{dist} (x, B_1) \le \epsilon\}$, $a_2(x)$ is the function in (F2) and $a_2^\epsilon(x)$ is an appropriate function such that $0\le v_\epsilon(x) \le a_2(x),~x \in \Omega$ and $v_\epsilon \in C^1_0(\overline \Omega)$. Then $F_2(x,v_\epsilon(x))=F(x,v_\epsilon(x))$, $x \in \Omega$ and \begin{equation} \label{170} \begin{aligned} &I(\lambda,v_\epsilon)\\ &= \int_\Omega \frac{1}{p(x)} |\nabla v_\epsilon(x)|^{p(x)} \,dx -\lambda\int_\Omega F(x,v_\epsilon(x))\,dx \\ &= \int_\Omega \frac{1}{p(x)} |\nabla v_\epsilon(x)|^{p(x)} \,dx -\lambda\int_{B_1} F(x,a_2(x))\,dx -\lambda\int_{B_1^\epsilon \setminus \overline B_1} F(x,a_2^\epsilon(x))\,dx \\ &\leq \int_\Omega \frac{1}{p(x)} |\nabla v_\epsilon(x)|^{p(x)} \,dx -\lambda\int_{B_1} F(x,a_2(x))\,dx+\lambda M |B_1^\epsilon \setminus \overline B_1|, \end{aligned} \end{equation} where $M:=\max \{|F(x,u)| : 0\le u \le a_2(x),\; x \in \overline \Omega \}$. By (F5), $\int_{B_1} F(x,a_2(x))\,dx>0$, and we can choose a sufficiently small constant $\epsilon_0>0$ so that $$ 00$ such that for all $\lambda>\lambda_k$, $u_k(\lambda) \not \in \mathcal{C}_{k-1}(\lambda)$ and $I(\lambda,u_k(\lambda)) < I(\lambda,u_{k-1}(\lambda))$. \end{lemma} \begin{proof} It is sufficient to show that there exist $\lambda_k>0$ and $w_k \in W_0^{1,p(x)}(\Omega)$ such that $w_k\ge0$, $\|w_k\|_\infty \le c_{k}$ and \begin{equation}\label{le4.3,1} I(\lambda,w_k) < I(\lambda,u_{k-1})~\text{for all}~\lambda>\lambda_k, \end{equation} to complete the proof. We first show that for all $x\in \Omega$, $$ F(x,u_{k-1}(x))\leq \max \{F(x,s): 0\leq s \leq a_{k-1}(x),\; x\in \overline{\Omega}\}. $$ The assertion is obvious if $u_{k-1}(x)\leq a_{k-1}(x)$. For the case $a_{k-1}(x)\leq u_{k-1}(x)\leq c_{k-1}$, we obtain that $f(x,u_{k-1}(x)) \le 0$ and \begin{align*} F(x,u_{k-1}(x)) &= \int_{0}^{a_{k-1}(x)}f(x,s)ds+\int_{a_{k-1}(x)}^{u_{k-1}(x)}f(x,s)ds\\ &\leq \int_{0}^{a_{k-1}(x)}f(x,s)ds\\ &= F(x,a_{k-1}(x))\\ &\leq \max \{F(x,s): 0\leq s \leq a_{k-1}(x),~x\in \overline{\Omega}\}. \end{align*} From this inequality and (F4k) it follows that $$ F(x,a_{k}(x))\geq F(x,u_{k-1}(x))+\alpha_{k}(x), \forall x\in \Omega, $$ and hence, \begin{equation}\label{le4.3,2} \int_{\Omega}F(x,a_{k}(x))\,dx \geq \int_{\Omega}F(x,u_{k-1}(x))\,dx +\int_{\Omega}\alpha_{k}(x)\,dx. \end{equation} For $\delta>0$, let $\Omega_{\delta}:=\{x\in \Omega: \operatorname{dist}(x,\partial \Omega)<\delta\}$. Then $|\Omega_{\delta}| \to 0$ as $\delta \to 0$. For each small $\delta>0$, there exists $w_{\delta}\in W_0^{1,p(x)}(\Omega) $ such that $w_{\delta}(x)=a_{k}(x)$ for $x \in \Omega \backslash \Omega_{\delta}$ and $0\leq w_{\delta}(x) \leq a_{k}(x)$ for $x \in \Omega$. Thus \begin{align*}\label{le4.3,3} \int_{\Omega}F(x,w_{\delta}(x))\,dx &= \int_{\Omega \backslash \Omega_{\delta}}F(x,a_{k}(x))\,dx +\int_{\Omega_{\delta}}F(x,w_{\delta}(x))\,dx \\ &= \int_{\Omega}F(x,a_{k}(x))\,dx-\int_{\Omega_{\delta}}[F(x,a_{k}(x)) -F(x,w_{\delta}(x))]\,dx \\ &\geq \int_{\Omega}F(x,a_{k}(x))\,dx-C_k|\Omega_{\delta}|, \end{align*} where $C_k:=2\max \{|F(x,s)|: 0\leq s \leq a_{k}(x),\; x\in \overline{\Omega}\}$. By \eqref{le4.3,2}, $$ \int_{\Omega}F(x,w_{\delta}(x))\,dx \geq\int_{\Omega}F(x,u_{k-1}(x))\,dx+ \int_{\Omega}\alpha_{k}(x)\,dx-C_k|\Omega_{\delta}|. $$ Fixing $\delta>0$ such that $$ \eta:=\int_{\Omega}\alpha_{k}(x)\,dx-C_k|\Omega_{\delta}|>0, $$ and setting $w_k:=w_{\delta}$, we obtain \begin{align*} &I(\lambda,w_k)-I(\lambda,u_{k-1})\\ &= \int_\Omega \frac{1}{p(x)}\left(|\nabla w_k|^{p(x)}-|\nabla u|^{p(x)}\right) \,dx-\lambda\int_{\Omega}(F(x,w_k(x))-F(x,u_{k-1}(x)))\,dx\\ &\leq \int_\Omega \frac{1}{p(x)}|\nabla w_k|^{p(x)} \,dx-\lambda \eta, \end{align*} which implies that there exists $\lambda_k>0$ such that \eqref{le4.3,1} is satisfied. \end{proof} Next we shall give some results by using the degree theory for $(S_+)$ type maps in the Banach space. For the basic properties of the degree of $(S_+)$ type maps, we refer to \cite{Drabek,Motreanu}. For each $k \in \{1,2,\dots, m\}$ and $\epsilon>0$, let $\mathcal{U}_\epsilon(\mathcal{C}_k(\lambda))$ be the $\epsilon$-neighborhood of $\mathcal{C}_k(\lambda)$ in $W_0^{1,p(x)}(\Omega)$. For $m \ge 2$, $\mathcal{C}_{k-1}(\lambda)\subsetneq \mathcal{C}_k(\lambda)$ for each $k \in \{2,\dots, m\}$. By Proposition~\ref{reg}, $\mathcal{C}_k(\lambda)$ is a compact set in $W_0^{1,p(x)}(\Omega)$. Let $B_R(0)$ denote the open ball in $W_0^{1,p(x)}(\Omega)$ with radius $R>0$ and center at the origin. By the boundedness of $f_k$, for sufficiently large $R=R(\lambda)>0$, $I'_k(\lambda,u)u>0$ for any $u \in \partial B_R(0)$. Thus, by the property for the degree of $(S_+)$ type operator, we have \begin{equation}\label{deg1} \deg (I'_k(\lambda,\cdot), B_R(0),0)=1. \end{equation} By the modified arguments which were used in \cite[Lemma 3]{Hess} for the Hilbert space, we have the following lemma. \begin{lemma} Fix $k \in \{2, \dots, m\}$ and assume that {\rm (F1), (F2), (F6), (F4k)} hold. Then there exists $\epsilon_k=\epsilon_k(\lambda)>0$ such that for any $\epsilon \in (0,\epsilon_k)$, \begin{equation}\label{deg2} \deg (I'_k(\lambda,\cdot), \mathcal{U}_\epsilon(\mathcal{C}_{k-1}(\lambda)),0)=1. \end{equation} \end{lemma} \begin{proof} By \eqref{deg1} and the excision property of the degree, for any $\epsilon>0$, $$ \deg (I'_{k-1}(\lambda,\cdot), \mathcal{U}_\epsilon(\mathcal{C}_{k-1}(\lambda)),0)=1. $$ We claim that there exists $\epsilon_{k-1}>0$ such that, for all $\epsilon \in (0,\epsilon_{k-1})$ and all $\mu \in [0,1]$, $$ \mu I'_{k-1}(\lambda,v)+(1-\mu)I'_{k}(\lambda,v) \neq 0\quad \text{for } v \in \partial \mathcal{U}_\epsilon(\mathcal{C}_{k-1}(\lambda)). $$ Indeed, if the assertion were false then there are a sequences of positive numbers $\delta_n$ approaching 0, and sequences $\{\mu_n\}_{n=1}^\infty \subset [0,1]$ and $\{v_n\}_{n=1}^\infty \subset W_0^{1,p(x)}(\Omega)$ such that \begin{equation}\label{e-1092} \operatorname{dist} (v_n,\mathcal{C}_{k-1}(\lambda))=\delta_n, \end{equation} and $$ \mu_n I'_{k-1}(\lambda,v_n)+(1-\mu_n)I'_{k}(\lambda,v_n) = 0. $$ Thus $v_n$ satisfies %\label{011} \begin{gather*} -\Delta_{p(x)} v_n = \lambda (\mu_n f_{k-1}(x,v_n)+(1-\mu_n)f_k(x,v_n)), \quad x \in \Omega, \\ v_n(x)=0, \quad x \in \partial \Omega. \end{gather*} Since \begin{align*} &\mu_n f_{k-1}(x,s)+(1-\mu_n)f_k(x,s)\\ &=\begin{cases} f(x,0), &(x,s) \in \overline \Omega \times (-\infty,0],\\ f(x,s), &(x,s) \in \overline \Omega \times (0,c_{k-1}],\\ \mu_n f(x,c_{k-1})+(1-\mu_n)f(x,c_{k}) , &(x,s) \in \overline \Omega \times (c_{k},\infty), \end{cases} \end{align*} by Lemma \ref{Lem1}, $0\le v_n(x) \le c_{k}$ for a.e $x \in \Omega$ and all $n \in \mathbb{N}$, and thus by Proposition~\ref{reg}, $\{v_n\}_{n=1}^\infty$ is relatively compact in $C^{1}(\overline \Omega)$. Then, there exist a subsequence of $\{v_n\}_{n=1}^\infty$, still denote by $\{v_n\}_{n=1}^\infty$, and $v \in C^1(\overline\Omega)$ such that $v_n \to v$ in $C^1(\overline\Omega)$. It follows from \eqref{e-1092} that $v \in \mathcal{C}_{k-1}(\lambda)$. Hence, by Lemma~\ref{regularity}, $0\le v(x) \le c_{k-1}$ for all $x \in \Omega$. Next, we show that $\|v\|_\inftyc_{k-1}$, letting $n \to \infty$, we get a contradiction. Thus \eqref{deg2} holds by the homotopy invariance property of the degree. \end{proof} \section{Proofs of main results and an example} Now we give the proofs of Theorems \ref{main1}, \ref{main3}, \ref{main5}, \ref{main6} and \ref{main8}. \begin{proof}[Proof of Theorem \ref{main1}] Fix $\lambda > \max \{\lambda_k: k=2,\dots,m\}$, where $\lambda_k$ are taken as in Lemma~\ref{minimizer}. Also as in Lemma~\ref{minimizer}, denote by $u_k(\lambda)$ the global minimizer of $I_k(\lambda,\cdot)$. Then, by Lemma~\ref{regularity} and Lemma \ref{minimizer}, we have $0\leq u_k(\lambda)\leq c_k$ and \begin{gather*} 0\leq \|u_1(\lambda)\|_\infty \leq c_1<\|u_2(\lambda)\| \leq c_2<\dots\lambda_2$, and $u_2(\lambda)\not \equiv 0$. Hence, $u_2(\lambda)$ is positive by Proposition~\ref{maximum}. In the case $m\geq 3$, fix $\lambda > \max \{\lambda_k: k=2,\dots,m\}$, where $\lambda_2$ is taken as in Lemma~\ref{lem-124} whereas $\lambda_k$ ($k=3,\dots,m$) are taken as in Lemma~\ref{minimizer}. Using the same argument as in the proof of Theorem \ref{main1} with noting that $I_2(\lambda,u_2(\lambda))<0$, it follows that problem \eqref{0} has $m-1$ positive solutions $u_2(\lambda),\dots, u_{m}(\lambda)$ such that $\|u_k(\lambda)\|_\infty \in (c_{k-1},c_{k}]$ and $I(\lambda,u_k(\lambda))<0$ for $k \in \{2,\dots,m\}$. If we assume in addition that (F6) holds, then by the same argument as in the proof of Theorem \ref{main3}, there exists other $m-2$ positive solutions $\hat u_3(\lambda),\dots, \hat u_{m}(\lambda)$ such that $\|\hat u_k(\lambda)\|_\infty \in (c_{k-1},c_{k})$ and $\hat u_k(\lambda)\neq u_k(\lambda)$ for $k \in \{3,\dots,m\}$. \end{proof} \begin{proof}[Proof of Theorem \ref{main6}] Since $p^+ < p^*(x)$ for all $x\in \overline \Omega$, we can choose a constant $q$ such that $q \in (p^+,p^*(x))$ for all $x \in \overline \Omega$. From the fact that $a_1(x) =0$ for all $x \in \Omega$, there exists a constant $C(q)>0$ such that \begin{gather*} %\label{growth2} f_2(x,s) \le C(q) |s|^{q-1},\quad (x,s) \in \Omega \times \mathbb{R},\\ F_2(x,s) \le C(q)\frac{|s|^{q}}{q},\quad (x,s) \in \Omega \times \mathbb{R}. \end{gather*} Let $0<\delta<\min\{1,1/C_q\}$, where $C_q$ is the imbedding constant such that $\|u\|_{q}\leq C_q\|u\|$ for $u \in W_0^{1,p(x)}(\Omega)$. For $\|u\| < \delta$, we estimate \begin{align*} I_2(\lambda,u) &\geq \int_\Omega \frac{1}{p(x)} |\nabla u(x)|^{p(x)} \,dx -\lambda \frac{C(q)}{q}\int_\Omega |u(x)|^{q}\,dx\\ &\geq \big[\frac{1}{p^+}-\lambda \frac{C(q)C_q^q}{q} \|u\|^{q-p^+}\big]\|u\|^{p^+}. \end{align*} Thus, for each $\lambda >0$, there exists $\rho\in (0,\delta)$ such that $I_2(\lambda,u) > 0= I_2(\lambda,0)$ if $0<\|u\| \le \rho$. Fix $\lambda >0$ such that $I_2(\lambda,u_2(\lambda))<0$. It follows from Mountain pass Theorem that $I_2(\lambda,\cdot)$ has another critical point $\hat u_1$ such that $$ I_2(\lambda,\hat u_1(\lambda)) >0 >I_2(\lambda,u_2(\lambda)), $$ and thus, for sufficiently large $\lambda$, problem \eqref{0} has other positive solution $\hat u_1(\lambda)$, which is different from $2m-3$ positive solutions $u_2,\dots,u_m,\hat{u}_3,\dots,\hat{u}_m$ obtained in Theorem ~\ref{main5}, satisfying $\|\hat u_1(\lambda)\|_\infty0$. \end{proof} \begin{remark}\label{rem3.100} \rm If we replace (F3) by (F1) as in Loc-Schmitt's work \cite{Loc}, the conclusions of Theorems \ref{main1}, \ref{main3}, \ref{main5}, \ref{main6}, and Corollary \ref{co} remain valid with the non-negativity of solutions not the positivity. \end{remark} \begin{proof}[Proof of Theorem \ref{main8}] By contradiction, assume that $\{(\lambda_n,u_n)\}_{n=1}^\infty$ is a sequence such that $u_n$ is a positive solution of \eqref{0} with $\lambda=\lambda_n$ for each $n \in \mathbb{N}$, and $\lambda_n \to 0$ as $n \to \infty$. Then $\|u_n\|_\infty>C_1$ for all $n \in \mathbb{N}$, since $f(x,s) \le 0$ for all $x \in \overline\Omega$ and $0\le s\le C_1$. Indeed, assume on the contrary that $\|u_n\|_\infty \le C_1$ for some $n \in \mathbb{N}$. It follows from the comparison principle \cite[Proposition 2.3] {Fan5} that $u_n \le0$, which contradicts the fact that $u_n$ is a positive solution of problem \eqref{0} with $\lambda=\lambda_n$. By Lemma \ref{Lem1}, $\|u_n\|_\infty\le C_2$ for all $n \in \mathbb{N}$. Let $h_n=\lambda_n f(\cdot,u_n)$, then $h_n \to 0$ as $n \to \infty$ in $L^\infty (\Omega)$. By Proposition~\ref{reg}, $u_n:=K(h_n)\to 0$ as $n\to \infty$ in $C^1(\Omega)$ which contradicts the fact that $\|u_n\|_\infty>C_1$ for all $n \in \mathbb{N}$. \end{proof} \begin{example} \rm To illustrate Corollary \ref{co} in the case $m=2$, let us consider the nonautonomous cubic nonlinearity $$ f(x,s)=s^{p(x)-1}(s-b(x))(c(x)-s), $$ where $p \in C^1(\overline \Omega)$ with $p^+ < p^*(x)$ for all $x\in \overline \Omega$, and $b,c\in C(\overline \Omega)$ such that $0< b(x)0$, and it has no positive solutions for small $\lambda>0$. \end{example} \subsection*{Acknowledgements} The third author was supported by the National Research Foundation of Korea, Grant funded by the Korea Government (MEST) (NRF-2012R1A1A2000739). \begin{thebibliography}{99} \bibitem{Dinca} G. Dinca, P. Jebelean, J. Mawhin; \emph{Variational and topological methods for Dirichlet problems with p-Laplacian}, Port. Math. (N.S.), 58 (3) (2001), 339--378. \bibitem{Drabek} P. Dr\'{a}bek, A. Kufner, F. 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