\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 180, pp. 1--8.\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-2013/180\hfil Existence of positive solutions] {Existence of positive solutions for Kirchhoff type equations} \author[G. A. Afrouzi, N. T. Chung, S. Shakeri \hfil EJDE-2013/180\hfilneg] {Ghasem A. Afrouzi, Nguyen Thanh Chung, Saleh Shakeri} % in alphabetical order \address{Ghasem Alizadeh Afrouzi \newline Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran} \email{afrouzi@umz.ac.ir} \address{Nguyen Thanh Chung, \newline Dept. Science Management and International Cooperation, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Vietnam} \email{ntchung82@yahoo.com} \address{Saleh Shakeri \newline Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran} \email{s.shakeri@umz.ac.ir} \thanks{Submitted April 23, 2013. Published August 7, 2013.} \subjclass[2000]{35D05, 35J60} \keywords{Kirchhoff type problems; semipositone; positive solution; \hfill\break\indent sub-supersolution method} \begin{abstract} In this article, we are interested in the existence of positive solutions for the Kirchhoff type problems \begin{gather*} -M\Big(\int_{\Omega}|\nabla u|^p\,dx\Big)\Delta_pu = \lambda f(u) \quad \text{in } \Omega,\\ u > 0 \quad \text{in } \Omega, \quad u =0 \quad \text{on } \partial\Omega, \end{gather*} where $1 0 \quad \text{in } \Omega, \quad u =0 \quad \text{on } \partial\Omega, \end{gathered} where$ 1\alpha>0$such that$ f(s)(s-\alpha) \geq 0 $. Since the first equation in \eqref{e1.1} contains an integral over$\Omega $, it is no longer a pointwise identity; therefore it is often called nonlocal problem. This problem models several physical and biological systems, where$ u $describes a process which depends on the average of itself, such as the population density, see \cite{ChLo}. Moreover, problem \eqref{e1.1} is related to the stationary version of the Kirchhoff equation $$\label{e1.2} \rho\frac{\partial^2u}{\partial t^2}-\Big(\frac{P_0}{h}+\frac{E}{2L}\int_0^L\Big|\frac{\partial u}{\partial x}\Big|^2dx\Big)\frac{\partial^2 u}{\partial x^2}=0$$ presented by Kirchhoff in 1883, see \cite{Kirchhoff}. This equation is an extension of the classical d'Alembert's wave equation by considering the effects of the changes in the length of the string during the vibrations. The parameters in \eqref{e1.2} have the following meanings:$L$is the length of the string,$ h $is the area of the cross-section,$ E $is the Young modulus of the material,$\rho$is the mass density, and$ P_{0}$is the initial tension. In recent years, problems involving Kirchhoff type operators have been studied in many papers, we refer to \cite{AlCoMa,BeBo,Chung1,Chung2,CoFi, TFMa, Ricceri, SuTa, YaHa}, in which the authors have used variational method and topological method to get the existence of solutions for \eqref{e1.1} in the cases when$ f$could satisfy$ p $-superlinear,$p$-sublinear or$p$-linear growth condition at infinity. In this paper, motivated by the ideas introduced in \cite{ChSh} and the properties of Kirchhoff type operators in \cite{Dai,DaMa,HaDa}, we study problem \eqref{e1.1} in the semipositone case; i.e.,$ f(0)<0 $. Using the sub- and supersolutions techniques, we prove the existence of a positive solution for the problem in a range of$\lambda $without assuming any condition on$ f $at infinity. To our best knowledge, this is a new research topic for nonlocal problems, see \cite{HaDa}. In order to state precisely our main result we first consider the eigenvalue problem for the$p$-Laplace operator$-\Delta_pu$: $$\label{e1.3} \begin{gathered} -\Delta_pu = \lambda |u|^{p-2}u \quad \text{in } \Omega,\\ u =0 \quad \text{on } \partial\Omega. \end{gathered}$$ Let$\phi_1\in C^1(\overline\Omega)$be the eigenfunction corresponding to the first eigenvalue$\lambda_1$of \eqref{e1.3} such that$ \phi_1>0 $in$ \Omega $and$\|\phi_1\|_\infty=1 $. It can be shown that$ \frac{\partial\phi_1}{\partial \eta}<0 $on$\partial\Omega$and hence, depending on$ \Omega $, there exist positive constants$ m, \delta, \sigma$such that $$\label{e1.4} \begin{gathered} |\nabla\phi_1|^p -\lambda_1 \phi_1^p \geq m \quad \text{in } \overline\Omega_\delta,\\ \phi_1 \geq \sigma \quad \text{in } \Omega\setminus\overline\Omega_\delta, \end{gathered}$$ where$\overline\Omega_\delta :=\{x\in \Omega: d(x,\partial\Omega)\leq \delta\}$. We will also consider the unique solution$e \in C^1(\overline\Omega)$of the boundary value problem $$\label{e1.5} \begin{gathered} -\Delta_p e = 1 \quad \text{in } \Omega,\\ e =0 \quad \text{on } \partial\Omega \end{gathered}$$ to discuss our result. It is known that$ e>0 $in$\Omega $and$\frac{\partial e}{\partial \eta}<0 $on$\partial\Omega$. For our main result we assume that there exist positive constants$ M_0, M_\infty $and$l_1, l_2\in (\alpha,r]$satisfying \begin{itemize} \item[(H1)]$M_0 \leq M(t) \leq M_\infty$for all$t \in \mathbb{R}^+$; \item[(H2)]$l_2 \geq kl_1$, where$k=k(\Omega)=\frac{p}{p-1}\lambda_1^\frac{1}{p-1} \sigma^\frac{p}{1-p}\|e\|_{L^\infty(\Omega)}$; \item[(H3)]$\frac{M_\infty\lambda_1}{f(l_1)} < \frac{mM_0}{|f(0)|}$; \item[(H4)]$\frac{l_2^{p-1}}{f(l_2)} > \mu \frac{l_1^{p-1}}{f(l_1)}$, where$\mu = \mu(\Omega) = \frac{M_\infty\lambda_1}{M_0\sigma^p} \left(\frac{p\|e\|_{L^\infty (\Omega)}}{p-1} \right)^{p-1}$. \end{itemize} Our main results reads as follows. \begin{theorem}\label{thm1.1} Under assumptions {\rm (H1)--(H4)}, there exist two positive constants$\lambda_\ast $and$ \lambda^\ast $such that \eqref{e1.1} has a positive solution for all$\lambda \in (\lambda_\ast,\lambda^\ast)$. \end{theorem} \section{Preliminaries} We will prove our result by using the method of sub- and supersolutions, we refer the readers to a recent paper \cite{HaDa} on the topic. A function$ \psi $is said to be a subsolution of \eqref{e1.1} if it is in$ W^{1,p}(\Omega)\cap C^0(\overline\Omega)$such that$\psi = 0 $on$\partial\Omega $and satisfies $$\label{e2.1} M\Big(\int_\Omega |\nabla \psi|^p\,dx\Big)\int_\Omega |\nabla \psi|^{p-2}\nabla \psi\cdot\nabla w\,dx \leq \lambda\int_\Omega f(\psi)w\,dx, \quad \forall w \in W,$$ where$W:= \{w\in C_0^\infty(\Omega): w\geq 0 \text{ in } \Omega\}$. A function$\phi \in W^{1,p}(\Omega)\cap C^0(\overline\Omega)$is said to be a supersolution if$\phi = 0$on$\partial \Omega$and satisfies $$\label{e2.2} M\Big(\int_\Omega |\nabla \phi|^p\,dx\Big)\int_\Omega |\nabla \phi|^{p-2}\nabla \phi\cdot\nabla w\,dx \geq \lambda\int_\Omega f(\phi)w\,dx, \quad \forall w \in W.$$ The following result plays an important role in our arguments. For the readers' convenience, we present its proof in detail. \begin{lemma}\label{lem2.1} Assume that$M: \mathbb{R}^+\to \mathbb{R}^+$is a continuous and increasing function satisfying $$M(t) \geq M_0 > 0 \text{ for all } t\in \mathbb{R}^+.$$ If the functions$u,v \in W^{1,p}_0(\Omega)$satisfy $$\label{e2.3} \begin{split} &M\Big(\int_{\Omega}|\nabla u|^p\,dx\Big)\int_\Omega |\nabla u|^{p-2}\nabla u\cdot\nabla \varphi\,dx\\ & \leq M\Big(\int_{\Omega}|\nabla v|^p\,dx\Big)\int_\Omega |\nabla v|^{p-2}\nabla v\cdot\nabla \varphi\,dx \end{split}$$ for all$\varphi \in W^{1,p}_0(\Omega)$,$\varphi \geq 0$, then$u \leq v$in$\Omega$. \end{lemma} \begin{proof} Our proof is based on the arguments presented in \cite{Dai,DaMa}. Define the functional$ \Phi: W^{1,p}_0 (\Omega) \to \mathbb{R} $by the formula $$\Phi(u):= \frac{1}{p}\widehat{M}\Big(\int_\Omega |\nabla u|^p\,dx\Big), \quad u \in W^{1,p}_0(\Omega).$$ It is obviously that the functional$ \Phi $is a continuously G\^{a}teaux differentiable whose G\^{a}teaux derivative at the point$u \in W^{1,p}_0(\Omega)$is the functional$\Phi'\in W^{-1,p}_0(\Omega)$, given by $$\Phi'(u)(\varphi) = M\Big(\int_\Omega |\nabla u|^p\,dx\Big)\int_\Omega |\nabla u|^{p-2}\nabla u\cdot\nabla \varphi\,dx, \quad \varphi \in W^{1,p}_0(\Omega).$$ It is obvious that$\Phi'$is continuous and bounded since the function$ M $is continuous. We will show that$ \Phi'$is strictly monotone in$W^{1,p}_0(\Omega)$. Indeed, for any$ u,v \in W^{1,p}_0(\Omega)$,$ u \ne v $, without loss of generality, we may assume that $$\int_\Omega |\nabla u|^p\,dx \geq \int_\Omega |\nabla v|^p\,dx.$$ (otherwise, changing the role of u and v in the following proof). Therefore, we have $$\label{e2.4} M\Big(\int_\Omega |\nabla u|^p\,dx\Big) \geq M\Big(\int_\Omega |\nabla v|^p\,dx\Big)$$ since$M(t)$is a monotone function. Using Cauchy's inequality, we have $$\label{e2.5} \nabla u\cdot\nabla v \leq |\nabla u||\nabla v| \leq \frac{1}{2}(|\nabla u|^2+|\nabla v|^2).$$ Using \eqref{e2.5} we obtain \begin{gather}\label{e2.6} \int_\Omega |\nabla u|^p\, dx - \int_\Omega |\nabla u|^{p-2}\nabla u\cdot\nabla v\,dx \geq \frac{1}{2}\int_{ \Omega} |\nabla u|^{p-2}(|\nabla u|^2-|\nabla v|^2)\,dx, \\ \label{e2.7} \int_\Omega |\nabla v|^p\, dx - \int_\Omega |\nabla v|^{p-2}\nabla v\cdot\nabla u\,dx \geq \frac{1}{2}\int_{\Omega} |\nabla v|^{p-2}(|\nabla v|^2-|\nabla u|^2)\, dx. \end{gather} If$|\nabla u| \geq |\nabla v|$, using \eqref{e2.4}-\eqref{e2.7}, we have $$\label{e2.8} \begin{split} I_1 &:= \Phi'(u)(u)-\Phi'(u)(v)-\Phi'(v)(u)+\Phi'(v)(v) \\ & = M\Big(\int_\Omega |\nabla u|^p\,dx\Big)\Big(\int_\Omega |\nabla u|^p\,dx-\int_\Omega |\nabla u|^{p-2}\nabla u\cdot\nabla v\,dx\Big) \\ &\quad - M\Big(\int_\Omega |\nabla v|^p\,dx\Big)\Big(\int_\Omega |\nabla v|^{p-2}\nabla v\cdot \nabla u\,dx-\int_\Omega |\nabla v|^p\,dx\Big) \\ &\geq\frac{1}{2}M\Big(\int_\Omega |\nabla u|^p\,dx\Big)\int_\Omega |\nabla u|^{p-2}(|\nabla u|^2-|\nabla v|^2)\, dx \\ &\quad -\frac{1}{2}M\Big(\int_\Omega |\nabla v|^p\,dx\Big) \int_\Omega |\nabla u|^{p-2}(|\nabla u|^2-|\nabla v|^2)\, dx \\ & = \frac{1}{2}M\Big(\int_\Omega |\nabla v|^p\,dx\Big) \int_\Omega (|\nabla u|^{p-2}-|\nabla v|^{p-2})(|\nabla u|^2-|\nabla v|^2)\,dx \\ & \geq \frac{M_0}{2}\int_\Omega (|\nabla u|^{p-2}-|\nabla v|^{p-2})(|\nabla u|^2-|\nabla v|^2)\,dx. \end{split}$$ If$|\nabla v| \geq |\nabla u|$, changing the role of$u$and$vin \eqref{e2.4}-\eqref{e2.7}, we have \label{e2.9} \begin{aligned} I_2 &:= \Phi'(v)(v)-\Phi'(v)(u)-\Phi'(u)(v)+\Phi'(u)(u) \\ & = M\Big(\int_\Omega |\nabla v|^p\,dx\Big) \Big(\int_\Omega |\nabla v|^p\,dx-\int_\Omega |\nabla v|^{p-2}\nabla v\cdot\nabla u\,dx\Big) \\ &\quad- M\Big(\int_\Omega |\nabla u|^p\,dx\Big) \Big(\int_\Omega |\nabla u|^{p-2}\nabla u\cdot \nabla v\,dx -\int_\Omega |\nabla u|^p\,dx\Big) \\ & \geq \frac{1}{2}M\Big(\int_\Omega |\nabla v|^p\,dx\Big) \int_\Omega |\nabla v|^{p-2}(|\nabla v|^2-|\nabla u|^2)\, dx \\ &\quad -\frac{1}{2}M\Big(\int_\Omega |\nabla u|^p\,dx\Big) \int_\Omega |\nabla u|^{p-2}(|\nabla v|^2-|\nabla u|^2)\, dx \\ & = \frac{1}{2}M\Big(\int_\Omega |\nabla v|^p\,dx\Big) \int_\Omega (|\nabla v|^{p-2}-|\nabla u|^{p-2})(|\nabla v|^2-|\nabla u|^2)\,dx \\ & \geq \frac{M_0}{2}\int_\Omega (|\nabla v|^{p-2}-|\nabla u|^{p-2})(|\nabla v|^2-|\nabla u|^2)\,dx. \end{aligned} From \eqref{e2.8} and \eqref{e2.9}, we have $$\label{e2.10} \Big(\Phi'(u)-\Phi'(v)\Big)(u-v) = I_1=I_2 \geq 0, \quad \forall u,v \in W^{1,p}_0(\Omega).$$ Moreover, ifu\ne v$and$\Big(\Phi'(u)-\Phi'(v)\Big)(u-v)=0$, then we have $$\int_\Omega (|\nabla u|^{p-2}-|\nabla v|^{p-2})(|\nabla u|^2-|\nabla v|^2)\,dx = 0,$$ so$|\nabla u|=|\nabla v|$in$\Omega. Thus, we deduce that \label{e2.11} \begin{aligned} \Big(\Phi'(u)-\Phi'(v)\Big)(u-v) &= \Phi'(u)(u-v)-\Phi'(v)(u-v) \\ & = M\Big(\int_\Omega |\nabla u|^p\,dx\Big) \int_\Omega |\nabla u|^{p-2}|\nabla u-\nabla v|^2\,dx = 0; \end{aligned} i.e.,u-v$is a constant. In view of$ u=v=0 $on$\partial\Omega $we have$u\equiv v $which is contrary with$ u\ne v $. Therefore$\big(\Phi'(u)-\Phi'(v)\big)(u-v)>0$and$\Phi'$is strictly monotone in$W^{1,p}_0 (\Omega)$. Let$u,v$be two functions such that \eqref{e2.3} is satisfied. Taking$\varphi=(u-v)^+$, the positive part of$u-v, as a test function of \eqref{e2.3}, we have \label{e2.12} \begin{aligned} (\Phi'(u)-\Phi'(v))(\varphi) &= M\Big(\int_\Omega |\nabla u|^p\,dx\Big) \int_\Omega |\nabla u|^{p-2}\nabla u\cdot \nabla\varphi \,dx \\ & \quad -M\Big(\int_\Omega |\nabla v|^p\,dx\Big) \int_\Omega |\nabla v|^{p-2}\nabla v\cdot \nabla\varphi \,dx \leq 0. \end{aligned} Relations \eqref{e2.11} and \eqref{e2.12} imply thatu \leq v$. \end{proof} From Lemma \ref{lem2.1} we obtain the following basic principle of the sub- and super-solutions method. \begin{theorem}[\cite{HaDa}]\label{thm2.2} Let$ M: \mathbb{R}^+\to \mathbb{R}^+$be a continuous and increasing function satisfying $$M(t) \geq M_0 > 0 \quad \text{for all } t\in \mathbb{R}^+.$$ Assume that$ f $satisfies the subcritical growth condition $$|f(x,t)| \leq C(1+|t|^{q-1}), \quad \forall x\in \Omega, \; \forall t\in R,$$ where$1 0 \quad \text{for all } t\in \mathbb{R}^+.  Assume that $\underline u, \overline u$ are a subsolution and a super-solution of problem \eqref{e1.1} such that $W^{1,p}(\Omega)\cap L^\infty(\Omega)$ and $\underline u \leq \overline u$ in $\Omega$. If $f\in C(\overline\Omega \times R,R)$ is nondecreasing in $t \in [\inf_\Omega\underline u, \sup_\Omega \overline u]$ then the conclusion of Theorem \ref{thm2.2} is valid. \end{theorem} \section{Proof of main result} In this section, we prove Theorem \ref{thm1.1} by using the sub- and super-solutions method. Our arguments are similar to those presented in \cite{ChSh}. First we construct a positive subsolution of problem \eqref{e1.1}. For this purpose, we let $\psi = l_1 \sigma^\frac{p}{1-p}\phi_1^\frac{p}{p-1}$. Since $\nabla \psi=\frac{pl_1}{p-1}\sigma^\frac{p}{1-p} \phi_1^\frac{1}{p-1}\nabla \phi_1$, we deduce that $$\label{e3.1} \begin{split} & M\Big(\int_\Omega |\nabla \psi|^p\,dx\Big) \int_\Omega |\psi|^{p-2}\nabla\psi\cdot\nabla w\,dx \\ &= \Big(\frac{pl_1}{p-1}\sigma^\frac{p}{1-p}\Big)^{p-1} M\Big(\int_\Omega |\nabla \psi|^p\,dx\Big) \int_\Omega \phi_1|\nabla\phi_1|^{p-2}\nabla\phi_1\cdot\nabla w\,dx \\ & = \Big(\frac{pl_1}{p-1}\sigma^\frac{p}{1-p}\Big)^{p-1} M\Big(\int_\Omega |\nabla \psi|^p\,dx\Big) \int_\Omega |\nabla\phi_1|^{p-2}\nabla\phi_1\cdot [\nabla(\phi_1w)-w\nabla\phi_1]\,dx \\ & = \Big(\frac{pl_1}{p-1}\sigma^\frac{p}{1-p}\Big)^{p-1} M\Big(\int_\Omega |\nabla \psi|^p\,dx\Big) \int_\Omega |\nabla\phi_1|^{p-2}\nabla\phi_1\cdot\nabla(\phi_1w)\,dx \\ & \quad -\Big(\frac{pl_1}{p-1}\sigma^\frac{p}{1-p}\Big)^{p-1} M\Big(\int_\Omega |\nabla \psi|^p\,dx\Big)\int_\Omega |\nabla\phi_1|^pw\,dx \\ & = \Big(\frac{pl_1}{p-1}\sigma^\frac{p}{1-p}\Big)^{p-1} M\Big(\int_\Omega |\nabla \psi|^p\,dx\Big) \int_\Omega \lambda_1|\phi_1|^{p-2}\phi_1(\phi_1w)\,dx \\ & \quad -\Big(\frac{pl_1}{p-1}\sigma^\frac{p}{1-p}\Big)^{p-1} M\Big(\int_\Omega |\nabla \psi|^p\,dx\Big) \int_\Omega |\nabla\phi_1|^pw\,dx \\ & =\Big(\frac{pl_1}{p-1}\sigma^\frac{p}{1-p}\Big)^{p-1} M\Big(\int_\Omega|\nabla \psi|^p\,dx\Big) \int_\Omega [\lambda_1\phi_1^p-|\nabla\phi_1|^p]w\,dx. \end{split}$$ Thus $\psi$ is a subsolution of problem \eqref{e1.1} if $$\label{e3.2} \Big(\frac{pl_1}{p-1}\sigma^\frac{p}{1-p}\Big)^{p-1}M\Big(\int_\Omega |\nabla \psi|^p\,dx\Big) \int_\Omega [\lambda_1\phi_1^p-|\nabla\phi_1|^p]w\,dx \leq \lambda\int_\Omega f(\psi)w\,dx$$ On $\overline\Omega_\delta$, we have $$\label{e3.3} |\nabla\phi_1|^p - \lambda_1\phi_1^p \geq m$$ and therefore, by (H1), $$\label{e3.4} \begin{split} &\Big(\frac{pl_1}{p-1}\sigma^\frac{p}{1-p}\Big)^{p-1} M\Big(\int_\Omega |\nabla \psi|^p\,dx\Big) [\lambda_1\phi_1^p-|\nabla\phi_1|^p] \\ & \leq -mM_0\Big(\frac{pl_1}{p-1}\sigma^\frac{p}{1-p}\Big)^{p-1} \leq \lambda f(\psi) \end{split}$$ if $$\label{e3.5} \lambda \leq \overline\lambda := \frac{m M_0}{|f(0)|}.\Big(\frac{pl_1}{p-1}\sigma^\frac{p}{1-p}\Big)^{p-1}.$$ On $\Omega\setminus\Omega_\delta$ we have $\phi_1 \geq \sigma$ and therefore, $$\label{e3.6} \psi = l_1 \sigma^\frac{p}{1-p}\phi_1^\frac{p}{p-1} \geq l_1\sigma^\frac{p}{1-p}\sigma^\frac{p}{p-1}=l_1.$$ Thus, by (H1), $$\label{e3.7} \begin{split} &\Big(\frac{pl_1}{p-1}\sigma^\frac{p}{1-p}\Big)^{p-1} M\Big(\int_\Omega |\nabla \psi|^p\,dx\Big) [\lambda_1\phi_1^p-|\nabla\phi_1|^p] \\ &\leq \Big(\frac{pl_1}{p-1}\sigma^\frac{p}{1-p}\Big)^{p-1}M_\infty\lambda_1 \leq \lambda f(\psi) \end{split}$$ if $$\label{e3.8} \lambda \geq \lambda_\ast :=\frac{M_\infty\lambda_1}{f(l_1)}\Big(\frac{pl_1}{p-1} \sigma^\frac{p}{1-p}\Big)^{p-1}.$$ By condition (H3), we have $\lambda_\ast < \overline\lambda$. Therefore, $\psi$ is a subsolution of problem \eqref{e1.1} for all $\lambda_\ast \leq\lambda \leq \overline\lambda$. Next, we construct a supersolution of \eqref{e1.1}. Let $\phi=\frac{l_2}{\|e\|_{L^\infty(\Omega)}}e$, in which $e$ is defined by \eqref{e1.5}. Then, by (H1), $\phi$ is a supersolution of problem \eqref{e1.1} if $$\label{e3.9} \begin{split} & M\Big(\int_\Omega |\nabla \phi|^p\,dx\Big) \int_\Omega |\nabla\phi|^{p-2}\nabla\phi\cdot\nabla w\,dx \\ & = M\Big(\int_\Omega |\nabla \phi|^p\,dx\Big) \Big(\frac{l_2}{\|e\|_{L^\infty(\Omega)}}\Big)^{p-1} \int_\Omega |\nabla e|^{p-2}\nabla e\cdot\nabla w\,dx\\ & = M\Big(\int_\Omega |\nabla \phi|^p\,dx\Big) \Big(\frac{l_2}{\|e\|_{L^\infty(\Omega)}}\Big)^{p-1}\int_\Omega w\,dx\\ & \geq M_0\Big(\frac{l_2}{\|e\|_{L^\infty(\Omega)}}\Big)^{p-1}\int_\Omega w\,dx \\ & \geq \lambda \int_\Omega f(\phi)w\,dx, \quad \forall w\in W. \end{split}$$ But $f(\phi) \leq f(l_2)$ and hence $\phi$ is a supersolution of problem \eqref{e1.1} if $$\label{e3.10} \lambda \leq \widehat{\lambda} := \frac{M_0l_2^{p-1}}{\|e\|_{L^\infty(\Omega)}^{p-1}f(l_2)}.$$ By (H4), we have $\widehat\lambda >\lambda_\ast$. We have $$\label{e3.11} \begin{split} -\Delta_p \phi & = -\nabla (|\nabla \phi|^{p-2}\nabla\phi) \\ &= -\Big(\frac{l_2}{\|e\|_{L^\infty(\Omega)}}\Big)^{p-1} \nabla (|\nabla e|^{p-2}\nabla e) \\ &=\frac{l_2^{p-1}}{\|e\|_{L^\infty(\Omega)}^{p-1}} \end{split}$$ and $$\label{e3.12} \begin{split} -\Delta_p \psi & = -\nabla (|\nabla \psi|^{p-2}\nabla\psi) \\ &= \Big(\frac{pl_1}{p-1}\sigma^\frac{p}{1-p}\Big)^{p-1} [\lambda_1\phi_1^p-|\nabla\phi_1|^p] \\ & \leq \lambda_1\Big(\frac{pl_1}{p-1}\sigma^\frac{p}{1-p}\Big)^{p-1}. \end{split}$$ By condition (H2), using the weak comparison principle for the $p$-Laplace operator $-\Delta_p u$, we see that $\psi \leq \phi$ in $\Omega$. Set $\lambda^\ast:= \min\{\overline\lambda,\widehat \lambda\}$. By Theorem \ref{thm2.3}, we conclude that problem \eqref{e1.1} has a positive solution for any $\lambda \in (\lambda_\ast,\lambda^\ast)$. \begin{thebibliography}{99} \bibitem{AlCoMa} {C. O. Alves, F. J. S. A. Corr\^{e}a, T. M. 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