\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<p< N $, $M : \mathbb{R}^+\to \mathbb{R}^+$ is a continuous and increasing
 function, $ \lambda $ is a parameter, $ f: [0,+\infty) \to \mathbb{R} $ is a
 $ C^1 $ nondecreasing function satisfying $ f(0)<0 $ (semipositone).
 Our proof is based on the sub- and super-solutions techniques.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks

\section{Introduction}

In this article, we are interested in the existence of positive
solutions for  Kirchhoff type problems of the form
\begin{equation}\label{e1.1}
\begin{gathered}
-M \Big(\int_{\Omega}|\nabla u|^p\,dx\Big)\Delta_p u
= \lambda f(u) \quad \text{in } \Omega,\\
u > 0 \quad \text{in } \Omega, \quad u =0 \quad \text{on } \partial\Omega,
\end{gathered}
\end{equation}
where $ 1<p<N $, $ M: \mathbb{R}^+\to \mathbb{R}^+ $ is a continuous and increasing function, $ f: [0,+\infty) \to \mathbb{R} $ is a $ C^1 $ nondecreasing function such that $ f(0)<0 $ (semipositone) and there exist
$r>\alpha>0$ 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
\begin{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
\end{equation}
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
$-\Delta_pu$:
\begin{equation}\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}
\end{equation}
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
\begin{equation}\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}
\end{equation}
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
\begin{equation}\label{e1.5}
\begin{gathered}
-\Delta_p e = 1 \quad \text{in } \Omega,\\
e =0 \quad \text{on }  \partial\Omega
\end{gathered}
\end{equation}
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
\begin{equation}\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,
\end{equation}
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
\begin{equation}\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.
\end{equation}

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
\begin{equation}\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}
\end{equation}
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
\begin{equation}\label{e2.4}
M\Big(\int_\Omega |\nabla u|^p\,dx\Big)
\geq M\Big(\int_\Omega |\nabla v|^p\,dx\Big)
\end{equation}
since $M(t)$ is a monotone function. Using Cauchy's inequality, we
have
\begin{equation}\label{e2.5}
\nabla u\cdot\nabla v \leq |\nabla u||\nabla v| \leq
\frac{1}{2}(|\nabla u|^2+|\nabla v|^2).
\end{equation}
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
\begin{equation}\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}
\end{equation}
If $|\nabla v| \geq |\nabla u|$, changing the role of $u$ and $v$ in
\eqref{e2.4}-\eqref{e2.7}, we have
\begin{equation} \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}
\end{equation}
From \eqref{e2.8} and \eqref{e2.9}, we have
\begin{equation}\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).
\end{equation}
Moreover, if $u\ne v$ 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
\begin{equation} \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}
\end{equation}
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
\begin{equation} \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}
\end{equation}
Relations \eqref{e2.11} and \eqref{e2.12} imply that $u \leq v$.
\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 $1<q< p^\ast = \frac{Np}{N-p}$, and the function $f(x,t)$ is
nondecreasing in $t\in R$. If there exist a subsolution 
$\underline u \in W^{1,p}(\Omega)$ and a supersolution
$\overline u \in W^{1,p}(\Omega)$ of problem \eqref{e1.1}, 
then \eqref{e1.1} has a minimal solution $u_\ast$ and a maximal solution 
$u^\ast$ in the order interval $[u_\ast,u^\ast]$; i.e., 
$\underline u \leq u_\ast \leq u^\ast \leq \overline u$ and if $u$ is 
any solution of \eqref{e1.1} such that $\underline u \leq u \leq \overline u$, 
then $u_\ast \leq u \leq u^\ast$.
\end{theorem}

In practice problems, it is often known that the subsolution
$\underline u$ and the supersolution $\overline u $ are in 
$ L^\infty(\Omega)$, so the restriction on the growth condition of $f$
is needless. Hence, the following theorem is more suitable for our
framework.

\begin{theorem}[\cite{HaDa}]\label{thm2.3}
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 $\underline u, \overline u$ 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
\begin{equation}\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}
\end{equation}
Thus $ \psi $ is a subsolution of problem \eqref{e1.1} if
\begin{equation}\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
\end{equation}
On $\overline\Omega_\delta$, we have
\begin{equation}\label{e3.3}
|\nabla\phi_1|^p - \lambda_1\phi_1^p \geq m
\end{equation}
and therefore, by (H1),
\begin{equation} \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}
\end{equation}
if
\begin{equation}\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}.
\end{equation}
On $\Omega\setminus\Omega_\delta$ we have $\phi_1 \geq \sigma$ and
therefore,
\begin{equation}\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.
\end{equation}
Thus, by (H1),
\begin{equation} \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}
\end{equation}
if
\begin{equation}\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}.
\end{equation}
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
\begin{equation} \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}
\end{equation}
But $f(\phi) \leq f(l_2)$ and hence $\phi$ is a supersolution of
problem \eqref{e1.1} if
\begin{equation}\label{e3.10}
\lambda \leq \widehat{\lambda} :=
\frac{M_0l_2^{p-1}}{\|e\|_{L^\infty(\Omega)}^{p-1}f(l_2)}.
\end{equation}
By (H4), we have $\widehat\lambda >\lambda_\ast $.
We have
\begin{equation} \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}
\end{equation}
and
\begin{equation} \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}
\end{equation}
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 $\lambda \in (\lambda_\ast,\lambda^\ast)$.

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\end{document}