\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 75, pp. 1--14.\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-2013/75\hfil Existence and multiplicity of solutions]
{Existence and multiplicity of solutions for nonlocal elliptic problems}

\author[M. Massar \hfil EJDE-2013/75\hfilneg]
{Mohammed Massar}  % in alphabetical order

\address{Mohammed Massar \newline
University Mohamed I, Faculty of Sciences, Department of
Mathematics, Oujda, Morocco}
\email{massarmed@hotmail.com}

\thanks{Submitted November 24, 2012. Published March 18, 2013.}
\subjclass[2000]{35J20, 35J60}
\keywords{Neumann problem; p-Kirchhoff problem; positive solutions;
\hfill\break\indent variational method}

\begin{abstract}
 This article concerns the existence and multiplicity  solutions
 for a class of p-Kirchhoff type equations with Neumann  boundary conditions.
 Our technical approach is based on variational  methods.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

In this work, we study the existence and multiplicity of solutions
for the nonlocal elliptic problem under Neumann boundary condition:
\begin{equation}\label{E11}
\begin{gathered}
\Big[M\Big(\int_\Omega(|\nabla
u|^p+a(x)|u|^p)dx\Big)\Big]^{p-1}\Big(-\Delta_pu+a(x)|u|^{p-2}u\Big)
=\lambda f(x,u)\quad \text{in }\Omega\\
\frac{\partial u}{\partial\nu}=0\quad \text{on }\partial\Omega,
\end{gathered}
\end{equation}
where $p>N$, $\Omega$ is a nonempty bounded open subset of
$\mathbb{R}^N$ with a boundary of class $C^1$,
$\frac{\partial u}{\partial\nu}$ is the outer unit normal derivative,
$a\in L^\infty(\Omega)$, with
$\operatorname{ess\,inf}_\Omega a\geq0$, $a\neq0$,
$\lambda\in(0,\infty)$, $f:\Omega\times\mathbb{R}\to\mathbb{R}$
and $M:\mathbb{R}^+\to\mathbb{R}^+$ are two functions that
satisfy conditions which will be stated later.

The problem \eqref{E11}) is related to the stationary problem of a
model introduced by Kirchhoff \cite{K}. More precisely, Kirchhoff
introduced a model given by the  equation
\begin{equation}\label{E12}
\rho\frac{\partial^2u}{\partial
t^2}-\Big(\frac{\rho_0}h+\frac{E}{2L}\int_0^L|\frac{\partial
u}{\partial x}|^2dx\Big)\frac{\partial^2u}{\partial x^2}=0,
\end{equation}
which extends the classical D'Alembert's wave equation by
considering the effects of the changes in the length of the strings
during the vibrations. Latter \eqref{E12} was developed to form
\begin{equation}\label{E13}
u_{tt}-M\Big(\int_\Omega|\nabla u|^2dx\Big)\Delta
u=f(x,u)\quad\text{in}\,\Omega.
\end{equation}
After that, many authors studied the following nonlocal elliptic
boundary value problem
\begin{equation}\label{E14}
-M\Big(\int_\Omega|\nabla u|^2dx\Big)\Delta
u=f(x,u)\quad\text{in }\Omega,\quad u=0\quad\text{on }\partial\Omega.
\end{equation}
Problems like \eqref{E14} can be used for modeling several physical
and biological systems where $u$ describes a process which depends
on the average of it self, such as the population density , see
\cite{ACM}. Many interesting results for problems of Kirchhoff type
were obtained and we refer to \cite{ACM,A,APS,HMT, HZ,MZ} and
references therein for an overview on these subjects.

The main purpose of the present paper is to establish the existence
of at least one solution and, as a consequence, existence results of
two and three solutions for the nonlocal problem \eqref{E11}, by
adopting the framework of Bonanno and Sciammetta \cite{BS}.

\section{Preliminaries and basic notation}

Our main tools are two consequences of a local minimum theorem
\cite[Theorem 3.1]{B} which are recalled below. Given $X$ a set and
two functionals $\Phi,\Psi : X\to\mathbb{R}$, put
\begin{gather}\label{E21}
\beta(r_1,r_2)=\inf_{v\in\Phi^{-1}((r_1,r_2))}
\frac{\sup_{u\in\Phi^{-1}((r_1,r_2))}\Psi(u)-\Psi(v)}
{r_2-\Phi(v)},
\\
\label{E22}
\rho_2(r_1,r_2)=\sup_{v\in\Phi^{-1}((r_1,r_2))}
\frac{\Psi(v)-\sup_{u\in\Phi^{-1}((-\infty,r_1])}\Psi(u)}
{\Phi(v)-r_1},
\end{gather}
for all $r_1,r_2\in\mathbb{R}$, with $r_1<r_2$, and
\begin{equation}\label{E23}
\rho(r)=\sup_{v\in\Phi^{-1}((r,+\infty))}\frac{\Psi(v)
-\sup_{u\in\Phi^{-1}((-\infty,r])}\Psi(u)}
{\Phi(v)-r},
\end{equation}
for all $r\in\mathbb{R}$.

\begin{theorem}[{\cite[Theorem 5.1]{B}}]\label{theo21}
Let $X$ be a reflexive real Banach
space, $\Phi: X\to\mathbb{R}$ be a sequentially weakly lower
semicontinuous, coercive and continuously G\^{a}teaux differentiable
function whose G\^{a}teaux derivative admits a continuous inverse on
$X^*,\Psi: X\to\mathbb{R}$ be a continuously G\^{a}teaux
differentiable function whose G\^{a}teaux derivative is compact. Put
$I_\lambda=\Phi-\lambda\Psi$ and assume that there are
$r_1,r_2\in\mathbb{R}$, with $r_1<r_2$, such that
\begin{equation}\label{E24}
\beta(r_1,r_2)<\rho_2(r_1,r_2),
\end{equation}
where $\beta$ and $\rho_2$ are given by \eqref{E21} and
\eqref{E22}.
Then, for each
$\lambda\in\big(\frac1{\rho_2(r_1,r_2)},\frac1{\beta(r_1,r_2)}\big)$
there is $u_{0,\lambda}\in\Phi^{-1}((r_1,r_2))$ such that
$I_\lambda(u_{0,\lambda})\leq I_\lambda(u)$ for all
$u\in\Phi^{-1}((r_1,r_2))$ and $I_\lambda'(u_{0,\lambda})=0$.
\end{theorem}

\begin{theorem}[{\cite[Theorem 5.3]{B}}] \label{theo22}
Let $X$ be a real Banach space, $\Phi : X \to\mathbb{R}$ be a
continuously G\^{a}teaux differentiable function whose G\^{a}teaux
derivative admits a continuous inverse on $X^*,\Psi:
X\to\mathbb{R}$ be a continuously G\^{a}teaux differentiable
function whose G\^{a}teaux derivative is compact. Fix
$\inf_X\Phi<r<\sup_X\Phi$ and assume that
\begin{equation}\label{E25}
\rho(r)>0,
\end{equation}
where $\rho$ is given by \eqref{E23} and for each
$\lambda>1/\rho(r)$ the function $I_\lambda=\Phi-\lambda\Psi$
is coercive.

Then, for each $\lambda>1/\rho(r)$ there is
$u_{0,\lambda}\in\Phi^{-1}((r,+\infty))$ such that
$I_\lambda(u_{0,\lambda})\leq I_\lambda(u)$ for all
$u\in\Phi^{-1}((r,+\infty))$ and $I_\lambda'(u_{0,\lambda})=0$.
\end{theorem}
Theorems \ref{theo21} and \ref{theo22} are consequences of a local
minimum theorem \cite[Theorem 3.1]{B} which is a more general
version of the Ricceri Variational Principle (see \cite{R1}).

Let $X$ be the Sobolev space $W^{1,p}(\Omega)$ endowed with the norm
$$
\|u\|:=\Big(\int_\Omega(|\nabla u|^p+a(x)|u|^p)dx\Big)^{1/p}.
$$
Let
\begin{equation}\label{E26}
k:=\sup_{u\in X\setminus\{0\}}
\frac{\max_{x\in\overline{\Omega}} |u(x)|}{\|u\|}.
\end{equation}
Since $p>N$, $X$ is compactly embedded in $C^0(\overline{\Omega})$,
so that $k<\infty$. We have
\begin{equation}\label{E27}
|u(x)|\leq k\|u\|\,\quad\text{for all }x\in\Omega,\;u\in X.
\end{equation}
Therefore, taking $u\equiv1$ in \eqref{E27},
$$
k^p\|a\|_1\geq1,\quad \text{where }\|a\|_1=\int_\Omega|a(x)|dx.
$$

We assume that $f:\Omega\times\mathbb{R}\to\mathbb{R}$ is
$L^1$-Carath\'{e}odory; that is, $x\mapsto f(x,t)$ is measurable for
every $t\in\mathbb{R}$, $t\mapsto f(x,t)$ is continuous for almost
every $x\in\Omega$ and for all $s>0$ there is $l_s\in L^1(\Omega)$
such that
$$
\sup_{|t|\leq s}|f(x,t)|\leq l_s(x)\quad \text{for a.e. }x\in\Omega,
$$
and $M:\mathbb{R}^+\to\mathbb{R}^+$ is a nondecreasing
continuous function with the following condition:
\begin{itemize}
\item[(M0)]
$m_0:=\inf_{t\geq0}M(t)>0$.
\end{itemize}
We say that $u\in X$ is a weak solution of problem \eqref{E11} if
$$
\left[M\left(\|u\|^p\right)\right]^{p-1}\int_\Omega\left(|\nabla
u|^{p-2}\nabla u \nabla
v+a(x)|u|^{p-2}uv\right)dx-\lambda\int_\Omega f(x,u)vdx=0,
$$
for all $v\in X$.

We introduce the functionals $\Phi,\Psi: X\to\mathbb{R}$,
defined by
\begin{equation}\label{E29}
\Phi(u)=\frac1p\widehat{M}\left(\|u\|^p\right),\quad
\Psi(u)=\int_\Omega F(x,u)dx,
\end{equation}
for all $u\in X$, where
\begin{gather*}
\widehat{M}(t)=\int_0^t [M(s)]^{p-1}ds\,\;\text{for all}\;t\geq0,\\
F(x,\xi)=\int_0^\xi f(x,s)ds\;\,\text{for
all}\;(x,\xi)\in\Omega\times\mathbb{R}.
\end{gather*}
It is well known that $\Phi$ and $\Psi$ are well defined and
continuously G\^{a}teaux differentiable whose G\^{a}teaux
derivatives at point $u\in X$ are given by
\begin{gather*}
\langle\Phi'(u),v\rangle = \left[M\left(\|u\|^p\right)\right]^{p-1}\int_\Omega\left(|\nabla
u|^{p-2}\nabla u \nabla v+a(x)|u|^{p-2}uv\right)dx\\
\langle\Psi'(u),v\rangle = \int_\Omega f(x,u)vdx,
\end{gather*}
for all $v\in X$. Moreover, $\Psi'$ is compact.

\begin{proposition}\label{prop}
Assume that {\rm (M0)} holds. Then
\begin{itemize}
\item[(i)] $\Phi$ is sequentially weakly lower semicontinuous;
\item[(ii)] $\Phi$ is coercive;
\item[(iii)] $\Phi': X\to X^*$ is strictly monotone;

\item[(iv)] $\Phi'$ is of type $(S_+)$, i.e. if $u_n\rightharpoonup u$ in $X$
and
$$
\overline{\lim}_{n\to+\infty}\langle\Phi'(u_n)-\Phi'(u),u_n-u\rangle=0,
$$
then $u_n\to u$ in $X$;
\item[(v)] $\Phi'$ admits a continuous inverse on $X^*$.
\end{itemize}
\end{proposition}

\begin{proof}
(i) Let $u_n\rightharpoonup u$ weakly in $X$. By the weakly lower
semicontinuity of norm, it follows that
$$
\|u\|\leq\liminf_{n\to+\infty}\|u_n\|.
$$
In view of the continuity and monotonicity of $\widehat{M}$, we
deduce that
$$
\widehat{M}\left(\|u\|^p\right)\leq
\widehat{M}\Big(\liminf_{n\to+\infty}\|u_n\|^p\Big)\leq\liminf_{n\to+\infty}
\widehat{M}\left(\|u_n\|^p\right),
$$
and hence $\Phi$ is sequentially weakly lower semicontinuous.

$(ii)$ Thanks to (M0), we have
\begin{equation}\label{E210}
\Phi(u)=\frac1p\widehat{M}\left(\|u\|^p\right)
\geq\frac{m_0^{p-1}}p\|u\|^p.
\end{equation}
So, $\Phi$ is coercive.

(iii) Consider the functional $T: X\to \mathbb{R}$,
defined by
$$
T(u)=\int_\Omega\left(|\nabla u|^p+a(x)|u|^p\right)dx\,\quad
\text{for all } u\in X,
$$
whose G\^{a}teaux derivative at point $u\in X$ is given by
$$
\langle T'(u),v\rangle=p\int_\Omega\left(|\nabla u|^{p-2}\nabla u
\nabla v+a(x)|u|^{p-2}uv\right)dx,\quad
\text{for all}\;v\in X,
$$
Taking into account \cite[(2.2)]{S} for $p>1$ there exists a
positive constant $C_p$ such that
\begin{equation}\label{E211}
\langle|x|^{p-2}x-|y|^{p-2}y,x-y\rangle
\geq \begin{cases}
C_p|x-y|^p &\text{if }p\geq2\\
C_p\frac{|x-y|^2}{(|x|+|y|)^{p-2}},\;(x,y)\neq(0,0) &\text{if }1<p<2,
\end{cases}
\end{equation}
for all $x,y\in\mathbb{R}^N$. Therefore,
\begin{align*}
\langle T'(u)-T'(v),u-v\rangle
&\geq \begin{cases}
C_p\int_\Omega\left(|\nabla u-\nabla v|^p+a(x)|u-v|^p\right)dx
 &\text{if } p\geq2\\
C_p\int_\Omega\left(\frac{|\nabla u-\nabla v|^2}{(|\nabla u|+|\nabla
v|)^{2-p}}+\frac{a(x)|u-v|^2}{(|u|+|v|)^{2-p}}\right)dx
 &\text{if }1<p<2
\end{cases}\\
&>0,
\end{align*}
for all $u\neq v\in X$, which means that $T'$ is strictly monotone.
So, by  \cite[Prop. 25.10]{Z}, $T$ is strictly convex.
Moreover, since $M$ is nondecreasing, $\widehat{M}$ is convex in
$[0,+\infty[$. Thus, for every $u,v\in X$ with $u\neq v$, and every
$s,t\in (0,1)$ with $s+t=1$, one has
$$
\widehat{M}(T(su+tv))<\widehat{M}(sT(u)+tT(v))\leq
s\widehat{M}(T(u))+t\widehat{M}(T(v)).
$$
This shows $\Phi$ is strictly convex, and, as already said, that
$\Phi'$ is strictly monotone.

(iv) From (iii), if $u_n\rightharpoonup u$ in $X$ and
$\limsup_{n\to+\infty}\langle\Phi'(u_n)-\Phi'(u),u_n-u\rangle=0$,
then
$$
\lim_{n\to+\infty}\langle\Phi'(u_n)-\Phi'(u),u_n-u\rangle=0,
$$
and so,
\begin{equation}\label{E212}
\lim_{n\to+\infty}\langle\Phi'(u_n),u_n-u\rangle=0,
\end{equation}
that is,
\begin{equation}\label{E213}
\lim_{n\to+\infty}[M\left(\|u_n\|^p\right)]^{p-1}\int_\Omega|\nabla
u_n|^{p-2}\nabla u_n\nabla(u_n-u)+a(x)|u_n|^{p-2}u_n(u_n-u)dx=0.
\end{equation}
Since $(u_n)$ is bounded in $X$ and $M$ is continuous, up to
subsequence, there is $t_0\geq0$ such that
$$
M\left(\|u_n\|^p\right)\to M\left(t_0^p\right)\geq
m_0,\;\,\text{as}\;n\to+\infty.
$$
This and \eqref{E213} imply
\begin{equation}\label{E214}
\lim_{n\to+\infty}\int_\Omega|\nabla u_n|^{p-2}\nabla
u_n\nabla(u_n-u)+a(x)|u_n|^{p-2}u_n(u_n-u)dx=0.
\end{equation}
In a same way,
\begin{equation}\label{E215}
\lim_{n\to+\infty}\int_\Omega|\nabla u|^{p-2}\nabla
u\nabla(u_n-u)+a(x)|u|^{p-2}u(u_n-u)dx=0.
\end{equation}
Now, by using again inequality \eqref{E211}, we obtain by
\eqref{E214} and \eqref{E215},
\begin{equation}\label{E216}
\begin{aligned}
o_n(1)&= \int_\Omega\left(|\nabla u_n|^{p-2}\nabla u_n-|\nabla
u|^{p-2}\nabla u\right)\nabla(u_n-u)dx \\
&\quad +\int_\Omega a(x)\left(|u_n|^{p-2}u_n-|u|^{p-2}u\right)
(u_n-u)dx \\
&\geq \begin{cases}
C_p\int_\Omega\left(|\nabla u_n-\nabla u|^p+a(x)|u_n-u|^p\right)dx
&\text{if }p\geq 2\\
C_p\int_\Omega\left(\frac{|\nabla u_n-\nabla u|^2}{(|\nabla
u_n|+|\nabla u|)^{2-p}}+\frac{a(x)|u_n-u|^2}{(|u_n|+|u|)^{2-p}}\right)dx
&\text{if }1<p<2.
\end{cases}
\end{aligned}
\end{equation}
If $p\geq2$, we have
$$
\lim_{n\to+\infty}\int_\Omega\left(|\nabla u_n-\nabla
u|^p+a(x)|u_n-u|^p\right)dx=0.
$$
If $1<p<2$, by H\"{o}lder's inequality, it follows that By applying
H\"{o}lder's inequality, we obtain
\begin{equation}\label{E217}
\begin{aligned}
\int_\Omega|\nabla u_n-\nabla u|^pdx
&\leq \Big(\int_\Omega\frac{|\nabla u_n-\nabla
u|^2}{(|\nabla u_n|+|\nabla
u|)^{2-p}}dx\Big)^{p/2} \Big(\int_\Omega(|\nabla
u_n|+|\nabla u|)^pdx\Big)^{\frac{2-p}2} \\
&\leq \Big(\int_\Omega\frac{|\nabla u_n-\nabla u|^2}{\left(|\nabla
 u_n|+|\nabla u|\right)^{2-p}}dx\Big)^{p/2}
 \left(\|u_n\|+\|u\|\right)^{\frac{2-p}2p} \\
&\leq C\Big(\int_\Omega\frac{|\nabla u_n-\nabla u|^2}{(|\nabla
u_n|+|\nabla u|)^{2-p}}dx\Big)^{p/2}.
\end{aligned}
\end{equation}
and
\begin{equation}\label{E218}
\begin{aligned}
\int_\Omega a(x)|u_n-u|^pdx
&\leq \Big(\int_\Omega\frac{a(x)|u_n-u|^2}{\left(|u_n|+|u|\right)^{2-p}}dx
\Big)^{p/2}\Big(\int_\Omega a(x)\left(|u_n|+|u|\right)^pdx\Big)^{\frac{2-p}2} \\
&\leq \Big(\int_\Omega\frac{a(x)|u_n-u|^2}{\left(|u_n|+|u|\right)^{2-p}}dx
\Big)^{p/2}\left(\|u_n\|+\|u\|\right)^{\frac{2-p}2p} \\
&\leq C\Big(\int_\Omega\frac{a(x)|u_n-u|^2}{\left(|u_n|+|u|\right)^{2-p}}dx
\Big)^{p/2}.
\end{aligned}
\end{equation}
From \eqref{E216}-\eqref{E218}, it follows that
\begin{align*}
\|u_n-u\|^2
&= \Big(\int_\Omega\left(|\nabla u_n-\nabla
u|^p+a(x)|u_n-u|^p\right)dx\Big)^{2/p}\\
&\leq C'\Big[\Big(\int_\Omega|\nabla
u_n-\nabla u|^pdx\Big)^{2/p}+\Big(\int_\Omega a(x)|u_n-u|^pdx\Big)^{2/p}
\Big] \\
&\leq C'C^{2/p}\int_\Omega\Big(\frac{|\nabla u_n-\nabla
u|^2}{\left(|\nabla u_n|+|\nabla
u|\right)^{2-p}}+\frac{a(x)|u_n-u|^2}{\left(|u_n|+|u|\right)^{2-p}}\Big)dx\\
&\leq o_n(1).
\end{align*}
Therefore, in both cases we have
$$
\lim_{n\to+\infty}\|u_n-u\|=0,
$$
and this completes the proof of (iv).

(v) Since $\Phi'$ is a strictly monotone operator in $X$, $\Phi'$
is an injection. For $u\in X$ with $\|u\|>1$, we have
\[
\frac{\langle\Phi'(u),u\rangle}{\|u\|}
= \frac{[M(\|u\|^p)]^{p-1}\|u\|^p}{\|u\|}
\geq  m_0^{p-1}\|u\|^{p-1},
\]
therefore, $\Phi'$ is coercive. Clearly $\Phi'$ is also
demicontinuous. On account of the well-known Minty-Browder theorem
\cite[Theorem 26A]{Z}, the operator $\Phi'$ is a surjection, and
hence the inverse $(\Phi')^{-1}: X^*\to X$ of $\Phi'$
exists. It suffices then to show the continuity of $(\Phi')^{-1}$.
Let $(g_n)$ be a sequence of $X^*$ such that $g_n\to g$ in
$X^*$. Let $u_n=(\Phi')^{-1}(g_n),\,u=(\Phi')^{-1}(g)$, then
$\Phi'(u_n)=g_n,\,\Phi'(u)=g$. By the coercivity of $\Phi'$, we
deduces that $(u_n)$ is bounded in $X$, up to subsequence, we can
assume that $u_n\rightharpoonup u$. Since $g_n\to g$,
$$
\underset{n\to+\infty}\lim\langle\Phi'(u_n)-\Phi'(u),u_n-u\rangle=
\underset{n\to+\infty}\lim\langle g_n-g,u_n-u\rangle=0.
$$
Since $\Phi'$ is of type $(S_+),\,u_n\to u$, so
$(\Phi')^{-1}$ is continuous.
\end{proof}

\section{Main results}

In this section we present our main results. To be precise, we
establish an existence result of at least one solution, Theorem
\ref{theo31}, which is based on Theorem \ref{theo21}, and we point
out some consequences, Theorems \ref{theo32}, \ref{theo33} and
\ref{theo34}. Finally, we present an other existence result of at
least one solution, Theorem \ref{theo35}, which is based in turn on
Theorem \ref{theo22}.

Given two nonnegative constants $c, d$ with $c\neq
k\|a\|^{1/p}d$, put
$$
\gamma(c):=\frac{\int_\Omega\max_{|\xi|\leq\sigma(c)}F(x,\xi)dx-\int_\Omega
F(x,d)dx}{\widehat{M}(\frac{c^p}{k^p})-\widehat{M}(d^p\|a\|_1)},
$$
where
$$
\sigma(c):=k\Big(\frac1{m_0^{p-1}}\widehat{M}\big(\frac{c^p}{k^p}\big)\Big)^{1/p}.
$$

\begin{theorem}\label{theo31}
Assume that there exist three constants $c_1, c_2, d$ with
 $0\leq c_1<k\|a\|^{1/p}d<c_2$, such that
$$
\gamma(c_2)<\gamma(c_1).
$$
Then, for each
$\lambda\in\big(\frac1{p\gamma(c_1)},\frac1{p\gamma(c_2)}\big)$,
problem \eqref{E11} admits at least one nontrivial weak solution
$\overline{u}$ such that
$\frac{c_1}{k}<\|\overline{u}\|<\frac{c_2}{k}$.
\end{theorem}

\begin{proof}
Let $\Phi,\,\Psi$ be the functionals defined in Section 2. It is
well known that they satisfy all regularity assumptions requested in
Theorem \ref{theo21} and that the critical points of the functional
$\Phi-\lambda\Psi$ in X are exactly the weak solutions of problem
\eqref{E11}. So, our aim is to verify condition \eqref{E24} of
Theorem \ref{theo21}. To this end, put
$$
r_1=\frac1p\widehat{M}\Big(\frac{c_1^p}{k^p}\Big),\quad
r_2=\frac1p\widehat{M}\Big(\frac{c_2^p}{k^p}\Big),\quad u_0(x)=d
\quad \text{for all }x\in\Omega.
$$
Clearly $u_0\in X$,
$$
\Phi(u_0)=\frac1p\widehat{M}(\|u_0\|^p)
=\frac1p\widehat{M}(\|a\|_1d^p),
$$
and
\begin{equation}\label{E31}
\Psi(u_0)=\int_\Omega F(x,u_0)dx=\int_\Omega F(x,d)dx.
\end{equation}
It follows from $c_1<k\|a\|^{1/p}d<c_2$ and the strict
monotonicity of $\widehat{M}$ that
$$
\widehat{M}\Big(\frac{c_1^p}{k^p}\Big)<\widehat{M}(\|a\|_1d^p)<
\widehat{M}\Big(\frac{c_2^p}{k^p}\Big),
$$
and so
\begin{equation}\label{E32}
r_1<\Phi(u_0)<r_2.
\end{equation}
Let $u\in X$ such that $u\in \Phi^{-1}((-\infty,r_2))$. By
\eqref{E210}, one has
$$
\frac{m_0^{p-1}}p\|u\|^p\leq\Phi(u)<r_2.
$$
Therefore,
$$
\|u\|<\Big(\frac{pr_2}{m_0^{p-1}}\Big)^{1/p}
$$
This together with \eqref{E27}, yields
\begin{equation}\label{E33}
|u(x)|\leq
k\|u\|<k\Big(\frac{pr_2}{m_0^{p-1}}\Big)^{1/p}
=\sigma(c_2) \quad \text{for all }x\in\Omega.
\end{equation}
So
$$
\Psi(u)=\int_\Omega
F(x,u)dx\leq\int_\Omega\max_{|\xi|\leq\sigma(c_2)} F(x,\xi)dx,
$$
for all $u\in X$ such that $u\in \Phi^{-1}((-\infty,r_2))$. Thus
\begin{equation}\label{E34}
\sup_{u\in\Phi^{-1}((-\infty,r_2))}\Psi(u)\leq
\int_\Omega\max_{|\xi|\leq\sigma(c_2)} F(x,\xi)dx.
\end{equation}
On the other hand, arguing as before we obtain
\begin{equation}\label{E35}
\sup_{u\in\Phi^{-1}((-\infty,r_1))}\Psi(u)\leq
\int_\Omega\max_{|\xi|\leq\sigma(c_1)} F(x,\xi)dx.
\end{equation}
In view of \eqref{E31}-\eqref{E32} and \eqref{E34}-\eqref{E35}, one
has
\begin{align*}
\beta(r_1,r_2)
&\leq \frac{\sup_{u\in\Phi^{-1}((-\infty,r_2))}\Psi(u)-\Psi(u_0)}{r_2-\Phi(u_0)}\\
&\leq p\frac{\int_\Omega\max_{|\xi|\leq\sigma(c_2)}
F(x,\xi)dx-\int_\Omega F(x,d)dx}
{\widehat{M}(\frac{c_2^p}{k^p})-\widehat{M}(\|a\|_1d^p)}\\
&= p\gamma(c_2)
\end{align*}
and
\begin{align*}
\rho_2(r_1,r_2)
&\geq \frac{\Psi(u_0)-\sup_{u\in\Phi^{-1}((-\infty,r_1])}
 \Psi(u)}{\Phi(u_0)-r_1}\\
&\geq p\frac{\int_\Omega\max_{|\xi|\leq\sigma(c_1)}F(x,\xi)dx-\int_\Omega
F(x,d)dx}{\widehat{M}(\frac{c_1^p}{k^p})-\widehat{M}(\|a\|_1d^p)}\\
&= p\gamma(c_1).
\end{align*}
So, by our assumption it follows that
$$
\beta(r_1,r_2)<\rho_2(r_1,r_2).
$$
Hence, from Theorem \ref{theo21} for each
$\lambda\in\big(\frac1{p\gamma(c_1)},\frac1{p\gamma(c_2)}\big)
\subset\big(\frac1{\rho_2(r_1,r_2)}, \frac1{\beta(r_1,r_2)}\big)$, 
$I_\lambda:=\Phi-\lambda\Psi$ admits at
least one critical point $\overline{u}$ such that
$$
\widehat{M}\big(\frac{c_1^p}{k^p}\big)
<\widehat{M}\big(\|\overline{u}\|^p\big)<
\widehat{M}\big(\|\overline{u}\|^p\big).
$$
Taking in to account that the function $\widehat{M}$ is  increasing,
it follows that
$$
\frac{c_1}{k}<\|\overline{u}\|<\frac{c_2}{k},
$$
and the proof of Theorem \ref{theo31} is achieved.
\end{proof}

Now we point out the following consequence of Theorem \ref{theo31}.

\begin{theorem}\label{theo32}
Assume that there exist two positive constants $c, d$, with
$k\|a\|^{1/p}d<c$, such that
\begin{equation}\label{E36}
\frac{\int_\Omega\max_{|\xi|\leq\sigma(c)}F(x,\xi)dx}
{\widehat{M}\big(\frac{c^p}{k^p}\big)}
<\frac{\int_\Omega F(x,d)dx}{\widehat{M}(\|a\|_1d^p)}.
\end{equation}
Then, for each
\[
\lambda\in\big(\frac{\widehat{M}(\|a\|_1d^p)}{p\int_\Omega
F(x,d)dx},\frac{\widehat{M}\left(\frac{c^p}{k^p}\right)}
{p\int_\Omega\max_{|\xi|\leq\sigma(c)}F(x,\xi)dx}\big),
\]
problem \eqref{E11} admits at least one nontrivial weak solution
$\overline{u}$ such that $|\overline{u}(x)|<c$ for all $x\in\Omega$.
\end{theorem}

\begin{proof}
Our aim is to apply Theorem \ref{theo31}. To this end we pick
$c_1=0$ and $c_2=c$. From \eqref{E36}, one has
\begin{align*}
\gamma(c)&= \frac{\int_\Omega\max_{|\xi|\leq\sigma(c)}F(x,\xi)dx-\int_\Omega
F(x,d)dx}{\widehat{M}(\frac{c^p}{k^p})-\widehat{M}(\|a\|_1d^p)}\\
&< \frac{\int_\Omega\max_{|\xi|\leq\sigma(c)}F(x,\xi)dx-\frac{\widehat{M}(\|a\|_1d^p)}
{\widehat{M}(\frac{c^p}{k^p})}\int_\Omega\max_{|\xi|\leq\sigma(c)}F(x,\xi)dx}
{\widehat{M}(\frac{c^p}{k^p})-\widehat{M}(\|a\|_1d^p)}\\
&= \frac{\int_\Omega\max_{|\xi|\leq\sigma(c)}F(x,\xi)dx}
 {\widehat{M}(\frac{c^p}{k^p})}\\
&<\frac{\int_\Omega F(x,d)dx}{\widehat{M}(\|a\|_1d^p)}
= \gamma(0).
\end{align*}
Hence, Theorem \eqref{theo31} ensures the existence of weak solution
$\overline{u}$ of problem \eqref{E11}, such that
$\|\overline{u}\|<\frac{c}{k}$, and clearly by \eqref{E27},
$|\overline{u}(x)|< c$ for all $x\in\Omega$.
\end{proof}

Now, we point out  a previous result when the nonlinear term has
separable variables. To be precise, let $\alpha\in L^1(\Omega)$ such
that $\alpha(x)\geq0$ a.e. $x\in\Omega,\alpha\neq0$, and let
$g:\mathbb{R}\to\mathbb{R}$ be a continuous function.
Consider the  Neumann boundary-value problem
\begin{equation}\label{E37}
\begin{gathered}
\Big[M\Big(\int_\Omega\big(|\nabla
u|^p+a(x)|u|^p\big)dx\Big)\Big]^{p-1}
\big(-\Delta_pu+a(x)|u|^{p-2}u\big)=\lambda\alpha(x)g(u)
\quad \text{in }\Omega\\
\frac{\partial u}{\partial\nu}=0 \quad \text{on }\partial\Omega.
\end{gathered}
\end{equation}
Let
$$
G(\xi):=\int_0^\xi g(t)dt\quad \text{for all }\xi\in\mathbb{R}.
$$

\begin{theorem}\label{theo33}
Assume that $g$ is nonnegative and there exist two positive constants
$c, d$, with $k\|a\|^{1/p}d<c$, such that
\begin{equation}\label{E38}
\frac{G(\sigma(c))}{\widehat{M}(\frac{c^p}{k^p})}
<\frac{G(d)}{\widehat{M}(\|a\|_1d^p)}.
\end{equation}
Then, for each
$\lambda\in\big(\frac1{p\|\alpha\|_1}\frac{\widehat{M}(\|a\|_1d^p)}{G(d)},
\frac1{p\|\alpha\|_1}\frac{\widehat{M}(\frac{c^p}{k^p})}
{G(\sigma(c))}\big)$,
problem \eqref{E37} admits at least one positive weak solution
$\overline{u}$ such that $\overline{u}(x)<c$ for all $x\in\Omega$.
\end{theorem}

\begin{proof}
Put $f(x,t)=\alpha(x)g(t)$ for all $(x,t)\in\Omega\times\mathbb{R}$,
thus $F(x,\xi)=\alpha(x)G(\xi)$ for all
$(x,\xi)\in\Omega\times\mathbb{R}$. Therefore taking into account
that $G$ is nondecreasing, Theorem \ref{theo32} ensures the
existence of a nontrivial weak solution $\overline{u}$. We claim
that $\overline{u}$ is nonnegative. In fact, let
$\overline{u}_-:=\max\{-\overline{u},0\}$ and setting
$$
\Omega^-=\{x\in\Omega: \overline{u}(x)<0\}.
$$
So, taking into account that $\overline{u}$ is a weak solution and
$\overline{u}_-\in X$, we have
\[
[M(\|\overline{u}\|^p)]^{p-1}\int_{\Omega^-}(|\nabla
\overline{u}|^p+a(x)|\overline{u}|^p)dx
= \lambda\int_{\Omega^-}f(x,\overline{u})\overline{u}dx
\leq 0.
\]
Therefore,
$$
\int_{\Omega^-}\left(|\nabla
\overline{u}|^p+a(x)|\overline{u}|^p\right)dx=0.
$$
It follows that $\Omega^-=\emptyset$, and hence $u\geq0$ in
$\Omega$. By the strong maximum principle (see, for instance,
\cite[Theorem 11.1]{PS}) the weak solution $\overline{u}$, being
nontrivial, is positive and the conclusion is achieved.
\end{proof}

\begin{theorem}\label{theo34}
Assume that $g$ is nonnegative such that
\begin{equation}\label{E39}
\lim_{t\to0^+} \frac{g(t)}{t^{p-1}}=+\infty,
\end{equation}
and put
$$
\lambda^*=\frac{1}{p\|\alpha\|_1}\sup_{c>0}
\frac{\widehat{M}(\frac{c^p}{k^p})}{G(\sigma(c))}.
$$
Then, for each $\lambda\in (0,\lambda^*)$, problem \eqref{E37}
admits at least one positive weak solution.
\end{theorem}
\begin{proof}
Fix $\lambda\in (0,\lambda^*)$. Then, there exists $c>0$ such that
$\lambda<\frac{1}{p\|\alpha\|_1}\frac{\widehat{M}
(\frac{c^p}{k^p})}{G(\sigma(c))}$.
By \eqref{E39}, one has
$$
\lim_{t\to0^+} \frac{g(t)}{t^{p-1}[M(t^p\|a\|_1)]^{p-1}}=+\infty,
$$
and hence, there exists $0<d<\frac{c}{k\|a\|_1^{1/p}}$ such that
$$
\frac{\|a\|_1}{\lambda
\|\alpha\|_1}<\frac{g(t)}{t^{p-1}[M(t^p\|a\|_1)]^{p-1}}\quad
\text{for all } t\in (0,d).
$$
Thus
$$
\frac{\|a\|_1}{\lambda \|\alpha\|_1}\int_0^d
t^{p-1}\left[M\left(t^p\|a\|_1\right)\right]^{p-1}dt<\int_0^dg(t)dt.
$$
Using the change of variables $s=\|a\|_1t^p$, we get
$$
\frac{1}{\lambda p\|\alpha\|_1}\int_0^{\|a\|_1d^p}
\left[M(s)\right]^{p-1}ds<G(d),
$$
that is,
$$
\frac{1}{p\|\alpha\|_1}\frac{\widehat{M}(\|a\|_1d^p)}{G(d)}<\lambda.
$$
Hence, Theorem \ref{theo33} ensures the conclusion.
\end{proof}

\begin{remark} \rm
Let $g: \mathbb{R}\to\mathbb{R}$ such \eqref{E39} holds
(that is, without any assumption of sign). By \eqref{E39}, there
is $\delta>0$ such that $g(t)>0$ for all $t\in (0,\delta)$. Then Put
$$
\overline{\lambda}_0:=\frac{1}{p\|\alpha\|_1}\sup_{c\in (0,\delta)}
\frac{\widehat{M}\left(\frac{c^p}{k^p}\right)}{G(\sigma(c))}.
$$
Clearly $\overline{\lambda}_0\leq\lambda^*$, if $g$ is nonnegative.
Now, fixed $\lambda\in (0,\overline{\lambda}_0)$ and arguing as in
the proof of Theorem \ref{theo34}, there are $c\in (0,\delta)$ and
$0<d<\frac{c}{k\|a\|_1^{1/p}}$ such that
$$
\frac{1}{p\|\alpha\|_1}\frac{\widehat{M}(\|a\|_1d^p)}{G(d)}
<\lambda<\frac{1}{p\|\alpha\|_1}\frac{\widehat{M}
\left(\frac{c^p}{k^p}\right)}{G(\sigma(c))}.
$$
Hence, Theorem \ref{theo33} ensures that, for each
$\lambda\in (0, \overline{\lambda}_0)$, problem \eqref{E37}
admits at least one positive weak solution $\overline{u}(x)<\delta$ for all
$x\in\Omega$.
\end{remark}

Finally, we also give an application of Theorem \ref{theo22} which
we will use in next section to obtain multiple solutions.

\begin{theorem}\label{theo35}
Assume that there exist two constants $\overline{c}, \overline{d}$,
with $0<\overline{c}<k\|a\|_1^{1/p}\overline{d}$, such that
\begin{gather}\label{E310}
\int_\Omega
\max_{|\xi|\leq\sigma(\overline{c})}F(x,\xi)dx<\int_\Omega
F(x,\overline{d})dx,\\
\label{E311}
\limsup_{|\xi|\to+\infty}\frac{F(x,\xi)}{|\xi|^p}\leq0\quad
\text{uniformly in } x.
\end{gather}
Then, for each $\lambda>\overline{\lambda}$, where
$$
\overline{\lambda}=\frac{\widehat{M}\left(\frac{\overline{c}^p}{k^p}\right)
-\widehat{M}(\|a\|_1\overline{d}^p)}
{p\left(\int_\Omega
\max_{|\xi|\leq\sigma(\overline{c})}F(x,\xi)dx-\int_\Omega
F(x,\overline{d})dx\right)},
$$
problem \eqref{E11} admits at least one nontrivial weak solution
$\overline{u}$ such that $\|u\|>\overline{c}/k$.
\end{theorem}

\begin{proof}
The functionals $\Phi$ and $\Psi$ given by \eqref{E29} satisfy all
regularity assumptions requested in Theorem \ref{theo22}. By
\eqref{E311}, for every $\varepsilon>0$ one has
$$
F(x,\xi)\leq \varepsilon |\xi|^p+l_\varepsilon(x)\quad
\text{for all } (x,\xi)\in\Omega\times\mathbb{R},
$$
where $l_\varepsilon\in L^1(\Omega)$. This implies that
$$
\int_\Omega F(x,u)dx\leq\varepsilon C_1\|u\|^p+\int_\Omega
l_\varepsilon(x)dx\quad \text{for all } u\in X,
$$
where $C_1$ is a Sobolev constant. Therefore,
\[
I_\lambda(u)= \Phi(u)-\lambda\Psi(u)
\geq \Big(\frac{m_0^{p-1}}p-C_1\varepsilon\Big)\|u\|^p-\int_\Omega
l_\varepsilon(x)dx.
\]
So, choosing $\varepsilon$ small enough we deduce that $I_\lambda$
is coercive. To apply Theorem \ref{theo22}, it suffices to verify
condition \eqref{E25}. Indeed, put
$$
r=\frac1p\widehat{M}\Big(\frac{\overline{c}^p}{k^p}\Big),\quad
u_0(x)=\overline{d}\quad \text{for all }x\in\Omega.
$$
Arguing as in the proof of Theorem \ref{theo31}, we obtain
$$
\rho(r)\geq p\frac{\int_\Omega\max_{|\xi|\leq
\sigma(\overline{c})}F(x,\xi)dx-\int_\Omega
F(x,\overline{d})dx}{\widehat{M}(\frac{\overline{c}^p}{k^p})
-\widehat{M}\big(\|a\|_1\overline{d}^p\big)}.
$$
So, from our assumption it follows that $\rho(r)>0$. Hence, in view
of Theorem \ref{theo22} for each
$\lambda>\overline{\lambda},\,I_\lambda$ admits at least one local
minimum $\overline{u}$ such that
$$
\widehat{M}\Big(\frac{\overline{c}^p}{k^p}\Big)
<\widehat{M}\big(\|\overline{u}\|^p\big).
$$
Therefore,
$$
\frac{\overline{c}}{k}<\|\overline{u}\|,
$$
and our conclusion is achieved.
\end{proof}

\section{Applications}

The main aim of this section is to present multiplicity results.
First, as a consequence of Theorems \ref{theo32}, and \ref{theo35}
the following theorem of the existence of three solutions is
obtained and its consequence for the nonlinearity with separable
variables is presented.

\begin{theorem}\label{theo41}
Assume that \eqref{E311} holds. Moreover, assume that there exist
four positive constants $c, d, \overline{c}, \overline{d}$, with
$k\|a\|_1^{1/p}d<c\leq
\overline{c}<k\|a\|_1^{1/p}\overline{d}$, such that
\eqref{E36}, \eqref{E310} and
\begin{equation}\label{E41}
\frac{\int_\Omega\max_{|\xi|\leq\sigma(c)}
F(x,\xi)dx}{\widehat{M}\left(\frac{c^p}{k^p}\right)}
<\frac{\int_\Omega\max_{|\xi|\leq
\sigma(\overline{c})} F(x,\xi)dx-\int_\Omega
F(x,\overline{d})dx}{\widehat{M}\left(\frac{\overline{c}^p}{k^p}\right)
-\widehat{M}\big(\|a\|_1\overline{d}^p\big)}
\end{equation}
are satisfied.
Then, for each
\[
\lambda\in\Lambda:=\Big(\max\big\{\overline{\lambda},
\frac{\widehat{M}(\|a\|_1d^p)}
{p\int_\Omega F(x,d)dx}\big\},
\frac{\widehat{M}\big(\frac{c^p}{k^p}\big)}{p\int_\Omega\max_{|\xi|\leq
\sigma(c)}F(x,\xi)dx}\Big),
\]
 with $\overline{\lambda}$ is given in
Theorem \ref{theo35}, problem \eqref{E11} admits at least three
weak solutions.
\end{theorem}

\begin{proof}
By assumptions, we see that $\Lambda\neq\emptyset$. Fix
$\lambda\in\Lambda$. Theorem \ref{theo32} ensures a nontrivial weak
solution $\overline{u}_1$ such that $\|\overline{u}_1\|<\frac{c}{k}$
which is a local minimum for $I_\lambda$, as well as Theorem
\ref{theo35} guarantees a nontrivial weak solution $\overline{u}_2$
such that $\|\overline{u}_2\|>\frac{\overline{c}}{k}$ which is a
local minimum for $I_\lambda$. Hence $I_\lambda$ has two different
local minimum points. By standard arguments, we see that $I_\lambda$
satisfies the Palais-Smale condition. Hence, the theorem given by
Pucci and Serrin \cite[Corollary 1]{PS0} ensures the third weak
solution and the proof is achieved.
\end{proof}

\begin{theorem}\label{theo42}
Assume that $g$ is a nonnegative function such that
\begin{gather}\label{E42}
\limsup_{\xi\to0^+}\frac{G(\xi)}{\widehat{M}\big(\|a\|_1\xi^p\big)}=+\infty,\\
\label{E43}
\limsup_{\xi\to+\infty}\frac{G(\xi)}{\xi^p}=0.
\end{gather}
Further, assume that there exist two positive constants
$\overline{c}, \overline{d}$, with
$\overline{c}<k\left(\|a\|\right)^{1/p}\overline{d}$, such that
\begin{equation}\label{E44}
\frac{G(\sigma(\overline{c}))}{\widehat{M}\left(\frac{\overline{c}^p}{k^p}\right)}<
\frac{G(\overline{d})}{\widehat{M}\big(\|a\|_1\overline{d}^p\big)}.
\end{equation}
Then, for each
$\lambda\in\Big(\frac{1}{p\|\alpha\|_1}\frac{\widehat{M}\big(\|a\|_1\overline{d}^p\big)}{G(\overline{d})},
\frac1{p\|\alpha\|_1}\frac{\widehat{M}\left(\frac{\overline{c}^p}{k^p}\right)}{G(\sigma(\overline{c}))}\Big)$,
problem \eqref{E37} admits at least three nonnegative weak
solutions.
\end{theorem}

\begin{proof}
Put $f(x,t)=\alpha(x)g(t)$ for all $(x,t)\in\Omega\times\mathbb{R}$,
then $F(x,\xi)=\alpha(x)G(\xi)$ for all
$(x,\xi)\in\Omega\times\mathbb{R}$. By \eqref{E43} and taking into
account that $g$ is nonnegative, it is easy to verify condition
\eqref{E311}. Choosing $c=\overline{c}$, condition \eqref{E42}
ensures the existence of positive constant $d$, with
$d<\frac{c}{k\|a\|_1^{1/p}}$ such that
\begin{equation}\label{E45}
\frac{G(\sigma(c))}{\widehat{M}\left(\frac{c^p}{k^p}\right)}
<\frac{G(\overline{d})}
{\widehat{M}\big(\|a\|_1\overline{d}^p\big)}
<\frac{G(d)}{\widehat{M}(\|a\|_1d^p)}.
\end{equation}
This implies \eqref{E36}. Since
$\frac{\overline{c}^p}{k^p}<\|a\|_1\overline{d}^p$ and the function
$\widehat{M}$ is increasing,
$$
\widehat{M}\Big(\frac{\overline{c}^p}{k^p}\Big)
<\widehat{M}\big(\|a\|_1\overline{d}^p\big).
$$
Therefore, from \eqref{E44}, we deduce
$G(\sigma(\overline{c}))<G(\overline{d})$, and hence \eqref{E310}
follows. Using again \eqref{E44}, one has
\begin{align*}
\frac{\int_\Omega\max_{|\xi|\leq \sigma(\overline{c})}
F(x,\xi)dx-\int_\Omega
F(x,\overline{d})dx}{\widehat{M}\left(\frac{\overline{c}^p}{k^p}\right)
-\widehat{M}\big(\|a\|_1\overline{d}^p\big)}
&= \|\alpha\|_1\frac{G(\sigma(\overline{c}))
-G(\overline{d}))}{\widehat{M}\left(\frac{\overline{c}^p}{k^p}\right)
-\widehat{M}\big(\|a\|_1\overline{d}^p\big)}\\
&> \|\alpha\|_1\frac{G(\sigma(\overline{c}))
\big(1-\frac{\widehat{M}\big(\|a\|_1\overline{d}^p\big)}
{\widehat{M}\big(\frac{\overline{c}^p}{k^p}\big)}\big)}
{\widehat{M}\left(\frac{\overline{c}^p}{k^p}\right)
-\widehat{M}\big(\|a\|_1\overline{d}^p\big)}\\
&= \|\alpha\|_1\frac{G(\sigma(\overline{c}))}{\widehat{M}\left(\frac{\overline{c}^p}{k^p}\right)}\\
&= \frac{\int_\Omega\max_{|\xi|\leq\sigma(\overline{c})}
F(x,\xi)dx}{\widehat{M}\left(\frac{\overline{c}^p}{k^p}\right)}\\
&= \frac{\int_\Omega\max_{|\xi|\leq\sigma(c)}
F(x,\xi)dx}{\widehat{M}\left(\frac{c^p}{k^p}\right)},
\end{align*}
so, \eqref{E41} holds. Also, by \eqref{E45} one has
\begin{align*}
\overline{\lambda}
&= \frac{\widehat{M}\left(\frac{\overline{c}^p}{k^p}\right)
-\widehat{M}\big(\|a\|_1\overline{d}^p\big)}
{p\left(\int_\Omega\max_{|\xi|\leq\sigma(\overline{c})}F(x,\xi)dx-\int_\Omega
F(x,\overline{d})dx\right)}\\
&< \frac1{p\|\alpha\|_1}\frac{\widehat{M}\big(\|a\|_1\overline{d}^p\big)}
{G(\overline{d})}.
\end{align*}
Therefore,
$$
\max\big\{\overline{\lambda},\frac{\widehat{M}(\|a\|_1d^p)}
{p\|\alpha\|_1G(d)}\big\}<\frac1{p\|\alpha\|_1}
\frac{\widehat{M}\big(\|a\|_1\overline{d}^p\big)}{G(\overline{d})},
$$
thus
$\Big(\frac{1}{p\|\alpha\|_1}\frac{\widehat{M}\big(\|a\|_1\overline{d}^p\big)}{G(\overline{d})},
\frac1{p\|\alpha\|_1}\frac{\widehat{M}
\left(\frac{\overline{c}^p}{k^p}\right)}{G(\sigma(\overline{c}))}\Big)
\subset\Lambda$, and hence, Theorem \ref{theo41} ensures three
nonnegative weak solutions.
\end{proof}

\begin{remark}\rm
If $g(0)\neq0$, Theorem \ref{theo42} ensures three positive weak
solutions (see proof of Theorem \ref{theo33}).
\end{remark}

\begin{remark} \rm
In applying Theorem \ref{theo34}, it is enough to known an
explicit upper bound for constant $k$ defined in \eqref{E26}). If
$\Omega$ is convex, we have the following estimate
(see \cite[Remark 1]{BC})
\begin{equation}\label{E46}
k\leq
2^{\frac{p-1}p}\max\Big\{\frac1{\|a\|_1^{1/p}},
\frac{\operatorname{diam}(\Omega)}{N^{1/p}}\Big(\frac{p-1}{p-N}
\operatorname{meas}(\Omega)\Big)^\frac{p-1}{p}
\frac{\|a\|_\infty}{\|a\|_1}\Big\}.
\end{equation}
\end{remark}

\subsection*{Example} Let $b_0,b_1>0$. Due to  Theorem
\ref{theo34}, for each
\[
\lambda \in \Big(0,\frac12\frac{b_0+\frac{b_1}2}
{\frac13\big(2+\frac{b_1}{b_0}\big)^{3/2}
+\big(2+\frac{b_1}{b_0}\big)^{1/2}}\Big),
\]
 the Neumann problem
\begin{equation}\label{E47}
\begin{gathered}
\Big(b_0+b_1\int_0^1(|\nabla u|^2+|u|^2)dx\Big)
\left(-u''+u\right)=\lambda\left(u^2+1\right)\quad \text{in } (0,1)\\
u'(0)=u'(1),
\end{gathered}
\end{equation}
admits at least one positive weak solution. In fact, set
$M(t)=b_0+b_1t$ for all $t\geq0$, then
$\widehat{M}(t)=b_0t+\frac{b_1}2t^2$ for all $t\geq0$ and (M0)
holds. Observe that
$$
\lim_{u\to0^+}\frac{g(u)}u=\lim_{u\to0^+}\frac{u^2+1}u=+\infty.
$$
Moreover, one has
$$
\sigma(k)=k\Big(\frac1{b_0}\widehat{M}(1)\Big)^{1/2}
=k\Big(1+\frac{b_1}{2b_0}\Big)^{1/2}
$$
Therefore,
\begin{align*}
\lambda^*&= \frac{1}{p\|\alpha\|_1}\sup_{c>0}\frac{\widehat{M}\left(\frac{c^p}{k^p}\right)}{G(\sigma(c))}\\
&\geq \frac{1}{p\|\alpha\|_1}\frac{\widehat{M}\left(1\right)}{G(\sigma(k))}\\
&= \frac12\frac{b_0+\frac{b_1}2}
{G\big(k\big(1+\frac{b_1}{2b_0}\big)^{1/2}\big)}
\end{align*}
Taking into account that  estimate \eqref{E46} implies
$k\leq\sqrt2$, we deduce that
\[
\lambda^*
\geq \frac12\frac{b_0+\frac{b_1}2}
{G\big(\big(2+\frac{b_1}{b_0}\big)^{1/2}\big)}
= \frac12\frac{b_0+\frac{b_1}2}
{\frac13\big(2+\frac{b_1}{b_0}\big)^{3/2}
+\big(2+\frac{b_1}{b_0}\big)^{1/2}},
\]
and Theorem \ref{theo34} ensures the conclusion.

\subsection*{Acknowledgements}
The author would like to thank the anonymous referee for his/her
helpful comments and suggestions.

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