\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: $$\label{E11} \begin{gathered} \Big[M\Big(\int_\Omega(|\nabla u|^p+a(x)|u|^p)dx\Big)\Big]^{p-1}\Big(-\Delta_pu+a(x)|u|^{p-2}u\Big) =\lambda f(x,u)\quad \text{in }\Omega\\ \frac{\partial u}{\partial\nu}=0\quad \text{on }\partial\Omega, \end{gathered}$$ 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 $$\label{E12} \rho\frac{\partial^2u}{\partial t^2}-\Big(\frac{\rho_0}h+\frac{E}{2L}\int_0^L|\frac{\partial u}{\partial x}|^2dx\Big)\frac{\partial^2u}{\partial x^2}=0,$$ 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 $$\label{E13} u_{tt}-M\Big(\int_\Omega|\nabla u|^2dx\Big)\Delta u=f(x,u)\quad\text{in}\,\Omega.$$ After that, many authors studied the following nonlocal elliptic boundary value problem $$\label{E14} -M\Big(\int_\Omega|\nabla u|^2dx\Big)\Delta u=f(x,u)\quad\text{in }\Omega,\quad u=0\quad\text{on }\partial\Omega.$$ 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 $r_10, where$\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 $$\label{E26} k:=\sup_{u\in X\setminus\{0\}} \frac{\max_{x\in\overline{\Omega}} |u(x)|}{\|u\|}.$$ Since$p>N$,$X$is compactly embedded in$C^0(\overline{\Omega})$, so that$k<\infty$. We have $$\label{E27} |u(x)|\leq k\|u\|\,\quad\text{for all }x\in\Omega,\;u\in X.$$ 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 $$\label{E29} \Phi(u)=\frac1p\widehat{M}\left(\|u\|^p\right),\quad \Psi(u)=\int_\Omega F(x,u)dx,$$ 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 $$\label{E210} \Phi(u)=\frac1p\widehat{M}\left(\|u\|^p\right) \geq\frac{m_0^{p-1}}p\|u\|^p.$$ 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$C_psuch that \label{E211} \langle|x|^{p-2}x-|y|^{p-2}y,x-y\rangle \geq \begin{cases} C_p|x-y|^p &\text{if }p\geq2\\ C_p\frac{|x-y|^2}{(|x|+|y|)^{p-2}},\;(x,y)\neq(0,0) &\text{if }10, \end{align*} for allu\neq v\in X$, 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, $$\label{E212} \lim_{n\to+\infty}\langle\Phi'(u_n),u_n-u\rangle=0,$$ that is, $$\label{E213} \lim_{n\to+\infty}[M\left(\|u_n\|^p\right)]^{p-1}\int_\Omega|\nabla u_n|^{p-2}\nabla u_n\nabla(u_n-u)+a(x)|u_n|^{p-2}u_n(u_n-u)dx=0.$$ Since$(u_n)$is bounded in$X$and$M$is continuous, up to subsequence, there is$t_0\geq0such 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 $$\label{E214} \lim_{n\to+\infty}\int_\Omega|\nabla u_n|^{p-2}\nabla u_n\nabla(u_n-u)+a(x)|u_n|^{p-2}u_n(u_n-u)dx=0.$$ In a same way, $$\label{E215} \lim_{n\to+\infty}\int_\Omega|\nabla u|^{p-2}\nabla u\nabla(u_n-u)+a(x)|u|^{p-2}u(u_n-u)dx=0.$$ Now, by using again inequality \eqref{E211}, we obtain by \eqref{E214} and \eqref{E215}, \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 }11, 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_10} \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, $$and hence, there exists 00 such that g(t)>0 for all t\in (0,\delta). Then Put$$ \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))}. $$Clearly \overline{\lambda}_0\leq\lambda^*, if g is nonnegative. Now, fixed \lambda\in (0,\overline{\lambda}_0) and arguing as in the proof of Theorem \ref{theo34}, there are c\in (0,\delta) and 0\overline{\lambda}, where$$ \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\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} \|\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})}, $$thus \Big(\frac{1}{p\|\alpha\|_1}\frac{\widehat{M}\big(\|a\|_1\overline{d}^p\big)}{G(\overline{d})}, \frac1{p\|\alpha\|_1}\frac{\widehat{M} \left(\frac{\overline{c}^p}{k^p}\right)}{G(\sigma(\overline{c}))}\Big) \subset\Lambda, and hence, Theorem \ref{theo41} ensures three nonnegative weak solutions. \end{proof} \begin{remark}\rm If g(0)\neq0, Theorem \ref{theo42} ensures three positive weak solutions (see proof of Theorem \ref{theo33}). \end{remark} \begin{remark} \rm In applying Theorem \ref{theo34}, it is enough to known an explicit upper bound for constant k defined in \eqref{E26}). If \Omega is convex, we have the following estimate (see \cite[Remark 1]{BC}) $$\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\}.$$ \end{remark} \subsection*{Example} Let b_0,b_1>0. Due to Theorem \ref{theo34}, for each $\lambda \in \Big(0,\frac12\frac{b_0+\frac{b_1}2} {\frac13\big(2+\frac{b_1}{b_0}\big)^{3/2} +\big(2+\frac{b_1}{b_0}\big)^{1/2}}\Big),$ the Neumann problem $$\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}$$ admits at least one positive weak solution. In fact, set M(t)=b_0+b_1t for all t\geq0, then \widehat{M}(t)=b_0t+\frac{b_1}2t^2 for all t\geq0 and (M0) holds. Observe that$$ \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} impliesk\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. \begin{thebibliography}{00} \bibitem{ACM} C. O. Alves, F. J. S. A. Corr\^{e}a, T. F. MA; \emph{Positive solutions for a quasilinear elliptic equation of Kirchhoff type}, Comput. Math. Appl, 49 (2005), 85-93. \bibitem{A} G. Anello; \emph{A uniqueness result for a nonlocal equation of Kirchhoff type and some related open problems}, J. Math. Anal. Appl. 373 (2011) 248-251. \bibitem{APS} G. Autuori, P. Pucci, M. C. Salvatori; \emph{Asymptotic Stability for anisotropic Kirchhoff system}, J. Math. Anal. Appl. 352 (2009) 149-165. \bibitem{BS} G. Bonanno, A. Sciammetta; \emph{Existence and multiplicity results to Neumann problems for elliptic equations involving the p-Laplacian}, J. Math. Anal. Appl. 390 (2012) 59-67. \bibitem{B} G. Bonanno; \emph{A critical point theorem via the Ekeland variational principle}, Nonlinear Anal. 75 (2012) 2992-3007. \bibitem{BC} G. Bonanno, P. Candito; \emph{Three solutions to a Neumann problem for elliptic equations involving the p-Laplacian}, Arch. Math. (Basel) 80 (2003) 424-429. \bibitem{HMT} M. Massar, A. Hamydy, N. Tsouli; \emph{Existence of solutions for p-Kirchhoff type problems with critical exponent}, Electronic Journal of Differential Equations, Vol. 2011 (2011), No. 105, pp. 1-8. \bibitem{HZ} X. He, W. Zou; \emph{Infinitely many positive solutions for Kirchhoff-type problems}, Nonlinear Analysis. Theory, Methods and Applications, 70, 3, (2009), 1407-1414. \bibitem{K} G. Kirchhoff; \emph{Mechanik}, Teubner, leipzig, Germany, 1883. \bibitem{MZ} A. Mao, Z. Zhang; \emph{Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition}, Nonlinear Analysis. Theory, Methods Applications, 70, 3, (2009), 1275-1287. \bibitem{PS0} P. Pucci, J. Serrin; \emph{A mountain pass theorem}, J. Differential Equations 63 (1985) 142-149. \bibitem{PS} P. Pucci, J. Serrin; \emph{The strong maximum principle revisited}, J. Differential Equations 196 (2004) 1-66. \bibitem{R} B. Ricceri; \emph{On an elliptic Kirchhoff-type problem depending on two parameters}, J. Global Optim. 46 (2010) 543-549. \bibitem{R1} B. Ricceri; \emph{A general variational principle and some of its applications}, J. Comput. Appl. Math. 113 (2000) 401–410. \bibitem{S} J. Simon; \emph{R\'egularit\'e de la solution d'une \'equationnon lin\'eaire dans$\mathbb{R}^N\$}, in Journ\'ees d'Analyse Non Lin\'eaire (Proc. Conf., Besançon, 1977), in: Lecture Notes in Math., vol.665, Springer, Berlin, 1978, pp. 205-227. \bibitem{Z} E. Zeidler; \emph{Nonlinear Functional Analysis and its Applications}, vol. II/B, Berlin, Heidelberg, New York, 1985. \end{thebibliography} \end{document}