\documentclass[11pt]{amsart}
%\usepackage{graphicx, color}
\usepackage{amscd}
\usepackage{amsmath,empheq}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{mathrsfs}
\usepackage{color}

\textwidth=6in \textheight=9.5in \topmargin=-0.5cm
\oddsidemargin=0.5cm \evensidemargin=0.5cm
%\usepackage[notref,notcite]{showkeys}

\newtheorem{theorem}{Theorem}
\newtheorem{ex}{Example}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{prop}[theorem]{Proposition}
\newtheorem{remark}{Remark}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{claim}{Claim}
\newtheorem{step}{Step}
\newtheorem{case}{Case}
\newtheorem{definition}[theorem]{Definition}

%\newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
\newenvironment{proof-sketch}{\noindent{\bf Sketch of Proof}\hspace*{1em}}{\qed\bigskip}

\newcommand{\eqname}[1]{\tag*{#1}}% Name of equation


%\numberwithin{equation}{section}
\newcommand{\RR}{\mathbb R}
\newcommand{\NN}{\mathbb N}
\newcommand{\PP}{\mathbb P}
\newcommand{\ZZ}{\mathbb Z}
\newcommand{\di}{\displaystyle}
\renewcommand{\le}{\leqslant}
\renewcommand{\leq}{\leqslant}
\renewcommand{\ge}{\geqslant}
\renewcommand{\geq}{\geqslant}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\baselineskip=16pt plus 1pt minus 1pt

\begin{document}
%\hfill\today\bigskip

\title[Nonlinear Robin eigenvalue problems]{Positive solutions of nonlinear Robin eigenvalue problems}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\author[N.S. Papageorgiou]{Nikolaos S. Papageorgiou}
\address{National Technical University, Department of Mathematics,
				Zografou Campus, Athens 15780, Greece}
\email{\tt npapg@math.ntua.gr}

\author[V.D. R\u{a}dulescu]{Vicen\c{t}iu D. R\u{a}dulescu}
\address{Institute of Mathematics ``Simion Stoilow" of the Romanian Academy, P.O. Box 1-764,
          014700 Bucharest, Romania \& Department of Mathematics, University of Craiova, Street A.I. Cuza 13,
          200585 Craiova, Romania}
\email{\tt vicentiu.radulescu@math.cnrs.fr}

\keywords{Superdiffusive reaction, bifurcation type theorem, extremal positive solutions, $p$-Laplacian, nonlinear regularity.\\
\phantom{aa} 2010 AMS Subject Classification:
35J66, 35J92}


\begin{abstract}
We consider a nonlinear eigenvalue problem driven by the $p$-Laplacian with Robin boundary condition. Using variational methods and truncation techniques, we prove a bifurcation-type result describing the set of positive solutions as the positive parameter $\lambda$ varies. We also produce extremal positive solutions and study their properties.
\end{abstract}

\maketitle

%\tableofcontents

\section{Introduction}

Let $\Omega\subseteq \RR^N$ be a bounded domain with a $C^2$-boundary $\partial\Omega$. In this paper, we study the following nonlinear parametric Robin problem
\begin{equation}
		\left\{\begin{array}{ll}
			-\Delta_{p}u(z)=\lambda f(z,u(z))&\mbox{in}\ \Omega,\\
			\di\frac{\partial u}{\partial n_p}+\beta(z)u(z)^{p-1}=0&\mbox{on}\ \partial\Omega,\\
			u>0, \lambda>0,\ 1<p<\infty.
		\end{array}\right\}\tag{$ P_{\lambda}$} \label{eqP}
\end{equation}

Here $\Delta_p$ denotes the $p$-Laplacian differential operator defined by
$$\Delta_p u=\mbox{div}\,(|Du|^{p-2}Du)\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$

Also, $\frac{\partial u}{\partial n_p}$ denotes the generalized normal derivative defined by
$$\frac{\partial u}{\partial n_p}=|Du|^{p-2}(Du,n)_{\RR^N},$$
with $n(\cdot)$ being the outward unit normal on $\partial\Omega$.

The reaction $f(z,x)$ is a Carath\'eodory function (that is, for all $x\in\RR$, the mapping $z\mapsto f(z,x)$ is measurable and for almost all $z\in\Omega$, the function $x\mapsto f(z,x)$ is continuous), which satisfies certain asymptotic conditions as $x\rightarrow+\infty$ and as $x\rightarrow 0^+$. These conditions incorporate as a special case the $p$-logistic equation of superdiffusive type.

{\color{red}Our aim in this paper is to produce a precise description of the dependence of positive solutions on the parameter $\lambda>0$.
When $f(z,x)=x^{\tau -1}-x^{r-1}$ ($x\geq 0$), we have the classical $p$-logistic equation, which is important in problems of population dynamics (mathematical biology) and in the study of reaction-diffusion processes. Depending on the relation of the exponents $1<\tau,\, p,\, r<\infty$ we distinguish different types of logistic equations. More precisely we have:\\
(i) {\it subdiffusive equations} when $\tau<p<r$;\\
(ii) {\it equidiffusive equations} when $\tau =p<r$;\\
(iii) {\it superdiffusive equations} when $p<\tau<r$.

Our hypotheses on the reaction term $f(z,x)$ incorporate as a very special case the superdiffusive $p$-logistic equation and we show that it exhibits bifurcation phenomena for large values of the parameter $\lambda>0$.
So, we prove that there exists a critical parameter $\lambda^*>0$ such that for all $\lambda>\lambda^*$, problem \eqref{eqP} admits at least two positive solutions, for $\lambda=\lambda^*$ there is at least one positive solution and for $\lambda\in(0,\lambda^*)$ problem \eqref{eqP} has no positive solutions. We also show that for every admissible parameter $\lambda\geq\lambda^*$, problem \eqref{eqP} has a smallest positive solution $\underline{u}_\lambda$ and establish the continuity and monotonicity properties of the map $\lambda\mapsto\underline{u}_\lambda$.  The two main results of this paper read as follows.

\smallskip
{\bf Theorem A.} {\sl
	Assume that hypotheses $H(\beta)$ and $H$ hold. Then there exists $\lambda^*>0$ such that
	\begin{itemize}
		\item[(i)] for all $\lambda>\lambda^*$, problem \eqref{eqP} admits at least two positive solutions
		$$u_0,\hat{u}\in {\rm int}\, C_+,\ u_0\neq\hat{u};$$
		\item[(ii)] for $\lambda=\lambda^*$, problem $(P_{\lambda_*})$ has at least one positive solution $u_*\in {\rm int}\, C_+;$
		\item[(iii)] for all $\lambda\in(0,\lambda^*)$, problem \eqref{eqP} has no positive solution.
	\end{itemize}
}


\smallskip
{\bf Theorem B.} {\sl
	\begin{itemize}
		\item[(a)] If hypotheses $H(\beta)$ and $H$ hold and $\lambda\in\mathcal{L}=\left[\lambda^*,+\infty\right)$, then problem \eqref{eqP} has a smallest positive solution $\underline{u}_{\lambda}\in {\rm int}\, C_+$.
		\item[(b)] If hypotheses $H(\beta)$ and $H'$ hold and $\lambda\in\mathcal{L}=\left[\lambda^*,+\infty\right)$, then problem \eqref{eqP} has a biggest positive solution $\overline{u}_{\lambda}\in {\rm int}\, C_+$ and the map $\lambda\mapsto\overline{u}_{\lambda}$ from $\mathcal{L}=\left[\lambda^*,+\infty\right)$ into $C^1(\overline{\Omega})$ is nondecreasing and right continuous.
	\end{itemize}
}}


Recently, Papageorgiou \& R\u{a}dulescu \cite{13} consider a more general class of nonlinear parametric Robin problems and proved a bifurcation result near the origin. In \cite{13}, the differential operator is in general nonhomogeneous (the $p$-Laplacian is included as a special case) and the parameter $\lambda>0$ enters in the reaction in general in a nonlinear fashion. Our work here partially complements that paper. For Dirichlet $p$-logistic equations of superdiffusive type, we mention the works of Dong \cite{3} and Takeuchi \cite{14}, \cite{15}. Extensions to equations with more general reactions of superdiffusive type, can be found in Gasinski \& Papageorgiou \cite{6} (Dirichlet problems) and Cardinali, Papageorgiou \& Rubbioni \cite{2}, Papageorgiou \& R\u{a}dulescu \cite{11} (Neumann problems).

Our approach uses variational methods based on the critical point theory, coupled with suitable truncation techniques. In the next section, for the convenience of the reader, we recall the main mathematical tools which we will use in the sequel.


\section{Mathematical Background}

Let $X$ be a Banach space and $X^*$ its topological dual. By $\left\langle \cdot,\cdot\right\rangle$ we denote the duality brackets for the pair $(X^*,X)$. Given $\varphi\in C^1(X)$, we say that $\varphi$ satisfies the ``Palais-Smale condition" (the ``PS-condition" for short), it the following holds:
\begin{center}
	``Every sequence $\{u_n\}_{n\geq 1}\subseteq X$ such that $\{\varphi(u_n)\}_{n\geq 1}\subseteq\RR$ is bounded and
	$$\varphi'(u_n)\rightarrow 0\ \mbox{in}\ X^*\ \mbox{as}\ n\rightarrow \infty,$$
	admits a strongly convergent subsequence".
\end{center}

This is a compactness-type condition on the functional $\varphi$ and leads to a deformation theorem, from which one can derive the minimax theory of the critical values of $\varphi$. 

Throughout this work, the hypotheses on the boundary coefficient $\beta(\cdot)$, are the following:

\smallskip
$H(\beta):$ $\beta\in C^{1,\alpha}(\partial\Omega)$ with $\alpha\in(0,1),\ \beta(z)\geq 0$ for all $z\in\partial\Omega,\ \beta\neq 0$.

\smallskip
In the analysis of problem \eqref{eqP}, in addition to the Sobolev space $W^{1,p}(\Omega)$, we will also use the Banach space $C^1(\overline{\Omega})$. This is an ordered Banach space with positive cone
$$C_+=\{u\in C^1(\overline{\Omega}):u(z)\geq 0\ \mbox{for all}\ z\in\overline{\Omega}\}.$$

This cone has a nonempty interior given by
$${\rm int}\, C_+=\{u\in C_+:u(z)>0\ \mbox{for all}\ z\in\overline{\Omega}\}.$$

By $||\cdot||$ we denote the norm of the Sobolev space $W^{1,p}(\Omega)$, given by
$$||u||=\left[||u||^p_p+||Du||^p_p\right]^{1/p}\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$

On $\partial\Omega$ we consider the $(N-1)$-dimensional Hausdorff measure (the surface measure on $\partial\Omega$). Using this measure on $\partial\Omega$, we can define the Lebesgue spaces $L^q(\partial\Omega),\ 1\leq q\leq \infty$. We know that there exists a unique continuous, linear map $\gamma_0:W^{1,p}(\Omega)\rightarrow L^p(\partial\Omega)$, known as the trace map such that $\gamma_0(u)=u|_{\partial\Omega}$ for all $u\in C^1(\overline{\Omega})$. This map is compact (more precisely, $\gamma_0:W^{1,p}(\Omega)\rightarrow L^q(\partial\Omega)$ is compact for any $1\leq q<\frac{Np-p}{N-p}$). Moreover, we have
$$\mbox{im}\,\gamma_0=W^{\frac{1}{p'},p}(\partial\Omega)\left(\frac{1}{p}+\frac{1}{p'}=1\right)\ \mbox{and}\ \mbox{ker}\,\gamma_0=W^{1,p}_{0}(\Omega).$$

In what follows, for the sake of notational simplicity, we drop the explicit use of the trace map. All restrictions of Sobolev functions from $W^{1,p}(\Omega)$ on $\partial\Omega$ are defined in the sense of traces.

{\color{red}We say that a map $A:W^{1,p}(\Omega)\rightarrow W^{1,p}(\Omega)^*$ is of type $(S)_+$, if the following property holds:
$$``u_n\stackrel{w}{\rightarrow}u\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{and}\ \limsup_{n\rightarrow\infty}\langle A(u_n),u_n-u\rangle\leq 0\Rightarrow u_n\rightarrow u\ \mbox{in}\ W^{1,p}(\Omega)."$$}

In what follows, by $|\cdot|_N$ we denote the Lebesgue measure on $\RR^N$. Also, if $x\in\RR$, then $x^{\pm}=\max\{\pm x,0\}$. Given $u\in W^{1,p}(\Omega)$, we set $u^{\pm}(\cdot)=u(\cdot)^{\pm}$. We know that
$$u^{\pm}\in W^{1,p}(\Omega),\ u=u^+-u^-,\ |u|=u^++u^-.$$


\section{Bifurcation Type Theorem}

In this section, we prove a bifurcation type theorem describing in a precise way the dependence of positive solutions of \eqref{eqP} on the parameter $\lambda>0$.

We introduce the following conditions on the reaction $f(z,x)$.

\smallskip
$H:$ $f:\Omega\times\RR\rightarrow\RR$ is a Carath\'eodory function such that $f(z,0)=0$ for almost all $z\in \Omega$ and
\begin{itemize}
	\item[(i)] $|f(z,x)|\leq a(z)(1+x^{r-1})$ for almost all $z\in\Omega$, all $x\geq 0$, with $a\in L^{\infty}(\Omega)_+,\ p<r< p^*$;
	\item[(ii)] $\limsup\limits_{x\rightarrow+\infty}\frac{f(z,x)}{x^{p-1}}\leq 0$ uniformly for almost all $z\in\Omega$;
	\item[(iii)] $-\eta_0\leq\liminf\limits_{x\rightarrow 0^+}\frac{f(z,x)}{x^{p-1}}\leq\limsup\limits_{x\rightarrow 0^+}\frac{f(z,x)}{x^{p-1}}\leq 0$ uniformly for almost all $z\in\Omega$, with $\eta_0>0$;
	\item[(iv)] if $F(z,x)=\int^{x}_{0}f(z,s)ds$, then there exists $\tilde{u}\in L^r(\Omega)$ such that $\int_{\Omega}F(z,\tilde{u})dz>0;$
	\item[(v)] there exists $\mu>p$ such that $x\mapsto\frac{f(z,x)}{x^{\mu-1}}$ is strictly decreasing uniformly for almost all $z\in\Omega$, that is, if $x-y\geq\eta>0$, there exists $c_{\eta}>0$ such that $c_{\eta}\leq\frac{f(z,y)}{y^{\mu-1}}-\frac{f(z,x)}{x^{\mu-1}}$ for almost all $z\in\Omega$
	\item[(vi)] for every $\rho>0$, there exists $\xi_{\rho}>0$ such that for almost all $z\in\Omega$, the mapping $x\mapsto f(z,x)+\xi_{\rho}x^{p-1}$ is nondecreasing on $[0,\rho]$.
\end{itemize}

\begin{remark}
	Since we are looking for positive solutions and the above hypotheses concern the positive semi-axis, without any loss of generality, we may assume that $f(z,x)=0$ for almost all $z\in\Omega$ and all $x\leq 0$.
\end{remark}

\begin{ex}
	The following function satisfies hypotheses $H$ above. For the sake of simplicity, we drop the $z$-dependence:
	$$f(x)=x^{\tau-1}-x^{r-1}\ \mbox{for all}\ x\geq 0\ \mbox{and with}\ p<\tau<r<p^*.$$
\end{ex}

	This function is the classical superdiffusive reaction for the $p$-logistic equation and was used by Dong \cite{3} and Takeuchi \cite{14,15}.
	
{\color{red}The reader interested in applications can use in what follows as a reaction term the above function $f(x)$.}

	We introduce the following two sets:
	\begin{center}
		$\mathcal{L}=\{\lambda>0:\ \mbox{problem \eqref{eqP} admits a positive solution}\}$,\\
		$S(\lambda)$ is the set of positive solutions for problem \eqref{eqP}.
	\end{center}
	
	Let $\lambda^*=\inf \mathcal{L}\geq 0$.
\begin{prop}\label{prop4}
	If hypotheses $H(\beta)$ and $H$ hold, then for every $\lambda>0$ we have $S(\lambda)\subseteq {\rm int}\, C_+$ and $\lambda^*>0$.
\end{prop}

\begin{proof}
	Let $\lambda\in\mathcal{L}$. Then we can find $u_{\lambda}\in S(\lambda)$ such that
	$$-\Delta_p u_{\lambda}(z)=\lambda f(z,u_{\lambda}(z))\ \mbox{for almost all}\ z\in\Omega,\ \frac{\partial u_{\lambda}}{\partial n_p}+\beta(z)u_{\lambda}(z)^{p-1}=0\ \mbox{on}\ \partial\Omega$$
(see Papageorgiou \& R\u{a}dulescu \cite{10}). From Winkert \cite{16}, we know that $u_{\lambda}\in L^{\infty}(\Omega)$. So, we apply Theorem 2 of Lieberman \cite{8} and we have $u_{\lambda}\in C_+\backslash\{0\}$. Let $\rho=||u_{\lambda}||_{\infty}$ and let $\xi_{\rho}>0$ be as postulated by hypothesis $H(vi)$. We have
\begin{eqnarray*}
	&&-\Delta_pu_{\lambda}(z)+\lambda\xi_{\rho}u_{\lambda}(z)^{p-1}=\lambda f(z,u_{\lambda}(z))+\lambda\xi_{\rho}u_{\lambda}(z)^{p-1}\geq 0\ \mbox{for almost all}\ z\in\Omega,\\
	&\Rightarrow&\Delta_pu_{\lambda}(z)\leq\lambda\xi_{\rho}u_{\lambda}(z)^{p-1}\ \mbox{for almost all}\ z\in\Omega,\\
	&\Rightarrow&u_{\lambda}\in {\rm int}\, C_+\ (\mbox{by the nonlinear maximum principle})\\
	&\Rightarrow&S(\lambda)\subseteq {\rm int}\, C_+.
\end{eqnarray*}

Hypotheses $H(i),(ii),(iii)$ imply that we can find $c_1>0$ such that
\begin{equation}\label{eq2}
	f(z,x)\leq c_1x^{p-1}\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\geq 0.
\end{equation}

Let $\hat{\lambda}_1$ denote the principal eigenvalue of $-\Delta_p$ with Robin boundary condition (see Papageorgiou \& R\u{a}dulescu \cite{10}). We know that $\hat{\lambda}_1>0$. We choose $\lambda\in\left(0,\frac{\hat{\lambda}_1}{c_1}\right)$. Then we have
\begin{equation}\label{eq3}
	\lambda f(z,x)\leq \lambda c_1 x^{p-1}<\hat{\lambda}_1 x^{p-1}\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x>0\ (\mbox{see (\ref{eq2})}).
\end{equation}

Suppose that $\lambda\in\mathcal{L}$. Then there exists $u_{\lambda}\in S(\lambda)\subseteq {\rm int}\, C_+$ such that
\begin{equation}\label{eq4}
	\left\langle A(u_{\lambda},h)\right\rangle+\int_{\partial\Omega}\beta(z)u_{\lambda}^{p-1} hd\sigma=\lambda\int_{\Omega}f(z,u_{\lambda})hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega).
\end{equation}

In (\ref{eq4}) we choose $h=u_{\lambda}\in W^{1,p}(\Omega)$ and use (\ref{eq3}). Then
$$||Du_{\lambda}||^p_p+\int_{\partial\Omega}\beta(z)u_{\lambda}^{p}d\sigma<\hat{\lambda}_1||u_{\lambda}||^p_p,$$
which contradicts the variational characterization of $\hat{\lambda}_1>0$ (see Papageorgiou \& R\u{a}dulescu \cite{10}). Therefore $\lambda\notin\mathcal{L}$ and so $\lambda^*\geq\frac{\hat{\lambda}_1}{c_1}>0$.
\end{proof}
\begin{prop}\label{prop5}
	If hypotheses $H(\beta)$ and $H$ hold, then $\mathcal{L}\neq\varnothing$ and $(\lambda^*,+\infty)\subseteq\mathcal{L}.$
\end{prop}
\begin{proof}
	We introduce the following Carath\'eodory function
	\begin{equation}\label{eq5}
		\hat{f}_{\lambda}(z,x)=\lambda f(z,x)+(x^+)^{p-1}\ \mbox{for all}\ (z,x)\in\Omega\times\RR,\ \mbox{all}\ \lambda>0.
	\end{equation}
	
	We set $\hat{F}_{\lambda}(z,x)=\int^{x}_{0}\hat{f}_{\lambda}(z,s)ds$ and for every $\lambda>0$, we consider the $C^1$-functional $\hat{\varphi}_{\lambda}:W^{1,p}(\Omega)\rightarrow\RR$ defined by
	 $$\hat{\varphi}_{\lambda}(u)=\frac{1}{p}||Du||^p_p+\frac{1}{p}||u||^p_p+\frac{1}{p}\int_{\partial\Omega}\beta(z)(u^+)^pd\sigma-\int_{\partial\Omega}\hat{F}_{\lambda}(z,u)dz\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$
	
	Hypotheses $H(i),(ii)$ imply that given $\epsilon\in\left(0,\frac{\hat{\lambda}_1}{\lambda}\right)$, we can find $c_2=c_2(\epsilon)>0$ such that
	\begin{equation}\label{eq6}
		F(z,x)\leq\frac{\epsilon}{p}x^p+c_2\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\geq 0.
	\end{equation}
	
	Then for all $u\in W^{1,p}(\Omega)$, we have
	\begin{eqnarray*}
		 &&\hat{\varphi}_{\lambda}(u)\geq\frac{1}{p}||Du||^p_p+\frac{1}{p}||u^-||^p_p+\frac{1}{p}\int_{\partial\Omega}\beta(z)(u^+)^pd\sigma-\frac{\lambda\epsilon}{p}||u^+||^p_p-\lambda c_2|\Omega|_N\ (\mbox{see (\ref{eq5}), (\ref{eq6})})\\
		 &&=\frac{1}{p}\left[||Du^+||^p_p+\int_{\partial\Omega}\beta(z)(u^+)^pd\sigma-\lambda\epsilon||u^+||^p_p\right]+\frac{1}{p}||u^-||^p_p-\lambda c_2|\Omega|_N\\
		&&\geq\frac{c_3}{p}||u^+||^p+\frac{1}{p}||u^-||^p_p-\lambda c_2|\Omega|_N\ \mbox{for some}\ c_3>0\\
		&&\hspace{4cm}(\mbox{see Proposition \ref{prop4} of Papageorgiou \& R\u{a}dulescu \cite{10}})\\
		&&\geq c_4||u||^p-\lambda c_2|\Omega|_N\ \mbox{for some}\ c_4>0,\\
		&\Rightarrow&\hat{\varphi}_{\lambda}\ \mbox{is coercive}.
	\end{eqnarray*}
	
	Also, using the Sobolev embedding theorem and the trace theorem, we see that $\hat{\varphi}_{\lambda}$ is sequentially weakly lower semicontinuous. So, by the Weierstrass theorem we can find $u_0\in W^{1,p}(\Omega)$ such that
	\begin{equation}\label{eq7}
	  	\hat{\varphi}_{\lambda}(u_0)=\inf\left[\hat{\varphi}_{\lambda}(u):u\in W^{1,p}(\Omega)\right].
	\end{equation}
	
	Consider the integral functional $I_F:L^r(\Omega)\rightarrow\RR$
	$$I_F(u)=\int_{\Omega}F(z,u(z))dz\ \mbox{for all}\ u\in L^r(\Omega).$$
	
	From hypothesis $H(i)$ and the dominated convergence theorem, we see that $I_F$ is continuous. By hypothesis $H(iv)$, we have
	$$I_F(\tilde{u})>0.$$
	
	Exploiting the density of $W^{1,p}(\Omega)$ in $L^r(\Omega)$, we can find $\bar{u}\in W^{1,p}(\Omega)$ such that
	\begin{equation}\label{eq8}
		I_F(\tilde{u})>0.
	\end{equation}
	
	So, because of (\ref{eq8}) and by choosing $\lambda>0$ appropriately big, we have
	\begin{eqnarray*}
		&&\hat{\varphi}_{\lambda}(\bar{u})<0,\\
		&\Rightarrow&\hat{\varphi}_{\lambda}(u_0)<0=\hat{\varphi}_{\lambda}(0)\ (\mbox{see (\ref{eq7})}),\ \mbox{hence}\ u_0\neq 0.
	\end{eqnarray*}
	
	From (\ref{eq7}), we have
	\begin{eqnarray}\label{eq9}
		&&\hat{\varphi}'_{\lambda}(u_0)=0,\nonumber\\
		&\Rightarrow&\left\langle A(u_0),h\right\rangle+\int_{\Omega}|u_0|^{p-2}u_0hdz+\int_{\partial\Omega}\beta(z)(u_0^+)^{p-1}hd\sigma
=\int_{\Omega}\hat{f}_{\lambda}(z,u_0)hdz\\
		&&\hspace{5cm}\mbox{for all}\ h\in W^{1,p}(\Omega).\nonumber
	\end{eqnarray}
	
	In (\ref{eq9}) we choose $h=-u_0^-\in W^{1,p}(\Omega)$. Then
	\begin{eqnarray*}
		&&||Du_0^-||^p_p+||u_0^-||^p_p=0\ (\mbox{see (\ref{eq5})}),\\
		&\Rightarrow&||u_0^-||^p=0\ \mbox{and so}\ u_0\geq 0,\ u_0\neq 0.
	\end{eqnarray*}
	
	Therefore (\ref{eq9}) becomes
	\begin{equation}\label{eq10}
		\left\langle A(u_0),h\right\rangle+\int_{\partial\Omega}\beta(z)u_0^{p-1}hd\sigma=\int_{\Omega}f(z,u_0)hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega)\ (\mbox{see (\ref{eq5})}).
	\end{equation}
	
	From (\ref{eq10}), reasoning as in Papageorgiou \& R\u{a}dulescu \cite{10}, using the nonlinear Green's identity, we infer that $u_{\lambda}\in S(\lambda)\subseteq {\rm int}\, C_+$ for $\lambda>0$ big, hence $\mathcal{L}\neq\varnothing$.
	
	Next, let $\lambda\in\mathcal{L}$ and let $\vartheta>\lambda$. Consider $u_{\lambda}\in S(\lambda)\subseteq {\rm int}\, C_+$ and let $\xi\in(0,1)$ such that
	\begin{equation}\label{eq11}
		\lambda=\xi^{\mu-p}\vartheta\ (\mbox{recall that}\ \mu>p,\ \mbox{see hypothesis}\ H(v)).
	\end{equation}
	
	We set $\underline{u}=\xi u_{\lambda}\in {\rm int}\, C_+$. We have
	\begin{eqnarray}
		-\Delta_p\underline{u}(z)=\xi^{p-1}(-\Delta_pu_{\lambda}(z))&=&\xi^{p}\lambda f(z,u_{\lambda}(z))\ (\mbox{since}\ u_{\lambda}\in S(\lambda))\nonumber\\
		&=&\xi^{\mu-1}\vartheta f(z,u_{\lambda}(z))\ \mbox{for almost all}\ z\in\Omega\ (\mbox{see (\ref{eq11})})\label{eq12}
	\end{eqnarray}
	\begin{equation}
		\frac{\partial \underline{u}}{\partial n_p}+\beta(z)\underline{u}^{p-1}=0\ \mbox{on}\ \partial\Omega\tag{12'}\label{eq12'}
	\end{equation}
	
	Let $\eta=(1-\xi)\min\limits_{\overline{\Omega}}u_{\lambda}>0$ (recall $u_{\lambda}\in {\rm int}\, C_+,\ \xi\in(0,1)$). So, by virtue of hypothesis $H(v)$, we can find $c_{\eta}>0$ such that
	\begin{eqnarray}\label{eq13}
		&&c_{\eta}\leq\frac{f(z,\xi u_{\lambda}(z))}{\xi^{\mu-1}u_{\lambda}(z)^{\mu-1}}-\frac{f(z,u_{\lambda}(z))}{u_{\lambda}(z)^{\mu-1}}\ \mbox{for almost all}\ z\in\Omega,\nonumber\\
		&\Rightarrow&c_{\eta}\underline{u}(z)^{\mu-1}+\xi^{\mu-1}f(z,u_{\lambda}(z))\leq f(z,\underline{u}(z))\ \mbox{for almost all}\ z\in\Omega.
	\end{eqnarray}
	
	Using (\ref{eq13}) in (\ref{eq12}), we obtain
	\begin{equation}\label{eq14}
		-\Delta_p\underline{u}(z)\leq\vartheta f(z,\underline{u}(z))-c_{\eta}\underline{u}(z)^{\mu-1}\ \mbox{for almost all}\ z\in\Omega.
	\end{equation}
	
	We introduce the following truncations:
	\begin{eqnarray}
		&&\hat{g}_{\vartheta}(z,x)=\left\{
		\begin{array}{ll}
			\hat{f}_0(z,\underline{u}(z))&\mbox{if}\ x\leq\underline{u}(z)\\
			\hat{f}_0(z,x)&\mbox{if}\ \underline{u}(z)<x
		\end{array}\right.\ \mbox{for all}\ (z,x)\in\Omega\times\RR\label{eq15}\\
		&&\hat{k}(z,x)=\left\{
		\begin{array}{ll}
			\beta(z)\underline{u}(z)^{p-1}&\mbox{if}\ x\leq\underline{u}(z)\\
			\beta(z)x^{p-1}&\mbox{if}\ \underline{u}(z)<x
		\end{array}\right.\ \mbox{for all}\ (z,x)\in\partial\Omega\times\RR\,.\label{eq16}
	\end{eqnarray}

	Both are Carath\'eodory functions. We set
	$$\hat{G}_{\vartheta}(z,x)=\int^{x}_{0}\hat{g}_{\vartheta}(z,s)ds\ \mbox{and}\ \hat{K}(z,x)=\int^{x}_{0}\hat{k}(z,s)ds$$
	and consider the $C^1$-functional $\hat{\psi}_{\vartheta}:W^{1,p}(\Omega)\rightarrow\RR$ defined by
	 $$\hat{\psi}_(\vartheta)(u)=\frac{1}{p}||Du||^p_p+\frac{1}{p}||u||^p_p+\int_{\partial\Omega}\hat{K}(z,u)d\sigma-\int_{\Omega}\hat{G}_{\vartheta}(z,u)dz\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$
	
	From (\ref{eq15}) and (\ref{eq16}) it is clear that the functional $\hat{\psi}_{\vartheta}$ has essentially the same structure as the functional $\hat{\varphi}_{\vartheta}$. In particular, $\hat{\psi}_{\vartheta}$ is coercive and sequentially weakly lower semicontinuous. So, we can find $u_{\vartheta}\in W^{1,p}(\Omega)$ such that	
		\begin{eqnarray}\label{eq17}
			&&\hat{\psi}_{\vartheta}(u_{\vartheta})=\inf[\hat{\psi}_{\vartheta}(u):u\in W^{1,p}(\Omega)],\nonumber\\
			&\Rightarrow&\hat{\psi}'_{\vartheta}(u_{\vartheta})=0,\nonumber\\
			&\Rightarrow&\left\langle A(u_{\vartheta}),h\right\rangle+\int_{\Omega}|u_{\vartheta}|^{p-2}u_{\vartheta}hdz+\int_{\partial\Omega}\hat{k}(z,u_{\vartheta})hd\sigma=\int_{\Omega}\hat{g}_{\vartheta}(z,u_{\vartheta})hdz\\
			&&\mbox{for all}\ h\in W^{1,p}(\Omega).\nonumber
		\end{eqnarray}
		
		In (\ref{eq17}) we choose $h=(\underline{u}-u_{\vartheta})^+\in W^{1,p}(\Omega)$ and obtain $\underline{u}\leq u_{\vartheta}$.
		
		
		Using (\ref{eq15}) and (\ref{eq16}), we see that equation (\ref{eq17}) becomes
		\begin{eqnarray*}
			&&\left\langle A(u_{\vartheta}),h\right\rangle+\int_{\partial\Omega}\beta(z)u^{p-1}_{\vartheta}hd\sigma-\int_{\Omega}\vartheta(z,u_{\vartheta})dz\ \mbox{for all}\ h\in W^{1,p}(\Omega)\\
			&\Rightarrow&u_{\vartheta}\in S(\vartheta)\subseteq {\rm int}\, C_+\  \mbox{and so}\ \vartheta\in\mathcal{L},\\
			&\Rightarrow&(\lambda^*,\infty)\subseteq\mathcal{L}.
		\end{eqnarray*}
This completes the proof.
\end{proof}

\begin{prop}\label{prop6}
	If hypotheses $H(\beta)$ and $H$ hold and $\vartheta>\lambda^*$, then problem ($P_{\vartheta}$) admits at least two positive solutions
	$$u_0,\hat{u}\in {\rm int}\, C_+,\ u_0\neq 0.$$
\end{prop}
\begin{proof}
	Fix $\lambda\in(\lambda^*,\vartheta)$. From Proposition \ref{prop5} we know that $\lambda\in\mathcal{L}$ and so there exists $u_{\lambda}\in S(\lambda)\subseteq {\rm int}\, C_+$. Let $\xi\in(0,1)$ such that $\lambda=\xi^{\mu-p}\vartheta$ (recall $\mu >sp$, see hypothesis $H(v)$). We set $\underline{u}=\xi u_{\lambda}\in {\rm int}\, C_+$. Reasoning as in the proof of Proposition \ref{prop5} and using the truncated functional $\hat{\psi}_{\vartheta}\in C^1(W^{1,p}(\Omega))$, we can find $u_0\in W^{1,p}(\Omega)$ such that
	\begin{eqnarray}
		&&u_0\in S(\vartheta)\subseteq {\rm int}\, C_+,\ \underline{u}\leq u_0,\label{eq18}\\
		&&\hat{\psi}_{\vartheta}(u_0)=\inf[\hat{\psi}_{\vartheta}(u):u\in W^{1,p}(\Omega)].\label{eq19}
	\end{eqnarray}
	
	Let $\rho=||u_0||_{\infty}$ and let $\xi_{\rho}>0$ be as postulated by hypothesis $H(vi)$. For $\delta>0$, let $\underline{u}^{\delta}=\underline{u}+\delta\in {\rm int}\, C_+$. We have
	\begin{eqnarray}\label{eq20}
		 &&-\Delta_p\underline{u}^{\delta}+\vartheta\xi_{\rho}(\underline{u}^{\delta})^{p-1}\nonumber\\
		&\leq&-\Delta_p\underline{u}+\vartheta\xi_{\rho}\underline{u}^{p-1}+\gamma(\delta)\ \mbox{with}\ \gamma(\delta)\rightarrow 0^+\ \mbox{as}\ \delta\rightarrow 0^+\nonumber\\
		&\leq&\vartheta f(z,\underline{u})+\vartheta\xi_{\rho}\underline{u}^{p-1}-c_{\eta}\underline{u}^{\mu-1}+\gamma(\delta)\ (\mbox{see (\ref{eq14})})\nonumber\\
		&\leq&\vartheta f(z,u_0)+\vartheta\xi_{\rho}u_0^{p-1}-c_{\eta}\overline{m}^{\mu-1}+\gamma(\delta)\ \mbox{with}\ \overline{m}=\min\limits_{\overline{\Omega}}\underline{u}>0\nonumber\\
		&&\hspace{2cm}(\mbox{see (\ref{eq18}), hypothesis $H(vi)$ and recall}\ \underline{u}\in {\rm int}\, C_+),\nonumber\\
		&\leq&\vartheta f(z,u_0)+\vartheta \xi_{\rho}u_0^{p-1}\ \mbox{for}\ \delta>0\ \mbox{small}\ (\mbox{recall}\ \gamma(\delta)\rightarrow 0^+)\nonumber\\
		&=&-\Delta_pu_0+\vartheta\xi_{\rho}u_0^{p-1}\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ \delta>0\ \mbox{small},\nonumber\\
		\Rightarrow&&\underline{u}^{\delta}\leq u_0\ \mbox{for}\ \delta>0\ \mbox{small},\nonumber\\
		\Rightarrow&&u_0-\underline{u}\in {\rm int}\, C_+.
	\end{eqnarray}
	
	Let $$\left[\underline{u}\right)=\{u\in W^{1,p}(\Omega):\underline{u}(z)\leq u(z)\ \mbox{for almost all}\ z\in \Omega\}.$$ From (\ref{eq15}) and (\ref{eq16}), we see that
	\begin{equation}\label{eq21}
		 \hat{\psi}_{\vartheta}|_{\left[\underline{u}\right)}=\hat{\varphi}_{\vartheta}|_{\left[\underline{u}\right)}+\xi^*_{\vartheta}\ \mbox{with}\ \xi^*_{\vartheta}\in\RR.
	\end{equation}
	
	From (\ref{eq19}), (\ref{eq20}) and (\ref{eq21}), it follows that
	\begin{eqnarray}\label{eq22}
		&&u_0\ \mbox{is a local}\ C^1(\overline{\Omega})-\mbox{minimizer of}\ \hat{\varphi}_{\vartheta},\nonumber\\
		&\Rightarrow&u_0\ \mbox{is a local}\ W^{1,p}(\Omega)-\mbox{minimizer of}\ \hat{\varphi}_{\vartheta}\ (\mbox{see \cite{10}}).
	\end{eqnarray}
	
	Hypothesis $H(iii)$ implies that given $\epsilon>0$, we can find $\delta=\delta(\epsilon)>0$ such that
	\begin{equation}\label{eq23}
		F(z,x)\leq\frac{\epsilon}{p}x^p\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\in[0,\delta].
	\end{equation}
	
	Then for all $u\in C^1(\overline{\Omega})$ with $||u||_{C^1(\overline{\Omega})}\leq\delta$, we have
	\begin{eqnarray}\label{eq24}
		 &\hat{\varphi}_{\vartheta}(u)&=\frac{1}{p}||Du||^p_p+\frac{1}{p}||u^-||^p_p+\frac{1}{p}\int_{\partial\Omega}\beta(z)(u^+)^pd\sigma-\int_{\Omega}F(z,u^+)dz\ (\mbox{see (\ref{eq5})})\nonumber\\
		 &&\geq\frac{1}{p}\left[||Du^+||^p_p+\int_{\partial\Omega}\beta(z)(u^+)^pd\sigma-\epsilon||u^+||^p_p\right]+\frac{1}{p}||u^-||^p\ (\mbox{see (\ref{eq23})})\nonumber\\
		&&\geq c_5||u||^p\ \mbox{for some}\ c_5>0\ (\mbox{see \cite[Proposition 4]{12}}),\nonumber\\
		\Rightarrow&&u=0\ \mbox{is a local}\ C^1(\overline{\Omega})-\mbox{minimizer of}\ \hat{\varphi}_{\vartheta},\nonumber\\
		\Rightarrow&&u=0\ \mbox{is a local}\ W^{1,p}(\Omega)-\mbox{minimizer of}\ \hat{\varphi}_{\vartheta}\ (\mbox{see \cite{10}}).
	\end{eqnarray}
	
	We may assume that $\hat{\varphi}_{\vartheta}(0)=0\leq\hat{\varphi}_{\vartheta}(u_0)$ (the reasoning is similar if the opposite inequality holds). Since $K_{\hat{\varphi}_{\vartheta}}\backslash\{0\}\subseteq {\rm int}\, C_+$, we may assume that $K_{\hat{\varphi}_{\vartheta}}$ is finite or otherwise we already have an infinity of distinct positive solutions for problem \eqref{eqP}. Because of (\ref{eq22}) (if the opposite inequality holds, we use (\ref{eq24})), we can find $\rho\in(0,1)$, small such that
	\begin{equation}\label{eq25}	 \hat{\varphi}_{\vartheta}(0)=0\leq\varphi_{\vartheta}(u_0)<\inf\left[\hat{\varphi}_{\vartheta}(u):||u-u_0||=\rho\right]=\hat{m}_{\rho}\ ||u_0||>\rho
	\end{equation}
	(see Aizicovici, Papageorgiou \& Staicu \cite{1}, proof of Proposition 29). The functional $\hat{\varphi}_{\vartheta}$ being coercive, it satisfies the $PS$-condition. This fact and (\ref{eq25}) permit the use of the mountain pass-theorem, see \cite[p. 648]{5}. So, we can find $\hat{u}\in W^{1,p}(\Omega)$ such that
	\begin{equation}\label{eq26}
		\hat{u}\in K_{\hat{\varphi}_{\vartheta}}\ \mbox{and}\ \hat{m}_{\rho}\leq\hat{\varphi}_{\vartheta}(\hat{u}).
	\end{equation}
	
	From (\ref{eq25}) and (\ref{eq26}) it follows that $\hat{u}\in S(\vartheta)\subseteq {\rm int}\, C_+,\ \hat{u}\neq u_0.$
\end{proof}

Next we examine what happens at the critical parameter $\lambda=\lambda^*$.
\begin{prop}\label{prop7}
	If hypotheses $H(\beta)$ and $H$ hold, then $\lambda^*\in \mathcal{L}$ and so $\mathcal{L}=\left[\lambda^*,+\infty\right)$.
\end{prop}
\begin{proof}
	Let $\lambda_p>\lambda^*$ for all $n\geq 1$ and assume that $\lambda_n\downarrow \lambda^*$ as $n\rightarrow\infty$. From Proposition \ref{prop5}, we can find $u_n\in S(\lambda_n)\subseteq {\rm int}\, C_+$ for all $n\geq 1$. We have
	\begin{equation}\label{eq27}
		\left\langle A(u_n),h\right\rangle+\int_{\partial\Omega}\beta(z)u_n^{p-1}hd\sigma=\lambda_n\int_{\Omega}f(z,u_n)hdz\ \mbox{for all}\ n\geq 1,\ \mbox{all}\ h\in W^{1,p}(\Omega).
	\end{equation}
	
	In (\ref{eq27}) we choose $h=u_n\in {\rm int}\, C_+$ and have
	\begin{equation}\label{eq28}
		 ||Du_n||^p_p+\int_{\partial\Omega}\beta(z)u_n^pd\sigma=\lambda_n\int_{\Omega}f(z,u_n)u_ndz\ \mbox{for all}\ n\geq 1.
	\end{equation}
	
	Hypotheses $H(i),(ii)$ imply that given $\epsilon>0$, we can find $c_6=c_6(\epsilon)>0$ such that
	\begin{equation}\label{eq29}
		f(z,x)x\leq \epsilon x^p+c_6x\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\geq 0.
	\end{equation}
	
	We use (\ref{eq29}) in (\ref{eq28}) and obtain
	 $$||Du_n||^p_p+\int_{\partial\Omega}\beta(z)u^p_nd\sigma\leq\lambda_1\epsilon||u_n||^p_p+\lambda_1c_6||u_n||_1\ \mbox{for all}\ n\geq 1.$$
	
	Choosing $\epsilon\in\left(0,\frac{\hat{\lambda}_1}{\lambda_1}\right)$ and using Proposition 4 of Papageorgiou \& R\u{a}dulescu \cite{12}, we have
	\begin{eqnarray*}
		&&||u_n||^p\leq c_7||u_n||\ \mbox{for some}\ c_7>0,\ \mbox{all}\ n\geq 1,\\
		&\Rightarrow&\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)\ \mbox{is bounded}.
	\end{eqnarray*}
	
	So, by passing to a suitable subsequence if necessary, we may assume that
	\begin{equation}\label{eq30}
		u_n\stackrel{w}{\rightarrow}u_*\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{and}\ u_n\rightarrow u_*\ \mbox{in}\ L^r(\Omega)\ \mbox{and in}\ L^p(\partial\Omega)\ \mbox{as}\ n\rightarrow\infty\,.
	\end{equation}
	
	In (\ref{eq27}) we choose $h=u_n-u_*\in W^{1,p}(\Omega)$, pass to the limit as $n\rightarrow\infty$ and use (\ref{eq30}). Then we have
	\begin{eqnarray}\label{eq31}
		&&\lim\limits_{n\rightarrow\infty}\left\langle A(u_n),u_n-u_*\right\rangle=0,\nonumber\\
		&\Rightarrow&u_n\rightarrow u_*\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{as}\ n\rightarrow\infty\ (\mbox{see \cite[p. 314]{9}}).
	\end{eqnarray}
	
	So, if in (\ref{eq27}) we pass to the limit as $n\rightarrow\infty$ and use (\ref{eq31}), then
	\begin{eqnarray*}
		&&\left\langle A(u_*),h\right\rangle+\int_{\partial\Omega}\beta(z)u_*^{p-1}hd\sigma=\lambda^*\int_{\Omega}f(z,u_*)hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega),\\
		&\Rightarrow&u_*\geq 0\ \mbox{solves problem}\ (P_{\lambda_*})\,.
	\end{eqnarray*}
	
	If we can show that $u_*\neq 0$, then $u_*\in S(\lambda^*)\subseteq {\rm int}\, C_+$ and so $\lambda^*\in\mathcal{L}$.
	
	We argue indirectly. So, suppose that $u_*=0$. We set $y_n=\frac{u_n}{||u_n||}$, $n\geq 1$. Then $||y_n||=1,\ y_n\geq 0,\ n \geq 1$. By passing to a subsequence if necessary, we may assume that
	\begin{equation}\label{eq32}
		y_n\stackrel{w}{\rightarrow}y\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{and}\ y_n\rightarrow y\ \mbox{in}\ L^p(\Omega)\ \mbox{and in}\ L^p(\partial\Omega)\ \mbox{as}\ n\rightarrow\infty.
	\end{equation}
	
	Hypotheses $H(i),(iii)$ imply that
	\begin{eqnarray}\label{eq33}
		&&|f(z,x)|\leq c_8(|x|^{p-1}+|x|^{r-1})\ \mbox{for almost all}\ z\in \Omega,\ \mbox{all}\ x\in\RR,\ \mbox{with}\ c_8>0,\nonumber\\
		&\Rightarrow&\left\{\frac{f(\cdot,u_n(\cdot))}{||u_n||^{p-1}}\right\}_{n\geq 1}\subseteq L^{p'}(\Omega)\ \mbox{is bounded}\ \left(\frac{1}{p}+\frac{1}{p'}=1\right).
	\end{eqnarray}
	
	Thus, by passing to a subsequence if necessary and using hypothesis $H(iii)$ we obtain
	\begin{equation}\label{eq34}
		\frac{f(\cdot,u_n(\cdot))}{||u_n||^{p-1}}\stackrel{w}{\rightarrow} gy^{p-1}\ \mbox{in}\ L^{p'}(\Omega)\ \mbox{with}\ -\eta_0\leq g(z)\leq 0\ \mbox{for almost all}\ z\in\Omega
	\end{equation}
	(see Aizicovici, Papageorgiou \& Staicu \cite{1}, proof of Proposition 14).
	
	From (\ref{eq27}) we have
	\begin{equation}\label{eq35}
		\left\langle A(y_n),h\right\rangle+\int_{\partial\Omega}\beta(z)y_n^{p-1}hd\sigma=\lambda_n\int_{\Omega}\frac{f(z,u_n)}{||u_n||^{p-1}}hdz\ \mbox{for all}\ n\geq 1.
	\end{equation}
	
	Choosing $h=y_n-y\in W^{1,p}(\Omega)$, passing to the limit as $n\rightarrow\infty$ and using (\ref{eq32}) and (\ref{eq34}), we obtain
	\begin{eqnarray}\label{eq36}
		&&\lim\limits_{n\rightarrow\infty}\left\langle A(y_n),y_n-y\right\rangle=0,\nonumber\\
		&\Rightarrow&y_n\rightarrow y\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{as}\ n\rightarrow\infty,\  ||y||=1,\ y\geq 0.
	\end{eqnarray}
	
	So, if in (\ref{eq35}) we pass to the limit as $n\rightarrow\infty$ and use (\ref{eq36}), we obtain
	$$\left\langle A(y),h\right\rangle+\int_{\partial\Omega}\beta(z)y^{p-1}hd\sigma=\lambda^*\int_{\Omega}gy^{p-1}hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega).$$
	
	We choose $h=y\in W^{1,p}(\Omega)$. Then
	 $$0<\hat{\lambda}_1\leq||Dy||^p_p+\int_{\partial\Omega}\beta(z)y^pd\sigma=\lambda_*\int_{\Omega}gy^pdz\leq 0\ \mbox{(see \cite{10} and (\ref{eq34}), (\ref{eq36}))},$$
	a contradiction.
	
	Therefore $u_*\neq 0$ and so $u_*\in S(\lambda^*)\subseteq {\rm int}\, C_+$, hence $\lambda^*\in\mathcal{L}$ and so $\mathcal{L}= \left[\lambda^*,+\infty\right)$.
\end{proof}

The proof of Theorem A is now complete.



\section{Extremal Positive Solutions}

In this section we show that for all $\lambda\geq\lambda^*$, problem \eqref{eqP} has extremal positive solutions, that is, a smallest positive solution $\underline{u}_{\lambda}\in {\rm int}\, C_+$ and a biggest positive solution $\overline{u}_{\lambda}\in {\rm int}\, C_+$. We also investigate the monotonicity and continuity properties of the map $\lambda\mapsto\overline{u}_{\lambda}$.

To produce the smallest positive solution $\underline{u}_{\lambda}\in {\rm int}\, C_+$, we do not need to introduce any additional conditions on the reaction $f(z,x)$.

\begin{prop}\label{prop9}
	If hypotheses $H(\beta)$ and $H$ hold and $\lambda\in\mathcal{L}=\left[\lambda^*,\infty\right)$, then problem \eqref{eqP} admits a smallest positive solution $\underline{u}_{\lambda}\in {\rm int}\, C_+$.
\end{prop}

\begin{proof}
	The set $S(\lambda)$ is downward directed, that is, if $u,\hat{u}\in S(\lambda)$, then there exists $\tilde{u}\in S(\lambda)$ such that $\tilde{u}\leq u,\ \tilde{u}\leq\hat{u}.$
	
	Since we are looking for the smallest positive solution, thanks to the above property of $S(\lambda)$, without any loss of generality, we may assume that there exists $c_9>0$ such that
	\begin{equation}\label{eq37}
		||u||_{\infty}\leq c_9\ \mbox{for all}\ u\in S(\lambda).
	\end{equation}
	
	From Hu \& Papageorgiou \cite[p. 178]{7}, we know that we can find $\{u_n\}_{n\geq 1}\subseteq S(\lambda)$ such that
	$$\inf S(\lambda)=\inf\limits_{n\geq 1}u_n.$$
	
	Then we have
	\begin{equation}\label{eq38}
		\left\langle A(u_n),h\right\rangle+\int_{\partial\Omega}\beta(z)u_n^{p-1}hd\sigma=\lambda\int_{\Omega}f(z,u_n)hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega).
	\end{equation}
	
	Choosing $h=u_n\in W^{1,p}(\Omega)$ in (\ref{eq38}), we obtain
	\begin{eqnarray*}
		 &&\hat{\lambda}_1||u_n||^p_p\leq||Du_n||^p_p+\int_{\partial\Omega}\beta(z)u_n^pd\sigma=\lambda\int_{\Omega}f(z,u_n)u_ndz\ \mbox{for all}\ n\geq 1,\\
		&\Rightarrow&\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)\ \mbox{is bounded (see (\ref{eq37}) and hypothesis $H(i)$)}.
	\end{eqnarray*}
	
	So, we may assume that
	\begin{equation}\label{eq39}
		u_n\stackrel{w}{\rightarrow}\underline{u}_{\lambda}\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{and}\ u_n\rightarrow\underline{u}_{\lambda}\ \mbox{in}\ L^p(\Omega)\ \mbox{and in}\ L^p(\partial\Omega).
	\end{equation}
	
	In (\ref{eq38}), we choose $h=u_n-\underline{u}_{\lambda}\in W^{1,p}(\Omega)$, pass to the limit as $n\rightarrow\infty$ and use (\ref{eq39}). Then
	\begin{eqnarray*}
		&&\lim\limits_{n\rightarrow\infty}\left\langle A(u_n),u_n-\underline{u}_{\lambda}\right\rangle=0,\\
		&\Rightarrow&u_n\rightarrow\underline{u}_{\lambda}\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{(see Proposition \ref{prop3})}.
	\end{eqnarray*}
	
	Arguing as in the proof of Proposition \ref{prop7}, via a contradiction argument and using hypothesis $H(iii)$, we show that $\underline{u}_{\lambda}\neq 0$. Hence $\underline{u}_{\lambda}\in S(\lambda)\subseteq {\rm int}\, C_+$ and $\underline{u}_{\lambda}=\inf S(\lambda).$
\end{proof}

To produce the biggest positive solution for \eqref{eqP}, we need an extra condition on the reaction $f(z,\cdot)$.
So, we assume the following on the function $f(z,x)$.

\smallskip
$H':$ $f:\Omega\times\RR\rightarrow\RR$ is a Carath\'eodory function such that $f(z,0)=0$ for almost all $z\in\Omega$, hypotheses $H'(i)\rightarrow(vi)$ are the same as the corresponding hypotheses $H(i)\rightarrow (vi)$ and

$(vii)$ there exists $w\in W^{1,p}(\Omega)\cap C(\overline{\Omega})$ with $w(z)\geq 0$, for all $z\in\overline{\Omega}$, $w\neq 0$ and
\begin{eqnarray*}
	&&A(w)\geq 0\ \mbox{in}\ W^{1,p}(\Omega)^*\\
	&&f(z,x)\geq 0\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\in[0,w(z)]\\
	&&f(z,x)\leq 0\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\geq w(z).
\end{eqnarray*}

\begin{remark}
	Evidently, $f(z,w(z))=0$ for almost all $z\in\Omega$. If there exists $\xi>0$ such that
	$$f(z,x)\geq 0\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\in[0,\xi]\ \mbox{and}\ f(z,x)\leq 0\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\geq\xi,$$
	then hypothesis $H(vii)$ is satisfied. In particular, the classical superdiffusive reaction
	$$f(x)=x^{\tau-1}-x^{r-1}\ \mbox{for all}\ x\geq 0\ \mbox{with}\ 1<p<\tau<r<p^*,$$
	satisfies this new condition.
\end{remark}

\begin{prop}\label{prop10}
  If hypotheses $H(\beta)$ and $H'$ hold and $\lambda\in\mathcal{L}$, then $u_{\lambda}(z)\leq w(z)$ for all $z\in\overline{\Omega}$ and all $u_{\lambda}\in S(\lambda)$.
\end{prop}
\begin{proof}
	Let $u_{\lambda}\in S(\lambda)\subseteq {\rm int}\, C_+$ (see Proposition \ref{prop4}). We have
	$$\left\langle A(u_{\lambda}),h\right\rangle+\int_{\partial\Omega}\beta(z)u_{\lambda}^{p-1}hd\sigma=\lambda\int_{\Omega}f(z,u_{\lambda})hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega).$$
	
	Let $h=(u_{\lambda}-w)^+\in W^{1,p}(\Omega)$. Then
	\begin{eqnarray*}
		&&\left\langle A(u_{\lambda}),(u_{\lambda}-w)^+\right\rangle+\int_{\partial\Omega}\beta(z)u_{\lambda}^{p-1}(u_{\lambda}-w)^+d\sigma\\
		&&=\lambda\int_{\Omega}f(z,u_{\lambda})(u_{\lambda}-w)^+dz\\
		&&\leq\left\langle A(w),(u_{\lambda}-w)^+\right\rangle+\int_{\partial\Omega}\beta(z)w^{p-1}(u_{\lambda}-w)^+d\sigma\ (\mbox{see}\ H'(vii)\ \mbox{and}\ H(\beta)),\\
		 &\Rightarrow&\int_{\{u_{\lambda},w\}}(|Du_{\lambda}|^{p-2}Du_{\lambda}-|Dw|^{p-2}Dw,Du_{\lambda}-Dw)_{\RR^N}dz\\
		 &&\hspace{3cm}+\int_{\partial\Omega}\beta(z)(u_{\lambda}^{p-1}-w^{p-1})(u_{\lambda}-w)^+d\sigma\leq 0,\\
		&\Rightarrow&|\{u_{\lambda}>w\}|_N=0,\ \mbox{hence}\ u_{\lambda}(z)\leq w(z)\ \mbox{for all}\ z\in\overline{\Omega}.
	\end{eqnarray*}
This completes the proof.
\end{proof}

\begin{remark}
	This proposition implies that $w(z)>0$ for all $z\in\overline{\Omega}$ and hypothesis $H'(iii)$ becomes
	$$\lim\limits_{x\rightarrow 0^+}\frac{f(z,x)}{x^{p-1}}=0\ \mbox{uniformly for almost all}\ z\in\Omega.$$
\end{remark}

Using Proposition \ref{prop10} and reasoning similarly as in the proof of Proposition \ref{prop9}, we obtain:
\begin{prop}\label{prop11}
	If hypotheses $H(\beta)$ and $H'$ hold and $\lambda\in\mathcal{L}$, then problem \eqref{eqP} admits a biggest positive solution $\overline{u}_{\lambda}\in {\rm int}\, C_+$.
\end{prop}

Now we investigate the properties of the map $\lambda\mapsto\overline{u}_{\lambda}$ from $(0,+\infty)$ into ${\rm int}\, C_+$. First we show the monotonicity of this map.
\begin{prop}\label{prop12}
	If hypotheses $H(\beta)$ and $H'$ hold and $\lambda,\vartheta\in\mathcal{L}$, then $\overline{u}_{\lambda}\leq\overline{u}_{\vartheta}$.
\end{prop}
\begin{proof}
	We have
	\begin{eqnarray}\label{eq40}
		\left\langle A(\overline{u}_{\lambda},h)\right\rangle+\int_{\partial\Omega}\beta(z)\overline{u}_{\lambda}^{p-1}hd\sigma&=&\lambda\int_{\Omega}f(z,\overline{u}_{\lambda})hdz\nonumber\\
		&\leq&\vartheta\int_{\Omega}f(z,\overline{u}_{\lambda})hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega)\ \mbox{with}\ h\geq 0\\
		&&\hspace{0.8cm}(\mbox{see hypothesis $H'(vii)$ and Proposition \ref{prop10}})\nonumber
	\end{eqnarray}
	
	We introduce the following truncations-perturbations of the reaction and of the boundary term:
	\begin{eqnarray}
		&&\hat{g}_{\vartheta}(z,x)=\left\{
		\begin{array}{ll}
			\vartheta f(z,\overline{u}_{\lambda}(z))+\overline{u}_{\lambda}(z)^{p-1}&\mbox{if}\ x\leq\overline{u}_{\lambda}(z)\\
			\vartheta f(z,x)+x^{p-1}&\mbox{if}\ \overline{u}_{\lambda}(z)<x
		\end{array}\right.\ \mbox{for all}\ (z,x)\in\Omega\times\RR\label{eq41}\\
		&&\hat{k}(z,x)=\left\{\begin{array}{ll}
			\beta(z)\overline{u}_{\lambda}(z)^{p-1}&\mbox{if}\ x\leq\overline{u}_{\lambda}(z)\\
			\beta(z)x^{p-1}&\mbox{if}\ \overline{u}_{\lambda}(z)\leq x
		\end{array}\right.\ \mbox{for all}\ (z,x)\in\partial\Omega\times\RR.\label{eq42}
	\end{eqnarray}
	
	Both are Carath\'eodory functions. We set $\hat{G}_{\vartheta}(z,x)=\int^{x}_{0}\hat{g}_{\vartheta}(z,s)ds$ and $\hat{K}(z,x)=\int^{x}_{0}\hat{k}(z,s)ds$ and consider the $C^1$-functional $\hat{\psi}_{\vartheta}:W^{1,p}(\Omega)\rightarrow\RR$ defined by
	 $$\hat{\psi}_{\vartheta}(u)=\frac{1}{p}||Du||^p_p+\frac{1}{p}||u||^p_p+\int_{\partial\Omega}\hat{K}(z,u(z))d\sigma-\int_{\Omega}\hat{G}_{\vartheta}(z,u(z))dz\ \mbox{for all}\ u\in W^{1,p}(\Omega)$$
	
	We know that $\hat{\psi}_{\vartheta}$ is coercive and sequentially weakly lower semicontinuous (see the proof of Proposition \ref{prop5}). So, by the Weierstrass theorem, we can find $u_{\vartheta}\in W^{1,p}(\Omega)$ such that
	\begin{eqnarray}\label{eq43}
		&&\hat\psi_{\vartheta}(u_{\vartheta})=\inf[\psi_{\vartheta}(u):u\in W^{1,p}(\Omega)],\nonumber\\
		&\Rightarrow&\hat\psi'_{\vartheta}(u_{\vartheta})=0,\nonumber\\
		&\Rightarrow&\left\langle A(u_{\vartheta}),h\right\rangle+\int_{\Omega}|u_{\vartheta}|^{p-2}u_{\vartheta}hdz+\int_{\partial\Omega}\hat{k}(z,u_{\vartheta})hd\sigma=\int_{\Omega}\hat{g}_{\vartheta}(z,u_{\vartheta})hdz\\
		&&\hspace{8cm}\mbox{for all}\ h\in W^{1,p}(\Omega).\nonumber
	\end{eqnarray}
	
	In (\ref{eq43}) we choose $h=(\overline{u}_{\lambda}-u_{\vartheta})^+\in W^{1,p}(\Omega)$. We have
	\begin{eqnarray*}
		&&\left\langle A(u_{\vartheta}),(\overline{u}_{\lambda}-u_{\vartheta})^+\right\rangle+\int_{\Omega}|u_{\vartheta}|^{p-2}u_{\vartheta}(\overline{u}_{\lambda}-u_{\vartheta})^+dz+\int_{\partial\Omega}\beta(z)\overline{u}_{\lambda}^{p-1}(\overline{u}_{\lambda}-u_{\vartheta})^+d\sigma\\
		&&=\int_{\Omega}(\vartheta f(z,\overline{u}_{\lambda})+\overline{u}_{\lambda}^{p-1})(\overline{u}_{\lambda}-u_{\vartheta})^+dz\ (\mbox{see (\ref{eq41}), (\ref{eq42})})\\
		&&\geq\left\langle A(\overline{u}_{\lambda}),(\overline{u}_{\lambda}-u_{\vartheta})^+\right\rangle+\int_{\Omega}\overline{u}_{\lambda}^{p-1}(\overline{u}_{\lambda}-u_{\vartheta})^+dz+\int_{\partial\Omega}\beta(z)\overline{u}_{\lambda}^{p-1}(\overline{u}_{\lambda}-u_{\lambda})^+d\sigma\ (\mbox{see (\ref{eq40}}))\\
		 &\Rightarrow&\int_{\{\overline{u}_{\lambda},u_{\vartheta}\}}\left(|D\overline{u}_{\lambda}|^{p-2}D\overline{u}_{\lambda}-|Du_{\vartheta}|^{p-2}Du_{\vartheta},D\overline{u}_{\lambda}-Du_{\vartheta}\right)_{\RR^N}dz\\
		 &&\hspace{4cm}+\int_{\{\overline{u}_{\lambda}>u_{\vartheta}\}}(\overline{u}_{\lambda}^{p-1}-|u_{\vartheta}|^{p-2}u_{\vartheta})(u_{\lambda}-u_{\vartheta})dz\leq 0\\
		&\Rightarrow&|\{\overline{u}_{\lambda}>u_{\vartheta}\}|_N=0,\ \mbox{hence}\ \overline{u}_{\lambda}\leq u_{\vartheta}.
	\end{eqnarray*}
	
	So, (\ref{eq43}) becomes
	\begin{eqnarray*}
		&&\left\langle A(u_{\vartheta},h)\right\rangle+\int_{\partial\Omega}\beta(z)u_n^{p-1}hd\sigma=\int_{\Omega}\vartheta f(z,u_{\vartheta})hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega)\ \mbox{(see (\ref{eq41}), (\ref{eq42}))}\\
		&\Rightarrow&u_{\vartheta}\in S(\vartheta)\subseteq {\rm int}\, C_+\,.
	\end{eqnarray*}
	
	We conclude that
	$$\overline{u}_{\lambda}\leq u_{\vartheta}\leq \overline{u}_{\vartheta},$$
which completes the proof.
\end{proof}

Next we examine the continuity of the map $\lambda\mapsto \overline{u}_{\lambda}$ from $\left[\lambda^*,+\infty\right)$ into $C^1(\overline{\Omega})$.
\begin{prop}\label{prop13}
	If hypotheses $H(\beta)$ and $H'$ hold, then the map $\lambda\mapsto \overline{u}_{\lambda}$ is right continuous from $\left[\lambda^*,+\infty\right)$ into $C^1(\overline{\Omega})$.
\end{prop}
\begin{proof}
	Let $\{\lambda_n\}_{n\geq 1}\subseteq\mathcal{L}=\left[\lambda^*,+\infty\right)$ and assume that $\lambda_n\downarrow\lambda\in\mathcal{L}$ as $n\rightarrow\infty$. Let $\overline{u}_{n}=\overline{u}_{\lambda_n}\in S(\lambda_n)\subseteq {\rm int}\, C_+$ be the maximal positive solution of problem $(P_{\lambda_n})\ n\geq 1$ (see Proposition \ref{prop11}). We have
	\begin{eqnarray}
		&&-\Delta_pu_n(z)=\lambda_nf(z,u_n(z))\ \mbox{for almost all}\ z\in\Omega,\ \frac{\partial u_n}{\partial n_p}+\beta(z)u_n^{p-1}=0\ \mbox{on}\ \partial\Omega,\ n\geq 1\label{eq44}\\
		&&u_n\leq w\ \mbox{for all}\ n\geq 1\ (\mbox{see Proposition \ref{prop10}}).\label{eq45}
	\end{eqnarray}
	
	Then (\ref{eq44}), (\ref{eq45}) and Theorem 2 of Lieberman \cite{8} imply that there exist $\alpha\in(0,1)$ and $c_{10}>0$ such that
	\begin{equation}\label{eq46}
			\overline{u}_{n}\in C^{1,\alpha}(\overline{\Omega})\ \mbox{and}\ ||\overline{u}_{n}||_{C^{1,\alpha}(\overline{\Omega})}\leq c_{10}\ \mbox{for all}\ n\geq 1.
	\end{equation}
	
	Moreover, from Proposition \ref{prop12} we have that $\{\overline{u}_{n}\}_{n\geq 1}$ is decreasing. From (\ref{eq46}) and the compact embedding of $C^{1,\alpha}(\overline{\Omega})$ into $C^1(\overline{\Omega})$, we have
	\begin{equation}\label{eq47}
		\overline{u}_{n}\downarrow\tilde{u}\ \mbox{in}\ C^1(\overline{\Omega})\ \mbox{as}\ n\rightarrow \infty.
	\end{equation}
	
	We show that $\tilde{u}=\overline{u}_{\lambda}$. Indeed, if this is not true, then we can find $z_0\in\Omega$ such that
	$$\tilde{u}(z_0)<\overline{u}_{\lambda}(z_0).$$
	
	Because of (\ref{eq47}), we can find $n_0\in\NN$ such that
	\begin{equation}\label{eq48}
		\overline{u}_{n}(z_0)<\overline{u}_{\lambda}(z_0)\ \mbox{for all}\ n\geq n_0.
	\end{equation}
	
	On the other hand, from Proposition \ref{prop12} we have
	\begin{equation}\label{eq49}
		\overline{u}_{\lambda}(z_0)\leq \overline{u}_{n}(z_0)\ \mbox{for all}\ n\geq 1.
	\end{equation}
	
	Comparing (\ref{eq48}) and (\ref{eq49}) we reach a contradiction. This proves that $\tilde{u}=\overline{u}_{\lambda}$ and so we have established the right continuity of the map $\lambda\mapsto\overline{u}_{\lambda}$ from $\mathcal{L}=\left[\lambda^*,+\infty\right)$ into $C^1(\overline{\Omega})$.
\end{proof}

The proof of Theorem B is now complete.

\medskip
{\bf Acknowledgement}. The authors wish to thank a knowledgeable referee for his/her remarks and constructive criticism.





\begin{thebibliography}{99}

\bibitem{1} {\sc S. Aizicovici, N.S. Papageorgiou, V. Staicu}, {\em Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints}, Memoirs Amer. Math. Soc., Vol. 196, No. \textbf{915} (November 2008).

\bibitem{2} {\sc T. Cardinali, N.S. Papageorgiou, P. Rubbioni}, {\em Bifurcation phenomena for nonlinear superdiffusive Neumann equations of logistic type}, Annali. Mat. Pura Appl. \textbf{193} (2013), 1-21.

\bibitem{3} {\sc W. Dong}, {\em A priori estimates and existence of positive solutions for a quasilinear elliptic equation}, J. London Math. Soc. \textbf{72} (2005), 645-662.

%\bibitem{4} {\sc M. Filippakis, A. Kristaly, N.S. Papageorgiou}, {\em Existence of five nonzero solutions with exact %sign for a $p$-Laplacian operator}, Discr. Cont. Dynam. Systems \textbf{24} (2009), 405-440.

\bibitem{5} {\sc L. Gasinski, N.S. Papageorgiou}, {\em Nonlinear Analysis}, Chapman \& Hall / CRC, Boca Raton, FL (2006).

\bibitem{6} {\sc L. Gasinski, N.S. Papageorgiou}, {\em Bifurcation-type results for nonlinear parametric elliptic equations}, Proc. Royal Soc. Edinburgh, Sect. A \textbf{142} (2006), 595-623.

\bibitem{7} {\sc S. Hu, N.S. Papageorgiou}, {\em Handbook of Multivalued Analysis. Volume I: Theory}, Kluwer Academic Publishers, Dordrecht, The Netherlands (1997).

\bibitem{8} {\sc G. Lieberman}, {\em Boundary regularity for solutions of degenerate elliptic equations}, Nonlinear Anal. \textbf{12} (1988), 1203-1219.

\bibitem{9} {\sc N.S. Papageorgiou, S. Kyritsi}, {\em Handbook of Applied Analysis}, Springer, New York (2009).

\bibitem{10} {\sc N.S. Papageorgiou, V.D. R\u{a}dulescu}, {\em Multiple solutions with precise sign information for parametric Robin problems}, J. Differential Equations \textbf{256} (2014), 2449-2479.

\bibitem{11} {\sc N.S. Papageorgiou, V.D. R\u{a}dulescu}, {\em Positive solutions for nonlinear nonhomogeneous Neumann equations of superdiffusive type}, J. Fixed Point Theory Appl. {\bf 15} (2014), 519-535

\bibitem{12} {\sc N.S. Papageorgiou, V.D. R\u{a}dulescu}, {\em Positive solutions for perturbations of the eigenvalue problem for the Robin $p$-Laplacian}, Annales Acad. Scient. Fennicae Math. {\bf 40} (2015), 255-277.
    {\bf 40} (2015).

\bibitem{13} {\sc N.S. Papageorgiou, V.D. R\u{a}dulescu}, {\em Bifurcation of positive solutions for nonlinear nonhomogeneous Robin and Neumann problems with competing nonlinearities},  Discrete and Continuous Dynamical Systems {\bf 35} (2015), 5003-5036.

\bibitem{14} {\sc S. Takeuchi}, {\em Positive solutions of a degenerate elliptic equation with a logistic reaction}, Proc. Amer. Math. Soc. \textbf{129} (2001), 433-441.

\bibitem{15} {\sc S. Takeuchi}, {\em Multiplicity result for a degenerate elliptic equation with a logistic reaction}, J. Differential Equations \textbf{173} (2001), 138-144.

\bibitem{16} {\sc P. Winkert}, {\em $L^{\infty}$ estimates for nonlinear Neumann boundary value problems}, Nonlin. Differ. Equations Appl. (NoDEA) \textbf{17} (2010), 289-302.


\end{thebibliography}


\end{document} 