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\title[Bifurcation near infinity for the Robin $p$-Laplacian]{Bifurcation near infinity for the Robin $p$-Laplacian}


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\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. 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 \\
          and Department of Mathematics, University of Craiova, Street A.I.Cuza 13,
          200585 Craiova, Romania}
\email{\tt vicentiu.radulescu@imar.ro}

\keywords{$p$-Laplacian, Robin problem, truncation and perturbation of the reaction, positive solutions, bifurcation-type result\\
\phantom{aa} 2010 AMS Subject Classification:
35J66, 35J70, 35J92}


\begin{abstract}
We consider a parametric Robin problem driven by the $p$-Laplacian and with a Carath\'eodory reaction. Our hypotheses on the reaction incorporate a special case $p$-logistic equations with a superdiffusive reaction. Using variational methods coupled with suitable truncation, perturbation and comparison techniques, we prove a bifurcation near infinity result.
\end{abstract}

\maketitle

%\tableofcontents

\section{Introduction}\label{sec1}

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_pu(z)=\lambda f(z,u(z))&\mbox{in}\ \Omega\\
		\di\frac{\partial u}{\partial n_p}+\beta(z)u^{p-1}=0&\mbox{on}\ \partial\Omega\\
 u>0&\mbox{in}\ \Omega,
	\end{array}\right.
		\tag{$P_{\lambda}$} \label{eqP}
\end{equation}
where $1<p<\infty$.
Here $\Delta_p$ denotes the $p$-Laplace differential operator defined by
$$\Delta_p u=\mbox{div}\,(|Du|^{p-2}Du)\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$ 

We are looking for solutions in the Sobolev space $W^{1,p}(\Omega)$. The weak distributional formulation of \eqref{eqP} is
$$\int_\Omega |Du|^{p-2}(Du,Dh)_{\RR^N}dz+\int_{\partial\Omega}\beta(z)|u|^{p-2}uhd\sigma
=\lambda \int_\Omega f(z,u)hdz\quad\mbox{for all}\ h\in W^{1,p}(\Omega)$$
(or, equivalently, for all $h\in C^\infty (\overline\Omega)$; recall that since $\partial\Omega$ is a $C^2$-manifold, the space $C^\infty (\overline\Omega)$ is dense in $W^{1,p}(\Omega)$, see for example Gasinski and Papageorgiou \cite[p. 189]{7}). In the above equation, $d\sigma$ denotes the $(N-1)$-dimensional  Hausdorff measure on $\partial\Omega$ (the surface measure on $\partial\Omega$). 

The Robin boundary condition is interpreted using the nonlinear Green's identity (see Casas and Fernandez \cite{casa}, Kenmochi \cite{kenm}, and Gasinski and Papageorgiou \cite[p. 211]{7}) as is the case in Lieberman \cite{10}. So, according to that identity, there exists a unique element 
$$\frac{\partial u}{\partial n_p}\in W^{-1/p',p'}(\partial\Omega)\qquad \left(\frac 1p+\frac{1}{p'}=1\right),$$
which by extension we denote by
$$\frac{\partial u}{\partial n_p}=|Du|^{p-2}(Du,n)_{\RR^N}$$
with $n(\cdot)$ being the outward unit normal on $\partial\Omega$ such that if $\Delta_pu\in L^{p'}(\Omega)$, then
$$\int_\Omega |Du|^{p-2}(Du,Dh)_{\RR^N}dz+\int_\Omega (\Delta_pu)hdz=\langle \frac{\partial u}{\partial n_p},\gamma_0(h)\rangle\quad \mbox{for all}\ h\in W^{1,p}(\Omega).$$ 
Here $\gamma_0$ is the trace map and we have denoted by $\langle\,\cdot\, ,\, \cdot\,\rangle$ the duality brackets for the pair $\left( W^{-1/p',p'}(\partial\Omega),W^{1/p',p}(\partial\Omega)\right)$.

The reaction $f(z,x)$ is a Carath\'eodory function (that is, for all $x\in\RR$ the mapping $z\longmapsto f(z,x)$ is measurable and for a.a. $z\in\Omega$, $ x\longmapsto f(z,x)$ is continuous). On $f(z,\cdot)$ we impose conditions near $+\infty$ and near $0^+$. Our conditions are general and incorporate as a special case, the so-called {\it superdiffusive reaction} of the {\it $p$-logistic equation}. Finally $\lambda>0$ is a parameter.

Our goal is to study the existence, nonexistence and multiplicity of positive solutions as the parameter $\lambda>0$ varies. More precisely, we prove a bifurcation-type result for large values of the parameter $\lambda>0$ (bifurcation near $+\infty$). So, we establish the existence of a critical parameter value $\lambda_*>0$ such that  for every $\lambda>\lambda_*$ problem \eqref{eqP} admits at least two positive solutions, when $\lambda=\lambda_*$ problem \eqref{eqP} has at least one positive solution and finally for all $\lambda\in(0,\lambda_*)$ problem \eqref{eqP} has no positive solutions.

Such bifurcation type results, were proved by Brock, Itturiaga and Ubilla \cite{3}, Filippakis, O'Regan and Papageorgiou \cite{6}, Gasinski and Papageorgiou \cite{8}, R\u adulescu and Repovs \cite{14}, Takeuchi \cite{16}, \cite{17} (semilinear or nonlinear Dirichlet problems) and by Cardinali, Papageorgiou and Rubbioni \cite{4}, Papageorgiou and R\u adulescu \cite{12} (for nonlinear Neumann problems). All the aforementioned results impose more restrictive conditions on the reaction $f(z,\cdot)$. Moreover, our work here complements the recent one by Papageorgiou and R\u adulescu \cite{13}, where the authors prove for Robin problems a bifurcation theorem for small values of $\lambda>0$ (bifurcation near zero). However, it should be pointed out that in \cite{13} the differential operator is considerably more general and nonhomogeneous. It is an interesting open problem whether our work here can be extended to equations driven by such operators.

An important role in our analysis is played by the regularity theory of Lieberman \cite{10}, who established regularity up to the boundary (global regularity) for solutions of equations driven by a broad class of nonhomogeneous differential operators, which includes as a special case the $p$-Laplacian. The results of Liberman \cite{10} extend local regularity results of DiBenedetto \cite{dibe} and Tolksdorf \cite{tolk}.

Our approach is variational and it is based on the critical point theory combined with suitable truncation, perturbation and comparison techniques.

\section{Mathematical Background}

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

This compactness-type condition, is in general weaker than the usual Palais-Smale condition (``$PS$-condition" for short). The two conditions coincide if $\varphi$ is bounded below (see, for example, Denkowski, Migorski and Papageorgiou \cite[p. 174]{5}). The $C$-condition leads to a deformation theorem, from which one can derive the minimax theory of the critical values of $\varphi$. Prominent in that theory, is the so-called ``mountain pass theorem" of Ambrosetti and Rabinowitz \cite{2}, which here we state in a slightly more general form (see, for example, Denkowski, Migorski and Papageorgiou \cite[p. 179]{5}).

\begin{theorem}\label{th1}
	Assume that $\varphi\in C^1(X)$ satisfies the $C$-condition, $u_0,u_1\in X$, $\rho>0,\ ||u_1-u_0||>\rho$,
	$$\max\{\varphi(u_0),\varphi(u_1)\}<\inf[\varphi(u):||u-u_0||=\rho]=\eta_{\rho}$$
	and $c=\inf\limits_{\gamma\in\Gamma}{\underset{\mathrm{0\leq t\leq 1}}\max}\varphi(\gamma(t))$ where $\Gamma=\{\gamma\in C([0,1],X):\gamma(0)=u_0,\ \gamma(1)=u_1\}$. Then $c\geq \eta_{\rho}$ and $c$ is a critical value of $\varphi$.
\end{theorem}

The analysis of problem \eqref{eqP} will involve the Sobolev space $W^{1,p}(\Omega)$ and the Banach space $C^1(\overline{\Omega})$. We will also make use of the fact that $C^1(\overline{\Omega})$ 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
$$\mbox{int}\, C_+=\{u\in C_+:u(z)>0\ \mbox{for all}\ z\in \overline{\Omega}\}.$$

In what follows by $||\cdot||$ we denote the norm of the Sobolev space $W^{1,p}(\Omega)$. So, we have
$$||u||=\left(||u||^p_p+||Du||^p_p\right)^{1/p}\ \mbox{for all}\ u\in W^{1,p}(\Omega)$$

On $\partial\Omega$ we use the $(N-1)$-dimensional Hausdorff (surface) measure $\sigma(\cdot)$. Using this measure, we can define the Lebesgue spaces $L^p(\partial\Omega)$, $1\leq p\leq \infty$. 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", which satisfies $\left.\gamma_0(u)=u\right|_{\partial\Omega}$ for all $u\in C^1(\overline{\Omega})$. In fact $\gamma_0$ is compact and we have
\begin{eqnarray*}
&&\mbox{im}\,\gamma_0=W^{\frac{1}{p'},p}(\partial\Omega)\ \left(\mbox{with}\ \frac{1}{p}+\frac{1}{p'}=1\right)\\
&&\mbox{ker}\,\gamma_0=W^{1,p}_0(\Omega)
\end{eqnarray*}

In what follows, for the sake of notational simplicity, we drop the use of the trace map. All the restrictions of functions on $\partial\Omega$, are understood in the sense of traces.

Let $f_0:\Omega\times \RR\rightarrow\RR$ be a Carath\'eodory function with subcritical growth in the $x\in\RR$ variable, that is
$$|f_0(z,x)|\leq a_0(z)(1+|x|^{r-1})\ \mbox{for a.a}\ z\in\Omega,\ \mbox{all}\ x\in\RR$$
with $a_0\in L^{\infty}(\Omega)_+$ and $1<r<p^*=\left\{\begin{array}{ll}
\frac{Np}{N-p}&\mbox{if}\ p<N\\
+\infty&\mbox{if}\ N\leq p
\end{array}\right.$. We set $F_0(z,x)=\int^x_0f_0(z,s)ds$ and consider the $C^1$-functional $\varphi_0:W^{1,p}(\Omega)\rightarrow\RR$ defined by
$$\varphi_0(u)=\frac{1}{p}||Du||^p_p+\frac{1}{p}\int_{\partial\Omega}\beta(z)|u(z)|^pd\sigma-\int_{\Omega}F_0(z,u(z))dz\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$

The next proposition can be found in Papageorgiou and R\u adulescu \cite{13} and it is essentially a consequence of the nonlinear regularity theory of Lieberman \cite{10}.

\begin{prop}\label{prop2}
	Assume that $u_0\in W^{1,p}(\Omega)$ is a local $C^1(\overline{\Omega})$-minimizer of $\varphi_0$, that is, there exists $\rho_0>0$ such that
	$$\varphi_0(u_0)\leq \varphi_0(u_0+h)\ \mbox{for all}\ h\in C^1(\overline{\Omega})\ \mbox{with}\ ||h||_{C^1(\overline{\Omega})}\leq \rho_0.$$
	Then $u_0\in C^{1,\alpha}(\overline{\Omega})$ for some $\alpha\in(0,1)$ and it is also a local $W^{1,p}(\Omega)$-minimizer of $\varphi_0$, that is, there exists $\rho_1>0$ such that
	$$\varphi_0(u_0)\leq\varphi_0(u_)+h)\ \mbox{for all}\ h\in W^{1,p}(\Omega)\ \mbox{with}\ ||h||\leq \rho_1.$$
\end{prop}

Let $A:W^{1,p}(\Omega)\rightarrow W^{1,p}(\Omega)^*$ be the nonlinear map defined by
\begin{eqnarray}\label{eq1}
	\left\langle A(u),y\right\rangle=\int_{\Omega}|Du|^{p-2}(Du,Dy)_{\RR^N}dz\ \mbox{for all}\ u,y\in W^{1,p}(\Omega).
\end{eqnarray}

The next result about the map $A$ is well-known (see, for example, Gasinski and Papageorgiou \cite[p. 745]{7}).
\begin{prop}\label{prop3}
	The map $A:W^{1,p}(\Omega)\rightarrow W^{1,p}(\Omega)^*$ defined by (\ref{eq1}) is demicontinuous, monotone (hence maximal monotone too) and of type $(S)_+$, that is
	\begin{center}
		``if $u_n\stackrel{w}{\rightarrow}u$ in $W^{1,p}(\Omega)$ and $\limsup\limits_{n\rightarrow\infty}\left\langle A(u_n),u_n-u\right\rangle\leq 0$,
		then $u_n\rightarrow u$ in $W^{1,p}(\Omega)$."
	\end{center}
\end{prop}

Finally, let us fix our notation. In what follows, by $|\cdot|_N$ we denote the Lebesgue measure on $\RR^N$. If $x\in \RR^N$, then we set $x^{\pm}=\max\{\pm x,0\}$. So, if $u\in W^{1,p}(\Omega)$, we define $u^{\pm}(\cdot)=u(\cdot)^{\pm}$. We know that
$$u^{\pm}\in W^{1,p}(\Omega),\ u=u^+-u^-,\ |u|=u^++u^-$$

If $h:\Omega\times\RR\rightarrow\RR$ is a measurable function (for example, a Carath\'eodory function), then we define
$$N_h(u)(\cdot)=h(\cdot,u(\cdot))\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$
Evidently, the mapping $z\longmapsto N_h(u)(z)$ is measurable.

\section{Bifurcation Near Infinity}

The hypotheses in the reaction $f(z,x)$ are the following:

\smallskip
\textbf{$H:$} $f:\Omega\times\RR\rightarrow\RR$ is a Carath\'eodory function such that  $f(z,0)=0$ for a.a. $z\in\Omega$ and
\begin{itemize}
	\item[(i)] $|f(z,x)|\leq a(z)\ (1+x^{r-1})$ for a.a. $z\in\Omega$, all $x\geq 0$, with $a\in L^{\infty}(\Omega)_+,\ p\leq r<p^*$;
	\item[(ii)] if $F(z,x)=\int^x_0f(z,s)ds$, then $F(z,x)\rightarrow-\infty$ uniformly for a.a. $z\in\Omega$ as $x\rightarrow+\infty$ and there exists $\tilde{u}\in L^1(\Omega)$ such that  $\int_{\Omega}F(z,\tilde{u}(z))dz>0$;
	\item[(iii)] there exist $\tau>p$ and $x\geq M$ such that  $\tau F(z,x)\leq f(z,x)x$ for a.a. $z\in\Omega$, all $x\geq M$;
	\item[(iv)] $-\hat{c}\leq\liminf\limits_{x\rightarrow 0^+}\frac{f(z,x)}{x^{p-1}}\leq\limsup\limits_{x\rightarrow0^+}\frac{f(z,x)}{x^{p-1}}\leq 0$ uniformly for a.a. $z\in\Omega$;
	\item[(v)] there exists $\mu>p$ such that  for all $\eta>0$, we can find $\tilde{\xi}_{\eta}>0$ for which we have
	$$y-x\geq \eta\Rightarrow \frac{f(z,x)}{x^{\mu-1}}-\frac{f(z,y)}{y^{\mu -1}}\geq \tilde{\xi}_{\eta}\ \mbox{for a.a.}\ z\in\Omega.$$
\end{itemize}
\begin{remark}
	Since we are interested on positive solutions and all the above hypotheses concern the positive semi-axis $\RR_+=\left[0,+\infty\right)$, without any loss of generality we may assume that $f(z,x)=0$ for a.a. $z\in\Omega$, all $x\leq 0$.
\end{remark}

\begin{ex}
	The following functions satisfy hypotheses $H$. For the sake of simplicity, we drop the $z$-dependence:
	\begin{eqnarray*}
	&&f_1(x)=x^{q-1}-x^{r-1}\ \mbox{for all}\ x\geq 0,\ \mbox{with}\ p<q<r<p^*,\\
	&&f_2(x)=x^{\tau-1}\left(\frac{1}{\tau}-\ln x\right)\ \mbox{for all}\ x\geq 0,\ \mbox{with}\ p<\tau<p^*
	\end{eqnarray*}
	
\end{ex}
The function $f_1(x)$ corresponds to the superdiffusive reaction of the $p$-logistic equation.

Our condition on the boundary weight $\beta(\cdot)$ is the following:

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

	%%%%%%%%%%%%%%%%%%%%%%%   pag 8  %%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Under this hypothesis, we know that the eigenvalue problem
\begin{eqnarray}\label{eq2}
-\Delta_p u(z)=\lambda(u(z))^{p-2}u(z)\ \mbox{in}\ \Omega,\ \frac{\partial u}{\partial n_p}+\beta(z)|u|^{p-2}u=0\ \mbox{on}\ \partial\Omega,
\end{eqnarray}
has a first eigenvalue $\hat{\lambda}_1>0$ which is isolated, simple and  admits the following variational characterization
\begin{eqnarray}\label{eq3}
\hat{\lambda}_1=\inf\left[\frac{||Du||^p_p+\int_{\partial\Omega}\beta(z)|u|^{p}d\sigma}{||u||^p_p}:\ u\in W^{1,p}(\Omega),\ u\neq 0\right].
\end{eqnarray}

This infimum is realized on the corresponding one dimensional eigenspace whose elements are in $C^1(\overline{\Omega})$ and have constant sign. For details, we refer to Papageorgiou and R\u adulescu \cite{13}.

We introduce the following truncation-perturbation of the term $\lambda f(z,\cdot)$:
$$\hat{f}_{\lambda}(z,x)=\left\{
\begin{array}{ll}
	0&\mbox{if}\ x\leq 0\\
	\lambda f(z,x)+x^{p-1}&\mbox{if}\ 0<x.
\end{array}\right.$$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

This is a Carath\'eodory function. We set $\hat{F}_{\lambda}(z,x)=\int^x_0\hat{f}_{\lambda}(z,s)ds$ and consider the $C^1$-functional $\hat{\varphi}_{\lambda}:W^{1,p}(\Omega)\rightarrow\RR$ defined by
\begin{eqnarray*}
	 &&\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^+(z)^pd\sigma-\int_{\Omega}\hat{F}_{\lambda}(z,u(z))dz\\
	&&\hspace{7cm}\mbox{for all}\ u\in W^{1,p}(\Omega).
\end{eqnarray*}

\begin{prop}\label{prop4}
	If hypotheses $H,H(\beta)$ hold and $\lambda>0$, then the functional $\hat{\varphi}_{\lambda}$ is bounded below.
\end{prop}
\begin{proof}
	Hypotheses $H(i),(ii)$, imply that we can find $c_1>0$ such that
	\begin{eqnarray}\label{eq4}
		F(z,x)\leq c_1\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\geq 0.
	\end{eqnarray}
	
	Then for all $u\in W^{1,p}(\Omega)$, we have
	\begin{eqnarray*}
		 \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_{\Omega}\hat{F}_{\lambda}(z,u)dz\\
		&\geq&\frac{1}{p}||Du||^p_p-c_1|\Omega|_N\ (\mbox{see (\ref{eq3}), (\ref{eq4}) and hypothesis}\ H(\beta)),\\
		\Rightarrow&&\hat{\varphi}_{\lambda}\ \mbox{is bounded below.}
	\end{eqnarray*}
The proof is complete.
\end{proof}

\begin{prop}\label{prop5}
	If hypotheses $H,H(\beta)$ hold and $\lambda>0$, then the functional $\hat{\varphi}_{\lambda}$ satisfies the $C$-condition.
\end{prop}

\begin{proof}
	Let $\{u_n\}_{n\geq 1}\subset W^{1,p}(\Omega)$ be a sequence such that
	\begin{eqnarray}
		&&|\hat{\varphi}_{\lambda}(u_n)|\leq M_1\ \mbox{for some}\ M_1>0,\ \mbox{all}\ n\geq 1\label{eq5}\\
		&&(1+||u_n||)\hat{\varphi}^{'}_{\lambda}(u_n)\rightarrow 0\ \mbox{in}\ W^{1,p}(\Omega)^*\ \mbox{as}\ n\rightarrow\infty\,.\label{eq6}
	\end{eqnarray}
	
	From (\ref{eq6}), we have
	\begin{eqnarray}
		&&\left|\left\langle \hat\varphi^{'}_{\lambda}(u_n),h\right\rangle\right|\leq\frac{\epsilon_n||h||}{1+||u_n||}\ \mbox{for all}\ h\in W^{1,p}(\Omega)\ \mbox{with}\ \epsilon_n\rightarrow 0^+,\nonumber\\
		\Rightarrow&&\left|\left\langle A(u_n),h\right\rangle+\int_{\Omega}|u_n|^{p-2}u_nhdz+\int_{\partial\Omega}\beta(z)(u_n^+)^{p-1}hd\sigma-\int_{\Omega}\hat{f}_{\lambda}(z,u_n)hdz \right|\nonumber\\
		&&\hspace{9cm}\leq \frac{\epsilon_n||h||}{1+||u_n||}\label{eq7}\\
		&&\hspace{6cm}\ \mbox{for all}\ h\in W^{1,p}(\Omega)\ \mbox{with}\ \epsilon_n\rightarrow 0^+.\nonumber
	\end{eqnarray}
	
	In (\ref{eq7}) we choose $h=-u_n^-\in W^{1,p}(\Omega)$ and obtain
	\begin{eqnarray}\label{eq8}
		&&||u_n^-||^p\leq \epsilon_n\ \mbox{for all}\ n\geq 1\ (\mbox{see (\ref{eq3})}),\nonumber\\
		&\Rightarrow&u_n^-\rightarrow 0\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{as}\ n\rightarrow\infty.
	\end{eqnarray}
	
	Next in (\ref{eq7}) we choose $h=u_n^+\in W^{1,p}(\Omega)$. Then
	\begin{eqnarray}\label{eq9}
		-||Du_n^+||^p_p-\int_{\partial\Omega}\beta(z)(u_n^+)^pd\sigma+\int_{\Omega}\lambda f(z,u_n^+)u_n^+dz\leq\epsilon_n\ \mbox{for all}\ n\geq 1\ \mbox{(see (\ref{eq3}))}.
	\end{eqnarray}
	
	On the other hand from (\ref{eq5}) and (\ref{eq8}), we have
	\begin{eqnarray}\label{eq10}
		&&||Du_n^+||^p_p+\int_{\partial\Omega}\beta(z)(u_n^+)^pd\sigma-\lambda\int_{\Omega}pF(z,u_n^+)dz\leq M_2\\
		&&\hspace{4.5cm}\mbox{for some}\ M_2>0,\ \mbox{all}\ n\geq 1.\nonumber
	\end{eqnarray}
	
	Adding (\ref{eq9}) and (\ref{eq10}), we obtain
	\begin{eqnarray}\label{eq11}
		&&\lambda\int_{\Omega}\left[f(z,u_n^+)u_n^+-pF(z,u_n^+)\right]dz\leq M_3\ \mbox{for some}\ M_3>0,\ \mbox{all}\ n\geq 1,\nonumber\\
		&\Rightarrow&\lambda(\tau-p)\int_{\Omega}F(z,u_n^+)dz+\lambda\int_{\Omega}\left[f(z,u_n^+)u_n^+-\tau F(z,u_n^+)\right]dz\leq M_3\ \mbox{for all}\ n\geq 1,\nonumber\\
		&\Rightarrow&\lambda(\tau-p)\int_{\Omega}F(z,u_n^+)dz\leq M_4\ \mbox{for some}\ M_4>0,\ \mbox{all}\ n\geq 1\\
		&&\hspace{7cm}(\mbox{see hypotheses}\ H(i),(iii)).\nonumber
	\end{eqnarray}
	
	From (\ref{eq5}) and (\ref{eq8}), we have
	\begin{eqnarray}
		 &&\frac{1}{p}||Du_n^+||^p_p+\frac{1}{p}\int_{\partial\Omega}\beta(z)(u_n^+)^pd\sigma-\lambda\int_{\Omega}F(z,u_n^+)dz\leq M_5\label{eq12}\\
		&&\hspace{4cm}\mbox{for some}\ M_5>0,\ \mbox{all}\ n\geq 1\ (\mbox{see (\ref{eq3})}),\nonumber\\
		&\Rightarrow&||Du_n^+||^p_p\leq M_6\ \mbox{for some}\ M_6>0,\ \mbox{all}\ n\geq 1\label{eq13}\\
		&&\hspace{2cm}(\mbox{see hypothesis}\ H(\beta),\ \mbox{(\ref{eq11}) and recall}\ \tau>p\ (\mbox{see}\ H(iii))).\nonumber
	\end{eqnarray}
	
	Let $V=\{u\in W^{1,p}(\Omega):\int_{\Omega}u(z)dz=0\}$. We have
	$$W^{1,p}(\Omega)=\RR\oplus V.$$
	
	Then every $u_n\in W^{1,p}(\Omega)\ n\geq 1$, can be written in a unique way as
	$$u_n=\bar{u}_n+\hat{u}_n\ \mbox{with}\ \bar{u}_n\in\RR,\ \hat{u}_n\in V\ \mbox{for all}\ n\geq 1.$$
	
	From (\ref{eq8}) and (\ref{eq13}), we see that
	\begin{eqnarray*}
		&&\{Du_n\}_{n\geq 1}\subseteq L^p(\Omega,\RR^N)\ \mbox{is bounded},\\
		&\Rightarrow&\{D\hat{u}_n\}_{n\geq 1}\subseteq L^p(\Omega,\RR^N)\ \mbox{is bounded}.
	\end{eqnarray*}
	
	By virtue of the Poincar\'e-Wirtinger inequality (see, for example, Gasinski and Papageorgiou \cite[p. 84]{7}), we have that
	\begin{eqnarray}\label{eq14}
		\{\hat{u}_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)\ \mbox{is bounded}.
	\end{eqnarray}
	
	Suppose that $|\bar{u}_n|\rightarrow\infty$ as $n\rightarrow\infty$. We know that there exists $c_2>0$ such that
	\begin{eqnarray*}
		&&||\bar{u}_n||\leq c_2||u_n||\ (\mbox{see Goldberg \cite[p. 48]{9}})\\
		&\Rightarrow&||u_n||\rightarrow+\infty\ \mbox{as}\ n\rightarrow\infty.
	\end{eqnarray*}
	
	Also, we have
	\begin{eqnarray*}
		&&u_n^-=(\bar{u}_n+\hat{u}_n)^-=-\min\{\bar{u}_n+\hat{u}_n,0\}\\
		&\geq&\bar{u}_n^-+\hat{u}_n^-\ \mbox{for all}\ n\geq 1\ (\mbox{see Schaefer \cite[p. 53]{15}}).
	\end{eqnarray*}
	
	From (\ref{eq8}) we see that $\bar{u}_n^-\rightarrow 0$ and so $\bar{u}_n^+\rightarrow+\infty$ as $n\rightarrow\infty$. Therefore $u_n(z)\rightarrow+\infty$ for a.a. $z\in\Omega$, hence $u_n^+(z)\rightarrow\infty$ for a.a. $z\in\Omega$.
	
	Let $y_n=\frac{u_n}{||u_n||}\ n\geq 1$. Then $||y_n||=1$ for all $n\geq 1$ and so by passing to a suitable subsequence if necessary, we may assume that
	\begin{eqnarray}\label{eq15}
		y_n\stackrel{w}{\rightarrow} y\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{and}\ y_n\rightarrow y\ \mbox{in}\ L^r(\Omega)\ \mbox{as}\ n\rightarrow\infty.
	\end{eqnarray}
	
	We have
	$$y_n=\bar{y}_n+\hat{y}_n\ \mbox{with}\ \bar{y}_n=\frac{\bar{u}_n}{||u_n||},\ \hat{y}_n=\frac{\hat{u}_n}{||u_n||}\ \mbox{for all}\ n\geq 1.$$
	
	From (\ref{eq14}) and (\ref{eq15}), we infer that
	$$y_n\rightarrow y=\bar{y}\in\RR\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{as}\ n\rightarrow\infty\ \mbox{and}\ ||y||=1.$$
	
	Since $u_n^+(z)\rightarrow+\infty$ for a.a. $z\in\Omega$ as $n\rightarrow\infty$, from hypothesis $H_2(ii)$ and Fatou's lemma, we have
	\begin{eqnarray}\label{eq16}
		\int_{\Omega}F(z,u_n^+)dz\rightarrow-\infty\ \mbox{as}\ n\rightarrow\infty.
	\end{eqnarray}
	
	On the other hand from (\ref{eq12}), we have
	\begin{eqnarray}\label{eq17}
		-\lambda\int_{\Omega}F(z,u_n^+)dz\leq M_5\ \mbox{for all}\ n\geq 1.
	\end{eqnarray}
	
	Comparing (\ref{eq16}) and (\ref{eq17}), we reach a contradiction (recall that $\lambda>0$). So, we have that
	\begin{eqnarray*}
		&&\{u_n^+\}_{n\geq 1}\subseteq W^{1,p}(\Omega)\ \mbox{is bounded},\\
		&\Rightarrow&\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)\ \mbox{is bounded (see (\ref{eq8}))}.
	\end{eqnarray*}
	
	Therefore, we may assume that
	\begin{eqnarray}\label{eq18}
		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{eqnarray}
	
	In (\ref{eq7}) we choose $h=u_n-u\in W^{1,p}(\Omega)$, pass to the limit as $n\rightarrow\infty$ and use (\ref{eq18}). Then we have
	\begin{eqnarray*}
		&&\lim\limits_{n\rightarrow\infty}\left\langle A(u_n),u_n-u\right\rangle=0,\\
		&\Rightarrow&u_n\rightarrow u\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{as}\ n\rightarrow\infty\ (\mbox{see Proposition \ref{prop3}}),\\
		&\Rightarrow&\hat{\varphi}_{\lambda}\ \mbox{satisfies the}\ C-\mbox{condition}.
	\end{eqnarray*}
This completes the proof.
\end{proof}

\begin{remark}
	Proposition \ref{prop4} and \ref{prop5} imply that for all $\lambda>0$, the functional $\hat{\varphi}_{\lambda}$ satisfies the $PS$-condition too (see \cite[p. 174]{5}).
\end{remark}

We introduce the following sets
\begin{center}
	$\mathcal{L}=\{\lambda>0:\ \mbox{problem \eqref{eqP} admits a positive solution}\}$\\
	$S(\lambda)=$ set of positive solutions of problem \eqref{eqP}.
\end{center}
\begin{prop}\label{prop6}
	If hypotheses $H$ and $H(\beta)$ hold, then for every $\lambda>0$, $S(\lambda)\subseteq \mbox{int}\, C_+$ and $\lambda_*=\inf\mathcal{L}>0$.
\end{prop}
\begin{proof}
	We may assume that $\lambda\in\mathcal{L}$ (otherwise $S(\lambda)=\varnothing$). Let $u\in S(\lambda)$. Then we have
	\begin{eqnarray}\label{eq19}
		\left\langle A(u),h\right\rangle+\int_{\partial\Omega}\beta(z)|u|^{p-2}uhd\sigma=\int_{\sigma}\lambda f(z,u)hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega).
	\end{eqnarray}
	
	Let $\left\langle \cdot,\cdot\right\rangle_0$ denote the duality brackets for the pair $(W^{-1,p'}(\Omega)=W^{1,p}_{0}(\Omega)^*,W^{1,p}_{0}(\Omega))$. From the representation theorem for the elements of the dual space $W^{-1,p'}(\Omega)$ (see, for example, Gasinski and Papageorgiou \cite[p. 212]{7}), we have
	$$\Delta_pu=\mbox{div}\,(|Du|^{p-2}Du)\in W^{-1,p'}(\Omega).$$
	
	Integration by parts (Green's identity), shows that
	$$\left\langle A(u),h\right\rangle=\left\langle -\Delta_pu,h\right\rangle_0\ \mbox{for all}\ h\in W^{1,p}_{0}(\Omega)\subseteq W^{1,p}(\Omega).$$
	
	Using this equation in (\ref{eq19}) and recalling that, if $h\in W^{1,p}_{0}(\Omega)$, then $h|_{\partial\Omega}=0$ in the sense of trace. So, we have 
		$$\left\langle -\Delta_pu,h\right\rangle_0=\int_{\Omega}\lambda f(z,u)hdz\ \mbox{for all}\ h\in W^{1,p}_{0}(\Omega).$$
		
Note that	hypothesis $H(i)$ implies that $N_f(u)\in L^{r'}(\Omega)\subseteq L^{p'}(\Omega)$ (recall that $p\leq r$, hence $r'\leq p'$; $\frac{1}{p}+\frac{1}{p'}=1$, $\frac{1}{r}+\frac{1}{r'}=1$). So, we have
\begin{eqnarray}\label{eq20}
		&&\int_{\Omega}\lambda f(z,u)hdz=\langle \lambda N_f(u),h\rangle_0\ \mbox{for all}\ h\in W^{1,p}_{0}(\Omega),\nonumber\\
	&\Rightarrow& \langle -\Delta_pu-\lambda N_f(u),h\rangle_0=0\ \mbox{for all}\ h\in W^{1,p}_{0}(\Omega),\nonumber\\
		&\Rightarrow&-\Delta_pu(z)=\lambda f(z,u(z))\ \mbox{for a.a.}\ z\in\Omega.
	\end{eqnarray}


Then using the nonlinear Green's identity mentioned in the Introduction (see also Casas and Fernandez \cite{casa}, Gasinski and Papageorgiou \cite[p. 210]{7}, and Kenmochi \cite{kenm}), we have
	\begin{eqnarray}\label{eq21}
		\left\langle A(u),h\right\rangle+\int_{\Omega}(\Delta_pu)hdz=\left\langle \frac{\partial u}{\partial n_p},h\right\rangle_{\partial\Omega}\ \mbox{for all}\ h\in W^{1,p}(\Omega).
	\end{eqnarray}
	
	Here by $\left\langle \cdot,\cdot\right\rangle_{\partial\Omega}$ we denote the duality brackets for the pair $(W^{-\frac{1}{p'},p'}(\partial\Omega),W^{\frac{1}{p'},p}(\partial\Omega))$. Returning to (\ref{eq19}) and using (\ref{eq20}) and (\ref{eq21}), we obtain
	\begin{eqnarray}\label{eq22}
		\left\langle \frac{\partial u}{\partial n_p}+\beta(z)u^{p-1},h\right\rangle_{\partial\Omega}=0\ \mbox{for all}\ h\in W^{1,p}(\Omega).
	\end{eqnarray}
	
	Let $\gamma_0$ denote the trace map on $W^{1,p}(\Omega)$. We know that
	$$\mbox{im}\, \gamma_0=W^{\frac{1}{p'},p}(\partial\Omega).$$
	
	So, from (\ref{eq22}) it follows that
	\begin{eqnarray}\label{eq23}
		\frac{\partial u}{\partial n_p}+\beta(z)u^{p-1}=0\ \mbox{in}\ W^{-1/p',p'}(\partial\Omega).
	\end{eqnarray}
	
	In fact the nonlinear regularity theory will help us to interpret this boundary condition in a pointwise sense. More precisely,
	from (\ref{eq20}), (\ref{eq23}) and Winkert \cite{18}, we know that $u\in L^{\infty}(\Omega)$. Thus
	we can apply Theorem 2 of Lieberman \cite{10} and have that $u\in C_+\backslash\{0\}$.
	So, it follows that relation \eqref{eq23} holds for all $z\in\partial\Omega$. 
	
	Hypotheses $H(i), (iv)$ imply that given $\rho>0$, we can find $\xi_{\rho}>0$ such that
	\begin{eqnarray}\label{eq24}
		f(z,x)+\xi_{\rho}x^{p-1}\geq 0\ \mbox{for a.a}\ z\in\Omega,\ \mbox{all}\ x\in[0,\rho].
	\end{eqnarray}
	
	Let $\rho=||u||_{\infty}$ and let $\xi_{\rho}>0$ as in (\ref{eq24}). Then
	\begin{eqnarray*}
		&&-\Delta_pu(z)+\lambda\xi_{\rho}u(z)^{p-1}=\lambda f(z,u(z))+\lambda\xi_{\rho}u(z)^{p-1}\geq 0\ \mbox{for a.a.}\ z\in\Omega\ (\mbox{see (\ref{eq20}), (\ref{eq24})})\\
		&\Rightarrow&\Delta_p u(z)\leq \lambda\xi_{\rho}u(z)^{p-1}\ \mbox{for a.a.}\ z\in\Omega,\\
		&\Rightarrow&u\in \mbox{int}\, C_+\ (\mbox{see, for example, Gasinski and Papageorgiou \cite[p. 738]{7}}).
	\end{eqnarray*}
	
Therefore, we conclude that $S(\lambda)\subseteq \mbox{int}\, C_{+}$.

Note that hypotheses $H,(i),(ii),(iv)$ imply that
\begin{eqnarray}\label{eq25}
f(z,x)\leq c_{3} x^{p-1}\ \mbox{for\ a.a.}\ z\in \Omega,\ \mbox{all}\ x\geq 0\ \mbox{with}\ c_{3}>0.
\end{eqnarray}

Let $\lambda \in \left(0,\frac{\hat{\lambda}_{1}}{c_{2}}\right)$. Recall that $\hat{\lambda}_{1}>0$ is the principal eigenvalue of the negative Robin $p$-Laplacian (see (\ref{eq2})). We have
\begin{eqnarray}\label{eq26}
\lambda f(z,x) < \hat{\lambda}_{1}x^{p-1}\ \mbox{for\ a.a.}\ z\in \Omega,\ \mbox{all}\ x>0\ (\mbox{see}\ (\ref{eq25})).
\end{eqnarray}

Suppose that $\lambda \in \mathcal{L}$. Then we can find $u_{\lambda}\in S(\lambda)\subseteq \mbox{int}\, C_{+}$ and we have
\begin{eqnarray*}
\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{eqnarray*}

Choosing $h=u_{\lambda}\in \mbox{int}\, C_{+}$, we have
\begin{eqnarray*}
||Du_{\lambda}||^{p}_{p} + \int_{\partial \Omega} \beta(z)u_{\lambda}^{p} d\sigma < \hat{\lambda}_{1} ||u_{\lambda}||^{p}_{p}\ (\mbox{see (\ref{eq26})\ and\ recall}\ u_{\lambda}\in\ \mbox{int}\, C_{+}),
\end{eqnarray*}
which contradicts (\ref{eq2}). Therefore $\lambda_{*}\geq \frac{\hat{\lambda}_{1}}{c_{2}}>0.$
\end{proof}

Next we show the nonemptiness and a structural property of the admissible set $\mathcal{L}$.

\begin{prop}\label{prop7}
If hypotheses $H$ and $H(\beta)$ hold, then $\mathcal{L} \neq \emptyset$ and $\lambda \in \mathcal{L}$, $\eta > \lambda$ imply $\eta \in \mathcal{L}$.
\end{prop}

\begin{proof}
>From Proposition \ref{prop4} and \ref{prop5} and Theorem 2.1.14  of Denkowski, Migorski and Papageorgiou \cite[p. 184]{5}, we know that there exists $u_{0}\in W^{1,p}(\Omega)$ such that
\begin{eqnarray}\label{eq27}
\hat{\varphi}_{\lambda} (u_{0})=\inf\left[\hat{\varphi}_{\lambda}(u):u\in W^{1,p}(\Omega)\right].
\end{eqnarray}

The integral functional $u \longmapsto I_{F}(u)=\int_{\Omega}F(z,u(z))dz$ is continuous on $L^{1}(\Omega)$ and, by hypothesis $H(ii)$, $I_{F}(\tilde{u})>0$. Exploiting the density of $W^{1,p}(\Omega)$ in $L^{1}(\Omega)$, we can find $\tilde{u}_{0}\in W^{1,p}(\Omega)$ such that
\begin{eqnarray}\label{eq28}
I_{F}(\tilde{u}_{0})=\int_{\Omega}F(z,\tilde{u}_{0}(z))dz >0.
\end{eqnarray}

Recalling that $F(z,x)=0$ for a.a. $z\in \Omega$, all $x\leq 0$, we see that we can replace $\tilde{u}_{0}$ by $\tilde{u}_{0}^{+}\in W^{1,p}(\Omega)$. So, without any loss of generality, we may assume that $\tilde{u}_{0}\geq 0$.

We have
\begin{eqnarray*}
\hat{\varphi}_{\lambda}(\tilde{u}_{0})=\frac{1}{p}||D\tilde{u}_{0}||^{p}_{p}+\frac{1}{p}\int_{\partial \Omega}\beta(z)\tilde{u}_{0}^{p}d\sigma - \lambda\int_{\Omega}F(z,\tilde{u}_{0})dz.
\end{eqnarray*}

Because of (\ref{eq28}), we see that we can find $\tilde{\lambda} >0$ such that
\begin{eqnarray*}
&&\hat{\varphi}_{\lambda}(\tilde{u}_{0})<0\ \mbox{for\ all}\ \lambda > \tilde{\lambda},\\
&\Rightarrow&\hat{\varphi}_\lambda(u_{0})<0=\hat{\varphi}_{\lambda}(0)\ \mbox{(see\ (\ref{eq27})),\ hence}\ u_{0}\neq 0.
\end{eqnarray*}


>From (\ref{eq27}) we have
\begin{eqnarray*}
\hat{\varphi}_{\lambda}^{'}(u_{0})=0,
\end{eqnarray*}
\begin{eqnarray}\label{eq29}
\Rightarrow \left\langle A(u_{0}),h\right\rangle+\int_{\Omega}|u_{0}|^{p-2}u_{0} hdz + \int_{\partial \Omega}\beta(z)(u_{0}^{+})^{p-1}hd \sigma = \\
\int_{\Omega} \hat{f}_{\lambda} (z,u_{0})hdz\ \mbox{for\ all}\ h\in W^{1,p} (\Omega). \nonumber
\end{eqnarray}

In (\ref{eq29}) we choose $h=-u_{0}^{-}\in W^{1,p}(\Omega)$ and obtain
\begin{eqnarray*}
||Du_{0}^{-}||_{p}^{p}+||u_{0}^{-}||_{p}^{p}=0\ \mbox{(see (\ref{eq3})),\ hence}\ u_{0}\geq 0,\ u_{0}\neq 0.
\end{eqnarray*}

So, (\ref{eq29}) becomes
\begin{eqnarray*}
\left\langle A(u_{0}),h \right\rangle + \int_{\partial \Omega} \beta (z)u_{0}^{p-1} hd \sigma = \lambda \int_{\Omega}f(z,u_{0}) hdz\ \mbox{for\ all}\ h \in W^{1,p}(\Omega)\ \mbox{(see (\ref{eq3}))}.
\end{eqnarray*}

>From this equality, as in the proof of Proposition \ref{prop6}, using the nonlinear Green's identity, we obtain
\begin{eqnarray*}
&&-\Delta_{p}u_{0}(z)=\lambda f(z,u_{0}(z))\ \mbox{for\ a.a.}\ z\in \Omega,\ \frac{\partial u_{0}}{\partial n_{p}}+ \beta (z)u_{0}^{p-1}=0\ \mbox{on}\ \partial \Omega,\\
&\Rightarrow&u_{0}\in S(\lambda)\ \mbox{and\ so}\ (\tilde{\lambda}, \infty) \subseteq \mathcal{L},\ \mbox{hence}\ \mathcal{L}\neq \emptyset .
\end{eqnarray*}

Next let $\lambda \in \mathcal{L}$ and $\eta > \lambda$. Choose $\vartheta \in (0,1)$ such that  $\lambda=\vartheta^{\tau -p}\eta$ (hence $\tau > p$ is as in hypothesis $H(v)$). Since $\lambda \in \mathcal{L}$, we can find $u_{\lambda}\in S(\lambda)\subseteq \mbox{int}\, C_{+}$ (see Proposition \ref{prop6}).

We have
\begin{eqnarray}\label{eq30}
-\Delta_{p}(\vartheta u_{\lambda})(z) & = & \vartheta^{p+1}\lambda f (z,u_{\lambda}(z))\ (\mbox{since}\ u_{\lambda}\in S(\lambda))\\ \nonumber
& = & \vartheta^{\tau -1}\eta f (z, u_{\lambda}(z))\ (\mbox{since}\ \lambda = \vartheta^{\tau - p}\eta)\\ \nonumber
& \leq & \eta f(z, \vartheta u_{\lambda}(z))\ \mbox{for\ a.a.}\ z\in \Omega\ \\
& &\hspace{1cm}(\mbox{since}\ \vartheta\in(0,1),\ \mbox{see\ hypothesis}\ H(v)). \nonumber
\end{eqnarray}

Also, we have
\begin{eqnarray}\label{eq31}
\frac{\partial(\vartheta u_{\lambda})}{\partial n_{p}} + \beta(z)(\vartheta u_{\lambda})^{p-1}=0\ \mbox{on}\ \partial \Omega.
\end{eqnarray}

Let $\underline{u}=\vartheta u_\lambda \in \mbox{int}\, C_{+}$ and consider the following truncation-perturbation of the reaction in problem $(P_{\eta}):$
\begin{eqnarray}\label{eq32}
\hat{g}_{\eta}(z,x)=\begin{cases}\eta f(z,\underline{u}(z))+\underline{u}(z)^{p-1} & \mbox{if}\ x \leq \underline{u}(z) \\ \eta f(z,x)+x^{p-1} & \mbox{if}\ \underline{u}(z)<x.\end{cases}
\end{eqnarray}

This is a Carath\'eodory function. We set $\hat{G}_{\eta}(z,x)=\int_{0}^{x}\hat{g}_{\eta}(z,s)ds$ and consider the $C^{1}-$functional $\hat{\psi}_{\eta}:W^{1,p}(\Omega)\rightarrow \RR$ defined by
$$
\hat{\psi}_{\eta}(u)=\frac{1}{p}||Du||_{p}^{p}+\frac{1}{p}||u||^{p}_{p}+\frac{1}{p} \int_{\partial \Omega}\beta(z)u^{+}(z)^{p}d\sigma - \int_{\Omega}\hat{G}_{\eta}(z,u(z))dz.$$

>From (\ref{eq32}) it is clear that
\begin{eqnarray*}
\hat{\psi}_{\eta}=\hat{\varphi}_{\eta}+\xi^{*}_{\eta}\ \mbox{for\ some}\ \xi^{*}_{\eta}\in \RR.
\end{eqnarray*}

So, it follows that
\begin{itemize}
	\item[$\bullet$] $\hat{\psi}_{\eta}$ is bounded below (see Proposition \ref{prop4}).
	\item[$\bullet$] $\hat{\psi}_{\eta}$ satisfies the C-condition (see Proposition \ref{prop5}).
\end{itemize}

Therefore we can find $u_{\eta}\in W^{1,p}(\Omega)$ such that
\begin{eqnarray}\label{eq33}
&&\hat{\psi}_{\eta}(u_{\eta})=\inf\left[\hat{\psi}_{\eta}(u):u\in W^{1,p}(\Omega)\right]\ (\mbox{see\ \mbox{\cite[p. 184]{5}}}),\\ \nonumber
& \Rightarrow & \hat{\psi}^{'}_{\eta}(u_{\eta})=0, \\ \nonumber
& \Rightarrow & \left\langle A(u_{\eta}),h \right\rangle + \int_{\Omega} |u_{\eta}|^{p-2} u_{\eta}hdz+\int_{\partial\Omega}\beta(z)(u_{\eta}^{+})^{p-1}hd\sigma = \int_{\Omega}\hat{g}_{\eta}(z,u_{\eta})hdz\\
&&\hspace{8cm}\mbox{ for\ all}\ h\in W^{1,p}(\Omega). \nonumber
\end{eqnarray}

Choosing $h=(\underline{u}-u_{\eta})^{+}\in W^{1,p}(\Omega)$, we obtain
\begin{eqnarray*}
&&\left\langle A(u_{\eta}), (\underline{u}-u_{\eta})^+ \right\rangle + \int_{\Omega} |u_{\eta}|^{p-2}u_{\eta}(\underline{u}-u_{\eta})^{+}dz+\int_{\partial \Omega}\beta(z)(u_{\eta}^{+})^{p-1}(\underline{u}-u_{\eta})^{+}d\sigma\\
&=&\int_{\Omega}\hat{g}_{\eta}(z,u_{\eta})(\underline{u}-u_{\eta})^{+}dz\\
&=&\int_{\Omega}\left[\eta f(z, \underline{u})+\underline{u}^{p-1}\right](\underline{u}-u_{\eta})^{+}dz\ \mbox{(see (\ref{eq32}))}\\
&\geq&\left\langle A(\underline{u}), (\underline{u}-u_{\eta})^+ \right\rangle + \int_{\Omega} \underline{u}^{p-1}(\underline{u}-u_{\eta})^{+}dz+\int_{\partial \Omega} \beta(z)\underline{u}^{p-1}(\underline{u}-u_{\eta})^{+}d\sigma\\
&&\hspace{4cm}\mbox{(see\ (\ref{eq30}), (\ref{eq31})\ and\ use\ the\ nonlinear\ Green's\ identity)}
\end{eqnarray*}
\begin{eqnarray}\label{eq34}
\Rightarrow \left\langle A(\underline{u}) - A(u_{\eta}), (\underline{u}-u_{\eta})^+ \right\rangle & + & \int_{\Omega}(\underline{u}^{p-1}-|u_{\eta}|^{p-2}u_{\eta})(\underline{u}-u_{\eta})^+dz\\ \nonumber
& + & \int_{\partial\Omega}\beta(z)(\underline{u}^{p-1}-(u^{+}_{\eta})^{p-1})(\underline{u}-u_{\eta})^{+}d \sigma \leq 0.
\end{eqnarray}

Note that
\begin{eqnarray*}
&&\underline{u}(z)>u_{\eta}(z)\ \mbox{for\ all}\ z\in\{\underline{u}>u_{\eta}\},\\
&\Rightarrow&\underline{u}(z)\geq u_{\eta}^{+}(z)\ \mbox{for\ all}\ z \in \{\underline{u}>u_{\eta}\}\ (\mbox{recall}\ \underline{u}=\vartheta u_{\lambda} \in \mbox{int}\, C_{+})\\
&\Rightarrow&\int_{\partial\Omega}\beta(z)(\underline{u}^{p-1}-(u_{\eta}^{+})^{p-1})(\underline{u}-u_{\eta})^{+}d \sigma \geq 0\ (\mbox{see\ hypothesis}\ H(\beta)).
\end{eqnarray*}

Using this last inequality in (\ref{eq34}), we obtain
\begin{eqnarray*}
&&\left\langle A(\underline{u})-A(u_{\eta}), (\underline{u}-u_{\eta})^{+} \right\rangle + \int_{\Omega}(\underline{u}^{p-1}-|u_{\eta}|^{p-2}u_{\eta})(\underline{u}-u_{\eta})^{+}dz \leq 0,\\
&\Rightarrow&|\{\underline{u}>u_{\eta}\}|_{N}=0,\ \mbox{hence}\ \underline{u}\leq u_{\eta}.
\end{eqnarray*}

So, relation (\ref{eq33}) becomes
\begin{eqnarray*}
&&\left\langle A(u_{\eta}),h\right\rangle + \int_{\partial \Omega}\beta(z)u_{\eta}^{p-1}hd\sigma = \int_{\Omega}\eta f(z,u_{\eta})hdz\ \mbox{for\ all}\ h\in W^{1,p}(\Omega)\ \mbox{(see\ (\ref{eq32}))}\\
&\Rightarrow&u_{\eta}\in S(\eta)\subseteq \mbox{int}\, C_{+}\ \mbox{(see\ Proposition\ \ref{prop6}\ its\ proof),}\\
&\Rightarrow&\eta \in \mathcal{L}\ \mbox{and\ so}\ [\lambda, +\infty) \subseteq \mathcal{L}.
\end{eqnarray*}
This completes the proof.
\end{proof}

\begin{remark}
According to Proposition \ref{prop7}, the admissible set $\mathcal{L}$ is an upper half line.
\end{remark}

Next we prove a multiplicity result for the positive solutions of problem $(p_{\lambda})$. To do this we need to strengthen the conditions on $f(z, \cdot)$.

The new hypotheses on the reaction  function $f(z,x)$, are the following:

\smallskip
$H^{'}:f:\Omega \times \RR \rightarrow \RR$ is a Carath\'eodory function such that  $f(z,0)=0$ for a.a. $z\in \Omega$, hypotheses $H^{'}(i)\rightarrow(v)$ are the same as the corresponding hypotheses $H(i)\rightarrow(v)$ and\\ $(vi)$ for every $\rho >0$, there exists $\xi_{\rho}>0$ such that  for a.a. $z\in \Omega$, the function
\begin{eqnarray*}
x\longmapsto f(z,x)+\xi_{\rho}x^{p-1}
\end{eqnarray*}
is nondecreasing on $[0,\rho]$.

\begin{remark}
If $p=2$ and $f(z,\cdot)$ is differentiable with $f^{'}_{x}(z,\cdot)$ being $L^{\infty}(\Omega)$-bounded on bounded sets in $\RR$, then hypothesis $H^{'}(vi)$ is satisfied. Also, the two examples given after hypotheses $H$ (the functions $f_{1}(x)$ and $f_{2}(x)$), satisfy the new hypothesis $H^{'}(vi)$.
\end{remark}

\begin{prop}\label{prop8}
If hypotheses $H,H(\beta)$ hold and $\lambda > \lambda_{*}$, then problem \eqref{eqP} has at least two positive solutions
\begin{eqnarray*}
u_{0},\hat{u}\in \mbox{int}\,C_{+}, u_{0}\neq\hat{u}.
\end{eqnarray*}
\end{prop}

\begin{proof}
Let $\nu \in(\lambda_{*},\lambda)$. By virtue of Proposition \ref{prop7}, $\nu \in \mathcal{L}$ and so we can find $u_{\nu}\in S(\nu)\subseteq \mbox{int}\, C_{+}$. We choose $\vartheta \in(0,1)$ such that  $\nu=\vartheta^{\tau-p}\lambda$ (see hypothesis $H^{'}(iii)$). We set $\underline{u}=\vartheta u_{\nu}\in \mbox{int}\, C_{+}$ and define
\begin{eqnarray}\label{eq35}
\hat{g}_{\lambda}(z,x)=\begin{cases}\lambda f(z,\underline{u}(z))+\underline{u}(z)^{p-1} & \mbox{if}\ x \leq \underline{u}(z)\\
\lambda f(z,x)+x^{p-1} & \mbox{if}\ \underline{u}(z)<x.  \end{cases}
\end{eqnarray}

This is a Carath\'eodory function. We set $\hat{G}_{\lambda}(z,x)=\int_{0}^{x}\hat{g}_{\lambda}(z,s)ds$ and consider the $C^{1}-$functional $\hat{\psi}_{\lambda}:W^{1,p}(\Omega)\rightarrow \RR$ defined by
\begin{eqnarray*}
\hat{\psi}_{\lambda}(u)=\frac{1}{p}||Du||_{p}^{p}+\frac{1}{p}||u||_{p}^{p}+\frac{1}{p}\int_{\partial \Omega}\beta(z)u^{+}(z)^{p}dz-\int_{\Omega}\hat{G}_{\lambda}(z,u(z))dz\\
\hspace{5cm}\mbox{for\ all}\ u\in W^{1,p}(\Omega).
\end{eqnarray*}

Reasoning as in the second half of the proof of Proposition \ref{prop7}, we can find $u_{0}\in W^{1,p}(\Omega)$ such that
\begin{eqnarray}\label{eq36}
\hat{\psi}_{\lambda}(u_{0})=\inf[\hat{\psi}_{\lambda}(u):u\in W^{1,p}(\Omega)].
\end{eqnarray}

>From (\ref{eq36}) it follows that
$$u_{0}\in S(\lambda)\subseteq \mbox{int}\, C_{+}\ \mbox{and}\ \underline{u}\leq u_{0}\ \mbox{(see the proof of Proposition \ref{prop7}).}$$

Let $\rho=||u_{0}||_{\infty}$ and let $\xi_{\rho}>0$ be as postulated by hypothesis $H^{'}(vi)$. Let $\delta >0$ and set $\underline{u}^{\delta}=\underline{u}+\delta\in \mbox{int}\, C_{+}$. We have
\begin{eqnarray}\label{eq37}
&&-\Delta_{p}\underline{u}^{\delta}(z)+\lambda\xi_{\rho}\underline{u}^{\delta}(z)^{p-1} \nonumber \\
&\leq&-\Delta_{p}\underline{u}(z)+\lambda\xi_{\rho}\underline{u}(z)^{p-1}+\gamma(\delta)\ \mbox{with}\ \gamma(\delta)\rightarrow0^{+}\ \mbox{as}\ \delta\rightarrow 0^{+}\nonumber \\
&=&\vartheta^{p-1}(-\Delta_{p}u_{\nu}(z))+\lambda\xi_{\rho}\underline{u}(z)^{p-1}+\gamma(\delta) \nonumber\\
&=&\vartheta^{p-1}\nu f(z,u_{\nu}(z))+\lambda\xi_{\rho} \underline{u}(z)^{p-1}+\gamma(\delta)\ (\mbox{since}\ u_{\nu}\in S(\nu)) \nonumber\\
&=&\vartheta^{\tau-1}\lambda f(z,u_{\nu}(z))+\lambda\xi_{\rho} \underline{u} (z)^{p-1}+\gamma(\delta)\ (\mbox{since}\ \nu=\vartheta^{\tau-p}\lambda).
\end{eqnarray}

Let $m_{\nu}=\underset{\overline{\Omega}}\min u_{\nu}>0$ (recall that $u_{\nu}\in \mbox{int}\, C_{+}$). By virtue of hypothesis $H^{'}(v)$ for $\eta=(1-\vartheta)m_{\nu}$, we can find $\overset{N}\xi_{\eta}>0$ such that
\begin{eqnarray}\label{eq38}
&&\frac{f(z,\underline{u}(z))}{\underline{u}(z)^{\tau -1}} - \frac{f(z,u_{\nu}(z))}{u_{\nu}(z)^{\tau-1}} \geq \tilde{\xi}_{\eta}\ (\mbox{recall}\ \underline{u}=\vartheta u_{\nu},\vartheta\in(0,1))\nonumber\\
&\Rightarrow&\frac{f(z,\underline{u}(z))}{\vartheta^{\tau-1}} - f(z,u_{\nu}(z)) \geq \tilde{\xi}_{\eta} m_{\nu}^{\tau-1}, \nonumber\\
&\Rightarrow& f(z,\underline{u}(z))-\vartheta^{\tau-1} f(z,u_{\nu}(z)) \geq \vartheta ^{\tau-1}\tilde{\xi}_{\eta} m_{\nu}^{\tau-1} >0.
\end{eqnarray}

Using (\ref{eq38}) in (\ref{eq37}), we obtain
\begin{eqnarray}\label{eq39}
&&-\Delta_{p}\underline{u}^{\delta}(z)+\lambda\xi_{\rho}\underline{u}^{\delta}(z)^{p-1}\nonumber\\
&\leq&\lambda f(z,\underline{u}(z))-\vartheta^{\tau-1}\tilde{\xi}_{\eta}m_{\nu}^{\tau-1}+\lambda\xi_{\rho}\underline{u}(z)^{p-1}+\gamma(\delta)\nonumber\\
&\leq&\lambda f(z,u_0(z))+\lambda\xi_{\eta} u_{0}(z)^{p-1}-\vartheta^{\tau-1}\tilde{\xi}_{\eta}m_{\nu}^{\tau-1}+\gamma(\vartheta)\nonumber\\
&&\hspace{2cm} (\mbox{see\ hypothesis}\ H^{'}(vi)\ \mbox{and\ recall}\ \underline{u}\leq u_{0}) \nonumber\\
&\leq&-\Delta_{p}u_{0}(z)+\lambda\xi_{\rho}u_{0}(z)^{p-1}\ \mbox{for\ a.a.}\ z\in \Omega,\ \mbox{for}\ \delta>0\ \mbox{small}\\
&&\hspace{2cm}(\mbox{recall}\ \gamma(\delta)\rightarrow 0^{+}\ \mbox{as}\ \delta\rightarrow 0^{+}). \nonumber
\end{eqnarray}

Note that
\begin{eqnarray}\label{eq40}
\frac{\partial \underline{u}}{\partial n_{\eta}}+\beta(z)\underline{u}^{p-1}=0\ \mbox{and}\ \frac{\partial u_{0}}{\partial n_{p}}+\beta(z)u_{0}^{p-1}=0\ \mbox{on}\ \partial\Omega .
\end{eqnarray}

>From (\ref{eq39}), (\ref{eq40}) and using the nonlinear Green's identity, as before we obtain
\begin{eqnarray}\label{eq41}
&&u_{0}-\underline{u}^{\delta}\geq 0\ \mbox{for}\ \delta>0\ \mbox{small},\nonumber\\
&\Rightarrow&u_{0}-\underline{u}\in \mbox{int}\, C_{+}.
\end{eqnarray}

Let $\left[ \underline{u} \right) = \left\{u \in W^{1,p}(\Omega):\underline{u}(z)\leq u(z)\ \mbox{for\ a.a.}\ z \in \Omega\right\}$. From (\ref{eq35}) it is clear that
\begin{eqnarray*}
\hat{\psi}_{\lambda} = \hat{\varphi}_{\lambda}+\xi^{*}_{\lambda}\ \mbox{with}\ \xi^{*}_{\lambda}\in \RR .
\end{eqnarray*}

Then (\ref{eq41}) implies that $u_{0}$ is a local $C^{1}(\overline{\Omega})$-minimizer of $\hat{\psi}_{\lambda}$. Invoking Proposition \ref{prop2}, we infer that $u_{0}$ is a local $W^{1,p}(\Omega)$-minimizer of $\hat{\psi}_{\lambda}$.

By virtue of hypothesis $H^{'}(iv)$, given $\epsilon > 0$, we can find $\delta=\delta(\epsilon)>0$ such that
\begin{eqnarray}\label{eq42}
F(z,x)\leq\frac{\epsilon}{p}x^{p}\ \mbox{for\ a.a.}\ z\in \Omega,\ \mbox{all}\ x \in[0,\delta].
\end{eqnarray}

Then for $u\in C^{1}(\overline{\Omega})$ with $||u||_{C^{1}(\overline{\Omega})}\leq \delta$, we have
\begin{eqnarray*}
\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^{+})^{p}d\sigma - \int_{\Omega}\lambda F(z,u^{+})dz\\
& \geq & \frac{1}{p}\left[||Du^{+}||_{p}^{p}+\int_{\partial\Omega}\beta(z)(u^{+})^{p}d\sigma\right] + \frac{1}{p}||u^{-}||_{p}^{p}-\frac{\epsilon}{p}\lambda||u^{+}||_{p}^{p}\ \mbox{(see\ (\ref{eq42}))}\\
& \geq & \frac{\hat{\lambda_{1}}-\lambda \epsilon}{p}||u^{+}||_{p}^{p}+\frac{1}{p}||u^{-}||_{p}^{p}\ \mbox{(see\ (\ref{eq21}))}\\
& \geq & c_{4}||u||_{p}^{p}\ \mbox{for\ some}\ c_{4}>0\ (\mbox{choosing}\ \epsilon\in ( 0,\hat{\lambda}_{1}/\lambda )).
\end{eqnarray*}

So, we have
\begin{eqnarray*}
\hat{\varphi}_{\lambda}(0)& = & 0< \hat{\varphi}_{\lambda}(u)\ \mbox{for\ all}\ u\in C^{1}(\overline{\Omega}),\ 0<||u||_{C^{1}(\overline{\Omega})}\leq \delta,\\
\Rightarrow u & = & 0\ \mbox{is\ a\ (strict)\ local}\ C^{1}(\overline{\Omega})-\mbox{minimizer\ of}\ \hat{\varphi}_{\lambda},\\
\Rightarrow u & = & 0\ \mbox{is\ a\ local}\ W^{1,p}(\Omega)-\mbox{minimizer\ of}\ \hat{\varphi}_{\lambda}\ \mbox{(see\ Proposition\ \ref{prop2}).}
\end{eqnarray*}

Without any loss of generality, we may assume that $\hat{\varphi}_{\lambda}(0)=0\leq \hat{\varphi}_{\lambda}(u_{0})$ (the analysis is similar if the opposite inequality holds). We may assume that the critical set of $\hat{\varphi}_{\lambda}$ is finite (otherwise, we already have infinitely many positive solutions for problem \eqref{eqP}). Since $u_{0}\in \mbox{int}\, C_{+}$ is a local minimizer of $\hat{\varphi}_{\lambda}$, we can find $\rho \in (0,1)$ small such that
\begin{eqnarray}\label{eq43}
0=\hat{\varphi}_{\lambda}(0)\leq \hat{\varphi}_{\lambda}(u_{0})< \inf[\hat{\varphi_{\lambda}}(u):||u-u_{0}||=\rho]=m_{\rho},\ ||u_{0}||>\rho
\end{eqnarray}
(see Aizicovici, Papageorgiou and Staicu \cite{1} (proof of Proposition 29). Recall that $\hat{\varphi}_{\lambda}$ satisfies the C-condition (see Proposition \ref{prop5}). This fact and (\ref{eq43}) permit the use of Theorem \ref{th1} (the mountain pass theorem). So, se can find $\hat{u}\in W^{1,p}(\Omega)$ such that
\begin{eqnarray}\label{eq44}
\hat{\varphi}_{\lambda}'(\hat{u})=0\ \mbox{and}\ m_{\rho}\leq \hat{\varphi}_{\lambda}(\hat{u}).
\end{eqnarray}

>From (\ref{eq43}) and (\ref{eq44}) it follows that $\hat{u} \notin \{0,u_{0}\}$ and $\hat{u}\in S(\lambda)\subseteq \mbox{int}\, C_{+}$.
\end{proof}

Next we examine what happens in the critical case $\lambda = \lambda_{*}$.

\begin{prop}\label{prop9}
If hypotheses $H$ and $H(\beta)$ hold, then $\lambda_{*}\in \mathcal{L}$.
\end{prop}
\begin{proof}
Le $\{\lambda_{n}\}_{n\geq 1}$ $\subseteq \mathcal{L}$ such that  $\lambda_{n}\downarrow \lambda_{*}$ as $n \rightarrow \infty$ and let $u_{n}\in S(\lambda_{n})$ for all $n\geq 1$.

We have
\begin{eqnarray}\label{eq45}
\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}\ h\in W^{1,p}(\Omega),\\
 \mbox{all}\ n\geq 1 \nonumber.
\end{eqnarray}

Hypotheses $H(i),\ (ii),\ (iv)$, imply that given $\epsilon > 0$, we can find $c_{5}=c_{5}(\epsilon)>0$\ such that
\begin{eqnarray}\label{eq46}
f(z,x) \leq \epsilon x^{p-1}+c_{5}\ \mbox{for\ a.a.}\ z\in \Omega,\ \mbox{all}\ x\geq 0.
\end{eqnarray}

In (\ref{eq45}) we choose $h=u_{n}\in W^{1,p}(\Omega)$ and use (\ref{eq46}). Then
\begin{eqnarray}\label{eq47}
||Du_{n}||_{p}^{p}+\int_{\partial \Omega}\beta(z)u_{n}^{p}d\sigma & \leq & \lambda_{n} \epsilon ||u_{n}||_{p}^{p}+\lambda_{n} c_{5}|\Omega|_{N}\nonumber\\
& \leq & \lambda_{1}\epsilon ||u_{n}||_{p}^{p}+\lambda_{1}c_{5}|\Omega|_{N}\\
&&\hspace{2cm}(\mbox{since}\ \lambda_{n}\leq\lambda_{1}\ \mbox{for\ all}\ n\geq 1), \nonumber\\
\Rightarrow(\hat{\lambda}_{1}-\lambda_{1}\epsilon)||u_{n}||_{p}^{p} &\leq& \lambda_{1} c_{5} |\Omega|_{N}.\nonumber
\end{eqnarray}

Choosing $\epsilon \in \left( 0, \frac{\hat{\lambda}_{1}}{\lambda_{1}} \right)$, we see that $\{u_{n}\}_{n\geq 1}\subseteq L^{p}(\Omega)$ is bounded. Using this in (\ref{eq47}) and since $\beta \geq 0$ (see hypothesis $H(\beta)$), we infer that
\begin{eqnarray*}
&&\{Du_{n}\}_{n\geq 1} \subseteq L^{p}(\Omega, \RR^{N})\ \mbox{is\ bounded,}\\
&\Rightarrow&\{u_{n}\}_{n\geq 1} \subseteq W^{1,p}(\Omega)\ \mbox{is\ bounded}
\end{eqnarray*}

So, we may assume that
\begin{eqnarray}\label{eq48}
u_{n}\overset{w}\rightarrow u_{*}\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{and}\ u_{n}\rightarrow u_{*}\ \mbox{in}\ L^{r}(\Omega)\ \mbox{and}\ \mbox{in}\ L^{p}(\partial \Omega)\ \mbox{as}\ n\rightarrow \infty .
\end{eqnarray}

In (\ref{eq45}) we choose $h=u_{n}-u_{*}\in W^{1,p}(\Omega)$, pass to the limit as $n \rightarrow \infty$ and use (\ref{eq48}). Then
\begin{eqnarray}\label{eq49}
&&\lim_{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\ Proposition \ref{prop3})}.
\end{eqnarray}

So, if in (\ref{eq45}) we pass to the limit as $n \rightarrow \infty$ and use (\ref{eq49}), 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_{*}\ \mbox{is\ a\ solution\ of\ problem}\ (P_{\lambda_{*}}).
\end{eqnarray*}

We need to show that $u_{*}\neq 0 $ in order to conclude that $u_{*}\in S(\lambda_{*})$, hence that $\lambda_{*}\in \mathcal{L}$. Arguing by contradiction, suppose $u_{*}=0$. Then
\begin{eqnarray*}
u_{n}\rightarrow 0\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{as}\ n\rightarrow\infty\ \mbox{(see\ (\ref{eq49}))}.
\end{eqnarray*}

Let $y_{n}=\frac{u_{n}}{||u_{n}||}$ $n\geq 1$. Then $||y_{n}||=1$, $y_{n}\geq 0$, for all $n \geq 1$. So, we may assume that
\begin{eqnarray}\label{eq50}
y_{n}\overset{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{eqnarray}

>From (\ref{eq45}), we have
\begin{eqnarray}\label{eq51}
\left\langle A(y_{n}),h \right\rangle + \int_{\partial \Omega}\beta(z)y_{n}^{p-1}hd\sigma = \int_{\Omega}\frac{N_{f}(u_{n})}{||u_{n}||^{p-1}} hdz\ \mbox{for\ all}\ h\in W^{1,p}(\Omega),\\
\hspace{5cm}\mbox{all}\ n \geq 1.\nonumber
\end{eqnarray}

Hypotheses $H(i),\ (iv)$ imply that
\begin{eqnarray*}
&&|f(z,x)|\leq c_{6}(x^{p-1}+x^{r-1})\ \mbox{for\ a.a.}\ z\in\Omega\ \mbox{with}\ c_{6}>0\\
&\Rightarrow&\left\{\frac{N_{f}(u_{n})}{||u_{n}||^{p-1}}\right\}_{n \geq 1} \subseteq L^{p'}(\Omega)\ \mbox{is\ bounded.}
\end{eqnarray*}

Passing to a subsequence if necessary and using hypothesis $H(iv)$ we obtain
\begin{eqnarray}\label{eq52}
\frac{N_{f}(u_{n})}{||u_{n}||^{p-1}} \overset{w}\rightarrow \xi y^{p-1}\ \mbox{in}\ L^{p^{'}}(\Omega)\ \mbox{with}\ -\hat{c}\leq\xi(z)\leq 0\ \mbox{for\ a.a.}\ z\in\Omega .
\end{eqnarray}

In (\ref{eq51}), we choose $h=y_{n}-y\in W^{1,p}(\Omega)$, pass to the limit as $n\rightarrow \infty$ and use (\ref{eq52}). Then
\begin{eqnarray}\label{eq53}
&&\lim_{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,\ \mbox{hence}\ ||y||=1,\ y\geq 0.
\end{eqnarray}

If in (\ref{eq51}) we pass to the limit as $n\rightarrow\infty$ and use (\ref{eq52}) and (\ref{eq53}), we obtain
\begin{eqnarray*}
\left\langle A(y),h \right\rangle + \int_{\partial\Omega}\beta(z)y^{p-1}hd\sigma = \lambda_{*}\int_{\Omega}\xi(z)y^{p-1}hdz\ \mbox{for\ all}\ h\in W^{1,p}(\Omega).
\end{eqnarray*}

Choose $h=y\in W^{1,p}(\Omega)$. Then using (\ref{eq2}), we have
\begin{eqnarray*}
0 < \hat{\lambda}_{1}||y||_{p}^{p}\leq||Dy||_{p}^{p}+\int_{\partial\Omega}\beta(z)y^{p}d\sigma = \lambda_{*}\int_{\Omega}\xi(z)y^{p}dz\leq 0\ \mbox{(see\ (\ref{eq52})),}
\end{eqnarray*}
a contradiction. This proves that $u_{*}\neq 0$ and so $u_{*}\in S(\lambda_{*})$, that is, $\lambda_{*}\in\mathcal{L}$.
\end{proof}

So, we can summarize the situation for problem \eqref{eqP}, by stating the following bifurcation near infinity result.
\begin{theorem}\label{th10}
If hypotheses $H^{'}$ and $H(\beta)$ hold, then there exists $\lambda_{*}>0$ such that
\begin{itemize}
	\item[(a)] for all $\lambda > \lambda_{*}$ problem \eqref{eqP} admits at least two positive solutions
\begin{eqnarray*}
u_{0},\hat{u}\in \mbox{int}\, C_{+},\ u_{0}\neq \hat{u};
\end{eqnarray*}
	\item[(b)] for $\lambda = \lambda_{*}$ problem \eqref{eqP} admits at least one positive solution
\begin{eqnarray*}
u_{*}\in \mbox{int}\, C_{+};
\end{eqnarray*}
\item[(c)] for all $\lambda\in(0,\lambda_{*})$ problem \eqref{eqP} has no positive solutions.
\end{itemize}
\end{theorem}

\medskip
{\bf Acknowledgement.} The authors wish to thank a knowledgeable referee for his/her remarks and corrections, which helped to improve the presentation.

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\end{document} 