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%% Place the running title of the paper with 40 letters or less in []
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\title[Bifurcation of positive solutions] %Use the shortened version of the full title
      {Bifurcation of positive solutions for nonlinear nonhomogeneous Robin and Neumann problems with competing nonlinearities}

% Place all authors' names in [ ] shown as running head, Leave { } empty
% Please use `and' to connect the last two names if applicable
% Use FirstNameInitial.  MiddleNameInitial. LastName, or only last names of authors if there are too many authors
\author[Nikolaos S. Papageorgiou and Vicen\c{t}iu D. R\u{a}dulescu]{}

% It is required to enter 2010 MSC.
\subjclass{35J66, 35J70, 35J92}
% Please provide minimum  5 keywords.
 \keywords{Competing nonlinearities, nonhomogeneous differential operator, positive solutions, bifurcation phenomena, Robin and Neumann problems.}

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 \email{npapg@math.ntua.gr}
 \email{vicentiu.radulescu@imar.ro}


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%please go to the last acknowlegments section.
%\thanks{The first author is supported by NSF grant xx-xxxx}

\begin{document}
\maketitle

% Enter the first author's name and address:
\centerline{\scshape Nikolaos S. Papageorgiou }
\medskip
{\footnotesize
% please put the address of the first author
 \centerline{National Technical University, Department of Mathematics,}
   \centerline{Zografou Campus, Athens 15780, Greece}
} % Do not forget to end the {\footnotesize by the sign }

\medskip

\centerline{\scshape Vicen\c{t}iu D. R\u{a}dulescu}
\medskip
{\footnotesize
 % please put the address of the second  and third author
 \centerline{ Department of Mathematics, Faculty of Sciences,}
	\centerline{ King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia }
   \centerline{\& Institute of Mathematics ``Simion Stoilow" of the Romanian Academy,}
	\centerline{ P.O. Box 1-764, 014700 Bucharest, Romania}
}

\bigskip

% The name of the associate editor will be entered by an editorial staff
% "Communicated by the associate editor name" is not needed for special issue.
% \centerline{(Communicated by the associate editor name)}


%The abstract of your paper
\begin{abstract}
In this paper we deal with Robin and Neumann parametric elliptic equations driven by a nonhomogeneous differential operator and with a reaction that exhibits competing nonlinearities (concave-convex nonlinearities). For the Robin problem and without employing the Ambrosetti-Rabinowitz condition, we prove a bifurcation theorem for the positive solutions for small values of the parameter $\lambda>0$. For the Neumann problem with a different geometry and using the Ambrosetti-Rabinowitz condition we prove bifurcation for large values of $\lambda>0$.
\end{abstract}

%The title of your section 1
\section{Introduction}

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

Hence $a:\mathbb{R}^N\rightarrow\mathbb{R}^N$ is a continuous and strictly monotone map, which satisfies certain other regularity and growth conditions, listed in hypotheses $H(a)$ below. These conditions are general enough, to incorporate in our setting various differential operators of interest, such as the $p$-Laplacian $(1<p<\infty)$. Also, $\frac{\partial u}{\partial n_a}$ denotes the conormal derivative defined by $\frac{\partial u}{\partial n_a}=(a(Du),n)_{\mathbb{R}^N}$ with $n(z)$ being the outward unit normal at $z\in\partial\Omega$. The reaction $f(z,x,\lambda)$ is a parametric function with $\lambda>0$ being the parameter and $(z,x)\rightarrow f(z,x,\lambda)$ is Carath\'eodory (that is, for all $x\in\mathbb{R}$ the mapping $z\longmapsto f(z,x,\lambda)$ is measurable and for a.a. $z\in\Omega$ the map $x\longmapsto f(z,x,\lambda)$ is continuous). We assume that $f(z,\cdot,\lambda)$ exhibits competing nonlinearities, namely near the origin, it has a ``concave" term (
 that is, a strictly $(p-1)$-
 sublinear term), while near $+\infty$, the reaction is ``convex" term (that is, $x\longmapsto f(z,x,\lambda)$ is ($p-1$)-superlinear). A special case of our reaction, is the following function:
$$f(z,x,\lambda)=f(x,\lambda)=\lambda x^{q-1}+x^{r-1}\ \mbox{for all}\ x\geq 0,$$
with $1<q<p<r<p^*=\left\{
	\begin{array}{cl}
		\frac{Np}{N-p}&\ \mbox{if}\ p<N\\
		+\infty&\ \mbox{if}\ N\leq p
	\end{array}
	\right.$ This reaction is encountered in the literature in the context of equations driven by the Laplacian (that is, $p=2$) or by the $p$-Laplacian ($1<p<\infty$).
	
	Our aim is to investigate the existence, nonexistence and multiplicity of positive solutions as the parameter $\lambda>0$ varies. So, we prove two bifurcation type results, describing the set of positive solutions of \eqref{eqP} as the parameter $\lambda>0$ changes, when the reaction exhibits the competing effects of concave (that is, sublinear) and convex (that is, superlinear) nonlinearities. In the first theorem the bifurcation occurs near zero.
More precisely, under general hypotheses we show that  there exists $\lambda^*>0$ such that the following properties hold:
	\begin{itemize}
		\item[(a)] for all $\lambda\in(0,\lambda^*)$, problem $(P_{\lambda})$ has at least two positive solutions;
		\item[(b)] for $\lambda=\lambda^*$ problem $(P_{\lambda^*})$ has at least one positive solution;
		\item[(c)] for all $\lambda>\lambda^*$ problem $(P_{\lambda})$ has no positive solution.
\end{itemize}

In the second case, we assume that $\beta\equiv 0$ (Neumann boundary condition) and we consider the problem
 \begin{equation}
	\left\{\begin{array}{ll}
		-\mbox{div}\, a(Du(z))=f_0(z,u(z))-\lambda u(z)^{p-1}\ &\mbox{in}\ \Omega,\\
		\frac{\partial u}{\partial n}(z)=0\ &\mbox{on}\ \partial\Omega,\\
		u>0\ &\mbox{in}\ \Omega\,.
	\end{array}
	\right\}\tag{$S_{\lambda}$} \label{eqS}
\end{equation}
 We obtain a different geometry and we establish that the bifurcation occurs for large values of the parameter $\lambda>0$. More precisely, under natural assumptions on $f_0$ we show that there exists $\lambda_*>0$ such that
	\begin{itemize}
		\item[(a)] for every $\lambda>\lambda_*$ problem \eqref{eqS} has at least two positive solutions;
		\item[(b)] for $\lambda=\lambda_*$ problem $(S_{\lambda_*})$ has at least one positive solution;
		\item[(c)] for every $\lambda\in(0,\lambda_*)$ problem \eqref{eqS} has no positive solution.
	\end{itemize}
	
	The first work concerning positive solutions for problems with concave and convex nonlinearities, was that of Ambrosetti, Brezis and Cerami \cite{2}. They studied semilinear equations driven by the Dirichlet Laplacian and with a reaction of the form (\ref{eq1}). Their work was extended to equations driven by the Dirichlet $p$-Laplacian by Garcia Azorero, Manfredi and Peral Alonso \cite{8} and by Guo and Zhang \cite{12}.
We also refer to the contributions of de Figueiredo,  Gossez and Ubilla \cite{dja1}, \cite{dja2} to concave-convex type problems and general nonlinearities
for the Laplacian, resp. $p$-Laplacian case.
Extensions to equations involving more general reactions, were obtained by Gasinski and Papageorgiou \cite{11}, Hu and Papageorgiou \cite{13} and R\u{a}dulescu and Repov\v{s} \cite{20}. Other problems with competition phenomena, can be found in the works of C\^{i}rstea, Ghergu and R\u{a}dulescu \cite{4} (problems with singular terms) and of Kristaly and Moro\c sanu \cite{14} (problems with oscillating reaction). Finally we mention the recent work of Papageorgiou and R\u{a}dulescu \cite{18}, who studied a Robin problem driven by the $p$-Laplacian and with a logistic
  reaction and proved multiplicity theorems for all large values of the parameter $\lambda>0$, producing also nodal solutions.
	
	We stress that the differential operator in \eqref{eqP} is not homogeneous and this is a source of difficulties in the analysis of the problem, since many of the methods and techniques developed in the aforementioned papers do not work here. It appears that our results in the present paper are the first bifurcation-type theorems for nonhomogeneous elliptic equations.
	


\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 Cerami condition (the $C$-condition), if the following is true:
\begin{center}
	``Every sequence $\{u_n\}_{n\geq 1}\subseteq X$ such that  $\{\varphi(u_n)\}_{n\geq 1}\subseteq\mathbb{R}$ 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 is a compactness-type condition on the function $\varphi$ which compensates for the fact that the space $X$ need not be locally compact (being in general infinite dimensional). It is more general than the more common Palais-Smale condition. Nevertheless, the $C$-condition suffices to prove a deformation theorem and from it derive the minimax theory of the critical values of $\varphi$. One of the main results in that theory, is the so-called mountain pass theorem of Ambrosetti and Rabinowitz \cite{3}. Here we state it in a slightly more general form.
\begin{theorem}\label{th1}
	Let $X$ be a Banach space, $\varphi\in C^1(X)$ satisfies the $C$-condition, $u_0$, $u_1\in X$ with $||u_1-u_0||>\rho>0$
	$$\max\{\varphi(u_0),\varphi(u_1)\}<\inf[\varphi(u):||u-u_0||=\rho]=m_{\rho}$$
	and $c=\inf\limits_{\gamma\in\Gamma}\max\limits_{0\leq t\leq 1}\varphi(\gamma(t))$ with $\Gamma=\{\gamma\in C([0,1],X):\gamma(0)=u_0,\gamma(1)=u_1\}$. Then $c\geq m_{\rho}$ and $c$ is a critical value of $\varphi$.
\end{theorem}

Let $\eta\in C^1(0,\infty)$ and assume that
\begin{eqnarray}\label{eq1}
	&&0<\hat{c}\leq\frac{t\eta'(t)}{\eta(t)}\leq c_0\ \mbox{and}\ c_1t^{p-1}\leq\eta(t)\leq c_2(1+t^{p-1})\ \mbox{for all}\ t>0\\
	&&\hspace{6cm}\mbox{with}\ c_1,c_2>0,\ 1<p<\infty.\nonumber
\end{eqnarray}

The hypotheses on the map $a(\cdot)$ are the following:

\smallskip
\textbf{$H(a):$} $a(y)=a_0(|y|)y$ for all $y\in \mathbb{R}^N$, with $a_0(t)>0$ for all $t>0$ and
\begin{itemize}
	\item[(i)] $a_0\in C^1(0,\infty),\ t\longmapsto a_0(t)t$ is strictly increasing on $(0,\infty)$, $a_0(t)t\rightarrow 0$\\ as $t\rightarrow 0^+$ and
	$$\lim\limits_{t\rightarrow 0^+}\frac{a'_0(t)t}{a_0(t)}>-1;$$
	\item[(ii)] $ |\nabla a(y)|\leq c_3\frac{\eta(|y|)}{|y|}$ for some $c_3>0$, all $y\in\mathbb{R}^N\backslash\{0\}$;
	\item[(iii)] $\frac{\eta(|y|)}{|y|}|\xi|^2\leq(\nabla a(y)\xi,\xi)_{\mathbb{R}^N}$ for all $y\in\mathbb{R}^N\backslash\{0\}$, all $\xi\in\mathbb{R}^N$;
	\item[(iv)] if $G_0(t)=\int^{t}_{0}a_0(s)sds$ for all $t\geq 0$, then $pG_0(t)-a_0(t)t^2\geq -\hat\xi$\\ for all $t\geq 0$, some $\hat\xi>0$;
	\item[(v)] there exists $\tau\in(1,p)$ such that  $t\longmapsto G_0(t^{1/\tau})$ is convex on $(0,\infty)$,\\ $\lim\limits_{t\rightarrow 0^+}\frac{G_0(t)}{t^{\tau}}=0$ and
	$$a_0(t)t^2-\tau G_0(t)\geq \tilde{c}t^p\ \mbox{for some}\ \tilde{c}>0,\ \mbox{all}\ t>0.$$
\end{itemize}
\begin{remark}
	These conditions on $a(\cdot)$ are motivated by the regularity results of Lieberman \cite{15} and the nonlinear maximum principle of Pucci and Serrin \cite{19}. According to the above conditions, the potential function $G_0(\cdot)$ is strictly convex and strictly increasing. We set $G(y)=G_0(|y|)$ for all $y\in\mathbb{R}^N$. Then the function $y\longmapsto G(y)$ is convex and differentiable on $\mathbb{R}^N\backslash\{0\}$. We have
	$$\nabla G(y)=G'_0(|y|)\frac{y}{|y|}=a_0(|y|)y=a(y)\ \mbox{for all}\ y\in\mathbb{R}^N\backslash\{0\},\ \nabla G(0)=0.$$
	
	So, $G(\cdot)$ is the primitive of the map $a(\cdot)$. Because $G(0)=0$ and $y\longmapsto G(y)$ is convex, from the properties of convex functions, we have
	\begin{eqnarray}\label{eq2}
		G(y)\leq(a(y),y)_{\mathbb{R}^N}\ \mbox{for all}\ y\in\mathbb{R}^N.
	\end{eqnarray}
\end{remark}	

The next lemma summarizes the main properties of the map $a(\cdot)$. They follow easily from hypotheses $H(a)$ above.
\begin{lemma}\label{lem2}
	If hypotheses $H(a)(i),(ii),(iii)$ hold, then
	\begin{itemize}
		\item[(a)] $y\longmapsto a(y)$ is continuous and strictly monotone, hence maximal monotone too;
		\item[(b)] $|a(y)|\leq c_4(1+|y|^{p-1})$ for some $c_4>0$, all $y\in\mathbb{R}^N;$
		\item[(c)] $(a(y),y)_{\mathbb{R}^N}\geq\frac{c_1}{p-1}|y|^p$ for all $y\in\mathbb{R}^N$.
	\end{itemize}
\end{lemma}

Lemma \ref{lem2} together with (\ref{eq1}) and (\ref{eq2}), lead to the following growth estimates for the primitive $G(\cdot)$.
\begin{corollary}\label{cor3}
	If hypotheses $H(a)(i),(ii),(iii)$ hold, then $\frac{c_1}{p(p-1)}|y|^p\leq G(y)\leq c_5(1+|y|^p)$ for some $c_5>0$, all $y\in\mathbb{R}^N$.
\end{corollary}
\begin{example}
	The following maps $a(y)$, satisfy hypotheses $H(a)$ above:
	\begin{itemize}
		\item[(a)] $a(y)=|y|^{p-2}y$ with $1<p<\infty$.\\
		This map corresponds to the $p$-Laplace operator defined by
		$$\Delta_pu={\rm div}\,(|Du|^{p-2}Du)\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$
		\item[(b)] $a(y)=|y|^{p-2}y+\mu|y|^{q-2}y$ with $1<q<p<\infty$ and $\mu>0$.\\
		This map corresponds to the $(p,q)$-differential operator defined by
		$$\Delta_pu+\mu\Delta_qu\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$
		Such differential operators arise in many physical applications (see Papageorgiou and R\u{a}dulescu \cite{16}, \cite{17} and the references therein).
		\item[(c)] $a(y)=(1+|y|^2)^{\frac{p-2}{2}}y$ with $1<p<\infty$.\\
		This map corresponds to the generalized $p$-mean curvature differential operator defined by
		$${\rm div}\left[(1+|Du|^2)^{\frac{p-2}{2}}Du\right]\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$
		\item[(d)] $a(y)=|y|^{p-2}y+\frac{|y|^{p-2}y}{1+|y|^p}$ with $1<p<\infty$.
	\end{itemize}
\end{example}

The hypotheses on the boundary weight map $\beta(\cdot)$ are the following:

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

\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
$$\mbox{int}\, C_+=\{u\in C_+:u(z)>0\ \mbox{for all}\ z\in\overline{\Omega}\}.$$

In the Sobolev space $W^{1,p}(\Omega)$, we use the norm
$$||u||=\left[||u||^{p}_{p}+||Du||^{p}_{p}\right]^{1/p}\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$

To distinguish, we use $|\cdot|$ to denote the norm of $\mathbb{R}^N$.

If on $\partial\Omega$ we use the $(N-1)$-dimensional Hausdorff measure $\sigma(\cdot)$ (the surface measure on $\partial\Omega$), then 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 $u\in C^1(\overline{\Omega})$. In fact $\gamma_0$ is compact. 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 the sequel, for the sake of notational simplicity, we drop the use of the trace map $\gamma_0$, with the understanding that all restrictions of elements of $W^{1,p}(\Omega)$ on $\partial\Omega$, are defined in the sense of traces.

Suppose $f_0:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carath\'eodory function with subcritical growth in the $x\in\mathbb{R}$ variable, that is
$$|f_0(z,x)|\leq a_0(z)(1+|x|^{r-1})\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\in\mathbb{R},$$
with $a_0\in L^{\infty}(\Omega)_+$, $1<r<p^*$. We set $F_0(z,x)=\int^{x}_{0}f_0(z,s)ds$  and consider the $C^1$-functional $\varphi_0:W^{1,p}(\Omega)\rightarrow\mathbb{R}$ defined by
\begin{eqnarray*}
	 &&\varphi_0(u)=\int_{\Omega}G(Du)dz+\frac{1}{p}\int_{\partial\Omega}\beta(z)|u|^{p}d\sigma-\int_{\Omega}F_0(z,u)dz\\
	&&\hspace{6cm}\mbox{for all}\ u\in W^{1,p}(\Omega).
\end{eqnarray*}

The next proposition, was proved by Papageorgiou and R\u{a}dulescu \cite{18} for $G(y)=\frac{1}{p}|y|^p$ for all $y\in\mathbb{R}^N$. The proof remains valid in the present more general setting, using Lemma \ref{lem2}, Corollary \ref{cor3} and the regularity result of Lieberman \cite{15} [p. 320].
\begin{proposition}\label{prop4}
	Assume that $u_0\in W^{1,p}(\Omega)$ is a local $C^1(\overline{\Omega})$-minimizer of $\varphi_0$, that is, there exist $\rho_0>0$ 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 $u_0\in C^{1,\eta}(\overline{\Omega})$ for some $\eta\in(0,1)$ and it is also a local $W^{1,p}(\Omega)$-minimizer of $\varphi_0$, that is, there exists $\rho_1>0$ such that
	$$\varphi_0(u_0)\leq\varphi_0(u_0+h)\ \mbox{for all}\ h\in W^{1,p}(\Omega)\ \mbox{with}\ ||h||\leq\rho_1.$$
\end{proposition}

Let $A:W^{1,p}(\Omega)\rightarrow W^{1,p}(\Omega)^*$ be the nonlinear map defined by
\begin{eqnarray}\label{eq3}
	\left\langle A(u),y\right\rangle=\int_{\Omega}(a(Du),Dy)_{\mathbb{R}^N}dz\ \mbox{for all}\ u,y\in W^{1,p}(\Omega)
\end{eqnarray}

The following, is a particular case of a more general result due to Gasinski and Papageorgiou \cite{10}.
\begin{proposition}\label{prop5}
	If $A:W^{1,p}(\Omega)\rightarrow W^{1,p}(\Omega)^*$ is defined by (\ref{eq3}), then $A$ is demicontinuous and of type $(S)_+$, that is, 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{proposition}

In the sequel, by $|\cdot|_N$ we denote the Lebesgue measure on $\mathbb{R}^N$. Also, if $x\in\mathbb{R}$, then we set $x^{\pm}=\max\{\pm x,0\}$ and  for $u\in W^{1,p}(\Omega)$, we define $u^{\pm}(\cdot)=u(\cdot)^{\pm}$. We know that
$$u^{\pm}\in W^{1,p}(\Omega)\ \mbox{and}\ |u|=u^++u^-,\ u=u^+-u^-.$$

Also, if $h:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a measurable function (for example a Carath\'eodory function), then we define
$$N_h(u)(\cdot)=h(\cdot,u(\cdot))\ \mbox{for all}\ u\in W^{1,p}(\Omega),$$
(the Nemytskii operator corresponding to the function $h$).

\section{Bifurcation Near Zero for the Robin Problem}\label{sec3}

In this section, we deal with competition phenomena that give rise to bifurcation of the problem solutions, when the parameter $\lambda>0$ is near zero. This situation includes the classical equations with concave and convex nonlinearities.

The hypotheses on the reaction $f(z,x,\lambda)$ are the following:

\smallskip
\textbf{$H_1:$} $f:\Omega\times\mathbb{R}\times(0,\infty)\rightarrow\mathbb{R}$ is a function such that  for all $(z,x)\in\Omega\times\left[0,+\infty\right)$, $\lambda\longmapsto f(z,x,\lambda)$ is nondecreasing, for all $\lambda>0$ $f(z,0,\lambda)=0$ for a.a. $z\in\Omega$ and
\begin{itemize}
	\item[(i)] $(z,x)\longmapsto f(z,x,\lambda )$ is a Carath\'eodory function on $\Omega\times\left[0,+\infty\right)$;
	\item[(ii)] $|f(z,x,\lambda)|\leq a_{\lambda}(z)(1+x^{r-1})$ for a.a. $z\in\Omega$, all $x\geq 0$, with $a_{\lambda}\in L^{\infty}(\Omega)_+$, $p<r<p^*$;
	\item[(iii)] if $F(z,x,\lambda)=\int^{x}_{0}f(z,s,\lambda)ds$, then $\lim\limits_{x\rightarrow+\infty}\frac{F(z,x,\lambda)}{x^p}=+\infty$ uniformly for a.a. $z\in\Omega$;
	\item[(iv)] there exists $\vartheta=\vartheta(\lambda)\in((r-p)\max\left\{\frac{N}{p},1\right\},p^*)$ such that
	 $$0<\gamma_0\leq\liminf_{x\rightarrow+\infty}\frac{f(z,x,\lambda)x-pF(z,x,\lambda)}{x^{\vartheta}}\ \mbox{uniformly for a.a.}\ z\in\Omega;$$
	\item[(v)] there exists $1<\mu=\mu(\lambda)<q=q(\lambda)<\tau$ (see hypothesis $H(a)(v)$) and $\gamma=\gamma(\lambda)>\mu$, $\delta_0=\delta_0(\lambda)\in\left(0,1\right]$ such that
	$$c_6x^p\leq f(z,x,\lambda)x\leq qF(z,x,\lambda)\leq\xi_{\lambda}(z)x^{\mu}+\overline{c}x^{\gamma}\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\in[0,\delta_0],$$
	with $c_6=c_6(\lambda)>0$, $\tau=\tau(\lambda)>0$, $\xi_{\lambda}\in L^{\infty}(\Omega)_+$ and $||\xi_{\lambda}||_{\infty}\rightarrow 0$ as $\lambda\rightarrow 0^+$.
\end{itemize}

\begin{remark}
	Since we are interested to find positive solutions and the above hypotheses concern the positive semiaxis $\mathbb{R}_+=\left[0,+\infty\right)$, without any loss of generality we may assume that $f(z,x,\lambda)=0$ for a.a. $z\in\Omega$, all $x\leq 0$ and all $\lambda>0$. Note that hypotheses $H_1(ii),(iii)$ imply that
	$$\lim\limits_{x\rightarrow+\infty}\frac{f(z,x,\lambda)}{x^{p-1}}=+\infty\ \mbox{uniformly for a.a.}\ z\in\Omega.$$
	
	Therefore, $f(z,\cdot,\lambda)$ is $(p-1)$-superlinear near $+\infty$. However, we do not employ the $AR$-condition (unilateral version). We recall (see \cite{3}), that $f(z,\cdot,\lambda)$ satisfies the (unilateral) $AR$-condition, if there exist $\eta=\eta(\lambda)>p$ and $M=M(\lambda)>0$ such that
\begin{eqnarray}\label{eq4}
\left.\begin{array}{cl}
		(a)&0<\eta F(z,x,\lambda)\leq f(z,x,\lambda)x\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\geq M,\\
		(b)&{\rm ess\,inf}_{\Omega}\ F(\cdot,M,\lambda)>0.
		\end{array}\right.
	\end{eqnarray}
\end{remark}

Integrating (\ref{eq4}a) and using (\ref{eq4}b), we obtain a weaker condition, namely that
\begin{eqnarray}\label{eq5}
	c_7 x^{\eta}\leq F(z,x,\lambda)\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ z\geq M\ \mbox{and some}\ c_7>0.
\end{eqnarray}

Evidently (\ref{eq5}) implies the much weaker hypothesis $H_1(iii)$. In (\ref{eq4}) we may assume that $\eta>(r-p)\max\left\{\frac{N}{p},1\right\}$. Then we have
\begin{eqnarray*}
	&&\frac{f(z,x,\lambda)x-pF(z,x,\lambda)}{x^{\eta}}\\
	&=&\frac{f(z,x,\lambda)x-\eta F(z,x,\lambda)}{x^{\eta}}+\frac{(\eta-p)F(z,x,\lambda)}{x^{\eta}}\\
	&\geq&(\eta-p)c_7\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\geq M\ (\mbox{see (\ref{eq4}a) and (\ref{eq5})}).
\end{eqnarray*}

So, we see that the $AR$-condition implies hypothesis $H_1(iv)$. This weaker ``superlinearity" condition, incorporates in our setting $(p-1)$-superlinear nonlinearities with ``slower" growth near $+\infty$, which fail to satisfy the $AR$-condition (see the examples below). Finally note that hypothesis $H_1(v)$ implies the presence of a concave nonlinearity near zero.
\begin{example}
	The following functions satisfy hypotheses $H_1$. For the sake of simplicity, we drop the $z$-dependence:
	\begin{eqnarray*}
		&&f_1(x,\lambda)=\lambda x^{q-1}+x^{r-1}\ \mbox{for all}\ x\geq 0,\ \mbox{with}\ 1<q<p<r<p^*\\
		&&f_2(x,\lambda)=\left\{
		\begin{array}{ll}
			\lambda x^{q-1}-x^{\eta-1}&\ \mbox{if}\ x\in[0,1]\\
			x^{p-1}\left(\ln x+\frac{1}{p}\right)+\left(\lambda-\frac{1}{p}\right)x^{\nu-1}&\ \mbox{if}\ 1<x
		\end{array}\right.\\
		&&\hspace{1.8cm}\mbox{with}\ q,\nu\in(1,p)\ \mbox{and}\ \eta>p\\
		&&f_3(x,\lambda)=\left\{
		\begin{array}{cl}
			x^{q-1}&\ \mbox{if}\ x\in[0,\rho(\lambda)]\\
			x^{r-1}+\eta(\lambda)&\ \mbox{if}\ \rho(\lambda)<x
		\end{array}\right.\\
		&&\hspace{1.8cm}\mbox{with}\ 1<q<p<r<p^*,\ \eta(\lambda)=\rho(\lambda)^{p-1}-\rho(\lambda)^{r-1}\\
		&&\hspace{1.8cm}\mbox{and}\ \rho(\lambda)\rightarrow 0^+\ \mbox{as}\ \lambda\rightarrow 0^+.
	\end{eqnarray*}
\end{example}

Note that $f_2(\cdot,\lambda)$ does not satisfy the $AR$-condition.

We introduce the following Carath\'eodory function
$$\hat{f}(z,x,\lambda)=f(z,x,\lambda)+(x^+)^{p-1}\ \mbox{for all}\ (z,x,\lambda)\in\Omega\times\mathbb{R}\times(0,+\infty).$$

Let $\hat{F}(z,x,\lambda)=\int^{x}_{0}\hat{f}(z,s,\lambda)ds$ and consider the $C^1$-functional $\hat{\varphi}_{\lambda}:W^{1,p}(\Omega)\rightarrow\mathbb{R}$ defined by
\begin{eqnarray*}
	 &&\hat{\varphi}_{\lambda}(u)=\int_{\Omega}G(Du)dz+\frac{1}{p}||u||^{p}_{p}+\frac{1}{p}\int_{\partial\Omega}\beta(z)(u^+)^pd\sigma-\int_{\Omega}\hat{F}(z,u,\lambda)dz\\
	&&\hspace{8cm}\mbox{for all}\ u\in W^{1,p}(\Omega).
\end{eqnarray*}
\begin{proposition}\label{prop6}
	If hypotheses $H(a),\ H(\beta)$ and $H_1$ hold and $\lambda>0$, then the functional $\hat{\varphi}_{\lambda}$ satisfies the $C$-condition.
\end{proposition}
\begin{proof}
	Let $\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)$ 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{eq6}\\
		&&(1+||u_n||)\hat{\varphi}'_{\lambda}(u_n)\rightarrow 0\ \mbox{in}\ W^{1,p}(\Omega)^*\ \mbox{as}\ n\rightarrow\infty.\label{eq7}
	\end{eqnarray}
	
	From (\ref{eq7}) we have
	\begin{eqnarray}\label{eq8}
		&&\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{all}\ n\geq 1,\nonumber\\
		&&\hspace{5cm}\mbox{with}\ \epsilon_n\rightarrow 0^+\ \mbox{as}\ n\rightarrow\infty,\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-\right.\nonumber\\
		 &&\hspace{2cm}\left.\int_{\Omega}\hat{f}(z,u_n,\lambda)hdz\right|\leq\frac{\epsilon_n||h||}{1+||u_n||}\ \mbox{for all}\ n\geq 1.
	\end{eqnarray}
	
	In (\ref{eq8}), first we choose $h=-u^{-}_{n}\in W^{1,p}(\Omega)$. Using Lemma \ref{lem2}, we have
	\begin{eqnarray}\label{eq9}
		&&\frac{c_1}{p-1}||Du^{-}_{n}||^{p}_{p}+||u^{-}_{n}||^{p}_{p}\leq\epsilon_n\ \mbox{for all}\ n\geq 1,\nonumber\\
		&\Rightarrow&u^{-}_{n}\rightarrow 0\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{as}\ n\rightarrow\infty.
	\end{eqnarray}
	
	From (\ref{eq6}), (\ref{eq9}) and hypothesis $H_1(i)$, we have
	\begin{eqnarray}\label{eq10} &&\int_{\Omega}pG(Du^{+}_{n})dz+\int_{\Omega}\beta(z)(u^{+}_{n})^pd\sigma-\int_{\Omega}pF(z,u^{+}_{n},\lambda)dz\leq M_2\\
		&&\hspace{6cm}\mbox{for some}\ M_2>0,\ \mbox{all}\ n\geq 1.\nonumber
	\end{eqnarray}
	
	Also, in (\ref{eq8}) we choose $h=u^{+}_{n}\in W^{1,p}(\Omega)$ and obtain
	\begin{eqnarray}\label{eq11}
		 &&-\int_{\Omega}(a(Du^{+}_{n}),Du^{+}_{n})_{\mathbb{R}^N}dz-\int_{\partial\Omega}\beta(z)(u_n^+)^pd\sigma+\int_{\Omega}f(z,u_n^+,\lambda)u_n^+dz\leq\epsilon_n\nonumber\\
		&&\hspace{8cm}\quad\mbox{for all}\ n\geq 1.
	\end{eqnarray}
	
	Adding (\ref{eq10}) and (\ref{eq11}), we have
	\begin{eqnarray}\label{eq12}
		 &&\int_{\Omega}\left[pG(Du_n^+)-(a(Du_n^+),Du_n^+)_{\mathbb{R}^N}\right]dz+\int_{\Omega}\left[f(z,u_n^+,\lambda)u_n^+-\right.\nonumber\\
		 &&\hspace{1cm}\left.-pF(z,u_n^+,\lambda)\right]dz\leq M_3\mbox{for some}\ M_3>0,\ \mbox{all}\ n\geq 1,\nonumber\\
		&& \Rightarrow\int_{\Omega}\left[f(z,u_n^+,\lambda)u_n^+-pF(z,u_n^+,\lambda)\right]dz\leq M_3+\hat\xi\ \mbox{for all}\ n\geq 1\\
		&&\hspace{6cm}(\mbox{see hypothesis}\ H(a)(iv)).\nonumber
	\end{eqnarray}
	
	By virtue of hypotheses $H_1(ii),(iv)$, we can find $\gamma_1\in(0,\gamma_0)$ and $c_8=c_8(\gamma_1,\lambda)>0$ such that
	$$f(z,x,\lambda)x-pF(z,x,\lambda)\geq\gamma_1 x^{\vartheta}-c_8\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\geq 0.$$
	
	We use this unilateral growth estimate in (\ref{eq12}) and obtain
	\begin{eqnarray}\label{eq13}
		&&\gamma_1||u_n^+||^{\vartheta}_{\vartheta}\leq M_4\ \mbox{for some}\ M_4>0,\ \mbox{all}\ n\geq 1,\nonumber\\
		&\Rightarrow&\{u_n^+\}_{n\geq 1}\subseteq L^{\vartheta}(\Omega)\ \mbox{is bounded.}
	\end{eqnarray}
	
First assume that $N\neq p$. From hypothesis $H_1(iv)$ it is clear that without any loss of generality, we may assume that $\vartheta\leq r<p^*$. Then we can find $t\in\left[0,1\right)$ such that
\begin{eqnarray}\label{eq14}
	\frac{1}{r}=\frac{1-t}{\vartheta}+\frac{t}{p^*}.
\end{eqnarray}

From the interpolation inequality (see, for example, Gasinski and Papageorgiou \cite{9} [p. 905]), we have
\begin{eqnarray}\label{eq15}
	&||u_n^+||_r&\leq||u_n^+||^{1-t}_{\vartheta}||u_n^+||^{t}_{p^*}\nonumber\\
	&&\leq c_9||u_n^+||^t\ \mbox{for some}\ c_9>0,\ \mbox{all}\ n\geq 1\nonumber\\
	&&\mbox{(see (\ref{eq13}) and use the Sobolev embedding theorem),}\nonumber\\
	\Rightarrow&||u_n^+||^{r}_{r}&\leq c_{10}||u_n^+||^{tr}\ \mbox{for all}\ n\geq 1,\ \mbox{with}\ c_{10}=c_9^p>0.
\end{eqnarray}

By virtue of hypothesis $H_1(ii)$ we have
\begin{eqnarray}\label{eq16}
	f(z,x,\lambda)x\leq a_{\lambda}(z)(x+x^r)\ \mbox{for a.a}\ z\in\Omega,\ \mbox{all}\ x\geq 0.
\end{eqnarray}

In (\ref{eq8}) we choose $h=u_n^+\in W^{1,p}(\Omega)$. Then
\begin{eqnarray}\label{eq17}
	 &||Du_n^+||^{p}_{p}&+\int_{\partial\Omega}\beta(z)(u_n^+)^pd\sigma-\int_{\Omega}f(z,u_n^+,\lambda)u_n^+dz\leq\epsilon_n\ \mbox{for all}\ n\geq 1,\nonumber\\
	\Rightarrow&||Du_n^+||^{p}_{p}&\leq c_{11}(1+||u_n^+||^{r}_{r})\ \mbox{for some}\ c_{11}>0,\ \mbox{all}\ n\geq 1\ (\mbox{see (\ref{eq16}) and}\ H(\beta)),\nonumber\\
	&&\leq c_{12}(1+||u_n^+||^{tr})\ \mbox{for some}\ c_{12}>0,\ \mbox{all}\ n\geq 1\ (\mbox{see (\ref{eq15})}),\nonumber\\
	\Rightarrow&||Du_n^+||^{p}_{p}&+||u_n^+||^{p}_{\vartheta}\leq c_{13}(1+||u_n^+||^{tr})\ \mbox{for some}\ c_{13}>0,\ \mbox{all}\ n\geq 1\\
	&&\hspace{7cm}(\mbox{see (\ref{eq13})})\nonumber
\end{eqnarray}

Since $\vartheta\leq r<p^*$, we know that
$$u\longmapsto ||u||_{\vartheta}+||Du||_p$$
is an equivalent norm on $W^{1,p}(\Omega)$ (see, for example, Gasinski and Papageorgiou \cite{9} [p. 227]). So, from (\ref{eq17}) we obtain
\begin{eqnarray}\label{eq18}
	||u_n^+||^p\leq c_{14}(1+||u_n^+||^{tr})\ \mbox{for some}\ c_{14}>0,\ \mbox{all}\ n\geq 1.
\end{eqnarray}

The hypothesis on $\vartheta$ (see $H_1(iv)$) and (\ref{eq14}), imply that $tr<p$. So, from (\ref{eq18}) we infer that
\begin{eqnarray}\label{eq19}
	\{u_n^+\}_{n\geq 1}\subseteq W^{1,p}(\Omega)\ \mbox{is bounded}.
\end{eqnarray}

If $N=p$, then $p^*=\infty$, while from the Sobolev embedding theorem, we know that $W^{1,p}(\Omega)$ is embedded (compactly) in $L^s(\Omega)$ for all $s\in\left[1,\infty\right)$. So, in the above argument, we need to replace $p^*=\infty$ by $s>r$ large such that
$$tr=\frac{s(r-\mu)}{s-\mu}<p\ (\mbox{see (\ref{eq14}) with}\ p^*\ \mbox{replaced by}\ s>r).$$

Then the previous argument works and leads again to (\ref{eq19}).

From (\ref{eq9}) and (\ref{eq19}) it follows that $\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)$ is bounded. So, we may assume that
\begin{eqnarray}\label{eq20}
	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).
\end{eqnarray}

In (\ref{eq8}) we choose $h=u_n-u\in W^{1,p}(\Omega)$, pass to the limit as $n\rightarrow\infty$ and use (\ref{eq20}). Then
\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),\\
	&\Rightarrow&\hat{\varphi}_{\lambda}\ \mbox{satisfies the}\ C-\mbox{condition}.
\end{eqnarray*}	
\end{proof}
\begin{proposition}\label{prop7}
	If hypotheses $H(a),\ H(\beta)$ and $H_1$ hold, then there exists $\lambda_+>0$ such that  for every $\lambda\in(0,\lambda_+)$ there exists $\rho_{\lambda}>0$ for which we have $$\inf\left[\hat{\varphi}_{\lambda}(u):||u||=\rho_{\lambda}\right]=\hat{m}_{\lambda}>0=\hat{\varphi}_{\lambda}(0).$$
\end{proposition}
\begin{proof}
	Hypotheses $H_1(ii),(v)$ imply that for every $\lambda>0$, we can find $c_{15}=c_{15}(\lambda)>0$ such that
	\begin{eqnarray}\label{eq21}
		F(z,x,\lambda)\leq\frac{\xi_{\lambda}(z)}{\mu}(x^+)^{\mu}+c_{15}[(x^+)^{\gamma}+(x^+)^{r}]\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\in\mathbb{R}.
	\end{eqnarray}
	
	Then for $u\in W^{1,p}(\Omega)$, we have
	\begin{eqnarray}\label{eq22}
		 &\hat{\varphi}_{\lambda}(u)&=\int_{\Omega}G(Du)dz+\frac{1}{p}||u||^{p}_{p}+\frac{1}{p}\int_{\partial\Omega}\beta(z)(u^+)^pd\sigma-\int_{\Omega}\hat{F}(z,u,\lambda)dz\nonumber\\
		 &&\geq\frac{c_1}{p(p-1)}||Du||^{p}_{p}+\frac{1}{p}||u||^{p}_{p}+\frac{1}{p}\int_{\partial\Omega}\beta(z)(u^+)^pd\sigma-\frac{||\xi_{\lambda}||_{\infty}}{\mu}||u^+||^{\mu}_{\mu}-\nonumber\\
		 &&\hspace{0.5cm}-c_{15}||u^+||^{r}_{r}-c_{15}||u^+||^{\gamma}_{\gamma}-\frac{1}{p}||u^+||^{p}_{p}\ (\mbox{see Corollary \ref{cor3} and (\ref{eq21})}).
	\end{eqnarray}
	
	It is clear that in hypothesis $H_1(v)$ we can always assume $\gamma\leq p^*$ and that $\mu>1$ is small enough so that $\frac{p-1}{\mu-1}\mu\geq r$ and $\frac{\gamma-1}{\mu-1}\mu\geq r$. By Young's inequality with $\epsilon>0$ (see, for example, Gasinski and Papageorgiou \cite{9} [p. 913]), we have
	\begin{eqnarray*}
		&||u||^p=||u||\ ||u||^{p-1}&\leq\frac{\epsilon}{\mu}||u||^{\mu}+\frac{1}{\epsilon\mu'}||u||^{\frac{(p-1)\mu}{\mu-1}}\ \ \left(\frac{1}{\mu}+\frac{1}{\mu'}=1\right)\\
		&&\leq\frac{\epsilon}{\mu}||u||^{\mu}+\frac{\mu-1}{\epsilon\mu}||u||^r\\
		&||u||^{\gamma}=||u||\ ||u||^{\gamma-1}&\leq\frac{\epsilon}{\mu}||u||^{\mu}+
\frac{1}{\epsilon\mu'}||u||^{\frac{(\gamma-1)\mu}{\mu-1}}\\
		&&\leq\frac{\epsilon}{\mu}||u||^{\mu}+\frac{\mu-1}{\epsilon\mu}||u||^r\ \mbox{for all}\ u\in W^{1,p}(\Omega)\ \mbox{with}\ ||u||\leq 1
	\end{eqnarray*}
	(recall that $\mu<\gamma,p\leq p^*$). Using these bounds in (\ref{eq22}), we obtain
	\begin{eqnarray}\label{eq23}
		&\hat{\varphi}_{\lambda}(u)&\geq c_{16}||u||^p-c_{17}\left[\left(||\xi_{\lambda}||_{\infty}+\epsilon\right)||u||^{\mu}+(1+c_{\epsilon})||u||^r\right]\nonumber\\
		&&\hspace{4.7cm}\mbox{for some}\ c_{16},c_{17},c_{\epsilon}>0\nonumber\\
		 &&=\left[c_{16}-c_{17}\left(\left(||\xi_{\lambda}||_{\infty}+\epsilon\right)||u||^{\mu-p}+(1+c_{\epsilon})||u||^{r-p}\right)\right]||u||^p\\
		&&\hspace{4.7cm}\mbox{for all}\ u\in W^{1,p}(\Omega)\ \mbox{with}\ ||u||\leq 1.\nonumber
	\end{eqnarray}
	
	Let $k^{\lambda}_{\epsilon}(t)=\left(||\xi_{\lambda}||_{\infty}+\epsilon\right)t^{\mu-p}+(1+c_{\epsilon})t^{r-p}$. Evidently $k^{\lambda}_{\epsilon}\in C^1(0,\infty)$ and since $\mu<p<r$ we have
	$$k^{\lambda}_{\epsilon}(t)\rightarrow+\infty\ \mbox{as}\ t\rightarrow 0^+\ \mbox{and as}\ t\rightarrow+\infty.$$
	
	Therefore we can find $t_0>0$ such that
	\begin{eqnarray*}
		&&k^{\lambda}_{\epsilon}(t_0)=\min\limits_{t>0}k^{\lambda}_{\epsilon}(t),\\
		 &\Rightarrow&(k^{\lambda}_{\epsilon})'(t_0)=(\mu-p)(||\xi_{\lambda}||_{\infty}+\epsilon)t_0^{\mu-p-1}+(r-p)(1+c_{\epsilon})t_0^{r-p-1},\\
		 &\Rightarrow&t_0=t_0(\lambda)=\left[\frac{(p-\mu)(||\xi_{\lambda}||_{\infty}+\epsilon)}{(r-p)(1+c_{\epsilon})}\right]^{\frac{1}{r-\mu}}.
	\end{eqnarray*}
	
	Then we have
	$$k^{\lambda}_{\epsilon}(t)\rightarrow\chi(\epsilon)\ \mbox{as}\ \lambda\rightarrow 0^+\ \mbox{with}\ \chi(\epsilon)\rightarrow 0^+\ \mbox{as}\ \epsilon\rightarrow 0^+.$$
	
	We choose $\epsilon>0$ small such that  $\chi(\epsilon)<\frac{1}{2}\ \frac{c_{16}}{c_{17}}$. Then for such an $\epsilon>0$, we can find $\lambda_+=\lambda_+(\epsilon)>0$ such that
	$$k^{\lambda}_{\epsilon}(t_0)<\frac{c_{16}}{c_{17}}\ \mbox{and}\ t_0(\lambda)\leq 1\ \mbox{for all}\ \lambda\in(0,\lambda_+)\ (\mbox{see hypothesis}\ H_1(v))$$
	
	Then by virtue of (\ref{eq23}), we have
	$$\hat{\varphi}_{\lambda}(u)\geq\hat{m}_{\lambda}>0=\hat{\varphi}_{\lambda}(0)\ \mbox{for all}\ u\in W^{1,p}(\Omega)\ \mbox{with}\ ||u||=\rho_{\lambda}=t_0(\lambda)\leq 1.$$
\end{proof}

Note that as a direct consequence of hypothesis $H_1(iii)$, we have:
\begin{proposition}\label{prop8}
	If hypotheses $H(a),H(\beta)$ and $H_1$ hold, $\lambda>0$ and $u\in {\rm int}\, C_+$, then $\hat{\varphi}_{\lambda}(tu)\rightarrow-\infty$ as $t\rightarrow\infty$.
\end{proposition}

We introduce the following sets:
\begin{eqnarray*}
	&&\mathcal{S}=\{\lambda>0:\mbox{problem}\ \eqref{eqP}\ \mbox{admits a positive solution}\},\\
	&&S(\lambda)=\mbox{the set of positive solutions of \eqref{eqP}}.
\end{eqnarray*}

We can show that $\mathcal{S}$ is nonempty, as well as a useful structural property of the solution set $S(\lambda)$.
\begin{proposition}\label{prop9}
	If hypotheses $H(a),H(\beta)$ and $H_1$ hold, then $\mathcal{S}\neq\varnothing$ and for every $\lambda\in\mathcal{S}$ $\varnothing\neq S(\lambda)\subseteq {\rm int}\, C_+$.
\end{proposition}
\begin{proof}
	Let $\lambda_+>0$ be as postulated by Proposition \ref{prop7} and let $\lambda\in(0,\lambda_+)$. Propositions \ref{prop6}, \ref{prop7} and \ref{prop8} permit the use of Theorem \ref{th1} (the mountain pass theorem) on the functional $\hat{\varphi}_{\lambda}$. So, we can find $u_0\in W^{1,p}(\Omega)$ such that
	\begin{eqnarray}\label{eq24}
		\hat{\varphi}'_{\lambda}(u_0)=0\ \mbox{and}\ \hat{\varphi}_{\lambda}(0)=0<\hat{m}_{\lambda}\leq\hat{\varphi}_{\lambda}(u_0).
	\end{eqnarray}
	
	From the inequality in (\ref{eq24}) we see that $u_0\neq 0$. From the inequality in (\ref{eq24}), we have
	\begin{eqnarray}\label{eq25}
		&&\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}(z,u_0,\lambda)hdz\\
		&&\hspace{6cm}\mbox{for all}\ h\in W^{1,p}(\Omega).\nonumber
	\end{eqnarray}
	
	In (\ref{eq25}) we choose $h=-u_0^-\in W^{1,p}(\Omega)$. Using Lemma \ref{lem2}, we have
	\begin{eqnarray*}
		&&\frac{c_1}{p-1}||Du_0^-||^{p}_{p}+||u_0^-||^{p}_{p}\leq 0,\\
		&\Rightarrow&u_0\geq 0,\ u_0\neq 0.
	\end{eqnarray*}
	
	Therefore (\ref{eq25}) becomes
	\begin{eqnarray}\label{eq26}
		\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,\lambda)dz\ \mbox{for all}\ h\in W^{1,p}(\Omega).
	\end{eqnarray}
	
	In what follows by $\left\langle \cdot,\cdot\right\rangle_0$ we denote the duality brackets for the pair $(W^{-1,p'}(\Omega)$, $W_0^{1,p}(\Omega))$ (recall that $\frac{1}{p}+\frac{1}{p'}=1$ and $W^{-1,p'}(\Omega)=W^{1,p}_{0}(\Omega)^*$). From the representation theorem for the elements of the dual space $W^{-1,p'}(\Omega)$ (see, for example, Gasinski and Papageorgiou \cite{9} [p. 212]), we have
	$$\mbox{div}\, a(Du_0)\in W^{-1,p'}(\Omega)\ (\mbox{see Lemma \ref{lem2}}).$$
	
	Performing integration by parts, we have
	$$\left\langle A(u_0),h\right\rangle=\left\langle -\mbox{div}\, a(Du_0),h\right\rangle_0\ \mbox{for all}\ h\in W^{1,p}_{0}(\Omega)\subseteq W^{1,p}(\Omega).$$
	
	Using this equation in (\ref{eq26}) and recalling that $h|_{\partial\Omega}=0$ for all $h\in W^{1,p}_{0}(\Omega)$, we obtain
	\begin{eqnarray}\label{eq27}
		&&\left\langle -\mbox{div}\, a(Du_0),h\right\rangle=\int_{\Omega}f(z,u_0,\lambda)hdz\ \mbox{for all}\ h\in W^{1,p}_{0}(\Omega)\subseteq W^{1,p}(\Omega),\nonumber\\
		&\Rightarrow&-\mbox{div}\, a(Du_0(z))=f(z,u_0(z),\lambda)\ \mbox{for a.a.}\ z\in\Omega.
	\end{eqnarray}
	
	Note that $f(\cdot,u_0(\cdot),\lambda)\in L^{r'}(\Omega)$ where $\frac{1}{r}+\frac{1}{r'}=1$ (see hypothesis $H_1(ii)$). Since $p<r$, we have $W^{1,r}_{0}(\Omega)\hookrightarrow W^{1,p}_{0}(\Omega)$ continuously and densely. Then $W^{-1,p'}(\Omega)\hookrightarrow W^{-1,r'}(\Omega)$ continuously and densely (see, for example, Gasinski and Papageorgiou \cite{9} [p. 141]). Then from (\ref{eq27}) we see that we can apply the nonlinear Green's identity (see, for example, Gasinski and Papageorgiou \cite{9} [p. 210]) and have
	\begin{eqnarray}\label{eq28}
		&&\left\langle A(u_0),h\right\rangle+\int_{\Omega}(\mbox{div}\, a(Du_0))hdz=\left\langle \frac{\partial u_0}{\partial n_a},h\right\rangle_{\partial\Omega}\\
		&&\hspace{5cm}\mbox{for all}\ h\in W^{1,r}(\Omega)\subseteq W^{1,p}(\Omega).\nonumber
	\end{eqnarray}
	
	Here by $\left\langle \cdot,\cdot\right\rangle_{\partial\Omega}$ we denote the duality brackets for the pair $(W^{-\frac{1}{r'},r'}(\partial\Omega)$,\\  $W^{\frac{1}{r'},r}(\partial\Omega))$. Returning to (\ref{eq26}) and using (\ref{eq28}), we obtain
	\begin{eqnarray}\label{eq29}
		&&\left\langle -\mbox{div}\, a(Du_0),h\right\rangle+\left\langle \frac{\partial u_0}{\partial n_a},h\right\rangle_{\partial\Omega}+\int_{\partial\Omega}\beta(z)u_0^{p-1}hd\sigma=\int_{\Omega}f(z,u_0,\lambda)hdz\nonumber\\
		&&\hspace{6cm}\mbox{for all}\ h\in W^{1,r}(\Omega)\nonumber\\
		&\Rightarrow&\left\langle \frac{\partial u_0}{\partial n_a},h\right\rangle_{\partial\Omega}+\int_{\partial\Omega}\beta(z)u_0^{p-1}hd\sigma=0\ \mbox{for all}\ h\in W^{1,r}(\Omega)\ (\mbox{see (\ref{eq27})}).
	\end{eqnarray}
	
	But we know that if $\gamma_0$ is the trace map on $W^{1,p}(\Omega)$, then $\mbox{im}\, \left(\gamma_0|_{W^{1,r}(\Omega)}\right)=W^{\frac{1}{r'},r}(\partial\Omega)$. So, from (\ref{eq29}), it follows that
	\begin{eqnarray}\label{eq30}
		\frac{\partial u_0}{\partial n_a}+\beta(z)u_0^{p-1}=0\ \mbox{on}\ \partial\Omega.
	\end{eqnarray}
	
	From (\ref{eq27}) and (\ref{eq30}) it follows that $u_0\in S(\lambda)$ and so $(0,\lambda_+)\subseteq\mathcal{S}$.
	
	From Winkert \cite{21} we have that $u_0\in L^{\infty}(\Omega)$. Then we can apply the regularity result of Lieberman \cite{15} [p. 320] and infer that $u_0\in C_+,\ u_0\neq 0$.
	
	Hypotheses $H_1(ii),(v)$ imply that given $\rho>0$, we can find $\xi_{\rho}>0$ such that
	\begin{eqnarray}\label{eq31}
		f(z,x,\lambda)+\xi_{\rho}x^{p-1}\geq 0\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\in[0,\rho].
	\end{eqnarray}
	
	Let $\rho=||u_0||_{\infty}$ and let $\xi_{\rho}>0$ be as postulated by (\ref{eq31}). Then
	\begin{eqnarray*}
		&&-\mbox{div}\, a(Du_0(z))+\xi_{\rho}u_0(z)^{p-1}\\
		&&=f(z,u_0(z),\lambda)+\xi_{\rho}u_0(z)^{p-1}\geq 0\ \mbox{for a.a.}\ z\in\Omega\ (\mbox{see (\ref{eq27}) and (\ref{eq31})}),\\
		&\Rightarrow&\mbox{div}\, a(Du_0(z))\leq\xi_{\rho}u_0(z)^{p-1}\ \mbox{a.e. in}\ \Omega,\\
		&\Rightarrow&u_0\in \mbox{int}\, C_+\ (\mbox{see Pucci and Serrin \cite{19} [pp. 111, 120]})\\
		&\Rightarrow&S(\lambda)\subseteq \mbox{int}\, C_+.
	\end{eqnarray*}
\end{proof}

The next proposition establishes a useful property of the set $\mathcal{S}$.
\begin{proposition}\label{prop10}
	If hypotheses $H(a),\ H(\beta)$ and $H_1$ hold and $\lambda\in\mathcal{S}$, then $\left(0,\lambda\right]\subseteq\mathcal{S}$.
\end{proposition}
\begin{proof}
	Since $\lambda\in\mathcal{S}$, we can find $u_{\lambda}\in S(\lambda)\subseteq \mbox{int}\, C_+$. Let $\eta\in(0,\lambda)$ and consider the following truncation-perturbation of the reaction in problem $(P_{\eta})$:
	\begin{eqnarray}\label{eq32}
		k_{\eta}(z,x)=\left\{
		\begin{array}{ll}
			0&\mbox{if}\ x<0\\
			f(z,x,\eta)+x^{p-1}&\mbox{if}\ 0\leq x\leq u_{\lambda}(z)\\
			f(z,u_{\lambda}(z),\eta)+u_{\lambda}(z)^{p-1}&\mbox{if}\ u_{\lambda}(z)<x.
		\end{array}
		\right.
	\end{eqnarray}
	
	This is a Carath\'eodory function. We set $K_{\eta}(z,x)=\int^{x}_{0}k_{\eta}(z,s)ds$ and consider the $C^1$-functional $\hat{\psi}_{\eta}:W^{1,p}(\Omega)\rightarrow\mathbb{R}$ defined by
	\begin{eqnarray*}
		 &&\hat{\psi}_{\eta}(u)=\int_{\Omega}G(Du)dz+\frac{1}{p}||u||^{p}_{p}+\frac{1}{p}\int_{\partial\Omega}\beta(z)(u^+)^pd\sigma-\int_{\Omega}K_{\eta}(z,u)dz\\
		&&\hspace{6cm}\mbox{for all}\ u\in W^{1,p}(\Omega).
	\end{eqnarray*}
	
	From Corollary \ref{cor3}, hypothesis $H(\beta)$ and (\ref{eq32}), it is clear that $\hat{\psi}_{\eta}$ is coercive. Also, from the Sobolev embedding theorem and the compactness of the trace map $\gamma_0$ into $L^p(\partial\Omega)$, we see that $\hat{\psi}_{\eta}$ is sequentially weakly lower semicontinuous. So, from the Weierstrass theorem 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}(u): u\in W^{1,p}(\Omega)\right].
	\end{eqnarray}
	
	Let $\xi\in\left(0,\delta_0(\eta)\right]$ and $\xi\leq\min\limits_{\overline{\Omega}}u_{\lambda}$ (see hypothesis $H_1(v)$ and recall that $u_{\lambda}\in \mbox{int}\, C_+$). Then
	$$\hat{\psi}_{\eta}(\xi)\leq\frac{\xi^p}{p}||\beta||_{L^{\infty}(\partial\Omega)}-\frac{\xi^q c_6}{q}|\Omega|_N\ (\mbox{see (\ref{eq32})}).$$
	
	Since $q<p$ (see hypothesis $H_1(v)$), by taking $\xi\in(0,1)$ even smaller if necessary, we will have
	\begin{eqnarray*}
		&&\hat{\psi}_{\eta}(\xi)<0\\
		&\Rightarrow&\hat{\psi}_{\eta}(u_{\eta})<0=\hat{\psi}_{\eta}(0)\ (\mbox{see (\ref{eq33})}),\ \mbox{hence}\ u_{\eta}\neq 0.
	\end{eqnarray*}
	
	From (\ref{eq33}) we have
	\begin{eqnarray}\label{eq34}
		&&\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}k_{\eta}(z,u_{\eta})hdz\\
		&&\hspace{8cm}\mbox{for all}\ h\in W^{1,p}(\Omega).\nonumber
	\end{eqnarray}
	
	In (\ref{eq34}), first we choose $h=-u^{-}_{\eta}\in W^{1,p}(\Omega)$. Using Lemma \ref{lem2} and (\ref{eq32}), we have
	\begin{eqnarray*}
		&&\frac{c_1}{p-1}||Du^{-}_{\eta}||^{p}_{p}+||u^{-}_{\eta}||^{p}_{p}\leq 0,\\
		&\Rightarrow&u_{\eta}\geq 0,\ u_{\eta}\neq 0.
	\end{eqnarray*}
	
	Next, in (\ref{eq34}), we choose $h=(u_{\eta}-u_{\lambda})^+\in W^{1,p}(\Omega)$. Then
	\begin{eqnarray*}
		&&\left\langle A(u_{\eta}),(u_{\eta}-u_{\lambda})^+\right\rangle+\int_{\Omega}u^{p-1}_{\eta}(u_{\eta}-u_{\lambda})^+dz+\int_{\partial\Omega}\beta(z)u^{p-1}_{\eta}(u_{\eta}-u_{\lambda})^+d\sigma\\
		&=&\int_{\Omega}[f(z,u_{\lambda},\eta)+u^{p-1}_{\lambda}](u_{\eta}-u_{\lambda})^+dz\ (\mbox{see (\ref{eq32})})\\
		&\leq&\int_{\Omega}[f(z,u_{\lambda},\lambda)+u^{p-1}_{\lambda}](u_{\eta}-u_{\lambda})^+dz\ (\mbox{since}\ f(z,u_{\lambda}(z),\cdot)\ \mbox{is nondecreasing})\\
		&=&\left\langle A(u_{\lambda}),(u_{\eta},u_{\lambda})^+\right\rangle+\int_{\Omega}u^{p-1}_{\lambda}(u_{\eta}-u_{\lambda})^+dz+\int_{\partial\Omega}\beta(z)u^{p-1}_{\lambda}(u_{\eta}-u_{\lambda})^+d\sigma\\
		\Rightarrow&&\left\langle A(u_{\eta})-A(u_{\lambda}),(u_{\eta}-u_{\lambda})^+\right\rangle+\int_{\Omega}(u^{p-1}_{\eta}-u^{p-1}_{\lambda})(u_{\eta}-u_{\lambda})^+dz\leq 0\\
		&&\hspace{6cm}(\mbox{see hypothesis}\ H(\beta))\\
		\Rightarrow&&|\{u_{\eta}>u_{\lambda}\}|_N=0,\ \mbox{hence}\ u_{\eta}\leq u_{\lambda}.
	\end{eqnarray*}
	
	So, we have proved that
	$$u_{\eta}\in[0,u_{\lambda}]=\{u\in W^{1,p}(\Omega):0\leq u(z)\leq u_{\lambda}(z)\ \mbox{for a.a.}\ z\in\Omega\},\ u_{\eta}\neq 0.$$
	
	Then because of (\ref{eq32}), equation (\ref{eq34}) becomes
	$$\left\langle A(u_{\eta}),h\right\rangle+\int_{\partial\Omega}\beta(z)u^{p-1}_{\eta}hd\sigma=\int_{\Omega}f(z,u_{\eta},\eta)hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega).$$
	
	From this, as in the proof of Proposition \ref{prop9}, using the nonlinear Green's identity, we infer that
	\begin{eqnarray*}
		&&u_{\eta}\in S(\eta)\subseteq \mbox{int}\, C_+,\ \mbox{hence}\ \eta\in\mathcal{S},\\
		&\Rightarrow&\left(0,\lambda\right]\subseteq\mathcal{S}.
	\end{eqnarray*}
\end{proof}

Let $\lambda^*=\sup\mathcal{S}$. We show that $\lambda^*$ is finite by strengthening the conditions on the reaction $f(z,x,\lambda)$. So, the new stronger hypotheses on $f$ are the following:

\smallskip
\textbf{$H_2:$} $f:\Omega\times\mathbb{R}\times(0,\infty)\rightarrow\mathbb{R}$ is a function such that  for a.a. $z\in\Omega$ and all $\lambda>0$ $f(z,0,\lambda)=0$ and
\begin{itemize}
	\item[(i)] for all $(x,\lambda)\in\mathbb{R}\times(0,\infty)$, $z\longmapsto f(z,x,\lambda)$ is measurable, while for a.a. $z\in\Omega$, $(x,\lambda)\longmapsto f(z,x,\lambda)$ is continuous;
	\item[(ii)] $|f(z,x,\lambda)|\leq a_{\lambda}(z)(1+x^{r-1})$ for a.a. $z\in\Omega$, all $x\geq 0$, all $\lambda>0$, with $a_{\lambda}\in L^{\infty}(\Omega)$, $\lambda\longmapsto||a_{\lambda}||_{\infty}$ bounded on bounded sets in $(0,\infty)$ and $p<r<p^*$;
	\item[(iii)] if $F(z,x,\lambda)=\int^{x}_{0}f(z,s,\lambda)ds$, then $\lim\limits_{x\rightarrow+\infty}\frac{F(z,x,\lambda)}{x^p}=+\infty$ uniformly for a.a. $z\in\Omega$;
	\item[(iv)]there exists $\vartheta=\vartheta(\lambda)\in\left((r-p)\max\left\{\frac{N}{p},1\right\},p^*\right)$ such that
 $$0<\gamma_0\leq\liminf\limits_{x\rightarrow+\infty}\frac{f(z,x,\lambda)x-pF(z,x,\lambda)}{x^{\vartheta}}\ \mbox{uniformly for a.a.}\ z\in\Omega;$$
	\item[(v)] there exists $1<\mu=\mu(\lambda)<q=q(\lambda)<\tau$ (see hypothesis $H(a)(v)$) and $\gamma=\gamma(\lambda)>\mu$, $\delta_0=\delta_0(\lambda)\in(0,1)$ such that
	$$c_6x^q\leq f(z,x,\lambda)x\leq qF(z,x,\lambda)\leq\xi_{\lambda}(z)x^{\mu}+\tau x^{\gamma}\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ 0\leq x\leq \delta_0$$
	with $c_6=c_6(\lambda)>0$, $c_6(\lambda)\rightarrow+\infty$ as $\lambda\rightarrow+\infty$, $\overline{c}=\overline{c}(\lambda)>0$, $\xi_{\lambda}\in L^{\infty}(\Omega)_+$ with $||\xi_{\lambda}||_{\infty}\rightarrow 0$ as $\lambda\rightarrow 0^+$;
	\item[(vi)] for every $\rho>0$, there exists $\xi_{\rho}=\xi_{\rho}(\lambda)>0$ such that  for a.a. $z\in\Omega$, $x\longmapsto f(z,x,\lambda)+\xi_{\rho}x^{p-1}$ is nondecreasing on $[0,\rho]$;
	\item[(vii)] for every interval $K=[x_0,\hat{x}]$ with $x_0>0$ and every $\lambda>\lambda'>0$, there exists $d_{K}(x_0,\lambda)$ nondecreasing in $\lambda$ with $d_K(x_0,\lambda)\rightarrow+\infty$ as $\lambda\rightarrow+\infty$ and $\hat{d}_K(x_0,\lambda,\lambda')$ such that
	\begin{eqnarray*}
		&&f(z,x,\lambda)\geq d_K(x_0,\lambda)\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\in K\\
		&&f(z,x,\lambda)-f(z,x,\lambda')\geq\hat{d}_K(x_0,\lambda,\lambda')\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\in K.
	\end{eqnarray*}
\end{itemize}
\begin{remark}
	Suppose that $f(z,x,\lambda)=\lambda g(x)+h(z,x)$ with $g(\cdot)$ continuous, nondecreasing, positive on $(0,\infty)$ and $h\geq 0$, $h(z,\cdot)\in C^1(\mathbb{R})$ for a.a. $z\in \Omega$ and $h'_x(z,x)\geq-\xi^*x^{\eta-2}$ for a.a. $z\in\Omega$, all $x>0$ and some $\xi^*>0$, $\eta\geq p$. Then hypotheses $H_2(vi),(vii)$ are satisfied. Also, the examples presented after hypotheses $H_1$, satisfy also the new conditions.
\end{remark}
\begin{proposition}\label{prop11}
	If hypotheses $H(a),\ H(\beta)$ and $H_2$ hold, then $\lambda^*<\infty$.
\end{proposition}
\begin{proof}
	We claim that there exists $\overline{\lambda}>0$ such that
	\begin{eqnarray}\label{eq35}
		f(z,x,\overline{\lambda})\geq x^{p-1}\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\geq 0.
	\end{eqnarray}
	
	Indeed by virtue of hypothesis $H_2(v)$, we have $f(z,x,\lambda)\geq c_6(\lambda)x^{q-1}$ for a.a. $z\in\Omega$ all $x\in[0,\delta_0(\lambda)]$. The hypothesis on $c_6(\cdot)$ implies that we can find $\lambda_0>0$ and $0<\delta_1\leq\delta_0(\lambda_0)$ such that
	\begin{eqnarray}\label{eq36}
		f(z,x,\lambda_0)\geq c_6(\lambda_0)x^{q-1}\geq x^{p-1}\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\in[0,\delta_1]
	\end{eqnarray}
	
	Hypotheses $H_1(iii),(iv)$ imply that we can find $M_5>0$ such that
	\begin{eqnarray}\label{eq37}
		f(z,x,\lambda_0)\geq x^{p-1}\ \mbox{for all a.a.}\ z\in \Omega,\ \mbox{all}\ x\geq M_5.
	\end{eqnarray}
	
	Finally, from hypothesis $H_2(vii)$, for $K=[\delta_1,M_5]$ we have
	$$f(z,x,\lambda)\geq d_K(z,\lambda)\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\in[\delta_1,M_5],\ \mbox{all}\ \lambda>0.$$
	
	Since $d_K(x,\lambda)\rightarrow+\infty$ as $\lambda\rightarrow+\infty$, we can find $\overline{\lambda}\geq\lambda_0$ such that
	\begin{eqnarray}\label{eq38}
		f(z,x,\overline{\lambda})\geq d_K(x,\overline{\lambda})\geq M_5^{p-1}\geq x^{p-1}\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\in[\delta_1,M_5].
	\end{eqnarray}
	
	Recalling that $f(z,x,\cdot)$ and $c_6(\cdot)$ are nondecreasing in $\lambda>0$, from (\ref{eq36}), (\ref{eq37}) and (\ref{eq38}) we conclude that (\ref{eq35}) is true.
	
	Now, let $\lambda>\overline{\lambda}$ and assume that $\lambda\in\mathcal{S}$. Then we can find $u_{\lambda}\in S(\lambda)\subseteq \mbox{int}\, C_1$ (see Proposition \ref{prop9}). Let $m_{\lambda}=\min\limits_{\overline{\Omega}}u_{\lambda}>0$. For $\delta>0$ we set $m^{\delta}_{\lambda}=m_{\lambda}+\delta\in \mbox{int}\, C_+$. Also, let $\rho=||u_{\lambda}||_{\infty}$ and let $\xi_{\rho}>0$ be as postulated by hypothesis $H_2(vi)$. We have
	\begin{eqnarray*}
		&&-\mbox{div}\, a(Dm^{\delta}_{\lambda})+\xi_{\rho}(m^{\delta}_{\lambda})^{p-1}\\
		&\leq&\xi_{\rho}m^{p-1}_{\lambda}+\chi(\delta)\ \mbox{with}\ \chi(\delta)\rightarrow 0^+\ \mbox{as}\ \delta\rightarrow 0^+\\
		&\leq&(1+\xi_{\rho})m^{p-1}_{\lambda}+\chi(\delta)\\
		&\leq&f(z,m_{\lambda},\overline{\lambda})+\xi_{\rho}m^{p-1}_{\lambda}+\chi(\delta)\ (\mbox{see (\ref{eq35})})\\
		 &=&f(z,m_{\lambda},\lambda)+\xi_{\rho}m^{p-1}_{\lambda}+[f(z,m_{\lambda},\overline{\lambda})-f(z,m_{\lambda},\lambda)]+\chi(\delta)\\
		 &\leq&f(z,m_{\lambda},\lambda)+\xi_{\rho}m^{p-1}_{\lambda}-\hat{d}_K(m_{\lambda},\lambda,\overline{\lambda})+\chi(\delta)\ \mbox{with}\ K=\{m_{\lambda}\}\\
		&&\hspace{6cm}(\mbox{see hypothesis}\ H_2(vii))\\
		&\leq&f(z,m_{\lambda},\lambda)+\xi_{\rho}m^{p-1}_{\lambda}\ \mbox{for all}\ \delta>0\ \mbox{small}\\
		&\leq&f(z,u_{\lambda},\lambda)+\xi_{\rho}u_{\lambda}(z)^{p-1}\ (\mbox{since}\ m_{\lambda}\leq u_{\lambda}(z)\ \mbox{for all}\ z\in\overline{\Omega},\\
		&&\hspace{6cm}\mbox{see hypothesis}\ H_2(vii))\\
		&=&-\mbox{div}\, a(Du_{\lambda}(z))+\xi_{\rho}u_{\lambda}(z)^{p-1}\ \mbox{for a.a.}\ z\in\Omega\ (\mbox{since}\ u_{\lambda}\in S(\lambda)),\\
		\Rightarrow&&m^{\delta}_{\lambda}\leq u_{\lambda}(z)\ \mbox{for all}\ z\in\overline{\Omega},\ \mbox{all}\ \delta>0\ \mbox{small, a contradiction}.
	\end{eqnarray*}
	
	This means that $\lambda\notin \mathcal{S}$ and so $\lambda^*\leq\overline{\lambda}<\infty$.
\end{proof}
\begin{proposition}\label{prop12}
	If hypotheses $H(a),\ H(\beta)$ and $H_2$ hold and $\eta\in(0,\lambda^*)$, then problem $(P_{\eta})$ admits at least two distinct positive solutions
	$$u_0,\hat{u}\in {\rm int}\, C_+,\ u_0\leq\hat{u}.$$
\end{proposition}
\begin{proof}
	Let $\eta,\lambda\in(0,\lambda^*)$ with $\eta<\lambda$ and let $u_{\lambda}\in S(\lambda)\subseteq \mbox{int}\, C_+$. From the proof of Proposition \ref{prop10}, we know that by using a suitable truncation-perturbation of the reaction of problem $(P_{\eta})$ (see (\ref{eq32})), we can find $u_0\in[0,u_{\lambda}]\cap S(\eta)$, which is a minimizer of the corresponding truncated energy functional $\hat{\psi}_{\eta}$ (see the proof of Proposition \ref{prop10}).
	
	For $\delta>0$, let $u^{\delta}_{0}=u_0+\delta\in \mbox{int}\, C_+$ and for $\rho=||u_{\lambda}||_{\infty}$, let $\xi_{\rho}>0$ be as postulated by hypothesis $H_2(vi)$. We have
	\begin{eqnarray*}
		&&-\mbox{div}\, a(Du^{\delta}_{0}(z))+\xi_{\rho}u^{\delta}_{0}(z)^{p-1}\\
		&\leq&-\mbox{div}\, a(Du_0(z))+\xi_{\rho}u_0(z)^{p-1}+\chi(\delta)\ \mbox{with}\ \chi(\delta)\rightarrow 0^+\ \mbox{as}\ \delta\rightarrow 0^+\\
		&=&f(z,u_0(z),\eta)+\xi_{\rho}u_0(z)^{p-1}+\chi(\delta)\ (\mbox{since}\ u_0\in S(\eta))\\
		 &=&f(z,u_0(z),\lambda)+\xi_{\rho}u_0(z)^{p-1}+[f(z,u_0(z),\eta)-f(z,u_0(z),\lambda)]+\chi(\delta)\\
		 &\leq&f(z,u_{\lambda}(z),\lambda)+\xi_{\rho}u_{\lambda}(z)^{p-1}-\hat{d}_K(m_0,\lambda,\eta)+\chi(\delta)\\
		&&(\mbox{since}\ u_0\leq u_{\lambda},\ \mbox{see hypothesis}\ H_2(vi)\ \mbox{and with}\ K=u_0(\overline{\Omega}),\ m_0=\inf K)\\
		&\leq&f(z,u_{\lambda}(z),\lambda)+\xi_{\rho}u_{\lambda}(z)^{p-1}\ \mbox{for}\ \delta>0\ \mbox{small},\\
		&=&-\mbox{div}\,a(Du_{\lambda}(z))+\xi_{\rho}u_{\lambda}(z)^{p-1}\ \mbox{a.e. in}\ \Omega\ (\mbox{since}\ u_{\lambda}\in S(\lambda)),\\
		\Rightarrow&&u_0^{\delta}\leq u_{\lambda}\ \mbox{for}\ \delta>0\ \mbox{small},\\
		\Rightarrow&&u_{\lambda}-u_0\in \mbox{int}\, C_+.
	\end{eqnarray*}
	
	So, we have proved that
	\begin{eqnarray}\label{eq39}
		u_0\in \mbox{int}_{C^1(\overline{\Omega})}[0,u_{\lambda}].
	\end{eqnarray}
	
	Recall that $u_0$ is a minimizer of the functional $\hat{\psi}_{\lambda}$ (see the proof of Proposition \ref{prop10}). Note that
	\begin{eqnarray*}
		&&\hat{\psi}_{\lambda}|_{[0,u_{\lambda}]}=\hat{\varphi}_{\lambda}|_{[0,u_{\lambda}]}\ (\mbox{see (\ref{eq32})})\\
		&\Rightarrow&u_0\ \mbox{is a local}\ C^1(\overline{\Omega})-\mbox{minimizer of}\ \hat{\varphi}_{\lambda}\ (\mbox{see (\ref{eq39})}),\\
		&\Rightarrow&u_0\ \mbox{is a local}\ W^{1,p}(\Omega)-\mbox{minimizer of}\ \hat{\varphi}_{\lambda}\ (\mbox{see Proposition \ref{prop4}}).
	\end{eqnarray*}
	
	Next, we consider the following truncation-perturbation of the reaction in problem $(P_{\eta})$:
	\begin{eqnarray}\label{eq40}
		\gamma_{\eta}(z,x)=\left\{
		\begin{array}{ll}
			f(z,u_0(z),\eta)+u_0(z)^{p-1}&\mbox{if}\ x\leq u_0(z)\\
			f(z,x,\eta)+x^{p-1}&\mbox{if}\ u_0(z)<x.
		\end{array}
		\right.
	\end{eqnarray}
	
	This is a Carath\'eodory function. Let $\Gamma_{\eta}(z,x)=\int^{x}_{0}\gamma_{\eta}(z,s)ds$ and consider the $C^1$-functional $\sigma_{\eta}:W^{1,p}(\Omega)\rightarrow\mathbb{R}$ defined by
	\begin{eqnarray*}
		 &&\sigma_{\eta}(u)=\int_{\Omega}G(Du)dz+\frac{1}{p}||u||^{p}_{p}+\frac{1}{p}\int_{\partial\Omega}\beta(z)(u^+)^pd\sigma=\int_{\Omega}\Gamma_{\eta}(z,u)dz\\
		&&\hspace{7cm}\mbox{for all}\ u\in W^{1,p}(\Omega).
	\end{eqnarray*}
	
	Note that
	\begin{eqnarray}\label{eq41}
		&&\sigma_{\eta}=\hat{\varphi}_{\eta}+\hat{\xi}_{\eta}\ \mbox{with}\ \hat{\xi}_{\eta}\in\mathbb{R}\ (\mbox{see (\ref{eq40})}),\nonumber\\
		&\Rightarrow&\sigma_{\eta}\ \mbox{satisfies the}\ C-\mbox{condition (see Proposition \ref{prop6})}.
	\end{eqnarray}
	
	Moreover, Proposition \ref{prop8} implies that if $u\in \mbox{int}\, C_+$, then
	\begin{eqnarray}\label{eq42}
		\sigma_{\eta}(tu)\rightarrow -\infty\ \mbox{as}\ t\rightarrow+\infty.
	\end{eqnarray}
	
	\begin{claim}
		We may assume that $u_0$ is a local minimizer of $\sigma_{\eta}$.
	\end{claim}
	
	Recall that $u_0\leq u_{\lambda}$. Then using $u_{\lambda}$, we truncate $\gamma_{\eta}(z,\cdot)$ as follows:
	\begin{eqnarray}\label{eq43}
		\hat{\gamma}_{\eta}(z,x)=\left\{
		\begin{array}{ll}
			\gamma_{\eta}(z,x)&\mbox{if}\ x\leq u_{\lambda}(z)\\
			\gamma_{\eta}(z,u_{\lambda}(z))&\mbox{if}\ u_{\lambda}(z)<x
		\end{array}
		\right.
	\end{eqnarray}
	
	This is a Carath\'eodory function. We set $\hat{\Gamma}_{\eta}(z,x)=\int^{x}_{0}\hat{\gamma}_{\eta}(z,s)ds$ and consider the $C^1$-functional $\hat{\sigma}_{\eta}:W^{1,p}(\Omega)\rightarrow\mathbb{R}$ defined by
	\begin{eqnarray*}
		 &&\hat{\sigma}_{\eta}(u)=\int_{\Omega}G(Du)dz+\frac{1}{p}||u||^{p}_{p}+\frac{1}{p}\int_{\partial\Omega}\beta(z)(u^+)^pd\sigma-\int_{\Omega}\hat{\Gamma}_{\eta}(z,u)dz\\
		&&\hspace{7cm}\mbox{for all}\ u\in W^{1,p}(\Omega).
	\end{eqnarray*}
	
	From (\ref{eq43}), Corollary \ref{cor3} and hypothesis $H(\beta)$, we see that the functional $\hat{\sigma}_{\eta}$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, by the Weierstrass theorem, we can find $\hat{u}_0\in W^{1,p}(\Omega)$ such that
	\begin{eqnarray}\label{eq44}
		&&\hat{\sigma}_{\eta}(\hat{u}_0)=\inf[\hat{\sigma}_{\eta}(u):u\in W^{1,p}(\Omega)],\nonumber\\
		&\Rightarrow&\hat{\sigma}'_{\eta}(\hat{u}_0)=0,\nonumber\\
		&\Rightarrow&\left\langle A(\hat{u}_0),h\right\rangle+\int_{\Omega}|\hat{u}_0|^{p-2}\hat{u}_0hdz+\int_{\partial\Omega}\beta(z)(\hat{u}^{+}_{0})^{p-1}hd\sigma=\int_{\Omega}\hat{\gamma}_{\eta}(z,\hat{u}_0)hdz\\
		&&\hspace{6cm}\mbox{for all}\ h\in W^{1,p}(\Omega).\nonumber
	\end{eqnarray}
	
	In (\ref{eq44}), first we choose $h=(u_0-\hat{u}_0)^+\in W^{1,p}(\Omega)$. Then
	\begin{eqnarray*}
		&&\left\langle A(\hat{u}_0),(u_0-\hat{u}_{0})^+\right\rangle+\int_{\Omega}|\hat{u}_{0}|^{p-2}\hat{u}_0(u_0-\hat{u}_0)^+dz+\\
		&&\hspace{6cm}+\int_{\partial\Omega}\beta(z)(\hat{u}_0^+)^{p-1}(u_0-\hat{u}_{0})^+d\sigma\\
		&=&\int_{\Omega}\left[f(z,u_0,\eta)+u_0^{p-1}\right](u_0-\hat{u}_0)^+dz\ (\mbox{recall that}\ u_0\leq u_{\lambda}\\
		&&\hspace{6cm}\mbox{and see (\ref{eq43}) and (\ref{eq40})})\\
		&=&\left\langle A(u_0),(u_0-\hat{u}_{0})^+\right\rangle+\int_{\Omega}u_{0}^{p-1}(u_0-\hat{u}_0)^+dz+\int_{\partial\Omega}\beta(z)u_0^{p-1}(u_0-\hat{u}_{0})^+d\sigma\\
		&&\hspace{7cm}(\mbox{since}\ u_0\in S(\eta)),\\
		&\Rightarrow&\left\langle A(u_0)-A(\hat{u}_0),(u_0-\hat{u}_0)^+\right\rangle+\int_{\Omega}(u_0^{p-1}-|\hat{u}_0|^{p-2}\hat{u}_0)\ (u_0-\hat{u}_0)^+dz\leq 0\\
		&&\hspace{7cm}(\mbox{see hypothesis}\ H(\beta)),\\
		&\Rightarrow&|\{u_0>\hat{u}_0\}_N|=0,\ \mbox{hence}\ u_0\leq\hat{u}_0.
	\end{eqnarray*}
	
	Next in (\ref{eq44}) we choose $(\hat{u}_0-u_{\lambda})^+\in W^{1,p}(\Omega)$. We obtain
	\begin{eqnarray*}
		&&\left\langle A(\hat{u}_0),(\hat{u}_0-u_{\lambda})^+\right\rangle+\int_{\Omega}\hat{u}^{p-1}_{0}(\hat{u}_0-u_{\lambda})^+dz+\int_{\partial\Omega}\beta(z)\hat{u}^{p-1}_{0}(\hat{u}_0-u_{\lambda})^+d\sigma\\
		&=&\int_{\Omega}[f(z,u_{\lambda},\eta)+u_{\lambda}^{p-1}](\hat{u}_0-u_{\lambda})^+dz\ (\mbox{see\ (\ref{eq43})\ and\ (\ref{eq40})})\\
		 &\leq&\int_{\Omega}\left[f(z,u_{\lambda},\lambda)+u_{\lambda}^{p-1}\right](\hat{u}_0-u_{\lambda})^+dz\ (\mbox{see hypothesis}\ H_2(vii))\\
		&=&\left\langle A(u_{\lambda}),(\hat{u}_0-u_{\lambda})^+\right\rangle+\int_{\Omega}u_{\lambda}^{p-1}(\hat{u}_0-u_{\lambda})^+dz+\int_{\partial\Omega}\beta(z)u_{\lambda}^{p-1}(\hat{u}_0-u_{\lambda})^+d\sigma\\
		&&\hspace{7cm}(\mbox{since}\ u_{\lambda}\in S(\lambda)),\\
		&\Rightarrow&\left\langle A(\hat{u}_0)-A(u_{\lambda}),(\hat{u}_0-u_{\lambda})^+\right\rangle+\int_{\Omega}(\hat{u}_0^{p-1}-u_{\lambda}^{p-1})(\hat{u}_0-u_{\lambda})^+dz\leq 0\\
		&&\hspace{7cm}(\mbox{see hypothesis}\ H(\beta)),\\
		&\Rightarrow&|\{\hat{u}_0>u_{\lambda}\}|_N=0,\ \mbox{hence}\ \hat{u}_0\leq u_{\lambda}.
	\end{eqnarray*}
	
	So, we have proved that
	$$\hat{u}_0\in[u_0,u_{\lambda}]=\{u\in W^{1,p}(\Omega):u_0(z)\leq u(z)\leq u_{\lambda}(z)\ \mbox{a.e. in}\ \Omega\}.$$
	
	If $\hat{u}_0\neq u_0$, then by virtue of (\ref{eq43}) and (\ref{eq40}), we see that
	$$\hat{u}_0\in S(\eta)\subseteq \mbox{int}\, C_+,\ u_0\leq\hat{u}_0,\ u_0\neq\hat{u}_0$$
	and so we are done, since this is the desired second positive solution of problem $(P_{\eta})$.
	
	Hence, we may assume that $\hat{u}_0=u_0\in \mbox{int}\, C_+$. Recall that $u_{\lambda}-u_0\in \mbox{int}\, C_+$ (see (\ref{eq39})) and $\hat{\sigma}_{\eta}|_{[0,u_{\lambda}]}=\sigma_{\eta}|_{[0,u_{\lambda}]}$ (see (\ref{eq43})). Therefore
	\begin{eqnarray*}
		&&u_0\ \mbox{is a local}\ C^1(\overline{\Omega})-\mbox{minimizer of}\ \sigma_{\eta},\\
		&\Rightarrow&u_0\ \mbox{is a local}\ W^{1,p}(\Omega)-\mbox{minimizer of}\ \sigma_{\eta}\ (\mbox{see Proposition \ref{prop4}}).
	\end{eqnarray*}
	
	This proves the Claim.
	
	Reasoning as above, we can show that
	\begin{eqnarray}\label{eq45}
		K_{\sigma_{\eta}}\subseteq\left[u_0,\infty \right)=\{u\in W^{1,p}(\Omega):u_0(z)\leq u(z)\ \mbox{a.e. in}\ \Omega\}.
	\end{eqnarray}
	
	Then from (\ref{eq40}) we see that the elements of $K_{\sigma_{\eta}}$ are positive solutions of problem $(P_{\eta})$. Therefore, we may assume that $K_{\sigma_{\eta}}$ is finite of otherwise we already have an infinity of positive solutions for problem $(P_{\eta})$.
	
	The finiteness of $K_{\sigma_{\eta}}$ and the Claim imply that we can find $\rho\in(0,1)$ small such that
	\begin{eqnarray}\label{eq46}
		\sigma_{\eta}(u_0)<\inf[\sigma_{\eta}(u):||u-u_0||=\rho]=m^{\eta}_{\rho}
	\end{eqnarray}
	(see Aizicovici, Papageorgiou and Staicu \cite{1} (proof of Proposition 29)). Then (\ref{eq41}), (\ref{eq42}) and (\ref{eq46}) imply that we can use Theorem \ref{th1} (the mountain pass theorem). So, we can find $\hat{u}\in W^{1,p}(\Omega)$ such that
	\begin{eqnarray}\label{eq47}
		\hat{u}\in K_{\sigma_{\eta}}\ \mbox{and}\ \sigma_{\eta}(u_0)<m^{\eta}_{\rho}\leq\sigma_{\eta}(\hat{u}).
	\end{eqnarray}
	
	From (\ref{eq47}) it follows that $\hat{u}\neq u_0$ and $\hat{u}\in S(\eta)\subseteq \mbox{int}\, C_+,\ u_0\leq \hat{u}$ (see (\ref{eq45})).
\end{proof}

Next we examine what happens in the critical case $\lambda=\lambda^*$. To this end, note that hypotheses $H_2(ii),(v)$ imply that we can find $c_{18}=c_{18}(\lambda)>0$ such that
\begin{eqnarray}\label{eq48}
	f(z,x,\lambda)\geq c_6 x^{q-1}-c_{18}x^{r-1}\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ z\geq 0.
\end{eqnarray}

This unilateral growth estimate on the reaction $f(z,\cdot,\lambda)$ leads to the following auxiliary Robin problem:
\begin{eqnarray}\label{eq49}
	\left\{\begin{array}{ll}
		-\mbox{div}\, a(Du(z))=c_6 u(z)^{q-1}-c_{18}u(z)^{r-1}\ &\mbox{in}\ \Omega,\\
		\frac{\partial u}{\partial n_0}(z)+\beta(z)u(z)^{p-1}=0\ &\mbox{on}\ \partial\Omega,\ u>0.
	\end{array}\right\}
\end{eqnarray}

For this problem we have the following existence and uniqueness result.
\begin{proposition}\label{prop13}
	If hypotheses $H(a)$ and $H(\beta)$ hold, the problem (\ref{eq49}) admits a unique positive solution $\bar{u}\in {\rm int}\, C_+$.
\end{proposition}
\begin{proof}
	First we show the existence of a positive solution for problem (\ref{eq49}). To this end let $\xi_+:W^{1,p}(\Omega)\rightarrow\mathbb{R}$ be the $C^1$-functional defined by
	\begin{eqnarray*}
		 &&\xi_+(u)=\int_{\Omega}G(Du)dz+\frac{1}{p}||u^-||^{p}_{p}+\frac{1}{p}\int_{\partial\Omega}\beta(z)(u^+)^pd\sigma+\frac{c_{18}}{r}||u^+||^{r}_{r}-\frac{c_{6}}{q}||u^+||^{q}_{q}\\
		&&\hspace{7cm}\mbox{for all}\ u\in W^{1,p}(\Omega).
	\end{eqnarray*}
	
	Using Corollary \ref{cor3} and hypothesis $H(\beta)$, we have
	\begin{eqnarray*}
		 &&\xi_+(u)\geq\frac{c_1}{p(p-1)}||Du||^{p}_{p}+\frac{1}{p}||u||^{p}_{p}+\frac{c_{18}}{r}||u^+||^{r}_{r}-c_{19}(||u^+||^{q}_{r}+||u^+||^{p}_{r})\\
		&&\hspace{6cm}\mbox{for some}\ c_{19}>0\ (\mbox{recall}\ q<p<r).
	\end{eqnarray*}
	
	Because $q<p<r$, it follows that $\xi_+$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find $\bar{u}\in W^{1,p}(\Omega)$ such that
	\begin{eqnarray}\label{eq50}
		\xi_+(\bar{u})=\inf[\xi_+(u):u\in W^{1,p}(\Omega)].
	\end{eqnarray}
	
	Exploiting the fact $q<p<r$, by choosing $\xi^*\in(0,1)$ small, we have
	\begin{eqnarray*}
		&&\xi_+(\xi^*)<0,\\
		&\Rightarrow&\xi_+(\bar{u})<0=\xi_+(0)\ (\mbox{see (\ref{eq50})}),\ \mbox{hence}\ \bar{u}\neq 0.
	\end{eqnarray*}
	
	From (\ref{eq50}) we have
	\begin{eqnarray}\label{eq51}
		&&\xi'_+(\bar{u})=0,\nonumber\\
		&\Rightarrow&\left\langle A(\bar{u}),h\right\rangle-\int_{\Omega}(\bar{u}^-)^{p-1}hdz+\int_{\partial\Omega}\beta(z)(\bar{u}^+)^{p-1}hd\sigma=c_6\int_{\Omega}(\bar{u}^+)^{q-1}hdz-\\
		&&\hspace{5cm}-c_{18}\int_{\Omega}(\bar{u}^+)^{r-1}hdz\ \mbox{for all}\ u\in W^{1,p}(\Omega)\nonumber.
	\end{eqnarray}
	
	Let, $h=-\bar{u}^-\in W^{1,p}(\Omega)$ in (\ref{eq51}). Then, we see that $\bar{u}\geq 0$, $\bar{u}\neq 0$. So, (\ref{eq51}) becomes
	\begin{eqnarray*}
		&&\left\langle A(\bar{u}),h\right\rangle+\int_{\partial\Omega}\beta(z)\bar{u}^{p-1}hd\sigma=c_6\int_{\Omega}\bar{u}^{q-1}hdz-c_{18}\int_{\Omega}\bar{u}^{r-1}hdz\\
		&&\hspace{7cm}\mbox{for all}\ h\in W^{1,p}(\Omega),\\
		&\Rightarrow&\bar{u}\in \mbox{int}\, C_+\ \mbox{is a solution of (\ref{eq49}) (see the proof of Proposition \ref{prop9})}.
	\end{eqnarray*}
	
	So, we have established the existence of positive solutions for problem (\ref{eq49}).
	
	Next we show the uniqueness of this positive solution.
	
	To this end, let $e:L^{\tau}(\Omega)\rightarrow\overline{\mathbb{R}}=\mathbb{R}\cup\{+\infty\}$ be the integral functional defined by
	\begin{eqnarray*}
		e(u)=\left\{
		\begin{array}{ll}
			 \int_{\Omega}G(Du^{1/\tau})dz+\frac{1}{p}\int_{\partial\Omega}\beta(z)u^{p/\tau}d\sigma&\mbox{if}\ u\geq 0,\ u^{1/\tau}\in W^{1,p}(\Omega)\\
			+\infty&\mbox{otherwise}
		\end{array}\right.
	\end{eqnarray*}
	
	Let $u_1,u_2\in \mbox{dom}\, e=\{u\in W^{1,p}(\Omega):e(u)<\infty\}$ (the effective domain of the functional $e$) and let $t\in[0,1]$. We define
	$$y=((1-t)u_1+tu_2)^{1/\tau},\ v_1=u_1^{1/\tau},\ v_2=u_2^{1/\tau}.$$
	
	Using Lemma 1 of Diaz and Saa \cite{5}, we have
$$	\begin{array}{lll}
		&\ \ \ \ |Dy(z)|&\leq\left[(1-t)|Dv_1(z)|^{\tau}+t|Dv_2(z)|^{\tau}\right]^{1/\tau}\ \mbox{for a.a.}\ z\in\Omega,\\
		\Rightarrow & G_0(|Dy(z)|)&\leq G_0\left(\left((1-t)|Dv_1(z)|^{\tau}+t|Dv_2(z)|^{\tau}\right)\right)\\
		&&\hspace{5cm}(\mbox{since}\ G_0\ \mbox{is increasing})\\
		&&\leq(1-t)G_0(|Dv_1(z)|)+tG_0(|Dv_2(z)|)\ \mbox{for a.a.}\ z\in\Omega\\
		&&\hspace{5cm}(\mbox{see hypothesis}\ H(a)(v)),\\
		\Rightarrow &\ \ \ G(Dy(z))&\leq(1-t)G(Du_1(z)^{1/\tau})+tG(Du_2(z)^{1/\tau})\ \mbox{for a.a.}\ z\in\Omega,\\
		\Rightarrow && u\longmapsto\int_{\Omega}G(Du^{1/\tau})dz\ \mbox{is convex}.
	\end{array}$$
	
	Since $p>\tau$ and $\beta\geq 0$ (see hypothesis $H(\beta)$), we see that $u\longmapsto\frac{1}{p}\int_{\partial\Omega}\beta(z)u^{p/\tau}d\sigma$ is a convex functional. Therefore, $e$ is convex and also via Fatou's lemma, we have that $e$ is lower semicontinuous.
	
	We already have $\bar{u}\in \mbox{int}\, C_+$ a positive solution of problem (\ref{eq49}). Let $\bar{y}\in W^{1,p}(\Omega)$ be another positive solution. As above, we can show that $\bar{y}\in \mbox{int}\,C_+$. Then for all $h\in C^1(\overline{\Omega})$ and for $|t|$ small, we have
	$$\bar{u}^{\tau}+th,\ \bar{y}^{\tau}+th\in \mbox{dom}\, e.$$
	
	Then $e(\cdot)$ is G\^ateaux differentiable at $\bar{u}^{\tau}$ and $\bar{y}^{\tau}$ in the direction $h$. Moreover, via the chain rule and the nonlinear Green's identity, we obtain
	\begin{eqnarray*}
		&&e'(\bar{u}^p)(h)=\int_{\Omega}\frac{-\mbox{div}\, a(D\bar{u})}{\bar{u}^{\tau-1}}hdz\\
		 &&e'(\bar{y}^p)(h)=\int_{\Omega}\frac{-\mbox{div}\,a(D\bar{y})}{\bar{y}^{\tau-1}}hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega)
	\end{eqnarray*}
	(recall that $C^1(\overline{\Omega})$ is dense in $W^{1,p}(\Omega)$). The convexity of $e$ implies the monotonicity of $e'$. Then
	\begin{eqnarray*}
		 &&0\leq\int_{\Omega}-\mbox{div}\,a(D\bar{u})\left(\frac{\bar{u}^p-\bar{y}^p}{\bar{u}^{\tau-1}}\right)dz
-\int_{\Omega}(-\mbox{div}\, a(D\bar{y}))\left(\frac{\bar{u}^p-\bar{y}^p}{\bar{y}^{p-1}}\right)dz
\\
		 &=&\int_{\Omega}\frac{c_6\bar{u}^{q-1}-
c_{18}\bar{u}^{r-1}}{\bar{u}^{\tau-1}}(\bar{u}^p-\bar{y}^p)dz-
\int_{\Omega}\frac{c_6\bar{y}^{q-1}-c_{18}\bar{y}^{r-1}}{\bar{y}^{\tau-1}}(\bar{u}^p-\bar{y}^p)dz\\
	\\
		 &=&\int_{\Omega}c_6(\bar{u}^{q-\tau}-\bar{y}^{q-\tau})(\bar{u}^p-\bar{y}^p)dz-\int_{\Omega}c_{18}(\bar{u}^{r-\tau}-\bar{y}^{r-\tau})(\bar{u}^p-\bar{y}^p)dz\\
		 \\ &\leq & 0\ (\mbox{since}\ q<\tau<p<r)\\
		\Rightarrow&&\bar{u}=\bar{y}.
	\end{eqnarray*}
	
	This proves the uniqueness of the positive solution $\bar{u}\in \mbox{int}\, C_+$ of problem (\ref{eq49}).
\end{proof}
\begin{proposition}\label{prop14}
	If hypotheses $H(a),\ H(\beta)$ and $H_2$ hold and $\lambda\in\mathcal{S}$, then $\bar{u}\leq u$ for all $u\in S(\lambda)$.
\end{proposition}
\begin{proof}
	Let $u\in S(\lambda)$ and consider the following Carath\'eodory function:
	\begin{eqnarray}\label{eq52}
		w(z,x)=\left\{
		\begin{array}{ll}
			0&\mbox{if}\ x<0\\
			c_6x^{q-1}-c_{18}x^{r-1}+x^{p-1}&\mbox{if}\ 0\leq x\leq u(z)\\
			c_6u(z)^{q-1}-c_{18}u(z)^{r-1}+u(z)^{p-1}&\mbox{if}\ u(z)<x.
		\end{array}
		\right.
	\end{eqnarray}
	
	Let $W(z,x)=\int^{x}_{0}w(z,s)ds$ and consider the $C^1$-functional $\hat{\gamma}:W^{1,p}(\Omega)\rightarrow\mathbb{R}$ defined by
	\begin{eqnarray*}
		 &&\hat{\gamma}(u)=\int_{\Omega}G(Du)dz+\frac{1}{p}||u||^{p}_{p}+\frac{1}{p}\int_{\partial\Omega}\beta(z)(u^+)^pdz-\int_{\Omega}W(z,u)dz\\
		&&\hspace{7cm}\mbox{for all}\ u\in W^{1,p}(\Omega).
	\end{eqnarray*}
	
	From hypothesis $H(\beta)$ and (\ref{eq52}) it is clear that $\hat{\gamma}$ is coercive. Also, it is sequentially weakly lower semicontinuous. Therefore, we can find $\bar{u}_*\in W^{1,p}(\Omega)$ such that
	\begin{eqnarray}\label{eq53}
		\hat{\gamma}(\bar{u}_*)=\inf[\hat{\gamma}(u):u\in W^{1,p}(\Omega)].
	\end{eqnarray}
	
	Since $q<p<r$, for $\xi\in(0,\min\limits_{\overline{\Omega}}u)$ (recall that $u\in \mbox{int}\, C_+$) small, we have
	\begin{eqnarray*}
		&&\hat{\gamma}(\xi)<0,\\
		&\Rightarrow&\hat{\gamma}(\bar{u}_*)<0=\hat{\gamma}(0)\ (\mbox{see (\ref{eq53})}),\ \mbox{hence}\ \bar{u}_*\neq 0.
	\end{eqnarray*}
	
	From (\ref{eq53}) we have
	\begin{eqnarray}\label{eq54}
		&\hat{\gamma}'(\bar{u}_*)=0,&\nonumber\\
		\Rightarrow&\left\langle A(\bar{u}_*),h\right\rangle+\int_{\Omega}|\bar{u}_*|^{p-2}\bar{u}_*hdz+
\int_{\partial\Omega}\beta(z)(\bar{u}_*^+)^{p-1}hd\sigma&=\int_{\Omega}w(z,\bar{u}_*)hdz\\
		&&\mbox{for all}\ h\in W^{1,p}(\Omega).\nonumber
	\end{eqnarray}
	
	In (\ref{eq54}) we choose first $h=-\bar{u}_*^-\in W^{1,p}(\Omega)$ and then $h=(\bar{u}_*-u)^+\in W^{1,p}(\Omega)$ and as in the proof of Proposition \ref{prop10}, we show that
	$$\bar{u}_*\in[0,u],\ \bar{u}_*\neq 0$$
	
	So, (\ref{eq54}) becomes
	\begin{eqnarray*}
		&&\left\langle A(\bar{u}_*),h\right\rangle+\int_{\partial\Omega}\beta(z)\bar{u}_*^{p-1}hd\sigma=c_6\int_{\Omega}\bar{u}_*^{q-1}hdz-c_{18}\int_{\Omega}\bar{u}_*^{r-1}hdz\\
		&&\hspace{6cm}\mbox{for all}\ h\in W^{1,p}(\Omega),\\
		&\Rightarrow&\bar{u}_*\ \mbox{is a positive solution of the auxiliary problem (\ref{eq49})},\\
		&\Rightarrow&\bar{u}_*=\bar{u}\in \mbox{int}\, C_+\ (\mbox{see Proposition \ref{prop13}})\\
		&\Rightarrow&\bar{u}\leq u\ \mbox{for all}\ u\in S(\lambda).
	\end{eqnarray*}
\end{proof}

Now we can show that the critical value $\lambda^*$ is admissible, that is $\lambda^*\in\mathcal{S}$.
\begin{proposition}\label{prop15}
	If hypotheses $H(a),\ H(\beta)$ and $H_2$ hold, then $\lambda^*\in\mathcal{S}$ and so $\mathcal{S}=\left(0,\lambda^*\right]$.
\end{proposition}
\begin{proof}
	Let $\{\lambda_n\}_{n\geq 1}\subseteq\mathcal{S}$ such that  $\lambda_n\rightarrow(\lambda^*)^-$. Then we can find $u_n\in S(\lambda_n)\subseteq \mbox{int}\, C_+$ and from the proof of Proposition \ref{prop10} we know that we can assume that
	\begin{eqnarray}\label{eq55}
		&&\hat{\varphi}_{\lambda_n}(u_n)<0\ \mbox{for all}\ n\geq 1,\nonumber\\
		 &\Rightarrow&\int_{\Omega}pG(Du_n)dz+\int_{\partial\Omega}\beta(z)u_n^pd\sigma-\int_{\Omega}pF(z,u_n,\lambda_n)dz<0\\
		&&\hspace{6.8cm}\mbox{for all}\ n\geq 1.\nonumber
	\end{eqnarray}
	
	Also, we have
	\begin{eqnarray}\label{eq56}
		-\left\langle A(u_n),u_n\right\rangle-\int_{\partial\Omega}\beta(z)u_n^pd\sigma+\int_{\Omega}f(z,u_n,\lambda_n)u_ndz=0\ \mbox{for all}\ n\geq 1.
	\end{eqnarray}
	
	Adding (\ref{eq55}) and (\ref{eq56}), we obtain
	\begin{eqnarray}\label{eq57}
		 &&\int_{\Omega}\left[pG(Du_n)-(a(Du_n),Du_n)_{\mathbb{R}^N}\right]dz+\int_{\Omega}\left[f(z,u_n,\lambda_n)u_n-\right.\nonumber\\
		&&\hspace{5cm}-\left.pF(z,u_n,\lambda_n)\right]dz<0\ \ \mbox{for all}\ n\geq 1,\nonumber\\
		&\Rightarrow&\int_{\Omega}\left[f(z,u_n,\lambda_n)u_n-pF(z,u_n,\lambda_n)\right]dz<0\ \mbox{for all}\ n\geq 1\\
		&&\hspace{5cm}(\mbox{see hypothesis}\ H(a)(iv)).\nonumber
	\end{eqnarray}
	
	From (\ref{eq57}), as in the proof of Proposition \ref{prop6}, using hypothesis $H_2(iv)$, we show that $\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)$ is bounded. So, we may assume that
	\begin{eqnarray}\label{eq58}
		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}
	
	Since $u_n\in S(\lambda)$ for all $n\geq 1$, we have
	\begin{eqnarray}\label{eq59}
		\left\langle A(u_n),h\right\rangle+\int_{\partial\Omega}\beta(z)u_n^{p-1}hd\sigma-\int_{\Omega}f(z,u_n,\lambda_n)hdz=0\ \mbox{for all}\ h\in W^{1,p}(\Omega).
	\end{eqnarray}
	
	Choosing $h=u_n-u_*\in W^{1,p}(\Omega)$ in (\ref{eq59}), passing to the limit as $n\rightarrow\infty$ and using (\ref{eq58}), we obtain
	\begin{eqnarray}\label{eq60}
		&&\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).
	\end{eqnarray}
	
	So, if in (\ref{eq59}) we pass to the limit as $n\rightarrow\infty$ and use (\ref{eq60}), then
	\begin{eqnarray*}
		&&\left\langle A(u_*),h\right\rangle+\int_{\partial\Omega}\beta(z)u_*^{p-1}hd\sigma=\int_{\Omega}f(z,u_*,\lambda_*)hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega),\\
		&\Rightarrow&u_*\geq 0\ \mbox{is a solution of problem}\ (P_{\lambda^*}).
	\end{eqnarray*}
	
	From Proposition \ref{prop14} we have $\bar{u}\leq u_n$ for all $n\geq 1$. Hence $\bar{u}\leq u_*$ and so $u_*\in S(\lambda^*)\subseteq \mbox{int}\, C_+$. Therefore $\lambda^*\in\mathcal{S}$ and so $\mathcal{S}=\left(0,\lambda^*\right]$.
\end{proof}

Summarizing the situation for problem $(P_{\lambda})$, we can state the following\\ bifurcation-type result.
\begin{theorem}\label{th16}
	If hypotheses $H(a),\ H(\beta)$ and $H_2$ hold, then there exists $\lambda^*>0$ such that
	\begin{itemize}
		\item[(a)] for all $\lambda\in(0,\lambda^*)$, problem $(P_{\lambda})$ has at least two positive solutions\\
		$u_0,\hat{u}\in {\rm int}\, C_+,\ u_0\leq\hat{u},\ u_0\neq\hat{u};$
		\item[(b)] for $\lambda=\lambda^*$ problem $(P_{\lambda^*})$ has at least one positive solution $u_*\in {\rm int}\, C_+$;
		\item[(c)] for all $\lambda>\lambda^*$ problem $(P_{\lambda})$ has no positive solution.
	\end{itemize}
\end{theorem}

\section{Bifurcation Near Infinity for the Neumann Problem}

In this section we deal with the Neumann problem (that is, $\beta\equiv 0$) and with a parametric reaction of the form
$$f(z,x,\lambda)=f_0(z,x)-\lambda x^{p-1}\ \mbox{for all}\ (z,x)\in\Omega\times\left[0,\infty\right).$$

Here $f_0$ is a Carath\'eodory function which as before exhibits competing nonlinearities, namely it is $(p-1)$-superlinear near $+\infty$ and admits a concave term near zero. This time the superlinearity of $f(z,\cdot)$ is expressed via the $AR$-condition. The presence of the term $-\lambda x^{p-1}$ changes the geometry of the problem and hypotheses $H_1$ and $H_2$ do not hold anymore. In fact, we will show that in this case the bifurcation occurs at large values of the parameter $\lambda>0$ (bifurcation near infinity).

The problem under consideration, is the following:
\begin{equation}
	\left\{\begin{array}{ll}
		-\mbox{div}\, a(Du(z))=f_0(z,u(z))-\lambda u(z)^{p-1}&\mbox{in}\ \Omega,\\
		\frac{\partial u}{\partial n}(z)=0\ \mbox{on}\ \partial\Omega,\ u>0&
	\end{array}
	\right\}\tag{$S_{\lambda}$} \label{eqS'}
\end{equation}

For the differential operator, we keep hypotheses $H(a)$ as in Section \ref{sec3}. On the nonparametric nonlinearity $f_0(z,x)$, we impose the following conditions:

\smallskip
\textbf{$H_3:$} $f_0:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carath\'eodory function such that  $f_0(z,0)=0$ for a.a. $z\in\Omega$ and
\begin{itemize}
	\item[(i)] $|f_0(z,x)|\leq a(z)(1+x^{r-1})$ for a.a. $z\in\Omega$, all $x\geq 0$, with $a\in L^{\infty}(\Omega)_+$, $p<r<p^*$;
	\item[(ii)] if $F_0(z,x)=\int^{x}_{0}f_0(z,s)ds$, then there exist $c_{19}>0$ and $\eta>p$ such that
	$$c_{19}x^{\eta}\leq \eta F_0(z,x)\leq f_0(z,x)x\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\geq 0;$$
	\item[(iii)] there exists $q\in(1,\tau)$ (see hypothesis $H(a)(v)$) such that
	$$0<c_{20}\leq\liminf\limits_{x\rightarrow 0^+}\frac{f_0(z,x)}{x^{q-1}}\leq\limsup\limits_{x\rightarrow 0^+}\frac{f_0(z,x)}{x^{q-1}}\leq c_{21}<\infty\ \mbox{uniformly for a.a.}\ z\in\Omega;$$
	\item[(iv)] for every $\rho>0$, there exists $\xi_{\rho}>0$ such that  for a.a. $z\in\Omega$, $x\longmapsto f_0(z,x)+\xi_{\rho}x^{p-1}$ is nondecreasing on $[0,\rho]$.
\end{itemize}
\begin{remark}
	As in Section \ref{sec3}, without any loss of generality, we may assume that $f_0(z,x)=0$ for all $(z,x)\in\Omega\times\left(-\infty,0\right]$. Hypotheses $H_3(ii),(iii)$ reveal the competing nonlinearities (concave-convex nonlinearities). Observe that in this case the superlinearity of $f_0(z,\cdot)$ is expressed using a global version of the unilateral $AR$-condition.
\end{remark}
\begin{example}
	The model for the nonlinearity $f_0(z,\cdot)$, is the function
	$$f_0(z,x)=f_0(x)=x^{q-1}+x^{r-1}\ \mbox{for all}\ x\geq 0$$
	with $1<q<\tau<p<r<p^*.$
\end{example}

As before, we introduce the following two sets
\begin{eqnarray*}
	&&\mathcal{S}_0=\{\lambda>0:\mbox{problem \eqref{eqS'} admits a positive solution}\}\\
	&&S_0(\lambda)=\mbox{the set of positive solutions of problem \eqref{eqS'}}.
\end{eqnarray*}
\begin{proposition}\label{prop17}
	Assume that hypotheses $H(a)$ and $H_3$ hold. Then $\mathcal{S}_0\neq\varnothing$ and for all $\lambda>0$, $S_0(\lambda)\subseteq {\rm int}\, C_+$ and for $\lambda\in\mathcal{S}_0$, we have $\left[\lambda,+\infty\right)\subseteq\mathcal{S}_0.$
\end{proposition}
\begin{proof}
	We consider the following auxiliary Neumann problem
	\begin{equation}\label{eq61}
		-\mbox{div}\, a(Du(z))+|u(z)|^{p-2}u(z)=1\ \mbox{in}\ \Omega,\ \frac{\partial u}{\partial n}=0\ \mbox{on}\ \partial\Omega.
	\end{equation}
	
	Let $K_p:L^p(\Omega)\rightarrow L^{p'}(\Omega)$ $\left(\frac{1}{p}+\frac{1}{p'}=1\right)$ be the nonlinear map defined by
	$$K_p(u)(\cdot)=|u(\cdot)|^{p-2}u(\cdot)$$
	
	This is bounded (maps bounded sets to bounded sets) and continuous. Moreover, by the Sobolev embedding theorem $\hat{K}_p=K_p|_{W^{1,p}(\Omega)}$ is completely continuous (that is, if $u_n\stackrel{w}{\rightarrow} u$ in $W^{1,p}(\Omega)$, then $\hat{K}_p(u_n)\rightarrow K_p(u)$ in $L^{p'}(\Omega)$). So, the map $u\longmapsto V(u)=A(u)+\hat{K}_p(u)$ is demicontinuous and of type $(S)_+$, hence pseudomonotone (see \cite{9}). Moreover, we have
	\begin{eqnarray*}
		&&\left\langle V(u),u\right\rangle\geq \frac{c_1}{p-1}||Du||^{p}_{p}+||u||^{p}_{p}\ \mbox{for all}\ u\in W^{1,p}(\Omega)\ (\mbox{see Lemma \ref{lem2}}),\\
		&\Rightarrow&V\ \mbox{is coercive}.
	\end{eqnarray*}
	
	But a pseudomonotone coercive operator is surjective (see, for example, Gasinski and Papageorgiou \cite{9} [p. 336]). So, we can find $\bar{u}\in W^{1,p}(\Omega)$ such that
	\begin{eqnarray}\label{eq62}
		V(\bar{u})+\hat{K}_p(\bar{u})=1.
	\end{eqnarray}
	
	In fact, using Lemma \ref{lem2} and the strict monotonicity of the map $x\longmapsto |x|^{p-2}x,$\\ $x\in\mathbb{R}$, we see that $V$ is strictly monotone and so $\bar{u}\in W^{1,p}(\Omega)$ is unique. Acting on (\ref{eq62}) with $-\bar{u}^-\in W^{1,p}(\Omega)$ and using Lemma \ref{lem2}, we have
	\begin{eqnarray*}
		&&\frac{c_1}{p-1}||D\bar{u}^-||^{p}_{p}+||\bar{u}^-||^{p}_{p}\leq 0,\\
		&\Rightarrow&\bar{u}\geq 0,\ \bar{u}\neq 0
	\end{eqnarray*}
	
	So, $\bar{u}\geq 0$ is the unique solution of the auxiliary problem (\ref{eq61}) and as before the nonlinear regularity theory (see \cite{15}) and the nonlinear maximum principle (see \cite{19}), imply $\bar{u}\in \mbox{int}\, C_+$.
	
	Let $0<\bar{m}=\min\limits_{\overline{\Omega}}\bar{u}$ and let $\lambda_0=1+\frac{||N_{f_0}(\bar{u})||_{\infty}}{\bar{m}^{p-1}}$. We have
	\begin{eqnarray}\label{eq63}
		&&\left\langle A(\bar{u}),h\right\rangle+\lambda_0\int_{\Omega}\bar{u}^{p-1}hdz\nonumber\\
		&=&\left\langle A(\bar{u}),h\right\rangle+\int_{\Omega}\bar{u}^{p-1}hdz+||N_{f_0}(\bar{u})||_{\infty}\int_{\Omega}\frac{\bar{u}^{p-1}}{\bar{m}^{p-1}}hdz\nonumber\\
		&=&\int_{\Omega}hdz+\int_{\Omega}f_0(z,\bar{u})hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega)\ \mbox{with}\ h\geq 0\\
		&&\hspace{3cm}(\mbox{see (\ref{eq61}) and recall}\ \bar{m}=\min\limits_{\overline{\Omega}}\bar{u}>0).\nonumber
	\end{eqnarray}
	
	We introduce the following truncation of $f_0(z,\cdot)$:
	\begin{eqnarray}\label{eq64}
		\hat{f}_0(z,x)=\left\{
		\begin{array}{cl}
			0&\mbox{if}\ x<0\\
			f_0(z,x)&\mbox{if}\ 0\leq x\leq\bar{x}(z)\\
			f_0(z,\bar{u}(z))&\mbox{if}\ \bar{u}(z)<x.
		\end{array}
		\right.
	\end{eqnarray}
	
	This is a Carath\'eodory function. We set $\hat{F}_0(z,x)=\int^{x}_{0}\hat{f}_0(z,s)ds$ and consider the $C^1$-conditional $\hat{\varphi}_0:W^{1,p}(\Omega)\rightarrow\mathbb{R}$ defined by
	 $$\hat{\varphi}_0(u)=\int_{\Omega}G(Du)dz+\frac{\lambda_0}{p}||u||^{p}_{p}-\int_{\Omega}\hat{F}_0(z,u)dz\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$
	
	From (\ref{eq64}) it is clear that $\hat{\varphi_0}$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find $u_0\in W^{1,p}(\Omega)$ such that
	\begin{eqnarray}\label{eq65}
		\hat{\varphi}_0(u_0)=\inf[\hat{\varphi}_0(u):u\in W^{1,p}(\Omega)].
	\end{eqnarray}
	
	Using hypothesis $H_3(iii)$ and since $1<q<p$, we have that for $\xi\in(0,\min\{1,\bar{m}\})$ small
	\begin{eqnarray*}
		&&\hat{\varphi}_0(\xi)<0=\hat{\varphi}_0(0),\\
		&\Rightarrow&\hat{\varphi}_0(u_0)<0=\hat{\varphi}_0(0)\ (\mbox{see (\ref{eq65})}),\ \mbox{hence}\ u_0\neq 0.
	\end{eqnarray*}
	
	From (\ref{eq65}) we have
	\begin{eqnarray}\label{eq66}
		&&\hat{\varphi}'_0(u_0)=0,\nonumber\\
		&\Rightarrow&A(u_0)+\lambda_0|u_0|^{p-2}u_0=N_{\hat{f}_0}(u_0)\ \mbox{in}\ W^{1,p}(\Omega)^*.
	\end{eqnarray}
	
	On (\ref{eq66}) we act with $-u_0^-\in W^{1,p}(\Omega)$. Then
	\begin{eqnarray*}
		&&\frac{c_1}{p-1}||Du_0^-||^{p}_{p}+\lambda_0||u_0^-||^{p}_{p}\leq 0\ (\mbox{see Lemma \ref{lem2} and (\ref{eq64})}),\\
		&\Rightarrow&u_0\geq 0,\ u_0\neq 0.
	\end{eqnarray*}
	
	Next on (\ref{eq66}) we act with $(u_0-\bar{u})^+\in W^{1,p}(\Omega)$. Then
	\begin{eqnarray*}
		&&\left\langle A(u_0),(u_0-\bar{u})^+\right\rangle+\lambda_0\int_{\Omega}u_0^{p-1}(u_0-\bar{u})^+dz\\
		&=&\int_{\Omega}f_0(z,\bar{u})(u_0-\bar{u})^+dz\ (\mbox{see (\ref{eq64})})\\
		&\leq&\left\langle A(\bar{u}),(u_0-\bar{u})^+\right\rangle+\lambda_0\int_{\Omega}\bar{u}^{p-1}(u_0-\bar{u})^+dz\ (\mbox{see (\ref{eq63})})\\
		\Rightarrow&&\left\langle A(u_0)-A(\bar{u}),(u_0-\bar{u})^+\right\rangle+\lambda_0\int_{\Omega}(u_0^{p-1}-\bar{u}^{p-1})(u_0-\bar{u})^+dz\leq 0,\\
		\Rightarrow&&|\{u_0>\bar{u}\}|_N=0,\ \mbox{hence}\ u_0\leq\bar{u}.
	\end{eqnarray*}
	
	So, we have proved that
	$$u_0\in[0,\bar{u}]=\{u\in W^{1,p}(\Omega):0\leq u(z)\leq\bar{u}(z)\ \mbox{a.e. in}\ \Omega\}.$$
	
	Then by virtue of (\ref{eq64}), equation (\ref{eq66}) becomes
	\begin{eqnarray*}
		&&A(u_0)+\lambda_0u_0^{p-1}=N_{f_0}(u_0),\\
		&\Rightarrow&u_0\ \mbox{is a positive solution of}\ (P_{\lambda_0})\ \mbox{and so}\ \lambda_0\in\mathcal{S}_0\neq\varnothing
	\end{eqnarray*}
	
	Moreover, as before the nonlinear regularity theory and the nonlinear maximum principle imply that $u_0\in \mbox{int}\, C_+$. Therefore, for every $\lambda\in\mathcal{S}_0$ $S_0(\lambda)\subseteq \mbox{int}\, C_+$.
	
	Next, let $\lambda\in\mathcal{S}_0$ and $\mu>\lambda$. Let $u_{\lambda}\in S_0(\lambda)\subseteq \mbox{int}\, C_+$. Then
	\begin{eqnarray}\label{eq67}
		&&\left\langle A(u_{\lambda}),h\right\rangle+\mu\int_{\Omega}u_{\lambda}^{p-1}hdz\geq\left\langle A(u_{\lambda}),h\right\rangle+\lambda\int_{\Omega}u_{\lambda}^{p-1}hdz\\
		&&\hspace{4cm}\mbox{for all}\ h\in W^{1,p}(\Omega)\ \mbox{with}\ h\geq 0.\nonumber
	\end{eqnarray}
	
	Then truncating $f_0(z,\cdot)$ at $\{0,u_{\lambda}(z)\}$ and reasoning as above, via the direct method and using this time (\ref{eq67}), we obtain $u_{\mu}\in[0,u_{\lambda}]\cap S_0(\mu)$, hence $\mu\in\mathcal{S}_0$. Therefore we infer that $\left[\lambda,+\infty\right)\subseteq\mathcal{S}_0$.
\end{proof}
\begin{remark}
	Note that in the above proof we have proved that, if $\lambda\in\mathcal{S}_0,\ u_{\lambda}\in S_0(\lambda)\subseteq {\rm int}\, C_+$ and $\mu>\lambda$, then $\mu\in\mathcal{S}_0$ and we can find $u_{\mu}\in S_0(\mu)\subseteq {\rm int}\, C_+$ such that  $u_{\mu}\leq u_{\lambda}$. In fact in the next proposition, we improve this conclusion.
\end{remark}
\begin{proposition}\label{prop18}
	If hypotheses $H(a)$ and $H_3$ hold, $\lambda\in\mathcal{S}_0,\ u_{\lambda}\in S(\lambda)\subseteq {\rm int}\, C_+$ and $\mu>\lambda$, then we can find $u_{\mu}\in S_0(\mu)\subseteq {\rm int}\, C_+$ such that  $u_{\lambda}-u_{\mu}\in {\rm int}\, C_+$.
\end{proposition}
\begin{proof}
	From Proposition \ref{prop17}, we already know that we can find $u_{\mu}\in S_0(\mu)\subseteq {\rm int}\, C_+$ such that  $u_{\mu}\leq u_{\lambda}$.
	
	Let $m_{\mu}=\min\limits_{\overline{\Omega}}u_{\mu}>0$ and let $\delta\in\left(0,\frac{m_{\mu}}{2}\right)$. We set $u^{\delta}_{\lambda}=u_{\lambda}-\delta\in \mbox{int}\, C_+$. Also, for $\rho=||u_{\lambda}||_{\infty}$, we let $\xi_{\rho}>0$ be as postulated by hypothesis $H_3(iv)$. Then
	\begin{eqnarray*}
		&&-\mbox{div}\,a(Du^{\delta}_{\lambda})+(\mu+\xi_{\rho})(u^{\delta}_{\lambda})^{p-1}\\
		&\geq&-\mbox{div}\,a (Du_{\lambda})+(\mu+\xi_{\rho})u_{\lambda}^{p-1}-\chi(\delta)\ \mbox{with}\ \chi(\delta)\rightarrow 0^+\ \mbox{as}\ \delta\rightarrow 0^+\\
	&=&f_0(z,u_{\lambda})+(\mu-\lambda)u_{\lambda}^{p-1}+\xi_{\rho}u_{\lambda}^{p-1}-\chi(\delta)\ (\mbox{since}\ u_{\lambda}\in S_0(\lambda))\\
		&\geq&f_0(z,u_{\mu})+\xi_{\rho}u_{\mu}^{p-1}+(\mu-\lambda)m_{\mu}-\chi(\delta)\\
		&&\hspace{2cm}(\mbox{since}\ m_{\mu}\leq u_{\mu}\leq u_{\lambda}\ \mbox{and use hypothesis}\ H_3(iv))\\
		&\geq&f_0(z,u_{\mu})+\xi_{\rho}u_{\mu}^{p-1}\ \mbox{for}\ \delta>0\ \mbox{small}\\
		&=&-\mbox{div}\, a(Du_{\mu})+(\mu+\xi_{\rho})u_{\mu}^{p-1}\ (\mbox{since}\ u_{\mu}\in S_0(\mu)),\\
		\Rightarrow&&u_{\mu}\leq u^{\delta}_{\lambda},\ \mbox{for all}\ \delta>0\ \mbox{small},\\
		\Rightarrow&&u_{\lambda}-u_{\mu}\in \mbox{int}\, C_+.
	\end{eqnarray*}
\end{proof}

Let $\lambda_*=\inf\mathcal{S}_0$.
\begin{proposition}\label{prop19}
	If hypotheses $H(a)$ and $H_3$ hold, then $\lambda_*>0$.
\end{proposition}
\begin{proof}
	Consider a sequence $\{\lambda_n\}_{n\geq}\subseteq\mathcal{S}_0$ such that  $\lambda_n\downarrow\lambda_*$. We can find a corresponding sequence $\{u_n\}_{n\geq 1}$ such that  $u_n\in S_0(\lambda_n)\subseteq \mbox{int}\, C_+$ for all $n\geq 1$. We claim that $\{u_n\}_{n\geq 1}$ can be chosen to be increasing. To see this, note that since $\lambda_2<\lambda_1$, the function $u_{\lambda_2}\in S_0(\lambda_2)\subseteq \mbox{int}\, C_+$ satisfies
	\begin{eqnarray}\label{eq68}
		&&\left\langle A(u_2),h\right\rangle+\lambda_1\int_{\Omega}u_2^{p-1}hdz\geq\left\langle A(u_2),h\right\rangle+\lambda_2\int_{\Omega}u_2^{p-1}hdz\\
		&&\hspace{4cm}\mbox{for all}\ h\in W^{1,p}(\Omega)\ \mbox{with}\ h\geq 0\nonumber
	\end{eqnarray}
	
	Considering problem $(P_{\lambda_1})$ and truncating $f_0(z,\cdot)$ at $\{0,u_2(z)\}$, via the direct method and using (\ref{eq68}), as in the proof of Proposition \ref{prop17}, we obtain $u_1\in[0,u_2]\cap S_0(\lambda_1)$.
	
	Then we have
	\begin{eqnarray}\label{eq69}
		&&\left\langle A(u_1),h\right\rangle+\lambda_2\int_{\Omega}u_1^{p-1}hdz\nonumber\\
		&=&\int_{\Omega}f_0(z,u_1)hdz+(\lambda_2-\lambda_1)\int_{\Omega}u_1^{p-1}hdz\ (\mbox{since}\ u_1\in S_0(\lambda_1))\nonumber\\
		&\leq&\int_{\Omega}f_0(z,u_1)hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega)\ \mbox{with}\ h\geq 0\ (\mbox{recall}\ \lambda_1>\lambda_2)
	\end{eqnarray}
	
	Also, we have
	\begin{eqnarray}\label{eq70}
		&&\left\langle A(u_3),h\right\rangle+\lambda_2\int_{\Omega}u_3^{p-1}hdz\geq\left\langle A(u_3),h\right\rangle+\lambda_3\int_{\Omega}u_3^{p-1}hdz\\
		&&\hspace{4cm}\mbox{for all}\ h\in W^{1,p}(\Omega)\ \mbox{with}\ h\geq 0.\nonumber
	\end{eqnarray}
	
	Truncating $f_0(z,\cdot)$ at $\{u_1(z),u_3(z)\}$ and using the direct method and (\ref{eq69}), (\ref{eq70}), we produce $u_2\in[u_1,u_3]\cap S_0(\lambda_2)$. Continuing this way, we see that we can choose $\{u_n\}_{n\geq 1}$ to be increasing.
	
	We have
	$$\left\langle A(u_n),h\right\rangle+\lambda_n\int_{\lambda}u_n^{p-1}hdz=\int_{\Omega}f_0(z,u_n)hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega),\ \mbox{all}\ n\geq 1.$$
	
	Choose $h\equiv 1\in W^{1,p}(\Omega)$. Then for all $n\geq 1$ we have
	\begin{eqnarray*}
		&&\lambda_n\int_{\Omega}u_n^{p-1}dz=\int_{\Omega}f_0(z,u_n)dz,\\
		&\Rightarrow&\lambda_n||u_n||^{p-1}_{p-1}\geq c_{22}||u_n||^{\eta-1}_{p-1}\ \mbox{for some}\ c_{22}>0,\\
		&&\hspace{4cm}(\mbox{see hypothesis}\ H_3(ii)\ \mbox{and recall}\ p<\eta),\\
		&\Rightarrow&\lambda_n\geq c_{22}||u_1||^{\eta-p}_{p-1}\ (\mbox{recall}\ u_n\geq u_1,\ \mbox{for all}\ n\geq 1),\\
		&\Rightarrow&\lambda_*\geq c_{22}||u_1||^{\eta-p}_{p-1}>0.
	\end{eqnarray*}
\end{proof}
\begin{proposition}\label{prop20}
	If hypotheses $H(a)$ and $H_3$ hold and $\lambda>\lambda_*$ then problem \eqref{eqS'} admits at least two positive solutions
	$$u_{\lambda},\hat{u}_{\lambda}\in {\rm int}\, C_+,\ u_{\lambda}\neq\hat{u}_{\lambda}.$$
\end{proposition}
\begin{proof}
	Let $\lambda_*<\lambda_1<\lambda<\lambda_2$. We know that we can find $u_{\lambda_1}\in S_0(\lambda_1)\subseteq \mbox{int}\, C_+$ and $u_{\lambda_2}\in S_0(\lambda_2)\subseteq \mbox{int}\, C_+$ such that  $u_{\lambda_1}-u_{\lambda_2}\in \mbox{int}\, C_+$ (see Proposition \ref{prop18}). We introduce the Carath\'eodory function $e(z,x)$ defined by
	\begin{eqnarray}\label{eq71}
		e(z,x)=\left\{
		\begin{array}{ll}
			f_0(z,u_{\lambda_2}(z))&\mbox{if}\ x<u_{\lambda_2}(z)\\
			f_0(z,x)&\mbox{if}\ u_{\lambda_2}(z)\leq x\leq u_{\lambda_1}(z)\\
			f_0(z,u_{\lambda_1}(z))&\mbox{if}\ u_{\lambda_1}(z)<x.
		\end{array}
		\right.
	\end{eqnarray}
	
	We set $E(z,x)=\int^{x}_{0}e(z,s)ds$ and consider the $C^1$-functional $\psi_{\lambda}:W^{1,p}(\Omega)\rightarrow\mathbb{R}$ defined by
	$$\psi_{\lambda}(u)=\int_{\Omega}G(Du)dz+\frac{\lambda}{p}||u||^{p}_{p}-\int_{\Omega}E(z,u)dz\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$
	
	Evidently $\psi_{\lambda}$ is coercive (see (\ref{eq71})) and sequentially weakly lower semicontinuous. So, we can find $u_{\lambda}\in W^{1,p}(\Omega)$ such that
	\begin{eqnarray}\label{eq72}
		&&\psi_{\lambda}(u_{\lambda})=\inf[\psi_{\lambda}(u):u\in W^{1,p}(\Omega)],\nonumber\\
		&\Rightarrow&\psi'_{\lambda}(u_{\lambda})=0,\nonumber\\
		&\Rightarrow&\left\langle A(u_{\lambda}),h\right\rangle+\lambda\int_{\Omega}|u_{\lambda}|^{p-2}u_{\lambda}hdz=\int_{\Omega}e(z,u_{\lambda})hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega).
	\end{eqnarray}
	
	If in (\ref{eq72}) we use $h=(u_{\lambda}-u_{\lambda_1})^+\in W^{1,p}(\Omega)$ and $h=(u_{\lambda_2-u_{\lambda}})^+\in W^{1,p}(\Omega)$, we show that $u_{\lambda}\in[u_{\lambda_2},u_{\lambda_1}]=\{u\in W^{1,p}(\Omega):u_{\lambda_2}(z)\leq u(z)\leq u_{\lambda_1}(z)\ \mbox{a.e. in}\ \Omega\}$. In fact reasoning as in the proof of Proposition \ref{prop18}, we obtain
	\begin{eqnarray}\label{eq73}
		u_{\lambda}\in \mbox{int}_{C'(\overline{\Omega})}[u_{\lambda_2},u_{\lambda_1}].
	\end{eqnarray}
	
	From (\ref{eq71}) and (\ref{eq72}) we have
	\begin{eqnarray*}
		&&\left\langle A(u_{\lambda}),h\right\rangle+\lambda\int_{\Omega}u_{\lambda}^{p-1}hdz=\int_{\Omega}f_0(z,u_{\lambda})hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega),\\
		&\Rightarrow&u_{\lambda}\in S_0(\lambda)\subseteq \mbox{int}\, C_+.
	\end{eqnarray*}
	
	Next, using $u_{\lambda_2}$, we introduce the following truncation of $f_0(z,\cdot)$:
	\begin{eqnarray}\label{eq74}
		\gamma(z,x)=\left\{
		\begin{array}{ll}
			f_0(z,u_{\lambda_2}(z))&\mbox{if}\ x\leq u_{\lambda_2}(z)\\
			f_0(z,x)&\mbox{if}\ u_{\lambda_2}(z)<x.
		\end{array}
		\right.
	\end{eqnarray}
	
	This is a Carath\'eodory function. Let $\Gamma(z,x)=\int^{x}_{0}\gamma(z,s)ds$ and consider the $C^1$-functional $\sigma_{\lambda}:W^{1,p}(\Omega)\rightarrow\mathbb{R}$ defined by
		 $$\sigma_{\lambda}(u)=\int_{\Omega}G(Du)dz+\frac{\lambda}{p}||u||^{p}_{p}-\int_{\Omega}\Gamma(z,u)dz\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$
		
	Hypothesis $H_3(ii)$ implies that the $AR$-condition (see (\ref{eq5})) is satisfied by $\gamma(z,x)$. It follows that
	\begin{eqnarray}\label{eq75}
		\sigma_{\lambda}\ \mbox{satisfied the}\ C-\mbox{condition}.
	\end{eqnarray}
	
	Note that
	 $$\psi_{\lambda}|_{[u_{\lambda_2},u_{\lambda_1}]}=\sigma_{\lambda}|_{[u_{\lambda_2},u_{\lambda_1}]}\ (\mbox{see (\ref{eq71}) and (\ref{eq74})}).$$
	
	From this equality and (\ref{eq73}), we infer that $u_{\lambda}$ is a local $C^1(\overline{\Omega})$ minimizer of $\sigma_{\lambda}$, hence $u_{\lambda}$ is a local $W^{1,p}(\Omega)$-minimizer of $\sigma_{\lambda}$ (see Proposition \ref{prop4}). We can easily check that the critical points of $\sigma_{\lambda}$ are in $\left[u_{\lambda_2}\right)=\{u\in W^{1,p}(\Omega):u_{\lambda_2}(z)\leq u(z)\ \mbox{for a.a}\ z\in\Omega\}$. So, we may assume that the critical points are finite or otherwise we already have an infinity of positive solutions for problem \eqref{eqS'} (see (\ref{eq74})) and so we are done. Then we can find $\rho\in\left(0,1\right]$ small such that
	\begin{eqnarray}\label{eq76}
		 \sigma_{\lambda}(u_{\lambda})<\inf\left[\sigma_{\lambda}(u):||u-u_{\lambda}||=\rho\right]=m^{\lambda}_{\rho}
	\end{eqnarray}
	
	Hypothesis $H_3(ii)$ and (\ref{eq74}) imply that
	\begin{eqnarray}\label{eq77}
		\sigma_{\lambda}(\xi)\rightarrow-\infty\ \mbox{as}\ \xi\rightarrow+\infty.
	\end{eqnarray}
	
	Then (\ref{eq75}), (\ref{eq76}) and (\ref{eq77}) permit the use of Theorem \ref{th1} (the mountain pass theorem) and so we can find $\hat{u}_{\lambda}\in W^{1,p}(\Omega)$ such that
	\begin{eqnarray*}
		&&\sigma_{\lambda}(u_{\lambda})<m^{\lambda}_{\rho}\leq\sigma_{\lambda}(\hat{u}_{\lambda}),\ \mbox{hence}\ \hat{u}_{\lambda}\neq u_{\lambda},\\
		&&\sigma'_{\lambda}(\hat{u}_{\lambda})=0.
	\end{eqnarray*}
	
	From the last equality and some $\hat{u}_{\lambda}\geq u_{\lambda_2}$, we infer that $\hat{u}_{\lambda}\in S_0(\lambda)\subseteq \mbox{int}\, C_+$ (see (\ref{eq74})).
\end{proof}

Note that hypotheses $H_3(i),(iii)$ imply that we can find $c_{23},c_{24}>0$ such that
\begin{eqnarray}\label{eq78}
	f_0(z,x)\geq c_{23}x^{q-1}-c_{24}x^{r-1}\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\geq 0.
\end{eqnarray}

This leads to the following auxiliary Neumann problem
\begin{equation}
	-\mbox{div}\,a(Du(z))+\lambda u(z)^{p-1}=c_{23}u(z)^{q-1}-c_{24}u(z)^{r-1}\ \mbox{in}\ \Omega,\ \frac{\partial u}{\partial n}=0\ \mbox{on}\ \partial\Omega,\ u>0\tag{$Q_{\lambda}$} \label{eqQ}
\end{equation}

Reasoning as in the proof of Proposition \ref{prop13}, we have the following existence and uniqueness result for problem \eqref{eqQ}.
\begin{proposition}\label{prop21}
	If hypotheses $H(a)$ hold and $\lambda>0$ then problem \eqref{eqQ} admits a unique positive solution $u^*_{\lambda}\in {\rm int}\, C_+$.
\end{proposition}

In fact the map $\lambda\rightarrow u^*_{\lambda}$ has useful monotonicity and continuity properties.
\begin{proposition}\label{prop22}
	If hypotheses $H(a)$ hold, then $\lambda\rightarrow u^*_{\lambda}$ is nonincreasing and continuous from $(0,\infty)$ into $C^1(\overline{\Omega})$.
\end{proposition}
\begin{proof}
	The monotonicity of $\lambda\rightarrow u^*_{\lambda}$ is established as in the first part of the proof of Proposition \ref{prop19}, by exploiting the uniqueness result of Proposition \ref{prop21}.
	
	Next, let $\lambda_n\rightarrow\lambda>0$ and let $0<\hat{\lambda}<\lambda_n$ for all $n\geq 1$. Then $u^*_{\lambda_n}\leq u^*_{\hat{\lambda}}$ for all $n\geq 1$. Also, we have
	\begin{eqnarray}\label{eq79}
		 A(u^*_{\lambda_n})+\lambda_n(u^*_{\lambda_n})^{p-1}=c_{23}(u^*_{\lambda_n})^{q-1}-c_{24}(u^*_{\lambda_n})^{r-1}\ \mbox{for all}\ n\geq 1.
	\end{eqnarray}
	
	Since $u^*_{\lambda_n}\leq u^*_{\hat{\lambda}}\in \mbox{int}\, C_+$, from Lieberman \cite{15}, we know that we can find $\vartheta\in(0,1)$ and $M_6>0$ such that
	$$u^*_{\lambda_n}\in C^{1,\vartheta}(\overline{\Omega}),\ ||u^*_{\lambda_{n}}||_{C^{1,\vartheta}(\overline{\Omega})}\leq M_6\ \mbox{for all}\ n\geq 1$$
	
	Exploiting the compact embedding of $C^{1,\vartheta}(\overline{\Omega})$ into $C^1(\overline{\Omega})$, we may assume that
	\begin{eqnarray}\label{eq80}
		u^*_{\lambda_n}\rightarrow u^*\ \mbox{in}\ C^1(\overline{\Omega}).
	\end{eqnarray}
	
	So, if in (\ref{eq79}) we pass to the limit as $n\rightarrow\infty$ and we use (\ref{eq80}), then
	\begin{eqnarray*}
		&&A(u^*)+\lambda(u^*)^{p-1}=c_{23}(u^*)^{p-1}-c_{24}(u^*)^{r-1},\\
		&\Rightarrow&u^*=u^*_{\lambda}\ (\mbox{see Proposition \ref{prop21}}).
	\end{eqnarray*}
	
	This proves the desired continuity of $\lambda\rightarrow u^*_{\lambda}$.
\end{proof}

Moreover, as in Proposition \ref{prop14}, we have:
\begin{proposition}\label{prop23}
	If hypotheses $H(a)$ hold and $\lambda>0$, then $u^*_{\lambda}\leq u$ for all $u\in S_0(\lambda)$.
\end{proposition}

Using these facts we can treat the critical case $\lambda=\lambda_*>0$. In what follows for $\lambda>0$, $\varphi_{\lambda}:W^{1,p}(\Omega)\rightarrow\mathbb{R}$ is the energy functional for problem \eqref{eqS'} defined by
$$\varphi_{\lambda}(u)=\int_{\Omega}G(Du)dz+\frac{\lambda}{p}||u||^{p}_{p}-\int_{\Omega}F_0(z,u)dz\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$

Evidently $\varphi_{\lambda}\in C^1(W^{1,p}(\Omega))$.
\begin{proposition}\label{prop24}
	If hypotheses $H(a)$ and $H_3$ hold, then $\lambda_*\in\mathcal{S}_0$ (that is, $\mathcal{S}=\left[\lambda_*,\infty\right)$).
\end{proposition}
\begin{proof}
	Let $\{\lambda_n\}_{n\geq 1}\subseteq\mathcal{S}_0$ such that  $\lambda_N\downarrow\lambda_*$. There exists a corresponding sequence $\{u_n\}_{n\geq 1}$ such that  $u_n\in S_0(\lambda_n)$ for all $n\geq 1$. We claim that this sequence of solutions can be chosen so that
	\begin{eqnarray}\label{eq81}
		\varphi_{\lambda_n}(u_n)<0\ \mbox{for all}\ n\geq 1.
	\end{eqnarray}
	
	To see this note that
	\begin{eqnarray}\label{eq82}
		&&\left\langle A(u_2),h\right\rangle+\lambda_1\int_{\Omega}u_2^{p-1}hdz\nonumber\\
		&\geq&\left\langle A(u_2),h\right\rangle+\lambda_2\int_{\Omega}u_2^{p-1}hdz=\int_{\Omega}f_0(z,u_2)hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega)\\
		&&\hspace{8cm}\mbox{with}\ h\geq 0\nonumber.
	\end{eqnarray}
	
	Truncating $f_0(z,\cdot)$ at $\{0,u_2(z)\}$ and reasoning as in the proof of Proposition \ref{prop17} via the direct method and using hypothesis $H_2(iii)$ and (\ref{eq82}), we obtain $u_1\in S_0(\lambda_1)$ such that
	$$\varphi_{\lambda_1}(u_1)<0$$
	
	Next note that
	\begin{eqnarray}\label{eq83}
		&&\left\langle A(u_3),h\right\rangle+\lambda_2\int_{\Omega}u_3^{p-1}hdz\nonumber\\
		&\geq&\left\langle A(u_3),h\right\rangle+\lambda_3\int_{\Omega}u_3^{p-1}hdz=\int_{\Omega}f_0(z,u_3)hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega)\\
		&&\hspace{8cm}\mbox{with}\ h\geq 0.\nonumber
	\end{eqnarray}
	
	As above truncating $f_0(z,\cdot)$ at $\{0,u_3(z)\}$ and using this time (\ref{eq83}), we produce $u_2\in S_0(\lambda_2)$ such that
	$$\varphi_{\lambda}(u_2)<0.$$
	
	So, continuing this way, we see that we can have $u_n\in S_0(\lambda_n)$ $n\geq 1$ such that  (\ref{eq81}) holds. Then it follows that
	\begin{eqnarray}\label{eq84}
		\int_{\Omega}\eta G(Du_n)dz+\frac{\lambda_n\eta}{p}||u_n||^{p}_{p}\leq\int_{\Omega}\eta F_0(z,u_n)dz\ \mbox{for all}\ n\geq 1.
	\end{eqnarray}
	
	Also, since $u_n\in S_0(\lambda_n)\ n\geq1$, we have
	\begin{eqnarray}\label{eq85}
		 -\int_{\Omega}(a(Du_n),Du_n)_{\mathbb{R}^N}dz-\lambda_n||u_n||^{p}_{p}=-\int_{\Omega}f_0(z,u_n)u_ndz\ \mbox{for all}\ n\geq 1.
	\end{eqnarray}
	
	Adding (\ref{eq84}) and (\ref{eq85}) and using hypothesis $H(a)(iv)$, we obtain
	\begin{eqnarray*}
		 &&(\eta-p)\int_{\Omega}G(Du_n)dz+\int_{\Omega}\left[f_0(z,u_n)u_n-pF_0(z,u_n)\right]dz+\\
		&&\hspace{2cm}+\lambda_n\left(\frac{\eta}{p}-1\right)||u_n||^{p}_{p}\leq 0\ \ \mbox{for all}\ n\geq 1,\\
		 &\Rightarrow&\frac{(\eta-p)c_1}{p(p-1)}||Du_n||^{p}_{p}+\lambda_*\left(\frac{\eta}{p}-1\right)||u_n||^{p}_{p}\leq 0\ \mbox{for all}\ n\geq 1\\
		&&\hspace{1cm}(\mbox{see Corollary \ref{cor3}, hypothesis}\ H_3(ii)\ \mbox{and recall}\ \lambda_*\leq \lambda_n\ \mbox{for all}\ n\geq 1)\\
		&\Rightarrow&\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)\ \mbox{is bounded}\ (\mbox{recall}\ \eta>p).
	\end{eqnarray*}
	
	So, we may assume that
	\begin{eqnarray}\label{eq86}
		u_n\stackrel{w}{\rightarrow} u_*\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{and}\ u_n\rightarrow u_*\ \mbox{in}\ L^r(\Omega).
	\end{eqnarray}
	
	Recall that
	\begin{eqnarray}\label{eq87}
		\left\langle A(u_n),h\right\rangle+\lambda_n\int_{\Omega}u_n^{p-1}hdz=\int_{\Omega}f_0(z,u_n)hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega),\ \mbox{all}\ n\geq 1.
	\end{eqnarray}
	
	If in (\ref{eq87}) we choose $h=u_n-u_*\in W^{1,p}(\Omega)$, pass to the limit as $n\rightarrow\infty$ and use (\ref{eq86}), then
	\begin{eqnarray}\label{eq88}
		&&\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 Proposition \ref{prop5} and (\ref{eq86})}).
	\end{eqnarray}
	
	Passing to the limit as $n\rightarrow\infty$ in (\ref{eq87}) and using (\ref{eq88}), we obtain
	\begin{eqnarray}\label{eq89}
		\left\langle A(u_*),h\right\rangle+\lambda_*\int_{\Omega}u_*^{p-1}hdz=\int_{\Omega}f_0(z,u_*)hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega).
	\end{eqnarray}
	
	From Proposition \ref{prop23}, we have
	\begin{eqnarray}\label{eq90}
		&&u^*_{\lambda_n}\leq u_n\ \mbox{for all}\ n\geq 1,\nonumber\\
		&\Rightarrow&u^*_{\lambda_*}\leq u_*\ \mbox{(see Proposition \ref{prop22} and (\ref{eq88}))}.
	\end{eqnarray}
	
	From (\ref{eq89}) and (\ref{eq90}) it follows that $u_*\in S_0(\lambda_*)$, hence $\lambda_*\in\mathcal{S}_0.$
\end{proof}

Summarizing, for problem \eqref{eqS'} we have the following bifurcation-type result.
\begin{theorem}\label{th25}
	If hypotheses $H(a)$ and $H_3$ hold, then exists $\lambda_*>0$ such that
	\begin{itemize}
		\item[(a)] for every $\lambda>\lambda_*$ problem \eqref{eqS'} has at least two positive solutions
		$$u_{\lambda},\hat{u}_{\lambda}\in {\rm int}\, C_+,\ u_{\lambda}\neq\hat{u}_{\lambda};$$
		\item[(b)] for $\lambda=\lambda_*$ problem $(S_{\lambda_*})$ has at least one positive solution $u_*\in {\rm int}\, C_+$;
		\item[(c)] for every $\lambda\in(0,\lambda_*)$ problem \eqref{eqS'} has no positive solution.
	\end{itemize}
\end{theorem}
\begin{remark}
	This bifurcation-type theorem leaves open two interesting questions:
	\begin{itemize}
		\item[(a)] Is it possible in hypothesis $H_3(ii)$ to replace the global $AR$-condition by the usual local one (see (\ref{eq5})) or even better by the more general ``superlinearity" condition used in Section \ref{sec3}? The difficulties can be traced in Proposition \ref{prop19}, which shows that $\lambda_*>0$. It is not clear how this can be proved in the aforementioned two more general settings.
		\item[(b)] Can we have Theorem \ref{th25} for the Robin problem (that is, for $\beta\neq 0,\ \beta\geq 0$)? Again the difficulty is in Proposition \ref{prop19}. The proof of that proposition fails since we can not control the boundary term $\int_{\partial\Omega}\beta(z)u_n^{p-1}d\sigma$.
	\end{itemize}
\end{remark}

Next, we carry the study of problem \eqref{eqS'} ($\lambda\geq\lambda_*$) a little further and produce a smallest positive solution $\hat{w}_{\lambda}\in \mbox{int}\, C_+$ and show that the map $\lambda\longmapsto\hat{w}_{\lambda}$ is strictly decreasing from $(0,\infty)$ into $C^1(\overline{\Omega})$ and $||\hat{w}_{\lambda}||\rightarrow 0$ as $\lambda\rightarrow+\infty$.
\begin{proposition}\label{prop26}
	If hypotheses $H(a)$ and $H_3$ hold and $\lambda\geq \lambda_*$, then problem \eqref{eqS'} admits a smallest positive solution $\hat{w}_{\lambda}\in {\rm int}\, C_+$.
\end{proposition}
\begin{proof}
	As in Filippakis, Kristaly and Papageorgiou \cite{7}, exploiting the monotonicity of $A$ (see Lemma \ref{lem2}), we show that $S_0(\lambda)$ is downward directed (that is, if $u_1,u_2\in S_0(\lambda)$), then we can find $u\in S_0(\lambda)$ such that  $u\leq u_1,\ u\leq u_2$. Therefore, since we are looking for the smallest positive solution, without any loss of generality, we may assume that
	$$u^*_{\lambda}(z)\leq u(z)\leq c_{25}\ \mbox{for some}\ c_{25}>0,\ \mbox{all}\ u\in S_0(\lambda),\ \mbox{all}\ z\in\overline{\Omega}\ (\mbox{see Proposition \ref{prop23}})$$
	
	Then from Dunford and Schwartz \cite{3} [p. 336], we know that we can find\\ $\{u_n\}_{n\geq 1}\subseteq S_0(\lambda)$ such that
	$$\inf S_0(\lambda)=\inf\limits_{n\geq 1} u_n.$$
	
	We have
	\begin{eqnarray}\label{eq91}
		&&A(u_n)+\lambda u_n^{p-1}=N_f(u_n)\ \mbox{and}\ u^*_{\lambda}\leq u_n\leq c_{25}\ \mbox{for all}\ n\geq 1,\\
		&\Rightarrow&\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)\ \mbox{is bounded}.\nonumber
	\end{eqnarray}
	
	Thus we may assume that
	\begin{eqnarray}\label{eq92}
		u_n\stackrel{w}{\rightarrow} \hat{w}_{\lambda}\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{and}\ u_n\rightarrow\hat{w}_{\Lambda}\ \mbox{in}\ L^r(\Omega).
	\end{eqnarray}
	
	On (\ref{eq91}) we act with $u_n-\hat{w}_{\lambda}\in W^{1,p}(\Omega)$, pass to the limit as $n\rightarrow\infty$ and use (\ref{eq92}). Then
	\begin{eqnarray*}
		&&\lim\limits_{n\rightarrow\infty}\left\langle A(u_n),u_n-\hat{w}_{\lambda}\right\rangle=0,\\
		&\Rightarrow&u_n\rightarrow\hat{w}_{\lambda}\ \mbox{in}\ W^{1,p}(\Omega)\ (\mbox{see Proposition \ref{prop5}}),\\
		&\Rightarrow&A(\hat{w}_{\lambda})+\lambda\hat{w}^{p-1}_{\lambda}=N_f(\hat{w}_{\lambda}),\ u^*_{\lambda}\leq \hat{w}_{\lambda}\leq c_{25}\ (\mbox{see (\ref{eq91})}),\\
		&\Rightarrow&\hat{w}_{\lambda}\in S(\lambda)\ \mbox{and}\ \hat{w}_{\lambda}=\inf S_0(\lambda.)
	\end{eqnarray*}
\end{proof}

We examine the map $\lambda\longmapsto\hat{w}_{\lambda}$.
\begin{proposition}\label{prop27}
	If hypotheses $H(a)$ and $H_3$ hold then $\lambda\longmapsto\hat{w}_{\lambda}$ is strictly decreasing from $\left[\lambda_*,\infty\right)$ into $C^1(\overline{\Omega})$, that is, if $\lambda_*\leq\lambda<\mu$, then $\hat{w}_{\lambda}-\hat{w}_{\mu}\in {\rm int}\, C_+$.
\end{proposition}
\begin{proof}
	Note that
	\begin{eqnarray}\label{eq93}
		&&\left\langle A(\hat{w}_{\lambda}),h\right\rangle+\mu\int_{\Omega}\hat{w}^{p-1}_{\lambda}hdz\geq\left\langle A(\hat{w}_{\lambda}),h\right\rangle+\lambda\int_{\Omega}\hat{w}_{\lambda}^{p-1}hdz=\nonumber\\
		&&\hspace{1cm}=\int_{\Omega}f_0(z,\hat{w}_{\lambda})hdz\ \ \mbox{for all}\ h\in W^{1,p}(\Omega)\ \mbox{with}\ h\geq 0.
	\end{eqnarray}
	
	We consider the problem $(S_{\mu})$ and truncate the reaction $f_0(z,\cdot)$ at $\{0,\hat{w}_{\lambda}(z)\}$. Then reasoning as in the proof of Proposition \ref{prop17}, via the direct method and using (\ref{eq93}), we produce $u_{\mu}\in[0,\hat{w}_{\lambda}]\cap S_0(\mu)$, hence $\hat{w}_{\mu}\leq u_{\mu}\leq \hat{w}_{\lambda}$. Moreover, from Proposition \ref{prop18}, we have $\hat{w}_{\lambda}-\hat{w}_{\mu}\in \mbox{int}\, C_+$.
\end{proof}

\begin{proposition}\label{prop28}
	If hypotheses $H(a)$ and $H_3$ hold, then $\hat{w}_{\lambda}\rightarrow 0$ in $W^{1,p}(\Omega)$ as $\lambda\uparrow+\infty$.
\end{proposition}
\begin{proof}
	Let $\{\lambda_n\}_{n\geq 1}\subseteq(\lambda_*,\infty)$ such that  $\lambda_n\uparrow+\infty$. Let $\hat{w}_{\lambda_n}\in S_0(\lambda_n)\subseteq \mbox{int}\, C_+$ be the smallest positive solution of $(S_{\lambda_n})$ $n\geq 1$. From Proposition \ref{prop26}, we know that $\{\hat{w}_{\lambda_n}\}_{n\geq 1}$ is (strictly) decreasing. We have
	\begin{eqnarray}
		&&A(\hat{w}_{\lambda_n})+\lambda_n\hat{w}^{p-1}_{\lambda_n}=N_{f_0}(\hat{w}_{\lambda_n}),\ \hat{w}_{\lambda_n}\leq\hat{w}_1\ \mbox{for all}\ n\geq 1,\label{eq94}\\
		 &\Rightarrow&\frac{c_1}{p-1}||D\hat{w}_{\lambda_n}||^{p}_{p}+\lambda_n||\hat{w}_{\lambda_n}||\leq\int_{\Omega}f_0(z,\hat{w}_{\lambda_n})\hat{w}_{\lambda_n}dz\leq c_{26}\nonumber\\
		&&\hspace{4cm}\mbox{for some}\ c_{26}>0,\ \mbox{all}\ n\geq 1\ (\mbox{see Lemma \ref{lem2}}),\nonumber\\
		&\Rightarrow&\{\hat{w}_{\lambda_n}\}_{n\geq 1}\subseteq W^{1,p}(\Omega)\ \mbox{is bounded}.\label{eq95}
	\end{eqnarray}
	
	From (\ref{eq94}) and (\ref{eq95}) is follows that
	$$\{A(\hat{w}_{\lambda_n})-N_{f_0}(\hat{w}_{\lambda_n})\}_{n\geq 1}=\{\lambda_n\hat{w}^{p-1}_{\lambda_n}\}_{n \geq 1}\subseteq W^{1,p}(\Omega)^*\ \mbox{is bounded}.$$
	
	So, for every $h\in W^{1,p}(\Omega)$ we have
	\begin{eqnarray}\label{eq96}
		\left|\lambda_n\int_{\Omega}\hat{w}_{\lambda_n}^{p-1}hdz\right|\leq c_{27}||h||\ \mbox{for some}\ c_{27}>0,\ \mbox{all}\ n\geq 1.
	\end{eqnarray}
	
	In (\ref{eq96}) we choose $h-\hat{w}_{\lambda_n}$. Then
	\begin{eqnarray}\label{eq97}
		&&\lambda_n||\hat{w}_{\lambda_n}||^{p}_{p}\leq c_{27}||\hat{w}_{\lambda_n}||\leq c_{28}\ \mbox{for some}\ c_{28}>0\ \mbox{all}\ n\geq 1\ (\mbox{see (\ref{eq95})}),\nonumber\\
		&\Rightarrow&||\hat{w}_{\lambda_n}||_p\rightarrow 0\ \mbox{as}\ n\rightarrow\infty\ (\mbox{recall}\ \lambda_n\uparrow\infty).
	\end{eqnarray}
	
	Recall that
	$$ \frac{c_1}{p-1}||D\hat{w}_{\lambda_n}||^{p}_{p}+\lambda_n||\hat{w}_{\lambda_n}||^{p}_{p}\leq\int_{\Omega}f_0(z,\hat{w}_{\lambda_n})\hat{w}_{\lambda_n}dz\rightarrow 0\ \mbox{as}\ n\rightarrow\infty\ (\mbox{see (\ref{eq97})}).$$
We conclude that
		$\hat{w}_{\lambda_n}\rightarrow 0$ in $W^{1,p}(\Omega)$.
\end{proof}

\begin{remark}
	An interesting open question is whether $\lambda\longmapsto\hat{w}_{\lambda}$ is continuous from $\left[\lambda_*,+\infty\right)$ into $C^1(\overline{\Omega})$.
\end{remark}

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