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\title[Perturbations of the eigenvalue problem for the Robin $p$-Laplacian]{Positive solutions for perturbations of the eigenvalue problem for the Robin $p$-Laplacian}


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

\author[V. R\u{a}dulescu]{Vicen\c{t}iu D. R\u{a}dulescu}
\address{Department of Mathematics, Faculty of Science, King Abdulaziz University,
Jeddah, Saudi Arabia}
\email{\tt vicentiu.radulescu@math.cnrs.fr}

\keywords{Robin boundary condition, nonlinear regularity, $(p-1)$-sublinear and $(p-1)$ superlinear perturbation, maximum principle.\\
\phantom{aa} 2010 AMS Subject Classification:
35J66, 35J70, 35J92}

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\begin{abstract}
We study perturbations of the eigenvalue problem for the Robin $p$-Laplacian. First we consider the case of a $(p-1)$-sublinear perturbation and prove existence, nonexistence and uniqueness of positive solutions. Then we deal with the case of a $(p-1)$-superlinear perturbation which need not satisfy the Ambrosetti-Rabinowitz condition and prove a multiplicity result for positive solutions. Our approach uses variational methods together with suitable truncation and perturbation techniques.
\end{abstract}

\maketitle

%\tableofcontents

\section{Introduction}\label{sec1}

Let $\Omega\subseteq \RR^{N}$ be a bounded domain with a $C^2$-boundary $\partial\Omega$. In this paper, we study the following nonlinear parametric Robin problem:
\begin{center}
$\left\{  \begin{array}{ll}
\di    -\Delta_p u(z)=\lambda u(z)^{p-1}+f(z,u(z))\ \mbox{in}\ \Omega,\\
   \di \frac{\partial u}{\partial n_p}+\beta(z)u(z)^{p-1}=0\ \mbox{on}\ \partial\Omega,\ u>0,\ 1<p<\infty.
    \end{array}\right. \hspace{4cm}(P_{\lambda})$
\end{center}
Here $\Delta_p$ denotes the $p$-Laplace differential operator defined by
$$\Delta_p u=\mbox{div}\, (|Du|^{p-2}Du)\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$
Also $\frac{\partial u}{\partial n_p}=|Du|^{p-2}(Du,n)_{\RR^N}$ with $n(z)$ being the outward unit normal at $z\in\partial\Omega$. Moreover, $\lambda\in\RR$ is a parameter and $f(z,x)$ is a Carath\'eodory perturbation (that is, for all $x\in\RR$, the mapping $z\longmapsto f(z,x)$ is measurable on $\Omega$ and for a.a. $z\in\Omega$, $x\longmapsto f(z,x)$ is continuous).

We are interested in the existence, nonexistence and uniqueness of positive solutions for problem $(P_{\lambda})$ as the parameter $\lambda\in\RR$ varies. We can view problem $(P_{\lambda})$ as a perturbation of the classical eigenvalue problem for the Robin $p$-Laplacian, investigated by L\^{e} \cite{12} and Papageorgiou and R\u adulescu \cite{15}. Similar studies concerning positive solutions, were conducted by Brezis and Oswald \cite{5} (for problems driven by the Dirichlet Laplacian) and by Diaz and Saa \cite{6} (for problems driven by the Dirichlet $p$-Laplacian). More recently, Gasinski and Papageorgiou \cite{11} produced analogous results for the Neumann $p$-Laplacian. Multiplicity results concerning perturbed
Robin problems involving the $p$-Laplacian were investigated recently by Winkert \cite{wink}. We also mention the recent work of Papageorgiou and R\u adulescu \cite{15}, who studied a class of parametric equations driven by the Robin $p$-Laplacian and proved multiplicity results with precise sign information for all the solutions produced.

Here we first examine the case where $f(z,\cdot)$ is $(p-1)$-sublinear near $+\infty$, which leads to uniqueness results. Next, we consider the case where $f(z,\cdot)$ is $(p-1)$-superlinear (but without employing the Ambrosetti-Rabinowitz condition), which leads to multiplicity results.

\section{Mathematical Background}\label{sec2}
Our approach uses variational methods based on the critical point theory as well as suitable truncation and perturbation techniques.

So, let $X$ be a Banach space and $X^*$ be its topological dual. By $\langle\cdot,\cdot\rangle$ 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 for short), if the following is true:

``Every sequence $\{u_n\}_{n\geq 1}\subseteq X$ such that  $\{\varphi(u_n)\}_{n\geq 1}\subseteq\RR$ is bounded and
$$(1+\|u_n\|)\varphi'(u_n)\to 0\ \mbox{in}\ X^*\ \mbox{as}\ n\to\infty,$$
admits a strongly convergent subsequence."

This is a compactness type condition on $\varphi$ needed to offset the fact that the space $X$ is not necessarily locally compact (being in general infinite dimensional). It is a basic tool in proving a deformation theorem which in turn leads to a minimax theory for the critical values of $\varphi$. Prominent in this theory, is the so-called ``mountain pass theorem" due to Ambrosetti and Rabinowitz \cite{3}, stated here in a slightly more general form.
\begin{theorem}\label{th1}
Assume that $\varphi\in C^1(X)$ satisfies the $C$-condition, $u_0,u_1\in X$, $\|u_1-u_0\|>\rho>0$,
$$\max\{\varphi(u_0),\varphi(u_1)\}<\inf[\varphi(u):\|u-u_0\|=\rho]=m_{\rho}$$
and $c=\inf\limits_{\gamma\in\Gamma}\max\limits_{0\leq t\leq 1}\varphi(\gamma(t))$ where $\Gamma=\{\gamma\in C([0,1],X):\gamma(0)=u_0,\gamma(1)=u_1\}$. Then $c\geq m_{\rho}$ and $c$ is a critical value of $\varphi$.
\end{theorem}
In the analysis of problem $(P_{\lambda})$, 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 consider the usual norm given by
$$\|u\|=\left[\|u\|_p^p+\|Du\|_p^p\right]^{1/p}\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$
To distinguish, we denote by $|\cdot|$ the Euclidean norm on $\RR^N$. On $\partial\Omega$ we use the $(N-1)$-dimensional surface (Hausdorff) measure. So, we can define the Lebesgue spaces $L^q(\partial\Omega)$, $1\leq q\leq \infty$. We know that there is a unique, continuous linear map $\gamma_0:W^{1,p}(\Omega)\to L^p(\partial\Omega)$, known as the ``trace map", such that $\gamma_0(u)=u|_{\partial\Omega}$ for all $u\in C^1(\overline{\Omega})$. We have $\gamma_0(W^{1,p}(\Omega))=W^{\frac{1}{p'},p}(\partial\Omega)$ $\left(\frac{1}{p}+\frac{1}{p'}=1\right)$ and $\ker\gamma_0=W_0^{1,p}(\Omega)$. In the sequel, for the sake of notational simplicity, we drop the use of the trace map $\gamma_0$ to denote the restriction of a Sobolev function on $\partial\Omega$. All such restrictions are understood in the sense of traces.

For every $x\in\RR$, we set $x^{\pm}=\max\{\pm x,0\}$. Then for $u\in W^{1,p}(\Omega)$ we define $u^{\pm}(\cdot)=u(\cdot)^{\pm}$. We know that
$$u^{\pm}\in W^{1,p}(\Omega),\ u=u^{+}-u^{-}\ \mbox{and}\ |u|=u^{+}+u^{-}.$$
Given a measurable function $h:\Omega\times\RR\to\RR$ (for example, a Carath\'eodory function), we define
$$N_h(u)(\cdot)=h(\cdot,u(\cdot))\ \mbox{for all}\ u\in W^{1,p}(\Omega)$$
and by $|\cdot|_N$ we denote the Lebesgue measure on $\RR^N$.

Let $A:W^{1,p}(\Omega)\to W^{1,p}(\Omega)^*$ be the nonlinear map defined by
\begin{eqnarray}\label{eq1}
\langle A(u),v\rangle=\int\limits_{\Omega}|Du|^{p-2}(Du,Dv)_{\RR^N}dz\ \mbox{for all}\ u,v\in W^{1,p}(\Omega).
\end{eqnarray}
\begin{prop}\label{prop2}
The map $A:W^{1,p}(\Omega)\to W^{1,p}(\Omega)^*$ defined by (\ref{eq1}) is bounded (maps bounded sets to bounded sets), demicontinuous monotone (hence maximal monotone too) and of type $(S)_{+}$, that is, if $u_n\overset{w}{\to} u$ in $W^{1,p}(\Omega)$ and
$$\limsup\limits_{n\to\infty}\langle A(u_n),u_n-u\rangle\leq 0,$$
then $u_n\to u$ in $W^{1,p}(\Omega)$.
\end{prop}
Suppose that $f_0:\Omega\times\RR\to\RR$ is a Carath\'eodory function such that
$$|f_0(z,x)|\leq a(z)(1+|x|^{r-1})\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\in\RR,$$
with $a\in L^{\infty}(\Omega)_{+}$ and $1<r<p^*=\left\{  \begin{array}{ll}
    \frac{Np}{N-p}\ \mbox{if}\ p<N\\
    +\infty\ \mbox{if}\ p\geq N.
    \end{array}\right.$

We set $F_0(z,x)=\int\limits_0^x f_0(z,s)ds$ and consider the $C^1$-functional $\varphi_0:W^{1,p}(\Omega)\to\RR$ defined by
$$\varphi_0(u)=\frac{1}{p}\|Du\|_p^p+\frac{1}{p}\int\limits_{\partial\Omega}\beta(z)|u(z)|^p d\sigma-\int\limits_{\Omega}F_0(z,u(z))dz\ \mbox{for\ all}\ u\in W^{1,p}(\Omega).$$
We assume that $\beta\in C^{0,\tau}(\partial\Omega)$ with $0<\tau<1$ and $\beta\geq 0$, $\beta\neq 0$. From Papageorgiou and R\u adulescu \cite{15}, we have the following result, which is a consequence of the nonlinear regularity theory.
\begin{prop}\label{prop3}
Let $u_0\in W^{1,p}(\Omega)$ be a local $C^1(\overline{\Omega})$-minimizer of $\varphi_0$, that is, there exists $\rho_0>0$ such that
$$\varphi_0(u_0)\leq\varphi_0(u_0+h)\ \mbox{for all}\ h\in C^1(\overline{\Omega}),\ \|h\|_{C^1(\overline{\Omega})}\leq \rho_0.$$
Then $u_0\in C^{1,\gamma}(\overline{\Omega})$ for some $\gamma\in (0,1)$ and it is also a local $W^{1,p}(\Omega)$-minimizer of $\varphi_0$, that is, there exists $\rho_1>0$ such that
$$\varphi_0(u_0)\leq\varphi_0(u_0+h)\ \mbox{for all}\ h\in W^{1,p}(\Omega),\ \|h\|\leq\rho_1.$$
\end{prop}
\begin{remark}
We mention that the first such result was proved by Brezis and Nirenberg \cite{4} for the space $H_0^1(\Omega)$.
\end{remark}
Finally consider the nonlinear eigenvalue problem
\begin{center}
$\left\{  \begin{array}{ll}
   \di -\Delta_p u(z)=\lambda |u(z)|^{p-2}u(z)\ \mbox{in}\ \Omega,\\
    \di\frac{\partial u}{\partial n_p}+\beta(z)|u(z)|^{p-2}u(z)=0\ \mbox{on}\ \partial\Omega.
    \end{array}\right. \hspace{7cm}(E_{\lambda})$
\end{center}
This eigenvalue problem was studied by L\^{e} \cite{12} and Papageorgiou and R\u adulescu \cite{15}.

We say that $\lambda\in\RR$ is an eigenvalue of the negative Robin $p$-Laplacian (denoted by $-\Delta_p^R$), if problem $(E_{\lambda})$ admits a nontrivial solution $u$, known as an eigenfunction corresponding to the eigenvalue $\lambda$.

Suppose that $\beta\in C^{0,\tau}(\partial\Omega)$, $0<\tau<1$ and $\beta(z)\geq 0$ for all $z\in\partial\Omega$, $\beta\neq 0$. Then we know that $(E_{\lambda})$ admits a smallest eigenvalue $\hat{\lambda}_1$ such that
\begin{itemize}
\item{} $\hat{\lambda}_1>0$;
\item{} $\hat{\lambda}_1$ is simple and isolated (that is, if $u,v$ are eigenfunctions corresponding to $\hat{\lambda}_1$, then $u=\xi v$ for some $\xi\in\RR\setminus\{0\}$ and there exists $\varepsilon>0$ such that  $(\hat{\lambda}_1,\hat{\lambda}_1+\varepsilon)$ contains no eigenvalue);
\item{} we have \begin{eqnarray}\label{eq2}
\hat{\lambda}_1=\inf\left[\frac{\|Du\|_p^p+\int\limits_{\partial\Omega}\beta(z)|u|^p d\sigma}{\|u\|_p^p}:\ u\in W^{1,p}(\Omega),\ u\neq 0\right].
\end{eqnarray}
\end{itemize}
The infimum in (\ref{eq2}) is realized on the one dimensional eigenspace corresponding to $\hat{\lambda}_1$. From (\ref{eq2}) it is clear that the elements of this eigenspace, do not change sign. Let $\hat{u}_1\in W^{1,p}(\Omega)$ be the positive, $L^p$-normalized (that is, $\|\hat{u}_1\|_p=1$) eigenfunction corresponding to $\hat{\lambda}_1$. The nonlinear regularity theory (see Lieberman \cite{13}) and the nonlinear maximum principle (see Vazquez \cite{16}), imply $\hat{u}_1\in\mbox{int}\, C_{+}$. We mention that $\hat{\lambda}_1$ is the only eigenvalue with eigenfunctions of constant sign. All the other eigenvalues have nodal (sign-changing) eigenfunctions. For more about the higher parts of the spectrum of $-\Delta_p^R$, we refer to L\^{e} \cite{12} and Papageorgiou and R\u adulescu \cite{15}.

As an easy consequence of the above properties, we have the following result (see for example, Papageorgiou and R\u adulescu \cite{14}).
\begin{prop}\label{prop4}
If $\vartheta\in L^{\infty}(\Omega)$, $\vartheta(z)\leq\hat{\lambda}_1$ a.e. in $\Omega$, $\vartheta\neq\hat{\lambda}_1$, then there exists $\xi_0>0$ such that
$$\|Du\|_p^p+\int\limits_{\partial\Omega}\beta(z)|u|^p d\sigma-\int\limits_{\Omega}\vartheta(z)|u|^p dz\geq\xi_0\|u\|^p,$$
for all $u\in W^{1,p}(\Omega)$.
\end{prop}
In the next section, we study the case in which the perturbation $f(z,\cdot)$ is $(p-1)$-sublinear.

\section{Sublinear Perturbations}\label{sec3}
Our hypotheses on the data of problem $(P_{\lambda})$, are the following:

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

\smallskip
$H(f)$: $f:\Omega\times\RR\to\RR$ is a Carath\'eodory function such that  for a.a. $z\in\Omega$, $f(z,0)=0$, $f(z,x)>0$ for all $x>0$ and
\begin{itemize}
\item [(i)] $f(z,x)\leq a(z)(1+x^{p-1})$ for a.a. $z\in\Omega$, all $x\geq 0$, with $a\in L^{\infty}(\Omega)_{+}$;
\item [(ii)] $\lim\limits_{x\to +\infty}\frac{f(z,x)}{x^{p-1}}=0$ uniformly for a.a. $z\in\Omega$;
\item [(iii)] $\lim\limits_{x\to 0^{+}}\frac{f(z,x)}{x^{p-1}}=+\infty$ uniformly for a.a. $z\in\Omega$.
\end{itemize}
\begin{remark}
Since we are interested in positive solutions and the above hypotheses concern the positive semiaxis $(0,+\infty)$, without any loss of generality, we assume that $f(z,x)=0$ for a.a. $z\in\Omega$, all $x\leq 0$. Hypothesis $H(f)(ii)$ implies that the perturbation $f(z,\cdot)$ is strictly $(p-1)$-sublinear near $+\infty$, while hypothesis $H(f)(iii)$ dictates a similar polynomial growth near $0^{+}$. A simple example illustrating such a perturbation, is given by the function $f(x)=x^{q-1}$ for all $x\geq 0$, with $q\in (1,p)$. In the sequel $F(z,x)=\int\limits_0^x f(z,s)ds$.
\end{remark}
We introduce the following two sets related to problem $(P_{\lambda})$:
\begin{eqnarray*}
&&\mathcal{P}=\{\lambda\in\RR:\ \mbox{problem}\ (P_{\lambda})\ \mbox{admits a positive solution}\},\\
&&S(\lambda)=\mbox{the set of positive solutions for problem}\ (P_{\lambda}).
\end{eqnarray*}
Note that as in Filippakis, Kristaly and Papageorgiou \cite{8}, exploiting the monotonicity of the operator $A$ (see Proposition \ref{prop2}), we have that $S(\lambda)$ is downward directed, that is, if $u_1,u_2\in S(\lambda)$, then we can find $u\in S(\lambda)$ such that  $u\leq u_1$, $u\leq u_2$.
\begin{prop}\label{prop5}
If hypotheses $H(\beta)$ and $H(f)$ hold, then $\mathcal{P}\neq\emptyset$ and for every $\lambda\in \mathcal{P}$, we have $S(\lambda)\subseteq \mbox{int}\, C_{+}$.
\end{prop}
\begin{proof}
For every $\lambda\in \RR$, we consider the $C^1$-functional $\hat{\varphi}_{\lambda}:W^{1,p}(\Omega)\to\RR$ defined by
\begin{eqnarray*}
&&\hat{\varphi}_{\lambda}(u)=\frac{1}{p}\|Du\|_p^p+\frac{1}{p}\|u^{-}\|_p^p+\frac{1}{p}\int\limits_{\partial\Omega}\beta(z)(u^{+})^p d\sigma-\frac{\lambda}{p}\|u^{+}\|_p^p-\int\limits_{\Omega}F(z,u^{+})dz\\
&&\hspace{8cm} \mbox{for all}\ u\in W^{1,p}(\Omega).
\end{eqnarray*}
Hypotheses $H(f)(i),(ii)$ imply that given $\varepsilon>0$, we can find $c_1=c_1(\varepsilon)>0$ such that
\begin{eqnarray}\label{eq3}
F(z,x)\leq\frac{\varepsilon}{p}x^p+c_1\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\geq 0.
\end{eqnarray}
Let $\lambda<\hat{\lambda}_1$. Then for all $u\in W^{1,p}(\Omega)$, we have
\begin{eqnarray*}
\hat{\varphi}_{\lambda}(u)&\geq&\frac{1}{p}\|Du^{+}\|_p^p+\frac{1}{p}\int\limits_{\partial\Omega}\beta(z)(u^{+})^p d\sigma-\frac{\lambda+\varepsilon}{p}\|u^{+}\|_p^p+\frac{1}{p}\|Du^{-}\|_p^p+\frac{1}{p}\|u^{-}\|_p^p-c_1|\Omega|_N\\
&&\hspace{10cm} (\mbox{see}\ (\ref{eq3}))\\
&\geq&\frac{1}{p}[c_2-\varepsilon]\|u^{+}\|^p+\frac{1}{p}\|u^{-}\|^p-c_1|\Omega|_N\ (\mbox{see Proposition}\ \ref{prop4}\ \mbox{and recall}\ \lambda<\hat{\lambda}_1).
\end{eqnarray*}
Choosing $\varepsilon\in (0,c_2)$, we see that
\begin{eqnarray*}
&&\hat{\varphi}_{\lambda}(u)\geq\frac{c_3}{p}\|u\|^p-c_1|\Omega|_N\ \mbox{with}\ c_3=\min\{1,c_2-\varepsilon\}>0,\\
&\Rightarrow& \hat{\varphi}_{\lambda}\ \mbox{is coercive}.
\end{eqnarray*}
Also, using the Sobolev embedding theorem and the continuity of the trace map, we see that $\hat{\varphi}_{\lambda}$ is sequentially weakly lower semicontinuous. So, by the Weierstrass theorem, we can find $\hat{u}_{\lambda}\in W^{1,p}(\Omega)$ such that
\begin{eqnarray}\label{eq4}
\hat{\varphi}_{\lambda}(\hat{u}_{\lambda})=\inf\left[\hat{\varphi}_{\lambda}(u):u\in W^{1,p}(\Omega)\right].
\end{eqnarray}
By virtue of hypothesis $H(f)(iii)$, given any $\xi>\hat{\lambda}_1-\lambda$, we can find $\delta=\delta(\xi)>0$ such that
\begin{eqnarray}\label{eq5}
F(z,x)\geq\frac{\xi}{p}x^p\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\in [0,\delta].
\end{eqnarray}
Choose $t\in (0,1)$ small such that  $t\hat{u}_1(z)\in (0,\delta]$ for all $z\in\overline{\Omega}$ (recall that $\hat{u}_1\in\mbox{int}\, C_{+}$). We have
\begin{eqnarray*}
\hat{\varphi}_{\lambda}(t\hat{u}_1)&\leq&\frac{t^p}{p}\|D\hat{u}_1\|_p^p+\frac{t^p}{p}\int\limits_{\partial\Omega}\beta(z)\hat{u}_1^p d\sigma-\frac{\lambda t^p}{p}\|\hat{u}_1\|_p^p-\frac{\xi t^p}{p}\|\hat{u}_1\|_p^p\ (\mbox{see}\ (\ref{eq5}))\\
&=&\frac{t^p}{p}[\hat{\lambda}_1-\lambda-\xi]\ (\mbox{recall}\ \|\hat{u}_1\|_p=1).
\end{eqnarray*}
Since $\xi>\hat{\lambda}_1-\lambda$, it follows that
\begin{eqnarray*}
&&\hat{\varphi}_{\lambda}(t\hat{u}_1)<0,\\
&\Rightarrow& \hat{\varphi}_{\lambda}(\hat{u}_{\lambda})<0=\hat{\varphi}_{\lambda}(0)\ (\mbox{see}\ (\ref{eq4})),\ \mbox{hence}\ \hat{u}_{\lambda}\neq 0.
\end{eqnarray*}
From (\ref{eq4}), we have
\begin{eqnarray}\label{eq6}
&&\hat{\varphi}_{\lambda}'(\hat{u}_{\lambda})=0\nonumber\\
&\Rightarrow& \langle A(\hat{u}_{\lambda}),h\rangle+\int\limits_{\partial\Omega}\beta(z)(\hat{u}_{\lambda}^{+})^{p-1}h d\sigma-\int\limits_{\Omega}(\hat{u}_{\lambda}^{-})^{p-1}h dz=\lambda\int\limits_{\Omega}(\hat{u}_{\lambda}^{+})^{p-1}h dz\\
&&\hspace{1.5cm}+\int\limits_{\Omega}f(z,\hat{u}_{\lambda}^{+})h dz\ \mbox{for all}\ h\in W^{1,p}(\Omega)\nonumber.
\end{eqnarray}
In (\ref{eq6}) we choose $h=-\hat{u}_{\lambda}^{-}\in W^{1,p}(\Omega)$. Then
\begin{eqnarray*}
&&\|D\hat{u}_{\lambda}^{-}\|_p^p+\|\hat{u}_{\lambda}^{-}\|_p^p=0,\\
&\Rightarrow& \hat{u}_{\lambda}\geq 0,\ \hat{u}_{\lambda}\neq 0.
\end{eqnarray*}
Therefore (\ref{eq6}) becomes
\begin{eqnarray}\label{eq7}
&&\langle A(\hat{u}_{\lambda}),h\rangle+\int\limits_{\partial\Omega}\beta(z)\hat{u}_{\lambda}^{p-1}h d\sigma=\lambda\int\limits_{\Omega}\hat{u}_{\lambda}^{p-1}h dz+\int\limits_{\Omega}f(z,\hat{u}_{\lambda})h dz\\
&&\hspace{8cm} \mbox{for all}\ h\in W^{1,p}(\Omega).\nonumber
\end{eqnarray}

By $\langle\cdot,\cdot\rangle_0$ we denote the duality brackets for the pair $(W^{-1,p'}(\Omega),W_0^{1,p}(\Omega))$. From the representation theorem for the elements of $W^{-1,p'}(\Omega)=W^{1,p}_0(\Omega)^*$ (see, for example, Gasinski and Papageorgiou \cite[p. 212]{9}), we have
$$\mbox{div}\, \left(|D\hat{u}_{\lambda}|^{p-2}D\hat{u}_{\lambda}\right)\in W^{-1,p'}(\Omega)\ \left(\frac{1}{p}+\frac{1}{p'}=1\right).$$
Integrating by parts, we have
$$\langle A(\hat{u}_{\lambda}),h\rangle=\langle -\mbox{div}\, \left(|D\hat{u}_{\lambda}|^{p-2}D\hat{u}_{\lambda}\right),h\rangle_0\ \mbox{for all}\ h\in W_0^{1,p}(\Omega)\subseteq W^{1,p}(\Omega).$$
We use this in (\ref{eq7}) and recall that $h|_{\partial\Omega}=0$ for all $h\in W_0^{1,p}(\Omega)$. We obtain
\begin{eqnarray}\label{eq8}
&&\langle -\mbox{div}\, \left(|D\hat{u}_{\lambda}|^{p-2}D\hat{u}_{\lambda}\right),h\rangle_0=\lambda\int\limits_{\Omega}\hat{u}_{\lambda}^{p-1}h dz+\int\limits_{\Omega}f(z,\hat{u}_{\lambda})h dz\ \mbox{for all}\ h\in W_0^{1,p}(\Omega),\nonumber\\
&\Rightarrow& -\Delta_p \hat{u}_{\lambda}(z)=\lambda\hat{u}_{\lambda}(z)^{p-1}+f(z,\hat{u}_{\lambda}(z))\ \mbox{a.e. in}\ \Omega.
\end{eqnarray}
From the nonlinear Green's identity (see, for example, Gasinski and Papageorgiou \cite[p. 210]{9}), we have
$$\langle A(\hat{u}_{\lambda}),h\rangle+\int\limits_{\Omega}(\Delta_p\hat{u}_{\lambda})h dz=\left\langle\frac{\partial\hat{u}_{\lambda}}{\partial n_p},h\right\rangle_{\partial\Omega}\ \mbox{for all}\ h\in W^{1,p}(\Omega)\ (\mbox{see}\ (\ref{eq8}))$$
where by $\langle\cdot,\cdot\rangle_{\partial\Omega}$ we denote the duality brackets for the pair
\begin{eqnarray}\label{eq9}
\left(W^{-\frac{1}{p'},p'}(\partial\Omega),W^{\frac{1}{p},p}(\partial\Omega)\right)\ \left(\frac{1}{p}+\frac{1}{p'}=1\right).
\end{eqnarray}
We return to (\ref{eq7}) and use (\ref{eq9}) above. We obtain
\begin{eqnarray}\label{eq10}
&&\int\limits_{\Omega}(-\Delta_p\hat{u}_{\lambda})h dz+\left\langle\frac{\partial\hat{u}_{\lambda}}{\partial n_p},h\right\rangle_{\partial\Omega}+\int\limits_{\partial\Omega}\beta(z)\hat{u}_{\lambda}^{p-1}h d\sigma=\lambda\int\limits_{\Omega}\hat{u}_{\lambda}^{p-1}h dz\nonumber\\
&&\hspace{1.5cm}+\int\limits_{\Omega}f(z,\hat{u}_{\lambda})h dz\ \mbox{for all}\ h\in W^{1,p}(\Omega),\nonumber\\
&\Rightarrow&\left\langle\frac{\partial\hat{u}_{\lambda}}{\partial n_p},h\right\rangle_{\partial\Omega}+\int\limits_{\partial\Omega}\beta(z)\hat{u}_{\lambda}^{p-1}h d\sigma=0\ \mbox{for all}\ h\in W^{1,p}(\Omega)\ (\mbox{see}\ (\ref{eq8})),\nonumber\\
&\Rightarrow& \frac{\partial\hat{u}_{\lambda}}{\partial n_p}+\beta(z)\hat{u}_{\lambda}^{p-1}=0\ \mbox{on}\ \partial\Omega.
\end{eqnarray}

From (\ref{eq8}) and (\ref{eq10}) it follows that $\hat{u}_{\lambda}\in S(\lambda)$ and so $\lambda\in {\mathcal P}$ for every $\lambda<\hat{\lambda}_1$. From Winkert \cite{17}, we have that $\hat{u}_{\lambda}\in L^{\infty}(\Omega)$. So, we can apply Theorem $2$ of Lieberman \cite{13} and obtain that $\hat{u}_{\lambda}\in C_{+}\setminus\{0\}$.

Hypotheses $H(f)(i),(iii)$ imply that given $\rho>0$, we can find $\xi_{\rho}>0$ such that
\begin{eqnarray}\label{eq11}
f(z,x)+\xi_{\rho}x^{p-1}\geq 0\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\in [0,\rho].
\end{eqnarray}
Let $\rho=\|\hat{u}_{\lambda}\|_{\infty}$ and let $\xi_{\rho}>0$ be as in (\ref{eq11}) above. Then
\begin{eqnarray*}
&&-\Delta_p\hat{u}_{\lambda}(z)+\xi_{\rho}\hat{u}_{\lambda}(z)^{p-1}=\lambda\hat{u}_{\lambda}(z)^{p-1}+f(z,\hat{u}_{\lambda}(z))+\xi_{\rho} \hat{u}_{\lambda}(z)^{p-1}\geq 0\ \mbox{a.e. in}\ \Omega\\
&&\hspace{11cm} (\mbox{see}\ (\ref{eq11}))\\
&\Rightarrow&\Delta_p\hat{u}_{\lambda}(z)\leq\xi_{\rho}\hat{u}_{\lambda}(z)^{p-1}\ \mbox{a.e. in}\ \Omega,\\
&\Rightarrow& \hat{u}_{\lambda}\in\mbox{int}\, C_{+}\ (\mbox{see Vazquez \cite{16}}).
\end{eqnarray*}
So, we have proved that $S(\lambda)\subseteq\mbox{int}\, C_{+}$.
\end{proof}

\begin{prop}\label{prop6}
If hypotheses $H(\beta)$ and $H(f)$ hold and $\lambda\in \mathcal{P}$, then $(-\infty,\lambda]\subseteq \mathcal{P}$.
\end{prop}
\begin{proof}
Since $\lambda\in \mathcal{P}$, we can find $u_{\lambda}\in S(\lambda)\subseteq\mbox{int}\, C_{+}$ (see Proposition \ref{prop5}). Let $\mu\in(-\infty,\lambda]$. Using $u_{\lambda}\in\mbox{int}\, C_{+}$, we introduce the following truncation-perturbation of the reaction in problem $(P_{\mu})$:
\begin{equation}\label{eq12}
e_{\mu}(z,x)=\left\{  \begin{array}{ll}
    0\ &\mbox{if}\ x<0\\
    (\mu+1)x^{p-1}+f(z,x)\ &\mbox{if}\ 0\leq x\leq u_{\lambda}(z)  \\
    (\mu+1)u_{\lambda}(z)^{p-1}+f(z,u_{\lambda}(z))\ &\mbox{if}\ u_{\lambda}(z)<x.
    \end{array}\right.
\end{equation}
This is a Carath\'eodory function. We set $E_{\mu}(z,x)=\int\limits_0^x e_{\mu}(z,s)ds$ and consider the $C^1$-functional $\tau_{\mu}:W^{1,p}(\Omega)\to\RR$ defined by
\begin{eqnarray*}
&&\tau_{\mu}(u)=\frac{1}{p}\|Du\|_p^p+\frac{1}{p}\|u\|_p^p+\frac{1}{p}\int\limits_{\partial\Omega}\beta(z)(u^{+})^p d\sigma-\int\limits_{\Omega}E_{\mu}(z,u)dz\\
&&\hspace{8cm} \mbox{for all}\ u\in W^{1,p}(\Omega),\\
&\Rightarrow&\tau_{\mu}(u)\geq\frac{1}{p}\|u\|^p-c_4\ \mbox{for some}\ c_4>0\ (\mbox{see}\ H(\beta)\ \mbox{and}\ (\ref{eq12})),\\
&\Rightarrow&\tau_{\mu}\ \mbox{is coercive}.
\end{eqnarray*}
Also $\tau_{\mu}$ is sequentially weakly lower semicontinuous. Hence we can find $u_{\mu}\in W^{1,p}(\Omega)$ such that
\begin{eqnarray}\label{eq13}
\tau_{\mu}(u_{\mu})=\inf[\tau_{\mu}(u):u\in W^{1,p}(\Omega)].
\end{eqnarray}
As in the proof of Proposition \ref{prop5} for $t\in (0,1)$ small (at least such that $t\hat{u}_1(z)\leq\min\limits_{\overline{\Omega}}u_{\lambda}$ for all $z\in\overline{\Omega}$; recall that $\hat{u}_{\lambda}\in\mbox{int}\, C_{+}$), we have
\begin{eqnarray*}
&&\tau_{\mu}(t\hat{u}_1)<0,\\
&\Rightarrow& \tau_{\mu}(u_{\mu})<0=\tau_{\mu}(0)\ (\mbox{see}\ (\ref{eq13})),\ \mbox{hence}\ u_{\mu}\neq 0.
\end{eqnarray*}
From (\ref{eq13}) we have
\begin{eqnarray}\label{eq14}
&&\tau_{\mu}'(u_{\mu})=0,\nonumber\\
&\Rightarrow&\langle A(u_{\mu}),h\rangle+\int\limits_{\Omega}|u_{\mu}|^{p-2}u_{\mu}h dz+\int\limits_{\partial\Omega}\beta(z)(u_{\mu}^{+})^{p-1}h d\sigma=\int\limits_{\Omega}e_{\mu}(z,u_{\mu})h dz\\
&&\hspace{8cm} \mbox{for all}\ h\in W^{1,p}(\Omega)\nonumber.
\end{eqnarray}
In (\ref{eq14}) we choose $h=-u_{\mu}^{-}\in W^{1,p}(\Omega)$. Then
\begin{eqnarray*}
&&\|D u_{\mu}^{-}\|_p^p+\|u_{\mu}^{-}\|_p^p=0\ (\mbox{see}\ (\ref{eq12}),\\
&\Rightarrow& u_{\mu}\geq 0,\ u_{\mu}\neq 0.
\end{eqnarray*}
Next in (\ref{eq14}) we choose $(u_{\mu}-u_{\lambda})^{+}\in W^{1,p}(\Omega)$. Then
\begin{eqnarray*}
&&\langle A(u_{\mu}),(u_{\mu}-u_{\lambda})^{+}\rangle+\int\limits_{\Omega}u_{\mu}^{p-1}(u_{\mu}-u_{\lambda})^{+} dz+\int\limits_{\partial\Omega}\beta(z)u_{\mu}^{p-1}(u_{\mu}-u_{\lambda})^{+} d\sigma\\
&=&\int\limits_{\Omega}e_{\mu}(z,u_{\mu})(u_{\mu}-u_{\lambda})^{+} dz\\
&=&\int\limits_{\Omega}\left[\mu u_{\lambda}^{p-1}+f(z,u_{\lambda})\right](u_{\mu}-u_{\lambda})^{+}dz+\int\limits_{\Omega}u_{\lambda}^{p-1}(u_{\mu}-u_{\lambda})^{+}dz\\
&\leq&\int\limits_{\Omega}\left[\lambda u_{\lambda}^{p-1}+f(z,u_{\lambda})\right](u_{\mu}-u_{\lambda})^{+}dz+\int\limits_{\Omega}u_{\lambda}^{p-1}(u_{\mu}-u_{\lambda})^{+}dz\\
&=&\langle A(u_{\lambda}),(u_{\mu}-u_{\lambda})^{+}\rangle+\int\limits_{\Omega}u_{\lambda}^{p-1}(u_{\mu}-u_{\lambda})^{+} dz+\int\limits_{\partial\Omega}\beta(z)u_{\lambda}^{p-1}(u_{\mu}-u_{\lambda})^{+} d\sigma,\\
&\Rightarrow&\langle A(u_{\mu})-A(u_{\lambda}),(u_{\mu}-u_{\lambda})^{+}\rangle+\int\limits_{\Omega}(u_{\mu}^{p-1}-u_{\lambda}^{p-1})(u_{\mu}-u_{\lambda})^{+} dz\\
& &+\int\limits_{\partial\Omega}\beta(z)(u_{\mu}^{p-1}-u_{\lambda}^{p-1})(u_{\mu}-u_{\lambda})^{+} d\sigma\leq 0,\\
&\Rightarrow& |\{u_{\mu}>u_{\lambda}\}|_{N}=0,\ \mbox{hence}\ u_{\mu}\leq u_{\lambda}.
\end{eqnarray*}
So, we have proved that
$$u_{\mu}\in [0,u_{\lambda}]\setminus\{0\},$$
where $[0,u_{\lambda}]=\{u\in W^{1,p}(\Omega):0\leq u(z)\leq u_{\lambda}(z)\ \mbox{a.e. in}\ \Omega\}$. Then (\ref{eq14}) becomes
\begin{eqnarray*}
&&\langle A(u_{\mu}),h\rangle+\int\limits_{\Omega}u_{\mu}^{p-1}h dz+\int\limits_{\partial\Omega}\beta(z)u_{\mu}^{p-1}h d\sigma= (\mu +1)\int\limits_{\Omega}u_{\mu}^{p-1}h dz+\int\limits_{\Omega}f(z,u_{\mu})h dz\\
&&\hspace{8cm} \mbox{for all}\ h\in W^{1,p}(\Omega).
\end{eqnarray*}
As in the proof of Proposition \ref{prop5}, using the nonlinear Green's identity, we obtain
$$u_{\mu}\in S(\mu)\subseteq\mbox{int}\, C_{+}\ \mbox{and so}\ \mu\in\mathcal{P}.$$
Therefore $(-\infty,\lambda]\subseteq \mathcal{P}$.
\end{proof}

Hypotheses $H(f)(i),(iii)$, imply that given any $\xi>0$ and $r\in (p,p^*)$, we can find $c_5=c_5(\xi,r)>0$ such that
\begin{eqnarray}\label{eq15}
f(z,x)\geq\xi x^{p-1}-c_5 x^{r-1}\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\geq 0.
\end{eqnarray}
This unilateral growth constraint on the perturbation $f(z,x)$, leads to the following auxiliary Robin problem:
\begin{equation}\label{eq16}
\left\{  \begin{array}{ll}
 \di   -\Delta_p u(z)=\xi u(z)^{p-1}-c_5 u(z)^{r-1}\ \mbox{in}\ \Omega,\\
    \di\frac{\partial u}{\partial n_p}+\beta(z)u(z)^{p-1}=0\ \mbox{on}\ \partial\Omega,\ u>0.
    \end{array}\right.
\end{equation}
\begin{prop}\label{prop7}
If hypotheses $H(\beta)$ hold, then for $\xi>0$ big problem (\ref{eq16}) has a unique positive solution $\overline{u}\in\mbox{int}\, C_{+}$.
\end{prop}

\begin{proof}
First we establish the existence of a positive solution for problem (\ref{eq16}). To this end, we consider the $C^1$-functional $\psi:W^{1,p}(\Omega)\to\RR$ defined by
\begin{eqnarray*}
&&\psi(u)=\frac{1}{p}\|Du\|_p^p+\frac{1}{p}\|u^{-}\|_p^p+\frac{1}{p}\int\limits_{\partial\Omega}\beta(z)(u^{+})^p d\sigma+\frac{c_5}{r}\|u^{+}\|_r^r-\frac{\xi}{p}\|u^{+}\|_p^p\\
&&\hspace{8cm} \mbox{for all}\ u\in W^{1,p}(\Omega).
\end{eqnarray*}
We have
\begin{eqnarray}\label{eq17}
\psi(u)\geq\frac{1}{p}\|u\|^p+\left[\frac{c_5}{r}\|u^{+}\|_r^{r-p}-\left(\frac{\xi}{p}+1\right)c_6\right]\|u^{+}\|_r^p\ \mbox{for some}\ c_6>0.
\end{eqnarray}
Since $r>p$, from (\ref{eq17}) it follows that $\psi$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find $\overline{u}\in W^{1,p}(\Omega)$ such that
\begin{eqnarray}\label{eq18}
\psi(\overline{u})=\inf[\psi(u):\ u\in W^{1,p}(\Omega)].
\end{eqnarray}
Choosing $\xi>\hat{\lambda}_1$ and since $r>p$, we see that for $t\in (0,1)$ small, we have
\begin{eqnarray*}
&&\psi(t\hat{u}_1)<0,\\
&\Rightarrow&\psi(\overline{u})<0=\psi(0)\ (\mbox{see}\ (\ref{eq18})),\ \mbox{hence}\ \overline{u}\neq 0.
\end{eqnarray*}
From (\ref{eq18}) we have
\begin{eqnarray}\label{eq19}
&&\psi'(\overline{u})=0,\nonumber\\
&\Rightarrow&\langle A(\overline{u}),h\rangle-\int\limits_{\Omega}(\overline{u}^{\,-})^{p-1}h dz+\int\limits_{\partial\Omega}\beta(z)(\overline{u}^{+})^{p-1}h d\sigma\nonumber\\
& & =\xi\int\limits_{\Omega}(\overline{u}^{+})^{p-1}h dz-c_5\int\limits_{\Omega}(\overline{u}^{+})^{r-1}h dz\ \mbox{for all}\ h\in W^{1,p}(\Omega).
\end{eqnarray}
Choose $h=-\overline{u}^{-}\in W^{1,p}(\Omega)$. Then we obtain $\overline{u}\geq 0,\ \overline{u}\neq 0$ and so (\ref{eq19}) becomes
\begin{eqnarray*}
&&\langle A(\overline{u}),h\rangle+\int\limits_{\partial\Omega}\beta(z)\overline{u}^{p-1}h d\sigma=\xi\int\limits_{\Omega}\overline{u}^{p-1}h dz-c_5\int\limits_{\Omega}\overline{u}^{r-1}h dz\ \mbox{for all}\ h\in W^{1,p}(\Omega),\\
&\Rightarrow& \overline{u}\ \mbox{is a positive solution of}\ (\ref{eq16})\ (\mbox{as in the proof of Proposition}\ \ref{prop5}).
\end{eqnarray*}
The nonlinear regularity theory (see \cite{17}, \cite{13}) implies that $\overline{u}\in C_{+}\setminus\{0\}$. We have
\begin{eqnarray*}
&&-\Delta_p\overline{u}(z)\geq -c_5\overline{u}(z)^{r-1}\ \mbox{a.e. in}\ \Omega,\\
&\Rightarrow& \Delta_p\overline{u}(z)\leq c_5\|\overline{u}\|_{\infty}^{r-p}\overline{u}(z)^{p-1}\ \mbox{a.e. in}\ \Omega,\\
&\Rightarrow&\overline{u}\in\mbox{int}\, C_{+}\ (\mbox{see Vazquez \cite{16}}).
\end{eqnarray*}
Next we show the uniqueness of this positive solution. For this purpose, we introduce the integral functional $\vartheta: L^p(\Omega)\to\overline{\RR}=\RR\cup\{+\infty\}$ defined by
\begin{eqnarray*}
\vartheta(u)=\left\{  \begin{array}{ll}
 \di   \frac{1}{p}\|Du^{1/p}\|_p^p+\frac{1}{p}\int_{\partial\Omega}\beta (z)ud\sigma\ &\mbox{if}\ u\geq 0,\ u^{1/p}\in W^{1,p}(\Omega),\\
    +\infty\ &\mbox{otherwise}.
    \end{array}\right.
\end{eqnarray*}
Lemma $1$ of Diaz and Saa \cite{6} implies that $\vartheta$ is convex and lower semicontinuous. Suppose that $\overline{u},\ v$ are two positive solutions of the auxiliary problem (\ref{eq16}). From the first part of the proof, we have
\begin{eqnarray*}
&&\overline{u}, v\in \mbox{int}\, C_{+}\\
&\Rightarrow&\overline{u}^p, v^p\in \mbox{dom}\ \vartheta=\{y\in W^{1,p}(\Omega):\vartheta(y)<\infty\}\\
&&\hspace{3cm}(\mbox{the effective domain of}\ \vartheta).
\end{eqnarray*}
Then for every $h\in C^1(\overline{\Omega})$ and for $|t|\leq 1$ small, we have
$$\overline{u}^p+th,\ v+th\in\mbox{dom}\ \vartheta.$$
It follows that $\vartheta$ is G\^ateaux differentiable at $\overline{u}^p$ and at $v^p$ in the direction $h$. Using the chain rule, we have
\begin{eqnarray*}
&&\vartheta'(\overline{u}^p)(h)=\frac{1}{p}\int\limits_{\Omega}\frac{-\Delta_p\overline{u}}{\overline{u}^{p-1}}h dz+\frac{1}{p}\int_{\partial\Omega}\beta (z)hd\sigma\\
&&\vartheta'(v^p)(h)=\frac{1}{p}\int\limits_{\Omega}\frac{-\Delta_p v}{v^{p-1}}h dz+\frac{1}{p}\int_{\partial\Omega}\beta (z)hd\sigma\ \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 $\vartheta$ implies the monotonicity of $\vartheta'$. So, we have
\begin{eqnarray*}
0&\leq&\frac{1}{p}\int\limits_{\Omega}\left[\frac{-\Delta_p\overline{u}}{\overline{u}^{p-1}}-\frac{-\Delta_p v}{v^{p-1}}\right](\overline{u}^p-v^p)dz\\
&\leq&\frac{1}{p}\int\limits_{\Omega}c_5(v^{r-p}-\overline{u}^{r-p})(\overline{u}^p-v^p)dz\leq 0\ (\mbox{see}\ (\ref{eq16})),
\end{eqnarray*}
\begin{eqnarray*}
&& \Rightarrow \overline{u}=v,\\
&& \Rightarrow\overline{u}\in\mbox{int}\, C_{+}\ \mbox{is the unique positive solution of}\ (\ref{eq16}).
\end{eqnarray*}
\end{proof}

\begin{prop}\label{prop8}
If hypotheses $H(\beta)$ and $H(f)$ hold and $\lambda\in \mathcal{P}$, then $\overline{u}\leq u$ for all $u\in S(\lambda)$.
\end{prop}
\begin{proof}
Let $u\in S(\lambda)$. We introduce the following Carath\'eodory function
\begin{eqnarray}\label{eq20}
\gamma(z,x)=\left\{  \begin{array}{ll}
    0\ &\mbox{if}\ x<0\\
    (\xi+1)x^{p-1}-c_5 x^{r-1}\ &\mbox{if}\ 0\leq x\leq u(z)\\
    (\xi+1)u(z)^{p-1}-c_5 u(z)^{r-1}\ &\mbox{if}\ u(z)<x.
    \end{array}\right.
\end{eqnarray}
Let $\Gamma(z,x)=\int\limits_0^x\gamma(z,s)ds$ and consider the $C^1$-functional $\chi:W^{1,p}(\Omega)\to\RR$ defined by
$$\chi(u)=\frac{1}{p}\|Du\|_p^p+\frac{1}{p}\|u\|_p^p+\frac{1}{p}\int\limits_{\partial\Omega}\beta(z)(u^{+})^p d\sigma-\int\limits_{\Omega}\Gamma(z,u)dz\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$
Using hypothesis $H(\beta)$ and (\ref{eq20}), we see that
\begin{eqnarray*}
&&\chi(u)\geq\frac{1}{p}\|u\|^p-c_6\ \mbox{for some}\ c_6>0,\\
&\Rightarrow& \chi\ \mbox{is coercive}.
\end{eqnarray*}
In addition, $\chi$ is sequentially weakly lower semicontinuous. So, we can find $\overline{u}_*\in W^{1,p}(\Omega)$ such that
\begin{eqnarray}\label{eq21}
\chi(\overline{u}_*)=\inf[\chi(u):u\in W^{1,p}(\Omega)].
\end{eqnarray}
As before, since $r>p$, for $t\in (0,1)$ small, we have
\begin{eqnarray*}
&&\chi(t\hat{u}_1)<0,\\
&\Rightarrow&\chi(\overline{u}_*)<0=\chi(0)\ (\mbox{see}\ (\ref{eq21})),\ \mbox{hence}\ \overline{u}_*\neq 0.
\end{eqnarray*}
From (\ref{eq21}) we have
\begin{eqnarray}\label{eq22}
&&\chi'(\overline{u}_*)=0,\nonumber\\
&\Rightarrow&\langle A(\overline{u}_*),h\rangle+\int\limits_{\Omega}|\overline{u}_*|^{p-2}\overline{u}_*h dz+\int\limits_{\partial\Omega}\beta(z)(\overline{u}_*^{+})^{p-1}h d\sigma=\int\limits_{\Omega}\gamma(z,\overline{u}_*)h dz\\
&&\hspace{8cm} \mbox{for all}\ h\in W^{1,p}(\Omega).\nonumber
\end{eqnarray}
In (\ref{eq22}) we choose $h=-\overline{u}_*^{-}\in W^{1,p}(\Omega)$. Then
\begin{eqnarray*}
&&\|D\overline{u}_*^{-}\|_p^p+\|\overline{u}_*^{-}\|_p^p=0\ (\mbox{see}\ (\ref{eq20}))\\
&\Rightarrow& \overline{u}_*\geq 0,\ \overline{u}_*\neq 0.
\end{eqnarray*}
Next in (\ref{eq22}) we choose $h=(\overline{u}_*-u)^{+}\in W^{1,p}(\Omega)$. Then
\begin{eqnarray*}
&&\langle A(\overline{u}_*),(\overline{u}_*-u)^{+}\rangle+\int\limits_{\Omega}\overline{u}_*^{p-1}(\overline{u}_*-u)^{+} dz+\int\limits_{\partial\Omega}\beta(z)\overline{u}_*^{p-1}(\overline{u}_*-u)^{+} d\sigma\\
&=&\int\limits_{\Omega}[\xi u^{p-1}-c_5u^{r-1}](\overline{u}_*-u)^{+}dz+\int\limits_{\Omega}u^{p-1}(\overline{u}_*-u)^{+}dz\ (\mbox{see}\ (\ref{eq20}))\\
&\leq&\int\limits_{\Omega}[\lambda u^{p-1}+f(z,u)](\overline{u}_*-u)^{+}dz+\int\limits_{\Omega}u^{p-1}(\overline{u}_*-u)^{+}dz\ (\mbox{see}\ (\ref{eq15}))\\
&=&\langle A(u),(\overline{u}_*-u)^{+}\rangle+\int\limits_{\Omega}u^{p-1}(\overline{u}_*-u)^{+} dz+\int\limits_{\partial\Omega}\beta(z)u^{p-1}(\overline{u}_*-u)^{+} d\sigma\ (\mbox{since}\ u\in S(\lambda)),\\
&\Rightarrow& |\{\overline{u}_*>u\}|_N=0\ (\mbox{as before}),\ \mbox{hence}\ \overline{u}_*\leq u.
\end{eqnarray*}
So, we have proved that
$$\overline{u}_*\in [0,u]\setminus\{0\}.$$
Then from (\ref{eq20}) and (\ref{eq22}) it follows that $\overline{u}_*\in\mbox{int}\, C_{+}$ is a positive solution of (\ref{eq16}) and so by virtue of Proposition \ref{prop7}, we have
\begin{eqnarray*}
&&\overline{u}_*=\overline{u},\\
&\Rightarrow&\overline{u}\leq u\ \mbox{for all}\ u\in S(\lambda).
\end{eqnarray*}
\end{proof}

In the proof of Proposition \ref{prop5} we have seen that $(-\infty,\hat{\lambda}_1)\subseteq \mathcal{P}$. Next we show that in fact we have $\mathcal{P}=(-\infty,\hat{\lambda}_1)$.
\begin{prop}\label{prop9}
If hypotheses $H(\beta)$ and $H(f)$ hold, then $\hat{\lambda}_1\notin \mathcal{P}$.
\end{prop}
\begin{proof}
Arguing by contradiction, suppose that $\hat{\lambda}_1\in \mathcal{P}$. Then we can find $u_0\in S(\hat{\lambda}_1)\subseteq\mbox{int}\, C_{+}$. Recall that $\hat{u}_1\in\mbox{int}\, C_{+}$ too. Invoking Lemma $3.3$ of Filippakis, Kristaly and Papageorgiou \cite{8} we can find $c_7,c_8>0$ such that
\begin{eqnarray}\label{eq23}
&&c_7 u_0\leq\hat{u}_1\leq c_8 u_0,\nonumber\\
&\Rightarrow& c_7\leq \frac{\hat{u}_1}{u_0}\leq c_8\ \mbox{and}\ \frac{1}{c_8}\leq\frac{u_0}{\hat{u}_1}\leq\frac{1}{c_7},\nonumber\\
&\Rightarrow& \frac{\hat{u}_1}{u_0}\ \mbox{and}\ \frac{u_0}{\hat{u}_1}\ \mbox{belong in}\ L^{\infty}(\Omega).
\end{eqnarray}
We have
\begin{eqnarray}\label{eq24}
-\Delta_p u_0(z)=\hat{\lambda}_1u_0(z)^{p-1}+f(z,u_0(z))\ \mbox{a.e. in}\ \Omega,\ \frac{\partial u_0}{\partial n_p}+\beta(z)u_0^{p-1}=0\ \mbox{on}\ \partial\Omega.
\end{eqnarray}
Let
\begin{eqnarray}\label{eq25}
R(\hat{u}_1,u_0)(z)=|D\hat{u}_1(z)|^p-|Du_0(z)|^{p-2}\left(Du_0(z),D\left(\frac{\hat{u}_1^p}{u_0^{p-1}}\right)(z)\right)_{\RR^N}.
\end{eqnarray}
From the nonlinear Picone's identity of Allegretto and Huang \cite{2}, we have
\begin{eqnarray}\label{eq26}
0&\leq&\int\limits_{\Omega}R(\hat{u}_1,u_0)dz\nonumber\\
&=&\|D\hat{u}_1\|_p^p-\int\limits_{\Omega}|Du_0|^{p-2}\left(Du_0,D\left(\frac{\hat{u}_1^p}{u_0^{p-1}}\right)\right)_{\RR^N}dz.
\end{eqnarray}
From (\ref{eq23}), (\ref{eq24}) and the nonlinear Green's identity (see, for example, Gasinski and Papageorgiou \cite[p. 211]{9}), we have
\begin{eqnarray}\label{eq27}
&&\int\limits_{\Omega}|Du_0|^{p-2}\left(Du_0,D\left(\frac{\hat{u}_1^p}{u_0^{p-1}}\right)\right)_{\RR^N}dz=\int\limits_{\Omega}(-\Delta_p u_0)\left(\frac{\hat{u}_1^p}{u_0^{p-1}}\right)dz+\left\langle\frac{\partial u_0}{\partial n_p},\frac{\hat{u}_1^p}{u_0^{p-1}}\right\rangle_{\partial\Omega}.
\end{eqnarray}
Returning to (\ref{eq26}) and using (\ref{eq24}) and (\ref{eq27}), we obtain
\begin{eqnarray*}
0&\leq&\|D\hat{u}_1\|_p^p-\hat{\lambda}_1\|\hat{u}_1\|_p^p-\int\limits_{\Omega}f(z,u_0)\frac{\hat{u}_1^p}{u_0^{p-1}}dz+\int\limits_{\partial\Omega} \beta(z)\hat{u}_1^pd\sigma\\
&=&-\int\limits_{\Omega}f(z,u_0)\frac{\hat{u}_1^p}{u_0^{p-1}}dz<0\ (\mbox{see}\ H(f)),
\end{eqnarray*}
a contradiction. So, $\hat{\lambda}_1\notin \mathcal{P}$.
\end{proof}

From Propositions \ref{prop6} and \ref{prop9} it follows that
$$\mathcal{P}=(-\infty,\hat{\lambda}_1)$$
(recall that in the proof of Proposition \ref{prop5} we established that $(-\infty,\hat{\lambda}_1)\subseteq \mathcal{P}$).
\begin{prop}\label{prop10}
If hypotheses $H(\beta)$ and $H(f)$ hold, $\lambda\in \mathcal{P}$ and $u_{\lambda}\in S(\lambda)\subseteq\mbox{int}\, C_{+}$, then for every $\mu<\lambda$, we can find $u_{\mu}\in S(\mu)\subseteq\mbox{int}\, C_{+}$ such that  $u_{\mu}\leq u_{\lambda}$.
\end{prop}
\begin{proof}
We consider the following truncation-perturbation of the reaction in problem $(P_{\mu})$:
\begin{eqnarray}\label{eq28}
\gamma_{\mu}(z,x)=\left\{  \begin{array}{ll}
    0\ &\mbox{if}\ x<0\\
    (\mu+1)x^{p-1}+f(z,x)\ &\mbox{if}\ 0\leq x\leq u_{\lambda}(z)\\
    (\mu+1)u_{\lambda}(z)^{p-1}+f(z,u_{\lambda}(z))\ &\mbox{if}\ u_{\lambda}(z)<x.
    \end{array}\right.
\end{eqnarray}
This is a Carath\'eodory function. We set $\Gamma_{\mu}(z,x)=\int\limits_0^x\gamma_{\mu}(z,s)ds$ and consider the $C^1$-functional $\eta:W^{1,p}(\Omega)\to\RR$ defined by
$$\eta(u)=\frac{1}{p}\|Du\|_p^p+\frac{1}{p}\|u\|_p^p+\frac{1}{p}\int\limits_{\partial\Omega}\beta(z)(u^{+})^pd\sigma-\int\limits_{\Omega} \Gamma_{\mu}(z,u)dz\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$
From hypothesis $H(\beta)$ and (\ref{eq28}) it is clear that $\eta$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find $u_{\mu}\in W^{1,p}(\Omega)$ such that
\begin{eqnarray}\label{eq29}
\eta(u_{\mu})=\inf [\eta(u):\ u\in W^{1,p}(\Omega)].
\end{eqnarray}
As before (see the proof of Proposition \ref{prop5}), using hypothesis $H(f)(iii)$, we show that for $t\in (0,1)$ small (at least such that $t\hat{u}_1(z)\leq\min\limits_{\overline{\Omega}}u_{\lambda}$, recall $u_{\lambda}\in\mbox{int}\, C_{+}$), we have
\begin{eqnarray*}
&&\eta(t\hat{u}_1)<0,\\
&\Rightarrow&\eta(u_{\mu})<0=\eta(0)\ (\mbox{see}\ (\ref{eq29})),\ \mbox{hence}\ u_{\mu}\neq 0.
\end{eqnarray*}
From (\ref{eq29}), we have
\begin{eqnarray*}
&&\eta'(u_{\mu})=0,\\
&\Rightarrow&\langle A(u_{\mu}),h\rangle+\int\limits_{\Omega}|u_{\mu}|^{p-2}u_{\mu} h dz+\int\limits_{\partial\Omega}\beta(z)(u_{\mu}^{+})^{p-1}h d\sigma=\int\limits_{\Omega}\gamma_{\mu}(z,u_{\mu})h dz\\
&&\hspace{8cm} \mbox{for all}\ h\in W^{1,p}(\Omega).
\end{eqnarray*}
As in the proof of Proposition \ref{prop8}, choosing first $h=-u_{\mu}^{-}\in W^{1,p}(\Omega)$ and then $h=(u_{\mu}-u_{\lambda})^{+}\in W^{1,p}(\Omega)$, we show that
$$u_{\mu}\in [0,u_{\lambda}]\setminus\{0\}.$$
From (\ref{eq28}) it follows that $u_{\mu}\in S(\mu)\subseteq\mbox{int}\, C_{+}$ and $u_{\mu}\leq u_{\lambda}$.
\end{proof}

\begin{prop}\label{prop11}
If hypotheses $H(\beta)$ and $H(f)$ hold, $\lambda\in \mathcal{P}=(-\infty,\hat{\lambda}_1)$, then problem $(P_{\lambda})$ admits a smallest positive solution $u_{\lambda}^*\in S(\lambda)\subseteq\mbox{int}\, C_{+}$.
\end{prop}
\begin{proof}
From Dunford and Schwartz \cite[p. 336]{7}, we know that we can find $\{u_n\}_{n\geq 1}\subseteq S(\lambda)$ such that
$$\inf S(\lambda)=\inf\limits_{n\geq 1}u_n.$$
From Proposition \ref{prop10} and since $S(\lambda)$ is downward directed, we may assume that
\begin{eqnarray}\label{eq30}
u_n\leq \hat{u}\ \mbox{for all}\ n\geq 1,\ \mbox{with}\ \hat{u}\in S(\hat{\lambda})\subseteq\mbox{int}\, C_{+},\ \hat{\lambda}\in \mathcal{P},\ \lambda_n<\hat{\lambda},\ n\geq 1.
\end{eqnarray}
We have
\begin{eqnarray}\label{eq31}
&&\langle A(u_n),h\rangle+\int\limits_{\partial\Omega}\beta(z)u_n^{p-1}h d\sigma=\lambda\int\limits_{\Omega}u_n^{p-1}h dz+\int\limits_{\Omega}f(z,u_n)h dz\\
&&\hspace{7cm} \mbox{for all}\ h\in W^{1,p}(\Omega),\mbox{all}\ n\geq 1.\nonumber
\end{eqnarray}
In (\ref{eq31}) we choose $h=u_n\in W^{1,p}(\Omega)$. Then using hypotheses $H(\beta),H(f)(i)$ and (\ref{eq30}) we see that
$$\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)\ \mbox{is bounded}.$$
So, we may assume that
\begin{eqnarray}\label{eq32}
u_n\overset{w}{\to} u_{\lambda}^*\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{and}\ u_n\to u_{\lambda}^*\ \mbox{in}\ L^p(\Omega)\ \mbox{and in}\ L^p(\partial\Omega).
\end{eqnarray}
In (\ref{eq31}) we choose $h=u_n-u_{\lambda}^*\in W^{1,p}(\Omega)$, pass to the limit as $n\to\infty$ and use (\ref{eq32}). We obtain
\begin{eqnarray}\label{eq33}
&&\lim\limits_{n\to\infty}\langle A(u_n),u_n-u_{\lambda}^*\rangle=0,\nonumber\\
&\Rightarrow& u_n\to u_{\lambda}^*\ \mbox{in}\ W^{1,p}(\Omega)\ (\mbox{see Proposition}\ \ref{prop2}\ \mbox{and}\ (\ref{eq32})).
\end{eqnarray}
So, if in (\ref{eq31}) we pass to the limit as $n\to\infty$ and use (\ref{eq33}) and Proposition \ref{prop2}, then
\begin{eqnarray}\label{eq34}
&&\langle A(u_{\lambda}^*),h\rangle+\int\limits_{\partial\Omega}\beta(z)(u_{\lambda}^*)^{p-1}h d\sigma=\lambda\int\limits_{\Omega}(u_{\lambda}^*)^{p-1}h dz+\int\limits_{\Omega}f(z,u_{\lambda}^*)h dz\nonumber\\
 &&\hspace{8cm}\mbox{for all}\ h\in W^{1,p}(\Omega).\nonumber\\
&\Rightarrow& -\Delta_p u_{\lambda}^*(z)=\lambda (u_{\lambda}^*)(z)^{p-1}+f(z,u_{\lambda}^*(z))\ \mbox{a.e. in}\ \Omega,\ \frac{\partial u_{\lambda}^*}{\partial n_p}+\beta(z)(u_{\lambda}^*)^{p-1}=0\ \mbox{on}\ \partial\Omega
\end{eqnarray}
(as in the proof of Proposition \ref{prop5}). Moreover, from Proposition \ref{prop8}, we have
\begin{eqnarray}\label{eq35}
&&\overline{u}\leq u_n\ \mbox{for all}\ n\geq 1.\nonumber\\
&\Rightarrow& \overline{u}\leq u_{\lambda}^*\ (\mbox{see}\ (\ref{eq33})).
\end{eqnarray}
Then (\ref{eq34}) and (\ref{eq35}) imply that
$$u_{\lambda}^*\in S(\lambda)\ \mbox{and}\ u_{\lambda}^*=\inf S(\lambda).$$
\end{proof}
If we strengthen the conditions on the perturbation $f(z,\cdot)$, we can guarantee the uniqueness of the positive solution of problem $(P_{\lambda})$.

The new stronger conditions on $f(z,x)$ are the following:

\smallskip
$H(f)'$: $f:\Omega\times\RR\to\RR$ is a Carath\'eodory function such that  for a.a. $z\in\Omega$, $f(z,0)=0$, $f(z,x)>0$ for all $x>0$, hypotheses $H(f)'(i),(ii),(iii)$ are the same as the corresponding hypotheses $H(f)(i),(ii),(iii)$ and

$(iv)$ for a.a. $z\in\Omega$, $x\to\frac{f(z,x)}{x^{p-1}}$ is decreasing, strictly for all $z\in\Omega_0\subseteq\Omega$ with $|\Omega_0|_N>0$.
\begin{prop}\label{prop12}
If hypotheses $H(\beta)$ and $H(f)'$ hold and $\lambda\in \mathcal{P}=(-\infty,\hat{\lambda}_1)$, then $S(\lambda)$ is a singleton $\{u_{\lambda}\}$ and the map $\lambda\longmapsto u_{\lambda}$ is continuous from $(-\infty,\hat{\lambda}_1)$ into $C^1(\overline{\Omega})$ and increasing (that is, if $\mu<\lambda$, then $u_{\lambda}-u_{\mu}\in C_{+})$.
\end{prop}
\begin{proof}
We already know that for all $\lambda\in (-\infty,\hat{\lambda}_1)$, $S(\lambda)\neq\emptyset$.

Let $u,v\in S(\lambda)\subseteq\mbox{int}\, C_{+}$. Then as in the proof of Proposition \ref{prop7}, we have
\begin{eqnarray*}
0&\leq&\frac{1}{p}\int\limits_{\Omega}\left[\frac{-\Delta_p u}{u^{p-1}}-\frac{-\Delta_p v}{v^{p-1}}\right](u^p-v^p)dz\\
&=&\frac{1}{p}\int\limits_{\Omega}\left[\frac{f(z,u)}{u^{p-1}}-\frac{f(z,v)}{v^{p-1}}\right](u^p-v^p)dz\leq 0,\\
&\Rightarrow& u=v\ (\mbox{see hypothesis}\ H(f)'(iv)),\\
&\Rightarrow& S(\lambda)=\{u_{\lambda}\}\ (\mbox{a singleton}).
\end{eqnarray*}

Next we show the continuity of $\lambda\longmapsto u_{\lambda}$. To this end, suppose $\{\lambda_n\}_{n\geq 1}\subseteq(-\infty,\hat{\lambda}_1)$ and assume that $\lambda_n\to\lambda\in (-\infty,\hat{\lambda}_1)$. Let $u_n=u_{\lambda_n}\in S(\lambda_n)\subseteq\mbox{int}\, C_{+}$, $n\geq 1$. We can find $\hat{\lambda}\in (-\infty,\hat{\lambda}_1)$ such that  $\lambda_n\leq\hat{\lambda}$ for all $n\geq 1$. Let $\hat{u}\in S(\hat{\lambda})\subseteq\mbox{int}\, C_{+}$. Proposition \ref{prop8} and \ref{prop10} imply that
\begin{eqnarray}\label{eq36}
\overline{u}\leq u_n\leq\hat{u}\ \mbox{for all}\ n\geq 1.
\end{eqnarray}
Also, we have
\begin{eqnarray}\label{eq37}
&&\langle A(u_n),h\rangle+\int\limits_{\partial\Omega}\beta(z)u_n^{p-1}h d\sigma=\lambda\int\limits_{\Omega}u_n^{p-1}h dz+\int\limits_{\Omega}f(z,u_n)h dz\ \mbox{for all}\ h\in W^{1,p}(\Omega).
\end{eqnarray}
Choosing $h=u_n\in W^{1,p}(\Omega)$ and using hypotheses $H(\beta),H(f)(i)$ and (\ref{eq36}), we see that
$$\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)\ \mbox{is bounded}.$$
So, we may assume that
\begin{eqnarray}\label{eq38}
u_n\overset{w}{\to} u_{\lambda}\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{and}\ u_n\to u_{\lambda}\ \mbox{in}\ L^p(\Omega)\ \mbox{and in}\ L^p(\partial\Omega).
\end{eqnarray}
If in (\ref{eq37}) we choose $h=u_n-u_{\lambda}\in W^{1,p}(\Omega)$, pass to the limit as $n\to\infty$ and use (\ref{eq38}), then
\begin{eqnarray}\label{eq39}
&&\lim\limits_{n\to\infty}\langle A(u_n),u_n-u_{\lambda}\rangle=0,\nonumber\\
&\Rightarrow&u_n\to u_{\lambda}\ \mbox{in}\ W^{1,p}(\Omega).
\end{eqnarray}
So, if in (\ref{eq37}) we pass to the limit as $n\to\infty$ and use (\ref{eq39}) and Proposition \ref{prop2}, then
\begin{eqnarray*}
&&\langle A(u_{\lambda}),h\rangle+\int\limits_{\partial\Omega}\beta(z)u_{\lambda}^{p-1}h d\sigma=\lambda\int\limits_{\Omega}u_{\lambda}^{p-1}h dz+\int\limits_{\Omega}f(z,u_{\lambda})h dz\ \mbox{for all}\ h\in W^{1,p}(\Omega),\\
&\Rightarrow& u_{\lambda}\in S(\lambda)\subseteq \mbox{int}\, C_{+}.
\end{eqnarray*}
Since $S(\lambda)$ is a singleton, we have
\begin{eqnarray}\label{eq40}
u_n\to u_{\lambda}\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{for the original sequence}.
\end{eqnarray}
From Theorem $2$ of Lieberman \cite{13}, we know that we can find $\alpha\in (0,1)$ and $c_9>0$ such that
\begin{eqnarray}\label{eq41}
u_n\in C^{1,\alpha}(\overline{\Omega})\ \mbox{and}\ \|u_n\|_{C^{1,\alpha}(\overline{\Omega})}\leq c_9\ \mbox{for all}\ n\geq 1.
\end{eqnarray}
Exploiting the compact embedding of $C^{1,\alpha}(\overline{\Omega})$ into $C^1(\overline{\Omega})$, from (\ref{eq40}) and (\ref{eq41}) it follows that
\begin{eqnarray*}
&&u_n\to u_{\lambda}\ \mbox{in}\ C^1(\overline{\Omega}),\\
&\Rightarrow& \lambda\longmapsto u_{\lambda}\ \mbox{is continuous from}\ (-\infty,\hat{\lambda}_1)\ \mbox{into}\ C^1(\overline{\Omega}).
\end{eqnarray*}
Finally the monotonicity of $\lambda\longmapsto u_{\lambda}$ follows from Proposition \ref{prop10}.
\end{proof}

In fact the monotonicity conclusion in the above proposition, can be improved provided we strengthen further the conditions on $f(z,\cdot)$.

The new stronger conditions on the perturbation $f(z,x)$ are the following:

\smallskip
$H(f)''$: $f:\Omega\times\RR\to\RR$ is a Carath\'eodory function such that  for a.a. $z\in\Omega$, $f(z,0)=0$, $f(z,x)>0$ for all $x>0$, hypotheses $H(f)''(i),(ii),(iii),(iv)$ are the same as the corresponding hypotheses $H(f)'(i),(ii),(iii),(iv)$ and

$(v)$ for every $\rho>0$, there exists $\xi_{\rho}>0$ such that  for a.a. $z\in\Omega$, the mapping $x\longmapsto f(z,x)+\xi_{\rho}x^{p-1}$ is nondecreasing on $[0,\rho]$.

\smallskip
Under these new conditions on the perturbation $f(z,x)$, we have the following result.
\begin{prop}\label{prop13}
If hypotheses $H(\beta)$ and $H(f)''$ hold, then the mapping $\lambda\longmapsto u_{\lambda}$ from $(-\infty,\hat{\lambda}_1)$ into $C^1(\overline{\Omega})$ is strictly increasing, that is, if $\lambda<\vartheta\in (-\infty,\hat{\lambda}_1)$, then $u_{\vartheta}-u_{\lambda}\in\mbox{int}\, C_{+}$.
\end{prop}
\begin{proof}
From Proposition \ref{prop12}, we know that $u_{\vartheta}-u_{\lambda}\in C_{+}$. Let $\rho=\|u_{\vartheta}\|_{\infty}$ and let $\xi_{\rho}>0$ be as postulated by hypothesis $H(f)''(v)$. Also, for $\delta>0$, let $u_{\lambda}^{\delta}=u_{\lambda}+\delta\in \mbox{int}\, C_{+}$. We have
\begin{eqnarray*}
&&-\Delta_p u_{\lambda}^{\delta}(z)+\xi_{\rho}u_{\lambda}^{\delta}(z)^{p-1}\\
&\leq&-\Delta_p u_{\lambda}(z)+\xi_{\rho}u_{\lambda}(z)^{p-1}+\gamma(\delta)\ \mbox{with}\ \gamma(\delta)\to 0^{+}\ \mbox{as}\ \delta\to 0^{+}\\
&=&\lambda u_{\lambda}(z)^{p-1}+f(z,u_{\lambda}(z))+\xi_{\rho}u_{\lambda}(z)^{p-1}+\gamma(\delta)\\
&\leq&\lambda u_{\vartheta}(z)^{p-1}+f(z,u_{\vartheta}(z))+\xi_{\rho}u_{\vartheta}(z)^{p-1}+\gamma(\delta)\\
&&\hspace{5cm}(\mbox{see}\ H(f)''(v)\ \mbox{and recall}\ u_{\lambda}\leq u_{\vartheta})\\
&=&\vartheta u_{\vartheta}(z)^{p-1}+f(z,u_{\vartheta}(z))+\xi_{\rho}u_{\vartheta}(z)^{p-1}-(\vartheta-\lambda)u_{\vartheta}(z)^{p-1}+\gamma(\delta)\\
&\leq&\vartheta u_{\vartheta}(z)^{p-1}+f(z,u_{\vartheta}(z))+\xi_{\rho}u_{\vartheta}(z)^{p-1}-(\vartheta-\lambda)\hat{m}_{\vartheta}^{p-1}+\gamma(\delta)\\
&&\hspace{5cm}\mbox{with}\ \hat{m}_{\vartheta}=\min\limits_{\overline{\Omega}}u_{\vartheta}>0\\
&\leq&-\Delta_p u_{\vartheta}(z)+\xi_{\rho}u_{\vartheta}(z)^{p-1}\ \mbox{for a.a.}\ z\in\Omega\ \mbox{and for}\ \delta>0\ \mbox{small}, \\
&\Rightarrow& u_{\lambda}^{\delta}\leq u_{\vartheta},\\
&\Rightarrow& u_{\vartheta}-u_{\lambda}\in\mbox{int}\, C_{+}.
\end{eqnarray*}
\end{proof}

The next theorem summarizes the situation for problem $(P_{\lambda})$ when the perturbation $f(z,x)$ is $(p-1)$-sublinear in $x\in\RR$.
\begin{theorem}\label{th14}
\begin{itemize}
\item[(a)] If hypotheses $H(\beta)$ and $H(f)$ hold, then for all $\lambda\in (-\infty,\hat{\lambda}_1)$, $S(\lambda)\neq\emptyset$, $S(\lambda)\subseteq\mbox{int}\, C_{+}$ and $S(\lambda)$ admits a smallest element $u_{\lambda}^*\in\mbox{int}\, C_{+}$; if $\lambda\geq\hat{\lambda}_1$, then $S(\lambda)=\emptyset$.
\item[(b)] If hypotheses $H(\beta)$ and $H(f)'$ hold, then for all $\lambda\in (-\infty,\hat{\lambda}_1)$, $S(\lambda)=\{u_{\lambda}\}$ and the map $\lambda\longmapsto u_{\lambda}$ is continuous and increasing (that is, $\lambda\leq\vartheta\Rightarrow u_{\vartheta}-u_{\lambda}\in C_{+}$).
\item[(c)] If hypotheses $H(\beta)$ and $H(f)''$ hold, then the map $\lambda\longmapsto u_{\lambda}$ is strictly increasing (that is, $\lambda<\vartheta\in (-\infty,\hat{\lambda}_1)\Rightarrow u_{\vartheta}-u_{\lambda}\in\mbox{int}\, C_{+}$).
\end{itemize}
\end{theorem}

\section{Superlinear Perturbation}\label{sec4}
In this section, we examine problem $(P_{\lambda})$ when the perturbation $f(z,\cdot)$ is $(p-1)$-superlinear, but without satisfying the usual in such cases Ambrosetti-Rabinowitz condition ($AR$-condition for short). Now we can not hope for uniqueness and we have multiplicity of positive solutions.

The hypotheses on the perturbation $f(z,x)$, are the following:

\smallskip
$H(f)_1$: $f:\Omega\times\RR\to\RR$ is a Carath\'eodory function such that  for a.a. $z\in\Omega$, $f(z,0)=0$, $f(z,x)>0$ for all $x>0$ and
\begin{itemize}
\item[(i)] $f(z,x)\leq a(z)(1+x^{r-1})$ for a.a. $z\in\Omega$, all $x\geq 0$, with $a\in L^{\infty}(\Omega)_{+}$ and $p<r<p^*$;
\item[(ii)] if $F(z,x)=\int\limits_0^x f(z,s)ds$, then $\lim\limits_{x\to +\infty}\frac{F(z,x)}{x^p}=+\infty$ uniformly for a.a. $z\in\Omega$;
\item[(iii)] there exists $\mu\in \left((r-p)\max\left\{1,\frac{N}{p}\right\},p^*\right)$ such that
$$0<\eta_0\leq\liminf\limits_{x\to +\infty}\frac{f(z,x)x-pF(z,x)}{x^{\mu}}\ \mbox{uniformly for a.a.}\ z\in\Omega;$$
\item[(iv)] $\lim\limits_{x\to 0^{+}}\frac{f(z,x)}{x^{p-1}}=0$ uniformly for a.a. $z\in\Omega;$
\item[(v)] for every $\rho>0$, there exists $\xi_{\rho}>0$ such that  for a.a. $z\in\Omega$, the map $x\longmapsto f(z,x)+\xi_{\rho}x^{p-1}$ is nondecreasing on $[0,\rho]$.
\end{itemize}
\begin{remark}
As before, since we are interested on positive solutions and the above hypotheses concern the positive semiaxis $\RR_{+}=[0,+\infty)$, without any loss of generality, we assume that $f(z,x)=0$ for a.a. $z\in\Omega$, all $x\leq 0$. From hypotheses $H(f)_1(ii),(iii)$ it follows that
$$\lim\limits_{x\to +\infty}\frac{f(z,x)}{x^{p-1}}=+\infty\ \mbox{uniformly for a.a.}\ z\in\Omega.$$
\end{remark}
So, for a.a. $z\in\Omega$, $f(z,\cdot)$ is $(p-1)$-superlinear. However, we do not employ the usual in such cases $AR$-condition (unilateral version) which says that there exist $q>p$ and $M>0$ such that
$$0<qF(z,x)\leq f(z,x)x\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\geq M\ (\mbox{see \cite{3}}).$$
This implies that
$$c_{10}x^q\leq F(z,x)\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\geq M,\ \mbox{some}\ c_{10}>0.$$
Here instead, we employ the weaker condition $H(f)(iii)$ which incorporates in our framework $(p-1)$-superlinear perturbations, with ``slower" growth near $+\infty$ (see the examples below). A similar polynomial growth is assumed near $0^{+}$ by virtue of hypothesis $H(f)_1(iv)$.
\begin{ex}
The following functions satisfy hypotheses $H(f)_1$. For the sake of simplicity, we drop the $z$-dependence:
\begin{eqnarray*}
&&f_1(x)=x^{r-1}\ \mbox{for all}\ x\geq 0\ \mbox{with}\ p<r<p^*,\\
&&f_2(x)=x^{p-1}\left(\ln x+\frac{1}{p}\right)\ \mbox{for all}\ x\geq 0.
\end{eqnarray*}
\end{ex}
Note that $f_2$ does not satisfy the $AR$-condition.

The sets $\mathcal{P}$ and $S(\lambda)$ have the same meaning as in Section \ref{sec3}.
\begin{prop}\label{prop15}
If hypotheses $H(\beta)$ and $H(f)_1$ hold, then $\mathcal{P}\neq\emptyset$ and $S(\lambda)\subseteq\mbox{int}\, C_{+}$.
\end{prop}
\begin{proof}
	For $\lambda\in \RR$, we consider the $C^1$-functional $\varphi_{\lambda}:W^{1,p}(\Omega)\rightarrow\RR$ defined by
	\begin{eqnarray*}
		 &&\varphi_{\lambda}(u)=\frac{1}{p}||Du||^{p}_{p}+\frac{1}{p}\int\limits_{\partial\Omega}\beta(z)(u^+)^pd\sigma+\frac{1}{p}||u^-||^{p}_{p}-\frac{\lambda}{p}||u^+||^{p}_{p}-\int_{\Omega}F(z,u^+)dz\\
		&&\hspace{7.3cm}\mbox{for all}\ u\in W^{1,p}(\Omega).
	\end{eqnarray*}
	
	Hypotheses $H(f)_1(i),(iv)$ imply that  given $\varepsilon>0$, we can find $c_{11}=c_{11}(\varepsilon)>0$ such that
	\begin{eqnarray}\label{eq42}
		F(z,x)\leq\frac{\varepsilon}{p}x^p+c_{11}x^r\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\geq 0.
	\end{eqnarray}
	
	Let $\lambda<\hat{\lambda}_1$. Then for any $u\in W^{1,p}(\Omega)$ we have
	\begin{eqnarray*}
		 &\varphi_{\lambda}(u)&=\frac{1}{p}||Du^+||^{p}_{p}+\frac{1}{p}\int\limits_{\partial\Omega}\beta(z)(u^+)^pd\sigma-\frac{\lambda}{p}||u^+||^{p}_{p}+\frac{1}{p}||Du^-||^{p}_{p}+\\
		&&\hspace{1.5cm}+\frac{1}{p}||u^-||^{p}_{p}-\frac{\varepsilon}{p}||u^+||^{p}_{p}-c_{12}||u||^r\ \mbox{for some}\ c_{12}>0\ (\mbox{see (\ref{eq42})}).\\
		&&\geq\left(c_{13}-\frac{\varepsilon}{p}\right)||u^+||^p+\frac{1}{p}||u^-||^p-c_{12}||u||^r\ \mbox{for some}\ c_{13}>0\\
		&&\hspace{3cm}(\mbox{see Proposition \ref{prop4} and recall}\ \lambda<\hat{\lambda}_1).
	\end{eqnarray*}
	
Choosing $\varepsilon\in(0,p\ c_{13})$, we have
$$\varphi_{\lambda}(u)\geq c_{14}||u||^p-c_{12}||u||^r\ \mbox{for some}\ c_{14}>0.$$

Since $r>p$, if we choose $\rho\in(0,1)$ small, we have
\begin{eqnarray*}
	&&\varphi_{\lambda}(u)>0=\varphi_{\lambda}(0)\ \mbox{for all}\ u\in W^{1,p}(\Omega)\ \mbox{with}\ 0<||u||\leq\rho,\\
	&\Rightarrow&u=0\ \mbox{is a (strict) local minimizer of}\ \varphi_{\lambda}.
\end{eqnarray*}	

So, we can find $\rho\in(0,1)$ small such that
\begin{eqnarray}\label{eq43}
	\varphi_{\lambda}(0)=0<\inf\left[\varphi_{\lambda}(u):||u||=\rho\right]=m_{\rho}
\end{eqnarray}
(see, for example, Aizicovici, Papageorgiou and Staicu \cite{1}, proof of Proposition 29).

By virtue of hypothesis $H(f)_1(ii)$, we see that for every $u\in \mbox{int}\, C_+$, we have
\begin{eqnarray}\label{eq44}
	\varphi_{\lambda}(tu)\rightarrow-\infty\ \mbox{as}\ t\rightarrow+\infty.
\end{eqnarray}

Moreover, as in Gasinski and Papageorgiou \cite{10}, we can check that
\begin{eqnarray}\label{eq45}
	\varphi_{\lambda}\ \mbox{satisfies the}\ C\mbox{-condition}.
\end{eqnarray}

Because of (\ref{eq43}), (\ref{eq44}) and (\ref{eq45}), we can apply Theorem \ref{th1} (the mountain pass theorem) and obtain $u_{\lambda}\in W^{1,p}(\Omega)$ such that
\begin{eqnarray}\label{eq46}
	\varphi_{\lambda}(0)=0<m_{\rho}\leq\varphi_{\lambda}(u_{\lambda})\ \mbox{and}\ \varphi'_{\lambda}(u_{\lambda})=0.
\end{eqnarray}

From (\ref{eq46}) we have $u_{\lambda}\neq 0$ and
\begin{eqnarray}\label{eq47}
	&&\left\langle A(u_{\lambda}),h\right\rangle+\int\limits_{\partial\Omega}\beta(z)(u^{+}_{\lambda})^{p-1}hd\sigma-\int\limits_{\Omega}(u^{-}_{\lambda})^{p-1}hdz=\lambda\int\limits_{\Omega}(u^{+}_{\lambda})^{p-1}hdz+\nonumber\\
	&&\hspace{1.5cm}+\int\limits_{\Omega}f(z,u^+)hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega).
\end{eqnarray}

In (\ref{eq47}) we choose $h=-u^{-}_{\lambda}\in W^{1,p}(\Omega)$ and we infer that $u_{\lambda}\geq 0$, $u_{\lambda}\neq 0$. So, we have
\begin{eqnarray*}
	&&\left\langle A(u_{\lambda}),h\right\rangle+\int\limits_{\partial\Omega}\beta(z)u_{\lambda}^{p-1}hd\sigma=\lambda\int\limits_{\Omega}u_{\lambda}^{p-1}hdz+\int\limits_{\Omega}f(z,u)hdz\\
	&&\hspace{8cm}\mbox{for all}\ h\in W^{1,p}(\Omega),\\
	&\Rightarrow&u_{\lambda}\in S(\lambda)\ (\mbox{see the proof of Proposition \ref{prop5}}).
\end{eqnarray*}

The nonlinear regularity theory implies $u_{\lambda}\in C_+\backslash\{0\}$. Let $\rho=||u_{\lambda}||_{\infty}$ and let $\xi_{\rho}>0$ be as postulated by hypothesis $H(f)_1(v)$. Then
\begin{eqnarray*}
	&&-\Delta_pu_{\lambda}(z)+\xi_{\rho}u_{\lambda}(z)^{p-1}=\lambda u_{\lambda}(z)^{p-1}+f(z,u_{\lambda}(z))+\xi_{\rho}u_{\lambda}(z)^{p-1}\geq 0\ \mbox{a.e. in}\ \Omega,\\
	&\Rightarrow&\Delta_p u_{\lambda}(z)\leq\xi_{\rho}u_{\lambda}(z)^{p-1}\ \mbox{a.e. in}\ \Omega,\\
	&\Rightarrow&u_{\lambda}\in \mbox{int}\, C_+\ (\mbox{see Vazquez \cite{16}}).
\end{eqnarray*}

Therefore we have proved that $\mathcal{P}\neq\varnothing$ (in fact $(-\infty,\hat{\lambda}_1)\subseteq\mathcal{P}$) and that $S(\lambda)\subseteq \mbox{int}\, C_+$.
\end{proof}

The proof of the next proposition is identical to the proof of Proposition \ref{prop6}.
\begin{prop}\label{prop16}
	If hypotheses $H(\beta)$ and $H(f)_1$ hold and $\lambda\in\mathcal{P}$, then $\left(-\infty,\lambda\right]\subseteq\mathcal{P}$.
\end{prop}

Moreover, as in the proof of Proposition \ref{prop9}, using the nonlinear Picone's identity (see \cite{2}), we have:
\begin{prop}\label{prop17}
	If hypotheses $H(\beta)$ and $H(f)_1$ hold, then $\hat{\lambda}_1\not\in\mathcal{P}$ and so $\mathcal{P}=(-\infty,\hat{\lambda}_1)$.
\end{prop}

In fact, as we already mentioned, in this case for every $\lambda\in\mathcal{P}=(-\infty,\hat{\lambda}_1)$ problem $(P_{\lambda})$ has at least two positive solutions.
\begin{prop}\label{prop18}
	If hypotheses $H(\beta)$ and $H(f)_1$ hold and $\lambda\in\mathcal{P}=(-\infty,\hat{\lambda}_1)$, then problem $(P_{\lambda})$ has at least two positive solutions
	$$u_{\lambda},v_{\lambda}\in \mbox{int}\, C_+,\ u_{\lambda}\leq v_{\lambda},\ u_{\lambda}\neq v_{\lambda}.$$
\end{prop}
\begin{proof}
	Since $\lambda\in\mathcal{P}$, we can find $u_{\lambda}\in S(\lambda)\subseteq \mbox{int}\, C_+$. We introduce the following Carath\'eodory function:
	\begin{eqnarray}\label{eq48}
		k_{\lambda}(z,x)=\left\{
		\begin{array}{cl}
			(\lambda+1)u_{\lambda}(z)^{p-1}+f(z,u_{\lambda}(z))&\mbox{if}\ x\leq u_{\lambda}(z)\\
			(\lambda+1)x^{p-1}+f(z,x)&\mbox{if}\ u_{\lambda}(z)<x.
		\end{array}
		\right.
	\end{eqnarray}
	
	In addition we consider the following truncation of the boundary term (recall that $u_{\lambda}\in \mbox{int}\, C_+$):
	\begin{eqnarray}\label{eq49}
		d_{\lambda}(z,x)=\left\{
		\begin{array}{cl}
			u_{\lambda}(z)^{p-1}&\mbox{if}\ x\leq u_{\lambda}(z)\\
			x^{p-1}&\mbox{if}\ u_{\lambda}(z)<x
		\end{array}
		\right.\ \mbox{for all}\ (z,x)\in\partial\Omega\times\RR.
	\end{eqnarray}
	
	This is also a Carath\'eodory function on $\partial\Omega\times\RR$.
	
	Let $K_{\lambda}(z,x)=\int\limits^{x}_{0}k_{\lambda}(z,s)ds$ and $D_{\lambda}(z,x)=\int\limits^{x}_{0}d_{\lambda}(z,s)ds$ and consider the $C^1$-functional $\psi_{\lambda}:W^{1,p}(\Omega)\rightarrow\RR$ defined by
	 $$\psi_{\lambda}(u)=\frac{1}{p}||Du||^{p}_{p}+\frac{1}{p}||u||^{p}_{p}+\frac{1}{p}\int\limits_{\partial\Omega}\beta(z)D_{\lambda}(z,u)d\sigma-\int\limits_{\Omega}K_{\lambda}(z,u)dz\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$
	
	\begin{claim}\label{claim1} We have
		$$K_{\psi_{\lambda}}=\{u\in W^{1,p}(\Omega):\psi'_{\lambda}(u)=0\}\subseteq\left[u_{\lambda}\right)= \{u\in W^{1,p}(\Omega):u_{\lambda}(z)\leq u(z)\ \mbox{a.e. in}\ \Omega\}.$$
	\end{claim}
	
	To this end, let $u\in K_{\psi_{\lambda}}$. Then
	\begin{eqnarray}\label{eq50}
		&&\left\langle A(u),h\right\rangle+\int\limits_{\Omega}|u|^{p-2}uhdz+\int\limits_{\partial\Omega}\beta(z)d_{\lambda}(z,x)d\sigma=\int\limits_{\Omega}k_{\lambda}(z,u)hdz\\
		&&\hspace{7.5cm}\mbox{for all}\ h\in W^{1,p}(\Omega)\nonumber.
	\end{eqnarray}
	
	In (\ref{eq50}) we choose $h=(u_{\lambda}-u)^+\in W^{1,p}(\Omega)$. Then
	\begin{eqnarray*}
		&&\left\langle A(u),(u_{\lambda}-u)^+\right\rangle+\int\limits_{\Omega}|u|^{p-2}u(u_{\lambda}-u)^+dz+\int\limits_{\partial\Omega}\beta(z)u_{\lambda}^{p-1}(u_{\lambda}-u)^+d\sigma\\
		&&=\int\limits_{\Omega}[\lambda u_{\lambda}^{p-1}+f(z,u_{\lambda})](u_{\lambda}-u)^+dz+\int\limits_{\Omega}u_{\lambda}^{p-1}(u_{\lambda}-u)^+dz\ (\mbox{see (\ref{eq48}) and (\ref{eq49})})\\
		&&=\left\langle A(u_{\lambda}),(u_{\lambda}-u)^+\right\rangle+\int\limits_{\Omega}u_{\lambda}^{p-1}(u_{\lambda}-u)^+dz+\int\limits_{\partial\Omega}\beta(z)u_{\lambda}^{p-1}(u_{\lambda}-u)^+d\sigma,\\
		&\Rightarrow&\left\langle A(u_{\lambda})-A(u),(u_{\lambda}-u)^+\right\rangle+\int\limits_{\Omega}(u_{\lambda}^{p-1}-|u|^{p-2}u)(u_{\lambda}-u)^+dz\leq 0\ (\mbox{see}\ H(\beta)),\\
		&\Rightarrow&|\{u_{\lambda}>u\}|_N=0,\ \mbox{hence}\ u_{\lambda}\leq u\ \mbox{and so}\ u\in\left[u_{\lambda}\right).
	\end{eqnarray*}
	
	This proves Claim \ref{claim1}.
	\begin{claim}\label{claim2}
		Every $u\in K_{\psi_{\lambda}}$ belongs in $S(\lambda)$.
	\end{claim}
	
	From (\ref{eq50}) and Claim \ref{claim1}, we have
	\begin{eqnarray*}
		&&\left\langle A(u),h\right\rangle+\int\limits_{\Omega}u^{p-1}hdz+\int\limits_{\partial\Omega}\beta(z)u^{p-1}hd\sigma=\int\limits_{\Omega}[\lambda u^{p-1}+f(z,u)]hdz+\int\limits_{\Omega}u^{p-1}hdz\\
		&&\hspace{7cm}\mbox{for all}\ h\in W^{1,p}(\Omega)\ (\mbox{see (\ref{eq48}) and (\ref{eq49})}),\\
		&\Rightarrow&\left\langle A(u),h\right\rangle+\int\limits_{\partial\Omega}\beta(z)u^{p-1}hd\sigma=\int\limits_{\Omega}[\lambda u^{p-1}+f(z,u)]hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega).
	\end{eqnarray*}
	
	From this as in the proof of Proposition \ref{prop5}, we infer that $u\in S(\lambda)$.
	
	This proves Claim \ref{claim2}.
	\begin{claim}\label{claim3}
		We may assume that $u_{\lambda}\in \mbox{int}\, C_+$ is a local minimizer of $\psi_{\lambda}$.
	\end{claim}
	
	Let $\vartheta\in(\lambda,\hat{\lambda}_1)\subseteq\mathcal{P}$. We can find $u_{\vartheta}\in S(\vartheta)$. In fact as in the proof of Proposition \ref{prop6} we can have $u_{\lambda}\leq u_{\vartheta}$. Then we introduce the following truncation of $k_{\lambda}(z,\cdot)$:
	\begin{eqnarray}\label{eq51}
		\hat{k}_{\lambda}(z,x)=\left\{
		\begin{array}{cl}
			k_{\lambda}(z,x)&\mbox{if}\ x< u_{\vartheta}(z)\\
			k_{\lambda}(z,u_{\vartheta}(z))&\mbox{if}\ u_{\vartheta}(z)\leq x.
		\end{array}
		\right.
	\end{eqnarray}
	
	We also consider the corresponding truncation of the boundary term $d_{\lambda}(z,\cdot)$:
	\begin{eqnarray}\label{eq52}
		\hat{d}_{\lambda}(z,x)=\left\{
		\begin{array}{cl}
			d_{\lambda}(z,x)&\mbox{if}\ x< u_{\vartheta}(z)\\
			d_{\lambda}(z,u_{\vartheta}(z))&\mbox{if}\ u_{\vartheta}(z)\leq x
		\end{array}
		\right.\ \mbox{for all}\ (z,x)\in\partial\Omega\times\RR.
	\end{eqnarray}	
	
	Both are Carath\'eodory functions. We set
	$$\hat{K}_{\lambda}(z,x)=\int\limits^{x}_{0}\hat{k}_{\lambda}(z,s)ds\ \mbox{and}\ \hat{D}_{\lambda}(z,x)=\int\limits^{x}_{0}\hat{d}_{\lambda}(z,s)ds$$
	and consider the $C^1$-functional $\hat{\psi}_{\lambda}:W^{1,p}(\Omega)\rightarrow\RR$ defined by
	 $$\hat{\psi}_{\lambda}(u)=\frac{1}{p}||Du||^{p}_{p}+\frac{1}{p}||u||^{p}_{p}+\int\limits_{\partial\Omega}\beta(z)\hat{D}_{\lambda}(z,u)d\sigma-\int\limits_{\Omega}\hat{K}_{\lambda}(z,u)dz\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$
	
	From (\ref{eq51}) and (\ref{eq52}) it is clear that $\hat{\psi}_{\lambda}$ is coercive. Also, is is sequentially weakly lower semicontinuous. So, we can find $\hat{u}_{\lambda}\in W^{1,p}(\Omega)$ such that
	\begin{eqnarray}\label{eq53}
		 &&\hat{\psi}_{\lambda}(\hat{u}_{\lambda})=\inf\left[\hat{\psi}_{\lambda}(u):W^{1,p}(\Omega)\right]\nonumber\\
		&\Rightarrow&\hat{\psi}'_{\lambda}(\hat{u}_{\lambda})=0,\nonumber\\
		&\Rightarrow&\left\langle A(\hat{u}_{\lambda}),h\right\rangle+
\int\limits_{\Omega}|\hat{u}_{\lambda}|^{p-2}\hat{u}_{\lambda}hdz+
\int\limits_{\partial\Omega}\beta(z)\hat{d}_{\lambda}(z,u_\lambda )hd\sigma=\int\limits_{\Omega}\hat{k}_{\lambda}(z,u_\lambda )hdz\\
		&&\hspace{7cm}\mbox{for all}\ h\in W^{1,p}(\Omega).\nonumber
	\end{eqnarray}
	
	As in the proof of Claim \ref{claim1} earlier, choosing $h=(u_{\lambda}-\hat{\lambda}_{\lambda})^+\in W^{1,p}(\Omega)$ in (\ref{eq53}), we obtain
	$$u_{\lambda}\leq\hat{u}_{\lambda}.$$
	
	Next in (\ref{eq53}), we choose $h=(\hat{u}_{\lambda}-u_{\vartheta})^+\in W^{1,p}(\Omega)$. Then
	\begin{eqnarray*}
		&&\left\langle A(\hat{u}_{\lambda}),(\hat{u}_{\lambda}-u_{\vartheta})^+\right\rangle+\int\limits_{\Omega}\hat{u}_{\lambda}^{p-1}(\hat{u}_{\lambda}-u_{\vartheta})^+dz+\int\limits_{\partial\Omega}\beta(z)u_{\vartheta}^{p-1}(\hat{u}_{\lambda}-u_{\vartheta})^+d\sigma\\
		&=&\int\limits_{\Omega}[\lambda u_{\vartheta}^{p-1}+f(z,u_{\vartheta})](\hat{u}_{\lambda}-u_{\vartheta})^+dz+\int\limits_{\Omega}u_{\vartheta}^{p-1}(\hat{u}_{\lambda}-u_{\vartheta})^+dz\ (\mbox{see (\ref{eq51}) and (\ref{eq52})})\\
		&\leq&\int\limits_{\Omega}[\vartheta u_{\vartheta}^{p-1}+f(z,u_{\vartheta})](\hat{u}_{\lambda}-u_{\vartheta})^+dz+\int\limits_{\Omega}u_{\vartheta}^{p-1}(\hat{u}_{\lambda}-u_{\vartheta})^+dz\ (\mbox{since}\ \lambda<\vartheta)\\		
		&=&\left\langle A(u_{\vartheta}),(\hat{u}_{\lambda}-u_{\vartheta})^+\right\rangle+\int\limits_{\Omega}u_{\vartheta}^{p-1}(\hat{u}_{\lambda}-u_{\vartheta})^+dz+\int\limits_{\partial\Omega}\beta(z)u_{\vartheta}^{p-1}(\hat{u}_{\lambda}-u_{\vartheta})^+d\sigma,\\
		&\Rightarrow&\left\langle A(\hat{u}_{\lambda})-A(u_{\vartheta}),(\hat{u}_{\lambda}-u_{\vartheta})^+\right\rangle+\int\limits_{\Omega}(\hat{u}_{\lambda}^{p-1}-u_{\vartheta}^{p-1})(\hat{u}_{\lambda}-u_{\vartheta})^+dz\leq 0,\\
		&\Rightarrow&|\{\hat{u}_{\lambda}>u_{\vartheta}\}|_N=0,\ \mbox{hence}\ \hat{u}_{\lambda}\leq u_{\vartheta}.
	\end{eqnarray*}
	
	So, we have proved that
	$$\hat{u}_{\lambda}\in[u_{\lambda},u_{\vartheta}]=\{u\in W^{1,p}(\Omega):u_{\lambda}(z)\leq u(z)\leq u_{\vartheta}(z)\ \mbox{a.e. in}\ \Omega\}.$$
	
	Then from (\ref{eq51}), (\ref{eq52}) and Claim \ref{claim2}, it follows that $\hat{u}_{\lambda}\in S(\lambda)$. If $\hat{u}_{\lambda}\neq u_{\lambda}$, then this is desired second positive solution of problem $(P_{\lambda})$ and so we are done. Therefore, we may assume that $\hat{u}_{\lambda}=u_{\lambda}$.
	
	Note that $\hat{\psi}_{\lambda}|_{[0,u_{\vartheta}]}=\psi_{\lambda}|_{[0,u_{\vartheta}]}$ (see (\ref{eq51}) and (\ref{eq52})). Also as in the proof of Proposition \ref{prop13}, using $u^{\delta}_{\lambda}=u_{\lambda}+\delta\in \mbox{int}\, C_+\ (\delta>0)$ and hypothesis $H(f)_1(v)$, we show that
	\begin{eqnarray*}
		&&u_{\vartheta}-u_{\lambda}\in \mbox{int}\, C_+,\\
		&\Rightarrow&u_{\lambda}\in \mbox{int}_{C^1(\overline{\Omega})}[0,u_{\vartheta}],\\
		&\Rightarrow&u_{\lambda}\ \mbox{is a local}\ C^1(\overline{\Omega})-\mbox{minimizer of}\ \psi_{\lambda},\\
		&\Rightarrow&u_{\lambda}\ \mbox{is a local}\ W^{1,p}(\Omega)-\mbox{minimizer of}\ \psi_{\lambda}\ (\mbox{see Proposition \ref{prop3}}).
	\end{eqnarray*}
	
	This proves Claim \ref{claim3}.
	
	We assume that $K_{\psi_{\lambda}}$ is finite (or otherwise we are done since we already have an infinity of solutions (see Claim \ref{claim1} and (\ref{eq48}), (\ref{eq49}))). By virtue of Claim \ref{claim3}, we can find $\rho>0$ small such that
	\begin{eqnarray}\label{eq54}
		\psi_{\lambda}(u_0)<\inf\left[\psi_{\lambda}(u):||u-u_0||=\rho\right]=m_{\lambda}
	\end{eqnarray}
	(see Aizicovici, Papageorgiou and Staicu \cite{1}, proof of Proposition 29). If $\varphi_{\lambda}$ is as in the proof of Proposition \ref{prop15}, then
	$$\varphi_{\lambda}=\psi_{\lambda}+\xi^{*}_{\lambda}\ \mbox{with}\ \xi^{*}_{\lambda}\in\RR.$$
	
	So, if $u\in \mbox{int}\, C_+$, then
	\begin{eqnarray*}
		&&\psi_{\lambda}(tu)\rightarrow-\infty\ \mbox{as}\ t\rightarrow-\infty\ (\mbox{see (\ref{eq44})}),\\
		&&\psi_{\lambda}\ \mbox{satisfies the}\ C-\mbox{condition (see (\ref{eq45}))}.
	\end{eqnarray*}
	
	These two facts and (\ref{eq54}), permit the use of Theorem \ref{th1} (the mountain pass theorem). Hence we obtain $v_{\lambda}\in W^{1,p}(\Omega)$ such that
	\begin{eqnarray}\label{eq55}
		m_{\lambda}\leq\psi_{\lambda}(v_{\lambda})\ \mbox{and}\ v_{\lambda}\in K_{\psi_{\lambda}}.
	\end{eqnarray}
	
	From (\ref{eq54}), (\ref{eq55}) and Claims \ref{claim1} and \ref{claim2}, we infer that
	$$v_{\lambda}\in S(\lambda)\subseteq \mbox{int}\, C_+,\ u_{\lambda}\leq v_{\lambda},\ u_{\lambda}\neq v_{\lambda}.$$
\end{proof}

We can also establish the existence of a smallest positive solution.
\begin{prop}\label{prop19}
	If hypotheses $H(\beta)$ and $H(f)_1$ hold and $\lambda\in\mathcal{P}=(-\infty,\hat{\lambda}_1)$, then problem $(P_{\lambda})$ admits a smallest positive solution $u_{\lambda}^*\in \mbox{int}\, C_+$ and the map $\lambda\longmapsto u_{\lambda}^*$ is strictly increasing (that is, $\lambda<\vartheta\in(-\infty,\hat{\lambda}_1)\Rightarrow u_{\vartheta}^*-u_{\lambda}^*\in \mbox{int}\, C_+$).
\end{prop}
\begin{proof}
	As in the proof of Proposition \ref{prop11}, we can find $\{u_n\}_{n\geq 1}\subseteq S(\lambda)$ such that
	$$\inf S(\lambda)=\inf\limits_{n\geq 1}u_n$$
	
	Since $S(\lambda)$ is downward directed, we may assume that $\{u_n\}_{n\geq 1}$ is decreasing. So, we have
	\begin{eqnarray}\label{eq56}
		u_n\leq u_1\in \mbox{int}\, C_+\ \mbox{for all}\ n\geq 1.
	\end{eqnarray}
	
	We have
	\begin{eqnarray}\label{eq57}
		&&\left\langle A(u_n),h\right\rangle+\int\limits_{\partial\Omega}\beta(z)u_{n}^{p-1}hd\sigma=\lambda\int\limits_{\Omega}u_{n}^{p-1}hdz+\int\limits_{\Omega}f(z,u_n)hdz\\
		&&\hspace{8cm}\mbox{for all}\ h\in W^{1,p}(\Omega).\nonumber
	\end{eqnarray}
	
	Choosing $h=u_n\in W^{1,p}(\Omega)$ in (\ref{eq57}) and using (\ref{eq56}), we infer that
	$$\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)\ \mbox{is bounded}.$$
	
	So, we may assume that
	\begin{eqnarray}\label{eq58}
		u_n\stackrel{w}{\rightarrow}u_{\lambda}^*\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{and}\ u_n\rightarrow u_{\lambda}^*\ \mbox{in}\ L^r(\Omega)\ \mbox{and in}\ L^p(\partial\Omega).
	\end{eqnarray}
	
	Suppose that $u_{\lambda}^*\equiv 0$. Let $y_n=\frac{u_n}{||u_n||}\ n\geq 1$. Then $||y_n||=1$ for all $n\geq 1$ and so we may assume that
	\begin{eqnarray}\label{eq59}
		y_n\stackrel{w}{\rightarrow}y\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{and}\ y_n\rightarrow y\ \mbox{in}\ L^r(\Omega)\ \mbox{and in}\ L^p(\partial\Omega).
	\end{eqnarray}
	
	From (\ref{eq57}) we have
	\begin{eqnarray}\label{eq60}
		&&\left\langle A(y_n),h\right\rangle+\int\limits_{\partial\Omega}\beta(z)y_{n}^{p-1}hd\sigma=\lambda\int\limits_{\Omega}y_n^{p-1}hdz+\int\limits_{\Omega}\frac{f(z,u_n)}{||u_n||^{p-1}}hdz\\
		&&\hspace{7cm}\mbox{for all}\ h\in W^{1,p}(\Omega).\nonumber
	\end{eqnarray}
	
	In (\ref{eq60}) we choose $h=y_n-y\in W^{1,p}(\Omega)$, pass to the limit as $n\rightarrow\infty$ and use (\ref{eq59}). Then
	\begin{eqnarray}\label{eq61}
		&&\lim\limits_{n\rightarrow\infty}\left\langle A(y_n),y_n-y\right\rangle=0,\nonumber\\
		&\Rightarrow&y_n\rightarrow y\ \mbox{in}\ W^{1,p}(\Omega),\ \mbox{and so}\ ||y||=1,\ y\geq 0.
	\end{eqnarray}
	
	Note that since we have assumed that $u_{\lambda}^*\equiv 0$, by virtue of hypothesis $H(f)_1(iv)$, we have (at least for a subsequence) that
	\begin{eqnarray}\label{eq62}
		\frac{N_f(u_n)}{||u_n||^{p-1}}\stackrel{w}{\rightarrow}0\ \mbox{in}\ L^{r'}(\Omega).
	\end{eqnarray}
	
	So, if in (\ref{eq60}) we pass to the limit as $n\rightarrow\infty$ and use (\ref{eq61}) and (\ref{eq62}), then
	\begin{eqnarray*}
		&&\left\langle A(y),h\right\rangle+\int\limits_{\partial\Omega}\beta(z)y^{p-1}hd\sigma=\lambda\int\limits_{\Omega}y^{p-1}hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega),\\
		&\Rightarrow&-\Delta_py(z)=\lambda y(z)^{p-1}\ \mbox{a.e. in}\ \Omega,\ \frac{\partial y}{\partial n_p}+\beta(z)y^{p-1}=0\ \mbox{on}\ \partial\Omega.
	\end{eqnarray*}
	
	Since $\lambda<\hat{\lambda}_1$, it follows that $y\equiv 0$, a contradiction to (\ref{eq61}). Therefore $u_{\lambda}^*\neq 0$.
	
	In (\ref{eq57}) we choose $h=u_n-u_{\lambda}^*\in W^{1,p}(\Omega)$, pass to the limit as $n\rightarrow\infty$ and use (\ref{eq58}). Then
	\begin{eqnarray}\label{eq63}
		&&\lim\limits_{n\rightarrow\infty}\left\langle A(u_n),u_n-u_{\lambda}^*\right\rangle=0,\nonumber\\
		&\Rightarrow&u_n\rightarrow u_{\lambda}^*\ \mbox{in}\ W^{1,p}(\Omega)\ (\mbox{see Proposition \ref{prop2}}).
	\end{eqnarray}
	
	So, if in (\ref{eq57}) we pass to the limit as $n\rightarrow\infty$ and use (\ref{eq63}), then
	\begin{eqnarray*}
		&&\left\langle A(u_{\lambda}^*),h\right\rangle+\int\limits_{\partial\Omega}\beta(z)(u_{\lambda}^*)^{p-1}hd\sigma=\lambda\int\limits_{\Omega}(u_{\lambda}^*)^{p-1}hdz+\int\limits_{\Omega}f(z,u_{\lambda}^*)hdz\\
		&&\hspace{8cm}\mbox{for all}\ h\in W^{1,p}(\Omega),\\
		&\Rightarrow&u_{\lambda}^*\in S(\lambda)\ \mbox{and}\ u_{\lambda}^*=\inf S(\lambda).
	\end{eqnarray*}
	
	Therefore $u_{\lambda}^*\in \mbox{int}\, C_+$ is the smallest positive solution of problem $(P_{\lambda})$.
	
	Suppose that $\lambda<\vartheta\in\mathcal{P}=(-\infty,\hat{\lambda}_1)$ and let $u_{\vartheta}\in S(\vartheta)$. Then
	\begin{eqnarray*}
		&&u_{\lambda}^*\leq u_{\vartheta}\ (\mbox{see Proposition \ref{prop10}}),\\
		&\Rightarrow&u_{\lambda}^*\leq u_{\vartheta}^*.
	\end{eqnarray*}
	
	In fact, by considering $(u_{\lambda}^*)^{\delta}=u_{\lambda}^*+\delta\in \mbox{int}\, C_+\ (\delta>0)$ as in the proof of Proposition \ref{prop13}, via hypothesis $H(f)_1(v)$, we show that
	$$u_{\vartheta}^*-u_{\lambda}^*\in \mbox{int}\, C_+.$$
\end{proof}

Summarizing the situation in the case of superlinear perturbations, we can state the following theorem.
\begin{theorem}\label{th20}
	If hypotheses $H(\beta)$ and $H(f)_1$ hold, then for every $\lambda\in(-\infty,\hat{\lambda}_1)$ problem $(P_{\lambda})$ has at least two positive solutions
	$$u_{\lambda},v_{\lambda}\in \mbox{int}\, C_+,\ u_{\lambda}\leq v_{\lambda},\ u_{\lambda}\neq v_{\lambda};$$
	also it admits a smallest positive solution $u_{\lambda}^*\in \mbox{int}\, C_+$ and the map $\lambda\rightarrow u_{\lambda}^*$ is strictly increasing, that is,
	$$\lambda<\vartheta\in(-\infty,\hat{\lambda}_1)\Rightarrow u_{\vartheta}^*-u_{\lambda}^*\in \mbox{int}\, C_+;$$
	finally for $\lambda\geq\hat{\lambda}_1$ problem $(P_{\lambda})$ has no positive solution.
\end{theorem}

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\end{document} 