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\title[Parametric nonlinear elliptic problems]{Nonlinear elliptic problems with superlinear reaction and parametric concave boundary condition}


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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.D. R\u{a}dulescu]{Vicen\c{t}iu D. R\u{a}dulescu}
\address{Institute of Mathematics ``Simion Stoilow" of the Romanian Academy, P.O. Box 1-764,
          014700 Bucharest, Romania \&
          Topology, Geometry and Nonlinear Analysis Group, Institute of Mathematics, Physics and Mechanics, University of Ljubljana, 1000 Ljubljana, Slovenia }
\email{\tt vicentiu.radulescu@imar.ro}

\keywords{Concave-convex problem, nonlinear boundary condition, minimal positive solution, bifurcation-type result, nodal solution.\\
\phantom{aa} 2010 AMS Subject Classification:
35J70, 35B32}


\begin{abstract}
We study parametric nonlinear elliptic boundary value problems driven by the $p$-Laplacian with convex and concave terms. The convex term appears in the reaction and the concave in the boundary condition (source). We study the existence and nonexistence of positive solutions as the parameter $\lambda>0$ varies. For the semilinear problem $(p=2)$, we prove a bifurcation type result. Finally we show the existence of nodal (sign changing) solutions.
\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 boundary value problem
\begin{equation}
\left\{\begin{array}{ll}
	-\Delta_pu(z)=f(z,u(z))\ \mbox{in}\ \Omega,&1<p<\infty,\\
	\di\frac{\partial u}{\partial n_p}=\beta(z)u(z)^{p-1}\ \mbox{on}\ \partial\Omega,&u>0.
\end{array}\right. \tag{$P_{\lambda}$} \label{eqP}
\end{equation}

In this equation, by $\Delta_p$ we denote the $p$-Laplacian differential operator defined by
$$\Delta_pu=\mbox{div}\,(|Du|^{p-2}Du)\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$

Also $\frac{\partial u}{\partial n_p}$ denotes the generalized normal derivative corresponding to the $p$-Laplacian and defined by $\frac{\partial u}{\partial n_p}=|Du|^{p-2}(Du,n)_{\RR^N}$, with $n(\cdot)$ being the outward unit normal on $\partial\Omega$. The reaction $f(z,x)$ is a Carath\'eodory function (that is, for all $x\in\RR$, $z\longmapsto f(z,x)$ is measurable and for almost all $z\in\Omega$, $x\longmapsto f(z,x)$ is continuous), which is $(p-1)$-superlinear near $+\infty$. In the boundary condition, $\lambda>0$ is parameter, $\beta\in L^{\infty}(\Omega)_+,\ \beta\neq 0$ and $1<q<p$. So, problem \eqref{eqP} is an alternative version of the well-known ``concave-convex" problem (problem with competing nonlinearities) in which a ``convex" (superlinear) reaction $f(z,x)$ is coupled with a ``superlinear" parametric source term. The original ``concave-convex" problem, had both the competing nonlinearities in the reaction, which had the form $\lambda x^{q-1}+x^{r-1}$ for all $x\geq 0$, with $\lambda>0$ being the parameter and $$1<q<p<r<p^*=\left\{\begin{array}{ll}
	\displaystyle\frac{Np}{N-p}&\mbox{if}\ p<N\\
	+\infty&\mbox{if}\ N\leq p,
\end{array}\right.  $$
where $p^*$ is the critical Sobolev exponent.

The study of such problems was launched with the pioneering works of Garcia Azorero and Peral \cite{garcia}, and  Ambrosetti, Brezis and Cerami \cite{2}. In the first paper, among other results, the critical case is considered for small values of the parameter.
Ambrosetti, Brezis and Cerami \cite{2} investigated the following semilinear Dirichlet problem
$$-\Delta u(z)=\lambda u(z)^{q-1}+u(z)^{r-1}\ \mbox{in}\ \Omega,\ u|_{\partial\Omega}=0,\ u>0,$$
with $1<q<2<r<2^*$. They proved bifurcation-type results describing the set of positive solutions as the parameter $\lambda>0$ varies. Their work was extended to equations driven by the $p$-Laplacian, by Garcia Azorero, Manfredi and Peral Alonso \cite{7} and Guo and Zhang \cite{11}. Further extensions with more general reactions, can be found in Filippakis, Kristaly and Papageorgiou \cite{5} and Iannizzotto and Papageorgiou \cite{13}. We also refer to Boccardo, Escobedo and Peral \cite{bocc}, who studied the branch of minimal solutions without growth hypotheses.
Problems in which the competing nonlinearities come from both the reaction (the convex term) and the source (the concave term), were first considered by Garcia Azorero, Peral and Rossi \cite{8} for semilinear problems with a reaction of the form $f(x)=x^{r-1}$ for all $x\geq 0$, where $1<2<r<2^*$. Semiliear problems with a more general reaction, were studied recently by Furtado and Ruviaro \cite{6}. Generalizations to $p$-Laplacian equations with a reaction of the form $f(x)=x^{r-1}$ for all $x\geq 0$ where $1<p<q<p^*$, can be found in the work of Sabina de Lis \cite{19}. We stress that in all there of the aforementioned works, the differential operators (left hand-side of the equation) has the form $-\Delta_pu+u^{p-1}$ (with $p=2$ in \cite{6}, \cite{8}). This operator is coercive and this facilitates the analysis. In contrast, in problem \eqref{eqP} the differential operator is not coercive.

In Section \ref{sec3}, for problem \eqref{eqP}, we prove a theorem concerning the existence and nonexistence of positive solutions, depending on the value of the parameter $\lambda>0$. We also show the existence of a minimal positive solution $\underline{u}_{\lambda}$ and investigate the properties of the map $\lambda\longmapsto\underline{u}_{\lambda}$. If $p=2$ (semilinear problem), then we prove a bifurcation result describing in a more precise way the existence and multiplicity of positive solutions as the parameter $\lambda>0$ varies. It is an interesting open problem whether such a bifurcation result is also possible for the $p$-Laplacian equation. In Section \ref{sec4} we prove the existence of nodal (sign changing) solutions.

In the next section, for easy reference we recall the main mathematical tools which we will use in the sequel and fix our notation.

\section{Mathematical Background}\label{sec2}

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

This is a compactness-type condition on the functional $\varphi$. It is needed because the ambient space is not in general locally compact (since $X$ in general is infinite dimensional). With this compactness-type condition on $\varphi$, one can prove a deformation theorem which leads to a minimax theory for the critical values of $\varphi$. One of the main results in this theory, is the so-called ``mountain pass theorem" due ti Ambrosetti and Rabinowitz \cite{3}. Here we state it in a slightly more general form (see Gasinski and Papageorgiou \cite[p. 648]{9}).
\begin{theorem}\label{th1}
	Assume that $\varphi\in C^1(X)$ satisfies the $C$-condition, $u_0,u_1\in X,\ ||u_1-u_0||<p<0$,
	$$\max\{\varphi(u_0),\varphi(u_1)\}<\inf\left[\varphi(u):||u-u_0||=\rho\right]=m_{\rho}$$
	and $c=\inf\limits_{\gamma\in\Gamma}\ \max\limits_{0\leq t\leq 1}\ \varphi(\gamma(t))$, where $\Gamma=\left\{\gamma\in C\left([0,1],X\right):\gamma(0)=u_0,\gamma(1)=u_1\right\}$. Then $c\geq m_{\rho}$ and $c$ is a critical value of $\varphi$.
\end{theorem}

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

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

On $\partial\Omega$ we consider the $(N-1)$-dimensional Hausdorff (surface) measure $\sigma(\cdot)$. With this measure, we can define the Lebesgue spaces $L^p(\partial\Omega)\ (1\leq p\leq\infty)$. From the trace theorem, we know that there exists a unique continuous linear map $\gamma_0:W^{1,p}(\Omega)\rightarrow L^p(\partial\Omega)$, with the property that $\gamma_0(u)=u|_{\partial\Omega}$ for all $u\in C^1(\overline{\Omega})$. The trace map is compact into $L^q(\partial\Omega)$ with $1\leq q<\frac{Np-p}{N-p}$ and we have
$$\mbox{im}\,\gamma_0=W^{\frac{1}{p'},p}(\partial\Omega)\ \left(\frac{1}{p}+\frac{1}{p'}=1\right),\ \mbox{ker}\,\gamma_0=W^{1,p}_{0}(\Omega).$$

In the sequel, for notational simplicity, we drop the use of the trace map $\gamma_0$. Every Sobolev function defined on $\partial\Omega$ is understood in the sense of traces.

Let $f_0:\Omega\times\RR\rightarrow\RR$ be a Carath\'eodory function such that
$$|f_0(z,x)|\leq a_0(z)(1+|x|^{r-1})\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\in\RR$$
with $a_0\in L^{\infty}(\Omega)_+$ and $1<r<p^*$. We set $F_0(z,x)=\int^{x}_{0}f_0(z,s)ds$. Let $\beta\in L^{\infty}(\partial\Omega)$ and consider the $C^1$-functional $\varphi_0:W^{1,p}(\Omega)\rightarrow\RR$ defined by
$$\varphi_0(u)=\frac{1}{p}||Du||^{p}_{p}-\frac{1}{q}\int_{\partial\Omega}\beta(z)|u(z)|^qd\sigma-\int_{\Omega}F_0(z,u(z))dz\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$

From Papageorgiou and R\u{a}dulescu \cite{17}, we have the following result.
\begin{prop}\label{prop2}
	If $u_0\in W^{1,p}(\Omega)$ is a local $C^1(\overline{\Omega})$-minimizer of $\varphi_0$, that is, there exists $\rho_0>0$ such that
	$$\varphi_0(u_0)\leq\varphi_0(u_0+h)\ \mbox{for all}\ h\in C^1(\overline{\Omega})\ \mbox{with}\ ||h||_{C^1(\overline{\Omega})}\leq\rho_0,$$
	then $u_0\in C^{1,\alpha}(\overline{\Omega})$ for some $\alpha\in(0,1)$ and $u_0$ 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{prop}

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

From Papageorgiou and Kyritsi \cite[p. 314]{16}, we have:
\begin{prop}\label{prop3}
	The map $A:W^{1,p}(\Omega)\rightarrow W^{1,p}(\Omega)^*$ is continuous, strictly monotone (hence maximal monotone too) 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{prop}

Finally let us fix our notation. For $\varphi\in C^1(X)$, by $K_{\varphi}$ we denote set of critical points of $\varphi$, that is, $K_{\varphi}=\{u\in X:\varphi'(u)=0\}$. Also, if $x\in\RR$, then $x^{\pm}=\max\{0,\pm x\}$. Given $u\in W^{1,p}(\Omega)$, we set $u^{\pm}(\cdot)=u(\cdot)^{\pm}$ and we have
$$u^{\pm}\in W^{1,p}(\Omega),\ u=u^+-u^-,\ |u|=u^++u^-.$$

By $|\cdot|_N$ we denote the Lebesgue measure on $\RR^N$. Also, if $h:\Omega\times\RR\rightarrow\RR$ is a measurable function (for example a Carath\'eodory function), then
$$N_h(u)(\cdot)=h(\cdot,u(\cdot))$$
(the Nemytskii map corresponding to $h$). Evidently, the mapping $z\longmapsto N_h(u)(z)=f(z,u(z))$ is measurable.


\section{Positive Solutions}\label{sec3}

In this section, we study the existence and nonexistence of positive solutions for problem \eqref{eqP} as $\lambda>0$ varies. We also prove the existence of a minimal positive solution $\underline{u}_{\lambda}$ and examine the properties of the map $\lambda\longmapsto\underline{u}_{\lambda}$.

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

\smallskip
$H_1:$ $f:\Omega\times\RR\rightarrow\RR$ is a Carath\'eodory function such that for almost all $z\in\Omega$, $f(z,0)=0$ and
\begin{itemize}
	\item[(i)] $|f(z,x)|\leq a(z)(1+x^{r-1})$ for almost all $z\in\Omega$, all $x\geq 0$, with $a\in L^{\infty}(\Omega)_+,\ p<r<p^*$;
	\item[(ii)] if $F(z,x)=\int_{x}^{0}f(z,s)ds$, then there exists $\eta>p$ and $M>0$ such that
	\begin{eqnarray*}
		&&0<\eta F(z,x)\leq f(z,x)x\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\geq M,\\
		&&\mbox{essinf}\, F(\cdot,M)>0;
	\end{eqnarray*}
	\item[(iii)] $\lim\limits_{x\rightarrow 0^+}\frac{f(z,x)}{x^{p-1}}=0$ uniformly for almost all $z\in\Omega$ and for all $\tau>0,\ f(z,x)\geq\mu_{\tau}>0$ for almost all $z\in\Omega$, all $x\geq\tau$;
	\item[(iv)] if $p=2$, then for every $\rho>0$, there exists $\xi_{\rho}>0$ such that for almost all $z\in\Omega$, the map $x\longmapsto f(z,x)+\xi_{\rho}x$ is nondecreasing on $[0,\rho]$.
\end{itemize}
\begin{remark}
	Since we are interested on positive solutions and the above hypotheses concern the positive semi-axis $\RR_+=\left[0,+\infty\right)$, without any loss of generality, we may assume that $f(z,x)=0$ for almost all $z\in\Omega$, all $x\leq 0$. Hypothesis $H_1(ii)$ is the well-known Ambrosetti-Rabinowitz condition ($AR$-condition for short) (see \cite{3}). It implies that
	\begin{equation}\label{eq1}
		c_1x^\eta\leq F(z,x)\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\geq M,\ \mbox{some}\ c_1>0.
	\end{equation}
	(see for example, Papageorgiou and Kyritsi \cite[p. 424]{16}). It is an interesting open problem whether we can replace the $AR$-condition by a more general superlinearity condition, like the one employed by Gasinski and Papageorgiou \cite{10} and Iannizzotto and Papageorgiou \cite{13}. The noncoercivity of the differential operator together with the boundary term $\lambda\beta(z)x^{q-1}$ for all $(z,x)\in\partial\Omega\times\RR_+$ raise serious technical difficulties and make the use of more general superlinearity conditions problematic. Hypothesis $H_1(iv)$ is satisfied, if for almost all $z\in\Omega$, $f(z,\cdot)\in C^1(\RR)$ and $f'_x(z,x)$ is locally $L^{\infty}(\Omega)$-bounded.
\end{remark}
\begin{ex}
	The following functions satisfy hypotheses $H_1$:
	\begin{eqnarray*}
		f_1(x)&=&x^{r-1}\ \mbox{for all}\ x\geq 0,\ \mbox{with}\ p<r<p^*,\\
		f_2(x)&=&\left\{\begin{array}{ll}
				0&\mbox{if}\ x<0\\
				cx^{s-1}-x^{\tau-1}&\mbox{if}\ 0\leq x\leq 1\\
				(c-1)x^{\eta-1}&\mbox{if}\ 1<x
			\end{array}\right.\ \mbox{with}\ c>1,\ p<s<\tau\ \mbox{and}\ p<\eta<p^*,\\
		f_3(x)&=&\left\{\begin{array}{ll}
				0&\mbox{if}\ x<0\\
				x^{\tau-1}&\mbox{if}\ 0\leq x \leq 1\\
				x^{\eta-1}&\mbox{if}\ 1<x.
			\end{array}\right.\ \mbox{with}\ p<\tau,\eta\ \mbox{and}\ \eta<p^*.
	\end{eqnarray*}
\end{ex}

The hypotheses on the boundary term are:

\smallskip
$\widehat{H}:$ $\beta\in C^{0,\alpha}(\partial\Omega)$ with $\alpha\in\left(0,1\right]$, $\beta\geq 0,\ \beta\neq 0$ and $1<q<p$.

\smallskip
We introduce the following Carath\'eodory function
\begin{eqnarray}\label{eq2}
	\hat{f}(z,x)=\left\{\begin{array}{ll}
		0&\mbox{if}\ x\leq 0\\
		f(z,x)+x^{p-1}&\mbox{if}\ 0<x.
	\end{array}\right.
\end{eqnarray}


Let $\widehat{F}(z,x)=\int^{x}_{0}\hat{f}(z,s)ds$ and for every $\lambda>0$, we consider the $C^1$-functional $\hat{\varphi}_{\lambda}:W^{1,p}(\Omega)\rightarrow\RR$ defined by
$$\hat{\varphi}_{\lambda}(u)=\frac{1}{p}||Du||^{p}_{p}+\frac{1}{p}||u||^{p}_{p}-\frac{\lambda}{q}\int_{\partial\Omega}\beta(z)u^+(z)^qd\sigma-\int_{\Omega}\widehat{F}(z,u(z))dz\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$
\begin{prop}\label{prop4}
	If hypotheses $H_1(i),(ii),(iii)$ and $\widehat{H}$ hold, then the functional $\hat{\varphi}_{\lambda}$ satisfies the $C$-condition.
\end{prop}
\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{eq3}\\
		&&(1+||u_n||)\hat{\varphi}'_{\lambda}(u_n)\rightarrow 0\ \mbox{in}\ W^{1,p}(\Omega)^*\ \mbox{as}\ n\rightarrow\infty.\label{eq4}
	\end{eqnarray}
	
	From (\ref{eq4}) we have
	\begin{eqnarray}\label{eq5}
		&&\left|\left\langle \hat{\varphi}'_{\lambda}(u_n),h\right\rangle\right|\leq\frac{\epsilon_n||h||}{1+||u_n||}\ \mbox{for all}\ h\in W^{1,p}(\Omega),\ \mbox{with}\ \epsilon_n\rightarrow 0^+,\nonumber\\
		&\Rightarrow&\left|\left\langle A(u_n),h\right\rangle+\int_{\Omega}|u_n|^{p-2}u_nhdz-\lambda\int_{\partial\Omega}\beta(z)(u_n^+)^{q-1}hd\sigma-\int_{\Omega}\hat{f}(z,u_n)hdz\right|\leq\\
		&&\hspace{6cm}\frac{\epsilon_n||h||}{1+||u_n||}\ \mbox{for all}\ h\in W^{1,p}(\Omega),\ \mbox{all}\ n\geq 1.\nonumber
	\end{eqnarray}
	
	In (\ref{eq5}) we choose $h=-u^-_n\in W^{1,p}(\Omega)$. Then
	\begin{eqnarray}\label{eq6}
		&&\frac{1}{p}||Du^-_n||^{p}_{p}+\frac{1}{p}||u^-_n||^{p}_{p}\leq\epsilon_n\ \mbox{for all}\ n\geq 1\ (\mbox{see (\ref{eq2})}),\nonumber\\
		&\Rightarrow&u^-_n\rightarrow 0\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{as}\ n\rightarrow\infty.
	\end{eqnarray}
	
	From (\ref{eq3}), (\ref{eq6}) and hypothesis $H_1(i)$, we have
	\begin{eqnarray}\label{eq7}
		&&||Du^+_n||^{p}_{p}-\frac{\lambda p}{q}\int_{\partial\Omega}\beta(z)(u^+_n)^qd\sigma-\int_{\Omega}pF(z,u^+_n)dz\leq M_2\\
		&&\hspace{4.7cm}\mbox{for some}\ M_2>0,\ \mbox{all}\ n\geq 1.\nonumber
	\end{eqnarray}
	
	Also, in (\ref{eq5}) we choose $h=u^+_n\in W^{1,p}(\Omega)$ and obtain
	\begin{equation}\label{eq8}
		 -||Du^+_n||^{p}_{p}+\lambda\int_{\partial\Omega}\beta(z)(u^+_n)^qd\sigma+\int_{\Omega}f(z,u^+_n)u^+_ndz\leq\epsilon_n\ \mbox{for all}\ n\geq 1\ (\mbox{see (\ref{eq4})}).
	\end{equation}
	
	Adding (\ref{eq7}) and (\ref{eq8}), we obtain
	\begin{eqnarray*}
		\int_{\Omega}\left[f(z,u^+_n)u^+_n-pF(z,u^+_n)\right]dz&\leq& M_3+\lambda\left(\frac{p}{q}-1\right)\int_{\partial\Omega}\beta(z)(u^+_n)^qd\sigma\\
		&&\hspace{1.7cm}\mbox{for some}\ M_3>0,\ \mbox{all}\ n\geq 1,\\
		&\leq&c_2(1+||u^+_n||^q)\ \mbox{for some}\ c_2>0,\ \mbox{all}\ n\geq 1\\
		&&\hspace{1.7cm}(\mbox{use the trace theorem}),
	\end{eqnarray*}
	\begin{eqnarray}\label{eq9}
		&\Rightarrow&\int_{\Omega}\left[f(z,u^+_n)u^+_n-\eta F(z,u^+_n)\right]dz+(\eta-p)\int_{\Omega}F(z,u^+_n)dz\leq c_2(1+||u^+_n||^q)\nonumber\\
		&&\hspace{9.7cm}\mbox{for all}\ n\geq 1,\nonumber\\
		&\Rightarrow&(\eta-p)\int_{\Omega}F(z,u^+_n)dz\leq c_2(1+||u^+_n||^q)\ \mbox{for all}\ n\geq 1\\
		&&\hspace{5cm}(\mbox{see hypothesis}\ H_1(ii)).\nonumber
	\end{eqnarray}
	
	From (\ref{eq1}) and hypothesis $H_1(i)$, we have
	\begin{equation}\label{eq10}
		c_1x^{\eta}-c_3\leq F(z,x)\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\geq 0,\ \mbox{some}\ c_3>0.
	\end{equation}
	
	Using (\ref{eq10}) in (\ref{eq9}), we obtain
	\begin{eqnarray}\label{eq11}
		||u^+_n||^{p}_{p}&\leq&c_4(1+||u^+_n||^q)^{p/\eta}\ \mbox{for some}\ c_4>0,\ \mbox{all}\ n\geq 1\nonumber\\
		&\leq&c_4(1+||u^+_n||^{qp/eta})\ (\mbox{recall}\ p<\eta)\nonumber\\
		&\leq&c_5(1+||u^+_n||^q)\ \mbox{for some}\ c_5>0,\ \mbox{all}\ n\geq 1.
	\end{eqnarray}
	
	From (\ref{eq3}), (\ref{eq6}) and hypothesis $H_1(i)$ we have
	\begin{eqnarray}\label{eq12}
		 &&\frac{\eta}{p}||Du^+_n||^{p}_{p}-\frac{\lambda\eta}{q}\int_{\partial\Omega}\beta(z)(u^+_n)^qd\sigma-\int_{\Omega}\eta F(z,u^+_n)dz\leq M_4\\
		&&\hspace{5cm}\mbox{for some}\ M_4>0,\ \mbox{all}\ n\geq 1.\nonumber
	\end{eqnarray}
	
	Adding (\ref{eq8}) and (\ref{eq12}), we obtain
	\begin{eqnarray}\label{eq13}
		&&\left(\frac{\eta}{p}-1\right)||Du^+_n||^{p}_{p}+\int_{\Omega}[f(z,u^+_n)u^+_n-\eta F(z,u^+_n)]dz\leq\nonumber\\
		 &&\hspace{3cm}M_5+\lambda\left(\frac{\eta}{q}-1\right)\int_{\partial\Omega}\beta(z)(u^+_n)^qd\sigma\nonumber\\
		&&\hspace{4cm}\mbox{for some}\ M_5>0,\ \mbox{all}\ n\geq 1,\nonumber\\
		&\Rightarrow&||Du^+_n||^{p}_{p}\leq c_6(1+||u^+_n||^q)\ \mbox{for some}\ c_6>0,\ \mbox{all}\ n\geq 1\\
		&&(\mbox{see hypotheses\ $H_1(i),(ii)$, recall that $p<\eta$ and use the trace theorem})\nonumber
	\end{eqnarray}
	
	From (\ref{eq11}) and (\ref{eq13}) and recalling that $u\longmapsto ||u||_{\eta}+||Du||_{p}$ is an equivalent norm on $W^{1,p}(\Omega)$ (see, for example, Gasinski and Papageorgiou \cite[p. 227]{9}), we infer that
	$$||u^+_n||^p\leq c_7(1+||u^+_n||^q)\ \mbox{for some}\ c_7>0,\ \mbox{all}\ n\geq 1.$$
	
	Since $q<p$ (see hypotheses $\widehat{H}$), we conclude that
	\begin{equation}\label{eq14}
		\{u^+_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)\ \mbox{is bounded}.
	\end{equation}
	
	From (\ref{eq6}) and (\ref{eq14}) it follows that $\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)$ is bounded. Using the Sobolev embedding theorem and the trace theorem and by passing to a subsequence if necessary, we may assume that
	\begin{equation}\label{eq15}
		u_n\stackrel{w}{\rightarrow} u\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{and}\ u_n\rightarrow u\ \mbox{in}\ L^r(\Omega)\ \mbox{and in}\ L^p(\partial\Omega)\ \mbox{as}\ n\rightarrow\infty.
	\end{equation}
	
	In (\ref{eq5}) we choose $h=u_n-u\in W^{1,p}(\Omega)$, pass to the limit as $n\rightarrow\infty$ and use (\ref{eq15}). 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)\ \mbox{as}\ n\rightarrow\infty\ (\mbox{see Proposition \ref{prop3}}),\\
		&\Rightarrow&\hat{\varphi}_{\lambda}\ \mbox{satisfies the $C$-condition}.
	\end{eqnarray*} This completes the proof.
\end{proof}

\begin{prop}\label{prop5}
	If hypotheses $H_1(i),(ii),(iii)$ and $\widehat{H}$ 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{prop}
\begin{proof}
	Hypotheses $H_1(i),(iii)$ imply that given $\epsilon>0$, we can find $c_8=c_8(\epsilon)>0$ such that
	\begin{equation}\label{eq16}
		F(z,x)\leq\epsilon x^p+c_8x^r\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\geq 0.
	\end{equation}
	
	Then for every $u\in W^{1,p}(\Omega)$, we have
	\begin{equation}\label{eq17}
		\hat{\varphi}_{\lambda}(u)\geq\frac{1}{p}||Du||^{p}_{p}+\frac{1}{p}||u||^{p}_{p}-\lambda c_9||u||^q-c_{10}||u||^r-\epsilon||u||^p-\frac{1}{p}||u||^p\ (\mbox{see (\ref{eq2})}).
	\end{equation}
	
	Since $q<p<r$, we have
	$$||u||^p\leq\lambda||u||^q+c_{11}||u||^r\ \mbox{for some}\ c_{11}=c_{11}(\lambda)>0.$$
	
	Returning to (\ref{eq17}) and choosing $\epsilon>0$ small, we have
	\begin{eqnarray}\label{eq18}
		\hat{\varphi}_{\lambda}(u)&\geq&c_{12}||u||^p-\lambda c_{13}||u||^q-c_{14}||u||^r\nonumber\\
		&&\mbox{with}\ c_{12}=c_{12}(\epsilon)>0,\ c_{13}=c_9+\frac{1}{p}>0,\ c_{14}=c_{10}+\frac{1}{p}>0\nonumber\\
		&=&\left[c_{12}-\left(\lambda c_{13}||u||^{q-p}+c_{14}||u||^{r-p}\right)\right]||u||^p.
	\end{eqnarray}
	
	Let $\vartheta_{\lambda}(t)=\lambda c_{13}t^{q-p}+c_{14}t^{r-p}$ for all $t>0$. Evidently $\vartheta_{\lambda}\in C^1(0,+\infty)$ and since $q<p<r$, we have
	$$\vartheta_{\lambda}(t)\rightarrow+\infty\ \mbox{as}\ t\rightarrow 0^+\ \mbox{and as}\ t\rightarrow+\infty.$$
	
	So, we can find $t_0\in(0,+\infty)$ such that
	\begin{eqnarray*}
		&&\vartheta_{\lambda}(t_0)=\inf\limits_{t>0}\vartheta_{\lambda}(t),\\
		&\Rightarrow&\vartheta'_{\lambda}(t_0)=0,\\
		&\Rightarrow&\lambda(p-q)c_{13}=(r-p)c_{14}t_0^{r-q},\\
		 &\Rightarrow&t_0=t_0(\lambda)=\left[\frac{\lambda(p-q)c_{13}}{(r-p)c_{14}}\right]^{\frac{1}{r-q}}.
	\end{eqnarray*}
	
	Evidently, we have
	$$\vartheta_{\lambda}(t_0)\rightarrow 0^+\ \mbox{as}\ \lambda\rightarrow 0^+.$$
	
	So, from (\ref{eq18}) we see that we can find $\lambda_+>0$ such that
	$$\vartheta_{\lambda}(t_0)<c_{12}\ \mbox{for all}\ \lambda\in(0,\lambda_+).$$
	
	Therefore
	$$\hat{\varphi}_{\lambda}(u)\geq\hat{m}_{\lambda}>0=\hat{\varphi}_{\lambda}(0)\ \mbox{for all}\ u\in W^{1,p}(\Omega)\ \mbox{with}\ ||u||=\rho_{\lambda}=t_0(\lambda).$$
\end{proof}

An immediate consequence of the $AR$-condition (see (\ref{eq1})), is the following proposition.
\begin{prop}\label{prop6}
	If hypotheses $H_1(i),(ii),(iii)$ and $\widehat{H}$ hold $\lambda>0$ and $u\in \mbox{int}\, C_+$, then $\hat{\varphi}_{\lambda}(tu)\rightarrow-\infty$ as $t\rightarrow+\infty$.
\end{prop}

We introduce the following sets
\begin{center}
	$\mathcal{L}=\{\lambda>0:\ \mbox{problem}\ \eqref{eqP}\ \mbox{admits a positive solution}\}$\\
	$S(\lambda)=\mbox{set of positive solutions for problem}\ \eqref{eqP}.$
\end{center}
\begin{prop}\label{prop7}
	If hypotheses $H_1(i),(ii),(iii)$ and $\widehat{H}$ hold, then $\mathcal{L}\neq\varnothing$ and every $\lambda>0$, $S(\lambda)\subseteq \mbox{int}\, C_+$.
\end{prop}
\begin{proof}
	Let $\lambda_+>0$ be as in Proposition \ref{prop5}. We fix $\lambda\in(0,\lambda_+)$. Then Proposition \ref{prop5} and \ref{prop6} imply that the functional $\hat{\varphi}_{\lambda}$ satisfies the mountain pass geometry. This fact, together with Proposition \ref{prop4}, permit the use of Theorem \ref{th1} (the mountain pass theorem). So, we can find $u_0\in W^{1,p}(\Omega)$ such that
	\begin{equation}\label{eq19}
		u_0\in K_{\hat{\varphi}_{\lambda}}\ \mbox{and}\ \hat{\varphi}_{\lambda}(0)=0<\hat{m}_{\lambda}\leq\hat{\varphi}_{\lambda}(u_0).
	\end{equation}
	
	From (\ref{eq19}) we see that $u_0\neq 0$ and
	\begin{eqnarray}\label{eq20}
		&&\hat{\varphi}'_{\lambda}(u_0)=0\nonumber\\
		&\Rightarrow&\left\langle A(u_0),h\right\rangle+\int_{\Omega}|u_0|^{p-2}u_0hdz=\lambda\int_{\Omega}\beta(z)(u^+_0)^{q-1}hd\sigma+\int_{\Omega}\hat{f}(z,u_0)hdz\\
		&&\hspace{8.4cm}\mbox{for all}\ h\in W^{1,p}(\Omega).\nonumber
	\end{eqnarray}
	
	In (\ref{eq20}) we choose $h=-u^-_0\in W^{1,p}(\Omega)$. Using (\ref{eq2}), we have
	\begin{eqnarray*}
		&&||Du^-_0||^{p}_{p}+||u^-_0||^{p}_{p}=0,\\
		&\Rightarrow&u_0\geq 0,\ u_0\neq 0.
	\end{eqnarray*}
	
	So, (\ref{eq20}) becomes
	\begin{equation}\label{eq21}
		\left\langle A(u_0),h\right\rangle=\lambda\int_{\partial\Omega}\beta(z)u^{q-1}_{0}hd\sigma+\int_{\Omega}f(z,u_0)hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega)\ (\mbox{see (\ref{eq2})}).
	\end{equation}
	
	In what follows, by $\left\langle \cdot,\cdot\right\rangle_0$ we denote the duality brackets for the pair $$(W^{-1,p'}(\Omega)=W^{1,p}_{0}(\Omega)^*,\, W^{1,p}_{0}(\Omega))\qquad \left(\frac{1}{p}+\frac{1}{p'}=1\right).$$ From the representation theorem for the elements of the dual space $W^{-1,p'}(\Omega)=W^{1,p}_{0}(\Omega)^*$ (see for example Gasinski and Papageorgiou \cite[p. 212]{9}), we have
	$$\Delta_pu_0\in W^{-1,p'}(\Omega).$$
	
	Then integration by parts gives
	\begin{equation}\label{eq22}
		\left\langle A(u_0),h\right\rangle=\left\langle -\Delta_pu_0,h\right\rangle_0\ \mbox{for all}\ h\in W^{1,p}_{0}(\Omega)\subseteq W^{1,p}(\Omega).
	\end{equation}
	
	We return to (\ref{eq21}) and use (\ref{eq22}). Recall that $\mbox{ker}\,\gamma_0=W^{1,p}_{0}(\Omega)$. So, we have
	\begin{eqnarray}\label{eq23}
		&&\left\langle -\Delta_pu_0,h\right\rangle_0=\int_{\Omega}f(z,u_0)hdz\ \mbox{for all}\ h\in W^{1,p}_{0}(\Omega)\nonumber\\
		&\Rightarrow&-\Delta_pu_0(z)=f(z,u_0(z))\ \mbox{for almost all}\ z\in\Omega\\
		&&\hspace{3cm}(\mbox{recall}\ L^{r'}(\Omega)\hookrightarrow W^{-1,p'}(\Omega)).\nonumber
	\end{eqnarray}
	
	Hypothesis $H_1(i)$ implies that $f(\cdot,u_0(\cdot))\in L^{r'}(\Omega)$. Since $W^{1,r}_{0}(\Omega)\hookrightarrow W^{1,p}_{0}(\Omega)$ continuously and densely (recall $p<r$), we have $W^{-1,p'}(\Omega)\hookrightarrow W^{-1,r'}(\Omega)$ continuously and densely (see, for example, Gasinski and Papageorgiou \cite[p. 141]{9}). Then because of (\ref{eq23}) we can apply the nonlinear Green's identity (see, for example, Gasinski and Papageorgiou \cite[p. 210]{9}) and have
	\begin{eqnarray}\label{eq24}
			&&\left\langle A(u_0),h\right\rangle+\int_{\Omega}(\Delta_pu_0)hdz=\left\langle \frac{\partial u_0}{\partial n_p},h\right\rangle_{\partial\Omega}\ \mbox{for all}\ h\in W^{1,r}(\Omega)\hookrightarrow W^{1,p}(\Omega)\\
			&&\hspace{10.4cm}(\mbox{see (\ref{eq23})}).\nonumber
		\end{eqnarray}
		
		Here by $\left\langle \cdot,\cdot\right\rangle_{\partial\Omega}$ we denote the duality brackets for the pair $\left(W^{-\frac{1}{p'},p'}(\partial\Omega),\ W^{\frac{1}{p'},p}(\partial\Omega)\right)$. If we use (\ref{eq21}) and (\ref{eq23}) in (\ref{eq24}), we obtain
	\begin{eqnarray}\label{eq25}
		&&\lambda\int_{\partial\Omega}\beta(z)u^{q-1}_{0}hd\sigma=\left\langle \frac{\partial u_0}{\partial n_p},h\right\rangle_{\partial \Omega}\ \mbox{for all}\ h\in W^{1,p}(\Omega)\\
		&&\hspace{3cm}(\mbox{recall}\ W^{1,r}(\Omega)\ \mbox{is dense in}\ W^{1,p}(\Omega)).\nonumber
	\end{eqnarray}
	
	Recall that
	$$\gamma_0(W^{1,p}(\Omega))=W^{\frac{1}{p'},p}(\partial\Omega).$$
	
	So, from (\ref{eq25}) it follows that
	\begin{eqnarray*}
		&&\frac{\partial u_0}{\partial n_p}=\lambda\beta(z)u^{q-1}_{0}\ \mbox{on}\ \partial\Omega,\\
		&\Rightarrow&u_0\in S(\lambda)\ \mbox{and so}\ (0,\lambda_+)\subseteq\mathcal{L},\ \mbox{hence}\ \mathcal{L}\neq\varnothing.
	\end{eqnarray*}
	
	From Winkert \cite{20} we have $u_0\in L^{\infty}(\Omega)$ and then Theorem 2 of Lieberman \cite{14}, implies that $u_0\in C_+\backslash\{0\}$.
	
	Let $\rho=||u_0||_{\infty}$. Hypotheses $H_1(i),(iii)$ imply that we can find $\hat{x}_{\rho}>0$ such that
	\begin{equation}\label{eq26}
		f(z,x)+\hat{\xi}_{\rho}x^{p-1}\geq 0\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\in[0,\rho].
	\end{equation}
	
	From (\ref{eq23}) we obtain
	\begin{eqnarray*}
		&&\Delta_pu_0(z)\leq\hat{\xi}_{\rho}u_0(z)^{p-1}\ \mbox{for almost all}\ z\in\Omega,\\
		&\Rightarrow&u_0\in \mbox{int}\, C_+\ (\mbox{by the nonlinear maximum principle, see \cite[p. 738]{9}}).
	\end{eqnarray*}
	
	The above argument shows that
	$$S(\lambda)\subseteq \mbox{int}\, C_+\ \mbox{for all}\ \lambda>0$$
	(of course, if $\lambda\notin\mathcal{L}$, then $S(\lambda)=\varnothing$).
\end{proof}

Next we prove a useful structural property of the set $\mathcal{L}$, which shows that $\mathcal{L}$ is an interval.
\begin{prop}\label{prop8}
	If hypotheses $H_1(i),(ii),(iii)$ and $\widehat{H}$ hold, $\lambda\in\mathcal{L}$ and $\nu\in\left(0,\lambda\right)$, then $\nu\in\mathcal{L}$.
\end{prop}
\begin{proof}
	Since $\lambda\in\mathcal{L}$, we can find $u_{\lambda}\in S(\lambda)\subseteq \mbox{int}\, C_+$ (see Proposition \ref{prop7}). We introduce the following Carath\'eodory functions
	\begin{eqnarray}
		k(z,x)=\left\{\begin{array}{ll}
			f(z,x)+(x^+)^{p-1}&\mbox{if}\ x\leq u_{\lambda}(z)\\
			f(z,u_{\lambda}(z))+u_{\lambda}(z)^{p-1}&\mbox{if}\ u_{\lambda}(z)<x
		\end{array}\right.\ \mbox{for all}\ (z,x)\in\Omega\times\RR\label{eq27}\\
		\gamma_{\nu}(z,x)=\left\{\begin{array}{ll}
			\nu\beta(z)(x^+)^{q-1}&\mbox{if}\ x\leq u_{\lambda}(z)\\
			\nu\beta(z)u_{\lambda}(z)^{q-1}&\mbox{if}\ u_{\lambda}(z)<x
		\end{array}\right.\ \mbox{for all}\ (z,x)\in\partial\Omega\times\RR.\label{eq28}
	\end{eqnarray}
	
	Let $K(z,x)=\int^{x}_{0}k(z,s)ds$ and $\Gamma_{\nu}(z,x)=\int^{x}_{0}\gamma_{\nu}(z,s)ds$. We consider the $C^1$-functional $\hat{\psi}_{\nu}:W^{1,p}(\Omega)\rightarrow\RR$ defined by
	 $$\hat{\psi}_{\nu}(u)=\frac{1}{p}||Du||^{p}_{p}+\frac{1}{p}||u||^{p}_{p}-\int_{\partial\Omega}\Gamma_{\nu}(z,u(z))d\sigma-\int_{\Omega}K(z,u(z))dz\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$
	
	From (\ref{eq27}) and (\ref{eq28}), it is clear that $\hat{\psi}_{\nu}(\cdot)$ is coercive. Also, using the Sobolev embedding theorem and the compactness of the trace map, we have that $\hat{\psi}_{\nu}$ is sequentially weakly lower semicontinuous. So, the Weierstrass theorem implies that we can find $u_{\nu}\in W^{1,p}(\Omega)$ such that
	\begin{equation}\label{eq29}
		\hat{\psi}_{\nu}(u_{\nu})=\inf\left[\hat{\psi}_{\nu}(u):u\in W^{1,p}(\Omega)\right].
	\end{equation}
	
	Let $m_{\lambda}=\min\limits_{\overline{\Omega}}u_{\lambda}>0$ (recall that $u_{\lambda}\in \mbox{int}\, C_+$). Because of hypothesis $H_1(iii)$, given $\epsilon>0$, we can find $\delta=\delta(\epsilon)\in\left(0,m_{\lambda}\right)$ such that
	\begin{equation}\label{eq30}
		F(z,x)\geq-\frac{\epsilon}{p}x^p\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\in[0,\delta].
	\end{equation}
	
	Let $u\in \mbox{int}\, C_+$ and choose $t\in(0,1)$ small such that $tu(z)\in\left(0,\delta\right]$ for all $z\in\overline{\Omega}$. Then we have
	\begin{eqnarray}\label{eq31}
		&&\hat{\psi}_{\nu}(tu)\leq\frac{t^p}{p}||Du||^{p}_{p}-\frac{\nu t^q}{q}\int_{\partial\Omega}\beta(z)u^qd\sigma+\frac{\epsilon t^p}{p}||u||^{p}_{p}\\
		&&\hspace{4cm}(\mbox{see (\ref{eq27}), (\ref{eq28}) and (\ref{eq30})})\nonumber
	\end{eqnarray}
	
	Since $q<p$, from (\ref{eq31}) we see that by choosing $t\in(0,1)$ even smaller if necessary, we can have
	\begin{eqnarray*}
		&&\hat{\psi}_{\nu}(tu)<0,\\
		&\Rightarrow&\hat{\psi}_{\nu}(u_{\nu})<0=\hat{\psi}_{\nu}(0)\ (\mbox{see (\ref{eq29})}),\ \mbox{hence}\ u_{\nu}\neq 0.
	\end{eqnarray*}
	
	From (\ref{eq29}) we have
	\begin{eqnarray}\label{eq32}
		&&\hat{\psi}'_{\nu}(u_{\nu})=0,\nonumber\\
		&\Rightarrow&\left\langle A(u_{\nu}),h\right\rangle+\int_{\Omega}|u_{\nu}|^{p-2}u_{\nu}hdz=\int_{\partial\Omega}\gamma_{\nu}(z,u_{\nu})hd\sigma+\int_{\Omega}k(z,u_{\nu})dz\\
		&&\hspace{7.3cm}\mbox{for all}\ h\in W^{1,p}(\Omega).\nonumber
	\end{eqnarray}
	
	In (\ref{eq32}) first we choose $h=-u_{\nu}^-\in W^{1,p}(\Omega)$. We obtain
	\begin{eqnarray*}
		&&||Du_{\nu}^-||^{p}_{p}+||u^-_{\nu}||^{p}_{p}=0\ (\mbox{see (\ref{eq27}), (\ref{eq28})}),\\
		&\Rightarrow&u_{\nu}\geq 0,\ u_{\nu}\neq 0.
	\end{eqnarray*}
	
	Next in (\ref{eq32}) we choose $(u_{\nu}-u_{\lambda})^+\in W^{1,p}(\Omega)$. We have
	\begin{eqnarray*}
		&&\left\langle A(u_{\nu}),(u_{\nu}-u_{\lambda})^+\right\rangle+\int_{\Omega}u_{\nu}^{p-1}(u_{\nu}-u_{\lambda})^+dz\\
		 &=&\int_{\partial\Omega}\nu\beta(z)u_{\lambda}^{q-1}(u_{\nu}-u_{\lambda})^+d\sigma+\int_{\Omega}f(z,u_{\lambda})(u_{\nu}-u_{\lambda})^+dz+\int_{\Omega}u^{p-1}_{\lambda}(u_{\nu}-u_{\lambda})^+dz\\
		&&\hspace{9cm}(\mbox{see (\ref{eq27}),(\ref{eq28})})\\
		 &\leq&\int_{\partial\Omega}\lambda\beta(z)u_{\lambda}^{q-1}(u_{\nu}-u_{\lambda})^+d\sigma+\int_{\Omega}f(z,u_{\lambda})(u_{\nu}-u_{\lambda})^+dz+\int_{\Omega}u_{\lambda}^{p-1}(u_{\nu}-u_{\lambda})^+dz\\
		&&\hspace{9cm}(\mbox{since}\ \nu<\lambda\ \mbox{and see}\ \widehat{H})\\
		&=&\left\langle A(u_{\lambda}),(u_{\nu}-u_{\lambda})^+\right\rangle+\int_{\Omega}u_{\lambda}^{p-1}(u_{\nu}-u_{\lambda})^+dz,\\
		&\Rightarrow&\left\langle A(u_{\nu})-A(u_{\lambda}),(u_{\nu}-u_{\lambda})^+\right\rangle+\int_{\Omega}(u_{\nu}^{p-1}-u_{\lambda}^{p-1})(u_{\nu}-u_{\lambda})^+dz\leq 0,\\
		&\Rightarrow&\left|\{u_{\nu}>u_{\lambda}\}\right|_N=0,\ \mbox{hence}\ u_{\nu}\leq u_{\lambda}.
	\end{eqnarray*}
	
	So, we have proved that
	$$u_{\nu}\in[0,u_{\lambda}]=\{u\in W^{1,p}(\Omega):0\leq u(z)\leq u_{\lambda}(z)\ \mbox{for almost all}\ z\in\Omega\}.$$
	
	Using (\ref{eq27}) and (\ref{eq28}), equation (\ref{eq32}) becomes
	\begin{eqnarray*}
		&&\left\langle A(u_{\nu}),h\right\rangle=\nu\int_{\partial\Omega}\beta(z)u_{\nu}^{q-1}hd\sigma+\int_{\Omega}f(z,u_{\nu})hdz,\\
		&\Rightarrow&u_{\nu}\in S(\nu)\subseteq \mbox{int}\, C_+\ (\mbox{see the proof of Proposition \ref{prop7}}),\\
		&\Rightarrow&\nu\in\mathcal{L}\ \mbox{and so}\ \left(0,\lambda\right]\subseteq\mathcal{L}.
	\end{eqnarray*}
The proof is now complete.
\end{proof}
\begin{remark}
	As a consequence of Proposition \ref{prop8}, we see that $\mathcal{L}$ is an interval.
\end{remark}

An interesting byproduct of the above proof, is the following corollary.
\begin{corollary}\label{cor9}
	If hypotheses $H_1(i),(ii),(iii)$ and $\widehat{H}$ hold, $\lambda\in\mathcal{L},\ u_{\lambda}\in S(\lambda)\subseteq \mbox{int}\, C_+$ and $\nu\in(0,\lambda)$, then there exists $u_{\nu}\in S(\nu)\subseteq \mbox{int}\, C_+$ such that $u_{\nu}\leq u_{\lambda}$.
\end{corollary}

In the semilinear case $(p=2)$, we can improve the above corollary by bringing into play hypothesis $H_1(iv)$. We will need this result in order to produce a second positive solution for problem \eqref{eqP} when $\lambda\in(0,\lambda^*=\sup\mathcal{L})$.
\begin{prop}\label{prop10}
	If $p=2$ (semilinear problem), hypotheses $H_1$ and $\widehat{H}$ hold, $\lambda\in\mathcal{L}$, $u_{\lambda}\in S(\lambda)\subseteq \mbox{int}\, C_+$ and $\nu\in(0,\lambda)$, then there exists $u_{\nu}\in S(\nu)\subseteq \mbox{int}\, C_+$ such that $u_{\lambda}-u_{\nu}\in \mbox{int}\, C_+$.
\end{prop}
\begin{proof}
	From Corollary \ref{cor9}, we already have a solution $u_{\nu}\in S(\nu)\subseteq \mbox{int}\,C_+$ such that
	\begin{equation}\label{eq33}
		u_{\nu}\leq u_{\lambda}.
	\end{equation}
	
	Let $\rho=||u_{\lambda}||_{\infty}$ and let $\xi_{\rho}>0$ be as postulated by hypothesis $H_1(iv)$. Then
	\begin{eqnarray*}
		&-\Delta_pu_{\nu}(z)+\xi_{\rho}u_{\nu}(z)&=f(z,u_{\nu}(z))+\xi_{\rho}u_{\nu}(z)\\
		&&\leq f(z,u_{\lambda}(z))+\xi_{\rho}u_{\lambda}(z)\ (\mbox{see (\ref{eq33}) and hypothesis}\ H_1(iv))\\
		&&=-\Delta u_{\lambda}(z)+\xi_{\rho}u_{\lambda}(z)\ \mbox{for almost all}\ z\in\Omega\ (\mbox{since}\ u_{\lambda}\in S(\lambda)),\\
		\Rightarrow&\Delta(u_{\lambda}-u_{\nu})(z)&\leq\xi_{\rho}(u_{\lambda}-u_{\nu})(z)\ \mbox{for almost all}\ z\in\Omega,\\
		\Rightarrow&u_{\lambda}-u_{\nu}\in \mbox{int}\,C_+&(\mbox{from the maximum principle, see \cite[p. 738]{9}}).
	\end{eqnarray*}
\end{proof}
\begin{remark}
	In the nonlinear case ($p\in(1,\infty)$), it is this strong comparison result that we are missing in order to have a bifurcation-type theorem. It is an interesting open problem whether Proposition \ref{prop10} is still valid when $1<p<\infty$. Such a result will lead to a bifurcation-type theorem for the general nonlinear problem \eqref{eqP}.
\end{remark}
	
	Let $\lambda^*=\sup\mathcal{L}$. We will show that $\lambda^*<\infty$. To this end we will need some preparation. Note that hypotheses $H_1(i),(ii),(iii)$ imply that there exists $c_{15}>0$ such that
	\begin{equation}\label{eq34}
		f(z,x)\geq -c_{15}x^{p-1}\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\geq 0.
	\end{equation}
	
	We consider the following auxiliary parameter nonlinear problem:
	\begin{eqnarray}\label{eq35}
		\left\{\begin{array}{ll}
			\di -\Delta_pu(z)+c_{15}u(z)^{p-1}=0&\mbox{in}\ \Omega,\ 1<p<\infty,\\
		\di	\frac{\partial u}{\partial n_p}=\lambda\beta(z)u(z)^{q-1}&\mbox{on}\ \partial\Omega,\ u>0.
		\end{array}\right\}
	\end{eqnarray}
	
	For this problem we have the following existence and uniqueness result (see also Sabina de Lis \cite[p. 472]{19}).
	\begin{prop}\label{prop11}
		If hypotheses $\widehat{H}$ hold, then for every $\lambda>0$ problem (\ref{eq35}) admits a unique solution $\tilde{u}_{\lambda}\in \mbox{int}\,C_+$
		$$\tilde{u}_{\lambda}=\lambda^{\frac{1}{p-q}}\tilde{u}_1,\ \tilde{u}_{\lambda}\rightarrow 0\ \mbox{in}\ C^1(\overline{\Omega})\ \mbox{as}\ \lambda\rightarrow 0^+\ \mbox{and}\ \tilde{u}_{\lambda}\leq u\ \mbox{for all}\ u\in S(\lambda)\subseteq \mbox{int}\,C_+.$$
	\end{prop}
	\begin{proof}
	Let $\psi_{\lambda}:W^{1,p}(\Omega)\rightarrow\RR$ be the $C^1$-functional defined by
	 $$\psi_{\lambda}(u)=\frac{1}{p}||Du||^{p}_{p}+\frac{c_{15}}{p}||u||^{p}_{p}-\frac{\lambda}{q}\int_{\partial\Omega}\beta(z)u^+(z)^qd\sigma\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$
	
	Since $q<p$, the functional $\psi_{\lambda}$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find $(\tilde{u})_{\lambda}\in W^{1,p}(\Omega)$ such that
	\begin{equation}\label{eq36}
		\psi_{\lambda}(\tilde{u}_{\lambda})=\inf[\psi_{\lambda}(u):u\in W^{1,p}(\Omega)].
	\end{equation}
	
	Let $u\in \mbox{int}\,C_+$ and $t>0$. We have
	\begin{eqnarray}\label{eq37}
		 \psi_{\lambda}(tu)&=&\frac{t^p}{p}||Du||^{p}_{p}+\frac{c_{15}t^p}{p}||u||^{p}_{p}-\frac{\lambda}{q}t^q\int_{\partial\Omega}\beta(z)u(z)d\sigma\nonumber\\
		&=&\frac{t^p}{p}\left[||Du||^{p}_{p}+c_{15}||u||^{p}_{p}\right]-t^q\frac{\lambda}{q}c_{16}
	\end{eqnarray}
	where $c_{16}=c_{16}(u)=\int_{\partial\Omega}\beta(z)u(z)d\sigma>0$ (see hypotheses $\widehat{H}$). Since $q<p$, choosing $t\in(0,1)$ small from (\ref{eq37}) we have
	\begin{eqnarray*}
		&&\psi_{\lambda}(tu)<0,\\
		&\Rightarrow&\psi_{\lambda}(\tilde{u}_{\lambda})<0=\psi_{\lambda}(0),\\
		&\Rightarrow&\tilde{u}_{\lambda}\neq 0.
	\end{eqnarray*}
	
	From (\ref{eq36}) we have
	\begin{eqnarray}\label{eq38}
		&&\psi'_{\lambda}(\tilde{u}_{\lambda})=0,\nonumber\\
		&\Rightarrow&\left\langle A(\tilde{u}_{\lambda}),h\right\rangle+c_{15}\int_{\Omega}|\tilde{u}_{\lambda}|^{p-2}\tilde{u}_{\lambda}hdz=\lambda\int_{\partial\Omega}\beta(z)(\tilde{u}^+_{\lambda})^{q-1}hd\sigma\\
		&&\hspace{6.5cm}\mbox{for all}\ h\in W^{1,p}(\Omega).\nonumber
	\end{eqnarray}
	
	In (\ref{eq38}) we choose $h=-\tilde{u}^-_{\lambda}\in W^{1,p}(\Omega)$. Then
	\begin{eqnarray*}
		&&||D\tilde{u}^-_{\lambda}||^{p}_{p}+c_{15}||\tilde{u}^-_{\lambda}||^{p}_{p}=0,\\
		&\Rightarrow&\tilde{u}_{\lambda}\geq 0,\ \tilde{u}_{\lambda}\neq 0.
	\end{eqnarray*}
	
	Therefore $\tilde{u}_{\lambda}$ is a positive solution of problem (\ref{eq35}) (see the proof of Proposition \ref{prop7}) and as before the nonlinear regularity theory (see Lieberman \cite{14}) and the nonlinear maximum principle (see \cite[p. 738]{9}), imply $\tilde{u}_{\lambda}\in \mbox{int}\,C_+$.
	
	Next we prove the uniqueness of this positive solution. So, we consider the integral functional $J_{\lambda}:L^1(\Omega)\rightarrow\RR$ defined by
	$$J_{\lambda}(u)=\left\{\begin{array}{ll}
		 \di\frac{1}{p}||Du^{1/p}||^{p}_{p}-\frac{\lambda}{q}\int_{\partial\Omega}\beta(z)u(z)^{q/p}d\sigma&\mbox{if}\ u\geq 0,\ u^{1/p}\in W^{1,p}(\Omega)\\
\di		+\infty&\mbox{otherwise}.
	\end{array}\right.$$
	
	Let $u_1,u_2\in \mbox{dom}\, J_{\lambda}=\{u\in W^{1,p}(\Omega):J_{\lambda}(u)<\infty\}$. From Lemma 1 of Diaz and Saa \cite{4} we see that $u\longmapsto \frac{1}{p}||Du||^{1/p}||^{p}_{p}$ is convex. Since $q<p$, the map $ u\longmapsto -\frac{\lambda}{q}\int_{\partial\Omega}\beta(z)u^{q/p}d\sigma$ id convex. Therefore $J_{\lambda}$ is convex and by Fatou's lemma it is also lower semicontinuous.
	
	Suppose $\tilde{u}_{\lambda},\tilde{v}_{\lambda}$ are two positive solutions of (\ref{eq35}). From the first part of the proof we have $\tilde{u}_{\lambda},\tilde{v}_{\lambda}\in \mbox{int}\,C_+$. Hence $\tilde{u}^p_{\lambda},\tilde{v}^p_{\lambda}\in \mbox{dom}\, J_{\lambda}$. Also, if $h\in C^1(\overline{\Omega})$, then we all $t\in(-1,1)$ with $|t|$ small, we have
	$$\tilde{u}_{\lambda}^p+th\in \mbox{dom}\, J_{\lambda}\ \mbox{and}\ \tilde{v}_{\lambda}^p+th\in \mbox{dom}\, J_{\lambda.}$$
	
	Moreover, via the chain rule and Green's identity, we have
	\begin{eqnarray*}
		 &&J'_{\lambda}(\tilde{u}_{\lambda}^p)(h)=\frac{1}{p}\int_{\Omega}\frac{-\Delta_p\tilde{u}_{\lambda}}{\tilde{u}_{\lambda}^{p-1}}hdz\\
		 &&J'_{\lambda}(\tilde{v}_{\lambda}^p)(h)=\frac{1}{p}\int_{\Omega}\frac{-\Delta_p\tilde{v}_{\lambda}}{\tilde{v}_{\lambda}^{p-1}}hdz\ \mbox{for all}\ h\in C^1(\overline{\Omega}).
	\end{eqnarray*}
	
	The convexity of $J_{\lambda}$ implies  the monotonicity of $J'_{\lambda}$. Hence
	\begin{eqnarray*}
		 0&\leq&\int_{\Omega}\left(\frac{-\Delta_{p}\tilde{u}_{\lambda}}{\tilde{u}_{\lambda}^{p-1}}+\frac{\Delta_p \tilde{v}_{\lambda}}{\tilde{v}_{\lambda}^{p-1}}\right)(\tilde{u}_{\lambda}^{p-1}-\tilde{v}_{\lambda}^{p-1})dz\\
		 &=&\int_{\Omega}c_{15}(\tilde{v}_{\lambda}-\tilde{u}_{\lambda})(\tilde{u}_{\lambda}^{p-1}-\tilde{v}_{\lambda}^{p-1})dz\ (\mbox{see (\ref{eq35})}),\\
		\Rightarrow&&\tilde{u}_{\lambda}=\tilde{v}_{\lambda}.
	\end{eqnarray*}
	
	This proves the uniqueness of the positive solution $\tilde{u}_{\lambda}\in \mbox{int}\,C_+$ of problem (\ref{eq35}).
	
	Clearly $\tilde{u}_{\lambda}=\lambda^{\frac{1}{p-q}}\tilde{u}_1$ for all $\lambda>0$.
	
	Let $\lambda_n\rightarrow 0^+$ and let $\tilde{u}_n=\tilde{u}_{\lambda_n}\in \mbox{int}\,C_+$ be the corresponding positive solution of (\ref{eq35}). Then from Lieberman \cite[Theorem 2]{14}  we can find $\alpha\in(0,1)$ and $c_{16}>0$ such that
		\begin{equation}\label{eq39}
			\tilde{u}_n\in C^{1,\alpha}(\overline{\Omega})\ \mbox{and}\ ||\tilde{u}_n||_{C^{1,\alpha}(\overline{\Omega})}\leq c_{16}\ \mbox{for all}\ n\geq 1.
		\end{equation}
		
		Exploiting the compact embedding of $C^{1,\alpha}(\overline{\Omega})$ into $C^1(\overline{\Omega})$, we obtain
		\begin{equation}\label{eq40}
			\tilde{u}_n\rightarrow\tilde{u}\ \mbox{in}\ C^1(\overline{\Omega}).
		\end{equation}
		
		We have
		\begin{eqnarray*}
			&&\left\langle A(\tilde{u}_n),h\right\rangle+c_{15}\int_{\Omega}\tilde{u}_n^{p-1}dz=\lambda_n\int_{\partial\Omega}\beta(z)\tilde{u}_n^{q-1}hd\sigma\ \mbox{for all}\ h\in W^{1,p}(\Omega),\ \mbox{all}\ n\geq 1,\\
			&\Rightarrow&\left\langle A(\tilde{u}),h\right\rangle+c_{15}\int_{\Omega}\tilde{u}^{p-1}hdz=0\ (\mbox{see (\ref{eq40})}).
		\end{eqnarray*}
		
		Let $h=\tilde{u}\in W^{1,p}(\Omega)$. Then
		\begin{eqnarray*}
			&&||D\tilde{u}||^{p}_{p}+c_{15}||\tilde{u}||^{p}_{p}=0,\\
			&\Rightarrow&\tilde{u}=0.
		\end{eqnarray*}
		
		So, we conclude that $\tilde{u}_{\lambda}\rightarrow 0$ in $C^1(\overline{\Omega})$ as $\lambda\rightarrow 0^+.$
		
		Finally, let $u\in S(\lambda)$ and consider the following Carath\'eodory functions
		\begin{eqnarray}\label{eq41}
			&&e(z,x)=\left\{\begin{array}{ll}
				(c_{15}-1)(x^+)^{p-1}&\mbox{if}\ x\leq u(z)\\
				(c_{15}-1)u(z)^{p-1}&\mbox{if}\ u(z)<x
			\end{array}\right.\ \mbox{and}\nonumber\\ &&d_{\lambda}(z,x)=\left\{\begin{array}{ll}
				\lambda\beta(z)(x^+)^{q-1}&\mbox{if}\ x\leq u(z)\\
				\lambda\beta(z)u(z)^{q-1}&\mbox{if}\ u(z)<x
			\end{array}\right.
		\end{eqnarray}
		
		We set $E(z,x)=\int^{x}_{0}e(z,s)ds$ and $D_{\lambda}(z,x)=\int^{x}_{0}d_{\lambda}(z,s)ds$ and consider the $C^1$-functional $\psi_{\lambda}:W^{1,p}(\Omega)\rightarrow \RR$ defined by
		\begin{eqnarray*}
			 &&\psi_{\lambda}(u)=\frac{1}{p}||Du||^{p}_{p}+\frac{1}{p}||u||^{p}_{p}+\int_{\Omega}E(z,u(z))dz-\int_{\partial\Omega}D_{\lambda}(z,u(z))d\sigma\\
			&&\hspace{7cm}\mbox{for all}\ u\in W^{1,p}(\Omega).
		\end{eqnarray*}
		
		It is clear from (\ref{eq41}) that $\psi_{\lambda}$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find $\tilde{u}\in W^{1,p}(\Omega)$ such that
		\begin{equation}\label{eq42}
			\psi_{\lambda}(\tilde{u})=\inf[\psi_{\lambda}(u):u\in W^{1,p}(\Omega)].
		\end{equation}
		
		Since $q<p$, as before we have that $\psi_{\lambda}(\tilde{u})<0=\psi_{\lambda}(0)$, hence $\tilde{u}\neq 0$. From (\ref{eq42}) we have
		\begin{eqnarray*}
			&&\psi'_{\lambda}(\tilde{u})=0,\\
			&\Rightarrow&\left\langle A(\tilde{u}),h\right\rangle+\int_{\Omega}|\tilde{u}|^{p-2}\tilde{u}hdz+\int_{\Omega}e(z,\tilde{u})hdz=\int_{\partial\Omega}\lambda\beta(z)\tilde{u}^{q-1}hd\sigma\\
			&&\hspace{7cm}\mbox{for all}\ h\in W^{1,p}(\Omega).
		\end{eqnarray*}
		
		Choosing $h=-\tilde{u}^-\in W^{1,p}(\Omega)$ and $h=(\tilde{u}-u)^+\in W^{1,p}(\Omega)$, using (\ref{eq34}) and reasoning as in the proof of Proposition \ref{prop8}, we show that
		\begin{eqnarray*}
			&&\tilde{u}\in[0,u],\ \tilde{u}\neq 0\\
			&\Rightarrow&\tilde{u}\in \mbox{int}\,C_+\ \mbox{is a positive solution of (\ref{eq35})},\\
			&\Rightarrow&\tilde{u}=\tilde{u}_{\lambda}\leq u.
		\end{eqnarray*}
	\end{proof}
	
	Now let $\hat{a}\in L^{\infty}(\Omega)$ with $\mbox{essinf}_{\Omega}\hat{a}>0$ and $\hat{b}\in L^{\infty}(\partial\Omega)$. We consider the following nonlinear eigenvalue problem:
	\begin{equation}\label{eq43}
		\left\{\begin{array}{ll}
		\di	-\Delta_pu(z)=\vartheta\hat{a}(z)|u(z)|^{p-2}u(z)&\mbox{in}\ \Omega,\\
			\di\frac{\partial u}{\partial n_p}=\hat{b}(z)|u(z)|^{p-2}u(z)&\mbox{on}\ \partial\Omega.
		\end{array}\right\}
	\end{equation}
\begin{prop}\label{prop12}
	Problem (\ref{eq43}) has a similar eigenvalue $\theta_1$ with positive eigenfunctions. No other eigenvalue has positive eigenfunctions and $\theta_1$ is simple (that is, if $u_1,u_2$ are both eigenfunctions for $\vartheta_1$, then $u_1=\xi u_2$ with $\xi\in\RR\backslash\{0\}$).
\end{prop}	
\begin{proof}
	Let $M=\{u\in W^{1,p}(\Omega):\int_{\Omega}\hat{a}(z)|u(z)|^pd\sigma=1\}$. The Sobolev embedding theorem implies that $M$ is weakly closed in $W^{1,p}(\Omega)$.
	
	Let $j:W^{1,p}(\Omega)\rightarrow\RR$ be the $C^1$-functional defined by
	\begin{equation}\label{eq44}
		j(u)=\frac{1}{p}||Du||^{p}_{p}-\frac{1}{p}\int_{\partial\Omega}\hat{b}(z)|u(z)|^pd\sigma\ \mbox{for all}\ u\in W^{1,p}(\Omega).
	\end{equation}
	
	From Ehrling's inequality (see \cite[p. 695]{16}  and \cite{19}), we have that given $\epsilon>0$, we can find $c_{\epsilon}>0$ such that
	\begin{equation}\label{eq45}
		\int_{\partial\Omega}|u|^pd\sigma\leq\epsilon||Du||^{p}_{p}+c_{\epsilon}||u||^{p}_{p}.
	\end{equation}
	
	Also note that, if $\hat{m}=\mbox{essinf}_{\Omega}\hat{a}>0$, then
	\begin{equation}\label{eq46}
		1=\int_{\Omega}\hat{a}(z)|u|^pdz\geq \hat{m}||u||^{p}_{p}\ \mbox{for all}\ u\in M.
	\end{equation}
	
	From (\ref{eq44}), (\ref{eq45}) and (\ref{eq46}) it follows that $j|_M$ is coercive. Also, by the Sobolev embedding theorem and the compactness of the trace map, we see that $j(\cdot)$ is sequentially weakly lower semicontinuous. Since $M$ is weakly closed, we can find $\hat{u}_1\in M$ such that
	$$\vartheta_1=j(\hat{u}_1)=\inf[j(u):u\in M].$$
	
	Replacing $\hat{u}_1$ by $|\hat{u}_1|\in M$, we see that $j(|\hat{u}_1|)=\theta_1$ and so without any lose of generality we may assume that $\hat{u}_1\geq 0$ and of course $\hat{u}_1\neq 0$ since $\hat{u}_1\in M$.
	
	From the Lagrange multiplier rule (see, for example, Papageorgiou and Kyritsi \cite[p. 361]{16}), we have
	\begin{eqnarray*}
		&&\left\langle A(\hat{u}_1),h\right\rangle-\int_{\partial\Omega}\hat{b}(z)\hat{u}_1^{p-1}hd\sigma=\eta_1\int_{\Omega}\hat{a}(z)\hat{u}_1^{p-1}hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega)\\
		&\Rightarrow&-\Delta_p\hat{u}_1(z)=\eta_1\hat{a}(z)\hat{u}_1(z)^{p-1}\ \mbox{for almost all}\ z\in\Omega,\ \frac{\partial\hat{u}_1}{\partial n_p}=\hat{b}(z)\hat{u}_1^{p-1}\ \mbox{on}\ \partial\Omega\\
		&&\hspace{6cm}(\mbox{see the proof of Proposition \ref{prop7}}).
	\end{eqnarray*}
	
	As before, the nonlinear regularity theory and the nonlinear maximum principle imply $\hat{u}_1\in \mbox{int}\, C_+$.
	
	That every other eigenvalue $\vartheta>\vartheta_1$ has nodal (sign changing) eigenfunctions and that $\vartheta_1$ is simple, follows exactly as in Gasinski and Papageorgiou \cite[pp. 741, 743]{9}.
\end{proof}

Now we are ready to show the finiteness of $\lambda^*=\sup \mathcal{L}.$
\begin{prop}\label{prop13}
	If hypotheses $H_1(i),(ii),(iii)$ and $\widehat{H}$ hold, then $\lambda^*<\infty$.
\end{prop}
\begin{proof}
	Let $\lambda\in\mathcal{L}$. We show that we can find $u_{\lambda}\in S(\lambda)\subseteq \mbox{int}\,C_+$ (see Proposition \ref{prop7}). From Proposition \ref{prop11}, we know that
	\begin{equation}\label{eq47}
		\tilde{u}_{\lambda}\leq u_{\lambda}.
	\end{equation}
	
	We have
	\begin{eqnarray}\label{eq48}
		\left\{\begin{array}{ll}
			 \di -\Delta_pu_{\lambda}(z)=f(z,u_{\lambda}(z))=\frac{f(z,u_{\lambda}(z))}{u_{\lambda}(z)^{p-1}}u_{\lambda}(z)^{p-1}\ \mbox{for almost all}\ z\in\Omega,\\
\di			\frac{\partial u_{\lambda}}{\partial n_p}=\lambda \beta(z)u_{\lambda}^{q-1}=\lambda\beta(z)u_{\lambda}^{q-p}u_{\lambda}^{p-1}\ \mbox{on}\ \partial\Omega.
		\end{array}\right\}
	\end{eqnarray}
	
	We set $\hat{a}(z)=\frac{f(z,u_{\lambda}(z))}{u_{\lambda}(z)^{p-1}}$ and $\hat{b}(z)=\lambda\beta(z)u_{\lambda}(z)^{q-p}$. Then
	\begin{eqnarray*}
		&&\hat{a}\in L^{\infty}(\Omega)\ \mbox{and}\ \mbox{essinf}_{\Omega}\hat{a}\geq\frac{\mu_{\tau_{\lambda}}}{||u_{\lambda}||^{p-1}_{\infty}}\ \mbox{with}\ \tau_{\lambda}=\min\limits_{\overline{\Omega}}u_{\lambda}>0\\
		&&\hspace{4cm}(\mbox{recall}\ u_{\lambda}\in \mbox{int}\,C_+\ \mbox{and see}\ H_1(iii)),\\
		&&\hat{b}\in L^{\infty}(\Omega)\ (\mbox{see hypotheses}\ \widehat{H}\ \mbox{and recall}\ u_{\lambda}\in \mbox{int}\, C_+).
	\end{eqnarray*}
	
	So, problem (\ref{eq48}) has the form of problem (\ref{eq43}). Since $u_{\lambda}\in \mbox{int}\, C_+$ solves problem (\ref{eq48}), according to Proposition \ref{prop12} we must have $\vartheta_1=1$. Moreover, from the proof of Proposition \ref{prop12}, we have
	\begin{equation}\label{eq49}\di		 1\leq\frac{||Du||^{p}_{p}-\int_{\partial\Omega}\hat{b}(z)|u|^pd\sigma}{\int_{\Omega}\hat{a}(z)|u|^pdz}\ \mbox{for all}\ u\in W^{1,p}(\Omega).
	\end{equation}
	
	From (\ref{eq1}) and hypotheses $H_1(ii),(iii)$ we see that we can find $c_{17}>0$ such that
	\begin{eqnarray}\label{eq50}
		&&f(z,x)\geq c_{17}x^{\eta-1}\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ z\geq\tau_{\lambda}>0\\
		&&\hspace{1cm}(\mbox{recall that}\ \tau_{\lambda}=\min\limits_{\overline{\Omega}}u_{\lambda}\ \mbox{and that}\ u_{\lambda}\in \mbox{int}\,C_+)\nonumber
	\end{eqnarray}
	
	Then we have
	\begin{eqnarray}\label{eq51}
	\hat{a}(z)=\frac{f(z,u_{\lambda}(z))}{u_{\lambda}(z)^{p-1}}&\geq&c_{17}u_{\lambda}(z)^{\eta-p}\ (\mbox{see (\ref{eq50})})\nonumber\\
		&\geq&c_{17}\tilde{u}_{\lambda}(z)^{\eta-p}\ (\mbox{see (\ref{eq47})})\nonumber\\
		&\geq&\lambda^{\frac{\eta-p}{p-q}}c_{17}\tilde{u}_{\lambda}(z)^{\eta-p}\ \mbox{for almost all}\ z\in\Omega\ (\mbox{see Proposition \ref{prop11}}).
	\end{eqnarray}
	
	We return to (\ref{eq49}) and use (\ref{eq51}). Then since $\hat{b}\geq 0$, we have
	\begin{eqnarray}\label{eq52}		 &&\di\lambda^{\frac{\eta-p}{p-q}}\leq\frac{||Du||^{p}_{p}+||u||^{p}_{p}}{\int_{\Omega}c_{17}\tilde{u}_1^{\eta-p}|u|^pdz}\ \mbox{for all}\ u\in W^{1,p}(\Omega),\nonumber\\
		&\di\Rightarrow&\lambda^{\frac{\eta-p}{p-q}}\leq\gamma_1,
	\end{eqnarray}
	where $\gamma_1>0$ is the principle eigenvalue of
	\begin{eqnarray*}
		\left\{\begin{array}{ll}
		\di	-\Delta_pu(z)+|u(z)|^{p-2}u(z)=\gamma c_{17}\tilde{u}_1(z)^{\eta-p}|u(z)|^{p-2}u(z)\ \mbox{in}\ \Omega\\
		\di	\frac{\partial u}{\partial n_p}=0.
		\end{array}\right\}
	\end{eqnarray*}
		(see Mugnai and Papageorgiou \cite{15}). Since $\lambda\in\mathcal{L}$ is arbitrary, from (\ref{eq52}) we conclude that $\lambda^*<\infty.$
\end{proof}
\begin{prop}\label{prop14}
	If hypotheses $H_1(i),(ii),(iii)$ and $\widehat{H}$ hold, then $\lambda^*\in\mathcal{L}$ and so $\mathcal{L}=\left(0,\lambda^*\right]$.
\end{prop}
\begin{proof}
	Let $\{\lambda_n\}_{n\geq 1}\subseteq\mathcal{L}$ such that $\lambda_n\uparrow\lambda^*$ and let $u_n\in S(\lambda_n)\subseteq \mbox{int}\,C_+$. From the proof on Proposition \ref{prop8} and Corollary \ref{cor9}, we know that we can assume that $\{u_n\}_{n\geq 1}$ is increasing (that is, $u_n\leq u_{n+1}$ for all $n\geq 1$) and
	\begin{equation}\label{eq53}
		\hat{\varphi}_{\lambda_n}(u_n)<0\ \mbox{for all}\ n\geq 1.
	\end{equation}
	
	We have
	\begin{equation}\label{eq54}
		\left\langle A(u_n),h\right\rangle=\int_{\Omega}f(z,u_n)hdz+\lambda_n\int_{\partial\Omega}\beta(z)u_n^{q-1}hd\sigma\ \mbox{for all}\ h\in W^{1,p}(\Omega),\ \mbox{all}\ n\geq 1.
	\end{equation}
	
	Using (\ref{eq53}) and (\ref{eq54}) and reasoning as in the proof of Proposition \ref{prop4}, we can show that $\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)$ is bounded. So, by passing to a subsequence if necessary, we may assume that
	\begin{equation}\label{eq55}
		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{equation}
	
	In (\ref{eq54}) we choose $h=u_n-u_*\in W^{1,p}(\Omega)$, pass to the limit as $n\rightarrow\infty$ and use (\ref{eq55}). Then
	\begin{eqnarray}\label{eq56}
		&&\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{see Proposition \ref{prop3}}).
	\end{eqnarray}
	
	Evidently $u_*$ (recall $\{u_n\}_{n\geq 1}\subseteq \mbox{int}\,C_+$) is increasing.
	
	If in (\ref{eq54}) we pass to the limit as $n\rightarrow\infty$ and use (\ref{eq56}), then
	\begin{eqnarray*}
		&&\left\langle A(u_*),h\right\rangle=\int_{\Omega}f(z,u_*)hdz+\lambda_*\int_{\partial\Omega}\beta(z)u_*^{q-1}hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega),\\
		&\Rightarrow&u_*\in S(\lambda_*)\subseteq \mbox{int}\,C_+\ (\mbox{see the proof of Proposition \ref{prop7}})\  \mbox{and so}\ \lambda_*\in\mathcal{L}.
	\end{eqnarray*}
	Therefore by virtue of Proposition \ref{prop8} and \ref{prop13}, we have $\mathcal{L}=\left(0,\lambda^*\right]$.	
\end{proof}

Next we show that for every $\lambda\in\mathcal{L}$ problem \eqref{eqP} admits a minimal positive solution.
\begin{prop}\label{prop15}
	If hypotheses $H_1(i),(ii),(iii)$ and $\widehat{H}$ hold and $\lambda\in\mathcal{L}=\left(0,\lambda^*\right]$, then problem \eqref{eqP} admits a smallest positive solution $\underline{u}_{\lambda}\in \mbox{int}\,C_+$.
\end{prop}
\begin{proof}
	From Filippakis, Kristaly and Papageorgiou \cite{5}, we have that $S(\lambda)$ is downward directed that this, if $u_1,u_2\in S(\lambda)$,  then we can find $u\in S(\lambda)$ such that $u\leq u_1,u\leq u_2$. Since we are looking for the smallest positive solution of $S(\lambda)$, without any loss of generality, we may assume that
	\begin{equation}\label{eq57}
		||u||_{\infty}\geq c_{18}\ \mbox{for some}\ c_{18}>0\ \mbox{all}\ u\in S(\lambda).
	\end{equation}
	
	From Hu and Papageorgiou \cite[p. 178]{12}, we know that there exist $\{u_n\}_{n\geq 1}\subseteq S(\lambda)$ such that
	\begin{equation}\label{eq58}
		\inf S(\lambda)=\inf\limits_{n\geq 1}\ u_n\ \mbox{and}\ \tilde{u}_{\lambda}\leq u_n\ \mbox{for all}\ n\geq 1
\ (\mbox{see Proposition \ref{prop11}}).
	\end{equation}
	We have
	\begin{equation}\label{eq59}
		\left\langle A(u_n),h\right\rangle=\int_{\Omega}f(z,u_n)hdz+\lambda\int_{\partial\Omega}\beta(z)u_n^{q-1}hd\sigma\ \mbox{for all}\ h\in W^{1,p}(\Omega).
	\end{equation}
	
	In (\ref{eq59}) we choose $h=u_n\in W^{1,p}(\Omega)$. Then using (\ref{eq57}) and hypothesis $\widehat{H}$, we have
	\begin{equation}\label{eq60}
		||Du_n||^{p}_{p}\leq c_{19}(1+||u_n||^q)\ \mbox{for some}\ c_{19}>0,\ \mbox{all}\ n\geq 1.
	\end{equation}
	
	From (\ref{eq57}) and (\ref{eq60}) it follows that $\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)$ is bounded. Hence, we may assume that
	\begin{equation}\label{eq61}
		u_n\stackrel{w}{\rightarrow}\underline{u}_{\lambda} \mbox{in}\ W^{1,p}(\Omega)\ \mbox{and}\ u_n\rightarrow\underline{u}_{\lambda}\ \mbox{in}\ L^r(\Omega)\ \mbox{and in}\ L^p(\partial\Omega).
	\end{equation}
	
	If in (\ref{eq59}) we choose $h=u_n-\underline{u}_{\lambda}\in W^{1,p}(\Omega)$, pass to the limit as $n\rightarrow\infty$ and use (\ref{eq61}), then
	\begin{eqnarray}\label{eq62}
		&&\lim\limits_{n\rightarrow\infty}\left\langle A(u_n),u_n-\underline{u}_{\lambda}\right\rangle=0,\nonumber\\
		&\Rightarrow&u_n\rightarrow\underline{u}_{\lambda}\ \mbox{in}\ W^{1,p}(\Omega)\ (\mbox{see Proposition \ref{prop3}})\ \mbox{and}\ \tilde{u}_{\lambda}\leq\underline{u}_{\lambda}\ (\mbox{see (\ref{eq58})}).
	\end{eqnarray}
	
	So, if in (\ref{eq59}) we pass to the limit as (\ref{eq59}), then
	\begin{eqnarray*}
		&&\left\langle A(\underline{u}_{\lambda}),h\right\rangle=\int_{\Omega}f(z,\underline{u}_{\lambda})hdz+\lambda\int_{\partial\Omega}\beta(z)\underline{u}_{\lambda}^{q-1}hd\sigma\ \mbox{for all}\ h\in W^{1,p}(\Omega),\\
		&\Rightarrow&\underline{u}_{\lambda}\in S(\lambda)\subseteq \mbox{int}\,C_+\ (\mbox{see (\ref{eq62})})\ \mbox{and}\ \underline{u}_{\lambda}=\inf S(\lambda).
	\end{eqnarray*}
\end{proof}

We examine the map $\lambda\longmapsto\underline{u}_{\lambda}$.
\begin{prop}\label{prop16}
	If hypotheses $H_1(i),(ii),(iii)$ and $\widehat{H}$ hold, then the map $\lambda\rightarrow\underline{u}_{\lambda}$ from $\mathcal{L}=\left(0,\lambda^*\right]$ into $C^1(\overline{\Omega})$ is
	\begin{itemize}
		\item[$\bullet$] increasing (that is, if $\nu<\lambda$, then $\underline{u}_{\nu}\leq\underline{u}_{\lambda}$)
		\item[$\bullet$] left continuous.
	\end{itemize}
\end{prop}
\begin{proof}
	Let, $\lambda,\nu\in[0,\lambda^*]$ with $\nu<\lambda$. From Corollary \ref{cor9} we know that there exist $u_{\nu}\in S(\nu)$ such that
	\begin{eqnarray*}
		&&u_{\nu}\leq\underline{u}_{\lambda},\\
		&\Rightarrow&\underline{u}_{\nu}\leq\underline{u}_{\lambda},\\
		&\Rightarrow&\lambda\longmapsto\underline{u}_{\lambda}\ \mbox{is increasing}.
	\end{eqnarray*}
	
	Next let $\{\lambda_n\}_{n\geq 1}\subseteq\mathcal{L}$ such that $\lambda_n\uparrow\lambda\in\mathcal{L}$. We can find $u_n\in S(\lambda_n)\ n\geq 1$ such that
	\begin{equation}\label{eq63}
		\{u_n\}_{n\geq 1}\subseteq \mbox{int}\,C_+\ \mbox{is increasing and}\ \hat{\varphi}_{\lambda_n}(u_n)<0\ \mbox{for all}\ n\geq 1
	\end{equation}
	(see the proof of Proposition \ref{prop7}). We have
	\begin{equation}\label{eq64}
		\left\langle A(u_n),h\right\rangle=\int_{\Omega}f(z,u_n)hdz+\lambda_n\int_{\partial\Omega}\beta(z)u_n^{q-1}hd\sigma\ \mbox{for all}\ h\in W^{1,p}(\Omega),\ \mbox{all}\ n\geq 1.
	\end{equation}
	
	Using (\ref{eq63}) and (\ref{eq64}) and reasoning as in the proof of Proposition \ref{prop4}, we have
	$$\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)\ \mbox{is bounded}.$$
	
	From Lieberman \cite{14}, we know that there exist $\alpha\in(0,1)$ and $c_{20}>0$ such that
	$$\underline{u}_n\in C^{1,\alpha}(\overline{\Omega})\ \mbox{and}\ ||\underline{u}_n||_{C^{1,\alpha}(\overline{\Omega})}\leq c_{20}\ \mbox{for all}\ n\geq 1.$$
	
	Because of the compact embedding of $C^{1,\alpha}(\overline{\Omega})$ into $C^1(\overline{\Omega})$ and since $\{\underline{u}_n\}_{n\geq}\subseteq \mbox{int}\,C_+$ is increasing (we have already established that $\lambda\rightarrow\underline{u}_{\lambda}$ is increasing), we have
	\begin{equation}\label{eq65}
		\underline{u}_n\rightarrow\tilde{u}\ \mbox{in}\ C^1(\overline{\Omega})\ \mbox{as}\ n\rightarrow\infty.
	\end{equation}
	
	By passing to the limit as $n\rightarrow\infty$ in (\ref{eq64}), we see that $\tilde{u}\in S(\lambda)$. Suppose $\tilde{u}\neq\underline{u}_{\lambda}$. Then we can find $z_0\in\overline{\Omega}$ such that
	\begin{eqnarray*}
		&&\underline{u}_{\lambda}(z_0)<\tilde{u}(z_0)\\
		&\Rightarrow&\underline{u}_{\lambda}(z_0)<\underline{u}_n(z_0)\ \mbox{for all}\ n\geq n_0\ (\mbox{see (\ref{eq65})}),
	\end{eqnarray*}
	which contradicts the monotonicity of $\lambda\rightarrow\underline{u}_{\lambda}$ established in the first part of the proof. This proves the left continuity of $\lambda\rightarrow\underline{u}_{\lambda}$ from $\mathcal{L}=\left(0,\lambda^*\right]$ into $C^1(\overline{\Omega})$.
\end{proof}

In the semilinear case $(p=2)$ and is $\lambda\in(0,\lambda^*)$, we can prove a multiplicity result for the positive solutions of \eqref{eqP}.
\begin{prop}\label{prop17}
	If $p=2$, hypotheses $H_1$ and $\widehat{H}$ hold and $\lambda\in(0,\lambda^*)$, then  problem \eqref{eqP} has at least two positive solutions
	$$u_n,\hat{u}\in \mbox{int}\,C_+\ \mbox{and}\ u_0\neq\hat{u}.$$
\end{prop}
\begin{proof}
	Let $\nu<\lambda<\mu<\lambda^*$. From Proposition \ref{prop8} we know that $\nu,\lambda,\mu\in\mathcal{L}$. Also, from Proposition \ref{prop10}, we know that we can find $u_{\nu}\in S(\nu)\subseteq \mbox{int}\,C_+,\ u_0\in S(\lambda)\subseteq \mbox{int}\,C_+$ and $u_{\mu}\in S(\mu)\subseteq \mbox{int}\,C_+$ such that
	\begin{equation}\label{eq66}
		u_0\in \mbox{int}_{C^1(\overline{\Omega})}[u_{\nu},u_{\mu}].
	\end{equation}
	
	From the proof of Proposition \ref{prop8}, we know that if we consider the truncated functional $\hat{\psi}_{\mu}\in C^1(H^1(\Omega))$ (see (\ref{eq27}), (\ref{eq28}) with $u_{\lambda}$ replaced by $u_{\mu}$ and $\nu$ by $\lambda$), then
	\begin{equation}\label{eq67}
		u_0\ \mbox{is a minimizer of}\ \hat{\psi}_{\mu}.
	\end{equation}
	
	We consider the following Carath\'eodory functions
	\begin{eqnarray}
		g(z,x)=\left\{\begin{array}{ll}
			f(z,u_{\nu}(z))+u_{\nu}(z)&\ \mbox{if}\ x\leq u_{\nu}(z)\\
			f(z,x)+x&\mbox{if}\ u_{\nu}(z)<x
		\end{array}\right.\ \mbox{for all}\ (z,x)\in\Omega\times\RR,\label{eq68}\\
		w_{\lambda}(z,x)=\left\{\begin{array}{ll}
			\lambda\beta(z)u_{\nu}(z)^{q-1}&\mbox{if}\ x\leq u_{\nu}(z)\\
			\lambda\beta(z)x^{q-1}&\mbox{if}\ u_{\nu}(z)<x
		\end{array}\right.\ \mbox{for all}\ (z,x)\in\partial\Omega\times\RR.\label{eq69}
	\end{eqnarray}
	
	We set $G(z,x)=\int^{x}_{0}g(z,s)ds$ and $W_{\lambda}(z,s)=\int^{x}_{0}w_{\lambda}(z,s)ds$ and introduce the $C^1$-functional $\hat{\sigma}_{\lambda}:H^1(\Omega)\rightarrow \RR$ defined by
	\begin{eqnarray*}	 &&\hat{\sigma}_{\lambda}(u)=\frac{1}{2}||Du||^{2}_{2}+\frac{1}{2}||u||^{2}_{2}-
\int_{\Omega}G(z,u(z))dz-\int_{\partial\Omega}W_{\lambda}(z,u(z))d\sigma\\
		&&\hspace{8cm}\mbox{for all}\ u\in H^1(\Omega).
	\end{eqnarray*}
	
	From (\ref{eq68}) and (\ref{eq69}), we have
	\begin{equation}\label{eq70}
		\hat{\sigma}_{\lambda}=\hat{\varphi}_{\lambda}+\hat{\xi}_{\lambda}^*\ \mbox{with}\ \hat{\xi}_{\lambda}^*\in\RR.
	\end{equation}
	
	From (\ref{eq70}) it follows that
	\begin{eqnarray}
		&\bullet&\hat{\sigma}_{\lambda}\ \mbox{satisfies the}\ C-\mbox{condition (see Proposition \ref{prop4})};\label{eq71}\\
		&\bullet&\mbox{for all}\ u\in \mbox{int}\,C_+,\ \hat{\sigma}_{\lambda}(tu)\rightarrow-\infty\ \mbox{as}\ t\rightarrow+\infty\ (\mbox{see Proposition \ref{prop6}}).\label{eq72}
	\end{eqnarray}
	
	Moreover, note that
	\begin{eqnarray}\label{eq73}	 &&\left.\hat{\sigma}_{\lambda}\right|_{[u_{\nu},u_{\mu}]}=\left.\hat{\varphi}_{\lambda}\right|_{[u_{\nu},u_{\mu}]}\ (\mbox{see (\ref{eq68}), (\ref{eq69}) and the proof of Proposition \ref{prop8}}),\nonumber\\
		&\Rightarrow&u_0\ \mbox{is a local}\ C^1(\overline{\Omega})-\mbox{minimizer of}\ \hat{\sigma}_{\lambda}\nonumber\\
		&&\hspace{3cm}(\mbox{see (\ref{eq68}), (\ref{eq69}) and the proof of Proposition \ref{prop8}})\nonumber\\
		&\Rightarrow&u_0\ \mbox{is a local}\ H^1(\Omega)-\mbox{minimizer of}\ \hat{\sigma}_{\lambda}\ (\mbox{see Proposition \ref{prop2}}).
	\end{eqnarray}
	
	Let $\left[u_{\nu}\right)=\{u\in H^1(\Omega):u_{\nu}(z)\leq u(z)\ \mbox{for almost all}\ z\in\Omega\}$. We can easily check that
	\begin{equation}\label{eq74}
		K_{\hat{\sigma}_{\lambda}}\subseteq\left[u_{\nu}\right)
	\end{equation}
	
	We may assume that $K_{\hat{\sigma}_{\lambda}}$ is finite or otherwise we already have infinitely many distinct positive solutions (see (\ref{eq68}), (\ref{eq69}), (\ref{eq74})). Then because of (\ref{eq73}), we can find $\rho\in\left[0,1\right)$ small such that
	\begin{equation}\label{eq75}
		 \hat{\sigma}_{\lambda}(u_0)<\inf\left[\hat{\sigma}_{\lambda}(u):||u-u_0||=\rho\right]=\hat{m}_{\lambda}
	\end{equation}
	(see Aizicovici, Papageorgiou and Staicu \cite{1} (proof of Proposition 29)). The (\ref{eq71}), (\ref{eq72}) and (\ref{eq75}) permit the use of Theorem \ref{th1} (the mountain pass theorem). So, we can find $\hat{u}\in W^{1,p}(\Omega)$ such that
	\begin{equation}\label{eq76}
		\hat{u}\in K_{\hat{\sigma}_{\lambda}}\ \mbox{and}\ \hat{m}_{\lambda}\leq\hat{\sigma}_{\lambda}(\hat{u}).
	\end{equation}
	
	From (\ref{eq68}), (\ref{eq69}), (\ref{eq74}) and (\ref{eq76}), we see that $\hat{u}\in S(\lambda)\subseteq \mbox{int}\,C_+$, while from (\ref{eq75}) and (\ref{eq76}) it follows that $\hat{u}\neq u_0$.
\end{proof}

We can summarize our investigation of the positive solutions for problem \eqref{eqP}, with two theorems. The first concerns the nonlinear equation $(1<p<\infty)$.
\begin{theorem}\label{th18}
	If hypotheses $H_1(i),(ii),(iii)$ and $\widehat{H}$ hold, then there exists $\lambda^*\in(0,+\infty)$ such that for every $\lambda\in\left(0,\lambda^*\right]$, problem \eqref{eqP} has a positive solution, in fact it has a smallest positive solution $\underline{u}_{\lambda}\in \mbox{int}\,C_+$ and the map $\lambda\rightarrow \underline{u}_{\lambda}$ from $\mathcal{L}=\left(0,\lambda^*\right]$ into $C^1(\overline{\Omega})$ is increasing and left continuous. For $\lambda>\lambda^*$ problem \eqref{eqP} has no positive solution.
\end{theorem}

For the semilinear equation $(p\equiv 2)$, we have a bifurcation-type result.
\begin{theorem}\label{th19}
	If $p=2$ and hypotheses $H_1$ and $\widehat{H}$ hold, then there exists $\lambda^*\in(0,+\infty)$ such that
	\begin{itemize}
		\item[(a)] for every $\lambda\in(0,\lambda^*)$ problem \eqref{eqP} has at least two positive solutions
		$$u_0,\hat{u}\in \mbox{int}\,C_+,\ u_0\neq\hat{u};$$
		\item[(b)] for $\lambda=\lambda^*$ problem ($P_{\lambda^*}$) has at least one positive solution $u_*\in \mbox{int}\,C_+$;
		\item[(c)] for $\lambda>\lambda^*$ problem \eqref{eqP} has no positive solution;
		\item[(d)] for every $\lambda\in\mathcal{L}=\left(0,\lambda^*\right]$ problem \eqref{eqP} has a smallest positive solution $\underline{u}_{\lambda}\in \mbox{int}\,C_+$ and the map $\lambda\rightarrow\underline{u}_{\lambda}$ from $\mathcal{L}=\left(0,\lambda^*\right]$ into $C^1(\overline{\Omega})$ is increasing and left-continuous.
	\end{itemize}
\end{theorem}


\section{Nodal Solutions}\label{sec4}

In this section we produce nodal (sign changing) solutions, by imposing bilateral conditions on $f(z,\cdot)$.

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

$H_2:$ $f:\Omega\times\RR\rightarrow\RR$ is a Carath\'eodory function such that for almost all $z\in\Omega$, $f(z,0)=0$ and
\begin{itemize}
	\item[(i)] $|f(z,x)|\leq a(z)(1+|x|^{r-1})$ for almost all $z\in\Omega$, all $z\in\RR$, with $z\in L^{\infty}(\Omega)_+,\ p<r<p^*$;
	\item[(ii)] if $F(z,x)=\int^{x}_{0}f(z,s)ds$ then there exist $\eta>p$ and $M>0$ such that
		\begin{eqnarray*}
		&&0<\eta F(z,x)\leq f(z.x)x\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ |x|\geq M,\\
		&&\mbox{essinf}_{\Omega}\ F(\cdot,\pm M)>0;
		\end{eqnarray*}
	\item[(iii)] $\lim\limits_{x\rightarrow 0}\frac{f(z,x)}{|x|^{p-2}x}=0$ uniformly for almost all $z\in\Omega$ and for all $\tau>0,\ f(z,x)x\geq \mu_{\tau}>0$ for almost all $z\in\Omega$, all $|x|\geq\tau$;
	\item[(iv)] if $p=2$, then for every $\rho>0$, there exists $\xi_{\rho}>0$ such that for almost all $z\in\Omega$, the function $x\longmapsto f(z,x)+\xi_{\rho}x$ is increasing on $[-\rho,\rho]$.
\end{itemize}
	
	Arguing as in Section \ref{sec3}, this time on the negative semi-axis $\left(-\infty,0\right]$, we can produce a critical parameter value $\hat{\lambda}^*>0$ such that for all $\lambda\in\left(0,\hat{\lambda}^*\right]$ problem \eqref{eqP} has a maximal negative solution $\bar{v}_{\lambda}\in-\mbox{int}\,C_+$. So, for $\lambda\in(0,\lambda^*]$ with $\lambda_0^*=\min\{\lambda^*,\hat{\lambda}^*\}$, problem \eqref{eqP} admits extremal constant sign solutions $\underline{u}_{\lambda}\in \mbox{int}\,C_+$ and $\bar{v}_{\lambda}\in-\mbox{int}\,C_+$. Using then, we can generate a nodal solution.
	\begin{prop}\label{prop20}
		If hypotheses $H_2(i),(ii),(iii)$ hold and $\lambda\in\left(0,\lambda_0^*\right]$, then problem \eqref{eqP} admits a nodal solution $y_{\lambda}\in[\bar{v}_{\lambda},\underline{u}_{\lambda}]\cap C^1(\overline{\Omega})$.
	\end{prop}
	\begin{proof}
		Let $\underline{u}_{\lambda}\in \mbox{int}\,C_+$ and $\bar{v}_{\lambda}\in-\mbox{int}\,C_+$ be the two extremal constant sign solutions. We introduce the following Carath\'eodory functions
		\begin{eqnarray}
			j(z,x)=\left\{\begin{array}{ll}
				 f(z,\bar{v}_{\lambda}(z))+|\bar{v}_{\lambda}(z)|^{p-2}\bar{v}_{\lambda}(z)&\mbox{if}\ x<\bar{v}_{\lambda}(z)\\
				f(z,x)+|x|^{p-2}x&\mbox{if}\ \bar{v}_{\lambda}(z)\leq x\leq\underline{u}_{\lambda}(z)\\
				f(z,\underline{u}_{\lambda}(z))+\underline{u}_{\lambda}(z)^{p-1}&\mbox{if}\ \underline{u}_{\lambda}(z)<x
			\end{array}\right.\label{eq77}\\
			\hspace{6cm}\mbox{for all}\ (z,x)\in\Omega\times\RR,\nonumber\\
			\tau_{\lambda}(z,x)=\left\{\begin{array}{ll}
				\lambda\beta(z)|\bar{v}_{\lambda}(z)|^{q-2}\bar{v}_{\lambda}(z)&\mbox{if}\ x<\bar{v}_{\lambda}(z)\\
				\lambda\beta(z)|x|^{q-2}x&\mbox{if}\ \bar{v}_{\lambda}(z)\leq x \leq\underline{u}_{\lambda}(z)\\
				\lambda\beta(z)\underline{u}_{\lambda}(z)^{q-1}&\mbox{if}\ \underline{u}_{\lambda}(z)<x
			\end{array}\right.\label{eq78}\\
			\hspace{6cm}\mbox{for all}\ (z,x)\in\partial\Omega\times\RR.\nonumber
		\end{eqnarray}
		We set $J(z,x)=\int^{x}_{0}j(z,s)ds$ and $T_{\lambda}(z,x)=\int^{x}_{0}\tau_{\lambda}(z,s)ds$ and consider the $C^1$-functional $\psi_{\lambda}:W^{1,p}(\Omega)\rightarrow\RR$ defined by
		 $$\psi_{\lambda}(u)=\frac{1}{p}||Du||^{p}_{p}+\frac{1}{p}||u||^{p}_{p}-\int_{\Omega}J(z,u(z))dz-\int_{\partial\Omega}T_{\lambda}(z,u(z))d\sigma\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$
		
		Also, we consider the positive and negative truncations of $j(z,\cdot),\ \tau_{\lambda}(z,\cdot)$, namely the Carath\'eodory functions
		$$j_{\pm}(z,x)=j(z,\pm x^{\pm})\ \mbox{and}\ \tau^{\pm}_{\lambda}(z,x)=\tau_{\lambda}(z,\pm x^{\pm}).$$
		
		We set $J_{\pm}(z,x)=\int^{x}_{0}j_{\pm}(z,s)ds$ and $T^{\pm}_{\lambda}(z,x)=\int^{x}_{0}\tau^{\pm}_{\lambda}(z,s)$ and introduce the $C^1$-functionals $\psi_{\lambda}^{\pm}:W^{1,p}(\Omega)\rightarrow\RR$ defined by
		\begin{eqnarray*}
			 &&\psi^{\pm}_{\lambda}(u)=\frac{1}{p}||Du||^{p}_{p}+\frac{1}{p}||u||^{p}_{p}-\int_{\Omega}J_{\pm}(z,u(z))dz-\int_{\partial\Omega}T^{\pm}_{\lambda}(z,u(z))d\sigma\\
			&&\hspace{6cm}\mbox{for all}\ u\in W^{1,p}(\Omega).
		\end{eqnarray*}
		
		As in the proof of Proposition \ref{prop8}, we can see that
		$$K_{\psi_{\lambda}}\subseteq[\bar{v}_{\lambda},\underline{u}_{\lambda}],\ K_{\psi^{+}_{\lambda}}\subseteq[0,\underline{u}_{\lambda}],\ K_{\psi^{-}_{\lambda}}\subseteq[\bar{v}_{\lambda},0]\ (\mbox{see (\ref{eq77}) and (\ref{eq78})}).$$
		
		The extremality of $\bar{v}_{\lambda}$ and $\underline{u}_{\lambda}$, implies that
		\begin{equation}\label{eq79}
			K_{\psi_{\lambda}}\subseteq[\bar{v}_{\lambda},\underline{u}_{\lambda}],\ K_{\psi^{+}_{\lambda}}=\{0,\underline{u}_{\lambda}\},\ K_{\psi_{\lambda}}=\{0,\bar{v}_{\lambda}\}.
		\end{equation}
		
		\begin{claim}
			$\bar{v}_{\lambda}\in-\mbox{int}\,C_+$ and $\underline{u}_{\lambda}\in \mbox{int}\,C_+$ are both local minimizers of $\psi_{\lambda}$.
		\end{claim}
		
		Note that (\ref{eq77}) and (\ref{eq78}) imply that $\psi^{+}_{\lambda}$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find $\tilde{\underline{u}}_{\lambda}\in W^{1,p}(\Omega)$ such that
		$$\psi^{+}_{\lambda}(\tilde{\underline{u}}_{\lambda})=\inf\left[\psi^{+}_{\lambda}(u):u\in W^{1,p}(\Omega)\right].$$
		
		As in the proof of Proposition \ref{prop8} and since $q<p$, we show that
		\begin{eqnarray*}
			&&\psi^{+}_{\lambda}(\tilde{\underline{u}}_{\lambda})<0=\psi^{+}_{\lambda}(0),\\
			&\Rightarrow&\tilde{\underline{u}}_{\lambda}\neq 0.
		\end{eqnarray*}
		
		Since $\tilde{\underline{u}}_{\lambda}\in K_{\psi^{+}_{\lambda}}$, from (\ref{eq79}) we have $\tilde{\underline{u}}_{\lambda}=\underline{u}_{\lambda}\in \mbox{int}\,C_+$. But
		\begin{eqnarray*}
			&&\psi_{\lambda}|_{C_+}=\psi^{+}_{\lambda}|_{C_+},\\
			&\Rightarrow&\underline{u}_{\lambda}\in \mbox{int}\,C_+\ \mbox{is a local}\ C^1(\overline{\Omega})-\mbox{minimizer of}\ \psi_{\lambda},\\
			&\Rightarrow&\underline{u}_{\lambda}\in \mbox{int}\,C_+\ \mbox{is a local}\ W^{1,p}(\Omega)-\mbox{minimizer of}\ \psi_{\lambda}\ (\mbox{see Proposition \ref{prop2}}).
		\end{eqnarray*}
		
		Similarly for $\bar{v}_{\lambda}\in-\mbox{int}\,C_+$ using this time the functional $\psi^{-}_{\lambda}$. This proves the Claim.
		
		Without any loss of generality, we may assume that
		$$\psi_{\lambda}(\bar{v}_{\lambda})\leq\psi_{\lambda}(\underline{u}_{\lambda})$$
		(the reasoning is similar, if the opposite inequality holds). By virtue of the Claim, we can find $\rho\in(0,1)$ small such that
		\begin{equation}\label{eq80}
			 \psi_{\lambda}(\bar{v}_{\lambda})\leq\psi_{\lambda}(\underline{u}_{\lambda})<\inf[\psi_{\lambda}(u):||u-\underline{u}_{\lambda}||=\rho]=m_{\lambda},\ ||\bar{v}_{\lambda}-\underline{u}_{\lambda}||>\rho
		\end{equation}
		(see \cite{1}). Since $\psi_{\lambda}$ is coercive (see (\ref{eq77}) and (\ref{eq78})), we have
		\begin{equation}\label{eq81}
			\varphi_{\lambda}\ \mbox{satisfies the}\ C-\mbox{condition.}
		\end{equation}
		
		Then (\ref{eq80}) and (\ref{eq81}) permit the use of Theorem \ref{th1} (the mountain pass theorem). So, we can find $y_{\lambda}\in W^{1,p}(\Omega)$ such that
		\begin{equation}\label{eq82}
			y_{\lambda}\in K_{\psi_{\lambda}}\ \mbox{and}\ m_{\lambda}\leq \psi_{\lambda}(y_{\lambda}).
		\end{equation}
		
		From (\ref{eq79}), (\ref{eq80}) and (\ref{eq82}) it follows that
		$$y_{\lambda}\notin\{\bar{v}_{\lambda},\underline{u}_{\lambda}\}\ \mbox{and}\ y_{\lambda}\ \mbox{is solution of \eqref{eqP}}.$$
		
		Note that $y_{\lambda}$ is a critical point of mountain pass type for $\psi_{\lambda}$ . If $\hat{\varphi}_{\lambda}(u)=\frac{1}{p}||Du||^{p}_{p}-\int_{\Omega}F(z,u(z))dz-\frac{1}{q}\int_{\partial\Omega}\beta(z)|u(z)|^qd\sigma$ for all $u\in W^{1,p}(\Omega)$, then
		 $$\left.\psi_{\lambda}\right|_{[\bar{v}_{\lambda},\underline{u}_{\lambda}]}=\left.\hat{\varphi}_{\lambda}\right|_{[\bar{v}_{\lambda},\underline{u}_{\lambda}]}\ (\mbox{see (\ref{eq77}), (\ref{eq78})})$$
		
		Since $\bar{v}_{\lambda}\in-\mbox{int}\,C_+$, $\underline{u}_{\lambda}\in \mbox{int}\,C_+$ and $C^1(\overline{\Omega})$ is dense in $W^{1,p}(\Omega)$, we see that $y_{\lambda}$ is a critical point of mountain pass type for $\hat{\varphi}_{\lambda}$ too (see Gasinski and Papageorgiou \cite[p. 645]{10}). On the other hand, since $q<p,\ u=0$ can not be a critical point mountain pass type for $\hat{\varphi}_{\lambda}$. Therefore $y_{\lambda}\neq 0$ and since $y_{\lambda}\in[\bar{v}_{\lambda},\underline{u}_{\lambda}]$ (see (\ref{eq79})), the extremality of $\bar{v}_{\lambda}$ and $\underline{u}_{\lambda}$ implies that $y_{\lambda}$ is nodal. Finally the nonlinear regularity theory of Lieberman \cite{14} implies $y_{\lambda}\in C^1(\overline{\Omega})$.
	\end{proof}

So, we can state two multiplicity theorems for problem \eqref{eqP}. First the nonlinear case $(1<p<\infty)$.
\begin{theorem}\label{th21}
	If hypotheses $H_2(i),(ii),(iii)$ and $\widehat{H}$ hold, then there exists $\lambda_0^*>0$ such that for all $\lambda\in\left(0,\lambda^*_0\right]$ problem \eqref{eqP} has at least there nontrivial solutions $u_0\in \mbox{int}\,C_+,\ v_0\in-\mbox{int}\,C_+$, $y_0\in[v_0,u_0]\cap C^1(\overline{\Omega})$ nodal.
\end{theorem}

In the semilinear case $(p=2)$, we can improve this multiplicity result.
\begin{theorem}\label{th22}
	If $p=2$ and hypotheses $H_1$ and $\widehat{H}$ hold, then there exists $\lambda^*_0>0$ such that for all $\lambda\in(0,\lambda^*_0)$ problem \eqref{eqP} has at least five nontrivial solutions
	\begin{eqnarray*}
		&&u_0,\hat{u}\in \mbox{int}\,C_+,\ u_0\leq\hat{u},\ u_0\neq\hat{u}\\
		&&u_0,\hat{v}\in-\mbox{int}\,C_+,\ \hat{v}\leq v_0,\ v_0\neq\hat{v}\\
		&\mbox{and}&y_0\in int_{C^1(\overline{\Omega})}[v_0,u_0]\ \mbox{nodal.}
	\end{eqnarray*}
\end{theorem}
\begin{remark}
	An interesting open problems is whether we can extend the work of this paper to equations driven by the more general nonlinear, nonhomogeneous differential operators used by Papageorgiou and R\u adulescu \cite{18}. Such an extension will require new methods and techniques.
\end{remark}



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\end{document} 