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\begin{document}


\title{
\begin{flushleft}
{\small{\it Advanced Nonlinear Studies} {\bf 1} (2001), xxx--xxx}
\end{flushleft}
\vspace{0.5in}
{\bf\Large Infinitely Many Nodal Solutions For Nonlinear Nonhomogeneous Robin Problems}
}

\author{{\bf\large Nikolaos S. Papageorgiou$\,^1$}\hspace{2mm}
{\bf\large Vicen\c{t}iu D. R\u{a}dulescu$^{2,3}$}\footnote{The authors would like to thank Professor Zhi-Qiang Wang for his interest in this work
and for
bringing to our attention some relevant related works. V. R\u adulescu acknowledges the support through the grant CNCS-UEFISCDI PCCA-23/2014.}\vspace{1mm}\\
{\it\small $^1\,$Department of Mathematics, National Technical University,}\\
{\it\small Zografou Campus, Athens 15780, Greece}\\
{\it\small e-mail: npapg@math.ntua.gr}\\
{\it\small $^2\,$Department of Mathematics,}\\ {\it\small Faculty of Sciences, King Abdulaziz University,}\\
{\it\small P.O. Box 80203, Jeddah 21589, Saudi Arabia}\\
{\it\small $^3\,$ Institute of Mathematics ``Simion Stoilow" of the Romanian Academy,}\\ {\it\small 014700 Bucharest, Romania}\\
{\it\small e-mail: vicentiu.radulescu@imar.ro}}
\vspace{1mm}
\date{\small{Received August 1, 2015} \\
{\it\small Communicated by Zhi-Qiang Wang}}
\maketitle

\begin{center}
{\bf\small Abstract}
\vspace{3mm}
\hspace{.05in}\parbox{4.5in}
{{\small We consider a nonlinear Robin problem driven by a nonhomogenous differential operator which incorporates as special case the $p$--Laplacian. The reaction $f(z,x)$ is a Carath\'eodory function which need not satisfy a global growth condition and it is assumed only to be odd near zero. Using variational tools, we show that the problem has a whole sequence of distinct nodal (that is, sign changing) solutions.}}
\end{center}
\noindent
{\it \footnotesize 2001 Mathematics Subject Classification}. {\scriptsize 35J20, 35J60}.\\
{\it \footnotesize Key words}. {\scriptsize Nonhomogenous differential operator, nodal solution, nonlinear regularity theory, nonlinear maximum principle}


\section{\bf Introduction}

Let $\Omega\subseteq\RR^N$ be a bounded domain with a $C^2$-boundary $\partial\Omega$. We consider the following nonlinear nonhomogeneous Robin problem:
\begin{eqnarray}\label{eq1}
	\left\{\begin{array}{ll}
		-\mbox{div}\, a(Du(z))=f(z,u(z))&\mbox{in}\ \Omega,\\
		\displaystyle\frac{\partial u}{\partial n_a}+\beta(z)|u|^{p-2}u=0&\mbox{on}\ \partial\Omega.
	\end{array}\right\}
\end{eqnarray}

In this problem, $a:\RR^N\rightarrow\RR^N$ is a strictly monotone, continuous map which satisfies certain other regularity and growth  conditions listed in hypotheses $H(a)$ below. These conditions are general enough to incorporate in our framework many differential operators of interest, such as the $p$--Laplace operator $(1<p<\infty)$ and the sum of a $p$--Laplacian with a $q$--Laplacian $(1<q<p<\infty)$. The reaction term $f(z,x)$ is a Carath\'eodory function (that is, for all $x\in\RR$ the mapping $z\mapsto f(z,x)$ is measurable, while for almost all $z\in\Omega$, $x\mapsto f(z,x)$ is continuous).

The interesting feature of our work here, is that we do not impose any global growth condition on $f(z,\cdot)$. Instead, we assume a local symmetry condition, namely we require that for almost all $z\in\Omega$ the function $x\mapsto f(z,x)$ is odd on the bounded interval $[-\eta,\eta]$. In the boundary condition, $\frac{\partial u}{\partial n_a}$ denotes the generalized normal derivative corresponding to the differential operator $\mbox{div}\, a(Du)$ and is defined by
$$\frac{\partial u}{\partial n_a}=(a(Du),n)_{\RR^N}\ \mbox{for all}\ u\in W^{1,p}(\Omega),$$
with $n(\cdot)$ being the outward unit normal on $\partial\Omega$. This kind of normal derivative is dictated by the nonlinear Green identity (see, for example, Gasinski \& Papageorgiou \cite[p. 210]{7}) and can be found also in the work of Lieberman \cite{12}. The boundary weight function $\beta\in C^{0,\alpha}(\partial\Omega)$ with $\alpha\in(0,1)$ satisfies $\beta(z)\geq 0$ for all $z\in\partial\Omega$. When $\beta\equiv 0$, then we have the Neumann problem.

Under these general hypotheses on the data of problem (\ref{eq1}), we show that there exists a whole sequence $\{u_n\}_{n\geq 1}\subseteq C^1(\overline{\Omega})$ of distinct nodal (that is, sign changing) solutions. Our approach uses variational tools together with suitable truncation-perturbation techniques. Recently nodal solutions for nonlinear, nonhomogenous Robin problems, were produced by Papageorgiou \& R\u adulescu \cite{18}, \cite{20}. However, in the aforementioned works the authors establish the existence of only one nodal solution.

\section{\bf Mathematical Background and Hypotheses}

Let $X$ be a Banach space and $\varphi\in C^1(X,\RR)$. We say that $\varphi$ satisfies the Palais-Smale condition (PS-condition for short), if the following holds:
\begin{center}
``Every sequence $\{u_n\}_{n\geq 1}\subseteq X$ such that $\{\varphi(u_n)\}_{n\geq 1}\subseteq\RR$ is bounded and
$$\varphi'(u_n)\rightarrow 0\ \mbox{in}\ X^*\ \mbox{as}\ n\rightarrow\infty,$$
admits a strongly convergent subsequence''.
\end{center}

Our main variational tool will be a variant due to Heinz \cite{10} of a classical result of Clark \cite{4}.
The next result is essentially due to Heinz \cite{10} and can be found
in Wang \cite{25}. Further extensions with applications to semilinear elliptic Dirichlet problems and to Hamiltonian systems can be found in the works of Liu \& Wang \cite{liuwang1} and Kajikiya \cite{kaji}.

\begin{theorem}\label{th1}
	Let $X$ be a Banach space and assume that $\varphi\in C^1(X,\RR)$ satisfies the PS-condition, it is even bounded from below, $\varphi(0)=0$ and for every $n\in\NN$ there exist an $n$-dimensional subspace $Y_n$ of $X$ and $\rho_n>0$ such that
	$$\sup[\varphi(u):u\in Y_n\cap\partial B_{\rho_n}]<0,$$
	where $\partial B_{\rho_n}=\{u\in X:||u||=\rho_n\}$.

Then there exists a sequence $\{u_n\}_{n\geq 1}$ of critical points of $\varphi$ such that
	$$\varphi(u_n)<0\ \mbox{for all}\ n\in\NN\ \mbox{and}\ \varphi(u_n)\rightarrow 0\ \mbox{as}\ n\rightarrow\infty.$$
\end{theorem}

Let $\vartheta\in C^1(0,+\infty)$ with $\vartheta(t)>0$ for all $t>0$ and assume that there exists $p>1$ such that
\begin{eqnarray}\label{eq2}
	&&0<\hat{c}\leq\frac{\vartheta'(t)t}{\vartheta(t)}\leq c_0\ \mbox{and}\ c_1t^{p-1}\leq\vartheta(t)\leq c_2(1+t^{p-1})\\
	&&\hspace{4cm}\mbox{for all}\ t>0,\ \mbox{some}\ c_1,c_2>0.\nonumber
\end{eqnarray}

Then our hypotheses on the map $a(\cdot)$ involved in the definition of the differential operator, are the following:

\smallskip
$H(a)$: $a(y)=a_0(|y|)y$ for all $y\in\RR^N$ with $a_0(t)>0$ for all $t>0$ and
\begin{itemize}
	\item[(i)] $a_0\in C^1(0,+\infty),\ t\mapsto a_0(t)t$ is strictly increasing on $(0,+\infty)$, $a_0(t)t\rightarrow 0$ as $t\rightarrow 0^+$ and
	$$\lim\limits_{t\rightarrow 0^+}\frac{a'_0(t)t}{a_0(t)}>-1;$$
	\item[(ii)] there exists $c_3>0$ such that $|\nabla a(y)|\leq c_3\frac{\vartheta(|y|)}{|y|}$ for all $y\in\RR^N\backslash\{0\}$;
	\item[(iii)] $(\nabla a(y)\xi,\xi)_{\RR^N}\geq\frac{\vartheta (|y|)}{|y|}|\xi|^2$ for all $y\in\RR^N\backslash\{0\}$, all $\xi\in\RR^N$;
	\item[(iv)] if $G_0(t)=\int^{t}_{0}a_0(s)sds$ for $t>0$, then there exists $q\in(1,p)$ such that
	\begin{eqnarray*}
		&&\limsup\limits_{t\rightarrow 0^+}\frac{qG_0(t)}{t^q}\leq c^*\\
		\mbox{and}&&t\mapsto G_0(t^{1/q})\ \mbox{is convex.}
	\end{eqnarray*}
\end{itemize}

{\bf Remark.}
	Hypotheses $H(a)(i),(ii),(iii)$ come from the nonlinear regularity theory of Lieberman \cite{12} and the nonlinear maximum principle of Pucci \& Serrin \cite{23}. Hypothesis $H(a)(iv)$ serves the needs of our problem, but it is a mild condition which is satisfied in all the main cases of interest, as the examples which follow illustrate.	

\smallskip
>From the above hypotheses it is clear that the primitive $G_0(\cdot)$ is strictly convex and strictly increasing. We set $G(y)=G_0(|y|)$ for all $y\in\RR^N$. Then $G(\cdot)$ is convex, $G(0)=0$ and
$$\nabla G(0)=0,\ \nabla G(y)=G'_0(|y|)\frac{y}{|y|}=a_0(|y|)y=a(y)\ \mbox{for all}\ y\in\RR^N\backslash\{0\}.$$

So, $G(\cdot)$ is the primitive of $a(\cdot)$. The convexity of $G(\cdot)$ and because $G(0)=0$, imply that
\begin{equation}\label{eq3}
	G(y)\leq (a(y),y)_{\RR^N}\ \mbox{for all}\ y\in\RR^N.
\end{equation}

The next lemma summarizes the main properties of the map $a(\cdot)$. It is a straightforward consequence of hypotheses $H(a)(i),(ii),(iii)$ and of (\ref{eq2}).
\begin{lemma}\label{lem2}
	If hypotheses $H(a)(i),(ii),(iii)$ hold, then
	\begin{itemize}
		\item[(a)] $y\mapsto a(y)$ is continuous and strictly monotone, hence maximal monotone too;
		\item[(b)] $|a(y)|\leq c_4(1+|y|^{p-1})$ for all $y\in\RR^N$ and some $c_4>0;$
		\item[(c)] $(a(y),y)_{\RR^N}\geq \frac{c_1}{p-1}|y|^p$ for all $y\in\RR^N$.
	\end{itemize}
\end{lemma}

This lemma and (\ref{eq3}), lead to the following growth estimates for the primitive $G(\cdot)$.
\begin{corollary}\label{cor3}
	If hypotheses $H(a)(i),(ii),(iii)$ hold, then $\frac{c_1}{p(p-1)}|y|^p\leq G(y)\leq c_5(1+|y|^p)$ for all $y\in\RR^N$ and some $c_5>0$.
\end{corollary}

The examples that follow illustrate that our conditions on the map $a(\cdot)$ cover many cases of interest.

{\bf Example.} 
	The following maps satisfy hypotheses $H(a)$:
	\begin{itemize}
		\item[(a)] $a(y)=|y|^{p-2}y$ with $1<p<\infty$.
		
		This map corresponds to the $p$-Laplacian differential operator defined by
		$$\Delta_pu={\rm div}\,(|Du|^{p-2}Du)\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$
		\item[(b)] $a(y)=|y|^{p-2}y+|y|^{q-2}y$ with $1<q<p<\infty$.
		
		This map corresponds to the $(p,q)$--differential operator defined by
		$$\Delta_pu+\Delta_qu\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$
		
		Such operator arise in problems of mathematical physics. We mention the works of Benci, D'Avenia, Fortunato \& Pisani \cite{1} (quantum physics) and Cherfils \& Ilyasov \cite{2} (plasma physics). Recently existence and multiplicity results for such equations with Dirichlet boundary conditions, were proved by Cingolani \& Degiovanni \cite{3}, Gasinski \& Papageorgiou \cite{9}, Mugnai \& Papageorgiou \cite{14}, Papageorgiou \& R\u adulescu \cite{15}, \cite{16}, \cite{19} and Sun, Zhang \& Su \cite{24}.
		\item[(c)] $a(y)=(1+|y|^2)^{\frac{p-2}{2}}y$ with $1<p<\infty$.
		
		This map corresponds to the generalized $p$-mean curvature differential operator defined by
		$${\rm div}\,\left((1+|Du|^2)^{\frac{p-2}{2}}Du\right)\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$
		\item[(d)] $a(y)=|y|^{p-2}y\left(1+\frac{1}{1+|y|^p}\right)$ with $1<p<\infty$.
		
		This map corresponds to the differential operator
		$$\Delta_pu+{\rm div}\,\left(\frac{|Du|^{p-2}Du}{1+|Du|^p}\right)\ \mbox{for all}\ u\in W^{1,p}(\Omega),$$
		which is used in problems of plasticity.
	\end{itemize}


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

\smallskip
Finally our hypotheses on the reaction term $f(z,x)$ are the following:

\smallskip
$H(f)$: $f:\Omega\times\RR\rightarrow\RR$ is a Carath\'eodory function such that for almost all $z\in\Omega$, $f(z,0)=0$, $f(z,\cdot)$ is odd on $[-\eta,\eta]$ for some $\eta>0$ with $f(z,\eta)\leq 0\leq f(z,-\eta)$ and
\begin{itemize}
	\item[(i)] there exists $a_{\eta}\in L^{\infty}(\Omega)_+$ such that $|f(z,x)|\leq a_{\eta}(z)$ for almost all $z\in\Omega$, all $|x|\leq\eta$;
	\item[(ii)] if $q\in(1,p)$ is as in $H(a)(iv)$, then we have
	$$\lim\limits_{x\rightarrow 0}\frac{f(z,x)}{|x|^{q-2}x}=+\infty\ \mbox{uniformly for almost all}\ z\in\Omega.$$
\end{itemize}

{\bf Remark.}
	We stress that the above hypotheses do not impose any global growth condition on $f(z,\cdot)$. Instead we assume that $f(z,\cdot)$ has a kind of oscillatory behavior near zero and it is symmetric in that interval. Hypothesis $H(f)(ii)$ implies the presence of a ``concave'' term near zero. We mention the work of Liu \& Wang \cite{liuwang2}
who produced infinitely many nodal solutions for a semilinear Schr\"odinger equation without assuming the existence of zeros. We should point out that the idea of using cut-off techniques to produce an infinity of solutions converging to zero, goes back to the work of Wang \cite{25} who modified the reaction term in the interval $[-\eta,\eta]$ and applied the result of Clark and Heinz to the modified functional (see Lemma 2.3 of Wang \cite{25}).

\smallskip
Using hypothesis $H(f)(ii)$ we see that given any $\xi>0$, we can find $\delta=\delta(\xi)\in(0,\hat{\eta})$ with $\hat{\eta}=\min\{1,\eta\}$ such that
\begin{eqnarray}\label{eq4}
	f(z,x)x\geq \xi|x|^q\geq \xi|x|^p\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ |x|\leq\delta\ (\mbox{recall that}\ q< p).
\end{eqnarray}

Then given $r\in(p,+\infty)$, we can find $c_6=c_6(r,\delta)>0$ such that
\begin{eqnarray}\label{eq5}
	f(z,x)x\geq\xi|x|^q-c_6|x|^r\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\in[-\eta,\eta]
\end{eqnarray}

>From (\ref{eq4}) we have
\begin{equation}\label{eq6}
	F(z,x)\geq\frac{\xi}{q}|x|^q\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ |x|\leq\delta.
\end{equation}

In our analysis of problem (\ref{eq1}), 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 and if $u\in C_+$ with $u(z)>0$ for all $z\in\overline{\Omega}$, then $u\in \mbox{int}\, C_+$. On $\partial\Omega$ we consider the $(N-1)$-dimensional Hausdorff (surface) measure $\sigma(\cdot)$. Using this measure, we can define the ``boundary'' Lebesgue spaces $L^q(\partial\Omega)$, $1\leq q\leq\infty$. From the theory of Sobolev spaces, we know that there exists a unique continuous linear map $\gamma_0:W^{1,p}(\Omega)\rightarrow L^p(\partial\Omega)$ known as the ``trace map'' such that $\gamma_0(u)=u|_{\partial\Omega}$ for all $u\in W^{1,p}(\Omega)\cap C(\overline{\Omega})$. We have
$${\rm im}\,\gamma_0=W^{\frac{1}{p'},p}(\partial\Omega)\left(\frac{1}{p}+\frac{1}{p'}=1\right)\ \mbox{and}\ {\rm ker}\,\gamma_0=W^{1,p}_{0}(\Omega).$$

Moreover, the trace map $\gamma_0$ is compact into $L^q(\partial\Omega)$ for $q\in\left[1,\frac{(N-1)p}{N-p}\right)$. Hereafter, for the sake of notational simplicity, we will drop the use of the trace map $\gamma_0$. The restrictions of Sobolev functions on $\partial\Omega$, are understood in the sense of traces.

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

For every $x\in\RR$, let $x^{\pm}=\max\{\pm x,0\}$. Then for $u\in W^{1,p}(\Omega)$ se wet $u^{\pm}(\cdot)=u(\cdot)^{\pm}$. We know that
$$u=u^+-u^-,\ |u|=u^++u^-\ \mbox{and}\ u^+,u^-\in W^{1,p}(\Omega).$$

Also, by $|\cdot|_N$ we denote the Lebesque measure on $\RR^N$ and $A:W^{1,p}(\Omega)\rightarrow W^{1,p}(\Omega)^*$ is the nonlinear map defined by
$$\left\langle A(u),h\right\rangle=\int_{\Omega}(a(Du),Dh)_{\RR^N}dz\ \mbox{for all}\ u,h\in W^{1,p}(\Omega).$$

The map $A:W^{1,p}(\Omega)\rightarrow W^{1,p}(\Omega)^*$ is continuous, monotone and of type $(S)_+$, that is,
\begin{center}
``$u_n\stackrel{w}{\rightarrow}u$ in $W^{1,p}(\Omega)$ and $\limsup\limits_{n\rightarrow\infty}\left\langle A(u_n),u_n-u\right\rangle\leq 0\Rightarrow u_n\rightarrow u\ \mbox{in}\ W^{1,p}(\Omega)$.''
\end{center}

Here by $\left\langle \cdot,\cdot\right\rangle$ we denote the duality brackets for the pair $(W^{1,p}(\Omega)^*,W^{1,p}(\Omega))$ (see Gasinski \& Papageorgiou \cite{8}).

Finally for any $\varphi\in C^1(X,\RR)$, by $K_{\varphi}$ we denote the critical set of $\varphi$, that is,
$$K_{\varphi}=\{u\in X:\varphi'(u)=0\}.$$


\section{\bf Nodal Solutions}

Using (\ref{eq5}), we introduce the following truncations of the right-hand side of (\ref{eq5}) and of the boundary term $\beta(z)|x|^{p-2}x$:
\begin{eqnarray}
	&&e(z,x)=\left\{\begin{array}{ll}
		-\xi\eta^{q-1}+c_6\eta^{r-1}&\mbox{if}\ x<-\eta\\
		\xi|x|^{q-2}x-c_6|x|^{r-2}x&\mbox{if}\ -\eta\leq x\leq \eta\\
		\xi\eta^{q-1}-c_6\eta^{r-1}&\mbox{if}\ \eta<x,
	\end{array}\right.\label{eq7}\\
	\mbox{and}&&b(z,x)=\left\{\begin{array}{ll}
		-\beta(z)\eta^{p-1}&\mbox{if}\ x<-\eta\\
		\beta(z)|x|^{p-2}x&\mbox{if}\ -\eta\leq x\leq \eta\\
		\beta(z)\eta^{p-1}&\mbox{if}\ \eta<x
	\end{array}\right.\ \mbox{for all}\ (z,x)\in\partial\Omega\times\RR\label{eq8}
\end{eqnarray}

Both are Carath\'eodory functions. We consider the following auxiliary nonlinear, nonhomogeneous Robin problem:
\begin{eqnarray}\label{eq9}
	\left\{\begin{array}{ll}
		-\mbox{div}\, a(Du(z))=e(z,u(z))&\mbox{in}\ \Omega,\\
	\displaystyle	\frac{\partial u}{\partial n_a}+b(z,u)=0&\mbox{on}\ \partial\Omega.
	\end{array}\right\}
\end{eqnarray}
\begin{proposition}\label{prop4}
	If hypotheses $H(a),H(\beta)$ and $H(f)$ hold, then problem (\ref{eq9}) admits a unique positive solution $\bar{u}\in {\rm int}\, C_+$ and $\bar{v}=-\bar{u}\in-{\rm int}\, C_+$ is the unique negative solution of problem (\ref{eq9}).
\end{proposition}
\proof
	We introduce the Carath\'eodory function $\tau:\Omega\times\RR\rightarrow\RR$ defined by
	\begin{eqnarray}\label{eq10}
		\tau(z,x)=\left\{\begin{array}{ll}
			-\eta^{p-1}&\mbox{if}\ x<-\eta\\
			|x|^{p-2}x&\mbox{if}\ -\eta\leq x\leq\eta\\
			\eta^{p-1}&\mbox{if}\ \eta<x.
		\end{array}\right.
	\end{eqnarray}
	
	Let $T(z,x)=\int^x_0\tau(z,s)ds,\ E(z,x)=\int^x_0e(z,s)ds$ and $B(z,x)=\int^x_0b(z,s)ds$ and consider the $C^1$-functional $\psi_+:W^{1,p}(\Omega)\rightarrow\RR$ defined by
	\begin{eqnarray*}
		 &&\psi_+(u)=\int_{\Omega}G(Du)dz+\frac{1}{p}||u||^p_p+\int_{\partial\Omega}B(z,u^+)d\sigma-\int_{\Omega}E(z,u^+)dz-\int_{\Omega}T(z,u^+)dz\\
		&&\hspace{7cm}\mbox{for all}\ u\in W^{1,p}(\Omega).
	\end{eqnarray*}
	
	From Corollary \ref{cor3} and (\ref{eq7}), (\ref{eq8}), (\ref{eq10}) it is clear that $\psi_+$ is coercive. Also, using the Sobolev embedding theorem and the trace theorem, we see that $\psi_+$ is sequentially weakly lower semicontinuous. So, by the Weierstrass theorem, we can find $\bar{u}\in W^{1,p}(\Omega)$ such that
	\begin{equation}\label{eq11}
		\psi_+(\bar{u})=\inf[\psi_+(u):u\in W^{1,p}(\Omega)].
	\end{equation}
	
	Hypothesis $H(a)(iv)$ implies that we can find $c^*_1>c^*$ and $\delta_1\in(0,\hat{\eta})$ such that
	\begin{equation}\label{eq12}
		G_0(t)\leq\frac{c^*_1}{q}t^q\ \mbox{for all}\ t\in[0,\delta_1].
	\end{equation}
	
	Let $u\in \mbox{int}\, C_+$ and choose $t\in(0,1)$ small such that
	\begin{equation}\label{eq13}
		tu(z)\leq\delta_1\ \mbox{and}\ t|Du(z)|\leq\delta_1\ \mbox{for all}\ z\in\overline{\Omega}.
	\end{equation}
	
	Using (\ref{eq12}), (\ref{eq13}), (\ref{eq7}), (\ref{eq8}), (\ref{eq10}), we have
	\begin{eqnarray*}
		\psi_+(tu)&\leq&\frac{t^p c^*_1}{p}||Du||^p_p+\frac{t^p}{p}\int_{\partial\Omega}\beta(z)u^pd\sigma-\frac{t^q\xi}{q}||u||^q_q+\frac{t^rc_6}{r}||u||^r_r\\
		 &\leq&\left(\frac{t^{p-q}c^*_1}{p}||Du||^p_p+\frac{t^{p-q}}{p}c_8||u||^p-\frac{\xi}{q}||u||^q_q+\frac{t^{r-q}}{r}c_8||u||^r_r\right)t^q\\
		&&\mbox{for some}\ c_8>0\ (\mbox{see hypothesis $H(\beta)$ and the trace theorem}.)
	\end{eqnarray*}
	
	Since $1<q<p<r$, choosing $t\in(0,1)$ even smaller if necessary, we have
	\begin{eqnarray*}
		&&\psi_+(tu)<0\\
		&\Rightarrow&\psi_+(\bar{u})<0=\psi_+(0)\ (\mbox{see (\ref{eq11})}),\ \mbox{hence}\ \bar{u}\neq 0.
	\end{eqnarray*}
	
	From (\ref{eq11}) we have
	\begin{eqnarray}\label{eq14}
		&&\psi'_+(\bar{u})=0,\nonumber\\
		&\Rightarrow&\left\langle A(\bar{u}),h\right\rangle+\int_{\Omega}|\bar{u}|^{p-2}\bar{u}hdz+\int_{\partial\Omega}b(z,u^+)hd\sigma=\int_{\Omega}(e(z,u^+)+\tau(z,u^+))hdz\\
		&&\hspace{8cm}\mbox{for all}\ h\in W^{1,p}(\Omega).\nonumber
	\end{eqnarray}
	
	In (\ref{eq14}) first we choose $h=-\bar{u}^-\in W^{1,p}(\Omega)$. Then
	\begin{eqnarray*}
		&&\frac{c_1}{p-1}||D\bar{u}^-||^p_p+||u^-||^p_p\leq 0\ \mbox{(see Lemma \ref{lem2} and (\ref{eq7}), (\ref{eq8}), (\ref{eq10}))},\\
		&\Rightarrow&\bar{u}\geq 0,\ \bar{u}\neq 0.
	\end{eqnarray*}
	
	Also in (\ref{eq14}) we choose $h=(\bar{u}-\eta)^+\in W^{1,p}(\Omega)$. Then we have
	\begin{eqnarray*}
		&&\left\langle A(\bar{u}),(\bar{u}-\eta)^+\right\rangle+\int_{\Omega}\bar{u}^{p-1}(\bar{u}-\eta)^+dz+\int_{\partial\Omega}\beta(z)\eta^{p-1}(\bar{u}-\eta)^+d\sigma\\
		&=&\int_{\Omega}(\xi\eta^{q-1}-c_6\eta^{r-1}+\eta^{p-1})(\bar{u}-\eta)^+dz\ (\mbox{see (\ref{eq7}), (\ref{eq8}), (\ref{eq10})})\\
		&\leq&\int_{\Omega}(f(z,\eta)+\eta^{p-1})(\bar{u}-\eta)^+dz\ (\mbox{see (\ref{eq5})})\\
		&\leq&\left\langle A(\eta),(\bar{u}-\eta)^+\right\rangle+\int_{\Omega}\eta^{p-1}(\bar{u}-\eta)^+dz+\int_{\partial\Omega}\beta(z)\eta^{p-1}(\bar{u}-\eta)^+d\sigma\\
		&&\hspace{3cm}(\mbox{since}\ A(\eta)=0\ \mbox{and}\ f(z,\eta)\leq 0\ \mbox{for almost all}\ z\in\Omega),\\
		\Rightarrow&&\left\langle A(\bar{u})-A(\eta),(\bar{u}-\eta)^+\right\rangle+\int_{\Omega}(\bar{u}^{p-1}-\eta^{p-1})(\bar{u}-\eta)^+dz\leq 0,\\
		\Rightarrow&&|\{\bar{u}>\eta\}|_N=0,\ \mbox{hence}\ \bar{u}\leq\eta.
	\end{eqnarray*}
	
	Therefore, we have proved that
	\begin{equation}\label{eq15}
		\bar{u}\in[0,\eta]=\{u\in W^{1,p}(\Omega):0\leq u(z)\leq\eta\ \mbox{for almost all}\ z\in\Omega\},\ \bar{u}\neq 0.
	\end{equation}
	
	Then using (\ref{eq7}), (\ref{eq8}), (\ref{eq10}) and (\ref{eq15}), we see that equation (\ref{eq14}) becomes
	\begin{eqnarray*}
		&&\left\langle A(\bar{u}),h\right\rangle+\int_{\partial\Omega}\beta(z)\bar{u}^{p-1}hd\sigma=\int_{\Omega}e(z,\bar{u})hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega),\\
		&\Rightarrow&-\mbox{div}\, a(D\bar{u}(z))=e(z,\bar{u}(z))\ \mbox{for almost all}\ z\in\Omega,\\
		&&\frac{\partial\bar{u}}{\partial n_a}+\beta(z)\bar{u}^{p-1}=0\ \mbox{on}\ \partial\Omega\ (\mbox{see Papageorgiou \& R\u adulescu \cite{17}}),\\
		&\Rightarrow&\bar{u}\ \mbox{is a positive solution of problem (\ref{eq9})}.
	\end{eqnarray*}
	
	From Papageorgiou \& R\u adulescu \cite{21}, we have that $\bar{u}\in L^{\infty}(\Omega)$ and then the nonlinear regularity result of Lieberman \cite[p. 320]{12}) implies that $\bar{u}\in C_+\backslash\{0\}$. Because of (\ref{eq15}) we have
	\begin{eqnarray*}
		&&-\mbox{div}\, a (D\bar{u}(z))=\xi\bar{u}(z)^{q-1}-c_6\bar{u}(z)^{r-1}\ \mbox{for almost all}\ z\in\Omega,\\
		&\Rightarrow& \mbox{div}\, a (D\bar{u}(z))\leq c_6\eta^{r-p}\bar{u}(z)^{p-1}\ \mbox{for almost all}\ z\in\Omega,\\
		&\Rightarrow&\bar{u}\in \mbox{int}\, C_+\ (\mbox{see Pucci \& Serrin \cite[pp. 111, 120]{23}}).
	\end{eqnarray*}
	
	Next we show the uniqueness of this positive solution. To this end, we consider the integral functional $j:L^1(\Omega)\rightarrow\overline{\RR}=\RR\cup\{+\infty\}$ defined by
	$$j(u)=\left\{\begin{array}{ll}
		\int_{\Omega}G(Du^{1/q})dz+\frac{1}{p}\int_{\partial\Omega}\beta(z)u^{p/q}d\sigma&\mbox{if}\ u\geq 0,u^{1/q}\in W^{1,p}(\Omega)\\
		+\infty&\mbox{otherwise}.
	\end{array}\right.$$
	
	Let $u_1,u_2\in {\rm dom}\, j=\{u\in L^1(\Omega):j(u)<+\infty\}$ (the effective domain of $j$). We set
	$$u=((1-t)u_1+tu_2)^{1/q}\ \mbox{with}\ t\in[0,1].$$
	
	Using Lemma 1 of Diaz \& Saa \cite{5}, we have
	$$|Du(z)|\leq[(1-t)|Du_1(z)^{1/q}|^q+t|Du_2(z)^{1/q}|^q]^{1/q}\ \mbox{for almost all}\ z\in\Omega.$$
	
	We have
	\begin{eqnarray*}
		G_0(|Du(z)|)&\leq& G_0\left(\left[(1-t)|Du_1(z)^{1/q}|^q+t|Du_2(z)^{1/q}|^q\right]^{1/q}\right)\\
		&&\hspace{4cm}(\mbox{because}\ G_0(\cdot)\ \mbox{is increasing})\\
		&\leq&(1-t)G_0(|Du_1(z)^{1/q}|)+tG_0(|Du_2(z)^{1/q}|)\ \mbox{for almost all}\ z\in\Omega\\
		&&\hspace{4cm}(\mbox{see hypothesis}\ H(a)(iv)),\\
		\Rightarrow G(Du(z))&\leq&(1-t)G(Du_1(z)^{1/q})+tG(Du_2(z)^{1/q})\ \mbox{for almost all}\ z\in\Omega,\\
		\Rightarrow j(\cdot)\ \mbox{is convex}&&(\mbox{recall}\ q<p\ \mbox{and see hypothesis}\ H(\beta)).
	\end{eqnarray*}
	
	By Fatou's lemma, $j(\cdot)$ is lower semicontinuous.
	
	Let $\bar{y}\in W^{1,p}(\Omega)$ be another positive solution of problem (\ref{eq9}). As we did for $\bar{u}$ in the first part of the proof, we can show that
	$$\bar{y}\in[0,\eta]\cap \mbox{int}\, C_+.$$
	
	For any $h\in C^1(\overline{\Omega})$ and for $|t|<1$ small, we have
	$$\bar{u}^q+th,\ \bar{y}^q+th\in {\rm dom}\, j.$$
	
	Then we see that the functional $j(\cdot)$ is G\^ateaux differentiable at $\bar{u}^q,\ \bar{y}^q$ in the direction $h$. Moreover, via the chain rule and the nonlinear Green identity, we have
	$$j'(\bar{u}^q)(h)=\frac{1}{q}\int_{\Omega}\frac{-\mbox{div}\, a(D\bar{u})}{\bar{u}^{q-1}}hdz\ \mbox{and}\ j'(\bar{y}^q)(h)=\frac{1}{q}\int_{\Omega}\frac{-\mbox{div}\, a(D\bar{y})}{\bar{y}^{q-1}}hdz.$$
	
	Choose $h=\bar{u}^q-\bar{y}^q$. Since $j(\cdot)$ is convex, $j'(\cdot)$ is monotone and so we have
	\begin{eqnarray*}
		0&\leq&\int_{\Omega}\left(\frac{-\mbox{div}\, a(D\bar{u})}{\bar{u}^{q-1}}-\frac{-\mbox{div}\, a(D\bar{y})}{\bar{y}^{q-1}}\right)(\bar{u}^q-\bar{y}^q)dz\\
		&=&\int_{\Omega}c_6(\bar{y}^{r-q}-\bar{u}^{r-q})(\bar{u}^q-\bar{y}^q)dz\ (\mbox{see (\ref{eq7})}),\\
		\Rightarrow&&\bar{u}=\bar{y},\\
		\Rightarrow&&\bar{u}\in[0,\eta]\cap \mbox{int}\,C_+\ \mbox{is the unique positive solution of } (\ref{eq9}).
	\end{eqnarray*}
	
	Evidently since $x\mapsto\xi|x|^{q-2}x-c_6|x|^{r-2}x$ is odd, $\bar{v}=-\bar{u}\in[-\eta,0]\cap(-\mbox{int}\,C_+)$ is the unique negative solution of (\ref{eq9}).
This is the end of the proof.

We introduce the following two sets:
\begin{eqnarray*}
	S_+&=&\{u\in W^{1,p}(\Omega):u\ \mbox{is  a positive solution of (\ref{eq1}),}\ u\in[0,\eta]\},\\
	S_-&=&\{v\in W^{1,p}(\Omega):v\ \mbox{is a negative solution of (\ref{eq1}), }v\in[-\eta,0]\}.
\end{eqnarray*}

As before, the nonlinear maximum principle implies that
$$S_+\subseteq \mbox{int}\,C_+\ \mbox{and}\ S_-\subseteq-\mbox{int}\,C_+.$$

Moreover, as in Filippakis \& Papageorgiou \cite{6}, we have that
$$S_+\ \mbox{is downward directed}$$
(that is, if $u_1,u_2\in S_+$, then we can find $u\in S_+$ such that $u\leq \min\{u_1,u_2\}$) and
$$S_-\ \mbox{is upward directed}$$
(that is, if $v_1,v_2\in S_-$, then we can find $v\in S_-$ such that $v\geq \max\{v_1,v_2\}$) (see also Motreanu, Motreanu \& Papageorgiou \cite[p. 421]{13}).
\begin{proposition}\label{prop5}
	If hypotheses $H(a),\ H(\beta)$ and $H(f)$ hold, then $\bar{u}\leq u$ for all $u\in S_+$ and $v\leq\bar{v}$ for all $v\in S_-$.
\end{proposition}
\proof
	Let $u\in S_+$. We consider the Carath\'eodory functions $k_+(z,x),\ \hat{b}_+(z,x)$ and $\hat{\tau}_+(z,x)$ defined by
	\begin{eqnarray}
		k_+(z,x)&=&\left\{\begin{array}{ll}
			0&\mbox{if}\ x<0\\
			e(z,x)&\mbox{if}\ 0\leq x\leq u(z)\\
			e(z,u(z))&\mbox{if}\ u(z)<x,
		\end{array}\right.\label{eq16}\\
		\hat{b}_+(z,x)&=&\left\{\begin{array}{ll}
			0&\mbox{if}\ x<0\\
			\beta(z)x^{p-1}&\mbox{if}\ 0\leq x\leq u(z)\\
			\beta(z)u(z)^{p-1}&\mbox{if}\ u(z)<x
		\end{array}\right.\ \mbox{for all}\ (z,x)\in\partial\Omega\times\RR\label{eq17}\\
		\hat{\tau}_+(z,x)&=&\left\{\begin{array}{ll}
			0&\mbox{if}\ x<0\\
			x^{p-1}&\mbox{if}\ 0\leq x\leq u(z)\\
			u(z)^{p-1}&\mbox{if}\ u(z)<x.
		\end{array}\right.\label{eq18}
	\end{eqnarray}
	
	We set $$K_+(z,x)=\int^x_0k_+(z,s)ds,\ \hat{B}_+(z,x)=\int^x_0\hat{b}_+(d,s)ds,\ \hat{T}_+(z,x)= \int^x_0\hat{\tau}_+(z,s)ds.$$ Consider the $C^1$-functional $\gamma_+:W^{1,p}(\Omega)\rightarrow\RR$ defined by
	\begin{eqnarray*}
		 \gamma_+(u)&=&\int_{\Omega}G(Du)dz+\frac{1}{p}||u||^p_p+\int_{\partial\Omega}\hat{B}_+(z,u)d\sigma-\int_{\Omega}K_+(z,u)dz-\int_{\Omega}\hat{T}_+(z,u)dz\\
		&&\hspace{8cm}\mbox{for all}\ u\in W^{1,p}(\Omega).
	\end{eqnarray*}
	
	From Corollary \ref{cor3} and (\ref{eq16}), (\ref{eq17}), (\ref{eq18}), we see that $\gamma_+$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find $\bar{u}_0\in W^{1,p}(\Omega)$ such that
	\begin{equation}\label{eq19}
		\gamma_+(\bar{u}_0)=\inf[\gamma_+(u):u\in W^{1,p}(\Omega)].
	\end{equation}
	
	As before (see the proof of Proposition \ref{prop4}), since $1<q<p<r$, for $\tilde{u}\in \mbox{int}\,C_+$ and $t\in(0,1)$ small, we have
	\begin{eqnarray*}
		&&\gamma_+(t\tilde{u})<0=\gamma_+(0)\ (\mbox{see hypothesis}\ H(a)(iv)),\\
		&\Rightarrow&\gamma_+(\bar{u}_0)<0=\gamma_+(0)\ (\mbox{see (\ref{eq19})}),\ \mbox{hence}\ \bar{u}_0\neq 0.
	\end{eqnarray*}
	
	From (\ref{eq19}) we have
	\begin{eqnarray}\label{eq20}
		&&\gamma'_+(\bar{u}_0)=0,\nonumber\\
		&\Rightarrow&\left\langle A(\bar{u}_0),h\right\rangle+\int_{\Omega}|\bar{u}_0|^{p-2}\bar{u}_0hdz+
\int_{\partial\Omega}\hat{b}_+(z,\bar{u}_0)hdz=\nonumber\\
&&\int_{\Omega}(k_+(z,\bar{u}_0)+\hat{\tau}_+(z,\bar{u}_0))hdz
\hspace{1cm}\mbox{for all}\ h\in W^{1,p}(\Omega).
	\end{eqnarray}
	
	In (\ref{eq20}) we choose $h=-\bar{u}^-_0$. Using Lemma \ref{lem2} and (\ref{eq16}), (\ref{eq17}), (\ref{eq18}) we obtain
	\begin{eqnarray*}
		&&\frac{c_1}{p-1}||D\bar{u}^-_0||^p_p+||\bar{u}^-_0||^p_p\leq 0,\\
		&\Rightarrow&\bar{u}_0\geq 0,\ \bar{u}_0\neq 0.
	\end{eqnarray*}
	
	Also, in (\ref{eq20}) we choose $h=(\bar{u}_0-u)^+\in W^{1,p}(\Omega)$. Then
	\begin{eqnarray*}
		&&\left\langle A(\bar{u}_0),(\bar{u}_0-u)^+\right\rangle+\int_{\Omega}\bar{u}^{p-1}_{0}(\bar{u}_0-u)^+dz+\int_{\partial\Omega}\beta(z)u^{p-1}(\bar{u}_0-u)^+d\sigma\\
		&=&\int_{\Omega}(e(z,u)+u^{p-1})(\bar{u}_0-u)^+dz\ (\mbox{see (\ref{eq16}), (\ref{eq17}), (\ref{eq18})})\\
		&=&\int_{\Omega}(\xi u^{q-1}-c_6u^{r-1}+u^{p-1})(\bar{u}_0-u)^+dz\ (\mbox{see (\ref{eq7}) and recall $u\in[0,\eta]$})\\
		&\leq&\int_{\Omega}(f(z,u)+u^{p-1})(\bar{u}_0-u)^+dz\ (\mbox{see (\ref{eq5})})\\
		&=&\left\langle A(u),(\bar{u}_0-u)^+\right\rangle+\int_{\Omega}u^{p-1}(\bar{u}_0-u)^+dz+\int_{\partial\Omega}\beta(z)u^{p-1}(u_0-u)^+d\sigma\\
		&&\hspace{8cm}(\mbox{since}\ u\in S_+),\\
		\Rightarrow&&\left\langle A(\bar{u}_0)-A(u),(\bar{u}_0-u)^+\right\rangle+\int_{\Omega}(\bar{u}_0^{p-1}-u^{p-1})(\bar{u}_0-u)^+dz\leq 0,\\
		\Rightarrow&&|\{\bar{u}_0>u\}|_N=0\ \mbox{and so}\ \bar{u}_0\leq u.
	\end{eqnarray*}
	
	Thus we have
	\begin{equation}\label{eq21}
		\bar{u}_0\in[0,u]=\{y\in W^{1,p}(\Omega):0\leq y(z)\leq u(z)\ \mbox{for almost all}\ z\in\Omega\},\ \bar{u}_0\neq 0.
	\end{equation}
	
	Because of (\ref{eq16}), (\ref{eq17}), (\ref{eq18}) and (\ref{eq21}), equation (\ref{eq20}) becomes
	\begin{eqnarray*}
		&&\left\langle A(\bar{u}_0),h\right\rangle+\int_{\partial\Omega}\beta(z)\bar{u}_0^{p-1}hd\sigma=\int_{\Omega}e(z,\bar{u}_0)hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega),\\		
		&\Rightarrow&\bar{u}_0\ \mbox{is a positive solution of problem (\ref{eq9}) (see Papageorgiou \& R\u adulescu \cite{17})},\\
		&\Rightarrow&\bar{u}_0=\bar{u}\ (\mbox{see Proposition \ref{prop4}}),\\
		&\Rightarrow&\bar{u}\leq u\ \mbox{for all}\ u\in S_+.
	\end{eqnarray*}
	
	In a similar fashion, we show that $v\leq\bar{v}$ for all $v\in S_+$.
This is the end of the proof.

Next we produce extremal constant sign solutions, that is, a smallest positive solution and a biggest negative solution.
\begin{proposition}\label{prop6}
	If hypotheses $H(a),\ H(\beta)$ and $H(f)$ hold, then problem (\ref{eq1}) admits a smallest positive solution
	$$u_*\in[0,\eta]\cap {\rm int}\,C_+$$
	and a biggest negative solution
	$$v_*\in[-\eta,0]\cap(-{\rm int}\,C_+).$$
\end{proposition}
{\it Proof.}
	Evidently we restrict ourselves to the set $S_+$ and $S_-$.
	
	From Lemma 3.10 of Hu \& Papageorgiou \cite[p. 178]{11}, we know that we can find a decreasing sequence $\{u_n\}_{n\geq 1}\subseteq S_+$ such that
	$$\inf S_+=\inf\limits_{n\geq 1}u_n.$$
	
	For all $n\in\NN$, we have
	\begin{equation}\label{eq22}
		\left\langle A(u_n),h\right\rangle+\int_{\partial\Omega}\beta(z)u^{p-1}_nhdz=\int_{\Omega}f(z,u_n)hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega).
	\end{equation}
	
	Clearly $\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)$ is bounded and so we may assume that
	\begin{equation}\label{eq23}
		u_n\stackrel{w}{\rightarrow}u_*\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{and}\ u_n\rightarrow u_*\ \mbox{in}\ L^p(\Omega)\ \mbox{and in}\ L^p(\partial\Omega).
	\end{equation}
	
	In (\ref{eq22}) we choose $h=u_n-u_*\in W^{1,p}(\Omega)$, pass to the limit as $n\rightarrow\infty$ and use (\ref{eq23}). Then
	\begin{eqnarray}\label{eq24}
		&&\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{since }A(\cdot)\ \mbox{is of type}\ (S)_+).
	\end{eqnarray}
	
	So, if in (\ref{eq22}) we pass to the limit as $n\rightarrow\infty$ and use (\ref{eq24}), then
	\begin{equation}\label{eq25}
		\left\langle A(u_*),h\right\rangle+\int_{\partial\Omega}\beta(z)u^{p-1}_*hd\sigma=\int_{\Omega}f(z,u_*)hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega).
	\end{equation}
	
	Also
	\begin{equation}\label{eq26}
		\bar{u}\leq u_*\ (\mbox{see Proposition \ref{prop5}}).
	\end{equation}
	
	From (\ref{eq25}) and (\ref{eq26}) we infer that
	$$u_*\in S_+\subseteq \mbox{int}\,C_+\ \mbox{and}\ u_*=\inf S_+.$$
	
	Similarly we produce
	$$v_*\in S_-\ \mbox{and}\ v_*=\sup S_-.$$
This concludes the proof.

Using these two extremal constant sign solutions, we introduce the following Carath\'eodory functions
\begin{eqnarray}
	\mu(z,x)&=&\left\{\begin{array}{ll}
		f(z,v_*(z))+|v_*(z)|^{p-2}v_*(z)&\mbox{if}\ x<v_*(z)\\
		f(z,x)+|x|^{p-2}x&\mbox{if}\ v_*(z)\leq x\leq u_*(z)\\
		f(z,u_*(z))+u_*(z)^{p-1}&\mbox{if}\ u_*(z)<x
	\end{array}\right.\label{eq27}\\
	\bar{b}(z,x)&=&\left\{\begin{array}{ll}
		\beta(z)|v_*(z)|^{p-2}v_*(z)&\mbox{if}\ x<v_*(z)\\
		\beta(z)|x|^{p-2}x&\mbox{if}\ v_*(z)\leq x\leq u_*(z)\\
		\beta(z)u_*(z)^{p-1}&\mbox{if}\ u_*(z)<x.
	\end{array}\right.\label{eq28}
\end{eqnarray}

We set $M(z,x)=\int^x_0\mu(z,s)ds$ and $\bar{B}(z,x)=\int^x_0\bar{b}(z,s)ds$ and consider the $C^1$-functional $\bar{\varphi}:W^{1,p}(\Omega)\rightarrow\RR$ defined by
\begin{eqnarray*}
	 &&\bar{\varphi}(u)=\int_{\Omega}G(Du)dz+\frac{1}{p}||u||^p_p+\int_{\partial\Omega}\bar{B}(z,u)d\sigma-\int_{\Omega}M(z,u)dz\\
	&&\hspace{6cm}\mbox{for all}\ u\in W^{1,p}(\Omega).
\end{eqnarray*}
\begin{proposition}\label{prop7}
	If hypotheses $H(a),\ H(\beta)$ and $H(f)$ hold, then $\bar{\varphi}$ satisfies the PS-condition, it is bounded below, even, $\bar{\varphi}(0)=0$ and $K_{\bar{\varphi}}\subseteq[v_*,u_*]$.
\end{proposition}
{\it Proof.}
	From (\ref{eq27}) and (\ref{eq28}) it is clear that $\bar{\varphi}$ is coercive. So, it is bounded below and satisfies the PS-condition (see Papageorgiou \& Winkert \cite{22}). Hypotheses $H(f)$ imply that $\bar{\varphi}$ is even (recall $u_*\in S_+,\ v_*\in S_-$) and $\bar{\varphi}(0)=0$.
	
	Finally let $u\in K_{\bar{\varphi}}$. Then
	\begin{eqnarray}\label{eq29}
		&&\bar{\varphi}'(u)=0,\nonumber\\
		&\Rightarrow&\left\langle A(u),h\right\rangle+\int_{\Omega}|u|^{p-2}uhdz+\int_{\partial\Omega}\bar{b}(z,u)hd\sigma=\int_{\Omega}\mu(z,u)hdz\\
		&&\hspace{7cm}\mbox{for all}\ h\in W^{1,p}(\Omega).\nonumber
	\end{eqnarray}
	
	In (\ref{eq29}) first we choose $h=(u-u_*)^+\in W^{1,p}(\Omega)$. Then we have
	\begin{eqnarray*}
		&&\left\langle A(u),(u-u_*)^+\right\rangle+\int_{\Omega}u^{p-1}(u-u_*)^+dz+\int_{\partial\Omega}\beta(z)u^{p-1}_*(u-u_*)^+d\sigma\\
		&=&\int_{\Omega}(f(z,u_*)+u^{p-1}_*)(u-u_*)^+dz\ (\mbox{see (\ref{eq27}), (\ref{eq28})})\\
		&=&\left\langle A(u_*),(u-u_*)^+\right\rangle+\int_{\Omega}u^{p-1}_*(u-u_*)^+dz+\int_{\partial\Omega}\beta(z)u^{p-1}_*(u-u_*)^+d\sigma\\
		&&\hspace{8cm}(\mbox{since}\ u_*\in S_+)\\
		&\Rightarrow&\left\langle A(u)-A(u_*),(u-u_*)^+\right\rangle+\int_{\Omega}(u^{p-1}-u^{p-1}_*)(u-u_*)^+dz=0\\
		&\Rightarrow&|\{u>u_*\}|_N=0\ \mbox{and so}\ u\leq u_*.
	\end{eqnarray*}
	
	Similarly, if in (\ref{eq29}) we choose $h=(v_*-u)^+\in W^{1,p}(\Omega)$, then we obtain
	\begin{eqnarray*}
		&&v_*\leq u,\\
		&\Rightarrow&K_{\bar{\varphi}}\subseteq[v_*,u_*].
	\end{eqnarray*}
This ends the proof.

The extremality of $v_*\in-\mbox{int}\,C_+$ and of $u_*\in \mbox{int}\,C_+$ implies the following property.
\begin{corollary}\label{cor8}
	If hypotheses $H(a),\ H(\beta)$ and $H(f)$ hold, then the elements of $K_{\bar{\varphi}}\backslash\{0,v_*,u_*\}$ are nodal solutions of (\ref{eq1}).
\end{corollary}

Now we are ready to produce a whole sequence of distinct nodal solutions for problem (\ref{eq1}).
\begin{theorem}\label{th9}
	Assume that hypotheses $H(a),\ H(\beta)$ and $H(f)$ hold. Then problem (\ref{eq1}) has a whole sequence $\{u_n\}_{n\geq 1}\subseteq C^1(\overline{\Omega})$ of distinct nodal solutions.
\end{theorem}
{\it Proof.}
	Let $\bar{\eta}=\min\{\min\limits_{\overline{\Omega}}u_*,-\max\limits_{\overline{\Omega}}v_*\}$ (recall that $u_*\in \mbox{int}\,C_+$ and $v_*\in-\mbox{int}\,C_+$). Hypothesis $H(a)(iv)$ implies that we can find $\delta_0\in\left(0,\bar{\eta}\right]$ such that
	\begin{equation}\label{eq30}
		G(y)\leq c_9|y|^q\ \mbox{for all}\ y\in\RR^N\ \mbox{with}\ |y|\leq\delta_0\ \mbox{and some}\ c_9>0.
	\end{equation}
	
	Also, from (\ref{eq6}) we have
	\begin{equation}\label{eq31}
		F(z,x)\geq\frac{\xi}{q}|x|^q\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ |x|\leq\delta\ \mbox{with}\ \xi>0.
	\end{equation}
	
	Let $n\in\NN$ and let $Y_n\subseteq W^{1,p}(\Omega)$ be an $n$-dimensional subspace. Then on $Y_n$ all norms are equivalent. So, we can find $\rho_n>0$ such that
	\begin{equation}\label{eq32}
		u\in Y_n\ \mbox{and}\ ||u||\leq\rho_n\Rightarrow|u(z)|\leq\delta\ \mbox{for almost all}\ z\in\Omega.
	\end{equation}
	
	Using (\ref{eq30}), (\ref{eq31}) and (\ref{eq32}) together with (\ref{eq27}), (\ref{eq28}), we have
	\begin{eqnarray}\label{eq33}
		 \bar{\varphi}(u)&\leq&c_9||Du||^q_q+\frac{1}{p}\int_{\partial\Omega}\beta(z)|u|^{p}d\sigma-\frac{\xi}{q}||u||^q_q\nonumber\\
		&&\hspace{6cm}\mbox{for all}\ u\in Y_n\ \mbox{with}\ ||u||\leq\rho_n\nonumber\\
		&\leq&(c_{10}-\xi c_{11})||u||^q\ \mbox{with}\ c_{10},c_{11}>0\ (\mbox{independent of}\ \xi>0)
	\end{eqnarray}
	(use the trace theorem and recall that on $Y_n$ all norms are equivalent).
	
	Recall that $\xi>0$ is arbitrary (see (\ref{eq4})). So, we choose $\xi>\frac{c_{10}}{c_{11}}$ and have that
	$$\varphi(u)<0\ \mbox{for all}\ u\in Y_n\ \mbox{with}\ ||u||=\rho_n.$$
	
	Because of Proposition \ref{prop7}, we can apply Theorem \ref{th1} and find $\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)\backslash\{0\}$ such that
	$$u_n\in K_{\bar{\varphi}}\backslash\{0\}\ \mbox{for all}\ n\in\NN\ \mbox{and}\ \bar{\varphi}(u_n)\rightarrow 0\ \mbox{as}\ n\rightarrow\infty.$$
	
	Since $K_{\bar{\varphi}}\subseteq C^1(\overline{\Omega})$ (nonlinear regularity theory), we have
	$$u_n\in C^1(\overline{\Omega})\backslash\{0\}\ \mbox{for all}\ n\in\NN.$$
	
	Corollary \ref{cor8} implies that $\{u_n\}_{n\geq 1}\subseteq C^1(\overline{\Omega})$ is a sequence of distinct nodal solutions for problem (\ref{eq1}).
This concludes the proof.




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