% version 2013
\documentclass[11pt]{amsart}
\usepackage{graphicx, color}
\usepackage{amscd}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{mathrsfs}

\textwidth=6in \textheight=9.5in \topmargin=-0.5cm
\oddsidemargin=0.5cm \evensidemargin=0.5cm
%\usepackage[notref,notcite]{showkeys}

\newtheorem{theorem}{Theorem}
\newtheorem{ex}{Example}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{prop}[theorem]{Proposition}
\newtheorem{remark}{Remark}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{claim}{Claim}
\newtheorem{step}{Step}
\newtheorem{case}{Case}
\newtheorem{definition}[theorem]{Definition}

%\newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
\newenvironment{proof-sketch}{\noindent{\bf Sketch of Proof}\hspace*{1em}}{\qed\bigskip}


%\numberwithin{equation}{section}
\newcommand{\RR}{\mathbb R}
\newcommand{\NN}{\mathbb N}
\newcommand{\PP}{\mathbb P}
\newcommand{\ZZ}{\mathbb Z}
\newcommand{\di}{\displaystyle}
\renewcommand{\le}{\leqslant}
\renewcommand{\leq}{\leqslant}
\renewcommand{\ge}{\geqslant}
\renewcommand{\geq}{\geqslant}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\baselineskip=16pt plus 1pt minus 1pt

\begin{document}
%\hfill\today\bigskip

\title[Multiplicity theorems for semilinear Robin problems]{Multiplicity theorems for semilinear Robin problems}


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

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

\keywords{Robin boundary conditions, Green's identity, nodal solutions, maximum principle, local minimizers, critical groups.\\
\phantom{aa} 2010 AMS Subject Classification:
35J20, 35J60, Secondary 58E05}

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


\begin{abstract}
We consider a semilinear Robin problem driven by the Laplacian with a reaction which does not satisfy a global growth condition, only a local one. Using variational methods coupled with truncation and perturbation techniques and Morse theory, we prove two multiplicity theorems producing four and three nontrivial solutions respectively, all with precise sign. Also, we show that our results incorporate as a special case a semilinear parametric problem which has been considered primarily in the context of Dirichlet problems.
\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 semilinear Robin problem:
\begin{eqnarray}\label{eq1}
-\Delta u(z)=f(z,u(z))\ \mbox{in}\ \Omega,\ \frac{\partial u}{\partial n}(z)+\beta(z)u(z)=0\ \mbox{on}\ \partial\Omega.
\end{eqnarray}

In this problem, $n(\cdot)$ is the outward unit normal on $\partial\Omega$ and the reaction $f(z,x)$ is a Carath\'eodory function (that is, for all $x\in\RR, z\mapsto f(z,x)$ is measurable and for a.a. $z\in\Omega$, $x\mapsto f(z,x)$ is continuous). The interesting feature of our analysis of problem (\ref{eq1}), is that on the nonlineariry $x\mapsto f(z,x)$ we do not impose any global growth condition. Instead, we assume that $f(z,\cdot)$ admits $z$-dependent zeros of constant sign. Using variational methods coupled with suitable truncation and perturbation techniques and Morse theory (critical groups), we prove a multiplicity theorem producing four nontrivial solutions, all with sign information (two of constant sign and two nodal (sign changing)). A second multiplicity theorem establishing three nontrivial solutions is also proved.

Recently a similar problem was investigated by Zhang, Li and Xue \cite{20}, who deal with a semilinear Robin problem driven by the differential operator $u\mapsto -\Delta u+\alpha u$ with $\alpha>0$. Therefore their differentiable operator is coercive and this makes easier the use of the direct method. Also their reaction $f$ is $z$-independent (autonomous) and the zeros are constant functions. Their main multiplicity theorem (see Theorem $1.1$) produces four nontrivial solutions, but without providing sign information for all of them. On the other hand, we should mention that in \cite{20} it is assumed that $f\in C^{1}(\RR\setminus\{0\})$ and so $f'(\cdot)$ can have jump discontinuities at $x=0$. We should also mention the recent work of the authors \cite{14} who studied a parametric Robin problem driven by the $p$--Laplacian. They proved that if $\hat{\lambda}_{2}>0$ is the second eigenvalue of the negative Robin $p$--Laplacian and $\lambda>\hat{\lambda}_{2}$ ($\lambda>0$ b
 eing the parameter), then the problem has at least three nontrivial solutions and in the semilinear case ($p=2$) four nontrivial solutions, all with sign information. Our framework here is more general and recovers as a special case the setting of \cite{14} (see Section \ref{sec5}).

In the next section, for easy reference, we present the main mathematical tools which will be used in this paper.

\section{Mathematical Background}\label{sec2}

We start with critical point theory. Let $X$ be a Banach space and $X^{\ast}$ its topological dual. By $\langle \cdot,\cdot \rangle$ we denote the duality brackets for the pair $(X^{\ast},X)$. Let $\varphi\in C^{1}(X)$. We say that $\varphi$ satisfies the Palais-Smale condition (PS-condition for short), if the following holds:
$$\begin{array}{ll}
&\mbox{``Every sequence $\{x_n\}_{n\geq 1}\subseteq X$ such that $\{\varphi(x_n)\}_{n\geq 1}\subseteq \RR$ is bounded and}\\
&\varphi'(x_n)\rightarrow 0\ \mbox{in}\ X^{\ast}\ \mbox{as}\ n\rightarrow\infty,\ \mbox{
admits a strongly convergent subsequence".}\end{array}$$

This compactness-type condition on the functional $\varphi$, compensates for the fact that the ambient space is not in general locally compact. It leads to a deformation theorem, from which we can derive the minimax theory for the critical values of $\varphi$. One result in this direction is the so-called ``mountain pass theorem".

\begin{theorem}\label{th10}
Assume that $X$ is a Banach space, $\varphi\in C^1(X)$ and satisfies the PS-condition, $x_0, x_1\in X$, $\|x_1-x_0\|>r>0$,
\begin{eqnarray*}
&&\max\{\varphi(x_0),\varphi(x_1)\}<\inf\left[\varphi(x): \|x-x_0\|=r\right]=m_r.\\
& &c=\inf_{\gamma\in\Gamma}\max_{0\leq t\leq 1}\varphi(\gamma(t))\ \mbox{where}\ \Gamma=\{\gamma\in C([0,1],X): \gamma(0)=x_0,\ \gamma(1)=x_1\},
\end{eqnarray*}
Then $c\geq m_r$ and $c$ is a critical value of $\varphi$.
\end{theorem}

In the analysis of problem (\ref{eq1}), in addition to the Sobolev space $H^1(\Omega)$, we will also use the Banach space $C^1(\overline{\Omega})$, which is an ordered Banach space with positive cone
$$C_{+}=\{u\in C^1(\overline{\Omega}): u(z)\geq 0\ \mbox{for\ all}\ z\in\overline{\Omega}\}.$$
This cone has a nonempty interior given by
$$\mbox{int}\, C_{+}=\{u\in C_{+}: u(z)>0\ \mbox{for\ all}\ z\in\overline{\Omega}\}.$$
In what follows, by $\|\cdot\|$ we denote the norm of $H^1(\Omega)$. We have
$$\|u\|=\Big[\|u\|_2^2+\|Du\|_2^2\Big]^{1/2}\ \mbox{for\ all}\ u\in H^1(\Omega).$$
Let $\sigma(\cdot)$ denote the $(N-1)$-dimensional Hausdorff measure on $\partial\Omega$. Then we can consider the Lebesgue space $L^2(\partial\Omega)$ and the Sobolev space of fractional order $H^{\frac{1}{2},2}(\partial\Omega)$. From the trace theory, we know that there exists a unique continuous linear map $\gamma_0:H^1(\Omega)\rightarrow L^2(\partial\Omega)$, known as the trace map such that
$$\mbox{range}\ \gamma_0=H^{\frac{1}{2},2}(\partial\Omega)\ \mbox{and}\ \ker\gamma_0=H_0^1(\Omega).$$
Consider a Carath\'eodory function $f_0:\Omega\times\RR\rightarrow \RR$ such that
$$|f_0(z,x)|\leq a(z) (1+|x|^{r-1})\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\in\RR,$$
with $a\in L^{\infty}(\Omega)_{+}$ and $1<r<2^{\ast}=\left\{
\begin{array}{ll}
  \frac{2N}{N-2} &\mbox{if}\ N\geq 3\\
     +\infty &\mbox{if}\ N=1,2
    \end{array}\right.
.$

We set $F_0(z,x)=\int_0^x f_0(z,s)ds$ and consider the $C^1$-functional $\psi_{0}:H^1(\Omega)\rightarrow \RR$ defined by
$$\psi_{0}(u)=\frac{1}{2}\|Du\|_2^2+\frac{1}{2}\int_{\partial\Omega}\beta(z)u(z)^2 d\sigma-\int_{\Omega} F_0(z,u(z))dz\ \mbox{for\ all}\ u\in H^1(\Omega).$$
Hereafter, we assume the following for the boundary weight function $\beta(\cdot)$.

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

\medskip
The next result is a special case of a result in Papageorgiou and R\u adulescu \cite{14}. It is an outgrowth of the regularity theory for Robin problems (see Lieberman \cite{12}).
\begin{prop}\label{prop2}
 Assume that hypotheses $H(\beta)$ hold and $u_0\in H^1(\Omega)$ is a local $C^1(\overline{\Omega})$-minimizer of $\psi_{0}$, that is, there exists $\varrho_{0}>0$ such that
$$\psi_{0}(u_0)\leq \psi_{0}(u_0+h)\ \mbox{for\ all}\ h\in C^1(\overline{\Omega})\ \mbox{with}\ \|h\|_{C^1(\overline{\Omega})}\leq\varrho_{0}.$$
Then $u_0\in C^{1,\gamma}(\overline{\Omega})$ for some $\gamma\in (0,1)$ and $u_0$ is a local $H^1(\Omega)$-minimizer of $\psi_0$, that is, there exists $\varrho_{1}>0$ such that
$$\psi_{0}(u_0)\leq \psi_{0}(u_0+h)\ \mbox{for\ all}\ h\in H^1(\Omega)\ \mbox{with}\ \|h\|\leq\varrho_{1}.$$
\end{prop}
\begin{remark}\label{rem}
We should point out that the first such result relating local minimizers, was proved for the space $H_0^1(\Omega)$ by Brezis and Nirenberg \cite{6}.
\end{remark}
Next, we recall some basic facts about the spectrum of the negative Robin Laplacian. So, we consider the following linear eigenvalue problem
$$-\Delta u(z)=\hat{\lambda}u(z)\ \mbox{in}\ \Omega,\ \frac{\partial u}{\partial n}(z)+\beta(z)u(z)=0\ \mbox{on}\ \partial\Omega.$$
>From the spectral theory for compact self-adjoint operators, we know that this eigenvalue problem has a sequence of eigenvalues $\{\hat{\lambda}_k\}_{k\geq 1}$ such that  $\hat{\lambda}_k\rightarrow +\infty$ as $k\rightarrow \infty$ and this sequence exhausts the spectrum of the negative Robin Laplacian.

We know that:

$\hat{\lambda}_1>0$ and it is simple (that is, the corresponding eigenspace is one-dimensional);
\begin{eqnarray}\label{eq2}
\hat{\lambda}_1=\inf \left[\frac{\|Du\|_2^2+\int_{\partial\Omega}\beta(z)u^2 d\sigma}{\|u\|_2^2}: u\in H^1(\Omega), u\neq 0\right].
\end{eqnarray}
The infimum in (\ref{eq2}) is realized on the one-dimensional eigenspace $E(\hat{\lambda}_1)$ corresponding to $\hat{\lambda}_1>0$. It is clear from (\ref{eq2}) that the elements of this eigenspace do not change sign. By $\hat{u}_1$ we denote the $L^2$-normalized (that is, $\|\hat{u}_1\|_2=1$) nonnegative eigenfunction corresponding to $\hat{\lambda}_1>0$. The regularity theory of Lieberman \cite{12} and the maximum principle of Vazquez \cite{18}, imply $\hat{u}_1\in\ \mbox{int}\, C_{+}$. We mention that $\hat{\lambda}_1>0$ is the only eigenvalue with eigenfunctions of constant sign. All the other eigenvalues have nodal eigenfunctions.

For every integer $k\geq 1$, by $E(\hat{\lambda}_k)$ we denote the eigenspace corresponding to the eigenvalue $\hat{\lambda}_k$. From the regularity theory (see Lieberman \cite{12}), we have
$$E(\hat{\lambda}_k)\subseteq C^1(\overline{\Omega})\ \mbox{for\ all}\ k\geq 1.$$
Moreover, these spaces have the so-called unique continuation property (UCP for short), namely, if $u\in E(\hat{\lambda}_k)$ vanishes on a set of positive Lebesgue measure, then $u(z)=0$ for all $z\in\overline{\Omega}$. We set
$$\overline{H}_m=\oplus_{k=1}^m E(\hat{\lambda}_k)\ \mbox{and}\ \hat{H}_m=\overline{H}_m^{\perp}=\overline{\oplus_{k\geq m+1} E(\hat{\lambda}_k)}.$$
Using these spaces we have the following variational characterizations of the higher eigenvalues ($m\geq 2$):
\begin{equation}\label{eq3}\begin{array}{ll}
\di\hat{\lambda}_m&\di=\max\left[\frac{\|D\overline{u}\|_2^2+\int_{\partial\Omega}\beta(z)\overline{u}^2 d\sigma}{\|\overline{u}\|_2^2}: \overline{u}\in \overline{H}_m, \overline{u}\neq 0\right] \\
&\di=\min\left[\frac{\|D\hat{u}\|_2^2+\int_{\partial\Omega}\beta(z)\hat{u}^2 d\sigma}{\|\hat{u}\|_2^2}: \hat{u}\in \hat{H}_{m-1}, \hat{u}\neq 0\right].
\end{array}\end{equation}
Both the maximum and the minimum in (\ref{eq3}) are realized on $E(\hat{\lambda}_m)$.

The next lemma is an easy consequence of the UCP (see also Gasinski and Papageorgiou \cite{10}). In what follows $\xi(u)=\|Du\|_2^2+\int_{\partial\Omega}\beta(z)u^2 d\sigma$ for all $u\in H^1(\Omega).$
\begin{lemma}\label{lem3}
\begin{itemize}
\item [(a)] If $\vartheta\in L^{\infty}(\Omega)$ and $\vartheta(z)\leq \hat{\lambda}_{m+1}$ for a.a. $z\in\Omega$, $\vartheta\neq\hat{\lambda}_{m+1}, m\geq 0$ then there exists $\xi_0>0$ such that
    $$\xi(u)-\int_{\Omega}\vartheta(z)u^2dz\geq \xi_0\|u\|^2\ \mbox{for\ all}\ u\in\hat{H}_m\ (\hat{H}_0=H^1(\Omega)).$$
\item [(b)] If $\vartheta\in L^{\infty}(\Omega)$ and $\vartheta(z)\leq \hat{\lambda}_{m}$ for a.a. $z\in\Omega$, $\vartheta\neq\hat{\lambda}_{m}, m\geq 1$ then there exists $\xi_1>0$ such that
    $$\xi(u)-\int_{\Omega}\vartheta(z)u^2dz\leq -\xi_1\|u\|^2\ \mbox{for\ all}\ u\in\overline{H}_m.$$
\end{itemize}
\end{lemma}
Let $X$ be a Banach space and $\varphi\in C^1(X)$, $c\in\RR $. We introduce the following sets
$$\varphi^{c}=\{x\in X: \varphi(x)\leq c\},\ K_{\varphi}=\{x\in X: \varphi'(x)=0\}\ \mbox{and}\ K_{\varphi}^c=\{x\in K_{\varphi}: \varphi(x)=c\}.$$

Let $(Y_1,Y_2)$ be a topological pair such that  $Y_2\subseteq Y_1\subseteq X$. For every integer $k\geq 0$ by $H_k(Y_1,Y_2)$ we denote the $k$th relative singular homology group with integer coefficients for the pair $(Y_1,Y_2)$. Recall that for $k<0$, $H_k(Y_1,Y_2)=0$. The critical groups of $\varphi$ at an isolated $x\in K_{\varphi}^c$, are defined by
$$C_k(\varphi,x)=H_k\left(\varphi^{c}\cap U, \varphi^{c}\cap U\setminus \{x\}\right)\ \mbox{for\ all}\ k\geq 0,$$
where $U$ is a neighborhood of $x$ such that  $K_{\varphi}\cap\varphi^c\cap U=\{x\}.$  The excision property of singular homology, implies that the above definition of critical groups is independent of the particular choice of the neighborhood $U$.

Suppose that $\varphi\in C^1(X)$ satisfies the PS-condition and $\inf \varphi(K_{\varphi})>-\infty$. Let $c<\inf \varphi(K_{\varphi})$. The critical groups of $\varphi$ at infinity are defined by
$$C_k(\varphi,\infty)=H_k(X,\varphi^{c})\ \mbox{for\ all\ integers}\ k\geq 0.$$
The second deformation theorem (see, for example, Gasinski and Papageorgiou \cite[p. 628]{10}), implies that the above definition of critical groups at infinity is independent of the particular choice of the level $c<\inf \varphi(K_{\varphi})$.

Suppose that $K_{\varphi}$ is finite. We define
\begin{eqnarray*}
&&M(t,x)=\sum_{k\geq 0}\mbox{rank}\ C_k(\varphi,x)t^k\ \mbox{for\ all}\ t\in\RR,\ \mbox{all}\ x\in K_{\varphi},\\
&&P(t,\infty)=\sum_{k\geq 0}\mbox{rank}\ C_k(\varphi,\infty)t^k\ \mbox{for\ all}\ t\in\RR.
\end{eqnarray*}
Then the Morse relation says
\begin{eqnarray}\label{eq4}
\sum\limits_{x\in K_{\varphi}} M(t,x)=P(t,\infty)+(1+t)Q(t),
\end{eqnarray}
where $Q(t)=\sum\limits_{k\geq 0}\beta_{k}t^{k}$ is a formal series in $t\in\RR$ with nonnegative integer coefficients.

Suppose that $X=H$=Hilbert space, $x\in H$, $U$ is a neighborhood of $x$ and $\varphi\in C^2(U)$. If $x\in K_{\varphi}$, then the Morse index of $\varphi$ at $x$, denoted by $\mu=\mu(x)$, is defined as the supremum of the dimensions of the vector subspaces of $H$ for which $\varphi''(x)$ is negative definite. The nullity of $\varphi$ at $x$, denoted by $\nu=\nu(x)$, is defined to be the dimension of the subspace $\ker\varphi''(x)$.

Suppose that $H=V\oplus Y$ with $\dim V<+\infty$. We say that $\varphi\in C^1(X)$ has a ``local linking at $0$" with respect to the decomposition $(V,Y)$, if we can find $\varrho>0$ such that
$$\varphi(v)\leq 0\ \mbox{for\ all}\ v\in V\ \mbox{with}\ \|v\|\leq\varrho,$$
$$\varphi(y)\geq 0\ \mbox{for\ all}\ y\in Y\ \mbox{with}\ \|y\|\leq\varrho.$$
>From \cite{17} (Proposition $2.3$), we have the following result.
\begin{prop}\label{prop4}
If $H$ is a Hilbert space, $U$ is a neighborhood of $u=0$, $\varphi\in C^2(U)$, $u=0$ is an isolated critical point of $\varphi$, $\varphi''(0)$ is a Fredholm operator and $\varphi$ has a local linking at $0$ with respect to the decomposition $(V,Y)$ with $H=V\oplus Y$, $d=\dim V<+\infty$, then $C_k(\varphi,0)=\delta_{k,\mu}\ZZ$, for all $k\geq 0$ if $d=\mu$ with $\mu$ the Morse index of $\varphi$ at $0$, $C_k(\varphi,0)=\delta_{k,\mu+\nu}\ZZ$, for all $k\geq 0$ if $d=\mu+\nu$ with $\nu$ the nullity of $\varphi$ at $0$.
\end{prop}
In the sequel $A:H^1(\Omega)\rightarrow H^1(\Omega)^{\ast}$ denotes the bounded linear operator defined by
$$\langle A(u),y\rangle=\int_{\Omega} (Du,Dy)_{\RR^N}dz\ \mbox{for\ all}\ u,y\in H^1(\Omega).$$
Moreover, if $x\in\RR$, then we set $x^{\pm}=\max\{0,\pm x\}$. For $u\in H^1(\Omega)$, we define
$$u^{\pm}(\cdot)=u(\cdot)^{\pm}.$$
We know that
$$u^{\pm}\in H^1(\Omega),\ u=u^{+}-u^{-},\ |u|=u^{+}+u^{-}.$$
Also, given a measurable function $h:\Omega\times\RR\rightarrow\RR$ (for example a Carath\'eodory function), we set
$$N_h(u)(\cdot)=h(\cdot,u(\cdot))\ \mbox{for\ all}\ u\in H^1(\Omega).$$
(the Nemytskii operator corresponding to $h(\cdot,\cdot)$). In what follows, by $\|\cdot\|_p$ we denote the norm of the Lebesgue space $L^p(\Omega)$ or of $L^p(\Omega,\RR^N)$. Finally, by $|\cdot|_N$ we denote the Lebesgue measure on $\RR^N$.

\section{Constant Sign Solutions}\label{sec3}

In this section we establish the existence of solutions constant sign for problem. To this end, we introduce the following conditions on the reaction $f(z,x):$

\smallskip
$H_1:\ f:\Omega\times\RR\rightarrow\RR$ is a Carath\'eodory function such that  $f(z,0)=0$ for a.a. $z\in\Omega$ and
\begin{itemize}
\item [(i)] for every $\varrho>0$, there exists $a_{\varrho}\in L^{\infty}(\Omega)_{+}$ such that
$$|f(z,x)|\leq a_{\varrho}(z)\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\in[-\varrho,\varrho];$$
\item [(ii)] there exist functions $w_{\pm}\in H^1(\Omega)\cap L^{\infty}(\Omega)$ such that
\begin{eqnarray*}
&&w_{-}(z)\leq c_{-}<0<c_{+}\leq w_{+}(z)\ \mbox{for\ a.a.}\ z\in\Omega,\\
&&f(z,w_{+}(z))\leq 0\leq f(z,w_{-}(z))\ \mbox{a.e.\ in}\ \Omega,\\
&and& A(w_{-})+\beta(\cdot)u_{-}\leq 0\leq A(w_{+})+\beta(\cdot)u_{+}\ \mbox{in}\ H^1(\Omega)^{\ast};
\end{eqnarray*}
\item [(iii)] there exist an integer $m\geq 2$ and $\delta_{0}>0$ such that
$$\hat{\lambda}_m x^2\leq f(z,x)x\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\in [-\delta_0,\delta_0].$$
\end{itemize}
\begin{remark}
Note that the above hypotheses do not impose any growth restriction on $f(z,\cdot)$ near $\pm\infty$. Hypotheses $H_1 (ii), (iii)$ imply that near zero, $f(z,\cdot)$ exhibits an oscillatory behavior. Evidently, hypothesis $H_1 (ii)$ is satisfied if we can find $\xi_{-}<0<\xi_{+}$ such that
$$f(z,\xi_{+})\leq 0\leq f(z,\xi_{-})\ \mbox{a.e.\ in}\ \Omega.$$
Hypothesis $H_1 (iii)$ permits for resonance to occur asymptotically at zero but it also incorporates reactions which are concave near zero. Clearly, in $H_1 (iii)$ we can always assume that $\delta_0\in (0,\min\{c_{+},-c_{-},1\})$. As we will see in the proof of the next proposition, hypotheses $H_1\, (i),\, (iii)$ imply that given $\varrho>0,\ r>2$, we can find $\hat{\xi}>0$ such that
$$f(z,x)x+\hat{\xi} x^{r}\geq 0\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\in [-\varrho,\varrho].$$
For the boundary weight function $\beta(\cdot)$, we keep the hypothesis $H(\beta)$ introduced in Section \ref{sec2}.
\end{remark}
\begin{prop}\label{prop5}
Assume that hypotheses $H(\beta)$ and $H_1$ hold. Then problem (\ref{eq1}) admits at least two nontrivial constant sign solutions
$$u_0\in \mbox{int}\, C_{+}\ \mbox{and}\ v_0\in -\mbox{int}\, C_{+}.$$
\end{prop}
\begin{proof}
First we produce a positive solution.

To this end, we introduce the following truncation-perturbation of $f(z,\cdot)$:
\begin{eqnarray}\label{eq5}
g_{+}(z,x)=\left\{  \begin{array}{ll}
   0 &\mbox{if}\ x<0 \\
     f(z,x)+x &\mbox{if}\ 0\leq x\leq w_{+}(z)\\
     f(z,w_{+}(z))+w_{+}(z) &\mbox{if}\ w_{+}(z)<x.
    \end{array}\right.
\end{eqnarray}
This is a Carath\'eodory function. We set $G_{+}(z,x)=\int\limits_0^x g_{+}(z,s)ds$ and consider the $C^1$-functional $\varphi_{+}:H^1(\Omega)\rightarrow \RR$ defined by
$$\varphi_{+}(u)=\frac{1}{2}\xi(u)+\frac{1}{2}\|u\|_2^2-\int_{\Omega} G_{+}(z,u(z))dz\ \mbox{for\ all}\ u\in H^1(\Omega),$$
where
$$\xi(u)=\|Du\|_2^2+\int_{\partial\Omega}\beta(z)u(z)^2d\sigma\ \mbox{for\ all}\ u\in H^1(\Omega).$$
As before, for the sake of notational simplicity, we drop the use of the trace map $\gamma_0$ to denote the restriction of a Sobolev function on $\partial\Omega$.

>From (\ref{eq5}) it is clear that $\varphi_{+}$ is coercive. Also, using the Sobolev embedding theorem, we show easily that $\varphi_{+}$ is sequentially weakly lower semicontinuous. So, by the Weierstrass theorem, we can find $u_0\in H^1(\Omega)$ such that
\begin{eqnarray}\label{eq6}
\varphi_{+}(u_0)=\inf\left[\varphi_{+}(u): u\in H^1(\Omega)\right].
\end{eqnarray}
Let $t\in(0,1)$ be small such that  $t\hat{u}_1(z)\in (0,\delta_0]$ for all $z\in\overline{\Omega}$ (recall that $\hat{u}_1\in \ \mbox{int}\, C_{+})$. We have
$$\begin{array}{ll}
\varphi_{+}(t\hat{u}_1)&\di=\frac{t^2}{2}\xi(\hat{u}_1)-\int_{\Omega} F(z,t\hat{u}_1)dz\ (\mbox{see}\ (\ref{eq5}))\\
&\di\leq\frac{t^2}{2}[\hat{\lambda}_1-\hat{\lambda}_m]\ (\mbox{see}\ H_1\ (iii)\ \mbox{and\ recall}\ \|\hat{u}_1\|_2=1)\\
&\di <0\ (\mbox{since}\ m\geq 2)
\\
&\di\Rightarrow \varphi_{+}(u_0)<0=\varphi_{+}(0)\ (\mbox{see}\ (\ref{eq6})),\ \mbox{hence}\ u_0\neq 0.
\end{array}$$
>From (\ref{eq6}) we have
\begin{eqnarray}\label{eq7}
&&\varphi_{+}'(u_0)=0,\nonumber \\
&\Rightarrow& \langle A(u_0),h\rangle+\int_{\partial\Omega}\beta(z)u_0 h d\sigma+\int_{\Omega}u_0 h dz=\int_{\Omega}g_{+}(z,u_0)h dz\ \mbox{for\ all}\ h\in H^1(\Omega).
\end{eqnarray}
In (\ref{eq7}), first we choose $h=-u_0^{-}\in H^1(\Omega)$. Then
\begin{eqnarray*}
&&\|Du_0^{-}\|_2^2+\|u_0^{-}\|_2^2\leq 0\ (\mbox{see}\ H(\beta)\ \mbox{and}\ (\ref{eq5}))\\
&\Rightarrow& u_0\geq 0,\ u_0\neq 0.
\end{eqnarray*}
Next, in (\ref{eq7}) we choose $(u_0-w_{+})^{+}\in H^1(\Omega)$. Then
\begin{eqnarray*}
&&\langle A(u_0),(u_0-w_{+})^{+}\rangle+\int_{\partial\Omega}\beta(z)u_0 (u_0-w_{+})^{+} d\sigma+\int_{\Omega}u_0 (u_0-w_{+})^{+} dz\\
&=&\int_{\Omega} [f(z,w_{+})+w_{+}](u_0-w_{+})^{+}dz\ (\mbox{see}\ (\ref{eq5}))\\
&\leq&\langle A(w_{+}),(u_0-w_{+})^{+}\rangle+\int_{\partial\Omega}\beta(z)w_{+} (u_0-w_{+})^{+} d\sigma+\int_{\Omega}w_{+} (u_0-w_{+})^{+} dz\ (\mbox{see}\ H_1(ii)) \\
&\Rightarrow&\langle A(u_0-w_{+}),(u_0-w_{+})^{+}\rangle+\int_{\Omega}[u_0-w_{+}] (u_0-w_{+})^{+} dz\leq 0\\
&\Rightarrow& \|(u_0-w_{+})^{+}\|^2\leq 0,\ \mbox{hence}\ u_0\leq w_{+}.
\end{eqnarray*}
So, we have proved that
\begin{eqnarray}\label{eq8}
u_{0}\in [0,w_{+}]=\{u\in H^1(\Omega): 0\leq u(z)\leq w_{+}(z)\ \mbox{for\ a.a.}\ z\in\Omega\},\ u_0\neq 0.
\end{eqnarray}
Because of (\ref{eq5}) and (\ref{eq8}), (\ref{eq7}) becomes
\begin{eqnarray}\label{eq9}
\langle A(u_0),h\rangle+\int_{\partial\Omega}\beta(z)u_0 h d\sigma=\int_{\Omega}f(z,u_0) h dz\ \mbox{for\ all}\ h\in H^1(\Omega).
\end{eqnarray}
>From Green's identity (see, for example, Gasinski and Papageorgiou \cite[p. 211]{10}), we have
\begin{eqnarray}\label{eq10}
\langle A(u_0),h\rangle=\langle -\Delta u_0,h\rangle+\Big\langle\frac{\partial u_0}{\partial n},h\Big\rangle_{\partial\Omega}\ \mbox{for\ all}\ h\in H^1(\Omega),
\end{eqnarray}
with $\langle\cdot,\cdot\rangle_{\partial\Omega}$ denoting the duality brackets for the pair $\left(H^{-\frac{1}{2},2}(\partial\Omega),H^{\frac{1}{2},2}(\partial\Omega)\right)$. We know that $\Delta u_0=\mbox{div}\, Du_0\in H^{-1}(\Omega)=H_0^1(\Omega)^*$ (see, for example, Gasinski and Papageorgiou \cite[p. 212]{10}). Hence from (\ref{eq9}) and (\ref{eq10}) it follows that
\begin{eqnarray*}
&&\langle -\Delta u_0,h\rangle=\int_{\Omega}f(z,u_0) h dz\ \mbox{for\ all}\ h\in H_0^1(\Omega)\subseteq H^1(\Omega)\\
&\Rightarrow& -\Delta u_0(z)=f(z,u_0(z))\ \mbox{a.e.\ in}\ \Omega.
\end{eqnarray*}
Then from (\ref{eq9}) and (\ref{eq10}) it follows that
$$\Big\langle\frac{\partial u_0}{\partial n}+\beta(z)u,h\Big\rangle_{\partial\Omega}=0\ \mbox{for\ all}\ h\in H^1(\Omega).$$
Recall that the trace map is surjective on $H^{\frac{1}{2},2}(\partial\Omega)$. Hence
$$\frac{\partial u_0}{\partial n}+\beta(z)u_0=0\ \mbox{on}\ \partial\Omega.$$
Therefore $u_0$ is a nontrivial positive solution of (\ref{eq1}) such that  $0\leq u_0\leq w_{+}$. From the regularity result of Lieberman \cite{12}, we have $u_0\in C_{+}\setminus\{0\}$. Hypotheses $H_1(i),\ (iii)$ imply that we can find $c_1>0$,
$$f(z,x)x\geq\hat{\lambda}_m x^2-c_1|x|^r\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ |x|\leq\varrho_0=\max\{\|w_{+}\|_{\infty},\|w_{-}\|_{\infty}\},$$
with $r>2$. Then we have
\begin{eqnarray*}
&&-\Delta_p u_0(z)+c_1 u_0(z)^{r-1}=f(z,u_0(z))+c_1 u_0(z)^{r-1}\geq 0\ \mbox{a.e.\ in}\ \Omega\\
&\Rightarrow& \Delta u_0(z)\leq c_1\|u_0\|^{r-2}u_0(z)\leq c_1\varrho_0^{r-2}u_0(z)\ \mbox{a.e.\ in}\ \Omega\\
&\Rightarrow& u_0\in \mbox{int}\, C_{+}\ (\mbox{see\ Vazquez\ \cite{18}}).
\end{eqnarray*}
To produce a negative solution, we introduce the following Carath\'eodory function
\begin{eqnarray*}
g_{-}(z,x)=\left\{  \begin{array}{ll}
    f(z,w_{-}(z))+w_{-}(z) &\mbox{if}\ x<w_{-}(z) \\
     f(z,x)+x &\mbox{if}\ w_{-}(z)\leq x\leq w_{+}(z)\\
     0 &\mbox{if}\ 0<x.
    \end{array}\right.
\end{eqnarray*}
We set $G_{-}(z,x)=\int\limits_0^x g_{-}(z,s)ds$ and consider the $C^1$-functional $\varphi_{-}:H^1(\Omega)\rightarrow \RR$ defined by
$$\varphi_{-}(u)=\frac{1}{2}\xi(u)+\frac{1}{2}\|u\|_2^2-\int_{\Omega} G_{-}(z,u(z))dz\ \mbox{for\ all}\ u\in H^1(\Omega).$$
Reasoning as above, using this time the functional $\varphi_{-}$, we produce a negative solution
$$v_0\in [w_{-},0]\cap (-\mbox{int}\, C_{+}).$$
\end{proof}
Next we show that in fact we can produce extremal solutions of constant sign namely the smallest positive solution and the biggest negative solution.

To this end, we introduce the following solution sets
$$S_{+}=\{u\in H^1(\Omega):u\neq 0, u\in [0,w_{+}], u\ \mbox{is\ a\ solution\ of}\ (\ref{eq1})\},$$
$$S_{-}=\{v\in H^1(\Omega):v\neq 0, v\in [w_{-},0], v\ \mbox{is\ a\ solution\ of}\ (\ref{eq1})\}.$$
We introduce the following Carath\'eodory function
\begin{eqnarray}\label{eq11}
g(z,x)=\left\{  \begin{array}{ll}
    f(z,w_{-}(z))+w_{-}(z) &\mbox{if}\ x<w_{-}(z) \\
     f(z,x)+x &\mbox{if}\ w_{-}(z)\leq x\leq w_{+}(z)\\
     f(z,w_{+}(z))+w_{+}(z) &\mbox{if}\ w_{+}(z)<x.
    \end{array}\right.
\end{eqnarray}
>From (\ref{eq11}) and hypotheses $H_1(i),\ (iii)$ we see that given $r\in (2,2^{\ast})$ we can find $c_2>0$ big such that
\begin{eqnarray}\label{eq12}
g(z,x)x\geq (\hat{\lambda}_m+1)x^2-c_2|x|^r\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\in\RR.
\end{eqnarray}
This unilateral growth estimate for $g(z,\cdot)$ leads to the following auxiliary Robin problem
\begin{eqnarray}\label{eq13}
\displaystyle -\Delta u(z)=\hat{\lambda}_m u(z)-c_2|u(z)|^{r-2}u(z)\ \mbox{in}\ \Omega,\ \frac{\partial u}{\partial n}(z)+\beta(z)u=0\ \mbox{on}\ \partial\Omega.
\end{eqnarray}
For this auxiliary problem, we have the following existence and uniqueness result.
\begin{prop}\label{prop6}
 Assume that hypotheses $H(\beta)$ hold. Then problem (\ref{eq13}) has a unique positive solution $\tilde{u}\in \mbox{int}\, C_{+}$ and since (\ref{eq13}) is odd $\tilde{v}=-\tilde{u}\in -\mbox{int}\, C_{+}$ is the unique negative solution of (\ref{eq13}).
\end{prop}
\begin{proof}
First we establish the existence of a positive solution for problem (\ref{eq13}).

To this end, let $\tau_{+}:H^1(\Omega)\rightarrow\RR$ be the $C^1$-functional defined by
$$\tau_{+}(u)=\frac{1}{2}\xi(u)+\frac{1}{2}\|u\|_2^2-\frac{\hat{\lambda}_m}{2}\|u^{+}\|_2^2-\frac{1}{2}\|u^{+}\|_2^2+\frac{c_2}{r}\|u^{+}\|_r^r\ \mbox{for\ all}\ u\in H^1(\Omega).$$
Since $r>2$, it is clear that $\tau_{+}$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find $\tilde{u}\in H^1(\Omega)$ such that
\begin{eqnarray}\label{eq14}
\tau_{+}(\tilde{u})=\inf\ [\tau_{+}(u):u\in H^1(\Omega)].
\end{eqnarray}
For $t\in (0,1)$ we have
$$\begin{array}{ll}
\di\tau_{+}(t\hat{u}_1)&\di=\frac{t^2}{2}\xi(\hat{u}_1)+\frac{c_2 t^r}{r}\|\hat{u}_1\|_r^r-\frac{\hat{\lambda}_m t^2}{2}\|\hat{u}_1\|_2^2\\
&\di=\frac{t^2}{2}[\hat{\lambda}_1-\hat{\lambda}_m]+\frac{c_2 t^r}{2}\|\hat{u}_1\|_r^r\ (\mbox{recall\ that}\ \|\hat{u}_1\|_2=1).
\end{array}$$
Since $\hat{\lambda}_m>\hat{\lambda}_1$ (recall $m\geq 2$) and $r>2$, choosing $t\in (0,1)$ small we have
\begin{eqnarray*}
&&\tau_{+}(t\hat{u}_1)<0\\
&\Rightarrow& \tau_{+}(\tilde{u})<0=\tau_{+}(0),\ (\mbox{see\ (\ref{eq14})),\ hence}\ \tilde{u}\neq 0.
\end{eqnarray*}
>From (\ref{eq14}), we have
\begin{eqnarray}\label{eq15}
&&\tau_{+}'(\tilde{u})=0,\nonumber \\
&\Rightarrow& \langle A(\tilde{u}),h\rangle+\int_{\partial\Omega}\beta(z)\tilde{u}h d\sigma+\int_{\Omega}\tilde{u}h dz=\int_{\Omega}(\hat{\lambda}_m+1)\tilde{u}^{+}h dz-c_2\int_{\Omega}(\tilde{u}^{+})^{r-1}h dz\\
&& \mbox{for\ all}\ h\in H^1(\Omega)\nonumber.
\end{eqnarray}
If in (\ref{eq15}) we choose $h=-\tilde{u}^{-}\in H^1(\Omega)$, then we obtain $\tilde{u}\geq 0,\ \tilde{u}\neq 0$. Hence
\begin{eqnarray}\label{eq16}
\langle A(\tilde{u}),h\rangle+\int_{\partial\Omega}\beta(z)\tilde{u}h d\sigma=\hat{\lambda}_m\int_{\Omega}\tilde{u}h dz-c_2\int_{\Omega}\tilde{u}^{r-1}h dz\ \mbox{for\ all}\ h\in H^1(\Omega).
\end{eqnarray}
>From (\ref{eq16}) as in the proof of Proposition \ref{prop5}, via Green's identity, we obtain
$$-\Delta \tilde{u}(z)=\hat{\lambda}_m\tilde{u}(z)-c_2\tilde{u}(z)^{r-1}\ \mbox{a.e.\ in}\ \Omega,\ \frac{\partial\tilde{u}}{\partial n}+\beta(z)\tilde{u}=0\ \mbox{on}\ \partial\Omega.$$
So, $\tilde{u}$ is a positive solution for the auxiliary problem (\ref{eq13}). From Winkert \cite{19} we know that $\tilde{u}\in L^{\infty}(\Omega)$. Then applying the regularity result of Lieberman \cite[p. 320]{12}, we have that $\tilde{u}\in C_{+}\setminus\{0\}$. Also, we have
\begin{eqnarray*}
&&\Delta\tilde{u}(z)\leq c_2\|\tilde{u}\|_{\infty}^{r-2}\tilde{u}(z)\ \mbox{a.e.\ in}\ \Omega\\
&\Rightarrow& \tilde{u}\in \mbox{int}\, C_{+}\ (\mbox{see\ Vazquez\ \cite{18}}).
\end{eqnarray*}
Next we prove the uniqueness of this positive solution. To this end, we introduce the integral functional $\gamma_{+}: L^1(\Omega)\rightarrow\overline{\RR}=\RR\cup\{+\infty\}$ defined by
\begin{eqnarray*}
\gamma_{+}(u)=\left\{  \begin{array}{ll}
  \frac{1}{2}\|Du^{1/2}\|_2^2+\frac{1}{2}\int\limits_{\partial\Omega}\beta(z)u(z) d\sigma &\mbox{if}\ u\geq 0,\ u^{1/2}\in H^1(\Omega) \\
     +\infty &\mbox{otherwise}.
    \end{array}\right.
\end{eqnarray*}
>From Benguria, Brezis and Lieb \cite{5}, we have that $\gamma_{+}$ is convex. Also, via Fatou's lemma, we check that $\gamma_{+}$ is lower semicontinuous.

Let $u$ be a positive solution of the auxiliary problem (\ref{eq13}). From the first of the proof, we have that $u\in \mbox{int}\, C_{+}$. Hence $u^2\in\mbox{dom}\ \gamma_{+}$ and if $h\in C^1(\overline{\Omega})$, then for $t\in (-1,1)$ with $|t|$ small, we have $u^2+th\in\mbox{dom}\ \gamma_{+}$. So, the G\^ateaux derivative of $\gamma_{+}$ at $u$ in the direction $h$ exists. Moreover, via the chain rule, we have
$$\begin{array}{ll}
\gamma_{+}'(u^2)(h)&\di=\Big\langle A(u),\frac{h}{u}\Big\rangle+\int\limits_{\partial\Omega}\beta(z)h d\sigma\\
&\di=\Big\langle -\Delta u,\frac{h}{u}\Big\rangle+\int\limits_{\partial\Omega}\left[\frac{\partial u}{\partial n}+\beta(z)u\right]h d\sigma\ (\mbox{by\ Green's identity})\\
&\di=\int_{\Omega}\Big[\hat{\lambda}_m-c_2u^{r-2}\Big]h dz\ (\mbox{since\ u\ is\ a\ solution\ of\ (\ref{eq13})}).
\end{array}$$
Similarly, if $v\in H^1(\Omega)$ is another positive solution of (\ref{eq13}), then $v\in\mbox{int}\, C_{+}$ and we have
$$\gamma_{+}'(v^2)(h)=\int\limits_{\Omega} \Big[\hat{\lambda}_m-c_1 v^{r-2}\Big]h dz.$$
The convexity of $\gamma_{+}$, implies the monotonicity of $\gamma_{+}'$. Therefore
\begin{eqnarray*}
&&0\leq c_1\int_{\Omega}\Big[v^{r-2}-u^{r-2}\Big](u^2-v^2) dz\leq 0\\
&\Rightarrow& u=v\ \mbox{and\ this\ proves\ the\ uniqueness\ of}\ \tilde{u}\in\mbox{int}\, C_{+}.
\end{eqnarray*}
Since problem (\ref{eq13}) is odd, $\tilde{v}=-\tilde{u}\in -\mbox{int}\, C_{+}$ is the unique negative solution of (\ref{eq13}).
\end{proof}
\begin{prop}\label{prop7}
Assume that hypotheses $H(\beta)$ and $H_1$ hold. Then $\tilde{u}\leq u$ for all $u\in S_{+}$ and $v\leq \tilde{v}$ for all $v\in S_{-}$.
\end{prop}
\begin{proof}
Let $u\in S_{+}$ and consider the following Carath\'eodory function
\begin{eqnarray}\label{eq17}
k_{+}(z,x)=\left\{  \begin{array}{ll}
    0 &\mbox{if}\ x<0 \\
    (\hat{\lambda}_m+1)x-c_2 x^{r-1} &\mbox{if}\ 0\leq u(z)\leq x\\
    (\hat{\lambda}_m+1)u(z)-c_2 u(z)^{r-1}&\mbox{if}\ u(z)<x.
    \end{array}\right.
\end{eqnarray}
We set $K_{+}(z,x)=\int\limits_0^x k_{+}(z,s) ds$ and consider the $C^1$-functional $\mu_{+}: H^1(\Omega)\rightarrow \RR$ defined by
$$\mu_{+}(u)=\frac{1}{2}\xi(u)+\frac{1}{2}\|u\|_2^2-\int\limits_{\Omega} K_{+}(z,u(z)) dz\ \mbox{for\ all}\ u\in H^1(\Omega).$$
It is clear from (\ref{eq17}) that $\mu_{+}$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find $\tilde{u}_{\ast}\in H^1(\Omega)$,
\begin{eqnarray}\label{eq18}
\mu_{+}(\tilde{u}_{\ast})=\inf\ [\mu_{+}(u): u\in H^1(\Omega)].
\end{eqnarray}
As before (see the proof of Proposition \ref{prop5}), using hypothesis $H_1(iii)$, for $t\in (0,1)$ small we have
\begin{eqnarray*}
&&\mu_{+}(t\hat{u}_1)<0\\
&\Rightarrow& \mu_{+}(\tilde{u}_{\ast})<0=\mu_{+}(0)\ (\mbox{see}\ (\ref{eq18})),\ \mbox{hence}\ \tilde{u}_{\ast}\neq 0.
\end{eqnarray*}
>From (\ref{eq18}) we have
\begin{eqnarray}\label{eq19}
&&\mu_{+}'(\tilde{u}_{\ast})=0,\nonumber \\
&\Rightarrow& \langle A(\tilde{u}_{\ast}),h\rangle+\int_{\partial\Omega}\beta(z)\tilde{u}_{\ast}h d\sigma+\int_{\Omega}\tilde{u}_{\ast}h dz=\int_{\Omega}k_{+}(z,\tilde{u}_{\ast})h dz\ \mbox{for\ all}\ h\in H^1(\Omega).
\end{eqnarray}
In (\ref{eq19}), first we choose $h=-\tilde{u}_{\ast}^{-}\in H^1(\Omega)$. Then $\tilde{u}_{\ast}\geq 0,\ \tilde{u}_{\ast}\neq 0$. Also, in (\ref{eq19}) we choose $h=(\tilde{u}_{\ast}-u)^{+}\in H^1(\Omega)$. Then
\begin{eqnarray*}
&&\langle A(\tilde{u}_{\ast}),(\tilde{u}_{\ast}-u)^{+}\rangle+\int_{\partial\Omega}\beta(z)\tilde{u}_{\ast}(\tilde{u}_{\ast}-u)^{+} d\sigma+\int_{\Omega}\tilde{u}_{\ast}(\tilde{u}_{\ast}-u)^{+} dz\\
&=& \int_{\Omega}\Big[(\hat{\lambda}_m+1)u-c_2u^{r-1}\Big] (\tilde{u}_{\ast}-u)^{+} dz\ (\mbox{see}\ (\ref{eq17}))\\
&\leq& \int_{\Omega}g(z,u)(\tilde{u}_{\ast}-u)^{+} dz+\int_{\partial\Omega}\beta(z)\tilde{u}_{\ast}(\tilde{u}_{\ast}-u)^{+} d\sigma\ (\mbox{see}\ (\ref{eq12})\ \mbox{and}\ H(\beta))\\
&=& \int_{\Omega}f(z,u)(\tilde{u}_{\ast}-u)^{+} dz+\int_{\partial\Omega}\beta(z)\tilde{u}_{\ast}(\tilde{u}_{\ast}-u)^{+} d\sigma\\ &+&\int_{\Omega}u (\tilde{u}_{\ast}-u)^{+} dz\ (\mbox{see}\ (\ref{eq11})\ \mbox{and\ recall}\ u\leq w_{+})\\
&=&\langle A(u),(\tilde{u}_{\ast}-u)^{+}\rangle+\int_{\partial\Omega}\beta(z)\tilde{u}_{\ast}(\tilde{u}_{\ast}-u)^{+} d\sigma+\int_{\Omega}u(\tilde{u}_{\ast}-u)^{+} dz\ (\mbox{since}\ u\in S_{+})\\
&\Rightarrow& \langle A(\tilde{u}_{\ast})-A(u),(\tilde{u}_{\ast}-u)^{+}\rangle+\int_{\Omega}(\tilde{u}_{\ast}-u)(\tilde{u}_{\ast}-u)^{+} dz\leq 0\\
&\Rightarrow&\|(\tilde{u}_{\ast}-u)^{+}\|^2\leq 0,\ \mbox{hence}\ \tilde{u}_{\ast}\leq u.
\end{eqnarray*}
So, we have proved that
\begin{eqnarray}\label{eq20}
\tilde{u}_{\ast}\in [0,u]=\{y\in H^1(\Omega): 0\leq y(z)\leq u(z)\ \mbox{a.e.\ in}\ \Omega\},\ \tilde{u}_{\ast}\neq 0.
\end{eqnarray}
>From (\ref{eq17}) and (\ref{eq20}), we see that (\ref{eq19}) becomes
\begin{eqnarray}\label{eq21}
\langle A(\tilde{u}_{\ast}),h\rangle+\int_{\partial\Omega}\beta(z)\tilde{u}_{\ast}h d\sigma=\int_{\Omega}\Big[\hat{\lambda}_m\tilde{u}_{\ast}-c_2\tilde{u}_{\ast}^{r-1}\Big]h dz\ \mbox{for\ all}\ h\in H^1(\Omega).
\end{eqnarray}
>From (\ref{eq21}), as in the proof of Proposition \ref{prop5} via Green's identity, we obtain
\begin{eqnarray*}
&&-\Delta \tilde{u}_{\ast}(z)=\hat{\lambda}_m\tilde{u}_{\ast}(z)-c_2\tilde{u}_{\ast}(z)^{r-1}\ \mbox{a.e.\ in}\ \Omega,\ \frac{\partial\tilde{u}_{\ast}}{\partial n}+\beta(z)\tilde{u}_{\ast}=0\ \mbox{on}\ \partial\Omega\\
&\Rightarrow& \tilde{u}_{\ast}=\tilde{u}\in \mbox{int}\, C_{+}\ (\mbox{see\ Proposition}\ \ref{prop6})\\
&\Rightarrow& \tilde{u}\leq u\ (\mbox{see}\ (\ref{eq20})).
\end{eqnarray*}
Similarly we show that
$$v\leq\tilde{v}\ \mbox{for\ all}\ v\in S_{-}.$$
\end{proof}
Now we are ready to produce extremal nontrivial constant sign solutions for problem (\ref{eq1}).
\begin{prop}\label{prop8}
 Assume that hypotheses $H(\beta)$ and $H_1$ hold. Then problem (\ref{eq1}) admits a smallest positive solution $u_{\ast}\in\mbox{int}\, C_{+}$ and a biggest negative solution $v_{\ast}\in -\mbox{int}\, C_{+}$
\end{prop}
\begin{proof}
>From Dunford and Schwartz \cite[p. 336]{9}, we know that we can find $\{u_n\}_{n\geq 1}\subseteq C$ such that
$$\inf S_{+}=\inf_{n\geq 1}u_n.$$
We have
\begin{eqnarray}\label{eq22}
-\Delta u_n(z)=f(z,u_n(z))\ \mbox{a.e.\ in}\ \Omega, \frac{\partial u_n}{\partial n}+\beta(z)u_n=0\ \mbox{on}\ \partial\Omega,u_n\in\mbox{int}\, C_{+},n\geq 1.
\end{eqnarray}
>From Lieberman \cite[p. 320]{12}, we know that we can find $\eta\in (0,1)$ and $M_1>0$ such that
$$u_n\in C^{1,\eta}(\overline{\Omega})\ \mbox{and}\ \|u_n\|_{C^{1,\eta}(\overline{\Omega})}\leq M_1\ \mbox{for\ all}\ n\geq 1.$$
By virtue of the compact embedding of $C^{1,\eta}(\overline{\Omega})$ into $C^{1}(\overline{\Omega})$ and by passing to a subsequence if necessary, we may assume that
\begin{eqnarray}\label{eq23}
u_n\rightarrow u_{\ast}\ \mbox{in}\ C^1(\overline{\Omega}).
\end{eqnarray}
>From (\ref{eq22}) and Green's identity, we have
\begin{eqnarray}\label{eq24}
&&\langle A(u_n),h\rangle+\int_{\partial\Omega}\beta(z)u_n h d\sigma=\int_{\Omega}f(z,u_n)h dz,\nonumber \\
&\Rightarrow& \langle A(u_{\ast}),h\rangle+\int_{\partial\Omega}\beta(z)u_{\ast} h d\sigma=\int_{\Omega}f(z,u_{\ast})h dz\ (\mbox{see}\ (\ref{eq23})),\nonumber \\
&\Rightarrow& \langle -\Delta u_{\ast},h\rangle+\Big\langle\frac{\partial u_{\ast}}{\partial n}+\beta(z)u_{\ast},h\Big\rangle_{\partial\Omega}=\int_{\Omega}f(z,u_{\ast})h dz\\
&&(\mbox{using\ again\ Green's\ identity}).\nonumber
\end{eqnarray}
>From (\ref{eq24}) as in the proof of Proposition \ref{prop5}, we obtain
\begin{eqnarray}\label{eq25}
-\Delta u_{\ast}(z)=f(z,u_{\ast}(z))\ \mbox{a.e.\ in}\ \Omega,\ \frac{\partial u_{\ast}}{\partial n}+\beta(z)u_{\ast}=0\ \mbox{on}\ \partial\Omega.
\end{eqnarray}
Moreover, from Proposition \ref{eq7}, we have
\begin{eqnarray*}
&&\tilde{u}\leq u_n\ \mbox{for\ all}\ n\geq 1\\
&\Rightarrow& \tilde{u}\leq u_{\ast}\ (\mbox{see}\ (\ref{eq23})),\ \mbox{hence}\ u_{\ast}\neq 0\ \mbox{and\ so}\ u_{\ast}\in S_{+}\ (\mbox{see}\ (\ref{eq25})).
\end{eqnarray*}
Evidently $u_{\ast}=\inf S_{+}$.

Similarly, working with the set $S_{-}$ we produce $v_{\ast}\in -\mbox{int}\, C_{+}$ ($v_{\ast}\leq \tilde{v}$) the biggest negative solution of (\ref{eq1}).

\end{proof}
\begin{remark}
Suppose that instead of $H_1(iii)$ we assume that
\begin{eqnarray}\label{eq26}
\hat{\lambda}_m x^2\leq f(z,x)x\leq \hat{\eta}(z)x^2\ \mbox{for\ a.a}\ z\in\Omega,\ \mbox{all}\ |x|\leq\delta_0,
\end{eqnarray}
with $\hat{\eta}\in L^{\infty}(\Omega)_{+}$. Evidently (\ref{eq26}), in contrast to $H_1(iii)$, excludes from consideration reactions which are concave near zero. Using (\ref{eq26}) the proof of the nontriviality of $u_{\ast}$ is much easier and we do not need to go through Propositions \ref{prop6} and \ref{prop7}. Indeed, as before we argue by contradiction and assume that $u=0$. We set $y_n=\frac{u_n}{\|u_n\|},\ n\geq 1$. Then $\|y_n\|=1,\ y_n\geq 0$ for all $n\geq 1$ and so we may assume that
\begin{eqnarray}\label{eq27}
y_n\overset{w}{\rightarrow}y\ \mbox{in}\ H^1(\Omega)\ \mbox{and}\ y_n\rightarrow y\ \mbox{in}\ L^2(\Omega)\ \mbox{as}\ n\rightarrow\infty,\ \mbox{with}\ y\geq 0.
\end{eqnarray}
\end{remark}
>From the proof of Proposition \ref{eq8}, we have
\begin{eqnarray}\label{eq28}
\langle A(y_n),h\rangle+\int_{\partial\Omega}\beta(z)y_n h d\sigma=\int_{\Omega}\frac{f(z,u_n)}{\|u_n\|}h dz\ \mbox{for\ all}\ n\geq 1.
\end{eqnarray}
By virtue of (\ref{eq26}) and $H_1(i)$, we see that $\left\{\frac{f(\cdot,u_n(\cdot))}{\|u_n\|}\right\}_{n\geq 1}\subseteq L^2(\Omega)$ is bounded. So,we may assume that
\begin{eqnarray}\label{eq29}
\frac{f(\cdot,u_n(\cdot))}{\|u_n\|}\overset{w}{\rightarrow}\eta_0 y\ \mbox{in}\ L^2(\Omega)\ \mbox{with}\ \hat{\lambda}_m\leq\eta_0(z)\leq\hat{\eta}(z)\ \mbox{a.e.\ in}\ \Omega\ (\mbox{see}\ (\ref{eq26})).
\end{eqnarray}
In (\ref{eq28}) we choose $h=y_n-y\in H^1(\Omega)$, pass to the limit as $n\rightarrow \infty$ and use (\ref{eq27}) and (\ref{eq29}). Then we obtain
\begin{eqnarray}\label{eq30}
&&\lim_{n\rightarrow \infty}\langle A(y_n),y_n-y\rangle=0,\nonumber \\
&\Rightarrow& \|Dy_n\|_2\rightarrow\|Dy\|_2,\nonumber \\
&\Rightarrow& y_n\rightarrow y\ \mbox{in}\ H^1(\Omega)\ (\mbox{by\ the\ Kadec-Klee\ property,\ see}\ (\ref{eq27})),\ \mbox{hence}\ \|y\|=1.
\end{eqnarray}
So, if in (\ref{eq28}) we pass to the limit as $n\rightarrow \infty$ and use (\ref{eq29}) and (\ref{eq30}), then
\begin{eqnarray}\label{eq31}
&&\langle A(y),h\rangle+\int_{\partial\Omega}\beta(z)y h d\sigma=\int_{\Omega}\eta_0 y dz\ \mbox{for\ all}\ h\ \mbox{in}\ H^1(\Omega)\\
&&(\mbox{recall\ that\ the\ trace\ map\ is\ continuous}).\nonumber
\end{eqnarray}
>From (\ref{eq31}) it follows that
$$-\Delta y(z)=\eta_0(z)y(z)\ \mbox{a.e.\ in}\ \Omega,\ \frac{\partial y}{\partial n}+\beta(z)y=0\ \mbox{on}\ \partial\Omega,\ y\geq 0,\ y\neq 0,$$
a contradiction to the fact that $\eta_0(z)\geq \hat{\lambda}_m$ a.e. in $\Omega$, with $m\geq 2$ (see Section \ref{sec2}).

\section{Nodal Solutions}\label{sec4}
In this section, we produce nodal (that is, sign changing) solutions for problem (\ref{eq1}). To do this, we need to strengthen our hypotheses on the reaction $f(z,x)$. So, we assume the following:

\smallskip
$H_2:\ f:\Omega\times\RR\rightarrow\RR$ is measurable function such that  for a.a. $z\in\Omega,\ f(z,0)=0,\ f(z,\cdot)\in C^1(\RR)$ and
\begin{itemize}
\item [(i)] for every $\varrho>0$, there exists $a_{\varrho}\in L^{\infty}(\Omega)_{+}$ such that  $|f_x'(z,x)|\leq a_{\varrho}(z)$ for a.a. $z\in\Omega$, all $|x|\leq \varrho$;
\item [(ii)] there exist functions $w_{\pm}\in H^1(\Omega)\cap L^{\infty}(\Omega)$ such that
\begin{eqnarray*}
&&w_{-}(z)\leq c_{-}<0<c_{+}\leq w_{+}(z)\ \mbox{for\ a.a.}\ z\in\Omega,\\
&&f(z,w_{+}(z))\leq 0\leq f(z,w_{-}(z))\ \mbox{for\ a.a.}\ z\in\Omega,\\
&&A(w_{-})+\beta(\cdot)w_{-}\leq 0\leq A(w_{+})+\beta(\cdot)w_{+}\ \mbox{in}\ H^1(\Omega)^{\ast}.
\end{eqnarray*}
\item [(iii)] there exist an integer $m\geq 2$ and $\delta_0>0$ such that
$$\hat{\lambda}_m x^2\leq f(z,x)x\leq \hat{\eta}(z)x^2\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ |x|\leq \delta_0.$$
with $\hat{\eta}\in L^{\infty}(\Omega)_{+},\ \hat{\eta}(z)\leq \hat{\lambda}_{m+1}$ a.e. in $\Omega$, $\hat{\eta}\neq\hat{\lambda}_{m+1}$ and
$$f_x'(z,0)=\lim_{x\rightarrow 0}\frac{f(z,x)}{x}\ \mbox{uniformly\ for\ a.a.}\ z\in\Omega.$$
\end{itemize}
\begin{remark}
It is clear from hypothesis $H_2(iii)$ that
\begin{eqnarray}\label{eq32}
\hat{\lambda}_m\leq f_x'(z,0)\leq \hat{\eta}(z)\ \mbox{for\ a.a.}\ z\in\Omega.
\end{eqnarray}
We also observe, by hypotheses $H_2(i),\ (iii)$ and the differentiability of $f(z,\cdot)$, that if $\varrho= \max\{\|w_{+}\|_{\infty},\|w_{-}\|_{\infty}\}$, then we can find $\xi_0>0$ such that  for a.a. $z\in\Omega$, $x\mapsto f(z,x)+\xi_0 x$ is nondecreasing on $[-\varrho_0,\varrho_0]$.
\end{remark}
Let $g(z,x)$ be the Carath\'eodory function introduced in (\ref{eq11}). Let $G(z,x)=\int\limits_0^x g(z,s)ds$ and consider the functional $\varphi: H^1(\Omega)\rightarrow\RR$ defined by
$$\varphi(u)=\frac{1}{2}\xi(u)+\frac{1}{2}\|u\|_2^2-\int_{\Omega}G(z,u(z))dz\ \mbox{for\ all}\ u\in H^1(\Omega).$$
Recall that
$$\xi(u)=\|Du\|_2^2+\int_{\partial\Omega}\beta(z)u(z)^2 d\sigma\ \mbox{for\ all}\ u\in H^1(\Omega).$$
We have that $\varphi\in C^{2-0}(H^1(\Omega))$.
\begin{prop}\label{prop9}
 Assume that hypotheses $H(\beta)$ and $H_2$ hold. Then $C_k(\varphi,0)=\delta_{k,d_m}\ZZ$ for all $k\geq 0$ with $d_m=\dim \displaystyle\oplus_{i=1}^m E(\hat{\lambda}_i)$.
\end{prop}
\begin{proof}
Let $\psi:H^1(\Omega)\rightarrow \RR$ be the $C^2$-functional defined by
$$\psi(u)=\frac{1}{2}\xi(u)-\frac{1}{2}\int_{\Omega} f_x'(z,0)u(z)^2 dz\ \mbox{for\ all}\ u\in H^1(\Omega).$$
Note that by virtue of hypothesis $H_2(iii)$, given $\varepsilon>0$, we can find $\delta=\delta(\varepsilon)>0$ such that
\begin{eqnarray*}
&&-\varepsilon x\leq f(z,x)-f_x'(z,0)x\leq \varepsilon x\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ |x|\leq \delta\\
&\Rightarrow& -\frac{\varepsilon}{2}x^2\leq F(z,x)-\frac{1}{2}f_x'(z,0)x^2\leq \frac{\varepsilon}{2}x^2\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ |x|\leq \delta.
\end{eqnarray*}
So, if $\overline{B}_{\delta}^c=\{u\in C^1(\overline{\Omega}):\ \|u\|_{ C^1(\overline{\Omega})}\leq \delta\}$, then
$$\|\varphi-\psi\|_{C^1(\overline{B}_{\delta}^c)}\leq\varepsilon.$$
Choosing $\varepsilon>0$ sufficiently small and using the continuity of the critical groups with respect to the $C^1$-topology (see Chang \cite[p. 336]{7}), we have
\begin{eqnarray}\label{eq33}
C_k(\varphi\big|_{C^1(\overline{\Omega})},0)=C_k(\psi\big|_{C^1(\overline{\Omega})},0)\ \mbox{for\ all}\ k\geq 0.
\end{eqnarray}
But from Palais \cite{13} and Bartsch \cite{4}, we know that
\begin{eqnarray}\label{eq34}
C_k(\varphi\big|_{C^1(\overline{\Omega})},0)=C_k(\varphi,0)\ \mbox{and}\ C_k(\psi\big|_{C^1(\overline{\Omega})},0)=C_k(\psi,0)\ \mbox{for\ all}\ k\geq 0.
\end{eqnarray}
>From (\ref{eq33}) and (\ref{eq34}) it follows that
\begin{eqnarray}\label{eq35}
C_k(\varphi,0)=C_k(\psi,0)\ \mbox{for\ all}\ k\geq 0.
\end{eqnarray}
Let $\overline{H}=\displaystyle\oplus_{k=1}^m E(\hat{\lambda}_k)$ and $\hat{H}=\overline{\displaystyle\oplus_{k\geq m+1} E(\hat{\lambda}_k)}=\overline{H}^{\perp}$. We have the following orthogonal direct sum decomposition
$$H^1(\Omega)=\overline{H}\oplus\hat{H}.$$
Then for $\overline{u}\in \overline{H}$ we have
$$\begin{array}{ll}
\psi(u)&\di=\frac{1}{2}\xi(u)-\frac{1}{2}\int_{\Omega} f_x'(z,0)u^2 dz\\
&\di\leq \frac{1}{2}\xi(u)-\frac{\hat{\lambda}_m}{2}\|u\|_2^2\ (\mbox{see}\ (\ref{eq32}))\\
&\di\leq 0\ (\mbox{see}\ (\ref{eq3})).
\end{array}$$
Also, if $\hat{u}\in\hat{H}$, then
$$\begin{array}{ll}
\psi(u)&\di=\frac{1}{2}\xi(u)-\frac{1}{2}\int_{\Omega} f_x'(z,0)u^2 dz\\
&\di\geq\frac{1}{2}\xi(u)-\frac{1}{2}\int_{\Omega} \hat{\eta}(z)u^2 dz\ (\mbox{see}\ (\ref{eq32}))\\
&\di\geq \frac{\xi_0}{2}\|u\|^2\ \mbox{for\ some}\ \xi_0>0\ (\mbox{see\ Lemma}\ \ref{lem3}).
\end{array}$$
So, we can apply Proposition \ref{prop4} and infer that
\begin{eqnarray*}
&&C_k(\psi,0)=\delta_{k,d_m}\ZZ\  \mbox{for\ all}\ k\geq 0\ \mbox{with}\ d_m=\dim \displaystyle\oplus_{k=1}^m E(\hat{\lambda}_k)\\
&\Rightarrow& C_k(\varphi,0)=\delta_{k,d_m}\ZZ\  \mbox{for\ all}\ k\geq 0\ (\mbox{see}\ (\ref{eq35})).
\end{eqnarray*}
\end{proof}
Using this result, we can produce nodal solutions. In what follows, $u_{\ast}\in \mbox{int}\, C_{+}$ and  $v_{\ast}\in -\mbox{int}\, C_{+}$ are the two extremal constant sign solutions produced in Proposition \ref{prop8}.
\begin{prop}\label{prop10}
Assume that hypotheses $H(\beta)$ and $H_2$ hold. Then problem (\ref{eq1}) has at least two nodal solutions
$$y_0,\hat{y}\in \mbox{int}_{C^1(\overline{\Omega})}[v_{\ast},u_{\ast}].$$
\end{prop}
\begin{proof}
Using the extremal constant sign solutions $u_{\ast}\in \mbox{int}\, C_{+}$ and  $v_{\ast}\in -\mbox{int}\, C_{+}$ produced in Proposition \ref{prop8}, we introduce the following Carath\'eodory function
\begin{eqnarray}\label{eq36}
h(z,x)=\left\{  \begin{array}{ll}
    f(z,v_{\ast}(z))+v_{\ast}(z) &\mbox{if}\ x<v_{\ast}(z) \\
    f(z,x)+x &\mbox{if}\ v_{\ast}(z)\leq x\leq u_{\ast}(z)\\
    f(z,u_{\ast}(z))+u_{\ast}(z) &\mbox{if}\ u_{\ast}(z)<x.
    \end{array}\right.
\end{eqnarray}
Let $H(z,x)=\int\limits_0^x h(z,s)ds$ and consider a $C^1$-functional $\tau: H^1(\Omega)\rightarrow\RR$ defined by
$$\tau(u)=\frac{1}{2}\xi(u)+\frac{1}{2}\|u\|_2^2-\int_{\Omega}H(z,u(z))dz\ \mbox{for\ all}\ u\in H^1(\Omega),$$
where
$$\xi(u)=\|Du\|_2^2+\int_{\partial\Omega}\beta(z)u(z)^2 d\sigma\ \mbox{for\ all}\ u\in H^1(\Omega).$$
Also, we consider the positive and negative truncations of $h(z,\cdot)$, namely the  Carath\'eodory function
$$h_{\pm}(z,x)=h(z,\pm x^{\pm}).$$
Let $H_{\pm}(z,x)=\int\limits_0^x h_{\pm}(z,x)ds$ and introduce the $C^1$-functionals $\tau_{\pm}: H^1(\Omega)\rightarrow\RR$ defined by
$$\tau_{\pm}(u)=\frac{1}{2}\xi(u)+\frac{1}{2}\|u\|_2^2-\int_{\Omega}H_{\pm}(z,u(z))dz\ \mbox{for\ all}\ u\in H^1(\Omega).$$
As in the proof of Proposition \ref{prop5}, we can check that
$$K_{\tau}\subseteq[v_{\ast},u_{\ast}],\ K_{\tau_{+}}\subseteq[0,u_{\ast}],\ K_{\tau_{-}}\subseteq[v_{\ast},0].$$
The extremality of $u_{\ast}$ and $v_{\ast}$ implies that
\begin{eqnarray}\label{eq37}
K_{\tau}\subseteq[v_{\ast},u_{\ast}],\ K_{\tau_{+}}=\{0,u_{\ast}\},\ K_{\tau_{-}}=\{0,v_{\ast }\}.
\end{eqnarray}
\begin{claim}
$u_{\ast}\in \mbox{int}\, C_{+}$ and  $v_{\ast}\in -\mbox{int}\, C_{+}$ are local minimizers of $\tau$.
\end{claim}
>From (\ref{eq36}) it is clear that $\tau$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find $\overline{u}_{\ast}\in H^1(\Omega)$ such that
\begin{eqnarray}\label{eq38}
\tau_{+}(\overline{u}_{\ast})=\inf [\tau_{+}(u):\ u\in H^1(\Omega)].
\end{eqnarray}
As in the proof of Proposition \ref{prop5}, using hypothesis $H_2(iii)$ and the fact that $\hat{u}_1\in \mbox{int}\, C_{+}$, for $t\in (0,1)$ small, we have
\begin{eqnarray}\label{eq39}
&&\tau_{+}(t\hat{u}_1)<0,\nonumber\\
&\Rightarrow& \tau_{+}(\overline{u}_{\ast})<0=\tau_{+}(0)\ (\mbox{see}\ (\ref{eq38})),\ \mbox{hence}\ \overline{u}_{\ast}\neq 0.
\end{eqnarray}
>From (\ref{eq38}) we have
\begin{eqnarray*}
&&\overline{u}_{\ast}\in K_{\tau_{+}}=\{0,u_{\ast}\}\ (\mbox{see}\ (\ref{eq37}))\\
&\Rightarrow& \overline{u}_{\ast}=u_{\ast}\ (\mbox{see}\ (\ref{eq39})).
\end{eqnarray*}
Since $\tau\big|_{C_{+}}=\tau_{+}\big|_{C_{+}}$ and $u_{\ast}\in \mbox{int}\, C_{+}$, it follows that $u_{\ast}$ is a local $C^1(\overline{\Omega})$- minimizer of $\tau$, hence by Proposition \ref{prop2} it is also a local $H^1(\Omega)$-minimizer of $\tau$. Similarly, for $v_{\ast}\in -\mbox{int}\, C_{+}$ using this time the functional $\tau_{-}$. This proves the Claim.

Without any loss of generality, we may assume that $\tau(v_{\ast})\leq \tau(u_{\ast})$ (the analysis is similar if the opposite inequality holds). By virtue of the Claim, we can find $\varrho\in (0,1)$ small such that
\begin{eqnarray}\label{eq40}
\tau(v_{\ast})\leq \tau(u_{\ast})<\inf\ [\tau(u):\ \|u-u_{\ast}\|=\varrho]=m_{\varrho},\ \|v_{\ast}-u_{\ast}\|>\varrho.
\end{eqnarray}
(see Aizicovici, Papageorgiou and Staicu \cite{1}, proof of Proposition $29$). The functional $\tau$ is coercive (see (\ref{eq36})) and so it satisfies the PS-condition. This fact and (\ref{eq40}) permit the use of Theorem \ref{th10} (the mountain pass theorem). So, we can find $y_0\in H^1(\Omega)$ such that
\begin{eqnarray}\label{eq41}
y_0\in K_{\tau}\subseteq [v_{\ast},u_{\ast}]\ (\mbox{see}\ (\ref{eq37}))\ \mbox{and}\ m_{\varrho}\leq \tau(y_0).
\end{eqnarray}
>From (\ref{eq40}) and (\ref{eq41}) it follows that
$$y_0\notin \{v_{\ast},u_{\ast}\}\ \mbox{and}\ \tau'(y_0)=0.$$
We have
\begin{eqnarray}\label{eq42}
\langle A(y_0),v\rangle +\int_{\partial\Omega}\beta(z)y_0 v d\sigma+\int_{\Omega}y_0 v dz=\int_{\Omega}h(z,y_0)v dz\ \mbox{for\ all}\ v\in H^1(\Omega).
\end{eqnarray}
>From (\ref{eq36}), (\ref{eq41}), (\ref{eq42}) as before, we infer that
$$y_0\in [v_{\ast},u_{\ast}]\cap C^1(\overline{\Omega})\ \mbox{is\ a\ solution\ of}\ (\ref{eq1}).$$
Since $y_0$ is a critical point of $\tau$ of mountain pass type, we have
\begin{eqnarray}\label{eq43}
C_1(\tau,y_0)\neq 0.
\end{eqnarray}
On the other hand, since $\tau\big|_{[v_{\ast},u_{\ast}]}=\varphi\big|_{[v_{\ast},u_{\ast}]}$ (see (\ref{eq36})) and $v_{\ast}\in -\mbox{int}\, C_{+}$, $u_{\ast}\in \mbox{int}\, C_{+}$, we have
\begin{eqnarray}\label{eq44}
&&C_k(\tau,0)=C_k(\varphi,0)\ \mbox{for\ all}\ k\geq 0\ (\mbox{see\ the\ proof\ of\ Proposition}\ \ref{prop9}),\nonumber\\
&\Rightarrow& C_k(\tau,0)=\delta_{k,d_m}\ZZ\ \mbox{for\ all}\ k\geq 0\ (\mbox{see\ Proposition}\ \ref{prop9}).
\end{eqnarray}
Comparing (\ref{eq43}) and (\ref{eq44}) and since $d_m\geq 2$ (recall $m\geq 2$), we infer $y_0\neq 0$. Because $y_0\in [v_{\ast},u_{\ast}]\setminus \{v_{\ast},u_{\ast}\}$, by virtue of the extremality of $v_{\ast}$ and $u_{\ast}$, we conclude that $y_0\in C^1(\overline{\Omega})$ is nodal.

To produce a second nodal solution, we will use Morse theory. This requires that we improve (\ref{eq43}) and compute the critical groups of $\tau$ exactly. We assume that $K_{\tau}=\{0,u_{\ast},v_{\ast},y_0\}$ (otherwise we already have a second nodal solution of (\ref{eq1}) and so we are done, see (\ref{eq37})). Note that because of the truncations at $u_{\ast}(z)$ and $v_{\ast}(z)$, the functional $\tau$ is not $C^2$, it belongs to $C^{2-0}(H^1(\Omega))$. So, we can not apply directly the results of Morse theory for critical points of mountain pass-type. For this reason, we proceed as follows. We have
$$-\Delta y_0(z)=f(z,y_0(z))\ \mbox{a.e.\ in}\ \Omega,\ \frac{\partial y_0}{\partial n}+\beta(z)y_0=0\ \mbox{on}\ \partial\Omega.$$
Recall that $u_{\ast},v_{\ast}\in [w_{-},w_{+}]$ and let $\xi_0>0$ such that  for a.a. $z\in\Omega$, the function $x\mapsto f(z,x)+\xi_0x$ is nondecreasing on $[-\varrho_0,\varrho_0]$ with $ \varrho_0=\max\{\|w_{+}\|_{\infty},\|w_{-}\|_{\infty}\}$ (see the Remark after hypotheses $H_2$). Then we have
$$\begin{array}{ll}
\di-\Delta y_0(z)+\xi_0 y_0(z)&\di=f(z,y_0(z))+\xi_0 y_0(z)\\
&\di\leq f(z,u_{\ast}(z))+\xi_0 u_{\ast}(z)\ (\mbox{since}\ y_0\leq u_{\ast})\\
&\di=-\Delta u_{\ast}(z)+\xi_0 u_{\ast}(z)\ \mbox{a.e.\ in}\ \Omega\\
&\Rightarrow\di \Delta(u_{\ast}-y_0)(z)\leq \xi_0(u_{\ast}-y_0)(z)\ \mbox{a.e.\ in}\ \Omega\\
&\Rightarrow\di u_{\ast}-y_0\in \mbox{int}\, C_{+}\ (\mbox{see\ Vazquez\ \cite{18}}).
\end{array}$$
Similarly, we show that $y_0-v_{\ast}\in \mbox{int}\, C_{+}$, hence $y_0\in \mbox{int}_{C^1(\overline{\Omega})}[v_{\ast},u_{\ast}]$.

Let $0<\varepsilon<\min\left\{\min\limits_{\overline{\Omega}}(u_{\ast}-y_0),\min\limits_{\overline{\Omega}}(y_0-v_{\ast})\right\}$ and consider the multifunction $L_{\varepsilon}:\Omega\rightarrow 2^{C^1(\RR)}$ defined by
$$L_{\varepsilon}(z)=\{\vartheta\in C^1(\RR):\ \|h(z,\cdot)-\vartheta\|_{C(\RR)}\leq \varepsilon,\ h(z,x)=\vartheta(x)\ \mbox{for}\ |x-u_{\ast}(z)|\geq\varepsilon, \ |x-v_{\ast}(z)|\geq\varepsilon\}$$
Since $h(z,\cdot)$ need not be differentiable at $v_{\ast}(z)$ and $u_{\ast}(z)$, by a smooth modification of $h(z,\cdot)$ near  $v_{\ast}(z)$ and $u_{\ast}(z)$, we see that $L_{\varepsilon}(z)\neq \emptyset$ for all $z\in\Omega$. Clearly
$$GrL_{\varepsilon}\in \mathcal{L}(\Omega)\times \mathcal{B}(C^1(\RR)),$$
with $\mathcal{L}(\Omega)$ being the Lebesgue $\sigma$-field of $\Omega$ and $\mathcal{B}(C^1(\RR))$ the Borel $\sigma$-field of $C^1(\RR)$. Since $C^1(\RR)$ is a separable Fr\'echet space, we can apply the Yankov--von Neumann--Aumann selection theorem (see Hu and Papageorgiou \cite[pp. 158-159]{11}) and obtain a Borel measurable map $\hat{\vartheta}:\Omega\rightarrow C^1(\RR)$ such that  $\hat{\vartheta}(z)\in L_{\varepsilon}(z)$ for a.a. $z\in\Omega$. We set $\hat{h}(z,x)=\hat{\vartheta}(z)(x)$ for all $x\in\RR$. Then $\hat{h}(\cdot,\cdot)$ is jointly measurable and for a.a. $z\in\Omega$, $\hat{h}(z,\cdot)\in C^1(\RR)$. We set $\hat{H}(z,x)=\int\limits_0^x \hat{h}(z,x)ds$ and consider the functional $\hat{\tau}:H^1(\Omega)\rightarrow\RR$ defined by
$$\hat{\tau}(u)=\frac{1}{2}\xi(u)+\frac{1}{2}\|u\|_2^2-\int_{\Omega}\hat{H}(z,u(z))dz\ \mbox{for\ all}\ u\in H^1(\Omega).$$
Evidently $\hat{\tau}\in C^2(H^1(\Omega))$ and $y_0\in K_{\hat{\tau}}$. Choosing $\varepsilon>0$ small, by virtue of the continuity of the critical groups in the $C^1$-topology, we have
\begin{eqnarray}\label{eq45}
C_k(\tau,y_0)=C_k(\hat{\tau},y_0)\ \mbox{for\ all}\ k\geq 0,
\end{eqnarray}
\begin{eqnarray}\label{eq46}
&\Rightarrow& C_1(\hat{\tau},y_0)\neq 0\ (\mbox{see}\ (\ref{eq43})),\nonumber \\
&\Rightarrow& C_k(\hat{\tau},y_0)=\delta_{k,1}\ZZ\ \mbox{for\ all}\ k\geq 0\ (\mbox{see\ Bartsch\ \cite{4}}),\nonumber \\
&\Rightarrow& C_k(\tau,y_0)=\delta_{k,1}\ZZ\ \mbox{for\ all}\ k\geq 0\ (\mbox{see}\ (\ref{eq45})).
\end{eqnarray}
By virtue of the Claim, we have
\begin{eqnarray}\label{eq47}
 C_k(\tau,u_{\ast})=C_k(\tau,v_{\ast})=\delta_{k,0}\ZZ\ \mbox{for\ all}\ k\geq 0.
\end{eqnarray}
Moreover, since $\tau$ is coercive (see (\ref{eq36})), we have
\begin{eqnarray}\label{eq48}
 C_k(\tau,\infty)=\delta_{k,0}\ZZ\ \mbox{for\ all}\ k\geq 0.
\end{eqnarray}
Recall that $K_{\tau}=\{0,u_{\ast},v_{\ast},y_0\}$. From (\ref{eq44}), (\ref{eq46}), (\ref{eq47}), (\ref{eq48}) and the Morse relation (see (\ref{eq4})) with $t=-1$, we have
\begin{eqnarray*}
&&(-1)^{d_m}+2(-1)^0+(-1)^1=(-1)^0\\
&\Rightarrow& (-1)^{d_m}=0,\ \mbox{a\ contradiction}.
\end{eqnarray*}
So, we can find $\hat{y}\in K_{\tau}\setminus \{0,u_{\ast},v_{\ast},y_0\}$. Then $\hat{y}\in [v_{\ast},u_{\ast}]\cap C^1(\overline{\Omega})$ (see (\ref{eq37})) and so it is the second nodal solution of (\ref{eq1}). As we did for $y_0$, we show that $\hat{y}\in\mbox{int}_{C^1(\overline{\Omega})}[v_{\ast},u_{\ast}]$.

\end{proof}
Summarizing the situation for problem (\ref{eq1}), we can state the following multiplicity theorem.
\begin{theorem}\label{th11}
Assume that hypotheses $H(\beta)$ and $H_2$ hold. Then problem (\ref{eq1}) has at least four nontrivial solutions
$$u_0\in\mbox{int}\, C_{+},\ v_0\in -\mbox{int}\, C_{+},\ \mbox{and}\ y_0,\ \hat{y}\in\mbox{int}_{C^1(\overline{\Omega})}[v_0,u_0]\ \mbox{nodal}.$$
\end{theorem}
A careful reading of the above proof, leads to another multiplicity theorem producing three nontrivial solutions all with sign information and without imposing any differentiability condition on the reaction $f(z,\cdot)$.

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

\smallskip
$H_3$: $f:\Omega\times\RR\rightarrow\RR$ is a Carath\'eodory function such that  $f(z,0)=0$ for a.a. $z\in\Omega$ and
\begin{itemize}
\item [(i)] for every $\varrho>0$, there exists $a_{\varrho}\in L^{\infty}(\Omega)_{+}$ such that  $|f(z,x)|\leq a_{\varrho}(z)$ for a.a. $z\in\Omega$, all $|x|\leq \varrho$;
\item [(ii)] there exist functions $w_{\pm}\in H^1(\Omega)\cap L^{\infty}(\Omega)$ such that
\begin{eqnarray*}
&&w_{-}(z)\leq c_{-}<0<c_{+}\leq w_{+}(z)\ \mbox{for\ a.e.}\ z\in\Omega,\\
&&f(z,w_{+}(z))\leq 0\leq f(z,w_{-}(z))\ \mbox{for\ a.e.}\ z\in\Omega,\\
&&A(w_{-})+\beta(\cdot)w_{-}\leq 0\leq A(w_{+})+\beta(\cdot)w_{+}\ \mbox{in}\ H^1(\Omega)^{\ast}.
\end{eqnarray*}
\item [(iii)] there exist an integer $m\geq 2$ and functions $\eta,\hat{\eta}\in L^{\infty}(\Omega)_{+}$ such that
\begin{eqnarray*}
&&\hat{\lambda}_m\leq \eta(z)\leq\hat{\eta}(z)\leq \hat{\lambda}_{m+1}\ \mbox{a.e.\ in}\ \Omega,\ \eta\neq\hat{\lambda}_m,\ \hat{\eta} \neq\hat{\lambda}_{m+1};\\
&&\eta(z)\leq\liminf_{x\rightarrow 0}\frac{f(z,x)}{x}\leq\limsup_{x\rightarrow 0}\frac{f(z,x)}{x}\leq\hat{\eta}(z)\ \mbox{uniformly\ for\ a.a.}\ z\in\Omega;
\end{eqnarray*}
\item [(iv)] if $\varrho_0=\max\{\|w_{+}\|_{\infty},\|w_{-}\|_{\infty}\}$, then there exists $\xi_0>0$ such that  for a.a. $z\in\Omega$, the function $x\mapsto f(z,x)+\xi_0 x$ is nondecreasing on $[-\varrho_0,\varrho_0]$.
\end{itemize}
\begin{remark}
Now, in contrast to the previous setting, we do not allow resonance at zero, only nonuniform nonresonance.
\end{remark}
In this case, hypothesis $H_3(iii)$, the homotopy invariance of critical groups and the results of Dancer \cite{8}, imply
$$C_{d_m}(\tau,0)\neq 0\ \mbox{and}\ C_1(\tau,0)=0,$$
with $\tau$ as in the proof of Proposition \ref{prop10} and with $d_m=\dim \oplus_{k=1}^m E(\hat{\lambda}_k)\geq 2$.
Then the argument in the proof of Proposition \ref{prop10}, remains valid and we have.

\begin{theorem}\label{th12}
Assume that hypotheses $H(\beta)$ and $H_1$ hold. Then problem (\ref{eq1}) has at least three nontrivial solutions
$$u_0\in\mbox{int}\, C_{+},\ v_0\in -\mbox{int}\, C_{+},\ \mbox{and}\ y_0\in\mbox{int}_{C^1(\overline{\Omega})}[v_0,u_0]\ \mbox{nodal}.$$
\end{theorem}

\section{Parametric Equations}\label{sec5}

We consider the following parametric semilinear Robin problem
\begin{center}
$\begin{array}{cl}%
\displaystyle	-\Delta u(z)=\lambda u(z)-f(z,u(z))\ \mbox{in}\ \Omega,\ \frac{\partial u}{\partial n}(z)+\beta(z)u(z)=0\ \mbox{on}\ \partial\Omega.
\end{array}\; \hspace{4cm}(P_{\lambda})$
\end{center}
Here $\lambda>0$ is a parameter and $f(z,x)$ is a Carath\'eodory perturbation. The precise conditions on the nonlinearity $f(z,x)$ are the following:

\smallskip
$H_4$: $f:\Omega\times\RR\rightarrow\RR$ is a measurable function such that  for a.a. $z\in\Omega$, $f(z,0)=0$, $f(z,\cdot)\in C^1(\RR)$  and
\begin{itemize}
\item [(i)] for every $\varrho>0$, there exists $a_{\varrho}\in L^{\infty}(\Omega)_{+}$ such that  $|f_x'(z,x)|\leq a_{\varrho}(z)$ for a.a. $z\in\Omega$, all $|x|\leq \varrho$;
\item [(ii)] $\lim\limits_{x\rightarrow\pm\infty}\frac{f(z,x)}{x}=+\infty\ \mbox{uniformly\ for\ a.a.}\ z\in\Omega$;
\item [(iii)] $\lim\limits_{x\rightarrow 0}\frac{f(z,x)}{x}=0\ \mbox{uniformly\ for\ a.a.}\ z\in\Omega$;
\item [(iv)] there exists $\hat{\delta}>0$ such that  $f(z,x)x\geq 0$ for a.a. $z\in\Omega$, all $|x|\leq \hat{\delta}$.
\end{itemize}
\begin{remark}
The above hypotheses imply that given $\varrho>0$, we can find $\xi_{\varrho}>0$ such that  for a.a. $z\in\Omega$ the function $x\mapsto (\lambda+\xi_{\varrho})x-f(z,x)$ is nondecreasing on $[-\varrho,\varrho]$.
\end{remark}
\begin{theorem}\label{th13}
 Assume that hypotheses $H(\beta)$ and $H_4$ hold and $\lambda>\hat{\lambda}_2$. Then problem $(P_{\lambda})$ admits at least four nontrivial solutions
$$u_0\in\mbox{int}\, C_{+},\ v_0\in -\mbox{int}\, C_{+},\ \mbox{and}\ y_0,\ \hat{y}\in\mbox{int}_{C^1(\overline{\Omega})}[v_0,u_0].$$
\end{theorem}
\begin{proof}
Hypothesis $H_4(ii)$ implies that given $\vartheta>0$, we can find $M=M(\vartheta)>0$ such that
$$f(z,x)\geq \vartheta x\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\geq M.$$
Since $\hat{u}_1\in\mbox{int}\, C_{+}$, for $t>0$ large we will have $t\hat{u}_1(z)\geq M$ for all $z\in\overline{\Omega}$. Then
\begin{eqnarray}\label{eq49}
f(z,t\hat{u}_1(z))\geq \vartheta (t\hat{u}_1(z))\ \mbox{for\ a.a.}\ z\in\Omega.
\end{eqnarray}
We have
\begin{eqnarray}\label{eq50}
-\Delta(t\hat{u}_1)(z)=\hat{\lambda}_1(t\hat{u}_1)(z)\ \mbox{a.e.\ in}\ \Omega,\ \frac{\partial (t\hat{u}_1)}{\partial n}+\beta(z)(t\hat{u}_1)=0\ \mbox{on}\ \partial\Omega.
\end{eqnarray}
For every $h\in H^1(\Omega),\ h\geq 0$, we have
\begin{eqnarray}\label{eq51}
&&\langle -\Delta(t\hat{u}_1),h\rangle= \int_{\Omega}\hat{\lambda}_1(t\hat{u}_1)h dz,\nonumber \\
&\Rightarrow& \langle A(t\hat{u}_1),h\rangle-\big\langle\frac{\partial (t\hat{u}_1)}{\partial n},h\big\rangle_{\partial\Omega}=\int_{\Omega}\hat{\lambda}_1(t\hat{u}_1)h dz\ (\mbox{by\ Green's\ identity}),\nonumber \\
&\Rightarrow&\langle A(t\hat{u}_1),h\rangle+\int_{\partial\Omega}\beta(z)(t\hat{u}_1)d\sigma=\int_{\Omega}\hat{\lambda}_1(t\hat{u}_1)h dz\ (\mbox{see}\ (\ref{eq50})).
\end{eqnarray}
Choosing $\vartheta=\lambda-\hat{\lambda}_1>0$, from (\ref{eq49}) and (\ref{eq51}), we have
\begin{eqnarray*}
&&\int_{\Omega}[\lambda(t\hat{u}_1)-f(z,t\hat{u}_1)]h dz\\
&\leq& \int_{\Omega}\hat{\lambda}_1(t\hat{u}_1)h dz\\
&=&\langle A(t\hat{u}_1),h\rangle+\int_{\partial\Omega}\beta(z)(t\hat{u}_1)h d\sigma\ \mbox{for\ all}\ h\in H^1(\Omega)\ \mbox{with}\ h\geq 0.
\end{eqnarray*}
Then $w_{+}=t\hat{u}_1\in\mbox{int}\, C_{+}$ satisfies hypothesis $H_2(ii)$. Similarly, we produce $w_{-}=-t\hat{u}_1\in -\mbox{int}\, C_{+}$ which also satisfies  hypothesis $H_2(ii)$.

Finally note that by virtue of hypothesis $H_4(iii)$ given $\varepsilon>0$, we can find $\delta=\delta(\varepsilon)\in (0,\hat{\delta}]$ (see $H_4(iv)$) such that
\begin{eqnarray}\label{eq52}
|f(z,x)|\leq \varepsilon|x|\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ |x|\leq\delta_0.
\end{eqnarray}
Then for a.a. $z\in\Omega$ and all $|x|\leq\delta_0$, we have
$$\lambda x^2-f(z,x)x\geq (\lambda-\varepsilon)x^2.$$
Choosing $\varepsilon\in (0,\lambda-\hat{\lambda}_2]$, we see that
$$\lambda x^2-f(z,x)x\geq \hat{\lambda}_2 x^2\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ |x|\leq\delta_0.$$
Also by hypothesis $H_4(iv)$ we have
$$\lambda x^2-f(z,x)x\leq \lambda x^2\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ |x|\leq\delta_0.$$
So, we have satisfied hypothesis $H_2(iii)$. Therefore, we are within the framework of Theorem \ref{th12}. Applying that theorem, we produce four nontrivial solutions
$$u_0\in\mbox{int}\, C_{+},\ v_0\in -\mbox{int}\, C_{+},\ \mbox{and}\ y_0,\ \hat{y}\in\mbox{int}_{C^1(\overline{\Omega})}[v_0,u_0]\ \mbox{nodal}.$$
\end{proof}
\begin{remark}
Theorem \ref{th13} above extends Theorem $14$ of Papageorgiou and R\u{a}dulescu \cite{14}, since here we do not impose a global growth condition on $f_x'(z,\cdot)$ (see $H_4(i)$). Moreover, it extends analogous results for the semilinear Dirichlet problem by Ambrosetti and Lupo \cite{2}, Ambrosetti and Mancini \cite{3} and Struwe \cite{15}, \cite{16}, where the authors produce only three nontrivial solutions, without obtaining nodal solutions.
\end{remark}

\begin{thebibliography}{99}
\bibitem{1} {\sc S. Aizicovici, N.S. Papageorgiu, V. Staicu}, {\em Degree Theory for Operators of Monotone Type and Nonlinear Elliptic Equations with Inequality Constraints}, Memoirs Amer. Math. Soc. \textbf{196} (2008), No.~\textbf{915}.
\bibitem{2} {\sc A. Ambrosetti, D. Lupo}, {\em On a class of nonlinear Dirichlet problems with multiple solutions}, Nonlinear Analysis \textbf{8} (1984), 1145-1150.
\bibitem{3} {\sc A. Ambrosetti, G. Mancini}, {\em Sharp nonuniqueness results for some nonlinear problems}, Nonlinear Analysis \textbf{3} (1979), 635-645.
\bibitem{4} {\sc T. Bartsch}, {\em Critical point theory on partially ordered Hilbert spaces}, J. Functional Analysis \textbf{186} (2001), 117-152.
\bibitem{5} {\sc R. Benguria, H. Brezis, E.H. Lieb}, {\em The Thomas--Fermi--von Weizs\"{a}cker theory of atoms and molecules}, Comm. Math. Physics \textbf{79} (1981), 167-180.
\bibitem{6} {\sc H. Brezis, L. Nirenberg}, {\em $H^1$ versus $C^1$ local minimizers}, C.R. Acad. Sci. Paris S\'er.~I \textbf{317} (1993), 465-472.
\bibitem{7} {\sc K.-C. Chang}, {\em Methods in Nonlinear Analysis}, Springer, Berlin, 2005.
\bibitem{8} {\sc E.N. Dancer}, {\em Remark on jumping nonlinearity} in {\em Topics in Nonlinear Analysis}, Birkh\"auser, Basel (1998), 101-116.
\bibitem{9} {\sc N. Dunford, J. Schwartz}, {\em Linear Operators I}, Wiley-Interscience, New York, 1958.
\bibitem{10} {\sc L. Gasinski, N.S. Papageorgiou}, {\em Nonlinear Analysis}, Chapman, Hall/CRC, Boca Raton, 2006.
\bibitem{11} {\sc S. Hu, N.S. Papageorgiou}, {\em Handbook of Multivalued Analysis. Volume I: Theory}, Kluwer, Dordrecht, The Netherlands, 1997.
\bibitem{12} {\sc G. Lieberman}, {\em The natural generalization of the natural conditions of Ladyzhenskaya and Uraltseva}, Comm. Partial Differential Equations \textbf{16} (1991), 311-361.
\bibitem{13} {\sc R. Palais}, {\em Homotopy theory of infinite dimensional manifolds}, Topology \textbf{5} (1966), 1-16.
\bibitem{14} {\sc N.S. Papageorgiou, V. R\u adulescu}, {\em Multiple solutions with precise sign for nonlinear parametric Robin problems},  Journal of Differential Equations {\bf 256} (2014), 2449-2479.
\bibitem{15} {\sc M. Struwe}, {\em A note on a result of Ambrosetti and Mancini}, Annal. Mat. Pura Appl. \textbf{131} (1982), 107-115.
\bibitem{16} {\sc M. Struwe}, {\em Variational Methods}, Springer-Verlag, Berlin (1990).
\bibitem{17} {\sc J. Su}, {\em Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues}, Nonlinear Analysis \textbf{48} (2002), 881-895.
\bibitem{18} {\sc J. Vazquez}, {\em A strong maximum principle for some quasilinear elliptic equations}, Appl. Math. Optim. \textbf{12} (1984), 191-202.
\bibitem{19} {\sc P. Winkert}, {\em $L^{\infty}$-estimates for nonlinear elliptic Neumann boundary value problems}, NoDEA (Nonlin. Diff. Equations Appl.) \textbf{17} (2010), 289-302.
\bibitem{20} {\sc J. Zhang, S. Li, X. Xue}, {\em Multiple solutions for a class of semilinear elliptic problems with Robin boundary condition}, J. Math. Anal. Appl. \textbf{388} (2012), 435-442.

\end{thebibliography}


\end{document} 