% version 2013
\documentclass[12pt]{amsart}
%\usepackage{graphicx, color}
\usepackage{amscd}
\usepackage{amsmath,empheq}
\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}

\newcommand{\eqname}[1]{\tag*{#1}}% Name of equation


%\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[Robin problems with indefinite, unbounded potential]{Robin problems with indefinite, unbounded potential and reaction of arbitrary growth}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 Sciences, King Abdulaziz University,
 P.O. Box 80203, Jeddah 21589, Saudi Arabia \& Institute of Mathematics ``Simion Stoilow" of the Romanian Academy, P.O. Box 1-764,
          014700 Bucharest, Romania }
\email{\tt vicentiu.radulescu@imar.ro}

\keywords{Indefinite and unbounded potential, Robin boundary condition, constant sign and nodal solutions, multiplicity theorem, critical groups.\\
\phantom{aa} 2010 AMS Subject Classification:
35J20, 35J60, 58E05}.


\begin{abstract}
We study an elliptic Robin problem driven by the negative Laplacian plus an indefinite and unbounded potential and with a reaction of arbitrary growth which exhibits $z$-dependent zeros of constant sign. We prove multiplicity theorems producing three or four nontrivial solutions, all with precise sign information. As a particular case we consider a generalized equidiffusive logistic equation with potential.
\end{abstract}

\maketitle

%\tableofcontents

\section{Introduction}

Let $\Omega\subseteq\RR^N$ be a bounded with a $C^2$-boundary $\partial\Omega$. In this paper, we study the following Robin problem
\begin{equation}\label{eq1}
	-\Delta u(z)+\xi(z)u(z)=f(z,u(z))\ \mbox{in}\ \Omega,\ \frac{\partial u}{\partial n}+\beta(z)u=0\ \mbox{on}\ \partial\Omega.
\end{equation}

Here $\xi\in L^s(\Omega)$ with $s>N$ and in general it is sign changing. Also, $\beta\in W^{1,\infty}(\partial\Omega)$, $\beta\geq 0$. When $\beta\equiv 0$, then we have the Neumann problem. The reaction $f(z,x)$ is a Carath\'eodory function (that is, for all $x\in\RR,\ z\longmapsto f(z,x)$ is measurable and for almost all $z\in\Omega$, $x\longmapsto f(z,x)$ is continuous). The interesting feature of our work here, is that we do not impose any global growth condition on $x\longmapsto f(z,x)$. Instead, we assume that $f(z,\cdot)$ admits $z$-dependent zeros of constant sign. Our aim is to prove a multiplicity theorem providing precise sign information for all the solutions. Using variational methods based on the critical point theory, together with suitable truncation and perturbation techniques and Morse theory (critical groups), we prove two multiplicity theorems producing three nontrivial solutions (two of constant sign and the third nodal (sign changing)). The two multiplic
 ity results differ on the behavior of the reaction $f(z,\cdot)$ near zero. Subsequently, by improving the regularity condition on $x\longmapsto f(z,x)$ and using tools from Morse theory, we prove a third multiplicity theorem, producing four nontrivial solutions, two of constant sign and two nodal. Our work here extends the semilinear part of the recent work of Papageorgiou and R\u{a}dulescu \cite{19}.

Semilinear equations with indefinite and bounded potential, were studied recently under different conditions on the reaction and under different boundary conditions. We mention the works of Kyritsi and Papageorgiou \cite{10}, Papageorgiou and Papalini \cite{17} (Dirichlet problems), and Papageorgiou and R\u{a}dulescu \cite{18}, \cite{20}, Papageorgiou and Smyrlis \cite{21} (Neumann problems).

None of the aforementioned works addresses the general boundary condition used in this paper (which incorporates as a special case the Neumann problem for $\beta\equiv 0$) and all assumed that the reaction term $f(z,\cdot)$ has subcritical polynomial growth. In contrast here, the behavior of $f(z,\cdot)$ near $\pm\infty$ is irrelevant and instead we assume a kind of oscillatory behavior near zero for the nonlinearity $x\longmapsto f(z,x)-\xi (z)x$, by requiring the presence of $z$-dependent zeros for the function. In this way we can focus our analysis on an interval $[-\rho,\rho]$ ignoring the structure of the reaction term outside it.

\section{Mathematical Background}

In this section, we briefly review the main mathematical tools which we will use in this paper.

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

This is a compactness-type condition on the functional $\varphi$, which leads to a deformation theorem, from which one can derive the minimax theory for the critical values of $\varphi$. A basic result in that theory, is the so-called ``mountain pass theorem".
\begin{theorem}\label{th1}
	Assume that $\varphi\in C^1(X)$ satisfies the $PS$-condition, $u_0,u_1\in X,\ ||u_1-u_0||>r>0$
	$$\max\{\varphi(u_0),\varphi(u_1)\}<\inf[\varphi(u):||u-u_0||=r]=m_r$$
	and $c=\inf\limits_{\gamma\in\Gamma}\max\limits_{0\leq t\leq 1}\varphi(\gamma(t))$ with $\Gamma=\{\gamma\in C([0,1],X):\gamma(0)=u_0,\gamma(1)=u_1\}$. Then $c\geq m_{\rho}$ and $c$ is critical value of $\varphi$.
\end{theorem}

In the study of problem (\ref{eq1}) we will use the Sobolev space $H^1(\Omega)$, the Banach space $C^1(\overline{\Omega})$ and the Lebesgue spaces $L^p(\partial\Omega)$ $(1\leq p\leq\infty)$. By $||\cdot||$ we denote the norm of the Sobolev space $H^1(\Omega)$, defined by
$$||u|=\left[||u||^2_2+||Du||^2_2\right]^{1/2}\ \mbox{for all}\ u\in H^1(\Omega).$$

The space $C^1(\overline{\Omega})$ 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
$$\mbox{int}\, C_+=\{u\in C_+:u(z)>0\ \mbox{for all}\ z\in \overline{\Omega}\}.$$

The Lebesgue spaces $L^p(\partial\Omega)\ (1\leq p\leq\infty)$ are defined as follows. On $\partial\Omega$ we consider the $(N-1)$-dimensional Hausdorff (surface) measure $\sigma_0(\cdot)$. Then using $\sigma_0(\cdot)$ we can introduce the spaces $L^p(\partial\Omega)\ (1\leq p\leq\infty)$ in the usual way. From the trace theorem, 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 $\gamma_0(u)=u|_{\partial\Omega}$ for all $u\in C^1(\overline{\Omega})$. This map is compact into $L^{\eta}(\partial\Omega)$ for $1\leq\eta<\frac{2(N-1)}{N-2}$. Moreover, we know that
$$\mbox{im}\,\gamma_0=H^{\frac{1}{2},2}(\partial\Omega)\ \mbox{and}\ \mbox{ker}\,\gamma_0=H^1_0(\Omega)$$
(for details see, for example, Gasinski and Papageorgiou \cite{7}). In what follows, for the sake of notational simplicity, we drop the use of the trace map $\gamma_0$. Every Sobolev function defined on $\partial\Omega$ is understood in the sense of traces.

For $x\in\RR$, we set $x^{\pm}=\max\{\pm x,0\}$. Then given $u\in H^1(\Omega)$, we set $u^{\pm}(\cdot)=u(\cdot)^{\pm}$. We know that
$$u^{\pm}\in H^1(\Omega),\ |u|=u^++u^-\ \mbox{and}\ u=u^+-u^-.$$

We will also use some aspects of the spectrum of $-\Delta u+\xi(z)u$ with Robin boundary condition. So, we consider the following eigenvalue problem
\begin{equation}\label{eq2}
	-\Delta u(z)+\xi(z)u(z)=\hat{\lambda}u(z)\ \mbox{in}\ \Omega,\ \frac{\partial u}{\partial n}+\beta(z)u=0\ \mbox{on}\ \partial\Omega.
\end{equation}

This eigenvalue problem was studied for the Neumann boundary condition (that is, $\beta\equiv 0$), in Papageorgiou and R\u{a}dulescu \cite{18} and Papageorgiou and Smyrlis \cite{21}. For the $p$-Laplacian and Neumann boundary condition, it was investigated by Mugnai and Papageorgiou \cite{14} and for the $p$-Laplacian with Robin boundary condition and $\xi\equiv 0$ by Papageorgiou and R\u{a}dulescu \cite{19}. An analogous study can be conducted for problem (\ref{eq2}) and leads to similar results. More, precisely, assume that $\xi\in L^{\frac{N}{2}}(\Omega)$ if $N\geq 3$, $\xi\in L^r(\Omega)$ for $r>1$ when $N=2$ and $\xi\in L^1(\Omega)$ when $N=1,\ \beta\in W^{1,\infty}(\partial\Omega)$, $\beta\geq 0$ and let $\sigma:H^1(\Omega)\rightarrow\RR$ be the functional defined by
$$\sigma(u)=||Du||^2_2+\int_{\Omega}\xi(z)u^2dz+\int_{\partial\Omega}\beta(z)u^2d\sigma_0\ \mbox{for all}\ u\in H^1(\Omega).$$

Then the eigenvalue problem (\ref{eq2}) admits a smallest eigenvalue $\hat{\lambda}_1(\beta)>-\infty$ given by
\begin{equation}\label{eq3}
	\hat{\lambda}_1(\beta)=\inf\left[\frac{\sigma(u)}{||u||^2_2}:u\in H^1(\Omega),u\neq 0\right].
\end{equation}

So, we can find $\mu>\max\{-\hat{\lambda}_1(\beta),1\}$ such that
\begin{equation}\label{eq4}
	\sigma(u)+\mu||u||^2_2\geq c_0||u||^2\ \mbox{for all}\ u\in H^1(\Omega),\ \mbox{with}\ c_0>0.
\end{equation}

Using (\ref{eq4}) and the spectral theorem for compact self-adjoint operators, exactly as in \cite{18} and \cite{21}, we have a sequence $\{\hat{\lambda}_{k}(\beta)\}_{k\geq 1}$ such that $\hat{\lambda}_k(\beta)\rightarrow+\infty$ as $k\rightarrow\infty$ which are all the eigenvalues of (\ref{eq2}). Let $E(\hat{\lambda}_k(\beta))$ be the corresponding eigenspace. If $\xi\in L^s(\Omega)$ with $s>N$, then using the regularity result of Wang \cite{25}, we have that $E(\hat{\lambda}_k(\beta))\subseteq C^1(\overline{\Omega})$. For the eigenvalues $\hat{\lambda}_k(\beta)\ k\geq 2$, we have the following variational characterization
\begin{eqnarray}\label{eq5}
	\hat{\lambda}_k(\beta)&=&\inf\left[\frac{\sigma(u)}{||u||^2_2}:u\in \overline{{\underset{\mathrm{i\geq k}}\oplus}E(\hat{\lambda}_i(\beta))},u\neq 0\right]\nonumber\\
	&=&\sup\left[\frac{\sigma(u)}{||u||^2_2}:u\in \overset{k}{\underset{\mathrm{i=1}}\oplus}E(\hat{\lambda}_i(\beta)),u\neq 0\right],k\geq 2.
\end{eqnarray}

In both (\ref{eq3}) and (\ref{eq5}) the infimum (and the case of (\ref{eq5}) also the supremum), is realized on the corresponding eigenspace $E(\hat{\lambda}_k(\beta))$. The first eigenvalue $\hat{\lambda}_1(\beta)$ is simple (that is, $\mbox{dim}\, E(\hat{\lambda}_1(\beta)=1)$, Krein-Rutman theorem) and from (\ref{eq3}) it is clear that the nontrivial elements of $E(\hat{\lambda}_1(\beta))$ do not change sign. All the other eigenvalues have nodal (sign changing) eigenfunctions. By $\hat{u}_1(\beta)\in H^1(\Omega)$ we denote the $L^2$-normalized (that is, $||\hat{u}_1(\beta)||_2=1$) positive eigenfunction corresponding to $\hat{\lambda}_1(\beta)$. If $\xi\in L^s(\Omega)$ with $s>N$, then $\hat{u}_1(\beta)\in C_+\backslash\{0\}$ (see Wang \cite{25}). Moreover, using the Harnack inequality of Pucci and Serrin \cite[p. 163]{22}, we have $u_1(z)>0$ for all $z\in\Omega$. Also, if $\xi^+\in L^{\infty}(\Omega)$, then by the Hopf theorem (see, for example, Pucci and Serrin \cite[p. 1
 20]{22}), we have $\hat{u}_1(\beta)\in \mbox{int}\, C_+.$ Finally when $\xi\in L^s(\Omega)$ with $s>\frac{N}{2}$, then all the eigenspaces $E(\hat{\lambda}_k(\beta))$ have the ``unique continuation property" (UCP for short), that is, if $u\in E(\hat{\lambda}_k(\beta))$ and vanishes on a set of positive measure, then $u\equiv 0$ (see de Figueiredo and Gossez \cite{5}).

For the second eigenvalue $\hat{\lambda}_2(\beta)$, in addition to the variational characterization (\ref{eq5}) we have a minimax expression (see \cite{14}, \cite{19}) which we will need in the sequel. So, let
$$\partial B^{L^2}_{1}=\{u\in L^2(\Omega):||u||_2=1\}\ \mbox{and}\ M=H^1(\Omega)\cap\partial B^{L^2}_{1}.$$
\begin{prop}\label{prop2}
We have	$\hat{\lambda}_2(\beta)=\inf\limits_{\hat{\gamma}\in\hat{\Gamma}}\max\limits_{-1\leq t\leq 1}\sigma(\hat{\gamma}(t))$, where $$\hat{\Gamma}=\{\hat{\gamma}\in C([-1,1],M):\hat{\gamma}(-1)=-\hat{u}_1(\beta),\ \hat{\gamma}(1)=\hat{u}_1(\beta)\}.$$
\end{prop}

Let $f_0:\Omega\times\RR\rightarrow\RR$ be a Carath\'eodory function such that
$$|f_0(z,x)|\leq a_0(z)(1+|x|^{r-1})\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\in\RR,$$
with $a_0\in L^{\infty}(\Omega)_+$ and $$1<r<2^*=\left\{\begin{array}{ll}
\frac{2N}{N-2}&\mbox{if}\ N\geq 3\\
+\infty&\mbox{if}\ N=1,2.
\end{array}\right.$$
We set $F_0(z,x)=\int^{x}_{0}f_0(z,s)ds$ and consider the $C^1$-functional $\varphi_0:H^1(\Omega)\rightarrow\RR$ defined by
$$\varphi_0(u)=\frac{1}{2}\sigma(u)-\int_{\Omega}F_0(z,u)dz\ \mbox{for all}\ u\in H^1(\Omega).$$

%From Papageorgiou and R\u{a}dulescu \cite{19}, we have
%\begin{prop}\label{prop3}
%	Assume that $\xi\in L^s(\Omega)$ with $s>N$, $\xi^+\in L^{\infty}(\Omega)$, $\beta\in %W^{1,\infty}(\partial\Omega)$ with $\beta\geq 0$ and $u_0\in H^1(\Omega)$ is a local %$C^1(\overline{\Omega})$-minimizer of $\varphi_0$, that is, there exists $\rho_0>0$ such that
%	$$\varphi_0(u_0)\leq\varphi_0(u_0+h)\ \mbox{for all}\ h\in C^1(\overline{\Omega})\ \mbox{with}\ %||h||_{C^1(\overline{\Omega})}\leq\rho_0.$$
%	Then $u_0\in C^{1,\alpha}(\overline{\Omega})$ for some $\alpha\in(0,1)$ and $u_0$ is a local %$H^1(\Omega)$-minimizer of $\varphi_0$, that is, there exists $\rho_1>0$ such that
%	$$\varphi_0(u_0)\leq\varphi_0(u_0+h)\ \mbox{for all}\ h\in H^1(\Omega)\ \mbox{with}\ ||h||\leq\rho_1.$$
%\end{prop}

Next we recall some basic definitions and facts from Morse theory (critical groups), which we will use in the sequel.

Let $X$ be a Banach space and $\varphi\in C^1(X),\ c\in\RR$. We introduce the following sets:
$$\varphi^c =\{u\in X:\varphi(u)\leq c\},\ K_{\varphi}=\{u\in X:\varphi'(u)=0\}\ \mbox{and}\ K^c_{\varphi}=\{u\in K_{\varphi}:\varphi(u)=c\}.$$

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 for the pair $(Y_1,Y_2)$ with integer coefficients. Recall that for $k<0,\ H_k(Y_1,Y_2)=0$. Consider an isolated critical point $u_0\in K^c_{\varphi}$. Then the critical groups of $\varphi$ at $u_0$ are defined by
$$C_k(\varphi,u_0)=H_k(\varphi^c\cap U,\varphi^c\cap U\backslash\{u_0\})\ \mbox{for every integer}\ k\geq 0.$$

Here $U$ is a neighborhood of $u_0$ such that $K_{\varphi}\cap\varphi^c\cap U=\{u_0\}$. The excision property of singular homology implies that this definition of critical groups is independent of the choice of the neighborhood $U$.

Suppose that $\varphi\in C^1(X)$ satisfies the $PS$-condition and $\inf\varphi(K_{\varphi})>-\infty$. Let $c<\inf\varphi(K_{\varphi})$. Then the critical groups of $\varphi$ at infinity are defined by
$$C_k(\varphi,\infty)=H_k(X,\varphi^c)\ \mbox{for all}\ k\in\NN_0.$$

The second deformation theorem (see, for example, Gasinski and Papageogiou \cite[p. 628]{7}), implies that the above definition of critical groups at infinity, is independent of the choice of the level $c<\inf\varphi(K_{\varphi})$.

Suppose that $K_{\varphi}$ is finite. We introduce the following quantities:
\begin{eqnarray*}
	&&M(t,u)=\sum_{k\geq 0}\mbox{rank}\, C_k(\varphi,u)t^k\ \mbox{for all}\ t\in\RR,\ \mbox{all}\ u\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*}

The ``Morse relation" says that
\begin{equation}\label{eq6}
	{\underset{\mathrm{u\in K_{\varphi}}}\sum}M(t,u)=P(t,\infty)+(1+t)Q(t),
\end{equation}
where $Q(t)={\underset{\mathrm{k\geq 0}}\sum}\beta_k t^k$ is a formal series in $t\in\RR$ with nonnegative integer coefficients $\beta_k$.

Suppose that $X=Y\oplus V$ with $\mbox{dim}\, Y<\infty$ and $\varphi\in C^1(X)$. We say that $\varphi\in C^1(X)$ has a ``local linking" at the origin, if we can find $\rho>0$ such that
\begin{eqnarray*}
	&&\varphi(y)\leq 0\ \mbox{for all}\ y\in Y,\ ||y||_X\leq\rho\\
	&&\varphi(v)>0\ \mbox{for all}\ v\in V,\ 0<||v||_X\leq\rho\,.
\end{eqnarray*}

In that case, we know that
$$C_{d_Y}(\varphi,0)\neq 0,\ \mbox{where}\ d_Y=\mbox{dim}\, Y.$$

By $C^{2-0}(X)$ we denote the $C^1(X)$-functionals whose derivative is locally Lipschitz. The so-called ``shifting theorem" which is known to hold for $C^2$-functionals, was extended to $C^{2-0}$ functionals by Li, Li and Liu \cite{11}. We present a particular case of their result suitable for our purposes. Let $f_0:\Omega\times\RR\rightarrow\RR$ be a measurable function such that for almost all $z\in\Omega$, $f_0(z,\cdot)\in C^{2-0}(\RR)$. Set $F_0(z,x)=\int^{x}_{0}f_0(z,s)ds$ and consider the functional $\varphi_0:H^1(\Omega)\rightarrow\RR$ defined by
$$\varphi_0(u)=\frac{1}{2}\sigma(u)-\int_{\Omega}F_0(z,u)dz\ \mbox{for all}\ u\in H^1(\Omega).$$

Then $\varphi_0\in C^{2-0}(H^1(\Omega))$. Let $X=C^1(\overline{\Omega})$ and suppose that $u\in X$ is a critical point of $\varphi_0$. Then $\varphi'_0\in C^1(D,H^1(\Omega))$ and $\varphi''_{0}(u)\in\mathcal{L}(X,H^1(\Omega))$, with $D$ an $X$-neighborhood of $u$. The Morse index $\mu(u)$ of $u$ is the dimension of the maximal subspace of $X$ on which $\varphi''_0(u)$ is negative definite. The nullity of $u$, denoted by $\nu(u)$, is the dimension of the kernel of $\varphi''_{0}(u)$. The extended shifting theorem of Li, Li and Liu \cite{11} says:
\begin{prop}\label{prop4}
	If $\xi\in L^{\frac{N}{2}}(\Omega),\ \beta\in L^{\infty}(\partial\Omega)$ and $u\in K_{\varphi}$ has finite Morse index $\mu=\mu(u)$ and nullity $\nu=\nu(u)$, then either
	\begin{itemize}
		\item[(a)] $C_k(\varphi_0,u)=0$ for $k\leq\mu$ and $k\geq\mu+\nu$, or
		\item[(b)] $C_k(\varphi_0,u)=\delta_{k,\mu}\ZZ$ for all $k\geq 0$, or
		\item[(c)] $C_k(\varphi,u)=\delta_{k,\mu+\nu}\ZZ$ for all $k\geq 0$.
	\end{itemize}
\end{prop}

Finally by $A\in\mathcal{L}(H^1(\Omega),H^1(\Omega)^*)$ we denote the operator
$$\left\langle A(u),y\right\rangle=\int_{\Omega}(Du,Dy)_{\RR^N}dz\ \mbox{for all}\ u,y\in H^1(\Omega).$$

Recall that a Banach space $X$ has the ``Kadec-Klee property", if the following is true:
$$``u_n\stackrel{w}{\rightarrow}u\ \mbox{in}\ X\ \mbox{and}\ ||u_n||_X\rightarrow||u||_X\Rightarrow u_n\rightarrow u\ \mbox{in}\ X".$$

We know that locally uniformly convex Banach spaces, in particular Hilbert spaces, have the Kadec-Klee property.

\section{Three Nontrivial Solutions}

In this section, we prove two multiplicity theorems producing three nontrivial solutions, two of constant sign and the third nodal. The two multiplicity results differ on the conditions on $f(z,\cdot)$ near zero. In the first, it is assumed that $f(z,\cdot)$ is superlinear near zero, while in the second $f(z,\cdot)$ is linear near zero.

First let us state our conditions on the data of problem (\ref{eq1}).

\smallskip
$H(\xi):$ $\xi\in L^s(\Omega)$ with $s>N$ and $\xi^+\in L^{\infty}(\Omega)_+.$

\smallskip
$H(\beta):$ $\beta\in W^{1,\infty}(\partial\Omega),\ \beta\geq 0$.

\smallskip
For the first multiplicity theorem, our hypotheses on the reaction $f(z,x)$ are the following:

\smallskip
$H_1:$ $f:\Omega\times\RR\rightarrow\RR$ is a Carath\'eodory function such that $f(z,0)=0$ for almost all $z\in\Omega$ and
\begin{itemize}
	\item[(i)] there exist functions $w_{\pm}\in H^1(\Omega)\cap C(\overline{\Omega})$ such that
	\begin{eqnarray*}
		&&w_-(z)\leq c_-<0<c_+\leq w_+(z)\ \mbox{for all}\ z\in \overline{\Omega},\\
		&&f(z,w_+(z))-\xi(z)w_+(z)\leq 0\leq f(z,w_-(z))-\xi(z)w_-(z)\ \mbox{for almost all}\ z\in\Omega,\\
		&&A(w_-)\leq 0\leq A(w_+)\ \mbox{in}\ H^1(\Omega)^*;
	\end{eqnarray*}
	\item[(ii)] if $\rho=\max\{||w_+||_{\infty},||w_-||_{\infty}\}$, then there exists $a_{\rho}\in L^{\infty}(\Omega)_+$ such that
	$$|f(z,x)|\leq a_{\rho}(z)\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ |x|\leq\rho;$$
	\item[(iii)] if $F(z,x)=\int^{x}_{0}f(z,s)ds$, then there exist $\delta\in(0,\min\{c_{\pm},1\})$ and $q\in(1,2)$ such that
	$$c_1|x|^q\leq f(z,x)x\leq qF(z,x)\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ |x|\leq\delta.$$
\end{itemize}
\begin{remark}
	Note that no global condition is imposed on $f(z,\cdot)$. In fact the behavior of $f(z,\cdot)$ beyond $w_{\pm}(z)$ is irrelevant. Hypothesis $H_1(i)$ is satisfied if for example we have
	$$\lim\limits_{x\rightarrow\pm\infty}\frac{f(z,x)}{x}=-\infty\ \mbox{uniformly for almost all}\ z\in\Omega.$$
\end{remark}

Hypothesis $H_1(iii)$ implies the presence of a concave term near zero.

By hypotheses $H_1(i),(ii)$ and since $q<2$, we have
\begin{equation}\label{eq7}
	f(z,x)x\geq c_2 x^2-c_3|x|^r\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\in[-\rho,\rho]
\end{equation}
with $c_2>\hat{\lambda}_1(\beta),\ c_3>0,\ r\in(2,2^*)$ and $\rho=\max[||w_+||_{\infty},||w_-||_{\infty}]$. We choose the unilateral growth condition (\ref{eq7}) (instead of the more natural one involving a term with $|x|^q$ due to hypothesis $H_1(iii)$), because it can be used also in the second multiplicity theorem and also facilitates our arguments in the existence results of Proposition \ref{prop7}.

Motivated from (\ref{eq7}), we can consider the following auxiliary Robin problems
\begin{eqnarray}\label{eq8}
	\left\{\begin{array}{ll}
		-\Delta u(z)+\xi(z) u=c_2 u(z)-c_3|u(z)|^{r-2}u(z)&\mbox{in}\ \Omega,\\
	\displaystyle	\frac{\partial u}{\partial n}+\beta(z)u=0&\mbox{on}\ \partial\Omega.
	\end{array}\right\}
\end{eqnarray}
\begin{prop}\label{prop5}
	If hypotheses $H(\xi)$ and $H(\beta)$ hold, then problem (\ref{eq8}) admits a unique positive solution $\bar{u}\in \mbox{int}\, C_+$ and since (\ref{eq8}) is odd, $\bar{v}=-\bar{u}\in-\mbox{int}\, C_+$ is the unique negative solution.
\end{prop}
\begin{proof}
	Let $\psi:H^1(\Omega)\rightarrow\RR$ be the $C^1$-functional defined by
	\begin{eqnarray}\label{eq9}
		 \psi(u)&=&\frac{1}{2}\sigma(u)+\frac{\mu}{2}||u||^2_2-\frac{c_2+\mu}{2}||u^+||^2_2+\frac{c_3}{r}||u^+||^r_r\ \mbox{for all}\ u\in H^1(\Omega)\nonumber\\
		 &\geq&\frac{c_0}{2}||u^-||^2+\frac{1}{2}\sigma(u^+)+\frac{c_3}{r}||u^+||^r_r-\frac{c_2}{\mu}||u^+||^2_2\ (\mbox{see (\ref{eq4})}).
	\end{eqnarray}
	
	Recall that $2<r$. Then using Young's inequality with $\epsilon>0$, we have
	$$\frac{c_2}{2}||u^+||^2_2\leq\hat{c}||u^+||^2_r\leq c_{\epsilon}+\epsilon||u^+||^r_r\ \mbox{for some}\ \hat{c},c_{\epsilon}>0.$$
	
	Using this estimate in (\ref{eq9}) with $\epsilon\in\left(0,\frac{c_3}{r}\right)$ and because of hypothesis $H(\xi)$ and since $r>2$, we see that $\psi$ is coercive. Moreover, using the Sobolev embedding theorem and the trace theorem, we see that $\psi$ is sequentially weakly lower semicontinuous. So, we can find $\bar{u}\in H^1(\Omega)$ such that
	\begin{equation}\label{eq10}
		\psi(\bar{u})=\inf[\psi(u):u\in H^1(\Omega)].
	\end{equation}
	
	Let $t>0$ and $\hat{u}_1(\beta)\in \mbox{int}\, C_+$ is the $L^2$-normalized principal eigenfunction of (\ref{eq2}). We have $$\psi(t\hat{u}_1(\beta))=\frac{t^2}{2}(\hat{\lambda}_1(\beta)-c_2)+\frac{t^rc_3}{r}||\hat{u}_1(\beta)||^r_r\ (\mbox{see (\ref{eq3})}).$$
	
	Since $c_2>\hat{\lambda}_1(\beta)$ and $r>2$, for $t\in(0,1)$ small we have
	\begin{eqnarray*}
		&&\psi(t\hat{u}_1(\beta))<0,\\
		&\Rightarrow&\psi(\bar{u})<0=\psi(0)\ (\mbox{see (\ref{eq10})}),\ \mbox{hence}\ \bar{u}\neq 0.
	\end{eqnarray*}
	
	From (\ref{eq10}), we have
	\begin{eqnarray}\label{eq11}
		&&\psi'(\bar{u})=0,\nonumber\\
		&\Rightarrow&\left\langle A(\bar{u},h)\right\rangle+\int_{\Omega}(\xi(z)+\mu)\bar{u}hdz+\int_{\partial\Omega}\beta(z)\bar{u}hd\sigma_0=(c_2+\mu)\int_{\Omega}\bar{u}^+hdz-\nonumber\\
		&&\hspace{6cm}-c_3\int_{\Omega}(\bar{u}^+)^{r-1}hdz\ \mbox{for all}\ h\in H^1(\Omega).
	\end{eqnarray}
	
	In (\ref{eq11}), we choose $h=-\bar{u}^-\in H^1(\Omega)$. Then
	\begin{eqnarray*}
		&&\sigma(\bar{u}^-)+\mu||\bar{u}^-||^2_2=0,\\
		&\Rightarrow&c_0||\bar{u}^-||^2\leq 0\ \mbox{see (\ref{eq4})},\\
		&\Rightarrow&\bar{u}\geq 0,\ \bar{u}\neq 0.
	\end{eqnarray*}
	
	Then relation (\ref{eq11}) becomes, for all $h\in H^1(\Omega)$,
	\begin{eqnarray*}
		&&\left\langle A(\bar{u}),h\right\rangle+\int_{\Omega}\xi(z)\bar{u}hdz+\int_{\partial\Omega}\beta(z)\bar{u}hd\sigma_0=
\int_{\Omega}(c_2\bar{u}-c_3\bar{u}^{r-1})hdz\,,\\
		&\Rightarrow&\bar{u}\ \mbox{is a positive solution of (\ref{eq8})}.
	\end{eqnarray*}
	
	From the regularity result of Wang \cite[Section 5]{25}  and the strong maximum principle, we have $\bar{u}\in \mbox{int}\, C_+$.
	
	Next we show the uniqueness of this positive solutions. So, let $\bar{y}$ be another positive solution of (\ref{eq8}). As above, we show that $\bar{y}\in \mbox{int}\, C_+$. From Lemma 3.3 of Filippakis, Kristaly and Papageorgiou \cite{6}, we know that there exists $t>0$ such that
	$$t\bar{y}\leq\bar{u}.$$
	
	Consider the biggest such real number and assume that $t\in(0,1)$ (if $t=1$, then $\bar{u}=\bar{y}$). Note that there exists $\bar{\eta}>0$ such that $x\longmapsto (c_2+\bar{\eta})x-c_3x^{r-1}$ is nondecreasing on $[0,\rho]$. We have
	\begin{eqnarray*}
		&&-\Delta(t\bar{y})+(\xi(z)+\bar{\eta})(t\bar{y})\\
		&&=t[-\Delta\bar{y}+(\xi(z)+\bar{\eta})\bar{y}]\\
		&&=t[(c_2+\bar{\eta})\bar{y}-c_3\bar{y}^{r-1}]\ (\mbox{since}\ \bar{y}\in \mbox{int}\,C_+\ \mbox{is a solution of (\ref{eq8})})\\
		&&<(c_2+\bar{\eta})(t\bar{y})-c_3(t\bar{y})^{r-1}\ (\mbox{since}\ t\in(0,1)\ \mbox{and}\ 2<r)\\
		&&\leq(c_2+\bar{\eta})\bar{u}-c_3\bar{u}^{r-1}\ (\mbox{since}\ t\bar{y}\leq\bar{u})\\
		&&=-\Delta\bar{u}+(\xi(z)+\bar{\eta})\bar{u}\ (\mbox{since}\ \bar{u}\in \mbox{int}\,C_+\ \mbox{is a solution of (\ref{eq8})}),\\
		&\Rightarrow&-\Delta(\bar{u}-t\bar{y})+(\xi(z)+\bar{\eta})(\bar{u}-t\bar{y})\geq 0,\\
		 &\Rightarrow&\Delta(\bar{u}-t\bar{y})\leq(||\xi^+||_{\infty}+\bar{\eta})(\bar{u}-t\bar{y})\ (\mbox{see hypothesis}\ H(\xi)),\\
		&\Rightarrow&\bar{u}-t\bar{y}\in \mbox{int}\,C_+\ (\mbox{by the strong maximum principle}).
	\end{eqnarray*}
	
	This contradicts the maximality of $t>0$. Hence $t>1$ and so
	$$\bar{y}\leq\bar{u}.$$
	
	Interchanging the roles of $\bar{u}$ and $\bar{y}$ in the above argument, we can also have
	\begin{eqnarray*}
		&&\bar{u}\leq\bar{y},\\
		&\Rightarrow&\bar{u}=\bar{y}.
	\end{eqnarray*}
	This proves the uniqueness of the positive solutions $\bar{u}\in \mbox{int}\,C_+$ of (\ref{eq8}). Since (\ref{eq8}) is odd, it follows that $\bar{v}=-\bar{u}\in-\mbox{int}\,C_+$ is the unique negative solution of problem (\ref{eq8}).
\end{proof}

\begin{remark}
	To prove the uniqueness of the positive solution $\bar{u}\in \mbox{int}\,C_+$ of (\ref{eq8}), one can alternatively use Picone's identity (see, for example, Gasinski and Papageorgiou \cite[p. 783]{7}).
\end{remark}

Now let
\begin{eqnarray*}
	&&S_+=\{u\in H^1(\Omega):u\in[0,w_+],u\neq 0,u\ \mbox{is a solution of (\ref{eq1})}\}\\
	&&S_-=\{v\in H^1(\Omega):v\in[w_-,0],v\neq 0,v\ \mbox{is a solution of (\ref{eq1})}\}.
\end{eqnarray*}

Eventually we will establish the nonemptiness of the sets $S_+$ and $S_-$. For the moment, we establish some a priori bounds for the elements of $S_+$ and $S_-$.
\begin{prop}\label{prop6}
	If hypotheses $H(\xi),H(\beta)$ and $H_1$ hold, then $\bar{u}\leq u$ for all $u\in S_+$ and $v\leq\bar{v}$ for all $v\in S_-$.
\end{prop}
\begin{proof}
	Let $u\in S_+$ and consider the following Carath\'eodory function
	\begin{eqnarray}\label{eq12}
		g_+(z,x)=\left\{\begin{array}{ll}
			0&\mbox{if}\ x<0\\
			(c_2+\mu)x-c_3x^{r-1}&\mbox{if}\ 0\leq x\leq u(z)\\
			(c_2+\mu)u(z)-c_3u(z)^{r-1}&\mbox{if}\ u(z)<x.
		\end{array}\right.
	\end{eqnarray}
	
	Let $G_+(z,x)=\int^{x}_{0}g_+(z,s)ds$ and consider the $C^1$-functional $\psi_+:H^1(\Omega)\rightarrow\RR$ defined by
	$$\psi_+(u)=\frac{1}{2}\sigma(u)+\frac{\mu}{2}||u||^2_2-\int_{\Omega}G_+(z,u)dz\ \mbox{for all}\ u\in H^1(\Omega).$$
	
	From (\ref{eq4}) and (\ref{eq11}), we see that $\psi_+$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find $\bar{u}_*\in H^1(\Omega)$ such that
	\begin{equation}\label{eq13}
		\psi_+(\bar{u}_*)=\inf[\psi_+(u):u\in H^1(\Omega)].
	\end{equation}
	
	As in the proof of Proposition \ref{prop5} and because $2<r$ and $c_2>\hat{\lambda}_1(\beta)$, we have
	$$\psi_+(\bar{u}_*)<0=\psi_+(0),\ \mbox{hence}\ \bar{u}_*\neq 0.$$
	
	From (\ref{eq13}) we have
	\begin{eqnarray}\label{eq14}
		&&\psi'_{+}(\bar{u}_*)=0,\nonumber\\
		&\Rightarrow&\left\langle A(\bar{u}_*),h\right\rangle+\int_{\Omega}(\xi(z)+\mu)\bar{u}_*hdz+\int_{\partial\Omega}\beta(z)\bar{u}_*hd\sigma_0=\int_{\Omega}g_+(z,\bar{u}_*)hdz\\
		&&\hspace{8cm}\mbox{for all}\ h\in H^1(\Omega).\nonumber
	\end{eqnarray}
	
	In (\ref{eq14}) first we choose $h=-\bar{u}^{-}_{*}\in H^1(\Omega)$. Then
	\begin{eqnarray*}
		&&\sigma(\bar{u}^{-}_{*})+\mu||\bar{u}^{-}_{*}||^2_2=0,\\
		&\Rightarrow&c_0||\bar{u}^{-}_{*}||^2\leq 0\ (\mbox{see (\ref{eq4})}),\\
		&\Rightarrow&\bar{u}_*\geq 0,\bar{u}_*\neq 0.
	\end{eqnarray*}
	
	Next, in (\ref{eq14}) we choose $h=(\bar{u}_*-u)^+\in H^1(\Omega)$. We have
	\begin{eqnarray*}
		&&\left\langle A(\bar{u}_*),(\bar{u}_*-u)^+\right\rangle+\int_{\Omega}(\xi(z)+\mu)\bar{u}_*(\bar{u}_*-u)^+dz+\int_{\partial\Omega}\beta(z)\bar{u}_*(\bar{u}_*-u)^+d\sigma_0\\
		&=&\int_{\Omega}[(c_2+\mu)u-c_3u^{r-1}](\bar{u}_*-u)^+dz\ (\mbox{see (\ref{eq11})})\\
		&\leq&\int_{\Omega}[f(z,u)+\mu u](\bar{u}_*-u)^+dz\ (\mbox{see (\ref{eq7})})\\
		&=&\left\langle A(u),(\bar{u}_*-u)^+\right\rangle+\int_{\Omega}(\xi(z)+\mu)u(\bar{u}_*-u)^+dz+\int_{\partial\Omega}\beta(z)u(\bar{u}_*-u)^+d\sigma\\
		&&\hspace{7cm}(\mbox{since}\ u\in S_+)\\
		\Rightarrow&&\sigma((\bar{u}_*-u)^+)+\mu||(\bar{u}_*-u)^+||^2_2\leq 0,\\
		\Rightarrow&&c_0||(\bar{u}_*-u)^+||^2\leq 0\ (\mbox{see (\ref{eq4})}),\\
		\Rightarrow&&\bar{u}_*\leq u.
	\end{eqnarray*}
	
	So, we have proved that
	\begin{equation}\label{eq15}
		\bar{u}_*\in[0,u]=\{y\in H^1(\Omega):0\leq y(z)\leq u(z)\ \mbox{for almost all}\ z\in\Omega\},\ \bar{u}_*\neq 0.
	\end{equation}
	
	Using (\ref{eq12}) and (\ref{eq15}), equation (\ref{eq14}) becomes
	\begin{eqnarray*}
		&&\left\langle A(\bar{u}_*),h\right\rangle+\int_{\Omega}\xi(z)\bar{u}_*hdz+\int_{\partial\Omega}\beta(z)\bar{u}_*hd\sigma_0=\int_{\Omega}(c_2\bar{u}_*-c_3\bar{u}_*^{r-1})hdz\\
		&&\hspace{7cm}\mbox{for all}\ h\in H^1(\Omega),\\
		&\Rightarrow&\bar{u}_*\ \mbox{is a positive solution of problem (\ref{eq8})},\\
		&\Rightarrow&\bar{u}_*=\bar{u}\in \mbox{int}\,C_+\ (\mbox{see Proposition \ref{prop5}}),\\
		&\Rightarrow&\bar{u}\leq u\ \mbox{for all}\ u\in S_+.
	\end{eqnarray*}
	
	For the {\it a priori} bound on the negative solutions, given $v\in S$, we consider the Carath\'eodory function
	\begin{eqnarray*}
		g_-(z,x)=\left\{\begin{array}{ll}
		c_2v(z)-c_3|v(z)|^{r-2}v(z)&\mbox{if}\ x<v(z)\\
			c_2x-c_3|x|^{r-2}x&\mbox{if}\ v(z)\leq x\leq 0\\
			0&\mbox{if}\ 0<x.
		\end{array}\right.
	\end{eqnarray*}
	
	We set $G_-(z,x)=\int^{x}_{0}g_-(z,s)ds$ and consider the $C^1$-functional $\psi_-:H^1(\Omega)\rightarrow\RR$ defined by
	$$\psi_-(u)=\frac{1}{2}\sigma(u)+\frac{\mu}{2}||u||^2_2-\int_{\Omega}G_-(z,u)dz\ \mbox{for all}\ u\in H^1(\Omega).$$
	
	Working as above, this time with the functional $\psi_-$, we show that
	$$v\leq\bar{v}\ \mbox{for all}\ v\in S_-.$$ The proof is now complete.
\end{proof}

\begin{remark}
	These a priori bounds will be useful in producing extremal constant sign solutions which will lead to nodal solutions.
\end{remark}
\begin{prop}\label{prop7}
	If hypotheses $H(\xi),H(\beta)$ and $H_1$ hold, then problem (\ref{eq1}) has at least two solutions of constant sign
	$$u_0\in \mbox{int}\,C_+\ \mbox{and}\ v_0\in-\mbox{int}\,C_+.$$
\end{prop}
\begin{proof}
	First we produce the positive solution.
	
	So, let $\hat{f}_{+}(z,x)$ be the Carath\'eodory function defined by
	\begin{eqnarray}\label{eq16}
		\hat{f}_+(z,x)=\left\{\begin{array}{ll}
			0&\mbox{if}\ x<0\\
			f(z,x)+\mu x&\mbox{if}\ 0\leq x\leq w_+(z)\\
			f(z,w_+(z))+\mu w_+(z)&\mbox{if}\ w_+(z)<x.
		\end{array}\right.
	\end{eqnarray}
	
	We set $\hat{F}_+(z,x)=\int^{x}_{0}\hat{f}_+(z,s)ds$ and consider the $C^1$-functional $\hat{\varphi}_+:H^1(\Omega)\rightarrow\RR$ defined by $$\hat{\varphi}_+(u)=\frac{1}{2}\sigma(u)+\frac{\mu}{2}||u|^2_2-\int_{\Omega}\hat{F}_+(z,u)dz\ \mbox{for all}\ u\in H^1(\Omega).$$
	
	From (\ref{eq4}) and (\ref{eq16}), it is clear that $\hat{\varphi}_+$ is coercive. Moreover, using the Sobolev embedding theorem and the trace theorem, we see that $\hat{\varphi}_+$ is sequentially weakly lower semicontinuous. So, by the Weierstrass theorem, we can find $u_0\in H^1(\Omega)$ such that
	\begin{equation}\label{eq17}
		\hat{\varphi}_+(u_0)=\inf[\hat{\varphi}_+(u):u\in H^1(\Omega)].
	\end{equation}
	
	As before (see the proof of Proposition \ref{prop5}) for $t\in(0,1)$ small such that $t\bar{u}_1(\beta)(z)\leq\delta$, since $q<2<r$, we have
	\begin{eqnarray*}
		&&\hat{\varphi}_+(t\hat{u}_1(\beta))<0,\\
		&\Rightarrow&\hat{\varphi}_+(u_0)<0=\hat{\varphi}_+(0)\ (\mbox{see (\ref{eq17})}),\ \mbox{hence}\ u_0\neq 0.
	\end{eqnarray*}
	
	From (\ref{eq17}) we have
	\begin{eqnarray}\label{eq18}
		&&\hat{\varphi}'_+(u_0)=0,\nonumber\\
		&\Rightarrow&\left\langle A(u_0),h\right\rangle+\int_{\Omega}(\xi(z)+\mu)u_0hdz+\int_{\partial\Omega}\beta(z)u_0hd\sigma_0=\int_{\Omega}\hat{f}_+(z,u_0)hdz\\
		&&\hspace{8cm}\mbox{for all}\ h\in H^1(\Omega).\nonumber
	\end{eqnarray}
	
	In (\ref{eq18}), first we choose $h=-u_0^-\in H^1(\Omega)$. Then
	\begin{eqnarray*}
		&&\sigma(u_0^-)+\mu||u_0^-||^2_2=0,\\
		&\Rightarrow&c_0||u_0^-||^2\leq 0\ (\mbox{see (\ref{eq4})}),\ \mbox{hence}\ u_0\geq 0,\ u_0\neq 0.
	\end{eqnarray*}
	
	Next in (\ref{eq17}) we choose $h=(u_0-w_+)^+\in H^1(\Omega)$. Then
	\begin{eqnarray*}
		&&\left\langle A(u_0),(u_0-w_+)^+\right\rangle+\int_{\Omega}(\xi(z)+\mu)u_0(u_0-w_+)^+dz\\
&+&\int_{\partial\Omega}\beta(z)u_0(u_0-w_+)^+d\sigma_0\\
		&=&\int_{\Omega}[f(z,w_+)+\mu w_+](u_0-w_+)^+dz\ (\mbox{see (\ref{eq16})}),\\
		&\leq&\left\langle A(w_+),(u_0-w_+)^+\right\rangle+\int_{\Omega}(\xi(z)+\mu)w_+(u_0-w_+)^+dz\\
&+&\int_{\partial\Omega}\beta(z)u_0(u_0-w_+)^+d\sigma_0
		\hspace{1cm}(\mbox{see hypotheses}\ H_1(i))\\
		\Rightarrow&&\sigma((u_0-w_+)^+)+\mu||(u_0-w_+)^+||^2_2\leq 0,\\
		\Rightarrow&&c_0||(u_0-w_+)^+||^2\leq 0\ (\mbox{see (\ref{eq4})}),\\
		\Rightarrow&&u_0\leq w_+.
	\end{eqnarray*}
	
	So, we have proved that
	\begin{equation}\label{eq19}
		u_0\in[0,w_+]=\{u\in H^1(\Omega):0\leq u(z)\leq w_+(z)\ \mbox{for almost all}\ z\in\Omega\}.
	\end{equation}
	
	Because of (\ref{eq16}) and (\ref{eq19}), equation (\ref{eq18}) becomes
	\begin{eqnarray*}
		&&\left\langle A(u_0),h\right\rangle+\int_{\Omega}\xi(z)u_0hdz+\int_{\partial\Omega}\beta(z)u_0hd\sigma_0=\int_{\Omega}f(z,u_0)hdz\\
		&&\hspace{8cm}\mbox{for all}\ h\in H^1(\Omega),\\
		&\Rightarrow&u_0\ \mbox{is a positive solution of problem (\ref{eq1})}\\
		&&\hspace{6cm}(\mbox{see Papageorgiou and R\u{a}dulescu \cite{19}}).
	\end{eqnarray*}
	
	We set
	$$k_0(z)=\left\{\begin{array}{ll}
	\displaystyle	\frac{f(z,u_0(z))}{u_0(z)}&\mbox{if}\ u_0(z)\neq 0\\
		0&\mbox{if}\ u_0(z)=0.
	\end{array}\right.$$
	
	From (\ref{eq19}), Proposition \ref{prop6} and hypothesis $H_1(i)$, we see that $k_0\in L^{\infty}(\Omega)$. We have
	\begin{equation}\label{eq20}
		\left\{\begin{array}{ll}
			-\Delta u_0(z)=(k_0(z)-\xi(z))u_0(z)&\mbox{in}\ \Omega,\\
		\displaystyle	\frac{\partial u_0}{\partial n}+\beta(z)u_0=0&\mbox{on}\ \partial\Omega.
		\end{array}\right\}
	\end{equation}
		
		Note that $k_0-\xi\in L^s(\Omega)$ (see hypothesis $H(\xi)$). Then Lemma 5.1 in Wang \cite{25} implies that $u_0\in L^{\infty}(\Omega)$ and so $\Delta u_0\in L^s(\Omega)$. Then Lemma 5.2. of Wang \cite{25} implies that $u_0\in W^{2,s}(\Omega)$. Since $s>N$ (see hypothesis $H(\xi)$), from the Sobolev embedding theorem, we have
		\begin{eqnarray*}
			&&W^{2,s}(\Omega)\hookrightarrow C^{1+\alpha}(\overline{\Omega})\ \mbox{with}\ \alpha=1-\frac{N}{s}>0,\\
			&\Rightarrow&u_0\in C_+\backslash\{0\}.
		\end{eqnarray*}
	
	Hypotheses $H_1(ii),(iii)$ imply that there exists $\eta_+>0$ such that
	\begin{equation}\label{eq21}
		f(z,x)+\eta_+x\geq 0\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\in[0,\rho].
	\end{equation}
	
	Then we have
	\begin{eqnarray*}
		&&-\Delta u_0(z)+(\xi(z)+\eta_+)u_0(z)\geq 0\ \mbox{for almost all}\ z\in\Omega\ (\mbox{see (\ref{eq21})}),\\
		&\Rightarrow&\Delta u_0(z)\leq(||\xi^+||_{\infty}+\eta_+)u_0(z)\ \mbox{for almost all}\ z\in\Omega\ (\mbox{see hypothesis}\ H(\xi)),\\
		&\Rightarrow&u_0\in \mbox{int}\,C_+\ (\mbox{by the strong maximum principle}).
	\end{eqnarray*}
	
	To produce the negative solution, we consider the Carath\'eodory function
	\begin{eqnarray}\label{eq22}
		\hat{f}_-(z,x)=\left\{\begin{array}{ll}
			f(z,w_-(z))+\mu w_-(z)&\mbox{if}\ x<w_-(z)\\
			f(z,x)+\mu x&\mbox{if}\ w_-(z)\leq x\leq 0\\
			0&\mbox{if}\ 0<x.
		\end{array}\right.
	\end{eqnarray}
	
	We set $\hat{F}_-(z,x)=\int^{x}_{0}\hat{f}_-(z,s)ds$ and consider the $C^1$-functional $\hat{\varphi}_-:H^1(\Omega)\rightarrow\RR$ defined by
	 $$\hat{\varphi}_-(u)=\frac{1}{2}\sigma(u)+\frac{\mu}{2}||u||^2_2-\int_{\Omega}\hat{F}_-(z,u)dz\ \mbox{for all}\ u\in H^1(\Omega).$$
	
	Reasoning as in the first part of the proof, using this time (\ref{eq22}) and the functional $\hat{\varphi}_-$, we produce a negative solution $v_0\in-\mbox{int}\,C_+$.
	\end{proof}
\begin{remark}
	So, from this proposition and its proof, we have
	$$\emptyset\neq S_+\subseteq[0,w_+]\cap \mbox{int}\,C_+\ \mbox{and}\ \emptyset \neq S_-\subseteq[w_-,0]\cap(-\mbox{int}\,C_+).$$

As in Filippakis and Papageorgiou \cite{fipa} (see also Motreanu, Motreanu and Papageorgiou \cite[p. 421]{mmpa}), we can show that the set $S_+$ is downward directed (that is, if $u_1,\, u_2\in S_+$, then we can find $u\in S_+$ such that $u\leq u_1$, $u\leq u_2$) and the set $S_-$ is upward directed (that is, if $v_1,\, v_2\in S_-$, then we can find $v\in S_-$ such that $v_1\leq v$, $v_2\leq v$). Moreover, since both sets are bounded, the infimum of $S_+$ and the supremum of $S_-$ can be taken over countable sets (see Dunford and Schwartz \cite[p. 336]{dusc} and Hu and Papageorgiou \cite[p. 178]{9}).
\end{remark}

Next we will produce extremal constant sign solutions, that is, the smallest positive solution $u_*\in \mbox{int}\,C_+$ and the biggest negative solution $v_*\in-\mbox{int}\,C_+$. Subsequently these extremal constant sign solutions will lead to the existence of a nodal solution.
\begin{prop}\label{prop8}
	If hypotheses $H(\xi),H(\beta)$ and $H_1$ hold, then problem (\ref{eq1}) has a smallest positive solution $u_*\in \mbox{int}\,C_+$ and a biggest negative solution $v_*\in-\mbox{int}\,C_+$.
\end{prop}
\begin{proof}
	From Dunford and Schwartz \cite[p. 336]{dusc} and Hu and Papageorgiou \cite[p. 178]{9}, we know that we can find $\{u_n\}_{n\geq 1}\subseteq S_+$ such that
	$$\inf S_+=\inf\limits_{n\geq 1}u_n.$$
	
	We have for all $n\geq 1$ and for all $h\in H^1(\Omega)$
	\begin{equation}\label{eq23}
		\left\langle A(u_n),h\right\rangle+\int_{\Omega}\xi(z)u_n hdz+\int_{\partial\Omega}\beta(z)u_nhd\sigma=\int_{\Omega}f(z,u_n)hdz\,.
	\end{equation}
	
	In (\ref{eq23}) we choose $h=u_n\in H^1(\Omega)$. Then
	\begin{eqnarray*}
		&&\sigma(u_n)\leq c_4\ \mbox{for all}\ n\geq 1\ \mbox{and some}\ c_4>0\\
		&&\hspace{2cm}(\mbox{recall}\ 0\leq u_n\leq w_+\ \mbox{for all}\ n\geq 1\ \mbox{and see hypothesis}\ H_1(ii))\\
		&\Rightarrow&\sigma(u_n)+\mu||u_n||^2_2\leq c_5\ \mbox{for some}\ c_5>0,\ \mbox{all}\ n\geq 1,\\
		&\Rightarrow&c_0||u_n||^2\leq c_5\ \mbox{for all}\ n\geq 1\ (\mbox{see (\ref{eq4})}),\\
		&\Rightarrow&\{u_n\}_{n\geq 1}\subseteq H^1(\Omega)\ \mbox{is bounded.}
	\end{eqnarray*}
	
	So, we may assume that
	\begin{equation}\label{eq24}
		u_n\stackrel{w}{\rightarrow}u_*\ \mbox{in}\ H^1(\Omega)\ \mbox{and}\ u_n\rightarrow u_*\ \mbox{in}\ L^{\frac{2s}{s-1}}(\Omega)\ \mbox{and in}\ L^2(\partial\Omega).
	\end{equation}
	
	Passing to the limit as $n\rightarrow\infty$ in (\ref{eq23}) and using (\ref{eq24}), we obtain
	\begin{equation}\label{eq25}
		\left\langle A(u_*),h\right\rangle+\int_{\Omega}\xi(z)u_*hdz+\int_{\partial\Omega}\beta(z)u_*hd\sigma_0=\int_{\Omega}f(z,u_*)hdz\ \mbox{for all}\ h\in H^1(\Omega).
	\end{equation}
	
	From Proposition \ref{prop6}, we know that
	\begin{eqnarray}\label{eq26}
		&&\bar{u}\leq u_n\ \mbox{for all}\ n\geq 1,\nonumber\\
		&\Rightarrow&\bar{u}\leq u_*.
	\end{eqnarray}
	
	From (\ref{eq25}) and (\ref{eq26}), it follows that
	\begin{eqnarray*}
		&&u_*\in S_+\ \mbox{and}\ u_*=\inf S_+\ (\mbox{see Papageorgiou and R\u{a}dulescu \cite{19}}),\\
		&\Rightarrow&u_*\in \mbox{int}\,C_+\ \mbox{is the smallest positive solution of problem (\ref{eq1})}.
	\end{eqnarray*}
	
	Similarly we produce
	$$v_*\in S_-\ \mbox{with}\ v_*=\sup S_-,$$
	the biggest negative solution of problem (\ref{eq1}).
\end{proof}

Using these extremal constant solutions, we can produce nodal (sign changing) solutions. To this end, we consider the following truncation-perturbation of the reaction $f(z,\cdot)$:
\begin{eqnarray}\label{eq27}
	k(z,x)=\left\{\begin{array}{ll}
		f(z,w_-(z))+\mu w_-(z)&\mbox{if}\ x< w_-(z)\\
		f(z,x)+\mu x&\mbox{if}\ w_-(z)\leq x\leq w_+(z)\\
		f(z,w_+(z))+\mu w_+(z)&\mbox{if}\ w_+(z)<x.
	\end{array}\right.
\end{eqnarray}

This is a Carath\'eodory function. Let $K(z,x)=\int^{x}_{0}k(z,s)ds$ and consider the $C^1$-functional $\vartheta:H^1(\Omega)\rightarrow\RR$ defined by
$$\vartheta(u)=\frac{1}{2}\sigma(u)+\frac{\mu}{2}||u||^2_2-\int_{\Omega}K(z,u)dz\ \mbox{for all}\ u\in H^1(\Omega).$$

Next we compute the critical groups of $\vartheta$ at the origin. Our result extends that of Moroz \cite{13}, who did a similar computation but under stronger hypotheses on the function $f(z,x)$ and for the space $H^{1}_{0}(\Omega)$. In the space $H^{1}_{0}(\Omega)$ the Poincar\'e inequality simplifies the argument.
\begin{prop}\label{prop9}
	If hypotheses $H(\xi),H(\beta)$ and $H_1$ hold and the critical set $K_{\vartheta}$ is finite, then $C_k(\vartheta,0)=0$ for all $k\geq 0$.
\end{prop}
\begin{proof}
	From hypothesis $H_1(iii)$ and (\ref{eq27}), we have
	\begin{equation}\label{eq28}
		K(z,x)\geq\frac{c_1}{q}|x|^q-c_6|x|^r\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\in\RR,\ \mbox{with}\ c_6>0.
	\end{equation}
	
	Let $u\in H^1(\Omega)$ and $t\in(0,1)$. Then
	\begin{eqnarray}\label{eq29}
		 \vartheta(tu)&=&\frac{t^2}{2}\sigma(u)+\frac{\mu t^2}{2}||u||^2_2-\int_{\Omega}K(z,tu)dz\nonumber\\
		&\leq&\frac{t^2}{2}\sigma(u)+\frac{\mu t^2}{2}||u||^2_2-\frac{c_1 t^q}{q}||u||^q_q+c_6 t^r||u||^r_r\ (\mbox{see (\ref{eq28})}).
	\end{eqnarray}
	
	Since $1<q<2<r$, from (\ref{eq29}) it follows that we can find $t^*=t^*(u)\in(0,1)$ small such that
	\begin{equation}\label{eq30}
		\vartheta(tu)<0\ \mbox{for all}\ t\in (0,t^*).
	\end{equation}
	
	Let $u\in H^1(\Omega)$ with $0<||u||\leq 1$ and $\vartheta(u)=0$. Then
	\begin{eqnarray}\label{eq31}
		\frac{d}{dt}\vartheta(tu)|_{t=1}&=&\left\langle \vartheta'(u),u\right\rangle\ (\mbox{by the chain rule})\nonumber\\
		&=&\left\langle A(u),u\right\rangle+\int_{\Omega}(\xi(z)+\mu)u^2 dz+\int_{\partial\Omega}\beta(z)u^2d\sigma_0-\int_{\Omega}k(z,u)udz\nonumber\\
		&=&\left(1-\frac{q}{2}\right)\sigma(u)+\int_{\Omega}[q K(z,u)-k(z,u)u]dz+\left(1-\frac{q}{2}\right)\mu||u||^2_2\\
		&&\hspace{7.5cm}(\mbox{since}\ \vartheta(u)=0)\nonumber
	\end{eqnarray}
	
	Hypotheses $H_1(ii),(iii)$ imply that
	\begin{equation}\label{eq32}
		qK(z,x)-k(z,x)x\geq-c_7|x|^r\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\in\RR.
	\end{equation}
	
	Using (\ref{eq32}) in (\ref{eq31}), we obtain
	\begin{eqnarray}\label{eq33}
		 \frac{d}{dt}\vartheta(u)|_{t=1}&\geq&\left(1-\frac{q}{2}\right)\left(\sigma(u)+\mu||u||^2_2\right)-c_8||u||^r\ \mbox{for some}\ c_8>0\nonumber\\
		&&\geq c_9||u||^2-c_8||u||^r\ \mbox{with}\ c_9=\left(1-\frac{q}{2}\right)c_0>0\  (\mbox{recall}\ q<2).
	\end{eqnarray}
	
	Since $2<r$, from (\ref{eq33}) we see that we can find $\rho\in(0,1)$ small such that
	\begin{equation}\label{eq34}
		\frac{d}{dt}\vartheta(tu)|_{t=1}>0\ \mbox{for all}\ u\in H^1(\Omega)\ \mbox{with}\ 0<||u||\leq\rho,\ \vartheta(u)=0.
	\end{equation}
	
	We fix $u\in H^1(\Omega)$ with $0<||u||\leq\rho$ and $\varphi(u)=0$. We claim that
	\begin{equation}\label{eq35}
		\vartheta(tu)\leq 0\ \mbox{for all}\ t\in[0,1].
	\end{equation}
	
	We argue by contradiction. So, suppose that we can find $t_0\in(0,1)$ such that $\vartheta(t_0u)>0$. Since $\vartheta(u)=0$ and $\vartheta(\cdot)$ is continuous, from Bolzano's theorem we have
	$$t_*=\min\{t\in[t_0,1]:\vartheta(tu)=0\}>t_0>0.$$
	
	Then
	\begin{equation}\label{eq36}
		\vartheta(tu)>0\ \mbox{for all}\ t\in\left[t_0,t_*\right).
	\end{equation}
	
	We set $v=t_*u$. Then $0<||v||\leq||u||\leq\rho$ and $\vartheta(v)=0$. Then from (\ref{eq34}) we have
	\begin{equation}\label{eq37}
		\frac{d}{dt}\vartheta(tv)|_{t=1}>0.
	\end{equation}
	
	From (\ref{eq36}) we have
	\begin{eqnarray}\label{eq38}
		&&\vartheta(v)=\vartheta(t_*u)=0<\vartheta(tu)\ \mbox{for all}\ t\in\left[t_0,t_*\right),\nonumber\\
		 &\Rightarrow&\frac{d}{dt}\vartheta(tv)|_{t=1}=t_*\frac{d}{dt}\vartheta(tu)|_{t=t_*}=t_*\lim\limits_{t\rightarrow t^-_*}\frac{\vartheta(tu)}{t-t_*}\leq 0.
	\end{eqnarray}
	
	Comparing (\ref{eq37}) and (\ref{eq38}), we reach a contradiction. This proves (\ref{eq35}) for all $u\in H^1(\Omega)$ with $0<||u||\leq\rho$ and $\vartheta(u)=0$.
	
	Also, we have
	\begin{equation}\label{eq39}
		\vartheta(tu)<0\ \mbox{for all}\ t\in(0,1)\ \mbox{and all}\ u\in H^1(\Omega),\ 0<||u||\leq\rho,\ \vartheta(u)<0.
	\end{equation}
	
	Indeed, note that due to the continuity of $\vartheta$, we can find $s\in(0,1)$ such that
	$$\vartheta(tu)<0\ \mbox{for all}\ t\in\left(1-s,1\right].$$
	
	Suppose that there exists $t_0\in\left(0,1-s\right]$ such that $\vartheta(t_0u)=0$ and $\vartheta(tu)<0$ for all $t\in\left(t_0,1\right]$. Let $u_0=t_0u$. Then $0<||u_0||\leq\rho$ and $\vartheta(u_0)=0$. So, from (\ref{eq34}) we have
	\begin{equation}\label{eq40}
		\frac{d}{dt}\vartheta(tu_0)|_{t=1}>0
	\end{equation}
	
	On the other hand, we have
	\begin{eqnarray*}
		&&\vartheta(tu)=\vartheta(tu)-\vartheta(t_0u)<0\ \mbox{for all}\ t\in\left(t_0,1\right],\\
		&\Rightarrow&\frac{d}{dt}\vartheta(tu)|_{t=t_0}=\frac{d}{dt}\vartheta(tu_0)|_{t=1}\leq 0,
	\end{eqnarray*}
	which contradicts (\ref{eq40}). Therefore (\ref{eq39}) holds.
	
	We can always choose $\rho\in(0,1)$ small such that $K_{\vartheta}\cap\overline{B}_{\rho}=\{0\}$. Let $h:[0,1]\times(\vartheta^0\cap \overline{B}_{\rho})\rightarrow\vartheta^0\cap \overline{B}_{\rho}$ be the deformation defined by
	$$h(t,u)=(1-t)u\ \mbox{for all}\ t\in[0,1],\ \mbox{all}\ u\in\vartheta^0\cap \overline{B}_{\rho}.$$
	
	From (\ref{eq35}) and (\ref{eq39}), we see that this deformation is well-defined. Moreover, with this deformation we show that
	\begin{equation}\label{eq41}
		\vartheta^0\cap \overline{B}_{\rho}\ \mbox{is contractible in itself}.
	\end{equation}
	
	Let $u\in \overline{B}_{\rho}$ with $\vartheta(u)>0$. We claim that there exists unique $t(u)\in(0,1)$ such that
	\begin{equation}\label{eq42}
		\vartheta(t(u)u)=0.
	\end{equation}
	
	From (\ref{eq30}) and using Bolzano's theorem, we see that such a $t(u)\in(0,1)$ exists. So, we have to show the uniqueness of $t(u)$. Arguing by contradiction suppose we can find $0<t_1=t(u)_1<t_2=t(u)_2<1$ such that $\vartheta(t_1u)=\vartheta(t_2u)=0$.
	
	From (\ref{eq35}), we have
	\begin{eqnarray*}
		&&\vartheta(tt_2u)\leq 0\ \mbox{for all}\ t\in[0,1],\\
		&\Rightarrow&\frac{t_1}{t_2}\in(0,1)\ \mbox{is a maximizer of}\ t\longmapsto\vartheta(tt_2u),\\
		&\Rightarrow&\frac{t_1}{t_2}\ \frac{d}{dt}\vartheta(tt_2,u)|_{t=\frac{t_1}{t_2}}=\frac{d}{dt}\vartheta(tt_1u)|_{t=1}=0,
	\end{eqnarray*}
	which contradicts (\ref{eq34}). This proves the uniqueness of $t(u)\in(0,1)$ in (\ref{eq42}). We have
	$$\vartheta(tu)<0\ \mbox{for all}\ t\in(0,t(u))\ \mbox{and}\ \vartheta(tu)>0\ \mbox{for all}\ t\in\left(t(u),1\right].$$
	
	We consider the function $\eta_1:\overline{B}_{\rho}\backslash\{0\}\rightarrow\left(0,1\right]$ defined by
	$$\eta_1(u)=\left\{\begin{array}{ll}
		1&\mbox{if}\ u\in \overline{B}_{\rho}\backslash\{0\},\vartheta(u)\leq 0\\
		t(u)&\mbox{if}\ u\in \overline{B}_{\rho}\backslash\{0\},\vartheta(u)>0.
	\end{array}\right.$$
	
	It is straightforward to check that $\eta_1$ is continuous. Then consider the map $\eta_2:\overline{B}_{\rho}\backslash\{0\}\rightarrow(\vartheta^0\cap \overline{B}_{\rho})\backslash\{0\}$ defined by
	\begin{eqnarray*}\eta_2(u)=\left\{\begin{array}{ll}
		u&\mbox{if}\ u\in \overline{B}_{\rho}\backslash\{0\},\vartheta(u)\leq 0\\
		\eta_1(u)u&\mbox{if}\ u\in \overline{B}_{\rho}\backslash\{0\},\vartheta(u)>0.
	\end{array}\right.
	\end{eqnarray*}
	
	Then $\eta_2$ is continuous and
	$$\eta_2|_{(\vartheta^0\cap \overline{B}_{\rho})\backslash\{0\}}=id|_{(\vartheta^0\cap \overline{B}_{\rho})\backslash\{0\}}.$$
	
	So, we have that $(\vartheta^0\cap \overline{B}_{\rho})\backslash\{0\}$ is a retract of $\overline{B}_{\rho}\backslash\{0\}$ and the latter is contractible. Therefore $(\vartheta^0\cap \overline{B}_{\rho})\backslash\{0\}$ is contractible. This fact, (\ref{eq41}) and Proposition 4.9 and 4.10 of Granas and Dugundji \cite[p. 389]{8}, imply
	\begin{eqnarray*}
		&&H_k(\vartheta^0\cap \overline{B}_{\rho},(\vartheta^0\cap \overline{B}_{\rho})\backslash\{0\})=0\ \mbox{for all}\ k\geq 0,\\
		&\Rightarrow&C_k(\vartheta,0)=0\ \mbox{for all}\ k\geq 0.
	\end{eqnarray*}
The proof is now complete.
\end{proof}

Now we are ready to produce nodal solutions. In what follows, $u_*\in \mbox{int}\,C_+$ and $v_*\in-\mbox{int}\,C_+$ are the extremal constant sign solutions of problem (\ref{eq1}) produced in Proposition \ref{prop8}.
\begin{prop}\label{prop10}
	If hypotheses $H(\xi),H(\beta)$ and $H_1$ hold, then problem (\ref{eq1}) admits a nodal solution
	$$y_0\in[v_*,u_*]\cap C^1(\overline{\Omega}).$$
\end{prop}

\begin{proof}
	Let $u_*\in \mbox{int}\,C_+$ and $v_*\in-\mbox{int}\,C_+$ be the two extremal constant sign solutions of (\ref{eq1}) produced in Proposition \ref{prop8}. We introduce the following truncation-perturbation of the reaction $f(z,\cdot)$:
	\begin{eqnarray}\label{eq43}
		\tau(z,x)=\left\{
		\begin{array}{ll}
			f(z,v_*(z))+\mu v_*(z)&\mbox{if}\ x<v_*(z)\\
			f(z,x)+\mu x&\mbox{if}\ v_*(z)\leq x\leq u_*(z)\\
			f(z,u_*(z))+\mu u_*(z)&\mbox{if}\ u_*(z)<x.
		\end{array}\right.
	\end{eqnarray}
	
	This is a Carath\'eodory function. Let $T(z,x)=\int^{x}_{0}\tau(z,s)ds$ and consider the $C^1$-functional $\psi:H^1(\Omega)\rightarrow\RR$ defined by
	$$\psi(u)=\frac{1}{2}\tau(u)+\frac{\mu}{2}||u||^2_2-\int_{\Omega}T(z,u)dz\ \mbox{for all}\ u\in H^1(\Omega).$$
	
	In addition, we consider the positive and negative truncation of $\tau(z,\cdot)$, that is, the Carath\'eodory functions
	$$\tau_{\pm}(z,x)=\tau(z,\pm x^{\pm}).$$
	
	We set $T_{\pm}(z,x)=\int^{x}_{0}\tau_{\pm}(z,s)ds$ and consider the $C^1$-functionals $\psi_{\pm}:H^1(\Omega)\rightarrow\RR$ defined by
	$$\psi_{\pm}(u)=\frac{1}{2}\sigma(u)+\frac{\mu}{2}||u||^2_2-\int_{\Omega}T_{\pm}(z,u)dz\ \mbox{for all}\ u\in H^1(\Omega).$$
	\begin{claim}\label{cl1} We have
		$$K_{\psi}\subseteq[v_*,u_*],\ K_{\psi_+}=\{0,u_*\},K_{\psi_-}=\{0,v_*\}.$$
	\end{claim}
	
	Let $u\in K_{\psi}$. Then we have
	\begin{eqnarray}\label{eq44}
		&&\psi'(u)=0\Rightarrow\nonumber\\
		&&\left\langle A(u),h\right\rangle+\int_{\Omega}(\xi(z)+\mu)uhdz+\int_{\partial\Omega}\beta(z)uhd\sigma_0=
 \int_{\Omega}\tau(z,u)hdz,\ \forall\, h\in H^1(\Omega).
	\end{eqnarray}
	
	In (\ref{eq44}) we choose $h=(u-u_*)^+\in H^1(\Omega)$. Then
	\begin{eqnarray*}
		&&\left\langle A(u),(u-u_*)^+\right\rangle+\int_{\Omega}(\xi(z)+\mu)u(u-u_*)^+dz+\int_{\partial\Omega}\beta(z)u(u-u_*)^+d\sigma_0\\
		&=&\int_{\Omega}[f(z,u_*)+\mu u_*](u-u_*)^+dz\ (\mbox{see (\ref{eq43})})\\
		&=&\left\langle A(u_*),(u-u_*)^+\right\rangle+\int_{\Omega}(\xi(z)+\mu)u_*(u-u_*)^+dz+\int_{\partial\Omega}\beta(z)u_*(u-u_*)^+d\sigma_0\\
		&&\hspace{6cm}(\mbox{recall that}\ u_*\in S_+)\\
		\Rightarrow&&\sigma((u-u_*)^+)+\mu||(u-u_*)^+||^2_2=0,\\
		\Rightarrow&&c_0||(u-u_*)^+||^2\leq 0\ (\mbox{see (\ref{eq4})}),\\
		\Rightarrow&&u\leq u_*.
	\end{eqnarray*}
	
	Similarly, choosing $h=(v_*-u)^+\in H^1(\Omega)$ in (\ref{eq44}), we show that
	\begin{eqnarray*}
		&&v_*\leq x,\\
		&\Rightarrow&K_{\psi}\subseteq[v_*,u_*]\ (\mbox{since}\ u\in K_{\psi}\ \mbox{is arbitrary}).
	\end{eqnarray*}
	
	In a similar fashion, we show that
	$$K_{\psi_+}\subseteq[0,u_*]\ \mbox{and}\ K_{\psi_-}\subseteq[v_*,0].$$
	
	The extremality of the constant sign solutions $u_*\in \mbox{int}\,C_+$ and $v_*\in-\mbox{int}\, C_+$ implies that
	$$K_{\psi_+}=\{0,u_*\}\ \mbox{and}\ K_{\psi_-}=\{0,v_*\}.$$
	
	This proves Claim \ref{cl1}.
	\begin{claim}\label{cl2}
		$u_*\in \mbox{int}\,C_+$ and $v_*\in-\mbox{int}\,C_+$ are local minimizers of $\psi$.
	\end{claim}
	
	From (\ref{eq4}) and (\ref{eq39}), it is clear that $\psi_+$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we ca find $\hat{u}_*\in H^1(\Omega)$ such that
	\begin{equation}\label{eq45}
		\psi_+(\hat{u}_*)=\inf[\psi_+(u):u\in H^1(\Omega)].
	\end{equation}
	
	Using hypothesis $H_1(iii)$ and since $q<2<r$, we see that for $t\in(0,1)$ small we have
	\begin{eqnarray*}
		&&\psi_+(t\hat{u}_1(\beta))<0,\\
		&\Rightarrow&\psi_+(\hat{u}_*)<0=\psi_+(0)\ (\mbox{see (\ref{eq45})}),\ \mbox{hence}\ \hat{u}_*\neq 0.
	\end{eqnarray*}
	
	From (\ref{eq45}) we have
	\begin{eqnarray*}
		&&\hat{u}_*\in K_{\psi_+}\backslash\{0\},\\
		&\Rightarrow&\hat{u}_*=u_*\in \mbox{int}\,C_+\ (\mbox{see Claim \ref{cl1}}).
	\end{eqnarray*}
	
	Note that $\psi|_{C_+}=\psi_+|_{C_+}$. So, $u_*\in \mbox{int}\,C_+$ is a local $C^1(\overline{\Omega})$-minimizer of $\psi$. Then using Proposition 3 in Papageorgiou and R\u{a}dulescu \cite{19} (generalized version of the classical result established by Brezis and Nirenberg \cite{brenir}), $u_0$ is also a local $H^1(\Omega)$-minimizer of $\psi$.
	
	Similarly for $v_*\in -\mbox{int}\,C_+$ using this time the functional $\psi_-$.
	
	This proves Claim \ref{cl2}.
	
\smallskip
	Due to (\ref{eq43}) and the extremality of $u_*\in \mbox{int}\,C_+$ and $v_*\in -\mbox{int}\,C_+$, every nontrivial critical point of $\psi$ distinct from $u_*$ and $v_*$ is necessarily a nodal solution of (\ref{eq1}) (see Claim \ref{cl1}). So, we may assume that $K_{\varphi}$ is finite. Also, without any loss of generality, we may assume that $\psi(v_*)\leq \psi(u_*)$ (the reasoning is similar if the opposite inequality holds). By Claim \ref{cl2}, $u_*\in \mbox{int}\,C_+$ is a local minimizers of $\psi$. So, we can find $\rho\in(0,1)$ small such that
	\begin{eqnarray}\label{eq46}
		\psi(v_*)\leq\psi(u_*)<\inf[\psi(u):||u-u_*||=\rho]=m_{\rho},\ ||v_*-u_*||>\rho
	\end{eqnarray}
	(see Aizicovici, Papageorgiou and Staicu \cite{1}, proof of Proposition 29). Recall that $\psi$ is coercive. So, it satisfies the $PS$-condition. This fact and (\ref{eq46}) permit the use of Theorem \ref{th1} (the mountain pass theorem). So, we can find $y_0\in H^1(\Omega)$ such that
	\begin{equation}\label{eq47}
		y_0\in K_{\psi}\subseteq[v_*,u_*]\ (\mbox{see Claim \ref{cl1}})\ \mbox{and}\ m_{\rho}\leq\psi(y_0).
	\end{equation}
	
	From (\ref{eq46}) and (\ref{eq47}), we see that $y_0\notin\{v_*,u_*\}$. Therefore, if we can show that $y_0\neq 0$, then $y_0$ is a nodal solution of (\ref{eq1}). Since $y_0$ is a critical point of $\psi$ of mountain pass type, we have
	\begin{equation}\label{eq48}
		C_1(\psi,y_0)\neq 0\ (\mbox{see Chang \cite[p. 89]{4}}).
	\end{equation}
	
	On the other hand, from Proposition \ref{prop9}, we know that
	\begin{equation}\label{eq49}
		C_k(\psi,0)=0\ \mbox{for all}\ k\geq 0.
	\end{equation}
	
	Comparing (\ref{eq48}) and (\ref{eq49}), we conclude that $y_0\neq 0$. So, $y_0$ is a nodal solution of (\ref{eq1}) and as before $y_0\in C^1(\overline{\Omega})$.
\end{proof}

In fact we can improve the conclusion of Proposition \ref{prop10}, proved we strengthen a little the conditions on $f(z,\cdot)$. So, now we assume the following:

\smallskip
$H_2:\ f:\Omega\times\RR\rightarrow\RR$ is Carath\'eodory function such that $f(z,0)=0$, hypotheses $H_2(i),(ii),(iii)$ are the same as the corresponding hypotheses $H_1(i),(ii),(iii)$ and
\begin{itemize}
	\item[(iv)] there exists $\hat{\vartheta}>0$ such that for almost all $z\in\Omega$, $x\longmapsto f(z,x)+\hat{\vartheta}x$ is nondecreasing on $[-\rho,\rho]$.
\end{itemize}

\begin{remark}
	Evidently this extra condition on $f(z,\cdot)$ is satisfied if for example, for almost all $z\in\Omega$, $f(z,\cdot)\in C^1(\RR)$ and $f'_x(z,\cdot)$ is $L^{\infty}(\Omega)$-bounded on $[-\rho,\rho]$.
\end{remark}

\begin{prop}\label{prop11}
	If hypotheses $H(\xi), H(\beta)$ and $H_2$ hold, then problem (\ref{eq1}) admits a nodal solution
	$$y_0\in int_{C^1(\overline{\Omega})}[v_*,u_*].$$
\end{prop}

\begin{proof}
	From Proposition \ref{prop10}, we already have a nodal solution $y_0\in[v_*,u_*]\cap C^1(\overline{\Omega})$. Let $\hat{\vartheta}>0$ be as postulated by hypothesis $H_2(iv)$. Then
	\begin{eqnarray*}
		&&-\Delta y_0(z)+(\xi(z)+\hat{\vartheta})y_0(z)\\
		&=&f(z,u_*(z))+\hat{\vartheta}y_0(z)\\
		&\leq&f(z,u_*(z))+\hat{\vartheta}u_*(z)\ (\mbox{since}\ y_0\leq x_*,\ \mbox{see hypothesis}\ H_2(iv))\\
		&=&-\Delta u_*(z)+(\xi(z)+\hat{\vartheta})u_*(z)\ (\mbox{since}\ u_*\in S_+),\\
		\Rightarrow&&\Delta(u_*-y_0)(z)\leq(||\xi^+||_{\infty}+\hat{\varphi})(u_*-y_0)(z)\ \mbox{for almost all}\ z\in\Omega\\
		&&\hspace{6cm}(\mbox{see hypothesis}\ H(\xi)),\\
		\Rightarrow&&u_*-y_0\in \mbox{int}\,C_+\ (\mbox{by the strong maximum principle}).
	\end{eqnarray*}
	
	In a similar fashion, we show that
	$$y_0-v_*\in \mbox{int}\,C_+.$$
	
	Therefore finally we have $y_0\in \mbox{int}_{C^1(\overline{\Omega})}[v_*,u_*]$.
\end{proof}

Now we can formulate our first multiplicity result.
\begin{theorem}\label{th12}
Assume that hypotheses $H(\xi),H(\beta)$ and $H_1$ hold. Then problem (\ref{eq1}) admits at least three nontrivial solutions
	$$u_0\in \mbox{int}\,C_+,\ v_0\in-\mbox{int}\,C_+\ \mbox{and}\ y_0\in[v_0,u_0]\cap C^1(\overline{\Omega})\ \mbox{nodal}.$$
	
	Moreover, if hypotheses $H_2$ hold, then $y_0\in \mbox{int}_{C^1(\overline{\Omega})}[v_0,u_0]$.
\end{theorem}

Next we modify the behavior of $f(z,\cdot)$ near zero and assume that $f(z,\cdot)$ is linear near zero. In this way we change the geometry of the problem. Nevertheless, for the new setting we prove again a three solutions theorem proving sign information for all the solutions.

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 almost all $z\in\Omega$ and
\begin{itemize}
	\item[(i)] there exist functions $w_{\pm}\in H^1(\Omega)\cap C(\overline{\Omega})$ such that
	\begin{eqnarray*}
		&&w_-(z)\leq c_-<0<c_+\leq w_+(z)\ \mbox{for all}\ z\in\overline{\Omega},\\
		&&f(z,w_+(z))-\xi(z)w_+(z)\leq 0\leq f(z,w_-(z))-\xi(z)w_-(z)\ \mbox{for almost all}\ z\in\Omega,\\
		&&A(w_-)\leq 0\leq A(w_+)\ \mbox{in}\ H^1(\Omega)^*;
	\end{eqnarray*}
	\item[(ii)] if $\rho=\max\{||w_+||_{\infty},||w_-||_{\infty}\}$, then there exists $a_{\rho}\in L^{\infty}(\Omega)_+$ such that
	$$|f(z,x)|\leq a_{\rho}(z)\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ |x|\leq\rho;$$
	\item[(iii)] there exist functions $\eta,\hat{\eta}\in L^{\infty}(\Omega)$ such that
	\begin{eqnarray*}
		&&\hat{\lambda}_1(\beta)\leq \eta(z)\ \mbox{for almost all}\ z\in\Omega,\ \mbox{strictly on a set of positive measure,}\\
		&&\eta(z)\leq \liminf\limits_{x\rightarrow 0}\frac{f(z,x)}{x}\leq\limsup\limits_{x\rightarrow 0}\frac{f(z,x)}{x}\leq \hat{\eta}(z)\ \mbox{uniformly for almost all}\ z\in\Omega.
	\end{eqnarray*}
\end{itemize}

Under this new geometry near zero for the reaction $f(z,\cdot)$, the previous results on the existence of constant sign solutions remain valid with very minor changes in their proofs. So, we have:
\begin{prop}\label{prop13}
	If hypotheses $H(\xi),H(\beta)$ and $H_3$ hold, then problem (\ref{eq1}) has at least two solutions of constant sign
	$$u_0\in \mbox{int}\,C_+\ \mbox{and}\ v_0\in-\mbox{int}\,C_+.$$
\end{prop}
\begin{proof}
	The proof is similar to that of Proposition \ref{prop7}. Again we consider the $C^1$-functional $\hat{\varphi}_+$ (see the proof of Proposition \ref{prop7}) and use the direct method. The only thing that differs in the present proof, is how we show that
	\begin{equation}\label{eq50}
		\hat{\varphi}_+(u_0)=\inf[\hat{\varphi}_+(u):u\in H^1(\Omega)]<0=\hat{\varphi}_+(0).
	\end{equation}
	
	By virtue of hypothesis $H_3(iii)$, given $\epsilon>0$, we can find $\delta=\delta(\epsilon)\in\left(0,c_+\right]$ such that
	\begin{eqnarray}\label{eq51}
		&&f(z,x)\geq(\eta(z)-\epsilon)x\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\in[0,\delta],\nonumber\\
		&\Rightarrow&F(z,x)\geq\frac{1}{2}(\eta(z)-\epsilon)x^2\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\in[0,\delta].
	\end{eqnarray}
	
	For $t\in(0,1)$ small such that $t\hat{u}_1(\beta)(z)\in\left(0,\delta\right]$ for all $z\in\overline{\Omega}$. Then
	\begin{eqnarray*}
		 \hat{\varphi}_+(t\hat{u}_1(\beta))&\leq&\frac{t^2}{2}\sigma(\hat{u}_1(\beta))-\frac{t^2}{2}\int_{\Omega}\eta(z)\hat{u}_1(\beta)^2dz+\frac{t^2\epsilon}{2}||\hat{u}_1(\beta)||^2_2\ (\mbox{see (\ref{eq51})})\\
		 &=&\frac{t^2}{2}\left(\int_{\Omega}[\hat{\lambda}_1(\beta)-\eta(z)]\hat{u}_1(\beta)^2dz+\epsilon\right)\\
		&&\hspace{5cm}(\mbox{recall that}\ ||\hat{u}_1(\beta)||_2=1).
	\end{eqnarray*}
	
	Since $\hat{u}_1(\beta)\in \mbox{int}\,C_+$, we have
	$$\mathcal{I}=\int_{\Omega}(\eta(z)-\hat{\lambda}_1(\beta))\hat{u}_1(\beta)^2dz>0.$$
	
	Then
	$$\hat{\varphi}_1(t\hat{u}_1(\beta))\leq\frac{t^2}{2}[-\mathcal{I}+\epsilon].$$
	
	So, choosing $\epsilon\in(0,\mathcal{I})$, we infer that
	\begin{eqnarray*}
		&&\hat{\varphi}_+(t\hat{u}_1(\beta))<0,\\
		&\Rightarrow&\hat{\varphi}_+(u_0)<0=\hat{\varphi}_+(0)\ (\mbox{see (\ref{eq50})}),\ \mbox{hence}\ u_0\neq 0.
	\end{eqnarray*}
	
	Then as in proof of Proposition \ref{prop7}, we show that $u_0\in \mbox{int}\,C_+$ is a solution of problem (\ref{eq1}).
	
	Similarly, working with the functional $\hat{\varphi}_-$ (see the proof of Proposition \ref{prop7}), we produce a negative solution $v_0\in -\mbox{int}\,C_+$.
\end{proof}

In the present setting, we see that given $\epsilon>0$, we can find $c_{10}=c_{10}(\epsilon)>0$ such that
\begin{equation}\label{eq52}
	f(z,x)x\geq(\eta(z)-\epsilon)x^2-c_{10}|x|^r\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\in[-\rho,\rho].
\end{equation}

This leads to the following auxiliary problem
\begin{equation}\label{eq53}
	\left\{\begin{array}{ll}
		-\Delta u(z)+\xi(z)u(z)=(\eta(z)-\epsilon)u(z)-c_{10}|u(z)|^{r-2}u(z)&\mbox{in}\ \Omega,\\
	\displaystyle	\frac{\partial u}{\partial n}+\beta(z)u=0&\mbox{on}\ \partial\Omega.
	\end{array}\right\}
\end{equation}

With essentially identical proofs to those of Proposition \ref{prop5} and \ref{prop6}, we have the following result.
\begin{prop}\label{prop14}
	If hypotheses $H(\xi),H(\beta)$ and $H_3$ hold, then problem (\ref{eq53}) admits a unique positive solution $\bar{u}\in \mbox{int}\,C_+$, a unique negative solution $\bar{v}=-\bar{u}\in-\mbox{int}\,C_+$ and
	$$\bar{u}\leq u\ \mbox{for all}\ u\in S_+\ \mbox{and}\ v\leq\bar{v}\ \mbox{for all}\ v\in S_-.$$
\end{prop}

As in Proposition \ref{prop8}, using this time Proposition \ref{prop14}, we generate extremal constant sign solutions.
\begin{prop}\label{prop15}
	If hypotheses $H(\xi),H(\beta)$ and $H_3$ hold, then problem (\ref{eq1}) admits extremal constant sign solutions, that is, there exists a smallest positive solution $u_*\in \mbox{int}\,C_+$ and a biggest negative solution $v_*\in-\mbox{int}\,C_+$.
\end{prop}
	
	Then we can produce a nodal solution. To do this we need to strengthen further the condition on $f(z,\cdot)$ near zero, without altering the geometry of the problem.


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

\smallskip
$H_4:\ f:\Omega\times\RR\rightarrow\RR$ is a Carath\'eodory function such that $f(z,0)=0$ for almost all $z\in\Omega$, hypotheses $H_3(i),(ii)$ are the same as the corresponding hypotheses $H_3(i),(ii)$ and
\begin{itemize}
	\item[(iii)] there exist functions $\eta,\hat{\eta}\in L^{\infty}(\Omega)$ such that
	\begin{eqnarray*}
		&&\hat{\lambda}_2(\beta)<\eta(z)\ \mbox{for almost all}\ z\in\Omega,\\
		&&\eta(z)\leq\liminf\limits_{x\rightarrow 0}\frac{f(z,x)}{x}\leq\limsup\limits_{x\rightarrow 0}\frac{f(z,x)}{x}\leq\hat{\eta}(z)\ \mbox{uniformly for almost all}\ z\in\Omega.
	\end{eqnarray*}
\end{itemize}
\begin{prop}\label{prop16}
	If hypotheses $H(\xi),H(\beta)$ and $H_4$ hold, then problem (\ref{eq1}) admits a nodal solution
	$$y_0\in [v_*,u_*]\cap C^1(\overline{\Omega}).$$
\end{prop}
\begin{proof}
	We use the extremal constant sign solutions $u_*\in \mbox{int}\,C_+$ and $v_*\in-\mbox{int}\,C_+$ produced in Proposition \ref{eq15} and argue as in the proof of Proposition \ref{prop10}. Then via the mountain pass theorem (see Theorem \ref{th1}), we produce a solution
	$$y_0\in[v_*,u_*]\cap C^1(\overline{\Omega}),y_0\neq\{v_*,u_*\}.$$
	
	We need to show that $y_0\neq 0$ to conclude that $y_0$ is nodal.
	
	From Theorem \ref{th1}, we have
	\begin{equation}\label{eq54}
		m_{\rho}\leq\inf\limits_{\gamma\in\Gamma}\max\limits_{0\leq t\leq 1}\psi(\gamma(t))=\psi(y_0),
	\end{equation}
	where $\Gamma=\{\gamma\in C([0,1],H^1(\Omega)):\gamma(0)=v_*,\gamma(1)=u_*\}$ (see also (\ref{eq46}). According to (\ref{eq54}), in order to establish the nontriviality of $y_0$ and therefore conclude that $y_0$ is nodal, it suffices to produce a path $\gamma_*\in\Gamma$ such that $\psi|_{\gamma_*}<0=\psi(0)$. To this end, we consider the following Banach manifolds
	$$M=H^1(\Omega)\cap\partial B^{L^2}_{1}\ \mbox{and}\ M_c=M\cap C^1(\overline{\Omega}).$$
	
	Here $\partial B^{L^2}_{1}=\{u\in L^2(\Omega):||u||_2=1\}$. Evidently $M_c$ is dense in $M$. We introduce the following sets of paths
	\begin{eqnarray*}
		&&\hat{\Gamma}=\{\hat{\gamma}\in C([-1,1],M):\hat{\gamma}(-1)=-\hat{u}_1(\beta),\hat{\gamma}(1)=\hat{u}_1(\beta)\}\\
		&&\hat{\Gamma}_c=\{\hat{\gamma}\in C([-1,1],M_c):\hat{\gamma}(-1)=-\hat{u}_1(\beta),\hat{\gamma}(1)=\hat{u}_1(\beta)\}\\
		&&\hspace{5cm}(\mbox{recall that}\ \hat{u}_1(\beta)\in \mbox{int}\,C_+).
	\end{eqnarray*}
	\begin{claim}\label{cl3}
		$\hat{\Gamma}_c$ is dense in $\hat{\Gamma}$.
	\end{claim}
	
	Let $\hat{\gamma}\in\hat{\Gamma}$ and $\epsilon>0$. We consider the multifunction $R_{\epsilon}:[-1,1]\rightarrow 2^{C^1(\overline{\Omega})}$ defined by
	\begin{eqnarray*}
		R_{\epsilon}(t)=\left\{\begin{array}{ll}
			\{u\in C^1(\overline{\Omega}):||u-\hat{\gamma}(t)||<\epsilon\}&\mbox{if}\ -1<t<1\\
			\{\pm\hat{u}_1(\beta)\}&\mbox{if}\ t=\pm 1.
		\end{array}\right.
	\end{eqnarray*}
	
	Evidently $R_{\epsilon}(\cdot)$ has nonempty and convex values. In addition
	\begin{eqnarray*}
		&&R_{\epsilon}(t)\ \mbox{is open for all}\ t\in(-1,1),\\
		&&R_{\epsilon}(\pm 1)\ \mbox{are singletons.}
	\end{eqnarray*}
	
	From Papageorgiou and Kyritsi \cite[p. 458]{16}, we have that the multifunction $R_{\epsilon}(\cdot)$ is lower semicontinuous. So, we can apply Theorem 3.1''' of Michael \cite{12} (see also Hu and Papageorgiou \cite[p. 97]{9}) and find a continuous path $\hat{\gamma}_{\epsilon}:[-1,1]\rightarrow C^1(\overline{\Omega})$ such that
	$$\hat{\gamma}_{\epsilon}(t)\in R_{\epsilon}(t)\ \mbox{for all}\ t\in[-1,1].$$
	
	Let $\epsilon_n=\frac{1}{n}$ and let $\hat{\gamma}_n=\hat{\gamma}_{\epsilon_n}$, $n\geq 1$ be as above. We have
	\begin{equation}\label{eq55}
		||\hat{\gamma}_n(t)-\hat{\gamma}(t)||<\frac{1}{n}\ \mbox{for all}\ t\in[-1,1],\ \mbox{all}\ n\geq 1.
	\end{equation}
	
	Since $\hat{\gamma}(t)\in \partial B^{L^2}_{1}$ for all $t\in[-1,1]$, from (\ref{eq55}) we see that $n\geq 1$ big, we have $||\hat{\gamma}_n(t)||_2\neq 0$ for all $t\in[-1,1]$. So, we may assume that $||\hat{\gamma}_n(t)||_2\neq 0$ for all $t\in[-1,1]$, all $n\geq 1$. We set
	\begin{equation}\label{eq56}
		\hat{\gamma}^{0}_{n}(t)=\frac{\hat{\gamma}_n(t)}{||\gamma_n(t)||_2}\ \mbox{for all}\ t\in[-1,1],\ \mbox{all}\ n\geq 1.
	\end{equation}
	
	We have $\hat{\gamma}^0_n\in C([-1,1],M_c)$ and $\hat{\gamma}^0_n(\pm \hat{u}_1(\beta))=\pm\hat{u}_1(\beta)$. From (\ref{eq55}) and (\ref{eq56}) we have
	\begin{eqnarray}\label{eq57}
		 ||\hat{\gamma}^0_n(t)-\hat{\gamma}(t)||&\leq&||\hat{\gamma}^0_n(t)-\hat{\gamma}_n(t)||+||\hat{\gamma}_n(t)-\hat{\gamma}(t)||\nonumber\\
		 &\leq&\frac{|1-||\hat{\gamma}_n(t)||_2|}{||\hat{\gamma}_n(t)||_2}||\hat{\gamma}_n(t)||+\frac{1}{n}\ (\mbox{see (\ref{eq55})}).
	\end{eqnarray}
	
	We have
	\begin{eqnarray}\label{eq58}
		&&\max\limits_{-1\leq t\leq 1}|1-||\hat{\gamma}_n(t)||_2|=\max\limits_{-1\leq t\leq 1}|||\hat{\gamma}(t)||_2-||\hat{\gamma}_n(t)||_2|\ (\mbox{recall}\ \hat{\gamma}(t)\in\partial B^{L^1}_{1}\nonumber\\
		&&\hspace{9cm}\mbox{for all}\ t\in[-1,1])\nonumber\\
		&&\leq\max\limits_{-1\leq t\leq 1}||\hat{\gamma}(t)-\hat{\gamma}_n(t)||_2\ (\mbox{by the triangle inequality})\nonumber\\
		&&\leq\max\limits_{-1\leq t\leq 1}||\hat{\gamma}(t)-\hat{\gamma}_n(t)||\leq\frac{1}{n}\ \mbox{for all}\ n\geq 1\ (\mbox{see (\ref{eq55})}).
	\end{eqnarray}
	
	From (\ref{eq57}) and (\ref{eq58}) it follows that
	\begin{eqnarray*}
		&&\hat{\gamma}^0_n\rightarrow\hat{\gamma}\ \mbox{in}\ C([-1,1],M)\ \mbox{as}\ n\rightarrow\infty,\\
		&\Rightarrow&\hat{\Gamma}_c\ \mbox{is dense in}\ \hat{\Gamma}.
	\end{eqnarray*}
	
	This proves Claim \ref{cl3}.
	
	Invoking Proposition \ref{prop2} and the Claim \ref{cl3}, given $\delta_0>0$, we can find $\hat{\gamma}_0\in\hat{\Gamma}_c$ such that
	\begin{equation}\label{eq59}
		\max\limits_{-1\leq t\leq 1}\sigma(\hat{\gamma}_0(t))\leq\hat{\lambda}_2(\beta)+\delta_0.
	\end{equation}
	
	Since $\hat{\gamma}_0\in\hat{\Gamma}_c$ and $u_*\in \mbox{int}\,C_+,\ v_*\in-\mbox{int}\,C_+$, we can find $\vartheta\in(0,1)$ small such that
	\begin{eqnarray}\label{eq60}
		&&\vartheta\hat{\gamma}_0(t)\in[v_*,u_*]\ \mbox{and}\ \vartheta|\hat{\gamma}_0(t)(z)|,\ \vartheta|D\hat{\gamma}_0(t)(z)|\leq\delta\\
		&&\hspace{6cm}\mbox{for all}\ t\in[-1,1],\ \mbox{all}\ z\in\overline{\Omega}.\nonumber
	\end{eqnarray}
	
	Here for the first inclusion in (\ref{eq60}) we have used Lemma 3.3 of Filippakis, Kristaly and Papageorgiou \cite{6} and $\delta>0$ is as in (\ref{eq51}). Of course we can always take $\delta>0$ such that $\delta\leq \min\{\min\limits_{\overline{\Omega}}u_*,\min\limits_{\overline{\Omega}}v_*\}$. We have
	\begin{eqnarray}\label{eq61}
		 \psi(\vartheta\hat{\gamma}_0(t))&=&\frac{\vartheta^2}{2}\sigma(\hat{\gamma}_0(t))+\frac{\mu\vartheta^2}{2}||\hat{\gamma}_0(t)||^2_2-\int_{\Omega}T(z,\vartheta\hat{\gamma}_0(z))dz\nonumber\\
		 &=&\frac{\vartheta^2}{2}\sigma(\hat{\gamma}_0(t))-\int_{\Omega}F(z,\vartheta\hat{\gamma}_0(t))dz\ (\mbox{see (\ref{eq43}) and (\ref{eq60})})\nonumber\\
		 &\leq&\frac{\vartheta^2}{2}\sigma(\hat{\gamma}_0(t))-\frac{\vartheta^2}{2}\int_{\Omega}\eta(z)\hat{\gamma}_0(t)^2dz+\frac{\vartheta^2\epsilon}{2}\nonumber\\
		&&\hspace{0.5cm}(\mbox{see (\ref{eq51}) and (\ref{eq60}) and recall that}\ ||\hat{\gamma}_0(t)||_2=1\ \mbox{for all}\ t\in[-1,1])\nonumber\\
		 &\leq&\frac{\vartheta^2}{2}\left[\int_{\Omega}(\hat{\lambda}_2(\beta)-\eta(z))\hat{\gamma}_0(t)^2dz+\epsilon\right]\ \mbox{for all}\ t\in[-1,1].
	\end{eqnarray}
	
	From hypothesis $H_4(iii)$, we have
	$$\mathcal{I}=\int_{\Omega}[\eta(z)-\hat{\lambda}_2(\beta)]\hat{\gamma}_0(t)^2dz>0.$$
	
	So, choosing $\epsilon\in\left(0,\mathcal{I}\right]$ from (\ref{eq61}), we have
	$$\psi(\vartheta\hat{\gamma}_0(t))<0\ \mbox{for all}\ t\in[-1,1].$$
	
	We set $\hat{\gamma}=\vartheta\hat{\gamma}_0$. Then this is a continuous path in $H^1(\Omega)$ joining $-\vartheta\hat{u}_1(\beta)$ and $\vartheta\hat{u}_1(\beta)$ and
	\begin{equation}\label{eq62}
		\psi|_{\hat{\gamma}}<0.
	\end{equation}
	
	Next we produce a continuous path in $H^1(\Omega)$ joining $\vartheta\hat{u}_1(\beta)$ and $u_*$ and along which the functional $\psi$ is negative. To this end, recall that
	\begin{eqnarray}
		&&\tau=\psi_+(u_*)=\inf\limits_{H^1(\Omega)^+}\psi_+<0=\psi_+(0),\label{eq63}\\
		&&K_{\psi_+}=\{0,u_*\}\ \mbox{(see Claim \ref{cl1} in the proof of Proposition \ref{prop10})},\label{eq64}\\
		&&\psi_+\ \mbox{satisfies the}\ PS-\mbox{condition (being coercive, see (\ref{eq4}) and (\ref{eq43}))}.\nonumber
	\end{eqnarray}
	
	So, we can apply the second deformation theorem (see Gasinski and Papageorgiou \cite[p. 628]{7}) and produce a deformation $h:[0,1]\times(\psi^0_+\backslash K^0_{\psi_+})\rightarrow\psi^0_+$ such that
	\begin{eqnarray}
		&&h(0,u)=u\ \mbox{for all}\ u\in \psi^0_+\backslash K^0_{\psi_+},\label{eq65}\\
		&&h(1,\psi^0_+\backslash K^0_{\psi_+})\subseteq\psi^{\tau}_{+}=\{u_*\}\ \mbox{(see (\ref{eq63}) and (\ref{eq64}))}\label{eq66}\\
		&&\psi_+(h(t,u))\leq\psi_+(h(s,u))\ \mbox{for all}\ s,t\in[0,1],s\leq t,\ \mbox{all}\ u\in \psi^0_+\backslash K^0_{\psi_+}.\label{eq67}
	\end{eqnarray}
	
	We have
	\begin{eqnarray}\label{eq68}
		 &&\psi_+(\vartheta\hat{u}_1(\beta))=\psi(\vartheta\hat{u}_1(\beta))=\psi(\hat{\gamma}(1))<0\ (\mbox{see (\ref{eq62})}),\\
		&\Rightarrow&\vartheta\hat{u}_1(\beta)\in\psi^0_+\backslash K^0_{\psi_+}\nonumber.
	\end{eqnarray}
	
	Therefore we can define
	\begin{equation}\label{eq69}
		\hat{\gamma}_+(t)=h(t,\vartheta\hat{u}_1(\beta))^+\ \mbox{for all}\ t\in[0,1].
	\end{equation}
	
	We have
	\begin{eqnarray*}
		&&\hat{\gamma}_+(0)=h(0,\vartheta\hat{u}_1(\beta))^+=\vartheta\hat{u}_1(\beta)\ (\mbox{see (\ref{eq65}) and recall that $\hat{u}_1(\beta)\in \mbox{int}\,C_+$}),\\
		&&\hat{\gamma}_+(1)=h(1,\vartheta\hat{u}_1(\beta))^+=u_*\ (\mbox{see (\ref{eq66}) and recall that $u_*\in \mbox{int}\,C_+$}),\\
		&&\psi_+(\hat{\gamma}_+(t))\leq\psi_+(\vartheta\hat{u}_1(\beta))<0\ \mbox{for all}\ t\in[0,1]\ (\mbox{see (\ref{eq67}) and (\ref{eq68})}).
	\end{eqnarray*}
	
	So, $\hat{\gamma}_+$ is a continuous path in $H^1(\Omega)$ joining $\vartheta \hat{u}_1(\beta)$ and $u_*$ and such that
	$$\psi_+|_{\hat{\gamma}_+}<0.$$
	
	From (\ref{eq69}) we see that $\hat{\gamma}_+(t)(z)\geq 0$ for almost all $z\in\Omega$, all $t\in[0,1]$. Hence
	\begin{equation}\label{eq70}
		\psi|_{\hat{\gamma}_+}<0.
	\end{equation}
	
	Similarly, we produce another continuous path $\hat{\gamma}_-$ in $H^1(\Omega)$ which joins $-\vartheta\hat{u}_1(\beta)$ and $v_*$ and such that
	\begin{equation}\label{eq71}
		\psi|_{\hat{\gamma}_-}<0.
	\end{equation}
	
	We concatenate $\hat{\gamma}_-,\hat{\gamma},\hat{\gamma}_+$ and generate $\gamma_*\in\Gamma$ such that
	\begin{eqnarray*}
		&&\psi|_{\gamma_*}<0\ (\mbox{see (\ref{eq62}), (\ref{eq70}), (\ref{eq71})}),\\
		&\Rightarrow&y_0\neq 0\ \mbox{and so}\ y_0\in[v_*,u_*]\cap C^1(\overline{\Omega})\ \mbox{is a nodal solution of (\ref{eq1})}.
	\end{eqnarray*}
This completes the proof.
\end{proof}

Again, if we strengthen the conditions on $f(z,\cdot)$, we can improve the conclusion of the above proposition.

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

\smallskip
$H_5:$ $f:\Omega\times\RR\rightarrow\RR$ is a Carath\'eodory function such that $f(z,0)=0$ for almost all $z\in\Omega$, hypotheses $H_5(i),(ii),(iii)$ are the same as the corresponding hypotheses $H_4(i),(ii),(iii)$ and
\begin{itemize}
	\item[(iv)] there exists $\hat{\vartheta}>0$ such that for almost all $z\in\Omega$, the mapping $x\rightarrow f(z,x)+\hat{\vartheta}x$ is nondecreasing on $[-\rho,\rho]$.
\end{itemize}

\smallskip
This is a mild extra requirement on the reaction $f(z,\cdot)$ and it is satisfied if for example $f(z,\cdot)$  is differentiable and $f_x'(z,\cdot)$ is $L^\infty$-bounded on $[-\rho,\rho]$. This hypothesis leads to strong comparison results for $v_*$, $y_0$, $u_*$. Indeed we have
$$\begin{array}{ll}
&\di -\Delta u_*(z)+(\xi(z)+\hat\vartheta)u_*(z)=f(z,u_*(z))+\hat\vartheta u_*(z)\quad\mbox{for a.a.}\ z\in\Omega ,\\
&\di -\Delta y_0(z)+(\xi(z)+\hat\vartheta)y_0(z)=f(z,y_0(z))+\hat\vartheta y_0(z)\quad\mbox{for a.a.}\ z\in\Omega ,\\
\Longrightarrow &\di \Delta (u_*-y_0)(z)\leq (\xi(z)+\hat\vartheta)(u_*-y_0)(z)\quad\mbox{for a.a.}\ z\in\Omega,\\
\Longrightarrow &\di \Delta (u_*-y_0)(z)\leq (\|\xi\|_\infty +\hat\vartheta)(u_*-y_0)(z)\quad\mbox{for a.a.}\ z\in\Omega,\\
\Longrightarrow &\di u_*-y_0\in\mbox{int}\, C_+.
\end{array}
$$
Similarly, we show that $y_0-v_*\in\mbox{int}\, C_+$.

Therefore we can improve the conclusion of Proposition \ref{prop16}. We will need this stronger result in Section 4.

\begin{prop}\label{prop17}
	If hypotheses $H(\xi),H(\beta)$ and $H_4$ hold, then problem (\ref{eq1}) admits a nodal solution
	$$y_0\in int_{C^1(\overline{\Omega})}[v_*,u_*].$$
\end{prop}

So, now we can formulate our second multiplicity theorem.
\begin{theorem}\label{th18}
	If hypotheses $H(\xi),H(\beta)$ and $H_5$ hold, then problem (\ref{eq1}) admits at least three nontrivial solutions
	$$u_0\in \mbox{int}\,C_+,\ v_0\in-\mbox{int}\,C_+\ \mbox{and}\ y_0\in[v_0,u_0]\cap C^1(\overline{\Omega})\ \mbox{nodal}.$$
	Moreover, if hypotheses $H_4$ hold, then $y_0\in \mbox{int}_{C^1(\overline{\Omega})}[v_0,u_0]$.
\end{theorem}

Note that in this multiplicity result at zero we avoid any interaction with $\hat{\lambda}_2(\beta)$ (see hypothesis $H_4(iii)$). We can allow partial interaction (nonuniform nonresonance) at the expense of strengthening the behavior of $f(z,\cdot)$ near $x=0$.

So, we impose the following conditions on the reaction $f(z,x)$.

\smallskip
$H_6:$ $f:\Omega\times\RR\rightarrow\RR$ is a Carath\'eodory function such that $f(z,0)=0$ for almost all $z\in\Omega$ and
\begin{itemize}
	\item[(i)] there exist functions $w_{\pm}\in H^1(\Omega)\cap C(\overline{\Omega})$ such that
	\begin{eqnarray*}
		&&w_-(z)\leq c_-<0<c_+\leq w_+(z)\ \mbox{for all}\ z\in\overline{\Omega},\\
		&&f(z,w_+(z))-\xi(z)w_+(z)\leq 0\leq f(z,w_-(z))-\xi(z)w_-(z)\ \mbox{for almost all}\ z\in\Omega\\
		&&A(w_-)\leq 0\leq A(w_+)\ \mbox{in}\ H^1(\Omega)^*;
	\end{eqnarray*}
	\item[(ii)] if $\rho=\max\{||w_+||_{\infty},||w_-||_{\infty}\}$, then there exists $a_{\rho}\in L^{\infty}(\Omega)_+$ such that
	$$|f(z,x)|\leq a_{\rho}(z)\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ |x|\leq\rho;$$
	\item[(iii)] $f(z,\cdot)$ is locally Lipschitz and differentiable at $x=0$ and
	$$\eta(z)\leq f'_x(z,0)=\lim\limits_{x\rightarrow 0}\frac{f(z,x)}{x}\ \mbox{uniformly for almost all}\ z\in\Omega$$
	with $\eta\in L^{\infty}(\Omega),\eta(z)\geq\hat{\lambda}_2(\beta)$ for almost all $z\in\Omega$ and the inequality is strict on a set of positive measure.
\end{itemize}

\smallskip
So, we see from hypothesis $H_6(iii)$ that now at zero we allow partial interaction with $\hat\lambda_2(\beta)$ (nonuniform nonresonance), while in hypothesis $H_4(iii)$ we required uniform nonresonance (recall that in that hypothesis we had $\eta(z)>\hat\lambda_2(\beta)$ for a.a. $z\in\Omega$). The proof now changes and uses tools from Morse theory.

\begin{prop}\label{prop19}
	If hypotheses $H(\xi),H(\beta)$ and $H_6$ hold, then problem (\ref{eq1}) admits a nodal solution
	$$y_0\in[v_*,u_*]\cap C^1(\overline{\Omega}).$$
\end{prop}
\begin{proof}
	In this case there is a $C^1(\overline{\Omega})$-neighborhood $D$ of $u=0$ such that $\psi'\in C^1(D,H^1(\Omega))$ and $\psi''(0)\in\mathcal{L}(C^1(\Omega),H^1(\Omega))$. Using hypothesis $H_6(iii)$, we see that we can find $\delta\in(0,1)$ small such that
	$$\psi(u)\leq 0\ \mbox{for all}\ u\in \overset{2}{\underset{\mathrm{i=1}}\oplus}E(\hat{\lambda}_i(\beta)),\ ||u||\leq\delta.$$
	
	On the other hand from (\ref{eq7}) we see that we can have
	$$\psi(u)>0\ \mbox{for all}\ u\in\overline{{\underset{\mathrm{i\geq 3}}\oplus}E(\hat{\lambda}_i(\beta))},||u||\leq\delta.$$
	
	So, $\psi$ has a local linking at $u=0$, hence
	$$C_{d_2}(\psi,0)\neq 0\ \mbox{with}\ d_2=\mbox{dim}\, \overset{2}{\underset{\mathrm{i=1}}\oplus}E(\hat{\lambda}_i(\beta))\geq 2.$$
	
	Invoking Proposition \ref{prop4}, we infer that
	\begin{equation}\label{eq72}
		C_1(\psi,0)=0.
	\end{equation}
	
	On the other hand, from the proof of Proposition \ref{prop10} (see (\ref{eq48})), we have
	\begin{equation}\label{eq73}
		C_1(\psi,y_0)\neq 0
	\end{equation}
	
	Comparing (\ref{eq72}) and (\ref{eq73}), we conclude that $y_0\neq 0$ and so $y_0\in[v_*,u_*]\cap C^1(\overline{\Omega})$ is a nodal solution for problem (\ref{eq1}).
\end{proof}

\section{Four Nontrivial Solutions}

In this section we improve the regularity on $f(z,\cdot)$ and using Morse theory we are able to produce a second nodal solution, for a total of four nontrivial solutions all with precise sign information.

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

\smallskip
$H_7:$ $f:\Omega\times\RR\rightarrow\RR$ is a measurable function such that for almost all $z\in\Omega$ $f(z,0)=0,\ f(z,\cdot)\in C^1(\RR)$ and
\begin{itemize}
	\item[(i)] there exist functions $w_{\pm}\in H^1(\Omega)\cap C(\overline{\Omega})$ such that
	\begin{eqnarray*}
		&&w_-(z)\leq c_-<0<c_+\leq w_+(z)\ \mbox{for all}\ z\in\overline{\Omega},\\
		&&f(z,w_+(z))-\xi(z)w_+(z)\leq 0\leq f(z,w_-(z))-\xi(z)w_-(z)\ \mbox{for almost all}\ z\in\Omega,\\
		&&A(w_-)\leq 0\leq A(w_+)\ \mbox{in}\ H^1(\Omega)^*;
	\end{eqnarray*}
	\item[(ii)] if $\rho=\max\{||w_+||_{\infty},||w_-||_{\infty}\}$, then there exists $a_{\rho}\in L^{\infty}(\Omega)_+$ such that
	$$|f'_x(z,x)|\leq a_{\rho}(z)\ \mbox{for almost all}\ z\in \Omega,\ |x|\leq\rho;$$
	\item[(iii)] there exist an integer $m\geq 2$ and $\delta_0>0$ such that $\hat{\lambda}_m(\beta)x^2\leq f(z,x)x$ for almost all $z\in\Omega$, all $|x|\leq\delta_0$ if $m=2$, then the inequality is strict on a set of positive measure and $f'_x(z,0)=\lim\limits_{x\rightarrow 0}\frac{f(z,x)}{x}\leq\hat{\lambda}_{m+1}(\beta)$ uniformly for almost all $z\in\Omega$ and the inequality is strict on a set of positive measure.
	\end{itemize}

\begin{remark}
	From hypothesis $H_7(ii)$ and the mean value theorem, we see that we can find $\hat{\vartheta}>0$ such that for almost all $z\in\Omega$, the mapping $x\longmapsto f(z,x)+\hat{\vartheta}x$ is nondecreasing on $[-\rho,\rho]$. We know that Morse theory is more effective in the framework of $C^2$-functionals. For this reason we strengthened the regularity of $f(z,\cdot)$.
\end{remark}

\begin{theorem}\label{th20}
	If hypotheses $H(\xi),H(\beta)$ and $H_7$ hold, then problem (\ref{eq1}) admits at least four nontrivial solutions
	$$u_0\in \mbox{int}\,C_+,v_0\in-\mbox{int}\,C_+\ \mbox{and}\ y_0,\hat{y}\in int_{C^1(\overline{\Omega})}[v_0,u_0]\ \mbox{nodal}.$$
\end{theorem}
\begin{proof}
	From Theorem \ref{th18} and Proposition \ref{prop19}, we know that there are at least three nontrivial solutions
	$$u_0\in \mbox{int}\,C_+,v_0\in-\mbox{int}\,C_+\ \mbox{and}\ y_0\in \mbox{int}\, C_{C^1(\overline{\Omega})}[v_0,u_0]\ \mbox{nodal}.$$
	
	Let $\psi$ be the functional introduced in the proof of Proposition \ref{prop16}. We have that $\psi\in C^{2-0}(H^1(\Omega))$. Hypothesis $H_7(iii)$ implies that given $\epsilon>0$, we can find $\delta=\delta(\epsilon)\left(-,\delta_0\right]$ such that
	\begin{equation}\label{eq74}
		F(z,x)\leq\frac{1}{2}(f'_x(z,0)+\epsilon)x^2\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ |x|\leq\delta.
	\end{equation}
	
	Without any loss of generality, we may assume that $\delta_0\leq \min\{\min\limits_{\overline{\Omega}}u^*,\min\limits_{\overline{\Omega}}(-v_*)\}$ (recall that $u_*\in \mbox{int}\,C_+$ and $v_*\in-\mbox{int}\,C_+$). Let $\widehat{H}_m=\overline{{\underset{\mathrm{i\geq m+1}}\oplus}E(\hat{\lambda}_i(\beta))}$ and let $u\in C^1(\overline{\Omega})\cap \widehat{H}_m$ with $||u||_{C^1(\overline{\Omega})}\leq\delta$. Then
	\begin{eqnarray}\label{eq75}
		\psi(u)&=&\frac{1}{2}\sigma(u)-\int_{\Omega}F(z,u)dz\ (\mbox{see (\ref{eq43})})\nonumber\\
		 &\geq&\frac{1}{2}\sigma(u)-\frac{1}{2}\int_{\Omega}f'_x(z,0)u^2dz-\frac{\epsilon}{2}||u||^2_2\ (\mbox{see (\ref{eq74})})\nonumber\\
		&\geq&\frac{1}{2}\left(c_{11}-\frac{\epsilon}{\hat{\lambda}_1(\beta)}\right)||u||^2\ \mbox{for some}\ c_{11}>0\\
		&&\hspace{2.5cm}(\mbox{see hypothesis}\ H_7(iii)\ \mbox{and (\ref{eq3})}).\nonumber
	\end{eqnarray}
	
	Also, if $\bar{H}_m=\overset{m}{\underset{\mathrm{i=1}}\oplus}E(\hat{\lambda}_i(\beta))$ and $u\in C^1(\overline{\Omega})\cap\bar{H}_m$ with $||u||_{C^1(\overline{\Omega})}\leq 0$, then
	\begin{equation}\label{eq76}
		\psi(u)\leq\frac{1}{2}\sigma(u)-\frac{\hat{\lambda}_m(\beta)}{2}||u||^2_2\leq 0\ (\mbox{see hypothesis}\ H_7(iii)\ \mbox{and (\ref{eq5})}).
	\end{equation}
	
	From (\ref{eq75}) and (\ref{eq76}) it follows that $\psi$ has a local linking at the origin and so
	\begin{equation}\label{eq77}
		C_{d_m}\left(\psi|_{c^1(\overline{\Omega})},0\right)\neq 0\ \mbox{with}\ d_m=\mbox{dim}\, \bar{H}_m.
	\end{equation}
	
	From Palais \cite{15} (see also Chang \cite[p. 14]{4}), we have
	\begin{eqnarray*}
		&&C_k\left(\psi|_{C^1(\overline{\Omega})},0\right)\neq 0=C_k(\psi,0)\ \mbox{for all}\ k\geq 0,\\
		&\Rightarrow&C_{d_m}(\psi,0)\neq 0\ (\mbox{see (\ref{eq77})}).
	\end{eqnarray*}
	
	Since $\mu(0)=\mbox{dim}\,\overline{H}_{m-1}$ and $\nu(0)=\mbox{dim}\, E(\hat{\lambda}_m(\beta))$, from Proposition \ref{prop4} we have
	\begin{equation}\label{eq78}
		C_k(\psi,0)=\delta_{k,d_m}\ZZ\ \mbox{for all}\ k\geq 0.
	\end{equation}
	
	Suppose that $K_{\psi}=\{0,u_0,v_0,y_0\}$. We can always assume that $u_0$ and $v_0$ are the extremal constant solutions (that is, $u_0=u_*\in \mbox{int}\,C_+$ and $v_0=v_*\in-\mbox{int}\, C_+$, see Proposition \ref{prop8}). From the proof of Proposition \ref{prop16} (see Claim \ref{cl2}), we know that $u_0$ and $v_0$ are local minimizers of $\psi$. Hence
	\begin{equation}\label{eq79}
		C_k(\psi,u_0)=C_k(\psi,v_0)=\delta_{k,0}\ZZ\ \mbox{for all}\ k\geq 0.
	\end{equation}
	
	Recall that $y_0$ is a critical point of $\psi$ of mountain pass type (see the proof of Proposition \ref{prop16}). So, from Theorem 2.7 of Li, Li and Liu \cite{11}, we have
	\begin{equation}\label{eq80}
		C_k(\psi,y_0)=\delta_{k,1}\ZZ\ \mbox{for all}\ k\geq 0.
	\end{equation}
	
	Finally recall that $\psi$ is coercive (see (\ref{eq4})) and (\ref{eq43}). Hence
	\begin{equation}\label{eq81}
		C_k(\psi,\infty)=\delta_{k,0}\ZZ\ \mbox{for all}\ k\geq 0.
	\end{equation}
	
	From (\ref{eq78}), (\ref{eq79}), (\ref{eq80}), (\ref{eq81}) and the Morse relation with $t=-1$ (see (\ref{eq6})), 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, there exists $\hat{y}\in K_{\psi},\ \hat{y}\notin\{0,u_0,v_0,y_0\}$. We have
	$$\hat{y}\in[v_0,u_0]\cap C^1(\overline{\Omega})\ (\mbox{see Claim \ref{cl1} in the proof of Proposition \ref{prop16}}).$$
	
	Therefore $\hat{y}$ is a nodal solution of (\ref{eq1}). Moreover, as before (see the proof of Proposition \ref{prop17}), using the strong maximum principle, we show that $\hat{y}\in \mbox{int}_{C^1(\overline{\Omega})}[v_0,u_0]$.
\end{proof}

\section{A special case}

Consider the following Robin problem:
\begin{equation}\label{eq82}
	\left\{\begin{array}{ll}
		-\Delta u(z)+\xi(z)u(z)=\lambda u(z)-g(z,u(z))&\mbox{in}\ \Omega,\\
	\displaystyle	\frac{\partial u}{\partial n}+\beta(z)u(z)=0&\mbox{on}\ \partial\Omega,\lambda\in\RR
	\end{array}\right\}
\end{equation}

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

\smallskip
$H_8:$ $g:\Omega\times\RR\rightarrow\RR$ is a Carath\'eodory function such that $g(z,0)=0$ for almost all $z\in\Omega$ and
\begin{itemize}
	\item[(i)] for every $\rho>0$, there exists $a_{\rho}\in L^{\infty}(\Omega)_+$ such that
	$$|f(z,x)|\leq a_{\rho}(z)\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ |x|\leq\rho;$$
	\item[(ii)] $\lim\limits_{x\rightarrow+\infty}\frac{g(z,x)}{x}=+\infty$ uniformly for almost all $z\in\Omega$;
	\item[(iii)] $\lim\limits_{x\rightarrow 0}\frac{g(z,x)}{x}=0$ uniformly for almost all $z\in\Omega$;
	\item[(iv)] for every $\rho>0$, there exists $\hat{\vartheta}_{\rho}>0$ such that for almost all $z\in\Omega$, $x\longmapsto\hat{\vartheta}_{\rho}x-g(z,x)$ is nondecreasing on $[-\rho,\rho]$.
\end{itemize}
\begin{remark}
	If $g(z,x)=g(z)=|x|^{r-2}x$ with $r>2$, then we have the equidiffusive logistic equation with an indefinite and unbounded potential.
\end{remark}

Using Theorem \ref{th18}, we have:
\begin{theorem}\label{th21}
	If hypotheses $H(\xi),H(\beta)$ and $H_8(i),(ii),(iii)$ hold and $\lambda>\hat{\lambda}_2(\beta)$, then problem (\ref{eq82}) has at least three nontrivial solutions
	$$u_0\in \mbox{int}\,C_+,v_0\in-\mbox{int}\,C_+\ \mbox{and}\ y_0\in[v_0,u_0]\cap C^1(\overline{\Omega})\ \mbox{nodal}.$$
	Moreover, if in addition hypothesis $H_8(iv)$ holds, then
	$$y_0\in int_{C^1(\overline{\Omega})}[v_0,u_0].$$
\end{theorem}
\begin{remark}
	Such a multiplicity result, was first proved by Ambrosetti and Mancini \cite{3} with subsequent improvements by Ambrosetti and Lupo \cite{2} and Struwe \cite{23}, \cite{24}, for the Dirichlet problem, with $\xi\equiv 0$ and with stronger conditions on the perturbation $g$. None of the aforementioned works produced a nodal solution. The extension to Neumann problems (that is, $\beta\equiv 0$) with a potential term, was proved by Papageorgiou and Smyrlis \cite{21}. The extension to $p$-Laplacian equations with $\xi\equiv 0$ and Robin boundary condition, can be found in the recent work of Papageorgiou and R\u{a}dulescu \cite{19}.
\end{remark}

By strengthening the regularity on $f(z,\cdot)$, we can improve Theorem \ref{th21} by producing a second nodal solution.

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

\smallskip
$H_9:$ $g:\Omega\times\RR\rightarrow\RR$ is a measurable function such that for almost all $z\in\Omega$ $g(z,0)=0,\ g(z,\cdot)\in C^1(\RR)$ and
\begin{itemize}
	\item[(i)] for every $\rho>0$, there exists $a_{\rho}\in L^{\infty}(\Omega)_+$ such that
	$$|g'_{x}(z,x)|\leq a_{\rho}(z)\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ |x|\leq\rho;$$
	\item[(ii)] $\lim\limits_{x\rightarrow\pm \infty}\frac{g(z,x)}{x}=+\infty$ uniformly for almost all $z\in\Omega$;
	\item[(iii)] $g'_x(z,0)=\lim\limits_{x\rightarrow 0}\frac{g(z,x)}{x}=0$ uniformly for almost all $z\in\Omega.$
\end{itemize}

Using Theorem \ref{th20}, we have:
\begin{theorem}\label{th22}
	If hypotheses $H(\xi),H(\beta)$ and $H_9$ hold and $\lambda>\hat{\lambda}_2(\beta)$, then problem (\ref{eq82}) admits at least four nontrivial solutions
	$$u_0\in \mbox{int}\,C_+,v_0\in-\mbox{int}\,C_+\ \mbox{and}\ y_0,\hat{y}\in int_{C_1(\overline{\Omega})}[v_0,u_0]\ \mbox{nodal}.$$
\end{theorem}
\begin{remark}
	This theorem extends Theorem 14 of Papageorgiou and R\u{a}dulescu \cite{19}, where $\xi\equiv 0$ and the conditions on the perturbation $g(z,x)$ are a little stronger.
\end{remark}

\medskip
{\bf Acknowledgement}. The authors wish to thank to very knowledgeable referees for their corrections and helpful remarks which improved the paper considerably.





\begin{thebibliography}{99}

\bibitem{1} {\sc S. Aizicovici, N.S, Papageorgiou, V. Staicu}, {\em Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints}, Memoirs Amer. Math. Soc, Vol. 196, No. \textbf{915} (November 2008).

\bibitem{2} {\sc A. Ambrosetti, G. Mancini}, {\em Sharp nonuniqueness results for some nonlinear problems}, Nonlinear Anal. \textbf{3} (1979), 635-645.

\bibitem{3} {\sc A. Ambrosetti, D. Lupo}, {\em On a class of nonlinear Dirichlet problems with multiple solutions}, Nonlinear Anal. \textbf{8} (1984), 1145-1150.

\bibitem{brenir} {\sc H. Brezis, L. Nirenberg}, {\em $H^1$ versus $C^1$ local minimizers},  C. R. Acad. Sci. Paris, S\'er. I Math. {\bf 317} (1993),  465-472.

\bibitem{4} {\sc K.C. Chang}, {\em Infinite Dimensional Morse Theory and Multiple Solution Problems}, Birkh\"auser Boston (1993).

\bibitem{dusc} {\sc N. Dunford, J. Schwartz}, {\em Linear Operators I}, Wiley-Interscience, New York (1958).

\bibitem{5} {\sc D. de Figueiredo, J.P. Gossez}, {\em Strong monotonicity of eigenvalues and unique continuation}, Comm. Partial Diff. Equations \textbf{17} (1992), 339-346.

\bibitem{6} {\sc M. Filippakis, A. Kristaly, N.S. Papageorgiou}, {\em Existence of five nonzero solutions with exact sign for a $p$-Laplacian operator}, Discr. Cont. Dynam. Systems \textbf{24} (2009), 405-440.

\bibitem{fipa} {\sc M. Filippakis, N.S. Papageorgiou}, {\em Multiple constant sign and nodal solutions for nonlinear elliptic equations with the $p$-Laplacian}, J. Differential Equations {\bf 245} (2008), 1883-1922.

\bibitem{7} {\sc L. Gasinski, N.S. Papageorgiou}, {\em Nonlinear Analysis}, Chapman \& Hall/CRC, Boca Raton, Fl (2006).

\bibitem{8} {\sc A. Granas, J. Dugundji}, {\em Fixed Point Theory}, Springer-Verlag, New York (2003).

\bibitem{9} {\sc S. Hu, N.S. Papageorgiou} {\em Handbook of Multivalued Analysis. Volume I: Theory}, Kluwer Academic Publishers, Dordrecht, The Netherlands (1997).

\bibitem{10}  {\sc S. Kyritsi, N.S. Papageorgiou}, {\em Multiple solutions for superlinear Dirichlet problems with an indefinite potential}, Ann. Mat. Pura Appl. \textbf{192} (2013), 297-315.

\bibitem{11} {\sc C. Li, S. Li, J. Liu}, {\em Splitting theorem, Poincar\'e-Hopf theorem and jumping nonlinear problems}, J. Functional Anal. \textbf{221} (2005), 439-455.

\bibitem{12} {\sc E. Michael}, {\em Continuous selections I}, Ann. Math. \textbf{63} (1956), 361-382.

\bibitem{13} {\sc V. Moroz}, {\em Solutions of superlinear at zero elliptic equations via Morse theory}, Topol. Methods Nonlin. Anal. \textbf{10} (1997), 1-11.

\bibitem{mmpa} {\sc D. Motreanu, V. Motreanu, N.S. Papageorgiou}, {\em Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems}, Springer, New York (2014).

\bibitem{14} {\sc D. Mugnai, N.S. Papageorgiou}, {\em Resonant nonlinear Neumann problems with indefinite weight}, Ann. Sc. Norm. Super. Pisa \textbf{11} (2012), 729-788.

\bibitem{15} {\sc R. Palais}, {\em Homotopy theory of infinite dimensional manifolds}, Topology \textbf{5} (1966),  1-16.

\bibitem{16} {\sc N.S. Papageorgiou, S. Kyritsi}, {\em Handbook of Applied Analysis}, Springer, New York (2009).

\bibitem{17} {\sc N.S. Papageorgiou, F. Papalini}, {\em Seven solutions with sign information for sublinear equations with unbounded and indefinite potential and no symmetries}, Israel J. Math. {\bf 201} (2014), 761-796.

\bibitem{18} {\sc N.S. Papageorgiou, V.D. R\u{a}dulescu}, {\em Semilinear Neumann problems with indefinite and unbounded potential and crossing nonlinearity}, Contemp. Math \textbf{595} (2013), 293-315.

\bibitem{19} {\sc N.S. Papageorgiou, V.D. R\u{a}dulescu}, {\em Multiple solutions with precise sign information for parametric Robin problems}, J. Differential Equations \textbf{256} (2014), 2449-2479.

\bibitem{20} {\sc N.S. Papageorgiou, V.D. R\u{a}dulescu}, {\em Multiplicity of solutions for resonant Neumann problems with an indefinite and unbounded potential}, Trans. Amer. Math. Soc.
    (DOI: http://dx.doi.org/10.1090/S0002-9947-2014-06518-5).

\bibitem{21} {\sc N.S. Papageorgiou, G. Smyrlis}, {\em On a class of parametric Neumann problems with indefinite and unbounded potetial}, Forum Math. (DOI:10.1515/forum-2012-0042).

\bibitem{22} {\sc P. Pucci, J. Serrin}, {\em The Maximum Principle}, Birkh\"auser, Basel (2007).

\bibitem{23} {\sc M. Struwe}, {\em A note on a result of Ambrosetti and Mancini}, Ann. Mat. Pura. Appl. \textbf{81} (1982), 107-115.

\bibitem{24} {\sc M. Struwe}, {\em Variational Methods}, Springer-Verlag, Berlin (1990).

\bibitem{25} {\sc X. Wang}, {\em Neumann problems of semilinear elliptic equations involving critical Sobolev exponents}, J. Differential Equations \textbf{93} (1991), 283-310.

\end{thebibliography}


\end{document} 