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\title[Multiple solutions for Robin problems]{Multiple solutions with precise sign for \\nonlinear parametric Robin problems}


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

\author[V. R\u{a}dulescu]{Vicen\c{t}iu D. R\u{a}dulescu}
\address{Institute of Mathematics ``Simion Stoilow" of the Romanian Academy, P.O. Box 1-764,
          014700 Bucharest, Romania \&
 Department of Mathematics, University of Craiova, Street A.I. Cuza 13,
          200585 Craiova, Romania}
\email{\tt vicentiu.radulescu@imar.ro}

\keywords{Nonlinear regularity, nonlinear maximum principle, Robin $p$-Laplacian, eigenvalues, nodal and constant sign solutions, extremal solutions, Morse theory.\\
\phantom{aa} 2010 AMS Subject Classification:
35J20, 35J60, 58E05}

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\begin{abstract}
We consider a parametric nonlinear Robin problem driven by the $p$-Laplacian. We show that if the parameter $\lambda>\hat{\lambda}_2=$ the second eigenvalue of the Robin $p$-Laplacian, then the problem has at least three nontrivial solutions, two of constant sign and the third nodal. In the semilinear case $(p=2)$, we show that we can generate a second nodal solution. Our approach uses variational methods, truncation and perturbation techniques, and Morse theory. In the process we produce two useful remarks about the first two eigenvalues of the Robin $p$-Laplacian.
\end{abstract}

\maketitle

\tableofcontents


\section{Introduction}
Let $\Omega\subseteq \RR^N$ be a bounded domain with a $C^2$--boundary $\partial\Omega$. In this paper we study the following nonlinear parametric Robin problem:

$$\left\{
\begin{array}{lll}
&\di	-\Delta_p u(z)=\lambda|u(z)|^{p-2}u(z)-f(z,u(z))&\quad\mbox{in}\ \Omega,\\
&\di	\frac{\partial u}{\partial n_p}(z)+\beta(z)|u(z)|^{p-2}u(z)=0&\quad\mbox{on}\ \partial\Omega.
\end{array}
\right.  \eqno(P_{\lambda})$$

In this problem, $\Delta_p\ (1<p<\infty)$ denotes the $p$-Laplacian differential operator defined by
$$\Delta_p u=\mbox{div}(||Du||^{p-2}Du)\quad \mbox{for\ all}\ u\in W^{1,p}(\Omega).$$
Also, $\frac{\partial u}{\partial n_p}=||Du||^{p-2}(Du,n)_{\RR^N}$ with $n(z)$ being the outward unit normal at $z\in\partial\Omega$. In addition, $\lambda>0$ is a parameter and $f(z,x)$ is a Carath\'eodory perturbation (that is, for all $x\in\RR$,\ $z\longmapsto f(z,x)$ is measurable and for a.a. $z\in\Omega$, $x\longmapsto f(z,x)$ is continuous), which exhibits $(p-1)$-superlinear growth near $\pm\infty$.

Our aim in this paper is to prove a multiplicity theorem for problem $(P_{\lambda})$ for all $\lambda>0$ big. More precisely, we show that, if $\hat{\lambda}_2$ is the second eigenvalue of $-\Delta_p$ with Robin boundary conditions (denoted by $-\Delta^{R}_{p}$) and $\lambda>\hat{\lambda}_2$ then problem $(P_{\lambda})$ admits at least three nontrivial solutions, two of constant sign (the first positive and the second negative) and the third solution is nodal (sign changing). Moreover, in the semilinear case $(p=2)$, we show the existence of a second nodal solution, for a total of four nontrivial solutions all with sign information. Our approach uses variation methods coupled with suitable truncation and perturbation techniques and Morse theory.

This kind of problem was studied for semilinear (that is, $p=2$) Dirichlet equations by Ambrosetti \& Lupo \cite{2}, Ambrosetti \& Mancini \cite{3} and Struwe \cite{20}, \cite[p. 133]{21}. Extensions to Dirichlet $p$-Laplacian equations can be found in Papageorgiou \& Papageorgiou \cite{18}. However, none of the aforementioned works produced nodal solutions and the hypotheses on the data of the problem are more restrictive. Another class of Robin eigenvalue problems was investigated by Duchateau \cite{7}, who proved multiplicity results producing two solutions with no sign information.

\section{Mathematical Background - Auxiliary Results}

Let $X$ be a Banach space and let $X^*$ be 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 (PS-condition for short), if the following is true
\begin{eqnarray}
	&&``Every\ sequence\ \{x_n\}_{n\geq 1}\subseteq X\ such\ that\ \{\varphi(x_n)\}_{n\geq 1}\subseteq\RR\ is\ bounded\ and\nonumber\\
	&&\varphi'(x_n)\rightarrow 0\ in\ X^*,
admits\ a\ strongly\ convergent\ subsequence."\nonumber
\end{eqnarray}

This is a compactness type condition, which compensates for the fact that the underlying space $X$ being in general infinite dimensional, need not be locally compact. It leads to the following minimax theorem, known in the literature as the ``mountain pass theorem". It characterizes certain critical values of $\varphi \in C^1(X)$.

\begin{theorem}\label{th1}
If $\varphi\in C^1(X)$ satisfies the PS-condition, $x_0,x_1\in X,\ ||x_1-x_0||>\rho>0$
$$\max\{\varphi(x_0),\varphi(x_1)\}<\inf\left[\varphi(x):||x-x_0||=\rho\right]=\eta_{\rho}$$
and $c=\inf\limits_{\gamma\in \Gamma}\max\limits_{0\leq t\leq 1}\ \varphi(\gamma(t))$ where $\Gamma=\{\gamma\in C([0,1],X):\gamma(0)=x_0,\ \gamma(1)=x_1\}$,
then $c\geq\eta_{\rho}$ and $c$ is a critical value of $\varphi$.
\end{theorem}

In the analysis of problem $(P_{\lambda})$, in addition to the Sobolev space $W^{1,p}(\Omega)$, we will also use the Banach space $C^1(\bar{\Omega})$, which is an ordered Banach space with positive cone
$$C_+=\{u\in C^1(\bar{\Omega}):u(z)\geq 0\ \mbox{for\ all}\ z\in\bar{\Omega}\}.$$
This cone has a nonempty interior given by
$$\mbox{int}\,C_+=\{u\in C_+:u(z)>0\ \mbox{for\ all}\ z\in\bar{\Omega}\}.$$
In the sequel by $||\cdot||$ we denote the norm of the Sobolev space $W^{1,p}(\Omega)$, that is,
$$||u||=\left[\ ||u||^{p}_{p}+||Du||^{p}_{p}\ \right]^{1/p}\ \mbox{for\ all}\ u\in W^{1,p}(\Omega).$$
To distinguish, by $|\cdot|$ we denote the norm in $\RR^m\ (m\geq 1)$. Also, given $x\in\RR$, we set $x^{\pm}=\max\{\pm x,0\}$. Then for $u\in W^{1,p}(\Omega)$, we define $u^{\pm}(\cdot)=u(\cdot)^{\pm}$. We know that
$$u^{\pm}\in W^{1,p}(\Omega),\ |u|=u^++u^-,\ u=u^+-u^-$$
If on $\partial \Omega$ we employ the $(N-1)$-dimensional surface (Hausdorff) measure $\sigma(\cdot)$, we can define the Lebesgue space $L^p(\partial\Omega)$. Recall that there is a unique continuous, linear map $\gamma_0:W^{1,p}(\Omega)\rightarrow L^p(\partial\Omega)$ such that $\gamma_0(u)=u|_{\partial\Omega}$ for all $u\in C^1(\bar{\Omega})$. This map is known as the ``trace map". Recall that $\mbox{im}\,\gamma_0=W^{\frac{1}{p'},p}(\partial\Omega)\ (\frac{1}{p}+\frac{1}{p'}=1)$ and $\mbox{ker}\,\gamma_0=W^{1,p}_{0}(\Omega)$. In the sequel, for the sake of notational simplicity, we will drop the use of the map $\gamma_0$ to denote the restriction of a Sobolev function on $\partial\Omega$. All such restrictions are understood in the sense of traces.

If $h:\Omega\times\RR\rightarrow\RR$ is a measurable function (for example, a Carath\'eodory function), then we set
$$N_h(u)(\cdot)=h(\cdot,u(\cdot))\ \mbox{for\ all}\ u\in W^{1,p}(\Omega)$$
(the Nemytskii map corresponding to $h$)

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

\begin{prop}\label{prop1}
	The map $A:W^{1,p}(\Omega)\rightarrow W^{1,p}(\Omega)^*$ is bounded (that is, maps bounded sets to bounded sets), continuous, monotone (hence maximal monotone, too) and of type $(S)_+$, that is, if $u_n\stackrel{w}{\rightarrow}u\ in\ W^{1,p}(\Omega)$ and $\limsup\limits_{n\rightarrow\infty}\left\langle A(u_n),u_n-u\right\rangle\leq 0$, then $u_n\rightarrow u\ in\ W^{1,p}(\Omega).$
\end{prop}

Suppose that $f_0(z,x)$ is a Carath\'eodory function with subcritical growth in $x\in\RR$, that is,
$$|f_0(z,x)|\leq a(z)(1+|x|^{r-1})\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\in\RR,$$
with $a_0\in L^{\infty}(\Omega)_+$ and $1<r<p^*=\left\{\begin{array}{cl}\frac{Np}{N-p}&\mbox{if}\ p<N\\+\infty&\mbox{otherwise}.
\end{array}\right.$

We set $F_0(z,x)=\int^{x}_{0}f_0(z,s)ds$ and consider the $C^1$-functional $\varphi_0:W^{1,p}(\Omega)\rightarrow\RR$ defined by
$$\varphi_0(u)=\frac{1}{p}||Du||^{p}_{p}+\frac{1}{p}\int_{\partial\Omega}\beta(z)|u(z)|^p d\sigma-\int_{\Omega}F_0(z,u(z))dz\ \mbox{for\ all}\ u\in W^{1,p}(\Omega).$$

As a consequence of the nonlinear regularity theory (see Lieberman \cite{15}), we show that local $C^1(\bar{\Omega})$-minimizers and local $W^{1,p}(\Omega)$-minimizers of $\varphi_0$ coincide. The first such result is due to Brezis \& Nirenberg \cite{5} for the space $H^{1}_{0}(\Omega)$. It was extended to the space $W^{1,p}(\Omega)$ with $\beta\equiv 0$, see Motreanu \& Papageorgiou \cite{16}.

We impose the following conditions on the boundary weight $\beta(\cdot)$:
$$H(\beta):\ \beta\in C^{0,\tau}(\bar{\Omega})\ \mbox{with}\ \tau \in(0,1),\ \beta(z)\geq 0\ \mbox{for\ all}\ z\in\bar{\Omega},\ \beta\neq 0.$$

\begin{prop}\label{prop2}
 Assume that $u_0\in W^{1,p}(\Omega)$ is a local $C^1(\bar{\Omega})$-minimizer of $\varphi_0$, that is, there exists $\rho_0>0$ such that
$$\varphi_0(u_0)\leq \varphi_0(u_0+h)\ for\ all\ h\in C^1(\bar{\Omega})\ with\ ||h||_{C^1(\bar{\Omega})}\leq\rho_0.$$
Then $u_0\in C^{1,\alpha}(\bar{\Omega})$ for same $\alpha\in(0,1)$ and $u_0$ is a local $W^{1,p}(\Omega)$-minimizer of $\varphi_0,$
that is, there exists $\rho_1>0$ such that
$$\varphi_0(u_0)\leq\varphi_0(u_0+h)\ for\ all\ h\in W^{1,p}(\Omega)\ with\ ||h||\leq\rho_1.$$
\end{prop}

\begin{proof}
Let $h\in C^1(\bar{\Omega})$ and $t>0$ small. Then by hypothesis we have
\begin{eqnarray}\label{eq1}
&&\varphi_0(u_0)\leq\varphi_0(u_0+h),\nonumber\\
\Rightarrow&&0\leq\langle \varphi'_{0}(u_0),h\rangle\ \mbox{for\ all}\ h\in C^1(\bar{\Omega}),\nonumber\\
\Rightarrow&&\varphi'_{0}(u_0)=0\ (\mbox{since}\ C^1(\bar{\Omega})\ \mbox{is\ dense\ in}\ W^{1,p}(\Omega)),\nonumber\\
\Rightarrow&&\left\langle A(u_0,h)\right\rangle+\int_{\partial\Omega}\beta(z)|u_0|^{p-2}u_0 h\ d\sigma=\int_{\Omega}f_0(z,u_0)h\ dz\\
&&\hspace{5cm}\mbox{for\ all}\ h\in W^{1,p}(\Omega).\nonumber
\end{eqnarray}
From the nonlinear Green's identity (see, for example, Gasinski \& Papageorgiou \cite[p. 210]{12}), we have
\begin{eqnarray}\label{eq2}
\left\langle A(u_0),h\right\rangle=\left\langle -\Delta_p u_0, h\right\rangle+\left\langle \frac{\partial u_0}{\partial n_p},h\right\rangle_{\partial\Omega}\ \mbox{for\ all}\ h\in W^{1,p}(\Omega)
\end{eqnarray}
(see Gasinski \& Papageorgiou \cite[p. 211]{12}). Here by $\left\langle \cdot,\cdot\right\rangle_{\partial\Omega}$ we denote the duality brackets for the pair $\left(W^{-\frac{1}{p'},p'}(\partial\Omega),\ W^{\frac{1}{p'},p}(\partial\Omega)\right)\left(\frac{1}{p}+\frac{1}{p'}=1\right)$. From the representation theorem for the dual space $W^{1,p}_{0}(\Omega)^*=W^{-1,p'}(\Omega)$ (see Gasinski \& Papageorgiou \cite[p. 212]{12}), we have $\Delta_p u_0\in W^{-1,p'}(\Omega)$. Then using $h\in W^{1,p}_{0}(\Omega)\subseteq W^{1,p}(\Omega)$ in (\ref{eq1}), we have
\begin{eqnarray}
	&&\left\langle -\Delta_p u_0,h\right\rangle=\int_{\Omega}f_0(z,u_0)hdz\ \mbox{for\ all}\ h\in W^{1,p}_{0}(\Omega)\ (\mbox{see}\ (\ref{eq2})),\nonumber\\
	\Rightarrow&&-\Delta_p u_0(z)=f_0(z,u_0(z))\ \mbox{a.e.\ in}\ \Omega.\nonumber
\end{eqnarray}
Then from (\ref{eq1}) and (\ref{eq2}) we have
$$\left\langle \frac{\partial u_0}{\partial n_p}+\beta(z)|u_0|^{p-2}u_0,h\right\rangle_{\partial\Omega}=0\ \mbox{for\ all}\ h\in W^{1,p}(\Omega).$$
Recall that the image of the trace map is $W^{\frac{1}{p'},p}(\partial\Omega)$. So, from this last equality it follows that
$$\frac{\partial u_0}{\partial n_p}+\beta(z)|u_0|^{p-2}u_0=0\ \mbox{on}\ \partial\Omega.$$
From Winkert \cite{24}, we know that $u_0\in L^{\infty}(\Omega)$. So, we can apply Theorem 2 of Lieberman \cite{15} and have that
$$u_0\in C^{1,\alpha}(\bar{\Omega})\ \mbox{for\ some}\ \alpha\in(0,1).$$
Next we show that $u_0$ is a local $W^{1,p}(\Omega)$-minimizer of $\varphi_0$. We argue by contradiction. So, suppose that $u_0$ is not a local $W^{1,p}(\Omega)$-minimizer of $\varphi_0$. Let $\epsilon>0$ and consider the set $\bar{B}^{r}_{\epsilon}=\{h\in W^{1,p}(\Omega):||h||_r\leq\epsilon\}$. We have
\begin{eqnarray}\label{eq3}
-\infty<m^{\epsilon}_{0}=\inf\left[\varphi_0(u_0+h):h\in\bar{B}^{r}_{\epsilon}\right].
\end{eqnarray}
By virtue of the contradiction hypothesis, we have
\begin{eqnarray}\label{eq4}
m^{\epsilon}_{0}<\varphi_0(u_0).
\end{eqnarray}
Let $\{h_n\}_{n\geq 1}\subseteq\bar{B}^{r}_{\epsilon}$ be a minimizing sequence for problem (\ref{eq3}). Recalling that $u\rightarrow||u||_r+||Du||_p$ is an equivalent norm on $W^{1,p}(\Omega)$ (see \cite[p. 227]{12}), we see that $\{h_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)$ is bounded and so we may assume that
$$h_n\stackrel{w}{\rightarrow}h_{\epsilon}\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{and}\ h_n\rightarrow h_{\epsilon}\ \mbox{in}\ L^r(\Omega).$$
Clearly $\varphi_0$ is sequentially weakly lower semicontinuous. So, we have
\begin{eqnarray}
&&\varphi_0(u_0+h_{\epsilon})\leq\liminf\limits_{n\rightarrow\infty}\varphi_0(u_0+h_n),\nonumber\\
\Rightarrow&&\varphi_0(u_0+h_{\epsilon})=m^{\epsilon}_{0}\ \mbox{and}\ h_{\epsilon}\neq 0\ (see\ (\ref{eq4})).\nonumber
\end{eqnarray}
From  the Lagrange multiplier rule, we can find $\lambda_{\epsilon}\leq 0$ such that
\begin{eqnarray}
&&\varphi'_{0}(u_0+h_{\epsilon})=\lambda_{\epsilon}|h_{\epsilon}|^{r-2}h_{\epsilon}\nonumber\\
\Rightarrow&&\left\langle A(u_0+h_{\epsilon}),v\right\rangle+\int_{\partial\Omega}\beta(z)|u_0+h_{\epsilon}|^{p-2}(u_0+h_{\epsilon})v\ d\sigma\nonumber\\
&&=\int_{\Omega}f_0(z,u_0+h_{\epsilon})vdz+\lambda_{\epsilon}\int_{\Omega}|h_{\epsilon}|^{r-2}h_{\epsilon}v\ dz\ \mbox{for\ all}\ v\in W^{1,p}(\Omega).\nonumber
\end{eqnarray}
From this equality, as in the first part of the proof, we infer that
\begin{equation}\label{eq5}
\left\{
\begin{array}{cl}
-\Delta_p(u_0+h_{\epsilon})(z)=f_0(z,(u_0+h_{\epsilon})(z))+
\lambda_{\epsilon}|h_{\epsilon}(z)|^{r-2}h_{\epsilon}(z)&\mbox{a.e.\ in}\ \Omega,\\
\frac{\partial(u_0+h_{\epsilon})}{\partial n_p}+\beta(z)|u_0+h_{\epsilon}|^{p-2}(u_0+h_{\epsilon})=0&\mbox{on}\ \partial\Omega.
\end{array}
\right\}
\end{equation}
Recall that
\begin{eqnarray}\label{eq6}
	-\Delta_p u_0(z)=f_0(z,u_0(z))\ \mbox{a.e.\ in}\ \Omega,\ \frac{\partial u_0}{\partial n_p}+\beta(z)|u_0|^{p-2}u_0=0\ \mbox{on}\ \partial\Omega.
\end{eqnarray}
From (\ref{eq5}) and (\ref{eq6}) it follows that
\begin{equation}\label{eq7}
\left\{
\begin{array}{rl}
-\Delta_p(u_0+h_{\epsilon})(z)+\Delta_p u_0(z)=f_0(z,(u_0+h_{\epsilon})(z))-&\\
f_0(z,u_0(z))+\lambda_{\epsilon}|h_{\epsilon}(z)|^{r-2}h_{\epsilon}(z)&\mbox{a.e.\ in}\ \Omega,\\
\displaystyle\frac{\partial (u_0+ h_{\epsilon})}{\partial n_p}-\frac{\partial u_0}{\partial n_p}+\beta(z)\left[|u_0+h_{\epsilon}|^{p-2}(u_0+h_{\epsilon})-|u_0|^{p-2}u_0\right]=0&\mbox{on}\ \partial\Omega.
\end{array}
\right\}
\end{equation}
We consider two distinct cases:
\begin{case}\label{case1}
	$\lambda_{\epsilon}\in[-1,1]$ for all $\epsilon\in\left(0,1\right]$.
\end{case}

Let $v_{\epsilon}(z)=(u_0+h_{\epsilon})(z)$ and set $\sigma_{\epsilon}(z,y)=||y||^{p-2}y+||Du_0(z)||^{p-2}Du_0(z)$. Then we can rewrite (\ref{eq7}) as follows
\begin{center}
$\left\{
\begin{array}{rl}
-\mbox{div}\sigma_{\epsilon}(z,Dv_{\epsilon}(z))=f_0(z,v_{\epsilon}(z))-f_0(z,u_0(z))+&\\
\lambda_{\epsilon}|(v_{\epsilon}-u_0)(z)|^{r-2}(v_{\epsilon}-u_0)(z)& \mbox{a.e.\ in}\ \Omega\\
 \frac{\partial v_{\epsilon}}{\partial n_p}-\frac{\partial u_0}{\partial n_p}+\beta(z)\left[|v_{\epsilon}(z)|^{p-2}v_{\epsilon}(z)-|u_0(z)|^{p-2}u_0(z)\right]&=0.
\end{array}
\right\}$
\end{center}

Note that $\frac{\partial u_0}{\partial n_p}\in C^{0,\alpha}(\partial\Omega)$ (recall that $u_0\in C^{1,\alpha}(\bar{\Omega})$) and also on $\RR^m(m\geq 1)$ the map $y\rightarrow |y|^{p-2}y$ is locally Lipschitz if $p> 2$ and H\"{o}lder continuous if $1<p<2$. From Winkert \cite{24}, we know that we can find $M_1>0$ such that $||v_{\epsilon}||_{\infty}\leq M_1$ for all $\epsilon\in\left(0,1\right]$. Therefore, we can apply Theorem 2 of Lieberman \cite{15} and find $\gamma\in(0,1),\ M_2>0$ such that
$$v_{\epsilon}\in C^{1,\gamma}(\bar{\Omega}),\ ||v_{\epsilon}||_{C^{1,\gamma}(\bar{\Omega})}\leq M_2\ \mbox{for\ all}\ \epsilon\in\left(0,1\right].$$

\begin{case}\label{case2}
$\lambda_{\epsilon_n}<-1$ for all $n\geq 1$ with $\epsilon_n\downarrow 0$.
\end{case}

In this case, we set
$$\hat{\sigma}_{\epsilon_n}(z,y)=\frac{1}{|\lambda_{\epsilon_n}|}\left[\ |y|^{p-2}y-|Du_0(z)|^{p-2}Du_0(z)\right]$$
Then we can rewrite (\ref{eq7}) as follows
\begin{equation}\label{eq8}
\left\{
	\begin{array}{rl}
		-\mbox{div}\ \hat{\sigma}_{\epsilon_n}(z,Dv_{\epsilon_n}(z))=\frac{1}{|\lambda_{\epsilon_n}|}\left[\ f_0(z,v_{\epsilon_n}(z))-f_0(z,u_0(z))\ \right]-&\\
		|(v_{\epsilon_n}-u_0)(z)|^{r-2}(v_{\epsilon_n}-u_0)(z)& \mbox{a.e.\ in}\ \Omega\\
		\frac{\partial v_{\epsilon_n}}{\partial n_p}-\frac{\partial u_0}{\partial n_p}+\beta(z)\left[|v_{\epsilon_n}(z)|^{p-2}v_{\epsilon_n}(z)-|u_0(z)|^{p-2}u_0(z)\right]=0&\mbox{on}\ \partial\Omega.
	\end{array}
\right\}
\end{equation}

Recall that for all $y\in W^{1,p}(\Omega)$, we have
\begin{eqnarray}
&&\left\langle A(u_0),y\right\rangle+\int_{\partial\Omega}\beta(z)|u_0|^{p-2}u_0\ y\ d\sigma=\int_{\Omega}f_0(z,u_0)y\ dz,\label{eq9}\\
&&\left\langle A(v_{\epsilon_n}),y\right\rangle+\int_{\partial\Omega}\beta(z)|v_{\epsilon_n}|^{p-2}v_{\epsilon_n}y\ d\sigma=\int_{\Omega}f_0(z,v_{\epsilon_n})y\ dz+\nonumber\\
&&\hspace{2cm}\lambda_{\epsilon_n}\int_{\Omega}|v_{\epsilon_n}-u_0|^{r-2}(v_{\epsilon_n}-u_0)dz\ \mbox{for\ all}\ n\geq 1.\label{eq10}
\end{eqnarray}

Let $\mu>1$ and consider the function
$$|v_{\epsilon_n}-u_0|^{\mu}(v_{\epsilon_n}-u_0).$$

We have
\begin{eqnarray}
	&&D(|v_{\epsilon_n}-u_0|^{\mu}(v_{\epsilon_n}-u_0))\nonumber\\
	 =&&|v_{\epsilon_n}-u_0|^{\mu}D(v_{\epsilon_n}-u_0)+\mu(v_{\epsilon_n}-u_0)\frac{v_{\epsilon_n}-u_0}{|v_{\epsilon_n}-u_0|}|v_{\epsilon_n}-u_0|^{\mu-1}D(v_{\epsilon_n}-u_0)\nonumber\\
	=&&(\mu+1)|v_{\epsilon_n}-u_0|^{\mu}D(v_{\epsilon_n}-u_0),\nonumber\\
	\Rightarrow&&|v_{\epsilon_n}-u_0|^{\mu}(v_{\epsilon_n}-u_0)\in W^{1,p}(\Omega)\ (\mbox{recall\ that}\ v_{\epsilon_n},u_0\in C^1(\bar{\Omega}),\ \mbox{see}\ (\ref{eq7})).\nonumber
\end{eqnarray}
So, we can use $|v_{\epsilon_n}-u_0|^{\mu}(v_{\epsilon_n},u_0)$ as a test function in (\ref{eq9}) and (\ref{eq10}). We have
\begin{eqnarray}\label{eq11}
0&&\leq\left\langle A(v_{\epsilon_n})-A(u_0),|v_{\epsilon_n}-u_0|^{\mu}(v_{\epsilon_n}-u_0)\right\rangle+\nonumber\\
&&\hspace{2cm}\int_{\partial\Omega}\beta (z)\left[\ |v_{\epsilon_n}|^{p-2}v_{\epsilon_n}-|u_0|^{p-2}u_0\right]|v_{\epsilon_n}-u_0|^{\mu}(v_{\epsilon_n}-u_0)d\sigma\nonumber\\
&&=\int_{\Omega}\left[f_0(z,v_{\epsilon_n})-f_0(z,u_0)\right]|v_{\epsilon_n}-u_0|^{\mu}(v_{\epsilon_n}-u_0)dz+\lambda_{\epsilon_n}\int_{\Omega}|v_{\epsilon_n}-u_0|^{r+\mu}dz.
\end{eqnarray}
As before, from Winkert \cite{24}, we have $||v_{\epsilon_n}||_{\infty}\leq M_3$ for some $M_3>0$, all $n\geq 1.$ Hence
\begin{eqnarray}\label{eq12}
	 &&\left|\int_{\Omega}\left[f_0(z,v_{\epsilon_n})-f_0(z,u_0)\right]|v_{\epsilon_n}-u_0|^{\mu}(v_{\epsilon_n}-u_0)dz\right|\nonumber\\
	&&\leq\ M_4\ \int_{\Omega}|v_{\epsilon_n}-u_0|^{\mu+1}dz\ \mbox{for\ some}\ M_4>0,\ \mbox{all}\ n\geq 1\nonumber\\
	&&\leq\ M_4\ |\Omega|^{\frac{r-1}{\mu+r}}_{N}||v_{\epsilon_n}-u_0||^{\mu+1}_{\mu+r}
\end{eqnarray}
(here we have used H\"{o}lder's inequality with conjugate exponents $\frac{\mu+r}{\mu+1},\ \frac{\mu+r}{r-1}$ )

From (\ref{eq11}) and (\ref{eq12}) it follows that
\begin{eqnarray}
	&&-\lambda_{\epsilon_n}||v_{\epsilon_n}-u_0||^{\mu+r}_{\mu+r}\leq M_4\ |\Omega|^{\frac{r-1}{\mu+r}}_{N}||v_{\epsilon_n}-u_0||^{\mu+1}_{\mu+r},\nonumber\\
	\Rightarrow&&|\lambda_{\epsilon_n}|\ ||v_{\epsilon_n}-u_0||^{r-1}_{\mu+r}\leq M_4|\Omega|^{\frac{r-1}{\mu+r}}_{N}\ \mbox{for\ all}\ n\geq 1.\nonumber
\end{eqnarray}
Recall that $\mu>1$ is arbitrary. So, we let $\mu\rightarrow +\infty$ and obtain
\begin{eqnarray}\label{eq13}
	||h_{\epsilon_n}||_{\infty}\leq\left[\frac{M_4}{|\lambda_{\epsilon_n}|}\right]^{\frac{1}{r-1}}\ \mbox{for\ all}\ n\geq 1.
\end{eqnarray}
In (\ref{eq8}) we denote the reaction (right hand side) by $\vartheta_{\epsilon_n}(z,x)$. Using (\ref{eq13}) we have
$$|\vartheta_{\epsilon_n}(z,h_{\epsilon_n}(z))|\leq\frac{M_5}{|\lambda_{\epsilon_n}|^{\frac{1}{r-1}}}\leq M_5\ \mbox{for\ all}\ n\geq 1\ \mbox{and\ some}\ M_5>0.$$
From this, as before, the nonlinear regularity theory (see \cite{15}) implies the existence of $\gamma_0\in(0,1),\ M_6>0$ such that
$$h_{\epsilon_n}\in C^{1,\gamma_0}(\bar{\Omega})\ \mbox{and}\ ||h_{\epsilon_n}||_{C^{1,\gamma_0}(\bar{\Omega})}\leq M_6\ \mbox{for\ all}\ n\geq 1.$$
So, in both Case \ref{case1} and Case \ref{case2}, we reached similar uniform bounds for the sequence $\left\{h_{\epsilon_n}\right\}_{n\geq 1}\subseteq C^{1,\mu}(\bar{\Omega})$ for some $\mu\in(0,1)$. Therefore, exploiting the compact embedding of $C^{1,\mu}(\bar{\Omega})$ into $C^1(\bar{\Omega})$, we may assume that
$$u_0+h_{\epsilon_n}\rightarrow u_0\ \mbox{in}\ C^1(\bar{\Omega})\ (\mbox{recall\ that}\ ||h_{\epsilon_n}||_r\leq\epsilon_n\ \mbox{for\ all}\ n\geq 1).$$
But by hypothesis $u_0$ is a local $C^1(\bar{\Omega})$-minimizer of $\varphi_0$. So, we can find $n_0\geq 1$ such that
$$\varphi_0(u_0)\leq \varphi_0(u_0+h_{\epsilon_n})\ \mbox{for\ all}\ n\geq n_0.$$
On the other hand, from the choice of the $h'_{n}s$ we have
$$\varphi_0(u_0+h_{\epsilon_n})<\varphi_0(u_0)\ \mbox{for\ all}\ n\geq 1\ (\mbox{see}\ (\ref{eq4})),$$
a contradiction. This proves that $u_0$ is also a local $W^{1,p}(\Omega)$ minimizer of $\varphi_0$.
\end{proof}

Finally, we recall some basic definitions and facts from Morse theory (critical groups). So, let $X$ be a Banach space and let $(Y_1,Y_2)$ be a topological pair with $Y_2\subseteq Y_1\subseteq X$. For every integer $k\geq 0$, by $H_k(Y_1,Y_2)$ we denote the $k$th relative singular homology group with integer coefficients for the pair $(Y_1,Y_2)$. Recall that $H_k(Y_1,Y_2)=0$ for all integers $k<0$.

Let $X$ be a Banach space and $\varphi\in C^1(X),\ c\in\RR$. We introduce the following sets
$$\varphi^c=\{x\in X:\varphi(x)\leq c\},\ K_{\varphi}=\{x\in X: \varphi'(x)=0\},\ K^{c}_{\varphi}=\{x\in K_{\varphi}:\varphi(x)=c\}.$$
Then the critical groups of $\varphi$ at an isolated critical point $x\in X$ with $\varphi(x)=c$, are defined by
$$C_k(\varphi,x)=H_k(\varphi^c\cap U,\ \varphi^c\cap U\backslash\{x\})\ \mbox{for\ all}\ k\geq 0.$$
Here $U$ is a neighborhood of $x$ such that $K_{\varphi}\cap \varphi^c\cap U=\{x\}$. The excision property of the singular homology theory implies that the above definition of critical groups is independent of the particular choice of the neighborhood $U$.

Suppose that $\varphi\in C^1(X)$ satisfies the PS-condition and $\inf \varphi(K_{\varphi})>\infty$. Let $c<\inf \varphi(K_{\varphi})$. The critical groups of $\varphi$ at infinity, are defined by
$$C_k(\varphi,\infty)=H_k(X,\varphi^c)\ \mbox{for\ all}\ k\geq 0.$$
The second deformation theorem (see, for example, Gasinski \& Papageorgiou \cite[p. 628]{12}), implies that the above definition of critical groups, is independent of the particular choice of the level $c<\inf \varphi(K_{\varphi})$.

Assume that $K_{\varphi}$ is finite and introduce the following items:
\begin{eqnarray}
	&&M(t,x)=\sum\limits_{k\geq 0}\mbox{rank}\ C_k(\varphi,x)t^k\ \mbox{for\ all}\ t\in\RR,\ \mbox{all}\ x\in K_{\varphi}\nonumber\\
	&&P(t,\infty)=\sum\limits_{k\geq 0}\mbox{rank}\ C_k(\varphi,\infty)t^k\ \mbox{for\ all}\ t\in\RR.\nonumber
\end{eqnarray}
The Morse relation says that
\begin{equation}\label{eq14}
\sum\limits_{x\in K_{\varphi}}\ M(t,x)=P(t,\infty)+(1+t)Q(t),
\end{equation}
where $Q(t)=\sum\limits_{k\geq 0}\beta_k t^k$ is a formal series in $t\in\RR$ with nonnegative integer coefficients.

Suppose that $X=H$ is a Hilbert space, $x\in H,\ U$ a neighborhood of $x$ and $\varphi\in C^2(U)$. Suppose that $\varphi\in C^2(U)$ and assume that $x\in K_{\varphi}$.
Then, the Morse index of $x\in K_{\varphi}$ is defined to be the supremum of the dimensions of the subspaces of $H$ on which $\varphi''(x)$ is negative definite. The nullity of $x\in K_{\varphi}$, is the dimension of $\mbox{ker}\, \varphi''(x)$. We say that $x\in K_{\varphi}$ is nondegenerate, if $\varphi''(x)$ is invertible, that is, the nullity of $x$ is zero. If $x\in K_{\varphi}$ is nondegenerate with Morse index $m$, then $C_k(\varphi,x)=\delta_{k,m}\ \mathbb Z$ for all $k\geq 0$, where

$\delta_{k,m}=
\left\{
\begin{array}{cl}
1&\mbox{if}\ k=m\\
0&\mbox{if}\ k\neq m
\end{array}
\right.$ (the Kronecker symbol)

Suppose that $H=Y\oplus V$ with $\mbox{dim}\, Y<+\infty$ and $\varphi\in C^1(H)$. We say that $\varphi$ admits a local linking at the origin with respect to the decomposition $(Y,V)$, if there exists $\rho>0$ such that
$$
\varphi(u)\leq \varphi(0)\ \mbox{for\ all}\ u\in Y,\ ||u||\leq \rho$$
$$\varphi(u)\geq \varphi(0)\ \mbox{for\ all}\ u\in V,\ ||u||\leq \rho\,.
$$


\section{Some Remarks on the Spectrum of $-\Delta^{R}_{p}$.}

We consider the following nonlinear eigenvalue problem
\begin{equation}\label{eq15}
\left\{
\begin{array}{cl}
-\Delta_p u(z)=\lambda|u(z)|^{p-2}u(z)& \mbox{in}\ \Omega,\\
\frac{\partial u}{\partial n_p}(z)+\beta(z)|u(z)|^{p-2}u(z)=0& \mbox{on}\ \partial\Omega.
\end{array}
\right\}
\end{equation}
We say that $\lambda\in\RR$ is an eigenvalue of $-\Delta^{R}_{p}$, if problem (\ref{eq15}) admits a nontrivial solution. This eigenvalue problem, was investigated by Le \cite{14}, who proved many important facts concerning the first two eigenvalues of $-\Delta^{R}_{p}$. Here, we prove two additional results concerning the first two eigenvalues of $-\Delta^{R}_{p}$.

We introduce the following quantity
\begin{eqnarray}\label{eq16}
	\hat{\lambda}_1=\inf \left[\frac{||Du||^{p}_{p}+\int_{\partial \Omega}\beta(z)|u|^pd\sigma}{||u||^{p}_{p}}:u\in W^{1,p}(\Omega),\ u\neq 0\right].
\end{eqnarray}
This is the first eigenvalue of $-\Delta^{R}_{p}$ (see \cite{14}). We also have:
\begin{prop}\label{prop3}
	If $\beta\in L^{\infty}(\partial\Omega)\backslash\{0\}$ and $\beta(z)\geq 0$\ $\sigma$--a.e. on $\partial\Omega$, then $\hat{\lambda}_1>0$.
\end{prop}

\begin{proof}
	Evidently $\hat{\lambda}_1\geq 0$. Suppose that $\hat{\lambda}_1=0$ and let $\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)$ be such that
	\begin{equation}\label{eq17}
		||Du_n||^{p}_{p}+\int_{\partial\Omega}\beta(z)|u_n|^p d\sigma\rightarrow 0^+\ \mbox{as}\ n\rightarrow\infty,\ \mbox{with}\ ||u_n||_p=1\ \mbox{for\ all}\ n\geq 1.
	\end{equation}
	Clearly $\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)$ is bounded and so we any assume that
	\begin{equation}\label{eq18}
		u_n\stackrel{w}{\rightarrow} u\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{and}\ u_n\rightarrow u\ \mbox{in}\ L^p(\Omega).
	\end{equation}
	From the weak lower semicontinuity of the norm functional in a Banach space, we have
	$$||Du||^{p}_{p}\leq \liminf\limits_{n\rightarrow\infty}||Du_n||^{p}_{p}.$$
	Moreover, the continuity of the trace map and (\ref{eq18}), imply that
	 $$\int_{\partial\Omega}\beta(z)|u_n|^pd\sigma\rightarrow\int_{\partial\Omega}\beta(z)|u|^pd\sigma.$$
	Therefore in the limit as $n\rightarrow\infty$, we have
	$$||Du||^{p}_{p}+\int_{\partial\Omega}\beta(z)|u|^pd\sigma\leq 0
		\Rightarrow u\equiv 0,$$\ a\ contradiction\ to\ the\ fact\ that\ $||u||_p=1$\ (see\ (\ref{eq17}),\ (\ref{eq18})).
\end{proof}

From Le \cite{14}, we know that $\hat{\lambda}_1>0$ is a simple eigenvalue (that is, if $u,y$ are eigenfunctions corresponding to $\hat{\lambda}_1$, then $u=\vartheta y$ for some $\vartheta\in\RR\backslash\{0\}$) and it isolated (that is, if $\sigma_R(p)$ denotes the set of eigenvalues of $-\Delta^{R}_{p}$, then there exists $\epsilon>0$ such that ($\hat{\lambda}_1,\ \hat{\lambda}_1+\epsilon)\cap\sigma_R(p)=\O$). Let $\hat{u}_1\in W^{1,p}(\Omega)$ be the $L^p$-normalized (that is, $||\hat{u}_1||_p=1$) eigenfunction corresponding to $\hat{\lambda}_1>0$. It is clear from (\ref{eq16}) that $\hat{u}_1$ does not change sign and so we may assume that $\hat{u}_1\geq 0$. If hypothesis $H(\beta)$ holds, then Theorem 2 of Lieberman \cite{15} implies that $\hat{u}_1\in C_+\backslash\{0\}$. Finally the nonlinear strong maximum principle of Vazquez \cite{23} implies that $\hat{u}_1\in\mbox{int}\, C_+$.

The Ljusternik-Schnirelmann minimax scheme, implies that $-\Delta^{R}_{p}$ admits a whole strictly increasing sequence of eigenvalues $\{\hat{\lambda}_k\}_{k\geq 1}$ such that $\hat{\lambda}_k\rightarrow+\infty$. These eigenvalues are known as the LS-eigenvalues (or variational eigenvalues) of $-\Delta^{R}_{p}$. If $p=2$ (linear eigenvalue problem), then $\sigma_R(p)=\{\hat{\lambda}_k\}_{k\geq 1}$. If $p\neq 2$ (nonlinear eigenvalue problem), then we do not know if this is the case. We can easily see that $\sigma_R(p)$ is closed. Since $\hat{\lambda}_1>0$ is isolated, we can define
$$\hat{\lambda}^{*}_{2}=\inf [\lambda\in\sigma_R(p):\lambda>\hat{\lambda}_1].$$
The closedness of $\sigma_R(p)$ implies that $\hat{\lambda}^{*}_{2}$ is the second eigenvalue of $-\Delta^{R}_{p}$. We have (see Le \cite{14})
$$\hat{\lambda}^{*}_{2}=\hat{\lambda}_2,$$
that is, the second eigenvalue and the second LS-eigenvalue of $-\Delta^{R}_{p}$ coincide. For $\hat{\lambda}_2$ we have the minimax characterization provided by the Ljustenik-Schnirelmann theory. In the next proposition, we produce an alternative minimax characterization of $\hat{\lambda}_2$, which is more suitable to our purposes. Analogous characterizations for the Dirichlet and Neumann $p$-Laplacians, were produced by Cuesta, de Figueiredo \& Gossez \cite{6} and by Aizicovici, Papageorgiou \& Staicu \cite{1} respectively.

\begin{prop}\label{prop4}
	Assume that hypotheses $H(\beta)$ hold. Then $\hat{\lambda}_2=\inf\limits_{\hat{\gamma}\in\hat{\Gamma}}\max\limits_{-1\leq t\leq 1}\varphi(\hat{\gamma}(t))$, where
	\begin{eqnarray}
		&&\hat{\Gamma}=\left\{\hat{\gamma}\in C([-1,1],M):\hat{\gamma}(-1)=-\hat{u}_1,\ \hat{\gamma}(1)=\hat{u}_1\right\}\nonumber\\
		&&M=W^{1,p}(\Omega)\cap\partial B^{L^p}_{1},\ \partial B^{L^p}_{1}=\{u\in L^p(\Omega):||u||_p=1\}\nonumber\\
		and&&\varphi(u)=||Du||^{p}_{p}+\int_{\partial\Omega}\beta(z)|u|^p d\sigma\ for\ all\ u\in W^{1,p}(\Omega).\nonumber
	\end{eqnarray}
\end{prop}

\begin{proof}
By Ljusternik's theorem (see, for example, Papageorgiou \& Kyritsi \cite[p. 74]{17}), we know that $M$ is a $C^1$-Banach manifold and
$$T_uM=\left\{h\in W^{1,p}(\Omega):\ \int_{\Omega}|u|^{p-2}uh\ dz=0\right\}\ \mbox{for\ all}\ u\in M$$
(the tangent space to $M$ at $u$).

\begin{claim}
$\varphi|_{M}$ satisfies the PS-condition.
\end{claim}

Let $\{u_n\}_{n\geq 1}\subseteq M$ such that
\begin{eqnarray}
&&|\varphi(u_n)|\leq M_1\ \mbox{for\ some}\ M_1>0,\ \mbox{all}\ n\geq 1\ \ \ \ \mbox{and}\label{eq19}\\
&&\left|\left\langle A(u_n),h\right\rangle+\int_{\partial\Omega}\beta|u_n|^{p-2}u_n h\ d\sigma\right|\leq\epsilon_n||h||\ \mbox{for\ all}\ h\in T_{u_n}M\ \mbox{with}\ \epsilon_n\rightarrow 0^+.\label{eq20}
\end{eqnarray}
Given any $y\in W^{1,p}(\Omega)$, we define
$$h=y-\left(\int_{\Omega}|u_n|^{p-2}u_ny\ dz\right)u_n$$
Evidently $h\in T_{u_n}M$ and so we can use it as a test function in (\ref{eq20}). We have
\begin{eqnarray}
&&\left|\left\langle A(u_n),y\right\rangle-\left(\int_{\Omega}|u_n|^{p-2}u_ny\ dz\right)||Du||^{p}_{p}+\int_{\partial\Omega}\beta|u_n|^{p-2}u_ny\ d\sigma-\right.\nonumber\\
&&\hspace{2cm}\left.\left(\int_{\Omega}|u_n|^{p-2}u_ny\ dz\right)\ \int_{\partial\Omega}\beta|u_n|^{p}d\sigma\right|\leq\epsilon_n||h||\ \mbox{for\ all}\ n\geq 1,\nonumber\\
\Rightarrow&&\left|\left\langle A(u_n),y\right\rangle+\int_{\partial\Omega}\beta|u_n|^{p-2}u_ny\ d\sigma-\left(\int_{\Omega}|u_n|^{p-2}u_ny\ dz\right)\varphi(u_n)\right|\nonumber\\
&&\hspace{2cm}\leq\epsilon_n c_1||y||\ \mbox{for\ some}\ c_1>0,\ \mbox{all}\ n\geq 1\ (\mbox{see\ Goldberg\ \cite[p. 48]{13} }),\nonumber\\
\qquad \Rightarrow&&\left|\left\langle A(u_n),y\right\rangle+\int_{\partial\Omega}\beta|u_n|^{p-2}u_ny\ d\sigma\right|\leq c_2||y||\ \mbox{for\ some}\ c_2>0,\ \mbox{all}\ n\geq 1\label{eq21}\\
&&\hspace{5cm}(\mbox{see}\ (\ref{eq19})\ \mbox{and\ recall}\ ||u_n||_p=1\ \mbox{for\ all}\ n\geq 1).\nonumber
\end{eqnarray}
From (\ref{eq19}) and since $\int_{\partial\Omega}\beta|u_n|^{p}d\sigma\geq 0$ for all $n\geq 1$, we have that $\{Du_n\}_{n\geq 1}\subseteq L^p(\Omega,\ \RR^N)$ is bounded. Recall that $\{u_n\}_{n\geq 1}\subseteq M$, hence $||u_n||_p=1$ for all $n\geq 1$. Therefore $\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)$ is bounded and so we may assume that
$$u_n\stackrel{w}{\rightarrow}u\ \mbox{in}\ W^{1,p}(\Omega).$$
Since $y\in W^{1,p}(\Omega)$ is arbitrary, in (\ref{eq21}) we may choose $y=u_n-u$. We pass to the limit and exploit the continuity of the trace map. We obtain
$$\lim\limits_{n\rightarrow\infty}\left\langle A(y_n),y_n-y\right\rangle=0
\Rightarrow u_n\rightarrow u\ \mbox{in}\ W^{1,p}(\Omega).
$$
This proves the Claim.

Note that
$$\varphi(\pm\hat{u}_1)=\hat{\lambda}_1\ \mbox{and\ both}\ \pm\hat{u}_1\ \mbox{are\ local\ minimizers\ of}\ \varphi.$$
From Filippakis, Kristaly \& Papageorgiou \cite{10} (see the proof of Proposition 3.2) or from de Figueiredo \cite[p. 42]{9}, we know that we can find $\rho_{\pm}\in(0,1)$ such that
\begin{eqnarray}\label{eq22}
	&&\varphi(\pm\hat{u}_1)<\inf\left[\varphi(u):u\in M,\ ||u-(\pm\hat{u}_1)||=\rho_{\pm}\right],\ \rho_{\pm}<2||\hat{u}_1||
\end{eqnarray}
Let
\begin{eqnarray}\label{eq23}
	&&\hat{\lambda}=\inf\limits_{\hat{\gamma}\in\hat{\Gamma}}\max\limits_{-1\leq t\leq 1}\varphi(\hat{\gamma}(t))
\end{eqnarray}
Every path connecting $-\hat{u}_1$ and $\hat{u}_1$ crosses $\partial B_{\rho_{\pm}}(\pm\hat{u}_1)$ (see (\ref{eq22})) and $\varphi(\pm\hat{u}_1)=\hat{\lambda}_1$, from (\ref{eq23}) we see that $\hat{\lambda}>\hat{\lambda}_1$. It is well-known that $\hat{\lambda}$ is a critical value of $\varphi|_{M}$, hence an eigenvalue of $-\Delta^{R}_{p}$ distinct from $\hat{\lambda}_1$.

Suppose that $\lambda\in(\hat{\lambda}_1,\hat{\lambda})$ is an eigenvalue of $-\Delta^{R}_{p}$ with $\hat{u}\in M$ a corresponding eigenfunction. From Le \cite{14}, we know that $\hat{u}$ must be nodal (sign changing) and so, we have $\hat{u}^+\neq 0,\ \hat{u}^-\neq 0$. We consider the following two paths in the manifold $M$
\begin{eqnarray}\label{eq24}
	&&\gamma_1(t)=\frac{\hat{u}^{+}_{1}-t\hat{u}^{-}_{1}}{||\hat{u}^+-t\hat{u}^-||_p}\ and\ \gamma_2(t)=\frac{-\hat{u}^-+(1-t)\hat{u}^+}{||-\hat{u}^-+(1-t)\hat{u}^+||_p}\ \mbox{for\ all}\ t\in[0,1].
\end{eqnarray}
Note that $\gamma_1$ connects $\frac{\hat{u}^{+}_{+}}{||\hat{u}^+||_p}$ with $\hat{u}$, while $\gamma_2$ connects $\hat{u}$ with $\frac{-\hat{u}^-}{||\hat{u}^-||_p}$. So, if concatenate the two paths, we produce a path $\gamma$ in $M$ connecting $\frac{\hat{u}^+}{||\hat{u}^+||_p}$ with $\frac{-\hat{u}^-}{||\hat{u}^-||_p}$.

Recall that
\begin{eqnarray}\label{eq25}
	&&-\Delta_p\hat{u}(z)=\lambda|\hat{u}(z)|^{p-2}\hat{u}(z)\ \mbox{a.e.\ in}\ \Omega,\ \frac{\partial\hat{u}}{\partial n_p}+\beta(z)|\hat{u}|^{p-2}\hat{u}=0\ \mbox{on}\ \partial\Omega.
\end{eqnarray}
On (\ref{eq25}) we act with $\hat{u}^+$. Using the nonlinear Green's identity (see, for example Gasinski \& Papageorgiou \cite[p. 211]{12}), we have
\begin{eqnarray}
	&&\int_{\Omega}|D\hat{u}|^{p-2}(D\hat{u},D\hat{u}^+)_{\RR^N}\ dz-\int_{\partial\Omega}\frac{\partial\hat{u}}{\partial n_p}\hat{u}^{+}d\sigma=\lambda||\hat{u}^{+}||^{p}_{p},\nonumber\\
	 \Rightarrow&&||D\hat{u}^+||^{p}_{p}+\int_{\partial\Omega}\beta(\hat{u}^+)^pd\sigma=\lambda||\hat{u}^+||^{p}_{p}.\label{eq26}
\end{eqnarray}
Similarly, acting on (\ref{eq25}) with $-\hat{u}^-\in W^{1,p}(\Omega)$, we obtain
\begin{eqnarray}\label{eq27}
	 &&||D\hat{u}^-||^{p}_{p}+\int_{\partial\Omega}\beta(\hat{u}^-)^pd\sigma=\lambda||\hat{u}^-||^{p}_{p}\ .
\end{eqnarray}
From (\ref{eq24}), (\ref{eq26}), (\ref{eq27}) and since $\hat{u}^+$ and $\hat{u}^-$ have disjoint interior supports, we have
\begin{eqnarray}\label{eq28}
	\varphi(\gamma_1(t))=\varphi(\gamma_2(t))=\lambda\ \mbox{for\ all}\ t\in[0,1].
\end{eqnarray}
Let $\hat{L}=\{u\in M:\varphi(u)<\hat{\lambda}\}$. Since $\hat{u}_1,\ -\hat{u}_1\in\hat{L}$, this set cannot be path connected or otherwise we violate relation (\ref{eq23}). Moreover, using the Ekeland variational principle and the fact that $\varphi|_M$ satisfies the PS-condition (see the Claim), we see that every path component of $\hat{L}$ contains a critical point of $\varphi|_M$. Since $\pm\hat{u}_1$, are the only critical points of $\varphi|_M$ in $\hat{L}$, we infer that $\hat{L}$ has two path components.

Since $\frac{\hat{u}^+}{||\hat{u}^+||_p}\in M\cap(\mbox{int}\, C_+)$ and $\varphi\left(\frac{\hat{u}^+}{||\hat{u}^+||_p}\right)=\lambda$ (see (\ref{eq26})), we see that $\frac{\hat{u}^+}{||\hat{u}^+||_p}$ can not be a critical point of $\varphi|_M$. Hence we can find a path $s:[-\epsilon,\epsilon]\rightarrow M$ such that
$$s(0)=\frac{\hat{u}^+}{||\hat{u}^+||_p}\ \mbox{and}\ \frac{d}{dt}\left(\varphi|_M\right)\ (s(t))\neq 0\ \mbox{for\ all}\ t\in[-\epsilon,\epsilon].$$
Moving along this path, we can start from $\frac{\hat{u}^+}{||\hat{u}^+||_p}$ and reach a point $y\in M$ staying in the set $\hat{L}$ with the exception of the starting point $\frac{\hat{u}^+}{||\hat{u}^+||_p}$. Let $U_1$ be the path-component of $\hat{L}$ containing $y$. Without any loss of generality, we may assume that $\hat{u}_1\in U_1$. Then $y$ and $\hat{u}_1$ can be connected by a path which stays in $U_1$. Concatenating this path with $s$ introduced above, we have a path $\gamma_+:[0,1]\rightarrow U_1$ such that
$$\gamma_+(0)=\hat{u}_1,\ \gamma_+(1)=\frac{\hat{u}^+}{||\hat{u}^+||_p}\ \mbox{and}\ \gamma_+(t)\in \hat{L}\ \mbox{for\ all}\ t\in\left[0,1\right).$$
Similarly, if $U_2$ is the other path component of $\hat{L}$ containing $-\hat{u}_1$, then we produce a path $\gamma_-:[0,1]\rightarrow U_2$ such that
$$\gamma_-(0)=\frac{-\hat{u}^-}{||\hat{u}^-||_p},\ \gamma_-(1)=-\hat{u}_1\ \mbox{and}\ \gamma_-(t)\in\hat{L}\ \mbox{for\ all}\ t\in\left(0,1\right].$$
Finally, we concatenate $\gamma_-,\ \gamma,\ \gamma_+$ and have $\hat{\gamma}_*\in\hat{\Gamma}$ such that
\begin{eqnarray}
	&&\varphi(\gamma_*(t))\leq\lambda\ \mbox{for\ all}\ t\in[-1,1],\nonumber\\
	\Rightarrow&&\hat{\lambda}\leq\lambda\ (\mbox{see}\ (\ref{eq23})),\ \mbox{a\ contradiction}.\nonumber
\end{eqnarray}
This means that $(\hat{\lambda}_1,\hat{\lambda})\cap\sigma_R(\rho)=\O$ and so we conclude that $\hat{\lambda}=\hat{\lambda}_2$.
\end{proof}

\section{Nonlinear Equations}

We introduce the following conditions on the perturbation $f(z,x)$:

\textbf{$H_1$:} $f:\Omega\times\RR\rightarrow\RR$ is a Carath\'eodory function such that $f(z,0)=0$ for a.a. $z\in\Omega$ and
\begin{description}
	\item[(i)]  for every $\rho>0$ there exists $a_\rho\in L^{\infty}(\Omega)_+$ such that $|f(z,x)|\leq a_\rho (z)$ for a.a. $z\in \Omega$, all $|x|\leq \rho$;
	\item[(ii)] $\lim\limits_{x\rightarrow\pm\infty}\frac{f(z,x)}{|x|^{p-2}x}=+\infty$ uniformly for a.a. $z\in\Omega$;
	\item[(iii)] $\lim\limits_{x\rightarrow 0}\frac{f(z,x)}{|x|^{p-2}x}=0$ uniformly for a.a. $z\in\Omega$.
\end{description}

\begin{remark}
	We stress that no global growth restriction is imposed on $f(z,\cdot)$. So, the function $x\longmapsto f(z,x)$ can have any growth faster than $|x|^{p-2}x$ near $\pm\infty$
\end{remark}

First we produce two nontrivial constant sign solutions.

\begin{prop}\label{prop5}
	Assume that hypotheses $H(\beta)$ and $H_1$ hold and $\lambda>\hat{\lambda}_1$. Then problem $(P_{\lambda})$ has at least two nontrivial constant sign solutions
	$$u_0\in{\rm int}\, C_+\ and\ v_0\in-{\rm int}\, C_+.$$
\end{prop}

\begin{proof}
First we produce a nontrivial positive solution.

By virtue of hypothesis $H_1(ii)$, given $\xi>0$, we can find $M_7=M_7(\xi)>0$ such that
$$f(z,x)\geq\xi x^{p-1}\ \mbox{for\ a.a.}\ z\in\Omega \ \mbox{all}\ x\geq M.$$
Since $\hat{u}_1\in\mbox{int}\, C_+$, we can find $t>0$ big such that $t\hat{u}_1\geq M_7$. Then we have
\begin{eqnarray}\label{eq28'}
 &&f(z,t\hat{u}_1(z))\geq\xi(t\hat{u}_1(z))^{p-1}\ \mbox{a.e.\ in}\ \Omega.
\end{eqnarray}
Also, we have
$$-\Delta_p(t\hat{u}_1)(z)=\hat{\lambda}_1(t\hat{u}_1)(z)^{p-1}\ \mbox{a.e.\ in}\ \Omega,\ \frac{\partial(t\hat{u}_1)}{\partial n_p}+\beta(z)(t\hat{u}_1)^p=0\ \mbox{on}\ \partial\Omega.$$
Then for every $h\in W^{1,p}(\Omega),\ h\geq 0$, we have
\begin{eqnarray}\label{eq29}
	&&\left\langle -\Delta_p(t\hat{u}_1),h\right\rangle=\int_{\Omega}\hat{\lambda}_1(t\hat{u}_1)^{p-1}h\ dz,\nonumber\\
	\Rightarrow&&\left\langle A(t\hat{u}_1),h\right\rangle-\left\langle \frac{\partial(t\hat{u}_1)}{\partial n_p},h\right\rangle_{\partial\Omega}=\int_{\Omega}\hat{\lambda}_1(t\hat{u}_1)^{p-1}h\ dz\nonumber\\
	&&\hspace{3cm}(\mbox{by\ the\ nonlinear\ Green's\ identity,\ see\ \cite[p. 211]{12}}),\nonumber\\
	\Rightarrow&&\left\langle A(t\hat{u}_1),h\right\rangle+\int_{\partial\Omega}\beta(z)(t\hat{u}_1)^{p-1}h\ d\sigma=\int_{\Omega}\hat{\lambda}_1(t\hat{u}_1)^{p-1}h\ dz.
\end{eqnarray}
Choosing $\xi=\lambda-\hat{\lambda}_1>0$, from (\ref{eq28'}) and (\ref{eq29}), we have
\begin{eqnarray}\label{eq30}
	&&\int_{\Omega}\left[\lambda(t\hat{u}_1)^{p-1}-f(z,t\hat{u}_1)\right]h\ dz\nonumber\\
	\leq&&\int_{\Omega}\hat{\lambda}_1(t\hat{u}_1)^{p-1}h\ dz\nonumber\\
	=&&\left\langle A(t\hat{u}_1),h\right\rangle+\int_{\partial\Omega}\beta(z)(t\hat{u}_1)^{p-1}h\ d\sigma\ \mbox{for\ all}\ h\in W^{1,p}(\Omega),\ h\geq 0.
\end{eqnarray}
Setting $\bar{u}=t\hat{u}_1\in\mbox{int}\, C_+$, we introduce the following truncation-perturbation of the reaction in problem $(P_{\lambda})$
\begin{equation}\label{eq31}
h^{+}_{\lambda}(z,x)=
\left\{
	\begin{array}{ll}
		0&\mbox{if}\ x<0\\
		(\lambda+1)x^{p-1}-f(z,x)&\mbox{if}\ 0\leq x\leq \bar{u}(z)\\
		(\lambda+1)\bar{u}(z)^{p-1}-f(z,\bar{u}(z))&\mbox{if}\ \bar{u}(z)<x\ .
	\end{array}
\right.
\end{equation}
This is a Carath\'eodory function. We set $H^{+}_{\lambda}(z,x)=\int^{x}_{0}h^{+}_{\lambda}(z,s)ds$ and consider the $C^1$-functional $\Psi^{+}_{\lambda}:W^{1,p}(\Omega)\rightarrow\RR$ defined by
\begin{eqnarray} &&\Psi^{+}_{\lambda}(u)=\frac{1}{p}||Du||^{p}_{p}+\frac{1}{p}||u||^{p}_{p}+\frac{1}{p}\int_{\partial\Omega}\beta(z)|u(z)|^pd\sigma-\int_{\Omega}H^{+}_{\lambda}(z,u(z))dz\nonumber\\
	&&\hspace{8cm}\mbox{for\ all}\ u\in W^{1,p}(\Omega).\nonumber
\end{eqnarray}
From (\ref{eq31}) it is clear that $\Psi^{+}_{\lambda}$ is coercive. Also, using the Sobolev embedding theorem and the continuity of the trace map, we see that $\Psi^{+}_{\lambda}$ is sequentially weakly lower semicontinuous. So, we can find $u_0\in W^{1,p}(\Omega)$ such that
\begin{equation}\label{eq32}
	\Psi^{+}_{\lambda}(u_0)=\inf\left[\Psi^{+}_{\lambda}(u):u\in W^{1,p}(\Omega)\right].
\end{equation}
By virtue of hypothesis $H_1(iii)$ given $\epsilon>0$, we can find $\delta=\delta(\epsilon)\in\left(0,\min\limits_{\bar{\Omega}}\bar{u}\right]$ such that
\begin{equation}\label{eq33}
	F(z,x)\leq\frac{\epsilon}{p}|x|^p\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\in[0,\delta].
\end{equation}
Choose $\vartheta\in(0,1)$ small such that $\vartheta\hat{u}_1(z)\in\left(0,\delta\right]$ for all $z\in\bar{\Omega}$. Then
\begin{eqnarray} \Psi^{+}_{\lambda}(\vartheta\hat{u}_1)&=&\frac{\vartheta^p}{p}||D\hat{u}_1||^{p}_{p}+\frac{\vartheta^p}{p}\int_{\partial\Omega}\beta(z)|\hat{u}_1|^pd\sigma-\frac{\lambda\vartheta^p}{p}||\hat{u}_1||^{p}_{p}+\int_{\Omega}F(z,t\hat{u}_1)dz\ (\mbox{see}\ (\ref{eq31}))\nonumber\\
	&\leq&\frac{\vartheta^p}{p}[(\hat{\lambda}_1+\epsilon)-\lambda]\ (\mbox{see\ (\ref{eq33})\ and\ recall\ that}\ ||\hat{u}_1||_p=1)\nonumber
\end{eqnarray}
Choosing $\epsilon\in(0,\lambda-\hat{\lambda}_1)$ (recall $\lambda>\hat{\lambda}_1$), we have
\begin{eqnarray}
	&&\Psi^{+}_{\lambda}(\vartheta\hat{u}_1)<0,\nonumber\\
	\Rightarrow&&\Psi^{+}_{\lambda}(u_0)<0=\Psi^{+}_{\lambda}(0)\ (\mbox{see}\ (\ref{eq32})),\ \mbox{hence}\ u_0\neq 0.\nonumber
\end{eqnarray}
From (\ref{eq32}) we have
\begin{eqnarray}\label{eq34}
	&&(\Psi^{+}_{\lambda})'(u_0)=0,\nonumber\\
	\Rightarrow&&\left\langle A(u_0),v\right\rangle+\int_{\Omega}|u_0|^{p-2}u_0v\ dz+\int_{\partial\Omega}\beta(z)|u_0|^{p-2}u_0\ v\ d\sigma\nonumber\\
	=&&\int_{\Omega}h^{+}_{\lambda}(z,u_0)\ v\ dz\ \mbox{for\ all}\ v\in W^{1,p}(\Omega).
\end{eqnarray}
In (\ref{eq34}) first we choose $v=-u^{-}_{0}\in W^{1,p}(\Omega)$. Then
\begin{eqnarray}
	&&||Du^{-}_{0}||^{p}_{p}+||u^{-}_{0}||^{p}_{p}\leq 0\ (\mbox{see}\ (\ref{eq31})\ \mbox{and}\ H(\beta)),\nonumber\\
	\Rightarrow&&u_0\geq 0,\ u_0\neq 0.\nonumber
\end{eqnarray}
Next in (\ref{eq34}) we choose $v=(u_0-\bar{u})^+\in W^{1,p}(\Omega)$. Then
\begin{eqnarray}
	&&\left\langle A(u_0),(u_0-\bar{u})^+\right\rangle+\int_{\Omega}u^{p-1}_{0}(u_0-\bar{u})^+dz+\int_{\partial\Omega}\beta(z)u^{p-1}_{0}(u_0-\bar{u})^+d\sigma\nonumber\\
	=&&\int_{\Omega}\left[(\lambda+1)\bar{u}^{p-1}-f(z,\bar{u})\right](u_0-\bar{u})^+dz\ (\mbox{see}\ (\ref{eq31}))\nonumber\\
	\leq&&\left\langle A(\bar{u}),(v_0-\bar{u})^+\right\rangle+\int_{\Omega}\bar{u}^{p-1}(u_0-\bar{u})^+dz+\int_{\partial\Omega}\beta(z)\bar{u}^{p-1}(u_0-\bar{u})^+d\sigma\nonumber\\
	&&\hspace{10cm}(\mbox{see}\ (\ref{eq30})),\nonumber\\
	\Rightarrow&&\left\langle A(u_0)-A(\bar{u}),(u_0-\bar{u})^+\right\rangle+\int_{\Omega}(u^{p-1}_{0}-\bar{u}^{p-1})(u_0-\bar{u})^+dz\nonumber\\
	 &&\hspace{5cm}\int_{\partial\Omega}\beta(z)[u^{p-1}_{0}-\bar{u}^{p-1}](u_0-\bar{u})^+d\sigma\leq 0,\nonumber\\
	\Rightarrow&&|\{u_0>\bar{u}\}|_N=0,\ \mbox{hence}\ u_0\leq\bar{u}.\nonumber
\end{eqnarray}
So, we have proved that
$$u_0\in [0,\bar{u}]=\{u\in W^{1,p}(\Omega):0\leq u(z)\leq\bar{u}(z)\ \mbox{a.e.\ in}\ \Omega\},\ u_0\neq 0.$$
Therefore from (\ref{eq31}) and (\ref{eq34}), we have
$$\left\langle A(u_0),h\right\rangle+\int_{\partial\Omega}\beta(z)|u_0|^{p-2}u_0h\ d\sigma=\int_{\Omega}[\lambda u^{p-1}_{0}-f(z,u_0)]h\ dz\ \mbox{for\ all}\ h\in W^{1,p}(\Omega).$$
As before (see the proof of Proposition \ref{prop2}), via the nonlinear Green's identity we have
\begin{eqnarray}
	&&-\Delta_pu_0(z)=\lambda u_0(z)^{p-1}-f(z,u_0(z))\ \mbox{a.e.\ in}\ \Omega,\ \frac{\partial u_0}{\partial n_p}+\beta(z)u^{p-1}_{0}=0\ \mbox{on}\ \partial\Omega,\nonumber\\
	\Rightarrow&&u_0\ \mbox{is\ a\ nontrivial\ positive\ solution\ of\ problem}\ (P_{\lambda}).\nonumber
\end{eqnarray}
The nonlinear regularity theory, implies that $u_0\in C_+\backslash\{0\}$. Hypotheses $H_1(i),\ (iii)$ imply that we can find $c_3>0$ such that
$$f(z,x)\leq c_3x^{p-1}\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\in[0,||\bar{u}||_{\infty}].$$
Then
\begin{eqnarray}
	&&-\Delta_pu_0(z)\geq-f(z,u_0(z))\geq-c_3u_0(z)^{p-1}\ \mbox{a.e.\ in}\ \Omega,\nonumber\\
	\Rightarrow && \Delta_pu_0(z)\leq c_3u_0(z)^{p-1}\ \mbox{a.e.\ in}\ \Omega,\nonumber\\
	\Rightarrow && u_0\in\mbox{int}\, C_+ \ \text{(see Vazquez \cite{23})}.
	\nonumber
\end{eqnarray}
Similarly, we produce a nontrivial negative solution $v_0\in-\mbox{int}\, C_+$. Using this time $\bar{v}=-\hat{t}\hat{u}_1$ for $\hat{t}>0$ big, for which we have
\begin{eqnarray}
	&&\left\langle A(\bar{v}),h\right\rangle+\int_{\partial\Omega}\beta(z)|\bar{v}|^{p-2}\bar{v}h\ d\sigma\leq\int_{\Omega}\left[\lambda|\bar{v}|^{p-2}\bar{v}-f(z,\bar{v})\right]h\ dz\nonumber\\
	&&\hspace{7cm}\mbox{for\ all}\ h\in W^{1,p}(\Omega),\ h\geq 0.\nonumber
\end{eqnarray}
 Truncating and perturbing the reaction of $(P_{\lambda})$ at $\{\bar{v}(z),0\}$, as above we produce $v_0\in[\bar{v},0]\cap(-\mbox{int}\, C_+)$, a solution of $(P_{\lambda})$, $\lambda>\hat{\lambda}_1$.
\end{proof}

In fact, we can produce extremal nontrivial constant sign solutions for problem $(P_{\lambda})$ $\lambda>\hat{\lambda}_1$, that is, there exist $u_*\in\mbox{int}\, C_+$ the smallest nontrivial positive solution of $(P_{\lambda})$ and $v_*\in-\mbox{int}\, C_+$ the biggest nontrivial negative solution of $(P_{\lambda})$

For $\lambda>\hat{\lambda}_1$ we define
\begin{eqnarray}
	&&S_+(\lambda)=\{u\in W^{1,p}(\Omega):\ u\neq 0,\ u\in[0,\bar{u}],\ u\ \mbox{is\ a\ solution\ of}\ (P_{\lambda})\}.\nonumber\\
	&&S_-(\lambda)=\{v\in W^{1,p}(\Omega):\ v\neq 0,\ v\in[\bar{v},0],\ v\ \mbox{is\ a\ solution\ of}\ (P_{\lambda})\}\nonumber
\end{eqnarray}
From Proposition \ref{prop5} and its proof we have
$$\O\neq S_+(\lambda)\subseteq\mbox{int}\, C_+\ \mbox{and}\ \O\neq S_-(\lambda)\subseteq\mbox{int}\, C_+.$$
Moreover, as in Filippakis, Kristaly \& Papageorgiou \cite{10}, we have that the set of nontrivial positive (resp. negative) solutions of $(P_{\lambda})$ is downward directed, that is, if $u,\hat{u}$ are nontrivial positive solutions of $(P_{\lambda})$, then there exists $y$ a nontrivial positive solution of $(P_{\lambda})$ such that $y\leq u,\ y\leq\hat{u}$ (resp. upward directed, that is, if $v,\hat{v}$ are nontrivial negative solutions of $(P_{\lambda})$, there exists $w$ a nontrivial negative solution of $(P_{\lambda})$, such that $v\leq w,\ \hat{v}\leq w$).

\begin{prop}\label{prop6}
 Assume that hypotheses $H(\beta)$ and $H_1$ hold and $\lambda>\hat{\lambda}_1$. Then problem $(P_{\lambda})$ admits smallest nontrivial positive solution $u^{\lambda}_{*}\in {\rm int}\, C_+$ and a biggest nontrivial negative solution $v^{\lambda}_{*}\in{\rm int}\, C_+$.
\end{prop}

\begin{proof}
	We consider a chain $C\subseteq S_+(\lambda)$ (that is, a totally ordered subset of $S_+(\lambda)$). Then from Dunford \& Schwartz \cite[p. 336]{8}, we know that we can find $\{u_n\}_{n\geq 1}\subseteq C$ such that $\inf C=\inf\limits_{n\geq 1}u_n$. We have
	$$-\Delta_pu_n(z)=\lambda u_n(z)^{p-1}-f(z,u_n(z))\ \mbox{a.e.\ in}\ \Omega,\ \frac{\partial u_n}{\partial n_p}+\beta(z)u^{p-1}_{n}=0\ \mbox{on}\ \partial\Omega.$$
Using the nonlinear Green's identify, we obtain
\begin{eqnarray}\label{eq35}
&&\left\langle A(u_n),h\right\rangle+\int_{\partial\Omega}\beta(z)u^{p-1}_{n}h\ d\sigma=\ \int_{\Omega}\lambda u^{p-1}_{n}h\ dz-\int_{\Omega}f(z,u_n)h\ dz\\
&&\hspace{8cm} \mbox{for\ all}\ h\in W^{1,p}(\Omega).\nonumber
\end{eqnarray}
We choose $h=u_n\in W^{1,p}(\Omega)$. Then
\begin{eqnarray}
	&&||Du_n||^{p}_{p}\leq M_8\ \mbox{for\ some}\ M_8>0,\ \mbox{all}\ n\geq 1,\nonumber\\
	\Rightarrow&&\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)\ \mbox{is\ bounded}\ (\mbox{recall}\ 0\leq u_n\leq\bar{u}\ \mbox{for\ all}\ n\geq 1).\nonumber
\end{eqnarray}
So, we may assume that
$$u_n\stackrel{w}{\rightarrow}u\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{and}\ u_n\rightarrow u\ \mbox{in}\ L^p(\Omega).$$
In (\ref{eq35}) we choose $h=u_n-u\in W^{1,p}(\Omega)$ and pass to the limit as $n\rightarrow\infty$. Using the continuity of the trace map (hence $u_n|_{\partial\Omega}\stackrel{w}{\rightarrow} u|_{\partial\Omega}$ in $L^p(\partial\Omega)$), we obtain
\begin{eqnarray}
	&&\lim\limits_{n\rightarrow\infty}\left\langle A(u_n),u_n-u\right\rangle=0,\nonumber\\
	\Rightarrow&&u_n\rightarrow u\ \mbox{in}\ W^{1,p}(\Omega)\ (\mbox{see\ Proposition}\ \ref{prop1}),\nonumber\\
	\Rightarrow&&\left\langle A(u),h\right\rangle+\int_{\partial\Omega}\beta(z)u^{p-1}h\ d\sigma=\int_{\Omega}\lambda u^{p-1}h\ dz-\int_{\Omega}f(z,u)h\ d\sigma\nonumber\\
	&&\hspace{8cm}\mbox{for\ all}\ h\in W^{1,p}(\Omega).\nonumber
	\end{eqnarray}
	Hence $u$ is a positive solution of $(P_{\lambda})$ So, if we show that $u\neq 0$, then $u\in S_+(\lambda)$. Arguing by contradiction, suppose that $u=0$ and let $y_n=\frac{u_n}{||u_n||}n\geq 1$. Then $||y_n||=1$ for all $n\geq 1$ and so we may assume that
	$$y_n\stackrel{w}{\rightarrow}y\ in\ W^{1,p}(\Omega)\ \mbox{and}\ y_n\rightarrow y\ \mbox{in}\ L^p(\Omega).$$
From (\ref{eq35}) we have
\begin{equation}\label{eq36}
	\left\langle A(y_n),h\right\rangle+\int_{\partial\Omega}\beta(z)y^{p-1}_{n}h\ d\sigma=\int_{\Omega}\lambda y^{p-1}_{n}h\ dz-\int_{\Omega}\frac{N_f(u_n)}{||u_n||^{p-1}}h\ dz.
\end{equation}
By virtue of hypotheses $H_1(i),\ (iii)$, we have that
\begin{equation}\label{eq37}
	\frac{N_f(u_n)}{||u_n||^{p-1}}\stackrel{w}{\rightarrow}0\ \mbox{in}\ L^{p'}(\Omega)\ \left(\frac{1}{p}+\frac{1}{p'}=1\right).
\end{equation}
So, if in (\ref{eq36}) we choose $h=y_n-y$ and pass to the limit as $n\rightarrow\infty$, then
\begin{eqnarray}\label{eq38}
	&&\lim\limits_{n\rightarrow\infty}\left\langle A(y_n),y_n-y\right\rangle=0,\nonumber\\
	\Rightarrow&&y_n\rightarrow y\ \mbox{in}\ W^{1,p}(\Omega)\ (\mbox{see\ Proposition}\ \ref{prop1}),\ \mbox{hence}\ ||y||=1,\ y\geq 0.
\end{eqnarray}
	If in (\ref{eq36}) we pass to the limit as $n\rightarrow\infty$ and use (\ref{eq37}) and (\ref{eq38}), then
	\begin{eqnarray}
		&&\left\langle A(y),h\right\rangle+\int_{\partial\Omega}\beta(z)y^{p-1}h\ d\sigma=\int_{\Omega}\lambda y^{p-1}h\ dz\ \mbox{for\ all}\ h\in W^{1,p}(\Omega),\nonumber\\
		\Rightarrow&&-\Delta_p y(z)=\lambda y(z)^{p-1}\ \mbox{a.e.\ in}\ \Omega,\ \frac{\partial y}{\partial n_p}+\beta(z)y^{p-1}=0\ \mbox{on}\ \partial\Omega\nonumber\\
		&&\hspace{8cm}(\mbox{see\ the\ proof\ of\ Proposition}\ \ref{prop2}).\nonumber
	\end{eqnarray}
Since $\lambda>\hat{\lambda}_1,\ y=0$ or $y$ is a nodal, a contradiction to (\ref{eq38}). Therefore $u\neq 0$ and so
$$u\in C_+(\lambda)\ \mbox{and}\ u=\inf C.$$
Because $C\subseteq S_+(\lambda)$ is a arbitrary chain, invoking the Kuratowski-Zorn lemma, we can find $u^{\lambda}_{*}\in S_+(\lambda)\subseteq\mbox{int}\, C_+$ a minimal element. If $u$ is a nontrivial positive solution, then we know that we can find $\tilde{u}\in S_+(\lambda)$ such that $\tilde{u}\leq u^{\lambda}_{*},\ \tilde{u}\leq u$. The minimality of $u^{\lambda}_{*}$ implies that $\tilde{u}=u^{\lambda}_{*}$ and so $u^{\lambda}_{*}\in\mbox{int}\, C_+$ is the smallest nontrivial positive solution of $(P_{\lambda})$

Similarly, working with $S_-(\lambda)\subseteq-\mbox{int}\, C_+$ and using again the Kuratowski-Zorn lemma, we produce $v^{\lambda}_{*}\in-\mbox{int}\, C_+$ the biggest nontrivial negative solution of $(P_{\lambda})$
\end{proof}

These extremal nontrivial constant sign solutions, will lead to a nodal (sign changing solution). To this end, fix $\lambda>\hat{\lambda}_1$ and let
$$\eta=\max\{||u^{\lambda}_{*}||_{\infty},\ ||v^{\lambda}_{*}||_{\infty}\}.$$
Hypotheses $H_1(i),\ (iii)$ imply that we can find $\xi>0$ such that
\begin{eqnarray}
	&&(\lambda+\xi)x^{p-1}\geq f(z,x)\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\in[0,\eta].\label{eq39}\\
	&&f(z,x)\geq(\lambda+\xi)|x|^{p-2}x\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\in[-\eta,0]\label{eq40}
\end{eqnarray}
From (\ref{eq39}) and (\ref{eq40}), after integration, we obtain
\begin{equation}\label{eq41}
	F(z,x)\geq\frac{\lambda+\xi}{p}|x|^p\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ |x|\leq \eta.
\end{equation}
We introduce the following Carath\'eodory functions
\begin{equation}\label{eq42}
k^{+}_{\lambda}(z,x)=
\left\{
	\begin{array}{ll}
	0&\mbox{if}\ x<0\\
	(\lambda+\xi)x^{p-1}-f(z,x)&\mbox{if}\ 0\leq x\leq u^{\lambda}_{*}(z)\\
	(\lambda+\xi)u^{\lambda}_{*}(z)^{p-1}-f(z,u^{\lambda}_{*}(z))&\mbox{if}\ u^{\lambda}_{*}(z)<x
	\end{array}
\right.
\end{equation}

\begin{equation}\label{eq43}
k^{-}_{\lambda}(z,x)=
\left\{
	\begin{array}{ll}
	(\lambda+\xi)|v^{\lambda}_{*}(z)|^{p-2}v^{\lambda}_{*}(z)-f(z,v^{\lambda}_{*}(z))&\mbox{if}\ x<v^{\lambda}_{*}(z)\\
	(\lambda+\xi)|x|^{p-2}x-f(z,x)&\mbox{if}\ v^{\lambda}_{*}(z)\leq x\leq 0\\
	0&\mbox{if}\ 0<x
	\end{array}
\right.
\end{equation}

\begin{equation}\label{eq44}
k_{\lambda}(z,x)=
\left\{
	\begin{array}{ll}
	(\lambda+\xi)|v^{\lambda}_{*}(z)|^{p-2}v^{\lambda}_{*}(z)-f(z,v^{\lambda}_{*}(z))&\mbox{if}\ x<v^{\lambda}_{*}(z)\\
	(\lambda+\xi)|x|^{p-2}x-f(z,x)&\mbox{if}\ v^{\lambda}_{*}(z)\leq x\leq u^{\lambda}_{*}(z)\\
	(\lambda+\xi)u^{\lambda}_{*}(z)^{p-1}-f(z,u^{\lambda}_{*}(z))&\mbox{if}\ u^{\lambda}_{*}(z)<x
	\end{array}
\right.
\end{equation}

We set $K^{\pm}_{\lambda}(z,x)=\int^{x}_{0}k^{\pm}_{\lambda}(z,s)ds,\ K_{\lambda}(z,x)=\int^{x}_{0}k_{\lambda}(z,s)ds$ and consider the $C^1$-functionals $\hat{\varphi}^{\pm}_{\lambda},\ \hat{\varphi}_{\lambda}:\ W^{1,p}(\Omega)\rightarrow\RR$ defined by
\begin{eqnarray} &&\hat{\varphi}^{\pm}_{\lambda}(u)=\frac{1}{p}||Du||^{p}_{p}+\frac{\xi}{p}||u||^{p}_{p}+\frac{1}{p}\int_{\partial\Omega}\beta(z)|u|^pd\sigma-\int_{\Omega}K^{\pm}_{\lambda}(z,u)dz\nonumber\\
	 &&\hat{\varphi}_{\lambda}(u)=\frac{1}{p}||Du||^{p}_{p}+\frac{\xi}{p}||u||^{p}_{p}+\frac{1}{p}\int_{\partial\Omega}\beta(z)|u|^pd\sigma-\int_{\Omega}K_{\lambda}(z,u)dz\ \mbox{for\ all}\ u\in W^{1,p}(\Omega).\nonumber
\end{eqnarray}

\begin{prop}\label{prop7}
	Assume that hypotheses $H(\beta)$ and $H_1$ hold and $\lambda>\hat{\lambda}_1$. Then $K_{\hat{\varphi}_{\lambda}}\subseteq[v^{\lambda}_{*},\ u^{\lambda}_{*}],\ K_{\hat{\varphi}^{+}_{\lambda}}=\{0, u^{*}_{\lambda}\},\ K_{\hat{\varphi}^{-}_{\lambda}}=\{v^{*}_{\lambda}, 0\}.$
\end{prop}

\begin{proof}
	Let $u\in K_{\hat{\varphi}_{\lambda}}$. Then
	\begin{eqnarray}
		&&\hat{\varphi}'_{\lambda}(u)=0,\nonumber\\
		\Rightarrow&&\left\langle A(u),h\right\rangle+\int_{\Omega}\xi|u|^{p-2}uh\ dz+\int_{\partial\Omega}\beta(z)|u|^{p-2}uh\ d\sigma\nonumber\\
		&&=\int_{\Omega}k_{\lambda}(z,u)h\ dz\ \mbox{for\ all}\ h\in W^{1,p}(\Omega).\nonumber
	\end{eqnarray}
First we choose $h=(u-u^{\lambda}_{*})^+\in W^{1,p}(\Omega)$. Then
\begin{eqnarray}\label{eq45}
	&&\left\langle A(u), (u-u^{\lambda}_{*})^+\right\rangle+\int_{\Omega}\xi u^{p-1}(u-u^{\lambda}_{*})^+dz+\int_{\partial\Omega}\beta(z)u^{p-1}(u-u^{\lambda}_{*})^+d\sigma\nonumber\\ &&=\int_{\Omega}\left[(\lambda+\xi)(u^{\lambda}_{*})^{p-1}-f(z,u^{\lambda}_{*})\right](u-u^{\lambda}_{*})^+dz\ (\mbox{see}\ (\ref{eq44}))
\end{eqnarray}
Recall that
\begin{eqnarray}\label{eq46}
	&&-\Delta_pu^{\lambda}_{*}(z)=\lambda u^{\lambda}_{*}(z)^{p-1}-f(z,u^{\lambda}_{*}(z))\ \mbox{a.e.\ in}\ \Omega,\ \frac{\partial u^{\lambda}_{*}}{\partial n_p}+\beta(z)(u^{\lambda}_{*})^{p-1}=0\ \mbox{on}\ \partial\Omega,\nonumber\\
	\Rightarrow&&\left\langle A(u^{\lambda}_{*}),\ (u-u^{\lambda}_{*})^+\right\rangle+\int_{\partial\Omega}\beta(z)(u^{\lambda}_{*})^{p-1}(u-u^{\lambda}_{*})^+d\sigma\nonumber\\
	&&=\int_{\Omega}[\lambda(u^{\lambda}_{*})^{p-1}-f(z,u^{\lambda}_{*})](u-u^{\lambda}_{*})^+dz.
\end{eqnarray}
From (\ref{eq45}) and (\ref{eq46}) it follows that
\begin{eqnarray}
	&&\left\langle A(u)-A(u^{\lambda}_{*}),\ (u-u^{\lambda}_{*})^+\right\rangle+\xi\int_{\Omega}(u^{p-1}-(u^{\lambda}_{*})^{p-1})(u-u^{\lambda}_{*})^+
dz+\nonumber\\
&&\hspace{5cm}\int_{\partial\Omega}\beta(z)(u^{p-1}-(u^{\lambda}_{*})^{p-1})(u-u^{\lambda}_{*})^+d\sigma=0,\nonumber\\
\Rightarrow&&|\{u>u^{\lambda}_{*}\}|_N=0,\ \mbox{hence}\ u\leq u^{\lambda}_{*}.\nonumber
\end{eqnarray}
Similarly, using the test function $(v^{\lambda}_{*}-u)^+\in W^{1,p}(\Omega)$, we show that $v^{\lambda}_{*}\leq u$. So, we have proved that
\begin{eqnarray}
	&&u\in[v^{\lambda}_{*}, u^{\lambda}_{*}]=\{y\in W^{1,p}(\Omega):\ v^{\lambda}_{*}(z)\leq y(z)\leq u^{\lambda}_{*}(z)\ \mbox{a.e.\ in}\ \Omega\},\nonumber\\
	\Rightarrow&&K_{\hat{\Psi}_{\lambda}}\subseteq[v^{\lambda}_{*}, u^{\lambda}_{*}].\nonumber
\end{eqnarray}
In a similar fashion, using (\ref{eq42}) (resp. (\ref{eq43})), we show that
\begin{eqnarray}
	&&K_{\hat{\varphi}^{+}_{\lambda}}\subseteq[0,u^{\lambda}_{*}]=\{y\in W^{1,p}(\Omega):\ 0\leq y(z)\leq u^{\lambda}_{*}(z)\ \mbox{a.e.\ in}\ \Omega\}\nonumber\\
	&&(resp.\ K_{\hat{\varphi}^{-}_{\lambda}}\subseteq[v^{\lambda}_{*},0]=\{y\in W^{1,p}(\Omega):\ v^{\lambda}_{*}(z)\leq y(z)\leq 0\ \mbox{a.e.\ in}\ \Omega\}).\nonumber
\end{eqnarray}
The extremality of $u^{\lambda}_{*}\in\mbox{int}\, C_+$ and $v^{\lambda}_{*}\in-\mbox{int}\, C_+$ (see Proposition \ref{prop6}), implies that
$$K_{\hat{\varphi}^{+}_{\lambda}}=\{0,u^{\lambda}_{*}\}\ \mbox{and}\ K_{\hat{\varphi}^{-}_{\lambda}}=\{v^{\lambda}_{*},0\}.$$
\end{proof}

\begin{prop}\label{prop8}
	Assume that hypotheses $H(\beta)$ and $H_1$ hold and $\lambda>\hat{\lambda}_1$. Then $u^{\lambda}_{*}\in {\rm int}\, C_+$ and $v^{\lambda}_{*}\in-{\rm int}\, C_+$ are both local minimizers of $\hat{\varphi}_{\lambda}$.
\end{prop}

\begin{proof}
	It is clear from (\ref{eq42}) that $\hat{\varphi}_{\lambda}$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find $\tilde{u}^{\lambda}_{*}\in W^{1,p}(\Omega)$ such that
	\begin{equation}\label{eq47}
		\hat{\varphi}^{+}_{\lambda}(\tilde{u}^{\lambda}_{*})=\inf [\hat{\varphi}^{+}_{\lambda}(u):\ u\in W^{1,p}(\Omega)].
	\end{equation}
As before (see the proof of Proposition \ref{prop5}), for $\vartheta\in(0,1)$ small we have
\begin{eqnarray}
	&&\hat{\varphi}^{+}_{\lambda}(\vartheta\hat{u}_1)<0,\nonumber\\
	 \Rightarrow&&\hat{\varphi}^{+}_{\lambda}(\tilde{u}^{\lambda}_{*})<0=\hat{\varphi}^{+}_{\lambda}(0)\ (\mbox{see}\ (\ref{eq47})),\ \mbox{hence}\ \tilde{u}^{\lambda}_{*}\neq 0.\nonumber
\end{eqnarray}
Since $\tilde{u}^{\lambda}_{*}\in K_{\hat{\varphi}^{+}_{\lambda}}\backslash\{0\}$, from Proposition \ref{prop7} it follows $\tilde{u}^{\lambda}_{*}=u^{\lambda}_{*}\in \mbox{int}\, C_+$. Note that
\begin{eqnarray}
	&&\hat{\varphi}_{\lambda}|_{C_+}=\hat{\varphi}^{+}_{\lambda}|_{C_+}\ (\mbox{see}\ (\ref{eq42}),\ (\ref{eq44})),\nonumber\\
	\Rightarrow&&u^{\lambda}_{*}\in \mbox{int}\, C_+\ \mbox{is\ a\ local\ $C^1(\bar{\Omega})$-minimizer}\ \hat{\varphi}_{\lambda},\nonumber\\
	\Rightarrow&&u^{\lambda}_{*}\in \mbox{int}\, C_+\ \mbox{is\ a\ local\ $W^{1,p}(\Omega)$-minimizer}\ \hat{\varphi}_{\lambda}\ (\mbox{see\ Proposition}\ \ref{prop2}).\nonumber
\end{eqnarray}
Similarly for $v^{\lambda}_{*}\in-\mbox{int}\, C_+$, using this time the functional $\hat{\varphi}^{-}_{\lambda}$ and (\ref{eq43}).
\end{proof}

To produce a nodal solution, we need to restrict further the range of the parameter $\lambda$.

\begin{prop}\label{prop9}
	Assume that hypotheses $H(\beta)$ and $H_1$ hold and $\lambda>\hat{\lambda}_2$. Then problem $(P_{\lambda})$ admits solution $y_{\lambda}\in[v^{\lambda}_{*},u^{\lambda}_{*}]\cap C^1(\bar{\Omega})$.
\end{prop}

\begin{proof}
	Let $u^{\lambda}_{*}\in\mbox{int}\, C_+$ and $v^{\lambda}_{*}\in\mbox{int}\, C_+$ be the two extremal nontrivial constant sign solutions of problem $(P_{\lambda})$ produced in Proposition \ref{prop6}. Without any loss of generality, we may assume that $\hat{\varphi}_{\lambda}(v^{\lambda}_{*})\leq \hat{\varphi}_{\lambda}(u^{\lambda}_{*})$ (the analysis is similar if the opposite inequality holds). From Proposition \ref{prop8}, we know that $u^{\lambda}_{*}\in\mbox{int}\, C_+$ is a local minimizer of $\hat{\varphi}_{\lambda}$. So, we can find $\rho\in(0,1)$ small such that
	\begin{equation}\label{eq48}
		\hat{\varphi}_{\lambda}(v^{\lambda}_{*})\leq\hat{\varphi}_{\lambda}(u^{\lambda}_{*})<\inf [\hat{\varphi}_{\lambda}(u):\ ||u-u^{\lambda}_{*}||=\rho]=\eta^{\lambda}_{\rho},\ ||v^{\lambda}_{*}-u^{\lambda}_{*}||>\rho.
	\end{equation}
Recall that $\hat{\varphi}_{\lambda}$ is coercive (see (\ref{eq44})), hence it satisfies the PS-condition. This fact and (\ref{eq48}) permit the use of Theorem \ref{th1} (the mountain pass theorem). So, there exists $y_{\lambda}\in W^{1,p}(\Omega)$ such that
\begin{equation}\label{eq49}
	y_{\lambda}\in K_{\hat{\varphi}_{\lambda}}\subseteq[v^{\lambda}_{*},u^{\lambda}_{*}]\ (\mbox{see\ Proposition}\ \ref{prop7})\ \mbox{and}\ \eta^{\lambda}_{\rho}\leq\hat{\varphi}_{\lambda}(y_{\lambda})\ (\mbox{see}\ (\ref{eq48})).
\end{equation}
From (\ref{eq48}) and (\ref{eq49}) it follows that $y_{\lambda}\notin\{v^{\lambda}_{*},u^{\lambda}_{*}\}$ and it solves problem $(P_{\lambda})$ see (\ref{eq44}), hence $y_{\lambda}\in C^1(\bar{\Omega})$ (nonlinear regularity theory).

We need to show that $y_{\lambda}\neq 0$ and then by virtue of the extremality of the solutions $u^{\lambda}_{*}$ and $v^{\lambda}_{*}$, we will have that $y_{\lambda}$ is nodal. From the mountain pass theorem, we have
$$\hat{\varphi}_{\lambda}(y_0)=\inf\limits_{\gamma\in\Gamma}\max\limits_{0\leq t\leq 1}\hat{\varphi}_{\lambda}(\gamma(t)),$$
where $\Gamma=\{\gamma\in C([0,1], W^{1,p}(\Omega)):\ \gamma(0)=v^{\lambda}_{*},\ \gamma(1)=u^{\lambda}_{*}\}.$

Recall that $M=W^{1,p}(\Omega)\cap\partial B^{L^p}$, (see Proposition \ref{prop4}). Let $M_c=M\cap C^1(\bar{\Omega})$. We consider the following two sets of paths
\begin{eqnarray}
	&&\hat{\Gamma}=\left\{\hat{\gamma}\in C([-1,1],M):\ \hat{\gamma}(-1)=-\hat{u}_1,\ \hat{\gamma}(1)=\hat{u}_1\right\}\ (\mbox{see\ Proposition}\ \ref{prop4})\nonumber\\
	&&\hat{\Gamma}_c=\left\{\gamma\in C([-1,1],M_c):\ \hat{\gamma}(-1)=-\hat{u}_1,\ \hat{\gamma}(1)=\hat{u}_1\right\}.\nonumber
\end{eqnarray}
From Papageorgiou \& R\u{a}dulescu \cite{19}, we know that $\hat{\Gamma}_c$ is dense in $\hat{\Gamma}$. Since $u^{\lambda}_{*}\in\mbox{int}\, C_+,\ v^{\lambda}_{*}\in-\mbox{int}\, C_+$, we have
$$m_0=\mbox{min}\ \left\{\min\limits_{\bar{\Omega}}u^{\lambda}_{*},\ \min\limits_{\bar{\Omega}}(-v^{\lambda}_{*})\right\}>0.$$
Hypothesis $H(iii)$ implies that given $\epsilon>0$, we can find $\delta\in(0,m_0)$ such that
\begin{equation}\label{eq50}
	|F(z,x)|\leq\frac{\epsilon}{p}|x|^p\ \mbox{for\ a.a.}\ z\in \Omega,\ \mbox{all}\ |x|\leq\delta.
\end{equation}
(recall $F(z,x)=\int^{x}_{0}f(z,s)ds$). From (\ref{eq44}) and (\ref{eq50}) we have
$$K_{\lambda}(z,x)=\frac{\lambda+\xi}{p}|x|^p-F(z,x)\geq\frac{\lambda+\xi-\epsilon}{p}|x|^p\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ |x|\leq\delta.$$
From Proposition \ref{prop4} and the density of $\hat{\Gamma}_c$ in $\hat{\Gamma}$, we see that given $\epsilon\in(0,\frac{\lambda-\hat{\lambda}_2}{2})$ (recall that $\lambda>\hat{\lambda}_2$), we can find $\hat{\gamma}_0\in\hat{\Gamma}_c$ such that
\begin{equation}\label{eq51}
	\varphi(\hat{\gamma}_0(t))\leq\hat{\lambda}_2+\epsilon\ \mbox{for\ all}\ t\in[-1,1].
\end{equation}
Recall that $\varphi: W^{1,p}(\Omega)\rightarrow\RR$ is defined by
$$\varphi(u)=||Du||^{p}_{p}+\int_{\partial\Omega}\beta(z)|u|^pd\sigma\ \mbox{for\ all}\ u\in W^{1,p}(\Omega),$$
(see Proposition \ref{prop4}). Evidently $\hat{\gamma}_0([-1,1])\subseteq C^1(\bar{\Omega})$ and recall that $u^{\lambda}_{*}\in\mbox{int}\, C_+, v^{\lambda}_{*}\in-\mbox{int}\, C_+$. So, we can find $\tau\in(0,1)$ small such that for all $u\in\hat{\gamma}_0([-1,1])$ we have
\begin{equation}\label{eq52}
	|\tau u(z)|\leq\delta\ \mbox{for\ all}\ z\in\bar{\Omega}\ \mbox{and}\ \tau u\in[v^{\lambda}_{*},u^{\lambda}_{*}]
\end{equation}
Then for every $u\in\hat{\gamma}_0([-1,1])$ we have
\begin{eqnarray}
	\hat{\varphi}_{\lambda}(\tau u)&=&\frac{\tau^p}{p}||Du||^{p}_{p}+\frac{\xi\tau^p}{p}||u||^{p}_{p}+\frac{\tau p}{p}\int_{\partial\Omega}\beta(z)|u|^pd\sigma-\int_{\Omega}K_{\lambda}(z,\tau u)dz\nonumber\\
	&\leq&\frac{\tau^p}{p}[\hat{\lambda}_2+\epsilon]-\frac{\tau^p}{p}[\lambda-\epsilon]\ (\mbox{see}\ (\ref{eq44}),\ (\ref{eq50}),\ (\ref{eq51}))\nonumber\\
	&=&\frac{\tau^p}{p}[\hat{\lambda}_2+2\epsilon-\lambda]<0\ (\mbox{recall\ that}\ \epsilon<\frac{\lambda-\hat{\lambda}_2}{2}).\nonumber
\end{eqnarray}
So, if we set $\gamma_0=\tau\hat{\gamma}_0$, then $\gamma_0$ is a continuous path in $W^{1,p}(\Omega)$ which connects $-\tau\hat{u}_1$ and $\tau\hat{u}_1$ and such that
\begin{equation}\label{eq53}
		\hat{\varphi}_{\lambda}|_{\gamma_0}<0.
\end{equation}
Recall that $\hat{\varphi}^{+}_{\lambda}(u^{\lambda}_{*})<0=\hat{\varphi}^{+}_{\lambda}(0)$ and $K_{\hat{\varphi}^{+}_{\lambda}}=\{0,u^{\lambda}_{*}\}$ (see Propositions \ref{prop7},\ \ref{prop8} and the proof of the latter). Applying the second deformation theorem (see, foe example, Gasinski \& Papageorgiou \cite[p. 628]{12}), we produce a deformation $h:[0,1]\times((\hat{\varphi}^{+}_{\lambda})^0\backslash\{0\})\rightarrow(\hat{\varphi}^{+}_{\lambda})^0$ such that
\begin{eqnarray}
	&&h(1,(\hat{\varphi}^{+}_{\lambda})^0\backslash\{0\})=u^{\lambda}_{*}\label{eq54}\\
	&&\hat{\varphi}^{+}_{\lambda}(h(t,u))\leq\hat{\varphi}^{+}_{\lambda}(u)\ \mbox{for\ all}\ t\in[0,1]\label{eq55}
\end{eqnarray}
Let $\gamma_+(t)=h(t,\tau\hat{u}_1)^+$ for all $t\in[0,1]$. Then $\gamma_+$ is a continuous path in $W^{1,p}(\Omega)$ such that $\gamma_+(0)=\tau\hat{u}_1$ ($h$ is a deformation), $\gamma_+(1)=u^{\lambda}_{*}$ (see (\ref{eq54}) and recall $u^{\lambda}_{*}\in \mbox{int}\, C_+$) and
$$\hat{\varphi}^{+}_{\lambda}|_{\gamma_+}<0\ (\mbox{see\ (\ref{eq55})\ and}\ \eqref{eq53}).$$
Since $\mbox{im}\,\gamma_+\subseteq W_+=\{u\in W^{1,p}(\Omega):\ u(z)\geq 0\ \mbox{a.e.\ in}\ \Omega\}$ and\ $\hat{\varphi}^{+}_{\lambda}|_{W_+}=\hat{\varphi}_{\lambda}|_{W_+}$\ (see\ (\ref{eq42}),\ (\ref{eq44})), we have
\begin{equation}\label{eq56}
\hat{\varphi}_{\lambda}|_{\gamma_+}<0.
\end{equation}
In a similar fashion, using this time the functional $\hat{\varphi}^{-}_{\lambda}$, we produce another continuous path $\gamma_-$ in $W^{1,p}(\Omega)$ which connects $-\tau\hat{u}_1$ and $v^{\lambda}_{*}$ and such that
\begin{equation}\label{eq57}
	\hat{\varphi}|_{\gamma_-}<0.
\end{equation}
Concatenating $\gamma_-,\ \gamma_0,\ \gamma_+$, we produce a path $\gamma_*\in\Gamma$ such that
\begin{eqnarray}
	&&\hat{\varphi}_{\lambda}|_{\gamma_*}<0\ (\mbox{see}\ (\ref{eq53}),\ (\ref{eq56}),\ (\ref{eq57})),\nonumber\\
	\Rightarrow&&y_{\lambda}\neq 0\ \mbox{and\ so}\ y_{\lambda}\in C^1(\bar{\Omega})\ \mbox{is\ nodal\ solution\ of}\ (P_{\lambda}).
\end{eqnarray}
\end{proof}

So, we can state the following multiplicity theorem for problem $(P_{\lambda})\ (\lambda>\hat{\lambda}_2)$.

\begin{theorem}\label{th2}
	Assume that hypotheses $H(\beta)$ and $H_1$ hold. Then for every $\lambda>\hat{\lambda}_2$ problem $(P_{\lambda})$ has at least three nontrivial solutions
	$u^{\lambda}_{0}\in{\rm int}\, C_+$, $v^{\lambda}_{0}\in-{\rm int}\, C_+$,
		and $y_{\lambda}\in[v^{\lambda}_{0},u^{\lambda}_{0}]\cap C^1(\bar{\Omega})$\ nodal.
\end{theorem}


\section{Semilinear Problems}

In this section, we deal with the semilinear problem (that is, $p=2$). So, the problem under consideration is the following:
$$
-\Delta u(z)=\lambda u(z)-f(z,u(z))\ \mbox{in}\ \Omega,\ \frac{\partial u}{\partial n}+\beta(z)u=0\ \mbox{on}\ \partial\Omega. \eqno(S_{\lambda})
$$

For this problem, under additional regularity conditions on $f(z,\cdot)$ and with a global growth restriction this time, we show that for all $\lambda>\hat{\lambda}_2$ problem $(S_{\lambda})$ admits a second nodal solution, for a total of four nontrivial solutions all with sign information.

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

\textbf{$H_2:$} $f:\Omega\times\RR\rightarrow\RR$ is a measurable function such that for a.a. $z\in\Omega\ f(z,0)=0,\\ f(z,\cdot)\in C^1(\RR)$ and

\begin{description}
	\item[(i)]$|f'_{x}(z,x)|\leq a(z)(1+|x|^{r-2})$ for a.a. $z\in\Omega$, all $x\in\RR$, with $a\in L^{\infty}(\Omega)_+,\\ 2<r<2^*$;
	\item[(ii)]$\lim\limits_{x\rightarrow\pm\infty}\frac{f(z,x)}{x}=+\infty$ uniformly for a.a. $z\in\Omega$;
	\item[(iii)]$f'_{x}(z,0)=\lim\limits_{x\rightarrow 0}\frac{f(z,x)}{x}=0$ uniformly for a.a. $z\in\Omega$;
	\item[(iv)]there exists $\delta>0$ such that $f(z,x)x\geq 0$ for a.a. $z\in\Omega$, all $|x|\leq\delta$.
\end{description}

\begin{remark}
	It is clear that hypothesis $H_2(i)$ implies that given $\rho>0$, we can find $\xi_{\rho}>0$ such that for a.a. $z\in\Omega$, the function $x\longmapsto(\lambda+\xi)x-f(z,x)$ is nondecreasing on $[-\rho,\rho].$
\end{remark}

We have the following multiplicity theorem for problem $(S_{\lambda})$.

\begin{theorem}\label{th3}
	Assume that hypotheses $H(\beta)$ and $H_2$ hold. Then for every $\lambda>\hat{\lambda}_2$ problem $(S_{\lambda})$ has at least four nontrivial solutions
	$$u^{\lambda}_{0}\in{\rm int}\, C_+,\ v^{\lambda}_{0}\in-{\rm int}\, C_+$$
	and $y_{\lambda},\hat{y}_{\lambda}\in{\rm int}_{C^1(\bar{\Omega})}[v^{\lambda}_{0},u^{\lambda}_{0}]$ nodal.
\end{theorem}

\begin{proof}
	From Theorem \ref{th2}, we already have three nontrivial solutions
	$$u^{\lambda}_{0}\in\mbox{int}\, C_+,\ v^{\lambda}_{0}\in-\mbox{int}\, C_+\ \mbox{and}\ y_{\lambda}\in[v^{\lambda}_{0},u^{\lambda}_{0}]\cap C^1(\bar{\Omega})\ \mbox{nodal}.$$
	Without any loss of generality, we may assume that $u^{\lambda}_{0}$ and $v^{\lambda}_{0}$ are extremal (see Proposition \ref{prop6}), that is, $u^{\lambda}_{0}=u^{\lambda}_{*}\in\mbox{int}\, C_+$. Let $\rho=\max\{||u^{\lambda}_{0}||_{\infty},\ ||v^{\lambda}_{0}||_{\infty}\}$ and let $\xi_{\rho}>0$ be such that for a.a. $z\in\Omega\ x\rightarrow(\lambda+\xi_{\rho})x-f(z,x)$ is nondecreasing on $[-\rho,\rho]$. Then
	\begin{eqnarray}
		-\Delta y_{\lambda}(z)+\xi_{\rho}y_{\lambda}(z)&=&(\lambda+\xi_{\rho})y_{\lambda}(z)-f(z,y_{\lambda}(z))\nonumber\\
		&\leq&(\lambda+\xi_{\rho})u^{\lambda}_{0}(z)-f(z,u^{\lambda}_{0}(z))\ (\mbox{since}\ y_{\lambda}\leq u^{\lambda}_{0})\nonumber\\
		&=&-\Delta u^{\lambda}_{0}(z)+\xi_{\rho}u^{\lambda}_{0}(z)\ \mbox{a.e.\ in}\ \Omega,\nonumber
	\end{eqnarray}
	\begin{eqnarray}
	 \Rightarrow&&\Delta(u^{\lambda}_{0}-y_{\lambda})(z)\leq\xi_{\rho}(u^{\lambda}_{0}-y_{\lambda})(z)\ \mbox{a.e.\ in}\ \Omega,\nonumber\\
	\Rightarrow&&u^{\lambda}_{0}-y_{\lambda}\in\mbox{int}\, C_+\ \text{(see\ Vazquez\ \cite{23})}.\nonumber
	\end{eqnarray}
	Similarly, we show that
	$$y_{\lambda}-v^{\lambda}_{0}\in\mbox{int}\, C_+$$
	Therefore, we have
	\begin{equation}\label{eq58}
		y_{\lambda}\in\mbox{int}_{C^1(\bar{\Omega})}[v^{\lambda}_{0},u^{\lambda}_{0}].
	\end{equation}
	Next let $\sigma_{\lambda}: W^{1,p}(\Omega)\rightarrow\RR$ be the functional defined by	 $$\sigma_{\lambda}(u)=\frac{1}{2}||Du||^{2}_{2}+\frac{1}{2}\int_{\partial\Omega}\beta(z)u^2d\sigma-\frac{\lambda}{2}||u||^{2}_{2}+\int_{\Omega}F(z,u)dz\ \mbox{for\ all}\ u\in H^{1}(\Omega)$$
	(recall $F(z,x)=\int^{x}_{0}f(z,s)ds$). Evidently $\sigma_{\lambda}\in C^2(H^{1}(\Omega))$.
	
%%%%%%

We consider the following orthogonal direct sum decomposition of $H^{1}(\Omega)$
$$H^{1}(\Omega)=\bar{H}\oplus E (\hat{\lambda}_{k})\oplus \hat{H}$$
with $k\geq 3,\ \bar{H}=\overset{k-1}{\underset{\mathrm{i=1}}\oplus} E (\hat{\lambda}_{i}),\ \hat{H}= \overline{\underset{i\geq k +1}\oplus E (\hat{\lambda}_{i})}$. Set $Y=E(\hat{\lambda}_{k})\oplus \hat{H}$.

Recall that $\lambda > \hat{\lambda}_{2}$. First we assume that $\lambda \in \sigma_{R}$ (2) (problem resonant at zero). Then $\lambda = \hat{\lambda}_{k}$ for some $k \geq 3$.

\begin{claim} The energy functional
$\sigma_{\lambda}$ admits a local linking at $u=0$, with respect to the orthogonal direct sum
$$H^{1}(\Omega)=\bar{H}\oplus Y.$$\end{claim}

By virtue of hypothesis $H_{2}$(iii), given $\epsilon > 0$, we can find $\delta_{0}=\delta_{0}(\epsilon)>0$ such that
\begin{equation}\label{eq59}
F(z,x) \leq \frac{\epsilon}{2} x^{2}\ \mbox{for\ a.a.}\ z\in \Omega,\ \mbox{all}\ |x|\leq \delta_{0}.
\end{equation}

Since $\bar{H}$ is finite dimensional, all norms are equivalent and so we can find $\rho_{1}=\rho_{1}(\epsilon)>0$ such that
\begin{equation}\label{eq60}
``||u||\leq \rho_{1} \Rightarrow |u(z)|\leq \delta_{0}\ \mbox{for\ all}\ z\in \bar{\Omega}"\ \mbox{for\ all}\ u \in \bar{H}.
\end{equation}

Therefore, for $u \in \bar{H}$ with $||u||\leq \rho_{1}$, we have
\begin{eqnarray}
\sigma_{\lambda}(u) & = & \frac{1}{2}||Du||_{2}^{2}+\frac{1}{2}\int_{\partial \Omega}  \beta(z) u^{2}d\sigma - \frac{\hat{\lambda}_{k}}{2}||u||^{2}_{2}+\int_{\Omega}F(z,u)dz\nonumber\\
										& \leq & \frac{\hat{\lambda}_{k-1}-\hat{\lambda}_{k}}{2} ||u||^{2}_{2} + \frac{\epsilon}{2}||u||^{2}_{2}\ \mbox{see (\ref{eq59}) (\ref{eq60})}\nonumber\\
										& \leq & -c_{4} ||u||^{2}\ \mbox{for some}\ c_{4}>0\ (\mbox{choosing}\ \epsilon \in (0,\hat{\lambda}_{k}-\hat{\lambda}_{k-1})).\nonumber
\end{eqnarray}

So, we have proved that
\begin{equation}\label{eq61}
\sigma_{\lambda}(u)\leq 0\ \mbox{for all}\ u \in \bar{H}\ \mbox{with}\ ||u||\leq \rho_{1}.
\end{equation}

Next let $u \in Y$. Then we have $u=u^{0}+\hat{u}$ with $u^{0}\in E(\hat{\lambda}_{k})$, $\hat{u}\in \hat{H}$. Hence
\begin{eqnarray}
\sigma_{\lambda}(u) &  =  & \frac{1}{2}||Du||^{2}_{2} + \frac{1}{2}\int_{\partial\Omega}\beta(z)u^{2}d\sigma - \frac{\hat{\lambda}_{k}}{2}||u||^{2}_{2}+\int_{\Omega}F(z,u)dz  \nonumber\\
										& \geq & \frac{c_{5}}{2} ||\hat{u}||^{2} + \int_{\{|u|\leq \delta\}} F(z,u) dz + \int_{\{|u|>\delta\}}F(z,u)dz\ \mbox{for\ some}\ c_5>0 \\ \label{eq62}
										&			 &												 \mbox{(exploiting the orthogonality of the component spaces}\ E(\hat{\lambda}_{k}), \hat{H})  \nonumber \\
										& \geq & \frac{c_{5}}{2}||\hat u||^{2}  + \int_{\{|u|>\delta\}}F(z,u)dz\ (\mbox{see}\ H_{2}(iv)). \nonumber
\end{eqnarray}

Since $E(\hat{\lambda}_{k})$ is finite dimensional, we can find $\rho_{2}>0$ small such that
$$``||u^{0}||\leq\rho_{2} \Rightarrow|u^{0}(z)|\leq \frac{\delta}{2}\ \mbox{for all}\ z\ \in \bar{\Omega}"\ \mbox{for all}\ u^{0}\in E(\hat{\lambda}_{k})$$

So, if $C_{\delta}=\{z\in\Omega : |u(z)|\leq \delta\}$, then for $u\in Y$ with $||u||\leq \rho_{2}$, we have $||u^{0}||\leq \rho_{2}$ hence $|u^{0}(z)|\leq \frac{\delta}{2}$ for all $z\in \bar{\Omega}$. Therefore, for $u\in Y$ with $||u||\leq \rho_{2}$, we have
\begin{equation}\label{eq63}
|\hat{u}(z)|\geq |u(z)|-|u^{0}(z)|\geq |u(z)|-\frac{\delta}{2} \geq \frac{1}{2} |u(z)|\ \mbox{a.e.\ on}\ C_{\delta}
\end{equation}

Moreover, it is clear from hypotheses $H_{2}$(i),(iii) that gives $\epsilon > 0$, we can find $c_{6}=c_{6}(\epsilon)>0$ such that
\begin{equation}\label{eq64}
F(z,x)\geq-\frac{\epsilon}{2}x^{2}-c_{6}|x|^{r}\ \mbox{for a.a}\ z\in \Omega,\ \mbox{all}\ x\in\RR,\ \mbox{with\ $2 < r$}.
\end{equation}

Then
\begin{eqnarray}
\int_{C_{\delta}}F(z,u)dz & \geq & - \frac{\epsilon}{2}\int_{C_{\delta}}u^{2}dz - c_{6}\int_{C_{\delta}} |u|^{r}dz\ \mbox{(see (\ref{eq64}))} \nonumber \\
													& \geq & - \epsilon \int_{C_{\delta}}\hat{u}^{2}dz-c_{7}||\hat{u}||^{r}\ \mbox{for some}\ c_{7}>0\ \mbox{(see (\ref{eq63}))} \nonumber \\
													& \geq & - \epsilon ||\hat{u}||^{2}-c_{7}||\hat{u}||^{r} \label{eq65}
\end{eqnarray}

We return to (\ref{eq62}) and use (\ref{eq65}). Then
$$\sigma_{\lambda}(u)\geq \left( \frac{c_{5}}{2} - \epsilon \right) ||\hat{u}||^{2}-c_{7}||\hat{u}||^{r}.$$

Choosing $\epsilon \in (0, \frac{c_{5}}{2})$, we have
$$\sigma_{\lambda}(u) \geq c_{8} ||\hat{u}||^{2}-c_7||\hat{u}||^r\ \mbox{for some}\ c_{8}>0.$$

Since $r>2$, choosing $\rho_{2}\in(0,1)$ small, for $u\in Y$ with $||u||\leq \rho_{2}$, we have $||\hat{u}||\leq \rho_{2}$ and so
\begin{equation}\label{eq66}
\sigma_{\lambda}(u) \geq 0\ \mbox{for all}\ u\in Y\ \mbox{with}\ ||u||\leq \rho_{2}.
\end{equation}

Let $\rho=\min\{\rho_{1}, \rho_{2}\}$. Then from (\ref{eq61}), (\ref{eq66}) we infer that $\sigma_{\lambda}$ admits a local linking at $u=0$ with respect to the orthogonal direct sum decomposition $H^{1}(\Omega)=\bar{H}\oplus Y$. This proves the Claim.

Then by virtue of the Claim and Proposition 2.3 of Su \cite{22}, we have
\begin{equation}\label{eq67}
C_{i}(\sigma_{\lambda},0)=\delta_{i, d_{k}}\ZZ\ \mbox{for all}\ i\geq 0,\ \mbox{with}\ d_{k}=\dim\bar{H}\geq 2.
\end{equation}

Note that $\hat\varphi_{\lambda}|_{\left[v^{\lambda}_{0}, u^{\lambda}_{0}\right]}=\sigma_{\lambda}|_{\left[v^{\lambda}_{0}, u^{\lambda}_{0}\right]}$ (see (\ref{eq44})) and recall that $u^{\lambda}_{0}\in\ \mbox{int}\, C_{+}$, $v^{\lambda}_{0}\in - \mbox{int}\, C_{+}$. Hence
\begin{eqnarray}
&  &  C_{i}(\hat{\varphi}_{\lambda}|_{C^{1}(\bar{\Omega})}, 0) = C_{i}(\sigma_{\lambda}|_{C^{1}(\bar{\Omega})},0)\ \mbox{for all}\ i\geq 0,\nonumber  \\
&\Rightarrow & C_{i} (\hat{\varphi}_{\lambda}, 0)= C_{i}(\sigma_{\lambda},0)=\delta_{i,d_{k}} \ZZ\,,\  \mbox{for all}\ i\geq 0\ \label{eq68} \mbox{(see \eqref{eq67} and Bartsch \cite{4})}.
\end{eqnarray}

From Proposition \ref{prop7}, we have
\begin{equation}\label{eq69}
c_{i}(\hat{\varphi}_{\lambda},u^{\lambda}_{0})=c_{i}(\hat{\varphi}_{\lambda},v^{\lambda}_{0})=\delta_{i,0}\ \ZZ\ \mbox{for all}\ i\geq 0.
\end{equation}

From the proof of Proposition \ref{prop8}, we know that $y_{\lambda}$ is a critical point of mountain pass type the functional $\hat{\varphi}_{\lambda}$. Hence
\begin{eqnarray}
& & C_{1}(\hat{\varphi}_{\lambda}, y_{\lambda}) \neq 0 \nonumber \\
&\Rightarrow & C_{1}(\sigma_{\lambda},y_{\lambda}) \neq 0\ \mbox{(see (\ref{eq58}) and recall }\ \hat{\varphi}_{\lambda}|_{\left[v^{\lambda}_{0},u^{\lambda}_{0}\right]} = \sigma_{\lambda}|_{\left[v^{\lambda}_{0},u^{\lambda}_{0}\right]} ) \nonumber \\
&\Rightarrow &C_{i}(\sigma_{\lambda},y_{\lambda})=\delta_{i,1} \ZZ\ \mbox{for all}\ i\geq 0\ \mbox{see Bartsch \cite{4}} \nonumber \\
&\Rightarrow &C_{i}(\varphi_{\lambda},y_{\lambda})=\delta_{i,1} \ZZ\ \mbox{for all}\ i\geq 0\ \mbox{see (\ref{eq68})}. \label{eq70}
\end{eqnarray}

Finally, recall that $\hat{\varphi}_{\lambda}$ is coercive (see (\ref{eq44})). So, we have
\begin{equation}\label{eq71}
C_{i}(\hat{\varphi}_{\lambda}, \infty) = \delta_{i,0} \ZZ\ \mbox{for all}\ i\geq 0.
\end{equation}

Suppose that $K_{\hat{\varphi}_{\lambda}}=\{0,u^{\lambda}_{0},v^{\lambda}_{0},y_{\lambda}\}$. Then from (\ref{eq68}),(\ref{eq69}),(\ref{eq70}),(\ref{eq71}) and the Morse relation with $t=-1$ (see (\ref{eq14})), we have
\begin{eqnarray}
& &(-1)^{d_{k}}+2(-1)^{0}+(-1)^{1}=(-1)^{0}\nonumber \\
&\Rightarrow&(-1)^{d_{k}}=0,\ \mbox{a contradiction}.\nonumber
\end{eqnarray}

So, there exists $\hat{y}_{\lambda}\in K_{\hat{\varphi}_{\lambda}} \backslash \{0,u^{\lambda}_{0},v^{\lambda}_{0},y_{\lambda}\}$. From Proposition \ref{prop6} and (\ref{eq44}) we see that $\hat{y}_{\lambda}\in C^{1}(\bar{\Omega})$ is a second nodal solution of $(S_{\lambda})$ and $\hat{y}_{\lambda}\in \mbox{int}_{C^{1}(\bar{\Omega})}[v^{\lambda}_{0},u^{\lambda}_{0}]$

Next suppose $\lambda \notin \sigma_{R}$ (2).Then $\lambda \in (\hat{\lambda}_{k}, \hat{\lambda}_{k+1})$ for some $k\geq 2$. In this case $u=0$ is a nondegenerate critical point of $\sigma_{\lambda}$ with Morse index $d_{k}=\dim \overset{k}{\underset{\mathrm{i=1}}\oplus} E(\hat{\lambda}_{i})\geq 2$. Therefore
$$c_{i}(\sigma_{\lambda},0)=\delta_{i,d_{k}} \ZZ\ \mbox{for all}\ i \geq 0.$$

Then reasoning as above, we produce a second nodal solution $\hat{y}_{\lambda}\in \mbox{int}_{c^{1}(\Omega)}[v^{\lambda}_{0},u^{\lambda}_{0}]$ for problem $(S_{\lambda})$ $(\lambda > \hat{\lambda}_{2})$.
\end{proof}

\medskip
\textbf{Acknowledgements.} V.~R\u adulescu acknowledges the support through Grant CNCS PCE-47/2011.

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\end{document}


\item[(i)] for every $\rho>0$, there exists $a_{\rho}\in L^{\infty}(\Omega)_+$ such that $|f(z,x)|\leq a_{\rho}(z)$ for a.a. $z\in \Omega$, all $x\in \RR$; 