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\begin{document}

\title{\sc Solutions with sign information for nonlinear nonhomogeneous elliptic equations}

\author{Nikolaos S. Papageorgiou\footnote{National Technical University, Department of Mathematics, Zografou Campus, Athens 15780, Greece, e-mail: {\tt npapg@math.ntua.gr}} \ and Vicen\c tiu D. R\u adulescu\footnote{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 No. 13, 200585 Craiova, Romania, e-mail: {\tt vicentiu.radulescu@imar.ro}}}
%\address{, }

\date{}

%\baselineskip16pt
\maketitle

\begin{abstract}
We consider a class of nonlinear, coercive elliptic equations driven by a nonhomogeneous differential operator. Using variational methods together with truncation and comparison techniques, we show that the problem has at least three nontrivial solutions, all with sign information. In the special case of $(p,2)$-equations, using tools from Morse theory, we show the existence of four nontrivial solutions, all with sign information. Finally, for a special class of parametric equations, we obtain multiplicity theorems that  substantially extend earlier results on the subject.\\
\noindent \textbf{AMS 2010 Subject Classification:} 35J20, 35J60, 35J92, 58E05.\\
\textbf{Keywords:} nonlinear nonhomogeneous differential operator, nonlinear maximum principle, strong comparison principle, Morse theory, parametric equations, nodal solutions.
\end{abstract}

%\maketitle

\section{Introduction}
\label{sec:1}
Let $\Omega \subseteq \RR^{N}$ be a bounded domain with a $C^{2}$--boundary $\partial \Omega$.
In this paper we study the existence of multiple nontrivial solutions for the following nonlinear Dirichlet problem:
\begin{equation}\label{eq1}
-\mbox{div}\, a (Du(z))=f(z,u(z)) \quad \mbox{in}\ \Omega ,\qquad u\left|_{\partial \Omega}=0. \right.
\end{equation}

The map $a:\RR^{N}\rightarrow \RR^{N}$ involved in the differential operator of (\ref{eq1}) is strictly monotone and satisfies certain other regularity conditions listed in hypotheses $H(a)$ below. Special cases of the differential operator in (\ref{eq1}), are the $p$--Laplacian, the $(p, q)$--Laplace operator (the sum of a $p$--Laplacian and of a $q$--Laplacian), and the generalized $p$--mean curvature differential operator. It is important to notice that the differential operator in (\ref{eq1}) is not necessarily homogeneous. The reaction term $f(z,x)$ is a Carath\'eodory function (that is, for all $x \in \RR$, the mapping $z \longmapsto f(z,x)$ is measurable and for a.a. $z \in \Omega$, the map\ $x \longmapsto f(z,x)$ is continuous). Our hypotheses on the growth of $f(z, \cdot)$ ensure that the energy functional of the problem is coercive.

We prove a ``three solutions theorem" providing sign information for all the solutions produced. In the particular case where $a(y)=||y||^{p-2}y+y$ with $2 \leq p < \infty$ (the differential operator is the sum of a $p$--Laplacian and the Laplace operator), we show that we can have four nontrivial solutions, all with sign information (one positive, one negative, and two nodal (that is, sign changing)).

Multiplicity results for coercive elliptic problems were proved for semilinear equations driven by the Laplace operator by Ambrosetti \& Lupo \cite{1}, Ambrosetti \& Mancini \cite{2}, and Struwe \cite{30, 29}. For equations driven by the $p$--Laplacian, we refer to the works by Liu \cite{19}, Liu \& Liu \cite{20}, and Papageorgiou \& Papageorgiou \cite{24}. None of the aforementioned works produces nodal solutions.

Our approach is variational and it is based on the critical point theory in combination with suitable truncation and comparison techniques and Morse theory. We show that problem (\ref{eq1}) has at least three nontrivial smooth solutions, one positive, one negative, and a third nodal. In the particular case when the differential operator is the sum of a $p$--Laplacian $(p\geq 2)$ and a Laplacian, we show that we can have a second nodal solution for a total of four nontrivial smooth solutions, all with sign information. Finally, we show that for a special case of parametric nonlinear equations driven by the $p$--Laplacian, our multiplicity theorems lead to a substantial improvement of the results of Ambrosetti \& Lupo \cite{1}, Ambrosetti \& Mancini \cite{2}, Struwe \cite{30,29}, and Papageorgiou \& Papageorgiou \cite{24}.

\section{Mathematical Background and Auxiliary Results}
\label{sec:2}

Let $X$ be a Banach space and $X^{*}$ 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 (the ``PS-condition" for short), if the following is true:

\begin{center}
``{\it Every sequence $\left\{x_{n}\right\}_{n\geq 1}\subseteq X$ such that\ $\left\{\varphi(x_{n})\right\}_{n\geq 1} \subseteq \RR$
 is bounded and\\ $\varphi{'}(x_{n})\rightarrow 0$\ in\ $X^{*}$\ as\ $n\rightarrow \infty$,
 admits a strongly convergent subsequence}".
\end{center}

This compactness-type condition which compensates for the fact that the ambient space $X$ need not be locally compact, leads to a deformation theorem for the functional $\varphi$, from which we can derive the minimax theory of certain critical values of $\varphi$. One such minimax theorem is the so-called ``mountain pass theorem", which we recall here for future use. For details we refer to  Gasinski \& Papageorgiou \cite{13}, Kristaly, R\u adulescu \& Varga \cite{16}, and R\u adulescu \cite{28}.

\begin{theorem}\label{th1}
Assume that $\varphi \in C^{1}(X)$ satisfies the PS-condition, $x_{0},\, x_{1}\in X$, $||x_{1}-x_{0}||>\rho>0$,
 $$\max\,\{\varphi(x_{0}), \varphi(x_{1})\} < \inf\,\{\varphi(x):\ ||x-x_{0}||=\rho\}=\eta_{\rho}\,$$
and $c=\inf_{\gamma\in\Gamma}\,\max_{t\in [0,1]} \varphi (\gamma(t))$, where $$\Gamma = \{\gamma \in C([0,1],X):\ \varphi(0)=x_{0},\ \varphi(1)=x_{1}\}\,.$$ Then $c\geq \eta_{\rho}$ and $c$ is a critical value of $\varphi$.
\end{theorem}

The analysis of problem (\ref{eq1}) will use the Sobolev space $W^{1,p}_{0}(\Omega)$ and the Banach space $$C^{1}_{0}(\bar{\Omega})=\{u\in C^{1}(\bar{\Omega}):\ u=0\ \mbox{on}\ \partial \Omega\}\,.$$ The latter is an ordered Banach space with positive cone given by
$$C_{+}=\{u \in C_{0}^{1}(\bar{\Omega}):u(z)\geq 0\ \mbox{for\ all}\ z\in \bar{\Omega}\}\,.$$
This cone has a nonempty interior given by
$$\mbox{int}\, C_{+}=\{u \in C_{+}:u(z)>0\ \mbox{for\ all}\ z\in \Omega,\ \frac{\partial u}{\partial n}(z)<0\ \mbox{for\ all}\ z \in \partial \Omega\},$$
where $n(\cdot)$ is the outward unit normal on $\partial \Omega$.

Let $\vartheta \in C^{1}(0,\infty)$ and assume that
\begin{equation}\label{eq2}\begin{array}{ll}
&\di0<\widehat{c} \leq \frac{t\vartheta^{'}(t)}{\vartheta(t)}\leq c_{0}\ \mbox{for\ all}\ t>0\ \mbox{and some}\ c_{0},\ \widehat{c}>0\,;\\
&\di c_{1}t^{p-1}\leq \vartheta(t) \leq c_{2} (1+t^{p-1})\ \mbox{for\ all}\ t>0 \quad
\mbox{and\ some}\ c_{1}, c_{2}>0,\ 1<p<\infty.\end{array}
\end{equation}

Our hypotheses on the map $y\longmapsto a(y)$ are the following:
\begin{eqnarray}
H(a):&&a(y)=a_{0}(||y||)y\ \mbox{for\ all}\ y \in \RR^{N},\ \mbox{with}\ a_{0}(t)>0\ \mbox{for\ all}\ t>0\ \mbox{and}\nonumber\\
(i)&&a_{0} \in C^{1}(0,\infty),\ t \longmapsto ta_{0}(t)\ \mbox{is\ strictly\ increasing\ in}\ (0, \infty),\
ta_{0}(t)\rightarrow 0\ \mbox{as}\ t\rightarrow 0^{+}\nonumber\\
&&  \mbox{and}\ \lim_{t\rightarrow 0^{+}} \frac{ta^{'}_{0}(t)}{a_{0}(t)}=c>-1;\nonumber\\
(ii)&& \mbox{for\ every}\ y\in \RR^{N}\backslash\{0\},\ \mbox{we\ have}
\ ||\nabla a(y)||\leq c_{3}\frac{\vartheta(||y||)}{||y||}\ \mbox{for\ some}\ c_{3}>0;\nonumber\\
(iii)&& \mbox{for\ every}\ y\in \RR^{N}\backslash\{0\},\ \mbox{we\ have}\
(\nabla a(y)\xi,\xi)_{\RR^{N}}\geq\frac{\vartheta(||y||)}{||y||}||\xi||^{2}\ \mbox{for\ all}\ \xi \in \RR^{N};\nonumber\\
(iv) &&\mbox{if}\ G_{0}(t)=\int^{t}_{0}sa_{0}(s)ds\ \mbox{for\ all}\ t>0,\ \mbox{then\ there\ exists}\ \tau\in (1,p] \nonumber\\
&&\mbox{such that}\ t\longmapsto G_{0}(t^{1/\tau})\ \mbox{is\ convex\ in}\ (0, +\infty)\ \mbox{and}\
\lim\limits_{t\rightarrow 0^{+}}\frac{\tau G_{0}(t)}{t^{\tau}}=\tilde{c}>0\,.\nonumber
\end{eqnarray}

\begin{remark}\label{re:1}
The above assumptions show that the function $G_{0}(\cdot)$ is strictly convex and strictly increasing. We set $G(y)=G_{0}(||y||)$ for all $y\in \RR^{N}$. Then $G(\cdot)$ is convex, $G(0)=0$ and
$$\nabla G(y)=G_{0}'(||y||)\frac{y}{||y||}=a_{0}(||y||)y=a(y)\qquad\mbox{for all $y\in \RR^{N}\backslash\{0\}$ }\,.$$
Therefore $G(\cdot)$ is the primitive of $a(\cdot)$. The convexity of $G(\cdot)$ and the fact that $G(0)=0$ imply that
\begin{equation}\label{eq3}
G(y)\leq(a(y),y)_{\RR^{N}}\qquad \mbox{for\ all}\ y\in \RR^{N}.
\end{equation}
\end{remark}

Using hypotheses $H(a)$ and (\ref{eq2}), (\ref{eq3}), we deduce the following lemma summarizing the properties of $a(\cdot)$.


\begin{lemma}\label{lem1}
Assume that hypotheses $H(a)$ hold. Then the following properties hold:\\
(a) the operator $y\longmapsto a(y)$ is maximal monotone and strictly monotone;\\
(b) $||a(y)||\leq c_{4}(1+||y||^{p-1})$ for all $y\in \RR^{N}$ and some $c_{4}>0$;\\
(c) $(a(y),y)_{\RR^{N}}\geq \frac{c_{1}}{p-1}||y||^{p}$ for all $y\in \RR^{N}$.
\end{lemma}

Using this lemma and the integral form of the mean value theorem, we obtain the following growth conditions for the primitive $G(\cdot)$.


\begin{corollary}\label{col:1}
Assume that hypotheses H(a) hold. Then
 $$\frac{c_{1}}{p(p-1)}||y||^{p}\leq G(y)\leq c_{5} (1+||y||^{p})\qquad \mbox{for all $y \in \RR^{N}$ and some $c_{5}>0$}.$$
\end{corollary}

\textbf{Examples.} The following maps satisfy hypotheses H(a):

(a) $a(y)=||y||^{p-2}y$ with $1<p<\infty$.
Then the corresponding differential operator is the $p$--Laplacian
$\Delta_{p} u=\mbox{div}\,(||Du||^{p-2}Du)$\ for\ all\ $u\in W^{1,p}_{0}(\Omega).$
This quasilinear operator arises in many applications such as non-Newtonian fluid flows and turbulent filtration in porous media.

(b) $a(y)=||y||^{p-2}y+\mu||y||^{q-2}y$ with $1<q<p,\ p\geq 2$ and $\mu \geq 0$.
Then the corresponding differential operator is the $(p,q)$--differential operator
$$u\longmapsto \Delta_p u+\mu\Delta_q u\qquad \mbox{for\ all\ $u\in W^{1,p}_{0}(\Omega).$}$$

Equations driven by such differential operators are important in mathematical physics. We refer to Benci, D'Avenia, Fortunato \& Pisani \cite{5} (existence of solitons for problems of quantum physics) and Cherfils \& Ilyasov \cite{7} (problems in plasma physics). Recently, multiplicity results for resonant equations of this type were proved by  Papageorgiou \& R\u adulescu \cite{25}.

(c) $a(y)=(1+||y||^{2})^{\frac{p-2}{2}}y$ with $1<p<\infty$.
Then the corresponding differential operator is the generalized $p$--mean curvature differential operator
$$u\longmapsto \mbox{div}\,((1+||Du||^{2})^{\frac{p-2}{2}}Du)\qquad \mbox{for\ all}\ u\in W^{1,p}_{0}(\Omega).$$

(d) $a(y)=||y||^{p-2}y+\frac{||y||^{p-2}y}{1+||y||^{p}}$ with $1<p<\infty$.

\smallskip
Let $A:W^{1,p}_{0}(\Omega) \rightarrow W^{-1,p'}(\Omega)=W^{1,p}_{0}(\Omega)^{*}\left(\frac{1}{p}+\frac{1}{p^{'}}=1\right)$ be the nonlinear map defined by

\begin{equation}\label{eq4}
\left\langle A(u),y\right\rangle = \int_{\Omega}(a(Du(z)), Dy(z))_{\RR^{N}}\ dz\ for\ all\ u,y \in W^{1,p}_{0}(\Omega).
\end{equation}

According to Gasinski \& Papageorgiou \cite{14}, we have:

\begin{prop}\label{prop1}
Assume that hypotheses H(a) hold.
Then the operator $A:W^{1,p}_{0}(\Omega)\rightarrow W^{-1,p'}(\Omega)$ defined by (\ref{eq4}) is bounded (that is, maps
bounded sets to bounded sets), continuous, maximal monotone, and of type $(S)_{+}$, that is, if $u_{n}\stackrel{w}{\rightarrow}\ u$ in $W^{1,p}_{0}(\Omega)$
and $\lim\limits_{n\rightarrow\infty}\left\langle A(u_n),u_n-u\right\rangle\leq 0$, then $u_n\rightarrow u$ in $W^{1,p}_{0}(\Omega)$.
\end{prop}

Let $f_{0}:\Omega\times\RR\rightarrow \RR$ be a Carath\'eodory function with subcritical growth in the $x \in \RR$ variable, that is,
$$|f_{0}(z,x)|\leq a(z) (1+|x|^{r-1})\qquad \mbox{for\ a.a.}\ z\in \Omega\ \mbox{and all}\ x \in \RR\,,$$
with $a\in L^{\infty} (\Omega)_{+}$ and $$1<r<p^{*}=\left\{\begin{array}{lll}\di\frac{Np}{N-p} & \quad\mbox{if}\ &p<N \\+\infty & \quad\mbox{if}\ & p\geq N.\end{array}\right.$$

We\ set\ $F_{0}(z,x)=\int^{x}_{0}f_{0}(z,s)ds$
and consider the $C^{1}$-functional $\varphi_{0}:W^{1,p}_{0}(\Omega)\rightarrow \RR$ defined by
$$\varphi_{0}(u)=\int_{\Omega}G(Du(z))dz-\int_{\Omega}F_{0}(z,u(z))dz\qquad \mbox{for\ all}\ u \in W^{1,p}_{0}(\Omega).$$

The next result is concerned with $C^{1}_{0}(\bar{\Omega})$ and $W^{1,p}_{0}(\Omega)$ local minimizers of $\varphi_{0}$. The first such property was proved by Brezis \& Nirenberg \cite{6} when $G(y)=\frac{1}{2}\,||y||^{2}$ and it was extended by Garcia Azorero, Manfredi \& Peral Alonso \cite{12} to the case $G(y)=\frac{1}{p}\,||y||^{p}$, $1<p<\infty$ (see also Guo \& Zhang \cite{15} where $2\leq p < \infty$). The following result can be found in Gasinski \& Papageorgiou \cite{14}. In the sequel we denote by $||\cdot||$  the norm in the Sobolev space $W^{1,p}_{0}(\Omega)$. By virtue of the Poincar\'e inequality, $||u||=||Du||_{p}$ for all $u\in W^{1,p}_{0}(\Omega)$.

\begin{prop}\label{prop2}
 Assume that hypotheses H(a)(i), (ii), (iii) hold and $u_{0}\in W^{1,p}_{0}(\Omega)$ is a local $C^{1}_{0}(\bar{\Omega})$-minimizer of $\varphi_{0}$, that is, there exists $\rho_{0}>0$ such that
$$\varphi_{0}(u_{0})\leq \varphi_{0}(u_{0}+h)\qquad \mbox{for\ all}\ h\in C^{1}_{0}(\bar{\Omega})\ \mbox{with}\ ||h||_{C^{1}_{0}(\bar{\Omega})}\leq \rho_{0}\,.$$
Then $u_{0}\in C^{1,\beta}_{0}(\bar{\Omega})$ with $\beta \in(0,1)$ and $u_{0}$ is a local $W^{1,p}_{0}(\Omega)$--minimizer of $\varphi_{0}$, that is, there exists $\rho_{1}>0$ such that
$$\varphi_{0}(u_{0})\leq \varphi_{0}(u_{0}+h)\qquad \mbox{for\ all}\ h\in W^{1,p}_{0}(\Omega)\ \mbox{with}\ ||h||\leq \rho_{1}.$$
\end{prop}

 For $h_{1}, h_{2}\in L^{\infty}(\Omega)$, we write $h_{1} \prec h_{2}$ if for every compact set $K\subseteq \Omega$ there exists $\epsilon=\epsilon(K)>0$ such that
$$h_{1}(z)+\epsilon \leq h_{2}(z)\qquad \mbox{for\ a.a.}\ z\in K.$$
Evidently, if $h_{1}, h_{2} \in C(\Omega)$ and $h_{1}(z)<h_{2}(z)$ for all $z\in\Omega$, then $h_{1}\prec h_{2}$.

>From Papageorgiou \& R\u adulescu \cite{25}, we  recall the following strong comparison principle (see also Arcoya \& Ruiz \cite[Proposition 2.6]{3}).

\begin{prop}\label{prop3}
 Assume $\xi\geq 0$, $h_{1}, h_{2}\in L^{\infty}(\Omega),\ h_{1} \prec h_{2}$, and $u_{1}, u_{2}\in C^{1}_{0}(\bar{\Omega})$ with $u_{2}\in {\rm int}\, C_{+}$ are solutions of
$$-\Delta_{p}u_1(z)-\Delta u_1(z)+\xi\,|u_1(z)|^{p-2}u_1(z)=h_{1}(z)\qquad \mbox{in}\ \Omega$$
$$-\Delta_{p}u_2(z)-\Delta u_2(z)+\xi\,u_2(z)^{p-1}=h_{2}(z)\qquad \mbox{in}\ \Omega\,.$$
Then $u_{2}-u_{1}\in {\rm  int}\, C_{+}$.
\end{prop}

Next, for $q \in (1,\infty)$, we recall some basic facts concerning the spectrum  of $(-\Delta_{q},W^{1,q}_{0}(\Omega))$. So, we consider the following nonlinear eigenvalue problem
$$-\Delta_{q}u(z)=\hat{\lambda}|u(z)|^{q-2}u(z)\qquad\mbox{in}\ \Omega,\qquad u|_{\partial\Omega}=0.$$
A number $\hat{\lambda}\in \RR$ is an eigenvalue of $(-\Delta_{q},W^{1,q}_{0}(\Omega))$ if the above equation admits a nontrivial solution $\hat{u}\in W^{1,q}_{0}(\Omega)$, which is an eigenfunction corresponding to the eigenvalue $\hat{\lambda}$. We know that $(-\Delta_{q}, W^{1,q}_{0}(\Omega))$ admits a smallest eigenvalue $\hat{\lambda}_1(q)$, which has the following properties:\\
(i) $\hat{\lambda}_{1}(q)>0$;\\
(ii) $\hat{\lambda}_{1}(q)$ is isolated, that is there exists $\epsilon > 0$ such that $[\hat{\lambda}_{1}(q),\hat{\lambda}_{1}(q)+\epsilon)$ does not contain any other eigenvalue of $(-\Delta_{q},W^{1,q}_{0}(\Omega))$;\\
(iii) $\hat{\lambda}_{1}(q)$ is simple, that is, if $u,\,v$ are eigenfunctions corresponding to the eigenvalue $\hat{\lambda}_1(q)>0$, then $u=\xi v$ for some $\xi \neq 0$;\\
(iv) $\hat{\lambda}_1(q)$ admits the following variational characterization
\begin{equation}\label{eq5}
\hat{\lambda}_{1}(q)=\inf\, \left\{\frac{||Du||^{q}_{q}}{||u||^{q}_{q}}:\ u \in W^{1,q}_{0}(\Omega), u\neq 0\right\}.
\end{equation}

In relation (\ref{eq5}) the infimum is attained on the one-dimensional eigenspace of $\hat{\lambda}_{1}(q)$. It is clear from (\ref{eq5}) that the elements of this eigenspace do not change sign. By $\hat{u}_{1}(q)$ we denote the $L^{q}$--normalized (that is, $||\hat{u}_{1}(q)||_{q}=1$) positive eigenfunction corresponding to $\hat{\lambda}_{1}(q)$. In fact, $\hat{\lambda}_{1}(q)>0$ is the only eigenvalue with eigenfunctions of constant sign. All the other eigenvalues  have nodal eigenfunctions. The nonlinear regularity theory (see for example, Gasinski \& Papageorgiou \cite[pp. 737-738]{13})  implies that $\hat{u}_{1}(q)\in C_{+} \backslash \{0\}$. The nonlinear maximum principle of Vazquez \cite{31} implies that $\hat{u}_{1}(q)\in \mbox{int}\ C_{+}$.

Let $\hat{\sigma}{(q)}$ denote the spectrum (that is, the set of eigenvalues) of $(-\Delta_{q},W^{1,q}_{0}(\Omega))$. We can easily check that $\hat{\sigma}{(q)}$ is closed. Since $\hat{\sigma}{(q)}$ is closed and $\hat{\lambda}_{1}(q)>0$ is isolated, the second eigenvalue $\hat{\lambda}_{2}(q)$ of $(-\Delta_{q},W^{1,q}_{0}(\Omega))$ is well defined by
$$\hat{\lambda}_{2}(q)=\inf\, \{\hat{\lambda}\in \hat{\sigma}(q):\ \hat{\lambda}>\hat{\lambda}_{1}(q)\}.$$

The Ljusternik-Schnirelmann minimax scheme gives a whole sequence $\{\hat{\lambda}_{k}(q)\}_{k\geq 1}$ of distinct eigenvalues such that $\hat{\lambda}_{k}(q)\rightarrow + \infty$ as $k \rightarrow + \infty$. If $p=2$ or $N=1$ these are all the elements of $\hat{\sigma}(q)$. Otherwise we do not know if this is the complete list of eigenvalues.

The Ljusternik-Schnirelmann scheme provides a minimax characterization of $\hat{\lambda}_{2}(q)$. For our purposes, this characterization is not convenient. Instead we will use an alternative one due to Cuesta, de Figueiredo \& Gossez \cite{8}. So, let
$$\partial B^{L^{q}}_{1}=\{u \in L^{q}(\Omega):\ ||u||_{q}=1\},\quad M=W^{1,p}_{0}(\Omega) \cap \partial B^{L^{q}}_{1}$$
$$\hat{\Gamma}=\{\hat{\gamma}\in C([-1,1],M):\ \hat{\gamma}(-1)=-\hat{u}_{1}(q),\ \hat{\gamma}(1)=\hat{u}_{1}(q)\}.$$

\begin{prop}\label{prop4} We have
$$\hat{\lambda}_{2}(q)=\inf_{\hat{\lambda}\in \hat{\Gamma}} \max_{-1\leq t \leq 1} ||D\hat{\gamma}(t)||^{q}_{q}\,.$$
\end{prop}

As we already mentioned, in the particular case of $(p,2)$--equations, using also tools from Morse theory (critical groups), we will be able to generate two nodal solutions, for a total of four nontrivial solutions. So, let us briefly recall some basic relevant definitions and facts from Morse theory.

Let $X$ be a Banach space and $Y_{2} \subseteq Y_{1} \subseteq X$. For any integer $k \geq 0$, we denote by $H_{k}(Y_{1},Y_{2})$  the $k$th relative singular homology group for the pair $(Y_{1},Y_{2})$ with integer coefficients. Recall that $H_{k}(Y_{1},Y_{2})=0$ for all integers $k<0$.

Given $\varphi\in C^{1}(X)$ and $c \in \RR$, we introduce the following sets:
$$\varphi^{c}=\{x\in X:\ \varphi(x)\leq c\},\qquad K_{\varphi}=\{x\in X:\ \varphi'(x)=0\},\qquad K^{c}_{\varphi}=\{x\in K_{\varphi}:\ \varphi (x)=c\}.$$

The critical groups of $\varphi$ at an isolated critical point $x_{0}\in X$ with $\varphi(x_{0})=c $ (that is, $x_{0}\in K^{c}_{\varphi}$) are defined by
$$C_{k}(\varphi , x_{0})=H_{k}(\varphi^{c} \cap U, \varphi^{c} \cap U \backslash \{x_{0}\})\qquad \mbox{for\ all}\ k\geq 0,$$
where $U$ is a neighborhood of $x_{0}$ such that $\varphi^{c} \cap K_{\varphi}\cap U=\{x_{0}\}$. The excision property of 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 $c<\inf\, \varphi(K_{\varphi})$. Then the critical groups of $\varphi$ at infinity are defined by
$$C_{k}(\varphi, \infty)=H_{k}(X, \varphi^{c})\qquad \mbox{for\ all}\ k\geq 0.$$

The second deformation theorem (see, for example, Gasinski \& Papageorgiou \cite[p. 628]{14}) implies that the above definition of critical groups at infinity is independent of the choice of the level $c < \inf\, \varphi(K_{\varphi})$.

Assume that $K_{\varphi}$ is finite. We introduce the following polynomials in $t\in \RR$:
$$M(t,x)=\sum\limits_{k\geq 0} \mbox{rank}\, C_{k}(\varphi,x)t^{k}\qquad
P(t,\infty)=\sum\limits_{k \geq 0} \mbox{rank}\, C_{k}(\varphi, \infty)t^{k}\,.$$
Then the Morse relation says that
\begin{equation}\label{eq6}
\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 with nonnegative integer coefficients $\beta_{k}$.

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

Given a measurable function $h:\Omega \times \RR \rightarrow \RR$ (for example, a Carath\'eodory function), we define
$$N_{h}(u)(\cdot)=h(\cdot,u(\cdot))\qquad \mbox{for\ all}\ u\in W^{1,p}_{0}(\Omega).$$

Recall that $||\cdot||$ denotes the norm of the Sobolev space $W^{1,p}_{0}(\Omega)$, hence $||u||=||Du||_{p}$ for all $u\in W^{1,p}_{0}(\Omega)$. The same notation will also be used to denote the norm of $\RR^{N}$. However, no confusion is possible, since it will always be clear from the context what norm we mean. Finally, we denote by $|\cdot|_{N}$  the Lebesgue measure on $\RR^{N}$.

\section{Three Nontrivial Solutions}
In this section we prove the existence of three nontrivial solutions for problem (\ref{eq1}) and provide sign information for all of them.

We assume that the reaction term $f(z,x)$ is a Carath\'eodory\ function $f:\Omega \times \RR \rightarrow \RR$ such that $f(z,0)=0$ for a.a. $z\in\Omega$ and the following hypotheses are fulfilled:
\begin{eqnarray}
H_{1}:
(i) &&\mbox{for\ every}\ \rho > 0,\ \mbox{there\ exists}\ a_{\rho}\in L^{\infty}(\Omega)_{+}\ \mbox{such that}\
|f(z,x)|\leq a_{\rho}(z)\ \mbox{for\ a.a.}\ z\in \Omega\ \mbox{and all}\ |x|\leq\rho\,;\nonumber\\
(ii) &&\lim\limits_{x\rightarrow \pm \infty} \frac{f(z,x)}{|x|^{p-2}x}=-\infty\ \mbox{uniformly\ for\ a.a.}\ z\in \Omega;\nonumber\\
(iii) &&\mbox{if}\ \tau \in (1,p]\ \mbox{and}\ \tilde{c}>0\ \mbox{are\ as\ in\ hypothesis\ H(a)(iv)},\ \mbox{then\ there\ exist}\ \delta_{0}>0\ \mbox{and}\ c_{6}>\tilde{c}\hat{\lambda}_{1}(\tau)\nonumber\\
&&   \mbox{such that}\
f(z,x)x \geq c_{6}|x|^{\tau}\ \mbox{for\ a.a.}\ z\in\Omega\ \mbox{and all}\ |x|\leq\delta_{0}.\nonumber
\end{eqnarray}


\begin{remark} Note that the growth restriction on $f(z,\cdot)$ (see $H_{1}(i)$) is only local. In fact, the only growth restriction on $f(z,\cdot)$ is hypothesis $H_{1}(ii)$ and the growth to $-\infty$ as $x\rightarrow +\infty$ and to $+\infty$ as $x\rightarrow - \infty$ can be arbitrary. When $\tau <p$, hypothesis $H_{1}(iii)$ reveals the presence of a concave term near zero.
\end{remark}

\textbf{Example.} The following function satisfies hypotheses $H_{1}$ (for the sake of simplicity, we drop the $z$--dependence):
$$f(x)=|x|^{\tau-2}x-|x|^{r-2}x\qquad \mbox{with}\ 1<\tau\leq p<r<\infty.$$

First, we produce two nontrivial constant sign smooth solutions. To this end, note that we can find $M_{1}>\delta_{0}$ (see hypothesis $H_{1}(iii)$) such that
$$f(z,x)x\leq -1<0\qquad \mbox{for\ a.a.}\ z\in\Omega\ \mbox{and all}\ |x|\geq M_{1}.$$

Let $\xi >M_{1}$. Then
\begin{equation}\label{eq7}
0=A(\xi)\geq N_{f}(\xi)\qquad \mbox{in}\ W^{-1,p'}(\Omega)
\end{equation}
\begin{equation}\label{eq8}
0=A(-\xi)\leq N_{f}(-\xi)\qquad \mbox{in}\ W^{-1,p'}(\Omega).
\end{equation}

We define the following order intervals in the Sobolev space $W^{1,p}_{0}(\Omega)$:
$$[0,\xi]=\{u\in W^{1,p}_{0}(\Omega):\ 0\leq u(z)\leq\xi\ \mbox{for\ a.a.}\ z\in\Omega\}$$
$$[-\xi,0]=\{u\in W^{1,p}_{0}(\Omega):\ -\xi\leq u(z)\leq 0\ \mbox{for\ a.a.}\ z\in\Omega\}\,.$$

We introduce the following truncations of the reaction $f(z,\cdot)$:
\begin{equation}\label{eq9}
k_{+}(z,x)=\left\{\begin{array}{cl}0&\mbox{if}\ x<0\\ f(z,x)&\mbox{if}\ 0\leq x\leq\xi\\ f(z,\xi)&\mbox{if}\ \xi<x\end{array}\right.\qquad k_{-}(z,x)=\left\{\begin{array}{cl}f(z,-\xi)&\mbox{if}\ x<-\xi\\ f(z,x)& \mbox{if}\ -\xi\leq x\leq 0\\ 0 & \mbox{if}\ 0<x.\end{array}\right.
\end{equation}

Both $k_+$ and $k_-$ are Carath\'eodory functions. We set
$$K_{\pm}(z,x)=\int^{x}_{0}k_{\pm}(z,s)ds$$
and consider the $C^{1}$-functions $\psi_{\pm}:W^{1,p}_{0}(\Omega)\rightarrow \RR$ defined by
$$\psi_{\pm}(u)=\int_{\Omega}G(Du(z))dz-\int_{\Omega}K_{\pm}(z,u(z))dz\qquad \mbox{for\ all}\ u\in W^{1,p}_{0}(\Omega).$$

Let $\varphi : W^{1,p}_{0}(\Omega)\rightarrow \RR$ be the energy functional for problem (\ref{eq1}) defined by
$$\varphi(u)=\int_{\Omega}G(Du(z))dz-\int_{\Omega}F(z,u(z))dz\qquad \mbox{for\ all}\ u\in W^{1,p}_{0}(\Omega),$$
where $F(z,x)=\int^{x}_{0}f(z,s)ds$.

\begin{prop}\label{prop5}
Assume that hypotheses H(a) and $H_{1}$ hold.
Then problem (\ref{eq1}) has at least two nontrivial solutions of constant sign
$u_{0}\in[0,\xi]\cap {\rm int}\, C_{+}$ and
$v_{0}\in[-\xi,0]\cap(-{\rm int}\, C_{+}).$
\end{prop}

{\it Proof.}\label{proof1}
First we produce the nontrivial positive solution. To this end, we consider the functional $\psi_{+}$. It is clear from (\ref{eq9}) that $\psi_{+}$ is coercive. Also, using the Sobolev embedding theorem, we see that $\psi_{+}$ is sequentially weakly lower semi-continuous. So, by the Weierstrass theorem, we can find $u_{0}\in W^{1,p}_{0}(\Omega)$ such that
\begin{equation}\label{eq10}
\psi_{+}(u_{0})=\inf\,[\psi_{+}(u):\ u\in W^{1,p}_{0}(\Omega)]\,.
\end{equation}

Fix $\epsilon > 0$. By virtue of hypothesis H(a)(iv), there exists $\delta=\delta(\epsilon)\leq\delta_{0}$ (see hypothesis $H_{1}(iii)$) such that
$$G_{0}(t)\leq\frac{\tilde{c}+\epsilon}{\tau}t^{\tau}\qquad \mbox{for\ all}\ t\in[0,\delta]\,.$$
Therefore
\begin{equation}\label{eq11}
 G(y)\leq\frac{\tilde{c}+\epsilon}{\tau}||y||^{\tau}\qquad \mbox{for\ all}\ ||y||\leq\delta , \ y\in\RR^{N}.
\end{equation}


Let $t\in(0,1)$ be small such that $t\hat{u_{1}}(\tau)(z)$, $t||D\hat{u_{1}}(\tau)(z)||\leq\delta$ for all $z\in\overline{\Omega}$ (recall that $\hat{u}_{1}(\tau)\in \mbox{int}\, C_{+}$). Then
\begin{equation}\label{eq12}\begin{array}{ll}
\psi_{+}(t\hat{u}_1(\tau)) &  = \di  \int_{\Omega}G(tD\hat{u_{1}}(\tau)) dz - \int_{\Omega}K_{+}(z,t\hat{u}_{1}(\tau))dz
													 \leq  \frac{\tilde{c}+\epsilon}{\tau}t^{\tau} ||D\hat{u}_{1}||^{\tau}_{\tau}-\int_{\Omega}F(z,t\hat{u}_{1}(\tau))dz\\
													&       (\mbox{see}\ (\ref{eq11})\ \mbox{and\
recall that}\ \delta\leq\delta_{0}<\xi)\\
													& \leq \di \frac{\tilde{c}+\epsilon}{\tau}t^{\tau}\hat{\lambda},(\tau)-\frac{c_{6}}{\tau}t^{\tau}\\
													&			  (\mbox{see}\ (\ref{eq5}),\ \mbox{hypothesis}\ H_{1}(iii)\ \mbox{and\ recall\ that}\ ||\hat{u}_{1}(\tau)||_{\tau}=1)\\
													&   =  \di \frac{t^{\tau}}{\tau}[(\tilde{c}+\epsilon)\hat{\lambda}_{1}(\tau)-c_{6}]\,.
\end{array}\end{equation}

Since $c_{6}>\tilde{c}\hat{\lambda}_{1}(\tau)$ (see hypothesis $H_{1}$(iii)), by choosing $\epsilon \in (0,1)$ small, from (\ref{eq12}) we infer that
$$\psi_{+}(t\hat{u}_{1}(\tau))<0
\Rightarrow \psi_{+}(u_{0})<0=\psi_{+}(0)\quad (\mbox{see}\ (\ref{eq10}))\,,$$
hence\ $u_{0}\neq 0$.

Relation (\ref{eq10}) yields
\begin{eqnarray}\label{eq13}
\psi'_{+}(u_{0})=0
\Rightarrow A(u_{0})=N_{k_{+}}(u_{0}).
\end{eqnarray}

On (\ref{eq13}) first we act with $-u^{-}_{0}\in W^{1,p}_{0}(\Omega)$ and obtain
$$\frac{c_{1}}{p-1}||Du^{-}_{0}||^{p}_{p}\leq 0\qquad \mbox{see\ Lemma}\ \ref{lem1}(c)\ \mbox{and}\ \eqref{eq9},$$
hence $u_{0}\geq 0$, $u_{0}\neq 0$.


Next, we act on (\ref{eq13}) with $(u_{0}-\xi)^{+}\in W^{1,p}_{0}(\Omega)$. Thus, by (\ref{eq9}) and (\ref{eq7}),
$$
\left\langle A(u_{0}),(u_{0}-\xi)^{+} \right\rangle    =  \int_{\Omega} k_{+}(z,u_{0})(u_{0}-\xi)^{+}dz
																										    =  \int_{\Omega} f(z,\xi)(u_{0}-\xi)^{+}dz
											 \leq  0\,.$$
It follows that
$$ \int_{\{u_{0}>\xi\}}(a(Du_{0}),Du_{0})_{\RR^N} dz \leq 0
\Rightarrow |\{u_{0}>\xi\}|_{N}=0\,,$$ hence\ $u_{0}\leq \xi$.

So, we have proved that $u_{0}\in[0,\xi]$. Relation (\ref{eq13}) becomes
\begin{equation}\label{eq14}
A(u_{0})=N_{f}(u_{0})\quad (\mbox{see}\ \eqref{eq9})
\Rightarrow -\mbox{div}\, a(Du_{0}(z)) = f(z,u_{0}(z))\quad \mbox{a.e.\ in}\ \Omega,\qquad u_{0}|_{\partial\Omega}=0.
\end{equation}

>From Ladyzhenskaya \& Uraltseva \cite[p. 286]{17}, we have  $u_{0}\in L^{\infty}(\Omega)$. Then, using the regularity results of Lieberman \cite[p. 320]{18}, we deduce that $u_{0}\in C_{+}\backslash\{0\}$. Since $u_{0}\in[0,\xi]$, using hypotheses $H_{1}$(i), (iii), for $r\in(p,p^{*})$ we can find $c_{7}=c_{7}(r,\xi)>0$
such that
$$f(z,x)x\geq c_{6}|x|^{\tau}-c_{7}|x|^{r}\qquad \mbox{for\ a.a.}\ z\in\Omega\ \mbox{and all}\ |x|\leq \xi\,.$$

Therefore from (\ref{eq14}) and since $u_{0}\in C_{+}\backslash\{0\}$, we have
$$-\mbox{div}\,a(Du_{0}(z))+c_{7}u_{0}(z)^{r-1}
=f(z,u_{0}(z))+c_{6}u_{0}(z)^{\tau-1} \geq c_{6}u_{0}(z)^{\tau-1}\geq 0\qquad \mbox{a.e.\ in}\ \Omega\,.$$
It follows that
$$ \mbox{div}\, a(Du_{0}(z))\leq c_{7}||u_{0}||_{\infty}^{r-p}u_{0}(z)^{p-1}\qquad \mbox{a.e.\ in}\ \Omega\,.
$$

The nonlinear strong maximum principle of Pucci \& Serrin \cite[p. 111]{27}  implies that $u_{0}(z)>0$ for all $z\in\Omega$. Thus, by the boundary point theorem of Pucci \& Serrin \cite[p. 120]{27} we conclude that $u_{0}\in \mbox{int}\, C_{+}$.
Similarly, working this time with the functional $\psi_{-}$, we produce a second nontrivial constant sign solution $v_{0}\in [-\xi,0]\cap(-\mbox{int}\ C_{+})$.
\qed

If we strengthen a little the conditions on the reaction term $f(z,\cdot)$, we can say more about the two constant sign smooth solutions produced in Proposition \ref{prop5}.
The new stronger hypotheses on the nonlinearity $f(z,x)$ are the following:

\smallskip
$H_{2}$: $f:\Omega \times \RR\rightarrow \RR$ is a Carath\'eodory function such that $f(z,0)=0$ for a.a. $z\in\Omega$, hypotheses $H_{2}$(i), (ii), (iii) are the same as the corresponding hypotheses $H_{1}$(i)(ii)(iii), and \\
\noindent (iv) for every $\rho>0$, there exists $\xi_{\rho}>0$ such that for a.a. $z\in \Omega$ the function $x\longmapsto f(z,x)+\xi_{\rho}|x|^{p-2}x$ is nondecreasing on $[-\rho, \rho]$.

\begin{prop}\label{prop6}
Assume that hypotheses $H(a)$ and $H_{2}$ hold.
Then problem (\ref{eq1}) has at least two nontrivial solutions of constant sign
$u_{0}\in[0,\xi]\cap {\rm int}\, C_{+}$ \ and\ $ v_{0}\in[-\xi,0]\cap({\rm int}\, C_{+})$. Moreover, both $u_0$ and $v_0$ are local minimizers of the energy functional $\varphi$.
\end{prop}

{\it Proof.}\label{proof2}
By virtue of Proposition \ref{prop5}, we already have two nontrivial constant sign solutions
$u_{0}\in[0,\xi]\cap \mbox{int}\, C_{+}$\ \ and\ $v_{0}\in[-\xi,0]\cap(-\mbox{int}\ C_{+}).$

Hypothesis $H_{2}$(iv) implies that we can find $\hat{\xi}>0$ such that for a.a. $z\in \Omega$, the function $x\longmapsto f(z,x)+\hat{\xi}|x|^{p-2}x$ is nondecreasing on $[-\xi,\xi]$.

Let $\delta >0$ and set $u_{\delta}(z)=u_{0}(z)+\delta$ for all $z\in\bar{\Omega}$. Then $u_{\delta}\in C^{1}(\bar{\Omega})$ and
$$\begin{array}{ll}
&\di -\mbox{div}\,a(Du_{\delta}(z))+\hat{\xi}u_{\delta}(z)^{p-1}
\leq\di -\mbox{div}\,a(Du_{0}(z))+\hat{\xi}u_{0}(z)^{p-1}+\gamma(\delta)\quad \mbox{with}\ \gamma(\delta)\rightarrow 0^{+}\ \mbox{as}\ \delta\rightarrow 0^{+}\\
&\di=f(z,u_{0}(z))+\hat{\xi}u_{0}(z)^{p-1}+\gamma(\delta)
\leq\di f(z,\xi)+\hat{\xi}\xi^{p-1}+\gamma(\delta)\ (\mbox{since}\ u_{0}\in[0,\xi])\\
&\leq\di \hat{\xi}\xi^{p-1}\quad \mbox{for}\ \delta>0\ \mbox{small}\ (\mbox{recall that}\ f(z,\xi)\leq-\frac{1}{\xi}\ \mbox{for\ a.a.}\ z\in\Omega)\\
&=\di -\mbox{div}\,a(D\xi)+\hat{\xi}\xi^{p^{-1}}\quad \mbox{a.e.\ in}\ \Omega\,.\end{array}$$
Thus,  $u_{\delta}(z)\leq\xi$ for\ all\ $z\in\bar{\Omega}$ (by the the\ weak\ comparison\ principle). Hence we conclude that
$u_{0}(z)<\xi$ for\ all\ $z\in\bar{\Omega}$.
Since $u_{0}\in \mbox{int}\ C_{+}$, it follows that
$$u_{0}\in \mbox{int}_{C^{1}_{0}(\bar{\Omega})}[0,\xi].$$

Notice that $\psi_{+}|_{[0,\xi]}=\varphi|_{[0,\xi]}$. So, it follows that $u_{0}$ is a local $C^{1}_{0}(\overline{\Omega})$-minimizer of $\varphi$. Invoking Proposition \ref{prop2} it follows that $u_{0}$ is a local $W^{1,p}_{0}(\Omega)$-minimizer of $\varphi_-$.
Similarly for $v_{0}\in[-\xi,0]\cap(-\mbox{int}\ C_{+})$, using this time the functional $\psi$.
\qed

In fact, we can produce extremal nontrivial constant sign solutions for problem (\ref{eq1}), that is, the smallest nontrivial positive solution and the biggest nontrivial negative solution. We follow the reasoning on Papageorgiou \& R\u adulescu \cite{25}.

>From the proof of Proposition \ref{prop5}, we know that
\begin{equation}\label{eq15}
f(z,x)x\geq c_{6}|x|^{\tau}-c_{7}|x|^{r}\qquad \mbox{for\ a.a.}\ z\in\Omega\ \mbox{and all}\ |x|\leq\xi\,.
\end{equation}
This growth estimate leads to the following auxiliary Dirichlet problem:
\begin{equation}\label{eq16}
-\mbox{div}\,a(Du(z))=c_{6}|u(z)|^{\tau-2}u(z)-c_{7}|u(z)|^{r-2}u(z)\ \ \mbox{in}\ \Omega,\qquad u|_{\partial\Omega}=0.
\end{equation}

\begin{prop}\label{prop7}
Assume that hypotheses H(a) hold.
Then problem (\ref{eq16}) has a unique nontrivial positive solution $\tilde{u}\in {\rm int}\, C_{+}$ and since problem (\ref{eq16}) is odd, $\tilde{v}=-\tilde{u}\in -{\rm int}\, C_{+}$ is the unique nontrivial negative solution of (\ref{eq16}).
\end{prop}

{\it Proof.}\label{proof3}
First we establish the existence of a nontrivial positive solution for problem (\ref{eq16}). To this end, let $\sigma_{+}:W^{1,p}_{0}(\Omega)\rightarrow \RR$ be the $C^{1}$-functional defined by
$$\sigma_{+}(u)=\int_{\Omega}G(Du(z))dz-\frac{c_{6}}{\tau}||u^{+}||^{\tau}_{\tau}+\frac{c_{7}}{r}||u^{+}||^{r}\qquad \mbox{for\ all}\ u\in W^{1,p}_{0}(\Omega).$$

Since $r>p\geq\tau$, from Corollary \ref{col:1} we deduce that $\sigma_{+}(\cdot)$ is coercive and sequentially weakly lower semi-continuous. Thus, we can find $\tilde{u}\in W^{1,p}_{0}(\Omega)$ such that
\begin{equation}\label{eq17}
\sigma_{+}(\tilde{u})=\inf\,\{\sigma_{+}(u):u\in W^{1,p}_{0}(\Omega)\}\,.
\end{equation}

As before (see the proof of Proposition \ref{prop5}), we choose $t\in(0,1)$ small such that
$$t\hat{u}_{1}(\tau)(z),\ t||D\hat{u}_{1}(\tau)(z)||\leq\delta\qquad \mbox{for\ all}\ z\in\bar{\Omega}$$
with $\delta\in(0,\delta_{0}]$ as in (\ref{eq11}). Then we have
\begin{eqnarray}\label{eq18}
\sigma_{+}(t\hat{u}_{1}(\tau))\leq\frac{\tilde{c}+\epsilon}{\tau}t^{\tau}\hat{\lambda}_{1}(\tau)-
\frac{c_{6}}{\tau}t^{\tau}+\frac{c_{7}}{r}t^r||\hat{u}_{1}(\tau)||^{r}_{r}
														     = \frac{t^{\tau}}{\tau}[(\tilde{c}+\epsilon)\hat{\lambda}_{1}(\tau)-c_{6}]+\frac{c_{7}}{r}t^{r}||
\hat{u}_{1}(\tau)||^{r}_{r}\,.
\end{eqnarray}

Since $c_{6}>\tilde{c}\hat{\lambda}_{1}(\tau)$, by choosing $\epsilon\in(0,1)$ small and since $r>\tau$, by choosing $t\in(0,1)$ even smaller if necessary, from (\ref{eq18}) we see that
$\sigma_{+}(t\hat{u}_{1}(\tau))<0$. Taking into account (\ref{eq17}) and since $\sigma_{+}(0)=0$, we deduce that
$\tilde{u}\neq 0$.


>From (\ref{eq18}) we have
\begin{eqnarray}\label{eq19}
\sigma'_{+}(\tilde{u})=0
\Rightarrow A(\tilde{u})=c_{6}(\tilde{u}^{+})^{\tau-1}-c_{7}(\tilde{u}^{+})^{r-1}\,.
\end{eqnarray}
On (\ref{eq19}) we act with $-\tilde{u}^{-}\in W^{1,p}_{0}(\Omega)$. Then using Lemma \ref{lem1}(c), we have
$$\frac{c_1}{p-1}||D\tilde{u}^{-}||^{p}_{p}\leq 0\,,$$
hence\ $\tilde{u}\geq 0$, $\tilde{u}\neq 0$.
Then relation (\ref{eq19}) becomes
$$
A(\tilde{u})=c_{6}\tilde{u}^{\tau -1}-c_{7}\tilde{u}^{r-1}
\Rightarrow -\mbox{div}\,a(D\tilde{u}(z))=c_{6}\tilde{u}(z)^{\tau-1}-c_{7}\tilde{u}(z)^{r-1}\ \mbox{a.e.\ in}\ \Omega,\ \ \tilde{u}|_{\partial\Omega}=0.
$$

As before, the nonlinear regularity theory (see \cite{17}, \cite{18}), implies that $\tilde{u}\in C_{+}\backslash\{0\}$. We have
\begin{eqnarray}
&&\mbox{div}\,a(D\tilde{u}(z))\leq c_{7}||\tilde{u}||_{\infty}^{r-p}\tilde{u}(z)^{p-1}\qquad \mbox{a.e.\ in}\ \Omega\nonumber\\
&\Rightarrow& \tilde{u}\in \mbox{int}\, C_{+}\quad (\mbox{see\ Pucci \& Serrin\ \cite[pp. 111, 120]{27}})\,.\nonumber
\end{eqnarray}

Now we show the uniqueness of this positive solution. To this end, we introduce the integral functional $\gamma_{+}:L^{1}(\Omega)\rightarrow \bar{\RR}=\RR \cup\{+\infty\}$ defined by
\begin{equation}\label{eq20}
\gamma_{+}(u)=\left\{
\begin{array}{cl}
\di\int_{\Omega}G(Du^{1/\tau})dt &\quad \mbox{if}\ u \geq 0,\ u^{1/\tau}\in W^{1,p}_{0}(\Omega)\\
+\infty &\quad \mbox{otherwise}.
\end{array}\right.
\end{equation}

Let $u_{1}, u_{2}\in \mbox{dom}\,\gamma_{+}$ and set $v_{1}=u_{1}^{1/\tau}, v_{2}=u_{2}^{1/\tau}$. We have $v_{1}, v_{2}\in W^{1,p}_{0}(\Omega)$, see (\ref{eq20}). We set
$$v=(tu_{1}+(1-t)u_{2})^{1/\tau}\qquad \mbox{with}\ t\in[0,1]\,.$$

Invoking Lemma 1 of Diaz \& Saa \cite{9}, we have
$$||Dv(z)||\leq(t||Dv_{1}(z)||^{\tau}+(1-t)||Dv_{2}(z)||^{\tau})^{1/\tau}\qquad \mbox{for\ a.a.}\ z\in\Omega\,.$$
Since $G_{0}$ is increasing, we have
$$G_{0}(||Dv(z)||)\leq G_{0}((t||Dv_{1}(z)||^{\tau}+(1-t)||Dv_{2}(z)||^{\tau})^{1/\tau})\qquad \mbox{for\ a.a.}\ z\in \Omega\,.$$


>From hypothesis $H(a)$(iv), we know that $t\longmapsto G_{0}(t^{1/\tau})$ is convex in $(0,+\infty)$. It follows that
$$\begin{array}{ll}
&\di G_{0}((t||Dv_{1}(z)||^{\tau}+(1-t)||Dv_{2}(z)||^{\tau})^{1/\tau}) \\
&\leq\di t\ G_{0}(||Dv_{1}(z))||+(1-t)G_{0}(||Dv_{2}(z)||)\quad \mbox{a.e.\ in}\ \Omega \\
&\Rightarrow\di G(Dv(z))\leq t\ G(Dv_{1}(z))+(1-t)G(Dv_{2}(z))\quad \mbox{a.e.\ in}\ \Omega \\
&\Rightarrow\di \gamma_{+}\ \mbox{is\ convex}.\end{array}
$$

Also, by Fatou's lemma, we see that $\gamma_{+}$ is lower semi-continuous.

Let $u\in W^{1,p}_{0}(\Omega)$ be a nontrivial positive solution of problem (\ref{eq16}). From the first part of the proof we know that $u$ exists and $u\in \mbox{int}\, C_{+}$. We have $u^{\tau}\geq 0,\ (u^{\tau})^{1/\tau}=u\in W^{1,p}_{0}(\Omega)$. Hence $u^{\tau}\in \mbox{dom}\,\gamma_{+}$. Let $h\in C^{1}_{0}(\bar{\Omega})$. Then for $t\in(-1,1)$ with $|t|$ small, we have $u^{\tau}+th\in \mbox{int}\, C_{+}$ and so $u^{\tau}+th\in \mbox{dom}\, \gamma_{+}$. Therefore, the G\^ateaux derivative of $\gamma_{+}$ at $u^{\tau}$ in the direction $h$ exists and by the chain rule,
\begin{equation}\label{eq21}
\gamma'_{+}(u^{\tau})(h)=\frac{1}{\tau}\int_{\Omega}\frac{-\mbox{div}\,a(Du)}{u^{\tau-1}}hdz\,.
\end{equation}

If $v\in W^{1,p}_{0}(\Omega)$ is another nontrivial positive solution of (\ref{eq16}), then as above we have $v\in \mbox{int}\, C_{+}$ and
\begin{equation}\label{eq22}
\gamma'_{+}(v^{\tau})(h)=\frac{1}{\tau}\int_{\Omega}\frac{-\mbox{div}\, a(Dv)}{v^{\tau-1}}hdz\,.
\end{equation}

The convexity of $\gamma_{+}$ implies the monotonicity of $\gamma'_{+}$. Hence
$$\begin{array}{ll}
&0\di\leq\frac{1}{\tau}\int_{\Omega}\left(\frac{-\mbox{div}\,a(Du)}{u^{\tau -1}}+\frac{\mbox{div}\,a(Dv)}{v^{\tau-1}}\right)(u^{\tau}-v^{\tau})dz \\
&\di =\frac{c_{7}}{\tau}\int_{\Omega}(v^{r-\tau}-u^{r-\tau})(u^{\tau}-v^{\tau})dz \leq 0,\end{array}$$
hence
$ u=v$. It follows that
$\tilde{u}\in \mbox{int}\, C_{+}$\ is\ the\ unique\ nontrivial\ positive\ solution\ of\ (\ref{eq16}).
Since (\ref{eq16}) is odd, $\tilde{v}=-\tilde{u}\in-\mbox{int}\, C_{+}$ is the unique nontrivial negative solution of (\ref{eq16}).
\qed

Using the solutions produced in Proposition \ref{prop7}, we can generate extremal constant sign solutions of problem (\ref{eq1}).

\begin{prop}\label{prop8}
Assume that hypotheses $H(a)$ and $H_{1}$ hold.
Then problem (\ref{eq1}) has a smallest nontrivial positive solution $u_{*}\in {\rm int}\, C_{+}$
and a biggest nontrivial negative solution $v_{*}\in -{\rm int}\, C_{+}$.
\end{prop}

{\it Proof.}\label{proof4}
First we produce $u_{*}\in \mbox{int}\, C_{+}$, the smallest nontrivial positive solution of (\ref{eq1}).
Let $S_{+}$ be the set of nontrivial positive solutions of (\ref{eq1}) in the order interval $[0,\xi]$.
>From Proposition \ref{prop5} and its proof we have
$S_{+}\neq \O$\ and\ $S_{+}\subseteq [0,\xi]\cap \mbox{int}\, C_{+}$.
Moreover, as in Filippakis, Kristaly \& Papageorgiou \cite{11}, we have that $S_{+}$ is downward directed, that is, if $u_{1},u_{2}\in S_{+}$, then we can find $u\in S_{+}$ such that $u\leq u_{1}$, $u\leq u_{2}$.

\begin{claim}\label{claim:1}
$\tilde{u} \leq u$ for\ all\ $u\in S_{+}$.\end{claim}

Let $u\in S_{+}$ and let
\begin{equation}\label{eq23}
\eta_{+}(z,x)=\left\{
\begin{array}{ll}
0 & \mbox{if}\ x<0\\
c_{6}x^{\tau-1}-c_{7}x^{r-1} & \mbox{if}\ 0\leq x\leq u (z)\\
c_{6}(z)^{\tau-1}-c_{7}u(z)^{r-1} & \mbox{if}\ u(z)<x.
\end{array}
\right.
\end{equation}


Then $\eta_{+}$ is a Carath\'eodory function. We set $H_{+}(z,x)=\int_{0}^x\eta_{+}(z,s)ds$ and consider the $C^{1}$-functional $\beta_{+}: W^{1,p}_{0}(\Omega)\rightarrow \RR$ defined by
$$\beta_{+}(y)=\int_{\Omega}G(Dy(z))dz-\int_{\Omega}H_{+}(z,y(z))dz\qquad \mbox{for\ all}\ y\in W^{1,p}_{0}(\Omega)\,.$$

It is clear from (\ref{eq23}) and Corollary \ref{col:1} that $\beta_{+}$ is coercive. Also, it is sequentially weakly lower semi-continuous. So, we can find $\tilde{u}_{0}\in W^{1,p}_{0}(\Omega)$ such that
\begin{eqnarray}\label{eq24}
	\beta_+(\tilde{u}_0)=\inf\,\left\{\beta_+(u):\ u\in W^{1,p}_{0}(\Omega)\right\}\,.
\end{eqnarray}
As before, via hypothesis $H(a)(iv)$, we show that
$\beta_{+}(\tilde{u}_{0}) <0=\beta_{+}(0)$, hence\ $\tilde{u}_{0} \neq 0.$


>From (\ref{eq24}) we have
\begin{eqnarray}\label{eq25}
\beta'_{+}(\tilde{u}_{0})=0
\Rightarrow A(\tilde{u}_{0})=N_{\eta_{+}}(\tilde{u}_{0}).
\end{eqnarray}

Acting on (\ref{eq25}) with $-\tilde{u}^{-}_{0},\ (\tilde{u}_{0}-u)^{+}\in W^{1,p}_{0}(\Omega)$, we show that
$$\tilde{u}_{0}\in [0,u]=\{y\in W^{1,p}_{0}(\Omega):\ 0\leq y(z)\leq u(z)\ \mbox{a.e.\ in}\ \Omega\}\,.$$
Therefore (\ref{eq25}) becomes
\begin{eqnarray}
&&A(\tilde{u}_{0})=c_{6}\tilde{u}^{\tau -1}_{0}-c_{7}\tilde{u}^{r-1}_{0}\nonumber\\
&\Rightarrow& \tilde{u}_{0}\ \mbox{is\ a\ positive\ solution of}\ (\ref{eq16})\nonumber\\
&\Rightarrow& \tilde{u}_{0}=\tilde{u}\ (\mbox{see\ Proposition}\ \ref{prop7}) \nonumber\\
&\Rightarrow& \tilde{u} \leq u\ \mbox{for\ all}\ u\in S_{+}.\nonumber
\end{eqnarray}
This proves Claim \ref{claim:1}.

Let $C\subseteq S_{+}$ be a chain (that is, a totally ordered subset of $S_{+}$). From Dunford \& Schwartz \cite[p. 336]{10}, we know we can find $\{u_{n}\}_{n\geq 1} \subseteq C$ such that
$$\inf\, C=\inf_{n \geq 1}\, u_{n}.$$
We have
\begin{equation}\label{eq26}
A(u_{n})=N_{f}(u_{n}),\quad \tilde{u}\leq u_{n} \leq \xi\qquad \mbox{for\ all}\ n\geq 1 \ \ (\mbox{see\ the\ Claim}).
\end{equation}
Evidently, $\{u_{n}\}_{n\geq 1}\subseteq W^{1,p}_{0}(\Omega)$ is bounded. So, we may assume that
\begin{equation}\label{eq27}
u_{n} \stackrel{w}{\rightarrow}u\quad \mbox{in}\ W^{1,p}_{0}(\Omega)\qquad \mbox{and}\ u_{n}\rightarrow u\quad \mbox{in}\ L^{\rho}(\Omega).
\end{equation}

We act with $u_{n}-u\in W^{1,p}_{0}(\Omega)$, pass to the limit as $n\rightarrow \infty$, and use (\ref{eq27}).
We obtain
\begin{eqnarray}\label{eq28}
\lim_{n\rightarrow \infty} \left\langle A(u_{n}),u_{n}-u\right\rangle = 0
\Rightarrow u_{n}\rightarrow u\quad \mbox{in}\ W^{1,p}_{0}(\Omega)\ (\mbox{see\ Proposition}\ \ref{prop1}).
\end{eqnarray}

So, if  we pass to the limit in (\ref{eq26}) as $n\rightarrow \infty$ and use (\ref{eq28}), we deduce that
\begin{eqnarray}
A(u)=N_{f}(u),\ \tilde{u}\leq u\leq \xi
\Rightarrow u\in S_{+}\ \mbox{and}\ u=\inf\, C.\nonumber
\end{eqnarray}

Since $C$ is an arbitrary chain, from the Kuratowski-Zorn lemma we can find $u_{*}\in S_{+} \subseteq \mbox{int}\ C_{+}$ a minimal element. The fact that $S_{+}$ is downward directed implies that $u_{*}\in \mbox{int}\, C_{+}$ is the smallest nontrivial positive solution of problem (\ref{eq1}).

Similarly, let $S_{-}$ be the set of nontrivial negative solutions of (\ref{eq1}) in $[-\xi,0]$. We have that $S_{-}\neq \emptyset$ and $S_{-}\subseteq -\mbox{int}\, C_{+}$. Also, $S_{-}$ is upward directed (that is, if $v_{1},\ v_{2}\in S_{-}$ then there exists $v\in S_{-}$ such that $v_{1}\leq v,\ v_{2}\leq v$). The previous argument based on the Kuratowski-Zorn lemma leads to the biggest nontrivial negative solution $v_{*}\in -\mbox{int}\, C_{+}$ of problem (\ref{eq1}).
\qed


The extremal nontrivial constant sign solutions of (\ref{eq1}) lead to the existence of a nodal solution. For this purpose we need to slightly strengthen hypotheses $H_{1}$.
The new conditions on $f(z,x)$ are the following:

\medskip
$H'_{1}:$ $f:\Omega \times \RR \rightarrow \RR$ is a Carath\'eodory function such that $f(z,0)=0$ for a.a. $z\in \Omega$, hypotheses $H'_{1}$(i), (ii) are the same as the corresponding hypotheses $H_{1}$(i)(ii), and \\
\noindent (iii) if $\tau\in(1,p]$ and $\tilde{c}>0$ are as in hypothesis $H(a)$(iv), then there are $\delta_{0}>0$ and $c_{6}>\tilde{c}\hat{\lambda}_{2}(\tau)$ such that
$$f(z,x)x\geq c_{6}|x|^{\tau}\qquad \mbox{for\ a.a.}\ z\in \Omega\ \mbox{and all}\ |x|\leq\delta_{0}.$$

\begin{prop}\label{prop9}
Assume that hypotheses $H(a)$ and $H'_{1}$ hold.
Then problem (\ref{eq1}) admits a nodal solution $y_{0}\in[v_{*},u_{*}]\cap C^{1}_{0}(\bar{\Omega})$.
\end{prop}


{\it Proof.}\label{proof5}
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 of $f(z,\cdot)$:
\begin{eqnarray}\label{eq29}
h(z,x)=
\left\{
\begin{array}{ll}
f(z,v_{*}(z)) &\quad \mbox{if}\ x<v_{*}(z)\\
f(z,x) & \quad\mbox{if}\ v_{*}(z)\leq x \leq u_{*}(z)\\
f(z,u_{*}(z)) & \quad\mbox{if}\ u_{*}(z)<x.
\end{array}
\right.
\end{eqnarray}

Set $h_{\pm}(z,x)=h(z, \pm x^{\pm})$. Both are Carath\'eodory functions. We also define
$$H(z,x)=\int^{x}_{0}h(z,s)ds\qquad \mbox{and}\qquad  H_{\pm}(z,x)=\int^{x}_{0}h_{\pm}(z,s)ds$$
and introduce the $C^1$--functionals $\hat{\varphi},\hat{\varphi}_{\pm}:W^{1,p}_{0}(\Omega)\rightarrow \RR$ defined by
$$\begin{array}{ll}
\hat{\varphi}(u)&=\di\int_{\Omega}G(Du(z))dz - \int_{\Omega} H(z,u(z))dz\\
\hat{\varphi_{\pm}}(u)&\di=\int_{\Omega}G(Du(z))dz-\int_{\Omega} H_{\pm}(z,u(z))dz\qquad \mbox{for\ all}\ u\in W^{1,p}_{0}(\Omega).\end{array}
$$

We can easily check as before that
\begin{equation}\label{eq30}
K_{\hat{\varphi}}\subseteq [v_{*},u_{*}],\quad K_{\hat{\varphi}_{+}}=\{0,u_{*}\},\quad K_{\hat{\varphi}_{-}}=\{v_{*},0\}
\end{equation}
(recall the extremality of $u_{*}\in \mbox{int}\, C_{+}$\ and\ $v_{*}\in -\mbox{int}\, C_{+}$).
Note that $\hat{\varphi}_{+}$ is coercive (see (\ref{eq29})) and sequentially weakly lower semi-continuous. So, there exists $\hat{u}_{*}\in W^{1,p}_{0}(\Omega)$ such that
$$\hat{\varphi}_{+}(\hat{u}_{*})=\inf\,\{\hat{\varphi}_{+}(u):u\in W^{1,p}_0 (\Omega)\}<0=\hat{\varphi}_{+}(0)\quad\mbox{(as\ before,\ see\ H(a)(iv))}
\Rightarrow \hat{u}_{*}\in K_{\hat{\varphi}_{+}}\backslash\{0\},$$ hence\ $\hat{u}_{*}=u_{*}\in\ \mbox{int}\, C_{+}$,
see \eqref{eq30}.

Since $\hat{\varphi}|_{C_{+}}=\hat{\varphi}_{+}|_{C_{+}}$, it follows that $u_{*}$ is a local $C^{1}_{0}(\bar{\Omega})$-minimizer of $\hat{\varphi}$. Invoking Proposition \ref{prop2},
we infer that $u_{*}$ is a local $W^{1,p}_{0}(\Omega)$--minimizer of $\hat{\varphi}$. Similarly for $v_{*}\in - \mbox{int}\, C_{+}$.

Without any loss of generality, we may assume that $\hat{\varphi}(v_{*})\leq \hat{\varphi}(u_{*})$ (the analysis is similar if the opposite inequality holds).
Since $u_{*}\in\ \mbox{int}\, C_{+}$ is a local minimizer of $\hat{\varphi}$, we can find $\rho\in (0,1)$ small such that
\begin{equation}\label{eq31}
	\hat{\varphi}(v_{*})\leq \hat{\varphi}(u_{*})< \inf\,\{\hat{\varphi}(u): || u-u_{*} ||=\rho\}=\hat{\eta}_{\rho}\, ,\qquad || v_{*}-u_{*}||>\rho\,.
\end{equation}
Recall that $\hat{\varphi}$ is coercive (see (29)), hence it satisfies the PS-condition. This fact and (31) permit the use of Theorem 1 (the mountain pass theorem). So, we can find $y_{0}\in W^{1,p}_{0}(\Omega)$ such that
\begin{equation}\label{eq32}
	{\hat{\varphi}}'(y_0)=0 \qquad \mbox{and} \qquad \hat{\eta}_{\rho}\leq\hat{\varphi}(y_0)
\end{equation}
\begin{equation}\label{eq33}
	\hat{\varphi}(y_0)=\inf_{\gamma \in \Gamma}\max_{0\leq t\leq 1} \hat{\varphi}(\gamma (t)),
\end{equation}
where $\Gamma=\{\gamma\in C([0,1], W^{1,p}_0 (\Omega)):\ \gamma(0)=v_{*}, \gamma(1)=u_{*} \}$. Since $y_0\in K_{\hat{\varphi}}\subseteq [v_{*},u_{*}]$, it follows from (29) that $y_0$ is a solution of (1) and $y_0\in C^{1}_{0} (\bar{\Omega})$ (nonlinear regularity, see [17], [18]). If we show that $y_0$ is nontrivial, then the extremality of $u_{*},\,v_{*}$ and (32) imply that $y_0$ is nodal. To show the nontriviality of $y_{0}$ we use (33). According to (33), in order to show that $y_{0}$ is nonzero, it suffices to produce a path $\gamma_{*}\in\Gamma$ such that $\hat{\varphi}|_{\gamma_{*}}<0$. To this end, let
$$ M=W^{1,\tau}_{0} (\Omega)\cap\partial B^{L^{\tau}}_{1}\quad \mbox{endowed\ with\ the\ relative}\ W^{1,p}_{0}(\Omega)-\mbox{topology} $$
$$ M_c=M\cap C^1_0 (\bar{\Omega})\quad \mbox{endowed\ with\ the\ relative}\ C^1_0 (\bar{\Omega})-\mbox{topology}. $$

Recall that $\partial B_1^{L^{\tau}}=\{u\in L^{\tau}(\Omega): || u ||_{\tau}=1\}$. Clearly, $M_c$ is dense in $M$. We introduce the following two sets of paths
$$ \Gamma=\{\hat{\gamma}\in C([-1,1],M):\ \hat{\gamma}(-1)=-\hat{u}_1(\tau), \hat{\gamma}(1)=\hat{u}_1(\tau) \}$$
$$ \Gamma_c=\{\hat{\gamma}\in C([-1,1],M_c):\ \hat{\gamma}(-1)=-\hat{u}_1(\tau), \hat{\gamma}(1)=\hat{u}_1(\tau) \}\,. $$

\textbf{Claim 2:} {\it $\hat{\Gamma}_c$ is dense in $\hat{\Gamma}$.}

Let $\hat{\gamma}\in \hat{\Gamma}$ and $\epsilon\in (0,1)$. We introduce the multi-function $L_\epsilon :[-1,1]\rightarrow 2^{C^1_0(\bar{\Omega})}$ defined by
$$ L_{\epsilon}(t)=\{u\in C^1_0 (\bar{\Omega}): || u-\hat{\gamma}(t) || <\epsilon\}\qquad \mbox{for\ all}\ t\in(-1,1);$$
$$ L_{\epsilon}(-1)=\{-\hat{u}_1 (\tau)\},\quad L_\epsilon (1)=\{\hat{u}_1 (\tau)\}.$$

Clearly $L_{\epsilon}$ has nonempty and convex values. Note that for $t\in(-1,1)$, $L_{\epsilon}(t)$ is open, while both $L_{\epsilon}(-1)$ and $L_{\epsilon}(1)$ are finite dimensional.
Therefore $L_{\epsilon}$ has values in the family ${\mathcal D}(C^1_0 (\bar{\Omega}))$ of Michael [21]. Moreover, the continuity of $\hat{\gamma}$ implies that the multifunction $L_{\epsilon}$ is lower semi-continuous (see Papageorgiou \& Kyritsi \cite[p. 458]{23}). So, we can apply Theorem 3.1''' of Michael [21] and produce a continuous map $\hat{\gamma}^{\epsilon}:[-1,1]\rightarrow C^1_0 (\bar{\Omega})$ such that $\hat{\gamma}^\epsilon (t)\in L_{\epsilon}(t)$ for all $t\in[-1,1]$. Let $\epsilon_n=\frac{1}{n}$ and $\{\hat{\gamma}^n=\hat{\gamma}^{\epsilon_n}\}_{n \geq 1} \subseteq C([-1,1], C^1_0 (\bar{\Omega}))$ be the selectors produced above. Then
\begin{equation}\label{eq34}
\di||\hat{\gamma}^n (t)-\hat{\gamma}(t)||<\frac{1}{n}\quad \mbox{for\ all}\ t\in(-1,1),\quad \hat{\gamma}^n (-1)=-\hat{u}_1 (\tau),\quad\mbox{and}\quad
\hat{\gamma}^n (1)=\hat{u}_1 (\tau)\quad \mbox{for\ all}\ n\geq 1.
\end{equation}

Recall that $\hat{\gamma}(t)\in \partial B^{L^{\tau}}_1$ for all $t\in[-1,1]$. So from (34) we see that we may assume that $|| \hat{\gamma}^n (t) || \neq 0$ for all $t\in[-1,1]$. Then we set
$$ \hat{\gamma}^n_0 (t)=\frac{\hat{\gamma}^n (t)}{|| \hat{\gamma}^n (t) ||_{\tau}}\qquad \mbox{for\ all}\ t\in [-1,1]\,.$$
Then $\hat{\gamma}^n_0\in C([-1,1],M_c)$ and $\hat{\gamma}_0 (-1)=-\hat{u}_1 (\tau),\ \hat{\gamma}_0 (1)=\hat{u}_1 (\tau)$. Also, for all $t\in [-1,1]$ and all $n\geq 1$,
\begin{equation}\label{eq35}
	|| \hat{\gamma}^n_0 (t)-\hat{\gamma} (t) || \leq || \hat{\gamma}^n_0 (t) - \hat{\gamma}^n (t) || + || \hat{\gamma}^n (t)- \hat{\gamma}(t)||
= \frac{| 1- || \hat{\gamma}^n (t)||_{\tau} |}{||\hat{\gamma}^n (t)||_\tau}\, || \hat{\gamma}^n (t) || +\frac{1}{n}\quad (\mbox{see}\ (34))\,.
\end{equation}

Note that
\begin{eqnarray}
\max_{-1\leq t\leq 1}|1-||\hat{\gamma}^n (t) ||_{\tau} |&&=\max_{-1\leq t\leq 1}|\ ||\hat{\gamma}(t)||_{\tau}-||\hat{\gamma}^n (t) ||_{\tau} |\quad (\mbox{recall that}\ \hat{\gamma}(t)\in \partial B^{L^\tau}_1)\nonumber\\
 &&\leq\max_{-1\leq t \leq 1} || \hat{\gamma}(t)-\hat{\gamma}^n (t) ||_{\tau}\nonumber\\
&&\leq C_8 \max_{-1\leq t\leq 1} || \hat{\gamma} (t)-\hat{\gamma}^n (t) ||\quad \mbox{for\ some}\ C_8>0\nonumber\\
&& \leq \frac{C_8}{n}\ (\mbox{see}\ (\ref{eq26})),\nonumber\\
\Rightarrow&& \max_{-1\leq t \leq 1} || \hat{\gamma}^n_0 (t)-\hat{\gamma}(t)||\rightarrow 0 \quad \mbox{as} \ n\rightarrow \infty\nonumber\\
\Rightarrow&&\hat{\Gamma}_C\ \mbox{is\ dense\ in}\ \hat{\Gamma}.\nonumber
\end{eqnarray}
This proves Claim 2.

Using Claim 2 and Proposition \ref{prop4}, we see that given $\hat{\delta} > 0$, we can find $\hat{\gamma}_0\in \hat{\Gamma}_c$ such that
\begin{equation}\label{eq36}
	\max_{-1\leq t \leq 1} || D\hat{\gamma}_0 (t) ||_{\tau}^{\tau}\leq \hat{\lambda}_2 (\tau)+ \hat{\delta}.
\end{equation}
Also, from hypothesis $H^{'}_{1} (iii)$ we have
\begin{equation}\label{eq37}
	F(z,x)\geq \frac{C_6}{\tau}|x|^{\tau}\qquad \mbox{for\ a.a.}\ z\in\Omega \ \mbox{and all}\ |x|\leq \delta_0.
\end{equation}
Moreover, hypothesis $H(a)(iv)$ implies that given $\epsilon > 0$, we can find $\delta=\delta(\epsilon)>0$ such that
\begin{equation}\label{eq38}
	G(y)\leq \frac{\tilde{c}+\epsilon}{\tau}||y||^{\tau}\qquad \mbox{for\ all}\ ||y||\leq \delta.
\end{equation}

Let $\delta_1=\min\{\hat{\delta},\delta\}$ (see (37) and (38)). 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 for all $t\in[-1,1]$ we have
\begin{equation}\label{eq39}
	\vartheta\hat{\gamma}_{0}(t)\in[v_{*},u_{*}],\ \vartheta |\hat{\gamma}_{0} (t)(z)|,\ \vartheta || D\hat{\gamma}_{0}(t)(z) ||\leq \delta_1\qquad \mbox{for\ all}\ z\in \bar{\Omega}.
\end{equation}
Thus, for all $t\in[-1,1]$,
\begin{equation}\label{eq40}\begin{array}{ll}
	\hat{\varphi}(\vartheta\hat{\gamma}_{0} (t))&=\di\int_{\Omega}{G(\vartheta D \hat{\gamma}_{0} (t))dz}-\int_{\Omega}{F(z,\vartheta \hat{\gamma}_{0} (t)) dz}\quad (\mbox{see}\ (\ref{eq29}),\ (\ref{eq39}))\\
	&\di\leq \frac{\tilde{c}+\epsilon}{\tau}\vartheta^{\tau}|| D\hat{\gamma}_{0} (t) ||_{\tau}^{\tau}-\frac{C_6}{\tau}||\hat{\gamma}_{0} (t) ||_{\tau}^{\tau}\quad (\mbox{see}\ (\ref{eq37}),\ (\ref{eq38}),\ (\ref{eq39}))\\
	&\di\leq \frac{\vartheta^{\tau}}{\tau}[(\tilde{c}+\epsilon)(\hat{\lambda}_2(\tau)+\hat{\delta})-C_6]\,.\end{array}
\end{equation}
Since $c_6>\tilde{c}\hat{\lambda}_2(\tau)$ (see $H^{'}_{1}(iii)$) by choosing $\epsilon>0$ and $\hat{\delta}>0$ small, relation \eqref{eq40} yields
$$ \hat{\varphi}(\vartheta\hat{\gamma}_{0})(t)<0\qquad \mbox{for\ all}\ t\in[-1,1]\,.$$
Therefore, if $\hat{\gamma}=\vartheta\hat{\gamma}_{0}$, then $\hat{\gamma}$ is a continuous path in the Sobolev space $W^{1,p}_0(\Omega)$ which connects $-\vartheta\hat{u}_1(\tau)$ and $\vartheta\hat{u}_1(\tau)$, and
\begin{equation}\label{eq41}
	\hat{\varphi}|_{\hat{\gamma}}<0.
\end{equation}

Next, we produce a continuous path in $W^{1,p}_{0}(\Omega)$ connecting $\vartheta\hat{u}_1 (r)$ and $u_*$ and along this path the functional $\hat{\varphi}$ is strictly negative. To this end, let
\begin{equation}\label{eq42}
	a=\hat{\varphi}_{+}(u_*)=\inf\,\{\hat{\varphi}_{+}(u):\ u\in W^{1,p}_0(\Omega)\}<0=\hat{\varphi}_{+}(0)
\end{equation}
(see the first part of the proof).

Applying the second deformation theorem (see, for example, Gasinski \& Papageorgiou \cite[p. 628]{13}), we find a continuous map $h:[0.1]\times(\hat{\varphi}_{+}^0\backslash K^0_{\hat{\varphi}_{+}})\rightarrow \hat{\varphi}^0_{+}$ such that
\begin{equation}\label{eq43}
h(0,u)=u\qquad \mbox{for\ all}\ u\in \hat{\varphi}^{0}_{+} \backslash K^{0}_{\hat{\varphi}_{+}}
\end{equation}
\begin{equation}\label{eq44}
h(1,\hat{\varphi}^{0}_{+}\backslash K^{0}_{\hat{\varphi}_{+}})\subseteq \hat{\varphi}^{a}_{+}
\end{equation}
and
\begin{equation}\label{eq45}
\hat{\varphi}_{+}(h(t,u)) \leq \hat{\varphi}_{+} (h(s,u))\quad \mbox{for\ all}\ t,s\in[0,1],\ s\leq t\ \mbox{and all}\  u\in\hat{\varphi}_{+}^{0}\backslash K^{0}_{\hat{\varphi}_{+}}.
\end{equation}

>From (\ref{eq30}) we have $K_{\hat{\varphi}_{+}}=\{0,u_{*}\}$. Hence $\hat{\varphi}^{a}_{+}=\{u_{*}\}$ (see (\ref{eq42})). We deduce that
$$\hat{\varphi}_{+}(\vartheta\hat{u}_{1}(\tau))=\hat{\varphi}(\vartheta\hat{u}_{1}(\tau))=\hat{\varphi}
(\hat{\gamma}(\ref{eq1}))<0\quad (\mbox{see}\ (\ref{eq41}))
\Rightarrow\vartheta\hat{u}_{1}(\tau)\in\hat{\varphi}^{0}_{+}\backslash K^{0}_{\hat{\varphi}_{+}}=\hat{\varphi}^{0}_{+}\backslash\{0\}\,.
$$
Hence we can define
$$\hat{\gamma}_{+}(t)=h(t,\vartheta\hat{u}_{1}(\tau))^{+}\qquad \mbox{for\ all}\ t\in[0,1].$$
Clearly, this defines a continuous path in $W^{1,p}_{0}(\Omega)$ and we have
\begin{eqnarray}
&&\hat{\gamma}_{+}(0)=h(0,\vartheta\hat{u}_{1}(\tau))^{+}=\vartheta\hat{u}_{1}(\tau)\quad (\mbox{see}\ (\ref{eq43}));\nonumber\\
&&\hat{\gamma}_{+}(1)=h(1,\vartheta\hat{u}_{1}(\tau))^{+}=u_{*}\quad (\mbox{see}\ (\ref{eq44}));\nonumber\\
&&\hat{\varphi}(\hat{\gamma}_{+}(t))=\hat{\varphi}_{+}(\hat{\gamma}_{+}(t))\leq\hat{\varphi}_{+}(\vartheta\hat{u}_{1}(\tau))=\hat{\varphi}(\vartheta\hat{u}_1(\tau)) < 0\quad (\mbox{see}\ (\ref{eq45})\ \mbox{and}\ (\ref{eq41})).\nonumber
\end{eqnarray}
Therefore $\hat{\gamma}_{+}$ is a continuous path in the Sobolev space $W^{1,p}_{0}(\Omega)$ which connects $\vartheta\hat{u}_{1}(\tau)$ and $u_{*}$ and
\begin{equation}\label{eq46}
\hat{\varphi}|_{\hat{\gamma}_{+}}<0.
\end{equation}

In a similar fashion we produce $\hat{\gamma}_{-}$ a continuous path in $W^{1,p}_{0}(\Omega)$ which connects $-\vartheta\hat{u}_{1}(\tau)$ and $v_{*}$ such that
\begin{equation}\label{eq47}
\hat{\varphi}|_{\hat{\gamma}_{-}}<0.
\end{equation}

We concatenate the paths $\hat{\gamma}_{-}$, $\hat{\gamma}$, $\hat{\gamma}_{+}$ and produce a path $\gamma_{*}\in\Gamma$ such that
$$\hat{\varphi}|_{\gamma_{*}}<0\quad (\mbox{see}\ (\ref{eq41}),\ (\ref{eq46}),\ (\ref{eq47}))
\Rightarrow y_{0}\in C^{1}_{0}(\bar{\Omega})\backslash\{0\}\quad \mbox{is\ a\ nodal\ solution\ of}\ (\ref{eq1})\,.
$$
\qed

So, we can conclude this section with the following multiplicity theorem for problem (\ref{eq1}).

\begin{theorem}\label{th2}
Assume that hypotheses $H(a)$ and $H'_{1}$ hold.
Then problem (\ref{eq1}) admits at least three nontrivial solutions
$u_{0}\in {\rm int}\,C_{+}$,\ $v_{0}\in- {\rm int}\,C_{+}$, and
$y_{0}\in[v_{0},u_{0}]\cap C^{1}_{0}(\bar{\Omega})$\ nodal.
\end{theorem}

\section{The $(p,2)$--Problem}
In this section we deal with the particular case of problem (\ref{eq1}) in which $a(y)=||y||^{p-2}y+y$ for all $y\in\RR^{N}$, with $2\leq p<\infty$. Thus, the nonlinear nonhomogeneous Dirichlet problem under consideration is the following:
\begin{equation}\label{eq48}
-\Delta_{p}u(z)-\Delta u(z)=f(z,u(z))\quad  \mbox{in}\ \Omega,\qquad u|_{\partial \Omega}=0.
\end{equation}

We have
\begin{equation}\label{eq49}
\nabla a(y)=||y||^{p-2}\left[I+(p-2)\frac{y\otimes y}{||y||^{2}}\right]+I\qquad\mbox{
for\ all}\ y\in\RR^{N}\backslash\{0\},\ \nabla a(0)=0.
\end{equation}


For this problem, if on the reaction term $f(z,x)$ we impose the stronger hypotheses $H_{2}$, we can localize more precisely the nodal solution $y_{0}$.

In what follows, $u_{*}\in\mbox{int}\,C_{+}$ and $v_{*}\in\mbox{-int}\,C_{+}$ are the extremal constant sign solutions from Proposition \ref{prop8}.

\begin{prop}\label{prop10}
Assume that hypotheses $H_{2}$ hold.
Then problem (\ref{eq48}) has a nodal solution $y_{0}\in C^{1}_{0}(\bar{\Omega})$\ such that
$y_{0}\in {\rm int}_{C^{1}_{0}(\bar{\Omega})}[v_{*},u_{*}]$.
\end{prop}


{\it Proof.}
>From Proposition \ref{prop9}, we already have the existence of a nodal solution
$y_{0}\in[v_{*},u_{*}]\cap C^{1}_{0}(\bar{\Omega})$.

>From (\ref{eq49}) we see that $\nabla a(u_{*}(z))$ and $\nabla a(v_*(z))$ are both positive definite in $\Omega$. Hence we can apply the tangency principle of Pucci \& Serrin \cite[p. 35]{27}. Therefore
$$v_{*}(z)< y_{0}(z)<u_{*}(z)\qquad \mbox{for\ all}\ z\in\Omega.$$

Let $\rho=\max\{||v_{*}||_{\infty},||u_{*}||_{\infty}\}$ and let $\xi_{\rho}>0$ be as postulated by hypothesis
	$H_2(iv)$. Let\ $\hat{\xi}_{\rho}>\xi_{\rho}$.\ Then
	$$\begin{array}{ll}-\Delta_p y_0(z)-\Delta y_0(z)+\hat{\xi}_{\rho}|y_0(z)|^{p-2}y_0(z)
	&\di=f(z,y_0(z))+\hat{\xi}_{\rho}|y_0(z)|^{p-2}y_0(z)\\
	 &\di=f(z,y_0(z))+\xi_{\rho}|y_0(z)|^{p-2}y_0(z)+(\hat{\xi}_{\rho}-\xi_{\rho})|y_0(z)|^{p-2}y_0(z)\\
	&\di\leq f(z,u_*(z))+\xi_{\rho}u_*(z)^{p-1}+(\hat{\xi}_{\rho}-\xi_{\rho})u_*(z)^{p-1}\\
	&\di=-\Delta_p u_*(z)-\Delta u_*(z)+\hat{\xi}_{\rho}u_*(z)^{p-1}\quad \mbox{a.e.\ in}\ \Omega,\end{array}$$
hence
	$u_*-y_0\in\mbox{int}\,C_+$\ (see\ Proposition\ \ref{prop3}).
In a similar fashion we show that
$y_0-v_*\in\mbox{int}\,C_+.$
Therefore we conclude that
$y_0\in\mbox{int}_{C^{1}_{0}(\bar{\Omega})}[v_*,u_*]$.
\qed

Using this result and  improving the regularity of $f(z,\cdot)$, we can produce a second nodal solution for problem (\ref{eq48}) (for a total of four nontrivial smooth solutions). The new hypotheses on $f(z,x)$ are the following:

\medskip
	\noindent $H_3$:\quad $f:\Omega\times\RR\rightarrow\RR\ \mbox{is\ a\ measurable\ function\ such that
	for\ a.a.}\ z\in\Omega,\ f(z,0)=0,\ f(z,\cdot)\in C^1(\RR)\ \mbox{and}$\\
	\noindent (i)\quad $|f'_x(z,x)|\leq a(z)(1+|x|^{r-2})\ \mbox{for\ a.a.}\ z\in\Omega\ \mbox{and
all}\ x\in\RR\ \mbox{with}\ a\in L^{\infty}(\Omega)_+,\ p\leq r<p^*$;\\
(ii)\quad$\lim\limits_{x\rightarrow \pm\infty}\frac{f(z,x)}{|x|^{p-2}x}=-\infty\ \mbox{uniformly\ for\ a.a.}\ z\in\Omega$;\\
	\noindent (iii)\quad$\mbox{there\ exists\ an integer}$\ $m\geq 2$\ \mbox{such that}\
	$\hat{\lambda}_m(2)\leq f'_x(z,0)\leq \hat{\lambda}_{m+1}(2)$
$\mbox{a.e.\ in}\ \Omega,\ f'_x(\cdot,0)\neq \hat{\lambda}_m(2)$, $f'_x(\cdot,0)\neq\hat{\lambda}_{m+1}(2)$,
$f'_x(z,0)=\lim\limits_{x\rightarrow 0}\frac{f(z,x)}{x}$\ \mbox{uniformly\ for\ a.a.}\ $z\in\Omega$.


\begin{remark}
	Hypotheses $H_3(i),(iii)$ imply that for every $\rho>0$ we can find $\xi_{\rho}>0$ such that
	\begin{eqnarray}
		&&f^{'}_{x}(z,x)+\xi_{\rho}(p-1)|x|^{p-2}\geq 0\quad \mbox{for\ a.a.}\ z\in\Omega\ \mbox{and all}\ |x|\leq\rho\nonumber\\
		\Rightarrow&&\frac{\partial}{\partial x}\left[f(z,x)+\xi_{\rho}|x|^{p-2}x\right]\geq 0\quad \mbox{for\ a.a.}\ z\in\Omega\  \mbox{and all}\ |x|\leq\rho\nonumber\\
		\Rightarrow&&x\longmapsto f(z,x)+\xi_{\rho}|x|^{p-2}x\quad \mbox{is\ nondecreasing\ on}\ [-\rho,\rho].\nonumber
	\end{eqnarray}
	Thus, hypotheses $H_2(iv)$ is automatically satisfied due to the stronger regularity of $f(z,\cdot)$.
\end{remark}

Under these stronger conditions on the reaction $f(z,x)$ we can prove the following multiplicity theorem.

\begin{theorem}\label{th3}
	Assume that hypotheses $H_3$ hold.
Then problem (\ref{eq48}) admits at least four nontrivial solutions
	$u_0\in{\rm int}\,C_+$,\ $v_0\in- {\rm int}\,C_+$, and
$y_0,\, \hat{y}\in{\rm int}_{C^{1}_{0}(\bar{\Omega})}[v_0,u_0]$\ nodal.
\end{theorem}

{\it Proof.}
	From Theorem \ref{th2} and Proposition \ref{prop10}, we already have three solutions
	$u_0\in\mbox{int}\,C_+$,\ $v_0\in-\mbox{int}\,C_+$ and
$y_0\in\mbox{int}_{C^{1}_{0}(\bar{\Omega})}[v_0,u_0]$\ nodal.
	
	Without any loss of generality we may assume that $u_0,\ v_0$ we are extremal (that is, $u_0=u_*$,\ $v_0=v_*$). Using the notation introduced in Proposition \ref{prop9}, we obtain that $u_0\in\mbox{int}\,C_+$ and $v_0\in-\mbox{int}\,C_+$ are both local minimizer of $\hat{\varphi}$. Hence
	\begin{eqnarray}\label{eq50}
		C_k(\hat{\varphi},u_0)=C_k(\hat{\varphi},v_0)=\delta_{k,0}\ZZ\qquad \mbox{for\ all}\ k\geq 0.
	\end{eqnarray}
	Since $y_0\in\mbox{int}_{C^{1}_{0}(\bar{\Omega})}[v_0,u_0]$ and $\varphi\left|_{[v_0,u_0]}=\hat{\varphi}\right|_{[v_0,u_0]}$, we have
	\begin{eqnarray}\label{eq51} &&C_k(\varphi |_{C^{1}_{0}(\bar{\Omega})},
y_0)=C_k(\hat\varphi |_{C^{1}_{0}(\bar{\Omega})},
y_0)
   \qquad \mbox{for\ all}\ k\geq 0\nonumber\\
		\Rightarrow&&C_k(\varphi,y_0)=C_k(\hat{\varphi},y_0)\qquad \mbox{for\ all}\ k\geq 0\quad
(\mbox{see\ Palais\ \cite{22}\ and\ Bartsch\ \cite{4}}).
	\end{eqnarray}
	From the proof of Proposition \ref{prop9}, we know that $y_0$ is a critical point of $\hat{\varphi}$. Hence
	\begin{eqnarray}\label{eq52}
		C_1(\hat{\varphi},y_0)\neq 0
		\Rightarrow C_1(\varphi,y_0)\neq 0\qquad (\mbox{see}\ (\ref{eq51})).
	\end{eqnarray}
	Since $\varphi\in C^2(W^{1,p}_{0}(\Omega))$, using (\ref{eq52}) as in Papageorgiou \& Smyrlis \cite{26} (see the proof of Proposition 13), we obtain for all $k\geq 0$
	\begin{eqnarray}\label{eq53}
		C_k(\varphi,y_0)=\delta_{k,1}\ZZ\
		\Rightarrow \ C_k(\hat{\varphi},y_0)=\delta_{k,1}\ZZ\,.
	\end{eqnarray}
	Also, by virtue of hypothesis $H_3(iii)$, from Papageorgiou \& Smyrlis \cite[Proposition 10]{26}  we have
	\begin{eqnarray}\label{eq54}
		C_k(\hat{\varphi},0)=\delta_{k,d_m}\ZZ\quad \mbox{for\ all}\ k\geq 0,\quad \mbox{with}\ d_m=\mbox{dim}\,\overset{m}{\underset{\mathrm{i=1}}\oplus} E(\hat{\lambda}_i(2)).
	\end{eqnarray}
	Finally, recall that $\hat{\varphi}$ is coercive (see (\ref{eq29})). Hence
	\begin{eqnarray}\label{eq55}
		C_k(\hat{\varphi},\infty)=\delta_{k,0}\ZZ\qquad \mbox{for\ all}\ k\geq 0.
	\end{eqnarray}
	Suppose that $K_{\hat{\varphi}}=\{0,u_0,v_0,y_0\}$. Then from (\ref{eq50}), (\ref{eq53}), (\ref{eq54}), (\ref{eq55}) and the Morse relation with $t=-1$ (see \eqref{eq6}), we have
	$(-1)^{d_m}+2(-1)^0+(-1)^1=(-1)^0$,
	 a\ contradiction.
	So, we can find $\hat{y}\in K_{\hat{\varphi}},\ \hat{y}\notin\{0,u_0,v_0,y_0\}$. Since $\hat{y}\in[v_0,u_0]$ (see (\ref{eq30})), we conclude that $\hat{y}\in C^{1}_{0}(\bar{\Omega})$ is nodal. Moreover, as in the proof of Proposition \ref{prop10}, we show that $\hat{y}\in\mbox{int}_{C^{1}_{0}(\bar{\Omega})}[v_0,u_0]$.
\qed

\section{Parametric Equations}

In this section we deal with the following parametric $p$--Laplacian Dirichlet equation
\begin{eqnarray}\label{eq56}
	-\Delta_p u(z)=\lambda|u(z)|^{p-2}u(z)-f(z,u(z))\quad \mbox{in}\ \Omega,\qquad u\left|_{\partial\Omega}=0\right.,\ 1<p<\infty.
\end{eqnarray}
Here, $\lambda>0$ is a parameter. The hypotheses on the perturbation term $f(z,x)$ are the following:

\medskip
\noindent	$H_4$:\quad$f:\Omega\times\RR\rightarrow\RR$\ is\ a\ Carath\'eodory\ function
such that\ $f(z,0)=0$\ for\ a.a.\ $z\in\Omega$\ and\\
	\noindent (i)\quad for\ every\ $\rho>0$,\ there\ exists\ $a_{\rho}\in L^{\infty}(\Omega)_+$
such that\ $|f(z,x)|\leq a_{\rho}(z)$\ for\ a.a.\ $z\in\Omega$ and\ all\ $|x|\leq\rho$;\\
\noindent	 (ii)\quad$\lim\limits_{x\rightarrow\pm\infty}\frac{f(z,x)}{|x|^{p-2}x}=+\infty$\ uniformly\ for\ a.a.\ $z\in\Omega$;\\
\noindent	(iii)\quad$\lim\limits_{x\rightarrow 0}\frac{f(z,x)}{|x|^{p-2}x}=0$\ uniformly\ for\ a.a.\ $z\in\Omega$.

\medskip
For problem (\ref{eq56}), $a(y)=||y||^{p-2}y$ for all $y\in\RR^N,\ \tau=p$ and $\tilde{c}=1$ (see hypothesis $H(a)(iv)$). By virtue of hypothesis $H_4(iii)$, we see that given $\epsilon>0$, we can find $\delta_0=\delta_0(\epsilon)>0$ such that
\begin{eqnarray}\label{eq57}
	f(z,x)x\leq \epsilon|x|^p\qquad \mbox{for\ a.a.}\ z\in\Omega\ \mbox{and all}\ |x|\leq\delta_0.
\end{eqnarray}
Hence we have
$$\lambda|x|^p-f(z,x)x\geq(\lambda-\epsilon)|x|^p\qquad \mbox{for\ a.a.}\ z\in\Omega\ \mbox{and all}\ |x|\leq\delta_0.$$
If $\lambda>\hat{\lambda}_2(p)$, then we choose $\epsilon\in(0,\lambda-\hat{\lambda}_2(p))$ and we can apply Theorem \ref{prop2}. Thus, we obtain

\begin{theorem}\label{th4}
	Assume that hypotheses $H_3$ hold and $\lambda>\hat{\lambda}_2(p)$.
Then problem (\ref{eq56}) has at least three nontrivial solutions
	$u_0\in{\rm int}\,C_+$,\ $v_0\in-{\rm int}\,C_+$,\ and
$y_0\in[v_0,u_0]\cap C^{1}_{0}(\bar{\Omega})$\ nodal.
\end{theorem}
\begin{remark}
	Such a multiplicity result was first proved by Ambrosetti \& Mancini \cite{2}, with subsequent improvements by Ambrosetti \& Lupo \cite{1} and Struwe \cite{30}, \cite[p. 132]{29}, when $p=2$ and $f(z,\cdot)=f(\cdot)\in C^1(\RR)$ or even Lipschitz continuous. In their multiplicity theorems, they do not show that the third solution is nodal. This result was extended to $p$--Laplacian equations by Papageorgiou \& Papageorgiou \cite{24}. Again, it is not shown that the third solution is nodal.
\end{remark}

In the semilinear case $(p=2)$, with a more regular reaction $f(z,\cdot)$ and with additional restrictions on the parameter $\lambda>0$, we can improve Theorem \ref{th4} and produce a second nodal solution.
So, we consider the following semilinear parametric problem:
\begin{eqnarray}\label{eq58}
	-\Delta u(z)=\lambda u(z)-f(z,u(z))\quad \mbox{in}\ \Omega,\qquad u\left|_{\partial\Omega}=0.\right.
\end{eqnarray}
The hypotheses on the perturbation $f(z,x)$ are the following:

\medskip
\noindent $H_4$:\quad $f:\Omega\times\RR\rightarrow\RR$\ is\ a\ measurable\ function\ such that
 for\ a.a.\ $z\in\Omega$,\ $f(z,0)=0$,\ $f(z,\cdot)\in C^1(\RR)$\ and\\
\noindent (i)\quad $|f'_x(z,x)|\leq a(z)(1+|x|^{r-2})$\ for\ a.a.\ $z\in\Omega$ and\ all\ $x\in\RR$,
 with\ $a\in L^{\infty}(\Omega)_+$,\ $2\leq r<2^*$;\\
\noindent (ii)\quad $\lim\limits_{x\rightarrow\pm\infty}\frac{f(z,x)}{x}=+\infty$\ uniformly\ for\ a.a.\ $z\in\Omega$;\\
\noindent (iii)\quad$\lim\limits_{x\rightarrow 0}\frac{f(z,x)}{x}=0$\ uniformly\ for\ a.a.\ $z\in\Omega$.

\medskip
In what follows, $\hat{\sigma}(2)$ denotes the spectrum of $(-\Delta,H^{1}_{0}(\Omega))$, that is, $\hat{\sigma}(2)=
\{\hat{\lambda}_k(2)\}_{k\geq 1}$ (see Section 2). Using Theorem \ref{th3}, we obtain the following multiplicity property.
\begin{theorem}\label{th5}
	Assume that hypotheses $H_4$ hold and $\lambda>\hat{\lambda}_2(2),\ \lambda\notin\hat{\sigma}(2)$.
Then problem (\ref{eq58}) has at least four nontrivial solutions
	$u_0\in{\rm int}\,C_+$,\ $v_0\in-{\rm int}\,C_+$,\ and
$y_0,\ \hat{y}\in{\rm int}_{C^{1}_{0}(\bar{\Omega})}[v_0,u_0]$\ nodal.
\end{theorem}

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\end{document} 