%----------------------------------------------------------------------------% Beginning of tmna-l2e.tex
%----------------------------------------------------------------------------%
% LaTeX sample file for TMNA volumes.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\documentclass{tmna}

\newcommand{\RR}{\mathbb R}
\newcommand{\NN}{\mathbb N}
\newcommand{\PP}{\mathbb P}
\newcommand{\ZZ}{\mathbb Z}
\renewcommand{\le}{\leqslant}
\renewcommand{\leq}{\leqslant}
\renewcommand{\ge}{\geqslant}
\renewcommand{\geq}{\geqslant}


%  The following items give publication information for the TMNA logo

\nvol{2}
\cvol{43}
\cyear{2014}
\cvolyear{2014}
\pages{1--99}
\date{May 28, 2014}

 

\begin{document}

\title[Resonant and superlinear
nonhomogeneous elliptic equations]{Multiplicity
theorems for resonant and superlinear nonhomogeneous elliptic equations}

\author[N.S. Papageorgiou --- V.D. R\u adulescu]{Nikolaos S. Papageorgiou --- Vicen\c{t}iu D. R\u{a}dulescu}

\address
{\textsc{Nikolaos S. Papageorgiou}\\
National Technical University\\ Department of Mathematics\\
				Zografou Campus\\ Athens 15780\\ Greece}

% Research address for author one - current address

%\curraddr{Department of Mathematics and Statistics\\	
%Case Western Reserve \\
%University, Cleveland, OH 43403}

\email{npapg@math.ntua.gr}

%address for  author two

\address{\textsc{Vicen\c{t}iu D. R\u{a}dulescu}\\
Department of Mathematics\\ Faculty of Sciences\\ King Abdulaziz University\\
P.O. Box 80203\\ Jeddah 21589\\ SAUDI ARABIA \\ \& \\ Institute of Mathematics ``Simion Stoilow" of the Romanian Academy\\ P.O. Box 1-764\\
          014700 Bucharest\\ ROMANIA}
          
          \email{vicentiu.radulescu@math.cnrs.fr}

\submittedby {M.~Clapp}

%\dedicatory {This is dedicated to a dear friend}

%  Math Subject Classifications
\subjclass[2010]{Primary 35J20. Secondary 35J60, 35J92, 58E05\hfill}

\keywords{Resonance, multiple solution, superlinear reaction, nodal solutions, critical groups.}

%  \thanks will become a 1st page footnote.

\thanks{The authors wish to thank a knowledgeable referee for his/her corrections and helpful remarks.}

\thanks{V.D. R\u adulescu was supported by
Grant CNCS-PCCA-23/2014.}


\begin{abstract}%
We consider nonlinear elliptic equations driven by the sum of a $p$-Laplacian ($p>2$) and a Laplacian. We
consider two distinct cases. In the first one, the reaction $f(z,\cdot)$ is $(p-1)$-linear near $\pm\infty$ and
resonant with respect to a nonprincipal variational eigenvalue of $(-\Delta_{p},W_{0}^{1,p}(\Omega))$.
We prove a multiplicity theorem producing three nontrivial solutions.
In the
second case, the reaction $f(z,\cdot)$ is $(p-1)$-superlinear but does not satisfy the Ambrosetti-Rabinowitz
condition. We prove two multiplicity theorems. In the first main result we produce six nontrivial solutions all with sign
information and in the second theorem we have five nontrivial solutions. Our approach uses variational methods
combined with Morse theory, truncation methods, and comparison techniques.
\end{abstract}


\maketitle


\section{Introduction}\label{sec1}
Let $\Omega\subseteq\RR^{N}$ be a bounded domain with a $C^{2}$-boundary $\partial\Omega$. In this paper we
study the following nonlinear nonhomogeneous Dirichlet problem
\begin{eqnarray}\label{eq1}
-\Delta_{p}u(z)-\Delta u(z)=f(z,u(z))\ \mbox{in}\ \Omega,\ u_{|_{\partial\Omega}}=0,\ 2<p<\infty.
\end{eqnarray}
Here $\Delta_{p}$ denotes the $p$-Laplace differential operator defined by
\begin{eqnarray*}
\Delta_{p}u=\mbox{div}(|Du|^{p-2}Du)\ \mbox{for all}\ u\in W_{0}^{1,p}(\Omega).
\end{eqnarray*}
If $p=2$, then $\Delta_{2}=\Delta=$\ the Laplace differential operator.

The reaction $f(z,x)$ 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. $z\in\Omega$, \ $x\longmapsto f(z,x)$ is continuous). Additional regularity conditions
on $f(z,\cdot)$ are introduced in order to produce extra solutions. We consider two distinct cases. In the
first (Section \ref{sec3}), we assume that $f(z,\cdot)$ is $(p-1)$-linear near $\pm\infty$ and interacts with a
nonprincipal variational eigenvalue of $(-\Delta_{p}, W_{0}^{1,p}(\Omega))$ (resonant problem). We prove a
multiplicity theorem, producing three nontrivial solutions two of which have constant sign. In the second case
(Section \ref{sec4}), we deal with reaction $f(z,\cdot)$ which is $(p-1)$-superlinear near $\pm\infty$ but
without satisfying the usual in such cases Ambrosetti-Rabinowitz condition (AR-condition for short). We also
assume that $f(z,\cdot)$ has $z$-dependent zeros of constant sign. We prove a multiplicity theorem producing
six nontrivial solutions, four of constant sign and two nodal (sign changing).

Recently nonhomogeneous nonlinear equations driven by the sum of a $p$-Laplacian and a Laplacian (a
$(p,2)$-equation for short), were studied by Cingolani and Degiovanni \cite{7},
Papageorgiou and R\u{a}dulescu \cite{21}, Papageorgiou and Smyrlis \cite{22} and Sun \cite{24}. All these
works deal with equations that have a $(p-1)$-linear reaction and either they do not allow resonance (see
\cite{7}) or the resonance is with respect to the principal eigenvalue of $(-\Delta_{p},W_{0}^{1,p}(\Omega))$ (see
\cite{21}, \cite{22}, \cite{24}). Recall that for $p\neq2$, we do not have a complete knowledge of the
spectrum of $(-\Delta_{p},W_{0}^{1,p}(\Omega))$, the eigenspaces are not linear subspaces and we do not have a
direct sum decomposition of $W_{0}^{1,p}(\Omega)$ in terms of them. All these facts, make problems resonant at
higher parts of the spectrum difficult to deal with. On the other hand, problems with reactions which have zeros
of constant sign, were investigated only in the context of $p$-Laplacian equations, by Bartsch, Liu and Weth
\cite{5} (constant zeros) and by Iturriaga, Massa, Sanchez and Ubilla \cite{17} (variable zeros for a class of
parametric equations). None of the aforementioned works produces six nontrivial solutions all with sign
information.

Our approach is a combination of variational methods based on the critical point theory together with Morse
theory (critical groups) and truncation and comparison techniques. In the next section, for the convenience of
the reader, we recall the main mathematical tools which we will use in the sequel.

\section{Mathematical Background}\label{sec2}
Let $X$ be a Banach space and $X^{*}$ its topological dual. By $\langle\cdot,\cdot\rangle$ we denote the duality
brackets for the pair $(X,X^{*})$. Let $\varphi\in C^{1}(X)$. We say that $\varphi$ satisfies the Cerami
condition (the C-condition for short), if the following holds:

\smallskip
``Every sequence $\{u_{n}\}_{n\geq1}\subseteq X$ such that $\{\varphi(u_{n})\}_{n\geq1}\subseteq\RR$ is bounded and
$$(1+\|u_{n}\|)\varphi'(u_{n})\rightarrow0\ \mbox{in}\ X^{*},$$
admits a strongly convergent subsequence".

\smallskip
This is a compactness type condition on the functional $\varphi$ which compensates for the fact that the
underlying space $X$ is not locally compact (being in general infinite dimensional). The C-condition is more
general than the more familiar Palais-Smale condition (PS-condition for short) and the two are equivalent if
$\varphi$ is bounded below. The C-condition is a basic tool in proving a deformation theorem from which we can
derive the minimax theory of certain critical values of $\varphi$. Prominent in that theory is the so-called
``mountain pass theorem" due to Ambrosetti and Rabinowitz \cite{3}. Here the theorem is presented in a slightly
more general form with the C-condition replacing the PS-condition (see, for example, Gasinski and Papageorgiou
\cite[p. 648]{15}).

\begin{theorem}\label{th1}
Assume that $\varphi\in C^{1}(X)$ satisfies the C-condition, $u_{0},u_{1}\in X$, $\|u_{1}-u_{0}\|>\rho>0$,
$$\max\{\varphi(u_{0}),\varphi(u_{1})\}<\inf[\varphi(u): \|u-u_{0}\|=\rho]=m_{\rho}$$
and $c=\displaystyle\inf_{\gamma\in\Gamma}\max_{0\leq t\leq1}\varphi(\gamma(t))$ with $\Gamma=\{\gamma\in
C([0,1],X): \gamma(0)=u_{0}, \gamma(1)=u_{1}\}.$\\
Then $c\geq m_{\rho}$ and $c$ is a critical value of $\varphi$.
\end{theorem}
In what follows by $|\cdot|$ we denote the norm of $\RR^{N}$ and by $\|\cdot\|$ the norm of the Sobolev space
$W_{0}^{1,p}(\Omega)$. By virtue of the Poincar\'e inequality, we have
$$\|u\|=\|Du\|_{p}\ \mbox{for all}\ \ u\in W_{0}^{1,p}(\Omega).$$

Let $1<p<\infty$ and let $A_{p}:W_{0}^{1,p}(\Omega)\rightarrow W^{-1,p'}(\Omega)=W_{0}^{1,p}(\Omega)^{*}$
\big($\frac{1}{p}+\frac{1}{p'}=1$\big) be the nonlinear map defined by
\begin{eqnarray*}
\langle A_{p}(u),y\rangle=\int_{\Omega}|Du|^{p-2}(Du,Dy)_{\RR^{N}}dz\ \ \mbox{for all}\ u,y\in
W_{0}^{1,p}(\Omega).
\end{eqnarray*}
When $p=2$, then we write $A_{2}=A$ and this map is linear. The next proposition summarizes the main properties
of this map (see, for example, Papageorgiou and Kyritsi \cite[p. 314]{20}).
\begin{prop}\label{prop2}
The map $A_{p}:W_{0}^{1,p}(\Omega)\rightarrow W^{-1,p'}(\Omega)$ is bounded (that is, maps bounded sets to bounded
sets), demicontinuous, strictly monotone (hence maximal monotone too) and of type $(S)_{+}$, that is, if
$u_{n}\overset{w}\rightarrow u$ in $W_{0}^{1,p}(\Omega)$ and
\begin{eqnarray*}
\limsup_{n\rightarrow\infty}\langle A_{p}(u_{n}), u_{n}-u\rangle\leq0,
\end{eqnarray*}
then $u_{n}\rightarrow u$ in $W_{0}^{1,p}(\Omega)$.
\end{prop}
Also, we can view $A_{p}$ defined on the bigger Sobolev space $W^{1,p}(\Omega)$, that is
$A_{p}:W^{1,p}(\Omega)\rightarrow W^{1,p}(\Omega)^{*}$. With a slight abuse of notation, we still denote the
duality brackets by $\langle\cdot,\cdot\rangle$ and we have
\begin{eqnarray*}
\langle A(u),y\rangle=\int_{\Omega}|Du|^{p-2}(Du,Dy)_{\RR^{N}}dz\ \ \mbox{for all}\ u,y\in W^{1,p}(\Omega).
\end{eqnarray*}
This map too is bounded, demicontinuous, maximal monotone and of  type $(S)_{+}$.

In addition to the Sobolev space $W_{0}^{1,p}(\Omega)$, we will also use the Banach space
$C_{0}^{1}(\overline{\Omega})=\{u\in C^{1}(\overline{\Omega}):u_{|_{\partial\Omega}}=0\}$. This is an ordered
Banach space with positive cone $C_{+}=\{u\in C_{0}^{1}(\overline{\Omega}): u(z)\geq0\ \mbox{for all}\
z\in\overline{\Omega}\}$. This cone has a nonempty interior, given by
\begin{eqnarray*}
\mbox{int}\, C_{+}=\left\{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\right\},
\end{eqnarray*}
where $n(\cdot)$ denotes the outward unit normal on $\partial\Omega$.

By $p^{*}$ we denote the critical Sobolev exponent. So, $
p^{*}=\left\{
\begin{array}{rl}
\frac{Np}{N-p} & \mbox{if}\ p<N \\
+\infty & \mbox{if}\ N\leq p
\end{array} \right.
$.

Let $f_{0}:\Omega\times\RR\rightarrow\RR$ be a Carath\'eodory function with subcritical growth in the
$x$-variable, that is,
\begin{eqnarray*}
|f_{0}(z,x)|\leq a_{0}(z)(1+|x|^{r-1})\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\in\RR
\end{eqnarray*}
with $a_{0}\in L^{\infty}(\Omega)_{+}$, $1<r<p^{*}$. Let $F_{0}(z,x)=\int_{0}^{x}f_{0}(z,s)ds$ and  let
$\varphi_{0}: W_{0}^{1,p}(\Omega)\rightarrow\RR$ ($2<p<\infty$) be the $C^{1}$-functional defined by
\begin{eqnarray*}
\varphi_{0}(u)=\frac{1}{p}\|Du\|_{p}^{p}+\frac{1}{2}\|Du\|_{2}^{2}-\int_{\Omega}F_{0}(z,u(z))dz\ \mbox{for all}\
u\in W_{0}^{1,p}(\Omega).
\end{eqnarray*}

The following proposition is a special case of a more general result of Gasinski and Papageorgiou \cite{16} and
essentially is a consequence of the regularity  results of Lieberman \cite{19}.
\begin{prop}\label{prop3}
 Assume that $u_{0}\in W_{0}^{1,p}(\Omega)$ is a local $C_{0}^{1}(\overline\Omega)$-minimizer of $\varphi_{0}$, that is,
there exists  $\rho_{0}>0$ such that
\begin{eqnarray*}
\varphi_{0}(u_{0})\leq\varphi_{0}(u_{0}+h)\ \mbox{for all}\ h\in C_{0}^{1}(\overline{\Omega}),\
\|h\|_{C_{0}^{1}(\overline{\Omega})}\leq\rho_{0}.
\end{eqnarray*}
Then $u_{0}\in C_{0}^{1,\alpha}(\overline{\Omega})$ for some $\alpha\in(0,1)$ and $u_{0}$ is also a local
$W_{0}^{1,p}(\Omega)$-minimizer of $\varphi_{0}$, that is, there exists $\rho_{1}>0$ such that
\begin{eqnarray*}
\varphi_{0}(u_{0})\leq\varphi_{0}(u_{0}+h)\ \mbox{for all}\ h\in W_{0}^{1,p}(\Omega),\ \|h\|\leq\rho_{1}.
\end{eqnarray*}
\end{prop}
Let $\eta,\hat{\eta}\in L^{\infty}(\Omega)$. We say $\eta\prec\hat{\eta}$ if and only if for every compact
$K\subseteq\Omega$, we can find $\varepsilon=\varepsilon_{K}>0$ such that
\begin{eqnarray*}
\eta(z)+\varepsilon\leq\hat{\eta}(z)\ \mbox{for a.a.}\ z\in K.
\end{eqnarray*}
Clearly, if $\eta,\hat{\eta}\in C(\Omega)$ and $\eta(z)<\hat{\eta}(z)$ for all $z\in\Omega$, then
$\eta\prec\hat{\eta}$.

A simple modification of the proof of Proposition 2.6 of Arcoya and Ruiz \cite{4}, in order to accommodate the
presence of the extra linear term $-\Delta u$, leads to the following strong comparison property.

\begin{prop}\label{prop4}
Assume that $\beta\geq0,\ \eta,\hat{\eta}\in L^{\infty}(\Omega)$, $\eta\prec\hat{\eta}$ and
$u\in C_{0}^{1}(\overline{\Omega})$, $v\in\mbox{int}\, C_{+}$ satisfy
\begin{align*}
&-\Delta_{p}u(z)-\Delta u(z)+\beta|u(z)|^{p-2}u(z)=\eta(z)\ \mbox{in}\ \Omega,\\
&-\Delta_{p}v(z)-\Delta v(z)+\beta v(z)^{p-1}=\hat{\eta}(z)\ \mbox{in}\ \Omega.
\end{align*}
Then $v-u\in\mbox{int}\, C_{+}$.
\end{prop}
As we already mentioned in the Introduction, our approach will use also Morse theory (critical groups). So, next
we recall some basic definitions and facts from Morse theory. Given a topological pair $(Y_{1}, Y_{2})$ with
$Y_{2}\subseteq Y_{1}\subseteq X$, for every integer $k\geq0$ by $H_{k}(Y_{1},Y_{2})$ we denote the
$k^{th}$-relative singular homology group with integer coefficients for the pair $(Y_{1}, Y_{2})$. Given
$\varphi\in C^{1}(X)$ and $c\in\RR$, we introduce the following sets:
\begin{eqnarray*}
\varphi^{c}=\{u\in X: \varphi(u)\leq c\},\ K_{\varphi}=\{u\in X:\varphi'(u)=0\},\ K_{\varphi}^{c}=\{u\in
K_{\varphi}: \varphi(u)=c\}.
\end{eqnarray*}
If $u\in K_{\varphi}^{c}$ is isolated, then the critical groups of $\varphi$ at $u$ are defined by
\begin{eqnarray*}
C_{k}(\varphi,u)=H_{k}(\varphi^{c}\cap U,\  \varphi^{c}\cap U\setminus\{u\})\ \mbox{for all}\ k\geq0,
\end{eqnarray*}
where $U$ is a neighborhood of $u$ such that $K_{\varphi}\cap\varphi^{c}\cap U=\{u\}$. The excision property of
singular homology implies that the above definition of critical groups is independent of the choice of the
neighborhood $U$.

Suppose that $\varphi\in C^{1}(X)$ satisfies the C-condition and $\inf \varphi(K_{\varphi})>-\infty$. Let
$c<\inf\varphi(K_{\varphi})$. The critical groups of $\varphi$ at infinity are defined by
\begin{eqnarray*}
C_{k}(\varphi,\infty)=H_{k}(X,\varphi^{c}), \mbox{for all}\ k\geq0.
\end{eqnarray*}
The second deformation theorem (see, for example, Gasinski and Papageorgiou \cite[p. 628]{15}), implies that the
above definition of critical groups at infinity is independent of the choice of the level
$c<\inf\varphi(K_{\varphi})$.

Suppose that $K_{\varphi}$ is finite. We introduce
\begin{eqnarray*}
M(t,u)=\sum_{k\geq0}\mbox{rank}\, C_{k}(\varphi,u)t^{k}\ \mbox{for all}\ t\in\RR,\ \mbox{all}\ u\in K_{\varphi}
\end{eqnarray*}
and
\begin{eqnarray*}
P(t,\infty)=\sum_{k\geq0}\mbox{rank}\, C_{k}(\varphi,\infty)t^{k}\ \mbox{for all}\ t\in\RR.
\end{eqnarray*}
The Morse relation says that
\begin{eqnarray}\label{eq2}
\sum_{u\in K_{\varphi}}M(t,u)=P(t,\infty)+(1+t)Q(t),
\end{eqnarray}
where $Q(t)=\displaystyle\sum_{k\geq0}\beta_{k}t^{k}$ is a formal series in $t\in\RR$ with nonnegative integer
coefficients.

Next, let us recall some basic facts concerning the spectrum of $(-\Delta_{p}, W_{0}^{1,p}(\Omega))$. So, we
consider the following nonlinear eigenvalue problem
\begin{eqnarray}\label{eq3}
-\Delta_{p}u(z)=\hat{\lambda}|u(z)|^{p-2}u(z)\ \mbox{in}\ \Omega,\ u_{|_{\partial\Omega}}=0.
\end{eqnarray}
A number $\hat{\lambda}\in\RR$ is an eigenvalue of $(-\Delta_{p}, W_{0}^{1,p}(\Omega))$, if problem \eqref{eq3}
admits a nontrvial solution $\hat{u}$. The nontrivial solution $\hat{u}$ is an eigenfunction corresponding to
the eigenvalue $\hat{\lambda}$. The smallest eigenvalue is denoted by $\hat{\lambda}_{1}(p)$ and has the
following properties:
\begin{itemize}
  \item $\hat{\lambda}_{1}(p)>0$ and it is isolated;
  \item $\hat{\lambda}_{1}(p)$ is simple (that is, if $\hat{u}, \hat{v}$ are eigenfunctions corresponding to
      the eigenvalue $\hat{\lambda}_{1}(p)$, then $\hat{u}=\xi\hat{v}$ for some $\xi\neq0$);
  \item we have \begin{eqnarray}\label{eq4}
  \hat{\lambda}_{1}(p)=\inf\bigg[\frac{\|Du\|_{p}^{p}}{\|u\|_{p}^{p}}:u\in W_{0}^{1,p}(\Omega), u\neq0\bigg].
  \end{eqnarray}
\end{itemize}
In \eqref{eq4} the infimum is realized on the one-dimensional eigenspace corresponding to
$\hat{\lambda}_{1}(p)>0$. In what follows, by $\hat{u}_{1}(p)$ we denote the $L^{p}$-normalized (that is,
$\|\hat{u}_{1}(p)\|_{p}=1$) positive eigenfunction corresponding to $\hat{\lambda}_{1}(p)$. The nonlinear
regularity theory and the nonlinear maximum principle (see, for example, Gasinski and Papageorgiou \cite[pp. 737-738]{15}), imply that $\hat{u}_{1}(p)\in\mbox{int}\ C_{+}$. Since the spectrum $\sigma (p)$ of $(-\Delta_{p},
W_{0}^{1,p}(\Omega))$ is closed and $\hat{\lambda}_{1}(p)>0$ is isolated, then the second eigenvalue is
well-defined by
\begin{eqnarray*}
\hat{\lambda}_{2}(p)=\inf\big[\hat{\lambda}\in\sigma (p):\hat{\lambda}>\hat{\lambda}_{1}(p)\big].
\end{eqnarray*}

Let $\mbox{ind}\,(A)$ be the $\ZZ_{2}$-cohomological index introduced by Fadell and Rabinowitz \cite{13}. Employing
the Ljusternik-Schnirelmann minimax scheme, we can define a whole divergent sequence
$\{\hat{\lambda}_{k}(p)\}_{k\geq1}$ of distinct eigenvalues of $(-\Delta_{p}, W_{0}^{1,p}(\Omega))$ by setting
\begin{eqnarray*}
\hat{\lambda}_{k}(p)=\inf\big[\sup_{u\in A}\|Du\|_{p}^{p}: A\subseteq M\ \mbox{symmetric},\ \mbox{ind}(A)\geq
k\big],\ k\geq1,
\end{eqnarray*}
where $M=\{u\in W_{0}^{1,p}(\Omega):\|u\|_{p}=1\}$ is a $C^{1}$-Banach manifold. For $k=1,2$, these eigenvalues
are as above. We do not know if the sequence $\{\hat{\lambda}_{k}(p)\}_{k\geq1}$ exhausts $\sigma (p)$. This is the
case if $N=1$ (ordinary differential equation) or if $p=2$ and $N\geq2$ (linear eigenvalue problem). In the
latter case, we have
\begin{eqnarray*}
H_{0}^{1}(\Omega)=\overline{\underset{k\geq1}\oplus E(\hat{\lambda}_{k}(2))}\ ,
\end{eqnarray*}
where $E(\hat{\lambda}_{k}(2))$ denotes the eigenspace corresponding to the eigenvalue $\hat{\lambda}_{k}(2)$.
We know that for every $k\geq1$, $E(\hat{\lambda}_{k}(2))$ is finite dimensional,
$E(\hat{\lambda}_{k}(2))\subseteq C_{0}^{1}(\overline{\Omega})$ and it has the ``Unique Continuation Property"
(UCP for short), that is, if $u\in E(\hat{\lambda}_{k}(2))$ and vanishes on a set of positive measure, then
$u\equiv0$.

We mention that if $\hat{u}$ is an eigenfunction corresponding to an eigenvalue $\hat{\lambda}_{m}(p)$ with
$m\geq2$, then $\hat{u}\in C_{0}^{1}(\overline{\Omega})$ (nonlinear regularity theory) and $\hat{u}$ is nodal
(that is, sign changing). In fact, if $p=2$, $\{\hat{u}_{n}\}_{n\geq1}\subseteq H_{0}^{1}(\Omega)$ are the
eigenfunctions corresponding to the eigenvalues $\{\hat{\lambda}_{k}(2)\}_{k\geq1}$ and
$S_{n}=\{z\in\Omega:\hat{u}_{n}(z)=0\}$, then $\Omega\setminus S_{n}$ has at most $n$-components (Courant nodal
domain theorem, see, for example, Gasinski and Papageorgiou \cite[p. 797]{15}).

For $x\in\RR$, let $x^{\pm}=\max\{\pm x,0\}$. Then for $u\in W_{0}^{1,p}(\Omega)$, we set
$u^{\pm}(\cdot)=u(\cdot)^{\pm}$. We know that
\begin{eqnarray*}
u^{\pm}\in W_{0}^{1,p}(\Omega),\ u=u^{+}-u^{-},\ |u|=u^{+}+u^{-}.
\end{eqnarray*}
By $|\cdot|_{N}$ we denote the Lebesgue measure on $\RR^{N}$. Finally, if $h:\Omega\times\RR\rightarrow\RR$ is a
measurable function (for example, a Carath\'eodory function), then we set
\begin{eqnarray*}
N_{h}(u)(\cdot)=h(\cdot, u(\cdot))\ \mbox{for all}\ u\in W_{0}^{1,p}(\Omega),
\end{eqnarray*}
the Nemytskii (superposition) operator corresponding to the function $h(z,x)$. Evidently, $z\longmapsto
N_{h}(u)(z)=h(z,u(z))$ is measurable.

\section{Resonant Problems}\label{sec3}
In this section we study problem \eqref{eq1} when the reaction $f(z,\cdot)$ is $(p-1)$-linear near $\pm\infty$
and resonant with respect to a nonprincipal eigenvalue of $(-\Delta_{p}, W_{0}^{1,p}(\Omega))$. So, we impose
the following conditions on the function $f(z,x)$:

\smallskip
$H_{1}$: $f:\Omega\times\RR\rightarrow\RR$ is a Carath\'eodory function such that $f(z,0)=0$ for a.a. $z\in\Omega$ and
\begin{itemize}
\item[(i)] $|f(z,x)|\leq a(z)(1+|x|^{p-1})$ for a.a. $z\in\Omega$, all $x\in\RR$, with $a\in
    L^{\infty}(\Omega)_{+}$;
\item[(ii)] $\displaystyle\lim_{x\rightarrow\pm\infty}\frac{f(z,x)}{|x|^{p-2}x}=\hat{\lambda}_{m}(p)$
    uniformly for a.a. $z\in\Omega$ and with $m\geq2$;
\item[(iii)] if $F(z,x)=\displaystyle\int_{0}^{x}f(z,s)ds$, then there exists $\tau\in(2,p)$ such that\\
$0<\eta_{0}\leq\displaystyle\liminf_{x\rightarrow\pm\infty}\frac{p F(z,x)-f(z,x)x}{|x|^{\tau}}$ uniformly
for a.a. $z\in\Omega$;
\item[(iv)] there exists $\vartheta\in L^{\infty}(\Omega)_{+}$ such that\\
\begin{align*}
&\vartheta(z)\leq\hat{\lambda}_{1}(2)\ \mbox{a.e. in}\ \Omega,\ \vartheta\not\equiv\hat{\lambda}_{1}(2)\\
&\limsup_{x\rightarrow0}\frac{2F(z,x)}{x^{2}}\leq\vartheta(z)\ \mbox{uniformly for a.a.}\ z\in\Omega;
\end{align*}
\item[(v)] for every $\rho>0$, there exists $\xi_{\rho}>0$ such that $f(z,x)x+\xi_{\rho}|x|^{p}\geq0$ for a.a.
    $z\in\Omega$, all $|x|\leq\rho$.
\end{itemize}
\begin{remark}\label{remark}
If there exists $\delta>0$ such that $f(z,x)x\geq0$ for a.a. $z\in\Omega$, all $|x|\leq\delta$, then hypothesis
$H_{1}(v)$ is automatically satisfied.
\end{remark}
\begin{example}
The following function satisfies hypotheses $H_{1}$. For the sake of simplicity we drop the
$z$-dependence:
\[f(x)=\left\{
\begin{array}{rl}
\vartheta x+\hat{\lambda}_{m}(p)|x|^{q-2}x & \mbox{if}\ |x|\leq1 \\
\hat{\lambda}_{m}(p)|x|^{p-2}x+\vartheta|x|^{\tau-2}x & \mbox{if}\ |x|>1
\end{array} \right.\]
with $\tau,q>2$, $\tau<p$ and $\vartheta\in(0,\hat{\lambda}_{1}(2))$.
\end{example}

We set $f_{\pm}(z,x)=f(z,\pm x^{\pm})$. These are Carath\'eodory functions. Let
$F_{\pm}(z,x)=\displaystyle\int_{0}^{x}f_{\pm}(z,s)ds$ and let $\varphi_{\pm}:W_{0}^{1,p}(\Omega)\rightarrow\RR$
be the $C^{1}$-functionals defined by
\begin{eqnarray*}
\varphi_{\pm}(u)=\frac{1}{p}\|Du\|_{p}^{p}+\frac{1}{2}\|Du\|_{2}^{2}-\int_{\Omega}F_{\pm}(z,u(z))dz\ \mbox{for
all}\ u\in W_{0}^{1,p}(\Omega).
\end{eqnarray*}
Also, let $\varphi:W_{0}^{1,p}(\Omega)\rightarrow\RR$ be the energy functional for problem \eqref{eq1} defined
by
\begin{eqnarray*}
\varphi(u)=\frac{1}{p}\|Du\|_{p}^{p}+\frac{1}{2}\|Du\|_{2}^{2}-\int_{\Omega}F(z,u(z))dz\ \mbox{for all}\ u\in
W_{0}^{1,p}(\Omega).
\end{eqnarray*}
Evidently $\varphi\in C^{1}(W_{0}^{1,p}(\Omega))$.

\begin{prop}\label{prop5}
Assume that hypotheses $H_{1}(i), (ii)$ hold. Then the functionals $\varphi_{\pm}$ satisfy the C-condition.
\end{prop}
\begin{proof}
We do the proof for $\varphi_{+}$, the proof for $\varphi_{-}$ being similar. So, let
$\{u_{n}\}_{n\geq1}\subseteq W_{0}^{1,p}(\Omega)$ be such that
\begin{eqnarray}\label{eq5}
|\varphi_{+}(u_{n})|\leq M_{1}\ \mbox{ for some }\ M_{1}>0,\ \mbox{all}\ n\geq1
\end{eqnarray}
\begin{eqnarray}\label{eq6}
(1+\|u_{n}\|)\varphi'(u_{n})\rightarrow0\ \mbox{ in }\ W^{-1,p'}(\Omega)\ \mbox{ as }\ n\rightarrow\infty.
\end{eqnarray}
From \eqref{eq6} we have
\begin{eqnarray}\label{eq7}
\bigg|\langle A_{p}(u_{n}),h\rangle+\langle A(u_{n}),h\rangle-\int_{\Omega}f_{+}(z,u_{n})h\,
dz\bigg|\leq\frac{\varepsilon_{n}\|h\|}{1+\|u_{n}\|}
\end{eqnarray}
\hspace{8cm} for all $h\in W_{0}^{1,p}(\Omega)$ with $\varepsilon_{n}\rightarrow0^{+}$.

In \eqref{eq7} we choose $h=-u_{n}^{-}\in W_{0}^{1,p}(\Omega)$. Then
\begin{align}\label{eq8}
&\|Du_{n}^{-}\|_{p}^{p}+\|Du_{n}^{-}\|_{2}^{2}\leq\varepsilon_{n}\ \mbox{ for all }\ n\geq1,\nonumber\\
\Rightarrow&u_{n}^{-}\rightarrow0\ \mbox{ in }\ W_{0}^{1,p}(\Omega)\ \mbox{ as }\ n\rightarrow\infty.
\end{align}

{\bf Claim.} {\it
$\{u_{n}^{+}\}_{n\geq1}\subseteq W_{0}^{1,p}(\Omega)$ is bounded.
}

We argue indirectly. So, suppose that $\{u_{n}^{+}\}_{n\geq1}\subseteq W_{0}^{1,p}(\Omega)$ is not bounded. We
may assume that $\|u_{n}^{+}\|\rightarrow\infty$. Let $y_{n}=\frac{u_{n}^{+}}{\|u_{n}^{+}\|}$, $n\geq1$. Then
$\|y_{n}\|=1$, $y_{n}\geq0$ for all $n\geq1$. So, by passing to a subsequence if necessary, we may assume that
\begin{eqnarray}\label{eq9}
y_{n}\overset{w}\rightarrow y\ \mbox{ in  }\ W_{0}^{1,p}(\Omega)\ \mbox{ and }\ y_{n}\rightarrow y\ \mbox{ in }\
L^{p}(\Omega),\ \mbox{ with }\ y\geq0.
\end{eqnarray}
From \eqref{eq7} and \eqref{eq8}, we have
\begin{eqnarray}\label{eq10}
\bigg|\langle A_{p}(y_{n}),h\rangle+\frac{1}{\|u_{n}^{+}\|^{p-2}}\langle
A(y_{n}),h\rangle-\int_{\Omega}\frac{f_{+}(z,u_{n})}{\|u_{n}^{+}\|^{p-1}}h\, dz\bigg|\leq\varepsilon_{n}'\|h\|
\end{eqnarray}
\hspace{10cm}with $\varepsilon_{n}'\rightarrow0^{+}$ as $n\rightarrow\infty$.

In \eqref{eq10} we choose $h=y_{n}-y\in W_{0}^{1,p}(\Omega)$, pass to the limit as $n\rightarrow\infty$ and use
\eqref{eq9}. Then
\begin{align}\label{eq11}
&\lim_{n\rightarrow\infty}\langle A_{p}(y_{n}),y_{n}-y\rangle=0\ (\mbox{recall } p>2),\nonumber\\
\Rightarrow&y_{n}\rightarrow y\ \mbox{ in }\ W_{0}^{1,p}(\Omega)\ \mbox{(see Proposition \ref{prop2}), hence}\
\|y\|=1,\ y\geq0.
\end{align}
Hypothesis $H_{1}(i)$ implies that
\begin{eqnarray*}
\bigg\{\frac{N_{f_{+}}(u_{n})}{\|u_{n}^{+}\|^{p-1}}
=\frac{N_{f}(u_{n}^{+})}{\|u_{n}^{+}\|^{p-1}}\bigg\}_{n\geq1}\subseteq L^{p'}(\Omega)\ \mbox{is bounded}.
\end{eqnarray*}
So, we may assume that
\begin{eqnarray}\label{eq12}
\frac{N_{f_{+}}(u_{n})}{\|u_{n}^{+}\|^{p-1}}\overset{w}\rightarrow g\ \mbox{ in }\ L^{p'}(\Omega)\ \mbox{ as }\
n\rightarrow\infty.
\end{eqnarray}
By virtue of hypothesis $H_{1}(ii)$, we have
\begin{eqnarray}\label{eq13}
g=\hat{\lambda}_{m}(p)y^{p-1}
\end{eqnarray}
(see Aizicovici, Papageorgiou and Staicu \cite{1}, proof of Proposition 30). Then, if in \eqref{eq10} we pass to
the limit  as $n\rightarrow\infty$ and use \eqref{eq11}, \eqref{eq12}, \eqref{eq13} and the fact that $p>2$, we
obtain
\begin{align}\label{eq14}
&\langle A_{p}(y),h\rangle=\hat{\lambda}_{m}(p)\int_{\Omega}y^{p-1}h\, dz\ \mbox{for all}\ h\in
W_{0}^{1,p}(\Omega),\nonumber\\
\Rightarrow&A_{p}(y)=\hat{\lambda}_{m}(p)y^{p-1}.
\end{align}
Since $m\geq2$, from \eqref{eq14} we infer that
\begin{eqnarray*}
y\equiv0\ \mbox{ or }\ y\ \mbox{ is nodal,}
\end{eqnarray*}
both cases contradict \eqref{eq11}. This proves the Claim.

From \eqref{eq8} and the Claim, it follows that $\{u_{n}\}_{n\geq1}\subseteq W_{0}^{1,p}(\Omega)$
is bounded and so we may assume that
\begin{eqnarray}\label{eq15}
u_{n}\overset{w}\rightarrow u\ \mbox{ in }\ W_{0}^{1,p}(\Omega)\ \mbox{ and }\ u_{n}\rightarrow u\ \mbox{ in }\
L^{p}(\Omega).
\end{eqnarray}
In \eqref{eq7} we choose $h=u_{n}-u\in W_{0}^{1,p}(\Omega)$, pass to the limit as $n\rightarrow\infty$ and use
\eqref{eq15}. Then
\begin{align*}
&\lim_{n\rightarrow\infty}[\langle A_{p}(u_{n}),u_{n}-u\rangle+\langle A(u_{n}),u_{n}-u\rangle]=0,\\
\Rightarrow&\limsup_{n\rightarrow\infty}[\langle A_{p}(u_{n}),u_{n}-u\rangle+\langle
A(u),u_{n}-u\rangle]\leq0\\
&\hspace{5cm}(\mbox{since A is monotone}),\\
\Rightarrow&\limsup_{n\rightarrow\infty}\langle
A_{p}(u_{n}),u_{n}-u\rangle\leq0,\\ \Rightarrow&u_{n}\rightarrow u\ \mbox{ in }\ W_{0}^{1,p}(\Omega)\ \mbox{
(see Proposition \ref{prop2})}.
\end{align*}
This proves that the functional $\varphi_{+}$ satisfies the C-condition. Similarly for the functional
$\varphi_{-}$.
\end{proof}

\begin{prop}\label{prop6}
If hypotheses $H_{1}(i),(ii),(iii)$ hold, then the functional $\varphi$ satisfies the C-condition.
\end{prop}
\begin{proof}
Let $\{u_{n}\}_{n\geq1}\subseteq W_{0}^{1,p}(\Omega)$ be a sequence such that
\begin{eqnarray}\label{eq16}
|\varphi(u_{n})|\leq M_{2}\ \mbox{ for some }\ M_{2}>0,\ \mbox{ all }\ n\geq1,
\end{eqnarray}
\begin{eqnarray}\label{eq17}
(1+\|u_{n}\|)\varphi'(u_{n})\rightarrow0\ \mbox{ in }\ W^{-1,p'}(\Omega)\ \mbox{ as }\ n\rightarrow\infty.
\end{eqnarray}
From \eqref{eq17} we have
\begin{eqnarray}\label{eq18}
\bigg|\langle A_{p}(u_{n}),h\rangle+\langle A(u_{n}),h\rangle|-\int_{\Omega}f(z,u_{n})h\,dz\bigg|\leq\frac{\varepsilon_{n}\|h\|}{1+\|u_{n}\|}
\end{eqnarray}
\hspace{9cm}for all $h\in W_{0}^{1,p}(\Omega)$ with $\varepsilon_{n}\rightarrow0^{+}$.

In \eqref{eq18} we choose $h=u_{n}\in W_{0}^{1,p}(\Omega)$ and obtain
\begin{eqnarray}\label{eq19}
\|Du_{n}\|_{p}^{p}+\|Du\|_{2}^{2}-\int_{\Omega}f(z,u_{n})u_{n}dz\leq\varepsilon_{n}\ \mbox{ for all }\ n\geq1.
\end{eqnarray}
Also, from \eqref{eq16} we have
\begin{eqnarray}\label{eq20}
-\|Du_{n}\|_{p}^{p}-\frac{p}{2}\|Du_{n}\|_{2}^{2}+\int_{\Omega}p F(z,u_{n})dz\leq p M_{2}\ \mbox{ for all }\ n\geq1.
\end{eqnarray}
Adding \eqref{eq19} and \eqref{eq20}, we obtain
\begin{eqnarray}\label{eq21}
\int_{\Omega}\big[p F(z,u_{n})-f(z,u_{n})u_{n}\big]dz\leq M_{3}+\bigg(\frac{p}{2}-1\bigg)\|Du_{n}\|_{2}^{2}
\end{eqnarray}
\hspace{8cm} for some $M_{3}>0$, all $n\geq1$.

By virtue of hypotheses $H_{1}(i),(iii)$, we can find $\eta_{1}\in(0,\eta_{0})$ and $c_{1}>0$ such that
\begin{align}\label{eq22}
&\eta_{1}|x|^{\tau}-c_{1}\leq p F(z,x)=f(z,x)x\ \mbox{ for a.a. }\ z\in\Omega,\ \mbox{ all }\ x\in\RR,\nonumber\\
\Rightarrow&\eta_{1}\|u_{n}\|_{\tau}^{\tau}\leq M_{4}+\bigg(\frac{p}{2}-1\bigg)\|Du_{n}\|_{2}^{2}\ \mbox{ for some }\ M_{4}>0,\ \mbox{ all }\ n\geq1
\end{align}
\hspace{12cm}(see \eqref{eq21}).

{\bf Claim} {\it
$\{u_{n}\}_{n\geq1}\subseteq W_{0}^{1,p}(\Omega)$ is bounded.
}

Arguing by contradiction, suppose that the Claim  is not true. Then by passing to a subsequence if necessary, we may assume that $\|u_{n}\|\rightarrow\infty$. Let $y_{n}=\frac{u_{n}}{\|u_{n}\|}$, $n\geq1$. Then $\|y_{n}\|=1$ for all $n\geq1$ and so we may assume that
\begin{eqnarray}\label{eq23}
y_{n}\overset{w}\rightarrow y\ \mbox{ in }\ W_{0}^{1,p}(\Omega)\ \mbox{ and }\ y_{n}\rightarrow y\ \mbox{ in }\ L^{p}(\Omega)\ \mbox{ as }\ n\rightarrow\infty.
\end{eqnarray}
From \eqref{eq18} we have
\begin{eqnarray}\label{eq24}
\bigg|\langle A_{p}(y_{n}),h\rangle+\frac{1}{\|u_{n}\|^{p-2}}\langle A(y_{n}),h\rangle-\int_{\Omega}\frac{f(z,u_{n})}{\|u_{n}\|^{p-1}}h\, dz\bigg|\leq\frac{\varepsilon_{n}\|h\|}{\|u_{n}\|^{p-1}(1+\|u_{n}\|)}
\end{eqnarray}
\hspace{8cm}for all $h\in W_{0}^{1,p}(\Omega)$ and with $\varepsilon_{n}\rightarrow0^{+}$.

Hypothesis $H_{1}(i)$ implies that $\bigg\{\frac{N_{f}(u_{n})}{\|u_{n}\|^{p-1}}\bigg\}_{n\geq1}\subseteq L^{p'}(\Omega)$ is bounded. So, if in \eqref{eq24} we choose $h=y_{n}-y\in W_{0}^{1,p}(\Omega)$, pass to the limit as $n\rightarrow\infty$ and use \eqref{eq23}, then
\begin{align}\label{eq25}
&\lim_{n\rightarrow\infty}\langle A_{p}(y_{n}),y_{n}-y\rangle=0\ (\mbox{recall }\ p>2),\nonumber\\
\Rightarrow&y_{n}\rightarrow y\ \mbox{ in }\ W_{0}^{1,p}(\Omega),\ \mbox{ hence }\ \|y\|=1.
\end{align}
From \eqref{eq22}, we have
\begin{align*}
&\eta_{1}\|y_{n}\|_{\tau}^{\tau}\leq\frac{M_{4}}{\|u_{n}\|^{\tau}}+\frac{c_{2}}{\|u_{n}\|^{\tau-2}}\ \mbox{ with }\ c_{2}=\bigg(\frac{p}{2}-1\bigg)>0\ (\mbox{recall }\ p>2),\\
\Rightarrow&\eta_{1}\|y\|_{\tau}^{\tau}\leq0,\ \mbox{ hence }\ y=0,\ \mbox{ contradicting }\ \eqref{eq25}.
\end{align*}
This proves the Claim.

By virtue of the Claim, we may assume that
\begin{eqnarray}\label{eq26}
u_{n}\overset{w}\rightarrow u\ \mbox{ in }\ W_{0}^{1,p}(\Omega)\ \mbox{ and }\ u_{n}\rightarrow u\ \mbox{ in }\ L^{p}(\Omega)\ \mbox{ as }\ n\rightarrow\infty.
\end{eqnarray}
In \eqref{eq18} we choose $h=u_{n}-u\in W_{0}^{1,p}(\Omega)$, pass to the limit as $n\rightarrow\infty$ and use \eqref{eq26}. Then
\begin{align*}
&\lim_{n\rightarrow\infty}[\langle A_{p}(u_{n}),u_{n}-u\rangle+\langle A(u_{n}),u_{n}-u\rangle]=0,\\
\Rightarrow&\limsup_{n\rightarrow\infty}[\langle A_{p}(u_{n}),u_{n}-u\rangle+\langle A(u),u_{n}-u\rangle]\leq0\\
&\hspace{6cm}(\mbox{since}\ A\ \mbox{ is monotone}),\\
\Rightarrow&\limsup_{n\rightarrow\infty}\langle A_{p}(u_{n}),u_{n}-u\rangle\leq0,\\
\Rightarrow&u_{n}\rightarrow u\ \mbox{ in }\ W_{0}^{1,p}(\Omega)\ \mbox{ (see Proposition \ref{prop2})}.
\end{align*}
Therefore $\varphi$ satisfies the C-condition.                                                                                                                  \end{proof}

\begin{prop}\label{prop7}
If hypotheses $H_{1}(i)\rightarrow(iv)$ hold, then $u=0$ is a local minimizer for the functionals $\varphi_{\pm}$ and $\varphi$.
\end{prop}
\begin{proof}
We do the proof for the functional $\varphi_{+}$, the proofs for $\varphi_{-}$ and $\varphi$ being similar.

By virtue of hypotheses $H_{1}(i),(iv)$, given $\varepsilon>0$ we can find $c_{3}=c_{3}(\varepsilon)>0$ such that
\begin{eqnarray}\label{eq27}
F(z,x)\leq\frac{1}{2}(\vartheta(z)+\varepsilon)x^{2}+c_{3}|x|^{p}\ \mbox{ for a.a. }\ z\in\Omega,\ \mbox{ all }\ x\in\RR.
\end{eqnarray}
Then for all $u\in W_{0}^{1,p}(\Omega)$, we have
\begin{align*}
\varphi_{+}(u)&=\frac{1}{p}\|Du\|_{p}^{p}+\frac{1}{2}\|Du\|_{2}^{2}-\int_{\Omega}F_{+}(z,u)dz\\
&\geq\frac{1}{p}\|Du\|_{p}^{p}+\frac{1}{2}\|Du\|_{2}^{2}-\frac{1}{2}\int_{\Omega}(\vartheta(z)+\varepsilon)(u^{+})^{2}dz
-c_{3}\|u^{+}\|_{p}^{p}\\
&\hspace{9cm}(\mbox{see \eqref{eq27}})\\
&\geq\frac{1}{2}\bigg[\bigg(\|Du\|_{2}^{2}-\int_{\Omega}\vartheta(z)u^{2}dz\bigg)
-\frac{\varepsilon}{\hat{\lambda}_{1}(2)}\|Du\|_{2}^{2}\bigg]-c_{4}\|u\|^{p}\\
&\hspace{7cm}\mbox{for some }\ c_{4}>0\ (\mbox{see \eqref{eq4}})\\
&\geq\frac{1}{2}\bigg[c_{5}-\frac{\varepsilon}{\hat{\lambda}_{1}(2)}\bigg]\|u\|^{2}-c_{4}\|u\|^{p}\ \mbox{ for some }\ c_{5}>0\\
&\hspace{6cm}(\mbox{see Papageorgiou and Kyritsi \cite[p. 356]{20}}).
\end{align*}
Choosing $\varepsilon\in(0,\hat{\lambda}_{1}(2)c_{5})$, we infer that
\begin{eqnarray}\label{eq28}
\varphi_{+}(u)\geq c_{6}\|u\|^{2}-c_{4}\|u\|^{p}\ \mbox{ for some }\ c_{6}>0.
\end{eqnarray}
Since $p>2$, from \eqref{eq28} we see that for $\rho\in(0,1)$ small, we have
\begin{align*}
&\varphi_{+}(u)\geq0=\varphi_{+}(0)\ \mbox{ for all }\ u\in W_{0}^{1,p}(\Omega)\ \mbox{ with }\ \|u\|\leq\rho,\\
\Rightarrow&u=0\ \mbox{ is a local minimizer for the functional }\ \varphi_{+}.
\end{align*}
Similarly we show that $u=0$ is also a local minimizer for the functionals $\varphi_{-}$ and $\varphi$.
\end{proof}

Now we are ready to produce constant sign solutions.
\begin{theorem}\label{th8}
If hypotheses $H_{1}$ hold, then problem \eqref{eq1} admits at least two nontrivial constant sign solutions
\begin{eqnarray*}
u_{0}\in\mbox{int}\ C_{+}\, \mbox{ and }\ v_{0}\in-\mbox{int}\, C_{+}.
\end{eqnarray*}
\end{theorem}
\begin{proof}
First we produce a positive solution.

From Proposition \ref{prop7} we see that we can find $\rho\in(0,1)$ small such that
\begin{eqnarray}\label{eq29}
\varphi_{+}(0)=0<\inf[\varphi_{+}(u):\|u\|=\rho]=m_{\rho}^{+}
\end{eqnarray}
(see Aizicovici, Papageorgiou and Staicu \cite{1}, proof of Proposition 29). By virtue of hypotheses $H_{1}(i),(ii)$, given $\varepsilon>0$, we can find $c_{7}=c_{7}(\varepsilon)>0$ such that
\begin{eqnarray}\label{eq30}
F(z,x)\geq\frac{1}{p}(\hat{\lambda}_{m}(p)-\varepsilon)|x|^{p}-c_{7}\ \mbox{ for a.a. }\ z\in\Omega,\ \mbox{ all }\ x\in\RR.
\end{eqnarray}
Then for $t>0$, we have
\begin{align*}
\varphi_{+}(t\hat{u}_{1}(p))&=\frac{t^{p}}{p}\hat{\lambda}_{1}(p)+\frac{t^{2}}{2}\|D\hat{u}_{1}(p)\|_{2}^{2}
=\int_{\Omega}F(z,t\hat{u}_{1})dz\\
&\leq\frac{t^{p}}{p}[\hat{\lambda}_{1}(p)-\hat{\lambda}_{m}(p)+\varepsilon]+\frac{t^{2}}{2}\|D\hat{u}_{1}(p)\|_{2}^{2}
+c_{7}|\Omega|_{N}\ \mbox{ (see \eqref{eq30})}.
\end{align*}
Choose $\varepsilon\in(0,\hat{\lambda}_{m}(p)-\hat{\lambda}_{1}(p))$ (recall that $m\geq2$). Then
\begin{align}\label{eq31}
&\varphi_{+}(t\hat{u}_{1}(p))\leq-\frac{c_{8}t^{p}}{p}+\frac{t^{2}}{2}\|D\hat{u}_{1}(p)\|_{2}^{2}+c_{7}|\Omega|_{N}\ \mbox{ for some }\ c_{8}>0,\nonumber\\
\Rightarrow&\frac{\varphi_{+}(t\hat{u}_{1}(p))}{t^{p}}\leq-\frac{c_{8}}{p}+\frac{1}{2t^{p-2}}\|D\hat{u}_{1}(p)\|_{2}^{2}
+\frac{c_{7}}{t^{p}}|\Omega|_{N},\nonumber\\
\Rightarrow&\limsup_{t\rightarrow+\infty}\frac{\varphi_{+}(t\hat{u}_{1}(p))}{t^{p}}\leq-\frac{c_{8}}{p}<0\ (\mbox{recall }\ p>2),\nonumber\\
\Rightarrow&\lim_{t\rightarrow+\infty}\varphi_{+}(t\hat{u}_{1}(p))=-\infty.
\end{align}
From \eqref{eq29}, \eqref{eq31} and Proposition \ref{prop5}, we see that we can apply Theorem \ref{th1} (the mountain pass theorem) and find $u_{0}\in W_{0}^{1,p}(\Omega)$ such that
\begin{eqnarray}\label{eq32}
u_{0}\in K_{\varphi_{+}}\ \mbox{ and }\ m_{\rho}^{+}\leq\varphi_{+}(u_{0}).
\end{eqnarray}
From \eqref{eq29} and \eqref{eq32} it follows that $u_{0}\neq0$. Also
\begin{align}\label{eq33}
&\varphi_{+}'(u_{0})=0\ (\mbox{see \eqref{eq32}}),\nonumber\\
\Rightarrow&A_{p}(u_{0})+A(u_{0})=N_{f_{+}}(u_{0}).
\end{align}
On \eqref{eq33} we act with $-u_{0}^{-}\in W_{0}^{1,p}(\Omega)$ and obtain
\begin{align*}
&\|Du_{0}^{-}\|_{p}^{p}+\|Du_{0}^{-}\|_{2}^{2}=0,\nonumber\\
\Rightarrow&u_{0}\geq0,\ u_{0}\neq0.
\end{align*}
Then from \eqref{eq33} we see that $u_{0}$ is a nontrivial solution of \eqref{eq1}. From Ladyzhenskaya and Uraltseva \cite[p. 286]{18}, we have $u_{0}\in L^{\infty}(\Omega)$. Then we can apply Theorem 1 of Lieberman \cite{19} and conclude that $u_{0}\in C_{+}\setminus\{0\}$. Let $\rho=\|u_{0}\|_{\infty}$ and let $\xi_{\rho}>0$ be as postulated by hypothesis $H_{1}(v)$. Then
\begin{align}\label{eq34}
&-\Delta_{p}u_{0}(z)-\Delta u_{0}(z)+\xi_{\rho}u_{0}(z)^{p-1}=f(z,u_{0}(z))+\xi_{\rho}u_{0}(z)^{p-1}\geq0\ \mbox{ for a.a.}\ z\in\Omega,\nonumber\\
\Rightarrow&\Delta_{p}u_{0}(z)+\Delta u_{0}(z)\leq\xi_{\rho}u_{0}(z)^{p-1}\ \mbox{ for a.a.}\ z\in\Omega.
\end{align}
Then from \eqref{eq34} and Pucci and Serrin \cite[pp. 111, 120]{23}, we infer that
$$u_{0}\in\mbox{int}\ C_{+}.$$
Similarly, working with the functional $\varphi_{-}$, via the mountain pass theorem, we produce $v_{0}\in-\mbox{int}\ C_{+}$ a negative solution of problem \eqref{eq1}.
\end{proof}

To produce a third nontrivial solution, we need to strengthen our hypotheses on the reaction $f(z,\cdot)$ and employ also tools from Morse theory.

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

\smallskip
$H_{2}$: $f:\Omega\times\RR\rightarrow\RR$ 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{itemize}
\item[(i)] $|f_{x}'(z,x)|\leq a(z)(1+|x|^{p-2})$ for a.a. $z\in\Omega$, all $x\in\RR$ and with $a\in L^{\infty}(\Omega)_{+}$;
\item[(ii)] $\displaystyle\lim_{x\rightarrow\pm\infty}\frac{f(z,x)}{|x|^{p-2}x}=\hat{\lambda}_{m}(p)$ uniformly for a.a. $z\in\Omega$ and with $m\geq2$;
\item[(iii)] if $F(z,x)=\displaystyle\int_{0}^{x}f(z,s)ds$, then there exists $\tau\in(2,p)$ such that
\begin{eqnarray*}
\lim_{x\rightarrow\pm\infty}\frac{p F(z,x)-f(z,x)x}{|x|^{\tau}}=+\infty\ \mbox{ uniformly for a.a.}\ z\in\Omega;
\end{eqnarray*}
\item[(iv)] $f_{x}'(z,0)=\displaystyle\lim_{x\rightarrow0}\frac{f(z,x)}{x}$ uniformly for a.a. $z\in\Omega$ and
\begin{eqnarray*}
f_{x}'(z,0)\leq\hat{\lambda}_{1}(2)\ \mbox{ for a.a.}\ z\in\Omega, f_{x}'(\cdot,0)\not\equiv\hat{\lambda}_{1}(2);
\end{eqnarray*}
\item[(v)] for every $\rho>0$, there exists $\xi_{\rho}>0$ such that
\begin{eqnarray*}
f(z,x)x+\xi_{\rho}|x|^{p}\geq0\ \mbox{ for a.a.}\ z\in\Omega,\ \mbox{ all}\ |x|\leq\rho.
\end{eqnarray*}
\end{itemize}

Under these stronger conditions on the reaction, we can produce a third nontrivial solution and have the following multiplicity theorem.
\begin{theorem}\label{th9}
If hypotheses $H_{2}$ hold, then problem \eqref{eq1} has at least three nontrivial solutions
\begin{eqnarray*}
u_{0}\in\mbox{int}\, C_{+},\ v_{0}\in-\mbox{int}\, C_{+}\ \mbox{ and }\ y_{0}\in C_{0}^{1}(\overline\Omega)\setminus\{0\}.
\end{eqnarray*}
\end{theorem}
\begin{proof}
The two nontrivial constant sign solutions
\begin{eqnarray*}
u_{0}\in\mbox{int}\, C_{+}\ \mbox{ and}\ v_{0}\in-\mbox{int}\, C_{+}
\end{eqnarray*}
are guaranteed by Theorem \ref{th8}.

From the proof of Theorem \ref{th8}, we know that $u_{0}\in\mbox{int}\, C_{+}$ is a critical point of mountain pass type for the functional $\varphi_{+}$, while $v_{0}\in-\mbox{int}\, C_{+}$ is a critical point of mountain pass type for the functional $\varphi_{-}$. Therefore we have
\begin{eqnarray}\label{eq35}
C_{1}(\varphi_{+},u_{0})\neq0\ \mbox{ and}\ C_{1}(\varphi_{-},v_{0})\neq0.
\end{eqnarray}

We consider the homotopy
\begin{eqnarray*}
h(t,u)=(1-t)\varphi_{+}(u)+t\varphi(u)\ \mbox{ for all}\ (t,u)\in[0,1]\times W_{0}^{1,p}(\Omega).
\end{eqnarray*}
We assume that $K_{\varphi}$ is finite (otherwise we already have infinitely many distinct solutions for problem \eqref{eq1}). Suppose that we can find $\{t_{n}\}_{n\geq1}\subseteq[0,1]$ and $\{u_{n}\}_{n\geq1}\subseteq W_{0}^{1,p}(\Omega)$ such that
\begin{eqnarray}\label{eq36}
t_{n}\rightarrow t,\ u_{n}\rightarrow u_{0}\ \mbox{ in}\ W_{0}^{1,p}(\Omega)\ \mbox{ and}\ h_{u}'(t_{n},u_{n})=0\ \mbox{ for all}\ n\geq1.
\end{eqnarray}
From \eqref{eq36}, we have
\begin{align*}
&A_{p}(u_{n})+A(u_{n})=(1-t_{n})N_{f_{+}}(u_{n})+t_{n}N_{f}(u_{n})\ \mbox{ for all}\ n\geq1,\\
\Rightarrow&-\Delta_{p}u_{n}(z)-\Delta u_{n}(z)=(1-t_{n})f_{+}(z,u_{n}(z))+t_{n}f(z,u_{n}(z))\ \mbox {for a.a.}\ z\in\Omega,\ \mbox{ all}\ n\geq1.
\end{align*}
From Ladyzhenskaya and Uraltseva \cite[p. 286]{18}, we know that we can find $M_{5}>0$ such that
\begin{eqnarray*}
\|u_n\|_{\infty}\leq M_{5}\ \mbox{ for all}\ n\geq1\ (\mbox{see also \eqref{eq36}}).
\end{eqnarray*}
Then Theorem 1 of Lieberman \cite{19} implies that there exist $\alpha\in(0,1)$ and $M_{6}>0$ such that
\begin{eqnarray}\label{eq37}
u_{n}\in C_{0}^{1,\alpha}(\overline\Omega)\ \mbox{ and}\ \|u_{n}\|_{C_{0}^{1,\alpha}(\overline\Omega)}\leq M_{6}\ \mbox{ for all}\ n\geq1.
\end{eqnarray}
From \eqref{eq37} and the compact embedding of $C_{0}^{1,\alpha}(\overline\Omega)$ into $C_{0}^{1}(\overline\Omega)$, we have
\begin{eqnarray*}
u_{n}\rightarrow u_{0}\ \mbox{ in}\ C_{0}^{1}(\overline\Omega)\ (\mbox{see \eqref{eq36}}).
\end{eqnarray*}
Since $u_{0}\in\mbox{int}\, C_{+}$ (see Theorem \ref{th8}), we infer that $u_{n}\in\mbox{int}\, C_{+}$ for all $n\geq1$ and so $\{u_{n}\}_{n\geq n_{0}}$ is a sequence of distinct positive solutions of problem \eqref{eq1}, a contradiction to our hypothesis that $K_{\varphi}$ is finite. So, \eqref{eq36} can not occur and then the homotopy  invariance of critical groups, implies that
\begin{eqnarray}\label{eq38}
C_{k}(\varphi_{+},u_{0})=C_{k}(\varphi,u_{0})\ \mbox{ for all}\ k\geq0.
\end{eqnarray}
Similarly we show that
\begin{eqnarray}\label{eq39}
C_{k}(\varphi_{-},v_{0})=C_{k}(\varphi,v_{0})\ \mbox{ for all}\ k\geq0\ (\mbox{recall}\ v_{0}\in-\mbox{int}\, C_{+}).
\end{eqnarray}
From \eqref{eq35}, \eqref{eq38} and \eqref{eq39}, we have
\begin{eqnarray}\label{eq40}
C_{1}(\varphi,u_{0})\neq0,\ C_{1}(\varphi,v_{0})\neq0.
\end{eqnarray}
Since $\varphi\in C^{2}(W_{0}^{1,p}(\Omega))$ (see hypotheses $H_{2}$), from \eqref{eq40} and Papageorgiou and Smyrlis \cite{22} (see also Papageorgiou and R\u{a}dulescu \cite{21}), it follows that
\begin{eqnarray}\label{eq41}
C_{k}(\varphi,u_{0})=C_{k}(\varphi,v_{0})=\delta_{k,1}\ZZ\ \mbox{ for all}\ k\geq0.
\end{eqnarray}
Also, Proposition \ref{prop7} implies that
\begin{eqnarray}\label{eq42}
C_{k}(\varphi,0)=\delta_{k,0}\ZZ\ \mbox{ for all}\ k\geq0.
\end{eqnarray}
Hypothesis $H_{2}(iii)$ implies that given $\xi>0$, we can find $M_{7}=M_{7}(\xi)>0$ such that
\begin{eqnarray}\label{eq43}
p F(z,x)-f(z,x)x\geq\xi|x|^{\tau}\ \mbox{ for a.a.}\ z\in\Omega,\ \mbox{ all}\ |x|\geq M_{7}.
\end{eqnarray}
For a.a. $z\in\Omega$ and for $s\neq0$, we have
\begin{align*}
\frac{d}{ds}\frac{F(z,s)}{|s|^{p}}&=\frac{f(z,s)|s|^{p}-p|s|^{p-2}s F(z,s)}{|s|^{2p}}\\
&\leq\frac{|s|^{p-2}s[f(z,s)s-p F(z,s)]}{|s|^{2p}}\\
&=\frac{f(z,s)s-p F(z,s)}{|s|^{p}s}.
\end{align*}
Then for $s>0$, we have
\begin{align}\label{eq44}
&\frac{d}{ds}\frac{F(z,s)}{s^{p}}=\frac{f(z,s)s-p F(z,s)}{s^{p+1}}\leq-\xi s^{\tau-p-1}\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ s\geq M_{7}\ (\mbox{see \eqref{eq43}}),\nonumber\\
\Rightarrow&\frac{F(z,y)}{y^{p}}-\frac{F(z,x)}{x^{p}}\leq\frac{\xi}{p-\tau}\bigg[\frac{1}{y^{p-\tau}}
-\frac{1}{x^{p-\tau}}\bigg]\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ y\geq x\geq M_{7}.
\end{align}
Note that hypothesis $H_{2}(ii)$ implies that
\begin{eqnarray}\label{eq45}
\lim_{s\rightarrow\pm\infty}\frac{p F(z,s)}{|s|^{p}}=\hat{\lambda}_{m}(p)\ \mbox{ uniformly for a.a.}\ z\in\Omega.
\end{eqnarray}
Recall that $\tau<p$ (see hypothesis $H_{2}(iii)$). So, if in \eqref{eq44} we let $y\rightarrow+\infty$ and use \eqref{eq45}, then
\begin{align*}
&\frac{\hat{\lambda}_{m}(p)}{p}-\frac{F(z,x)}{x^{p}}\leq-\frac{\xi}{p-\tau}\frac{1}{x^{p-\tau}}\ \mbox{ for a.a.}\ z\in\Omega,\ \mbox{ all}\ x\geq M_{7},\\
\Rightarrow&\frac{\hat{\lambda}_{m}(p)}{p}x^{p}-F(z,x)\leq-\frac{\xi}{p-\tau}x^{\tau}\ \mbox{ for a.a.}\ z\in\Omega,\ \mbox{ all}\ x\geq M_{7},\\
\Rightarrow&\frac{\hat{\lambda}_{m}(p)x^{p}-p F(z,x)}{x^{\tau}}\leq-\xi\frac{p}{p-\tau}\ \mbox{ for a.a.}\ z\in\Omega,\ \mbox{ all}\ x\geq M_{7}.
\end{align*}
Since $\xi>0$ is arbitrary, it follows that
\begin{eqnarray*}
\lim_{x\rightarrow+\infty}\frac{\hat{\lambda}_{m}(p)x^{p}-p F(z,x)}{x^{\tau}}=-\infty\ \mbox{ uniformly for a.a.}\ z\in\Omega.
\end{eqnarray*}
For $s<0$, we have
\begin{align*}
&\frac{d}{ds}\frac{F(z,s)}{|s|^{p}}=\frac{f(z,s)s-p F(z,s)}{|s|^{p}s}\geq-\frac{\xi}{|s|^{p-\tau}s}\ \mbox{ for a.a.}\ z\in\Omega,\ \mbox{ all}\ s\leq-M_{7}\\
&\hspace{7cm}(\mbox{see \eqref{eq43} and recall that}\ s<0),\\
\Rightarrow&\frac{F(z,x)}{|x|^{p}}-\frac{F(z,y)}{|y|^{p}}\geq-\frac{\xi}{p-\tau}\bigg[\frac{1}{|y|^{p-\tau}}
-\frac{1}{|x|^{p-\tau}}\bigg]\ \mbox{ for a.a.}\ z\in\Omega,\ \mbox{ all}\ y\leq x\leq-M_{7}.
\end{align*}
Letting $y\rightarrow-\infty$ and using \eqref{eq45}, we obtain
\begin{align*}
&\frac{F(z,x)}{|x|^{p}}-\frac{\hat{\lambda}_{m}(p)}{p}\geq\frac{\xi}{p-\tau}\frac{1}{|x|^{p-\tau}}\ \mbox{ for a.a.}\ z\in\Omega,\ \mbox{ all}\ x\leq-M_{7}\ \mbox{(see \eqref{eq45})},\\
\Rightarrow&\frac{\hat{\lambda}_{m}(p)|x|^{p}-p F(z,x)}{|x|^{\tau}}\leq-\xi\frac{p}{p-\tau}\ \mbox{ for a.a.}\ z\in\Omega,\ \mbox{ all}\ x\leq-M_{7}.
\end{align*}
Since $\xi>0$ is arbitrary, we infer that
\begin{eqnarray*}
\lim_{x\rightarrow-\infty}\frac{\hat{\lambda}_{m}(p)|x|^{p}-p F(z,x)}{|x|^{\tau}}=-\infty\ \mbox{ uniformly for a.a.}\ z\in\Omega.
\end{eqnarray*}
Therefore we have proved that
\begin{eqnarray}\label{eq46}
\lim_{x\rightarrow\pm\infty}\frac{\hat{\lambda}_{m}(p)|x|^{p}-p F(z,x)}{|x|^{\tau}}=-\infty\ \mbox{ uniformly for a.a.}\ z\in\Omega.
\end{eqnarray}

We introduce the following two sets
$$S_r=\big\{u\in W_{0}^{1,p}(\Omega):\ \|u\|^{p}=r^{p},\ \|Du\|_{p}^{p}\leq\hat{\lambda}_{m}(p)\|u\|_{p}^{p}\big\},$$
$$A=\big\{u\in W_{0}^{1,p}(\Omega):\ \|Du\|_{p}^{p}\geq\hat{\lambda}_{m+1}(p)\|u\|_{p}^{p}\big\}.$$
Note that $S_r$ is a $C^{1}$-Banach manifold with boundary.

From Degiovanni and Lancelotti \cite[Theorem 3.2]{9}, we have
$$\mbox{ind}\,(S_r)=\mbox{ind}(W_{0}^{1,p}(\Omega)\setminus A)=m.$$

Also, from Cingolani and Degiovanni \cite[Theorem 3.6]{7}  we know that the sets $S$ and $A$ homologically link in dimension $m$.

Hypotheses $H_{2}(i),(ii)$ imply that given $\varepsilon>0$, we can find $c_{9}=c_{9}(\varepsilon)>0$ such that
$$F(z,x)\leq\frac{\hat{\lambda}_{m}(p)+\varepsilon}{p}|x|^{p}+c_{9}\ \mbox{ for a.a.}\ z\in\Omega,\ \mbox{ all}\ x\in\RR.$$

Recall that $\hat{\lambda}_m(p)<\hat{\lambda}_{m+1}(p)$.
Then for all $u\in A$ we have
$$\varphi(u)\geq\frac{\hat{\lambda}_{m+1}(p)-\hat{\lambda}_{m}(p)-\varepsilon}{p}\|u\|_{p}^{p}-c_{9}|\Omega|_{N}.$$

Choosing $\varepsilon\in(0,\hat{\lambda}_{m+1}(p)-\hat{\lambda}_{m}(p))$, we obtain
$$\inf_{A}\varphi>-\infty.$$

Also, we claim that $\sup_{S_r}\varphi<\inf_A\varphi$ for $r>0$ big. Arguing by contradiction, suppose that we can find $\{u_{n}\}_{n\geq1}\subseteq S$ and $\xi>0$ such that
\begin{eqnarray}\label{eq47}
-\xi\leq\varphi(u_{n})\ \mbox{ for all}\ n\geq1\ \mbox{ and}\ \|u_{n}\|\rightarrow+\infty.
\end{eqnarray}
Let $y_{n}=\frac{u_{n}}{\|u_{n}\|}$,  $n\geq1$. Then $\|y_{n}\|=1$ for all $n\geq1$. So, we may assume that
$$y_{n}\overset{w}\rightarrow y\ \mbox{ in}\ W_{0}^{1,p}(\Omega)\ \mbox{ and}\ y_{n}\rightarrow y\ \mbox{ in}\ L^{p}(\Omega)\ \mbox{ as}\ n\rightarrow\infty.$$

We have
\begin{align*}
&1=\|Dy_{n}\|_{p}^{p}\leq\hat{\lambda}_{m}(p)\|y_{n}\|_{p}^{p}\ (\mbox{recall }\ u_{n}\in S\ \mbox{ for all}\ n\geq1),\\
\Rightarrow&1\leq\hat{\lambda}_{m}(p)\|y\|_{p}^{p}\ \mbox{ (see \eqref{eq46})},\\
\Rightarrow&y\neq0.
\end{align*}
Therefore $|u_{n}(z)|\rightarrow+\infty$ for a.a. $z\in\Omega_{0}=\{z\in\Omega:\ y(z)\neq0\}$ and $|\Omega_{0}|_{N}>0$. We have
$$-\frac{p\xi}{\|u_{n}\|^{\tau}}\leq\int_{\Omega}\frac{\hat{\lambda}_{m}(p)|u_{n}|^{p}-p F(z,u_{n})}{|u_{n}|^{\tau}}|y_{n}|^{\tau}dz+\frac{p}{2}\frac{1}{\|u_{n}\|^{\tau-2}}\|Dy_{n}\|_{2}^{2}.$$
\hspace{12cm} (see \eqref{eq47})

Since $\tau>2$, passing to the limit as $n\rightarrow\infty$ and since $\|u_{n}\|\rightarrow\infty$, we reach a contradiction to \eqref{eq46}. This proves the claim, namely that  $\varphi(u)\rightarrow-\infty$ as $\|u\|\rightarrow+\infty$ with $u\in S$.

Therefore choosing $r>0$ big, we can have
$$\sup_{S}\varphi<\inf_{A}\varphi.$$
So, we can apply Theorem 3.2 of Cingolani and Degiovanni \cite{7} and infer that there exists $y_{0}\in W_{0}^{1,p}(\Omega)$ such that
$$y_{0}\in K_{\varphi}\ \mbox{ and}\ C_{m}(\varphi,y_{0})\neq0.$$
Comparing with (\ref{eq41}) and (\ref{eq42}), we obtain
$$y_0\notin\{0,u_0,v_0\}.$$
Also, since $y_0\in K_{\varphi}$, $y_0$ is a nontrivial solution of problem (\ref{eq1}) and as before the nonlinear regularity theory implies that $y_0\in C_0^1(\overline{\Omega})$.
\end{proof}

\begin{remark}
Is it possible to show that $y_0$ is nodal?
\end{remark}

\section{Superlinear Problems}\label{sec4}
In this section, we assume that the reaction $f(z,\cdot)$ is $(p-1)$-superlinear near $\pm\infty$ without satisfying the $AR$-condition and has $z$-dependent zeros of constant sign. For such problems, we prove a multiplicity theorem producing six nontrivial solutions all with sign information and a second multiplicity theorem producing five solutions.

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

\smallskip
$H_3$: $f:\Omega\times\RR\to\RR$ is a measurable function such that for a.a. $z\in\Omega$, $f(z,0)=0$, $f(z,\cdot)\in C^1(\RR)$ and
\begin{itemize}
\item [(i)] $|f_x'(z,x)|\leq a(z)(1+|x|^{r-2})$ for a.a. $z\in\Omega$, all $x\in\RR$, with $a\in L^{\infty}(\Omega)_{+}$ and $r\in (p,p^*)$;
\item [(ii)] $\lim\limits_{x\to\pm\infty}\frac{F(z,x)}{|x|^p}=+\infty$ uniformly for a.a. $z\in\Omega$;
\item [(iii)] there exist $\tau\in\left(\max\{1,(r-p)\frac{N}{p}\},r\right)$ and $\eta_0>0$ such that
$$\eta_0\leq\liminf_{x\to\pm\infty}\frac{f(z,x)x-pF(z,x)}{|x|^{\tau}}\ \mbox{uniformly for a.a.}\ z\in\Omega;$$
\item [(iv)] $f_x'(z,0)=\lim_{x\to 0}\frac{f(z,x)}{x}$ uniformly for a.a. $z\in\Omega$ and for some $m\geq 2$, we have
$$f_x'(z,0)\in [\hat{\lambda}_m(2),\hat{\lambda}_{m+1}(2)]\ \mbox{for a.a.}\ z\in\Omega,\ f_x'(\cdot,0)\neq\hat{\lambda}_m(2),\ f_x'(\cdot,0)\neq\hat{\lambda}_{m+1}(2);$$
\item [(v)] there exist functions $w_{\pm}\in W^{1,p}(\Omega)\cap C(\overline{\Omega})$ such that
\begin{eqnarray*}
& & w_{-}(z)\leq c_{-}<0<c_{+}\leq w_{+}(z)\ \mbox{for all}\ z\in\overline{\Omega},\\
& & f(z,w_{+}(z))\leq 0\leq f(z,w_{-}(z))\ \mbox{for a.a.}\ z\in\Omega,\\
&\mbox{and}& \langle A_p(w_{-}),\vartheta\rangle+\langle A(w_{-}),\vartheta\rangle\leq 0\leq \langle A_p(w_{+}),\vartheta\rangle+\langle A(w_{+}),\vartheta\rangle\\
& &\hspace{4cm} \mbox{for all}\ \vartheta\in W^{1,p}(\Omega)\ \mbox{with}\ \vartheta\geq 0;
\end{eqnarray*}
\item [(vi)] for every $\rho>0$, there exists $\xi_{\rho}>0$ such that for a.a. $z\in\Omega$ the map
$$x\to f(z,x)+\xi_{\rho}|x|^{p-2}x$$
is nondecreasing on $[-\rho,\rho]$.
\end{itemize}
\begin{remark}
We stress that now the regularity condition on $f(z,\cdot)$ is stronger, namely $f(z,\cdot)\in C^1(\RR)$. Hypothesis $H_3(ii)$ implies that the potential $F(z,\cdot)$ is $p$-superlinear near $\pm\infty$. This hypothesis together with $H_3(iii)$ imply that the reaction $f(z,\cdot)$ is $(p-1)$-superlinear near $\pm\infty$. Note that, unlike most works on superlinear problems, we do not employ the $AR$-conditions. We recall that this condition says that there exist $q>p$ and $M>0$ such that
\begin{eqnarray*}
&&0<qF(z,x)\leq f(z,x)x\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ |x|\geq M,\\
&&0<\operatorname*{ess\,inf}\limits_{\Omega} F(\cdot, \pm M).
\end{eqnarray*}
\end{remark}

The $AR$-condition implies the following weaker unilateral growth condition
$$c_{10}|x|^q\leq F(z,x)\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ |x|\geq M\ \mbox{and some}\ c_{10}>0.$$
From this, we have the much weaker condition
$$\lim\limits_{x\to\pm\infty}\frac{F(z,x)}{|x|^p}=+\infty\ \mbox{uniformly for a.a.}\ z\in\Omega.$$

Here we employ the above asymptotic condition and hypothesis $H_3(iii)$, which together are weaker than the $AR$-condition and incorporate in our framework superlinear reactions with ``slower" growth near $\pm\infty$ which fail to satisfy the $AR$-condition (see the example below). Hypothesis $H_3(iv)$ implies that asymptotically at $0$ we have nonuniform nonresonance with respect to the spectral interval $[\hat{\lambda}_m(2),\hat{\lambda}_{m+1}(2)]$. Finally hypothesis $H_3(v)$ implies the existence of $z$-dependent zeros of constant sign. Evidently this hypothesis is satisfied if there exist $c_{-}<0<c_{+}$ such that
$$f(z,c_{+})\leq 0\leq f(z,c_{-})\ \mbox{for a.a.}\ z\in\Omega.$$
As a final remark concerning hypotheses $H_3$, we mention that by virtue of hypotheses $H_3(i),(iv)$, we can find $\xi_0>\hat{\lambda}_1(2)$ and $\xi_1>0$ such that
\begin{eqnarray}\label{eq48}
f(z,x)x\geq\xi_0 x^2-\xi_1|x|^r\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\in\RR.
\end{eqnarray}
\begin{example}
The following function satisfies hypotheses $H_3$. For the sake of simplicity we drop the $z$-dependence
$$f(x)=|x|^{p-2}x\left(\ln |x|+\frac{1}{p}\right)+\vartheta x-c|x|^{\tau-2}x$$
with $\tau\in (\hat{\lambda}_m,\hat{\lambda}_{m+1})$ $(m\geq 2)$, $c\geq \frac{1}{p}+\vartheta$, $2<\tau<p$. Note that this function does not satisfy the $AR$-condition.
\end{example}
First we will produce constant sign solutions. As before $\varphi: W^{1,p}_0(\Omega)\to\RR$ is the energy functional for problem (\ref{eq1}) defined by
$$\varphi(u)=\frac{1}{p}\|Du\|_p^p+\frac{1}{2}\|Du\|_2^2-\int\limits_{\Omega}F(z,u(z))dz\ \mbox{for all}\ u\in W^{1,p}_0(\Omega).$$

To produce the first two constant sign solutions, the asymptotic condition on $f(z,\cdot)$ at $\pm\infty$ is irrelevant (that is, we do not need hypotheses $H_3(ii),(iii)$).
\begin{prop}\label{prop10}
If hypotheses $H_3(i),(iv),(v),(vi)$ hold, then problem (\ref{eq1}) has two nontrivial constant sign solutions
$$u_0\in\mbox{int}\, C_{+}\ \mbox{and}\ v_0\in-\mbox{int}\, C_{+}$$
such that $-c_{-}\leq w_{-}(z)\leq v_0(z)\leq 0\leq u_0(z)\leq w_{+}(z)\leq c_{+}$ for all $z\in\overline{\Omega}$.
\end{prop}
\begin{proof}
First we produce the positive solution. To this end, we introduce the following truncation of $f(z,\cdot)$:
\begin{eqnarray}\label{eq49}
\hat{f}_{+}(z,x)=\left\{  \begin{array}{ll}
    0\ &\mbox{if}\ x<0\\
    f(z,x)\ &\mbox{if}\ 0\leq x\leq w_{+}(z)\\
    f(z,w_{+}(z))\ &\mbox{if}\ w_{+}(z)<x.
    \end{array}\right.
\end{eqnarray}
This is a Carath\'eodory function. Let $\hat{F}_{+}(z,x)=\int\limits_0^x\hat{f}_{+}(z,s)ds$ and consider the $C^1$-functional $\hat{\varphi}_{+}:W^{1,p}_0(\Omega)\to\RR$ defined by
$$\hat{\varphi}_{+}(u)=\frac{1}{p}\|Du\|_p^p+\frac{1}{2}\|Du\|_2^2-\int\limits_{\Omega}\hat{F}_{+}(z,u(z))dz\ \mbox{for all}\ u\in W^{1,p}_0(\Omega).$$
It is clear  from (\ref{eq49}) that $\hat{\varphi}_{+}$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, by the Weierstrass theorem, we can find $u_0\in W^{1,p}_0(\Omega)$ such that
\begin{eqnarray}\label{eq50}
\hat{\varphi}_{+}(u_0)=\inf [\hat{\varphi}_{+}(u):\ u\in W^{1,p}_0(\Omega)]=\hat{\eta}_{+}.
\end{eqnarray}
Hypothesis $H_3(iv)$ implies that given $\varepsilon>0$, we can find $\delta\in (0,c_{+})$ such that
\begin{eqnarray}\label{eq51}
& &f(z,x)\geq (f_x'(z,0)-\varepsilon)x\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\in [0,\delta],\nonumber\\
&\Rightarrow& F(z,x)\geq \frac{1}{2}(f_x'(z,0)-\varepsilon)x^2\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\in [0,\delta].
\end{eqnarray}

Let $\hat{u}_1(2)\in\mbox{int}\, C_{+}$ be the $L^2$-normalized principal eigenfunction of $(-\Delta,H_0^1(\Omega))$ and let $t\in (0,1)$ be small such that $t\hat{u}_1(2)(z)\in [0,\delta]$ for all $z\in\overline{\Omega}$. Then
\begin{eqnarray*}
&&\hat{\varphi}_{+}(t\hat{u}_1(2))\leq\frac{t^p}{p}\|D\hat{u}_1(2)\|_p^p+\frac{t^2}{2}\left[\int\limits_{\Omega}(\hat{\lambda}_1(2)-f_x'(z,0)) \hat{u}_1(2)^2dz+\varepsilon\right]\\
&&\hspace{5cm}(\mbox{see}\ (\ref{eq51})\ \mbox{and recall that}\ \|\hat{u}_1(2)\|_2=1).
\end{eqnarray*}
Note that
$$\xi_*=\int\limits_{\Omega}[f_x'(z,0)-\hat{\lambda}_1(2)]\hat{u}_1(2)^2 dz>0\ (\mbox{see hypothesis}\ H_3(iv)).$$
We have
$$\hat{\varphi}_{+}(t\hat{u}_1(2))\leq\frac{t^p}{p}\|D\hat{u}_1(2)\|_p^p-\frac{t^2}{2}[\xi_*-\varepsilon].$$
Choosing $\varepsilon\in (0,\xi_*)$ and since $p>2$, for $t\in (0,1)$ even smaller if necessary, we have
\begin{eqnarray*}
& &\hat{\varphi}_{+}(t\hat{u}_1(2))<0,\\
&\Rightarrow& \hat{\varphi}_{+}(u_0)<0=\hat{\varphi}_{+}(0)\ (\mbox{see}\ (\ref{eq49})),\ \mbox{hence}\ u_0\neq 0.
\end{eqnarray*}
From (\ref{eq49}), we have
\begin{eqnarray}\label{eq52}
& &\hat{\varphi}_{+}'(u_0)=0,\nonumber\\
&\Rightarrow& A_p(u_0)+A(u_0)=N_{\hat{f}_{+}}(u_0).
\end{eqnarray}
On (\ref{eq52}) we act with $-u_0^{-}\in W^{1,p}_0(\Omega)$ and obtain
$$\|Du_0^{-}\|_p^p+\|Du_0^{-}\|_2^2=0\ (\mbox{see}\ (\ref{eq49})),\ \mbox{hence}\ u_0\geq 0,\ u_0\neq 0.$$
Also, on (\ref{eq51}) we act with $(u_0-w_{+})^{+}\in W^{1,p}_0(\Omega)$ and obtain
\begin{eqnarray*}
& &\langle A_p(u_0),(u_0-w_{+})^{+}\rangle+\langle A(u_0),(u_0-w_{+})^{+}\rangle\\
&=&\int\limits_{\Omega}\hat{f}_{+}(z,u_0)(u_0-w_{+})^{+}dz\\
&=&\int\limits_{\Omega}f(z,w_{+})(u_0-w_{+})^{+}dz\ (\mbox{see}\ (\ref{eq49}))\\
&\leq&\langle A_p(w_{+}),(u_0-w_{+})^{+}\rangle+\langle A(w_{+}),(u_0-w_{+})^{+}\rangle\ (\mbox{see hypothesis}\ H_3(v)),
\end{eqnarray*}
\begin{eqnarray*}
&&\Rightarrow \langle A_p(u_0)-A_p(w_{+}),(u_0-w_{+})^{+}\rangle+\langle A(u_0)-A(w_{+}),(u_0-w_{+})^{+}\rangle\leq 0,\\
&&\Rightarrow |\{u_0>w_{+}\}|_N=0,\ \mbox{hence}\ u_0\leq w_{+}.
\end{eqnarray*}
So, we have proved that
$$u_0\in [0,w_{+}]=\{u\in W^{1,p}_0(\Omega):\ 0\leq u(z)\leq w_{+}(z)\ \mbox{a.e. in}\ \Omega\}.$$
Then from (\ref{eq49}) and (\ref{eq52}) it follows that $u_0$ is a positive solution of problem (\ref{eq1}). As before the nonlinear regularity theory (see Lieberman \cite{19}) and the nonlinear maximum principle (see Pucci and Serrin \cite[pp. 111, 120]{23}), imply $u_0\in\mbox{int}\, C_{+}$.

Similarly, we consider the following truncation of $f(z,\cdot)$
\begin{eqnarray*}
\hat{f}_{-}(z,x)=\left\{  \begin{array}{ll}
    f(z,w_{-}(z))\ &\mbox{if}\ x<w_{-}(z)\\
    f(z,x)\ &\mbox{if}\ w_{-}(z)\leq x\leq 0\\
    0\ &\mbox{if}\ 0<x.
    \end{array}\right.
\end{eqnarray*}
This is a Carath\'eodory function. We set $\hat{F}_{-}(z,x)=\int\limits_0^x \hat{f}_{-}(z,s)ds$ and consider the $C^1$-functional $\hat{\varphi}_{-}:W^{1,p}_0(\Omega)\to\RR$ defined by
$$\hat{\varphi}_{-}(u)=\frac{1}{p}\|Du\|_p^p+\frac{1}{2}\|Du\|_2^2-\int\limits_{\Omega}\hat{F}_{-}(z,u(z))dz\ \mbox{for all}\ u\in W^{1,p}_0(\Omega).$$
Reasoning as above, via the direct method, we produce a negative solution $v_0\in-\mbox{int}\, C_{+}$ with $c_{-}\leq w_{-}(z)\leq v_0(z)\leq 0$ for all $z\in\overline{\Omega}$.
\end{proof}

Using these two constant sign solutions and the asymptotic conditions at $\pm\infty$ (see hypotheses $H_3(ii),(iii)$), we can generate two more nontrivial solutions of constant sign.
\begin{prop}\label{prop11}
If hypotheses $H_3$ hold, then problem (\ref{eq1}) has two additional constant sign solutions
$$\hat{u}\in\mbox{int}\, C_{+}\ \mbox{and}\ \hat{v}\in-\mbox{int}\, C_{+}$$
such that $\hat{u}-u_0\in\mbox{int}\, C_{+}$ and $v_0-\hat{v}\in\mbox{int}\, C_{+}$.
\end{prop}
\begin{proof}
Again, first we show the existence of the second positive solution. To this end, let $u_0\in\mbox{int}\, C_{+}$ be the positive solution produced in Proposition \ref{prop10}. We introduce the following truncation of the reaction $f(z,\cdot)$:
\begin{eqnarray}\label{eq53}
g_{+}(z,x)=\left\{  \begin{array}{ll}
    f(z,u_0(z))\ &\mbox{if}\ x\leq u_0(z)\\
    f(z,x)\ &\mbox{if}\ u_0(z)<x.
    \end{array}\right.
\end{eqnarray}
This is a Carath\'eodory function. We set $G_{+}(z,x)=\int\limits_0^x g_{+}(z,s)ds$ and consider the $C^1$-functional $\psi_{+}:W^{1,p}_0(\Omega)\to\RR$ defined by
$$\psi_{+}(u)=\frac{1}{p}\|Du\|_p^p+\frac{1}{2}\|Du\|_2^2-\int\limits_{\Omega}G_{+}(z,u(z))dz\ \mbox{for all}\ u\in W^{1,p}_0(\Omega).$$
From Aizicovici, Papageorgiou and Staicu \cite{2} (proof of Proposition $4$), we know that
\begin{eqnarray}\label{eq54}
\psi_{+}\ \mbox{satisfies the}\ C\mbox{-condition}.
\end{eqnarray}

{\bf Claim.} {\it
We may assume that $u_0\in\mbox{int}\, C_{+}$ is a local minimizer of $\psi_{+}$.
}

We consider the following truncation of $g_{+}(z,\cdot)$ (recall that $u_0\leq w_{+}$, see Proposition \ref{prop10}):
\begin{eqnarray}\label{eq55}
\hat{g}_{+}(z,x)=\left\{  \begin{array}{ll}
    g_{+}(z,x)\ &\mbox{if}\ x\leq w_{+}(z)\\
    g_{+}(z,w_{+}(z))\ &\mbox{if}\ w_{+}(z)<x.
    \end{array}\right.
\end{eqnarray}
This is a Carath\'eodory function. We set $\hat{G}_{+}(z,x)=\int\limits_0^x\hat{g}_{+}(z,s)ds$ and consider the $C^1$-functional $\hat{\psi}:W^{1,p}_0(\Omega)\to\RR$ defined by
$$\hat{\psi}_{+}(u)=\frac{1}{p}\|Du\|_p^p+\frac{1}{2}\|Du\|_2^2-\int\limits_{\Omega}\hat{G}_{+}(z,u(z))dz\ \mbox{for all}\ u\in W^{1,p}_0(\Omega).$$
From (\ref{eq53}) and (\ref{eq55}) it is clear that $\hat{\psi}_{+}$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find $\hat{u}_0\in W^{1,p}_0(\Omega)$ such that
\begin{eqnarray}\label{eq56}
& &\hat{\psi}_{+}(\hat{u}_0)=\inf [\hat{\psi}_{+}(u):\ u\in W^{1,p}_0(\Omega)],\nonumber\\
&\Rightarrow& \hat{\psi}_{+}'(\hat{u}_0)=0,\nonumber\\
&\Rightarrow& A_p(\hat{u}_0)+A(\hat{u}_0)=N_{\hat{g}_{+}}(\hat{u}_0).
\end{eqnarray}
On (\ref{eq56}) first we act with $(u_0-\hat{u}_0)^{+}\in W^{1,p}_0(\Omega)$ and obtain
\begin{eqnarray*}
& &\langle A_p(\hat{u}_0),(u_0-\hat{u}_0)^{+}\rangle+\langle  A(\hat{u}_0),(u_0-\hat{u}_0)^{+}\rangle\\
&=& \int\limits_{\Omega}\hat{g}_{+}(z,\hat{u}_0)(u_0-\hat{u}_0)^{+}dz\\
&=&\int\limits_{\Omega} f(z,u_0)(u_0-\hat{u}_0)^{+}dz\ (\mbox{see}\ (\ref{eq55})\ \mbox{and}\ (\ref{eq53}))\\
&=&\langle A_p(u_0),(u_0-\hat{u}_0)^{+}\rangle+\langle  A(u_0),(u_0-\hat{u}_0)^{+}\rangle\ (\mbox{see Proposition \ref{prop10}}),
\end{eqnarray*}
\begin{eqnarray*}
&&\Rightarrow\langle A_p(u_0)-A_p(\hat{u}_0),(u_0-\hat{u}_0)^{+}\rangle+\langle  A(u_0)-A(\hat{u}_0),(u_0-\hat{u}_0)^{+}\rangle=0,\\
&&\Rightarrow |\{u_0>\hat{u}_0\}|_N=0,\ \mbox{hence}\ u_0\leq\hat{u}_0.
\end{eqnarray*}
Also, on (\ref{eq56}) we act with $(\hat{u}_0-w_{+})^{+}\in W^{1,p}_0(\Omega)$. Then
\begin{eqnarray*}
& &\langle A_p(\hat{u}_0),(\hat{u}_0-w_{+})^{+}\rangle+\langle  A(\hat{u}_0),(\hat{u}_0-w_{+})^{+}\rangle\\
&=& \int\limits_{\Omega}\hat{g}_{+}(z,\hat{u}_0)(\hat{u}_0-w_{+})^{+}dz\\
&=&\int\limits_{\Omega} f(z,w_{+})(\hat{u}_0-w_{+})^{+}dz\ (\mbox{see}\ (\ref{eq55})\ \mbox{and}\ (\ref{eq53}))\\
&\leq&\langle A_p(w_{+}),(\hat{u}_0-w_{+})^{+}\rangle+\langle  A(w_{+}),(\hat{u}_0-w_{+})^{+}\rangle\ (\mbox{see hypothesis}\ H_3(v)),
\end{eqnarray*}
\begin{eqnarray*}
&&\Rightarrow\langle A_p(\hat{u}_0)-A_p(w_{+}),(\hat{u}_0-w_{+})^{+}\rangle+\langle A(\hat{u}_0)-A(w_{+}),(\hat{u}_0-w_{+})^{+}\rangle\leq 0,\\
&&\Rightarrow |\{\hat{u}_0>w_{+}\}|_N=0,\ \mbox{hence}\ \hat{u}_0\leq w_{+}.
\end{eqnarray*}
So, we have proved that
$$\hat{u}_0\in [u_0,w_{+}]=\{u\in W^{1,p}_0(\Omega):\ u_0(z)\leq u(z)\leq w_{+}(z)\ \mbox{a.e. in}\ \Omega\}.$$
Then from (\ref{eq53}), (\ref{eq55}) and (\ref{eq56}) it follows that $\hat{u}_0$ is a positive solution of problem (\ref{eq1}) and as before the nonlinear regularity theory and the nonlinear maximum principle imply $\hat{u}_0\in\mbox{int}\, C_{+}$.

If $\hat{u}_0\neq u_0$, then this is the desired second positive solution of (\ref{eq1}).

If $\hat{u}_0= u_0\in\mbox{int}\, C_{+}$, then for $\rho=\|w_{+}\|_{\infty}$, let $\xi_{\rho}>0$ be as postulated by hypothesis $H_3(vi)$. We have
\begin{eqnarray*}
& &-\Delta_p u_0(z)-\Delta u_0(z)+\xi_{\rho}u_0(z)^{p-1}\\
&=& f(z,u_0(z))+\xi_{\rho}u_0(z)^{p-1}\\
&\leq& f(z,w_{+}(z))+\xi_{\rho}w_{+}(z)^{p-1}\ (\mbox{recall that}\ u_0\leq w_{+}\ \mbox{and see hypothesis}\ H_3(vi)).
\end{eqnarray*}
It follows that
\begin{eqnarray}\label{eq57}
& &\langle A_p(u_0),\vartheta\rangle+\langle A(u_0),\vartheta\rangle-\int\limits_{\Omega}f(z,u_0)\vartheta dz\leq\nonumber\\
& &0\leq\langle A_p(w_{+}),\vartheta\rangle+\langle A(w_{+}),\vartheta\rangle-\int\limits_{\Omega}f(z,w_{+})\vartheta dz\\
&&\hspace{2cm}\mbox{for all}\ \vartheta\in W^{1,p}(\Omega)\ \mbox{with}\ \vartheta\geq 0\ (\mbox{see hypothesis}\ H_3(v))\nonumber.
\end{eqnarray}
Let $a(y)=|y|^{p-2}y+y$ for all $y\in\RR^N$. Then $a\in C^1(\RR^N)$ (recall $p>2$) and
\begin{eqnarray*}
& &\nabla a(y)=|y|^{p-2}\left(I+(p-2)\frac{y\otimes y}{|y|^2}\right)+I,\ \mbox{for all}\ y\in\RR^N,\\
&\Rightarrow& (\nabla a(y)\xi,\xi)_{\RR^N}\geq (1+|y|^{p-2})|\xi|^2\ \mbox{for all}\ y,\xi\in\RR^N.
\end{eqnarray*}
So, by virtue of (\ref{eq57}), we can apply Theorem $2.5.2$ of Pucci and Serrin \cite[p. 35]{23} (the tangency principle) and infer that
\begin{eqnarray}\label{eq58}
& & u_0(z)<w_{+}(z)\ \mbox{for all}\ z\in\overline{\Omega},\nonumber\\
&\Rightarrow& u_0\in\mbox{int}_{C_0^1(\overline{\Omega})}[0,w_{+}].
\end{eqnarray}
But from (\ref{eq53}) and (\ref{eq55}) it is clear that $\hat{\psi}_{+}|_{[0,w_{+}]}=\psi_{+}|_{[0,w_{+}]}$. So, (\ref{eq58}) implies that $u_0\in\mbox{int}\, C_{+}$ is a local $C_0^1(\overline{\Omega})$-minimizer of $\psi_{+}$, hence it is also a local $W^{1,p}_0(\Omega)$-minimizer of $\psi_{+}$ (see Proposition \ref{prop3}). This proves the Claim.

By virtue of the Claim, we can find $\rho\in (0,1)$ small such that
\begin{eqnarray}\label{eq59}
&&\psi_{+}(u_0)<\inf [\psi_{+}(u):\ \|u-u_0\|=\rho]=m_{+}
\end{eqnarray}
(see Aizicovici, Papageorgiou and Staicu \cite{1}, proof of Proposition $29$).

Hypothesis $H_3(ii)$ implies that for any $\tilde{u}\in\mbox{int}\, C_{+}$, we have
\begin{eqnarray}\label{eq60}
\psi_{+}(t\tilde{u})\to-\infty\ \mbox{as}\ t\to+\infty.
\end{eqnarray}
Then (\ref{eq54}), (\ref{eq59}) and (\ref{eq60}), permit the use of Theorem \ref{th1} (the mountain pass theorem) and obtain $\hat{u}\in W^{1,p}_0(\Omega)$ such that
\begin{eqnarray}\label{eq61}
\hat{u}\in K_{\psi_{+}}\ \mbox{and}\ m_{+}\leq\psi_{+}(\hat{u}).
\end{eqnarray}
From (\ref{eq59}) and (\ref{eq61}), we see that $\hat{u}\neq u_0$. Also, since $\hat{u}\in K_{\psi_{+}}$ (see (\ref{eq61})), we have that
\begin{eqnarray}\label{eq62}
A_p(\hat{u})+A(\hat{u})=N_{g_{+}}(\hat{u}).
\end{eqnarray}
Acting on (\ref{eq62}) with $(u_0-\hat{u})^{+}\in W^{1,p}_0(\Omega)$ and using (\ref{eq53}), as before we obtain that $u_0\leq\hat{u}$ and so $\hat{u}$ is a positive solution of problem (\ref{eq1}) (see (\ref{eq53}) and (\ref{eq62})). As before the nonlinear regularity theory implies that $\hat{u}\in\mbox{int}\, C_{+}$.

Let $\rho=\|\hat{u}\|_{\infty}$ and let $\xi_{\rho}>0$ be a postulated by hypothesis $H_3(vi)$. Then for $\hat{\xi}_{\rho}>\xi_{\rho}$ we have
\begin{eqnarray}\label{eq63}
& &-\Delta_p u_0(z)-\Delta u_0(z)+\hat{\xi}_{\rho}u_0(z)^{p-1}\nonumber\\
&=& f(z,u_0(z))+\hat{\xi}_{\rho}u_0(z)^{p-1}\nonumber\\
&\leq& f(z,\hat{u}(z))+\hat{\xi}_{\rho}\hat{u}(z)^{p-1}\ (\mbox{since}\ u_0\leq\hat{u},\ \mbox{see hypothesis}\ H_3(vi))\nonumber\\
&=&-\Delta_p \hat{u}(z)-\Delta\hat{u}(z)+\hat{\xi}_{\rho}\hat{u}(z)^{p-1}\ \mbox{a.e. in}\ \Omega.
\end{eqnarray}

Once again the tangency principle of Pucci and Serrin \cite[Theorem 2.5.2, p. 35]{23} implies that $u_0(z)<\hat{u}(z)$ for all $z\in\Omega$. Hence, from (\ref{eq63}) and Proposition \ref{prop4} (recall $\hat{\xi}_{\rho}>\xi_{\rho}$) we infer that $\hat{u}-u_0\in\mbox{int}\, C_{+}$.

Similarly, for the second negative solution we introduce the truncation
\begin{eqnarray*}
g_{-}(z,x)=\left\{  \begin{array}{ll}
  f(z,x)\ &\mbox{if}\ x\leq v_0(z)\\
    f(z,v_0(z))\ &\mbox{if}\ v_0(z)<x.
    \end{array}\right.
\end{eqnarray*}
This is a Carath\'eodory function. We set $G_{-}(z,x)=\int\limits_0^x g_{-}(z,s)ds$ and introduce the $C^1$-functional $\psi_{-}:W^{1,p}_0(\Omega)\to\RR$ defined by
$$\psi_{-}(u)=\frac{1}{p}\|Du\|_p^p+\frac{1}{2}\|Du\|_2^2-\int\limits_{\Omega}G_{-}(z,u(z))dz\ \mbox{for all}\ u\in W^{1,p}_0(\Omega).$$
Working as above, this time using the functional $\psi_{-}$, we produce a second negative solution $\hat{v}\in-\mbox{int}\, C_{+}$ such that $v_0-\hat{v}\in\mbox{int}\, C_{+}$.
\end{proof}

Next we look for nodal (sign changing) solutions. We will employ tools from Morse theory.

Let $d_m=\dim \oplus_{i=1}^{m}E(\hat{\lambda}_i(2))$ (recall that $E(\hat{\lambda}_i(2))$ denotes the eigenspace of $(-\Delta,H_0^1(\Omega))$ corresponding to the eigenvalue $\hat{\lambda}_i(2)$; see Section \ref{sec2}).
\begin{prop}\label{prop12}
If hypotheses $H_3$ hold, then $C_k(\varphi,0)=\delta_{k,d_m}\ZZ$ for all $k\geq 0$.
\end{prop}
\begin{proof}
Let $\tau:W^{1,p}_0(\Omega)\to\RR$ be the $C^2$-functional defined by
$$\tau(u)=\frac{1}{p}\|Du\|_p^p+\frac{1}{2}\|Du\|_2^2-\frac{1}{2}\int\limits_{\Omega}f_x'(z,0)u(z)^2dz\ \mbox{for all}\ u\in W^{1,p}_0(\Omega).$$
We consider the affine homotopy
$$h(t,u)=(1-t)\varphi(u)+t\tau(u)\ \mbox{for all}\ (t,u)\in [0,1]\times W^{1,p}_0(\Omega).$$
Suppose that we can find $\{t_n\}_{n\geq 1}\subseteq [0,1]$ and $\{u_n\}_{n\geq 1}\subseteq W^{1,p}_0(\Omega)$ such that
\begin{eqnarray}\label{eq64}
t_n\to t\in [0,1],\ u_n\to 0\ \mbox{in}\ W^{1,p}_0(\Omega)\ \mbox{and}\ h_u'(t_n,u_n)=0\ \mbox{for all}\ n\geq 1.
\end{eqnarray}
We have
\begin{eqnarray}\label{eq65}
A_p(u_n)+A(u_n)=(1-t_n)N_f(u_n)+t_n f_x'(\cdot,0)u_n\ \mbox{for all}\ n\geq 1.
\end{eqnarray}
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
\begin{eqnarray}\label{eq66}
y_n\to y\ \mbox{in}\ W^{1,p}_0(\Omega)\ \mbox{and}\ y_n\to y\ \mbox{in}\ L^r(\Omega).
\end{eqnarray}
From (\ref{eq65}) we have
\begin{eqnarray}\label{eq67}
\|u_n\|^{p-2}A_p(y_n)+A(y_n)=(1-t_n)\frac{N_f(u_n)}{\|u_n\|}+t_n f_x'(\cdot,0)y_n\ \mbox{for all}\ n\geq 1.
\end{eqnarray}
From hypotheses $H_3(i),(iv)$, we have
$$|f(z,x)|\leq c_{10}(|x|+|x|^{r-1})\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\in\RR,\ \mbox{some}\ c_{10}>0.$$
Note that (\ref{eq65}) and Ladyzhenskaya and Uraltseva \cite[p. 286]{18}, imply that $\{u_n\}_{n\geq 1}\subseteq L^{\infty}(\Omega)$ is bounded. So, it follows that $\left\{\frac{N_f(u_n)}{\|u_n\|}\right\}_{n\geq 1}\subseteq L^r(\Omega)$ is bounded. Thus by passing to a subsequence if necessary and using hypothesis $H_3(iv)$, we have
\begin{eqnarray}\label{eq68}
\frac{N_f(u_n)}{\|u_n\|}\overset{w}{\to}f_x'(\cdot,0)y\ \mbox{in}\ L^r(\Omega)\ \mbox{as}\ n\to\infty.
\end{eqnarray}
Since $\{A_p(y_n)\}_{n\geq 1}\subseteq W^{-1,p'}(\Omega)$ is bounded (see Proposition \ref{prop2}) and $\|u_n\|^{p-2}\to 0$ (recall $p>2$ and see (\ref{eq64})), we have
\begin{eqnarray}\label{eq69}
\|u_n\|^{p-2}A_p(y_n)\to 0\ \mbox{in}\ W^{-1,p'}(\Omega)\ \mbox{as}\ n\to\infty.
\end{eqnarray}
So, if in (\ref{eq67}) we pass to the limit as $n\to\infty$ and use (\ref{eq68}) and (\ref{eq69}), then
\begin{eqnarray*}
& &A(y)=f_x'(\cdot,0)y,\\
&\Rightarrow&-\Delta y(z)=f_x'(z,0)y(z)\ \mbox{a.e. in}\ \Omega,\ y\Big|_{\partial\Omega}=0.
\end{eqnarray*}
Hypothesis $H_3(iv)$ implies that $y=0$. On the other hand, from (\ref{eq67}) and Theorem \ref{th1} of Lieberman \cite{19}, we know that there exist $\alpha\in (0,1)$ and $M>0$ such that
\begin{eqnarray}\label{eq70}
y_n\in C_0^{1,\alpha}(\overline{\Omega})\ \mbox{and}\ \|y_n\|_{C_0^{1,\alpha}(\overline{\Omega})}\leq M\ \mbox{for all}\ n\geq 1.
\end{eqnarray}
Exploiting the compact embedding of $C_0^{1,\alpha}(\overline{\Omega})$ into $C_0^1(\overline{\Omega})$ and using (\ref{eq66}), we have
\begin{eqnarray*}
& & y_n\to y\ \mbox{in}\ C_0^1(\overline{\Omega})\\
&\Rightarrow&\|y\|=1,\ \mbox{contradiction since}\ y=0.
\end{eqnarray*}
So, (\ref{eq64}) can not occur and then the homotopy invariance of the critical groups implies that
\begin{eqnarray}\label{eq71}
C_k(\varphi,0)=C_k(\tau,0)\ \mbox{for all}\ k\geq 0.
\end{eqnarray}
From Theorem 1 of Cingolani and Vannella \cite{8}, we have
\begin{eqnarray*}
& &C_k(\tau,0)=\delta_{k,d_m}\ZZ\ \mbox{for all}\ k\geq 0,\\
&\Rightarrow&  C_k(\varphi,0)=\delta_{k,d_m}\ZZ\ \mbox{for all}\ k\geq 0\ (\mbox{see}\ (\ref{eq71})).
\end{eqnarray*}
The proof is now complete.
\end{proof}

\begin{prop}\label{prop13}
 Assume that hypotheses $H_3(i),(ii),(iii)$ hold and $\inf\varphi(K_{\varphi})>-\infty$. Then $C_k(\varphi,\infty)=0$ for all $k\geq 0$.
\end{prop}

\begin{proof}
Note that $\varphi$ satisfies the $C$-condition (see Aizicovici, Papageorgiou and Staicu \cite{2}). Hypotheses $H_3(i),(ii)$ imply that given $\xi>0$, we can find $c_{11}=c_{11}(\xi)>0$ such that
\begin{eqnarray}\label{eq72}
F(z,x)\geq\frac{\xi}{p}|x|^p-c_{11}\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\in\RR.
\end{eqnarray}
Let $u\in\partial B_1=\{u\in W^{1,p}_0(\Omega):\ \|u\|=1\}$ and $t>0$. Then
$$\varphi(tu)\leq\frac{t^p}{p}\left[1-\xi\|u\|_p^p\right]+\frac{t^2}{2}\|Du\|_2^2+c_{11}|\Omega|_{N}.$$
Choosing $\xi>\frac{1}{\|u\|_p^p}$ and since $p>2$, we infer that
\begin{eqnarray}\label{eq73}
\varphi(tu)\to-\infty\ \mbox{as}\ t\to+\infty.
\end{eqnarray}
By virtue of hypotheses $H_3(i),(iii)$, we can find $\eta_1\in (0,\eta_0)$ and $c_{12}>0$ such that
\begin{eqnarray}\label{eq74}
f(z,x)x-p F(z,x)\geq \eta_1|x|^{\tau}-c_{12}\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\in\RR.
\end{eqnarray}
Then for all $v\in W^{1,p}_0(\Omega)$, we have
\begin{eqnarray}\label{eq75}
\int\limits_{\Omega}[p F(z,v)-f(z,v)v]dz\leq -\eta_1\|v\|_{\tau}^{\tau}+c_{12}|\Omega|_N\ (\mbox{see}\ (\ref{eq74})).
\end{eqnarray}
Let $\vartheta<\min[-c_{12}|\Omega|_N-1,\inf\varphi (K_{\varphi})]$. From (\ref{eq73}), we see that for every $u\in W^{1,p}_0(\Omega)\setminus\{0\}$ we can find $t\geq 1$ such that
\begin{eqnarray}\label{eq76}
\varphi(tu)\leq\frac{\vartheta}{p}.
\end{eqnarray}
Then for such big $t\geq 1$, we have
\begin{eqnarray}\label{eq77}
\frac{d}{dt}\varphi(tu)&=&\langle\varphi'(tu),u\rangle\nonumber\\
&=&\frac{1}{t}\left[t^p\|Du\|_p^p+t^2\|Du\|_2^2-\int\limits_{\Omega}f(z,tu)(tu)dz\right]\nonumber\\
&\leq&\frac{1}{t}\left[t^p\|Du\|_p^p+t^2\|Du\|_2^2-\int\limits_{\Omega}p F(z,tu)dz+c_{12}|\Omega|_N\right]\nonumber\\
& &\hspace{7cm} (\mbox{see}\ (\ref{eq75}))\nonumber\\
&\leq&\frac{1}{t}\left[p\varphi(tu)+c_{12}|\Omega|_N\right]\ (\mbox{since}\ p>2)\nonumber\\
&\leq&\frac{1}{t}\left[\vartheta+c_{12}|\Omega|_N\right]<0.
\end{eqnarray}
Given $u\in W^{1,p}_0(\Omega)\setminus\{0\}$, let $\tau(u)\geq 1$ be the first $t\geq 1$ such that (\ref{eq76}) holds. Then by virtue of (\ref{eq77}) and the implicit function theorem, we have $\tau\in C^1(W^{1,p}_0(\Omega)\setminus\{0\})$. Let $h\in C([0,1]\times(W^{1,p}_0(\Omega)\setminus\{0\}),W^{1,p}_0(\Omega)\setminus\{0\})$ be defined by
$$h(t,u)=(1-t+t\tau(u))u\ \mbox{for all}\ (t,u)\in [0,1]\times (W^{1,p}_0(\Omega)\setminus\{0\}).$$
Note that
\begin{eqnarray*}
& & h(0,u)=u,\ h(1,u)=\tau(u)u\in\varphi^{\vartheta}\ \mbox{for all}\ u\in W^{1,p}_0(\Omega)\setminus\{0\}\\
&\mbox{and}&\ h(t,u)=u\ \mbox{for all}\ (t,u)\in [0,1]\times\varphi^{\vartheta}.
\end{eqnarray*}
These facts imply that $\varphi^{\vartheta}$ is a strong deformation retract of $W^{1,p}_0(\Omega)\setminus\{0\}$. Using the radial retraction and  Dugundji \cite[Theorem 6.5, p. 325]{11}, we see that $\partial B_1$ is a deformation retract of $W^{1,p}_0(\Omega)\setminus\{0\}$. It follows that the sets $\varphi^{\vartheta}$ and $\partial B_1$ are homotopy equivalent and so we have
\begin{eqnarray}\label{eq78}
H_k(W^{1,p}_0(\Omega),\partial B_1)=H_k(W^{1,p}_0(\Omega),\varphi^{\vartheta})\ \mbox{for all}\ k\geq 0.
\end{eqnarray}
The choice of $\vartheta$ implies that
\begin{eqnarray}\label{eq79}
H_k(W^{1,p}_0(\Omega),\varphi^{\vartheta})=C_k(\varphi,\infty)\ \mbox{for all}\ k\geq 0.
\end{eqnarray}
Also, since $W^{1,p}_0(\Omega)$ is infinite dimensional, the boundary $\partial B_1$ is contractible in itself (see, for example, Gasinski and Papageorgiou \cite[p. 691]{15}). Therefore
\begin{eqnarray}\label{eq80}
H_k(W^{1,p}_0(\Omega),\partial B_1)=0\ \mbox{for all}\ k\geq 0.
\end{eqnarray}
From (\ref{eq78}), (\ref{eq79}) and (\ref{eq80}) we conclude that
$$C_k(\varphi,\infty)=0\ \mbox{for all}\ k\geq 0.$$
This completes the proof.
\end{proof}

Next we compute the critical groups of the functional $\varphi$ at $\hat{u}\in K_{\varphi}$ and at $\hat{v}\in K_{\varphi}$.
\begin{prop}\label{prop14}
If hypotheses $H_3$ hold and $K_{\varphi}$ is finite, then $C_k(\varphi,\hat{u})=C_k(\varphi,\hat{v})=\delta_{k,1}\ZZ$ for all $k\geq 0$.
\end{prop}
\begin{proof}
We do the proof for the pair $(\varphi,\hat{u})$, the proof for the pair $(\varphi,\hat{v})$ being similar. Let $\psi_{+}:W^{1,p}_0(\Omega)\to\RR$ be the functional introduced in the proof of Proposition \ref{prop11}. We consider the homotopy $h_{+}:[0,1]\times W^{1,p}_0(\Omega)\to W^{1,p}_0(\Omega)$ defined by
$$h_{+}(t,u)=(1-t)\varphi(u)+t\psi_{+}(u)\ \mbox{for all}\ (t,u)\in [0,1]\times W^{1,p}_0(\Omega).$$
Suppose we can find $\{t_n\}_{n\geq 1}\subseteq [0,1]$ and $\{u_n\}_{n\geq 1}\subseteq W^{1,p}_0(\Omega)$ such that
\begin{eqnarray}\label{eq81}
t_n\to t\in [0,1],\ u_n\to\hat{u}\ \mbox{in}\ W^{1,p}_0(\Omega)\ \mbox{and}\ (h_{+})_u'(t_n,u_n)=0\ \mbox{for all}\ n\geq 1.
\end{eqnarray}
From (\ref{eq81}) we have
\begin{eqnarray}\label{eq82}
-\Delta_p u_n(z)-\Delta u_n(z)=(1-t_n)f(z,u_n(z))+t_n g_{+}(z,u_n(z))\ \mbox{for a.a.}\ z\in\Omega.
\end{eqnarray}
From Ladyzhenskaya and Uraltseva \cite[p. 286]{18}, we know that there exists $M_8>0$ such that $\|u_n\|_{\infty}\leq M_8$ for all $n\geq 1$. Then applying Theorem \ref{th1} of Lieberman \cite{19}, we can find $\alpha\in (0,1)$ and $M_9>0$ such that
$$u_n\in C_0^{1,\alpha}(\overline{\Omega})\ \mbox{and}\ \|u_n\|_{C_0^{1,\alpha}(\overline{\Omega})}\leq M_9\ \mbox{for all}\ n\geq 1.$$
The compact embedding of $C_0^{1,\alpha}(\overline{\Omega})$ into $C_0^1(\overline{\Omega})$ and (\ref{eq81}), imply
$$u_n\to\hat{u}\ \mbox{in}\ C_0^1(\overline{\Omega})\ \mbox{as}\ n\to\infty.$$
From Proposition \ref{prop11} we know that $\hat{u}-u_0\in\mbox{int}\, C_{+}$. Therefore we can find $n_0\geq 1$ such that $u_n\geq u_0$ for all $n\geq n_0$. Then from (\ref{eq53}) and (\ref{eq82}), it follows that $\{u_n\}_{n\geq n_0}\subseteq K_{\varphi}$, which contradicts our hypothesis that $K_{\varphi}$ is finite. Therefore, we can find $\rho\in (0,1)$ small such that $\hat{u}\in\mbox{int}\, C_{+}$ is the only critical point of $\{h_{+}(t,\cdot)\}_{t\in [0,1]}$ in $\overline{B}_{\rho}(\hat{u})=\{u\in W^{1,p}_0(\Omega):\ \|u-\hat{u}\|\leq \rho\}$. Then the homotopy invariance of critical groups implies that
\begin{eqnarray}\label{eq83}
C_k(\varphi,\hat{u})=C_k(\psi_{+},\hat{u})\ \mbox{for all}\ k\geq 0.
\end{eqnarray}
From the proof of Proposition \ref{prop11}, we know that $\hat{u}\in\mbox{int}\, C_{+}$ is a critical point of mountain pass type for the functional $\psi_{+}$. Therefore
\begin{eqnarray}\label{eq84}
& &C_1(\psi_{+},\hat{u})\neq 0,\nonumber\\
&\Rightarrow& C_1(\varphi,\hat{u})\neq 0\ (\mbox{see}\ (\ref{eq83})).
\end{eqnarray}
Since $\varphi\in C^2(W^{1,p}_0(\Omega))$ (see hypotheses $H_3$), from (\ref{eq84}) and Papageorgiou and Smyrlis \cite{22} (see also Papageorgiou and R\u adulescu \cite{21}), we have
$$C_k(\varphi,\hat{u})=\delta_{k,1}\ZZ\ \mbox{for all}\ k\geq 0.$$
In a similar fashion, we show that $C_k(\varphi,\hat{v})=\delta_{k,1}\ZZ\ \mbox{for all}\ k\geq 0$.
\end{proof}

Our strategy in order to generate nodal solutions is the following. First we show that problem (\ref{eq1}) admits extremal constant sign solutions, that is there exist a smallest positive solution $u_{+}\in\mbox{int}\, C_{+}$ and a biggest negative solution $v_{-}\in-\mbox{int}\, C_{+}$. Having the extremal constant sign solutions, we focus on the order interval $[v_{-},u_{+}]=\{u\in W^{1,p}_0(\Omega): v_{-}(z)\leq u(z)\leq u_{+}(z)\ \mbox{a.e. in}\ \Omega\}$. Through a combination of variational methods and of Morse theory, we show that the problem has a nontrivial solution in $[v_{-},u_{+}]$ distinct from $v_{-}$ and $u_{+}$. The extremality of $v_{-}$ and $u_{+}$, implies that this solution must be nodal. The nonhomogeneity of the differential operator, is the source of difficulties in the execution of this solution strategy. In this respect, the unilateral growth estimate in (\ref{eq48}), will help us to overcome these difficulties. So, motivated by (\ref{eq48}), we consi
 der the following auxiliary Dirichlet problem
\begin{eqnarray}\label{eq85}
-\Delta_p u(z)-\Delta u(z)=\xi_0 u(z)-\xi_1 |u(z)|^{r-2}u(z)\ \mbox{in}\ \Omega,\ u\Big|_{\partial\Omega}=0.
\end{eqnarray}
\begin{prop}\label{prop15}
Problem (\ref{eq85}) admits a unique positive solution $u_*\in\mbox{int}\, C_{+}$ and since the equation in (\ref{eq85}) is odd, $v_*=-u_*\in-\mbox{int}\, C_{+}$ is the unique negative solution of (\ref{eq85}).
\end{prop}
\begin{proof}
Let $\mu_{+}:W^{1,p}_0(\Omega)\to\RR$ be the $C^1$-functional defined by
$$\mu_{+}(u)=\frac{1}{p}\|Du\|_p^p+\frac{1}{2}\|Du\|_2^2-\frac{\xi_0}{2}\|u^{+}\|_2^2+\frac{\xi_1}{p}\|u^{+}\|_r^r\ \mbox{for all}\ u\in W^{1,p}_0(\Omega).$$
Since $r>p>2$, it is clear that $\mu_{+}$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find $u_*\in W^{1,p}_0(\Omega)$ such that
\begin{eqnarray}\label{eq86}
\mu_{+}(u_*)=\inf [\mu_{+}(u):\ u\in W^{1,p}_0(\Omega)].
\end{eqnarray}
Since $\xi_0>\hat{\lambda}_1(2)$ and $p>2$, we see that for $t\in (0,1)$ small, we have
\begin{eqnarray*}
& &\mu_{+}(t\hat{u}_1(2))<0,\\
&\Rightarrow&\mu_{+}(u_*)<0=\mu_{+}(0)\ (\mbox{see}\ (\ref{eq86})),\ \mbox{hence}\ u_*\neq 0.
\end{eqnarray*}
From (\ref{eq86}) we have
\begin{eqnarray}\label{eq87}
& & \mu_{+}'(u_*)=0,\nonumber\\
&\Rightarrow& A_p(u_*)+A(u_*)=\xi_0 u_*^{+}-\xi_1(u_*^{+})^{r-1}.
\end{eqnarray}
On (\ref{eq87}) we act with $-u_*^{-}\in W^{1,p}_0(\Omega)$ and obtain $u_*\geq 0$, $u_*\neq 0$. So, it follows that $u_*$ is a positive solution of the auxiliary problem (\ref{eq85}). The nonlinear regularity theory (see \cite{19}) and the nonlinear maximum principle (see Pucci and Serrin \cite[pp. 111, 120]{23}), imply that $u_*\in\mbox{int}\, C_{+}$.

We need to show the uniqueness of this positive solution. To this end, let $G_0(t)=\frac{t^p}{p}+\frac{t^2}{2}$ for all $t\geq 0$. We set $G(y)=G_0(|y|)$ and consider the integral functional $j:L^1(\Omega)\to\overline{\RR}=\RR\cup\{+\infty\}$ defined by
\begin{eqnarray*}
j(u)=\left\{  \begin{array}{ll}
  \int\limits_{\Omega} G(Du^{1/2})dz\ &\mbox{if}\ u\geq 0,\ u^{1/2}\in W^{1,p}_0(\Omega)\\
    +\infty\ &\mbox{otherwise}.
    \end{array}\right.
\end{eqnarray*}
We claim that $j(\cdot)$ is convex. So, let $u_1,u_2\in\mbox{dom}\ j=\{u\in W^{1,p}_0(\Omega):j(u)<+\infty\}$ and let $v_1=u_1^{1/2}$, $v_2=u_2^{1/2}\in W^{1,p}_0(\Omega)$. We define
$$v_3=((1-t)u_1+tu_2)^{1/2}\ \mbox{with}\ t\in [0,1].$$
From Lemma $1$ of Benguria, Brezis and Lieb \cite{6} (see also Diaz and Saa \cite[Lemma 1]{10}), we have
$$|Dv_3(z)|\leq [(1-t)|Dv_1(z)|^2+t|Dv_2(z)|^2]^{1/2}\ \mbox{for a.a.}\ z\in\Omega.$$
Note that $t\to G_0(t)$ is increasing and $t\to G_0(t^{1/2})$ is convex on $[0,+\infty]$. So, we obtain
\begin{eqnarray*}
& & G_0(|Dv_3(z)|)\\
&\leq& G_0 ([(1-t)|Dv_1(z)|^2+t|Dv_2(z)|^2]^{1/2})\ (\mbox{the monotonicity of}\ G_0(\cdot))\\
&\leq& (1-t)G_0(|Dv_1(z)|^2)+t G_0(|Dv_2(z)|)\ (\mbox{the convexity of}\ t\to G_0(t^{1/2})),\\
&\Rightarrow& G(Dv_3(z))\leq (1-t)G(Dv_1(z))+t G(Dv_2(z))\ \mbox{a.e. in}\ \Omega,\\
&\Rightarrow& j(\cdot)\ \mbox{is convex}.
\end{eqnarray*}

Let $u,v$ be two positive solutions of the auxiliary problem (\ref{eq85}). Then from the first part of the proof we have $u,v\in\mbox{int}\, C_{+}$. So, for all $h\in C_0^1(\overline{\Omega})$ and for $t\in [0,1]$ small, we have $u^2\pm th$, $v^2\pm th\in\mbox{dom}\, j$. Then by the chain rule, we have
\begin{eqnarray*}
&&j'(u^2)(h)=\langle A_p(u)+A(u),\frac{h}{u}\rangle\\
&&j'(v^2)(h)=\langle A_p(v)+A(v),\frac{h}{v}\rangle\ \mbox{for all}\ h\in W^{1,p}_0(\Omega)\\
&&\hspace{3cm} (\mbox{recall}\ C_0^1(\overline{\Omega})\ \mbox{is dense in}\ W^{1,p}_0(\Omega)).
\end{eqnarray*}
The convexity of $j$, implies the monotonicity of $j'(y)(\cdot)$ for all $y\in\mbox{dom}\, j$. Hence
\begin{eqnarray*}
0&\leq&\langle j'(u^2)-j'(v^2),u^2-v^2\rangle\\
&=&\int\limits_{\Omega}\left(\frac{-\Delta_p u-\Delta u}{u}-\frac{-\Delta_p v-\Delta v}{v}\right)(u^2-v^2)dz\\
&=&\int\limits_{\Omega}\xi_1 (v^{r-2}-u^{r-2})(u^2-v^2)dz\leq 0,
\end{eqnarray*}
\begin{eqnarray*}
&&\Rightarrow u=v,\\
&&\Rightarrow u_*\in\mbox{int}\, C_{+}\ \mbox{is the unique positive solution of}\ (\ref{eq85}).
\end{eqnarray*}
Since the equation is odd, we deduce that $v_*=-u_*\in-\mbox{int}\, C_{+}$ is the unique negative solution of problem (\ref{eq85}).
\end{proof}

Now let $S_{+}$ (resp. $S_{-}$) denote the set of positive (resp. negative) solutions of problem (\ref{eq1}) belonging in the order interval $[0,w{+}]$ (resp. $[w_{-},0]$). From Proposition \ref{prop10} we know that
$$S_{+},S_{-}\neq\emptyset\ \mbox{and}\ S_{+}\subseteq\mbox{int}\, C_{+},\ S_{-}\subseteq-\mbox{int}\, C_{+}.$$
\begin{prop}\label{prop16}
If hypotheses $H_3$ hold and $u\in S_{+}$ (resp. $v\in-S_{-}$), then $u_*\leq u$ and $v\leq v_*$.
\end{prop}
\begin{proof}
We do the proof for $u\in S_{+}$, the proof for $v\in S_{-}$ being similar.

We introduce the following Carath\'eodory function:
\begin{eqnarray}\label{eq88}
\gamma_{+}(z,x)=\left\{  \begin{array}{ll}
  0\ &\mbox{if}\ x<0\\
    \xi_0 x-\xi_1 x^{r-1}\ &\mbox{if}\ 0\leq x\leq u(z)\\
     \xi_0 u(z)-\xi_1 u(z)^{r-1}\ &\mbox{if}\ u(z)<x.
    \end{array}\right.
\end{eqnarray}
Let $\Gamma_{+}(z,x)=\int\limits_0^x\gamma_{+}(z,s)ds$ and consider the $C^1$-functional $\xi_{+}:W^{1,p}_0(\Omega)\to\RR$ defined by
$$\xi_{+}(u)=\frac{1}{p}\|Du\|_p^p+\frac{1}{2}\|Du\|_2^2-\int\limits_{\Omega}\Gamma_{+}(z,u(z))dz\ \mbox{for all}\ u\in W^{1,p}_0(\Omega).$$
It is clear from (\ref{eq88}) that $\xi_{+}$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find $\tilde{u}_*\in W^{1,p}_0(\Omega)$ such that
\begin{eqnarray}\label{eq89}
\xi_{+}(\tilde{u}_*)=\inf [\xi_{+}(u):\ u\in W^{1,p}_0(\Omega)].
\end{eqnarray}
As before (see the proof of Proposition \ref{prop15}), for $t\in (0,1)$ small (at least such that $t\hat{u}_1(2)\leq u$; recall that $u\in\mbox{int}\, C_{+}$ and see Lemma $3.3$ of Filippakis, Kristaly and Papageorgiou \cite{14}), we have $\xi_{+}(t\hat{u}_1(2))<0$. Therefore
$$\xi_{+}(\tilde{u}_*)<0\ (\mbox{see}\ (\ref{eq89})),\ \mbox{hence}\ \tilde{u}_*\neq 0.$$
Also, from (\ref{eq89}) we have
\begin{eqnarray}\label{eq90}
& &\xi_{+}'(\tilde{u}_*)=0,\nonumber\\
&\Rightarrow& A_p(\tilde{u}_*)+A(\tilde{u}_*)=N_{\gamma_{+}}(\tilde{u}_*).
\end{eqnarray}
On (\ref{eq90}) we act with $-\tilde{u}_*^{-}\in  W^{1,p}_0(\Omega)$ and obtain $\tilde{u}_*\geq 0$, $\tilde{u}_*\neq 0$. Also, we act with $(\tilde{u}_*-u)^{+}\in  W^{1,p}_0(\Omega)$ and have
\begin{eqnarray*}
& &\langle A_p(\tilde{u}_*),(\tilde{u}_*-u)^{+}\rangle+\langle A(\tilde{u}_*),(\tilde{u}_*-u)^{+}\rangle\\
&=&\int\limits_{\Omega}\gamma_{+}(z,\tilde{u}_*)(\tilde{u}_*-u)^{+}dz\\
&=&\int\limits_{\Omega}[\xi_0 u-\xi_1 u^{r-1}](\tilde{u}_*-u)^{+}dz\ (\mbox{see}\ (\ref{eq88}))\\
&\leq&\int\limits_{\Omega}f(z,u)(\tilde{u}_*-u)^{+}dz\ (\mbox{see}\ (\ref{eq48}))\\
&=&\langle A_p(u),(\tilde{u}_*-u)^{+}\rangle+\langle A(u),(\tilde{u}_*-u)^{+}\rangle\ (\mbox{since}\ u\in S_{+}),
\end{eqnarray*}
\begin{eqnarray*}
&&\Rightarrow \langle A_p(\tilde{u}_*)-A_p(u),(\tilde{u}_*-u)^{+}\rangle+\langle A(\tilde{u}_*)-A(u),(\tilde{u}_*-u)^{+}\rangle\leq 0,\\
&&\Rightarrow |\{\tilde{u}_*>u\}|_N=0,\ \mbox{hence}\ \tilde{u}_*\leq u.
\end{eqnarray*}
So, we have proved that
\begin{eqnarray*}
& &\tilde{u}_*\in [0,u]=\{y\in W^{1,p}_0(\Omega):\ 0\leq y(z)\leq u(z)\ \mbox{for a.a.}\ z\in\Omega\},\ \tilde{u}_*\neq 0,\\
&\Rightarrow& \tilde{u}_*\ \mbox{is a nontrivial positive solution of problem}\ (\ref{eq85})\\
& &\hspace{4cm} (\mbox{see}\ (\ref{eq88})\ \mbox{and}\ (\ref{eq90})),\\
&\Rightarrow& \tilde{u}_*=u_*\in\mbox{int}\, C_{+}\ (\mbox{see Proposition}\ \ref{prop15}),\\
&\Rightarrow& u_*\leq u.
\end{eqnarray*}
Similarly we show that $v\leq v_*$.
\end{proof}

Now we can establish the existence of extremal constant sign solutions for problem (\ref{eq1}).
\begin{prop}\label{prop17}
If hypotheses $H_3$ hold, then problem (\ref{eq1}) admits a smallest positive solution $u_{+}\in\mbox{int}\, C_{+}$ and a biggest negative solution $v_{-}\in-\mbox{int}\, C_{+}$.
\end{prop}
\begin{proof}
As in Filippakis, Kristaly and Papageorgiou \cite{14}, we can show that the set of positive solutions of (\ref{eq1}) is downward directed, that is, if $u,y$ are two positive solutions, then there exists a third positive solution of (\ref{eq1}) such that $w\leq u$, $w\leq y$. So, in order to produce the smallest positive solution, we can restrict ourselves to the set $S_{+}$. Then from Dunford and Schwartz \cite[p. 336]{12}, we know that we can find $\{u_n\}_{n\geq 1}\subseteq S_{+}$ such that
$$\inf S_{+}=\inf\limits_{n\geq 1}u_n.$$
We have
\begin{eqnarray}\label{eq91}
& & A_p(u_n)+A(u_n)=N_f(u_n),\ u_n\in [0,w_{+}]\ \mbox{for all}\ n\geq 1,\\
&\Rightarrow& \{u_n\}_{n\geq 1}\subseteq  W^{1,p}_0(\Omega)\ \mbox{is bounded}\nonumber.
\end{eqnarray}
So, we may assume that
\begin{eqnarray}\label{eq92}
u_n\overset{w}{\to}u_{+}\ \mbox{in}\ W^{1,p}_0(\Omega)\ \mbox{and}\ u_n\to u_{+}\ \mbox{in}\ L^r(\Omega)\ \mbox{as}\ n\to\infty.
\end{eqnarray}
Acting on the equation in (\ref{eq91}) with $u_n-u_{+}\in W^{1,p}_0(\Omega)$, passing to the limit as $n\to\infty$ and using (\ref{eq92}), we obtain
\begin{eqnarray}\label{eq93}
& &\lim\limits_{n\to\infty}[\langle A_p(u_n),u_n-u_{+}\rangle+\langle A(u_n),u_n-u_{+}\rangle]=0,\nonumber\\
&\Rightarrow& u_n\to u_{+}\ \mbox{in}\ W^{1,p}_0(\Omega)\ (\mbox{as before, see the proof of Proposition}\ \ref{prop5}).
\end{eqnarray}
So, if in (\ref{eq91}) we pass to the limit as $n\to\infty$ and use (\ref{eq93}), then
$$A_p(u_{+})+A(u_{+})=N_f(u_{+}).$$

From Proposition \ref{prop16}, we have $u_*\leq u_n$ for all $n\geq 1$, hence $u_*\leq u_{+}$ (see (\ref{eq93})) and so $u_{+}\in S_{+}$ and $u_{+}=\inf S_{+}$, that is $u_{+}\in\mbox{int}\, C_{+}$ is the smallest positive solution of problem (\ref{eq1}).

Similarly we produce $v_{-}\in-\mbox{int}\, C_{+}$ the biggest negative solution of problem (\ref{eq1}).
\end{proof}

Now we are ready for the second multiplicity theorem. It generates six nontrivial solutions, all with sign information.
\begin{theorem}\label{th18}
If hypotheses $H_3$ hold, then problem (\ref{eq1}) admits at least six nontrivial solutions
\begin{eqnarray*}
& & u_0,\hat{u}\in\mbox{int}\, C_{+},\ \hat{u}-u_0\in\mbox{int}\, C_{+}\\
& & v_0,\hat{v}\in-\mbox{int}\, C_{+},\ v_0-\hat{v}\in\mbox{int}\, C_{+}\\
&\mbox{and}& y_0,\hat{y}\in C_0^1(\overline{\Omega})\setminus\{0\}\ \mbox{nodal}.
\end{eqnarray*}
\end{theorem}
\begin{proof}
The four nontrivial constant sign solutions
\begin{eqnarray*}
&& u_0,\hat{u}\in\mbox{int}\, C_{+}\ \mbox{and}\  v_0,\hat{v}\in-\mbox{int}C_{+}\\
&&\hat{u}-u_0\in\mbox{int}\, C_{+},\ v_0-\hat{v}\in\mbox{int}\, C_{+}
\end{eqnarray*}
come from Propositions \ref{prop10} and \ref{prop11}.

So, we need to produce the two nodal solutions. By virtue of Proposition \ref{prop17} without any loss of generality, we may assume that $u_0\in\mbox{int}\, C_{+}$ and $v_0\in-\mbox{int}\, C_{+}$ are extremal.

Let $g_*(z,x)$ be the Carath\'eodory function defined by
\begin{eqnarray}\label{eq94}
g_*(z,x)=\left\{  \begin{array}{ll}
  f(z,v_0(z))\ &\mbox{if}\ x<v_0(z)\\
    f(z,x)\ &\mbox{if}\ v_{-}(z)\leq x\leq u_0(z)\\
    f(z,u_0(z))\ &\mbox{if}\ u_0(z)<x.
    \end{array}\right.
\end{eqnarray}
Let $G_*(z,x)=\int\limits_0^x g_*(z,s)ds$ and consider the $C^1$-functional $\tau_*:W^{1,p}_0(\Omega)\to\RR$ defined by
$$\tau_*(u)=\frac{1}{p}\|Du\|_p^p+\frac{1}{2}\|Du\|_2^2-\int\limits_{\Omega}G_*(z,u(z))dz\ \mbox{for all}\ u\in W^{1,p}_0(\Omega).$$
Also, we consider the positive and negative truncations of $g_*(z,\cdot)$, namely the Carath\'eodory functions
$$g_*^{\pm}(z,x)=g_*(z,\pm x^{\pm}).$$
We set $G_*^{\pm}(z,x)=\int\limits_0^x g_*^{\pm}(z,s)ds$ and consider the $C^1$-functionals $\tau_*^{\pm}:W^{1,p}_0(\Omega)\to\RR$ defined by
$$\tau_*^{\pm}(u)=\frac{1}{p}\|Du\|_p^p+\frac{1}{2}\|Du\|_2^2-\int\limits_{\Omega}G_*^{\pm}(z,u(z))dz\ \mbox{for all}\ u\in W^{1,p}_0(\Omega).$$
It is easy to see that
\begin{eqnarray*}
&&K_{\tau_*}\subseteq [v_0,u_0],\ K_{\tau_*^{+}}\subseteq [0,u_0],\ K_{\tau_*^{-}}\subseteq [v_0,0]\\
&&\hspace{4cm}(\mbox{see the proof of Proposition}\ \ref{prop10}).
\end{eqnarray*}
The extremality of $u_0$ and $v_0$, implies that
\begin{eqnarray}\label{eq95}
&&K_{\tau_*}\subseteq [v_0,u_0],\ K_{\tau_*^{+}}=\{0,u_0\},\ K_{\tau_*^{-}}=\{v_0,0\}.
\end{eqnarray}

{\bf Claim.} {\it
$u_0\in\mbox{int}\, C_{+}$ and $v_0\in-\mbox{int}\, C_{+}$ are local minimizers of the functional $\tau_*$.
}

From (\ref{eq94}) it is clear that $\tau_*^{+}$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find $\tilde{u}_0\in W^{1,p}_0(\Omega)$ such that
\begin{eqnarray}\label{eq96}
&&\tau_*^{+}(\tilde{u}_0)=\inf [\tau_*^{+}(u):\ u\in W^{1,p}_0(\Omega)].
\end{eqnarray}
As before (see the proof of Proposition \ref{prop10}), we have
$$\tau_*^{+}(\tilde{u}_0)<0=\tau_*^{+}(0),\ \mbox{hence}\ \tilde{u}_*\neq 0.$$
Since $\tilde{u}_0\in K_{\tau_*^{+}}$ (see (\ref{eq96})), we have $\tilde{u}_0=u_0$ (see (\ref{eq95})). Recall that $u_0\in\mbox{int}\, C_{+}$. Because $\tau_*^{+}|_{C_{+}}=\tau_*|_{C_{+}}$, it follows that $u_0\in\mbox{int}\, C_{+}$ is a local $C_0^1(\overline{\Omega})$-minimizer of $\tau_*$. Invoking Proposition \ref{prop3}, we infer that $u_0\in\mbox{int}\, C_{+}$ is a local $W^{1,p}_0(\Omega)$-minimizer of $\tau_*$.

Similarly for $v_0\in-\mbox{int}\, C_{+}$ using this time the functional $\tau_*^{-}$. This proves the Claim.

Without any loss of generality, we may assume that $\tau_*(v_0)\leq\tau_*(u_0)$ (the analysis is similar, if the opposite inequality holds). By virtue of the Claim, we can find $\rho\in (0,1)$ small ($\rho<\|u_0-v_0\|$) such that
\begin{eqnarray}\label{eq97}
&&\tau_*(v_0)\leq\tau_*(u_0)<\inf[\tau_*(u):\|u-u_0\|=\rho]=m_{\rho}\ (\mbox{see \cite{1}}).
\end{eqnarray}
Since $\tau_*$ is coercive (see (\ref{eq94})), it satisfies the $C$-condition. This fact and (\ref{eq97}), permit the use of Theorem \ref{th1} (the mountain pass theorem). So, we can find $y_0\in W^{1,p}_0(\Omega)$ such that
\begin{eqnarray}\label{eq98}
y_0\in K_{\tau_*}\ \mbox{and}\ m_{\rho}\leq \tau_*(y_0).
\end{eqnarray}
From (\ref{eq97}) and (\ref{eq98}) it follows that $y_0\notin\{v_0,u_0\}$. If we can show that $y_0\neq 0$, then we infer that $y_0$ is nodal. Reasoning as in the proof of Proposition \ref{prop11}, using hypothesis $H_3(vi)$ and Proposition \ref{prop4}, we show that
\begin{eqnarray}\label{eq99}
u_0-y_0\in\mbox{int}\, C_{+}\ \mbox{and}\ y_0-v_0\in\mbox{int}\, C_{+}.
\end{eqnarray}
Since $y_0$ is a critical point of $\tau_*$ of mountain pass type, we have
\begin{eqnarray}\label{eq100}
C_1(\tau_*,y_0)\neq 0.
\end{eqnarray}
Note that $\tau_*|_{[v_0,u_0]}=\varphi|_{[v_0,u_0]}$ (see (\ref{eq94})). From this fact and (\ref{eq99}), via a homotopy invariance argument, as in the proof of Proposition \ref{prop14}, we obtain
\begin{eqnarray}\label{eq101}
& & C_k(\tau_*,y_0)=C_k(\varphi,y_0)\ \mbox{for all}\ k\geq 0,\\
&\Rightarrow& C_1(\varphi,y_0)\neq 0\ (\mbox{see}\ (\ref{eq100})).\nonumber
\end{eqnarray}
But $\varphi\in C^2(W^{1,p}_0(\Omega))$. Hence
\begin{eqnarray}\label{eq102}
C_k(\varphi,y_0)=\delta_{k,1}\ZZ\ \mbox{for all}\ k\geq 0\ (\mbox{see \cite{22}, \cite{21}}).
\end{eqnarray}
From Proposition \ref{prop12}, we know that
\begin{eqnarray}\label{eq103}
&&C_k(\varphi,0)=\delta_{k,d_m}\ZZ\ \mbox{for all}\ k\geq 0,\ \mbox{with}\ d_m\geq 2\\
&&\hspace{5cm} (\mbox{recall}\ m\geq 2).\nonumber
\end{eqnarray}
Comparing (\ref{eq102}) and (\ref{eq103}), we have $y_0\neq 0$, hence $y_0$ is nodal and $y_0\in C_0^1(\overline{\Omega})$ (nonlinear regularity theory). From (\ref{eq101}), (\ref{eq102}) and (\ref{eq103}), we have
\begin{eqnarray}\label{eq104}
C_k(\tau_*,y_0)=\delta_{k,1}\ZZ\ \mbox{for all}\ k\geq 0,
\end{eqnarray}
\begin{eqnarray}\label{eq105}
C_k(\tau_*,0)=\delta_{k,d_m}\ZZ\ \mbox{for all}\ k\geq 0.
\end{eqnarray}
By virtue of the Claim, we have
\begin{eqnarray}\label{eq106}
C_k(\tau_*,u_0)=C_k(\tau_*,v_0)=\delta_{k,0}\ZZ\ \mbox{for all}\ k\geq 0.
\end{eqnarray}
Finally the coercivity of $\tau_*$ implies
\begin{eqnarray}\label{eq107}
C_k(\tau_*,\infty)=\delta_{k,0}\ZZ\ \mbox{for all}\ k\geq 0.
\end{eqnarray}
Suppose $K_{\tau_*}=\{v_0,u_0,0,y_0\}$. Then from (\ref{eq104}), (\ref{eq105}), (\ref{eq106}), (\ref{eq107}) and the Morse relation with $t=-1$ (see (\ref{eq2})), we have
\begin{eqnarray*}
& & 2(-1)^0+(-1)^{d_m}+(-1)^1=(-1)^0,\\
&\Rightarrow& (-1)^{d_m}=0,\ \mbox{a contradiction}.
\end{eqnarray*}
So, there exists $\hat{y}\in K_{\tau_*}$, $\hat{y}\notin\{v_0,u_0,0,y_0\}$. From (\ref{eq95}) it follows that $\hat{y}$ is a second nodal solution and by nonlinear regularity, we have $\hat{y}\in C_0^1(\overline{\Omega})$.
\end{proof}

It is natural to ask what happens if in $H_3(iv)$, $m\geq 1$. In this case the unilateral growth estimate (\ref{eq48}) is not valid. Therefore, we can not generate extremal solutions and so Theorem \ref{th18} fails. Nevertheless, we still have a multiplicity theorem producing five nontrivial solutions, but we are unable to determine the sign of the fifth solution.

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

\smallskip
$H_4$: $f:\Omega\times\RR\to\RR$ is a measurable function such that for a.a. $z\in\Omega$, $f(z,0)=0$, $f(z,\cdot)\in C^1(\RR)$, hypotheses $H_4(i),(ii),(iii),(iv),(v)$ are the same as the corresponding hypotheses $H_3(i),(ii),(iii),(v),(vi)$

$(iv)$ $f_x'(z,0)=\lim\limits_{x\to 0}\frac{f(z,x)}{x}$ uniformly for a.a. $z\in\Omega$ and for some $m\geq 1$, we have
\begin{eqnarray*}
& &f_x'(z,0)\in [\hat{\lambda}_m(2),\hat{\lambda}_{m+1}(2)]\ \mbox{for a.a.}\ z\in\Omega\\
&\mbox{and}& f_x'(\cdot,0)\not\equiv\hat{\lambda}_m(2),\ f_x'(\cdot,0)\not\equiv\lambda_{m+1}(2).
\end{eqnarray*}
\begin{theorem}\label{th19}
If hypotheses $H_4$ hold, then problem (\ref{eq1}) has at least five nontrivial solutions
\begin{eqnarray*}
& & u_0,\hat{u}\in\mbox{int}\, C_{+}\ \mbox{with}\ \hat{u}-u_0\in\mbox{int}\, C_{+},\\
& & v_0,\hat{v}\in-\mbox{int}\, C_{+}\ \mbox{with}\ v_0-\hat{v}\in\mbox{int}\, C_{+}\\
&\mbox{and}& y_0\in C_0^1(\overline{\Omega}).
\end{eqnarray*}
\end{theorem}
\begin{proof}
A careful reading of the proofs of Propositions \ref{prop10} and \ref{prop11}, reveals that by virtue of the UCP, the results remain valid under the new set of hypotheses $H_4$. Therefore we can say that problem (\ref{eq1}) has at least four nontrivial solutions of constant sign
\begin{eqnarray*}
& & u_0,\hat{u}\in\mbox{int}\, C_{+}\ \mbox{with}\ \hat{u}-u_0\in\mbox{int}\, C_{+},\\
& \mbox{and}& v_0,\hat{v}\in-\mbox{int}\, C_{+}\ \mbox{with}\ v_0-\hat{v}\in\mbox{int}\, C_{+}
\end{eqnarray*}
From the proof of Proposition \ref{prop10}, we know that $u_0$ is a minimizer of $\hat{\varphi}_{+}$ (see (\ref{eq50})). From (\ref{eq58}) we have $u_0\in\mbox{int}_{C_0^1(\overline{\Omega})}[0,w_{+}]$. Since $\hat{\varphi}_{+}|_{[0,w_{+}]}=\varphi|_{[0,w_{+}]}$ (see (\ref{eq49})), it follows that $u_0$ is a local $C_0^1(\overline{\Omega})$-minimizer of $\varphi$, hence by Proposition \ref{prop3} it is also a local $W_0^{1,p}(\Omega)$-minimizer of $\varphi$. Similarly we show that $v_0\in-\mbox{int}\, C_{+}$ too is a local minimizer of $\varphi$. Therefore, we have
\begin{eqnarray}\label{eq108}
C_k(\varphi,u_0)=C_k(\varphi,v_0)=\delta_{k,0}\ZZ\ \mbox{for all}\ k\geq 0.
\end{eqnarray}
From the proof of Proposition \ref{prop11}, we know that $\hat{u}\in\mbox{int}\, C_{+}$ is a critical point of $\psi_{+}$ of mountain pass type. Hence
\begin{eqnarray}\label{eq109}
C_1(\psi_{+},\hat{u})\neq 0.
\end{eqnarray}
Since $\hat{u}-u_0\in\mbox{int}\, C_{+}$, via a homotopy invariance argument as in the proof of Proposition \ref{prop14}, we show that
\begin{eqnarray}\label{eq110}
& &C_k(\psi_{+},\hat{u})=C_k(\varphi,\hat{u})\ \mbox{for all}\ k\geq 0,\nonumber\\
&\Rightarrow& C_1(\varphi,\hat{u})\neq 0\ (\mbox{see}\ (\ref{eq109})),\nonumber\\
&\Rightarrow& C_k(\varphi,\hat{u})=\delta_{k,1}\ZZ\ \mbox{for all}\ k\geq 0\ (\mbox{see \cite{22}, \cite{21}}).
\end{eqnarray}
Similarly we show that
\begin{eqnarray}\label{eq111}
C_k(\varphi,\hat{v})=\delta_{k,1}\ZZ\ \mbox{for all}\ k\geq 0.
\end{eqnarray}
From Propositions \ref{prop12} and \ref{prop13}, we have
\begin{eqnarray}\label{eq112}
C_k(\varphi,0)=\delta_{k,d_m}\ZZ\ \mbox{for all}\ k\geq 0,
\end{eqnarray}
\begin{eqnarray}\label{eq113}
C_k(\varphi,\infty)=0\ \mbox{for all}\ k\geq 0.
\end{eqnarray}
Suppose $K_{\varphi}=\{v_0,u_0,\hat{v},\hat{u},0\}$. Then from (\ref{eq108}), (\ref{eq110}), (\ref{eq111}), (\ref{eq112}), (\ref{eq113}) and the Morse relation with $t=-1$ (see (\ref{eq2})), we have
\begin{eqnarray*}
& & 2(-1)^0+2(-1)^1+(-1)^{d_m}=0,\\
&\Rightarrow& (-1)^{d_m}=0,\ \mbox{a contradiction}.
\end{eqnarray*}
So, there exists $y_0\in K_{\varphi}$, $y_0\notin\{v_0u_0,\hat{v},\hat{u},0\}$, hence $y_0$ is a fifth nontrivial solution for problem (\ref{eq1}) and $y_0\in C_0^1(\overline{\Omega})$ (nonlinear regularity theory).
\end{proof}

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


\begin{thebibliography}{99}

\bibitem{1} {\sc S. Aizicovici, N.S. Papageorgiou, and V. Staicu}, {\em Degree Theory for Operators of Monotone Type and Nonlinear Elliptic Equations with Inequality Constraints}, Memoirs Amer. Math. Soc., Vol \textbf{196}, No. \textbf{915} (2008).

\bibitem{2} {\sc S. Aizicovici, N.S. Papageorgiou, and V. Staicu}, {\em On $p$-superlinear equations with a nonhomogeneous differential operator}, Nonlinear Differential Equations Appl. (NoDEA) \textbf{20} (2013), 151-175.

\bibitem{3} {\sc A. Ambrosetti and P. Rabinowitz}, {\em Dual variational methods in critical point theory and applications}, J. Functional Anal. \textbf{14} (1973), 349-381.

\bibitem{4} {\sc D. Arcoya and D. Ruiz}, {\em The Ambrosetti-Prodi problem for the $p$-Laplace operator}, Comm. Partial Differential Equations \textbf{31} (2006), 849-865.

\bibitem{5} {\sc T. Bartsch, Z. Liu, and T. Weth}, {\em Nodal solutions of a $p$-Laplacian equation}, Proc. London Math. Soc. \textbf{91} (2005), 129-152.

\bibitem{6} {\sc R. Benguria, H. Brezis, and E.H. Lieb}, {\em The Thomas-Fermi-von Weizs\"{a}cker theory of atoms and molecules}, Comm. Math. Physics \textbf{79} (1981), 167-180.

\bibitem{7} {\sc S. Cingolani and M. Degiovanni}, {\em Nontrivial solutions for $p$-Laplace equations with right hand side having $p$-linear growth}, Comm. Partial Differential Equations \textbf{30} (2005), 1191-1203.

\bibitem{8} {\sc S. Cingolani and G. Vannella}, {\em Critical groups computations on a class of Sobolev Banach spaces via Morse index}, Ann. Inst. H. Poincar\'e Analyse Non Lin\'eaire \textbf{20} (2003), 271-292.

\bibitem{9} {\sc M. Degiovanni and S. Lancelotti}, {\em Linking over cones and nontrivial solutions for $p$-Laplace equations with $p$-superlinear nonlinearity}, Ann. Inst. H. Poincar\'e Analyse Non Lin\'eaire \textbf{24} (2007), 907-919.

\bibitem{10} {\sc J.I. Diaz and J.E. Saa}, {\em Existence et unicit\'{e} de solutions positives pour certaines \'equations elliptiques quasilin\'eaires}, C.R. Acad. Sci. Paris \textbf{305} (1987), 521-524.

\bibitem{11} {\sc J. Dugundji}, {\em Topology}, Allyn and Bacon Inc, Boston (1966).

\bibitem{12} {\sc N. Dunford and J. Schwartz}, {\em Linear Operators I. General Theory}, Wiley-Interscience, New York (1958).

\bibitem{13} {\sc E. Fadell and P. Rabinowitz}, {\em Generalized cohomological index theories for Lie group actions with an application to bifurcation questions in Hamiltonian systems}, Invent. Math. \textbf{45} (1978), 139-174.

\bibitem{14} {\sc M. Filippakis, A. Kristaly, and N.S. Papageorgiou}, {\em Existence of five nonzero solutions with constant sign for a $p$-Laplacian equation}, Discrete Cont. Dyn. Systems \textbf{24} (2009), 405-440.

\bibitem{15} {\sc L. Gasinski and N.S. Papageorgiou}, {\em Nonlinear Analysis}, Chapman, Hall/CRC, Boca Raton, Fl. (2006).

\bibitem{16} {\sc L. Gasinski and N.S. Papageorgiou}, {\em Multiple solutions for nonlinear coercive problems with a nonhomogeneous differential operator and a nonsmooth potential}, Set. Valued Var. Anal. \textbf{20} (2012), 417-443.

\bibitem{17} {\sc L. Iturriaga, E. Massa, J. Sachez, and P. Ubilla}, {\em Positive solutions of the $p$-Laplacian involving a superlinear nonlinearity with zeros}, J. Differential Equations \textbf{248} (2010), 309-327.

\bibitem{18} {\sc O. Ladyzhenskaya and N. Uraltseva}, {\em Linear and Quasilinear Elliptic Equations}, Academic Press, New York (1968).

\bibitem{19} {\sc G. Lieberman}, {\em Boundary regularity for solutions of degenerate elliptic equations}, Nonlinear Anal. \textbf{12} (1988), 1203-1219.

\bibitem{20} {\sc N.S. Papageorgiou and S. Kyritsi}, {\em Handbook of Applied Analysis}, Springer, New York (2009).

\bibitem{21} {\sc N.S. Papageorgiou and V.D. R\u adulescu}, {\em Qualitative phenomena for some classes of quasilinear elliptic equations with multiple resonance}, Appl. Math. Optim. {\bf 69} (2014), 393-430.

\bibitem{22} {\sc N.S. Papageorgiou and G. Smyrlis}, {\em On nonlinear nonhomogeneous resonant Dirichlet equations}, Pacific J. Math. \textbf{264} (2013), 421-453.

\bibitem{23} {\sc P. Pucci and J. Serrin}, {\em The Maximum Principle}, Birkh\"{a}user, Basel (2007).

\bibitem{24} {\sc M. Sun}, {\em Multiplicity of solutions for a class of quasilinear elliptic equations at resonance}, J. Math. Anal. Appl. \textbf{386} (2012), 661-668.

\end{thebibliography}

\end{document}

%------------------------------------------------------------------------------
% End of tmna-l2e.tex
%------------------------------------------------------------------------------ 