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%\title[Multiplicity of solutions for nonhomogeneous elliptic equations]
%{Multiplicity of solutions for nonhomogeneous elliptic equations}
\title[Elliptic equations with multiple resonance]
{Qualitative phenomena for some classes of quasilinear elliptic equations with multiple resonance}
\author{Nikolaos S. Papageorgiou and Vicen\c{t}iu D. R\u{a}dulescu}
\address[Nikolaos S. Papageorgiou]{National Technical University, Department of Mathematics, Zografou Campus, Athens 15780, Greece}
\email{npapg@@math.ntua.gr}
\address[Vicen\c tiu D. R\u{a}dulescu]{Institute of Mathematics ``Simion Stoilow" of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania \& Department of Mathematics, University of Craiova,  Street A.I. Cuza No. 13, 200585 Craiova,
Romania} \email{vicentiu.radulescu@@imar.ro}

\keywords{Strong comparison principle, nonlinear maximum principle, critical group, nodal and constant sign solutions, resonant equations, Lyapunoff-Schmidt reduction method}

\thanks{{\em 2010 Mathematics Subject Classification.} Primary 35J20; Secondary 35J60, 35J92,58E05}

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\begin{document}


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\begin{abstract}
We consider nonlinear nonhomogeneous Dirichlet problems driven by the sum of a $p$-Laplacian and a Laplacian. The hypotheses on the reaction term incorporate problems resonant at both $\pm \infty$ and zero. We consider both cases $p>2$ and $1<p<2$ (singular case) and we prove four multiplicity theorems producing three or four nontrivial solutions. For the case $p>2$ we provide precise sign information for all the solutions. Our approach uses critical point theory, truncation and comparison techniques, Morse theory and the Lyapunoff-Schmidt reduction method.
\end{abstract}
%\footnote{Corresponding author}
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\maketitle
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\section{Introduction}
Let $\Omega \subseteq \RR^{N}$ be a bounded domain with a $C^{2}$-boundary $\partial \Omega$. In this paper, we study the following nonlinear nonhomogeneous Dirichlet problem
\begin{equation}
\label{eq:1}
	- \Delta_{p}u(z)-\Delta u (z)=f(z,u(z)) \quad\mbox{in}\ \Omega,\quad u|_{\partial \Omega} = 0.
 \end{equation}
Here $\Delta_{p}$ denotes the $p$-Laplace differential operator defined by
\begin{equation}
\label{eq:2}
	\Delta_{p} u=\mbox{div}\,(||Du||^{p-2}Du)\ \mbox{for\ all}\ u\ \in W^{1,p}_{0}(\Omega)\ (1<p<\infty).
\end{equation}

We consider separately the cases $p>2$ and $1<p<2$ and prove multiplicity theorems for both of them. A similar study was conducted recently by Sun \cite{ref33} but under stronger conditions on the reaction $f(z,x)$ and the results proved there are weaker than ours. We stress that in (\ref{eq:1}) the differential operator is nonhomogeneous and this is the source of difficulties which require new techniques. Such nonhomogeneous elliptic equation were investigated recently by Cingolani \& Degiovanni \cite{ref11}, Cingolani \& Vannella \cite{ref13}, and He \& Li \cite{ref20}, who proved existence theorems. Multiplicity theorems can be found in Papageorgiou \& Smyrlis \cite{ref30}. We mention that equations like (\ref{eq:1}) (we call them $(p,2)$--equations for short) are important in quantum physics in the search for solitons, see Benci, D'Avenia, Fortunato, \& Pisani \cite{ref6}.


Compared with \cite{ref30} our setting here is different. In \cite{ref30} the authors deal only with the case $2<p<\infty$ and assume that the reaction $f(z,\cdot)$ exhibits a kind of oscillatory behavior near zero and so the geometry is different and leads to the existence of more nontrivial solutions of constant sign.

Compared with \cite{ref33}, our results here are considerably stronger. In \cite{ref33}, for the case $p>2$, the reaction $f(z,x)$ is a $C^1$--function on $\overline\Omega\times\RR$, which satisfies stronger asymptotic conditions (see ($f_2$)). The author proves a multiplicity theorem (Theorem 1.1) producing three solutions. However, no nodal solution is obtained. For the case $1<p<2$ only an existence theorem is proved (Theorem 1.2) under the hypothesis that $f\in C(\overline\Omega\times\RR)$.

Our approach uses critical point theory, combined with suitable truncation and comparison techniques, Morse theory and in the case where $1<p<2$, we also employ the so-called Lyapunoff-Schmidt reduction technique. In the next section for the convenience of the reader we recall some of the main mathematical tools which we will use in this paper.

\section{Mathematical Background}

Let $X$ be a Banach space and $X^{*}$ its topological dual. By $\langle\cdot,\cdot\rangle$ we denote the duality brackets for the pair $(X^{*},X)$. Given $\varphi \in C^{1}(X)$, we say that $\varphi$ satisfies the ``Cerami condition" (the ``C-condition" for short), if the following is true:


\begin{center}
``Every sequence $\{x_{n}\}_{n\geq 1} \subseteq X$ such that $\{\varphi (x_{n})\}_{n\geq 1} \subseteq \RR$ is bounded and
\end{center}

\begin{center}
$(1+||x_{n}||) \varphi'(x_{n}) \rightarrow 0 \in X^{*}$ as $n \rightarrow \infty$,
\end{center}

\begin{center}
then $\{x_{n}\}_{n\geq 1}$ admits a strongly convergent subsequence".
\end{center}

This compactness-type condition is in general weaker than the more common Palais-Smale condition. Nevertheless, the C-condition suffices to prove a deformation theorem and from it derive the minimax theory of certain critical values of $\varphi$ (see, for example Gasinski \& Papageorgiou \cite{ref18}, Kristaly, V. R\u{a}dulescu \& Varga \cite{ref22} and R\u adulescu \cite{ref32}). In particular, we have the following result known in the literature as the ``mountain pass theorem".

\begin{theorem}
\label{th:2.1}
 Assume $\varphi \in C^{1}(X)$ satisfies the C-condition, $x_{0}$, $x_{1}\in X$, $||x_{1}-x_{0}||>\rho$
$$\max\{\varphi(x_{0}), \varphi(x_{1})\}\ <\inf\left[\varphi(x):||x-x_{0}||=\rho\right]=\eta_{\rho}\,, $$
and $C=\inf_{\gamma\in \Gamma}\max_{t\in [0,1]} \varphi (\gamma(t))$, where $\Gamma = \left\{\gamma \in C([0,1],X):\gamma(0)=x_{0}, \gamma(1)=x_{1}\right\}$.

Then\ $C\ \geq \eta_{\rho}$ and $C$ is a critical value of $\varphi$.
\end{theorem}

In the analysis of problem (\ref{eq:1}), in addition to the Sobolev spaces $W^{1,p}_{0}(\Omega)$ and $H^{1}_{0}(\Omega)$, we will also use the Banach space
$C^{1}_{0}(\overline{\Omega})=\left\{u \in C^{1}(\overline{\Omega}):u |_{\partial \Omega}=0\right\}$.

This is an ordered Banach space with positive cone $C_{+}=\left\{u \in C^{1}_{0}(\overline{\Omega}):u\geq\ 0\ \mbox{in}\ \overline{\Omega}\right\}$. This cone has a nonempty interior given by
$$\mbox{int}\, C_{+}=\left\{  u \in C_{+}:\ u(z)>0\ \mbox{for all}\ z\in\Omega\ \mbox{and}\ \frac{\partial u}{\partial n} (z)<0\ \mbox{for\ all}\ z \in \partial \Omega \right\},$$
where $n$ stands for the outward unit normal on $\partial \Omega$.

Suppose $f_{0}:\Omega \times \RR \rightarrow \RR$ is a Carath\'eodory function (that is, for all $x \in \RR$, the map $z \longmapsto f_0(z,x)$ is measurable and for a.a. $z \in \Omega$, the map $x \longmapsto f_0(z,x)$ is continuous) with subcritical growth in $x \in \RR$, that is,
$$|f_{0}(z,x)|\leq a_0(z)(1+|x|^{r-1})\ \mbox{for\ a.a.}\ z \in \Omega\ \mbox{and all}\ x \in \RR,$$
\begin{eqnarray}
\mbox{with}\ a_{0} \in L^{\infty} (\Omega)_{+}\ \mbox{and}\ 1<r<p^{*}= \left\{
	\begin{array}{ll}
	\frac{Np}{N-p} & \mbox{if}\ N>p\\
	+\infty & \mbox{if}\ N\leq p.
	\end{array} \right. \nonumber
\end{eqnarray}

We set $F_{0}(z,x)=\int_{0}^{x}f_{0}(z,s)\ ds$ and consider the $C^{1}$-functional $\Psi_{0}:X \rightarrow \RR$ defined by
$$\Psi_{0}(u)=\frac{1}{p}||Du||^{p}_{p}+ \frac{1}{2}||Du||^{2}_{2}-\int_{\Omega}F_{0}(z,u(z))\ dz\qquad \mbox{for\ all}\ u \in X,$$
where $X=W^{1,p}_{0}(\Omega)$ if $2<p<\infty$ and\ $X=H^{1}_{0}(\Omega)$ if $p \in (1,2)$.

The next result is a special case of Proposition 2 of Aizicovici, Papageorgiou \& Staicu \cite{ref1} and essentially it is a consequence of the nonlinear regularity results of Ladyzhenskaya \& Uraltseva \cite{ref23} (p.~286) and  Lieberman \cite[p. 320]{ref24}.

\begin{proposition}
\label{prop:2.2}
 Assume that $u_{0} \in W^{1,p}_{0}(\Omega)$ is a local $C^{1}_{0}(\overline\Omega)$-minimizer of $\Psi_{0}$, that is, there exists $\rho_{0}>0$ such that
$\Psi_{0}(u_{0})\leq \Psi_{0}(u_{0}+h)$ for\ all $h \in C^{1}_{0}(\overline{\Omega})$ with $||h||_{C^{1}_{0}(\overline{\Omega})}\leq \rho_{0}.$

Then\ $u_{0} \in C^{1, \beta}_{0}(\overline{\Omega})$ with $\beta \in (0,1)$ and\ $u_0$ is\ a\ local\ $X$-minimizer\ of $\Psi_{0}$, that is,\ there\ exists\ $\rho_{1}>0$ such that
$\Psi_{0}(u_{0})\leq \Psi_{0}(u_{0}+h)$ for\ all $h \in X$ with $||h||_{X} \leq \rho_{1}.$
\end{proposition}

\textbf{Remark.} We should mention that first such a result was proved by Brezis \& Nirenberg \cite{ref8} for semilinear problems (that is, $p=2$).


Next let $h,\ g \in L^{\infty}(\Omega)$. We write $h \prec g$, if for every compact $K \subseteq \Omega$ we can find $\epsilon=\epsilon (K)>0$ such that
$$h(z)+\epsilon \leq g(z)\ \mbox{for\ a.a.}\ z \in K.$$


Note that if $h,\ g \in C(\Omega)$ and $h(z)<g(z)$ for all $z \in \Omega$, then $h \prec g$.

The next proposition is essentially due to Arcoya \& Ruiz \cite[Proposition 2.6]{ref3}. The presence of the extra linear term $-\Delta u$ does not affect their proof.

\begin{proposition}
	\label{prop:2.3}
		Assume that $\xi \geq 0,\ h,\ g \in L^{\infty}(\Omega),\ h \prec g$. Let $u,\ v \in C^{1}_{0}(\overline{\Omega})$ with $v \in {\mathrm int}\, C_{+}$ be solutions of
		$$-\Delta_{p}u(z)-\Delta u(z)+\xi |u(z)|^{p-2}u(z)=h(z)\ \mbox{in}\ \Omega$$
		$$-\Delta_{p}v(z)-\Delta v(z)+\xi v(z)^{p-1} = g(z)\ \mbox{in}\ \Omega\,.$$
Then		$v-u\ \in {\mathrm int}\, C_{+}$.
\end{proposition}

We will also need some basic facts concerning the first eigenvalue of $(-\Delta_{p},\,W^{1,p}_{0}(\Omega))$ for $1<p<\infty$. So, we consider the following nonlinear eigenvalue problem:
$$-\Delta_{p}u(z)= \hat{\lambda}|u(z)|^{p-2}u(z)\ \mbox{in}\ \Omega ,\ u|_{\partial \Omega}=0\,.$$

A number $\hat{\lambda} \in \RR$ is an eigenvalue of $(-\Delta_{p},\, W^{1,p}_{0}(\Omega))$ if the above problem  admits a nontrivial solution $\hat{u} \in W^{1,p}_{0}(\Omega)$ which is an eigenfunction corresponding to the eigenvalue $\hat{\lambda}$. We know that there exists a smallest eigenvalue $\hat{\lambda}_{1}(p)$ with the following properties: (i) $ \hat{\lambda}_{1}(p) > 0$; (ii)
$\hat{\lambda}_{1}(p)$ is isolated, that is, there exists $\epsilon > 0$ such that the interval $\left[\hat{\lambda}_{1}(p),\ \hat{\lambda}_{1}(p)+ \epsilon \right)$ contains no other eigenvalue of $(-\Delta_{p},\, W^{1,p}_{0}(\Omega))$; (iii)
$ \hat{\lambda}_{1}(p)$ is simple, that is, if $u,\, v$ are eigenfunctions corresponding to the eigenvalue $\hat{\lambda}_{1}(p)$, then $u=\xi v$ for some $\xi \in \RR$;
(iv)
the eigenvalue $\hat{\lambda}_{1}(p)>0$ admits the following variational characterization:
\begin{equation}
	\label{eq:3}
		\hat{\lambda}_{1}(p)=\inf \left\{ \frac{||Du||^{p}_{p}}{||u||_{p}^{p}}: u \in W^{1,p}_{0}(\Omega),\ u\neq 0 \right\}.
\end{equation}
Moreover, in
 relation (\ref{eq:3}) the infimum is realized on the one-dimensional eigenspace corresponding to $\hat{\lambda}_{1}(p)$. From (\ref{eq:3}) it is clear that the elements of this eigenspace do not change sign. By $\hat{u}_{1,p}$ we denote the $L^{p}$-normalized (that is, $||\hat{u}_{1,p}||_{p}=1$) positive eigenfunction corresponding to $\hat{\lambda}_{1}(p)$. In fact $\hat{\lambda}_{1}(p)>0$ is the only eigenvalue with eigenfunctions of constant sign. All the other eigenvalues have nodal (that is, sign-changing) eigenfunctions. The nonlinear regularity theory (see \cite{ref23}, \cite{ref24}) implies that $\hat{u}_{1,p}\in C_{+} \backslash \{0\}$ and the nonlinear maximum principle of Vazquez \cite{ref35} says that $\hat{u}_{1,p} \in \mbox{int}\, C_{+}$.

When $p=2$ (linear eigenvalue problem), then the spectrum of $(-\Delta,\, H^{1}_{0}(\Omega))$ is a sequence $\left\{\hat{\lambda}_{k}(2)\right\}_{k\geq 1} \subseteq (0,+\infty)$ of eigenvalues such that $\hat{\lambda}_{k}(2) \rightarrow +\infty$ as $k \rightarrow +\infty$. By $E(\hat{\lambda}_{k}(2))$ we denote the eigenspace corresponding to $\hat{\lambda}_{k}(2)$, We have the following orthogonal direct sum decomposition $H^{1}_{0}(\Omega)= \overline{{\underset{\mathrm{k\geq 1}}\oplus}\ E(\hat{\lambda}_{k}(2))}$. These eigenspaces have the so-called ``Unique Continuation Property" (UCP for short), namely if $u \in E(\hat{\lambda}_{k}(2))$ and $u$ vanishes on a set of positive Lebesgue measure, then $u=0$. The eigenvalues $\left\{\hat{\lambda}_{k}(2)\right\}_{k \geq 1}$ have the following variational characterizations:
\begin{eqnarray}
     \hat{\lambda}_{1}(2) & = & \inf \left\{ \frac{||Du||^{2}_2}{||u||^{2}_{2}}: u \in H^{1}_{0}(\Omega),\ u\neq 0\right\} \label{eq:4} \\
 \hat{\lambda}_{n}(2) & = & \inf \left\{ \frac{||Du||^{2}_{2}}{||u||^{2}_{2}}: u \in \overline{{\underset{\mathrm{k\geq n}}\oplus}\ E(\hat{\lambda}_{k}(2))},\ u\neq 0\right\}\nonumber \\
                          & = & \sup \left\{ \frac{||Du||^{2}_{2}}{||u||^{2}_{2}}: u \in \overset{n}{\underset{\mathrm{k =1}}\oplus}\ E(\hat{\lambda}_{k}(2)),\ u\neq 0 \right\}\ \mbox{for}\ n\geq 2. \label{eq:5}
\end{eqnarray}

The infimum in (\ref{eq:4}) and both the infimum and the supremum in (\ref{eq:5}), are realized on the corresponding eigenspaces $E(\hat{\lambda}_{n}(2))$. Similar remarks can be made for a weighted version of the linear eigenvalue problem. So, let $m \in L^{\infty}(\Omega)$, $m \geq 0$, $m \neq 0$ and consider the  following weighted linear eigenvalue problem
\begin{equation}
	\label{eq:6}	
		-\Delta u(z)=\widetilde{\lambda} m (z)u(z)\quad \mbox{in}\ \Omega,\ u|_{\partial \Omega} = 0\,.	
\end{equation}

Problem (\ref{eq:6}) has a sequence $\left\{ \widetilde{\lambda}_{k} (2,m) \right\}_{k\geq 1}$ of eigenvalues such that $\widetilde{\lambda}_{k}(2, m) \rightarrow +\infty$ as $k \rightarrow +\infty$. These eigenvalues still have the unique continuation property and have variational characterizations analogous to (\ref{eq:4}) and (\ref{eq:5}), in terms of the Rayleigh quotient $\frac{||Du||^{2}_{2}}{\int_{\Omega} mu^{2}\ dz}$. As a consequence of these variational characterizations and of the UCP, we have the following monotonicity of the eigenvalues $\widetilde{\lambda}_{k}(2,m)$ with respect to the weight function $m \in L^{\infty}(\Omega)_{+}$.

\begin{proposition}
	\label{prop:2.4}
		If $m_1,\, m_2 \in L^{\infty}(\Omega)$, $0\leq m_1 (z) \leq m_2 (z)$ a.e. in\ $\Omega$, $m_1 \neq 0$,  $m_1\neq m_2,$
		then\ $\widetilde{\lambda}_{k}(2,m_2)< \widetilde{\lambda}_{k}(2,m_1)$, for all $k\geq 1$.
\end{proposition}

Next, let us recall some basic definitions and facts from Morse theory especially concerning critical groups. So, let $H_k(Y_{1},\, Y_{2})$  denote the $k$th relative singular homology group with integer coefficients for the pair $(Y_{1},\, Y_{2})$. For all integers $k<0$, we have $H_{k}(Y_{1},\, Y_{2})=0$.

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

The critical groups of $\varphi$ at an isolated critical point $x \in X$ of $\varphi$ with $\varphi(x)=c$  (that is, $x \in K^{c}_{\varphi}$), are defined by
$$C_{k}(\varphi,x)=H_{k}(\varphi^{c}\cap {\mathcal U}, \varphi^{c}\cap {\mathcal U} \backslash \{x\})\qquad \mbox{for\ all}\ k\geq 0,$$
with ${\mathcal U}$ being a neighborhood of $x$ such that $K_{\varphi}\cap \varphi^{c}\cap {\mathcal U} = \{x\}$. The excision property of singular homology implies that the above definition of critical groups is independent of the particular choice of the neighborhood ${\mathcal U}$.

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
$$C_{k}(\varphi, \infty)= H_{k}(X,\varphi^{c})\qquad \mbox{for\ all}\ k\geq 0\quad (\mbox{see\ Bartsch \&\  Li}\ \cite{ref5}).$$
The second deformation theorem (see, for example, Gasinski \& Papageorgiou \cite[p. 628]{ref18}), implies that the above definition of critical groups at infinity, is independent of the particular choice of the level $c< \inf \varphi(K_{\varphi})$.

Suppose that $K_{\varphi}$ is finite. We define
\begin{eqnarray}
	M(t,x) 					 & = & \sum_{k\geq 0} \mbox{rank}\, C_{k}(\varphi,x)t^{k}\qquad \mbox{for\ all}\ t\in \RR \ \mbox{and for all}\ x\in K_{\varphi} \nonumber \\
	P(t, \infty) & = & \sum_{k\geq 0} \mbox{rank}\, C_{k}(\varphi, \infty)t^{k}\qquad \mbox{for\ all}\ t \in \RR. \nonumber
\end{eqnarray}

The Morse relation says that
\begin{equation}
	\label{eq:7}
		\sum_{x \in K_{\varphi}} M(t, x)=P(t,\infty)+(1+t)Q(t)\qquad \mbox{for\ all}\ t \in \RR,
\end{equation}
where $Q(t)=\sum_{k \geq 0} \beta_{k}t^{k}$ is a formal series in $t \in \RR$ with nonnegative integer coefficients (see Chang \cite[p. 337]{changk} and Mawhin \& Willem \cite[p. 184]{mawhinj}).

For $r \in (1,+\infty)$, let $A_{r}:W^{1,r}_{0}(\Omega)\rightarrow W^{1,r}_{0}(\Omega)^{*}=W^{-1,r'}(\Omega)$ $(\frac{1}{r}+\frac{1}{r'}=1)$ be the nonlinear map defined by
\begin{equation}
	\label{eq:8}
		\left\langle A_{r}(u),y \right\rangle = \int_{\Omega}||Du||^{r-2}(Du,Dy)_{\RR^{N}}dz\qquad \mbox{for all}\ u,y \in W^{1,r}_{0}(\Omega).
\end{equation}

If $r=2$, then $A_{2}=A \in $$\mathcal{L}$ $(H^{1}_{0}(\Omega),\ H^{-1}(\Omega))$. Concerning the map $A_{r}$ we can state the following well-known result (see Aizicovici, Papageorgiou \&  Staicu \cite{aizi}).

\begin{proposition}
	\label{prop:2.5}
	Let $ A_{r}:W^{1,r}_{0}(\Omega)\rightarrow W^{-1,r'}(\Omega)$ be the nonlinear map defined by (\ref{eq:8}).	 Then $A_{r}$ is continuous, strictly monotone (hence, maximal monotone) and of type $(S)_{+}$.
\end{proposition}

We recall (see Brezis \cite{brethese}) that if $X$ is a real Banach space and $A:X\ri X^*$ is a nonlinear operator, then $A$ is said to be of type $(S)_+$ if for any sequence $\{u_n\}$ converging weakly to $u$ in $X$ and $\limsup_{n\ri\infty} \left\langle A(u_{n}),u_{n}-u \right\rangle \leq 0$, then\ $u_{n} \rightarrow u$.




We will also make use of some results of Cingolani \& Vannella \cite{ref12, ref13}, which we recall here for easy reference. In those works, the authors showed that $(p,2)$-equations can be embedded in a Hilbert space setting and this makes possible the use of Morse theoretic methods. So, let $f(z,x)$ be a measurable function such that $f(z,\cdot )\in C^1(\RR)$ for a.a. $z\in\Omega$ and assume that
$$|f_x'(z,x)|\leq a(z)(1+|x|^{r-1})\quad\mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\in\RR,$$
with $a\in L^\infty (\Omega)_+$ and $2<p\leq r<p^*$. We set $F(z,x)=\int_0^xf(z,s)ds$ and consider the $C^2$-functional $\varphi:W_0^{1,p}(\Omega)\ri\RR$ defined by
$$\varphi (u)=\frac 1p\,\| Du\|^p_p+\frac 12\,\|Du\|^2_2-\int_\Omega F(z,u(z))dz\quad\mbox{for all}\ u\in W_0^{1,p}(\Omega).$$
We have that for all $u,\, y,\, v\in W_0^{1,p}(\Omega)$
$$\begin{array}{ll}\di \langle\varphi''(u)y,v\rangle &\di =\int_\Omega (1+\| Du\|^{p-2})(Dy,Dv)_{\RR^N}dz\\ &\di +(p-2)\int_\Omega \|Du\|^{p-4}
(Du,Dy)_{\RR^N}(Du,Dv)_{\RR^N}dz-\int_\Omega f_x'(z,u)yvdz.\end{array}$$

The nonlinear regularity theory implies that if $u_0\in K_\varphi$, then $u_0\in C^1_0(\overline\Omega)$ (see \cite{ref20, ref21}). Therefore
$$b=\|Du_0\|^{(p-4)/2}Du_0\in L^\infty (\Omega,\RR^N).$$
Let $H_b$ be the closure of $C^\infty_c(\Omega)$ under the inner product
$$(y,v)_b=\int_\Omega\left[(1+\|b\|^2)(Dy,Dv)_{\RR^N}+(p-2)(b,Dy)_{\RR^N}(b,Dv)_{\RR^N}\right]dz.$$
Let $\|\,\cdot\,\|_b$ be the corresponding Hilbert norm, which is evidently equivalent to the Sobolev norm
$\|\,\cdot\,\|_{H^1_0(\Omega)}$. Therefore, we have that $W_0^{1,p}(\Omega)$ is embedded continuously into $H_b$.
Defining $L_b\in {\mathcal L}(H_b,H_b^*)$ by
$$\langle L_b(u),v\rangle = (u,v)_b-\int_\Omega f_x'(z,u_0)uvdz\quad\mbox{for all}\ u,v\in H_b,$$
we note that $L_b$ is a Fredholm operator of index zero and in fact is the extension of $\varphi''(u_0)$ on $H_b$. We consider the orthogonal direct sum decomposition
$$H_b=H^-\oplus H^0\oplus H^+,$$
where $H^-$, $H^0$, $H^+$ are respectively the negative, null and positive subspaces according to the spectral decomposition of $L_b$. The spaces $H^-$ and $H^0$ are finite dimensional. Since $u_0\in C^1_0(\overline\Omega)$, from standard regularity theory, we have
$$H^-\oplus H^0\subseteq W_0^{1,p}(\Omega)\cap L^\infty (\Omega)\,.$$
Set $V=H^-\oplus H^0$ and $W=H^+\cap W_0^{1,p}(\Omega)$. Then
$$W_0^{1,p}(\Omega)=V\oplus W\quad\mbox{and}\quad \langle\varphi''(u_0)y,y\rangle\geq\beta\,\|y\|^2_b\ \ \mbox{for all}\ y\in W,$$
for some $\beta>0$ (see Cingolani \& Vannella \cite{ref12}).




Hereafter, by $||\cdot||$ we denote the norm of $W^{1,p}_{0}(\Omega)$, where $1<p<\infty$. By virtue of the Poincar\'e inequality, we have
$||u||=||Du||_{p}$ for all $u \in W^{1,p}_{0}(\Omega)$.

By $||\cdot||$ we also denote the norm of $\RR^{N}$. No confusion is possible, since it will always be clear from the context which norm is used.

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

Finally, if $h:\Omega \times \RR \rightarrow\RR$ is a measurable function (for example, a Carath\'eodory function), then we define
$$N_{h}(u)(\cdot)=h(\cdot,u(\cdot))\ \mbox{for all}\ u \in W^{1,p}_{0}(\Omega)\,.$$
We denote by $|\cdot|_{N}$ the Lebesgue measure on $\RR^{N}$.

\section{The Case $2<p<\infty$}

Throughout this section we assume that $2<p<\infty$.

Let $ f: \Omega \times \RR \rightarrow \RR$ be  a Carath\'eodory function such that $f(z,0)=0$ for a.a. $ z \in \Omega$.
We impose the following conditions on the reaction $f(z,x)$:

\medskip
$H_{1}:\  (i)\ |f(z,x)|\leq a(z)(1+|x|^{r-1})\ \mbox{for\ a.a.}\ z \in \Omega\ \mbox{and for all}\ x \in \RR,\ \mbox{with}\ a \in L^{\infty}(\Omega)_{+}\ \mbox{and}\
			 p\leq r<p^{*}$;

\smallskip
			$ (ii)\ \mbox{if}\ F(z,x)= \int_{0}^{x} f(z,s)ds,\ \mbox{then}\ \limsup_{x\rightarrow \pm \infty} \frac{pF(z,x)}{|x|^p} \leq \hat{\lambda}_{1}(p)\ \mbox{uniformly\ for\
			 a.a.}\ z \in \Omega\ \mbox{and\ there\ exists}\ \xi>0\ \mbox{such \ that}\ f(z,x)x-pF(z,x)\geq-\xi\ \mbox{for\ a.a.}\
		 z \in \Omega \ \mbox{and for all}\ x \in \RR$;

\smallskip
			 $(iii)\ \mbox{there\ are\ an\ integer}\ m\geq 2,\ \delta_{0}>0\ \mbox{and}\ \eta \in L^{\infty}(\Omega)\ \mbox{with}\ \eta(z)\geq \hat{\lambda}_{m}(2)
		\	  \mbox{a.e.\ in}\ \Omega, \eta \neq \hat{\lambda}_{m}(2)\ \mbox{such that}\
			 \eta(z)x^{2}\leq f(z,x)x\leq\hat{\lambda}_{m+1}(2)x^{2}\ \mbox{for\ a.a.}\ z \in \Omega\  \mbox{and for}\ |x|\leq \delta_{0}$;

\smallskip
			$ (iv)\ \mbox{for\ every}\ \rho > 0,\ \mbox{there\ exists}\ \xi_{\rho}>0\ \mbox{such that\ for\ a.a.}\ z \in \Omega,\ x \longmapsto f(z,x)+
			 \xi_{\rho}|x|^{p-2}x\ \mbox{is\ nondecreasing\ in}\ \left[-\rho, \rho\right]$.

\medskip
\textbf{Remarks:} Hypothesis $H_{1}(ii)$ allows for resonance to occur at $\pm \infty$ with respect to the principal eigenvalue $\hat{\lambda}_{1}(p)$. Such resonant $p$--Laplacian equations were studied by Jiu \& Su \cite{ref21}, Liu \& Liu \cite{ref25}, Liu \& Su \cite{ref27} and Zhang, Li, Liu \& Feng \cite{ref36}, using the additional condition that
$$\lim_{x\rightarrow \pm \infty} \left[ f(z,x)x-pF(z,x) \right]=+\infty\ \mbox{uniformly\ for\ a.a.}\ z \in \Omega\,.$$

Evidently our hypothesis $H_{1}(ii)$ is weaker. Hypothesis $H_{1}(iii)$ implies that we can have resonance at zero with respect to $\hat{\lambda}_{m+1}(2).$ So, we have a ``double resonance" situation. Clearly, hypothesis $H_{1}(iv)$ is much weaker than assuming the monotonicity of $f(z,\cdot)$.

We consider the positive and negative truncations of $f(z,\cdot)$, namely
$$f_{\pm}(z,x)=f(z,\pm x^{\pm}).$$

Both are Carath\'eodory functions. We set $F_{\pm}(z,x)=\int_{0}^{x}f_{\pm}(z,s)ds$ and consider the $C^{1}$-functionals $\varphi,\,\varphi_{\pm}:W^{1,p}_{0}(\Omega)\rightarrow \RR$ defined by
$$\varphi (u) = \frac{1}{p}||Du||^{p}_{p}+\frac{1}{2}||Du||^{2}_{2}-\int_{\Omega} F(z,u(z)) dz\qquad \mbox{for all}\ u \in W^{1,p}_{0}(\Omega)$$
$$\varphi_{\pm}(u)=\frac{1}{p}||Du||^{p}_{p}+\frac{1}{2}||Du||^{2}_{2}-\int_{\Omega} F_{\pm}(z,u(z))dz\qquad \mbox{for all}\ u \in W^{1,p}_{0}(\Omega).$$



\begin{proposition}
	\label{prop:3.1}
		If hypotheses $H_{1}$ hold, then the functionals $\varphi$ and $\varphi_{\pm}$ are coercive.
\end{proposition}

\textit{Proof.} We do the proof for $\varphi_{+}$, the proofs for $\varphi_{-}$ and $\varphi$ being similar.

Note that for a.a. $z \in \Omega$ and all $x>0$, we have
\begin{eqnarray*}
	\frac{d}{dx} \frac{F_{+}(z,x)}{x^{p}} &   =  & \frac{f_{+}(z,x)x^{p}-px^{p-1}F_{+}(z,x)}{x^{2p}}\\
																				&   =  & \frac{f_{+}(z,x)x-pF_{+}(z,x)}{x^{p+1}}\\
																				& \geq & - \frac{\xi}{x^{p+1}}\quad \mbox{(see\ hypothesis\ $H_{1}(ii)$)}
\end{eqnarray*}
$$\Rightarrow \frac{F_{+}(z,x)}{x^{p}}- \frac{F_{+}(z,u)}{u^{p}} \geq \frac{\xi}{p} \left[ \frac{1}{x^{p}} -\frac{1}{u^{p}} \right]\ \mbox{for\ a.a.}\ z \in \Omega\ \mbox{ and fo all}\ x\geq u \geq 0. $$

Let $x \rightarrow + \infty$. Then by virtue of hypothesis $H_{1}(ii)$, we have
\begin{eqnarray}
	 && \frac{\hat{\lambda}_{1}(p)}{p} - \frac{F_{+}(z,u)}{u^{p}} \geq - \frac{\xi}{p} \frac{1}{u^{p}}\ \mbox{for\ a.a.}\ z \in \Omega\ \mbox{and for all}\ u>0 \nonumber\\
  \Rightarrow &&	pF_{+}(z,u)-\hat{\lambda}(p)(u^{+})^{p} \leq \xi\ \mbox{for all}\ u \in \RR \label{eq:9}.
\end{eqnarray}


Arguing by contradiction, suppose that $\varphi_{+}$ is not coercive. Then we can find $\left\{ u_{n} \right\}_{n\geq 1}\subseteq W^{1,p}_{0}(\Omega)$\ and\ $M_{1}>0$ such that
\begin{equation}
	\label{eq:10}
	||u_{n}||\rightarrow + \infty\ \mbox{and}\ \varphi_{+}(u_{n})\leq M_{1}\ \mbox{for all}\ n\geq 1.
\end{equation}

>From relations (\ref{eq:10}), (\ref{eq:9}) and (\ref{eq:3}), we deduce that
\begin{equation}
	\label{eq:11}
		\left\{ u^{-}_{n} \right\}_{n\geq 1} \subset W^{1,p}_{0}(\Omega)\quad \mbox{is\ bounded}.
\end{equation}

So, from (\ref{eq:10}) it follows that $||u_{n}^{+}||\rightarrow \infty$. Let $y_{n}=\frac{u_{n}^{+}}{||u_{n}^{+}||}\,.$ Then $||y_{n}||=1$ for all $n\geq 1$ and so we may assume that
\begin{equation}
	\label{eq:12}
		y_{n}\overset{w}{\rightarrow}y\ \mbox{in}\ W^{1,p}_{0}(\Omega)\ \mbox{and}\ y_{n}\rightarrow y\ \mbox{in}\ L^{r}(\Omega).
\end{equation}

We have
$$\frac{1}{p}||Dy_{n}||^{p}_{p} + \frac{1}{2||u^{+}_{n}||^{p-2}} ||Dy_{n}||^{2}_{2} - \int_{\Omega} \frac{F_{+}(z,u_{n}^{+})}{||u_{n}^{+}||^{p}}dz\leq \frac{M_{2}}{||u_{n}^{+}||^{p}}$$
for some $M_{2}>M_{1}>0$\quad (see (\ref{eq:11}))
\begin{eqnarray*}
	\Rightarrow && \frac{1}{p}||Dy_{n}||^{p}_{p}+\frac{1}{2||u_{n}^{+}||^{p-2}} ||Dy_{n}||^{2}_{2}-\frac{\hat{\lambda}_{1}(p)}{p}||y_{n}||^{p}_{p}-\frac{\xi |\Omega|_{N}}{p||u_{n}^{+}||^{p}} \leq \frac{M_{2}}{||u_{n}^{+}||^{p}}\ \mbox{for all}\ n\geq 1 \\
							&& \hspace{12cm}(see\ \eqref{eq:9}) \\
	\Rightarrow && ||Dy||^{p}_{p}\leq \hat{\lambda}_{1}(p)||y||^{p}_{p}\quad \mbox{(see\ \eqref{eq:12}\ and\ recall\ $2<p$)}\\
	\Rightarrow && y = \mu \hat{u}_{1,p}\ \mbox{with}\ \mu \geq 0\ \mbox{(recall\ $y\geq 0$)}\,.
\end{eqnarray*}

If $\mu=0$, then $y=0$ and so we have $y_{n}\rightarrow 0$ in $W^{1,p}_{0}(\Omega)$
(since $\| Dy_n\|_p\rightarrow 0$), a contradiction to the fact that $||y_{n}||=1$ for all $n\geq 1$.
So, $\mu > 0$ and we have $y(z)>0$ for all $z \in \Omega$ (recall $\hat{u}_{1,p} \in \mbox{int}\, C_{+}$). This implies that $u_{n}^{+}(z) \rightarrow +\infty$ for a.a. $z \in \Omega$ as $n\rightarrow \infty$. Recall that
\begin{eqnarray}
						&& \frac{1}{p}||Du_{n}^{+}||^{p}_{p}+\frac{1}{2}||Du_{n}^{+}||^{2}_{2}-\int_{\Omega}F(z,u_{n}^{+})dz \leq M_{1}\ \mbox{for all}\ n\geq 1 \quad\mbox{(see \eqref{eq:10})}\nonumber \\
\Rightarrow && \int_{\Omega}\left[ \frac{\hat{\lambda}_{1}(p)}{p} (u_{n}^+)^{p}-F(z,u_{n}^{+}) \right] dz + \int_{\Omega}\frac{\hat{\lambda}_{1}(2)}{2}(u_{n}^{+})^{2}dz\leq M_{1}\quad \mbox{(see \eqref{eq:3})}\nonumber \\
\Rightarrow &&	\frac{\hat{\lambda}_{1}(2)}{2} \int_{\Omega}(u_{n}^{+})^{2}dz\leq M_{1}+\xi|\Omega|_{N}\quad\mbox{(see \eqref{eq:9})}. \label{eq:13}
\end{eqnarray}
But $u_{n}^{+}(z)\rightarrow + \infty$\ for\ a.a.\ $z \in \Omega$ and so, by Fatou's lemma,
$\int_{\Omega}(u_{n}^{+})^{2}dz\rightarrow + \infty$\ as\ $n\rightarrow \infty,$
which contradicts (\ref{eq:13}). This proves the coercivity of $\varphi_{+}$. Similarly we show the coercivity of the functionals $\varphi_-$ and $\varphi$.
	\qed



	Using this proposition and the direct method, we can produce two nontrivial solutions of constant sign.
	
	\begin{proposition}
		\label{prop:3.2}
			Assume that hypotheses\ $H_1$\ hold. Then\ problem\ (1)\ has\ at\ least\ two\ nontrivial\ solutions\ of\ constant\ sign $u_0 \in {\mathrm int}\,C_+,\, v_0 \in -{\mathrm int}\,C_+$
				 and\ both\ are\ local\ minimizers\ of\ the\ functional\ $\varphi$.
	\end{proposition}
	
	\textit{Proof.} From Proposition \ref{prop:3.1}, we know that the functional $\varphi_+$ is coercive. Also, using the Sobolev embedding theorem, we can easily check that $\varphi_+$ is sequentially weakly lower semi-continuous. So, by the Weierstrass theorem we can find $u_0 \in W^{1,p}_{0}(\Omega)$ such that	 
\begin{equation}
		\label{eq:14}
				\varphi_+(u_0)\ =\inf\left\{\varphi_+(u): u \in W^{1,p}_{0}(\Omega)\right\}.
	\end{equation}
	
Let $\delta_0>0$ be as in hypotheses $H_1(iii)$ and let $t \in(0,1)$ be small such that $t\hat{u}_{1,2}(z) \in [0,\delta_0]$ for all $z \in \bar{\Omega}$ (recall that $\hat{u}_{1,2}\in \mbox{int}\, C_+$). Using $H_1(iii)$ and\  $||\hat{u}_{1,2}||_2=1$ we obtain
\begin{eqnarray}
	\varphi_+(t \hat{u}_{1,2}) && =\ \frac{t^p}{p}||D\hat{u}_{1,2}||^{p}_{p}+\frac{t^2}{2}||D\hat{u}_{1,2}||^{2}_{2}-\int_{\Omega}\ F_+(z,t \hat{u}_{1,2})dz\nonumber \\
	 && \leq	 \frac{t^p}{p}||D\hat{u}_{1,2}||^{p}_{p}+\frac{t^2}{2}\left[\hat{\lambda}_1(2)-\hat{\lambda}_m(2)\right] \label{eq:15} .
\end{eqnarray}

We have $\hat{\lambda}_1(2)<\hat{\lambda}_m(2)$ (recall $m\geq 2$). Since $p>2$, by choosing $t \in (0,1)$ even smaller if necessary, from (\ref{eq:15}) we have
$$ \varphi_+(t\hat{u}_{1,2})<0
		\Rightarrow  \varphi_+(u_0)<0 = \varphi_+(0)\quad \mbox{(see\ (\ref{eq:14}))},$$ hence\ $u_0\neq 0$.
	From (\ref{eq:14}) we have
	\begin{equation}\label{eq:16}\varphi'_{+}(u_0) = 0
		\Rightarrow  A_p(u_0)+A(u_0) = N_{f_+}(u_0).
	\end{equation}
	
	On (\ref{eq:16}) we act with $-u^{-}_{0}\in W^{1,p}_{0}(\Omega)$. Then
	$||Du^{-}_{0}||^{p}_{p}+||Du^{-}_{0}||^{2}_{2} = 0$, hence\ $u_0\geq 0$,\ $u_0\neq 0.$
	Then (\ref{eq:16}) becomes
	\begin{eqnarray}
		&& A_p(u_0)+A(u_0)=N_f(u_0)\nonumber \\
		\Rightarrow && -\Delta_p u_0(z)-\Delta u_0(z)=f(z,u_0(z))\ \mbox{a.e.\ in}\ \Omega,\ u_0|_{\partial \Omega}=0.\label{eq:17}
	\end{eqnarray}
	
	From Ladyzhenskaya \& Uraltseva \cite[p. 286]{ref23} and Fan \& Zhao \cite[Theorem 4.1]{fanzhao}, we have $u_0 \in L^{\infty}(\Omega)$. Then we can apply the regularity result of Lieberman \cite[p. 320]{ref24} and infer that $u_0 \in C_+ \backslash \{0\}$. Let $\rho=||u_0||_{\infty}$ and let $\xi_{\rho}>0$ be as postulated by hypothesis $H_1(iv)$. Then from (\ref{eq:17}) we have
	$$\Delta_p u_0(z)+\Delta u_0(z)\ \leq\ \xi_{\rho}u_0(z)^{p-1}\quad \mbox{a.e.\ in}\ \Omega.$$
	
	From the strong maximum principle of Pucci \& Serrin \cite[p. 111]{ref31}, we have $u_0(z)>0$ for all $z \in \Omega$. Then we can apply the boundary point theorem of Pucci \& Serrin \cite[p. 120]{ref31} and conclude that $u_0 \in\ \mbox{int}\,C_+$. Note that $\varphi_+{_{|C_+}}=\varphi_{|C_+}$. So, $u_0 \in \mbox{int}\, C_+$ is a local $C^{1}_{0}(\bar{\Omega})$-minimizer of $\varphi$, hence by virtue of Proposition \ref{prop:2.2} it is also a local $W^{1,p}_{0}(\Omega)$-minimizer of $\varphi$.
	Similarly, working with the functional $\varphi_-$, we produce a second nontrivial constant sign solution $v_0 \in -\mbox{int}\,C_+$, which  is also a local minimizer of $\varphi$.
\qed

	In fact, we can show that problem (1) has extremal constant sign solutions, that is, a smallest nontrivial positive solution and a biggest nontrivial negative solution. The existence of such extremal constant sign solutions will lead to nodal (sign--changing) solutions.
	
	From hypotheses $H_1(i), (iii)$ we see that we can find $c_1>\hat{\lambda}_1(2)$ and $c_2>0$ such that
	\begin{equation}
		\label{eq:18}
			f(z,x)x\geq c_1x^2 - c_2|x|^r\quad \mbox{for\ a.a.}\ z \in \Omega\ \mbox{ and for all}\ x \in \RR.
	\end{equation}
	
	This unilateral growth estimate leads to the following auxiliary problem
	\begin{equation}
		\label{eq:19}
			-\Delta_p u(z)-\Delta u(z)=c_1u(z)-c_2|u(z)|^{r-2}u(z)\ \mbox{in}\ \Omega,\ u|_{\partial \Omega}=0.
	\end{equation}
	
	\begin{proposition}
		\label{prop:3.3}
			Problem (\ref{eq:19}) has a unique nontrivial positive solution $u_* \in {\mathrm int}\, C_+$ and because the problem is odd we have that $v_*=-u_* \in -{\mathrm int}\, C_+$ is the unique nontrivial negative solution of (\ref{eq:19}).
	\end{proposition}

	\textit{Proof.} We start by proving the existence of a nontrivial positive solution for problem \eqref{eq:19}.
	
	Let $\Psi_+:W^{1,p}_{0}(\Omega)\rightarrow \RR$ be the $C^1$-functional defined by	 $$\Psi_+(u)=\frac{1}{p}||Du||^{p}_{p}+\frac{1}{2}||Du||^{2}_{2}-\frac{c_1}{2}||u^+||^{2}_{2}+\frac{c_2}{r}||u^+||^{r}_{r}\ \qquad\mbox{for all}\ u \in W^{1,p}_{0}(\Omega).$$
	
	Since $p>2$, then $\Psi_+$ is coercive. Also, it is sequentially weakly lower semi-continuous. So, we can find $u_* \in W^{1,p}_{0}(\Omega)$ such that
	\begin{equation}
		\label{eq:20}
			\Psi_+(u_*)=\inf\left\{\Psi_+(u):u \in W^{1,p}_{0}(\Omega)\right\}\,.
	\end{equation}
	
	Since $c_1>\hat{\lambda}_1(2)$ and $2<p\leq r$, as before (see the proof of Proposition \ref{prop:3.2}), for $t \in (0,1)$ small, we have
		$ \Psi_+(t\hat{u}_{1,2})<0$. Thus, by (\ref{eq:20}),
$\Psi_+(u_*)<0=\Psi_+(0)$, hence $u_*\neq 0.$

	
Relation (\ref{eq:20}) yields
		\begin{equation}\label{eq:21}
		\Psi'_{+}(u_*)\ = 0
		\Rightarrow  A_p(u_*)+A(u_*)=c_1 u^{+}_{*}-c_2(u^{+}_{*})^{r-1}.
	\end{equation}
	On (\ref{eq:21}) we act with $-u^{-}_{*} \in W^{1,p}_{0}(\Omega)$ and obtain $u_*\geq 0$ and $u_* \neq 0$. Then
		\begin{eqnarray}
		&& A_{p}(u_*)\ +\ A(u_*)=c_1u_*-c_2u^{r-1}_{*}\nonumber \\
		\Rightarrow && -\Delta_{p}u_*(z)-\Delta u_*(z)=c_1u_*(z)-c_2u_*(z)^{r-1}\ \mbox{a.e.\ in}\ \Omega,\ u_*|_{\partial \Omega}=0\,. \nonumber
	\end{eqnarray}
	
	Hence $u_*$ is a nontrivial positive solution of the auxiliary problem (\ref{eq:19}). As in the proof of Proposition \ref{prop:3.2}, using the nonlinear regularity theory (see \cite{fanzhao}, \cite{ref23}, \cite{ref24}) and the results of Pucci \& Serrin \cite[pp. 111 and 120]{ref31}, we show that $u_* \in {\rm int}\, C_+$.
	
	Next, we show the uniqueness of this nontrivial positive solution. To this end, let $G_0(t) = \frac{t^{p}}{p} + \frac{t^2}{2}$ for all $t\geq 0$ and set $G(y) = G_0(||y||)$ for all $y \in \RR^N$. Then $G\in C^1(\RR^N)$ and
$\nabla G(y)=a(y)=||y||^{p-2}y\ +\ y$ for all $y \in \RR^N.$
	The mapping $G_0(\cdot)$ is increasing on $\RR_+$ and $t\longmapsto G_0(t^{1/2})$ is convex and we have
	$$\mbox{div}\, a(Du)=\Delta_{p}u + \Delta u\qquad \mbox{for all}\ u\in W^{1,p}_{0}(\Omega).$$
	
	We consider the integral functional $\mu_+ : L^{1}(\Omega)\rightarrow \bar{\RR} = \RR \cup\{+\infty\}$ defined by
	\begin{center}
		$\mu_+(u)=
			\left\{
				\begin{array}{ll}
					\di\int_{\Omega}G(Du^{1/2})dz		& \mbox{if}\ u\geq 0\ \mbox{and}\ u^{1/2} \in W^{1,p}_{0}(\Omega)\\
					\\
					+\infty                       & \mbox{otherwise}.
				\end{array}
			\right.$
	\end{center}
	
	Let $u_1,\, u_2 \in \mbox{dom}\, \mu_+$ and let $y=(tu_1+(1-t)u_2)^{1/2}\in W^{1,p}_{0}(\Omega)$ for $t \in [0,1]$. From Benguria, Brezis \& Lieb \cite[Lemma 4]{ref7} (see also Diaz \& Saa \cite[Lemma 1]{ref14})  we have
		\begin{eqnarray}
		&& ||Dy(z)|| \leq\ (t||Du_1(z)^{1/2}||^2 + (1-t)||Du_2(z)^{1/2}||^2)^{1/2}\ \mbox{a.e.\ in}\ \Omega\nonumber \\
		\Rightarrow && G_0(||Dy(z)||) \leq G_0((t||Du_1(z)^{1/2}||^2 + (1-t)||Du_2(z)^{1/2}||^2)^{1/2})\nonumber\\
		&& \hspace{70 mm} (\mbox{since\ $G_0$\ is\ increasing})\nonumber \\
		&& \hspace{27 mm} \leq t G_0(||Du_1(z)^{1/2}||) + (1-t) G_0\, (||Du_2(z)^{1/2}||)\nonumber \\
		&& \hspace{70 mm} (\mbox{since\ $t\longmapsto G_0(t^{1/2})$\ is\ convex})\nonumber \\
		\Rightarrow && G(Dy(z)) \leq t G(Du_1(z)^{1/2}) + (1-t) G(Du_2(z)^{1/2})
		\Rightarrow  \mu_+\ \mbox{is\ convex}.\nonumber
	\end{eqnarray}
	%Moreover, via Fatou's lemma we check that $\mu_+$ is lower semi-continuous.
	
	Suppose $u,y\in W^{1,p}_{0}(\Omega)$ are two nontrivial positive solutions of (\ref{eq:19}). From the first part of the proof, we have $u,y \in \mbox{int}\,C_+$ and so $u^2,y^2 \in \mbox{dom}\, \mu_+$. Let $h \in C^{1}_{0}(\bar{\Omega})$. For $t \in [-1,1]$ small in absolute value, we have $u^2+th, y^2+th \in \mbox{dom}\, \mu_+$. The G\^ateaux derivative of $\mu_+$ at $u^2,\, y^2$ in the direction $h$ exists and by the chain rule and the density of $C^{1}_{0}(\bar{\Omega})$ in $W^{1,p}_{0}(\Omega)$, we have for all $h\in W^{1,p}_{0}(\Omega)$
		\begin{eqnarray}
		&& \mu'_{+}(u^2)(h)=\frac{1}{2}\int_{\Omega}\frac{-\Delta_{p}u-\Delta u}{u}hdz \label{eq:22}\\
		&& \mu'_{+}(y^2)(h)=\frac{1}{2}\int_{\Omega}\frac{-\Delta_{p}y-\Delta y}{y}hdz\,. \label{eq:23}
	\end{eqnarray}
	
	Since $\mu_+$ is convex, $\mu_+'$ is monotone, we have
		\begin{eqnarray}
		&& 0\ \leq\ \left\langle \mu'_{+}(u^2)-\mu'_{+}(y^2),\ u^2-y^2\right\rangle_{L^{1}(\Omega)}=\nonumber \\
		&& \frac{1}{2}\ \int_{\Omega}\ \left(\frac{-\Delta_{p}u-\Delta u}{u}-\frac{-\Delta_{p}y-\Delta y}{y}\right)\ (u^2-y^2)dz=\nonumber \\
		&& \frac{c_2}{2}\ \int_{\Omega}\ (y^{r-2}-u^{r-2})\ (u^2-y^2)dz\ \leq\ 0\nonumber \\
		\Rightarrow && u=y.\nonumber
	\end{eqnarray}
	This proves the uniqueness of $u_*$.
	
	Since the auxiliary problem (\ref{eq:19}) is odd, then $v_*=-u_* \in -\mbox{int}\,C_+$ is the unique nontrivial negative solution of (\ref{eq:19}).
	\qed

\smallskip
	Using $u_* \in \mbox{int}\, C_+$ and $v_* \in -\mbox{int}\, C_+$, we can produce extremal constant sign solutions for problem (1).
	
	\begin{proposition}
		\label{prop:3.4}
			Assume that hypotheses\ $H_1$\ hold.
Then\ problem\ (1)\ has\ a\ smallest\ nontrivial\ positive\ solution\ $u_+ \in {\mathrm int}\, C_+$
 and\ a\ biggest\ nontrivial\ negative\ solution\ $v_- \in -{\mathrm int}\, C_+$.
	\end{proposition}

	\textit{Proof.} Let $S_+$ be the set of nontrivial positive solutions for problem (1). From Proposition \ref{prop:3.2} and its proof, we have
	$S_+ \neq \emptyset$  and $S_+\subseteq \mbox{int}\,C_+.$
	
	\textbf{Claim:} If $\tilde{u} \in S_+$, then $u_* \leq \tilde{u}$.

	We consider the following Carath\'eodory function
	\begin{equation}
		h(z,x)=
			\left\{
				\begin{array}{ll}
					0																			 & if\ x<0\\
					c_1x-c_2x^{r-1}  											 & if\ 0\leq x\leq \tilde{u}(z) \label{eq:24}\\
					c_1\tilde{u}(z)-c_2\tilde{u}(z)^{r-1}  & if\ \tilde{u}(z)<x.
				\end{array}
			\right.
	\end{equation}

	We set $H(z,x)=\int^{x}_{0}h(z,s)ds$ and consider the $C^{1}$-functional $\Psi:W^{1,p}_{0}(\Omega)\rightarrow \RR$ defined by
	$$\Psi(u)=\frac{1}{p}||Du||^{p}_{p}\ +\ \frac{1}{2}||Du||^{2}_{2}-\int_{\Omega}\ H(z,u(z))dz\ \mbox{for all}\ u \in W^{1,p}_{0}(\Omega).$$
	
	It is clear from (\ref{eq:24}) that $\Psi$ is coercive. Also, it is sequentially weakly lower semi-continuous. So, we can find $\tilde{u}_*\in W^{1,p}_{0}(\Omega)$ such that
		\begin{equation}
		\label{eq:25}
			\Psi(\tilde{u}_*)=\inf\, \{\Psi(u):u\in W^{1,p}_{0}(\Omega)\}.
	\end{equation}

	Recall that $\tilde{u} \in \mbox{int}\,C_+$. So we can find $t \in (0,1)$ small such that $t \tilde{u}_{1,2}\leq \tilde{u}$. Then we have
	\begin{eqnarray}
		\Psi(t\hat{u}_{1,2})&& =\ \frac{t^{p}}{p}||D\hat{u}_{1,2}||^{p}_{p}\ +\ \frac{t^2}{2}||D\hat{u}_{1,2}||^{2}_{2}-\int_{\Omega}H(z,t\hat{u}_{1,2})dz\nonumber \\
		&& \leq\ \frac{t^{p}}{p}||D\hat{u}_{1,2}||^{p}_{p}\ +\ \frac{t^2}{2}[\hat{\lambda}_1(2)-c_1]\ +\ \frac{t^r c_2}{r}||\hat{u}_{1,2}||^{r}_{r}\nonumber \\
		&& \hspace{60 mm} (\mbox{see\ (\ref{eq:24})\ and\ recall}\ ||\hat{u}_{1,2}||_2=1).\nonumber
	\end{eqnarray}
	
	Since $c_1>\hat{\lambda}_1(2)$ and $2< p \leq r$, choosing $t \in (0,1)$ even smaller if necessary, we have
		\begin{eqnarray}
		&& \Psi(t\hat{u}_{1,2})\ <\ 0\nonumber \\
		\Rightarrow && \Psi(\tilde{u}_*)\ <\ 0=\Psi(0)\ (\mbox{see\ (\ref{eq:25})),\ hence}\ \tilde{u}_*\neq 0.\nonumber
	\end{eqnarray}
	
	From (\ref{eq:25}) we have
		\begin{equation}\label{eq:26}
 \Psi'(\tilde{u}_*)=0
		\Rightarrow  A_{p}(\tilde{u}_*) + A(\tilde{u}_*)=N_h(\tilde{u}_*).
	\end{equation}
	
	On (\ref{eq:26}) 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 $(\tilde{u}_* - \tilde{u})^+ \in W^{1,p}_{0}(\Omega)$. Then
		\begin{eqnarray}
		&&	\ \ \left\langle A_{p}(\tilde{u}_*),\ (\tilde{u}_*-\tilde{u})^+\right\rangle\ +\ \left\langle A(\tilde{u}_*),\ (\tilde{u}_*-\tilde{u})^+\right\rangle\nonumber \\
		&&	=\ \int_{\Omega}\ h(z,\tilde{u}_*)\ (\tilde{u}_* -\tilde{u})^+\ dz\nonumber \\
		&&	=\ \int_{\Omega}\ (c_1\tilde{u}-c_2\tilde{u}^{r-1})\ (\tilde{u}_* -\tilde{u})^+\ dz\ (\mbox{see}\ (\ref{eq:24}))\nonumber \\
		&&	\leq\ \int_{\Omega}\ h(z,\tilde{u})\ (\tilde{u}_* -\tilde{u})^+\ dz\ (\mbox{see}\ (\ref{eq:18}))\nonumber \\
		&&	=\ \left\langle A_{p}(\tilde{u}),\ (\tilde{u}_*-\tilde{u})^+\right\rangle\ +\ \left\langle A(\tilde{u}),\ (\tilde{u}_*-\tilde{u})^+\right\rangle\ (\mbox{since}\ \tilde{u}\in S_+)\nonumber \\
		\Rightarrow &&	||D(\tilde{u}_*-\tilde{u})^+||^{2}_{2}\ \leq\ 0\ (\mbox{see\ Proposition}\ \ref{prop:2.5}),\nonumber
\end{eqnarray}
hence $\tilde{u}_*\leq\tilde{u}$.

	So, we have proved that
	$$\tilde{u}_* \in [0,\tilde{u}]=\{u \in W^{1,p}_{0}(\Omega):\ 0\leq u(z)\leq \tilde{u}(z)\ \mbox{a.e.\ in}\ \Omega\},\ \tilde{u}_* \neq 0.$$

	Then (\ref{eq:26}) becomes
	\begin{eqnarray}
		&&	A_{p}(\tilde{u}_*)\ +\ A(\tilde{u}_*)=c_1\tilde{u}_*-c_2\tilde{u}^{r-1}_{*}\ (\mbox{see}\ (\ref{eq:24}))\nonumber \\
		\Rightarrow &&	-\Delta_{p}\tilde{u}_*(z) - \Delta\tilde{u}_*(z)=c_1\tilde{u}_*(z) - c_2\tilde{u}_*(z)^{r-1}\ \mbox{a.e.\ in}\ \Omega,\ \tilde{u}_*|_{\partial \Omega}\ = 0\nonumber \\
		\Rightarrow &&	\tilde{u}_*=u_*\ (\mbox{see\ Proposition\ \ref{prop:3.3}})\nonumber \\
		\Rightarrow && u_*\ \leq \tilde{u}.\nonumber
	\end{eqnarray}
	
	This proves the Claim.
	
	From Filippakis, Kristaly \& Papageorgiou \cite{ref17} (Proposition 4.2 and Lemma 4.3) we have that $S_+$ is downward directed (that is, if $u_1,u_2 \in S_+$, then we can find $u \in S_+$ such that $u\leq u_1, u\leq u_2$). So, without any loss of generality we may assume that there exists $M_3>0$ such that $u(z)\leq M_3$ for all $z \in\bar{\Omega}$ and all $u \in S_+$. Let $C\subseteq S_+$ be a chain (that is, a totally ordered subset of $S_+$). From Dunford \& Schwartz \cite{ref16}, we know that we can find $\{u_n\}_{n\geq 1}\subseteq C$ such that
	$\inf C=\inf_{n\geq 1} u_n.$
	
	We have
		\begin{eqnarray}
		&& A_{p}(u_n) + A(u_n)=N_f(u_n),\ u_*\leq u_n\leq M_3\ \mbox{for all}\ n\geq 1\ (\mbox{see\ the\ Claim})	 \label{eq:27} \\
		\Rightarrow &&	\{u_n\}_{n\geq 1}\ \subseteq\ W^{1,p}_{0}(\Omega)\ \mbox{is\ bounded}.\nonumber
	\end{eqnarray}

	We may assume that
		\begin{equation}
		\label{eq:28}
			u_n\ \stackrel{w}{\rightarrow}\ u\ \mbox{in}\ W^{1,p}_{0}(\Omega)\ \mbox{and}\ u_n\ \rightarrow\ u\ \mbox{in}\ L^r(\Omega).
	\end{equation}
	
	On (\ref{eq:27}) we act with $u_n-u \in W^{1,p}_{0}(\Omega)$, pass to the limit as $n \rightarrow \infty$ and use (\ref{eq:28}). Then
		\begin{eqnarray}
		&&	\lim_{n\rightarrow\infty}\left[\left\langle A_{p}(u_n),u_n-u\right\rangle\ +\ \left\langle A(u_n),u_n-u\right\rangle\right]=0\nonumber \\
		\Rightarrow &&	\limsup_{n\ri\infty}\left[\left\langle A_{p}(u_n),u_n-u\right\rangle\ +\ \left\langle A(u),u_n-u\right\rangle\right] \leq\ 0\nonumber \\
		&&	\hspace {70 mm} (\mbox{due\ to\ the\ monotonicity\ of}\ A)\nonumber \\
		\Rightarrow &&	\limsup_{n\ri\infty}\left\langle A_{p}(u_n),u_n-u\right\rangle\ \leq\ 0\ (\mbox{see}\ (\ref{eq:28}))\nonumber \\
		\Rightarrow &&	u_n\rightarrow u\ \mbox{in}\ W^{1,p}_{0}(\Omega)\ (\mbox{see\ Proposition}\ \ref{prop:2.5})\ \mbox{and}\ u_*\leq u\ (\mbox{see}\ (\ref{eq:27})).\label{eq:29}
	\end{eqnarray}
	
	So, if in (\ref{eq:27}) we pass to the limit as $n \rightarrow \infty$ and use (\ref{eq:29}), then
		\begin{eqnarray}
		&&	A_{p}(u)+A(u)=N_f(u),\ u_* \leq u\nonumber \\
		\Rightarrow &&	u \in S_+\ \mbox{and}\ u=\inf C\nonumber
	\end{eqnarray}
	
	Since $C$ is an arbitrary chain in $S_+$, from the Kuratowski--Zorn lemma we know that we can find a minimal element $u_+ \in S_+ \subseteq \mbox{int}\, C_+$. Since $S_+$ is downward directed, if $u \in S_+$ we can find $\hat{u} \in S_+$ such that $\hat{u}\leq u_+$ and $\hat{u}\leq u$. The minimality of $u_+$ implies that $\hat{u}=u_+$ and so $u_+\leq u$ for all $u \in S_+$.
		
	Let $S_-$ be the set of nontrivial negative solutions of (1). We have
		$$S_-\neq \O\ \mbox{and}\ S_-\subset -\mbox{int}\,C_+\quad (\mbox{see\ Proposition}\ \ref{prop:3.2}).$$
		
	The set $S_-$ is upward directed (that is, if $v_1,v_2 \in S_-$, then we can find such that $v_1\leq v, v_2\leq v$; see \cite{ref17}). Reasoning as above, via the Kuratowski-Zorn lemma, we produce $v_- \in -\mbox{int}\,C_+$ the biggest nontrivial negative solution of (1).
		\qed

	Using these extremal constant sign solutions, we can produce nodal solutions. To do this, we need to strengthen the conditions on the reaction $f(z,x)$. The new hypotheses on $f(z,x)$, are the following:
	\\
	
	\textbf{$H_2:$} $f:\Omega \times \RR\rightarrow \RR$ is a measurable function such that for a.a. $z \in \Omega, f(z,0)=0, f(z,\cdot) \in C^1(\RR)$ and
	\\

	\textbf{(i)} $|f'_{x}(z,x)|\leq a(z)(1+|x|^{r-2})$ for a.a. $z \in \Omega$, all $x \in \RR$, with $a\in L^\infty (\Omega)$, $p\leq r< p^{*}$;
	\\

	\textbf{(ii)} $\limsup_{x\rightarrow \pm \infty}\frac{p F(z,x)}{|x|^{p}}\ \leq\ \hat{\lambda}_1(p)$ uniformly for a.a. $x \in \Omega$ and there exists $\xi > 0$ such that
	$$f(z,x)x-p F(z,x)\geq -\xi\ \mbox{for\ a.a.}\ z\in \Omega,\ \mbox{all}\ x\in \RR;$$

	\textbf{(iii)} there are $\delta_0>0$, an integer $m\geq 2$ and $\eta \in L^{\infty}(\Omega)$, such that $\eta(z)\geq\hat{\lambda}_m(2)$ a.e. in $\Omega, \eta\neq\hat{\lambda}_m(2)$ and
	$$\eta(z) x^2\ \leq\ f(z,x)x\ \leq\ \hat{\lambda}_{m+1}(2)x^2\ \mbox{for\ a.a.}\ z \in \Omega,\ \mbox{all}\ |x|\leq \delta_0;$$
	
\textbf{(iv)} for every $\rho >0$, there exists $\xi_\rho>0$ such that for a.a. $z\in\Omega$, the map $x\longmapsto f(z,x)+\xi_\rho |x|^{p-2}x$ is nondecreasing on $[-\rho ,\rho]$.

	\textbf{Remark.} In this setting hypothesis $H_2$(iv) is satisfied, if for example there exists $\delta_0>0$ such that $f'_x(z,x)\geq 0$ for a.a. $z\in\Omega$, all $|x|\leq\delta_0$ (that is, $f(z,\cdot )$ is increasing near zero).
	
	\begin{proposition}
		\label{prop:3.5} {\sl
			Assume that hypotheses\ H$_2$\ are fulfilled.
Then\ problem\ (1)\ has\ a\ nodal\ solution
				$y_0 \in {\mathrm int}_{C^{1}_{0}(\bar{\Omega})}[v_-,u_+]$,
				that is,\ $y_0 \in C^{1}_{0}(\bar{\Omega})$\ and\ $u_+-y_0,\ y_0-v_- \in {\mathrm int}\,C_+$.
}
	\end{proposition}
	
	\textit{Proof.} Let $u_+\in \mbox{int}\,C_+$ and $v_- \in \mbox{int}\,C_+$ be the two extremal constant sign solutions produced in Proposition \ref{prop:3.4}. We introduce the following Carath\'eodory function
		\begin{equation}
		\hat{g}(z,x)=
			\left\{
				\begin{array}{ll}
					f(z,v_-(z))	 & \mbox{if}\ x<v_-(z)\\
					f(z,x)  		 & \mbox{if}\ v_-(z)\leq x\leq u_+(z) \label{eq:30}\\
					f(z,u_+(z))  & \mbox{if}\ u_+(z)<x.
				\end{array}
			\right.
	\end{equation}
	
	We set $\hat{G}(z,x) = \int^{x}_{0}\ \hat{g}(z,s)ds$ and consider the $C^1$-functional $\hat{\varphi}:W^{1,p}_{0}(\Omega)\rightarrow \RR$ defined by
	$$\hat{\varphi}(u)=\frac{1}{p}||Du||^{p}_{p}\ +\ \frac{1}{2}||Du||^{2}_{2}-\int_{\Omega}\ \hat{G}(z,u(z))dz\quad \mbox{for all}\ u \in W^{1,p}_{0}(\Omega).$$
	
	Also let $\hat{g}_{\pm}(z,x)=\hat{g}(z,\pm x^{\pm}),\ \hat{G}_{\pm}(z,x)=\int^{x}_{0}\ \hat{g}_\pm (z,s)ds$ and consider the $C^1$-functionals $\hat{\varphi}_{\pm}:W^{1,p}_{0}(\Omega)\rightarrow \RR$ defined by
	$$\hat{\varphi}_{\pm}(u)=\frac{1}{p}||Du||^{p}_{p}\ +\ \frac{1}{2}||Du||^{2}_{2}-\int_{\Omega}\ \hat{G}_{\pm}(z,u(z))dz\quad \mbox{for all}\ u \in W^{1,p}_{0}(\Omega).$$

	\textbf{Claim 1:} $K_{\hat{\varphi}}\subseteq[v_-,u_+]=\{u \in W^{1,p}_{0}(\Omega):v_-(z)\leq u(z)\leq u_+(z)\ \mbox{a.e.}\ in\ \Omega\},\ K_{\hat{\varphi}_+}=\{0,u_+\},\ K_{\hat{\varphi}_-}=\{0,v_-\}.$
	
	Let $u \in K_{\hat{\varphi}}$. Then we have
	\begin{equation}
		\label{eq:31}
			A_p(u)+A(u)=N_{\hat{g}}(u).
	\end{equation}
	
	On (\ref{eq:31}) we act with $(u-u_+)^+ \in W^{1,p}_{0}(\Omega)$. Then
		\begin{eqnarray}
		&&	\ \ \ \left\langle A_p(u),(u-u_+)^+\right\rangle\ +\ \left\langle A(u),(u-u_+)^+\right\rangle\nonumber \\
		&&	=\ \int_{\Omega}\ \hat{g}(z,u)(u-u_+)^+\ dz\nonumber \\
		&&	=\ \int_{\Omega}\ f(z,u_+)(u-u_+)^+\ dz\ (\mbox{see}\ (\ref{eq:30}))\nonumber \\
		&&	=\ \left\langle A_p(u_+),(u-u_+)^+\right\rangle\ +\ \left\langle A(u_+),(u-u_+)^+\right\rangle\nonumber \\
		\Rightarrow && ||D(u-u_+)^+||^{2}_{2}\ \leq\ 0\ (\mbox{since}\ A_p\ \mbox{is\ monotone,\ see\ Proposition}\ \ref{prop:2.5})	 \nonumber \\
		\Rightarrow && u\leq u_+.\nonumber
	\end{eqnarray}
	
	Similarly, acting on (\ref{eq:31}) with $(v_- -u)^+ \in W^{1,p}_{0}(\Omega)$, we show that $v_-\leq u$. It follows that
$K_{\hat{\varphi}}\subseteq [v_-,u_+].$
	
	Reasoning in a similar way, we show that
		\begin{eqnarray}
		&&	K_{\hat{\varphi}_+}\subset[0,u_+]=\{u \in W^{1,p}_{0}(\Omega): 0\leq u(z)\leq u_+(z)\ \mbox{a.e.\ in}\ \Omega\}\nonumber \\
		&&	K_{\hat{\varphi}_-}\subset[v_-,0]=\{u \in W^{1,p}_{0}(\Omega): v_-(z)\leq u(z)\leq 0\ \mbox{a.e.\ in}\ \Omega\}.\nonumber
	\end{eqnarray}

	The extremality of $v_- \in -\mbox{int}\,C_+$ and $u_+ \in \mbox{int}\,C_+$ (see Proposition \ref{prop:3.4}), implies that
	$$K_{\hat{\varphi}_+}=\{0,u_+\}\quad \mbox{and}\quad K_{\hat{\varphi}_-}=\{0,v_-\}.$$
	This proves Claim 1.
	
	\textbf{Claim 2:} $u_+ \in \mbox{int}\,C_+$ and $v_- \in -\mbox{int}\,C_+$ are local minimizers of the functional $\hat{\varphi}$.
	
	From (\ref{eq:30}) it is clear that $\hat{\varphi}$ is coercive. Also, it is sequentially weakly lower semi-continuous. So, we can find $\tilde{u}\in W^{1,p}_{0}(\Omega)$ such that
		\begin{equation}
		\label{eq:32}
			\hat{\varphi}_+(\tilde{u})=\inf\left\{\hat{\varphi}_+(u):u \in W^{1,p}_{0}(\Omega)\right\}.
	\end{equation}
	
	Integrating hypothesis in $H_2(iii)$, we have
	$$\frac{\eta(z)}{2}x^2\leq F(z,x)\leq \frac{\hat{\lambda}_{m+1}(2)}{2}x^2\ \mbox{for\ a.a.}\ z \in \Omega,\ \mbox{all}\ |x|\leq \delta_0.$$
	
	As before (see the proof of Proposition \ref{prop:3.4}), for $t \in (0,1)$ small we have by \eqref{eq:32}
		$$	\hat{\varphi}_+(t\hat{u}_{1,2})<0
		\Rightarrow 	\hat{\varphi}_+(\tilde{u})<0=\hat{\varphi}_+(0)\,,$$ hence $\tilde{u}\neq 0$.

	
	From (\ref{eq:32}) we have
		$$\tilde{u} \in K_{\hat{\varphi}_+},\ \tilde{u}\neq 0
		\Rightarrow 	\tilde{u}=u_+\ (\mbox{see\ Claim}\ 1).
$$
	
	Note that $\hat{\varphi}_+|_{C_+}=\hat{\varphi}|_{C_+}$. So, $u_+$ is a local $C^{1}_{0}(\bar{\Omega})$-minimizer of $\hat{\varphi}$. Hence by virtue of Proposition \ref{prop:2.2}, it is also a local $W^{1,p}_{0}(\Omega)$-minimizer of $\hat{\varphi}$.
	
	Similarly for $v_- \in -\mbox{int}\,C_+$ using this time the functional $\hat{\varphi}_-$. This proves Claim 2.
	
	Without any loss of generality we may assume that $\hat{\varphi}(v_-)\leq \hat{\varphi}(u_+)$ (the analysis is similar if the opposite inequality is true). From Claim 2 we know that $u_+ \in \mbox{int}\, C_+$ is a local minimizer of the functional $\hat{\varphi}$. So, as in Filippakis, Kristaly \& Papageorgiou \cite{ref17} (proof of Proposition 3.2) or from de Figueiredo \cite[Theorem 5.10]{figu0}, we can find $\rho \in(0,1)$ small such that
		\begin{equation}
		\label{eq:33}
			\hat{\varphi}(v_-)\leq \hat{\varphi}(u_+)< \inf[\hat{\varphi}(u):||u-u_+||=\rho]= \hat{\eta}^{+}_{\rho},\ ||v_- -u_+||>\rho\,.
	\end{equation}

	Since $\hat{\varphi}$ is coercive (see (\ref{eq:30})), it satisfies the $C$-condition. This fact and (\ref{eq:33}) permit the use of Theorem \ref{th:2.1} (the mountain pass theorem). So, we can find $y_0 \in W^{1,p}_{0}(\Omega)$ such that
		\begin{equation}
		\label{eq:34}
			y_0 \in K_{\hat{\varphi}}\subset [v_-,u_+]\ (\mbox{see\ Claim\ 1})\ \mbox{and}\ \hat{\eta}^{+}_{\rho}\leq \hat{\varphi}(y_0).
	\end{equation}
	
	From (\ref{eq:33}) and (\ref{eq:34}) it follows that
		\begin{equation}
		\label{eq:35}
				y_0 \notin \{v_-,u_+\}.
	\end{equation}
	
	We have
		\begin{eqnarray}
		&&	A_p(y_0)+A(y_0)=N_f(y_0)\ (\mbox{see}\ (\ref{eq:30}))\nonumber \\
		\Rightarrow &&	-\Delta_p y_0(z) - \Delta y_0(z)=f(z,y_0(z))\ \mbox{a.e.\ in}\ \Omega,\ y_0|_{\partial \Omega}=0.\nonumber
	\end{eqnarray}
	
	As before the nonlinear regularity theory implies that $y_0 \in C^{1}_{0}(\bar{\Omega})$. Set $$\rho=\max\{||u_+||_{\infty},||v_-||_{\infty}\}$$ and let $\xi_{\rho}>0$ be such that for a.a. $z\in \Omega$,
the mapping $x\rightarrow f(z,x)+\xi_{\rho}|x|^{p-2}x$ is nondecreasing on $[-\rho,\rho]$. Then
		\begin{eqnarray}
		&&	-\Delta_{p}y_0(z)-\Delta y_0(z) + \xi_{p}|y_0(z)|^{p-2}y_0(z)\nonumber \\
		&&	=\ f(z,y_0(z))+\xi_{\rho}|y_0(z)|^{p-2}y_0(z)\nonumber \\
		&&	\leq f(z,u_+(z))+\xi_{\rho}u_+(z)^{p-1}\ (\mbox{since}\ y_0\leq u_+)\nonumber \\
		&&	= -\Delta_p u_+(z)-\Delta u_+(z) + \xi_{\rho}u_+(z)^{p-1}\ \mbox{a.e.\ in}\ \Omega\,.\label{eq:36}
	\end{eqnarray}
	
	As in the proof of Proposition \ref{prop:3.3}, let $a:\RR^N\rightarrow \RR^N$ be the $C^{1}$-map, defined by
	$$a(y) = ||y||^{p-2}y+y\ \mbox{for all}\ y \in \RR^N.$$
	
	We have
	$$\mbox{div}\, a(Du)=\Delta_{p}u + \Delta u\ \mbox{for\ all}\ u\in W^{1,p}_{0}(\Omega).$$
	
	We have
	\begin{eqnarray}
		&&\ \ \ \ \nabla a(y)=||y||^{p-2}\ \left[I+(p-2)\frac{y \otimes y}{||y||}\right]\ +\ I\nonumber \\
		&&\Rightarrow (\nabla a(y)\xi,\ \xi)_{\RR^N}\ \geq\ ||\xi||^2\ \mbox{for all}\ y,\ \xi \in \RR^N.\label{eq:37}
	\end{eqnarray}

	%As we already remarked, via the fact that $f(z,\cdot)\in C^1(\RR)$ and hypothesis $H_2$(i), we see that for %every $\rho>0$, we can find $\xi_{\rho}>0$ such that for a.a. $z \in \Omega$, $x \rightarrow f(z,x)\ +\ %\xi_{\rho}|x|^{p-2}x$ is increasing on $[-\rho,\ \rho]$.
Then relation \eqref{eq:37} permits the use of the tangency principle of Pucci \& Serrin \cite[p.~35]{ref31}  and we have
		$$y_0(z)<u_+(z)\ \mbox{for all}\ z \in \Omega.$$
	
	Then from \eqref{eq:36} and Proposition \ref{prop:2.3}, we have
	$$u_+ - y_0 \in \mbox{int}\, C_+\,.$$
	
	In a similar manner, we show that
	$$y_0 - v_-\in \mbox{int}\, C_+\,.$$
	
	Therefore
	$$y_0 \in \mbox{int}_{C^{1}_{0}(\Omega)}[v_-,u_+]\,.$$
	
	So, we have
		\begin{eqnarray}
		&& \ \ \ \ C_k(\hat{\varphi}|_{C^{1}_{0}(\bar{\Omega})}, y_0)= C_k(\varphi|_{C^{1}_{0}(\bar{\Omega})}, y_0)\nonumber \\
		&&\Rightarrow C_k(\hat{\varphi},y_0)=C_k(\varphi, y_0)\ \mbox{for all}\ k\geq 0 \label{eq:38}\\
		&& \hspace{60 mm}(\mbox{see\ Bartsch\ \cite[Proposition 2.6]{ref4}\ and\ Palais}\ \cite{ref29})\nonumber
	\end{eqnarray}
	
	Recall that $y_0 \in C^{1}_{0}(\bar{\Omega})$ is a critical point of mountain pass type for the functional $\hat{\varphi}$. Therefore $C_1(\hat{\varphi}, y_0)\ \neq\ 0$, hence $C_1(\varphi, y_0)\ \neq\ 0$.
		

%\begin{eqnarray}
%		&&\ \ \ \ C_1(\hat{\varphi}, y_0)\ \neq\ 0\nonumber \\
%		&&\Rightarrow C_1(\varphi, y_0)\ \neq\ 0\ (\mbox{see}\ \ref{eq:38})\nonumber \\
%		&&\Rightarrow C_k(\varphi, y_0)=\delta_{k,1}\ \ZZ\ \mbox{for all}\ k\geq 0 \nonumber \\
%		&& \hspace{30 mm}(\mbox{see\ Papageorgiou\ \& \ Smyrlis\ \cite{ref30}\ and\ recall}\ \varphi\in %C^2(W^{1,p}_{0}(\Omega)))\nonumber \\

The next Claim can be found in \cite{ref30}. For completeness and the convenience of the reader, we present the detailed proof.

\textbf{Claim 3:}  $C_k(\varphi ,y_0)=\delta_{k,1}\ZZ$ for all $k\geq 0$.

It is known from \cite[Lemma 2.2]{ref13} that there are $\rho>0$ and $\xi:V\cap \overline{B_\rho}\ri\RR$ (with $V$ as in Section 2 and $\overline{B_\rho}=\{u\in W_0^{1,p}(\Omega):\ \|u\|\leq\rho\}$) such that
$$\langle\xi ''(0)v,w\rangle =\langle\varphi''(y_0)v,w\rangle\qquad\mbox{for all}\ v,w\in V.$$
Moreover, $\xi''(0)$ is Fredholm and ${\rm ker}\, \xi''(0)=H^0$ (see Section 2). From \cite[p. 286]{ref12}, we have
$$C_k(\varphi,y_0)=C_k(\xi ,0)\qquad\mbox{for all}\ k\geq 0.$$
Hence it follows that
$$C_1(\xi, 0)\not=0.$$
Therefore we have $d={\rm dim}\, H^-\leq 1$ (see for example \cite[Theorem 2.5]{ref12}). Let $d_0={\rm dim}\, H^0$.

First we assume that $d_0=0$. In this case the origin is a nondegenerate critical point of $\xi$ with Morse index $d$. Hence
$$C_k(\xi,0)=\delta_{k,d}\ZZ\qquad\mbox{for all}\ k\geq 0.$$
It follows that $d=1$ and so we have
$$C_k(\xi,0)=\delta_{k,1}\ZZ\qquad\mbox{for all}\ k\geq 0.$$

Next assume that $d_0>0$. In this case the origin is a degenerate critical point of $\xi$. Invoking the Shifting Theorem (see Chang \cite{ref10}) we have
$$C_k(\xi,0)=C_{k-d}(\hat\xi,0)\qquad\mbox{for all}\ k\geq 0,$$
where $\hat\xi=\xi|_{H^0}$.

Assume $d=1$. Then we have
$$C_0(\hat\xi ,0)\not=0$$
and so from Chang \cite[Theorem 5.1.20]{ref10}, we have
$$C_k(\xi,y_0)=\delta_{k,1}\ZZ\qquad\mbox{for all}\ k\geq 0.$$
Next we assume that $d=0$. We have
$$C_k(\xi,0)=C_k(\hat\xi,0)\qquad\mbox{for all}\ k\geq 0,$$
hence $C_1(\hat\xi,0)\not=0$.

We show  that ``if $\sigma (\hat\xi''(0))\subseteq [0,\infty)$ (the spectrum of $\hat\xi''(0)$ is in $\RR_+$), then ${\rm dim}\,{\rm ker}\, \xi ''(0)\leq 1$".

Under the hypothesis on the spectrum of $\hat\xi''(0)$, for ${\rm ker}\, \xi ''(0)$ to be nontrivial it amounts to saying that 1 is the first eigenvalue of the weighted linear eigenvalue problem (see Section 2)
$$-{\rm div}\, \left((1+\|b\|^2)Du+(p-2)(b,Du)_{\RR^N}\right)=\lambda f_x'(z,y_0)u\ \ \mbox{in}\ \Omega,\quad u|_{\partial\Omega}=0.$$
But as it is well known (see, for example, de Figueiredo \cite{figu0} and Gasinski \& Papageorgiou \cite[Section 6.1]{ref18}), this first eigenvalue is simple. So, we can apply Theorem 5.1.20 of Chang \cite{ref10} and have
$$C_k(\varphi, y_0)=\delta_{k,1}\ZZ\qquad \mbox{for all}\ k\geq 0.
	$$
This proves Claim 4.

>From Claim 4 we have
$$C_k(\hat\varphi, y_0)=\delta_{k,1}\ZZ\qquad \mbox{for all}\ k\geq 0.
	$$

	\textbf{Claim 4:}  $\ C_{d_m}(\hat{\varphi},0) \neq 0$ where $d_m = \mbox{dim}\, \overset{m}{\underset{\mathrm{i=1}}\oplus} E(\hat{\lambda_i}(2))\geq 2 $.

	Set $$Y = W^{1,p}_{0}(\Omega)\cap\left[\overset{m}{\underset{\mathrm{i=1}}\oplus}E(\hat{\lambda_i}(2))\right]\quad \mbox{and} \quad V=W^{1,p}_{0}(\Omega)\cap\overline{\left[{\underset{\mathrm{i\geq m+1}}\oplus}E(\hat{\lambda_i}(2))\right]}.$$ Then $W^{1,p}_{0}(\Omega) = Y \oplus V.$
	
	Note that $Y$ is finite dimensional and $Y \subseteq C^{1}_{0}(\bar{\Omega})$. So, we can find $\rho_0 \in (0,1)$ such that
		$$||y||\leq \rho_0\ \Rightarrow\ ||y||_{C^{1}_{0}(\bar{\Omega})}\ \leq\ \delta_0\ \mbox{for all}\ y \in Y.$$

	Here $\delta_0>0$ is as postulated by hypothesis $H_2(iii)$. Then for $y \in Y$ with $||y||\leq \rho_0$ we have
		\begin{eqnarray}
		&& \hspace{10 mm}\varphi(y)=\frac{1}{p}||Dy||^{p}_{p}\ +\ \frac{1}{2}||Dy||^{2}_{2}- \int_{\Omega}F(z,y(z))\ dz\nonumber \\
		&& \ \ \ \ \leq \frac{1}{p}||Dy||^{p}_{p}\ +\ \frac{1}{2}||Dy||^{2}_{2}-\frac{1}{2}\int_{\Omega} \eta y^2\ dz\ \ (\mbox{see\ hypothesis\ H$_2(iii)$})\nonumber \\
		&& \ \ \ \ \leq \frac{1}{p}||y||^{p}-\frac{c_3}{2}||y||^{2}_{H^{1}_{0}(\Omega)}\ \mbox{for\ some}\ c_3>0\ \ (\mbox{see}\ (\ref{eq:5})) \nonumber \\
		&& \ \ \ \ \leq \frac{1}{p}||y||^{p}-c_4||y||^2\ \mbox{for\ some}\ c_4>0 \nonumber \\
		&& \hspace{40 mm}(Y\ \mbox{is\ finite\ dimensional\ so\ all\ norms\ are\ equivalent})\nonumber \\
		&& \Rightarrow\ \varphi(y)\ \leq\ 0\ \mbox{for all}\ ||y||\leq \hat{\rho_0}\ \mbox{with}\ \hat{\rho_0}\ \leq\ \rho_0\ \ (\mbox{since}\ p>2).\label{eq:40}
	\end{eqnarray}
	
	Next, let $v \in V$. From hypotheses $H_2(i), (iii)$ we have
		\begin{equation}
		\label{eq:41}
			F(z,x)\ \leq\ \frac{\hat{\lambda}_{m+1}(2)}{2}x^2\ +\ c_5|x|^q\ \mbox{for\ a.a.}\ z \in \Omega,\ \mbox{all}\ x \in \RR\ \mbox{and\ some}\ c_5>0,\ p<q.
	\end{equation}
	
	Then for $v \in V$, we have
		\begin{eqnarray}
		\varphi(v)\ && = \frac{1}{p}||Dv||^{p}_{p}\ +\ \frac{1}{2}||Dv||^{2}_{2}-\int_{\Omega}\ F(z,v)dz \nonumber \\
		&& \geq \frac{1}{p}||v||^{p}-c_6||v||^{q}\ \mbox{for\ some}\ c_6>0\ \ (\mbox{see}\ (\ref{eq:41})\ \mbox{and}\ (\ref{eq:5})). \nonumber
	\end{eqnarray}
	
	Because $q>p$, we can find $\tilde{\rho}_0 \in (0,\hat{\rho}_0]$ such that
		\begin{equation}
		\label{eq:42}
			\varphi(v)>0\ \mbox{for all}\ v \in V\ \mbox{with}\ 0<||v||\leq \tilde{\rho}_0.
	\end{equation}
	
	From (\ref{eq:40}) and (\ref{eq:42}) we see that $\varphi$ has local linking at the origin and we can apply Proposition \ref{prop:2.3} of Bartsch \& Li \cite{ref5} and infer that
		$$C_{d_m}(\varphi,\ 0)\ \neq\ 0
		\Rightarrow\ C_{d_m}(\hat{\varphi},\ 0)\ \neq\ 0\ \ (\mbox{see}\ (\ref{eq:38})).$$

	This proves Claim 3.
	
	From Claim 3 and  since $d_m\ \geq\ 2$, we have that $y_0\ \neq\ 0$. Since $y_0 \in [v_-,u_+], y_0 \notin \{v_-,u_+\}$, we see that $y_0$ is a solution of (1) (see (\ref{eq:30})) and the extremality of $v_-$ and $u_+$ implies that $y_0$ is nodal. Finally from the nonlinear regularity theory (see \cite{ref23}, \cite{ref24}) we deduce that $y_0 \in C^{1}_{0}(\bar{\Omega}).$
	\qed

\smallskip
	Now, we can state our first multiplicity theorem for problem (1). Our theorem improves Theorem 1.1 of Sun \cite{ref33}, where the hypotheses on the reaction $f(z,x)$ are more restrictive, no sign information is given for the third solution, no regularity properties are established for the solutions and, finally, no location information is given for them.

	\begin{theorem}
		\label{th:3.6}
			Assume $H_2$ and\ $2<p<\infty.$
				Then\ problem\ (1)\ has\ at\ least\ three\ nontrivial\ solutions $u_0\in {\mathrm int}\, C_+$, $v_0 \in -{\mathrm int}\, C_+$
and $y_0 \in {\mathrm int}_{C^{1}_{0}(\bar{\Omega})}[v_0,\ u_0]$ nodal.
	\end{theorem}
	
	In fact, by strengthening the condition on $f(z,\cdot)$ near the origin (see hypothesis $H_2(iii)$), we can improve the conclusion of the above multiplicity theorem and produce a second nodal solution for a total of four nontrivial solutions, two of constant sign and two nodal (sign changing).
	
	The new hypotheses on the reaction $f(z,x)$ are the following:

	\smallskip
\textbf{$H_3$:} $f:\Omega \times \RR \rightarrow \RR$ is a measurable function such that for a.a. $z \in \Omega$, $f(z,0)=0, f(z,\cdot)\in C^1(\RR)$, hypotheses $H_3(i),(ii)$ are the same as the corresponding hypotheses $H_2(i)(ii)$, and $(iii)$ there exists an integer $m \geq 2$ such that
\begin{eqnarray}
	&& f'_{x}(z,0) \in [\hat{\lambda}_m(2),\ \hat{\lambda}_{m+1}(2)]\ \mbox{a.e.\ in}\ \Omega,\ f'_{x}(\cdot,0)\neq \hat{\lambda}_m(2),\ f'_{x}(\cdot,0)\neq \hat{\lambda}_{m+1}(2)\nonumber \\
	\mbox{and} && f'_{x}(z,0)=\lim_{x\rightarrow 0}\frac{f(z,x)}{x}\ \mbox{uniformly\ for\ a.a.}\ z \in \Omega.\nonumber
\end{eqnarray}
	
	\textbf{Remark.} Now we do not allow for resonance to occur at the origin. Instead we have nonuniform non-resonance in the spectral interval $[\hat{\lambda}_m(2), \hat{\lambda}_{m+1}(2)]$.
	
	\begin{theorem}
		\label{th:3.7}
			Assume $H_3$ and\ $2<p<\infty$. Then\ problem\ (1)\ has\ at\ least\ four\ nontrivial\ solutions
$u_0\in {\mathrm int}\, C_+$,\ $v_0 \in -{\mathrm int}\, C_+$\ and\ $y_0,\ \hat{y} \in {\mathrm int}_{C^{1}_{0}(\bar{\Omega})}[v_0,\ u_0]$\ nodal.
	\end{theorem}
	
	\textit{Proof.}	From Theorem \ref{th:3.6} we already have three nontrivial solutions
		$$u_0 \in \mbox{int}\, C_+,\ v_0 \in -\mbox{int}\, C_+,\ \mbox{and}\ y_0 \in \mbox{int}_{C^{1}_{0}(\bar{\Omega})}[v_0,\ u_0]\ \mbox{nodal}.$$
	
	By virtue of Proposition \ref{prop:3.4}, we can always assume that $u_0 \in \mbox{int}\,C_+$ and $v_0 \in -\mbox{int}\,C_+$ are the extremal nontrivial constant sign solutions of (1) (that is, $u_0=u_+, v_0=v_-$). From Claim 2 in the proof of Proposition \ref{prop:3.5}, we know that $u_0$ and $v_0$ are local minimizers of the functional $\hat{\varphi}$, hence
		\begin{equation}
		\label{eq:43}
			C_k(\hat{\varphi}, u_0)=C_k(\hat{\varphi}, v_0)=\delta_{k,0}\ \ZZ\quad \mbox{for all}\ k\geq 0.
	\end{equation}
	
	We have
		\begin{equation}
		\label{eq:44}
			C_k(\hat{\varphi}, y_0)=\delta_{k,1}\ \ZZ\quad \mbox{for all}\ k\geq 0.
	\end{equation}	
	Using the new stronger condition near the origin (see $H_3(iii)$), we can improve Claim 4 in the proof Proposition \ref{prop:3.5}.
	\\
	
	\textbf{Claim:} $C_k(\hat{\varphi},0) = \delta_{k,d_m} \ZZ$ for all $k \geq 0$, with $d_m = \mbox{dim} \overset{m}{\underset{\mathrm{i=1}}\oplus}E(\hat{\lambda}_i(2))\geq 2.$
	\\
	
	Let $\epsilon \in (0,\hat{\lambda}_m(2))$. By virtue of hypothesis $H_3(iii)$ we can find $\delta = \delta(\epsilon)>0$ such that
		\begin{equation}\label{eq:45}
 \frac{1}{2}[f'_{x}(z,0)-\epsilon]x^2\leq F(z,x) \leq \frac{1}{2}[f'_{x}(z,0)+\epsilon]x^2\ \mbox{for\ a.a.}\ z \in \Omega,\ \mbox{all}\ |x|\leq \delta.
	\end{equation}
	
%%%%%%%%% &&\ \ \ \ \ 0 \leq f(z,x)\leq (f'_{x}(z,0)+\epsilon)|x|\ \mbox{for\ a.a.}\ z \in \Omega,\ \mbox{all}\ &&&&&&&&&&&&|x|\leq \delta \nonumber \\
		

	Let $\Psi: W^{1,p}_{0}(\Omega) \rightarrow \RR$ be the $C^2$-functional defined by
	$$\Psi(u)=\frac{1}{p}||Du||^{p}_{p}+ \frac{1}{2}||Du||^{2}_{2}- \frac{1}{2}\int_{\Omega}f'_{x}(z,0)u^2 dz\quad \mbox{for all}\ u \in W^{1,p}_{0}(\Omega).$$
	
	Evidently $\Psi$ is coercive (recall $p>2$). Also from (\ref{eq:45}) and Chang \cite[p. 336]{ref10}  we have
		\begin{equation}
		\label{eq:46}
			C_k(\varphi,0)=C_k(\Psi,0)\ \mbox{for all}\ k\geq 0.
	\end{equation}
	
	From $H_3(iii)$ and Cingolani \& Vannella \cite{ref12} (see Theorem 1.1), we have
		\begin{eqnarray}
		&& \ \ \ \ \ C_k(\Psi,0)=\delta_{k,d_m}\ \ZZ\ \mbox{for all}\ k \geq 0\nonumber \\
		&& \Rightarrow\ C_k(\varphi,0)=\delta_{k,d_m}\ \ZZ\ \mbox{for all}\ k \geq 0\ (\mbox{see}\ (\ref{eq:46}))\nonumber \\
		&& \Rightarrow\ C_k(\hat{\varphi},0)=\delta_{k,d_m}\ \ZZ\ \mbox{for all}\ k \geq 0\nonumber \\
		&& \hspace{40 mm}(\mbox{recall}\ \varphi|_{[v_0,u_0]}=\hat{\varphi}|_{[v_0,u_0]}\ \mbox{and}\ v_0 \in -\mbox{int}\,C_+,\ u_0 \in \mbox{int}\,C_+).\nonumber
	\end{eqnarray}
	
	This proves the Claim.
	
	Recall that $\hat{\varphi}$ is coercive (see (\ref{eq:30})). Hence
		\begin{equation}
		\label{eq:47}
			C_k(\hat{\varphi},\infty)=\delta_{k,0}\ \ZZ\ \mbox{for all}\ k \geq 0.
	\end{equation}
	
	Suppose $K_{\hat{\varphi}} = \{0, u_0, v_0, y_0\}$. From (\ref{eq:43}), (\ref{eq:44}), (\ref{eq:47}), the Claim and the Morse relation (see (\ref{eq:7})) with $t=-1$, we have
		$2(-1)^0 + (-1)^1 + (-1)^{d_m}=(-1)^0$,
 a\ contradiction.	
	So, we can find $\hat{y} \in K_{\hat{\varphi}} \subseteq [v_0, u_0] \cap C^{1}_{0}(\bar{\Omega})$ such that $\hat{y} \notin \{0, v_0, u_0, y_0\}$. The extremality of $u_0, v_0$ implies that $\hat{y} \in C^{1}_{0}(\bar{\Omega})$ is the second nodal solution of (1).
	\qed



	\section{The Case $1< p < 2$}	
	
	In this section we deal with the case in which $1 < p < 2$. Now the ambient space is the Hilbert Sobolev space $H^{1}_{0}(\Omega)$, which creates more possibilities in the analysis of problem (1) and compensates for the fact that $-\Delta_{p}$ is singular.

	We impose the following conditions on the reaction $f(z,x)$:
	\\
	
	\textbf{$H_4:$} $f:\Omega \times \RR \rightarrow \RR$ is a Carath\'eodory function such that for a.a. $z \in \Omega$, $f(z,0) = 0$, $|f(z,x)|\leq a(z)(1+|x|)$ for all $x\in\RR$ with $a\in L^\infty (\Omega)_+$ and
	\\
	
	\textbf{(i)} there exist an integer $m \geq 2$ and a function $\eta \in L^{\infty}(\Omega)$ such that
		\begin{eqnarray}
		&& \eta(z) \leq \hat{\lambda}_{m+1}(2)\ \mbox{for\ a.a.}\ z \in \Omega,\ \eta \neq \hat{\lambda}_{m+1}(2)\nonumber \\
		&& (f(z,x)-f(z,y))(x-y) \leq \eta(z)(x-y)^2\ \mbox{for\ a.a.}\ z \in \Omega,\ \mbox{all}\ x,y \in \RR;\nonumber
	\end{eqnarray}
	
	\textbf{(ii)} $\hat{\lambda}_m(2)\ \leq\ \liminf_{x\rightarrow \pm \infty}\frac{f(z,x)}{x}$ uniformly for a.a. $z \in \Omega$;
	
	\textbf{(iii)} $\lim_{x\rightarrow \pm \infty} \frac{f(z,x)x - 2F(z,x)}{|x|^{p}}=- \infty$ uniformly for a.a. $z \in \Omega$;
	
	\textbf{(iv)} for every $\rho > 0$, there exists $\xi_{\rho} > 0$ such that	
	$$f(z,x)x\ +\ \xi_{\rho}x^2 \geq 0\ \mbox{for\ a.a.}\ z \in \Omega,\ \mbox{all}\ |x| \leq \rho.$$
	
	As we already mentioned in the Introduction, by employing the  Lyapunoff--Schmidt reduction technique, we will prove two multiplicity theorems for problem (1) producing three and four nontrivial solutions respectively.
	
	Let $\varphi: H^{1}_{0}(\Omega)\rightarrow \RR$ be the $C^{1}$-energy functional for problem (1) defined by
	$$\varphi(u)=\frac{1}{p}||Du||^{p}_{p}\ +\ \frac{1}{2}||Du||^{2}_{2}-\int_{\Omega} F(z,u(z))dz\ \mbox{for all}\ u \in H^{1}_{0}(\Omega).$$
	
	Also as before (see Section 3), let $f_{\pm}(z,x) = f(z, \pm x^{\pm}), F_{\pm}(z,x) = \int^{x}_{0} f_{\pm}(z,s)ds$ and consider the $C^{1}$-functionals $\varphi_{\pm}: H^{1}_{0}(\Omega) \rightarrow \RR$ defined by
	
	$$\varphi_{\pm}(u)=\frac{1}{p}||Du||^{p}_{p}\ +\ \frac{1}{2}||Du||^{2}_{2}- \int_{\Omega}F_{\pm}(z,u(z))dz\ \mbox{for all}\ u \in H^{1}_{0}(\Omega).$$
	
	\begin{proposition}
		\label{prop:4.1} {\sl
	Assume that hypotheses\ $H_4$\ hold.
Then\ the\ functionals\ $\varphi_{\pm}$\ satisfy\ the\ C-condition.}
	\end{proposition}
	
	
		\textit{Proof.}  We\ do\ the\ proof\ for\ the\ functional\ $\varphi_+$,\ the\ proof\ for\ $\varphi_-$\ being\ similar.
		
% Note\ that\ by\ virtue\ of\ hypothesis\ H$_4$(i)\ we\ have
%	\begin{equation}
%		\label{eq:48}
%			|f(z,x)|\ \leq\ \eta(z)|x|\ \mbox{for\ a.a.}\ z \in \Omega,\ \mbox{all}\ x \in \RR.
%	\end{equation}
	
	Let $\{u_n\}_{n\geq 1}\subseteq H^{1}_{0}(\Omega)$ be a sequence such that
		\begin{eqnarray}
		&& |\varphi_+(u_n)|\ \leq\ M_4\ \mbox{for\ some}\ m_4>0,\ all\ n\geq 1,\label{eq:49}\\
		&& (1+||u_n||)\varphi'_{+}(u_n)\rightarrow 0\ \mbox{in}\ H^{-1}(\Omega)=H^{1}_{0}(\Omega)^{*}\ \mbox{as}\ n\rightarrow \infty\,. \label{eq:50}
	\end{eqnarray}
	
	Now $||\cdot||$ is the norm of $H^{1}_{0}(\Omega)$ (that is, $||u||=||Du||_2$ for all $u \in H^{1}_{0}(\Omega)$). From (\ref{eq:50}) we have for all\ $h \in H^{1}_{0}(\Omega)$
		\begin{equation}
		\label{eq:51}
			\left|\left\langle A_{p}(u_n),h\right\rangle + \left\langle A(u_n),h\right\rangle - \int_{\Omega}f_+(z,u_n)hdz\right| \leq \frac{\epsilon_n||h||}{1+||u_n||}\ \ \mbox{with}\ \epsilon_n \rightarrow 0^+.
	\end{equation}
	
	In (\ref{eq:51}) we choose $h=-u^{-}_{n} \in H^{1}_{0}(\Omega)$. Then
		\begin{eqnarray}
		&& ||Du^{-}_{n}||^{p}_{p}\ +\ ||Du^{-}_{n}||^{2}_{2}\ \leq\ \epsilon_n\ \mbox{for all}\ n\geq 1\nonumber \\
		\Rightarrow && u^{-}_{n}\rightarrow 0\ \mbox{in}\ H^{1}_{0}(\Omega).\label{eq:52}
	\end{eqnarray}
	
	Using (\ref{eq:51}) and (\ref{eq:52}), we have
		\begin{equation}
		\label{eq:53}
			\left|\left\langle A_{p}(u^{+}_{n}),h\right\rangle\ +\ \left\langle A(u^{+}_{n}),h\right\rangle-\int_{\Omega}f(z,u^{+}_{n})hdz\right|\ \leq\ \epsilon'_{n}||h||\ \mbox{with}\ \epsilon'_{n}\rightarrow 0^+.
	\end{equation}
	
	Suppose that $||u^{+}_{n}||\rightarrow \infty$. Let $y_n=\frac{u^{+}_{n}}{||u^{+}_{n}||} n\geq 1$. Then $||y_n||=1$ for all $n\geq 1$. So, we may assume that
		\begin{equation}
		\label{eq:54}
			y_n \stackrel{w}{\rightarrow} y\ \mbox{in}\ H^{1}_{0}(\Omega)\ \mbox{and}\ y_n \rightarrow y\ \mbox{in}\ L^2(\Omega).
	\end{equation}
	
	From (\ref{eq:53}) we have for all $n\geq 1$
	\begin{equation}
		\label{eq:55}
			\left|\frac{1}{||u^{+}_{n}||^{2-p}}\left\langle A_{p}(y_n),h\right\rangle\ +\ \left\langle A(y_n),h\right\rangle-\int_{\Omega}\frac{f(z,u^{+}_{n})}{||u^{+}_{n}||}hdz\right|\ \leq\ \epsilon'_{n}||h||\,.
	\end{equation}
	
	From the growth condition on $f(z,\cdot)$, it is clear that $\left\{\frac{N_f(u^{+}_{n})}{||u^{+}_{n}||}\right\}_{n\geq 1}\subseteq L^2(\Omega)$ is bounded. So, we may assume that
		\begin{equation}
		\label{eq:56}
			\frac{N_f(u^{+}_{n})}{||u^{+}_{n}||}\stackrel{w}{\rightarrow}g\ \mbox{in}\ L^2(\Omega).
	\end{equation}
	
	Moreover, using   $H_4(i),\,(ii)$, we have
		\begin{equation}
		\label{eq:57}
			g=\hat{\xi}y\ \mbox{with}\ \hat{\lambda}_m(2)\leq \hat{\xi}(z) \leq \eta(z)\ \mbox{a.e.\ in}\ \Omega.
	\end{equation}
	
	In (\ref{eq:50}) we choose $h=y_n-y \in H^{1}_{0}(\Omega)$, pass to the limit as $n\rightarrow \infty$ and use (\ref{eq:54}) and (\ref{eq:56}). Then
		\begin{eqnarray}
		&& \lim_{n\rightarrow \infty}\left\langle A(y_n),y_n-y\right\rangle=0\ (\mbox{recall\ that}\ p<2).\nonumber \\
		\Rightarrow && y_n \rightarrow y\ \mbox{in}\ H^{1}_{0}(\Omega)\ (\mbox{see\ Proposition}\ \ref{prop:2.5}),\ \mbox{hence}\ ||y||=1.\label{eq:58}
	\end{eqnarray}
	
	If in (\ref{eq:55}) we pass to the limit as $n\rightarrow \infty$ and use (\ref{eq:56}), (\ref{eq:57}) and (\ref{eq:58}), then
		\begin{eqnarray}
		&& \left\langle A(y),h\right\rangle=\int_{\Omega}\hat{\xi}yhdz\ \mbox{for all}\ h \in H^{1}_{0}(\Omega)
		\Rightarrow  A(y)=\hat{\xi}y\nonumber \\
		\Rightarrow && -\Delta y(z)=\hat{\xi}(z)y(z)\ \mbox{a.e.\ in}\ \Omega,\ y|_{\partial \Omega}= 0.\label{eq:59}
	\end{eqnarray}
	
	If $\hat{\xi} \neq \hat{\lambda}_m(2)$ (see (\ref{eq:57})), then by  Proposition \ref{prop:2.4}, we have
		\begin{equation}
		\label{eq:60}
			\tilde{\lambda}_m(2, \hat{\xi})\ <\ \tilde{\lambda}_m(2,\hat{\lambda}_m(2))=1\ \&\ \tilde{\lambda}_{m+1}(2,\hat{\lambda}_{m+1}(2))=1\ < \tilde{\lambda}_{m+1}(2,\eta)\ \leq\ \tilde{\lambda}_{m+1}(2,\hat{\xi}).
	\end{equation}
	
	From (\ref{eq:59}) and (\ref{eq:60}), it follows that $y=0$, which contradicts (\ref{eq:58}).
	
	Now suppose that $\hat{\xi}(z)=\hat{\lambda}_m(2)$ a.e. in $\Omega$. Then $y \in E(\hat{\lambda}_m(2))\backslash \{0\}$ and so $y(z)>0$ for all $z \in \Omega$ (by the UCP and since $y \geq 0$). This implies that $u^{+}_{n}(z)\rightarrow +\infty$ for a.a. $z \in \Omega$. Then hypothesis $H_4(iii)$ and Fatou's Lemma imply that
		\begin{equation}
		\label{eq:61}
			\frac{1}{||u^{+}_{n}||^{p}}\int_{\Omega}\left[f(z,u^{+}_{n})u^{+}_{n}- 2F(z,u^{+}_{n})\right]dz\rightarrow - \infty\ \mbox{as}\ n\rightarrow \infty.
	\end{equation}
	
	From (\ref{eq:49}) and (\ref{eq:52}), we have  for\ some\ $M_5>0$ and for \ all\ $n \geq 1$
		\begin{eqnarray}
		\label{eq:62}
			&& \frac{2}{p}||Du^{+}_{n}||^{p}_{p}\ +\ ||Du^{+}_{n}||^{2}_{2}- 2\int_{\Omega}F(z,u^{+}_{n})dz\ \geq\ -M_5 \nonumber
	\end{eqnarray}
	
	Also from (\ref{eq:53}) with $h=u^{+}_{n}$, we have
		\begin{equation}
		\label{eq:63}
			-||Du^{+}_{n}||^{p}_{p}-||Du^{+}_{n}||^{2}_{2}\ +\ \int_{\Omega}f(z,u^{+}_{n})u^{+}_{n} \geq - \epsilon'_{n}||u^{+}_{n}||
	\end{equation}
	
	Adding (\ref{eq:61}) and (\ref{eq:63}), we obtain
		\begin{eqnarray}
		&& \int_{\Omega}\left[f(z,u^{+}_{n})u^{+}_{n}-2F(z,u^{+}_{n})\right]dz\ \geq- M_6(1+||Du^{+}_{n}||^{p}_{p}\ +\ ||u^{+}_{n}||)\nonumber \\
		\Rightarrow && \frac{1}{||u^{+}_{n}||^{p}}\int_{\Omega}\left[f(z,u^{+}_{n})u^{+}_{n}- 2F(z,u^{+}_{n})\right]dz\ \geq-M_6\left(\frac{1}{||u^{+}_{n}||^{p}}\ +\ ||Dy_n||^{p}_{p}\ +\ \frac{1}{||u^{+}_{n}||^{p-1}}\right)\nonumber \\
		&& \mbox{for all}\ n\geq 1\label{eq:64}
	\end{eqnarray}
	
	Comparing (\ref{eq:61}) and (\ref{eq:64}) and since $p>1$, we reach a contradiction. This proves that $\{u^{+}_{n}\}_{n\geq 1}\subseteq H^{1}_{0}(\Omega)$ is bounded, hence $\{u_n\}_{n \geq 1}\subseteq H^{1}_{0}(\Omega)$ is bounded (see (\ref{eq:52})). So, we may assume that
		\begin{equation}
		\label{eq:65}
			u_n\stackrel{w}{\rightarrow} u\ \mbox{in}\ H^{1}_{0}(\Omega)\ \mbox{and}\ u_n \rightarrow u\ \mbox{in}\ L^2(\Omega).
	\end{equation}
	
	If in (\ref{eq:51}) we choose $h=u_n-u \in H^{1}_{0}(\Omega)$ and pass to the limit as $n \rightarrow \infty$, then recalling that $\{N_f(u_n)\}_{n \geq 1}\subseteq L^2(\Omega)$ is bounded, we have
		\begin{eqnarray}
		&& \lim_{n\ri\infty}\left[\left\langle A_{p}(u_n),u_n-u\right\rangle\ +\ \left\langle A(u_n),u_n-u\right\rangle\right]=0\nonumber \\
		\Rightarrow	&&  \limsup_{n\ri\infty}\left[\left\langle A_{p}(u),u_n-u\right\rangle\ +\ \left\langle A(u_n),u_n-u\right\rangle\right]\ \leq\ 0\ \ (\mbox{since}\ A_{p}\ \mbox{is\ monotone})\nonumber \\
		\Rightarrow	&& \limsup_{n\ri\infty}\ \left\langle A(u_n),u_n-u\right\rangle\ \leq\ 0\nonumber \\
		\Rightarrow	&& u_n \rightarrow u\ \mbox{in}\ H^{1}_{0}(\Omega)\ \ (\mbox{see\ Proposition}\ \ref{prop:2.5}).\nonumber
	\end{eqnarray}
	This proves that the functional $\varphi_+$ satisfies the C-condition.
	Similarly we show that the functional $\varphi_-$ satisfies the C-condition.
	\qed

\smallskip
	Straightforward changes in the above proof lead to the same result for the functional $\varphi$.
	
	\begin{proposition}
		\label{prop:4.2}
			{\sl Assume that hypotheses\ H$_4$\ hold.
Then\ the\ functional\ $\varphi$\ satisfies\ the\ C-condition.}
	\end{proposition}
	
	Next we verify the mountain pass geometry for the functionals $\varphi_{\pm}$.
	
	\begin{proposition}
		\label{prop:4.3}
			{\sl Assume that hypotheses\ H$_4$\ hold.
Then\ $u=0$\ is\ a\ local\ minimizer\ for\ the\ functionals\ $\varphi_{\pm}$\ and\ $\varphi$.
		}
	\end{proposition}
	
	\textit{Proof.} We do the proof for the functional $\varphi_+$, the proofs for $\varphi_-$ and $\varphi$ being similar.
	
	Hypotheses $H_4(i),\, (iv)$ imply that
		\begin{equation}
		\label{eq:66}
			\lim_{x\rightarrow 0}\frac{f(z,x)}{|x|^{p-2}x}=0\ \mbox{uniformly\ for\ a.a.}\ z \in \Omega\ \ (\mbox{recall}\ p<2).
	\end{equation}
	
	So, given $\epsilon>0$, we can find $\delta=\delta(\epsilon)>0$ such that
		\begin{eqnarray}
		&& |f(z,x)|\ \leq\ \epsilon|x|^{p-1}\ \mbox{for\ a.a.}\ z \in \Omega,\ \mbox{all}\ |x|\leq \delta \nonumber \\
		\Rightarrow && F_+(z,x)\ \leq\ \frac{\epsilon}{p}(x^+)^{p}\ \mbox{for\ a.a.}\ z \in \Omega,\ \mbox{all}\ |x|\leq \delta.\label{eq:67}
	\end{eqnarray}
	
	Let $u \in C^{1}_{0}(\bar{\Omega})$ with $||u||_{C^{1}_{0}(\bar{\Omega})}\leq \delta$. Then
		\begin{equation}\label{eq:68}
		\varphi_+(u) =\ \frac{1}{p}||Du||^{p}_{p}\ +\ \frac{1}{2}||Du||^{2}_{2}-\int_{\Omega}F_+(z,u)dz \geq\ \frac{1-\epsilon}{p}||u||^{p}\ (\mbox{see}\ (\ref{eq:67})).
	\end{equation}
	
	Choosing $\epsilon \in (0,1)$, from (\ref{eq:68}) we see that $u=0$ is a local $C^{1}_{0}(\bar{\Omega})$-minimizer of $\varphi_+$. Invoking Proposition \ref{prop:2.2}, we deduce that $u=0$ is a local $H^{1}_{0}(\Omega)$-minimizer of $\varphi_+$.
	Similarly for the functionals $\varphi_-$ and $\varphi$.
	\qed
	
	\begin{proposition}
		\label{prop:4.4}
			{\sl Assume that hypotheses\ H$_4$\ hold\ and\ $u \in E(\hat{\lambda}_{m-1}(2))\backslash\{0\}$\ with\ $||u||_2=1$.
				Then\ $\varphi_{\pm}(tu)\rightarrow - \infty$\ as\ $t\rightarrow \pm \infty$.
		}
	\end{proposition}
	
	\textit{Proof.} By virtue hypothesis $H_4(ii)$, given $\epsilon \in (0,\hat{\lambda}_m(2) - \hat{\lambda}_{m-1}(2))$, we can find $M_7 = M_7(\epsilon)>0$ such that
		\begin{equation}
		\label{eq:69}
			F(z,x)\geq\frac{1}{2}(\hat{\lambda}_m(2)-\epsilon)x^2\ \mbox{for\ a.a.}\ z\in \Omega,\ \mbox{all}\ |x|\geq M_7.
	\end{equation}
	
	For $t>0$, we have
		\begin{eqnarray}
		\varphi_+(tu) && =\ \frac{t^{p}}{p}||Du||^{p}_{p}\ +\ \frac{t^2}{2}||Du||^{2}_{2}- \int_{\Omega}F_+(z,tu)dz \nonumber \\
		&& =\ \frac{t^{p}}{p}||Du||^{p}_{p}\ +\ \frac{t^2}{2}||Du||^{2}_{2}-\int_{\{|tu|\geq M_7\}}F_+(z,tu)dz-\int_{\{0\leq |tu| < M_7\}}F_+(z,tu)dz \nonumber \\
		&& \leq\  \frac{t^{p}}{p}||Du||^{p}_{p}\ +\ \frac{t^2}{2}\hat{\lambda}_{m-1}(2)- \frac{t^2}{2}\left[\hat{\lambda}_m(2)-\epsilon\right]\ +\ \xi_*(t)\nonumber \\
		&& \hspace{70 mm} \mbox{with}\ \xi_*(t)\ \mbox{bounded}\ \ \ (\mbox{see}\ (\ref{eq:69}))\nonumber \\
		&& =\ \frac{t^{p}}{p}||Du||^{p}_{p}\ +\ \frac{t^2}{2}\left[\hat{\lambda}_{m-1}(2)\ +\ \epsilon- \hat{\lambda}_m(2)\right]\ +\ \xi_*(t). \label{eq:70}
	\end{eqnarray}
	
	Since $\hat{\lambda}_{m-1}(2)+\epsilon < \hat{\lambda}_m(2)$ and $p<2$, from (\ref{eq:70}) we infer that		 $$\varphi_+(tu)\rightarrow - \infty\quad \mbox{as}\ t\rightarrow + \infty.$$
	
	Similarly for the functional $\varphi_-$.
	\qed
	
	Now we are ready to produce two nontrivial constant sign solutions for problem (1).
	
	\begin{proposition}
		\label{prop:4.5}
			{\sl Assume that hypotheses\ H$_4$\ hold.
Then\ problem\ (1)\ has\ at\ least\ two\ nontrivial\ constant\ sign\ solutions
	$$u_0 \in {\mathrm int}\,C_+\qquad \mbox{and}\qquad v_0 \in -{\mathrm int}\,C_+.$$}\end{proposition}
	
	\textit{Proof.} By virtue of Proposition \ref{prop:4.3}, $u=0$ is a critical point of $\varphi_+$. Then this is an isolated critical point of $\varphi_+$. Otherwise, because $K_{\varphi_+}\subseteq C_+$, we will have a whole sequence of distinct nontrivial positive solutions of (1). So, we can find $\rho \in (0,1)$ small such that
		\begin{equation}
		\label{eq:71}
			\varphi_+(0)=0\ <\inf\left[\varphi_+(u):||u||=\rho\right]=\eta^{+}_{\rho}.
	\end{equation}
	
	Combing with Propositions \ref{prop:4.1} and \ref{prop:4.4}, we see that we can apply Theorem \ref{th:2.1} (the mountain pass theorem) and find $u_0 \in H^{1}_{0}(\Omega)$ such that
		\begin{equation}
		\label{eq:72}
			u_0 \in K_{\varphi_+}\quad \mbox{and}\quad \eta^{+}_{\rho}\leq \varphi_+(u_0).
	\end{equation}
	
	From (\ref{eq:71}) and (\ref{eq:72}) we deduce that
$u_0\geq 0$, $u_0 \neq 0$ and $\varphi'_{+}(u_0)=0$, hence
$u_0$\ is\ a\ nontrivial\ solution\ of\ (1)\ and\ so\ $u_0 \in C_+ \backslash\{0\}$
 (nonlinear\ regularity).
	
	Let $\rho=||u_0||_{\infty}$ and let $\xi_{\rho}>0$ be as postulated by hypothesis $H_4(iv)$. Then
		\begin{eqnarray}
		&& -\Delta_{p}u_0(z)-\Delta u_0(z)+ \xi_{\rho}u_0(z)=f(z,u_0(z))+\xi_{\rho}u_0(z)\geq 0\ \mbox{a.e.\ in}\ \Omega\nonumber \\
		\Rightarrow && \Delta_{p}u_0(z)+\Delta u_0(z)\leq \xi_{\rho}u_0(z)\ \mbox{a.e.\ in}\ \Omega.\nonumber
	\end{eqnarray}
	
	Invoking the boundary point theorem of Pucci \& Serrin \cite[p. 120]{ref31}, we deduce that $u_0 \in \mbox{int}\,C_+$.
	Similarly, working with the functional $\varphi_-$, we produce a nontrivial negative solution $v_0 \in -\mbox{int}\,C_+$ for problem (1).
	\qed
	
	Let $Y=\overset{m}{\underset{\mathrm{i=1}}\oplus}E(\hat{\lambda}_i(2))$ and $\hat{H}=Y^\perp$, hence
$H^{1}_{0}(\Omega)=Y \oplus \hat{H}.$	
	The Lyapunoff-Schmidt reduction technique will be based on this decomposition. We should mention that the reduction technique was first developed for elliptic equations with a $C^2$-energy functional by Amann \cite{ref2}, Castro \& Lazer \cite{ref9} and Thews \cite{ref34}. The next proposition is a crucial step in the implementation of the reduction technique.
	
	\begin{proposition}
		\label{prop:4.6}
			{\sl Assume that  hypotheses\ H$_4$\ hold.
Then\ there\ exists\ a\ continuous\ map\ $\gamma_0:Y\rightarrow \hat{H}$\ such\ that
  $$\varphi(y+\gamma_0(y))=\inf\left[\varphi(y+\hat{u}):\hat{u} \in \hat{H}\right]\ \mbox{for all}\ y \in Y.$$
			}
	\end{proposition}
	
	\textit{Proof.} Let $y \in Y$ and consider the $C^{1}$-functional $\varphi_y(u):H^{1}_{0}(\Omega)\rightarrow \RR$ defined by
	$$\varphi_y(u)=\varphi(y+u)\ \mbox{for all}\ u \in H^{1}_{0}(\Omega).$$
Let $i:\hat{H}\rightarrow H^{1}_{0}(\Omega)$ be the inclusion map. We set $\hat{\varphi}_y=\varphi_y \circ i:\hat{H}\rightarrow \RR$. From the chain rule we have
		\begin{equation}
		\label{eq:73}
			\hat{\varphi}'_{y}(\hat{u})=p_{\hat{H}^*}\varphi'_{y}(\hat{u})\ \mbox{for all}\ \hat{u}\in \hat{H},
\end{equation}
where $p_{\hat{H}^*}$ is the orthogonal projection of $H^{-1}(\Omega)=H^{1}_{0}(\Omega)^*$ onto $\hat{H}^*$. Let $\hat u_1,\, \hat u_2 \in \hat{H}$. We have
\begin{eqnarray}
	&&\ \ \ \ \left\langle \hat{\varphi}'_{y}(\hat{u}_1)-\hat{\varphi}'_{y}(\hat{u}_2),\ \hat{u}_1-\hat{u}_2\right\rangle_{\hat{H}}\nonumber \\
	&& =\ \left\langle \varphi'_{y}(\hat{u}_1)-\varphi'_{y}(\hat{u}_2),\ \hat{u}_1-\hat{u}_2\right\rangle\nonumber \\
	&& =\ \left\langle A_{p}(y+\hat{u}_1)-A_{p}(y+\hat{u}_2),\ \hat{u}_1-\hat{u}_2\right\rangle\ +\ \left\langle A(\hat{u}_1-\hat{u}_2),\hat{u}_1-\hat{u}_2\right\rangle\ -\nonumber \\
	&& \ \ \ -\int_{\Omega}\left[f(z,y+\hat{u}_1)-f(z,y+\hat{u}_1)\right]\ (\hat{u}_1-\hat{u}_2)dz\nonumber \\
	&& \geq\ ||D(\hat{u}_1-\hat{u}_2)||^{2}_{2}-\int_{\Omega}\eta(\hat{u}_1-\hat{u}_2)^2 dz\nonumber \\
	&& \hspace{60 mm} (\mbox{since}\ A_{p}\ \mbox{is\ monotone\ and\ see}\ H_4(i))\nonumber \\
	&& \geq\ C_7||\hat{u}_1-\hat{u}_2||^2\ \mbox{for\ some}\ C_7>0\ \ (\mbox{see}\ H_4(i)\ \mbox{and\ recall}\ \hat{u}_1,\hat{u}_2 \in \hat{H})\nonumber\\
	\Rightarrow && \hat{\varphi}'_{y}\ \mbox{is\ strongly\ monotone}
	\Rightarrow  \hat{\varphi}_y\ \mbox{is\ strictly\ convex}.\nonumber
\end{eqnarray}

Also, note that
$$ \left\langle \hat{\varphi}'_{y}(\hat{u}),\hat{u}\right\rangle=\left\langle \hat{\varphi}'_{y}(\hat{u})-\hat{\varphi}'_{y}(0),\hat{u}\right\rangle + \left\langle \hat{\varphi}'_{y}(0),\hat{u}\right\rangle  \geq C_7||\hat{u}||^2-C_8||\hat{u}||\ \mbox{for\ some}\ C_8>0.
$$ Thus, $\hat{\varphi}'_{y}$ is\ coercive.
	The map $\hat{\varphi}'_{y}$ is continuous and strongly monotone, hence it is maximal monotone. This fact combined with the coercivity of $\hat{\varphi}'_{y}$ implies that the map is surjective (see, for example, Gasinski \& Papageorgiou \cite[p. 320]{ref18}). So, we can find $\hat{u}_0 \in \hat{H}$ such that	 
	$\hat{\varphi}'_{y}(\hat{u}_0) = 0.$
	The strong monotonicity of $\hat{\varphi}'_{y}$ implies that this $\hat{u}_0$ is unique and in fact is the unique global minimizer of the strictly convex functional $\hat{u}\rightarrow \hat{\varphi}_y(\hat{u})$, $ \hat{u} \in \hat{H}$. So, we can define the single valued map $\gamma_0:Y\rightarrow \hat{H}$ which to each $y \in Y$ assigns this unique global minimizer of $\hat{\varphi}_y(\cdot)$. We have
		\begin{equation}
		\label{eq:74}
			0=\hat{\varphi}'_{y}(\gamma_0(y))=p_{\hat{H}^*}\varphi'(y+\gamma_0(y))\ \&\ \varphi(y+\gamma_0(y))=\inf[\varphi(y+\hat{u}):\hat{u} \in \hat{H}].
	\end{equation}
	
		Next we show the continuity of the map $\gamma_0:Y\rightarrow \hat{H}$. To this end, let $y_n\rightarrow y$ in $Y$. The coercivity of $\hat{\varphi}'_{y}$ and (\ref{eq:74}) imply that
				$$\{\gamma_0(y_n)\}_{n\geq 1}\subseteq \hat{H} \subseteq H^{1}_{0}(\Omega)\ \mbox{is\ bounded}.$$
		So, we may assume that
		$$\gamma_0(y_n)\stackrel{w}{\rightarrow} h\ \mbox{in}\ H^{1}_{0}(\Omega)\quad \mbox{and}\quad h \in \hat{H}.$$
	
		Using the Sobolev embedding theorem, we can easily check that $\varphi$ is sequentially weakly lower semi-continuous. Hence
				\begin{equation}
			\label{eq:75}
				\varphi(y+h)\leq \liminf_{n\ri\infty}\varphi(y_n+\gamma_0(y_n)).
		\end{equation}
		
		From (\ref{eq:74}) we have
				\begin{eqnarray}
			&& \varphi(y_n+\gamma_0(y_n))\leq \varphi(y_n+\hat{u})\ \mbox{for all}\ \hat{u}\in\hat{H} \nonumber \\
			\Rightarrow && \limsup_{n\ri\infty} \varphi(y_n+\gamma_0(y_n))\leq \varphi(y+\hat{u})\ \mbox{for all}\ \hat{u}\ \mbox{in}\ \hat{H}\ (\mbox{since}\ y_n \rightarrow y\ \mbox{in}\ Y)\nonumber \\
			\Rightarrow && \varphi(y+h)\leq\varphi(y+\hat{u})\ \mbox{for all}\ \hat{u}\in\hat{H}\ (\mbox{see}\ (\ref{eq:75}))\nonumber \\
			\Rightarrow && h=\gamma_0(y)\ (\mbox{see}\ (\ref{eq:74}))\nonumber \\
			\Rightarrow && \gamma_0(y_n)\stackrel{w}{\rightarrow}\gamma_0(y)\ \mbox{in}\ H^{1}_{0}(\Omega).\label{eq:76}
		\end{eqnarray}
		
		Moreover, again from (\ref{eq:74}), we have
				\begin{eqnarray}
			&& p_{\hat{H}^*}\varphi'(y_n+\gamma_0(y_n))=0\ \mbox{for all}\ n\geq 1 \nonumber \\
			\Rightarrow &&  p_{\hat{H}^*}\left[A_{p}(y_n+\gamma_0(y_n)) + A(y_n+\gamma_0(y_n))\right]= p_{\hat{H}^*}N_f(y_n+\gamma_0(y_n))\ \mbox{for all}\ n\geq 1.\nonumber
		\end{eqnarray}
		
		Acting on this equation with $\gamma_0(y_n)-\gamma_0(y)$ and passing to the limit as $n \rightarrow \infty$, as before (see the proof of Proposition \ref{prop:3.4}), exploiting the monotonicity of $A_{p}$, we obtain
				\begin{eqnarray}
			&& \limsup_{n\ri\infty}\left\langle A(y_n+\gamma_0(y_n)),\ \gamma_0(y_n)-\gamma_0(y)\right\rangle\ \leq\ 0 \nonumber \\
			\Rightarrow && \gamma_0(y_n)\rightarrow \gamma_0(y)\ \mbox{in}\ H^{1}_{0}(\Omega)\ (\mbox{see\ Proposition}\ \ref{prop:2.5}) \nonumber \\
			\Rightarrow && \gamma_0(\cdot)\ \mbox{is\ continuous}.\nonumber
		\end{eqnarray}
	\qed
	
	We consider the functional $\Psi:Y\rightarrow \RR$ defined by
		$$\Psi(y)=\varphi(y+\gamma_0(y))\ \mbox{for all}\ y \in Y.$$
	
	The next lemma is not immediately clear, since $\gamma_0$ is only continuous.
	
	
	
	\begin{lemma}
		\label{lem:4.7}
		Assume that hypotheses $H_4$ hold. Then $\Psi \in C^1(Y).$
	\end{lemma}
	
	\textit{Proof.} Let $y,v \in Y$ and $t>0$ (the analysis is similar if $t<0$). Then
		\begin{eqnarray}
		&& \frac{\Psi(y+tv)-\Psi(y)}{t}\ \leq\ \frac{\varphi(y+tv+\gamma_0(y))- \varphi(y+\gamma_0(y))}{t}\nonumber \\
		\Rightarrow && \limsup_{t\rightarrow 0}\frac{\Psi(y+tv)-\Psi(y)}{t}\ \leq\ \left\langle \varphi'(y+\gamma_0(y)),v\right\rangle.\label{eq:77}
	\end{eqnarray}
	
	Also, we have
		\begin{eqnarray}
		&& \frac{\Psi(y+tv)-\Psi(y)}{t}\ \geq\ \frac{\varphi(y+tv+\gamma_0(y+tv))- \varphi(y+\gamma_0(y+tv))}{t}\nonumber \\
		\Rightarrow && \liminf_{t\rightarrow 0}\frac{\Psi(y+tv)-\Psi(y)}{t}\ \geq\ \left\langle \varphi'(y+\gamma_0(y)),v\right\rangle.\label{eq:78}\\
		&& \hspace{60 mm}(\mbox{since}\ \varphi \in C^1(H^{1}_{0}(\Omega))\ \mbox{and}\ \gamma_0\ \mbox{is\ continuous}).\nonumber
	\end{eqnarray}
	
	From (\ref{eq:77}) and (\ref{eq:78}) it follows that $\Psi$ is G\^ateaux differentiable at $y \in Y$ in the direction $v \in Y$ and if by $i_Y:Y\rightarrow H^{1}_{0}(\Omega)$ we denote the inclusion map, then
		\begin{eqnarray}
		&& \left\langle \Psi'_{G}(y),v\right\rangle_Y=\left\langle \varphi'(y+\gamma_0(y)), i_Y(v)\right\rangle\ \mbox{for all}\ v \in Y\nonumber \\
		\Rightarrow && \Psi'_{G}(y)=p_{Y^*}\varphi'(y+\gamma_0(y)),\nonumber
	\end{eqnarray}
	where $p_{Y^*}$ is the orthogonal projection of $H^{-1}(\Omega)=H^{1}_{0}(\Omega)^*$ onto $Y^*$. Hence by virtue of the continuity of $\gamma_0(\cdot)$ (see Proposition \ref{prop:4.6}), we see that $y \rightarrow \Psi'_{G}(y)$ is continuous and this proves that $\Psi \in C^1(Y)$.
	\qed
	
	\begin{proposition}
		\label{prop:4.8}
			{\sl Assume that hypotheses\ H$_4$\ hold.
 Then\ $\Psi$\ is\ anticoercive\ (that\ is,\ if\ $||y||\rightarrow +\infty$,\ $y \in Y$,\ then\ $ \Psi(y)\rightarrow -\infty$).
			}
	\end{proposition}
	
	\textit{Proof.} We argue by contradiction. So, suppose we can find $\{y_n\}_{n\geq 1}\subseteq Y$ and $M_8>0$ such that
	$||y_n||\rightarrow\infty$ and $\Psi(y_n)\geq -M_8$ for all $n\geq 1.$
	
	We have
		\begin{equation}
		\label{eq:79}
			-M_8\leq \Psi(y_n)\ \leq\ \varphi(y_n)=\frac{1}{p}||Dy_n||^{p}_{p}\ +\ \frac{1}{2}||Dy_n||^{2}_{2}-\int_{\Omega}F(z,y_n)dz.
	\end{equation}
	
	Let $v_n = \frac{y_n}{||y_n||}\ n\geq 1$. Then $v_n \in Y$ and $||v_n||=1$ for all $n \geq 1$. The finite dimensionality of $Y$ implies that at least for a subsequence, we have
		\begin{equation}
		\label{eq:80}
			v_n \rightarrow v\ \mbox{in}\ H^{1}_{0}(\Omega)\ \mbox{and}\ v \in Y,\ ||v||=1.
	\end{equation}

	From (\ref{eq:79}) he have
	\begin{equation}
		\label{eq:81}
			-\frac{M_8}{||y_n||^2}\leq \frac{1}{p}\ \frac{1}{||y_n||^{2-p}}||Dv_n||^{p}_{p}\ +\ \frac{1}{2}||Dv_n||^{2}_{2}-\int_{\Omega}\frac{F(z,y_n)}{||y_n||^2}dz.
	\end{equation}	
	Hypothesis $H_4(i)$ implies that	
	$$\left\{\frac{F(\cdot,y_n(\cdot))}{||y_n||^2}\right\}_{n\geq 1}\ \subseteq\ L^{1}(\Omega)\ \mbox{is\ uniformly\ integrable}.$$
	So, by the Dunford-Pettis theorem and hypothesis $H_4(i)$, we have
		\begin{equation}
		\label{eq:82}
			\frac{F(\cdot,y_n(\cdot))}{||y_n||^2} \stackrel{w}{\rightarrow} \frac{1}{2}\xi^* v^2\ \mbox{in}\ L^{1}(\Omega)\ \mbox{with}\ \xi^*(z)\leq \eta(z)\ \mbox{a.e.\ in}\ \Omega.
	\end{equation}
	
	Passing to the limit as $n \rightarrow \infty$ in (\ref{eq:81}) and using (\ref{eq:80}) and (\ref{eq:82}), we obtain
	$$0 \leq \frac{1}{2}||Dv||^{2}_{2}-\frac{1}{2}\int_{\Omega}\xi^* v^2 dz<0\quad \mbox{when $\xi^*\not=\hat\lambda_m(2)$}\ (\mbox{recall}\ p<2),$$
	a contradiction.  If $\xi^*=\hat\lambda_m(2)$, then $v\in E(\hat\lambda_m(2))\setminus\{0\}$ and so the argument of Proposition \ref{prop:4.1} leads again to a contradiction.
This proves the anti-coercivity of $\Psi$.
	\qed
	
	In particular the above proposition implies that $\Psi$ satisfies the C-condition (since $\Psi$ is coercive). Now we can state the first multiplicity results for problem (1) when $1 < p <2.$
	
	\begin{theorem}
		\label{th:4.9}
			Assume $H_4$ and\ $1<p<2$.
			Then\ problem\ (1)\ has\ at\ least\ three\ nontrivial\ solutions
$u_0\in {\mathrm int}\,C_+$,\ $v_0 \in -{\mathrm int}\,C_+$\ and\ $y_0 \in C^{1}_{0}(\bar{\Omega})$.
	\end{theorem}
	
	\textit{Proof.} From Proposition \ref{prop:4.5}, we already have two nontrivial solutions of constant sign, namely
	$u_0 \in {\rm int}\,C_+$ and\ $v_0 \in -\mbox{int}\,C_+.$
	From the proof of Proposition \ref{prop:4.5} 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 $\varphi_-$. We know that
		\begin{eqnarray}
		&& \varphi_+{_{|{C_+}}}=\varphi{_{|{C_+}}}\ \mbox{and}\ \varphi_-{_{|{-C_+}}}=\varphi{_{|{-C_+}}}\nonumber \\
		\Rightarrow && C_k(\varphi_+|_{C^{1}_{0}(\bar{\Omega})},u_0)= C_k(\varphi|_{C^{1}_{0}(\bar{\Omega})},u_0)\ \mbox{and}\ C_k(\varphi_-|_{C^{1}_{0}(\bar\Omega)},v_0)= C_k(\varphi|_{C^{1}_{0}(\bar\Omega)},v_0)\nonumber \\
		&& \hspace{100 mm} \mbox{for all}\ k\geq 0.\label{eq:83}
	\end{eqnarray}
	
	From Bartsch \cite[Proposition 2.6]{ref4}  and Palais \cite{ref29} we have for all $k\geq 0$
		\begin{eqnarray}
		&& C_k(\varphi_+|_{C^{1}_{0}(\bar{\Omega})},u_0)=C_k(\varphi_+,u_0)\ \mbox{and}\ C_k(\varphi|_{C^{1}_{0}(\bar{\Omega})},u_0)=C_k(\varphi,u_0)\ \label{eq:84}\\
		&& C_k(\varphi_-|_{C^{1}_{0}(\bar{\Omega})},v_0)=C_k(\varphi_-,v_0)\ \mbox{and}\ C_k(\varphi|_{C^{1}_{0}(\bar{\Omega})},v_0)=C_k(\varphi,v_0).\ \label{eq:85}
	\end{eqnarray}
	
	From (\ref{eq:83}), (\ref{eq:84}), (\ref{eq:85}) and since $u_0$ and $v_0$ are critical points of mountain pass type for $\varphi_+$ and $\varphi_-$ respectively, we have
		\begin{equation}
		\label{eq:86}
			C_1(\varphi,u_0)\neq 0,\ C_1(\varphi,v_0)\neq 0.
	\end{equation}
	
	Let $\bar{u}_0=p_{Y}(u_0)$ and $\bar{v}_0=p_Y(v_0)$. From Liu \& Li \cite{ref27}, we have
		\begin{eqnarray}
		&& C_k(\varphi,u_0)=C_k(\Psi,\bar{u}_0)\ \mbox{and}\ C_k(\varphi,v_0)=C_k(\Psi,\bar{v}_0)\ \mbox{for all}\ k\geq 0\nonumber \\
		\Rightarrow && C_1(\Psi,\bar{u}_0)\neq 0\ \mbox{and}\ C_1(\Psi,\bar{v}_0)\neq 0\ (\mbox{see}\ (\ref{eq:86})).\label{eq:87}
	\end{eqnarray}
		
	From Proposition \ref{prop:4.8} we know that $\Psi$ is anticoercive on $Y$. Hence by the Weierstrass theorem, we can find $\bar{y}_0 \in Y$ such that
			\begin{eqnarray}
		&& \Psi(\bar{y}_0)=\max [\Psi(y):y \in Y]\nonumber \\
		\Rightarrow && C_k(\Psi,\bar{y}_0)=\delta_{k,d_m}\ \ZZ\ \mbox{for all}\ k\geq 0\ \mbox{with}\ d_m=\mbox{dim}\, \overset{m}{\underset{\mathrm{i=1}}\oplus}\ E(\hat{\lambda}_i(2))\geq 2,\label{eq:88}
	\end{eqnarray}
see\ Chang\ \cite{ref10}.

	Finally from Proposition \ref{prop:4.3}, we know that $u=0$ is a local minimizer of $\varphi$, hence
			\begin{equation}
		\label{eq:89}
			C_k(\Psi,0)=C_k(\varphi,0)=\delta_{k,0}\ \ZZ\ \mbox{for all}\ k\geq 0\ (\mbox{see\ Liu\ \&\  Li}\ \cite{ref27}).
	\end{equation}
		
	From (\ref{eq:87}), (\ref{eq:88}) and (\ref{eq:89}) we infer that
$\bar{y}_0\ \notin\ \{0,\bar{u}_0,\bar{v}_0\}.$		
	Therefore, if $y_0=\bar{y}_0+\gamma_0(\bar y_0)$, then $y_0$ is a critical point of $\varphi$ distinct from $\{0,u_0,v_0\}$. This is the third nontrivial solution of (1) and the nonlinear regularity theory (see \cite{ref23}, \cite{ref24}) implies that $y_0 \in C^{1}_{0}(\bar{\Omega})$.
		\qed
	
	Next, by strengthening the regularity of $f(z,\cdot)$, we can improve the above multiplicity theorem and produce four nontrivial solutions.
		
	To this end, first we compute the critical groups of $\Psi$ at infinity. To do this we do not need the stronger conditions on $f(z,\cdot)$ and in the proof we use some ideas of Liu \cite{ref25bis}.
	
	\begin{proposition}
		\label{prop:4.10}
			If hypotheses $H_4$ hold, then $C_k(\Psi, \infty) = \delta_{k,d_m}\ \ZZ$ for all $k \geq 0$.
	\end{proposition}
	
	\textit{Proof.} Let $m_0< \inf \Psi(K_{\Psi})$. Since $\Psi$ is anti-coercive (see Proposition \ref{prop:4.8}), we can find $\eta <\vartheta < m_0$ and $0 <\rho < R$ such that
			$C_R\subseteq \Psi^{\eta}\subseteq C_{\rho}\subseteq \Psi^{\vartheta},$
	where for every $r>0, C_r=\{y \in Y:||y||\geq r\}$.
	
	For the triples $(C_R, C_{\rho},Y)$ and $(\Psi^{\eta},\Psi^{\vartheta}, Y)$ we consider the corresponding long exact sequences of homology groups. So, we have
		\begin{eqnarray}
		&& \cdots\ \rightarrow\ H_k(C_{\rho}, C_R)\ \stackrel{i_*}{\rightarrow}\ H_k(Y,C_R)\ \stackrel{j_*}{\rightarrow}\ H_k(Y,C_{\rho})\ \stackrel{\partial_*}{\rightarrow}\ H_{k-1}(C_{\rho},C_R)\ \rightarrow\ \cdots\nonumber \\
		&& \hspace{25 mm} \downarrow h_*{_{|{C_{\rho}}}} \hspace{13 mm} \downarrow h_* \hspace{20 mm} \downarrow h_* \hspace{23 mm} \downarrow h_*{_{|{C_{\rho}}}}\label{eq:90} \\
		&& \cdots\ \rightarrow\ H_k(\Psi^{\vartheta},\Psi^{\eta})\ \stackrel{\hat{i}_*}{\rightarrow}\ H_k(Y,\Psi^{\eta})\ \stackrel{\hat{j}_*}{\rightarrow}\ H_k(Y,\Psi^{\vartheta})\ \stackrel{\hat{\partial}_*}{\rightarrow}\ H_{k-1}(\Psi^{\vartheta},\Psi^{\eta})\nonumber
	\end{eqnarray}
	
	In (\ref{eq:90}) all the squares are commutative (see Granas \& Dugundji \cite{ref19} (p.377)) and the map $i_*,\,j_*,\,\hat{i}_*,\,\hat{j}_*,\,h_*$ are the group homomorphisms induced by the corresponding inclusion maps. Also, $\partial_*$ and $\hat{\partial}_*$ are the corresponding boundary homomorphisms. Since $\eta < \vartheta < m_0 < \inf \Psi(K_{\Psi})$, we have that $\Psi^{\eta}$ is a strong deformation retract of $\Psi^{\vartheta}$ (by the second deformation theorem, see \cite[p. 628]{ref18}) and so
		\begin{equation}
			\label{eq:91}
				H_k(\Psi^{\vartheta}, \Psi^{\eta})=0\ \mbox{for all}\ k \geq 0.
	\end{equation}
		
	Consider the map $\sigma:C_{\rho}\rightarrow C_R$ defined by
	\begin{center}
		$\sigma(u) =
			\left\{
				\begin{array}{ll}
					R\ \frac{u}{||u||}	\qquad & \mbox{if}\ \rho \leq ||u|| \leq R\\
					u               	 \qquad & \mbox{if}\ R<||u||.
			\end{array}
		\right.$
	\end{center}
	
	Clearly $\sigma$ is continuous and $\sigma|_{C_R} = id|_{C_R}$, So, $C_R$ is a retract of $C_{\rho}$. Moreover, if $h:[0,1]\times C_{\rho}\rightarrow Y$ is defined by
	
	$$h(t,u)=(1-t)u + tR\frac{u}{||u||}\ \mbox{for all}\ t \in [0,1],\ \mbox{all}\ u \in C_{\rho},$$	
	then we see that $C_{\rho}$ is deformable into $C_R$ in $Y$. Therefore, invoking Theorem 6.5 of Dugundji \cite[p. 325]{ref15}, we infer that $C_R$ is a deformation retract of $C_{\rho}$. Hence
		\begin{equation}
		\label{eq:92}
			H_k(C_{\rho},C_R)=0\ \mbox{for all}\ k\geq 0.
	\end{equation}
		(see Granas \& Dugundji \cite{ref19} (p.387)). From the exactness of the long homology sequences in (\ref{eq:90}), we have
		\begin{eqnarray}
		&& 0=\mbox{im}\, i_*=\mbox{ker}\, j_*\ \mbox{and}\ \mbox{im}\, j_*=\mbox{ker}\, \partial_*=H_k(Y,C_{\rho}),\quad \mbox{see}\ (\ref{eq:91})\nonumber \\
		&& 0=\mbox{im}\, \hat{i}_*=\mbox{ker}\, \hat{j}_*\ \mbox{and}\ \mbox{im}\, \hat{j}_*=\mbox{ker}\, \hat{\partial}_*= H_k(Y,\Psi^{\vartheta}),\quad \mbox{see}\ (\ref{eq:92}).\nonumber
	\end{eqnarray}
	
	If follows that both $j_*$ and $\hat{j}_*$ are group isomorphisms. Then invoking Lemma D.1 of Granas \& Dugundji \cite[p. 610]{ref19}, we deduce that $h_*$ is an isomorphism. So, for all $k\geq 0$
		\begin{equation}\label{eq:93}
		 H_k(Y,C_{\rho})=H_k(Y,\Psi^{\vartheta}))
		\Rightarrow  H_k(Y,C_{\rho})=C_k(\Psi,\infty)\,.
	\end{equation}
	
	As before, using the radical retraction and Theorem 6.5 of Dugundji \cite[p. 325]{ref15}, we show that $\partial B_{\rho} =\{y \in Y:||y||=\rho\}$ is a deformation retract of $C_{\rho}$. Therefore
		\begin{eqnarray}
		&& H_k(Y,C_{\rho})=H_k(Y,\partial B_{\rho})\ \mbox{for all}\ k\geq 0 \nonumber \\
		\Rightarrow && H_k(Y,C_{\rho})=\delta_{k,d_m}\ \ZZ\ \mbox{for all}\ k\geq 0\ (\mbox{see\ Maunder\ \cite[p. 121]{ref28}})\nonumber \\
		\Rightarrow && C_k(\Psi,\infty)=\delta_{k,d_m}\ \ZZ\ \mbox{for all}\ k\geq 0\ (\mbox{see}\ (\ref{eq:93})).\nonumber
	\end{eqnarray}
	\qed
	
	The new stronger conditions on $f(z,x)$ which we will need in order to prove a four solutions theorem for problem (1) when $1<p<2$, are the following:
	
	\textbf{$H_5$:} $f:\Omega \times \RR \rightarrow \RR$ is a measurable function such that for a.a. $z \in \Omega, f(z,0)=0, f(z,\cdot)\in C^1(\RR)$ and
	
	\textbf{(i)} there exist an integer $m\geq 2$ and a function $\eta \in L^{\infty}(\Omega)$ such that
		\begin{eqnarray}
		&& \eta(z)\ \leq\ \hat{\lambda}_{m+1}(2)\ \mbox{a.e.\ in}\ \Omega,\ \eta\neq\hat{\lambda}_{m+1}(2)\nonumber \\
		\mbox{and} && |f'_{x}(z,x)|\ \leq\ \eta(z)\ \mbox{a.e.\ in}\ \Omega,\ \mbox{for all}\ x \in \RR;\nonumber
	\end{eqnarray}
		
	\textbf{(ii)} $\hat{\lambda}_m\ \leq\ \liminf_{x\rightarrow \pm \infty}\frac{f(z,x)}{x}$ uniformly for a.a. $z \in \Omega;$
	
	\textbf{(iii)} $\lim_{x\rightarrow \pm \infty}\frac{f(z,x)x-2F(z,x)}{|x|^{p}}= -\infty$ uniformly for a.a. $z \in \Omega.$
	
	\textbf{Remark.} From hypothesis $H_5(i)$ and the mean value theorem we have
	
	$$\left|\frac{f(z,x)}{x}\right|\ \leq\ \eta(z)\ \mbox{for\ a.a.}\ z \in \Omega,\ \mbox{all}\ x \in \RR\backslash\{0\}.$$
		
	\textbf{Remark.} Similarly, we see for all $\rho>0$, there exists $\xi_{\rho}>0$ such that for a.a. $z \in \Omega, x\rightarrow f(z,x)+\xi_{\rho}x$ is nondecreasing on $[-\rho,\rho]$.
	
	Then we can state the following multiplicity theorem for problem (1) (case $1<p<2$).
	
	\begin{theorem}
		\label{th:4.11}
			Assume $H_5$ and\ $1<p<2$.
				Then\ problem\ (1)\ has\ at\ least\ four\ nontrivial\ solutions.
		$u_0 \in {\mathrm int}\,C_+$,\ $v_0 \in -{\mathrm int}\,C_+$\ and\ $y_0,\,\hat{y}\in C^{1}_{0}(\bar{\Omega})$.\end{theorem}
	
	\textit{Proof.} From Theorem \ref{th:4.9}, we already have three nontrivial solutions
	$u_0 \in \mbox{int}\,C_+$, $v_0 \in -\mbox{int}\,C_+$ and $y_0 \in C^{1}_{0}(\bar{\Omega}).$
	
	We know that $\varphi \in C^2(H^{1}_{0}(\Omega)\backslash\{0\})$. Then as in Claim 3 in the proof of Proposition 3.5 (see also Motreanu, Motreanu \& Papageorgiou \cite{momopa}, proof of Theorem 4.2) we can apply
 Proposition 2.5 of  Bartsch \cite{ref5} and have that
		\begin{equation}
		\label{eq:94}
			C_k(\varphi,u_0)=C_k(\varphi,v_0)=\delta_{k,1}\ \ZZ\ \mbox{for all}\ k\geq 0
	\end{equation}
(see relation \eqref{eq:88} and recall that $u_0\in{\rm int}\, C_+$ and $v_0\in -{\rm int}\, C_+$ and $\varphi$ is $C^2$
in neighborhoods $\tilde{\mathcal U}$, $\tilde{\mathcal V}$ of $u_0$ and $v_0$ in the $C^1_0(\overline\Omega)$ space).
	
	From (\ref{eq:88}) and since $C_k(\varphi,y_0)=C_k(\Psi,\bar{y}_0)$ for all $k\geq 0$ (see \cite{ref27}), it follows that
		\begin{equation}
		\label{eq:95}
			C_k(\varphi,y_0)=\delta_{k,d_m}\ \ZZ\ \mbox{for all}\ k\geq 0.
	\end{equation}
	
	From Proposition \ref{prop:4.10}, we have for all $k\geq 0$
		\begin{equation}\label{eq:96}
		 C_k(\Psi, \infty)=\delta_{k,d_m}\, \ZZ\
		\Rightarrow  C_k(\varphi, \infty)=\delta_{k,d_m}\,\ZZ\quad \mbox{(\mbox{see} \cite{ref27})}.
	\end{equation}
	
	Finally from Proposition \ref{prop:4.3}, we have
	\begin{equation}
		\label{eq:97}
			C_k(\varphi,0)=\delta_{k,0}\ \ZZ\ \mbox{for all}\ k\geq 0.
	\end{equation}
	
	Suppose $K_{\varphi}=\{0, u_0, v_0, y_0\}$. then from (\ref{eq:94}), (\ref{eq:95}), (\ref{eq:96}), (\ref{eq:97}) and the Morse relation (see (\ref{eq:7})) with $t=-1$, we have
		$2(-1)^1\ +\ (-1)^0\ +\ (-1)^{d_m}=(-1)^{d_m}$, a\ contradiction.
Thus, there exists $\hat{y}\in K_{\varphi}$ such that $\hat{y}\notin \{0, u_0, v_0, y_0\}$. This the fourth nontrivial solution of problem (1) and by the nonlinear regularity theory we have $\hat{y}\in C^{1}_{0}(\bar{\Omega})$.
	\qed
		
	
	
\medskip
\textbf{Acknowledgements.} The authors wish to thank the two referees for their corrections and remarks that improved the paper considerably. V. R\u adulescu acknowledges the support through Grant CNCS PCE-47/2011.
	


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\end{document} 