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\title[Asymmetric, noncoercive, superlinear $(p,2)$-equations]{Asymmetric, Noncoercive, Superlinear $(p,2)$-Equations}


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

\author[V.D. R\u{a}dulescu]{Vicen\c{t}iu D. R\u{a}dulescu}
\address{Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia \& Institute of Mathematics ``Simion Stoilow" of the Romanian Academy, P.O. Box 1-764,
          014700 Bucharest, Romania }
\email{\tt vicentiu.radulescu@imar.ro}

\keywords{$(p,2)$-equation, asymmetric reaction, superlinear growth, multiple solutions, nonlinear regularity, critical groups.\\
\phantom{aa} 2010 AMS Subject Classification: 35J20, 35J60, 58E05}


\begin{abstract}
We examine a nonlinear nonhomogeneous Dirichlet problem driven by the sum of a $p$-Laplacian $(p\geq 2)$ and a Laplacian (a $(p,2)$-equation). The reaction term is asymmetric and it is superlinear in the positive direction and sublinear in the negative direction. The superlinearity is not expressed using the Ambrosetti-Rabinowitz condition, while the asymptotic behavior as $x\rightarrow-\infty$ permits resonance with respect to any nonprincipal eigenvalue of $(-\Delta_p,W^{1,p}_{0}(\Omega))$. Using variational methods based on the critical point theory and Morse theory (critical groups), we prove a multiplicity theorem producing three nontrivial solutions.
\end{abstract}

\maketitle

%\tableofcontents

\section{Introduction}

Let $\Omega\subseteq\RR^N$ be a bounded domain with a $C^2$-boundary $\partial\Omega$. In this paper, we study the following nonlinear Dirichlet problem driven by the sum of a $p$-Laplacian $(p\geq 2)$ and a Laplacian (a $(p,2)$-equation):
\begin{equation}\label{eq1}
	-\Delta_pu(z)-\Delta u(z)=f(z,u(z))\ \mbox{in}\ \Omega,\ u|_{\partial\Omega}=0,\ 2\leq p.
\end{equation}

By $\Delta_p$ we denote the $p$-Laplace differential operator defined by
$$\Delta_p u=\mbox{div}\,(|Du|^{p-2}Du)\ \mbox{for all}\ u\in W^{1,p}_{0}(\Omega).$$

In this problem the reaction term $f(z,x)$ is a measurable function which is $C^1$ in the $x\in\RR$ variable and exhibits an asymmetric behavior as $x\rightarrow\pm\infty$. More precisely, $x\longmapsto f(z,x)$ is $(p-1)$-superlinear near $+\infty$, but it is $(p-1)$-sublinear near $-\infty$. The superlinearity in the positive direction is not expressed using the Ambrosetti-Rabinowitz condition (the AR-condition for short). Instead we employ a weaker condition which incorporates in our framework superlinear nonlinearities with slower growth near $+\infty$ which fail to satisfy the AR-condition. In the negative direction where $f(z,\cdot)$ is sublinear, our hypothesis permits resonance with respect to any nonprincipal eigenvalue of $(-\Delta_p,W^{1,p}_{0}(\Omega))$. So, problem (\ref{eq1}) is asymmetric, superlinear and at resonance.

Recently such problems were studied by Recova and Rumbos \cite{24}, \cite{25} for semilinear Dirichlet problems driven by the Laplacian and with more restrictive conditions on the reaction term (see Theorem 1.1 of \cite{24} and Theorem 1.2 of \cite{25}). We also mention the semilinear works of de Paiva and Presoto \cite{18} (they study a parametric equation driven by the Laplacian) and Motreanu, Motreanu and Papageorgiou \cite{13} (they study an equation driven by the Laplacian, no resonance is allowed as $x\rightarrow-\infty$ and they produce only two nontrivial solutions). For equations driven by the $p$-Laplacian, we mention the work of Motreanu, Motreanu and Papageorgiou \cite{14}, who deal with a parametric problem involving concave nonlinearities.

We mention that $(p,2)$-equations arise in many physical applications. We refer to the works of Benci, D'Avenia, Fortunato and Pisani \cite{3} (quantum physics) and Cherfils and Ilyasov \cite{4} (diffusion problems). Recently there have been some existence and multiplicity results for such equations under different settings. We mention the works of Cingolani and Degiovanni \cite{5}, Mugnai and Papageorgiou \cite{16}, Papageorgiou and R\u adulescu \cite{19}, Papageorgiou and Smyrlis \cite{21} and Papageorgiou and Winkert \cite{22}.

Our approach combines variational methods based on the critical point theory with Morse theory (critical groups).

\section{Mathematical Background}

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

This is a compactness-type condition on the functional $\varphi$ and it is more general than the more common Palais-Smale condition. The $C$-condition leads to a deformation theorem from which one can derive the min-max theory for the critical values of $\varphi$. Prominent in this theory is the so-called ``mountain pass theorem" due to Ambrosetti and Rabinowitz \cite{2}, which we state here in a slightly more general form (see, for example, Gasinski and Papageorgiou \cite[p. 648]{7}).
\begin{theorem}\label{th1}
	Let $X$ be a Banach space and assume that $\varphi\in C^1(X,\RR)$ satisfies the $C$-condition, $u_0,u_1\in X$, $||u_1-u_0||>\rho>0,$
	$$\max\{\varphi(u_0),\varphi(u_1)\}<\inf[\varphi(u):||u-u_0||=\rho]=m_{\rho}$$
	and $c=\inf\limits_{\gamma\in\Gamma}\max\limits_{0\leq t\leq 1}\varphi(\gamma(t))$ with $\Gamma=\{\gamma\in C([0,1],X):\gamma(0)=u_0,\gamma(1)=u_1\}$. Then $c\geq m_{\rho}$ and $c$ is a critical value of $\varphi$.
\end{theorem}

In the analysis of problem (\ref{eq1}), we will use the Sobolev spaces $W^{1,p}_{0}(\Omega)$ and $H^{1}_{0}(\Omega)$. Since $p\geq 2$, we have $W^{1,p}_{0}(\Omega)\subseteq H^{1}_{0}(\Omega)$. We will also use the Banach space $C^{1}_{0}(\overline{\Omega})=\{u\in C^1(\overline{\Omega}):u|_{\partial\Omega}=0\}$. This is an ordered Banach space with positive cone
$$C_+=\{u\in C^1_0(\overline{\Omega}):u(z)\geq 0\ \mbox{for all}\ z\in\overline{\Omega}\}.$$

This cone has a nonempty interior given by
$$\mbox{int}\, C_+=\{u\in C_+:u(z)>0\ \mbox{for all}\ z\in\Omega,\ \frac{\partial u}{\partial n}(z)<0\ \mbox{for all}\ z\in\partial\Omega\}.$$

Here $\frac{\partial u}{\partial n}=(Du,n)_{\RR^N}$ with $n(z)$ being the outward unit normal at $z\in\partial\Omega$.

We will also need some facts about the spectrum of $(-\Delta_p,W^{1,p}_{0}(\Omega))$. So, we consider the following nonlinear eigenvalue problem:
\begin{equation}\label{eq2}
	-\Delta_pu(z)=\hat{\lambda}|u(z)|^{p-2}u(z)\ \mbox{in}\ \Omega,\ u|_{\partial\Omega}=0,\ 1<p<\infty.
\end{equation}

We say that $\hat{\lambda}\in\RR$ is an eigenvalue of $(-\Delta_p,W^{1,p}_{0}(\Omega))$, if problem (\ref{eq2}) admits a nontrivial solution $\hat{u}\in W^{1,p}_{0}(\Omega)$ which is an eigenfunction corresponding to the eigenvalue $\hat{\lambda}$. There exists a smallest eigenvalue $\hat{\lambda}_1(p)>0$ which has the following properties:
\begin{itemize}
	\item $\hat{\lambda}_1(p)$ is isolated (that is, there exists $\epsilon>0$ such that open interval ($\hat{\lambda}_1(p),\hat{\lambda}_1(p)+\epsilon$) contains no eigenvalues of $(-\Delta_p,W^{1,p}_{0}(\Omega))$).
	\item $\hat{\lambda}_1(p)$ is simple (that is, if $\hat{u},\hat{v}\in W^{1,p}_{0}(\Omega)$ are eigenfunctions corresponding to the eigenvalue $\hat{\lambda}_1(p)$, then $\hat{u}=\xi\hat{v}$ for some $\xi\in\RR\backslash\{0\}$).
	\begin{equation}\label{eq3}
		\bullet\hspace{0.4cm}\hat{\lambda}_1(p)=\inf\left[\frac{||Du||^p_p}{||u||^p_p}:u\in W^{1,p}_{0}(\Omega),u\neq 0\right].\hspace{6cm}
	\end{equation}
\end{itemize}

The infimum in (\ref{eq3}) is realized at the corresponding one-dimensional eigenspace. From (\ref{eq3}) it is clear that the elements of this eigenspace do not change sign. Let $\hat{u}_1(p)$ be the $L^p-$normalized (that is, $||\hat{u}_1(p)||_p=1$) positive eigenfunction corresponding to $\hat{\lambda}_1(p)$. The nonlinear regularity theory (see Lieberman \cite{10}) and the nonlinear maximum principle (see Pucci and Serrin \cite{23}), imply that $\hat{u}_1(p)\in \mbox{int}\,C_+$.

The Ljusternik-Schnirelmann minimax scheme gives a whole strictly increasing sequence $\{\hat{\lambda}_k(p)\}_{k\geq 1}$ of distinct eigenvalues such that $\hat{\lambda}_k(p)\rightarrow+\infty$. However, we do not know if this sequence exhausts the spectrum of $(-\Delta_p, W^{1,p}_{0}(\Omega))$. This is the case if $p=2$ (linear eigenvalue problem) or if $N=1$ (ordinary differential equations).

The following lemma can be found in Motreanu, Motreanu and Papageorgiou \cite[p. 305]{15}.
\begin{lemma}\label{lem2}
 Assume that $\vartheta\in L^{\infty}(\Omega)$ satisfies $\vartheta(z)\leq\hat{\lambda}_1(p)$ $(1<p<\infty)$ for almost all $z\in\Omega$, with strict inequality on a set of positive measure. Then there exists $\hat{c}>0$ such that
$$||Du||^p_p-\int_{\Omega}\vartheta(z)|u|^pdz\geq\hat{c}||Du||^p_p\ \mbox{for all}\ u\in W^{1,p}_{0}(\Omega).$$
\end{lemma}

The same results are also true for the following weighted version of problem (\ref{eq2}):
$$-\Delta_pu(z)=\tilde{\lambda}m(z)|u(z)|^{p-2}u(z)\ \mbox{in}\ \Omega,\ u|_{\partial\Omega}=0,$$
with $m\in L^{\infty}(\Omega),m\geq 0,m\not\equiv 0$. In this case
$$\tilde{\lambda}_1(p,m)=\inf\left[\frac{||Du||^p_p}{\int_{\Omega}m(z)|u|^pdz}:u\in W^{1,p}_{0}(\Omega),u\neq 0\right]\,.$$

We have the following monotonicity property for the map $m\rightarrow\tilde{\lambda}_1(p,m)$.
\begin{prop}\label{prop3}
	Assume that $m,m'\in L^{\infty}(\Omega)$, $0\leq m(z)\leq m'(z)$ for almost all $z\in\Omega$ and $m\not\equiv m'$. Then $\tilde{\lambda}_1(p,m')<\tilde{\lambda}_1(p,m)$.
\end{prop}

We mention that only the first eigenvalue has eigenfunctions of constant sign. All the other eigenvalues have nodal (that is, sign-changing) eigenfunctions. For further details on these and related issues, we refer to Gasinski and Papageorgiou \cite{7}.

For $1<p<\infty$, let $A_p:W^{1,p}_{0}(\Omega)\rightarrow W^{-1,p'}(\Omega)=W^{1,p}_{0}(\Omega)^*\left(\frac{1}{p}+\frac{1}{p'}=1\right)$ be the map defined by
$$\left\langle A_p(u),h\right\rangle=\int_{\Omega}|Du|^{p-2}(Du,Dh)_{\RR^N}dz\ \mbox{for all}\ u,h\in W^{1,p}_{0}(\Omega).$$

When $p=2$, we write $A_2=A$ and we have $A\in\mathcal{L}(H^1_0(\Omega),H^{-1}(\Omega))$. For $p\neq 2$, $A_p$ is nonlinear and $(p-1)$-homogeneous. Also we have (see Gasinski and Papageorgiou \cite[p. 746]{7}).
\begin{prop}\label{prop4}
	The map $A_p:W^{1,p}_{0}(\Omega)\rightarrow W^{-1,p'}(\Omega)$ $(1<p<\infty)$ is continuous, strictly monotone (hence maximal monotone too) and of type $(S)_+$, that is
	$$``u_n\stackrel{w}{\rightarrow}u\ \mbox{in}\ W^{1,p}_{0}(\Omega)\ \mbox{and}\ \limsup\limits_{n\rightarrow\infty}\left\langle A_p(u_n),u_n-u\right\rangle\leq 0\Rightarrow u_n\rightarrow u\ \mbox{in}\ W^{1,p}_{0}(\Omega)."$$
\end{prop}

Next we recall some basic facts about critical groups (Morse theory). For further details we refer to the book of Motreanu, Motreanu and Papageorgiou \cite{15} (see also Cingolani, Degiovanni and Vannella \cite{cingo1}
and Cingolani and Vannella \cite{cingo2}.

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

Let $(Y_1,Y_2)$ be a topological pair such that $Y_2\subseteq Y_1\subseteq X$ and $k\in \NN_0$. By $H_k(Y_1,Y_2)$ we denote the $k$th relative singular homology group for the topological pair $(Y_1,Y_2)$ with integer coefficients. The critical groups of $\varphi$ at an isolated $u\in K^c_{\varphi}$ are defined by
$$C_k(\varphi,u)=H_k(\varphi^c\cap U,\varphi^c\cap U\backslash\{0\})\ \mbox{for all}\ k\in\NN_0,$$
with $U$ being a neighborhood of $u$ such that $K_{\varphi}\cap\varphi^c\cap U=\{u\}$. The excision property of singular homology implies that the above definition of critical groups is independent of the choice of the neighborhood $U$ of $u$.

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

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

Suppose that $\varphi\in C^1(X,\RR)$, satisfies the $C$-condition and $K_{\varphi}$ is finite. We define
\begin{eqnarray*}
	&&M(t,u)=\sum\limits_{k\geq 0} \mbox{rank}\, C_k(\varphi,u)t^k\ \mbox{for all}\ t\in\RR,\ \mbox{all}\ u\in K_{\varphi},\\
	&&P(t,\infty)=\sum\limits_{k\geq 0} \mbox{rank}\, C_k(\varphi,\infty)t^k\ \mbox{for all}\ t\in\RR.
\end{eqnarray*}

The Morse relation says that
\begin{equation}\label{eq4}
	\sum\limits_{u\in K_{\varphi}}M(t,u)=P(t,\infty)+(1+t)Q(t)\ \mbox{for all}\ t\in\RR,
\end{equation}
where $Q(t)=\sum\limits_{k\geq 0}\beta_kt^k$ is a formal series in $t\in\RR$ with nonnegative integer coefficients.

Finally we fix our notation. By $||\cdot||$ we denote the norm of $W^{1,p}_{0}(\Omega)$. From the Poincar\'e inequality (see, for example, Gasinski and Papageorgiou \cite[p. 216]{7}), we have
$$||u||=||Du||_p\ \mbox{for all}\ u\in W^{1,p}_{0}(\Omega).$$

Let $x\in\RR$. We define $x^{\pm}=\max\{\pm x,0\}$. Then for $u\in W^{1,p}_{0}(\Omega)$ we set
$$u^{\pm}(\cdot)=u(\cdot)^{\pm}.$$

We know that $u^{\pm}\in W^{1,p}_{0}(\Omega)$, $u=u^+-u^-,|u|=u^++u^-$. Given a measurable function $g:\Omega\times\RR\rightarrow\RR$, we set
$$N_g(u)(\cdot)=g(\cdot,u(\cdot))\ \mbox{for all}\ u\in W^{1,p}_{0}(\Omega).$$

Then $z\longmapsto N_g(u)(z)=g(z,u(z))$ is measurable. By $|\cdot|_N$ we denote the Lebesgue measure on $\RR^N$ and by $p^*=\left\{\begin{array}{ll}
\frac{Np}{N-p}&\mbox{if}\ p<N\\
+\infty&\mbox{if}\ N\leq p
\end{array}\right.$ the critical Sobolev exponent.


\section{Multiplicity Theorem}

In this section we prove a multiplicity theorem for problem (\ref{eq1}) producing three nontrivial solutions. Our hypotheses on the reaction term $f(z,x)$ are the following:

$H:$ $f:\Omega\times\RR\rightarrow\RR$ is a measurable function such that for almost all $z\in\Omega$, $f(z,0)=0,f(z,\cdot)\in C^1(\RR)$ and
\begin{itemize}
	\item[(i)] $|f'_x(z,x)|\leq a(z)(1+|x|^{r-2})$ for almost all $z\in\Omega$, all $x\in\RR$, with $a\in L^{\infty}(\Omega)_+,\ p<r<p^*$;
	\item[(ii)] if $F(z,x)=\int^x_0f(z,s)ds$, then
	$$\lim\limits_{x\rightarrow+\infty}\frac{F(z,x)}{x^p}=+\infty\ \mbox{uniformly for almost all}\ z\in\Omega;$$
	\item[(iii)] if $\xi(z,x)=f(z,x)x-pF(z,x)$, then there exists $\gamma_0\in L^1(\Omega)$ such that
	$$\xi(z,x)\leq\xi(z,y)+\gamma_0(z)\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ 0\leq x\leq y;$$
	\item[(iv)] there exist functions $\eta,\hat{\eta}\in L^{\infty}(\Omega)$ and $c_0>0$ such that
	\begin{eqnarray*}
		&&\eta(z)\geq\hat{\lambda}_1(p)\ \mbox{for almost all}\ z\in\Omega,\ \mbox{strictly on a set of positive measure,}\\
		&&\eta(z)\leq\liminf\limits_{x\rightarrow +\infty}\frac{f(z,x)}{|x|^{p-2}x}\leq \limsup\limits_{x\rightarrow +\infty}\frac{f(z,x)}{|x|^{p-2}x}\leq\hat{\eta}(z)\ \mbox{uniformly for almost all}\ z\in\Omega;\\
		&&-c_0\leq f(z,x)x-pF(z,x)\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\leq 0;
	\end{eqnarray*}
		\item[(v)] $f'_x(z,0)=\lim\limits_{x\rightarrow 0}\frac{f(z,x)}{x}$ uniformly for almost all $z\in\Omega$, $f'_x(z,0)\leq\hat{\lambda}_1(2)$ for almost all $z\in\Omega$ and the inequality is strict on a set of positive measure;
		\item[(vi)] for every $\rho>0$, there exists $\hat{\xi}_{\rho}>0$ such that $f(z,x)+\hat{\xi}_{\rho}x^{p-1}\geq 0$ for almost all $z\in\Omega$, all $0\leq x\leq\rho.$
\end{itemize}
\begin{remark}
Hypothesis $H(ii)$ implies that for almost all $z\in\Omega$, the primitive $F(z,\cdot)$ is $p$-superlinear near $+\infty$. This fact and hypothesis $H(iii)$, imply that for almost all $z\in\Omega$, $f(z,\cdot)$ is $(p-1)$-superlinear near $+\infty$ (see Li and Yang \cite[Lemma 2.4]{11}). Hypothesis $H(iii)$ replaces the AR-condition which says that there exist $q>p$ and $M>0$ such that
\begin{equation}
 0<qF(z,x)\leq f(z,x)x\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\geq M \tag{5a} \label{eq5a}
\end{equation}
\begin{equation}
 0<{\rm ess}\,\inf\limits_{\Omega}F(\cdot,M) \tag{5b} \label{eq5b}
\end{equation}
\end{remark}

\stepcounter{equation}%Custom Equation Numbering

An easy integration of \eqref{eq5a} and the use of \eqref{eq5b}, imply the weaker condition
\begin{equation}\label{eq6}
	c_1 x^q\leq F(z,x)\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\geq M\ \mbox{with}\ c_1>0.
\end{equation}

So, the AR-condition restricts $F(z,\cdot)$ to have at least $q$-polynomial growth near $+\infty$. With $H(iii)$ we avoid this (see the examples which follow). Condition $H(iii)$ also extends earlier ones used by Li and Yang \cite{11} and Miyagaki and Souto \cite{12}. Hypothesis $H(iv)$ implies that for almost all $z\in\Omega$, $f(z,\cdot)$ is $(p-1)$-sublinear near $-\infty$. Note that this hypothesis does not exclude resonance with respect to a nonprincipal eigenvalue.
\begin{ex}
The following functions satisfy hypotheses $H$. For the sake of simplicity we drop the $z$-dependence:
$$f_1(x)=\left\{\begin{array}{ll}
\eta|x|^{p-2}x+(\eta-\vartheta)&\mbox{if}\ x<-1\\
\vartheta x&\mbox{if}\ -1\leq x\leq 1\\
x^{r-1}+(\vartheta-1)&\mbox{if}\ 1\leq x
\end{array}\right.,$$
$$ f_2(x)=\left\{\begin{array}{ll}
\eta|x|^{p-2}x+(\eta-\vartheta)&\mbox{if}\ x<-1\\
\vartheta x&\mbox{if}\ -1\leq x\leq 1\\
x^{p-1}\left(\ln x+\frac{1}{p}\right)+\left(\vartheta-\frac{1}{p}\right)&\mbox{if}\ 1\leq x,
\end{array}\right.$$
with $\vartheta<\hat\lambda_1(2)$.
Note that $f_2$ does not satisfy the AR-condition (see (\ref{eq6})).
\end{ex}
Let $\varphi: W^{1,p}_{0}(\Omega)\rightarrow\RR$ be the energy functional for problem (\ref{eq1}) defined by
$$\varphi(u)=\frac{1}{p}||Du||^p_p+\frac{1}{2}||Du||^2_2-\int_{\Omega}F(z,u)dz\ \mbox{for all}\ u\in W^{1,p}_{0}(\Omega).$$

We have $\varphi\in C^2(W^{1,p}_{0}(\Omega))$.
\begin{prop}\label{prop5}
	If hypotheses $H$ hold, then the functional $\varphi$ satisfies the $C$-condition.
\end{prop}
\begin{proof}
	Let $\{u_n\}_{n\geq 1}\subseteq W^{1,p}_{0}(\Omega)$ be a sequence such that
	\begin{eqnarray}
		&&|\varphi(u_n)|\leq M_1\ \mbox{for some}\ M_1>0,\ \mbox{all}\ n\geq 1\label{eq7}\\
		&&(1+||u_n||)\varphi'(u_n)\rightarrow 0\ \mbox{in}\ W^{-1,p'}(\Omega)\ \mbox{as}\ n\rightarrow\infty\,.\label{eq8}
	\end{eqnarray}
	
	From (\ref{eq8}) we have
	\begin{eqnarray}\label{eq9}
		&&|\left\langle A_p(u_n),h\right\rangle+\left\langle A(u_n),h\right\rangle-\int_{\Omega}f(z,u_n)hdz|\leq\frac{\epsilon_n||h||}{1+||u_n||}\\
		&&\hspace{4cm}\mbox{for all}\ h\in W^{1,p}_{0}(\Omega)\ \mbox{with}\ \epsilon_n\rightarrow 0^+.\nonumber
	\end{eqnarray}
	
	Recall that $u_n=u^+_n-u^-_n$ for all $n\geq 1$. So, we have
	\begin{eqnarray}\label{eq10}
		&&\frac{1}{p}||Du^+_n||^p_p+\frac{1}{2}||Du^+_n||^2_2\nonumber\\
		 &=&\frac{1}{p}||Du_n||^p_p+\frac{1}{2}||Du_n||^2_2-\frac{1}{p}||Du^-_n||^p_p-\frac{1}{2}||Du^-_n||^2_2+\int_{\Omega}F(z,u_n)dz-\int_{\Omega}F(z,u_n)dz\nonumber\\
		&=&\varphi(u_n)-\frac{1}{p}||Du^-_n||^p_p-\frac{1}{2}||Du^-_n||^2_2+\int_{\Omega}F(z,u_n)dz\nonumber\\
		&\leq&M_1+\frac{1}{p}\left[\int_{\Omega}pF(z,u_n)dz-||Du^-_n||^p_p-||Du^-_n||^2_2\right]\ \mbox{for all}\ n\geq 1\ \\
		&&\hspace{9cm}(\mbox{see (\ref{eq7}) and recall}\ p\geq 2).\nonumber
	\end{eqnarray}
	
	In (\ref{eq9}) we choose $h=-u^-_n\in W^{1,p}_{0}(\Omega)$ and obtain
	\begin{eqnarray}\label{eq11}
		&&\left|||Du^-_n||^p_p+||Du^-_n||^2_2-\int_{\Omega}f(z,-u^-_n)(-u^-_n)dz\right|\leq\epsilon_n\ \mbox{for all}\ n\geq 1,\nonumber\\
		&\Rightarrow&-||Du^-_n||^p_p-||Du^-_n||^2_2\leq\epsilon_n-\int_{\Omega}f(z,-u^-_n)(-u^-_n)dz\ \mbox{for all}\ n\geq 1.
	\end{eqnarray}
		
	We return to (\ref{eq10}) and use (\ref{eq11}). Then
	\begin{eqnarray}\label{eq12}
		&&\frac{1}{p}||Du^+_n||^p_p+\frac{1}{2}||Du^+_n||^2_2\leq M_2+\frac{1}{p}\int_{\Omega}\left[pF(z,u_n)-f(z,-u^-_n)(-u^-_n)\right]dz\\
		&&\hspace{7cm}\mbox{for some}\ M_2>0,\ \mbox{all}\ n\geq 1\nonumber
	\end{eqnarray}
	
	We have
	\begin{equation}\label{eq13}
		pF(z,u_n)=pF(z,u^+_n)+pF(z,-u^-_n)\ \mbox{for all}\ n\geq 1
	\end{equation}
	and from hypothesis $H(iv)$, we have
	\begin{equation}\label{eq14}
		pF(z,-u^-_n)-f(z,-u^-_n)(-u^-_n)\leq c_0\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ n\geq 1.
	\end{equation}
	
	Returning to (\ref{eq12}) and using (\ref{eq13}) and (\ref{eq14}) we obtain
	\begin{eqnarray}
		&&\frac{1}{p}||Du^+_n||^p_p+\frac{1}{2}||Du^+_n||^2_2\leq M_3+\int_{\Omega}F(z,u^+_n)dz\nonumber\\
		&&\hspace{1cm}\mbox{with}\ M_3=M_2+c_0|\Omega|_N>0,\ \mbox{for all}\ n\geq 1,\nonumber\\
		&\Rightarrow&\varphi(u^+_n)\leq M_3\ \mbox{for all}\ n\geq 1.\label{eq15}
	\end{eqnarray}
	
	In (\ref{eq9}) we choose $h=u^+_n\in W^{1,p}_{0}(\Omega)$ and have
	\begin{equation}\label{eq16}
		-||Du^+_n||^p_p-||Du^+_n||^2_2+\int_{\Omega}f(z,u^+_n)u^+_ndz\leq \epsilon_n\ \mbox{for all}\ n\geq 1.
	\end{equation}
	
	From (\ref{eq15}) and since $p\geq 2$, we have
	\begin{equation}\label{eq17}
		||Du^+_n||^p_p+||Du^+_n||^2_2-\int_{\Omega}pF(z,u^+_n)dz\leq pM_3\ \mbox{for all}\ n\geq 1.
	\end{equation}
	
	Adding (\ref{eq16}) and (\ref{eq17}), we obtain
	\begin{eqnarray}\label{eq18}
		&&\int_{\Omega}\left[f(z,u^+_n)u^+_n-pF(z,u^+_n)\right]dz\leq M_4\ \mbox{for some}\ M_4>0,\ \mbox{all}\ n\geq 1,\nonumber\\
		&\Rightarrow&\int_{\Omega}\xi(z,u^+_n)dz\leq M_4\ \mbox{for all}\ n\geq 1.
	\end{eqnarray}
	\begin{claim}
		$\{u^+_n\}_{n\geq 1}\subseteq W^{1,p}_{0}(\Omega)$ is bounded.
	\end{claim}
	
	We argue indirectly. So, suppose that $\{u^+_n\}_{n\geq 1}\subseteq W^{1,p}_{0}(\Omega)$ is not bounded. Then we may assume that $||u^+_n||\rightarrow\infty$ as $n\rightarrow\infty$. We set $y_n=\frac{u^+_n}{||u^+_n||}n\geq 1$. We have
	$$||y_n||=1\ \mbox{and}\ y_n\geq 0\ \mbox{for all}\ n\geq 1.$$
	
	Hence we may assume that
	\begin{equation}\label{eq19}
		y_n\stackrel{w}{\rightarrow}y\ \mbox{in}\ W^{1,p}_{0}(\Omega)\ \mbox{and}\ y_n\rightarrow y\ \mbox{in}\ L^r(\Omega)\ \mbox{as}\ n\rightarrow\infty,\ y\geq 0.
	\end{equation}
	
	Suppose that $y\neq 0$. Then $|\{y>0\}|_N=0$ (recall that $y\geq 0$, see (\ref{eq14})) and we have
	$$u^+_n(z)\rightarrow+\infty\ \mbox{for almost all}\ z\in\{y>0\}.$$
	
	Hypothesis $H(ii)$ implies that
	$$\frac{F(z,u^+_n(z))}{||u^+_n||^p}=\frac{F(z,u^+_n(z))}{u^+_n(z)^p}y_n(z)^p\rightarrow+\infty\ \mbox{for almost all}\ z\in\{y>0\}.$$
	
	This fact and Fatou's lemma (see hypothesis $H(ii)$), imply that
	\begin{equation}\label{eq20}
		\lim\limits_{n\rightarrow\infty}\int_{\Omega}\frac{F(z,u^+_n)}{||u^+_n||^p}dz=+\infty.
	\end{equation}
	
	Since $\xi(z,0)=0$ for almost all $z\in\Omega$, from hypothesis $H(iii)$ we have
	\begin{eqnarray}\label{eq21}
		&\hspace{1cm}pF(z,u^+_n)&\leq f(z,u^+_n)u^+_n+\gamma_0(z)\ \mbox{for almost all}\ z\in\Omega,\nonumber\\
		\Rightarrow&\int_{\Omega}pF(z,u^+_n)dz&\leq\int_{\Omega}f(z,u^+_n)u^+_ndz+||\gamma_0||_1\nonumber\\
		&&\leq M_5+||Du^+_n||^p_p+||Du^+_n||^2_2\ \mbox{for some}\ M_5>0,\ \mbox{all}\ n\geq 1\ (\mbox{see (\ref{eq16})})\nonumber\\
		&&\leq M_5+||Du^+_n||^p_p+\frac{p}{2}||Du^+_n||^2_2\ \mbox{since}\ p\geq 2\nonumber\\
		 \Rightarrow&\int_{\Omega}\frac{F(z,u^+_n)}{||u^+_n||^p}dz&\leq\frac{M_5}{p||u^+_n||^p}+\frac{1}{p}||Dy_n||^p_p+\frac{1}{2||u^+_n||^{p-2}}||Dy_n||^2_2\nonumber\\
		&&\leq M_6\ \mbox{for some}\ M_6>0,\ \mbox{all}\ n\geq 1.
	\end{eqnarray}
	
	Comparing (\ref{eq20}) and (\ref{eq21}) we reach a contradiction.
	
	So, suppose that $y=0$. For $k\geq 1$, we set
	$$v_n=(pk)^{1/p}y_n\in W^{1,p}_{0}(\Omega).$$
	
	We have
	$$v_n\rightarrow 0\ \mbox{in}\ L^r(\Omega)\ (\mbox{see (\ref{eq19}) and recall that}\ y=0).$$
	
	Hypothesis $H(i)$ implies that
	$$|F(z,x)|\leq c_1(1+|x|^r)\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\in\RR\ \mbox{with}\ c_1>0.$$
	
	Using the Krasnoselskii theorem (see, for example, Gasinski and Papageorgiou \cite[p. 407]{7}), we have
	\begin{equation}\label{eq22}
		\int_{\Omega}F(z,v_n)dz\rightarrow 0\ \mbox{as}\ n\rightarrow\infty.
	\end{equation}
	
	Since $||u^+_n||\rightarrow\infty$, we can find $n_0\in\NN$ such that
	\begin{equation}\label{eq23}
		0<(pk)^{1/p}\frac{1}{||u^+_n||}\leq 1\ \mbox{for all}\ n\geq n_0.
	\end{equation}
	
	Let $\hat{\varphi}(u)=\frac{1}{p}||Du||^p_p-\int_{\Omega}F(z,u)dz$ for all $u\in W^{1,p}_{0}(\Omega)$. Let $t_n\in[0,1]$ be such that
	\begin{equation}\label{eq24}
		\hat{\varphi}(t_nu^+_n)=\max[\hat{\varphi}(tu^+_n):0\leq t\leq 1].
	\end{equation}
	
	From (\ref{eq23}) and (\ref{eq24}) we see that
	\begin{eqnarray*}
		&\hat{\varphi}(t_nu^+_n)&\geq\hat{\varphi}(v_n)\\
		&&=k||Dy_n||^p_p-\int_{\Omega}F(z,v_n)dz\\
		&&=k-\int_{\Omega}F(z,v_n)dz\ \mbox{for all}\ n\geq n_0,\\
		\Rightarrow&\hat{\varphi}(t_n,u^+_n)&\geq\frac{k}{2}\ \mbox{for all}\ n\geq n_1\geq n_0\ (\mbox{see (\ref{eq22})})\,.
	\end{eqnarray*}
	
	But $k>0$ is arbitrary. So, we infer that
	\begin{equation}\label{eq25}
		\hat{\varphi}(t_nu^+_n)\rightarrow+\infty\ \mbox{as}\ n\rightarrow\infty.
	\end{equation}
	
	We have
	$$\hat{\varphi}(0)=0\ \mbox{and}\ \hat{\varphi}(u^+_n)\leq M_3\ \mbox{for all}\ n\geq 1\ (\mbox{see (\ref{eq15}) and note that}\ \hat{\varphi}\leq\varphi).$$
	
	Because of (\ref{eq25}), we see that we can find $n_2\in\NN$ such that
	$$t_n\in(0,1)\ \mbox{for all}\ n\geq n_2.$$
	
	Then from (\ref{eq24}) it follows that
	\begin{eqnarray}\label{eq26}
		&&\left.\frac{d}{dt}\hat{\varphi}(tu^+_n)\right|_{t=t_n}=0\ \mbox{for all}\ n\geq n_2,\nonumber\\
		&\Rightarrow&\left\langle \hat{\varphi}'(t_nu^+_n),u^+_n\right\rangle=0\ \mbox{for all}\ n\geq n_2\ (\mbox{by the chain rule}),\nonumber\\
		&\Rightarrow&\left\langle \hat{\varphi}'(t_nu^+_n),t_nu^+_n\right\rangle=0\ \mbox{for all}\ n\geq n_2,\nonumber\\
		&\Rightarrow&||D(t_nu^+_n)||^p_p-\int_{\Omega}f(z,t_nu^+_n)(t_nu^+_n)dz\ \mbox{for all}\ n\geq n_2.
	\end{eqnarray}
	
	Hypothesis $H(iii)$ implies that
	\begin{eqnarray}\label{eq27}
		&&\int_{\Omega}\xi(z,t_nu^+_n)dz\leq\int_{\Omega}\xi(z,u^+_n)dz+||\gamma_0||_1\leq M_7\\
		&&\hspace{3cm}\mbox{for some}\ M_7>0,\ \mbox{all}\ n\geq n_2\ (\mbox{see (\ref{eq18}) and recall}\ t_n\in(0,1))\nonumber
	\end{eqnarray}
	
	We return to (\ref{eq26}) and use (\ref{eq27}). Then
	\begin{eqnarray}\label{eq28}
		&&||D(t_nu^+_n)||^p_p\leq M_7+\int_{\Omega}pF(z,t_nu^+_n)dz\ \mbox{for all}\ n\geq n_2,\nonumber\\
		&\Rightarrow&\hat{\varphi}(t_nu^+_n)\leq\frac{M_7}{p}\ \mbox{for all}\ n\geq n_2.
	\end{eqnarray}
	
	Comparing (\ref{eq25}) and (\ref{eq28}) we reach a contradiction.
	
	This proves the Claim.
	
	From (\ref{eq9}) and using the Claim, we have that
	\begin{eqnarray}\label{eq29}
		&&\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 M_8||h||\\
		&&\hspace{8cm}\mbox{for some}\ M_8>0,\ \mbox{all}\ n\geq 1.\nonumber
	\end{eqnarray}
	
	Suppose that $||u^-_n||\rightarrow\infty$ and set $w_n=\frac{u^-_n}{||u^-_n||}n\geq 1$. Then
	$$||w_n||=1\ \mbox{and}\ w_n\geq 0\ \mbox{for all}\ n\geq 1.$$
	
	So, we may assume that
	\begin{equation}\label{eq30}
		w_n\stackrel{w}{\rightarrow}w\ \mbox{in}\ W^{1,p}_{0}(\Omega)\ \mbox{and}\ w_n\rightarrow w\ \mbox{in}\ L^p(\Omega),\ w\geq 0.
	\end{equation}
	
	From (\ref{eq29}) we have
	\begin{eqnarray}\label{eq31}
		&&\left|\left\langle A_p(-w_n),h\right\rangle+\frac{1}{||w_n||^{p-2}}\left\langle A(-w_n),h\right\rangle-\int_{\Omega}\frac{N_f(-u^-_n)}{||u^-_n||^{p-1}}hdz\right|\leq\frac{M_8||h||}{||u^-_n||^{p-1}}\\
		&&\hspace{10cm}\mbox{for all}\ n\geq 1\nonumber
	\end{eqnarray}
	
	Hypotheses $H(i),(iv)$ imply that
	\begin{eqnarray*}
		&&|f(z,x)|\leq c_2(1+|x|^{p-1})\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\leq 0\ \mbox{and some}\ c_2>0,\\
		&\Rightarrow&\left\{\frac{N_f(-u^-_n)}{||u^-_n||^{p-1}}\right\}_{n\geq 1}\subseteq L^{p'}(\Omega)\ \mbox{is bounded}\ \left(\frac{1}{p}+\frac{1}{p'}=1\right)\,.
	\end{eqnarray*}
	
	Using this fact and hypothesis $H(iv)$ we have, at least for a subsequence, that
	\begin{equation}\label{eq32}
		\frac{N_f(-u^-_n)}{||u^-_n||^{p-1}}\stackrel{w}{\rightarrow}-\vartheta w^{p-1}\ \mbox{in}\ L^{p'}(\Omega)\ \mbox{with}\ \eta(z)\leq\vartheta(z)\leq\hat{\eta}(z)\ \mbox{for almost all}\ z\in\Omega
	\end{equation}
	(see Aizicovici, Papageorgiou and Staicu \cite{1}, proof of Proposition 16). In (\ref{eq31}) we use $h=w_n-w\in W^{1,p}_{0}(\Omega)$, pass to the limit as $n\rightarrow\infty$ and use (\ref{eq30}) and (\ref{eq32}). We obtain
	\begin{eqnarray}\label{eq33}
		&&\lim\limits_{n\rightarrow\infty}\left\langle A_p(w_n),w_n-w\right\rangle=0\ (\mbox{recall}\ p\geq 2),\nonumber\\
		&\Rightarrow&w_n\rightarrow w\ \mbox{in}\ W^{1,p}_{0}(\Omega)\ \mbox{(see Proposition \ref{prop4}), hence}\ ||w||=1,w\geq 0.
	\end{eqnarray}
	
	Therefore, if in (\ref{eq31}) we pass to the limit as $n\rightarrow\infty$ and use (\ref{eq32}) and (\ref{eq33}), then
	\begin{eqnarray}\label{eq34}
		&&\left\langle A_p(w),h\right\rangle=\int_{\Omega}\vartheta(z)w^{p-1}hdz\ \mbox{for all}\ h\in W^{1,p}_{0}(\Omega),\nonumber\\
		&\Rightarrow&-\Delta_pw(z)=\vartheta(z)w(z)^{p-1}\ \mbox{for almost all}\ z\in\Omega,\ w|_{\partial\Omega}=0.
	\end{eqnarray}
	
	Recall that
	$$\hat{\lambda}_1(p)\leq\eta(z)\leq\vartheta(z)\ \mbox{for almost all}\ z\in\Omega$$
	and the first inequality is strict on a set of positive measure. So, using Proposition \ref{prop3}, we have
	$$\tilde{\lambda}_1(p,\vartheta)<\tilde{\lambda}_1(p,\hat{\lambda}_1)=1.$$
	
	Then returning to (\ref{eq34}) we infer that $w(\cdot)$ must be nodal or zero, a contradiction (see (\ref{eq33})). Therefore
	\begin{eqnarray*}
		&&\{u^-_n\}_{n\geq 1}\subseteq W^{1,p}_{0}(\Omega)\ \mbox{is bounded},\\
		&\Rightarrow&\{u_n\}_{n\geq 1}\subseteq W^{1,p}_{0}(\Omega)\ \mbox{is bounded (see the Claim)}.
	\end{eqnarray*}
	
	So, we may assume that
	\begin{equation}\label{eq35}
		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}
	
	In (\ref{eq9}) we choose $h=u_n-u\in W^{1,p}_{0}(\Omega)$, pass to the limit as $n\rightarrow\infty$ and use (\ref{eq35}). Then
	\begin{eqnarray*}
		&&\lim\limits_{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,\\
		&\Rightarrow&\limsup\limits_{n\rightarrow\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\ \mbox{(since $A$ is monotone)},\\
		&\Rightarrow&\limsup\limits_{n\rightarrow\infty}\left\langle A_p(u_n),u_n-u\right\rangle\leq 0,\\
		&\Rightarrow&u_n\rightarrow u\ \mbox{in}\ W^{1,p}_{0}(\Omega)\ (\mbox{see Proposition \ref{prop4}}).
	\end{eqnarray*}
	
	This proves that $\varphi$ satisfies the $C$-condition.
\end{proof}

Having established that $\varphi$ satisfies the $C$-condition, we can compute the critical groups of $\varphi$ at infinity.
\begin{prop}\label{prop6}
	If hypotheses $H$ hold and $\varphi(K_{\varphi})$ is bounded below, then $C_k(\varphi,\infty)=0$ for all $k\in\NN_0$.
\end{prop}
\begin{proof}
	Let $\varphi_c=\varphi|_{C^1_0(\overline{\Omega})}$. From the nonlinear regularity theory (see Lieberman \cite{10}), we have that $K_{\varphi_c}=K_{\varphi}=K$. Moreover, since $C^1_0(\overline{\Omega})$ is dense in $W^{1,p}_{0}(\Omega)$, from Palais \cite[Theorem 16]{17}, we have
	\begin{equation}\label{eq36}
		H_k(W^{1,p}_{0}(\Omega),\dot{\varphi}^a)=H_k(C^1_0(\overline{\Omega}),\dot{\varphi}^a_c)\ \mbox{for all}\ a\in\RR,\ \mbox{all}\ k\in\NN,
	\end{equation}
	with $\dot{\varphi}^a=\{u\in W^{1,p}_{0}(\Omega):\varphi(u)<a\},\ \dot{\varphi}^a_c=\{u\in C^1_0(\overline{\Omega}):\varphi_c(u)<a\}.$
	
	Choosing $a<\inf\varphi(K)=\inf\varphi_c(K)$ (this is possible by hypothesis), we have
	\begin{eqnarray}
		&&H_k(W^{1,p}_{0}(\Omega),\dot{\varphi}^a)=H_k(W^{1,p}_{0}(\Omega),\varphi^a)=C_k(\varphi,\infty)\ \mbox{for all}\ k\in\NN_0,\label{eq37}\\
		 &&H_k(C^1_0(\overline{\Omega}),\dot{\varphi}^a_c)=H_k(C^1_0(\overline{\Omega}),\varphi^a)=C_k(\varphi_c,\infty)\ \mbox{for all}\ k\in\NN_0\label{eq38}
	\end{eqnarray}
	(see Granas and Dugundji \cite[p. 407]{8}). From (\ref{eq36}), (\ref{eq37}), (\ref{eq38}) we see that in order to prove the proposition, we need to show that
	$$H_k(C^1_0(\overline{\Omega}),\varphi^a_c)=0\ \mbox{for all}\ k\in\NN_0.$$
	
	To this end, let $C\subseteq\varphi^a_c$ be a compact set.
	\begin{claim}
		For $a<0$ with $|a|>0$ big, the set $C$ is contractible in $\varphi^a_c$.
	\end{claim}
	
	In what follows by $\left\langle \cdot,\cdot\right\rangle_0$ we denote the duality brackets for the pair $(C^1_0(\overline{\Omega})^*,C^1_0(\overline{\Omega}))$. Also, let $i:C^1_0(\overline{\Omega})\rightarrow W^{1,p}_{0}(\Omega)$ be the continuous embedding map. We have
	\begin{eqnarray}\label{eq39}
		&&\varphi_c=\varphi\circ i\nonumber\\
		&\Rightarrow&\varphi'_c(u)=i^*\varphi'(u)\ \mbox{for all}\ u\in C^1_0(\overline{\Omega}).
	\end{eqnarray}
	
	Let $u\in\varphi^a_c$. Then for $t>0$ we have
	\begin{eqnarray*}
		\frac{d}{dt}\varphi_c(tu)&=&\left\langle \varphi'_c(tu),u\right\rangle_0\ (\mbox{by the chain rule})\\
		&=&\left\langle \varphi'(tu),u\right\rangle\ (\mbox{see (\ref{eq39})})\\
		&=&\frac{1}{t}\left\langle \varphi'(tu),tu\right\rangle\\
		 &=&\frac{1}{t}\left[t^p||Du||^p_p+t^2||Du||^2_2-\int_{\Omega}f(z,tu^+)(tu^+)dz-\int_{\Omega}f(z,-tu^-)(-tu^-)dz\right]\\
		 &\leq&\frac{1}{t}\left[t^p||Du||^p_p+t^2||Du||^2_2-\int_{\Omega}pF(z,tu^+)dz-\int_{\Omega}pF(z,-tu^-)dz+c_3\right]\\
		&&\hspace{2cm}\mbox{with}\ c_3=||\gamma_0||_1+c_0|\Omega|_N>0\ (\mbox{see hypotheses}\ H(iii),(iv))\\
		&\leq&\frac{1}{t}\left[t^p||Du||^p_p+\frac{p}{2}t^2||Du||^2_2-\int_{\Omega}pF(z,tu)dz+c_3\right]\ (\mbox{since}\ p\geq 2)\\
		&=&\frac{1}{2}\left[p\varphi_c(tu)+c_3\right],\\
		\Rightarrow\left.\frac{d}{dt}\varphi_c(tu)\right|_{t=1}&\leq&p\varphi_c(u)+c_3\leq pa+c_3\ (\mbox{recall}\ u\in\varphi^a_c).
	\end{eqnarray*}
	
	Therefore
	$$a<-\frac{c_3}{p}\Rightarrow\left.\frac{d}{dt}\varphi_c(tu)\right|_{t=1}<0.$$
	
	So, if $\varphi_c(u)\in\left(a-1,a\right]$, then we can find a unique $k(u)>0$ such that $\varphi_c(k(u)u)=a-1$. If $u\in\varphi^{a-1}_{c}$, then we set $k(u)=1$. The implicit function theorem implies that $k\in C(\varphi^a_c,\left(0,1\right])$. We consider the deformation $h_1:[0,1]\times C\rightarrow\varphi^a_c$ defined by
	$$h_1(t,u)=((1-t)+tk(u))u\,.$$
	
	Let $C_1=h_1(1,C)\subseteq\varphi^{a-1}_{c}$. The set $C_1\subseteq C^1_0(\overline{\Omega})$ is compact. So, we can find $M_9>0$ such that
	\begin{equation}\label{eq40}
		\left|\frac{\partial u}{\partial n}(z)\right|\leq M_9\ \mbox{for all}\ z\in\overline{\Omega},\ \mbox{all}\ u\in C_1.
	\end{equation}
	
	Given $\epsilon>0$, we can find $\tilde{h}_{\epsilon}\in \mbox{int}\, C_+$ such that
	$$\frac{\partial\tilde{h}\epsilon}{\partial n}(z)<-M_9\ \mbox{for all}\ z\in\partial\Omega\ \mbox{and}\ (u+\tilde{h}_{\epsilon})^+\neq 0.$$
	
	To see this, set $\hat{d}(z)=d(z,\partial\Omega)$ and define
	$$\hat{h}_{\epsilon}(z)=\left\{\begin{array}{ll}
		\hat{M}\hat{d}(z)&\mbox{if}\ \hat{d}(z)\leq\epsilon\\
		\hat{M}\epsilon&\mbox{if}\ \epsilon<\hat{d}(z)
	\end{array}\right.\ \mbox{with}\ \hat{M}>0.$$
	
	Approximate $\hat{h}_{\epsilon}$ by a $C^1_0(\overline{\Omega})$-function $\tilde{h}_{\epsilon}$ and choose $\hat{M}>0$ big enough so that $\tilde{h}_{\epsilon}\in \mbox{int}\, C_+$ has the desired properties.
	
	We have $C_1\subseteq \varphi^{a-1}_{c}$. Hence, if we choose $\epsilon>0$ small, then the deformation $h_2:[0,1]\times C_1\rightarrow\varphi^a_c$ defined by
	$$h_2(t,u)=u+t\tilde{h}_{\epsilon}\ \mbox{for all}\ (t,u)\in[0,1]\times C_1,$$
	is well-defined.
	
	Let $C_2=h_2(1,C_1)$ and pick $u\in C_2$.Then $u^+\neq 0$ and we have
	$$\varphi_c(u)=\varphi_c(u^+)+\varphi_c(-u^-)\leq a.$$
	
	From the previous considerations we know that $t\longmapsto\varphi_c(tu)$ is decreasing on $\left[1,\infty\right)$. Because $C_2\subseteq C^1_0(\overline{\Omega})$ is compact, we can find $t_*\geq 1$ such that
	\begin{equation}\label{eq41}
		\varphi_c(tu^+)\leq a\ \mbox{for all}\ t\geq t_*,\ \mbox{all}\ u\in C_2.
	\end{equation}
	
	We introduce the deformation $h_3:[0,1]\times C_2\rightarrow \varphi^a_c$ defined by
	$$h_3(t,u)=(1-t+tt_*)u\ \mbox{for all}\ (t,u)\in[0,1]\times C_2.$$
	
	Evidently this is a well-defined deformation and if $C_3=h_3(1,C_2)$, then
	\begin{equation}\label{eq42}
		\varphi_c(u^+)\leq a\ \mbox{for all}\ u\in C_3\ (\mbox{see (\ref{eq41})}).
	\end{equation}
	
	The set $C_3=h_3(1,C_2)\subseteq C^1_0(\overline{\Omega})$ is compact. So, we can find $M_{10}>0$ such that
	\begin{equation}\label{eq43}
		\varphi_c(s(-u^-))\leq M_{10}\ \mbox{for all}\ u\in C_3,\ \mbox{all}\ s\in[0,1].
	\end{equation}
	
	From (\ref{eq42}) and since $t\longmapsto\varphi_c(tu^+)$ is decreasing on $\left[1,\infty\right)$, we can find $\hat{t}_*\geq 1$ big such that
	$$\varphi_c(\hat{t}_*u^+)\leq a-M_{10}\ \mbox{for all}\ u\in C_3.$$
	
	We consider the deformation $h_4:[0,1]\times C_3\rightarrow \varphi^a_c$ defined by
	$$h_4(t,u)=(1-t+t\hat{t}_*)u^++u^-.$$
	
	This deformation too is well-defined. We set $C_4=h_3(1,C_3)$ and have
	\begin{eqnarray}\label{eq44}
		&&C_4\subseteq C^1_0(\overline{\Omega})\ \mbox{is compact}\nonumber\\
		&&C_4\subseteq\varphi^a_c\cap\{u\in C^1_0(\overline{\Omega}):\varphi_c(u^+)\leq a-M_{10}\}\ (\mbox{see (\ref{eq43})}).
	\end{eqnarray}
	
	Using $C_4\subseteq C^1_0(\overline{\Omega})$, we will deform to a compact subset of positive functions in $\varphi^a_c$. To this end, let $h_5:[0,1]\times C_4\rightarrow\varphi^a_c$ be the deformation defined by
	$$h_5(t,u)=u^++(1-t)(-u^-)\ \mbox{for all}\ (t,u)\in[0,1]\times C_4.$$
	
	We have
	\begin{eqnarray*}
		&\varphi_c(h_5(t,u))&=\varphi_c(u^++(1-t)(-u^-))\\
		&&=\varphi_c(u^+)+\varphi_c((1-t)(-u^-))\\
		&&\leq a-M_{10}+M_{10}=a\ (\mbox{see (\ref{eq44}) and (\ref{eq43})}),\\
		&\Rightarrow&h_5\ \mbox{is well defined.}
	\end{eqnarray*}
	
	So, if $C_5=h(1,C_4)$, then we have
	\begin{eqnarray}\label{eq45}
		&&C_5\subseteq\varphi^a_c\ \mbox{and}\ C_5\subseteq C_+,\nonumber\\
		&\Rightarrow&C_5\subseteq\varphi^a_c\cap C_+=C^a_+.
	\end{eqnarray}
	
	Let $\partial B^c_+=\{u\in C^1_0(\overline{\Omega}):||u||_{C^1_0(\overline{\Omega})}=1\}\cap C_+$. From the first part of the proof we have
	$$C^a_+=\{tu:u\in\partial B^c_+,t\geq\hat{k}(u)\}$$
	with $\hat{k}(u)>0$ being the unique real such that $\varphi_c(\hat{k}(u)u)=a.$
	
	Using the radial retraction, we see that $C^a_+$ and $\partial B^c_+$ are homotopy equivalent. We consider the deformation $h_+:[0,1]\times\partial B^c_+\rightarrow\partial B^c_+$ defined by
	$$h_+(t,u)=\frac{(1-t)u+t\hat{u}_1(p)}{||(1-t)u+t\hat{u}_1(p)||_{C^1_0(\overline{\Omega})}}\ \mbox{for all}\ (t,u)\in[0,1]\times\partial B^c_+.$$
	
	Note that
	\begin{eqnarray*}
		&&h_+(1,u)=\frac{\hat{u}_1(p)}{||\hat{u}_1(p)||_{C^1_0(\overline{\Omega})}}\in \partial B^c_+,\\
		&\Rightarrow&\partial B^c_+\ \mbox{is contractible,}\\
		&\Rightarrow&C^a_+\ \mbox{is contractible.}
	\end{eqnarray*}
	
	Then from (\ref{eq45}) we infer that $C_5$ is contractible. Since $C$ was deformed to $C_5$ by successive deformations, we conclude that $C$ is contractible in $\varphi^a_c$ for $a<0$ with $|a|>0$ big. This proves the Claim.
	
	Let $*\in\dot{\varphi}^a_c$. For $a<\inf\varphi(K_{\varphi})$, we have
	\begin{eqnarray}\label{eq46}
		&&H_k(\varphi^a_c,*)=H_k(\dot{\varphi}^a_c,*)\ \mbox{for all}\ k\in\NN_0\\
		&&\hspace{1.5cm}(\mbox{see Granas and Dugundji \cite[p. 407]{8}}).\nonumber
	\end{eqnarray}
	
	The Banach space $C^1_0(\overline{\Omega})$ is separable. So, we can find a sequence $\{V_n\}_{n\geq 1}$ of increasing finite dimensional subspaces of $C^1_0(\overline{\Omega})$ such that
	$$C^1_0(\overline{\Omega})=\overline{{\underset{\mathrm{n\geq 1}}\bigcup}V_n}\,.$$
	
	From the Claim we have
	\begin{equation}\label{eq47}
		H_k(\dot{\varphi}^a_c,*)=H_k(\dot{\varphi}^a_c,\dot{\varphi}^a_c\cap\bar{B}^{V_n}_{n})\ \mbox{for all}\ k\in\NN_0,
	\end{equation}
	where $\bar{B}^{V_n}_{n}=\{u\in V_n:||u||_{C^1_0(\overline{\Omega})}\leq n\},\ *\in\bar{B}^{V_n}_{n}$. From Palais \cite{17} (Corollary p.5) (see also Granas and Dugundji \cite[Theorem D.6, p. 615]{8}), we have
	 $$0=H_k(\dot{\varphi}^a_c,\dot{\varphi}^a_c)=\lim\limits_{\stackrel{\longrightarrow}{n}}H_k(\dot{\varphi}^a_c,\dot{\varphi}^a_c\cap\bar{B}^{V_n}_{n})$$
	where $\lim\limits_{\stackrel{\longrightarrow}{n}}$ denotes the inductive limit. So, from (\ref{eq47}), we infer that
	\begin{equation}\label{eq48}
		H_k(\varphi^a_c,*)=0\ \mbox{for all}\ k\in\NN_0.
	\end{equation}
	
	Consider the following triple of sets:
	$$\{*\}\subseteq\varphi^a_c\subseteq C^1_0(\overline{\Omega}).$$
	
	For this triple, we introduce corresponding long exact sequence of singular homology groups
	\begin{equation}\label{eq49}
		\cdots\rightarrow H_k(\varphi^a_c,*)\stackrel{i_*}{\longrightarrow}H_k(C^1_0(\overline{\Omega}),\varphi^a_c)\stackrel{\partial_*}{\longrightarrow}H_{k-1}(\varphi^a_c,*)\rightarrow\cdots
	\end{equation}
	
	Here $i_*$ is the homomorphism induced by the inclusion $i:(\varphi^a_c,*)\rightarrow(C^1_0(\overline{\Omega}),\varphi^a_c)$ and $\partial_*$ is the boundary homomorphism. From (\ref{eq48}) and the exactness of (\ref{eq49}), we see that
	\begin{eqnarray*}
		&&H_k(C^1_0(\overline{\Omega}),\varphi^a_c)=0\ \mbox{for all}\ k\in\NN_0,\\
		&\Rightarrow&C_k(\varphi_c,\infty)=0\ \mbox{for all}\ k\in\NN_0,\\
		&\Rightarrow&C_k(\varphi,\infty)=0\ \mbox{for all}\ k\in\NN_0.
	\end{eqnarray*}
\end{proof}
\begin{prop}\label{prop7}
	If hypotheses $H$ hold, then $u=0$ is a local minimizer of the functional $\varphi$.
\end{prop}
\begin{proof}
	Hypotheses $H(i),(iv)$ imply that given $\epsilon>0$, we can find $c_{\epsilon}>0$ such that
	\begin{equation}\label{eq50}
		F(z,x)\leq\frac{1}{2}(f'_x(z,0)+\epsilon)x^2+\frac{c_{\epsilon}}{r}|x|^r\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\in\RR.
	\end{equation}
	
	Then for all $u\in W^{1,p}_{0}(\Omega)$ we have
	\begin{eqnarray*}
		&&\varphi(u)\geq\frac{1}{p}||Du||^p_p+\frac{1}{2}\left[||Du||^2_2-\int_{\Omega}f'_x(z,0)u^2dz\right]-
\frac{\epsilon}{2}\frac{1}{\hat{\lambda}_1(2)}||Du||^2_2-c_4||u||^r\\
		&&\hspace{2cm}\mbox{for some}\ c_4=c_4(\epsilon)>0\ (\mbox{see (\ref{eq50}) and (\ref{eq3})})\\
		 &&\geq\frac{1}{p}||Du||^p_p+\frac{1}{2}\left[\hat{c}-\frac{\epsilon}{\hat{\lambda}_1(2)}\right]||Du||^2_2-c_4||u||^r\ (\mbox{see Lemma \ref{lem2}}).
	\end{eqnarray*}
	
	Choosing $\epsilon\in(0,\hat{\lambda}_1(2)c_6)$, we have
	$$\varphi(u)\geq\frac{1}{p}||u||^p+c_5||u||^2-c_4||u||^r\ \mbox{for some}\ c_5>0,\ \mbox{all}\ u\in W^{1,p}_{0}(\Omega)$$
	
	Because $2\leq p<r$, for $\rho\in(0,1)$ small we have
	\begin{eqnarray*}
		&&\varphi(u)>0=\varphi(0)\ \mbox{for all}\ u\in W^{1,p}_{0}(\Omega)\ \mbox{with}\ 0<||u||\leq\rho,\\
		&\Rightarrow&u=0\ \mbox{is a (strict) local minimizer of}\ \varphi.
	\end{eqnarray*}
\end{proof}

Now we are ready to produce two nontrivial constant sign solutions.
\begin{prop}\label{prop8}
	If hypotheses $H$ hold, then problem (\ref{eq1}) has at least two constant sign solutions
	$$u_0\in \mbox{int}\, C_+\ \mbox{and}\ v_0\in-\mbox{int}\, C_+.$$
\end{prop}
\begin{proof}
	Let $\varphi_+:W^{1,p}_{0}(\Omega)\rightarrow\RR$ be the $C^1$-functional defined by
	$$\varphi_+(u)=\frac{1}{p}||Du||^p_p+\frac{1}{2}||Du||^2_2-\int_{\Omega}F(z,u^+)dz\ \mbox{for all}\ u\in W^{1,p}_{0}(\Omega).$$
	\begin{claim}\label{cl3}
	The functional $\varphi_+$ satisfies the $C$-condition.
	\end{claim}
	
	Let $\{u_n\}_{n\geq 1}\subseteq W^{1,p}_{0}(\Omega)$ be a sequence such that
	\begin{eqnarray}
		&&|\varphi_+(u_n)|\leq M_{11}\ \mbox{for some}\ M_{11}>0,\ \mbox{all}\ n\geq 1\label{eq51}\\
		&&(1+||u_n||)\varphi'(u_n)\rightarrow 0\ \mbox{in}\ W^{-1,p'}(\Omega)\ \mbox{as}\ n\rightarrow\infty\,.\label{eq52}
	\end{eqnarray}
	
	From (\ref{eq52}) we have
	\begin{eqnarray}\label{eq53}
		&&\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||}\\
		&&\hspace{3cm}\mbox{for all}\ h\in W^{1,p}_{0}(\Omega)\ \mbox{with}\ \epsilon_n\rightarrow 0^+\nonumber.
	\end{eqnarray}
	
	In (\ref{eq53}) we choose $h=-u^-_n\in W^{1,p}_{0}(\Omega)$. Then
	\begin{eqnarray}\label{eq54}
		&&||Du^-_n||^p_p+||Du^-_n||^2_2\leq\epsilon_n\ \mbox{for all}\ n\in\NN,\nonumber\\
		&\Rightarrow&u^-_n\rightarrow 0\ \mbox{in}\ W^{1,p}_{0}(\Omega)\ \mbox{as}\ n\rightarrow\infty.
	\end{eqnarray}
	
	From (\ref{eq51}) and (\ref{eq54}) it follows that
	\begin{equation}\label{eq55}
		\varphi_+(u^+_n)\leq M_{12}\ \mbox{for some}\ M_{12}>0,\ \mbox{for all}\ n\in\NN.
	\end{equation}
	
	In (\ref{eq53}) we choose $h=u^+_n\in W^{1,p}_{0}(\Omega)$. Then
	\begin{equation}\label{eq56}
		-||Du^+_n||^p_p-||Du^+_n||^2_2+\int_{\Omega}f(z,u^+_n)u^+_ndz\leq\epsilon_n\ \mbox{for all}\ n\in\NN.
	\end{equation}
	
	From (\ref{eq55}) and since $2\leq p$, we have
	\begin{equation}\label{eq57}
		||Du^+_n||^p_p+||Du^+_n||^2_2-\int_{\Omega}pF(z,u^+_n)dz\leq pM_{12}\ \mbox{for all}\ n\in\NN.
	\end{equation}
	
	Adding (\ref{eq56}) and (\ref{eq57}), we obtain
	\begin{equation}\label{eq58}
		\int_{\Omega}\xi(z,u^+_n)dz\leq M_{13}\ \mbox{for some}\ M_{13}>0,\ \mbox{all}\ n\in\NN.
	\end{equation}
	
	Using (\ref{eq58}) and reasoning as in the Claim in the proof of Proposition \ref{prop5}, we obtain that
	\begin{eqnarray*}
		&&\{u^+_n\}_{n\geq 1}\subseteq W^{1,p}_{0}(\Omega)\ \mbox{is bounded},\\
		&\Rightarrow&\{u_n\}_{n\geq 1}\subseteq W^{1,p}_{0}(\Omega)\ \mbox{is bounded (see (\ref{eq54})).}
	\end{eqnarray*}
	
	From this, as in the proof of Proposition \ref{prop5}, via the $(S)_+$-property of the map $A_p$ (see Proposition \ref{prop4}), we conclude that $\varphi_+$ satisfies the $C$-condition. This proves Claim \ref{cl3}.
	
	It is straightforward to check that
	$$u\in K_{\varphi_+}\Rightarrow u\geq 0.$$
	
	So, we may assume that $K_{\varphi_+}$ is finite or otherwise we already have a sequence of distinct positive solutions for problem (\ref{eq1}).
	
	A careful reading of the proof of Proposition \ref{prop7}, reveals that $u=0$ is also a local minimizer for $\varphi_+$. So, we can find $\rho\in(0,1)$ small such that
	\begin{equation}\label{eq59}
		\varphi_+(0)=0<\inf[\varphi_+(u):||u||=\rho]=m^+_{\rho}
	\end{equation}
	(see Aizicovici, Papageorgiou and Staicu \cite{1}, proof of Proposition 29).
	
	Finally note that hypothesis $H(ii)$ implies that
	\begin{equation}\label{eq60}
		\varphi_+(t\hat{u}_1(p))\rightarrow-\infty\ \mbox{as}\ t\rightarrow+\infty.
	\end{equation}
	
	The Claim and (\ref{eq59}) and (\ref{eq60}), permit the use of Theorem \ref{th1} (the mountain pass theorem). So, we can find $u_0\in W^{1,p}_{0}(\Omega)$ such that
	\begin{equation}\label{eq61}
		u_0\in K_{\varphi_+}\ \mbox{and}\ m^+_{\rho}\leq\varphi_+(u_0).
	\end{equation}
	
	From (\ref{eq59}) and (\ref{eq61}), we see that $u_0\neq 0,\ u_0\geq 0$. Also, we have
	\begin{eqnarray}\label{eq62}
		&&A_p(u_0)+A(u_0)=N_f(u_0)\ \mbox{in}\ W^{-1,p'}(\Omega),\nonumber\\
		&\Rightarrow&-\Delta_p(u_0)(z)-\Delta u_0(z)=f(z,u_0(z))\ \mbox{for almost all}\ z\in\Omega,\ u_0|_{\partial\Omega}=0.
	\end{eqnarray}
	
	From Ladyzhenskaya and Uraltseva \cite[Theorem 7.1, p. 286]{9}, we have that $u_0\in L^{\infty}(\Omega)$. So, we can apply Theorem 1 of Lieberman \cite{10} and infer that $u_0\in C_+\backslash\{0\}$.
	
	Let $a(y)=|y|^{p-2}+y$ for all $y\in\RR^N$. Evidently $a\in C^1(\RR^N,\RR^N)$ and
	\begin{eqnarray*}
		&&\nabla a(y)=|y|^{p-2}\left[I+(p-2)\frac{y\otimes y}{|y|^2}\right]+I\ \mbox{for all}\ y\in\RR^N,\\
		&\Rightarrow&(\nabla a(y)\xi,\xi)_{\RR^N}\geq|\xi|^2\ \mbox{for all}\ y,\xi\in\RR^N.
	\end{eqnarray*}
	
	Note that
	$$\mbox{div}\, a(Du)=\Delta_pu+\Delta u\ \mbox{for all}\ u\in W^{1,p}_{0}(\Omega).$$
	
	So, we can use the tangency principle of Pucci and Serrin \cite[Theorem 2.5.2, p. 35]{23} and have
	$$u_0(z)>0\ \mbox{for all}\ z\in\Omega.$$
	
	For $\rho=||u_0||_{\infty}$, let $\hat{\xi}_{\rho}>0$ be as postulated by hypothesis $H(iv)$. From (\ref{eq62}) we have
	\begin{eqnarray*}
		&&-\Delta_pu_0(z)-\Delta u_0(z)+\hat{\xi}_{\rho}u_0(z)^{p-1}\geq 0\ \mbox{for almost all}\ z\in\Omega,\\
		&\Rightarrow&\Delta_pu_0(z)+\Delta u_0(z)\leq\hat{\xi}_{\rho}u_0(z)^{p-1}\ \mbox{for almost all}\ z\in\Omega.
	\end{eqnarray*}
	
	Then the boundary point theorem of Pucci and Serrin \cite[Theorem 5.5.1, p. 120]{23}  implies that $u_0\in \mbox{int}\, C_+$.
	
	Next we produce a negative solution. For this purpose let
	$$f_-(z,x)=f(z,-x^-),\ F_-(z,x)=\int^x_0f_-(z,s)ds$$
	and let $\varphi_-:W^{1,p}_{0}(\Omega)\rightarrow\RR$ be the $C^1$-functional defined by
	$$\varphi_-(u)=\frac{1}{p}||Du||^p_p+\frac{1}{2}||Du||^2_2-\int_{\Omega}F_-(z,u)dz\ \mbox{for all}\ u\in W^{1,p}_{0}(\Omega).$$
	\begin{claim}\label{cl4}
		The functional $\varphi_-$ satisfies the $C$-condition.
	\end{claim}
	
	Let $\{u_n\}_{n\geq 1}\subseteq W^{1,p}_{0}(\Omega)$ be a sequence such that
	\begin{eqnarray}
		&&|\varphi_-(u_n)|\leq M_{14}\ \mbox{for some}\ M_{14}>0,\ \mbox{all}\ n\in\NN,\label{eq63}\\
		&&(1+||u_n||)\varphi'_-(u_n)\rightarrow 0\ \mbox{in}\ W^{-1,p'}(\Omega)\ \mbox{as}\ n\rightarrow\infty.\label{eq64}
	\end{eqnarray}
	
	From (\ref{eq64}) we have
	\begin{eqnarray}\label{eq65}
		&&\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||}\\
		&&\hspace{4cm}\mbox{for all}\ h\in W^{1,p}(\Omega),\ \mbox{with}\ \epsilon_n\rightarrow 0^+.\nonumber
	\end{eqnarray}
	
	In (\ref{eq65}) we choose $h=u^+_n\in W^{1,p}_{0}(\Omega)$. Then
	\begin{eqnarray}\label{eq66}
		&&||Du^+_n||^p_p+||Du^+_n||^2_2\leq\epsilon_n\ \mbox{for all}\ n\in\NN,\nonumber\\
		&\Rightarrow&u^+_n\rightarrow 0\ \mbox{in}\ W^{1,p}_{0}(\Omega)\ \mbox{as}\ n\rightarrow\infty.
	\end{eqnarray}
	
	Then using (\ref{eq66}), inequality (\ref{eq65}) becomes
	\begin{eqnarray*}
		&&\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||\\
		&&\hspace{4cm}\mbox{for all}\ h\in W^{1,p}_{0}(\Omega),\ \mbox{with}\ \epsilon'_n\rightarrow 0^+.
	\end{eqnarray*}
	
	Reasoning as in the last part of the proof of Proposition \ref{prop5} (see the part of the proof after (\ref{eq29})), we obtain
	\begin{eqnarray*}
		&&\{u^-_n\}_{n\geq 1}\subseteq W^{1,p}_{0}(\Omega)\ \mbox{is bounded},\\
		&\Rightarrow&\{u_n\}_{n\geq 1}\subseteq W^{1,p}_{0}(\Omega)\ \mbox{is bounded (see (\ref{eq66}))},\\
		&\Rightarrow&\varphi_-\ \mbox{satisfies the $C$-condition (as before using Proposition \ref{prop4})}.
	\end{eqnarray*}
	
	This proves Claim \ref{cl4}.
	
	As we did for $\varphi_+$, a critical inspection of the proof of Proposition \ref{prop7}, reveals that $u=0$ is a local minimizer of $\varphi_-$. Also, it is easy to see that $K_{\varphi_-}\subseteq -C_+$ and so we may assume that $K_{\varphi_-}$ is finite or otherwise we already have a whole sequence of distinct negative solutions of (\ref{eq1}). These facts imply that we can find $\rho\in(0,1)$ small such that
	\begin{equation}\label{eq67}
		\varphi_-(0)=0<\inf[\varphi_-(u):||u||=\rho]=m^-_{\rho}
	\end{equation}
	(see Aizicovici, Papageorgiou and Staicu \cite{1}, proof of Proposition 29).
	
	Note that hypothesis $H(iv)$ implies that
	\begin{equation}\label{eq68}
		\varphi_-(t\hat{u}_1(p))\rightarrow-\infty\ \mbox{as}\ t\rightarrow-\infty.
	\end{equation}
	
	Then Claim \ref{cl4} and (\ref{eq67}), (\ref{eq68}) permit the use of Theorem \ref{th1} (the mountain pass theorem). So, we can find $v_0\in W^{1,p}_{0}(\Omega)$ such that
	\begin{equation}\label{eq69}
		v_0\in K_{\varphi_-}\ \mbox{and}\ m^-_{\rho}\leq\varphi_-(v_0).
	\end{equation}
	
	From (\ref{eq67}) and (\ref{eq69}) we see that
	$$v_0\in(-C_+)\backslash\{0\}\ (\mbox{see Lieberman \cite{10}}).$$
	
	In fact as we did for $u_0$, using the tangency principle and the boundary point theorem of Pucci and Serrin \cite[pp. 35 and 120]{23}, we have
	$$v_0\in -\mbox{int}\, C_+.$$
\end{proof}

Next we compute the critical groups of $\varphi$ at these solutions.
\begin{prop}\label{prop9}
	If hypotheses $H$ hold and $K_{\varphi}$ is finite, then $C_k(\varphi,u_0)=C_k(\varphi,v_0)=\delta_{k,1}\ZZ$ for all $k\in\NN_0$.
\end{prop}
\begin{proof}
	Let $h_+(t,u)=(1-t)\varphi_+(u)+t\varphi(u)$ for all $(t,u)\in[0,1]\times W^{1,p}_{0}(\Omega)$. Suppose that we can find $\{t_n\}_{n\geq 1}\subseteq[0,1]$ and  $\{u_n\}_{n\geq 1}\subseteq W^{1,p}_{0}(\Omega)$ such that
	\begin{equation}\label{eq70}
		t_n\rightarrow t,u_n\rightarrow u_0\ \mbox{in}\ W^{1,p}_{0}(\Omega)\ \mbox{and}\ (h_+)'_u(t_n,u_n)=0\ \mbox{for all}\ n\in\NN.
	\end{equation}
	
	From (\ref{eq70}) we have
	\begin{eqnarray}\label{eq71}
		&&A_p(u_n)+A(u_n)=N_f(u^+_n)+t_nN_f(-u^-_n),\nonumber\\
		&\Rightarrow&-\Delta_pu_n(z)-\Delta u_n(z)=f(z,u^+_n(z))+t_nf(z,-u^-_n(z))\\
		&&\hspace{4cm}\mbox{for almost all}\ z\in\Omega,u_n|_{\partial\Omega}=0.\nonumber
	\end{eqnarray}
	
	From Theorem 7.1, p. 286 of Ladyzhenskaya and Uraltseva \cite{9}, we can find $M_{15}>0$ such that
	$$||u_n||_{\infty}\leq M_{15}\ \mbox{for all}\ n\in\NN.$$
	
	Invoking Theorem 1 of Lieberman \cite{10}, we can find $\beta\in(0,1)$ and $M_{16}>0$ such that
	\begin{equation}\label{eq72}
		u_n\in C^{1,\beta}_{0}(\overline{\Omega})\ \mbox{and}\ ||u_n||_{C^{1,\beta}_{0}(\overline{\Omega})}\leq M_{16}\ \mbox{for all}\ n\in\NN.
	\end{equation}
	
	Since $C^{1,\beta}_{0}(\overline{\Omega})$ is embedded compactly into $C^1_0(\overline{\Omega})$, from (\ref{eq70}) and (\ref{eq72}) we infer that
	$$u_n\rightarrow u_n\ \mbox{in}\ C^1_0(\overline{\Omega})$$
	
	Recall that $u_0\in \mbox{int}\, C_+$ (see Proposition \ref{prop8}). So, we have
	\begin{eqnarray*}
		&&u_n\in \mbox{int}\, C_+\ \mbox{for all}\ n\geq n_0,\\
		&\Rightarrow&\{u_n\}_{n\geq n_0}\subseteq K_{\varphi}\ (\mbox{see (\ref{eq71})}),
	\end{eqnarray*}
	which contradicts our hypothesis that $K_{\varphi}$ is finite. So, (\ref{eq65}) cannot hold. Since for every $t\in[0,1]$ and every bounded set $D\subseteq W^{1,p}_{0}(\Omega)$, $h_+(t,\cdot)$ satisfies the $C$-condition on $D$ (see Proposition \ref{prop4}), using Theorem 5.2 of Corvellec and Hantoute \cite{6} (the homotopy invariance of the critical groups), we have
	\begin{equation}\label{eq73}
		C_k(\varphi,u_0)=C_k(\varphi_+,u_0)\ \mbox{for all}\ k\in\NN_0.
	\end{equation}
	
	From the proof of Proposition \ref{prop8}, we know that $u_0$ is a critical point of $\varphi_+$ of mountain pass-type. Then from Proposition 6.10, p.176 of Motreanu, Motreanu and Papageorgiou \cite{15}, we have
	\begin{eqnarray*}
		&&C_1(\varphi_+,u_0)\neq 0,\\
		&\Rightarrow&C_1(\varphi,u_0)\neq 0\ (\mbox{see (\ref{eq73})}).
	\end{eqnarray*}
	
	But $\varphi\in C^2(W^{1,p}_{0}(\Omega))$. So, from Papageorgiou and Smyrlis \cite{21} (see also Papageorgiou and R\u adulescu \cite{19}), we have
	$$C_k(\varphi,u_0)=\delta_{k,1}\ZZ\ \mbox{for all}\ k\in\NN_0.$$
	
	Similarly for $v_0\in -\mbox{int}\, C_+$, using this time the functional $\varphi_-$.
\end{proof}

Now we are ready for the multiplicity theorem concerning problem (\ref{eq1}).
\begin{theorem}\label{th10}
	If hypotheses $H$ hold, then problem (\ref{eq1}) has at least three nontrivial solutions
	$$u_0\in \mbox{int}\, C_+,v_0\in-\mbox{int}\, C_+\ \mbox{and}\ y_0\in C^1_0(\overline{\Omega}).$$
\end{theorem}
\begin{proof}
	From Proposition \ref{prop8}, we already have two constant sign solutions
	$$u_0\in \mbox{int}\, C_+\ \mbox{and}\ v_0\in-\mbox{int}\, C_+.$$
	
	Suppose $K_{\varphi}=\{0,u_0,v_0\}$. From Proposition \ref{prop9}, we have
	\begin{equation}\label{eq74}
		C_k(\varphi,u_0)=C_k(\varphi,v_0)=\delta_{k,1}\ZZ\ \mbox{for all}\ k\in\NN_0.
	\end{equation}
	
	From Proposition \ref{prop7} we know that $u=0$ is a local minimizer of $\varphi$. Hence
	\begin{equation}\label{eq75}
		C_k(\varphi,u)=\delta_{k,0}\ZZ\ \mbox{for all}\ k\in\NN_0.
	\end{equation}
	
	Moreover, from Proposition \ref{prop6} we have
	\begin{equation}\label{eq76}
		C_k(\varphi,\infty)=0\ \mbox{for all}\ k\in\NN_0.
	\end{equation}
	
	From (\ref{eq74}), (\ref{eq75}), (\ref{eq76}) and the Morse relation with $t=-1$ (see (\ref{eq4})), we have
	\begin{eqnarray*}
		&&(-1)^0+2(-1)^1=0,\\
		&\Rightarrow&(-1)^1=0\ \mbox{a contradiction}.
	\end{eqnarray*}
	
	So, there exists $y_0\in K_{\varphi}$, $y_0\notin\{0,u_0,v_0\}$. Then $y_0$ is a third nontrivial solution of problem (\ref{eq1}) and the nonlinear regularity theory (see Lieberman \cite{10}), implies that $y_0\in C^1_0(\overline{\Omega})$.
\end{proof}

\begin{remark}
	When $p=2$, Theorem \ref{th10} is related to the multiplicity theorems of Recova and Rumbos \cite{24}, \cite{25} who produce three nontrivial solutions under more restrictive regularity conditions on the reaction $f(z,x)$ and using the Ambrosetti-Rabinowitz condition to express the superlinearity condition in the positive direction. A precise improvement of the works of Recova and Rumbos \cite{24}, \cite{25}, in fact to Robin problems with an indefinite potential, can be found in the paper of Papageorgiou and R\u adulescu \cite{20}.
\end{remark}

\medskip
{\bf Acknowledgments.} V.D. R\u adulescu was partially supported
 by a grant of the Romanian National Authority for Scientific Research  UEFISCDI, project number PCCA-23/2014.


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\end{thebibliography}


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