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\title[Resonant $(p,2)$--equations with asymmetric reaction]{Resonant $(p,2)$--equations with asymmetric reaction}


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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{Institute of Mathematics "Simion Stoilow" of the Romanian Academy, P.O. Box 1-764,
          014700 Bucharest, Romania
          \& Department of Mathematics, University of Craiova, Street A.I. Cuza 13,
          200585 Craiova, Romania}
\email{\tt vicentiu.radulescu@imar.ro}

\keywords{$(p,2)$-equation, resonance, nonlinear regularity, critical groups, Picone's identity\\
\phantom{aa} 2010 AMS Subject Classification:
35B34, 35J20, 35J60, 58E05.}

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\begin{abstract}
We consider a nonlinear, nonhomogeneous Dirichlet problem driven by the sum of a $p$--Laplacian and a Laplacian, $2<p<\infty$ (($p,2$)-equation) and with a reaction which exhibits asymmetric behavior at $+\infty$ and at $-\infty$ and is resonant. Using variational methods together with Morse theoretic arguments, we prove the existence of two and three nontrivial solutions.
\end{abstract}

\maketitle

%\tableofcontents

\section{Introduction}

Let $\Omega\subset\RR^N$ be a bounded domain with a $C^2$--boundary $\partial\Omega$ and let $p>2$ be a real number.

In this paper we study the following nonlinear nonhomogeneous elliptic equation (($p,2$)-equation):
\begin{eqnarray}\label{eq1}
	-\Delta_pu(z)-\Delta u(z)=f(z,u(z))\quad \mbox{in}\ \Omega,\ u|_{\partial\Omega}=0\,.
\end{eqnarray}
Here $\Delta_p$ denotes the $p$--Laplacian differential operator defined by
$$\Delta_pu=\mbox{div}(||Du||^{p-2}Du)\ \mbox{for\ all}\ u\in W^{1,p}_{0}(\Omega).$$
Also $f:\Omega\times\RR\rightarrow\RR$ is a Carath\'eodory reaction (that is, for all $x\in\RR$, the mapping $z\longmapsto f(z,x)$ is measurable and for a.a. $z\in\Omega$, $ x\longmapsto f(z,x)$ is continuous). The aim of this work is to prove a multiplicity theorem (in particular, a three solutions theorem), when the reaction $f(z,\cdot)$ is $(p-1)$-linear near $\pm\infty$, but exhibits asymmetric behavior at $+\infty$ and at $-\infty$. More precisely, we assume that the quotient $\frac{f(z,x)}{|x|^{p-2}x}$ crosses the principal eigenvalue $\hat{\lambda}_1(p)>0$ of $(-\Delta_p,W^{1,p}_{0}(\Omega))$ as $x\in\RR$ moves from $-\infty$ to $+\infty$. Another interesting feature of our framework is that we allow for resonance to occur at both $+\infty$ and $-\infty$. At $+\infty$ the resonance can occur with respect to the principal eigenvalue $\hat{\lambda}_1(p)>0$, while at $-\infty$ with respect to the second eigenvalue $\hat{\lambda}_2(p)>\hat{\lambda}_1(p)$.

Problems with an asymmetric nonlinearity were studied by Chabrowski and Yang \cite{5}, Chang \cite{6}, de Paiva and Massa \cite{10}, de Paiva and Presoto \cite{11}, Motreanu, Motreanu and Papageorgiou \cite{17}, Perera \cite{24} (for semilinear Dirichlet problems), by Motreanu, Motreanu and Papageorgiou \cite{18} (for nonlinear equations driven by the Dirichlet $p$-Laplacian) and by Papageorgiou and R\u{a}dulescu \cite{20} (for semilinear Neumann problems with an indefinite and unbounded potential). None of the aforementioned works permits resonance.

We mention that $(p,2)$--equations (that is, equations driven by the sum of a $p$-Laplacian and a Laplacian, with $2<p<\infty$) arise in mathematical physics (see Benci, D'Avenia, Fortunato and Pisani \cite{4} (quantum physics) and Cherfils and Ilyasov \cite{7} (plasma physics)). Recently some existence and multiplicity results  for such equations were proved by Cingolani and Degiovanni \cite{8}, Cingolani and Vannella \cite{9}, Papageorgiou and R\u{a}dulescu \cite{21,22}, and Sun \cite{27}. However, none of the aforementioned works treats the asymmetric resonant case. We also refer to the recent book by Ciarlet \cite{ciarletsiam} for the rigorous qualitative analysis of many models described by nonlinear partial differential equations.

Our approach combines variational methods based on the critical point theory with Morse theoretic arguments (critical groups). In the next section, for the convenience of the reader, we recall the main mathematical tools that will be used in the sequel.

\section{Mathematical Background}

Let $X$ be a Banach space and $X^{\ast}$ be its topological dual. By $\left\langle \cdot,\cdot\right\rangle$ we denote the duality brackets for the pair $(X^{\ast},X)$. Given $\varphi\in C^1(X)$, we say that $\varphi$ satisfies the ``Cerami condition" (the ``$C$-condition" for short) if the following holds:
\begin{center}
	"Every sequence $\{x_n\}_{n\geq 1}\subseteq X$ such that  $\{\varphi(x_n)\}_{n\geq 1}\subseteq\RR$ is bounded and
	$$(1+||x_n||)\ \varphi^{'}(x_n)\rightarrow 0\ \mbox{in}\ X^{\ast}\ \mbox{as}\ n\rightarrow\infty,$$
	admits a strongly convergent subsequence."
\end{center}
This is a compactness-type condition on the functional $\varphi$, which is more general than the usual Palais-Smale condition. Both conditions compensate for the fact that the ambient space $X$ need not be locally compact. Using the $C$-condition, we can have the following minimax characterization of certain critical values of $\varphi$. The result is know in the literature as the ``mountain pass theorem" (see, for example, Gasinski and Papageorgiou \cite{12} and R\u{a}dulescu \cite{26}).

\begin{theorem}\label{th1}
Let $X$ be a Banach space. Assume that $\varphi\in C^1(X)$ satisfies the $C$-condition, $x_0,x_1\in X$ with $||x_1-x_0||>\rho>0$,
	$$\max\{\varphi(x_0),\varphi(x_1)\}<\inf \{\varphi(x):||x-x_0||=\rho\}=\eta_{\rho}\,,$$
	and $c=\inf\limits_{\gamma\in \Gamma}\max\limits_{0\leq t\leq 1}\ \varphi(\gamma(t))$, where $\Gamma=\{\gamma\in C([0,1],W^{1,p}_{0}(\Omega)):\gamma(0)=x_0,\ \gamma(1)=x_1\}$. Then $c\geq\eta_{\rho}$ and $c$ is a critical value of $\varphi$.
\end{theorem}

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

Suppose that $f_0:\Omega\times\RR\rightarrow\RR$ is a Carath\'eodory function with subcritical growth in the $x\in\RR$ variable, that is,
$$|f_0(z,x)|\leq a(z)\ (1+|z|^{r-1})\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\in\RR,$$
with $a\in L^{\infty}(\Omega)_+,\ 1<r<p^{\ast}=\left\{
			\begin{array}{ll}
				\frac{Np}{N-p}&\ \mbox{if}\ p<N\\
				+\infty&\ \mbox{if}\ p\geq N
			\end{array}
			\right.$.
We set $F_0(z,x)=\int^{x}_{0}f_0(z,s)ds$ and consider the $C^1$-functional $\varphi_0:W^{1,p}_{0}(\Omega)\rightarrow\RR$ defined by
$$\varphi_0(u)=\frac{1}{p}||Du||^{p}_{p}+\frac{1}{2}||Du||^{2}_{2}-\int_{\Omega}F_0(z,u(z))dz\ \mbox{for\ all}\ u\in W^{1,p}_{0}(\Omega).$$

The next proposition is a special case of a more general result due to Aizicovici, Papageorgiou and Staicu \cite{2} and it relates local $C^{1}_{0}(\bar{\Omega})$ and $W^{1,p}_{0}(\Omega)$-minimizers of $\varphi_0$. The result is essentially a consequence of the nonlinear regularity theory (see Ladyzhenskaya and Uraltseva \cite{14} and Lieberman \cite{15}).
\begin{prop}\label{prop2}
	Let $\hat{u}\in W^{1,p}_{0}(\Omega)$ be a local $C^{1}_{0}(\bar{\Omega})$-minimizer of $\varphi_0$, that is, there exists $\rho_0>0$ such that
	$$\varphi_0(\hat{u})\leq \varphi_0(\hat{u}+h)\ \mbox{for\ all}\ h\in C^{1}_{0}(\bar{\Omega}),\ ||h||_{C^{1}_{0}(\bar{\Omega})}\leq\rho_0.$$
Then $\hat{u}\in C^{1,\beta}_{0}(\bar{\Omega})$ with $\beta\in(0,1)$ and it is also a local $W^{1,p}_{0}(\Omega)$-minimizer of $\varphi_0$, that is, there exists $\rho_1>0$ such that
	$$\varphi_0(\hat{u})\leq\varphi_0(\hat{u}+h)\ \mbox{for\ all}\ h\in W^{1,p}_{0}(\Omega),\ ||h||\leq\rho_1.$$
\end{prop}

In the above result and in the sequel we denote by $||\cdot||$ the norm of $W^{1,p}_{0}(\Omega)$. Using Poincar\'e's inequality, we have
$$||u||=||Du||_{p}\ \mbox{for\ all}\ u\in W^{1,p}_{0}(\Omega).$$
Note that by $||\cdot||$ we also denote the norm in $\RR^N$. However, no confusion is possible since it will always be clear from the context which norm is used.

For every $x\in\RR$, we set $X^{\pm}=\max\{0,\pm x\}$. Then for $u\in W^{1,p}_{0}(\Omega)$ we can define $u^{\pm}(\cdot)=u(\cdot)^{\pm}$. We know that
$$u^{\pm}\in W^{1,p}_{0}(\Omega),\ u=u^{+}-u^{-}\ \mbox{and}\ |u|=u^{+}+u^{-}.$$
Given a measurable function $h:\Omega\times\RR\rightarrow\RR$ (for example, a Carath\'eodory function), we set
$$N_h(u)(\cdot)=h(\cdot,u(\cdot))\ \mbox{for\ all}\ u\in W^{1,p}_{0}(\Omega).$$
This is the Nemytskii map corresponding to $h$.

For every $r\in(1,\infty)$, let $A_r:W^{1,r}_{0}(\Omega)\rightarrow W^{-1,r'}(\Omega)=W^{1,r}_{0}(\Omega)^{\ast}\left(\frac{1}{r}+\frac{1}{r'}=1\right)$ be the nonlinear map defined by
\begin{eqnarray}\label{eq2}
	\left\langle A_r(u),v\right\rangle=\int_{\Omega}||Du||^{r-2}(Du,Dv)_{\RR^N}dz\ \mbox{for\ all}\ u,v\in W^{1,r}_{0}(\Omega).
\end{eqnarray}
If $r=2$, then we write $A_2=A\in\mathcal{L}(H^{1}_{0}(\Omega),H^{-1}(\Omega))$. The next proposition summarizes the basic properties of this map (see, for example, Gasinski and Papageorgiou \cite{12}).
\begin{prop}\label{prop3}
	The nonlinear map $A_r:W^{1,r}_{0}(\Omega)\rightarrow W^{-1,r'}(\Omega)$ defined by (\ref{eq2}) is bounded (that is, maps bounded sets to bounded sets), continuous, strictly monotone (strongly monotone if $r\geq 2$) hence maximal monotone too and of type $(S)_+$, that is, if $u_n\stackrel{w}{\longrightarrow}u$ in $W^{1,p}_{0}(\Omega)$ and $\limsup\limits_{n\rightarrow\infty}\left\langle A_r(u_n),u_n-u\right\rangle\leq 0$, then $u_n\rightarrow u$ in $W^{1,p}_{0}(\Omega)$.
\end{prop}

Next we recall some basic facts about the spectrum of the Dirichlet $r$-Laplace operator. So, let $m\in L^{\infty}(\Omega),\ m\geq 0,\ m\neq 0$ and consider the following nonlinear weighted eigenvalue problem
\begin{eqnarray}\label{eq3}
	-\Delta_r u(z)=\hat{\lambda}m(z)|u(z)|^{r-2}u(z)\ \mbox{in}\ \Omega,\ u|_{\partial\Omega}=0.
\end{eqnarray}
A number $\hat{\lambda}\in\RR$ is an eigenvalue if problem (\ref{eq3}) admits a nontrivial solution $\hat{u}$, which is an eigenfunction corresponding to $\hat{\lambda}$. Problem (\ref{eq3}) admits a smallest eigenvalue denoted by $\hat{\lambda}_1(r,m)$. The following facts are known about this first eigenvalue:
\begin{itemize}
	\item $\hat{\lambda}_1(r,m)>0;$\\
	\item $\hat{\lambda}_1(r,m)$ is isolated, that is, we can find $\epsilon>0$ such that  the interval $(\hat{\lambda}_1(r,m), \hat{\lambda}_1(r,m)+\epsilon)$ contains no eigenvalues;\\
	\item $\hat{\lambda}_1(r,m)$ is simple, that is, if $\hat{u}_1,\hat{u}_2$ are eigenfunctions corresponding to $\hat{\lambda}_1(r,m)$ then $\hat{u}_1=\xi\hat{u}_2$ for some $\xi\neq 0$;\\
	\item $\hat{\lambda}_1(r,m)>0$ admits the following variational characterization:
	\begin{eqnarray}\label{eq4}
		\hat{\lambda}_1(r,m)=\inf\left\{\frac{||Du||^{r}_{r}}{\int_{\Omega}m(z)|u|^rdz}:u\in W^{1,r}_{0}(\Omega),\ u\neq 0\right\}.
	\end{eqnarray}
\end{itemize}

The infimum in (\ref{eq4}) is attained on the one dimensional eigenspace corresponding to $\hat{\lambda}_1(r,m)>0$. It is clear from (\ref{eq4}) that the elements of this one-dimensional eigenspace do not change sign. In what follows by $\hat{u}_1(r,m)$ we denote the positive $L^{r}$-normalized (that is, $||\hat{u}_1(r,m)||_r=1$) eigenfunction corresponding to $\hat{\lambda}_1(r,m)>0$. The nonlinear regularity theory and the nonlinear maximum principle (see, for example, Gasinski and Papageorgiou \cite[pp. 737-738]{12}) imply that $\hat{u}_1(r,m)\in\mbox{int}\, C_+$. The first eigenvalue $\hat{\lambda}_1(r,m)>0$ as a function of the weight $m$, exhibits the following strict monotonicity property.
\begin{prop}\label{prop4}
	Assume that $m,\hat{m}\in L^{\infty}(\Omega)_+\backslash\{0\},\ m(z)\leq\hat{m}(z)$ a.e. in $\Omega,\ m\neq\hat{m}$. Then
 $\hat{\lambda}_1(r,\hat{m})<\hat{\lambda}_1(r,m)$.
\end{prop}
If $\sigma(r,m)$ denotes the set of eigenvalues of (\ref{eq3}), then $\sigma(r,m)$ is closed. This fact and since $\hat{\lambda}_1(r,m)>0$ is isolated, imply that the second eigenvalue is well-defined by
$${\hat{\lambda}}^{\ast}_{2}(r,m)=\inf \left\{\hat{\lambda}:\hat{\lambda}\in\sigma(r,m),\ \hat{\lambda}>\hat{\lambda}_1(r,m)\right\}.$$
The Ljusternik-Schnirelmann minimax scheme produces a whole strictly increasing sequence $\{\hat{\lambda}_k(r,m)\}_{k\geq 1}$ of eigenvalues such that  $\hat{\lambda}_k(r,m)\rightarrow +\infty$. We know that ${\hat{\lambda}}^{\ast}_{2}(r,m)=\hat{\lambda}_2(r,m)$, that is, the second eigenvalue and the second Ljusternik-Schnirelmann eigenvalue coincide. However, we do not know if the Ljusternik-Schnirelmann exhausts the spectral set $\sigma(r,m)$. This is the case if $r=2$ (linear eigenvalue problem) or if $N=1$ (ordinary differential equation). We point out that $\hat{\lambda}_1(r,m)>0$ is the only eigenvalue with eigenfunctions of constant sign. All the other eigenvalues have nodal (that is, sign changing) eigenfunctions.

When $r=2$, the eigenvalue problem is linear and so we can define the eigenspace for every eigenvalue. By $E(\hat{\lambda}_k(2,m))$ (for all $k\geq 1$) we denote the eigenspace corresponding to the eigenvalue $\hat{\lambda}_k(2,m)$. The elements of this eigenspace have the ``unique continuation property", namely if $u\in E(\hat{\lambda}_k(2,m))$ and $u$ vanishes on a set of positive Lebesgue measure, then $u\equiv 0$. The regularity theory implies that $E(\hat{\lambda}_k(2,m))\subseteq C^{1}_{0}(\bar{\Omega}),\ k\geq 1$. Also, we have the following orthogonal direct sum decomposition
$$H^{1}_{0}(\Omega)=\overline{\underset{\mathrm{k\geq 1}}\oplus E(\hat{\lambda}_k(2,m))}\,.$$
If $m\equiv 1$, then we write $\hat{\lambda}_1(r,m)=\hat{\lambda}_1(r),\ \hat{\lambda}_2(r,m)=\hat{\lambda}_2(r)$, and $\hat{u}_1(r,m)=\hat{u}_1(r)$.

As a straightforward consequence of (\ref{eq4}) and of the fact that $\hat{u}_1(r)\in\mbox{int}\, C_+$, we have the following easy lemma (see Papageorgiou and Kyritsi \cite[p. 356]{19})
\begin{lemma}\label{lem5}
	If $\vartheta\in L^{\infty}(\Omega),\ \vartheta(z)\leq\hat{\lambda}_1(r)$ a.e. in $\Omega$ and $\vartheta\neq\hat{\lambda}_1(r)$, then there exists $c_0>0$ such that
	$$||Du||^{p}_{p}-\int_{\Omega}\vartheta(z)|u|^pdz\geq c_0||u||^p\ \mbox{for\ all}\ u\in W^{1,p}_{0}(\Omega).$$
\end{lemma}

We conclude this section by recalling some basic facts from Morse theory (critical groups) which we will use in the sequel. So, let ($Y_1,Y_2$) be a topological pair such that  $Y_2\subseteq Y_1\subseteq X$. For every integer $k\geq 0$ we denote by $H_k(Y_1,Y_2)$  the $k$th relative singular homology group with integer coefficients for the pair $(Y_1,Y_2)$.

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

Suppose that $\varphi\in C^1(X)$ satisfies the $C$-condition and $\inf \varphi(K_{\varphi})>-\infty$. Let $c<\inf \varphi(K_{\varphi})$. The critical groups of $\varphi$ at infinity are defined by
$$C_k(\varphi,\infty)=H_k(X,\varphi^c)\ \mbox{for\ all}\ k\geq 0.$$
The second deformation theorem (see, for example, Gasinski and Papageorgiou \cite[p. 628]{12}) 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 set
\begin{eqnarray}
	&&M(t,x)=\sum\limits_{k\geq 0}\mbox{rank}\ C_k(\varphi,x)t^k\ \mbox{for\ all}\ t\in\RR,\ \mbox{all}\ x\in K_{\varphi}\nonumber\\
	&&P(t,\infty)=\sum\limits_{k\geq 0}\mbox{rank}\ C_k(\varphi,\infty)t^k\ \mbox{for\ all}\ t\in\RR\,.\nonumber
\end{eqnarray}
The Morse relation says that
\begin{eqnarray}\label{eq5}
	\sum\limits_{x\in K_{\varphi}}M(t,x)=P(t,\infty)+(1+t)Q(t),
\end{eqnarray}
where $Q(t)=\sum\limits_{k\geq 0}\beta_k t^k$ is a formal series in $t\in\RR$ with nonnegative integer coefficients.

The next proposition is a useful tool in the computation of critical groups at infinity and it extends an earlier result for Hilbert spaces by Liang and Su \cite{16}.
\begin{prop}\label{prop6}
	Let $X$ be a Banach space and assume that $(t,x)\longmapsto h_t(x)$ is a $C^1([0,1]\times X)$-function which maps bounded sets to bounded sets, the maps $x\longmapsto (h_t)'(x)$ and $t\longmapsto \partial_t h_t(x)$ are both locally Lipschitz, $h_0$ and $h_1$ satisfy the $C$-condition
	$$|\partial_t h_t(x)|\leq c_1(||u||^{q}_{X}+||u||^{p}_{X})\ \mbox{for\ all}\ x\in X$$
	with $c_1>0,\ 1<q<p<\infty$, and there exist $\gamma_0\in\RR$ and $\delta_0>0$ such that
	 $$h_t(x)\leq\gamma_0\Rightarrow(1+||x||_X)||(h_t)'(x)||_{X^{\ast}}\geq\delta_0(||x||^{q}_{X}+||x||^{p}_{X})\ \mbox{for\ all}\ t\in[0,1].$$
	Then $C_k(h_0,\infty)=C_k(h_1,\infty)$ for all $k\geq 0$.
\end{prop}

\begin{proof}
	Since $h\in C^1([0,1]\times X)$, we know that it admits a pseudo-gradient vector field $\hat{v}_t(x)$ (see, for example, Gasinski and Papageorgiou \cite[p. 616]{12}). From the construction of the pseudo-gradient vector field we deduce that
	$$\hat{v}_t(x)=(\partial_t h_t(x),\ v_t(x)),$$
	with $(t,x)\longmapsto v_t(x)$ locally Lipschitz and for all $t\in[0,1],\ v_t(\cdot)$ is the pseudo-gradient vector field corresponding to $h_t(\cdot)$. So, for all $t\in[0,1]$ and all $x\in X$, we have
	\begin{eqnarray}\label{eq6}
		||(h_t)'(x)||^{2}_{X^{\ast}}\leq\left\langle (h_t)'(x),v_t(x)\right\rangle\ \mbox{and}\ ||v_t(x)||_{X}\leq 2||(h_t)'(x)||_{X^{\ast}}\,.
	\end{eqnarray}
	Given $t\in[0,1]$, we consider the map $w_t:X\rightarrow X$ defined by
	$$w_t(x)=-\frac{|\partial_t h_t(x)|}{||(h_t)'(x)||^{2}_{X^{\ast}}}v_t(x)\ \mbox{for\ all}\ x\in X.$$
	Evidently, $[t,x]\longmapsto w_t(x)$ is well-defined and locally Lipschitz. Let $\gamma\leq \gamma_0$ be such that
	$${h_0}^\gamma\neq\O\ \mbox{or}\ {h_1}^\gamma\neq\O\,.$$
	If we can not find such a $\gamma\leq\gamma_0$, then $C_k(h,\infty)=C_k(h,\infty)=\delta_{k,0}\ZZ\ \mbox{for\ all}\ k\geq 0$.
	
	Assume that $h^{\gamma}_{0}\neq\O$ and let $y\in h^{\gamma}_{0}$. We consider the following abstract Cauchy problem
	\begin{eqnarray}\label{eq7}
		\frac{d\sigma}{dt}=w_t(\sigma)\ \mbox{on}\ [0,1],\ \sigma(0)=y.
	\end{eqnarray}
	Problem (\ref{eq7}) admits a local flow $\sigma(t,y)$ (see, for example, Gasinski and Papageorgiou \cite[p. 618]{12}). In what follows, for notational simplicity, we drop $y$ from the description of $\sigma$. Using the chain rule, we have
$$	\begin{array}{ll}
	\displaystyle	\frac{d}{dt}\ h_t(\sigma)&=\displaystyle\left\langle (h_t)'(\sigma),\ \frac{d\sigma}{dt}\right\rangle+\partial_t h_t(\sigma)\\
		&\displaystyle=\left\langle (h_t)'(\sigma),\ \frac{-|\partial_t h_t(\sigma)|}{||(h_t)'(\sigma)||^{2}_{X^{\ast}}}\ v_t(\sigma)\right\rangle+\partial_t h_t(\sigma)\\
		&\displaystyle\leq -|\partial_t h_t(\sigma)|+\partial_t h_t(\sigma)\ (\mbox{see}\ (\ref{eq6}))\\
		&\displaystyle\leq 0\\
		\Rightarrow&\displaystyle t\longmapsto h_t(\sigma)\ \mbox{is\ nonincreasing}.
	\end{array}$$
	Hence for $t\geq 0$ small, we have
	\begin{eqnarray}\label{eq8}
		&&h_t(\sigma(t))\leq h_0(\sigma(0))=h_0(y)\leq\gamma\leq\gamma_0\nonumber\\
		\Rightarrow&&(1+||\sigma(t)||_X)\ ||(h_t)'(\sigma(t))||_{X^{\ast}}\ \geq\ \delta_0(||\sigma(t)||^{q}_{X}+||\sigma(t)||^{p}_{X}).
	\end{eqnarray}
	Then
	$$\begin{array}{ll}
\displaystyle		|w_t(\sigma(t))|&\displaystyle\leq\ \frac{|\partial_t h_t(\sigma(t))|}{||(h_t)'(\sigma(t))||^{2}_{X^{\ast}}}\ ||v_t(\sigma(t))||_{X}\\
		&\displaystyle\leq\ \frac{C_1(||\sigma(t)||^{q}_{X}+||\sigma(t)||^{p}_{X})}{||(h_t)'(\sigma(t))||^{2}_{X^{\ast}}}\ 2||(h_t)'(x)||_{X^{\tau}}\ (\mbox{see}\ (\ref{eq6}))\\
		&\displaystyle\leq\ \frac{C_1(||\sigma(t)||^{q}_{X}+||\sigma(t)||^{p}_{X})}{\delta_0\left(||\sigma(t)||^{q}_{X}+||\sigma(t)||^{p}_{X}\right)}\ (1+||\sigma(t)||_X)\ (\mbox{see}\ (\ref{eq8}))\\
		&\displaystyle=\ \frac{C_1}{\delta_0}(1+||\sigma(t)||_X)\ \mbox{for\ all}\ t\in[0,1]\ \mbox{small}.
	\end{array}$$
	This means that the flow in (\ref{eq7}) is global on $[0,1]$.
	
	Then $\sigma(t,\cdot)$ is a homeomorphism of $h^{\gamma}_{0}$ onto a subset $D_0$ of $h^{\gamma}_{1}$. Also, reversing the time $t\rightarrow 1-t$ and using the corresponding global flow $\sigma_{\ast}(\cdot,v)$ (here $v\in h^{\gamma}_{1}$), we deduce that $h^{\gamma}_{1}$ is homeomorphic to a subset $D_1$ of $h^{\gamma}_{0}$.
	
	Let
	$$\eta(t,y)=\sigma_{\ast}(t,\sigma(1,y))\ \mbox{for\ all}\ (t,y)\in[0,1]\times h^{\gamma}_{0}.$$
	Then we have
	\begin{eqnarray}\label{eq9}
		\eta(0,\cdot)\ \mbox{is\ homotopy \ equivalent\ to\ id}|_{D_0}(\cdot)\ \mbox{and}\ \eta(1,\cdot)=(\sigma_{\ast})_1\circ\sigma_1(\cdot).
	\end{eqnarray}
	Similarly, if
	$$\eta_{\ast}(t,v)=\sigma(t,\sigma_{\ast}(1,v))\ \mbox{for\ all}\ (t,v)\in[0,1]\times h^{\gamma}_{1},$$
	then
	\begin{eqnarray}\label{eq10}
		\eta_{\ast}(0,\cdot)\ \mbox{is\ homotopy\ equivalent\ to\ id}|_{D_1}(\cdot)\ \mbox{and}\ \eta_{\ast}(1,\cdot)=\sigma_1\circ(\sigma_{\ast})_1(\cdot)\,.
	\end{eqnarray}
	Recall that $D_0$ and $H^{\gamma}_{0}$ are homeomorphic. Similarly $D_1$ and $h^{\gamma}_{1}$ are homeomorphic. Combining these facts with (\ref{eq9}) and (\ref{eq10}), we infer that the level sets $h^{\gamma}_{0}$ and $h^{\gamma}_{1}$ are homotopy equivalent. Therefore
	\begin{eqnarray}
		&&H_k(X,h^{\gamma}_{0})=H_k(X,h^{\gamma}_{1})\ \mbox{for\ all}\ k\geq 0\ (\mbox{see\ Granas\ and\ Dugundji\ \cite[p. 387]{13}})\nonumber\\
		\Rightarrow&&C_k(h_0,\infty)=C_k(h,\infty)\ \mbox{for\ all}\ k\geq 0.\nonumber
	\end{eqnarray}
		This completes the proof.
\end{proof}

\section{Two Nontrivial Solutions}

In this section we establish the existence of two nontrivial solutions for problem (\ref{eq1}) without imposing any differentiability condition on $f(z,\cdot)$.

First we produce a positive solution. To this end, we impose the following conditions on the reaction $f(z,x)$:

$\underline{H_1}:\ f:\Omega\times\RR\rightarrow\RR$ is a Carath\'eodory function such that  $f(z,0)=0$ for a.a. $z\in\Omega$ and
\begin{itemize}
	\item[(i)]$|f(z,x)|\leq a(z)(1+|x|^{p-1})$ for a.a. $z\in\Omega$, all $x\in\RR$ with $a\in L^{\infty}(\Omega)_+$;
	\item[(ii)]$\limsup\limits_{x\rightarrow+\infty}\frac{f(z,x)}{x^{p-1}}\leq\hat{\lambda}_1(p)$ uniformly for a.a. $z\in\Omega$ and if $F(z,x)=\int^{x}_{0}f(z,s)ds$, then
		$$\lim\limits_{x\rightarrow+\infty}[f(z,x)x-pF(z,x)]=+\infty\ \mbox{uniformly\ for\ a.a.}\ z\in\Omega;$$
	\item[(iii)]there exists a function $\eta\in L^{\infty}(\Omega)_+$ such that
	\begin{eqnarray}
		&&\hat{\lambda}_1(p)\leq\eta(z)\ \mbox{for\ a.a.}\ z\in\Omega,\ \hat{\lambda}_1(p)\neq\eta\ \mbox{and}\nonumber\\
		 &&\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{\lambda}_2(p)\ \mbox{uniformly\ for\ a.a.}\ z\in\Omega;\nonumber
	\end{eqnarray}
	\item[(iv)]there exist functions $\beta,\hat{\beta}\in L^{\infty}(\Omega)_+$ such that
	\begin{eqnarray}
		&&\hat{\lambda}_1(2)\leq\beta(z)\ \mbox{for\ a.a.}\ z\in\Omega,\ \hat{\lambda}_1(2)\neq\beta\ \mbox{and}\nonumber\\
		&&\beta(z)\leq\liminf\limits_{x\rightarrow 0}\frac{f(z,x)}{x}\leq\limsup\limits_{x\rightarrow 0}\frac{f(z,x)}{x}\leq\hat{\beta}(z)\ \mbox{uniformly\ for\ a.a.}\ z\in\Omega;\nonumber
	\end{eqnarray}
	\item[(v)]$f(z,x)x-pF(z,x)\geq 0$ for a.a. $z\in\Omega$, all $x\leq 0$, and $f(z,\cdot)$ is lower locally Lipschitz on $[0,+\infty)$.
\end{itemize}

\smallskip
Let $\varphi:W^{1,p}_{0}(\Omega)\rightarrow\RR$ be the energy functional for problem (\ref{eq1}) defined by
$$\varphi(u)=\frac{1}{p}||Du||^{p}_{p}+\frac{1}{2}||Du||^{2}_{2}-\int_{\Omega}F(z,u(z))dz\ \mbox{for\ all}\ u\in W^{1,p}_{0}(\Omega).$$
\begin{prop}\label{prop7}
	Assume that hypotheses $H_1$ 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 such that
	\begin{eqnarray}
		&&|\varphi(u_n)|\leq M_1\ \mbox{for\ all}\ n\geq 1,\ \mbox{some}\ M_1>0\label{eq11}\\
		&&(1+||u_n||)\varphi'(u_n)\rightarrow 0\ \mbox{in}\ W^{-1,p'}(\Omega)\ \mbox{as}\ n\rightarrow\infty.\label{eq12}
	\end{eqnarray}
	From (\ref{eq12}) we have
	\begin{eqnarray}\label{eq13}
		&&\left|\left\langle A_p(u_n),h\right\rangle-\int_{\Omega}f(z,u_n)h\ dz\right|\leq\frac{\epsilon_n||h||}{1+||u_n||}\ \mbox{for\ all}\ h\in W^{1,p}_{0}(\Omega)\ \mbox{with}\ \epsilon_n\rightarrow 0^+.
	\end{eqnarray}
	In (\ref{eq13}) we choose $h=u^{+}_{n}\in W^{1,p}_{0}(\Omega)$. Then
	\begin{eqnarray}\label{eq14}
		 &&\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.
	\end{eqnarray}
	Using (\ref{eq14}) we will show that the sequence $\{u^{+}_{n}\}_{n\geq 1}\subset W^{1,p}_{0}(\Omega)$ is bounded. Arguing indirectly, suppose that the sequence is not bounded in $W^{1,p}_{0}(\Omega)$. Then by passing to a subsequence if necessary, we may assume that $||u^{+}_{n}||\rightarrow\infty$. Let $y_n=\frac{u^{+}_{n}}{||u^{+}_{n}||}\ n\geq 1$. Then $||y_n||=1$ and $y_n\geq 0$ for all $n\geq 1$. We may assume that
	\begin{eqnarray}\label{eq15}
		y_n\stackrel{w}{\longrightarrow}y\ \mbox{in}\ W^{1,p}_{0}(\Omega)\ \mbox{and}\ y_n\rightarrow y\ \mbox{in}\ L^{p}(\Omega).
	\end{eqnarray}
	From (\ref{eq14}), we have
	\begin{eqnarray}\label{eq16}
		 ||Dy_n||^{p}_{p}\leq\frac{\epsilon_n}{||u^{+}_{n}||^p}+\int_{\Omega}\frac{f(z,u^{+}_{n})}{||u^{+}_{n}||^{p-1}}y_ndz\ \mbox{for\ all}\ n\geq 1.
	\end{eqnarray}
	From hypothesis $H_1(i)$ it is clear that
	$$\left\{\frac{N_f(u^{+}_{n})}{||u^{+}_{n}||^{p-1}}\right\}_{n\geq 1}\subseteq L^{p'}(\Omega)\ \mbox{is\ bounded.}$$
So, we may assume that
\begin{eqnarray}\label{eq17}
	\frac{N_f(u^{+}_{n})}{||u^{+}_{n}||^{p-1}}\ \stackrel{w}{\longrightarrow}g\ \mbox{in}\ L^{p'}(\Omega)\ \mbox{as}\ n\rightarrow\infty.
\end{eqnarray}
Using hypothesis $H_1(ii)$, as in Aizicovici, Papageorgiou and Staicu \cite{1}, we show that
\begin{eqnarray}\label{eq18}
	g(z)=\vartheta(z)y(z)^{p-1}\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{with}\ \vartheta\in L^{\infty}(\Omega),\ \vartheta(z)\leq\hat{\lambda}_1(p)\ \mbox{a.e.\ in}\ \Omega.
\end{eqnarray}
Hence, if in (\ref{eq16}) we pass to the limit as $n\rightarrow\infty$ and use (\ref{eq15}), (\ref{eq17}),\ (\ref{eq18}) we obtain
\begin{eqnarray}\label{eq19}
	||Dy||-\int_{\Omega}\vartheta(z)y^p dz\leq 0.
\end{eqnarray}
If $\vartheta\neq\hat{\lambda}_1(p)$, then from (\ref{eq19}) and Lemma \ref{lem5}, we have
$$c_0||y||^p\leq 0,\ \mbox{hence}\ y=0.$$
Then from (\ref{eq16}) it follows that $Dy_n\rightarrow 0$ in $L^p(\Omega,\RR^N)$ and so $y_n\rightarrow 0$ in $W^{1,p}_{0}(\Omega)$, a contradiction to the fact that $||y_n||=1$ for all $n\geq 1$.

Next suppose that $\vartheta(z)=\hat{\lambda}_1(p)$ a.e. in $\Omega$. Then from (\ref{eq19}) and (\ref{eq4}) we have
\begin{eqnarray}
	&&||Dy||^{p}_{p}=\hat{\lambda}_1(p)||y||^{p}_{p}\nonumber\\
	\Rightarrow&&y=\xi\hat{u}_1(p)\ \mbox{for\ some}\ \xi>0\ \ (\mbox{see\  \eqref{eq4}}).\nonumber
\end{eqnarray}
Since $y\in\mbox{int}\, C_+$, we have $u^{+}_{n}(z)\rightarrow+\infty$ for a.a. $z\in\Omega$ and so by virtue of hypothesis $H_1(ii)$ we have
\begin{eqnarray}\label{eq20}
	&&f(z,u^{+}_{n}(z))u^{+}_{n}(z)-pF(z,u^{+}_{n}(z))\rightarrow\infty\ \mbox{for\ a.a.}\ z\in\Omega\nonumber\\
	\Rightarrow&&\int_{\Omega}\left[f(z,u^{+}_{n})u^{+}_{n}-pF(z,u^{+}_{n})\right]dz\rightarrow+\infty\ (\mbox{by\ Fatou's\ lemma}).
\end{eqnarray}
On the other hand from (\ref{eq11}) we have
\begin{eqnarray}\label{eq21}
	||Du_n||^{p}_{p}+\frac{p}{2}||Du_n||^{2}_{2}-\int_{\Omega}pF(z,u_n)dz\leq p\ M_1\ \mbox{for\ all}\ n\geq 1.
\end{eqnarray}
Also from (\ref{eq13}) with $h=u_n\in W^{1,p}_{0}(\Omega)$, we obtain
\begin{eqnarray}\label{eq22}
	-||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{eqnarray}
Adding (\ref{eq21}) and (\ref{eq22}) we have
\begin{eqnarray}\label{eq23}
	&&\left(\frac{p}{2}-1\right)||Du_n||^{2}_{2}+\int_{\Omega}\left[f(z,u_n)u_n-pF(z,u_n)\right]dz\leq M_2\ \mbox{for\ some}\ M_2>0\nonumber\\
	&&\hspace{10cm}\mbox{all}\ n\geq 1\nonumber\\
	&\Rightarrow&\int_{\Omega}\left[f(z,u_n)u_n-pF(z,u_n)\right]dz\leq M_2\ \mbox{for\ all}\ n\geq 1\ (\mbox{recall}\ p>2)\nonumber\\
	 &\Rightarrow&\int_{\Omega}\left[f(z,u^{+}_{n})u^{+}_{n}-pF(z,u^{+}_{n})\right]dz+
\int_{\Omega}\left[f(z,-u^{-}_{n})(-u^{-}_{n})-pF(z,-u^{-}_{n})\right]dz\leq M_2\nonumber\\
	&&\hspace{10cm}\mbox{for\ all}\ n\geq 1\nonumber\\
	&\Rightarrow&\int_{\Omega}\left[f(z,u^{+}_{n})u^{+}_{n}-pF(z,u^{+}_{n})\right]dz\leq M_2\ \mbox{for\ all}\ n\geq 1\ (\mbox{see}\ H_1(v)).
\end{eqnarray}
Comparing (\ref{eq20}) and (\ref{eq23}), we reach a contradiction which proves that
\begin{eqnarray}\label{eq24}
	\{u^{+}_{n}\}_{n\geq 1}\subseteq W^{1,p}_{0}(\Omega)\ \mbox{is\ bounded}.
\end{eqnarray}
Next we show that $\{u^{-}_{n}\}_{n\geq 1}\subseteq W^{1,p}_{0}(\Omega)$ is bounded. Again we argue by contradiction. So, assume that $||u^{-}_{n}||\rightarrow\infty$ and let $v_n=\frac{u^{-}_{n}}{||u^{-}_{n}||}\ n\geq 1$. Then $||v_n||=1,\ v_n\geq 0$ for all $n\geq 1$ and so we may assume that
\begin{eqnarray}\label{eq25}
	v_n\stackrel{w}{\longrightarrow} v\ \mbox{in}\ W^{1,p}_{0}(\Omega)\ \mbox{and}\ v_n\rightarrow v\ \mbox{in}\ L^{p}(\Omega),\ v\geq 0.
\end{eqnarray}
From (\ref{eq13}) and (\ref{eq24}), we have
\begin{eqnarray}\label{eq26}
	&&\left|\left\langle A_p(-u^{-}_{n}),h\right\rangle+\left\langle A(-u^{-}_{n}),h\right\rangle-\int_{\Omega}f(z,-u^{-}_{n})h\ dz\right|\leq M_3||h||\nonumber\\
	&&\hspace{8cm}\mbox{for\ some}\ M_3>0,\ \mbox{all}\ n\geq 1\nonumber\\
	&\Rightarrow&\left|\left\langle A_p(-v_{n}),h\right\rangle+\frac{1}{||u^{-}_{n}||^{p-2}}\left\langle A(-v_n),h\right\rangle-\int_{\Omega}\frac{f(z,-u^{-}_{n})}{||u^{-}_{n}||^{p-1}}h\ dz\right|\leq\frac{M_3||h||}{||u^{-}_{n}||^{p-1}}\\
	&&\hspace{10.5cm}\mbox{for\ all}\ n\geq 1.\nonumber
\end{eqnarray}
Hypothesis $H_1(i)$ implies that
$$\left\{\frac{N_f(-u^{-}_{n})}{||u^{-}_{n}||^{p-1}}\right\}_{n\geq 1}\subseteq L^{p'}(\Omega)\ \mbox{is\ bounded}.$$
Passing to a subsequence if necessary and using hypothesis $H_1(iii)$ we have
\begin{eqnarray}\label{eq27}
	\frac{N_f(-u^{-}_{n})}{||u^{-}_{n}||^{p-1}}\ \stackrel{w}{\longrightarrow}\ -\hat{\eta}v^{p-1}\ \mbox{in}\ L^{p'}(\Omega)\ \mbox{with}\ \eta(z)\leq\hat{\eta}(z)\leq\hat{\lambda}_2(p)\ \mbox{for\ a.a.}\ z\in\Omega.
\end{eqnarray}
In (\ref{eq26}) we choose $h=v-v_n\in W^{1,p}_{0}(\Omega)$, pass to the limit as $n\rightarrow\infty$ and use (\ref{eq25}) and (\ref{eq27}). Since $p>2$, we obtain
\begin{eqnarray}\label{eq28}	
	&&\lim\limits_{n\rightarrow\infty}\left\langle A_p(-v_n),\ v-v_n\right\rangle=0\nonumber\\
	&\Rightarrow&v_n\rightarrow v\ \mbox{in}\ W^{1,p}_{0}(\Omega)\ (\mbox{see\ Proposition\ \ref{prop3}}),\ \mbox{hence}\ ||v||=1,\ v\geq 0.
\end{eqnarray}
Therefore, if in (\ref{eq26}) we pass to the limit as $n\rightarrow\infty$ and use (\ref{eq27}) and (\ref{eq28}) and the fact that $p>2$, we deduce that
\begin{eqnarray}\label{eq29}
	\left\langle A_p(-v),h\right\rangle&=&\int_{\Omega}-\hat{\eta}v^{p-1}h\ dz\ \mbox{for\ all}\ h\in W^{1,p}_{0}(\Omega)\nonumber\\
	&\Rightarrow&A_p(v)=\hat{\eta}v^{p-1}\nonumber\\
	&\Rightarrow&-\Delta_p v(z)=\hat{\eta}(z)v(z)^{p-1}\ \mbox{a.e.\ in}\ \Omega,\ v|_{\partial\Omega}=0.
\end{eqnarray}
From Proposition \ref{prop4}, we have
$$\hat{\lambda}_1(p,\hat{\eta})<\hat{\lambda}_1(p,\hat{\lambda}_1(p))=1.$$
So, from (\ref{eq29}) it follows that $v$ must be nodal, which contradicts (\ref{eq28}). This proves that
\begin{eqnarray}
	&&\{u^{-}_{n}\}_{n\geq 1}\subseteq W^{1,p}_{0}(\Omega)\ \mbox{is\ bounded}\nonumber\\
	&\Rightarrow&\{u_n\}_{n\geq 1}\subseteq W^{1,p}_{0}(\Omega)\ \mbox{is\ bounded\ (see\ (\ref{eq24})).}\nonumber
\end{eqnarray}
Hence, we may assume that
\begin{eqnarray}\label{eq30}
	u_n\stackrel{w}{\longrightarrow} u\ \mbox{in}\ W^{1,p}_{0}(\Omega)\ \mbox{and}\ u_n\rightarrow u\ \mbox{in}\ L^{p}(\Omega).
\end{eqnarray}
In (\ref{eq13}) we choose $h=u_n-u\in W^{1,p}_{0}(\Omega)$, pass to the limit as $n\rightarrow\infty$ and use (\ref{eq30}). 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\nonumber\\
	&\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\ \mbox{is\ monotone})\nonumber\\
	&\Rightarrow&\limsup\limits_{n\rightarrow\infty}\left\langle A_p(u_n),u_n-u\right\rangle\leq 0\nonumber\\
	&\Rightarrow&u_n\rightarrow u\ \mbox{in}\ W^{1,p}_{0}(\Omega)\ (\mbox{see\ Proposition\ \ref{prop3}}).\nonumber
\end{eqnarray}
This proves that $\varphi$ satisfies the $C$-condition.
\end{proof}

We consider the positive truncation of $f(z,\cdot)$ defined by
$$f_+(z,x)=f(z,x^+).$$
This is a Carath\'eodory function. We set $F_+(z,x)=\int^{x}_{0}f_+(z,s)ds$ and consider the $C^1$-functional $\varphi_+:W^{1,p}_{0}(\Omega)\rightarrow\RR$ defined by
$$\varphi_+(u)=\frac{1}{p}||Du||^{p}_{p}+\frac{1}{2}||Du||^{2}_{2}-\int_{\Omega}F_+(z,u(z))dz\ \mbox{for\ all}\ u\in W^{1,p}_{0}(\Omega).$$
\begin{prop}\label{prop8}
	Assume that hypotheses $H_1$ hold. Then the functional $\varphi_+$ is coercive.
\end{prop}

\begin{proof}
	We argue indirectly. So, suppose that  $\varphi_+$ is not coercive. Then we can find $\{u_n\}_{n\geq 1}\subseteq W^{1,p}_{0}(\Omega)$ and $M_4>0$ such that
	$$||u_n||\rightarrow\infty\ \mbox{as}\ n\rightarrow\infty\ \mbox{and}\ \varphi_+(u_n)\leq M_4\ \mbox{for\ all}\ n\geq 1.$$
	Let $y_n=\frac{u_n}{||u_n||},\ n\geq 1$. Then $||y_n||=1$ for all $n\geq 1$ and so we may assume that
	\begin{eqnarray}\label{eq31}
		y_n\stackrel{w}{\longrightarrow} y\ \mbox{in}\ W^{1,p}_{0}(\Omega)\ \mbox{and}\ y_n\rightarrow y\ \mbox{in}\ L^{p}(\Omega).
	\end{eqnarray}
	We have
	\begin{eqnarray}\label{eq32}
		&&\frac{1}{p}||Du_n||^{p}_{p}+\frac{1}{2}||Du_n||^{2}_{2}-\int_{\Omega}F_+(z,u_n)dz\leq M_4\ \mbox{for\ all}\ n\geq 1\nonumber\\
		 &\Rightarrow&\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_4}{||u_n||^p}\ \mbox{for\ all}\ n\geq 1.
	\end{eqnarray}
	Hypothesis $H_1(ii)$ implies that given $\epsilon>0$, we can find $M_5=M_5(\epsilon)>0$ such that
	\begin{eqnarray}
		&&f(z,x)\leq(\hat{\lambda}_1(p)+\epsilon)x^{p-1}\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\geq M_5\nonumber\\
		&\Rightarrow&F(z,x)\leq\frac{1}{p}(\hat{\lambda}_1(p)+\epsilon)x^p\ \mbox{for\ a.a.}\ x\in\Omega,\ \mbox{all}\ x\geq M_5\nonumber\\
		&\Rightarrow&\frac{pF(z,x)}{x^p}\leq\hat{\lambda}_1(p)+\epsilon\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\geq M_5\nonumber\\
		 &\Rightarrow&\limsup\limits_{x\rightarrow+\infty}\frac{pF(z,x)}{x^p}\leq\hat{\lambda}_1(p)+\epsilon\ \mbox{uniformly\ for\ a.a.}\ z\in\Omega.\nonumber
	\end{eqnarray}
	Since $\epsilon>0$, is arbitrary, we let $\epsilon\downarrow 0$ to conclude that
	\begin{eqnarray}\label{eq33}
		\limsup\limits_{x\rightarrow+\infty}\ \frac{pF_+(z,x)}{x^p}\leq\hat{\lambda}_1(p)\ \mbox{uniformly\ for\ a.a.}\ z\in\Omega.
	\end{eqnarray}
	Hypothesis $H_1(i)$ implies that
	$$\Rightarrow\left\{\frac{F_+(\cdot,u_n(\cdot))}{||u_n||^p}\right\}_{n\geq 1}\subseteq L^1(\Omega)\ \mbox{uniformly\ integrable.}$$
	Then from the Dunford-Pettis theorem and using (\ref{eq33}), at least for a subsequence, we have
	\begin{eqnarray}\label{eq34}
		&&\frac{F_+(\cdot,u_n(\cdot))}{||u_n||^p}\ \stackrel{w}{\longrightarrow}\ \frac{1}{p}\vartheta(y^+)^p\ \mbox{in}\ L^1(\Omega)\ \mbox{with}\ \vartheta\in L^{\infty}(\Omega),\ \vartheta(z)\leq\hat{\lambda}_1(p)\ \mbox{a.a.\ in}\ \Omega.
	\end{eqnarray}
	We return to (\ref{eq32}), pass to the limit as $n\rightarrow\infty$ and use (\ref{eq31}) and (\ref{eq34}). Since $2<p$, we obtain
	\begin{eqnarray}
		&&\frac{1}{p}||Dy||^{p}_{p}\leq\frac{1}{p}\int_{\Omega}\vartheta(y^+)^pdz\label{eq35}\\
		&\Rightarrow&||Dy^+||^{p}_{p}\leq\int_{\Omega}\vartheta(y^+)^pdz.\label{eq36}
	\end{eqnarray}
	If $\vartheta\neq\hat{\lambda}_1(p)$, then from (\ref{eq36}) and Lemma \ref{lem5}, we have
	$$c_0||y^+||^p\leq 0,\ \mbox{hence}\ y\leq 0.$$
	Then from (\ref{eq35}) it follows that $y=0$ and this from (\ref{eq32}) implies that
	\begin{eqnarray}
		&&Dy_n\rightarrow 0\ \mbox{in}\ L^p(\Omega,\RR^N)\nonumber\\
		&\Rightarrow&y_n\rightarrow 0\ \mbox{in}\ W^{1,p}_{0}(\Omega).\nonumber
	\end{eqnarray}
	This contradicts the fact that $||y_n||=1$ for all $n\geq 1$.
	
	Next we assume that $\vartheta(z)=\hat{\lambda}_1(p)$ a.e. in $\Omega$. In this case from (\ref{eq36}), we have $y^+=\xi \hat{u}_1(p)$ with $\xi\geq 0$. If $\xi=0$, then as above we reach a contradiction. So, we assume that $\xi>0$. We have
	\begin{eqnarray}\label{eq37}
		u^{+}_{n}(z)\rightarrow+\infty\ \mbox{for\ a.a.}\ z\in\Omega.
	\end{eqnarray}
	By virtue of hypothesis $H_1(ii)$, given $\xi>0$, we can find $M_6=M_6(\xi)>0$ such that
	\begin{eqnarray}\label{eq38}
		f(z,x)x-pF(z,x)\geq\xi\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\geq M_6.
	\end{eqnarray}
	For a.a. $z\in\Omega$ and all $x\geq M_6$, we have
	\begin{eqnarray}\label{eq39}
		\frac{d}{dx}\ \frac{F(z,x)}{x^p}&=&\frac{f(z,x)x^p-pF(z,x)x^{p-1}}{x^{2p}}\nonumber\\
		&=&\frac{f(z,x)x-pF(z,x)}{x^{p+1}}\nonumber\\
		&\geq&\frac{\xi}{x^{p+1}}\ (\mbox{see\ (\ref{eq38})})\nonumber\\
		\Rightarrow&&\frac{F(z,x)}{v^p}-\frac{F(z,x)}{x^p}\geq-\frac{\xi}{p}\ \left[\frac{1}{v^p}-\frac{1}{x^p}\right]\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ v\geq x\geq M_6.
	\end{eqnarray}
	So, if in (\ref{eq39}) we let $v\rightarrow+\infty$ and use (\ref{eq33}), we obtain
	\begin{eqnarray}
		&&\frac{\hat{\lambda}_1(p)}{p}-\frac{F(z,x)}{x^p}\geq\frac{\xi}{p}\ \frac{1}{x^p}\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\geq M_6\nonumber\\
		&\Rightarrow&\frac{\hat{\lambda}_1(p)}{p}\ x^p-F(z,x)\geq\frac{\xi}{p}\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\geq M_6\nonumber\\
		 &\Rightarrow&\lim\limits_{x\rightarrow+\infty}\left[\frac{\hat{\lambda}_1(p)}{p}x^p-F(z,x)\right]
\geq\frac{\xi}{p}\ \mbox{uniformly\ for\ a.a.}\ z\in\Omega.\nonumber
	\end{eqnarray}
	But $\xi>0$ is arbitrary. So, we conclude that
	$$\lim\limits_{x\rightarrow+\infty}\left[\frac{\hat{\lambda}_1(p)}{p}x^p-F(z,x)\right]=+\infty\ \mbox{uniformly\ for\ a.a.}\ z\in\Omega.$$
	Thus, using (\ref{eq37}), we have
	\begin{eqnarray}\label{eq40}
		&&\frac{\hat{\lambda}_1(p)}{p}u^{+}_{n}(z)^p-F(z,u^{+}_{n}(z))\rightarrow+\infty\ \mbox{for\ a.a.}\ z\in\Omega\nonumber\\
		 &\Rightarrow&\int_{\Omega}\left[\frac{\hat{\lambda}_1(p)}{p}u^{+}_{n}(z)^p-F_+(z,u^{+}_{n}(z))\right]dz\rightarrow+\infty\ (\mbox{by\ Fatou's\ Lemma}).
	\end{eqnarray}
	Recall that
	\begin{eqnarray}\label{eq41}
		&&\frac{1}{p}||Du_n||^{p}_{p}+\frac{1}{2}||Du_n||^{2}_{2}-\int_{\Omega}F_+(z,u^{+}_{n})dz\leq M_4\ \mbox{all}\ n\geq 1\nonumber\\
		&&\int_{\Omega}\left[\frac{\hat{\lambda}_1(p)}{p}(u^{+}_{n})^p-F_+(z,u^{+}_{n})\right]dz\leq M_4\ \mbox{for\ all}\ n\geq 1\ (\mbox{see\ (\ref{eq4})}).
	\end{eqnarray}
	Comparing (\ref{eq40}) and (\ref{eq41}), we have a contradiction which proves that $\varphi_+$ is coercive.
\end{proof}

Now we can produce a first solution for problem (\ref{eq1}) which is positive.

\begin{prop}\label{prop9}
	Assume that hypotheses $H_1$ hold. Then problem (\ref{eq1}) admits a positive solution $u_0\in{\rm int}\, C_+$ which is a local minimizer of the energy functional $\varphi$.
\end{prop}

\begin{proof}
	From Proposition \ref{prop8}, we know that $\varphi_+$ is coercive. Also, using the Sobolev embedding theorem, we see that $\varphi_+$ is sequentially weakly lower semicontinuous. So, by the Weierstrass theorem, we can find $u_0\in W^{1,p}_{0}(\Omega)$ such that
	\begin{eqnarray}\label{eq42}
		\varphi_+(u_0)=\inf\left\{\varphi_+(u):u\in W^{1,p}_{0}(\Omega)\right\}.
	\end{eqnarray}
	Hypothesis $H_1(iv)$ implies that given $\epsilon>0$, we can find $\delta=\delta(\epsilon)>0$ such that
	\begin{eqnarray}\label{eq43}
		F(z,x)\geq\frac{1}{2}(\beta(z)-\epsilon)x^2\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ |x|\leq\delta.
	\end{eqnarray}
	Since $\hat{u}_1(2)\in\mbox{int}\, C_+$, for $\lambda\in(0,1)$ small we  have $\lambda\hat{u}_1(2)(z)\in[0,\delta]$ for all $z\in\bar{\Omega}$. Then
	\begin{eqnarray}
		 \varphi_+(\lambda\hat{u}_1(2))&=&\frac{\lambda^p}{p}||D\hat{u}_1(2)||^{p}_{p}+\frac{\lambda^2}{2}\hat{\lambda}_1(2)-\int_{\Omega}F_+(z,\lambda\hat{u}_1(2))dz\nonumber\\
		 &\leq&\frac{\lambda^p}{p}||D\hat{u}_1(2)||^{p}_{p}-\frac{\lambda^2}{2}\left[\int_{\Omega}(\beta(z)-\hat{\lambda}_1(2))\hat{u}_1(2)^2dz+\epsilon\hat{\lambda}_1(2)\right]\ \mbox{(see\ (\ref{eq43})).}\nonumber
	\end{eqnarray}
	The hypothesis on $\beta(\cdot)$ (see $H_1(iv)$) and since $\hat{u}_1(2)\in\mbox{int}\, C_+$, imply that
	$$\xi_{\ast}=\int_{\Omega}(\beta(z)-\hat{\lambda}_1(2))\hat{u}_1(2)^2dz>0.$$
	So, if we choose $\epsilon\in(0,\frac{\xi_{\ast}}{\hat{\lambda}_1(2)})$, then
	\begin{eqnarray}
		&&\varphi_+(\lambda\hat{u}_1(2))<0\nonumber\\
		\Rightarrow&&\varphi_+(u_0)<0=\varphi_+(0)\ (\mbox{see\ (\ref{eq42})}), \mbox{hence}\ u_0\neq 0.\nonumber
	\end{eqnarray}
	From (\ref{eq42}) we have
	\begin{eqnarray}\label{eq44}
		&&\varphi'_{+}(u_0)=0\nonumber\\
		\Rightarrow&&A_p(u_n)+A(u_n)=N_f(u_0).
	\end{eqnarray}
	On (\ref{eq44}) we act with $-u^{-}_{n}\in W^{1,p}_{0}(\Omega)$. We obtain
	$$||Du^{-}_{n}||^{p}_{p}+||Du^{-}_{n}||^{2}_{2}=0,\ \mbox{hence}\ u_0\geq 0,\ u_0\neq 0.$$
	
	Then from (\ref{eq44}) we have
	$$-\Delta_pu_0(z)-\Delta u_0(z)=f(z,u_0(z))\ \mbox{a.e.\ in}\ \Omega,\ u_0|_{\partial\Omega}=0.$$
	
	From Ladyzhenskaya and Uraltseva \cite[p. 286]{14}, we know that $u_0\in L^{\infty}(\Omega)$. Then Theorem \ref{th1} of Lieberman \cite{15} implies that $u_0\in C_+\backslash\{0\}$.
	
	Evidently hypotheses $H_1(i),\ (iv)$ imply that for every $\rho>0$, we can find $\hat{\xi}_{\rho}>0$ such that  $f(z,x)x+\xi_{\rho}|x|^p\geq 0$ for a.a. $z\in\Omega$, all $|x|\leq\rho$. Let $\rho=||u_0||_{\infty}$ and let $\hat{\xi}_{\rho}>0$ as just mentioned. We have
	\begin{eqnarray}\label{eq45}
		&&-\Delta_pu_0(z)-\Delta u_0(z)+\hat{\xi}_{\rho}u_0(z)^{p-1}\nonumber\\
		&&=f(z,u_0(z))+\hat{\xi}_{\rho}u_0(z)^{p-1}\geq 0\ \mbox{for\ a.a.}\ z\in\Omega\nonumber\\
		\Rightarrow&&\Delta_pu_0(z)+\Delta u_0(z)\leq\xi_{\rho}u_0(z)^{p-1}\ \mbox{for\ a.a.}\ z\in\Omega.
	\end{eqnarray}
	Let $a:\RR^N\rightarrow\RR^N$ be the $C^1$-map defined by
	$$a(y)=||y||^{p-2}y+y\ (\mbox{recall}\ p>2).$$
	We have $\mbox{div}\ a(Du)=\Delta_pu+\Delta u$ for all $u\in W^{1,p}_{0}(\Omega)$ 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\nonumber\\
		&\Rightarrow&(\nabla a(y)\xi,\xi)_{\RR^N}\geq||\xi||^2\ \mbox{for\ all}\ y,\xi\in\RR^N.\nonumber
	\end{eqnarray}
	Then the tangency principle of Pucci and Serrin \cite[p. 35]{25} implies that
	$$u_0(z)>0\ \mbox{for\ all}\ z\in\Omega.$$
	So, from (\ref{eq45}) and the boundary point theorem of Pucci and Serrin \cite[p. 120]{25}, we 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 $\varphi$. Invoking Proposition \ref{prop2}, we conclude that $u_0$ is a local $W^{1,p}_{0}(\Omega)$-minimizer of $\varphi$.
\end{proof}

To produce a second nontrivial solution, we need to restrict the behavior of $f(z,\cdot)$ near zero. More precisely, the new hypotheses on the reaction $f(z,x)$ are the following:

\smallskip
$\underline{H_2}:\ f:\Omega\times\RR\rightarrow\RR$ is a Carath\'eodory function such that  $f(z,0)=0$ for a.a. $z\in\Omega$, hypotheses $H_2(i),\ (ii),\ (iii),\ (v)$ are the same as the corresponding hypotheses $H_1(i),\ (ii),\ (iii),\ (v)$ and
\begin{itemize}
	\item[(iv)]there exist an integer $m\geq 2$ and functions $\beta,\hat{\beta}\in L^{\infty}(\Omega)_+$ such that
	\begin{eqnarray}
		&&\hat{\lambda}_m(2)\leq\beta(z)\leq\hat{\beta}(z)\leq\hat{\lambda}_{m+1}(2)\ \mbox{a.e.\ in}\ \Omega,\ \hat{\lambda}_m(2)\neq\beta,\ \hat{\lambda}_{m+1}(2)\neq\hat{\beta}\ \mbox{and}\nonumber\\
		&&\beta(z)\leq\liminf\limits_{x\rightarrow 0}\frac{f(z,x)}{x}\leq\limsup\limits_{x\rightarrow 0}\frac{f(z,x)}{x}\leq\hat{\beta}(z)\ \mbox{uniformly\ for\ a.a.}\ z\in\Omega.\nonumber
	\end{eqnarray}
\end{itemize}

\begin{theorem}\label{th10}
	Assume that hypotheses $H_2$ hold. Then problem (\ref{eq1}) admits at least two nontrivial solutions
	$$u_0\in{\rm int}\, C_+\ \mbox{and}\ \hat{u}\in C^{1}_{0}(\bar{\Omega}),\ u_0\neq\hat{u}.$$
\end{theorem}

\begin{proof}
	From Proposition \ref{prop9} we already have one nontrivial solution $u_0\in\mbox{int}\, C_+$, which is a local minimizer of $\varphi$. Hence as in Aizicovici, Papageorgiou and Staicu \cite{1} (see the proof of Proposition 29), we can find $\rho\in(0,1)$ small such that
	\begin{eqnarray}\label{eq46}
		\varphi(u_0)<\inf \left\{\varphi(u):||u-u_0||=\rho\right\}=m_{\rho}\,.
	\end{eqnarray}
	Hypothesis $H_2(iii)$ implies that
	\begin{eqnarray}\label{eq47}
		\varphi(t\hat{u}_1(p))\rightarrow-\infty\ \mbox{as}\ t\rightarrow-\infty.
	\end{eqnarray}
	Recall that $\varphi$ satisfies the $C$-condition (see Proposition \ref{prop7}). This fact together with (\ref{eq46}) and (\ref{eq47}) permit the use of Theorem \ref{th1} (the mountain pass theorem). So, we can find $\hat{u}\in W^{1,p}_{0}(\Omega)$ such that
	\begin{eqnarray}\label{eq48}
		\hat{u}\in K_{\varphi}\ \mbox{and}\ \varphi(u_0)<m_{\rho}\leq\varphi(\hat{u}).
	\end{eqnarray}
	From (\ref{eq48}) it is clear that $\hat{u}\neq u_0$ and it is a solution of problem (\ref{eq1}). We need to show that $\hat{u}\neq 0$.
	
	Since $\hat{u}$ is a critical point of $\varphi$ of mountain pass type, we have
	\begin{eqnarray}\label{eq49}
		C_1(\varphi,\hat{u})\neq 0.
	\end{eqnarray}
	
	\begin{claim}
		$C_k(\varphi,0)=\delta_{k,d_m}\ZZ$ for all $k\geq 0$, with $d_m={\rm dim}\,\overset{m}{\underset{{i=1}}\oplus}E(\hat{\lambda}_i(2))$.
	\end{claim}
	
	Let $\mu\in(\hat{\lambda}_m,\hat{\lambda}_{m+1})$ and consider the $C^2$-functional $\psi:W^{1,p}_{0}(\Omega)\rightarrow\RR$ defined by
	$$\psi(u)=\frac{1}{p}||Du||^{p}_{p}+\frac{1}{2}||Du||^{2}_{2}-\frac{\mu}{2}||Du||^{2}_{2}\quad \mbox{for\ all}\ u\in W^{1,p}_{0}(\Omega).$$
	
	Evidently $\psi$ is coercive (recall that $p>2$) and so it satisfies the $C$-condition.
	
	We consider the homotopy $h:[0,1]\times W^{1,p}_{0}(\Omega)\rightarrow\RR$ defined by
	$$h(t,u)=(1-t)\varphi(u)+t\psi(u)\ \mbox{for\ all}\ (t,u)\in[0,1]\times W^{1,p}_{0}(\Omega).$$
	Suppose that we can find $\{t_n\}_{n\geq 1}\subseteq[0,1]$ and $\{u_n\}_{n\geq 1}\subseteq W^{1,p}_{0}(\Omega)$ such that
	\begin{eqnarray}\label{eq50}
		t_n\rightarrow t\ \mbox{in}\ [0,1],\ u_n\rightarrow 0\ \mbox{in}\ W^{1,p}_{0}(\Omega)\ \mbox{and}\ h'_{u}(t_n,u_n)=0\ \mbox{for\ all}\ n\geq 1.
	\end{eqnarray}
	We have
	\begin{eqnarray}\label{eq51}
		A_p(u_n)+A(u_n)=(1-t_n)N_f(u_n)+t_n\mu u_n\ \mbox{for\ all}\ n\geq 1.
	\end{eqnarray}
	Let $y_n=\frac{u_n}{||u_n||}\ n\geq 1$. Then $||y_n||=1$ for all $n\geq 1$ and so we may assume that
	\begin{eqnarray}\label{eq52}
		y_n\stackrel{w}{\longrightarrow} y\ \mbox{in}\ W^{1,p}_{0}(\Omega)\ \mbox{and}\ y_n\rightarrow y\ \mbox{in}\ L^2(\Omega).
	\end{eqnarray}
	From (\ref{eq51}) we have
	\begin{eqnarray}\label{eq53}
		||u_n||^{p-2}A_p(y_n)+A(y_n)=(1-t_n)\frac{N_f(u_n)}{||u_n||}+t_n\mu y_n\ \mbox{for\ all}\ n\geq 1.
	\end{eqnarray}
	Evidently $\left\{\frac{N_f(u_n)}{||u_n||}\right\}_{n\geq 1}\subseteq L^2(\Omega)$ is bounded (see $H_2(i),\ (iv)$) and by virtue of hypothesis $H_2(iv)$ and (\ref{eq50}), we have (at least for a subsequence)
	\begin{eqnarray}\label{eq54}
		\frac{N_f(u_n)}{||u_n||}\stackrel{w}{\longrightarrow}\ gy\ \mbox{in}\ L^2(\Omega)\ \mbox{with}\ \beta(z)\leq g(z)\leq \hat{\beta}(z)\ \mbox{a.e.\ in}\ \Omega.
	\end{eqnarray}
	Since $A_p$ is bounded (see Proposition \ref{prop3}) and $A\in\mathcal{L}(H^{1}_{0}(\Omega),\ H^{-1}(\Omega))$, if in (\ref{eq53}) we pass to the limit as $n\rightarrow\infty$ and use (\ref{eq52}) and (\ref{eq54}), we obtain
	\begin{eqnarray}\label{eq55}
		&&A(y)=[(1-t)g+t\mu]y\nonumber\\
		\Rightarrow&&-\Delta y(z)=[(1-t)g(z)+t\mu]y(z)\ \mbox{a.e.\ in}\ \Omega,\ y|_{\partial\Omega}=0.
	\end{eqnarray}
	Note that
	$$\hat{\lambda}_m(2)\leq(1-t)g(z)+t\mu=g_t(z)\leq\hat{\lambda}_{m+1}(2)\ \mbox{a.e.\ in}\ \Omega,\ \hat{\lambda}_m(2)\neq g_t,\ \hat{\lambda}_{m+1}(2)\neq g_t.$$
	
	By virtue of the unique continuation property, we have
	$$\hat{\lambda}_{m}(2,g_t)<\hat{\lambda}_{m}(2,\hat{\lambda}_m(2))=1\ \mbox{and}\ 1=\hat{\lambda}_{m+1}(2,\hat{\lambda}_{m+1}(2))<\hat{\lambda}_{m+1}(2,g_t).$$
	
	Then from (\ref{eq55}) it follows that $y=0$.
	
	From (\ref{eq53}), we have
	$$-||u_n||^{p-2}\Delta_py_n(z)-\Delta y_n(z)=(1-t)n)\frac{f(z,u_n(z))}{||u_n||}+t_n\mu y_n(z)\ \mbox{a.e.\ in}\ \Omega,\ y_n|_{\partial\Omega}=0.$$
	
	From Ladyzhensaya and Uraltseva \cite{14}, we know that we can find $M_7>0$ such that
	$$||y_n||_{\infty}\leq M_7.$$
	
	Then the regularity result of Liebermann \cite{15} implies that we can find $\gamma\in(0,1)$ and $M_8>0$ such that
	$$y_n\in C^{1,\gamma}_{0}(\bar{\Omega})\ \mbox{and}\ ||y_n||_{C^{1,\gamma}_{0}(\bar{\Omega})}\leq M_8\ \mbox{for\ all}\ n\geq 1.$$
	
	Exploiting the compact embedding of $C^{1,\gamma}_{0}(\bar{\Omega})$ into $C^{1}_{0}(\bar{\Omega})$, we have
	\begin{eqnarray}
		&&y_n\rightarrow 0\ \mbox{in}\ C^{1}_{0}(\bar{\Omega})\ (\mbox{see\ (\ref{eq52})\ and\ recall}\ y=0)\nonumber\\
		\Rightarrow&&y_n\rightarrow 0\ \mbox{in}\ W^{1,p}_{0}(\Omega)\nonumber
	\end{eqnarray}
	which contradicts the fact that $||y_n||=1$ for all $n\geq 1$. This implies that (\ref{eq50}) can not happen. Then the homotopy invariance property of critical groups implies that
	$$C_k(\varphi,0)=C_k(\psi,0)\ \mbox{for\ all}\ k\geq 0.$$
	
	But since $\mu\in(\hat{\lambda}_m(2),\hat{\lambda}_{m+1}(2))$, by Theorem 1 of Cingolani and Vannella \cite{9} we deduce that
	$$C_k(\psi,0)=\delta_{k,d_m}\ZZ\ \mbox{for\ all}\ k\geq 0.$$
	This proves the Claim.
	
	From (\ref{eq49}) and the Claim, we conclude that $\hat{u}\neq 0$. Then $\hat{u}$ is the second nontrivial solution of (\ref{eq1}) and by the nonlinear regularity theory (see Ladyzhenskaya and Uraltseva \cite{14}, Lieberman \cite{15}), we have $\hat{u}\in C^{1}_{0}(\bar{\Omega}).$
\end{proof}

\section{Three Solutions Theorem}

In this section, we produce a third nontrivial solution for problem (\ref{eq1}) (three solutions theorem). To do this we need to improve the regularity of $f(z,\cdot)$ and also avoid complete resonance at $+\infty$. So, the new hypotheses on the reaction $f(z,x)$ are the following:

\smallskip
$\underline{H_3}:\ f:\Omega\times\RR\rightarrow\RR$ is a measurable function such that  for a.a. $z\in\Omega\ f(z,0)=0,\ f(z,\cdot)\in C^1(\RR)$ and
\begin{itemize}
	\item[(i)]$\left|f'_{x}(z,x)\right|\leq a(z)(1+|x|^{p-2})$ for a.a. $z\in\Omega$, all $x\in\RR$, with $a\in L^{\infty}(\Omega)_+$;
	\item[(ii)]there exists a function $\vartheta\in L^{\infty}(\Omega)_+$ such that  $\vartheta(z)\leq\hat{\lambda}_1(p)$ a.e. in $\Omega,\ \vartheta\neq\hat{\lambda}_1(p)$ and
	$$\limsup\limits_{x\rightarrow+\infty}\frac{f(z,x)}{x^{p-1}}\leq \vartheta(z)\ \mbox{uniformly\ for\ a.a.}\ z\in\Omega;$$
	\item[(iii)]there exists a function $\eta\in L^{\infty}(\Omega)_+$ such that
	\begin{eqnarray}
		&&\hat{\lambda}_1(p)\leq\eta(z)\ \mbox{for\ a.a.}\ z\in\Omega,\ \hat{\lambda}_1(p)\neq\eta\ \mbox{and}\nonumber\\
		 &&\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{\lambda}_2(p)\ \mbox{uniformly\ for\ a.a.}\ z\in\Omega;\nonumber
	\end{eqnarray}
	\item[(iv)]there exists integer $m\geq 2$ such that
	$$f'_{x}(z,0)\in[\hat{\lambda}_m(2),\hat{\lambda}_{m+1}(2)]\ \mbox{a.e.\ in}\ \Omega,\ f'_{x}(\cdot,0)\neq\hat{\lambda}_m(2),\ f'_{x}(\cdot,0)\neq\hat{\lambda}_{m+1}(2);$$
	\item[(v)]$f(z,x)x-pF(z,x)\geq 0$ for a.a. $z\in\Omega$, all $x\leq 0.$
\end{itemize}

\begin{remark}
	Now at $+\infty$ we allow only nonuniform nonresonance with respect to the principal eigenvalue $\hat{\lambda}_1(p)>0$ (see hypothesis $H_3(ii)$). The reason for this is the computation of the critical groups of $\varphi$ at infinity based on Proposition \ref{prop6} (see Proposition \ref{prop11} below).
\end{remark}

\begin{prop}\label{prop11}
	Assume that hypotheses $H_3$ hold. Then $C_k(\varphi,\infty)=0$ for all $k\geq 0$.
\end{prop}

\begin{proof}
	Let $\beta\in L^{\infty}(\Omega)_+,\ \beta\neq 0$ and $\mu\in(\hat{\lambda}_1(p),\hat{\lambda}_2(p))$. We consider the following one parameter family of $C^1$-functionals defined by
	\begin{eqnarray}
		 h_t(u)&=&\frac{1}{p}||Du||^{p}_{p}+\frac{1}{2}||Du||^{2}_{2}-t\int_{\Omega}F(z,u(z))dz-\frac{1-t}{p}\mu||u^-||^{p}_{p}+(1-t)\int_{\Omega}\beta(z)u(z)dz\nonumber\\
		&&\hspace{8cm}\mbox{for\ all}\ (t,u)\in[0,1]\times W^{1,p}_{0}(\Omega).\nonumber
	\end{eqnarray}
	We have
	\begin{eqnarray}
		 &&h_0(u)=\psi(u)=\frac{1}{p}||Du||^{p}_{p}-\frac{\mu}{p}||u^-||^{p}_{p}+\int_{\Omega}\beta(z)u(z)dz\ \mbox{for\ all}\ u\in W^{1,p}_{0}(\Omega)\nonumber\\
		&&h_1(u)=\varphi(u)\ \mbox{for\ all}\ u\in W^{1,p}_{0}(\Omega).\nonumber
	\end{eqnarray}
	Since $\mu\in(\hat{\lambda}_1(p),\hat{\lambda}_2(p))$, we can easily check that $\psi$ satisfies the $C$-condition. Also, from Proposition \ref{prop7} we know that $\varphi$ satisfies the $C$-condition.
	
	\begin{claim}
		There exists $\gamma_0\in\RR$ and $\delta_0>0$ such that
		$$h_t(u)\leq\gamma_0\Rightarrow(1+||u||)||(h_t)'(u)||_{\ast}\geq\delta_0(||u||^2+||u||^p)\ \mbox{for\ all}\ t\in\left[0,1\right).$$
	\end{claim}
	
	We proceed by contradiction. So, suppose that the Claim is not true. Since the function $(t,u)\longmapsto h_t(u)$ maps bounded sets to bounded sets, we can find $\{t_n\}_{n\geq 1}\subseteq[0,1]$ and $\{u_n\}_{n\geq 1}\subseteq W^{1,p}_{0}(\Omega)$ such that
	\begin{eqnarray}\label{eq56}
		&&t_n\rightarrow t,\ ||u_n||\rightarrow\infty,\ h_{t_n}(u_n)\rightarrow-\infty\ \ \mbox{and}\nonumber\\
		&&(1+||u_n||)||(h_{t_n})'(u_n)||_{\ast}\leq\frac{1}{n}(||u_n||^2+||u_n||^p)\\
		&&\hspace{5.5cm}\mbox{for\ all}\ n\geq 1.\nonumber
	\end{eqnarray}
Let $y_n=\frac{u_n}{||u_n||}\ n\geq 1$. Then $||y_n||=1$ for all $n\geq 1$ and so we may assume that
\begin{eqnarray}\label{eq57}
	y_n\stackrel{w}{\longrightarrow} y\ \mbox{in}\ W^{1,p}_{0}(\Omega)\ \mbox{and}\ y_n\rightarrow y\ \mbox{in}\ L^p(\Omega).
\end{eqnarray}
From (\ref{eq56}) we have
\begin{eqnarray}\label{eq58}
	&&\left|\left\langle A_p(u_n),v\right\rangle+\left\langle A(u_n),v\right\rangle-t_n\int_{\Omega}f(z,u_n)v\ dz+(1-t_n)\mu\int_{\Omega}(u^{-}_{n})^{p-1}v\ dz+\right.\nonumber\\
	&&\hspace{2cm}\left.+(1-t_n)\int_{\Omega}\beta(z)v\ dz\right|\leq\frac{1}{n}\ \frac{||v||}{1+||u_n||}(||u_n||^2+||u_n||^p)\nonumber\\
	\Rightarrow&&\left|\left\langle A_p(y_n),v\right\rangle+\frac{1}{||u_n||^{p-2}}\left\langle A(y_n),v\right\rangle-t_n\int_{\Omega}\frac{f(z,u_n)}{||u_n||^{p-1}}v\ dz\right.\nonumber\\
	&&\hspace{2cm}\left.+(1-t_n)\mu\int_{\Omega}(y^{-}_{n})^{p-1}v\ dz+\frac{1-t_n}{||u_n||^{p-1}}\int_{\Omega}\beta(z)v\ dz\right|\leq\frac{||v||}{n}\ \mbox{for\ all}\ n\geq 1.
\end{eqnarray}
In (\ref{eq58}) we choose $v=y_n-y\in W^{1,p}_{0}(\Omega)$, pass to the limit as $n\rightarrow \infty$ and use (\ref{eq57}). Since $p>2$, we obtain
\begin{eqnarray}\label{eq59}
	&&\lim\limits_{n\rightarrow\infty}\left\langle A_p(y_n),y_n-y\right\rangle=0\nonumber\\
	\Rightarrow&&y_n\rightarrow y\ \mbox{in}\ W^{1,p}_{0}(\Omega)\ \mbox{and\ so}\ ||y||=1.
\end{eqnarray}
Hypotheses $H_3(i),\ (iv)$ imply that
$$\begin{array}{ll}
\displaystyle	|f(z,x)|&\displaystyle\leq c_1(|x|+|x|^{p-1})\\
	&\displaystyle\leq c_2(1+|x|^{p-1})\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\in\RR\ \mbox{and\ some}\ c_2>0\ (\mbox{recall}\ p>2)\\
	\Rightarrow&\displaystyle\left\{\frac{N_f(u_n)}{||u_n||^{p-1}}\right\}_{n\geq 1}\subseteq L^{p'}(\Omega)\ \mbox{is\ bounded}.
\end{array}$$
Hence we may assume that
\begin{eqnarray}\label{eq60}
	\frac{N_f(u_n)}{||u_n||^{p-1}}\stackrel{w}{\longrightarrow}\ g\ \mbox{in}\ L^{p'}(\Omega)
\end{eqnarray}
Hypotheses $H_3(ii),\ (iii)$ imply that
\begin{eqnarray}\label{eq61}
	g(z)=\tilde{\vartheta}(z)y^+(z)^{p-1}-\tilde{\xi}(z)y^-(z)^{p-1}\ \mbox{for\ a.a.}\ z\in\Omega.
\end{eqnarray}
So, if we return to (\ref{eq58}), pass to the limit as $n\rightarrow\infty$ and use (\ref{eq59}), (\ref{eq60}), (\ref{eq61}), we obtain
\begin{eqnarray}
	&&\left\langle A_p(y),v\right\rangle=t\int_{\Omega}\tilde{\vartheta}(z)(y^+)^{p-1}v\ dz-\int_{\Omega}[t\tilde{\xi}(z)+(1-t)\mu](y^-)^{p-1}v\ dz\nonumber\\
	&&\hspace{8.5cm}\mbox{for\ all}\ v\in W^{1,p}_{0}(\Omega)\nonumber\\
	&\Rightarrow&A_p(y)=t\tilde{\vartheta}(y^+)^{p-1}-\tilde{\xi}_t(y^-)^{p-1}\ \mbox{where}\ \tilde{\xi}_t=t\tilde{\xi}+(1-t)\mu,\label{eq62}\\
	&\Rightarrow&-\Delta_py(z)=t\tilde{\vartheta}(z)y^+(z)^{p-1}-\tilde{\xi}_t(z)y^-(z)^{p-1}\ \mbox{a.e.\ in}\ \Omega,\ y|_{\partial\Omega}=0.\label{eq63}
\end{eqnarray}
On (\ref{eq62}) first we act with $y^+\in W^{1,p}_{0}(\Omega)$. Then
\begin{eqnarray}
	 &&||Dy^+||^{p}_{p}=t\int_{\Omega}\tilde{\vartheta}(z)(y^+)^pdz\leq\int_{\Omega}\vartheta(z)(y^+)^pdz\nonumber\\
	\Rightarrow&&c_0||Dy^+||^{p}_{p}\leq 0\ (\mbox{see\ Lemma\ \ref{lem5}}),\ \mbox{hence}\ y^+=0.\nonumber
\end{eqnarray}
From (\ref{eq63}) and since $\tilde{\xi}_t(z)\in\left[\eta(z),\ \hat{\lambda}_2(p)\right]$ a.e. in $\Omega$, it follows that $y^-$ must be nodal, a contradiction. This proves the Claim.

Invoking Proposition \ref{prop6}, we infer that
\begin{eqnarray}\label{eq64}
	C_k(\varphi,\infty)=C_k(\psi,\infty)\ \mbox{for\ all}\ k\geq 0.
\end{eqnarray}
Now let $u\in K_{\psi}$. Then
\begin{eqnarray}\label{eq65}
	A_p(u)=-\mu(u^-)^{p-1}-\beta.
\end{eqnarray}
On (\ref{eq65}) we act with $u^+\in W^{1,p}_{0}(\Omega)$ and obtain
\begin{eqnarray}
	&&||Du^+||^{p}_{p}=-\int_{\Omega}\beta(z)u^+dz\leq 0\nonumber\\
	\Rightarrow&&u^+=0,\ \mbox{hence}\ u\leq 0\nonumber
\end{eqnarray}
Equation (\ref{eq65}) becomes
\begin{eqnarray}\label{eq66}
	&&A_p(u)=\mu|u|^{p-2}u-\beta\nonumber\\
	\Rightarrow&&-\Delta_pu(z)=\mu|u(z)|^{p-2}u(z)-\beta(z)\ \mbox{a.e.\ in}\ \Omega,\ u|_{\partial\Omega}=0.
\end{eqnarray}
The nonlinear regularity theory implies that $u\in(-C_+)\backslash\{0\}$. Moreover, as before from the tangency principle and the boundary point theorem of Pucci and Serrin \cite[pp. 35 and 120]{25}, we have
$$u\in-\mbox{int}\, C_+\,.$$

Let $v\in\mbox{int}\, C_+$ and consider the function
$$R(v,-u)(z)=||Dv(z)||^p-||D(-u)(z)||^p\left(D(-u)(z),\ D\left(\frac{v^p}{(-u)^{p-1}}\right)(z)\right)_{\RR^N}\,.$$

From the nonlinear Picone's identity of Allegretto and Huang \cite{3}, we have
\begin{eqnarray}
	0&&\leq\int_{\Omega}R(v_1-u)dz\nonumber\\
	&&=||Dv||^{p}_{p}-\int_{\Omega}-\Delta_p(-u)\frac{v^p}{-u^{p-1}}dz\ (\mbox{by\ Green's\ theorem})\nonumber\\
	 &&=||Dv||^{p}_{p}-\int_{\Omega}\mu(-u)^{p-1}\frac{v^p}{-u^{p-1}}dz-\int_{\Omega}\beta\frac{v^p}{(-u)^{p-1}}dz\ (\mbox{see\ (\ref{eq66})})\nonumber\\
	&&\leq||Dv||^{p}_{p}-\mu||v||^{p}_{p}\,.\nonumber
\end{eqnarray}
Choose $v=\hat{u}_1(p)\in\mbox{int}\, C_+$, to reach a contradiction (recall that $\mu\in(\hat{\lambda}_1(p),\hat{\lambda}_2(p))$). Hence $K_{\psi}=\O$ and so we have
\begin{eqnarray}
	&&C_k(\psi,\infty)=0\ \mbox{for\ all}\ k\geq 0\nonumber\\
	\Rightarrow&&C_k(\varphi,\infty)=0\ \mbox{for\ all}\ k\geq 0\ (\mbox{see\ (\ref{eq64})}).\nonumber
\end{eqnarray}
This completes the proof.
\end{proof}

Now we are ready for the three solutions theorem.

\begin{theorem}\label{th12}
	Assume that hypotheses $H_3$ hold. Then problem (\ref{eq1}) has at least three nontrivial solutions
	$$u_0\in{\rm int}\, C_+\ \mbox{and}\ \hat{u},\tilde{u}\in C^{1}_{0}(\bar{\Omega})\backslash\{0\}.$$
\end{theorem}

\begin{proof}
	From Theorem \ref{th10}, we already have two nontrivial solutions
	$$u_0\in\mbox{int}\, C_+\ \mbox{and}\ \hat{u}\in C^{1}_{0}(\bar{\Omega})\backslash\{0\}.$$
	
	From Proposition \ref{prop9} we know that $u_0$ is a local minimizer of the energy functional $\varphi$. Therefore
	\begin{eqnarray}\label{eq67}
		C_k(\varphi,u_0)=\delta_{k,0}\ZZ\ \mbox{for\ all}\ k\geq 0.
	\end{eqnarray}
	From the proof of Theorem \ref{th10}, we know that $\hat{u}$ is a critical point $\varphi$ of mountain pass type. Then from Papageorgiou and R\u{a}dulescu \cite{21} and Papageorgiou and Smyrlis \cite{23}, we have
	\begin{eqnarray}\label{eq68}
		C_k(\varphi,\hat{u})=\delta_{k,1}\ZZ\ \mbox{for\ all}\ k\geq 0.
	\end{eqnarray}
	From the proof of the Theorem \ref{th10} (see the Claim), we have
	\begin{eqnarray}\label{eq69}
		C_k(\varphi,0)=\delta_{k,d_m}\ZZ\ \mbox{for\ all}\ k\geq 0.
	\end{eqnarray}
	Finally, Proposition \ref{prop11} implies that
	\begin{eqnarray}\label{eq70}
		C_k(\varphi,\infty)=0\ \mbox{for\ all}\ k\geq 0.
	\end{eqnarray}
	Suppose that $K_{\varphi}=\{0,u_0,\hat{u}\}$. Then from (\ref{eq67}), (\ref{eq68}), (\ref{eq69}), (\ref{eq70}) and the Morse relation (see (\ref{eq5})) with $t=-1$, we have
	\begin{eqnarray}
		&&(-1)^{d_m}+(-1)^0+(-1)^1=0\nonumber\\
		\Rightarrow&&(-1)^{d_m}=0,\ \mbox{a\ contradiction}.\nonumber
	\end{eqnarray}
	So, we can find $\tilde{u}\in K_{\varphi},\ \tilde{u}\notin\{0,u_0,\hat{u}\}$. Then $\tilde{u}$ is the third nontrivial solution of (\ref{eq1}) and $\tilde{u}\in C^{1}_{0}(\bar{\Omega})$ (nonlinear regularity).	 \end{proof}

\begin{remark}
	It is an interesting open problem if this three solutions theorem remains valid when we allow complete resonance at $+\infty$.
\end{remark}

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




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\end{document} 