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\begin{document}

\title{\sc Coercive and Noncoercive Nonlinear Neumann Problems With  Indefinite Potential}

\author{Nikolaos S. Papageorgiou\footnote{National Technical University, Department of Mathematics, Zografou Campus, Athens 15780, Greece. Email: {\tt npapg@math.ntua.gr}} \ and Vicen\c tiu D. R\u adulescu\footnote{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@math.cnrs.fr}}}
%\address{, }


\date{}

%\baselineskip16pt
\maketitle



\begin{abstract}
We consider nonlinear Neumann problems driven by a nonhomogeneous differential operator and an indefinite potential. In this paper we are concerned with two distinct cases. We first consider the case where the reaction is $(p-1)$--sublinear near $\pm \infty$ and $(p-1)$--superlinear near zero. In this setting the energy functional of the problem is coercive. In the second case, the reaction is $(p-1)$--superlinear near $\pm \infty$ (without satisfying the Ambrosetti-Rabinowitz condition) and has a $(p-1)$--sublinear growth near zero. Now, the energy functional is indefinite. For both cases we prove ``three solutions theorems" and in the coercive setting we provide sign information for all of them. Our approach combines variational methods, truncation and perturbation techniques, and Morse theory (critical groups).\\
\noindent \textbf{2010 Mathematics Subject Classification:} 35J20, 35J60, 35J92, 58E05.\\
\textbf{Keywords:} Concave terms, superlinear reaction, nodal solution, critical groups, nonlinear regularity theory, indefinite potential.
\end{abstract}



%\maketitle

\section{Introduction}

Let $\Omega\subseteq \mathbb R^N$ be a bounded domain with a $C^2$-boundary $\partial \Omega$. In this paper we study the following nonlinear Neumann problem
\begin{eqnarray}\label{eq1}
	-\mbox{div}\, a(Du(z))+\beta(z)|u(z)|^{p-2}u(z)=f(z,u(z))\ \mbox{in}\ \Omega,\qquad \frac{\partial u}{\partial n}=0\ \mbox{on}\ \partial \Omega.
\end{eqnarray}
Here $n(\cdot)$ stands for the outward unit normal on $\partial \Omega$. Also, $a:\mathbb R^N\rightarrow \mathbb R^N$ is a continuous strictly monotone map that satisfies certain other regularity conditions. The precise conditions on the map $a(\cdot)$, are listed in hypotheses $H(a)_1$. These assumptions are general enough to include some important classes of nonlinear differential operators. In particular, they incorporate the $p$--Laplace differential operator. However, we stress that in contrast to the $p$-Laplacian, the differential operator in (\ref{eq1}) is not necessarily homogeneous and this is a source of difficulties, especially when we look for nodal (that is, sign changing) solutions. The potential (weight) function $\beta(\cdot)$ belongs in $L^{\infty}(\Omega)$ and may change sign (indefinite potential). Finally, the reaction $f(z,x)$ is a Carath\'eodory function (that is, for all $x\in \mathbb R$, the mapping $z\longmapsto f(z,x)$ is measurable and for a.a. $z\
 in \Omega$, $x\longmapsto f(z,x)$ is continuous).

Our aim is to prove a ``three solutions theorem" for problem (\ref{eq1}), providing if possible, sign information for all the solutions. We present two such multiplicity theorems under complementary conditions on the reaction $f(z,x)$. In the first multiplicity theorem we assume that $f(z,\cdot)$ is $(p-1)$--linear near $\pm \infty$, while near zero it exhibits a ``concave" term (that is, a $(p-1)$--superlinear term). In the second multiplicity theorem, $f(z,\cdot)$ is $(p-1)$--superlinear near $\pm \infty$, while near zero it is $(p-1)$--linear. In the first case, the energy functional of the problem is coercive, while in the second case it is indefinite.

In the past, such multiplicity results were proved for equations driven by the $p$--Laplacian. We refer to the works of Liu \cite{22}, Liu \& Liu \cite{24} (Dirichlet problems) and Aizicovici, Papageorgiou \& Staicu \cite{3}, Kyritsi \& Papageorgiou \cite{20} (Neumann problems) for the coercive case and by Bartsch \& Liu \cite{4}, Bartsch, Liu \& Weth \cite{5}, Filippakis, Kristaly \& Papageorgiou \cite{12}, Liu \cite{23}, Sun \cite{35} (Dirichlet problems) and Aizicovici, Papageorgiou \& Staicu \cite{1,2} (Neumann equations) for the noncoercive case with superlinear reaction. In the aforementioned works, the hypotheses on the reaction $f(z,x)$ are in general more restrictive and they do not always provide sign information for all the solutions produced. We mention that another class of coercive Dirichlet equations with a nonlinear nonhomogeneous differential operator was studied recently by the authors in \cite{31}. Finally, resonant semilinear equations with an indefinite a
 nd unbounded potential were investigated by Papageorgiou \& R\u adulescu \cite{30}.

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

\section{Mathematical Background-Auxiliary Results}

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

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

This compactness-type condition is more general than the usual Palais-Smale condition. Nevertheless, the $C$-condition suffices to prove a deformation theorem that develops the minimax theory for certain critical values of $\varphi$. In particular, we can have the following version of the well-known  ``mountain pass theorem" (see, for example, Gasinski \& Papageorgiou \cite{14}, Kristaly, R\u adulescu \& Varga \cite{19}, and R\u adulescu \cite{34}).

\begin{theorem}\label{th1}
	Let $X$ be a Banach space, $\varphi\in C^1(X)$ satisfies the C-condition, $x_0,x_1\in X$ with $||x_1-x_0||>r$,
	$$\max\left\{\varphi(x_0),\varphi(x_1)\right\}< \inf\left\{\varphi(x):||x-x_0||=r\right\}=\eta_r\,,$$
	and $c=\inf\limits_{\gamma\in\Gamma}\ \max\limits_{0\leq t\leq 1}\ \varphi(\gamma(t))$ with $\Gamma=\left\{\gamma\in C([0,1],X):\gamma(0)=x_0,
\ \gamma(1)=x_1\right\}$.

Then $c\geq\eta_r$ and $c$ is a critical value of $\varphi.$
\end{theorem}

The analysis of problem (\ref{eq1}) will use the Sobolev space $W^{1,p}(\Omega)$ and the Banach space $C^1(\overline{\Omega}).$ The latter function space is an ordered Banach space with
positive cone $C_+=\left\{u\in C^1(\overline{\Omega}):u(z)\geq 0\ \mbox{for\ all}\ z\in \overline{\Omega}\right\}$.This cone has a nonempty interior given by
$$\mbox{int}\,C_+=\{u\in C_+:u(z)>0\ \mbox{for\ all}\ z\in\overline{\Omega}\}.$$

We also use in the following some facts about the spectrum of $(-\Delta_p+\hat{\beta}I, W^{1,p}(\Omega))$ with $\hat{\beta}\in L^q(\Omega), 1< q\leq\infty.$ So, we consider the following nonlinear Neumann eigenvalue
\begin{eqnarray}\label{eq2}
	-\Delta_p u(z)+\hat{\beta}(z)|u(z)|^{p-2}u(z)=\lambda|u(z)|^{p-2}u(z)\ \mbox{in}\ \Omega, \qquad\frac{\partial u}{\partial n}=0\ \mbox{on}\ \partial\Omega.
\end{eqnarray}
This eigenvalue problem was studied recently by Mugnai \& Papageorgiou \cite{28}. Among other qualitative properties, they proved that if $\hat{\beta}\in L^q(\Omega)$ with $q>Np'$ $(\frac{1}{p}+\frac{1}{p'}=1)$, then problem (\ref{eq2}) has a smallest eigenvalue $\hat{\lambda}_1(p,\hat{\beta})$ which is simple, isolated and admits the following characterization
\begin{eqnarray}\label{eq3}
	\hat{\lambda}_1(p,\hat{\beta})=\inf\,\left[\frac{{\mathcal E}(u)}{||u||^{p}_{p}}:u\in W^{1,p}(\Omega),u\neq 0\right],
\end{eqnarray}
where ${\mathcal E}(u)=||Du||^{p}_{p}+\int_{\Omega}\beta(z)|u(z)|^p\ dz$ for all $u\in W^{1,p}(\Omega)$. The infimum in (\ref{eq3}) is realized on the one-dimensional eigenspace corresponding to $\hat{\lambda}_1(p,\hat{\beta})$. From (\ref{eq3}) it is clear that the elements of this eigenspace have constant sign. By $\hat{u}_1(p,\hat{\beta})\in W^{1,p}(\Omega)$ we denote the positive $L^p$-normalized (that is, $||\hat{u}_1(p,\hat{\beta})||_p=1$) eigenfunction corresponding to $\hat{\lambda}_1(p,\hat{\beta})$. The interior regularity theory implies that $\hat{u}_1(p,\hat{\beta})\in C^{1,\alpha}(\Omega)$ with $\alpha\in(0,1)$. If $\hat{\beta}\in L^{\infty}(\Omega)$, then $\hat{u}_1(p,\hat{\beta})\in \mbox{int}\, C_+$ (see \cite{28}).

Let $\eta\in C^1(0,\infty)$ and assume that
\begin{eqnarray}\label{eq4}
	&&0<\hat{c}\leq\frac{t\eta'(t)}{\eta(t)}\leq c_0\quad \mbox{for\ all}\ t>0,\nonumber\\
	&&c_1t^{p-1}\leq\eta(t)\leq c_2(1+t^{p-1})\quad \mbox{for\ all}\ t>0,\ \mbox{with}\ c_1,\,c_2>0.
\end{eqnarray}
The hypotheses on the map $a(\cdot)$ are the following:
\begin{eqnarray}
	H(a)_1:&&a(y)=a_0(||y||)y\ \mbox{for\ all}\ y\in \mathbb R^N\ \mbox{with}\ a_0(t)>0\ \mbox{for\ all}\ t>0\ \mbox{and}\nonumber\\
	(i)&&a_0\in C^1(0,\infty),\ t\longmapsto ta_0(t)\ \mbox{is strictly increasing},\ ta_0(t)\rightarrow 0\ \mbox{as}\ t\rightarrow 0^+\ \mbox{and}\ \lim_{t\rightarrow 0^+}\frac{ta^{'}_{0}(t)}{a_0(t)}>-1;\nonumber\\
	(ii)&&||\nabla a(y)||\leq c_3\frac{\eta(||y||)}{||y||}\ \mbox{for\ all}\ y\in \mathbb R^N\backslash\{0\}\ \mbox{and\ some}\ c_3>0;\nonumber\\
	(iii)&&\frac{\eta(||y||)}{||y||}||\xi||^2\leq(\nabla a(y)\xi,\xi)_{\mathbb R^N}\ \mbox{for\ all}\ y\in \mathbb R^N\backslash\{0\}\ and\ all\ \xi\in\mathbb R^N;\nonumber\\
	(iv)&&\mbox{if}\ G_0(t)=\int^{t}_{0}s a_0(s)ds\ (t>0),\ \mbox{then\ there\ exists}\ \tau\in(1,p)\ \mbox{such\ that}\nonumber\\
	&&t\longmapsto G_0(t^{1/ \tau})\ \mbox{is\ convex\ on}\ (0,+\infty),\ \lim_{t\rightarrow 0^+}\frac{\tau G_0(t)}{t^{\tau}}<+\infty\ \mbox{and}\nonumber\\
	&&t^2 a_0(t)-\tau G_0(t)\geq \tilde{c}\ t^p\ \mbox{for\ all}\ t>0\ \mbox{and\ some}\ \tilde{c}>0.\nonumber
\end{eqnarray}

\medskip
\begin{remark}
	Let $G(y)=G_0(||y||),\ y\in\mathbb R^N.$ Then for all $y\in \mathbb R^N\backslash\{0\}$, we have
	$$\nabla G(y)=G^{'}_{0}(||y||)\,\frac{y}{||y||}=a_0(||y||)y=a(y)\,.$$
\end{remark}

Hence, $G(\cdot)$ is primitive of $a(\cdot)$. Hypotheses $H(a)_1$ have some interesting consequences, which we present below. We first observe that $a(\cdot)$ is strictly monotone. Indeed, for all $y,\,y'\in\mathbb R^N$
\begin{eqnarray}
(a(y)-a(y'),y-y')_{\mathbb R^N}&&=\int^{1}_{0}\left(\frac{d}{dt}a(y'+t(y-y')),y-y'\right)_{\mathbb R^N}dt\nonumber \\
&&=\int^{1}_{0}\left(\nabla a(y'+t(y-y'))(y-y'),y-y'\right)_{\mathbb R^N}dt\nonumber\\
&&\geq c_1||y'+t(y-y')||^{p-2}||y-y'||^2\nonumber\\
&&(\mbox{see}\ H(a)_1(iii)\ \mbox{and}\ (\ref{eq4})).\nonumber
\end{eqnarray}
It follows that the primitives $G(\cdot),\, G_0(\cdot)$ are strictly convex functions and $G_0(\cdot)$ is strictly increasing, too.
In addition, we have for\ all $y\in\mathbb R^N$ and\ some $c_4>0$
\begin{eqnarray}\label{eq5}
 a(y)=\int^{1}_{0}\frac{d}{dt}a(ty)dt=\int^{1}_{0}\nabla a(ty)y\ dt
 \Rightarrow||a(y)||\leq\int^{1}_{0}||\nabla a(ty)||\ ||y||\ dt
 \leq c_4(1+||y||^{p-1})\,.
\end{eqnarray}
Moreover, using $H(a)_1(iii)$ and \eqref{eq4}, we obtain for all $y\in\mathbb R^N$
\begin{equation}\label{eq6}
 (a(y),y)=\int^{1}_{0}(\frac{d}{dt}a(ty),y)_{\mathbb R^N}dt
=\int^{1}_{0}(\nabla a(ty)y,y)_{\mathbb R^N}dt
\geq\frac{c_1}{p-1}||y||^p\,.
\end{equation}

Since $\nabla G(y)=a(y)$ for all $y\in\mathbb R^N$ (recall\ $a(0)=0,\ G(0)=0$), we have
$$G(y)=\int^{1}_{0}\frac{d}{dt}G(ty)dt=\int^{1}_{0}(a(ty),y)_{\mathbb R^N}dt.$$
Then using (\ref{eq5}) and (\ref{eq6}), we have for\ all\ $y\in\mathbb R^N$ and\ some\ $c_5>0$
\begin{eqnarray}\label{eq7}
	\frac{c_1}{p(p-1)}||y||^p\leq G(y)\leq c_5(1+||y||^p)\,.
\end{eqnarray}
Hypotheses $H(a)_1$ are general enough to incorporate in our framework differential operators of interest.

\begin{exam}
	The\ following\ maps\  satisfy\ hypotheses\ H(a):\\
		(i)\ $a(y)=||y||^{p-2}y$\ for\ all\ $y\in\mathbb R^N$\ with\ $1<p<\infty$;
(ii)\ $a(y)=||y||^{p-2}y+\mu||y||^{q-2}y$\ with\ $1<q<p,\ \mu\geq 0$; (iii)\ $a(y)=(1+||y||^2)^{(p-2)/2}y$ with $1<p<\infty$.
\end{exam}

In case (i) we take $G_0(t)=\frac{t^p}{p}$ for all $t\geq 0$ and the corresponding differential operator is the $p$-Laplacian defined by
$\Delta_p u=\mbox{div}\,(||Du||^{p-2}Du)$\ for\ all\ $u\in W^{1,p}(\Omega).$
For this operator we take
$\eta(t)=(p-1)t^{p-1}$\ for\ all\ $t>0$\
if $1<p\leq 2$ and $\eta (t)=t^{p-1}$ for all $t>0$ if $2<p$
and\ $1<\tau<p$\ (see\ $H(a)_1(iv)$).


In  case (ii) we take $G_0(t)=\frac{t^p}{p}+\frac{\mu t^q}{q}$ for all $t\geq 0$ and the corresponding differential operator is the $(p,q)$-Laplacian defined by
$\Delta_p u+\mu\Delta_q u$\ for\ all\ $u\in W^{1,p}(\Omega).$
For this operator we have for\ all\ $t\geq 0$
\begin{eqnarray}
	&&\eta(t)=(p-1)t^{p-1}+\mu(q-1)t^{q-1}\qquad \mbox{when}\ 1<q<p<2,\nonumber\\
	&&\eta(t)=t^{p-1}+\mu(q-1)t^{q-1}\qquad \mbox{when}\ 1<q<2\leq p\ \nonumber\\
	&&\eta(t)=t^{p-1}+\mu t^{q-1}\qquad \mbox{when}\ 2\leq q<p\,.\nonumber
\end{eqnarray}
Indeed, for\ all\ $y\in\mathbb R^N\backslash\{0\}$
$$
\nabla a(y)=||y||^{p-2}\left(I+(p-2)\frac{y\otimes y}{||y||^2}\right)+
\mu||y||^{q-2}\left(I+(q-2)\frac{y\otimes y}{||y||^2}\right)\, .
$$
First assume that $1<q<p<2$. Then
$$||\nabla a(y)||\leq(p-1)||y||^{p-2}+\mu(q-1)||y||^{q-2}\,.$$
Also, for all $\xi\in \mathbb R^N$, we have
$$(\nabla a(y)\xi,\xi)_{\mathbb R^N}\geq((p-1)||y||^{p-2}+\mu(q-1)||y||^{q-2})\,||\xi||^2\,.$$
Therefore with $\eta(t)=(p-1)t^{p-1}+\mu(q-1)t^{q-1}$ for $t\geq 0$, hypotheses $H(a)_1(ii),(iii)$ are fulfilled.

Next, we assume that $1<q<2\leq p$. Then for\ all\ $y\in\mathbb R^N\backslash\{0\}$ and \ all\ $\xi\in\mathbb R^N$
$$
	||\nabla a(y)||\leq(p-1)||y||^{p-2}+\mu(q-1)||y||^{q-2}
\leq(p-1)\left[||y||^{p-2}+\mu(q-1)||y||^{q-2}\right]$$
$$	(\nabla a(y)\xi,\xi)_{\mathbb R^N}\geq\left(||y||^{p-2}+\mu(q-1)||y||^{q-2}\right)\,||\xi||^2.$$
Therefore with $\eta(t)=t^{p-1}+\mu(q-1)t^{q-1}$ for $t\geq 0$, hypotheses $H(a)_1(ii),(iii)$ are satisfied.

Finally, assume that $2\leq q<p$. Then for\ all\ $y\in\mathbb R^N\backslash\{0\}$ and \ all\ $\xi\in\mathbb R^N$
$$
	||\nabla a(y)||\leq(p-1)||y||^{p-2}+\mu(q-1)||y||^{q-1}\leq
(p-1)\left[||y||^{p-2}+\mu||y||^{q-2}\right]$$
$$	(\nabla a(y)\xi,\xi)_{\mathbb R^N}\geq\left(||y||^{p-2}+\mu||y||^{q-2}\right)\,||\xi||^2.
$$
Therefore with $\eta(t)=t^{p-1}+\mu t^{q-1}$ for $t\geq 0$, hypotheses $H(a)_1(ii),(iii)$ are fulfilled.
Moreover, in hypothesis $H(a)_1(iv)$, we have $\tau=q$.

In  case (iii) we take $G_0(t)=\frac{1}{p}\left[(1+t^2)^{\frac{p}{2}}-1\right]$ for all $t\geq 0$ and the corresponding differential operator is the generalized $p$--mean curvature differential operator
$$(1+||Du||^2)^{\frac{p-2}{2}}Du\qquad \mbox{for\ all}\ u\in W^{1,p}(\Omega).$$
Note that
$$\nabla a(y)=(1+||y||^2)^{\frac{p-4}{2}}\left[(p-2)y\otimes y+(1+||y||^2)I\right]\qquad \mbox{for\ all}\ y\in\mathbb R^N\backslash\{0\}.$$
We first assume that $1<p<2$. Then for\ all\ $y\in\mathbb R^N\backslash\{0\}$ and for \ all\ $\xi\in \mathbb R^N$
$$
||\nabla a(y)||\leq(1+||y||^2)^{\frac{p-4}{2}}[(2-p)||y||^2+1+||y||^2]
\leq c_*(1+(p-1)||y||^2)^{\frac{p-2}{2}}\quad \mbox{for\ some}\ c_*>0$$
$$(\nabla a(y)\xi,\xi)_{\mathbb R^N}\geq(1+||y||^2)^{\frac{p-4}{2}}\left[(1+||y||^2)||\xi||^2+(p-2)||y||^2||\xi||^2\right]
\geq(1+(p-1)||y||^2)^{\frac{p-2}{2}}||\xi||^2.
$$
Therefore with $\eta(t)=(1+(p-1)t^2)^{\frac{p-2}{2}}t$ for $t\geq 0$, hypotheses $H(a)_1 (ii),(iii)$ are satisfied.

Next, assume that $2\leq p$. Then for\ all\ $y\in\mathbb R^N\backslash\{0\}$ and for \ all\ $\xi\in \mathbb R^N$
$$
	||\nabla a(y)||\leq c_*(1+||y||^2)^{\frac{p-2}{2}}$$
$$	(\nabla a(y)\xi,\xi)_{\mathbb R^N}\geq(1+||y||^2)^{\frac{p-2}{2}}||\xi||^2\,.$$
Therefore with $\eta(t)=(1+t^2)^{\frac{p-2}{2}}t$ hypotheses $H(a)_1(ii),(iii)$ are fulfilled.
Moreover, in hypotheses $H(a)_1(iv)$, we have $1<\tau<p$.

\smallskip
Let $A:W^{1,p}(\Omega)\rightarrow W^{1,p}(\Omega)^*$ be the nonlinear map defined by
\begin{eqnarray}\label{eq8}
	\left\langle A(u),y\right\rangle=\int_{\Omega}(a(Du),Dy)_{\mathbb R^N}dz\qquad \mbox{for\ all}\ u,y\in W^{1,p}(\Omega).
\end{eqnarray}
>From Gasinski \& Papageorgiou \cite{15}, we have:
\begin{prop}\label{prop1}
		Assume that hypotheses $H(a)_1(i),(ii),(iii)$ are fulfilled.
Then the map $A:W^{1,p}(\Omega)\rightarrow W^{1,p}(\Omega)^*$ defined by (\ref{eq8}) is bounded (that is, maps bounded sets into bounded sets), continuous,  maximal monotone, and of type $(S)_+$, that is, if $u_n\stackrel{w}{\rightarrow}u\ in\ W^{1,p}(\Omega)$ and $\limsup\limits_{n\rightarrow\infty}\left\langle A(u_n),u_n-u\right\rangle\leq 0$, then $u_n\rightarrow u\ in\ W^{1,p}(\Omega)$.
\end{prop}

Let $f_0:\Omega\times\mathbb R\rightarrow\mathbb R$ be a Carath\'eodory function with subcritical growth in $x\in\mathbb R$, that is,
$$|f_0(z,x)|\leq a(z)(1+|x|^{r-1})\qquad \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\in\mathbb R$$
with $a\in L^{\infty}(\Omega)_+,\ 1<r<p^*=\left\{\begin{array}{rl}
\frac{Np}{N-p}& \mbox{if}\ p<N\\
+\infty& \mbox{if}\ N\leq p\end{array}
\right.$ .

Let $F_0(z,x)=\int^{x}_{0}f_0(z,s)ds$ and let $\varphi_0:W^{1,p}(\Omega)\rightarrow\mathbb R$ be the $C^1$-functional defined by
$$\varphi_0(u)=\int_{\Omega}G(Du(z))dz-\int_{\Omega}F_0(z,u(z))dz\qquad \mbox{for\ all}\ u\in W^{1,p}(\Omega).$$
The following result relates H$\ddot{o}$lder and Sobolev local minimizers of a $C^1$-functional.

\begin{prop}\label{prop2}
 Assume that hypotheses $H(a)_1(i),(ii),(iii)$ hold and $u_0\in W^{1,p}(\Omega)$ is a local $C^1(\bar{\Omega})$-minimizer of $\varphi_0$, that is, there exists $\rho_0>0$ such that
$$\varphi_0(u_0)\leq\varphi_0(u_0+h)\qquad \mbox{for\ all}\ h\in C^1(\bar{\Omega}),\ ||h||_{C^1(\bar{\Omega})}\leq\rho_0.$$ Then $u_0\in C^{1,\beta}(\bar{\Omega})$ with $\beta\in(0,1)$ and $u_0$ is also a local $W^{1,p}(\Omega)$-minimizer of $\varphi_0$, that is, there exists $\rho_1>0$ such that $$\varphi_0(u_0)\leq\varphi_0(u_0+h)\qquad \mbox{for\ all}\ h\in W^{1,p}(\Omega),\ ||h||\leq\rho_1.$$
\end{prop}

\begin{remark}
	In the above result and in the sequel, $||\cdot||$ denotes the norm of the Sobolev space $W^{1,p}(\Omega)$, that is,
	$$||u||=(||u||^{p}_{p}+||Du||^{p}_{p})^{1/p}\qquad \mbox{for\ all}\ u\in W^{1,p}(\Omega).$$
\end{remark}

Proposition \ref{prop2} was first proved for the Dirichlet Sobolev space $H^{1}_{0}(\Omega)$ and for $G(y)=\frac{1}{2}||y||^2$ ($y\in\mathbb R^N$) by Brezis \& Nirenberg \cite{6} and it was extended to the Dirichlet Sobolev space $W^{1,p}_{0}(\Omega)\ (1<p<\infty)$ and for $G(y)=\frac{1}{p}||y||^p$ ($y\in\mathbb R^N$) by Garcia Azorero, Manfredi \& Peral Alonso \cite{13}. Proposition \ref{prop2} can be found in Motreanu \& Papageorgiou \cite{27}.

Another mathematical tool that we will use in the sequel is the Morse theory and in particular critical groups. So, let us recall some basic definitions and facts from that theory.

Let $X$ be a Banach space and $Y_2\subseteq Y_1\subseteq X$. For every integer $k\geq 0$ we denote by $H_k(Y_1,Y_2)$  the $k$th relative singular homology group for the pair $(Y_1,Y_2)$ with integer coefficients. We recall that $H_k(Y_1,Y_2)=0$ for all integers $k<0$.

Given $\varphi\in C^1(X)$ and $c\in \mathbb R$, we introduce the following sets
\begin{eqnarray}
&&\varphi^c=\{x\in X:\varphi(x)\leq c\},\qquad \dot{\varphi}^c=\{x\in X:\varphi(x)<c\},\nonumber \\
&&K_{\varphi}=\{x\in X:\varphi'(x)=0\}\qquad \mbox{and}\qquad K^{c}_{\varphi}=\{x\in K_{\varphi}:\varphi(x)=c\}.\nonumber
\end{eqnarray}
The critical points of $\varphi$ at an isolated critical point $x_0\in X$ with $\varphi(x_0)=c$ (that is, $x_0\in K^{c}_{\varphi}$) are defined by
$$C_k(\varphi,x_0)=H_k(\varphi^c\cap U, \varphi^c\cap U\backslash\{0\})\qquad \mbox{for\ all}\ k\geq 0.$$
Here $U$ is a neighborhood of $x_0\in X$ such that $K_{\varphi}\cap \varphi^{c}\cap U=\{x_0\}$. The excision property of singular homology theory implies that the above definition of critical groups is independent of the choice of the neighborhood $U$.

Suppose that $\varphi\in C^1(X)$ satisfies the $C$-condition and $\inf\,\ \varphi(K_{\varphi})>-\infty$. Let $c<\inf\, \varphi(K_{\varphi})$. Then the critical groups of $\varphi$ at infinity are defined by
$$C_k(\varphi,\infty)=H_k(X,\varphi^c)\qquad \mbox{for\ all}\ k\geq 0.$$
The second deformation theorem (see, for example, Gasinski \& Papageorgiou \cite[p. 628]{14}) implies that the above definition of critical groups of $\varphi$ at infinity is independent of the particular choice of the level $c<\inf\varphi (K_{\varphi}).$

Suppose that $K_{\varphi}$ is finite. We introduce the following polynomials in $t\in\mathbb R$:
\begin{eqnarray}
	&&M(t,x)=\sum_{k\geq 0}\mbox{rank}\ C_k(\varphi,x)t^k\qquad \mbox{for\ all}\ x\in K_{\varphi}\nonumber \\
	&&P(t,\infty)=\sum_{k\geq 0}\mbox{rank}\ C_k(\varphi,\infty)t^k.\nonumber
\end{eqnarray}
The Morse relation says that
\begin{eqnarray}\label{eq9}
\sum_{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 with nonnegative integer coefficients $\beta_k$.

As we already mentioned, by $||\cdot||$ we denote the norm of the Sobolev space $W^{1,p}(\Omega)$. The same notation will also be used to denote the norm of $\mathbb R^N$. However, no confusion is possible, since it will always be clear from the context which norm is used. For $x\in\mathbb R$, we set $x^{\pm}=\max\{\pm x,0\}$ and for $u\in W^{1,p}(\Omega)$ we define $u^{\pm}(\cdot)=u(\cdot)^{\pm}$. We know that
$$u^{\pm}(\cdot)\in W^{1,p}(\Omega),\ u=u^+-u^-\ \mbox{and}\ |u|=u^+=u^-.$$
Given a measurable function $h:\Omega\times\mathbb R\rightarrow\mathbb R$, we introduce the map
$$N_h(u)(\cdot)=h(\cdot,u(\cdot))\qquad \mbox{for\ all}\ u\in W^{1,p}(\Omega)$$
(the Nemytskii map corresponding to $h$). Finally, by $|\cdot|_N$ we denote the Lebesgue measure on $\mathbb R^N$.


\section{Coercive Problems}

In this section, we examine problem (1) under hypotheses on $f(z,x)$ that make the energy functional coercive. We prove a ``three solutions theorem" providing sign information for all the solutions. First we fix the hypotheses on the potential $\beta(\cdot)$:
$$
	\qquad\beta\in L^{\infty}(\Omega).\leqno(H_0)$$
	
Set\ $\hat{\beta}(z)=\frac{p-1}{c_1}\beta(z)$.

We assume that the reaction term $f(z,x)$ is\ a\ Carath\'eodory\ function\ such\ that $f(z,0)=0$\ a.e.\ in\ $\Omega$ and
$$\left\{ \begin{array}{ll}
	&(i)\quad |f(z,x)|\leq a(z)(1+|x|^{p-1})\ \mbox{a.e.\ in}\ \Omega,\ \mbox{for\ all}\ x\in\mathbb R\ \mbox{with}\ a\in L^{\infty}(\Omega)_+;\\
	&(ii)\quad\mbox{there\ exists\ a\ function}\ \vartheta\in L^{\infty}(\Omega),\vartheta(z)\leq\frac{c_1}{p-1}\hat{\lambda_1}(p,\hat{\beta})\ \mbox{a.e.\ in}\ \Omega,
\vartheta\neq\frac{c_1}{p-1}\hat{\lambda_1}(p,\hat{\beta})\ \mbox{and}\\
&\limsup_{x\rightarrow\pm\infty}\ \frac{f(z,x)}{|x|^{p-2}x}\leq\vartheta(z)\ \mbox{uniformly\ for\ a.a.}\ z\in\Omega;\\
	&(iii)\quad\mbox{there\ exist}\ q\in(1,\tau)\ (\mbox{see}\ H(a)_1(iv)),\ \tilde{c_0}>0\ \mbox{and}\ \delta>0\ \mbox{such that}\\
	&\tilde{c_0}|x|^q\leq f(z,x)x\leq qF(z,x)\ \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ 0<|x|\leq\delta;\\
	&(iv)\quad\mbox{for\ every}\ \rho<0,\ \mbox{there\ exists}\ \epsilon_{\rho}>0\ \mbox{such\ that\ for\ a.a.}\ z\in\Omega, \ \mbox{the mapping}\\
	&x\longmapsto f(z,x)+\epsilon_{\rho}|x|^{p-2}x\ \mbox{is\ nondecreasing\ on}\ [-\rho,\rho].\end{array}\right. \leqno(H_1)
$$
\begin{remark}
	Hypothesis $H_1(i)$ implies that asymptotically at $\pm\infty\ f(z,\cdot)$ is $(p-1)$-sublinear. Hypothesis $H_1(ii)$ will make the energy functional coercive. Hypothesis $H_1(iii)$ implies the existence of a concave term near zero. Finally, hypothesis $H_1(iv)$ is weaker than assuming the monotonicity of $f(z.\cdot)$ and is a one sided Lipschitz condition on $f(z,\cdot)$.
\end{remark}

\begin{exam}
	The following functions satisfy hypotheses $H_1$. For the sake of simplicity, we drop the $z$-dependence
	$$f_1(x)=\vartheta|x|^{p-2}x+|x|^{q-2}x\quad \mbox{with}\ \vartheta<\frac{c_1}{p-1}\hat{\lambda_1}(p,\hat{\beta}),\ 1<q<p.$$
\begin{eqnarray}
	f_2(x)=\left\{
		\begin{array}{rl}
		\vartheta(|x|^{q-2}x-|x|^{r-2}x)&\quad \mbox{if}\ |x|\leq 1\\
		\vartheta(|x|^{p-2}x-|x|^{\mu-2}x)&\quad\mbox{if}\ |x|>1,
	\end{array}
		\right.\nonumber
\end{eqnarray}
with\ $\theta<\frac{c_1}{p-1}\hat{\lambda_1}(p,\hat{\beta}),
		1<q,\mu<p,r$.
\end{exam}

The next auxiliary result is useful to establish the coercivity of the energy functional of problem (\ref{eq1}). This property can be found in Mugnai \& Papageorgiou \cite[Lemma 4.11]{28}.
\begin{lem}\label{lem1}
	Assume that $\tilde{\beta}, \tilde{\vartheta}\in L^{\infty}(\Omega)$ and $\tilde{\vartheta}(z)\leq\hat{\lambda_1}(p,\tilde{\beta})$ a.e. in $\Omega,\tilde{\vartheta}\neq\hat{\lambda_1}(p,\tilde{\beta})$.
Then there exists $c_6>0$ such\ that
	$$\tilde{\epsilon}(u)-\int_{\Omega}\tilde{\vartheta}(z)|u(z)|^p dz\geq c_6||u||^p\qquad \mbox{for\ all}\ u\in W^{1,p}(\Omega),$$
	where $\tilde{\epsilon}(u)=||Du||^{p}_{p}+\int_{\Omega}\tilde{\beta}(z)|u(z)|^p dz\ for\ all\ u\in W^{1,p}(\Omega)$.
\end{lem}

Let $\varphi:W^{1,p}(\Omega)\rightarrow\mathbb R$ be the energy functional associated to problem (\ref{eq1}), namely
$$\varphi(u)=\int_{\Omega}G(Du(z))dz-\int_{\Omega}F(z,u(z))dz\qquad \mbox{for\ all}\ u\in W^{1,p}(\Omega).$$
Evidently, $\varphi\in C^1(W^{1,p}(\Omega))$.

Let $\lambda>||\beta^-||_{\infty}$ and consider the following truncations-perturbations of $f(z,\cdot)$:
$$\hat{f_+}(z,x)=f(z,x^+)+\lambda(x^+)^{p-1}\qquad \mbox{and}\qquad \hat{f_-}(z,x)=f(z,-x^-)-\lambda(x^-)^{p-1}.$$
We set $\hat{F_{\pm}}(z,x)=\int^{x}_{0}\hat{f_{\pm}}(z,s)dz$ and consider the $C^1$-functional $\hat{\varphi_{\pm}}:W^{1,p}(\Omega)\rightarrow\mathbb R$ defined by
$$
	\hat{\varphi_{\pm}}(u)=\int_{\Omega}G(Du(z))dz+\frac{1}{p}\int_{\Omega}(\beta(z)+\lambda)|u(z)|^p\ dz-
\int_{\Omega}\hat{F_{\pm}}(z,u(z))dz\qquad \mbox{for\ all}\ u\in W^{1,p}(\Omega).$$

\begin{prop}\label{prop3}
	Assume that hypotheses $H(a)_1,\ H_0$ and $H_1$ are fulfilled.
Then problem (\ref{eq1}) has at least two nontrivial constant sign solutions
	$u_0\in{\rm int}\,C_+$\ and\ $v_0\in-{\rm int}\,C_+$,
	both local minimizers of the energy functional $\varphi$.
\end{prop}

{\it Proof.}
	By virtue of hypotheses $H_1(i),(ii)$, given $\epsilon>0$, we can find $c_+=c_+(\epsilon)>0$ such that
	\begin{eqnarray}\label{eq10}
		F(z,x)\leq\frac{1}{p}(\vartheta(z)+\epsilon)|x|^p+c_7\qquad \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\in\mathbb R.
	\end{eqnarray}
Thus, for all $u\in W^{1,p}(\Omega)$
\begin{equation}\label{eq11}\begin{array}{ll}
\di	\hat{\varphi_+}(u)&\di=\int_{\Omega}G(Du)dz+\frac{1}{p}\int_{\Omega}(\beta(z)+\lambda)|u|^p dz+\int_{\Omega}\hat{F_{\pm}}(z,u)dz\\
	&\di\geq\frac{c_1}{p(p-1)}||Du||^{p}_{p}+\frac{1}{p}\int_{\Omega}\beta(z)|u|^p dz-\frac{1}{p}\int_{\Omega}(\vartheta(z)+\epsilon)|u|^p dz+c_7|\Omega|_N\qquad (\mbox{see}\ \cite{9})\\
	&\di=\frac{c_1}{p(p-1)}\left[||Du||^{p}_{p}+\int_{\Omega}\hat{\beta}(z)|u|^p dz+\frac{p-1}{c_1}\int_{\Omega}\vartheta(z)|u|^p dz\right]\\
	&\di-\frac{\epsilon}{p}||u||^p+c_7|\Omega|_N\nonumber\\
	&\di\geq\frac{1}{p}\left[\frac{c_1 c_6}{p-1}-\epsilon\right]||u||^p+c_7|\Omega|_N\qquad (\mbox{see\ Lemma}\ \ref{lem1}).\end{array}
\end{equation}
Choosing $\epsilon\in\left(0,\frac{c_1 c_6}{p-1}\right)$, we deduce from (\ref{eq11})  that $\hat{\varphi_+}$ is coercive. Also, via the Sobolev embedding theorem, we see that $\hat{\varphi_+}$ is sequentially weakly lower semi-continuous. So, by the Weierstrass theorem, we can find $u_0\in W^{1,p}(\Omega)$ such\ that
\begin{eqnarray}\label{eq12}
	\hat{\varphi_+}(u_0)=\inf\,\left[\hat{\varphi_+}(u):u\in W^{1,p}(\Omega)\right]=\hat{m_+}\,.
\end{eqnarray}
By virtue of hypothesis $H(a)_1(iv)$, we can find $c_8>0$ and $\delta_0\in\left(0,\delta\right]$ such that
\begin{eqnarray}\label{eq13}
G(y)\leq\frac{c_8}{\tau}||y||^{\tau}\qquad \mbox{for\ all}\ ||y||\leq\delta_0.
\end{eqnarray}
Hypothesis $H_1(iii)$ yields
\begin{eqnarray}\label{eq14}
	\frac{\tilde{c_0}}{q}|x|^q\leq F(z,x)\qquad \mbox{for\ all}\ z\in\Omega,\ \mbox{all}\ |x|\leq\delta.
\end{eqnarray}
Recall that $\hat{u_1}(p,\beta)\in\mbox{int}\,C_+$. So, we can find $\eta\in(0,1)$ small such\ that
\begin{eqnarray}\label{eq15}
	\eta|\hat{u_1}(p,\beta)(z)|,\ \eta||D\hat{u_1}(p,\beta)(z)||\in\left(0,\delta_0\right]\qquad \mbox{for\ all}\ z\in\bar{\Omega}.
\end{eqnarray}
Therefore
\begin{equation}\label{eq16}\begin{array}{ll}
\hat{\varphi_+}(\eta\ \hat{u}_1(p,\beta))&\di=\int_{\Omega}G(\eta D\hat{u}_1(p,\beta))dz+\frac{\eta^p}{p}\int_{\Omega}\beta(z)|\hat{u}_1(p,\beta)|^p dz-
\int_{\Omega}F(z,\eta\ \hat{u_1}(p,\beta))dz\\
&\di\leq\frac{c_8}{\tau}\eta^{\tau}||D\hat{u}_1(p,\beta)||^{\tau}_{\tau}+
\frac{\eta^p}{p}||\beta||_{\infty}-\frac{\tilde{c_0}}{q}\eta^q||\hat{u_1}(p,\beta)||^{q}_{q}\\
&\di(\mbox{see}\ (\ref{eq13}),\ (\ref{eq14}),\ (\ref{eq15})\ \mbox{and\ recall}\ ||\hat{u_1}(p,\beta)||_p=1).
\end{array}\end{equation}
Since $1<q<\tau<p$, by choosing $\eta\in(0,1)$ even smaller if necessary, relation (\ref{eq16}) yields
$$
\hat{\varphi_+}(\eta\ \hat{u_1}(p,\beta))<0
\Rightarrow\hat{\varphi}_+(u_0)=\hat{m}_+<0=\hat{\varphi}_+(0)\ (\mbox{see}\ (\ref{eq12})),\ \mbox{hence}\ u_0\neq 0.
$$
>From (\ref{eq12}) we have
\begin{equation}\label{eq17}
	\hat{\varphi}^{'}_{+}(u_0)=0
	\Rightarrow A(u_0)+(\beta+\lambda)|u_0|^{p-2}u_0=N_{\hat{f_+}}(u_0).
\end{equation}
On (\ref{eq17}) we act with $-u_0^-\in W^{1,p}(\Omega)$ and using (\ref{eq6}) we obtain
$$\frac{c_1}{p-1}||Du^{-}_{0}||^{p}_{p}+\int_{\Omega}(\beta(z)+\lambda)u^{-}_{0}(z)^p dz\leq 0.$$
Since $\lambda>||\beta^-||_{\infty}$, we infer that $u_0\geq 0,\ u_0\neq 0$. Therefore relation (\ref{eq17}) becomes
\begin{eqnarray}
	&&A(u_0)+\beta u^{p-1}_{0}=N_f(u_0)
	\Rightarrow -\mbox{div}\, a(Du_0(z))+\beta(z)u_0(z)^{p-1}=f(z,u_0(z))\qquad \mbox{a.e.\ in}\ \Omega,\nonumber\\
	&&\frac{\partial u_0}{\partial n}=0\ on\ \partial\Omega\quad (\mbox{see}\ \cite{2}).\nonumber
\end{eqnarray}
>From Hu \& Papageorgiou \cite{17}, we know that $u_0\in L^{\infty}(\Omega)$ and so we can apply the regularity result of Lieberman \cite[p. 320]{21} and deduce that $u_0\in C_+\backslash\{0\}$. Let $\rho=||u_0||_{\infty}$ and let $\epsilon_{\rho}>0$ be as postulated by hypothesis $H_1(iv)$. Then
\begin{equation}\label{eq18}
	-\mbox{div}\, a(Du_0(z))+\epsilon_{\rho}(u_0)(z)^{p-1}
=f(z,u_0(z))+\epsilon_{\rho}u_0(z)^{p-1}\geq 0\qquad \mbox{a.e.\ in}\ \Omega.
\end{equation}
Let $\gamma_0(t)=ta_0(t)$. Hypothesis $H(a)_1(iii)$ implies the one-dimensional estimate
\begin{eqnarray}
&&t\gamma^{'}_{0}(t)=t^2a^{'}_{0}(t)+ta_0(t)\geq c_9 t^{p-1}\quad \mbox{for\ all}\ t>0,\ \mbox{some}\ c_9>0\nonumber\\
\Rightarrow&&\int^{t}_{0}s\gamma^{'}_{0}(s)ds=t\gamma_0(t)-\int^{t}_{0}\gamma_0(s)ds
=t^2 a_0(t)-G_0(t)\ \geq\frac{c_9}{p}t^p\quad \mbox{for\ all}\ t>0.\nonumber
\end{eqnarray}
This estimate and (\ref{eq18}) permit the use of the strong maximum principle of Pucci \& Serrin \cite[p. 111]{33} and so we have $u_0(z)>0$ for all $z\in\Omega$. Finally, we  apply the boundary point theorem of Pucci \& Serrin \cite[p. 120]{33} and conclude that $u_0\in\mbox{int}\,C_+$. Note that $\hat{\varphi}_+\left|_{C_+}=\varphi\right|_{C_+}$. So, $u_0\in\mbox{int}\,C_+$ is a local $C^1(\bar{\Omega})$-minimizer of $\varphi$. Invoking Proposition \ref{prop2} we conclude that $u_0\in\mbox{int}\, C_+$ is a local $W^{1,p}(\Omega)$ -minimizer of $\varphi$.

Similarly, working this time with $\hat{\varphi}_-$ we produce a nontrivial negative solution $v_0\in-\mbox{int}\,C_+$ of problem (\ref{eq1}), which is a local minimizer of $\varphi$.
\qed

In fact, we can show the existence of extremal nontrivial constant sign solutions for problem (\ref{eq1}). Namely, we show that there exists a smallest nontrivial positive solution and a biggest nontrivial negative solution. Our argument follows closely the reasoning of Papageorgiou \& R\u adulescu \cite{31}, where the authors deal with Dirichlet $(p,q)$-equations. For the convenience of the reader, we present the proofs in detail.

Note that hypotheses $H_1(i),(iii)$ imply that we can find $c_{10}>||\beta||_{\infty}$ and $\lambda$ such\ that
\begin{eqnarray}\label{eq19}
	f(z,x)x\geq\tilde{c}|x|^q-c_{10}|x|^p\qquad \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\in\mathbb R.
\end{eqnarray}
This unilateral growth condition on $f(z,\cdot)$ leads to the following auxiliary Neumann problem:
\begin{eqnarray}\label{eq20}
	-\mbox{div}\, a(Du(z))=\tilde{c}|u(z)|^{q-2}u(z)-c_{10}|u(z)|^{p-2}u(z)\ \ \mbox{in}\ \Omega,\ \frac{\partial u}{\partial n}=0\ \ \mbox{on}\ \partial\Omega\,.
\end{eqnarray}

\begin{prop}\label{prop4}
Assume that hypotheses $H(a)_1$ hold.
Then problem (\ref{eq20}) has a unique nontrivial positive solution $\tilde{u}\in {\rm int}\,C_+$ and
	since (\ref{eq20}) is add $\tilde{v}=-\tilde{u}\in- {\rm int}\,C_+$ is unique nontrivial negative solution of (\ref{eq20}).
\end{prop}

{\it Proof.}
	First we show that problem (\ref{eq20}) admits a nontrivial positive solution. To this end, let $\Psi_+:W^{1,p}(\Omega)\rightarrow\mathbb R$ be the $C^1$-functional defined by
	$$
		\Psi_+(u)=\int_{\Omega}G(Du(z))dz+\frac{1}{p}\int_{\Omega}(\beta(z)+\lambda)|u(z)|^p dz
-\frac{\tilde{c}}{q}||u^+||^{q}_{q}+\frac{c_{10}-\lambda}{p}||u^+||^{p}_{p}\qquad \mbox{for\ all}\ u\in W^{1,p}(\Omega).$$
	Recall that $\lambda>||\beta^-||_{\infty}$. From this fact and since $q<p$, we infer that $\hat{\Psi}_+$ is coercive. Also, it is sequentially weakly lower semi-continuous. Therefore, we can find $\tilde{u}\in W^{1,p}(\Omega)$ such\ that
	$$\Psi_+(\tilde{u})=\inf\,\left[\Psi_+(u):u\in W^{1,p}(\Omega)\right]\,.$$
	As before (see the proof of Proposition \ref{prop3}), using hypothesis $H(a)_1(iv)$, we show that
	$\Psi_+(\tilde{u})<0=\Psi_+(0)$, hence\ $\tilde{u}\neq 0$.
	Also, we have
	\begin{equation}\label{eq21}
		\Psi^{'}_{+}(\tilde{u})=0 \Rightarrow A(\tilde{u})+(\beta+\lambda)|\tilde{u}|^{p-2}\tilde{u}=\tilde{c}(\tilde{u}^+)^{q-1}-(c_{10}-\lambda)(\tilde{u}^+)^{p-1}.
	\end{equation}
	On (\ref{eq21}) we act with $-\tilde{u}^-\in W^{1,p}(\Omega)$. Since $\lambda>||\beta^-||_{\infty}$, we see that $\tilde{u}\geq0,\ \tilde{u}\neq 0$. Hence (\ref{eq21}) becomes
	$$A(\tilde{u})+\beta\tilde{u}^{p-1}=\tilde{c}\tilde{u}^{q-1}-c_{10}\tilde{u}^{p-1},$$ which shows that
$\tilde{u}$\ is\ a\ nontrivial\ positive\ solution\ of\ problem (\ref{eq20}).
	Moreover, as before using the nonlinear regularity theory we obtain $\tilde{u}\in C_+\backslash\{0\}$. Also, we have
	$$\mbox{div}\, a(D\tilde{u}(z))\leq(||\beta||_{\infty}+c_{10})\tilde{u}(z)^{p-1}\qquad \mbox{a.e.\ in}\ \Omega,$$
hence
	$\tilde{u}\in\mbox{int}\,C_+$ (see\ Pucci \& Serrin\ \cite[p. 120]{33}).

	Next, we show the uniqueness of this positive solution. For this purpose, we consider the integral functional $\sigma_+:L^1(\Omega)\rightarrow\bar{\mathbb R}=\mathbb R\cup\{+\infty\}$ defined by
	\begin{eqnarray}
		\sigma_+(u)=\left\{
			\begin{array}{cl}
				\int_{\Omega}G(Du^{1/\tau})dz&\quad \mbox{if}\ u\geq 0,\ u^{1/\tau}\in W^{1,p}(\Omega)\nonumber\\
				+\infty&\quad \mbox{otherwise}.\nonumber
			\end{array}
			\right.
	\end{eqnarray}
	Let $u_1,u_2\in\mbox{dom}\,\sigma_+$ and let $y=(tu_1+(1-t)u_2)^{1/\tau}$ with $t\in[0,1]$. From Diaz \& Saa \cite{8} (Lemma 1), we have
	$$||Dy(z)||\leq\left(t||Du_1(z)^{1/\tau}||^{\tau}+(1-t)||Du_2(z)^{1/\tau}||^{\tau}\right)^{1/\tau}.$$
	Recall that $G_0(\cdot)$ is increasing. Hence
	$$\begin{array}{ll}
			G_0(||Dy(z)||)&\di\leq G_0\left(\left(t||Du_1(z)^{1/\tau}||^{\tau}+(1-t)||Du_2(z)^{1/\tau}||^{\tau}\right)^{1/\tau}\right)\\
			&\di\leq tG_0\left(||Du_1(z)^{1/\tau}||\right)+(1-t)G_0\left(||Du_2(z)^{1/\tau}||\right)\quad \mbox{a.e.\ in}\ \Omega\\
			&(\mbox{see\ hypothesis}\ H(a)_1(iv))\\
			\Rightarrow G(Dy(z))&\di\leq tG(Du_1(z)^{1/\tau})+(1-t)G(Du_2(z)^{1/\tau})\quad \mbox{a.e.\ in}\ \Omega
\Rightarrow \sigma_+\ \mbox{is\ convex}.\end{array}
	$$
	Moreover, via Fatou's lemma, we see that $\sigma_+$ is lower semi-continuous.
	
	Suppose that $u,v\in W^{1,p}(\Omega)$ are two nontrivial positive solutions of (\ref{eq20}). From the first part of the proof, we have $u,v\in\mbox{int}\,C_+$. Therefore $u^{\tau},v^{\tau}\in\mbox{dom}\ \sigma_+$. Let $h\in C^1(\bar{\Omega})$. Then for $t\in[-1,1]$ with $|t|$ small, we have $u^{\tau}+th,\ v^{\tau}+th\in\mbox{dom}\ \sigma_+$ and so the G\^ateaux derivatives of $\sigma_+$ at $u^{\tau}$ and at $v^{\tau}$ in the direction $h$ do not exist. Moreover, via the chain rule, we have
	$$
		\sigma^{'}_{+}(u^{\tau})(h)=\frac{1}{\tau}\int_{\Omega}\frac{-\mbox{div}\, a(Du)}{u^{\tau-1}}h\ dz\qquad
		\sigma^{'}_{+}(v^{\tau})(h)=\frac{1}{\tau}\int_{\Omega}\frac{-\mbox{div}\, a(Dv)}{v^{\tau-1}}h\ dz\,.
	$$
	The convexity of $\sigma_+$ implies the monotonicity of $\sigma^{1}_{+}$. Therefore
	\begin{equation}\label{eq22}\begin{array}{ll}
	0&\di\leq\int_{\Omega}\left(\frac{-\mbox{div}\, a(Du)}{u^{\tau-1}}+\frac{\mbox{div}\, a(Dv)}{v^{\tau-1}}\right)(u^{\tau}-v^{\tau})dz\\ &=\di\int_{\Omega}\left(\frac{\tilde{c}u^{q-1}-c_{10}u^{p-1}}{u^{\tau-1}}-\frac{\tilde{c}v^{q-1}-
c_{10}v^{p-1}}{v^{\tau-1}}\right)(u^{\tau}-v^{\tau})dz\\ &=\di\int_{\Omega}\left[\tilde{c}\left(\frac{1}{u^{\tau-q}}-\frac{1}{v^{\tau-q}}\right)-c_{10}(u^{p-\tau}-
v^{p-\tau})\right](u^{\tau}-v^{\tau})dz.\end{array}
	\end{equation}
	Since $q<\tau<p$, the function $x\longmapsto\frac{\tilde{c}}{x^{\tau-q}}-c_{10}x^{p-\tau}$ is strictly decreasing on $(0,+\infty)$. So, from (\ref{eq22}) it follows that $u=v$ and this proves the uniqueness of the nontrivial positive solution $\tilde{u}\in\mbox{int}\,C_+$. Evidently $\tilde{v}=-\tilde{u}\in-\mbox{int}\,C_+$ is the unique nontrivial negative solution of problem (\ref{eq20}).
\qed

Using these unique constant sign solutions of (\ref{eq20}), we can generate extremal constant sign solutions of (\ref{eq1}).

\begin{prop}\label{prop5}
	Assume that hypotheses $H(a)_1$ and $H_1$ hold.
Then problem (\ref{eq1}) has a smallest nontrivial positive solution $u_*\in{\rm int}\,C_+$ and a
	biggest nontrivial negative solution $v_*\in-{\rm int}\,C_+$.
\end{prop}

{\it Proof.}
	Let $S_+$ be the set of nontrivial positive solutions of (\ref{eq1}). From Proposition \ref{prop3} we know that $S_+\neq \O$ and $S_+\subseteq{\rm int}\,C_+$. Moreover, as in Aizicovici, Papageorgiou \& Staicu \cite{1}, we have that $S_+$ is downward directed, that is, if $u_1,u_2\in S_+$, then we can find $u\in S_+$ such\ that $u\leq u_1,\ u\leq u_2$. So, without any loss of generality, we may assume that there exists $M>0$ such\ that
	\begin{eqnarray}\label{eq23}
		||u||_{\infty}\leq M\qquad \mbox{for\ all}\ u\in S_+\,.
	\end{eqnarray}

\textbf{Claim:} $\tilde{u}\leq u$ for all $u\in S_+$.

Let $u\in S_+$ and consider the Carath\'eodory function
\begin{eqnarray}\label{eq24}
	h_+(z,x)=\left\{
		\begin{array}{cl}
			0&\mbox{if}\ x<0\\
			\tilde{c}x^{q-1}-(c_{10}-\lambda)x^{p-1}&\mbox{if}\ 0\leq x\leq u(z)\\
			\tilde{c}u(z)^{q-1}-(c_{10}-\lambda)u(z)^{p-1}&\mbox{if}\ u(z)<x.
		\end{array}
		\right.
\end{eqnarray}
As before, $\lambda>||\beta^-||_{\infty}$. We set $H_+(z,x)=\int^{x}_{0}h_+(z,s)ds$ and consider the $C^1$-functional $\gamma_+:W^{1,p}(\Omega)\rightarrow\mathbb R$ defined by
$$
	\gamma_+(u)=\int_{\Omega}G(Du(z))dz+\frac{1}{p}\int_{\Omega}\left[\beta(z)+\lambda\right]|u(z)|^p dz
-\int_{\Omega}H_+(z_1u(z))dz\qquad \mbox{for\ all}\ u\in W^{1,p}(\Omega).
$$
Relation (\ref{eq24}) implies that $\gamma_+$ is coercive. Also, it is sequentially weakly lower semi-continuous. So, we can find $\tilde{u}_0\in W^{1,p}(\Omega)$ such that
\begin{eqnarray}\label{eq25}
	\gamma_+(\tilde{u}_0)=\inf\,\left[\gamma_+(u):u\in W^{1,p}(\Omega)\right]\,.
\end{eqnarray}
As before (see Proposition \ref{prop4}) and since $u\in\mbox{int}\,C_+$, we have
$\gamma_+(\tilde{u}_0)<0=\gamma_+(0)$, hence\ $\tilde{u}_0\neq 0$.
>From (\ref{eq20}) we have
\begin{eqnarray}\label{eq26}
\gamma^{'}_{+}(\tilde{u}_0)=0
	\Rightarrow A(\tilde{u}_0)+(\beta+\lambda)|\tilde{u}_0|^{p-2}\tilde{u}_0=N_{h_+}(\tilde{u}_0).
\end{eqnarray}
On (\ref{eq26})  we first act with $-\tilde{u}^{-}_{0}\in W^{1,p}(\Omega)$. Since $\lambda>||\beta^{-}||_{\infty}$, we obtain $\tilde{u}_0\geq 0,\ \tilde{u}_0\neq 0$. Then on (\ref{eq26}) we act with $(\tilde{u}_0-u)^+\in W^{1,p}(\Omega)$. We have
\begin{eqnarray}
	&&\left\langle A(\tilde{u}_0),(\tilde{u}_0-u)^+\right\rangle+\int_{\Omega}(\beta(z)+\lambda){\tilde{u}_0}^{p-1}(\tilde{u}_0-u)^+dz\nonumber\\
	=&&\int_{\Omega}h_+(z,\tilde{u}_0)(\tilde{u}_0-u)^+dz\nonumber\\
	=&&\int_{\Omega}\left[\tilde{c}u^{q-1}-c_{10}u^{p-1}\right](\tilde{u}_0-u)^+dz\ (\mbox{see}\ (\ref{eq24}))\nonumber\\
	\leq&&\int_{\Omega}f(z,u)(\tilde{u}_0-u)^+dz\ (\mbox{see}\ (\ref{eq19}))\nonumber\\
	=&&\left\langle A(u),(\tilde{u}_0-u)^+\right\rangle+\int_{\Omega}(\beta(z)+\lambda)u^{p-1}(\tilde{u}_0-u)^+dz,\nonumber\\
	\Rightarrow&&\left\langle A(\tilde{u}_0)-A(u),(\tilde{u}_0-u)^+\right\rangle+\int_{\Omega}(\beta(z)+\lambda)(\tilde{u}^{p-1}_{0}-u^{p-1})(\tilde{u}_0-u)^+dz\leq 0,\nonumber\\
	\Rightarrow&&\left|\{\tilde{u}_0>u\}\right|_N=0\quad (\mbox{since}\ \lambda>||\beta^-||_{\infty}),\nonumber
\end{eqnarray}
 hence\ $\tilde{u}_0\leq u$.
So, we have proved that
$$\tilde{u}_0\in[0,u]=\left\{v\in W^{1,p}(\Omega):0\leq v(z)\leq u(z)\ a.e.\ in\ \Omega\right\},\qquad \tilde{u}_0\neq 0.$$
Then relation (\ref{eq26}) becomes
\begin{eqnarray}
	&&A(\tilde{u}_0)+\beta\tilde{u}^{p-1}_{0}=\tilde{c}\tilde{u}^{q-1}_{0}-c_{10}\tilde{u}^{p-1}_{0}\ (\mbox{see}\ (\ref{eq24})),\nonumber\\
	\Rightarrow&&\tilde{u}_0\ \mbox{is\ a\ nontrivial\ positive\ solution\ of}\ (\ref{eq20}),\nonumber\\
	\Rightarrow&&\tilde{u}_0=\tilde{u}\in\mbox{int}\,C_+\ (\mbox{see\ Proposition}\ \ref{prop4}).\nonumber
\end{eqnarray}
Therefore $\tilde{u}\leq u$ for all $u\in S_+$ and this proves the Claim.

Now, let $C\subseteq S_+$ be a chain (that is, a totally ordered subset of $S_+$). We know that we can find $\{u_n\}_{n\geq 1}\subseteq C$ such\ that $\mbox{int}\,C=\inf\limits_{n\geq 1}u_n$ (see Dunford \& Schwartz \cite[p. 336]{10}). We have
\begin{eqnarray}\label{eq27}
	A(u_n)+\beta u_{n}^{p-1}=N_f(u_n),\ u_*\leq u_n\leq M\quad \mbox{for\ all}\ n\geq 1\ (\mbox{see}\ (\ref{eq23})).
\end{eqnarray}
So, $\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)$ is bounded and we may assume that
\begin{eqnarray}\label{eq28}
	u_n\stackrel{w}{\rightarrow}u\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{and}\ u_n\rightarrow u\ \mbox{in}\ L^p(\Omega)\ \mbox{as}\ n\rightarrow\infty.
\end{eqnarray}
On (\ref{eq27}) we act with $u_n-u\in W^{1,p}(\Omega)$, pass to the limit as $n\rightarrow\infty$ and use (\ref{eq28}). Then
\begin{eqnarray}\label{eq29}
	\lim_{n\rightarrow\infty}\left\langle A(u_n),u_n-u\right\rangle=0
	\Rightarrow u_n\rightarrow u\ in\ W^{1,p}(\Omega)\quad (\mbox{see\ Proposition}\ \ref{prop1}).
\end{eqnarray}
Passing to the limit as $n\rightarrow\infty$ in (\ref{eq27}) and using (\ref{eq29}), we obtain
$$
	A(u)+\beta u^{p-1}=N_f(u),\ \tilde{u}\leq u\leq M
	\Rightarrow u\in S_+,\ u=\inf\,C.
$$
Since $C$ is an arbitrary chain of $S_+$, from the Kuratowski-Zorn lemma, we infer that we can find $u_*\in S_+$ a minimal element. Since $S_+$ is downward directed, we conclude that $u_*\in\mbox{int}\,C_+$ is the smallest nontrivial positive solution of (\ref{eq1}).

Similarly, let $S_-$ be the set of nontrivial negative solutions of (\ref{eq1}). From Proposition \ref{prop3} we know that $S_-\neq \O$ and $S_-\subseteq-\mbox{int}\,C_+$. Also, $S_-$ is upward directed, that is, if $v_1,v_2\in S_-$, then we can find $v\in S_-$ such\ that $v_1\leq v_1,\ v_2\leq v$ (see \cite{1}). Reasoning as above, via the Kuratowski--Zorn lemma, we produce $v_*\in-\mbox{int}\,C_+$ the biggest nontrivial negative solution of (\ref{eq1}).
\qed

Using these extremal constant sign solutions of (\ref{eq1}), we can produce a nodal solution.  Via suitable truncation and perturbation techniques, we focus on the order interval $[v_*,u_*]=\left\{u\in W^{1,p}(\Omega):v_*\leq u\leq u_*\  \mbox{a.e. in}\ \Omega\right\}$. Then using variational methods coupled with Morse theory, we show that problem (\ref{eq1}) admits a solution in $[v_*,u_*]$ distinct from $0,u_*,v_*$. Evidently, this a nodal solution.

To execute this solution plan, we need to compute the critical groups of $\varphi$ at the origin.
\begin{prop}\label{prop6}
	Assume that hypotheses $H(a)_1,\ H_0,\ H_1$ hold.
Then $C_k(\varphi,0)=0$ for all $k\geq 0$.
\end{prop}

{\it Proof.}
Recall that hypotheses $H_1(i),(iii)$ imply that
	\begin{eqnarray}\label{eq30}
		F(z,x)\geq \tilde{c}_0|x|^q-c_{11}|x|^r\quad \mbox{for\ a.a.}\ z\in \Omega,\ \mbox{all}\ x\in\mathbb R,\ \mbox{with}\ c_{11}>0,\ r>p.
	\end{eqnarray}
Also, from hypothesis $H(a)_1(iv)$ and (\ref{eq7}), we have
	\begin{eqnarray}\label{eq31}
		G(y)\leq c_{12}(||y||^q+||y||^p)\quad \mbox{for\ all}\ y\in\mathbb R^N\ \mbox{and\ some}\ c_{12}>0.
	\end{eqnarray}
Then for all $u\in W^{1,p}(\Omega)$ and all $t>0$, we have
	\begin{equation}\label{eq32}\begin{array}{ll}
		\varphi(tu)&\di=\int_{\Omega}G(tDu)dz+\frac{t^p}{p}\int_{\Omega}\beta(z)|u|^p dz-\int_{\Omega}F(z,tu)dz\\
		&\di\leq c_{12}\left(t^q||Du||^{q}_{q}+t^p||Du||^{p}_{p}\right)+\frac{t^p}{p}||\beta||_{\infty}||u||^{p}_{p}\\
		&\di +c_{11}t^r||u||^{r}_{r}-\tilde{c}_0 t^q||u||^{q}_{q}\quad (\mbox{see}\ (\ref{eq30}),\ (\ref{eq31})).
	\end{array}\end{equation}
Since $q<p<r$, from (\ref{eq32}) it is clear that we can find $t^*=t^*(u)\in(0,1)$ such\ that
	\begin{eqnarray}\label{eq33}
		\varphi(tu)<0\quad \mbox{for\ all}\ t\in(0,t^*).
	\end{eqnarray}
Let $u\in W^{1,p}(\Omega),\ 0<||u||\leq 1$ and $\varphi(u)=0$. We have
	\begin{equation}\label{eq34}\begin{array}{ll}
		\left.\frac{d}{dt}\varphi(tu)\right|_{t=1}&\di=\left\langle \varphi^{'}(u),u\right\rangle\\
		&\di=\left\langle A(u),u\right\rangle+\int_{\Omega}\beta|u|^p dz-\int_{\Omega}f(z,u)u\ dz\\
		&\di=\int_{\Omega}\left((a(Du),Du)_{\mathbb R^N}-\tau G(Du)\right)dz+\left(1-\frac{\tau}{p}\right)\int_{\Omega}\beta(z)|u|^p dz\\
		&\di+(\tau-q)\int_{\Omega}F(z,u)dz+\int_{\Omega}\left[qF(z,u)-f(z,u)u\right]dz.\\
		&\hspace{5cm}(\mbox{since}\ \varphi(u)=0)\end{array}
	\end{equation}
By virtue of hypothesis $H(a)_1(iv)$, we have
	\begin{eqnarray}\label{eq35}
		\left(a(Du(z)),Du(z)\right)_{\mathbb R^N}-\tau G(Du(z))\geq\tilde{c}||Du(z)||^p\quad \mbox{for\ a.a.}\ z\in\Omega.
	\end{eqnarray}
Hypothesis $H_1(iii)$ implies that for\ a.a.\ $z\in\Omega$,\ all\ $|x|\leq\delta_1$\ with\ $\delta_1\in\left(0,\delta\right]$,
	\begin{eqnarray}\label{eq36}
		F(z,x)\geq\frac{\tilde{c}_0}{q}|x|^q
		\geq\frac{\tilde{c}_0}{\delta^{p-q}_{1}}|x|^p\,.
	\end{eqnarray}
On the other hand hypothesis $H_1(i)$ implies that we can find $c_{13}=c_{13}(\delta_1,r)>0$ such\ that
	\begin{eqnarray}\label{eq37}
		F(z,x)\geq-c_{13}|x|^r\quad \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ |x|>\delta_1.
	\end{eqnarray}
Combining (\ref{eq36}) and (\ref{eq37}), we  find $c_{14}=c_{14}(\delta_1,r)>0$ such\ that
	\begin{eqnarray}\label{eq38}
		F(z,x)\geq\frac{\tilde{c}_0}{\delta^{p-q}_{1}}|x|^p-c_{14}|x|^r\quad \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\in\mathbb R.
	\end{eqnarray}	
Moreover, from hypotheses $H_1(i),(iii)$ we have
		\begin{eqnarray}\label{eq39}
		qF(z,x)-f(z,x)x\geq-c_{15}|x|^r\quad \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\in\mathbb R\ \mbox{and\ some}\ c_{15}>0.
	\end{eqnarray}
Returning to (\ref{eq34}) and using (\ref{eq35}), (\ref{eq38}), and (\ref{eq39}), we obtain
	\begin{eqnarray}\label{eq40} \left.\frac{d}{dt}\varphi(tu)\right|_{t=1}\geq\tilde{c}||Du||^{p}_{p}+\left[\frac{\tilde{c}_0}{\delta^{p-q}_{1}}-
\left(1-\frac{\tau}{p}\right)||\beta||_{\infty}\right]||u||^{p}_{p}-
c_{16}||u||^r\quad \mbox{for\ some}\ c_{16}>0.
	\end{eqnarray}
We choose $\delta_1\in\left(0,\delta\right]$ small such that $\frac{\tilde{c}_0}{\delta^{p-q}_{1}}>\left(1-\frac{\tau}{p}\right)||\beta||_{\infty}$. Then from (\ref{eq40}) we see that
$$\left.\frac{d}{dt}\varphi(tu)\right|_{t=1}\geq c_{17}||u||^p-c_{16}||u||^r\quad \mbox{for\ some}\ c_{17}>0.$$
Since $p<r$, there exists $\rho\in(0,1)$ small such\ that
\begin{eqnarray}\label{eq41}
	\left.\frac{d}{dt}\varphi(tu)\right|_{t=1}>0\quad \mbox{for\ all}\ u\in W^{1,p}(\Omega)\ \mbox{with}\ 0<||u||\leq\rho,\ \varphi(u)=0.
\end{eqnarray}
Now, let $u\in W^{1,p}(\Omega)$, with $0<||u||\leq\rho$ and $\varphi(u)=0$. We  show in what follows that
\begin{eqnarray}\label{eq42}
	\varphi(tu)\leq 0\quad \mbox{for\ all}\ t\in[0,1].
\end{eqnarray}
We argue by contradiction. So, suppose that there is $t_0\in(0,1)$ such\ that $\varphi(t_0 u)>0$. Since $\varphi$ is continuous and $\varphi(u)=0$, by Bolzano's theorem, we can find $t_1\in\left(t_0,1\right]$ such\ that $\varphi(t_1u)=0$. Let
$$t_*=\mbox{min}\left\{t\in[t_0,1]:\varphi(tu)=0\right\}>t_0>0.$$
Then
\begin{eqnarray}\label{eq43}
	\varphi(tu)>0\quad \mbox{for\ all}\ t\in\left[t_0,t_*\right).
\end{eqnarray}
Let $y=t_*u$. We have $0<||y||\leq||u||\leq\rho$ and $\varphi(y)=0$. Therefore, from (\ref{eq41}) it follows that
\begin{eqnarray}\label{eq44}
	\left.\frac{d}{dt}\varphi(ty)\right|_{t=1}>0
\end{eqnarray}
Also, from (\ref{eq43}) we have
\begin{eqnarray}\label{eq45}
	\varphi(y)=\varphi(t_* u)=0<\varphi(tu)\quad \mbox{for\ all}\ t\in\left[t_0,t_*\right) \Rightarrow \left.\frac{d}{dt}\varphi(ty)\right|_{t=1}=t_*\left.\frac{d}{dt}\varphi(tu)\right|_{t=t_*}=t_{*}\lim_{t\rightarrow t^{-}_{*}}\frac{\varphi(tu)}{t-t_*}\leq 0.
\end{eqnarray}
Comparing (\ref{eq44}) and (\ref{eq45}), we reach a contradiction. This proves (\ref{eq42}).

By taking $\rho\in(0,1)$ even smaller if necessary, we may assume that
$K_{\varphi}\cap\bar{B}_{\rho}=\{0\},$
where\\ $\bar{B_{\rho}}=\left\{u\in W^{1,p}(\Omega):||u||\leq\rho\right\}$. Let $h:[0,1]\times(\varphi^0\cap\bar{B}_{\rho})\rightarrow \varphi^0\cap \bar{B}_{\rho}$ be the continuous function defined by
$$h(t,u)=(1-t)u\qquad \mbox{for\ all}\ (t,u)\in[0,1]\times(\varphi^0\cap\bar{B}_{\rho}).$$
>From (\ref{eq42}) we see that $h(\cdot,\cdot)$ is well-defined. This deformation shows that $\varphi^0\cap\bar{B}_{\rho}$ is contractible in itself.

Fix $u\in\bar{B}_{\rho}$ with $\varphi(u)>0$. We show that there exists a unique $t(u)\in(0,1)$ such\ that
\begin{eqnarray}\label{eq46}
	\varphi(t(u)u)=0.
\end{eqnarray}
Note that $\varphi(u)>0$ and $t\longmapsto\varphi(tu)$ is continuous. So, the existence of some $t(u)\in(0,1)$ follows from Bolzano's theorem. We need to show the uniqueness of $t(u)$. Suppose there are
$0<\hat{t}_1=t(u)_1<\hat{t}_2=t(u)_2<1$ such\ that\ $\varphi(\hat{t}_1,u)=\varphi(\hat{t}_2 u)=0.$
Then from (\ref{eq42}), we have
$$k(t)=\varphi(t\hat{t}_2 u)\leq 0\qquad \mbox{for\ all}\ t\in[0,1].$$
Hence $\frac{\hat{t}_1}{\hat{t}_2}\in(0,1)$ is a maximizer of $k(\cdot)$ and so
$$
	\left.\frac{d}{dt}k(t)\right|_{t=\frac{\hat{t}_1}{\hat{t}_2}}=0
	\Rightarrow\left.\frac{\hat{t}_1}{\hat{t}_2}\frac{d}{dt}\varphi(t\hat{t}_2 u)\right|_{t=\frac{\hat{t}_1}{\hat{t}_2}}=\left.\frac{d}{dt}\varphi(t\hat{t}_1 u)\right|_{t=1}=0,
$$
which contradicts (\ref{eq41}). This proves the uniqueness of $t(u)$.

>From the uniqueness of $t(u)\in(0,1)$ and (\ref{eq42}), we have
\begin{eqnarray}\label{eq47}
	\varphi(tu)<0\qquad \mbox{if}\ t\in(0,t(u))\ \mbox{and}\ \varphi(tu)>0\ \mbox{if}\ t\in\left(t(u),1\right].
\end{eqnarray}
Now let $\hat{\epsilon}_1:\bar{B}_{\rho}\backslash\{0\}\rightarrow\left(0,1\right]$ be defined by
\begin{eqnarray}\label{eq48}
	\hat{\epsilon}_1(u)=\left\{
		\begin{array}{cl}
			1&\qquad\mbox{if}\ u\in\bar{B}_{\rho}\backslash\{0\},\ \varphi(u)\leq 0\\
			t(u)&\qquad\mbox{if}\ u\in\bar{B}_{\rho}\backslash\{0\},\ \varphi(u)>0.
		\end{array}
	\right.
\end{eqnarray}
We claim that $\hat{\epsilon}_1$ is continuous. Evidently, we need to check the continuity at $u\in\bar{B}_{\rho}\backslash\{0\}$ with $\varphi(u)=0$. Let $u_n\rightarrow u$ with $\varphi(u_n)>0$ for all $n\geq 1$. Arguing by contradiction, suppose that by passing to a subsequence if necessary, we have $t(u_n)\leq\hat{t}<1$ for all $n\geq 1$. From (\ref{eq47}) we have
\begin{eqnarray}
	&&\varphi(tu_n)>0\ \mbox{for\ all}\ t\in\left(\tilde{t},1\right],\ \mbox{all}\ n\geq 1
	\Rightarrow\varphi(tu)\geq 0\ \mbox{for\ all}\ t\in\left(\tilde{t},1\right]\nonumber\\
	\Rightarrow&&\varphi(tu)=0\ \mbox{for\ all}\ t\in\left(\tilde{t},1\right]\ (\mbox{see}\ (\ref{eq42}))
	\Rightarrow\left.\frac{d}{dt}\varphi(tu)\right|_{t=1}=0,\nonumber
\end{eqnarray}
which\ contradicts\ (\ref{eq41}).
This proves the continuity of $\hat{\epsilon}_1$.

Next, consider the map $\hat{\epsilon}_1:\bar{B}_{\rho}\backslash\{0\}\rightarrow(\bar{B}_{\rho}\cap\varphi^0)\backslash\{0\}$ defined by
\begin{eqnarray}
	\hat{\epsilon}_2(u)=\left\{\nonumber
		\begin{array}{cl}
			u&\qquad\mbox{if}\ u\in\bar{B}_{\rho}\backslash\{0\},\ \varphi(u)\leq 0\nonumber\\
			\hat{\epsilon}_1(u)u&\qquad\mbox{if}\ u\in\bar{B}_{\rho}\backslash\{0\},\ \varphi(u)>0).\nonumber
		\end{array}
	\right.\nonumber
\end{eqnarray}
Evidently $\hat{\epsilon}_2$ is continuous and $\left.\hat{\epsilon}_2\right|_{(\bar{B}_{\rho}\cap\varphi^0)\backslash\{0\}}=\left.\mbox{id}\right|_{(\bar{B}_{\rho}\cap\varphi^0)\backslash\{0\}}$. Therefore $\hat{\epsilon}_2$ is a retraction of $\bar{B}_{\rho\backslash\{0\}}$ onto $(\bar{B}_{\rho}\cap\varphi^0)\backslash\{0\}$. But $\bar{B}_{\rho\backslash\{0\}}$ is contractible in itself. Hence the same holds for $(\bar{B}_{\rho}\cap\varphi^0)\backslash\{0\}$. Recall that we have seen that $\bar{B}_{\rho}\cap\varphi^0$ is contractible in itself. So, from Granas \& Dugundji \cite[p. 389]{16}, we deduce that for all $k\geq 0$,
$H_1(\bar{B}_{\rho}\cap\varphi^0,(\bar{B}_{\rho}\cap\varphi^0)\backslash\{0\})=0$, hence
$C_k(\varphi,0)=0$\ for\ all\ $k\geq 0$ (see\ Section 2).
\qed

\begin{remark}
	The first such computation of the critical groups of $\varphi$ for equations with concave nonlinearities near the origin was conducted by Moroz \cite{26} for Dirichlet problems driven by the Laplace operator (semilinear equations) with $\beta\equiv 0$. The conditions on $f(z,x)$ in Moroz \cite{26} were more restrictive. The result of Moroz \cite{26} was extended to Dirichlet problems driven by the $p$--Laplacian with $\beta\equiv 0$, by Jiu \& Su \cite{18}. Our proof here was inspired by these two works.
\end{remark}

Now, we are ready to produce a nodal solution for problem (\ref{eq1}).

\begin{prop}\label{prop7}
	Assume that hypotheses $H(a)_1,\ H_0,\ H_1$ hold.
Then problem (\ref{eq1}) admits a nodal solution $y_0\in[v_*,u_*]\cap C^1(\bar{\Omega})$
	(here $v_*\in-{\rm int}\,C_+$ and $u_*\in{\rm int}\,C_+$ are the two extremal nontrivial constant sign solutions of (\ref{eq1}) produced in Proposition \ref{prop5}).
\end{prop}

{\it Proof.}
As before, let $\lambda>||\beta^-||_{\infty}$ and consider the following truncation-perturbation of the reaction $f(z,\cdot)$:
	\begin{eqnarray}\label{eq49}
		\hat{f}(z,x)=\left\{
			\begin{array}{cl}
				f(z,v_*(z))+\lambda|v_*(z)|^{p-2}&\qquad\mbox{if}\ x<v_*(z)\\
				f(z,x)+\lambda|x|^{p-2}x&\qquad\mbox{if}\ v_*(z)\leq x\leq u_*(z)\\
				f(z,u_*(z))+\lambda u_*(z)^{p-1}&\qquad\mbox{if}\ u_*(z)<x.
			\end{array}
			\right.
	\end{eqnarray}
This is a Carath\'eodory function. We set $\hat{F}(z,x)=\int^{x}_{0}\hat{f}(z,s)ds$ and consider the $C^1$-functional $\hat{\Psi}:W^{1,p}(\Omega)\rightarrow\mathbb R$ defined by
\begin{eqnarray}
	\hat{\Psi}(u)=\int_{\Omega}G(Du(z))dz+\frac{1}{p}\int_{\Omega}[\beta(z)+\lambda]|u(z)|^p dz
-\int_{\Omega}\hat{F}(z,u(z))dz\qquad \mbox{for\ all}\ u\in W^{1,p}(\Omega).\nonumber
\end{eqnarray}
Also, we introduce the Carath\'eodory functions $\hat{f}_{\pm}(z,x)=\hat{f}(z,\pm x^{\pm})$, we set $\hat{F}_{\pm}(z,x)=\int^{x}_{0}\hat{f}_{\pm}(z,s)ds$ and consider the $C^1$-functionals $\hat{\Psi}_{\pm}:W^{1,p}(\Omega)\rightarrow\mathbb R$ defined by
\begin{eqnarray}
	\hat{\Psi}_{\pm}(u)=\int_{\Omega}G(Du(z))dz+\frac{1}{p}\int_{\Omega}[\beta(z)+\lambda]|u(z)|^p dz
-\int_{\Omega}\hat{F}_{\pm}(z,u(z))dz\qquad \mbox{for\ all}\ u\in W^{1,p}(\Omega).\nonumber
\end{eqnarray}
We can easily check that
$$K_{\hat{\Psi}}\subseteq[v_*,u_*],\ K_{\hat{\Psi}_+}\subseteq[0,u_*],\ K_{\hat{\Psi}_-}\subseteq[v_*,0].$$
The extremality of $v_*\in-\mbox{int}\,C_+$ and of $u_*\in\mbox{int}\,C_+$ implies that
\begin{eqnarray}\label{eq50}
	K_{\hat{\Psi}}\subseteq[v_*,u_*],\ K_{\hat{\Psi}_+}=\{0,u_*\},\ K_{\hat{\Psi}_-}=\{v_*,0\}.
\end{eqnarray}
\textbf{Claim:} $u_*\in\mbox{int}\,C_+$ and $v_*\in-\mbox{int}\,C_+$ are local minimizers of $\hat{\Psi}$.

Evidently $\hat{\Psi}_+$ is coercive (see (\ref{eq49}) and recall that $\lambda>||\beta^-||_{\infty}$). Also, it is sequentially weakly lower semi-continuous. Therefore, we can find $\hat{u}_*\in W^{1,p}(\Omega)$ such that
\begin{eqnarray}\label{eq51}
	\hat{\Psi}_+(\hat{u}_*)=\inf\,\left[\hat{\Psi}_+(u):u\in W^{1,p}(\Omega)\right].
\end{eqnarray}
As before (see the proof of Proposition \ref{prop3}), using hypotheses $H(a)_1(iv),\ H_1(iii)$ we show that
$\hat{\Psi}_+(\hat{u}_*)<0=\hat{\Psi}_+(0)$,\ hence\ $\hat{u}_*\neq 0.$
>From (\ref{eq51}) we have
\begin{eqnarray}\label{eq52}
	\hat{\Psi}_+'(\hat{u}_*)=0
	\Rightarrow A(\hat{u}_*)+(\beta+\lambda)|\hat{u}_*|^{p-2}\hat{u}_*=N_{\hat{f}_+}(\hat{u}_*).
\end{eqnarray}
On (\ref{eq52}) we act with $-u^-\in W^{1,p}(\Omega)$ and since $\lambda>||\beta^-||_{\infty}$, we obtain that $\hat{u}_*\geq 0,\ \hat{u}_*\neq 0$ (see (\ref{eq49})). Next on (\ref{eq52}) we act with $(\hat{u}_*-u_*)^+\in W^{1,p}(\Omega)$ and we have
\begin{eqnarray}
	&&\left\langle A(\hat{u}_*),(\hat{u}_*-u_*)^+\right\rangle+\int_{\Omega}(\beta(z)+\lambda)\hat{u}^{p-1}_{*}(\hat{u}_*-u_*)^+dz
=\int_{\Omega}\hat{f}_+(z,\hat{u}_*)(\hat{u}_*-u_*)^+dz=\nonumber\\
	&&\int_{\Omega}\left[f(z,u_*)+\lambda u^{p-1}_{*}\right](\hat{u}_*-u_*)^+dz\ (\mbox{see}\ (\ref{eq49}))\nonumber\\
	&&=\left\langle A(u_*),(\hat{u}_*-u_*)^+\right\rangle+\int_{\Omega}[\beta(z)+\lambda]u^{p-1}_{*}(\hat{u}_*-u_*)^+dz\nonumber\\
	\Rightarrow&&\int_{\{\hat{u}_*>u_*\}}(a(D\hat{u}_*)-a(Du_*),\ D\hat{u}_*-Du_*)_{\mathbb R^N}dz\nonumber\\
	&&+\int_{\{\hat{u}_*>u_*\}}(\beta(z)+\lambda)(\hat{u}^{p-1}_{*}-u^{p-1}_{*})(\hat{u}_*-u_*)dz=0
\Rightarrow\left|\{\hat{u}_*>u_*\}\right|_N=0,\nonumber\
\end{eqnarray}
hence\ $\hat{u}_*\leq u_*$.
Hence we have proved that
$\hat{u}_*\in K_{\hat{\Psi}_+}$\ and\ $\hat{u}_* \in[0,u_*]$,\ $\hat{u}_*\neq 0$, hence
$\hat{u}_*=u_*$\ (see\ (\ref{eq50})).
But $\hat{\Psi}\left|_{C_+}=\hat{\Psi}_+\right|_{C_+}$. Therefore $u_*$ is a local $C^1(\bar{\Omega})$-minimizer of $\hat{\Psi}$, hence it is also a local $W^{1,p}(\Omega)$-minimizer of $\hat{\Psi}$ (see Proposition \ref{prop2}).

Similarly for $v_*\in-\mbox{int}\,C_+$ using this time the functional $\hat{\Psi}_-$. This proves the Claim.

Without any loss of generality, we may assume that $\hat{\Psi}(v_*)\leq\hat{\Psi}(u_*)$ (the analysis is similar if the opposite inequality holds). From the Claim we know that $u_*\in\mbox{int}\,C_+$ is a local minimizer of $\hat{\Psi}$. Hence we can find $\rho\in(0,1)$ small such\ that
\begin{eqnarray}\label{eq53}
	\hat{\Psi}(v_*)\leq \hat{\Psi}(u_*)<\inf\left[\hat{\Psi}(u):||u-u_*||=\rho\right]=\hat{\eta}_{\rho},\quad ||v_*-u_*||>\rho.
\end{eqnarray}
Recall that $\hat{\Psi}$ is coercive, hence it satisfies the $C$-condition. Combining this fact and (\ref{eq53}) we see that we can apply Theorem (\ref{th1}) (the mountain pass theorem) and find $y_0\in W^{1,p}(\Omega)$ such\ that
\begin{eqnarray}\label{eq54}
	y_0\in K_{\hat{\Psi}}\quad \mbox{and}\quad \hat{\eta}_{\rho}\leq \hat{\Psi}(y_0).
\end{eqnarray}
>From (\ref{eq50}) and (\ref{eq54}) it follows that
$y_0\in[v_*,u_*]$, hence
$y_0$\ is\ a\ solution\ of\ problem (\ref{eq1})\ (see\ (\ref{eq49})).

Since $y_0$ is a critical point of $\hat{\Psi}$ of mountain pass type, we have
\begin{eqnarray}\label{eq55}
	C_1(\hat{\Psi},y_0)\neq 0.
\end{eqnarray}
Also because $u_*\in\mbox{int}\,C_+,\ v_*\in-\mbox{int}\,C_+$ and $\hat{\Psi}\left|_{[v_*,u_*]}=\varphi\right|_{[v_*,u_*]}$, from the homotopy invariance of critical groups, we have
\begin{eqnarray}\label{eq56}C_k(\hat{\Psi},0)=C_k(\varphi,0)=0\qquad \mbox{for\ all}\ k\geq 0\ (\mbox{see\ Proposition}\  \ref{prop6}).
\end{eqnarray}
Comparing (\ref{eq55}) and (\ref{eq56}), we see that $y_0\neq 0$. Hence $y_0$ is a nodal solution of (\ref{eq1}) and the nonlinear regularity theory implies $y_0\in C^1(\bar{\Omega})$.
\qed

Now we can state the following multiplicity theorem for problem (\ref{eq1}).

\begin{theorem}\label{th2}
	Assume that hypotheses $H(a)_1,\ H_0,\ H_1$ hold.
Then problem (\ref{eq1}) admits at least three nontrivial solutions
	$u_0\in{\rm int}\,C_+$,\ $v_0\in-{\rm int}\,C_+$ and $y_0\in[v_0,u_0]\cap C^1(\bar{\Omega})$ nodal.
\end{theorem}

\begin{remark}
	Three solutions theorems for coercive nonlinear equations driven by the $p$-Laplacian (that is, $a(y)=||y||^{p-2}y$ for all $y\in\mathbb R^N,\ 1<p<\infty$), were proved by Liu \cite{22}, Liu \& Liu \cite{24} and Kyritsi \& Papageorgiou \cite{20}. In \cite{22} and \cite{24}, the authors deal with Dirichlet problems with $\beta\equiv 0$ and the reaction $f(z,\cdot)$ satisfies a global sign condition. They prove a three solutions theorem, but they do not produce a nodal solution. In Kyritsi \& Papageorgiou \cite{20} the problem is Neumann, with $\beta(z)\equiv\beta\in(0,+\infty)$ for all $z\in\Omega$ and no nodal solution is obtained.
\end{remark}

\section{Noncoercive Problems}

In the previous section it was assumed that the reaction $f(z,\cdot)$ is $(p-1)$--sublinear near $\pm\infty$ and $(p-1)$-superlinear near zero (see hypotheses $H_1(i),(iii)$). In this section, we investigate the complementary situation. Namely, we consider nonlinearities which are $(p-1)$--superlinear near $\pm\infty$ and $(p-1)$--sublinear near $0$. In this case the energy functional of the problem is indefinite. To express the $(p-1)$--superlinearity of $f(z,\cdot)$ near $\pm\infty$, we do not employ the usual in such cases Ambrosetti-Rabinowitz condition (AR-condition for short). Instead we use a more general condition which incorporates in our setting ``superlinear" reactions with ``slower" growth. These nonlinearities fail to satisfy the AR-condition.

We need to modify a little the conditions on the map $a(y)$:
\begin{eqnarray}
	H(a)_2:&&a(y)=a_0(||y||)y\ \mbox{for\ all}\ y\in\mathbb R^N\ \mbox{with}\ a_0(t)>0\ \mbox{for\ all}\ t>0,\ \mbox{hypotheses}\nonumber\\
	&& H(a)_2(i),(ii),(iii)\ \mbox{are\ the\ same\ as\ the\ corresponding}\ \mbox{hypotheses}\ H(a_1)(i),(ii),(iii)\ \mbox{and}\nonumber\\
	&& (iv)\ \mbox{there\ exists}\ q\in(1,p)\ \mbox{such\ that}\ t\longmapsto G_0(t^{1/q})\ \mbox{is\ convex\ in}\ (0,+\infty)\ \mbox{and\ there\ exists}\ \gamma\in\mathbb R\nonumber\\
	&& \mbox{such that}\
	\gamma\leq p\ G_0(t)-t^2a_0(t)\ \mbox{for\ all}\ t>0.\nonumber
\end{eqnarray}
\begin{remark}
	The examples presented in Section 2 satisfy hypotheses $H(a)_2$.
\end{remark}

We assume that the reaction term $f(z,x)$ is\ a\ Carath\'eodory\ function\ $f:\Omega\times\mathbb R\rightarrow\mathbb R$ such\ that $ f(z,0)=0$\ a.e.\ in\ $\Omega$ satisfying the following conditions:
$$
	\left\{\begin{array}{ll}
	&(i) \ \di |f(z,x)|\leq a(z)(1+|x|^{r-1})\ \mbox{a.e.\ in}\ \Omega,\ \mbox{for\ all}\ x\in\mathbb R,\
	\mbox{with}\ a\in L^{\infty}(\Omega)_+\ \mbox{and}\ p<r<p^*;\\
	&(ii) \ \di \mbox{if}\ F(z,x)=\int^{x}_{0}f(z,s)ds,\ \mbox{then}\ \lim_{x\rightarrow\pm\infty}\frac{F(z,x)}{|x|^p}=\pm\infty\
	\mbox{uniformly\ for\ a.a.}\ z\in\Omega;\\
	&(iii) \di\ \mbox{there\ exists}\ \tau\in\left((r-p)\max\left\{\frac{N}{p},1\right\},p^*\right)\ \mbox{such that}\
	0<\beta_0\leq\liminf_{x\rightarrow\pm\infty}\frac{f(z,x)x-pF(z,x)}{|x^{\tau}|}\\ &\di\mbox{uniformly\ for\ a.a.}\ z\in\Omega;\\
	&(iv) \ \di\mbox{there\ exists}\ \vartheta\in L^{\infty}(\Omega),\ \vartheta(z)\leq\frac{c_1}{p-1}\hat{\lambda}_1(p,\hat{\beta})\ \mbox{a.e.\ in}\ \Omega,\ \vartheta\neq\frac{c_1}{p-1}\hat{\lambda}_1(p,\hat{\beta})\\
	&\di (\mbox{recall\ that}\ \hat{\beta}=\frac{p-1}{c_1}\beta\in L^{\infty}(\Omega))\ \mbox{such that}\
	\di\limsup_{x\rightarrow 0}\frac{p\ F(z,x)}{|x|^p}\leq\vartheta(z)\ \mbox{uniformly\ for\ a.a.}\ z\in\Omega;\\
	&(v)\ \mbox{for\ every}\ \rho>0,\ \mbox{there\ exists}\ \epsilon_{\rho}>0\ \mbox{such\ that\ for\ a.a.}\ z\in\Omega\,,\ \mbox{the mapping} \\
	&\di x\longmapsto f(z,x)+\epsilon_{\rho}|x|^{p-2}x\ \mbox{is\ nondecreasing\ on}\ [-\rho,\rho].\end{array}\right.\leqno(H_2)
$$
\begin{remark}
	Hypotheses $H_2(ii),(iii)$ imply that the reaction $f(z,\cdot)$ is $(p-1)$--superlinear near $\pm\infty$. In the literature, ``superlinear" problems are usually treated with the help of the so-called AR-condition. According to that condition, there exist $\eta>p$ and $M>0$ such that
	\begin{eqnarray}\label{eq57}
		0<\eta F(z,x)\leq f(z,x)x\quad \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ |x|\geq M\ \mbox{and}\ \inf_{\Omega}F(\cdot;\pm M)>0.
	\end{eqnarray}
\end{remark}
A straightforward integration of (\ref{eq57}) leads to
\begin{eqnarray}\label{eq58}
	c_{18}|x|^{\eta}\leq F(z,x)\quad \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ |x|\geq M\ \mbox{and\ some}\ c_{18}>0.
\end{eqnarray}
Clearly (\ref{eq58}) implies the much weaker condition $H_2(ii)$. Here, we replace (\ref{eq57}) (the AR-condition) by hypothesis $H_2(iii)$ which is weaker. Indeed, suppose that (\ref{eq57}). We may assume that $\eta>(r-p)\mbox{max}\left\{\frac{N}{p},1\right\}$. Then
$$
\frac{f(z,x)x-pF(z,x)}{|x|^{\eta}}=\frac{f(z,x)x-\eta F(z,x)}{|x|^{\eta}}+(\eta-p)\frac{F(z,x)}{|x|^{\eta}}
\geq(\eta-p)c_{18}\quad \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ |x|\geq M,
$$
hence $H_2(iii)$ holds. Similar superlinearity conditions were used by Costa \& Magalhaes \cite{7}, Fei \cite{11} and Li, Wu \& Zhou \cite{25}.

\begin{exam}
	The following functions satisfy hypotheses $H_2$. For the sake of simplicity, we drop the z-dependence:
	\begin{eqnarray}
		&&f_1(x)=\vartheta|x|^{p-2}x+|x|^{r-2}x\quad with\ \vartheta<\hat{\lambda}_1(p,\hat{\beta}),\ 1<p<r<p^*\nonumber\\
		&&f_2(x)=\left\{
			\begin{array}{cl}
				\vartheta|x|^{p-2}x&\quad\mbox{if}\ |x|<1\nonumber\\
				|x|^{p-2}x\ ln|x|+\vartheta|x|^{q-2}x&\quad\mbox{if}\ |x|\geq 1\nonumber
			\end{array}
		\right.
		with\ \vartheta<\hat{\lambda}_1(p,\hat{\beta}),\ 1<q<p.\nonumber
	\end{eqnarray}
\end{exam}	
Note that $f_1$ satisfies the AR-condition (see (\ref{eq57})), but $f_2$ does not.

	As before (see Section 3) with $\lambda>||\beta^-||_{\infty}$, we consider the truncations-perturbations of $f(z,\cdot),\ \hat{f}_{\pm}(z,x)$ and the corresponding $C^1$-functionals $\hat{\varphi}_{\pm}:W^{1,p}(\Omega)\rightarrow\mathbb R$. Also $\varphi:W^{1,p}(\Omega)\rightarrow\mathbb R$ is the $C^1$-energy functional of problem (\ref{eq1}) (see Section 3).
	\begin{prop}\label{prop8}
		Assume that hypotheses $H(a)_2,\ H_0,\ H_2$ hold.
Then the functionals $\hat{\varphi}_{\pm}$ satisfy the $C$-condition.
	\end{prop}

{\it Proof.}
We do the proof for the functional $\hat{\varphi}_+$, the arguments for $\hat{\varphi}_-$ being similar. So, let $\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)$ be such\ that
	\begin{eqnarray}
		&&|\hat{\varphi}_+(u_n)|\leq M_1\quad \mbox{for\ some}\ M_1>0,\ \mbox{all}\ n\geq 1\label{eq59}\\
		\mbox{and}&&(1+||u_n||)\hat{\varphi}^{'}_{+}(u_n)\rightarrow 0\ \mbox{in}\ W^{1,p}(\Omega)^*\ \mbox{as}\ n\rightarrow\infty.\label{eq60}
	\end{eqnarray}
>From (\ref{eq60}), we have for\ all\ $h\in W^{1,p}(\Omega)$\ and some\ $\epsilon_n\rightarrow 0^+$
\begin{eqnarray}\label{eq61}
	\left|\left\langle A(u_n),h\right\rangle+\int_{\Omega}(\beta(z)+\lambda)|u_n|^{p-2}u_n h\ dz-\int_{\Omega}\hat{f}_+(z,u_n)h\ dz\right|\leq\frac{\epsilon_n||h||}{1+||u_n||}\,.
\end{eqnarray}
In (\ref{eq61}) we choose $h=-u^{-}_{n}\in W^{1,p}(\Omega)$. Then by (\ref{eq6}) we obtain for all $n\geq 1$
\begin{eqnarray}\label{eq62}
	\frac{c_1}{p-1}||Du^{-}_{n}||^{p}_{p}+\int_{\Omega}(\beta(z)+\lambda)(u^{-}_{n})^p dz\leq\epsilon_n\ \Rightarrow u^{-}_{n}\rightarrow 0\ \mbox{in}\ W^{1,p}(\Omega)\ (\mbox{recall}\ \lambda>||\beta^-||_{\infty}).
\end{eqnarray}
Next in (\ref{eq61}) we choose $h=u^{+}_{n}\in W^{1,p}(\Omega)$. It follows that
\begin{eqnarray}\label{eq63}
	-\int_{\Omega}(a(Du^{+}_{n}),Du^{+}_{n})_{\mathbb R^N}dz-\int_{\Omega}\beta(z)(u^{+}_{n})^p dz+
\int_{\Omega}f(z,u^{+}_{n})u^{+}_{n}dz\leq\epsilon_n\qquad \mbox{for\ all}\ n\geq 1.
\end{eqnarray}
On the other hand from (\ref{eq59}) and (\ref{eq62}), we have for all $n\geq 1$
\begin{eqnarray}\label{eq64}
	\int_{\Omega}pG(Du^{+}_{n})dz+\int_{\Omega}\beta(z)(u^{+}_{n})^p dz-\int_{\Omega}pF(z,u^{+}_{n})dz\leq M_2\qquad
\mbox{for\ some}\ M_2>0.
\end{eqnarray}
Adding (\ref{eq63}) and (\ref{eq64}), we obtain for all $n\geq 1$
\begin{equation}\label{eq65}
	\begin{array}{ll}&\di\int_{\Omega}\left[pG(Du^{+}_{n})-(a(Du^{+}_{n}),Du^{+}_{n})_{\mathbb R^N}\right]dz+\int_{\Omega}\left[f(z,u^{+}_{n})u^{+}_{n}-pF(z,u^{+}_{n})\right]dz\leq M_3\qquad \mbox{for\ some}\ M_3>0,\\
	&\di\int_{\Omega}\left[f(z,u^{+}_{n})u^{+}_{n}-pF(z,u^{+}_{n})\right]dz\leq M_4
\qquad\mbox{for\ some}\ M_4>0\ (\mbox{see}\ H(a)_2(iv)).\end{array}
\end{equation}
By virtue of hypotheses $H_2(i),(iii)$, we can find $\beta_1\in(0,\beta_0)$ and $c_{19}>0$ such\ that
\begin{eqnarray}\label{eq66}
	\beta_1x^{\tau}-c_{19}\leq f(z,x)x-pF(z,x)\qquad \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\geq 0.
\end{eqnarray}
Using (\ref{eq66}) in (\ref{eq65}) we deduce that there is some $M_5>0$ such that for all $n\geq 1$
\begin{eqnarray}\label{eq67}
	\beta_1||u^{+}_{n}||^{\tau}_{\tau}\leq M_5,\end{eqnarray} hence
	$\{u^{+}_{n}\}_{n\geq 1}\subseteq L^{\tau}(\Omega)$\ is\ bounded.

First suppose that $N\neq p$. From hypothesis $H_2(iii)$ it is clear that without any loss of generality we may assume that $\tau<r<p^*$. Hence, we can find $t\in(0,1)$ such\ that
$$\frac{1}{r}=\frac{1-t}{\tau}+\frac{t}{p^*}\,.$$
Invoking the interpolation inequality (see, for example, Gasinski \& Papageorgiou \cite[p. 905]{14}), we have
\begin{eqnarray}\label{eq68}
	&&||u^{+}_{n}||\leq||u^{+}_{n}||^{1-t}_{\tau}\,||u^{+}_{n}||^{t}_{p^*}
	\Rightarrow ||u^{+}_{n}||^{r}_{r}\leq M_6||u^{+}_{n}||^{tr}\quad \mbox{for\ some}\ M_6>0,\ \mbox{all}\ n\geq 1\\
	&&(\mbox{see}\ (\ref{eq54})\ \mbox{and\ use\ the\ Sobolev\ embedding\ theorem}).\nonumber
\end{eqnarray}
Hypothesis $H_2(i)$ implies that
\begin{eqnarray}\label{eq69}
	f(z,x)x\leq c_{20}(1+x^r)\quad \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\geq 0\ \mbox{with}\ c_{20}>0.
\end{eqnarray}
>From (\ref{eq61}) with $h=u^{+}_{n}\in W^{1,p}(\Omega)$, we have for all $n\geq 1$
\begin{equation}\label{eq70}\begin{array}{ll}
	&
\di\int_{\Omega}(a(Du^{+}_{n}),Du^{+}_{n})_{\mathbb R^N}dz+\int_{\Omega}\beta(u^{+}_{n})^p dz-\int_{\Omega}f(z,u^{+}_{n})u^{+}_{n}dz\leq\epsilon_n\\
	\Rightarrow&\frac{c_1}{p-1}||Du^{+}_{n}||^{p}_{p}\leq c_{21}(1+||u^{+}_{n}||^{r}_{r})\ \mbox{for\ some}\ c_{21}>0\\
	&\hspace{3cm}(\mbox{see}\ (\ref{eq6}),\ (\ref{eq69}),\ \mbox{hypothesis}\ H_0\ \mbox{and\ recall\ that}\ r>p).\end{array}
\end{equation}
We know that $u\rightarrow||u||_{\tau}+||Du||_p$ is on equivalent norm on $W^{1,p}(\Omega)$ (see, for example, Gasinski \& Papageorgiou \cite[p. 227]{14}). So, from (\ref{eq67}) and (\ref{eq70}) and since $\tau<r$, we have for all $n\geq 1$
\begin{equation}\label{eq71}
	||u^{+}_{n}||^p\leq c_{22}(1+||u^{+}_{n}||^{r}_{r})\leq c_{23}(1+||u^{+}_{n}||^{tr})\,.
\end{equation}
The hypothesis on $\tau$ (see $H_2(iii)$) implies that $tr<p$. So, from (\ref{eq71}) it follows that
\begin{eqnarray}\label{eq72}
	\{u^{+}_{n}\}_{n\geq 1}\subseteq W^{1,p}(\Omega)\quad \mbox{is\ bounded}
	\Rightarrow \{u_{n}\}_{n\geq 1}\subseteq W^{1,p}(\Omega)\quad \mbox{is\ bounded}\ (\mbox{see}\ (\ref{eq62})).
\end{eqnarray}
If $N=p$, then $p^*=+\infty$ and from the Sobolev embedding theorem, we have that $W^{1,p}(\Omega)\hookrightarrow L^s(\Omega)$ for all $1\leq s<\infty$. Then for the previous argument to work we replace $p^*$ by $\eta>r>\tau$ and choose $t\in(0,1)$ such that
$$
\frac{1}{r}=\frac{1-t}{\tau}+\frac{t}{\eta}
\Rightarrow tr=\frac{\eta(r-\tau)}{\eta-\tau}\,.
$$
Note that $\frac{\eta(r-\tau)}{\eta-\tau}\rightarrow r-\tau$ as $\eta\rightarrow p^*=+\infty$. But $r-\tau<p$ (see $H_2(iii)$). Therefore for $\eta>r$ large, we can have $tr<p$ and so (\ref{eq72}) holds.

By virtue of (\ref{eq72}), we may assume that
\begin{eqnarray}\label{eq73}
	u_n\stackrel{w}{\rightarrow}u\quad \mbox{in}\ W^{1,p}(\Omega)\quad \mbox{and}\ u_n\rightarrow u\quad \mbox{in}\ L^r(\Omega)\quad \mbox{as}\ n\rightarrow\infty.
\end{eqnarray}
In (\ref{eq61}) we choose $h=u_n-u\in W^{1,p}(\Omega)$, pass to the limit as $n\rightarrow\infty$ and use (\ref{eq73}). Thus, by  Proposition \ref{prop1},
$$
	\lim_{n\rightarrow\infty}\left\langle A(u_n),u_n-u\right\rangle=0
	\Rightarrow u_n\rightarrow u\quad  \mbox{in}\ W^{1,p}(\Omega)
	\Rightarrow \hat{\varphi}_+\quad \mbox{satisfies\ the\ C-condition}.
$$
Similarly for the functional $\hat{\varphi}_-$.	
\qed

With some minor straightforward changes in the above proof, we can also have the following result.
\begin{prop}\label{prop}
	Assume that hypotheses $H(a)_2,\ H_0,\ H_2$ hold.
Then the functional $\varphi$ satisfies the $C$-condition.
\end{prop}
First we will produce two nontrivial constant sign solutions. This will be done by using Theorem \ref{th1} (the mountain pass theorem). To this end, we check the mountain pass geometry for the functionals $\hat{\varphi}_{\pm}$.
\begin{prop}\label{prop10}
	Assume that hypotheses $H(a)_2,\ H_0,\ H_1$ hold.
Then $u=0$ is a local minimizer for the functionals $\hat{\varphi}_{\pm}$ and $\varphi$.
\end{prop}

{\it Proof.}
	We do the proof for the functional $\hat{\varphi}_+$, the arguments for $\hat{\varphi}_-$ and $\varphi$ being similar. By virtue of hypothesis $H_2(iv)$, given $\epsilon>0$, we can find $\delta=\delta(\epsilon)>0$ such\ that
	\begin{eqnarray}\label{eq74}
		F(z,x)\leq\frac{1}{p}[\vartheta(z)+\epsilon]|x|^p\quad \mbox{for\ a.a.}\ z\in\Omega,\ \  \mbox{all}\ |x|\leq\delta.
	\end{eqnarray}
	Let $u\in C^1(\bar{\Omega})$ such\ that $||u||_{C^1(\bar{\Omega})}\leq\delta$. Then
	\begin{equation}\label{eq75}\begin{array}{ll}
		\hat{\varphi}_+(u)&\di=\int_{\Omega}G(Du)dz+\frac{1}{p}\int_{\Omega}(\beta(z)+\lambda)|u|^p dz+\int_{\Omega}\hat{F}_+(z,u)dz\\
		&\di\geq\frac{c_1}{p(p-1)}||Du||^{p}_{p}+\frac{1}{p}\int_{\Omega}\beta(z)|u|^p dz-\int_{\Omega}F(z,u)dz\ (\mbox{see}\ (\ref{eq7}))\\
		&\di\geq\frac{c_1}{p(p-1)}||Du||^{p}_{p}+\frac{1}{p}\int_{\Omega}\beta(z)|u|^p dz-\frac{1}{p}\int_{\Omega}\vartheta(z)|u|^p dz-\frac{\epsilon}{p}||u||^p\\
		&\di=\frac{c_1}{p(p-1)}\left[||Du||^{p}_{p}+\int_{\Omega}\hat{\beta}(z)|u|^p dz-\int_{\Omega}\frac{p-1}{c_1}\vartheta(z)|u|^p dz\right]-\frac{\epsilon}{p}||u||^p\\
		&\di\geq\frac{1}{p}\left[\frac{c_1c_{25}}{p-1}-\epsilon\right]||u||^p\ \ \mbox{for\ some}\ c_{25}>0\ (\mbox{see\ Lemma}\ \ref{lem1}).
	\end{array}\end{equation}
	Choosing $\epsilon\in\left(0,\frac{c_1c_{25}}{p-1}\right)$ from (\ref{eq75}) we infer that $u=0$ is a local $C^1(\bar{\Omega})$-minimizer of $\hat{\varphi}_+$, hence $u=0$ is a local $W^{1,p}(\Omega)$-minimizer of $\hat{\varphi}_+$ (see Proposition \ref{prop2}).
	
	Similarly for the functionals $\hat{\varphi}_-$ and $\varphi$.
\qed

The superlinearity of $F(z,\cdot)$ (see hypothesis $H_2(ii)$) leads to the following result.
\begin{prop}\label{prop11}
	Assume that hypotheses $H(a)_2,\ H_0,\ H_2$ hold and $u\in W^{1,p}(\Omega),\ u\geq 0,\ u\neq 0$.
Then $\hat{\varphi}_{\pm}(tu)\rightarrow-\infty$ as $t\rightarrow\pm\infty$.
\end{prop}

{\it Proof.}
By virtue of hypotheses $H_2(i),(ii)$, given any $\eta>0$, we can find $c_{26}=c_{26}(\eta)>0$ such\ that
	\begin{eqnarray}\label{eq76}
		F(z,x)\geq\eta|x|^p-c_{26}\qquad \mbox{for\ a.a.}\ z\in\Omega,\ \ \mbox{all}\ x\in\mathbb R.
	\end{eqnarray}
Then for all $t\geq 1$, we have
\begin{equation}\label{eq77}\begin{array}{ll}
	\hat{\varphi}_+(tu)&\di=\int_{\Omega}G(tDu)dz+\frac{t^p}{p}\int_{\Omega}\beta(z)|u|^p dz-\int_{\Omega}F(z,tu)dz\\
	&\di\leq c_{12}t^p\left(||Du||^{q}_{q}+||Du||^{p}_{p}\right)+c_{27}t^p||u||^{p}_{p}-\eta t^p||u||^{p}_{p}+c_{26}|\Omega|_N\\
	&\di\mbox{for\ some}\ c_{27}>0\ (\mbox{see}\ (\ref{eq31}),\ (\ref{eq76})\ \mbox{and\ recall}\ t\geq 1\ \mbox{and}\ q<p)\\
	&\di\leq t^p\left[c_{28}||u||^p-\eta||u||^{p}_{p}\right]+c_{26}|\Omega|_N\ \ \mbox{for\ some}\ c_{28}>0.
\end{array}\end{equation}
Choosing $\eta>\frac{c_{28}||u||^p}{||u||^{p}_{p}}$, from (\ref{eq77}) we infer that
$\hat{\varphi}_+(tu)\rightarrow-\infty$ as\ $t\rightarrow+\infty.$
In a similar fashion we also show that
$\hat{\varphi}_-(tu)\rightarrow-\infty$ as\ $t\rightarrow-\infty.$
\qed

Now we have the mountain pass geometry for the functionals $\hat{\varphi}_{\pm}$ and we can produce two nontrivial constant sign solutions of (\ref{eq1}).
\begin{prop}\label{prop12}
	Assume that hypotheses $H(a)_2,\ H_0,\ H_2$ hold.
Then problem (\ref{eq1}) has at least two nontrivial constant sign solutions
	$u_0\in{\rm int}\,C_+$\ and\ $v_0\in-{\rm int}\,C_+\,.$
\end{prop}

{\it Proof.}
Proposition \ref{prop10} implies that we can find $\rho\in(0,1)$ small such\ that
	$$\hat{\varphi}_+(0)=0<\inf\,\left[\hat{\varphi}_+(u):||u||=\rho\right]=\hat{\eta}_+.$$
This fact together with Propositions \ref{prop8} and \ref{prop11} permit the use of Theorem \ref{th1} (the mountain pass theorem). So, we can find $u_0\in W^{1,p}(\Omega)$ such\ that
\begin{eqnarray}\label{eq78}
	&u_0\in K_{\hat{\varphi}_+}\quad \mbox{and}\quad \hat{\eta}_+\leq\hat{\varphi}_+(u_0)
	\Rightarrow u_0\neq 0\quad \mbox{and}\quad A(u_0)+(\beta+\lambda)|u_0|^{p-2}u_0=N_{\hat{f}_+}(u_0).
\end{eqnarray}	
On (\ref{eq78}) we act with $-u^{-}_{0}\in W^{1,p}(\Omega)$ and since $\lambda>||\beta^-||_{\infty}$, we obtain $u_0\geq 0,\ u_0\neq 0$. Therefore (\ref{eq78}) yields
$$
	A(u_0)+\beta u^{p-1}_{0}=N_f(u_0),
$$
hence $u_0$\ is\ a\ nontrivial\ positive\ solution\ of\ (\ref{eq1}) and $u_0\in C_+\backslash \{0\}$
(by\ the\ nonlinear\ regularity\ theory).

Let $\rho=||u_0||_{\infty}$ and let $\epsilon_{\rho}>0$ be as postulated by hypothesis $H_2(iv)$. Then
\begin{eqnarray}
	&&-\mbox{div}\, a(Du_0(z))+(\beta(z)+\epsilon_{\rho})u_0(z)^{p-1}\geq f(z,u_0(z))+\epsilon_{\rho}u_0(z)^{p-1}\geq 0\quad \mbox{a.e.\ in}\ \Omega\nonumber\\
	\Rightarrow&&u_0\in\mbox{int}\,C_+\ (\mbox{see\ the\ proof\ of\ Proposition}\ \ref{prop3}\ \mbox{and\ Pucci \& Serrin}\ \cite[\mbox{pp. }111,120]{33}.
\end{eqnarray}
	Similarly, working with the functional $\hat{\varphi}_-$, we produce $v_0\in-\mbox{int}\,C_+$ a nontrivial negative solution of (\ref{eq1}).
\qed

To produce a third nontrivial solution, we will use Morse theory (critical groups). In the next two propositions, we compute the critical groups at infinity for the functionals $\varphi$ and $\hat{\varphi}_+$.
\begin{prop}\label{prop13}
	Assume that hypotheses $H(a)_2,\ H_0,\ H_2$ hold.
Then $C_k(\varphi,\infty)=0$ for all $k\geq 0$.
\end{prop}

{\it Proof.}
	As in the proof of Proposition \ref{prop11}, we show that for every $u\in W^{1,p}(\Omega),\ u\neq 0$, we have
	\begin{eqnarray}\label{eq79}
		\varphi(tu)\rightarrow-\infty\quad \mbox{as}\quad t\rightarrow+\infty.
	\end{eqnarray}
	By virtue of hypotheses $H_2(i),(iii)$, we have for some $c_{29}>0$ and\ $\beta_1\in(0,\beta_0)$
	\begin{eqnarray}\label{eq80}
		pF(z,x)-f(z,x)x\leq c_{29}-\beta_1|x|^{\tau}\qquad \mbox{for\ a.a.}\ z\in\Omega,\ \mbox{all}\ x\in\mathbb R.\nonumber
	\end{eqnarray}
	For $u\in W^{1,p}(\Omega)$ and $t>0$, we have
	\begin{equation}\label{eq81}\begin{array}{ll}
		\di\frac{d}{dt}\varphi(tu)&\di=\left\langle \varphi^{'}(tu),u\right\rangle
		=\frac{1}{t}\left\langle \varphi^{'}(tu),tu\right\rangle\\
		&\di=\frac{1}{t}\left[\int_{\Omega}(a(tDu),tDu)_{\mathbb R^N}dz+\int_{\Omega}\beta|tu|^p dz-\int_{\Omega}f(z,tu)(tu)dz\right]\\
		&\di\leq\frac{1}{t}\left[\int_{\Omega}pG(tDu)dz+|\gamma||\Omega|_N+\int_{\Omega}\beta|tu|^p dz\right]\\
		&\di+c_{29}|\Omega|_N-\int_{\Omega}pF(z,tu)dz-\beta_1||tu||^{\tau}_{\tau}\quad (\mbox{see}\ H(a)_2(iv)\ \mbox{and}\ (\ref{eq79}))\\
		&\di\leq\frac{1}{t}\left[p\varphi(tu)+(|\gamma|+c_{29})|\Omega|_N\right].
	\end{array}\end{equation}
	From (\ref{eq79}) and (\ref{eq81}), we deduce that
	\begin{eqnarray}\label{eq82}
		\frac{d}{dt}\varphi(tu)<0\qquad \mbox{for\ all}\ t>0\ \mbox{big enough}.
	\end{eqnarray}
	Then by virtue of the implicit function theorem, we can find a unique function $\epsilon\in C(\partial B_1)$ such that
	\begin{eqnarray}\label{eq83}
		\epsilon>0\ and\ \varphi(\epsilon(u)u)=\rho_*<-\frac{|\gamma|+c_{29}}{p}\quad (\mbox{see}\ (\ref{eq81})).		
	\end{eqnarray}
	We extend $\epsilon$ on $W^{1,p}(\Omega)\backslash\{0\}$ as
	$$\epsilon_0(u)=\frac{1}{||u||}\epsilon\left(\frac{u}{||u||}\right)\qquad \mbox{for\ all}\ u\in W^{1,p}(\Omega)\backslash\{0\}.$$
	Clearly $\epsilon_0\in C(W^{1,p}(\Omega)\backslash\{0\})$ and $\varphi(\epsilon_0(u)u)=\rho_*$ (see (\ref{eq83})). Moreover, if $\varphi(u)=\rho_*$, then $\epsilon_0(u)=1$. Therefore, if we set
	\begin{eqnarray}\label{eq84}
		\epsilon^*(u)=\left\{
			\begin{array}{cl}
				1&\quad\mbox{if}\ \varphi(u)<\rho_*\\
				\epsilon_0(u)&\quad\mbox{if}\ \varphi(u)\geq \rho_*,
			\end{array}
		\right.
	\end{eqnarray}
	then $\epsilon^*\in C(W^{1,p}(\Omega)\backslash\{0\})$.
	
	We consider the homotopy $h:[0,1]\times(W^{1,p}(\Omega)\backslash\{0\})\rightarrow W^{1,p}(\Omega)\backslash\{0\}$ defined by
	$$h(t,u)=(1-t)u+t\epsilon^*(u)u\qquad \mbox{for\ all}\ (t,u)\in[0,1]\times(W^{1,p}(\Omega)\backslash\{0\}).$$
	Note that
	\begin{eqnarray}
		&&h(0,u)=u,\ h(1,u)=\epsilon^*(u)u\in\varphi^{\rho_*}\qquad \mbox{for\ all}\ u\in W^{1,p}(\Omega)\backslash\{0\}\ \mbox{and}\nonumber\\
		&&h(t,\cdot)\left|_{\varphi^{\rho_*}}=\mbox{id}\right|_{\varphi^{\rho_*}}\qquad \mbox{for\ all}\
t\in[0,1]\ (\mbox{see}\ (\ref{eq84})).\nonumber
	\end{eqnarray}
	This shows that
	\begin{eqnarray}\label{eq85}
		\varphi^{\rho_*}\quad \mbox{is\ a\ strong\ deformation\ retract\ of}\ W^{1,p}(\Omega)\backslash\{0\}.
	\end{eqnarray}
	If we use the radial retraction $r_0(u)=\frac{u}{||u||}$ for all $u\in W^{1,p}(\Omega)\backslash\{0\}$, we see that $\partial B_1$ is a retract of $W^{1,p}(\Omega)\backslash\{0\}$ and $W^{1,p}(\Omega)\backslash\{0\}$ is deformable onto $\partial B_1$. Therefore Theorem 6.5  of Dugundji \cite[p. 325]{9} implies that
	\begin{eqnarray}\label{eq86}
		\partial B_1\ \mbox{is\ a\ deformation\ retract\ of}\ W^{1,p}(\Omega)\backslash\{0\}.
	\end{eqnarray}
	From (\ref{eq85}) and (\ref{eq86}) it follows that for\ all\ $k\geq 0$
	\begin{eqnarray}\label{eq87}
		\varphi^{\rho_*}\ \mbox{and}\ \partial B_1\ \mbox{are\ homotopically\ equivalent}
		\Rightarrow H_k(W^{1,p}(\Omega),\varphi^{\rho_*})=H_k(W^{1,p}(\Omega),\partial B_1)\,.
	\end{eqnarray}
	Since $W^{1,p}(\Omega)$ is infinite dimensional, we know that $\partial B_1$ is contractible in itself. Thus, by Granas \& Dugundji \cite[p. 389]{16},
	\begin{eqnarray}\label{eq88}
		H_k(W^{1,p}(\Omega),\partial B_1)=0\qquad \mbox{for\ all}\ k\geq 0\,.
	\end{eqnarray}
	From (\ref{eq87}) and (\ref{eq88}) it follows that
	$H_k(W^{1,p}(\Omega),\varphi^{\rho_*})=0$\ for\ all\ $k\geq 0$.
	Choosing $\rho_*< -\frac{|\gamma|+c_{29}}{p}$ with $|\rho_*|$ large enough, we have for all $k\geq 0$,
	$C_k(\varphi,\infty)=H_k(W^{1,p}(\Omega),\varphi^{\rho_*})$, hence
$C_k(\varphi,\infty)=0$.
	This completes the proof.
\qed

\begin{remark}
	The first computation of the critical groups of the energy functional for problems with superlinear reaction was developed by Wang \cite{36}. In that case the problem is Dirichlet, driven by the Laplacian, $\beta\equiv 0$, and the superlinear reaction $f$ is autonomous (that is, $f(z,\cdot)=f(\cdot)$), $f\in C^1(\mathbb R)$ and satisfies the AR-condition (see (\ref{eq57})). Our proof uses ideas from the proof of Wang \cite{36}.
\end{remark}
	
	We can obtain an analogous result for the functionals $\hat{\varphi}_{\pm}$.
\begin{prop}\label{prop14}
Assume that hypotheses $H(a)_2,\ H_0,\ H_2$ hold.
Then $C_k(\hat{\varphi}_{\pm,\infty})=0$ for all $k\geq 0$.
\end{prop}	

{\it Proof.}
	Let $\hat{\sigma}_+=\hat{\varphi}_+\left.\right|_{C^1(\bar{\Omega})}$. From the nonlinear regularity theory (see Lieberman \cite{21}), we have that $K_{\hat{\varphi}_+}\subseteq C^1(\bar{\Omega})$ and in fact $K_{\hat{\varphi}_+}\subseteq C_+$. Hence $K_{\hat{\varphi}_+}=K_{\hat{\sigma}_+}=K\subseteq C_+$. Since $C^1(\bar{\Omega})$ is dense in $W^{1,p}(\Omega)$, from Palais \cite{29} we have for $a<\inf\limits_{K}\hat{\varphi}_+=\inf\limits_{K}\hat{\sigma}_+$
	\begin{eqnarray}\label{eq89}
		H_k(W^{1,p}(\Omega),\dot{\hat{\varphi}}^{a}_+)=H_k(W^{1,p}(\Omega),\dot{\hat{\sigma}}^{a}_+)
		\Rightarrow C_k(\hat{\varphi}_+,\infty)=C_k(\hat{\sigma}_+,\infty)\quad \mbox{for\ all}\ k\geq 0.
	\end{eqnarray}
	So, by virtue of (\ref{eq89}), in order to prove the proposition, we need to show that
	\begin{eqnarray}\label{eq90}
		H_k(C^1(\bar{\Omega}),\hat{\sigma^{a}_{+}})=0\quad \mbox{for\ all}\ k\geq 0.
	\end{eqnarray}
	To this end, let
	$$\partial B^{C}_{1}=\left\{u\in C^1(\bar{\Omega}):||u||_{C^1(\bar{\Omega})=1}\right\}\quad \mbox{and}\quad \partial B^{C}_{1,+}=\{u\in \partial B^{C}_{1}:u^+\neq 0\}.$$
	We consider the homotopy $h_+:[0,1]\times\partial B^{C}_{1,+}\rightarrow\partial B^{C}_{1,+}$ defined by
	$$h_+(t,u)=\frac{(1-t)u+t\hat{u}_1(p,\beta)}{||(1-t)u+t\hat{u}_1(p,\beta)||_{C^1(\bar{\Omega})}}\quad \mbox{for\ all}\ (t,u)\in[0,1]\times W^{1,p}(\Omega).$$
	We have
	$$h_+(1,u)=\frac{\hat{u}_1(p,\beta)}{||\hat{u}_1(p,\beta)||_{C^1(\bar{\Omega})}}\in\ \partial B^{C}_{1,+},$$ hence
		$\partial B^{C}_{1,+}$\ is\ contractible\ in\ itself.
	As a consequence of hypothesis $H_2(ii)$ for every $u\in\partial B^{C}_{1,+}$, we have
	\begin{eqnarray}\label{eq91}
		\hat{\sigma}_+(tu)\rightarrow-\infty\qquad \mbox{as}\ t\rightarrow+\infty\,.
	\end{eqnarray}
	For all $u\in\partial B^{C}_{1,+}$, we have
	\begin{eqnarray}
		&&\frac{d}{dt}\ \hat{\sigma}_+(tu)=\frac{1}{t}\left[\int_{\Omega}(a(D(tu)),D(tu))_{\RR^N}dz+\int_{\Omega}(\beta(z)+\lambda)|tu|^p dz-\int_{\Omega}\hat{f}_+(z,tu)tu\ dz\right]\nonumber\\
		&&\leq\frac{1}{t}\left[\int_{\Omega}pG(D(tu))dz+\int_{\Omega}(\beta(z)+\lambda)|tu|^p dz-\int_{\Omega}pF(z,tu)dz+c_{30}\right]\nonumber\\
		&&\mbox{for\ some}\ c_{30}>0\ (\mbox{see}\ H(a)_2(iv)\ \mbox{and}\ (\ref{eq80}))\nonumber\\
	 &&=\frac{1}{t}\left[\hat{\varphi}_+(tu)+c_{30}\right]=\frac{1}{t}\left[\hat{\sigma}_+(tu)+c_{30}\right]\,.\nonumber
	\end{eqnarray}
	From (\ref{eq91}) we see that for $t>0$ big enough we have $\hat{\sigma}_+(tu)<-\frac{c_{30}}{p}$. Hence
	\begin{eqnarray}\label{eq92}
		\frac{d}{dt}\hat{\sigma}_+(tu)<0\quad \mbox{for\ all}\ t>0\ \mbox{large enough}.
	\end{eqnarray}
	Let $\bar{B}^{C}_{1}=\{u\in C^1(\bar{\Omega}):||u||_{C^1(\bar{\Omega})}\leq 1\}$ and choose $a\in \RR$ such\ that
	\begin{eqnarray}\label{eq93}
		a<\mbox{min}\left\{-\frac{c_{30}}{p},\ \inf\limits_{\bar{B}^{C}_{1}}\hat{\sigma}_+\right\}.
	\end{eqnarray}
	As before (see the proof of Proposition \ref{prop13}), from (\ref{eq93}) and the implicit function theorem, we can find a unique $\mu \in C(\partial B^{C}_{1})$,\ $\mu\geq 1$ such\ that
	\begin{eqnarray}\label{eq94}
		\hat{\sigma}_+(tu)\left\{
			\begin{array}{cl}
				>a&\quad\mbox{if}\ t\in\left[0,\mu(u)\right)\\
				=a&\quad\mbox{if}\ t=\mu(u)\\
				<a&\quad\mbox{if}\ t>\mu(u).
			\end{array}
			\right.
	\end{eqnarray}
	From (\ref{eq93}) and (\ref{eq94}), we have
	\begin{eqnarray}\label{eq95}
		\hat{\sigma}^{a}_{+}=\{tu:u\in\partial B^{C}_{1,+},t\geq \mu(u)\}\,.
	\end{eqnarray}	
	Let $E_+=\left\{tu:u\in\partial B^{C}_{1,+},t\geq 1\right\}$. From (\ref{eq95}) we have $\bar{\sigma}^{a}_{+}\subseteq E_+$. We consider the deformation $\hat{h}_+:[0,1]\times E_+\rightarrow E_+$ defined by
	\begin{eqnarray}
		\hat{h}_+(s,tu)=\left\{\nonumber
			\begin{array}{cl}
				(1-s)tu+s\mu(u)u&\quad\mbox{if}\ t\in[1,\mu(u)]\nonumber\\
				tu&\quad\mbox{if}\ t>\mu(u).\nonumber
			\end{array}
		\right.\nonumber
	\end{eqnarray}
Then we have
\begin{eqnarray}
\hat{h}_+(0,tu)=tu,\ \hat{h}_+(1,tu)\in\hat{\sigma}^{a}_{+}\quad (\mbox{see}\ (\ref{eq95}))\ \mbox{and}\quad
 \hat{h}_+(s,\cdot)\left|_{\hat{\sigma}^{a}_{+}}=\mbox{id}\right|_{\hat{\sigma}^{a}_{+}}\quad \mbox{for\ all}\ s\in[0,1].\nonumber
\end{eqnarray}
This means that $\hat{\sigma}^{a}_{+}$ is a strong deformation retract of $E_+$. Hence
\begin{eqnarray}\label{eq96}
	H_k(C^1(\bar{\Omega}),E_+)=H_k(C^1(\bar{\Omega}),\hat{\sigma}^{a}_{+})\quad \mbox{for\ all}\ k\geq 0.
\end{eqnarray}
Let $h^{*}_{+}:[0,1]\times E_+\rightarrow E_+$ be the homotopy defined by
$$h^{*}_{+}(s,tu)=(1-s)tu+s\frac{tu}{||tu||_{C^1(\bar{\Omega})}}\,.$$
>From Dugundji \cite[p. 325]{9}, we obtain that $\partial B^{C}_{1,+}$ is a deformation retract of $E_+$. Therefore
\begin{eqnarray}\label{eq97}
	&&H_k(C^1(\bar{\Omega}),E_+)=H_k(C^1(\bar{\Omega}),\partial B^{C}_{1,+})\qquad \mbox{for\ all}\ k\geq 0\nonumber\\
	\Rightarrow&&H_k(C^1(\bar{\Omega}),\hat{\sigma}^{a}_{+})=H_k(C^1(\bar{\Omega}),\partial B^{C}_{1,+})\qquad \mbox{for\ all}\ k\geq 0\ (\mbox{see}\ (\ref{eq96})).
\end{eqnarray}
We have seen earlier in the proof that $\partial B^{C}_{1,+}$ is contractible in itself. Thus, by Granas \& Dugundji\ \cite[p. 389]{16},
$H_k(C^1(\bar{\Omega}),\partial B^{C}_{1,+})=0$, hence $H_k(C^1(\bar{\Omega}),\hat{\sigma}^{a}_{+})=0$\ for\ all\ $k\geq 0$ (see\ (\ref{eq97})).

So, we have proved relation (\ref{eq90}) and from this it follows that for\ all\ $k\geq 0$,
$C_k(\hat{\varphi}_+,\infty)=C_k(\hat{\sigma}_+,\infty)=0$.
Similarly we show that $C_k(\hat{\varphi}_-,\infty)=0$ for all $k\geq 0.$
\qed

Using this result we can compute the critical groups of $\varphi$ at $u_0\in\mbox{int}\,C_+$ and $v_0\in-\mbox{int}\,C_+$.

\begin{prop}\label{prop15}
	Assume that hypotheses $H(a)_2,\ H_0,\ H_2$  hold and $K_{\varphi}=\{0,u_0,v_0\}$.
Then $C_k(\varphi,u_0)=C_k(\varphi,v_0)=\delta_{k,1}\ZZ$ for all $k\geq 0$.
\end{prop}

{\it Proof.}
Note that $\hat{\varphi}^{'}_{+}\left|_{C_+}=\varphi^{'}\right|_{C_+}$ and so $K_{\hat{\varphi}_+}=\{0,u_0\}$.
	
Let $\eta<0<\lambda<\hat{\varphi}_+(u_0)=\varphi(u_0)$ (since $u_0\in\mbox{int}\,C_+$). We consider the following triple of sets
	$$\hat{\varphi}^{\eta}_{+}\subseteq\hat{\varphi}^{\lambda}_{+}\subseteq W^{1,p}(\Omega)=W.$$
For this triple of sets, we consider the corresponding long exact sequence of homology groups
	\begin{eqnarray}\label{eq98}
		\cdots\rightarrow H_k(W,\hat{\varphi}^{\eta}_{+})\stackrel{i_*}{\longrightarrow} H_k(W,\hat{\varphi}^{\lambda}_{+})\stackrel{\partial_*}{\longrightarrow} H_k(\hat{\varphi}^{\lambda}_{+},\hat{\varphi}^{\eta}_{+})\rightarrow\cdots,
	\end{eqnarray}
where $i_*$ is the homomorphism induced by the inclusion $(W,\hat{\varphi}^{\eta}_{+})\hookrightarrow(W,\hat{\varphi}^{\lambda}_{+})$ and $\partial_*$ is the boundary homomorphism. From the rank theorem and using the\ exactness\ of\ (\ref{eq98}), we have
\begin{eqnarray}\label{eq99}
	\mbox{rank}\ H_k(W,\hat{\varphi}^{\lambda}_{+})=\mbox{rank}\, \mbox{ker}\,\partial_*+\mbox{rank}\, \mbox{im}\,\partial_*
	=\mbox{rank}\, \mbox{im}\, i_*+\mbox{rank}\, \mbox{im}\,\partial_*\,.
\end{eqnarray}
>From the choice of $\lambda$, we have
\begin{eqnarray}\label{eq100}
	H_k(W,\hat{\varphi}^{\lambda}_{+})=C_k(\hat{\varphi}_+,u_0)\qquad \mbox{for\ all}\ k\geq 0.
\end{eqnarray}
Also, since $\eta<0=\hat{\varphi}^{\eta}_{+}(0)<\hat{\varphi}^{\eta}_{+}(u_0)$ and $K_{\hat{\varphi}_+}=\{0,u_0\}$, we have for all $k\geq 0$
\begin{eqnarray}\label{eq101}
	H_k(W,\hat{\varphi}^{\eta}_{+})=C_k(\hat{\varphi}_{+},\infty)
	\Rightarrow H_k(W,\hat{\varphi}^{\eta}_{+})=0\  (\mbox{see\ Proposition} \ref{prop14})
	\Rightarrow\mbox{im}\,i_*=\{0\}.
\end{eqnarray}
Similarly, we have
\begin{eqnarray}\label{eq102}
	 H_{k-1}(\hat{\varphi}^{\lambda}_{+},\hat{\varphi}^{\eta}_{+})=C_{k-1}(\hat{\varphi}_{+},0)=\delta_{k-1,0}\ZZ\quad \mbox{for\ all}\ k\geq 0\ (\mbox{see\ Proposition}\ \ref{prop10}).
\end{eqnarray}
We return to (\ref{eq99}) and use (\ref{eq100}), (\ref{eq101}), (\ref{eq102}). We obtain
\begin{eqnarray}\label{eq103}
	\mbox{rank}\ C_k(\hat{\varphi}_+,u_0)\leq 1.
\end{eqnarray}
But recall that $u_0$ is a critical point of $\hat{\varphi}_{+}$ of mountain pass type (see the proof of Proposition \ref{prop12}). Therefore
\begin{eqnarray}\label{eq104}
	C_1(\hat{\varphi}_{+},u_0)\neq 0.
\end{eqnarray}
>From (\ref{eq103}) and (\ref{eq104}) and since in (\ref{eq98}) only the tail (that is, $k=1$) is nontrivial, we have
\begin{eqnarray}\label{eq105}
	C_k(\hat{\varphi}_+,u_0)=\delta_{k,1}\ZZ\qquad \mbox{for\ all}\ k\geq 0.
\end{eqnarray}
\textbf{Claim:} $C_k(\hat{\varphi}_+,u_0)=C_k(\varphi,u_0)$ for all $k\geq 0$.

We consider the homotopy
$$h(t,u)=(1-t)\varphi(u)+t\hat{\varphi}_+(u)\qquad \mbox{for\ all}\ (t,u)\in[0,1]\times W^{1,p}(\Omega).$$
Suppose that we can find $\{t_n\}_{n\geq 1}\subseteq[0,1]$ and $\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)$ such\ that
\begin{eqnarray}\label{eq106}
	t_n\rightarrow t,\ u_n\rightarrow u_0\ \ \mbox{in}\ W^{1,p}(\Omega)\ \ \mbox{and}\ h^{'}_{u}(t_n,u_n)=0\qquad \mbox{for\ all}\ n\geq 1.
\end{eqnarray}
>From (\ref{eq106}), we have
\begin{eqnarray}
	&&A(u_n)+\beta|u_n|^{p-2}u_n+t_n\lambda|u_n|^{p-2}u_n=(1-t_n)N_f(u_n)+t_n N_{\hat{f}_+}(u_n),\nonumber\\
	\Rightarrow&&-\mbox{div}\, a(Du_n(z))+(\beta(z)+t_n\lambda)|u_n(z)|^{p-2}u_n(z)\nonumber\\
	&&=(1-t_n)f(z,u_n(z))+t_n\hat{f}_+(z,u_n(z))\ \ \mbox{a.e.\ in}\ \Omega,\quad \frac{\partial u_n}{\partial n}=0\ \mbox{on}\ \partial\Omega\,.\nonumber
\end{eqnarray}
>From Hu \& Papageorgiou \cite{17}, we know that we can find $M_7>0$ such\ that
$$||u_n||_{\infty}\leq M_7\qquad \mbox{for\ all}\ n\geq 1.$$
Then from Lieberman \cite[p. 320]{21}, there are $\gamma\in(0,1)$ and $M_8>0$ such that
\begin{eqnarray}\label{eq107}
	u_n\in C^{1,\gamma}(\bar{\Omega})\ \ \mbox{and}\ \ ||u_n||_{C^{1,\gamma}(\bar{\Omega})}\leq M_8\qquad \mbox{for\ all}\ n\geq 1.
\end{eqnarray}
Recall that $C^{1,\gamma}(\bar{\Omega})$ is embedded compactly in $C^1(\bar{\Omega})$. So, from (\ref{eq106}) and (\ref{eq107}) it follows that
\begin{eqnarray}
u_n\rightarrow u_0\ \mbox{in}\ C^1(\bar{\Omega})
	\Rightarrow u_n\in\mbox{int}\, C_+\ \mbox{for\ all}\ n\geq n_0\ (\mbox{since}\ u_0\in\mbox{int}\,C_+).
\end{eqnarray}
We deduce that $\{u_n\}_{n\geq n_0}\subseteq K_{\varphi}$,\ a\ contradiction.

Invoking the homotopy invariance property of critical groups, we have
$$C_k(\hat{\varphi}_+,u_0)=C_k(\varphi,u_0)\qquad \mbox{for\ all}\ k\geq 0.$$
This proves the Claim.

>From the Claim and (\ref{eq105}), we have
$$C_k(\varphi,u_0)=\delta_{k,1}\ZZ\qquad \mbox{for\ all}\ k\geq 0.$$
In a similar fashion, using this time $\hat{\varphi}_-$, we show that
$$C_k(\varphi,v_0)=\delta_{k,1}\ZZ\qquad \mbox{for\ all}\ k\geq 0.$$
This completes the proof.
\qed

\begin{prop}\label{prop16}
	Assume that hypotheses $H(a)_2,\ H_0,\ H_2$ are fulfilled.
	Then problem (\ref{eq1}) admits a third nontrivial solution $y_0\in C^1(\bar{\Omega})$.
\end{prop}

{\it Proof.}
	Arguing by contradiction, suppose that $K_{\varphi}=\{0,u_0,v_0\}$.
	From Proposition \ref{prop15}, we have
	\begin{eqnarray}\label{eq108}
		C_k(\varphi,u_0)=C_k(\varphi,v_0)=\delta_{k,1}\ZZ\qquad \mbox{for\ all}\ k\geq 0.
	\end{eqnarray}
	Next, Proposition \ref{prop10} yields
	\begin{eqnarray}\label{eq109}
		C_k(\varphi,0)=\delta_{k,0}\ZZ\qquad \mbox{for\ all}\ k\geq 0.
	\end{eqnarray}
Finally, from Proposition \ref{prop13}
	\begin{eqnarray}\label{eq110}
		C_k(\varphi,\infty)=0\qquad \mbox{for\ all}\ k\geq 0.
	\end{eqnarray}
>From (\ref{eq108}), (\ref{eq109}), (\ref{eq110}) and the Morse relation with $t=-1$ (see (\ref{eq9})), we have
$2(-1)^1+(-1)^0=0,$
a contradiction. So, we can find $y_0\in K_{\varphi},\ y_0\notin \{0,u_0,v_0\}$. This is the third nontrivial solution of (\ref{eq1}) and the nonlinear regularity theory implies that $y_0\in C^1(\bar{\Omega})$.
\qed

Therefore, we can state the following multiplicity theorem (three solutions theorem) for the noncoercive version of problem  (\ref{eq1}).
\begin{theorem}\label{th3}
	Assume that hypotheses $H(a)_2,\ H_0,\ H_2$ hold.
Then problem (\ref{eq1}) has at least three nontrivial solutions
	$u_0\in{\rm int}\,C_+$, $v_0\in-{\rm int}\,C_+$, and $y_0\in C^1(\bar{\Omega}).$
\end{theorem}
\begin{remark}
	It is an interesting open question, whether we can have the third nontrivial solution $y_0\in C^1(\bar{\Omega})$ to be nodal. Nodal solutions for superlinear Neumann problems driven by the $p$-Laplacian with $\beta(\cdot)\equiv\beta\in(0,+\infty)$ and a reaction satisfying the AR-condition, were obtained by Aizicovici, Papageorgiou \& Staicu \cite{1}, under stronger conditions. Theorem \ref{th3} extends the multiplicity theorem of Wang \cite{36}, where the problem is semilinear (driven by the Laplacian), with Dirichlet boundary condition, $\beta\equiv 0$ and a superlinear reaction satisfying the AR-condition.
\end{remark}



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