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\title[Nonlinear nonhomogeneous Robin problems with superlinear reaction term]{Nonlinear nonhomogeneous Robin problems with superlinear reaction term}


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

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

\keywords{Critical growth, almost critical growth, boundedness of weak solutions, nonlinear regularity and nonlinear maximum principle, superlinear reaction, three nontrivial solutions, critical groups.\\
\phantom{aa} 2010 AMS Subject Classification: 35J20, 35J60, 58E05}


\begin{abstract}
We consider a nonlinear Robin problem driven by  a nonlinear, nonhomogeneous differential operator and with a Carath\'eodory reaction term which is $(p-1)$-superlinear near $\pm\infty$ without satisfying the Ambrosetti-Rabinowitz condition and which does not have a standard subcritical polynomial growth. Using a combination of variational methods and Morse theoretic techniques, we prove a multiplicity theorem producing three nontrivial solutions (two of which have constant sign). In the process we establish some useful facts about the boundedness of the weak solutions of critical equations and the relation of Sobolev and H\"{o}lder local minimizers for functionals with a critical perturbation term.
\end{abstract}

\maketitle

%\tableofcontents

\section{Introduction}

Let $\Omega\subseteq\RR^N$ be a bounded domain with a $C^2$-boundary $\partial \Omega$ and consider the following semilinear Dirichlet problem
\begin{equation}\label{eq1}
	-\Delta u(z)=f(u(z))\ \mbox{in}\ \Omega,\ u|_{\partial\Omega}=0.
\end{equation}

Suppose that the reaction term $f:\RR\rightarrow\RR$ satisfies the following conditions: $f\in C^1(\RR,\RR)$, $f(0)=f'(0)=0,\ |f'(x)|\leq c_1(1+|x|^{r-2})$ for all $x\in\RR$, with $c_1>0,\ 2<r<2^*=\left\{\begin{array}{ll}
	\frac{2N}{N-2}&\mbox{if}\ N\geq 3\\
	+\infty&\mbox{if}\ N=1,2
\end{array}\right.$ and there exist $\mu>2$ and $M>0$ such that
\begin{equation}\label{eq2}
	0<\mu F(x)\leq f(x)x\ \mbox{for all}\ |x|\geq M\ \mbox{with}\ F(x)=\int^x_0f(s)ds.
\end{equation}

In (\ref{eq2}) we recognize the Ambrosetti-Rabinowitz condition (AR-condition for short). Integrating (\ref{eq2}) we obtain the weaker condition that
\begin{equation}\label{eq3}
	c_2|x|^{\mu}\leq F(x)\ \mbox{for all}\ |x|\geq M\ \mbox{and some}\ c_2>0.
\end{equation}

From (\ref{eq2}) and (\ref{eq3}) it follows that $f(\cdot)$ is superlinear near $\pm\infty$, that is
$$\lim\limits_{x\rightarrow\pm\infty}\frac{f(x)}{x}=+\infty.$$

Under (\ref{eq2}), in a well-known paper, Wang \cite{40} proved that problem (\ref{eq1}) admits at least three nontrivial solutions.

The multiplicity result of Wang \cite{40} was extended to Dirichlet problems driven by the $p$-Laplacian by Liu \cite{24}. More recent works, relaxed the AR-condition. In this direction, we mention the papers of Li and Yang \cite{23}, Miyagaki and Souto \cite{27}, Sun \cite{36} for Dirichlet problems and Aizicovici, Papageorgiou and Staicu \cite{2} for Neumann problems always with the $p$-Laplacian as differential operator. Very recently Mugnai and Papageorgiou \cite{30} extended the aforementioned result of Wang to Dirichlet $(p,q)$-equations (that is, equations driven by the sum of a $p$-Laplacian and a $q$-Laplacian, $1<q<p<\infty$), without assuming the AR-condition.

The aim of this paper is to prove such a ``three solutions theorem" for a larger class of differential equations in which the differential operator need not be homogeneous and covers as a special case the $p$-Laplacian $(1<p<\infty)$. So, as above, let $\Omega\subseteq\RR^N$ be a bounded domain with a $C^2$-boundary $\partial\Omega$. The problem under consideration is the following:
\begin{eqnarray}\label{eq4}
	\left\{\begin{array}{ll}
		-{\rm div}\, a(Du(z))=f(z,u(z))&\mbox{in}\ \Omega,\\
		\frac{\partial u}{\partial n_a}+\beta(z)|u|^{p-2}u=0&\mbox{on}\ \partial\Omega
	\end{array}\right\}
\end{eqnarray}

In this problem $a:\RR^N\rightarrow\RR^N$ is continuous and strictly monotone and satisfies certain other regularity and growth conditions. The precise requirements on the map $a(\cdot)$ are listed in hypotheses $H(a)$ below. These hypotheses are quite general and incorporate in our framework many differential operators of interest such as the $p$-Laplacian and the $(p,q)$-Laplacian. In the boundary condition $\frac{\partial u}{\partial n_a}$ denotes the generalized normal derivative defined by
$$\frac{\partial u}{\partial n_a}=(a(Du),n)_{\RR^N},$$
with $n(\cdot)$ being the outward unit normal on $\partial\Omega$. This particular normal derivative is dictated by the nonlinear Green's identity (see, for example, Gasinski and Papageorgiou \cite[p. 210]{13}) and is also used by Lieberman \cite{21}. The reaction term $f(z,x)$ is a Carath\'eodory function (that is, for all $x\in\RR\ z\rightarrow f(z,x)$ is measurable and for almost all $z\in\Omega$ $x\rightarrow f(z,x)$ is continuous), which is $(p-1)$-superlinear in the $x$-variable but without satisfying the AR-condition. In this way we can fit in our analysis superlinear nonlinearities with ``slower" growth near $\pm\infty$ which fail to satisfy the AR-condition. In addition, $f(z,\cdot)$ needs not to satisfy a polynomial subcritical growth and it grows in an almost critical fashion (see hypothesis $H(f)(i)$). The nonhomogeneity of the differential operator and the failure of the Poincare inequality in the ambient Sobolev space $W^{1,p}(\Omega)$, as well as the almost cri
 tical growth of the reaction term, are sources of difficulties which require new methods and techniques in order to overcome them.

Our approach uses variational tools based on the critical point theory together with Morse theory (critical groups). Also, the almost critical growth of $f(z,\cdot)$, requires a careful analysis of the boundedness of the weak solutions of (\ref{eq4}).


\section{Mathematical Background}

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

This is a compactness-type condition on the functional $\varphi$ which compensates for the fact the ambient space $X$ need not be locally compact (usually $X$ is infinite dimensional). It is more general than the usual Palais-Smale condition. Nevertheless, the $C$-condition leads to a deformation theorem from which one can derive the minimax theory of the critical values of $\varphi$. Prominent in this theory is the so-called ``mountain pass theorem", due to Ambrosetti and Rabinowitz \cite{4}. Here we state it in a slightly more general form (see, for example, Gasinski and Papageorgiou \cite[p. 648]{13}).
\begin{theorem}\label{th1}
	If $X$ is a Banach space, $\varphi\in C^1(X,\RR)$, it satisfies the $C$-condition, $u_0,u_1\in X,\ ||u_1-u_0||>\rho>0$
	$$\max\{\varphi(u_0),\varphi(u_1)\}<\inf[\varphi(u):||u-u_0||=\rho]=m_{\rho}$$
	and $c=\inf\limits_{\gamma\in\Gamma}\max\limits_{0\leq t\leq 1}\varphi(\gamma(t))$ where $\Gamma=\{\gamma\in C([0,1],X):\gamma(0)=u_0,\ \gamma(1)=u_1\}$, then $c\geq m_{\rho}$ and $c$ is a critical value of $\varphi$.
\end{theorem}

Let $\vartheta\in C^1(0,\infty)$ and assume that it satisfies the following growth conditions:
\begin{eqnarray}\label{eq5}
	&&0<\hat{c}\leq\frac{t\vartheta'(t)}{\vartheta(t)}\leq c_0\ \mbox{and}\ c_1t^{p-1}\leq\vartheta(t)\leq c_2(1+t^{p-1})\\
	&&\mbox{for all}\ t>0\ \mbox{and some}\ c_1,c_2>0,1<p<\infty\nonumber
\end{eqnarray}

We introduce the precise conditions on the map $y\rightarrow a(y),y\in\RR^N$, involved in the definition of the differential operator.

$H(a):$ $a(y)=a_0(|y|)y$ for all $y\in\RR^N$ with $a_0(t)>0$ for all $t>0$ and
\begin{itemize}
	\item[(i)] $a_0\in C^1(0,\infty),t\rightarrow a_0(t)t$ is strictly increasing on $(0,\infty),a_0(t)t\rightarrow 0^+$ as $t\rightarrow 0^+$ and
	$$\lim\limits_{t\rightarrow 0^+}\frac{a'_0(t)t}{a_0(t)}>-1;$$
	\item[(ii)] $|\nabla a(y)|\leq c_3\frac{\vartheta(|y|)}{|y|}$ for some $c_3>0$, all $y\in\RR^N\backslash\{0\};$
	\item[(iii)] $\frac{\vartheta(|y|)}{|y|}|\xi|^2\leq(\nabla a(y)\xi,\xi)_{\RR^N}$ for all $y\in\RR^N\backslash\{0\}$, all $\xi\in\RR^N;$
	\item[(iv)] if $G_0(t)=\int^t_0a_0(s)sds$ for all $t\geq 0$\\
	then we have
	$$-\tilde{c}\leq pG_0(t)-a_0(t)t^2$$
	for all $t\geq 0$ with $\tilde{c}>0.$
\end{itemize}
\begin{remark}
	Hypotheses $H(a)(i),(ii),(iii)$ are dictated by the nonlinear regularity theory of Lieberman \cite{22} and the nonlinear maximum principle of Pucci and Serrin \cite{35}. Hypothesis $H(a)(iv)$ corresponds to the particular features of our problem, but it is very mild and it is satisfied in all the major cases of interest as the examples below illustrate.
\end{remark}

Set $G(y)=G_0(|y|)$ for all $y\in\RR^N$. We have
$$\nabla G(y)=G'_0(|y|)\frac{y}{|y|}=a_0(|y|)y=a(y)\ \mbox{for all}\ y\in\RR^N\backslash\{0\},\nabla G(0)=0.$$

So, $G(\cdot)$ is the primitive of $a(\cdot)$.

Hypotheses $H(a)$ imply that the functions $G(\cdot),G_0(\cdot)$ are both strictly convex and $G_0(\cdot)$ is also strictly increasing. The convexity of $G(\cdot)$ and since $ G(0)=0$, imply
\begin{equation}\label{eq6}
	G(y)\leq(a(y),y)_{\RR^N}\ \mbox{for all}\ y\in\RR^N.
\end{equation}

The next lemma summarizes the main properties of the map $a(\cdot)$ and it is a straightforward consequence of hypotheses $H(a)$.
\begin{lemma}\label{lem2}
	If hypotheses $H(a)(i),(ii),(iii)$ hold, then
	\begin{itemize}
		\item[(a)] $y\rightarrow a(y)$ is continuous and strictly monotone, hence maximal monotone too;
		\item[(b)] $|a(y)|\leq c_4(1+|y|^{p-1})$ for some $c_4>0$, all $y\in\RR^N$;
		\item[(c)] $(a(y),y)_{\RR^N}\geq\frac{c_1}{p-1}|y|^p$ for all $y\in\RR^N$.
	\end{itemize}
\end{lemma}

This lemma together with (\ref{eq5}) and (\ref{eq6}), leads to the following growth estimates for the primitive $G(\cdot)$.
\begin{corollary}\label{cor3}
	If hypotheses $H(a)(i),(ii),(iii)$ hold, then $\frac{c_1}{p-1}|y|^p\leq G(y)\leq c_5(1+|y|^p)$ for all $y\in\RR^N$ and some $c_5>0$.
\end{corollary}
\begin{ex}
	The following maps $a(\cdot)$ satisfy hypotheses $H(a)$:
	\begin{itemize}
		\item[(a)] $a(y)=|y|^{p-2}y$ with $1<p<\infty$.\\
		This map corresponds to the $p$-Laplace differential operator defined by
		$$\Delta_pu={\rm div}\,(|Du|^{p-2}Du)\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$
		\item[(b)] $a(y)=|y|^{p-2}y+|y|^{q-2}y$ with $1<p<q<\infty$.\\
		This map corresponds to the $(p,q)$-differential operator defined by
		$$\Delta_pu+\Delta_qu\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$
		Such differential operators arise in many physical applications, see Benci, D'Avenia, Fortunato and Pisani \cite{6} (quantum physics) and Cherfils and Ilyasov \cite{8} (plasma physics). Recently there have been some existence and multiplicity results for such equations. We refer to the works of Barile and Figueiredo \cite{5}, Cingolani and Degiovanni \cite{9}, Gasinski and Papageorgiou \cite{15}, Marano and Papageorgiou \cite{26}, Papageorgiou and R\u adulescu \cite{32},\cite{33}, Sun \cite{37}, Sun, Zhang and Su \cite{38}.
		\item[(c)] $a(y)=(1+|y|^2)^{\frac{p-2}{2}}y$ with $1<p<\infty.$\\
		This map corresponds to the generalized $p$-mean curvature differential operator defined by
		$${\rm div}\,\left((1+|Du|^2)^{\frac{p-2}{2}}Du\right)\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$
		\item[(d)] $a(y)=|y|^{p-2}y\left(1+\frac{1}{(1+|y|^{2p})^{1/2}}\right)$ with $1<p<\infty$.
	\end{itemize}
\end{ex}

Our hypotheses on the boundary weight function $\beta(\cdot)$ are:

$H(\beta):$ $\beta\in C^{1,\alpha}(\partial\Omega)$ with $\alpha\in(0,1),\beta\geq 0$.
\begin{remark}
	If $\beta\equiv0$, then we have the Neumann problem.
\end{remark}

In the analysis of problem (\ref{eq4}), in addition to the Sobolev space $W^{1,p}(\Omega)$ we will also use the Banach space $C^1(\overline{\Omega})$. This is an ordered Banach space with positive cone given by
$$C_+=\{u\in C^1(\overline{\Omega}):u(z)\geq 0\ \mbox{for all}\ z\in\overline{\Omega}\}.$$

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

On $\partial\Omega$ we use the $(N-1)$-dimensional surface (Hausdorff) measure $\sigma(\cdot)$ and using this measure we can define the Lebesgue spaces $L^p(\partial\Omega)\ (1\leq p\leq\infty)$. We know that there exists a unique continuous linear map $\gamma_0:W^{1,p}(\Omega)\rightarrow L^p(\partial\Omega)$, known as the ``trace map" such that $\gamma_0(u)=u|_{\partial\Omega}$ for all $u\in W^{1,p}(\Omega)\cap C(\overline{\Omega})$. Recall that, $im\gamma_0=W^{\frac{1}{p'},p}(\partial\Omega)\left(\frac{1}{p}+\frac{1}{p'}=1\right)$ and $ker\gamma_0=W^{1,p}_0(\Omega)$. Moreover, the trace map $\gamma_0$ is compact into $L^q(\partial\Omega)$ for all $q\in\left[1,\frac{Np-p}{N-p}\right)$. In the sequel for the sake of notational simplicity, we will drop the use of the trace map $\gamma_0$. It is understood that all restrictions of Sobolev functions on the boundary $\partial\Omega$ are defined in the sense of traces.

In what follows, by $||\cdot||$ we denote the norm of the Sobolev space $W^{1,p}(\Omega)$ defined by
$$||u||=\left[||u||^p_p+||Du||^p_p\right]^{1/p}\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$

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

Also, by $|\cdot|_N$ we denote the Lebesgue measure on $\RR^N$ and by $A_p:W^{1,p}(\Omega)\rightarrow W^{1,p}(\Omega)^*$ the nonlinear map defined by
$$\left\langle A_p(u),h\right\rangle=\int_{\Omega}|Du|^{p-2}(Du,Dh)_{\RR^N}dz\ \mbox{for all}\ u,h\in W^{1,p}(\Omega).$$

Let $A:W^{1,p}(\Omega)\rightarrow W^{1,p}(\Omega)^*$ be the nonlinear map defined by
\begin{equation}\label{eq7}
	\left\langle A(u),h\right\rangle=\int_{\Omega}(a(Du),Dh)_{\RR^N}dz\ \mbox{for all}\ u,h\in W^{1,p}(\Omega).
\end{equation}

The next proposition is a special case of a more general result of Gasinski and Papageorgiou \cite{14}.
\begin{prop}\label{prop4}
	The map $A:W^{1,p}(\Omega)\rightarrow W^{1,p}(\Omega)^*$ defined by (\ref{eq7}) is bounded (that is, maps bounded sets to bounded sets), continuous, monotone (hence maximal monotone too) 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 $W^{1,p}(\Omega)$ as $n\rightarrow\infty$.
\end{prop}

We consider the following nonlinear Robin problem:
\begin{eqnarray}\label{eq8}
	\left\{\begin{array}{ll}
		-\mbox{div}\,a(Du(z))=f_0(z,u(z))&\mbox{in}\ \Omega,\\
		\frac{\partial u}{\partial n_a}=\beta(z)|u|^{p-2}u&\mbox{on}\ \partial\Omega.
	\end{array}\right\}
\end{eqnarray}

In this problem $f_0:\Omega\times\RR\rightarrow\RR$ is a Carath\'eodory function with critical growth in the $x$-variable, that is,
\begin{equation}\label{eq9}
	|f_0(z,x)|\leq a_0(z)(1+|x|^{p^*-1})\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ z\in\RR,
\end{equation}
with $a_0\in L^{\infty}(\Omega)_+,p^*=\left\{\begin{array}{ll}
	\frac{Np}{N-p}&\mbox{if}\ p<N\\
	+\infty&\mbox{if}\ N\leq p
\end{array}\right.$.

By a ``weak solution" of problem (\ref{eq8}) we understand a function $u\in W^{1,p}(\Omega)$ such that
$$\int_{\Omega}(a(Du),Dh)_{\RR^N}dz+\int_{\partial\Omega}\beta(z)|u|^{p-2}uhd\sigma=\int_{\Omega}f_0(z,u)hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega).$$

Next we establish the boundedness of weak solutions. Due to the critical growth of $f(z,\cdot)$, the Moser iteration technique used by Hu and Papageorgiou \cite{18} and Winkert \cite{41} does not work. Instead, we follow the approach of Garcia Azorero and Peral Alonso \cite{11} (see also Wang \cite{39} for semilinear equations). An alternative method can be based on the work of Guedda and V\'eron \cite{17}.
\begin{prop}\label{prop5}
	If hypotheses $H(a),H(\beta)$ hold and $u\in W^{1,p}(\Omega)$ is a weak solution of (\ref{eq8}), then $u\in L^q(\Omega)$ for all $q\in\left[1,\infty\right)$.
\end{prop}
\begin{proof}
	Recalling that $u=u^+-u^-$ and performing the argument on $u^+$ and $u^-$ separately we see that without any loss of generality, we may assume that $u\geq 0$.
	
	For $\beta>1$ and $\lambda>0$, we introduce the following Lipschitz continuous functions
	$$H(t)=\left\{\begin{array}{ll}
		t^{\beta}&\mbox{if}\ 0\leq t\leq \lambda\\
		\beta\lambda^{\beta-1}(t-\lambda)+\lambda^{\beta}&\mbox{if}\ \lambda<t
	\end{array}\right.$$
	$$S(t)=\left\{\begin{array}{ll}
		t^{(\beta-1)p+1}&\mbox{if}\ 0\leq t\leq \lambda\\
		\beta((\beta-1)p+1)\lambda^{(\beta-1)p}(t-\lambda)+\lambda^{(\beta-1)p+1}&\mbox{if}\ \lambda<t.
	\end{array}\right.$$
	
	It is easy to check (see also Garcia Azorero and Peral Alonso \cite{11}), that these two functions satisfy the following properties:
	\begin{eqnarray}
		&&S(t)\leq tS'(t)\ \mbox{for all}\ t\geq 0\label{eq10}\\
		&&c_5H'(t)\leq S'(t)\ \mbox{for all}\ t\geq 0\ \mbox{with}\ c_5>0\ \mbox{independent of}\ \lambda>0\label{eq11}\\
		&&t^{p-1}S(t)\leq c_7H(t)^p\ \mbox{for all}\ t\geq 0,\ \mbox{with}\ c_7>0\ \mbox{independent of}\ \lambda>0\label{eq12}\\
		&&\mbox{for every}\ y\in W^{1,p}(\Omega),H(y),S(y)\in W^{1,p}(\Omega)\label{eq13}\\
		&&\mbox{(see, for example, Gasinski and Papageorgiou \cite[p. 194]{13}.}\nonumber
	\end{eqnarray}
	
	We fix $\beta>1$ such that $\beta p<p^*$ and let $\vartheta\in C^{\infty}_c(\RR^N,\RR)$ with $0\leq\vartheta\leq 1$ to be fixed more precisely in the process of the proof. We use the test function
	$$h=\vartheta^p S(u)\in W^{1,p}(\Omega),h\geq 0.$$
	
	We have
	\begin{equation}\label{eq14} \int_{\Omega}(a(Du),Dh)_{\RR^N}dz+\int_{\partial\Omega}\beta(z)u^{p-1}hd\sigma=\int_{\Omega}f_0(z,u)hdz.
	\end{equation}
	
	Note that
	$$Dh=p\vartheta^{p-1}S(u)D\vartheta+\vartheta^pG'(u)Du.$$
	
	So, we have
	\begin{eqnarray}\label{eq15}		 &&\int_{\Omega}(a(Du),Dh)_{\RR^N}dz=p\int_{\Omega}\vartheta^{p-1}S(u)(a(Du),D\vartheta)_{\RR^N}dz\nonumber\\
		&&\hspace{4cm}+\int_{\Omega}\vartheta^pG'(u)(a(Du),Du)_{\RR^N}dz.
	\end{eqnarray}
	
	Using (\ref{eq15}) in (\ref{eq14}), we have
	\begin{eqnarray}\label{eq16}		 &&\int_{\Omega}\vartheta^pG'(u)(a(Du),Du)_{\RR^N}dz+\int_{\partial\Omega}\beta(z)u^{p-1}hd\sigma\nonumber\\
		&\leq&\int_{\Omega}f_0(z,u)hdz-p\int_{\Omega}\vartheta^{p-1}S(u)(a(Du),D\vartheta)_{\RR^N}dz.
	\end{eqnarray}
	
	From Lemma \ref{lem2} and since $\vartheta^pS'(u)\geq 0$ (see (\ref{eq10})), we have
	\begin{equation}\label{eq17}
		 \frac{c_1}{p-1}\int_{\Omega}\vartheta^pS'(u)|Du|^pdz\leq\int_{\Omega}\vartheta^pS'(u)(a(Du),Du)_{\RR^N}dz.
	\end{equation}
	
	Also we have
	\begin{eqnarray}\label{eq18}
		&&\left|p\int_{\Omega}\vartheta^{p-1}S(u)(a(Du),D\vartheta)_{\RR^N}dz\right|\nonumber\\
		&\leq&p\int_{\Omega}\vartheta^{p-1}S(u)|a(Du)||D\vartheta|dz\nonumber\\
		 &=&p\int_{\Omega}\vartheta^{p-1}S(u)^{\frac{1}{p}}S(u)^{\frac{p-1}{p}}|a(Du)||D\vartheta|dz\nonumber\\
		&\leq&p\int_{\Omega}\vartheta^{p-1}|a(Du)|S(u)^{\frac{1}{p}}(uS'(u))^{\frac{p-1}{p}}|D\vartheta|dz\ (\mbox{see (\ref{eq10})})\nonumber\\
		 &\leq&\epsilon\int_{\Omega}\vartheta^{p}|a(Du)|^{\frac{p}{p-1}}S'(u)dz+c_{\epsilon}\int_{\Omega}u^{p-1}S(u)|D\vartheta|dz\nonumber\\
		&&\hspace{2cm}\mbox{with}\ c_{\epsilon}>0\ (\mbox{use Young's inequality with}\ \epsilon>0)\nonumber\\
		 &\leq&\epsilon\int_{\Omega}c_8(1+|Du|^p)S'(u)\vartheta^pdz+c_{\epsilon}\int_{\Omega}u^{p-1}S(u)|D\vartheta|^pdz\\
		&&\hspace{2cm}\mbox{for some}\ c_8>0\ (\mbox{see Lemma \ref{lem2}}).\nonumber
	\end{eqnarray}
	
	We return to (\ref{eq16}) and use (\ref{eq17}),(\ref{eq18}). Then choosing $\epsilon\in(0,1)$ small and since\\ $\int_{\partial\Omega}\beta(z)u^{p-1}hd\sigma\geq 0$, we have
	\begin{eqnarray}\label{eq19}
		&&\int_{\Omega}\vartheta^pS'(u)|Du|^pdz\nonumber\\
		&\leq&\epsilon c_9\int_{\Omega}\vartheta^pS'(u)dz+c_9c_{\epsilon}\int_{\Omega}u^{p-1}S(u)|D\vartheta|^pdz+c_9\int_{\Omega}f_0(z,u)\vartheta^pS(u)dz\nonumber\\
		&&\hspace{6cm}\mbox{for some}\ c_9>0\nonumber\\
		&\leq&\epsilon c_9\int_{\Omega}\vartheta^pS'(u)dz+c_9c_{\epsilon}\int_{\Omega}H(u)^p|D\vartheta|^pdz\nonumber\\
		 &&+c_9||a_0||_{\infty}\int_{\Omega}\vartheta^pS(u)dz+c_9||a_0||_{\infty}\int_{\Omega}u^{p^*-1}\vartheta^pS(u)dz\\
		&&\hspace{6cm}(\mbox{see (\ref{eq9}) and (\ref{eq12})}).\nonumber
	\end{eqnarray}
	
	Using (\ref{eq11}) we obtain
	\begin{equation}\label{eq20}
		c_{10}\int_{\Omega}(\vartheta H'(u))^p|Du|^pdz\leq\int_{\Omega}\vartheta^pS'(u)|Du|^pdz\ \mbox{for some}\ c_{10}>0.
	\end{equation}
	
	Then we can have the following estimate
	\begin{eqnarray}\label{eq21}
		\int_{\Omega}|D(\vartheta H(u))|^pdz&=&\int_{\Omega}|\vartheta H'(u)Du+H(u)D\vartheta|^pdz\nonumber\\
		&\leq&c_{11}\left[\int_{\Omega}|\vartheta H'(u)Du|^pdz+\int_{\Omega}H(u)^p|D\vartheta|^pdz\right]\ \mbox{for some}\ c_{11}>0\nonumber\\
		 &\leq&c_{12}\left[\int_{\Omega}\vartheta^pS'(u)|Du|^pdz+\int_{\Omega}H(u)^p|D\vartheta|^pdz\right]\nonumber\\
		&&\hspace{4cm}\mbox{for some}\ c_{12}>0\ (\mbox{see (\ref{eq20})})\nonumber\\
		&\leq&c_{13}\left[\int_{\Omega}H(u)^p|D\vartheta|^pdz+\int_{\Omega}u^{p^*-p}(\vartheta H(u))^pdz+\int_{\Omega}\vartheta^pS(u)dz\right]\nonumber\\
		&&\hspace{4cm}\mbox{for some}\ c_{13}>0\ (\mbox{see (\ref{eq19}) and (\ref{eq12})})\nonumber\\
		&\leq&c_{14}\left[\int_{\Omega}H(u)^p|D\vartheta|^pdz+\int_{\Omega}u^{p^*-p}(\vartheta H(u))^pdz+1\right]\\
		&&\hspace{1cm}\mbox{for some}\ c_{14}>0\ (\mbox{since}\ 0\leq\vartheta\leq 1\ \mbox{and}\ |D\vartheta(\cdot)|\ \mbox{is bounded}).\nonumber
	\end{eqnarray}
	
	We choose $\rho>0$ such that for any ball $B_{\rho}$ of radius $\rho>0$ with $B_{\rho}\cap\Omega\neq\varnothing$. We have
	\begin{equation}\label{eq22}
		||u||^{p^*-p}_{L^{p^*}(B_{\rho}\cap\Omega)}\leq\frac{1}{\eta c_{14}}\ \mbox{with}\ \eta>0
	\end{equation}
	(recall that $W^{1,p}(\Omega)\hookrightarrow L^{p^*}(\Omega)$).
	
	Given $z_0\in\Omega$ choose $\vartheta\in C^{\infty}_{c}(\RR^N)$ with $0\leq\vartheta\leq 1$, ${\rm supp}\,\vartheta=\bar{B}_{\rho}(z_0)$ and $\vartheta\equiv 1$ on $\bar{B}_{\rho/2}(z_0)$. Using H\"{o}lder's inequality, we have
	\begin{eqnarray}\label{eq23}
		&&\int_{\Omega}u^{p^*-p}\vartheta^pH(u)^pdz\nonumber\\
		&=&\int_{B_{\rho}(z_0)\cap\Omega}u^{p^*-p}\vartheta^pH(u)^pdz\nonumber\\
		 &\leq&\left(\int_{\Omega}\vartheta^{p^*}H(u)^{p^*}dz\right)^{\frac{p}{p^*}}\left(\int_{B_{\rho}(z_0)\cap\Omega}u^{p^*}dz\right)^{\frac{p^*-p}{p^*}}\nonumber\\
		&\leq&\frac{1}{\eta c_{14}}\left(\int_{\Omega}\vartheta ^{p^*} H(u)^{p^*}dz\right)^{\frac{p}{p^*}}\ (\mbox{see (\ref{eq22})}).
	\end{eqnarray}
	
	Note that for $\delta>0,u\rightarrow\delta||u||_{p^*}+||Du||_p$ is an equivalent norm on the Sobolev space $W^{1,p}(\Omega)$ (see, for example, Gasinski and Papageorgiou \cite[p. 227]{13}). So, by choosing $\delta>0$ small, we can find $c_{15}>0$ such that
	\begin{equation}\label{eq24}
		\left(\int_{\Omega}\vartheta^{p^*}H(u)^{p^*}dz\right)^{\frac{p}{p^*}}\leq c_{15}\int_{\Omega}|D(\vartheta H(u))|^pdz.
	\end{equation}
	
	Using (\ref{eq24}) in (\ref{eq23}), we obtain
	\begin{equation}\label{eq25}
		\int_{\Omega}u^{p^*-p}\vartheta^p H(u)^pdz\leq\frac{c_{15}}{\eta c_{14}}\int_{\Omega}|D(\vartheta H(u))|^pdz.
	\end{equation}
	
	Returning to (\ref{eq21}) and using (\ref{eq25}) with $\eta>\frac{c_{15}}{c_{14}}$, we have
	\begin{eqnarray}\label{eq26}
		&&\int_{\Omega}|D(\vartheta H(u))|^pdz\leq c_{16}\left[\int_{\Omega}H(u)^pdz+1\right]\ \mbox{for some}\ c_{16}>0,\nonumber\\
		&\Rightarrow&\left(\int_{\Omega}\vartheta^{p^*}H(u)^{p^*}dz\right)^{\frac{p}{p^*}}\leq c_{17}\left[\int_{\Omega}H(u)^pdz+1\right]\ \mbox{for some}\ c_{17}>0\ \mbox{(see (\ref{eq24}))}.
	\end{eqnarray}
	
	We let $\lambda\rightarrow+\infty$ (see the definition of $H(\cdot)$ in the beginning of the proof). It follows that
	\begin{equation}\label{eq27}
		\left(\int_{B_{\rho/2}(z_0)\cap\Omega}u^{\beta p^*}dz\right)^{\frac{p}{p^*}}\leq c_{17}\left[\int_{\Omega}u^{\beta p}dz+1\right]\ (\mbox{see (\ref{eq26})}).
	\end{equation}
	
	Since $\beta p<p^*$ and $u\in W^{1,p}(\Omega)$, we have that $u\in L^{\beta p^*}(B_{\rho/2}(z_0)\cap\Omega)$. Then from (\ref{eq27}) and since $\Omega$ is totally bounded, we infer that $u\in L^{\beta p^*}(\Omega)$. Fix $\epsilon_0>0$ such that $\beta-\epsilon_0>1$. Then  repeating the above argument we can generate a sequence $\{\beta_n\}_{n\geq 1}$ such that $\beta_1p^*<\beta p^*,\beta_n\geq(\beta-\epsilon_0)^n$ and $u\in L^{\beta_np^*}(\Omega)$ for all $n\in\NN$. Since $(\beta-\epsilon_0)^n\rightarrow+\infty$, we conclude that $u\in L^q(\Omega)$ for all $q\in\left[1,\infty\right)$.
\end{proof}

Next we will establish the essential boundedness of $u$ and produce a useful bound for its $L^{\infty}$-norm. We start with a lemma, which is essentially Lemma 5.1,p.71, of Ladyzhenskaya and Uraltseva \cite{19}. For completeness in our argument we include it here.
\begin{lemma}\label{lem6}
	If $u\in W^{1,p}(\Omega),0\leq u,q\in(1,p^*),k_0>1$ and $\bar{c}>0$ are such that
	\begin{equation}\label{eq28}
		\int_{E_k}|Du|^pdz\leq\bar{c}k^p|E_k|^{\frac{p}{q}}_{N}\ \mbox{for all}\ k\geq k_0
	\end{equation}
where $E_k=\{z\in\Omega:u(z)\geq k\}$, then there exists $M_1=M_1(\Omega,\bar{c},q,k_0)>0$ such that $||u||_{\infty}\leq M.$
\end{lemma}
\begin{proof}
	From Ladyzhenskaya and Uraltseva \cite[p. 45]{19}, we know that
	\begin{equation}\label{eq29}
		\left(\int_{E_k}(u-k)^pdz\right)^{1/p}\leq c_{18}\left(\int_{E_k}|Du|^pdz\right)^{1/p}|E_k|^{\frac{1}{p}-\frac{1}{p^*}}_{N}\ \mbox{for some}\ c_{18}>0.
	\end{equation}
	
	Using H\"{o}lder's inequality, we have
	\begin{eqnarray}\label{eq30}
		&&\int_{E_k}(u-k)dz\leq\left(\int_{E_k}(u-k)^pdz\right)^{1/p}|E_k|^{1-\frac{1}{p}}_{N}\nonumber\\
		 &\leq&c_{18}\left(\int_{E_k}|Du|^pdz\right)^{1/p}|E_k|^{\frac{1}{p}-\frac{1}{p^*}}_{N}|E_k|^{1-\frac{1}{p}}\ \mbox{(see (\ref{eq29}))}\nonumber\\
		&\leq&c_{19}k|E_k|^{1+\frac{1}{q}-\frac{1}{p^*}}\ \mbox{for all}\ k\geq k_0\ (\mbox{see (\ref{eq28})}).
	\end{eqnarray}
	
	Let $\vartheta=\frac{1}{q}-\frac{1}{p^*}>0$ (recall that $\vartheta\in(1,p^*)$). Then from (\ref{eq30}) we have
	\begin{equation}\label{eq31}
		\int_{E_k}(u-k)dz\leq c_{19}k|E_k|^{1+\vartheta}_{N}\ \mbox{for all}\ k\geq k_0.
	\end{equation}
	
	We set $\xi(k)=\int_{E_k}(u-k)dz=\int^{\infty}_{k}|E_s|_Nds$ (see Ziemer \cite[p. 19]{42}). We have
	\begin{equation}\label{eq32}
		-\xi'(k)=|E_k|_N.
	\end{equation}
	
	From (\ref{eq31}) we have
	\begin{eqnarray}\label{eq33}
		&&\xi(k)^{-\frac{1}{1+\vartheta}}\geq(c_{19}k)^{-\frac{1}{1+\vartheta}}|E_k|^{-1}_{N},\nonumber\\
		&\Rightarrow&-\xi'(k)\xi(k)^{-\frac{1}{1+\vartheta}}\geq(c_{19}k)^{-\frac{1}{1+\vartheta}}\ (\mbox{see (\ref{eq32})}).
	\end{eqnarray}
	
	Let $k^*=\esssup\limits_{\Omega}u$ and integrate (\ref{eq33}) from $k_0$ to $k^*$. Then
	$$(k^*)^{\frac{\vartheta}{1+\vartheta}}\leq k_0^{\frac{\vartheta}{1+\vartheta}}+c_{19}\xi(k_0)^{\frac{\vartheta}{1+\vartheta}}=M_1^{\frac{1+\vartheta}{\vartheta}}.$$
\end{proof}
Now we are ready to establish the essential boundedness of the weak solutions of problem (\ref{eq8}) and provide a useful description of their bound.
\begin{prop}\label{prop7}
	If hypotheses $H(a),H(\beta)$ hold and $u\in W^{1,p}(\Omega)$ is a weak solution of problem (\ref{eq8}), then there exists $M_2=M_2(p,N,||u||_{p^*},\Omega)>0$ such that $||u||_{\infty}\leq M_2$.
\end{prop}
\begin{proof}
	As in the proof of Proposition \ref{prop5}, without any loss of generality, we may assume that $u\geq 0$.
	
	Let $u_k=(u-k)^+\in W^{1,p}(\Omega)$ and $E_k={\rm supp}\, u_k\ k\in\NN$. Since $u\in W^{1,p}(\Omega)$ is a weak solution of the Robin problem (\ref{eq8}), we have
	\begin{equation}\label{eq34}
		 \int_{\Omega}(a(Du),Dh)_{\RR^N}dz+\int_{\partial\Omega}\beta(z)u^{p-1}hd\sigma=\int_{\Omega}f_0(z,u)hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega).
	\end{equation}
	
	In (\ref{eq34}) we choose $h=u_k\in W^{1,p}(\Omega)$. We obtain
	\begin{eqnarray}\label{eq35}
		&&\int_{E_k}(a(Du),Du)_{\RR^N}dz\leq\int_{E_k}f_0(z,u)udz\ (\mbox{see hypothesis}\ H(\beta)),\nonumber\\
		&\Rightarrow&\frac{c_1}{p-1}\int_{E_k}|Du|^pdz\leq\int_{E_k}f_0(z,u)udz\ (\mbox{see Lemma \ref{lem2}}).
	\end{eqnarray}
	
	Note that
	\begin{eqnarray}\label{eq36}
		\left|\int_{E_k}f_0(z,u)udz\right|&\leq&\int_{E_k}|f_0(z,u)||u|dz\nonumber\\
		&\leq&c_{20}\int_{E_k}(1+u^{p^*-1})udz\ \mbox{for some}\ c_{20}>0\ (\mbox{see (\ref{eq9})})\nonumber\\
		&\leq&2c_{20}\int_{E_k}u^{p^*-1}udz\ (\mbox{since}\ k\in\NN)\nonumber\\
		&=&2c_{20}\int_{E_k}u^pu^{p^*-p}dz.
	\end{eqnarray}
	
	We choose $q\in(p,p^*)$. Using Proposition \ref{prop5} we have
	$$u^p\in L^{\frac{q}{p}}(\Omega)\ \mbox{and}\ u^{p^*-p}\in L^{\frac{q}{q-p}}(\Omega).$$
	
	Note that $\frac{p}{q}+\frac{q-p}{q}=1$. So, using H\"{o}lder's inequality, in (\ref{eq36}), we have
	\begin{eqnarray*}
		 \left|\int_{E_k}f_0(z,u)udz\right|&\leq&2c_{20}\left(\int_{E_k}u^qdz\right)^{\frac{p}{q}}\left(\int_{E_k}u^{(p^*-p)\frac{q}{p-q}}dz\right)^{\frac{q-p}{q}}\\
		&\leq&c_{21}\left(\int_{E_k}u^qdz\right)^{\frac{p}{q}}\ \mbox{for some}\ c_{21}=c_{21}(||u||_{p^*})>0\\
		&&\hspace{3cm}(\mbox{see Proposition \ref{prop5} and (\ref{eq27})})\\
		&=&c_{21}\left(\int_{E_k}(u-k+k)^qdz\right)^{\frac{p}{q}}\\
		&\leq&c_{22}\left(\int_{E_k}(u-k)^qdz\right)^{\frac{p}{q}}+c_{22}k^p|E_k|^{\frac{p}{q}}_{N}\ \mbox{for some}\ c_{22}>0\\
		 &\leq&c_{23}|E_k|^{\frac{1}{q}-\frac{1}{p^*}}_{N}\int_{E_k}|Du|^pdz+c_{22}k^p|E_k|^{\frac{p}{q}}_{N}\ \mbox{for some}\ c_{23}>0\\
		&&\hspace{2cm}(\mbox{see Ladyzhenskaya and Uraltseva \cite[p. 45]{19}})
	\end{eqnarray*}
	
	Returning to (\ref{eq35}) and choosing $k\in\NN$ big so that $|E_k|_N$ is small, we have
	\begin{equation}\label{eq37}
		\int_{E_k}|Du|^pdz\leq c_{24}k^p|E_k|^{\frac{p}{q}}_{N}\ \mbox{for some}\ c_{24}>0
	\end{equation}
	(note that all the above estimation constants depend only on $(p,N,||u||_{p^*},\Omega)$). Then from (\ref{eq37}) and Lemma \ref{lem6}, we see that we can find $M_2=M_2(p,N,||u||_{p^*},\Omega)>0$ such that
	$$u\in L^{\infty}(\Omega)\ \mbox{with}\ ||u||_{\infty}\leq M_2.$$
\end{proof}
\begin{remark}
	As we already said an alternative approach can be based on the work of Guedda and V\'eron \cite{17}. Indeed, let $K(z)=\frac{\mbox{sign}\,(u)f_0(z,u(z))}{1+|u(z)|^{p-1}}$. Then
	\begin{eqnarray*}
		|K(z)|&\leq&\frac{c_{25}(1+|u(z)|^{p^*}-1)}{1+|u(z)|^{p-1}}\ \mbox{for some}\ c_{25}>0\ (\mbox{see (\ref{eq9})})\\
		&\leq&c_{26}(1+|u(z)|^{p^*-p})\ \mbox{for almost all}\ z\in\Omega\ \mbox{and some}\ c_{26}>0.
	\end{eqnarray*}
\end{remark}

	Note that $p^*-p=\frac{p^2}{N-p}$ for $p<N$ and recall that $u\in L^{p^*}(\Omega)$. Hence $K\in L^{\frac{N}{p}}(\Omega)$. We have
	\begin{eqnarray*}
		-\mbox{div}\, a(Du(z))&=&K(z)|u(z)|^{p-2}u(z)+\mbox{sign}\,(u)K(z)\\
		&=&K(z)(|u(z)|^{p-2}u(z)+\mbox{sign}\,(u))\\
		&=&\frac{f_0(z,u(z))}{1+|u(z)|^{p-1}}(1+|u(z)|^{p-1})=f_0(z,u(z)).
	\end{eqnarray*}
	
	So, keeping in mind that for every $\epsilon>0$, $u\mapsto\epsilon||u||_{p^*}+||Du||_p$ is an equivalent norm on $W^{1,p}(\Omega)$, we can follow the proof of Proposition 2.1 of \cite{17} (with suitable modifications to accommodate the more general differential operator and the boundary term), to prove that $u\in L^q(\Omega)$ for all $q\in\left[1,+\infty\right)$. Then we can continue with Lemma \ref{lem6} and Proposition \ref{prop7} to reach the desired conclusion.
	
	We can use Proposition \ref{prop7} to prove a result comparing Sobolev and H\"{o}lder local minimizers of certain $C^1$-functionals. Such a result was first proved by Brezis and Nirenberg \cite{7} for functionals defined on $H^1_0(\Omega)$ and its was extended to functionals defined on $W^{1,p}_{0}(\Omega)$ by Azorero-Peral, Peral Alonso and Manfredi \cite{12} and to functionals defined on $W^{1,p}(\Omega)$ by Motreanu and Papageorgiou \cite{29} and Papageorgiou and R\u adulescu \cite{34}. All these works involve perturbation terms with subcritical growth. Our result here is more general, since the functional is more general and the perturbation has critical growth.
	
	So, let $F_0(z,x)=\int^x_0f_0(z,s)ds$ and consider the $C^1$-functional $\varphi_0:W^{1,p}(\Omega)\rightarrow\RR$ defined by
	 $$\varphi_0(u)=\int_{\Omega}G(Du)dz+\frac{1}{p}\int_{\partial\Omega}\beta(z)|u|^pd\sigma-\int_{\Omega}F_0(z,u)dz\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$
	\begin{prop}\label{prop8}
		If $u_0\in W^{1,p}(\Omega)$ is a local $C^1(\overline{\Omega})$ minimizer of $\varphi_0$, that is, we can find $\rho_0>0$ such that
		$$\varphi_0(u_0)\leq\varphi_0(u_0+h)\ \mbox{for all}\ h\in C^1(\overline{\Omega})\ \mbox{with}\ ||h||_{C^1(\overline{\Omega})}\leq\rho_0,$$
		then $u_0\in C^{1,\eta}(\overline{\Omega})$ for some $\eta\in(0,1)$ and $u_0$ is also a local $W^{1,p}(\Omega)$-minimizer of $\varphi_0$, that is, we can find $\rho_1>0$ such that
		$$\varphi_0(u_0)\leq\varphi_0(u_0+h)\ \mbox{for all}\ h\in W^{1,p}(\Omega)\ \mbox{with}\ ||h||\leq\rho_1.$$
	\end{prop}
	\begin{proof}
		Since by hypothesis $u_0$ is a local $C^1(\overline{\Omega})$-minimizer of $\varphi_0$, for every $h\in C^1(\overline{\Omega})$ and for $t>0$ small we have
		\begin{eqnarray}\label{eq38}
			&&\varphi_0(u_0)\leq\varphi_0(u_0+th),\nonumber\\
			&\Rightarrow&0\leq\left\langle \varphi'_0(u_0),h\right\rangle\ \mbox{for all}\ h\in C^1(\overline{\Omega}).
		\end{eqnarray}
		
		Recalling that $C^1(\overline{\Omega})$ is dense in $W^{1,p}(\Omega)$, from (\ref{eq38}) we infer that
		\begin{eqnarray}\label{eq39}
			&&\varphi'_0(u_0)=0\nonumber\\
			&\Rightarrow&\left\langle A(u_0),h\right\rangle+\int_{\partial\Omega}\beta(z)|u_0|^{p-2}u_0hd\sigma=\int_{\Omega}f_0(z,u_0)hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega).
		\end{eqnarray}
		
		From the nonlinear Green's identity, we have
		\begin{equation}\label{eq40}
			\left\langle A(u_0),h\right\rangle=\left\langle -\mbox{div}\, a(Du_0),h\right\rangle+\left\langle \frac{\partial u_0}{\partial n_a},h\right\rangle_{\partial\Omega}\ \mbox{for all}\ h\in W^{1,p}(\Omega),
		\end{equation}
		where by $\left\langle \cdot,\cdot\right\rangle_{\partial\Omega}$ we denote the duality brackets for the pair $(W^{-\frac{1}{p'},p'}(\partial\Omega),W^{\frac{1}{p'},p}(\partial\Omega))$. Note that
		$$\mbox{div}\, a(Du_0)\in W^{-1,p'}(\Omega)=W^{1,p}_0(\Omega)^*.$$
		 So, if by $\left\langle \cdot,\cdot\right\rangle_0$ we denote the duality brackets for the pair $(W^{-1,p'}(\Omega),W^{1,p}_{0}(\Omega))$ we have
		\begin{eqnarray}\label{eq41}
			&&\left\langle -\mbox{div}\,a(Du_0),h\right\rangle_0=\left\langle A(u_0),h\right\rangle_0=\left\langle A(u_0),h\right\rangle\nonumber\\
			&&\hspace{2cm}\mbox{for all}\ h\in W^{1,p}_0(\Omega)\subseteq W^{1,p}(\Omega)\ (\mbox{see (\ref{eq40})}),\nonumber\\
			&\Rightarrow&\left\langle -\mbox{div}\,a(Du_0),h\right\rangle_0=\int_{\Omega}f_0(z,u_0)hdz\ \mbox{for all}\ h\in W^{1,p}_0(\Omega)\ (\mbox{see (\ref{eq39})}),\nonumber\\
			&\Rightarrow&-\mbox{div}\,a(Du_0(z))=f_0(z,u_0(z))\ \mbox{for almost all}\ z\in\Omega.
		\end{eqnarray}
		
		From (\ref{eq39}),(\ref{eq40}) and (\ref{eq41}), we obtain
		\begin{equation}\label{eq42}
			\left\langle \frac{\partial u_0}{\partial n_a}+\beta(z)|u_0|^{p-2}u_0,h\right\rangle_{\partial\Omega}=0\ \mbox{for all}\ h\in W^{1,p}(\Omega).
		\end{equation}
		
		Recall that, if $\gamma_0$ is the trace map, then $im\gamma_0=W^{\frac{1}{p'},p}(\partial\Omega)$. So, from (\ref{eq42}) it follows that
		$$\frac{\partial u_0}{\partial n_a}+\beta(z)|u_0|^{p-2}u_0=0\ \mbox{in}\ W^{-\frac{1}{p'},p'}(\partial\Omega).$$
		
		From Proposition \ref{prop7}, we have that $u_0\in L^{\infty}(\Omega)$. So, the nonlinear regularity result of Lieberman \cite[p. 320]{22} implies that
		$$u_0\in C^{1,\eta}(\overline{\Omega})\ \mbox{for some}\ \eta\in(0,1).$$
		
		Next we show that $u_0$ is also a local $W^{1,p}(\Omega)$-minimizer of $\varphi_0$. We argue indirectly. So, we assume that $u_0$ is not a local $W^{1,p}(\Omega)$-minimizer of $\varphi_0$. Given $\epsilon>0$, we consider the set
		$$\bar{B}^*_{\epsilon}=\{h\in W^{1,p}(\Omega):||h||_{p^*}\leq \epsilon\}$$
		and define
		\begin{equation}\label{eq43}
			m^*_{\epsilon}=\inf[\varphi_0(u_0+h):h\in\bar{B}^*_{\epsilon}].
		\end{equation}
		
		By our contradiction hypothesis we have
		\begin{equation}\label{eq44}
			m^*_{\epsilon}<\varphi_0(u_0).
		\end{equation}
		
		Let $\{h_n\}_{n\geq 1}\subseteq \bar{B}^*_{\epsilon}$ be a minimizing sequence for (\ref{eq43}). Then since $u\rightarrow||u||_{p^*}+||Du||$ is an equivalent norm on the Sobolev space $W^{1,p}(\Omega)$, we see that $\{h_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)$ is bounded and so we may assume that
		\begin{eqnarray}\label{eq45}
			&&h_n\stackrel{w}{\rightarrow}\hat{h}_{\epsilon}\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{and in}\ L^{p^*}(\Omega),\nonumber\\
			&&h_n(z)\rightarrow\hat{h}_{\epsilon}(z)\ \mbox{for almost all}\ z\in\Omega.
		\end{eqnarray}
		
		Using the extended Fatou's lemma, we see that $\varphi_0$ is sequentially weakly lower semicontinuous. So, we have
		$$\varphi_0(u_0+\hat{h}_{\epsilon})\leq\liminf\limits_{n\rightarrow\infty}\varphi_0(u_0+h_n).$$
		
		Since $||h_{\epsilon}||_{p^*}\leq\epsilon$ (see (\ref{eq45})), it follows that
		\begin{eqnarray*}
			&&m^*_{\epsilon}=\varphi(u_0+\hat{h}_{\epsilon}),\\
			&\Rightarrow&\hat{h}_{\epsilon}\neq 0\ (\mbox{see (\ref{eq44})}).
		\end{eqnarray*}
		
		By the Lagrange multiplier rule (see for example, Papageorgiou and Kyritsi \cite[p. 35]{31}) we can find $\lambda_{\epsilon}\leq 0$ such that
		\begin{eqnarray}\label{eq46}
			&&\left\langle \varphi'_0(u_0+\hat{h}_{\epsilon}),v\right\rangle=\lambda_{\epsilon}\int_{\Omega}
|\hat{h}_{\epsilon}|^{p^*-2}\hat{h}_{\epsilon}vdz\ \mbox{for all}\ v\in W^{1,p}(\Omega),\nonumber\\
			&\Rightarrow&\left\langle A(u_0,\hat{h}_{\epsilon}),v\right\rangle+\int_{\partial\Omega}\beta(z)|u_0+\hat{h}_{\epsilon}|^{p-2}(u_0+\hat{h}_{\epsilon})vd\sigma\nonumber\\
			 &&=\int_{\Omega}f_0(z,u_0+\hat{h}_{\epsilon})vdz+\lambda_{\epsilon}\int_{\Omega}|\hat{h}_{\epsilon}|^{p^*-2}\hat{h}_{\epsilon}vdz\ \mbox{for all}\ v\in W^{1,p}(\Omega).
		\end{eqnarray}
		
		From (\ref{eq46}) as above, using the nonlinear Green's identity, we obtain
		\begin{eqnarray}\label{eq47}
			\left\{\begin{array}{ll}
				-\mbox{div}\, a(D(u_0+\hat{h}_{\epsilon})(z))=f_0(z,(u_0+\hat{h}_{\epsilon})(z))+\lambda_{\epsilon}|\hat{h}_{\epsilon}(z)|^{p^*-2}\hat{h}_{\epsilon}(z)&\\
				\hspace{9cm}\mbox{for almost all}&z\in\Omega,\\
				\frac{\partial(u_0+\hat{h}_{\epsilon})}{\partial n_a}+\beta(z)|u_0+\hat{h}_{\epsilon}|^{p-2}(u_0+\hat{h}_{\epsilon})=0&\mbox{on}\ \partial\Omega.
			\end{array}\right\}
		\end{eqnarray}
		
		First assume that $\lambda_{\epsilon}\in[-1,0]$ for all $\epsilon\in\left(0,1\right]$. Then from (\ref{eq47}) and Proposition \ref{prop7}, we can find $M_3>0$ such that
		\begin{equation}\label{eq48}
			||u_0+\hat{h}_{\epsilon}||_{\infty}\leq M_3\ \mbox{for all}\ \epsilon\in\left(0,1\right].
		\end{equation}
		
		Invoking the regularity result of Lieberman \cite{22}, we can find $\eta\in(0,1)$ and $M_4>0$ such that
		\begin{equation}\label{eq49}
			u_0+\hat{h}_{\epsilon}\in C^{1,\eta}(\overline{\Omega}),||u_0+\hat{h}_{\epsilon}||_{C^1(\overline{\Omega})}\leq M_4\ \mbox{for all}\ \epsilon\in\left(0,1\right].
		\end{equation}
		
		Next suppose that there exists $\epsilon_n\downarrow 0$ such that $\lambda_n=\lambda_{\epsilon_n}<-1$ for all $n\in\NN$.
		
		From (\ref{eq47}) with $\hat{h}_n=\hat{h}_{\epsilon_n}$, we have
		\begin{eqnarray}\label{eq50}
			&&-\frac{1}{|\lambda_n|}\mbox{div}\, a(D(u_0+\hat{h}_{n})(z))=\frac{1}{|\lambda_n|}f_0(z,(u_0+\hat{h}_n)(z))+|\hat{h}_n(z)|^{p^*-2}\hat{h}_n(z)\\
			&&\hspace{8.5cm}\mbox{for almost all}\ z\in\Omega.\nonumber
		\end{eqnarray}
		
		Also, from the first part of the proof, we have
		\begin{equation}\label{eq51}
			-\frac{1}{|\lambda_n|}\mbox{div}\,a(Du_0(z))=\frac{1}{|\lambda_n|}f_0(z,u_0(z))\ \mbox{for almost all}\ z\in\Omega.
		\end{equation}
		
		Let $\mu>1$ and consider the function $|\hat{h}_n|^{\mu}\hat{h}_n\ n\in\NN$. We have
		\begin{eqnarray*}
			&&D(|\hat{h}_n|^{\mu}\hat{h}_n)\\
			 &=&|\hat{h}_n|^{\mu}D\hat{h}_n+\mu\hat{h}_n\frac{\hat{h}_n}{|\hat{h}_n|}|\hat{h}_n|^{\mu-1}D\hat{h}_n\\
			&=&(\mu+1)|\hat{h}_n|^{\mu}D\hat{h}_n\\
			\Rightarrow&&|\hat{h}_n|^{\mu}\hat{h}_n\in W^{1,p}(\Omega)\ (\mbox{see (\ref{eq49}) and recall that}\ u_0\in C^{1,\eta}(\overline{\Omega})).
		\end{eqnarray*}
		
		Using this as test function, from (\ref{eq50}) and (\ref{eq51}), we have
		\begin{eqnarray*}
			&0&\leq\left\langle A(u_0+h_n)-A(u_0),|\hat{h}_n|^{\mu}\hat{h}_n\right\rangle+\int_{\partial\Omega}\beta(z)\left[|u_0+\hat{h}_n|^{p-2}(u_0+\hat{h}_n)-|u_0|^{p-2}u_0\right]d\sigma\\
			 &&=\int_{\Omega}[f_0(z,u_0+\hat{h}_n)-f_0(z,u_0)]|\hat{h}_n|^{\mu}\hat{h}_ndz+\lambda_n\int_{\Omega}|\hat{h}_n|^{p^*+\mu}dz\\
			&&\leq M_5\int_{\Omega}|\hat{h}_n|^{\mu+1}dz+\lambda_n\int_{\Omega}|\hat{h}_n|^{p^*+\mu}dz\ \mbox{for some}\ M_5>0,\ \mbox{all}\ n\in\NN\\
			&&\hspace{8cm}(\mbox{see (\ref{eq48}) and (\ref{eq9})})\\
			&&\leq M_5|\Omega|^{\frac{p^*-1}{p^*+\mu}}_{N}||\hat{h}_n||^{\mu+1}_{p^*+\mu}+\lambda_n||\hat{h}_n||^{p^*+\mu}_{p^*+\mu}\\
			&&\hspace{1cm}\left(\mbox{here we have used H\"{o}lder's inequality with exponents}\ \frac{p^*+\mu}{\mu+1},\frac{p^*+\mu}{p^*-1}\right)\\
			\Rightarrow&&|\lambda_n|||\hat{h}_n||^{p^*-1}_{p^*+\mu}\leq M_5|\Omega|^{\frac{p^*-1}{p^*+\mu}}_{N},\\
			\Rightarrow&&||\hat{h}_n||^{p^*-1}_{p^*+\mu}\leq M_6\ \mbox{for some}\ M_6>0\ (\mbox{independent of}\ \mu>1),\ \mbox{all}\ n\in\NN\\
			&&\hspace{8cm}(\mbox{recall that}\ |\lambda_n|>1).
		\end{eqnarray*}
		
		Since $\mu>1$ is arbitrary, we let $\mu\rightarrow\infty$ and obtain that
		$$||\hat{h}_n||_{\infty}\leq M_7\ \mbox{for some}\ M_7>0,\ \mbox{all}\ n\in\NN.$$
		 So, the nonlinear regularity theory of Lieberman \cite{22} implies that for some $\eta\in(0,1)$ and some $M_8>0$, we have
		$$\hat{h}_n\in C^{1,\eta}(\overline{\Omega}),||\hat{h}_n||_{C^1(\overline{\Omega})}\leq M_8\ \mbox{for all}\ n\in\NN$$
		(see (\ref{eq50}) and recall that $u_0\in C^{1,\eta}(\overline{\Omega})$).
		
		Therefore in both cases (case 1, $\lambda_{\epsilon}\in[-1,0]$ for all $\epsilon\in\left(0,1\right]$ and case 2, $\lambda_{\epsilon_n}<-1$ for some $\epsilon_n\downarrow 0$), we reach the same uniform $C^{1,\eta}(\overline{\Omega})$ bounds for the sequence $\{\hat{h}_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)$ such that $\varphi_0(u_0+h_n)<\varphi_0(u_0)$ for all $n\in\NN$ (see (\ref{eq44})).
		
		Recalling that $||\hat{h}_n||_{p^*}\leq\epsilon_n$ for all $n\in\NN$ and exploiting the compact embedding of $C^{1,\eta}(\overline{\Omega})$ into $C^1(\overline{\Omega})$, we have
		\begin{eqnarray*}
			&&\hat{h}_n\rightarrow 0\ \mbox{in}\ C^1(\overline{\Omega}),\\
			&\Rightarrow&u_0+\hat{h}_n\rightarrow u_0\ \mbox{in}\ C^1(\overline{\Omega}),\\
			&\Rightarrow&\varphi_0(u_0)\leq\varphi_0(u_0+\hat{h}_n)\ \mbox{for all}\ n\geq n_0.
		\end{eqnarray*}
		
		But recall that
		$$\varphi_0(u_0+\hat{h}_n)<\varphi_0(u_0)\ \mbox{for all}\ n\in\NN,$$
		a contradiction. This proves that $u_0\in C^{1,\eta}(\overline{\Omega})$ is also a local $W^{1,p}(\Omega)$-minimizer of $\varphi_0$.
	\end{proof}
	\begin{remark}
		A careful reading of the proof of Proposition \ref{prop5} reveals that the result remains valid if instead we use the more general nonlinear boundary condition
		$$\frac{\partial u}{\partial n_a}=\xi(z,u)\ \mbox{on}\ \partial\Omega$$
		with $\xi\in C^{0,\eta}(\partial\Omega\times\RR),0<\eta<1$ such that
		$$|\xi(z,x)|\leq c_{25}|x|^{\tau}\ \mbox{for all}\ (z,x)\in\partial\Omega\times\RR,$$
		with $c_{25}>0$ and $\tau\in\left(1,p\right]$. For simplicity in our presentation, we have used in problem (\ref{eq8}) the Robin boundary condition from problem (\ref{eq1}), simplifying this way a little the necessary estimates.
	\end{remark}
	
	As we already mentioned in the Introduction, we will also use tools from Morse theory (critical groups). So, let us recall some basic definitions and facts from that theory.
	
	Given a Banach space $X$, a function $\varphi\in C^1(X,\RR)$ and $c\in\RR$, we introduce the following sets
	$$\varphi^c=\{u\in X:\varphi(u)\leq c\},K_{\varphi}=\{u\in X:\varphi'(u)=0\},K^c_{\varphi}=\{u\in K_{\varphi}:\varphi(u)=c\}.$$
	
	Let $(Y_1,Y_2)$ be a topological pair such that $Y_2\subseteq Y_1\subseteq X$ and $k\in\NN_0$. By $H_k(Y_1,Y_2)$ we denote the $k^{th}$-relative singular homology group for the topological pair $(Y_1,Y_2)$ with integer coefficients. The critical groups of $\varphi$ at an isolated $u\in K^c_{\varphi}$ are defined by
	$$C_k(\varphi,u)=H_k(\varphi^c\cap U,\varphi^c\cap U\backslash\{u\})\ \mbox{for all}\ k\in\NN_0.$$
	
	Here $U$ is a neighborhood of $u$ such that $K_{\varphi}\cap\varphi^c\cap U=\{u\}$. The excision property of singular homology implies that the above definition of critical groups is dependent of the choice of the neighborhood $U$ of $u$.
	
	Suppose that $\varphi$ satisfies the $C$-condition and that $\inf\varphi(K_{\varphi})>-\infty$. Let $c<\inf\varphi(K_{\varphi})$. The critical groups of $\varphi$ at infinity are defined by
	$$C_k(\varphi,\infty)=H_k(X,\varphi^c)\ \mbox{for all}\ k\in\NN_0.$$
	
	The second deformation theorem implies that this definition is independent of the choice of the level $c<\inf\varphi(K_{\varphi})$.
	
	Suppose that $\varphi\in C^1(X,\RR)$ satisfies the $C$-condition and that $K_{\varphi}$ is finite. We define
	\begin{eqnarray*}
		&&M(t,u)=\sum\limits_{k\in\NN_0}\mbox{rank}\, C_k(\varphi,u)t^k\ \mbox{for all}\ t\in\RR,\ \mbox{all}\ u\in K_{\varphi},\\
		&&P(t,\infty)=\sum\limits_{k\in\NN_0}\mbox{rank}\, C_k(\varphi,\infty)t^k\ \mbox{for all}\ t\in\RR.
	\end{eqnarray*}
	
	The Morse relation says that
	\begin{equation}\label{eq52}
		\sum\limits_{u\in K_{\varphi}}M(t,u)=P(t,\infty)+(1+t)Q(t)\ \mbox{for all}\ t\in\RR,
	\end{equation}
	where $Q(t)=\sum\limits_{k\in\NN_0}\beta_kt^k$ is a formal series in $t\in\RR$ with nonnegative integer coefficients $\beta_k$.
	
	Finally from Papageorgiou and R\u adulescu \cite{34}, we recall that the nonlinear eigenvalue problem
	$$\left\{\begin{array}{ll}
		-\Delta_pu(z)=\hat{\lambda}|u(z)|^{p-2}u(z)&\mbox{in}\ \Omega,\\
		\frac{\partial u}{\partial n_p}+\beta(z)|u|^{p-2}u=0&\mbox{on}\ \partial\Omega
	\end{array}\right\}$$
	has a smallest eigenvalue $\hat{\lambda}_1(p,\beta)\geq 0$. If $\beta\not\equiv0$, then $\hat{\lambda}_1(p,\beta)>0$, while if $\beta\equiv0$ then $\hat{\lambda}_1(p,0)=\hat{\lambda}_1(p)=0$ (Neumann problem). The eigenfunctions corresponding to this eigenvalue have constant sign and
	 $$\hat{\lambda}_1(p,\beta)=\inf\left[\frac{||Du||^p_p+\int_{\partial\Omega}\beta(z)|u|^pd\sigma}{||u||^p_p}:u\in W^{1,p}(\Omega),u\neq 0\right].$$
	
	By $\hat{u}_1(p,\beta)$ we denote the $L^p$-normalized (that is $||\hat{u}_1(p,\beta)||_p=1$) positive eigenfunction corresponding to $\hat{\lambda}_1(p,\beta)$. We have
	 $$\hat{\lambda}_1(p,\beta)=||D\hat{u}_1(p,\beta)||^p_p+\int_{\partial\Omega}\beta(z)|\hat{u}_1(p,\beta)|^pd\sigma$$
	and from the nonlinear regularity theory and the nonlinear maximum principle, we have
	$$\hat{u}_1(p,\beta)\in \mbox{int}\, C_+.$$
	
	
	\section{Three Solutions Theorem}
	
	The hypotheses on the reaction $f(z,x)$ are the following:
	
	$H(f):$ $f:\Omega\times\RR\rightarrow\RR$ is a Carath\'eodory function such that
	\begin{itemize}
		\item[(i)] $\lim\limits_{x\rightarrow\pm\infty}\frac{f(z,x)}{|x|^{p^*-2}x}=0$ uniformly for almost all $z\in\Omega$ and for every $\rho>0$ there exists $a_{\rho}\in L^{\infty}(\Omega)_+$ such that $|f(z,x)|\leq a_{\rho}(z)$ for almost all $z\in\Omega$, all $|x|\leq \rho$;
		\item[(ii)] if $F(z,x)=\int^x_0f(z,s)ds$, then $\lim\limits_{x\rightarrow\pm\infty}\frac{F(z,x)}{|x|^p}=+\infty$ uniformly for almost all $z\in\Omega$;
		\item[(iii)] if $\xi(z,x)=f(z,x)x-pF(z,x)$, then there exists $\eta\in L^1(\Omega)_+$ such that
		$$\xi(z,x)\leq\xi(z,y)+\eta(z)\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ 0\leq x\leq y\ \mbox{or}\ y\leq x\leq 0;$$
		\item[(iv)] there exist $\delta>0$ and $\gamma_{\delta}>0$ such that $-\gamma_{\delta}|x|^p\leq f(z,x)x$ for almost all $z\in\Omega$, all $|x|\leq\delta$ and \\
		if $\beta\not\equiv 0$, then there exists $\Im\in L^{\infty}(\Omega)_+,\ \Im(z)\leq\hat{\lambda}_1(p,\hat{\beta})$ for almost all $z\in\Omega$, $\Im\not\equiv\hat{\lambda}_1(p,\hat{\beta})$ with $\hat{\beta}=\frac{p-1}{c_1}\beta$ and $\limsup\limits_{x\rightarrow 0}\frac{f(z,x)}{|x|^{p-2}x}\leq \Im(x)$ uniformly for almost all $z\in\Omega$,\\
		if $\beta\equiv0$, then $f(z,x)x\leq 0$ for almost all $z\in\Omega$, all $|x|\leq\delta$.
	\end{itemize}
\begin{remark}
	Hypothesis $H(f)(i)$ is more general than the usual polynomial subcritical growth condition which says that
	\begin{equation}\label{eq53}
		|f(z,x)|\leq c_{26}(1+|x|^{r-1})\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\in\RR,
	\end{equation}
	with $c_{26}>0$ and $1<r<p^*$.
For example the function (for the sake of simplicity we drop the $z$-dependence)
$$f(x)=\frac{|x|^{p^*-2}x}{\ln ((1+|x|^p))}-\frac{p}{p^*}\,\frac{|x|^{p^*}|x|^{p-2}x}{\ln (1+|x|^p)^2(1+|x|^p)}$$
with primitive
$$F(x)=\frac{1}{p^*}\,\frac{|x|^{p^*}}{\ln (1+|x|^p)}\,,$$
satisfies hypothesis $H(f)(i)$ but fails to satisfy the subcritical polynomial growth \eqref{eq53}. 
The lack of compactness in the embedding of $W^{1,p}(\Omega)$ into $L^{p^*}(\Omega)$ is a source of difficulties which we have to overcome. We do this without any appeal to the concentration-compactness principle (see Ambrosetti and Malchiodi \cite[p. 252]{3}).
 It is not clear how hypothesis $H(f)(i)$ can lead to concentration phenomena and for this reason our approach avoids the use of the concentration-compactness method of P.-L. Lions. Instead we show that despite the almost critical growth of the reaction term $f(z,\cdot )$ (see hypothesis $H(f)(i)$), the compactness condition is still valid for the energy functional of the problem and so we can proceed with the usual variational methods of critical point theory.  
Hypothesis $H(f)(iv)$ implies that
	$$f(z,0)=0\ \mbox{for almost all}\ z\in\Omega.$$
	
	
	Then hypothesis $H(f)(iii)$ implies
	\begin{eqnarray*}
		&&\xi(z,0)=0\leq \xi(z,x)+\eta(z)\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\in\RR,\\
		&\Rightarrow&pF(z,x)\leq f(z,x)x+\eta(z)\ \mbox{for almost all}\ z\in\Omega,\\
		&\Rightarrow&\lim\limits_{x\rightarrow\pm\infty}\frac{f(z,x)}{|x|^{p-2}x}=+\infty\ \mbox{uniformly for almost all}\ z\in\Omega\ (\mbox{see hypothesis}\ H(f)(ii)).
	\end{eqnarray*}
	
Hypotheses $H(f)(ii),(iii)$ replace the AR-condition and allow in our framework superlinear reactions with ``slower" growth near $\pm\infty$ which fail to satisfy the AR-condition (see the examples below). Hypothesis $H(f)(iii)$ is a quasimonotonicity condition on $\xi(z,\cdot)$ and it is satisfied if, for example, we can find $M_9>0$ such that for almost all $z\in\Omega$
\begin{eqnarray*}
	&&x\rightarrow\frac{f(z,x)}{x^{p-1}}\ \mbox{is nondecreasing on}\ \left[M_9,+\infty\right)\\
	\mbox{and}&&x\rightarrow\frac{f(z,x)}{|x|^{p-2}x}\ \mbox{is nonincreasing on}\ \left(-\infty,-M_9\right].
\end{eqnarray*}

More restrictive versions of hypothesis $H(f)(iii)$ were used by Li and Yang \cite{23}, Liu \cite{25}, Miyagaki and Souto \cite{27} and Sun \cite{36}.
We should mention that all these conditions originate from the important work of Jeanjean \cite{jean} (see also Struwe \cite{struwe}), who was the first to employ an alternative to the AR-condition. So, Jeanjean \cite{jean} assumed (for $p=2$) that there exists $\vartheta\geq 1$ such that$$\xi (z,sx)\leq\vartheta \xi(z,x)\quad\mbox{for almost all $z\in\Omega$, all $x\in\RR$ and $s\in [0,1]$}.$$

The disadvantage of this condition is that it is global. In contrast, the previous remarks show that condition $H(f)(iii)$ avoids this global character and so it is a quite generic condition. For a further discussion and comparison of these extensions of the AR-condition, we refer to the paper by Li and Yang \cite{23}. 
\end{remark}

\begin{ex}
	The following primitive functions satisfy hypotheses $H(f)$ (for the sake of simplicity we drop the $z$-dependence)
	\begin{eqnarray*}
		&&F_1(x)=\frac{1}{q}|x|^q-\frac{1}{p}|x|^p\ \mbox{with}\ 1<p<q<p^*\\
		&&F_2(x)=\frac{1}{p^*}\ \frac{|x|^{p^*}}{ln(1+|x|^{p^*})}+\left\{\begin{array}{ll}
			-\frac{1}{p}|x|^p&\mbox{if}\ |x|\leq 1\\
			\frac{1}{p}|x|^pln|x|-\frac{1}{p}&\mbox{if}\ 1<|x|
		\end{array}\right. .
	\end{eqnarray*}
	
	Note that $f_2(x)=\frac{d}{dx}F_2(x)$ fails to satisfy (\ref{eq53}) and the AR-condition.
\end{ex}
	
	We introduce the following truncations-perturbations of the reaction term $f(z,\cdot)$:
	\begin{eqnarray}
		&&\hat{f}_+(z,x)=\left\{\begin{array}{ll}
			0&\mbox{if}\ x\leq 0\\
			f(z,x)+x^{p-1}&\mbox{if}\ 0<x
		\end{array}\right.\label{eq54}\\
		&&\hat{f}_-(z,x)=\left\{\begin{array}{ll}
			f(z,x)+|x|^{p-2}x&\mbox{if}\ x<0\\
			0&\mbox{if}\ 0\leq x
		\end{array}\right. .\label{eq55}
	\end{eqnarray}
	
	Both are Carath\'eodory functions. We set $\hat{F}_{\pm}(z,x)=\int^x_0\hat{f}_{\pm}(z,s)ds$ and consider the $C^1$-functionals $\hat{\varphi}_{\pm}:W^{1,p}(\Omega)\rightarrow\RR$ defined by
	\begin{eqnarray*}
		 &&\hat{\varphi}_{\pm}(u)=\int_{\Omega}G(Du)dz+\frac{1}{p}||u||^p_p\pm\frac{1}{p}\int_{\partial\Omega}\beta(z)(u^{\pm})^pd\sigma-\int_{\Omega}\hat{F}_{\pm}(z,u)dz\\
		&&\hspace{7.5cm}\mbox{for all}\ u\in W^{1,p}(\Omega).
	\end{eqnarray*}
	
	Also, let $\varphi:W^{1,p}(\Omega)\rightarrow\RR$ be the energy functional for problem (\ref{eq4}) defined by
	$$\varphi(u)=\int_{\Omega}G(Du)dz-\int_{\Omega}F(z,u)dz\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$
	
	Evidently $\varphi\in C^1(W^{1,p}(\Omega))$.
\begin{prop}\label{prop9}
		If hypotheses $H(a),H(\beta),H(f)$ hold, then the functionals $\hat{\varphi}_{\pm}$ satisfy the $C$-condition.
\end{prop}
\begin{proof}
We give the proof (similarly,
in two other occurrences) for the functional $\hat{\varphi}_+$, the proof for $\hat{\varphi}_-$ being similar.
	
	Consider a sequence $\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)$ such that
	\begin{eqnarray}
		&&|\hat{\varphi}_+(u_n)|\leq M_{10}\ \mbox{for some}\ M_{10}>0,\ \mbox{all}\ n\in\NN\label{eq56}\\
		&&(1+||u_n||)\hat{\varphi}'_+(u_n)\rightarrow 0\ \mbox{in}\ W^{1,p}(\Omega)^*\ \mbox{as}\ n\rightarrow\infty.\label{eq57}
	\end{eqnarray}
	
	From (\ref{eq57}) we have
	\begin{eqnarray}\label{eq58}
		&&\left|\left\langle A(u_n),h\right\rangle+\int_{\Omega}|u_n|^{p-2}u_nhdz+\int_{\partial\Omega}\beta(z)(u^+_n)^{p-1}hd\sigma-\int_{\Omega}\hat{f}_+(z,u_n)hdz\right|\leq\frac{\epsilon_n||h||}{1+||u_n||}\\
		&&\hspace{7cm}\mbox{for all}\ h\in W^{1,p}(\Omega)\ \mbox{with}\ \epsilon_n\rightarrow 0^+.\nonumber
	\end{eqnarray}
	
	In (\ref{eq58}) we choose $h=-u^-_n\in W^{1,p}(\Omega)$. Then
	\begin{eqnarray}\label{eq59}
		&&\frac{c_1}{p-1}||Du^-_n||^p_p+||u^-_n||^p_p\leq\epsilon_n\ \mbox{for all}\ n\in\NN\ (\mbox{see Lemma \ref{lem2} and (\ref{eq54})})\nonumber\\
		&\Rightarrow&u^-_n\rightarrow 0\ \mbox{in}\ W^{1,p}(\Omega).
	\end{eqnarray}
	
	We use (\ref{eq59}) in (\ref{eq56}). Then because of Corollary \ref{cor3} and (\ref{eq54}), we have
	\begin{eqnarray}\label{eq60}
		 &&\left|\int_{\Omega}pG(Du^+_n)dz+\int_{\partial\Omega}\beta(z)(u^+_n)^pd\sigma-\int_{\Omega}pF(z,u^+_n)dz\right|\leq M_{11}\\
		&&\hspace{7cm}\mbox{for some}\ M_{11}>0,\ \mbox{all}\ n\in\NN\nonumber.
	\end{eqnarray}
	
	In (\ref{eq58}) we choose $h=u^+_n\in W^{1,p}(\Omega)$. Then
	\begin{equation}\label{eq61}
		 -\int_{\Omega}(a(Du^+_n),Du^+_n)_{\RR^N}dz-\int_{\partial\Omega}\beta(z)(u^+_n)^pd\sigma+\int_{\Omega}f(z,u^+_n)u^+_ndz\leq\epsilon_n\ \mbox{for all}\ n\in\NN.
	\end{equation}
	
	We add (\ref{eq60}) and (\ref{eq61}) and use hypothesis $H(a)(iv)$. We obtain
	\begin{equation}\label{eq62}
		\int_{\Omega}\xi(z,u^+_n)dz\leq M_{12}\ \mbox{for some}\ M_{12}>0,\ \mbox{all}\ n\in\NN.
	\end{equation}
	\begin{claim}\label{Claim1}
		$\{u^+_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)$ is bounded.
	\end{claim}
	
	We argue indirectly. So, suppose that the Claim \ref{Claim1} is not true. By passing to a subsequence if necessary, we may assume that $||u^+_n||\rightarrow\infty$. Let $y_n=\frac{u^+_n}{||u^+_n||}\ n\in\NN$. Then $||y_n||=1,y_n\geq 0$ for all $n\in\NN$ and so we may assume that
	\begin{equation}\label{eq63}
		y_n\stackrel{w}{\rightarrow}y\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{and}\ y_n\rightarrow y\ \mbox{in}\ L^p(\Omega)\ \mbox{and in}\ L^p(\partial\Omega),\ y\geq 0.
	\end{equation}
	
	Suppose that $y\neq 0$ and let $\Omega_+(y)=\{y>0\}$. Then $|\Omega_+(y)|_N>0$ and we have
	$$u^+_n(z)\rightarrow+\infty\ \mbox{for almost all}\ z\in\Omega_+(y).$$
	
	Hypothesis $H(f)(ii)$ implies that
	$$\frac{F(z,u^+_n(z))}{||u^+_n||^p}=\frac{F(z,u^+_n(z))}{u^+_n(z)^p}y_n(z)^p\rightarrow+\infty\ \mbox{for almost all}\ z\in\Omega_+(y).$$
	
	From this fact and Fatou's lemma (see hypothesis $H(f)(ii)$ and (\ref{eq63})), we have
	\begin{equation}\label{eq64}
		\int_{\Omega}\frac{F(z,u^+_n)}{||u^+_n||^p}dz\rightarrow+\infty.
	\end{equation}
	
	From (\ref{eq60}) we have
	\begin{eqnarray}\label{eq65}
		 \int_{\Omega}\frac{F(z,u^+_n)}{||u^+_n||^p}&\leq&M_{11}+\frac{1}{||u^+_n||^p}\int_{\Omega}G(Du^+_n)dz+\int_{\partial\Omega}\beta(z)y^p_nd\sigma\nonumber\\
		&&\hspace{1cm}\mbox{for all}\ n\in\NN\ (\mbox{recall}\ p>1)\nonumber\\
		&\leq&c_{27}(1+||y_n||^p)\ \mbox{for some}\ c_{27}>0,\ \mbox{all}\ n\in\NN\nonumber\\
		&&\hspace{1cm}(\mbox{see Corollary \ref{cor3} and hypothesis}\ H(\beta))\nonumber\\
		&\leq&c_{28}\ \mbox{for all}\ n\in\NN\ (\mbox{see (\ref{eq63})}).
	\end{eqnarray}
	
	Comparing (\ref{eq64}) and (\ref{eq65}) we reach a contradiction.
	
	So, we assume that $y\equiv 0$. Let $k>0$ and set $v_n=(kp)^{1/p}y_n$ for all $n\in\NN$. We have
	\begin{eqnarray}\label{eq66}
		&&v_n\stackrel{w}{\rightarrow}0\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{and}\ v_n\rightarrow 0\ \mbox{in}\ L^p(\Omega)\ \mbox{and in}\ L^p(\partial\Omega)\\
		&&\hspace{4cm}(\mbox{see (\ref{eq63}) and recall that}\ y=0).\nonumber
	\end{eqnarray}
	
	Let $c_{29}=\sup\limits_{n\geq 1}||v_n||^{p^*}_{p^*}<+\infty$ (see (\ref{eq66})). Hypothesis $H(f)(i)$ implies that given $\epsilon>0$ we can find $c_{\epsilon}>0$ such that
	\begin{equation}\label{eq67}
		|F(z,x)|\leq\frac{\epsilon}{2c_{29}}|x|^{p^*}+c_{\epsilon}\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\in\RR.
	\end{equation}
	
	For every measurable set $E\subseteq\Omega$ with $|E_N|\leq\frac{\epsilon}{2c_{\epsilon}}$, we have
	\begin{eqnarray*}
		\left|\int_{E}F(z,v_n)dz\right|&\leq&\int_E|F(z,v_n)|dz\\
		&\leq&\frac{\epsilon}{2c_{29}}||v_n||^{p^*}_{p^*}+c_{\epsilon}|E|_N\ (\mbox{see (\ref{eq67})})\\
		&\leq&\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon\ \mbox{for all}\ n\in\NN,\\
		\Rightarrow\{F(\cdot,v_n(\cdot))\}_{n\geq 1}&\subseteq&L^1(\Omega)\ \mbox{is uniformly integrable.}
	\end{eqnarray*}
	
	Since $F(z,v_n(z))\rightarrow 0$ for almost all $z\in\Omega$, from the extended dominated convergence theorem (Vitali's theorem) we have
	\begin{equation}\label{eq68}
		\int_{\Omega}F(z,v_n)dz\rightarrow 0\ \mbox{as}\ n\rightarrow\infty.
	\end{equation}
	
	Recall that we have assumed that $||u^+_n||\rightarrow\infty$. So, we can find $n_0\in\NN$ such that
	\begin{equation}\label{eq69}
		0<(kp)^{1/p}\frac{1}{||u^+_n||}\leq 1\ \mbox{for all}\ n\geq n_0.
	\end{equation}
	
	Consider the $C^1$-functional $\hat{\psi}_+:W^{1,p}(\Omega)\rightarrow\RR$ defined by
	 $$\hat{\psi}_+(u)=\frac{c_1}{p(p-1)}||Du||^p_p+\frac{1}{p}||u||^p_p+\frac{1}{p}\int_{\partial\Omega}\beta(z)(u^+)^pd\sigma-\int_{\Omega}\hat{F}_+(z,u)dz\ \mbox{for all}\ u\in W^{1,p}(\Omega).$$
	
	Let $t_n\in[0,1]$ be such that
	$$\hat{\psi}_+(t_nu^+_n)=\max\limits_{0\leq t\leq 1}\hat{\psi}_+(tu^+_n)\ \mbox{for all}\ n\in\NN.$$
	
	From (\ref{eq68}) we see that we can find $n_1\in\NN,\ n_1\geq n_0$ such that
	\begin{equation}\label{eq70}
		\int_{\Omega}F(z,v_n)dz\leq\frac{c_1}{2(p-1)}k\ \mbox{for all}\ n\geq n_1.
	\end{equation}
	
	Using (\ref{eq69}) and (\ref{eq70}) we have
	\begin{eqnarray}\label{eq71}
		 &&\hat{\psi}_+(t_nu^+_n)\geq\hat{\psi}_+(v_n)\geq\frac{c_1k}{p-1}-\frac{c_1k}{2(p-1)}-\|v_n\|_p^p=\frac{c_1k}{3(p-1)}\ \mbox{for all}\ n\geq n_1\\
		&&\hspace{4cm}(\mbox{see (\ref{eq70}) and hypothesis}\ H(\beta)).\nonumber
	\end{eqnarray}
	
	Recall that $k>0$ is arbitrary. So, from (\ref{eq71}) it follows that
	\begin{equation}\label{eq72}
		\hat{\psi}_+(t_nu^+_n)\rightarrow+\infty\ \mbox{as}\ n\rightarrow\infty.
	\end{equation}
	
	From (\ref{eq56}) and (\ref{eq59}) and since $\hat{\psi}_+\leq\hat{\varphi}_+$ (see Corollary \ref{cor3}), we see that
	\begin{equation}\label{eq73}
		\{\hat{\psi}_+(u^+_n)\}_{n\geq 1}\subseteq\RR\ \mbox{is bounded}.
	\end{equation}
	
	Also, we have
	\begin{equation}\label{eq74}
		\hat{\psi}_+(0)=0.
	\end{equation}
	
	From (\ref{eq72}), (\ref{eq73}), (\ref{eq74}) it follows that we can find $n_2\in\NN$ such that
	\begin{equation}\label{eq75}
		t_n\in(0,1)\ \mbox{for all}\ n\geq n_2.
	\end{equation}
	
	Then for $n\geq n_2$ we have
	\begin{eqnarray}\label{eq76}
		&&\left.\frac{d}{dt}\hat{\psi}_+(tu^+_n)\right|_{t=t_n}=0,\nonumber\\
		&\Rightarrow&\frac{c_1}{p-1}\left\langle A_p(t_nu^+_n),u^+_n\right\rangle+\int_{\partial\Omega}\beta(z)(t_nu^+_n)^{p-1}u^+_nd\sigma=\int_{\Omega}f(z,t_n(u^+_n)u^+_n)dz\nonumber\\
		&&\hspace{6cm}(\mbox{from the chain rule})\nonumber\\
		 &\Rightarrow&\frac{c_1}{p-1}||D(t_nu^+_n)||^p_p+\int_{\partial\Omega}\beta(z)(t_nu^+_n)^pd\sigma=\int_{\Omega}f(z,t_nu^+_n)(t_nu^+_n)dz\\
		&&\hspace{6cm}\mbox{for all}\ n\geq n_2.\nonumber
	\end{eqnarray}
	
	From hypothesis $H(f)(iii)$ we have
	\begin{eqnarray}\label{eq77}
		 &&\int_{\Omega}f(z,t_nu^+_n)(t_nu^+_n)dz\leq\int_{\Omega}\xi(z,u^+_n)dz+\int_{\Omega}pF(z,t_nu^+_n)dz+||\eta||_1\\
		&&\hspace{6cm}\mbox{for all}\ n\geq n_2\ (\mbox{see (\ref{eq75})}).\nonumber
	\end{eqnarray}
	
	Using (\ref{eq77}) in (\ref{eq76}), we obtain
	\begin{equation}\label{eq78}
		\hat{\psi}_+(t_nu^+_n)\leq M_{12}+||\eta||_1=M_{13}\ \mbox{for all}\ n\geq n_2\ (\mbox{see (\ref{eq62})}).
	\end{equation}
	
	Comparing (\ref{eq72}) and (\ref{eq78}), we reach a contradiction. This proves the Claim \ref{Claim1}.
	
	From (\ref{eq59}) and the Claim \ref{Claim1} it follows that
	$$\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)\ \mbox{is bounded}.$$
	
	So, we way assume that
	\begin{equation}\label{eq79}
		u_n\stackrel{w}{\rightarrow}u\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{and}\ u_n\rightarrow u\ \mbox{in}\ L^p(\Omega)\ \mbox{and in}\ L^p(\partial\Omega).
	\end{equation}
	
	Let $c_{30}=\sup\limits_{n\geq 1}||u_n||^{p^*}_{p^*}<+\infty$ (see (\ref{eq79})). Hypothesis $H(f)(i)$ implies that given $\epsilon>0$, we can find $\hat{c}_{\epsilon}>0$ such that
	\begin{equation}\label{eq80}
		|f(z,x)|\leq\frac{\epsilon}{2c_{30}}|x|^{p^*-1}+\hat{c}_{\epsilon}\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\in\RR.
	\end{equation}
	
	For $E\subseteq\Omega$ measurable, we have
	\begin{eqnarray}\label{eq81}
		&&\left|\int_{E}f(z,u_n)(u_n-u)dz\right|\nonumber\\
		&&\leq\int_E|f(z,u_n)||u_n-u|dz\nonumber\\
		&&\leq\frac{\epsilon}{2c_{30}}\int_E|u_n|^{p^*-1}|u_n-u|dz+\hat{c}_{\epsilon}\int_E|u_n-u|dz.
	\end{eqnarray}
	
	Using H\"{o}lder inequality, we have
	\begin{eqnarray}
		\hat{c}_{\epsilon}\int_E|u_n-u|dz&\leq&\hat{c}_{\epsilon}|E|^{\frac{1}{(p^*)'}}_{N}||u_n-u||_{p^*}\ (\mbox{recall}\ \frac{1}{p^*}+\frac{1}{(p^*)'}=1)\nonumber\\
		&\leq&2\hat{c}_{\epsilon}|E|^{\frac{1}{(p^*)'}}_{N}c_{30}^{1/p^*}\label{eq82}\\
		 \frac{\epsilon}{2c_{30}}\int_E|u_n|^{p^*-1}|u_n-u|dz&\leq&\frac{\epsilon}{2c_{30}}||u_n||^{p^*-1}_{p^*}||u_n-u||_{p^*}\leq\frac{\epsilon}{2}\ \mbox{for all}\ n\in\NN.\label{eq83}
	\end{eqnarray}
	
	Choose $E\subseteq\Omega$ measurable with $|E|_N\leq\frac{\epsilon}{2(2\hat{c}_{\epsilon})^{(p^*)'}c_{30}^{\frac{1}{p^*-1}}}$. Then from (\ref{eq82}) we have
	\begin{equation}\label{eq84}
		\hat{c}_{\epsilon}\int_E|u_n-u|dz\leq\frac{\epsilon}{2}\ \mbox{for all}\ n\in\NN.
	\end{equation}
	
	From (\ref{eq81}), (\ref{eq83}) and (\ref{eq84}) it follows that
	\begin{eqnarray*}
		&&\sup\limits_{n\geq 1}\int_E|f(z,u_n)||u_n-u|dz\leq\epsilon,\\
		&\Rightarrow&\{f(\cdot,u_n(\cdot))(u_n-u)(\cdot)\}_{n\geq 1}\subseteq L^1(\Omega)\ \mbox{is uniformly integrable}.
	\end{eqnarray*}
	
	From (\ref{eq79}) we have (at least for a subsequence) that
	$$f(z,u_n(z))(u_n-u)(z)\rightarrow 0\ \mbox{for almost all}\ z\in\Omega.$$
	
	So, employing the extended dominated convergence theorem (Vitali's theorem), we have
	\begin{equation}\label{eq85}
		\int_{\Omega}f(z,u_n)(u_n-u)dz\rightarrow 0\ \mbox{as}\ n\rightarrow\infty.
	\end{equation}
	
	In (\ref{eq58}) we choose $h=u_n-u\in W^{1,p}(\Omega)$, pass to the limit as $n\rightarrow\infty$ and use (\ref{eq85}),(\ref{eq79}) and hypothesis $H(\beta)$. Then
	\begin{eqnarray*}
		&&\lim\limits_{n\rightarrow\infty}\left\langle A(u_n),u_n-u\right\rangle=0,\\
		&\Rightarrow&u_n\rightarrow u\ \mbox{in}\ W^{1,p}(\Omega)\ (\mbox{see Proposition \ref{prop4}}),\\
		&\Rightarrow&\hat{\varphi}_+\ \mbox{satisfies the $C$-condition}.
	\end{eqnarray*}
	
	Similarly for $\hat{\varphi}_-$ using (\ref{eq55}).
\end{proof}

A careful reading of the above proof shows with minor and straightforward changes, we can have the same result for the energy functional $\varphi$. Therefore we can state the following proposition.
\begin{prop}\label{prop10}
	If hypotheses $H(a),H(\beta),H(f)$ hold, then the energy functional $\varphi$ satisfies the $C$-condition.
\end{prop}

Hypothesis $H(f)(ii)$ leads easily to the following result.

\begin{prop}\label{prop11}
	If hypotheses $H(a),H(\beta),H(f)$ hold and $u\in \mbox{int}\, C_+$, then $\hat{\varphi}_{\pm}(tu)\rightarrow-\infty$ as $t\rightarrow+\infty$.
	\end{prop}

The next result establishes the mountain pass geometry (see Theorem \ref{th1}) for the functionals $\hat{\varphi}_{\pm}$. Also, this result will be useful in generating a third nontrivial solution for problem (\ref{eq4}), since it identifies the nature of $u=0\in K_{\varphi}$.

	\begin{prop}\label{prop12}
		If hypotheses $H(a),H(\beta),H(f)$ hold, then $u=0$ is a local minimizers of the functional $\hat{\varphi}_{\pm}$ and $\varphi$.
	\end{prop}
\begin{proof}
	We do the proof for the functional $\hat{\varphi}_+$, the proofs for $\hat{\varphi}_-$ and $\varphi$ being similar.
	
	First suppose $\beta\not\equiv 0$. Hypothesis $H(f)(iv)$ implies that given $\epsilon>0$, we can find $\delta_1=\delta_1(\epsilon)>0$ such that
	\begin{equation}\label{eq86}
		F(z,x)\leq\frac{1}{p}(\Im(z)+\epsilon)|x|^p\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ |x|\leq\delta_1.
	\end{equation}
	
	Let $u\in C^1(\overline{\Omega})$ with $||u||_{C^1(\overline{\Omega})}\leq\delta_1$. Then
	\begin{eqnarray*}
		 \hat{\varphi}_+(u)&=&\int_{\Omega}G(Du)dz+\frac{1}{p}||u^-||^p_p+\frac{1}{p}\int_{\partial\Omega}\beta(z)(u^+)^pd\sigma-\int_{\Omega}F(z,u^+)dz\ (\mbox{see (\ref{eq54})})\\
		 &\geq&\frac{c_1}{p(p-1)}\left[||Du^+||^p_p+\int_{\partial\Omega}\hat{\beta}(z)(u^+)^pd\sigma-\int_{\Omega}\Im(z)(u^+)^pdz-\epsilon||u^+||^p\right]\\
		&&+\frac{1}{p}\left[\frac{c_1}{p-1}||Du^-||^p_p+||u^-||^p_p\right]\ \mbox{with}\ \hat{\beta}=\frac{p-1}{c_1}\beta\ (\mbox{see (\ref{eq86})})\\
		&\geq&(c_{31}-\epsilon)||u^+||^p+c_{32}||u^-||^p\ \mbox{for some}\ c_{31},c_{32}>0\\
		&&\hspace{1cm}(\mbox{see Papageorgiou and R\u adulescu \cite{34} and}\ H(f)(iv))
	\end{eqnarray*}
	
	Choosing $\epsilon\in(0,c_{31})$ from (\ref{eq86}) we infer that
	\begin{eqnarray*}
		&&\hat{\varphi}_+(u)\geq c_{33}||u||^p\ \mbox{for all}\ u\in C^1(\overline{\Omega})\ \mbox{with}\ ||u||_{C^1(\overline{\Omega})}\leq\delta_1,\\
		&\Rightarrow&u=0\ \mbox{is a local}\ C^1(\overline{\Omega})-\mbox{minimizer of}\ \hat{\varphi}_+,\\
		&\Rightarrow&u=0\ \mbox{is a local}\ W^{1,p}(\Omega)-\mbox{minimizer of}\ \hat{\varphi}_+\ (\mbox{see Proposition \ref{prop8}}).
	\end{eqnarray*}
	
	Next suppose that $\beta\equiv 0$. Let $\delta>0$ be as postulated by hypothesis $H(f)(iv)$ and let $u\in C^1(\overline{\Omega})$ with $||u||_{C^1(\overline{\Omega})}\leq\delta$. Then hypothesis $H(f)(iv)$ implies
	$$-\int_{\Omega}F(z,u)dz\geq 0.$$
	
	So, we have
	\begin{eqnarray*}
		&&\hat{\varphi}_+(u)\geq 0=\hat{\varphi}_+(0)\ \mbox{for all}\ u\in C^1(\overline{\Omega})\ \mbox{with}\ ||u||_{C^1(\overline{\Omega})}\leq\delta,\\
		&\Rightarrow&u=0\ \mbox{is a local}\ W^{1,p}(\Omega)-\mbox{minimizer of}\ \hat{\varphi}_+\ (\mbox{see Proposition \ref{prop8}}).
	\end{eqnarray*}
	
	Similarly for the functionals $\hat{\varphi}_-$ and $\varphi$.
\end{proof}
\begin{prop}\label{prop13}
	If hypotheses $H(a),H(\beta),H(f)$ hold, then $K_{\hat{\varphi}_+}\subseteq C_+$ and $K_{\hat{\varphi}_-}\subseteq C_+$.
\end{prop}
\begin{proof}
	Let $u\in K_{\hat{\varphi}_+}$. Then
	\begin{eqnarray}\label{eq87}
		&&\hat{\varphi}'_+(u)=0,\nonumber\\
		&\Rightarrow&\left\langle A(u),h\right\rangle+\int_{\Omega}|u|^{p-2}uhdz+\int_{\partial\Omega}\beta(z)(u^+)^{p-1}hd\sigma=\int_{\Omega}[f(z,u^+)+(u^+)^{p-1}]hdz\\
		&&\hspace{6cm}\mbox{for all}\ h\in W^{1,p}(\Omega)\ (\mbox{see (\ref{eq54})}).\nonumber
	\end{eqnarray}
	
	In (\ref{eq87}) we choose $h=-u^-\in W^{1,p}(\Omega)$. Then
	\begin{eqnarray*}
		&&\frac{c_1}{p-1}||Du^-||^p_p+||u^-||^p_p\leq 0\ (\mbox{see Lemma \ref{lem2}})\\
		&\Rightarrow&u\geq 0.
	\end{eqnarray*}
	
	From Proposition \ref{prop7}, we have that $u\in L^{\infty}(\Omega)$. So, we can use the regularity theory of Lieberman \cite[p. 320]{22} and have that $u\in C_+$. Therefore
	$$K_{\hat{\varphi}_+}\subseteq C_+.$$
	
	Similarly for the functional $\hat{\varphi}_-$, using this time (\ref{eq55}). We show that
	$$K_{\hat{\varphi}_-}\subseteq -C_+.$$
\end{proof}

Now we are ready to produce two constant sign solutions for problem (\ref{eq4}).
\begin{prop}\label{prop14}
	If hypotheses $H(a),H(\beta),H(f)$ hold, then problem (\ref{eq4}) has at least two constant sign solutions
	$$u_0\in \mbox{int}\, C_+\ \mbox{and}\ v_0\in-\mbox{int}\, C_+.$$
\end{prop}
\begin{proof}
	Proposition \ref{prop13} together with (\ref{eq54}),(\ref{eq55}) indicate that we may assume that
	$$K_{\hat{\varphi}_+}\ \mbox{and}\ K_{\hat{\varphi}_-}\ \mbox{are infinite},$$
	or otherwise we already have a whole sequence of distinct solutions of constant sign.
	
	From Proposition \ref{prop12} we know that $u=0$ is a local minimizer of $\hat{\varphi}_+$. So, we can find $\rho\in(0,1)$ small such that
	\begin{equation}\label{eq88}
		\hat{\varphi}_+(0)=0<\inf[\hat{\varphi}_+(u):||u||=\rho]=\hat{m}^+_{\rho}
	\end{equation}
	(see Aizicovici, Papageorgiou and Staicu \cite{1}, proof of Proposition 29).
	
	Combining (\ref{eq88}) with Proposition \ref{prop9} and \ref{prop11}, we see that we can apply Theorem \ref{th1} (the mountain pass theorem). So, we can find $u_0\in W^{1,p}(\Omega)$ such that
	\begin{equation}\label{eq89}
		u_0\in K_{\hat{\varphi}_+}\subseteq C_+\ \mbox{(see Proposition \ref{prop13}) and}\ \hat{m}^+_{\rho}\leq\hat{\varphi}_+(u_0).
	\end{equation}
	
	From (\ref{eq88}) and (\ref{eq89}) we have that $u_0\neq 0$. Also since $u_0\geq 0$ (see (\ref{eq89})), we have
	\begin{eqnarray}\label{eq90}
		&&\left\langle A(u_0),h\right\rangle+\int_{\partial\Omega}\beta(z)u^{p-1}_0hd\sigma=\int_{\Omega}f(z,u_0)hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega)\ (\mbox{see (\ref{eq54})}),\nonumber\\
		&\Rightarrow&-\mbox{div}\,a(Du_0(z))=f(z,u_0(z))\ \mbox{for almost all}\ z\in\Omega,\nonumber\\
		&&\frac{\partial u_0}{\partial n_a}+\beta(z)u^{p-1}_0=0\ \mbox{on}\ \partial\Omega.
	\end{eqnarray}
	
	Hypothesis $H(f)(iv)$ implies that given $\rho>0$, we can find $\hat{\xi}_{\rho}>0$ such that
	\begin{equation}\label{eq91}
		f(z,x)x+\hat{\xi}_{\rho}|x|^p\geq 0\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ |x|\leq\rho.
	\end{equation}
	
	Let $\rho=||u_0||_{\infty}$ (recall $u_0\in C_+\backslash\{0\}$) and let $\hat{\xi}_{\rho}>0$ as in (\ref{eq91}). Then from (\ref{eq90}) we have
	\begin{equation}\label{eq92}
		\mbox{div}\,a(Du_0(z))\leq\hat{\xi}_{\rho}u_0(z)^{p-1}\ \mbox{for almost all}\ z\in\Omega.
	\end{equation}
	
	Let $\gamma(t)=a_0(t)t$ for $t>0$. Then  (\ref{eq2}) and hypothesis $H(a)(ii)$ ensure that
	$$\gamma'(t)t=a'_0(t)t^2+a_0(t)t\geq c_1t^{p-1}.$$
	
	Integrating we obtain
	\begin{eqnarray}\label{eq93}
		\int^t_0\gamma'(s)sds&=&\gamma(t)t-\int^t_0\gamma (s)ds\ (\mbox{integration by parts})\nonumber\\
		&=&a_0(t)t^2-G_0(t)\nonumber\\
		&\geq&\frac{c_1}{p}t^p\ \mbox{for all}\ t>0.
	\end{eqnarray}
	
	Let $\hat{d}(t)=a_0(t)t^2-G_0(t)$ and $\hat{d}_0(t)=\frac{c_1}{p}t^p$ for all $t>0$. Let $s>0$ and consider the following two sets
	$$C_1=\{t\in(0,1):\hat{d}(t)\geq s\},\ C_2=\{t\in(0,1):\hat{d}_0(t)\geq s\}.$$
	
	From (\ref{eq93}) we see that $C_2\subseteq C_1$ and so $\inf C_1\leq \inf C_2$. Therefore
	$$\hat{d}^{-1}(s)\leq\hat{d}^{-1}_0(s)\ (\mbox{see, for example, Leoni \cite[p. 6]{20}}).$$
	
	Then we have for $\delta>0$
	 $$\int^{\delta}_0\frac{1}{\hat{d}^{-1}\left(\frac{\hat{\xi}_{\rho}}{p}s^p\right)}ds\geq\int^{\delta}_0\frac{1}{\hat{d}^{-1}_0\left(\frac{\hat{\xi}_{\rho}}{p}s^p\right)}ds=\frac{\hat{\xi}_{\rho}}{p}\int^{\delta}_0\frac{ds}{s}=+\infty.$$
	
	Hence, because of (\ref{eq92}) we can apply the nonlinear strong maximum principle of Pucci and Serrin \cite[p. 111]{35}  and have that
	$$u_0(z)>0\ \mbox{for all}\ z\in\Omega.$$
	
	Then the boundary point theorem of Pucci and Serrin \cite[p. 120]{35}  implies that $u_0\in \mbox{int}\, C_+$.
	
	Similarly, working with the functional $\hat{\varphi}_-$ we produce a second constant sign solution $v_0\in -\mbox{int}\, C_+$.
\end{proof}

To produce a third nontrivial solution, we will use Morse theoretical tools (critical groups). To this end we compute the critical groups of $\varphi$ at infinity.
\begin{prop}\label{prop15}
	If hypotheses $H(a),H(\beta),H(f)$ hold and $\inf\varphi(K_{\varphi})>-\infty$, then $C_k(\varphi,\infty)=0$ for all $k\in\NN_0$.
\end{prop}
\begin{proof}
	From hypotheses $H(f)(i),(ii)$, we see that given $\gamma>0$, we can find $c_{34}=c_{34}(\gamma)>0$ such that
	\begin{equation}\label{eq94}
		F(z,x)\geq\gamma|x|^p-c_{34}\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\in\RR.
	\end{equation}
	
	Let $u\in\partial B_1=\{u\in W^{1,p}(\Omega):||u||=1\}$ and $t>0$. We have
	\begin{equation}\label{eq95}
		\varphi(tu)\leq t^p\left[c_{35}||Du||^p_p+c_{36}||u||^p_{L^p(\partial\Omega)}-\gamma||u||^p_p\right]+c_{37}
	\end{equation}
	for some $c_{35},c_{36},c_{37}>0$ (see Corollary \ref{cor3}, (\ref{eq94}) and hypothesis $H(\beta)$).
	
	Because $\gamma>0$ is arbitrary, from (\ref{eq95}) we see that
	\begin{equation}\label{eq96}
		\varphi(tu)\rightarrow-\infty\ \mbox{as}\ t\rightarrow-\infty.
	\end{equation}
	
	Also, we have
	\begin{eqnarray*}
		\frac{d}{dt}\varphi(tu)&=&\left\langle \varphi'(tu),u\right\rangle\ (\mbox{by the chain rule})\\
		&=&\frac{1}{t}\left\langle \varphi'(tu),tu\right\rangle\\
		 &=&\frac{1}{t}\left[\int_{\Omega}(a(tDu),tDu)_{\RR^N}dz+\int_{\partial\Omega}\beta(z)|tu|^pd\sigma-\int_{\Omega}f(z,tu)tudz\right]\\
		 &\leq&\frac{1}{t}\left[\int_{\Omega}pG(tDu)dz+\int_{\partial\Omega}\beta(z)|tu|^pd\sigma-\int_{\Omega}pF(z,tu)dz+c_{38}\right]\\
		&&\hspace{1cm}\mbox{for some}\ c_{38}>0\ (\mbox{see hypotheses}\ H(a)(iv)\ \mbox{and}\ H(f)(iii))\\
		&=&\frac{1}{t}[p\varphi(tu)+c_{38}].
	\end{eqnarray*}
	
	Then (\ref{eq96}) implies that for $t>0$ big, we have
	\begin{eqnarray*}
		&&\varphi(tu)\leq\Im_0<-c_{38},\\
		&\Rightarrow&\frac{d}{dt}\varphi(tu)<0\ \mbox{for}\ t>0\ \mbox{big}.
	\end{eqnarray*}
	
	Therefore we can find a unique $r(u)>0$ such that $\varphi(r(u)u)=\Im_0$. The implicit function theorem implies that $r\in C(\partial B_1)$. We extend $r(\cdot)$ to all of $W^{1,p}(\Omega)\backslash\{0\}$ by
	$$r_0(u)=\frac{1}{||u||}r\left(\frac{u}{||u||}\right)\ \mbox{for all}\ u\in W^{1,p}(\Omega)\backslash\{0\}.$$
	
	Then $r_0\in C(W^{1,p}(\Omega)\backslash\{0\})$ and $\varphi(r_0(u)u)=\Im_0$. Also, if $\varphi(u)=\Im_0$, then $r_0(u)=1$. So, we set
	\begin{eqnarray}\label{eq97}
		\hat{r}_0(u)=\left\{\begin{array}{ll}
			1&\mbox{if}\ \varphi(u)\leq\Im_0\\
			r_0(u)&\mbox{if}\ \Im_0<\varphi(u)
		\end{array}\right. .
	\end{eqnarray}
	
	Evidently $\hat{r}_0\in C(W^{1,p}(\Omega)\backslash\{0\})$.
	
	Consider the deformation $h(t,u)$ defined by
	$$h(t,u)=(1-t)u+t\hat{r}_0(u)u\ \mbox{for all}\ (t,u)\in[0,1]\times(W^{1,p}(\Omega)\backslash\{0\}).$$
	
	We have
	$$h(0,u)=u,h(1,u)=\hat{r}_0(u)u\in\varphi^{\Im_0}\ \mbox{and}\ h(t,\cdot)|_{\varphi^{\Im_0}}=id|_{\varphi^{\Im_0}}\ \mbox{for all}\ t\in[0,1]\ (\mbox{see (\ref{eq97})}).$$
	
	It follows that
	\begin{equation}\label{eq98}
		\varphi^{\Im_0}\ \mbox{is a strong deformation retract of}\ W^{1,p}(\Omega)\backslash\{0\}.
	\end{equation}
	
	We consider the radial retraction $\tilde{r}:W^{1,p}(\Omega)\backslash\{0\}\rightarrow\RR$ defined by
	$$\tilde{r}(u)=\frac{u}{||u||}\ \mbox{for all}\ u\in W^{1,p}(\Omega)\backslash\{0\}.$$
	
	This map is continuous and $\tilde{r}|_{\partial B_1}=id|_{\partial B_1}$. Therefore $\partial B_1$ is a retract of $W^{1,p}(\Omega)\backslash\{0\}$. We consider the deformation $\tilde{h}(t,u)$ defined by
	$$\tilde{h}(t,u)=(1-t)u+t\tilde{r}(u)\ \mbox{for all}\ (t,u)\in[0,1]\times(W^{1,p}(\Omega)\backslash\{0\}).$$
	
	Then
	$$\tilde{h}(0,u)=u,\tilde{h}(1,u)=\tilde{r}(u)\in\partial B_1\ \mbox{and}\ \tilde{h}(1,\cdot)|_{\partial B_1}=id|_{\partial B_1}.$$
	
	Hence we infer that
	\begin{equation}\label{eq99}
		\partial B_1\ \mbox{is a deformation retract of}\ W^{1,p}(\Omega)\backslash\{0\}.
	\end{equation}
	
	From (\ref{eq98}) and (\ref{eq99}) it follows that
	\begin{eqnarray}\label{eq100}
		&&\varphi^{\Im_0}\ \mbox{and}\ \partial B_1\ \mbox{are homotopy equivalent},\nonumber\\
		&\Rightarrow&H_k(W^{1,p}(\Omega),\varphi^{\Im_0})=H_k(W^{1,p}(\Omega),\partial B_1)\ \mbox{for all}\ k\in\NN_0,\nonumber\\
		&\Rightarrow&C_k(\varphi,\infty)=H_k(W^{1,p}(\Omega),\partial B_1)\ \mbox{for all}\ k\in\NN_0\\
		&&\hspace{1cm}(\mbox{by choosing $\Im_0<0$ even more negative if necessary}).\nonumber
	\end{eqnarray}
	
	The space $W^{1,p}(\Omega)$ is infinite dimensional and so $\partial B_1$ is contractible. Hence
	\begin{eqnarray*}
		&&H_k(W^{1,p}(\Omega),\partial B_1)=0\ \mbox{for all}\ k\in\NN_0\\
		&&\hspace{1cm}(\mbox{see Motreanu, Motreanu and Papageorgiou \cite[p. 147]{28}}),\\
		&\Rightarrow&C_k(\varphi,\infty)=0\ \mbox{for all}\ k\in\NN_0\ (\mbox{see (\ref{eq100})}).
	\end{eqnarray*}
\end{proof}

With suitable changes in the above proof, we can also compute the critical groups at infinity for the functionals $\hat{\varphi}_{\pm}$. So, we have:
\begin{prop}\label{prop16}
	Assume that hypotheses $H(a),H(\beta),H(f)$  hold and $\inf\hat{\varphi}_{\pm}(K_{\hat{\varphi}_{\pm}})>-\infty$. Then $C_k(\hat{\varphi}_{\pm},\infty)=0$ for all $k\in\NN_0$.
\end{prop}
\begin{proof}
	We do the proof for $\hat{\varphi}_+$, the proof for the functional $\hat{\varphi}_-$ being similar.
	
	Let $\partial B^+_1=\{u\in\partial B_1:u^+\neq 0\}$. Consider the deformation $h_+:[0,1]\times\partial B^+_1\rightarrow\partial B^+_1$ defined by
	$$h_+(t,u)=\frac{(1-t)u+t\hat{u}_1(p,\beta)}{||(1-t)u+t\hat{u}_1(p,\beta)||}\ \mbox{for all}\ (t,u)\in[0,1]\times\partial B^+_1.$$
	
	We have
	\begin{eqnarray*}
		&&h_+(1,u)=\frac{\hat{u}_1(p,\beta)}{||\hat{u}_1(p,\beta)||}\in\partial B^+_1,\\
		&\Rightarrow&\partial B^+_1\ \mbox{is contractible.}
	\end{eqnarray*}
	
	Hypotheses $H(f)(ii),(iii)$ imply that for every $u\in\partial B^+_1$, we have
	\begin{equation}\label{eq101}
		\hat{\varphi}_+(tu)\rightarrow-\infty\ \mbox{as}\ t\rightarrow+\infty.
	\end{equation}
	
	For $u\in\partial B^+_1$ and $t>0$, we have
	\begin{eqnarray}\label{eq102}
		\frac{d}{dt}\hat{\varphi}_+(tu)&=&\left\langle \hat{\varphi}_+'(tu),u\right\rangle\ (\mbox{by the chain rule})\nonumber\\
		&=&\frac{1}{t}\left\langle \hat{\varphi}'_+(tu),tu\right\rangle\nonumber\\
		 &=&\frac{1}{t}\left[\int_{\Omega}(a(tDu),tDu)_{\RR^N}dz+||tu^-||^p_p+\int_{\partial\Omega}\beta(z)(tu^+)^pd\sigma-\int_{\Omega}f(z,tu^+)tu^+dz\right]\nonumber\\
		&&\hspace{9cm}(\mbox{see (\ref{eq54})})\nonumber\\
		 &\leq&\frac{1}{t}\left[pG(tDu)dz+||tu^-||^p_p+\int_{\partial\Omega}\beta(z)(tu^+)^pdz-\int_{\Omega}pF(z,tu^+)dz+c_{39}\right]\nonumber\\
		&&\hspace{3cm}\mbox{for some}\ c_{39}>0\ (\mbox{see hypotheses}\ H(a)(iv),H(f)(iii))\nonumber\\
		&=&\frac{1}{t}[p\hat{\varphi}_+(tu)+c_{39}].
	\end{eqnarray}
	
	From (\ref{eq101}) and (\ref{eq102}) it follows that
	\begin{equation}\label{eq103}
		\frac{d}{dt}\hat{\varphi}_+(tu)<-\frac{c_{39}}{p}<0\ \mbox{for all}\ t>0\ \mbox{big}.
	\end{equation}
	
	Choose
	$$\xi_0<min\left\{-\frac{c_{39}}{p},\inf\limits_{\bar{B}_1}\hat{\varphi}_+\right\}$$
	(recall that $\bar{B}_1=\{u\in W^{1,p}(\Omega):||u||\leq 1\}$). Given $u\in\partial B^+_1$, because of (\ref{eq103}) we see that there is unique $s_0(u)\geq 1$ such that
	\begin{eqnarray}\label{eq104}
		\hat{\varphi}_+(tu)\left\{\begin{array}{ll}
			>\xi_0&\mbox{if}\ t\in\left[0,s_0(u)\right)\\
			=\xi_0&\mbox{if}\ t=s_0(u)\\
			<\xi_0&\mbox{if}\ s_0(u)<t
		\end{array}\right. .
	\end{eqnarray}
	
	The implicit function theorem implies that $s_0\in C(\partial B^+_1)$. Note that
	$$\hat{\varphi}^{\xi_0}_+=\{tu:u\in\partial B^+_1,t\geq\gamma_0(u)\}\ \mbox{(see (\ref{eq104}))}.$$
	
	We define $E_+=\{tu:u\in\partial B^+_1,t\geq 1\}$. We have $\hat{\varphi}^{\xi_0}_+\subseteq E_+$. We consider the deformation $\hat{h}_+(r,tu)$ defined by
	$$\hat{h}_+(r,tu)=\left\{\begin{array}{ll}
		(1-r)tu+rs_0(u)u&\mbox{if}\ t\in[0,s_0(u)]\\
		tu&\mbox{if}\ s_0(u)<t
	\end{array}\right.\ \mbox{for all}\ (r,tu)\in[0,1]\times E_+.$$
	
	We have
	$$\hat{h}_+(0,tu)=tu,\hat{h}_+(1,tu)\in \hat{\varphi}^{\xi_0}_+\ \mbox{and}\ \hat{h}_+(r,\cdot)|_{\hat{\varphi}^{\xi_0}_+}=id|_{\hat{\varphi}^{\xi_0}_+}\ \mbox{for all}\ r\in[0,1]\ (\mbox{see (\ref{eq104})}).$$
	
	Therefore
	\begin{eqnarray}\label{eq105}
		&&\hat{\varphi}^{\xi_0}_+\ \mbox{is a strong deformation retract of}\ E_+\nonumber,\\
		&\Rightarrow&H_k(W^{1,p}(\Omega),\hat{\varphi}^{\xi_0}_+)=H_k(W^{1,p}(\Omega),E_+)\ \mbox{for all}\ k\in\NN_0\nonumber\\
		&\Rightarrow&C_k(\hat{\varphi}_+,\infty)=H_k(W^{1,p}(\Omega),E_+)\ \mbox{for all}\ k\in\NN_0\\
		&&\hspace{3cm}(\mbox{by choosing}\ \xi_0<0\ \mbox{even more negative if necessary}).\nonumber
	\end{eqnarray}
	
	Consider the deformation
	$$h^*_+(r,tu)=(1-r)tu+r\frac{tu}{||tu||}\ \mbox{for all}\ (r,tu)\in[0,1]\times E_+.$$
	
	We see that
	$$h^*_+(0,tu)=tu,h^*_+(1,tu)\in\partial B^+_1\ \mbox{and}\ h^*_+(1,\cdot)|_{\partial B^+_1}=id|_{\partial B^+_1}.$$
	
	Therefore
	\begin{eqnarray*}
		&&\partial B^+_1\ \mbox{is a deformation retract of}\ E_+,\\
		&\Rightarrow&H_k(W^{1,p}(\Omega),\partial B^+_1)=H_k(W^{1,p}(\Omega),E_+)\ \mbox{for all}\ k\in\NN_0,\\
		&\Rightarrow&H_k(W^{1,p}(\Omega),E_+)=0\ \mbox{for all}\ k\in\NN_0\\
		&&\hspace{3cm}(\mbox{recall that $\partial B^+_1$, is contractible}),\\
		&\Rightarrow&C_k(\hat{\varphi}_+,\infty)=0\ \mbox{for all}\ k\in\NN_0\ (\mbox{see (\ref{eq105})}).
	\end{eqnarray*}
	
	Similarly for the functional $\hat{\varphi}_-$.
\end{proof}

Using Propositions \ref{prop15} and \ref{prop16}, we can compute precisely the critical groups of the energy functional $\varphi$ at the two constant sign solutions $u_0\in \mbox{int}\, C_+$ and $v_0\in-\mbox{int}\, C_+$ produced in Proposition \ref{prop14}.

First we relate the critical groups of $\varphi$ with those of $\hat{\varphi}_{\pm}$. In what follows we assume that the critical sets
$$K_{\varphi}\ \mbox{and}\ K_{\hat{\varphi}_{\pm}}\ \mbox{are finite}.$$

Otherwise we already have a whole sequence of distinct solutions of (\ref{eq4}) (see Proposition \ref{prop13} and (\ref{eq54}), (\ref{eq55})).
\begin{prop}\label{prop17}
	If hypotheses $H(a),H(\beta),H(f)$ hold, then $C_k(\varphi,u_0)=C_k(\hat{\varphi}_+,u_0)$ and $C_k(\varphi,u_0)=C_k(\hat{\varphi}_-,v_0)$ for all $k\in\NN_0$.
\end{prop}
\begin{proof}
	We do the proof for the triple $(\varphi,\hat{\varphi}_+,u_0)$, the proof for the other triple $(\varphi,\hat{\varphi}_-,v_0)$ being similar.
	
	We consider the homotopy
	$$h(t,u)=(1-t)\varphi(u)+t\hat{\varphi}_+(u)\ \mbox{for all}\ (t,u)\in[0,1]\times W^{1,p}(\Omega).$$
	
	Suppose 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{equation}\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\ \mbox{for all}\ n\in\NN.
	\end{equation}
	
	Then from the equation in (\ref{eq106}) we have
	\begin{eqnarray}\label{eq107}
		&&\left\langle A(u_n),v\right\rangle+\int_{\partial\Omega}\beta(z)(u^+_n)^{p-1}vd\sigma-t_n\int_{\partial\Omega}\beta(z)(u^-_n)^{p-1}vd\sigma-t_n\int_{\Omega}(u^-_n)^{p-1}vdz\nonumber\\
		&&=\int_{\Omega}f(z,u^+_n)vdz+(1-t_n)\int_{\Omega}f(z,-u^-_n)dz\ \mbox{for all}\ v\in W^{1,p}(\Omega)\ (\mbox{see (\ref{eq54})})\nonumber\\
		 &\Rightarrow&-\mbox{div}\,a(Du_n(z))-t_nu^-_n(z)^{p-1}=f(z,u^+_n(z))+(1-t_n)f(z,-u^-_n(z))\nonumber\\
		&&\hspace{8cm}\mbox{for almost all}\ z\in\Omega,\nonumber\\
		&&\frac{\partial u_n}{\partial n_a}+\beta(z)((u^+_n)^{p-1}-t_n(u^-_n)^{p-1})=0\ \mbox{on}\ \partial\Omega.
	\end{eqnarray}
	
	From Proposition \ref{prop7} we know that there exists $M_{14}>0$ such that
	$$||u_n||_{\infty}\leq M_{14}\ \mbox{for all}\ n\in\NN.$$
	
	So, from Lieberman \cite{22}, we know that there exist $\alpha\in(0,1)$ and $M_{15}>0$ such that
	\begin{equation}\label{eq108}
		u_n\in C^{1,\alpha}(\overline{\Omega})\ \mbox{and}\ ||u_n||_{C^{1,\alpha}(\overline{\Omega})}\leq M_{14}\ \mbox{for all}\ n\in\NN.
	\end{equation}
	
	Because of (\ref{eq106}) and since $C^{1,\alpha}(\overline{\Omega})$ si embedded compactly into $C^1(\overline{\Omega})$, from (\ref{eq108}) we have
	$$u_n\rightarrow u_0\ \mbox{in}\ C^1(\overline{\Omega}).$$
	
	Recall that $u_0\in \mbox{int}\, C_+$ (see Proposition \ref{prop14}). So, we can find $n_0\in\NN$ such that
	\begin{eqnarray*}
		&&u_n\in intC_+\ \mbox{for all}\ n\geq n_0,\\
		&\Rightarrow&\{u_n\}_{n\geq n_0}\ \mbox{are distinct (positive) solutions of (\ref{eq4}) (see (\ref{eq106}))},
	\end{eqnarray*}
	a contradiction (recall that we have assumed $K_{\hat{\varphi}_+}$ is finite).
	
	Therefore (\ref{eq106}) can not happen. Then invoking Theorem 5.2 of Corvellec and Hantoute \cite{10} (the homotopy invariance of critical groups), we have
	$$C_k(\varphi,u_0)=C_k(\hat{\varphi}_+,u_0)\ \mbox{for all}\ k\in\NN_0.$$
	
	In a similar fashion we show that
	$$C_k(\varphi,v_0)=C_k(\hat{\varphi}_-,v_0)\ \mbox{for all}\ k\in\NN_0.$$
\end{proof}
\begin{prop}\label{prop18}
	If hypotheses $H(a),H(\beta),H(f)$ hold, then $C_k(\hat{\varphi}_+,u_0)=C_k(\hat{\varphi}_-,v_0)=\delta_{k,1}\ZZ$ for all $k\in\NN_0$.
\end{prop}
\begin{proof}
	We do the proof for the pair $(\hat{\varphi}_+,u_0)$, the proof for the pair $(\hat{\varphi}_-,v_0)$ being similar.
	
	From Proposition \ref{prop13} we know that $K_{\hat{\varphi}_+}\subseteq C_+$. So, we may assume that
	\begin{equation}\label{eq109}
		K_{\hat{\varphi}_+}=\{0,u_0\}
	\end{equation}
	or otherwise we already have a third nontrivial solution for problem (\ref{eq4}) which in fact is positive.
	
	From the proof of Proposition \ref{prop14} (see (\ref{eq88}),(\ref{eq89})), we have
	$$0=\hat{\varphi}_+(0)<m^+_{\rho}\leq\hat{\varphi}_+(u_0).$$
	
	Let $\xi_-<0<\xi_+<m^+_{\rho}$ and consider the following triple of sets
	$$\hat{\varphi}^{\xi_-}_+\subseteq\hat{\varphi}^{\xi_+}_+\subseteq W^{1,p}(\Omega).$$
	
	For this triple of sets, we consider the corresponding long exact sequence of singular homology groups (see Motreanu, Motreanu and Papageorgiou \cite[p. 143]{28}).
	\begin{equation}\label{eq110}
		\cdots\rightarrow H_k(W^{1,p}(\Omega),\hat{\varphi}^{\xi_-}_+)\stackrel{i_*}{\longrightarrow}H_k(W^{1,p}(\Omega),\hat{\varphi}^{\xi_+}_+)\stackrel{\partial_*}{\longrightarrow}H_{k-1}(\hat{\varphi}^{\xi_+}_+,\hat{\varphi}^{\xi_-}_+)\rightarrow\cdots
	\end{equation}
	with $i_*$ being the homomorphism induced by the inclusion $i:(W^{1,p}(\Omega),\hat{\varphi}^{\xi_-}_+)\rightarrow$\\ $(W^{1,p}(\Omega),\hat{\varphi}^{\xi_+}_+)$ and $\partial_*$ is the boundary homomorphism.
	
	From (\ref{eq109}) and since $\xi_-<0=\hat{\varphi}_+(0)$, we have
	\begin{equation}\label{eq111}
		H_k(W^{1,p}(\Omega),\hat{\varphi}_+^{\xi_-})=C_k(\hat{\varphi}_+,\infty)=0\ \mbox{for all}\ k\in\NN_0\ (\mbox{see Proposition \ref{prop16}}).
	\end{equation}
	
	Also, we have
	$$0=\hat{\varphi}_+(0)<\xi_+<\hat{\varphi}_+(u_0).$$
	
	Then from (\ref{eq109})  we have
	\begin{equation}\label{eq112}
		H_k(W^{1,p}(\Omega),\hat{\varphi}_+^{\xi_+})=C_k(\hat{\varphi}_+,u_0)\ \mbox{for all}\ k\in\NN_0.
	\end{equation}
	
	Similarly, we have
	\begin{equation}\label{eq113}
		 H_{k-1}(\hat{\varphi}_+^{\xi_+},\hat{\varphi}_+^{\xi_-})=C_{k-1}(\hat{\varphi}_+,0)=\delta_{k-1,0}\ZZ=\delta_{k,1}\ZZ\ \mbox{for all}\ k\in\NN_0\ (\mbox{see Proposition \ref{prop12}}).
	\end{equation}
	
	From (\ref{eq111}), (\ref{eq112}), (\ref{eq113}) and the exactness of (\ref{eq110}), we see that only the tail of that chain (that is, $k=1$) is nontrivial. From the rank theorem, we have
	\begin{eqnarray}\label{eq114}
		\mbox{rank}\, H_1(W^{1,p}(\Omega),\hat{\varphi}_+^{\xi_+})&=&\mbox{rank}\, \mbox{ker}\,\partial_*+\mbox{rank}\, \mbox{im}\,\partial_*\nonumber\\
		&=&\mbox{rank}\, \mbox{im}\, i_*+\mbox{rank}\, \mbox{im}\,\partial_*\ (\mbox{by the exactness of (\ref{eq110})})\nonumber\\
		&\leq&1\ (\mbox{see (\ref{eq111}), (\ref{eq113})}).
	\end{eqnarray}
	
	From the proof of Proposition \ref{prop14}, we know that $u_0$ is a critical point of $\hat{\varphi}_+$ of mountain pass type. Therefore
	\begin{equation}\label{eq115}
		C_1(\hat{\varphi}_+,u_0)\neq 0.
	\end{equation}
	
	From (\ref{eq112}), (\ref{eq114}), (\ref{eq115}) and recalling that only for $k=1$, the chain (\ref{eq110}) is nontrivial, we conclude that
	$$C_k(\hat{\varphi}_+,u_0)=\delta_{k,1}\ZZ\ \mbox{for all}\ k\in\NN_0.$$
	
	Similarly for the pair $(\hat{\varphi}_-,v_0)$.
\end{proof}

From Proposition \ref{prop17} and \ref{prop18}, we infer the following corollary.
\begin{corollary}\label{cor19}
	If hypotheses $H(a),H(\beta),H(f)$ hold, then $C_k(\varphi,u_0)=C_k(\varphi,v_0)=\delta_{k,1}\ZZ$ for all $k\in\NN_0$.
\end{corollary}

Now we ready for the ``three solutions theorem" for problem (\ref{eq4}).
\begin{theorem}\label{th20}
	If hypotheses $H(a),H(\beta),H(f)$ hold, then problem (\ref{eq4}) has at least three nontrivial solutions
	$$u_0\in \mbox{int}\, C_+,v_0\in-\mbox{int}\, C_+\ \mbox{and}\ y_0\in C^1(\overline{\Omega}).$$
\end{theorem}
\begin{proof}
	From Proposition \ref{prop14} we already have two constant sign solutions
	$$u_0\in \mbox{int}\, C_+\ \mbox{and}\ v_0\in-\mbox{int}\, C_+.$$
	
	Suppose that these are the only nontrivial solutions of problem (\ref{eq4}) (that is, $K_{\varphi}=\{0,u_0,v_0\}$).
	
	From Corollary \ref{cor19}, we have
	\begin{equation}\label{eq116}
		C_k(\varphi,u_0)=C_k(\varphi,v_0)=\delta_{k,1}\ZZ\ \mbox{for all}\ k\in\NN_0.
	\end{equation}
	
	From Proposition \ref{prop12}, we have
	\begin{equation}\label{eq117}
		C_k(\varphi,0)=\delta_{k,0}\ZZ\ \mbox{for all}\ k\in\NN_0.
	\end{equation}
	
	Finally Proposition \ref{prop15} implies that
	\begin{equation}\label{eq118}
		C_k(\varphi,\infty)=0\ \mbox{for all}\ k\in\NN_0
	\end{equation}
	
	From (\ref{eq116}), (\ref{eq117}), (\ref{eq118}) and the Morse relation with $t=-1$ (see (\ref{eq52})), we have
	\begin{eqnarray*}
		&&2(-1)^1+(-1)^0=0,\\
		&\Rightarrow&(-1)^1=0,\ \mbox{a contradiction}.
	\end{eqnarray*}
	
	So, we can find $y_0\in K_{\varphi},\ y_0\notin\{0,u_0,v_0\}$. This is the third nontrivial solution of problem (\ref{eq4}) and as before the nonlinear regularity theory implies $y_0\in C^1(\overline{\Omega})$.
\end{proof}

\medskip
 {\bf Acknowledgement}. The authors wish to thank a knowledgeable referee for his/her constructive remarks.


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\end{document} 