% version 1 April 2014
\documentclass{elsart3-1}

\usepackage{amssymb}

\usepackage[english,francais]{babel}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{e-proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{e-definition}[theorem]{Definition\rm}
\newtheorem{remark}{\it Remark\/}
\newtheorem{example}{\it Example\/}
%French
\newtheorem{theoreme}{Th\'eor\`eme}[section]
\newtheorem{lemme}[theoreme]{Lemme}
\newtheorem{proposition}[theoreme]{Proposition}
\newtheorem{corollaire}[theoreme]{Corollaire}
\newtheorem{definition}[theoreme]{D\'efinition\rm}
\newtheorem{remarque}{\it Remarque}
\newtheorem{exemple}{\it Exemple\/}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\di}{\displaystyle}

\renewcommand{\theequation}{\arabic{equation}}
\setcounter{equation}{0}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% GUILLEMETS (FRENCH QUOTES) %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\og{\leavevmode\raise.3ex\hbox{$\scriptscriptstyle\langle\!\langle$~}}
\def\fg{\leavevmode\raise.3ex\hbox{~$\!\scriptscriptstyle\,\rangle\!\rangle$}}
\def\ri{\rightarrow}

\begin{document}

\begin{frontmatter}


\selectlanguage{english}
\title{Bifurcation near the origin for the Robin problem with concave-convex nonlinearities}

\vspace{-2.6cm}

\selectlanguage{french}
\title{Bifurcation autour de l'origine pour le probl\`eme de Robin avec terme concave-convexe}

\selectlanguage{english}
\author[NP]{Nikolaos S. Papageorgiou}
\ead{npapg@math.ntua.gr}
\address[NP]{National Technical University, Department of Mathematics,
				Zografou Campus, Athens 15780, Greece}
\author[VR]{Vicen\c tiu D. R\u{a}dulescu}
\ead{vicentiu.radulescu@math.cnrs.fr}
%\ead[url]{http://www.math.ucv.ro/\~{}radulescu}
\address[VR]{Department of Mathematics, Faculty of Science, King Abdulaziz University,
Jeddah, Saudi Arabia}


\begin{abstract}
% Text of abstract in English
In this Note, we deal with the Robin parametric elliptic equation driven by a nonhomogeneous differential operator and with a reaction that exhibits competing terms (concave-convex nonlinearities). Without employing the Ambrosetti-Rabinowitz condition, we prove a bifurcation theorem for small positive values of the real parameter.
\vskip 0.5\baselineskip

\selectlanguage{francais}
% Text of abstract in French
\noindent{\bf R\'esum\'e} \vskip 0.5\baselineskip \noindent
Dans cette Note, nous  \'etudions le probl\`eme elliptique param\'etrique de Robin pour un op\'erateur diff\'erentiel non homog\`ene et avec une r\'eaction qui pr\'esente des termes concurrents (du type concave-convexe). Sans utiliser la condition d'Ambrosetti-Rabinowitz, nous prouvons un th\'eor\`eme de bifurcation pour de petites valeurs positives du param\`etre r\'eel.
\end{abstract}

\end{frontmatter}

% now the Version fran\c{c}aise abr\'eg\'ee, if it exists
\selectlanguage{francais}
\section*{Version fran\c{c}aise abr\'eg\'ee}
Soit $p>1$ et supposons que $\Omega\subset \RR^N$ est un domaine born\'e et r\'egulier. Soit $\beta\in C^{1,\alpha}(\partial\Omega)$ telle que $\beta\geq 0$ sur $\partial\Omega$ et supposons que $a:\RR^N\rightarrow\RR^N$ est continue, strictement monotone et satisfait les hypoth\`eses suivantes:

 \smallskip
\textbf{$H(a):$} $a(y)=a_0(||y||)y$ pour tout $y\in \RR^N$, avec $a_0>0$ sur $(0,\infty)$ et
\begin{itemize}
	\item[(i)] $a_0\in C^1(0,\infty),\ t\longmapsto a_0(t)t$ est strictement croissante sur $(0,\infty)$, $a_0(t)t\rightarrow 0$ si $t\rightarrow 0^+$ et
	$$\lim\limits_{t\rightarrow 0^+}\frac{a'_0(t)t}{a_0(t)}>-1;$$
	\item[(ii)] il existe $c_1>0$ tel que $\di |\nabla a(y)|\leq c_1\frac{\eta(|y|)}{|y|}$  pour tout $y\in\RR^N\backslash\{0\}$;
	\item[(iii)] $\di\frac{\eta(|y|)}{|y|}|\xi|^2\leq(\nabla a(y)\xi,\xi)_{\RR^N}$ pour tout $y\in\RR^N\backslash\{0\}$ et $\xi\in\RR^N$;
	\item[(iv)] si $G_0(t)=\int^{t}_{0}a_0(s)sds$ pour $t\geq 0$, alors $pG_0(t)-a_0(t)t^2\geq -\hat\xi$ pour tout $t\geq 0$, o\`u $\hat\xi>0$;
	\item[(v)] il existe $\tau\in(1,p)$ tel que  $t\longmapsto G_0(t^{1/\tau})$ est convexe sur $(0,\infty)$, $\lim\limits_{t\rightarrow 0^+}\frac{G_0(t)}{t^{\tau}}=0$ et il existe $\tilde{c}>0$ tel que
	$a_0(t)t^2-\tau G_0(t)\geq \tilde{c}t^p$ pour tout $t>0.$
\end{itemize}

\smallskip
Supposons que $f:\Omega\times\RR\times(0,\infty)\rightarrow\RR$ est une fonction qui satisfait les conditions suivantes:

\smallskip
\textbf{$H(f):$}
\begin{itemize}
 	\item[(i)] pour tout $(x,\lambda)\in\RR\times(0,\infty)$, la fonction $z\longmapsto f(z,x,\lambda)$ est mesurable et pour presque tout $z\in\Omega$, l'application $(x,\lambda)\longmapsto f(z,x,\lambda)$ est continue;
	\item[(ii)] $|f(z,x,\lambda)|\leq a_{\lambda}(z)(1+x^{r-1})$ p.p. $z\in\Omega$, pour tout $x\geq 0$ et tout $\lambda>0$, o\`u $a_{\lambda}\in L^{\infty}(\Omega)$, $\lambda\longmapsto||a_{\lambda}||_{\infty}$ est born\'e sur les ensembles born\'es de $(0,\infty)$ et $p<r<p^*$;
	\item[(iii)] si $F(z,x,\lambda)=\int^{x}_{0}f(z,s,\lambda)ds$, alors $\lim\limits_{x\rightarrow+\infty}\frac{F(z,x,\lambda)}{x^p}=+\infty$ uniform\'ement p.p. pour $z\in\Omega$;
	\item[(iv)] il existe $\vartheta=\vartheta(\lambda)\in\left((r-p)\max\left\{\frac{N}{p},1\right\},p^*\right)$ tel que
 $$0<\gamma_0\leq\liminf\limits_{x\rightarrow+\infty}\frac{f(z,x,\lambda)x-pF(z,x,\lambda)}{x^{\vartheta}}\ \mbox{uniform\'ement p.p. pour}\ z\in\Omega;$$
	\item[(v)] il existe $1<\mu=\mu(\lambda)<q=q(\lambda)<\tau$ et $\gamma=\gamma(\lambda)>\mu$, $\delta_0=\delta_0(\lambda)\in(0,1)$ tels que
	$$c_3x^q\leq f(z,x,\lambda)x\leq qF(z,x,\lambda)\leq\xi_{\lambda}(z)x^{\mu}+\tau x^{\gamma}\ \mbox{p.p.}\ z\in\Omega,\ \mbox{pour tout}\ 0\leq x\leq \delta_0,$$
	avec $c_3=c_3(\lambda)>0$, $c_3(\lambda)\rightarrow+\infty$ si $\lambda\rightarrow+\infty$, $\overline{c}=\overline{c}(\lambda)>0$, $\xi_{\lambda}\in L^{\infty}(\Omega)_+$ o\`u $||\xi_{\lambda}||_{\infty}\rightarrow 0$ si $\lambda\rightarrow 0^+$;
	\item[(vi)] pour tout $\rho>0$, il existe $\xi_{\rho}=\xi_{\rho}(\lambda)>0$ tel que pour p.p. $z\in\Omega$, l'application $x\longmapsto f(z,x,\lambda)+\xi_{\rho}x^{p-1}$ est croissante sur $[0,\rho]$;
	\item[(vii)] pour chaque intervalle $K=[x_0,\hat{x}]$ avec $x_0>0$ et pour tout $\lambda>\lambda'>0$, il existe $d_{K}(x_0,\lambda)$ croissante par rapport \`a $\lambda$ et telle que $d_K(x_0,\lambda)\rightarrow+\infty$ si $\lambda\rightarrow+\infty$ et il existe $\hat{d}_K(x_0,\lambda,\lambda')$ tel que
	\begin{eqnarray*}
		&&f(z,x,\lambda)\geq d_K(x_0,\lambda)\ \mbox{p.p.}\ z\in\Omega,\ \mbox{pour tout}\ x\in K\\
		&&f(z,x,\lambda)-f(z,x,\lambda')\geq\hat{d}_K(x_0,\lambda,\lambda')\ \mbox{p.p.}\ z\in\Omega,\ \mbox{pour tout}\ x\in K.
	\end{eqnarray*}
\end{itemize}


Dans cette Note nous \'etudions
le probl\`eme de Robin non lin\'eaire suivante:
$$
	\left\{
	\begin{array}{lll}
		&-\mbox{div}\, a(Du(z))=f(z,u(z),\lambda)\quad & \mbox{dans}\ \Omega,\\
		&\di\frac{\partial u}{\partial n_a}(z)+\beta(z)u(z)^{p-1}=0\quad & \mbox{sur}\ \partial\Omega,\\
		&u>0.&\
	\end{array}\right.\eqno(P)
$$

Soit
$C_+=\{u\in C^1(\overline{\Omega}):u(z)\geq 0\ \mbox{pour tout}\ z\in\overline{\Omega}\}$ et
$\mbox{int}\, C_+=\{u\in C_+:u(z)>0\ \mbox{pour tout}\ z\in\overline{\Omega}\}.$

Le r\'esultat principal de cette Note est le suivant.

\begin{theoreme}\label{th0}
Supposons que les hypoth\`eses  $H(a)$ et $H(f)$ sont satisfaites. Alors il existe $\lambda^*>0$ tel que
	\begin{itemize}
		\item[(a)] pour tout $\lambda\in(0,\lambda^*)$, le probl\`eme $(P)$ a au moins deux solutions
		$u_0,\hat{u}\in {\rm int}\, C_+,\ u_0\leq\hat{u},\ u_0\neq\hat{u};$
		\item[(b)] si $\lambda=\lambda^*$, le probl\`eme  $(P)$ admet au moins une solution $u_*\in {\rm int}\, C_+$;
		\item[(c)] pour tout $\lambda>\lambda^*$, le probl\`eme  $(P)$ n'a pas de solutions.
	\end{itemize}
\end{theoreme}




\vspace{0.05cm}
$$\line(1,0){300}$$
\selectlanguage{english}

\vspace{0.05cm}
Let $p>1$ be a real number and let $\Omega\subset\RR^N$ be a bounded domain with $C^2$-boundary $\partial\Omega$.
In this Note, we study the following nonlinear, nonhomogeneous parametric Robin problem:
$$
	\left\{
	\begin{array}{lll}
		&-\mbox{div}\, a(Du(z))=f(z,u(z),\lambda)\quad & \mbox{in}\ \Omega,\\
		&\di\frac{\partial u}{\partial n_a}(z)+\beta(z)u(z)^{p-1}=0\quad & \mbox{on}\ \partial\Omega,\\
		&u>0.&\
	\end{array}\right.\eqno(P)
$$

 We assume that $a:\RR^N\rightarrow\RR^N$ is a continuous and strictly monotone map, which satisfies the following hypotheses:

 \smallskip
\textbf{$H(a):$} $a(y)=a_0(||y||)y$ for all $y\in \RR^N$, with $a_0(t)>0$ for all $t>0$ and
\begin{itemize}
	\item[(i)] $a_0\in C^1(0,\infty),\ t\longmapsto a_0(t)t$ 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)] $\di |\nabla a(y)|\leq c_1\frac{\eta(|y|)}{|y|}$ for some $c_1>0$, all $y\in\RR^N\backslash\{0\}$;
	\item[(iii)] $\di\frac{\eta(|y|)}{|y|}|\xi|^2\leq(\nabla a(y)\xi,\xi)_{\RR^N}$ for all $y\in\RR^N\backslash\{0\}$, all $\xi\in\RR^N$;
	\item[(iv)] if $G_0(t)=\int^{t}_{0}a_0(s)sds$ for all $t\geq 0$, then $pG_0(t)-a_0(t)t^2\geq -\hat\xi$ for all $t\geq 0$, some $\hat\xi>0$;
	\item[(v)] there exists $\tau\in(1,p)$ such that  $t\longmapsto G_0(t^{1/\tau})$ is convex on $(0,\infty)$, $\lim\limits_{t\rightarrow 0^+}\frac{G_0(t)}{t^{\tau}}=0$ and
	$$a_0(t)t^2-\tau G_0(t)\geq \tilde{c}t^p\ \mbox{for some}\ \tilde{c}>0,\ \mbox{all}\ t>0.$$
\end{itemize}

These conditions on $a(\cdot)$ are motivated by the regularity results of Lieberman \cite{15} and the nonlinear maximum principle of Pucci and Serrin \cite{19}. According to the above conditions, the potential function $G_0(\cdot)$ is strictly convex and strictly increasing. We set $G(y)=G_0(|y|)$ for all $y\in\RR^N$. Then  $G$ is convex and differentiable, $\nabla G(0)=0$, and
	$$\nabla G(y)=G'_0(|y|)\frac{y}{|y|}=a_0(|y|)y=a(y)\ \mbox{for all}\ y\in\RR^N\backslash\{0\}.$$
	So, $G(\cdot)$ is the primitive of the map $a(\cdot)$. Because $G(0)=0$ and $y\longmapsto G(y)$ is convex, from the properties of convex functions, we have
		$G(y)\leq(a(y),y)_{\RR^N}$ for all $y\in\RR^N$. By direct computation we deduce that hypotheses $H(a)(i),(ii),(iii)$ imply the following properties of $a(\cdot)$:
	\begin{itemize}
		\item[(a)] $y\longmapsto a(y)$ is continuous and strictly monotone, hence maximal monotone too;
		\item[(b)] $|a(y)|\leq c_2(1+|y|^{p-1})$ for some $c_2>0$, all $y\in\RR^N;$
		\item[(c)] $(a(y),y)_{\RR^N}\geq\frac{c}{p-1}|y|^p$ for all $y\in\RR^N$.
	\end{itemize}


	The following maps  satisfy hypotheses $H(a)$:\\
		(a) $a(y)=|y|^{p-2}y$ with $1<p<\infty$.
		This map corresponds to the $p$-Laplacian operator defined by
		$$\Delta_pu={\rm div}\,(|Du|^{p-2}Du)\ \mbox{for all}\ u\in W^{1,p}(\Omega);$$\\
		(b) $a(y)=|y|^{p-2}y+\mu|y|^{q-2}y$ with $1<q<p<\infty$ and $\mu>0$.
		This map corresponds to the $(p,q)$-differential operator defined by
		$$\Delta_pu+\mu\Delta_qu\ \mbox{for all}\ u\in W^{1,p}(\Omega);$$\\
		(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
		$${\rm div}\left[(1+|Du|^2)^{\frac{p-2}{2}}Du\right]\ \mbox{for all}\ u\in W^{1,p}(\Omega);$$\\
		(d) $a(y)=|y|^{p-2}y+\frac{|y|^{p-2}y}{1+|y|^p}$ with $1<p<\infty$.
	

 We denote by  $\frac{\partial u}{\partial n_a}$  the conormal derivative, that is, $\frac{\partial u}{\partial n_a}=(a(Du),n)_{\RR^N}$ with $n(z)$ being the outward unit normal at $z\in\partial\Omega$.

 Throughout this paper, we assume that the boundary weight map $\beta\in C^{1,\alpha}(\partial\Omega)$ (for some $\alpha\in(0,1)$) satisfies $\beta(z)\geq 0$ for all $z\in\partial\Omega$.


 The reaction term $f(z,x,\lambda)$ is a parametric function with $\lambda>0$ being the parameter and $(z,x)\longmapsto f(z,x,\lambda)$ is Carath\'eodory (that is, for all $x\in\RR$ the mapping $z\longmapsto f(z,x,\lambda)$ is measurable and for a.a. $z\in\Omega$ the map $x\longmapsto f(z,x,\lambda)$ is continuous). We assume that $f(z,\cdot,\lambda)$ exhibits competing nonlinearities, namely near the origin, it has a ``concave" term (that is, a strictly $(p-1)$-sublinear term), while near $+\infty$, the reaction is ``convex"  (that is, $x\longmapsto f(z,x,\lambda)$ is ($p-1$)-superlinear). More precisely, the hypotheses on $f$ are the following:

\smallskip
\textbf{$H(f):$} $f:\Omega\times\RR\times(0,\infty)\rightarrow\RR$ is a function such that  for a.a. $z\in\Omega$ and all $\lambda>0$, $f(z,0,\lambda)=0$ and
\begin{itemize}
	\item[(i)] for all $(x,\lambda)\in\RR\times(0,\infty)$, $z\longmapsto f(z,x,\lambda)$ is measurable, while for a.a. $z\in\Omega$, $(x,\lambda)\longmapsto f(z,x,\lambda)$ is continuous;
	\item[(ii)] $|f(z,x,\lambda)|\leq a_{\lambda}(z)(1+x^{r-1})$ for a.a. $z\in\Omega$, all $x\geq 0$, all $\lambda>0$, with $a_{\lambda}\in L^{\infty}(\Omega)$, $\lambda\longmapsto||a_{\lambda}||_{\infty}$ bounded on bounded sets in $(0,\infty)$ and $p<r<p^*$;
	\item[(iii)] if $F(z,x,\lambda)=\int^{x}_{0}f(z,s,\lambda)ds$, then $\lim\limits_{x\rightarrow+\infty}\frac{F(z,x,\lambda)}{x^p}=+\infty$ uniformly for a.a. $z\in\Omega$;
	\item[(iv)]there exists $\vartheta=\vartheta(\lambda)\in\left((r-p)\max\left\{\frac{N}{p},1\right\},p^*\right)$ such that
 $$0<\gamma_0\leq\liminf\limits_{x\rightarrow+\infty}\frac{f(z,x,\lambda)x-pF(z,x,\lambda)}{x^{\vartheta}}\ \mbox{uniformly for a.a.}\ z\in\Omega;$$
	\item[(v)] there exists $1<\mu=\mu(\lambda)<q=q(\lambda)<\tau$ (see hypothesis $H(a)(v)$) and $\gamma=\gamma(\lambda)>\mu$, $\delta_0=\delta_0(\lambda)\in(0,1)$ such that
	$$c_3x^q\leq f(z,x,\lambda)x\leq qF(z,x,\lambda)\leq\xi_{\lambda}(z)x^{\mu}+\tau x^{\gamma}\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ 0\leq x\leq \delta_0$$
	with $c_3=c_3(\lambda)>0$, $c_3(\lambda)\rightarrow+\infty$ as $\lambda\rightarrow+\infty$, $\overline{c}=\overline{c}(\lambda)>0$, $\xi_{\lambda}\in L^{\infty}(\Omega)_+$ with $||\xi_{\lambda}||_{\infty}\rightarrow 0$ as $\lambda\rightarrow 0^+$;
	\item[(vi)] for every $\rho>0$, there exists $\xi_{\rho}=\xi_{\rho}(\lambda)>0$ such that  for a.a. $z\in\Omega$, $x\longmapsto f(z,x,\lambda)+\xi_{\rho}x^{p-1}$ is nondecreasing on $[0,\rho]$;
	\item[(vii)] for every interval $K=[x_0,\hat{x}]$ with $x_0>0$ and every $\lambda>\lambda'>0$, there exists $d_{K}(x_0,\lambda)$ nondecreasing in $\lambda$ with $d_K(x_0,\lambda)\rightarrow+\infty$ as $\lambda\rightarrow+\infty$ and $\hat{d}_K(x_0,\lambda,\lambda')$ such that
	\begin{eqnarray*}
		&&f(z,x,\lambda)\geq d_K(x_0,\lambda)\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\in K\\
		&&f(z,x,\lambda)-f(z,x,\lambda')\geq\hat{d}_K(x_0,\lambda,\lambda')\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\in K.
	\end{eqnarray*}
\end{itemize}

Since we are interested to find positive solutions and the above hypotheses concern the positive semiaxis $\RR_+=\left[0,+\infty\right)$, without any loss of generality we may assume that $f(z,x,\lambda)=0$ for a.a. $z\in\Omega$, all $x\leq 0$ and all $\lambda>0$. Note that hypotheses $H(f)(ii),(iii)$ imply that
	$$\lim\limits_{x\rightarrow+\infty}\frac{f(z,x,\lambda)}{x^{p-1}}=+\infty\ \mbox{uniformly for a.a.}\ z\in\Omega.$$	
	Therefore, $f(z,\cdot,\lambda)$ is $(p-1)$-superlinear near $+\infty$. However, we do not employ the Ambrosetti-Rabinowitz condition (unilateral version). We recall (see \cite{3}), that $f(z,\cdot,\lambda)$ satisfies the (unilateral) Ambrosetti-Rabinowitz condition, if there exist $\eta=\eta(\lambda)>p$ and $M=M(\lambda)>0$ such that
\begin{eqnarray}\label{eq4}
\left.\begin{array}{cl}
		(a)&0<\eta F(z,x,\lambda)\leq f(z,x,\lambda)x\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\geq M,\\
		(b)&\di{\rm ess\,inf}_{\Omega}\ F(\cdot,M,\lambda)>0.
		\end{array}\right.
	\end{eqnarray}

Integrating (\ref{eq4}a) and using (\ref{eq4}b), we obtain a weaker condition, namely
\begin{eqnarray}\label{eq5}
	c_4 x^{\eta}\leq F(z,x,\lambda)\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ z\geq M\ \mbox{and some}\ c_4>0.
\end{eqnarray}

 Relation (\ref{eq5}) implies the much weaker hypothesis $H(f)(iii)$. Next, by direct computation, we deduce that the Ambrosetti-Rabinowitz condition implies hypothesis $H(f)(iv)$. This weaker ``superlinearity" condition, incorporates in our setting $(p-1)$-superlinear nonlinearities with ``slower" growth near $+\infty$, which fail to satisfy the Ambrosetti-Rabinowitz condition. Finally, we notice that hypothesis $H(f)(v)$ implies the presence of a concave nonlinearity near zero.

The following functions satisfy hypotheses $H(f)$. For the sake of simplicity, we drop the $z$-dependence:
	\begin{eqnarray*}
		&&f_1(x,\lambda)=\lambda x^{q-1}+x^{r-1}\ \mbox{for all}\ x\geq 0,\ \mbox{with}\ 1<q<p<r<p^*;\\
		&&f_2(x,\lambda)=\left\{
		\begin{array}{ll}
			\lambda x^{q-1}-x^{\eta-1}&\ \mbox{if}\ x\in[0,1]\\
			x^{p-1}\left(\ln x+\frac{1}{p}\right)+\left(\lambda-\frac{1}{p}\right)x^{\nu-1}&\ \mbox{if}\ x>1
		\end{array}\right.\\
		&&\hspace{1.8cm}\mbox{with}\ q,\nu\in(1,p)\ \mbox{and}\ \eta>p;\\
		&&f_3(x,\lambda)=\left\{
		\begin{array}{cl}
			x^{q-1}&\ \mbox{if}\ x\in[0,\rho(\lambda)]\\
			x^{r-1}+\eta(\lambda)&\ \mbox{if}\ \rho(\lambda)<x
		\end{array}\right.\\
		&&\hspace{1.8cm}\mbox{with}\ 1<q<p<r<p^*,\ \eta(\lambda)=\rho(\lambda)^{p-1}-\rho(\lambda)^{r-1}\ \mbox{and}\ \rho(\lambda)\rightarrow 0\ \mbox{as}\ \lambda\rightarrow 0^+.
	\end{eqnarray*}

Note that $f_2(\cdot,\lambda)$ does not satisfy the Ambrosetti-Rabinowitz condition.

In the analysis of problem (P), 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 $$C_+=\{u\in C^1(\overline{\Omega}):u(z)\geq 0\ \mbox{for all}\ z\in\overline{\Omega}\}.$$ This cone has a nonempty interior given by
$$\mbox{int}\, C_+=\{u\in C_+:u(z)>0\ \mbox{for all}\ z\in\overline{\Omega}\}.$$

In the Sobolev space $W^{1,p}(\Omega)$, we use the norm
$||u||=\left[||u||^{p}_{p}+||Du||^{p}_{p}\right]^{1/p}$ for all $u\in W^{1,p}(\Omega).$
To distinguish, we use $|\cdot|$ to denote the norm of $\RR^N$.

We point out that the first work concerning positive solutions for problems with concave and convex nonlinearities is due to Ambrosetti, Brezis and Cerami \cite{2}.
We also refer to the books by Brezis \cite{brebook} and Ciarlet \cite{ciarlet} for related results and complements.

The main result in this Note is the following bifurcation property.
\begin{theorem}\label{thorigin}
Assume that hypotheses $H(a)$ and $H(f)$ are fulfilled. Then there exists $\lambda^*>0$ such that
	\begin{itemize}
		\item[(a)] for all $\lambda\in(0,\lambda^*)$, problem $(P)$ has at least two positive solutions
		$u_0,\hat{u}\in {\rm int}\, C_+,\ u_0\leq\hat{u},\ u_0\neq\hat{u};$
		\item[(b)] for $\lambda=\lambda^*$ problem $(P)$ has at least one positive solution $u_*\in {\rm int}\, C_+$;
		\item[(c)] for all $\lambda>\lambda^*$ problem $(P)$ has no positive solution.
	\end{itemize}
\end{theorem}

\smallskip
{\it Sketch of the proof.} We introduce the following Carath\'eodory function
$$\hat{f}(z,x,\lambda)=f(z,x,\lambda)+(x^+)^{p-1}\ \mbox{for all}\ (z,x,\lambda)\in\Omega\times\RR\times(0,+\infty).$$
Let $\hat{F}(z,x,\lambda)=\int^{x}_{0}\hat{f}(z,s,\lambda)ds$ and consider the $C^1$-functional $\hat{\varphi}_{\lambda}:W^{1,p}(\Omega)\rightarrow\RR$ defined by
$$\hat{\varphi}_{\lambda}(u)=\int_{\Omega}G(Du)dz+\frac{1}{p}||u||^{p}_{p}+\frac{1}{p}\int_{\partial\Omega}\beta(z)(u^+)^pd\sigma-
\int_{\Omega}\hat{F}(z,u,\lambda)dz\,
	.$$

{\it Step 1.} The functional $\hat{\varphi}_{\lambda}$ satisfies the Cerami compactness condition.

{\it Step 2.} There exists $\lambda_+>0$ such that  for every $\lambda\in(0,\lambda_+)$ there is $\rho_{\lambda}>0$ for which we have $$\inf\left[\hat{\varphi}_{\lambda}(u):||u||=\rho_{\lambda}\right]=\hat{m}_{\lambda}>0=\hat{\varphi}_{\lambda}(0).$$

{\it Step 3.}  If $\lambda>0$ and $u\in {\rm int}\, C_+$, then $\varphi_{\lambda}(tu)\rightarrow-\infty$ as $t\rightarrow\infty$. This is a direct consequence of $H(f)(iii)$.

Next, we introduce the following sets:
\begin{eqnarray*}
	&&\mathcal{S}=\{\lambda>0:\mbox{problem}\ (P)\ \mbox{admits a positive solution}\},\\
	&&S(\lambda)=\mbox{the set of positive solutions of (P)}.
\end{eqnarray*}

{\it Step 4.} We have $\mathcal{S}\neq\emptyset$ and for every $\lambda\in\mathcal{S}$, $\emptyset\neq S(\lambda)\subseteq {\rm int}\, C_+$.

{\it Step 5.} If $\lambda\in\mathcal{S}$, then $\left(0,\lambda\right]\subseteq\mathcal{S}$.

{\it Step 6.} Let $\lambda^*=\sup\mathcal{S}$. Then $\lambda^*$ is finite.

{\it Step 7.} If $\lambda\in(0,\lambda^*)$, then problem (P) admits at least two distinct positive solutions
	$u_0,\,\hat{u}\in {\rm int}\, C_+$, with $u_0\leq\hat{u}.$

Next, we examine what happens in the critical case $\lambda=\lambda^*$. To this end, note that hypotheses $H(f)(ii),(v)$ imply that we can find $M_1,\, M_2>0$ such that
\begin{eqnarray}\label{eq48}
	f(z,x,\lambda)\geq M_1 x^{q-1}-M_{2}x^{r-1}\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ z\geq 0.
\end{eqnarray}
This unilateral growth estimate on the reaction $f(z,\cdot,\lambda)$ leads to the following auxiliary Robin problem:
\begin{eqnarray}\label{eq49}
	\left\{\begin{array}{lll}
	&\di	-\mbox{div}\, a(Du(z))=M_1 u(z)^{q-1}-M_{2}u(z)^{r-1}\ &\mbox{in}\ \Omega,\\
&		\di\frac{\partial u}{\partial n_0}(z)+\beta(z)u(z)^{p-1}=0\ &\mbox{on}\ \partial\Omega,\\
& u>0.&\
	\end{array}\right.
\end{eqnarray}

{\it Step 8.} Problem (\ref{eq49}) admits a unique positive solution $\bar{u}\in {\rm int}\, C_+$.

{\it Step 9.} If $\lambda\in\mathcal{S}$, then $\bar{u}\leq u$ for all $u\in S(\lambda)$.

{\it Step 10.} We have $\lambda^*\in\mathcal{S}$ and so $\mathcal{S}=\left(0,\lambda^*\right]$. Indeed, let $\{\lambda_n\}_{n\geq 1}\subseteq\mathcal{S}$ be such that  $\lambda_n\rightarrow(\lambda^*)^-$. Then we can find $u_n\in S(\lambda_n)\subseteq \mbox{int}\, C_+$ and we can assume that
$\hat{\varphi}_{\lambda_n}(u_n)<0$ for all $n\geq 1$. Therefore
	\begin{equation}\label{eq55}
		 \int_{\Omega}pG(Du_n)dz+\int_{\partial\Omega}\beta(z)u_n^pd\sigma-\int_{\Omega}pF(z,u_n,\lambda_n)dz<0\ \mbox{for all}\ n\geq 1.
	\end{equation}
	We also have
	\begin{eqnarray}\label{eq56}
		-\left\langle A(u_n),u_n\right\rangle-\int_{\partial\Omega}\beta(z)u_n^pd\sigma+\int_{\Omega}f(z,u_n,\lambda_n)u_ndz=0\ \mbox{for all}\ n\geq 1.
	\end{eqnarray}
	Adding relations (\ref{eq55}) and (\ref{eq56}), we obtain
$$\int_{\Omega}\left[pG(Du_n)-(a(Du_n),Du_n)_{\RR^N}\right]dz+\int_{\Omega}\left[f(z,u_n,\lambda_n)u_n-pF(z,u_n,\lambda_n)\right]dz<0\,.$$
Thus, by hypothesis H(a)(iv),
	\begin{equation}\label{eq57}
		 \int_{\Omega}\left[f(z,u_n,\lambda_n)u_n-pF(z,u_n,\lambda_n)\right]dz<c\ \mbox{for all}\ n\geq 1,\ \mbox{some $c>0$}\,.
	\end{equation}
	From (\ref{eq57}) and  using hypothesis $H(f)(iv)$, we show that $\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)$ is bounded. So, we may assume that
	\begin{eqnarray}\label{eq58}
		u_n\stackrel{w}{\rightarrow}u_*\ \mbox{in}\ W^{1,p}(\Omega)\ \mbox{and}\ u_n\rightarrow u_*\ \mbox{in}\ L^r(\Omega)\ \mbox{and in}\ L^p(\partial\Omega)\ \mbox{as}\ n\rightarrow\infty.
	\end{eqnarray}
	
	Since $u_n\in S(\lambda)$ for all $n\geq 1$, we have
	\begin{eqnarray}\label{eq59}
		\left\langle A(u_n),h\right\rangle+\int_{\partial\Omega}\beta(z)u_n^{p-1}hd\sigma-\int_{\Omega}f(z,u_n,\lambda_n)hdz=0\ \mbox{for all}\ h\in W^{1,p}(\Omega).
	\end{eqnarray}	
	Choosing $h=u_n-u_*\in W^{1,p}(\Omega)$ in (\ref{eq59}), passing to the limit as $n\rightarrow\infty$ and using (\ref{eq58}), we obtain
$\lim\limits_{n\rightarrow\infty}\left\langle A(u_n),u_n-u_*\right\rangle=0$, hence
	\begin{equation}\label{eq60}
		u_n\rightarrow u_*\ \mbox{in}\ W^{1,p}(\Omega).
	\end{equation}
	
	So, if in (\ref{eq59}) we pass to the limit as $n\rightarrow\infty$ and use (\ref{eq60}), we obtain
	$$\left\langle A(u_*),h\right\rangle+\int_{\partial\Omega}\beta(z)u_*^{p-1}hd\sigma=\int_{\Omega}f(z,u_*,\lambda_*)hdz\ \mbox{for all}\ h\in W^{1,p}(\Omega).$$
This shows that
		$u_*\geq 0$ is a solution of problem (P).
	
	From Step~9 we have $\bar{u}\leq u_n$ for all $n\geq 1$. Hence $\bar{u}\leq u_*$ and so $u_*\in S(\lambda^*)\subseteq \mbox{int}\, C_+$. Therefore $\lambda^*\in\mathcal{S}$ and so $\mathcal{S}=\left(0,\lambda^*\right]$.\qed

We refer to Papageorgiou and R\u adulescu \cite{npvr} for more details, complete proofs, and related problems.

\begin{thebibliography}{99}
%\bibitem{1} {\sc S. Aizicovici, N.S. Papageorgiu, V. Staicu}, {\em Degree Theory for Operators of Monotone Type and Nonlinear Elliptic Equations with Inequality %Constraints}, Memoirs Amer. Math. Soc., Vol. \textbf{196}, No. \textbf{915} (2008).

\bibitem{2} {\sc A. Ambrosetti, H. Brezis, G. Cerami},  Combined effects of concave-convex nonlinearities in some elliptic problems, {\it J. Functional Anal.} \textbf{122} (1994), 519-543.

\bibitem{3} {\sc A. Ambrosetti, P. Rabinowitz},  Dual variational methods in critical point theory and applications, {\it J. Functional Anal.} \textbf{14} (1973), 349-381.

\bibitem{brebook}  {\sc H. Brezis}, {\it Functional Analysis, Sobolev Spaces and Partial Differential Equations}, Universitext, Springer, New York, 2011.

\bibitem{ciarlet} {\sc P.G. Ciarlet}, {\it Linear and Nonlinear Functional Analysis with Applications},
Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013.


%\bibitem{4} {\sc F. C\^{\i}rstea, M. Ghergu, V.D. R\u adulescu},  Combined effects of asymptotically linear and singular nonlinearities in bifurcation problems %of Lane-Emden-Fowler type, {\it J. Math. Pures Appl.} \textbf{84} (2005), 493-508.

%\bibitem{5} {\sc J.I. Diaz, J.E. Saa}, Existence et unicit\'{e} de solutions positives pour certaines \'equations elliptiques quasilin\'eaires, {\it C.R. Acad. %Sci. Paris} \textbf{305} (1987), 521-524.

%\bibitem{6} {\sc N. Dunford, J. Schwartz}, {\em Linear Operators I}, Wiley-Interscience, New York (1958).

%\bibitem{7} {\sc M. Filippakis, A. Kristaly, N.S. Papageorgiou},  Existence of five nonzero solutions with constant sign for a $p$-Laplacian equation, {\it %Discrete Cont. Dyn. Systems} \textbf{24} (2009), 405-440.

%\bibitem{8} {\sc J. Garcia Azero, J. Manfredi, I. Peral Alonso},  Sobolev versus H\"{o}lder local minimizers and global multiplicity for some quasilinear %elliptic equations, {\it Commun. Contemp. Math.} \textbf{2} (2000), 385-404.

%\bibitem{9} {\sc L. Gasinski, N.S. Papageorgiou}, {\em Nonlinear Analysis}, Chapman, Hall/CRC, Boca Raton, Fl. (2006).

%\bibitem{10} {\sc L. Gasinski, N.S. Papageorgiou},  Existence and multiplicity of solutions for Neumann $p$-Laplacian-type equations, {\it Adv. Nonlinear %Studies} \textbf{8} (2008), 843-870.

%\bibitem{11} {\sc L. Gasinski, N.S. Papageorgiou}, Bifurcation-type results for nonlinear parametric elliptic equations, {\it Proc. Royal. Soc. Edinburgh, Sect. %A} \textbf{142} (2012), 595-623.

%\bibitem{12} {\sc Z. Guo, Z. Zhang},  $W^{1,p}$ versus $C^1$ local minimizers and multiplicity results for quasilinear elliptic equations, {\it J. Math. Anal. %Appl.} \textbf{286} (2003), 32-50.

%\bibitem{13} {\sc S. Hu, N.S. Papageorgiou},  Multiplicity of solutions for parametric $p$-Laplacian equations with nonlinearity concave near the origin, {\it %T\^{o}hoku Math. J.} \textbf{62} (2010), 137-162.

%\bibitem{14} {\sc A. Kristaly, G. Moro\c sanu}, New competition phenomena in Dirichlet problems, {\it J. Math. Pures Appl.} \textbf{94} (2010), 555-570.

\bibitem{15} {\sc G. Lieberman},  The natural generalization of the conditions of Ladyzhenskaya and Uraltseva for elliptic equations, {\it Commun. Partial Diff. Equations} \textbf{16} (1991), 311-361.

%\bibitem{16} {\sc N.S. Papageorgiou, V.D. R\u adulescu}, Qualitative phenomena for some classes of quasilinear elliptic equations with multiple resonance, {\it %Appl. Math. Optim.}, in press (DOI: 10.1007/s 00245-013-9227-z).

%\bibitem{17} {\sc N.S. Papageorgiou, V.D. R\u adulescu},  Solutions with sign information for nonlinear nonhomogeneous elliptic equations, {\it Topol. Methods %Nonlin. Anal.}, to appear.

%\bibitem{18} {\sc N.S. Papageorgiou, V.D. R\u adulescu},  Multiple solutions with precise sign for nonlinear parametric Robin problems, {\it J. Differential %Equations} \textbf{256} (2014), 2449-2479.

\bibitem{npvr} {\sc N.S. Papageorgiou, V.D. R\u adulescu},  Bifurcation of positive solutions for nonlinear nonhomogeneous Robin and Neumann problems with competing nonlinearities, submitted.

\bibitem{19} {\sc P. Pucci, J. Serrin}, {\em The Maximum Principle}, Birkh\"auser, Basel, 2007.

%\bibitem{20} {\sc V.D. R\u adulescu, D. Repovs},  Combined effects in nonlinear problems arising in the study of anisotropic continuous media, {\it Nonlinear %Analysis} \textbf{75} (2012), 1524-1530.

%\bibitem{21} {\sc P. Winkert}, $L^{\infty}$ estimates for nonlinear elliptic Neumann boundary value problems, {\it Nonlinear Diff. Equ. Appl. (NoDEA)} %\textbf{17} (2010), 289-302.


\end{thebibliography}

\end{document} 