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\title[Bifurcation analysis for nonhomogeneous Robin problems]{Bifurcation analysis for nonhomogeneous Robin problems with competing nonlinearities}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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@math.cnrs.fr}

\keywords{Competing nonlinearities, nonhomogeneous differential operator, bifurcation analysis, Robin problem.\\
\phantom{aa} 2010 AMS Subject Classification:
35J66, 35J70, 35J92.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{abstract}
In this paper, we report on some recent results obtained in our joint paper \cite{prdcds}.
We consider a Robin problem driven by a nonhomogeneous differential operator and with a reaction that exhibits competing
effects of concave (that is, sublinear) and convex (that is, superlinear)
 nonlinearities. 
Without employing the Ambrosetti-Rabinowitz condition, we establish a bifurcation property  of the positive solutions near the origin.
The approach relies on variational methods and elliptic estimates.
\end{abstract}

\maketitle

\section{Introduction}\label{sec1}

Let $\Omega\subseteq \RR^{N}$ be a bounded domain with  $C^2$-boundary $\partial\Omega$. Let
$a:\mathbb{R}^N\rightarrow\mathbb{R}^N$ be a continuous strictly monotone map. Let $\partial u/\partial n_a$ denote the conormal derivative defined by $\partial u/\partial n_a:=(a(Du),n)_{\mathbb{R}^N}$, where $n(z)$ is the outward unit normal at $z\in\partial\Omega$.

In this paper we study the following  nonlinear Robin problem:
\begin{equation}
	\left\{
	\begin{array}{cl}
		-\mbox{div}\, a(Du(z))=f(z,u(z),\lambda)& \mbox{in}\ \Omega,\\
		\displaystyle\frac{\partial u}{\partial n_a}(z)+\beta(z)u(z)^{p-1}=0& \mbox{on}\ \partial\Omega,\\
		u>0,\ 1<p<\infty.&
	\end{array}\right.
\tag{$P_{\lambda}$} \label{eqP}
\end{equation}

The reaction $f(z,x,\lambda)$ is a parametric function with $\lambda>0$ being the parameter and $(z,x)\rightarrow f(z,x,\lambda)$ is a Carath\'eodory function. 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 a ``convex" term (that is, $x\longmapsto f(z,x,\lambda)$ is ($p-1$)-superlinear). A special case of our reaction is the function
$f(z,x,\lambda)=f(x,\lambda)=\lambda x^{q-1}+x^{r-1}$, for all $ x\geq 0$
with $$1<q<p<r<p^*:=\left\{
	\begin{array}{cl}
		\frac{Np}{N-p}&\ \mbox{if}\ p<N\\
		+\infty&\ \mbox{if}\ N\leq p.
	\end{array}
	\right.$$ 
	
	The first work concerning positive solutions for problems with concave and convex nonlinearities, was that of Ambrosetti, Brezis and Cerami \cite{2}. They studied semilinear equations driven by the Dirichlet Laplacian and with a reaction of the form (\ref{eq1}). Their work was extended to equations driven by the Dirichlet $p$-Laplacian by Garcia Azorero, Manfredi and Peral Alonso \cite{8} and by Guo and Zhang \cite{12}.
We also refer to the contributions of de Figueiredo,  Gossez and Ubilla \cite{dja1}, \cite{dja2} to concave-convex type problems and general nonlinearities
for the Laplacian, resp. $p$-Laplacian case.
Extensions to equations involving more general reactions, were obtained by Gasinski and Papageorgiou \cite{11}, Hu and Papageorgiou \cite{13} and R\u{a}dulescu and Repov\v{s} \cite{20}.

Let $\eta\in C^1(0,\infty)$ and assume that
\begin{eqnarray}\label{eq1}
	&&0<\hat{c}\leq\frac{t\eta'(t)}{\eta(t)}\leq c_0\ \mbox{and}\ c_1t^{p-1}\leq\eta(t)\leq c_2(1+t^{p-1})\ \mbox{for all}\ t>0\\
	&&\hspace{6cm}\mbox{with}\ c_1,c_2>0,\ 1<p<\infty.\nonumber
\end{eqnarray}

The hypotheses on the map $a(\cdot)$ are the following:

\smallskip
\textbf{$H(a):$} $a(y)=a_0(|y|)y$ for all $y\in \mathbb{R}^N$, 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)] $ |\nabla a(y)|\leq c_3\frac{\eta(|y|)}{|y|}$ for some $c_3>0$, all $y\in\mathbb{R}^N\backslash\{0\}$;
	\item[(iii)] $\frac{\eta(|y|)}{|y|}|\xi|^2\leq(\nabla a(y)\xi,\xi)_{\mathbb{R}^N}$ for all $y\in\mathbb{R}^N\backslash\{0\}$, all $\xi\in\mathbb{R}^N$;
	\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}


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 $y\in\mathbb{R}^N$. Then the function $y\longmapsto G(y)$ is convex and differentiable on $\mathbb{R}^N\backslash\{0\}$. We have
	$$\nabla G(y)=G'_0(|y|)\frac{y}{|y|}=a_0(|y|)y=a(y)\ \mbox{for all}\ y\in\mathbb{R}^N\backslash\{0\},\ \nabla G(0)=0.$$
	
	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
$G(y)\leq(a(y),y)_{\mathbb{R}^N}$ for all $y\in\mathbb{R}^N$.
	

The following properties follow by straightforward arguments.
\begin{lemma}\label{lem2}
	Assume that hypotheses $H(a)(i),(ii),(iii)$ hold. Then
	\begin{itemize}
		\item[(a)] the mapping $y\longmapsto a(y)$ 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 $y\in\mathbb{R}^N;$
		\item[(c)] $(a(y),y)_{\mathbb{R}^N}\geq\frac{c_1}{p-1}|y|^p$ for all $y\in\mathbb{R}^N$;
\item[(d)] for all $y\in\mathbb{R}^N$ we have $\frac{c_1}{p(p-1)}|y|^p\leq G(y)\leq c_5(1+|y|^p)$ with $c_5>0$.
	\end{itemize}
\end{lemma}

The hypotheses on the boundary weight map $\beta(\cdot)$ are the following:

\smallskip
\textbf{$H(\beta):$} $\beta\in C^{1,\alpha}(\partial\Omega)$ with $\alpha\in(0,1)$ and $\beta(z)\geq 0$ for all $z\in\partial\Omega$.


Throughout this paper we assume that the reaction $f$ satisfies the following hypotheses.

\textbf{$H(f):$} $f:\Omega\times\mathbb{R}\times(0,\infty)\rightarrow\mathbb{R}$ 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 $(x,\lambda)\in\mathbb{R}\times(0,\infty)$, $z\longmapsto f(z,x,\lambda)$ 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 exist $1<\mu=\mu(\lambda)<q=q(\lambda)<\tau$ (see hypothesis $H(a)(v)$) and $\gamma=\gamma(\lambda)>\mu$, $\delta_0=\delta_0(\lambda)\in(0,1)$ such that
	$$c_6x^q\leq f(z,x,\lambda)x\leq qF(z,x,\lambda)\leq\xi_{\lambda}(z)x^{\mu}+\tau x^{\gamma}\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ 0\leq x\leq \delta_0$$
	with $c_6=c_6(\lambda)>0$, $c_6(\lambda)\rightarrow+\infty$ 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}

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}\ x>\rho(\lambda)
		\end{array}\right.\\
		&&\hspace{1.8cm}\mbox{with}\ 1<q<p<r<p^*,\ \eta(\lambda)=\rho(\lambda)^{p-1}-\rho(\lambda)^{r-1}\\
		&&\hspace{1.8cm}\mbox{and}\ \rho(\lambda)\rightarrow 0^+\ \mbox{as}\ \lambda\rightarrow 0^+.
	\end{eqnarray*}



Since we are interested to find positive solutions and the above hypotheses concern the positive semiaxis $\mathbb{R}_+=\left[0,+\infty\right)$, 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
	$$\lim\limits_{x\rightarrow+\infty}\frac{f(z,x,\lambda)}{x^{p-1}}=+\infty\ \mbox{uniformly for a.a.}\ z\in\Omega.$$
	Thus $f(z,\cdot,\lambda)$ is $(p-1)$-superlinear near $+\infty$. However, we do not employ the Ambrosetti-Rabinowitz (AR) condition (unilateral version). Cf. \cite{3}), we say that $f(z,\cdot,\lambda)$ satisfies the (unilateral) (AR)-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)&{\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 that
\begin{eqnarray}\label{eq5}
	c_7 x^{\eta}\leq F(z,x,\lambda)\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ z\geq M\ \mbox{and some}\ c_7>0.
\end{eqnarray}

Evidently (\ref{eq5}) implies the much weaker hypothesis $H(f)\,(iii)$. In (\ref{eq4}) we may assume that $\eta>(r-p)\max\left\{\frac{N}{p},1\right\}$. Then we have
\begin{eqnarray*}
	&&\frac{f(z,x,\lambda)x-pF(z,x,\lambda)}{x^{\eta}}
	=\frac{f(z,x,\lambda)x-\eta F(z,x,\lambda)}{x^{\eta}}+\frac{(\eta-p)F(z,x,\lambda)}{x^{\eta}}\\
	&\geq&(\eta-p)c_7\ \mbox{for a.a.}\ z\in\Omega,\ \mbox{all}\ x\geq M\ (\mbox{see (\ref{eq4}a) and (\ref{eq5})}).
\end{eqnarray*}

So, we see that the (AR)-condition implies hypothesis $H_1(iv)$. This weaker ``superlinearity" condition incorporates in our setting $(p-1)$-superlinear nonlinearities with ``slower" growth near $+\infty$, which fail to satisfy the (AR)-condition (see the function $f_2(\cdot,\lambda)$ defined above). Finally note that hypothesis $H(f)\,(v)$ implies the presence of a concave nonlinearity near zero.


The main result of this paper establishes the following bifurcation property.

\begin{theorem}\label{th1}
	Assume that hypotheses $H(a),\ H(\beta)$ and $H(f)$ hold. Then there exists $\lambda^*>0$ such that
	\begin{itemize}
		\item[(a)] for all $\lambda\in(0,\lambda^*)$, problem $(P_{\lambda})$ 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 $(P_{\lambda^*})$ has at least one positive solution $u_*\in {\rm int}\, C_+$;
		\item[(c)] for all $\lambda>\lambda^*$ problem $(P_{\lambda})$ has no positive solution.
	\end{itemize}
\end{theorem}

{\bf Sketch of the proof.} We introduce the following Carath\'eodory function
$$\hat{f}(z,x,\lambda)=f(z,x,\lambda)+(x^+)^{p-1}\ \mbox{for all}\ (z,x,\lambda)\in\Omega\times\mathbb{R}\times(0,+\infty).$$

Let $\hat{F}(z,x,\lambda)=\int^{x}_{0}\hat{f}(z,s,\lambda)ds$ and consider the $C^1$-functional $\hat{\varphi}_{\lambda}:W^{1,p}(\Omega)\rightarrow\mathbb{R}$ defined by
$$\hat{\varphi}_{\lambda}(u)=\int_{\Omega}G(Du)dz+\frac{1}{p}||u||^{p}_{p}+\frac{1}{p}\int_{\partial\Omega}\beta(z)(u^+)^pd\sigma-\int_{\Omega}\hat{F}(z,u,\lambda)dz\,.
$$

We split the proof into several steps.

\smallskip
{\it Step 1.} For all $\lambda>0$,  the energy functional $\hat{\varphi}_{\lambda}$ satisfies the Cerami compactness condition.

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

\smallskip
{\it Step 3.} If $\lambda>0$ and $u\in {\rm int}\, C_+:=\{v\in C^1(\overline{\Omega}):\ v(z)>0\ \mbox{for all}\ z\in\overline{\Omega}\}$, then $\hat{\varphi}_{\lambda}(tu)\rightarrow-\infty$ as $t\rightarrow\infty$.
This property is a direct consequence of hypothesis
 $H(f)\,(iii)$.

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

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

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

\smallskip
{\it Step 6.} Set $\lambda^*=\sup\mathcal{S}$. We have $\lambda^*<\infty$.

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

\smallskip
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 $c_{8}=c_{8}(\lambda)>0$ such that
\begin{eqnarray}\label{eq48}
	f(z,x,\lambda)\geq c_6 x^{q-1}-c_{8}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}{ll}
		-\mbox{div}\, a(Du(z))=c_6 u(z)^{q-1}-c_{8}u(z)^{r-1}\ &\mbox{in}\ \Omega,\\
	\displaystyle	\frac{\partial u}{\partial n_0}(z)+\beta(z)u(z)^{p-1}=0\ &\mbox{on}\ \partial\Omega,\\ 
	u>0\ &\mbox{in}\ \Omega .
	\end{array}\right.
\end{eqnarray}

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

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

\smallskip
{\it Step 10.} We have $\lambda^*\in\mathcal{S}$ and so $\mathcal{S}=\left(0,\lambda^*\right]$.

\medskip
We refer to \cite{prdcds} for detailed arguments of the proof, as well as for related results on Neumann problems with competing nonlinearities.

\medskip
{\bf Acknowledgements}.  V. R\u{a}dulescu acknowledges the support through Grant
 Advanced Collaborative
Research Projects CNCS-PCCA-23/2014.

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\end{document}