% version 2013 \documentclass[11pt]{amsart} \usepackage{graphicx, color} \usepackage{amscd} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{mathrsfs} \textwidth=6in \textheight=9.5in \topmargin=-0.5cm \oddsidemargin=0.5cm \evensidemargin=0.5cm %\usepackage[notref,notcite]{showkeys} \newtheorem{theorem}{Theorem} \newtheorem{ex}{Example} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{prop}[theorem]{Proposition} \newtheorem{remark}{Remark} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{claim}{Claim} \newtheorem{step}{Step} \newtheorem{case}{Case} \newtheorem{definition}[theorem]{Definition} %\newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \newenvironment{proof-sketch}{\noindent{\bf Sketch of Proof}\hspace*{1em}}{\qed\bigskip} %\numberwithin{equation}{section} \newcommand{\RR}{\mathbb R} \newcommand{\NN}{\mathbb N} \newcommand{\PP}{\mathbb P} \newcommand{\ZZ}{\mathbb Z} %\def\di{\displaystyle} \renewcommand{\le}{\leqslant} \renewcommand{\leq}{\leqslant} \renewcommand{\ge}{\geqslant} \renewcommand{\geq}{\geqslant} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \baselineskip=16pt plus 1pt minus 1pt \begin{document} %\hfill\today\bigskip \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
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 $$1 0$ for all $t>0$ and
\begin{itemize}
\item[(i)] $a_0\in C^1(0,\infty),\ t\longmapsto a_0(t)t$ is strictly increasing on $(0,\infty)$, $a_0(t)t\rightarrow 0$\\ as $t\rightarrow 0^+$ and
$$\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 $p0\\
&&\hspace{6cm}\mbox{with}\ c_1,c_2>0,\ 1
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
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}