\documentclass[10pt]{studiamnew} \usepackage{graphicx} \usepackage{amsmath} \usepackage{amssymb} \sloppy \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \theoremstyle{definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{problem}[theorem]{Problem} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{solution}[theorem]{Solution} \newtheorem{notation}[theorem]{Notation} \renewcommand{\theequation}{\thesection.\arabic{equation}} \numberwithin{equation}{section} \begin{document} % \setcounter{page}{1} \setcounter{firstpage}{1} \setcounter{lastpage}{4} \renewcommand{\currentvolume}{??} \renewcommand{\currentyear}{??} \renewcommand{\currentissue}{??} % \title{Existence and multiplicity of positive radial solutions to the Dirichlet problem for nonlinear elliptic equations on annular domains} \author{Noureddine Bouteraa} \address{``Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO)''\\ University of Oran1, Ahmed Benbella, \\ Algeria} \email{bouteraa-27@hotmail.fr} % \author{Slimane Benaicha} \address{``Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO)''\\ University of Oran1, Ahmed Benbella, \\ Algeria} \email{slimanebenaicha@yahoo.fr} % \subjclass{35J25; 34B18.} \keywords{Positive solution, Elliptic equations, Existence, Multiplicity, local boundary, Green's function} \begin{abstract} In this paper, we study the existence and nonexistence of monotone positive radial solutions of elliptic boundary value problems on bounded annular domains subject to local boundary condition. By using Krasnoselskii's fixed point theorem of cone expansion-compression type we show that there exists $\lambda^{*}\geq\lambda_{*}>0$ such that the elliptic equation has at least two, one and no radial positive solutions for $0<\lambda\leq\lambda_{*},\;\lambda_{*}<\lambda\leq\lambda^{*}$ and $\lambda>\lambda^{*}$ respectively. We include an example to illustrate our results. \end{abstract} \maketitle \section{ Introduction} In this paper, we are interested in the existence of radial positive solutions to the following, boundary value problem BVP \begin{equation} \begin{cases} \begin{array}{c} -\triangle u\left(x\right)=\lambda f\left(\left|x\right|,u\left(x\right)\right),\ x\in\varOmega,\\ u\left(x\right)=0,\,\qquad\qquad\qquad\quad x\in\partial\varOmega, \end{array}\end{cases}\label{eq:1} \end{equation} \noindent where $\varOmega=\left\{ x\in\mathbb{R}^{N}\thinspace:\thinspace a<\left|x\right|2\right\} $. That result was extended for the systems of elliptic equations by Ma {[}24{]}. We quote also the research of Ovono and Rougirel. $\left[29\right]$ where the diffusion at each point depends on all the values of the solutions in a neighborhood of this point and Chipot et al. $\left[12,14\right]$. For example in $\left[12\right]$ considred the solvability of a class of nonlocal problems which admit a formulation in term of quasi-variational inequalities. There is a wide literature that deals with existence multiplicity results for various second-order, fourth-order and higher-order boundary value problems by different approaches, see $\left[6,7,8,9,12,17,22,23\right]$. \noindent In 2011, Bohneure et al. $\left[4\right]$ Studied the existence of positive increasing radial solutions for superlinear Neumann problem in the unit ball $B$ in $\mathbb{R}^{N},\;N\geq2$, \[ \begin{cases} \begin{array}{c} -\Delta u+u=a\left(\left|x\right|\right)f\left(u\right),\;\;\;\;in\;B,\\ u>0,\qquad\quad\qquad\qquad\qquad in\,B,\\ \partial_{t}u=0,\qquad\qquad\qquad\quad\;on\;\partial B, \end{array}\end{cases} \] \noindent where $a\in C^{1}\left(\left[0,1\right],\mathbb{R}\right),\:a\left(0\right)>0$ is nondecreasing, $f\in C^{1}\left(\left[0,1\right],\mathbb{R}\right),\:f\left(0\right)=0,\;\underset{s\rightarrow0^{+}}{lim}\frac{f\left(s\right)}{s}=0$ and $\underset{s\rightarrow+\infty}{lim}\frac{f\left(s\right)}{s}>\frac{1}{a\left(0\right)}$. \noindent In 2011, Hakimi and Zertiti, $\left[17\right]$ studied the nonexistence of radial positive solutions for a nonpositone problem when the nonliearity is superlinear and has more than one zero, \[ \begin{cases} \begin{array}{c} -\triangle u\left(x\right)=\lambda f\left(u\left(x\right)\right),\;\ x\in\varOmega,\\ u\left(x\right)=0,\;\,\qquad\qquad\quad x\in\partial\varOmega, \end{array}\end{cases} \] \noindent where $f\in C\left(\left[0,+\infty\right),\mathbb{R}\right)$. \noindent In 2014, Sfecci $\left[31\right]$, obtained the existence result by introduced the $limsup$ and $liminf$ types of nonresonance condition below the first positive eigenvalue for the following Neumann problems defined on the ball $B_{R}=\left\{ x\in\mathbb{R}^{N},\;\left|x\right|2,\,\left|x\right|>r_{0}\,with\,r_{0}>0\right\} $, $\lambda$ is a positive parameter, $a\in C\left(\left[r_{0},\infty\right),\mathbb{R}^{+}\right)$ such that $\underset{r\rightarrow\infty}{lim}a\left(r\right)=0$, $\frac{\partial}{\partial u}$ is the outward normal derivative and $\overline{c}\in C\left(\left[0,\infty\right),\left(0,\infty\right)\right)$. Instead of working directly with $\left(\ref{eq:1}\right)$, we note that the change of variable $u\left(x\right)=u\left(\left|x\right|\right),\:t=\left|x\right|$ transforms $\left(\ref{eq:1}\right)$ into the following boundary value problem (for details, see $\left[14\right]$ ): \[ \begin{cases} \begin{array}{c} -u''\left(t\right)-\frac{N-1}{t}u\left(t\right)=\lambda f\left(t,u\left(t\right)\right),\ t\in\left(a,b\right),\\ u\left(a\right)=u\left(b\right)=0,\;\qquad\qquad\qquad\qquad\qquad\quad \end{array}\end{cases} \] \noindent where where $\lambda\geq0$ is a positive parameter and $f\in C\left(\left[a,b\right]\times\left[0,\infty\right),\left[0,\infty\right)\right)$. Inspired and motivated by the works mentioned above, we deal with existence and nonexistence of radial positive solutions to the BVP $\left(\ref{eq:1}\right)$ i.e., an equivalant problem $\left(\ref{eq:2}\right)$ by using of the fixed point theorem together with the properties of Green's function and we impose certain conditions on $f$. The paper is organized as follows. In Section 2, we present that a nontrivial and nonnegative solution of BVP $\left(\ref{eq:2}\right)$ is monotone positive solution. In Section 3, we obtain some results of the existence, multiplicity and nonexistence positive solutions for BVP $\left(\ref{eq:2}\right)$ depends on the parameter $\lambda$ and we give an exemple to illustrate our results. \section{Preliminaries} We shall consider the Banach space $E=C\left[a,b\right]$ equipped with sup norm $\left\Vert u\right\Vert =\underset{a \leq t\leq b}{max}\left|u\left(t\right)\right|$ and $C^{+}\left[a,b\right]$ is the cone of nonnegative functions in $C\left[a,b\right]$, where $10$ implies $ru\in P,$ \noindent $\left(ii\right)$ $u\in P,\;-u\in P$ implies $u=\theta$, where $\theta$ denote the zero element of $E$. \end{definition} \begin{definition}\label{def2.2} A cone $P$ is said to be normal if there exists a positive number $N$ called the normal constant of $P$, such that $\theta\leq u\leq v$ implies $\left\Vert u\right\Vert \leq N\left\Vert v\right\Vert $. \end{definition} We are interested in finding radial solutions for problem $\left(\ref{eq:1}\right)$. We proceed as in introduction, setting $u\left(x\right)=u\left(\left|x\right|\right)\:t=\left|x\right|$, we have the following equivalent boundary value problem \begin{equation} \begin{cases} \begin{array}{c} -u''\left(t\right)-\frac{N-1}{t}u\left(t\right)=\lambda f\left(t,u\left(t\right)\right),\ t\in\left(a,b\right),\\ u\left(a\right)=u\left(b\right)=0.\qquad\qquad\qquad\qquad\qquad\quad \end{array}\end{cases}\label{eq:2} \end{equation} We observe that the existence and nonexistence of radial positive solutions of $\left(\ref{eq:1}\right)$ is equivalent to the existence and nonexistence of positive solutions of the problem $\left(\ref{eq:2}\right)$.\\ In arriving our results, we need the following six preliminary lemmas. The first one is well known. \begin{lemma}\label{lem2.1} (see, $\left[13\right]$) Let $y\left(\cdot\right)\in C\left[a,b\right]$. If $u\in C^{4}\left[a,b\right]$, then the BVP \[ \begin{cases} \begin{array}{c} -u''\left(t\right)-\frac{N-1}{t}u\left(t\right)=y\left(t\right),\ t\in\left(a,b\right),\\ u\left(a\right)=u\left(b\right)=0,\qquad\qquad\qquad\qquad \end{array}\end{cases} \] has a unique solution \[ u\left(t\right)=\intop_{a}^{b}s^{N-1}G\left(t,s\right)y\left(s\right)ds,\;N>2, \] \noindent where \begin{equation} G\left(t,s\right)=\begin{cases} \begin{array}{c} \frac{\left(1-\left(\frac{a}{s}\right)^{N-2}\right)\left(\left(\frac{b}{t}\right)^{N-2}-1\right)}{\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)},\quad a\leq t\leq s\leq b,\\ \frac{\left(1-\left(\frac{a}{t}\right)^{N-2}\right)\left(\left(\frac{b}{s}\right)^{N-2}-1\right)}{\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)},\quad a\leq s\leq t\leq b. \end{array}\end{cases}\label{eq:3} \end{equation} \end{lemma} \begin{lemma}\label{lem2.2} For any $\left(t,s\right)\in\left[a,b\right]\times\left[a,b\right]$, we have \begin{equation} \frac{\left(1-\left(\frac{a}{t}\right)^{N-2}\right)}{\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)}\leq G\left(t,s\right)\leq\frac{\left(\left(\frac{b}{t}\right)^{N-2}-1\right)}{\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)},\label{eq:4} \end{equation} \noindent and \begin{equation} 0\leq\frac{\partial G}{\partial t}\left(t,s\right)\leq\frac{\left(\left(\frac{b}{s}\right)^{N-2}-1\right)\left(\frac{\left(N-2\right)b}{a^{N-1}}\right)}{\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)},\;\left(t,s\right)\in\left[a,b\right]\times\left[a,b\right].\label{eq:4-1} \end{equation} \end{lemma} \begin{proof} The proof is evident, we omit it. \end{proof} \begin{lemma} (see $\left[12\right]$) For $y\left(\cdot\right)\in C^{+}\left[a,b\right]$. Then the unique solution $u\left(t\right)$ of BVP \[ \begin{cases} \begin{array}{c} -u''\left(t\right)-\frac{N-1}{t}u\left(t\right)=y\left(t\right),\ t\in\left(a,b\right),\\ u\left(a\right)=u\left(b\right)=0.\qquad\qquad\qquad\qquad \end{array}\end{cases} \] is nonnegative and satisfies \[ \underset{a_{1}\leq t\leq b_{1}}{min}u\left(t\right)\geq c\left\Vert u\right\Vert , \] \noindent where $c=\frac{min\left\{ \left(\frac{b}{b_{1}}\right)^{N-2}-1,1-\left(\frac{a}{a_{1}}\right)^{N-2}\right\} }{max\left\{ \left(\frac{b}{a}\right)^{N-2}-1,1-\left(\frac{a}{b}\right)^{N-2}\right\} }$ and $a_{1},b_{1}\in\left(a,b\right)$ with $a_{1}0$, we have \[ \left|G\left(t_{1},s\right)-G\left(t_{2},s\right)\right|\leq\frac{\varepsilon N}{\lambda\left(b^{N}-a^{N}\right)M_{2}},\quad for\;t_{1},t_{2},s\in\left[a,b\right],\;\left|t_{1}-t_{2}\right|<\delta. \] \noindent Then \[ \left|y_{k}\left(t_{1}\right)-y_{k}\left(t_{2}\right)\right|=\left|T_{\lambda}u\left(t_{1}\right)-T_{\lambda}u\left(t_{2}\right)\right|\quad\qquad\qquad\qquad \] \[ \qquad\qquad\qquad\qquad\leq\lambda M_{2}\intop_{a}^{b}s^{N-1}\left|G\left(t_{1},s\right)-G\left(t_{2},s\right)\right|ds \] \[ \leq\varepsilon.\qquad\qquad\qquad\quad \] \noindent Therefore, $A$ is equicontinuous. By the Ascoli-Arzela Theorem, we know that $A$ is completely continuous. \end{proof} By Lemmas $2.8$ and $2.9$, we know that if $u\in P\setminus\theta$ is solution for BVP $\left(\ref{eq:2}\right)$, then $u$ is positive solution for BVP $\left(\ref{eq:2}\right)$ and it is obvious from Lemma $2.8$ that if $u\in P\setminus\left\{ \theta\right\} $ is a solution for BVP $\left(\ref{eq:2}\right)$ then $u\in K\setminus\left\{ \theta\right\} $. \section{Existence and nonexistence results} In this section we will apply theorem due Krasnoselskii to study the existence, multiplicity and nonexistence of solutions for BVP $\left(\ref{eq:2}\right)$ in $K\setminus\left\{ \theta\right\} $. \begin{theorem}\label{thm3.2} (See $\left[19\right]$) Let $E$ be a Banach space and $K\subset E$ be a cone in $E$. Assume $\Omega_{1}$ and $\Omega_{2}$ are open subset of $E$ with $0\in\Omega_{1}$ and $\overline{\Omega_{1}}\subset\Omega_{2}$, $T:K\cap\left(\bar{\Omega}_{2}\setminus\Omega_{1}\right)\rightarrow K$ be a completely continuous operator such that \noindent (A) $\left\Vert Tu\right\Vert \leq\left\Vert u\right\Vert $, $\forall u\in K\cap\partial\Omega_{1}$ and $\left\Vert Tu\right\Vert \geq\left\Vert u\right\Vert $, $\forall u\in K\cap\partial\Omega_{2}$; or \noindent (B) $\left\Vert Tu\right\Vert \geq\left\Vert u\right\Vert $, $\forall u\in K\cap\partial\Omega_{1}$ and $\left\Vert Tu\right\Vert \leq\left\Vert u\right\Vert $, $\forall u\in K\cap\partial\Omega_{2}$ \noindent Then $T$ has a fixed point in $K\cap\left(\bar{\Omega}_{2}\setminus\Omega_{1}\right)$. \end{theorem} We adopt the following assumptions: \noindent ($H_{1}$) $f\left(t,u\left(t\right)\right)\in C\left(\left(a,b\right),\left[0,\infty\right)\right)$ is nondecreasing in $u\in\left[0,\infty\right)$ for fixed $t\in\left[a,b\right]$. \noindent $\left(H_{2}\right)$ $F_{a}=\intop_{a}^{b}s^{N-1}f\left(s,0\right)ds>0$, \noindent $\left(H_{3}\right)$ $f_{\infty}=\underset{u\rightarrow\infty}{lim}\underset{t\in\left[\frac{a}{a+b},b\right]}{min}\frac{f\left(t,u\right)}{u}=+\infty$. \noindent Set \[ \Lambda=\left\{ \lambda>0/there\;exists\;u_{\lambda}\in K\setminus\left\{ \theta\right\} \;such\;that\;T_{\lambda}u_{\lambda}=u_{\lambda}\right\} , \] \noindent and \[ \lambda^{*}=sup\Lambda. \] \begin{lemma} Suppose that $\left(H_{1}\right)-\left(H_{3}\right)$ hold. If $\lambda^{'}\in\Lambda$, then $\left(0,\lambda'\right]\subset\Lambda$. \end{lemma} \begin{proof} $\lambda^{'}\in\Lambda$ means that there exists $u_{\lambda'}\in K\setminus\left\{ \theta\right\} $ such that $T_{\lambda'}u_{\lambda'}=u_{\lambda'}$. Therefore, for any $\lambda\in\left(0,\lambda'\right]$ we have \[ T_{\lambda}u_{\lambda'}\leq T_{\lambda'}u_{\lambda'}=u_{\lambda'}, \] Set \[ w_{0}=u_{\lambda'},\;w_{n}=T_{\lambda}w_{n-1},\;n=1,2,... \] From $\left(H_{1}\right)$, we obtain \[ w_{0}\left(t\right)\geq w_{1}\left(t\right)\geq...\geq w_{n}\left(t\right)\geq...\geq\frac{F_{a}\lambda}{\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)}\left(1-\left(\frac{a}{t}\right)^{N-2}\right), \] by Lemma $2.9$ and $\left(H_{2}\right)$, $\left\{ w_{n}\right\} $ converges to fixed point of $T_{\lambda}$ in $K\setminus\left\{ \theta\right\} $. Thus $\left(0,\lambda'\right]\subset\Lambda$. The proof is complete. \end{proof} Let \[ \lambda_{*}<\frac{\left(b^{N-2}-a^{N-2}\right)}{F_{b}},\;\;\;\;F_{b}=\intop_{a}^{b}s^{N-1}f\left(s,\left(\frac{b}{a}\right)^{N-2}-1\right)ds, \] \[ u_{0}\left(t\right)=\frac{\lambda F_{a}}{\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)}\left(1-\left(\frac{a}{t}\right)^{N-2}\right),\;v_{0}\left(t\right)=\left(\left(\frac{b}{t}\right)^{N-2}-1\right), \] \noindent and \[ F_{\infty}=\underset{u\rightarrow\infty}{lim}sup\underset{a\leq t\leq b}{max}\frac{f\left(t,u\right)}{u}. \] \begin{theorem} Suppose that $\left(H_{1}\right)-\left(H_{3}\right)$ hold. Then $T_{\lambda}$ has minimal and maximal fixed point in $\left[u_{0},v_{0}\right]$ for $\lambda\in\left(0,\lambda_{*}\right]$. Moreover, there exists $\lambda^{*}\geq\lambda_{*}>0$ such that $T_{\lambda}$ has at least one and has no fixed points in $K\setminus\left\{ \theta\right\} $ for $0<\lambda<\lambda^{*}$ and $\lambda>\lambda^{*}$, respectively. \end{theorem} \begin{proof} From $\left(H_{1}\right)-\left(H_{3}\right)$ and $\left(\ref{eq:4}\right)$, we have $\lambda_{*}>0$. For any $\lambda\in\left(0,\lambda_{*}\right]$, we obtain \[ \left(T_{\lambda}u_{0}\right)\left(t\right)=\lambda\intop_{a}^{b}s^{N-1}G\left(t,s\right)f\left(s,u_{0}\left(s\right)\right)ds\qquad\qquad\qquad\qquad\;\; \] \[ \geq\lambda\intop_{a}^{b}s^{N-1}G\left(t,s\right)f\left(s,u_{0}\left(a\right)\right)ds\qquad\qquad \] \[ \;\;\qquad\qquad\qquad\quad\geq\frac{\lambda}{\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)}\left(1-\left(\frac{a}{t}\right)^{N-2}\right)\intop_{a}^{b}s^{N-1}f\left(s,0\right)ds \] \[ \;\qquad\qquad\geq\frac{\lambda F_{a}}{\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)}\left(1-\left(\frac{a}{t}\right)^{N-2}\right)=u_{0}\left(t\right), \] \noindent and \[ \left(T_{\lambda}v_{0}\right)\left(t\right)=\lambda\intop_{a}^{b}s^{N-1}G\left(t,s\right)f\left(s,v_{0}\left(s\right)\right)ds\qquad\qquad\qquad\qquad\qquad\qquad \] \[ \;\;\leq\lambda_{\text{*}}\intop_{a}^{b}s^{N-1}G\left(t,s\right)f\left(s,v_{0}\left(b\right)\right)ds\qquad\qquad\qquad\qquad \] \[ \;\qquad\qquad\quad\leq\frac{\lambda_{*}}{\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)}\left(\left(\frac{b}{t}\right)^{N-2}-1\right)\intop_{a}^{b}s^{N-1}f\left(s,v_{0}\left(b\right)\right)ds \] \[ \;\;\leq\frac{\lambda_{*}F_{b}}{\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)}\left(\left(\frac{b}{t}\right)^{N-2}-1\right)=v_{0}\left(t\right), \] \noindent Set \[ u_{n}=T_{\lambda}u_{n-1},\;v_{n}=T_{\lambda}v_{n-1},\;n=1,2,..., \] then from $\left(H_{1}\right)$, we have \begin{equation} u_{0}\left(t\right)\leq u_{1}\left(t\right)\leq...\leq u_{n}\left(t\right)\leq...\leq v_{1}\left(t\right)\leq v_{0}\left(t\right).\label{eq:6} \end{equation} \noindent Lemma $2.9$ implies that $\left\{ u_{n}\right\} $ and $\left\{ v_{n}\right\} $ converge to fixed points $u_{\lambda}$ and $v_{\lambda}$ of $T_{\lambda}$, respectively. \noindent From $\left(\ref{eq:6}\right)$ it is evident that $u_{\lambda},v_{\lambda}\in K\setminus\left\{ \theta\right\} $ are the mimimal fixed point and maximal fixed point of $T_{\lambda}$ in $\left[u_{0},v_{0}\right]$, respectively. \noindent By the definition of $\lambda^{*}$, there exists a nondecreasing sequence $\left\{ \lambda_{n}\right\} _{1}^{+\infty}$ such that $\underset{n\rightarrow+\infty}{lim}\lambda_{n}=\lambda^{*}$. Let $\left\{ u_{\lambda_{n}}\right\} _{1}^{+\infty}$ is bounded subset in $K$. Then there exists a constant $M>0$ such that \[ \left\Vert u_{\lambda_{n}}\right\Vert \leq M,\;for\;n\in\mathbb{N}^{*}, \] \noindent which implies that $\left\{ u_{\lambda_{n}}\right\} _{1}^{+\infty}$ is uniformly bounded. \noindent Now, we show that $\left\{ u_{\lambda_{n}}\right\} _{1}^{+\infty}$ is equicontinuous. For any $u_{\lambda_{n}}\in K$,$\;n\in\mathbb{N}^{*}$ and $t_{1},t_{2}\in\left[a,b\right]$, with $\left|t_{1}-t_{2}\right|<\delta$, we have \[ \left|x_{\lambda_{n}}\left(t_{1}\right)-x_{\lambda_{n}}\left(t_{2}\right)\right|\leq\lambda^{*}\intop_{a}^{b}s^{N-1}\left|G\left(t_{1},s\right)-G\left(t_{2},s\right)\right|f\left(s,M\right)ds \] \[ \qquad\qquad\qquad\quad\qquad\leq\lambda^{*}\intop_{a}^{b}s^{N-1}\left|G\left(t_{1},s\right)-G\left(t_{2},s\right)\right|f\left(s,M\right)ds, \] \noindent which implies that $\left\{ x_{\lambda_{n}}\right\} _{1}^{+\infty}$ is equicontinuous subset in $K$. Consequently, by an application of the Arzela-Ascoli theorem we conclude that $\left\{ x_{\lambda_{n}}\right\} _{1}^{+\infty}$ is a relatively compact set in $K$. So, there exists a subsequence $\left\{ x_{\lambda_{n_{i}}}\right\} \subset\left\{ x_{\lambda_{n}}\right\} $ converging to $x^{*}\in K$. Note that \[ \left(x_{\lambda_{n_{i}}}\right)\left(t\right)=\lambda_{n_{i}}\intop_{a}^{b}s^{N-1}G\left(t,s\right)f\left(s,x_{\lambda_{n_{i}}}\left(s\right)\right)ds. \] \noindent By taking the limit we have $x^{*}\left(t\right)=\left(T_{\lambda^{*}}x^{*}\right)\left(t\right)$. Therefore $T_{\lambda}$ has at least one fixed point for $0<\lambda<\lambda^{*}$. Finaly, for $T_{\lambda}$ has no fixed point for $\lambda>\lambda^{*}$. The proof is complete. \end{proof} \begin{theorem} Suppose that $\left(H_{1}\right),\left(H_{3}\right)$ and $\left(\ref{eq:4}\right)$ hold. If $\left(F_{+\infty}<+\infty\right)$, then when $F_{\infty}>0$, there exists $\lambda^{*}\geq\frac{N\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)\left(b^{N}-a^{N}\right)}{F_{\infty}}>0$ such that $T_{\lambda}$ has at least one and has no fixed points in $K\setminus\left\{ \theta\right\} $ for $0<\lambda<\lambda^{*}$ and $\lambda>\lambda^{*}$, respectively. When $F_{\infty}=0$, $T_{\lambda}$ has at least one fixed points in $K\setminus\left\{ \theta\right\} $ for $\lambda>0$. \end{theorem} \begin{proof} Since $F_{\infty}<\infty$, for any $\varepsilon>0$, there exists $N_{0}>0$ such that $f\left(t,u\right)\leq\left(F_{\infty}+\varepsilon\right)u$ for $u>N_{0},\;t\in\left[a,b\right]$. Let $w_{0}\left(t\right)=N_{0}\left(\left(\frac{b}{t}\right)^{N-2}-1\right)$ and $\lambda_{0}=\frac{N\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)\left(b^{N}-a^{N}\right)}{\left(\left(\frac{b}{a}\right)^{N-2}\right)\left(F_{\infty}+\varepsilon\right)}$, then $\lambda_{0}>0$ and \noindent \[ \left(T_{\lambda_{0}}w_{0}\right)\left(t\right)=\lambda_{0}\intop_{a}^{b}s^{N-1}G\left(t,s\right)f\left(s,w_{0}\left(s\right)\right)ds\qquad\qquad\qquad\qquad\qquad\qquad\qquad \] \[ \leq\frac{\lambda_{0}}{\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)}\left(\left(\frac{b}{t}\right)^{N-2}-1\right)\intop_{a}^{b}s^{N-1}\left(F_{\infty}+\epsilon\right)w_{0}\left(t\right)ds \] \[ \leq\frac{\lambda_{0}w_{0}\left(t\right)\left(F_{\infty}+\epsilon\right)}{\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)}\left(\left(\frac{b}{t}\right)^{N-3}-1\right)\intop_{a}^{b}s^{N-1}ds\qquad\qquad\qquad\quad \] \[ \;\leq\frac{\lambda_{0}w_{0}\left(t\right)\left(F_{\infty}+\epsilon\right)}{N\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)\left(b^{N}-a^{N}\right)}\left(\left(\frac{b}{a}\right)^{N-2}-1\right)\qquad\qquad\qquad \] \[ \leq w_{0}\left(t\right),\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \] \noindent Now, set $w_{0}\left(t\right)=N_{0}\left(\left(\frac{b}{t}\right)^{N-2}-1\right)$, \[ w_{n}=T_{\lambda_{n-1}}w_{n-1},\;n=1,2,.... \] From $\left(H_{1}\right)$, we obtain \begin{equation} w_{0}\left(t\right)\geq w_{1}\left(t\right)\geq...\geq w_{n}\left(t\right)\geq...\geq\frac{F_{a}\lambda}{\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)}\left(1-\left(\frac{a}{t}\right)^{N-2}\right).\label{eq:7} \end{equation} \noindent Therefore, the sequence $\left\{ w_{n}\right\} $ is bounded in $K\setminus\left\{ \theta\right\} $. By Lemma $2.9$ and the definition of $\lambda^{*}$, the operator $T_{\lambda_{n}}$ completely continuous. Hence the sequence $\left\{ w_{n}\right\} $ is compact in $K\setminus\left\{ \theta\right\} $ , its also monotone. Then it is uniformly convergent to fixed points $u^{*}$ of $T_{\lambda_{n}}$ in $K\setminus\left\{ \theta\right\} $. When we pass to the limit we get \[ u^{*}=T_{\lambda^{*}}u^{*} \] \noindent For $\lambda>\lambda^{*}$, there exists $\left\{ \lambda_{n}\right\} _{1}^{\infty},\;with\;\underset{n\rightarrow\infty}{lim}\lambda_{n}=\lambda$, we prove that problem has no positive solution. suppose the contrary that the problem has a positive solution $x_{\lambda_{n}}$, then we get \[ \left\Vert u_{\lambda_{n}}\right\Vert =\left(T_{\lambda_{n}}u_{\lambda_{n}}\right)\left(\left(\frac{b}{a}\right)^{N-2}-1\right)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \] \[ \leq\frac{\lambda_{n}}{\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)}\left(\left(\frac{b}{a}\right)^{N-2}-1\right)\intop_{a}^{b}s^{N-1}f\left(s,u_{\lambda_{n}}\left(s\right)\right)ds\qquad \] \[ \leq\frac{\lambda_{n}}{\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)}\left(\left(\frac{b}{a}\right)^{N-2}-1\right)\intop_{a}^{b}s^{N-1}\left(F_{\infty}+\epsilon\right)u_{\lambda_{n}}\left(b\right)ds\;\; \] \[ \leq\frac{\lambda_{n}\left(b^{N}-a^{N}\right)}{N\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)}\left(\left(\frac{b}{a}\right)^{N-2}-1\right)\left(F_{\infty}+\epsilon\right)u_{\lambda_{n}}\left(b\right)\qquad\qquad \] \[ \leq\frac{\lambda_{n}\left(b^{N}-a^{N}\right)}{N\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)}\left(\left(\frac{b}{a}\right)^{N-2}-1\right)\left(F_{\infty}+\epsilon\right)\left\Vert u_{\lambda_{n}}\right\Vert <\left\Vert u_{\lambda^{*}}\right\Vert . \] \noindent Taking the limit we obtain \[ \left\Vert u_{\lambda}\right\Vert <\left\Vert u_{\lambda}\right\Vert , \] \noindent which is a contradiction. The proof is complete. \end{proof} \begin{lemma}\label{lem2.3} Assume that $\left(H_{1}\right),\left(H_{2}\right)$ and $\left(H_{3}\right)$ hold. If $\Lambda$ is nonempty, then \noindent $\left(i\right)$ $\Lambda$ is bounded from above, that $\lambda^{*}<+\infty$. \noindent $\left(ii\right)$ $\lambda^{*}\in\Lambda$. \end{lemma} \begin{proof} Suppose to the contrary that there exists an increasing sequence $\left\{ \lambda_{n}\right\} _{1}^{+\infty}\subset\Lambda$ such that $\underset{n\rightarrow+\infty}{lim}\lambda_{n}=+\infty$. Set $x_{\lambda_{n}}\in K/\left\{ \theta\right\} $ is a fixed point of $T_{\lambda_{n}}$ that is , \[ T_{\lambda_{n}}u_{\lambda_{n}}=u_{\lambda_{n}}. \] There are two cases to be considered. Case 1. $\left\{ u_{\lambda_{n}}\right\} _{1}^{+\infty}$ is bounded, that is there exists a constant $M>0$ such that \[ \left\Vert u_{\lambda_{n}}\right\Vert \leq M,\;for\;n=1,2,\ldots. \] Hence, from $\left(H_{1}\right),\left(H_{2}\right),\;and\;\left(H_{3}\right)$ and Lemma $\ref{eq:4}$, we have \[ M\geq\left\Vert u_{\lambda_{n}}\right\Vert \geq\left(T_{\lambda_{n}}u_{\lambda_{n}}\right)\left(t\right)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \] \[ \qquad\geq\frac{\lambda_{n}}{\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)}\left(1-\left(\frac{a}{b}\right)^{N-2}\right)\intop_{a}^{b}s^{N-1}f\left(s,0\right)ds \] \[ =\frac{\lambda_{n}}{\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)}\left(1-\left(\frac{a}{b}\right)^{N-2}\right)F_{a}\rightarrow+\infty,\quad \] \noindent which is a contradiction. \noindent Case 2. $\left\{ u_{\lambda_{n}}\right\} _{1}^{+\infty}$ is unbounded, that is there exists subsequence of $\left\{ u_{\lambda_{n}}\right\} _{1}^{+\infty}$ still denoted by $\left\{ u_{\lambda_{n}}\right\} _{1}^{+\infty}$ such that $\underset{n\rightarrow+\infty}{lim}\left\Vert u_{\lambda_{n}}\right\Vert =+\infty$. \noindent When $\left(H_{3}\right)$, take $L>\frac{N\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)}{\left(1-\left(\frac{a}{b}\right)^{N-2}\right)\lambda_{1}}$ there exists $N_{1}>0$ such that $f\left(t,u\right)\geq Lu,\;for\;u\geq N_{1},\;t\in\left[a,b\right]$. Choose $n_{1}$ such that $\left\Vert u_{\lambda_{n_{1}}}\right\Vert >NN_{1}$.\\ Thus, for $t\in\left[a,b\right]$ , we have \[ f\left(t,\frac{1}{N}\left\Vert u_{\lambda_{n_{1}}}\right\Vert \right)\geq \frac{1}{N}L\left\Vert u_{\lambda_{n_{1}}}\right\Vert. \] Moreover, from $\left(H_{1}\right)$ and the definition of $K$, we have \[ \left\Vert x_{\lambda_{n_{1}}}\right\Vert \geq\left(T_{\lambda_{n_{1}}}u_{\lambda_{n_{1}}}\right)\left(t\right)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \] \[ \geq\frac{\lambda_{n_{1}}}{\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)}\left(1-\left(\frac{a}{b}\right)^{N-2}\right)\intop_{a}^{b}s^{N-1}f\left(s,u_{\lambda_{n_{1}}}\left(s\right)\right)ds \] \[ \qquad\geq\frac{\lambda_{n_{1}}}{\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)}\left(1-\left(\frac{a}{b}\right)^{N-2}\right)\intop_{a}^{b}s^{N-1}f\left(s,\frac{1}{6}\left\Vert u_{\lambda_{n_{1}}}\left(s\right)\right\Vert \right)ds \] \[ =\frac{\lambda_{n_{1}}L\left(1-\left(\frac{a}{b}\right)^{N-2}\right)}{N\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)}\left\Vert u_{\lambda_{n_{1}}}\right\Vert >\left\Vert u_{\lambda_{n_{1}}}\right\Vert,\qquad\qquad\qquad \qquad\quad \] \noindent which is a contradiction. \noindent Consequently, we find that $\Lambda$ is bounded from above. $\left(ii\right)$ From the definition of $\lambda^{*}$, there exists a nondecreasing sequence $\left\{ \lambda_{n}\right\} _{1}^{+\infty}$ such that $\underset{n\rightarrow+\infty}{lim}\lambda_{n}=\lambda^{*}$. Let $\left\{ u_{\lambda_{n}}\right\} _{1}^{+\infty}\in K\setminus\left\{ \theta\right\} $ be a fixed point of $T_{\lambda_{n}}$. Arguing similarly as above in Case 2, we can show that $\left\{ u_{\lambda_{n}}\right\} _{1}^{+\infty}$ is bounded subset in $K$, that is there exists a constant $M>0$. Hence from $\left(H_{1}\right),\left(H_{2}\right),\;and\;\left(H_{3}\right)$, we have \[ \left\Vert u_{\lambda_{n}}\right\Vert =\left(T_{\lambda_{n}}u_{\lambda_{n}}\right)\left(\left(\frac{b}{a}\right)^{N-2}-1\right)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \] \[ \leq\frac{\lambda_{n}}{\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)}\left(\left(\frac{b}{a}\right)^{N-2}-1\right)\intop_{a}^{b}s^{N-1}f\left(s,u_{\lambda_{n}}\left(s\right)\right)ds\ \] \[ \leq\frac{\lambda_{n}}{\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)}\left(\left(\frac{b}{a}\right)^{N-2}-1\right)\intop_{a}^{b}s^{N-1}f\left(s,u_{\lambda_{n}}\left(b\right)\right)ds \] \[ \leq\frac{\lambda_{n}}{\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)}\left(\left(\frac{b}{a}\right)^{N-2}-1\right)\intop_{a}^{b}s^{N-1}f\left(s,v_{\lambda_{n}}\left(b\right)\right)ds \] \[ =\frac{\lambda_{n}}{\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)}\left(\left(\frac{b}{a}\right)^{N-2}-1\right)\intop_{a}^{b}s^{N-1}f\left(s,0\right)ds\qquad\qquad\qquad\qquad \] \[ =\frac{\lambda_{n}F_{a}}{\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)}\left(\left(\frac{b}{a}\right)^{N-2}-1\right)\rightarrow\frac{\lambda_{*}F_{a}\left(\left(\frac{b}{a}\right)^{N-2}\right)}{\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)}=M, \] as $n\rightarrow\infty$.\\ \noindent Therefore \[ \left\Vert u_{\lambda_{n}}\right\Vert \leq M,\;n=1,2,... \] \noindent which shows that $\left\{ u_{\lambda_{n}}\right\} _{1}^{+\infty}$ is uniformly bounded.\\ \noindent From the proof of Theorem \ref{eq:8} we know that $\left\{ u_{\lambda_{n}}\right\} _{1}^{+\infty}$ is equicontinuous subset in $K$ and by an application of the Arzela-Ascoli theorem we conclude that $\left\{ u_{\lambda_{n}}\right\} _{1}^{+\infty}$ is a relatively compact set in $K$. So, there exists a subsequence $\left\{ u_{\lambda_{n_{i}}}\right\} \subset\left\{ u_{\lambda_{n}}\right\} $ converging to $u^{*}\in K$. Note that \[ \left(u_{\lambda_{n_{i}}}\right)\left(t\right)=\lambda_{n_{i}}\intop_{0}^{1}s^{N-1}G\left(t,s\right)f\left(s,u_{\lambda_{n_{i}}}\left(s\right)\right)ds. \] \noindent By taking the limit we have\\ $u^{*}\left(t\right)=\left(T_{\lambda^{*}}u^{*}\right)\left(t\right)\geq\frac{\lambda_{1}F_{a}}{\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)}\left(1-\left(\frac{a}{t}\right)^{N-2}\right)$,\\ that is $\lambda^{*}\in\Lambda$. The proof is complete. \end{proof} \begin{theorem} Suppose that $\left(H_{1}\right)-\left(H_{3}\right)$ holds. Then there exists $\lambda^{*}\geq\lambda_{*}>0$ such that BVP $\left(\ref{eq:2}\right)$ has at least two, one and no positive solutions for $0<\lambda\leq\lambda_{*},\;\lambda_{*}<\lambda\leq\lambda^{*}$ and $\lambda>\lambda^{*}$ respectively. \end{theorem} \begin{proof} \noindent From $\left(H_{1}\right),\left(H_{2}\right)$ and $\left(H_{3}\right)$ we have $\left(0,\lambda_{*}\right]\subset\Lambda$. So $\lambda^{*}\geq\lambda_{*}>0$ . \noindent From Lemma $3.2$ and $3.5$, we have $\left(0,\lambda^{*}\right]=\Lambda$. Therefore, from the definition of $\lambda^{*}$ we only to prove that $T_{\lambda}$ has at least two fixed points in $K\setminus\left\{ \theta\right\} $ for $\lambda\in\left(0,\lambda_{*}\right]$. Now, given $\lambda\in\left(0,\lambda_{*}\right]$. Theorem \ref{eq:8} means that $T_{\lambda}$ has at least one fixed point $u_{\lambda,1}\in K\setminus\left\{ \theta\right\} $ which satisfies $\left\Vert u_{\lambda,1}\right\Vert \leq\left(\frac{b}{a}\right)^{N-2}-1$. \noindent Let \[ K_{1}=\left\{ x\in K\mid\left\Vert u\right\Vert <\left(\frac{b}{a}\right)^{N-2}-1\right\}. \] For $t\in\left[a,b\right]$, so for $u\in K$ with $\left\Vert u\right\Vert =\left(\frac{b}{a}\right)^{N-2}-1$, i.e $u\in\partial K_{1}$, we have \[ \left\Vert u\right\Vert =\left\Vert T_{\lambda}u\right\Vert =\left(T_{\lambda}u\right)\left(\left(\frac{b}{a}\right)^{N-2}-1\right)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \] \[ \leq\frac{\lambda}{\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)}\left(\left(\frac{b}{a}\right)^{N-2}-1\right)\intop_{a}^{b}s^{N-1}f\left(s,u\left(s\right)\right)ds\qquad \] \[ \qquad\leq\frac{\lambda_{*}}{\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)}\left(\left(\frac{b}{a}\right)^{N-2}-1\right)\intop_{a}^{b}s^{N-1}f\left(s,\left(\frac{b}{a}\right)^{N-2}-1\right)ds \] \begin{equation} \;\;\;<\frac{\left(\left(\frac{b}{a}\right)^{N-2}-1\right)}{N-2}<\left\Vert u\right\Vert .\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\label{eq:8} \end{equation} \noindent When $\left(H_{3}\right)$, take $L>\frac{N\left(N-2\right)\left(b^{N-2}-a^{N-2}\right)}{\left(1-\left(\frac{a}{b}\right)^{N-2}\right)\lambda_{1}}$ there exists $N_{1}>0$ such that $f\left(t,u\right)\geq Lu,\;for\;u\geq N_{1},\;t\in\left[a,b\right]$. Set $K_{2}=\left\{ u\thinspace:\thinspace\left\Vert u\right\Vert \left\Vert u\right\Vert.\qquad\qquad\qquad \qquad\qquad\;\: \] \noindent Consequently, Applying Theorem \ref{eq:6} that $T_{\lambda}$ has a fixed point $u_{\lambda,2}\in\overline{K_{2}}\setminus K_{1}$. \noindent Equation $\left(\ref{eq:8}\right)$ implies that $T_{\lambda}$ has no fixed point in $\partial K_{1}$. In conclusion, for $\lambda\in\left(0,\lambda_{*}\right]$, $T_{\lambda}$ has at least two fixed points $u_{\lambda,1}$ and $u_{\lambda,2}$ in $K$. The proof is complete. \end{proof} We present an example to illustrate the applicability of the results shown before. \begin{example} Consider in $\mathbb{R}^{3}$ the elliptic boundary value problem \begin{equation} \begin{cases} \begin{array}{c} -\triangle u\left(x\right)=\lambda\left(\left|x\right|+u+ln\left(1+u\right)\right),\ x\in\varOmega,\\ u\left(x\right)=0,\qquad\qquad\qquad\qquad\quad\qquad x\in\partial\varOmega, \end{array}\end{cases}\label{eq:10} \end{equation} \noindent To the system $\left(\ref{eq:10}\right)$ we associate the the second order boundary value problem \[ \begin{cases} \begin{array}{c} -u''\left(t\right)=\lambda\left(t+u+ln\left(1+u\right)\right),\ t\in\left[a,b\right],\\ u\left(a\right)=u\left(b\right)=0,\qquad\qquad\qquad\qquad\qquad\;\: \end{array}\end{cases} \] \noindent By direct computation, we have \[ F_{\infty}=2,\;F_{0}=\frac{1}{4},\;F_{1}=\frac{1}{2}+\frac{2}{3}\left(1+ln\left(2\right)\right)\,and\,\lambda_{*}=\frac{48-9\pi}{6+8\left(1+ln\left(2\right)\right)}. \] \noindent So, the assumptions $\left(H_{1}\right),\,\left(H_{2}\right)$ and $\left(H_{3}\right)$ are satisfied, it follows from Theorem \ref{eq:10} there exists $\lambda^{*}=3\geq\lambda_{*}$ such that boundary value problem $\left(\ref{eq:10}\right)$ has at least one positive solution for $0<\lambda\leq3$ and has no positive solution for $\lambda>\lambda^{*}$. \end{example} \begin{thebibliography}{99} \bibitem{Bernstein} Bernstein. 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