\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 154, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/154\hfil Positive solutions] {Positive solutions for third-order Sturm-Liouville boundary-value problems with $p$-Laplacian} \author[C. B. Zhai,C. M. Guo \hfil EJDE-2009/154\hfilneg] {Chengbo Zhai, Chunmei Guo} % in alphabetical order \address{Chengbo Zhai\newline School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi, China} \email{cbzhai@sxu.edu.cn} \address{Chunmei Guo \newline School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi, China} \email{guocm@sxu.edu.cn} \thanks{Submitted September 1, 2008. Published November 28, 2009.} \subjclass[2000]{34K10} \keywords{Positive solution; Sturm-Liouville boundary value problem;\hfill\break\indent $p$-Laplacian operator; concave functional; fixed point} \begin{abstract} In this article, we consider the third-order Sturm-Liouville boundary value problem, with $p$-Laplacian, \begin{gather*} (\phi_p(u''(t)))'+f(t,u(t))=0, \quad t\in (0,1),\\ \alpha u(0)-\beta u'(0)=0,\quad \gamma u(1)+\delta u'(1)=0,\quad u''(0)=0, \end{gather*} where $\phi_p(s)=|s|^{p-2}s$, $p>1$. By means of the Leggett-Williams fixed-point theorems, we prove the existence of multiple positive solutions. As an application, we give an example that illustrates our result. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} In this paper, we study the existence of multiple positive solutions for the following third-order Sturm-Liouville boundary value problem with $p$-Laplacian \begin{gather} (\phi_p(u''(t)))'+f(t,u(t))=0, \quad t\in (0,1),\label{e1.1}\\ \alpha u(0)-\beta u'(0)=0,\quad \gamma u(1)+\delta u'(1)=0,\quad u''(0)=0,\label{e1.2} \end{gather} where $\phi_p(s)=|s|^{p-2}s$, $p>1$, $(\phi_p)^{-1} = \phi_q$, $\frac 1p + \frac 1q = 1$, $\alpha, \beta, \gamma, \delta\geq 0$. During the past decades, wide attention has been paid to the study equations with $p$-Laplacian operator, which arises in the modelling of different physical and natural phenomena, non-Newtonian mechanics \cite{d1,j1}, combustion theory \cite{r1}, population biology \cite{o2,o3}, nonlinear flow laws \cite{g1,l4,l5}, and system of Monge-Kantorovich partial differential equations \cite{e1}. There exist a very large number of papers devoted to the existence of solutions of the $p$-Laplacian operator. The second-order problem, $$(\phi_p(u'(t)))'+f(t,u(t))=0, \quad t\in (0,1),$$ with various boundary conditions has been studied by many authors, see \cite{h2,l3,m1,o1,w1,w2,z1,z2} and the references therein. However, to the best of our knowledge, few papers can be found in the literature on the existence of multiple positive solutions for the third-order Sturm-Liouville boundary value problem \eqref{e1.1}, \eqref{e1.2}. The purpose here is to fill this gap in the literature. Motivated by the works \cite{a1} and \cite{h1}, we shall establish the existence of at least two or at least three positive solutions to third-order Sturm-Liouville boundary value problem with $p$-Laplacian \eqref{e1.1}, \eqref{e1.2} by using fixed point theorems in cones. By a positive solution of \eqref{e1.1} and \eqref{e1.2} we understand a function $u(t)\in C^2[0,1]$ which is positive on $00$, $0<\sigma:=\min\big\{\frac{4\delta+\gamma}{4(\delta+\gamma)},\frac {\alpha+4\beta}{4(\alpha+\beta)}\big\}<1$. \item[(A2)] $G(t,s)$ is the Green's function of the differential equation $u''(t)=0$, $t\in (0,1)$ with respect to the boundary value condition \eqref{e1.2}, i.e., $$G(t,s)= \begin{cases} \frac 1\rho (\gamma+\delta-\gamma t)(\beta +\alpha s), &0\leq s\leq t\leq 1.\\ \frac 1\rho (\beta+\alpha t)(\gamma+\delta-\gamma s), & 0\leq t\leq s\leq 1. \end{cases}$$ Evidently $G(t,s)\leq G(s,s)$, $0\leq t,s\leq 1$. \item[(A3)] $f\in C([0,1]\times [0,\infty);[0,\infty))$. \end{itemize} For convenience, we denote \begin{gather*} \zeta(a)=\max\{f(t,u):0\leq t\leq 1,\ 0\leq u\leq a\},\\ \psi(b)=\min\{f(t,u):\frac 14\leq t\leq \frac 34,\ b\leq u\leq {\frac b{\sigma^2}}\}. \end{gather*} Where $\sigma$ is given as in (A1). Our main results are the following. \begin{theorem} \label{thm1.1} Assume {(A1)--(A3)}, and that there exist constants $0a$, where \begin{gather*} m=\Big(\int^1_0 G(s,s)ds\Big)^{-1} =\frac {6\rho}{\alpha\gamma+3\alpha\delta+3\beta\gamma+6\beta\delta},\\ l= \frac 2{\sigma {4}^{1-q}} \Big(\int^{3/4}_{1/4}G(\frac 12,s)ds\Big)^{-1}= \frac 2{\sigma {4}^{1-q}}\cdot \frac {32 \rho}{3\alpha\gamma+7\alpha\delta+7\beta\gamma+16\beta\delta}. \end{gather*} \end{theorem} \begin{theorem} \label{thm1.2} Assume {\rm (A1)--(A3)} and that there exist constants $a,b,c$ such that $0b$, $\|u_3\|>a$ and $\min_{t\in[\frac 14,\frac 34]}u_3(t)b\}\neq\emptyset$ and $\varphi (Ty)>b$ for $y\in P(\varphi,b,d)$; \item[(b')] $\|Ty\|b$ for $y\in P(\varphi,b,c)$ with $\|Ty\|>d$. \end{itemize} Then $T$ has at least three fixed points $y_1, y_2, y_3$ in $\bar{P_c}$ satisfying $\|y_1\|b, \|y_3\|>a$ and $\varphi(y_3)b\}\neq\emptyset$, and $\varphi (Ty)>b$ for $y\in P(\varphi,b,c)$; \item[(b'')] $\|Ty\|\frac bc \|Ty\|$ for $y\in \bar{ P_c}$ with $\|Ty\|>c.$ \end{itemize} Then $T$ has at least two fixed points $y_1, y_2$ in $\bar{P_c}$ satisfying $\|y_1\|a$ and $\varphi(y_2)0$ such that $\|u_n\|\leq M_0$, $\|\bar{u}\|\leq M_0$. Let $M_1=\max\{f(t,u)| t\in [0,1],\ u\in [0,M_0]\}$. Then for $t\in [0,1]$ we have \begin{align*} |Tu_n(t)-T\bar{u}(t)| &\leq \int^1_0G(t,v)\big|\phi_q \Big(\int^v_0 f(s,u_n(s))ds\Big)-\phi_q \Big(\int^v_0 f(s,\bar{u}(s))ds \Big)\big|dv\\ &\leq \int^1_0G(v,v)\big|\phi_q \Big(\int^v_0 f(s,u_n(s))ds\Big)-\phi_q \Big(\int^v_0 f(s,\bar{u}(s))ds\Big)\big|dv\\ &\leq \int^1_0 2\phi_q(M_1)G(v,v)dv. \end{align*} Note that $f(t,u)$ is continuous. We know that $\phi_q(\int^v_0 f(s,u)ds)$ is continuous in $u$ on $[0,\infty)$. Then for for each $\varepsilon>0$, there exists $\delta_1>0$, such that $|u_1-u_2|<\delta_1$ and we $$\big|\phi_q \Big(\int^v_0 f(s,u_1(s))ds\Big)-\phi_q \Big(\int^v_0 f(s,u_2(s))ds\Big)\big|<\frac \varepsilon{G(v,v)}.$$ In view of $u_n(s)\to \bar{u}(s)$, as $n\to \infty$, there exists a natural number $N>0$, for $n>N$ with $|u_n(s)-\bar{u}(s)|<\delta_1$, we have $$\big|\phi_q \Big(\int^v_0 f(s,u_n(s))ds\Big)-\phi_q \Big(\int^v_0 f(s,\bar{u}(s))ds\Big)\big|<\frac \varepsilon{G(v,v)}.$$ Thus for $\varepsilon>0$, there exists $N>0$, such that when $n>N$, $$G(v,v)\big|\phi_q \Big(\int^v_0 f(s,u_n(s))ds\Big) -\phi_q \Big(\int^v_0 f(s,\bar{u}(s))ds\big)\big|< \varepsilon,\quad \text{a.e. } [0,1].$$ An application of Lebesgue's dominated convergence theorem implies $$|Tu_n(t)-T\bar{u}(t)|\to 0 (\text{as }n\to\infty),\; t\in [0,1].$$ So operator $T:C^+[0,1]\to C^+[0,1]$ is continuous. Next we prove that $T$ is compact. Let $\Omega\subset C^+[0,1]$ be a bounded set. Then there exists $R>0$ such that $\Omega\subset \{u\in C^+[0,1]| \|u\|\leq R\}$. Set $M=\max\{f(t,u)|t\in [0,1],\,u\in \Omega\}$. For any $u\in \Omega$, we have $$|(Tu)(t)|=\big|\int^1_0G(t,v)\phi_q \Big(\int^v_0 f(s,u(s))ds\Big)dv \big|\leq \int^1_0G(v,v)\phi_q (M)dv,$$ which implies that $T(\Omega)$ is uniformly bounded. Furthermore, for any $u\in \Omega$ and $t\in [0,1]$, we have \begin{align*} |(Tu)'(t)|& = \Big|-\frac \gamma\rho\int^t_0(\beta+\alpha v)\phi_q \Big(\int^v_0 f(s,u(s))ds\Big)dv\\ &\quad + \frac \alpha\rho\int^1_t(\gamma+\delta-\gamma v)\phi_q \Big(\int^v_0 f(s,u(s))ds\Big)dv\Big|\\ &\leq \phi_q(M)\Big[\frac \gamma\rho\int^t_0(\beta+\alpha v) dv+\frac \alpha\rho\int^1_t(\gamma+\delta-\gamma v) dv\Big]\\ &= \phi_q(M)t\leq \phi_q(M). \end{align*} Hence $\|(Tu)'\|\leq \phi_q(M)$. So we can easily prove that $T(\Omega)$ is equicontinuous. The Arzela-Ascoli Theorem guarantee that $T(\Omega)$ is relatively compact and therefore that $T$ is compact. \end{proof} \section{Proofs of main results} In this section, we prove the existence of multiplicity results. Let $E=C[0,1]$ be endowed with the maximum norm $\|y\|=\max_{t\in[0,1]}| y(t)|$, and the ordering $x\leq y$ if $x(t)\leq y(t)$ for all $t\in[0,1]$. Define the cone $P\subset E$ by $$P=\{u\in C^+[0,1] : \min_{t\in [\frac 14,\frac 34]}u(t)\geq \sigma\|u\|\},$$ where $\sigma$ is given as in (A1). Next we show that $T(P)\subset P$. For any $u\in P$ and $t\in [0,1]$, from Lemma \ref{lem2.7} we have $$Tu(t)=\int^1_0G(t,v)\phi_q \Big(\int^v_0 f(s,u(s))ds\Big)dv \leq \int^1_0G(v,v)\phi_q \Big(\int^v_0 f(s,u(s))ds\Big)dv.$$ Consequently, $$\|Tu\|\leq \int^1_0G(v,v)\phi_q \Big(\int^v_0 f(s,u(s))ds\Big)dv.$$ Further, for $u\in P$ and $t\in [\frac 14,\frac 34]$, from Lemma \ref{lem2.7} we obtain \begin{align*} \min_{t\in [\frac 14, \frac 34]}Tu(t) &= \min_{t\in [\frac 14, \frac 34]}\int^1_0G(t,v)\phi_q \Big(\int^v_0 f(s,u(s))ds\Big)dv\\ &\geq \sigma \int^1_0G(v,v)\phi_q \Big(\int^v_0 f(s,u(s))ds\Big)dv \geq \sigma\|Tu\|. \end{align*} From Lemma \ref{lem2.8}, we know that $T:P\to P$ is completely continuous. Let $\varphi :P \to [0,\infty)$ be the nonnegative continuous concave functional defined by $$\varphi (u)=\min_{t\in [\frac 14,\frac 34]} u(t) , \quad u\in P.$$ Evidently, for each $u\in P$, we have $\varphi (u)\leq \|u\|$. We are now in a position to proving the main results. \begin{proof}[Proof of Theorem \ref{thm1.1}] It is easy to see that $T:\bar{P_{\frac b{\sigma^2}}}\to P$ is completely continuous and 0b. $$So \{u\in P(\varphi,b,\frac b{\sigma^2}): \varphi (u)>b\}\neq \emptyset. Hence, if u\in P(\varphi,b,\frac b{\sigma^2}), then b\leq u(t)\leq\frac b{\sigma^2} for t\in [\frac 14,\frac 34]. Thus for t\in [\frac 14,\frac 34], from assumption \eqref{eii}, we have$$ f(t,u(t))\geq \psi(b)\geq (lb)^{p-1},\ t\in [\frac 14,\frac 34]. Hence \begin{align*} Tu(\frac 12) &= \int^1_0G(\frac 12,v)\phi_q \Big(\int^v_0 f(s,u(s)\Big)ds)dv\\ & \geq \int^{3/4}_{1/4}G(\frac 12,v)\phi_q \Big(\int^v_0 f(s,u(s))ds\Big)dv\\ &\geq \int^{3/4}_{1/4} G(\frac 12,v)lb v^{q-1}dv \\ &\geq (\frac 14)^{q-1} lb\int^{3/4}_{1/4}G(\frac 12,v) dv = \frac {2b}{\sigma}>\frac b\sigma. \end{align*} Consequently, \min_{t\in [\frac 14,\frac 34]}Tu(t)\geq \sigma\|Tu\|> \sigma\times \frac b{\sigma}=b\ \ \mbox{for}\ b\leq u(t) \leq \frac b{\sigma^2},\ t\in [\frac 14,\frac 34]. $$That is,$$ \varphi(Tu)>b, \forall\ u\in P\big(\varphi,b,\frac b{\sigma^2}\big). Therefore, condition (a'') of Theorem \ref{thm2.5} is satisfied. Now if u\in \bar{P_a}, then \|u\|\leq a. By assumption \eqref{ei}, we have f(t,u(t))\leq \zeta(a)<(ma)^{p-1}, t\in[0,1]. Consequently, \begin{align*} \|Tu\|&= \max_{t\in [0,1]}|Tu(t)|=\max_{t\in [0,1]}\int^1_0G(t,v)\phi_q \Big(\int^v_0 f(s,u(s))ds\Big)dv\\ &< ma\max_{t\in [0,1]}\int^1_0 G(t,v)dv \leq ma \int^1_0 G(v,v)dv=a. \end{align*} This shows that T:\bar{P_a}\to P_a. That is, \|Tu\|< a for u\in \bar{P_a}. This shows that condition (b'') of Theorem \ref{thm2.5} is satisfied. Finally, we show that (c'') of Theorem \ref{thm2.5} also holds. Assume that u\in \bar {P_{\frac b{\sigma^2}}} with \|Tu\|>\frac b{\sigma^2}, then by the definition of cone P, we have \varphi (Tu)=\min_{t\in [\frac 14,\frac 34]}Tu(t)\geq \sigma\|Tu\| >\sigma^2\|Tu\| =b/ {\frac b{\sigma^2}}\|Tu\|. $$So condition (c'') of Theorem \ref{thm2.5} is satisfied. Thus using Theorem \ref{thm2.5}, T has at least two fixed points. That is to say, problem \eqref{e1.1},\eqref{e1.2} has at least two positive solutions u_1, u_2 in \bar{P_{\frac b{\sigma^2}}} satisfying \|u_1\|a. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1.2}] It follows from the conditions \eqref{ei'}-\eqref{eiii'} in Theorem \ref{thm1.2} that ab\}\neq \emptyset,\quad \varphi(Tu)>b \; \forall u\in P\big(\varphi,b,\frac b{\sigma^2}\big).$$ Moreover, foru\in P(\varphi,b,c)$and$\|Tu\|>\frac b{\sigma^2}$, we have $$\varphi (Tu)=\min_{t\in [\frac 14,\frac 34]}Tu(t) \geq \sigma\|Tu\|>\frac b\sigma>b.$$ So all the conditions of Theorem \ref{thm2.4} are satisfied. Thus using Theorem \ref{thm2.4},$T$has at least three fixed points. That is to say, the boundary value problem \eqref{e1.1},\eqref{e1.2} has at least three positive solutions$u_1, u_2\, u_3$with$\|u_1\|b$,$\|u_3\|>a$and$\min_{t\in[\frac 14,\frac 34]}u_3(t)\frac 5{512}$,$\|u_3\|>\frac 3{512}$and$\min_{t\in[\frac 14,\frac 34]}u_3(t)<\frac 5{512}$. \begin{thebibliography}{99} \bibitem{a1} R. P. Agarwal, D. O'Regan; A multiplicity result for second order impulsive differential equations via the Leggett Williams fixed point theorem, \emph{ Appl. Math. Comput.}, \textbf{161} (2005), 433-439. \bibitem{a2} D. Anderson, R. I. Avery and A. C. Peterson; Three positive solutions to a discrete focal boundary value problem, \emph{J. Comput. Appl. Math.}, \textbf{88}(1998), 103-118. \bibitem{d1} J. Diaz, F. de Thelin; On a nonlinear parabolic problem arising in some models related to turbulent flows,\emph{ SIAM J. Math. Anal.}, \textbf{25} (1994), (4), 1085-1111. \bibitem{e1} L. Evans, W. Gangbo; Differential equations methods for the Monge-Kantorovich mass transfer problem, \emph{ Mem. Amer. Math. Soc.}, \textbf{137} (653), (1999). \bibitem{g1} R. Glowinski, J. Rappaz; Approximation of a nonlinear elliptic problem arising in a non-Newtonian fluid flow model in glaciology, \emph{Math. Model. Numer. Anal.}, \textbf{37} (2003) (1), 175-186. \bibitem{g2} D. J. Guo, V. Lakshmikantham; \emph{Nonlinear Problems in Abstract Cones}, New York: Academic Press, 1988. \bibitem{h1} J. Henderson and H. B. Thompson; Multiple symmetric positive solutions for a second order boundary value problem, \emph{Proc. Amer. Math. Soc.}, \textbf{128} (2000), 2373-2379. \bibitem{h2} X. He, W. Ge; Existence of positive solutions for the one-dimension$p$-Laplacian equation, \emph{Acta Math.Sinica.}, \textbf{46} (2003) (4), 805-810(in Chinese). \bibitem{j1} U. Janfalk; \emph{On Certain Problem Concerning the$p$-Laplace Operator, in: Likping Studies in Sciences and Technology}, Dissertations, vol. 326, 1993. \bibitem{l1} R. W. Leggett and L. R. Williams; Multiple positive fixed points of nonlinear operaters on ordered Banach space, \emph{Indiana. univ. Math. J.}, \textbf{28} (1979), 673-688. \bibitem{l2} W. C. Lian, F. H. Wong, C. C. Yeh; On the existence of positive solutions of nonlinear second order differential equations, \emph{Proc. Amer. Math. Soc.}, \textbf{124} (1996), 1117-1126. \bibitem{l3} B. Liu; Positive solutions of singular three-point boundary value problems for the one-dimensional$p$-Laplacian, \emph{Comput. Math. Appl.}, \textbf{48} (2004), 913-925. \bibitem{l4} C. Liu; Weak solutions for a viscous$p$-Laplacian equation, \emph{Electron. J. Differential Equations}, \textbf{2003} (2003), no. 63, 1-11. \bibitem{l5} I. Ly, D. Seck; Isoperimetric inequality for an interior free boundary problem with$p$-Laplacian operator, \emph{Electron. J. Differential Equations}, \textbf{2004} (2004), Nov. 109, 1-12. \bibitem{m1} D. Ma, Z. Du, W. Ge; Existence and iteration of monotone positive solutions for multipoint boundary value problem with$p$-Laplacian operator, \emph{Comput. Math. Appl.}, \textbf{50} (2005), 729-739. \bibitem{o1} D. O'Regan; Some general existence principles results for$(\psi_{p}(y'))'=q(t)f(t, y, y'),0