\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 05, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2007/05\hfil Four-point boundary-value problems] {Positive solutions of four-point boundary-value problems for higher-order with $p$-Laplacian operator} \author[Y. Zhou, H. Su\hfil EJDE-2007/05\hfilneg] {Yunming Zhou, Hua Su} \address{Yunming Zhou \newline School of Mathematics and Information Science, Shandong University of Technology, Zibo Shandong, 255049, China} \email{zym571219@126.com} \address{Hua Su \newline School of Mathematics and System Sciences, Shandong University, Jinan Shandong, 250100, China \hfill\break School of Mathematical Sciences, Qufu Normal University, Qufu Shandong, 273165, China} \email{jnsuhua@163.com} \thanks{Submitted September 12, 2006. Published January 2, 2007.} \subjclass[2000]{34B18} \keywords{Higher-order $p$-Laplacian operator; four-point; positive solutions; \hfill\break\indent singular boundary-value; fixed-point index theory} \begin{abstract} In this paper, we study the existence of positive solutions for nonlinear four-point singular boundary-value problems for higher-order equation with the $p$-Laplacian operator. Using the fixed-point index theory, we find conditions for the existence of one solution, and of multiple solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{example}[theorem]{Example} %\allowdisplaybreaks \section{Introduction} In this paper, we study the quasi-linear equation, with $p$-Laplacian operator, (\phi_{p}(u^{(n-1)}))'+g(t)f(u(t),u'(t), \dots, u^{(n-2)}(t))=0,\quad 01$,$\phi_{q}=\phi_{p}^{-1}$,$\frac{1}{p}+\frac{1}{q}=1$.$\xi,\eta\in(0,1)$is prescribed and$\xi<\eta$,$g:(0,1)\to[0,\infty)$,$B_0, B_1$are both nondecreasing continuous odd functions defined on$(-\infty,+\infty)$. In recent years, the existence of positive solutions for nonlinear boundary-value problems with$p$-Laplacian operator received wide attention. Recently, for the existence of positive solutions of multi-points boundary-value problems for second-order ordinary differential equation, some authors have obtained the existence results \cite{a3,a1,a2,e1,m1}. However, the multi-points boundary-value problems treated in the above mentioned references do not discuss the problems with singularities and the higher-order$p$-Laplacian operator. For the singular case of multi-point boundary-value problems for higher-order$p$-Laplacian operator, with the author's acknowledge, no one has studied the existence of positive solutions in this case. Therefore this paper mainly studies the existence of positive solutions for nonlinear singular boundary-value problem \eqref{e1.1}, \eqref{e1.2}. In this paper, by constructing an integral equation which is equivalent to the problem \eqref{e1.1}, \eqref{e1.2}, we research the existence of positive solutions when$g$and$f$satisfy some suitable conditions. For the rest of this paper, we make the following assumptions: \begin{itemize} \item[(H1)]$f\in C([0,+\infty)^{n-1},[0,+\infty))$; \item[(H2)]$g:(0,1)\to[0,+\infty)$and$0<\int_{0}^{1}g(t)dt<\infty$; \item[(H3)]$B_0, B_1$are both increasing, continuous, odd functions defined on$(-\infty,+\infty)$and at least one of them satisfies the condition that there exists one$b>0such that $$00 such that A\geq L, t\in[\theta, 1-\theta]. \end{lemma} \begin{lemma}\label{lem2.2} Let u\in K and \theta\in (0,1/2) in Lemma \ref{lem2.1}. Then$$ u^{(n-2)}(t)\geq \theta\|u\|, \quad t\in[\theta,1-\theta]. $$\end{lemma} The proof of the above lemma is similar to the proof of in \cite[Lemma 2.2]{s1}, so we omit it. \begin{lemma}\label{lem2.3} Suppose that conditions (H1)--(H3) hold. Then u(t)\in K\cap C^{n-1}(0,1) is a solution of boundary-value problem \eqref{e1.1}, \eqref{e1.2} if and only if u(t)\in B is a solution of the integral equation$$ u(t)=\int_0^{t}\int_0^{s_1}\dots\int_0^{s_{n-3}}w(s_{n-2}) ds_{n-2}ds_{n-3}\dots ds_{1}, where $$w(t)=\begin{cases} B_0\circ\phi_{q}\big(\int_{\xi}^{\delta}g(s)f(u(s),u'(s),\dots, u^{(n-2)}(s))ds\big) \\ +\int_{0}^{t}\phi_{q}\big(\int_{s}^{\delta}g(r)f(u(r),u'(r),\dots, u^{(n-2)}(r))dr\big)ds &0\leq t\leq\delta, \\[4pt] B_1\circ\phi_{q}\big(\int_{\delta}^{\eta}g(s)f(u(s),u'(s),\dots, u^{(n-2)}(s))ds\big) \\ +\int_{t}^{1}\phi_{q}\big(\int_{\delta}^{s}g(r)f(u(r),u'(r),\dots, u^{(n-2)}(r))dr\big))ds &\delta\leq t\leq 1. \end{cases} \label{e2.1}$$ Here \delta is unique solution of the equation g_{1}(t)=g_{2}(t), where \begin{align*} g_{1}(t)&=B_0\circ\phi_{q}\big(\int_{\delta}^{\eta}g(s)f(u(s),u'(s),\dots, u^{(n-2)}(s))ds\big) \\ &\quad +\int_{t}^{1}\phi_{q}\big(\int_{\delta}^{s}g(r)f(u(r),u'(r),\dots, u^{(n-2)}(r))dr\big)ds, \\ g_{2}(t)&=B_1\circ\phi_{q}(\int_{\delta}^{\eta}g(s)f(u(s),u'(s),\dots, u^{(n-2)}(s))ds) \\ &\quad +\int_{t}^{1}\phi_{q}(\int_{\delta}^{s}g(r)f(u(r),u'(r),\dots, u^{(n-2)}(r))dr)ds. \end{align*} The equation g_{1}(t)=g_{2}(t) has unique solution in (0,1) because g_{1}(t) is strictly increasing on [0,1), and g_{1}(0)=0, while g_{2}(t) is strictly decreasing on (0,1], and g_{2}(1)=0. \end{lemma} \begin{proof} Necessity. By the equation of the boundary condition and (H3), we have u^{(n-1)}(\xi)\geq 0, u^{(n-1)}(\eta)\leq0, then there exist a constant \delta\in [\xi,\eta]\subset(0,1) such that u^{(n-1)}(\delta)=0. Firstly, by integrating the equation of the problems \eqref{e1.1} on (\delta,t), we have \phi_{p}(u^{(n-1)}(t))=\phi_{p}(u^{(n-1)}(\delta))-\int_{\delta}^t g(s)f\big(u(s),u'(s),\dots, u^{(n-2)}(s)\big)ds, $$then $$u^{(n-1)}(t)=-\phi_{q}\Big(\int_{\delta}^t g(s)f\big(u(s),u'(s),\dots, u^{(n-2)}(s)\big)ds\Big),\label{e2.2}$$ thus $$u^{(n-2)}(t)=u^{(n-2)}(\delta)-\int_{\delta}^t\phi_{q}\Big(\int_{\delta}^s g(r)f\big(u(r),u'(r),\dots, u^{(n-2)}(r)\big)dr\Big)ds.\label{e2.3}$$ By u^{(n-1)}(\delta)=0 and condition \eqref{e1.2}, letting t=\eta on \eqref{e2.2}, we have$$ u^{(n-2)}(1)=-B_1\big(u^{(n-1)}(\eta )\big) =B_1\circ\phi_{q}\Big(\int_{\delta}^\eta g(s)f\big(u(s),u'(s),\dots, u^{(n-2)}(s)\big)ds\Big). Then by \eqref{e2.3}, we have \begin{aligned} u^{(n-2)}(\delta)&=B_1\circ\phi_{q}\Big(\int_{\delta}^\eta g(s)f\big(u(s),u'(s),\dots, u^{(n-2)}(s)\big)ds\Big))\\ &\quad +\int_{\delta}^1\phi_{q}\Big(\int_{\delta}^s g(r)f\big(u(r),u'(r),\dots, u^{(n-2)}(r)\big)dr\Big)ds. \end{aligned} \label{e2.4} Then \begin{aligned} u^{(n-2)}(t)&=B_1\circ\phi_{q}\Big(\int_{\delta}^\eta g(s)f\big(u(s),u'(s),\dots, u^{(n-2)}(s)\big)ds\Big)\\ &\quad +\int_{t}^1\phi_{q}\Big(\int_{\delta}^s g(r)f\big(u(r),u'(r),\dots, u^{(n-2)}(r)\big)dr\Big)ds. \end{aligned}\label{e2.5} Integrating \eqref{e2.5} for n-2 times on (0,t), we have \begin{align*} u(t)&=\int_{0}^t\int_{0}^{s_1}\dots\int_{0}^{s_{n-3}} B_1\circ\phi_{q}\Big(\int_{\delta}^\eta g(s)f\big(u(s),u'(s),\\ &\quad \dots, u^{(n-2)}(s)\big)ds\Big)ds_{s_{n-2}}\dots ds_2\,ds_1\\ &\quad +\int_{0}^t\int_{0}^{s_1}\dots\int_{0}^{s_{n-3}} (\int_{s_{n-2}}^{1}\phi_{q}\Big(\int_{\delta}^s g(r)f\big(u(r),u'(r),\\ &\quad \dots, u^{(n-2)}(r)\big)dr\Big)ds)ds_{s_{n-2}}\dots ds_2\,ds_1. \end{align*} Similarly, for t\in(0,\delta), integrating problems \eqref{e1.1} on (0,\delta), we have \begin{align*} u(t)&=\int_{0}^t \int_{0}^{s_1}\dots\int_{0}^{s_{n-3}} B_0\circ\phi_{q}\Big(\int_{\xi}^{\delta} g(s)f\big(u(s),u'(s),\\ &\quad \dots, u^{(n-2)}(s)\big)ds\Big)ds_{s_{n-2}}\dots ds_2\,ds_1\\ &\quad +\int_{0}^t\int_{0}^{s_1}\dots\int_{0}^{s_{n-3}} (\int_{0}^{s_{n-2}}\phi_{q}\Big(\int_{s}^\delta g(r)f\big(u(r),u'(r),\\ &\quad \dots, u^{(n-2)}(r)\big)dr\Big)ds)ds_{s_{n-2}}\dots ds_2\,ds_1. \end{align*} Therefore, for any t\in [0,1], u(t) can be expressed as u(t)=\int_0^{t}\int_0^{s_1}\dots\int_0^{s_{n-3}}w(s_{n-2}) ds_{n-2}ds_{n-3}\dots ds_{1}, where w(t) is expressed as \eqref{e2.1}. Sufficiency. Suppose that u(t)=\int_0^{t}\int_0^{s_1}\dots\int_0^{s_{n-3}}w(s_{n-2}) ds_{n-2}ds_{n-3}\dots ds_{1}. Then by \eqref{e2.1}, we have $$u^{(n-1)}(t)=\begin{cases} \phi_{q}\Big(\int_{t}^{\delta}g(s)f\big(u(s),u'(s),\dots, u^{(n-2)}(s)\big)ds\Big)ds\geq0, &0\leq t<\delta, \\ -\phi_{q}\Big(\int_{\delta}^{t}g(s)f\big(u(s),u'(s),\dots, u^{(n-2)}(s)\big)ds\Big)ds\leq0, &\delta< t\leq1, \end{cases}\label{e2.6}$$ So that (\phi_{p}(u^{(n-1)}))'+g(t)f(u(t),u'(t), \dots, u^{(n-2)}(t))=0, 02r=2\|u\|. \end{align*} \noindent(ii) If \delta\in(1-\theta, 1], thus for u\in\partial\Omega_{1}, by (A1) and Lemma \ref{lem2.3}, we have \begin{align*} \|Tu\|&=(Tu)^{(n-2)}(\delta) \\ & \geq B_0\circ \phi_{q}\Big(\int_{\xi}^{\delta}g(r)f\big(u(r),u'(r),\dots, u^{(n-2)}(r)\big)dr\Big) \\ &\quad +\int_{0}^{\delta}\phi_{q} (\int_{s}^{\delta}g(r)f(u(r),u'(r),\dots, u^{(n-2)}(r))dr)ds \\ &\geq \int_{\theta}^{1-\theta}\phi_{q} \Big(\int_{s}^{1-\theta}g(r)f\big(u(r),u'(r),\dots, u^{(n-2)}(r)\big)dr\Big)ds \\ &\geq mrA(1-\theta)\geq mrL\\ &>2r>r=\|u\|. \end{align*} \noindent (iii) If \delta\in(0,\theta), thus for u\in\partial\Omega_{1}, by (A1) and Lemma \ref{lem2.3}, we have \begin{align*} \|Tu\|&=(Tu)^{(n-2)}(\delta) \\ & \geq B_1\circ \phi_{q}\Big(\int_{\delta}^{\eta}g(r)f\big(u(r),u'(r),\dots, u^{(n-2)}(r)\big)dr\Big) \\ &\quad +\int_{\delta}^{1}\phi_{q} \Big(\int_{\delta}^{s}g(r)f\big(u(r),u'(r),\dots, u^{(n-2)}(r)\big)dr\Big)ds \\ &\geq \int_{\theta}^{1-\theta}\phi_{q} \Big(\int_{\theta}^{s}g(r)f\big(u(r),u'(r),\dots, u^{(n-2)}(r)\big)dr\Big)ds \\ &\geq mrA(\theta)\geq mrL\\ &>2r>r=\|u\|. \end{align*} Therefore, under all condition, we have \|Tu\|>\|u\| for all u\in\partial\Omega_{1}. Then by Theorem \ref{thm2.6}, $$i(T,\Omega_{1},K)=0.\label{e3.2}$$ On the other hand, for u\in\partial\Omega_{2}, we have u^{(n-2)}(t)\leq\|u\|=R, by (A2), \begin{align*} \|Tu\|&=(Tu)^{(n-2)}(\delta) \\ & \leq B_0\circ \phi_{q}\Big(\int_{0}^{1}g(r)f\big(u(r),u'(r),\dots, u^{(n-2)}(r)\big)dr\Big) \\ &\quad +\int_{0}^{1}\phi_{q} \Big(\int_{s}^{\delta}g(r)f\big(u(r),u'(r),\dots, u^{(n-2)}(r)\big)dr\Big)ds \\ &\leq bMR\phi_{q} \big(\int_{0}^{1}g(r)dr\big)+ MR\phi_{q} \big(\int_{0}^{1}g(r)dr\big) \\ & =(b+1)MR\phi_{q} \big(\int_{0}^{1}g(r)dr\big)\\ &\leq R=\|u\|. \end{align*} Thus \|Tu\|<\|u\| for all u\in\partial\Omega_{2}. Then by Theorem \ref{thm3.1}, we have $$i(T,\Omega_{2},K)=1.\label{e3.3}$$ Therefore, by \eqref{e3.2}, \eqref{e3.3}, r\theta^*, thus by \eqref{e3.5}, condition (A1) holds. Therefore by Theorem \ref{thm3.1} we know that the results of Theorem \ref{thm3.2} hold. The proof is complete. \end{proof} \begin{proof}[Proof of Theorem \ref{thm3.3}] First, by condition (A6), f_0=\varphi\in ((2\theta^*/\theta)^{p-1},\infty), then for \epsilon=\varphi-(2\theta^*/\theta)^{p-1}, there exists an appropriately small positive number r, such that 0\leq u_{n-1}\leq r, u_{n-1}\neq0, we have f(u_1,u_2,\dots,u_{n-1})\geq(\varphi-\epsilon)(u_{n-1})^{p-1} =(2\theta^*/\theta)^{p-1}(u_{n-1})^{p-1}, $$thus when \theta r\leq u_{n-1}\leq r, we have $$f(u_1,u_2,\dots,u_{n-1})\geq(2\theta^*/\theta)^{p-1}(\theta r)^{p-1} =(2\theta^*r)^{p-1}.\label{e3.6}$$ Let m=2\theta^*>\theta^*, so by \eqref{e3.6}, condition (A1) holds. Next, by condition (A5): f^\infty=\lambda\in[0,(\theta_*/4)^{p-1}), then for \epsilon=(\theta_*/4)^{p-1}-\lambda, there exists an suitably big positive number \rho\neq r, such that u_{n-1}\geq\rho, we have $$f(u_1,u_2,\dots,u_{n-1})\leq(\lambda+\epsilon)(u_{n-1})^{p-1}\leq (\theta_*/4)^{p-1}(u_{n-1})^{p-1}.\label{e3.7}$$ If f is unbounded, by the continuation of f on [0, \infty)^{n-1}, then exists constant R\geq\rho, R\neq r, and a point (u_{01},u_{02},\dots,u_{0(n-1)})\in[0, \infty)^{n-1} such that$$\rho\leq u_{0(n-1)}\leq R$$and$$ f(u_1,u_2,\dots,u_{n-1})\leq f(u_{01},u_{02},\dots,u_{0(n-1)}), \quad 0\leq u_{n-1}\leq R. Thus, by \rho\leq u_{0(n-1)}\leq R, we know \begin{align*} f(u_1,u_2,\dots,u_{n-1}) &\leq f(u_{01},u_{02},\dots,u_{0(n-1)})\\ &\leq(\theta_*/4)^{p-1}(u_{0(n-1)})^{p-1}\\ &\leq(\theta_*R/4)^{p-1}. \end{align*} Choose M=\frac{\theta_*}{4}\in(0, \theta_*). Then, we have \begin{gather*} f(u_1,u_2,\dots,u_{n-1})\leq (MR)^{p-1}, \\ 0\leq u_{n-1}\leq R,\quad 0\leq u_1\leq\dots\leq u_{n-2}\leq u_{n-1}/\theta . \end{gather*} If f is bounded, we suppose f(u_1,u_2,\dots,u_{n-1})\leq \overline{M}^{p-1}, u_{n-1}\in[0, \infty), \overline{M}^{p-1} \in R_+, there exists an adequately big positive number R>4\overline{M}/\theta_*, then choose M=\theta_*/4\in(0, \theta_*), for 0\leq u_1\leq\dots\leq u_{n-2}\leq u_{n-1}/\theta , we have f(u_1,u_2,\dots,u_{n-1})\leq \overline{M}^{p-1}\leq(\theta_*R/4)^{p-1}=(MR)^{p-1}, \quad 0\leq u_{n-1}\leq R $$Therefore, condition (A2) holds. Therefore, by Theorem \ref{thm3.1}, we know that the results of Theorem \ref{thm3.3} holds. The proof is complete. \end{proof} \section{Existence of Many Positive Solutions } Next, we discuss the existence of many positive solutions. \begin{theorem}\label{thm4.1} Suppose that conditions (H1)--(H3) and (A2) hold. Assume that f also satisfies \begin{itemize} \item[(A7)] f_0=+\infty; \item[(A8)] f_\infty=+\infty. \end{itemize} Then the boundary-value problem \eqref{e1.1}, \eqref{e1.2} has at least two solutions u_1, u_2 such that$$0<\|u_1\|\frac{2}{\theta L}, there exists a constant $\rho_*\in(0,R)$ such that \begin{gathered} f(u_1,u_2,\dots,u_{n-1})\geq(Mu_{n-1})^{p-1},\\ 0\frac{2}{\theta L}$, there exists a constant$\rho_0>0$such that $$\begin{gathered} f(u_1,u_2,\dots,u_{n-1})\geq(\overline{M}u_{n-1})^{p-1},\\ u_{n-1}>\rho_0,\quad 0\leq u_1\leq\dots\leq u_{n-2}\leq u_{n-1}/\theta . \end{gathered}\label{e4.3}$$ We choose a constant$\rho^*>\max\{R, \frac{\rho_0}{\theta}\}$, obviously$\rho_*\rho_0,\quad t\in[\theta,1-\theta]. $$Then by \eqref{e4.3} and also similar to the proof of Theorem \ref{thm3.1}, we have from the three perspectives,$$ \|Tu\|\geq\|u\| \quad \forall u\in\partial\Omega_{\rho^*}. Then by Theorem \ref{thm2.6}, we have $$i(T,\Omega_{\rho^*},K)=0.\label{e4.4}$$ Finally, set \Omega_{R}=\{u\in K:\|u\|r such that $$f^*(x)\leq(\eta_2 x)^{p-1},\quad x\geq\rho^*.\label{e4.8}$$ Set \Omega_{\rho^*}=\{u\in K:\|u\|<\rho^*\}, for each u\in\partial\Omega_{\rho^*}, by \eqref{e4.8}, we have \begin{align*} \|Tu\|&=(Tu)^{(n-2)}(\delta) \\ &\leq B_0\circ \phi_{q}\Big(\int_{0}^{1}g(r)f\big(u(r),u'(r),\dots, u^{(n-2)}(r)\big)dr\Big) \\ &\quad +\int_{0}^{1}\phi_{q} \Big(\int_{s}^{\delta}g(r)f\big(u(r),u'(r),\dots, u^{(n-2)}(r)\big)dr\Big)ds \\ &\leq B_0\circ \phi_{q}\Big(\int_{0}^{1}g(r)f\big(u(r),u'(r),\dots, u^{(n-2)}(r)\big)dr\Big) \\ &\quad +\phi_{q} \Big(\int_{0}^{1}g(r)f\big(u(r),u'(r),\dots, u^{(n-2)}(r)\big)dr\Big) \\ &\leq (b+1)\phi_{q} \Big(\int_{0}^{1}g(r)f^*(\rho^*)dr\Big) \\ &\leq (b+1)\eta_2\rho^*\phi_{q} \Big(\int_{0}^{1}g(r)dr\Big)\\ &\leq \rho^*=\|u\|. \end{align*} i.e., \|Tu\|\leq\|u\| for all u\in\partial\Omega_{\rho^*}. Then by Theorem \ref{thm2.6}, we have $$i(T,\Omega_{\rho^*},K)=1.\label{e4.9}$$ Finally, set \Omega_{r}=\{u\in K:\|u\|\frac{51}{20}, i.e. \varphi\in[0, (\theta_*/4)^{p-1}), so conditions (A3) holds. For \theta=1/4, by calculating, it is easy see that L=\min_{t\in[\theta,1-\theta]}A(t)=\frac{1}{16} (\frac{7}{36}+\frac{\sqrt{3}}{3}). $$Because$$ (2\theta^*/\theta)^{p-1}=96\times(\frac{1}{7+12\sqrt{3}})^{1/2} <\frac{95}{5},$$we have$$ l\in ((2\theta^*/\theta)^{p-1}, \infty),  so conditions (A4) holds. Then by Theorem \ref{thm3.2}, \eqref{e5.1} has at least a positive solution. \end{example} \begin{example} \label{exam5.2} \rm Consider the following third-order singular boundary-value problem, with $p$-Laplacian, \begin{gathered} (\phi_{p}(u''))'+\frac{1}{64\pi^4}t^{-\frac{1}{2}}(1-t) [u+(u')^{2}+(u')^{4}]=0 \quad 0