\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 45, pp. 1--8.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2011 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2011/45\hfil Existence of multiple positive solutions] {Existence of multiple positive solutions for discrete problems with p-Laplacian via variational methods} \author[Y. Tian, W. Ge\hfil EJDE-2011/45\hfilneg] {Yu Tian, Weigao Ge} \address{Yu Tian \newline School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China} \email{tianyu2992@163.com} \address{Weigao Ge \newline Department of Applied Mathematics, Beijing Institute of Technology, Beijing 100081, China} \email{gew@bit.edu.cn} \thanks{Submitted February 22, 2011. Published April 4, 2011.} \thanks{Supported by grants 11001028 from the National Science Foundation for Young Scholars, \hfill\break\indent and BUPT2009RC0704 from the Chinese Universities Scientific Fund} \subjclass[2000]{39A10, 34B18, 58E30} \keywords{Discrete boundary value problem; variational methods; \hfill\break\indent mountain pass theorem} \begin{abstract} Using critical point theory, we prove the existence of multiple positive solutions for second-order discrete boundary-value problems with p-Laplacian. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \section{Introduction} In recent years, a great deal of work has been done in the study of the existence of multiple positive solutions for discrete boundary value problems describing physical and biological phenomena. For the background and summary of results, we refer the reader to the monograph by Agarwal et al \cite{a2}, and for some recent contributions to \cite{a1,a3}. Various fixed point theorems have been applied for obtaining solutions, among them, Krasnosel'skii fixed point theorem, Leggett-Williams fixed point theorem, fixed point theorem in cones; see \cite{a4,a5,h1,m1,t1} and the references therein. There is also a trend to study difference equation using variational methods which lead to many interesting results; see for example \cite{a3,g1,l1,y1}. Li \cite{l1} studied the existence of solutions for the problem $$\begin{gathered} \Delta(p(k)\Delta x(k-1))+f(k, x(k))=g(k)\\ x(0)=x(T+1)=0, \end{gathered}$$ where $f\in C(\mathbb{R}^2, \mathbb{R})$, $p, g\in C(\mathbb{R},\mathbb{R})$. Using variational methods, the existence of at least one non-trivial solution was obtained. Agarwal et al \cite{a3} show the existence of multiple positive solutions for the discrete boundary-value problem $$\begin{gathered} \Delta^2 y(k-1)+f(k, y(k))=0, \quad k\in [1, T],\\ y(0)=0=y(T+1), \end{gathered} \label{discrete}$$ where $[1, T]$ is the discrete interval $\{1, 2, \dots, T\}$, $\Delta y(k)=y(k+1)-y(k)$, $f\in C([1, T]\times [0, \infty), \mathbb{R})$ satisfies $f(k, 0)\ge0$, for all $k\in [1, T]$. They applied critical point theory under the following conditions: \begin{itemize} \item[(a)] $\min_{k\in[1, T]}\liminf_{u\to\infty} \frac{f(k,u)}{u}>\lambda_1$, where $\lambda_1$ is the smallest eigenvalue of $\Delta^2y(k-1)+\lambda y(k)=0$, $y\in H$; \item[(b)] there is a positive constant $M$, independent of $\lambda$, such that $\|y\|\neq M$ for every solution $y\ge 0$ of the equation $\Delta^2 y(k-1)+\lambda f(k, y(k))=0, \quad y\in H,\; \lambda\in( 0, 1].$ \end{itemize} We remark that is not easy to verify Condition (b) in applications. To the best of our knowledge, very few authors have studied the existence of multiple positive solutions for discrete boundary value problem with a p-Laplacian by using variational methods. As a result the goal of this paper is to fill the gap in this area. It is well known that positive solutions are very important in applications. Motivated by the above results, in this paper, we study the existence of multiple positive solutions for the second-order discrete boundary-value problem (BVP) $$\label{000} \begin{gathered} \Delta(\Phi_p(\Delta y(k-1)))+f(k, y(k))=0, \quad k\in[1, T],\\ y(0)=0=y(T+1), \end{gathered}$$ where $T$ is a positive integer, $[1, T]$ is the discrete interval $\{1, \dots, T\}$ and $\Delta y(k)=y(k+1)-y(k)$ is the forward difference operator, $p>1$, $\Phi_p(y):=|y|^{p-2}y$, $f\in C([1, T]\times [0, +\infty), [0, +\infty))$, $f(k, 0)\not\equiv 0$ for $k\in [1, T]$, $F(k, x)=\int_{0}^{x}f(k, s)ds$. For a review of variational methods, we refer the reader to \cite{m2,r1}. Our aim of this paper is to apply critical point theory to \eqref{000} and prove the existence of two positive solutions. We impose some conditions on the nonlinearity $f$ that are different from those in \cite{a2} for $p=2$, and are easy to verify. In this article, we assume the following conditions: \begin{itemize} \item[(C1)] there exist $\mu>p$, $h\in C([1, T]\times[0, +\infty), [0,+\infty))$, $l:[1, T]\to (0, +\infty)$, $\min_{k\in[1, T]}l(k)>0$ such that $f(k, y)=l(k)\Phi_{\mu}(y)+h(k, y);$ \item[(C2)] there exist functions $c, d: [1, T]\to [0, +\infty)$ such that $h(k, y)\le c(k)+d(k)\Phi_p(y).$ \end{itemize} \section{Related Lemmas} Here, and in the sequel, we denote $Y=W_{0}^{1, p}[0,T+1]=\{y:[0, T+1]\to R:y(0)=y(T+1)=0\}$ whihc is a $T$-dimensional Banach space with the norm $\|y\|=\Big(\sum_{k=1}^{T+1}|\Delta y(k-1)|^p\Big)^{1/p}\,.$ \begin{lemma}\label{lem0} Let $y^{\pm}=\max\{\pm y, 0\}$, then the following five properties hold: \begin{itemize} \item[(i)] $y=y^+ -y^-$; \item[(ii)] $\|y^+\|\le \|y\|$; \item[(iii)] $y^+(t)y^-(t)=0, (y^+)'(t)(y^-)'(t)=0$ for $t\in[0, T+1]$; \item[(iv)] $\Phi_p(y)y^+=|y^{+}|^p$, $\Phi_p(y)y^-=-|y^{-}|^p$. \end{itemize} \end{lemma} \begin{lemma}\label{lem1} If $y$ is a solution of the equation $$\Delta(\Phi_p(\Delta y(k-1)))+f(k, y^+(k))=0, \quad y\in Y, \label{eq1+}$$ then $y\ge 0$, $y(k)\not\equiv 0$, $k\in[0, T+1]$ and hence it is a solution of \eqref{000}. \end{lemma} \begin{proof} If $y$ is a solution of \eqref{eq1+}, then \label{12} \begin{aligned} 0&=\sum_{k=1}^{T}\left[\Delta(\Phi_p(\Delta y(k-1))) +f(k, y^+(k))\right]y^-(k)\\ &=\Phi_p(\Delta y(k-1))y^-(k)|_{k=1}^{T+1} -\sum_{i=1}^{T}\Phi_p(\Delta y(k))\Delta y^-(k) +\sum_{k=1}^{T}f(k, y^+(k))y^-(k)\\ &\geq -\Phi_p(y(1))y^-(1)+\sum_{k=1}^{T}|\Delta y^-(k)|^p\\ &=|y^-(1)|^p+\sum_{k=2}^{T+1}|\Delta y^-(k-1)|^p, \end{aligned} so $\Delta y^-(k)=0$, $k\in[1, T]$ and $y^-(1)=0$, which yield that $y^-(k)=0, k\in[1, T+1]$; that is, $y\ge0$. If $y(k)= 0$ for every $k\in[0, T+1]$, the fact $f(k, 0)\not\equiv0$ for every $k\in [1, T]$ gives a contradiction. \end{proof} \begin{remark}\label{rem1}\rm By Lemma \ref{lem1}, to find positive solutions of \eqref{000} it suffices to obtain solutions of \eqref{eq1+}. \end{remark} For $y\in Y$, put $$\varphi(y) :=\sum_{k=1}^{T+1}\big[\frac{1}{p}|\Delta y(k-1)|^p-F(k, y^+(k)) +f(k, 0)y^-(k)\big].\label{phi}$$ Clearly, the functional $\varphi$ is $C^1$ with $$\langle\varphi'(y), z\rangle =\sum_{k=1}^{T+1}\left[\Phi_p(\Delta y(k-1))\Delta z(k-1)-f(k,y^+(k))z(k)\right]\label{phi'}$$ for every $z\in Y$. So the solutions of \eqref{eq1+} are precisely the critical points of the functional $\varphi$. \begin{lemma}\label{inequality} For $y\in Y$, we have $\|y\|_{\infty}\le (T+1)^{1/q}\|y\|$, where $\|y\|_{\infty}=\max_{i\in[0, T+1]}|y(i)|.$ \end{lemma} \begin{proof} For $y\in Y$, it follows from H\"older's inequality, that \begin{align*} |y(k)|&=\big|y(0)+\sum_{i=0}^{k-1}\Delta y(i)\big| \le\sum_{i=0}^{T}|\Delta y(i)|\\ &\le(T+1)^{1/q}\Big(\sum_{i=0}^{T}|\Delta y(i)|^p\Big)^{1/p}=(T+1)^{1/q}\|y\|, \end{align*} which completes the proof. \end{proof} \begin{lemma}[{\cite[Theorem 38.A]{z1}}] \label{th38.a} For the functional $F: M\subseteq X\to[-\infty, +\infty]$ with $M\neq \emptyset$, $\min_{u\in M}F(u)=\alpha$ has a solution when the following conditions hold: \begin{itemize} \item[(i)] $X$ is a real reflexive Banach space; \item[(ii)] $M$ is bounded and weak sequentially closed; i.e., by definition, for each sequence $(u_n)$ in $M$ such that $u_n\rightharpoonup u$ as $n\to\infty$, we always have $u\in M$; \item[(iii)] $F$ is weak sequentially lower semi-continuous on $M$. \end{itemize} \end{lemma} \begin{lemma}[\cite{g1}]\label{guo} Let $E$ be a Banach space and $\varphi\in C^1(E, R)$ satisfy Palais-Smale condition. Assume there exist $x_0, x_1\in E$, and a bounded open neighborhood $\Omega$ of $x_0$ such that $x_1\not\in \overline{\Omega}$ and $\max\{\varphi(x_0), \varphi(x_1)\} <\inf_{x\in \partial\Omega}\varphi(x).$ Let $\Gamma=\{h:\, h:[0, 1]\to E \text{ is continuous}, h(0)=x_0, h(1)=x_1\}$ and $c=\inf_{h\in \Gamma}\max_{s\in[0, 1]}\varphi(h(s)).$ Then $c$ is a critical value of $\varphi$; that is, there exists $x^*\in E$ such that $\varphi'(x^*)=\Theta$ and $\varphi(x^*)=c$, where $c>\max\{\varphi(x_0), \varphi(x_1)\}$. \end{lemma} \begin{lemma}\label{ps} Suppose that {\rm (C1), (C2)} hold. Furthermore, we assume \begin{itemize} \item[(C3)] $(T+1)^{p/q}\sum_{k=1}^{T+1}d(k)<\frac{\mu}{p}-1$. \end{itemize} Then the functional $\varphi$ satisfies Palais-Smale condition; i.e., every sequence $\{y_n\}$ in $Y$ satisfying $\varphi(y_n)$ is bounded and $\varphi'(y_n)\to 0$ has a convergent subsequence. \end{lemma} \begin{proof} Since $Y$ is a finite dimensional Banach space, we only need to show that $(y_n)$ is a bounded sequence in $Y$. For this, by Lemma \ref{lem0} (iv) and \eqref{phi'} we have \label{23} \begin{aligned} \langle \varphi'(y_n),y_{n}^{-}\rangle &= \sum_{k=1}^{T+1}\left[\Phi_p(\Delta y_n(k-1))\Delta y_{n}^{-}(k-1)-f(k, y_{n}^{+}(k))y_{n}^{-}(k)\right]\\ &\leq -\sum_{k=1}^{T+1}|\Delta y_{n}^{-}(k-1)|^p=-\|y_{n}^{-}\|^p. \end{aligned} Set $w_{n}^{-}=\frac{y_{n}^{-}}{\|y_{n}^{-}\|}$. Dividing by $\|y_{n}^{-}\|$ on the both sides of the above inequality, we have $\|y_{n}^{-}\|^{p-1}\le-\langle\varphi'(y_n), w_{n}^{-}\rangle\to 0\quad \text{as }n\to\infty.$ So $y_{n}^{-}\to 0$ in $Y$. Now we show that $(y_{n}^{+})$ is bounded. By \eqref{phi} \eqref{phi'} we have \label{34} \begin{aligned} \frac{\mu}{p}\|y_n\|^p-\|y_{n}^{+}\|^p &=\mu\varphi(y_n)-\langle\varphi'(y_n), y_{n}^{+}\rangle-\sum_{k=1}^{T+1}\mu f(k,0)y_{n}^{+}(k)\\ &\quad +\sum_{k=1}^{T+1}\left[\mu F(k, y_{n}^{+}(k))- f(k, y_{n}^{+}(k))y_{n}^{+}(k)\right].\end{aligned} By (C1) (C2) Lemma \ref{inequality} one has \label{67} \begin{aligned} &\sum_{k=1}^{T+1}\left[\mu F(k, y_{n}^{+}(k))- f(k, y_{n}^{+}(k))y_{n}^{-}(k)\right]\\ &\le \sum_{k=1}^{T+1}\left[c(k)y_{n}^{+}(k)+d(k)|y_{n}^{+}(k)|^p\right]\\ &\le (T+1)^{1/q}\|y_{n}^{+}\|\sum_{k=1}^{T+1}c(k)+ (T+1)^{p/q}\|y_{n}^{+}\|^p\sum_{k=1}^{T+1}d(k). \end{aligned} Substituting \eqref{67} into \eqref{34}, in view of Lemma \ref{lem0} (ii), one has \label{67890} \begin{aligned} \big(\frac{\mu}{p}-1\big)\|y_{n}^{+}\|^p &\le\mu\varphi(y_n)-\langle\varphi'(y_n), y_{n}^{+}\rangle +(T+1)^{1/q}\|y_{n}^{+}\|\sum_{k=1}^{T+1}c(k)\\ &\quad + (T+1)^{p/q}\|y_{n}^{+}\|^p\sum_{k=1}^{T+1}d(k). \end{aligned} Suppose that $(y_{n}^{+})$ is unbounded. Passing to a subsequence, we may assume if necessary, that $\|y_{n}^{+}\|\to\infty$ as $n\to\infty$. Dividing the both sides of \eqref{67890} by $\|y_{n}^{+}\|^p$, denoting $w_{n}^{+}=\frac{y_{n}^{+}}{\|y_{n}^{+}\|}$, we have \label{e.29} \begin{aligned} \frac{\mu}{p}-1 &\le\frac{\mu\varphi(y_n)}{\|y_{n}^{+}\|^p}-\frac{\langle\varphi'(y_n), w_{n}^{+}\rangle}{\|y_{n}^{+}\|^{p-1}} +(T+1)^{1/q}\|y_{n}^{+}\|^{1-p}\sum_{k=1}^{T+1}c(k)\\ &\quad + (T+1)^{p/q}\sum_{k=1}^{T+1}d(k). \end{aligned} Since $\varphi(y_n)$ is bounded and $\varphi'(y_n)\to 0, y_{n}^{-}\to0$ in $Y$, let $n\to\infty$, we have $\frac{\mu}{p}-1\le(T+1)^{p/q}\sum_{k=1}^{T+1}d(k),$ which contradicts to (C3). Therefore, $(y_n)$ is bounded in $Y$. \end{proof} \section{Main Results} \begin{theorem}\label{thm1} Suppose that {\rm (C1)--(C3)} hold. Furthermore, we assume \begin{itemize} \item[(C4)] $(T+1)^{\frac{\mu}{q}}\sum_{k=1}^{T}b(k) +(T+1)^{1/q}\sum_{k=1}^{T}c(k) +(T+1)^{p/q}\sum_{k=1}^{T}d(k)<1$. \end{itemize} Then \eqref{000} has two positive solutions $x_0, x^*$. \end{theorem} \begin{proof} By Lemma \ref{ps} the functional $\varphi$ satisfies Palais-Smale condition. Now we shall show that there exists $R>0$ such that the functional $\varphi$ has a local minimum $x_0\in B_R:=\{x\in X: \|x\|R=1$ such that $\varphi(y_1)<\min_{y\in \partial B_R}\varphi(y)$. Let $\widetilde{e}(k)=1\in Y$. Then \label{2323} \begin{aligned} \varphi(\overline{\lambda}\widetilde{ e})&\leq -\sum_{k=1}^{T}[F(k, \overline{\lambda})-f(k, 0)\overline{\lambda}] \\ &=-\sum_{k=1}^{T}\big[\frac{l(k)\overline{\lambda}^{\mu}}{\mu}+H(k, \overline{\lambda})-f(k, 0)\overline{\lambda}\big]\\ &\leq -\sum_{k=1}^{T}\frac{l(k)\overline{\lambda}^{\mu}}{\mu}+\sum_{k=1}^{T}\left[ c(k)\overline{\lambda} +d(k)\overline{\lambda}^p+f(k, 0)\overline{\lambda}\right]. \end{aligned} Since $\mu>p$, we have $\lim_{\overline{\lambda}\to+\infty}\varphi(\overline{\lambda}\widetilde{e})=-\infty$. So there exists sufficiently large $\overline{\lambda}_0$ with $\|\overline{\lambda}_0\widetilde{e}\|>R$ such that $\varphi(\overline{\lambda}_0\widetilde{e})<\min_{y\in \partial B_R}\varphi(y)$. Lemma \ref{guo} now gives the critical value $c=\inf_{h\in \Gamma}\max_{t\in[0, 1]}\varphi(h(t)),$ where $\Gamma=\{h:\,h:[0, 1]\to E \text{ is continuous, } h(0)=y_0, h(1)=y_1\}$; that is, there exists $y^*\in Y$ such that $\varphi'(y^*)=0$. Therefore, $y_0, y^*$ are two critical points of $\varphi$, and hence they are classical solutions of \eqref{eq1+}. Lemma \ref{lem1} means $y_0, y^*$ are positive solutions of problem \eqref{000}. \end{proof} \begin{corollary} Suppose that {\rm (C1) (C4)} hold. Moreover we assume \begin{itemize} \item[(C2')] there exists \$0\le s