\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 27, 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 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/27\hfil Non-trivial solutions] {Non-trivial solutions for two-point boundary-value problems of fourth-order Sturm-Liouville type equations} \author[S. Heidarkhani \hfil EJDE-2012/27\hfilneg] {Shapour Heidarkhani} \address{Shapour Heidarkhani \newline Department of Mathematics, Faculty of Sciences\\ Razi University, 67149 Kermanshah, Iran \newline School of Mathematics, Institute for Research in Fundamental Sciences (IPM)\\ P.O. Box: 19395-5746, Tehran, Iran} \email{sh.heidarkhani@yahoo.com, s.heidarkhani@razi.ac.ir} \thanks{Submitted September 12, 2011. Published February 15, 2012.} \subjclass[2000]{35J35, 47J10, 58E05} \keywords{Fourth-order Sturm-Liouville type problem; multiplicity results; \hfill\break\indent critical point theory} \begin{abstract} Using critical point theory due to Bonanno \cite{B}, we prove the existence of at least one non-trivial solution for a class of two-point boundary-value problems for fourth-order Sturm-Liouville type equations. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{example}[theorem]{example} \section{Introduction} In this note, we prove the existence of at least one non-trivial solution for the two-point boundary-value problem of fourth-order Sturm-Liouville type: \begin{gather}\label{e1} (p_i(x) u_i''(x))'' - (q_i(x) u_i'(x))' + r_i(x) u_i(x) =\lambda F_{u_i}(x,u_1,\dots,u_n) \quad x \in (0,1), \notag\\ u_{i}(0)=u_{i}(1)=u_{i}''(0)=u_{i}''(1)=0 \end{gather} for $1\leq i\leq n$, where $n\geq 1$ is an integer, $p_i$, $q_i$, $r_i \in L^{\infty}([0,1])$ with $p_i^-:= \operatorname{ess\,inf}_{x \in [0,1]} p_i(x)> 0$ for $1\leq i\leq n$, $\lambda$ is a positive parameter, $F:[0,1]\times \mathbb{R}^n\to \mathbb{R}$ is a function such that $F(.,t_1,\dots,t_n)$ is measurable in $[0,1]$ for all $(t_1,\dots,t_n)\in \mathbb{R}^n$, $F(x,.,\dots,.)$ is $C^1$ in $\mathbb{R}^n$ for every $x\in [0,1]$ and for every $\varrho>0$, $$\sup_{|(t_1,\dots,t_n)|\leq \varrho} \sum_{i=1}^n |F_{t_i}(x,t_1,\dots,t_n)|\in L^1([0,1]),$$ and $F_{u_{i}}$ denotes the partial derivative of $F$ with respect to $u_{i}$ for $1\leq i\leq n$. Due to importance of fourth-order two-point boundary-value problems in describing a large class of elastic deflection, many authors have studied the existence and multiplicity of solutions for such a problem; we refer the reader to \cite{AHO,Ba,BD1,BD2,BD3,BDO,C,L,WHL} and references therein. In \cite{BD1}, the authors, employing a three critical point theorem due to Bonanno and Marano \cite[Theorem 2.6]{BM}, determined an exact open interval of the parameter $\lambda$ for which system \eqref{e1} in the case $n=1$, admits at least three distinct weak solutions. The aim of this article is to prove the existence of at least one non-trivial weak solution for \eqref{e1} for appropriate values of the parameter $\lambda$ belonging to a precise real interval, which extend the results in \cite{BDO}. Our motivation comes from the recent paper \cite{BD2}. For basic notation and definitions, and also for a thorough account on the subject, we refer the reader to \cite{BMO,GR,KRV}. \section{Preliminaries and basic notation} First we recall for the reader's convenience \cite[Theorem 2.5]{R} as given in \cite[Theorem 5.1]{B} (see also \cite[Proposition 2.1]{B} for related results) which is our main tool to transfer the question of existence of at least one weak solution of \eqref{e1} to the existence of a critical point of the Euler functional: For a given non-empty set $X$, and two functionals $\Phi,\Psi:X\to\mathbb{R}$, we define the following two functions: \begin{gather*} \beta(r_1,r_2)=\inf_{v\in \Phi^{-1}(]r_1,r_2[)} \frac{\sup_{u\in \Phi^{-1}(]r_1,r_2[)}\Psi(u)-\Psi(v)}{r_2-\Phi(v)}, \\ \rho(r_1,r_2)=\sup_{v\in \Phi^{-1}(]r_1,r_2[)} \frac{\Psi(v)-\sup_{u\in \Phi^{-1}(]-\infty,r_1[)}\Psi(u)}{\Phi(v)-r_1} \end{gather*} for all $r_1,r_2\in\mathbb{R}$, $r_1 - p_i^-, where $$p_i^- := \operatorname{ess\,inf}_{x \in [0,1]} p_{i}(x) > 0,\quad q_i^- := \operatorname{ess\,inf}_{x \in [0,1]} q_i(x), \quad r_i^- := \operatorname{ess\,inf}_{x \in [0,1]} r_i(x),$$ for$1\leq i\leq n$. Moreover, set $\sigma_i := \min \big\{ \frac{q_i^-}{\pi^2}, \frac{r_i^-}{\pi^4}, \frac{q_i^-}{\pi^2} + \frac{r_i^-}{\pi^4},0 \big\}, \quad \delta_i := \sqrt{ p_i^- + \sigma_i},$ for$1\leq i\leq n$. Let$Y:= H^{2}([0,1])\cap H_0^{1}([0,1])$be the Sobolev space endowed with the usual norm. We recall the following Poincar\'{e} type inequalities (see, for instance, \cite[Lemma 2.3]{PTV}): \begin{gather}\label{e3} \|u_i'\|_{L^2([0,1])}^2 \leq \frac{1}{\pi^2} \|u_i''\|_{L^2([0,1])}^2,\\ \label{e4} \|u_i\|_{L^2([0,1])}^2 \leq \frac{1}{\pi^4} \|u_i''\|_{L^2([0,1])}^2 \end{gather} for all$u_i \in Y$for$1\leq i\leq n$. Therefore, taking into account \eqref{e2}-\eqref{e4}, the norm $$\|u_i\|= \Big( \int_0^1 ( p_i(x) |u_i''(x)|^2 + q_i(x) |u_i'(x)|^2 + r_i(x) |u_i(x)|^2) dx \Big)^{1/2}$$ for$1\leq i\leq n$is equivalent to the usual norm and, in particular, one has $$\label{e5} \|u_i''\|_{L^2([0,1])} \leq \frac{1}{\delta_i} \|u_i\|$$ for$1\leq i\leq n$. We need the following proposition in the proof of Theorem \ref{thm2}. \begin{proposition}[{\cite[Proposition 2.1]{BD1}}] \label{prop1} Let$u_i \in Y$for$1\leq i\leq n$. Then $$\|u_i\|_{\infty} \leq \frac{1}{2 \pi \delta_i} \|u_i\|$$ for$1\leq i\leq n$. \end{proposition} Put $$k_i:= \Big( \|p_i\|_\infty + \frac{1}{\pi^2} \|q_i\|_\infty + \frac{1}{\pi^4}\|r_i\|_\infty \Big)^{1/2}$$ for$1\leq i\leq n$. It is easy to see that$k_i > 0$and$ \delta_i < k_i$for$1\leq i\leq n$. Set$\underline{\delta}:=\min\{\delta_{i};\ 1\leq i\leq n\}$and$\overline{k}:=\max\{k_{i};\ 1\leq i\leq n\}$. Here and in the sequel,$X:=Y\times\dots\times Y$. We say that$u=(u_1,\dots,u_n)$is a weak solution to the \eqref{e1} if$u=(u_1,\dots,u_n)\in Xand \begin{align*} &\sum_{i=1}^n\int_0^1 \left( p_i(x) u_i''(x) v_i''(x)+q_i(x)u_i'(x)v_i'(x)+r_i(x)u_i(x)v_i(x)\right)dx\\ &-\lambda\sum_{i=1}^n\int_0^1 F_{u_i}(x,u_1,\dots,u_n)v_i(x)dx=0 \end{align*} for everyv=(v_1,\dots,v_n)\in X$. For$\gamma>0$we denote the set $$\label{e6} K(\gamma)=\big\{(t_1,\dots,t_n)\in \mathbb{R}^n:\ \sum_{i=1}^n|t_i|\leq \gamma\big\}.$$ \section{Results} For a given non-negative constant$\nu$and a positive constant$\tau$with$2(\frac{\underline{\delta}\pi\nu}{n})^{2}\neq \frac{4096}{54}n(\overline{k}\tau)^2$, put $$a_\tau(\nu):=\frac{\int_0^{1}\sup_{(t_1,\dots,t_n)\in K(\nu)}F(x,t_1,\dots,t_n)dx-\int_{\frac{3}{8}}^{\frac{5}{8}}F(x,\tau,\dots,\tau)dx}{2(\frac{\underline{\delta}\pi\nu}{n})^{2}- \frac{4096}{54}n(\overline{k}\tau)^2}$$ where$K(\nu)=\big\{(t_1,\dots,t_n)\in \mathbb{R}^n: \sum_{i=1}^n|t_i|\leq \nu\big\}$(see \eqref{e6}). We formulate our main result as follows: \begin{theorem} \label{thm2} Assume that there exist a non-negative constant$\nu_1$and two positive constants$\nu_2$and$\tau$with$\pi\nu_10}\frac{\nu^{2}}{\int_0^{\nu}g(\xi)d\xi} [$, the problem \begin{gather*} (p(x) u''(x))'' - (q(x) u'(x))' + r(x) u(x) =\lambda g(u) \quad x \in (0,1), \\ u(0)=u(1)=u''(0)=u''(1)=0 \end{gather*} admits at least one non-trivial weak solution in$Y$. \end{theorem} \begin{proof} For fixed$\lambda$as in the conclusion, there exists positive constant$\nu$such that $$\lambda<2(\delta\pi)^{2}\frac{\nu^{2}}{\int_0^{\nu}g(\xi)d\xi}.$$ Moreover,$\lim_{t\to 0^{+}}\frac{g(t)}{t}=+\infty$implies$\lim_{t\to 0^{+}}\frac{\int_0^{t}g(\xi)d\xi}{t^{2}}=+\infty$. Therefore, a positive constant$\tau$satisfying$\sqrt{\frac{4096}{108}}k\tau<\delta\pi\nu$can be chosen such that $$\frac{1}{\lambda}(\frac{4\times 4096}{54}k^2)<\frac{\int_0^{\tau}g(\xi)d\xi}{\tau^{2}}.$$ Hence, arguing as in the proof of Theorem \ref{thm3}, the conclusion follows from Theorem \ref{thm5} with$\nu_1=0$,$\nu_2=\nu$and$f(t)=g(t)$for every$t\in \mathbb{R}$. \end{proof} \begin{remark}\label{rmk1} \rm For fixed$\rho$put$\lambda_\rho:=2(\delta\pi)^{2}\sup_{\nu\in]0,\rho[} \frac{\nu^{2}}{\int_0^{\nu}g(\xi)d\xi}$. The result of Theorem \ref{thm6} for every$\lambda\in]0,\lambda_\rho[$holds with$|u_0(x)|<\rho$for all$x\in [0,1]$where$u_0$is the ensured non-trivial weak solution in$Y$(see \cite[Remark 4.3]{BDO}). \end{remark} We close this article by presenting the following examples to illustrate our results. \begin{example} \rm Consider the problem $$\label{e9} \begin{gathered} (3e^xu'')''-((x^2-\pi^2)u')'+(x^2-\pi^4) u =\lambda (1+e^{-u^+}(u^+)^2(3-u^+))\quad x\in(0,1),\\ u(0)=u(1)=0,\quad u''(0)=u''(1)=0 \end{gathered}$$ where$u^+=\max\{u,0\}$. Let $$g(t)=1+e^{-t^+}(t^+)^2(3-t^+)$$ for all$t\in\mathbb{R}$where$t^+=\max\{t,0\}$. It is clear that$\lim_{t\to 0^{+}}\frac{g(t)}{t}=+\infty$. Note that$p^-=3$,$q^-=-\pi^2$and$r^-=-\pi^4$, we have$\sigma=-2$, and so$\delta=1$. Hence, taking Remark \ref{rmk1} into account, since$\sup_{\nu\in]0,1[}\frac{\nu^{2}}{\int_0^{\nu}g(\xi)d\xi}=\sup_{\nu\in]0,1[} \frac{\nu^{2}}{\nu+e^{-\nu}\nu^3}=\frac{e}{1+e}$, by applying Theorem \ref{thm6}, for every$\lambda\in]0,\frac{2\pi^2e}{1+e}[$the problem \eqref{e9} has at least one non-trivial classical solution$u_0\in Y$such that$\|u_0\|_{\infty}<1$. \end{example} \begin{example} \rm Put$p(x)=1$,$q(x)=\pi^2$,$r(x)=x-\pi$for all$x\in[0,1]$and$g(t)=(1+t)e^t$for every$t\in\mathbb{R}$. Clearly, one has$\sigma=(1-\frac{1}{\pi^3})^{\frac{1}{2}}$. Hence, since $$\sup_{\nu\in]0,1[}\frac{\nu^{2}}{\int_0^{\nu}g(\xi)d\xi} =\sup_{\nu\in]0,1[}\frac{\nu^{2}}{\nu e^\nu}=\frac{1}{e},$$ from Theorem \ref{thm6}, taking Remark \ref{rmk1} into account, for every$\lambda\in]0,\frac{2\pi^2(1-\frac{1}{\pi^3})}{e}[$the problem $$\label{e10} \begin{gathered} u^{iv}-\pi^2u''+(x-\pi)u=\lambda (1+u)e^u\quad x\in(0,1),\\ u(0)=u(1)=0,\quad u''(0)=u''(1)=0 \end{gathered}$$ has at least one non-trivial classical solution$u_0\in Y$such that$\|u_0\|_{\infty}<1\$. \end{example} \subsection*{Acknowledgements} The author expresses his sincere gratitude to the referees for reading this paper very carefully and specially for valuable suggestions concerning improvement of the manuscript. This research was partially supported by a grant 90470020 from IPM. \begin{thebibliography}{99} \bibitem{AHO} G. A. Afrouzi, S. Heidarkhani, D. O'Regan; \emph{Existence of three solutions for a doubly eigenvalue fourth-order boundary value problem,} Taiwanese J. 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