%\pdfoutput=1\relax\pdfpagewidth=8.26in\pdfpageheight=11.69in\pdfcompresslevel=9 \documentclass[reqno]{amsart} \usepackage{graphicx} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 41, 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 2004 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2004/41\hfil Existence of trivial and nontrivial solutions] {Existence of trivial and nontrivial solutions of a fourth-order ordinary differential equation} \author[Tihomir Gyulov \& Stepan Tersian\hfil EJDE-2004/41\hfilneg] {Tihomir Gyulov \& Stepan Tersian} % in alphabetical order \address{Tihomir Gyulov \hfill\break C.A.M.I., University of Rousse, 8 ``Studentska'' Str., 7017 Rousse, Bulgaria} \email{tgyulov@ecs.ru.acad.bg} \address{Stepan Tersian\hfill\break C.A.M.I., University of Rousse, 8 ``Studentska'' Str., 7017 Rousse, Bulgaria} \email{tersian@ami.ru.acad.bg} \date{} \thanks{Submitted January 5, 2004. Published March 23, 2004.} \subjclass[2000]{34B15, 34C25, 35K35} \keywords{Fourth-order ordinary differential equation, variational method, \hfill\break\indent Brezis-Nirenberg's theorem} \begin{abstract} We study the multiplicity of nontrivial solutions for a semilinear fourth-order ordinary differential equation arising in spatial patterns for bistable systems. In the proof of our results, we use minimization theorems and Brezis-Nirenberg's linking theorem. We obtain also estimates on the minimizers of the corresponding functionals. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \numberwithin{equation}{section} \section{Introduction} In this paper, we study existence and multiplicity of solutions to the boundary-value problem for the fourth-order ordinary differential equation \begin{equation} \label{P} \begin{gathered} u^{iv}+Au''+Bu+f(x,u)=0, \\ u(0)=u(L)=u''(0)=u''(L)=0, \end{gathered} \end{equation} \noindent where $A$ and $B$ are constants and $f(x,u)$ is a continuous function, defined in $\mathbb{R}^2$, whose potential $F(x,u)=\int_0^uf(x,t)dt$ satisfies suitable assumptions. The problem is motivated by the study of formation of spatial periodic patterns in bistable systems. In the study of spatial patterns an important role is played by a model equation, which is simpler than full equations describing the process. Recently, interest has turned to fourth-order parabolic differential equation, involving bistable dynamics, such as the extended Fisher-Kolmogorov (EFK) equation proposed by Coullet, Elphick \& Repaux in 1987 and Dee \& VanSaarlos in 1988. Another well known equation of this type is the Swift-Hohenberg (SH) equation proposed in 1977. With appropriate changes of variables, stationary solutions of these equations lead to the equation \begin{equation} u^{iv}-pu''-u+u^{3}=0, \label{e13} \end{equation} in which $p>0$ corresponds to EFK equation and $p<0$ to the SH equation. Solutions of Eq. (\ref{e13}) which are bounded on the real line have been recently studied by a variety of methods such as topological shooting method and variational methods \cite{b1,c1,c2,p2,p3,p4,t1}. When $f$ is an even $2L$ periodic function with respect to $x$, and odd with respect to $u$, the $2L$ periodic extension $\bar{u}$ of the odd extension of the solution $u$ of the problem \eqref{P} to the interval $[-L,L] $ yields a $2L$ periodic solution of \eqref{P}. The solvability of \eqref{P} for some extension of (\ref{e13}) was studied in \cite{c1,c2,p2,p3,t1} by variational methods. We suppose that $f(x,0)=0$, $\forall x\in\mathbb{R}$ and the potential \[ F(x,u)=\int_0^uf(x,s)\,ds \] satisfies following assumptions: \begin{itemize} \item[(H1)] There is a number $p>2$ and for each bounded interval $I$ there is a constant $c>0$ such that \[ F(x,u)\geq c|u| ^p,\ \forall x\in I , \forall u \in {\bf R} \] \item[(H2)] $F(x,u)=o(u^2)$ as $u\rightarrow 0$, uniformly with respect to $x $ in bounded intervals. \end{itemize} A typical function that satisfies (H1) and (H2) is \[ f(x,u)=b(x)u|u| ^{p-2},\quad p>2, \] where $b(x)$ is a continuous, positive function. Problem \eqref{P} has a variational structure and its solutions can be found as critical points of the functional \begin{equation} I(u;L):=\frac 12\int_0^L(u''{}^2 -Au'^2+Bu^2)dx+\int_0^LF(x,u)dx \label{e16} \end{equation} in the Sobolev space \[ X(L):=H^2(0,L)\cap H_0^1(0,L). \] In this work we obtain nontrivial critical points of the functional $I$ using Brezis-Nirenberg's linking theorem \cite{b2,g1}. Recall its statement. Let $E$ be a Banach space with a direct sum decomposition $ E=X\oplus Y$. The functional $J\in C^1(E,\mathbb{R})$ has a local linking at $0$ if, for some $r>0$ \begin{gather*} J(x)\leq 0, \quad x\in X, \quad \| x\| \leq r\,, \\ J(y)\geq 0, \quad y\in Y, \quad \| y\| \leq r\,. \end{gather*} \begin{theorem}[Brezis and Nirenberg \cite{b2}] \label{thm1} Suppose that $J\in C^1(E,\mathbb{R})$ satisfies the $(PS)$ condition and has a local linking at 0. Assume that $J$ is bounded below and $\inf_EJ<0$. Then $J$ has at least two nontrivial critical points. \end{theorem} It is easy to see that if $4B\geq A^2$ and $f(x,u)u>0$ for $x\geq0$ and $ u\neq0$ the problem \eqref{P} has only the trivial solution. We shall assume $ 4B0$ (SH equation). Our main results are as follows. \begin{theorem}[Nontrivial solutions] \label{thm2} Let the function $F(x,u)$ satisfy (H1) and (H2). \begin{itemize} \item[(i)] Let $4BL_1$, then problem \eqref{P} has at least two nontrivial solutions. \item[(ii)] Let $4B0$, and set $L_1:=\pi \sqrt{2}/\sqrt{A+\sqrt{A^2-4B}}$. Then, problem \eqref{P} has at least two nontrivial solutions if either \begin{itemize} \item[(a)] $B\leq 0$ and $L>L_1$, or \item[(b)] $B>0 $, and $L\in ]nL_1,nM_1[$, where $M_1:=\pi \sqrt{2}/\sqrt{A-\sqrt{A^2-4B}}$. \end{itemize} \end{itemize} \end{theorem} \begin{theorem}[trivial solutions] \label{thm3} Let the continuous function $f(x,u)$ satisfy the assumption $f(x,0)=0$ and $f(x,u)u>0, u\neq 0$ for $x\in [0,L]$. \begin{itemize} \item[(i)] Let $4B0$, set $L_1:=\pi \sqrt{2}/\sqrt{A+\sqrt{A^2-4B}}$ and $$ h_n=\big(\frac{(n^2+n)A}{2n^2+2n+1}\big)^2, $$ $n\in \mathbb{N}\cup \{0\}$. Then problem \eqref{P} has only the trivial solution provided that one of the following holds: (a) $B \leq 0$ and $00$, where $L_1$ and $M_1$ are depended on $A$ and $B$. One of these nontrivial solutions is a nontrivial minimizer $u_0$ of the functional $J$. In this section we will estimate the average of $L^2$-norm of the minimizer $u_0$. Let for $u\in X(L)$, let \[ |u|^2:=\frac{1}{L}\int_0^{L}u^2(x)dx. \] Let $P(\xi )=\xi ^{4}-A\xi ^2+B$ be the symbol of the linear operator $\mathcal{L}u=u^{iv}+Au''+Bu$. By the proof of Theorem \ref{thm2}, if $L\in \Delta _n$, where $\Delta _n$ is an interval which is the set of solutions of the inequality $P_n(L)<0$, \eqref{Q} has at least two nontrivial solutions. Moreover if $L\in \Delta_n$, there exist natural numbers $m,m+1,\ldots ,m+k$, $m\geq 1,k\geq 0$ depending on $L$ such that $P_{j}(L)<0$ if and only if $j\in S=\{m,m+1,\ldots ,m+k\} $ and $P_{j}(L)\geq 0$ if and only if $j\notin S$. Let $E_{k+1}(L)$ be the finite dimensional subspace of $X(L)$ \[ E_{k+1}(L)=\mathop{\rm span}\big\{ \sin (\frac{m\pi x}{L}) ,\ldots ,\sin (\frac{(m+k)\pi x}{L})\big\} , \] and for $u\in X(L)$, $u=\overline{u}+\widetilde{u}$, $\overline{u}\in E_{k+1}$, $\widetilde{u}\in E_{k+1}^{\bot }$ be the orthogonal decomposition of $u$. \begin{theorem} \label{thm4} Let for a fixed $n\in \mathbb{N}$, let $\Delta _n$ be the set of solutions of the inequality $P_n(L)<0$. For $L\in \Delta _n$ let \[ P_{j}(L)<0\quad \mbox{if }j\in S=\{ m,m+1,\dots,m+k\} , \] and \[ p_n=P_{m_n}(L)=\min \{ P_{j}(L): j\in S\} <0. \] Then, if $L\in \Delta _n$, problem \eqref{Q} has a nontrivial solution $u_0$, which is a minimizer of the functional $J$, and the following estimates hold: \begin{itemize} \item[(i)] $-\frac 14p_n^2\leq J(u_0;L)\leq -\frac 16p_n^2$ \item[(ii)] $\frac 23|p_n|\leq |\overline{u}_0|^2\leq |u_0|^2\leq |p_n|$ \item[(iii)] $|\widetilde{u}_0|^2\leq (-\frac 23+\sqrt{\frac 23})|p_n|$ \item[(iv)] $J_1(\widetilde{u}_0;L)\leq \frac 29p_n^2$. \end{itemize} \end{theorem} This paper is organized as follows: In Section 2 we prove some auxiliary lemmas. In Section 3 we prove Theorem \ref{thm2}, and Theorem \ref{thm3}. In Section 3 we prove Theorem \ref{thm4}. \section{Preliminaries} We study the nonautonomous fourth-order ordinary differential equation \[ u^{iv}+Au''+Bu+f(x,u)=0,\quad 0\frac 12\big(B-\frac{A^2}4\big) \Vert u\Vert_{L^2}^2+C_1(L)\Vert u\Vert _{L^2}^p. \label{e24} \end{equation} From the elementary inequality \[ -ax^2+bx^p\geq -a\frac{p-2}p(\frac{2a}{pb})^{\frac 2{p-2}} \] for $a>0$, $b>0$, $x>0$ and $p>2$, it follows that the right hand side of (\ref{e24}) is bounded from below by a negative constant. Suppose now that $(u_n)_n$ is a $(PS)$ sequence, i.e. there exists $c_1>0$ such that \begin{equation} c_1>| I(u_n;L)| \quad \mbox{and} \quad I' (u_n;L)\rightarrow 0. \label{e27} \end{equation} In what follows $c_{j}$ will denote various positive constants. We have \[ I(u;L)=\frac 14\int_0^Lu''{}^2 dx+\frac 12 \bar{I}(u;L), \] where \[ \bar{I}(u;L)=\frac 12\int_0^L(u''{}^2 -2Au'^2 +2Bu^2)dx+2\int_0^LF(x,u)dx. \] As before the functional $\bar{I}$ is bounded from below and we have \[ c_1\geq \frac{1}{4}\int_0^{L}u_n''{}^2 dx-c_{2}. \] The sequence $(u_n)_n$ is a bounded sequence in $X( L)$ in view of Lemma \ref{lm1}. There exists a subsequence still denoted by $ (u_n)_n$ and a function $u_0\in X(L)$ such that \begin{equation} u_n\rightharpoonup u_0\quad \mbox{in} \quad X(L), \label{e25} \end{equation} and by Sobolev's embedding theorem \begin{equation} \label{e26} \begin{gathered} u_n \rightarrow u_0\quad \mbox{in} \quad C^{1}[0,L], \\ u_n \rightarrow u_0\quad \mbox{in} \quad L^2(0,L). \end{gathered} \end{equation} Since $f(x,u)$ is continuous and $\{ |u_n( x)|\} $ uniformly bounded in $[0,L]$, and letting $ n\rightarrow \infty $ in \[ (I'(u_n;L),u_0)=\int_0^L( u_n''u_0''-Au_n'u_0' +Bu_nu_0+f(x,u_n)u_0)dx \] we obtain \begin{equation} \int_0^L(u_0''{}^2 -Au_0'^2 +Bu_0^2+f(x,u_0)u_0)dx=0. \label{e28} \end{equation} From the boundedness of $(u_n)_n$ in $X(L)$ and (\ref{e28}) it follows $(I'(u_n;L),u_n)\rightarrow 0$ and \begin{align*} \int_0^{L}u_n''{}^2 dx &=(I'(u_n),u_n)+\int_0^{L}(Au_n'^2-Bu_n^2-f(x,u_n)u_n)dx \\ &\rightarrow \int_0^{L}(Au_0'^2-Bu_0^2-f(x,u_0)u_0)dx =\int_0^{L}u_0''{}^2 dx, \end{align*} which implies that $\| u_n\|\rightarrow \| u_0\|$ and then $\| u_n-u_0\|\rightarrow 0$, which completes the proof of Lemma \ref{lm2}. \end{proof} \section{Existence results} The polynomial \[ p(\xi )=\xi ^4-A\xi ^2 \nonumber \] and the real functions \[ p_n(L)=p(\frac{n\pi }L) \nonumber \] play an important role in the sequel. Let $A\leq 0$. The polynomial $p(\xi )$ is a positive increasing and convex function for $\xi >0$. The functions $p_n(L)$ are positive decreasing functions for every $n\in \mathbb{N}$ and \begin{gather*} p_n(L)\rightarrow +\infty ,\quad \mbox{as } L\rightarrow 0,\\ p_n(L)\rightarrow 0,\quad \mbox{as}{\quad }L\rightarrow +\infty . \end{gather*} These functions are ordered as \[ 00$, and some of their graphs are showm in Figure 1. \begin{figure}[ht] \begin{center} \includegraphics[width=0.5\textwidth]{fig1.eps} % figure=gst11.eps \end{center} \caption{Graphs of functions $p_n(L)= (\frac{n\pi }L)^4+ (\frac{n\pi }L)^2$, $n=1,2,3,4$} \end{figure} Let $B<0$. Then the equation $p_n(L)+B=0$ has the unique solution \begin{equation} L_n=nL_1,\quad L_1:=\frac{\pi \sqrt{2}}{\sqrt{A+\sqrt{A^2-4B}}} \label{e32} \end{equation} and \begin{gather} p_n(L)+B \geq 0\quad \mbox{if }L\leq nL_1,\label{f1} \\ p_n(L)+B < 0\quad \mbox{if } L>nL_1. \label{f2} \end{gather} Let $A>0$. Then the polynomial $p(\xi )=\xi ^{4}-A\xi ^2$ is positive for $\xi >\sqrt{A}$ and it has a negative minimum $p_0=-A^2/4$ at $\xi _0=\sqrt{A/2}$. The functions $p_n(L)$ are decreasing if $0n\pi \sqrt{2/A}$, $p_n(L)>0$ if $0n\pi /\sqrt{A}$. The graphs of functions $p_n(L)$ with $A=1$ and $n=1,2,3,4$ are presented on Figure 2. \begin{figure}[ht] \begin{center} \includegraphics[width=0.5\textwidth]{fig2.eps} % gst22.eps \end{center} \caption{Graphs of functions $p_n(L)= (\frac{n\pi }L)^4- (\frac{n\pi }L)^2$, $n=1,2,3,4$} \end{figure} \begin{lemma} \label{lm3} Let $l_n:=\frac \pi {\sqrt{A}}\sqrt{2n^2+2n+1}$ and $L_1:=\pi \sqrt{2}/\sqrt{A+\sqrt{A^2-4B}}$. Then we have the following results: \begin{itemize} \item[(a)] \begin{equation} \label{e41} \begin{aligned} p_n(L)=p_{n+1}(L)\Leftrightarrow L=l_n,\\ p_n(L)p_{n+1}(L)\Leftrightarrow L>l_n, \end{aligned} \end{equation} and \begin{equation} q(L)=\inf \{ p_n(L): n\in \mathbb{N} \} = \begin{cases} p_1(L),& 0nL_1$. \item[(c)] Let $B>0$ and $M_1:=\pi \sqrt{2}/\sqrt{A-\sqrt{A^2-4B}}$. Then $p_n(L)+B<0$ if and only if $nL_1nL_1$. \noindent (c) To solve the same inequality in the case $B>0$ we compute \[ (nx)^2\in \big] \frac{A-\sqrt{A^2-4B}}2,\frac{A+\sqrt{A^2-4B}}2\big[ \] which is equivalent to $nL_1L_1$. There exists a natural number $n$ such that $nL_10$ such that if $|u|\leq \delta $ then $F(x,u)\leq \varepsilon |u|^2$, $x\in[ 0,L] $. Let us take $\rho ,0<\rho \leq \delta /\sqrt{n}$. Then by \[ |\varphi _n(x)|\leq \sum_{k=1}^n|c_k|\leq \sqrt{n}( \sum_{k=1}^nc_k^2)^{1/2}=\sqrt{n}\rho \leq \delta \] it follows that $F(x,\varphi _n(x))\leq \varepsilon |\varphi_n(x)|^2$ and \begin{align*} \int_0^LF(x,\varphi _n(x))dx &\leq \varepsilon \int_0^L|\varphi _n(x)|^2dx\\ &=\varepsilon \int_0^L\sum_{k=1}^nc_k^2\sin ^2(\frac{k\pi x}L)dx\\ &=\varepsilon \frac L2\sum_{k=1}^nc_k^2=\varepsilon \frac L2\rho ^2. \end{align*} We have \begin{equation} \label{e34} \begin{aligned} I(\varphi _n;L) &=\frac L4\sum_{k=1}^n(p_k(L)+B)c_k^2 +\int_0^LF(x,\varphi _n(x))dx \\ &\leq \frac L4\alpha _n\rho ^2+\varepsilon \frac L2\rho ^2 \\ &=\frac L2\rho ^2(\frac 12\alpha _n+\varepsilon )<0, \end{aligned} \end{equation} if $0<\rho \leq \delta /\sqrt{n}$. The functional $I$ has a local linking at $0$. Indeed, by (\ref{e34}), for sufficiently small $\rho >0$ we have \[ I(u;L)\leq0,\quad u\in E_n,\|u\|<\rho . \] Let $u\in E_n^{\perp }$ and $\|u\|\leq \rho $. It follows that $ p_{n+1}(L)+B\geq 0$ if $nL_10$ such that \begin{align*} I(u;L)&\geq \frac 12\min ((p_k(L)+B):k\geq n+1)\Vert u\Vert_{L^2}^2 +C(L)\Vert u\Vert _{L^2(0,L)}^p \\ &\geq \frac 12(p_{n+1}(L)+B)\Vert u\Vert_{L^2}^2+C(L)\Vert u\Vert _{L^2(0,L)}^p \geq 0, \end{align*} if $u\in E_n^{\perp }$. The functional $I$ satisfies the $(PS)$ condition. In view of Theorem \ref{thm1}, for $L>L_1$ the functional $I$ has at least two nontrivial critical points. \noindent(ii). By Lemma \ref{lm3}, (b) $p_k(L)+B<0$ iff $ L>kL_1$. If $L>L_1$ there exists a natural number $n$ such that $nL_10$.} Let $\Delta _n=] nL_1,nM_1[ $. \medskip\ Observe that for a fixed $L\in \Delta _n$ there exist finite number of intervals $\Delta _j$ numbered as $\Delta _m, \Delta _{m+1},\dots ,\Delta _{m+k}$ such that $L\in \Delta _j \cup \Delta _n $ if and only if $j\in S:=\{m, m+1,\dots ,m+k \}$ and \begin{gather*} p_j(L)+B<0,\quad j\in S,\\ p_j(L)+B\geq 0,\quad j\notin S. \end{gather*} Let \[ E_{k+1}:=\mathop{\rm span}\big\{ \sin \big(\frac{m\pi x}L\big), \sin \big(\frac{(m+1)\pi x}L\big),\dots ,\sin \big(\frac{(m+k) \pi x}L\big)\big\} . \] With a computation similar to the one in the proof of Theorem \ref{thm3} we observe that \[ I(u,L)<0,\quad u\in E_{k+1},\; 0<\|u\|\leq r, \] if $r$ is sufficiently small and \[ I(u,L)\geq 0,\quad u\in E_{k+1}^{\bot }, \] which implies that $I$ has a local linking at $0$. Then $I$ has at least two nontrivial critical points. \end{proof} \begin{proof}[Proof of Theorem \ref{thm3}] By Lemma \ref{lm1}, for $u\in X(L)$ we have: \begin{gather*} u=\sum_{k=1}^\infty c_k\sin \frac{k\pi x}L, \\ I(u;L)=\frac L4\sum_{k=1}^\infty c_k^2P(\frac{k\pi}L)+\int_0^LF(x,u)dx, \\ \mathcal{B}(u,u):=\langle I'(u;L),u\rangle =\frac L2 \sum_{k=1}^\infty c_k^2P(\frac{k\pi }L) +\int_0^Lf(x,u)u \, dx, \end{gather*} where $P(\xi )=\xi ^4-A\xi ^2+B=p(\xi)+B$ is the symbol of the linear differential operator \[ \mathcal{L}(u):=u^{i v}+Au''+Bu. \] \noindent {Case (i).} Let $B<0$ and $00$ if $u\neq 0$ which means that the functional $I$ has only the trivial critical point. If $B\geq0$ the same argument applies for every $L>0$. \noindent {Case (ii).} We consider the solvability of the inequality \begin{equation} q(L)+B\geq 0, \label{e44} \end{equation} where $q(L)= \inf \{ p_n(L):n\in \mathbb{N}\} $. Let $00$ if $u\neq 0$. Then the functional $I$ has only the trivial critical point which completes the proof of Theorem \ref{thm3}. \end{proof} \section{Bounds for the minimizer} Let us consider the problem \begin{gather*} u^{iv}+Au''+Bu+u^3=0,\quad 0