\documentclass[twoside]{article} \usepackage{amsfonts} % used for R in Real numbers \pagestyle{myheadings} \markboth{ Existence of doubly periodic solutions } { Chen Chang?} \begin{document} \setcounter{page}{325} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent Nonlinear Differential Equations, \newline Electron. J. Diff. Eqns., Conf. 05, 2000, pp. 325--333\newline http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)} \vspace{\bigskipamount} \\ % A condition on the potential for the existence of doubly periodic solutions of a semi-linear fourth-order partial differential equation % \thanks{ {\em Mathematics Subject Classifications:} 35J30, 35B10. \hfil\break\indent {\em Key words:} periodic solutions, elliptic, fourth-order PDE. \hfil\break\indent \copyright 2000 Southwest Texas State University. \hfil\break\indent Published October 31, 2000. } } \date{} \author{ Chen Chang } \maketitle \begin{abstract} We study the existence of solutions to the fourth order semi-linear equation $$\Delta ^2u=g(u)+h(x)\,.$$ We show that there is a positive constant $C_*$, such that if $g(\xi )\xi \geq 0$ for $|\xi |\geq \xi _0$ and $\limsup _{|\xi |\to \infty } 2G(\xi )/\xi^20$ such that if $g\in \Sigma$ and (1.6) holds, then for all $h$ satisfying (1.5), $h\in L^2(Q)$, (1.7) has a weak solution. We have shown that if $$C_*={ \frac{1}{4\pi ^2a^2_*+1}}, \quad\mbox{where}\quad a^2_*={ \frac{1}{\pi ^2}} { \sum ^\infty _{i=1}} { \sum ^\infty _{j=1}} { \frac{1}{(i^2+j^2)^2}}$$ then this statement will be true. However, we feel that this is far from the optimal value of $C_*$. It is clear that the optimal value must be less than 1, since it can be shown that if $g(\xi )=\xi$, $h(x_1,x_2)=\sin x_1$, then (1.7) does not have a weak solution, because of resonance. \section{Definitions and preliminary lemmas} In this section we state some preliminary lemmas. These results follow more or less from known results (see for example [1]). Full details will be given elsewhere. Let $Q=\{ (x_1,x_2) | 0\leq x_1\leq 2\pi, 0\leq x_2\leq 2\pi \}$. Let $L^2_{2\pi }({\mathbb R} ^2)$ denote the set of real-valued measurable functions defined in ${\mathbb R} ^2$ such that if $u\in L^2_{2\pi } ({\mathbb R} ^2)$, then $u(x_1+2\pi ,x_2)=u(x_1,x_2+2\pi )=u(x_1,x_2)$ and such that $u$ restricted to $Q$ is in $L^2(Q)$. We denote $C_{2\pi }$ and $C^\infty _{2\pi }$ the real-valued functions defined on ${\mathbb R} ^2$ which are $2\pi$-periodic in each variable, which are continuous and of class $C^\infty$ respectively. We denote by $H^2_{2\pi }({\mathbb R} ^2)$ the set of $u\in L^2_{2\pi }({\mathbb R}^2)$ such that for $p=1,2$ there exists $v_p\in L^2_{2\pi }({\mathbb R}^2)$ such that for all $\phi \in C^\infty _{2\pi }$, $$-{ \int _Q} (D_p\phi )u dx ={ \int _Q}v_p\phi dx$$ and for $1\leq p$, $q\leq 2$ there exists $v_{pq}\in L^2_{2\pi }({\mathbb R}^2)$ such that for all $\phi \in C^\infty _{2\pi }$, $${ \int _Q}(D_pD_q\phi )udx ={ \int _Q} \phi v_{pq}dx.$$ Here $D_p=\partial /\partial x_p$, $p=1,2$. It is clear that $v_p$, $p=1,2$, and $v_{pq}$, $p,q=1,2$, are determined uniquely and we write $v_p=D_pu$, $p=1,2,$ and $v_{pq}=D_pD_qu$, $p,q=1,2$. The space $H^2_{2\pi }({\mathbb R} ^2)$ is a real Hilbert space with inner product given by $$\langle u,v\rangle = { \int _Q} \left[ uv +{ \sum ^2_{p=1}}(D_pu)(D_pv)+{ \sum^2_{p,q=1}} (D_pD_qu)(D_pD_qv)\right] dx$$ In the following we denote the Hilbert space $H^2_{2\pi }$ by ${\mathbb E}$ and $\|\cdot \| _{{\mathbb E} }$ will denote the norm given by the inner product defined above. \begin{lemma} % Lemma 2.1 If $u\in {\mathbb E}$ then $u$ is equal almost everywhere to a unique function in $C_{2\pi }$. If this function is again denoted by $u$, then there exists a constant $a_0$ such that for all $u\in {\mathbb E}$, $\| u\| _{C_{2\pi }}={ \max _{x\in {\mathbb R} ^2}}|u(x)|\leq a_0\| u\| _{{\mathbb E} }$. (see [1, p 167]). \end{lemma} We denote by $\hat{{\mathbb E} }$ the set of $u\in {\mathbb E}$ such that ${ \int _Q}udx=0$. The following result can be proved using multiple Fourier series. \begin{lemma} % Lemma 2.2 An inner product on $\hat{{\mathbb E} }$ which is equivalent to the ${\mathbb E}$-inner product is given by $$\langle u,v\rangle _{\hat{{\mathbb E}}}={ \int _Q}(\Delta u)(\Delta v) dx$$ where, as usual $\Delta u=D^2_1u+D^2_2u$.\end{lemma} \begin{lemma} %Lemma 2.3 The best possible constant $a_*$ such that for all $u\in \hat{{\mathbb E} }$, $$\| u\| _{ c_{2\pi}}= \max _{x\in {\mathbb R}^2} |u(x)|\leq a_* \|u\|_{\hat{\mathbb E}} \,,$$ where $\| u\|_{\hat{\mathbb E}}=\| \Delta u\|_{L^2(Q)}$, is $$a_*= \frac{1}{2\pi} \Big( \sum _{\stackrel{k\in {\bf Z}^2}{k\neq (0,0)}} \frac{1}{|k|^4}\Big) ^{1/2} \eqno (2.1)$$ it where ${\bf Z}^2={\bf Z}\times {\bf Z}$, ${\bf Z}=\{0,\pm 1,\pm 2, \pm 3,\ldots \}$, and if $k=(k_1,k_2)\in {\bf Z}^2$, $|k|=\sqrt{k^2_1+k^2_2}$. \end{lemma} This lemma and the next are proved using multiple Fourier series. \begin{lemma} % Lemma 2.4 If $u\in \hat{\mathbb E}$, then ${ \int _Q}u^2dx \leq { \int _Q}(\Delta u)^2dx.$ \end{lemma} The following result is proved using the idea of the proof given in [5, p. 216] except Fourier series are used instead of Fourier transform. \begin{lemma} % Lemma 2.5 Let $0<\alpha <1$. There exists $M(\alpha)$ such that if $u\in {\mathbb E}$, then for $x\in {{\mathbb R} }^2$ and $y\in {\mathbb R} ^2$ $$|u(x)-u(y)|\leq M_{(\alpha )}\| u\| _{\mathbb E}|x-y|^\alpha \,.$$ Here, for $x=(x_1,x_2)$ and $y=(y_1,y_2)$, $$|x-y|=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2}\,.$$ \end{lemma} The final preliminary lemma follows from Lemma 2.1, Lemma 2.5 and Ascoli's Lemma. \begin{lemma} % Lemma 2.6 The injection from ${\mathbb E}$ to $C_{2\pi }$ is compact, that is, if $\{u_n\}^\infty _{n=1}$ is a bounded sequence in ${\mathbb E}$, then there exists a subsequence $\{u_{n_i}\}^\infty _{i=1}$ such that $\{u_{n_i}\}^\infty _{i=1}$ converges uniformly on ${\mathbb R}^2$. \end{lemma} \section {Periodic solutions of a semi-linear elliptic fourth-order partial differential equation} In this section $g$ will always denote a real-valued function defined and continuous on ${\mathbb R}$, and $G$ will denote the function such that $G'(\xi)=g(\xi )$ for $\xi \in {\mathbb R}$ with $G(0)=0$. $\hat{L}^2_{2\pi}$ will denote the closed subspace of $L^2_{2\pi }({\mathbb R} ^2)$ such that for all $h\in\hat{L}^2_{2\pi }$, ${ \int _Qh(x)dx=0}$ . We consider the question of existence of {\it weak} solution of the problem $$\displaylines{ \hfill \Delta ^2u=g(u)+h(x) \hfill\llap(3.1)\cr u\in H^2_{2\pi}({\mathbb R}^2) }$$ where $h\in \hat{L}^2_{2\pi }$. This is defined to be a function $u\in H^2_{2\pi }({\mathbb R} ^2)$ such that for all $v\in {\mathbb E}$ $(=H^2_{2\pi}({\mathbb R}^2))$, $$\int _Q [(\Delta u)(\Delta v)-g(u)v-h(x)v]dx=0\,. \eqno(3.2)$$ If $u$ is a function of class $C^4$ which is $2\pi$-periodic in each variable, then (3.1) holds if and only if (3.2) holds. Let $f:{\mathbb E} \to {\mathbb R}$ be the function $$f(u)={ \int _Q}\left[ { \frac{|\Delta u|^2}{2}}-G(u)-h(x)u\right] dx\,.$$ Since ${\mathbb E}\subset C_{2\pi }$, standard arguments (see, for example, [4]) show that $f\in C^1$. For $v\in {\mathbb E}$, $$f'(u)(v)={ \int _Q}[(\Delta u)(\Delta v)-g(u)v-h(x)v]dx\,.$$ Therefore, weak solutions of (3.1) coincide with critical points of $f$. Let $\Sigma$ denote the set of continuous $g:{\mathbb R} \to {\mathbb R}$ such that there exists some $\xi_0$, depending on $g$, such that $$g(\xi )\xi \geq 0\quad\mbox{for}\quad |\xi |\geq \xi _0\,. \eqno(3.3)$$ \begin{theorem} % Theorem Let $a_*$ be as in (2.1) and let $$C_*= \frac{1}{4\pi ^2a^2_*+1} \eqno(3.4)$$ If $g\in \Sigma$ and $${\limsup _{|\xi |\to \infty }}{\frac{2G(\xi )}{\xi ^2}} < C^* \eqno(3.5)$$ then, for all $h\in \hat{L} ^2_{2\pi }$, there exists a weak solution of (3.1). \end{theorem} \paragraph{Sketch of Proof:} The proof is an application of Rabinowitz's Saddle-Point Theorem [4]. Assume first that $g$ satisfies the stronger condition: There exist $\delta >0$ and $\xi _0\geq 0$ such that $$|\xi |\geq \xi _0 \mbox{ implies } \mathop{\rm sgn}(\xi )g(\xi )\geq \delta . \eqno (3.6)$$ Assuming that (3.5) holds there exist constants $C_2\geq 0$ and $C_1$ with $$C_1 0 for all x\in {\mathbb R} ^2 or w(x)<0 for all x\in {\mathbb R} ^2. Since u_n(x)=\|u_n\|_{L^2}w_n either u_n(x)\to \infty  uniformly with respect to x\in {\mathbb R} ^2 or u_n(x)\to -\infty  uniformly with respect to x\in {\mathbb R} ^2. From (3.6) it follows that in the first case $\int _Q g(u_n(x))dx \geq 4\pi ^2 \delta$ for n sufficiently large, and in the second case $\int _Q g(u_n(x))dx\leq -4\pi ^2\delta$ for n sufficiently large. But since h\in \hat{L} ^2_{2\pi }, f' (u_n)(1)={\int _Q}-[g(u_n(x))+h(x)]dx=-{\int _Q}g(u_n(x))dx. Since f' (u_n)(1)\to 0 as n\to \infty , therefore we have a contradiction. This contradiction proves the sequence \{ u_n\} ^\infty _{n=1} is bounded in L^2(Q). From the condition f(u_n)\leq C_3 for all n and the condition (3.7), it follows from Lemma 2.2, that \{u_n\} ^\infty _{n=1} is bounded in {\mathbb E}. Therefore, from the form of f'  and Lemma 2.6, standard arguments (see for example [4]) shows that f'  satisfies the Palais-Smale condition. The existence of a critical point of f follows from Rabinowitz's Saddle Point Theorem [4] corresponding to the direct sum decomposition {\mathbb E}=\hat{{\mathbb E} }\oplus {\mathbb R} . Since, according to Lemma 2.4, for all z\in \hat{{\mathbb E}}, \|z\|_{L^2}\leq \|\Delta z\|_{L^2}, it follows that for all z\in \hat{{\mathbb E}} , \begin{eqnarray*} \lefteqn{\int _Q \left[ {\frac{(\Delta z)^2}{2}}-G(z)-h(x)z\right] dx }\\ &\geq& {\int _Q} \left[ {\frac{(\Delta z)^2}{2}} - \frac{C_1}{2} z^2-C_2\right] dx -\|h\|_{L^2}\|z\|_{L^2} \\ & \geq& \left( {\frac{1-C_1}{2}}\right) {\int _Q} {\frac {(\Delta z)^2}{2}} dx -C_24\pi ^2-\|h\|_{L^2}\|\Delta z\|_{L^2}\,. \end{eqnarray*} Since, as shown above, C_1<1 it follows that $\inf _{z\in \hat{{\mathbb E}}} f(z)>-\infty .$ The condition g(\xi )sgn \; \xi \geq \delta  for |\xi |\geq \xi _0 implies that G(\xi )\to \infty  as |\xi |\to \infty . Therefore, since h\in \hat{L} ^2_{2\pi } , it follows that for \xi \in {\mathbb R} , $f(\xi )=\int _Q[-G(\xi )-\xi h(x)]dx \leq -4\pi ^2G(\xi )\to -\infty$ as |\xi |\to \infty . Thus there exists b>0 such that $\max \{ f(b),f(-b)\} <\inf _{z\in \hat{{\mathbb E}} } f(z).$ Since f satisfies the Palais-Smale condition, it follows that if \Gamma denotes the set of all continuous mappings \gamma :[-b,b]\to {\mathbb E}  with \gamma (\pm \,b)=\pm \,b, $C_0={\inf _{\gamma \in \Gamma }}\;\;\; {\max _{\xi \in [-b,b]}}f(\gamma (\xi )),$ then there exits u_0\in {\mathbb E} such that f(u_0)=C_0 and f' (u_0)=0. This u_0 is a solution of problem (3.1). To prove that (3.1) has a solution when it is only assumed that g(\xi )\mathop{\rm sgn} \xi \geq 0 for |\xi |\geq \xi _0. We can use a perturbation argument. We define$$r(\xi ) = \left\{ \begin{array}{ll} -1 & \mbox{ if } \xi \leq -\xi _0\,, \\ -1 + {\frac{2(\xi +\xi _0)}{2\xi _0}} & \mbox{ if } |\xi |\leq \xi _0\,, \\ 1 & \mbox{ if } \xi \geq \xi _0\,, \end{array} \right.  For $m=1,2,3,\dots$, set $g_m(\xi )=g(\xi ) +{\frac{r(\xi )}{m}}$. Then $g_m(\xi )\xi \geq {\frac{1}{m}}$ for $|\xi | \geq \xi _0$ and we still have \[ \limsup _{|\xi |\to \infty } {\frac{2G_m(\xi )}{\xi ^2}}