\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 162, 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 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/162\hfil Uniqueness of a symmetric positive solution] {Uniqueness of a symmetric positive solution to an ODE system} \author[O. Lopes\hfil EJDE-2009/162\hfilneg] {Orlando Lopes} \dedicatory{In memory of Jack K. Hale (1928--2009)} \address{Orlando Lopes \newline IMEUSP- Rua do Matao, 1010, Caixa postal 66281\\ CEP: 05315-970, Sao Paulo, SP, Brazil} \email{olopes@ime.usp.br} \thanks{Submitted October 9, 2009. Published December 21, 2009.} \subjclass[2000]{34A34} \keywords{Symmetric positive solutions; variational ODE systems} \begin{abstract} In this article, we prove uniqueness of symmetric positive solutions of the variational ODE system \begin{gather*} -w''+a w-wv=0 \\ -v''+b v -\frac{w^2}{2}=0, \end{gather*} where $a$ and $b$ are positive constants. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction and Statement of the Result} In this article, we prove uniqueness of symmetric positive solutions of the variational ODE system $$\label{e1} \begin{gathered} -w''+a w-wv = 0\\ -v''+b v -\frac{w^2}{2}=0 \end{gathered}$$ where $a$ and $b$ are positive constants. The solutions under consideration are defined for all $x\in \mathbb{R}$ and have finite energy. To show how \eqref{e1} arises, we consider the so-called $\chi^2$ SHG equations $$\label{e2} \begin{gathered} i\frac{\partial w}{\partial t}+r\frac{\partial^2w}{\partial x^2}-\theta w +w^*v=0\\ i\sigma \frac{\partial v}{\partial t}+s\frac{\partial^2v}{\partial x^2} -\alpha v+\frac{w^2}{2}=0 \end{gathered}$$ where $r,s,\sigma,\theta$ are positive real parameters and $w(x)$ and $v(x)$ are complex functions. This system governs phenomena in nonlinear optics (see \cite{torner} for instance). A solitary wave is a solution of \eqref{e2} of the form $$(w(x)e^{i\gamma t},v(x)e^{2i\gamma t}).$$ Hence, $(w,v)$ satisfies $$\label{e3} \begin{gathered} -rw''+(\theta +\gamma)w-w^*v=0\\ -sv''+(\alpha+2\sigma\gamma) v -\frac{w^2}{2}=0. \end{gathered}$$ The solutions of \eqref{e3} are critical points of $E+\gamma I$ where $E$ and $I$ are the following conserved quantities for \eqref{e2} \begin{gather} \label{e4} E(w,v)=\int_{-\infty}^{+\infty} (r|w'|^2+s|v'|^2+\theta |w|^2 +\alpha|v|^2-{\rm Re}(w^2v^*))\,dx, \\ \label{e5} I(w,v)=\int_{-\infty}^{+\infty} (|w|^2+2\sigma |v|^2)\,dx. \end{gather} If $w$ and $v$ are real solutions of \eqref{e3} then it solves $$\label{e6} \begin{gathered} -rw''+(\theta +\gamma)w-wv = 0\\ -sv''+(\alpha+2\sigma\gamma) v -\frac{w^2}{2}=0. \end{gathered}$$ Replacing $(w,v)$ by $(k_1w,k_2v)$ in \eqref{e6}, with $k_2=r$ and $k_1^2=rs$, we get \begin{gather*} -w''+\frac{(\theta +\gamma)}{r} w-wv =0\\ -v''+\frac{(\alpha+2\sigma\gamma)}{s} v -\frac{w^2}{2}=0. \end{gather*} Therefore, we consider the real variational ODE system \begin{gather} -w''+a w-wv=0 \label{e7} \\ -v''+b v -\frac{w^2}{2}=0 \label{e8} \end{gather} and we will be interested in solutions that have finite energy (or equivalently, tend to zero as $|x|$ tends to infinity). The existence of positive solutions of \eqref{e7}-\eqref{e8} has been proved in \cite{yew}. Briefly the argument goes as follows. We define $H=H^1(\mathbb{R})\times H^1(\mathbb{R})$ equipped with the norm $$\int_{-\infty}^{+\infty} (w'^2(x) +v'^2(x) +aw^2(x) +bv^2(x))\, dx.$$ We consider the functionals \begin{gather*} E(w,v)=\int_{-\infty}^{+\infty}(w'^2(x) +v'^2(x)-w^2(x)v(x))\,dx,\\ I(w,v)=\int_{-\infty}^{+\infty} (aw^2(x) +bv^2(x))\, dx. \end{gather*} Using the method of concentration-compactness (\cite{lions}), we minimize $E(w,v)$ under $I(w,v)=1$ in the space $H$. If we replace $(w(x),v(x))$ by $(|w(x)|,|v(x)|)$ then $E$ does not increase. Therefore, any minimizer is nonnegative and solves the Euler-Lagrange system \begin{gather} -w''+\mu a w-wv=0 \label{e9}\\ -v''+\mu b v -\frac{w^2}{2}=0 \label{e10} \end{gather} with $\mu\geq 0$ (because $(w,v)$ is a minimizer). On the other hand, it is easy to see that any solution $(w,v)\in H$ of \eqref{e9}-\eqref{e10} with $\mu=0$ is the solution identically zero. Therefore, we must have $\mu >0$. Defining a new pair $(k_1w(k_3x), k_2v(k_3 x))$ with $k_3^2=1/\mu,k_1=k_2=1/\mu$, we see that this new pair satisfies \eqref{e7}-\eqref{e8}. In \cite{lopes} the symmetry of any positive solution of \eqref{e7}-\eqref{e8} has been proved using a result of \cite{busca}. However, as pointed out in \cite{busca}, their proof works for $N\geq 2$. Since we are in dimension one, we need the following modified version given in \cite{ikoma}. \begin{theorem} \label{thm1.1} Consider the system $$\label{e11} \begin{gathered} w''+f(w,v)=0 \\ v'' +g(w,v)=0 \end{gathered}$$ where $f(w,v)$ and $g(w,v)$ are $C^1$ functions satisfying the conditions: $$f(0,0)=0=g(0,0),\quad \frac{\partial f(w,v)}{\partial v}, \frac{\partial g(w,v)}{\partial w}\geq 0.$$ Suppose that there exist $\epsilon>0$ and $\delta>0$ such that $w>0$, $v>0$, $w^2+v^2<\epsilon$ imply $$\frac{\partial f(w,v)}{\partial w},\frac{\partial g(w,v)}{\partial v} <-\delta, \quad 0<\frac{\partial f(w,v)}{\partial v}, \frac{\partial g(w,v)}{\partial w}<\delta.$$ Then, except for translations, any positive solution of \eqref{e11} is even and decreasing. \end{theorem} We conclude that, except for translations, any positive solution of \eqref{e7}-\eqref{e8} is symmetric and decreasing. In \cite{lopes} we have also proved the following result. \begin{theorem} \label{thm1.2} The linearized operator of \eqref{e7}-\eqref{e8} at any positive symmetric solution has zero as a simple eigenvalue with odd eigenfunctions $(w_x,v_x)$ and it has exactly one negative eigenvalue. \end{theorem} The fact that zero is a simple eigenvalue of the linearized operator is not a proof of uniqueness of symmetric positive solution, but it may suggest it. Our main result is that this is indeed the case. \begin{theorem} \label{thm1.3} For $a,b>0$, the positive symmetric decreasing solution of \eqref{e7}-\eqref{e8} is unique. \end{theorem} Several interesting numerical experiments concerning system \eqref{e7}-\eqref{e8} are presented in [6]. They indicate uniqueness of positive solution (which is confirmed by Theorem \ref{thm1.3}) and that \eqref{e7}-\eqref{e8} may have solutions that change sign. \section{Proof of main result} First we establish the following abstract uniqueness result. \begin{theorem} \label{thm2.1} Let $X$ be a Banach space and $F: X\times [0,1]\to X$ be a continuous functions with continuous Frechet derivative with respect to the first variable. Also assume that \begin{itemize} \item[(i)] the set of the solutions $(u,\lambda)$ of $F(u,\lambda)=0$, $u\in X, \lambda \in [0,1]$ is precompact; \item[(ii)] for any solution of $F(u,\lambda)=0$, the derivative $F_u(u,\lambda)$ is invertible; \item[(iii)] the equation $F(u,0)=0$ has a unique solution. \end{itemize} Then the equation $F(u,\lambda)=0$ has a unique solution for $\lambda \in [0,1]$. \end{theorem} \begin{proof} First we claim that there is a $\lambda_0 >0$ such that the solution of $F(u,\lambda)=0$ is unique for $0\leq \lambda < \lambda_0$. In fact, otherwise, there is a sequence $0<\lambda_n \to 0$ such that $F(u,\lambda_n)=0$ has at least two distinct solution $u_n$ and $v_n$. In view of assumption (i) and passing to a subsequence if necessary, we can assume that $u_n$ converges to $u$ and $v_n$ converges to $v$. In view of (iii), we must have $u=v$. However, by (ii) and the implicit function theorem, in a neighborhood of $u$, for small $\lambda$, the solution of $F(u,\lambda)=0$ is unique. This contradiction proves the claim. The same argument shows that the set $A$ of $\lambda$, $0\leq \lambda \leq 1$, for which the solution of $F(u,\mu)=0$ is unique for $0\leq \mu \leq \lambda$ is open. Since by ii) $A$ is clearly closed, $A$ has to be the whole interval $[0,1]$ and the theorem is proved. \end{proof} \subsection*{Remark} If we take $u\in \mathbb{R}$ and $F(u,\lambda)=u(\lambda u-1)=\lambda u^2-u$, we have $F_u(u,\lambda)=2\lambda -1$. We see that, except for assumption i), all the others are satisfied but the conclusion of the theorem does not hold. This is so because there is the branch $u=1/\lambda$ of solutions bifurcating from infinity. Theorem \ref{thm1.3} will be a consequence of Theorem \ref{thm2.1}. To verify all its assumptions, we start with the following result. \begin{lemma} \label{lem2.1} The system $$\label{e12} \begin{gathered} -w''+ aw -wv=0 \\ -v''+ av -\frac{w^2}{2}=0 \end{gathered}$$ ($a=b$ in \eqref{e7}-\eqref{e8}) has a unique positive solution with finite energy. \end{lemma} \begin{proof} Defining $z(x)=w(x)-\sqrt{2}v(x)$, multiplying the second equation by $\sqrt{2}$ and subtracting we get $$-z''+z+ \frac{w}{\sqrt{2}}z=0.$$ Multiplying this last equation by $z$ and integrating we get $$\int_{-\infty}^{+\infty} (z'^2(x) +z^2(x)+ \frac{w}{\sqrt{2}} z(x)^2)\,dx=0$$ and this implies $z\equiv 0$ (because $w$ is a positive). Therefore, each component of the solution of \eqref{e12} solves a single second order equation and this implies uniqueness and the lemma is proved. \end{proof} To verify the other assumptions of Theorem \ref{thm2.1}, we establish a chain of estimates. Since we wish to find estimates for solutions of \eqref{e7}-\eqref{e8} which remain uniform for $a$ and $b$ in a certain interval, we fix two constants $0a$. Moreover, $v''(0)\leq 0$ (because $v(x)$ has a maximum at $x=0$) and then the second equation \eqref{e8} yields $$\label{e17} b v(0)\leq \frac{w^2(0)}{2}\,.$$ This together with \eqref{e16} implies $$\label{e18} b v(0)\leq \frac{1}{2}\frac{b v^2(0)}{(v(0)-a)}$$ and finally %\label{e} $v(0)\leq 2a$ because $v(0)>a$. \subsection*{Bound for $v'(x)$} Multiplying the second equation \eqref{e8} by $v'(x)$, then for $x\geq 0$ we get: $$\frac{d}{dx}(-v'(x)^2+b v^2(x))=w^2(x)v'(x) \leq 0.$$ Therefore $-v'(x)^2+b v^2(x)$ is decreasing and, since it vanishes at $+\infty$, we get $$-v'(x)^2+b v^2(x)\geq 0$$ and then $$\label{e20} v'(x)^2\leq b v^2(x)\leq b v^2(0)\leq 4a^2b.$$ \subsection*{Bound for $w'(x)$} We know $w'(x)\leq 0$ and that $w'(x)$ reaches its minimum when $w''(x)=0$. By the first equation \eqref{e7}, this occurs when $v(x)=a$ and then, from \eqref{e15}, $$w'(x)^2+v'(x)^2=b v^2(x)\leq b v^2(0)\leq 4a^2b.$$ We conclude $$\label{e21} |w'(x)|=-w'(x)\leq 2a\sqrt{b}.$$ \subsection*{Bound for $w(0)$} Suppose $w(0)=M$ and $w(x_0)=M/2$ for some $x_0>0$. Since $$w(0)-w(x_0)=- \int_0^{x_0} w'(s)\, ds,$$ then, in view of \eqref{e21}, we have $\frac{M}{2}\leq 2a\sqrt{b} x_0$ and this implies $$\label{e22} x_0\geq \frac{M}{4a\sqrt{b}}.$$ Moreover, the solution of the linear equation $$\label{e23} -v''(x)+b v(x)=h(x)$$ is given by $$\label{e24} v(x)= \frac{1}{2\sqrt{b}} \int_{-\infty}^{+\infty} e^{-\sqrt{b}|x-y|}h(y)\,dy,$$ and then, the second equation \eqref{e8} and \eqref{e22} give \begin{align*} v(0) &=\frac{1}{4\sqrt{b}}\int_{-\infty}^{+\infty} e^{-\sqrt{b}|y|}w^2(y) \,dy\\ &= \frac{1}{2\sqrt{b}}\int_0^{+\infty} e^{-\sqrt{b}y}w^2(y) \,dy \\ &\geq \frac{1}{2\sqrt{b}}\int_0^{x_0} e^{-\sqrt{b}y}w^2(y) \,dy \\ & \geq \frac{M^2}{8\sqrt{b}}\int_0^{x_0}e^{-\sqrt{b}y}\,dy\\ &=\frac{M^2}{8b}(1-e^{-\sqrt{b} x_0})\\ &\geq \frac{M^2}{8b} (1-e^{- \frac{M}{4a}}). \end{align*} Therefore, $$2a \geq v(0) \geq \frac{M^2}{8b}(1-e^{- \frac{M}{4a}})$$ and this gives that $M=w(0)\leq d_1$, for some constant $d_1$. In view of \eqref{e16}, this gives also that $v(0)\geq d_2>a$, for some constant $d_2$, and also gives a lower bound for $w(0)\geq d_3$. \subsection*{Bound for the length of the interval for which $v(x) \geq a$} By the first equation in \eqref{e7} and the previous estimates for $v(0)$ and $w(0)$, we have $w''(0)\leq -d_4<0$ and $|w'''(x)|\leq d_5$. Defining $X=-\frac{w''(0)}{2d_5}$ then, for $0\leq x \leq X$ we have $w''(x)-w''(0)=\int_0^x w'''(s)\,ds\leq d_5 X=-w''(0)/2,$ and then $w''(x)\leq w''(0)/2\leq -d_4/2$ for $0\leq x \leq X$. Moreover, $w'(X)=w'(0)+\int_0^X w''(s)\,ds \leq \int_0^X \frac{w''(0)}{2} \,ds=X\frac{w''(0)}{2} =- \frac{w''(0)^2}{4d_5}\leq -d_6.$ Since, by \eqref{e7}, $w''(x)\leq 0$ whenever $v(x) \geq a$, we have $w'(x) \leq -d_6$ whenever $v(x) \geq a$ and $x\geq X$. Furthermore, $$-w(0)\leq -w(X)\leq w(x)-w(X)= \int_X^xw'(s)\, ds\leq -d_6(x-X).$$ Therefore, defining $X_1=w(0)/d_6+X$, we see that we must have $v(X_1)\leq a$. \subsection*{Estimate for the time $v(x)$ stays close (and less) than $a$} Let $x_0\leq X_1$ be such that $v(x_0)=a$ and let $d_7>0$ and $d_80$, there is an $x(\epsilon)>0$ such that for all $u\in K$ we have $$\int _{|x| \geq x(\epsilon)} (|u'|^2(x) + |u(x)|^2)\, dx <\epsilon.$$ \end{itemize} To verify these conditions we first notice that we have obtained uniform bound for the $H^1(\mathbb{R})$ norm of the solution $(w,v)$ of \eqref{e7}-\eqref{e8}. This implies uniform bound for the $H^2$ norm of such solutions and this verifies condition (1) for precompactness. The uniform exponential decay \eqref{e25} and \eqref{e26} for $w(x)$ and $v(x)$ together with \eqref{e14} gives the uniform exponential decay also for the derivatives. This implies that condition (2) for precompactness is satisfies; therefore, Theorem \ref{thm1.3} is proved. \end{proof} \begin{thebibliography}{00} \bibitem{busca} J. Busca and B. Sirakov; \emph{Symmetry results for semilinear elliptic systems in the whole space}, J. Diff. Eq., 163(2000),41-56. \bibitem{ikoma} N. Ikoma; \emph{Uniqueness of Positive Solution for a Nonlinear Elliptic systems, Nonlinear Differential Equations and Applications}, 2009, online. \bibitem{lions} P. L. Lions; \emph{The concentration compactness principle in the Calculus of Variations}, AIHP, Analyse Nonlineaire, part I, vol. 1, no. 2, 1984, 109-145; part II: vol. 1, no4, 1984, 223-283. \bibitem{lopes} Lopes, Orlando; \emph{Stability of solitary waves of some coupled systems}. Nonlinearity 19 (2006), no. 1, 95-113 . \bibitem{torner} Torner, L. et al; \emph{Stability of spatial solitary waves in quadratic media}, Optics Letters, (1995), vol.20, No. 21, 2183-2185. \bibitem{yew} A. C. Yew, A. R. Champneys and P. 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