\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 148, pp. 1--6.\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 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/148\hfil Nonlinear Schr\"odinger systems]
{A note on radial nonlinear Schr\"odinger systems
 with nonlinearity spatially modulated}

\author[J. Belmonte-Beitia\hfil EJDE-2008/148\hfilneg]
{Juan Belmonte-Beitia}

\address{Departamento de Matem\'aticas,
 E. T. S. de Ingenieros Industriales and Instituto de
 Matem\'atica Aplicada a la Ciencia y la Ingenier\'{\i}a (IMACI), \\
Universidad de Castilla-La Mancha s/n, 13071 Ciudad Real, Spain}
\email{juan.belmonte@uclm.es}

\thanks{Submitted May 13, 2008. Published October 29, 2008.}
\subjclass[2000]{35Q55, 34B15, 35Q51}
\keywords{Nonlinear Schr\"odinger equation; nonlinearity spatially modulated;
 Ermakov-Pinney equation; fixed point theorem}

\begin{abstract}
 First, we prove that for Schr\"odinger radial systems
 the polar angular coordinate must satisfy  $\theta'= 0$.
 Then using  radial symmetry, we transform the system
 into a generalized Ermakov-Pinney equation and prove the existence
 of positive  periodic solutions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{Lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

This note concerns the existence of solutions for the nonlinear
Schr\"odinger systems with nonlinearity spatially modulated and
radial symmetry in $1D$
\begin{subequations}\label{sist}
\begin{eqnarray}\label{sistema}
u_{1}''(x)+a(x)u_{1}(x)=b(x)f(u_{1}^2+u_{2}^2)u_{1}\\ \label{sistema1b}
u_{2}''(x)+a(x)u_{2}(x)=b(x)f(u_{1}^2+u_{2}^2)u_{2}
\end{eqnarray}
\end{subequations}
%
where $f(u_{1}^2+u_{2}^2)$ is a positive continuous function with
radial symmetry, and $a$ and $b$ are positive, continuous and
$L$-periodic functions; i.e.,
%
\begin{equation}
a(x)=a(x+L),\quad b(x)=b(x+L).
\end{equation}
Such solutions satisfy the  boundary conditions
%
\begin{subequations}\label{conditions}
\begin{eqnarray}\label{condiciones}
\lim_{|x|\to\infty}u_{1}(x)=\lim_{|x|\to\infty}u_{2}(x)=0,\\
\label{condicionesb}
\lim_{|x|\to\infty}u_{1}'(x)=\lim_{|x|\to\infty}u_{2}'(x)=0
\end{eqnarray}
\end{subequations}
%
The study of the existence of positive solutions for systems
like \eqref{sistema}, \eqref{sistema1b} with one coupled lineal term
has gained the interest of many mathematicians in recent years.
We refer to the surveys \cite{Ambrosetti,Eduardo2,Eduardo}.
In these papers, the authors show the existence of positive solutions
for different systems, using critical point theory or a variational approach.
Another different approximation to this kind of problems can be
found in Ref. \cite{Nuestro}.

From of physical point of view, this kind of systems has gained a
lot  of interest in the last years, in particular  in the context
of systems for the mean field dynamics of Bose-Einstein
condensates \cite{Williams} and in applications to fields as
nonlinear and fibers optics \cite{Malomed}.

On the other hand, the existence of positive solutions for the
nonlinear Schr\"odinger equation
%
\begin{equation}
u''+a(x)u=b(x)f(u(x))
\end{equation}
%
was proved in Ref. \cite{Pedro}. Thus, the existence of
semitrivial  solutions $(u_{1},0)$ and $(0,u_{2})$ of the system
\eqref{sistema} is guaranteed by the Ref. \cite{Pedro}.

We can transform the system \eqref{sistema} in a equation,  doing
$y=(u_{1},u_{2})$
%
\begin{equation}
\label{auxiliar}
y''+a(x)y=b(x)f(I)y
\end{equation}
%
with $I=u_{1}^2+u_{2}^2$.

With the change of variable
%
\begin{equation}\label{cambio}
u_{1}=\rho\cos\theta,\quad u_{2}=\rho\sin\theta.
\end{equation}
%
equation (\ref{auxiliar}) becomes
%
\begin{equation}
\label{mean}
\left[\rho''-\rho(\theta')^{2}+a(x)\rho \right]
\cos\theta-\left[2\rho'\theta'+\rho\theta''\right]
\sin\theta=b(x)f(\rho^{2})\rho\cos\theta
\end{equation}

The aim of this paper is to show that for Schr\"odinger radial
systems,  as  \eqref{sistema}, \eqref{sistema1b} with conditions
\eqref{condiciones}, \eqref{condicionesb}, can only exist
solutions with $\theta'=0$, specifically, we are thinking in the
semitrivial solutions $(u_{1},0)$ and $(0,u_{2})$. On the other
hand, for $\theta'\neq 0$, there not exist solutions of the system
\eqref{sistema}, \eqref{sistema1b} with conditions
\eqref{condiciones}, \eqref{condicionesb}.

Moreover, we can transform the system, by using the radial
symmetry,  to a generalized Ermakov-Pinney equation and study
positive  periodic solutions for this equation.

The rest of the papers is organized as follows. In section 2 we
prove that the only solutions of system \eqref{sistema},
\eqref{sistema1b} with conditions \eqref{condiciones},
\eqref{condicionesb}, if they exist, are given by solutions which
verify $\theta'=0$. In section 3, we prove the existence of
positive periodic solutions of the system \eqref{sistema},
\eqref{sistema1b}, with periodic conditions.

In this note, $\|\cdot\|$  denotes the supremum norm.

\section{Nonexistence of solutions for $\theta'\neq 0$ and
existence  for $\theta'=0$}

Physically, when a physical system possesses a symmetry, it means
that a physical quantity is conserved. As the system
\eqref{sistema}, \eqref{sistema1b} has radial symmetry, the
conserved quantity is the angular momentum. In polar coordinates,
the conservation of the angular momentum is given by
%
\begin{equation}
\label{conservacion}
\rho^{2}\theta'=\mu,
\end{equation}
%
where $\mu$ is a constant. Using this fact, \eqref{mean} becomes
%
\begin{equation}
\label{EP}
\rho''+a(x)\rho=b(x)f(\rho^{2})\rho+\frac{\mu^{2}}{\rho^{3}},
\end{equation}
%
which can be taken as a generalized Ermakov-Pinney \cite{Ermakov,Pinney}.

Now, it is easy to prove that, if there exist solutions of the
system \eqref{sistema}, \eqref{sistema1b}, with the boundary
conditions \eqref{condiciones}, \eqref{condicionesb}, they must
satisfy the condition $\theta'=0$: for these solutions, $\theta$
is constant and these solutions can be solutions of
(\ref{auxiliar}). In fact, we can find two examples of solutions
for this case: the semitrivial solutions $(u_{1},0)$ and
$(0,u_{2})$ are solutions of the system \eqref{sistema},
\eqref{sistema1b}, with conditions \eqref{condiciones},
\eqref{condicionesb} (see Ref. \cite{Pedro}).

On the other hand, for one solution $(u_{1},u_{2})$ with
$\theta'\neq 0$, one has $\mu\neq 0$. Thus, if  would exist a
solution $(u_{1},u_{2})$ of the system \eqref{sistema},
\eqref{sistema1b} with the boundary conditions
\eqref{condiciones}, \eqref{condicionesb} it would exist a
solution $\rho$ that would verify $\rho\to 0$ as $|x|\to\infty$.
But it is impossible, by the singularity of (\ref{EP}).

Thus, we are in disposition to formulate the following theorem.
%
\begin{theorem} \label{thm1}
Let system \eqref{sist} be with conditions \eqref{conditions}
where $a(x)$ and $b(x)$ are positive, continuous and $L$-periodic
functions. Then, if there exist solutions of the system
\eqref{sist}, with the conditions \eqref{conditions}, different of
the trivial solution, they must satisfy the condition $\theta'=0$,
where $\theta$ is the polar angular coordinate in \eqref{cambio}.
\end{theorem}

\begin{remark} \label{rmk1} \rm
Specifically, for $\theta=k\pi$ or $\theta=\frac{k}{2}\pi$, for
any $k\in\mathbb{Z}$, we obtain the semitrivial solutions. These
solutions are called bright solitons in the physical literature.
The dark solitons are also solutions of the system
\eqref{sistema}, \eqref{sistema1b} but with different boundary
conditions \cite{Kivshar}. It is straightforward to prove that,
for this case, the only solutions are the former with $\theta'=0$,
provided that $a(x)$ is different to $b(x)$.
\end{remark}

\begin{remark} \label{rmk2} \rm
We can use another approximation, where one can see the
universality  of the method exposed here. Thus, let the nonlinear
Schr\"odinger equation be
%
\begin{equation}
\label{NLSE}
iu_{t}+u_{xx}+b(x)f(|u|^2)u+V(x)u=0
\end{equation}
%
with $V(x)$ a $L$-periodic function. If we have the change  of
variable  $u(t,x)=\left(v(x)+iw(x)\right)e^{i\lambda t}$ and if we
separate in real and imaginary part, we obtain
%
\begin{gather*}
v''+\left(V(x)-\lambda\right)v+b(x)f(v^2+w^2)v=0\\
w''+\left(V(x)-\lambda\right)w+b(x)f(v^2+w^2)w=0
\end{gather*}
%
which is similar to the system \eqref{sistema}, \eqref{sistema1b}
for $a(x)=V(x)-\lambda$.
\end{remark}

\section{Periodic Solutions}

As we showed in the previous section, system \eqref{sistema}
\eqref{sistema1b}, or equation (\ref{NLSE}), can be reduced to
(\ref{EP}). Thus, we can describe the behaviour of solutions of
\eqref{sistema}--\eqref{sistema1b} (or  (\ref{NLSE})) using
(\ref{EP})

Then, the aim of this section is to provide some existence result
for the periodic boundary-value problem
%
\begin{equation}
\label{EP2}
\rho''+a(x)\rho=b(x)f(\rho^{2})\rho+\frac{\mu^{2}}{\rho^{3}},
\end{equation}
%
 with $\rho(0)=\rho(L)$, $\rho'(0)=\rho'(L)$, where $a(x)$ and
$b(x)$ are positive, continuous and $L$-periodic functions. To do
it, we will use the following fixed-point theorem for a completely
continuous operator in a Banach space, due to Krasnoselskii
\cite{Krasnoselskii}.

\begin{theorem}\label{KN}
Let $X$ be a Banach space, and let $P\subset X$ be a cone in $X$.
Assume $\Omega_{1}, \Omega_{2}$ are open subsets of $X$ with
$0\in\Omega_{1},\overline{\Omega}_{1}\subset\Omega_{2}$ and let
$T: P\cap(\overline{\Omega}_{2}\backslash\Omega_{1})\to P$ be a
completely continuous operator such that one of the following
conditions is satisfied
%
\begin{enumerate}
\item $\|Tu\|\leq\|u\|$, if $u\in P\cap\partial\Omega_{1}$,
   and $\|Tu\|\geq\|u\|$, if $u\in P\cap\partial\Omega_{2}$.
\item $\|Tu\|\geq\|u\|$, if $u\in P\cap\partial\Omega_{1}$,
   and $\|Tu\|\leq\|u\|$, if $u\in P\cap\partial\Omega_{2}$.
\end{enumerate}
%
Then, $T$ has at least one fixed point in
$P\cap(\overline{\Omega}_{2}\backslash\Omega_{1})$.
\end{theorem}

From the physical explanation, \eqref{EP2} has a repulsive
singularity at $x=0$. In order to apply Theorem \ref{KN}, we need
some information about the properties of the Green's function.
Thus, let us consider the linear equation
%
\begin{equation}\label{eqhom}
\rho''+a(x)\rho=0,
\end{equation}
%
with periodic conditions
%
\begin{equation}\label{condition}
\rho(0)=\rho(L),\quad \rho'(0)=\rho'(L)
\end{equation}
%
In this section, we assume conditions under which the only solution
of problem (\ref{eqhom})-(\ref{condition}) is the trivial one.
As a consequence of Fredholm's alternative, the nonhomogeneous equation
%
\begin{equation}\label{eqnonhom}
\rho''+a(x)\rho=h(x),
\end{equation}
%
admits a unique $T$-periodic solution which can be written as
%
\begin{equation}
\rho(x)=\int_{0}^{L}G(x,s)h(s)ds,
\end{equation}
%
 where $G(x,s)$ is the Green's function of problem
 (\ref{eqhom})-(\ref{condition}). Following \cite{Multiplicity},
we assume that problem (\ref{eqhom}) satisfies that the Green
function, $G(x,s)$, associated with  problem (\ref{eqnonhom}), is
positive for all $(x,s)\in[0,L]\times[0,L]$. Moreover, following
\cite{Pedro2}, we denote
 %
 \begin{equation}
 M=\max_{x,s\in[0,L]}G(x,s),\quad m=\min_{x,s\in[0,L]}G(x,s)
 \end{equation}
 %
 where $M>m>0$.
\begin{theorem}
Let us assume the following hypotheses
\begin{itemize}

\item[(i)] $a(x)$ and $b(x)$ are continuous and $L$-periodic functions
     with $a>0, b>0$.

\item[(ii)] $f(s)\geq 0$ for every $s\geq 0$.

\item[(iii)] There exists $r>0$ such that
\begin{equation*}
A_{r}\max_{x\in[0,L]}\int_{0}^{L}G(x,s)b(s)ds+B_{r}
 \max_{x\in[0,L]}\int_{0}^{L}G(x,s)ds\leq r
\end{equation*}
%
for $A_{r}=\max_{s\in[0,r]}f(s^2)s$ and $B_{r}=\max_{s\in[0,r]}\mu^{2}/s^3$.

\item[(iv)] There exist $R>r>0$ such that
\begin{equation*}
A_{R}\min_{x\in[0,L]}\int_{0}^{L}G(x,s)b(s)ds
 +B_{R}\min_{x\in[0,L]}\int_{0}^{L}G(x,s)ds\geq \frac{M}{m}R
\end{equation*}
for $A_{R}=\min_{s\in[R,(M/m)R]}f(s^2)s$ and
    $B_{R}=\min_{s\in[R,(M/m)R]}\mu^{2}/s^3$.
\end{itemize}
Then,  \eqref{EP2} has a positive periodic solution $\rho$ with
$\frac{m}{M}r\leq\rho(x)\leq\frac{M}{m}R$.
\end{theorem}

\begin{proof}
Let $X=C[0,L]$ with the supremum norm $\|\cdot\|$.
We define the open sets
%
\begin{gather*}
\Omega_{1}=\{\rho\in X: \|\rho\|<r\}\\
\Omega_{2}=\{\rho\in X: \|\rho\|<\frac{M}{m}R\}
\end{gather*}
%
Define the cone
%
\begin{equation*}
P=\{\rho\in X:\rho\geq 0
\min_{x\in[0,L]}\rho\geq\frac{m}{M}\|\rho\|\}\,.
\end{equation*}
%
It is easy to prove that if $\rho\in P\cap
(\overline{\Omega}_{2}\backslash\Omega_{1})$, then
%
\begin{equation*}
\frac{m}{M}r\leq\rho(x)\leq\frac{M}{m}R, \quad \forall x
\end{equation*}
%
Let us define the operator
%
\begin{equation}\label{operator}
T\rho=\int_{0}^{L}G(x,s)
\big[b(s)f(\rho^2(s))\rho(s)+\frac{\mu^{2}}{\rho^{3}(s)}\big]
\end{equation}
We note that such operator is completely continuous. Clearly, a
solution of problem \eqref{EP2} is just a fixed point of this
operator.

If $\rho\in P\cap (\overline{\Omega}_{2}\backslash\Omega_{1})$, then
%
\[
T\rho\geq \frac{m}{M}\int_{0}^{L}\max_{x\in[0,L]}G(x,s)
\big[b(s)f(\rho^2(s))\rho(s)+\frac{\mu^{2}}{\rho^{3}(s)}\big]ds
=\frac{m}{M}\|T\rho\|
\]
%
that is, $T\left(P\cap(\overline{\Omega}_{2}\backslash\Omega_{1})\right)\subset P$.

Now, if $\rho\in \partial\Omega_{1}\cap P$, then $\|\rho\|=r$ and
$(m/M)r\leq\rho(x)\leq r$ for all $x$. Therefore,  using (iii),
%
\[
\|T\rho\|=\max_{x\in[0,L]}T\rho(x)\leq A_{r}\max_{x\in[0,L]}
\int_{0}^{L}G(x,s)b(s)ds+B_{r}\max_{x\in[0,L]}\int_{0}^{L}G(x,s)ds\leq r
\]
%
Similarly, if $x\in\partial\Omega_{2}\cap P$, then
$\|\rho\|=(M/m)R$ and $R\leq\rho(x)\leq(M/m)R$, for all x. Then,
using the hypotheses (iv),
%
\begin{align*}
\|T\rho\|&=\max_{x\in[0,L]}T\rho(x)\\
&=\max_{x\in[0,L]}\int_{0}^{L}G(x,s)
 \big[b(s)f(\rho^2(s))\rho(s)+\frac{\mu^{2}}{\rho^{3}(s)}\big]ds\\
&\geq A_{R}\min_{x\in[0,L]}\int_{0}^{L}G(x,s)b(s)ds
 +B_{R}\min_{x\in[0,L]}\int_{0}^{L}G(x,s)ds\geq
\frac{M}{m}R
\end{align*}
%
Now, from Theorem \ref{KN} there exists $\rho\in P\cap
(\overline{\Omega}_{2}\backslash\Omega_{1})$ which is a solution
of problem \eqref{EP2}. Therefore,
%
\[
\frac{m}{M}r\leq\rho(x)\leq\frac{M}{m}R
\]
%
\end{proof}

\begin{corollary} \label{coro1}
Under the conditions of Theorem 2, system \eqref{sistema}--\eqref{sistema1b}
 and equation \eqref{NLSE}, with periodic conditions, have positive
 periodic solutions.
\end{corollary}

In the framework of Bose-Einstein condensates \cite{Williams}
or nonlinear optics \cite{Malomed}, such positive periodic solutions
are called periodic matter waves.


\subsection*{Acknowledgements}
The author would like to thank Professor Pedro Torres for a first
critical reading of the manuscript.
The author is also indebted to the anonymous referee for pointing
out some inaccuracies in the first version of the paper.
This work has been supported by grant PCI08-093
from Consejer\'{\i}a de Educaci\'on y Ciencia de la Junta
de Comunidades de Castilla-La Mancha, Spain, and
grant FIS2006-04190 from Ministerio de Educaci\'on y Ciencia, Spain.


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\end{document}