# Existence of Optical Vortices

Yisong Yang
Department of Mathematics
Polytechnic Institute of New York University
Brooklyn, New York 11201, USA

Ruifeng Zhang
Institute of Contemporary Mathematics
School of Mathematics
Henan University
Kaifeng, Henan 475004, PR China

###### Abstract

Optical vortices arise as phase singularities of the light fields and are of
central interest in modern optical physics. In this paper, some existence
theorems are established for stationary vortex wave solutions of a general
class of nonlinear Schrödinger equations. There are two types of results. The
first type concerns the existence of positive-radial-profile solutions which
are obtained through a constrained minimization approach. The second type
addresses the existence of saddle-point solutions through a mountain-pass-
theorem or min-max method so that the wave propagation constant may be
arbitrarily prescribed in an open interval. Furthermore some explicit
estimates for the lower bound and sign of the wave propagation constant with
respect to the light beam power and vortex winding number are also derived for
the first type solutions.

##  1 Introduction

Vortices have important applications in many areas of modern physics including
condensed matter systems, particle interactions, cosmology, and superfluids.
Research on vortices in optics also has a long history and was initiated in as
early as 1964 by Chiao, Garmire, and Townes [5] who explored some conditions
under which a light beam can produce its own waveguide and propagate without
spreading. They described such phenomenon as self-trapping, attributed it to
light propagation in materials whose dielectric coefficient increases with
field intensity in the context of high-intensity light beams such as lasers,
predicted marked optical and physical effects, and suggested the occurrence of
optical vortices. Such vortices have since then been observed in numerous
studies [2, 3, 27, 29, 32] and become a much pursued subject in optics [6, 9,
11, 19, 22, 31, 33] both theoretically and experimentally. (For a vast
literature up to 2005 and for a description of the rich features and profound
applications of optical vortices, see the nice survey article by Desyatnikov,
Kivshar, and Torner [10]. See also [16] for a more recent survey of the
subject in a broader perspective.) As waves, light propagation may be
described by a wave function. At certain spots of space, the intensity of the
waves vanishes and the phase of the waves cannot be defined. Thus, such spots
are phase singularities which were recognized in the comprehensive work of Nye
and Berry [23] as crucial characteristics of general wave motions in which
vortices are present. These phase singularities, also referred to as
dislocations or defects of waves, are the centers of vortices, around which
energy concentrates. In the context of light waves, vortices are centered
around vortexlines and light waves are twisted around the vortexlines. The
twisting arises from the phase ambiguity around a vortexline and is of a
topological nature. The twisting centers are exactly the vortex cores where
light waves cancel out leading to darkness so that light intensity measured at
any cross section vertical to a light beam axis should display concentric
ring-like patterns around the dark core. Light beams of such structure are
also vividly termed “helical beams” [4]. In optics research, a fundamental
prototype situation is when the light waves are described by a complex-valued
wave function governed by nonlinear Schrödinger equations [1, 7, 17, 18, 20,
21, 26, 28]. These theoretical studies provide a broad range of interesting
analytic problems related to the existence and properties of optical vortices
for mathematical investigation.

Our aim in the present work is to obtain some existence theorems for the
optical vortex solutions explored by Salgueiro and Kivshar in [28] through a
study of the normalized nonlinear Schrödinger equation

$\mbox{i}\frac{\partial\psi}{\partial
z}+\nabla^{2}_{\perp}\psi+(V+s|\psi|^{2})\psi=0,$ (1.1)

where $\psi$ is a complex-valued optical field propagating in the
(longitudinal) $z$-direction, $\nabla^{2}_{\perp}$ is the Laplace operator
over the (transverse) plane of coordinates $(x,y)$ which is perpendicular to
the $z$-axis, $V$ is an external potential function (cf. [18, 30]), and $s=\pm
1$ is the sign symbol indicating either a focusing or defocusing situation
[17] which is taken to be $+1$ (focusing) in [28] and will be our main focus
(the defocusing case $s=-1$ will be seen to be straightforward). The interest
of (1.1) actually goes beyond nonlinear optics. For example, it also arises in
the study of the Bose–Einstein condensates [1, 11, 15, 30] and is referred to
as the Gross–Pitaevskii equation. An important simplified situation that
allows optical vortices to present is when $V$ depends on the radial variable
only, $V=V(r),r=\sqrt{x^{2}+y^{2}}$. In this situation one may expect to find
an $n$-vortex solution of (1.1) of the form [28]

$\psi=\psi(r,\theta,z)=u(r)\mbox{e}^{\mbox{i}(n\theta+\beta z)},$ (1.2)

where $r,\theta$ are polar coordinates over ${\mathbb{R}}^{2}$, $u(r)$ is the
radial profile function which gives rise to the intensity of light waves,
integer $n\in{\mathbb{Z}}$ is the winding number or vortex charge of the
vortex solution, and $\beta\in{\mathbb{R}}$ is the wave propagation constant
[28]. This ansatz describes a vortex wave centered around the $z$-axis where
$r=0$ and propagating along the $z$-axis. Inserting (1.2) into (1.1), we
arrive at the following $n$-vortex equation

$(ru_{r})_{r}-\left(\frac{n^{2}}{r}+\beta r\right)u+r(V+su^{2})u=0,$ (1.3)

of cubic nonlinearity. The presence of the vortex core at $r=0$ requires
$u(0)=0$. As in [28] (for $n=1$), we are interested in ring-shaped vortices so
that light intensity concentrates around the vortex core which suggests that
$u(r)$ may be assumed to vanish at a sufficiently large distance.
Mathematically, this indicates that we may impose another ‘boundary’
condition, say $u(R)=0$, at a certain distance $R>0$ away from the core of the
vortex as seen in the numerical results of the work [28]. Thus, the problem of
the existence of optical vortices is reduced into a two-point boundary value
problem with undetermined parameter $\beta$ and prescribed $R$, for any given
$n\in{\mathbb{Z}}$. To tackle this problem, we shall use the methods of
calculus of variations. Our methods allow us to obtain two types of results
for the nontrivial focusing case $s=+1$. The first type of results rely on a
constrained minimization approach. The nature of minimization leads us to
obtaining positive-valued solutions in the open interval $(0,R)$ and that the
propagation constant $\beta$ arises as a Lagrange multiplier due to the
constraint. The second type of results are obtained from searching for saddle
points of the action functional associated to the problem. We will see that,
in this latter case, there is no assurance that the solutions must stay
positive-valued but the propagation constant $\beta$ arises as a prescribed
quantity.

In the next two sections, we shall concentrate on the focusing case when
$s=+1$. In Section 2, we formulate the problem of existence of optical
vortices as a constrained minimization problem, state the main existence
results regarding positive solutions, and then present the proofs. We will see
that the propagation constant $\beta$ arises as a Lagrange multiplier which is
ensured to be negative when the vortex charge $n$ is sufficiently large. We
will also derive some lower estimate for $\beta$. In Section 3, we treat
$\beta$ as a prescribed quantity and use a mountain-pass theorem approach to
establish the existence of solutions for any $R$ and vortex charge $n$. In
particular, we show that the propagation constant $\beta$ may assume any
prescribed value in an explicitly given interval. In Section 4, we briefly
discuss the defocusing case when $s=-1$.

##  2 Vortices via constrained minimization

As described in the previous section, we will be interested in ‘ring vortices’
such that (1.3) is considered over a bounded interval $(0,R)$ ($R>0$) so that
$u$ vanishes at the two endpoints of the interval. As mentioned earlier, we
will mostly concentrate on the nontrivial case, $s=+1$. Thus, our problem is a
two-point boundary value problem

$\displaystyle(ru_{r})_{r}-\frac{n^{2}}{r}u+r(V+u^{2})u$ $\displaystyle=$
$\displaystyle\beta ru,$ (2.1) $\displaystyle u(0)$ $\displaystyle=$
$\displaystyle u(R)=0,$ (2.2)

for which the parameter $\beta\in{\mathbb{R}}$ arises as an eigenvalue of the
problem.

In order to approach the problem consisting of (2.1) and (2.2), we write down
the action functional

$I(u)=\frac{1}{2}\int_{0}^{R}\left\\{ru^{2}_{r}+\frac{n^{2}}{r}u^{2}-rV(r)u^{2}-\frac{r}{2}u^{4}\right\\}\,\mbox{d}r,$
(2.3)

and the constraint functional

$P(u)=\int|\psi|^{2}r\,\mbox{d}r\mbox{d}\theta=2\pi\int_{0}^{R}ru^{2}\,\mbox{d}r,$
(2.4)

which measures the beam power [28] of the vortex wave. Thus, to get a solution
of (2.1)–(2.2), it suffices to prove the existence of a solution to the
following constrained minimization problem

$\min\left\\{I(u)\,|\,u\in{\cal A},P(u)=P_{0}\right\\},\quad P_{0}>0,$ (2.5)

where the admissible class $\cal A$ is defined by

${\cal A}=\left\\{u(r)\mbox{ is absolutely continuous over
}[0,R],\,u(0)=u(R)=0,\,E(u)<\infty\right\\},$ (2.6)

with

$E(u)=\frac{1}{2}\int_{0}^{R}\left\\{ru^{2}_{r}+\frac{1}{r}u^{2}+\frac{r}{2}u^{4}\right\\}\,\mbox{d}r,$
(2.7)

being the ‘energy’ functional, $P_{0}$ is a prescribed value for the beam
power, and $\beta$ arises as the Lagrange multiplier. Note that the finite-
energy condition $E(u)<\infty$ is imposed only to ensure that all the terms in
the indefinite action functional (2.3) stay finite.

For convenience, for a function $f$ of the variable $r$, we interchangeably
use $f_{r}$ and $f^{\prime}$ to denote its derivative. We will also need the
following decomposition and notation

$\left.\begin{array}[]{rll}V&=&V^{+}-V^{-},\quad V^{\pm}=\max\\{\pm V,0\\},\\\
&&\\\ V_{0}^{\pm}&=&\max\\{V^{\pm}(r)\,|\,r\in[0,R]\\},\\\ &&\\\
V_{0}&=&\max\\{|V(r)|\,|\,r\in[0,R]\\}=\max\\{V^{+}_{0},V^{-}_{0}\\}.\end{array}\right\\}$
(2.8)

The main results of this section may be stated as follows.

###### Theorem 2.1

. For any nonzero integer $n$ and a given continuous potential function $V(r)$
over the interval $[0,R]$ ($R>0$), consider the two-point boundary value
problem (2.1)–(2.2) governing an $n$-vortex wave solution of the nonlinear
Schrödinger equation (1.1), propagating along the $z$-axis with a propagation
constant $\beta$.

(i) The problem always has a solution pair $(u,\beta)$ with $u(r)>0$,
$r\in(0,R)$, and $\beta\in{\mathbb{R}}$, so that the associated beam power
enjoys the bound $P(u)<4\pi|n|$. In fact, such a solution may be obtained
through solving the constrained minimization problem (2.5) assuming
$P_{0}<4\pi|n|$, from which $\beta$ arises as a Lagrange multiplier.

(ii) Let $(u,\beta)$ be the solution pair of the problem (2.1)–(2.2) obtained
in part (i). Then $\beta$ has the lower bound

$\beta\geq\frac{7P_{0}}{5\pi R^{2}}-V_{0}^{-}-\frac{12}{R^{2}}(1+n^{2}[2\ln
2-1]).$ (2.9)

(iii) Let $(u,\beta)$ be the solution pair of the problem (2.1)–(2.2) obtained
in part (i). Then $\beta<0$ if $|n|$ is sufficiently large so that

$|n|>\left\\{\frac{P_{0}^{2}}{4\pi^{2}}+\max\\{r^{2}V^{+}(r)\,|\,r\in[0,R]\\}\right\\}^{\frac{1}{2}}.$
(2.10)

(iv) The problem (2.1)–(2.2) has no nontrivial small-beam-power solution
satisfying $P(u)\leq\frac{1}{2}$ if the condition

$n^{2}>r^{2}(V^{+}(r)-\beta),\quad r\in[0,R],$ (2.11)

is fulfilled. So, roughly speaking, the problem has no nontrivial small-power
$P$ and small-propagation-constant (i.e., $|\beta|$ is sufficiently small)
solution over a small interval $[0,R]$.

We now establish these results.

(i) For any function $u$ satisfying $u(0)=0$, the Schwartz inequality implies
that

$u^{2}(r)=\int^{r}_{0}2u(\rho)u_{\rho}(\rho)\,\mbox{d}\rho\leq
2\left(\int^{r}_{0}\rho
u_{\rho}^{2}(\rho)\,\mbox{d}\rho\right)^{\frac{1}{2}}\left(\int_{0}^{r}\frac{u^{2}(\rho)}{\rho}\,\mbox{d}\rho\right)^{\frac{1}{2}}.$
(2.12)

Thus, multiplying (2.12) by $ru^{2}$, integrating, and using $P(u)=P_{0}$, we
have

$\displaystyle\int_{0}^{R}ru^{4}\,\mbox{d}r$ $\displaystyle\leq$
$\displaystyle\frac{1}{\pi}P_{0}\left(\int^{R}_{0}\rho
u_{\rho}^{2}(\rho)\,\mbox{d}\rho\right)^{\frac{1}{2}}\left(\int_{0}^{R}\frac{u^{2}(\rho)}{\rho}\,\mbox{d}\rho\right)^{\frac{1}{2}}$
(2.13) $\displaystyle\leq$ $\displaystyle\varepsilon\int^{R}_{0}\rho
u_{\rho}^{2}(\rho)\,\mbox{d}\rho+\frac{1}{\varepsilon}\left(\frac{P_{0}}{2\pi}\right)^{2}\int_{0}^{R}\frac{u^{2}(\rho)}{\rho}\,\mbox{d}\rho.$

Inserting (2.13) into (2.3), we obtain

$I(u)\geq\frac{1}{2}\left(1-\frac{\varepsilon}{2}\right)\int_{0}^{R}ru_{r}^{2}\,\mbox{d}r+\frac{1}{2}\left(n^{2}-\frac{P_{0}^{2}}{8\pi^{2}\varepsilon}\right)\int_{0}^{R}\frac{u^{2}}{r}\,\mbox{d}r-\frac{1}{4\pi}P_{0}V_{0}.$
(2.14)

In order to be able to find a suitable $\varepsilon>0$ such that in (2.14) we
have

$1-\frac{\varepsilon}{2}>0,\quad
n^{2}-\frac{P_{0}^{2}}{8\pi^{2}\varepsilon}>0,$ (2.15)

simultaneously, it suffices to assume that $P_{0}$ satisfies the condition

$P_{0}<4\pi|n|.$ (2.16)

In this situation, we can find two positive constants $C_{1},C_{2}$, depending
on $\varepsilon,n,P_{0}$ but independent of $u$, such that

$I(u)\geq
C_{1}\int_{0}^{R}ru_{r}^{2}\,\mbox{d}r+C_{2}\int_{0}^{R}\frac{u^{2}}{r}\,\mbox{d}r-\frac{1}{4\pi}P_{0}V_{0}.$
(2.17)

Assume (2.16) and let $\\{u_{m}\\}$ be a minimizing sequence of (2.5). Then
the coercive inequality (2.17) gives us the bound

$\int_{0}^{R}r([u_{m}]_{r})^{2}\,\mbox{d}r+\int_{0}^{R}\frac{1}{r}u_{m}^{2}\,\mbox{d}r\leq
C,$ (2.18)

where $C>0$ is a constant independent of $m$.

Since both functionals $I$ and $P$ are even, we have $I(u_{m})\geq I(|u_{m}|)$
and $P(u_{m})=P(|u_{m}|)$, where we have also used the basic fact [13] that
for any function $u$ its distributional derivative must satisfy
$||u|_{r}|\leq|u_{r}|$. In other words, we see that the sequence $\\{u_{m}\\}$
may be modified so that each $u_{m}$ is nonnegative, $u_{m}\geq 0$. Thus we
may assume that the sequence $\\{u_{m}\\}$ consists of nonnegative-valued
functions. It is clear that these functions may be viewed as radially
symmetric functions over the disk
$D_{R}=\\{(x,y)\in{\mathbb{R}}^{2}\,|\,x^{2}+y^{2}\leq R^{2}\\}$ which all
vanish on $\partial D_{R}$. Moreover, since (2.18) holds, we see immediately
that $\\{u_{m}\\}$ is bounded under the radially symmetrically reduced norm
$\|\,\|$ where

$\|u\|^{2}=\int_{0}^{R}ru^{2}\,\mbox{d}r+\int_{0}^{R}ru_{r}^{2}\,\mbox{d}r,$
(2.19)

for the standard Sobolev space $W^{1,2}_{0}(D_{R})$ since

$\int_{0}^{R}ru^{2}\,\mbox{d}r\leq
R^{2}\int_{0}^{R}\frac{1}{r}u^{2}\,\mbox{d}r.$ (2.20)

Hence we may assume without loss of generality that $\\{u_{m}\\}$ converges
weakly to an element $u\in W^{1,2}_{0}(D_{R})$ as $m\to\infty$. Applying the
compact embedding $W^{1,2}(D_{R})\to L^{p}(D_{R})$ ($p\geq 1$), we see that
$u_{m}\to u$ strongly in $L^{p}(D_{R})$ as $m\to\infty$. Of course, $u$ is
radially symmetric as well. Thus we may write it as $u=u(r)$ which satisfies
$u(R)=0$. Moreover, it is clear that for any $\varepsilon\in(0,R)$,
$\\{u_{m}\\}$ is a bounded sequence in the space $W^{1,2}(\varepsilon,R)$.
Thus, using the compact embedding $W^{1,2}(\varepsilon,R)\to
C[\varepsilon,R]$, we see that $u_{m}\to u$ as $m\to\infty$ uniformly over
$[\varepsilon,R]$. Besides, similar to (2.12), we have for any pair
$r_{1},r_{2}\in(0,R)$, $r_{1}<r_{2}$, the inequality

$\displaystyle|u_{m}^{2}(r_{2})-u_{m}^{2}(r_{1})|$ $\displaystyle\leq$
$\displaystyle
2\left(\int^{r_{2}}_{r_{1}}r([u_{m}]_{r})^{2}\,\mbox{d}r\right)^{\frac{1}{2}}\left(\int_{r_{1}}^{r_{2}}\frac{u_{m}^{2}}{r}\,\mbox{d}r\right)^{\frac{1}{2}}$
(2.21) $\displaystyle\leq$ $\displaystyle
2C^{\frac{1}{2}}\left(\int_{r_{1}}^{r_{2}}\frac{u_{m}^{2}}{r}\,\mbox{d}r\right)^{\frac{1}{2}},$

where the constant $C>0$ is as given in (2.18). Letting $m\to\infty$ in
(2.21), we arrive at

$|u^{2}(r_{2})-u^{2}(r_{1})|\leq
2C^{\frac{1}{2}}\left(\int_{r_{1}}^{r_{2}}\frac{u^{2}}{r}\,\mbox{d}r\right)^{\frac{1}{2}}.$
(2.22)

However, in view of (2.18) and Fatou’s lemma, we have

$\displaystyle\int_{0}^{R}ru_{r}^{2}\,\mbox{d}r$ $\displaystyle\leq$
$\displaystyle\liminf_{m\to\infty}\int_{0}^{R}r([u_{m}]_{r})^{2}\,\mbox{d}r,$
(2.23) $\displaystyle\int_{0}^{R}\frac{1}{r}u^{2}\,\mbox{d}r$
$\displaystyle\leq$
$\displaystyle\liminf_{m\to\infty}\int_{0}^{R}\frac{1}{r}u_{m}^{2}\,\mbox{d}r.$
(2.24)

In particular, in view of (2.18) again, we see that $\frac{1}{r}u^{2}\in
L(0,R)$. Therefore the right-hand side of (2.22) tends to zero as
$r_{1},r_{2}\to 0$, which implies that the limit

$\eta_{0}=\lim_{r\to 0}u^{2}(r)$ (2.25)

exists. Since $\frac{1}{r}u^{2}\in L(0,R)$, we must have $\eta_{0}=0$. Hence
the boundary condition $u(0)=0$ is achieved.

Summarizing the above results, we conclude that the function $u$ obtained as
the limit of the minimizing sequence $\\{u_{m}\\}$ for the problem (2.5)
satisfies $u(0)=u(R)=0$, $u(r)\geq 0$ for all $r\in[0,R]$, $E(u)<\infty$, and

$I(u)\leq\liminf_{m\to\infty}I(u_{m}),\quad
P(u)=\lim_{m\to\infty}P(u_{m})=P_{0}.$ (2.26)

Thus, $u$ is a solution to (2.5). Consequently, there is some
$\beta\in{\mathbb{R}}$ such that $(u,\beta)$ verify the boundary value problem
(2.1)–(2.2).

If there is a point $r_{0}\in(0,R)$ such that $u(r_{0})=0$, then
$u_{r}(r_{0})=0$ since $r_{0}$ is a minimum point for the function $u(r)$.
Applying the uniqueness theorem for the initial value problem of ordinary
differential equations, we have $u(r)=0$ for all $r\in(0,R)$, which
contradicts the fact $P(u)=P_{0}>0$. This proves $u(r)>0$ for all $r\in(0,R)$.

(ii) Let $(u,\beta)$ be a solution pair just obtained. We next study the
quantity $\beta$ in (2.1). As a preparation, we establish

$\liminf_{r\to 0}\\{ru(r)|u_{r}(r)|\\}=0.$ (2.27)

Suppose otherwise that (2.27) is not valid. Then there are some
$\varepsilon_{0}>0$ and $r_{0}\in(0,R]$ such that

$ru(r)|u_{r}(r)|\geq\varepsilon_{0},\quad r\in(0,r_{0}),$ (2.28)

which leads to

$\infty=\int_{0}^{r_{0}}\frac{\varepsilon_{0}}{r}\,\mbox{d}r\leq\int_{0}^{r_{0}}u|u_{r}|\,\mbox{d}r\leq\left(\int_{0}^{r_{0}}\frac{1}{r}u^{2}\,\mbox{d}r\right)^{\frac{1}{2}}\left(\int_{0}^{r_{0}}ru_{r}^{2}\,\mbox{d}r\right)^{\frac{1}{2}},$
(2.29)

which contradicts with $E(u)<\infty$. So (2.27) is valid.

From (2.27), we can find a sequence $\\{r_{m}\\}$ so that $r_{m}\to 0$ as
$m\to\infty$ and

$\lim_{m\to\infty}\\{r_{m}u(r_{m})u_{r}(r_{m})\\}=0.$ (2.30)

Multiplying (2.1) by $u$, integrating over $[r_{m},R]$, letting $m\to\infty$,
and applying (2.30), we obtain

$\beta\int_{0}^{R}ru^{2}\,\mbox{d}r=\int_{0}^{R}(rVu^{2}+ru^{4})\,\mbox{d}r-\int_{0}^{R}\left(\frac{n^{2}}{r}u^{2}+ru_{r}^{2}\right)\mbox{d}r.$
(2.31)

Let $u_{0}$ be any absolutely continuous function satisfying
$E(u_{0})<\infty$, the boundary condition (2.2), and $P(u_{0})=P_{0}$. Since
$u$ solves (2.5), we have $I(u)\leq I(u_{0})$. As a consequence of this
observation, we have the bound

$\int_{0}^{R}\left(\frac{n^{2}}{r}u^{2}+ru_{r}^{2}\right)\mbox{d}r\leq\int_{0}^{R}\left(rVu^{2}+\frac{r}{2}u^{4}\right)\mbox{d}r+2I(u_{0}).$
(2.32)

Inserting (2.32) into (2.31), we obtain

$\frac{1}{2\pi}\beta{P_{0}}\geq-2I(u_{0})+\frac{1}{2}\int_{0}^{R}ru^{4}\,\mbox{d}r.$
(2.33)

To estimate the right-hand side of (2.33), we set $R=2a$ for convenience and
define

$u_{0}(r)=\left\\{\begin{array}[]{lrl}\frac{b}{a}r,&&0\leq r\leq a,\\\ &&\\\
\frac{b}{a}(2a-r),&&a\leq r\leq 2a.\end{array}\right.$ (2.34)

Therefore, after some calculation we have

$\displaystyle P_{0}=2\pi\int_{0}^{2a}ru_{0}^{2}\,\mbox{d}r$ $\displaystyle=$
$\displaystyle\frac{4\pi}{3}a^{2}b^{2},$ (2.35)
$\displaystyle\int_{0}^{2a}r(u_{0}^{\prime}(r))^{2}\,\mbox{d}r$
$\displaystyle=$ $\displaystyle 2b^{2},$ (2.36)
$\displaystyle\int_{0}^{2a}\frac{1}{r}u_{0}^{2}\,\mbox{d}r$ $\displaystyle=$
$\displaystyle 2b^{2}(2\ln 2-1),$ (2.37)
$\displaystyle\int_{0}^{2a}ru^{4}_{0}\,\mbox{d}r$ $\displaystyle=$
$\displaystyle\frac{2}{5}a^{2}b^{4}.$ (2.38)

Using (2.35)–(2.38) and (2.8), we get

$\displaystyle I(u_{0})$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{0}^{R}\left\\{r(u_{0}^{\prime}(r))^{2}+\frac{n^{2}}{r}u_{0}^{2}-\frac{r}{2}u_{0}^{4}\right\\}\,\mbox{d}r+\frac{1}{2}\int_{0}^{R}rV^{-}u^{2}\,\mbox{d}r$
(2.39) $\displaystyle\leq$ $\displaystyle b^{2}\left(1+n^{2}(2\ln
2-1)+\frac{1}{3}V_{0}^{-}a^{2}-\frac{1}{10}a^{2}b^{2}\right).$

Besides, applying the Schwartz inequality, we have

$\left(\int_{0}^{R}ru^{2}\,\mbox{d}r\right)^{2}\leq\frac{R^{2}}{2}\int_{0}^{R}ru^{4}\,\mbox{d}r.$
(2.40)

Thus, in view of (2.33), (2.39), and (2.40), we arrive at

$\frac{1}{2\pi}\beta P_{0}\geq
2b^{2}\left(\frac{7}{45}a^{2}b^{2}-\left[\frac{1}{12}V_{0}^{-}R^{2}+1+n^{2}(2\ln
2-1)\right]\right).$ (2.41)

Inserting $R=2a$ and $a^{2}b^{2}=3P_{0}/4\pi$, we obtain the lower estimate
for $\beta$:

$\beta\geq\frac{12}{R^{2}}\left(\frac{7}{60\pi}P_{0}-\left[\frac{1}{12}V_{0}^{-}R^{2}+1+n^{2}(2\ln
2-1)\right]\right).$ (2.42)

(iii) We next derive the sufficient condition stated to ensure $\beta<0$.

In fact, inserting (2.13) into (2.31), we have

$\frac{1}{2\pi}\beta
P_{0}\leq-(1-\varepsilon)\int_{0}^{R}ru^{2}_{r}\,\mbox{d}r-\int_{0}^{R}\left(\frac{n^{2}}{r}-\frac{P_{0}^{2}}{4\pi^{2}\varepsilon
r}-rV\right)u^{2}\,\mbox{d}r.$ (2.43)

For convenience, we may set $\varepsilon=1$ in (2.43). Thus, whenever the
inequality

$n^{2}>\frac{1}{4\pi^{2}}P_{0}^{2}+r^{2}V^{+}(r),\quad r\in[0,R],$ (2.44)

is fulfilled, we can conclude with $\beta<0$ since $u(r)>0$ for $r\in(0,R)$.

(iv) We now consider nonexistence. For any admissible function $u$, we may
view $u$ as a radially symmetric function defined over ${\mathbb{R}}^{2}$ with
support contained in the disk
$D_{R}=\\{(x,y)\in{\mathbb{R}}^{2}\,|\,x^{2}+y^{2}\leq R^{2}\\}$. Hence,
applying the classical Gagliardo–Nirenberg inequality over ${\mathbb{R}}^{2}$,
we deduce

$\int_{0}^{R}ru^{4}\,\mbox{d}r\leq
4\pi\int_{0}^{R}ru^{2}\,\mbox{d}r\int_{0}^{R}ru_{r}^{2}\,\mbox{d}r,$ (2.45)

From (2.31) and inserting (2.45) with $P(u)=P_{0}$, we have

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int_{0}^{R}(rVu^{2}+ru^{4})\,\mbox{d}r-\int_{0}^{R}\left(\frac{n^{2}}{r}u^{2}+ru_{r}^{2}+\beta
ru^{2}\right)\mbox{d}r$ (2.46) $\displaystyle\leq$
$\displaystyle(2P_{0}-1)\int_{0}^{R}ru^{2}_{r}\,\mbox{d}r-\int_{0}^{R}\left(\frac{n^{2}}{r^{2}}+\beta-V\right)ru^{2}\,\mbox{d}r.$

Therefore, when the conditions

$P_{0}\leq\frac{1}{2},\quad\frac{n^{2}}{r^{2}}-V(r)+\beta>0,\quad r\in(0,R],$
(2.47)

are imposed, we arrive at $u\equiv 0$, as anticipated. Thus, in this
situation, the problem consisting of (2.1)–(2.2) has no nontrivial solution.

The proof of Theorem 2.1 is complete.

##  3 Vortices as saddle points

In this section, we study the existence of optical vortices which are the
solutions of the boundary value problem (2.1)–(2.2) as the saddle points of
the action functional

$I_{\beta}(u)=\frac{1}{2}\int_{0}^{R}\left\\{ru^{2}_{r}+\frac{n^{2}}{r}u^{2}+(\beta-V(r))ru^{2}-\frac{r}{2}u^{4}\right\\}\,\mbox{d}r,$
(3.1)

with $|n|\geq 1$. We shall use a min-max theory approach. Suggested by the
discussion of the previous section, we introduce the function space $H$ which
is the completion of the space $X=\\{u\in C^{1}[0,R]\,|\,u(0)=u(R)=0\\}$ (the
set of differentiable functions over $[0,R]$ which vanish at the two endpoints
of the interval) equipped with the inner product

$\langle
u,v\rangle_{H}=\int_{0}^{R}\left(ru_{r}v_{r}+\frac{1}{r}uv\right)\,\mbox{d}r,\quad
u,v\in H.$ (3.2)

As seen in the discussion of the previous section, as a Hilbert space, $H$ may
be viewed as an embedded subspace of $W^{1,2}_{0}(D_{R})$ consisting of
radially symmetric functions such that any element $u\in H$ enjoys the desired
property $u(0)=0$.

In order to simplify the presentation of the study here, we assume that
$\beta$ satisfies

$\beta\geq V_{0}^{+}=\max\\{V^{+}(r)\,|\,r\in[0,R]\\}.$ (3.3)

Recall that a $C^{1}$-functional $I:H\to{\mathbb{R}}$ is said to enjoy the
Palais–Smale condition if for any sequence $\\{u_{m}\\}$ satisfying the
properties (i) $I(u_{m})\to\alpha$ as $m\to\infty$, and (ii)
$I^{\prime}(u_{m})\to 0$ as $m\to\infty$ as a sequence in the dual space of
$H$, one can extract a subsequence from $\\{u_{m}\\}$ which converges
(strongly) in $H$.

As an initial step, we have

###### Lemma 3.1

. The action functional $I_{\beta}$ given in (3.1) satisfies the Palais–Smale
condition.

Proof.It is straightforward to see that the functional (3.1) is $C^{1}$ over
$H$.

Let $\\{u_{m}\\}$ be a sequence in $H$ satisfying the properties

$\displaystyle I_{\beta}(u_{m})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{0}^{R}\left\\{r([u_{m}]_{r})^{2}+\frac{n^{2}}{r}u_{m}^{2}+\left(\beta-V(r)\right)ru_{m}^{2}-\frac{r}{2}u_{m}^{4}\right\\}\,\mbox{d}r$
(3.4) $\displaystyle\to$ $\displaystyle\alpha,\quad m\to\infty,$
$\displaystyle|I_{\beta}^{\prime}(u_{m})(v)|$ $\displaystyle=$
$\displaystyle\left|\int_{0}^{R}\left\\{r[u_{m}]_{r}v_{r}+\frac{n^{2}}{r}u_{m}v+(\beta-V(r))ru_{m}v-ru_{m}^{3}v\right\\}\,\mbox{d}r\right|$
(3.5) $\displaystyle\leq$
$\displaystyle\varepsilon_{m}\|v\|_{H},\quad\varepsilon_{m}\geq 0,\quad v\in
H,$

where $\varepsilon_{m}\to 0$ as $m\to\infty$. In (3.5), we may take $v=u_{m}$
to get

$\int_{0}^{R}ru_{m}^{4}\,\mbox{d}r\leq\int_{0}^{R}\left(r([u_{m}]_{r})^{2}+\frac{n^{2}}{r}u_{m}^{2}+(\beta-V(r))ru_{m}^{2}\right)\,\mbox{d}r+\varepsilon_{m}\|u_{m}\|_{H}.$
(3.6)

On the other hand, in view of (3.4), we may assume without loss of generality
that $I_{\beta}(u_{m})\leq\alpha+1$ for all $m=1,2,\cdots$. Hence, applying
(3.6), the assumption (3.3), and a simple interpolation inequality, we find

$\displaystyle 2(\alpha+1)$ $\displaystyle\geq$
$\displaystyle\frac{1}{2}\int_{0}^{R}\left(r([u_{m}]_{r})^{2}+\frac{n^{2}}{r}u_{m}^{2}+(\beta-V(r))ru_{m}^{2}\right)\,\mbox{d}r-\frac{1}{2}\varepsilon_{m}\|u_{m}\|_{H}$
(3.7) $\displaystyle\geq$
$\displaystyle\frac{1}{4}\|u_{m}\|^{2}_{H}-\frac{1}{4}\varepsilon_{m}^{2},\quad
m=1,2,\cdots.$

In other words, $\\{u_{m}\\}$ is a bounded sequence in $H$. Without loss of
generality, we may assume that $\\{u_{m}\\}$ weakly converges to an element
$u\in H$ as $m\to\infty$. It is clear that $u_{m}\to u$ as $m\to\infty$
strongly in any $L^{p}(D_{R})$ or $L^{p}((0,R),r\mbox{d}r)$ ($p\geq 1$). Thus,
letting $m\to\infty$ in (3.5), we arrive at

$\int_{0}^{R}\left\\{ru_{r}v_{r}+\frac{n^{2}}{r}uv+(\beta-V(r))ruv-
ru^{3}v\right\\}\,\mbox{d}r=0,\quad\forall v\in H.$ (3.8)

Let $v=u_{m}-u$ in (3.5) and (3.8) and insert the resulting (3.8) into the
resulting (3.5). We have

$\displaystyle\left|\int_{0}^{R}\left\\{r([u_{m}-u]_{r})^{2}+\frac{n^{2}}{r}(u_{m}-u)^{2}+(\beta-V)r(u_{m}-u)^{2}-r(u_{m}^{3}-u^{3})(u_{m}-u)\right\\}\,\mbox{d}r\right|$
$\displaystyle\leq\varepsilon_{m}\|u_{m}-u\|_{H}.$ (3.9)

As a consequence of (3.9) and $\beta\geq V_{0}^{+}$, we obtain

$\|u_{m}-u\|^{2}_{H}\leq\varepsilon_{m}\|u_{m}-u\|_{H}+\int_{0}^{R}r|(u_{m}^{3}-u^{3})(u_{m}-u)|\,\mbox{d}r,\quad
m=1,2,\cdots,$ (3.10)

which immediately implies that $u_{m}\to u$ strongly in $H$ as $m\to\infty$,
as desired.

We next identify a mountain-pass structure through the following two lemmas.

###### Lemma 3.2

. There are constants $K>0$ and $C_{0}>0$ such that

$\inf\\{I_{\beta}(u)\,|\,\|u\|_{H}^{2}=K\\}\geq C_{0}.$ (3.11)

Proof.For any constant $K>0$, let $u\in H$ satisfy $\|u\|^{2}_{H}=K>0$. From
(2.12), we have

$\int_{0}^{R}ru^{4}\,\mbox{d}r\leq
4\int_{0}^{R}r\,\mbox{d}r\int_{0}^{R}ru_{r}^{2}\,\mbox{d}r\int_{0}^{R}\frac{u^{2}}{r}\,\mbox{d}r\leq
2R^{2}K^{2}.$ (3.12)

Applying (3.12) in (3.1), we find

$I_{\beta}(u)\geq\frac{1}{2}\left(K-R^{2}K^{2}\right)\equiv f(K).$ (3.13)

However, the maximum of the function $f$ in (3.13) is attained at
$K_{0}=\frac{1}{2R^{2}}$ which gives us the value $f(K_{0})=\frac{1}{8R^{2}}$.
So, in conclusion, we have the lower bound

$I_{\beta}(u)\geq\frac{1}{8R^{2}},\quad\|u\|^{2}_{H}=\frac{1}{2R^{2}},$ (3.14)

which establishes (3.11).

###### Lemma 3.3

. For any constant $K>0$, there is an element $v\in H$ satisfying
$\|v\|_{H}^{2}>K$ and $I_{\beta}(v)<0$.

Proof.With $R=2a$, we will see that we can use the function $u_{0}$ defined in
(2.34) as a test function.

For this purpose, we first show that $u_{0}\in H$. To see this, we need to
prove that $u_{0}$ can be obtained in the limit from a sequence of functions
in $X$ under the norm of $H$. In fact, for any $0<\varepsilon<a$, we can
define

$u_{\varepsilon}(r)=\left\\{\begin{array}[]{rll}u_{0}(r),&r\in[0,a-\varepsilon)\cup(a+\varepsilon,2a],\\\
&\\\ Q_{\varepsilon}(r),&r\in[a-\varepsilon,a+\varepsilon],\end{array}\right.$
(3.15)

where $Q_{\varepsilon}(r)$ is taken to be a quadratic function satisfying

$Q_{\varepsilon}(a\pm\varepsilon)=u_{0}(a\pm\varepsilon),\quad
Q_{\varepsilon}^{\prime}(a\pm\varepsilon)=u_{0}^{\prime}(a\pm\varepsilon).$
(3.16)

Matching these conditions, we find

$Q_{\varepsilon}(r)=-\frac{b}{2a\varepsilon}(r^{2}-2ar+[a-\varepsilon]^{2}),\quad
r\in[a-\varepsilon,a+\varepsilon],$ (3.17)

which enjoys the bounds

$\frac{b}{a}(a-\varepsilon)\leq
Q_{\varepsilon}(r)\leq\frac{b}{a}\left(a-\frac{\varepsilon}{2}\right),\quad|Q^{\prime}_{\varepsilon}(r)|\leq\frac{b}{a},\quad
r\in[a-\varepsilon,a+\varepsilon].$ (3.18)

Consequently, from (3.18) we have

$\lim_{\varepsilon\to
0}\int_{a-\varepsilon}^{a+\varepsilon}\left(r[Q^{\prime}_{\varepsilon}(r)]^{2}+\frac{1}{r}Q_{\varepsilon}^{2}(r)\right)\,\mbox{d}r=0.$
(3.19)

Therefore $\\{u_{\varepsilon}\\}$ is a Cauchy sequence in $H$ as
$\varepsilon\to 0$ whose limit is clearly $u_{0}$ in view of the definition of
$\\{u_{\varepsilon}\\}$ given in (3.15). This proves $u_{0}\in H$.

Using the results (2.35)–(2.38), we have

$\displaystyle\|u_{0}\|^{2}_{H}$ $\displaystyle=$ $\displaystyle 4b^{2}\ln 2,$
(3.20) $\displaystyle I_{\beta}(u_{0})$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{0}^{R}\left\\{r([u_{0}]_{r})^{2}+\frac{n^{2}}{r}u_{0}^{2}+(\beta+V^{-})ru_{0}^{2}-\frac{r}{2}u_{0}^{4}\right\\}\,\mbox{d}r$
(3.21) $\displaystyle\leq$ $\displaystyle b^{2}\left(1+n^{2}(2\ln
2-1)+\frac{1}{3}a^{2}(\beta+V^{-}_{0})-\frac{1}{10}a^{2}b^{2}\right).$

From (3.20) and (3.21), we see that for any $K>0$ we may choose $b>0$
sufficiently large to get $I_{\beta}(u_{0})<0$ and $\|u_{0}\|^{2}_{H}>K$.

Thus the lemma follows.

It is interesting to note that (3.21) implies that
$I_{\beta}(u_{0})\to-\infty$ as $b\to\infty$. In other words, the action
functional (3.1) is not bounded from below over $H$ which prevents a direct
minimization approach to the problem. Indeed, we are now prepared to obtain a
nontrivial solution of the boundary value problem (2.1)–(2.2) as a saddle
point of the functional (3.1) in the following theorem.

###### Theorem 3.4

. For any $\beta\geq V_{0}^{+}$, $|n|\geq 1$, and $R>0$, the problem
(2.1)–(2.2) has a nontrivial solution over the interval $[0,R]$. Moreover,
such a solution may be obtained from a min-max approach applied to the action
functional (3.1).

Proof.Let $I_{\beta}$ be the action functional (3.1). Then Lemma 3.1 says that
$I_{\beta}$ satisfies the Palais–Smale condition. Let $K,C_{0}>0$ be the
constant stated in Lemma 3.2. Using Lemma 3.3, we can find some $u_{0}\in H$
such that $\|u_{0}\|_{H}^{2}>K$ and $I_{\beta}(u_{0})<0$. Denote by $\Gamma$
the set of all continuous paths in $H$ that link the zero element $0$ of $H$
to $u_{0}$:

$\Gamma=\left\\{g\in C([0,1];H)\,|\,g(0)=0,g(1)=u_{0}\right\\}.$ (3.22)

Therefore there is some point $t_{g}\in(0,1)$ such that
$\|g(t_{g})\|^{2}_{H}=K$. From the classical mountain-pass theorem (cf. Evans
[12]), we know that

$c=\inf_{g\in\Gamma}\max_{t\in[0,1]}I_{\beta}(g(t))\geq C_{0},$ (3.23)

is a critical value of $I_{\beta}$. In other words, there is an element $u\in
H$ satisfying $I_{\beta}(u)=c$ which is a critical point of $I_{\beta}$. Of
course, $u$ is nontrivial. That is, $u$ cannot be the zero element of $H$.

Recall that in Theorem 2.1 the condition (2.10) is obtained to ensure the wave
propagation constant $\beta$ to assume a negative value. Although (2.9) states
a lower estimate for $\beta$, no condition has been obtained to ensure
$\beta>0$ for the solution of the constrained minimization problem (2.5).
Theorem 3.4, however, complements Theorem 2.1 in that it gives us a family of
nontrivial solutions realizing arbitrarily prescribed parameter $\beta$ in the
entire interval $[V_{0}^{+},\infty)$ for any vortex charge $|n|\geq 1$ and
$R>0$. That is, our existence result indicates that $\beta$ may take any
positive value above or at $V^{+}_{0}$.

##  4 The defocusing case when $s=-1$

If we have $s=-1$ in (1.1) instead, then the action functional (2.3) is
replaced by

$I(u)=\frac{1}{2}\int_{0}^{R}\left\\{ru^{2}_{r}+\frac{n^{2}}{r}u^{2}-rV(r)u^{2}+\frac{r}{2}u^{4}\right\\}\,\mbox{d}r,$
(4.1)

the difficulty with the quartic term, which was negative before, disappears,
and the constrained minimization problem (2.5) is easily solved, which gives
us a solution to the associated equation

$(ru_{r})_{r}-\frac{n^{2}}{r}u+r(V-u^{2})u=\beta ru,$ (4.2)

for some $\beta\in{\mathbb{R}}$. As before, this equation leads us to the
relation

$\int_{0}^{R}(\beta-V^{+})ru^{2}\,\mbox{d}r=-\int_{0}^{R}\left(ru_{r}^{2}+\frac{n^{2}}{r}u^{2}+rV^{-}u^{2}+ru^{4}\right)\mbox{d}r.$
(4.3)

Thus, if $\beta$ satisfies $\beta\geq V^{+}_{0}$, then $u\equiv 0$. Therefore,
regardless of the value of $R$, the problem prevents the existence of a
nontrivial solution for sufficiently large propagation constant $\beta$. This
conclusion is in sharp contrast to that in the case when $s=+1$ stated in
Theorem 3.4.

In general, the simple relation (4.3) clearly indicates that it is natural for
$\beta$ to take negative rather than positive values. For example, using the
Poincaré inequality over $D_{R}$,

$\int_{0}^{R}ru^{2}\,\mbox{d}r\leq\left(\frac{R}{r_{0}}\right)^{2}\int_{0}^{R}ru^{2}_{r}\,\mbox{d}r,$
(4.4)

where $r_{0}$ ($\approx 2.404825$) is the first positive zero of the Bessel
function $J_{0}$, and (2.20), we obtain from (4.3) the result

$\displaystyle\beta\int_{0}^{R}ru^{2}\,\mbox{d}r$ $\displaystyle\leq$
$\displaystyle
V_{0}^{+}\int_{0}^{R}ru^{2}\,\mbox{d}r-\int_{0}^{R}\left(ru_{r}^{2}+\frac{n^{2}}{r}u^{2}+rV^{-}u^{2}+ru^{4}\right)\mbox{d}r$
(4.5) $\displaystyle\leq$
$\displaystyle-\left(\left[\frac{r_{0}}{R}\right]^{2}+\left[\frac{n}{R}\right]^{2}-V_{0}^{+}\right)\int_{0}^{R}ru^{2}\,\mbox{d}r-\int_{0}^{R}ru^{4}\,\mbox{d}r.$

Consequently, we obtain

$\beta<-\left(\frac{r_{0}^{2}+n^{2}}{R^{2}}-V_{0}^{+}\right).$ (4.6)

In particular, we have $\beta\to-\infty$ as $|n|\to\infty$.

An example of the defocusing case $s=-1$ is the study carried out in [18]
where $V$ takes the form $V=pJ^{2}_{1}(br)$ given in terms of the Bessel
function $J_{1}$ and positive parameters $p,b$. Thus $V^{-}\equiv 0$. Another
example of the case $s=-1$ is in the lines of the studies [24, 30]. There,
although $V$ is not radially symmetric, it is non-positive valued,
$V^{+}\equiv 0$. Here, assuming $V$ is radial as well as non-positive, then
(4.3) indicates that $\beta<0$ is the only possibility.

Note that the ($z$-direction) angular momentum of the obtained stationary
vortex wave in view of (1.2) has the simple but elegant expression [28]

$L_{z}=\mbox{Im}\int(\psi^{*}\partial_{\theta}\psi)\,r\mbox{d}r\mbox{d}\theta=2\pi
n\int_{0}^{R}ru^{2}\,\mbox{d}r=nP.$ (4.7)

Write $\phi(x,y)=u(r)\mbox{e}^{\mbox{i}n\theta}$, where $u$ solves (1.3) and
satisfies $u(0)=0$, and $\Delta=\nabla^{2}_{\perp}$. Then $\phi$ satisfies

$\Delta\phi+(V+s|\phi|^{2})\phi=\beta\phi,$ (4.8)

away from the origin of ${\mathbb{R}}^{2}$. The condition $u(0)=0$ ensures
that the origin is a removable singularity [35] such that when $V$ is an
analytic function of $(x,y)$, so is $\phi$. Consequently, in this situation
$u$ vanishes at $r=0$ like $r^{n}$ for an $n$-vortex solution as in the
classical Ginzburg–Landau equation case [8, 14, 25].

For the focusing case $s=+1$ with a non-positive potential (cf. [34]), we have
$V^{+}\equiv 0$ and the statements of our results simplify considerably. For
example, for the solution pair $(u,\beta)$ obtained in Theorem 2.1 to have the
property $\beta<0$, it suffices that the vortex number $n$ satisfies the
condition

$|n|>\frac{P_{0}}{2\pi}.$ (4.9)

Moreover, applying Theorem 2.1 (iv), we see that there is no nontrivial
solution satisfying

$P(u)\leq\frac{1}{2}\quad\mbox{and}\quad n^{2}>-r^{2}\beta,\quad r\in[0,R].$
(4.10)

In particular, we conclude that there is no nontrivial solution with
$P(u)\leq\frac{1}{2}$ and $\beta\geq 0$. Besides, in this case Theorem 3.4
becomes an existence theory for any prescribed propagation constant $\beta\geq
0$.

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