# Existence and Asymptotic Behavior of Ground States for Rotating Bose-
Einstein Condensates

Yujin Guo, Yong Luo and Shuangjie Peng School of Mathematics and Statistics,
and Hubei Key Laboratory of Mathematical Sciences, Central China Normal
University, P.O. Box 71010, Wuhan 430079, P. R. China. Email:
<EMAIL_ADDRESS>Y. Guo is partially supported by NSFC under Grant No.
11931012 and the Fundamental Research Funds for the Central Universities (No.
KJ02072020-0319). School of Mathematics and Statistics, and Hubei Key
Laboratory of Mathematical Sciences, Central China Normal University, P.O. Box
71010, Wuhan 430079, P. R. China. Email<EMAIL_ADDRESS>Y. Luo is
partially supported by the Project funded by China Postdoctoral Science
Foundation No. 2019M662680.School of Mathematics and Statistics, and Hubei Key
Laboratory of Mathematical Sciences, Central China Normal University, P.O. Box
71010, Wuhan 430079, P. R. China. Email<EMAIL_ADDRESS>S. Peng is
partially supported by the Key Project of NSFC under Grant No. 11831009.

###### Abstract

We study ground states of two-dimensional Bose-Einstein condensates with
repulsive ($a>0$) or attractive ($a<0$) interactions in a trap $V(x)$ rotating
at the velocity $\Omega$. It is known that there exist critical parameters
$a^{*}>0$ and $\Omega^{*}:=\Omega^{*}(V(x))>0$ such that if
$\Omega>\Omega^{*}$, then there is no ground state for any $a\in{\mathbb{R}}$;
if $0\leq\Omega<\Omega^{*}$, then ground states exist if and only if
$a\in(-a^{*},+\infty)$. As a completion of the existing results, in this
paper, we focus on the critical case where $0<\Omega=\Omega^{*}<+\infty$ and
classify the existence and nonexistence of ground states for
$a\in{\mathbb{R}}$. Moreover, for a suitable class of radially symmetric traps
$V(x)$, employing the inductive symmetry method, we prove that up to a
constant phase, the ground states must be real-valued, unique and free of
vortices as $\Omega\searrow 0$, no matter whether the interactions of the
condensates are repulsive or not.

Keywords: Bose-Einstein condensate; rotational velocity; ground states; free
of vortices

##  1 Introduction

Because Bose-Einstein condensates (BECs) can show the quantum effects at the
macroscopic scale, they have become an important subject in experimental
investigations since their first realization in dilute gases of alkali atoms
in 1995 [1, 2, 24, 10]. Various interesting quantum phenomena have been so far
observed in BECs experiments, such as the critical-mass collapse [10], the
appearance of quantized vortices [1], the center-of-mass rotation in rotating
traps [1, 30], and so on. In particular, the complex structures of BECs in
rotating traps have been observed and analyzed extensively (cf. [1, 2, 5, 9,
10, 12]) starting from the late 1990s.

As addressed in [12, 24, 27], the two-dimensional BECs in a rotating trap
$V(x)$ can be well described by the complex-valued wave functions minimizing
the following Gross-Pitaevskii (GP) energy functional:

$F_{\Omega,a}(u):=\int_{{\mathbb{R}}^{2}}\big{(}|\nabla
u|^{2}+V(x)|u|^{2}+\frac{a}{2}|u|^{4}\big{)}dx-\Omega\int_{{\mathbb{R}}^{2}}x^{\perp}\cdot(iu,\,\nabla
u)dx,\ \ u\in{\mathcal{H}},$ (1.1)

where the space ${\mathcal{H}}$ is defined as

${\mathcal{H}}:=\Big{\\{}u\in H^{1}({\mathbb{R}}^{2},\mathbb{C}):\
\int_{{\mathbb{R}}^{2}}V(x)|u|^{2}dx<\infty\Big{\\}},$ (1.2)

$x^{\perp}=(-x_{2},x_{1})$ with $x=(x_{1},x_{2})\in{\mathbb{R}}^{2}$, and
$(iu,\,\nabla u)=i(u\nabla\bar{u}-\bar{u}\nabla u)/2$. Here the two-body
interaction of the rotating BECs can be either repulsive ($a>0$) or attractive
($a<0$), and the parameter $\Omega\geq 0$ describes the rotational velocity of
the rotating trap. Consequently, the ground states of two-dimensional BECs in
a rotating trap $0\leq V(x)\in L^{\infty}_{loc}({\mathbb{R}}^{2})$ can be
equivalently described (see [12, 24, 27, 31]) by the minimizers of the
following variational problem:

$e(\Omega,a):=\inf_{\\{u\in{\mathcal{H}},\,\|u\|^{2}_{2}=1\\}}F_{\Omega,a}(u),\,\
\hbox{$\Omega\geq 0$, $a\in{\mathbb{R}}$},$ (1.3)

where the GP energy functional $F_{\Omega,a}(\cdot)$ is as in (1.1), and the
space ${\mathcal{H}}$ is defined by (1.2). Here the value $|a|$ characterizes
the absolute product for the scattering length of the two-body interaction
times the number $N$ of particles in the condensates.

We first introduce the non-rotational case $\Omega=0$ of $e(\Omega,a)$. As
illustrated soon after (1.17) (see also [17, Section 2]), in this case it
suffices to consider the real-valued minimizers of $e(a):=e(0,a)$, where we
define

$e(a):=\inf_{\\{u\in{\mathcal{H}},\,\|u\|^{2}_{2}=1\\}}F_{a}(u),$ (1.4)

and the GP energy functional $F_{a}(u)$ is given by

$F_{a}(u):=\int_{\mathbb{R}^{2}}\big{(}|\nabla
u|^{2}+V(x)|u|^{2}+\frac{a}{2}|u|^{4}\big{)}dx.$ (1.5)

When $a>0$, since the functional $F_{a}(u)$ is convex, one can obtain the
existence and uniqueness of real-valued minimizers for $e(a)$ in a direct way,
see for instance [27, Theorem 2.1]. However, when $a<0$, it was shown in [15,
18, 19] that there exists a critical constant $a^{*}>0$ such that $e(a)$
admits real-valued minimizers, if and only if $a>-a^{*}$. More precisely, the
critical constant $a^{*}$ is given by

$a^{*}:=\|w\|^{2}_{L^{2}({\mathbb{R}}^{2})}>0,$ (1.6)

where $w=w(|x|)>0$ is a positive solution of the following scalar field
equation

$-\Delta w+w-w^{3}=0\ \mbox{ in }\ {\mathbb{R}}^{2},\ \mbox{ where }\ w\in
H^{1}({\mathbb{R}}^{2},{\mathbb{R}}),$ (1.7)

which, up to translations, must be unique and radially symmetric (cf. [23,
35]). Moreover, the refined mass concentration and local uniqueness of real-
valued minimizers of $e(a)$ as $a\searrow-a^{*}$ were also studied in [15, 19]
and the references therein.

As for the rotational case $\Omega>0$ of $e(\Omega,a)$, we consider the
general trapping potential $0\leq V(x)\in L^{\infty}_{loc}({\mathbb{R}}^{2})$
satisfying

$\underline{\lim}_{|x|\to\infty}\frac{V(x)}{|x|^{2}}>0.$ (1.8)

Under the assumption (1.8), we define the critical rotational velocity
$\Omega^{*}=\Omega^{*}(V(x))$ by

$\Omega^{*}:=\sup\Big{\\{}\Omega>0:\ \
V(x)-\frac{\Omega^{2}}{4}|x|^{2}\to\infty\,\ \mbox{as}\,\
|x|\to\infty\Big{\\}},$ (1.9)

so that $\Omega^{*}>0$. Considering $V(x)=|x|^{s}$ $(s\geq 2)$ as an example,
we have

$\Omega^{*}:=\left\\{\begin{array}[]{ll}2,&{\rm if}\quad s=2;\\\\[5.69054pt]
\infty,&{\rm if}\quad s>2,\end{array}\right.$ (1.10)

which illustrates that both $0<\Omega^{*}<\infty$ and $\Omega^{*}=\infty$ can
happen. Recalling from [17, 24, 34], we have the following existence and non-
existence of minimizers:

Theorem A.  ([17, 24, 34]) Assume $V(x)\in L^{\infty}_{\rm
loc}({\mathbb{R}}^{2})$ satisfies (1.8), and let $\Omega^{*}>0$ and $a^{*}>0$
be defined in (1.9) and (1.6), respectively. Then we have

  1. 1.

If $\Omega\in(0,\Omega^{*})$ and $a\in(-a^{*},+\infty)$, then there exists at
least one minimizer for $e(\Omega,a)$.

  2. 2.

If $\Omega\in(0,\Omega^{*})$ and $a\in(-\infty,-a^{*}]$, then there is no
minimizer for $e(\Omega,a)$.

  3. 3.

If $\Omega>\Omega^{*}$, then there is no minimizer of $e(\Omega,a)$ for any
$a\in{\mathbb{R}}$.

One can note from Theorem A that it remains open to discuss the existence and
non-existence of minimizers for $e(\Omega^{*},a)$, provided that the critical
rotational velocity $\Omega^{*}=\Omega^{*}(V(x))$ defined in (1.9) satisfies
$\Omega^{*}<+\infty$. The first main purpose of this paper is to address this
issue.

When $0<\Omega<\Omega^{*}$, the detailed analytical properties of minimizers
for $e(\Omega,a)$ as either $a\nearrow+\infty$ or $a\searrow-a^{*}$ were
studied recently in [2, 21, 22, 9, 16, 17, 24] and the references therein.
Specially, if the interactions in the condensates are repulsive ($a>0$), all
kinds of quantized vortices for $e(\Omega,a)$ have been analyzed extensively
in the Thomas-Fermi regime where $a\nearrow+\infty$, see [1, 2, 4, 5, 9, 10,
21, 22] for instance. In particular, the nonexistence of vortices for
repulsive BECs under rotation were analyzed in [2, 4, 5]. However, as far as
we know, the above mentioned works of studying vortex structures focus more on
the analysis of the energy, which seems not enough for the attractive case
$a<0$ of $e(\Omega,a)$. Fortunately, the nonexistence of vortices for
$e(\Omega,a)$ in the attractive limiting case as $a\searrow-a^{*}$ was proved
in our recent work [17] by developing the inductive symmetry method. The
second main purpose of this paper is to address the nonexistence of vortices
for $e(\Omega,a)$ as $\Omega\searrow 0$ by a uniform approach, no matter
whether the fixed parameter $a>-a^{*}$ is positive or not.

###  1.1 Main results

In this subsection we shall introduce the main results of the present paper.
We are first concerned with the existence and non-existence of minimizers for
$e(\Omega^{*},a)$, provided that the critical rotational velocity
$\Omega^{*}=\Omega^{*}(V(x))$ defined in (1.9) satisfies $\Omega^{*}<+\infty$.
Towards this aim, we consider

$V(x)=A|x|^{2}+W(x),\,\ \hbox{where $A>0$ and $W(x)\in C({\mathbb{R}}^{2})$
satisfies $\lim_{|x|\to\infty}\frac{W(x)}{|x|^{2}}=0$,}$ (1.11)

so that the assumption (1.8) holds in a natural way. Under the assumption
(1.11), our first main result of this paper is devoted to the following
existence and non-existence of minimizers for $e(\Omega^{*},a)$:

######  Theorem 1.1.

Suppose that $a^{*}>0$ is defined by (1.6), and $V(x)$ satisfies (1.11) such
that $\Omega^{*}=2\sqrt{A}>0$ is defined by (1.9). Then we have

  1. (i).

If $W(x)$ satisfies $\lim_{|x|\to\infty}W(x)=+\infty$, then $e(\Omega^{*},a)$
admits minimizers if and only if $a\in(-a^{*},\infty)$.

  2. (ii).

If $W(x)\equiv C\in{\mathbb{R}}$, then $e(\Omega^{*},a)$ admits minimizers if
and only if $a\in(-a^{*},0]$.

When the non-constant function $W(x)$ is bounded uniformly in
${\mathbb{R}}^{2}$, the existence and non-existence of minimizers for
$e(\Omega^{*},a)$ are more complicated. The following theorem addresses this
case:

######  Theorem 1.2.

Suppose that $a^{*}>0$ is defined by (1.6), and $V(x)$ satisfies (1.11) such
that $\Omega^{*}=2\sqrt{A}>0$ is defined by (1.9). If $W(x)\not\equiv const.$
further satisfies $\lim_{|x|\to\infty}W(x)=B\in{\mathbb{R}}$, then we have

  1. (i).

If $W(x)>B$ in ${\mathbb{R}}^{2}$, then $e(\Omega^{*},a)$ admits no minimizer
for any $a\in{\mathbb{R}}$.

  2. (ii).

If $W(x)<B$ in ${\mathbb{R}}^{2}$, then there exists a constant
$a_{\ast}\in(0,+\infty]$ such that

    1. (a).

If $a\in(-a^{\ast},a_{\ast})$, there is at least one minimizer for
$e(\Omega^{*},a)$.

    2. (b).

If $a\in(-\infty,-a^{*}]\cup(a_{\ast},+\infty)$, there is no minimizer for
$e(\Omega^{*},a)$.

The proof of Theorem 1.2 (ii) shows that the constant $a_{\ast}\in(0,+\infty]$
of Theorem 1.2 (ii) can be characterized as

$a_{\ast}:=\sup\big{\\{}a\in{\mathbb{R}}:\,e(\Omega^{*},a)<2\sqrt{A}+B\,\big{\\}}.$
(1.12)

We remark that the energy $2\sqrt{A}+B$ is maximal to guarantee that vanishing
cannot occur for the minimizing sequences of $e(\Omega^{*},a)$. Theorem
1.2(ii) does not address the existence of minimizers for
$e(\Omega^{*},a_{\ast})$, which may depend on the shape of $W(x)<B$ in
${\mathbb{R}}^{2}$. Additionally, if $W(x)$ satisfies

$\liminf_{|x|\to\infty}\big{[}B-W(x)\big{]}|x|^{s}>0,\quad\hbox{where
$0<s<2$,}$ (1.13)

then the following corollary shows that $a_{\ast}=+\infty$ holds true.

######  Corollary 1.3.

Under the assumptions of Theorem 1.2 (ii), if $W(x)$ satisfies (1.13), then
$e(\Omega^{*},a)$ admits minimizers if and only if $a\in(-a^{*},+\infty)$.

Related to Corollary 1.3, we expect that if (1.13) holds for $s>2$, then it
might hold that $a_{\ast}<\infty$, which however we cannot address rigorously
at present.

As far as we know, most existing results focus mainly on the noncritical case
$e(\Omega,a)$, where $\Omega\not=\Omega^{*}$ and the energy functional is
coercive. But the existence and non-existence results of minimizers for the
critical case $e(\Omega^{*},a)$ seem very few, no matter whether the
interactions among the BEC system are repulsive or not. The proof of above
results shows actually that the analysis of the critical case
$e(\Omega^{*},a)$ is more challenging, compared with those involved in [27, 2,
17] for the noncritical case $e(\Omega,a)$ where $\Omega\not=\Omega^{*}$. The
above results also illustrate that whether the minimizers of $e(\Omega^{*},a)$
exist or not depends subtly on the shape of the trap $V(x)$ and the parameter
$a\in{\mathbb{R}}$ as well.

It should also be mentioned that the mass-subcritical version of
$e(\Omega,a)$, where the nonlinear term $|u|^{4}$ is replaced by $|u|^{p}$ for
$2<p<4$, was studied in the pioneering work of Esteban-Lions [11] by the
concentration-compactness lemma. However, those methods used in [11] are not
sufficient to prove Theorem 1.1, especially in the repulsive case where $a>0$.
To overcome this difficulty, we shall borrow some ideas from [2, 3], where the
properties of the first eigenpairs of

$-\Delta\psi+2i\,(x^{\perp}\cdot\nabla\psi)+|x|^{2}\psi=\lambda\psi\quad\hbox{in}\,\
{\mathbb{R}}^{2}$ (1.14)

were fully employed. Note from [29, Theorem 2] that the first eigenvalue of
(1.14) is $\lambda_{1}=2$ and all the first eigenfunctions are given by

$S:=\Big{\\{}e^{-\frac{|x|^{2}}{2}}f(x):f(x)\,\ \hbox{is any entire function
such that}\,\ e^{-\frac{|x|^{2}}{2}}f(x)\in
L^{2}({\mathbb{R}}^{2},\mathbb{C})\Big{\\}},$ (1.15)

where the entire function means the complex analytic function on the entire
plane. In Section 2 below, we shall seek for appropriate test functions in $S$
to derive the desired energy estimates, which then help us obtain the
existence and nonexistence of minimizers for the repulsive case where $a>0$.

We now recall the non-rotational case of $e(\Omega,a)$ at $\Omega=0$, i.e.,
$e(a):=e(0,a)$ defined in (1.4). By the variational theory, any minimizer
$u_{a}$ of $e(a)$ satisfies the following Euler-Lagrange equation

$-\Delta u_{a}+V(x)u_{a}=\mu_{a}u_{a}-a|u_{a}|^{2}u_{a}\ \ \hbox{in}\,\
{\mathbb{R}}^{2},$ (1.16)

where $\mu_{a}\in{\mathbb{R}}$ is the associated Lagrange multiplier
satisfying

$\mu_{a}=e(a)+\frac{a}{2}\int_{\mathbb{R}^{2}}|u_{a}|^{4}dx.$ (1.17)

Since $F_{a}(u)\geq F_{a}(|u|)$, we obtain that $|u_{a}|$ is also a minimizer
of $e(a)$ and $F_{a}(u_{a})=F_{a}(|u_{a}|)$ holds. Moreover, we can deduce
from (1.16) that $|u_{a}|>0$ by the maximal principle. Since
$\int_{\mathbb{R}^{2}}|\nabla
u_{a}|^{2}dx=\int_{\mathbb{R}^{2}}\big{|}\nabla|u_{a}|\big{|}^{2}dx$ and
$|u_{a}|>0$, we conclude that there exists a constant $\theta\in(0,2\pi]$ such
that $u_{a}(x)\equiv|u_{a}(x)|e^{i\theta}$ holds in ${\mathbb{R}}^{2}$. Hence,
up to a constant phase, any minimizer $u_{a}$ of $e(a)$ must be positive.

Under the assumption that the trap $V(x)$ is radially symmetric and satisfies

  1. ($V$).

$V(x)=V(|x|)\in C^{1,\alpha}({\mathbb{R}}^{2})$, $V^{\prime}(|x|)\geq 0$,
$\lim_{|x|\to\infty}V(x)=+\infty$ and there exists some constant $p\geq 2$
such that $|V(x)|,\,\ |\nabla V(x)|\leq C|x|^{p}$ as $|x|\to\infty$,

combining [27, Theorem 2.1] with [19, Corallary 1.1] yields immediately the
following uniqueness of positive minimizers for $e(a)$:

Theorem B. ([27, 19]) Suppose that $a^{*}>0$ is defined by (1.6), and $V(x)$
satisfies the assumption $(V)$. For any $a\in[0,+\infty)$ and a.e.
$a\in(-a^{*},0)$, then $e(a)$ admits a unique positive minimizer
$u_{0}=u(a,0)>0$, which must be radially symmetric.

Based on the uniqueness of Theorem B, the second main result of this paper is
the following nonexistence of vortices for any fixed $a\in[0,+\infty)$ and
a.e. $a\in(-a^{*},0)$.

######  Theorem 1.4.

Suppose that $a^{*}>0$ is defined by (1.6), $V(x)$ satisfies (1.8) and the
assumption $(V)$. Then for any fixed $a\in[0,+\infty)$ or a.e.
$a\in(-a^{*},0)$, there exists a sufficiently small $\delta:=\delta(a)>0$ such
that, up to a constant phase, all minimizers of $e(\Omega,a)$ are real-valued,
unique and free of vortices for any $\Omega\in[0,\delta)$.

The assumption $V^{\prime}(|x|)\geq 0$ of Theorem 1.4 can be removed in the
repulsive case, since it is used only in the attractive case to guarantee the
uniqueness of positive minimizers for $e(a)$. The proof of Theorem 1.4 gives
us that for any sufficiently small $\Omega\in[0,\delta)$, any minimizer
$u_{\Omega}$ of $e(\Omega,a)$ satisfies $u_{\Omega}\equiv u_{0}e^{i\theta}$ in
${\mathbb{R}}^{2}$, where $\theta:=\theta(\Omega)\in[0,2\pi)$ is a suitable
constant depending on $\Omega$. Thus $|u_{\Omega}|=|u_{0}|>0$ never vanishes
in ${\mathbb{R}}^{2}$ as $\Omega\searrow 0$, which implies that it is free of
vortices for ground states of $e(\Omega,a)$ as $\Omega\searrow 0$.

We next illustrate the general strategy of proving Theorem 1.4. Denote
$\\{\Omega_{n}>0\\}$ to be a sequence satisfying
$\lim_{n\to\infty}\Omega_{n}=0$. Choose some suitable constant
$\theta_{n}\in[0,2\pi)$ such that the minimizer $u_{n}$ of $e(\Omega_{n},a)$
satisfies the following orthogonal condition

$\int_{\mathbb{R}^{2}}u_{0}Im(u_{n}e^{i\theta_{n}})dx\equiv 0\,\ \hbox{for all
$n\in\mathbb{N}^{+}$,}$ (1.18)

where $u_{0}$ is the unique positive minimizer of $e(a)$, see Theorem B. As
the first step, we shall follow (1.18) to prove that

$\lim_{n\to\infty}u_{n}e^{i\theta_{n}}=u_{0}\,\ \hbox{uniformly in}\,\
L^{\infty}({\mathbb{R}}^{2},\mathbb{C}).$ (1.19)

Rewrite now $u_{n}e^{i\theta_{n}}=q_{n}+ir_{n}$, where $q_{n}$ and $r_{n}$
denote the real and imaginary parts of $u_{n}e^{i\theta_{n}}$, respectively.
One can show that $(q_{n},r_{n})$ satisfies the following elliptic system:

$\left\\{\begin{aligned} \mathcal{L}_{n}q_{n}&=\Omega_{n}(x^{\bot}\cdot\nabla
r_{n})\ \ &\mbox{in}\,\ {\mathbb{R}}^{2},\\\
\mathcal{L}_{n}r_{n}&=-\Omega_{n}(x^{\bot}\cdot\nabla q_{n})\ \ &\mbox{in}\,\
{\mathbb{R}}^{2},\\\ \end{aligned}\right.$ (1.20)

where the operator $\mathcal{L}_{n}$ is defined by

$\mathcal{L}_{n}:=-\Delta+V(x)-\mu_{n}+a|u_{n}|^{2},$ (1.21)

and $\mu_{n}\in{\mathbb{R}}$ is a suitable Lagrange multiplier. The refined
analysis gives that $\mu_{n}\to\mu_{0}$ as $n\to\infty$, where
$\mu_{0}\in{\mathbb{R}}$ is the Lagrange multiplier associated to $u_{0}$. By
applying (1.19), we then follow the system (1.20) to investigate the
linearized operators $\mathcal{L}:=-\Delta+V(x)-\mu_{0}+au_{0}^{2}$ and
$\mathcal{N}:=-\Delta+V(x)-\mu_{0}+3au_{0}^{2}$ in ${\mathbb{R}}^{2}$. As the
second step, this yields finally Corollary 4.1 on the first estimates of
$q_{n}-u_{0}$ and $r_{n}$ as $n\to\infty$.

Stimulated by the inductive symmetry method (cf. [17]), in the third step we
shall follow Corollary 4.1 to further prove the inductive process of Lemma 4.2
in the following sense: once we have an estimate on the non-radial part of
$q_{n}-u_{0}$, then a better estimate on the non-radial part of $q_{n}-u_{0}$
can be derived. As a consequence, we finally prove that the non-radial part of
$q_{n}-u_{0}$ is arbitrarily small and $r_{n}\equiv 0$ as $n\to\infty$. This
implies finally that $u_{n}e^{i\theta_{n}}\equiv q_{n}\equiv u_{0}$ in
${\mathbb{R}}^{2}$ as $\Omega_{n}\searrow 0$, and Theorem 1.4 is therefore
proved, see Section 4 for more details.

Similar nonexistence results of Theorem 1.4 in the repulsive case $a>0$ were
obtained in [2, 4, 5] and the references therein by applying vortex ball
constructions, jacobian estimates, and some other arguments, all of which make
full use of the refined energy analysis and the Ginzburg-Landau theory. As far
as we know, the above mentioned arguments however seem not applicable for the
attractive case of $e(\Omega,a)$, since it does not admit the variational
structure of the Ginzburg-Landau type. The above proof strategy shows that
Theorem 1.4 is established essentially by the inductive symmetry method, which
was first imposed in [17] to study the nonexistence of vortices for
$e(\Omega,a)$ in a different situation where $0<\Omega<\Omega^{*}$ is fixed
and $a\searrow-a^{*}$. This approach relies more on the refined analysis of
the governing equations of minimizers. The advantage of this approach lies in
the fact that Theorem 1.4 can be proved by a uniform approach, $i.e,$ no
matter whether the interactions of the condensates are repulsive ($a>0$) or
not. As a byproduct, Theorem 1.4 therefore yields the first result that the
nonexistence of vortices for repulsive BECs under rotation can be investigated
by the inductive symmetry method.

This paper is organized as follows. In Section 2, we shall prove Theorem 1.1
on the existence and non-existence of minimizers for $e(\Omega^{*},a)$. In
Section 3, we first investigate the limit behavior of $u_{n}$ as
$\Omega_{n}\searrow 0$, based on which we then address the detailed analysis
of the linearized problem (1.21). In Section 4 we shall complete the proof of
Theorem 1.4 by the inductive symmetry method.

##  2 Existence of minimizers for $e(\Omega^{*},a)$

The purpose of this section is to prove Theorem 1.1 on the existence and
nonexistence of minimizers for $e(\Omega^{*},a)$, where
$\Omega^{*}=2\sqrt{A}>0$ is defined by (1.9) under the assumption (1.11).
Towards this purpose, we first introduce the following Gagliardo-Nirenberg
inequality

$\int_{\mathbb{R}^{2}}|u(x)|^{4}dx\leq\frac{2}{a^{*}}\int_{\mathbb{R}^{2}}|\nabla
u(x)|^{2}dx\int_{\mathbb{R}^{2}}|u(x)|^{2}dx,\ \ u\in
H^{1}({\mathbb{R}}^{2},{\mathbb{R}}),$ (2.1)

where the equality is attained (cf. [35]) at the unique positive radial
solution $w$ of (1.7). Moreover, following [13, Proposition 4.1], we obtain
that $w=w(|x|)>0$ satisfies

$\int_{\mathbb{R}^{2}}|\nabla
w|^{2}dx=\int_{\mathbb{R}^{2}}w^{2}dx=\frac{1}{2}\int_{\mathbb{R}^{2}}w^{4}dx,$
(2.2)

and

$w(x)\,,\ |\nabla w(x)|=O(|x|^{-\frac{1}{2}}e^{-|x|})\quad\text{as \
$|x|\to\infty$.}$ (2.3)

Given any vector function $\mathcal{A}\in
L^{2}_{loc}({\mathbb{R}}^{2},{\mathbb{R}}^{2})$, we have the following
diamagnetic inequality [25]:

$|(\nabla-i\mathcal{A})u|^{2}\geq\big{|}\nabla|u|\big{|}^{2}\,\ \hbox{a.e. on
}\ {\mathbb{R}}^{2},\ \,u\in H^{1}({\mathbb{R}}^{2},\mathbb{C}).$ (2.4)

We also recall from [21, Lemma 2.1] the following compactness lemma:

######  Lemma 2.1.

Suppose $V(x)\in L_{\rm loc}^{\infty}({\mathbb{R}}^{2})$ satisfies
$\lim_{|x|\to\infty}V(x)=\infty$, then the embedding
${\mathcal{H}}\hookrightarrow L^{q}({\mathbb{R}}^{2},\mathbb{C})$ is compact
for any $2\leq q<\infty$.

Applying Lemma 2.1, we first prove Theorem 1.1 as follows.

Proof of Theorem 1.1. Without loss of generality, we may assume $A=1$ so that
$\Omega^{*}=2$. If $a\in(-\infty,-a^{*}]$, the argument of proving [17,
Theorem 2.3] then gives the nonexistence of minimizers for
$e(2,a)=e(\Omega^{*},a)$. Therefore, the rest is to consider the case where
$a\in(-a^{*},+\infty)$.

(i). Assume $a\in(-a^{*},+\infty)$ and $\lim_{|x|\to\infty}W(x)=+\infty$,
hence we get that $e(2,a)>-\infty$. By the constraint condition, we may assume
$W(x)\geq 0$ for simplicity. Let $\\{u_{n}\\}$ be a minimizing sequence of
$e(2,a)$. We then deduce from (2.1) and (2.4) that

$\displaystyle e(2,a)$
$\displaystyle=\lim_{n\to\infty}\Big{\\{}\int_{\mathbb{R}^{2}}\big{(}|(\nabla-
ix^{\perp})u_{n}|^{2}+W(x)|u_{n}|^{2}+\frac{a}{2}|u_{n}|^{4}\big{)}dx$
$\displaystyle\geq\int_{\mathbb{R}^{2}}\Big{(}\frac{a^{*}+sgn(a)|a|}{2}|u_{n}|^{4}+W(x)|u_{n}|^{2}\Big{)}dx,$

which implies that $\\{u_{n}\\}$ is bounded uniformly in
$L^{4}({\mathbb{R}}^{2},\mathbb{C})\cap
H^{1}_{A}({\mathbb{R}}^{2},\mathbb{C})$, where
$H^{1}_{A}({\mathbb{R}}^{2},\mathbb{C})$ is defined in (2.21).

Taking a subsequence if necessary, now we may assume that
$u_{n}\rightharpoonup u_{0}$ weakly in
$H^{1}_{A}({\mathbb{R}}^{2},\mathbb{C})$ as $n\to\infty$. On the other hand,
since $\lim_{|x|\to\infty}W(x)=+\infty$, we also obtain from Lemma 2.1 that up
to a subsequence if necessary, $\\{|u_{n}|\\}$ converges to $\\{|u_{0}|\\}$
strongly in $L^{p}({\mathbb{R}}^{2},{\mathbb{R}})$ $(2\leq p<\infty)$. We thus
have $\|u_{0}\|_{L^{2}({\mathbb{R}}^{2})}^{2}=1$ and
$\lim_{n\to\infty}\|u_{n}\|_{L^{4}({\mathbb{R}}^{2})}=\|u_{0}\|_{L^{4}({\mathbb{R}}^{2})}$.
By the weak lower semicontinuity, we further obtain that
$e(2,a)=F_{2,a}(u_{0})$, and hence $e(2,a)$ admits a minimizer $u_{0}$ if
$a\in(-a^{*},+\infty)$.

(ii). Without loss of generality, by the constraint we may assume $W(x)\equiv
C=0$ so that $\Omega^{*}=2$, and

$F_{2,a}(u):=\int_{\mathbb{R}^{2}}|(\nabla-
ix^{\perp})u|^{2}dx+\frac{a}{2}\int_{{\mathbb{R}}^{2}}|u|^{4}dx,\ \
u\in{\mathcal{H}}.$ (2.5)

The proof is then divided into two cases: $a\in(-a^{*},0]$ and
$a\in(0,+\infty)$.

(1). We first consider the case $a\in(-a^{*},0]$. The existence of minimizers
of $e(2,a)=e(\Omega^{*},a)$ at $a=0$ is trivial. Indeed, it follows from [29,
Theorem 2] (see also [11, Remark 2.5]) that if $u\in\mathcal{H}$ satisfies
$\|u\|^{2}_{2}=1$, then

$\int_{\mathbb{R}^{2}}|(\nabla-ix^{\perp})u|^{2}dx\geq
2\int_{\mathbb{R}^{2}}|u|^{2}dx=2,$ (2.6)

where the equality is attained at
$u=\frac{1}{\sqrt{\pi}}e^{-\frac{|x|^{2}}{2}}$. On the other hand, by choosing
$\frac{1}{\sqrt{\pi}}e^{-\frac{|x|^{2}}{2}}$ as a test function, one can
obtain that $e(2,0)\leq 2$, which hence yields that $e(2,0)=2$. Therefore,
$e(2,0)$ admits at least one minimizer which is
$\frac{1}{\sqrt{\pi}}e^{-\frac{|x|^{2}}{2}}$.

Next, we shall use the concentration-compactness lemma [28] to deal with the
case $a\in(-a^{*},0)$. Note from (2.1) and (2.4) that for any
$u\in\mathcal{H}$ satisfying $\|u\|^{2}_{2}=1$,

$F_{2,a}(u)\geq\int_{\mathbb{R}^{2}}\big{|}\nabla|u|\big{|}^{2}dx+\frac{a}{2}\int_{\mathbb{R}^{2}}|u|^{4}dx\geq\frac{a^{*}-|a|}{2}\int_{\mathbb{R}^{2}}|u|^{4}dx.$
(2.7)

Therefore, $e(2,a)$ is bounded from below in this case. Let $\\{u_{n}\\}$ be a
minimizing sequence of $e(2,a)$, i.e., $\|u_{n}\|^{2}_{2}=1$ and

$e(2,a)=\lim_{n\to\infty}F_{2,a}(u_{n})=\lim_{n\to\infty}\Big{\\{}\int_{\mathbb{R}^{2}}\Big{(}|(\nabla-
ix^{\perp})u_{n}|^{2}-\frac{|a|}{2}|u_{n}|^{4}\Big{)}dx\Big{\\}}.$

We then derive from (2.7) that there exists a constant $C(a)>0$, depending
only on $a$, such that $\\{u_{n}\\}$ satisfies

$\int_{\mathbb{R}^{2}}|u_{n}|^{4}dx\leq C(a)\,\ \hbox{and}\,\
\int_{\mathbb{R}^{2}}|(\nabla-ix^{\perp})u_{n}|^{2}dx\leq C(a).$ (2.8)

Denote

$f_{n}(x):=|(\nabla-ix^{\perp})u_{n}|^{2}+|u_{n}|^{2}\geq 0,$ (2.9)

so that the nonnegative sequence $\\{f_{n}\\}$ is bounded uniformly in
$L^{1}({\mathbb{R}}^{2})$ in view of (2.8). By the concentration-compactness
lemma [28, Lemma I.1], either the vanishing or the dichotomy or the
compactness holds for some subsequence of $\\{f_{n}\\}$.

We first claim that the vanishing

$\lim_{n\to\infty}\sup_{y\in{\mathbb{R}}^{2}}\int_{B_{R}(y)}f_{n}(x)dx=0,\quad\forall
R>0$ (2.10)

cannot occur. On the contrary, suppose that (2.10) holds true. We then get
from [36, Lemma 1.21] that up to a subsequence if necessary,

$\lim_{n\to\infty}\int_{\mathbb{R}^{2}}|u_{n}|^{4}dx=0.$ (2.11)

Following [29, Theorem 2], we thus obtain from (2.11) that

$e(2,a)=\lim_{n\to\infty}\Big{\\{}\int_{\mathbb{R}^{2}}\Big{(}|(\nabla-
ix^{\perp})u_{n}|^{2}-\frac{|a|}{2}|u_{n}|^{4}\Big{)}dx\Big{\\}}\geq 2,$

which however contradicts to the estimate

$e(2,a)\leq
F_{2,a}\Big{(}\frac{1}{\sqrt{\pi}}e^{-\frac{|x|^{2}}{2}}\Big{)}=2-\frac{|a|}{\pi^{2}}\int_{\mathbb{R}^{2}}e^{-2|x|^{2}}dx<2.$
(2.12)

Hence, the vanishing cannot occur.

To rule out the dichotomy, we first claim that

$e(2,a)<e_{\lambda}(2,a)+e_{1-\lambda}(2,a)\qquad\hbox{for any
$\lambda\in(0,1)$,}$ (2.13)

where $e_{\lambda}(2,a)$ is defined by

$e_{\lambda}(2,a):=\inf_{\\{u\in{\mathcal{H}},\,\|u\|^{2}_{2}=\lambda\\}}F_{2,a}(u),\quad
0<\lambda<1.$ (2.14)

Indeed, one can check from the definition of $F_{2,a}$ that

$\lambda
F_{2,a}(u)=F_{2,a}(\sqrt{\lambda}u)-\frac{|a|\lambda(1-\lambda)}{2}\int_{\mathbb{R}^{2}}|u|^{4}dx\qquad\hbox{for
any $\lambda\in(0,1)$.}$ (2.15)

Moreover, let $\\{\sqrt{\lambda}v_{1,n}\\}$ and
$\\{\sqrt{1-\lambda}v_{2,n}\\}$ be the minimizing sequences of
$e_{\lambda}(2,a)$ and $e_{1-\lambda}(2,a)$, respectively. The argument of
proving the above vanishing gives that

$\liminf_{n\to\infty}\int_{\mathbb{R}^{2}}|v_{1,n}|^{4}dx\geq\delta\quad\hbox{and}\quad\liminf_{n\to\infty}\int_{\mathbb{R}^{2}}|v_{2,n}|^{4}dx\geq\delta,$
(2.16)

where $\delta>0$ is independent of $n$. Combining (2.15) and (2.16) then
yields that

$\displaystyle e(2,a)$ $\displaystyle\leq\lim_{n\to\infty}\Big{[}\lambda
F_{2,a}(v_{1,n})+(1-\lambda)F_{2,a}(v_{2,n})\Big{]}$
$\displaystyle=\lim_{n\to\infty}\Big{[}F_{2,a}\big{(}\sqrt{\lambda}v_{1,n}\big{)}+F_{2,a}\big{(}\sqrt{1-\lambda}v_{2,n}\big{)}$
$\displaystyle\qquad\quad\quad-\frac{|a|\lambda(1-\lambda)}{2}\int_{\mathbb{R}^{2}}(|v_{1,n}|^{4}+|v_{2,n}|^{4})dx\Big{]}$
$\displaystyle\leq
e_{\lambda}(2,a)+e_{1-\lambda}(2,a)-|a|\lambda(1-\lambda)\delta$
$\displaystyle<e_{\lambda}(2,a)+e_{1-\lambda}(2,a),$

and the claim (2.13) is hence proved.

Based on (2.13), we now prove that the dichotomy cannot occur. Indeed, taking
a subsequence if necessary, we may assume that

$\lim_{n\to\infty}\int_{\mathbb{R}^{2}}f_{n}(x)dx=\alpha>0.$ (2.17)

By contradiction, suppose that the dichotomy occurs. Then there exist
$\beta\in(0,\alpha)$, $0<2R_{0}<R_{n}\to+\infty$ as $n\to+\infty$ and
$\\{y_{n}\\}\subset{\mathbb{R}}^{2}$ such that for any $\varepsilon>0$, we
have

$\Big{|}\int_{B_{R_{0}}(y_{n})}f_{n}(x)dx-\beta\Big{|}\leq\varepsilon,\,\
\Big{|}\int_{{\mathbb{R}}^{2}/B_{2R_{n}}(y_{n})}f_{n}(x)dx-(\alpha-\beta)\Big{|}\leq\varepsilon\,\
\hbox{as $n\to\infty$}.$ (2.18)

Set

$u_{1n}(x):=\psi\big{(}\frac{x-y_{n}}{R_{0}}\big{)}u_{n}(x),\,\ \,\
u_{2n}(x):=\phi\big{(}\frac{x-y_{n}}{R_{n}}\big{)}u_{n}(x),$

where $0\leq\psi(x)\leq 1$ and $0\leq\phi(x)\leq 1$ are smooth cut-off
functions and satisfy: $\psi\equiv 1$ if $|x|\leq 1$, and $\psi\equiv 0$ if
$|x|\geq 2$; $\phi\equiv 0$ if $|x|\leq 1$, and $\phi\equiv 1$ if $|x|\geq 2$.
By the definition of $\\{u_{1n}\\}$ and $\\{u_{2n}\\}$, we have

dist(supp $u_{1n}$, supp $u_{2n}$)$\to+\infty$ as $n\to\infty$. (2.19)

We claim that as $n\to\infty$,

$\displaystyle\Big{|}\|u_{1n}\|^{2}_{H_{A}^{1}}-\beta\Big{|}\leq
C\varepsilon,\quad\Big{|}\|u_{2n}\|^{2}_{H_{A}^{1}}-(\alpha-\beta)\Big{|}\leq
C\varepsilon,$ (2.20) $\displaystyle\|u_{n}-u_{1n}-u_{2n}\|_{H_{A}^{1}}\leq
C\varepsilon,$

where we denote

$H_{A}^{1}({\mathbb{R}}^{2},{\mathbb{C}}):=\big{\\{}u\in
L^{2}({\mathbb{R}}^{2},{\mathbb{C}}):\,(\nabla-ix^{\perp})u\in
L^{2}({\mathbb{R}}^{2},{\mathbb{C}})\big{\\}}$ (2.21)

associated with the norm

$\|u\|_{H^{1}_{A}}:=\Big{(}\int_{\mathbb{R}^{2}}(|(\nabla-
ix^{\perp})u|^{2}+|u|^{2})dx\Big{)}^{\frac{1}{2}}.$

We now prove the claim (2.20). Actually, we obtain from (2.17) and (2.18) that
as $n\to\infty$,

$\displaystyle\Big{|}\|u_{1n}\|^{2}_{H_{A}^{1}}-\beta\Big{|}$
$\displaystyle\leq$
$\displaystyle\Big{|}\|u_{1n}\|^{2}_{H_{A}^{1}}-\int_{B_{R_{0}}(y_{n})}f_{n}(x)dx\Big{|}+\Big{|}\int_{B_{R_{0}}(y_{n})}f_{n}(x)dx-\beta\Big{|}$
$\displaystyle\leq$
$\displaystyle\Big{|}\int_{B_{2R_{0}}(y_{n})/B_{R_{0}}(y_{n})}\Big{\\{}\big{|}(\nabla-
ix^{\perp})\big{[}\psi(\frac{x-y_{n}}{R_{0}})u_{n}\big{]}\big{|}^{2}+|\psi(\frac{x-y_{n}}{R_{0}})u_{n}|^{2}$
$\displaystyle-\big{|}(\nabla-
ix^{\perp})u_{n}\big{|}^{2}-|u_{n}|^{2}\Big{\\}}dx\Big{|}+\varepsilon$
$\displaystyle\leq$
$\displaystyle\int_{B_{2R_{0}}(y_{n})/B_{R_{0}}(y_{n})}\Big{(}3|(\nabla-
ix^{\perp})u_{n}|^{2}+2\Big{|}\nabla\psi\Big{(}\frac{x-y_{n}}{R_{0}}\Big{)}\Big{|}^{2}|u_{n}|^{2}+2|u_{n}|^{2}\Big{)}dx+\varepsilon$
$\displaystyle\leq$
$\displaystyle\int_{B_{2R_{n}}(y_{n})/B_{R_{0}}(y_{n})}C\big{(}|(\nabla-
ix^{\perp})u_{n}|^{2}+|u_{n}|^{2}\big{)}dx+\varepsilon\leq C\varepsilon.$

Similarly, we can also get that as $n\to\infty$,

$\big{|}\|u_{2n}\|^{2}_{H_{A}^{1}}-(\beta-\alpha)\big{|}\leq C\varepsilon.$

Following (2.17) and (2.18), we also have

$\displaystyle\|u_{n}-u_{1n}-u_{2n}\|_{H_{A}^{1}}^{2}$ $\displaystyle=$
$\displaystyle\int_{B_{2R_{n}}(y_{n})/B_{R_{0}}(y_{n})}\Big{|}\big{(}\nabla-
ix^{\perp}\big{)}\Big{(}1-\psi(\frac{x-y_{n}}{R_{0}})-\phi(\frac{x-y_{n}}{R_{n}})\Big{)}u_{n}\Big{|}^{2}$
$\displaystyle+\Big{|}\Big{(}1-\psi(\frac{x-y_{n}}{R_{0}})-\phi(\frac{x-y_{n}}{R_{n}})\Big{)}u_{n}\Big{|}^{2}dx$
$\displaystyle\leq$
$\displaystyle\int_{B_{2R_{n}}(y_{n})/B_{R_{0}}(y_{n})}C\Big{[}|(\nabla-
ix^{\perp})u_{n}|^{2}+\Big{|}\nabla\big{[}\psi(\frac{x-y_{n}}{R_{0}})+\phi(\frac{x-y_{n}}{R_{n}})\big{]}\Big{|}^{2}|u_{n}|^{2}+|u_{n}|^{2}\Big{]}dx$
$\displaystyle\leq$
$\displaystyle\int_{B_{2R_{n}}(y_{n})/B_{R_{0}}(y_{n})}Cf_{n}(x)dx\leq
C\varepsilon\ \ \hbox{as}\,\ n\to\infty.$

We now conclude from above that the claim (2.20) holds true.

Since the embedding $H_{A}^{1}({\mathbb{R}}^{2},\mathbb{C})\hookrightarrow
L^{p}({\mathbb{R}}^{2},\mathbb{C})$ $(2\leq p<\infty)$ is continuous, cf. [11,
Proposition 2.1], we derive from (2.20) that

$\big{|}F_{2,a}(u_{n})-F_{2,a}(u_{1n})-F_{2,a}(u_{2n})\big{|}\leq
C\varepsilon,$ (2.22)
$\Big{|}\int_{\mathbb{R}^{2}}\big{(}|u_{n}|^{p}-|u_{1n}|^{p}-|u_{2n}|^{p}\big{)}dx\Big{|}\leq
C\varepsilon\quad\hbox{for any $p\in[2,\infty)$}.$ (2.23)

Moreover, following (2.1), (2.6) and (2.20), there exists a constant
$\delta>0$ such that for $i=1,2$,

$\begin{split}F_{2,a}(u_{in})&\geq(1-\frac{a}{a^{*}})\int_{\mathbb{R}^{2}}|(\nabla-
ix^{\perp})u_{in}|^{2}dx\\\
&\geq\frac{2(a^{*}-a)}{3a^{*}}\|u_{in}\|^{2}_{H_{A}^{1}({\mathbb{R}}^{2})}\geq\delta\quad\hbox{for
all $n$, }\end{split}$ (2.24)

which then implies that there exist subsequences of $\\{u_{1n}\\}$ and
$\\{u_{2n}\\}$, still denoted by $\\{u_{1n}\\}$ and $\\{u_{2n}\\}$, satisfying
$\lim_{n\to\infty}\|u_{1n}\|^{2}_{L^{2}({\mathbb{R}}^{2})}=\theta$ and
$\lim_{n\to\infty}\|u_{2n}\|^{2}_{L^{2}({\mathbb{R}}^{2})}=1-\theta$ for some
constant $\theta\in(0,1)$ by applying (2.23). Since $e_{\lambda}(2,a)$ is
continuous in $\lambda$, we conclude from (2.23) and (2.24) that

$\lim_{n\to\infty}F_{2,a}(u_{1n})\geq
e_{\theta}(2,a),\,\lim_{n\to\infty}F_{2,a}(u_{2n})\geq e_{1-\theta}(2,a),$

which then yield that $e(2,a)\geq e_{\theta}(2,a)+e_{1-\theta}(2,a)$ holds for
some constant $\theta\in(0,1)$ in view of (2.22). This however contradict to
(2.13). Therefore, the dichotomy cannot occur.

Since both the vanishing and the dichotomy cannot occur, we now conclude that
only the compactness occurs for the sequence $\\{f_{n}\\}$, i.e., there exists
$\\{x_{n}\\}\subset{\mathbb{R}}^{2}$ such that for any fixed $\varepsilon>0$,

$\int_{{\mathbb{R}}^{2}/B_{R}(x_{n})}f_{n}(x)dx\leq\varepsilon,\quad\hbox{if
$R=R(\varepsilon)>0$ is large enough.}$ (2.25)

Since $\\{u_{n}\\}$ is bounded in $H_{A}^{1}({\mathbb{R}}^{2},\mathbb{C})$, we
obtain from (2.9) and (2.25) that $\\{u_{n}(\cdot+x_{n})\\}$ is relatively
compact in $L^{2}({\mathbb{R}}^{2})$ by [11, Proposition 2.1] and thus
relatively compact in $L^{p}({\mathbb{R}}^{2})$ $(2<p<+\infty)$ by Hölder
inequality. Note that

$\tilde{u}_{n}(x):=\hat{u}_{n}(x)e^{-ix_{n}\cdot
x^{\perp}}=u_{n}(x+x_{n})e^{-ix_{n}\cdot x^{\perp}}$

satisfies

$\int_{\mathbb{R}^{2}}|(\nabla-
ix^{\perp})\tilde{u}_{n}|^{2}dx=\int_{\mathbb{R}^{2}}\big{|}\big{(}\nabla-
ix^{\perp}\big{)}\big{(}\hat{u}_{n}e^{-ix_{n}\cdot
x^{\perp}}\big{)}\big{|}^{2}dx=\int_{\mathbb{R}^{2}}\big{|}(\nabla-
ix^{\perp})u_{n}\big{|}^{2}dx.$ (2.26)

Moreover, since $|\tilde{u}_{n}(x)|=|u_{n}(x+x_{n})|$ holds in
${\mathbb{R}}^{2}$, by (2.17) and (2.25) we now derive from above that there
exists a subsequence, still denoted by $\\{\tilde{u}_{n}\\}$, of
$\\{\tilde{u}_{n}\\}$ satisfying

$\lim_{n\to\infty}\tilde{u}_{n}=u_{0}\,\ \hbox{strongly in
$L^{p}({\mathbb{R}}^{2})$ ($2\leq p<\infty$) for some $u_{0}\in\mathcal{H}$,}$
(2.27)

and

$\liminf_{n\to\infty}\int_{\mathbb{R}^{2}}\big{|}(\nabla-
ix^{\perp})u_{n}\big{|}^{2}dx=\liminf_{n\to\infty}\int_{\mathbb{R}^{2}}\big{|}(\nabla-
ix^{\perp})\tilde{u}_{n}\big{|}^{2}dx\geq\int_{\mathbb{R}^{2}}\big{|}(\nabla-
ix^{\perp})u_{0}\big{|}^{2}dx.$

Therefore, $e(2,a)$ admits at least one minimizer $u_{0}$ in the case
$a\in(-a^{*},0)$.

(2). We now consider the case $a\in(0,\infty)$. By selecting a suitable
eigenvalue function of (1.14) as a test function, we first prove that

$e(2,a)\leq 2,\ \ a\in(0,\infty).$ (2.28)

Towards the proof of (2.28), motivated by [2, 3], we define the following
lattice

$\ell:=v(\mathbb{Z}\oplus
e^{\frac{2i\pi}{3}}\mathbb{Z})=v(m+ne^{\frac{2i\pi}{3}}),$ (2.29)

where $m,n\in\mathbb{Z}$ are arbitrary integers, and the constant
$v\in{\mathbb{R}}^{+}$ is chosen such that $\frac{\sqrt{3}v^{2}}{2}>\pi$. Let
$Q$ be a unit cell (i.e., a regular hexagon) centered at the origin, whose
area satisfies $|Q|=\frac{\sqrt{3}v^{2}}{2}$. Here $\frac{1}{|Q|}$ denotes an
average spatial density of the point in the lattice. Choose the following test
function

$\psi_{R}(z)=A_{R}e^{-\frac{1}{2}|z|^{2}}\prod_{j\in\ell\bigcap
B_{R}(0)}(z-j),$ (2.30)

where we identify any complex number $z=x_{1}+ix_{2}$ with the point
$x=(x_{1},x_{2})\in{\mathbb{R}}^{2}$, and the Lebesgue measure $dz$ with
$dxdy$, respectively. Here $A_{R}>0$ is chosen properly such that
$\|\psi_{R}\|_{L^{2}({\mathbb{R}}^{2})}=1$. Obviously, $\psi_{R}\in S$, where
$S$ is defined by (1.15), and $(2,\psi_{R})$ is the first eigenpair of (1.14).
We hence obtain from (1.14) that

$\int_{\mathbb{R}^{2}}\big{|}(\nabla-ix^{\perp})\psi_{R}\big{|}^{2}dx=2.$
(2.31)

Moreover, it follows from [2, Theorem 5.2] that as $R\to\infty$,

$|\psi_{R}(z)|\to\psi(z)=\frac{1}{\sqrt{\pi}\sigma}|\eta(z)|e^{-\frac{|z|^{2}}{2\sigma^{2}}}\quad\hbox{in}\,\
L^{p}\big{(}{\mathbb{R}}^{2},(1+|z|^{2})dz\big{)},\ \ p\geq 1,$ (2.32)

where

$\frac{1}{\sigma^{2}}=1-\frac{\pi}{|Q|}\to 0\,\ \hbox{as $|Q|\to\pi$},$ (2.33)

and $|\eta(z)|$ is periodic and vanishes at each point of the lattice $\ell$,
see more details in [2, Theorem 5.1]. We should remark that the main feature
of the periodic lattice $\ell$ is to modify the decay of the Gaussian from
$e^{-\frac{|z|^{2}}{2}}$ to $e^{-\frac{|z|^{2}}{2\sigma^{2}}}$, where
$\sigma>0$ depends on the area of $Q$ through the relation (2.33). Therefore,
we have

$\begin{split}\quad\lim_{R\to\infty}\int_{\mathbb{R}^{2}}|\psi_{R}(z)|^{4}dz&=\int_{\mathbb{R}^{2}}\frac{1}{\pi^{2}\sigma^{4}}|\eta(z)|^{4}e^{-\frac{2|z|^{2}}{\sigma^{2}}}dz\\\
&=\int_{\mathbb{R}^{2}}\frac{1}{\pi^{2}\sigma^{2}}|\eta(\sigma
z)|^{4}e^{-2|z|^{2}}dz\\\
&\leq\frac{C}{\pi^{2}\sigma^{2}}\int_{\mathbb{R}^{2}}e^{-2|z|^{2}}dz\leq\frac{C}{\sigma^{2}}\to
0\,\ \hbox{as}\,\ \sigma\to\infty,\end{split}$ (2.34)

and thus

$\displaystyle e(2,a)$
$\displaystyle\leq\lim_{\sigma\to\infty,\,R\to\infty}F_{2,a}(\psi_{R})$
$\displaystyle=2+\lim_{\sigma\to\infty,\,R\to\infty}\int_{\mathbb{R}^{2}}|\psi_{R}|^{4}dz\leq
2,\ \ \mbox{if}\ \ a\in(0,\infty),$

i.e., the estimate (2.28) holds true.

We now assume that $e(2,a)$ admits a minimizer $u_{a}$ for any
$a\in(0,\infty)$. We then obtain from (2.6) that

$\int_{\mathbb{R}^{2}}\big{|}(\nabla-ix^{\perp})u_{a}\big{|}^{2}dx\geq
2\int_{\mathbb{R}^{2}}|u_{a}|^{2}dx=2,$

which and (2.5) thus yield that

$e(2,a)>2,\ \ \mbox{if}\ \ a\in(0,\infty),$

a contradiction to (2.28). We therefore conclude that there is no minimizer
for $e(2,a)$ in the case $a\in(0,\infty)$. This therefore completes the proof
of Theorem 1.1 ∎

Proof of Theorem 1.2. Without loss of generality, as before we may assume
$A=1$ so that $\Omega^{*}=2$. If $a\in(-\infty,-a^{*}]$, the argument of
proving [17, Theorem 2.3] then gives the nonexistence of minimizers for
$e(2,a)=e(\Omega^{*},a)$. Therefore, we only consider the case where
$a\in(-a^{*},+\infty)$.

(i). Assume $a\in(-a^{*},+\infty)$ and $W(x)>B$ in ${\mathbb{R}}^{2}$. For
simplicity we may assume $B=0$ so that $W(x)>0$ in ${\mathbb{R}}^{2}$.

We first consider the case $a\in(-a^{*},0]$. In this case, we claim that

$e(2,a)=e^{\star}(2,a),\quad\hbox{where $e^{\star}(2,a)$ denotes the value of
$e(2,a)$ at $W\equiv 0$.}$ (2.35)

Recall from Theorem 1.1 (i) that $e^{\star}(2,a)$ admits a minimizer
$\hat{u}(x)\in\mathcal{H}$. Under the assumptions of $W(x)$, we have

$e(2,a)\geq e^{\star}(2,a).$ (2.36)

On the other hand, taking the following test function

$\hat{u}_{k}(x):=\hat{u}(x+x_{k})e^{-ix_{k}^{\perp}\cdot
x},\quad\hbox{where}\,\ |x_{k}|\to\infty\ \ \mbox{as}\ \ k\to\infty,$

we have

$e(2,a)\leq F_{2,a}(\hat{u}_{k})\to e^{\star}(2,a)\ \ \mbox{as}\ \
k\to\infty.$

We then obtain from (2.36) and above that the claim (2.35) is true. Following
this estimate, since $W(x)>0$ in ${\mathbb{R}}^{2}$, as in [18, Theorem 1] one
can further derive the nonexistence of minimizers for $e(2,a)$ in the case
$a\in(-a^{*},0]$.

We next consider the case $a\in(0,+\infty)$. Similar to Theorem 1.1 (ii), we
take a test function as follows:

$\psi_{R,\sigma}(z):=\psi_{R}(z+z_{\sigma})e^{-ix_{\sigma}^{\perp}\cdot x},$
(2.37)

where $\psi_{R}(z)$ and the parameter $\sigma$ are defined by (2.30) and
(2.33), respectively. Here $x_{\sigma}=(x_{1,\sigma},x_{2,\sigma})$ and
$z_{\sigma}=x_{1,\sigma}+ix_{2,\sigma}$ satisfy
$|x_{\sigma}|=|z_{\sigma}|=\sigma^{2}$. One can check that
$\psi_{R,\sigma}(z)\in S$ satisfies

$\int_{\mathbb{R}^{2}}\big{|}(\nabla-
ix^{\perp})\psi_{R,\sigma}\big{|}^{2}dx=\int_{\mathbb{R}^{2}}\big{|}(\nabla-
ix^{\perp})\psi_{R}\big{|}^{2}dx=2,$

and

$\int_{\mathbb{R}^{2}}|\psi_{R,\sigma}|^{2}dx=\int_{\mathbb{R}^{2}}|\psi_{R}|^{2}dx=1.$

Note from (2.34) that

$\lim_{R\to\infty}\int_{\mathbb{R}^{2}}|\psi_{R,\sigma}|^{4}dx=\lim_{R\to\infty}\int_{\mathbb{R}^{2}}|\psi_{R}|^{4}dx\to
0\quad\hbox{as}\,\ \sigma\to+\infty.$

Moreover, because $|z_{\sigma}|=\sigma^{2}$, we derive from (2.32) that

$\displaystyle\lim_{R\to\infty}\int_{\mathbb{R}^{2}}W(z)|\psi_{R}(z)|^{2}dz=\int_{\mathbb{R}^{2}}\frac{1}{\pi\sigma^{2}}W(z+z_{\sigma})|\eta(z)|^{2}e^{-\frac{|z|^{2}}{2\sigma^{2}}}dz$
$\displaystyle=$
$\displaystyle\int_{|z|<\sigma^{\frac{1}{2}}}\frac{1}{\pi}W(\sigma
z+z_{\sigma})|\eta(\sigma
z)|^{2}e^{-\frac{|z|^{2}}{2}}dz+\int_{|z|\geq\sigma^{\frac{1}{2}}}\frac{1}{\pi}W(\sigma
z+z_{\sigma})|\eta(\sigma z)|^{2}e^{-\frac{|z|^{2}}{2}}dz$ $\displaystyle:=$
$\displaystyle I+II.$

Since $\lim_{|x|\to\infty}W(x)=0$ and $|\eta(z)|\in
L^{\infty}({\mathbb{R}}^{2})$, we get that $|I|\leq C(\sigma)\to 0$ as
$\sigma\to\infty$. Moreover, the exponential decay of $e^{-\frac{|z|^{2}}{2}}$
yields that $|II|\leq C(\sigma)\to 0$ as $\sigma\to\infty$. Therefore, we
obtain from above that

$\displaystyle\lim_{R\to\infty}\int_{\mathbb{R}^{2}}W(x)|\psi_{R,\sigma}(x)|^{2}dx=\lim_{R\to\infty}\int_{\mathbb{R}^{2}}W(x+x_{\sigma})|\psi_{R}(x)|^{2}dx\to
0\,\ \hbox{as $\sigma\to\infty$}.$

We thus conclude from (2.36) and above that $e(2,a)=e^{\star}(2,a)$. Since
$W(x)>0$ in ${\mathbb{R}}^{2}$, as before one can further obtain the
nonexistence of minimizers for $e(2,a)$ in the case $a\in(0,+\infty)$. This
completes the proof of Theorem 1.1 (i).

(ii). Let $a\in(-a^{*},+\infty)$ and $W(x)<B$ in ${\mathbb{R}}^{2}$. For
simplicity we may assume $B=0$ so that $W(x)<0$ in ${\mathbb{R}}^{2}$.
Following the definition of $a_{\ast}$ in (1.12), we can show that
$a_{\ast}>0$ by using [29, Theorem 2]. Indeed, for $A=1$ and $B=0$, we can
take $u=e^{-\frac{|x|^{2}}{2}}$ as a test function to obtain that

$e(2,a)\leq
F_{2,a}(u)=2+\int_{\mathbb{R}^{2}}\frac{W(x)e^{-|x|^{2}}}{\pi}dx+\frac{a}{2}\int_{\mathbb{R}^{2}}\frac{e^{-2|x|^{2}}}{\pi^{2}}dx.$

Since $W(x)<0$ in ${\mathbb{R}}^{2}$, we get that $e(2,a)<2$ if $a>0$ is small
enough and thus $a_{\ast}>0$ holds.

(a). We first consider $a\in(-a^{*},a_{\ast})$. By the definition of
$a_{\ast}$, it is obviously that

$e(2,a)<2\quad\hbox{for any $a\in(-a^{*},a_{\ast})$.}$ (2.38)

Similar to Theorem 1.1 (ii) we shall prove the existence of minimizers for
$e(2,a)$ by the concentration-compactness lemma. Towards this aim, we denote
$\\{u_{n}\\}$ a minimizing sequence of $e(2,a)$, i.e., $\|u_{n}\|^{2}_{2}=1$
and $e(2,a)=\lim_{n\to\infty}F_{2,a}(u_{n})$. As in Theorem 1.1 (ii), we
denote

$f_{n}(x):=|(\nabla-ix^{\perp})u_{n}|^{2}+|u_{n}|^{2}\geq 0\ \ \mbox{in}\,\
{\mathbb{R}}^{2}.$ (2.39)

One can check that the nonnegative sequence $\\{f_{n}\\}$ is bounded uniformly
in $L^{1}({\mathbb{R}}^{2})$.

Following (2.38), the same argument of Theorem 1.1 (ii) yields that the
vanishing cannot occur for $\\{f_{n}(x)\\}$ defined in (2.39). Moreover,
applying (2.38), one can also derive that (2.13) holds for any
$a\in(-a^{*},+\infty)$. Following this estimate, the same argument of Theorem
1.1 (ii) yields that the dichotomy cannot occur for $\\{f_{n}(x)\\}$. By the
concentration-compactness lemma, we therefore conclude that the compactness
holds for $\\{f_{n}(x)\\}$.

Finally, the same argument of proving (2.27) yields that
$\tilde{u}_{n}=u_{n}(\cdot+x_{n})e^{ix_{n}\cdot x^{\perp}}$ converges to some
$u_{0}\in\mathcal{H}$ in $L^{p}({\mathbb{R}}^{2})$ $(2\leq p<\infty)$ as
$n\to\infty$. To obtain a true minimizer of $e(2,a)$, we next show that
$\\{x_{n}\\}$ is bounded uniformly in ${\mathbb{R}}^{2}$. On the contrary,
suppose

$|x_{n}|\to+\infty\,\ \hbox{as}\,\ n\to\infty.$

Then up to a subsequence if necessary, we have

$\lim_{n\to\infty}\int_{\mathbb{R}^{2}}W(x)|u_{n}(x)|^{2}dx=\lim_{n\to\infty}\int_{\mathbb{R}^{2}}W(x+x_{n})|u_{n}(x+x_{n})|^{2}dx=0.$

By the definition of $e^{*}(2,a)$ defined in (2.35), we thus have

$e(2,a)=\lim_{n\to\infty}F(2,a)(u_{n})\geq\int_{\mathbb{R}^{2}}\big{(}|(\nabla-
ix^{\perp})u_{0}|^{2}+\frac{a}{2}|u_{0}|^{4}\big{)}dx\geq e^{\star}(2,a),$

which however contradicts to (2.35). Once $\\{x_{n}\\}$ is bounded uniformly
in ${\mathbb{R}}^{2}$, we may assume that up to a subsequence if necessary,
$x_{n}\to y_{0}$ for some $y_{0}\in{\mathbb{R}}^{2}$ as $n\to\infty$.
Moreover, since $W(x)\in C({\mathbb{R}}^{2})$ and $\lim_{|x|\to\infty}W(x)=0$,
we obtain that $W(x+x_{n})$ converges to $W(x+y_{0})$ uniformly in
${\mathbb{R}}^{2}$ as $n\to\infty$. Therefore, we get that

$\displaystyle\lim_{n\to\infty}\int_{\mathbb{R}^{2}}W(x)|u_{n}(x)|^{2}dx=$
$\displaystyle\lim_{n\to\infty}\int_{\mathbb{R}^{2}}W(x+x_{n})|u_{n}(x+x_{n})|^{2}dx$
$\displaystyle=$
$\displaystyle\lim_{n\to\infty}\int_{\mathbb{R}^{2}}W(x+y_{0})|u_{n}(x+x_{n})|^{2}dx$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{2}}W(x+y_{0})|u_{0}(x)|^{2}dx,$

where we used the fact that $W(x)\in L^{\infty}({\mathbb{R}}^{2})$ and
$\lim_{n\to\infty}\int_{\mathbb{R}^{2}}|u_{n}(\cdot+x_{n})|^{2}dx=\int_{\mathbb{R}^{2}}|u_{0}|^{2}dx$
in the last equality. This further implies that

$\displaystyle e(2,a)=\lim_{n\to\infty}F_{2,a}(u_{n})$
$\displaystyle\geq\int_{\mathbb{R}^{2}}\big{(}|(\nabla-
ix^{\perp})u_{0}|^{2}+W(x+y_{0})|u_{0}|^{2}+\frac{a}{2}|u_{0}|^{4}\big{)}dx$
$\displaystyle=F_{2,a}\big{(}u_{0}(\cdot-y_{0})e^{iy_{0}\cdot
x^{\perp}}\big{)},$

i.e., $u_{0}(\cdot-y_{0})e^{iy_{0}\cdot x^{\perp}}$ is a minimizer of $e(2,a)$
for the case $a\in(-a^{*},a_{\ast})$, and we are done.

(b). We next consider $a\in(a_{\ast},+\infty)$. By the definition of
$a_{\ast}$, we obtain that $e(2,a)\geq 2$ for any $a\in(a_{\ast},+\infty)$. On
the other hand, taking the same test function as in (2.37), we can obtain that
$e(2,a)\leq 2$ and hence $e(2,a)\equiv 2$ for any $a\in(a_{\ast},+\infty)$.
Therefore, we can prove that there is no minimizers for $e(2,a)=2$ in this
case. Indeed, assume that there is some $u_{0}\in
H^{1}({\mathbb{R}}^{2},\mathbb{C})$ satisfying
$\|u_{0}\|^{2}_{L^{2}({\mathbb{R}}^{2})}=1$ such that $e(2,a)=2$. We then have
$2=e(2,a)\geq e(2,a_{\ast})+(a-a_{\ast})\int_{\mathbb{R}^{2}}|u_{0}|^{4}dx=2$,
and thus $u_{0}\equiv 0$, which contradicts with
$\|u_{0}\|^{2}_{L^{2}({\mathbb{R}}^{2})}=1$. This completes the proof of
Theorem 1.2. ∎

Proof of Corollary 1.3. Without loss of generality, as before we may assume
$A=1$ so that $\Omega^{*}=2$, and $B=0$. Following Theorem 1.2 (ii), we only
need to prove that $a_{\ast}=+\infty$, or equivalently, that $e(2,a)<2$ for
any $a>0$. Taking $\psi_{R}$ as a test function, where $\psi_{R}$ is defined
by (2.30), we have

$\displaystyle e(2,a)$
$\displaystyle\leq\lim_{R\to\infty}F_{2,a}(\psi_{R})=2+\lim_{R\to\infty}\int_{\mathbb{R}^{2}}\Big{(}W(z)|\psi_{R}(z)|^{2}+\frac{a}{2}|\psi_{R}(z)|^{4}\Big{)}dz$
$\displaystyle=2+\int_{\mathbb{R}^{2}}\Big{(}W(z)\frac{|\eta(z)|^{2}e^{-\frac{|z|^{2}}{\sigma^{2}}}}{\pi\sigma^{2}}+\frac{a}{2}\frac{|\eta(z)|^{4}e^{-\frac{2|z|^{2}}{\sigma^{2}}}}{\pi^{2}\sigma^{4}}\Big{)}dz$
$\displaystyle=2+\int_{\mathbb{R}^{2}}\Big{(}W(\sigma z)\frac{|\eta(\sigma
z)|^{2}e^{-|z|^{2}}}{\pi}+\frac{a}{2}\frac{|\eta(\sigma
z)|^{4}e^{-2|z|^{2}}}{\pi^{2}\sigma^{2}}\Big{)}dz,$

where $\eta(z)$ is as in (2.32). Since $|\eta(z)|$ is a periodic function, we
obtain from [6] that $|\eta(\sigma z)|^{2}$ and $|\eta(\sigma z)|^{4}$
converge $L^{\infty}-$weak star to the constants
${-\mkern-16.0mu\int}|\eta(z)|^{2}dz>0$ and
${-\mkern-16.0mu\int}|\eta(z)|^{4}dz>0$ as $\sigma\to\infty$, respectively,
where
${-\mkern-16.0mu\int}|\eta(z)|^{p}dz=\frac{1}{|Q|}\int_{Q}|\eta(z)|^{p}dz$
denotes the average of $|\eta(z)|^{p}$ in $Q$. Following (1.13),
$\limsup_{|x|\to\infty}W(x)|x|^{s}<0$ holds for $0<s<2$, we obtain that there
exists a small $\delta>0$ such that

$\displaystyle\quad\int_{\mathbb{R}^{2}}\Big{(}W(\sigma z)\frac{|\eta(\sigma
z)|^{2}e^{-|z|^{2}}}{\pi}+\frac{a}{2}\frac{|\eta(\sigma
z)|^{4}e^{-2|z|^{2}}}{\pi^{2}\sigma^{2}}\Big{)}dz$
$\displaystyle\leq\int_{{\mathbb{R}}^{2}/B_{1}(0)}\frac{-\delta|\eta(\sigma
z)|^{2}e^{-|z|^{2}}}{\pi\sigma^{s}|z|^{s}}dz+\frac{C}{\sigma^{2}}$
$\displaystyle\leq-\frac{C(\delta)}{\sigma^{s}}+\frac{C}{\sigma^{2}}<0\quad\hbox{as}\,\
\sigma\to\infty,$

which then gives that

$e(2,a)<2\quad\hbox{for any}\,\ a>0,$

and hence $a_{\ast}=+\infty$ and the proof is completed in view of Theorem 1.2
(ii). ∎

##  3 Limit behavior of minimizers as $\Omega\searrow 0$

For any fixed $a\in[0,+\infty)$ or a.e. $a\in(-a^{*},0)$, in this section we
analyze the refined limit behavior of minimizers for $e(\Omega,a)$ as
$\Omega>0$ approaches to zero, where the trap $V(x)\geq 0$ satisfies all
assumptions of Theorem 1.4.

Towards the above purpose, we denote $u_{n}$ a complex-valued minimizer of
$e(\Omega_{n},a)$, where $\Omega_{n}\searrow 0$ as $n\to\infty$, so that
$u_{n}$ satisfies the following Euler-Lagrange equation

$-\Delta
u_{n}+\big{(}V(x)-\mu_{n}\big{)}u_{n}+i\,\Omega_{n}\,(x^{\perp}\cdot\nabla
u_{n})+a|u_{n}|^{2}u_{n}=0\ \ \mbox{in}\,\ {\mathbb{R}}^{2},$ (3.1)

where $\mu_{n}\in{\mathbb{R}}$ is a Lagrange multiplier and satisfies

$\mu_{n}=e(\Omega_{n},a)+\frac{a}{2}\int_{\mathbb{R}^{2}}|u_{n}|^{4}dx.$ (3.2)

Under the assumptions of Theorem 1.4, recall from Theorem B that $u_{0}>0$ is
the unique positive minimizer of $e(a)$. Denote $\mu_{0}\in{\mathbb{R}}$ the
Lagrange multiplier associated to $u_{0}$, so that $(u_{0},\mu_{0})$ satisfies

$-\Delta u_{0}+V(x)u_{0}=\mu_{0}u_{0}-au_{0}^{3}\ \ \mbox{in}\,\
{\mathbb{R}}^{2}.$ (3.3)

We start with the following estimates of $e(\Omega_{n},a)$ and $\mu_{n}$ as
$n\to\infty$.

######  Lemma 3.1.

Under the assumptions of Theorem 1.4, let $u_{n}$ be a complex-valued
minimizer of $e(\Omega_{n},a)$, where $a\in(-a^{*},+\infty)$ is fixed, and
$\Omega_{n}\searrow 0$ as $n\to\infty$. Then we have

  1. 1.

The energy $e(\Omega_{n},a)$ satisfies

$\lim_{n\to\infty}e(\Omega_{n},a)=e(a),$ (3.4)

where the energy $e(a)$ is defined in (1.4).

  2. 2.

There exists $\theta_{n}\in[0,2\pi)$ such that $u_{n}$ satisfies

$\lim_{n\to\infty}u_{n}e^{i\theta_{n}}=u_{0}\,\ \hbox{strongly in}\,\
H^{1}({\mathbb{R}}^{2},\mathbb{C}),$ (3.5)

where $u_{0}>0$ is the unique positive minimizer of $e(a)$, and the Lagrange
multiplier $\mu_{n}=\mu(\Omega_{n},a)$ of (3.1) satisfies

$\lim_{n\to\infty}\mu_{n}=\mu_{0},$ (3.6)

where $\mu_{0}\in{\mathbb{R}}$ is the Lagrange multiplier of (3.3).

  3. 3.

$u_{n}$ satisfies the following exponential decay

$|u_{n}(x)|,\,|\nabla u_{n}(x)|\leq C(a)e^{-2|x|}\,\ \hbox{in}\,\
{\mathbb{R}}^{2},$ (3.7)

where the constant $C(a)>0$ depends only on $a$.

Proof. 1\. Taking $u_{0}$ as a test function, we obtain the following upper
bound estimate:

$\lim_{n\to\infty}e(\Omega_{n},a)\leq\lim_{n\to\infty}F_{\Omega_{n},a}(u_{0})=e(a).$
(3.8)

On the other hand, we claim that there exists a constant $C>0$, independent of
$n$, such that

$\int_{\mathbb{R}^{2}}\big{(}|\nabla u_{n}|^{2}+V(x)|u_{n}|^{2}\big{)}dx\leq
C\,\ \hbox{uniformly in}\,\ n.$ (3.9)

Indeed, following (3.8), we obtain from (2.1) and (2.4) that

$\displaystyle 1+e(a)$ $\displaystyle\geq e(\Omega_{n},a)$
$\displaystyle=\int_{\mathbb{R}^{2}}\Big{(}|\nabla
u_{n}|^{2}-\Omega_{n}x^{\perp}\cdot(iu_{n},\nabla
u_{n})+V(x)|u_{n}|^{2}+\frac{a}{2}|u_{n}|^{4}\Big{)}dx$
$\displaystyle\geq\int_{\mathbb{R}^{2}}\Big{(}\big{|}\nabla|u_{n}|\big{|}^{2}+\big{[}V(x)-\frac{\Omega_{n}^{2}}{4}|x|^{2}\big{]}|u_{n}|^{2}+\frac{a}{2}|u_{n}|^{4}\Big{)}dx$
$\displaystyle\geq\int_{\mathbb{R}^{2}}\Big{(}\big{|}\nabla|u_{n}|\big{|}^{2}+\frac{a}{2}|u_{n}|^{4}\Big{)}dx-C$
$\displaystyle\geq\Big{(}\frac{a^{*}+a}{2}\Big{)}\int_{\mathbb{R}^{2}}|u_{n}|^{4}dx-C\quad\hbox{uniformly
in $n$.}$

Since $a\in(-a^{*},\infty)$ is fixed, we deduce from above that

$\int_{\mathbb{R}^{2}}|u_{n}|^{4}dx\leq C(a)\quad\hbox{uniformly in $n$},$
(3.10)

which further implies that

$\displaystyle C(a)$ $\displaystyle\geq\int_{\mathbb{R}^{2}}\Big{(}|\nabla
u_{n}|^{2}-\Omega_{n}x^{\perp}\cdot(iu,\nabla u)+V(x)|u_{n}|^{2}\Big{)}dx$
$\displaystyle\geq\int_{\mathbb{R}^{2}}\Big{(}\frac{1}{2}|\nabla
u_{n}|^{2}+\big{[}V(x)-\frac{\Omega_{n}^{2}}{2}|x|^{2}\big{]}|u_{n}|^{2}\Big{)}dx$
$\displaystyle\geq\frac{1}{2}\int_{\mathbb{R}^{2}}\Big{[}|\nabla
u_{n}|^{2}+\big{(}V(x)-C\big{)}|u_{n}|^{2}\Big{]}dx\quad\hbox{uniformly in
$n$.}$

This proves the claim (3.9).

By the assumptions of $V(x)$, we now deduce from (3.9) that

$\displaystyle\frac{\Omega_{n}^{2}}{4}\int_{\mathbb{R}^{2}}|x|^{2}|u_{n}|^{2}dx$
$\displaystyle\leq$
$\displaystyle\frac{\Omega_{n}^{2}}{4}\Big{[}C\int_{\mathbb{R}^{2}}V(x)|u_{n}|^{2}dx+C\Big{]}$
$\displaystyle\leq$ $\displaystyle\frac{\Omega_{n}^{2}}{4}C(a)\to
0\quad\hbox{as $n\to\infty$,}$

which then gives that

$\begin{split}&\lim_{n\to\infty}e(\Omega_{n},a)=\lim_{n\to\infty}F_{\Omega_{n},a}(u_{n})\\\
=&\lim_{n\to\infty}\int_{\mathbb{R}^{2}}\Big{(}\Big{|}\nabla
u_{n}-\frac{i\Omega_{n}x^{\perp}}{2}u_{n}\Big{|}^{2}+\Big{[}V(x)-\frac{\Omega_{n}^{2}}{4}|x|^{2}\Big{]}|u_{n}|^{2}+\frac{a}{2}|u_{n}|^{4}\Big{)}dx\\\
\geq&\lim_{n\to\infty}\int_{\mathbb{R}^{2}}\Big{(}\big{|}\nabla|u_{n}|\big{|}^{2}+\Big{[}V(x)-\frac{\Omega_{n}^{2}}{4}|x|^{2}\Big{]}|u_{n}|^{2}+\frac{a}{2}|u_{n}|^{4}\Big{)}dx\\\
=&\lim_{n\to\infty}\Big{(}E_{a}(|u_{n}|)-\frac{\Omega_{n}^{2}}{4}\int_{\mathbb{R}^{2}}|x|^{2}|u_{n}|^{2}dx\Big{)}\geq
e(a).\end{split}$ (3.11)

Therefore, (3.4) is proved in view of (3.8) and (3.11).

2\. For the unique positive minimizer $u_{0}>0$ of (1.4), we choose
$\theta_{n}\in[0,2\pi)$ properly such that

$\big{\|}u_{n}e^{i\theta_{n}}-u_{0}\big{\|}_{L^{2}({\mathbb{R}}^{2})}=\min_{\theta\in[0,2\pi)}\big{\|}u_{n}e^{i\theta}-u_{0}\big{\|}_{L^{2}({\mathbb{R}}^{2})},$
(3.12)

which further implies the following orthogonal condition

$\int_{\mathbb{R}^{2}}u_{0}Im(u_{n}e^{i\theta_{n}})dx=0\,\ \hbox{for all
$n\in\mathbb{N}^{+}$,}$ (3.13)

where $Im(u)$ denotes the imaginary part of $u$. Taking a subsequence if
necessary, we deduce from (3.9) and Lemma 2.1 that there exists $\hat{u}\in
H^{1}({\mathbb{R}}^{2},\mathbb{C})$ such that
$u_{n}e^{i\theta_{n}}\rightharpoonup\hat{u}$ weakly in
$H^{1}({\mathbb{R}}^{2},\mathbb{C})$ and $u_{n}e^{i\theta_{n}}\to\hat{u}$
strongly in $L^{p}({\mathbb{R}}^{2})$ ($2\leq p<\infty$) as $n\to\infty$.
Thus, we get $\|\hat{u}\|_{L^{2}({\mathbb{R}}^{2})}=1$. Moreover, by the weak
lower semi-continuity and Fatou’s Lemma, we have

$\displaystyle e(a)$
$\displaystyle=\lim_{n\to\infty}F_{\Omega_{n}}(u_{n})\geq\int_{\mathbb{R}^{2}}\big{(}|\nabla\hat{u}|^{2}+V(x)|\hat{u}|^{2}+\frac{a}{2}|\hat{u}|^{4}\big{)}dx\geq
e(a),$

which therefore implies that $e(a)=E(\hat{u})$ and $\hat{u}=u_{0}$ in view of
(3.13).

Note from Theorem B that $u_{0}>0$ is the unique positive minimizer of $e(a)$.
We then conclude that the above convergence $u_{n}e^{i\theta_{n}}\to u_{0}$
strongly in $L^{p}({\mathbb{R}}^{2})$ ($2\leq p<\infty$) as $n\to\infty$ holds
for the whole sequence. Moreover, since
$\lim_{n\to\infty}F_{\Omega_{n}}(u_{n})=e(a)$ and $u_{n}\rightharpoonup u_{0}$
weakly in $H^{1}({\mathbb{R}}^{2})$, we obtain that
$\lim_{n\to\infty}\int_{\mathbb{R}^{2}}|\nabla
u_{n}|^{2}dx=\lim_{n\to\infty}\int_{\mathbb{R}^{2}}|\nabla u_{0}|^{2}dx$, and
thus (3.5) holds true. Finally, one can deduce from (3.2)–(3.5) that (3.6)
also holds true.

3\. Define $U_{n}(x)=|u_{n}(x)|^{2}\geq 0$ in ${\mathbb{R}}^{2}$. We then
derive from (3.1) that

$\begin{array}[]{lll}&&-\displaystyle\frac{1}{2}\Delta U_{n}+|\nabla
u_{n}|^{2}-\Omega_{n}\,x^{\bot}\cdot(iu_{n},\nabla
u_{n})+\displaystyle\Big{[}V(x)-\mu_{n}+aU_{n}\Big{]}U_{n}=0\quad\mbox{in}\,\
{\mathbb{R}}^{2}.\end{array}$ (3.14)

By Cauchy’s inequality,

$|\nabla u_{n}|^{2}-\Omega_{n}\,x^{\bot}\cdot(iu_{n},\nabla
u_{n})+\frac{\Omega_{n}^{2}}{4}|x|^{2}U_{n}\geq 0\ \ \mbox{in}\,\
{\mathbb{R}}^{2},$

we have

$-\frac{1}{2}\Delta
U_{n}+\Big{[}V(x)-\frac{\Omega_{n}^{2}|x|^{2}}{4}-\mu_{n}\Big{]}U_{n}+aU_{n}^{2}\leq
0\quad\mbox{in}\,\ {\mathbb{R}}^{2}.$ (3.15)

Note that
$\lim_{|x|\to\infty}\Big{(}V(x)-\frac{\Omega_{n}^{2}|x|^{2}}{4}\Big{)}=+\infty$
and $\lim_{n\to\infty}\mu_{n}=\mu_{0}$. By De Giorgi-Nash-Moser theory [20,
Theorem 4.1], we derive from above and (3.5) that

$0\leq U_{n}(x)\leq C(a)\quad\hbox{in ${\mathbb{R}}^{2}$ uniformly in $n$. }$
(3.16)

It follows from (3.16) that there exists a sufficiently large $R=R(a)>0$ such
that $V(x)-\frac{\Omega_{n}^{2}|x|^{2}}{4}-\mu_{n}-|a|U_{n}\geq 18$ holds in
${\mathbb{R}}^{2}/B_{R}(0)$ as $n\to\infty$. This then implies from (3.15)
that

$-\frac{1}{2}\Delta U_{n}+18U_{n}\leq 0\quad\mbox{in
${\mathbb{R}}^{2}/B_{R}(0)$ uniformly in $n$}.$

By the comparison principle, we obtain from above and (3.16) that

$|u_{n}(x)|\leq C(a)e^{-3|x|}\quad\mbox{in ${\mathbb{R}}^{2}$ uniformly in
$n$.}$ (3.17)

Based on (3.17), the same argument of proving [17, Lemma 4.2] then yields that
the gradient estimate of (3.7) also holds true, and we are done. ∎

Applying (3.5) and (3.7), by a standard bootstrap argument one can further
verify from (3.1) that

$\lim_{n\to\infty}u_{n}e^{i\theta_{n}}=u_{0}\,\ \hbox{uniformly in}\,\
L^{\infty}({\mathbb{R}}^{2},\mathbb{C}),$ (3.18)

where $u_{0}>0$ is the unique positive minimizer of $e(a)$, see [17,
Propostion 3.3] for the related argument. We now rewrite

$u_{n}e^{i\theta_{n}}:=q_{n}+ir_{n},$ (3.19)

where $\theta_{n}\in[0,2\pi)$ is as in (3.5), while $q_{n}$ and $r_{n}$ denote
the real and imaginary parts of $u_{n}e^{i\theta_{n}}$, respectively. It then
follows from (3.18) that

$q_{n}\to u_{0}\,\ \hbox{and}\,\ r_{n}\to 0\,\ \hbox{uniformly in}\,\
L^{\infty}({\mathbb{R}}^{2})\,\ \hbox{as}\,\ n\to\infty.$ (3.20)

Note from (3.1) that $(q_{n},r_{n})$ satisfies the following system:

$\left\\{\begin{aligned} \mathcal{L}_{n}q_{n}&=\Omega_{n}(x^{\bot}\cdot\nabla
r_{n})\,\ &\mbox{in}\,\ {\mathbb{R}}^{2},\\\
\mathcal{L}_{n}r_{n}&=-\Omega_{n}(x^{\bot}\cdot\nabla q_{n})\ \ &\mbox{in}\,\
{\mathbb{R}}^{2},\\\ \end{aligned}\right.$ (3.21)

where the operator $\mathcal{L}_{n}$ is defined by

$\mathcal{L}_{n}:=-\Delta+V(x)-\mu_{n}+a|u_{n}|^{2}.$ (3.22)

In order to obtain some refined estimates of $(q_{n},r_{n})$, we need to
analyze the associated linearized operators $\mathcal{L}$ and $\mathcal{N}$:
$M({\mathbb{R}}^{2},{\mathbb{R}})\longmapsto
L^{2}({\mathbb{R}}^{2},{\mathbb{R}})$ defined by

$\mathcal{L}:=-\Delta+V(x)-\mu_{0}+au_{0}^{2},$ (3.23)
$\mathcal{N}:=-\Delta+V(x)-\mu_{0}+3au_{0}^{2},$ (3.24)

where $M({\mathbb{R}}^{2},{\mathbb{R}})$ is denoted by

$M({\mathbb{R}}^{2},{\mathbb{R}}):=H^{2}({\mathbb{R}}^{2},{\mathbb{R}})\cap\Big{\\{}u\in
L^{2}({\mathbb{R}}^{2},{\mathbb{R}}):\int_{\mathbb{R}^{2}}V(x)u^{2}dx<\infty\Big{\\}}.$

Since $V(x)\geq 0$ is radially symmetric and satisfies the assumption ($V$),
we next address the following analytical properties of $\mathcal{L}$ and
$\mathcal{N}$.

######  Lemma 3.2.

Under the assumptions of Theorem 1.4, we have

  1. 1.

The linearized operator $\mathcal{L}$ satisfies

$ker\mathcal{L}=\\{u_{0}\\}\,\ \hbox{and}\,\ \langle\mathcal{L}u,u\rangle\geq
0\,\ \hbox{for all}\,\ u\in L^{2}({\mathbb{R}}^{2},{\mathbb{R}}),$ (3.25)

where $u_{0}>0$ is the unique positive minimizer of $e(a)$. Moreover, there
exists $\rho>0$ such that for all $u\in H^{1}({\mathbb{R}}^{2},{\mathbb{R}})$,

$\langle\mathcal{L}u,u\rangle\geq\rho\|u\|^{2}_{H^{1}({\mathbb{R}}^{2})},\,\
\hbox{if}\,\ \int_{\mathbb{R}^{2}}u(x)u_{0}(x)dx=0.$ (3.26)

  2. 2.

If $a\neq 0$, then the linearized operator $\mathcal{N}$ is non-degenerate, in
the sense that

$ker\mathcal{N}=\\{0\\},$ (3.27)

and $\mathcal{N}^{-1}$: $L^{2}({\mathbb{R}}^{2})\longmapsto
L^{2}({\mathbb{R}}^{2})$ exists and is a continuous linear operator. Moreover,
there exists $\hat{\rho}>0$ such that for all $u\in
H^{2}({\mathbb{R}}^{2},{\mathbb{R}})$,

$\|\mathcal{N}u\|_{L^{2}({\mathbb{R}}^{2})}\geq\hat{\rho}\|u\|_{H^{2}({\mathbb{R}}^{2})}.$
(3.28)

  3. 3.

If $a\neq 0$ and $\phi(|x|)\in L^{2}({\mathbb{R}}^{2})$ is radially symmetric,
then $\psi(x)=\mathcal{N}^{-1}\phi(|x|)\in M({\mathbb{R}}^{2},{\mathbb{R}})$
is also radially symmetric.

Proof. (1). Define the first eigenvalue $\lambda_{1}$ of $\mathcal{L}$ by

$\lambda_{1}:=\inf\Big{\\{}\int_{\mathbb{R}^{2}}|\nabla
u|^{2}+(V(x)-\mu_{0}+a|u_{0}|^{2})|u|^{2}dx:\,\ u\in
M({\mathbb{R}}^{2},{\mathbb{R}}),\,\
\|u\|_{L^{2}({\mathbb{R}}^{2})}=1\Big{\\}},$ (3.29)

and let $(\lambda_{1},\phi)$ be the first eigenpair of $\mathcal{L}$. One can
derive easily that $\phi$ does not change sign in ${\mathbb{R}}^{2}$. Since
$(0,u_{0})$, where $u_{0}>0$, is an eigenpair of $\mathcal{L}$, we claim that
$\lambda_{1}=0$. Otherwise, if $\lambda_{1}<0$, then the eigenfunction
$u_{0}>0$ must be orthogonal to $\phi$, which is impossible in view of the
fact that $\phi>0$. Therefore, we conclude from Theorem B that $u_{0}>0$ is
the unique eigenfunction associated to the first eigenvalue $\lambda_{1}=0$.

Following [32, Theorem XIII.47 and XIII.67], we deduce from above that
$\mathcal{L}$ has a non-degenerate ground state and purely discrete spectrum
$\sigma(\mathcal{L})=\sigma_{d}(\mathcal{L})$, which further imply that

$ker\mathcal{L}=\\{u_{0}\\}\,\ \hbox{and}\,\ 0\in\sigma_{d}(\mathcal{L}).$

Therefore, for any $u\in H^{1}({\mathbb{R}}^{2})$ satisfying
$\int_{\mathbb{R}^{2}}u(x)u_{0}(x)dx=0$, we have

$\langle\mathcal{L}u,u\rangle\geq\rho\|u\|^{2}_{L^{2}({\mathbb{R}}^{2})}.$

On the other hand, since $u_{0}\in L^{\infty}({\mathbb{R}}^{2})$, we also
reduce from (3.29) that for any $u\in H^{1}({\mathbb{R}}^{2})$ satisfying
$\int_{\mathbb{R}^{2}}u(x)u_{0}(x)dx=0$,

$\langle\mathcal{L}u,u\rangle\geq\|\nabla
u\|^{2}_{L^{2}({\mathbb{R}}^{2})}-C\|u\|^{2}_{L^{2}({\mathbb{R}}^{2})}.$

We conclude from above two estimates that (3.26) holds true.

(2). If $a\in(0,\infty)$, it is obvious that $\mathcal{N}>\mathcal{L}\geq 0$
and thus $ker\mathcal{N}=\\{0\\}$. On the other hand, if $a\in(-a^{*},0)$,
then the non-degenerate of $\mathcal{N}$ is a well-known result of [7,
Corollary 11.5], which thus implies that (3.27) holds true. Moreover, the same
argument of [17, Theorem 5.3] yields that $\mathcal{N}^{-1}$:
$L^{2}({\mathbb{R}}^{2})\longmapsto L^{2}({\mathbb{R}}^{2})$ exists and is a
continuous linear operator satisfying (3.28).

(3). Similar to [17, Theorem 5.3]), since the operator $\mathcal{N}^{-1}$
commutes with the angular momentum, one can get that if $a\neq 0$ and
$\phi(|x|)\in L^{2}({\mathbb{R}}^{2})$ is radially symmetric, then
$\psi(x)=\mathcal{N}^{-1}\phi(|x|)\in M({\mathbb{R}}^{2},{\mathbb{R}})$ is
also radially symmetric, and the proof is therefore complete. ∎

Applying Lemma 3.2, we next derive the following refined estimates.

######  Lemma 3.3.

Under the assumptions of Theorem 1.4, let $\\{\alpha_{n}\\}$ be a positive
bounded sequence, and consider positive constants $C>0$ and $K>0$, which are
independent of $n$.

  1. 1.

If $q_{n}$ satisfies

$|x^{\perp}\cdot\nabla{q}_{n}|\leq C\alpha_{n}e^{-K|x|}\ \ \mbox{in}\
\,{\mathbb{R}}^{2},$ (3.30)

then there exists a constant $C=C(a,K)>0$, depending only on $a$ and $K$, such
that

$|r_{n}(x)|\leq C(a,K)\Omega_{n}\alpha_{n}e^{-K|x|},\ \,|\nabla r_{n}(x)|\leq
C(a,K)\Omega_{n}\alpha_{n}e^{-\frac{K}{2}|x|}\,\ \hbox{in\,\
${\mathbb{R}}^{2}$.}$ (3.31)

  2. 2.

If $f_{n}(x)\in M({\mathbb{R}}^{2},{\mathbb{R}})$ and $g_{n}(x)\in
L^{2}({\mathbb{R}}^{2},{\mathbb{R}})$ satisfy

$\mathcal{N}f_{n}(x)=g_{n}(x),\,\ |g_{n}(x)|\leq C\alpha_{n}e^{-K|x|}\,\
\hbox{in ${\mathbb{R}}^{2}$,}$ (3.32)

then there exists a constant $C(a,K)>0$, depending only on $a$ and $K$, such
that

    1. (a)

If $a\neq 0$, then we have

$|f_{n}(x)|\leq C(a,K)\alpha_{n}e^{-K|x|},\,\ |\nabla f_{n}(x)|\leq
C(a,K)\alpha_{n}e^{-\frac{K}{2}|x|}\,\ \hbox{in ${\mathbb{R}}^{2}$.}$ (3.33)

    2. (b)

If $a=0$ and $f_{n}(x)$ satisfies $\int_{\mathbb{R}^{2}}f_{n}(x)u_{0}(x)dx=0$,
then we have

$|f_{n}(x)|\leq C(K)\alpha_{n}e^{-K|x|},\,\ |\nabla f_{n}(x)|\leq
C(K)\alpha_{n}e^{-\frac{K}{2}|x|}\,\ \hbox{in ${\mathbb{R}}^{2}$.}$ (3.34)

Proof. 1\. Recall from (3.21) that $r_{n}$ satisfies the following equation

$(-\Delta+V(x)-\mu_{n}+a|u_{n}|^{2})r_{n}=-\Omega_{n}(x^{\perp}\cdot\nabla
q_{n})\ \ \hbox{in}\,\ {\mathbb{R}}^{2}.$ (3.35)

Following (3.6), (3.20) and (3.26), we then have

$\displaystyle-\Omega_{n}\int_{\mathbb{R}^{2}}\big{[}(x^{\perp}\cdot\nabla
q_{n})\,r_{n}\big{]}dx$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{2}}\big{[}|\nabla
r_{n}|^{2}+(V(x)-\mu_{n}+a|u_{n}|^{2})r^{2}_{n}\big{]}dx$ $\displaystyle\geq$
$\displaystyle\int_{\mathbb{R}^{2}}(\mathcal{L}r_{n}\cdot
r_{n})dx+\int_{\mathbb{R}^{2}}\big{[}(\mu_{0}-\mu_{n})+a(|u_{n}|^{2}-u_{0}^{2})\big{]}r_{n}^{2}dx$
$\displaystyle\geq$
$\displaystyle\,\rho\|r_{n}\|^{2}_{H^{1}({\mathbb{R}}^{2})}-o(1)\int_{\mathbb{R}^{2}}r_{n}^{2}dx\geq\frac{\rho}{2}\|r_{n}\|^{2}_{H^{1}({\mathbb{R}}^{2})}.$

We then get from (3.30) and above that

$\frac{\rho}{2}\|r_{n}\|^{2}_{H^{1}({\mathbb{R}}^{2})}\leq\Big{|}\Omega_{n}\int_{\mathbb{R}^{2}}\big{[}(x^{\perp}\cdot\nabla
q_{n})\,r_{n}\big{]}dx\Big{|}\leq
C\Omega_{n}\alpha_{n}\|r_{n}\|_{L^{2}({\mathbb{R}}^{2})},$

which further implies that

$\|r_{n}\|_{H^{1}({\mathbb{R}}^{2})}\leq C\Omega_{n}\alpha_{n},$ (3.36)

where $\alpha_{n}>0$ is as in (3.30). On the other hand, we derive from (3.35)
that $r_{n}^{2}$ satisfies the following equation

$\displaystyle\Big{[}-\frac{1}{2}\Delta+\Big{(}V(x)-\mu_{n}+a|u_{n}|^{2}\Big{)}\Big{]}r_{n}^{2}+|\nabla
r_{n}|^{2}=-\Omega_{n}(x^{\bot}\cdot\nabla q_{n})\,r_{n}\ \ \hbox{in}\,\
{\mathbb{R}}^{2},$

which then implies that

$-\frac{1}{2}\Delta
r_{n}^{2}-\mu_{n}r_{n}^{2}+a|u_{n}|^{2}r_{n}^{2}\leq-\Omega_{n}(x^{\bot}\cdot\nabla
q_{n})r_{n}\ \ \hbox{in}\,\ {\mathbb{R}}^{2}.$ (3.37)

By De Giorgi-Nash-Moser theory [20, Theorem 4.1], we thus obtain from (3.37)
that for any $y\in{\mathbb{R}}^{2}$,

$\sup_{x\in B_{\frac{1}{2}}(y)}|r_{n}(x)|^{2}\leq
C\Big{(}\|r_{n}\|^{2}_{L^{2}(B_{1}(y))}+\|\Omega_{n}(x^{\bot}\cdot\nabla
q_{n})r_{n}\|_{L^{2}(B_{1}(y))}\Big{)}.$ (3.38)

Combining (3.36) and (3.38) yields that

$\|r_{n}\|_{L^{\infty}({\mathbb{R}}^{2})}\leq C\Omega_{n}\alpha_{n}.$ (3.39)

Moreover, since $\mu_{n}\to\mu_{0}$ as $n\to\infty$, there exists a
sufficiently large constant $R=R(a,K)>0$ such that

$V(x)-\mu_{n}+a|u_{n}|^{2}\geq K^{2}+1\,\ \hbox{in}\,\
{\mathbb{R}}^{2}/B_{R}(0).$

By the comparison principle, we then deduce from (3.30) and (3.35) that

$|r_{n}(x)|\leq C\Omega_{n}\alpha_{n}e^{-K|x|}\,\ \hbox{in}\,\
{\mathbb{R}}^{2}/B_{R}(0),$

from which and (3.39) we derive that

$|r_{n}(x)|\leq C\Omega_{n}\alpha_{n}e^{-K|x|}\,\ \hbox{in}\,\
{\mathbb{R}}^{2}.$ (3.40)

Furthermore, applying gradient estimates (see (3.15) in [14]) to the equation
(3.35), we conclude from above that

$|\nabla r_{n}(x)|\leq C\Omega_{n}\alpha_{n}e^{-\frac{K}{2}|x|}\ \ \hbox{in\,
${\mathbb{R}}^{2}$},$ (3.41)

which therefore implies that (3.31) holds true in view of (3.40).

2\. We first prove that (3.33) holds true, if $a\neq 0$. Since $f_{n}$
satisfies (3.32), we derive from (3.28) that

$\displaystyle\hat{\rho}\|f_{n}\|_{H^{2}({\mathbb{R}}^{2})}\leq\|g_{n}\|_{L^{2}({\mathbb{R}}^{2})}\leq
C(a,K)\alpha_{n},$

which then implies that

$\|f_{n}\|_{L^{\infty}({\mathbb{R}}^{2})}\leq C(a,K)\alpha_{n}.$ (3.42)

By the comparison principle, we then derive from (3.32) that

$|f_{n}(x)|\leq C(a,K)\alpha_{n}e^{-K|x|}\ \ \hbox{in}\,\ {\mathbb{R}}^{2}.$
(3.43)

Applying again gradient estimates (see (3.15) in [14]), we further obtain from
(3.43) that $|\nabla f_{n}|$ also satisfies the desired estimate (3.33).

Next, we prove that (3.34) holds true, if $a=0$ and $f_{n}(x)$ satisfies
$\int_{\mathbb{R}^{2}}f_{n}(x)u_{0}(x)dx=0$. Indeed, in this case, we have

$\mathcal{N}=\mathcal{L}=-\Delta+V(x)-\mu_{0}.$

Since $f_{n}$ satisfies (3.32) and
$\int_{\mathbb{R}^{2}}f_{n}(x)u_{0}(x)dx=0$, we derive from Lemma 3.2 (1) that

$\|f_{n}\|^{2}_{H^{1}({\mathbb{R}}^{2})}\leq
C\langle\mathcal{L}f_{n},f_{n}\rangle\leq
C\|g_{n}\|_{L^{2}({\mathbb{R}}^{2})}\|f_{n}\|_{L^{2}({\mathbb{R}}^{2})}\leq
C(K)\alpha_{n}\|f_{n}\|_{L^{2}({\mathbb{R}}^{2})},$ (3.44)

which implies that

$\|f_{n}\|_{H^{1}({\mathbb{R}}^{2})}\leq C(K)\alpha_{n}.$ (3.45)

Similar to the proof of (3.39), since $f_{n}$ satisfies (3.45), we obtain from
De Giorgi-Nash-Moser theory [20, Theorem 4.1] that

$\|f_{n}\|_{L^{\infty}({\mathbb{R}}^{2})}\leq C(K)\alpha_{n}.$ (3.46)

Following (3.46), the comparison principle applied to (3.32) then yields that

$|f_{n}(x)|\leq C(K)\alpha_{n}e^{-K|x|}\ \ \hbox{in}\,\ {\mathbb{R}}^{2},$
(3.47)

from which one can further obtain (3.34) on the similar exponential decay of
$|\nabla f_{n}|$. This completes the proof of Lemma 3.3. ∎

##  4 Proof of Theorem 1.4

For any fixed $a\in[0,+\infty)$ or a.e. $a\in(-a^{*},0)$, this section is
devoted to the complete proof of Theorem 1.4 on the nonexistence of vortices
for minimizers of $e(\Omega,a)$ in the case where $\Omega>0$ is small enough,
since the case $\Omega=0$ follows directly in view of Theorem B and those
explanations below (1.17). Under the assumptions of Theorem 1.4, consider the
minimizer $u_{n}$ of $e(\Omega,a)$, and let $(q_{n},r_{n})$ be as in (3.19).
We also recall from Theorem B that $u_{0}>0$ is the unique positive minimizer
of $e(a)$, and $(u_{0},\mu_{0})$ satisfies the elliptic equation (3.3).

Following the estimates of the previous section, we first establish the
following corollary.

######  Corollary 4.1.

Under the assumptions of Theorem 1.4, define

$w_{n}(x):=q_{n}(x)-u_{0}(x),\quad\varepsilon_{n}:=\max\big{\\{}\Omega_{n},|\mu_{n}-\mu_{0}|\big{\\}}>0,$
(4.1)

where $\mu_{n}$ and $\mu_{0}$ are as in (3.1) and (3.3), respectively. Then we
have

  1. 1.

There exists a constant $C>0$, independent of $n$, such that

$|r_{n}(x)|\leq C\Omega_{n}e^{-|x|},\ \,|\nabla r_{n}(x)|\leq
C\Omega_{n}e^{-\frac{1}{2}|x|}\ \ \mbox{in}\,\ {\mathbb{R}}^{2}.$ (4.2)

  2. 2.

There exists a constant $C>0$, independent of $n$, such that

    1. (a)

If $a\neq 0$, then $w_{n}$ satisfies

$|w_{n}(x)|\leq C\varepsilon_{n}e^{-|x|},\ \,|\nabla w_{n}(x)|\leq
C\varepsilon_{n}e^{-\frac{1}{2}|x|}\ \ \mbox{in}\,\ {\mathbb{R}}^{2}.$ (4.3)

    2. (b)

If $a=0$, then
$w^{\perp}_{n}=w_{n}-\big{(}{\int_{\mathbb{R}^{2}}w_{n}u_{0}dx}\big{)}u_{0}$
satisfies

$|w^{\perp}_{n}(x)|\leq C\varepsilon_{n}e^{-|x|},\ \,|\nabla
w^{\perp}_{n}(x)|\leq C\varepsilon_{n}e^{-\frac{1}{2}|x|}\ \ \mbox{in}\,\
{\mathbb{R}}^{2}.$ (4.4)

Proof. Following Lemma 3.1 (3), the estimates of (4.2) follow directly from
Lemma 3.3 by taking $\alpha_{n}\equiv 1$ and $K=1$.

Next, we prove that (4.3) holds if $a\neq 0$. Direct calculations give from
(3.21) that $w_{n}$ satisfies the following equation

$\mathcal{N}w_{n}+a(q_{n}+2u_{0})w_{n}^{2}=(\mu_{n}-\mu_{0})q_{n}-ar_{n}^{2}q_{n}+\Omega_{n}(x^{\perp}\cdot\nabla
r_{n})\ \ \mbox{in}\,\ {\mathbb{R}}^{2},$ (4.5)

where the linearized operator $\mathcal{N}$ is defined by (3.24). The left-
hand side of (4.5) satisfies

$\displaystyle\quad\|\mathcal{N}w_{n}+a(q_{n}+2u_{0})w_{n}^{2}\|_{L^{2}({\mathbb{R}}^{2})}$
$\displaystyle\geq\|\mathcal{N}w_{n}\|_{L^{2}({\mathbb{R}}^{2})}-C(a)\|w_{n}\|^{2}_{L^{4}({\mathbb{R}}^{2})}$
$\displaystyle\geq\hat{\rho}\|w_{n}\|_{H^{2}({\mathbb{R}}^{2})}-C(a)\|w_{n}\|^{2}_{H^{2}({\mathbb{R}}^{2})},$

where (3.28) is used in the second inequality. On the other hand, following
(4.2), the right-hand side of (4.5) satisfies

$\displaystyle\big{|}(\mu_{n}-\mu_{0})q_{n}-ar_{n}^{2}q_{n}+\Omega_{n}(x^{\perp}\cdot\nabla
r_{n})\big{|}\leq C\varepsilon_{n}e^{-|x|}\ \ \hbox{in}\,\ {\mathbb{R}}^{2}.$

We thus derive from above that

$\hat{\rho}\|w_{n}\|_{H^{2}({\mathbb{R}}^{2})}-C(a)\|w_{n}\|^{2}_{H^{2}({\mathbb{R}}^{2})}\leq
C\varepsilon_{n},$

where Lemma 3.2 (2) is also used. This further implies that

$\|w_{n}\|_{L^{\infty}({\mathbb{R}}^{2})}\leq
C\|w_{n}\|_{H^{2}({\mathbb{R}}^{2})}\leq C\varepsilon_{n}.$ (4.6)

Similar to the proofs of (3.39) and (3.40), one can further conclude from
(4.5) and (4.6) that (4.3) holds true.

Finally, we prove that (4.4) holds for the case $a=0$. In this case, the
operator $\mathcal{N}=\mathcal{L}=-\Delta+V(x)-\mu_{0}$, and
$w_{n}^{\perp}=w_{n}-\big{(}{\int_{\mathbb{R}^{2}}w_{n}u_{0}dx}\big{)}u_{0}$
satisfies $\int_{\mathbb{R}^{2}}w_{n}^{\perp}(x)u_{0}(x)dx=0$ and

$\mathcal{N}w_{n}^{\perp}=-\Delta
w_{n}^{\perp}+V(x)w_{n}^{\perp}-\mu_{0}w_{n}^{\perp}=(\mu_{n}-\mu_{0})q_{n}+\Omega_{n}(x^{\perp}\cdot\nabla
r_{n})\ \ \hbox{in}\,\ {\mathbb{R}}^{2}.$ (4.7)

Following Lemma 3.1 (3), we obtain that the right-hand side of (4.7) satisfies

$\big{|}(\mu_{n}-\mu_{0})q_{n}+\Omega_{n}(x^{\perp}\cdot\nabla
r_{n})\big{|}\leq C\varepsilon_{n}e^{-|x|}\,\ \hbox{in}\,\ {\mathbb{R}}^{2}.$

Thus, we derive from Lemma 3.3 (2) that (4.4) holds true, and we are done. ∎

######  Lemma 4.2.

Under the assumptions of Theorem 1.4, for some $m\geq 2$ suppose
$w_{n}=q_{n}-u_{0}$ can be decomposed as

$w_{n}(x)=\psi_{m,n}(|x|)+T_{m,n}(x),$ (4.8)

where $\psi_{m,n}(|x|)$ is radially symmetric, and $T_{m,n}$ satisfies

$|T_{m,n}(x)|\leq A^{m}\varepsilon_{n}^{m}e^{-|x|},\ \,|\nabla T_{m,n}(x)|\leq
A^{m}\varepsilon_{n}^{m}e^{-\frac{1}{2}|x|}\ \ \hbox{in\
\,${\mathbb{R}}^{2}$}\,\ \mbox{as}\ \,n\to\infty,$ (4.9)

for some constant $A>0$, independent of $m$ and $n$, where $\varepsilon_{n}>0$
is defined in (4.1). Then the decomposition of (4.8) and (4.9) holds for
$m+1$.

Proof. Note that $q_{n}$ is the real part of $u_{n}e^{i\theta_{n}}$, $u_{0}>0$
and $\varepsilon_{n}>0$ are as in (4.1). For some large constant $A>0$, we may
assume that $\varepsilon_{n}>0$ is small enough so that
$0<A\varepsilon_{n}^{\frac{1}{2}}<1$ for sufficiently large $n>0$. Moreover,
we always use the same symbol $C$ to denote various positive constants which
are independent of $A>0$, $m\geq 2$ and $\varepsilon_{n}>0$.

Since $u_{0}(|x|)+\psi_{m,n}(|x|)$ is radially symmetric, we derive from (4.8)
and (4.9) that for some $m\geq 2$,

$\big{|}x^{\perp}\cdot\nabla q_{n}\big{|}=\big{|}x^{\perp}\cdot\nabla
T_{m,n}\big{|}\leq CA^{m}\varepsilon_{n}^{m}e^{-\frac{1}{4}|x|}\ \ \mbox{in\,\
${\mathbb{R}}^{2}$},$

which and Lemma 3.3 (1) then yield that

$|r_{n}(x)|\leq CA^{m}\varepsilon_{n}^{m+1}e^{-\frac{1}{4}|x|},\ \ |\nabla
r_{n}(x)|\leq CA^{m}\varepsilon_{n}^{m+1}e^{-\frac{1}{8}|x|}\ \ \mbox{in\,\
${\mathbb{R}}^{2}$}.$ (4.10)

The rest proof is divided into two steps.

Step 1. We claim that there exists a decomposition
$w_{n}=\psi_{m+1,n}+T_{m+1,n}$, where $\psi_{m+1,n}(x)=\psi_{m+1,n}(|x|)$ is
radially symmetric and $T_{m+1,n}(x)$ satisfies

$|T_{m+1,n}|\leq CA^{m}\varepsilon_{n}^{m+1}e^{-\frac{1}{16}|x|},\,\ |\nabla
T_{m+1,n}|\leq CA^{m}\varepsilon_{n}^{m+1}e^{-\frac{1}{32}|x|}\ \ \hbox{in\,\
${\mathbb{R}}^{2}$.}$ (4.11)

To prove (4.11), we shall consider separately the following two cases:

Case 1: $a=0$. In this case, we define

$\psi_{m+1}\equiv\Big{(}\int_{\mathbb{R}^{2}}w_{n}u_{0}dx\Big{)}u_{0},\ \
\hbox{and}\ \ T_{m+1}\equiv
w_{n}^{\perp}=w_{n}-\Big{(}\int_{\mathbb{R}^{2}}w_{n}u_{0}dx\Big{)}u_{0}.$
(4.12)

Note from (4.7) that for $a=0$,
$\mathcal{N}=\mathcal{L}=-\Delta+V(x)-\mu_{0}$, $ker\mathcal{L}=\\{u_{0}\\}$,
and $w_{n}^{\perp}$ satisfies

$\mathcal{L}w_{n}^{\perp}=(\mu_{n}-\mu_{0})q_{n}+\Omega_{n}(x^{\perp}\cdot\nabla
r_{n})\ \ \hbox{in}\ \ {\mathbb{R}}^{2},\ \
\int_{\mathbb{R}^{2}}w_{n}^{\perp}u_{0}dx=0.$ (4.13)

Since

$0=\langle\mathcal{L}u_{0},w_{n}^{\perp}\rangle=\langle
u_{0},\mathcal{L}w_{n}^{\perp}\rangle=\langle
u_{0},(\mu_{n}-\mu_{0})q_{n}+\Omega_{n}(x^{\perp}\cdot\nabla r_{n})\rangle,$

and $q_{n}\to u_{0}$ strongly in $L^{\infty}({\mathbb{R}}^{2},{\mathbb{R}})$
as $n\to\infty$, we can deduce from (4.10) that

$\begin{split}\frac{1}{2}|\mu_{n}-\mu_{0}|\langle
u_{0},u_{0}\rangle&\leq|\langle u_{0},(\mu_{n}-\mu_{0})q_{n}\rangle|\\\
&=|\langle u_{0},\Omega_{n}(x^{\perp}\cdot\nabla r_{n})\rangle|\leq
CA^{m}\varepsilon_{n}^{m+1}.\end{split}$ (4.14)

Therefore, we deduce from (3.7), (4.10) and above that the right-hand side of
(4.13) satisfies

$|(\mu_{n}-\mu_{0})q_{n}+\Omega_{n}(x^{\perp}\cdot\nabla r_{n})|\leq
CA^{m}\varepsilon_{n}^{m+1}e^{-\frac{1}{16}|x|}\,\ \hbox{in\,\
${\mathbb{R}}^{2}$}.$

By Lemma 3.3 (2b), we thus obtain from (4.13) that

$|w_{n}^{\perp}(x)|\leq CA^{m}\varepsilon_{n}^{m+1}e^{-\frac{1}{16}|x|},\,\
|\nabla w_{n}^{\perp}(x)|\leq CA^{m}\varepsilon_{n}^{m+1}e^{-\frac{1}{32}|x|}\
\ \hbox{in}\,\ {\mathbb{R}}^{2},$

which shows that the claim (4.11) holds true in the case $a=0$.

Case 2: $a\neq 0$. In this case, for above $m\geq 2$ we obtain from (4.3),
(4.8) and (4.9) that

$\begin{split}|\psi_{m,n}(x)|\leq
A^{m}\varepsilon_{n}^{m}e^{-|x|}+C\varepsilon_{n}e^{-|x|}\leq
C\varepsilon_{n}e^{-|x|}\quad\hbox{in}\,\ {\mathbb{R}}^{2}.\end{split}$ (4.15)

Similarly, we have

$|\nabla\psi_{m,n}(x)|\leq
C\varepsilon_{n}e^{-\frac{1}{2}|x|}\quad\hbox{in}\,\ {\mathbb{R}}^{2}.$ (4.16)

Since $q_{n}=u_{0}+\psi_{m,n}+T_{m,n}$, we get from (4.5) that

$\begin{split}\mathcal{N}w_{n}:&=(\mu_{n}-\mu_{0})(u_{0}+w_{n})-a(q_{n}+2u_{0})w_{n}^{2}-ar_{n}^{2}q_{n}+\Omega_{n}(x^{\perp}\cdot\nabla
r_{n})\\\ &=\Omega_{n}\big{(}x^{\perp}\cdot\nabla
r_{n}\big{)}-ar_{n}^{2}q_{n}+B_{1,n}(x)+B_{2,n}(x)\quad\hbox{in}\,\
{\mathbb{R}}^{2},\end{split}$ (4.17)

where the radially symmetric function $B_{1,n}(x)$ satisfies

$\begin{split}B_{1,n}(x)&=(\mu_{n}-\mu_{0})(u_{0}+\psi_{m,n})-a\big{(}\psi_{m,n}^{3}+3u_{0}\psi_{m,n}^{2}\big{)},\end{split}$
(4.18)

and the non-radially symmetric function $B_{2,n}(x)$ satisfies

$\begin{split}B_{2,n}(x)=&(\mu_{n}-\mu_{0})T_{m,n}-a\big{(}3\psi_{m,n}^{2}+3\psi_{m,n}T_{m,n}\\\
&+T^{2}_{m,n}+6u_{0}\psi_{m,n}+3u_{0}T_{m,n}\big{)}T_{m,n}.\end{split}$ (4.19)

Following Lemma 3.2 (3) and (4.18), there exists a unique function
$\psi_{m+1,n}\in C^{2}({\mathbb{R}}^{2})\cap L^{\infty}({\mathbb{R}}^{2})$ of

$\begin{split}\mathcal{N}\psi_{m+1,n}=B_{1,n}(x)\quad\hbox{in\
${\mathbb{R}}^{2}$,}\end{split}$ (4.20)

and moreover, $\psi_{m+1,n}$ is also radially symmetric.

We now define

$T_{m+1,n}(x):=w_{n}(x)-\psi_{m+1,n}(x),$ (4.21)

where $\psi_{m+1,n}(x)$ is given by (4.20), so that $T_{m+1,n}(x)$ satisfies

$\mathcal{N}T_{m+1,n}=\Omega_{n}\big{(}x^{\perp}\cdot\nabla
r_{n}\big{)}-ar_{n}^{2}q_{n}+B_{2,n}(x)\ \ \mbox{in}\,\ {\mathbb{R}}^{2},$
(4.22)

due to (4.17). Following (4.9), (4.10) and (4.15), we deduce that there exists
a constant $C>0$ such that

$|\Omega_{n}(x^{\perp}\cdot\nabla r_{n})-ar_{n}^{2}q_{n}|\leq
CA^{m}\varepsilon_{n}^{m+1}e^{-\frac{1}{16}|x|}\ \ \hbox{in}\,\
{\mathbb{R}}^{2},$ (4.23)

and

$|B_{2,n}(x)|\leq CA^{m}\varepsilon_{n}^{m+1}e^{-|x|},\,\ \ |\nabla
B_{2,n}(x)|\leq CA^{m}\varepsilon_{n}^{m+1}e^{-\frac{1}{2}|x|}\ \ \hbox{in}\,\
{\mathbb{R}}^{2}.$ (4.24)

In view of (4.23) and (4.24), we conclude from Lemma 3.3 (2$a$) that the claim
(4.11) also holds true for the case $a\neq 0$.

Step 2. Under the decompositions (4.12) or (4.21), we shall prove that (4.9)
holds for $m+1$, which then completes the proof of the lemma. Towards this
aim, actually it suffices to show that $T_{m+1,n}$ decays faster than (4.11)
as $|x|\to\infty$.

Recall from (3.21), (4.13), (4.21) and (4.22) that $(r_{n},T_{m+1,n})$
satisfies the following system

$\left\\{\begin{aligned}
\mathcal{N}T_{m+1,n}&=\Omega_{n}\big{(}x^{\perp}\cdot\nabla
r_{n}\big{)}-ar_{n}^{2}q_{n}+\hat{B}_{2,n}(x)\ \ &\mbox{in}\ \
{\mathbb{R}}^{2},\\\ \mathcal{L}_{n}r_{n}&=-\Omega_{n}(x^{\bot}\cdot\nabla
T_{m+1,n})\ \ &\mbox{in}\,\ {\mathbb{R}}^{2},\end{aligned}\right.$ (4.25)

where $\hat{B}_{2,n}(x)=(\mu_{n}-\mu_{0})q_{n}$ if $a=0$, and
$\hat{B}_{2,n}(x)={B}_{2,n}(x)$ if $a\neq 0$. Multiplying the first equation
of (4.25) by $T_{m+1,n}$ and the second one of (4.25) by $r_{n}$,
respectively, we then have

$\left\\{\begin{aligned} &-\frac{1}{2}\Delta|T_{m+1,n}|^{2}+|\nabla
T_{m+1,n}|^{2}+\Big{[}V(x)-\mu_{0}+3au_{0}^{2}\Big{]}|T_{m+1,n}|^{2}\\\
&\quad\quad=\Big{[}\Omega_{n}\big{(}x^{\perp}\cdot\nabla
r_{n}\big{)}-ar_{n}^{2}q_{n}+\hat{B}_{2,n}\Big{]}T_{m+1,n}\quad\mbox{in\,\
${\mathbb{R}}^{2}$},\\\ &-\frac{1}{2}\Delta r_{n}^{2}+|\nabla
r_{n}|^{2}+\Big{[}V(x)-\mu_{n}+a|u_{n}|^{2}\Big{]}r_{n}^{2}=-\Omega_{n}(x^{\bot}\cdot\nabla
T_{m+1,n})r_{n}\quad\mbox{in\,\ ${\mathbb{R}}^{2}$}.\end{aligned}\right.$

By the diamagnetic inequality, we have

$|\nabla
T_{m+1,n}|^{2}+\frac{\Omega^{2}_{n}|x|^{2}}{4}|r_{n}|^{2}+\Omega_{n}(x^{\bot}\cdot\nabla
T_{m+1,n})r_{n}\geq 0\quad\mbox{in\,\ ${\mathbb{R}}^{2}$},$ (4.26)

and

$|\nabla
r_{n}|^{2}+\frac{\Omega^{2}_{n}|x|^{2}}{4}|T_{m+1,n}|^{2}-\Omega_{n}(x^{\bot}\cdot\nabla
r_{n})T_{m+1,n}\geq 0\quad\mbox{in\,\ ${\mathbb{R}}^{2}$}.$

By the exponential decay (3.7) of $|u_{n}|$, we also get from (4.10), (4.14)
and (4.24) that as $n\to\infty$,

$\displaystyle\big{(}-ar_{n}^{2}q_{n}+\hat{B}_{2,n}\big{)}T_{m+1,n}$
$\displaystyle\leq\frac{1}{2}|T_{m+1,n}|^{2}+\frac{1}{2}\big{(}-ar_{n}^{2}q_{n}+\hat{B}_{2,n}\big{)}^{2}$
$\displaystyle\leq\frac{1}{2}|T_{m+1,n}|^{2}+C\varepsilon_{n}^{2(m+1)}A^{2m}e^{-2|x|}\quad\mbox{in\,\
${\mathbb{R}}^{2}/B_{R}$},$

where and below $R>0$ is sufficiently large. We then obtain from above that as
$n\to\infty$,

$\frac{1}{2}V(x)-\mu_{0}+3a|u_{0}|^{2}+a|u_{n}|^{2}\geq 3\quad\hbox{in}\,\
{\mathbb{R}}^{2}/B_{R},$

which further implies that

$\begin{split}\quad&-\frac{1}{2}\Delta\big{(}|T_{m+1,n}|^{2}+r_{n}^{2}\big{)}+\frac{5}{2}\big{(}|T_{m+1,n}|^{2}+r_{n}^{2}\big{)}\\\
\leq&\,CA^{2m}\varepsilon_{n}^{2(m+1)}e^{-2|x|}\quad\mbox{in\ \
${\mathbb{R}}^{2}/B_{R}$}.\end{split}$ (4.27)

By the comparison principle, we thus derive from (4.11) and (4.27) that as
$n\to\infty$,

$|T_{m+1,n}(x)|,\ |r_{n}(x)|\leq
CA^{m}\varepsilon_{n}^{m+1}e^{-|x|}\quad\hbox{in\ \ ${\mathbb{R}}^{2}$.}$
(4.28)

We next address the desired exponential decay of $|\nabla T_{m+1,n}|$ and
$|\nabla r_{n}|$. Note from (4.25) that $\partial_{i}T_{m+1,n}:=\frac{\partial
T_{m+1,n}}{\partial x_{i}}$ satisfies

$\displaystyle\quad\mathcal{N}\partial_{i}T_{m+1,n}+\Big{[}\partial_{i}V(x)+6au_{0}\partial_{i}u_{0}\Big{]}T_{m+1,n}$
$\displaystyle=\partial_{i}\Big{[}\Omega_{n}\big{(}x^{\perp}\cdot\nabla
r_{n}\big{)}-ar_{n}^{2}q_{n}+\hat{B}_{2,n}(x)\Big{]}\quad\mbox{in\,\
${\mathbb{R}}^{2}$},\ i=1,\,2.$

Multiplying the above equation by $\partial_{i}T_{m+1,n}$ and summing it for
$i=1,2$, we obtain that

$\begin{split}&\quad-\frac{1}{2}\Delta|\nabla
T_{m+1,n}|^{2}+\Big{[}V(x)-\mu_{0}+3au_{0}^{2}\Big{]}|\nabla T_{m+1,n}|^{2}\\\
&\quad+\sum_{i=1}^{2}|\nabla\partial_{i}T_{m+1,n}|^{2}+\sum_{i=1}^{2}\Big{[}\partial_{i}V(x)+6au_{0}\partial_{i}u_{0}\Big{]}T_{m+1,n}\partial_{i}T_{m+1,n}\\\
&=\sum_{i=1}^{2}\partial_{i}\Big{[}\Omega_{n}\big{(}x^{\perp}\cdot\nabla
r_{n}\big{)}-ar_{n}^{2}q_{n}+\hat{B}_{2,n}(x)\Big{]}\partial_{i}T_{m+1,n}\quad\mbox{in\,\
${\mathbb{R}}^{2}$}.\end{split}$ (4.29)

By the exponential decay of $u_{0}$ and $|u_{n}|$, we calculate from (4.11)
and (4.28) that as $n\to\infty$,

$\displaystyle\quad\sum_{i=1}^{2}\Big{[}\partial_{i}V(x)+6au_{0}\partial_{i}u_{0}\Big{]}T_{m+1,n}\partial_{i}T_{m+1,n}$
(4.30)
$\displaystyle\leq\sum_{i=1}^{2}\Big{[}|\partial_{i}V(x)|^{2}+C\Big{]}|T_{m+1,n}|^{2}+\sum_{i=1}^{2}\frac{1}{4}|\partial_{i}T_{m+1,n}|^{2}$
$\displaystyle\leq CA^{2m}\varepsilon_{n}^{2(m+1)}e^{-|x|}+\frac{1}{4}|\nabla
T_{m+1,n}|^{2}\quad\hbox{in\,\ ${\mathbb{R}}^{2}/B_{R}$},$

where and below $R>0$ is as before sufficiently large, and the assumption
$(V)$ is used in the last inequality.

Similar to (4.30), we have

$\displaystyle\quad\sum_{i=1}^{2}\partial_{i}\Big{[}-ar_{n}^{2}q_{n}+\hat{B}_{2,n}(x)\Big{]}\partial_{i}T_{m+1,n}$
$\displaystyle\leq\frac{1}{4}\sum_{i=1}^{2}|\partial_{i}T_{m+1,n}|^{2}+\sum_{i=1}^{2}\Big{[}\partial_{i}\big{(}-ar_{n}^{2}q_{n}+\hat{B}_{2,n}\big{)}\Big{]}^{2}$
$\displaystyle\leq\frac{1}{4}|\nabla
T_{m+1,n}|^{2}+CA^{2m}\varepsilon_{n}^{2(m+1)}e^{-|x|}\quad\mbox{in\,\
${\mathbb{R}}^{2}/B_{R}$,}$

which further implies that as $n\to\infty$,

$\begin{split}&\quad-\frac{1}{2}\Delta|\nabla
T_{m+1,n}|^{2}+\Big{[}V(x)-\mu_{0}+3au_{0}^{2}-\frac{1}{2}\Big{]}|\nabla
T_{m+1,n}|^{2}+\sum_{i=1}^{2}|\nabla\partial_{i}T_{m+1,n}|^{2}\\\
&\leq\sum_{i=1}^{2}\partial_{i}\Big{[}\Omega_{n}\big{(}x^{\perp}\cdot\nabla
r_{n}\big{)}\Big{]}\partial_{i}T_{m+1,n}+CA^{2m}\varepsilon_{n}^{2(m+1)}e^{-|x|}\quad\mbox{in\,\
${\mathbb{R}}^{2}/B_{R}$},\end{split}$ (4.31)

in view of (4.29) and (4.30). Similar to (4.31), we also get that as
$n\to\infty$,

$\begin{split}&\quad-\frac{1}{2}\Delta|\nabla
r_{n}|^{2}+\Big{[}V(x)-\mu_{n}+a|u_{n}|^{2}-\frac{1}{2}\Big{]}|\nabla
r_{n}|^{2}+\sum_{i=1}^{2}|\nabla\partial_{i}r_{n}|^{2}\\\
&\leq-\sum_{i=1}^{2}\partial_{i}\Big{[}\Omega_{n}\big{(}x^{\perp}\cdot\nabla
T_{m+1,n}\big{)}\Big{]}\partial_{i}r_{n}+CA^{2m}\varepsilon_{n}^{2(m+1)}e^{-|x|}\quad\mbox{in\,\
${\mathbb{R}}^{2}/B_{R}$}.\end{split}$ (4.32)

Using the diamagnetic inequality as in (4.26), one can derive from (4.31) and
(4.32) that as $n\to\infty$,

$-\frac{1}{2}\Delta\Big{(}|\nabla r_{n}|^{2}+|\nabla
T_{m+1,n}|^{2}\Big{)}+\Big{(}|\nabla r_{n}|^{2}+|\nabla
T_{m+1,n}|^{2}\Big{)}\leq
CA^{2m}\varepsilon_{n}^{2(m+1)}e^{-|x|}\quad\mbox{in\,\
${\mathbb{R}}^{2}/B_{R}$.}$

By the comparison principle, we then derive from above that as $n\to\infty$,

$|\nabla r_{n}|,\ \ |\nabla T_{m+1,n}|\leq
CA^{m}\varepsilon_{n}^{(m+1)}e^{-\frac{1}{2}|x|}\quad\mbox{in\ \
${\mathbb{R}}^{2}/B_{R}$.}$ (4.33)

Therefore, we conclude from (4.10), (4.11) and (4.33) that there exists a
sufficiently large $C>0$ such that as $n\to\infty$,

$|\nabla r_{n}(x)|,\ \ |\nabla T_{m+1,n}(x)|\leq
CA^{m}\varepsilon_{n}^{(m+1)}e^{-\frac{1}{2}|x|}\quad\mbox{in\,\
${\mathbb{R}}^{2}$.}$ (4.34)

Because the positive constant $C$ in (4.34) is independent of the values
$A>0$, $m\geq 2$ and $n$, one can choose a sufficiently large constant $A$
such that $A>C$. We thus conclude that (4.9) holds for $m+1$ in view of (4.28)
and (4.34), and the proof of Lemma 4.2 is therefore complete. ∎

Proof of Theorem 1.4. Set
$T_{1,n}(x)=w^{\perp}_{n}(x)=w_{n}-\big{(}\int_{\mathbb{R}^{2}}w_{n}u_{0}dx\big{)}u_{0}$
and $\psi_{1,n}=\big{(}\int_{\mathbb{R}^{2}}w_{n}u_{0}dx\big{)}u_{0}$ if
$a=0$, while $T_{1,n}(x)=w_{n}(x)$ and $\psi_{1,n}\equiv 0$ if $a\neq 0$. We
then obtain from Corollary 4.1 (2) that

$|T_{1,n}(x)|\leq C_{1}\varepsilon_{n}e^{-|x|},\ \ |\nabla T_{1,n}(x)|\leq
C_{1}\varepsilon_{n}e^{-\frac{1}{2}|x|}\ \ \mbox{in}\ \,{\mathbb{R}}^{2},$
(4.35)

where $C_{1}>0$ is independent of $n>0$. Set $T_{2,n}(x)=T_{1,n}(x)$ if $a=0$,
and $T_{2,n}:=q_{n}-u_{0}-\psi_{2,n}$ if $a\neq 0$, where $\psi_{2,n}\in
C^{2}({\mathbb{R}}^{2})\cap L^{\infty}({\mathbb{R}}^{2})$ is the unique
solution of

$\mathcal{N}\psi_{2,n}=(\mu_{n}-\mu_{0})u_{0}\ \ \mbox{in}\
\,{\mathbb{R}}^{2}.$

It then follows from Lemma 3.2 (3) that $\psi_{2,n}(|x|)$ is radially
symmetric. Moreover, based on (4.35), the same argument of proving Lemma 4.2
gives that there exists a constant $C_{2}>0$, independent of $n>0$, such that

$|T_{2,n}(x)|\leq C_{2}\varepsilon_{n}^{2}e^{-|x|},\ \ |\nabla T_{2,n}(x)|\leq
C_{2}\varepsilon_{n}^{2}e^{-\frac{1}{2}|x|}\ \ \mbox{in}\
\,{\mathbb{R}}^{2}\quad\mbox{as}\ \ n\to\infty.$

Take $A>0$ large enough that $A^{2}>C_{2}$, from which we have

$|T_{2,n}(x)|\leq A^{2}\varepsilon_{n}^{2}e^{-|x|},\,\ |\nabla T_{2,n}(x)|\leq
A^{2}\varepsilon_{n}^{2}e^{-\frac{1}{2}|x|}\ \ \mbox{in}\
\,{\mathbb{R}}^{2}\quad\mbox{as}\ \ n\to\infty.$ (4.36)

By Lemma 4.2, we deduce from (4.36) that for any $m\geq 2$,

$|T_{m,n}(x)|\leq A^{m}\varepsilon_{n}^{m}e^{-|x|},\ \ |\nabla T_{m,n}(x)|\leq
A^{m}\varepsilon_{n}^{m}e^{-\frac{1}{2}|x|}\ \ \mbox{in}\
\,{\mathbb{R}}^{2}\quad\mbox{as}\ \ n\to\infty.$

Recall from (4.8) that $q_{n}(x)=[u_{0}(x)+\psi_{m,n}(|x|)]+T_{m,n}(x)$, where
$u_{0}(x)+\psi_{m,n}(|x|)$ is radially symmetric. Applying Lemma 3.3, we thus
derive from above that for any $m\geq 2$,

$|r_{n}(x)|\leq CA^{m}\varepsilon_{n}^{m+1}e^{-\frac{1}{4}|x|},\,\ |\nabla
r_{n}(x)|\leq CA^{m}\varepsilon_{n}^{m+1}e^{-\frac{1}{8}|x|}\ \ \mbox{in}\
\,{\mathbb{R}}^{2}\quad\mbox{as}\,\ n\to\infty.$ (4.37)

Therefore, we conclude from (4.37) that for any $m\geq 2$,

$\|r_{n}\|_{C^{1}({\mathbb{R}}^{2})}\leq C\varepsilon_{n}^{\frac{m+2}{2}}\ \
\mbox{in}\ \,{\mathbb{R}}^{2}\,\ \mbox{for sufficiently large}\,\ n,$

which further implies that $r_{n}\equiv 0$ as $n\to\infty$.

Since $r_{n}(x)\equiv 0$ as $n\to\infty$, $q_{n}$ must be a positive minimizer
of (1.4), and hence $q_{n}$ satisfies

$-\Delta q_{n}+\big{(}V(x)-\mu_{0}+aq_{n}^{2}\big{)}q_{n}=0\ \ \mbox{in}\,\
{\mathbb{R}}^{2},$ (4.38)

which implies the absence of vortices for $q_{n}$ as $n\to\infty$. Following
Theorem B, we also obtain the uniqueness of $q_{n}\equiv u_{0}$ for any fixed
$a\in[0,+\infty)$ or a.e. $a\in(-a^{*},0)$. This therefore completes the proof
of Theorem 1.4. ∎

## References

  * [1] J. R. Abo-Shaeer, C. Raman, J. M. Vogels and W. Ketterle, Observation of vortex lattices in Bose-Einstein condensate, Science 292 (2001), 476.
  * [2] A. Aftalion, Vortices in Bose-Einstein condensates, Progress in Nonlinear Differential Equations and their Applications 67, Birkh$\ddot{a}$user Boston, Inc., Boston, MA, 2006.
  * [3] A. Aftalion, X. Blanc and F. Nier, Lowest Landau level functional and Bargmann spaces for Bose-Einstein condensates, J. Funct. Anal. 241 (2006), 661–702.
  * [4] A. Aftalion, R. L. Jerrard and J. Royo-Letelier, Non-existence of vortices in the small density region of a condensate, J. Funct. Anal. 260 (2011), 2387–2406.
  * [5] A. Aftalion, B. Noris, and C. Sourdis, Thomas-Fermi approximation for coexisting two component Bose-Einstein condensates and nonexistence of vortices for small rotation, Comm. Math. Phys. 336 (2015), 509–579.
  * [6] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal. 23 (1992), 1482–1518.
  * [7] J. Byeon and Y. Oshita, Uniqueness of standing waves for nonlinear Schr$\ddot{o}$dinger equations, Proc. Roy. Soc. Edinburgh Sect. A 138 (2008), 975–987.
  * [8] K. Chandrasekharan, Elliptic functions, Grundlehren Math. Wiss., Springer, Berlin, 1985.
  * [9] M. Correggi, N. Rougerie and J. Yngvason, The transition to a giant vortex phase in a fast rotating Bose-Einstein condensate, Comm. Math. Phys. 303 (2011), 451–508.
  * [10] F. Dalfovo, S. Giorgini, L. P. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Math. Phys. 71 (1999), 463–512.
  * [11] M. J. Esteban and P. L. Lions, Stationary solutions of nonlinear Schr$\ddot{o}$dinger equations with an external magnetic field, Partial differential equations and the calculus of variations, Vol. I, 401–449, Progr. Nonlinear Differential Equations Appl. 1, Birkhuser Boston, Boston, MA, 1989.
  * [12] A. L. Fetter, Rotating trapped Bose-Einstein condensates, Rev. Math. Phys. 81 (2009), 647–691.
  * [13] B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in ${\mathbb{R}}^{n}$, Mathematical analysis and applications Part A, Adv. in Math. Suppl. Stud. Vol. 7, Academic Press, New York (1981), 369–402.
  * [14] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1997.
  * [15] Y. Guo, C. Lin and J. Wei, Local uniqueness and refined spike profiles of ground states for two-dimensional attractive Bose-Einstein condensates, SIAM J. Math. Anal. 49 (2017), 3671–3715.
  * [16] Y. Guo, Y. Luo and S. Peng, Local uniqueness of ground states for rotating Bose-Einstein Condensates with attractive interactions, submitted (2020), 28 pages, arXiv:2009.08013
  * [17] Y. Guo, Y. Luo and W. Yang, The nonexistence of vortices for rotating Bose-Einstein condensates with attractive interactions, Arch. Rational Mech. Anal. 238 (2020), 1231–1281.
  * [18] Y. Guo and R. Seiringer, On the mass concentration for Bose-Einstein condensates with attractive interactions, Lett. Math. Phys. 104 (2014), 141–156.
  * [19] Y. Guo, Z. Wang, X. Zeng and H. Zhou, Properties of ground states of attractive Gross-Pitaevskii equations with multi-well potentials, Nonlinearity 31 (2018), 957–979.
  * [20] Q. Han and F. Lin, Elliptic Partial Differential Equations, Courant Lecture Note in Math. 1, Courant Institute of Mathematical Science/AMS, New York, 2011.
  * [21] R. Ignat and V. Millot, The critical velocity for vortex existence in a two-dimensional rotating Bose-Einstein condensate, J. Funct. Anal. 233 (2006), 260–306.
  * [22] R. Ignat and V. Millot, Energy expansion and vortex location for a two-dimensional rotating Bose-Einstein condensate, Rev. Math. Phys. 18 (2006), 119–162.
  * [23] M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^{p}=0$ in ${\mathbb{R}}^{N}$, Arch. Rational Mech. Anal. 105 (1989), 243–266.
  * [24] M. Lewin, P. T. Nam and N. Rougerie, Blow-up profile of rotating 2D focusing Bose gases, Macroscopic Limits of Quantum Systems, Springer Verlag, 2018.
  * [25] E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics Vol. 14, Amer. Math. Soc., Providence, RI, 2001.
  * [26] E. H. Lieb and R. Seiringer, Derivation of the Gross-Pitaevskii equation for rotating Bose gases, Comm. Math. Phys. 264 (2006), 505–537.
  * [27] E. H. Lieb, R. Seiringer and J. Yngvason, Bosons in a trap: A rigorous derivation of the Gross-Pitaevskii energy functional, Phys. Rev. A 61 (2000), 043602.
  * [28] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, Part I: Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 109–145. Part II: Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 223–283.
  * [29] K. Lu, X. Pan, Gauge invariant eigenvalue problems in ${\mathbb{R}}^{2}$ and in ${\mathbb{R}}^{2}_{+}$, Trans. Amer. Math. Soc. 3 (2000), 1247–1276.
  * [30] E. Lundh, A. Collin and K.-A. Suominen, Rotational states of Bose gases with attractive interactions in anharmonic traps, Phys. Rev. Lett. 92 (2004), 070401.
  * [31] L. P. Pitaevskii, Vortex lines in an imperfect Bose gas, Sov. Phys. JETP. 13 (1961), 451–454.
  * [32] M. Reed and B. Simon, Methods of modern mathematical physics IV. Analysis of operators, Academic Press, New York-London, 1978.
  * [33] E. Sandier and S. Serfaty, Vortices in the Magnetic Ginzburg-Landau Model, Progress in Nonlinear Differential Equations and their Applications 70, Basel: Birkháuser, 2007.
  * [34] R. Seiringer, Gross-Pitaevskii theory of the rotating Bose gas, Comm. Math. Phys. 229 (2002), 491–509.
  * [35] M. I. Weinstein, Nonlinear Schr$\ddot{o}$dinger equations and sharp interpolations estimates, Comm. Math. Phys. 87 (1983), 567–576.
  * [36] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications 24, Birkhäuser Boston Inc, Boston, (1996).