# Concentrations for nonlinear Schrödinger equation with magnetic potential
and constant electric potential

Liping Wang  Liping Wang School of Mathematical Sciences, Shanghai Key
Laboratory of Pure Mathematics and Mathematical Practice, East China Normal
University, 200241, P.R. China<EMAIL_ADDRESS>and  Chunyi Zhao
Chunyi Zhao School of Mathematical Sciences, Shanghai Key Laboratory of Pure
Mathematics and Mathematical Practice, East China Normal University, 200241,
P.R. China<EMAIL_ADDRESS>
###### Abstract.

This paper studies the concentration phenomena to nonlinear Schrödinger
equations with magnetic potential and constant electric potential. The
existing results show that the magnetic field has no effect on the location of
point concentrations, as long as the electric potential is not constant. This
paper finds out what the role of the magnetic field plays in the location of
concentrations when the electric potential is constant.

###### Key words and phrases:

nonlinear magnetic Schrödinger equation, concentration, constant electric
potential, magnetic potential

###### 1991 Mathematics Subject Classification:

35J10, 35A01, 35B25

The two authors are partially supported by NSFC 11971169 and Natural Science
Foundation of Shanghai 22ZR1421600.

##  1\. Introduction

In this paper, we investigate the magnetic Schrödinger equation in
$\mathbb{R}^{N}$

$\mathrm{i}\varepsilon\frac{\partial\psi}{\partial
t}=(\mathrm{i}\varepsilon\nabla+\bm{A}(x))^{2}\psi+Q(x)\psi-|\psi|^{p-1}\psi,$
(1)

where $\varepsilon>0$ is a small parameter, $1<p<\frac{N+2}{N-2}$ if $N\geq
3$, $p>1$ if $N=2$ and $\mathrm{i}$ is the imaginary unit. The function $\psi$
is complex-valued. Vector
$\bm{A}=(A_{1},A_{2},\ldots,A_{N}):\mathbb{R}^{N}\to\mathbb{R}^{N}$ is the
magnetic potential and assumed to be smooth, bounded and
$Q:\mathbb{R}^{N}\to\mathbb{R}$ represents the electric potential. The
magnetic Laplacian is defined by

$(\mathrm{i}\varepsilon\nabla+\bm{A})^{2}\psi:=-\varepsilon^{2}\Delta\psi+2\mathrm{i}\varepsilon\bm{A}\cdot\nabla\psi+|\bm{A}|^{2}\psi+\mathrm{i}\varepsilon\psi\nabla\cdot\bm{A}.$

Terms involving $\bm{A}$ model the presence in some quantum model of the
magnetic field $\bm{B}$ given by

$\bm{B}=\begin{cases}\partial_{1}A_{2}-\partial_{2}A_{1},&\text{for }N=2,\\\
\text{curl}\bm{A},&\text{for }N=3,\\\
(\partial_{j}A_{k}-\partial_{k}A_{j})_{N\times N},&\text{for }N>3.\end{cases}$

For the discussion of this operator, one may refer to [22] and [25].

The equation (1) arises in various physical contexts such as Bose-Einstein
condensates and nonlinear optics, see [5]; or plasma physics where one can
simulate the interaction effect among many particles by introducing some
nonlinear term, see [27]. Concerning nonlinear Schrödinger equation with the
magnetic field, the pioneer work is by Esteban-Lions [15] where they studied
some minimization problems under suitable assumptions on the magnetic field by
concentrations and compactness arguments. For more results, one can refer to
[4, 19, 18, 21] and the references therein.

From now on we consider standing wave solutions to problem (1), namely
$\psi(t,x)=\mathrm{e}^{\mathrm{i}\lambda\varepsilon^{-1}t}u(x)$ for some
complex-valued function $u(x)$. Substituting this ansatz into problem (1),
$u(x)$ should satisfy

$(\mathrm{i}\varepsilon\nabla+\bm{A})^{2}u+V(x)u-|u|^{p-1}u=0\qquad\text{in
}\mathbb{R}^{N},$ (2)

where $V(x)=Q(x)+\lambda$. The potential $V(x)$ is usually assumed to be
smooth and

$\inf_{\mathbb{R}^{N}}\ V(x)>0.$

Problem (2) seems a very interesting problem since the Correspondence’s
Principle establishes that classical mechanics is, roughly speaking, contained
in quantum mechanics. The mathematical transition from quantum mechanics to
classical mechanics can be formally described by letting the Planck constant
$\varepsilon\rightarrow 0$, and thus the existence of solutions for
$\varepsilon$ small has physical interest. Standing waves for $\varepsilon$
small are usually referred as semi-classical bound states, see e.g. [19]. In
the linear case, Helffer et al. in [18, 20] have studied the asymptotic
behavior of the eigenfunctions of the Schrödinger operators with magnetic
fields in the semiclassical limit. Note that in these results, one can not see
the magnetic field in the definition of the well.

Concentration plays the important role in the problem (2). The pioneer work of
concentration to problem (2) with $\bm{A}\equiv 0$ in one dimension is given
by Floer and Weistein [16]. Really they proved that if the electric potential
$V(x)$ has a non-degenerate critical point, then $u(x)$ concentrates near this
critical point as $\varepsilon\rightarrow 0$. After that, it appears many
results concerning concentration phenomena about problem (2). Different
approaches are used to cover different cases, see [13, 24, 26, 33]. Cingolani
and Secchi [10] proved that for bounded vector potentials, concentration can
happen at any non-degenerate critical point, not necessarily a minimum of $V$,
as $\varepsilon\rightarrow 0$ using a perturbation approach given by
Ambrosetti, Malchiodi and Secchi in [3]. Later they [11] extended the result
to the case of a possibly unbounded vector potential by a penalization
procedure (see [12]). Semiclassical multi-peak solutions were found for
bounded vector potentials in [1, 9, 6, 17]. Li-Peng-Wang [23] constructed
infinitely many non-radial solutions if the potentials $\bm{A}$ and $V$ are
radially symmetric and suitably decay at infinity by using the number of bumps
as parameter motivated by [34], see also e.g. [30] and the references therein
for the application of such a very novel idea. For more recent results, we can
refer to [7, 8, 28, 29] and the references therein. Also readers can refer to
[2, 14, 31, 32] for high dimensional concentration. Till now, all these
concentrations, especially point concentrations, are very dependent on
critical points of $V(x)$ while the effect of the magnetic vector potential
$\bm{A}$ is always ignored as higher order. In other words, $\bm{A}$ has no
contribution to decide the location of point concentrations when $V(x)$ has
critical points. On the other hand, for the case of the constant $V(x)$, there
is no result on the concentration as far as we know.

Our aim in this paper is to exhibit how the magnetic potential $\bm{A}$ plays
the role in the concentration. For this purpose, we study the following
nonlinear magnetic Schrödinger equation

$(\mathrm{i}\varepsilon\nabla+\bm{A}(x))^{2}u+u-|u|^{p-1}u=0\qquad\mbox{in
}\mathbb{R}^{N}.$ (3)

Here $p>1$ and $\bm{A}=(A_{1},\ldots,A_{N})$:
$\mathbb{R}^{N}\to\mathbb{R}^{N}$ is assumed in
$W^{1,\infty}(\mathbb{R}^{N},\mathbb{R}^{N})$ and smooth everywhere for
simplicity. To state our main result, we introduce the Frobenius norm of a
matrix $\bm{\mathcal{M}}=(m_{ij})_{I\times J}$

$\|\bm{\mathcal{M}}\|_{F}=\left(\sum_{i=1}^{I}\sum_{j=1}^{J}|m_{ij}|^{2}\right)^{\frac{1}{2}}.$

And denote by $w(y)=w(|y|)$ the unique radial real-valued solution of

$\Delta w-w+w^{p}=0\quad\text{in }\mathbb{R}^{N},\qquad
w(0)=\max_{\mathbb{R}^{N}}w>0,\qquad w(\pm\infty)=0.$ (4)

Now our main results are the following.

######  Theorem 1.1.

Assume that the Forbenius norm $\|\bm{B}\|_{F}$ of the magnetic field $\bm{B}$
admits $K$ local maximum (minimum) points $\\{P_{m}\\}_{m=1}^{K}$ (which may
be degenerate) and $K$ disjoint, closed and bounded regions
$\\{\Omega_{m}\\}_{m=1}^{K}$ of $\mathbb{R}^{N}$ such that

$\|\bm{B}(P_{m})\|_{F}=\max_{\Omega_{m}}\|\bm{B}\|_{F}>\max_{\partial\Omega_{m}}\|\bm{B}\|_{F}.\qquad\left(\|\bm{B}(P_{m})\|_{F}=\min_{\Omega_{m}}\|\bm{B}\|_{F}<\min_{\partial\Omega_{m}}\|\bm{B}\|_{F}.\right)$

Then there exists an $\varepsilon_{0}>0$, such that for every
$0<\varepsilon<\varepsilon_{0}$, the problem (3) admits a solution
$u_{\varepsilon}$ with the form

$u_{\varepsilon}(x)=\sum_{m=1}^{K}\left[w\left(\frac{|x-\zeta_{m}|}{\varepsilon}\right)+\varepsilon\Psi_{m}\left(\frac{x}{\varepsilon}\right)\right]\mathrm{e}^{\mathrm{i}\sigma_{m}+\mathrm{i}\varepsilon^{-1}\bm{A}(\zeta_{m})\cdot
x}+O(\varepsilon^{2}),$

for some
$(\sigma_{1},\ldots,\sigma_{K})\in[0,2\pi)^{K},\zeta_{m}\in\Omega_{m}$. The
definition of $\Psi_{m}$ is given in (9).

For general critical points, i.e. not local extremum points, we also have the
following result.

######  Theorem 1.2.

Assume that $p>\frac{3}{2}$ and $P_{1},P_{2},\ldots,P_{K}$, $K\geq 1$ are all
non-degenerate critical points of $\|\bm{B}\|^{2}_{F}$. Then there exists an
$\varepsilon_{0}>0$ such that for any $0<\varepsilon<\varepsilon_{0},$ the
problem (3) admits a solution $u_{\varepsilon}$ with the form

$u_{\varepsilon}(x)=\sum_{m=1}^{K}\left[w\left(\frac{|x-\zeta_{m}|}{\varepsilon}\right)+\varepsilon\Psi_{m}\left(\frac{x}{\varepsilon}\right)\right]\mathrm{e}^{\mathrm{i}\sigma_{m}+\mathrm{i}\varepsilon^{-1}\bm{A}(\zeta_{m})\cdot
x}+O(\varepsilon^{2})$

for some $(\sigma_{1},\ldots,\sigma_{K})\in[0,2\pi)^{K}$ and
$\zeta_{m}=P_{m}+o(1)$. The definition of $\Psi_{m}$ is given in (9).

######  Remark 1.3.

Theorem 1.1 and Theorem 1.2 clearly show the principle of the magnetic vector
$\bm{A}$, or precisely the magnetic field $\bm{B}$ driving the location of
concentration if the electric potential is constant. This is a completely new
phenomenon up to now.

######  Remark 1.4.

Theorem 1.1 holds for any $p>1$. And Theorem 1.2 holds only for
$p>\frac{3}{2}$. The reason is that the critical point of the function deduced
by the energy functional is the local extremum point in Theorem 1.1, which may
be gotten by direct comparison. While in Theorem 1.2,
$P_{1},P_{2},\ldots,P_{K}$ may not be extremum points any more. Thus we have
to study the derivatives of the energy functional. Naturally, the
corresponding estimates should be higher accuracy for which the better
regularity of the nonlinear term $|u|^{p-1}u$ is required.

From the point view of physics, the magnetic field $\bm{B}$ is essential, not
the particular choice of magnetic potential $\bm{A}$. At the same time, there
is a gauge invariance for the magnetic Laplacian correspondingly, that is, the
magnetic field $\bm{B}$ is invariant under the transform of the potential
$\bm{A}\to\bm{A}+\nabla f$. Also it is easy to see that the energy of the
problem (3) is unchanged under the gauge invariance, with which our result
coincides (see Proposition 4.3).

When the electric potential $V(x)$ has some non-degeneracy, we just need to
make the first approximation by the limit equation, which is enough to deduce
the role of the function $V(x)$. But in order to see the role of magnetic
vector potential $\bm{A}$ in our case, we need the approximation up to order
$\varepsilon$ of the problem (3). Hence it is necessary to get a more accurate
expression. On the other hand, higher accuracy allows us to deal with extremum
points not necessary minimum points, which is different from the case of the
non-constant $V(x)$, see e.g. [17]. Finally we note that the solutions given
in Theorem 1.1 and Theorem 1.2 both show simple concentrations. In the
forthcoming paper, we will deal with multi-bump solutions for constant
electric potential.

The paper is organized as follows. In section 2, the ansatz and the estimate
of the error are given. The corresponding non-linear problem is solved in
section 3. In section 4, the original problem is reduced to the finite
dimension problem using variational reduction process and the expansion of the
energy functional is shown. Finally, in section 5, Theorem 1.1 and Theorem 1.2
are proved.

Notations.
1\. Constant $\beta=\min\\{p-1,1\\}$.
2\. The real part of $z\in\mathbb{C}$ will be denoted by $\operatorname{Re}z$.
3\. The complex conjugate of $z\in\mathbb{C}$ will be denoted by $\bar{z}$.
4\. $C$ denotes a generic positive constant, which may be different from lines
to lines.
5\. Landau symbols $O(\varepsilon)$ is a generic function such that
$|O(\varepsilon)|\leq C\varepsilon$ and $o(\varepsilon)$ means that
$\lim_{\varepsilon\rightarrow 0^{+}}\frac{|o(\varepsilon)|}{\varepsilon}=0$.

##  2\. Ansatz

In this section we will present the approximation of the problem and give the
corresponding error estimate.

Recall in [21] that the problem

$\Delta\tilde{w}-\tilde{w}+|\tilde{w}|^{p-1}\tilde{w}=0,\qquad\tilde{w}\in
H^{1}(\mathbb{R}^{N},\mathbb{C})$

possesses a unique ground state solution
$\tilde{w}(y)=w(y)\mathrm{e}^{\mathrm{i}\sigma}$, $\forall\ \sigma\in[0,2\pi]$
where $w(y)$ is the radial solution of problem (4). Thus by the gauge
invariance,

$\widetilde{U}(x)=w\left(\frac{|x-\zeta|}{\varepsilon}\right)\mathrm{e}^{\mathrm{i}\sigma+\mathrm{i}\varepsilon^{-1}\bm{A}(\zeta)\cdot
x},\qquad\forall\ \sigma\in[0,2\pi],$

is also the ground state solution to the constant magnetic potential problem

$(\mathrm{i}\varepsilon\nabla+\bm{A}(\zeta))^{2}\widetilde{U}+\widetilde{U}-|\widetilde{U}|^{p-1}\widetilde{U}=0,\qquad\widetilde{U}\in
H^{1}(\mathbb{R}^{N},\mathbb{C}).$

In the frame of large variable $y=x/\varepsilon$, the original problem (3) is
equivalent to

$(\mathrm{i}\nabla+\bm{A}(\varepsilon y))^{2}u+u-|u|^{p-1}u=0\qquad\mbox{in
}\mathbb{R}^{N}.$ (5)

Therefore, the function

$U(y)=w(|y-\zeta^{\prime}|)\mathrm{e}^{\mathrm{i}\sigma+\mathrm{i}\bm{A}(\zeta)\cdot
y},\qquad\zeta^{\prime}=\zeta/\varepsilon.$

formally approximates the solution at least near $y=\zeta^{\prime}$. Now it’s
time to give the first approximation

$W(y)=\sum_{m=1}^{K}U_{m}(y),\qquad\quad
U_{m}(y)=w(|y-\zeta_{m}^{\prime}|)\mathrm{e}^{\mathrm{i}\sigma_{m}+\mathrm{i}\bm{A}(\zeta_{m})\cdot
y},\qquad\sigma_{m}\in[0,2\pi],$ (6)

where $\zeta_{m}\in\Omega_{m},m=1,2,\ldots,K$. Denote

$\rho:=\min_{1\leq i\neq j\leq
K}\text{dist}\left(\Omega_{i},\Omega_{j}\right)>0$

and take a positive number $\delta$ such that $\delta\leq\frac{1}{4}\rho$. Let
$\widehat{R}(y)$ be the error caused by the first approximation $W$, which is

$\widehat{R}(y)=(\mathrm{i}\nabla+\bm{A}(\varepsilon y))^{2}W+W-|W|^{p-1}W.$

It is checked that for
$|y-\zeta^{\prime}_{1}|\leq\frac{\delta}{\sqrt{\varepsilon}},\quad|U_{m}|\leq|U_{1}|e^{-\frac{|\zeta^{\prime}_{m}-\zeta^{\prime}_{1}|}{3}},m=2,3,...,K$
and

$\displaystyle\widehat{R}(y)$
$\displaystyle=\widetilde{R}(y)+O\left(w(|y-\zeta^{\prime}_{1}|)\max_{2\leq
m\leq
K}e^{-\frac{|\zeta_{m}-\zeta_{1}|}{3\varepsilon}}+w^{p-1}(|y-\zeta^{\prime}_{1}|)\max_{2\leq
m\leq K}e^{-\frac{|\zeta_{m}-\zeta_{1}|}{2\varepsilon}}\right)$
$\displaystyle=\widetilde{R}(y)+O\left(w(|y-\zeta^{\prime}_{1}|)+w^{p-1}(|y-\zeta^{\prime}_{1}|)\right)e^{-\frac{\delta}{\varepsilon}}$

where

$\displaystyle\widetilde{R}(y)$
$\displaystyle=(\mathrm{i}\nabla+\bm{A}(\varepsilon
y))^{2}U_{1}+U_{1}-|U_{1}|^{p-1}U_{1}=(\mathrm{i}\nabla+\bm{A}(\varepsilon
y))^{2}U_{1}-(\mathrm{i}\nabla+\bm{A}(\zeta_{1}))^{2}U_{1}$
$\displaystyle=2\mathrm{i}(\bm{A}(\varepsilon y)-\bm{A}(\zeta_{1}))\cdot\nabla
U_{1}+\mathrm{i}\varepsilon(\nabla_{x}\cdot A)U_{1}+\left(|\bm{A}(\varepsilon
y)|^{2}-|\bm{A}(\zeta_{1})|^{2}\right)U_{1}$
$\displaystyle=2\mathrm{i}(\bm{A}(\varepsilon
y)-\bm{A}(\zeta_{1}))\cdot\left[\nabla
w(y-\zeta^{\prime}_{1})+\mathrm{i}\bm{A}(\zeta_{1})w(y-\zeta^{\prime}_{1})\right]\mathrm{e}^{\mathrm{i}\sigma_{1}+\mathrm{i}\bm{A}(\zeta_{1})\cdot
y}+\mathrm{i}\varepsilon(\nabla_{x}\cdot\bm{A})U_{1}$
$\displaystyle\quad+\left(|\bm{A}(\varepsilon
y)|^{2}-|\bm{A}(\zeta_{1})|^{2}\right)w(y-\zeta^{\prime}_{1})\mathrm{e}^{\mathrm{i}\sigma_{1}+\mathrm{i}\bm{A}(\zeta_{1})\cdot
y}$ $\displaystyle=\left[-2\bm{A}(\zeta_{1})\cdot(\bm{A}(\varepsilon
y)-\bm{A}(\zeta_{1}))+\left(|\bm{A}(\varepsilon
y)|^{2}-|\bm{A}(\zeta_{1})|^{2}\right)\right]U_{1}$
$\displaystyle\quad+\mathrm{i}[2(\bm{A}(\varepsilon
y)-\bm{A}(\zeta_{1}))\cdot\nabla
w(y-\zeta^{\prime}_{1})+\varepsilon(\nabla_{x}\cdot
A)w(y-\zeta^{\prime}_{1})]\mathrm{e}^{\mathrm{i}\sigma_{1}+\mathrm{i}\bm{A}(\zeta_{1})\cdot
y}$ $\displaystyle=\left(|\bm{A}(\varepsilon
y)-\bm{A}(\zeta_{1})|\right)^{2}w(y-\zeta^{\prime}_{1})\mathrm{e}^{\mathrm{i}\sigma_{1}+\mathrm{i}\bm{A}(\zeta_{1})\cdot
y}$ $\displaystyle\quad+\mathrm{i}\left[2(\bm{A}(\varepsilon
y)-\bm{A}(\zeta_{1}))\cdot\nabla
w(y-\zeta^{\prime}_{1})+\varepsilon(\nabla_{x}\cdot\bm{A})w(y-\zeta^{\prime}_{1})\right]\mathrm{e}^{\mathrm{i}\sigma_{1}+\mathrm{i}\bm{A}(\zeta_{1})\cdot
y}.$ (7)

As usual one writes
$\zeta^{\prime}_{m}=(\zeta^{\prime}_{m,1},\zeta^{\prime}_{m,2},\ldots,\zeta^{\prime}_{m,N})$.
Direct calculation shows that

$\displaystyle|\bm{A}(\varepsilon
y)-\bm{A}(\zeta_{1})|^{2}=\sum_{i=1}^{N}(A_{i}(\varepsilon
y)-A_{i}(\zeta_{1}))^{2}$ $\displaystyle=$ $\displaystyle\
\sum_{i=1}^{N}\left(\varepsilon\nabla
A_{i}(\zeta_{1})\cdot(y-\zeta^{\prime}_{1})+\frac{\varepsilon^{2}}{2}(y-\zeta^{\prime}_{1})^{\perp}\cdot\nabla^{2}A_{i}(\zeta_{1})\cdot(y-\zeta^{\prime}_{1})+O(\varepsilon^{3}|y-\zeta^{\prime}_{1}|^{3})\right)^{2}$
$\displaystyle=$ $\displaystyle\
\varepsilon^{2}\sum_{i,j,k=1}^{N}\partial_{j}A_{i}(\zeta_{1})\partial_{k}A_{i}(\zeta_{1})(y_{j}-\zeta^{\prime}_{1,j})(y_{k}-\zeta^{\prime}_{1,k})$
$\displaystyle+\varepsilon^{3}\sum_{i,j,k,\ell=1}^{N}\partial_{j}A_{i}(\zeta_{1})\partial_{k\ell}A_{i}(\zeta_{1})(y_{j}-\zeta^{\prime}_{1,j})(y_{k}-\zeta^{\prime}_{1,k})(y_{\ell}-\zeta^{\prime}_{1,\ell})+O(\varepsilon^{4}|y-\zeta^{\prime}_{1}|^{4}).$

Similarly, it is verified that

$\displaystyle(\bm{A}(\varepsilon y)-\bm{A}(\zeta_{1}))\cdot\nabla
w(|y-\zeta_{1}^{\prime}|)=\sum_{i=1}^{N}(A_{i}(\varepsilon
y)-A_{i}(\zeta_{1}))\partial_{i}w(|y-\zeta_{1}^{\prime}|)$ $\displaystyle=$
$\displaystyle\
\varepsilon\sum_{i,j=1}^{N}\partial_{j}A_{i}(\zeta_{1})(y_{j}-\zeta^{\prime}_{1,j})\partial_{i}w(|y-\zeta_{1}^{\prime}|)+\frac{\varepsilon^{2}}{2}\sum_{i,j,k=1}^{N}\partial_{jk}A_{i}(\zeta_{1})(y_{j}-\zeta^{\prime}_{1,j})(y_{k}-\zeta^{\prime}_{1,k})\partial_{i}w(|y-\zeta_{1}^{\prime}|)$
$\displaystyle+\frac{\varepsilon^{3}}{6}\sum_{i,j,k,\ell=1}^{N}\partial_{jk\ell}A_{i}(\zeta_{1})(y_{j}-\zeta^{\prime}_{1,j})(y_{k}-\zeta^{\prime}_{1,k})(y_{\ell}-\zeta^{\prime}_{1,\ell})\partial_{i}w(|y-\zeta_{1}^{\prime}|)+O\left(\varepsilon^{4}|y-\zeta_{1}^{\prime}|^{4}|\nabla
w(|y-\zeta_{1}^{\prime}|)|\right).$

As for the term $\nabla_{x}\cdot\bm{A}(\varepsilon
y)w(|y-\zeta_{1}^{\prime}|),$ we expand it similarly to

$\displaystyle\nabla_{x}\cdot\bm{A}(\varepsilon y)w(|y-\zeta_{1}^{\prime}|)$
$\displaystyle=$
$\displaystyle\nabla_{x}\cdot\bm{A}(\zeta_{1})w(|y-\zeta_{1}^{\prime}|)+\varepsilon\nabla(\nabla_{x}\cdot\bm{A})(\zeta_{1})\cdot(y-\zeta_{1}^{\prime})w(|y-\zeta_{1}^{\prime}|)$
$\displaystyle+\frac{1}{2}\varepsilon^{2}(y-\zeta^{\prime}_{1})^{\perp}\cdot\nabla^{2}(\nabla_{x}\cdot\bm{A})(\zeta_{1})\cdot(y-\zeta^{\prime}_{1})+O\left(\varepsilon^{3}|y-\zeta^{\prime}_{1}|^{3}w(|y-\zeta_{1}^{\prime}|)\right)$
$\displaystyle=$
$\displaystyle\nabla_{x}\cdot\bm{A}(\zeta_{1})w(|y-\zeta_{1}^{\prime}|)+\varepsilon\sum_{i,j=1}^{N}\partial_{ij}A_{i}(\zeta_{1})(y_{j}-\zeta^{\prime}_{1,j})w(|y-\zeta_{1}^{\prime}|)$
$\displaystyle+\frac{\varepsilon^{2}}{2}\sum_{i,j,k=1}^{N}\partial_{jki}A_{i}(\zeta_{1})(y_{j}-\zeta^{\prime}_{1,j})(y_{k}-\zeta^{\prime}_{1,k})w(|y-\zeta_{1}^{\prime}|)+O(\varepsilon^{3}|y-\zeta_{1}^{\prime}|^{3}w(|y-\zeta_{1}^{\prime}|)).$

Therefore, in the region
$|y-\zeta_{1}^{\prime}|<\frac{\delta}{\sqrt{\varepsilon}}$, we get that

$\displaystyle\
\widetilde{R}(y)\mathrm{e}^{-\mathrm{i}\sigma_{1}-\mathrm{i}\bm{A}(\zeta_{1})\cdot
y}$ $\displaystyle=$ $\displaystyle\
\varepsilon\mathrm{i}\left(2\sum_{i,j=1}^{N}\partial_{j}A_{i}(\zeta_{1})(y_{j}-\zeta^{\prime}_{1,j})\partial_{i}w(|y-\zeta^{\prime}_{1}|)+\nabla_{x}\cdot\bm{A}(\zeta_{1})w(|y-\zeta^{\prime}_{1}|)\right)$
$\displaystyle+\varepsilon^{2}\sum_{i,j,k=1}^{N}\partial_{j}A_{i}(\zeta_{1})\partial_{k}A_{i}(\zeta_{1})(y_{j}-\zeta^{\prime}_{1,j})(y_{k}-\zeta^{\prime}_{1,k})w(|y-\zeta^{\prime}_{1}|)$
$\displaystyle+\varepsilon^{2}\mathrm{i}\sum_{i,j,k=1}^{N}\partial_{jk}A_{i}(\zeta_{1})(y_{j}-\zeta^{\prime}_{1,j})(y_{k}-\zeta^{\prime}_{1,k})\partial_{i}w(|y-\zeta^{\prime}_{1}|)+\varepsilon^{2}\mathrm{i}\sum_{i,j=1}^{N}\partial_{ij}A_{i}(\zeta_{1})(y_{j}-\zeta^{\prime}_{1,j})w(|y-\zeta^{\prime}_{1}|)$
$\displaystyle+\varepsilon^{3}\sum_{i,j,k,\ell=1}^{N}\partial_{j}A_{i}(\zeta_{1})\partial_{k\ell}A_{i}(\zeta_{1})(y_{j}-\zeta^{\prime}_{1,j})(y_{k}-\zeta^{\prime}_{1,k})(y_{\ell}-\zeta^{\prime}_{1,\ell})w(|y-\zeta^{\prime}_{1}|)$
$\displaystyle+\frac{\varepsilon^{3}}{3}\mathrm{i}\sum_{i,j,k,\ell=1}^{N}\partial_{jk\ell}A_{i}(\zeta_{1})(y_{j}-\zeta^{\prime}_{1,j})(y_{k}-\zeta^{\prime}_{1,k})(y_{\ell}-\zeta^{\prime}_{1,\ell})\partial_{i}w(|y-\zeta^{\prime}_{1}|)$
$\displaystyle+\frac{\varepsilon^{3}}{2}\mathrm{i}\sum_{i,j,k=1}^{N}\partial_{jki}A_{i}(\zeta_{1})(y_{j}-\zeta^{\prime}_{1,j})(y_{k}-\zeta^{\prime}_{1,k})w(|y-\zeta^{\prime}_{1}|)$
$\displaystyle+O\left(\varepsilon^{4}|y-\zeta^{\prime}_{1}|^{4}|w(|y-\zeta^{\prime}_{1}|)|\right)+\mathrm{i}O\left(\varepsilon^{4}|y-\zeta^{\prime}_{1}|^{4}|\nabla
w(|y-\zeta^{\prime}_{1}|)|+\varepsilon^{4}|y-\zeta^{\prime}_{1}|^{3}|w(|y-\zeta^{\prime}_{1}|)|\right).$

Note that the imaginary part of
$\widetilde{R}(y)\mathrm{e}^{-\mathrm{i}\sigma_{1}-\mathrm{i}\bm{A}(\zeta_{1})\cdot
y}$ is $O(\varepsilon)$. It is of less accuracy for later application.
Complying with the guideline that the better approximation, the more
possibility to get a solution, we should improve the accuracy of the
approximation. To this purpose, we find that the real-valued function
$\Psi_{1,ij}(y)=\frac{1}{2}(y_{i}-\zeta^{\prime}_{1,i})(y_{j}-\zeta^{\prime}_{1,j})w(|y-\zeta^{\prime}_{1}|)$
$(i\neq j)$ is the solution to

$-\Delta\Psi_{1,ij}+\Psi_{1,ij}-w^{p-1}(|y-\zeta^{\prime}_{1}|)\Psi_{1,ij}=-2(y_{j}-\zeta^{\prime}_{1,j})\partial_{i}w(|y-\zeta^{\prime}_{1}|)=-2(y_{i}-\zeta^{\prime}_{1,i})\partial_{j}w(|y-\zeta^{\prime}_{1}|).$

Besides,
$\Psi_{1,ii}(y)=\frac{1}{2}(y_{i}-\zeta^{\prime}_{1,i})^{2}w(|y-\zeta^{\prime}_{1}|)$
satisfies the equation

$-\Delta\Psi_{1,ii}+\Psi_{1,ii}-w^{p-1}(|y-\zeta^{\prime}_{1}|)\Psi_{1,ii}=-2(y_{i}-\zeta^{\prime}_{1,i})\partial_{i}w(|y-\zeta^{\prime}_{1}|)-w.$

Then obviously the function

$\displaystyle\Psi_{1}(y)$ $\displaystyle=$
$\displaystyle\mathrm{i}\left(\sum_{i=1}^{N}\sum_{j\neq
i}\partial_{j}A_{i}(\zeta_{1})\Psi_{1,ij}(y)+\sum_{i=1}^{N}\partial_{i}A_{i}(\zeta_{1})\Psi_{1,ii}(y)\right)\mathrm{e}^{\mathrm{i}\sigma_{1}+\mathrm{i}\bm{A}(\zeta_{1})\cdot
y}$ $\displaystyle=$
$\displaystyle\mathrm{i}\left(\sum_{i,j=1}^{N}\partial_{j}A_{i}(\zeta_{1})\Psi_{1,ij}(y)\right)\mathrm{e}^{\mathrm{i}\sigma_{1}+\mathrm{i}\bm{A}(\zeta_{1})\cdot
y}:=\mathrm{i}\psi_{1}(y)\mathrm{e}^{\mathrm{i}\sigma_{1}+\mathrm{i}\bm{A}(\zeta_{1})\cdot
y}$

satisfies

$\displaystyle(\mathrm{i}\nabla+\bm{A}(\zeta_{1}))^{2}\Psi_{1}+\Psi_{1}-(p-1)|U_{1}|^{p-3}\operatorname{Re}(\overline{U}_{1}\Psi_{1})U_{1}-|U_{1}|^{p-1}\Psi_{1}$
$\displaystyle=$
$\displaystyle\mathrm{i}\left[-2\sum_{i,j=1}^{N}\partial_{j}A_{i}(\zeta_{1})(y_{j}-\zeta^{\prime}_{1,j})\partial_{i}w(|y-\zeta^{\prime}_{1}|)-\nabla_{x}\cdot\bm{A}(\zeta_{1})w(|y-\zeta^{\prime}_{1}|)\right]\mathrm{e}^{\mathrm{i}\sigma_{1}+\mathrm{i}\bm{A}(\zeta_{1})\cdot
y},$

since $\text{Re}(\overline{U}_{1}\Psi_{1})=0$. Moreover, the same computation
may also be carried out in the region
$|y-\zeta_{m}^{\prime}|<\frac{\delta}{\sqrt{\varepsilon}}$ for $m=2,\cdots,K$.
The ultimate approximation is then selected as

$\mathcal{W}(y)=W(y)+\varepsilon\Psi(y),$ (8)

where $\Psi(y)=\sum_{m=1}^{K}\Psi_{m}(y)$. Here

$\Psi_{m}(y)=\mathrm{i}\left(\sum_{i,j=1}^{N}\partial_{j}A_{i}(\zeta_{m})\Psi_{m,ij}(y)\right)\mathrm{e}^{\mathrm{i}\sigma_{m}+\mathrm{i}\bm{A}(\zeta_{m})\cdot
y}:=\mathrm{i}\psi_{m}(y)\mathrm{e}^{\mathrm{i}\sigma_{m}+\mathrm{i}\bm{A}(\zeta_{m})\cdot
y}$ (9)

and

$\Psi_{m,ij}(y)=\frac{1}{2}(y_{i}-\zeta^{\prime}_{m,i})(y_{j}-\zeta^{\prime}_{m,j})w(|y-\zeta^{\prime}_{m}|).$

Now the approximation $\mathcal{W}$ is good enough to help us find a solution.
Precisely our aim is to find a solution with the form $\mathcal{W}(y)+\phi(y)$
of problem (3), where $\phi$ is a small perturbation and satisfies the
equation

$L\phi:=(\mathrm{i}\nabla+\bm{A}(\varepsilon
y))^{2}\phi+\phi-(p-1)|\mathcal{W}|^{p-3}\operatorname{Re}(\overline{\mathcal{W}}\phi)\mathcal{W}-|\mathcal{W}|^{p-1}\phi=-R(y)+N(\phi).$
(10)

Here $R(y)$ denotes the error caused by $\mathcal{W}$, which is

$\displaystyle R(y)=(\mathrm{i}\nabla+\bm{A}(\varepsilon
y))^{2}\mathcal{W}+\mathcal{W}-|\mathcal{W}|^{p-1}\mathcal{W},$ and the
nonlinear term $\displaystyle
N(\phi)=|\mathcal{W}+\phi|^{p-1}(\mathcal{W}+\phi)-|\mathcal{W}|^{p-1}\mathcal{W}-(p-1)|\mathcal{W}|^{p-3}\operatorname{Re}(\overline{\mathcal{W}}\phi)\mathcal{W}-|\mathcal{W}|^{p-1}\phi.$

Obviously, with the notation $\beta=\min\\{p-1,\ 1\\}$,

$|N(\phi)|\leq C|\phi|^{1+\beta}.$

To solve problem (10), it is important to estimate the error $R(y)$.

######  Proposition 2.1.

We have that

$\displaystyle\|R(y)\|_{L^{2}}\leq C\varepsilon^{2},$ and
$\displaystyle\|\partial_{\zeta_{m,k}^{\prime}}R(y)\|_{L^{2}}\leq
C\varepsilon^{2},\qquad\|\partial_{\sigma_{m}}R(y)\|_{L^{2}}\leq
C\varepsilon^{2},\qquad\forall\ m=1,\ldots,K,\ k=1,\ldots,N.$

###### Proof.

First we consider the domain
$|y-\zeta^{\prime}_{m}|\leq\frac{\delta}{\sqrt{\varepsilon}},$ without loss of
generality, say $m=1$. Then

$\begin{array}[]{lll}R(y)&=&\widehat{R}(y)+\varepsilon\left[(\mathrm{i}\nabla+\bm{A}(\varepsilon
y))^{2}\Psi+\Psi-(p-1)|W|^{p-3}\operatorname{Re}(\overline{W}\Psi)W-|W|^{p-1}\Psi\right]\\\
\\\
&&-\left[|W+\varepsilon\Psi|^{p-1}(W+\varepsilon\Psi)-|W|^{p-1}W-\varepsilon(p-1)|W|^{p-3}\text{Re}(\overline{W}\Psi)W-\varepsilon|W|^{p-1}\Psi\right]\\\
\\\
&=&\widetilde{R}(y)-\varepsilon\mathrm{i}\left(2\sum_{i,j=1}^{N}\partial_{j}A_{i}(\zeta_{1})(y_{j}-\zeta^{\prime}_{1,j})\partial_{i}w(y-\zeta^{\prime}_{1})+(\nabla_{x}\cdot\bm{A})(\zeta_{1})w(|y-\zeta^{\prime}_{1}|)\right)\mathrm{e}^{\mathrm{i}\sigma_{1}+\mathrm{i}\bm{A}(\zeta_{1})\cdot
y}\\\ \\\ &&+\varepsilon\left[(\mathrm{i}\nabla+\bm{A}(\varepsilon
y))^{2}\Psi_{1}-(\mathrm{i}\nabla+\bm{A}(\zeta_{1}))^{2}\Psi_{1}\right]+\varepsilon\left[(\mathrm{i}\nabla+\bm{A}(\varepsilon
y))^{2}\sum_{m=2}^{K}\Psi_{m}+\sum_{m=2}^{K}\Psi_{m}\right]\\\ \\\
&&-\frac{p-1}{2}\varepsilon^{2}\psi_{1}^{2}(y)w^{p-2}(|y-\zeta^{\prime}_{1}|)\mathrm{e}^{\mathrm{i}\sigma_{1}+\mathrm{i}\bm{A}(\zeta_{1})\cdot
y}-\mathrm{i}\frac{p-1}{2}\varepsilon^{3}\psi^{3}_{1}(y)w^{p-3}(|y-\zeta^{\prime}_{1}|)\mathrm{e}^{\mathrm{i}\sigma_{1}+\mathrm{i}\bm{A}(\zeta_{1})\cdot
y}\\\ \\\
&&+O\left(\varepsilon^{4}|y-\zeta_{1}^{\prime}|^{8}|w^{p}(|y-\zeta^{\prime}_{1}|)+\mathrm{e}^{-\delta/\varepsilon}w(|y-\zeta_{1}^{\prime}|)+\mathrm{e}^{-\frac{\delta}{\varepsilon}}w^{p-1}(|y-\zeta^{\prime}_{1}|)\right).\end{array}$

Obviously we have that

$\displaystyle\varepsilon^{2}\psi_{1}(y)^{2}w^{p-2}(y-\zeta^{\prime}_{1})=\varepsilon^{2}w^{p-2}(|y-\zeta^{\prime}_{1}|)\sum_{i,j=1}^{N}\partial_{j}A_{i}(\zeta_{1})\Psi_{1,ij}(y)\sum_{k,\ell=1}^{N}\partial_{\ell}A_{k}(\zeta_{1})\Psi_{1,k\ell}(y)$
$\displaystyle=$
$\displaystyle\frac{\varepsilon^{2}}{4}w^{p}(|y-\zeta^{\prime}_{1}|)\sum_{i,j,k,\ell=1}^{N}\partial_{j}A_{i}(\zeta_{1})\partial_{\ell}A_{k}(\zeta_{1})(y_{i}-\zeta^{\prime}_{1,i})(y_{j}-\zeta^{\prime}_{1,j})(y_{k}-\zeta^{\prime}_{1,k})(y_{\ell}-\zeta^{\prime}_{1,\ell})$

and

$\varepsilon\left|\left[(\mathrm{i}\nabla+\bm{A}(\varepsilon
y))^{2}\sum_{m=2}^{K}\Psi_{m}+\sum_{m=2}^{K}\Psi_{m}\right]\right|=O\left(\sum_{m=2}^{K}w(y-\zeta^{\prime}_{m})\right)=O\left(\mathrm{e}^{-\frac{\delta}{\varepsilon}}w(y-\zeta^{\prime}_{1})\right).$

Moreover, it is checked as in (7) that

$\displaystyle\ (\mathrm{i}\nabla+\bm{A}(\varepsilon
y))^{2}\Psi_{1}-(\mathrm{i}\nabla+\bm{A}(\zeta_{1}))^{2}\Psi_{1}$
$\displaystyle=$ $\displaystyle\ 2\mathrm{i}(\bm{A}(\varepsilon
y)-\bm{A}(\zeta_{1}))\cdot\nabla\Psi_{1}+\mathrm{i}\varepsilon(\nabla_{x}\cdot\bm{A}(\varepsilon
y))\Psi_{1}+(|\bm{A}(\varepsilon y)|^{2}-|\bm{A}(\zeta_{1})|^{2})\Psi_{1}$
$\displaystyle=$ $\displaystyle\ -\left[2(\bm{A}(\varepsilon
y)-\bm{A}(\zeta_{1}))\cdot\nabla\psi_{1}+\varepsilon(\nabla_{x}\cdot\bm{A}(\varepsilon
y))\psi_{1}\right]e^{\mathrm{i}\sigma_{1}+\mathrm{i}\bm{A}(\zeta_{1})\cdot
y}+\mathrm{i}|\bm{A}(\varepsilon
y)-\bm{A}(\zeta_{1})|^{2}\psi_{1}e^{\mathrm{i}\sigma_{1}+\mathrm{i}\bm{A}(\zeta_{1})\cdot
y}.$ (11)

Let us estimate them one by one in (11) for $y\neq\zeta^{\prime}_{1}$. Note
that

$\displaystyle\ (\bm{A}(\varepsilon
y)-\bm{A}(\zeta_{1}))\cdot\nabla\psi_{1}=\sum_{k=1}^{N}(A_{k}(\varepsilon
y)-A_{k}(\zeta_{1}))\partial_{k}\psi_{1}$ $\displaystyle=$ $\displaystyle\
\varepsilon\sum_{k=1}^{N}\nabla
A_{k}(\zeta_{1})\cdot(y-\zeta^{\prime}_{1})\partial_{k}\psi_{1}+\frac{\varepsilon^{2}}{2}\sum_{k=1}^{N}(y-\zeta^{\prime}_{1})^{\perp}\cdot\nabla^{2}A_{k}(\zeta_{1})\cdot(y-\zeta^{\prime}_{1})\partial_{k}\psi_{1}+O(\varepsilon^{3}|y-\zeta^{\prime}_{1}|^{3}|\nabla\psi_{1}|)$
$\displaystyle=$ $\displaystyle\
\varepsilon\sum_{k,\ell=1}^{N}\partial_{\ell}A_{k}(\zeta_{1})(y_{\ell}-\zeta^{\prime}_{1,\ell})\left[\sum_{i,j=1}^{N}\partial_{j}A_{i}(\zeta_{1})\partial_{k}\Psi_{1,ij}\right]$
$\displaystyle\ +\frac{\varepsilon^{2}}{2}\sum_{k,\ell,s=1}^{N}\partial_{\ell
s}A_{k}(\zeta_{1})(y_{\ell}-\zeta^{\prime}_{1,\ell})(y_{s}-\zeta^{\prime}_{1,s})\left[\sum_{i,j=1}^{N}\partial_{j}A_{i}(\zeta_{1})\partial_{k}\Psi_{1,ij}\right]+O\left(\varepsilon^{3}|y-\zeta^{\prime}_{1}|^{5}w(|y-\zeta^{\prime}_{1}|)\right).$

It is easy to see that ($\delta_{ik}$ is the Kronecker symbol) for
$y\neq\zeta^{\prime}_{1}$,

$\displaystyle\partial_{k}\Psi_{1,ij}(y)$
$\displaystyle=\frac{1}{2}\left[\left(\delta_{ik}(y_{j}-\zeta^{\prime}_{1,j})+\delta_{jk}(y_{i}-\zeta^{\prime}_{1,i})\right)w(|y-\zeta^{\prime}_{1}|)+(y_{i}-\zeta^{\prime}_{1,i})(y_{j}-\zeta^{\prime}_{1,j})(y_{k}-\zeta^{\prime}_{1,k})\frac{w^{\prime}(|y-\zeta^{\prime}_{1}|)}{|y-\zeta^{\prime}_{1}|}\right].$

Thus we conclude that

$\displaystyle\ (\bm{A}(\varepsilon y)-\bm{A}(\zeta_{1}))\cdot\nabla\psi_{1}$
$\displaystyle=$ $\displaystyle\
\frac{\varepsilon}{2}\sum_{j,k,\ell=1}^{N}\partial_{\ell}A_{k}(\zeta_{1})\left[\partial_{j}A_{k}(\zeta_{1})+\partial_{k}A_{j}(\zeta_{1})\right](y_{j}-\zeta^{\prime}_{1,j})(y_{\ell}-\zeta^{\prime}_{1,\ell})w(|y-\zeta^{\prime}_{1}|)$
$\displaystyle\
+\frac{\varepsilon}{2}\sum_{i,j,k,\ell=1}^{N}\partial_{\ell}A_{k}(\zeta_{1})\partial_{j}A_{i}(\zeta_{1})(y_{i}-\zeta^{\prime}_{1,i})(y_{j}-\zeta^{\prime}_{1,j})(y_{k}-\zeta^{\prime}_{1,k})(y_{\ell}-\zeta^{\prime}_{1,\ell})\frac{w^{\prime}(|y-\zeta^{\prime}_{1}|)}{|y-\zeta^{\prime}_{1}|}$
$\displaystyle\
+\frac{\varepsilon^{2}}{4}\sum_{j,k,\ell,s=1}^{N}\partial_{\ell
s}A_{k}(\zeta_{1})\left[\partial_{j}A_{k}(\zeta_{1})+\partial_{k}A_{j}(\zeta_{1})\right](y_{j}-\zeta^{\prime}_{1,j})(y_{\ell}-\zeta^{\prime}_{1,\ell})(y_{s}-\zeta^{\prime}_{1,s})w(y-\zeta^{\prime}_{1})$
$\displaystyle\
+\frac{\varepsilon^{2}}{4}\sum_{i,j,k,\ell,s=1}^{N}\partial_{\ell
s}A_{k}(\zeta_{1})\partial_{j}A_{i}(\zeta_{1})(y_{i}-\zeta^{\prime}_{1,i})(y_{j}-\zeta^{\prime}_{1,j})(y_{k}-\zeta^{\prime}_{1,k})(y_{\ell}-\zeta^{\prime}_{1,\ell})(y_{s}-\zeta^{\prime}_{1,s})\frac{w^{\prime}}{|y-\zeta^{\prime}_{1}|}$
$\displaystyle\
+O\left(\varepsilon^{3}|y-\zeta^{\prime}_{1}|^{5}w(|y-\zeta^{\prime}_{1}|)\right).$
(12)

Also it can be obtained that

$\displaystyle(\nabla_{x}\cdot\bm{A}(\varepsilon y))\psi_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\nabla_{x}\cdot\bm{A})(\zeta_{1})\sum_{i,j=1}^{N}\partial_{j}A_{i}(\zeta_{1})(y_{i}-\zeta^{\prime}_{1,i})(y_{j}-\zeta^{\prime}_{1,j})w(|y-\zeta^{\prime}_{1}|)$
$\displaystyle+\frac{\varepsilon}{2}\sum_{i,j,k,\ell=1}^{N}\partial_{j}A_{i}(\zeta_{1})\partial_{\ell
k}A_{\ell}(\zeta_{1})(y_{i}-\zeta^{\prime}_{1,i})(y_{j}-\zeta^{\prime}_{1,j})(y_{k}-\zeta^{\prime}_{1,k})w(y-\zeta^{\prime}_{1})$
$\displaystyle+O\left(\varepsilon^{2}|y-\zeta^{\prime}_{1}|^{4}w(y-\zeta^{\prime}_{1})\right),$

and

$|\bm{A}(\varepsilon
y)-\bm{A}(\zeta_{1})|^{2}\psi_{1}=O\left(\varepsilon^{2}|y-\zeta^{\prime}_{1}|^{4}w(y-\zeta^{\prime}_{1})\right).$

Now (11) can be expanded as

$\displaystyle(\mathrm{i}\nabla+\bm{A}(\varepsilon
y))^{2}\Psi_{1}-(\mathrm{i}\nabla+\bm{A}(\zeta_{1}))^{2}\Psi_{1}$
$\displaystyle=$
$\displaystyle-\varepsilon\sum_{j,k,\ell=1}^{N}\partial_{\ell}A_{k}(\zeta_{1})\left[\partial_{j}A_{k}(\zeta_{1})+\partial_{k}A_{j}(\zeta_{1})\right](y_{j}-\zeta^{\prime}_{1,j})(y_{\ell}-\zeta^{\prime}_{1,\ell})w(|y-\zeta^{\prime}_{1}|)e^{\mathrm{i}\sigma_{1}+\mathrm{i}\bm{A}(\zeta_{1})\cdot
y}$
$\displaystyle-\varepsilon\sum_{i,j,k,\ell=1}^{N}\partial_{\ell}A_{k}(\zeta_{1})\partial_{j}A_{i}(\zeta_{1})(y_{i}-\zeta^{\prime}_{1,i})(y_{j}-\zeta^{\prime}_{1,j})(y_{k}-\zeta^{\prime}_{1,k})(y_{\ell}-\zeta^{\prime}_{1,\ell})\frac{w^{\prime}(|y-\zeta^{\prime}_{1}|)}{|y-\zeta^{\prime}_{1}|}\mathrm{e}^{\mathrm{i}\sigma_{1}+\mathrm{i}\bm{A}(\zeta_{1})\cdot
y}$
$\displaystyle-\frac{\varepsilon}{2}(\nabla_{x}\cdot\bm{A}(\zeta_{1}))\sum_{i,j=1}^{N}\partial_{j}A_{i}(\zeta_{1})(y_{i}-\zeta^{\prime}_{1,i})(y_{j}-\zeta^{\prime}_{1,j})w(|y-\zeta^{\prime}_{1}|)e^{\mathrm{i}\sigma_{1}+\mathrm{i}\bm{A}(\zeta_{1})\cdot
y}$
$\displaystyle-\frac{\varepsilon^{2}}{2}\sum_{j,k,\ell,s=1}^{N}\partial_{\ell
s}A_{k}(\zeta_{1})\left[\partial_{j}A_{k}(\zeta_{1})+\partial_{k}A_{j}(\zeta_{1})\right](y_{j}-\zeta^{\prime}_{1,j})(y_{\ell}-\zeta^{\prime}_{1,\ell})(y_{s}-\zeta^{\prime}_{1,s})w(y-\zeta^{\prime}_{1})\mathrm{e}^{\mathrm{i}\sigma_{1}+\mathrm{i}\bm{A}(\zeta_{1})\cdot
y}$
$\displaystyle-\frac{\varepsilon^{2}}{2}\sum_{i,j,k,\ell,s=1}^{N}\partial_{\ell
s}A_{k}(\zeta_{1})\partial_{j}A_{i}(\zeta_{1})(y_{i}-\zeta^{\prime}_{1,i})(y_{j}-\zeta^{\prime}_{1,j})(y_{k}-\zeta^{\prime}_{1,k})(y_{\ell}-\zeta^{\prime}_{1,\ell})(y_{s}-\zeta^{\prime}_{1,s})\frac{w^{\prime}}{|y-\zeta^{\prime}_{1}|}\mathrm{e}^{\mathrm{i}\sigma_{1}+\mathrm{i}\bm{A}(\zeta_{1})\cdot
y}$
$\displaystyle-\frac{\varepsilon^{2}}{2}\sum_{i,j,k,\ell=1}^{N}\partial_{j}A_{i}(\zeta_{1})\partial_{\ell
k}A_{\ell}(\zeta_{1})(y_{i}-\zeta^{\prime}_{1,i})(y_{j}-\zeta^{\prime}_{1,j})(y_{k}-\zeta^{\prime}_{1,k})w(|y-\zeta^{\prime}_{1}|)\mathrm{e}^{\mathrm{i}\sigma_{1}+\mathrm{i}\bm{A}(\zeta_{1})\cdot
y}$
$\displaystyle+\left[O(\varepsilon^{3}|y-\zeta^{\prime}_{1}|^{5}+\varepsilon^{3}|y-\zeta^{\prime}_{1}|^{4})+\mathrm{i}O(\varepsilon^{2}|y-\zeta^{\prime}_{1}|^{4})\right]w(|y-\zeta^{\prime}_{1}|))\mathrm{e}^{\mathrm{i}\sigma_{1}+\mathrm{i}\bm{A}(\zeta_{1})\cdot
y}.$

Therefore, in $|y-\zeta^{\prime}_{m}|\leq\frac{\delta}{\sqrt{\varepsilon}}$

$R(y)\mathrm{e}^{-\mathrm{i}\sigma_{m}-\mathrm{i}\bm{A}(\zeta_{m})\cdot
y}:=R_{m,1}(y)+\mathrm{i}R_{m,2}(y),$

where

$\displaystyle\ R_{m,1}(y)$ $\displaystyle=$ $\displaystyle\
\varepsilon^{2}\sum_{i,j,k=1}^{N}\partial_{j}A_{i}(\zeta_{m})\partial_{k}A_{i}(\zeta_{m})(y_{j}-\zeta^{\prime}_{m,j})(y_{k}-\zeta^{\prime}_{m,k})w(y-\zeta^{\prime}_{m})$
$\displaystyle\
-\varepsilon^{2}\sum_{j,k,\ell=1}^{N}\partial_{\ell}A_{k}(\zeta_{m})\left[\partial_{j}A_{k}(\zeta_{m})+\partial_{k}A_{j}(\zeta_{m})\right](y_{j}-\zeta^{\prime}_{m,j})(y_{\ell}-\zeta^{\prime}_{m,\ell})w(|y-\zeta^{\prime}_{m}|)$
$\displaystyle\
-\varepsilon^{2}\sum_{i,j,k,\ell=1}^{N}\partial_{\ell}A_{k}(\zeta_{m})\partial_{j}A_{i}(\zeta_{m})(y_{i}-\zeta^{\prime}_{m,i})(y_{j}-\zeta^{\prime}_{m,j})(y_{k}-\zeta^{\prime}_{m,k})(y_{\ell}-\zeta^{\prime}_{m,\ell})\frac{w^{\prime}(|y-\zeta^{\prime}_{m}|)}{|y-\zeta^{\prime}_{m}|}$
$\displaystyle\
-\frac{\varepsilon^{2}}{2}(\nabla_{x}\cdot\bm{A}(\zeta_{m}))\sum_{i,j=1}^{N}\partial_{j}A_{i}(\zeta_{m})(y_{i}-\zeta^{\prime}_{m,i})(y_{j}-\zeta^{\prime}_{m,j})w(|y-\zeta^{\prime}_{m}|)$
$\displaystyle\
-\frac{(p-1)\varepsilon^{2}}{8}\sum_{i,j,k,\ell=1}^{N}\partial_{j}A_{i}(\zeta_{m})\partial_{\ell}A_{k}(\zeta_{m})(y_{i}-\zeta^{\prime}_{m,i})(y_{j}-\zeta^{\prime}_{m,j})(y_{k}-\zeta^{\prime}_{m,k})(y_{\ell}-\zeta^{\prime}_{m,\ell})w^{p}(|y-\zeta^{\prime}_{m}|)$
$\displaystyle\
+\varepsilon^{3}\sum_{i,j,k,\ell=1}^{N}\partial_{j}A_{i}(\zeta_{m})\partial_{k\ell}A_{i}(\zeta_{m})(y_{j}-\zeta^{\prime}_{m,j})(y_{k}-\zeta^{\prime}_{m,k})(y_{\ell}-\zeta^{\prime}_{m,\ell})w(|y-\zeta^{\prime}_{m}|)$
$\displaystyle\
-\frac{\varepsilon^{3}}{2}\sum_{j,k,\ell,s=1}^{N}\partial_{\ell
s}A_{k}(\zeta_{m})\left[\partial_{j}A_{k}(\zeta_{m})+\partial_{k}A_{i}(\zeta_{m})\right](y_{j}-\zeta^{\prime}_{m,j})(y_{\ell}-\zeta^{\prime}_{m,\ell})(y_{s}-\zeta^{\prime}_{m,s})w(|y-\zeta^{\prime}_{m}|)$
$\displaystyle\
-\frac{\varepsilon^{3}}{2}\sum_{i,j,k,\ell,s=1}^{N}\partial_{\ell
s}A_{k}(\zeta_{m})\partial_{j}A_{i}(\zeta_{m})(y_{i}-\zeta^{\prime}_{m,i})(y_{j}-\zeta^{\prime}_{m,j})(y_{k}-\zeta^{\prime}_{m,k})(y_{\ell}-\zeta^{\prime}_{m,\ell})(y_{s}-\zeta^{\prime}_{m,s})\frac{w^{\prime}(|y-\zeta^{\prime}_{m}|)}{|y-\zeta^{\prime}_{m}|}$
$\displaystyle\
-\frac{\varepsilon^{3}}{2}\sum_{i,j,k,\ell=1}^{N}\partial_{j}A_{i}(\zeta_{m})\partial_{\ell
k}A_{\ell}(\zeta_{m})(y_{i}-\zeta^{\prime}_{m,i})(y_{j}-\zeta^{\prime}_{m,j})(y_{k}-\zeta^{\prime}_{m,k})w(|y-\zeta^{\prime}_{m}|)$
$\displaystyle\
+O\left(\varepsilon^{4}|y-\zeta^{\prime}_{m}|^{4}+\varepsilon^{4}|y-\zeta^{\prime}_{m}|^{5}\right)w(|y-\zeta^{\prime}_{m}|)+O(\varepsilon^{4}|y-\zeta^{\prime}_{m}|^{8}w^{p}(|y-\zeta^{\prime}_{m}|))$
(13)

and

$\begin{split}R_{m,2}(y)=&\
\varepsilon^{2}\sum_{i,j,k=1}^{N}\partial_{jk}A_{i}(\zeta_{m})(y_{j}-\zeta^{\prime}_{m,j})(y_{k}-\zeta^{\prime}_{m,k})\partial_{i}w(|y-\zeta^{\prime}_{m}|)\\\
&\
+\varepsilon^{2}\sum_{i,j=1}^{N}\partial_{ij}A_{i}(\zeta_{m})(y_{j}-\zeta^{\prime}_{m,j})w(|y-\zeta^{\prime}_{m}|)\\\
&\
+O\left(\varepsilon^{3}|y-\zeta^{\prime}_{m}|^{4}+\varepsilon^{3}|y-\zeta^{\prime}_{m}|^{2}\right)w(|y-\zeta^{\prime}_{m}|)+O\left(\varepsilon^{4}|y-\zeta^{\prime}_{m}|^{8}\right)w^{p}(|y-\zeta^{\prime}_{m}|).\end{split}$

Hence we get the estimate

$\sum_{m=1}^{K}\int_{B(\zeta^{\prime}_{m};\frac{\delta}{\sqrt{\varepsilon}})}|R(y)|^{2}dy\leq
C\varepsilon^{2}.$

As for the domain
$|y-\zeta^{\prime}_{m}|\geq\frac{\delta}{\sqrt{\varepsilon}},\forall\
m=1,2,\ldots,K$, using the asymptotic behaviour of
$w(|y-\zeta^{\prime}_{m}|)$, it is easy to see that

$\int_{\mathbb{R}^{N}\setminus\cup_{m=1}^{K}B(\zeta^{\prime}_{m};\frac{\delta}{\sqrt{\varepsilon}})}|R(y)|^{2}dy\leq
C\mathrm{e}^{-\frac{\delta}{\sqrt{\varepsilon}}}.$

The result for $R(y)$ is concluded.

As for the estimates of $\partial_{\zeta^{\prime}_{m,k}}R$ and
$\partial_{\sigma_{m}}R$, one may check it similarly. ∎

##  3\. The linear problem and the nonlinear problem

This section is devoted to the invertibility of the linear operator $L$ in
order to solve problem (10):

$L\phi=(\mathrm{i}\nabla+\bm{A}(\varepsilon
y))^{2}\phi+\phi-(p-1)|\mathcal{W}|^{p-3}\operatorname{Re}(\overline{\mathcal{W}}\phi)\mathcal{W}-|\mathcal{W}|^{p-1}\phi=-R(y)+N(\phi).$

Let $H$ be the Hilbert space as the closure of
$C_{0}^{\infty}(\mathbb{R}^{N},\mathbb{C})$ under the scalar product

$(u,v)=\operatorname{Re}\int_{\mathbb{R}^{N}}\left(\mathrm{i}\nabla
u+\bm{A}(\varepsilon y)u\right)\overline{\left(\mathrm{i}\nabla
v+\bm{A}(\varepsilon y)v\right)}+u\bar{v}.$

The norm deduced by the above scalar product is equivalent to the usual norm
of $H^{1}(\mathbb{R}^{N},\mathbb{C})$ due to the boundness of $|\bm{A}(x)|$,
see [10]. In $|y-\zeta^{\prime}_{m}|\leq\frac{\delta}{\sqrt{\varepsilon}}$,
the operator $L$ formally looks like

$(\mathrm{i}\nabla+\bm{A}(\zeta_{m}))^{2}\phi+\phi-(p-1)|U_{m}|^{p-3}\operatorname{Re}(\overline{U}_{m}\phi)U_{m}-|U_{m}|^{p-1}\phi,$

which is not invertible. Precisely, the null space of this limit operator is

$\operatorname{span}_{\mathbb{R}}\\{Z_{m,0},Z_{m,1},\cdots,Z_{m,N}\\}$

where

$Z_{m,0}=\mathrm{i}w(|y-\zeta_{m}^{\prime}|)\mathrm{e}^{\mathrm{i}\sigma_{m}+\mathrm{i}\bm{A}(\zeta_{m})\cdot
y}=\mathrm{i}U_{m}\qquad\text{and}\qquad Z_{m,i}=\frac{\partial
w(|y-\zeta_{m}^{\prime}|)}{\partial\zeta^{\prime}_{m,i}}\mathrm{e}^{\mathrm{i}\sigma_{m}+\mathrm{i}\bm{A}(\zeta_{m})\cdot
y},\quad 1\leq i\leq N.$

The symbol spanR means the linear combinations on real numbers, see for
instance [10, 11]. Therefore, we study the following linear problem with $h\in
L^{2}(\mathbb{R}^{N},\mathbb{C})$

$\begin{cases}\displaystyle
L\phi=h+\sum_{i=0}^{N}\sum_{m=1}^{K}c_{m,i}\chi_{m}Z_{m,i},\\\
\displaystyle\operatorname{Re}\int_{\mathbb{R}^{N}}\chi_{m}\overline{Z}_{m,i}\phi=0,\quad
i=0,1,\cdots,N,\quad m=1,\ldots,K\end{cases}$ (14)

where $\chi_{m}(y)=\chi(|y-\zeta^{\prime}_{m}|)$ is a smooth cut-off function
on the large ball $B_{R}(\zeta^{\prime}_{m})$, satisfying $\chi(s)=1$ for
$|s|\leq R$ and $\chi(s)=0$ for $|s|\geq R+1$.

Next, we will prove the following invertibility proposition which is the main
result in this section.

######  Proposition 3.1.

The linear problem (14) admits a unique solution
$(\phi,c_{m,i})=(\widetilde{T}(h),c_{m,i})$, $i=0,1,\cdots,N,\ m=1,\ldots,K$
satisfying

$\|\phi\|_{H^{2}}=\|\widetilde{T}(h)\|\leq C\|h\|_{L^{2}},\qquad|c_{m,i}|\leq
C\|h\|_{L^{2}}.$

Before giving the proof, it is necessary to get an apriori estimate.

######  Lemma 3.2.

If $(\phi,c_{m,i})$ is a solution of the problem (14), then

$\|\phi\|_{H^{2}}\leq C\|h\|_{L^{2}},\qquad|c_{m,i}|\leq C\|h\|_{L^{2}}.$

###### Proof.

The proof is very standard and we here prove it briefly for the completion.
First, we test the equation (14) by $\overline{Z}_{\ell,j}$, $1\leq\ell\leq
K$, $0\leq j\leq N$ and get that

$\langle
L\phi,\overline{Z}_{\ell,j}\rangle=\text{Re}\int_{\mathbb{R}^{N}}h\overline{Z}_{\ell,j}+c_{\ell,j}\int_{\mathbb{R}^{N}}|Z_{\ell,j}|^{2}+O\left(\mathrm{e}^{-\delta/{\varepsilon}}\sum_{i=0}^{N}\sum_{m=1}^{K}|c_{m,i}|\right).$
(15)

Note that

$\begin{split}\text{Re}\int_{\mathbb{R}^{N}}(\mathrm{i}\nabla+A(\varepsilon
y))\phi\overline{(\mathrm{i}\nabla+A(\varepsilon
y))Z_{\ell,j}}=&\text{Re}\int_{\mathbb{R}^{N}}(\mathrm{i}\nabla+A(\varepsilon
y))Z_{\ell,j}\overline{(\mathrm{i}\nabla+A(\varepsilon y))\phi}\\\
=&\text{Re}\int_{\mathbb{R}^{N}}(\mathrm{i}\nabla+A(\zeta_{\ell}))Z_{\ell,j}\overline{(\mathrm{i}\nabla+A(\zeta_{\ell}))\phi}+O(\varepsilon)\|\phi\|_{H^{1}},\end{split}$

and

$\left|\int_{\mathbb{R}^{N}}hZ_{\ell,j}\right|\leq C\|h\|_{L^{2}}.$

Thus, it holds from (15) and the equation of ${\overline{Z}_{\ell,j}}$ that

$c_{\ell,j}=O(\varepsilon\|\phi\|_{H^{1}}+\|h\|_{L^{2}}).$ (16)

Next we will prove $\|\phi\|_{H^{1}}\leq C\|h\|_{L^{2}}$ by contradiction.
Suppose that for some sequence $\\{\varepsilon_{n}\\}$, there always exist
$\phi_{n}$ and $h_{n}$ such that

$\|\phi_{n}\|_{H^{1}}=1\qquad\text{and}\qquad\|h_{n}\|_{L^{2}}=o(1)\ \text{as
}\varepsilon_{n}\to 0.$

Testing (14) against $\eta_{m}\varphi(\varphi\in
C_{c}^{\infty}(\mathbb{R}^{N},\mathbb{C}))$ where $\eta_{m}(y)\equiv 1$ in
$|y-\zeta_{m}^{\prime}|<\frac{\delta}{\sqrt{\varepsilon}}$ and
$\eta_{m}(y)\equiv 0$ in
$|y-\zeta_{m}^{\prime}|>\frac{2\delta}{\sqrt{\varepsilon}}$, one can obtain
that

$\displaystyle\text{Re}\int_{\mathbb{R}^{N}}(\mathrm{i}\nabla+A(\varepsilon_{n}y))\phi_{n}\overline{(\mathrm{i}\nabla+A(\varepsilon_{n}y))\eta_{m}\varphi}+\text{Re}\int_{\mathbb{R}^{N}}\phi_{n}\eta_{m}\overline{\varphi}$
$\displaystyle-(p-1)\text{Re}\int_{\mathbb{R}^{N}}|\mathcal{W}|^{p-3}(\operatorname{Re}(\overline{\mathcal{W}}\phi_{n}))\mathcal{W}\eta_{m}\overline{\varphi}-\text{Re}\int_{\mathbb{R}^{N}}|\mathcal{W}|^{p-1}\phi_{n}\eta_{m}\overline{\varphi}$
$\displaystyle=$
$\displaystyle\text{Re}\int_{\mathbb{R}^{N}}h_{n}\eta_{m}\overline{\varphi}+\sum_{i=0}^{N}c_{m,i}\text{Re}\int_{\mathbb{R}^{N}}\chi_{m}Z_{m,i}\overline{\varphi}.$

Note that $\phi_{n}\rightharpoonup\phi$ in
$H^{1}_{\text{loc}}(\mathbb{R}^{N},\mathbb{C})$ up to a subsequence. Thus
dominated convergence theorem tells us that

$\displaystyle\text{Re}\int_{\mathbb{R}^{N}}(\mathrm{i}\nabla+A(\zeta_{m}))\phi\overline{(\mathrm{i}\nabla+A(\zeta_{m}))\varphi}+\text{Re}\int_{\mathbb{R}^{N}}\phi\bar{\varphi}$
$\displaystyle-(p-1)\text{Re}\int_{\mathbb{R}^{N}}|U_{m}|^{p-3}(\operatorname{Re}(\overline{U}_{m}\phi))U_{m}\bar{\varphi}-\text{Re}\int_{\mathbb{R}^{N}}|U_{m}|^{p-1}\phi\bar{\varphi}=0.$

This means that $\phi$ is a solution of

$(\mathrm{i}\nabla+\bm{A}(\zeta_{m}))^{2}\phi+\phi-(p-1)|U_{m}|^{p-3}\text{Re}(\overline{U}_{m}\phi)U_{m}-|U_{m}|^{p-1}\phi=0\qquad\text{in
}\mathbb{R}^{N}.$

Then one gets that $\phi=0$ from the orthogonal conditions, which further
implies that

$\phi_{n}\to 0\quad\text{a.e. in }B_{R}(\zeta^{\prime}_{m}),\qquad\forall\
R>0,\ m=1,2,\ldots,K.$ (17)

On the other hand, note that

$\displaystyle\int_{\mathbb{R}^{N}}|(\mathrm{i}\nabla+\bm{A}(\varepsilon
y))\phi_{n}|^{2}+\int_{\mathbb{R}^{N}}|\phi_{n}|^{2}-(p-1)\int_{\mathbb{R}^{N}}|\mathcal{W}|^{p-3}(\operatorname{Re}(\overline{\mathcal{W}}\phi_{n}))^{2}-\int_{\mathbb{R}^{N}}|\mathcal{W}|^{p-1}|\phi_{n}|^{2}$
$\displaystyle=$
$\displaystyle\operatorname{Re}\int_{\mathbb{R}^{N}}\bar{h}_{n}\phi_{n}=o(1).$
(18)

From (17) and the exponential decay of $|U_{m}|$, we obviously have

$\int_{\mathbb{R}^{N}}|\mathcal{W}|^{p-1}|\phi_{n}|^{2}=\sum_{m=1}^{N}\int_{B_{R}(\zeta^{\prime}_{m})}|U_{m}|^{p-1}|\phi_{n}|^{2}+\int_{\mathbb{R}^{N}\setminus\cup_{m=1}^{K}B_{R}(\zeta^{\prime}_{m})}|\mathcal{W}|^{p-1}|\phi_{n}|^{2}+O(\varepsilon)=O(\mathrm{e}^{-R})+o(1).$

So is
$\int_{\mathbb{R}^{N}}|\mathcal{W}|^{p-3}(\operatorname{Re}(\overline{\mathcal{W}}\phi_{n}))^{2}$,
which together with (3) shows that

$\int_{\mathbb{R}^{N}}|\phi_{n}|^{2}=O(e^{-R})+o(1)\quad\text{and}\quad\int_{\mathbb{R}^{N}}|(\mathrm{i}\nabla+\bm{A}(\varepsilon
y))\phi_{n}|^{2}=O(\mathrm{e}^{-R})+o(1).$

Finally, it is derived that

$\displaystyle O(\mathrm{e}^{-R})+o(1)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}|(\mathrm{i}\nabla+\bm{A}(\varepsilon
y))\phi_{n}|^{2}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}|\nabla\phi_{n}|^{2}+\int_{\mathbb{R}^{N}}|\bm{A}(\varepsilon_{n}y)|^{2}|\phi_{n}|^{2}+2\text{Re}\int_{\mathbb{R}^{2}}\mathrm{i}\bm{A}(\varepsilon_{n}y)\cdot\nabla\phi_{n}\phi_{n}$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla\phi_{n}|^{2}-\int_{\mathbb{R}^{N}}|\bm{A}(\varepsilon_{n}y)|^{2}|\phi_{n}|^{2}=\frac{1}{2}\|\phi_{n}\|_{H^{1}}+O(\mathrm{e}^{-R})+o(1),$

since $\bm{A}(x)$ is bounded. This leads to a contradiction to
$\|\phi_{n}\|_{H^{1}}=1$. Hence

$\|\phi\|_{H^{1}}\leq C\|h\|_{L^{2}}.$

At last, the regularity theory gives $\|\phi\|_{H^{2}}\leq C\|h\|_{L^{2}}$. ∎

###### Proof of Proposition 3.1.

Denote the Hilbert space

$\mathcal{H}=\left\\{\phi\in
H(\mathbb{R}^{N})~{}\left|~{}\text{Re}\int_{\mathbb{R}^{N}}\chi_{m}\phi\overline{Z}_{m,i}=0\right.\right\\}.$

Then, from Riesz representation theorem the equation (14) is equivalent to

$\phi-T(\phi)=\tilde{h},\qquad\text{in }\mathcal{H},$

where $T$ is a compact operator on $\mathcal{H}$. Based on Proposition 3.1,
Fredholm alternative tells us the unique existence of $\phi$. And $c_{m,i}$
can be given by $\phi$ using integration. Their estimates were given in the
above proposition. ∎

Also for $\phi=\widetilde{T}(h)$, it is important for later purposes to
understand the differentiability of the operator $\widetilde{T}$ with respect
to $\zeta_{j}^{\prime}$ and $\sigma_{j},j=1,\ldots,K$. Recall that $\phi$
satisfies the equation

$L\phi=(\mathrm{i}\nabla+A(\varepsilon
y))^{2}\phi+\phi-(p-1)|\mathcal{W}|^{p-3}\text{Re}(\overline{\mathcal{W}}\phi)\mathcal{W}-|\mathcal{W}|^{p-1}\phi=h+\sum_{m,i}c_{m,i}\chi_{m}Z_{m,i}.$

Thus

$\displaystyle L(\partial_{\zeta^{\prime}_{j,k}}\phi)$ $\displaystyle=$
$\displaystyle(\mathrm{i}\nabla+A(\varepsilon
y))^{2}(\partial_{\zeta^{\prime}_{j,k}}\phi)+\partial_{\zeta^{\prime}_{j,k}}\phi-(p-1)|\mathcal{W}|^{p-3}\text{Re}(\overline{\mathcal{W}}\partial_{\zeta^{\prime}_{j,k}}\phi)\mathcal{W}-|\mathcal{W}|^{p-1}\partial_{\zeta^{\prime}_{j,k}}\phi$
$\displaystyle=$ $\displaystyle
O(|\mathcal{W}|^{p-2}|\phi||\partial_{\zeta^{\prime}_{j,k}}\mathcal{W}|)+\sum_{i}c_{j,i}\partial_{\zeta^{\prime}_{j,k}}(\chi_{j}Z_{j,i})+\sum_{m,i}(\partial_{\zeta^{\prime}_{j,k}}c_{m,i})\chi_{m}Z_{m,i}.$

Moreover, the derivative of the orthogonal condition is

$\displaystyle\text{Re}\int_{\mathbb{R}^{N}}\chi_{m}\overline{Z}_{m,i}(\partial_{\zeta^{\prime}_{j,k}}\phi)=0,\qquad\text{for
}m\neq j,$
$\displaystyle\text{Re}\int_{\mathbb{R}^{N}}\chi_{j}\overline{Z}_{j,i}(\partial_{\zeta^{\prime}_{j,k}}\phi)=-\text{Re}\int_{\mathbb{R}^{N}}\partial_{\zeta^{\prime}_{j,k}}(\chi_{j}\overline{Z}_{j,i})\phi.$

Set
$\varphi=\partial_{\zeta^{\prime}_{j,k}}\phi+\sum_{i=1}^{N}b_{jk,i}\chi_{j}Z_{j,i}$
and

$b_{jk,i}=\text{Re}\int_{\mathbb{R}^{N}}\partial_{\zeta^{\prime}_{j,k}}(\chi_{j}\overline{Z}_{j,i})\phi\Big{/}\int_{\mathbb{R}^{N}}\chi_{j}|Z_{j,i}|^{2}.$

Note that $|b_{jk,i}|\leq C\|\phi\|_{L^{2}}\leq C\|h\|_{L^{2}}$. Then
$\varphi$ satisfies all the orthogonal conditions and

$L\varphi=b_{ji}L(\chi_{j}Z_{j,i})+O(|\mathcal{W}|^{p-2}|\phi||\partial_{\zeta^{\prime}_{j,k}}\mathcal{W}|)+\sum_{i}c_{j,i}\partial_{\zeta^{\prime}_{j,k}}(\chi_{j}Z_{j,i})+\sum_{m,i}(\partial_{\zeta^{\prime}_{j,k}}c_{m,i})\chi_{m}Z_{m,i}.$

With Lemma 3.2 in hand,

$\|\varphi\|_{H^{2}}\leq
C\|b_{ji}L(\chi_{j}Z_{j,i})\|_{L^{2}}+C\||\mathcal{W}|^{p-2}\phi\partial_{\zeta^{\prime}_{j,k}}\mathcal{W}\|_{L^{2}}+C\sum_{i}\|c_{j,i}\partial_{\zeta^{\prime}_{j,k}}(\chi_{j}Z_{j,i})\|_{L^{2}}\leq
C\|h\|_{L^{2}}.$

Therefore, we conclude that

$\|\partial_{\zeta^{\prime}_{j,k}}\widetilde{T}(h)\|_{H^{2}}=\|\partial_{\zeta^{\prime}_{j,k}}\phi\|_{H^{2}}\leq\|\varphi\|_{H^{2}}+\|b_{ji}\chi_{j}Z_{j,i}\|_{H^{2}}\leq
C\|h\|_{L^{2}}.$

The same process may be carried out for $\partial_{\sigma_{j}}\phi$. Based on
the above discussion, the following proposition holds obviously.

######  Proposition 3.3.

For the unique solution $\phi=\widetilde{T}(h)$ in Proposition 3.1, it holds
that

$\|\partial_{\zeta^{\prime}_{j,k}}\widetilde{T}(h)\|_{H^{2}}\leq
C\|h\|_{L^{2}},\qquad\|\partial_{\sigma_{j}}\widetilde{T}(h)\|_{H^{2}}\leq
C\|h\|_{L^{2}},\quad\forall\ j=1,\ldots,K,k=1,\ldots,N.$

Now we can deal with the following nonlinear problem

$\begin{cases}\displaystyle
L\phi=-R(y)+N(\phi)+\sum_{i=0}^{N}\sum_{m=1}^{K}c_{m,i}\chi_{m}Z_{m,i},\\\
\displaystyle\operatorname{Re}\int_{\mathbb{R}^{N}}\chi_{m}\overline{Z}_{m,i}\phi=0,\quad
i=0,\cdots,N,m=1,\ldots,K.\end{cases}$ (19)

######  Proposition 3.4.

The nonlinear problem (19) admits a unique solution $\phi$ satisfying

$\|\phi\|_{H^{2}}=O(\varepsilon^{2}).$

Moreover, $(\bm{\sigma},\bm{\zeta}^{\prime})\to\phi$ is of class $C^{1}$ for
$\bm{\sigma}=(\sigma_{1},\ldots,\sigma_{K}),\bm{\zeta}^{\prime}=(\zeta_{1}^{\prime},\ldots,\zeta_{K}^{\prime})$,
and

$\|\partial_{\zeta^{\prime}_{j,k}}\phi\|_{H^{2}}=O(\varepsilon^{2\beta})\quad\text{and}\quad\|\partial_{\sigma_{j}}\phi\|_{H^{2}}=O(\varepsilon^{2\beta}),\qquad\forall\
j,k.$

###### Proof.

Recall that $\beta=\min\\{p-1,1\\}$. The proof is based on the contraction
mapping theorem. First, for a large enough number $\gamma_{0}>0$, we set

$\mathcal{S}=\left\\{\phi\in\mathcal{H}~{}\left|~{}\|\phi\|_{H^{2}}\leq\gamma_{0}\varepsilon^{2}\right.\right\\}.$

The nonlinear problem (19) is transferred to solving

$\phi=\widetilde{T}(-R(y)+N(\phi)):=\mathcal{A}(\phi),$

which means to find a fixed point of the operator $\mathcal{A}$.

First, the operator $\mathcal{A}$ is from $\mathcal{S}$ to itself. In fact,

$\|\mathcal{A}(\phi)\|_{H^{2}}=\|\widetilde{T}(-R(y)+N(\phi))\|_{H^{2}}\leq
C\|R(y)\|_{L^{2}}+C\|N(\phi)\|_{L^{2}}\leq
C\varepsilon^{2}+C\|\phi\|_{H^{2}}^{1+\beta}\leq\gamma_{0}\varepsilon^{2}.$

Next the operator $\mathcal{A}$ is a contraction mapping, since

$\displaystyle\|\mathcal{A}(\phi_{1})-\mathcal{A}(\phi_{2})\|_{H^{2}}$
$\displaystyle=$
$\displaystyle\|\widetilde{T}(N(\phi_{1})-N(\phi_{2}))\|_{H^{2}}\leq
C\|N(\phi_{1})-N(\phi_{2})\|_{L^{2}}$ $\displaystyle\leq$ $\displaystyle
C\left(\|\phi_{1}\|_{H^{2}}^{\beta}+\|\phi_{2}\|_{H^{2}}^{\beta}\right)\|\phi_{1}-\phi_{2}\|_{H^{2}}.$

Thus $\mathcal{A}$ has a unique fixed point in $\mathcal{S}$, which is the
unique solution of problem (19).

Next we come to $\partial_{\zeta^{\prime}_{j,k}}\phi$ and
$\partial_{\sigma_{j}}\phi$. The $C^{1}$-regularity in $\zeta^{\prime}_{j}$
and $\sigma_{j}$ is guaranteed by the implicit function theorem. One may refer
to the proof of Lemma 4.1 in [10]. It is easy to see

$\partial_{\zeta^{\prime}_{j,k}}\phi=\partial_{\zeta^{\prime}_{j,k}}\widetilde{T}(-R(y)+N(\phi))+\widetilde{T}(-\partial_{\zeta^{\prime}_{j,k}}R(y)+\partial_{\zeta^{\prime}_{j,k}}N(\phi)).$
(20)

Notice that
$\left|\partial_{\zeta^{\prime}_{j,k}}N(\phi)\right|=O\left(|\phi|^{\beta}|\partial^{\prime}_{jk}w|+|\phi|^{\beta}|\partial_{\zeta^{\prime}_{j,k}}\phi|\right)$.
So we get

$\|\partial_{\zeta^{\prime}_{j,k}}N(\phi)\|_{L^{2}}\leq
C\|\phi\|_{H^{2}}^{\beta}+C\|\phi\|_{H^{2}}^{\beta}\|\partial_{\zeta^{\prime}_{j,k}}\phi\|_{H^{2}}.$

Thus (20) and Proposition 3.3 lead to

$\|\partial_{\zeta^{\prime}_{j,k}}\phi\|_{H^{2}}\leq
C\|\partial_{\zeta^{\prime}_{j,k}}R(y)\|_{L^{2}}+C\|R(y)\|_{L^{2}}+C\|N(\phi)\|_{L^{2}}+C\|\partial_{\zeta^{\prime}_{j,k}}N(\phi)\|_{L^{2}}\leq
C\varepsilon^{2\beta}.$

The estimate for $\|\partial_{\sigma_{j}}\phi\|_{H^{2}}$ may be gotten by the
same process. ∎

##  4\. Variational Reduction

According to the above discussion, the remaining thing is to let $c_{m,i}=0$
in the nonlinear problem (19) in order to make $\mathcal{W}+\phi$ be a
solution of the original problem. It can be done by the variational reduction
process.

Note the energy functional of problem (5) is

$E(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}|\mathrm{i}\nabla u+\bm{A}(\varepsilon
y)u|^{2}\mathrm{d}y+\frac{1}{2}\int_{\mathbb{R}^{N}}|u|^{2}-\frac{1}{p+1}\int_{\mathbb{R}^{N}}|u|^{p+1}.$

Define

$F(\bm{\sigma},\bm{\zeta}^{\prime})=E(\mathcal{W}+\phi)(\bm{\sigma},\bm{\zeta}^{\prime}),$

then the existence of critical points to $E(u)$ may be reduced to find
critical points of the finite dimensional function
$F(\bm{\sigma},\bm{\zeta}^{\prime})$.

######  Proposition 4.1.

If $(\bm{\sigma},\bm{\zeta}^{\prime})$ is a critical point of
$F(\bm{\sigma},\bm{\zeta}^{\prime})$, then $c_{m,i}=0$ for all $m,i$.

###### Proof.

It is easy to see that

$\displaystyle\partial_{\zeta_{m,j}^{\prime}}F(\bm{\sigma},\bm{\zeta}^{\prime})$
$\displaystyle=\partial_{\zeta^{\prime}_{m,j}}E(\mathcal{W}+\phi)=E^{\prime}(\mathcal{W}+\phi)\left[\frac{\partial\mathcal{W}}{\partial{\zeta^{\prime}_{m,j}}}+\frac{\partial\phi}{\partial{\zeta^{\prime}_{m,j}}}\right]$
$\displaystyle=\sum_{i,\ell}\text{Re}\int_{\mathbb{R}^{N}}c_{\ell,i}\chi_{\ell}\overline{Z}_{\ell,i}\left[\frac{\partial
U_{m}}{\partial{\zeta^{\prime}_{m,j}}}+\varepsilon\frac{\partial\Psi_{m}}{\partial{\zeta^{\prime}_{m,j}}}+\frac{\partial\phi}{\partial{\zeta^{\prime}_{m,j}}}\right]=c_{m,j}\int_{\mathbb{R}^{N}}\chi_{m}|Z_{m,j}|^{2}+o(1).$

Similarly, it is also true that

$\partial_{\sigma_{m}}F(\bm{\sigma},\bm{\zeta}^{\prime})=c_{m,0}\int_{\mathbb{R}^{N}}\chi_{m}|Z_{m,0}|^{2}+o(1).$

Thus $c_{m,i}=0$ if $(\bm{\sigma},\bm{\zeta}^{\prime})$ is a critical point of
$F$ since the coefficient matrix of $c_{m,i}$ is diagonal dominant. ∎

Next we should calculate $F(\bm{\sigma},\bm{\zeta}^{\prime})$ in view of
Proposition 4.1.

######  Proposition 4.2.

It holds that

$\displaystyle
F(\bm{\sigma},\bm{\zeta}^{\prime})=E(\mathcal{W})+O(\varepsilon^{4}),$ and for
any $j,k$,
$\displaystyle\partial_{\zeta^{\prime}_{j,k}}F(\bm{\sigma},\bm{\zeta}^{\prime})=\partial_{\zeta^{\prime}_{j,k}}E(\mathcal{W})+O(\varepsilon^{2+2\beta}),\qquad\partial_{\sigma_{j}}F(\bm{\sigma},\bm{\zeta}^{\prime})=\partial_{\sigma_{j}}E(\mathcal{W})+O(\varepsilon^{2+2\beta}).$

###### Proof.

Direct computation leads to

$\begin{split}F(\bm{\sigma},\bm{\zeta}^{\prime})=&\frac{1}{2}\int_{\mathbb{R}^{N}}\left|\mathrm{i}\nabla(\mathcal{W}+\phi)+\bm{A}(\varepsilon
y)(\mathcal{W}+\phi)\right|^{2}+\frac{1}{2}\int_{\mathbb{R}^{N}}|\mathcal{W}+\phi|^{2}-\frac{1}{p+1}\int_{\mathbb{R}^{N}}|\mathcal{W}+\phi|^{p+1}\\\
=&\
E(\mathcal{W})+\frac{1}{2}\int_{\mathbb{R}^{N}}\text{Re}\left((R(y)-N(\phi))\bar{\phi}\right)-\int_{\mathbb{R}^{N}}\Big{[}\frac{1}{p+1}|\mathcal{W}+\phi|^{p+1}-\frac{1}{p+1}|\mathcal{W}|^{p+1}\\\
&-|\mathcal{W}|^{p-1}\text{Re}(\mathcal{W}\bar{\phi})-\frac{p-1}{2}|\mathcal{W}|^{p-3}(\text{Re}(\mathcal{W}\bar{\phi}))^{2}-\frac{1}{2}|\mathcal{W}|^{p-1}|\phi|^{2}\Big{]}.\end{split}$

Then the proposition follows from Proposition 2.1 and Proposition 3.4 easily.
By the computation in Proposition 3.4, it is easy to check that

$\partial_{\zeta^{\prime}_{j,k}}F(\bm{\sigma},\bm{\zeta}^{\prime})=\partial_{\zeta^{\prime}_{j,k}}E(\mathcal{W})+O(\|\phi\|^{1+\beta}).$

So is $\partial_{\sigma_{j}}F(\bm{\sigma},\bm{\zeta}^{\prime})$. ∎

Since $E(\mathcal{W})$ is the main part of
$F(\bm{\sigma},\bm{\zeta}^{\prime})$, it is important to get the expression of
$E(\mathcal{W})$. Elegant computation shows the following proposition.

######  Proposition 4.3.

It holds that for $\varepsilon$ small enough,

$E(\mathcal{W})=A_{0}K+B_{0}\varepsilon^{2}\sum_{m=1}^{K}\sum_{i,j=1}^{N}(\partial_{i}A_{j}(\zeta_{m})-\partial_{j}A_{i}(\zeta_{m}))^{2}+O(\varepsilon^{4}).$

Furthermore, the remainder term $O(\varepsilon^{4})$ also holds for the
derivatives in $\bm{\zeta}^{\prime},\bm{\sigma}$. Here
$A_{0}=\frac{p-1}{2(p+1)}\int_{\mathbb{R}^{N}}w^{p+1}(|y|)\mathrm{d}y$ and
$B_{0}=\frac{1}{4}\int_{\mathbb{R}^{N}}y_{1}^{2}w^{2}(|y|)\mathrm{d}y$ are
both universal positive constants.

###### Proof.

It is easy to see that

$\displaystyle E(\mathcal{W})$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}^{N}}|\mathrm{i}\nabla\mathcal{W}+\bm{A}(\varepsilon
y)\mathcal{W}|^{2}+\frac{1}{2}\int_{\mathbb{R}^{N}}|\mathcal{W}|^{2}-\frac{1}{p+1}\int_{\mathbb{R}^{N}}|\mathcal{W}|^{p+1}$
$\displaystyle=\frac{1}{2}\operatorname{Re}\int_{\mathbb{R}^{N}}R(y)\overline{\mathcal{W}}+\frac{p-1}{2(p+1)}\int_{\mathbb{R}^{N}}|\mathcal{W}|^{p+1}$
$\displaystyle=\sum_{m=1}^{K}\text{Re}\int_{B(\zeta^{\prime}_{m};\frac{\delta}{\sqrt{\varepsilon}})}\left[\frac{1}{2}R(y)\overline{\mathcal{W}}+\frac{p-1}{2(p+1)}|\mathcal{W}|^{p+1}\right]+\text{Re}\int_{\mathbb{R}^{N}\setminus\cup_{m=1}^{K}B(\zeta^{\prime}_{m};\frac{\delta}{\sqrt{\varepsilon}})}\left[\frac{1}{2}R(y)\overline{\mathcal{W}}+\frac{p-1}{2(p+1)}|\mathcal{W}|^{p+1}\right]$
$\displaystyle=\sum_{m=1}^{K}\int_{B(\zeta^{\prime}_{m};\frac{\delta}{\sqrt{\varepsilon}})}\Bigg{[}\frac{R_{m,1}}{2}w(|y-\zeta^{\prime}_{m}|)+\varepsilon\frac{R_{m,2}}{2}\psi_{m}(y)+\frac{p-1}{2(p+1)}|U_{m}|^{p+1}$
$\displaystyle\qquad+\frac{(p-1)\varepsilon^{2}}{4}|U_{m}|^{p-1}|\Psi_{m}|^{2}\Bigg{]}+O(\varepsilon^{4}),$
(21)

where $\psi_{m},\Psi_{m}$ are given in (9), and $R_{m,1}$, $R_{m,2}$ are
defined in Proposition 2.1. First, by the oddness of the terms in order
$\varepsilon^{3}$ in (13), it can be obtained that

$\displaystyle\
\int_{B(\zeta^{\prime}_{m};\frac{\delta}{\sqrt{\varepsilon}})}R_{m,1}w(|y-\zeta^{\prime}_{m}|)$
$\displaystyle=$ $\displaystyle\
\varepsilon^{2}\sum_{i,j,k=1}^{N}\partial_{j}A_{i}(\zeta_{m})\partial_{k}A_{i}(\zeta_{m})\int_{B(\zeta^{\prime}_{m};\frac{\delta}{\sqrt{\varepsilon}})}(y_{j}-\zeta^{\prime}_{m,j})(y_{k}-\zeta^{\prime}_{m,k})w^{2}(|y-\zeta^{\prime}_{m}|)$
$\displaystyle\
-\varepsilon^{2}\sum_{j,k,\ell=1}^{N}\partial_{\ell}A_{k}(\zeta_{m})\left[\partial_{j}A_{k}(\zeta_{m})+\partial_{k}A_{j}(\zeta_{m})\right]\int_{B(\zeta^{\prime}_{m};\frac{\delta}{\sqrt{\varepsilon}})}(y_{j}-\zeta^{\prime}_{m,j})(y_{\ell}-\zeta^{\prime}_{m,\ell})w^{2}(|y-\zeta^{\prime}_{m}|)$
$\displaystyle\
-\varepsilon^{2}\sum_{i,j,k,\ell}^{N}\partial_{\ell}A_{k}(\zeta_{m})\partial_{j}A_{i}(\zeta_{m})\int_{B(\zeta^{\prime}_{m};\frac{\delta}{\sqrt{\varepsilon}})}(y_{i}-\zeta^{\prime}_{m,i})(y_{j}-\zeta^{\prime}_{m,j})(y_{k}-\zeta^{\prime}_{m,k})(y_{\ell}-\zeta^{\prime}_{m,\ell})$
$\displaystyle\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\cdot\frac{w^{\prime}(|y-\zeta^{\prime}_{m}|)}{|y-\zeta^{\prime}_{m}|}w(|y-\zeta^{\prime}_{m}|)$
$\displaystyle\
-\frac{\varepsilon^{2}}{2}(\nabla_{x}\cdot\bm{A}(\zeta_{m}))\sum_{i,j=1}^{N}\partial_{j}A_{i}(\zeta_{m})\int_{B(\zeta^{\prime}_{m};\frac{\delta}{\sqrt{\varepsilon}})}(y_{i}-\zeta^{\prime}_{m,i})(y_{j}-\zeta^{\prime}_{m,j})w^{2}(|y-\zeta^{\prime}_{m}|)$
$\displaystyle\
-\frac{(p-1)\varepsilon^{2}}{8}\sum_{i,j,k,\ell}^{N}\partial_{j}A_{i}(\zeta_{m})\partial_{\ell}A_{k}(\zeta_{m})\int_{B(\zeta^{\prime}_{m};\frac{\delta}{\sqrt{\varepsilon}})}(y_{i}-\zeta^{\prime}_{m,i})(y_{j}-\zeta^{\prime}_{m,j})(y_{k}-\zeta^{\prime}_{m,k})(y_{\ell}-\zeta^{\prime}_{m,\ell})w^{p+1}$
$\displaystyle\ +O(\varepsilon^{4}).$

Since integration by parts gives

$\displaystyle\int_{\mathbb{R}^{N}}y_{i}y_{j}y_{k}y_{\ell}w(|y|)\frac{w^{\prime}(|y|)}{|y|}\mathrm{d}y$
$\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}y_{i}y_{j}y_{k}(\partial_{\ell}w^{2})$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\int_{\mathbb{R}^{N}}\delta_{i\ell}\delta_{jk}y_{j}y_{k}w^{2}-\frac{1}{2}\int_{\mathbb{R}^{N}}\delta_{j\ell}\delta_{ik}y_{i}y_{k}w^{2}-\frac{1}{2}\int_{\mathbb{R}^{N}}\delta_{k\ell}\delta_{ij}y_{i}y_{j}w^{2},$

one gets

$\displaystyle\sum_{i,j,k,\ell=1}^{N}\partial_{\ell}A_{k}(\zeta_{m})\partial_{j}A_{i}(\zeta_{m})\int_{\mathbb{R}^{N}}(y_{i}-\zeta^{\prime}_{m,i})(y_{j}-\zeta^{\prime}_{m,j})(y_{k}-\zeta^{\prime}_{m,k})(y_{\ell}-\zeta^{\prime}_{m,\ell})\frac{w^{\prime}(|y-\zeta^{\prime}_{m}|)}{|y-\zeta^{\prime}_{m}|}w(|y-\zeta^{\prime}_{m}|)$
$\displaystyle=$
$\displaystyle-2B_{0}\sum_{i,j}\left[\partial_{i}A_{i}(\zeta_{m})\partial_{j}A_{j}(\zeta_{m})+(\partial_{j}A_{i}(\zeta_{m}))^{2}+\partial_{i}A_{j}(\zeta_{m})\partial_{j}A_{i}(\zeta_{m})\right].$

It may be checked that

$\displaystyle\
\int_{B(\zeta^{\prime}_{m};\frac{\delta}{\sqrt{\varepsilon}})}\frac{R_{m,1}}{2}w(|y-\zeta^{\prime}_{m}|)$
$\displaystyle=$ $\displaystyle\
B_{0}\varepsilon^{2}\Bigg{\\{}2\sum_{i,j}(\partial_{j}A_{i}(\zeta_{m}))^{2}-2\sum_{i,j}\partial_{j}A_{i}(\zeta_{m})(\partial_{j}A_{i}(\zeta_{m})+\partial_{i}A_{j}(\zeta_{m}))$
$\displaystyle\qquad\quad+\sum_{i,j}\left[\partial_{i}A_{i}(\zeta_{m})\partial_{j}A_{j}(\zeta_{m})+(\partial_{j}A_{i}(\zeta_{m}))^{2}+\partial_{i}A_{j}(\zeta_{m})\partial_{j}A_{i}(\zeta_{m})\right]-\left(\sum_{i}\partial_{i}A_{i}(\zeta_{m})\right)^{2}\Bigg{\\}}$
$\displaystyle\
-\frac{(p-1)\varepsilon^{2}}{16}\sum_{i,j,k,\ell}^{N}\partial_{j}A_{i}(\zeta_{m})\partial_{\ell}A_{k}(\zeta_{m})\int_{B(\zeta^{\prime}_{m};\frac{\delta}{\sqrt{\varepsilon}})}(y_{i}-\zeta^{\prime}_{m,i})(y_{j}-\zeta^{\prime}_{m,j})(y_{k}-\zeta^{\prime}_{m,k})(y_{\ell}-\zeta^{\prime}_{m,\ell})w^{p+1}$
$\displaystyle\ +O(\varepsilon^{4}).$ (22)

Also

$\int_{B(\zeta^{\prime}_{m};\frac{\delta}{\sqrt{\varepsilon}})}|U_{m}|^{p-1}|\Psi_{m}|^{2}\\\
=\frac{1}{4}\sum_{i,j,k,\ell=1}^{N}\partial_{j}A_{i}(\zeta_{m})\partial_{\ell}A_{k}(\zeta_{m})\int_{B(\zeta^{\prime}_{m};\delta)}(y_{i}-\zeta^{\prime}_{m,i})(y_{j}-\zeta^{\prime}_{m,j})(y_{k}-\zeta^{\prime}_{m,k})(y_{\ell}-\zeta^{\prime}_{m,\ell})w^{p+1}(|y-\zeta^{\prime}_{m}|)+O(\mathrm{e}^{-\frac{\delta}{\varepsilon}}),$

and

$\int_{B(\zeta^{\prime}_{m};\frac{\delta}{\sqrt{\varepsilon}})}R_{m,2}\psi_{m}=O(\varepsilon^{3})$

by the oddness. Obviously one has

$\int_{B(\zeta^{\prime}_{m};\frac{\delta}{\sqrt{\varepsilon}})}|U_{m}|^{p+1}=\int_{\mathbb{R}^{N}}w^{p+1}(y)\mathrm{d}y+O(\mathrm{e}^{-\frac{\delta}{\varepsilon}}).$

Note that in (22) the term containing $w^{p+1}$ is canceled with
$\frac{p-1}{4}\varepsilon^{2}\int_{\mathbb{R}^{N}}|U_{m}|^{p-1}|\Psi_{m}|^{2}$.
So we conclude, from (21), that

$\displaystyle E(\mathcal{W})$
$\displaystyle=A_{0}K+\varepsilon^{2}B_{0}\sum_{m=1}^{K}\sum_{i,j=1}^{N}\left[(\partial_{j}A_{i}(\zeta_{m}))^{2}-\partial_{j}A_{i}(\zeta_{m})\partial_{i}A_{j}(\zeta_{m})\right]+O(\varepsilon^{4})$
$\displaystyle=A_{0}K+\varepsilon^{2}\frac{B_{0}}{2}\sum_{m=1}^{K}\sum_{i,j=1}^{N}\left((\partial_{j}A_{i}(\zeta_{m}))-\partial_{i}A_{j}(\zeta_{m})\right)^{2}+O(\varepsilon^{4}).$

The last equality is due to the symmetry of the indexes $i$ and $j$. The
remainder $O(\varepsilon^{4})$ also holds for the derivatives of
$E(\mathcal{W})$ in $(\bm{\sigma},\bm{\zeta}^{\prime})$ from directly checking
the expressions of $R_{m,1}$ and $R_{m,2}$ in the proof of Proposition 2.1.

∎

##  5\. Proof of the main theorems

This section devotes to the proof of main theorems.

###### Proof of Theorem 1.1.

Proposition 4.2 and Proposition 4.3 mean

$F(\bm{\sigma},\bm{\zeta}^{\prime})=A_{0}K+B_{0}\varepsilon^{2}\sum_{m=1}^{K}\|\bm{B}(\zeta_{m})\|_{F}^{2}+O(\varepsilon^{4}).$

We shall show that $F$ has a critical point under the assumption. Note that
for any fixed $\bm{\zeta}^{\prime}$, $F(\bm{\sigma},\bm{\zeta}^{\prime})$ is
periodic in $\bm{\sigma}\in([0,2\pi])^{K}$. So there always exists a
$\bm{\sigma}(\bm{\zeta}^{\prime})$ such that
$\partial_{\bm{\sigma}}F(\bm{\sigma}(\bm{\zeta}^{\prime}),\bm{\zeta}^{\prime})=0$.
Next consider the configuration set
$\bm{\Omega}^{\prime}=\Omega^{\prime}_{1}\times\Omega^{\prime}_{2}\times\cdots\times\Omega^{\prime}_{m}$
of
$\bm{\zeta}^{\prime}=(\zeta^{\prime}_{1},\zeta^{\prime}_{2},\cdots,\zeta^{\prime}_{m})$,
where $\Omega^{\prime}_{m}=\varepsilon^{-1}\Omega_{m}$. Obviously

$\max_{\bm{\Omega}^{\prime}}F(\bm{\sigma}(\bm{\zeta}^{\prime}),\bm{\zeta}^{\prime})\geq
A_{0}K+B_{0}\varepsilon^{2}\sum_{m=1}^{K}\|\bm{B}(P_{m})\|_{F}^{2}+O(\varepsilon^{4}).$

On the other hand, for any $\bm{\zeta}^{\prime}$ on the boundary
$\partial\bm{\Omega}^{\prime}$, i.e. at least
$\zeta_{1}^{\prime}\in\partial\Omega_{1}^{\prime}$ without loss of generality,
then $\|B(\zeta_{1})\|^{2}_{F}\leq\|B(P_{1})\|^{2}_{F}-\delta_{0}$ for some
fixed small $\delta_{0}>0$. Thus one finds that

$F(\bm{\sigma}(\bm{\zeta}^{\prime}),\bm{\zeta}^{\prime})\Big{|}_{\bm{\zeta}^{\prime}\in\partial\bm{\Omega}^{\prime}}\leq
A_{0}K+B_{0}\varepsilon^{2}\sum_{m=2}^{K}\|\bm{B}(P_{m})\|_{F}^{2}+B_{0}\varepsilon^{2}(\|\bm{B}(P_{1})\|_{F}^{2}-\delta_{0})+O(\varepsilon^{4}).$

Therefore
$\max_{\bm{\Omega}^{\prime}}F(\bm{\sigma}(\bm{\zeta}^{\prime}),\bm{\zeta}^{\prime})>\max_{\partial\bm{\Omega}^{\prime}}F(\bm{\sigma}(\bm{\zeta}^{\prime}),\bm{\zeta}^{\prime})$.
It implies that $F(\bm{\sigma},\bm{\zeta}^{\prime})$ admits a critical point.

The same procedure can be carried out for the case of $K$ local minimum
points. Theorem 1.1 concludes from Proposition 4.1. ∎

###### Proof of Theorem 1.2.

From Proposition 4.2 and Proposition 4.3, we see that

$\nabla_{\zeta_{m}^{\prime}}F(\bm{\sigma},\bm{\zeta}^{\prime})=B_{0}\varepsilon^{2}\nabla_{\zeta_{m}^{\prime}}(\|\bm{B}(\zeta_{m})\|_{F}^{2})+O(\varepsilon^{2+2\beta})=B_{0}\varepsilon^{3}\nabla_{\zeta_{m}}(\|\bm{B}(\zeta_{m})\|_{F}^{2})+O(\varepsilon^{2+2\beta}).$

Assume $m=1$ for simplicity. Choose
$\zeta_{1}=P_{1}+\varepsilon^{\alpha}\xi_{1}$ where $0<\alpha<2\beta-1$. Here
the assumption $p>\frac{3}{2}$ is used to let $\beta=\min\\{p-1,1\\}>1/2$.
Then it is equivalent to find a $|\xi_{1}|\leq 1$ such that

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\nabla_{\zeta_{1}}(\|\bm{B}(\zeta_{1})\|_{F}^{2})+O(\varepsilon^{2\beta-1})$
$\displaystyle=$
$\displaystyle\nabla_{\zeta_{1}}(\|\bm{B}(P_{1})\|_{F}^{2})+\varepsilon^{\alpha}\nabla^{2}_{\zeta_{1}\zeta_{1}}(\|\bm{B}(P_{1})\|_{F}^{2})\cdot\xi_{1}+O(\varepsilon^{2\alpha}|\xi_{1}|^{2})+O(\varepsilon^{2\beta-1}).$

Thus, the nondegeneracy of the critical point $P_{1}$ leads to the existence
of $\xi_{1},|\xi_{1}|=o(1)$ from the Brouwer fixed point theorem. Lastly, the
existence of critical $\bm{\sigma}$ is guaranteed by the periodicity just like
in the proof of Theorem 1.1. The proof is complete. ∎

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