# Helical vortices with small cross-section for 3D incompressible Euler
equation

Daomin Cao, Jie Wan  Institute of Applied Mathematics, Chinese Academy of
Sciences, Beijing 100190, and University of Chinese Academy of Sciences,
Beijing 100049, P.R. China<EMAIL_ADDRESS>School of Mathematics and
Statistics, Beijing Institute of Technology, Beijing 100081, P.R. China
<EMAIL_ADDRESS>
###### Abstract.

In this article, we construct traveling-rotating helical vortices with small
cross-section to the 3D incompressible Euler equations in an infinite pipe,
which tend asymptotically to singular helical vortex filament evolved by the
binormal curvature flow. The construction is based on studying a general
semilinear elliptic problem in divergence form

$\begin{cases}-\varepsilon^{2}\text{div}(K(x)\nabla
u)=(u-q|\ln\varepsilon|)^{p}_{+},\ \ &x\in\Omega,\\\ u=0,\ \
&x\in\partial\Omega,\end{cases}$

for small values of $\varepsilon.$ Helical vortex solutions concentrating near
several helical filaments with polygonal symmetry are also constructed.

Keywords: Incompressible Euler equation; Binormal curvature flow; Helical
symmetry; Semilinear elliptic equations; Variational method.

##  1\. Introduction and main results

The movement of an ideal incompressible flow confined in a 3D domain $D$ is
governed by the following Euler equation

$\begin{cases}\partial_{t}\mathbf{v}+(\mathbf{v}\cdot\nabla)\mathbf{v}=-\nabla
P,\ \ &D\times(0,T),\\\ \nabla\cdot\mathbf{v}=0,\ \ &D\times(0,T),\\\
\mathbf{v}(\cdot,0)=\mathbf{v}_{0}(\cdot),\ \ &D,\end{cases}$ (1.1)

where the domain $D\subseteq\mathbb{R}^{3}$, $\mathbf{v}=(v_{1},v_{2},v_{3})$
is the velocity field, $P$ is the scalar pressure and $\mathbf{v}_{0}$ is the
initial velocity field. If $D$ has a boundary, the following impermeable
boundary condition is usually assumed

$\mathbf{v}\cdot\mathbf{n}=0,\ \ \partial D\times(0,T),$

where $\mathbf{n}$ is the outward unit normal to $\partial D$.

Define the associated vorticity field
$\mathbf{w}=(w_{1},w_{2},w_{3})=curl\mathbf{v}=\nabla\times\mathbf{v}$, which
describes the rotation of the fluid. Then $\mathbf{w}$ satisfies the vorticity
equations

$\begin{cases}\partial_{t}\mathbf{w}+(\mathbf{v}\cdot\nabla)\mathbf{w}=(\mathbf{w}\cdot\nabla)\mathbf{v},\
\ &D\times(0,T),\\\ \mathbf{w}(\cdot,0)=\nabla\times\mathbf{v}_{0}(\cdot),\ \
&D.\end{cases}$ (1.2)

For background of the 3D incompressible Euler equation, see the classical
literature [32, 33].

Helmholtz [21] began the study of Euler equation in 1858, who first considered
the vorticity equations of the flow and found that the vortex rings, which are
toroidal regions in which the vorticity has small cross-section, translate
with a constant speed alone the axis of symmetry. The translating speed of
vortex rings was then studied by Kelvin and Hick [26] in 1899. Define the
circulation of a vortex

$c=\oint_{l}\mathbf{v}\cdot\mathbf{t}dl=\iint_{\sigma}\mathbf{w}\cdot\mathbf{n}d\sigma,$
(1.3)

where $l$ is any oriented curve with tangent vector field $\mathbf{t}$ that
encircles the vorticity region once and $\sigma$ is any surface with boundary
$l$. [26] showed that if the vortex ring has radius $r^{*}$, circulation $c$
and its cross-section $\varepsilon$ is small, then the vortex ring moves at
the velocity

$\frac{c}{4\pi r^{*}}\left(\ln\frac{8r^{*}}{\varepsilon}-\frac{1}{4}\right).$
(1.4)

Then Da Rios [10] in 1906, and Levi-Civita [27] in 1908, formally found the
general law of motion of a vortex filament with a small section of radius
$\varepsilon$ and a fixed circulation, uniformly distributed around an
evolving curve $\Gamma(t)$, which is well-known as the binormal curvature
flow, or the localized induction approximation (LIA). Roughly speaking, under
suitable assumptions on the solution, the curve evolves by the binormal flow,
with a large velocity of order $|\ln\varepsilon|$. More precisely, if
$\Gamma(t)$ is parameterized as $\gamma(s,t)$, where $s$ is the parameter of
arclength, then $\gamma(s,t)$ asymptotically obeys a law of the form (see
[29], p. 30, Eq. (8’) and [35] p. 260 Eq. (62))

$\partial_{t}\gamma=\frac{c}{4\pi}|\ln\varepsilon|(\partial_{s}\gamma\times\partial_{ss}\gamma)=\frac{c\bar{K}}{4\pi}|\ln\varepsilon|\mathbf{b}_{\gamma(t)},$
(1.5)

where $c$ is the circulation of the velocity field on the boundary of sections
to the filament, which is assumed to be a constant independent of
$\varepsilon$, $\mathbf{b}_{\gamma(t)}$ is the binormal unit vector and
$\bar{K}$ is its local curvature. If we scale $t=|\ln\varepsilon|^{-1}\tau$,
then

$\partial_{\tau}\gamma=\frac{c\bar{K}}{4\pi}\mathbf{b}_{\gamma(\tau)}.$ (1.6)

Hence, under LIA vortex filaments move simply in the binormal direction with
speed proportional to the local curvature and the circulation. It is
worthwhile to note that, when $\Gamma$ is a circular filament, the leading
term of (1.4) coincides with the coefficient of right hand side of (1.5) since
in this case the local curvature $\bar{K}=\frac{1}{r^{*}}.$ The localized
induction approximation found by Da Rios was applied to various physical
problems, for instance, the induction due to electric currents in a wire [11],
the gravitational effect associated with Saturnian rings [28] and vortex
motion [29], for more detail, see the survey papers by Ricca [34, 35].

From mathematical justification, Jerrad and Seis [22] first gave a precise
form to Da Rios’ computation under some mild conditions on a solution to (1.2)
which remains suitably concentrated around an evolving vortex filament. Their
result shows that under some conditions of a solution
$\mathbf{w}_{\varepsilon}$ of (1.2), there holds in the sense of distribution,

$\mathbf{w}_{\varepsilon}(\cdot,|\ln\varepsilon|^{-1}\tau)\to
c\delta_{\gamma(\tau)}\mathbf{t}_{\gamma(\tau)},\ \ \text{as}\ \varepsilon\to
0,$ (1.7)

where $\gamma(\tau)$ satisfies (1.6), $\mathbf{t}_{\gamma(\tau)}$ is the
tangent unit vector of $\gamma$ and $\delta_{\gamma(\tau)}$ is the uniform
Dirac measure on the curve. See [23] for more results of this problem.

Until now the existence of a family of solutions to (1.2) satisfying (1.7),
where $\gamma(\tau)$ is a given curve evolved by the binormal flow (1.6), is
still an open problem. This problem is well-known as the vortex filament
conjecture, which is unsolved except for the filament being several kinds of
special curves: the straight lines, the traveling circles and the traveling-
rotating helices. For the problem of vortex concentrating near straight lines,
it corresponds to the planar Euler equations concentrating near a collection
of given points governed by the 2D point vortex model, see [5, 6, 12, 31, 33,
37] for example. When the filament is a traveling circle with radius $r^{*}$,
by (1.6) the curve is

$\gamma(s,\tau)=\left(r^{*}\cos\left(\frac{s}{r^{*}}\right),r^{*}\sin\left(\frac{s}{r^{*}}\right),\frac{c}{4\pi
r^{*}}\tau\right)^{t},$ (1.8)

where $\mathbf{v}^{t}$ is the transposition of a vector $\mathbf{v}$. Fraenkel
[18] first gave a construction of vortex rings with small cross-section
without change of form concentrating near a traveling circle satisfying (1.8)
in sense of (1.7) and then many articles showed the desingularization results
under a variety of conditions, such as constructing vortex rings in different
kinds of domains with different vortex profiles, see [2, 4, 15, 19] for
instance.

For vortex filament being a helix satisfying (1.6), the curve is parameterized
as

$\gamma(s,\tau)=\left(r_{*}\cos\left(\frac{-s-a_{1}\tau}{\sqrt{k^{2}+r_{*}^{2}}}\right),r_{*}\sin\left(\frac{-s-a_{1}\tau}{\sqrt{k^{2}+r_{*}^{2}}}\right),\frac{ks-
b_{1}\tau}{\sqrt{k^{2}+r_{*}^{2}}}\right)^{t},$ (1.9)

where $r_{*}>0,k\neq 0$ are constants characterizing the distance between a
point in $\gamma(\tau)$ and the $x_{3}$-axis and the pitch of the helix, and

$a_{1}=\frac{ck}{4\pi(k^{2}+r_{*}^{2})},\
b_{1}=\frac{cr_{*}^{2}}{4\pi(k^{2}+r_{*}^{2})}.$

Note that the local curvature and torsion of the helix are
$\frac{r_{*}}{k^{2}+r_{*}^{2}}$ and $\frac{k}{k^{2}+r_{*}^{2}}$ respectively,
and the parametrization (1.9) satisfies (1.6). It should be noted that the
curve parameterized by (1.9) is a traveling-rotating helix. Let us define for
any $\theta\in[0,2\pi]$

$\bar{R}_{\theta}=\begin{pmatrix}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{pmatrix},\ \
\bar{Q}_{\theta}=\begin{pmatrix}\bar{R}_{\theta}&0\\\ 0&1\end{pmatrix}.$

One computes directly that

$\gamma(s,\tau)=\bar{Q}_{\frac{a_{1}\tau}{\sqrt{k^{2}+r_{*}^{2}}}}\gamma(s,0)+\left(0,0,-\frac{b_{1}\tau}{\sqrt{k^{2}+r_{*}^{2}}}\right)^{t}.$

We can readily check that (1.9) with $k>0$ and $k<0$ correspond to the left-
handed helix and the right-handed helix respectively (for consistency,
throughout this paper we always choose a right-handed Cartesian reference
centered at the origin). The problem of global well-posedness of solutions to
the vorticity equation (1.2) with helical symmetry was studied in many
articles, see [1, 3, 16, 24] for instance. For a helix $\gamma(\tau)$
satisfying (1.9), there are a few results of existence of true solutions of
(1.2) concentrating on this curve in sense of (1.7). The only result is shown
by D$\acute{\text{a}}$vila et al. [13], who considered traveling-rotating
invariant Euler flows with right-handed helical symmetry concentrating near a
single helix and multiple helices in the whole space $\mathbb{R}^{3}.$ In
their work, by considering

$\begin{split}-\text{div}(K_{H}(x)\nabla
u)=f_{\varepsilon}\left(u-\alpha|\ln\varepsilon|\frac{|x|^{2}}{2}\right)\ \
\text{in}\ \ \mathbb{R}^{2},\end{split}$

where $K_{H}$ is an elliptic operator in divergence form (defined by (2.8)),
$f_{\varepsilon}(t)=\varepsilon^{2}e^{t}$ and $\alpha$ is chosen properly, the
authors construct solutions concentrating near a helix in the distributional
sense. Note that by the choice of $f_{\varepsilon}$, the support set of
vorticity is still the whole plane.

The aim of this paper is to construct traveling-rotating solutions to Euler
equations (1.2) with helical symmetry in an infinite pipe, such that the
support set of vortex is a helical tube with small cross-section $\varepsilon$
without change of form, which tends asymptotically to a traveling-rotating
helix (1.9) in sense of (1.7). To get these result, we study the existence and
asymptotic behavior of solutions to a general semilinear elliptic equations
(see (2.16)). It should be noted that, Euler equations with helical symmetry
can be regarded as the general case of 2D and 3D axisymmetric Euler equations.
The cases $k\to+\infty$ and $k=0$ correspond to the 2D Euler equations and 3D
axisymmetric Euler equations, respectively. In contrast to the 2D and 3D
axisymmetric problems, the associated operator $\mathcal{L}_{H}$ in vorticity
equations (see (2.10)) is an elliptic operator in divergence form, which can
bring essential difficulty in the construction of solutions. First, it seems
impossible to reduce the second-order operator $\mathcal{L}_{H}$ to the
standard Laplace operator by means of a single change of coordinates. Second,
lack of understanding the properties of the Green’s function of a general
elliptic operator in divergence form is also a challenge. Moreover, since the
eigenvalues of $K_{H}$ are different, solutions of the associated limiting
equations are not radially symmetric functions, which is totally different
from the 2D and 3D axisymmetric cases.

To state our results, we need to introduce some notations first. Since the
helical vortices we are to construct is translating-rotating symmetric,
domains of the flow must be helical domains with rotating symmetry about
$x_{3}$ axis, which are the whole space and infinite pipes with circular cross
section. For any $R^{*}>0$, define
$B_{R^{*}}(0)\times\mathbb{R}=\\{(x_{1},x_{2},x_{3})\mid(x_{2},x_{2})\in
B_{R^{*}}(0),x_{3}\in\mathbb{R}\\}$ an infinite pipe in $\mathbb{R}^{3}$ whose
section is a disc with radius $R^{*}$. For two sets $A,B$, define
$dist(A,B)=\min_{x\in A,y\in B}|x-y|$ the distance between sets $A$ and $B$
and $diam(A)$ the diameter of the set $A$.

Our first result is concerned with the desingularization of traveling-rotating
helical vortices in $B_{R^{*}}(0)\times\mathbb{R}$, whose support set has
small cross-section $\varepsilon$ and concentrates near a single left-handed
helix (1.9) in sense of (1.7).

######  Theorem 1.1.

Let $k>0$, $c>0$ and $r_{*}\in(0,R^{*})$ be any given numbers. Let
$\gamma(\tau)$ be the helix parameterized by equation (1.9). Then for any
$\varepsilon\in(0,\varepsilon_{0}]$ for some $\varepsilon_{0}>0$, there exists
a classical solution pair $(\mathbf{v}_{\varepsilon},P_{\varepsilon})(x,t)\in
C^{1}(B_{R^{*}}(0)\times\mathbb{R}\times\mathbb{R}^{+})$ of (1.1) such that
the support set of $\mathbf{w}_{\varepsilon}$ is a topological traveling-
rotating helical tube that does not change form and concentrates near the
helix in sense of (1.7), that is for all $\tau$,

$\mathbf{w}_{\varepsilon}(\cdot,|\ln\varepsilon|^{-1}\tau)\to
c\delta_{\gamma(\tau)}\mathbf{t}_{\gamma(\tau)},\ \ \text{as}\ \varepsilon\to
0.$

Moreover, one has

  1. (1)

$\mathbf{v}_{\varepsilon}\cdot\mathbf{n}=0$ on $\partial
B_{R^{*}}(0)\times\mathbb{R}.$

  2. (2)

Define
$A_{\varepsilon}=supp(\mathbf{w}_{\varepsilon})\cap\mathbb{R}^{2}\times\\{0\\}$
the cross-section of $\mathbf{w}_{\varepsilon}$. Then there are
$R_{1},R_{2}>0$ such that

$R_{1}\varepsilon\leq diam(A_{\varepsilon})\leq R_{2}\varepsilon.$

######  Remark 1.2.

By the physical meaning of $k$, the sign of $k$ determines two different
helical structure. The curve parameterized by (1.9) with $k>0$ and $k<0$
correspond to the left-handed helical structure and right-handed helical
structure, respectively. Theorem 1.1 shows the desingularization of a left-
handed helix. For the case $k<0$, results are similar.

One can also construct multiple traveling-rotating helical vortices in
$B_{R^{*}}(0)\times\mathbb{R}$ with polygonal symmetry. Let us consider the
curve $\gamma(\tau)$ parameterized by (1.9). For any integer $m$, define for
$i=1\cdots,m$ the curves $\gamma_{i}(\tau)$ parameterized by

$\gamma_{i}(s,\tau)=\bar{Q}_{\frac{2\pi(i-1)}{m}}\gamma(s,\tau).$ (1.10)

The following result generalizes that of Theorem 1.1 to helical vortices
concentrating near multiple helices with polygonal symmetry.

######  Theorem 1.3.

Let $k>0$, $c>0$ and $r_{*}\in(0,R^{*})$ be any given numbers and $m\geq 2$ be
an integer. Let $\gamma_{i}(\tau)$ be the helix parameterized by (1.10). Then
for any $\varepsilon\in(0,\varepsilon_{0}]$ for some $\varepsilon_{0}>0$,
there exists a classical solution pair
$(\mathbf{v}_{\varepsilon},P_{\varepsilon})(x,t)\in
C^{1}(B_{R^{*}}(0)\times\mathbb{R}\times\mathbb{R}^{+})$ of (1.1) such that
the support set of $\mathbf{w}_{\varepsilon}$ is a collection of $m$
topological traveling-rotating helical tubes that does not change form and for
all $\tau$,

$\mathbf{w}_{\varepsilon}(\cdot,|\ln\varepsilon|^{-1}\tau)\to
c\sum_{i=1}^{m}\delta_{\gamma_{i}(\tau)}\mathbf{t}_{\gamma_{i}(\tau)},\ \
\text{as}\ \varepsilon\to 0.$

Moreover, one has

  1. (1)

$\mathbf{v}_{\varepsilon}\cdot\mathbf{n}=0$ on $\partial
B_{R^{*}}(0)\times\mathbb{R}.$

  2. (2)

Define $A_{i,\varepsilon}=supp(\mathbf{w}_{\varepsilon})\cap
B_{\bar{\rho}}\left(\bar{Q}_{\frac{2\pi(i-1)}{m}}\left(r_{*},0\right)\right)\times\\{0\\}$
for some small constant $\bar{\rho}>0$. Then there are $R_{1},R_{2}>0$ such
that

$R_{1}\varepsilon\leq diam(A_{i,\varepsilon})\leq R_{2}\varepsilon.$

The paper is organized as follows. In section 2, we deduce the 2D vorticity-
stream equations of left-handed helical solutions of (1.2) and the associated
semilinear elliptic equations. A generalized toy model (see (2.15)) and the
corresponding desingularization result (see Theorem 2.2) are introduced. In
section 3, we show the approximate solutions and some basic estimates. In
section 4 and section 5, we give proof of Theorem 2.2. The outline of proofs
for Theorem 1.1 and Theorem 1.3 are given in section 6.

##  2\. Solution with helical symmetry and a generalized model

Let us first define left-handed helical symmetric solutions and reduce (1.2)
to a 2D vorticity-stream model, see [12, 16, 17]. Let $k>0$. Define a one-
parameter group
$\mathcal{G}_{k}=\\{H_{\bar{\rho}}:\mathbb{R}^{3}\to\mathbb{R}^{3}\\}$, where

$H_{\rho}(x_{1},x_{2},x_{3})^{t}=(x_{1}\cos\rho+x_{2}\sin\rho,-x_{1}\sin\bar{\rho+x_{2}\cos\rho},x_{3}+k\rho)^{t}.$

So $H_{\rho}$ is a superposition of a rotation in $x_{1}Ox_{2}$ plane and a
translation in $x_{3}$ axis, that is,
$H_{\rho}(x)=\bar{Q}_{\rho}(x)+k\rho(0,0,1)$. Clearly,
$B_{R^{*}}(0)\times\mathbb{R}$ is invariant under the group $\mathcal{G}_{k}$.

Define a vector field

$\overrightarrow{\zeta}=(x_{2},-x_{1},k)^{t}.$

Then $\overrightarrow{\zeta}$ is the field of tangents of symmetry lines of
$\mathcal{G}_{k}$.

Let us define helical functions and vector fields. A scalar function $h$ is
called a $helical$ function, if $h(H_{\rho}(x))=h(x)$ for any
$\rho\in\mathbb{R},x\in B_{R^{*}}(0)\times\mathbb{R}.$ By direct computations
it is easy to see that a $C^{1}$ function $h$ is helical if and only if

$\overrightarrow{\zeta}\cdot\nabla h=0.$

A vector field $\mathbf{h}=(h_{1},h_{2},h_{3})$ is called a $helical$ field,
if $\mathbf{h}(H_{\rho}(x))=R_{\rho}\mathbf{h}(x)$ for any
$\rho\in\mathbb{R},x\in B_{R^{*}}(0)\times\mathbb{R}.$ Direct computation
shows that a $C^{1}$ vector field $\mathbf{h}$ is helical if and only if

$\overrightarrow{\zeta}\cdot\nabla\mathbf{h}=\mathcal{R}\mathbf{h},$

where $\mathcal{R}=\begin{pmatrix}0&1&0\\\ -1&0&0\\\ 0&0&0\end{pmatrix}$ (see
[17]). Helical solutions of (1.1) are then defined as follows.

######  Definition 2.1.

A function pair ($\mathbf{v},P$) is called a $helical$ solution of (1.1) in
$B_{R^{*}}(0)\times\mathbb{R}$, if ($\mathbf{v},P$) satisfies (1.1) and both
vector field $\mathbf{v}$ and scalar function $P$ are helical.

Throughout this paper, helical solutions also need to satisfy the
$orthogonality~{}condition$:

$\mathbf{v}\cdot\overrightarrow{\zeta}=0,$ (2.1)

that is, the velocity field and $\overrightarrow{\zeta}$ are orthogonal.

Under the condition (2.1), one can check that the vorticity field $\mathbf{w}$
satisfies (see [17])

$\mathbf{w}=\frac{w}{k}\overrightarrow{\zeta},$ (2.2)

where $w=w_{3}=\partial_{x_{1}}v_{2}-\partial_{x_{2}}v_{1}$, the third
component of vorticity field $\mathbf{w}$, is a helical function. Moreover,
the first equation of the vorticity equations (1.2) is equivalent to

$\partial_{t}\mathbf{w}+(\mathbf{v}\cdot\nabla)\mathbf{w}+\frac{1}{k}w\mathcal{R}\mathbf{v}=0.$

As a consequence, $w$ satisfies

$\partial_{t}w+(\mathbf{v}\cdot\nabla)w=0.$ (2.3)

We now introduce a $stream~{}function$ and reduce the system (1.2) to a 2D
vorticity-stream equation. Since $\mathbf{v}$ is a helical vector field, we
have $\overrightarrow{\zeta}\cdot\nabla\mathbf{v}=\mathcal{R}\mathbf{v}$,
which implies that

$x_{2}\partial_{x_{1}}v_{3}-x_{1}\partial_{x_{2}}v_{3}+k\partial_{x_{3}}v_{3}=0.$
(2.4)

The orthogonal condition shows that

$x_{2}v_{1}-x_{1}v_{2}+kv_{3}=0.$ (2.5)

It follows from the incompressible condition, (2.4) and (2.5) that

$\begin{split}0=&\partial_{x_{1}}v_{1}+\partial_{x_{2}}v_{2}+\partial_{x_{3}}v_{3}=\partial_{x_{1}}v_{1}+\partial_{x_{2}}v_{2}-\frac{x_{2}}{k}\partial_{x_{1}}v_{3}+\frac{x_{1}}{k}\partial_{x_{2}}v_{3}\\\
=&\partial_{x_{1}}v_{1}+\partial_{x_{2}}v_{2}-\frac{x_{2}}{k^{2}}\partial_{x_{1}}(-x_{2}v_{1}+x_{1}v_{2})+\frac{x_{1}}{k^{2}}\partial_{x_{2}}(-x_{2}v_{1}+x_{1}v_{2})\\\
=&\frac{1}{k^{2}}\partial_{x_{1}}[(k^{2}+x_{2}^{2})v_{1}-x_{1}x_{2}v_{2}]+\frac{1}{k^{2}}\partial_{x_{2}}[(k^{2}+x_{1}^{2})v_{2}-x_{1}x_{2}v_{1}].\end{split}$

Since $B_{R^{*}}(0)$ is simply-connected, we can define a stream function
$\varphi:B_{R^{*}}(0)\to\mathbb{R}$ such that
$\partial_{x_{2}}\varphi=\frac{1}{k^{2}}[(k^{2}+x_{2}^{2})v_{1}-x_{1}x_{2}v_{2}],\partial_{x_{1}}\varphi=-\frac{1}{k^{2}}[(k^{2}+x_{1}^{2})v_{2}-x_{1}x_{2}v_{1}]$,
that is,

$\begin{pmatrix}\partial_{x_{1}}\varphi\\\
\partial_{x_{2}}\varphi\end{pmatrix}=-\frac{1}{k^{2}}\begin{pmatrix}-x_{1}x_{2}&k^{2}+x_{1}^{2}\\\
-(k^{2}+x_{2}^{2})&x_{1}x_{2}\end{pmatrix}\begin{pmatrix}v_{1}\\\
v_{2}\end{pmatrix},$

or equivalently,

$\begin{pmatrix}v_{1}\\\
v_{2}\end{pmatrix}=-\frac{1}{k^{2}+x_{1}^{2}+x_{2}^{2}}\begin{pmatrix}x_{1}x_{2}&-k^{2}-x_{1}^{2}\\\
k^{2}+x_{2}^{2}&-x_{1}x_{2}\end{pmatrix}\begin{pmatrix}\partial_{x_{1}}\varphi\\\
\partial_{x_{2}}\varphi\end{pmatrix}.$ (2.6)

By the definition of $w$ and (2.6), we get

$\begin{split}w=&\partial_{x_{1}}v_{2}-\partial_{x_{2}}v_{1}=(-\partial_{x_{2}},\partial_{x_{1}})\begin{pmatrix}v_{1}\\\
v_{2}\end{pmatrix}\\\
=&(-\partial_{x_{2}},\partial_{x_{1}})\left(-\frac{1}{k^{2}+x_{1}^{2}+x_{2}^{2}}\begin{pmatrix}x_{1}x_{2}&-k^{2}-x_{1}^{2}\\\
k^{2}+x_{2}^{2}&-x_{1}x_{2}\end{pmatrix}\begin{pmatrix}\partial_{x_{1}}\varphi\\\
\partial_{x_{2}}\varphi\end{pmatrix}\right)\\\
=&-(\partial_{x_{1}},\partial_{x_{2}})\left(\frac{1}{k^{2}+x_{1}^{2}+x_{2}^{2}}\begin{pmatrix}k^{2}+x_{2}^{2}&-x_{1}x_{2}\\\
-x_{1}x_{2}&k^{2}+x_{1}^{2}\end{pmatrix}\begin{pmatrix}\partial_{x_{1}}\varphi\\\
\partial_{x_{2}}\varphi\end{pmatrix}\right)\\\
=&\mathcal{L}_{H}\varphi,\end{split}$ (2.7)

where $\mathcal{L}_{H}\varphi=-\text{div}(K_{H}(x_{1},x_{2})\nabla\varphi)$ is
a second order elliptic operator of divergence type with the coefficient
matrix

$K_{H}(x_{1},x_{2})=\frac{1}{k^{2}+x_{1}^{2}+x_{2}^{2}}\begin{pmatrix}k^{2}+x_{2}^{2}&-x_{1}x_{2}\\\
-x_{1}x_{2}&k^{2}+x_{1}^{2}\end{pmatrix}.$ (2.8)

Clearly from the definition of the matrix $K_{H}$, one has

  1. (1).

$K_{H}$ is a positive definite matrix and $(K_{H}(x))_{ij}\in
C^{\infty}(\overline{B_{R^{*}}(0)})$ for $i,j=1,2.$

  2. (2).

$\mathcal{L}_{H}$ is uniformly elliptic, namely,
$\lambda_{1}=1,\lambda_{2}=\frac{k^{2}}{k^{2}+|x|^{2}}$ are two eigenvalues of
$K_{H}$ which have positive lower and upper bounds.

From (2.3), (2.5) and (2.6), one has

$\begin{split}0=&\partial_{t}w+v_{1}\partial_{x_{1}}w+v_{2}\partial_{x_{2}}w+v_{3}\partial_{x_{3}}w\\\
=&\partial_{t}w+v_{1}\partial_{x_{1}}w+v_{2}\partial_{x_{2}}w+\frac{1}{k}(-x_{2}v_{1}+x_{1}v_{2})\cdot\frac{1}{k}(-x_{2}\partial_{x_{1}}w+x_{1}\partial_{x_{2}}w)\\\
=&\partial_{t}w+\frac{1}{k^{2}}(v_{1},v_{2})\begin{pmatrix}k^{2}+x_{2}^{2}&-x_{1}x_{2}\\\
-x_{1}x_{2}&k^{2}+x_{1}^{2}\end{pmatrix}\begin{pmatrix}\partial_{x_{1}}w\\\
\partial_{x_{2}}w\end{pmatrix}\\\
=&\partial_{t}w-\frac{1}{k^{2}(k^{2}+x_{1}^{2}+x_{2}^{2})}(\partial_{x_{1}}\varphi,\partial_{x_{2}}\varphi)\begin{pmatrix}x_{1}x_{2}&k^{2}+x_{2}^{2}\\\
-k^{2}-x_{1}^{2}&-x_{1}x_{2}\end{pmatrix}\begin{pmatrix}k^{2}+x_{2}^{2}&-x_{1}x_{2}\\\
-x_{1}x_{2}&k^{2}+x_{1}^{2}\end{pmatrix}\begin{pmatrix}\partial_{x_{1}}w\\\
\partial_{x_{2}}w\end{pmatrix}\\\
=&\partial_{t}w-(\partial_{x_{1}}\varphi,\partial_{x_{2}}\varphi)\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\begin{pmatrix}\partial_{x_{1}}w\\\
\partial_{x_{2}}w\end{pmatrix}\\\
=&\partial_{t}w+\partial_{x_{2}}\varphi\partial_{x_{1}}w-\partial_{x_{1}}\varphi\partial_{x_{2}}w\\\
=&\partial_{t}w+\nabla^{\perp}\varphi\cdot\nabla w,\end{split}$ (2.9)

where $\perp$ denotes the clockwise rotation through $\pi/2$, i.e.,
$(a,b)^{\perp}=(b,-a)$. As for the boundary condition of $\varphi$, it follows
from $\mathbf{v}\cdot\mathbf{n}=0$ on $\partial B_{R^{*}}(0)\times\mathbb{R}$
that (see (2.66), [17]) $\varphi$ is a constant on $\partial B_{R^{*}}(0).$
Without loss of generality, we set $\varphi|_{\partial B_{R^{*}}(0)}=0.$ Thus
the 2D vorticity-stream equations of (1.2) in $B_{R^{*}}(0)\times\mathbb{R}$
is

$\begin{cases}\partial_{t}w+\nabla^{\perp}\varphi\cdot\nabla w=0,\ \
&\text{in}\ B_{R^{*}}(0),\\\ w=\mathcal{L}_{H}\varphi,\ \ &\text{in}\
B_{R^{*}}(0),\\\ \varphi=0,\ \ &\text{on}\ \partial B_{R^{*}}(0).\end{cases}$
(2.10)

For a solution pair $(w,\varphi)$ of (2.10), one can recover left-handed
helical velocity field $\mathbf{v}$ and vorticity field by $\mathbf{w}$ of
(1.2) by using (2.6), (2.5),
$\mathbf{v}(x,t)=\bar{Q}_{\frac{x_{3}}{k}}\mathbf{v}\left(H_{-\frac{x_{3}}{k}}(x),t\right)$
and (2.2).

Let $\alpha$ be a constant. To construct traveling-rotating helical solutions
of (1.2), we look for solutions of (2.10) being of the form

$w(x^{\prime},t)=W(\bar{R}_{-\alpha|\ln\varepsilon|t}(x^{\prime})),\ \
\varphi(x^{\prime},t)=\varPhi(\bar{R}_{-\alpha|\ln\varepsilon|t}(x^{\prime})),$
(2.11)

where $x^{\prime}=(x_{1},x_{2})\in B_{R^{*}}(0)$. Then one computes directly
that $(W,\varPhi)$ satisfies

$\begin{cases}\nabla
W\cdot\nabla^{\perp}\left(\varPhi-\frac{\alpha}{2}|x^{\prime}|^{2}|\ln\varepsilon|\right)=0,\\\
W=\mathcal{L}_{H}\varPhi,\\\ \varPhi|_{\partial B_{R^{*}}(0)}=0.\end{cases}$
(2.12)

So formally if

$\mathcal{L}_{H}\varPhi=W=f_{\varepsilon}\left(\varPhi-\frac{\alpha}{2}|x^{\prime}|^{2}|\ln\varepsilon|\right)\
\ \text{in}\ B_{R^{*}}(0),$ (2.13)

for some function $f_{\varepsilon}$, then (2.12) automatically holds. In the
sequel we write $x^{\prime}$ as $x=(x_{1},x_{2})$ and look for solutions of a
semilinear elliptic equations

$\begin{cases}-\text{div}\cdot(K_{H}(x)\nabla\varPhi)=\frac{1}{\varepsilon^{2}}\left(\varPhi-\left(\frac{\alpha}{2}|x|^{2}+\beta\right)|\ln\varepsilon|\right)^{p}_{+},\
&x\in B_{R^{*}}(0),\\\ \varPhi(x)=0,\ &x\in\partial B_{R^{*}}(0),\end{cases}$
(2.14)

where $p>1$, $\alpha,\beta$ are constants to be determined later. For a
solution $\varPhi$ of (2.14), one can get a rotating-invariant solution pair
$(w,\varphi)$ of (2.10) with angular velocity $\alpha|\ln\varepsilon|$ by
simply using (2.13) and (2.11).

Inspired by equations (2.14), let us study the existence of solutions
concentrating around a couple of points to a more general model

$\begin{cases}-\varepsilon^{2}\text{div}(K(x)\nabla
u)=(u-q|\ln\varepsilon|)^{p}_{+},\ \ &x\in\Omega,\\\ u=0,\ \
&x\in\partial\Omega,\end{cases}$ (2.15)

where $\Omega\subset\mathbb{R}^{2}$ is a simply-connected bounded domain with
smooth boundary, $\varepsilon\in(0,1)$ and $p>1$. $K=(K_{i,j})_{2\times 2}$ is
a positive definite matrix satisfying

  1. ($\mathcal{K}$1).

$K_{i,j}(x)\in C^{\infty}(\overline{\Omega})$ for $1\leq i,j\leq 2.$

  2. ($\mathcal{K}$2).

$-\text{div}(K(x)\nabla\cdot)$ is a uniformly elliptic operator, that is,
there exist $\Lambda_{1},\Lambda_{2}>0$ such that

$\Lambda_{1}|\zeta|^{2}\leq(K(x)\zeta|\zeta)\leq\Lambda_{2}|\zeta|^{2},\ \ \ \
\forall\ x\in\Omega,\ \zeta\in\mathbb{R}^{2}.$

$q(x)$ is a function defined in $\overline{\Omega}$ satisfying

  1. (Q1).

$q(x)\in C^{\infty}(\overline{\Omega})$ and $q(x)>0$ for any
$x\in\overline{\Omega}.$

Denote $det(K)$ the determinant of $K$.

######  Theorem 2.2.

Let $K$ satisfy $(\mathcal{K}1)$-$(\mathcal{K}2)$ and $q$ satisfy $(Q1)$.
Then, for any given $m$ distinct strict local minimum (maximum) points
$x_{0,j}(j=1,\cdots,m)$ of $q^{2}\sqrt{det(K)}$ in $\Omega$, there exists
$\varepsilon_{0}>0$, such that for every $\varepsilon\in(0,\varepsilon_{0}]$,
(2.15) has a solution $u_{\varepsilon}$. Moreover, the following properties
hold

  1. (1)

Define the set
$\bar{A}_{\varepsilon,i}=\left\\{u_{\varepsilon}>q\ln\frac{1}{\varepsilon}\right\\}\cap
B_{\bar{\rho}}(x_{0,i})$, where $\bar{\rho}$ is small. Then there exist
$(z_{1,\varepsilon},\cdots,z_{m,\varepsilon})$ and $R_{1},R_{2}>0$ independent
of $\varepsilon$ satisfying

$\lim_{\varepsilon\to
0}(z_{1,\varepsilon},\cdots,z_{m,\varepsilon})=(x_{0,1},\cdots,x_{0,m}),$
$B_{R_{1}\varepsilon}(z_{i,\varepsilon})\subseteq\bar{A}_{\varepsilon,i}\subseteq
B_{R_{2}\varepsilon}(z_{i,\varepsilon}).$

  2. (2)

Define
$\kappa_{i}(u_{\varepsilon})=\frac{1}{\varepsilon^{2}}\int_{B_{\bar{\rho}}(x_{0,i})}\left(u_{\varepsilon}-q\ln\frac{1}{\varepsilon}\right)^{p}_{+}dx.$
Then

$\lim_{\varepsilon\to 0}\kappa_{i}(u_{\varepsilon})=2\pi
q\sqrt{det(K)}(x_{0,i}).$

To obtain good estimates for approximate solutions, we change (2.15) to the
following equivalent problem. Set
$\delta=\varepsilon|\ln\varepsilon|^{-\frac{p-1}{2}}$ and
$u=|\ln\varepsilon|w$, then (2.15) becomes

$\begin{cases}-\delta^{2}\text{div}(K(x)\nabla w)=(w-q)^{p}_{+},\ \
&x\in\Omega,\\\ w=0,\ \ &x\in\partial\Omega.\end{cases}$ (2.16)

We will construct multi-peak solutions of (2.16) in sections 3-5.

######  Remark 2.3.

Results of Theorem 2.2 can be regarded as a generalization of the
desingularization of classical planar vortex case (see [30, 37]) and the
vortex ring case (see [15]). Note that the cases of planar vortices and vortex
rings correspond to the coefficient matrix $K_{H}(x)=Id$ and
$\frac{1}{x_{1}}Id$, respectively. In [15], by considering solutions of

$\begin{cases}-\text{div}\left(\frac{1}{b}\nabla
u\right)=\frac{1}{\varepsilon^{2}}b\left(u-q\ln\frac{1}{\varepsilon}\right)^{p-1}_{+},\
\ &x\in\Omega,\\\ u=0,\ \ &x\in\partial\Omega,\end{cases}$

where $b$ is a scalar function, the authors constructed a family of $C^{1}$
solutions $u_{\varepsilon}$ with nonvanishing circulation concentrating near a
minimizer of $q^{2}/b$ as $\varepsilon\to 0$. Indeed, if we choose
$K_{H}(x)=\frac{1}{b}Id$, then by Theorem 2.2 solutions will shrink to
minimizers of $q^{2}\sqrt{det(K_{H})}=q^{2}/b$, which coincides with the
results in [15].

######  Remark 2.4.

Recently, [8] considered desingularization of steady solutions to 3D Euler
equation (1.2) with helical symmetry in helical domains. Using the critical
point theory and the estimates of capacity, [8] proved the existence and
asymptotic behavior of ground state solutions of (2.15) concentrating near a
single point. While in this paper, by using finite-dimensional reduction
method, we construct multi-peak solutions concentrating near a collection of
given points, which extends the results in [8].

##  3\. Approximate solutions

Our aim is to solve the following equations

$\begin{cases}-\delta^{2}\text{div}(K(x)\nabla w)=(w-q)^{p}_{+},\ \
&\text{in}\ \Omega,\\\ w=0,\ \ &\text{on}\ \partial\Omega.\end{cases}$

Note that $-\text{div}(K(x)\nabla\cdot)$ is a uniformly elliptic operator.
Throughout this paper, we denote $C,C_{1},C_{2}\cdots$ positive constants
independent of $\varepsilon$, whose values may change from line to line.

First, since $K$ is a $C^{\infty}$ positive definite matrix with all
eigenvalues having uniformly positive lower and upper bounds, by the Cholesky
decomposition one can find a matrix-valued function $T\in
C^{\infty}(\overline{\Omega})$ such that for any $x\in\Omega$, $T(x)$ is
invertible and

$(T(x)^{-1})(T(x)^{-1})^{t}=K(x).$ (3.1)

For simplicity, we denote $T_{x}=T(x)$.

Let $R>1$ be a large constant satisfying $\Omega\subseteq
T_{x}^{-1}(B_{R}(0))+x$ for any $x\in\Omega.$ Clearly by the positive
definiteness of $K$, such $R$ exists.

Consider

$\begin{cases}-\delta^{2}\Delta w=(w-a)^{p}_{+},\ \ &\text{in}\ B_{R}(0),\\\
w=0,\ \ &\text{on}\ \partial B_{R}(0),\end{cases}$ (3.2)

where $a>0$ is a constant. One computes directly that the unique $C^{1}$
positive solution of (3.2) is

$W_{\delta,a}(x)=\begin{cases}a+\delta^{\frac{2}{p-1}}s_{\delta}^{-\frac{2}{p-1}}\phi\left(\frac{|x|}{s_{\delta}}\right),\
\ &|x|\leq s_{\delta},\\\ a\ln\frac{|x|}{R}/\ln\frac{s_{\delta}}{R},\ \
&s_{\delta}\leq|x|\leq R,\end{cases}$

where $\phi\in H^{1}_{0}(B_{1}(0))$ satisfies

$-\Delta\phi=\phi^{p},\ \ \phi>0\ \ \text{in}\ B_{1}(0),$

and $s_{\delta}$ satisfy the relation

$\delta^{\frac{2}{p-1}}s_{\delta}^{-\frac{2}{p-1}}\phi^{\prime}(1)=a/\ln\frac{s_{\delta}}{R}.$
(3.3)

Hence (3.3) is uniquely solvable if $\delta>0$ is sufficiently small and

$\frac{s_{\delta}}{\delta|\ln\delta|^{\frac{p-1}{2}}}\to\left(\frac{|\phi^{\prime}(1)|}{a}\right)^{\frac{p-1}{2}}\
\ \ \ \text{as}\ \delta\to 0.$

The Pohazaev identity implies

$\int_{B_{1}(0)}\phi^{p+1}=\frac{\pi(p+1)}{2}|\phi^{\prime}(1)|^{2},\ \
\int_{B_{1}(0)}\phi^{p}=2\pi|\phi^{\prime}(1)|.$ (3.4)

Now for any $\hat{x}\in\Omega,\hat{q}>0$, let $V_{\delta,\hat{x},\hat{q}}$ be
a $C^{1}$ positive solution of the following equations

$\begin{cases}-\delta^{2}\text{div}(K(\hat{x})\nabla v)=(v-\hat{q})^{p}_{+},\
\ &\text{in}\ T_{\hat{x}}^{-1}(B_{R}(0)),\\\ v=0,\ \ &\text{on}\ \partial
T_{\hat{x}}^{-1}(B_{R}(0)).\end{cases}$ (3.5)

Thus one has $V_{\delta,\hat{x},\hat{q}}(x)=W_{\delta,\hat{q}}(T_{\hat{x}}x)$.
Indeed, let $u(x)=v(T_{\hat{x}}^{-1}x)\in H^{1}_{0}(B_{R}(0))$. Then $u$
satisfies (3.2) with $a=\hat{q}.$ So $u=W_{\delta,\hat{q}}$, which implies
that $v(x)=W_{\delta,\hat{q}}(T_{\hat{x}}x)$. Clearly
$V_{\delta,\hat{x},\hat{q}}$ has an explicit profile

$V_{\delta,\hat{x},\hat{q}}(x)=\begin{cases}\hat{q}+\delta^{\frac{2}{p-1}}s_{\delta}^{-\frac{2}{p-1}}\phi\left(\frac{|T_{\hat{x}}x|}{s_{\delta}}\right),\
\ &|T_{\hat{x}}x|\leq s_{\delta},\\\
\hat{q}\ln\frac{|T_{\hat{x}}x|}{R}/\ln\frac{s_{\delta}}{R},\ \
&s_{\delta}\leq|T_{\hat{x}}x|\leq R.\end{cases}$

For any $z\in\Omega,$ define

$V_{\delta,\hat{x},\hat{q},z}(x):=V_{\delta,\hat{x},\hat{q}}(x-z),\ \ \forall
x\in\Omega.$

Since $V_{\delta,\hat{x},\hat{q},z}$ is not 0 on $\partial\Omega$, we need to
make a projection of $V_{\delta,\hat{x},\hat{q},z}$ on $H^{1}_{0}(\Omega)$.
Let $PV_{\delta,\hat{x},\hat{q},z}$ be a solution of

$\begin{cases}-\delta^{2}\text{div}(K(\hat{x})\nabla
v)=(V_{\delta,\hat{x},\hat{q},z}-\hat{q})^{p}_{+},\ \ &\text{in}\ \Omega,\\\
v=0,\ \ &\text{on}\ \partial\Omega.\end{cases}$ (3.6)

We claim that for $\delta$ sufficiently small,

$PV_{\delta,\hat{x},\hat{q},z}(x)=V_{\delta,\hat{x},\hat{q},z}(x)-\frac{\hat{q}}{\ln\frac{R}{s_{\delta}}}g_{\hat{x}}(T_{\hat{x}}x,T_{\hat{x}}z),\
\ \forall x\in\Omega,$ (3.7)

where $g_{\hat{x}}(x,y)=2\pi h_{\hat{x}}(x,y)+\ln R$ for any $x,y\in
T_{\hat{x}}(\Omega)$, and $h_{\hat{x}}(x,y)$ is the regular part of Green’s
function of $-\Delta$ on $T_{\hat{x}}(\Omega)$, namely for any $y\in
T_{\hat{x}}(\Omega)$,

$\begin{cases}-\Delta h_{\hat{x}}(x,y)=0,\ \ &x\in T_{\hat{x}}(\Omega),\\\
h_{\hat{x}}(x,y)=\frac{1}{2\pi}\ln\frac{1}{|x-y|},\ \ &x\in\partial
T_{\hat{x}}(\Omega).\end{cases}$ (3.8)

Note that the Green’s function $G_{\hat{x}}(x,y)$ of $-\Delta$ in
$T_{\hat{x}}(\Omega)$ with Dirichlet zero boundary condition has the
decomposition

$G_{\hat{x}}(x,y)=\frac{1}{2\pi}\ln\frac{1}{|x-y|}-h_{\hat{x}}(x,y),\ \
\forall x,y\in T_{\hat{x}}(\Omega).$ (3.9)

Indeed, by (3.5) and (3.6) one has

$\begin{cases}-\delta^{2}\text{div}(K(\hat{x})\nabla(V_{\delta,\hat{x},\hat{q},z}-PV_{\delta,\hat{x},\hat{q},z}))(x)=0,\
\ &x\in\Omega,\\\
V_{\delta,\hat{x},\hat{q},z}-PV_{\delta,\hat{x},\hat{q},z}=\hat{q}\ln\frac{|T_{\hat{x}}(x-z)|}{R}/\ln\frac{s_{\delta}}{R},\
\ &x\in\partial\Omega.\end{cases}$

Define
$\check{u}(y)=(V_{\delta,\hat{x},\hat{q},z}-PV_{\delta,\hat{x},\hat{q},z})(T_{\hat{x}}^{-1}y),y\in
T_{\hat{x}}(\Omega)$. Then

$\begin{cases}-\delta^{2}\Delta\check{u}(y)=0,\ \ &y\in
T_{\hat{x}}(\Omega),\\\
\check{u}(y)=\hat{q}\ln\frac{|y-T_{\hat{x}}z|}{R}/\ln\frac{s_{\delta}}{R},\ \
&y\in\partial T_{\hat{x}}(\Omega).\end{cases}$

So $\check{u}(y)=\frac{\hat{q}}{\ln\frac{s_{\delta}}{R}}(-2\pi
h_{\hat{x}}(y,T_{\hat{x}}z)-\ln R)$, which implies that for any $x\in\Omega$

$(V_{\delta,\hat{x},\hat{q},z}-PV_{\delta,\hat{x},\hat{q},z})(x)=\check{u}(T_{\hat{x}}x)=\frac{\hat{q}}{\ln\frac{s_{\delta}}{R}}(-2\pi
h_{\hat{x}}(T_{\hat{x}}x,T_{\hat{x}}z)-\ln
R)=\frac{\hat{q}}{\ln\frac{R}{s_{\delta}}}g_{\hat{x}}(T_{\hat{x}}x,T_{\hat{x}}z).$

We get (3.7).

In the following, we will construct solutions of the form

$\sum_{j=1}^{m}PV_{\delta,\hat{x}_{j},\hat{q}_{j},z_{j}}+\omega_{\delta},$

where $\Sigma_{j=1}^{m}PV_{\delta,\hat{x}_{j},\hat{q}_{j},z_{j}}$ is the main
term and $\omega_{\delta}$ is an error term. To make the norm of
$\omega_{\delta}$ as small as possible, we need to choose $\hat{x}_{j}$ and
$\hat{q}_{j}$ suitably close to $z_{j}$ and $q(z_{j})$.

Let $(x_{0,1},\cdots,x_{0,m})$ be $m$ distinct strict local maximum points (or
minimum points) of $q^{2}\sqrt{det(K)}$ in $\Omega$. Hence we can choose
$\bar{\rho}>0$ sufficiently small such that

$B_{\bar{\rho}}(x_{0,i})\Subset\Omega,\
\overline{B_{\bar{\rho}}(x_{0,i})}\cap\overline{B_{\bar{\rho}}(x_{0,i})}=\varnothing,\
\ \forall 1\leq i\neq j\leq m.$

Define the admissible set $\mathcal{M}\subseteq\mathbb{R}^{(2m)}$ satisfying

$\mathcal{M}=\\{Z=(z_{1},z_{2},\cdots,z_{m})\in\mathbb{R}^{(2m)}\mid z_{i}\in
B_{\bar{\rho}}(x_{0,i}),\ i=1,\cdots,m\\}.$ (3.10)

Let

$\hat{x}_{i}=z_{i}$

and $\hat{q}_{i}=\hat{q}_{\delta,i}(Z)$, $i=1,\cdots,m$, be the solution the
equations

$\hat{q}_{i}=q(z_{i})+\frac{\hat{q}_{i}}{\ln\frac{R}{\varepsilon}}g_{z_{i}}(T_{z_{i}}z_{i},T_{z_{i}}z_{i})-\Sigma_{j\neq
i}\frac{\hat{q}_{j}}{\ln\frac{R}{\varepsilon}}\bar{G}_{z_{j}}(T_{z_{j}}z_{i},T_{z_{j}}z_{j}),$
(3.11)

where $\bar{G}_{z_{j}}(x,y)=\ln\frac{R}{|x-y|}-g_{z_{j}}(x,y)=2\pi
G_{z_{j}}(x,y)$ for any $x,y\in T_{z_{j}}(\Omega)$.

It follows from Lemma A.4 in Appendix that for any $Z$ satisfying (3.10) and
$\delta$ sufficiently small, there exist $\hat{q}_{\delta,i}(Z)$ satisfying
(3.11). Moreover, one has

$\hat{q}_{i}=\frac{q(z_{i})-\Sigma_{j\neq
i}\frac{\hat{q}_{j}}{\ln\frac{R}{\varepsilon}}\bar{G}_{z_{j}}(T_{z_{j}}z_{i},T_{z_{j}}z_{j})}{1-\frac{1}{\ln\frac{R}{\varepsilon}}g_{z_{i}}(T_{z_{i}}z_{i},T_{z_{i}}z_{i})}.$

For $Z=(z_{1},\cdots,z_{m})$, denote

$V_{\delta,Z,j}=PV_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}},\ \
V_{\delta,Z}=\sum_{j=1}^{m}V_{\delta,Z,j}.$

Let $s_{\delta,j}$ satisfy

$\delta^{\frac{2}{p-1}}s_{\delta,j}^{-\frac{2}{p-1}}\phi^{\prime}(1)=\hat{q}_{\delta,j}/\ln\frac{s_{\delta,j}}{R}.$

Then one can easily verify that

$\frac{1}{\ln\frac{R}{s_{\delta,j}}}=\frac{1}{\ln\frac{R}{\varepsilon}}+O\left(\frac{\ln|\ln\varepsilon|}{|\ln\varepsilon|^{2}}\right).$
(3.12)

By the choice of $\hat{x}_{j},\hat{q}_{\delta,j}$, we claim that for any fixed
constant $L>0$ and $x\in B_{Ls_{\delta,i}}(z_{i})$,

$\begin{split}V_{\delta,Z}(x)-q(x)=V_{\delta,z_{i},\hat{q}_{\delta,i},z_{i}}(x)-\hat{q}_{\delta,i}+O\left(\frac{\ln|\ln\varepsilon|}{|\ln\varepsilon|^{2}}\right).\end{split}$
(3.13)

Indeed, we have for any $x\in B_{Ls_{\delta,i}}(z_{i})$,

$\begin{split}&V_{\delta,Z,i}(x)-q(x)\\\
=&V_{\delta,z_{i},\hat{q}_{\delta,i},z_{i}}(x)-\frac{\hat{q}_{\delta,i}}{\ln\frac{R}{s_{\delta,i}}}g_{z_{i}}(T_{z_{i}}x,T_{z_{i}}z_{i})-q(x)\\\
=&V_{\delta,z_{i},\hat{q}_{\delta,i},z_{i}}(x)-q(z_{i})-\frac{\hat{q}_{\delta,i}}{\ln\frac{R}{s_{\delta,i}}}g_{z_{i}}(T_{z_{i}}z_{i},T_{z_{i}}z_{i})+O(s_{\delta,i})+O\left(\frac{s_{\delta,i}|\nabla
g_{z_{i}}(T_{z_{i}}z_{i},T_{z_{i}}z_{i})|}{\ln\frac{R}{s_{\delta,i}}}\right)\\\
=&V_{\delta,z_{i},\hat{q}_{\delta,i},z_{i}}(x)-q(z_{i})-\frac{\hat{q}_{\delta,i}}{\ln\frac{R}{\varepsilon}}g_{z_{i}}(T_{z_{i}}z_{i},T_{z_{i}}z_{i})+O\left(\frac{\ln|\ln\varepsilon|}{|\ln\varepsilon|^{2}}\right),\end{split}$

and for any $j\neq i$, $x\in B_{Ls_{\delta,i}}(z_{i})$,

$\begin{split}V_{\delta,Z,j}(x)=&V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}(x)-\frac{\hat{q}_{\delta,j}}{\ln\frac{R}{s_{\delta,j}}}g_{z_{j}}(T_{z_{j}}x,T_{z_{j}}z_{j})\\\
=&\frac{\hat{q}_{\delta,j}}{\ln\frac{R}{s_{\delta,j}}}\bar{G}_{z_{j}}(T_{z_{j}}x,T_{z_{j}}z_{j})\\\
=&\frac{\hat{q}_{\delta,j}}{\ln\frac{R}{s_{\delta,j}}}\bar{G}_{z_{j}}(T_{z_{j}}z_{i},T_{z_{j}}z_{j})+O\left(\frac{s_{\delta,j}|\nabla\bar{G}_{z_{j}}(T_{z_{j}}z_{i},T_{z_{j}}z_{j})|}{\ln\frac{R}{s_{\delta,j}}}\right)\\\
=&\frac{\hat{q}_{\delta,j}}{\ln\frac{R}{\varepsilon}}\bar{G}_{z_{j}}(T_{z_{j}}z_{i},T_{z_{j}}z_{j})+O\left(\frac{\ln|\ln\varepsilon|}{|\ln\varepsilon|^{2}}\bar{G}_{z_{j}}(T_{z_{j}}z_{i},T_{z_{j}}z_{j})\right)\\\
=&\frac{\hat{q}_{\delta,j}}{\ln\frac{R}{\varepsilon}}\bar{G}_{z_{j}}(T_{z_{j}}z_{i},T_{z_{j}}z_{j})+O\left(\frac{\ln|\ln\varepsilon|}{|\ln\varepsilon|^{2}}\right),\end{split}$

where we have used (3.12) and Lemma A.3 in Appendix. Adding up the above
inequalities and using (3.11), we get

$\begin{split}V_{\delta,Z}(x)-q(x)=V_{\delta,z_{i},\hat{q}_{\delta,i},z_{i}}(x)-\hat{q}_{\delta,i}+O\left(\frac{\ln|\ln\varepsilon|}{|\ln\varepsilon|^{2}}\right),\
\ \forall x\in B_{Ls_{\delta,i}}(z_{i}).\end{split}$

From (3.3), (3.11) and Lemma A.3, we get

$\frac{\partial\hat{q}_{\delta,i}}{\partial z_{i,h}}=O(1),$ (3.14)
$\frac{\partial s_{\delta,i}}{\partial
z_{i,h}}=O(\delta|\ln\delta|^{\frac{p-1}{2}}).$ (3.15)

Using the definition of $V_{\delta,z_{i},\hat{q}_{\delta,i},z_{i}}$, (3.14)
and (3.15), we obtain

$\begin{split}\frac{\partial
V_{\delta,z_{i},\hat{q}_{\delta,i},z_{i}}(x)}{\partial
z_{i,h}}=\begin{cases}-\frac{1}{s_{\delta,i}}(\frac{\delta}{s_{\delta,i}})^{\frac{2}{p-1}}\phi^{\prime}(\frac{|T_{z_{i}}(x-z_{i})|}{s_{\delta,i}})\frac{(T_{z_{i}})_{h}^{t}\cdot
T_{z_{i}}(x-z_{i})}{|T_{z_{i}}(x-z_{i})|}+O(1),\ \ &|T_{z_{i}}(x-z_{i})|\leq
s_{\delta,i},\\\
\frac{\hat{q}_{\delta,i}}{\ln\frac{R}{s_{\delta,i}}}\frac{(T_{z_{i}})_{h}^{t}\cdot
T_{z_{i}}(x-z_{i})}{|T_{z_{i}}(x-z_{i})|^{2}}+O\left(\frac{\ln\frac{R}{|T_{z_{i}}(x-z_{i})|}}{\ln\frac{R}{s_{\delta,i}}}\right),\
\ &|T_{z_{i}}(x-z_{i})|>s_{\delta,i},\end{cases}\end{split}$ (3.16)

where $(T_{z_{i}})_{h}^{t}$ is the h-th row of $(T_{z_{i}})^{t}$.

##  4\. The reduction

Now we find solution of (2.16) being of the form

$V_{\delta,Z}+\omega_{\delta}.$

First we prove that for any $Z$ satisfying (3.10), there exists
$\omega_{\delta,Z}$ such that $V_{\delta,Z}+\omega_{\delta,Z}$ solves (2.16)
in a co-dimensional $2m$ subspace of $H^{1}_{0}$. In the next section we
choose proper $Z=Z(\delta)$ such that $V_{\delta,Z}+\omega_{\delta}$ is a
solution.

Let us consider the following equation

$-\Delta w=w^{p}_{+},\ \ \text{in}\ \mathbb{R}^{2}.$ (4.1)

The unique $C^{1}$ solution is

$w(x)=\begin{cases}\phi(x),\ \ &|x|\leq 1,\\\ \phi^{\prime}(1)\ln|x|,\ \
&|x|>1.\end{cases}$

By the classical elliptic equation theory, $w\in C^{2,\alpha}(\mathbb{R}^{2})$
for any $\alpha\in(0,1)$. The linearized equation of (4.1) at $w$ is

$-\Delta v-pw^{p-1}_{+}v=0,\ \ v\in L^{\infty}(\mathbb{R}^{2}).$ (4.2)

Clearly, $\frac{\partial w}{\partial x_{h}}$ $(h=1,2)$ are solutions of (4.2).
It follows from [9] (see also [6]) that

######  Proposition 4.1 (Non-degeneracy).

$w$ is non-degenerate, i.e., the kernel of the linearized equation (4.2) is

$span\\{\frac{\partial w}{\partial x_{1}},\frac{\partial w}{\partial
x_{2}}\\}.$

Denote

$F_{\delta,Z}=\\{u\in L^{p}(\Omega)\mid\int_{\Omega}\frac{\partial
V_{\delta,Z,j}}{\partial z_{j,h}}u=0,\ \ \forall j=1,\cdots,m,\ h=1,2\\},$
(4.3)

and

$E_{\delta,Z}=\\{u\in W^{2,p}\cap
H^{1}_{0}(\Omega)\mid\int_{\Omega}\text{div}(K(x)\nabla\frac{\partial
V_{\delta,Z,j}}{\partial z_{j,h}})u=0,\ \ \forall j=1,\cdots,m,\ h=1,2\\}.$
(4.4)

So $F_{\delta,Z}$ and $E_{\delta,Z}$ are co-dimensional $2m$ subspaces of
$L^{p}$ and $W^{2,p}\cap H^{1}_{0}(\Omega)$, respectively.

For any $u\in L^{p}(\Omega)$, define the projection operator
$Q_{\delta}:L^{p}\to F_{\delta,Z}$

$Q_{\delta}u:=u-\sum_{j=1}^{m}\sum_{h=1}^{2}C_{j,h}\frac{\partial}{\partial
z_{j,h}}(-\delta^{2}\text{div}(K(z_{j})\nabla V_{\delta,Z,j})),$ (4.5)

where $C_{j,h}(j=1,\cdots,m,\ h=1,2)$ satisfies

$\sum_{j=1}^{m}\sum_{h=1}^{2}C_{j,h}\int_{\Omega}\frac{\partial}{\partial
z_{j,h}}(-\delta^{2}\text{div}(K(z_{j})\nabla V_{\delta,Z,j}))\frac{\partial
V_{\delta,Z,i}}{\partial z_{i,\hbar}}=\int_{\Omega}u\frac{\partial
V_{\delta,Z,i}}{\partial z_{i,\hbar}},\ \ \forall i=1,\cdots,m,\ \hbar=1,2.$
(4.6)

By Lemma A.5, we know that $Q_{\delta}$ is a well-defined linear operator from
$L^{p}$ to $F_{\delta,Z}$. Indeed, using (3.16) and Lemma A.3, the coefficient
matrix

$\begin{split}&\int_{\Omega}\frac{\partial}{\partial
z_{j,h}}(-\delta^{2}\text{div}(K(z_{j})\nabla V_{\delta,Z,j}))\frac{\partial
V_{\delta,Z,i}}{\partial z_{i,\hbar}}\\\
=&p\int_{\Omega}(V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}-\hat{q}_{\delta,j})^{p-1}_{+}\left(\frac{\partial
V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}}{\partial
z_{j,h}}-\frac{\partial\hat{q}_{\delta,j}}{\partial
z_{j,h}}\right)\frac{\partial V_{\delta,Z,i}}{\partial z_{i,\hbar}}\\\
=&p\int_{\Omega}(V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}-\hat{q}_{\delta,j})^{p-1}_{+}\frac{\partial
V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}}{\partial z_{j,h}}\frac{\partial
V_{\delta,z_{i},\hat{q}_{\delta,i},z_{i}}}{\partial
z_{i,\hbar}}+O\left(\frac{\varepsilon}{|\ln\varepsilon|^{p}}\right)\\\
=&\delta_{i,j}\frac{(M_{i})_{h,\hbar}}{|\ln\varepsilon|^{p+1}}+O\left(\frac{\varepsilon}{|\ln\varepsilon|^{p}}\right),\end{split}$
(4.7)

where $\delta_{i,j}=1$ if $i=j$; otherwise, $\delta_{i,j}=0$. $M_{i}$ are $m$
positive definite matrices and there exist positive constants
$\bar{c}_{1},\bar{c}_{2}$ independent of $\delta,Z$ such that all eigenvalues
of $M_{i}$ belong to $(\bar{c}_{1},\bar{c}_{2})$. So there exists the unique
$C_{j,h}$ satisfying (4.6). Note that for any $u\in L^{p}$, $Q_{\delta}u\equiv
u$ in $\Omega\backslash\cup_{i=1}^{m}B_{Ls_{\delta,i}}(z_{i})$ for some $L>1$.

The linearized operator of (2.16) at $V_{\delta,Z}$ is

$L_{\delta}\omega:=-\delta^{2}\text{div}(K(x)\nabla\omega)-p(V_{\delta,Z}-q)^{p-1}_{+}\omega.$

We have the following estimates of $L_{\delta}$.

######  Lemma 4.2.

There exist $\rho_{0}>0,\delta_{1}>0$ such that for any
$\delta\in(0,\delta_{1}],Z$ satisfying (3.10), $u\in E_{\delta,Z}$ satisfying
$Q_{\delta}L_{\delta}u=0$ in
$\Omega\backslash\cup_{j=1}^{m}B_{Ls_{\delta,j}}(z_{j})$ for some $L>1$ large,
then

$||Q_{\delta}L_{\delta}u||_{L^{p}}\geq\frac{\rho_{0}\varepsilon^{\frac{2}{p}}}{|\ln\varepsilon|^{p-1}}||u||_{L^{\infty}}.$

###### Proof.

We argue by contradiction. Suppose that there are $\delta_{N}\to 0$,
$Z_{N}=(z_{N,1},\cdots,z_{N,m})\to(z_{1},\cdots,z_{m})$ satisfying (2.5) and
$u_{N}\in E_{\delta_{N},Z_{N}}$ with $Q_{\delta_{N}}L_{\delta_{N}}u_{N}=0$ in
$\Omega\backslash\cup_{j=1}^{m}B_{Ls_{\delta_{N},j}}(z_{N,j})$ for some $L$
large and $||u_{N}||_{L^{\infty}}=1$ such that

$||Q_{\delta_{N}}L_{\delta_{N}}u_{N}||_{L^{p}}\leq\frac{1}{N}\frac{\varepsilon_{N}^{\frac{2}{p}}}{|\ln\varepsilon_{N}|^{p-1}}.$

Let

$Q_{\delta_{N}}L_{\delta_{N}}u_{N}=L_{\delta_{N}}u_{N}-\sum_{j=1}^{m}\sum_{h=1}^{2}C_{j,h,N}\frac{\partial}{\partial
z_{j,h}}(-\delta_{N}^{2}\text{div}(K(z_{N,j})\nabla V_{\delta_{N},Z_{N},j})).$
(4.8)

We now estimate $C_{j,h,N}$. For fixed $i=1,\cdots,m,\hbar=1,2$, multiplying
(4.8) by $\frac{\partial V_{\delta_{N},Z_{N},i}}{\partial z_{i,\hbar}}$ and
integrating on $\Omega$ we get

$\begin{split}&\int_{\Omega}u_{N}L_{\delta_{N}}\left(\frac{\partial
V_{\delta_{N},Z_{N},i}}{\partial
z_{i,\hbar}}\right)=\int_{\Omega}L_{\delta_{N}}u_{N}\frac{\partial
V_{\delta_{N},Z_{N},i}}{\partial z_{i,\hbar}}\\\
=&\sum_{j=1}^{m}\sum_{h=1}^{2}C_{j,h,N}\int_{\Omega}\frac{\partial}{\partial
z_{j,h}}(-\delta_{N}^{2}\text{div}(K(z_{N,j})\nabla
V_{\delta_{N},Z_{N},j}))\frac{\partial V_{\delta_{N},Z_{N},i}}{\partial
z_{i,\hbar}}.\end{split}$

We estimate $\int_{\Omega}u_{N}L_{\delta_{N}}\left(\frac{\partial
V_{\delta_{N},Z_{N},i}}{\partial z_{i,\hbar}}\right).$ Note that

$\begin{split}&\int_{\Omega}u_{N}L_{\delta_{N}}\left(\frac{\partial
V_{\delta_{N},Z_{N},i}}{\partial z_{i,\hbar}}\right)\\\
=&\int_{\Omega}u_{N}\left[-\delta_{N}^{2}\text{div}(K(x)\nabla\frac{\partial
V_{\delta_{N},Z_{N},i}}{\partial
z_{i,\hbar}})-p(V_{\delta_{N},Z_{N}}-q)^{p-1}_{+}\frac{\partial
V_{\delta_{N},Z_{N},i}}{\partial z_{i,\hbar}}\right]\\\
=&\int_{\Omega}u_{N}\frac{\partial}{\partial
z_{i,\hbar}}(-\delta_{N}^{2}\text{div}(K(z_{N,i})\nabla
V_{\delta_{N},Z_{N},i}))-\int_{\Omega}u_{N}\left(-\delta_{N}^{2}\text{div}\left(\frac{\partial
K(z_{N,i})}{\partial z_{i,\hbar}}\nabla
V_{\delta_{N},Z_{N},i}\right)\right)\\\
&+\int_{\Omega}u_{N}\left(-\delta_{N}^{2}\text{div}\left(\left(K(x)-K(z_{N,i})\right)\nabla\frac{\partial
V_{\delta_{N},Z_{N},i}}{\partial
z_{i,\hbar}}\right)\right)-p\int_{\Omega}u_{N}(V_{\delta_{N},Z_{N}}-q)^{p-1}_{+}\frac{\partial
V_{\delta_{N},Z_{N},i}}{\partial z_{i,\hbar}}\\\
=:&I_{1}+I_{2}+I_{3}+I_{4},\end{split}$ (4.9)

where the matrix $\frac{\partial K(z_{i})}{\partial z_{i,\hbar}}$ is defined
by $\frac{\partial K(z_{i})}{\partial
z_{i,\hbar}}=\begin{pmatrix}\frac{\partial K_{1,1}(z_{i})}{\partial
z_{i,\hbar}}&\frac{\partial K_{1,2}(z_{i})}{\partial z_{i,\hbar}}\\\
\frac{\partial K_{2,1}(z_{i})}{\partial z_{i,\hbar}}&\frac{\partial
K_{2,2}(z_{i})}{\partial z_{i,\hbar}}\end{pmatrix}.$

By (3.13), (3.14), Lemma A.3 and Lemma A.6, one has

$\begin{split}&I_{1}+I_{4}\\\
=&p\int_{\Omega}u_{N}\left(V_{\delta_{N},z_{N,i},\hat{q}_{\delta_{N},i},z_{N,i}}-\hat{q}_{\delta_{N},i}\right)^{p-1}_{+}\left(\frac{\partial
V_{\delta_{N},z_{N,i},\hat{q}_{\delta_{N},i},z_{N,i}}}{\partial
z_{i,h}}-\frac{\partial\hat{q}_{\delta_{N},i}}{\partial z_{i,h}}\right)\\\
&-p\int_{\Omega}u_{N}\left(V_{\delta_{N},z_{N,i},\hat{q}_{\delta_{N},i},z_{N,i}}-\hat{q}_{\delta_{N},i}+O\left(\frac{\ln|\ln\varepsilon_{N}|}{|\ln\varepsilon_{N}|^{2}}\right)\right)^{p-1}_{+}\frac{\partial
V_{\delta_{N},Z_{N},i}}{\partial z_{i,\hbar}}\\\
=&p\int_{\Omega}u_{N}\left(V_{\delta_{N},z_{N,i},\hat{q}_{\delta_{N},i},z_{N,i}}-\hat{q}_{\delta_{N},i}\right)^{p-1}_{+}\frac{\partial
V_{\delta_{N},z_{N,i},\hat{q}_{\delta_{N},i},z_{N,i}}}{\partial z_{i,h}}\\\
&-p\int_{\Omega}u_{N}\left(V_{\delta_{N},z_{N,i},\hat{q}_{\delta_{N},i},z_{N,i}}-\hat{q}_{\delta_{N},i}+O\left(\frac{\ln|\ln\varepsilon_{N}|}{|\ln\varepsilon_{N}|^{2}}\right)\right)^{p-1}_{+}\frac{\partial
V_{\delta_{N},z_{N,i},\hat{q}_{\delta_{N},i},z_{N,i}}}{\partial
z_{i,h}}+O\left(\frac{\varepsilon_{N}^{2}}{|\ln\varepsilon_{N}|^{p-1}}\right)\\\
=&O\left(\frac{\varepsilon_{N}\ln|\ln\varepsilon_{N}|}{|\ln\varepsilon_{N}|^{p+1}}\right).\end{split}$
(4.10)

For $I_{2}$ and $I_{3}$, direct computation shows that

$\begin{split}I_{2}=&\int_{\Omega}u_{N}\left(\delta_{N}^{2}\text{div}\left(\frac{\partial
K(z_{N,i})}{\partial z_{i,\hbar}}\nabla
V_{\delta_{N},Z_{N},i}\right)\right)\\\
=&\int_{\Omega}u_{N}\left(\delta_{N}^{2}\text{div}\left(\frac{\partial
K(z_{N,i})}{\partial z_{i,\hbar}}\nabla
V_{\delta_{N},z_{N,i},\hat{q}_{\delta_{N},i},z_{N,i}}\right)\right)+O\left(\frac{\delta_{N}^{2}}{|\ln\varepsilon_{N}|}\right)\\\
=&\int_{|T_{z_{N,i}}(x-z_{N,i})|\leq
s_{\delta_{N},i}}+\int_{|T_{z_{N,i}}(x-z_{N,i})|>s_{\delta_{N},i}}u_{N}\left(\delta_{N}^{2}\text{div}\left(\frac{\partial
K(z_{N,i})}{\partial z_{i,\hbar}}\nabla
V_{\delta_{N},z_{N,i},\hat{q}_{\delta_{N},i},z_{N,i}}\right)\right)+O\left(\frac{\delta_{N}^{2}}{|\ln\varepsilon_{N}|}\right)\\\
=&O\left(\frac{\delta_{N}^{2}}{|\ln\varepsilon_{N}|}\right)+O(\delta_{N}^{2})+O\left(\frac{\delta_{N}^{2}}{|\ln\varepsilon_{N}|}\right)\\\
=&O\left(\frac{\varepsilon_{N}^{2}}{|\ln\varepsilon_{N}|^{p-1}}\right),\end{split}$
(4.11)

and

$\begin{split}I_{3}=&\int_{\Omega}u_{N}\left(-\delta_{N}^{2}\text{div}\left((K(x)-K(z_{N,i}))\nabla\frac{\partial
V_{\delta_{N},Z_{N},i}}{\partial z_{i,\hbar}}\right)\right)\\\
=&\int_{\Omega}u_{N}\left(-\delta_{N}^{2}\text{div}\left((K(x)-K(z_{N,i}))\nabla\frac{\partial
V_{\delta_{N},z_{N,i},\hat{q}_{\delta_{N},i},z_{N,i}}}{\partial
z_{i,\hbar}}\right)\right)+O\left(\frac{\delta_{N}^{2}}{|\ln\varepsilon_{N}|}\right)\\\
=&\int_{|T_{z_{N,i}}(x-z_{N,i})|\leq
s_{\delta_{N},i}}+\int_{s_{\delta_{N},i}<|T_{z_{N,i}}(x-z_{N,i})|\leq\mu}+\int_{|T_{z_{N,i}}(x-z_{N,i})|>\mu}\\\
&u_{N}\left(-\delta_{N}^{2}\text{div}\left((K(x)-K(z_{N,i}))\nabla\frac{\partial
V_{\delta_{N},z_{N,i},\hat{q}_{\delta_{N},i},z_{N,i}}}{\partial
z_{i,\hbar}}\right)\right)+O\left(\frac{\delta_{N}^{2}}{|\ln\varepsilon_{N}|}\right)\\\
=&O\left(\frac{\delta_{N}^{2}}{|\ln\varepsilon_{N}|}\right)+O(\delta_{N}^{2})+O\left(\frac{\delta_{N}^{2}}{|\ln\varepsilon_{N}|}\right)\\\
=&O\left(\frac{\varepsilon_{N}^{2}}{|\ln\varepsilon_{N}|^{p-1}}\right).\end{split}$
(4.12)

Here $\mu>0$ is a small constant, and we have used (3.16), Lemmas A.3 and A.6.
Taking (4.10), (4.11) and (4.12) into (4.9), we get

$\begin{split}&\int_{\Omega}u_{N}L_{\delta_{N}}\left(\frac{\partial
V_{\delta_{N},Z_{N},i}}{\partial
z_{i,\hbar}}\right)=O\left(\frac{\varepsilon_{N}\ln|\ln\varepsilon_{N}|}{|\ln\varepsilon_{N}|^{p+1}}\right).\end{split}$

Combining with (4.7) we get

$C_{j,h,N}=O(\varepsilon_{N}\ln|\ln\varepsilon_{N}|).$

Hence

$\begin{split}&\sum_{j=1}^{m}\sum_{h=1}^{2}C_{j,h,N}\frac{\partial}{\partial
z_{j,h}}(-\delta_{N}^{2}\text{div}(K(z_{N,j})\nabla
V_{\delta_{N},Z_{N},j}))\\\
=&p\sum_{j=1}^{m}\sum_{h=1}^{2}C_{j,h,N}(V_{\delta_{N},z_{N,j},\hat{q}_{\delta_{N},j},z_{N,j}}-\hat{q}_{\delta_{N},j})^{p-1}_{+}\left(\frac{\partial
V_{\delta_{N},z_{N,j},\hat{q}_{\delta_{N},j},z_{N,j}}}{\partial
z_{j,h}}-\frac{\partial\hat{q}_{\delta_{N},j}}{\partial z_{j,h}}\right)\\\
=&O\left(\sum_{j=1}^{m}\sum_{h=1}^{2}\frac{\varepsilon_{N}^{\frac{2}{p}-1}|C_{j,h,N}|}{|\ln\varepsilon_{N}|^{p}}\right)+O\left(\sum_{j=1}^{m}\sum_{h=1}^{2}\frac{\varepsilon_{N}^{\frac{2}{p}}|C_{j,h,N}|}{|\ln\varepsilon_{N}|^{p}}\right)\\\
=&O\left(\frac{\varepsilon_{N}^{\frac{2}{p}}\ln|\ln\varepsilon_{N}|}{|\ln\varepsilon_{N}|^{p}}\right),\
\ \ \ \text{in}\ L^{p}(\Omega).\end{split}$

So by the assumption and (4.8) we have

$\begin{split}L_{\delta_{N}}u_{N}=&Q_{\delta_{N}}L_{\delta_{N}}u_{N}+\sum_{j=1}^{m}\sum_{h=1}^{2}C_{j,h,N}\frac{\partial}{\partial
z_{j,h}}(-\delta_{N}^{2}\text{div}(K(z_{N,j})\nabla
V_{\delta_{N},Z_{N},j}))\\\
=&O\left(\frac{1}{N}\frac{\varepsilon_{N}^{\frac{2}{p}}}{|\ln\varepsilon_{N}|^{p-1}}\right)+O\left(\frac{\varepsilon_{N}^{\frac{2}{p}}\ln|\ln\varepsilon_{N}|}{|\ln\varepsilon_{N}|^{p}}\right)\\\
=&o\left(\frac{\varepsilon_{N}^{\frac{2}{p}}}{|\ln\varepsilon_{N}|^{p-1}}\right),\
\ \ \ \text{in}\ L^{p}(\Omega).\end{split}$

For any fixed $i,$ define
$\tilde{u}_{N,i}(y)=u_{N}(s_{\delta_{N},i}y+z_{N,i})$ for
$y\in\Omega_{N,i}:=\\{y\in\mathbb{R}^{2}\mid
s_{\delta_{N},i}y+z_{N,i}\in\Omega\\}$. Define

$\tilde{L}_{N,i}u=-\text{div}(K(s_{\delta_{N},i}y+z_{N,i})\nabla
u)-p\frac{s_{\delta_{N},i}^{2}}{\delta_{N}^{2}}(V_{\delta_{N},Z_{N}}(s_{\delta_{N},i}y+z_{N,i})-q(s_{\delta_{N},i}y+z_{N,i}))^{p-1}_{+}u.$

Then direct computation shows that

$\begin{split}||\tilde{L}_{N,i}\tilde{u}_{N,i}||_{L^{p}(\Omega_{N,i})}=&\left(\int_{\Omega_{N,i}}\left(\tilde{L}_{N,i}\tilde{u}_{N,i}\right)^{p}dy\right)^{\frac{1}{p}}\\\
=&\left(\frac{1}{s_{\delta_{N},i}^{2}}\int_{\Omega}\left(-s_{\delta_{N},i}^{2}\text{div}(K(x)\nabla
u_{N})-p\frac{s_{\delta_{N},i}^{2}}{\delta_{N}^{2}}(V_{\delta_{N},Z_{N}}-q)^{p-1}_{+}u_{N}\right)^{p}dx\right)^{\frac{1}{p}}\\\
=&\frac{s_{\delta_{N},i}^{2}}{s_{\delta_{N},i}^{\frac{2}{p}}\delta_{N}^{2}}||L_{\delta_{N}}u_{N}||_{L^{p}(\Omega)},\end{split}$

i.e.,
$s_{\delta_{N},i}^{\frac{2}{p}}\cdot\frac{\delta_{N}^{2}}{s_{\delta_{N},i}^{2}}||\tilde{L}_{N,i}\tilde{u}_{N,i}||_{L^{p}(\Omega_{N,i})}=||L_{\delta_{N}}u_{N}||_{L^{p}(\Omega)}.$

By the fact that
$\frac{\delta_{N}^{2}}{s_{\delta_{N},i}^{2}}=O(\frac{1}{|\ln\varepsilon_{N}|^{p-1}})$
and $s_{\delta_{N},i}=O(\varepsilon_{N})$, we have

$\tilde{L}_{N,i}\tilde{u}_{N,i}=o(1)\ \ \ \ \text{in}\ \ L^{p}(\Omega_{N,i}).$

Since $||\tilde{u}_{N,i}||_{L^{\infty}(\Omega_{N,i})}=1$, by the classical
regularity theory of elliptic equations, $\tilde{u}_{N,i}$ is uniformly
bounded in $W^{2,p}_{loc}(\mathbb{R}^{2})$. Hence we may assume that

$\tilde{u}_{N,i}\to u_{i}\ \ \ \ \text{in}\ \ C^{1}_{loc}(\mathbb{R}^{2}).$

We claim that $u_{i}\equiv 0.$ On the one hand, by (3.13), the definition of
$V_{\delta_{N},z_{N,i},\hat{q}_{\delta_{N},i},z_{N,i}}$ and the fact that
$z_{N,i}\to z_{i}$ as $N\to\infty$, we obtain

$\begin{split}&\frac{s_{\delta_{N},i}^{2}}{\delta_{N}^{2}}(V_{\delta_{N},Z_{N}}(s_{\delta_{N},i}y+z_{N,i})-q(s_{\delta_{N},i}y+z_{N,i}))^{p-1}_{+}\\\
=&\frac{s_{\delta_{N},i}^{2}}{\delta_{N}^{2}}\left(V_{\delta_{N},z_{N,i},\hat{q}_{\delta_{N},i},z_{N,i}}(s_{\delta_{N},i}y+z_{N,i})-\hat{q}_{\delta_{N},i}+O(\frac{\ln|\ln\varepsilon_{N}|}{|\ln\varepsilon_{N}|^{2}})\right)^{p-1}_{+}\\\
\to&\phi(T_{z_{i}}y)^{p-1}_{+}\ \ \ \ \text{in}\ C^{0}_{loc}(\mathbb{R}^{2})\
\text{as}\ N\to\infty.\end{split}$

So $u_{i}$ satisfies

$-\text{div}(K(z_{i})\nabla u_{i}(x))-p\phi(T_{z_{i}}x)^{p-1}_{+}u_{i}(x)=0,\
\ x\in\mathbb{R}^{2}.$

Let $\hat{u}_{i}(x)=u_{i}(T_{z_{i}}^{-1}x)$. Note that
$T_{z_{i}}^{-1}(T_{z_{i}}^{-1})^{t}=K(z_{i})$. Then

$-\Delta\hat{u}_{i}(x)=-\text{div}(K(z_{i})\nabla
u_{i})(T_{z_{i}}^{-1}x)=p\phi(x)^{p-1}_{+}\hat{u}_{i}(x),\ \ \ \ \forall\
x\in\mathbb{R}^{2}.$

Since $\hat{u}_{i}(x)\in L^{\infty}(\mathbb{R}^{2})$, it follows from
Proposition 4.1 that there are $c_{1},c_{2}$ satisfying

$\hat{u}_{i}=c_{1}\frac{\partial\phi}{\partial
x_{1}}+c_{2}\frac{\partial\phi}{\partial x_{2}}.$ (4.13)

On the other hand, since $u_{N}\in E_{\delta_{N},Z_{N}}$, we have

$\int_{\Omega}-\delta_{N}^{2}\text{div}\left(K(x)\nabla\frac{\partial
V_{\delta_{N},Z_{N},i}}{\partial z_{i,h}}\right)u_{N}=0,\ \ \forall h=1,2,$

which implies that

$\begin{split}0=&\int_{\Omega}-\delta_{N}^{2}\text{div}\left(K(z_{N,i})\nabla\frac{\partial
V_{\delta_{N},Z_{N},i}}{\partial
z_{i,h}}\right)u_{N}+\int_{\Omega}-\delta_{N}^{2}\text{div}\left((K(x)-K(z_{N,i}))\nabla\frac{\partial
V_{\delta_{N},Z_{N},i}}{\partial z_{i,h}}\right)u_{N}\\\
=&\int_{\Omega}\frac{\partial}{\partial
z_{i,h}}\left[-\delta_{N}^{2}\text{div}(K(z_{N,i})\nabla
V_{\delta_{N},Z_{N},i})\right]u_{N}-\int_{\Omega}-\delta_{N}^{2}\text{div}\left(\frac{\partial
K(z_{N,i})}{\partial z_{i,h}}\nabla V_{\delta_{N},Z_{N},i}\right)u_{N}\\\
&+\int_{\Omega}-\delta_{N}^{2}\text{div}\left((K(x)-K(z_{N,i}))\nabla\frac{\partial
V_{\delta_{N},Z_{N},i}}{\partial z_{i,h}}\right)u_{N}.\end{split}$ (4.14)

By (4.11), we have

$\int_{\Omega}-\delta_{N}^{2}\text{div}\left(\frac{\partial
K(z_{N,i})}{\partial z_{i,h}}\nabla
V_{\delta_{N},Z_{N},i}\right)u_{N}=O\left(\frac{\varepsilon_{N}^{2}}{|\ln\varepsilon_{N}|^{p-1}}\right).$
(4.15)

By (4.12), we get

$\int_{\Omega}-\delta_{N}^{2}\text{div}\left((K(x)-K(z_{N,i}))\nabla\frac{\partial
V_{\delta_{N},Z_{N},i}}{\partial
z_{i,h}}\right)u_{N}=O\left(\frac{\varepsilon_{N}^{2}}{|\ln\varepsilon_{N}|^{p-1}}\right).$
(4.16)

Using the definition of $V_{\delta_{N},Z_{N},i}$, (3.14) and (3.16), we obtain

$\begin{split}&\int_{\Omega}\frac{\partial}{\partial
z_{i,h}}\left[-\delta_{N}^{2}\text{div}(K(z_{N,i})\nabla
V_{\delta_{N},Z_{N},i})\right]u_{N}\\\
=&p\int_{\Omega}(V_{\delta_{N},z_{N,i},\hat{q}_{\delta_{N},i},z_{N,i}}-\hat{q}_{\delta_{N},i})^{p-1}_{+}\left(\frac{\partial
V_{\delta_{N},z_{N,i},\hat{q}_{\delta_{N},i},z_{N,i}}}{\partial
z_{i,h}}-\frac{\partial\hat{q}_{\delta_{N},i}}{\partial
z_{i,h}}\right)u_{N}\\\
=&p\int_{\Omega}\left(\frac{\delta_{N}}{s_{\delta_{N},i}}\right)^{2}\phi\left(\frac{T_{z_{N,i}}(x-z_{N,i})}{s_{\delta_{N},i}}\right)^{p-1}_{+}\frac{1}{s_{\delta_{N},i}}\left(\frac{\delta_{N}}{s_{\delta_{N},i}}\right)^{\frac{2}{p-1}}\phi^{\prime}\left(\frac{T_{z_{N,i}}(x-z_{N,i})}{s_{\delta_{N},i}}\right)\frac{(T_{z_{N,i}})_{h}^{t}\cdot
T_{z_{N,i}}(x-z_{N,i})}{|T_{z_{N,i}}(x-z_{N,i})|}u_{N}\\\
&+O\left(\frac{\varepsilon_{N}^{2}}{|\ln\varepsilon_{N}|^{p-1}}\right)\\\
=&ps_{\delta_{N},i}\left(\frac{\delta_{N}}{s_{\delta_{N},i}}\right)^{\frac{2p}{p-1}}\int_{\mathbb{R}^{2}}\phi(T_{z_{N,i}}y)^{p-1}_{+}\phi^{\prime}(T_{z_{N,i}}y)\frac{(T_{z_{N,i}})_{h}^{t}\cdot
T_{z_{N,i}}y}{|T_{z_{N,i}}y|}\tilde{u}_{N,i}(y)dy+O\left(\frac{\varepsilon_{N}^{2}}{|\ln\varepsilon_{N}|^{p-1}}\right).\end{split}$
(4.17)

Hence taking (4.15), (4.16) and (4.17) into (4.14), we have

$\begin{split}0=&ps_{\delta_{N},i}\left(\frac{\delta_{N}}{s_{\delta_{N},i}}\right)^{\frac{2p}{p-1}}\int_{\mathbb{R}^{2}}\phi(T_{z_{N,i}}y)^{p-1}_{+}\phi^{\prime}(T_{z_{N,i}}y)\frac{(T_{z_{N,i}})_{h}^{t}\cdot
T_{z_{N,i}}y}{|T_{z_{N,i}}y|}\tilde{u}_{N,i}(y)dy+O\left(\frac{\varepsilon_{N}^{2}}{|\ln\varepsilon_{N}|^{p-1}}\right).\end{split}$
(4.18)

Dividing both sides of (4.18) into
$ps_{\delta_{N},i}(\frac{\delta_{N}}{s_{\delta_{N},i}})^{\frac{2p}{p-1}}$ and
passing $N$ to the limit, we get for $h=1,2$

$\begin{split}0=&\int_{\mathbb{R}^{2}}\phi(T_{z_{i}}y)^{p-1}_{+}\phi^{\prime}(T_{z_{i}}y)\frac{(T_{z_{i}})_{h}^{t}\cdot
T_{z_{i}}y}{|T_{z_{i}}y|}u_{i}(y)dy\\\
=&\int_{\mathbb{R}^{2}}\phi(x)^{p-1}_{+}\phi^{\prime}(x)\frac{(T_{z_{i}})_{h}^{t}\cdot
x}{|x|}\hat{u}_{i}(x)\sqrt{det(K(z_{i}))}dx,\end{split}$

which implies that

$0=\int_{B_{1}(0)}\phi^{p-1}_{+}\frac{\partial\phi}{\partial
x_{h}}\hat{u}_{i}.$ (4.19)

Combining (4.13) with (4.19), we have $c_{1}=c_{2}=0.$ That is, $u_{i}\equiv
0.$

So we conclude that $\tilde{u}_{N,i}\to 0$ in $C^{1}(B_{L}(0))$, which implies
that

$||u_{N}||_{L^{\infty}(B_{Ls_{\delta_{N},i}}(z_{N,i}))}=o(1).$ (4.20)

By the assumption that $Q_{\delta_{N}}L_{\delta_{N}}u_{N}=0$ in
$\Omega\backslash\cup_{i=1}^{m}B_{Ls_{\delta_{N},i}}(z_{N,i})$ and the
definition of $Q_{\delta_{N}}$, we have for $L$ large

$L_{\delta_{N}}u_{N}=0\ \ \text{in}\
\Omega\backslash\cup_{i=1}^{m}B_{Ls_{\delta_{N},i}}(z_{N,i}).$

It follows from Lemma A.6 that

$(V_{\delta_{N},Z_{N}}-q)_{+}=0\ \ \text{in}\
\Omega\backslash\cup_{i=1}^{m}B_{Ls_{\delta_{N},i}}(z_{N,i}).$

So we get $-\text{div}(K(x)\nabla u_{N})=0$ in
$\Omega\backslash\cup_{i=1}^{m}B_{Ls_{\delta_{N},i}}(z_{N,i}).$

Since $u_{N}=o(1)$ on $\cup_{i=1}^{m}\partial B_{Ls_{\delta_{N},i}}(z_{N,i})$
and $u_{N}=0$ on $\partial\Omega$, by the maximum principle, we get

$||u_{N}||_{L^{\infty}(\Omega\backslash\cup_{i=1}^{m}B_{Ls_{\delta_{N},i}}(z_{N,i}))}=o(1),$

which combined with (4.20) we have

$||u_{N}||_{L^{\infty}(\Omega)}=o(1).$

This is a contradiction since $||u_{N}||_{L^{\infty}(\Omega)}=1.$

∎

Then we can get

######  Proposition 4.3.

$Q_{\delta}L_{\delta}$ is a one to one and onto map from $E_{\delta,Z}$ to
$F_{\delta,Z}.$

###### Proof.

If $Q_{\delta}L_{\delta}u=0$, by Lemma 4.2, $u=0$. So $Q_{\delta}L_{\delta}$
is a one to one map from $E_{\delta,Z}$ to $F_{\delta,Z}.$

We prove that $Q_{\delta}L_{\delta}$ is an onto map from $E_{\delta,Z}$ to
$F_{\delta,Z}.$ Denote

$\hat{E}=\\{u\in H^{1}_{0}(\Omega)\mid\int_{\Omega}\left(K(x)\nabla
u|\nabla\frac{\partial V_{\delta,Z,i}}{\partial z_{i,h}}\right)=0,\ \ \ \
i=1,\cdots,m,\ h=1,2\\}.$

Then $E_{\delta,Z}=\hat{E}\cap W^{2,p}(\Omega)$. For any $\hat{h}\in
F_{\delta,Z}$, by the Riesz representation theorem there is a unique $u\in
H^{1}_{0}(\Omega)$ such that

$\delta^{2}\int_{\Omega}\left(K(x)\nabla
u|\nabla\varphi\right)=\int_{\Omega}\hat{h}\varphi,\ \ \ \ \forall\varphi\in
H^{1}_{0}(\Omega).$ (4.21)

Since $\hat{h}\in F_{\delta,Z}$, we have $u\in\hat{E}.$ Using the classical
$L^{p}$ theory, we conclude that $u\in W^{2,p}(\Omega)$, which implies that
$u\in E_{\delta,Z}.$ Thus
$-\delta^{2}\text{div}(K(x)\nabla)=Q_{\delta}(-\delta^{2}\text{div}(K(x)\nabla))$
is a one to one and onto map from $E_{\delta,Z}$ to $F_{\delta,Z}.$

For any $h\in F_{\delta,Z}$, $Q_{\delta}L_{\delta}u=h$ is equivalent to

$u=(Q_{\delta}(-\delta^{2}\text{div}(K(x)\nabla)))^{-1}pQ_{\delta}(V_{\delta,Z}-q)^{p-1}_{+}u+(Q_{\delta}(-\delta^{2}\text{div}(K(x)\nabla)))^{-1}h,\
\ u\in E_{\delta,Z}.$ (4.22)

Note that
$\mathcal{T}u:=(Q_{\delta}(-\delta^{2}\text{div}(K(x)\nabla)))^{-1}pQ_{\delta}(V_{\delta,Z}-q)^{p-1}_{+}u$
is a compact operator in $E_{\delta,Z}.$ By the Fredholm alternative, (4.22)
is solvable if and only if

$u=(Q_{\delta}(-\delta^{2}\text{div}(K(x)\nabla)))^{-1}pQ_{\delta}(V_{\delta,Z}-q)^{p-1}_{+}u$

has only trivial solution, which is true since $Q_{\delta}L_{\delta}$ is one
to one. The proof is thus complete. ∎

Now consider solutions of (2.16) being the form of
$V_{\delta,Z}+\omega_{\delta}$. Note that by (2.16), one has

$-\delta^{2}\text{div}(K(x)\nabla(V_{\delta,Z}+\omega_{\delta}))-(V_{\delta,Z}+\omega_{\delta}-q)^{p}_{+}=0,$

which is equivalent to

$\begin{split}L_{\delta}\omega_{\delta}=l_{1,\delta}+l_{2,\delta}+R_{\delta}(\omega_{\delta}),\end{split}$
(4.23)

where

$l_{1,\delta}=(V_{\delta,Z}-q)^{p}_{+}-\sum_{j=1}^{m}(V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}-\hat{q}_{\delta,j})^{p}_{+},$
$l_{2,\delta}=\delta^{2}\sum_{j=1}^{m}\text{div}((K(x)-K(z_{j}))\nabla
V_{\delta,Z,j}),$
$R_{\delta}(\omega_{\delta})=(V_{\delta,Z}+\omega_{\delta}-q)^{p}_{+}-(V_{\delta,Z}-q)^{p}_{+}-p(V_{\delta,Z}-q)^{p-1}_{+}\omega_{\delta}.$

We first solve the existence and uniqueness of $\omega\in E_{\delta,Z}$
satisfying

$Q_{\delta}L_{\delta}\omega=Q_{\delta}l_{1,\delta}+Q_{\delta}l_{2,\delta}+Q_{\delta}R_{\delta}(\omega),$
(4.24)

or equivalently,

$\begin{split}\omega=T_{\delta}(\omega):=(Q_{\delta}L_{\delta})^{-1}Q_{\delta}l_{1,\delta}+(Q_{\delta}L_{\delta})^{-1}Q_{\delta}l_{2,\delta}+(Q_{\delta}L_{\delta})^{-1}Q_{\delta}R_{\delta}(\omega).\end{split}$

And then we can reduce (4.23) to a finite dimensional problem.

######  Proposition 4.4.

There is $\delta_{0}>0,$ such that for any $0<\delta<\delta_{0}$ and $Z$
satisfying (3.10), (4.24) has the unique solution $\omega_{\delta,Z}\in
E_{\delta,Z}$ with

$||\omega_{\delta,Z}||_{L^{\infty}(\Omega)}=O\left(\frac{\ln|\ln\varepsilon|}{|\ln\varepsilon|^{2}}\right).$

###### Proof.

It follows from Lemma A.6 in Appendix that for $L$ sufficiently large and
$\delta$ small,

$(V_{\delta,Z}-q)_{+}=0,\ \ \ \ \text{in}\
\Omega\backslash\cup_{i=1}^{m}B_{Ls_{\delta,i}}(z_{i}).$

Let
$N=E_{\delta,Z}\cap\\{\omega\mid||\omega||_{L^{\infty}(\Omega)}\leq\frac{1}{|\ln\varepsilon|^{2-\theta_{0}}}\\}$
for some $\theta_{0}\in(0,1).$ Then $N$ is complete under $L^{\infty}$ norm
and $T_{\delta}$ is a map from $E_{\delta,Z}$ to $E_{\delta,Z}$. We now prove
that $T_{\delta}$ is a contraction map from $N$ to $N$.

First, we claim that $T_{\delta}$ is a map from $N$ to $N$. For any $\omega\in
N$, by Lemma A.6 we get that for $L>1$ large and $\delta$ small,

$(V_{\delta,Z}+\omega-q)_{+}=0,\ \ \ \ \text{in}\
\Omega\backslash\cup_{i=1}^{m}B_{Ls_{\delta,i}}(z_{i}).$

So $l_{1,\delta}=R_{\delta}(\omega)=0$ in
$\Omega\backslash\cup_{i=1}^{m}B_{Ls_{\delta,i}}(z_{i}).$ Note that for any
$u\in L^{\infty}(\Omega)$,

$Q_{\delta}u=u,\ \ \ \ \text{in}\
\Omega\backslash\cup_{i=1}^{m}B_{Ls_{\delta,i}}(z_{i}).$

Hence

$Q_{\delta}l_{1,\delta}+Q_{\delta}R_{\delta}(\omega)=0,\ \ \ \ \text{in}\
\Omega\backslash\cup_{i=1}^{m}B_{Ls_{\delta,i}}(z_{i}).$

So, we can apply Lemma 4.2 to obtain

$||(Q_{\delta}L_{\delta})^{-1}(Q_{\delta}l_{1,\delta}+Q_{\delta}R_{\delta}(\omega))||_{L^{\infty}}\leq
C\frac{|\ln\varepsilon|^{p-1}}{\varepsilon^{\frac{2}{p}}}||Q_{\delta}l_{1,\delta}+Q_{\delta}R_{\delta}(\omega)||_{L^{p}}.$

By Lemma A.5, we know that

$||Q_{\delta}l_{1,\delta}+Q_{\delta}R_{\delta}(\omega)||_{L^{p}}\leq
C(||l_{1,\delta}||_{L^{p}}+||R_{\delta}(\omega)||_{L^{p}}).$

It follows from (3.13), the definition of $l_{1,\delta},R_{\delta}(\omega)$
and Lemma A.6 that

$\begin{split}||l_{1,\delta}||_{L^{p}}=&||(V_{\delta,Z}-q)^{p}_{+}-\sum_{j=1}^{m}(V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}-\hat{q}_{\delta,j})^{p}_{+}||_{L^{p}}\\\
\leq&C\frac{\ln|\ln\varepsilon|}{|\ln\varepsilon|^{2}}\sum_{j=1}^{m}||(V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}-\hat{q}_{\delta,j})^{p-1}_{+}||_{L^{p}}\\\
\leq&C\frac{\ln|\ln\varepsilon|}{|\ln\varepsilon|^{2}}\sum_{j=1}^{m}\left(\frac{\delta}{s_{\delta,j}}\right)^{2}s_{\delta,j}^{\frac{2}{p}}\\\
\leq&C\frac{\varepsilon^{\frac{2}{p}}\ln|\ln\varepsilon|}{|\ln\varepsilon|^{p+1}},\end{split}$

and

$\begin{split}||R_{\delta}(\omega)||_{L^{p}}=&||(V_{\delta,Z}+\omega-q)^{p}_{+}-(V_{\delta,Z}-q)^{p}_{+}-p(V_{\delta,Z}-q)^{p-1}_{+}\omega||_{L^{p}}\\\
\leq&C||(V_{\delta,Z}-q)^{p-2}_{+}||_{L^{p}}||\omega||_{L^{\infty}}^{2}\\\
\leq&C\frac{\varepsilon^{\frac{2}{p}}}{|\ln\varepsilon|^{p-2}}||\omega||_{L^{\infty}}^{2}.\end{split}$

So

$\begin{split}||(Q_{\delta}L_{\delta})^{-1}(Q_{\delta}l_{1,\delta}+Q_{\delta}R_{\delta}(\omega))||_{L^{\infty}}\leq&C\varepsilon^{-\frac{2}{p}}|\ln\varepsilon|^{p-1}(||l_{1,\delta}||_{L^{p}}+||R_{\delta}(\omega)||_{L^{p}})\\\
\leq&C\varepsilon^{-\frac{2}{p}}|\ln\varepsilon|^{p-1}\left(\frac{\varepsilon^{\frac{2}{p}}\ln|\ln\varepsilon|}{|\ln\varepsilon|^{p+1}}+\frac{\varepsilon^{\frac{2}{p}}}{|\ln\varepsilon|^{p-2}}||\omega||_{L^{\infty}}^{2}\right).\end{split}$

By the definition of $l_{2,\delta}$, one computes directly that

$||(Q_{\delta}L_{\delta})^{-1}Q_{\delta}l_{2,\delta}||_{L^{\infty}}\leq
C||Q_{\delta}l_{2,\delta}||_{L^{\infty}}\leq\frac{C}{|\ln\delta|^{2}}.$

Hence by the definition of $N$,

$\begin{split}||T_{\delta}(\omega)||_{L^{\infty}}\leq&C\varepsilon^{-\frac{2}{p}}|\ln\varepsilon|^{p-1}(||l_{\delta}||_{L^{p}}+||R_{\delta}(\omega)||_{L^{p}})+\frac{C}{|\ln\delta|^{2}}\\\
\leq&C\varepsilon^{-\frac{2}{p}}|\ln\varepsilon|^{p-1}\left(\frac{\varepsilon^{\frac{2}{p}}\ln|\ln\varepsilon|}{|\ln\varepsilon|^{p+1}}+\frac{\varepsilon^{\frac{2}{p}}}{|\ln\varepsilon|^{p-2}}||\omega||_{L^{\infty}}^{2}\right)+\frac{C}{|\ln\delta|^{2}}\\\
\leq&\frac{1}{|\ln\varepsilon|^{2-\theta_{0}}}.\end{split}$ (4.25)

So $T_{\delta}$ is a map from $N$ to $N$.

Then we prove that $T_{\delta}$ is a contraction map. For any
$\omega_{1},\omega_{2}\in N$,

$T_{\delta}(\omega_{1})-T_{\delta}(\omega_{2})=(Q_{\delta}L_{\delta})^{-1}Q_{\delta}(R_{\delta}(\omega_{1})-R_{\delta}(\omega_{2})).$

Note that $R_{\delta}(\omega_{1})=R_{\delta}(\omega_{2})=0$ in
$\Omega\backslash\cup_{i=1}^{m}B_{Ls_{\delta,i}}(z_{i}).$ By Lemma 4.2 and the
definition of $N$, for $\delta$ sufficiently small

$\begin{split}||T_{\delta}(\omega_{1})-T_{\delta}(\omega_{2})||_{L^{\infty}}\leq&C\varepsilon^{-\frac{2}{p}}|\ln\varepsilon|^{p-1}||R_{\delta}(\omega_{1})-R_{\delta}(\omega_{2})||_{L^{p}}\\\
\leq&C\varepsilon^{-\frac{2}{p}}|\ln\varepsilon|^{p-1}\varepsilon^{\frac{2}{p}}\left(\frac{||\omega_{1}||_{L^{\infty}}+||\omega_{2}||_{L^{\infty}}}{|\ln\varepsilon|^{p-2}}\right)||\omega_{1}-\omega_{2}||_{L^{\infty}}\\\
\leq&\frac{1}{2}||\omega_{1}-\omega_{2}||_{L^{\infty}}.\end{split}$

So $T_{\delta}$ is a contraction map.

To conclude, $T_{\delta}$ is a contraction map from $N$ to $N$ and thus there
is a unique $\omega_{\delta,Z}\in N$ such that
$\omega_{\delta,Z}=T_{\delta}(\omega_{\delta,Z})$. Moreover, from (4.25), we
have
$||\omega_{\delta,Z}||_{L^{\infty}(\Omega)}=O\left(\frac{\ln|\ln\varepsilon|}{|\ln\varepsilon|^{2}}\right).$

∎

######  Remark 4.5.

Indeed, since $K,q\in C^{\infty}$ and $p>1$, we can also check that
$\omega_{\delta,Z}$ is a $C^{1}$ map about $Z$, see [6, 7] for example.

##  5\. Proof of Theorem 2.2

From Proposition 4.4, we know that for any $\delta$ small and $Z$ satisfying
(3.10), there exists the unique $\omega_{\delta,Z}\in E_{\delta,Z}$ satisfying

$Q_{\delta}L_{\delta}\omega_{\delta,Z}=Q_{\delta}l_{\delta}+Q_{\delta}R_{\delta}(\omega_{\delta,Z}),$

i.e., for some $C_{j,h}$ one has

$L_{\delta}\omega_{\delta,Z}=l_{\delta}+R_{\delta}(\omega_{\delta,Z})+\sum_{j=1}^{m}\sum_{h=1}^{2}C_{j,h}\frac{\partial}{\partial
z_{j,h}}(-\delta^{2}\text{div}(K(z_{j})\nabla V_{\delta,Z,j})).$

In the following we find proper $Z=Z(\delta)$ such that
$V_{\delta,Z}+\omega_{\delta,Z}$ is a solution of (2.16). Note that the
associated functional of (2.16) is

$I_{\delta}(u)=\frac{\delta^{2}}{2}\int_{\Omega}\left(K(x)\nabla u|\nabla
u\right)-\frac{1}{p+1}\int_{\Omega}(u-q)^{p+1}_{+}.$ (5.1)

Denote

$P_{\delta}(Z)=I_{\delta}(V_{\delta,Z}+\omega_{\delta,Z})$

Then by Remark 4.5, we know that $P_{\delta}(Z)$ is a $C^{1}$ function.

By the regularity of $K$ and $q$, it is not hard to check that if $Z$ is a
critical point of $P_{\delta}$, then $V_{\delta,Z}+\omega_{\delta,Z}$ is a
critical point of $I_{\delta}$, i.e., a solution of (2.16). We now prove that
$P_{\delta}$ has a critical point.

######  Proposition 5.1.

There holds

$P_{\delta}(Z)=I_{\delta}(V_{\delta,Z})+O\left(\frac{\varepsilon^{2}\ln|\ln\varepsilon|}{|\ln\varepsilon|^{p+2}}\right).$

###### Proof.

Note that

$\begin{split}P_{\delta}(Z)=&I_{\delta}(V_{\delta,Z})+\delta^{2}\int_{\Omega}\left(K(x)\nabla
V_{\delta,Z}|\nabla\omega_{\delta,Z}\right)+\frac{\delta^{2}}{2}\int_{\Omega}\left(K(x)\nabla\omega_{\delta,Z}|\nabla\omega_{\delta,Z}\right)\\\
&-\frac{1}{p+1}\left(\int_{\Omega}(V_{\delta,Z}+\omega_{\delta,Z}-q)^{p+1}_{+}-\int_{\Omega}(V_{\delta,Z}-q)^{p+1}_{+}\right).\end{split}$

By Proposition 4.4, we get

$\begin{split}&\int_{\Omega}(V_{\delta,Z}+\omega_{\delta,Z}-q)^{p+1}_{+}-\int_{\Omega}(V_{\delta,Z}-q)^{p+1}_{+}\\\
=&(p+1)\sum_{j=1}^{m}\int_{B_{Ls_{\delta,j}}(z_{j})}(V_{\delta,Z}-q)^{p}_{+}\omega_{\delta,Z}+O\left(\sum_{j=1}^{m}\int_{B_{Ls_{\delta,j}}(z_{j})}(V_{\delta,Z}-q)^{p-1}_{+}\omega_{\delta,Z}^{2}\right)\\\
=&O\left(\sum_{j=1}^{m}\frac{s_{\delta,j}^{2}||\omega_{\delta,Z}||_{L^{\infty}}}{|\ln\varepsilon|^{p}}\right)+O\left(\sum_{j=1}^{m}\frac{s_{\delta,j}^{2}||\omega_{\delta,Z}||_{L^{\infty}}^{2}}{|\ln\varepsilon|^{p-1}}\right)\\\
=&O\left(\frac{\varepsilon^{2}\ln|\ln\varepsilon|}{|\ln\varepsilon|^{p+2}}\right).\end{split}$

Using Proposition 4.4, we obtain

$\begin{split}&\delta^{2}\int_{\Omega}\left(K(x)\nabla
V_{\delta,Z}|\nabla\omega_{\delta,Z}\right)\\\
=&\sum_{j=1}^{m}\int_{B_{Ls_{\delta,j}}(z_{j})}(V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}-\hat{q}_{\delta,j})^{p}_{+}\omega_{\delta,Z}+\sum_{j=1}^{m}\delta^{2}\int_{\Omega}\left((K(x)-K(z_{j}))\nabla
V_{\delta,Z}|\nabla\omega_{\delta,Z}\right)\\\
=&O\left(\sum_{j=1}^{m}\frac{s_{\delta,j}^{2}||\omega_{\delta,Z}||_{L^{\infty}}}{|\ln\varepsilon|^{p}}\right)+O\left(\frac{\delta^{2}||\omega_{\delta,Z}||_{L^{\infty}}}{|\ln\varepsilon|}\right)\\\
=&O\left(\frac{\varepsilon^{2}\ln|\ln\varepsilon|}{|\ln\varepsilon|^{p+2}}\right).\end{split}$

Now we calculate the term
$\frac{\delta^{2}}{2}\int_{\Omega}\left(K(x)\nabla\omega_{\delta,Z}|\nabla\omega_{\delta,Z}\right).$
Since $\omega_{\delta,Z}\in E_{\delta,Z}$, we get

$\begin{split}Q_{\delta}L_{\delta}\omega_{\delta,Z}=&Q_{\delta}(-\delta^{2}\text{div}(K(x)\nabla\omega_{\delta,Z}))-Q_{\delta}(p(V_{\delta,Z}-q)^{p-1}_{+}\omega_{\delta,Z})\\\
=&-\delta^{2}\text{div}(K(x)\nabla\omega_{\delta,Z})-Q_{\delta}(p(V_{\delta,Z}-q)^{p-1}_{+}\omega_{\delta,Z}),\end{split}$

which combined with
$Q_{\delta}L_{\delta}\omega_{\delta,Z}=Q_{\delta}l_{\delta}+Q_{\delta}R_{\delta}(\omega_{\delta,Z})$
yields

$-\delta^{2}\text{div}(K(x)\nabla\omega_{\delta,Z})=Q_{\delta}(p(V_{\delta,Z}-q)^{p-1}_{+}\omega_{\delta,Z})+Q_{\delta}l_{\delta}+Q_{\delta}R_{\delta}(\omega_{\delta,Z}).$

Hence by Lemmas A.5, A.6 and Proposition 4.4, we get

$\begin{split}&\delta^{2}\int_{\Omega}\left(K(x)\nabla\omega_{\delta,Z}|\nabla\omega_{\delta,Z}\right)\\\
=&\int_{\Omega}Q_{\delta}(p(V_{\delta,Z}-q)^{p-1}_{+}\omega_{\delta,Z})\omega_{\delta,Z}+\int_{\Omega}Q_{\delta}l_{\delta}\omega_{\delta,Z}+\int_{\Omega}Q_{\delta}R_{\delta}(\omega_{\delta,Z})\omega_{\delta,Z}\\\
\leq&||Q_{\delta}(p(V_{\delta,Z}-q)^{p-1}_{+}\omega_{\delta,Z})||_{L^{1}}||\omega_{\delta,Z}||_{L^{\infty}}+||Q_{\delta}l_{\delta}||_{L^{1}}||\omega_{\delta,Z}||_{L^{\infty}}+||Q_{\delta}R_{\delta}(\omega_{\delta,Z})||_{L^{1}}||\omega_{\delta,Z}||_{L^{\infty}}\\\
\leq&C(||(p(V_{\delta,Z}-q)^{p-1}_{+}\omega_{\delta,Z})||_{L^{1}}+||l_{\delta}||_{L^{1}}+||R_{\delta}(\omega_{\delta,Z})||_{L^{1}})||\omega_{\delta,Z}||_{L^{\infty}}\\\
=&O\left(\frac{\varepsilon^{2}||\omega_{\delta,Z}||_{L^{\infty}}^{2}}{|\ln\varepsilon|^{p-1}}\right)+O\left(\frac{\varepsilon^{2}\ln|\ln\varepsilon|||\omega_{\delta,Z}||_{L^{\infty}}}{|\ln\varepsilon|^{p+1}}\right)+O\left(\frac{\varepsilon^{2}||\omega_{\delta,Z}||_{L^{\infty}}^{3}}{|\ln\varepsilon|^{p-2}}\right)\\\
=&O\left(\frac{\varepsilon^{2}\ln|\ln\varepsilon|}{|\ln\varepsilon|^{p+2}}\right).\end{split}$

To conclude, we get
$P_{\delta}(Z)=I_{\delta}(V_{\delta,Z})+O\left(\frac{\varepsilon^{2}\ln|\ln\varepsilon|}{|\ln\varepsilon|^{p+2}}\right).$

∎

######  Lemma 5.2.

There holds

$I_{\delta}(V_{\delta,Z})=\sum_{j=1}^{m}\frac{\pi\delta^{2}}{\ln\frac{R}{\varepsilon}}q^{2}(z_{j})\sqrt{det(K(z_{j}))}+O\left(\frac{\delta^{2}\ln|\ln\varepsilon|}{|\ln\varepsilon|^{2}}\right).$

###### Proof.

Note that by the definition of $V_{\delta,Z}$,

$\begin{split}I_{\delta}(V_{\delta,Z})=&\frac{\delta^{2}}{2}\int_{\Omega}\left(K(x)\nabla
V_{\delta,Z}|\nabla
V_{\delta,Z}\right)-\frac{1}{p+1}\int_{\Omega}(V_{\delta,Z}-q)^{p+1}_{+}\\\
=&\frac{1}{2}\sum_{j=1}^{m}\int_{\Omega}(V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}-\hat{q}_{\delta,j})^{p}_{+}V_{\delta,Z,j}+\frac{1}{2}\sum_{j=1}^{m}\delta^{2}\int_{\Omega}\left((K(x)-K(z_{j}))\nabla
V_{\delta,Z,j}|\nabla V_{\delta,Z,j}\right)\\\ &+\frac{1}{2}\sum_{1\leq i\neq
j\leq
m}\int_{\Omega}(V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}-\hat{q}_{\delta,j})^{p}_{+}V_{\delta,Z,i}+\frac{1}{2}\sum_{1\leq
i\neq j\leq m}\delta^{2}\int_{\Omega}\left((K(x)-K(z_{j}))\nabla
V_{\delta,Z,j}|\nabla V_{\delta,Z,i}\right)\\\
&-\frac{1}{p+1}\int_{\Omega}(V_{\delta,Z}-q)^{p+1}_{+}.\end{split}$ (5.2)

By (3.7), we have

$\begin{split}&\int_{\Omega}(V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}-\hat{q}_{\delta,j})^{p}_{+}V_{\delta,Z,j}\\\
=&\hat{q}_{\delta,j}\int_{\Omega}(V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}-\hat{q}_{\delta,j})^{p}_{+}+\int_{\Omega}(V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}-\hat{q}_{\delta,j})^{p+1}_{+}-\frac{\hat{q}_{\delta,j}}{\ln\frac{R}{s_{\delta,j}}}\int_{\Omega}(V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}-\hat{q}_{\delta,j})^{p}_{+}g_{z_{j}}(T_{z_{j}}x,T_{z_{j}}z_{j}).\end{split}$

By the definition of $V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}$, the fact
that $T_{z_{j}}^{-1}(T_{z_{j}}^{-1})^{t}=K(z_{j})$ and (3.4), we get

$\begin{split}&\hat{q}_{\delta,j}\int_{\Omega}(V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}-\hat{q}_{\delta,j})^{p}_{+}\\\
=&\hat{q}_{\delta,j}s_{\delta,j}^{2}(\frac{\delta}{s_{\delta,j}})^{\frac{2p}{p-1}}\int_{|T_{z_{j}}x|\leq
1}\phi(T_{z_{j}}x)^{p}dx\\\
=&\hat{q}_{\delta,j}s_{\delta,j}^{2}(\frac{\delta}{s_{\delta,j}})^{\frac{2p}{p-1}}\sqrt{det(K(z_{j}))}\cdot
2\pi|\phi^{\prime}(1)|\\\
=&\hat{q}_{\delta,j}\delta^{2}|\phi^{\prime}(1)|^{p-1}\left(\frac{\ln\frac{R}{s_{\delta,j}}}{\hat{q}_{\delta,j}}\right)^{p-1}|\phi^{\prime}(1)|^{-p}\left(\frac{\ln\frac{R}{s_{\delta,j}}}{\hat{q}_{\delta,j}}\right)^{-p}\sqrt{det(K(z_{j}))}\cdot
2\pi|\phi^{\prime}(1)|\\\
=&\frac{2\pi\delta^{2}}{\ln\frac{R}{s_{\delta,j}}}\hat{q}_{\delta,j}^{2}\sqrt{det(K(z_{j}))}.\end{split}$

Similarly, we have

$\begin{split}&\int_{\Omega}(V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}-\hat{q}_{\delta,j})^{p+1}_{+}\\\
=&s_{\delta,j}^{2}\left(\frac{\delta}{s_{\delta,j}}\right)^{\frac{2(p+1)}{p-1}}\sqrt{det(K(z_{j}))}\cdot\frac{(p+1)\pi}{2}|\phi^{\prime}(1)|^{2}\\\
=&\delta^{2}|\phi^{\prime}(1)|^{p-1}\left(\frac{\ln\frac{R}{s_{\delta,j}}}{\hat{q}_{\delta,j}}\right)^{p-1}|\phi^{\prime}(1)|^{-(p+1)}\left(\frac{\ln\frac{R}{s_{\delta,j}}}{\hat{q}_{\delta,j}}\right)^{-(p+1)}\sqrt{det(K(z_{j}))}\cdot\frac{(p+1)\pi}{2}|\phi^{\prime}(1)|^{2}\\\
=&\frac{(p+1)\pi\delta^{2}}{2\left(\ln\frac{R}{s_{\delta,j}}\right)^{2}}\hat{q}_{\delta,j}^{2}\sqrt{det(K(z_{j}))},\end{split}$

and

$\begin{split}&\frac{\hat{q}_{\delta,j}}{\ln\frac{R}{s_{\delta,j}}}\int_{\Omega}(V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}-\hat{q}_{\delta,j})^{p}_{+}g_{z_{j}}(T_{z_{j}}x,T_{z_{j}}z_{j})\\\
=&g_{z_{j}}(T_{z_{j}}z_{j},T_{z_{j}}z_{j})\frac{\hat{q}_{\delta,j}}{\ln\frac{R}{s_{\delta,j}}}\int_{\Omega}(V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}-\hat{q}_{\delta,j})^{p}_{+}+O\left(\frac{\varepsilon^{3}|\nabla
g_{z_{j}}(T_{z_{j}}z_{j},T_{z_{j}}z_{j})|}{|\ln\varepsilon|^{p+1}}\right)\\\
=&\frac{2\pi\delta^{2}g_{z_{j}}(T_{z_{j}}z_{j},T_{z_{j}}z_{j})}{\left(\ln\frac{R}{s_{\delta,j}}\right)^{2}}\hat{q}_{\delta,j}^{2}\sqrt{det(K(z_{j}))}+O\left(\frac{\varepsilon^{3}}{|\ln\varepsilon|^{p+1}}\right),\end{split}$

where we have used Lemma A.3. Thus we get

$\begin{split}\int_{\Omega}(V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}-\hat{q}_{\delta,j})^{p}_{+}V_{\delta,Z,j}=&\frac{2\pi\delta^{2}}{\ln\frac{R}{s_{\delta,j}}}\hat{q}_{\delta,j}^{2}\sqrt{det(K(z_{j}))}+\frac{(p+1)\pi\delta^{2}}{2\left(\ln\frac{R}{s_{\delta,j}}\right)^{2}}\hat{q}_{\delta,j}^{2}\sqrt{det(K(z_{j}))}\\\
&-\frac{2\pi\delta^{2}g_{z_{j}}(T_{z_{j}}z_{j},T_{z_{j}}z_{j})}{\left(\ln\frac{R}{s_{\delta,j}}\right)^{2}}\hat{q}_{\delta,j}^{2}\sqrt{det(K(z_{j}))}+O\left(\frac{\varepsilon^{3}}{|\ln\varepsilon|^{p+1}}\right).\end{split}$
(5.3)

Similarly for $1\leq i\neq j\leq m$,

$\begin{split}&\int_{\Omega}(V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}-\hat{q}_{\delta,j})^{p}_{+}V_{\delta,Z,i}\\\
=&\frac{\hat{q}_{\delta,i}}{\ln\frac{R}{s_{\delta,i}}}\int_{\Omega}(V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}-\hat{q}_{\delta,j})^{p}_{+}\bar{G}_{z_{i}}(T_{z_{i}}x,T_{z_{i}}z_{i})\\\
=&\frac{2\pi\delta^{2}\bar{G}_{z_{i}}(T_{z_{i}}z_{j},T_{z_{i}}z_{i})}{\ln\frac{R}{s_{\delta,i}}\ln\frac{R}{s_{\delta,j}}}\hat{q}_{\delta,i}\hat{q}_{\delta,j}\sqrt{det(K(z_{j}))}+O\left(\frac{\varepsilon^{3}}{|\ln\varepsilon|^{p+1}}\right).\end{split}$
(5.4)

For any $1\leq j\leq m$, using (3.16) and Lemma A.3 we have

$\begin{split}&\delta^{2}\int_{\Omega}\left((K(x)-K(z_{j}))\nabla
V_{\delta,Z,j}|\nabla V_{\delta,Z,j}\right)\\\
=&\int_{T_{z_{j}}^{-1}(B_{s_{\delta,j}}(0))+z_{j}}+\int_{T_{z_{j}}^{-1}(B_{\mu}(0))+z_{j}\backslash
T_{z_{j}}^{-1}(B_{s_{\delta,j}}(0))+z_{j}}+\int_{\Omega\backslash
T_{z_{j}}^{-1}(B_{\mu}(0))+z_{j}}\\\ &\delta^{2}\left((K(x)-K(z_{j}))\nabla
V_{\delta,Z,j}|\nabla V_{\delta,Z,j}\right)\\\
=&O\left(\frac{\delta^{2}\varepsilon}{|\ln\varepsilon|^{2}}\right)+O\left(\frac{\delta^{2}}{|\ln\varepsilon|^{2}}\right)+O\left(\frac{\delta^{2}}{|\ln\varepsilon|^{2}}\right)\\\
=&O\left(\frac{\delta^{2}}{|\ln\varepsilon|^{2}}\right).\end{split}$ (5.5)

Similarly for $1\leq i\neq j\leq m$,

$\begin{split}\delta^{2}\int_{\Omega}\left((K(x)-K(z_{j}))\nabla
V_{\delta,Z,j}|\nabla
V_{\delta,Z,i}\right)=O\left(\frac{\delta^{2}}{|\ln\varepsilon|^{2}}\right).\end{split}$
(5.6)

Finally by Lemma A.6 and (3.13), we get

$\begin{split}&\int_{\Omega}(V_{\delta,Z}-q)^{p+1}_{+}\\\
=&\sum_{j=1}^{m}\int_{B_{Ls_{\delta,j}}(z_{j})}\left(V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}-\hat{q}_{\delta,j}+O\left(\frac{\ln|\ln\varepsilon|}{|\ln\varepsilon|^{2}}\right)\right)^{p+1}_{+}\\\
=&\sum_{j=1}^{m}\int_{\Omega}(V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}-\hat{q}_{\delta,j})^{p+1}_{+}+O\left(\frac{\ln|\ln\varepsilon|}{|\ln\varepsilon|^{2}}\sum_{j=1}^{m}\int_{\Omega}(V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}-\hat{q}_{\delta,j})^{p}_{+}\right)\\\
=&\sum_{j=1}^{m}\frac{(p+1)\pi\delta^{2}}{2\left(\ln\frac{R}{s_{\delta,j}}\right)^{2}}\hat{q}_{\delta,j}^{2}\sqrt{det(K(z_{j}))}+O\left(\frac{\delta^{2}\ln|\ln\varepsilon|}{|\ln\varepsilon|^{3}}\right).\end{split}$
(5.7)

Taking (5.3), (5.4), (5.5), (5.6) and (5.7) into (5.2), one has

$\begin{split}I_{\delta}(V_{\delta,Z})=&\sum_{j=1}^{m}\frac{\pi\delta^{2}}{\ln\frac{R}{s_{\delta,j}}}\hat{q}_{\delta,j}^{2}\sqrt{det(K(z_{j}))}+\sum_{j=1}^{m}\frac{(p+1)\pi\delta^{2}}{4\left(\ln\frac{R}{s_{\delta,j}}\right)^{2}}\hat{q}_{\delta,j}^{2}\sqrt{det(K(z_{j}))}\\\
&-\sum_{j=1}^{m}\frac{\pi\delta^{2}g_{z_{j}}(T_{z_{j}}z_{j},T_{z_{j}}z_{j})}{\left(\ln\frac{R}{s_{\delta,j}}\right)^{2}}\hat{q}_{\delta,j}^{2}\sqrt{det(K(z_{j}))}\\\
&+\sum_{1\leq i\neq j\leq
m}\frac{\pi\delta^{2}\bar{G}_{z_{i}}(T_{z_{i}}z_{j},T_{z_{i}}z_{i})}{\ln\frac{R}{s_{\delta,i}}\ln\frac{R}{s_{\delta,j}}}\hat{q}_{\delta,i}\hat{q}_{\delta,j}\sqrt{det(K(z_{j}))}\\\
&-\sum_{j=1}^{m}\frac{\pi\delta^{2}}{2\left(\ln\frac{R}{s_{\delta,j}}\right)^{2}}\hat{q}_{\delta,j}^{2}\sqrt{det(K(z_{j}))}+O\left(\frac{\delta^{2}}{|\ln\varepsilon|^{2}}\right).\end{split}$

The result follows from (3.12) and the fact that
$\hat{q}_{\delta,j}=q(z_{j})+O\left(\frac{1}{|\ln\varepsilon|}\right).$

∎

Proof of Theorem 2.2: By Propositions 5.1 and 5.2, we obtain

$P_{\delta}(Z)=\sum_{j=1}^{m}\frac{\pi\delta^{2}}{\ln\frac{R}{\varepsilon}}q^{2}\sqrt{det(K)}(z_{j})+O\left(\frac{\delta^{2}\ln|\ln\varepsilon|}{|\ln\varepsilon|^{2}}\right).$

If each $x_{0,j}$ is a strict local maximum (minimum) point of
$q^{2}\sqrt{det(K)}$ for every $j=1,\cdots,m$, then for $\delta>0$
sufficiently small, there exist at least one point $z_{i,\delta}$ near
$x_{0,i}$ such that $Z_{\delta}=(z_{1,\delta},\cdots,z_{m,\delta})$ is a
maximum (minimum) point of $P_{\delta}$ and as $\delta\to 0$,

$(z_{1,\delta},\cdots,z_{m,\delta})\to(x_{0,1},\cdots,x_{0,m}).$

Hence, we get a solution $w_{\delta}=V_{\delta,Z}+\omega_{\delta,Z}$ of
(2.16). Let $u_{\varepsilon}=|\ln\varepsilon|w_{\delta}$ and
$\delta=\varepsilon|\ln\varepsilon|^{-\frac{p-1}{2}}$, we get solutions of
(2.15). Define the vortex set
$\bar{A}_{\varepsilon,i}=\\{u_{\varepsilon}>q\ln\frac{1}{\varepsilon}\\}\cap
B_{\bar{\rho}}(x_{0,i})$. From Lemma A.6, we can find constants
$R_{1},R_{2}>0$ such that

$B_{R_{1}\varepsilon}(z_{i,\delta})\subseteq\bar{A}_{\varepsilon,i}\subseteq
B_{R_{2}\varepsilon}(z_{i,\delta})\subseteq B_{\bar{\rho}}(x_{0,i}).$

It suffices to calculate the circulation of $u_{\varepsilon}.$ Define
$\kappa_{i}(u_{\varepsilon})=\frac{1}{\varepsilon^{2}}\int_{B_{\bar{\rho}}(x_{0,i})}(u_{\varepsilon}-q\ln\frac{1}{\varepsilon})^{p}_{+}dx.$
We have

######  Lemma 5.3.

There holds

$\lim_{\varepsilon\to 0}\kappa_{i}(u_{\varepsilon})=2\pi
q\sqrt{det(K)}(x_{0,i}).$

###### Proof.

It follows from (3.11), (3.13) and Proposition 4.4 that

$\begin{split}&\frac{1}{\varepsilon^{2}}\int_{B_{\bar{\rho}}(x_{0,i})}(u_{\varepsilon}-q\ln\frac{1}{\varepsilon})^{p}_{+}dx\\\
=&\frac{|\ln\varepsilon|^{p}}{\varepsilon^{2}}\int_{B_{\bar{\rho}}(x_{0,i})}(w_{\delta}-q)^{p}_{+}dx\\\
=&\frac{|\ln\varepsilon|^{p}}{\varepsilon^{2}}\int_{B_{Ls_{\delta,i}}(z_{i,\delta})}\left(V_{\delta,z_{i,\delta},\hat{q}_{\delta,i},z_{i,\delta}}(x)-\hat{q}_{\delta,i}+O\left(\frac{\ln|\ln\varepsilon|}{|\ln\varepsilon|^{2}}\right)\right)^{p}_{+}dx\\\
=&\frac{|\ln\varepsilon|^{p}}{\varepsilon^{2}}s_{\delta,i}^{2}\left(\frac{\delta}{s_{\delta,i}}\right)^{\frac{2p}{p-1}}\int_{|T_{z_{i,\delta}}x|\leq
1}\phi(T_{z_{i,\delta}}x)^{p}dx+o(1)\\\
=&\frac{|\ln\varepsilon|}{\delta^{2}}\delta^{2}|\phi^{\prime}(1)|^{p-1}\left(\frac{\ln\frac{R}{s_{\delta,i}}}{\hat{q}_{\delta,i}}\right)^{p-1}|\phi^{\prime}(1)|^{-p}\left(\frac{\ln\frac{R}{s_{\delta,i}}}{\hat{q}_{\delta,i}}\right)^{-p}\sqrt{det(K(z_{i,\delta}))}\cdot
2\pi|\phi^{\prime}(1)|+o(1)\\\
=&\frac{2\pi\hat{q}_{\delta,i}|\ln\varepsilon|}{\ln\frac{R}{s_{\delta,i}}}\cdot\sqrt{det(K(z_{i,\delta}))}+o(1)\\\
\to&2\pi q\sqrt{det(K)}(x_{0,i})\ \ \ \ \text{as}\ \delta\to 0.\end{split}$

∎

The proof of Theorem 2.2 is thus complete.

##  6\. Proof of Theorem 1.1 and 1.3

Consider the problem

$\begin{cases}-\varepsilon^{2}\text{div}(K_{H}(x)\nabla
u)=\left(u-\left(\frac{\alpha|x|^{2}}{2}+\beta\right)\ln\frac{1}{\varepsilon}\right)^{p}_{+},\
\ &x\in B_{R^{*}}(0),\\\ u=0,\ \ &x\in\partial B_{R^{*}}(0),\end{cases}$ (6.1)

where $\alpha,\beta$ are any given constants satisfying $\min_{x\in
B_{R^{*}}(0)}\frac{\alpha|x|^{2}}{2}+\beta>0$. Let
$v=u\backslash|\ln\varepsilon|$ and
$\delta=\varepsilon|\ln\varepsilon|^{-\frac{p-1}{2}}$, then

$\begin{cases}-\delta^{2}\text{div}(K_{H}(x)\nabla
v)=\left(v-\left(\frac{\alpha|x|^{2}}{2}+\beta\right)\right)^{p}_{+},\ \ &x\in
B_{R^{*}}(0),\\\ v=0,\ \ &x\in\partial B_{R^{*}}(0).\end{cases}$ (6.2)

Note that (6.2) coincides with (2.16) with $q=\frac{\alpha|x|^{2}}{2}+\beta$,
$K=K_{H}$ and $\Omega=B_{R^{*}}(0)$. However, results of Theorem 1.1 can not
be deduced directly from those of Theorem 2.2 since $q^{2}\sqrt{det(K_{H})}$
is a radial function and has no strict local extreme points in $B_{R^{*}}(0)$.
In this case one can also use the reduction procedure to construct solutions
of (6.2), by using the rotational symmetry of $K_{H},q$ and the domain
$B_{R^{*}}(0).$

Let $h(r)=h(|x|)=q^{2}\sqrt{det(K_{H})}(x)$ for any $x\in B_{R^{*}}(0)$. We
call $z^{*}$ is a $strict$ $local$ $maximum$ $(minimum)$ $point$ of
$q^{2}\sqrt{det(K_{H})}$ $up$ $to$ $ratation$ in $B_{R^{*}}(0)$, if $|z^{*}|$
is a strict local maximum (minimum) point of $h$ in $(0,{R^{*}})$.

Let $z_{1}$ be a strict local maximum (minimum) point of
$q^{2}\sqrt{det(K_{H})}$ up to rotation. Define
$\mathcal{N}=B_{\bar{\rho}}(z_{1})$. Then we can construct solutions of (6.2)
being of the form $v_{\delta}=V_{\delta,z}+\omega_{\delta,z}$, where $z$ is
near $z_{1}.$

Indeed, by Lemma 4.2 and Proposition 4.4, for any $z\in\mathcal{N}$ and
$\delta$ sufficiently small there exists a unique $\omega_{\delta,z}\in
E_{\delta,z}$ such that
$Q_{\delta}L_{\delta}\omega_{\delta,z}=Q_{\delta}l_{\delta}+Q_{\delta}R_{\delta}(\omega_{\delta,z}).$
So the final step is to prove the existence of $z=z_{\delta}$ near $z_{1}$
satisfying $\nabla_{z}P_{\delta}(z_{\delta})=0.$ We claim that

$P_{\delta}(z)=\frac{\pi\delta^{2}}{\ln\frac{R}{\varepsilon}}q^{2}\sqrt{det(K_{H})}(z)+N_{\delta}(z),$
(6.3)

where $N_{\delta}(z)$ is a
$O(\frac{\delta^{2}\ln|\ln\varepsilon|}{|\ln\varepsilon|^{2}})-$perturbation
term which is invariant under rotation. In fact, we can choose
$T_{x}^{-1}=\begin{pmatrix}\cos\theta_{x}&-\sin\theta_{x}\\\
\sin\theta_{x}&\cos\theta_{x}\end{pmatrix}\begin{pmatrix}\frac{k}{\sqrt{k^{2}+|x|^{2}}}&0\\\
0&1\end{pmatrix}$ in (3.1), where $(|x|,\theta_{x})$ is the polar coordinate
of $x$. By the rotational symmetry of $q$ and the domain $B_{R}(0)$, one can
prove that for every $\theta\in[0,2\pi]$, if we define
$\bar{z}=\bar{R}_{\theta}(z)$, then

$V_{\delta,\bar{z}}(\bar{R}_{\theta}(x))=V_{\delta,z}(x)\ \text{and}\
\omega_{\delta,\bar{z}}(\bar{R}_{\theta}(x))=\omega_{\delta,z}(x)\ \ \text{for
any}\ x\in B_{R}(0).$

Hence one computes directly that $P_{\delta}(\bar{z})=P_{\delta}(z)$, i.e.,
$P_{\delta}$ is a radially symmetric function. Note that
$q^{2}\sqrt{det(K_{H})}(x)=\left(\frac{\alpha|x|^{2}}{2}+\beta\right)^{2}\cdot\frac{k}{\sqrt{k^{2}+|x|^{2}}}$
is also radially symmetric. Thus by Proposition 5.1 and 5.2, we have (6.3).

Since $z_{1}$ is a strict local maximum (minimum) point of
$q^{2}\sqrt{det(K_{H})}$ up to rotation, it is not hard to prove the existence
of $z_{\delta}$ near $z_{1}$ satisfying $\nabla_{z}P_{\delta}(z_{\delta})=0$,
which yields a solution $v_{\delta}$ of (6.2). Let
$u_{\varepsilon}=v_{\delta}|\ln\varepsilon|$, then $u_{\varepsilon}$ is a
solution of (6.1). Moreover, by Lemma 5.3, one has

$\lim_{\varepsilon\to
0}\frac{1}{\varepsilon^{2}}\int_{B_{\bar{\rho}}(z_{1})}\left(u_{\varepsilon}-q\ln\frac{1}{\varepsilon}\right)^{p}_{+}dx=2\pi
q(z_{1})\sqrt{det(K_{H}(z_{1}))}=\frac{k\pi(\alpha|z_{1}|^{2}+2\beta)}{\sqrt{k^{2}+|z_{1}|^{2}}}.$

To conclude, we have

######  Theorem 6.1.

Let $\alpha,\beta$ be two constants satisfying $\min_{x\in
B_{R^{*}}(0)}\left(\frac{\alpha|x|^{2}}{2}+\beta\right)>0$ and $z_{1}\in
B_{R^{*}}(0)$ be a strict local maximum (minimum) point of
$\left(\frac{\alpha|x|^{2}}{2}+\beta\right)^{2}\cdot\frac{k}{\sqrt{k^{2}+|x|^{2}}}$
up to rotation. Then there exists $\varepsilon_{0}>0$, such that for any
$\varepsilon\in(0,\varepsilon_{0}]$, (6.1) has a solution $u_{\varepsilon}$
satisfying the following properties:

  1. (1)

Define
$A_{\varepsilon}=\left\\{u_{\varepsilon}>\left(\frac{\alpha|x|^{2}}{2}+\beta\right)\ln\frac{1}{\varepsilon}\right\\}$.
Then $\lim\limits_{\varepsilon\to 0}dist(A_{\varepsilon},z_{1})=0$.

  2. (2)

$\lim\limits_{\varepsilon\to
0}\frac{1}{\varepsilon^{2}}\int_{A_{\varepsilon}}\left(u_{\varepsilon}-\left(\frac{\alpha|x|^{2}}{2}+\beta\right)\ln\frac{1}{\varepsilon}\right)^{p}_{+}dx=\frac{k\pi(\alpha|z_{1}|^{2}+2\beta)}{\sqrt{k^{2}+|z_{1}|^{2}}}.$

  3. (3)

There exist $R_{1},R_{2}>0$ satisfying

$R_{1}\varepsilon\leq\text{diam}(A_{\varepsilon})\leq R_{2}\varepsilon.$

Proof of Theorem 1.1: To prove Theorem 1.1, we define for every
$r_{*}\in(0,R^{*})$, $c>0$ and

$\alpha=\frac{c}{4\pi k\sqrt{k^{2}+r_{*}^{2}}},\ \
\beta=\frac{\alpha}{2}(3r_{*}^{2}+4k^{2}).$ (6.4)

One computes directly that $(r_{*},0)$ is a strict minimum point of
$q^{2}\sqrt{det(K_{H})}(x)=\left(\frac{\alpha|x|^{2}}{2}+\beta\right)^{2}\cdot\frac{k}{\sqrt{k^{2}+|x|^{2}}}$
up to rotation and that

$2\pi q\sqrt{det(K_{H})}((r_{*},0))=\frac{k\pi(\alpha
r_{*}^{2}+2\beta)}{\sqrt{k^{2}+r_{*}^{2}}}=c.$

Hence by Theorem 6.1, for any $\varepsilon$ small there exists a solution
$u_{\varepsilon}$ of (6.1) concentrating near $(r_{*},0)$ with
$\alpha=\frac{c}{4\pi k\sqrt{k^{2}+r_{*}^{2}}}$ and
$\frac{1}{\varepsilon^{2}}\int_{A_{\varepsilon}}\left(u_{\varepsilon}-\left(\frac{\alpha|x|^{2}}{2}+\beta\right)\ln\frac{1}{\varepsilon}\right)^{p}_{+}dx$
tending to $c$. Define for any $(x_{1},x_{2},x_{3})\in
B_{R^{*}}(0)\times\mathbb{R},t\in\mathbb{R}$

$\mathbf{w}_{\varepsilon}(x_{1},x_{2},x_{3},t)=\frac{w_{\varepsilon}(x_{1},x_{2},x_{3},t)}{k}\overrightarrow{\zeta},$

where $w_{\varepsilon}(x_{1},x_{2},x_{3},t)$ is a helical function satisfying

$w_{\varepsilon}(x_{1},x_{2},0,t)=\frac{1}{\varepsilon^{2}}\left(u_{\varepsilon}(\bar{R}_{-\alpha|\ln\varepsilon|t}(x_{1},x_{2}))-\left(\frac{\alpha|(x_{1},x_{2})|^{2}}{2}+\beta\right)\ln\frac{1}{\varepsilon}\right)^{p}_{+}.$
(6.5)

Direct computations show that $w_{\varepsilon}(x_{1},x_{2},0,t)$ satisfies
(2.10) and $\mathbf{w}_{\varepsilon}$ is a left-handed helical vorticity field
of (1.2). Moreover, $w_{\varepsilon}(x_{1},x_{2},0,t)$ rotates clockwise
around the origin with angular velocity $\alpha|\ln\varepsilon|$. By (1.3),
the circulation of $\mathbf{w}_{\varepsilon}$ satisfies

$\iint_{A_{\varepsilon}}\mathbf{w}_{\varepsilon}\cdot\mathbf{n}d\sigma=\frac{1}{\varepsilon^{2}}\int_{A_{\varepsilon}}\left(u_{\varepsilon}-\left(\frac{\alpha|x|^{2}}{2}+\beta\right)\ln\frac{1}{\varepsilon}\right)^{p}_{+}dx\to
c,\ \ \text{as}\ \varepsilon\to 0.$

It suffices to prove that the vorticity field $\mathbf{w}_{\varepsilon}$ tends
asymptotically to (1.9) in sense of (1.7). Define $P(\tau)$ the intersection
point of the curve parameterized by (1.9) and the $x_{1}Ox_{2}$ plane. Note
that the helix (1.9) corresponds uniquely to the motion of $P(\tau)$. Taking
$s=\frac{b_{1}\tau}{k}$ into (1.9), one computes directly that $P(\tau)$
satisfies a 2D point vortex model

$\begin{split}P(\tau)=\bar{R}_{\alpha^{\prime}\tau}((r_{*},0)),\end{split}$

where

$\alpha^{\prime}=\frac{1}{\sqrt{k^{2}+r_{*}^{2}}}\left(a_{1}+\frac{b_{1}}{k}\right)=\frac{c}{4\pi
k\sqrt{k^{2}+r_{*}^{2}}},$

which is equal to $\alpha$ in (6.4). Thus by the construction, we readily
check that the support set of
$w_{\varepsilon}(x_{1},x_{2},0,|\ln\varepsilon|^{-1}\tau)$ defined by (6.5)
concentrates near $P(\tau)$ as $\varepsilon\to 0$, which implies that, the
vorticity field $\mathbf{w}_{\varepsilon}$ tends asymptotically to (1.9) in
sense of (1.7). The proof of Theorem 1.1 is thus complete.

Proof of Theorem 1.3: The proof of Theorem 1.3 is similar to that of Theorem
1.1, by constructing multiple concentration solutions $u_{\varepsilon}$ of
(6.1) (or equivalently, $v_{\delta}$ of (6.2)) with polygonal symmetry.
Indeed, let $m\geq 2$ be an integer, $r_{*}\in(0,R)$, $c>0$ and $\alpha,\beta$
satisfying (6.4). For any $z_{1}\in B_{\bar{\rho}}((r_{*},0))$, define
$z_{i}=\bar{R}_{\frac{2(i-1)\pi}{m}}(z_{1})$ for $i=2,\cdots,m.$ Our goal is
to construct solutions of (6.2) being of the form
$\sum_{i=1}^{m}V_{\delta,z_{i}}+\omega_{\delta}$. Using the symmetry of
$K_{H},q$ and $B_{R^{*}}(0)$, one can prove that

$P_{\delta}(z_{1},\cdots,z_{m})=\frac{m\pi\delta^{2}}{\ln\frac{R}{\varepsilon}}q^{2}\sqrt{det(K_{H})}(z_{1})+N_{\delta}(z_{1}),$

where $N_{\delta}(z_{1})$ is a
$O\left(\frac{\delta^{2}\ln|\ln\varepsilon|}{|\ln\varepsilon|^{2}}\right)-$perturbation
term which is invariant under rotation. The rest of the proof is exactly the
same as in that of Theorem 1.1 and we omit the details.

##  7\. Appendix: Some basic estimates

In this Appendix, we give some results which have been repeatedly used before.

First, we give estimates of the Green’s function. Let $G(x,y)$ be the Green’s
function of $-\Delta$ in $\Omega$ with zero-Dirichlet boundary condition and
$h(x,y):=\frac{1}{2\pi}\ln\frac{1}{|x-y|}-G(x,y)$ be the regular part of
$G(x,y)$, then

######  Lemma A.1 (Lemma 4.1, [14]).

For all $x,y\in\Omega$, there hold

$\begin{split}h(x,y)\leq\frac{1}{2\pi}\ln\frac{1}{\max\\{|x-y|,dist(x,\partial\Omega),dist(y,\partial\Omega)\\}},\end{split}$
(7.1)
$\begin{split}h(x,y)\geq\frac{1}{2\pi}\ln\frac{1}{|x-y|+2\max\\{dist(x,\partial\Omega),dist(y,\partial\Omega)\\}}.\end{split}$
(7.2)

Note that the Green’s function $G(x,y)$ is only determined by the domain. To
get (3.14) and (3.16), one must get estimates of $\frac{\partial
h_{z_{i}}(T_{z_{i}}z_{i},T_{z_{i}}z_{i})}{\partial z_{i,\hbar}}$ and
$\frac{\partial G_{z_{i}}(T_{z_{i}}z_{i},T_{z_{i}}z_{j})}{\partial
z_{i,\hbar}}$, which involve the $C^{1}-$dependence of the Green’s function on
the domain. The following $Hadamard$ $variational$ $formula$ gives qualitative
estimates of the first derivative of the Green’s function on the domain, see
[20, 25, 36] for instance.

More precisely, let $\Omega$ be a simply-connected bounded domain with smooth
boundary and $\Omega_{\varepsilon}$ be the perturbation of the domain $\Omega$
whose boundary $\partial\Omega_{\varepsilon}$ is expressed in such a way that

$\partial\Omega_{\varepsilon}=\\{x+\varepsilon\rho(x)\mathbf{n}_{x}\mid
x\in\partial\Omega\\},$ (7.3)

where $\rho\in C^{\infty}(\partial\Omega)$ and $\mathbf{n}_{x}$ is the unit
outer normal to $\partial\Omega$. For all $y,z\in\Omega$, we define

$\mathcal{\delta}_{\rho}G(y,z):=\lim_{\varepsilon\to
0}\varepsilon^{-1}(G_{\varepsilon}(y,z)-G(y,z)),$
$\delta_{\rho}h(y,z):=\lim_{\varepsilon\to
0}\varepsilon^{-1}(h_{\varepsilon}(y,z)-h(y,z)),$

where $G_{\varepsilon}(y,z)=-\frac{1}{2\pi}\ln|y-z|-h_{\varepsilon}(y,z)$ is
the Green’s function of $-\Delta$ in $\Omega_{\varepsilon}$ with Dirichlet
boundary condition and $h_{\varepsilon}(y,z)$ is the regular part of
$G_{\varepsilon}(y,z)$. We have

######  Lemma A.2 ([25]).

It holds that

$\delta_{\rho}G(y,z)=-\delta_{\rho}h(y,z)=\int_{\partial\Omega}\frac{\partial
G(y,x)}{\partial\mathbf{n}_{x}}\frac{\partial
G(z,x)}{\partial\mathbf{n}_{x}}\rho(x)d\sigma_{x},$ (7.4)

where $d\sigma_{x}$ denotes the surface element of $\partial\Omega.$

Let $h_{\hat{x}}(x,y)$ be the regular part of Green’s function
$G_{\hat{x}}(x,y)$ of $-\Delta$ in $T_{\hat{x}}(\Omega)$, where $T\in
C^{\infty}$ satisfies $(T_{\hat{x}}^{-1})(T_{\hat{x}}^{-1})^{t}=K(\hat{x})$
for any $\hat{x}\in\Omega$. Based on Lemmas A.1 and A.2, one can get the
following estimates.

######  Lemma A.3.

For any $Z\in\mathcal{M}$ and $\delta$ sufficiently small, there exists $C>0$
independent of $\varepsilon$ and $Z$ satisfying

$\begin{split}|h_{z_{i}}(T_{z_{i}}z_{i},T_{z_{i}}z_{j})|+|\nabla
h_{z_{i}}(T_{z_{i}}z_{i},T_{z_{i}}z_{j})|\leq C,\ \ 1\leq i,j\leq
m,\end{split}$ (7.5)
$\begin{split}|G_{z_{i}}(T_{z_{i}}z_{i},T_{z_{i}}z_{j})|+|\nabla
G_{z_{i}}(T_{z_{i}}z_{i},T_{z_{i}}z_{j})|\leq C,\ \ 1\leq i\neq j\leq
m,\end{split}$ (7.6) $\bigg{|}\frac{\partial
h_{z_{i}}(T_{z_{i}}z_{i},T_{z_{i}}z_{j})}{\partial z_{i,\hbar}}\bigg{|}\leq
C,\ \ 1\leq i,j\leq m,\hbar=1,2,$ (7.7)

and

$\bigg{|}\frac{\partial G_{z_{i}}(T_{z_{i}}z_{i},T_{z_{i}}z_{j})}{\partial
z_{i,\hbar}}\bigg{|}\leq C,\ \ 1\leq i\neq j\leq m,\hbar=1,2.$ (7.8)

###### Proof.

Since for any $Z\in\mathcal{M}$ all the eigenvalues of $T_{z_{i}}$ have
positive upper and lower bounds uniformly about $Z$, we can find $C_{1}>0$
such that $dist(T_{z_{i}}z_{i},T_{z_{i}}(\Omega))\geq C_{1}>0.$ Hence by Lemma
A.1, we have $|h_{z_{i}}(T_{z_{i}}z_{i},T_{z_{i}}z_{j})|\leq C.$ By the
interior gradient estimates for the harmonic functions, we get $|\nabla
h_{z_{i}}(T_{z_{i}}z_{i},T_{z_{i}}z_{j})|\leq C.$ So (7.5) holds. (7.6)
follows from (7.5) and the definition of $G_{z_{i}}$. Indeed, using the
interior gradient estimates, one can also get that for any integer $l\geq 1$
and $Z\in\mathcal{M}$,

$|\nabla^{l}h_{z_{i}}(T_{z_{i}}z_{i},T_{z_{i}}z_{j})|\leq C.$

Now we estimate $\frac{\partial
h_{z_{i}}(T_{z_{i}}z_{i},T_{z_{i}}z_{j})}{\partial z_{i,\hbar}}$. For $1\leq
i\neq j\leq m$, we have

$\begin{split}\frac{\partial
h_{z_{i}}(T_{z_{i}}z_{i},T_{z_{i}}z_{j})}{\partial
z_{i,\hbar}}=&\lim_{\varepsilon\to
0}\varepsilon^{-1}\left(h_{z_{i}+\varepsilon\mathbf{e}_{\hbar}}(T_{z_{i}+\varepsilon\mathbf{e}_{\hbar}}(z_{i}+\varepsilon\mathbf{e}_{\hbar}),T_{z_{i}+\varepsilon\mathbf{e}_{\hbar}}z_{j})-h_{z_{i}}(T_{z_{i}}z_{i},T_{z_{i}}z_{j})\right)\\\
=&\lim_{\varepsilon\to
0}\varepsilon^{-1}\left(h_{z_{i}+\varepsilon\mathbf{e}_{\hbar}}(T_{z_{i}+\varepsilon\mathbf{e}_{\hbar}}(z_{i}+\varepsilon\mathbf{e}_{\hbar}),T_{z_{i}+\varepsilon\mathbf{e}_{\hbar}}z_{j})-h_{z_{i}+\varepsilon\mathbf{e}_{\hbar}}(T_{z_{i}}z_{i},T_{z_{i}}z_{j})\right)\\\
&+\lim_{\varepsilon\to
0}\varepsilon^{-1}\left(h_{z_{i}+\varepsilon\mathbf{e}_{\hbar}}(T_{z_{i}}z_{i},T_{z_{i}}z_{j})-h_{z_{i}}(T_{z_{i}}z_{i},T_{z_{i}}z_{j})\right),\end{split}$

where $\mathbf{e}_{\hbar}$ is the unit vector of $x_{\hbar}-$axis. By (7.5),

$\big{|}\lim_{\varepsilon\to
0}\varepsilon^{-1}\left(h_{z_{i}+\varepsilon\mathbf{e}_{\hbar}}(T_{z_{i}+\varepsilon\mathbf{e}_{\hbar}}(z_{i}+\varepsilon\mathbf{e}_{\hbar}),T_{z_{i}+\varepsilon\mathbf{e}_{\hbar}}z_{j})-h_{z_{i}+\varepsilon\mathbf{e}_{\hbar}}(T_{z_{i}}z_{i},T_{z_{i}}z_{j})\right)\big{|}\leq
C.$

Note that $T_{z_{i}+\varepsilon\mathbf{e}_{\hbar}}(\Omega)$ is a perturbation
of $T_{z_{i}}(\Omega)$. Since $T$ is a $C^{\infty}$ matrix-valued function and
$\partial\Omega$ is a smooth curve, we find that
$\rho_{i,\hbar}(x):T_{z_{i}}(\Omega)\to\mathbb{R}$, the normal displacement
function defined by (7.3), is smooth about $x\in T_{z_{i}}(\Omega)$. Hence by
Lemmas A.1 and A.2, for any $Z\in\mathcal{M}$

$\begin{split}&\big{|}\lim_{\varepsilon\to
0}\varepsilon^{-1}\left(h_{z_{i}+\varepsilon\mathbf{e}_{\hbar}}(T_{z_{i}}z_{i},T_{z_{i}}z_{j})-h_{z_{i}}(T_{z_{i}}z_{i},T_{z_{i}}z_{j})\right)\big{|}\\\
=&\bigg{|}\int_{\partial(T_{z_{i}}(\Omega))}\frac{\partial
G_{z_{i}}(T_{z_{i}}z_{i},x)}{\partial\mathbf{n}_{x}}\frac{\partial
G_{z_{i}}(T_{z_{i}}z_{j},x)}{\partial\mathbf{n}_{x}}\rho_{i,\hbar}(x)d\sigma_{x}\bigg{|}\\\
\leq&C\max_{\partial(T_{z_{i}}(\Omega))}|\nabla
G_{z_{i}}(T_{z_{i}}z_{i},\cdot)|\max_{\partial(T_{z_{i}}(\Omega))}|\nabla
G_{z_{i}}(T_{z_{i}}z_{j},\cdot)|\max_{\partial(T_{z_{i}}(\Omega))}|\rho_{i,\hbar}(\cdot)|\leq
C.\end{split}$

So we get $\big{|}\frac{\partial
h_{z_{i}}(T_{z_{i}}z_{i},T_{z_{i}}z_{j})}{\partial z_{i,\hbar}}\big{|}\leq C$.
(7.8) can be proved similarly. Indeed, one can also get that for any integer
$l\geq 1$ and $Z\in\mathcal{M}$,

$\bigg{|}\nabla^{l}\frac{\partial
h_{z_{i}}(T_{z_{i}}z_{i},T_{z_{i}}z_{j})}{\partial z_{i,h}}\bigg{|}\leq C.$

∎

The following lemma shows the existence of $\hat{q}_{i}$ satisfying (3.11).

######  Lemma A.4.

For any $Z\in\mathcal{M}$ and $\delta$ sufficiently small, there exist
$\hat{q}_{i}=\hat{q}_{\delta,i}(Z)$ satisfying

$\hat{q}_{i}=q(z_{i})+\frac{\hat{q}_{i}}{\ln\frac{R}{\varepsilon}}g_{z_{i}}(T_{z_{i}}z_{i},T_{z_{i}}z_{i})-\Sigma_{j\neq
i}\frac{\hat{q}_{j}}{\ln\frac{R}{\varepsilon}}\bar{G}_{z_{j}}(T_{z_{j}}z_{i},T_{z_{j}}z_{j}),$
(7.9)

where $\bar{G}_{z_{j}}(x,y)=\ln\frac{R}{|x-y|}-g_{z_{j}}(x,y)=2\pi
G_{z_{j}}(x,y)$ for any $x,y\in T_{z_{j}}(\Omega)$.

###### Proof.

By Lemma A.3, it is not hard to prove that there exists the unique
$\hat{q}_{i}$ satisfying (7.9). Moreover, we have
$\hat{q}_{i}=q(z_{i})+O(\frac{1}{|\ln\varepsilon|})$ and
$\frac{\partial\hat{q}_{i}}{\partial z_{i,h}}=O(1)$ for $h=1,2.$ ∎

This lemma shows the well-definedness of $Q_{\delta}$ defined by (4.5).

######  Lemma A.5.

$Q_{\delta}$ is well-defined. Moreover, for any $q\in[1,+\infty)$, $u\in
L^{q}(\Omega)$ with $supp(u)\subseteqq\cup_{j=1}^{m}B_{Ls_{\delta,j}}(z_{j})$
for some $L>1$, there holds for some $C>0$

$||Q_{\delta}u||_{L^{q}}\leq C||u||_{L^{q}}.$

###### Proof.

By (4.7), we already know that there exists a unique $C_{j,h}$ satisfying
(4.6).

By the assumption, for any $q\in[1,+\infty)$, $l=1,\cdots,m,h=1,2,$ one has

$\begin{split}C_{l,h}=&O\left(|\ln\varepsilon|^{p+1}\sum_{i=1}^{m}\sum_{\hbar=1}^{2}\int_{\cup_{j=1}^{m}B_{Ls_{\delta,j}}(z_{j})}u\frac{\partial
V_{\delta,Z,i}}{\partial z_{i,\hbar}}\right)\\\
=&O\left(|\ln\varepsilon|^{p+1}\sum_{i=1}^{m}\sum_{\hbar=1}^{2}||u||_{L^{q}(\Omega)}||\frac{\partial
V_{\delta,Z,i}}{\partial
z_{i,\hbar}}||_{L^{q^{\prime}}(\cup_{j=1}^{m}B_{Ls_{\delta,j}}(z_{j}))}\right)\\\
=&O\left(|\ln\varepsilon|^{p+1}||u||_{L^{q}(\Omega)}\sum_{i=1}^{m}\sum_{\hbar=1}^{2}\frac{s_{\delta,j}^{\frac{2}{q^{\prime}}-1}}{|\ln\varepsilon|}\right)\\\
=&O(|\ln\varepsilon|^{p}\varepsilon^{1-\frac{2}{q}})||u||_{L^{q}(\Omega)},\end{split}$

where $q^{\prime}$ denotes the conjugate exponent of $q$. Hence we get

$\begin{split}&\sum_{j=1}^{m}\sum_{h=1}^{2}C_{j,h}\frac{\partial}{\partial
z_{j,h}}(-\delta^{2}\text{div}(K(z_{j})\nabla V_{\delta,Z,j}))\\\
=&\sum_{j=1}^{m}\sum_{h=1}^{2}C_{j,h}(V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}-\hat{q}_{\delta,j})^{p-1}_{+}\left(\frac{\partial
V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}}{\partial
z_{j,h}}-\frac{\partial\hat{q}_{\delta,j}}{\partial z_{j,h}}\right)\\\
=&O\left(\sum_{j=1}^{m}\sum_{h=1}^{2}C_{j,h}|\ln\varepsilon|^{-(p-1)}\cdot\frac{s_{\delta,j}^{\frac{2}{q}-1}}{|\ln\varepsilon|}\right)\\\
=&O(1)||u||_{L^{q}(\Omega)}\ \ \ \ \text{in}\ L^{q}(\Omega).\end{split}$

Thus $||Q_{\delta}u||_{L^{q}(\Omega)}\leq C||u||_{L^{q}(\Omega)}$.

∎

######  Lemma A.6.

There exists a constant $L>1$ such that for $\varepsilon$ small

$V_{\delta,Z}-q>0,\ \ \ \ \text{in}\ \
\cup_{j=1}^{m}\left(T_{z_{j}}^{-1}B_{\left(1-L\frac{\ln|\ln\varepsilon|}{|\ln\varepsilon|}\right)s_{\delta,j}}(0)+z_{j}\right),$
$V_{\delta,Z}-q<0,\ \ \ \ \text{in}\ \
\Omega\backslash\cup_{j=1}^{m}\left(T_{z_{j}}^{-1}B_{Ls_{\delta,j}}(0)+z_{j}\right).$

###### Proof.

The proof is similar to that of Lemma A.1 in [6]. If
$|T_{z_{j}}(x-z_{j})|\leq\left(1-L\frac{\ln|\ln\varepsilon|}{|\ln\varepsilon|}\right)s_{\delta,j}$,
then by (3.13) and $\phi^{\prime}(1)<0$ we have

$\begin{split}V_{\delta,Z}-q(x)=&V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}(x)-\hat{q}_{\delta,j}+O\left(\frac{\ln|\ln\varepsilon|}{|\ln\varepsilon|^{2}}\right)\\\
=&\frac{\hat{q}_{\delta,j}}{|\phi^{\prime}(1)|\ln\frac{R}{s_{\delta,j}}}\phi\left(\frac{|T_{z_{j}}(x-z_{j})|}{s_{\delta,j}}\right)+O\left(\frac{\ln|\ln\varepsilon|}{|\ln\varepsilon|^{2}}\right)>0,\end{split}$

if $L$ is sufficiently large.

On the other hand, if $\tau>0$ small and $|T_{z_{j}}(x-z_{j})|\geq
s_{\delta,j}^{\tau}$ for any $j=1,\cdots,m$, then

$\begin{split}V_{\delta,Z}-q(x)=&\sum_{j=1}^{m}\left(V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}(x)-\frac{\hat{q}_{\delta,j}}{\ln\frac{R}{s_{\delta,j}}}g_{z_{j}}(T_{z_{j}}x,T_{z_{j}}z_{j})\right)-q(x)\\\
\leq&\sum_{j=1}^{m}\frac{\hat{q}_{\delta,j}\ln\frac{R}{s_{\delta,j}^{\tau}}}{\ln\frac{R}{s_{\delta,j}}}-C\\\
\leq&\tau\sum_{j=1}^{m}\hat{q}_{\delta,j}-C<0.\end{split}$

If $Ls_{\delta,j}\leq|T_{z_{j}}(x-z_{j})|\leq s_{\delta,j}^{\tau}$, then by
Lemma A.3, we have

$\begin{split}&V_{\delta,Z}-q(x)\\\
=&V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}(x)-\frac{\hat{q}_{\delta,j}}{\ln\frac{R}{s_{\delta,j}}}g_{z_{j}}(T_{z_{j}}x,T_{z_{j}}z_{j})-q(x)+\sum_{i\neq
j}\left(\frac{\hat{q}_{\delta,i}}{\ln\frac{R}{s_{\delta,i}}}\bar{G}_{z_{i}}(T_{z_{i}}x,T_{z_{i}}z_{i})\right)\\\
=&V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}(x)-q(z_{j})-\frac{\hat{q}_{\delta,j}}{\ln\frac{R}{s_{\delta,j}}}g_{z_{j}}(T_{z_{j}}z_{j},T_{z_{j}}z_{j})+\sum_{i\neq
j}\left(\frac{\hat{q}_{\delta,i}}{\ln\frac{R}{s_{\delta,i}}}\bar{G}_{z_{i}}(T_{z_{i}}z_{j},T_{z_{i}}z_{i})\right)+O(\varepsilon^{\tau})\\\
=&V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}(x)-q(z_{j})-\frac{\hat{q}_{\delta,j}}{\ln\frac{R}{\varepsilon}}g_{z_{j}}(T_{z_{j}}z_{j},T_{z_{j}}z_{j})+\sum_{i\neq
j}\left(\frac{\hat{q}_{\delta,i}}{\ln\frac{R}{\varepsilon}}\bar{G}_{z_{i}}(T_{z_{i}}z_{j},T_{z_{i}}z_{i})\right)+O\left(\frac{\ln|\ln\varepsilon|}{|\ln\varepsilon|^{2}}\right)\\\
=&V_{\delta,z_{j},\hat{q}_{\delta,j},z_{j}}(x)-\hat{q}_{\delta,j}+O\left(\frac{\ln|\ln\varepsilon|}{|\ln\varepsilon|^{2}}\right)\\\
\leq&-\frac{\hat{q}_{\delta,j}\ln
L}{\ln\frac{R}{s_{\delta,j}}}+O\left(\frac{\ln|\ln\varepsilon|}{|\ln\varepsilon|^{2}}\right)<0,\end{split}$

if we choose $L$ sufficiently large.

∎

### Acknowledgments:

D. Cao was supported by NNSF of China (grant No. 11831009). J. Wan was
supported by NNSF of China (grant No. 12101045) and Beijing Institute of
Technology Research Fund Program for Young Scholars (No.3170011182016).

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