# On the construction of non-simple blow-up solutions
for the singular Liouville equation with a potential

Teresa D’Aprile  ,  Juncheng Wei  and  Lei Zhang  Dipartimento di Matematica,
Università di Roma “Tor Vergata”, via della Ricerca Scientifica 1, 00133 Roma,
Italy<EMAIL_ADDRESS>Department of Mathematics
University of British Columbia
Vancouver, BC V6T1Z2, Canada<EMAIL_ADDRESS>Department of Mathematics
University of Florida
1400 Stadium Rd
Gainesville FL 32611<EMAIL_ADDRESS>
###### Abstract.

We are concerned with the existence of blowing-up solutions to the following
boundary value problem

$-\Delta u=\lambda V(x)e^{u}-4\pi N{\bm{\delta}}_{0}\;\hbox{ in }B_{1},\quad
u=0\;\hbox{ on }\partial B_{1},$

where $B_{1}$ is the unit ball in $\mathbb{R}^{2}$ centered at the origin,
$V(x)$ is a positive smooth potential, $N$ is a positive integer ($N\geq 1$).
Here ${\bm{\delta}}_{0}$ defines the Dirac measure with pole at $0$, and
$\lambda>0$ is a small parameter. We assume that $N=1$ and, under some
suitable assumptions on the derivatives of the potential $V$ at $0$, we find a
solution which exhibits a non-simple blow-up profile as $\lambda\to 0^{+}$.

Mathematics Subject Classification 2010: 35J20, 35J57, 35J61

Keywords: singular Liouville equation, non-simple blow-up, finite-dimensional
reduction

The research of T. D’Aprile is partially supported by the MIUR Excellence
Department Project awarded to the Department of Mathematics, University of
Rome “Tor Vergata”, CUP E83C18000100006. The research of J. Wei is partially
supported by NSERC of Canada. The research of Lei Zhang is partially supported
by a Simons Foundation Collaboration Grant.

##  1\. Introduction

Given $\Omega$ a smooth and bounded domain in $\mathbb{R}^{2}$ containing the
origin, consider the following Liouville equation with Dirac mass measure

$\left\\{\begin{aligned} &-\Delta u=\lambda V(x)e^{u}-4\pi
N{\bm{\delta}}_{0}&\hbox{ in }&\Omega,\\\ &\ u=0&\hbox{ on
}&\partial\Omega.\end{aligned}\right.$ (1.1)

Here $\lambda$ is a positive small parameter, the potential $V$ is a positive
and smooth function, ${\bm{\delta}}_{0}$ denotes Dirac mass supported at $0$
and $N$ is a positive integer.

Problem (1.1) is motivated by its applications in conformal geometry and
several fields of physics, where quite a few semilinear elliptic equations
defined in two dimensional spaces with an exponential nonlinear term are very
commonly observed and studied. The well known prescribing Gauss curvature
equation, mean field equation, Liouville type equations from the Chern-Simons
self-dual theory, and systems of equations of the Toda system are a few
examples of this family. The analysis of these equations is usually
challenging as the interesting exponential nonlinear term is always related to
the lack of compactness in the variational approach. One important feature of
these equations is the blow-up phenomenon, the understanding of which is
closely related to results on existence, compactness, a-priori estimates, etc.

The asymptotic behaviour of family of blowing up solutions $u_{k}$ can be
referred to the papers [7], [9], [20], [21], [23], [25] for the regular
problem, i.e. when $N=0$. An extension to the singular case $N>0$ is contained
in [3]-[5]. If a blow-up point $p$ is either a regular point or a “non-
quantized” singular source, the asymptotic behavior of $u_{k}$ around $p$ is
well understood (see [3, 5, 8, 9, 16, 19, 32, 33]). As a matter of fact,
$u_{k}$ satisfies the spherical Harnack inequality around $0$, which implies
that, after scaling, the sequence $u_{k}$ behaves as a single bubble around
the maximum point. However, if $p$ happens to be a quantized singular source,
the so-called “non-simple” blow-up phenomenon does happen (see [18, 29, 30,
31]), which is equivalent to stating that $u_{k}$ violates the spherical
Harnack inequality around $p$. The study of non-simple blow-up solutions,
whether or not the blow-up point has to be a critical point of coefficient
functions, has been a major challenge for Liouville equations and its research
has intrigued people for years. Recently significant progress has been made by
Kuo-Lin, Bartolucci-Tarantello and other authors ([6, 11, 18, 29, 30, 31]. In
particular it is established in [4] and [18] that there are $N+1$ local
maximum points and they are evenly distributed on $\mathbb{S}^{1}$ after
scaling according to their magnitude. In [29] and [30] Harnack inequalities
and second order vanishing conditions for non-simple blow-ups are obtained.

The case $N\in\mathbb{N}$ is more difficult to treat, and at the same time the
most relevant to physical applications. Indeed, in vortex theory the number
$N$ represents vortex multiplicity, so that in that context the most
interesting case is precisely when it is a positive integer. The difference
between the case $N\in\mathbb{N}$ and $N\not\in\mathbb{N}$ is also
analytically essential. Indeed, as usual in problems involving concentration
phenomena like (1.1), after suitable rescaling of the blowing-up around a
concentration point one sees a limiting equation which, in this case, takes
the form of the planar singular Liouville equation:

$-\Delta U=e^{U}-4\pi N\delta_{0},\quad\int_{\mathbb{R}^{2}}e^{U}dx<\infty;$

only if $N\in\mathbb{N}$ the above limiting equation admits non-radial
solutions around $0$ since the family of all solutions extends to one carrying
an extra parameter (see [24]). This suggests that if $N\in\mathbb{N}$ and the
blow-up point happens to be the singular source, then solutions of (1.1) may
exhibit non-simple blow-up phenomenon.

So, from analytical viewpoints the study of non-simple blow-up solutions is
far more challenging than simple blow-up solutions, but the impact of this
study may be even more significant because they represent certain situations
in the blow-up analysis of systems of Liouville equations. Indeed, if local
maxima of blow-up solutions in a system tend to one point, the profile of
solutions can be described by a Liouville equation with quantized singular
source. For all this reasons, it is desirable to know exactly when non-simple
blow-up phenomenon happens.

However, the question on the existence of non-simple blowing-up solutions to
(1.1) concentrating at $0$ is far from being completely settled. A first
definite answer is provided by [12] which rules out the non-simple blow-up
phenomenon for (1.1) if the potential $V$ is constant: more precisely it is
established that there is no non-simple blow-up sequence for (1.1) with
$V=const.$, even if we are in the presence of multiples singularities
$\sum_{i}N_{i}{\bm{\delta}}_{p_{i}}$. Apart from this, only partial results
are known: in [11] the construction of solutions exhibiting a non simple blow-
up profile at $0$ is carried out for equation (1.1) with $V\equiv 1$ provided
that $\Omega$ is the unit ball and the weight of the source is a positive
number $N=N_{\lambda}$ close an integer $N$ from the right side. On the other
hand, in[13], for any fixed positive integer $N$, it is proved the existence
of a solution to (1.1) with $V\equiv 1$, where ${\bm{\delta}}_{0}$ is replaced
by ${\bm{\delta}}_{p_{\lambda}}$ for a suitable $p_{\lambda}\in\Omega$, with
$N+1$ blowing up points at the vertices of a sufficiently tiny regular polygon
centered in $p_{\lambda}$; moreover the location of $p_{\lambda}$ is
determined by the geometry of the domain in a $\lambda-$dependent way and does
not seem possible to be prescribed arbitrarily. To our knowledge, the
existence of non-simple blow-up phenomenon for (1.1) for a fixed $V$ and a
fixed $N$ independent of $\lambda$ is still open, even in the case of the
ball: the only example is constructed in [10] for a special class of
potentials of the form $V(|x|^{N+1})$.

In this paper we investigate the existence of non-simple blow-up solutions
when $\Omega$ is the unit ball $B_{1}$ centered at the origin, the potential
$V_{\lambda}=V$ is fixed and $N=1$:

$\left\\{\begin{aligned} &-\Delta u=\lambda
V(x)e^{u}-4\pi{\mathcal{\delta}}_{0}&\hbox{ in }&B_{1},\\\ &\ u=0&\hbox{ on
}&\partial B_{1}.\end{aligned}\right.$ (1.2)

Let us pass to enumerate the hypotheses on the potential $V$ that will be
steadily used throughout the paper.

  1. ${\mathrm{(H1)}}$

$\inf_{B_{1}}V(x)>c>0$ for a positive constant $c$ independent of $\lambda$
and, without loss of generality, we may assume $V(0)=1$;

  2. ${\mathrm{(H2)}}$

$V(x)$ is even, i.e.

$V(x)=V(-x)\quad\forall x\in B_{1}.$

Furthermore we will require sufficient regularity of $V$ at $0$ together with
crucial conditions on the derivatives of $V$ at $0$:

  1. ${\mathrm{(H3)}}$

$V(x)$ is of class $C^{1}$ in the closed unit ball $\overline{B}_{1}$ and the
following holds:

$\displaystyle V(x)=$
$\displaystyle\;1+A_{0}(x_{1}^{4}+x_{2}^{4})+A_{1}(x_{1}^{3}x_{2}-x_{1}x_{2}^{3})+A_{2}x_{1}^{2}x_{2}^{2}$
(1.3)
$\displaystyle+D_{0}(x_{1}^{6}-x_{2}^{6})+D_{1}(x_{1}^{5}x_{2}+x_{1}x_{2}^{5})+D_{2}(x_{1}^{4}x_{2}^{2}-x_{1}^{2}x_{2}^{4})+D_{3}x_{1}^{3}x_{2}^{3}+O(|x|^{7})$

for some constants $A_{0},A_{1},A_{2},D_{0},D_{1},D_{2},D_{3}\in\mathbb{R}$.

Let us comment on assumption ${\mathrm{(H3)}}$: in [29], [30], [31] the second
and the third authors proved that if non simple blow up scenarios occur for
equation (1.2), then the first derivatives as well as the Laplacian of
coefficient functions must tend to zero at the singular source; so the
vanishing of the second order terms in the expansion (1.3) is not surprising.
Moreover, the analysis reveals that the relation between the forth derivatives
and between the sixth derivatives plays a crucial role since it guarantees
that the non simple blow-up solutions can be accurately approximated by global
solutions by allowing an a priori estimate for the error which turns out to be
sufficiently small (see Remark 4.5 and Remark 6.1).

In order to provide the exact formulation of the result let us fix some
notation: in the following $G(x,y)$ is the Green’s function of $-\Delta$ over
$\Omega$ under Dirichlet boundary conditions and $H(x,y)$ denotes its regular
part:

$H(x,y):=G(x,y)-\frac{1}{2\pi}\log\frac{1}{|x-y|}.$

In the case of the unit ball we have the explicit formula for the regular part
of the Green function in $B_{1}$ which is given by

$H(x,y)=\frac{1}{2\pi}\log\bigg{(}|x|\bigg{|}y-\frac{x}{|x|^{2}}\bigg{|}\bigg{)},\quad
x,y\in B_{1}.$ (1.4)

Then the main result of this paper provides a sufficient condition on the
potential $V$, in addition to the assumptions $\mathrm{(H1)-(H2)-(H3)}$, which
implies that (1.2) admits a family of non-simple blowing-up solutions. Such a
sufficient condition is expressed in terms of the concept of stable zeroes for
a suitable vector field.

######  Theorem 1.1.

Assume that hypotheses $\mathrm{(H1)-(H2)-(H3)}$ hold and, in addition,

$A_{0}=2,\;\;A_{1}=0\;\;A_{2}=4.$ (1.5)

Let $\xi\in\mathbb{R}^{2}$, $\xi\neq 0$, be a zero for the following vector
field which is stable under uniform perturbations111Given
$F:\mathbb{R}^{2}\to\mathbb{R}^{2}$ a continuous vector filed, we say that
$\xi$ is a zero for $F$ which is stable with respect to uniform perturbations
if $F(\xi)=0$ and for any neighborhood $U$ of $\xi$ and $\epsilon>0$ there
exists $\eta>0$ such that if $\Psi:U\to\mathbb{R}^{2}$ is continuous and
$\|\Psi-F\|_{\infty}\leq\eta$, then $\Psi$ has a zero in $U$. A sufficient
condition which implies that $0$ is a stable zero of a vector field $F$ is
$deg(F,U,0)\neq 0$ for some neighborhood $U$ of $\xi$, where $deg$ denotes the
standard Brower degree.

$(\xi_{1},\xi_{2})\longmapsto\left(\begin{aligned}
&3D_{0}\xi_{1}^{2}+D_{1}\xi_{1}\xi_{2}+\frac{3D_{0}-D_{2}}{4}\xi_{2}^{2}+\frac{15D_{0}-D_{2}}{4}\\\
&\frac{D_{1}}{2}\xi_{1}^{2}+\frac{3D_{0}-D_{2}}{2}\xi_{1}\xi_{2}+3\frac{2D_{1}+D_{3}}{8}\xi_{2}^{2}+\frac{10D_{1}+3D_{3}}{8}\end{aligned}\right).$
(1.6)

Then, for $\lambda$ sufficiently small the problem (1.2) has a family of
solutions $u_{\lambda}$ blowing up at the origin as $\lambda\to 0^{+}$:

$\lambda e^{u_{\lambda}}\to 16\pi{\bm{\delta}}_{0}\;\;\hbox{ in the measure
sense.}$

More precisely there exist $\delta=\delta(\lambda)>0$ and $b=b(\lambda)\in
B_{1}$ in a neighborhood of $0$ such that $u_{\lambda}$ satisfies

$u_{\lambda}+4\pi G(x,0)=-2\log\big{(}\mu^{4}+|x^{2}-b|^{2}\big{)}+8\pi
H(x^{2},b)+o(1)$

in $H^{1}$-sense, where

$b(\lambda)=\frac{\xi_{0}}{4\sqrt{2}}\sqrt{\lambda\log\frac{1}{\lambda}}\big{(}1+o(1)\big{)},\qquad\mu^{2}(\lambda)=\frac{\sqrt{\lambda}}{4\sqrt{2}}(1+o(1)).$
(1.7)

In particular, $\mu^{2}=o(|b|).$

The solution constructed in Theorem 1.1 reveals a non-simple blow-up profile:
indeed, denoting by $\pm\beta$ the square complex roots of $b$, since the rate
of convergence $\beta\to 0$ is lower than the speed of the concentration
parameter $\mu\to 0$ (see estimate (1.7)), then $u_{\lambda}$ develops $2$
local maximum points concentrating at $0$ which are arranged close to two
opposite vertices. The analysis shows that the configuration of the limiting
local maxima is determined by the interaction of two crucial aspects: the
effect of the potential $V$, which tends to shrink the bubble to $0$, and the
boundary effect, represented by the Robin function $H(\xi,\xi)$, which tends
to repel the bubble from $0$. On the other hand, the existence of this kind of
non-simple blow-up is still open for more general potential $V$. Indeed, as we
will observe in Remark 6.1, if we apply our method for generic values
$A_{0},A_{1},A_{2}$ not satisfying (1.5), then we find out that the forces
exerted between the potential and the boundary may not balance and we are
unable to catch a solution different from the radially symmetric one.

######  Remark 1.2.

Let us observe that $F$ actually corresponds to a gradient field, precisely
$F(\xi)=\nabla J(\xi)$, where the potential $J$ is given by

$J(\xi)=D_{0}\xi_{1}^{3}+\frac{D_{1}}{2}\xi_{1}^{2}\xi_{2}+\frac{3D_{0}-D_{2}}{4}\xi_{1}\xi_{2}^{2}+\frac{2D_{1}+D_{3}}{8}\xi_{2}^{3}+\frac{15D_{0}-D_{2}}{4}\xi_{1}+\frac{10D_{1}+3D_{3}}{8}\xi_{2}.$

######  Example 1.3.

Let us provide explicit examples of coefficients $D_{0},D_{1},D_{2}$ for which
$0$ is a stable zero for $F$, so that, according to Theorem 1.1 the
corresponding $V$ will produce a non simple blowing up solution for eqation
(1.2). Indeed, if we take $D_{0}=0,$ $D_{1}=2$, $D_{2}=-4$, $D_{3}=-4$, then
the potential $J$ defined in Remark 1.2 becomes

$J(\xi)=\xi_{1}^{2}\xi_{2}+\xi_{1}\xi_{2}^{2}+\xi_{1}+\xi_{2}.$

It is immediate to check that $(1,-1)$ (respectively $(-1,1)$) is a critical
points for $J$, so

$F(1,-1)=\nabla J(1,-1)=0$

Moreover, the Hessian matrix of $J$ at $(1,-1)$ is given by

$\left(\begin{aligned} -&2&0&\\\ &0&2&\end{aligned}\right).$

Then $(1,-1)$ (respectively $(-1,1)$) turns out to be a nondegenerate critical
point for $J$ of saddle type, so $deg(F,U,0)\neq 0$ where $deg$ denotes the
standard Brower degree and $U$ is a sufficiently small neighbourhood of
$(1,-1)$. Consequently, $(1,-1)$ is a stable (with respect to uniform
perturbations) zero for $F$. Then, according to Theorem 1.1, if $V$ satisfies
$\mathrm{(H1)-(H2)}$ and

$V(x)=1+2x_{1}^{2}+4x_{1}^{2}x_{2}^{2}+2x_{2}^{4}+2(x_{1}^{5}x_{2}+x_{1}x_{2}^{5})+4(x_{1}^{4}x_{2}^{2}-x_{1}^{2}x_{2}^{4})-4x_{1}^{3}x_{2}^{3}+O(|x|^{7})$

then $\xi_{0}:=(1,-1)$ (respectively $\xi_{0}:=(-1,1)$) gives rise to a non-
simple blowing up family of solutions to (1.2).

The phenomena of non-simple bubbling solutions not only occur in single
equations, but also in systems. In a recent work of the second author and Gu
([17]) the non-simple blow-up behaviours are studied for singular Liouville
systems. In another work of the second, third authors and Wu [28] non-simple
blowup is ruled out for Toda systems. Examples of non-simple blow-up solutions
are available for other models: we recall, for instance, the Liouville
equation with anisotropic coefficients in [27] and the Toda system in [1].

The proofs use singular perturbation methods which combine the variational
approach with a Lyapunov-Schmidt type procedure. Roughly speaking, the first
step consists in the construction of an approximate solution, which should
turn out to be precise enough. In view of the expected asymptotic behaviour,
the shape of such approximate solution will resemble, after the change of
variables $x\mapsto x^{1/2}$, a bubble of the form (2.6) with a suitable
choice of the parameter $\delta=\delta(\lambda,b)$. Then we look for a
solution to (1.2) in a small neighborhood of the first approximation. As quite
standard in singular perturbation theory, a crucial ingredient is
nondegeneracy of the explicit family of solutions of the limiting Liouville
problem (2.5), as established in [2]. This allows us to study the
invertibility of the linearized operator associated to the problem (1.2) under
suitable orthogonality conditions. Next we introduce an intermediate problem
and a fixed point argument will provide a solution for an auxiliary equation,
which turns out to be solvable for any choice of $b$. Finally we test the
auxiliary equation on the elements of the kernel of the linearized operator
and we find out that, in order to find an exact solution of (1.2), the
location of the maximum points, which is detected by the parameter $b$, should
be a zero for a reduced finite dimensional map. The main technical difficulty
in the proof is that we need to expand the reduced map up to higher orders to
catch a nontrivial zero $b$, which will give rise to a non-simple blow-up
solution. Moreover the method fails for $N\geq 2$: indeed if we try to apply
our technique to $N\geq 2$, then the analogous of assumption $\mathrm{(H3)}$
would give that the potential has vanishing derivatives up to the order $N+1$
at $0$, and this implies that the approximation rate for the reduced finite
dimensional map is unfortunately not sufficiently small to carry out the
argument.

The rest of the paper is organized as follows. Section 2 is devoted to some
preliminary results, notation, and the definition of the approximating
solution. Moreover, a more general version of Theorems 1.1 is stated there
(see Theorem 2.1). In Section 3 we sketch the solvability of the linearized
problem by referring to [14] and [15] for the proof. The error up to which the
approximating solution solves problem (1.2) is estimated in Section 4. Section
5 considers the solvability of an auxiliary problem by a contraction argument.
In Section 6 we complete the proof of Theorem 1.1. In Appendix A we collect
some results, most of them well-known, which are usually referred to
throughout the paper.

NOTATION: In our estimates throughout the paper, we will frequently denote by
$C>0$, $c>0$ fixed constants, that may change from line to line, but are
always independent of the variables under consideration.

##  2\. Preliminaries and statement of the main results

We are going to provide an equivalent formulation of problem (1.2) and Theorem
1.1. Indeed, setting $v$ the regular part of $u$, namely

$v=u+4\pi G(x,0)=u+2\log\frac{1}{|x|},$ (2.1)

problem (1.2) is then equivalent to solving the following boundary value
problem

$\left\\{\begin{aligned} &-\Delta v=\lambda|x|^{2}V(x)e^{v}&\hbox{ in
}&B_{1}\\\ &v=0&\hbox{ on }&\partial B_{1}\end{aligned}\right..$ (2.2)

Here $G$ and $H$ are the Green’s function and its regular part as defined in
the introduction.

In what follows, we identify $x=(x_{1},x_{2})\in\mathbb{R}^{2}$ with
$x_{1}+{\rm i}x_{2}\in\mathbb{C}$ and we denote by $x\,y$ the multiplication
of the complex numbers $x,y$ and, analogously, by $x^{2}$ the square of the
complex number $x$.

Since $V$ and the solutions considered in the paper are even, we can rewrite
problem (2.2) as a regular Liouville problem: more precisely, denoting by
$x^{\frac{1}{2}}$ the complex $2$-roots of $x$, the change of variables

$w(x)=v\big{(}x^{\frac{1}{2}}\big{)}$ (2.3)

transforms problem (2.2) into a (regular) Liouville problem of the form

$\left\\{\begin{aligned} &-\Delta
w=\frac{\lambda}{4}V\big{(}x^{\frac{1}{2}}\big{)}e^{w}&\hbox{ in }&B_{1}\\\
&w=0&\hbox{ on }&\partial B_{1}\end{aligned}\right..$ (2.4)

Theorem 1.1 will be a consequence of a more general result concerning
Liouville-type problems. In order to provide such a result, we now give a
construction of a suitable approximate solution for (2.4). We can associate to
(2.4) a limiting problem of Liouville type which will play a crucial role in
the construction of blowing up solutions as $\lambda\to 0^{+}$:

$-\Delta
W=e^{W}\quad\hbox{in}\;\;\mathbb{R}^{2},\qquad\int_{\mathbb{R}^{2}}e^{W(x)}dx<+\infty.$
(2.5)

All solutions of this problem are given, in complex notation, by the three-
parameter family of functions

$W_{\delta,b}(x):=\log\frac{8\delta^{2}}{(\delta^{2}+|x-b|^{2})^{2}}\quad\delta>0,\,b\in\mathbb{C}.$
(2.6)

The following quantization property holds:

$\int_{\mathbb{R}^{2}}e^{W_{\delta,b}(x)}dx=8\pi.$ (2.7)

In the following we agree that

$W_{\lambda}(x)=W_{\delta,b}(x),\quad\delta>0,\,b\in\mathbb{C},$

where the value $\delta=\delta(\lambda,b)$ is defined by

$\delta^{{2}}:=\frac{\lambda}{32}{V}(b^{\frac{1}{2}})e^{8\pi
H(b,b)}=\frac{\lambda}{32}{V}(b^{\frac{1}{2}})(1-|b|^{2})^{4}.$ (2.8)

We point out that the diagonal $H(b,b)$ appearing in (2.8) is called the Robin
function of the domain and in the case of the ball it takes the form

$H(x,x)=\frac{1}{2\pi}\log(1-|x|^{2}),\quad x\in B_{1}$

according to (1.4). To obtain a better first approximation, we need to modify
the function $W_{\lambda}$ in order to satisfy the zero boundary condition.
Precisely, we consider the projection $PW_{\lambda}$ onto the space
$H^{1}_{0}(B_{1})$, where the projection $P:H^{1}(\mathbb{R}^{N})\to
H^{1}_{0}(B_{1})$ is defined as the unique solution of the problem

$\Delta Pv=\Delta v\quad\hbox{in}\ B_{1},\qquad Pv=0\quad\hbox{on}\ \partial
B_{1}.$

We recall that the regular part $H(x,b)$ of the Green function, defined in
(1.4), is harmonic in $B_{1}$ and satisfies $H(x,b)=\frac{1}{2\pi}\log|x-b|$
for $x\in\partial B_{1};$ a straightforward computation gives that for any
$x\in\partial B_{1}$

$\displaystyle PW_{\lambda}-W_{\lambda}+\log\left(8\delta^{2}\right)-8\pi
H(x,b)$
$\displaystyle=-W_{\lambda}+\log\left(8\delta^{2}\right)-4\log|x-b|=2\log\Big{(}1+\frac{\delta^{2}}{|x-b|^{2}}\Big{)}$
$\displaystyle=2\frac{\delta^{2}}{|x-b|^{2}}+O(\delta^{4})=2\frac{\delta^{2}}{1+O(|b|)}+O(\delta^{4})$
$\displaystyle=2\delta^{2}+O(\delta^{2}|b|)+O(\delta^{4})$

with uniform estimate for $x\in\partial B_{1}$ and $b$ in a small neighborhood
of $0$. Since the expressions
$PW_{\lambda}-W_{\lambda}+\log\left(8\delta^{2}\right)-8\pi H(x,b)$ and
$2\delta^{2}$ are harmonic in $B_{1}$, then the maximum principle applies and
implies the following asymptotic expansion

$\displaystyle PW_{\lambda}=$ $\displaystyle
W_{\lambda}-\log\left(8\delta^{2}\right)+8\pi
H(x,b)+2\delta^{2}+O(\delta^{2}|b|)+O(\delta^{4})$ (2.9)
$\displaystyle=-2\log\left(\delta^{{2}}+|x-b|^{2}\right)+8\pi
H(x,b)+2\delta^{2}+O(\delta^{2}|b|)+O(\delta^{4})$

uniformly for $x\in\overline{B}_{1}$ and $b$ in a small neighborhood of $0$.

We point out that, in order to simplify the notation, in our estimates
throughout the paper we will describe the asymptotic behaviors of quantities
under considerations in terms of $\delta=\delta(\lambda,b)$ instead of the
parameter $\lambda$ of the equation. Clearly according to (2.8) $\delta$ has
the same rate as $\lambda^{\frac{1}{2}}$, so at each step we can easily pass
to the analogous asymptotic in terms of $\lambda$: for instance, in (2.9) the
error term “$O(\delta^{4})$” can be equivalently replaced by
“$O(\lambda^{2})$”.

We shall look for a solution to (2.4) in a small neighborhood of the first
approximation, namely a solution of the form

$w_{\lambda}=PW_{\lambda}+{\phi}_{\lambda},$

where the rest term $\phi_{\lambda}$ is small in $H^{1}_{0}(B_{1})$-norm.

Let us reformulate the main theorem for problem (2.4), which prove that a non
symmetric blow-up occurs for problem (2.4). More precisely, we provide a
solution which develops a bubble centered at a point $b$; and since the rate
of convergence $b\to 0^{+}$ is lower than the speed of the concentration
parameter $\delta\to 0^{+}$ (see estimate (2.10)), then the blowing up turns
out to be non symmetric in the first approximation.

######  Theorem 2.1.

Assume that hypotheses ${\mathrm{(H1)-(H3)}}$ and (1.5) hold. Let
$b\in\mathbb{R}^{2}$ be a zero for the vector field (1.6) which is stable
under uniform perturbations. Then, for $\lambda$ sufficiently small the
problem (2.4) has a family of solutions $w_{\lambda}$ satisfying

$w_{\lambda}=-2\log\big{(}\delta^{2}+|x-b_{\lambda}|^{2}\big{)}+8\pi
H(x,b_{\lambda})+o(1)$

in $H^{1}$-sense, where

$b_{\lambda}=\frac{\xi_{0}}{4\sqrt{2}}\sqrt{\lambda\log\frac{1}{\lambda}}\big{(}1+o(1)\big{)}.$
(2.10)

In particular, by (2.8), $\delta^{2}=o(|b_{\lambda}|).$

In the remaining part of this paper we will prove Theorems 2.1 and at the end
of Section 6 we shall see how Theorems 1.1 follows quite directly as a
corollary.

We end this section by setting notation and basic well-known facts which will
be of use in the rest of the paper. Given $\Omega$ a bounded domain, we denote
by $\|\cdot\|$ and $\|\cdot\|_{p}$ the norms in the space $H^{1}_{0}(\Omega)$
and $L^{p}(\Omega)$, respectively, namely

$\|u\|:=\|u\|_{H^{1}_{0}(\Omega)},\qquad\|u\|_{p}:=\|u\|_{L^{p}(\Omega)}\quad\forall
u\in H^{1}_{0}(\Omega).$

In next lemma we recall the well-known Moser-Trudinger inequality ([22, 26]).

######  Lemma 2.2.

There exists $C>0$ such that for any bounded domain $\Omega$ in
$\mathbb{R}^{2}$

$\int_{\Omega}e^{\frac{4\pi u^{2}}{\|u\|^{2}}}dy\leq C|\Omega|\quad\forall
u\in{H}^{1}_{0}(\Omega),$

where $|\Omega|$ stands for the measure of the domain $\Omega$. In particular,
for any $q\geq 1$

$\|e^{u}\|_{q}\leq
C^{\frac{1}{q}}|\Omega|^{\frac{1}{q}}e^{{\frac{q}{16\pi}}\|u\|^{2}}\quad\forall
u\in{H}^{1}_{0}(\Omega).$

As commented in the introduction, our proof uses the singular perturbation
methods. For that, the nondegeneracy of the functions that we use to build our
approximating solution is essential. Next proposition is devoted to the
nondegeneracy of the finite mass solutions of the Liouville equation (see [2]
for the proof).

######  Proposition 2.3.

Assume that $\xi\in\mathbb{R}^{2}$ and $\phi:\mathbb{R}^{2}\to\mathbb{R}$
solves the problem

$-\Delta\phi=\frac{8}{(1+|z-\xi|^{2})^{2}}\phi\;\;\hbox{in}\
\mathbb{R}^{2},\quad\int_{\mathbb{R}^{2}}|\nabla\phi(z)|^{2}dz<+\infty.$
(2.11)

Then there exist $c_{0},\,c_{1},\,c_{2}\in\mathbb{R}$ such that

$\phi(z)=c_{0}Z_{0}+c_{1}Z_{1}+c_{2}Z_{2},$ $Z_{0}(z):={1-|z-\xi|^{2}\over
1+|z-\xi|^{2}},\ \;\;Z_{1}(z):={z_{1}-\xi_{1}\over 1+|z-\xi|^{2}},\
\;\;Z_{2}(z):={z_{2}-\xi_{2}\over 1+|z-\xi|^{2}}.$

##  3\. Analysis of the linearized operator

According to Proposition 2.3, by the change of variable $x=\delta z$, we
immediately get that all solutions $\psi$ of

$-\Delta\psi={8\delta^{2}\over(\delta^{2}+|x-b|^{2})^{2}}\psi=e^{W_{\lambda}}\psi\quad\hbox{in}\quad\mathbb{R}^{2},\qquad\int_{\mathbb{R}^{2}}|\nabla\phi(x)|^{2}dx<+\infty.$

are linear combinations of the functions

$Z^{0}_{\delta,b}(x)={\delta^{2}-|x-b|^{2}\over\delta^{2}+|x-b|^{2}},\
Z^{1}_{\delta,b}(x)={\delta(x_{1}-b_{1})\over\delta^{2}+|x-b|^{2}},\
Z^{2}_{\delta,b}(x)={\delta(x_{2}-b_{2})\over\delta^{2}+|x-b|^{2}}.$

We introduce the projections $PZ^{j}_{\delta,b}$ onto $H^{1}_{0}(B_{1})$. It
is immediate that

$PZ^{0}_{\delta,b}(x)=Z^{0}_{\delta,b}(x)+1+O\left(\delta^{2}\right)={2\delta^{2}\over\delta^{2}+|x-b|^{2}}+O(\delta^{2})$
(3.1)

and

$PZ^{j}_{\delta,b}(x)=Z^{j}_{\delta,b}(x)+O(\delta)\hbox{ for }j=1,2$ (3.2)

uniformly with respect to $x\in\overline{B}_{1}$ and $b>0$ in a small
neighborhood of $0.$

We agree that $Z_{\lambda}^{j}:=Z_{\delta,b}^{j}$ for any $j=0,1,2$, where
$\delta$ is defined in terms of $\lambda$ and $b$ according to (2.8). Let us
consider the following linear problem: given $h\in H^{1}_{0}(B_{1})$, find a
function $\phi\in H^{1}_{0}(B_{1})$ and constant $c_{1},c_{2}\in\mathbb{R}$
satisfying

$\left\\{\begin{aligned}
&-\Delta\phi-\frac{\lambda}{4}V\big{(}x^{\frac{1}{2}}\big{)}e^{P{W}_{\lambda}}\phi=\Delta
h+\sum_{j=1,2}c_{j}Z^{j}_{\lambda}e^{W_{\lambda}}\\\
&\int_{B_{1}}\nabla\phi\nabla
PZ^{j}_{\lambda}=0\;\;j=1,2\end{aligned}\right..$ (3.3)

In order to solve problem (3.3), we need to establish an a priori estimate.
For the proof we refer to [14] (Proposition 3.1) or [15] (Proposition 3.1).

######  Proposition 3.1.

There exist $\lambda_{0}>0$ and $C>0$ such that for any
$\lambda\in(0,\lambda_{0})$, any $b$ in a small neighborhood of $0$ and any
$h\in H^{1}_{0}(B_{1})$, if $(\phi,c_{1},c_{2})\in
H^{1}_{0}(B_{1})\times\mathbb{R}^{2}$ solves (3.3), then the following holds

$\|\phi\|\leq C|\log\delta|\|h\|.$

For any $p>1,$ let

$i^{*}_{p}:L^{p}(B_{1})\to H^{1}_{0}(B_{1})$ (3.4)

be the adjoint operator of the embedding
$i_{p}:H^{1}_{0}(B_{1})\hookrightarrow L^{p\over p-1}(B_{1}),$ i.e.
$u=i^{*}_{p}(v)$ if and only if $-\Delta u=v$ in $B_{1},$ $u=0$ on $\partial
B_{1}.$ We point out that $i^{*}_{p}$ is a continuous mapping, namely

$\|i^{*}_{p}(v)\|\leq c_{p}\|v\|_{p},\ \hbox{for any}\ v\in L^{p}(B_{1}),$
(3.5)

for some constant $c_{p}$ which depends on $p.$ Next let us set

${K}:=\hbox{span}\left\\{PZ^{1}_{\lambda},\,PZ_{\lambda}^{2}\right\\}$

and

${K^{\perp}}:=\left\\{\phi\in H^{1}_{0}(B_{1})\ :\
\int_{B_{1}}\nabla\phi\nabla PZ^{j}_{\lambda}dx=0\;\;j=1,2\right\\}$

and denote by

$\Pi:H^{1}_{0}(B_{1})\to{K},\qquad{\Pi^{\perp}}:H^{1}_{0}(B_{1})\to{K^{\perp}}$

the corresponding projections. Let $L:K^{\perp}\to K^{\perp}$ be the linear
operator defined by

$L(\phi):=\frac{1}{4}\Pi^{\perp}\Big{(}{i^{*}_{p}}\big{(}\lambda
V\big{(}x^{\frac{1}{2}}\big{)}e^{PW_{\lambda}}\phi\big{)}\Big{)}-\phi.$ (3.6)

Notice that problem (3.3) reduces to

$L(\phi)=\Pi^{\perp}h,\quad\phi\in K^{\perp}.$

As a consequence of Proposition 3.1 we derive the invertibility of $L$.

######  Proposition 3.2.

For any $p>1$ there exist $\lambda_{0}>0$ and $C>0$ such that for any
$\lambda\in(0,\lambda_{0})$, any $b$ in a small neighborhood of $0$ and any
$h\in K^{\perp}$ there is a unique solution $\phi\in K^{\perp}$ to the problem

$L(\phi)=h.$

In particular, $L$ is invertible; moreover,

$\|L^{-1}\|\leq C|\log\delta|.$

###### Proof.

Observe that the operator $\phi\mapsto\Pi^{\perp}\big{(}{i^{*}_{p}}(\lambda
V\big{(}x^{\frac{1}{2}}\big{)}e^{PW_{\lambda}}\phi)\big{)}$ is a compact
operator in $K^{\perp}$. Let us consider the case $h=0$, and take $\phi\in
K^{\perp}$ with $L(\phi)=0$. In other words, $\phi$ solves the system (3.3)
with $h=0$ for some $c_{1},c_{2}\in\mathbb{R}$. Proposition 3.1 implies
$\phi\equiv 0$. Then, Fredholm’s alternative implies the existence and
uniqueness result.

Once we have existence, the norm estimate follows directly from Proposition
3.1. ∎

##  4\. Estimate of the error term

The goal of this section is to provide an estimate of the error up to which
the approximate solution $PW_{\lambda}$ solves problem (2.4). First of all, we
perform the following estimates.

######  Lemma 4.1.

Let $\gamma=0,1,2$ and $p>1$ be fixed. The following holds:

$\||x-b|^{\gamma}e^{W_{\lambda}}\|_{p}\leq
C\delta^{\gamma}\delta^{-2\frac{p-1}{p}},\quad\||x-b|^{\gamma}\lambda
e^{PW_{\lambda}}\|_{p}\leq C\delta^{\gamma}\delta^{-2\frac{p-1}{p}}$ (4.1)

uniformly for $b$ in a small neighborhood of $0$.

###### Proof.

We compute

$\displaystyle\||x-b|^{\gamma}e^{W_{\lambda}}\|_{p}^{p}$
$\displaystyle=8^{p}\delta^{2p}\int_{B_{1}}\frac{|x-b|^{\gamma
p}}{(\delta^{2}+|x-b|^{2})^{2p}}dx\leq 8^{p}\delta^{\gamma
p-2(p-1)}\int_{\mathbb{R}^{2}}\frac{|z|^{\gamma p}}{(1+|z|^{2})^{2p}}dz.$

Taking into account that the last integral is finite for $\gamma=0,1,2$ and
$p>1$ we deduce the first part of (4.1). To prove the second part it is
sufficient to observe that by (2.9) and by the choice of $\delta$ in (2.8) we
derive

$\lambda
e^{PW_{\lambda}}=\frac{\lambda}{8\delta^{2}}e^{W_{\lambda}+O(1)}=e^{W_{\lambda}}(1+O(1)).$
(4.2)

∎

######  Lemma 4.2.

Assume that hypotheses ${\mathrm{(H1)-(H3)}}$ hold. There exists
$P:\mathbb{R}^{2}\to\mathbb{R}$ a homogeneous polynomial of degree 2 such that

$\displaystyle\frac{V(x^{\frac{1}{2}})}{V(b^{\frac{1}{2}})}$
$\displaystyle=1+2A_{0}b_{1}(x_{1}-b_{1})+\frac{A_{1}}{2}\Big{(}b_{1}(x_{2}-b_{2})+b_{2}(x_{1}-b_{1})\Big{)}+\Big{(}A_{0}+\frac{A_{2}}{2}\Big{)}b_{2}(x_{2}-b_{2})$
$\displaystyle\;\;\;\;+D_{0}\Big{(}(x_{1}-b_{1})^{3}+3b_{1}^{2}(x_{1}-b_{1})\Big{)}+\Big{(}\frac{D_{1}}{4}+\frac{D_{3}}{8}\Big{)}\Big{(}(x_{2}-b_{2})^{3}+3b_{2}^{2}(x_{2}-b_{2})\Big{)}$
$\displaystyle\;\;\;\;+\frac{D_{1}}{2}\Big{(}(x_{1}-b_{1})^{2}(x_{2}-b_{2})+b_{1}^{2}(x_{2}-b_{2})+2b_{1}b_{2}(x_{1}-b_{1})\Big{)}$
$\displaystyle\;\;\;\;+\frac{1}{4}\Big{(}3D_{0}-D_{2}\Big{)}\Big{(}(x_{1}-b_{1})(x_{2}-b_{2})^{2}+b_{2}^{2}(x_{1}-b_{1})+2b_{1}b_{2}(x_{2}-b_{2})\Big{)}$
$\displaystyle\;\;\;\;+P(x-b)+O(|b||x-b|^{2})+O(|b|^{3}|x-b|)+O(|x-b|^{\frac{7}{2}})+O(|b|^{\frac{7}{2}})$

uniformly for $b$ in a small neighborhood of $0$.

###### Proof.

Let us first consider a more general potential $V$ of the form

$\displaystyle V(x)$
$\displaystyle=1+\sum_{j=0}^{4}A_{j}x_{1}^{4-j}x_{2}^{j}+\sum_{j=0}^{6}D_{j}x_{1}^{6-j}x_{2}^{j}+O(|x|^{7}),\quad
A_{j},D_{j}\in\mathbb{R},$

and, using the polar coodinates $x=\rho e^{{\rm
i}\theta}=(\rho\cos\theta,\rho\sin\theta)$, we have

$\displaystyle V(x^{\frac{1}{2}})$
$\displaystyle=1+\rho^{2}\sum_{j=0}^{4}A_{j}\cos^{4-j}\frac{\theta}{2}\sin^{j}\frac{\theta}{2}+\rho^{3}\sum_{j=0}^{6}D_{j}\cos^{6-j}\frac{\theta}{2}\sin^{j}\frac{\theta}{2}+O(|x|^{\frac{7}{2}}).$

Now we use standard trigonometric identities to obtain:

$\cos^{4}\frac{\theta}{2}=\frac{\sin^{2}\theta+2\cos\theta+2\cos^{2}\theta}{4},\qquad\sin^{4}\frac{\theta}{2}=\frac{\sin^{2}\theta-2\cos\theta+2\cos^{2}\theta}{4}$
$\cos^{6}\frac{\theta}{2}=\frac{1+4\cos^{3}\theta+3\cos\theta\sin^{2}\theta+3\cos^{2}\theta}{8},\quad\sin^{6}\frac{\theta}{2}=\frac{1-4\cos^{3}\theta-3\sin^{2}\theta\cos\theta+3\cos^{2}\theta}{8}$
$\cos\frac{\theta}{2}\sin^{3}\frac{\theta}{2}=\sin\theta\frac{1-\cos\theta}{4},\qquad\sin\frac{\theta}{2}\cos^{3}\frac{\theta}{2}=\sin\theta\frac{1+\cos\theta}{4},\qquad\sin^{2}\frac{\theta}{2}\cos^{2}\frac{\theta}{2}=\frac{1}{4}\sin^{2}\theta,$
$\cos^{5}\frac{\theta}{2}\sin\frac{\theta}{2}=\sin\theta\frac{2\cos^{2}\theta+\sin^{2}\theta+2\cos\theta}{8},\qquad\sin^{5}\frac{\theta}{2}\cos\frac{\theta}{2}=\sin\theta\frac{2\cos^{2}\theta+\sin^{2}\theta-2\cos\theta}{8},$
$\sin^{3}\frac{\theta}{2}\sin^{3}\frac{\theta}{2}=\frac{1}{8}\sin^{3}\theta,\qquad\cos^{2}\frac{\theta}{2}\sin^{4}\frac{\theta}{2}=\sin^{2}\theta\frac{1-\cos\theta}{8},\qquad\sin^{2}\frac{\theta}{2}\cos^{4}\frac{\theta}{2}=\sin^{2}\theta\frac{1+\cos\theta}{8}.$

According to ${\mathrm{(H3)}}$ we get

$A_{0}=A_{4},\;\;A_{1}=-A_{3},\;\;D_{0}=-D_{6},\;\;D_{1}=D_{5},\;\;D_{4}=-D_{2},$
(4.3)

so we derive

$\displaystyle V(x^{\frac{1}{2}})$
$\displaystyle=1+A_{0}\Big{(}x_{1}^{2}+\frac{x_{2}^{2}}{2}\Big{)}+\frac{A_{1}}{2}x_{1}x_{2}+\frac{A_{2}}{4}x_{2}^{2}$
(4.4)
$\displaystyle\;\;\;+D_{0}\Big{(}x_{1}^{3}+\frac{3}{4}x_{1}x_{2}^{2}\Big{)}+D_{1}\Big{(}\frac{x_{1}^{2}x_{2}}{2}+\frac{x_{2}^{3}}{4}\Big{)}-\frac{D_{2}}{4}x_{1}x_{2}^{2}+\frac{D_{3}}{8}x_{2}^{3}+O(|x|^{3+\frac{1}{2}})$
$\displaystyle=1+A_{0}x_{1}^{2}+\frac{A_{1}}{2}x_{1}x_{2}+\Big{(}\frac{A_{0}}{2}+\frac{A_{2}}{4}\Big{)}x_{2}^{2}$
$\displaystyle\;\;\;+D_{0}x_{1}^{3}+\frac{D_{1}}{2}x_{1}^{2}x_{2}+\frac{1}{4}\Big{(}3D_{0}-D_{2}\Big{)}x_{1}x_{2}^{2}+\Big{(}\frac{D_{1}}{4}+\frac{D_{3}}{8}\Big{)}x_{2}^{3}+O(|x|^{3+\frac{1}{2}}).$

Next observe that, setting $x_{1}=(x_{1}-b_{1})+b_{1}$ and
$x_{2}=(x_{2}-b_{2})+b_{2}$ and making trivial computations we get

$\displaystyle
A_{0}x_{1}^{2}+\frac{A_{1}}{2}x_{1}x_{2}+\Big{(}\frac{A_{0}}{2}+\frac{A_{2}}{4}\Big{)}x_{2}^{2}$
$\displaystyle\;\;\;+D_{0}x_{1}^{3}+\frac{D_{1}}{2}x_{1}^{2}x_{2}+\frac{1}{4}\Big{(}3D_{0}-D_{2}\Big{)}x_{1}x_{2}^{2}+\Big{(}\frac{D_{1}}{4}+\frac{D_{3}}{8}\Big{)}x_{2}^{3}$
$\displaystyle=2A_{0}b_{1}(x_{1}-b_{1})+\frac{A_{1}}{2}\Big{(}b_{1}(x_{2}-b_{2})+b_{2}(x_{1}-b_{1})\Big{)}+\Big{(}A_{0}+\frac{A_{2}}{2}\Big{)}b_{2}(x_{2}-b_{2})$
$\displaystyle\;\;\;+D_{0}\Big{(}(x_{1}-b_{1})^{3}+3b_{1}^{2}(x_{1}-b_{1})\Big{)}+\frac{D_{1}}{2}\Big{(}(x_{1}-b_{1})^{2}(x_{2}-b_{2})+b_{1}^{2}(x_{2}-b_{2})+2b_{1}b_{2}(x_{1}-b_{1})\Big{)}$
$\displaystyle\;\;\;+\frac{1}{4}\Big{(}3D_{0}-D_{2}\Big{)}\Big{(}(x_{1}-b_{1})(x_{2}-b_{2})^{2}+b_{2}^{2}(x_{1}-b_{1})+2b_{1}b_{2}(x_{2}-b_{2})\Big{)}$
$\displaystyle\;\;\;+\Big{(}\frac{D_{1}}{4}+\frac{D_{3}}{8}\Big{)}\Big{(}(x_{2}-b_{2})^{3}+3b_{2}^{2}(x_{2}-b_{2})\Big{)}$
$\displaystyle\;\;\;+A_{0}b_{1}^{2}+\frac{A_{1}}{2}b_{1}b_{2}+\Big{(}\frac{A_{0}}{2}+\frac{A_{2}}{4}\Big{)}b_{2}^{2}+D_{0}b_{1}^{3}+\frac{D_{1}}{2}b_{1}^{2}b_{2}+\frac{1}{4}\Big{(}3D_{0}-D_{2}\Big{)}b_{1}b_{2}^{2}+\Big{(}\frac{D_{1}}{4}+\frac{D_{3}}{8}\Big{)}b_{2}^{3}$
$\displaystyle\;\;\;+P(x-b)+O(|b||x-b|^{2}).$

Finally, since by (4.4)

$\displaystyle V(b^{\frac{1}{2}})=$
$\displaystyle\;\,1+A_{0}b_{1}^{2}+\frac{A_{1}}{2}b_{1}b_{2}+\Big{(}\frac{A_{0}}{2}+\frac{A_{2}}{4}\Big{)}b_{2}^{2}$
$\displaystyle+D_{0}b_{1}^{3}+\frac{D_{1}}{2}b_{1}^{2}b_{2}+\frac{1}{4}\Big{(}3D_{0}-D_{2}\Big{)}b_{1}b_{2}^{2}+\Big{(}\frac{D_{1}}{4}+\frac{D_{3}}{8}\Big{)}b_{2}^{3}+O(|b|^{3+\frac{1}{2}})$

and, consequently, $\frac{1}{V(b^{\frac{1}{2}})}=1+O(|b|^{2})$, substituting
into (4.4) we obtain the thesis.

∎

######  Remark 4.3.

Let us observe that thanks to the symmetry of the coefficients (4.3) we obtain
that $V(x^{\frac{1}{2}})$ turns out to be three times differentiable at $0$:
indeed the choice of coefficients implies that the two sums
$\sum_{j=0}^{4}A_{j}\cos^{4-j}\frac{\theta}{2}\sin^{j}\frac{\theta}{2}$ and
$\sum_{j=0}^{6}D_{j}\cos^{6-j}\frac{\theta}{2}\sin^{j}\frac{\theta}{2}$ turn
out to be polynomials in the variables $\cos\theta,$ $\sin\theta$ of degree 2
and 3 respectively.

Now we are in the position to provide the error estimate.

######  Proposition 4.4.

Assume that hypotheses ${\mathrm{(H1)-(H2)-(H3)}}$ and (1.5) hold and define

$\displaystyle{R}_{\lambda}:=$
$\displaystyle\frac{\lambda}{4}V\big{(}x^{\frac{1}{2}}\big{)}e^{PW_{\lambda}}+\Delta
PW_{\lambda}=\frac{\lambda}{4}V\big{(}x^{\frac{1}{2}}\big{)}e^{PW_{\lambda}}-e^{W_{\lambda}}.$

Then the following holds

$\displaystyle R_{\lambda}=$
$\displaystyle\;2\delta^{2}e^{W_{\lambda}}+D_{0}e^{W_{\lambda}}\Big{(}(x_{1}-b_{1})^{3}+3b_{1}^{2}(x_{1}-b_{1})\Big{)}$
(4.5)
$\displaystyle+\frac{D_{1}}{2}e^{W_{\lambda}}\Big{(}(x_{1}-b_{1})^{2}(x_{2}-b_{2})+b_{1}^{2}(x_{2}-b_{2})+2b_{1}b_{2}(x_{1}-b_{1})\Big{)}$
$\displaystyle+\frac{1}{4}\Big{(}3D_{0}-D_{2}\Big{)}e^{W_{\lambda}}\Big{(}(x_{1}-b_{1})(x_{2}-b_{2})^{2}+b_{2}^{2}(x_{1}-b_{1})+2b_{1}b_{2}(x_{2}-b_{2})\Big{)}$
$\displaystyle+\Big{(}\frac{B_{1}}{4}+\frac{D_{3}}{8}\Big{)}e^{W_{\lambda}}\Big{(}(x_{2}-b_{2})^{3}+3b_{2}^{2}(x_{2}-b_{2})\Big{)}+P(x-b)e^{W_{\lambda}}$
$\displaystyle+O(\delta^{2}|x-b|)e^{W_{\lambda}}+O(|b||x-b|^{2})e^{W_{\lambda}}+O(|b|^{3}|x-b|)e^{W_{\lambda}}+O(|x-b|^{\frac{7}{2}})e^{W_{\lambda}}$
$\displaystyle+O(|b|^{\frac{7}{2}})e^{W_{\lambda}}+O(\delta^{2}|b|)e^{W_{\lambda}}+O(\delta^{4})e^{W_{\lambda}}$

uniformly for $b$ in a small neighborhood of $0$. Moreover for any $p>1$

$\|R_{\lambda}\|_{p}\leq C(\delta^{2}+|b|^{3})\delta^{-2\frac{p-1}{p}}$

uniformly for $b$ in a small neighborhood of $0$.

###### Proof.

By (2.9) and the choice of $\delta$ in (2.8) we derive

$\displaystyle\frac{\lambda}{4}V(x^{\frac{1}{2}})e^{PW_{\lambda}}$
$\displaystyle=\frac{\lambda}{32\delta^{2}}V(x^{\frac{1}{2}})e^{W_{\lambda}+8\pi
H(x,b)+2\delta^{2}+O(\delta^{2}|b|)+O(\delta^{4})}$ (4.6)
$\displaystyle=\frac{V(x^{\frac{1}{2}})}{V(b^{\frac{1}{2}})}e^{W_{\lambda}}e^{8\pi(H(x,b)-H(b,b))+2\delta^{2}+O(\delta^{2}|b|)+O(\delta^{4})}$
$\displaystyle=\frac{V(x^{\frac{1}{2}})}{V(b^{\frac{1}{2}})}e^{W_{\lambda}}e^{8\pi(H(x,b)-H(b,b))}\big{(}1+2\delta^{2}+O(\delta^{2}|b|)+O(\delta^{4})\big{)}.$

Using the expression of $H$ given in (1.4) we compute

$\displaystyle H(x,b)$
$\displaystyle=\frac{1}{4\pi}\log\Big{(}1+|x|^{2}|b|^{2}-2b_{1}x_{1}-2b_{2}x_{2}\Big{)}$
$\displaystyle=\frac{1}{4\pi}\log\Big{(}1+|x-b|^{2}|b|^{2}+|b|^{4}-2|b|^{2}-2b_{1}(x_{1}-b_{1})-2b_{2}(x_{2}-b_{2})+O(|b|^{3}|x-b|)\Big{)}$

by which

$\displaystyle e^{8\pi(H(x,b)-H(b,b))}$
$\displaystyle=\frac{(1+|x|^{2}|b|^{2}-2b_{1}x_{1}-2b_{2}x_{2})^{2}}{(1-|b|^{2})^{4}}$
$\displaystyle=\frac{\big{(}1+|x-b|^{2}|b|^{2}+|b|^{4}-2b_{1}(x_{1}-b_{1})-2b_{2}(x_{2}-b_{2})-2|b|^{2}+O(|b|^{3}|x-b|)\big{)}^{2}}{(1-|b|^{2})^{4}}$
$\displaystyle=\bigg{(}1+\frac{|x-b|^{2}|b|^{2}-2b_{1}(x_{1}-b_{1})-2b_{2}(x_{2}-b_{2})+O(|b|^{3}|x-b|)}{(1-|b|^{2})^{2}}\bigg{)}^{2}$
$\displaystyle=\Big{(}1-2b_{1}(x_{1}-b_{1})-2b_{2}(x_{2}-b_{2})+O(|b|^{3}|x-b|)+O(|b|^{2}|x-b|^{2})\Big{)}^{2}$
$\displaystyle=1-4b_{1}(x_{1}-b_{1})-4b_{2}(x_{2}-b_{2})+O(|b|^{2}|x-b|^{2})+O(|b|^{3}|x-b|).$

Then (4.6) becomes

$\displaystyle\frac{\lambda}{4}V(x^{\frac{1}{2}})e^{PW_{\lambda}}$
$\displaystyle=(1+2\delta^{2})\frac{V(x^{\frac{1}{2}})}{V(b^{\frac{1}{2}})}e^{W_{\lambda}}-4\frac{V(x^{\frac{1}{2}})}{V(b^{\frac{1}{2}})}e^{W_{\lambda}}(b_{1}(x_{1}-b_{1})+b_{2}(x_{2}-b_{2})$
(4.7)
$\displaystyle\;\;\;\;+e^{W_{\lambda}}\Big{(}O(|b|^{2}|x-b|^{2})+O(|b|^{3}|x-b|)+O(\delta^{2}|b|)+O(\delta^{4})\Big{)}.$

Using the expansion provided by Lemma 4.2 into (4.7), and the crucial
assumption (1.5), we get the estimate (4.5). Observe that (4.5) can be written
in more approximate way as

$R_{\lambda}=e^{W_{\lambda}}\Big{(}O(\delta^{2})+O(|b|^{2}|x-b|)+O(|x-b|^{2})+O(|b|^{\frac{7}{2}}\Big{)}.$

So, by applying Lemma 4.4 we obtain the $L^{p}$ estimate. ∎

######  Remark 4.5.

We observe that for general coefficients $A_{0},A_{1},A_{2}$, after
substituting the expansion of Lemma 4.2 into (4.7) we obtain that the
following term

$e^{W_{\lambda}}(2A_{0}-4)b_{1}(x_{1}-b_{1})+e^{W_{\lambda}}\Big{(}A_{0}+\frac{A_{2}}{2}-4\Big{)}b_{2}(x_{2}-b_{2})+e^{W_{\lambda}}\frac{A_{1}}{2}\Big{(}b_{1}(x_{2}-b_{2})+b_{2}(x_{1}-b_{1})\Big{)}$

does not vanish and actually shall represent the leading term in the estimate
of the error $R_{\lambda}$. This will explain later in Remark 6.1 why the
result of Theorem 1.1 fails in general without the assumption (1.5).

##  5\. The nonlinear problem: a contraction argument

In order to solve (2.4), let us consider the following intermediate problem:

$\left\\{\begin{aligned}
&-\Delta(PW_{\lambda}+\phi)-\frac{\lambda}{4}V(x^{\frac{1}{2}})e^{PW_{\lambda}+\phi}=\sum_{j=1,2}c_{j}Z_{\lambda}^{j}e^{W_{\lambda}},\\\
&\phi\in H^{1}_{0}(B_{1}),\;\;\;\;\int_{B_{1}}\nabla\phi\nabla
PZ_{\lambda}^{j}dx=0,\;\;j=1,2.\end{aligned}\right.$ (5.1)

Then it is convenient to solve as a first step the problem for $\phi$ as a
function of $b$.

Let us rewrite problem (5.1) in a more convenient way. In what follows we
denote by $N:H^{1}_{0}(B_{1})\to K^{\perp}$ the nonlinear operator

$\displaystyle N(\phi)$
$\displaystyle=\Pi^{\perp}\left({i^{*}_{p}}\bigg{(}\frac{\lambda}{4}V\big{(}x^{\frac{1}{2}}\big{)}e^{PW_{\lambda}}(e^{\phi}-1-\phi)\bigg{)}\right).$

Therefore problem (5.1) turns out to be equivalent to the problem

$L(\phi)+N(\phi)=\tilde{R},\quad\phi\in K^{\perp}$ (5.2)

where, recalling Lemma 4.1,

$\tilde{R}=\Pi^{\perp}\left({i^{*}_{p}}\big{(}R_{\lambda}\big{)}\right)=\Pi^{\perp}\left(PW_{\lambda}-{i^{*}_{p}}\bigg{(}\frac{\lambda}{4}V\big{(}x^{\frac{1}{2}}\big{)}e^{PW_{\lambda}}\bigg{)}\right).$

We need the following auxiliary lemma.

######  Lemma 5.1.

For any $p>1$ and any $\phi_{1},\phi_{2}\in H_{0}^{1}(B_{1})$ with
$\|\phi\|_{1},\,\|\phi_{2}\|<1$ the following holds

$\|e^{\phi_{1}}-\phi_{1}-e^{\phi_{2}}+\phi_{2}\|_{p}\leq
C(\|\phi_{1}\|+\|\phi_{2}\|)\|\phi_{1}-\phi_{2}\|,$ (5.3)
$\|N(\phi_{1})-N(\phi_{2})\|\leq
C\delta^{-2\frac{p^{2}-1}{p^{2}}}(\|\phi_{1}\|+\|\phi_{2}\|)\|\phi_{1}-\phi_{2}\|$
(5.4)

uniformly for $b$ in a small neighborhood of $0$.

###### Proof.

A straightforward computation gives that the inequality $|e^{a}-a-e^{b}+b|\leq
e^{|a|+|b|}(|a|+|b|)|a-b|$ holds for all $a,b\in\mathbb{R}$. Then, by applying
Hölder’s inequality with $\frac{1}{q}+\frac{1}{r}+\frac{1}{t}=1$, we derive

$\|e^{\phi_{1}}-\phi_{1}-e^{\phi_{2}}+\phi_{2}\|_{p}\leq
C\|e^{|\phi_{1}|+|\phi_{2}|}\|_{pq}(\|\phi_{1}\|_{pr}+\|\phi_{2}\|_{pr})\|\phi_{1}-\phi_{2}\|_{pt}$

and (5.3) follows by using Lemma 2.2 and the continuity of the embeddings
$H^{1}_{0}(B_{1})\subset L^{pr}(B_{1})$ and $H^{1}_{0}(B_{1})\subset
L^{pt}(B_{1})$. Let us prove (5.4). According to (3.5) we get

$\|{N}(\phi_{1})-{N}(\phi_{2})\|\leq C\|\lambda
V(x^{\frac{1}{2}})e^{PW_{\lambda}}(e^{\phi_{1}}-\phi_{1}-e^{\phi_{2}}+\phi_{2})\|_{p},$

and by Hölder’s inequality with $\frac{1}{p}+\frac{1}{q}=1$, we derive

$\displaystyle\|{N}(\phi_{1})-{N}(\phi_{2})\|$ $\displaystyle\leq C\|\lambda
V(x^{\frac{1}{2}})e^{PW_{\lambda}}\|_{p^{2}}\|e^{\phi_{1}}-\phi_{1}-e^{\phi_{2}}+\phi_{2}|\|_{pq}$
$\displaystyle\leq C\|\lambda
V(x^{\frac{1}{2}})e^{PW_{\lambda}}\|_{p^{2}}(\|\phi_{1}\|+\|\phi_{2}\|)\|\phi_{1}-\phi_{2}\|$

by (5.3), and the conclusion follows by Lemma 4.1. ∎

Problem (5.1) or, equivalently, problem (5.2) turns out to be solvable for any
choice of point $b$ in a small neighbourhood of $0$, provided that $\lambda$
is sufficiently small. Indeed we have the following result.

######  Proposition 5.2.

Assume ${\mathrm{(H1)-(H2)-(H3)}}$ and (1.5) hold and let $\varepsilon>0$ be a
fixed small number. Then there exists $\lambda_{0}>0$ such that for any
$\lambda\in(0,\lambda_{0})$ and any $b\in\mathbb{R}^{2}$ with
$|b|\leq\delta^{\frac{2}{3}}$ there is a unique
$\phi_{\lambda}=\phi_{\lambda,b}\in K^{\perp}$ satisfying (5.1) for some
$c_{1},c_{2}\in\mathbb{R}$ and

$\|\phi_{\lambda}\|\leq\delta^{2-\varepsilon}.$ (5.5)

Moreover the map $b\mapsto\phi_{\lambda,b}\in H^{1}_{0}(B_{1})$ is continuous.

###### Proof.

Since problem (5.2) is equivalent to problem (5.1), we will show that problem
(5.2) can be solved via a contraction mapping argument. Indeed, in virtue of
Proposition 3.2, let us introduce the map

$T:=L^{-1}(\tilde{R}-N(\phi)),\quad\phi\in K^{\perp}.$

Let us fix $p>1$ sufficiently close to 1. By (3.5) and Proposition 4.4, if
$|b|\leq\delta^{\frac{2}{3}}$ we get

$\|\tilde{R}\|\leq C\delta^{2-\frac{\varepsilon}{2}}.$ (5.6)

Next, by (5.4),

$\|N(\phi_{1})-N(\phi_{2})\|\leq
C\delta^{-\frac{\varepsilon}{2}}(\|\phi_{1}\|+\|\phi_{2}\|)\|\phi_{1}-\phi_{2}\|\quad\forall\phi_{1},\phi_{2}\in
H_{0}^{1}(B_{1}),\|\phi_{1}\|,\|\phi_{2}\|<1.$ (5.7)

In particular, by taking $\phi_{2}=0$,

$\|N(\phi)\|\leq
C\delta^{-\frac{\varepsilon}{2}}\|\phi\|^{2}\quad\forall\phi\in
H_{0}^{1}(B_{1}),\|\phi\|<1.$ (5.8)

We claim that $T$ is a contraction map over the ball

${\cal B}:=\Big{\\{}\phi\in
K^{\perp}\,\Big{|}\,\|\phi\|\leq\delta^{2-\varepsilon}\Big{\\}}$

provided that $\lambda$ is small enough. Indeed, combining Proposition 3.2,
(5.6), (5.7), (5.8), for any $\phi\in{\cal B}$ we have

$\displaystyle\|T(\phi)\|$ $\displaystyle\leq
C|\log\delta|\big{(}\|\tilde{R}\|+\|N(\phi)\|\big{)}\leq
C|\log\delta|\delta^{2-\frac{\varepsilon}{2}}<\delta^{2-\varepsilon}.$

Similarly, for any $\phi_{1},\phi_{2}\in{\cal B}$

$\|T(\phi_{1})-T(\phi_{2})\|\leq C|\log\delta|\|N(\phi_{1})-N(\phi_{2})\|\leq
C\delta^{-\frac{\varepsilon}{2}}|\log\delta|(\|\phi_{1}\|+\|\phi_{2}\|)\|\phi_{1}-\phi_{2}\|\leq\frac{1}{2}\|\phi_{1}-\phi_{2}\|.$

Uniqueness of solutions implies continuous dependence of
$\phi_{\lambda}=\phi_{\lambda,b}$ on $b$. ∎

##  6\. Proof of Theorems 1.1 and Theorem 2.1

During this section we assume that the crucial assumption
${\mathrm{(H1)-(H2)-(A3)}}$ and (1.5) of Theorem 1.1 hold.

After problem (5.1) has been solved according to Proposition 5.2, then we find
a solution to the original problem (2.4) if $b\in\mathbb{R}^{2}$ is such that
$|b|\leq\delta^{\frac{2}{3}}$ and

$c_{1}=c_{2}=0.$

Let us find the condition satisfied by $b$ in order to get $c_{1},c_{2}$ equal
to zero.

### Proof of Theorem 2.1

We multiply the equation in (5.1) by $PZ_{\lambda}^{i}$ and integrate over
$B_{1}$:

$\displaystyle\int_{B_{1}}\nabla(PW_{\lambda}+\phi_{\lambda})\nabla
PZ^{i}_{\lambda}dx$
$\displaystyle-\frac{\lambda}{4}\int_{B_{1}}V\big{(}x^{\frac{1}{2}}\big{)}e^{PW_{\lambda}+\phi_{\lambda}}PZ^{i}_{\lambda}dx$
(6.1)
$\displaystyle=\sum_{h=1,2}c_{h}\int_{B_{1}}Z_{\lambda}^{h}e^{W_{\lambda}}PZ_{\lambda}^{i}dx.$

The object is now to expand each integral of the above identity and analyze
the leading term. In the remaining part of the section all the estimates hold
uniformly for $|b|\leq\delta^{\frac{2}{3}}$, without further notice.

Let us begin by observing that the orthogonality in (5.1) gives

$\int_{B_{1}}\nabla\phi_{\lambda}\nabla
PZ^{i}_{\lambda}dx=\int_{B_{1}}e^{W_{\lambda}}\phi_{\lambda}Z_{\lambda}^{i}dx=0$
(6.2)

and, by (3.2),

$\displaystyle\int_{B_{1}}Z_{\lambda}^{h}e^{W_{\lambda}}PZ_{\lambda}^{i}dx=\int_{\mathbb{R}^{2}}\frac{8z_{i}z_{h}}{(1+|z|^{2})^{4}}dz+o(1)=\left\\{\begin{aligned}
&\frac{2}{3}\pi+o(1)&\hbox{ if }&h=i\\\ &o(1)&\hbox{ if }&h\neq
i\end{aligned}\right.$ (6.3)

where we have used that
$\int_{\mathbb{R}^{2}}\frac{z_{i}^{2}}{(1+|z|^{2})^{4}}dz=\frac{2}{3}\pi$ and
$\int_{\mathbb{R}^{2}}\frac{z_{1}z_{2}}{(1+|z|^{2})^{4}}dz=0.$ Using the
definition of $R_{\lambda}$ in Lemma 4.4, (6.2) and (6.3), then (6.1) becomes

$\displaystyle\int_{B_{1}}R_{\lambda}PZ^{i}_{\lambda}dx$
$\displaystyle+\frac{\lambda}{4}\int_{B_{1}}V(x^{\frac{1}{2}})e^{PW_{\lambda}}(e^{\phi_{\lambda}}-1)PZ^{i}_{\lambda}dx=\left\\{\begin{aligned}
&-\frac{2}{3}\pi+o(1)&\hbox{ if }&h=i\\\ &o(1)&\hbox{ if }&h\neq
i\end{aligned}\right..$ (6.4)

Let us first estimate the term containing the function $\phi_{\lambda}$:
recalling (6.2)

$\displaystyle\frac{\lambda}{4}\int_{B_{1}}V(x^{\frac{1}{2}})e^{PW_{\lambda}}(e^{\phi_{\lambda}}-1)PZ^{i}_{\lambda}dx$
$\displaystyle=\int_{B_{1}}R_{\lambda}(e^{\phi_{\lambda}}-1)PZ^{i}_{\lambda}dx$
(6.5)
$\displaystyle\;\;\;\;+\int_{B_{1}}e^{W_{\lambda}}(e^{\phi_{\lambda}}-1-\phi_{\lambda})PZ^{i}_{\lambda}dx$
$\displaystyle\;\;\;\;+\int_{B_{1}}e^{W_{\lambda}}\phi_{\lambda}(PZ^{i}_{\lambda}-Z_{\lambda}^{i})dx.$

Now, let us fix $\varepsilon>0$ sufficiently small and $p>1$ sufficiently
close to 1. Next let $1<q<\infty$ be such that $\frac{1}{p}+\frac{1}{q}=1$.
Then, (5.3) with $\phi_{2}=0$ and Proposition 5.2 give

$\|e^{\phi_{\lambda}}-1-\phi_{\lambda}\|_{q}\leq C\|\phi_{\lambda}\|^{2}\leq
C\delta^{4-2\varepsilon}$

and, consequently,

$\|e^{\phi_{\lambda}}-1\|_{q}\leq C\|\phi_{\lambda}\|\leq
C\delta^{2-\varepsilon}.$ (6.6)

Therefore, Lemma 4.1 implies

$\displaystyle\int_{B_{1}}e^{W_{\lambda}}(e^{\phi_{\lambda}}-1-\phi_{\lambda})PZ^{i}_{\lambda}dx$
$\displaystyle=O(\|e^{W_{\lambda}}(e^{\phi_{\lambda}}-1-\phi_{\lambda})\|_{1})=O(\|e^{W_{\lambda}}\|_{p}\|e^{\phi_{\lambda}}-1-\phi_{\lambda}\|_{q})$
(6.7) $\displaystyle=O\Big{(}\delta^{4-2\frac{p-1}{p}-2\varepsilon}\Big{)}.$

Now, by Lemma 4.4

$\displaystyle\int_{B_{1}}R_{\lambda}(e^{\phi_{\lambda}}-1)PZ^{i}_{\lambda}dx$
$\displaystyle=O\big{(}\big{\|}R_{\lambda}(e^{\phi_{\lambda}}-1)\big{\|}_{1}\big{)}=O\big{(}\big{\|}R_{\lambda}\|_{p}\|e^{\phi_{\lambda}}-1\|_{q}\big{)}$
(6.8) $\displaystyle=O\Big{(}\delta^{4-2\frac{p-1}{p}-\varepsilon}\Big{)}.$

Finally by Lemma A.3 and Lemma 4.1, using that $|b|\leq\delta^{\frac{2}{3}}$,

$\displaystyle\int_{B_{1}}e^{W_{\lambda}}\phi_{\lambda}(PZ^{i}_{\lambda}-Z_{\lambda}^{i})dx$
$\displaystyle=-\delta\int_{B_{1}}e^{W_{\lambda}}\phi_{\lambda}(x_{1}-b_{1})dx+O\bigg{(}\delta^{\frac{5}{3}}\int_{B_{1}}e^{W_{\lambda}}|\phi_{\lambda}|dx\bigg{)}$
(6.9)
$\displaystyle=O(\delta\||x-b|e^{W_{\lambda}}\|_{p}\|\phi_{\lambda}\|)+O(\delta^{\frac{5}{3}}\|e^{W_{\lambda}}\|_{p}\|\phi_{\lambda}\|)$
$\displaystyle=O(\delta^{4-\varepsilon-2\frac{p-1}{p}})+O(\delta^{\frac{11}{3}-2\frac{p-1}{p}-\varepsilon}).$

By inserting (6.7)-(6.8)-(6.9) into (6.5), we obtain

$\lambda\int_{B_{1}}V\big{(}x^{\frac{1}{\alpha}}\big{)}e^{PW_{\lambda}}(e^{\phi_{\lambda}}-1)PZ^{i}_{\lambda}dx=O(\delta^{3})$
(6.10)

provided that that $\varepsilon$ is chosen sufficiently close to $0$ and $p$
sufficiently close to $1$. Next, by (4.5), using Lemma A.2 and Lemma A.4, we
get

$\displaystyle\int_{B_{1}}R_{\lambda}PZ_{\lambda}^{1}dx$
$\displaystyle=2\pi\delta\bigg{(}3b_{1}^{2}D_{0}+D_{1}b_{1}b_{2}+\frac{15D_{0}-D_{2}}{4}\delta^{2}\log\frac{1}{\delta}+\frac{3D_{0}-D_{2}}{4}b_{2}^{2}\bigg{)}$
$\displaystyle\;\;\;\;+O(\delta^{3})+O(\delta^{2}|b|)+O(|b|^{\frac{7}{2}})+O(|b|^{3}\delta).$
$\displaystyle\int_{B_{1}}R_{\lambda}PZ_{\lambda}^{2}dx$
$\displaystyle=2\pi\delta\bigg{(}\frac{D_{1}}{2}b_{1}^{2}+\frac{3D_{0}-D_{2}}{2}b_{1}b_{2}+\frac{10D_{1}+3D_{3}}{8}\delta^{2}\log\frac{1}{\delta}+3\frac{2D_{1}+D_{3}}{8}b_{2}^{2}\bigg{)}$
$\displaystyle\;\;\;\;+O(\delta^{3})+O(\delta^{2}|b|)+O(|b|^{\frac{7}{2}})+O(|b|^{3}\delta).$

By inserting the above identity and (6.10) into (6.4) we deduce

$2\pi\delta^{2}\log\frac{1}{\delta}F\bigg{(}\frac{b}{\delta\sqrt{\log\frac{1}{\delta}}}\bigg{)}+O(\delta^{3})+O(\delta^{2}|b|)+O(|b|^{\frac{7}{2}})+O(|b|^{3}\delta)=-\frac{2}{3}\pi
c+o(|c|),$ (6.11)

where $c=(c_{1},c_{2})$ and $F:\mathbb{R}^{2}\to\mathbb{R}^{2}$ denotes the
vector field defined in (1.6).

Now let $\xi_{0}\neq 0$ be a zero for $F$ which is stable under uniform
perturbations according to Theorem 1.1; then (6.11) gives that the following
holds

$\displaystyle\delta^{2}\log\frac{1}{\delta}F\bigg{(}\frac{b}{\delta\sqrt{\log\frac{1}{\delta}}}\bigg{)}+o\bigg{(}\delta^{2}\log\frac{1}{\delta}\bigg{)}=-\frac{c}{3}+o(|c|)\;\hbox{
unif. for }|b|\leq 2|\xi_{0}|\delta\sqrt{\log\frac{1}{\delta}}.$ (6.12)

Now, setting

$\tilde{b}=\frac{b}{\delta\sqrt{\log\frac{1}{\delta}}},$

we rewrite (6.12) as

$\displaystyle\delta^{2}\log\frac{1}{\delta}\bigg{(}F(\tilde{b})+o(1)\bigg{)}=-\frac{c}{3}+o(|c|)\;\hbox{
unif. for }|\tilde{b}|\leq 2|\xi_{0}|.$ (6.13)

The continuity of the map $b\mapsto\phi_{\lambda}=\phi_{\lambda,b}$ guaranteed
by Proposition 5.2 implies that the left hand side of (6.13) is continuous
too. So, the uniform stability gives that, if $\eta>0$ is sufficiently small,
then for $\lambda$ small enough the left hand side of (6.13) has a zero
$\tilde{b}_{\lambda}$ with $|\tilde{b}_{\lambda}-\xi_{0}|\leq\eta$ or,
equivalently, the left hand side of (6.12) has a zero $b_{\lambda}$ with
$\big{|}\frac{b_{\lambda}}{\delta\sqrt{\log\frac{1}{\delta}}}-\xi_{0}\big{|}\leq\eta.$
The arbitrariness of $\eta$ implies

$b_{\lambda}=\xi_{0}\delta\sqrt{\log\frac{1}{\delta}}(1+o(1)).$

######  Remark 6.1.

We point put that, for general coefficients $A_{0},A_{1},A_{2}$, then the
error term $R_{\lambda}$ reduces to the expression in Remark 4.5 at the
leading part and, thanks to Lemma A.2, when we multiply it against
$PZ_{i}^{\lambda}$ we actually obtain

$2\pi\delta(2A_{0}-4)b_{1}+\pi A_{1}\delta b_{2}+h.o.t,\quad
2\pi\delta\Big{(}A_{0}+\frac{A_{2}}{2}-4\Big{)}b_{2}+\pi A_{1}\delta
b_{1}+h.o.t.$

which in general admits the only trivial zero $b=0$ for the leading term, so
we are unable to catch a non-simple blow-up solution without the assumption
(1.5).

###  6.1. Proof of Theorems 1.1.

Theorem 2.1 provides a solution to the problem (2.4) of the form

$w_{\lambda}=PW_{\lambda}+\phi_{\lambda}$

where $\phi_{\lambda}=\phi_{\lambda,b_{\lambda}}\in H^{1}_{0}(B_{1})$
satisfies (5.5) and $b=b_{\lambda}$ satisfies (1.7).

Moreover, using (6.6) and Lemma 4.1, by Hölder’s inequality with
$\frac{1}{p}+\frac{1}{q}=1$ we get

$\displaystyle\lambda\|V(y^{\frac{1}{2}})(e^{{w}_{\lambda}}-e^{{PW}_{\lambda}})\|_{1}$
$\displaystyle=\lambda\|V(y^{\frac{1}{2}})e^{{PW}_{\lambda}}(e^{{\phi}_{\lambda}}-1)\|_{1}$
$\displaystyle\leq\lambda\|e^{{PW}_{\lambda}}\|_{p}\|e^{\phi_{\lambda}}-1\|_{q}$
$\displaystyle=O(\delta^{2-2\frac{p-1}{p}-\varepsilon})=o(1),$

if $p$ is chosen sufficiently close to 1 and $\varepsilon$ sufficiently close
to $0$. Similarly, by Proposition 4.4,

$\displaystyle\Big{\|}\frac{\lambda}{4}V(y^{\frac{1}{2}})e^{{PW}_{\lambda}}-e^{W_{\lambda}}\Big{\|}_{1}$
$\displaystyle=\|R_{\lambda}\|_{1}=O(\delta^{2-2\frac{p-1}{p}})=o(1).$

Therefore

$\displaystyle\Big{\|}\frac{\lambda}{4}V(y^{\frac{1}{2}})e^{w_{\lambda}}-e^{W_{\lambda}}\Big{\|}_{1}=o(1).$

Clearly, by (2.1) and (2.3),

$u_{\lambda}(x)=w_{\lambda}(x^{2})-4\pi
G(x,0)=w_{\lambda}(x^{2})-2\log\frac{1}{|x|}$

solves equation (1.1) and

$\displaystyle\|\lambda
V(x)e^{u_{\lambda}(x)}-4|x|^{2}e^{W_{\lambda}(x^{2})}\|_{1}$
$\displaystyle=4\Big{\|}\frac{\lambda}{4}|x|^{2}V(x)e^{{w}_{\lambda}(x^{2})}-|x|^{2}e^{W_{\lambda}(x^{2})}\Big{\|}_{1}$
$\displaystyle=2\Big{\|}\frac{\lambda}{4}V(y^{\frac{1}{2}})e^{{w}_{\lambda}(y)}-e^{W_{\lambda}(y)}\Big{\|}_{1}=o(1)$

by Lemma A.5. Hence, recalling (2.7) and Lemma A.5,

$\displaystyle\lambda\int_{B_{1}}V(x)e^{u_{\lambda}}dx$
$\displaystyle=4\int_{\mathbb{R}^{2}}|x|^{2}V(x)e^{{W}_{\lambda}(x^{2})}dx+o(1)$
$\displaystyle=2\int_{\mathbb{R}^{2}}V(y^{\frac{1}{2}})e^{W_{\lambda}(y)}dy+o(1)=16\pi+o(1).$

Similarly for every neighborhood $U$ of $0$

$\lambda\int_{U}V(x)e^{u_{\lambda}}dx\to 16\pi.$

Theorem 1.1 is thus completely proved by setting $\mu^{2}=\delta$.

## Appendix A

In this appendix we derive some crucial integral estimates which arise in the
asymptotic expansion of the energy of approximate solution $PW_{\lambda}$.

######  Lemma A.1.

The following holds:

$\int_{B_{1}}e^{W_{\lambda}}|x-b|dx=O(\delta),\quad\int_{B_{1}}e^{W_{\lambda}}|x-b|^{2}dx=16\pi\delta^{2}|\log\delta|+O(\delta^{2}),\quad\int_{B_{1}}e^{W_{\lambda}}|x-b|^{3}dx=O(\delta^{2})$

uniformly for $b$ in a small neighborhood of $0$.

###### Proof.

We compute

$\displaystyle\int_{B_{1}}e^{W_{\lambda}}|x-b|dx$ $\displaystyle\leq
8\delta\int_{\mathbb{R}^{2}}\frac{1}{(1+|z-\delta^{-1}b|^{2})^{2}}|z-\delta^{-1}b|dz=8\delta\int_{\mathbb{R}^{2}}\frac{|z|}{(1+|z|^{2})^{2}}dz$

and the first estimate follows. In order to show the second estimate let us
observe that $B(b,1-|b|)\subset B(0,1)\subset B(b,1+|b|)$, so we compute

$\displaystyle\int_{B_{1}}e^{W_{\lambda}}|x-b|^{2}dx$
$\displaystyle=8\int_{B_{1}}\frac{\delta^{2}|x-b|^{2}}{(\delta^{2}+|x-b|^{2})^{2}}dx$
$\displaystyle=8\int_{B(b,1-|b|)}\frac{\delta^{2}|x-b|^{2}}{(\delta^{2}+|x-b|^{2})^{2}}dx+O\bigg{(}\int_{B(b,1+|b|)\setminus
B(b,1-|b|)}\frac{\delta^{2}|x-b|^{2}}{(\delta^{2}+|x-b|^{2})^{2}}dx\bigg{)}$
$\displaystyle=8\int_{B(0,1-|b|)}\frac{\delta^{2}|x|^{2}}{(\delta^{2}+|x|^{2})^{3}}dx+O\bigg{(}\int_{B(0,1+|b|)\setminus
B(0,1-|b|)}\frac{\delta^{2}|x|^{2}}{(\delta^{2}+|x|^{2})^{2}}dx\bigg{)}$
$\displaystyle=8\delta^{2}\int_{|z|\leq\frac{1-|b|}{\delta}}\frac{|z|^{2}}{(1+|z|^{2})^{2}}dz+O\bigg{(}\delta^{2}\int_{\frac{1-|b|}{\delta}\leq|z|\leq\frac{1+|b|}{\delta}}\frac{1}{|z|^{2}}dz\bigg{)}$
$\displaystyle=8\delta^{2}\int_{|z|\leq\frac{1-|b|}{\delta}}\frac{1}{1+|z|^{2}}dz+O(\delta^{2})$
$\displaystyle=16\pi\delta^{2}|\log\delta|+O(\delta^{2}).$

In order to prove the third estimate, let $R>1$ so that $B(0,1)\subset B(b,R)$
if $b$ lies in a small neighborhood of $0$. Then,

$\displaystyle\int_{B_{1}}e^{W_{\lambda}}|x-b|^{3}dx$
$\displaystyle=8\delta^{3}\int_{|z|\leq\frac{1}{\delta}}\frac{1}{(1+|z-\delta^{-1}b|^{2})^{2}}|z-\delta^{-1}b|^{3}dz$
$\displaystyle\leq
8\delta^{3}\int_{B(0,\frac{R}{\delta})}\frac{|z|^{3}}{(1+|z|^{2})^{2}}dz\leq
C\delta^{2}.$

∎

Since the key part in the proof of Theorem 2.1 relies in testing the equation
(5.1) with $PZ_{\lambda}^{i}$ in order to catch the leading terms, a crucial
step consists in the evaluation of some integral estimates, as provided by the
following lemma.

######  Lemma A.2.

The following holds for $i,j=1,2$:

$\int_{B_{1}}e^{W_{\lambda}}PZ^{i}_{\lambda}dx=O(\delta),$
$\int_{B_{1}}e^{W_{\lambda}}PZ^{i}_{\lambda}(x_{i}-b_{i})dx=2\pi\delta+O(\delta^{2}),\quad\int_{B_{1}}e^{W_{\lambda}}PZ^{i}_{\lambda}(x_{j}-b_{j})dx=O(\delta^{2})\quad
i\neq j,$
$\int_{B_{1}}e^{W_{\lambda}}|PZ^{i}_{\lambda}||x-b|^{2}dx=O(\delta^{2}),$
$\int_{B_{1}}e^{W_{\lambda}}PZ^{i}_{\lambda}(x_{i}-b_{i})^{3}dx=6\pi\delta^{3}\log\frac{1}{\delta}+O(\delta^{3})\quad\int_{B_{1}}e^{W_{\lambda}}PZ^{i}_{\lambda}(x_{j}-b_{j})^{3}dx=O(\delta^{3})\quad
i\neq j,$
$\quad\int_{B_{1}}e^{W_{\lambda}}PZ^{i}_{\lambda}(x_{j}-b_{j})^{2}(x_{i}-b_{i})dx=2\pi\delta^{3}\log\frac{1}{\delta}+O(\delta^{3})\quad
i\neq j,$
$\int_{B_{1}}e^{W_{\lambda}}PZ^{i}_{\lambda}(x_{i}-b_{i})^{2}(x_{j}-b_{j})dx=O(\delta^{3})\quad
i\neq j,$
$\int_{B_{1}}e^{W_{\lambda}}|PZ^{i}_{\lambda}||x-b|^{\frac{7}{2}}dx=O(\delta^{3})$

uniformly for $b$ in a small neighborhood of $0$.

###### Proof.

We compute

$\displaystyle\int_{B_{1}}e^{W_{\lambda}}Z^{i}_{\lambda}dx$
$\displaystyle=8\int_{|z|\leq\frac{1}{\delta}}\frac{1}{(1+|z-\delta^{-1}b|^{2})^{3}}(z_{i}-\delta^{-1}b_{i})dz$
$\displaystyle=8\int_{\mathbb{R}^{2}}\frac{1}{(1+|z-\delta^{-1}b|^{2})^{3}}(z_{i}-\delta^{-1}b_{i})dz+O(\delta^{3})$
$\displaystyle=8\int_{\mathbb{R}^{2}}\frac{z_{i}}{(1+|z|^{2})^{3}}dz+O(\delta^{3})=O(\delta^{3}),$

since $\int_{\mathbb{R}^{2}}\frac{z_{i}}{(1+|z||^{2})^{3}}dz=0$ by oddness.
Next

$\displaystyle\int_{B_{1}}e^{W_{\lambda}}Z^{i}_{\lambda}(x_{i}-b_{i})dx$
$\displaystyle=8\delta\int_{|z|\leq\frac{1}{\delta}}\frac{1}{(1+|z-\delta^{-1}b|^{2})^{3}}(z_{i}-\delta^{-1}b_{i})^{2}dz$
$\displaystyle=8\delta\int_{\mathbb{R}^{2}}\frac{1}{(1+|z-\delta^{-1}b|^{2})^{3}}(z_{i}-\delta^{-1}b_{i})^{2}dz+O(\delta^{3})$
$\displaystyle=8\delta\int_{\mathbb{R}^{2}}\frac{z_{i}^{2}}{(1+|z|^{2})^{3}}dz+O(\delta^{3})$
$\displaystyle=2\pi\delta+O(\delta^{3})$

where we have used the identity
$\int_{\mathbb{R}^{2}}\frac{(z_{i})^{2}}{(1+|z|^{2})^{3}}=\frac{1}{2}\int_{\mathbb{R}^{2}}\frac{|z|^{2}}{(1+|z|^{2})^{3}}=\frac{\pi}{4}$.
Similarly for $i\neq j$

$\displaystyle\int_{B_{1}}e^{W_{\lambda}}Z^{i}_{\lambda}(x_{j}-b_{j})dx$
$\displaystyle=8\delta\int_{\mathbb{R}^{2}}\frac{z_{i}z_{j}}{(1+|z|^{2})^{3}}dz+O(\delta^{3})=O(\delta^{3})$

since $\int_{\mathbb{R}^{2}}\frac{z_{i}z_{j}}{(1+|z|^{2})^{3}}dz=0.$ Next,

$\displaystyle\int_{B_{1}}e^{W_{\lambda}}|Z^{i}_{\lambda}||x-b|^{2}dx$
$\displaystyle\leq
8\delta^{2}\int_{\mathbb{R}^{2}}\frac{|x-b|^{3}}{(\delta^{2}+|x-b|^{2})^{3}}dx=8\delta^{2}\int_{\mathbb{R}^{2}}\frac{|z|^{3}}{(1+|z|^{2})^{3}}dz\leq
C\delta^{2}.$

Using that $B(b,1-|b|)\subset B(0,1)\subset B(b,1+|b|)$, we compute

$\displaystyle\int_{B_{1}}e^{W_{\lambda}}Z^{i}_{\lambda}(x_{i}-b_{i})^{3}dx$
$\displaystyle=8\delta^{3}\int_{B_{1}}\frac{(x_{i}-b_{i})^{4}}{(\delta^{2}+|x-b|^{2})^{3}}dx$
$\displaystyle=8\delta^{3}\int_{B(b,1-|b|)}\frac{(x_{i}-b_{i})^{4}}{(\delta^{2}+|x-b|^{2})^{3}}dx+O\bigg{(}\delta^{3}\int_{B(b,1+|b|)\setminus
B(b,1-|b|)}\frac{|x-b|^{4}}{(\delta^{2}+|x-b|^{2})^{3}}dx\bigg{)}$
$\displaystyle=8\delta^{3}\int_{B(0,1-|b|)}\frac{(x_{i})^{4}}{(\delta^{2}+|x|^{2})^{3}}dx+O\bigg{(}\delta^{3}\int_{B(0,1+|b|)\setminus
B(0,1-|b|)}\frac{|x|^{4}}{(\delta^{2}+|x|^{2})^{3}}dx\bigg{)}$
$\displaystyle=8\delta^{3}\int_{|z|\leq\frac{1-|b|}{\delta}}\frac{(z_{i})^{4}}{(1+|z|^{2})^{3}}dz+O(\delta^{3})$
$\displaystyle=6\pi\delta^{3}|\log\delta|+O(\delta^{3})$

where we have used the identity $\int_{|z|\leq
r}\frac{(z_{i})^{4}}{(1+|z|^{2})^{3}}dz=\frac{3}{8}\pi\log(1+r^{2})+\frac{3}{4}\frac{\pi}{1+r^{2}}-\frac{3}{16}\frac{\pi}{(1+r^{2})^{2}}-\frac{9}{16}\pi$.

Similarly, for $i\neq j$

$\displaystyle\int_{B_{1}}e^{W_{\lambda}}Z^{i}_{\lambda}(x_{j}-b_{j})^{3}dx$
$\displaystyle=8\delta^{3}\int_{|z|\leq\frac{1-|b|}{\delta}}\frac{z_{i}(z_{j})^{3}}{(1+|z|^{2})^{3}}dz+O(\delta^{3})=O(\delta^{3})$

since $\int_{|z|\leq r}\frac{z_{i}(z_{j})^{3}}{(1+|z|^{2})^{3}}dz=0$.

Next, for $i\neq j$

$\displaystyle\int_{B_{1}}e^{W_{\lambda}}Z^{i}_{\lambda}(x_{j}-b_{j})^{2}(x_{i}-b_{i})dx$
$\displaystyle=8\delta^{3}\int_{|z|\leq\frac{1-|b|}{\delta}}\frac{(z_{i})^{2}(z_{j})^{2}}{(1+|z|^{2})^{3}}dz+O(\delta^{3})$
$\displaystyle=2\pi\delta^{3}|\log\delta|+O(\delta^{3})$

where the last equality follows by $\int_{|z|\leq
r}\frac{(z_{i})^{2}(z_{j})^{2}}{(1+|z|^{2})^{3}}dz=\frac{\pi}{8}\log(1+r^{2})+\frac{1}{4}\frac{\pi}{1+r^{2}}-\frac{1}{16}\frac{\pi}{(1+r^{2})^{2}}-\frac{3}{16}\pi$.
Similarly for $i\neq j$

$\displaystyle\int_{B_{1}}e^{W_{\lambda}}Z^{i}_{\lambda}(x_{i}-b_{i})^{2}(x_{j}-b_{j})dx$
$\displaystyle=8\delta^{3}\int_{|z|\leq\frac{1-b}{\delta}}\frac{(z_{i})^{3}z_{j}}{(1+|z|^{2})^{3}}dz+O(\delta^{3})=O(\delta^{3})$

by $\int_{|z|\leq r}\frac{(z_{i})^{3}z_{j}}{(1+|z|^{2})^{3}}dz=0$. Finally

$\displaystyle\int_{B_{1}}e^{W_{\lambda}}|Z^{i}_{\lambda}||x-b|^{\frac{7}{2}}dx$
$\displaystyle\leq
8\delta^{2}\int_{B(b,1+|b|)}\frac{\delta|x-b|^{\frac{9}{2}}}{(\delta^{2}+|x-b|^{2})^{3}}dx=8\delta^{\frac{7}{2}}\int_{|z|\leq\frac{1+|b|}{\delta}}\frac{|z|^{\frac{9}{2}}}{(1+|z|^{2})^{3}}dz\leq
C\delta^{3}.$

Taking into account that $PZ^{i}_{\lambda}=Z^{i}_{\lambda}+O(\delta)$ by
(3.2), and recalling Lemma A.1, the above integral estimates give the thesis.
∎

In order to derive next integral estimate we need to expand the projections
$PZ_{\lambda}^{i}$ to a higher order with respect to (3.2).

######  Lemma A.3.

For $i=1,2$ the following holds:

$PZ_{\lambda}^{i}=Z_{\lambda}^{i}(x)-\delta(x_{i}-b_{i})+O(\delta^{3})+O(\delta|b|)\hbox{
in }B_{1}$

uniformly for $b$ in a small neighborhood of $0$.

###### Proof.

Let us consider $i=1$. Observe that

$\displaystyle\hbox{if }|x|=1:\;\;Z_{\lambda}^{1}(x)$
$\displaystyle=\frac{\delta(x_{1}-b_{1})}{\delta^{2}+|x-b|^{2}}=\frac{\delta(x_{1}-b_{1})}{1+\delta^{2}+O(|b|)}$
$\displaystyle=\delta(x_{1}-b_{1})\Big{(}1+O(|b|)+O(\delta^{2})\Big{)}$
$\displaystyle=\delta(x_{1}-b_{1})+O(|b|\delta)+O(\delta^{3}).$

Therefore, if we set

$\hat{Z}_{\lambda}^{1}:=Z_{\lambda}^{1}(x)-\delta(x_{1}-b_{1}),$

we get

$\hat{Z}_{\lambda}^{1}(x)=O(\delta^{3})+O(\delta|b|)\hbox{ if }|x|=1$

and

$-\Delta\hat{Z}_{\lambda}^{1}(x)=-\Delta Z_{\lambda}^{1}(x)=\Delta
PZ_{\lambda}^{1}(x)\hbox{ in }B_{1}.$

Hence, since by construction $PZ_{\lambda}^{1}=0$ for $|x|=1$, the maximum
principle applies and gives

$PZ_{\lambda}^{1}=\hat{Z}_{\lambda}^{1}+O(\delta^{3})+O(\delta|b|)=Z_{\lambda}^{1}(x)-\delta(x_{1}-b_{1})+O(\delta^{3})+O(\delta|b|)\hbox{
in }B_{1}.$

∎

######  Lemma A.4.

Let $P$ be a homogeneous polynomial of degree 2. Then the following holds:

$\int_{B_{1}}e^{W_{\lambda}}PZ^{i}_{\lambda}P(x-b)dx=O(\delta^{3})+O(\delta^{2}|b|)\quad
i=1,2$

uniformly for $b$ in a small neighborhood of $0$.

###### Proof.

We compute

$\displaystyle\int_{B_{1}}e^{W_{\lambda}}Z^{i}_{\lambda}P(x-b)dx$
$\displaystyle=8\delta^{2}\int_{|z|\leq\frac{1}{\delta}}\frac{P(z-\delta^{-1}b)}{(1+|z-\delta^{-1}b|^{2})^{3}}(z_{i}-\delta^{-1}b_{i})dz$
(A.1)
$\displaystyle=8\delta^{2}\int_{\mathbb{R}^{2}}\frac{P(z)}{(1+|z|^{2})^{3}}z_{i}dz+O(\delta^{3})=O(\delta^{3})$

where we have used that
$\int_{\mathbb{R}^{2}}\frac{P(z)}{(1+|z|^{2})^{3}}z_{i}dz=0$ by oddness.
Taking into account of Lemma A.3 we get

$\displaystyle\int_{B_{1}}e^{W_{\lambda}}PZ^{i}_{\lambda}P(x-b)dx$
$\displaystyle=\int_{B_{1}}e^{W_{\lambda}}Z^{i}_{\lambda}P(x-b)dx-\delta\int_{B_{1}}e^{W_{\lambda}}(x_{i}-b_{i})P(x-b)dx$
$\displaystyle\;\;\;\;+\Big{(}O(\delta^{3})+O(\delta|b|)\Big{)}\int_{B_{1}}e^{W_{\lambda}}|x-b|^{2}dx$
$\displaystyle=\int_{B_{1}}e^{W_{\lambda}}Z^{i}_{\lambda}P(x-b)dx+O(\delta)\int_{B_{1}}e^{W_{\lambda}}|x-b|^{3}dx$
$\displaystyle\;\;\;\;+\Big{(}O(\delta^{3})+O(\delta|b|)\Big{)}\int_{B_{1}}e^{W_{\lambda}}|x-b|^{2}dx$

and the thesis follows by (A.1), and recalling Lemma A.1. ∎

Finally we deduce some integral identities associated to the change of
variable $x\mapsto x^{\alpha}$ which appears frequently when dealing with
$\alpha$-symmetric functions.

######  Lemma A.5.

Let $\alpha\in\mathbb{N}$, $\alpha\geq 2$, and let $f\in L^{1}(B_{1})$. Then
we have that $|x|^{2(\alpha-1)}f(x^{\alpha})\in L^{1}(B_{1})$ and

$\int_{B_{1}}|x|^{2(\alpha-1)}f(x^{\alpha})dx=\frac{1}{\alpha}\int_{B_{1}}f(y)dy.$

###### Proof.

It is sufficient to prove the thesis for a smooth function $f$. Using the
polar coordinates $(\rho,\theta)$ and then applying the change of variables
$(\rho^{\prime},\theta^{\prime})=(\rho^{\alpha},\alpha\theta)$

$\displaystyle\int_{B_{1}}|x|^{2(\alpha-1)}f(x^{\alpha})dx$
$\displaystyle=\int_{0}^{+\infty}d\rho\int_{0}^{2\pi}\rho^{2\alpha-1}f(\rho^{\alpha}e^{{\rm
i}\alpha\theta})d\theta$
$\displaystyle=\frac{1}{\alpha^{2}}\int_{0}^{+\infty}d\rho^{\prime}\int_{0}^{2\alpha\pi}\rho^{\prime}f(\rho^{\prime}e^{{\rm
i}\theta^{\prime}})d\theta^{\prime}$
$\displaystyle=\frac{1}{\alpha}\int_{0}^{+\infty}d\rho^{\prime}\int_{0}^{2\pi}\rho^{\prime}|f(\rho^{\prime}e^{{\rm
i}\theta^{\prime}})|^{2}d\theta^{\prime}$
$\displaystyle=\frac{1}{\alpha}\int_{B_{1}}f(y)dy.$

∎

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