# Sharp stability of the logarithmic Sobolev inequality in the critical point
setting

Juncheng Wei  Department of Mathematics, University of British Columbia,
Vancouver, B.C., Canada, V6T 1Z2<EMAIL_ADDRESS>and  Yuanze Wu  School of
Mathematics, China University of Mining and Technology, Xuzhou, 221116, P.R.
China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we consider the Euclidean logarithmic Sobolev inequality

$\displaystyle\int_{\mathbb{R}^{d}}|u|^{2}\log|u|dx\leq\frac{d}{4}\log\bigg{(}\frac{2}{\pi
de}\|\nabla u\|_{L^{2}(\mathbb{R}^{d})}^{2}\bigg{)},$

where $u\in W^{1,2}(\mathbb{R}^{d})$ with $d\geq 2$ and
$\|u\|_{L^{2}(\mathbb{R}^{d})}=1$. It is well known that extremal functions of
this inequality are precisely the Gaussians

$\displaystyle\mathfrak{g}_{\sigma,z}(x)=(\pi\sigma)^{-\frac{d}{2}}\mathfrak{g}_{*}\bigg{(}\sqrt{\frac{\sigma}{2}}(x-z)\bigg{)}\quad\text{with}\quad\mathfrak{g}_{*}(x)=e^{-\frac{|x|^{2}}{2}}.$

We prove that if $u\geq 0$ satisfying
$(\nu-\frac{1}{2})c_{0}<\|u\|_{H^{1}(\mathbb{R}^{d})}^{2}<(\nu+\frac{1}{2})c_{0}$
and $\|-\Delta u+u-2u\log|u|\|_{H^{-1}}\leq\delta$, where
$c_{0}=\|\mathfrak{g}_{1,0}\|_{H^{1}(\mathbb{R}^{d})}^{2}$, $\nu\in\mathbb{N}$
and $\delta>0$ sufficiently small, then

$\displaystyle\text{dist}_{H^{1}}(u,\mathcal{M}^{\nu})\lesssim\|-\Delta
u+u-2u\log|u|\|_{H^{-1}}$

which is optimal in the sense that the order of the right hand side is sharp,
where

$\displaystyle\mathcal{M}^{\nu}=\\{(\mathfrak{g}_{1,0}(\cdot-
z_{1}),\mathfrak{g}_{1,0}(\cdot-z_{2}),\cdots,\mathfrak{g}_{1,0}(\cdot-
z_{\nu}))\mid z_{i}\in\mathbb{R}^{d}\\}.$

Our result provides an optimal stability of the Euclidean logarithmic Sobolev
inequality in the critical point setting.

Keywords: Euclidean logarithmic Sobolev inequality; Optimal stability;
Critical point setting.

AMS Subject Classification 2010: 35A23; 35B35.

##  1\. Introduction

A fundamental task in understanding functional inequalities, arise in the
calculus of variations, geometry, etc, is to study the best constants, the
classification of extremal functions, as well as their qualitative properties
for parameters in the full region, since such functional inequalities are
crucial in understanding nonlinear partial differential equations (Nonlinear
PDEs for short) by virtue of the complete knowledge of the best constants,
extremal functions, and qualitative properties. The most well-studied
functional inequality in the community of Nonlinear PDEs is the Sobolev
inequality, whose classical one with exponent $2$ states that for any $u\in
H^{1}(\mathbb{R}^{d})$ with $d\geq 3$, there holds

$\displaystyle
S\bigg{(}\int_{\mathbb{R}^{d}}|u|^{\frac{2d}{d-2}}dx\bigg{)}^{\frac{d-2}{d}}\leq\int_{\mathbb{R}^{d}}|\nabla
u|^{2}dx,$ (1.1)

where $S>0$ is a constant which is only dependent of the dimension and
$H^{1}(\mathbb{R}^{d})$ is the classical Sobolev space given by

$\displaystyle H^{1}(\mathbb{R}^{d})=\\{u\in
L^{\frac{2d}{d-2}}(\mathbb{R}^{d})\mid|\nabla u|\in L^{2}(\mathbb{R}^{d})\\}.$

It has been proved in [2, 55] that

$\displaystyle S=\pi
d(d-2)\bigg{(}\frac{\Gamma(\frac{d}{2})}{\Gamma(d)}\bigg{)}^{\frac{2}{d}}$

is optimal and the extremal functions, which are called the Aubin-Talenti
bubbles in the literature, are given by

$\displaystyle
U_{\lambda,z,c}(x)=c\bigg{(}\frac{1}{1+\lambda^{2}|x-z|^{2}}\bigg{)}^{\frac{d-2}{2}},$

where $\lambda>0$, $c\in\mathbb{R}$ and $z\in\mathbb{R}^{d}$.

Once a functional inequality is well understood for its best constant and
extremal functions, it is natural to concern its stability, which is growingly
interested in recent years by its important applications in understanding many
Nonlinear PDEs, such as the fast diffusion equation, the Keller-Segel equation
and so on. The basic question one wants to address in this aspect is the
following (cf. [30]):

  1. $(Q)$

Suppose we are given a functional inequality for which minimizers are known.
Can we prove, in some quantitative way, that if a function “almost attains the
equality” then it is close (in some suitable sense) to one of the minimizers?

Such study was first raised by Brezis and Lieb in [8] for the classical
Sobolev inequality (1.1) as an open question, which was settled by Bianchi and
Egnell in [6] by proving that

$\displaystyle dist_{H^{1}}^{2}(u,\mathcal{Z})\lesssim\|\nabla
u\|^{2}_{L^{2}(\mathbb{R}^{d})}-S\|u\|^{2}_{L^{\frac{2d}{d-2}}(\mathbb{R}^{N})},$

where $\|\cdot\|_{L^{p}(\mathbb{R}^{N})}$ is the usual norm in the Lebesgue
space $L^{p}(\mathbb{R}^{d})$ and

$\displaystyle\mathcal{Z}=\\{U_{\lambda,z,c}\mid(\lambda,z,c)\in\mathbb{R}_{+}\times\mathbb{R}^{d+1}\\}.$

Unlike Bianchi and Egnell’s study ([6]) in the functional inequality setting,
in recent years, Figalli et al. initiated the study on the stability of the
classical Sobolev inequality (1.1) in the critical point setting, that is,
studying the stability of the Euler-Lagrange equation of the classical Sobolev
inequality (1.1). The study on the stability of the classical Sobolev
inequality (1.1) in the critical point setting is more challenging since the
Euler-Lagrange equation of the classical Sobolev inequality (1.1) has sign-
changing solutions. Moreover, the “almost” solutions of the Euler-Lagrange
equation of the classical Sobolev inequality (1.1) may decompose into several
parts at infinity (cf. [54]). By setting the study in a suitable way, Figalli
et al. proved that

  1. $(1)$

(Ciraolo-Figalli-Maggi [16]) Let $d\geq 3$ and $u\in D^{1,2}(\mathbb{R}^{d})$
be positive such that $\|\nabla
u\|_{L^{2}(\mathbb{R}^{d})}^{2}\leq\frac{3}{2}S^{\frac{d}{2}}$ and $\|\Delta
u+|u|^{\frac{4}{d-2}}u\|_{H^{-1}}\leq\delta$ for some $\delta>0$ sufficiently
small. Then,

$\displaystyle dist_{D^{1,2}}(u,\mathcal{M}_{0})\lesssim\|\Delta
u+|u|^{\frac{4}{d-2}}u\|_{H^{-1}},$

where $\mathcal{M}_{0}=\\{(U[z,\lambda]\mid z\in\mathbb{R}^{d},\lambda>0\\}.$

  2. $(2)$

(Figalli-Glaudo [31]) Let $u\in D^{1,2}(\mathbb{R}^{d})$ be nonnegative such
that

$\displaystyle(\nu-\frac{1}{2})S^{\frac{d}{2}}<\|u\|_{D^{1,2}(\mathbb{R}^{N})}^{2}<(\nu+\frac{1}{2})S^{\frac{d}{2}}$

and $\|\Delta u+|u|^{\frac{4}{d-2}}u\|_{H^{-1}}\leq\delta$ for some $\delta>0$
sufficiently small. Then, for $3\leq d\leq 5$,

$\displaystyle dist_{D^{1,2}}(u,\mathcal{M}_{0}^{\nu})\lesssim\|\Delta
u+|u|^{\frac{4}{d-2}}u\|_{H^{-1}}$

where

$\displaystyle\mathcal{M}_{0}^{\nu}=\\{(U[z_{1},\lambda_{1}],U[z_{2},\lambda_{2}],\cdots,U[z_{\nu},\lambda_{\nu}])\mid
z_{i}\in\mathbb{R}^{d},\lambda_{i}>0\\}.$

All the above results are optimal in the sense that the orders of the right
hand sides in the above estimates are sharp, while the optimal stability of
the classical Sobolev inequality (1.1) in the critical point setting for the
case $N\geq 6$ was left in [31] as an open problem which was solved by Deng,
Sun and Wei in [21], very recently, by proving the following optimal
stability:

$\displaystyle
dist_{D^{1,2}}(u,\mathcal{M}_{0}^{\nu})\lesssim\left\\{\begin{aligned}
&\|\Delta u+|u|u\|_{H^{-1}}|\ln(\|\Delta u+|u|u\|_{H^{-1}})|^{\frac{1}{2}},\ \
d=6;\\\ &\|\Delta u+|u|^{\frac{4}{d-2}}u\|_{H^{-1}}^{\frac{d+2}{2(d-2)}},\ \
d\geq 7.\end{aligned}\right.$

According to the important applications in understanding many Nonlinear PDEs,
serval other famous functional inequalities, such as the Gagliardo-Nirenberg-
Sobolev inequality (cf. [11, 12, 25, 26, 49, 51, 52]), the Hardy-Littlewood-
Sobolev inequality (cf. [5, 18, 34, 37, 40, 43, 50]), the Caffarelli-Kohn-
Nirenberg inequality (cf. [9, 13, 17, 22, 23, 24, 28, 45, 57, 58, 61]), the
$L^{p}$-Sobolev inequality (cf. [14, 15, 32, 33, 35, 36, 48]) and so on, are
also wildly studied on the best constants, the classification of extremal
functions and stability. However, according to their multi-parameters or no-
Hilbert, most of these functional inequalities are far from well understood
except for some special cases.

It is worth pointing out that, besides the classical Sobolev inequality (1.1),
the Euclidean logarithmic Sobolev inequality,

$\displaystyle\int_{\mathbb{R}^{d}}|u|^{2}\log|u|dx\leq\frac{d}{4}\log\bigg{(}\frac{2}{\pi
de}\|\nabla u\|_{L^{2}(\mathbb{R}^{d})}^{2}\bigg{)}$ (1.2)

where $u\in W^{1,2}(\mathbb{R}^{d})$ with $d\geq 2$ and
$\|u\|_{L^{2}(\mathbb{R}^{d})}=1$, is also well understood for the best
constants and the classification of extremal functions. It is well known (cf.
[20, 60]) that this inequality is optimal and extremal functions of (1.2) are
precisely the Gaussians

$\displaystyle\mathfrak{g}_{\sigma,z}(x)=(\pi\sigma)^{-\frac{d}{2}}\mathfrak{g}_{*}\bigg{(}\sqrt{\frac{\sigma}{2}}(x-z)\bigg{)}\quad\text{with}\quad\mathfrak{g}_{*}(x)=e^{-\frac{|x|^{2}}{2}}.$
(1.3)

Moreover, it is equivalent to the Gross logarithmic inequality (cf. [38]) with
respect to Gaussian weight

$\displaystyle\int_{\mathbb{R}^{d}}|g|^{2}\log|g|d\mu\leq\int_{\mathbb{R}^{d}}|\nabla
g|^{2}d\mu\quad\text{with}\int_{\mathbb{R}^{d}}|g|^{2}d\mu=1\text{ and
}d\mu=(2\pi)^{-\frac{d}{2}}e^{-\frac{|x|^{2}}{2}}.$

The Euclidean logarithmic Sobolev inequality has another two different forms:

$\displaystyle\int_{\mathbb{R}^{d}}|v|^{2}\log|v|^{2}dx\leq\frac{a^{2}}{\pi}\|\nabla
v\|_{L^{2}(\mathbb{R}^{d})}^{2}+\bigg{(}\log\|v\|_{L^{2}(\mathbb{R}^{d})}^{2}-d(1+\log
a)\bigg{)}\|v\|_{L^{2}(\mathbb{R}^{d})}^{2}$ (1.4)

for all $a>0$ and any function $v\in W^{1,2}(\mathbb{R}^{d})$, and

$\displaystyle\int_{\mathbb{R}^{d}}|w|^{2}\log|w|^{2}dx+\frac{d}{2}(1+\frac{1}{2}\log(2\pi))\leq\|\nabla
w\|_{L^{2}(\mathbb{R}^{d})}^{2}$ (1.5)

for any function $w\in W^{1,2}(\mathbb{R}^{d})$, $d\geq 2$, with
$\|w\|_{L^{2}(\mathbb{R}^{d})}=1$. (1.4) and (1.5) are equivalent by the
relation $w(x)=v(\frac{a}{\sqrt{2\pi}}x)$ and they are established in [44,
Theorem 8.14] and [59, Theorem 1.2], respectively. However, (1.2) is optimal
since it can be obtained by minimizing the right hand side of (1.4) for $a>0$
(cf. [26]).

The stability of the Euclidean logarithmic Sobolev inequality (1.2) in the
functional inequality setting has also been widely studied, cf. [7, 10, 11,
26, 27, 29, 41, 42, 60] and the references therein. It is worth pointing out
that most of these results are devoted to more general version of the
logarithmic Sobolev inequality in the probability setting which contains the
Euclidean logarithmic Sobolev inequality (1.2) as a special case for the
Gaussians measure $d\mu=(2\pi)^{-\frac{d}{2}}e^{-\frac{|x|^{2}}{2}}$. Thus, it
is natural to consider the stability of the Euclidean logarithmic Sobolev
inequality (1.2) in the critical point setting, as that for the classical
Sobolev inequality (1.1). Let

$\displaystyle\mathcal{E}(u)=\frac{d}{4}\log\bigg{(}\frac{2}{\pi de}\|\nabla
u\|_{L^{2}(\mathbb{R}^{d})}^{2}\bigg{)}-\int_{\mathbb{R}^{d}}|u|^{2}\log|u|dx.$

Then critical points of $\mathcal{E}(u)$ in $H^{1}(\mathbb{R}^{d})$ with the
finite energy on the smooth manifold

$\displaystyle\mathcal{S}=\\{u\in
H^{1}(\mathbb{R}^{d})\mid\|u\|_{L^{2}(\mathbb{R}^{d})}=1\\}$

satisfy $\int_{\mathbb{R}^{d}}|u|^{2}|\log|u|^{2}|dx<+\infty$ and the
following Euler-Lagrange equation

$\displaystyle-(\frac{d}{2\|\nabla u\|_{L^{2}(\mathbb{R}^{d})}^{2}})\Delta
u-(1+2\sigma_{u})u=2u\log|u|\quad\text{in }\mathbb{R}^{d},$ (1.6)

where $\sigma_{u}$ is a part of unknown and appears in (1.6) as the Lagrange
multiplier. Thus, (1.6) is the Euler-Lagrange equation of (1.2). Let

$\displaystyle u_{\lambda}(x)=\lambda^{\frac{d}{2}}u(\lambda
x)\quad\text{where }\lambda=\bigg{(}\frac{d}{2\|\nabla
u\|_{L^{2}(\mathbb{R}^{d})}^{2}}\bigg{)}^{\frac{1}{2}},$

then by (1.6) and a direct calculation, $u_{\lambda}$ satisfies

$\displaystyle-\Delta
u_{\lambda}+(d\log\lambda-1-2\sigma_{u})u_{\lambda}=2u_{\lambda}\log|u_{\lambda}|\quad\text{in
}\mathbb{R}^{d}.$ (1.7)

Let

$\displaystyle u_{\lambda}^{*}(x)=\alpha_{u}u_{\lambda}(x)\quad\text{where
}\alpha_{u}=e^{\frac{d}{2}\log\lambda-\sigma_{u}-1},$

then by (1.7) and a direct calculation, $u_{\lambda}^{*}$ satisfies the
equation

$\displaystyle-\Delta u+u=2u\log|u|\quad\text{in }\mathbb{R}^{d}.$ (1.8)

Thus, the logarithmic Schrödinger equation (1.8) can be seen as the Euler-
Lagrange equation of the Euclidean logarithmic Sobolev inequality (1.2).

It has been proved in [19, 56] that the Gaussian

$\displaystyle\mathfrak{g}=e^{\frac{1+d}{2}}\mathfrak{g}_{*}$ (1.9)

is the unique positive solution of the logarithmic Schrödinger equation (1.8)
which satisfies $u(x)\to 0$ as $|x|\to+\infty$, where $\mathfrak{g}_{*}$ is
given by (1.3). We remark that any solution of (1.8) satisfying
$\int_{\mathbb{R}^{d}}|u|^{2}|\log|u|^{2}|dx<+\infty$ must exponentially decay
to zero as $|x|\to+\infty$ by the standard applications of the maximum
principle. Moreover, it has been proved in [19, Theorem 1.3] (see also [39,
Theorem 7.5]) that $\mathfrak{g}$ is nondegenerate in the sense that
$Ker(\mathcal{L})=span\\{\partial_{x_{j}}\mathfrak{g}\\}$ where

$\displaystyle\mathcal{L}=-\Delta+|x|^{2}-(d+2)$ (1.10)

is the linearized operator of (1.8) at $\mathfrak{g}$. Note that (1.8) is
invariant under translations, thus, the smooth manifold

$\displaystyle\mathcal{M}=\\{\mathfrak{g}(\cdot-z)\mid z\in\mathbb{R}^{d}\\}$

contains all positive solutions of (1.8) satisfying
$\int_{\mathbb{R}^{d}}|u|^{2}|\log|u|^{2}|dx<+\infty$.

Let $\\{u_{n}\\}$, $u_{n}\geq 0$ for all $n$, be bounded in
$H^{1}(\mathbb{R}^{d})$ and almost solves (1.8), that is, $\|-\Delta
u_{n}+u_{n}-2u_{n}\log|u_{n}|\|_{H^{-1}}\to 0$ as $n\to\infty$. Then it is
easy to see that $\\{|u_{n}|^{2}|\log|u_{n}|^{2}|\\}$ is bounded in
$L^{1}(\mathbb{R}^{d})$. Thanks to the Brezis-Lieb lemma [53, Lemma 3.1] and
the positivity of the energy of the unique positive solution of (1.8) (cf.
[53, Lemma 3.3]), it is standard (cf. [1, Proposition 3.1]) to prove the
following Struwe’s decomposition of $\\{u_{n}\\}$.

######  Proposition 1.1.

There exists $\nu\in\mathbb{N}$ and $y_{1,n},y_{2,n},\cdots
y_{\nu,n}\subset\mathbb{R}^{d}$ such that

$\displaystyle\|u_{n}-\sum_{i=1}^{\nu}\mathfrak{g}(\cdot-
y_{i,n})\|_{H^{1}(\mathbb{R}^{d})}\to 0\quad\text{as }n\to\infty.$

By Proposition 1.1, for nonnegative functions $u_{n}$ with uniform
$H^{1}$-bound, we know that there exists $\nu\in\mathbb{N}$ such that
$\\{u_{n}\\}$ will be close to the manifold

$\displaystyle\mathcal{M}^{\nu}=\\{(\mathfrak{g}(\cdot-
z_{1}),\mathfrak{g}(\cdot-z_{2}),\cdots,\mathfrak{g}(\cdot-z_{\nu}))\mid
z_{i}\in\mathbb{R}^{d}\\}$

in the $H^{1}$-topology. Thus, it is natural to ask, if $u\geq 0$ satisfy

$\displaystyle(\nu-\frac{1}{2})c_{0}<\|u\|_{H^{1}(\mathbb{R}^{d})}^{2}<(\nu+\frac{1}{2})c_{0}\quad\text{and}\quad\|-\Delta
u+u-2u\log|u|\|_{H^{-1}}\leq\delta,$

where $c_{0}=\|\mathfrak{g}\|_{H^{1}(\mathbb{R}^{d})}^{2}$, $\nu\in\mathbb{N}$
and $\delta>0$ sufficiently small, can we obtain a quantitative version of
Proposition 1.1 by optimally controlling $dist_{H^{1}}(u,\mathcal{M}^{\nu})$
by $\|-\Delta u+u-2u\log|u|\|_{H^{-1}}$? The purpose of this paper is to give
a positive answer to this natural question and our main result reads as
follows.

######  Theorem 1.1.

Let $u\geq 0$ satisfy

$\displaystyle(\nu-\frac{1}{2})c_{0}<\|u\|_{H^{1}(\mathbb{R}^{d})}^{2}<(\nu+\frac{1}{2})c_{0}\quad\text{and}\quad\|-\Delta
u+u-2u\log|u|\|_{H^{-1}}\leq\delta,$

where $c_{0}=\|\mathfrak{g}\|_{H^{1}(\mathbb{R}^{d})}^{2}$, $\nu\in\mathbb{N}$
and $\delta>0$ sufficiently small. Then

$\displaystyle\text{dist}_{H^{1}}(u,\mathcal{M}^{\nu})\lesssim\|-\Delta
u+u-2u\log|u|\|_{H^{-1}},$ (1.11)

where

$\displaystyle\mathcal{M}^{\nu}=\\{(\mathfrak{g}_{1,0}(\cdot-
z_{1}),\mathfrak{g}_{1,0}(\cdot-z_{2}),\cdots,\mathfrak{g}_{1,0}(\cdot-
z_{\nu}))\mid z_{i}\in\mathbb{R}^{d}\\}.$

Moreover, (1.1) is optimal in the sense that the order of the right hand side
is sharp.

######  Remark 1.1.

The main ideas in proving Theorem 1.1 is similar to that of [21, 61], that is,
we choose special $y_{1,\delta},y_{2,\delta},\cdots
y_{\nu,\delta}\subset\mathbb{R}^{d}$ and decompose
$u-\sum_{i=1}^{\nu}\mathfrak{g}(\cdot-y_{i,\delta})$ into two parts, where the
first part is very regular which can be well estimated and the second part is
much smaller than the first part. However, the logarithmic nonlinearity is
very different from the power-type one dealt with in [21, 61], thus, we need
to careful estimate it to make sure that the strategy in [21, 61] work for
(1.8).

Notations. Throughout this paper, $C$ and $C^{\prime}$ are indiscriminately
used to denote various absolutely positive constants.
$\sigma,\sigma^{\prime},\sigma^{\prime\prime}$ are indiscriminately used to
denote various absolutely positive constants which can be taken arbitrary
small. $a\sim b$ means that $C^{\prime}b\leq a\leq Cb$ and $a\lesssim b$ means
that $a\leq Cb$.

##  2\. Preliminaries

Let us consider the equation

$\displaystyle-\Delta u+u=2u\log|u|+f,\quad\text{in }\mathbb{R}^{d},$ (2.1)

where $f\in H^{-1}$ and $u\in H^{1}(\mathbb{R}^{d})$ is nonnegative and
satisfies

$\displaystyle(\nu-\frac{1}{2})c_{0}<\|u\|_{H^{1}(\mathbb{R}^{d})}^{2}<(\nu+\frac{1}{2})c_{0}\quad\text{for
some fixed }\nu\in\mathbb{N}$

with $c_{0}=\|\mathfrak{g}\|_{H^{1}(\mathbb{R}^{d})}^{2}$. By multiplying
(2.1) with $u$ and integrating by parts, it is easy to see that
$\int_{\mathbb{R}^{d}}|u|^{2}|\log|u|^{2}|dx\lesssim 1$. Moreover, by
Proposition 1.1, there exists $z_{1,f},z_{2,f},\cdots
z_{\nu,f}\subset\mathbb{R}^{d}$ such that

$\displaystyle\|u-\sum_{i=1}^{\nu}\mathfrak{g}(\cdot-
z_{i,f})\|_{H^{1}(\mathbb{R}^{d})}\to 0\quad\text{as }\|f\|_{H^{-1}}\to 0.$

It follows that

$\displaystyle\mathfrak{c}_{f}=\inf_{z_{i}\in\mathbb{R}^{d}}\|u-\sum_{i=1}^{\nu}\mathfrak{g}(\cdot-
z_{i})\|_{H^{1}(\mathbb{R}^{d})}^{2}\to 0\quad\text{as }\|f\|_{H^{-1}}\to 0.$
(2.2)

Thus, by solving the minimizing problem (2.2) in a standard way (cf. [4]), we
can write $u=\sum_{i=1}^{\nu}\mathfrak{g}(\cdot-y_{i,f})+\rho_{f}$, where
$\\{y_{i,f}\\}$ is the solution of (2.2) and the remaining term $\rho_{f}$
satisfies

$\displaystyle\|\rho_{f}\|_{H^{1}(\mathbb{R}^{d})}^{2}=\mathfrak{c}_{f}\to
0\quad\text{as }\|f\|_{H^{-1}}\to 0$

and the orthogonal conditions

$\displaystyle\langle\rho_{f},\partial_{x_{l}}\mathfrak{g}_{j,f}\rangle_{H^{1}(\mathbb{R}^{d})}=0,\quad
l=1,2,\cdots,d\text{ and }j=1,2,\cdots,\nu.$ (2.3)

For the sake of simplicity, we denote $\mathfrak{g}(\cdot-y_{i,f})$ by
$\mathfrak{g}_{i,f}$. Clearly, by (2.1), $\rho_{f}$ satisfies

$\displaystyle\left\\{\begin{aligned}
&-\Delta\rho_{f}+\rho_{f}=2(\sum_{i=1}^{\nu}\mathfrak{g}_{i,f}+\rho_{f})\log(\sum_{i=1}^{\nu}\mathfrak{g}_{i,f}+\rho_{f})-2\sum_{i=1}^{\nu}\mathfrak{g}_{i,f}\log\mathfrak{g}_{i,f}+f,\quad\text{in
}H^{-1},\\\
&\langle\rho_{f},\partial_{x_{l}}\mathfrak{g}_{i,f}\rangle_{H^{1}(\mathbb{R}^{d})}=0,\quad
l=1,2,\cdots,d\text{ and }i=1,2,\cdots,\nu.\end{aligned}\right.$ (2.4)

It is convenient to write (2.4) as follows:

$\displaystyle\left\\{\begin{aligned}
&\mathcal{L}_{f}(\rho_{f})=E+N(\rho_{f})+f,\quad\text{in }H^{-1},\\\
&\langle\rho_{f},\partial_{x_{l}}\mathfrak{g}_{i,f}\rangle_{H^{1}(\mathbb{R}^{d})}=0,\quad
l=1,2,\cdots,d\text{ and }i=1,2,\cdots,\nu,\end{aligned}\right.$ (2.5)

where

$\displaystyle\mathcal{L}_{f}=-\Delta-1-2\log(\sum_{i=1}^{\nu}\mathfrak{g}_{i,f})$
(2.6)

is the linear operator,

$\displaystyle
E=2(\sum_{i=1}^{\nu}\mathfrak{g}_{i,f})\log(\sum_{i=1}^{\nu}\mathfrak{g}_{i,f})-2\sum_{i=1}^{\nu}\mathfrak{g}_{i,f}\log\mathfrak{g}_{i,f}$
(2.7)

is the error and

$\displaystyle N(\rho_{f})$ $\displaystyle=$ $\displaystyle
2(\sum_{i=1}^{\nu}\mathfrak{g}_{i,f}+\rho_{f})\log(\sum_{i=1}^{\nu}\mathfrak{g}_{i,f}+\rho_{f})-2(\sum_{i=1}^{\nu}\mathfrak{g}_{i,f})\log(\sum_{i=1}^{\nu}\mathfrak{g}_{i,f})$
(2.8) $\displaystyle-2(1+\log(\sum_{i=1}^{\nu}\mathfrak{g}_{i,f}))\rho_{f}$

is the nonlinear part.

##  3\. Proof of (1.11)

Let

$\displaystyle\eta_{i}=\min_{j\not=i}\eta_{i,j}\quad\text{and}\quad\eta=\min_{i}\eta_{i},$
(3.1)

where $\eta_{i,j}=|y_{j,f}-y_{i,f}|$. Then by (2.1), (2.2) and the uniqueness
of $\mathfrak{g}$, we can easy to see that

$\displaystyle\eta=\min_{i\not=j}\\{|y_{i,f}-y_{j,f}|\\}\to+\infty\quad\text{as
}\delta\to 0,$ (3.2)

where we denote $\|f\|_{H^{-1}}$ by $\delta$ for the sake of simplicity. Let

$\displaystyle\Omega_{i}=\\{x\in\mathbb{R}^{d}\mid\mathfrak{g}_{i,f}\geq\mathfrak{g}_{j,f}\quad\text{for
all }j\not=i\\}.$ (3.3)

Then $\mathbb{R}^{d}=\cup_{i=1}^{\nu}\Omega_{i}$ and
$\sum_{j=1}^{\nu}\mathfrak{g}_{j,f}\sim\mathfrak{g}_{i,f}$ in $\Omega_{i}$.
Moreover, we also introduce

$\displaystyle\Pi_{c,i,j}:2(y_{i,f}-y_{j,f})x+c=0\quad\text{and}\quad\mathbb{L}_{i,j}:(x-y_{i,f})\times(y_{i,f}-y_{j,f})=0,$
(3.4)

where $c\in\mathbb{R}$ are constants. We take
$x_{c,i,j}\in\Pi_{c.,i,j}\cap\mathbb{L}_{i,j}\cap\Omega_{i}$ and denote

$\displaystyle|x_{c,i,j}-y_{i,f}|=\pm\alpha_{c,i,j}\eta_{i,j},$ (3.5)

where $\alpha_{c,i,j}>-\frac{1}{2}$ with

$\displaystyle\alpha_{c,i,j}\left\\{\begin{aligned} &>0,\quad\langle
x_{c,i,j}-y_{i,f},y_{i,f}-y_{j,f}\rangle>0,\\\ &<0,\quad\langle
x_{c,i,j}-y_{i,f},y_{i,f}-y_{j,f}\rangle<0.\end{aligned}\right.$

Then

$\displaystyle|x_{c,i,j}-y_{j,f}|=(1+\alpha_{c,i,j})\eta_{i,j}.$

######  Lemma 3.1.

We have,

$\displaystyle E$ $\displaystyle\sim$
$\displaystyle\sum_{i=1}^{\nu}\chi_{\Omega_{i}}\mathfrak{g}_{i,f}(\sum_{j\not=i}\chi_{\Pi_{c,i,j}}e^{-(\alpha_{c,i,j}+\frac{1}{2})\eta_{i,j}^{2}}\log(1+e^{(\alpha_{c,i,j}+\frac{1}{2})\eta_{i,j}^{2}}))$
(3.6)

as $\|f\|_{H^{-1}}\to 0$.

###### Proof.

By (2.7), we have

$\displaystyle
E=2\sum_{i=1}^{\nu}\mathfrak{g}_{i,f}\log(1+\sum_{j\not=i}\frac{\mathfrak{g}_{j,f}}{\mathfrak{g}_{i,f}}).$
(3.7)

By (3.7), we write

$\displaystyle E$ $\displaystyle=$ $\displaystyle
2\mathfrak{g}_{i,f}\log(1+\sum_{l\not=i}\frac{\mathfrak{g}_{l,f}}{\mathfrak{g}_{i,f}})+2\sum_{j\not=i}\mathfrak{g}_{j,f}\log(1+\sum_{l\not=j}\frac{\mathfrak{g}_{l,f}}{\mathfrak{g}_{j,f}})$
$\displaystyle=:$ $\displaystyle I+II$

in $\Omega_{i}$, where $\Omega_{i}$ is given by (3.3). For $I$, we have

$\displaystyle I\sim\sum_{j\not=i}\mathfrak{g}_{j,f}\quad\text{in
}\Omega_{i}.$

For
$\mathfrak{g}_{j,f}\log(1+\sum_{l\not=j}\frac{\mathfrak{g}_{l,f}}{\mathfrak{g}_{j,f}})$
with $j\not=i$, we write

$\displaystyle\mathfrak{g}_{j,f}\log(1+\sum_{l\not=j}\frac{\mathfrak{g}_{l,f}}{\mathfrak{g}_{j,f}})=\mathfrak{g}_{j,f}\log(1+\frac{\mathfrak{g}_{i,f}+\sum_{l\not=j,i}\mathfrak{g}_{l,f}}{\mathfrak{g}_{j,f}}).$

Then in $\Omega_{i}$, we have

$\displaystyle E\sim
II\sim\sum_{j\not=i}\mathfrak{g}_{j,f}\log(1+\frac{\mathfrak{g}_{i,f}}{\mathfrak{g}_{j,f}}).$
(3.8)

Let us consider the function

$\displaystyle\frac{\mathfrak{g}_{i,f}}{\mathfrak{g}_{j,f}},\quad\forall
j\not=i.$

By (1.3), (1.9) and direct calculations, we have

$\displaystyle\nabla(\frac{\mathfrak{g}_{i,f}}{\mathfrak{g}_{j,f}})=(y_{i,f}-y_{j,f})e^{-\frac{1}{2}(|x-y_{i,f}|^{2}-|x-y_{j,f}|^{2})}.$
(3.9)

Then by (3.9),

$\displaystyle\frac{\mathfrak{g}_{i,f}}{\mathfrak{g}_{j,f}}\equiv
const.\quad\text{on the hyperplane $\Pi_{c,i,j}$ for all $c\in\mathbb{R}$},$

where $\Pi_{c,i,j}$ is given by (3.4). Thus, we have

$\displaystyle\frac{\mathfrak{g}_{i,f}}{\mathfrak{g}_{j,f}}\equiv
e^{(\alpha_{c,i,j}+\frac{1}{2})|y_{i,f}-y_{j,f}|^{2}},\quad\forall
x\in\Pi_{c,i,j}\cap\Omega_{i},$ (3.10)

where $\alpha_{c,i,j}>-\frac{1}{2}$ is given by (3.5). It follows from (3.8),
(3.10) and the fact that $\Pi_{c,i,j}\perp\mathbb{L}_{i,j}$ for all
$c\in\mathbb{R}$ that (3.6) holds. ∎

As that in [21, 61], we decompose $\rho_{f}=\phi_{f}+\varphi_{f}$, where
$\phi_{f}$ is the solution of the following equation:

$\displaystyle\left\\{\begin{aligned}
&\mathcal{L}_{f}(\phi_{f})=E+N(\phi_{f})-\sum_{j=1}^{d}\sum_{i=1}^{\nu}a_{j,i}\partial_{x_{j}}\mathfrak{g}_{i,f},\quad\text{in
}H^{-1},\\\
&\langle\phi_{f},\partial_{x_{j}}\mathfrak{g}\rangle_{H^{1}(\mathbb{R}^{d})}=0,\quad
l=1,2,\cdots,d\end{aligned}\right.$ (3.11)

with $a_{j,i}$ being the Lagrange multipliers given by

$\displaystyle a_{j,i}\sim\langle
E+N(\phi_{f}),\partial_{x_{j}}\mathfrak{g}_{i,f}\rangle_{L^{2}}.$

To solve (3.11), let us first establish a good linear theory. Let

$\displaystyle\mathfrak{g}_{i,f,d-1,j}=e^{\frac{1+d}{2}-\frac{|z^{i,j}|^{2}}{2}}$

and

$\displaystyle
D_{R,i,j}=\\{x\in\Pi_{c,i,j}\mid|x_{c,i,j}-\frac{y_{i,f}+y_{j,f}}{2}|\leq
R\\},$

where $z^{i,j}\perp\mathbb{L}_{i,j}$ and $x_{c,i,j}$ is given by (3.5). We
define

$\displaystyle\Omega_{i,j,d-1}=\\{x\in\Omega_{i}\mid\mathfrak{g}_{i,f,d-1,j}\geq\mathfrak{g}_{i,f,d-1,l},\forall
l\not=i,j\\}.$

Then as above, $\Omega_{i}=\cup_{j\not=i}\Omega_{i,j,d-1}$ and
$\sum_{l\not=i}\mathfrak{g}_{i,f,d-1,l}\sim\mathfrak{g}_{i,f,d-1,j}$ in
$\Omega_{i,j,d-1}$ for all $j$, where $\Omega_{i}$ is given by (3.3). We
introduce the norms

$\displaystyle\|u\|_{\natural}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{\nu}\sum_{j\not=i}(\sup_{D_{\sigma\eta,i,j}^{c}\cap\Omega_{i,j,d-1}}\frac{|u|}{\mathfrak{g}_{i,f}^{1-\sigma}}+\sup_{D_{\sigma\eta,i,j}\cap\Omega_{i,j,d-1}}\frac{|u|}{e^{-\frac{1}{8}(\eta_{i,j}^{2}-\eta^{2})}\mathfrak{g}_{i,f,d-1,j}})$

and

$\displaystyle\|u\|_{\sharp}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{\nu}\sum_{j\not=i}(\sup_{D_{\sigma\eta-1,i,j}^{c}\cap\Omega_{i,j,d-1}}\frac{|u|}{\mathfrak{g}_{i,f}^{1-\sigma}}+\sup_{D_{\sigma\eta-1,i,j}\cap\Omega_{i,j,d-1}}\frac{\eta^{2}|u|}{e^{-\frac{1}{8}(\eta_{i,j}^{2}-\eta^{2})}\mathfrak{g}_{i,f,d-1,j}}).$

Then

$\displaystyle\mathbb{X}=\\{u\in
H^{1}(\mathbb{R}^{d})\mid\|u\|_{\sharp}<+\infty\\}\quad\text{and}\quad\mathbb{Y}=\\{u\in
L^{2}(\mathbb{R}^{d})\mid\|u\|_{\natural}<+\infty\\},$

are Banach spaces.

Let us consider the following linear equation:

$\displaystyle\left\\{\begin{aligned}
&\mathcal{L}_{f}(\psi)=h-\sum_{i=1}^{\nu}\sum_{j=1}^{d}b_{j,i}\partial_{x_{j}}\mathfrak{g}_{i,f},\quad\text{in
}\mathbb{R}^{d},\\\ &\phi\in\mathbb{X}^{\bot},\end{aligned}\right.$ (3.12)

where $\mathcal{L}_{f}$ is the linear operator given by (2.6),

$\displaystyle\mathbb{X}^{\bot}=\\{u\in\mathbb{X}\mid\langle
u,\partial_{x_{j}}\mathfrak{g}_{i,f}\rangle_{H^{1}(\mathbb{R}^{d})}=0,\quad
j=1,2,\cdots,d;\quad i=1,2,\cdots,\nu\\}$ (3.13)

and $b_{j,i}$ are the Lagrange multipliers given by

$\displaystyle b_{j,i}\sim\langle
h,\partial_{x_{j}}\mathfrak{g}_{i,f}\rangle_{L^{2}}.$ (3.14)

######  Lemma 3.2.

As $\|f\|_{H^{-1}}\to 0$, (3.12) is unique solvable for every $h\in\mathbb{Y}$
with
$\|\psi\|_{\sharp}+\sum_{i=1}^{\nu}\sum_{j=1}^{d}|b_{j,i}|\lesssim\|h\|_{\natural}$.

###### Proof.

By (1.3) and (1.9), it is easy to see that for $R>0$ sufficiently large,

$\displaystyle-(1+2\log(\sum_{j=1}^{\nu}\mathfrak{g}_{j,f}))\geq 0$

in $(\cup_{j=1}^{\nu}B_{R}(y_{j,f}))^{c}$. It follows from
$\sum_{j=1}^{\nu}\mathfrak{g}_{j,f}\sim\mathfrak{g}_{i,f}$ in $\Omega_{i}$
that

$\displaystyle-(1+2\log(\sum_{j=1}^{\nu}\mathfrak{g}_{j,f}))=|x-y_{i,f}|^{2}+O(1)\quad\text{in
}D_{\sigma\eta+2,i,j}\cap\Omega_{i}.$ (3.15)

Note that by the definition of $\mathbb{L}_{i,j}$ given by (3.4) and
rotations, $\mathfrak{g}_{i,f,d-1,j}$ is the unique solution of (1.8) in
$\mathbb{R}^{d-1}$. Thus, by (3.15), rotations, the definitions of
$\Pi_{c,i,j}$ and $\mathbb{L}_{i,j}$ given by (3.4),

$\displaystyle-\Delta\mathfrak{g}_{i,f,d-1,j}-(1+2\log(\sum_{j=1}^{\nu}\mathfrak{g}_{j,f}))\mathfrak{g}_{i,f,d-1,j}\gtrsim\eta^{2}\mathfrak{g}_{i,f,d-1,j}$
(3.16)

in $D_{\sigma\eta+2,i,j}\cap\Omega_{i,j,d-1}$. Similarly,

$\displaystyle-\Delta\mathfrak{g}_{i,f}^{1-\sigma}-(1+2\log(\sum_{j=1}^{\nu}\mathfrak{g}_{j,f}))\mathfrak{g}_{i,f}^{1-\sigma}\gtrsim\mathfrak{g}_{i,f}^{1-\sigma}$
(3.17)

in $(\cup_{j=1}^{\nu}B_{R}(y_{j,f}))^{c}$. For every $x\in\Omega_{i,j,d-1}$,
by the fact that $\Pi_{c,i,j}\perp\mathbb{L}_{i,j}$ for all $c\in\mathbb{R}$,
we can re-write $x=(\alpha_{c,i,j},z^{i,j})$, where
$\alpha_{c,i,j}>-\frac{1}{2}$ is given by (3.5) and
$z^{i,j}\perp\mathbb{L}_{i,j}$. Now, let

$\displaystyle\phi(\alpha_{c,i,j},z^{i,j})=\mathfrak{g}_{i,f}^{1-\sigma}\varphi(\alpha_{c,i,j})+\eta^{-2}e^{-\frac{1}{8}(\eta_{i,j}^{2}-\eta^{2})}\mathfrak{g}_{i,f,d-1,j}(1-\varphi(\alpha_{c,i,j}))$

where $\varphi(\alpha_{c,i,j})$ is the unique solution of the following
equation:

$\displaystyle\left\\{\begin{aligned}
&-\varphi^{\prime\prime}-\varphi^{\prime}+\varphi=1,\quad\text{in
}(\frac{\sigma\eta-1}{\eta_{i,j}}-\frac{1}{2},\frac{\sigma\eta}{\eta_{i,j}}-\frac{1}{2}),\\\
&\varphi^{\prime}(\frac{\sigma\eta-1}{\eta_{i,j}}-\frac{1}{2})=\varphi^{\prime}(\frac{\sigma\eta}{\eta_{i,j}}-\frac{1}{2})=0.\end{aligned}\right.$

Then by (3.16) and (3.17), in $(\cup_{j=1}^{\nu}B_{R}(y_{j,f}))^{c}$,

$\displaystyle-\Delta\phi-(1+2\log(\sum_{j=1}^{\nu}\mathfrak{g}_{j,f}))\phi\gtrsim\left\\{\begin{aligned}
&e^{-\frac{1}{8}(\eta_{i,j}^{2}-\eta^{2})}\mathfrak{g}_{i,f,d-1,j},\quad\alpha_{c,i,j}\leq\frac{\sigma\eta-1}{\eta_{i,j}}-\frac{1}{2},\\\
&\mathfrak{g}_{i,f}^{1-\sigma},\quad\alpha_{c,i,j}\geq\frac{\sigma\eta}{\eta_{i,j}}-\frac{1}{2},\end{aligned}\right.$

which, together with the maximum principle, implies that

$\displaystyle|\psi|\lesssim(\|h\|_{\natural}+\|\psi\|_{L^{\infty}(\partial(\cup_{j=1}^{\nu}B_{R}(y_{j,f}))^{c}})\phi(\alpha_{c,i,j},z^{i,j})$
(3.18)

in $(\cup_{j=1}^{\nu}B_{R}(y_{j,f}))^{c}$ for $R>0$ sufficiently large. Based
on the a-priori estimate (3.18), we shall prove the a-priori estimate

$\displaystyle\|\psi\|_{\sharp}\lesssim\|h\|_{\natural}\quad\text{uniformly as
}\delta\to 0.$ (3.19)

Since the proofs for (3.19), based on the blow-up arguments, are standard
nowadays (cf. [46, 47, 61]), we only sketch it here. We assume the contrary
that there exists $\delta_{n}\to 0$, $\\{\psi_{n}\\}$ solves (3.12) with
$\\{h_{n}\\}\subset L^{2}(\mathbb{R}^{d})$ satisfying
$\|\psi_{n}\|_{\sharp}=1$ and $\|h_{n}\|_{\natural}=o_{n}(1)$ as $n\to\infty$.
Since $\delta_{n}\to 0$, by (3.2),

$\displaystyle\log(\sum_{j=1}^{\nu}\mathfrak{g}_{j,f}(x+y_{i,f}))\to\log\mathfrak{g}=-\frac{1}{2}|x|^{2}+\frac{d+1}{2}$

in $\mathbb{R}^{d}$ as $n\to\infty$. Now, let

$\displaystyle\psi_{i,n}(x)=\phi_{n}(x+y_{i,f_{n}}),$

then by $\delta_{n}\to 0$ as $n\to\infty$ and (3.14), it is standard to prove
that $\psi_{i,n}\to\overline{\psi}_{i}$ uniformly in every compact set of
$\mathbb{R}^{d}$ for every $i$, where $\overline{\psi}_{i}$ are bounded
solutions of the following equation

$\displaystyle-\Delta\overline{\phi}+(|x|^{2}-d-2)\overline{\phi}=0\quad\text{in
}\mathbb{R}^{d}.$

By [19, Theorem 1.3] (see also [39, Theorem 7.5]),
$\overline{\psi}_{i}=\sum_{j=1}^{d}a_{j}\partial_{x_{j}}\mathfrak{g}$. On the
other hand, by (1.3) and (1.9), we can pass to the limit in the orthogonal
conditions of $\psi_{i,n}$ given by (3.13), which implies that

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\lim_{n\to+\infty}\langle\psi_{i,n},\partial_{x_{j}}\mathfrak{g}\rangle_{H^{1}(\mathbb{R}^{d})}$
$\displaystyle=$
$\displaystyle\lim_{n\to+\infty}\int_{\mathbb{R}^{d}}(2+2\log(\sum_{j=1}^{\nu}\mathfrak{g}_{j,f_{n}}))\psi_{n}\partial_{x_{j}}\mathfrak{g}_{i,f_{n}}dx$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}(3+d-|x|^{2})\overline{\psi}_{i}\partial_{x_{j}}\mathfrak{g}dx$
$\displaystyle=$
$\displaystyle\langle\overline{\psi}_{i},\partial_{x_{j}}\mathfrak{g}\rangle_{H^{1}(\mathbb{R}^{d})}$

for all $j=1,2,\cdots,d$. Since

$\displaystyle\langle\partial_{x_{i}}\mathfrak{g},\partial_{x_{j}}\mathfrak{g}\rangle_{H^{1}(\mathbb{R}^{d})}=\int_{\mathbb{R}^{d}}(d+3+|x|^{2})\partial_{x_{i}}\mathfrak{g}\partial_{x_{j}}\mathfrak{g}dx=0,$

we have $\overline{\psi}_{i}=0$, which implies that $\psi_{i,n}\to 0$
uniformly in every compact set of $\mathbb{R}^{d}$ for all $i$. Thus, by
(3.18), we have $\|\psi_{n}\|_{\natural}=o_{n}(1)$ which is a contradiction.
Thanks to the a-priori estimate (3.19), by the Fredholm alternative, we know
that the linear equation (3.12) is unique solvable for all $h\in\mathbb{Y}$.
The estimate $\sum_{i=1}^{\nu}\sum_{j=1}^{d}|b_{j,i}|\lesssim\|h\|_{\natural}$
comes from (3.1) and (3.14). ∎

By direct calculations,

$\displaystyle\max_{\alpha_{c,i,j}>-\frac{1}{2}}e^{-(\frac{\alpha_{c,i,j}^{2}}{2}+\alpha_{c,i,j}+\frac{1}{2})\eta_{i,j}^{2}}\log(1+e^{(\alpha_{c,i,j}+\frac{1}{2})\eta_{i,j}^{2}})\sim
e^{-\frac{1}{8}\eta_{i,j}^{2}}.$ (3.20)

Thus, by taking $\sigma>0$ sufficiently small and Lemma 3.1, we have
$\|E\|_{\natural}\lesssim e^{-\frac{1}{8}\eta^{2}}$. We define

$\displaystyle\mathbb{B}=\\{\phi\in\mathbb{X}^{\perp}\mid\|\phi\|_{\sharp}\leq
Me^{-\frac{1}{8}\eta^{2}}\\}$ (3.21)

where $M>0$ is a sufficiently large constant.

######  Lemma 3.3.

There exists $M>0$ sufficiently large such that (3.11) has a unique solution
$\phi_{f}\in\mathbb{B}$ with
$\|\phi_{f}\|_{\sharp}+\sum_{j=1}^{d}\sum_{i=1}^{\nu}|a_{j,i}|\lesssim
e^{-\frac{1}{8}\eta^{2}}$ as $\|f\|_{H^{-1}}\to 0$. Moreover,
$\|\phi_{f}\|_{H^{1}(\mathbb{R}^{d})}\lesssim e^{-\frac{1}{8}\eta^{2}}$ as
$\|f\|_{H^{-1}}\to 0$.

###### Proof.

The proof is standard nowadays, so we also sketch it here. For
$\phi\in\mathbb{B}$, by (2.8) and the Taylor expansion,

$\displaystyle
N(\phi_{+})=2\log(1+\frac{\theta\phi_{-}}{\sum_{j=1}^{\nu}\mathfrak{g}_{j,f}})\phi_{+}\text{
and
}N(\phi_{-})=-2\log(1-\frac{\theta\phi_{-}}{\sum_{j=1}^{\nu}\mathfrak{g}_{j,f}})\phi_{-},$
(3.22)

in $\Omega_{i}$, where $\Omega_{i}$ is given by (3.3) and
$\phi_{\pm}=\max\\{\pm\phi,0\\}$. Note that for $\phi\in\mathbb{B}$,

$\displaystyle\frac{\pm\phi_{\pm}}{\sum_{j=1}^{\nu}\mathfrak{g}_{j,f}}$
$\displaystyle\lesssim$
$\displaystyle\|\phi\|_{\sharp}\sum_{i=1}^{\nu}\sum_{j\not=i}(\sup_{D_{\sigma\eta-1,i,j}^{c}\cap\Omega_{i}}\mathfrak{g}_{i,f}^{-\sigma}+\sup_{D_{\sigma\eta-1,i,j}\cap\Omega_{i,j,d-1}}\frac{e^{-\frac{1}{8}(\eta_{i,j}^{2}-\eta^{2})}\mathfrak{g}_{i,f,d-1,j}}{\eta^{2}\mathfrak{g}_{i,f}})$
(3.23) $\displaystyle\lesssim$ $\displaystyle\eta^{-2}.$

Thus, by (3.22) and the symmetry of $\mathfrak{g}_{i,f}$,

$\displaystyle\|N(\phi_{\pm})\|_{\natural}$ $\displaystyle\lesssim$
$\displaystyle\eta^{-2}\sum_{i=1}^{\nu}\sum_{j\not=i}(\sup_{D_{\sigma\eta,i,j}^{c}\cap\Omega_{i}}\frac{|\phi_{\pm}|}{\mathfrak{g}_{i,f}^{1-\sigma}}+\sup_{D_{\sigma\eta,i,j}\cap\Omega_{i,j,d-1}}\frac{|\phi_{\pm}|}{e^{-\frac{1}{8}(\eta_{i,j}^{2}-\eta^{2})}\mathfrak{g}_{i,f,d-1,j}})$
$\displaystyle\lesssim$ $\displaystyle\eta^{-2}e^{-\frac{1}{8}\eta^{2}},$

which implies that
$\|N(\phi)\|_{\natural}\lesssim\eta^{-2}e^{-\frac{1}{8}\eta^{2}}$. Now, we can
solve (4.1) by the standard fix-point arguments in $\mathbb{B}$ by choosing a
sufficiently large $M>0$. The estimate
$\|\phi_{f}\|_{\sharp}+\sum_{j=1}^{d}\sum_{i=1}^{\nu}|a_{j,i}|\lesssim
e^{-\frac{1}{8}\eta^{2}}$ comes from Lemma 3.2. By multiplying (3.11) with
$\phi_{f}$ on both sides and integrating by parts and using the fact that
$\phi_{f}\in\mathbb{B}$, we have
$\|\phi_{f}\|_{H^{1}(\mathbb{R}^{d})}^{2}\lesssim e^{-\frac{1}{4}\eta^{2}}$ as
$\|f\|_{H^{-1}}\to 0$ which completes the proof. ∎

We recall that by (2.5) and (3.11), the remaining term
$\varphi_{f}=\rho_{f}-\phi_{f}$ satisfies

$\displaystyle\left\\{\begin{aligned}
&\mathcal{L}_{f}(\varphi_{f})=N(\phi_{f}+\varphi_{f})-N(\phi_{f})+\sum_{j=1}^{d}\sum_{i=1}^{\nu}a_{j,i}\partial_{x_{j}}\mathfrak{g}_{i,f}+f,\quad\text{in
}H^{-1},\\\
&\langle\varphi_{f},\partial_{x_{j}}\mathfrak{g}_{i,f}\rangle_{H^{1}(\mathbb{R}^{d})}=0,\quad
j=1,2,\cdots,d\quad\text{and}\quad i=1,2,\cdots,\nu.\end{aligned}\right.$
(3.24)

Moreover, it is well known (cf. [39, Theorem 7.5 and Remark 7.7]) that the
eigenfunctions of the linear operator $\mathcal{L}$ given by (1.10) forms an
orthogonal basis in $L^{2}(\mathbb{R}^{d})$, where the first eigenvalue is
$-2$ with eigenspace $span\\{\mathfrak{g}\\}$ and the second eigenvalue is $0$
with eigenspace $span\\{\partial_{x_{j}}\mathfrak{g}\\}$. Thus, we can write

$\displaystyle\varphi_{f}=\sum_{j=1}^{\nu}(c_{j}\mathfrak{g}_{j,f}+\sum_{l=1}^{d}b_{l,j}\partial_{x_{l}}\mathfrak{g}_{j,f})+\varphi_{f}^{\perp},$
(3.25)

where $\varphi_{f}^{\perp}$ is orthogonal to
$span_{j,l}\\{\mathfrak{g}_{j,f},\partial_{x_{l}}\mathfrak{g}_{j,f}\\}$ in
$L^{2}(\mathbb{R}^{d})$.

######  Lemma 3.4.

As $\|f\|_{H^{-1}}\to 0$, we have

$\displaystyle\|\varphi_{f}\|_{L^{2}(\mathbb{R}^{d})}^{2}\sim\sum_{j=1}^{\nu}|c_{j}|^{2}+\|\varphi_{f}^{\perp}\|_{L^{2}(\mathbb{R}^{d})}^{2}.$
(3.26)

###### Proof.

Note that we have

$\displaystyle\langle\mathfrak{g}_{j,f},\nabla\mathfrak{g}_{j,f}\rangle_{H^{1}}=-\frac{1}{2}\nabla_{y_{j,f}}\|\mathfrak{g}_{j,f}\|_{H^{1}(\mathbb{R}^{d})}^{2}=0$
(3.27)

for all $j=1,2,\cdots,\nu$. Thus, by (2.3), (3.25) and (3.27),

$\displaystyle
0=\sum_{i\not=j}(c_{i}\langle\mathfrak{g}_{i,f},\partial_{x_{l}}\mathfrak{g}_{j,f}\rangle_{H^{1}}+\sum_{m=1}^{d}b_{i,m}\langle\partial_{x_{m}}\mathfrak{g}_{i,f},\partial_{x_{l}}\mathfrak{g}_{j,f}\rangle_{H^{1}})+b_{l,j}$
(3.28)

for all $l=1,2,\cdots,d$ and $j=1,2,\cdots,\nu$. Since

$\displaystyle|\partial_{x_{l}}\mathfrak{g}_{i,f}|\lesssim
r_{i,f}\mathfrak{g}_{i,f}\lesssim\mathfrak{g}_{i,f}^{1-\sigma}$

for all $l$, where $\sigma>0$ can be taken arbitrary small if necessary and
$r_{i,f}=|x-y_{i,f}|$, by [3, Lemma 3.7],

$\displaystyle|\langle\mathfrak{g}_{i,f},\partial_{x_{l}}\mathfrak{g}_{j,f}\rangle_{H^{1}}|+|\langle\partial_{x_{m}}\mathfrak{g}_{i,f},\partial_{x_{l}}\mathfrak{g}_{j,f}\rangle_{H^{1}}|$
$\displaystyle\lesssim$
$\displaystyle\int_{\mathbb{R}^{d}}\mathfrak{g}_{i,f}^{1-\sigma^{\prime}}\mathfrak{g}_{j,f}^{1-\sigma}$
(3.29) $\displaystyle\lesssim$ $\displaystyle
e^{-\frac{1-\sigma^{\prime\prime}}{2}\eta^{2}}$

for all $i,j,l,m$, which, together with (3.28), implies that

$\displaystyle\sum_{j=1}^{\nu}\sum_{l=1}^{d}|b_{l,j}|\lesssim
e^{-\frac{1-\sigma^{\prime\prime}}{2}\eta^{2}}\sum_{j=1}^{\nu}|c_{j}|.$ (3.30)

Here, $\sigma^{\prime},\sigma^{\prime\prime}>0$ are constants which can be
taken arbitrary small. (3.26) then follows from (3.25) and (3.30). ∎

We denote $N(\phi_{f}+\varphi_{f})-N(\phi_{f})$ by $N_{\phi_{f}}(\varphi_{f})$
for the sake of simplicity. Then by Lemma 3.3 and by multiplying (3.24) with
$\varphi_{f}$ and integrating by parts, we have

$\displaystyle\langle\mathcal{L}_{f}(\varphi_{f}),\varphi_{f}\rangle_{L^{2}}\lesssim\int_{\mathbb{R}^{d}}N_{\phi_{f}}(\varphi_{f})\varphi_{f}dx+(\|f\|_{H^{-1}}+e^{-\frac{1}{8}\eta^{2}})\|\varphi_{f}\|_{H^{1}(\mathbb{R}^{d})}.$
(3.31)

######  Lemma 3.5.

As $\|f\|_{H^{-1}}\to 0$, we have

$\displaystyle\|\varphi_{f}\|_{H^{1}(\mathbb{R}^{d})}\lesssim\|f\|_{H^{-1}}+e^{-(\frac{3}{8}-\sigma)\eta^{2}}.$

###### Proof.

For the sake of clarity, we divide the proof into several parts.

Step. 1 The estimate of $N_{\phi_{f}}(\varphi_{f})\varphi_{f}$.

By the Taylor expansion and
$u=\sum_{j=1}^{\nu}\mathfrak{g}_{j,f}+\phi_{f}+\varphi_{f}\geq 0$,

$\displaystyle N_{\phi_{f}}(\varphi_{f})$ $\displaystyle=$ $\displaystyle
2(\sum_{j=1}^{\nu}\mathfrak{g}_{j,f}+\phi_{f}+\varphi_{f})\log(\sum_{j=1}^{\nu}\mathfrak{g}_{j,f}+\phi_{f}+\varphi_{f})$
(3.32)
$\displaystyle-2(\sum_{j=1}^{\nu}\mathfrak{g}_{j,f}+\phi_{f})\log(\sum_{j=1}^{\nu}\mathfrak{g}_{j,f}+\phi_{f})$
$\displaystyle-2(1+\log(\sum_{j=1}^{\nu}\mathfrak{g}_{j,f}))\varphi_{f}$
$\displaystyle=$ $\displaystyle
2\log(1+\frac{\phi_{f}+\theta\varphi_{f}}{\sum_{j=1}^{\nu}\mathfrak{g}_{j,f}}))\varphi_{f}$
(3.33) $\displaystyle=$ $\displaystyle
2\log(1+\frac{\phi_{f}}{\sum_{j=1}^{\nu}\mathfrak{g}_{j,f}}))\varphi_{f}+\frac{\varphi_{f}^{2}}{\sum_{i=1}^{\nu}\mathfrak{g}_{i,f}+\phi_{f}+\theta^{\prime}\varphi_{f}},$
(3.34)

where $\theta,\theta^{\prime}\in(0,1)$. By (3.23), we have

$\displaystyle|\frac{\phi_{f}}{\sum_{j=1}^{\nu}\mathfrak{g}_{j,f}}|\lesssim\eta^{-2}.$
(3.35)

Thus,

$\displaystyle\sum_{i=1}^{\nu}\mathfrak{g}_{i,f}+\phi_{f}=(1+O(\eta^{-2}))\sum_{i=1}^{\nu}\mathfrak{g}_{i,f}.$

For every $R>0$, let

$\displaystyle\Upsilon_{R,\alpha}=\\{x\in\partial(\cup_{j=1}^{\nu}B_{R}(y_{j,f}))\mid\varphi_{f}=\alpha\sum_{i=1}^{\nu}\mathfrak{g}_{i,f}\\}$

where $\alpha\geq-1+O(\eta^{-2})$. Since $0\leq\alpha\leq
e^{\frac{R^{2}-(d+5)}{2}}-1+O(\eta^{-2})$ implies that

$\displaystyle 2(1+\log(1+O(\eta^{-2})+\theta\alpha))\leq R^{2}-(d+3)$

and $-1+O(\eta^{-2})+e^{-\frac{R^{2}-(d+1)}{2}}\leq\alpha<0$ implies that

$\displaystyle-2(1+\log(1+O(\eta^{-2})+\theta\alpha))\leq R^{2}-(d+3)$

by $\theta\in(0,1)$, by (1.9), (3.33) and (3.35), for every $R>0$ and every
$x\in\partial(\cup_{j=1}^{\nu}B_{R}(y_{j,f}))$, one of the following cases
must happen:

  1. $(a)$

$N_{\phi_{f}}(\varphi_{f})\varphi_{f}\leq(R^{2}-(d+3))\varphi_{f}^{2}$,

  2. $(b)$

$\varphi_{f}\gtrsim 1$,

  3. $(c)$

$\varphi_{f}\sim-\sum_{i=1}^{\nu}\mathfrak{g}_{i,f}$.

On the other hand, by (3.32), (3.34) and (3.35), in $\Upsilon_{R,\alpha}$ with
$-1\leq\alpha<0$,

$\displaystyle
2((1+O(\eta^{-2})+\alpha)\log(1+O(\eta^{-2})+\alpha)-\alpha)=O(\eta^{-2})\alpha+\frac{\alpha^{2}}{1+O(\eta^{-2})+\theta^{\prime}\alpha}.$

It follows that $\theta^{\prime}\to\frac{1}{2}+O(\eta^{-\sigma})$ as
$\alpha\to-1$. Thus, by (1.9),

$\displaystyle
N_{\phi_{f}}(\varphi_{f})\varphi_{f}\leq(\max\\{-(2+2\log(\sum_{j=1}^{\nu}\mathfrak{g}_{j,f})),0\\}+O(\eta^{-\sigma}))|\varphi_{f}|^{2}+O(\varphi_{f}^{3}).$
(3.36)

Step. 2 The estimate of $\|\varphi_{f}^{\perp}\|_{H^{1}(\mathbb{R}^{d})}^{2}$.

By (3.25), (3.30), (3.31), (3.36) and Lemma 3.4,

$\displaystyle\langle\mathcal{L}_{f}(\varphi_{f}),\varphi_{f}\rangle_{L^{2}}$
$\displaystyle\lesssim$
$\displaystyle\int_{\mathbb{R}^{d}}N_{\phi_{f}}(\varphi_{f})\varphi_{f}dx+(\|f\|_{H^{-1}}+e^{-\frac{1-\sigma^{\prime\prime}}{2}\eta^{2}}\sum_{j=1}^{\nu}|c_{j}|)\|\varphi_{f}\|_{H^{1}(\mathbb{R}^{d})}$
$\displaystyle\leq$
$\displaystyle\int_{\mathbb{R}^{d}}\max\\{-(2+2\log(\sum_{j=1}^{\nu}\mathfrak{g}_{j,f})),0\\}\varphi_{f}^{2}dx$
$\displaystyle+O(\|\varphi_{f}\|^{3}_{H^{1}(\mathbb{R}^{d})})+(\|f\|_{H^{-1}}+e^{-\frac{1-\sigma^{\prime\prime}}{2}\eta^{2}}\sum_{j=1}^{\nu}|c_{j}|)\|\varphi_{f}\|_{H^{1}(\mathbb{R}^{d})}.$

It follows that

$\displaystyle\int_{\cup_{j=1}^{\nu}B_{R}(y_{j,f})}(|\nabla\varphi_{f}|^{2}-(1+2\log(\sum_{j=1}^{\nu}\mathfrak{g}_{j,f})))|\varphi_{f}|^{2}dx$
(3.37)
$\displaystyle+\int_{\mathbb{R}^{d}\backslash(\cup_{j=1}^{\nu}B_{R}(y_{j,f}))}|\nabla\varphi_{f}|^{2}+|\varphi_{f}|^{2}dx$
$\displaystyle\leq$ $\displaystyle
O(\|\varphi_{f}\|^{3}_{H^{1}(\mathbb{R}^{d})})+(\|f\|_{H^{-1}}+e^{-\frac{1-\sigma^{\prime\prime}}{2}\eta^{2}}\sum_{j=1}^{\nu}|c_{j}|)\|\varphi_{f}\|_{H^{1}(\mathbb{R}^{d})}.$

for a sufficiently large $R>0$. For every $j$,

$\displaystyle\int_{B_{R}(y_{j,f})}(|\nabla\varphi_{f}|^{2}-(1+2\log(\sum_{j=1}^{\nu}\mathfrak{g}_{j,f})))|\varphi_{f}|^{2}dx$
(3.38) $\displaystyle\geq$
$\displaystyle\int_{B_{R}(y_{j,f})}[(1-\sigma)(d+1)-r_{i,f}^{2}-2\log(\sum_{j=1}^{\nu}\mathfrak{g}_{j,f}))]|\varphi_{f}|^{2}dx)$
$\displaystyle+\int_{B_{R}(y_{j,f})}|\nabla\varphi_{f}|^{2}+(r_{i,f}^{2}-d-2)|\varphi_{f}|^{2}dx$

where $\sigma>0$ is sufficient small. By [39, Theorem 7.5 and Remark 7.7], it
is easy to show that

$\displaystyle\int_{B_{R}(y_{j,f})}|\nabla\widetilde{\varphi}_{f,R}^{\perp}|^{2}+(r_{j,f}^{2}-d-2)|\widetilde{\varphi}_{f,R}^{\perp}|^{2}dx\gtrsim\int_{B_{R}(y_{j,f})}|\nabla\widetilde{\varphi}_{f,R}^{\perp}|^{2}+|\widetilde{\varphi}_{f,R}^{\perp}|^{2}dx$
(3.39)

where $R>0$ sufficiently large and
$\widetilde{\varphi}_{f,R}^{\perp}=\varphi_{f}^{\perp}\psi_{R}$ with
$\psi_{R}$ being a smooth cut-off function such that $\psi_{R}=1$ in
$B_{\frac{R}{4}}(0)$ and $\psi_{R}=0$ in $B_{R}(0)\backslash
B_{\frac{R}{2}}(0)$. Note that

$\displaystyle\int_{B_{R}(y_{j,f})}|\nabla\widetilde{\varphi}_{f,R}^{\perp}|^{2}+(r_{j,f}^{2}-d-2)|\widetilde{\varphi}_{f,R}^{\perp}|^{2}dx$
$\displaystyle\lesssim$
$\displaystyle\int_{B_{R}(y_{j,f})}|\nabla\varphi_{f}^{\perp}|^{2}+(r_{i,f}^{2}-d-2)|\varphi_{f}^{\perp}|^{2}dx$
$\displaystyle+\int_{B_{R}(y_{j,f})\backslash
B_{\frac{R}{2}}(y_{j,f})}|\nabla\widetilde{\varphi}_{f,R}^{\perp}|^{2}+|\widetilde{\varphi}_{f,R}^{\perp}|^{2}dx$
$\displaystyle\lesssim$
$\displaystyle\int_{B_{R}(y_{j,f})}|\nabla\varphi_{f}^{\perp}|^{2}+(r_{i,f}^{2}-d-2)|\varphi_{f}^{\perp}|^{2}dx$

and

$\displaystyle\int_{B_{R}(y_{j,f})}|\nabla\widetilde{\varphi}_{f,R}^{\perp}|^{2}+|\widetilde{\varphi}_{f,R}^{\perp}|^{2}dx$
$\displaystyle=$
$\displaystyle\int_{B_{R}(y_{j,f})}|\nabla\varphi_{f}^{\perp}|^{2}+|\varphi_{f}^{\perp}|^{2}dx$
$\displaystyle+O(1)\int_{B_{R}(y_{j,f})}|\nabla\varphi_{f}^{\perp}|^{2}+(r_{i,f}^{2}-d-2)|\varphi_{f}^{\perp}|^{2}dx,$

thus, by (3.39),

$\displaystyle\int_{B_{R}(y_{j,f})}|\nabla\varphi_{f}^{\perp}|^{2}+(r_{i,f}^{2}-d-2)|\varphi_{f}^{\perp}|^{2}dx\gtrsim\int_{B_{R}(y_{j,f})}|\nabla\varphi_{f}^{\perp}|^{2}+|\varphi_{f}^{\perp}|^{2}dx.$
(3.40)

On the other hand, we have $|x-y_{i,f}|\geq\eta+O(1)$ for all $j\not=i$ in
$B_{R}(y_{j,f})$ which implies that

$\displaystyle\sum_{i\not=j}\mathfrak{g}_{i,f}\lesssim
e^{-\frac{\eta^{2}}{2}}\mathfrak{g}_{j,f}\quad\text{in }B_{R}(y_{j,f}).$

It follows that

$\displaystyle-2\log(\sum_{j=1}^{\nu}\mathfrak{g}_{j,f})=-(d+1)+r_{j,f}^{2}+O(e^{-\frac{\eta^{2}}{2}})\quad\text{in
}B_{R}(y_{j,f}),$

which implies that

$\displaystyle|\int_{B_{R}(y_{j,f})}[(1-\sigma)(d+1)-r_{i,f}^{2}-2\log(\sum_{j=1}^{\nu}\mathfrak{g}_{j,f}))]|\varphi_{f}|^{2}dx|\lesssim\sigma\|\varphi_{f}\|_{H^{1}(\mathbb{R}^{d})}^{2}.$

It follows from (3.37), (3.38), (3.40) and Lemma 3.4 that

$\displaystyle\|\varphi_{f}^{\perp}\|_{H^{1}(\mathbb{R}^{d})}^{2}\lesssim\|f\|_{H^{-1}}^{2}+e^{-(1-\sigma^{\prime\prime})\eta^{2}}+\sum_{j=1}^{\nu}|c_{j}|^{2}.$
(3.41)

Step. 3 The estimate of $\sum_{j=1}^{\nu}|c_{j}|^{2}$.

Let

$\displaystyle\widetilde{\psi}_{j}=\mathfrak{g}_{j,f}\widehat{\psi}_{j},$

where $\widehat{\psi}_{j}$ is a smooth cut-off function satisfying

$\displaystyle\widehat{\psi}_{j}=\left\\{\begin{aligned}
&1,\quad|x-y_{j,f}|\leq(\frac{1}{2}-\sigma)\eta,\\\
&0,\quad|x-y_{j,f}|\geq(\frac{1}{2}-\sigma)\eta+1.\end{aligned}\right.$

Note that $\mathfrak{g}_{j,f}$ is the unique positive solution of (1.8), thus,
by multiplying (3.24) with $-sgn(c_{j})\widetilde{\psi}_{j}$ and integrating
by parts, we have

$\displaystyle-
sgn(c_{j})\int_{\mathbb{R}^{d}}(N_{\phi_{f}}(\varphi_{f})+\sum_{j=1}^{d}\sum_{i=1}^{\nu}a_{j,i}\partial_{x_{j}}\mathfrak{g}_{i,f})\widetilde{\psi}_{j}+f\widetilde{\psi}_{j}dx$
(3.42) $\displaystyle=$ $\displaystyle-
sgn(c_{j})\int_{\mathbb{R}^{d}}-\Delta\mathfrak{g}_{j,f}\widehat{\psi}_{j}\varphi_{f}-2\varphi_{f}\nabla\widehat{\psi}_{j}\nabla\mathfrak{g}_{j,f}-\Delta\widehat{\psi}_{j}\mathfrak{g}_{j,f}\varphi_{f}dx$
$\displaystyle+sgn(c_{j})\int_{\mathbb{R}^{d}}(1+2\log(\sum_{j=1}^{\nu}\mathfrak{g}_{j,f}))\widehat{\psi}_{j}\mathfrak{g}_{j,f}\phi_{f}dx$
$\displaystyle=$ $\displaystyle
sgn(c_{j})\int_{\mathbb{R}^{d}}(r_{j,f}^{2}+1-d+2\log(\sum_{j=1}^{\nu}\mathfrak{g}_{j,f}))\widehat{\psi}_{j}\mathfrak{g}_{j,f}\phi_{f}dx$
$\displaystyle+sgn(c_{j})\int_{\mathbb{R}^{d}}2\phi_{f}\nabla\widehat{\psi}_{j}\nabla\mathfrak{g}_{j,f}+\Delta\widehat{\psi}_{j}\mathfrak{g}_{j,f}\phi_{f}dx.$

By (3.29) and similar estimates of (3.36),

$\displaystyle\int_{\mathbb{R}^{d}}N_{\phi_{f}}(\varphi_{f})\widetilde{\psi}_{j}dx$
$\displaystyle\leq$
$\displaystyle|c_{j}|\int_{\mathbb{R}^{d}}\max\\{-(2+2\log(\sum_{j=1}^{\nu}\mathfrak{g}_{j,f})),0\\}|\mathfrak{g}_{j,f}|^{2}\widehat{\psi}_{j}dx$
$\displaystyle+o(\sum_{i=1}^{\nu}|c_{i}|+\|\phi_{f}^{\perp}\|_{H^{1}(\mathbb{R}^{d})})+\sum_{i=1}^{\nu}|c_{i}|^{2}+\|\phi_{f}^{\perp}\|_{H^{1}(\mathbb{R}^{d})}^{2}.$

Note that supp$(\widehat{\psi}_{j})\subset\Omega_{j}$ with
$\sum_{i\not=j}\mathfrak{g}_{i,f}\lesssim
e^{-\sigma\eta^{2}}\mathfrak{g}_{j,f}$ in supp$(\widehat{\psi}_{j})$, thus, by
(1.9), (3.25) and Lemma 3.4,

$\displaystyle\int_{\mathbb{R}^{d}}2\phi_{f}\nabla\widehat{\psi}_{j}\nabla\mathfrak{g}_{j,f}+\Delta\widehat{\psi}_{j}\mathfrak{g}_{j,f}\phi_{f}dx=o(\sum_{i=1}^{\nu}|c_{i}|+\|\phi_{f}^{\perp}\|_{H^{1}(\mathbb{R}^{d})})$

and

$\displaystyle
sgn(c_{j})\int_{\mathbb{R}^{d}}(r_{j,f}^{2}+1-d+2\log(\sum_{j=1}^{\nu}\mathfrak{g}_{j,f}))\widehat{\psi}_{j}\mathfrak{g}_{j,f}\phi_{f}dx-|\int_{\mathbb{R}^{d}}N_{\phi_{f}}(\varphi_{f})\widetilde{\psi}_{j}dx|$
$\displaystyle\sim$
$\displaystyle|c_{j}|+o(\sum_{i=1}^{\nu}|c_{i}|+\|\phi_{f}^{\perp}\|_{H^{1}(\mathbb{R}^{d})}),$

which, together with (3.42) and Lemma 3.3, implies that

$\displaystyle|c_{j}|+o(\sum_{i\not=j}c_{i}+\|\phi_{f}^{\perp}\|_{H^{1}(\mathbb{R}^{d})})\lesssim
O(e^{-(\frac{3}{8}-\sigma)\eta^{2}})+\|f\|_{H^{-1}}.$ (3.43)

Since (3.43) holds for all $j=1,2,\cdots,\nu$ and the linear part of this
inequality is diagonally dominant, we have

$\displaystyle\sum_{j=1}^{\nu}|c_{j}|^{2}\lesssim
O(e^{-(\frac{3}{4}-2\sigma)\eta^{2}})+\|f\|_{H^{-1}}^{2}.$ (3.44)

Step. 4 The estimate of $\|\varphi_{f}\|_{H^{1}(\mathbb{R}^{d})}$.

(3.26) comes from (3.41), (3.44) and Lemma 3.4. ∎

We also need the following lemma.

######  Lemma 3.6.

As $\|f\|_{H^{-1}}\to 0$, we have
$e^{-\frac{1}{8}\eta^{2}}\lesssim\|f\|_{H^{-1}}$.

###### Proof.

Without loss of generality, we may assume that $\eta=|y_{1,f}-y_{2,f}|$. We
define

$\displaystyle\widehat{\psi}=\mathfrak{g}_{1,f}\overline{\psi},$ (3.45)

where $\overline{\psi}$ is a smooth cut-off function satisfying

$\displaystyle\overline{\psi}=\left\\{\begin{aligned} &1,\quad x\in
B_{1}(\frac{1}{2}(y_{1,f}+y_{2,f})+\frac{2\log\eta}{\eta^{2}}(y_{2,f}-y_{1,f})+5),\\\
&0,\quad
B_{2}^{c}(\frac{1}{2}(y_{1,f}+y_{2,f})+\frac{2\log\eta}{\eta^{2}}(y_{2,f}-y_{1,f})+5).\end{aligned}\right.$

By similar estimates of (3.34), we know that $N(\rho_{f})\geq 0$. Thus,
similar to (3.42), by multiplying (2.5) with $-\widehat{\psi}$ and integrating
by parts and (3.34), we have

$\displaystyle\|\widehat{\psi}\|_{H^{1}(\mathbb{R}^{d})}\|f\|_{H^{-1}}$
$\displaystyle\gtrsim$
$\displaystyle\int_{\mathbb{R}^{d}}(r_{1,f}^{2}+1-d+2\log(\sum_{j=1}^{\nu}\mathfrak{g}_{j,f}))\overline{\psi}\mathfrak{g}_{1,f}\rho_{f}dx$
(3.46)
$\displaystyle+\int_{\mathbb{R}^{d}}(E+N(\rho_{f}))\widehat{\psi}dx+\int_{\mathbb{R}^{d}}2\rho_{f}\nabla\overline{\psi}\nabla\mathfrak{g}_{1,f}+\Delta\overline{\psi}\mathfrak{g}_{1,f}\rho_{f}dx$
$\displaystyle\gtrsim$
$\displaystyle\int_{\mathbb{R}^{d}}(r_{1,f}^{2}+1-d+2\log(\sum_{j=1}^{\nu}\mathfrak{g}_{j,f}))\overline{\psi}\mathfrak{g}_{1,f}\rho_{f}dx+\int_{\mathbb{R}^{d}}E\overline{\psi}dx$
$\displaystyle\int_{\mathbb{R}^{d}}2\rho_{f}\nabla\overline{\psi}\nabla\mathfrak{g}_{1,f}+\Delta\overline{\psi}\mathfrak{g}_{1,f}\rho_{f}dx.$

By (1.9) and (3.20), $\int_{\mathbb{R}^{d}}E\widehat{\psi}dx\sim
e^{-\frac{1}{4}\eta^{2}}$. By Lemmas 3.3 and 3.5,

$\displaystyle|\int_{\mathbb{R}^{d}}2\rho_{f}\nabla\overline{\psi}\nabla\mathfrak{g}_{1,f}+\Delta\overline{\psi}\mathfrak{g}_{1,f}\rho_{f}dx|$
$\displaystyle+|\int_{\mathbb{R}^{d}}(r_{1,f}^{2}+1-d+2\log(\sum_{j=1}^{\nu}\mathfrak{g}_{j,f}))\overline{\psi}\mathfrak{g}_{1,f}\rho_{f}dx|$
$\displaystyle\lesssim$
$\displaystyle\eta^{-2}e^{-\frac{1}{4}\eta^{2}}+e^{-\sigma\eta^{2}}\|f\|_{H^{-1}}.$

Note that
$\|\widehat{\varphi}\mathfrak{g}_{1,f}\|_{H^{1}(\mathbb{R}^{d})}\lesssim
e^{-\frac{1}{8}\eta^{2}}$, thus, we obtain
$e^{-\frac{1}{8}\eta^{2}}\lesssim\|f\|_{H^{-1}}$. ∎

We close this section by the following proposition.

######  Proposition 3.1.

As $\|f\|_{H^{-1}}\to 0$, we have
$\|\rho_{f}\|_{H^{1}(\mathbb{R}^{d})}\lesssim\|f\|_{H^{-1}}$.

###### Proof.

The conclusion follows immediately from Lemmas 3.3, 3.5 and 3.6. ∎

##  4\. Optimality of (1.11)

In this section, we shall construct an example to show that (1.11) is optimal.
Let $\mathfrak{g}_{\pm L}=\mathfrak{g}(x\pm\frac{L}{2}e_{1})$ with
$e_{1}=(1,0,\cdots,0)$ and we consider the following equation:

$\displaystyle\left\\{\begin{aligned}
&\mathcal{L}_{L}(\rho_{L})=E_{L}+N(\rho_{L})-\sum_{j=1}^{d}(a_{j,+,L}\partial_{x_{j}}\mathfrak{g}_{L}+a_{j,-,L}\partial_{x_{j}}\mathfrak{g}_{-L}),\quad\text{in
}H^{-1},\\\ &\langle\rho_{L},\partial_{x_{j}}\mathfrak{g}_{\pm
L}\rangle_{H^{1}(\mathbb{R}^{d})}=0,\quad j=1,2,\cdots,d,\end{aligned}\right.$
(4.1)

where

$\displaystyle\mathcal{L}_{L}=-\Delta-1-2\log(\mathfrak{g}_{L}+\mathfrak{g}_{-L})$

is the linear operator,

$\displaystyle
E_{L}=2(\mathfrak{g}_{L}+\mathfrak{g}_{-L})\log(\mathfrak{g}_{L}+\mathfrak{g}_{-L})-2(\mathfrak{g}_{L}\log\mathfrak{g}_{L}+\mathfrak{g}_{-L}\log\mathfrak{g}_{-L})$

is the error,

$\displaystyle N(\rho_{L})$ $\displaystyle=$ $\displaystyle
2(\mathfrak{g}_{L}+\mathfrak{g}_{-L}+\rho_{L})\log(\mathfrak{g}_{L}+\mathfrak{g}_{-L}+\rho_{L})-2(\mathfrak{g}_{L}+\mathfrak{g}_{-L})\log(\mathfrak{g}_{L}+\mathfrak{g}_{-L})$
$\displaystyle-2(1+\log(\mathfrak{g}_{L}+\mathfrak{g}_{-L}))\rho_{L}$

is the nonlinear part, and $a_{j,\pm,L}$ are Lagrange multipliers given by

$\displaystyle a_{j,\pm,L}\sim\langle
E_{L}+N(\rho_{L}),\partial_{x_{j}}\mathfrak{g}_{\pm L}\rangle_{L^{2}}.$

By Lemma 3.3, (4.1) is unique solvable in $\mathbb{B}$, where $\mathbb{B}$ is
given by (3.21). Moreover, by Lemma 3.1,

$\displaystyle\|\rho_{L}\|_{\sharp}+\sum_{j=1}^{d}\sum_{i=1}^{\nu}|a_{j,\pm,L}|\lesssim
e^{-\frac{1}{8}L^{2}}.$ (4.2)

Let

$\displaystyle u_{L}=\mathfrak{g}_{L}+\mathfrak{g}_{-L}+\rho_{L}.$

Then by similar estimates for (3.35),
$u_{L}\sim\mathfrak{g}_{L}+\mathfrak{g}_{-L}>0$. Moreover, by the classical
regularity, $\rho_{L}\in C^{1,\alpha}(\mathbb{R}^{d})$ for all
$\alpha\in(0,1)$. It follows from (4.1) that

$\displaystyle f_{L}$ $\displaystyle:=$ $\displaystyle-\Delta
u_{L}+u_{L}-2u_{L}\log|u_{L}|$ (4.3) $\displaystyle=$
$\displaystyle-\sum_{j=1}^{d}(a_{j,+,L}\partial_{x_{j}}\mathfrak{g}_{L}+a_{j,-,L}\partial_{x_{j}}\mathfrak{g}_{-L}).$

######  Proposition 4.1.

As $L\to+\infty$, we have

$\displaystyle\inf_{z_{i}\in\mathbb{R}^{d}}\|u_{L}-\sum_{i=1}^{2}\mathfrak{g}(\cdot-
z_{i})\|_{H^{1}(\mathbb{R}^{d})}^{2}\sim\|f_{L}\|_{H^{-1}}^{2}.$

###### Proof.

Thanks to Lemma 3.6 and (4.2), we have

$\displaystyle\|f_{L}\|_{H^{-1}}\sim e^{-\frac{1}{8}L^{2}}.$ (4.4)

Now, we consider the minimizing problem

$\displaystyle\mathfrak{c}_{L}=\inf_{z_{i}\in\mathbb{R}^{d}}\|u_{L}-\sum_{i=1}^{2}\mathfrak{g}(\cdot-
z_{i})\|_{H^{1}(\mathbb{R}^{d})}^{2}.$ (4.5)

Clearly, by (4.2),
$\mathfrak{c}_{L}\lesssim\|\rho_{L}\|_{H^{1}(\mathbb{R}^{d})}^{2}\lesssim
e^{-\frac{1}{4}L^{2}}$. As before, we can also write

$\displaystyle u_{L}=\sum_{i=1}^{2}\mathfrak{g}(\cdot-y_{i,L})+\rho_{L}^{*}$
(4.6)

where $\\{y_{i,L}\\}$ is the solution of (4.5) and the remaining term
$\rho_{L}^{*}$ satisfies

$\displaystyle\|\rho_{L}^{*}\|_{H^{1}(\mathbb{R}^{d})}^{2}=\mathfrak{c}_{L}\lesssim
e^{-\frac{1}{4}L^{2}}.$ (4.7)

and the orthogonal conditions

$\displaystyle\langle\rho_{L}^{*},\partial_{x_{l}}\mathfrak{g}_{j,L}\rangle_{H^{1}(\mathbb{R}^{d})}=0,\quad
l=1,2,\cdots,d\text{ and }j=1,2,$

where $\mathfrak{g}_{j,L}=\mathfrak{g}(\cdot-y_{j,L})$. By (4.6) and (4.7), we
may assume that $y_{1,L}=\frac{L}{2}e_{1}+o(1)$ and
$y_{2,L}=-\frac{L}{2}e_{1}+o(1)$. By (4.3) and (4.6), $\rho_{L}^{*}$ satisfies
the equation

$\displaystyle\left\\{\begin{aligned}
&-\Delta\rho_{L}^{*}+\rho_{L}^{*}=2u_{L}\log
u_{L}-2(\sum_{j=1}^{2}\mathfrak{g}_{j,L}\log\mathfrak{g}_{j,L})+f_{L},\quad\text{in
}\mathbb{R}^{d},\\\
&\langle\rho_{L}^{*},\partial_{x_{j}}\mathfrak{g}_{j,L}\rangle_{H^{1}(\mathbb{R}^{d})}=0,\quad
l=1,2,\cdots,d\text{ and }j=1,2.\end{aligned}\right.$ (4.8)

As before, we can write

$\displaystyle 2u_{L}\log
u_{L}-2(\sum_{j=1}^{2}\mathfrak{g}_{j,L}\log\mathfrak{g}_{j,L})=E_{L}+N_{L}(\rho_{L}^{*})+2(1+\log(\sum_{j=1}^{2}\mathfrak{g}_{j,L}))\rho_{L}^{*},$

which, together with (4.3), implies that we can re-write (4.8) as follows:

$\displaystyle\left\\{\begin{aligned}
&-\mathcal{L}_{L}(\rho_{L}^{*})=E_{L}+N_{L}(\rho_{L}^{*})-\sum_{j=1}^{d}(a_{j,+,L}\partial_{x_{j}}\mathfrak{g}_{L}+a_{j,-,L}\partial_{x_{j}}\mathfrak{g}_{-L}),\quad\text{in
}\mathbb{R}^{d},\\\
&\langle\rho_{L}^{*},\partial_{x_{j}}\mathfrak{g}_{j,L}\rangle_{H^{1}(\mathbb{R}^{d})}=0,\quad
l=1,2,\cdots,d\text{ and }j=1,2.\end{aligned}\right.$

By Lemma 3.2 and similar estimates in the proof of Lemma 3.3, we have

$\displaystyle\|\rho_{L}^{*}\|_{\sharp}\lesssim e^{-\frac{1}{8}L^{2}}.$

Now, using $\widehat{\psi}$, given by (3.45), as the test function of (4.8)
and estimating as that for (3.46), we have
$\|\rho_{L}^{*}\|_{H^{1}(\mathbb{R}^{d})}\gtrsim e^{-\frac{1}{8}L^{2}}$. It
follows from (4.7) that $\|\rho_{L}^{*}\|_{H^{1}(\mathbb{R}^{d})}\sim
e^{-\frac{1}{8}L^{2}}$, which, together with (4.4) and (4.7), implies that
$\|\rho_{L}^{*}\|_{H^{1}(\mathbb{R}^{d})}\sim\|f_{L}\|_{H^{-1}}$. It completes
the proof. ∎

We close this section by the proof of Theorem 1.1.

Proof of Theorem 1.1: The conclusion follows immediately from Propositions 3.1
and 4.1. $\Box$

## References

  * [1] C. O. Alves, C. Ji, Multi-peak positive solutions for a logarithmic Schrödinger equation via variational methods, Isral J. Math., 2022, to appear.
  * [2] T. Aubin, Problèmes isopérimétriques de Sobolev. J. Differential Geometry, 11 (1976), 573–598.
  * [3] A. Ambrosetti, E. Colorado, D. Ruiz, Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations, Calc. Var., 30 (2007), 85–112.
  * [4] A. Bahri, J. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Commun. Pure Appl. Math., 41 (1988), 253–294.
  * [5] M. Badiale, G. Tarantello, A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics, Arch. Ration. Mech. Anal., 163 (2002), 259–293.
  * [6] G. Bianchi, H. Egnell, A note on the Sobolev inequality. J. Funct. Anal., 100 (1991), 18–24.
  * [7] S. Bobkov, N. Gozlan, C. Roberto P. Samson. Bounds on the deficit in the logarithmic Sobolev inequality, J. Funct. Anal., 267 (2014), 4110–4138.
  * [8] H. Brezis, E. Lieb, Sobolev inequalities with remainder terms. J. Funct. Anal., 62 (1985), 73–86.
  * [9] L. Caffarelli, R. Kohn, L. Nirenberg, First order interpolation inequalities with weights. Compos. Math., 53 (1984), 259–275.
  * [10] E. Carlen, Superadditivity of Fisher’s information and logarithmic Sobolev inequalities, J. Funct. Anal., 101 (1991), 194–211.
  * [11] E. Carlen, A. Figalli, Stability for a GNS inequality and the log-HLS inequality, with application to the critical mass Keller-Segel equation, Duke Math. J., 162 (2013), 579–625.
  * [12] E. Carlen, R. Frank, E. Lieb, Stability estimates for the lowest eigenvalue of a Schrödinger operator, Geom. Funct. Anal., 24 (2014), 63–84.
  * [13] F. Catrina, Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions. Comm. Pure Appl. Math., 54 (2001), 229–258.
  * [14] A. Cianchi, A quantitative Sobolev inequality in BV, J. Funct. Anal., 237 (2006), 466–481.
  * [15] A. Cianchi, N. Fusco, F. Maggi, A. Pratelli, The sharp Sobolev inequality in quantitative form, J. Eur. Math. Soc., 11 (2009), 1105–1139.
  * [16] G. Ciraolo, A. Figalli, F. Maggi, A quantitative analysis of metrics on $\mathbb{R}^{N}$ with almost constant positive scalar curvature, with applications to fast diffusion flows. Int. Math. Res. Not., 2018 (2017), 6780–6797.
  * [17] K. Chou, W. Chu, On the best constant for a weighted Sobolev-Hardy inequality. J. London Math. Soc., 48 (1993), 137–151.
  * [18] J. Dávila, L. Dupaigne, Hardy-type inequalities, J. Eur. Math. Soc., 6 (2004), 335–365.
  * [19] P. d’Avenia, E. Montefusco, M. Squassina, On the logarithmic Schrödinger equation, Commun. Contemp. Math., 16 (2014), 1350032.
  * [20] M. del Pino, J. Dolbeault, The optimal Euclidean $L^{p}$-Sobolev logarithmic inequality, J. Funct. Anal., 197 (2003), 151–161.
  * [21] B. Deng, L. Sun, J. Wei, Optimal quantitative estimates of Struew’s decomposition. Preprint, arXiv:2103.15360v1 [math.AP].
  * [22] J. Dolbeault, M. J. Esteban, M. Loss, and G. Tarantello, On the symmetry of extremals for the Caffarelli-Kohn-Nirenberg inequalities. Adv. Nonlinear Stud., 9 (2009), 713–726.
  * [23] J. Dolbeault, M. J. Esteban, M. Loss, Symmetry of extremals of functional inequalities via spectral estimates for linear operators. J. Math. Phys., 53 (2012), article 095204, 18 pp.
  * [24] J. Dolbeault, M. J. Esteban, M. Loss, Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces, Invent. math., 206 (2016), 397–440.
  * [25] J. Dolbeault, G. Toscani , Improved interpolation inequalities, relative entropy and fast diffusion equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 917–934.
  * [26] J. Dolbeault, G. Toscani, Stability results for logarithmic Sobolev and Gagliardo-Nirenberg inequalities, Int. Math. Res. Not., 2016, (2015), 473–498.
  * [27] M. Fathi, E. Indrei, M. Ledoux, Quantitative logarithmic Sobolev inequalities and stability estimates, Discrete Contin. Dyn. Syst., 36 (2016), 6835–6853.
  * [28] V. Felli, M. Schneider, Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type. J. Differential Equations, 191 (2003), 121–142.
  * [29] F. Feo, E. Indrei, M. R. Posteraro, C. Roberto, Some Remarks on the Stability of the Log-Sobolev Inequality for the Gaussian Measure, Potential Anal., 47 (2017), 37–52.
  * [30] A. Figalli, Stability in geometric and functional inequalities, European Congress of Mathematics, 585–599, Eur. Math. Soc., Zurich, 2013.
  * [31] A. Figalli, F. Glaudo, On the Sharp Stability of Critical Points of the Sobolev Inequality. Arch. Rational Mech. Anal., 237 (2020), 201–258.
  * [32] A. Figalli, R. Neumayer, Gradient stability for the Sobolev inequality: the case $p\geq 2$, J. Eur. Math. Soc., 21 (2019), 319–354.
  * [33] A. Figalli, Y. Zhang, Sharp gradient stability for the Sobolev inequality, preprint, arXiv:2003.04037v1.
  * [34] S. Filippas, A. Tertikas, Optimizing improved Hardy inequalities, J. Funct. Anal., 192 (2002), 186–233.
  * [35] N. Fusco, The quantitative isoperimetric inequality and related topics, Bull. Math. Sci., 5 (2015), 517–607.
  * [36] N. Fusco, F. Maggi, A. Pratelli, The sharp quantitative Sobolev inequality for functions of bounded variation, J. Funct. Anal., 244 (2007), 315–341.
  * [37] V. Glaser, H. Grosse, A. Martin, W. Thirring, A Family of Optimal Conditions for the Absence of Bound States in a Potential, Studies in Math. Phys., Princeton University Press, New Jersey, 1976, essays in Honor of Valentine Bargmann.
  * [38] L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 97 (1975), 1061–1083.
  * [39] S. J. Gustafson, I. M. Sigal, Mathematical concepts of quantum mechanics, Second edition. Springer, Heidelberg, (2011).
  * [40] T. Horiuchi, Best constant in weighted Sobolev inequality with weights being powers of distance from the origin, J. Inequal. Appl., 1 (1997), 275–292.
  * [41] E. Indrei, D. Marcon, A quantitative log-Sobolev inequality for a two parameter family of functions, Int. Math. Res. Not.,20 (2014), 5563–5580.
  * [42] E. Indrei, D. Kim, Deficit estimates for the logarithmic Sobolev inequality, Differ. Integral Equ., 34, (2021), 437–466.
  * [43] E.H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349–374.
  * [44] E.H. Lieb, M. Loss, Analysis, Graduate Studies in Mathematics, 14, Second edition. American Mathematical Society, Providence (2001).
  * [45] C.-S. Lin, Z.-Q. Wang, Symmetry of extremal functions for the Caffarrelli-Kohn-Nirenberg inequalities. Proc. Amer. Math. Soc., 132 (2004), 1685–1691.
  * [46] F. Lin, W. Ni, J. Wei, On the number of interior peak solutions for a singularly perturbed Neumann problem, Comm. Pure Appl. Math., 60 (2007), 252–281.
  * [47] M. Musso, F. Pacard, J. Wei, Finite energy sigh-changing solutions with dihedral symmetry for the stationary nonlinear schrodinger equation, J. Eur. Math. Soc., 14 (2012), 1923–1953.
  * [48] R. Neumayer, A note on strong-form stability for the Sobolev inequality, Calc. Var., 59 (2020), Paper No. 25.
  * [49] V. Nguyen, The sharp Gagliardo-Nirenberg-Sobolev inequality in quantitative form, J. Funct. Anal., 277 (2019), 2179–2208.
  * [50] V. Radulescu, D. Smets, M. Willem, Hardy-Sobolev inequalities with remainder terms. Topol. Methods Nonlinear Anal., 20 (2002), 145–149.
  * [51] B. Ruffini, Stability theorems for Gagliardo-Nirenberg-Sobolev inequalities: a reduction principle to the radial case, Rev. Mat. Complut., 27 (2014), 509–539.
  * [52] F. Seuffert, A stability result for a family of sharp Gagliardo-Nirenberg inequalities, preprint, arXiv:1610.06869v1.
  * [53] W. Shuai, Multiple solutions for logarithmic Schrödinger equations, Nonlinearity, 32 (2019), 2201.
  * [54] M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities. Math. Z., 187 (1984), 511–517.
  * [55] G. Talenti, Best constant in Sobolev inequality. Ann. Mat. Pura Appl. (4), 110 (1976), 353–372.
  * [56] W. Troy, Uniqueness of positive ground state solutions of the logarithmic Schrödinger equation, Arch. Ration. Mech. Anal., 222 (2016), 1581–1600.
  * [57] Z.-Q. Wang, M. Willem, Singular Minimization Problems, J. Differential Equations, 161 (2000), 307–320.
  * [58] Z.-Q. Wang, M. Willem, Caffarelli-Kohn-Nirenberg inequalities with remainder terms, J. Funct. Anal., 203 (2003), 550–568.
  * [59] Z.-Q. Wang, C. Zhang, Convergence from power-law to logarithm-law in nonlinear scalar field equations, Arch. Ration. Mech. Anal., 231 (2019), 45–61.
  * [60] F.B. Weissler, Logarithmic Sobolev inequalities for the heat-diffusion semigroup, Trans. Amer. Math. Soc., 237 (1978), 255–269.
  * [61] J. Wei, Y. Wu, On the stability of the Caffarelli-Kohn-Nirenberg inequality, Mathematische Annalen, 2022, Doi:10.1007/s00208-021-02325-0.