# On the stability of fractional Sobolev trace inequality and corresponding
profile decomposition††thanks: Supported NSFC(No.12171265,11025106). E-mail
addresses<EMAIL_ADDRESS>(Zhang), zhou-
yx22@mails.tsinghua.edu.cn(Zhou<EMAIL_ADDRESS>(Zou)

Yingfang Zhang, Yuxuan Zhou and Wenming Zou

Department of Mathematical Sciences, Tsinghua University, Beijing 100084,
China.

###### Abstract

In this paper, we study the stability of fractional Sobolev trace inequality
within both the functional and critical point settings.

In the functional setting, we establish the following sharp estimate:

$C_{\mathrm{BE}}(n,m,\alpha)\inf_{v\in\mathcal{M}_{n,m,\alpha}}{\lVert
f-v\rVert}_{D_{\alpha}(\mathbb{R}^{n})}^{2}\leq{\lVert
f\rVert}_{D_{\alpha}(\mathbb{R}^{n})}^{2}-S(n,m,\alpha){\lVert\tau_{m}f\rVert}_{L^{q}(\mathbb{R}^{n-m})}^{2},$

where $0\leq m<n$, $\frac{m}{2}<\alpha<\frac{n}{2},q=\frac{2(n-m)}{n-2\alpha}$
and $\mathcal{M}_{n,m,\alpha}$ denotes the manifold of extremal functions.
Additionally, We find an explicit bound for the stability constant
$C_{\mathrm{BE}}$. Furthermore, we establish a compactness result ensuring the
existence of minimizers for the Bianchi-Egnell type functional:

$S_{\mathrm{Tr}}(f)\coloneqq\frac{{\lVert
f\rVert}_{D_{\alpha}}^{2}-S(n,m,\alpha){\lVert\tau_{m}f\rVert}_{L^{q}}^{2}}{\inf_{v\in\mathcal{M}_{n,m,\alpha}}{\lVert
f-v\rVert}_{D_{\alpha}}^{2}},\quad\text{for }f\in
D_{\alpha}(\mathbb{R}^{n})\backslash\mathcal{M}_{n,m,\alpha}.$

Our stability results extend previous works on the Escobar trace inequality
and fractional Sobolev inequality. As a corollary, we derive some improved
trace inequalities for functions supported in general domains. Applying a
standard dual scheme, we also obtain a sharp a priori estimate for Neumann
problem on the half-space.

In the critical point setting, we investigate the validity of a sharp
quantitative profile decomposition related to the Escobar trace inequality and
establish a qualitative profile decomposition for the critical elliptic
equation

$\begin{cases}\Delta u=0&\text{in }\mathbb{R}_{+}^{n}\\\ \frac{\partial
u}{\partial t}=-|u|^{\frac{2}{n-2}}u&\text{on
}\partial\mathbb{R}_{+}^{n}.\end{cases}$

We then derive the sharp stability estimate:

$C_{\mathrm{CP}}(n,\nu)d(u,\mathcal{M}_{\mathrm{E}}^{\nu})\leq\left\|\Delta
u+|u|^{\frac{2}{n-2}}u\right\|_{H^{-1}(\mathbb{R}_{+}^{n})},$

where $\nu=1,n\geq 3$ or $\nu\geq 2,n=3$ and $\mathcal{M}_{\mathrm{E}}^{\nu}$
represents the manifold consisting of $\nu$ weak-interacting Escobar bubbles.
Through some refined estimates, we also give a strict upper bound for
$C_{\mathrm{CP}}(n,1)$, which is $\frac{2}{n+2}$.

###### Contents

  1. 1 Introduction
    1. 1.1 Background and motivations
    2. 1.2 Problem setup and main results
    3. 1.3 Structure of the paper
  2. 2 Preliminaries
  3. 3 Stability of fractional Sobolev trace inequality
  4. 4 Some consequences
  5. 5 Qualitative profile decomposition
  6. 6 Quantitative profile decomposition
  7. 7 Refined estimates for one bubble case

##  1 Introduction

###  1.1 Background and motivations

The classical Sobolev inequality with exponent $1<p<n$ states that, for any
$n\geq 2$ and $u\in W^{1,p}(\mathbb{R}^{n})$, it holds

$S_{p}\|u\|_{L^{p^{*}}}\leq\|\nabla u\|_{L^{p}},$ (1.1)

where $p^{*}=\frac{np}{n-p}$ and $S_{p}$ is the sharp constant. It is well
known that the extremal functions of (1.1) (see [2, 59]) form an
$(n+2)$-dimensional manifold:

$\mathcal{M}_{p}\coloneqq\Big{\\{}w_{a,b,x_{0}}\coloneqq\frac{a}{(1+b|x-x_{0}|^{\frac{p}{p-1}})^{\frac{n-p}{p}}}:a\in\mathbb{R}\setminus\\{0\\},\
b>0,\ x_{0}\in\mathbb{R}^{n}\Big{\\}}.$ (1.2)

When $p=2$ ,the Euler-Lagrange equation associated to the inequality (1.1) is,
up to suitable normalization, given by

$\Delta u+u|u|^{2^{*}-2}=0.$ (1.3)

It was shown in [40] that all the positive solutions of (1.3) are Talenti
bubbles:

$T[z,\lambda](x)\coloneqq(n(n-2))^{\frac{n-2}{4}}\frac{\lambda^{\frac{n-2}{2}}}{(1+\lambda^{2}{\lvert
x-z\rvert}^{2})^{\frac{n-2}{2}}},\quad\lambda>0,\ z\in\mathbb{R}^{n}.$ (1.4)

Once the rigidity results for (1.1) and (1.3) are established, it is natural
to consider stability versions. In the functional setting, a fundamental
question arises — does the deviation of a function from attaining equality in
(1.1) control its distance from the extremal manifold $\mathcal{M}_{p}$?
Similarly, in the critical point setting, does almost being a positive
solution of (1.3) imply almost being a Talenti bubble?

The question on the stability of functional inequalities was first raised by
Brézis and Lieb in [10]. In their work, they improved the Sobolev embedding on
bounded domains by introducing a remainder term, thereby raising an open
problem about whether the homogeneous inequality (1.1) can be reinforced. A
comprehensive answer was provided by Bianchi and Egnell in [7] for the case
$p=2$, presenting a complete result as follows:

$C_{\mathrm{BE}}\inf_{z\in\mathbb{R}^{n},\lambda>0,\alpha\in\mathbb{R}}\|\nabla(u-\alpha
U[z,\lambda])\|_{L^{2}}^{2}\leq\|\nabla
u\|_{L^{2}}^{2}-S_{2}^{2}\|u\|^{2}_{L^{2^{*}}}.$ (1.5)

The result is considered complete because both the exponent $2$ and the norm
in the left hand side are sharp. For the case $p\neq 2$, due to the absence of
Hilbertian structure, the stability of (1.1) remained an open problem for a
long time and was recently solved by Figalli and Zhang in [35] (see [35, 50,
34] and the references therein for earlier works).

Besides the classical Sobolev inequality, there are many other important
geometric and functional inequalities with well studied stability. Examples
include the fractional Sobolev inequality [5, 15], the Sobolev trace
inequality [52, 14], the isoperimetric inequality [39, 33, 46, 49, 17, 49,
28], the Hardy-Littlewood-Sobolev inequality [13, 12], the Gagliardo-
Nirenberg-Sobolev inequality [13, 23, 54, 8, 9] and the Caffarelli-Kohn-
Nirenberg inequality [61, 22, 38].

Once a complete stability result is established, a natural and interesting
question is to quantify the stability. For example, what can be said about the
constant $C_{\mathrm{BE}}$ in (1.5)? Traditional stability results of Sobolev-
type inequalities rely heavily on local spectrum analysis and global
concentration-compactness principle, which offer limited information about the
constants involved. Recent advancements by Dolbeault, Esteban, Figalli, Frank
and Loss [24] stated shedding light on explicit lower bound of
$C_{\mathrm{BE}}$ using competing symmetries and continuous Steiner
symmetrization. The subsequent work by König in [41, 42] gave an upper bound
of $C_{\mathrm{BE}}$ for general fractional Sobolev inequality, and showed
that $C_{\mathrm{BE}}$ can be attained by certain functions. In a parallel
vein, Chen, Lu and Tang [14] gave a lower bound for stability constants of the
Hardy-Littlewood-Sobolev inequality, the fractional Sobolev inequality and
some trace inequalities. Wei, Wu [62] and Deng, Tian [21] established
analogous results for the Caffarelli-Kohn-Nirenberg inequality.

The question on the stability of critical points traces back to the celebrated
global compactness principle of Struwe [57]. This principle states that a
bounded nonnegative sequence $\\{u_{n}\\}$ satisfying $\left\|\Delta
u+u|u|^{2^{*}-2}\right\|_{H^{-1}(\mathbb{R}^{n})}\rightarrow 0$ can be
approximated by several Talenti bubbles up to a subsequence. Ciraolo, Figalli
and Maggi [18] provided the first quantitative version of this principle,
yielding an optimal linear estimate when dealing with a single bubble:

$C\operatorname{d}(u,\mathcal{M}_{T})\leq\left\|\Delta
u+|u|^{2^{*}-2}u\right\|_{H^{-1}(\mathbb{R}^{n})},$

where $\mathcal{M}_{T}$ is the manifold of Talenti bubbles. When there are
more bubbles, it becomes complicated due to the interaction of distinct
bubbles. Figalli and Glaudo [29] tackled such case and found an interesting
phenomenon: linear estimates were possible for dimension $3\leq n\leq 5$, but
for $n>5$, counter-examples were constructed. The remaining case $n>5$ was
ultimately solved by Deng, Sun and Wei in [20] using finite dimensional
reduction method, highlighting that logarithmic ($n=6$) or sublinear ($n>6$)
estimates are sharp in high dimension.

In addition to the classical Yamabe equation (1.3), the stability of several
inequality-related critical equations have been extensively studied, such as
the fractional Sobolev inequality [1, 19], the Caffarelli-Kohn-Nirenberg
inequality [61] and the Hardy-Littlewood-Sobolev inequality [45, 53].

As in the functional setting, a natural question is to study the optimal
stability constant for critical equations. However, this question seems much
harder, for the possible reason that in general, dual norms do not behave as
well as Sobolev norms. Standard tools such as expansions and Brézis-Lieb type
arguments, which are useful in functional settings, are difficult to handle
dual norms. To our best knowledge, the first in this direction was given by De
Nitti and König in [19], which found an explicit upper bound for the stability
constant of the fractional Yamabe equation in one bubble case by treating the
dual norm carefully and using a third-order expansion.

###  1.2 Problem setup and main results

In this paper, we focus on the following sharp fractional Sobolev trace
inequality given by Einav and Loss in [25]:

$S(n,m,\alpha){\lVert\tau_{m}f\rVert}_{L^{\frac{2(n-m)}{n-2\alpha}}(\mathbb{R}^{n-1})}^{2}\leq{\lVert
f\rVert}_{D_{\alpha}(\mathbb{R}^{n})}^{2},$ (1.6)

where $0\leq m<n$, $\frac{m}{2}<\alpha<\frac{n}{2}$, $f\in
D_{\alpha}(\mathbb{R}^{n})$ and $\tau_{m}f$ is the restriction of $f$ on
$\mathbb{R}^{n-m}$. The space $D_{\alpha}(\mathbb{R}^{n})$ is the standard
fractional Sobolev space in $\mathbb{R}^{n}$. It is the closure of
$C_{c}^{\infty}(\mathbb{R}^{n})$ under the norm:

${\lVert
f\rVert}_{D_{\alpha}(\mathbb{R}^{n})}:=\left(\int_{\mathbb{R}^{n}}{\lvert\hat{f}(k)\rvert}{\lvert
2\pi k\rvert}^{2\alpha}\,dk\right)^{1/2}.$

The sharp constant $S(n,m,\alpha)$ and the extremal manifold
$\mathcal{M}_{n,m,\alpha}$ are explicit and given by

$S(n,m,\alpha)=2^{2\alpha}\pi^{\alpha}\frac{\Gamma(\alpha)\Gamma(n/2+\alpha-m)}{\Gamma(n/2-\alpha)\Gamma(\alpha-m/2)}\left(\frac{\Gamma(n-m)}{\Gamma((n-m)/2)}\right)^{\frac{m-2\alpha}{n-m}},$
$\mathcal{M}_{n,m,\alpha}=\left\\{\lambda\int_{\mathbb{R}^{n-m}}\frac{1}{(|x^{\prime}|^{2}+|x^{\prime\prime}-y^{\prime\prime}|^{2})^{(n-2\alpha)/2}}\frac{1}{(\gamma^{2}+|y^{\prime\prime}-a|^{2})^{(n+2\alpha-2m)/2}}\,dy^{\prime\prime}\right\\},$

Where
$\lambda\in\mathbb{R},\gamma>0,a\in\mathbb{R}^{n-m},x^{\prime}\in\mathbb{R}^{m},x^{\prime\prime}\in\mathbb{R}^{n-m}$.
When $m=0$, (1.6) reduces to the fractional Sobolev inequality. When
$m=\alpha=1$, (1.6) is equivalent to the Escobar trace inequality in [27]:

$S_{\mathrm{E}}(n){\lVert
f\rVert}_{L^{\frac{2(n-1)}{n-2}}(\mathbb{R}^{n-1})}^{2}\leq{\lVert
f\rVert}_{H^{1}(\mathbb{R}_{+}^{n})}^{2},$

where $S_{\mathrm{E}}(n)$ is the sharp constant. The stability of these two
inequalities are well studied (see [5, 15, 52, 18, 29, 20, 41, 19, 42, 14] for
examples). In this paper, we consider the stability of (1.6) in both
functional and critical point settings. In the functional setting, we prove
the following sharp estimate:

######  Theorem 1.1.

Let $0\leq m<n$ and $\frac{m}{2}<\alpha<\frac{n}{2}$. There exists a constant
$C=C(n,m,\alpha)$ such that for any $f\in D_{\alpha}(\mathbb{R}^{n})$, we have

$C{\lVert f-v\rVert}_{D_{\alpha}(\mathbb{R}^{n})}^{2}\leq{\lVert
f\rVert}_{D_{\alpha}(\mathbb{R}^{n})}^{2}-S(n,m,\alpha){\lVert\tau_{m}f\rVert}_{L^{\frac{2(n-m)}{n-2\alpha}}(\mathbb{R}^{n-m})}^{2}$
(1.7)

for some $v\in\mathcal{M}_{n,m,\alpha}$. We denote
$C_{\mathrm{BE}}(n,m,\alpha)$ to be the best stability constant $C$ above.

The inequality (1.7) is sharp in the sense that the exponent $2$ is optimal
and the norm $D_{\alpha}$ is the strongest. A crucial method employed in
proving (1.7) is a reduction principle (see Theorem 3.3) proved in [25]. This
reduction allows us to connect (1.6) with standard fractional Sobolev
inequality. Precisely, we show that the stability of (1.6) is equivalent to
that of the fractional Sobolev inequality.

With the establishment of (1.7), we are interested in the following
minimization problem:

$C_{\mathrm{BE}}(n,m,\alpha)=\inf_{f\in
D_{\alpha}(\mathbb{R}^{n})\backslash\mathcal{M}_{n,m,\alpha}}S_{\mathrm{Tr}}(f),$
(1.8)

where

$S_{\mathrm{Tr}}(f)\coloneqq\frac{{\lVert
f\rVert}_{D_{\alpha}(\mathbb{R}^{n})}^{2}-S(n,m,\alpha){\lVert\tau_{m}f\rVert}_{L^{\frac{2(n-m)}{n-2\alpha}}(\mathbb{R}^{n-m})}^{2}}{\operatorname{d}^{2}(f,\mathcal{M}_{n,m,\alpha})}.$

Here we prove, to our surprise, the optimal stability constant
$C_{\mathrm{BE}}(n,m,\alpha)$ is identical to the optimal constant
$C_{\mathrm{BE}}(n-m,0,\alpha-\frac{m}{2})$ for the fractional Sobolev
inequality (see Remark 3.5), which is well studied. Consequently, both the
upper bound and the lower bound of $C_{\mathrm{BE}}(n,m,\alpha)$ come directly
from [41, 42] and [14], respectively. Furthermore, following ideas from [41],
we derive the existence of minimizers which attain the infimum in (1.8)(see
Theorem 3.6). Based on the stability result (1.7), we derive enhanced Sobolev
embedding results for general domains. Specifically, we can add a remainder
term equipped with some weak $L^{q}$-norm $\|\cdot\|_{L_{w}^{q}}$ (see Theorem
4.1 and Theorem 4.4). Such type results were initially established by Brézis
and Lieb in [10]. Later, Bianchi and Egnell [7] presented an alternative proof
using gradient stability. Chen, Frank and Weth [15] generalized their results
to the fractional Sobolev inequality. We follow similar ideas in our proofs.
Another corollary of (1.7) is a stability result of a sharp a priori estimate
for the Neumann problem:

$\begin{cases}\Delta\mathcal{P}[u]=0&\text{in }\mathbb{R}_{+}^{n},\\\
\frac{\partial\mathcal{P}[u]}{\partial t}=-u&\text{on
}\partial\mathbb{R}_{+}^{n},\end{cases}$ (1.9)

where $\mathbb{R}_{+}^{n}=\left\\{(x,t)|x\in\mathbb{R}^{n-1},t>0\right\\}$.
Given any $u\in L^{\frac{2n-2}{n}}(\mathbb{R}^{n-1})$, we have

$\|u\|_{L^{\frac{2n-2}{n}}(\mathbb{R}^{n-1})}^{2}-\frac{2}{S(n,1,1)}\left\|\nabla\mathcal{P}[u]\right\|_{L^{2}(\mathbb{R}_{+}^{n})}^{2}\geq
C\inf_{v\in\mathcal{M}_{\mathrm{Neu}}}\|u-v\|_{L^{\frac{2n-2}{n}}(\mathbb{R}^{n-1})}^{2}.$
(1.10)

Here $\frac{2}{S(n,1,1)}$ is the sharp constant and
$\mathcal{M}_{\mathrm{Neu}}$ is the extremal manifold (see Theorem 4.6 in this
paper). The proof of (1.10) relies on a standard dual scheme developed by
Carlen in [12]. Precisely, we show that this a priori estimate is the dual of
the Escobar trace inequality, _i.e._ inequality (1.6) with $m=\alpha=1$.

In the critical point setting, we consider the Euler-Lagrange equation
associated with the Escobar trace inequality:

$\begin{cases}\Delta u=0&\text{in }\mathbb{R}_{+}^{n},\\\ \frac{\partial
u}{\partial t}=-|u|^{\frac{2}{n-2}}u&\text{on
}\partial\mathbb{R}_{+}^{n},\end{cases}$ (1.11)

It was shown by Ou in [51] that, up to a scaling, any nontrivial nonnegative
solution of (1.11) is an Escobar bubble:

$U[z,\lambda](x,t)=(n-2)^{\frac{n-2}{2}}\left(\frac{\lambda}{(1+\lambda
t)^{2}+\lambda^{2}|x-z|^{2})}\right)^{\frac{n-2}{2}}$

for some $z\in\mathbb{R}^{n-1},\lambda>0$. Motivated by Struwe’s work, we
initially establish a Struwe-type profile decomposition for (1.11) (see
Theorem 5.1). Specifically, we demonstrate that any almost solution of (1.11)
can be approximated by multiple exact solutions. Furthermore, we present a
nonnegative counterpart of this result (see Theorem 5.3): every nonnegative
almost solution of (1.11) can be approximated by several Escobar bubbles. The
transition from the general version to the nonnegative version requires a
Brézis-Lieb type lemma (see Lemma 5.2), which was previously used in [47].
Following ideas of Figalli and Glaudo in [29], we derive the following optimal
quantitative profile decomposition

######  Theorem 1.2.

For $n=3,\nu\in\mathbb{N}$ or $n\geq 3,\nu=1$, there exist a small constant
$\delta=\delta(n,\nu)>0$ and a large constant $C=C(n,\nu)>0$ such that the
following statement holds. Let $u\in H^{1}(\mathbb{R}^{n}_{+})$ be a function
such that

${\left\lVert\nabla
u-\sum_{i=1}^{\nu}\nabla\tilde{U}_{i}\right\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}\leq\delta,$
(1.12)

where $(\tilde{U}_{i})_{1\leq i\leq\nu}$ is a $\delta$-interacting family of
Escobar bubbles (see Definition 1.3). Then there exist $\nu$ Escobar bubbles
$U_{1},U_{2},\dots,U_{\nu}$ such that

${\left\lVert\nabla u-\sum_{i=1}^{\nu}\nabla
U_{i}\right\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}\leq C{\lVert\Delta u+{\lvert
u\rvert}^{p-1}u\rVert}_{H^{-1}}.$

where $p=\frac{n}{n-2}$. Furthermore, for any $i\not=j$, the interaction
between the bubbles can be estimated as

$\int_{\mathbb{R}^{n}}U_{i}^{p}U_{j}\leq C{\lVert\Delta u+{\lvert
u\rvert}^{p-1}u\rVert}_{H^{-1}}.$ (1.13)

The term in the right-hand side of (1.13) is defined by

${\lVert\Delta u+{\lvert
u\rvert}^{\frac{2}{n-2}}u\rVert}_{H^{-1}(\mathbb{R}_{+}^{n})}=\sup_{v\in
H^{1}(\mathbb{R}_{+}^{n})}\frac{\int_{\mathbb{R}_{+}^{n}}\nabla u\cdot\nabla
v-\int_{\mathbb{R}^{n-1}}|u|^{\frac{2}{n-2}}uv}{{\lVert
v\rVert}_{H^{1}(\mathbb{R}_{+}^{n})}}$ (1.14)

and the weak-interaction is defined by

######  Definition 1.3 (interaction of Escobar bubbles).

Let $U_{1}=U[z_{1},\lambda_{1}]$,…,$U_{\nu}=U[z_{\nu},\lambda_{\nu}]$ be a
family of Escobar bubbles. We say that the family is $\delta$-interacting for
some $\delta>0$ if

$\mu_{ij}\coloneqq\min\left(\frac{\lambda_{i}}{\lambda_{j}},\frac{\lambda_{j}}{\lambda_{i}},\frac{1}{\lambda_{i}\lambda_{j}|z_{i}-z_{j}|^{2}}\right)\leq\delta,\quad\forall\,1\leq
i\neq j\leq\nu.$ (1.15)

Moreover, we say the family with some positive coefficients
$\alpha_{1},\dots,\alpha_{\nu}$ is $\delta$-interacting, if (1.15) holds and

$\max_{1\leq i\leq\nu}{\lvert\alpha_{i}-1\rvert}\leq\delta.$

Thanks to qualitative results, the weak-interacting assumption in Theorem 1.2
can be substituted with nonnegativity, as demonstrated in Theorem 6.7. Similar
to (1.8), we study the following minimization problem in the critical point
setting:

$C_{\mathrm{CP}}(n,\nu)=\inf_{u}\frac{{\lVert\Delta u+{\lvert
u\rvert}^{\frac{2}{n-2}}u\rVert}_{H^{-1}(\mathbb{R}_{+}^{n})}}{\operatorname{d}(u,\mathcal{M}_{\mathrm{E}}^{\nu})},\quad\text{for
}u\in H^{1}(\mathbb{R}_{+}^{n})\backslash\mathcal{M}_{\mathrm{E}}^{\nu}.$
(1.16)

We are able to give the following explicit upper bound (see Theorem 7.2 and
7.4) for $C_{\mathrm{CP}}(n,1)$:

$C_{\mathrm{CP}}(n,1)<\frac{2}{n+2},$

using arguments inspired by De Nitti and König’s work in [19]. Our primary
methods involve a precise spectral gap inequality, a well-defined expression
for the $H^{-1}$ norm and a third-order expansion. To derive the spectral gap
inequality, we employ a conformal transformation, linking nondegeneracy
results on $\mathbb{R}_{+}^{n}$ to the classical Steklov eigenvalue problem on
the unit ball $\mathbb{B}^{n}$. We will use the operator $\mathcal{P}$ in
(1.9) to express the $H^{-1}$ norm, which makes it clear to do the third-order
expansion.

###  1.3 Structure of the paper

After a section of notations and preliminaries, in Section 3 we presents the
proof of the functional stability result (1.7). A discussion about the optimal
stability constant and the existence of minimizers will also be given. Based
on results in Section 3, we derive some corollaries in Section 4, including
some refined Sobolev embeddings and a stability result of a priori estimate.
Section 5 and Section 6 are devoted to qualitative and quantitative profile
decomposition respectively. We focus on both the general case and the
nonnegative case. Finally, in Section 7, we provide some improved estimates of
the stability constant for critical point in one bubble case. We begin with
the derivation of an explicit spectral gap inequality. Based on it, we can
proceed to give an upper bound of $C_{\mathrm{CP}}(n,1)$ in terms of specturm
constants.

##  2 Preliminaries

We begin with setting the notations and definitions we need in our paper.

For $n\geq 1$ and $0<\alpha<\frac{n}{2}$, we define the space
$D_{\alpha}(\mathbb{R}^{n})$ as the space of tempered distributions
$\mathcal{S}^{\prime}(\mathbb{R}^{n})$ whose Fourier transform belongs to
$L^{2}(\mathbb{R}^{n},|k|^{2\alpha}\,dk)$. That is to say, for any element
$T\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ of it, there exists a function
$\hat{f}\in L^{2}(\mathbb{R}^{n},|k|^{2\alpha}\,dk)$ such that for all
$\phi\in\mathcal{S}(\mathbb{R}^{n})$,

$\widehat{T}(\phi)=:T(\hat{\phi})=\int_{\mathbb{R}^{n}}\hat{f}(k)\phi(k)\,dk.$

Here the Fourier transform is defined by

$\hat{f}(k)=\int_{\mathbb{R}^{n}}f(x)e^{-2\pi ix\cdot k}\,dx.$

$D_{\alpha}(\mathbb{R}^{n})$ equipped with the following norm is a Hilbert
space, in which $\mathcal{S}(\mathbb{R}^{n})$ is dense:

${\lVert
f\rVert}_{D_{\alpha}(\mathbb{R}^{n})}:=\left(\int_{\mathbb{R}^{n}}{\lvert\hat{f}(k)\rvert}{\lvert
2\pi k\rvert}^{2\alpha}\,dk\right)^{1/2}.$

We should claim that, when $\alpha\in\mathbb{N}_{+}$, the norm above is equal
to classical Sobolev norm:

${\lVert\Delta^{\frac{\alpha}{2}}f\rVert}_{L^{2}(\mathbb{R}^{n})}=\left(\int_{\mathbb{R}^{n}}|\Delta^{\frac{\alpha}{2}}f(x)|^{2}\,dx\right)^{1/2}.$

For $f\in\mathcal{S}(\mathbb{R}^{n})$, we define the restriction of $f$ to the
$(n-m)$-dimensional hyperplane given by
$\\{x\in\mathbb{R}^{n}:x_{n-m+1}=\cdots=x_{n}=0\\}$ as

$(\tau_{m}f)(x_{1},\cdots,x_{n-m}):=f(x_{1},\cdots,x_{n-m},0,\cdots,0).$

For general $f\in D_{\alpha}(\mathbb{R}^{n})$, $\tau_{m}f$ can be uniquely
defined through a density argument.

For $n\geq 3$, in modern terminology, $2^{*}$ always denotes the critical
Sobolev exponent $\frac{2n}{n-2}$. In our settings, to avoid confusions, we
will always use $2^{\dagger}$ to denote the trace critical exponent
$\frac{2(n-1)}{n-2}$. We also define $p=2^{\dagger}-1=\frac{n}{n-2}$ for
simplicity.

In the whole paper, we denote by $S(n,m,\alpha)$ the sharp constant of the
fractional Sobolev trace inequality with respect to parameters $0\leq m<n$ and
$\frac{m}{2}<\alpha<\frac{n}{2}$, i.e.

$S(n,m,\alpha):=\inf\left\\{\frac{{\lVert
f\rVert}_{D_{\alpha}(\mathbb{R}^{n})}^{2}}{{\lVert\tau_{m}f\rVert}_{L^{s}(\mathbb{R}^{n-m})}^{2}}:f\in
D_{\alpha}(\mathbb{R}^{n})\backslash\\{0\\},\,s=\frac{2(n-m)}{n-2\alpha}\right\\}.$

Note that, under this notation, $S(n,0,\alpha)$ is the sharp constants of
fractional Sobolev inequality.

A useful remark is that, when $m=\alpha=1$, the trace inequality above is
equivalent to the original Escobar trace inequality in [27] via a simple
reflection:

$S_{\mathrm{E}}(n){\lVert
f\rVert}_{L^{2^{\dagger}}(\mathbb{R}^{n-1})}^{2}\leq{\lVert
f\rVert}_{H^{1}(\mathbb{R}_{+}^{n})}^{2},$

where $S_{\mathrm{E}}(n)=\frac{S(n,1,1)}{2}$ is the sharp constant, and
$H^{1}(\mathbb{R}_{+}^{n})$ is the closure of
$C_{c}^{\infty}(\mathbb{R}^{n})|_{\mathbb{R}_{+}^{n}}$ with respect to the
norm

${\lVert
f\rVert}_{H^{1}(\mathbb{R}_{+}^{n})}=\left(\int_{\mathbb{R}_{+}^{n}}|\nabla
f|^{2}\right)^{\frac{1}{2}}.$

The extremal functions, up to normalization, are precisely the so-called
Escobar bubbles below.

For any $z\in\mathbb{R}^{n-1}$ and $\lambda>0$, the Escobar bubble
$U[z,\lambda]$ is defined as

$U[z,\lambda](x,t)=(n-2)^{\frac{n-2}{2}}\left(\frac{\lambda}{(1+\lambda
t)^{2}+\lambda^{2}|x-z|^{2})}\right)^{\frac{n-2}{2}},\quad
x\in\mathbb{R}^{n-1},\,t>0.$ (2.1)

The corresponding manifold is denoted by $\mathcal{M}_{\mathrm{E}}$. We state
some important properties of Escobar bubbles $U=U[z,\lambda]$.

  * •

They are extremal functions of the Escobar trace inequality:

$\int_{\mathbb{R}^{n}_{+}}{\lvert\nabla
U\rvert}^{2}=\int_{\partial\mathbb{R}_{+}^{n}}U^{2^{\dagger}}=S_{\mathrm{E}}(n)^{n-1}.$

  * •

They and their derivatives satisfy

$\begin{cases}\Delta U=0&\text{in }\mathbb{R}_{+}^{n},\\\ \frac{\partial
U}{\partial t}=-U^{p}&\text{on
}\partial\mathbb{R}_{+}^{n},\end{cases}\quad\begin{cases}\Delta(\partial_{s}U)=0&\text{in
}\mathbb{R}_{+}^{n},\\\ \frac{\partial(\partial_{s}U)}{\partial
t}=-pU^{p-1}\partial_{s}U&\text{on }\partial\mathbb{R}_{+}^{n},\end{cases}$

for $s=\lambda,z_{1},\cdots,z_{n-1}$.

  * •

We have the following expression for $\partial_{\lambda}U$:

$\partial_{\lambda}U(x,t)=\frac{n-2}{2\lambda}U(x,t)\left(\frac{1-\lambda^{2}t^{2}-\lambda^{2}{\lvert
x-z\rvert}^{2}}{(1+\lambda t)^{2}+\lambda^{2}{\lvert x-z\rvert}^{2}}\right),$

which yields the estimate of ${\lvert\partial_{\lambda}U\rvert}$:

${\lvert\partial_{\lambda}U\rvert}\leq\frac{n-2}{2\lambda}{\lvert U\rvert}.$

  * •

Symmetries: for $k\in\mathbb{N}$ and exponents set $(\gamma_{i})_{i=1}^{k}$
with $\gamma_{1}+\cdots+\gamma_{k}=2^{\dagger}$, $k$ Escobar bubbles
$U[z_{1},\lambda_{1}],\dots,U[z_{k},\lambda_{k}]$ satisfy

$\int_{\partial\mathbb{R}_{+}^{n}}U[z_{1},\lambda_{1}]^{\gamma_{1}}\cdots
U[z_{k},\lambda_{k}]^{\gamma_{k}}=\int_{\partial\mathbb{R}_{+}^{n}}U[0,1]^{\gamma_{1}}\prod_{j=2}^{k}U[\lambda_{1}(z_{j}-z_{1}),\lambda_{j}/\lambda_{1}]^{\gamma_{j}}.$

In addition, when $k=2$, we have

$\int_{\mathbb{R}^{n}_{+}}\nabla U[z_{1},\lambda_{1}]\cdot\nabla
U[z_{2},\lambda_{2}]=\int_{\mathbb{R}^{n}_{+}}\nabla U[0,1]\cdot\nabla
U[\lambda_{1}(z_{2}-z_{1}),\lambda_{2}/\lambda_{1}].$

With these symmetries, for the stability problem, it is enough to consider
$U[0,1]$ instead of a generic Escobar bubble. When dealing with multiple
bubbles, we need to evaluate their interaction (recall Definition 1.3).

######  Lemma 2.1.

Given $n\geq 1$, let us fix $\alpha+\beta=2n$ with $\alpha,\beta\geq 0$,
$\lambda\in(0,1]$ and $z\in\mathbb{R}^{n}$, set $D\coloneqq|z|$. If
$|\alpha-\beta|\geq\epsilon$ for some $\epsilon>0$, then

$\int_{\mathbb{R}^{n}}\left(\frac{1}{1+|x|^{2}}\right)^{\alpha/2}\left(\frac{\lambda}{1+\lambda^{2}|x-z|^{2}}\right)^{\beta/2}\approx_{n,\epsilon}\mu^{\frac{\min(\alpha,\beta)}{2}},$

where $\mu=\min\left\\{\lambda,\frac{1}{\lambda D^{2}}\right\\}$, If instead
$\alpha=\beta=n$, then

$\int_{\mathbb{R}^{n}}\left(\frac{1}{1+|x|^{2}}\right)^{\alpha/2}\left(\frac{\lambda}{1+\lambda^{2}|x-z|^{2}}\right)^{\beta/2}\approx_{n}\ln(\mu)\mu^{n/2}.$

Here and in the following, the notation $A\approx_{n,\epsilon}B$ means
$0<c\leq\frac{A}{B}\leq C$ for some constants $c,C$ depending only on $n$ and
$\epsilon$, and the notation $A\approx_{n}B$ is similar. We also use
$A\lesssim B$ to mean that there exists a constant $C>0$ depending only on
$n,m,\alpha$ such that $A\leq CB$.

######  Remark 2.2.

It is a general version of [29, Lemma B.1] and the proof is similar. For
general $z_{1},z_{2}\in\mathbb{R}^{n}$ and $\lambda_{1}\geq\lambda_{2}$, we
have

$\displaystyle\int_{\mathbb{R}^{n}}\left(\frac{\lambda_{1}}{1+\lambda_{1}^{2}|x-z_{1}|^{2}}\right)^{\alpha/2}\left(\frac{\lambda_{2}}{1+\lambda_{2}^{2}|x-z_{2}|^{2}}\right)^{\beta/2}=$
$\displaystyle\int_{\mathbb{R}^{n}}\left(\frac{1}{1+|x|^{2}}\right)^{\alpha/2}\left(\frac{\lambda}{1+\lambda^{2}|x-z|^{2}}\right)^{\beta/2}$
$\displaystyle\approx$ $\displaystyle\ \mu^{\frac{\min(\alpha,\beta)}{2}}$

with $\lambda=\frac{\lambda_{2}}{\lambda_{1}}\in(0,1]$ and
$z=\lambda_{1}(z_{2}-z_{1})$. So $\lambda
D^{2}=\lambda_{1}\lambda_{2}|z_{1}-z_{2}|^{2}$ and
$\mu=\min\left\\{\frac{\lambda_{2}}{\lambda_{1}},\frac{1}{\lambda_{1}\lambda_{2}|z_{1}-z_{2}|^{2}}\right\\}$.
If instead $\lambda_{1}<\lambda_{2}$, the result is valid with
$\mu=\min\left\\{\frac{\lambda_{1}}{\lambda_{2}},\frac{1}{\lambda_{1}\lambda_{2}|z_{1}-z_{2}|^{2}}\right\\}$.

Thus, in order to identify whether the interactions of bubbles are weak, it is
natural to introduce the definition of weak-interaction (See Definition 1.15).

It is remarkable that the interaction of bubbles concentrates on some area
depending on the bubble:

######  Lemma 2.3.

Given $n\geq 3$ and two Escobar bubbles $U_{1}=U[z_{1},\lambda_{1}]$ and
$U_{2}=U[z_{2},\lambda_{2}]$ with $\lambda_{1}\geq\lambda_{2}$, it holds

$\int_{\partial\mathbb{R}_{+}^{n}}U_{1}^{p}U_{2}\approx_{n}\int_{B^{n-1}(z_{1},\lambda_{1}^{-1})\cap\partial\mathbb{R}_{+}^{n}}U_{1}^{p}U_{2}.$

This lemma can be obtained by [29, Corollary B.4].

For convenience, we will omit the domain of norms in the following sections
when no confusion arises.

##  3 Stability of fractional Sobolev trace inequality

In this section, we are devoted to analyzing the stability of the following
sharp fractional Sobolev trace inequality given by Einav and Loss in [25]:

######  Theorem 3.1.

Let $0\leq m<n$ and $\frac{m}{2}<\alpha<\frac{n}{2}$. For any $f\in
D_{\alpha}(\mathbb{R}^{n})$, we have

$S(n,m,\alpha){\lVert\tau_{m}f\rVert}_{L^{\frac{2(n-m)}{n-2\alpha}}(\mathbb{R}^{n-m})}^{2}\leq{\lVert
f\rVert}_{D_{\alpha}(\mathbb{R}^{n})}^{2},$ (3.1)

where

$S(n,m,\alpha)=2^{2\alpha}\pi^{\alpha}\frac{\Gamma(\alpha)\Gamma(n/2+\alpha-m)}{\Gamma(n/2-\alpha)\Gamma(\alpha-m/2)}\left(\frac{\Gamma(n-m)}{\Gamma((n-m)/2)}\right)^{\frac{m-2\alpha}{n-m}}$

and the equality holds if and only if $f(x)=f(x^{\prime\prime},x^{\prime})$,
$x^{\prime}\in\mathbb{R}^{m},\,x^{\prime\prime}\in\mathbb{R}^{n-m}$ is
proportional to

$\int_{\mathbb{R}^{n-m}}\frac{1}{(|x^{\prime}|^{2}+|x^{\prime\prime}-y^{\prime\prime}|^{2})^{(n-2\alpha)/2}}\frac{1}{(\gamma^{2}+|y^{\prime\prime}-a|^{2})^{(n+2\alpha-2m)/2}}\,dy^{\prime\prime}$
(3.2)

for some $a\in\mathbb{R}^{n-m}$ and $\gamma\neq 0$. We denote the set of
minimizers by $\mathcal{M}_{n,m,\alpha}$.

It was obtained by showing the two following inequalities and combining them
together:

######  Theorem 3.2 (fractional Sobolev inequality).

Let $0<\alpha<n/2$. For any $f\in D_{\alpha}(\mathbb{R}^{n})$,

$S(n,0,\alpha){\lVert f\rVert}_{L^{s}(\mathbb{R}^{n})}^{2}\leq{\lVert
f\rVert}_{D_{\alpha}(\mathbb{R}^{n})}^{2},$ (3.3)

where $s=\frac{2n}{n-2\alpha}$ and the equality holds if and only if

$f(x)=A(\gamma^{2}+|x-a|^{2})^{-\frac{n-2\alpha}{2}}$

for some $A\in\mathbb{R}$, $\gamma\neq 0$ and $a\in\mathbb{R}^{n}$. The set of
minimizers is denoted by $\mathcal{M}_{n,0,\alpha}$.

######  Theorem 3.3 (reduction principle).

Assume $0\leq m<n$ and $\frac{m}{2}<\alpha<\frac{n}{2}$. Then the trace
$\tau_{m}$ has a unique extension to a bounded operator
$\tau_{m}:D_{\alpha}(\mathbb{R}^{n})\to D_{\alpha-m/2}(\mathbb{R}^{n-m})$.
Moreover, for any $f\in D_{\alpha}(\mathbb{R}^{n})$,

$R(n,m,\alpha){\lVert\tau_{m}f\rVert}_{D_{\alpha-m/2}(\mathbb{R}^{n-m})}^{2}\leq{\lVert
f\rVert}_{D_{\alpha}(\mathbb{R}^{n})}^{2},$ (3.4)

where

$R(n,m,\alpha)=2^{m}\pi^{m/2}\frac{\Gamma(\alpha)}{\Gamma((2\alpha-m)/2)},$

and the equality holds if and only if

$\hat{f}(k_{1},k_{2})=\frac{\hat{g}(k_{1})}{(|k_{1}|^{2}+|k_{2}|^{2})^{\alpha}}$
(3.5)

with

$\int_{\mathbb{R}^{n-m}}\frac{{\lvert\hat{g}(k_{1})\rvert}^{2}}{{\lvert
k_{1}\rvert}^{2\alpha-m}}\,dk_{1}<\infty.$ (3.6)

Here $k_{1}\in\mathbb{R}^{n-m}$ and $k_{2}\in\mathbb{R}^{m}$.

To get the stability result (Theorem 1.1), we also need a stability result of
the fractional Sobolev inequality proved by Chen, Frank and Weth in [15]:

######  Theorem 3.4.

Let $0<\beta<N$, $\mathcal{M}=\mathcal{M}_{N,0,\beta}$, then
$C_{\mathrm{BE}}(N,0,\beta)\in(0,1)$ and we have

$\operatorname{d}^{2}(u,\mathcal{M})\geq{\lVert
u\rVert}_{D_{\beta}(\mathbb{R}^{N})}^{2}-S(N,0,\beta){\lVert
u\rVert}_{L^{\frac{2N}{N-2\beta}}(\mathbb{R}^{N})}^{2}\geq
C_{\mathrm{BE}}(N,0,\beta)\operatorname{d}^{2}(u,\mathcal{M})$ (3.7)

for all $u\in D_{\beta}(\mathbb{R}^{N})$, where
$\operatorname{d}(u,\mathcal{M})=\min\\{{\lVert
u-\varphi\rVert}_{D_{\beta}(\mathbb{R}^{N})}:\varphi\in\mathcal{M}\\}$.

The main idea of deriving Theorem 1.1 is using Theorems 3.3 and 3.4 to
construct $v$ directly. Precisely, any function $f\in
D_{\alpha}(\mathbb{R}^{n})$ can be orthogonally decomposed into two parts $g$
and $f-g$, where $g$ attains the equality of (3.4) with the same trace as $f$,
and ${\lVert f-g\rVert}_{D_{\alpha}}$ can be calculated directly. Then,
estimating $\operatorname{d}(\tau_{m}g,\mathcal{M}_{n-m,0,\alpha-m/2})$ by
(3.7) leads to the result. It is remarkable that the exponent 2 in (1.7),
coming from that of (3.7), is sharp due to the spectrum analysis in [15].

###### Proof of Theorem 1.1.

For a given $f\in D_{\alpha}(\mathbb{R}^{n})$, we claim that there exists a
function $g\in D_{\alpha}(\mathbb{R}^{n})$ such that $g$ takes equality of
(3.4) and $\tau_{m}g=\tau_{m}f$. Set

$\hat{g}(k_{1},k_{2})=C_{1}(m,\alpha)^{-1}\frac{\widehat{\tau_{m}f}(k_{1})|k_{1}|^{2\alpha-m}}{(|k_{1}|^{2}+|k_{2}|^{2})^{\alpha}},\quad
k_{1}\in\mathbb{R}^{n-m},k_{2}\in\mathbb{R}^{m},$

where $C_{1}(m,\alpha)=\int_{\mathbb{R}^{m}}(1+|x|^{2})^{-\alpha}\,dx$ is a
constant. Since $\tau_{m}f\in D_{\alpha-m/2}(\mathbb{R}^{n-m})$, if we take
$\hat{h}(k_{1})\coloneqq
C^{-1}(m,\alpha)\widehat{\tau_{m}f}(k_{1})|k_{1}|^{2\alpha-m}$, then $h\in
D_{\alpha-m/2}(\mathbb{R}^{n-m})$, and $g$ satisfies (3.5). On the other hand,

$\displaystyle\widehat{\tau_{m}g}(k_{1})=$
$\displaystyle\int_{\mathbb{R}^{m}}\hat{g}(k_{1},k_{2})\,dk_{2}=\int_{\mathbb{R}^{m}}\frac{\hat{h}(k_{1})}{(|k_{1}|^{2}+|k_{2}|^{2})^{\alpha}}\,dk_{2}$
$\displaystyle=$ $\displaystyle
C_{1}(m,\alpha)\hat{h(k_{1})}|k_{1}|^{-2\alpha+m}=\widehat{\tau_{m}f}(k_{1}),$

thus $\tau_{m}g=\tau_{m}f$. This proves the claim. Next, we compute the
distance from $f$ to $g$:

$\begin{split}{\lVert f-g\rVert}_{D_{\alpha}}^{2}={}&{\lVert
f\rVert}_{D_{\alpha}}^{2}+{\lVert
g\rVert}_{D_{\alpha}}^{2}-2\int_{\mathbb{R}^{n}}\overline{\hat{f}(k_{1},k_{2})}\hat{g}(k_{1},k_{2})(|2\pi
k_{1}|^{2}+|2\pi k_{2}|^{2})^{\alpha}\,dk_{1}dk_{2}\\\ ={}&{\lVert
f\rVert}_{D_{\alpha}}^{2}+R(n,m,\alpha){\lVert\tau_{m}f\rVert}_{D_{\alpha-m/2}}^{2}\\\
&-2C_{1}(m,\alpha)^{-1}(2\pi)^{m}\int_{R^{n-m}}\overline{\widehat{\tau_{m}f}(k_{1})}\widehat{\tau_{m}f}(k_{1})|2\pi
k_{1}|^{2\alpha-m}\,dk_{1}\\\ ={}&{\lVert
f\rVert}_{D_{\alpha}}^{2}-R(n,m,\alpha){\lVert\tau_{m}f\rVert}_{D_{\alpha-m/2}}^{2}.\end{split}$
(3.8)

Since $\tau_{m}f\in D_{\alpha-m/2}(\mathbb{R}^{n-m})$, picking $N=n-m$ and
$\beta=\alpha-\frac{m}{2}$ in Theorem 3.4, we get some
$\tilde{v}\in\mathcal{M}_{N,0,\beta}$ such that

$\begin{split}{\lVert\tau_{m}f-\tilde{v}\rVert}_{D_{\alpha-m/2}}^{2}\leq{}&C_{\mathrm{BE}}(N,0,\beta)^{-1}\left({\lVert\tau_{m}f\rVert}_{D_{\alpha-m/2}}^{2}-S(N,0,\beta){\lVert\tau_{m}f\rVert}_{L^{s}}^{2}\right),\end{split}$

where $s=\frac{2(n-m)}{n-2\alpha}$. Finally, taking $v\in
D_{\alpha}(\mathbb{R}^{n})$ that attains equality of (3.4) and
$\tau_{m}v=\tilde{v}$, we know that $g-v$ also attains the equality by
checking (3.5). Hence,

$\begin{split}{\lVert
g-v\rVert}_{D_{\alpha}}^{2}={}&R(n,m,\alpha){\lVert\tau_{m}g-\tau_{m}v\rVert}_{D_{\alpha-m/2}}^{2}\\\
={}&R(n,m,\alpha){\lVert\tau_{m}f-\tilde{v}\rVert}_{D_{\alpha-m/2}}^{2}\\\
\leq{}&C_{\mathrm{BE}}(N,0,\beta)^{-1}R(n,m,\alpha)\left({\lVert\tau_{m}f\rVert}_{D_{\alpha-m/2}}^{2}-S(N,0,\beta){\lVert\tau_{m}f\rVert}_{L^{s}}^{2}\right).\end{split}$
(3.9)

By similar computation as in (3.8), we know by $\tau_{m}f=\tau_{m}g$ that

$\begin{split}&\int_{\mathbb{R}^{n}}\overline{(\hat{f}-\hat{g})(k_{1},k_{2})}(\hat{g}-\hat{v})(k_{1},k_{2})({\lvert
2\pi k_{1}\rvert}^{2}+{\lvert 2\pi k_{2}\rvert}^{2})^{\alpha}\,dk_{1}dk_{2}\\\
={}&C_{1}(m,\alpha)^{-1}(2\pi)^{m}\int_{\mathbb{R}^{n-m}}\overline{(\widehat{\tau_{m}f}-\widehat{\tau_{m}g})(k_{1})}(\widehat{\tau_{m}g}-\widehat{\tau_{m}v})(k_{1}){\lvert
2\pi k_{1}\rvert}^{2\alpha-m}dk_{1}=0.\end{split}$

Thus, combining (3.8), (3.9), (3.4), and noticing that
$C_{\mathrm{BE}}(N,0,\beta)\in(0,1)$, we get the desired estimate (1.7):

$\displaystyle{\lVert f-v\rVert}_{D_{\alpha}}^{2}={}$ $\displaystyle{\lVert
f-g\rVert}_{D_{\alpha}}^{2}+{\lVert g-v\rVert}_{D_{\alpha}}^{2}$
$\displaystyle\leq{}$ $\displaystyle{\lVert
f\rVert}_{D_{\alpha}}^{2}-R(n,m,\alpha){\lVert\tau_{m}f\rVert}_{D_{\alpha-m/2}}^{2}$
$\displaystyle+C_{\mathrm{BE}}(N,0,\beta)^{-1}\,R(n,m,\alpha)\left({\lVert\tau_{m}f\rVert}_{D_{\alpha-m/2}}^{2}-S(N,0,\beta){\lVert\tau_{m}f\rVert}_{L^{s}}^{2}\right)$
$\displaystyle={}$ $\displaystyle{\lVert
f\rVert}_{D_{\alpha}}^{2}+\left(C_{\mathrm{BE}}(N,0,\beta)^{-1}-1\right)R(n,m,\alpha){\lVert\tau_{m}f\rVert}_{D_{\alpha-m/2}}^{2}$
$\displaystyle-
C_{\mathrm{BE}}(N,0,\beta)^{-1}\,S(n,m,\alpha){\lVert\tau_{m}f\rVert}_{L^{s}}^{2}$
$\displaystyle\leq{}$ $\displaystyle
C_{\mathrm{BE}}(N,0,\beta)^{-1}\left({\lVert
f\rVert}_{D_{\alpha}}^{2}-S(n,m,\alpha){\lVert\tau_{m}f\rVert}_{L^{s}}^{2}\right).\qed$

######  Remark 3.5 (estimate of stability constant).

We can go further to discuss the best constant $C_{\mathrm{BE}}(n,m,\alpha)$
of (1.7). Define

$S_{\mathrm{Tr}}(f)\coloneqq\frac{{\lVert
f\rVert}_{D_{\alpha}}^{2}-S(n,m,\alpha){\lVert\tau_{m}f\rVert}_{L^{s}}^{2}}{\operatorname{d}^{2}(f,\mathcal{M}_{n,m,\alpha})},\quad\text{for
}f\in D_{\alpha}(\mathbb{R}^{n})\backslash\mathcal{M}_{n,m,\alpha},$ (3.10)
$\mathcal{E}(f)\coloneqq\frac{{\lVert
f\rVert}_{D_{\alpha-m/2}}^{2}-S(N,0,\beta){\lVert\tau_{m}f\rVert}_{L^{s}}^{2}}{\operatorname{d}^{2}(f,\mathcal{M}_{N,0,\beta})},\quad\text{for
}f\in D_{\alpha-m/2}(\mathbb{R}^{n-m})\backslash\mathcal{M}_{N,0,\beta},$

with $N=n-m,\,\beta=\alpha-m/2,\,s=\frac{2(n-m)}{n-2\alpha}$. The above proof
shows that for any $f\in D_{\alpha}(\mathbb{R}^{n})$,

$\begin{split}\operatorname{d}^{2}(f,\mathcal{M}_{n,m,\alpha})={}&\inf_{h\in\mathcal{M}_{n,m,\alpha}}{\lVert
f-g\rVert}_{D_{\alpha}}^{2}+{\lVert g-h\rVert}_{D_{\alpha}}^{2}\\\ ={}&{\lVert
f\rVert}_{D_{\alpha}}^{2}-R(n,m,\alpha){\lVert\tau_{m}f\rVert}_{D_{\alpha-m/2}}^{2}+\inf_{h\in\mathcal{M}_{n,m,\alpha}}{\lVert
g-h\rVert}_{D_{\alpha}}^{2}\\\ ={}&{\lVert
f\rVert}_{D_{\alpha}}^{2}-R(n,m,\alpha){\lVert\tau_{m}f\rVert}_{D_{\alpha-m/2}}^{2}+R(n,m,\alpha)\inf_{h\in\mathcal{M}_{N,0,\beta}}{\lVert\tau_{m}f-h\rVert}_{D_{\alpha-m/2}}^{2}\\\
\coloneqq{}&\delta_{1}(f)+R(n,m,\alpha)\operatorname{d}^{2}(\tau_{m}f,\mathcal{M}_{N,0,\beta})\end{split}$

with $g$ defined as in the proof of Theorem 1.1. Hence, if
$\tau_{m}f\notin\mathcal{M}_{N,0,\beta}$, by

$\displaystyle S_{\mathrm{Tr}}(g)$
$\displaystyle=\frac{R(n,m,\alpha){\lVert\tau_{m}f\rVert}_{D_{\alpha-m/2}}^{2}-S(n,m,\alpha){\lVert\tau_{m}f\rVert}_{L^{s}}^{2}}{R(n,m,\alpha)\operatorname{d}^{2}(\tau_{m}f,\mathcal{M}_{N,0,\beta})}$
$\displaystyle=\frac{{\lVert\tau_{m}f\rVert}_{D_{\alpha-m/2}}^{2}-S(N,0,\beta){\lVert\tau_{m}f\rVert}_{L^{s}}^{2}}{\operatorname{d}^{2}(\tau_{m}f,\mathcal{M}_{N,0,\beta})}=\mathcal{E}(\tau_{m}f)\leq
1,$

we get

$S_{\mathrm{Tr}}(f)=\frac{\delta_{1}(f)+R(n,m,\alpha)\left({\lVert\tau_{m}f\rVert}_{D_{\alpha-m/2}}^{2}-S(N,0,\beta){\lVert\tau_{m}f\rVert}_{L^{s}}^{2}\right)}{\delta_{1}(f)+R(n,m,\alpha)\operatorname{d}^{2}(\tau_{m}f,\mathcal{M}_{N,0,\beta})}\geq\mathcal{E}(\tau_{m}f),$
(3.11)

and equality holds if and only if $f=g$. If
$\tau_{m}f\in\mathcal{M}_{N,0,\beta}$, then
$\operatorname{d}^{2}(f,\mathcal{M}_{n,m,\alpha})=\delta_{1}(f)$,
${\lVert\tau_{m}f\rVert}_{D_{\alpha-m/2}}^{2}=S(N,0,\beta){\lVert\tau_{m}f\rVert}_{L^{s}}^{2}$,
and so

$S_{\mathrm{Tr}}(f)=\frac{{\lVert
f\rVert}_{D_{\alpha}}^{2}-R(n,m,\alpha)S(N,0,\beta){\lVert\tau_{m}f\rVert}_{L^{s}}^{2}}{\delta_{1}(f)}=1.$

As a result,

$C_{\text{BE}}(n,m,\alpha)=\inf_{f\in
D_{\alpha}(\mathbb{R}^{n})\backslash\mathcal{M}_{n,m,\alpha}}S_{\mathrm{Tr}}(f)=\inf_{g\in
D_{\alpha-m/2}(\mathbb{R}^{n-m})\backslash\mathcal{M}_{N,0,\beta}}\mathcal{E}(g)=C_{\mathrm{BE}}(N,0,\beta),$

and the best constant of Theorem 1.1 is exactly that of Theorem 3.4.

Now based on the work in [41, 42] and [14], we have the bound:

$\frac{\min\\{K_{n-m,\alpha-m/2},1,2^{\frac{n+2\alpha-2m}{n-m}}-2\\}}{4}\leq
C_{\mathrm{BE}}(n,m,\alpha),$

and

$C_{\mathrm{BE}}(n,m,\alpha)\begin{cases}<\min\left\\{\frac{4\alpha-2m}{n+2\alpha+2-2m},2-2^{\frac{n-2\alpha}{n-m}}\right\\},&n-m\geq
2,\\\ \leq\frac{4\alpha-2m}{4+2\alpha},&n-m=1,\end{cases}$

where $K_{n,s}$ is a complicate number. We refer to [14, Theorem 1.1] for an
explicit expression of $K_{n,s}$.

At the end of this section, we consider the existence of minimizers for the
stability constant $C_{\mathrm{BE}}(n,m,\alpha)$ in the spirit of König’s work
in [42]. We show the following compactness result.

######  Theorem 3.6 (compactness of minimizing sequence).

Let $0\leq m<n-1$ and $\frac{m}{2}<\alpha<\frac{n}{2}$. Let $(u_{n})$ be a
minimizing sequence for (3.10) with ${\lVert
u_{n}\rVert}_{D_{\alpha}(\mathbb{R}^{n})}=1$. Then there exist a function
$u\in D_{\alpha}(\mathbb{R}^{n})\backslash\mathcal{M}_{n,m,\alpha}$ such that,
up to a subsequence, $u_{n}$ converge to $u$ strongly in
$D_{\alpha}(\mathbb{R}^{n})$. Moreover,
$S_{\mathrm{Tr}}(u)=C_{\mathrm{BE}}(n,m,\alpha)$, _i.e._ $u$ is a minimizer
for $C_{\mathrm{BE}}(n,m,\alpha)$.

###### Proof.

We still set $N=n-m$ and $\beta=\alpha-m/2$. By Remark 3.5, we may assume each
$S_{\mathrm{Tr}}(u_{n})<1$ to avoid the case
$\tau_{m}u_{n}\in\mathcal{M}_{N,0,\beta}$. Denote by $\tilde{u}_{n}$ the
minimizer of (3.4) with $\tau_{m}\tilde{u}_{n}=\tau_{m}u_{n}\in
D_{\alpha-m/2}(\mathbb{R}^{n-m})\backslash\mathcal{M}_{N,0,\beta}$, then
$\mathcal{E}(\tau_{m}u_{n})=S_{\mathrm{Tr}}(\tilde{u}_{n})\leq
S_{\mathrm{Tr}}(u_{n})$ and so
$\lim_{n}\mathcal{E}(\tau_{m}u_{n})=\lim_{n}S_{\mathrm{Tr}}(u_{n})=C_{\mathrm{BE}}(n,m,\alpha)$,
and
$\left(\frac{\tau_{m}u_{n}}{{\lVert\tau_{m}u_{n}\rVert}_{D_{\alpha-m/2}}}\right)$
is a minimizing sequence for $\mathcal{E}$ with norm 1. Using [42, Theorem
1.2], there exists a function $v\in
D_{\alpha-m/2}(\mathbb{R}^{n-m})\backslash\mathcal{M}_{N,0,\beta}$ such that,
up to a subsequence,
$\frac{\tau_{m}u_{n}}{{\lVert\tau_{m}u_{n}\rVert}_{D_{\alpha-m/2}}}$ converge
to $v$ strongly in $D_{\alpha-m/2}$, and in addition
$\mathcal{E}(v)=C_{\mathrm{BE}}(n,m,\alpha)$.

We now claim
$\lim_{n}{\lVert\tau_{m}u_{n}\rVert}_{D_{\alpha-m/2}}^{2}=R(n,m,\alpha)^{-1}$.
By (3.11),

$\displaystyle\left(1-S_{\mathrm{Tr}}(u_{n})\right)\delta_{1}(u_{n})={}$
$\displaystyle
R(n,m,\alpha)S_{\mathrm{Tr}}(u_{n})\operatorname{d}^{2}(\tau_{n}u_{n},\mathcal{M})$
$\displaystyle-R(n,m,\alpha)\left({\lVert\tau_{m}f\rVert}_{D_{\alpha-m/2}}^{2}-S(N,0,\beta){\lVert\tau_{m}f\rVert}_{L^{s}}^{2}\right)$
$\displaystyle={}$ $\displaystyle
R(n,m,\alpha)\operatorname{d}^{2}(\tau_{n}u_{n},\mathcal{M})\big{(}S_{\mathrm{Tr}}(u_{n})-\mathcal{E}(\tau_{m}u_{n})\big{)}.$

Since
$\lim_{n}S_{\mathrm{Tr}}(u_{n})=\lim_{n}\mathcal{E}(\tau_{m}u_{n})=C_{\mathrm{BE}}<1$,
the right side tends to 0 as $n\to\infty$. On the other hand, the left side is
larger than $\frac{1-C_{\mathrm{BE}}(n,m,\alpha)}{2}>0$ if $n$ is large
enough, and so

$\displaystyle\lim_{n\to\infty}\delta_{1}(u_{n})=0\implies\lim_{n}{\lVert\tau_{m}u_{n}\rVert}_{D_{\alpha-m/2}}^{2}=$
$\displaystyle\lim_{n}R(n,m,\alpha)^{-1}\big{(}{\lVert
u_{n}\rVert}_{D_{\alpha}}^{2}-\delta_{1}(u_{n})\big{)}$ $\displaystyle=$
$\displaystyle R(n,m,\alpha)^{-1}.$

Hence, up to a subsequence, $\tau_{m}u_{n}\to R(n,m,\alpha)^{-1}v$ strongly in
$D_{\alpha-m/2}$. As a result, if we set $u\in D_{\alpha}(\mathbb{R}^{n})$ to
be the minimizer of (3.4) with $\tau_{m}u=R(n,m,\alpha)^{-1}v$, then

$\displaystyle{\lVert u_{n}-u\rVert}_{D_{\alpha}}^{2}={}$
$\displaystyle{\lVert
u_{n}-\tilde{u}_{n}\rVert}_{D_{\alpha}}^{2}+{\lVert\tilde{u}_{n}-\tilde{v}\rVert}_{D_{\alpha}}^{2}$
$\displaystyle={}$
$\displaystyle\delta_{1}(u_{n})+R(n,m,\alpha){\lVert\tau_{m}u_{n}-R(n,m,\alpha)^{-1}v\rVert}_{D_{\alpha-m/2}}^{2}\to
0,\quad n\to\infty,$

yielding that $u_{n}\to u$ strongly in $D_{\alpha}$ up to a subsequence. The
only thing left is to show $S_{\mathrm{Tr}}(u)=C_{\mathrm{BE}}(n,m,\alpha)$.
By the definition of $u$, we have

$S_{\mathrm{Tr}}(u)=\mathcal{E}(\tau_{m}u)=\mathcal{E}(v)=C_{\mathrm{BE}}(n,m,\alpha).\qed$

##  4 Some consequences

In this section we give some corollaries of the gradient stability result
(1.7).

We first prove the following theorem concerning refined Sobolev embedding in
domains with finite measure.

######  Theorem 4.1.

Let $0\leq m<n,\frac{m}{2}<\alpha<\frac{n}{2}$ and
$s=\frac{2(n-m)}{n-2\alpha},q_{1}=\frac{n-m}{n-2\alpha},q_{2}=\frac{n-m}{n-\alpha-\frac{m}{2}}$.
Then there exists a constant $C$ depending only on $n,\alpha$ such that, for
any domain $\Omega\subset\mathbb{R}^{n}$ with
$\mathcal{H}^{m}(\Omega\cap\mathbb{R}^{m})<\infty$, and any $u\in
D_{\alpha}(\mathbb{R}^{n})$ with $\tau_{m}{u}=0$ in $\Omega^{c}$, we have

$\|u\|_{D_{\alpha}(\mathbb{R}^{n})}^{2}-S(n,m,\alpha)\|\tau_{m}{u}\|_{L^{s}(\mathbb{R}^{n-m})}^{2}\geq
C\mathcal{H}^{m}(\Omega\cap\mathbb{R}^{m})^{-\frac{1}{q_{1}}}\|\tau_{m}{u}\|_{L_{w}^{q_{1}}(\Omega\cap\mathbb{R}^{m})}^{2}.$
(4.1)

Here $\mathcal{H}$ denotes the Hausdorff measure. If
$\alpha-\frac{m}{2}\in\mathbb{N}$, then we also have

$\|u\|_{D_{\alpha}(\mathbb{R}^{n})}^{2}-S(n,m,\alpha)\|\tau_{m}{u}\|_{L^{s}(\mathbb{R}^{n-m})}^{2}\geq
C\mathcal{H}^{m}(\Omega\cap\mathbb{R}^{m})^{-\frac{1}{q_{1}}}\|\Delta^{\frac{2\alpha-m}{4}}(\tau_{m}{u})\|_{L_{w}^{q_{2}}(\Omega\cap\mathbb{R}^{m})}^{2}.$
(4.2)

Moreover, the weak norm on the right-hand side cannot be replaced by the
strong norm.

The proof of (4.1) and (4.2) can be reduced to the case $m=0$ using Theorem
3.3. Based on Theorem 1.1, it suffices to show the following lemma:

######  Lemma 4.2.

Let $0<\alpha<\frac{n}{2},q_{1}=\frac{n}{n-2\alpha},q_{2}=\frac{n}{n-\alpha}$.
Then there exists a constant $C$ depending only on $\alpha,n$ such that

$\left\|u\right\|_{L^{q_{1}}_{w}(\Omega)}\leq
C|\Omega|^{\frac{1}{2q_{1}}}d(u,\mathcal{M}_{n,0,\alpha})$ (4.3)

for all subdomains $\Omega\in\mathbb{R}^{n}$ with $|\Omega|<\infty$ and all
$u\in D_{\alpha}(\mathbb{R}^{n})$ with $u=0$ in $\Omega^{c}$.

If in addition $\alpha\in\mathbb{N}$, then the following estimate holds:

$\left\|\Delta^{\frac{\alpha}{2}}u\right\|_{L^{q_{2}}_{w}(\Omega)}\leq
C|\Omega|^{\frac{1}{2q_{1}}}d(u,\mathcal{M}_{n,0,\alpha}).$ (4.4)

######  Remark 4.3.

The inequality (4.3) has been proved by Chen, Frank and Weth in [15]. When
$\alpha=1$, (4.4) was proved by Brézis and Lieb in [10]. Although the proof of
(4.4) for general $\alpha$ are similar to that applied in [15], for
completeness, we give a detailed proof below.

###### Proof of Lemma 4.2.

Assume (4.4) fails and $|\Omega|=1$. Then there exists a sequence
$(u_{n})\subset H_{0}^{1}(\Omega)\backslash\\{0\\}$ such that

$\displaystyle\frac{d(u_{n},\mathcal{M}_{n,0,\alpha})}{\left\|\Delta^{\frac{\alpha}{2}}u_{n}\right\|_{L^{q_{2}}_{w}(\Omega)}}\rightarrow
0.$ (4.5)

By homogeneity, we can assume
$\left\|\Delta^{\frac{\alpha}{2}}u_{n}\right\|_{L^{2}}=1$. By the Hölder
inequality,
$\left\|\Delta^{\frac{\alpha}{2}}u_{n}\right\|_{L^{q_{2}}_{w}(\Omega)}$ is
bounded and hence $d(u_{n},\mathcal{M}_{n,0,\alpha})\to 0$. This shows that
there exist $c_{n}\rightarrow 1,z_{n}\in\Omega$ and
$\lambda_{n}\rightarrow\infty$ such that

$\displaystyle
d(u_{n},\mathcal{M}_{n,0,\alpha})=\left\|\Delta^{\frac{\alpha}{2}}(u_{n}-c_{n}U[z_{n},\lambda_{n}])\right\|_{L^{2}},$

with $U[z,\lambda]=\left(\frac{\lambda}{1+\lambda^{2}{\lvert
x-z\rvert}^{2}}\right)^{\frac{n-2\alpha}{2}}$. A direct computation yields

$\displaystyle d(u_{n},\mathcal{M}_{n,0,\alpha})^{2}$
$\displaystyle\geq\left\|\Delta^{\frac{\alpha}{2}}c_{n}U[0,\lambda_{n}]\right\|_{L^{2}(\Omega^{c})}^{2}$
(4.6)
$\displaystyle\gtrsim\left\|\Delta^{\frac{\alpha}{2}}U[0,\lambda_{n}]\right\|_{L^{2}(B_{2}(0)^{c})}^{2}$
$\displaystyle\gtrsim\int_{B_{2}(0)^{c}}\frac{\lambda_{n}^{2\alpha-n}}{|x|^{2(n-\alpha)}}\gtrsim\lambda_{n}^{2\alpha-n}.$

Therefore we have

$\displaystyle\left\|\Delta^{\frac{\alpha}{2}}u_{n}\right\|_{L^{q_{2}}_{w}(\Omega)}$
$\displaystyle\leq\left\|\Delta^{\frac{\alpha}{2}}(u_{n}-c_{n}U[z_{n},\lambda_{n}])\right\|_{L^{q_{2}}_{w}(\Omega)}+\left\|\Delta^{\frac{\alpha}{2}}c_{n}U[z_{n},\lambda_{n}]\right\|_{L^{q_{2}}_{w}(\Omega)}$
$\displaystyle\lesssim\left\|\Delta^{\frac{\alpha}{2}}(u_{n}-c_{n}U[z_{n},\lambda_{n}])\right\|_{L^{2}}+\lambda_{n}^{\frac{2\alpha-n}{2}}\left\|\Delta^{\frac{\alpha}{2}}U[0,1]\right\|_{L^{q_{2}}_{w}(\mathbb{R}^{n})}$
$\displaystyle\lesssim d(u_{n},\mathcal{M}_{n,0,\alpha}).$

This contradicts (4.5). ∎

By further exploiting the proof above, we can establish similar results for
strips.

######  Theorem 4.4.

Assume $m,n\in\mathbb{N}^{+}$ satisfy $\frac{n}{4}\leq m<\frac{n}{2}$. Let
$q=\frac{n}{n-2m}$. Then there exists a constant $C$ depending only on $m,n$
such that

$\|u\|_{D_{m}(\mathbb{R}^{n})}^{2}-S(n,0,m)\|u\|_{L^{2q}(\mathbb{R}^{n})}^{2}\geq
C\left\|u\right\|_{L^{q}_{w}(I\times\mathbb{R}^{n-1})}^{2}$ (4.7)

for all $u\in D_{m}(\mathbb{R}^{n})$ whose support belongs to
$I\times\mathbb{R}^{n-1}$; here $I=[-1,1]$ is an interval in $\mathbb{R}$.

######  Remark 4.5.

When $m=1$, (4.7) was proved by Wang and Willem in [60]. They in fact
considered the refined Caffarelli-Kohn-Nirenberg inequality in more general
domains. Their methods relied on a maximum principle. In our cases, due to the
absence of maximum principle for the polyharmonic operator, we restrict our
analysis to domains that are bounded in one direction.

###### Proof of Theorem 4.4.

Assume that (4.7) fails, then there exists a sequence $(u_{n})\subset
H_{0}^{1}(\Omega)$ such that

$\displaystyle\frac{d(u_{n},\mathcal{M}_{n,0,m})}{\left\|u_{n}\right\|_{L^{q}_{w}(I\times\mathbb{R}^{n-1})}}\rightarrow
0.$

Assume $\|u_{n}\|_{D_{m}}=1$. Note that

$\int_{I\times\mathbb{R}^{n-1}}|\Delta^{\frac{m}{2}}u_{n}|^{2}\geq\int_{\mathbb{R}^{n-1}}\int_{I}\left|\frac{\partial^{m}u_{n}}{\partial
x_{1}^{m}}\right|^{2}\gtrsim\int_{\mathbb{R}^{n-1}}\int_{I}|u_{n}|^{2}.$

The assumption $\frac{n}{4}\leq m<\frac{n}{2}$, the interpolation inequality
and the fractional Sobolev inequality yield

$\left\|u_{n}\right\|_{L^{q}_{w}(I\times\mathbb{R}^{n-1})}\leq\left\|u_{n}\right\|_{L^{2}}^{\lambda}\left\|u_{n}\right\|_{L^{2q}}^{1-\lambda}\lesssim\|u_{n}\|_{D_{m}}$
(4.8)

for some $0\leq\lambda\leq 1$. Hence
$\left\|u_{n}\right\|_{L^{q}_{w}(I\times\mathbb{R}^{n-1})}$ is bounded and
$d(u,\mathcal{M}_{n,0,m})\rightarrow 0$. This indicates the existence of
$c_{n}\rightarrow 1,z_{n}\in I\times\mathbb{R}^{n-1}$ and
$\lambda_{n}\rightarrow\infty$ such that

$\displaystyle
d(u_{n},\mathcal{M}_{n,0,m})=\left\|\Delta^{\frac{m}{2}}(u_{n}-c_{n}U[z_{n},\lambda_{n}])\right\|_{L^{2}}.$

Similarly to the computation in (4.6), we derive that
$d(u_{n},\mathcal{M}_{n,0,m})\gtrsim\lambda_{n}^{\frac{2m-n}{2}}$. Now fix a
cut-off function $\eta\in C_{c}^{\infty}(\mathbb{R})$ such that $0\leq\eta\leq
1$, $\eta=1$ in $2I$, $\eta=0$ outside $3I$. Using (4.8), we can estimate:

$\displaystyle\left\|u_{n}\right\|_{L^{q}_{w}(I\times\mathbb{R}^{n-1})}$
$\displaystyle\leq\left\|u_{n}-c_{n}\eta
U[z_{n},\lambda_{n}]\right\|_{L^{q}_{w}(I\times\mathbb{R}^{n-1})}+\left\|c_{n}\eta
U[z_{n},\lambda_{n}]\right\|_{L^{q}_{w}(I\times\mathbb{R}^{n-1})}$
$\displaystyle\lesssim\left\|u_{n}-c_{n}\eta
U[z_{n},\lambda_{n}]\right\|_{D_{m}}+\left\|c_{n}U[z_{n},\lambda_{n}]\right\|_{L^{q}_{w}(I\times\mathbb{R}^{n-1})}$
$\displaystyle\lesssim\left\|u_{n}-c_{n}U[z_{n},\lambda_{n}]\right\|_{D_{m}}+\left\|c_{n}(1-\eta)U[z_{n},\lambda_{n}]\right\|_{D_{m}}+\lambda_{n}^{\frac{2m-n}{2}}$
$\displaystyle\lesssim
d(u_{n},\mathcal{M}_{n,0,m})+\left\|(1-\eta)U[z_{n},\lambda_{n}]\right\|_{D_{m}}.$

It will lead to a contradiction if there exists a uniform constant $C$ such
that

$\left\|(1-\eta)U[z_{n},\lambda_{n}]\right\|_{D_{m}}\leq
C\lambda_{n}^{\frac{2m-n}{2}},\quad\forall n\in\mathbb{N}.$ (4.9)

Based on our choice of $z_{n},\lambda_{n}$ and $\eta$, (4.9) can be derived
from the following three simple observations:

$\displaystyle\left\|(1-\eta)U[z_{n},\lambda_{n}]\right\|_{D^{m}}\lesssim\sum_{i=0}^{m-1}\left\|\nabla^{i}U[z_{n},\lambda_{n}]\right\|_{L^{2}((3I-2I)\times\mathbb{R}^{n-1})}+\left\|\nabla^{m}U[z_{n},\lambda_{n}]\right\|_{L^{2}((2I)^{c}\times\mathbb{R}^{n-1})},$
$\displaystyle\left\||\nabla^{i}U[z_{n},\lambda_{n}]|\right\|_{L^{2}((3I-2I)\times\mathbb{R}^{n-1})}\lesssim\lambda_{n}^{\frac{2m-n}{2}},\quad
0\leq i\leq m-1,$
$\displaystyle\left\|\nabla^{m}U[z_{n},\lambda_{n}]\right\|_{L^{2}((2I)^{c}\times\mathbb{R}^{n-1})}\lesssim\lambda_{n}^{\frac{2m-n}{2}}.\qed$

Next we establish the dual stability results for the Escobar trace inequality:

$S_{\mathrm{E}}(n){\lVert
f\rVert}_{L^{\frac{2(n-1)}{n-2}}(\mathbb{R}^{n-1})}^{2}\leq{\lVert
f\rVert}_{H^{1}(\mathbb{R}_{+}^{n})}^{2}.$

The extremal functions are given by (2.1) up to normalizations. Consider the
pairing $(X,Y,\langle\cdot,\cdot\rangle)$:

$X=H^{1}(\mathbb{R}_{+}^{n}),\;Y=L^{\frac{2(n-1)}{n}}(\mathbb{R}^{n-1}),\;\langle
f,g\rangle=\int_{\mathbb{R}^{n-1}}fg,$

and two functionals $\Phi$ and $\Psi$ on $X$:

$\Phi(f)={\lVert
f\rVert}_{H^{1}(\mathbb{R}_{+}^{n})}^{2},\quad\Psi(f)=S_{\mathrm{E}}(n){\lVert
f\rVert}_{L^{\frac{2(n-1)}{n-2}}(\mathbb{R}^{n-1})}^{2}.$

It is easy to compute the Legendre transform $\Phi^{*}$ and $\Psi^{*}$ on $Y$:

$\displaystyle\Phi^{*}(g)$ $\displaystyle=\sup_{f\in X}\left\\{\langle
f,g\rangle-\Phi(f)\right\\}$ $\displaystyle=\sup_{f\in
X}\left\\{\int_{\mathbb{R}^{n-1}}fg-\int_{\mathbb{R}_{+}^{n}}|\nabla
f|^{2}\right\\}$
$\displaystyle=\frac{1}{4}\|\nabla\mathcal{P}[g]\|_{L^{2}(\mathbb{R}_{+}^{n})}^{2}$
$\displaystyle\Psi^{*}(g)$ $\displaystyle=\sup_{f\in X}\left\\{\langle
f,g\rangle-\Phi(f)\right\\}$ $\displaystyle=\sup_{f\in
X}\left\\{\int_{\mathbb{R}^{n-1}}fg-S_{\mathrm{E}}(n){\lVert
f\rVert}_{L^{\frac{2(n-1)}{n-2}}(\mathbb{R}^{n-1})}^{2}\right\\}$
$\displaystyle=\frac{1}{4S_{\mathrm{E}}(n)}{\lVert
g\rVert}_{L^{\frac{2(n-1)}{n}}}^{2}.$

The function $\mathcal{P}[g]$ in the above computation is the unique solution
of

$\begin{cases}\Delta\mathcal{P}[g]=0&\text{in }\mathbb{R}_{+}^{n},\\\
\frac{\partial\mathcal{P}[g]}{\partial t}=-g&\text{on
}\partial\mathbb{R}_{+}^{n}.\end{cases}$ (4.10)

Based on standard dual scheme developed by Carlen in [12], we can immediately
obtain the following stability result:

######  Theorem 4.6.

For any $n\geq 3$ and $u\in L^{\frac{2n-2}{n}}(\mathbb{R}^{n-1})$, it holds

$\|u\|_{L^{\frac{2n-2}{n}}(\mathbb{R}^{n-1})}^{2}\geq
S_{\mathrm{E}}(n)^{-1}\|\nabla\mathcal{P}[u]\|_{L^{2}(\mathbb{R}_{+}^{n})}^{2},$
(4.11)

where $\mathcal{P}[u]$ is the unique solution of (4.10). The equality holds
precisely when $u(x)=(1+|x|^{2})^{\frac{n}{2}}$ after suitable translation,
scaling and normalization.

Moreover, if denote by $\mathcal{M}_{\mathrm{Neu}}$ the extremal manifold, the
following sharp stability result holds:

$\|u\|_{L^{\frac{2n-2}{n}}(\mathbb{R}^{n-1})}^{2}-S_{\mathrm{E}}(n)^{-1}\|\nabla\mathcal{P}[u]\|_{L^{2}(\mathbb{R}_{+}^{n})}^{2}\geq
C\inf_{v\in\mathcal{M}_{\mathrm{Neu}}}\|u-v\|_{L^{\frac{2n-2}{n}}(\mathbb{R}^{n-1})}^{2}$
(4.12)

for a positive constant $C$ which can be writen explicitly in terms of
$C_{\mathrm{BE}}(n,1,1)$.

######  Remark 4.7.

By more calculations, the a priori estimate above is equivalent to the Hardy-
Littlewood-Sobolev inequality in $\mathbb{R}^{n-1}$ (see for example [6]).
However, this estimate has its own interest, and the operator $\mathcal{P}$
will be used in Section 7 to handle the dual Sobolev norm.

##  5 Qualitative profile decomposition

In this section we establish qualitative profile decomposition for the
following nonlinear critical Neumann boundary problem in the same spirit of
pioneer works of Struwe in [57]:

$\begin{cases}\Delta u=0&\text{in }\mathbb{R}_{+}^{n},\\\ \frac{\partial
u}{\partial t}=-|u|^{p-1}u&\text{on }\partial\mathbb{R}_{+}^{n}.\end{cases}$
(5.1)

Although we deal with the trace version here, our main methods are parallel to
those in [58], where (1.3) is treated. We first consider the global
compactness for general functions.

######  Theorem 5.1.

Let $n\geq 3$ and $(u_{k})_{k\in\mathbb{N}}\subset H^{1}(\mathbb{R}_{+}^{n})$
be a sequence of functions with $\int_{\mathbb{R}_{+}^{n}}|\nabla u_{k}|^{2}$
uniformly bounded. Assume that (recall the dual norm is defined in (1.14))

$\left\|\Delta u_{k}+|u_{k}|^{p-1}u_{k}\right\|_{H^{-1}}\rightarrow
0\quad\text{as}\;k\rightarrow\infty.$ (5.2)

Then there exist a number $\nu\in\mathbb{N}$, a subsequence of $(u_{k})_{k}$,
which we still denote by $(u_{k})_{k}$, a sequence
$(z_{1}^{(k)},...,z_{\nu}^{(k)})$ of $\nu$-tuples of points in
$\mathbb{R}^{n-1}$, a sequence $(\lambda_{1}^{(k)},...,\lambda_{\nu}^{k})$ of
$\nu$-tuples of positive real numbers and a sequence of functions
$(w_{i})_{1\leq i\leq\nu}$, which are nontrivial solutions of (5.1), such that

$\left\|\nabla\left(u_{k}-\sum_{i=1}^{\nu}w_{i}[z_{i}^{(k)},\lambda_{i}^{(k)}]\right)\right\|_{L^{2}}\rightarrow
0\quad\text{as}\;k\rightarrow\infty,$ (5.3) $\left\|\nabla
u_{k}\right\|_{L^{2}}^{2}-\sum_{i=1}^{\nu}\left\|\nabla
w_{i}[z_{i}^{(k)},\lambda_{i}^{(k)}]\right\|_{L^{2}}^{2}\rightarrow
0\quad\text{as}\;k\rightarrow\infty.$ (5.4)

Here $w_{i}[z,\lambda](x,t):=\lambda^{\frac{n-2}{2}}w_{i}(\lambda(x-z),\lambda
t)$ for every $z\in\mathbb{R}^{n-1},\lambda>0$.

###### Proof.

In the following proof, we will use $B_{r}(x,t)$ to denote the ball centered
at $(x,t)$ with radius $r$ and omit the subscript $r$ when $r=1$. We also
define $B_{r}^{+}(x,t)=B_{r}(x,t)\cap\mathbb{R}_{+}^{n}$,
$\partial_{b}B_{r}^{+}(x,t)=\partial B_{r}^{+}(x,t)\cap\mathbb{R}^{n-1}$.

Since $\int_{\mathbb{R}_{+}^{n}}|\nabla u_{k}|^{2}$ is uniformly bounded, we
may assume $\int_{\mathbb{R}_{+}^{n}}|\nabla u_{k}|^{2}\rightarrow
L\in[0,\infty)$. If $L=0$, then $u_{k}\rightarrow 0$ and the results hold
clearly. In the following we assume $L>0$ and
$\frac{L}{2}\leq\int_{\mathbb{R}_{+}^{n}}|\nabla u_{k}|^{2}\leq\frac{3L}{2}$
for any $k$. Due to translation and dilation invariance, we can take a
sequence $(z_{k})_{k}$ of points in $\mathbb{R}^{n-1}$, a sequence
$(\lambda_{k})_{k}$ of positive real numbers and a sequence $(t_{k})_{k}$ of
nonnegative numbers such that

$\int_{B^{+}(0,t_{k})}\left|\nabla
u_{k}\left[z_{k},\lambda_{k}\right]\right|^{2}=\sup_{(z,t)\in\mathbb{R}^{n}_{+}}\int_{B^{+}(z,t)}\left|\nabla
u_{k}\left[z_{k},\lambda_{k}\right]\right|^{2}=\epsilon$ (5.5)

for any $k$ and some $\epsilon$ sufficiently small that will be determined
later.

If $t_{k}\rightarrow\infty$ as $k\to\infty$, we may assume
$t_{k}\nearrow\infty$ and set
$v_{k}(x,t)=u_{k}\left[z_{k},\lambda_{k}\right](x,t+t_{k})$ and
$\Omega_{k}=\mathbb{R}^{n-1}\times(-t_{k},\infty)$. Then $v_{k}\subset
H^{1}(\Omega_{k})$ and $\Omega_{k}\rightarrow\mathbb{R}^{n}$. Since
$u_{k}\left[z_{k},\lambda_{k}\right]$ is uniformly bounded, there exist a
subsequence and a function $v\in H^{1}(\mathbb{R}^{n})$ such that

$v_{k}\rightharpoonup v\quad\text{in}\;H^{1}(\Omega_{m})\quad\text{for any
fixed }m,$

and

$v_{k}\rightarrow v\quad\text{in}\;L^{2}_{\mathrm{loc}}(\mathbb{R}^{n}).$

From (5.2) we also know that

$\left\|\Delta
v_{k}+|v_{k}|^{p-1}v_{k}\right\|_{H^{-1}(\Omega_{k})}\rightarrow
0\quad\text{as}\;k\rightarrow\infty.$ (5.6)

Hence $v$ is a global harmonic function in $\mathbb{R}^{n}$. But $v\in
H^{1}(\mathbb{R}^{n})$ and so $v$ must be zero function. Now take a cut-off
function $\eta\in C^{\infty}_{c}(\mathbb{R}^{n})$ such that $0\leq\eta\leq 1$,
$\eta=1$ in $B(0,0)$ and $\eta=0$ outside $B_{2}(0,0)$. Using (5.5) and (5.6)
and Hölder inequality we can obtain

$\displaystyle o(1)$ $\displaystyle=\left\langle v_{k}\eta,\Delta
v_{k}+|v_{k}|^{p-1}v_{k}\right\rangle$ (5.7)
$\displaystyle=\int_{B_{2}(0,0)}\nabla(v_{k}\eta)\cdot\nabla v_{k}$
$\displaystyle=\int_{B_{2}(0,0)}\eta|\nabla
v_{k}|^{2}+\int_{B_{2}(0,0)}v_{k}\nabla\eta\cdot\nabla v_{k}$
$\displaystyle\geq\epsilon+o(1),$

where $o(1)\rightarrow 0$ as $k\rightarrow\infty$. The computation above
yields a contradiction!

So $t_{k}$ must be uniformly bounded. Without loss of generality, we assume
$t_{k}\rightarrow\bar{t}$ for some $\bar{t}\in[0,\infty)$ and
$|t_{k}-\bar{t}|\leq\frac{1}{8}$. Choose cut-off functions $\eta_{i}\in
C^{\infty}_{c}(\mathbb{R}^{n})$ such that $0\leq\eta\leq 1$, $i=1,2$,
$\eta_{1}=1$ in $B_{2}(0,\bar{t})$, $\eta_{1}=0$ outside $B_{3}(0,\bar{t})$,
while $\eta_{2}=1$ in $B_{3}(0,\bar{t})$, $\eta_{2}=0$ outside
$B_{4}(0,\bar{t})$. Set $v_{k}(x,t)=u_{k}\left[z_{k},\lambda_{k}\right](x,t)$.
There exists a function $v\in H^{1}(\mathbb{R}^{n}_{+})$ such that

$v_{k}\rightharpoonup v\quad\text{in}\;H^{1}(\mathbb{R}^{n}_{+}),$
$v_{k}\rightarrow v\quad\text{in}\;L^{2}_{\mathrm{loc}}(\mathbb{R}^{n}_{+}),$

and

$v_{k}\rightarrow
v\quad\text{in}\;L^{q}_{\mathrm{loc}}(\partial\mathbb{R}^{n}_{+})\;\text{for
any }1\leq q<p+1.$

From (5.2) we know that

$\left\|\Delta v_{k}+|v_{k}|^{p-1}v_{k}\right\|_{H^{-1}}\rightarrow
0\quad\text{as}\;k\rightarrow\infty.$ (5.8)

Using $(v_{k}-v)\eta_{1}^{2}$ as a test function, we then obtain

$\displaystyle o(1)$ $\displaystyle=\left\langle(v_{k}-v)\eta_{1}^{2},\Delta
v_{k}+|v_{k}|^{p-1}v_{k}\right\rangle$ (5.9)
$\displaystyle=\int_{B^{+}_{3}(0,\bar{t})}\nabla\left((v_{k}-v)\eta_{1}^{2}\right)\cdot\nabla
v_{k}-\int_{\partial_{b}B^{+}_{3}(0,\bar{t})}v_{k}|v_{k}|^{p-1}(v_{k}-v)\eta_{1}^{2}$
$\displaystyle=\int_{B^{+}_{3}(0,\bar{t})}\left|\nabla((v_{k}-v)\eta_{1})\right|^{2}+o(1)-\int_{\partial_{b}B^{+}_{3}(0,\bar{t})}|v_{k}-v|^{p+1}\eta_{1}^{2}+o(1)$
$\displaystyle=\int_{B^{+}_{3}(0,\bar{t})}\left|\nabla((v_{k}-v)\eta_{1})\right|^{2}-\int_{\partial_{b}B^{+}_{3}(0,\bar{t})}((v_{k}-v)\eta_{1})^{2}|(v_{k}-v)\eta_{2}|^{p-1}+o(1)$
$\displaystyle\geq\int_{B^{+}_{3}(0,\bar{t})}\left|\nabla((v_{k}-v)\eta_{1})\right|^{2}+o(1)$
$\displaystyle\quad-\left(\int_{\partial_{b}B^{+}_{3}(0,\bar{t})}|(v_{k}-v)\eta_{1}|^{p+1}\right)^{\frac{2}{p+1}}\left(\int_{\partial_{b}B^{+}_{4}(0,\bar{t})}|(v_{k}-v)\eta_{2}|^{p+1}\right)^{\frac{p-1}{p+1}}$
$\displaystyle\geq\int_{B^{+}_{3}(0,\bar{t})}\left|\nabla((v_{k}-v)\eta_{1})\right|^{2}\left(1-S_{\mathrm{E}}(n)^{-\frac{p+1}{2}}\left(\int_{B^{+}_{4}(0,\bar{t})}\left|\nabla((v_{k}-v)\eta_{2})\right|^{2}\right)^{p-1}\right)+o(1),$

where $o(1)\rightarrow 0$ as $k\rightarrow\infty$. Note that

$\displaystyle\int_{B^{+}_{4}(0,\bar{t})}\left|\nabla((v_{k}-v)\eta_{2})\right|^{2}$
$\displaystyle=\int_{B^{+}_{4}(0,\bar{t})}\left|\nabla(v_{k}-v)\right|^{2}\eta_{2}^{2}+o(1)$
(5.10) $\displaystyle=\int_{B^{+}_{4}(0,\bar{t})}(|\nabla v_{k}|^{2}-|\nabla
v|^{2})\eta_{2}^{2}+o(1)$ $\displaystyle\leq\int_{B^{+}_{4}(0,\bar{t})}|\nabla
v_{k}|^{2}+o(1)$ $\displaystyle\leq C_{0}\epsilon+o(1).$

Here $C_{0}$ is a number such that $B_{4}(0,0)$ can be covered by $C_{0}$ half
unit balls. If we choose
$\epsilon<\min\left\\{\frac{L}{2},\frac{S_{\mathrm{E}}(n)^{\frac{p+1}{2(p-1)}}}{C_{0}}\right\\}$,
then (5.9) and (5.10) indicate that

$\int_{B^{+}_{3}(0,\bar{t})}\left|\nabla((v_{k}-v)\eta_{1})\right|^{2}\rightarrow
0\quad as\;k\rightarrow\infty.$

Hence $v_{k}\rightarrow v$ strongly in $H^{1}(B_{2}^{+}(0,\bar{t}))$. Since

$\int_{B^{+}_{2}(0,\bar{t})}|\nabla v_{k}|^{2}\geq\int_{B^{+}(0,t_{k})}|\nabla
v_{k}|^{2}=\epsilon,$

we know that $v\not\equiv 0$.

To conclude the proof, first note that from (5.8) and the weak convergence,
$v$ is indeed a nontrivial solution of (5.1). Moreover we have

$\left\|\nabla v_{k}\right\|_{L^{2}}^{2}=\left\|\nabla
v\right\|_{L^{2}}^{2}+\left\|\nabla(v_{k}-v)\right\|_{L^{2}}^{2}+o(1)$ (5.11)

and

$\displaystyle\left\langle\phi,\Delta(v_{k}-v)+|v_{k}-v|^{p-1}(v_{k}-v)\right\rangle$
(5.12) $\displaystyle={}$ $\displaystyle\left\langle\phi,\Delta
v_{k}+|v_{k}|^{p-1}v_{k}\right\rangle-\left\langle\phi,\Delta
v+|v|^{p-1}v\right\rangle+o(1)$ $\displaystyle={}$ $\displaystyle
o(1)+0+o(1)=o(1)$

for any $\phi\in H^{1}(\mathbb{R}^{n}_{+})$ with $\|\nabla\phi\|_{L^{2}}\leq
1$. Here $o(1)$ is independent of the choice of $\phi$ and $o(1)\rightarrow 0$
as $k\rightarrow\infty$. Now we can choose a sequence $(z_{1}^{(k)})_{k}$ of
points in $\mathbb{R}^{n-1}$ and a sequence $(\lambda_{1}^{(k)})_{k}$ of
positive numbers such that $u_{k}-v[z_{1}^{(k)},\lambda_{1}^{(k)}]$ is a
dilation of $v_{k}-v$. Define $w_{1}=v$, then the relations (5.11) and (5.12)
reduce to

$\left\|\nabla u_{k}\right\|_{L^{2}}^{2}=\left\|\nabla
w_{1}\right\|_{L^{2}}^{2}+\left\|\nabla\left(u_{k}-w_{1}\left[z_{1}^{(k)},\lambda_{1}^{(k)}\right]\right)\right\|_{L^{2}}^{2}+o(1)$

and

$\left\|\Delta\left(u_{k}-w_{1}\left[z_{1}^{(k)},\lambda_{1}^{(k)}\right]\right)+\left|u_{k}-w_{1}\left[z_{1}^{(k)},\lambda_{1}^{(k)}\right]\right|^{p-1}\left(u_{k}-w_{1}\left[z_{1}^{(k)},\lambda_{1}^{(k)}\right]\right)\right\|_{H^{-1}}\rightarrow
0$

as $k\rightarrow 0$. If
$\left\|\nabla\left(u_{k}-w_{1}\left[z_{1}^{(k)},\lambda_{1}^{(k)}\right]\right)\right\|_{L^{2}}\rightarrow
0$, then the proof is end.

If not, viewing $u_{k}-w_{1}\left[z_{1}^{(k)},\lambda_{1}^{(k)}\right]$ as new
$u_{k}$ and using the above arguments repeatedly, we may find a sequence of
nontrivial solutions $(w_{i})_{i}$, some suitable dilations
$(\lambda_{i}^{(k)})_{k,i}$ and translations $(z_{i}^{(k)})_{k,i}$ such that

$\left\|\nabla u_{k}\right\|_{L^{2}}^{2}=\sum_{i=1}\left\|\nabla
w_{i}[z_{i}^{(k)},\lambda_{i}^{(k)}]\right\|_{L^{2}}^{2}+\left\|\nabla\left(u_{k}-\sum_{i=1}w_{i}\left[z_{1}^{(k)},\lambda_{1}^{(k)}\right]\right)\right\|_{L^{2}}^{2}+o(1).$

Note that, for any nontrivial solution $v$ of (5.1), by the Sobolev inequality
we have

$\|\nabla v\|_{L^{2}}^{2}=\|v\|_{L^{2^{\dagger}}}^{2^{\dagger}}\leq
S_{\mathrm{E}}(n)^{-\frac{2^{\dagger}}{2}}\|\nabla v\|_{L^{2}}^{2^{\dagger}}.$

Hence we have the energy estimate

$\|\nabla v\|_{L^{2}}\geq S_{\mathrm{E}}(n)^{\frac{2^{\dagger}}{2(p-1)}}.$

Since $u_{k}$ are uniformly bounded in $H^{1}$, there must exists a finite
number $\nu$ such that

$\left\|\nabla\left(u_{k}-\sum_{i=1}^{\nu}w_{i}\left[z_{i}^{(k)},\lambda_{i}^{(k)}\right]\right)\right\|_{L^{2}}=o(1),$

and the proof is completed. ∎

To transition from the general case to the nonnegative case, we need the
following Brézis-Lieb type lemma, which has already been used by Mercuri and
Willem in [47] to obtain similar results for the p-Laplacian.

######  Lemma 5.2.

Let $n\geq 3$ and $(u_{k})_{k\in\mathbb{N}}\subset H^{1}(\mathbb{R}_{+}^{n})$,
assume that

  1. $(a)$

$\sup_{k}\left\|\nabla u_{k}\right\|_{L^{2}}\leq\infty$,

  2. $(b)$

$u_{k}\rightharpoonup u$ in $H^{1}(\mathbb{R}_{+}^{n})$,

  3. $(c)$

${\lVert(u_{k})_{-}\rVert}_{L^{2^{\dagger}}(\mathbb{R}^{n-1})}\rightarrow 0$
as $k\rightarrow\infty$.

Then $u\geq 0$ in $\mathbb{R}^{n-1}$ and

$\left\|(u_{k}-u)_{-}\right\|_{L^{2^{\dagger}}(\mathbb{R}^{n-1})}\rightarrow
0\quad as\;k\rightarrow\infty.$

###### Proof.

By the Sobolev compact embedding inequalities, we know that $u_{k}\rightarrow
u$ a.e. in $\mathbb{R}^{n-1}$. After passing to a subsequence, we may assume
that there exists a function $g\in L^{2^{\dagger}}$ such that

$|(u_{k})_{-}|\leq g$

for any $k$. Then we have

$|(u_{k}-u)_{-}|\leq u^{+}+g.$

By Lebesgue dominated convergence theorem, we immediately get that
$\left\|(u_{k}-u)_{-}\right\|_{L^{2^{\dagger}}}$ goes to 0 and $u\geq 0$ on
$\mathbb{R}^{n-1}$. Since the subsequence is arbitrary, the proof is complete.
∎

######  Theorem 5.3.

Let $n\geq 3$ and $\nu\geq 1$ be positive numbers. Let
$(u_{k})_{k\in\mathbb{N}}\subset H^{1}(\mathbb{R}_{+}^{n})$ be a sequence of
functions such that
$(\nu-\frac{1}{2})S_{\mathrm{E}}(n)^{n-1}\leq\int_{\mathbb{R}_{+}^{n}}|\nabla
u_{k}|^{2}\leq(\nu+\frac{1}{2})S_{\mathrm{E}}(n)^{n-1}$, and assume that

$\left\|\Delta u_{k}+u_{k}^{p}\right\|_{H^{-1}}\rightarrow 0\quad
as\;k\rightarrow\infty,$ (5.13)
$\left\|(u_{k})_{-}\right\|_{L^{2^{\dagger}}(\mathbb{R}^{n-1})}\rightarrow
0\quad as\;k\rightarrow\infty.$ (5.14)

Then there exist a subsequence of $(u_{k})_{k}$, which we still denote by
$(u_{k})_{k}$, a sequence $(z_{1}^{(k)},...,z_{\nu}^{(k)})$ of $\nu$-tuples of
points in $\mathbb{R}^{n-1}$ and a sequence
$(\lambda_{1}^{(k)},...,\lambda_{\nu}^{k})$ of $\nu$-tuples of positive real
numbers such that

$\left\|\nabla\left(u_{k}-\sum_{i=1}^{\nu}U[z_{i}^{(k)},\lambda_{i}^{(k)}]\right)\right\|_{L^{2}}\rightarrow
0\quad as\;k\rightarrow\infty,$ (5.15) $\left\|\nabla
u_{k}\right\|_{L^{2}}^{2}-\sum_{i=1}^{\nu}\left\|\nabla
U[z_{i}^{(k)},\lambda_{i}^{(k)}]\right\|_{L^{2}}^{2}\rightarrow 0\quad
as\;k\rightarrow\infty,$ (5.16)

and

$\min\left(\frac{\lambda_{i}}{\lambda_{j}},\frac{\lambda_{j}}{\lambda_{i}},\frac{1}{\lambda_{i}\lambda_{j}|z_{i}-z_{j}|^{2}}\right)\rightarrow
0\quad as\;k\rightarrow\infty.$ (5.17)

######  Remark 5.4.

We do not assume that $u_{k}$ is nonnegative in the statement, because
nonnegativity is not a property preserved under the repeating argument in the
proof of Theorem 5.1. In other words, we cannot ensure that
$u_{k}-w_{1}\left[z_{1}^{(k)},\lambda_{1}^{(k)}\right]$ is nonnegative!

###### Proof of Theorem 5.3.

By the proof of Theorem 5.1, there exist a number $\nu_{0}\in\mathbb{N}$, a
sequence $(z_{1}^{(k)},...,z_{\nu_{0}}^{(k)})$ of $\nu_{0}$-tuples of points
in $\mathbb{R}^{n-1}$, a sequence
$(\lambda_{1}^{(k)},...,\lambda_{\nu_{0}}^{k})$ of $\nu_{0}$-tuples of
positive numbers and a sequence $(w_{i})_{1\leq i\leq\nu_{0}}$ of nontrivial
solutions of (5.1) satisfying (5.3) and (5.4). Using Lemma (5.2) and the Hopf
maximum principle, it is easy to see that $w_{i},1\leq i\leq\nu_{0}$ are
nonnegative solutions of (5.1). Due to classification results, we may assume
$w_{i}=U$. Note that $\|\nabla U\|_{L^{2}}^{2}=S_{\mathrm{E}}(n)^{n-1}$. (5.4)
indicates that $\nu_{0}=\nu$. Hence (5.15) and (5.16) hold. (5.17) easily
follows from (5.15), (5.16) and (2.1). ∎

######  Remark 5.5.

In the above theorem, the condition (5.14) can be replaced by

$\limsup_{k\rightarrow\infty}\int_{\mathbb{R}_{+}^{n}}|\nabla(u_{k})_{+}|^{2}>0$

when we restrict to the case $\nu=1$. This argument is a direct consequence of
Theorem 5.1 and the following energy gap inequality:

${\lVert W\rVert}_{H^{1}(\mathbb{R}_{+}^{n})}^{2}\geq
2S_{\mathrm{E}}(n)^{n-1}$

for any sign-changing solution $W$ of (5.1). To prove this inequality, we
choose two test functions $W_{+}\coloneqq\max\\{W,0\\}$ and
$W_{-}\coloneqq\min\\{W,0\\}$. Using equation (5.1) and the Escobar trace
inequality, we have

${\lVert W_{+}\rVert}_{H^{1}(\mathbb{R}_{+}^{n})}^{2}={\lVert
W_{+}\rVert}_{L^{2^{\dagger}}(\mathbb{R}^{n-1})}^{2^{\dagger}}\leq
S_{\mathrm{E}}(n)^{-\frac{2^{\dagger}}{2}}{\lVert
W_{+}\rVert}_{H^{1}(\mathbb{R}_{+}^{n})}^{2^{\dagger}},$ ${\lVert
W_{-}\rVert}_{H^{1}(\mathbb{R}_{+}^{n})}^{2}={\lVert
W_{-}\rVert}_{L^{2^{\dagger}}(\mathbb{R}^{n-1})}^{2^{\dagger}}\leq
S_{\mathrm{E}}(n)^{-\frac{2^{\dagger}}{2}}{\lVert
W_{-}\rVert}_{H^{1}(\mathbb{R}_{+}^{n})}^{2^{\dagger}}.$

Hence ${\lVert W_{+}\rVert}_{H^{1}}^{2},{\lVert W_{-}\rVert}_{H^{1}}^{2}\geq
S_{\mathrm{E}}(n)^{n-1}$. Since ${\lVert W\rVert}_{H^{1}}^{2}={\lVert
W_{+}\rVert}_{H^{1}}^{2}+{\lVert W_{-}\rVert}_{H^{1}}^{2}$, we get the desired
inequality.

##  6 Quantitative profile decomposition

In this section, we follow the idea of Figalli and Glaudo in [29] to derive a
quantitative profile decomposition for (5.1), which is the Euler-Lagrange
equation related to the Escobar trace inequality:

${\left\lVert\nabla\varphi\right\rVert}_{L^{2}(\mathbb{R}^{n}_{+})}^{2}-S_{\mathrm{E}}(n){\lVert\varphi\rVert}_{L^{\frac{2(n-1)}{n-2}}(\partial\mathbb{R}^{n}_{+})}^{2}\geq
0,\quad\forall\,\varphi\in H^{1}(\mathbb{R}^{n}_{+}).$ (6.1)

###### Proof of Theorem 1.2.

Let $\sigma=\sum_{i=1}^{\nu}\alpha_{i}U[z_{i},\lambda_{i}]$ be the linear
combination of Escobar bubbles that is closest to $u$ in the $H^{1}$-norm,
that is

${\lVert\nabla
u-\nabla\sigma\rVert}_{L^{2}(\mathbb{R}^{n}_{+})}=\min_{\begin{subarray}{c}\tilde{\alpha}_{1},\dots,\tilde{\alpha}_{\nu}\in\mathbb{R}\\\
\tilde{z}_{1},\dots,\tilde{z}_{\nu}\in\mathbb{R}^{n-1}\\\
\tilde{\lambda}_{1},\dots,\tilde{\lambda}_{\nu}\end{subarray}}{\left\lVert\nabla
u-\nabla\left(\sum_{i=1}^{\nu}\tilde{\alpha}_{i}U[\tilde{z}_{i},\tilde{\lambda}_{i}]\right)\right\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}.$

Let $\rho\coloneqq u-\sigma$, and denote $U_{i}\coloneqq
U[z_{i},\lambda_{i}]$. From (1.12), it follows that
${\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}\leq\delta$. Furthermore,
the family $(\alpha_{i},U_{i})_{1\leq i\leq\nu}$ is
$\delta^{\prime}$-interacting for some $\delta^{\prime}$ that goes to zero as
$\delta$ goes to $0$. We can assume $(\alpha_{i})_{1\leq i\leq\nu}$ are
positive.

Since $\sigma$ minimizes the $H^{1}$-distance from $u$, $\rho$ is
$H^{1}$-orthogonal to the manifold composed of linear combinations of these
$\nu$ Escobar bubbles. That is, for any $1\leq i\leq\nu$, the following
orthogonal conditions hold:

$\displaystyle\int_{\mathbb{R}^{n}_{+}}\nabla\rho\cdot\nabla\phi=0\ \text{for
$\phi=U_{i},\,\partial_{\lambda}U_{i}$ and $\partial_{z_{j}}U_{i}$, $1\leq
j\leq n-1$.}$ (6.2)

Using the fact that each $U_{i}$ satisfies the equation (1.11), the above
conditions are equivalent to

$\displaystyle\int_{\partial\mathbb{R}_{+}^{n}}\rho\cdot U_{i}^{p-1}\phi=0\
\text{for $\phi=U_{i},\,\partial_{\lambda}U_{i}$ and $\partial_{z_{j}}U_{i}$,
$1\leq j\leq n-1$.}$ (6.3)

To estimate ${\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}$, applying
orthogonal condition (6.2), we obtain

$\begin{split}\int_{\mathbb{R}_{+}^{n}}|\nabla\rho|^{2}={}&\int_{\mathbb{R}_{+}^{n}}\nabla\rho\cdot\nabla
u-\int_{\partial\mathbb{R}_{+}^{n}}\rho{\lvert
u\rvert}^{p-1}u+\int_{\partial\mathbb{R}_{+}^{n}}\rho{\lvert
u\rvert}^{p-1}u\\\ \leq{}&{\lVert\Delta u+{\lvert
u\rvert}^{p-1}u\rVert}_{H^{-1}}{\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}^{n}_{+})}+\int_{\partial\mathbb{R}_{+}^{n}}\rho{\lvert
u\rvert}^{p-1}u.\end{split}$ (6.4)

To control the second term, we use the elementary estimates

$\displaystyle{\left\lvert{\lvert a+b\rvert}^{p-1}(a+b)-{\lvert
a\rvert}^{p-1}a-p{\lvert a\rvert}^{p-1}b\right\rvert}\leq{}$
$\displaystyle\begin{cases}C_{n}\big{(}{\lvert a\rvert}^{p-2}{\lvert
b\rvert}^{2}+{\lvert b\rvert}^{p}\big{)}&n=3,\\\ C_{n}{\lvert
b\rvert}^{p}&n\geq 4,\end{cases}$ (6.5)
$\displaystyle{\left\lvert\left(\sum_{i=1}^{\nu}a_{i}\right)\left|\sum_{i=1}^{\nu}a_{i}\right|^{p-1}-\sum_{i=1}^{\nu}{\lvert
a_{i}\rvert}^{p-1}a_{i}\right\rvert}\lesssim{}$ $\displaystyle\sum_{1\leq
i\not=j\leq\nu}{\lvert a_{i}\rvert}^{p-1}{\lvert a_{j}\rvert},$ (6.6)

that hold for any $a,b\in\mathbb{R}$ and for any
$a_{1},\dots,a_{\nu}\in\mathbb{R}$. Then

$\displaystyle{\left\lvert{\lvert
u\rvert}^{p-1}u-\sum_{i=1}^{\nu}\alpha_{i}^{p}U_{i}^{p}\right\rvert}\leq{}$
$\displaystyle{\left\lvert(\rho+\sigma){\lvert\rho+\sigma\rvert}^{p-1}-\sigma{\lvert\sigma\rvert}^{p-1}\right\rvert}+{\left\lvert\sigma{\lvert\sigma\rvert}^{p-1}-\sum_{i=1}^{\nu}\alpha_{i}^{p}U_{i}^{p}\right\rvert}$
$\displaystyle\leq{}$ $\displaystyle
p{\lvert\sigma\rvert}^{p-1}{\lvert\rho\rvert}+C_{n,\nu}\left({\lvert\sigma\rvert}^{p-2}\rho^{2}+{\lvert\rho\rvert}^{p}+\sum_{1\leq
i\neq j\leq\nu}U_{i}^{p-1}U_{j}\right).$

Combining (6.3) we obtain

$\displaystyle\int_{\partial\mathbb{R}_{+}^{n}}{\lvert
u\rvert}^{p-1}u\rho\leq{}$ $\displaystyle
p\int_{\partial\mathbb{R}_{+}^{n}}\sigma^{p-1}\rho^{2}+C_{n,\nu}\bigg{(}\chi_{\\{n=3\\}}\int_{\partial\mathbb{R}_{+}^{n}}{\lvert\sigma\rvert}^{p-2}{\lvert\rho\rvert}^{3}+{\lvert\rho\rvert}^{p+1}\bigg{)}$
$\displaystyle+C_{n,\nu}\sum_{1\leq i\neq
j\leq\nu}\int_{\partial\mathbb{R}_{+}^{n}}{\lvert\rho\rvert}U_{i}^{p-1}U_{j},$

where $\chi_{\\{n=3\\}}$ means that terms only appear when $n=3$. By Hölder
inequality and the Escobar trace inequality (6.1),

$\displaystyle\chi_{\\{n=3\\}}\int_{\partial\mathbb{R}_{+}^{n}}{\lvert\rho\rvert}^{3}{\lvert\sigma\rvert}^{p-2}\leq{\lVert\rho\rVert}_{L^{p+1}(\partial\mathbb{R}_{+}^{n})}^{3}\cdot{\lVert\sigma\rVert}_{L^{\frac{p+1}{p-2}}(\partial\mathbb{R}_{+}^{n})}\lesssim{\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}^{3},$
$\displaystyle\int_{\partial\mathbb{R}_{+}^{n}}\rho^{p+1}={\lVert\rho\rVert}_{L^{p+1}(\partial\mathbb{R}_{+}^{n})}^{p+1}\lesssim\|\nabla\rho\|_{L^{2}(\mathbb{R}_{+}^{n})}^{p+1},$
$\displaystyle\int_{\partial\mathbb{R}_{+}^{n}}{\lvert\rho\rvert}U_{i}^{p-1}U_{j}\leq{\lVert\rho\rVert}_{L^{p+1}(\partial\mathbb{R}_{+}^{n})}{\lVert
U_{i}^{p-1}U_{j}\rVert}_{L^{\frac{p+1}{p}}(\partial\mathbb{R}_{+}^{n})}\lesssim{\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}{\lVert
U_{i}^{p-1}U_{j}\rVert}_{L^{\frac{p+1}{p}}(\partial\mathbb{R}_{+}^{n})}.$

Thanks to Lemma 2.1, if $n=3$, then for any $i\neq j$ it holds

$\|U_{i}^{p-1}U_{j}\|_{L^{\frac{p+1}{p}}(\partial\mathbb{R}_{+}^{n})}\approx_{n}\mu_{ij}^{1/2}\approx_{n}\int_{\partial\mathbb{R}_{+}^{n}}U_{i}^{p}U_{j}.$
(6.7)

Hence,

$\displaystyle\int_{\partial\mathbb{R}_{+}^{n}}{\lvert
u\rvert}^{p-1}u\rho\leq{}$ $\displaystyle
p\int_{\partial\mathbb{R}_{+}^{n}}\sigma^{p-1}\rho^{2}+C_{n,\nu}\left({\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}^{3}+{\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}^{p+1}\right)$
(6.8) $\displaystyle+C_{n,\nu}\sum_{1\leq i\neq
j\leq\nu}{\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}\int_{\partial\mathbb{R}_{+}^{n}}U_{i}^{p}U_{j}..$

In the next chapter we will show

  * •

For $n\geq 3$, if $\delta^{\prime}$ is small enough, then

$\displaystyle\int_{\partial\mathbb{R}_{+}^{n}}\sigma^{p-1}\rho^{2}\leq\frac{c(n,\nu)}{p}{\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}^{2}$
(6.9)

for some $0<c(n,\nu)<1$.

  * •

For $n\geq 3$ and a given $\tilde{\epsilon}>0$, we have

$\displaystyle\int_{\partial\mathbb{R}_{+}^{n}}U_{i}^{p}U_{j}\leq\tilde{\epsilon}\|\nabla\rho\|_{L^{2}(\mathbb{R}_{+}^{n})}+C{\lVert\Delta
u+{\lvert
u\rvert}^{p-1}u\rVert}_{H^{-1}}+{\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}^{\min(2,p)}.$
(6.10)

With these estimates, we can choose $\tilde{\epsilon}$ such that
$c(n,\nu)+\nu^{2}\tilde{\epsilon}C_{n,\nu}<1$. Combining (6.9), (6.10) into
(6.8) and then into (6.4), we get

$\displaystyle\big{(}1-c(n,\nu)-\nu^{2}\tilde{\epsilon}c_{n,\nu}\big{)}{\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}\lesssim{\lVert\Delta
u+{\lvert
u\rvert}^{p-1}u\rVert}_{H^{-1}}+{\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}^{2}+{\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}^{3}$

for $n=3$. Since
${\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}\leq\delta$, by choosing
$\delta$ small enough we obtain the desired estimate

${\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}\lesssim{\lVert\Delta
u+{\lvert u\rvert}^{p-1}u\rVert}_{H^{-1}}$ (6.11)

for $n=3$. The only thing left is to check that all $\alpha_{i}$ can be
replaced by $1$ and (1.13) holds as well. Thanks to (6.11) and the proof of
(6.10) as in the next subsection, both facts are true. ∎

######  Remark 6.1.

If $n>3$, the proof fails because

${\lVert
U_{i}^{p-1}U_{j}\rVert}_{L^{\frac{p+1}{p}}(\partial\mathbb{R}_{+}^{n})}\approx_{n}\begin{cases}\ln(\mu_{ij})^{\frac{n}{2(n-1)}}\mu_{ij}^{\frac{n}{4}}&n=4\\\
\mu_{ij}&n\geq
5\end{cases}\gg\int_{\partial\mathbb{R}_{+}^{n}}U_{i}^{p}U_{j}\approx_{n}\mu_{ij}^{\frac{n-2}{2}}.$

But if $\nu=1$, the approximation of $\int_{\partial\mathbb{R}_{+}^{n}}{\lvert
u\rvert}^{p-1}u\rho$ will not contain the cross term
$\int_{\partial\mathbb{R}_{+}^{n}}{\lvert\rho\rvert}U_{i}^{p-1}U_{j}$. Since
(6.9) holds for all $n\geq 3$, we can still get the desired stability result:

For $n\geq 3$, there exist a small constant $\delta=\delta(n)>0$ and a large
constant $C=C(n)>0$ such that the following statement holds. Let $u\in
H^{1}(\mathbb{R}_{+}^{n})$ be a function such that

${\lVert\nabla
u-\nabla\tilde{U}\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}\leq\delta,$

where $\tilde{U}$ is a Escobar bubble, then there exists another Escobar
bubble $U$ such that

${\lVert\nabla u-\nabla U\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}\leq
C{\lVert\Delta u+{\lvert u\rvert}^{p-1}u\rVert}_{H^{-1}},$

where $p=\frac{n}{n-2}$.

Here we give proofs for general $n\geq 3$. The first estimate (6.9) follows
directly from the spectral properties of $\rho$. We need the following two
lemmas:

######  Lemma 6.2.

For $n\geq 3$, $x_{0}\in\mathbb{R}^{n-1}$, $\lambda_{0}>0$, there exists
$\Lambda_{3}>\Lambda_{2}=\frac{n}{n-2}$ such that

$\Lambda_{3}\leq\frac{\int_{\mathbb{R}_{+}^{n}}|\nabla
W|^{2}\,dxdt}{\int_{\partial\mathbb{R}_{+}^{n}}U[x_{0},\lambda_{0}]^{\frac{2}{n-2}}W^{2}\,dx}$

for all $W$ which is orthogonal to $U[x_{0},\lambda_{0}]$,
$\partial_{\lambda}U[x_{0},\lambda_{0}]$ and
$\partial_{z_{j}}U[x_{0},\lambda_{0}]$ ($1\leq j\leq n-1$).

For a proof of this lemma, see Ho [52, Lemma 3.1].

######  Remark 6.3.

A refined spectral gap inequality is given in Theorem 7.1 in Section 7.

######  Lemma 6.4.

For any $n\geq 3$, $\nu\in\mathbb{N}$ and $\epsilon>0$, there exists
$\delta=\delta(n,\nu,\epsilon)>0$ such that if
$\\{U_{i}=U[z_{i},\lambda_{i}]\\}_{i=1}^{\nu}$ is a $\delta$-interacting
family of $\nu$ Escobar bubbles, then there exist $\nu$ Lipschitz function
$\Phi_{i}:\mathbb{R}^{n}\to[0,1]$ satisfying

  1. 1.

Almost all mass of $U_{i}^{2^{\dagger}}$ on $\partial\mathbb{R}_{+}^{n}$ is in
the region $\\{\Phi_{i}=1\\}$, that is

$\int_{\\{\Phi_{i}=1\\}\cap\partial\mathbb{R}_{+}^{n}}U_{i}^{2^{\dagger}}\geq(1-\epsilon)S_{\mathrm{E}}(n)^{n-1}$

  2. 2.

In the region $\\{\Phi_{i}>0\\}\cap\partial\mathbb{R}_{+}^{n}$, we have
$\epsilon U_{i}>U_{j}$ for any $j\neq i$.

  3. 3.

The $L^{n}$-norm of $\Phi_{i}$ is small, that is

${\lVert\nabla\Phi_{i}\rVert}_{L^{n}(\mathbb{R}_{+}^{n})}\leq\epsilon.$

  4. 4.

For any $j\neq i$ such that $\lambda_{j}\leq\lambda_{i}$, we have

$\frac{\sup_{\\{\Phi_{i}>0\\}\cap\partial\mathbb{R}_{+}^{n}}U_{j}}{\inf_{\\{\Phi_{i}>0\\}\cap\partial\mathbb{R}_{+}^{n}}U_{j}}<1+\epsilon.$

The key point is to construct the following cut-off function:

$\varphi=\varphi_{x_{0},r,R}:\mathbb{R}^{n}\to[0,1],\quad\varphi=\begin{cases}1&|x-x_{0}|\leq
r,\\\ \frac{\ln R-\ln|x-x_{0}|}{\ln R-\ln r}&r<|x-x_{0}|<R,\\\
0&R<|x-x_{0}|.\end{cases}$

For a fixed $1\leq i\leq\nu$, we may assume $U_{i}=U[0,1]$ and take $\Phi_{i}$
as the following form:

$\Phi_{i}\coloneqq\varphi_{0,\epsilon R,R}\prod_{j\in
J}(1-\varphi_{(z_{j},0),R_{j},\epsilon^{-1}R_{j}}),$

where $J=\\{1\leq j\leq\nu:\lambda_{j}>1\text{ and }|z_{j}|<2R\\}$. Then, we
can choose suitable $R,\epsilon,R_{j}$ to ensure $\Phi_{i}$ satisfies all
conditions. Note that $\varphi_{(z,0),r,R}(x^{\prime},0)$ is exactly the cut-
off function in $\mathbb{R}^{n-1}=\partial\mathbb{R}_{+}^{n}$ with similar
expression, so the computation are similar to that of [29, Lemma 3.9] and we
skip the proof.

With these two lemmas, we can now prove (6.9):

######  Proposition 6.5.

Let $n\geq 3$ and $\nu\in\mathbb{N}$. There exists a constant
$\delta=\delta(n,\nu)>0$ such that if
$\sigma=\sum_{i=1}^{\nu}\alpha_{i}U[z_{i},\lambda_{i}]$ is a linear
combination of Escobar bubbles and $\rho\in H^{1}(\mathbb{R}_{+}^{n})$
satisfies (6.3) with $U_{i}=U[z_{i},\lambda_{i}]$, then

$\int_{\partial\mathbb{R}_{+}^{n}}\sigma^{p-1}\rho^{2}\leq\frac{c}{p}\int_{\mathbb{R}_{+}^{n}}|\nabla\rho|^{2},$

where $p=\frac{n}{n-2}$ and $c=c(n,\nu)<1$.

###### Proof.

Let $\Phi_{1},\dots,\Phi_{\nu}$ be the cut-off functions built in Lemma 6.4
for some $\epsilon=\epsilon(\delta)$. Thanks to Lemma 6.4-(2), it holds

$\displaystyle\int_{\partial\mathbb{R}_{+}^{n}}\sigma^{p-1}\rho^{2}\leq{}$
$\displaystyle\int_{\\{\sum\Phi^{2}_{i}\geq
1\\}\cap\partial\mathbb{R}_{+}^{n}}\left(\sum_{i=1}^{\nu}\Phi_{i}^{2}\right)\sigma^{p-1}\rho^{2}+\int_{\\{\sum\Phi^{2}_{i}<1\\}\cap\partial\mathbb{R}_{+}^{n}}\sigma^{p-1}\rho^{2}$
$\displaystyle\leq{}$
$\displaystyle(1+C\epsilon^{p-1})\sum_{i=1}^{\nu}\int_{\partial\mathbb{R}_{+}^{n}}\Phi_{i}^{2}\rho^{2}U_{i}^{p-1}+\int_{\\{\sum\Phi^{2}_{i}<1\\}\cap\partial\mathbb{R}_{+}^{n}}\sigma^{p-1}\rho^{2}.$

By Lemma 6.4-(1), using the Hölder and Escobar trace inequality we obtain

$\int_{\\{\sum\Phi^{2}_{i}<1\\}\cap\partial\mathbb{R}_{+}^{n}}\sigma^{p-1}\rho^{2}\leq\left(\int_{\\{\sum\Phi^{2}_{i}<1\\}\cap\partial\mathbb{R}_{+}^{n}}\sigma^{p+1}\right)^{\frac{p-1}{p+1}}{\lVert\rho\rVert}_{L^{p+1}(\partial\mathbb{R}_{+}^{n})}^{2}\leq
C\epsilon^{\frac{1}{n-1}}{\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}^{2}.$

Let $\psi:\mathbb{R}_{+}^{n}\to\mathbb{R}$ be, up to scaling, one of the
functions $U_{i},\,\partial_{\lambda}U_{i},\,\partial_{z_{j}}U_{i}$, with
$\int_{\partial\mathbb{R}_{+}^{n}}\psi^{2}U_{i}^{p-1}=1$. Then by the
orthogonal conditions (6.3) one gets

$\displaystyle\left|\int_{\partial\mathbb{R}_{+}^{n}}(\rho\Phi_{i})\psi
U_{i}^{p-1}\right|={}$
$\displaystyle\left|\int_{\partial\mathbb{R}_{+}^{n}}\rho\psi
U_{i}^{p-1}(1-\Phi_{i})\right|\leq\left|\int_{\\{\Phi_{i}<1\\}\cap\partial\mathbb{R}_{+}^{n}}\rho\psi
U_{i}^{p-1}\right|$ $\displaystyle\leq{}$
$\displaystyle{\lVert\rho\rVert}_{L^{p+1}}\left(\int_{\partial\mathbb{R}_{+}^{n}}\psi^{2}U_{i}^{p-1}\right)^{1/2}\left(\int_{\\{\Phi_{i}<1\\}\cap\partial\mathbb{R}_{+}^{n}}U_{i}^{p+1}\right)^{\frac{p-1}{p+1}}$
$\displaystyle\leq{}$ $\displaystyle
C\epsilon^{\frac{1}{n-1}}{\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})},$

Which means $\rho\Phi_{i}$ is almost orthogonal to $\psi$. Hence, by Lemma
6.2,

$\int_{\partial\mathbb{R}_{+}^{n}}(\rho\Phi_{i})^{2}U_{i}^{p-1}\leq\frac{1}{\Lambda_{3}}\int_{\mathbb{R}_{+}^{n}}|\nabla(\rho\Phi_{i})|^{2}+C\epsilon^{\frac{2}{n-1}}{\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}^{2}.$

Note that

$\int_{\mathbb{R}_{+}^{n}}|\nabla(\rho\Phi_{i})|^{2}=\int_{\mathbb{R}_{+}^{n}}|\nabla\rho|^{2}\Phi_{i}^{2}+\int_{\mathbb{R}_{+}^{n}}\rho^{2}|\nabla\Phi_{i}|^{2}+2\int_{\mathbb{R}_{+}^{n}}\rho\Phi_{i}\nabla\rho\cdot\nabla\Phi_{i}.$

Using the Hölder and Sobolev inequality (to use Sobolev inequality in the
whole space, we need to extend $\rho\in H^{1}(\mathbb{R}_{+}^{n})$ to
$H_{0}^{1}(\mathbb{R}^{n})$ by $\rho(x,t)\coloneqq\rho(x,|t|)$), we have

$\displaystyle\int_{\mathbb{R}_{+}^{n}}\rho^{2}|\nabla\Phi_{i}|^{2}\leq{}$
$\displaystyle{\lVert\rho\rVert}_{L^{2^{*}}(\mathbb{R}_{+}^{n})}^{2}{\lVert\nabla\Phi_{i}\rVert}_{L^{n}(\mathbb{R}_{+}^{n})}^{2}\lesssim\epsilon^{2}{\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}^{2},$
$\displaystyle\int_{\mathbb{R}_{+}^{n}}\rho\Phi_{i}\nabla\rho\cdot\nabla\Phi_{i}\leq{}$
$\displaystyle{\lVert\rho\rVert}_{L^{2^{*}}(\mathbb{R}_{+}^{n})}{\lVert\Phi_{i}\rVert}_{L^{\infty}(\mathbb{R}_{+}^{n})}{\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}{\lVert\nabla\Phi_{i}\rVert}_{L^{n}(\mathbb{R}_{+}^{n})}\lesssim\epsilon{\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}^{2}.$

As a consequence of Lemma 6.4-(2), $\\{\Phi_{i}\\}_{i=1}^{\nu}$ have disjoint
supports, and so

$\sum_{i=1}^{\nu}\int_{\mathbb{R}_{+}^{n}}|\nabla\rho|^{2}\Phi_{i}^{2}\leq\int_{\mathbb{R}_{+}^{n}}|\nabla\rho|^{2}.$

Combining all these estimates, we finally get

$\displaystyle\int_{\partial\mathbb{R}_{n}^{n}}\sigma^{p-1}\rho^{2}\leq\frac{1+o(1)}{\Lambda_{3}}\int_{\mathbb{R}_{+}^{n}}|\nabla\rho|^{2}+o(1){\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}^{2}=\left(\frac{1}{\Lambda_{3}}+o(1)\right){\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}^{2},$

where $o(1)$ denotes some small quantities going to zero as $\epsilon\to 0$.
Since $\Lambda_{3}>p$, we get the desired result. ∎

To prove the second estimate (6.10), we show the following stronger
proposition:

######  Proposition 6.6.

Let $n\geq 3$ and $\nu\in\mathbb{N}$. For any $\hat{\epsilon}>0$, there exists
$\delta=\delta(n,\nu,\hat{\epsilon})>0$ such that the following statement
holds. Let $u=\rho+\sum_{i=1}^{\nu}\alpha_{i}U_{i}$, where
$(\alpha_{i},U_{i})_{1\leq i\leq\nu}$ is $\delta$-interacting, and $\rho$
satisfies the orthogonal conditions (6.3) and
${\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}\leq 1$. Then for any
$1\leq i\leq\nu$,

$|\alpha_{i}-1|\lesssim\hat{\epsilon}{\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}+{\lVert\Delta
u+{\lvert
u\rvert}^{p-1}u\rVert}_{H^{-1}}+{\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}^{\min(2,p)},$
(6.12)

where $p=\frac{n}{n-2}$. And for any pair of indices $i\neq j$,

$\int_{\partial\mathbb{R}_{+}^{n}}U_{i}^{p}U_{j}\lesssim\hat{\epsilon}{\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}+{\lVert\Delta
u+{\lvert
u\rvert}^{p-1}u\rVert}_{H^{-1}}+{\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}^{\min(2,p)}.$
(6.13)

###### Proof.

Let $U_{i}=U[z_{i},\lambda_{i}]$ for $1\leq i\leq\nu$, and we can assume
$\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{\nu}$. Let
$\\{\Phi_{i}\\}_{i=1}^{\nu}$ be the cut-off functions built in Lemma 6.4 for
some fixed $\epsilon>0$ depending on $\delta$. We prove the statement by
induction on the index $i$. Fix $1\leq i\leq\nu$ and assume the statement
holds for any $1\leq j<\nu$, denote $U=U_{i},\,\alpha=\alpha_{i}$,
$V=\sum_{j\neq i}\alpha_{j}U_{j}$ and $\Phi=\Phi_{i}$. Without loss of
generality, we may assume $U=U[0,1]$.

First we have the following identity on the boundary
$\partial\mathbb{R}_{+}^{n}$:

$\displaystyle(\alpha-\alpha^{p})U^{p}-p(\alpha U)^{p-1}V$ $\displaystyle={}$
$\displaystyle-\frac{\partial u}{\partial t}-|u|^{p-1}u-\sum_{j\neq
i}\alpha_{j}U_{j}^{p}+\frac{\partial\rho}{\partial t}+p(\alpha
U)^{p-1}\rho+\left[|\sigma+\rho|^{p-1}(\sigma+\rho)-\sigma^{p}-p\sigma^{p-1}\rho\right]$
$\displaystyle+\left[(\alpha U+V)^{p}-(\alpha U)^{p}-p(\alpha
U)^{p-1}V\right]+\left[p\sigma^{p-1}\rho-p(\alpha U)^{p-1}\rho\right].$

By Lemma 6.4-(2), in the region $\\{\Phi>0\\}$, we have

$\displaystyle\sum_{j\neq i}\alpha_{j}U_{j}^{p}=o(U^{p-1}V),$
$\displaystyle\Big{|}|\sigma+\rho|^{p-1}(\sigma+\rho)-\sigma^{p}-p\sigma^{p-1}\rho\Big{|}\lesssim{\lvert\rho\rvert}^{p}+\chi_{n=3}U^{p-2}{\lvert\rho\rvert}^{2},$
$\displaystyle\Big{|}(\alpha U+V)^{p}-(\alpha U)^{p}-p(\alpha
U)^{p-1}V\Big{|}\lesssim{\lvert
V\rvert}^{p}+\chi_{n=3}U^{p-2}V^{2}=o(U^{p-1}V),$
$\displaystyle\Big{|}p\sigma^{p-1}\rho-p(\alpha
U)^{p-1}\rho\Big{|}=o(U^{p-1}{\lvert\rho\rvert}).$

Here we denote $o(E)$ for any expression that goes to zero when divided by $E$
if $\delta\to 0$, and $\chi_{n=3}$ means terms that only appears when $n=3$.
Applying these estimates into the identity we get

$\begin{split}&\left|(\alpha-\alpha^{p})U^{p}-(p\alpha^{p-1}+o(1))U^{p-1}V-\frac{\partial\rho}{\partial
t}+\left(\frac{\partial u}{\partial t}+|u|^{p-1}u\right)-p(\alpha
U)^{p-1}\rho\right|\\\
\lesssim{}&{\lvert\rho\rvert}^{p}+o(U^{p-1}{\lvert\rho\rvert})+\chi_{n=3}U^{p-2}\rho^{2}.\end{split}$
(6.14)

Let $\xi$ be either $U$ or $\partial_{\lambda}U$, then
$\int_{\partial\mathbb{R}_{+}^{n}}U^{p-1}\xi\rho=0$ by the orthogonal
conditions of $\rho$. Testing (6.14) by $\xi\Phi$, we get

$\begin{split}&\left|\int_{\partial\mathbb{R}_{+}^{n}}\Big{[}(\alpha-\alpha^{p})U^{p}-(p\alpha^{p-1}+o(1))U^{p-1}V\Big{]}\xi\Phi\right|\\\
\lesssim{}&\left|-\int_{\partial\mathbb{R}_{+}^{n}}\frac{\partial\rho}{\partial
t}\xi\Phi+\int_{\partial\mathbb{R}_{+}^{n}}\left(\frac{\partial u}{\partial
t}+|u|^{p-1}u\right)\xi\Phi\right|+\left|\int_{\partial\mathbb{R}_{+}^{n}}U^{p-1}\xi\rho\Phi\right|\\\
&+\int_{\partial\mathbb{R}_{+}^{n}}{\lvert\rho\rvert}^{p}{\lvert\xi\rvert}\Phi+o\left(\int_{\partial\mathbb{R}_{+}^{n}}U^{p-1}{\lvert\xi\rvert}{\lvert\rho\rvert}\Phi\right)+\chi_{n=3}\int_{\partial\mathbb{R}_{+}^{n}}U^{p-2}\rho^{2}{\lvert\xi\rvert}\Phi.\end{split}$
(6.15)

We now bound each term in the right-hand side of (6.15). Noticing that

$\begin{split}-\int_{\partial\mathbb{R}_{+}^{n}}\frac{\partial\rho}{\partial
t}\xi\Phi={}&\int_{\mathbb{R}_{+}^{n}}\nabla\rho\cdot\nabla(\xi\Phi)+\xi\Phi\Delta\rho=\int_{\mathbb{R}_{+}^{n}}\nabla\rho\cdot\nabla(\xi(\Phi-1))+\xi\Phi\Delta
u.\end{split}$ (6.16)
$\begin{split}\left|\int_{\partial\mathbb{R}_{+}^{n}}-\frac{\partial\rho}{\partial
t}\xi\Phi+\int_{\partial\mathbb{R}_{+}^{n}}\left(\frac{\partial u}{\partial
t}+|u|^{p-1}u\right)\xi\Phi\right|\leq{}&{\lVert\Delta u+{\lvert
u\rvert}^{p-1}u\rVert}_{H^{-1}}{\lVert\nabla(\xi\Phi)\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}\\\
&+{\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}{\lVert\nabla(\xi(\Phi-1))\rVert}_{L^{2}(\mathbb{R}_{+}^{n})},\\\
\left|\int_{\partial\mathbb{R}_{+}^{n}}U^{p-1}\xi\rho\Phi\right|=\left|\int_{\partial\mathbb{R}_{+}^{n}}U^{p-1}\xi\rho(\Phi-1)\right|\lesssim{}&{\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}\left(\int_{\\{\Phi<1\\}\cap\partial\mathbb{R}_{+}^{n}}(U^{p-1}{\lvert\xi\rvert})^{\frac{p+1}{p}}\right)^{\frac{p}{p+1}},\\\
\int_{\partial\mathbb{R}_{+}^{n}}{\lvert\rho\rvert}^{p}{\lvert\xi\rvert}\Phi\leq\int_{\partial\mathbb{R}_{+}^{n}}{\lvert\rho\rvert}^{p}{\lvert\xi\rvert}\lesssim{}&{\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}^{p}{\lVert\xi\rVert}_{L^{p+1}(\partial\mathbb{R}_{+}^{n})},\\\
\int_{\partial\mathbb{R}_{+}^{n}}U^{p-1}{\lvert\xi\rvert}{\lvert\rho\rvert}\Phi\leq\int_{\partial\mathbb{R}_{+}^{n}}U^{p-1}{\lvert\xi\rvert}{\lvert\rho\rvert}\lesssim{}&{\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}{\lVert
U^{p-1}\xi\rVert}_{L^{\frac{p+1}{p}}(\partial\mathbb{R}_{+}^{n})},\\\
\chi_{n=3}\int_{\partial\mathbb{R}_{+}^{n}}U^{p-2}\rho^{2}{\lvert\xi\rvert}\Phi\leq\int_{\partial\mathbb{R}_{+}^{n}}U^{p-2}\rho^{2}{\lvert\xi\rvert}\lesssim{}&{\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}^{2}{\lVert
U^{p-2}\xi\rVert}_{L^{\frac{p+1}{p-1}}(\partial\mathbb{R}_{+}^{n})}.\end{split}$
(6.17)

Again, thanks to Lemma 6.4-(1) and Lemma 6.4-(3), and the observation that
${\lvert\partial_{\lambda}U\rvert}\leq\frac{n-2}{2}U$, we have

$\displaystyle{\lVert\nabla(\xi(\Phi-1))\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}^{2}=\int_{\mathbb{R}_{+}^{n}}|\nabla(\xi(\Phi-1))|^{2}$
$\displaystyle={}$
$\displaystyle\int_{\mathbb{R}_{+}^{n}}|\nabla\xi|^{2}(1-\Phi)^{2}+\int_{\mathbb{R}_{+}^{n}}|\xi|^{2}|\nabla\Phi|^{2}+2(\Phi-1)\xi\nabla\xi\cdot\nabla\Phi$
$\displaystyle={}$
$\displaystyle-\int_{\mathbb{R}_{+}^{n}}\xi\,\nabla\cdot((1-\Phi)^{2}\nabla\xi)-\int_{\partial\mathbb{R}_{+}^{n}}(1-\Phi)^{2}\xi\frac{\partial\xi}{\partial
t}$
$\displaystyle+\int_{\mathbb{R}_{+}^{n}}|\xi|^{2}|\nabla\Phi|^{2}+\int_{\mathbb{R}_{+}^{n}}2(\Phi-1)\xi\nabla\xi\cdot\nabla\Phi$
$\displaystyle\lesssim{}$
$\displaystyle{\lVert\xi\rVert}_{L^{2^{*}}(\mathbb{R}_{+}^{n})}{\lVert\nabla\Phi\rVert}_{L^{n}(\mathbb{R}_{+}^{n})}{\lVert\nabla\xi\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}+\int_{\\{\Phi<1\\}\cap\partial\mathbb{R}_{+}^{n}}U^{p+1}+{\lVert\nabla\Phi\rVert}_{L^{n}(\mathbb{R}_{+}^{n})}^{2}{\lVert\xi\rVert}_{L^{2^{*}}(\mathbb{R}_{+}^{n})}^{2}$
$\displaystyle\lesssim{}$ $\displaystyle\epsilon+\epsilon^{2}=o(1).$

Thus,

$\displaystyle{\lVert\nabla(\xi(\Phi-1))\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}=o(1),\quad$
$\displaystyle{\lVert\nabla(\xi\Phi)\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}\lesssim
1,\quad\int_{\\{\Phi<1\\}\cap\partial\mathbb{R}_{+}^{n}}(U^{p-1}{\lvert\xi\rvert})^{\frac{2(n-1)}{n}}=o(1),$
(6.18)
$\displaystyle{\lVert\xi\rVert}_{L^{p+1}(\partial\mathbb{R}_{+}^{n})}\lesssim
1,$ $\displaystyle{\lVert
U^{p-1}\xi\rVert}_{L^{\frac{2(n-1)}{n}}(\partial\mathbb{R}_{+}^{n})}\lesssim
1,\quad\quad{\lVert
U^{p-2}\xi\rVert}_{L^{\frac{p+1}{p-1}}(\partial\mathbb{R}_{+}^{n})}\lesssim
1.$

Using (6.16), (6.17) and (6.18), it follows by (6.15) that

$\begin{split}&\left|\int_{\partial\mathbb{R}_{+}^{n}}\Big{[}(\alpha-\alpha^{p})U^{p}-(p\alpha^{p-1}+o(1))U^{p-1}V\Big{]}\xi\Phi\right|\\\
\lesssim{}&o(1){\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}+{\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}^{\min(2,p)}+{\lVert\Delta
u+{\lvert u\rvert}^{p-1}u\rVert}_{H^{-1}}.\end{split}$ (6.19)

Now, split $V$ into $V_{1}\coloneqq\sum_{j<i}\alpha_{j}U_{j}$ and
$V_{2}\coloneqq\sum_{j>i}\alpha_{j}U_{j}$. By induction, we may assume that
the statement holds for all $j<i$, and by
$\int_{\partial\mathbb{R}_{+}^{n}}U_{i}^{p}U_{j}=\int_{\mathbb{R}_{+}^{n}}\nabla
U_{i}\cdot\nabla U_{j}=\int_{\partial\mathbb{R}_{+}^{n}}U_{j}^{p}U_{i}$, we
get

$\int U^{p-1}V_{1}{\lvert\xi\rvert}\Phi\lesssim
o(1){\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}+{\lVert\Delta
u+{\lvert
u\rvert}^{p-1}u\rVert}_{H^{-1}}+{\lVert\nabla\rho\rVert}^{\min(2,p)}_{L^{2}(\mathbb{R}_{+}^{n})}.$
(6.20)

For $V_{2}$, thanks to Lemma 6.4-(4), $V_{2}(x)=(1+o(1))V_{2}(0)$ in the
region $\\{\Phi>0\\}\cap\partial\mathbb{R}_{+}^{n}$. To show (6.12), if
$\alpha=1$ then it’s done. If not, let
$\theta\coloneqq\frac{p\alpha^{p-1}V_{2}(0)}{\alpha-\alpha^{p}}$, then by
(6.19) and (6.20) we have

$\begin{split}&{\lvert\alpha-\alpha^{p}\rvert}\left|\int_{\partial\mathbb{R}_{+}^{n}}\big{(}U^{p}-(1+o(1))\theta
U^{p-1}\big{)}\xi\Phi\right|\\\
\lesssim{}&o(1){\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}+{\lVert\Delta
u+{\lvert
u\rvert}^{p-1}u\rVert}_{H^{-1}}+{\lVert\nabla\rho\rVert}^{\min(2,p)}_{L^{2}(\mathbb{R}_{+}^{n})}.\end{split}$
(6.21)

Since almost all the mass of $U$ on the boundary is in the region
$\\{\Phi=1\\}$, and $\Phi\equiv 1$ on a large ball centered at $0$, we get

$\int_{\partial\mathbb{R}_{+}^{n}}\big{(}U^{p}-(1+o(1))\theta
U^{p-1}\big{)}\xi\Phi=\int_{\partial\mathbb{R}_{+}^{n}}U^{p}\xi-\int_{\partial\mathbb{R}_{+}^{n}}\theta
U^{p-1}\xi+o(1).$ (6.22)

We can easily check that there exists $t>0$ such that for any
$\theta\in\mathbb{R}$,

$\max_{\xi\in\\{U,\partial_{\lambda}U\\}}\left(\left|\int_{\partial\mathbb{R}_{+}^{n}}U^{p}\xi-\int_{\partial\mathbb{R}_{+}^{n}}\theta
U^{p-1}\xi\right|\right)\geq t.$

Thus, choosing suitable $\xi$ in (6.21) we get (6.12). Choosing $\xi=U$ in
(6.21) implies

$\left|\int_{\partial\mathbb{R}_{+}^{n}}U^{p}V\Phi\right|\lesssim
o(1){\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}+{\lVert\Delta
u+{\lvert
u\rvert}^{p-1}u\rVert}_{H^{-1}}+{\lVert\nabla\rho\rVert}^{\min(2,p)}_{L^{2}(\mathbb{R}_{+}^{n})}.$

And in particular, since $\Phi\equiv 1$ in $B(0,1)$,

$\left|\int_{B(0,1)\cap\partial\mathbb{R}_{+}^{n}}U^{p}U_{j}\right|\lesssim
o(1){\lVert\nabla\rho\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}+{\lVert\Delta
u+{\lvert
u\rvert}^{p-1}u\rVert}_{H^{-1}}+{\lVert\nabla\rho\rVert}^{\min(2,p)}_{L^{2}(\mathbb{R}_{+}^{n})}.$

This, combining with Lemma 2.3, shows (6.13) for all $j>i$. Since the case
$j<i$ has been shown in (6.20), we get that (6.13) holds for all $j\neq i$,
and this concludes the proof by induction. ∎

As a direct consequence of Theorem 5.3 and Theorem 1.2, we have the following
stability result for nonnegative functions.

######  Theorem 6.7.

For $n\geq 3,\nu=1$ or $n=3,\nu\geq 2$, set $p=\frac{n}{n-2}$, there exists a
constant $C=C(n,\nu)$ such that, for any nonnegative function $u\in
H^{1}(\mathbb{R}^{n}_{+})$ with

$(\nu-\frac{1}{2})S_{\mathrm{E}}(n)^{n-1}\leq\int|\nabla
u|^{2}\leq(\nu+\frac{1}{2})S_{\mathrm{E}}(n)^{n-1},$

then there exist $\nu$ Escobar bubbles $U_{1},U_{2},\dots,U_{\nu}$ such that

${\left\lVert\nabla u-\sum_{i=1}^{\nu}\nabla
U_{i}\right\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}\leq C{\lVert\Delta
u+u^{p}\rVert}_{H^{-1}}.$

Furthermore, for any $i\not=j$, the interaction between the bubbles can be
estimated as

$\int_{\mathbb{R}^{n}}U_{i}^{p}U_{j}\leq C{\lVert\Delta
u+u^{p}\rVert}_{H^{-1}}.$

###### Proof.

Based on the qualitative result Theorem 5.1, there exists $\epsilon>0$ such
that, if ${\lVert\Delta u+u^{p}\rVert}_{H^{-1}}<\epsilon$, then there exist
$\nu$ Escobar bubbles $U_{1},U_{2},\dots,U_{\nu}$ such that

${\left\lVert\nabla u-\sum_{i=1}^{\nu}\nabla
U_{i}\right\rVert}_{L^{2}(\mathbb{R}_{+}^{n})}\leq\delta.$

This is exactly the hypothesis of Theorem 1.2. When ${\lVert\Delta
u+u^{p}\rVert}_{H^{-1}}\geq\epsilon$, Our results follows directly. ∎

In the end of this section, we show that linear decay is the sharpest in our
cases.

######  Theorem 6.8.

For any $n\geq 3,\nu\geq 1$ and $0<\delta<1$, there exist a nonnegative
function $u\in H^{1}(\mathbb{R}_{+}^{n})$, a constant $C$ depending only on
$n,\nu$ and a $\delta$-interacting family $U_{1},...,U_{\nu}$ of Escobar
bubbles such that

$\left\|\nabla\left(u-\sum_{i=1}^{\nu}U_{i}\right)\right\|_{L^{2}}\leq
2\inf_{W_{1},...,W_{\nu}}\left\|\nabla\left(u-\sum_{i=1}^{\nu}W_{i}\right)\right\|_{L^{2}}\leq\delta$
(6.23)

and

$\left\|\Delta u+u^{p}\right\|_{H^{-1}}\leq
C\left\|\nabla\left(u-\sum_{i=1}^{\nu}U_{i}\right)\right\|_{L^{2}}.$ (6.24)

###### Proof.

Fix $\nu$ distinct directions $\theta_{1},...,\theta_{\nu}$ in
$\mathbb{S}^{n-2}$. For any $0<\epsilon\ll 1\ll R<\infty$, we set
$u_{\epsilon,R}(x,t)=\sum_{i=1}^{\nu}U_{R,i}(x,t)+\rho_{\epsilon}(x,t)$, here
$U_{R,i}=U[R\theta_{i},1],\rho_{\epsilon}(x,t)=\left(1-\frac{\sqrt{|x|^{2}+|t-1|^{2}}}{\epsilon}\right)_{+}$.
It is easy to see that
$\|\nabla\rho_{\epsilon}\|_{L^{2}}=C_{n}\epsilon^{\frac{n-2}{2}}$ and
$|\nabla\rho|=\frac{1}{\epsilon}$ in $B^{n}_{\epsilon}(0,1)$. When $R$ is
sufficiently large, $\\{U_{R,i}\\}_{1\leq i\leq v}$ is clearly a
$\delta$-interacting family and $\|\nabla
U_{R,i}\|_{L^{\infty}(B^{n}_{\epsilon})}$ tend to $0$ uniformly as
$R\rightarrow\infty$.

Now fix $\epsilon=\left(\frac{\delta}{4C_{n}}\right)^{\frac{2}{n-2}}$, then
there exists $R_{\epsilon}\gg 1$ such that, when $R\geq R_{\epsilon}$, we have

$\frac{\delta}{8}\leq\inf_{W_{1},...,W_{\nu}}\left\|\nabla\left(u_{\epsilon,R}-\sum_{i=1}^{\nu}W_{i}\right)\right\|_{L^{2}}\leq\left\|\nabla\left(u_{\epsilon,R}-\sum_{i=1}^{\nu}U_{i}\right)\right\|_{L^{2}}\leq\frac{\delta}{2}$

and

$\displaystyle\left\|\Delta
u_{\epsilon,R}+u_{\epsilon,R}^{p}\right\|_{H^{-1}}$
$\displaystyle\leq\left\|\sum_{i=1}^{\nu}\left(\Delta
U_{i}+U_{i}^{p}\right)+\Delta\rho_{\epsilon}+\rho_{\epsilon}^{p}\right\|_{H^{-1}}+\delta$
$\displaystyle=\left\|\Delta\rho_{\epsilon}\right\|_{H^{-1}}+\delta=\frac{5}{4}\delta.$

Thus $u_{\epsilon,R}$ is a function satisfying the requirements. ∎

##  7 Refined estimates for one bubble case

In this section, we aim to give an explicit upper bound for the stability
constant $C_{\mathrm{CP}}(n,1)$. To achieve it, let us study the nondegeneracy
of (5.1) first.

Consider the following conformal map from the unit ball $\mathbb{B}^{n}$ to
$\mathbb{R}_{+}^{n}$ ($n\geq 3$):

$F(y^{\prime},y_{n})=\left(\frac{2y^{\prime}}{(1+y_{n})^{2}+|y^{\prime}|^{2}},\frac{1-|y|^{2}}{(1+y_{n})^{2}+|y^{\prime}|^{2}}\right),\quad
y^{\prime}=(y_{1},\dots,y_{n-1})\in\mathbb{R}^{n-1}.$ (7.1)

The inverse map is given by

$F^{-1}(x,t)=\left(\frac{2x}{(1+t)^{2}+|x|^{2}},\frac{1-t^{2}-|x|^{2}}{(1+t)^{2}+|x|^{2}}\right),\quad
x=(x_{1},\dots,x_{n-1})\in\mathbb{R}^{n-1}.$ (7.2)

For any function $\varphi\in H^{1}(\mathbb{R}_{+}^{n})$, we can define
$\mathcal{F}[\varphi]$ in $\mathbb{B}^{n}$ by

$\mathcal{F}[\varphi](y)=(U[0,1]^{-1}\varphi)\circ F(y)\quad\text{for
}y\in\mathbb{B}^{n}.$ (7.3)

It is not hard to verify following identities (for details, we refer to [27]
and [52]):

$\int_{\mathbb{R}^{n-1}}|\varphi|^{2^{\dagger}}\,dx=\left(\frac{n-2}{2}\right)^{n-1}\int_{\partial\mathbb{B}^{n}}|\mathcal{F}[\varphi]|^{2^{\dagger}}d\sigma,$
(7.4)
$\int_{\mathbb{R}_{+}^{n}}|\nabla\varphi|^{2}\,dxdt=\left(\frac{n-2}{2}\right)^{n-2}\left(\int_{\mathbb{B}^{n}}|\nabla\mathcal{F}[\varphi]|^{2}\,dy+\frac{n-2}{2}\int_{\partial\mathbb{B}^{n}}\mathcal{F}[\varphi]^{2}\,d\sigma\right),$
(7.5)
$\int_{\mathbb{R}^{n-1}}U[0,1]^{p-1}\varphi^{2}\,dx=\left(\frac{n-2}{2}\right)^{n-1}\int_{\partial\mathbb{B}^{n}}\mathcal{F}[\varphi]^{2}\,d\sigma,$
(7.6) $\Delta\varphi=0\quad\text{in
}\mathbb{R}^{n}_{+}\iff\Delta\mathcal{F}[\varphi]=0\quad\text{in
}\mathbb{B}^{n},$ (7.7)
$\mathcal{F}[U[0,1]]=1,\quad\mathcal{F}[\partial_{\lambda}U[0,1]]=-\frac{n-2}{2}y_{n},\quad\mathcal{F}[\partial_{z_{i}}U[0,1]]=-\frac{n-2}{2}y_{i},\quad
1\leq i\leq n-1.$ (7.8)

In the following we always equip $H^{1}(\mathbb{B}^{n})$ with the norm

${\lVert
u\rVert}_{H^{1}(\mathbb{B}^{n})}^{2}\coloneqq\left(\frac{n-2}{2}\right)^{n-2}\left(\int_{\mathbb{B}^{n}}|\nabla
u|^{2}\,dy+\frac{n-2}{2}\int_{\partial\mathbb{B}^{n}}u^{2}\,d\sigma\right).$
(7.9)

From Poincaré inequality, this norm is equivalent to the standard Sobolev norm
on $\mathbb{B}^{n}$. Under this norm, $H^{1}(\mathbb{B}^{n})$ is isometric to
$H^{1}(\mathbb{R}_{+}^{n})$ via map $\mathcal{F}$.

It is well known (see [3] for example) that the quotient

$\frac{\int_{\mathbb{B}^{n}}|\nabla
u|^{2}\,dy}{\int_{\partial\mathbb{B}^{n}}u^{2}\,d\sigma}$

is related to the Steklov eigenvalue problem on $\mathbb{B}^{n}$:

$\begin{cases}\Delta u=0&\text{in }\mathbb{B}^{n}\\\ \frac{\partial
u}{\partial\vec{n}}=\mu u&\text{on }\partial\mathbb{B}^{n},\end{cases}$ (7.10)

where $\vec{n}$ denotes the outward unit normal vector. The Steklov
eigenspaces $E_{k}$ are the restrictions of the spaces $H_{k}^{n}$ of
homogeneous harmonic polynomials of degree k in $\mathbb{R}^{n}$. The
corresponding eigenvalue $\mu_{k}=k$ has multiplicity

$\dim\,E_{k}=C_{n+k-1}^{n-1}-C_{n+k-3}^{n-1}.$

Note that $E_{0}$ is spanned by constant function $1=\mathcal{F}[U[0,1]]$ and
$E_{1}$ is spanned by $n$ coordinate functions:

$\left\\{y_{i}=-\frac{2}{n-2}\mathcal{F}[\partial_{z_{i}}U[0,1]],\ 1\leq i\leq
n-1;\ y_{n}=-\frac{2}{n-2}\mathcal{F}[\partial_{\lambda}U[0,1]]\right\\}.$

The eigenvalue problem provides an orthogonal decomposition of
$H^{1}(\mathbb{B}^{n})$:

$H^{1}(\mathbb{B}^{n})=\bigoplus_{i=-1}^{\infty}E_{i},$ (7.11)

where $E_{-1}\coloneqq\left\\{u\in H^{1}(\mathbb{B}^{n})|\;u=0\ \text{on
}\partial\mathbb{B}^{n}\right\\}=H^{1}_{0}(\mathbb{B}^{n})$.

Using the isometry $\mathcal{F}$ and identities above, we can transform the
equation (7.10) to

$\begin{cases}\Delta u=0&\text{in }\mathbb{R}^{n}_{+}\\\ \frac{\partial
u}{\partial t}=-\left(1+\frac{2\mu}{n-2}\right)U[0,1]^{p-1}u&\text{on
}\mathbb{R}^{n-1}.\end{cases}$ (7.12)

Set

$\kappa_{k}=1+\frac{2\mu_{k}}{n-2}=1+\frac{2k}{n-2}$ (7.13)

and $R_{i}=\mathcal{F}^{-1}(E_{i})$, then the decomposition (7.11) yields

$H^{1}(\mathbb{R}_{+}^{n})=\bigoplus_{i=-1}^{\infty}R_{i},$ (7.14)

where $R_{-1}=\left\\{u\in H^{1}(\mathbb{R}_{+}^{n})|\;u=0\ \text{on
}\mathbb{R}^{n-1}\right\\}$ and $R_{k}$ is the eigenspace of (7.12) with
eigenvalue $\kappa_{k}$ for $k\geq 0$. It is easy to see that

$R_{0}=\text{Span}\left\\{U[0,1]\right\\},\quad
R_{1}=\text{Span}\left\\{\partial_{z_{i}}U[0,1],1\leq i\leq
n-1;\partial_{\lambda}U[0,1]\right\\}.$

Let $T_{U}=R_{0}\oplus R_{1}$ be the tangent space at $U[0,1]$ in the manifold
$\mathcal{M}_{\mathrm{E}}$ of Escobar bubbles. Thanks to arguments above, we
can establish the following nondegeneracy result:

######  Theorem 7.1 (spectral gap).

Let $\rho\in T_{U}^{\perp}$. Then

${\lVert\rho\rVert}^{2}_{H^{1}(\mathbb{R}_{+}^{n})}\geq\frac{n+2}{n-2}\int_{\mathbb{R}^{n-1}}U[0,1]^{p-1}\rho^{2}.$
(7.15)

Moreover, the equality holds if and only if $\rho\in R_{2}$, i.e.
$\mathcal{F}[\rho]$ is a homogeneous harmonic polynomial of degree $2$.

Now we can state and proof our main results.

######  Theorem 7.2 (asymptotic behavior near $\mathcal{M}_{\mathrm{E}}$).

Let $n\geq 3$. Assume $(u_{k})_{k}\subset H^{1}(\mathbb{R}_{+}^{n})$ is a
sequence of functions such that

$\frac{1}{2}S_{\mathrm{E}}(n)^{n-1}\leq\int_{\mathbb{R}_{+}^{n}}|\nabla
u_{k}|^{2}\leq\frac{3}{2}S_{\mathrm{E}}(n)^{n-1},$

and

$\left\|\Delta u_{k}+|u_{k}|^{p-1}u_{k}\right\|_{H^{-1}}\rightarrow 0\quad
as\;k\rightarrow\infty.$

Then

$\liminf_{k\rightarrow\infty}\frac{\left\|\Delta
u_{k}+|u_{k}|^{p-1}u_{k}\right\|_{H^{-1}}}{d(u_{k},\mathcal{M}_{\mathrm{E}})}\geq\frac{2}{n+2}.$
(7.16)

Moreover, equality holds if and only if the following holds: Assume after
suitable normalizations, translations and dilations,
$d(u_{k},\mathcal{M}_{\mathrm{E}})={\lVert u_{k}-U[0,1]\rVert}_{H^{1}}$. Let
$u_{k}=\sum_{i=-1}^{\infty}\rho_{k,i}$ be the unique decomposition with
respect to (7.14). Then (up to some subsequence)

${\lVert
u_{k}-U[0,1]-\rho_{k,2}\rVert}_{H^{1}}=o({\lVert\rho_{k,2}\rVert}_{H^{1}}).$
(7.17)

######  Remark 7.3.

A natural corollary of above theorem is that
$C_{\mathrm{CP}}(n,1)\leq\frac{2}{n+2}$. It seems that the proof of Theorem
1.2 may only give the inequality (7.16). The proof there cannot give such
upper bound because the identification of equality case is not available
there. One possible reason is that (6.4) causes a loss.

######  Theorem 7.4 (strict upper bound).

Let $n\geq 3$ and $C_{\mathrm{CP}}(n,1)$ denotes the optimal constant in
(1.16). Then

$C_{\mathrm{CP}}(n,1)<\frac{2}{n+2}.$ (7.18)

######  Remark 7.5.

The combination of above two theorem indicates that: Any minimizing sequence
must be away from $\mathcal{M}_{\mathrm{E}}$. However, whether minimizers
exist is still unknown.

Before the proof, let us make two observations:

$\left\|\Delta u_{k}+|u_{k}|^{p-1}u_{k}\right\|_{H^{-1}}={\lVert
u_{k}-\mathcal{P}[|u_{k}|^{p-1}u_{k}]\rVert}_{H^{1}},$
$\int_{\mathbb{R}^{n-1}}u\mathcal{P}[v]=\int_{\mathbb{R}^{n-1}}v\mathcal{P}[u]\quad\text{for
any }u,v\in L^{\frac{2(n-1)}{n}}(\mathbb{R}^{n-1}).$

Recall that the operator $\mathcal{P}$ is defined in (4.10).

For simplicity, in the proofs below, we will write $U$ to denote $U[0,1]$.

###### Proof of Theorem 7.2.

From Remark 5.5, we can assume $d(u_{k},\mathcal{M}_{\mathrm{E}})={\lVert
u_{k}-U\rVert}_{H^{1}}$ without loss of generality. Take the decomposition
$u_{k}=\sum_{i=-1}^{\infty}\rho_{k,i}$ with respect to (7.14). It is easy to
see that $\rho_{k,1}=0$ and $\rho_{k,0}=\beta_{k}U$, where
$\beta_{k}\rightarrow 1$. Set
$v_{k}=\rho_{k,-1}+\sum_{i=3}^{\infty}\rho_{k,i}$ and
$w_{k}=\rho_{k,-1}+\sum_{i=2}^{\infty}\rho_{k,i}$. Then
$u_{k}=\beta_{k}U+w_{k}$. Let us expand the dual norm:

$\displaystyle\left\|\Delta u_{k}+|u_{k}|^{p-1}u_{k}\right\|_{H^{-1}}^{2}$
$\displaystyle={\lVert
u_{k}-\mathcal{P}[|u_{k}|^{p-1}u_{k}]\rVert}_{H^{1}}^{2}$
$\displaystyle={\lVert
u_{k}\rVert}_{H^{1}}^{2}+{\lVert\mathcal{P}[|u_{k}|^{p-1}u_{k}]\rVert}_{H^{1}}^{2}+2\left\langle
u_{k},\mathcal{P}[|u_{k}|^{p-1}u_{k}]\right\rangle_{H^{1}}$
$\displaystyle={\lVert
u_{k}\rVert}_{H^{1}}^{2}-2\int_{\mathbb{R}^{n-1}}|u_{k}|^{p+1}+\int_{\mathbb{R}^{n-1}}|u_{k}|^{p-1}u_{k}\mathcal{P}[|u_{k}|^{p-1}u_{k}]\,.$

Using the orthogonal decomposition, continuity of $\mathcal{P}$, observations
above and estimate (6.5), we have:

$\displaystyle{\lVert u_{k}\rVert}_{H^{1}}^{2}=\beta_{k}^{2}{\lVert
U\rVert}_{H^{1}}^{2}+{\lVert w_{k}\rVert}_{H^{1}}^{2},$
$\displaystyle\int_{\mathbb{R}^{n-1}}|u_{k}|^{p+1}=\beta_{k}^{p+1}\int_{\mathbb{R}^{n-1}}U^{p+1}+\frac{p(p+1)}{2}\beta_{k}^{p-1}\int_{\mathbb{R}^{n-1}}U^{p-1}w_{k}^{2}+o({\lVert
w_{k}\rVert}_{H^{1}}^{2}),$
$\displaystyle\int_{\mathbb{R}^{n-1}}|u_{k}|^{p-1}u_{k}\mathcal{P}[|u_{k}|^{p-1}u_{k}]$
$\displaystyle={}$
$\displaystyle\int_{\mathbb{R}^{n-1}}\left(\beta_{k}^{p}U^{p}+p\beta_{k}^{p-1}U^{p-1}w_{k}+A\right)\mathcal{P}\left[\beta_{k}^{p}U^{p}+p\beta_{k}^{p-1}U^{p-1}w_{k}+A\right]$
$\displaystyle={}$
$\displaystyle\int_{\mathbb{R}^{n-1}}\beta_{k}^{p}U^{p}\mathcal{P}[\beta_{k}^{p}U^{p}]+\int_{\mathbb{R}^{n-1}}p\beta_{k}^{p-1}U^{p-1}w_{k}\mathcal{P}[p\beta_{k}^{p-1}U^{p-1}w_{k}]+\int_{\mathbb{R}^{n-1}}A\mathcal{P}[A]$
$\displaystyle+2\int_{\mathbb{R}^{n-1}}\beta_{k}^{p}U^{p}\mathcal{P}[p\beta_{k}^{p-1}U^{p-1}w_{k}]+2\int_{\mathbb{R}^{n-1}}p\beta_{k}^{p-1}U^{p-1}w_{k}\mathcal{P}[A]+2\int_{\mathbb{R}^{n-1}}\beta_{k}^{p}U^{p}\mathcal{P}[A]$
$\displaystyle={}$
$\displaystyle\beta_{k}^{2p}\int_{\mathbb{R}^{n-1}}U^{p+1}+p^{2}\beta_{k}^{2p-2}\int_{\mathbb{R}^{n-1}}U^{p-1}w_{k}\mathcal{P}[U^{p-1}w_{k}]+2\beta_{k}^{p}\int_{\mathbb{R}^{n-1}}UA+o({\lVert
w_{k}\rVert}_{H^{1}}^{2})$ $\displaystyle={}$
$\displaystyle\beta_{k}^{2p}\int_{\mathbb{R}^{n-1}}U^{p+1}+p^{2}\beta_{k}^{2p-2}\int_{\mathbb{R}^{n-1}}U^{p-1}w_{k}\mathcal{P}[U^{p-1}w_{k}]$
$\displaystyle+p(p-1)\beta_{k}^{2p-2}\int_{\mathbb{R}^{n-1}}U^{p-1}w_{k}^{2}+o({\lVert
w_{k}\rVert}_{H^{1}}^{2}),$

where $A=|u_{k}|^{p-1}u_{k}-\beta_{k}^{p}U^{p}-p\beta_{k}^{p-1}U^{p-1}w_{k}$.
Combining all estimates above and using $\beta_{k}=1+o(1)$, we finally get

$\displaystyle\left\|\Delta u_{k}+|u_{k}|^{p-1}u_{k}\right\|_{H^{-1}}^{2}={}$
$\displaystyle(\beta_{k}-\beta_{k}^{p})^{2}{\lVert
U\rVert}_{H^{1}}^{2}+{\lVert
w_{k}\rVert}_{H^{1}}^{2}-2p\int_{\mathbb{R}^{n-1}}U^{p-1}w_{k}^{2}$
$\displaystyle+p^{2}\int_{\mathbb{R}^{n-1}}U^{p-1}w_{k}\mathcal{P}[U^{p-1}w_{k}]+o({\lVert
w_{k}\rVert}_{H^{1}}^{2}).$

Note that

${\lVert
w_{k}\rVert}_{H^{1}}^{2}={\lVert\rho_{k,-1}\rVert}_{H^{1}}^{2}+\sum_{i=2}^{\infty}{\lVert\rho_{k,i}\rVert}_{H^{1}}^{2},$
$\int_{\mathbb{R}^{n-1}}U^{p-1}w_{k}^{2}=\sum_{i=2}^{\infty}\kappa_{i}^{-1}{\lVert\rho_{k,i}\rVert}_{H^{1}}^{2},$
$\int_{\mathbb{R}^{n-1}}U^{p-1}w_{k}\mathcal{P}[U^{p-1}w_{k}]=\sum_{i=2}^{\infty}\kappa_{i}^{-2}{\lVert\rho_{k,i}\rVert}_{H^{1}}^{2},$

where $\kappa_{k}$ is defined in (7.13). Hence we obtain

$\displaystyle\left\|\Delta u_{k}+|u_{k}|^{p-1}u_{k}\right\|_{H^{-1}}^{2}={}$
$\displaystyle(\beta_{k}-\beta_{k}^{p})^{2}{\lVert
U\rVert}_{H^{1}}^{2}+{\lVert\rho_{k,-1}\rVert}_{H^{1}}^{2}$ (7.19)
$\displaystyle+\sum_{i=2}^{\infty}(1-p\kappa_{i}^{-1})^{2}{\lVert\rho_{k,i}\rVert}_{H^{1}}^{2}+o({\lVert
w_{k}\rVert}_{H^{1}}^{2})$ $\displaystyle\geq$
$\displaystyle(\beta_{k}-\beta_{k}^{p})^{2}{\lVert
U\rVert}_{H^{1}}^{2}+{\lVert\rho_{k,-1}\rVert}_{H^{1}}^{2}$
$\displaystyle+(1-p\kappa_{2}^{-1})^{2}\sum_{i=2}^{\infty}{\lVert\rho_{k,i}\rVert}_{H^{1}}^{2}+o({\lVert
w_{k}\rVert}_{H^{1}}^{2})$ $\displaystyle\geq$
$\displaystyle(\beta_{k}-\beta_{k}^{p})^{2}{\lVert
U\rVert}_{H^{1}}^{2}+(1-p\kappa_{2}^{-1})^{2}{\lVert
w_{k}\rVert}_{H^{1}}^{2}+o({\lVert w_{k}\rVert}_{H^{1}}^{2}).$

The denominator of the quotient can be evaluated by

$d(u_{k},\mathcal{M}_{\mathrm{E}})^{2}={\lVert
u_{k}-U\rVert}_{H^{1}}^{2}=(1-\beta_{k})^{2}{\lVert
U\rVert}_{H^{1}}^{2}+{\lVert w_{k}\rVert}_{H^{1}}^{2}.$ (7.20)

Now the full quotient holds

$\frac{\left\|\Delta
u_{k}+|u_{k}|^{p-1}u_{k}\right\|_{H^{-1}}^{2}}{d(u_{k},\mathcal{M}_{\mathrm{E}})^{2}}\geq\frac{(\beta_{k}-\beta_{k}^{p})^{2}{\lVert
U\rVert}_{H^{1}}^{2}+(1-p\kappa_{2}^{-1})^{2}{\lVert
w_{k}\rVert}_{H^{1}}^{2}+o({\lVert
w_{k}\rVert}_{H^{1}}^{2})}{(1-\beta_{k})^{2}{\lVert
U\rVert}_{H^{1}}^{2}+{\lVert w_{k}\rVert}_{H^{1}}^{2}}.$ (7.21)

Since $\beta_{k}-\beta_{k}^{p}=(p-1+o(1))(1-\beta_{k})$ and
$p-1>1-p\kappa_{2}^{-1}$, (7.16) follows directly from (7.21).

Assume that the equality in (7.16) holds, then (7.19) indicates

${\lVert\rho_{k,-1}\rVert}_{H^{1}}^{2}+\sum_{i=3}^{\infty}(1-p\kappa_{i}^{-1})^{2}{\lVert\rho_{k,i}\rVert}_{H^{1}}^{2}=o((1-p\kappa_{2}^{-1})^{2}{\lVert\rho_{k,2}\rVert}_{H^{1}}^{2})$

and (7.21) implies

$|1-\beta_{k}|=o({\lVert w_{k}\rVert}_{H^{1}}^{2}).$

These are equivalent to (7.17). The proof is complete. ∎

###### Proof of Theorem 7.4.

The main idea is to test the quotient with suitable function. Let
$u=U+\varepsilon\rho$ for some $\rho\in R_{2},{\lVert\rho\rVert}_{H^{1}}=1$ to
be determined and $\varepsilon$ sufficiently small. From implicit function
theorem, we know $d(u,\mathcal{M}_{\mathrm{E}})={\lVert u-U\rVert}_{H^{1}}$.
Since $|\rho|\leq CU$ by definition, here we can expand the dual norm up to
third order:

$\displaystyle\left\|\Delta u+|u|^{p-1}u\right\|_{H^{-1}}^{2}$
$\displaystyle={\lVert
u\rVert}_{H^{1}}^{2}-2\int_{\mathbb{R}^{n-1}}|u|^{p+1}+\int_{\mathbb{R}^{n-1}}|u|^{p-1}u\mathcal{P}[|u|^{p-1}u],$
$\displaystyle{\lVert u\rVert}_{H^{1}}^{2}$ $\displaystyle={\lVert
U\rVert}_{H^{1}}^{2}+\varepsilon^{2}{\lVert\rho\rVert}_{H^{1}}^{2}={\lVert
U\rVert}_{H^{1}}^{2}+\varepsilon^{2},$
$\displaystyle\int_{\mathbb{R}^{n-1}}|u|^{p+1}=$
$\displaystyle\int_{\mathbb{R}^{n-1}}U^{p+1}+\frac{p(p+1)}{2}\varepsilon^{2}\int_{\mathbb{R}^{n-1}}U^{p-1}\rho^{2}$
$\displaystyle+\frac{p(p+1)(p-1)}{6}\varepsilon^{3}\int_{\mathbb{R}^{n-1}}U^{p-2}\rho^{3}+o(\varepsilon^{3})$
$\displaystyle=$ $\displaystyle{\lVert
U\rVert}_{H^{1}}^{2}+\frac{p(p+1)}{2}\varepsilon^{2}\kappa_{2}^{-1}+\frac{p(p+1)(p-1)}{6}\varepsilon^{3}\int_{\mathbb{R}^{n-1}}U^{p-2}\rho^{3}+o(\varepsilon^{3}),$
$\displaystyle\int_{\mathbb{R}^{n-1}}|u|^{p-1}u\mathcal{P}[|u|^{p-1}u]$
$\displaystyle={}$
$\displaystyle\int_{\mathbb{R}^{n-1}}\left(U^{p}+pU^{p-1}\varepsilon\rho+\frac{p(p-1)}{2}U^{p-2}\varepsilon^{2}\rho^{2}+B\right)\mathcal{P}\left[U^{p}+pU^{p-1}\varepsilon\rho\right]$
$\displaystyle+\int_{\mathbb{R}^{n-1}}\left(U^{p}+pU^{p-1}\varepsilon\rho+\frac{p(p-1)}{2}U^{p-2}\varepsilon^{2}\rho^{2}+B\right)\mathcal{P}\left[\frac{p(p-1)}{2}U^{p-2}\varepsilon^{2}\rho^{2}+B\right]$
$\displaystyle={}$
$\displaystyle\int_{\mathbb{R}^{n-1}}U^{p}\mathcal{P}[U^{p}]+p^{2}\varepsilon^{2}\int_{\mathbb{R}^{n-1}}U^{p-1}\rho\mathcal{P}[U^{p-1}\rho]+\frac{p^{2}(p-1)^{2}}{4}\varepsilon^{4}\int_{\mathbb{R}^{n-1}}U^{p-2}\rho^{2}\mathcal{P}[U^{p-2}\rho^{2}]$
$\displaystyle+\int_{\mathbb{R}^{n-1}}B\mathcal{P}[B]+2\int_{\mathbb{R}^{n-1}}\left(\varepsilon
pU^{p-1}\rho+\varepsilon^{2}\frac{p(p-1)}{2}U^{p-2}\rho^{2}+B\right)\mathcal{P}[U^{p}]$
$\displaystyle+p^{2}(p-1)\varepsilon^{3}\int_{\mathbb{R}^{n-1}}U^{p-2}\rho^{2}\mathcal{P}[U^{p-1}\rho]+2\int_{\mathbb{R}^{n-1}}\left(\varepsilon
pU^{p-1}\rho+\varepsilon^{2}\frac{p(p-1)}{2}U^{p-2}\rho^{2}\right)\mathcal{P}[B]$
$\displaystyle={}$
$\displaystyle\int_{\mathbb{R}^{n-1}}U^{p+1}+\left(p^{2}\kappa_{2}^{-1}+p(p-1)\right)\varepsilon^{2}\int_{\mathbb{R}^{n-1}}U^{p-1}\rho^{2}++p^{2}(p-1)\kappa_{2}^{-1}\varepsilon^{3}\int_{\mathbb{R}^{n-1}}U^{p-2}\rho^{3}$
$\displaystyle+2\int_{\mathbb{R}^{n-1}}BU+o(\varepsilon^{3})$
$\displaystyle={}$
$\displaystyle\int_{\mathbb{R}^{n-1}}U^{p+1}+\left(p^{2}\kappa_{2}^{-1}+p(p-1)\right)\varepsilon^{2}\int_{\mathbb{R}^{n-1}}U^{p-1}\rho^{2}+p^{2}(p-1)\kappa_{2}^{-1}\varepsilon^{3}\int_{\mathbb{R}^{n-1}}U^{p-2}\rho^{3}$
$\displaystyle+\frac{p(p-1)(p-2)}{3}\varepsilon^{3}\int_{\mathbb{R}^{n-1}}U^{p-2}\rho^{3}+o(\varepsilon^{3})$
$\displaystyle={}$ $\displaystyle{\lVert
U\rVert}_{H^{1}}^{2}+\left(p^{2}\kappa_{2}^{-1}+p(p-1)\right)\varepsilon^{2}\kappa_{2}^{-1}+o(\varepsilon^{3})$
$\displaystyle+\varepsilon^{3}\left(p^{2}(p-1)\kappa_{2}^{-1}+\frac{p(p-1)(p-2)}{3}\right)\int_{\mathbb{R}^{n-1}}U^{p-2}\rho^{3},$

where
$B=|u|^{p-1}u-U^{p}-pU^{p-1}\varepsilon\rho-\frac{p(p-1)}{2}U^{p-2}\varepsilon^{2}\rho^{2}$.
Combining terms above, we can compute

$\displaystyle\left\|\Delta
u+|u|^{p-1}u\right\|_{H^{-1}}^{2}=\varepsilon^{2}(1-p\kappa_{2}^{-1})^{2}+\varepsilon^{3}\frac{p(p-1)}{3}(\kappa_{2}^{-1}-1)\int_{\mathbb{R}^{n-1}}U^{p-2}\rho^{3}+o(\varepsilon^{3}).$

Note that

$d(u,\mathcal{M}_{\mathrm{E}})^{2}={\lVert
u-U\rVert}_{H^{1}}^{2}=\varepsilon^{2}.$

The full quotient is

$\frac{\left\|\Delta
u+|u|^{p-1}u\right\|_{H^{-1}}^{2}}{d(u,\mathcal{M}_{\mathrm{E}})^{2}}=(1-p\kappa_{2}^{-1})^{2}+\varepsilon\frac{p(p-1)}{3}(\kappa_{2}^{-1}-1)\int_{\mathbb{R}^{n-1}}U^{p-2}\rho^{3}+o(\varepsilon).$

Since $\kappa_{2}=\frac{n+2}{n-2}$, it suffices to take suitable $\rho$ such
that $\int_{\mathbb{R}^{n-1}}U^{p-2}\rho^{3}<0$. Let

$\rho=-\mathcal{F}^{-1}[y_{1}y_{2}+y_{2}y_{3}+y_{3}y_{1}],$

then

$\displaystyle\int_{\mathbb{R}^{n-1}}U^{p-2}\rho^{3}$
$\displaystyle=-\left(\frac{n-2}{2}\right)^{n-1}\int_{\mathbb{B}^{n}}(y_{1}y_{2}+y_{2}y_{3}+y_{3}y_{1})^{3}\,dy$
$\displaystyle=-\left(\frac{n-2}{2}\right)^{n-1}\int_{\mathbb{B}^{n}}6y_{1}^{2}y_{2}^{2}y_{3}^{2}\,dy<0.$

The proof is complete. ∎

## References

  * [1] S. Aryan, Stability of Hardy Littlewood Sobolev inequality under bubbling, Calc. Var. Partial Differential Equations., 62 (2023), no. 8, Paper No. 223, 42 pp
  * [2] T. Aubin, Problemes isopérimétriques et espaces de Sobolev, J. Differential Geom., 11 (1976), 573-598.
  * [3] G. Alexandre, P. Iosif, Spectral geometry of the Steklov problem (survey article). J. Spectr. Theory, 7 (2017), no. 2, 321¨C359.
  * [4] A. Bahri, J. M. 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] T. Bartsch, T. Weth, M. Willem, A Sobolev inequality with remainder term and critical equations on domains with topology for the polyharmonic operator, Calc. Var. Partial Differential Equations, 18 (2003), 253-268.
  * [6] W. Beckner, Functionals for multilinear fractional embedding, Acta Math. Sin. (Engl. Ser.), 31 (2015), no. 1, 1¨C28
  * [7] G. Bianchi, H. Egnell, A note on the Sobolev inequality, J. Funct. Anal., 100 (1991), 18-24.
  * [8] M. Bonforte, J. Dolbeault, B. Nazaret, N. Simonov, Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method, arXiv preprint arXiv:2007.03674, 2020.
  * [9] M. Bonforte, J. Dolbeault, B. Nazaret, N. Simonov, Stability in Gagliardo-Nirenberg inequalities-Supplementary material, arXiv preprint arXiv:2007.03419, 2020.
  * [10] H. Brézis, E. Lieb, Sobolev inequalities with remainder terms,J. Funct. Anal., 62 (1985), 73-86.
  * [11] L. Caffarelli, B. Gidas, J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.
  * [12] E. Carlen, Duality and stability for functional inequalities, Ann. Fac. Sci. Toulouse Math., 26 (2017), 319-350.
  * [13] 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.
  * [14] L. Chen, G. Lu, H. Tang, Stability of Hardy-Littlewood-Sobolev inequalities with explicit lower bounds, arXiv preprint arXiv:2301.04097, 2023.
  * [15] S. Chen, R. L. Frank, T. Weth, Remainder terms in the fractional Sobolev inequality, Indiana Univ. Math. J., 62 (2013), 1381-1397.
  * [16] A. Cianchi, N. Fusco, F. Maggi, A. Pratelli, The sharp Sobolev inequality in quantitative form, J. Eur. Math. Soc. (JEMS), 11 (2009), 1105-1139.
  * [17] M. Cicalese, G. Leonardi, A selection principle for the sharp quantitative isoperimetric inequality, Arch. Ration. Mech. Anal., 206 (2012), 617-643.
  * [18] 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., 21 (2018), 6780-6797.
  * [19] N. De Nitti, T. König, Stability with explicit constants of the critical points of the fractional Sobolev inequality and applications to fast diffusion, J. Funct. Anal., 285 (2023), no. 9, Paper No. 110093, 30 pp
  * [20] B. Deng, L. Sun, J. Wei, Sharp quantitative estimates of Struwe’s Decomposition, arXiv preprint arXiv:2103.15360, 2021.
  * [21] S. Deng, X. Tian, On the stability of Caffarelli-Kohn-Nirenberg inequality in $\mathbb{R}^{2}$, arXiv preprint arxiv:2308.04111, 2023.
  * [22] A. Do, J. Flynn, N. Lam, G. Lu, Lp-Caffarelli-Kohn-Nirenberg inequalities and their stabilities, arXiv preprint arXiv:2310.07083, 2023.
  * [23] J. Dolbeault, G. Toscani, Stability results for logarithmic Sobolev and Gagliardo-Nirenberg inequalities, Int. Math. Res. Not., 2 (2016), 473-498.
  * [24] J. Dolbeault, J. Esteban, A. Figalli, L. Frank, M. Loss, Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence, arXiv preprint arXiv:2209.08651, 2022.
  * [25] A. Einav, M. Loss, Sharp trace inequalities for fractional Laplacians, Proc. Amer. Math. Soc.,  140(2012), no.12, 4209¨C4216.
  * [26] M. Engelstein, R. Neumayer, L. Spolaor, Quantitative stability for minimizing Yamabe metrics, Trans. Amer. Math. Soc., 9 (2022), 395-414.
  * [27] J. Escobar, Sharp constant in a Sobolev trace inequality, Indiana Univ. Math. J., 37 (1988), no.3, 687¨C698.
  * [28] A. Figalli, N. Fusco, F. Maggi, V. Millot, M. Morini, Isoperimetry and stability properties of balls with respect to nonlocal energies, Comm. Math. Phys., 336 (2015), 441-507.
  * [29] A. Figalli, F. Glaudo, On the Sharp Stability of Critical Points of the Sobolev Inequality, Arch. Ration. Mech. Anal., 237 (2020), 201-258.
  * [30] A. Figalli, D. Jerison, Quantitative stability for the Brunn-Minkowski inequality, Adv. Math., 314 (2017), 1-47.
  * [31] A. Figalli, F. Maggi, A. Pratelli, Sharp stability theorems for the anisotropic Sobolev and log-Sobolev inequalities on functions of bounded variation, Adv. Math., 242 (2013), 80-101.
  * [32] A. Figalli, F. Maggi, A. Pratelli, A refined Brunn-Minkowski inequality for convex sets,Ann. Inst. H. Poincare C Anal. Non Lineaire, 26 (2009), 2511-2519.
  * [33] A. Figalli, F. Maggi, A. Pratelli, A mass transportation approach to quantitative isoperimetric inequalities, Invent. Math., 182 (2010), 167-211.
  * [34] A. Figalli, R. Neumayer, Gradient stability for the Sobolev inequality: the case $p\geq 2$, J. Eur. Math. Soc. (JEMS), 21 (2018), 319-354.
  * [35] A. Figalli, Y. Zhang, Sharp gradient stability for the Sobolev inequality, Duke Math. J., 171 (2022), 2407-2459.
  * [36] A. Figalli, Y. Zhang, Strong stability for the Wulff inequality with a crystalline norm, Comm. Pure Appl. Math., 75 (2022), 422-446.
  * [37] R. Frank, Degenerate stability of some Sobolev inequalities, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 39 (2022), 1459-1484.
  * [38] R. Frank, J. Peteranderl, Degenerate Stability of the Caffarelli-Kohn-Nirenberg Inequality along the Felli-Schneider Curve, arXiv preprinted arXiv:2308.07917, 2023.
  * [39] N. Fusco, F. Maggi, A. Pratelli, The sharp quantitative isoperimetric inequality, Ann. of Math., 168 (2008), 941-980.
  * [40] B. Gidas, W. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.
  * [41] T. König, Stability for the Sobolev inequality: existence of a minimizer, arXiv preprint arXiv:2211.14185, 2022.
  * [42] T. König, On the sharp constant in the Bianchi-Egnell stability inequality, Bull. Lond. Math. Soc., 55 (2023), no. 4, 2070¨C2075.
  * [43] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374.
  * [44] P. Lions, The concentration-compactness principle in the calculus of variations. The limit case, I. Rev. Mat. Iberoamericana, 1 (1985), 145-201.
  * [45] K. Liu, Q. Zhang, W. Zou, On the stability of critical points of the Hardy-Littlewood-Sobolev inequality, arXiv preprint arXiv:2306.15862, 2023.
  * [46] F. Maggi, Some methods for studying stability in isoperimetric type problems, Bull. Amer. Math. Soc.,  45 (2008), 367-408.
  * [47] C. Mercuri, M. Willem, A global compactness result for the p-Laplacian involving critical nonlinearities. Discrete Contin. Dyn. Syst., 28 (2010), no. 2, 469¨C493
  * [48] R. Neumayer, A strong form of the quantitative Wulff inequality,SIAM J. Math. Anal., 48 (2016), 1727-1772.
  * [49] R. Neumayer, On minimizers and critical points for anisotropic isoperimetric problems, MATRIX Annals, 2018 (2020), 293-302.
  * [50] R. Neumayer, A note on strong-form stability for the Sobolev inequality, Calc. Var. Partial Differential Equations,  59 (2020), 25.
  * [51] B. Ou, Positive harmonic functions on the upper half space satisfying a nonlinear boundary condition, Differential Integral Equations, 9(1996), no.5, 1157¨C1164.
  * [52] H. Pak Tung, A note on the Sobolev trace inequality. Proc. Amer. Math. Soc., 150(2022), no.3, 1257¨C1267.
  * [53] P. Piccione, M. Yang, S. Zhao, Quantitative stability of a nonlocal Sobolev inequality, arxiv preprint arXiv:2306.16883, 2023.
  * [54] B. Ruffini, Stability theorems for Gagliardo-Nirenberg-Sobolev inequalities: a reduction principle to the radial case, Rev. Mat. Complut. 27 (2014), 509-539.
  * [55] F. Seuffert, An extension of the Bianchi-Egnell stability estimate to Bakry, Gentil, and Ledoux’s generalization of the Sobolev inequality to continuous dimensions, J. Funct. Anal., 273 (2017), 3094-3149.
  * [56] S. Sobolev, On a theorem of functional analysis, Mat. Sb. 4 (1938) 471-497; AMS Trans. 34 (1963) 39-68.
  * [57] M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517.
  * [58] M. Struwe, Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems. Fourth edition. A Series of Modern Surveys in Mathematics Results in Mathematics and Related Areas. 3rd.
  * [59] G. Talenti. Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.
  * [60] Z. Wang, W. Michel, Caffarelli-Kohn-Nirenberg inequalities with remainder terms, J. Funct. Anal., 203 (2003), no. 2, 550¨C568.
  * [61] J. Wei, Y. Wu, On the stability of the Caffarelli-Kohn-Nirenberg inequality, Math. Ann., 384 (2022), 1509-1546.
  * [62] J. Wei, Y. Wu, Stability of the Caffarelli-Kohn-Nirenberg inequality: the existence of minimizers, arXiv preprint arXiv:2308.04667, 2023.