# Degenerate stability of critical points of the Caffarelli-Kohn-Nirenberg
inequality along the Felli-Schneider curve††thanks: This work is supported by
National Key R$\&$D Program of China (Grant 2023YFA1010001) and
NSFC(12171265). E-mail addresses: zhou-yx22@mails.tsinghua.edu.cn(Zhou), zou-
<EMAIL_ADDRESS>(Zou)

Yuxuan Zhou, Wenming Zou
Department of Mathematical Sciences, Tsinghua University, Beijing 100084,
China.

###### Abstract

In this paper, we investigate the validity of a quantitative version of
stability for the critical Hardy-Hénon equation

$H(u):=\operatorname{div}(|x|^{-2a}\nabla u)+|x|^{-pb}|u|^{p-2}u=0,\quad u\in
D_{a}^{1,2}(\mathbb{R}^{n}),$ $n\geq 2,\quad a<b<a+1,\quad
a<\frac{n-2}{2},\quad p=\frac{2n}{n-2+2(b-a)},$

which is well known as the Euler-Lagrange equation of the classical
Caffarelli-Kohn-Nirenberg inequality. Establishing quantitative stability for
this equation usually refers to finding a nonnegative function $F$ such that
the following estimate

$\inf_{\begin{subarray}{c}U_{i}\in\mathcal{M}\\\ 1\leq
i\leq\nu\end{subarray}}{\left\lVert
u-\sum_{i=1}^{\nu}U_{i}\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}\leq
C(a,b,n)F({\left\lVert H(u)\right\rVert}_{D_{a}^{-1,2}(\mathbb{R}^{n})})$

holds for any nonnegative function $u$ satisfying

$\left(\nu-\frac{1}{2}\right)S(a,b,n)^{\frac{p}{p-2}}\leq\int_{\mathbb{R}^{n}}|x|^{-2a}|\nabla
u|^{2}\mathrm{d}x\leq\left(\nu+\frac{1}{2}\right)S(a,b,n)^{\frac{p}{p-2}}.$

Here $\nu\in\mathbb{N}_{+}$ and $\mathcal{M}$ denotes the set of positive
solutions to this equation. When $(a,b)$ falls above the Felli-Schneider
curve, Wei and Wu [30] found an optimal $F$. Their proof relies heavily on the
fact that $\mathcal{M}$ is non-degenerate. When $(a,b)$ falls on the Felli-
Schneider curve, due to the absence of the non-degeneracy condition, it
becomes complicated and technical to find a suitable $F$. In this paper, we
focus on this case. When $\nu=1$, we obtain an optimal $F$. When $\nu\geq 2$
and $u$ is not too degenerate, we also derive an optimal $F$. To our
knowledge, the results in this paper provide the first instance of degenerate
stability in the critical point setting. We believe that our methods will be
useful in other works on degenerate stability. Key words: Caffarelli-Kohn-
Nirenberg inequality, quantitative stability, critical point, Felli-Schneider
curve.

2020 Mathematics Subject Classification: Primary 46E35(Secondary 26D10);
35J61; 35B50; 35B06; 35J47;

##  1 Introduction

In this paper, we are concerned with the quantitative stability for the
critical Hardy-Hénon equation

$H(u):=\operatorname{div}(|x|^{-2a}\nabla u)+|x|^{-pb}|u|^{p-2}u=0,$ (1.1)

where $u\in D_{a}^{1,2}(\mathbb{R}^{n})$ and $n,a,b,p$ satisfy the relations

$n\geq 2,\quad a<b<a+1,\quad a<\frac{n-2}{2},\quad p=\frac{2n}{n-2+2(b-a)}.$
(1.2)

The space $D_{a}^{1,2}(\mathbb{R}^{n})$ is the completion of
$C_{c}^{\infty}(\mathbb{R}^{n})$ with respect to the norm

${\left\lVert
u\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}:=\left(\int_{\mathbb{R}^{n}}|x|^{-2a}|\nabla
u|^{2}\mathrm{d}x\right)^{\frac{1}{2}}.$

It is well known that the equation (1.1) is the Euler-Lagrange equation of the
classical Caffarelli-Kohn-Nirenberg inequality (see [6])

$S(a,b,n)\left(\int_{\mathbb{R}^{n}}|x|^{-bp}|u|^{p}\mathrm{d}x\right)^{\frac{2}{p}}\leq\int_{\mathbb{R}^{n}}|x|^{-2a}|\nabla
u|^{2}\mathrm{d}x.$ (1.3)

Here $S(a,b,n)$ denotes the sharp constant.

The classification of positive solutions to the equation (1.1) has been
extensively studied. Here, we provide a brief summary; for further background
and partial results, we refer the readers to [7, 15, 16, 17, 25] and their
references. When $a\geq 0$, Chou and Chu [9] found that the set of positive
solutions is given by

$\mathcal{M}_{a,b,n}:=\left\\{U_{\lambda}(x)\;\Big{|}\;U_{\lambda}(x):=\frac{\lambda^{\sqrt{\Lambda}}(2p\Lambda)^{\frac{1}{p-2}}}{(1+|\lambda
x|^{\sqrt{\Lambda}(p-2)})^{\frac{2}{p-2}}},\lambda>0\right\\},$ (1.4)

where $\Lambda:=\left(\frac{n-2-2a}{2}\right)^{2}$. Later, Catrina and Wang
[7] showed that for general $a,b,n$, the set $\mathcal{M}_{n,a,b}$ in (1.4)
gives all positive radially symmetric solutions. They also discovered that
there exists a curve $b=f(a),a<0$ such that if $b<f(a)$, then there exists a
positive non-radially symmetric solutions to the equation (1.1). Making use of
fine spectral analysis, Felli and Schneider [22] then proved that this curve
can be taken to be

$b=b_{\text{FS}}(a):=\frac{n(n-2-2a)}{2\sqrt{(n-2-2a)^{2}+4n-4}}-\frac{n-2-2a}{2},\;a<0.$
(1.5)

The curve $b=b_{\text{FS}}(a)$ is usually called the Felli-Schneider curve. It
was conjectured for a long time that when $a<0$ and $b\geq b_{\text{FS}}(a)$,
$\mathcal{M}_{a,b,n}$ contains all positive solutions to the equation (1.1).
This was finally confirmed by Dolbeault, Esteban and Loss [15] using the
method of nonlinear flows. In conclusion, when $a\geq 0$ or $a<0,b\geq
b_{\text{FS}}(a)$, the set $\mathcal{M}_{n,a,b}$ in (1.4) gives all the
positive solutions to the equation (1.1). In the following, we will call the
functions in $\mathcal{M}_{a,b,n}$ Talenti bubbles.

The stability of the equation (1.1), or in other words, the stability of the
Caffarelli-Kohn-Nirenberg inequality (1.3) in the critical point setting, can
be understood as a natural extension of the classification results.
Specifically, it refers to whether being almost a positive solution to the
equation (1.1) implies being close to an appropriate linear combination of
Talenti bubbles. This kind of stability traces back to the celebrated global
compactness principle of Struwe [28]. In [28], Struwe established the
following qualitative stability of the classical Yamabe equation in
$\mathbb{R}^{n}\;(n\geq 3)$: for any $\varepsilon>0$ and
$\nu\in\mathbb{N}_{+}$, there exists $\delta=\delta(n,\nu,\varepsilon)>0$ such
that

$\inf_{\begin{subarray}{c}W_{i}\in\mathcal{M}_{0}\\\ 1\leq
i\leq\nu\end{subarray}}{\left\lVert
u-\sum_{i=1}^{\nu}W_{i}\right\rVert}_{D_{0}^{1,2}(\mathbb{R}^{n})}\leq\varepsilon\quad\text{if
}\,{\left\lVert\Delta
u+u^{2^{*}-1}\right\rVert}_{D_{0}^{-1,2}(\mathbb{R}^{n})}\leq\delta.$

Here $\mathcal{M}_{0}$ denotes the set of positive solutions to the Yamabe
equation and $u$ is any nonnegative function satisfying

$\left(\nu-\frac{1}{2}\right)S(0,0,n)^{\frac{n}{2}}\leq\int_{\mathbb{R}^{n}}|\nabla
u|^{2}\mathrm{d}x\leq\left(\nu+\frac{1}{2}\right)S(0,0,n)^{\frac{n}{2}}.$

Ciraolo, Figalli and Maggi [10] were the first to quantify Struwe’s work. In
the case $\nu=1$, they proved the following sharp linear estimate

$\inf_{W\in\mathcal{M}_{0}}{\left\lVert
u-W\right\rVert}_{D_{0}^{1,2}(\mathbb{R}^{n})}\leq C(n){\left\lVert\Delta
u+u^{2^{*}-1}\right\rVert}_{D_{0}^{-1,2}(\mathbb{R}^{n})}.$

This estimate was later generalized by Figalli and Glaudo [19]. Assuming
$\nu\geq 2$ and $3\leq n\leq 5$, they proved

$\inf_{\begin{subarray}{c}W_{i}\in\mathcal{M}_{0}\\\ 1\leq
i\leq\nu\end{subarray}}{\left\lVert
u-\sum_{i=1}^{\nu}W_{i}\right\rVert}_{D_{0}^{1,2}(\mathbb{R}^{n})}\leq
C(n,\nu){\left\lVert\Delta
u+u^{2^{*}-1}\right\rVert}_{D_{0}^{-1,2}(\mathbb{R}^{n})}.$

Moreover, they constructed counterexamples to show that such a linear estimate
does not hold for $n\geq 6$. The remaining case $\nu\geq 2,n\geq 6$ was
eventually solved by Deng, Sun and Wei [12]. They obtained the following sharp
estimate

$\inf_{\begin{subarray}{c}W_{i}\in\mathcal{M}_{0}\\\ 1\leq
i\leq\nu\end{subarray}}{\left\lVert
u-\sum_{i=1}^{\nu}W_{i}\right\rVert}_{D_{0}^{1,2}(\mathbb{R}^{n})}\leq
C(n,\nu)F_{0}\left({\left\lVert\Delta
u+u^{2^{*}-1}\right\rVert}_{D_{0}^{-1,2}(\mathbb{R}^{n})}\right),$

where

$F_{0}(x):=\begin{cases}x|\ln{x}|^{\frac{1}{2}}+x,&n=6;\\\
x^{\frac{n+2}{2(n-2)}},&n\geq 7.\end{cases}$

Recently, Wei and Wu [30] generalized the results in [12] to the critical
Hardy-Hénon equation (1.1). In the case $a\geq 0$ or $a<0,b>b_{\text{FS}}(a)$,
they derived the following sharp estimate

$\inf_{\begin{subarray}{c}\lambda_{i}\in\mathbb{R}_{+}\\\ 1\leq
i\leq\nu\end{subarray}}{\left\lVert
u-\sum_{i=1}^{\nu}U_{\lambda_{i}}\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}\leq
C(n,a,b,\nu)F_{1}\left({\left\lVert
H(u)\right\rVert}_{D_{a}^{-1,2}(\mathbb{R}^{n})}\right),$ (1.6)

where

$F_{1}(x):=\begin{cases}x,&p>3\;\text{or }\nu=1;\\\
x|\ln{x}|^{\frac{1}{2}}+x,&p=3\;\text{and }\nu\geq 2;\\\
x^{\frac{p-1}{2}},&2<p<3\;\text{and }\nu\geq 2\end{cases}$

and $u$ satisfies

$\left(\nu-\frac{1}{2}\right)S(a,b,n)^{\frac{p}{p-2}}\leq\int_{\mathbb{R}^{n}}|x|^{-2a}|\nabla
u|^{2}\mathrm{d}x\leq\left(\nu+\frac{1}{2}\right)S(a,b,n)^{\frac{p}{p-2}}.$

Here $D_{a}^{-1,2}(\mathbb{R}^{n})$ denotes the dual space of
$D_{a}^{1,2}(\mathbb{R}^{n})$. However, as was noticed in [13, 24], there is
an inaccuracy in [30]: due to the absence of the nondegeneracy condition, the
stability results of [30] do not hold when $(a,b)$ falls on the Felli-
Schneider curve.

We refer the readers to [1, 4, 11, 26, 27, 31, 32] for various quantitative
stability results on other Sobolev-type inequalities.

In this paper, we aim to derive appropriate quantitative stability results of
the equation (1.1) in the case $a<0,b=b_{\text{FS}}(a)$. There are two main
motivations for our work.

The first motivation comes from the fact that Wei and Wu’s result (1.6) does
not cover the Felli-Schneider curve and its proof can hardly be applied to
this situation. One of the key tools utilized in their work is the non-
degeneracy of the equation (1.1). Indeed, to the best of our knowledge, all
such type quantitative stability results to date are dependent on the non-
degeneracy. However, it is well known that (see [22, 24] for example) the
equation (1.1) is non-degenerate when $a\geq 0$ or $a<0,b>b_{\text{FS}}(a)$,
but degenerate when $a<0,b=b_{\text{FS}}(a)$. Specifically, consider the
linearized equation of (1.1) near some Talenti bubble $U_{\lambda_{0}}$:

$\displaystyle\operatorname{div}(|x|^{-2a}\nabla
v)+(p-1)|x|^{-pb}U_{\lambda_{0}}^{p-2}v=0.$ (1.7)

We use $\bar{X}_{\lambda_{0}}$ to denotes the set of solutions to (1.7).
Clearly $\bar{X}_{\lambda_{0}}$ contains the trivial elements
$t\partial_{\lambda}U_{\lambda_{0}},t\in\mathbb{R}$. Define
$X_{\lambda_{0}}=\bar{X}_{\lambda_{0}}\setminus\text{Span}\\{\partial_{\lambda}U_{\lambda_{0}}\\}$.
When $a\geq 0$ or $a<0,b>b_{\text{FS}}(a)$, $X_{\lambda_{0}}=\emptyset$ and we
say (1.1) is non-degenerate. When $a<0$ and $b=b_{\text{FS}}(a)$,
$X_{\lambda_{0}}\neq\emptyset$ and we say (1.1) is degenerate (see Lemma 2.4
for an equivalent formulation). Due to the degeneracy, it is easy to construct
examples which do not satisfy (1.6). So a natural and challenging question is:
is it possible to replace $F_{1}$ by another function $F$ such that the
estimate (1.6) vaild in the case $a<0,b=b_{\text{FS}}(a)$?

The second motivation comes from the quantitative stability of the Caffarelli-
Kohn-Nirenberg inequality (1.3) in the functional setting. The stability of
functional inequalities was initially raised by Brézis and Lieb [5]. They
asked whether the homogeneous Sobolev inequality can be reinforced. This
question was answered affirmatively by Bianchi and Egnell [3]. After the work
[3], the stability of inequalities has drawn attention and has been addressed
by lots of researchers. We do not attempt a survey of the extensive
literature, but refer the readers to [8, 14, 20, 21, 23, 24, 30, 31, 32] and
the references therein for more background. Similar to the stability in the
critical point setting, many quantitative stability results of functional
inequalities depend on the non-degeneracy condition. Recently some interesting
degenerate stability results were discovered by Engelstein, Neumayer and
Spolaor [18]. Their results were later improved by Frank [23].

Let’s go back to our setting. In the case $a\geq 0$ or
$a<0,b>b_{\text{FS}}(a)$, Wei and Wu [30] established the following sharp
quadratic estimate

$\displaystyle{\left\lVert
u\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}^{2}-S(a,b,n){\left\lVert|x|^{-b}u\right\rVert}_{L^{p}(\mathbb{R}^{n})}^{2}\geq
c(a,b,n)\inf_{s\in\mathbb{R},\lambda>0}{\left\lVert
u-sU_{\lambda}\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}^{2}.$

In the case $a<0,b=b_{\text{FS}}(a)$, due to the degeneracy, the above
estimate no longer hold. Recently, Frank and Peteranderl [24] studied this
case and found the optimal exponent is $4$:

${\left\lVert
u\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}^{4}-S(a,b,n)^{2}{\left\lVert|x|^{-b}u\right\rVert}_{L^{p}(\mathbb{R}^{n})}^{4}\geq
c(a,b,n)\inf_{s\in\mathbb{R},\lambda>0}{\left\lVert
u-sU_{\lambda}\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}^{4}.$

Moreover, if the infimum is attained by $s_{0}U_{\lambda_{0}}$, their proof
yleids

$\displaystyle{\left\lVert
u\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}^{2}-S(a,b,n){\left\lVert|x|^{-b}u\right\rVert}_{L^{p}(\mathbb{R}^{n})}^{2}\geq
c(a,b,n){\left\lVert\Pi_{X_{\lambda_{0}}}^{\perp}(u)-s_{0}U_{\lambda_{0}}\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}^{2},$

where $\Pi_{X_{\lambda_{0}}}$ is the orthogonal projection in
$D_{a}^{1,2}(\mathbb{R}^{n})$ onto $X_{\lambda_{0}}$ and
$\Pi_{X_{\lambda_{0}}}^{\perp}=1-\Pi_{X_{\lambda_{0}}}$. Since in some sense
these two types of stability are closely related, we believe that there exist
corresponding degenerate stabilities in the critical point setting.

Let us state our main results.

######  Theorem 1.1.

Assume $n\geq 2$, $a<0$, $b=b_{\mathrm{FS}}(a)$ and $p=\frac{2n}{n-2+2(b-a)}$.
Then there exists a positive constant $C=C(a,n)$ such that, for any function
$u\in D_{a}^{1,2}(\mathbb{R}^{n})$ satisfying

$\frac{1}{2}S(a,b,n)^{\frac{p}{p-2}}\leq\int_{\mathbb{R}^{n}}|x|^{-2a}|\nabla
u|^{2}\mathrm{d}x\leq\frac{3}{2}S(a,b,n)^{\frac{p}{p-2}},$ (1.8)

it holds that

$\inf_{\lambda>0}{\left\lVert
u-U_{\lambda}\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}^{3}\leq C{\left\lVert
H(u)\right\rVert}_{D_{a}^{-1,2}(\mathbb{R}^{n})}$ (1.9)

The above estimate is sharp in the sense that there exists a sequence
$\\{w_{k}\\}_{k}\subset D_{a}^{1,2}(\mathbb{R}^{n})$ satisfying (1.8) such
that $w_{k}\notin\mathcal{M}_{a,b,n}$,

$\inf_{\lambda>0}{\left\lVert
w_{k}-U_{\lambda}\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}\rightarrow 0$

and

$\limsup_{k\rightarrow+\infty}\frac{{\left\lVert
H(w_{k})\right\rVert}_{D_{a}^{-1,2}(\mathbb{R}^{n})}}{\inf\limits_{\lambda>0}{\left\lVert
w_{k}-U_{\lambda}\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}^{3}}<+\infty.$

Moreover, if the infimum in (1.9) is attained by $U_{\lambda_{0}}$, then there
exists a positive constant $C_{1}=C_{1}(a,n)$ such that

$\displaystyle{\left\lVert\Pi_{X_{\lambda_{0}}}^{\perp}(u)-U_{\lambda_{0}}\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}\leq
C_{1}{\left\lVert H(u)\right\rVert}_{D_{a}^{-1,2}(\mathbb{R}^{n})}$ (1.10)

provided

$\displaystyle{\left\lVert\Pi_{X_{\lambda_{0}}}^{\perp}(u)-U_{\lambda_{0}}\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}\geq
C_{1}F_{2}\left({\left\lVert\Pi_{X_{\lambda_{0}}}(u)\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}\right),$
(1.11)

where

$\displaystyle F_{2}(x):=\begin{cases}x^{2},&p\geq 3,\\\
x^{p-1},&p<3.\end{cases}$

######  Theorem 1.2.

Assume $n\geq 2$, $\nu\geq 2$, $a<0$, $b=b_{\mathrm{FS}}(a)$ and
$p=\frac{2n}{n-2+2(b-a)}$. Then there exists a positive constant
$C=C(a,\nu,n)$ such that, for any nonnegative function $u\in
D_{a,s}^{1,2}(\mathbb{R}^{n})$ satisfying

$\left(\nu-\frac{1}{2}\right)S(a,b,n)^{\frac{p}{p-2}}\leq\int_{\mathbb{R}^{n}}|x|^{-2a}|\nabla
u|^{2}\mathrm{d}x\leq\left(\nu+\frac{1}{2}\right)S(a,b,n)^{\frac{p}{p-2}},$
(1.12)

if the infimum

$\displaystyle\inf_{\begin{subarray}{c}s_{i}\in\mathbb{R}_{+}\\\ 1\leq
i\leq\nu\end{subarray}}{\left\lVert
u-\sum_{i=1}^{\nu}U_{s_{i}}\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}$

is attained by $U_{\lambda_{i}},1\leq i\leq\nu$, then

${\left\lVert\Pi_{X}^{\perp}(u)-\sum_{i=1}^{\nu}U_{\lambda_{i}}\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}\leq
CF_{1}\left({\left\lVert
H(u)\right\rVert}_{D_{a}^{-1,2}(\mathbb{R}^{n})}\right)$ (1.13)

provided

$\displaystyle{\left\lVert\Pi_{X}^{\perp}(u)-\sum_{i=1}^{\nu}U_{\lambda_{i}}\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}\geq
CF_{1}\left({\left\lVert\Pi_{X}(u)\right\rVert}^{2}_{D_{a}^{1,2}(\mathbb{R}^{n})}\right).$
(1.14)

Here $X:=\oplus_{i=1}^{\nu}X_{\lambda_{i}}$.

######  Remark 1.3.

In the single bubble case (Theorem 1.1), we do not need the nonnegativity of
the function $u$. This is due to the fact that for each sign-changing solution
$u_{0}$ to the equation (1.1), we have the following energy estimate

$\int_{\mathbb{R}^{n}}|x|^{-2a}|\nabla u_{0}|^{2}\mathrm{d}x\geq
2S(a,b,n)^{\frac{p}{p-2}}.$

This estimate can be proved by testing the equation (1.1) with $(u_{0})^{+}$
and $(u_{0})^{-}$, and applying the inequality (1.3).

######  Remark 1.4.

In the case $\nu=1$, Theorem 1.1 completes the stability analysis of the
Caffarelli-Kohn-Nirenberg inequality within the critical point framework
initiated by Wei and Wu [30] (see (1.6)). It remains unclear whether the cubic
estimate (1.9) holds for $\nu\geq 2$.

######  Remark 1.5.

Unlike quantitative stabilities in the functional setting, we do not always
have (1.10) or (1.13). Indeed, in Section 3 we show that, for any $0<\delta\ll
1$, there exists a positive function $u$ (see (3.35)) such that the infimum in
(1.9) is attained by $U_{1}$ and

$\displaystyle{\left\lVert\Pi_{X_{1}}(u)\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}\approx\delta,\quad{\left\lVert\Pi_{X_{1}}^{\perp}(u)-U_{1}\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}\approx\delta^{2},\quad{\left\lVert
H(u)\right\rVert}_{D_{a}^{-1,2}(\mathbb{R}^{n})}\approx\delta^{3}.$

Thus in Theorem 1.1 and Theorem 1.2, we pose the restrictions (1.11) and
(1.14) which indicate that the function $u$ is not too degenerate. From the
example above, we see these restrictions are optimal when $\nu=1,p\geq 3$ or
$\nu\geq 2,p>3$. We wonder whether these restrictions are sharp in other
cases.

The following result is a direct consequence of the above theorems. To derive
it, one just need to note that, for any $u\in D_{a}^{1,2}(\mathbb{R}^{n})$
centrally symmetric and $\lambda\in\mathbb{R}$, we have
$\Pi_{X_{\lambda}}(u)=0$.

######  Corollary 1.6.

Assume $n\geq 2$, $\nu\geq 1$, $a<0$, $b=b_{\mathrm{FS}}(a)$ and
$p=\frac{2n}{n-2+2(b-a)}$. Then there exists a positive constant
$C=C(a,\nu,n)$ such that, for any nonnegative centrally symmetric function
$u\in D_{a}^{1,2}(\mathbb{R}^{n})$ satisfying (1.12), it holds that

$\displaystyle\inf_{\begin{subarray}{c}s_{i}\in\mathbb{R}_{+}\\\ 1\leq
i\leq\nu\end{subarray}}{\left\lVert
u-\sum_{i=1}^{\nu}U_{s_{i}}\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}\leq
CF_{1}\left({\left\lVert
H(u)\right\rVert}_{D_{a}^{-1,2}(\mathbb{R}^{n})}\right).$

Moreover, the above estimate is sharp.

Thanks to the global compactness principle (see [30, Proposition 3.2] for
example), to establish Theorem 1.1 and Theorem 1.2, we only need to prove
their local versions.

######  Theorem 1.7.

Assume $n\geq 2$, $a<0$, $b=b_{\mathrm{FS}}(a)$ and $p=\frac{2n}{n-2+2(b-a)}$.
Then there exists two positive constants $\delta=\delta(a,n)$ and $C=C(a,n)$
such that, for any function $u\in D_{a}^{1,2}(\mathbb{R}^{n})$ satisfying

$\displaystyle{\left\lVert
u-U_{1}\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}\leq\delta,$

the estimate (1.9) holds and is optimal. Moreover, if the infimum is attained
by $U_{\lambda_{0}}$ and (1.11) holds, then (1.10) holds.

######  Theorem 1.8.

Assume $n\geq 2$, $\nu\geq 2$, $a<0$, $b=b_{\mathrm{FS}}(a)$ and
$p=\frac{2n}{n-2+2(b-a)}$. Then there exists two positive constants
$\delta=\delta(a,\nu,n)$ and $C=C(a,\nu,n)$ such that, for any function $u\in
D_{a}^{1,2}(\mathbb{R}^{n})$ satisfying

$\displaystyle\inf_{\begin{subarray}{c}s_{i}\in\mathbb{R}_{+}\\\ 1\leq
i\leq\nu\end{subarray}}{\left\lVert
u-\sum_{i=1}^{\nu}U_{s_{i}}\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}\leq\delta$

and the infimum is attained by $U_{\lambda_{i}},1\leq i\leq\nu$ with

$\displaystyle\min\left\\{\frac{\lambda_{i}}{\lambda_{j}},\frac{\lambda_{j}}{\lambda_{i}}\right\\}\leq\delta,\quad\forall\;i\neq
j,$

the estimate (1.13) holds provided (1.14) holds.

######  Remark 1.9.

Note that in these local versions, we do not need $u$ to be positive. Instead,
we assume $u$ is close to a sum of Talenti bubbles.

######  Remark 1.10.

By further exploiting the proof of Theorem 1.7, we can obtain the following
asymptotic estimate: let $\\{\varphi_{k}\\}_{k}$ be a sequence in
$D_{a}^{1,2}(\mathbb{R}^{n})$ such that $\varphi_{k}\notin\mathcal{M}_{a,b,n}$
and
$\inf\limits_{\lambda>0}{\left\lVert\varphi_{k}-U_{\lambda}\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}\rightarrow
0$, then

$\liminf_{k\rightarrow+\infty}\frac{{\left\lVert
H(\varphi_{k})\right\rVert}_{D_{a}^{-1,2}(\mathbb{R}^{n})}}{\inf\limits_{\lambda>0}{\left\lVert\varphi_{k}-U_{\lambda}\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}^{3}}\geq
R(p,n).$ (1.15)

Here $R(p,n)$ is a complicated number defined in (3.43). Generally, this
number is not optimal. We will elaborate on (1.15) in Remark 3.3.

Let us briefly address our methods in deriving Theorem 1.7 and Theorem 1.8.

To establish Theorem 1.7, we adapt the ideas from [24, 30]. The estimate
(1.10) can be obtained by refining the arguments of [30, Proposition 5.1]. As
for the estimate (1.9), we first test the equation (1.1) with a suitable
function and then take advantage of a expansion technique presented in [23,
24] to deal with nonlinear terms. Finally we can obtain a unexplicit lower
bound of ${\left\lVert H(u)\right\rVert}_{D_{a}^{-1,2}(\mathbb{R}^{n})}$,
which is a quadratic form of $u$. The positivity of this lower bound has
already been investigated in [24] and thus (1.9) follows. It is worth
mentioning the following three points. The first point is that, unlike the
non-degenerate case ([30, Proposition 5.1]), the most canonical choice of the
test function does not allow us to estimate the
$D_{a}^{-1,2}(\mathbb{R}^{n})$-norm of the equation (1.1) well. The test
function we choose in this paper is in some sense optimal. We explain this
point further in Remark 3.1. The second point is that we need to expand the
equation (1.1) to the fourth order (note that in the non-degenerate case, a
second-order expansion is sufficient). Since in general $p<4$, we cannot
expand it directly. Here we employ a high order expansion technique from [23,
24] to overcome this problem. The third point is that, the construction of the
functions $\\{w_{k}\\}_{k}$ in Theorem 1.1 is technical. In the non-degenerate
case, the optimality can be easily proved. However, in our case, many natural
choices of $\\{w_{k}\\}_{k}$ can only yield a quadratic estimate. To achieve
the desired cubic estimate, we use the solutions of some special Schrödinger
equations to perturb certain natural choice of $\\{w_{k}\\}_{k}$. A further
explanation of this point can be found in Remark 3.2.

To establish Theorem 1.8, we first modify the finite-dimensional reduction
method employed in [30] to find an approximation function $u_{0}$ of
$\Pi_{X}^{\perp}(u)$. $u_{0}$ has a good pointwise estimate and its
$D_{a}^{1,2}(\mathbb{R}^{n})$-norm can be well controlled. Next by applying
careful estimates, we manage to work out the relations between
${\left\lVert\Pi_{X}(u)\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}$,
${\left\lVert\Pi_{X}^{\perp}(u)-u_{0}\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}$
and ${\left\lVert H(u)\right\rVert}_{D_{a}^{-1,2}(\mathbb{R}^{n})}$. Under the
restriction (1.14),
${\left\lVert\Pi_{X}^{\perp}(u)-u_{0}\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}$
can be controlled by ${\left\lVert
H(u)\right\rVert}_{D_{a}^{-1,2}(\mathbb{R}^{n})}$. Finally, combining above
two steps gives the estimate (1.13).

Our paper is organized as follows. In Section 2, we reformulate our main
results on the cylinder $\mathcal{C}=\mathbb{R}\times\mathbb{S}^{n-1}$ and
provide some technical lemmas. Concretely speaking, we employ the Emden-Fowler
transformation to transform the Caffarelli-Kohn-Nirenberg inequality (1.3) and
the equation (1.1) into certain Gagliardo-Nirenberg interpolation inequality
and its corresponding Euler-Lagrange equation on $\mathcal{C}$, respectively.
Based on this transformation, Theorem 1.7 and Theorem 1.8 are equivalent to
Theorem 2.1 and Theorem 2.2, respectively. We also list several useful lemmas
that play an important role in our study on $\mathcal{C}$. In Section 3, we
are devoted to the proofs of Theorem 1.1 and Theorem 2.1. We also explain some
key points of our proofs. In Section 4, we deal with Theorem 1.2 and Theorem
2.2.

Notations. Throughout this paper, for simplicity, we consistently use $C$ and
$c$ to denote positive quantities depending only on the parameters $n,a,b,p$
and possibly $\nu$. The values of $C$ and $c$ may vary from line to line.
Generally, $C$ is used to denote large numbers, while $c$ represents small
numbers. For any two quantities $a_{1}$ and $a_{2}$, we shall write
$a_{1}\lesssim a_{2}$ (resp. $a_{1}\gtrsim a_{2}$) if $a_{1}\leq Ca_{2}$
(resp. $a_{1}\geq ca_{2})$. Also we say $a_{1}\approx a_{2}$ if $a_{1}\lesssim
a_{2}$ and $a_{1}\gtrsim a_{2}$. Given any quantity $\varepsilon$, we say
$a_{1}\lesssim_{\varepsilon}a_{2}$ if $a_{1}\lesssim C(\varepsilon)a_{2}$,
where $C(\varepsilon)$ is a positive number depending only on $\varepsilon$.
We similarly define $\gtrsim_{\varepsilon}$ and $\approx_{\varepsilon}$. For
any two sequences $\\{A_{k}\\}_{k}$ and $\\{B_{k}\\}_{k}$ of numbers, we write
$A_{k}=\mathcal{O}(B_{k})$ if $|A_{k}|\leq C|B_{k}|,\;\forall
k\in\mathbb{N}_{+}$. When referring to a positive variable $A$, we will use
the notation $o_{A}(1)$ to indicate any quantity that tends towards zero as
$A$ approaches zero. For any two numbers $x$ and $y$, we define two new
numbers

$\chi({x>y}):=\begin{cases}1\quad\text{if }x>y\\\ 0\quad\text{if }x\leq
y,\end{cases}\quad\chi({x\geq y}):=\begin{cases}1\quad\text{if }x\geq y\\\
0\quad\text{if }x<y.\end{cases}$

Acknowledgement. The authors wish to thank R. Frank and J. Peteranderl for
their helpful discussions to improve this paper.

##  2 A reformulation and some technical lemmas

###  2.1 A reformulation

Consider the following so-called Emden-Fowler transformation

$u(r,\theta)=r^{\frac{2+2a-n}{2}}v(s,\theta),$

where $r=|x|$, $s=-\ln{r}$ and $\theta=\frac{x}{r}\in\mathbb{S}^{n-1}$. As was
observed in [7], with this transformation, the Caffarelli-Kohn-Nirenberg
inequality (1.3) can be rewritten as

$S(a,b,n){\left\lVert
v\right\rVert}^{2}_{L^{p}(\mathcal{C})}\leq{\left\lVert\partial_{s}v\right\rVert}^{2}_{L^{2}(\mathcal{C})}+{\left\lVert\nabla_{\theta}v\right\rVert}^{2}_{L^{2}(\mathcal{C})}+\Lambda{\left\lVert
v\right\rVert}^{2}_{L^{2}(\mathcal{C})}$ (2.1)

and the critical Hardy-Hénon equation (1.1) is equivalent to

$H_{1}(v):=\partial_{s}^{2}v+\Delta_{\theta}v-\Lambda v+|v|^{p-2}v=0.$ (2.2)

Here $\Lambda:=\left(\frac{n-2-2a}{2}\right)^{2}$ and
$\nabla_{\theta},\Delta_{\theta}$ denote the gradient and the Laplace-Beltrami
operator on $\mathbb{S}^{n-1}$, respectively. This transformation provides an
isometry between the spaces $D_{a}^{1,2}(\mathbb{R}^{n})$ and
$H^{1}(\mathcal{C})$. Here we equip $H^{1}(\mathcal{C})$ with the following
norm

${\left\lVert
v\right\rVert}_{H^{1}(\mathcal{C})}=\left({\left\lVert\partial_{s}v\right\rVert}^{2}_{L^{2}(\mathcal{C})}+{\left\lVert\nabla_{\theta}v\right\rVert}^{2}_{L^{2}(\mathcal{C})}+\Lambda{\left\lVert
v\right\rVert}^{2}_{L^{2}(\mathcal{C})}\right)^{\frac{1}{2}}.$

Note that in this case, the Felli-Schneider curve can be parametrized by
$n\geq 2$, $2<p<2^{*}$ and $\Lambda=\frac{4(n-1)}{p^{2}-4}$. Along the Felli-
Schneider curve, the set of positive solutions to the equation (2.2) is given
by

$\mathcal{M}_{p,\Lambda,n}=\left\\{V_{t}(s)\;\Big{|}\;V_{t}(s):=\beta(\cosh{(\alpha(s-t))})^{-\frac{2}{p-2}},t\in\mathbb{R}\right\\},$
(2.3)

where $\alpha:=\frac{p-2}{2}\sqrt{\Lambda}$ and
$\beta:=\left(\frac{p\Lambda}{2}\right)^{\frac{1}{p-2}}$. The elements in
$\mathcal{M}_{p,\Lambda,n}$ are also called Talenti bubbles. Similar to the
original setting on $\mathbb{R}^{n}$, for any $t\in\mathbb{R}$, we define
$Y_{t}\subset H^{1}(\mathcal{C})$ to be the space spanned by nontrivial
solutions to the linearized equation of (2.2) near the Talenti bubble $V_{t}$.
From Lemma 2.4 we see $Y_{t}=\text{Span}\\{V_{t}^{\frac{p}{2}}\theta_{i},1\leq
i\leq n\\}$. We also define $\Pi_{Y_{t}}$ and $\Pi_{Y_{t}}^{\perp}$ to be the
projections onto $Y_{t}$ and $Y_{t}^{\perp}$, respectively.

Now we can state the following exact counterparts of Theorem 1.7 and Theorem
1.8 on $\mathcal{C}$.

######  Theorem 2.1.

Assume $n\geq 2$, $2<p<2^{*}$ and $\Lambda=\frac{4(n-1)}{p^{2}-4}$. Then there
exists two positive constants $\delta=\delta(p,n)$ and $C=C(p,n)$ such that,
for any function $v\in H^{1}(\mathcal{C})$ satisfying

${\left\lVert v-V_{0}\right\rVert}_{H^{1}(\mathcal{C})}\leq\delta,$

it holds that

$\inf_{t\in\mathbb{R}}{\left\lVert
v-V_{t}\right\rVert}_{H^{1}(\mathcal{C})}^{3}\leq C{\left\lVert
H_{1}(v)\right\rVert}_{H^{-1}(\mathcal{C})}.$ (2.4)

The above cubic estimate is sharp. Moreover, if the infimum in (2.4) is
attained by $V_{t_{0}}$, then there exists a positive constant
$C_{1}=C_{1}(p,n)$ such that

$\displaystyle{\left\lVert\Pi_{Y_{t_{0}}}^{\perp}(v)-V_{t_{0}}\right\rVert}_{H^{1}(\mathcal{C})}\leq
C{\left\lVert H_{1}(v)\right\rVert}_{H^{-1}(\mathcal{C})}$ (2.5)

provided

$\displaystyle{\left\lVert\Pi_{Y_{t_{0}}}^{\perp}(v)-V_{t_{0}}\right\rVert}_{H^{1}(\mathcal{C})}\geq
C_{1}F_{2}\left({\left\lVert\Pi_{Y_{t_{0}}}(v)\right\rVert}_{H^{1}(\mathcal{C})}\right).$

######  Theorem 2.2.

Assume $n\geq 2$, $\nu\geq 2$, $2<p<2^{*}$ and
$\Lambda=\frac{4(n-1)}{p^{2}-4}$. Then there exists two positive constants
$\delta=\delta(p,\nu,n)$ and $C=C(p,\nu,n)$ such that, for any function $v\in
H^{1}(\mathcal{C})$ satisfying

$\displaystyle\inf_{\begin{subarray}{c}s_{i}\in\mathbb{R}\\\ 1\leq
i\leq\nu\end{subarray}}{\left\lVert
v-\sum_{i=1}^{\nu}V_{s_{i}}\right\rVert}_{H^{1}(\mathcal{C})}\leq\delta$

and the infimum is attained by $V_{t_{i}},1\leq i\leq\nu$ with

$\displaystyle|t_{i}-t_{j}|\geq\delta^{-1},\quad\forall\;i\neq j,$

the following estimate holds

$\displaystyle{\left\lVert\Pi_{Y}^{\perp}(v)-\sum_{i=1}^{\nu}V_{t_{i}}\right\rVert}_{H^{1}(\mathcal{C})}\leq
CF_{1}\left({\left\lVert H_{1}(v)\right\rVert}_{H^{-1}(\mathcal{C})}\right)$
(2.6)

provided

$\displaystyle{\left\lVert\Pi_{Y}^{\perp}(v)-\sum_{i=1}^{\nu}V_{t_{i}}\right\rVert}_{H^{1}(\mathcal{C})}\geq
CF_{1}\left({\left\lVert\Pi_{Y}(v)\right\rVert}^{2}_{H^{1}(\mathcal{C})}\right).$

Here $Y:=\oplus_{i=1}^{\nu}Y_{t_{i}}$.

In the following two sections we focus on the proofs of the above two new
theorems instead of the two original ones. It seems that dealing with the
equation (2.2) on $\mathcal{C}$ can simplify some expressions and
computations.

###  2.2 Some technical lemmas

In this subsection we provide some important lemmas. The first lemma contains
three elementary inequalities.

######  Lemma 2.3.

Assume $x,y\in\mathbb{R}$ and $p>2$. Then we have

$\left||x+y|^{p-2}(x+y)-|x|^{p-2}x-(p-1)|x|^{p-2}y\right|\lesssim\chi(p>3)|x|^{p-3}y^{2}+|y|^{p-1}$

and

$\displaystyle\left||x+y|^{p-2}-|x|^{p-2}\right|\cdot|x|\lesssim\begin{cases}|x||y|^{p-2}+|x|^{p-2}|y|,&p\geq
3,\\\ |xy|^{\frac{p-1}{2}},&p<3.\end{cases}$

Furthermore, if $|y|\leq\frac{1}{2}|x|$, we have

$\displaystyle\Big{|}$
$\displaystyle|x+y|^{p-2}(x+y)-|x|^{p-2}x-(p-1)|x|^{p-2}y-\frac{(p-1)(p-2)}{2}|x|^{p-3}y^{2}$
$\displaystyle-\frac{(p-1)(p-2)(p-3)}{6}|x|^{p-4}y^{3}\Big{|}\lesssim|x|^{p-5}y^{4}.$

The first and third inequalities are standard. The second inequality can be
obtained by treating the cases $2|x|\leq|y|$ and $2|x|\geq|y|$ separately. The
second lemma concerns with the spectrum of the linearized equation of (2.2)
near certain Talenti bubble. We refer the readers to [22, 24] for the proof of
this Lemma.

######  Lemma 2.4.

Assume $n\geq 2$, $2<p<2^{*}$ and $\Lambda=\frac{4(n-1)}{p^{2}-4}$. Denote by
$\\{\gamma_{i}\\}_{i}$ the eigenvalues of the equation

$-\partial_{s}^{2}v-\Delta_{\theta}v+\Lambda v=\gamma V_{0}^{p-2}v,\quad v\in
H^{1}(\mathcal{C}),$

then one has

$\gamma_{1}=1,\quad\gamma_{2}=p-1,\quad\gamma_{3}>p-1.$

Moreover, the first and second eigenspaces are given by

$\mathrm{span}\\{V_{0}\\}\;\;\mathrm{and}\;\;\mathrm{span}\left\\{\partial_{s}V_{0};V_{0}^{\frac{p}{2}}\theta_{i},1\leq
i\leq n\right\\},$

respectively. Here $\theta_{i}\;(1\leq i\leq n)$ denote the Cartesian
coordinates restricted to $\mathbb{S}^{n-1}$.

The degeneracy of (2.2) (and thus (1.1)) is caused by the nontrivial elements
$V_{0}^{\frac{p}{2}}\theta_{i},1\leq i\leq n$. Note that via the Emden-Fowler
transformation, $u\in D_{a}^{1,2}(\mathbb{R}^{n})$ is centrally symmetric if
and only if $v\in H^{1}(\mathcal{C})$ satisfies $v(s,\theta)=v(s,-\theta)$.
Obviously for such $v$ one has $\Pi_{Y_{t}}(v)=0,\forall t\in\mathbb{R}$.
Therefore, $\Pi_{X_{\lambda}}(u)=0,\forall\lambda>0$. The third lemma concerns
the power series solution of a special Schrödinger equation.

######  Lemma 2.5.

Assume $n\geq 2$, $2<p<2^{*}$ and $\Lambda=\frac{4(n-1)}{p^{2}-4}$. Then there
exists $\tau,\eta\in\mathbb{R}$ and
$\\{A_{k}\\}_{k},\\{B_{k}\\}_{k}\subset\mathbb{R}$, depending only on $p$ and
$n$, such that the unique $H^{1}(\mathbb{R})$-solution $g$ of

$-\partial_{s}^{2}g+(2n+\Lambda)g-(p-1)V_{0}^{p-2}g=V_{0}^{2p-3}$

has the following power series representation

$g(s)=\tau
V_{0}^{\sqrt{1+\frac{2n}{\Lambda}}}\sum_{k=0}^{\infty}A_{k}\cosh^{-2k}(\alpha
s)+\eta V_{0}^{2p-3}\sum_{k=0}^{\infty}B_{k}\cosh^{-2k}(\alpha s).$

Here $\alpha:=\frac{p-2}{2}\sqrt{\Lambda}$. Moreover,

$A_{k}=\mathcal{O}(k^{-\frac{3}{2}}),\quad
B_{k}=\mathcal{O}(k^{-\frac{3}{2}}).$

The uniqueness is a direct consequence of spectral analysis, and the power
series representation can be calculated concretely. We refer the readers to
[24, Lemma 12] for its detailed proof. The fourth lemma comes from the
discussions in [24, Section 4].

######  Lemma 2.6.

Assume $n\geq 2$, $2<p<2^{*}$ and $\Lambda=\frac{4(n-1)}{p^{2}-4}$. For any
$\varepsilon$ small, set

$E_{\varepsilon}:=\inf_{g\in\hat{Y}}\left((1-\varepsilon){\left\lVert
g\right\rVert}^{2}_{H^{1}(\mathcal{C})}-(p-1)\int_{\mathcal{C}}\left(V_{0}^{p-2}g^{2}+(p-2)V_{0}^{2p-3}\theta_{n}^{2}g\right)\right)$

and

$F:=\frac{(p-1)(p-2)}{4}\left(\frac{p-1}{{\left\lVert
V_{0}\right\rVert}^{p}_{L^{p}(\mathcal{C})}}\left(\int_{\mathcal{C}}V_{0}^{2p-2}\theta_{n}^{2}\right)^{2}-\frac{p-3}{3}\int_{\mathcal{C}}V_{0}^{3p-4}\theta_{n}^{4}\right).$

Here the space $\hat{Y}$ is the orthogonal complement of
$\mathrm{Span}\left\\{V_{0};\partial_{s}V_{0};V_{0}^{\frac{p}{2}}\theta_{i},1\leq
i\leq n\right\\}$ in $H^{1}(\mathcal{C})$. Then we have

$F>0,\quad E_{0}+F>0,\quad E_{\varepsilon}=E_{0}+\mathcal{O}(\varepsilon).$

Moreover, when $n\geq 3$,

$\frac{E_{0}}{F}+1\rightarrow 0\quad\text{as }p\rightarrow 2^{*}.$

Although a detailed proof of this lemma has been given in [24], we briefly
explain the main ideas here for the convenience of the readers, as this lemma
is crucial in our proof of Theorem 2.1. The value of $F$ can be explicitly
computed. As for the term $E_{\varepsilon}$, we expand the function $g$ in
spherical harmonics: $g(s,\theta):=\sum_{i,j}a_{i,j}(s)Y_{i,j}(\theta)$. Here
$\\{Y_{i,j}\\}_{j}$ is an $L^{2}(\mathbb{S}^{n-1})$-orthonormal basis of
spherical harmonics of degree $i$. We take $Y_{0,0}=c_{1}$ and
$Y_{2,0}=c_{2}(\theta_{n}^{2}-\frac{1}{n})$, where $c_{1},c_{2}$ are two
normalization factors. Substituting this expansion into $E_{\varepsilon}$ and
employing some spectrum conditions, we can transform $E_{\varepsilon}$ into a
sum of two quadratic forms involving $a_{0,0}$ and $a_{2,0}$. Again, using the
spectrum conditions, we can show that these two forms are non-degenerate and
can thus be minimized by the standard ‘completing the square’ argument. It is
worth mentioning that in this procedure one needs to explicitly solve some
special Schrödinger equations (see Lemma 2.5 for example). Even after
obtaining a concrete expression for $E_{\varepsilon}$, additional work is
required to verify that $E_{0}+F>0$.

Finally, we give some integral estimates involved Talenti bubbles and their
derivatives.

######  Lemma 2.7.

Assume $t_{1},t_{2}\in\mathbb{R}$, $q_{1},q_{2}\geq 0$ and $q_{1}+q_{2}=p$.
Then for any $\varepsilon>0$, if $|q_{1}-q_{2}|\geq\varepsilon$, it holds that

$\displaystyle\int_{\mathcal{C}}V_{t_{1}}^{q_{1}}V_{t_{2}}^{q_{2}}\approx_{\varepsilon}e^{-\sqrt{\Lambda}|t_{1}-t_{2}|\min\\{q_{1},q_{2}\\}}.$
(2.7)

When $q_{1}=q_{2}=\frac{p}{2}$, it holds that

$\displaystyle\int_{\mathcal{C}}V_{t_{1}}^{\frac{p}{2}}V_{t_{2}}^{\frac{p}{2}}\approx(|t_{1}-t_{2}|+1)e^{-\frac{p\sqrt{\Lambda}}{2}|t_{1}-t_{2}|}.$
(2.8)

######  Lemma 2.8.

Given $t_{1},t_{2}\in\mathbb{R}$. If $t_{1}\ll t_{2}$, it holds that

$\int_{\mathcal{C}}V_{t_{1}}^{p-1}\partial_{s}V_{t_{2}}\approx
e^{\sqrt{\Lambda}(t_{1}-t_{2})}.$ (2.9)

The proof of Lemma 2.7 is similar to [30, Lemma 5.1]. The estimate (2.9) can
be easily computed using the expressions of Talenti bubbles
$V_{t},t\in\mathbb{R}$.

##  3 Proofs of Theorem 1.1 and Theorem 2.1

In this section we first prove Theorem 2.1. Theorem 1.7 then follows from the
Emden-Fowler transformation. Combining Theorem 1.7 and the global compactness
principle [30, Proposition 3.2], we obtain Theorem 1.1. We also give some
remarks about our proof in the end of this section.

###### Proof of Theorem 2.1.

Let us prove (2.4) first. If (2.4) does not hold for any $\delta>0$, then
there exists a sequence $\\{v_{k}\\}_{k}$ of functions in $H^{1}(\mathcal{C})$
such that

$v_{k}\notin\mathcal{M}_{p,\Lambda,n},\quad{\left\lVert
v_{k}-V_{0}\right\rVert}_{H^{1}(\mathcal{C})}\rightarrow 0\;\;\text{as
}k\rightarrow+\infty$ (3.1)

and

$\frac{{\left\lVert
H_{1}(v_{k})\right\rVert}_{H^{-1}(\mathcal{C})}}{\inf\limits_{t\in\mathbb{R}}{\left\lVert
v_{k}-V_{t}\right\rVert}_{H^{1}(\mathcal{C})}^{3}}\rightarrow 0\;\;\text{as
}k\rightarrow+\infty.$ (3.2)

Up to a suitable translation on $\mathbb{R}$, we can assume

$\inf_{\alpha,t\in\mathbb{R}}{\left\lVert v_{k}-\alpha
V_{t}\right\rVert}_{H^{1}(\mathcal{C})}={\left\lVert
v_{k}-\alpha_{k}V_{0}\right\rVert}_{H^{1}(\mathcal{C})}.$ (3.3)

Set $\rho_{k}:=v_{k}-\alpha_{k}V_{0}$. From (3.1) it is easy to see that
$|\alpha_{k}-1|\rightarrow 0$ and
${\left\lVert\rho_{k}\right\rVert}_{H^{1}(\mathcal{C})}\rightarrow 0$. Since
$\alpha_{k}V_{0}$ minimizes the distance, by variation we can derive the
following two orthogonality conditions:

$\left\langle\rho_{k},V_{0}\right\rangle_{H^{1}(\mathcal{C})}=\left\langle\rho_{k},\partial_{s}V_{0}\right\rangle_{H^{1}(\mathcal{C})}=0.$
(3.4)

Since $H_{1}(V_{0})=0$, we can rewrite (3.4) as

$\left\langle\rho_{k},V^{p-1}_{0}\right\rangle_{L^{2}(\mathcal{C})}=\left\langle\rho_{k},V_{0}^{p-2}\partial_{s}V_{0}\right\rangle_{L^{2}(\mathcal{C})}=0.$
(3.5)

Consider the following identity:

$\displaystyle\partial_{s}^{2}v_{k}+\Delta_{\theta}v_{k}-\Lambda
v_{k}+|v_{k}|^{p-2}v_{k}=$
$\displaystyle\;\partial_{s}^{2}\rho_{k}+\Delta_{\theta}\rho_{k}-\Lambda\rho_{k}+\left(\alpha_{k}^{p-1}-\alpha_{k}\right)V_{0}^{p-1}$
(3.6)
$\displaystyle+\left(|v_{k}|^{p-2}v_{k}-\alpha_{k}^{p-1}V_{0}^{p-1}\right).$

Testing it with $V_{0}$, exploiting Lemma 2.3 and using the orthogonality
conditions in (3.4) and (3.5), we can deduce

$\displaystyle\left|\alpha_{k}^{p-1}-\alpha_{k}\right|\int_{\mathcal{C}}V_{0}^{p}\lesssim$
$\displaystyle\;{\left\lVert
H_{1}(v_{k})\right\rVert}_{H^{-1}(\mathcal{C})}+\chi(p>3)\int_{\mathcal{C}}V_{0}^{p-2}|\rho_{k}|^{2}+\int_{\mathcal{C}}V_{0}|\rho_{k}|^{p-1}$
$\displaystyle\lesssim$ $\displaystyle\;{\left\lVert
H_{1}(v_{k})\right\rVert}_{H^{-1}(\mathcal{C})}+{\left\lVert\rho_{k}\right\rVert}^{\min\\{2,p-1\\}}_{H^{1}(\mathcal{C})}.$

Since $|\alpha_{k}-1|\rightarrow 0$, we find

$|\alpha_{k}-1|\lesssim{\left\lVert
H_{1}(v_{k})\right\rVert}_{H^{-1}(\mathcal{C})}+{\left\lVert\rho_{k}\right\rVert}^{\min\\{2,p-1\\}}_{H^{1}(\mathcal{C})}.$
(3.7)

The combination of (3.1), (3.2), (3.3) and (3.7) indicates

$\frac{{\left\lVert
H_{1}(v_{k})\right\rVert}_{H^{-1}(\mathcal{C})}}{{\left\lVert\rho_{k}\right\rVert}_{H^{1}(\mathcal{C})}^{3}}\rightarrow
0\;\;\text{as }k\rightarrow+\infty.$ (3.8)

In the following we aim to show that (3.8) is impossible. Up to a suitable
rotation on $\mathbb{S}^{n-1}$, we assume

$\rho_{k}=\mu_{k}\left(\hat{\mu}_{k}V_{0}^{\frac{p}{2}}\theta_{n}+\eta_{k}\right),$

where $\\{\mu_{k}\\}_{k},\\{\hat{\mu}_{k}\\}_{k}$ are two sequences of numbers
such that

$(\mu_{k},\hat{\mu}_{k})=\begin{cases}\left(\left\langle\rho_{k},V^{\frac{p}{2}}_{0}\theta_{n}\right\rangle_{H^{1}(\mathcal{C})}{\left\lVert
V_{0}^{\frac{p}{2}}\theta_{n}\right\rVert}_{H^{1}(\mathcal{C})}^{-2},1\right)&\text{if
}\left\langle\rho_{k},V^{\frac{p}{2}}_{0}\theta_{n}\right\rangle_{H^{1}(\mathcal{C})}\neq
0,\\\
\left({\left\lVert\rho_{k}\right\rVert}_{H^{1}(\mathcal{C})},0\right)&\text{if
}\left\langle\rho_{k},V^{\frac{p}{2}}_{0}\theta_{n}\right\rangle_{H^{1}(\mathcal{C})}=0;\end{cases}$

and $\eta_{k}\in H^{1}(\mathcal{C})$ satisfies

$\left\langle\eta_{k},V_{0}\right\rangle_{H^{1}(\mathcal{C})}=\left\langle\eta_{k},\partial_{s}V_{0}\right\rangle_{H^{1}(\mathcal{C})}=\left\langle\eta_{k},V_{0}^{\frac{p}{2}}\theta_{i}\right\rangle_{H^{1}(\mathcal{C})}=0\quad\text{for
any }1\leq i\leq n.$ (3.9)

Thanks to Lemma 2.4, the relations in (3.9) can be rewritten as

$\left\langle\eta_{k},V^{p-1}_{0}\right\rangle_{L^{2}(\mathcal{C})}=\left\langle\eta_{k},V_{0}^{p-2}\partial_{s}V_{0}\right\rangle_{L^{2}(\mathcal{C})}=\left\langle\eta_{k},V_{0}^{\frac{3}{2}p-2}\theta_{i}\right\rangle_{L^{2}(\mathcal{C})}=0\quad\text{for
any }1\leq i\leq n.$ (3.10)

Without loss of generality, we suppose

$\lim_{k\rightarrow+\infty}{\left\lVert\eta_{k}\right\rVert}_{H^{1}(\mathcal{C})}=L\in[0,+\infty].$
(3.11)

We shall divide our analysis for (3.8) into two parts that are devoted to the
case $L>0$ and $L=0$, respectively.
_Case 1: $L>0$_.

Let us start by testing $H_{1}(v_{k})$ against $\rho_{k}$. Using Lemma 2.3 and
the orthogonality conditions in (3.4) and (3.5) yields

$\displaystyle{\left\lVert\rho_{k}\right\rVert}^{2}_{H^{1}(\mathcal{C})}=$
$\displaystyle-\left\langle
H_{1}(v_{k}),\rho_{k}\right\rangle_{L^{2}(\mathcal{C})}+\int_{\mathcal{C}}|v_{k}|^{p-2}v_{k}\rho_{k}$
$\displaystyle\leq$ $\displaystyle\;{\left\lVert
H_{1}(v_{k})\right\rVert}_{H^{-1}(\mathcal{C})}{\left\lVert\rho_{k}\right\rVert}_{H^{1}(\mathcal{C})}+(p-1)\alpha_{k}^{p-2}\int_{\mathcal{C}}V_{0}^{p-2}\rho^{2}_{k}$
(3.12)
$\displaystyle+C\chi(p>3)\int_{\mathcal{C}}V_{0}^{p-3}|\rho_{k}|^{3}+C\int_{\mathcal{C}}|\rho_{k}|^{p}.$

Exploiting the Hölder inequality and the estimate (3.7), we obtain

$\displaystyle{\left\lVert\rho_{k}\right\rVert}^{2}_{H^{1}(\mathcal{C})}-(p-1+o_{\delta}(1))\int_{\mathcal{C}}V_{0}^{p-2}\rho^{2}_{k}\lesssim{\left\lVert
H_{1}(v_{k})\right\rVert}_{H^{-1}(\mathcal{C})}{\left\lVert\rho_{k}\right\rVert}_{H^{1}(\mathcal{C})}.$
(3.13)

Thanks to the orthogonality conditions (3.9), (3.10) and Lemma 2.4, we find

$\displaystyle{\left\lVert\rho_{k}\right\rVert}^{2}_{H^{1}(\mathcal{C})}=$
$\displaystyle\;\mu_{k}^{2}\hat{\mu}_{k}^{2}{\left\lVert
V_{0}^{\frac{p}{2}}\theta_{n}\right\rVert}^{2}_{H^{1}(\mathcal{C})}+\mu_{k}^{2}{\left\lVert\eta_{k}\right\rVert}^{2}_{H^{1}(\mathcal{C})}$
$\displaystyle\geq$
$\displaystyle\;(p-1)\mu_{k}^{2}\hat{\mu}_{k}^{2}\int_{\mathcal{C}}V_{0}^{p-2}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{2}+(p-1)\mu_{k}^{2}\int_{\mathcal{C}}V_{0}^{p-2}\eta_{k}^{2}+\mu_{k}^{2}\left(1-\frac{p-1}{\gamma_{3}}\right){\left\lVert\eta_{k}\right\rVert}^{2}_{H^{1}(\mathcal{C})}$
$\displaystyle=$
$\displaystyle\;(p-1)\int_{\mathcal{C}}V_{0}^{p-2}\rho^{2}_{k}+\mu_{k}^{2}\left(1-\frac{p-1}{\gamma_{3}}\right){\left\lVert\eta_{k}\right\rVert}^{2}_{H^{1}(\mathcal{C})}.$
(3.14)

Recall that $\gamma_{3}>p-1$ is the third eigenvalue defined in Lemma 2.4.
Since $L>0$, we have

${\left\lVert\rho_{k}\right\rVert}_{H^{1}(\mathcal{C})}\lesssim_{L}\mu_{k}{\left\lVert\eta_{k}\right\rVert}_{H^{1}(\mathcal{C})}.$
(3.15)

Substituting (3) and (3.15) into (3.13) gives

${\left\lVert\rho_{k}\right\rVert}_{H^{1}(\mathcal{C})}\lesssim_{L}{\left\lVert
H_{1}(v_{k})\right\rVert}_{H^{1}(\mathcal{C})},$

which obviously contradicts (3.8).
_Case 2: $L=0$_.

In this case we have $\hat{\mu}_{k}\equiv 1$ and $\mu_{k}\neq 0$. Since the
$H^{1}(\mathcal{C})$-norm of $\rho_{k}$ is comparable to $\mu_{k}$, (3.8) is
equivalent to

$|\mu_{k}|^{-3}{\left\lVert
H_{1}(v_{k})\right\rVert}_{H^{-1}(\mathcal{C})}\rightarrow 0\;\;\text{as
}k\rightarrow+\infty.$ (3.16)

Let us test $H_{1}(v_{k})$ against
$\mu_{k}\left(V_{0}^{\frac{p}{2}}\theta_{n}+2\eta_{k}\right)$. Exploiting the
orthogonality conditions in (3.9), we deduce

$\displaystyle\mu_{k}^{2}{\left\lVert
V_{0}^{\frac{p}{2}}\theta_{n}\right\rVert}^{2}_{H^{1}(\mathcal{C})}+2\mu_{k}^{2}{\left\lVert\eta_{k}\right\rVert}^{2}_{H^{1}(\mathcal{C})}$
$\displaystyle=$ $\displaystyle-\mu_{k}\left\langle
H_{1}(v_{k}),V_{0}^{\frac{p}{2}}\theta_{n}+2\eta_{k}\right\rangle_{L^{2}(\mathcal{C})}+\mu_{k}\int_{\mathcal{C}}|v_{k}|^{p-2}v_{k}\left(V_{0}^{\frac{p}{2}}\theta_{n}+2\eta_{k}\right)$
(3.17) $\displaystyle\leq$ $\displaystyle\;C|\mu_{k}|{\left\lVert
H_{1}(v_{k})\right\rVert}_{H^{-1}(\mathcal{C})}+\mu_{k}\int_{\mathcal{C}}|v_{k}|^{p-2}v_{k}\left(V_{0}^{\frac{p}{2}}\theta_{n}+2\eta_{k}\right).$

To control the second term, we make use of a high order expansion technique
presented in [23, 24]. To be specific, we split the integration domain
$\mathcal{C}$ into two parts:
$\left\\{|\eta_{k}|<V_{0}^{\frac{p}{2}}\right\\}$ and $\left\\{|\eta_{k}|\geq
V_{0}^{\frac{p}{2}}\right\\}$. In the first part, using the facts that
$V_{0}\in L^{\infty}(\mathcal{C})$ and $p>2$, we know
$\left|V_{0}^{\frac{p}{2}}\theta_{n}+\eta_{k}\right|\leq
2V_{0}^{\frac{p}{2}}\lesssim V_{0}$. Therefore, we can apply the third
estimate in Lemma 2.3 to derive

$\displaystyle|v_{k}|^{p-2}v_{k}=$
$\displaystyle\;\left(\alpha_{k}V_{0}\right)^{p-1}+\mu_{k}(p-1)\left(\alpha_{k}V_{0}\right)^{p-2}\left(V_{0}^{\frac{p}{2}}\theta_{n}+\eta_{k}\right)$
$\displaystyle+\mu_{k}^{2}\frac{(p-1)(p-2)}{2}\left(\alpha_{k}V_{0}\right)^{p-3}\left(V_{0}^{\frac{p}{2}}\theta_{n}+\eta_{k}\right)^{2}$
(3.18)
$\displaystyle+\mu_{k}^{3}\frac{(p-1)(p-2)(p-3)}{6}\left(\alpha_{k}V_{0}\right)^{p-4}\left(V_{0}^{\frac{p}{2}}\theta_{n}+\eta_{k}\right)^{3}+\mathcal{O}\left(\mu_{k}^{4}V_{0}^{3p-5}\right).$

Note that

$\displaystyle
V_{0}^{p-2}\left(V_{0}^{\frac{p}{2}}\theta_{n}+\eta_{k}\right)\left(V_{0}^{\frac{p}{2}}\theta_{n}+2\eta_{k}\right)=$
$\displaystyle\;V_{0}^{p-2}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{2}+2V_{0}^{p-2}\eta_{k}^{2}+3V_{0}^{p-2}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)\eta_{k},$
$\displaystyle
V_{0}^{p-3}\left(V_{0}^{\frac{p}{2}}\theta_{n}+\eta_{k}\right)^{2}\left(V_{0}^{\frac{p}{2}}\theta_{n}+2\eta_{k}\right)=$
$\displaystyle\;V_{0}^{p-3}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{3}+4V_{0}^{p-3}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{2}\eta_{k}$
$\displaystyle+5V_{0}^{p-3}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)\eta_{k}^{2}+2V_{0}^{p-3}\eta_{k}^{3}$
$\displaystyle=$
$\displaystyle\;V_{0}^{p-3}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{3}+4V_{0}^{p-3}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{2}\eta_{k}+\mathcal{O}\left(\eta_{k}^{2}\right),$
$\displaystyle
V_{0}^{p-4}\left(V_{0}^{\frac{p}{2}}\theta_{n}+\eta_{k}\right)^{3}\left(V_{0}^{\frac{p}{2}}\theta_{n}+2\eta_{k}\right)=$
$\displaystyle\;V_{0}^{p-4}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{4}+5V_{0}^{p-4}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{3}\eta_{k}$
$\displaystyle+9V_{0}^{p-3}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{2}\eta_{k}^{2}+7V_{0}^{p-3}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)\eta_{k}^{3}$
$\displaystyle+2V_{0}^{p-4}\eta_{k}^{4}$ $\displaystyle=$
$\displaystyle\;V_{0}^{p-4}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{4}+\mathcal{O}\left(V_{0}^{\frac{5}{2}p-4}|\eta_{k}|\right).$

Substituting these three identities into (3) gives

$\displaystyle\;|v_{k}|^{p-2}v_{k}\left(V_{0}^{\frac{p}{2}}\theta_{n}+2\eta_{k}\right)$
$\displaystyle=$
$\displaystyle\;\left(\alpha_{k}V_{0}\right)^{p-1}\left(V_{0}^{\frac{p}{2}}\theta_{n}+2\eta_{k}\right)+\mu_{k}(p-1)\left(\alpha_{k}V_{0}\right)^{p-2}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{2}$
$\displaystyle+2\mu_{k}(p-1)\left(\alpha_{k}V_{0}\right)^{p-2}\eta_{k}^{2}+3\mu_{k}(p-1)\left(\alpha_{k}V_{0}\right)^{p-2}V_{0}^{\frac{p}{2}}\theta_{n}\eta_{k}$
$\displaystyle+\mu_{k}^{2}\frac{(p-1)(p-2)}{2}\left(\alpha_{k}V_{0}\right)^{p-3}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{3}+2\mu_{k}^{2}(p-1)(p-2)\left(\alpha_{k}V_{0}\right)^{p-3}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{2}\eta_{k}$
$\displaystyle+\mu_{k}^{3}\frac{(p-1)(p-2)(p-3)}{6}\left(\alpha_{k}V_{0}\right)^{p-4}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{4}$
(3.19)
$\displaystyle+\mathcal{O}\left(\mu_{k}^{2}\eta_{k}^{2}+|\mu_{k}|^{3}V_{0}^{\frac{5}{2}p-4}|\eta_{k}|+\mu_{k}^{4}V_{0}^{3p-5}\right).$

In the second part, since
$\left|V_{0}^{\frac{p}{2}}\theta_{n}+\eta_{k}\right|\leq 2|\eta_{k}|$ and
$p>2$, we can use the first estimate in Lemma 2.3 to deduce

$\displaystyle|v_{k}|^{p-2}v_{k}=$
$\displaystyle\;\left(\alpha_{k}V_{0}\right)^{p-1}+\mu_{k}(p-1)\left(\alpha_{k}V_{0}\right)^{p-2}\left(V_{0}^{\frac{p}{2}}\theta_{n}+\eta_{k}\right)$
$\displaystyle+\mathcal{O}\left(\chi(p>3)\mu_{k}^{2}V_{0}^{p-3}\left(V_{0}^{\frac{p}{2}}\theta_{n}+\eta_{k}\right)^{2}+\mu_{k}^{p-1}\left|V_{0}^{\frac{p}{2}}\theta_{n}+\eta_{k}\right|^{p-1}\right)$
$\displaystyle=$
$\displaystyle\;\left(\alpha_{k}V_{0}\right)^{p-1}+\mu_{k}(p-1)\left(\alpha_{k}V_{0}\right)^{p-2}\left(V_{0}^{\frac{p}{2}}\theta_{n}+\eta_{k}\right)$
$\displaystyle+\mathcal{O}\left(\chi(p>3)\mu_{k}^{2}\eta_{k}^{2}+|\mu_{k}|^{p-1}|\eta_{k}|^{p-1}\right)$

and thus

$\displaystyle\;|v_{k}|^{p-2}v_{k}\left(V_{0}^{\frac{p}{2}}\theta_{n}+2\eta_{k}\right)$
$\displaystyle=$
$\displaystyle\;\left(\alpha_{k}V_{0}\right)^{p-1}\left(V_{0}^{\frac{p}{2}}\theta_{n}+2\eta_{k}\right)+\mu_{k}(p-1)\left(\alpha_{k}V_{0}\right)^{p-2}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{2}$
(3.20)
$\displaystyle+2\mu_{k}(p-1)\left(\alpha_{k}V_{0}\right)^{p-2}\eta_{k}^{2}+3\mu_{k}(p-1)\left(\alpha_{k}V_{0}\right)^{p-2}V_{0}^{\frac{p}{2}}\theta_{n}\eta_{k}$
$\displaystyle+\mathcal{O}\left(\chi(p>3)\mu_{k}^{2}|\eta_{k}|^{3}+|\mu_{k}|^{p-1}|\eta_{k}|^{p}\right).$

Note that in this part, we also have

$\displaystyle\mu_{k}^{2}\frac{(p-1)(p-2)}{2}\left(\alpha_{k}V_{0}\right)^{p-3}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{3}=$
$\displaystyle\;\mathcal{O}\left(\mu_{k}^{2}\eta_{k}^{2}\right),$
$\displaystyle
2\mu_{k}^{2}(p-1)(p-2)\left(\alpha_{k}V_{0}\right)^{p-3}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{2}\eta_{k}=$
$\displaystyle\;\mathcal{O}\left(\mu_{k}^{2}\eta_{k}^{2}\right),$
$\displaystyle\mu_{k}^{3}\frac{(p-1)(p-2)(p-3)}{6}\left(\alpha_{k}V_{0}\right)^{p-4}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{4}=$
$\displaystyle\;\mathcal{O}\left(|\mu_{k}|^{3}\eta_{k}^{2}\right).$

Combining these three identities with (3) shows that the pointwise estimate
(3) still holds in this part, but with an additional error term
$\mathcal{O}\left(\chi(p>3)\mu_{k}^{2}|\eta_{k}|^{3}+|\mu_{k}|^{p-1}|\eta_{k}|^{p}\right)$.

Exploiting our arguments above and utilizing the orthogonality conditions in
(3.9) and (3.10) yields

$\displaystyle\int_{\mathcal{C}}|v_{k}|^{p-2}v_{k}\left(V_{0}^{\frac{p}{2}}\theta_{n}+2\eta_{k}\right)$
(3.21) $\displaystyle=$
$\displaystyle\;\mu_{k}(p-1)\int_{\mathcal{C}}\left(\alpha_{k}V_{0}\right)^{p-2}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{2}+2\mu_{k}(p-1)\int_{\mathcal{C}}\left(\alpha_{k}V_{0}\right)^{p-2}\eta_{k}^{2}$
(3.22)
$\displaystyle+2\mu_{k}^{2}(p-1)(p-2)\int_{\mathcal{C}}\left(\alpha_{k}V_{0}\right)^{p-3}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{2}\eta_{k}$
(3.23)
$\displaystyle+\mu_{k}^{3}\frac{(p-1)(p-2)(p-3)}{6}\int_{\mathcal{C}}\left(\alpha_{k}V_{0}\right)^{p-4}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{4}$
(3.24)
$\displaystyle+\mathcal{O}\left(|\mu_{k}|^{\min\\{2,p-1\\}}{\left\lVert\eta_{k}\right\rVert}^{2}_{H^{1}(\mathcal{C})}+\mu_{k}^{4}\right).$
(3.25)

In this computation we also use the Hölder inequality and the following two
inequalities

$\displaystyle
2|\mu_{k}|^{3}{\left\lVert\eta_{k}\right\rVert}_{H^{1}(\mathcal{C})}\leq\;\mu_{k}^{2}{\left\lVert\eta_{k}\right\rVert}^{2}_{H^{1}(\mathcal{C})}+\mu_{k}^{4},\quad|\mu_{k}|^{p-1}{\left\lVert\eta_{k}\right\rVert}^{p}_{H^{1}(\mathcal{C})}\leq$
$\displaystyle\;|\mu_{k}|^{p-1}{\left\lVert\eta_{k}\right\rVert}^{2}_{H^{1}(\mathcal{C})}.$

Now it follows from (3), (3.21) and Lemma 2.4 that the value of

$\displaystyle\;\mu_{k}^{2}{\left\lVert
V_{0}^{\frac{p}{2}}\theta_{n}\right\rVert}^{2}_{H^{1}(\mathcal{C})}+2\mu_{k}^{2}{\left\lVert\eta_{k}\right\rVert}^{2}_{H^{1}(\mathcal{C})}-\mu_{k}\int_{\mathcal{C}}|v_{k}|^{p-2}v_{k}\left(V_{0}^{\frac{p}{2}}\theta_{n}+2\eta_{k}\right)$
$\displaystyle=$
$\displaystyle\;\mu_{k}^{2}(p-1)\left(1-\alpha_{k}^{p-2}\right)\int_{\mathcal{C}}V_{0}^{p-2}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{2}+2\mu_{k}^{2}\left(1-C|\mu_{k}|^{\min\\{1,p-2\\}}\right){\left\lVert\eta_{k}\right\rVert}_{H^{1}(\mathcal{C})}^{2}$
$\displaystyle-2\mu_{k}^{2}(p-1)\int_{\mathcal{C}}\left(\alpha_{k}V_{0}\right)^{p-2}\eta_{k}^{2}-2\mu_{k}^{3}(p-1)(p-2)\int_{\mathcal{C}}\left(\alpha_{k}V_{0}\right)^{p-3}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{2}\eta_{k}$
(3.26)
$\displaystyle-\mu_{k}^{4}\frac{(p-1)(p-2)(p-3)}{6}\int_{\mathcal{C}}\left(\alpha_{k}V_{0}\right)^{p-4}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{4}+\mathcal{O}\left(|\mu_{k}|^{5}\right)$

can be bounded above by $C|\mu_{k}|{\left\lVert
H_{1}(v_{k})\right\rVert}_{H^{-1}(\mathcal{C})}$.

To simplify (3), we need to find appropriate estimates for
$\left|\alpha_{k}^{p-2}-1\right|$ and $\left|\alpha_{k}-1\right|$. Note that
(3.7) does not meet our requirements. Let us test the identity in (3.6) with
$V_{0}$ again and use the orthogonality conditions in (3.9) to derive

$\displaystyle\left|\left(\alpha_{k}^{p-1}-\alpha_{k}\right)\int_{\mathcal{C}}V_{0}^{p}+\int_{\mathcal{C}}\left(|v_{k}|^{p-2}v_{k}-\alpha_{k}^{p-1}V_{0}^{p-1}\right)V_{0}\right|\lesssim{\left\lVert
H_{1}(v_{k})\right\rVert}_{H^{-1}(\mathcal{C})}$ (3.27)

To control the second term, we apply the high order expansion technique and
Lemma 2.3 once again. In the first region
$\left\\{|\eta_{k}|<V_{0}^{\frac{p}{2}}\right\\}$, we have

$\displaystyle\;\left(|v_{k}|^{p-2}v_{k}-\alpha_{k}^{p-1}V_{0}^{p-1}\right)V_{0}$
$\displaystyle=$
$\displaystyle\;\mu_{k}(p-1)\alpha_{k}^{p-2}V_{0}^{p-1}\left(V_{0}^{\frac{p}{2}}\theta_{n}+\eta_{k}\right)$
$\displaystyle+\mu_{k}^{2}\frac{(p-1)(p-2)}{2}\alpha_{k}^{p-3}V_{0}^{p-2}\left(V_{0}^{\frac{p}{2}}\theta_{n}+\eta_{k}\right)^{2}+\mathcal{O}\left(|\mu_{k}|^{3}V_{0}^{\frac{5}{2}p-3}\right)$
$\displaystyle=$
$\displaystyle\;\mu_{k}(p-1)\alpha_{k}^{p-2}V_{0}^{p-1}\left(V_{0}^{\frac{p}{2}}\theta_{n}+\eta_{k}\right)+\mu_{k}^{2}\frac{(p-1)(p-2)}{2}\alpha_{k}^{p-3}V_{0}^{p-2}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{2}$
(3.28)
$\displaystyle+\mu_{k}^{2}(p-1)(p-2)\alpha_{k}^{p-3}V_{0}^{p-2}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)\eta_{k}+\mathcal{O}\left(|\mu_{k}|^{3}V_{0}^{\frac{5}{2}p-3}+\mu_{k}^{2}\eta_{k}^{2}\right).$

In the second region $\left\\{|\eta_{k}|\geq V_{0}^{\frac{p}{2}}\right\\}$, we
have

$\displaystyle\;\left(|v_{k}|^{p-2}v_{k}-\alpha_{k}^{p-1}V_{0}^{p-1}\right)V_{0}$
(3.29) $\displaystyle=$
$\displaystyle\;\mu_{k}(p-1)\alpha_{k}^{p-2}V_{0}^{p-1}\left(V_{0}^{\frac{p}{2}}\theta_{n}+\eta_{k}\right)+\mathcal{O}\left(\chi(p>3)\mu_{k}^{2}\eta_{k}^{2}+\mu_{k}^{p-1}V_{0}|\eta_{k}|^{p-1}\right)$
$\displaystyle=$
$\displaystyle\;\mu_{k}(p-1)\alpha_{k}^{p-2}V_{0}^{p-1}\left(V_{0}^{\frac{p}{2}}\theta_{n}+\eta_{k}\right)+\mu_{k}^{2}\frac{(p-1)(p-2)}{2}\alpha_{k}^{p-3}V_{0}^{p-2}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{2}$
$\displaystyle+\mu_{k}^{2}(p-1)(p-2)\alpha_{k}^{p-3}V_{0}^{p-2}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)\eta_{k}+\mathcal{O}\left(\mu_{k}^{2}\eta_{k}^{2}+\mu_{k}^{p-1}V_{0}|\eta_{k}|^{p-1}\right).$

The combination of (3), (3.29) and the orthogonality conditions in (3.10)
yields

$\displaystyle\int_{\mathcal{C}}\left(|v_{k}|^{p-2}v_{k}-\alpha_{k}^{p-1}V_{0}^{p-1}\right)V_{0}=$
$\displaystyle\;\mu_{k}^{2}\frac{(p-1)(p-2)}{2}\alpha_{k}^{p-3}\int_{\mathcal{C}}V_{0}^{p-2}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{2}$
(3.30)
$\displaystyle+\mathcal{O}\left(|\mu_{k}|^{3}+|\mu_{k}|^{\min\\{2,p-1\\}}{\left\lVert\eta_{k}\right\rVert}_{H^{1}(\mathcal{C})}^{\min\\{2,p-1\\}}\right).$

It follows from (3.27) and (3.30) that

$\displaystyle\;\left|\left(\alpha_{k}^{p-1}-\alpha_{k}\right)\int_{\mathcal{C}}V_{0}^{p}+\mu_{k}^{2}\frac{(p-1)(p-2)}{2}\alpha_{k}^{p-3}\int_{\mathcal{C}}V_{0}^{p-2}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{2}\right|$
$\displaystyle\lesssim$ $\displaystyle\;{\left\lVert
H_{1}(v_{k})\right\rVert}_{H^{-1}(\mathcal{C})}+\mathcal{O}\left(|\mu_{k}|^{3}+|\mu_{k}|^{\min\\{2,p-1\\}}{\left\lVert\eta_{k}\right\rVert}_{H^{1}(\mathcal{C})}^{\min\\{2,p-1\\}}\right).$
(3.31)

One can easily derive from (3) that

$\displaystyle\;\left|\left(\alpha_{k}^{p-2}-1\right)+\mu_{k}^{2}\frac{(p-1)(p-2)}{2}\alpha_{k}^{p-4}\frac{\int_{\mathcal{C}}V_{0}^{p-2}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{2}}{\int_{\mathcal{C}}V_{0}^{p}}\right|$
$\displaystyle\lesssim$ $\displaystyle\;{\left\lVert
H_{1}(v_{k})\right\rVert}_{H^{-1}(\mathcal{C})}+\mathcal{O}\left(|\mu_{k}|^{\min\\{3,2p-2\\}}+|\mu_{k}|^{\min\\{2,2p-4\\}}{\left\lVert\eta_{k}\right\rVert}_{H^{1}(\mathcal{C})}^{2}\right)$
(3.32)

and

$\displaystyle|\alpha_{k}-1|\lesssim{\left\lVert
H_{1}(v_{k})\right\rVert}_{H^{-1}(\mathcal{C})}+\mathcal{O}\left(\mu_{k}^{2}+|\mu_{k}|^{\min\\{2,2p-4\\}}{\left\lVert\eta_{k}\right\rVert}_{H^{1}(\mathcal{C})}^{2}\right).$
(3.33)

Combining (3) and (3.33) into (3) yields that the value of

$\displaystyle\;2\mu_{k}^{2}\left(1-C|\mu_{k}|^{\min\\{1,p-2\\}}\right){\left\lVert\eta_{k}\right\rVert}_{H^{1}(\mathcal{C})}^{2}-2\mu_{k}^{2}(p-1)\int_{\mathcal{C}}V_{0}^{p-2}\eta_{k}^{2}$
(3.34)
$\displaystyle-2\mu_{k}^{3}(p-1)(p-2)\int_{\mathcal{C}}V_{0}^{p-3}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{2}\eta_{k}+\mu_{k}^{4}\frac{(p-1)^{2}(p-2)}{2}\frac{\left(\int_{\mathcal{C}}V_{0}^{p-2}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{2}\right)^{2}}{\int_{\mathcal{C}}V_{0}^{p}}$
$\displaystyle-\mu_{k}^{4}\frac{(p-1)(p-2)(p-3)}{6}\int_{\mathcal{C}}V_{0}^{p-4}\left(V_{0}^{\frac{p}{2}}\theta_{n}\right)^{4}+\mathcal{O}\left(|\mu_{k}|^{5}\right)$

can be bounded above by $C|\mu_{k}|{\left\lVert
H_{1}(v_{k})\right\rVert}_{H^{-1}(\mathcal{C})}$. From Lemma 2.6 we know that
the value of (3.34) is larger than
$2\mu_{k}^{4}\left(E_{0}+F+\mathcal{O}\left(|\mu_{k}|^{\min\\{1,p-2\\}}\right)\right)$.
Thus

$\liminf_{k\rightarrow+\infty}|\mu_{k}|^{-3}{\left\lVert
H_{1}(v_{k})\right\rVert}_{H^{-1}(\mathcal{C})}\gtrsim E_{0}+F>0.$

This contradicts (3.16) and (3.8).

Now we show that the cubic estimate (2.4) is optimal. Choose a sequence
$\\{\mu_{k}\\}_{k}$ of numbers such that $\mu_{k}\neq 0$ and
$\mu_{k}\rightarrow 0$. Set

$w_{k}:=\left(1-C_{0}\mu_{k}^{2}\right)V_{0}+\mu_{k}\left(V_{0}^{\frac{p}{2}}\theta_{n}+\mu_{k}\eta\right).$
(3.35)

The constant $C_{0}=C_{0}(p,n)$ and the function $\eta\in H^{1}(\mathcal{C})$
will be determined later. We require $\eta$ to satisfy the orthogonality
conditions in (3.9) and (3.10). Moreover, we need $|\eta|\lesssim V_{0}$.
Thanks to Lemma 2.3 and our assumptions, we have the expansion

$\displaystyle\;\partial_{s}^{2}w_{k}+\Delta_{\theta}w_{k}-\Lambda
w_{k}+|w_{k}|^{p-2}w_{k}$ $\displaystyle=$
$\displaystyle-\left(1-C_{0}\mu_{k}^{2}\right)V_{0}^{p-1}-\mu_{k}(p-1)V_{0}^{p-2}V_{0}^{\frac{p}{2}}\theta_{n}+\mu_{k}^{2}\left(\partial_{s}^{2}\eta+\Delta_{\theta}\eta-\Lambda\eta\right)$
$\displaystyle+\left(1-C_{0}\mu_{k}^{2}\right)^{p-1}V_{0}^{p-1}+\mu_{k}(p-1)\left(1-C_{0}\mu_{k}^{2}\right)^{p-2}V_{0}^{p-2}V_{0}^{\frac{p}{2}}\theta_{n}$
$\displaystyle+\mu_{k}^{2}(p-1)\left(1-C_{0}\mu_{k}^{2}\right)^{p-2}V_{0}^{p-2}\eta+\mu_{k}^{2}\frac{(p-1)(p-2)}{2}\left(1-C_{0}\mu_{k}^{2}\right)^{p-3}V_{0}^{2p-3}\theta_{n}^{2}$
$\displaystyle+\mathcal{O}\left(|\mu_{k}|^{3}V_{0}^{p-1}\right)$
$\displaystyle=$
$\displaystyle-(p-2)C_{0}\mu_{k}^{2}V_{0}^{p-1}+\mu_{k}^{2}\left(\partial_{s}^{2}\eta+\Delta_{\theta}\eta-\Lambda\eta\right)+\mu_{k}^{2}(p-1)V_{0}^{p-2}\eta$
$\displaystyle+\mu_{k}^{2}\frac{(p-1)(p-2)}{2}V_{0}^{2p-3}\theta_{n}^{2}+\mathcal{O}\left(|\mu_{k}|^{3}V_{0}^{p-1}\right).$

Note that

$\inf_{t\in\mathbb{R}}{\left\lVert
w_{k}-V_{t}\right\rVert}_{H^{1}(\mathcal{C})}=|\mu_{k}|{\left\lVert
V_{0}^{\frac{p}{2}}\theta_{n}\right\rVert}_{H^{1}(\mathcal{C})}+\mathcal{O}\left(\mu_{k}^{2}\right).$

It suffices to choose suitable $C_{0}$ and $\eta$ such that

$-\partial_{s}^{2}\eta-\Delta_{\theta}\eta+\Lambda\eta-(p-1)V_{0}^{p-2}\eta+(p-2)C_{0}V_{0}^{p-1}=\frac{(p-1)(p-2)}{2}V_{0}^{2p-3}\theta_{n}^{2}.$
(3.36)

Assume $\eta$ has the following expansion in spherical harmonics

$\eta=\eta_{1}\left(\theta_{n}^{2}-\frac{1}{n}\right)+\eta_{2}.$

Then (3.36) is equivalent to

$\displaystyle\frac{(p-1)(p-2)}{2}V_{0}^{2p-3}\theta_{n}^{2}=$
$\displaystyle\;\left(-\partial_{s}^{2}\eta_{1}+(2n+\Lambda)\eta_{1}-(p-1)V_{0}^{p-2}\eta_{1}\right)\left(\theta_{n}^{2}-\frac{1}{n}\right)$
$\displaystyle-\partial_{s}^{2}\eta_{2}+\Lambda\eta_{2}-(p-1)V_{0}^{p-2}\eta_{2}+(p-2)C_{0}V_{0}^{p-1}.$

Thanks to Lemma 2.5, we can take $\eta_{1}$ to be the unique solution of

$-\partial_{s}^{2}\eta_{1}+(2n+\Lambda)\eta_{1}-(p-1)V_{0}^{p-2}\eta_{1}=\frac{(p-1)(p-2)}{2}V_{0}^{2p-3}.$

It remains to choose $C_{0}$ and $\eta_{2}$ such that

$-\partial_{s}^{2}\eta_{2}+\Lambda\eta_{2}-(p-1)V_{0}^{p-2}\eta_{2}+(p-2)C_{0}V_{0}^{p-1}=\frac{(p-1)(p-2)}{2n}V_{0}^{2p-3}.$

In fact, by direct computation we have

$-\partial_{s}^{2}V_{0}^{p-1}+\Lambda
V_{0}^{p-1}-(p-1)V_{0}^{p-2}V_{0}^{p-1}=\frac{2(p-1)(p-2)}{p}V_{0}^{2p-3}-p(p-2)\Lambda
V_{0}^{p-1}$

and

$-\partial_{s}^{2}V_{0}+\Lambda
V_{0}-(p-1)V_{0}^{p-2}V_{0}=-(p-2)V_{0}^{p-1}.$

Therefore, to achieve (3.36), we only need to take

$\eta_{2}:=\frac{p}{4n}V_{0}^{p-1}-\frac{p\int_{\mathcal{C}}V_{0}^{2p-2}}{4n\int_{\mathcal{C}}V_{0}^{p}}V_{0}$

and

$C_{0}:=\frac{p^{2}}{4n}\Lambda-\frac{p\int_{\mathcal{C}}V_{0}^{2p-2}}{4n\int_{\mathcal{C}}V_{0}^{p}}.$

Next we derive (2.5). Given $v\in H^{1}(\mathcal{C})$. Assume $V_{0}$ attains
the infimum in (2.4) and set $\rho:=v-V_{0}$. Define
$c_{0}=\left(\int_{\mathcal{C}}V_{0}^{p}\right)^{-1}\int_{\mathcal{C}}\rho
V_{0}^{p-1}$, $\rho_{1}=\Pi_{Y_{0}}(\rho)$ and
$\rho_{2}=\Pi_{Y_{0}}^{\perp}(\rho)-c_{0}V_{0}$. We have
$\rho=c_{0}V_{0}+\rho_{1}+\rho_{2}$. By testing $H_{1}(v)$ against $V_{0}$ and
using Lemma 2.3, we can obtain

$\displaystyle|c_{0}|\lesssim{\left\lVert
H_{1}(v)\right\rVert}_{H^{-1}(\mathcal{C})}+{\left\lVert\rho_{1}\right\rVert}^{\min\\{2,p-1\\}}_{H^{1}(\mathcal{C})}+{\left\lVert\rho_{2}\right\rVert}^{\min\\{2,p-1\\}}_{H^{1}(\mathcal{C})}.$
(3.37)

By testing $H_{1}(v)$ against $\rho_{2}$ and using orthogonality conditions,
we deduce

$\displaystyle{\left\lVert\rho_{2}\right\rVert}_{H^{1}(\mathcal{C})}^{2}\leq$
$\displaystyle\;{\left\lVert
H_{1}(v)\right\rVert}_{H^{-1}(\mathcal{C})}{\left\lVert\rho_{2}\right\rVert}_{H^{1}(\mathcal{C})}+(p-1)(1+c_{0})^{p-2}\int_{\mathcal{C}}V_{0}^{p-2}\rho_{2}^{2}$
$\displaystyle+C{\left\lVert\rho_{1}\right\rVert}^{\min\\{2,p-1\\}}_{H^{1}(\mathcal{C})}{\left\lVert\rho_{2}\right\rVert}_{H^{1}(\mathcal{C})}+C{\left\lVert\rho_{2}\right\rVert}^{\min\\{3,p\\}}_{H^{1}(\mathcal{C})}.$

It follows from Lemma 2.4 that

$\displaystyle{\left\lVert\rho_{2}\right\rVert}_{H^{1}(\mathcal{C})}^{2}\geq\gamma_{3}\int_{\mathcal{C}}V_{0}^{p-2}\rho_{2}^{2}$

and $\gamma_{3}>p-1$. Hence

$\displaystyle{\left\lVert\rho_{2}\right\rVert}_{H^{1}(\mathcal{C})}\lesssim{\left\lVert
H_{1}(v)\right\rVert}_{H^{-1}(\mathcal{C})}+{\left\lVert\rho_{1}\right\rVert}^{\min\\{2,p-1\\}}_{H^{1}(\mathcal{C})}.$
(3.38)

From the assumptions we have

$\displaystyle{\left\lVert\rho_{1}\right\rVert}^{\min\\{2,p-1\\}}_{H^{1}(\mathcal{C})}\ll|c_{0}|+{\left\lVert\rho_{2}\right\rVert}_{H^{1}(\mathcal{C})}.$
(3.39)

Combining (3.37), (3.38) and (3.39) yields

$\displaystyle|c_{0}|+{\left\lVert\rho_{2}\right\rVert}_{H^{1}(\mathcal{C})}\lesssim{\left\lVert
H_{1}(v)\right\rVert}_{H^{-1}(\mathcal{C})},$ (3.40)

which is equivalent to (2.5). ∎

###### Proof of Theorem 1.1.

From the global compactness principle (see [30, Proposition 3.2] for example),
we know that for any $\varepsilon>0$, if

${\left\lVert
H(u)\right\rVert}_{D_{a}^{-1,2}(\mathbb{R}^{n})}\leq\varepsilon,$

then $u$ is $o_{\varepsilon}(1)$-close to $\sum_{i=1}^{\nu}W_{i}$. Here
$\nu\in\mathbb{N}_{+}$ and $W_{i}\;(1\leq i\leq\nu)$ are nontrivial solutions
to the equation (1.1). Moreover, it holds that

${\left\lVert
u\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}^{2}=\sum_{i=1}^{\nu}{\left\lVert
W_{i}\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}^{2}+o_{\varepsilon}(1).$

Note that for any sign-changing solution $W$ to the equation (1.1), we have
the energy estimate

${\left\lVert W\right\rVert}^{2}\geq 2S(a,b,n)^{\frac{p}{p-2}}.$

It follows from (1.8) that when $\varepsilon$ is small, it must hold that
$\nu=1$ and $W_{1}$ is a Talenti bubble. Now we can take a number
$\varepsilon_{0}=\varepsilon_{0}(p,n)$ such that if

$\displaystyle{\left\lVert
H(u)\right\rVert}_{D_{a}^{-1,2}(\mathbb{R}^{n})}\leq\varepsilon_{0},$ (3.41)

then

$\inf_{\lambda>0}{\left\lVert
u-U_{\lambda}\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}\leq\delta.$

Here $\delta$ is the constant that appears in Theorem 1.7 and Theorem 2.1. It
follows from Theorem 1.7 that (1.9) and (1.10) hold when (3.41) holds. If
(3.41) does not hold, (1.9) and (1.10) clearly hold since the left sides of
them have trivial upper bounds. One can verify the optimality of (1.9) by
choosing the functions $w_{k}$ defined in (3.35). ∎

######  Remark 3.1.

In the _Case 2_ in our proof of Theorem 2.1, we choose
$\mu_{k}\left(V_{0}^{\frac{p}{2}}\theta_{n}+2\eta_{k}\right)$ as the test
function. A more natural choice, however, is
$\mu_{k}\left(V_{0}^{\frac{p}{2}}\theta_{n}+\eta_{k}\right)$, as it describes
the distance from the function $v_{k}$ to the manifold $\left\\{\lambda
V_{t}\;\big{|}\;\lambda,t\in\mathbb{R}\right\\}$. However, this choice hinders
the proof over the Felli-Schneider curve’s full parameter range. Indeed, for
any $\lambda>0$, if we use
$\mu_{k}\left(V_{0}^{\frac{p}{2}}\theta_{n}+\lambda\eta_{k}\right)$ as the
test function, we will finally deduce

$\liminf_{k\rightarrow+\infty}|\mu_{k}|^{-3}{\left\lVert
H_{1}(v_{k})\right\rVert}_{H^{-1}(\mathcal{C})}\gtrsim\frac{(\lambda+2)^{2}}{4\lambda}E_{0}+2F.$

If $\lambda\neq 2$, from Lemma 2.6 we know that our proof fails when $n\geq 3$
and $p$ is close to $2^{*}$. Therefore, our initial choice of the test
function appears to be optimal.

######  Remark 3.2.

The construction (3.35) of the sequence $\\{w_{k}\\}_{k}$ in the proof of
Theorem 2.1 is technical. Since the degeneracy of the equation (2.2) stems
from the nontrivial functions $V_{0}^{\frac{p}{2}}\theta_{i},1\leq i\leq n$,
the sequence
$\left\\{\hat{w}_{k}:=V_{0}+\mu_{k}V_{0}^{\frac{p}{2}}\theta_{n}\right\\}_{k}$
appears to be a natural choice. However, this sequence can only yield a
quadratic estimate. The sequence $\\{w_{k}\\}_{k}$ can be viewed as an
appropriate perturbation of $\\{\hat{w}_{k}\\}_{k}$. The process of
constructing the sequence $\\{w_{k}\\}_{k}$ demonstrates that our choices of
the number $C_{0}$ and the function $\eta$ are nearly optimal for obtaining a
cubic estimate.

######  Remark 3.3.

Here we derive the asymptotic estimate (1.15). Let us first give a
reformulation of it: let $\\{v_{k}\\}_{k}$ be a sequence in
$H^{1}(\mathcal{C})$ such that $v_{k}\notin\mathcal{M}_{p,\Lambda,n}$ and

$\inf_{t>0}{\left\lVert
v_{k}-V_{t}\right\rVert}_{H^{1}(\mathcal{C})}\rightarrow 0\;\;\text{as
}k\rightarrow+\infty,$

then there exists a positive number $R(p,n)$ such that

$\liminf_{k\rightarrow+\infty}\frac{{\left\lVert
H_{1}(v_{k})\right\rVert}_{H^{-1}(\mathcal{C})}}{\inf\limits_{t>0}{\left\lVert
v_{k}-V_{t}\right\rVert}_{H^{1}(\mathcal{C})}^{3}}\geq R(p,n).$ (3.42)

Without loss of generality, we assume that the limit of the left term exists
and is finite. Reviewing the proof of Theorem 2.1 with more attention to the
constant $C$, we can show that the _Case 2_ must hold and the value of (3.34)
can be bounded above by

$\left(|\mu_{k}|{\left\lVert
V_{0}^{\frac{p}{2}}\theta_{n}\right\rVert}_{H^{1}(\mathcal{C})}+\mathcal{O}\left(\mu_{k}^{2}\right)\right){\left\lVert
H_{1}(v_{k})\right\rVert}_{H^{-1}(\mathcal{C})}.$

Thanks to Lemma 2.6, we know that (3.34) has the lower bound

$2\mu_{k}^{4}\left(E_{0}+F+\mathcal{O}\left(|\mu_{k}|^{\min\\{1,p-2\\}}\right)\right).$

Thus we have

$\liminf_{k\rightarrow+\infty}|\mu_{k}|^{-3}{\left\lVert
H_{1}(v_{k})\right\rVert}_{H^{-1}(\mathcal{C})}\geq 2(E_{0}+F){\left\lVert
V_{0}^{\frac{p}{2}}\theta_{n}\right\rVert}_{H^{1}(\mathcal{C})}^{-1}.$

From (3) we get

$|\alpha_{k}-1|=\mathcal{O}(|\mu_{k}|^{\min\\{2,p-1\\}}).$

Therefore,

$\inf_{t>0}{\left\lVert
v_{k}-V_{t}\right\rVert}_{H^{1}(\mathcal{C})}\leq{\left\lVert
v_{k}-V_{0}\right\rVert}_{H^{1}(\mathcal{C})}=|\mu_{k}|\left({\left\lVert
V_{0}^{\frac{p}{2}}\theta_{n}\right\rVert}_{H^{1}(\mathcal{C})}+{\left\lVert\eta_{k}\right\rVert}_{H^{1}(\mathcal{C})}+\mathcal{O}(|\mu_{k}|^{\min\\{1,p-2\\}})\right).$

Combining all the arguments above, we can obtain

$\liminf_{k\rightarrow+\infty}\frac{{\left\lVert
H_{1}(v_{k})\right\rVert}_{H^{-1}(\mathcal{C})}}{\inf\limits_{t>0}{\left\lVert
v_{k}-V_{t}\right\rVert}_{H^{1}(\mathcal{C})}^{3}}\geq 2(E_{0}+F){\left\lVert
V_{0}^{\frac{p}{2}}\theta_{n}\right\rVert}_{H^{1}(\mathcal{C})}^{-4}=:R(p,n).$

Since the value of $E_{0}+F$ has been calculated in [24, Section 4], here we
can compute

$\displaystyle R(p,n)=$
$\displaystyle\;\alpha\beta^{-p}\pi^{-\frac{1}{2}}|\mathbb{S}^{n-1}|^{-1}\frac{2p(p-2)}{5p-6}\frac{\Gamma\left(\frac{2p-2}{p-2}+\frac{1}{2}\right)}{\Gamma\left(\frac{2p-2}{p-2}\right)}$
(3.43)
$\displaystyle\times\left(\frac{3p-4}{4p-4}-\frac{pn-3n}{pn+2p}-\frac{n-1}{n+2}\sum_{k=0}^{+\infty}\frac{P(k-\xi)-P(k)}{P(-1)}\right),$

where

$P(x):=\frac{\Gamma\left(x+\frac{3}{2}\right)\Gamma\left(x+2\xi_{1}-1\right)\Gamma\left(x+2\xi_{1}\right)}{\Gamma\left(x+\xi_{1}-\xi_{2}+1\right)\Gamma\left(x+\xi_{1}+\xi_{2}+1\right)\Gamma\left(x+2\xi_{1}+\frac{1}{2}\right)},$

$\alpha:=\frac{p-2}{2}\sqrt{\Lambda}$,
$\beta:=\left(\frac{p\Lambda}{2}\right)^{\frac{1}{p-2}}$,
$\xi_{1}:=\frac{2p-3}{p-2}$,
$\xi_{2}:=\frac{\sqrt{1+\frac{2n}{\Lambda}}}{p-2}$ and $\xi:=\xi_{1}-\xi_{2}$.

##  4 Proofs of Theorem 1.2 and Theorem 2.2

In this section we first establish Theorem 2.2. Theorem 1.8 can then be
obtained by using the Emden-Fowler transformation. Combining Theorem 1.8 and
the global compactness principle yields Theorem 1.2.

###### Proof of Theorem 2.2.

Assume

$\displaystyle\inf_{\begin{subarray}{c}s_{i}\in\mathbb{R}\\\ 1\leq
i\leq\nu\end{subarray}}{\left\lVert
v-\sum_{i=1}^{\nu}V_{s_{i}}\right\rVert}_{H^{1}(\mathcal{C})}={\left\lVert
v-\sum_{i=1}^{\nu}V_{t_{i}}\right\rVert}_{H^{1}(\mathcal{C})}\leq\delta$

and $|t_{i}-t_{j}|\geq\delta^{-1}$ for any $i\neq j$. For simplicity, we set

$\displaystyle\sigma:=\sum_{i=1}^{\nu}V_{t_{i}},\quad\rho:=v-\sigma,\quad
R:=\min_{i\neq j}\\{|t_{i}-t_{j}|\\},\quad Q:=e^{-\sqrt{\Lambda}R}.$

We also assume $-\infty=t_{0}<t_{1}<t_{2}<\cdots<t_{\nu}<t_{\nu+1}=+\infty$
and $V_{t_{0}}=V_{t_{\nu+1}}=0$. By variation we can obtain the following
orthogonality conditions:

$\left\langle\rho,\partial_{s}V_{t_{i}}\right\rangle_{H^{1}(\mathcal{C})}=\left\langle\rho,V_{t_{i}}^{p-2}\partial_{s}V_{t_{i}}\right\rangle_{L^{2}(\mathcal{C})}=0.$
(4.1)

Let us first provide a lower bound for ${\left\lVert
H_{1}(v)\right\rVert}_{H^{-1}(\mathcal{C})}$. Specifically, we prove that,
when $\delta$ is small, it holds that

${\left\lVert
H_{1}(v)\right\rVert}_{H^{-1}(\mathcal{C})}+{\left\lVert\rho\right\rVert}_{H^{1}(\mathcal{C})}^{2}+\max_{1\leq
k\leq\nu}\left|\int_{\mathcal{C}}\sigma^{p-2}\rho\;\partial_{s}V_{t_{k}}\right|\gtrsim
Q.$ (4.2)

We adapt the ideas in [12, 30] to derive (4.2). consider the following
identity:

$\displaystyle\partial_{s}^{2}v+\Delta_{\theta}v-\Lambda v+v^{p-1}=$
$\displaystyle\;\partial_{s}^{2}\rho+\Delta_{\theta}\rho-\Lambda\rho+\left(v^{p-1}-\sigma^{p-1}-(p-1)\sigma^{p-2}\rho\right)$
$\displaystyle+\left(\sigma^{p-1}-\sum_{i=1}^{\nu}V_{t_{i}}^{p-1}\right)+(p-1)\sigma^{p-2}\rho.$

Fix a number $1\leq k\leq\nu$. Testing this identity with
$\partial_{s}V_{t_{k}}$ and using the orthogonality conditions in (4.1), we
deduce

$\displaystyle\;{\left\lVert
H_{1}(v)\right\rVert}_{H^{-1}(\mathcal{C})}+\left|\int_{\mathcal{C}}\sigma^{p-2}\rho\;\partial_{s}V_{t_{k}}\right|$
(4.3) $\displaystyle\gtrsim$
$\displaystyle\;\int_{\mathcal{C}}\left(v^{p-1}-\sigma^{p-1}-(p-1)\sigma^{p-2}\rho\right)\partial_{s}V_{t_{k}}+\int_{\mathcal{C}}\left(\sigma^{p-1}-\sum_{i=1}^{\nu}V_{t_{i}}^{p-1}\right)\partial_{s}V_{t_{k}}.$

We need to estimate the two integrals on the right hand side. As for the first
integral, we split $\mathcal{C}$ into two regions: $\\{2|\rho|\leq\sigma\\}$
and $\\{2|\rho|>\sigma\\}$. In the first region, based on Lemma 2.3 and the
fact that $|\partial_{s}V_{t_{k}}|\lesssim V_{t_{k}}$, we have

$\left|\left(v^{p-1}-\sigma^{p-1}-(p-1)\sigma^{p-2}\rho\right)\partial_{s}V_{t_{k}}\right|\lesssim
C\sigma^{p-2}\rho^{2}.$

In the second region, we can directly derive

$\left|\left(v^{p-1}-\sigma^{p-1}-(p-1)\sigma^{p-2}\rho\right)\partial_{s}V_{t_{k}}\right|\lesssim|\rho|^{p}.$

Thus we can obtain

$\left|\int_{\mathcal{C}}\left(v^{p-1}-\sigma^{p-1}-(p-1)\sigma^{p-2}\rho\right)\partial_{s}V_{t_{k}}\right|\lesssim{\left\lVert\rho\right\rVert}_{H^{1}(\mathcal{C})}^{2}.$
(4.4)

As for the second integral, we decompose it into four parts:

$\displaystyle\int_{\mathcal{C}}\left(\sigma^{p-1}-\sum_{i=1}^{\nu}V_{t_{i}}^{p-1}\right)\partial_{s}V_{t_{k}}=\mathcal{I}_{1}+\mathcal{I}_{2}+\mathcal{I}_{3}+\mathcal{I}_{4},$

where

$\displaystyle\mathcal{I}_{1}=\int_{\\{\sigma_{k}\leq
V_{t_{k}}\\}}\left((p-1)V_{t_{k}}^{p-2}\sigma_{k}-\sum_{i=1,i\neq
k}^{\nu}V_{t_{i}}^{p-1}\right)\partial_{s}V_{t_{k}},$
$\displaystyle\mathcal{I}_{2}=\int_{\\{\sigma_{k}\leq
V_{t_{k}}\\}}\left(\sigma^{p-1}-V_{t_{k}}^{p-1}-(p-1)V_{t_{k}}^{p-2}\sigma_{k}\right)\partial_{s}V_{t_{k}},$
$\displaystyle\mathcal{I}_{3}=\int_{\\{\sigma_{k}>V_{t_{k}}\\}}\left((p-1)\sigma_{k}^{p-2}V_{t_{k}}+\sigma_{k}^{p-1}-\sum_{i=1}^{\nu}V_{t_{i}}^{p-1}\right)\partial_{s}V_{t_{k}},$
$\displaystyle\mathcal{I}_{4}=\int_{\\{\sigma_{k}>V_{t_{k}}\\}}\left(\sigma^{p-1}-\sigma_{k}^{p-1}-(p-1)\sigma_{k}^{p-2}V_{t_{k}}\right)\partial_{s}V_{t_{k}},$

and $\sigma_{k}:=\sigma-V_{t_{k}}$. From Lemma 2.3 and Lemma 2.7 we have

$\displaystyle|\mathcal{I}_{2}|\lesssim\int_{\\{\sigma_{k}\leq
V_{t_{k}}\\}}V_{t_{k}}^{p-2}\sigma_{k}^{2}\lesssim\int_{\mathcal{C}}V_{t_{k}}^{p-1-\varepsilon}\sigma_{k}^{1+\varepsilon}\lesssim
Q^{1+\varepsilon},$ (4.5)
$\displaystyle|\mathcal{I}_{4}|\lesssim\int_{\\{\sigma_{k}>V_{t_{k}}\\}}\sigma_{k}^{p-2}V_{t_{k}}^{2}\lesssim\int_{\mathcal{C}}\sigma_{k}^{p-1-\varepsilon}V_{t_{k}}^{1+\varepsilon}\lesssim
Q^{1+\varepsilon}.$

Here $\varepsilon=\varepsilon(p)$ is a small positive number such that
$\varepsilon<\frac{p-2}{3}$. As for $\mathcal{I}_{3}$, thanks to Lemma 2.7, we
have

$\displaystyle|\mathcal{I}_{3}|\lesssim$
$\displaystyle\;\int_{\\{\sigma_{k}>V_{t_{k}}\\}}\left(\sigma_{k}^{p-2}V_{t_{k}}^{2}+V_{t_{k}}^{p}\right)+\int_{\\{\sigma_{k}>V_{t_{k}}\\}}\left|\sigma_{k}^{p-1}-\sum_{i=1,i\neq
k}^{\nu}V_{t_{i}}^{p-1}\right|V_{t_{k}}$ (4.6) $\displaystyle\lesssim$
$\displaystyle\;\int_{\mathcal{C}}\sigma_{k}^{p-1-\varepsilon}V_{t_{k}}^{1+\varepsilon}+\sum_{\begin{subarray}{c}1\leq
i<j\leq\nu\\\ i,j\neq
k\end{subarray}}\int_{\mathcal{C}}V_{t_{i}}^{p-2}V_{t_{j}}V_{t_{k}}$
$\displaystyle\lesssim$ $\displaystyle\;Q^{1+\varepsilon}.$

In the last step of the above estimate, we use the Hölder inequality and the
observation that for any $i,j\neq k$,

$\max\\{|t_{i}-t_{j}|,|t_{i}-t_{k}|,|t_{j}-t_{k}|\\}\geq 2R.$

It remains to consider $\mathcal{I}_{1}$. Note that

$\displaystyle\mathcal{I}_{1}=$
$\displaystyle\;(p-1)\left(\int_{\mathcal{C}}-\int_{\\{\sigma_{k}>V_{t_{k}}\\}}\right)V_{t_{k}}^{p-2}\partial_{s}V_{t_{k}}\sigma_{k}-\sum_{i=1,i\neq
k}^{\nu}\int_{\\{\sigma_{k}\leq
V_{t_{k}}\\}}V_{t_{i}}^{p-1}\partial_{s}V_{t_{k}}$ $\displaystyle\geq$
$\displaystyle\;\sum_{i\neq
k}\int_{\mathcal{C}}\partial_{s}(V_{t_{k}})^{p-1}V_{t_{i}}-C\int_{\\{\sigma_{k}>V_{t_{k}}\\}}V_{t_{k}}^{p-1-\varepsilon}\sigma_{k}^{1+\varepsilon}-C\int_{\mathcal{C}}\sigma_{k}^{p-1-\varepsilon}V_{t_{k}}^{1+\varepsilon}$
(4.7) $\displaystyle\geq$ $\displaystyle\;\sum_{i\neq
k}\int_{\mathcal{C}}\partial_{s}V_{t_{k}}V_{t_{i}}^{p-1}-CQ^{1+\varepsilon}.$

From the expression of $V_{t}(s)$ (see (2.3)), we know that

$\displaystyle\int_{\mathcal{C}}\partial_{s}V_{t_{i}}V_{t_{j}}^{p-1}\geq
0\quad\text{if and only if}\;\;i\geq j.$

For any $j\geq k+2$, from Lemma 2.7 and the estimate
$|\partial_{s}V_{t_{k}}|\lesssim V_{t_{k}}$, we can deduce

$\displaystyle-\int_{\mathcal{C}}\partial_{s}V_{t_{k}}V_{t_{j}}^{p-1}\lesssim
Q^{2}.$

Based on Lemma 2.8 and the above arguments, we can find three positive
constants $C_{1},C_{2},C_{3}$ such that

$\sum_{i\neq k}\int_{\mathcal{C}}\partial_{s}V_{t_{k}}V_{t_{i}}^{p-1}\geq
C_{1}e^{-\sqrt{\Lambda}|t_{k-1}-t_{k}|}-C_{2}e^{-\sqrt{\Lambda}|t_{k}-t_{k+1}|}-C_{3}Q^{2}.$

Now set

$k_{0}=\sup\\{1\leq i\leq\nu\;|\;|t_{i}-t_{i-1}|=R\\}$

and

$k_{1}=\sup\left\\{k_{0}\leq
i<\nu\;\big{|}\;t_{j+1}-t_{j}<t_{j}-t_{j-1}+\Lambda^{-\frac{1}{2}}\ln(2C_{2}C_{1}^{-1})\;\text{for
any }k_{0}\leq j\leq i\right\\}.$

If $k_{0}=\nu$, choose $k=k_{0}$. If not, choose $k=k_{1}$. It is not hard to
check that

$\sum_{i\neq k}\int_{\mathcal{C}}\partial_{s}V_{t_{k}}V_{t_{i}}^{p-1}\gtrsim
Q$ (4.8)

provided $\delta$ is sufficiently small. Combining the estimates (4.3), (4.4),
(4.5), (4.6), (4) and (4.8) gives (4.2).

Next we need to find an upper bound of
${\left\lVert\rho\right\rVert}_{H^{1}(\mathcal{C})}$. As in [30] (see also
[12]), we define

$Q_{i,j}:=e^{-\sqrt{\Lambda}|t_{i}-t_{j}|},\quad\varphi_{t_{i}}(s):=e^{-\sqrt{\Lambda}|s-t_{i}|}\quad\text{for
}1\leq i,j\leq\nu,$ $\displaystyle W_{1}(s):=$
$\displaystyle\;\sum_{i=1}^{\nu-1}Q_{i,i+1}\varphi_{t_{i}}^{p-3}(s)\chi\left(t_{i}+1\leq
s\leq\frac{t_{i}+t_{i+1}}{2}\right)+Q_{\nu-1,\nu}\varphi_{t_{\nu}}^{1-\zeta}(s)\chi(t_{\nu}+1\leq
s)$
$\displaystyle+\sum_{i=1}^{\nu-1}Q_{i,i+1}\varphi_{t_{i+1}}^{p-3}(s)\chi\left(\frac{t_{i}+t_{i+1}}{2}\leq
s\leq t_{i+1}-1\right)+Q_{1,2}\varphi_{t_{1}}^{1-\zeta}(s)\chi(s\leq t_{1}-1)$
$\displaystyle+\sum_{i=1}^{\nu}Q\chi(t_{i}-1\leq s\leq t_{i}+1),$
$\displaystyle W_{2}(s):=$
$\displaystyle\sum_{i=1}^{\nu}Q\varphi_{t_{i}}^{1-\zeta}(s)\chi\left(\frac{t_{i}+t_{i-1}}{2}\leq
s\leq\frac{t_{i}+t_{i+1}}{2}\right),$ $\displaystyle W_{3}(s):=$
$\displaystyle\;\sum_{i=1}^{\nu-1}Q_{i,i+1}\varphi_{t_{i}}^{p-3}(s)\chi\left(t_{i}+2\leq
s\leq\frac{t_{i}+t_{i+1}}{2}\right)+Q_{\nu-1,\nu}\varphi_{t_{\nu}}^{1-\zeta}(s)\chi(t_{\nu}+2\leq
s)$
$\displaystyle+\sum_{i=1}^{\nu-1}Q_{i,i+1}\varphi_{t_{i+1}}^{p-3}(s)\chi\left(\frac{t_{i}+t_{i+1}}{2}\leq
s\leq t_{i+1}-2\right)+Q_{1,2}\varphi_{t_{1}}^{1-\zeta}(s)\chi(s\leq t_{1}-2)$
$\displaystyle+\sum_{i=1}^{\nu}Q\chi(t_{i}-2\leq s\leq t_{i}+2).$

Here $\zeta=\zeta(p,n,\nu)>0$ is a sufficiently small number. We also define
three new norms:

${\left\lVert
h\right\rVert}_{i}:=\sup_{\mathcal{C}}|h(s,\theta)|W_{i}^{-1}(s),\quad 1\leq
i\leq 3.$

From [30, Lemma 5.3] we know that

${\left\lVert\sigma^{p-1}-\sum_{i=1}^{\nu}V_{t_{i}}^{p-1}\right\rVert}_{1}\lesssim
1$ (4.9)

when $2<p<4$ and

${\left\lVert\sigma^{p-1}-\sum_{i=1}^{\nu}V_{t_{i}}^{p-1}\right\rVert}_{2}\lesssim
1$ (4.10)

when $p\geq 4$. By slightly modifying the finite-dimensional reduction method
used in [30, Lemma 5.4, 5.5] (see also [12, Proposition 3.8]), we get that,
when $\delta$ is small, there exists a function $\rho_{0}\in
H^{1}(\mathcal{C})$ and two sequences of numbers
$\\{c_{i}\\}_{1}^{\nu},\\{c_{ij}\\}^{\nu,n}_{i,j}$ such that

$\displaystyle-\partial_{s}^{2}\rho_{0}-\Delta_{\theta}\rho_{0}+\Lambda\rho_{0}-p\sigma^{p-2}\rho_{0}=$
$\displaystyle\;\sigma^{p-1}-\sum_{i=1}^{\nu}V_{t_{i}}^{p-1}$
$\displaystyle+|\sigma+\rho_{0}|^{p-2}(\sigma+\rho_{0})-\sigma^{p-1}-p\sigma^{p-2}\rho_{0}$
(4.11)
$\displaystyle+\sum_{i=1}^{\nu}c_{i}V_{t_{i}}^{p-2}\partial_{s}V_{t_{i}}+\sum_{i=1}^{\nu}\sum_{j=1}^{n}c_{ij}V_{t_{i}}^{p-2}V_{t_{i}}^{\frac{p}{2}}\theta_{j},$
$\left\langle\rho_{0},\partial_{s}V_{t_{i}}\right\rangle_{H^{1}(\mathcal{C})}=\left\langle\rho_{0},V^{\frac{p}{2}}_{t_{i}}\theta_{j}\right\rangle_{H^{1}(\mathcal{C})}=\left\langle\rho_{0},V_{t_{i}}^{p-2}\partial_{s}V_{t_{i}}\right\rangle_{L^{2}(\mathcal{C})}=\left\langle\rho_{0},V_{t_{i}}^{p-2}V^{\frac{p}{2}}_{t_{i}}\theta_{j}\right\rangle_{L^{2}(\mathcal{C})}=0$
(4.12)

and

${\left\lVert\rho_{0}\right\rVert}_{3}\chi(p<4)+{\left\lVert\rho_{0}\right\rVert}_{2}\chi(p\geq
4)\lesssim 1.$ (4.13)

Let us test (4) with $\partial_{s}V_{t_{i}}$,
$V_{t_{i}}^{\frac{p}{2}}\theta_{j}$ and $\rho_{0}$. Exploiting Lemma 2.3,
Lemma 2.7, the estimates (4.9), (4.10),(4.12), (4.13) and the fact
$|\partial_{s}V_{t_{i}}|\lesssim V_{t_{i}}$ yields

$\displaystyle|c_{i}|\int_{\mathcal{C}}V_{t_{i}}^{p-2}(\partial_{s}V_{t_{i}})^{2}+o_{\delta}(1)\sum_{j\neq
i}c_{j}\lesssim\chi(p<4)\int_{\mathcal{C}}(W_{1}+W_{3})V_{t_{i}}+\chi(p\geq
4)\int_{\mathcal{C}}W_{2}V_{t_{i}}\lesssim Q,$
$\displaystyle|c_{ij}|\int_{\mathcal{C}}V_{t_{i}}^{p-2}(V_{t_{i}}^{\frac{p}{2}}\theta_{j})^{2}+o_{\delta}(1)\sum_{k\neq
i}c_{kj}\lesssim\chi(p<4)\int_{\mathcal{C}}W_{3}V_{t_{i}}+\chi(p\geq
4)\int_{\mathcal{C}}W_{2}V_{t_{i}}\lesssim Q$

and

$\displaystyle{\left\lVert\rho_{0}\right\rVert}_{H^{1}(\mathcal{C})}^{2}\lesssim$
$\displaystyle\;\int_{\mathcal{C}}\sigma^{p-2}\rho_{0}^{2}+\chi(p<4)\int_{\mathcal{C}}W_{1}W_{3}+\chi(p\geq
4)\int_{C}W_{2}^{2}+{\left\lVert\rho_{0}\right\rVert}_{H^{1}(\mathcal{C})}^{\min\\{3,p\\}}$
$\displaystyle\lesssim$
$\displaystyle\;\sum_{i=1}^{\nu}\int_{\mathcal{C}}V_{t_{i}}^{p-2}\left(\chi(p<4)W^{2}_{3}+\chi(p\geq
4)W^{2}_{2}\right)+F_{1}^{2}(Q)+{\left\lVert\rho_{0}\right\rVert}_{H^{1}(\mathcal{C})}^{\min\\{3,p\\}}$
$\displaystyle\lesssim$
$\displaystyle\;F_{1}^{2}(Q)+{\left\lVert\rho_{0}\right\rVert}_{H^{1}(\mathcal{C})}^{\min\\{3,p\\}}.$

Therefore, when $\delta$ is sufficiently small, we have

$\sum_{i=1}^{\nu}|c_{i}|+\sum_{i=1}^{\nu}\sum_{j=1}^{n}|c_{ij}|\lesssim
Q,\quad{\left\lVert\rho_{0}\right\rVert}_{H^{1}(\mathcal{C})}\lesssim
F_{1}(Q).$ (4.14)

Define $\rho_{1}:=\rho-\rho_{0}\in H^{1}(\mathcal{C})$. It remains to estimate
${\left\lVert\rho_{1}\right\rVert}_{H^{1}(\mathcal{C})}$. Note that $\rho_{1}$
satisfies the equation

$\displaystyle-\partial_{s}^{2}\rho_{1}-\Delta_{\theta}\rho_{1}+\Lambda\rho_{1}=$
$\displaystyle\;|\sigma+\rho_{0}+\rho_{1}|^{p-2}(\sigma+\rho_{0}+\rho_{1})-|\sigma+\rho_{0}|^{p-2}(\sigma+\rho_{0})$
(4.15)
$\displaystyle-\sum_{i=1}^{\nu}c_{i}V_{t_{i}}^{p-2}\partial_{s}V_{t_{i}}-\sum_{i=1}^{\nu}\sum_{j=1}^{n}c_{ij}V_{t_{i}}^{p-2}V_{t_{i}}^{\frac{p}{2}}\theta_{j}-H_{1}(v).$

Consider

$\displaystyle\rho_{1}=\sum_{i=1}^{\nu}\beta_{i}V_{t_{i}}+\sum_{i=1}^{\nu}\hat{\beta}_{i}\partial_{s}V_{t_{i}}+\sum_{i=1}^{\nu}\sum_{j=1}^{n}\beta_{ij}V_{t_{i}}^{\frac{p}{2}}\theta_{j}+\rho_{2},$

where $\rho_{2}$ satisfies

$\displaystyle\left\langle\rho_{2},V_{t_{i}}\right\rangle_{H^{1}(\mathcal{C})}=\left\langle\rho_{2},\partial_{s}V_{t_{i}}\right\rangle_{H^{1}(\mathcal{C})}=\left\langle\rho_{2},V_{t_{i}}^{\frac{p}{2}}\theta_{j}\right\rangle_{H^{1}(\mathcal{C})}=0,\quad\forall
1\leq i\leq\nu,1\leq j\leq n.$ (4.16)

Since $\rho_{1}$ is orthogonal to $\partial_{s}V_{t_{i}}$ for any $1\leq
i\leq\nu$, we have

$\sum_{i=1}^{\nu}|\hat{\beta}_{i}|\lesssim Q\sum_{i=1}^{\nu}|\beta_{i}|.$
(4.17)

We also have

$\displaystyle\Pi_{Y}(v)=\sum_{i=1}^{\nu}\sum_{j=1}^{n}\beta_{ij}V_{t_{i}}^{\frac{p}{2}}\theta_{j}.$
(4.18)

By testing (4.15) with $V_{t_{i}}$, using Lemma 2.3, Lemma 2.7 and recalling
the estimates (4.13), (4.14), (4.16), (4.17), we can deduce

$\displaystyle\;(p-2)|\beta_{i}|{\left\lVert
V_{t_{i}}\right\rVert}_{H^{1}(\mathcal{C})}^{2}+o_{\delta}(1)\sum_{j\neq
i}\beta_{j}$ $\displaystyle\leq$
$\displaystyle\;(p-1)\left|\int_{\mathcal{C}}\left(|\sigma+\rho_{0}|^{p-2}-V_{t_{i}}^{p-2}\right)\rho_{1}V_{t_{i}}\right|+C{\left\lVert\rho_{1}\right\rVert}_{H^{1}(\mathcal{C})}^{\min\\{2,p-1\\}}+CQ^{2}+C{\left\lVert
H_{1}(v)\right\rVert}_{H^{-1}(\mathcal{C})}$ $\displaystyle\lesssim$
$\displaystyle\;\chi(p\geq 3)\sum_{j\neq
i}\int_{\mathcal{C}}\left(V_{t_{j}}^{p-2}V_{t_{i}}+V_{t_{j}}V_{t_{i}}^{p-2}\right)|\rho_{1}|+\chi(p\geq
3)\int_{\mathcal{C}}\left(|\rho_{0}|^{p-2}V_{t_{i}}+|\rho_{0}|V_{t_{i}}^{p-2}\right)|\rho_{1}|$
$\displaystyle+\chi(p<3)\sum_{j\neq
i}\int_{\mathcal{C}}V_{t_{j}}^{\frac{p-1}{2}}V_{t_{i}}^{\frac{p-1}{2}}|\rho_{1}|+\chi(p<3)\int_{\mathcal{C}}|\rho_{0}|^{\frac{p-1}{2}}V_{t_{i}}^{\frac{p-1}{2}}|\rho_{1}|$
$\displaystyle+\sum_{j=1}^{\nu}|\beta_{j}|^{\min\\{2,p-1\\}}+\sum_{j=1}^{\nu}\sum_{k=1}^{n}|\beta_{jk}|^{\min\\{2,p-1\\}}+{\left\lVert\rho_{2}\right\rVert}_{H^{1}(\mathcal{C})}^{\min\\{2,p-1\\}}+{\left\lVert
H_{1}(v)\right\rVert}_{H^{-1}(\mathcal{C})}+Q^{2}$ $\displaystyle\lesssim$
$\displaystyle\;o_{\delta}(1)\sum_{j=1}^{\nu}|\beta_{j}|+o_{\delta}(1){\left\lVert\rho_{2}\right\rVert}_{H^{1}(\mathcal{C})}+Q^{2}+{\left\lVert
H_{1}(v)\right\rVert}_{H^{-1}(\mathcal{C})}$ (4.19)
$\displaystyle+\sum_{j=1}^{\nu}\sum_{k=1}^{n}|\beta_{jk}|^{\min\\{2,p-1\\}}+F_{3}(Q)\sum_{j=1}^{\nu}\sum_{k=1}^{n}|\beta_{jk}|,$

where

$\displaystyle F_{3}(x):=\begin{cases}x,&\text{if }p>3,\\\
x^{\frac{p-1}{2}}(-\ln{x})^{\frac{p-1}{p}},&\text{if }p\leq 3.\end{cases}$

Note that (4) is equivalent to

$\displaystyle|\beta_{i}|\lesssim
o_{\delta}(1){\left\lVert\rho_{2}\right\rVert}_{H^{1}(\mathcal{C})}+Q^{2}+{\left\lVert
H_{1}(v)\right\rVert}_{H^{-1}(\mathcal{C})}+\sum_{j=1}^{\nu}\sum_{k=1}^{n}|\beta_{jk}|^{\min\\{2,p-1\\}}+F_{3}(Q)\sum_{j=1}^{\nu}\sum_{k=1}^{n}|\beta_{jk}|.$
(4.20)

Similarly, by testing (4.15) with $\rho_{2}$, we can obtain

$\displaystyle{\left\lVert\rho_{2}\right\rVert}_{H^{1}(\mathcal{C})}^{2}-(p-1)\int_{\mathcal{C}}\sigma^{p-2}\rho_{2}^{2}$
$\displaystyle\leq$
$\displaystyle\;(p-1)\left|\int_{\mathcal{C}}|\sigma+\rho_{0}|^{p-2}\rho_{1}\rho_{2}-\sigma^{p-2}\rho_{2}^{2}\right|+C\left({\left\lVert\rho_{1}\right\rVert}_{H^{1}(\mathcal{C})}^{\min\\{2,p-1\\}}+{\left\lVert
H_{1}(v)\right\rVert}_{H^{-1}(\mathcal{C})}\right){\left\lVert\rho_{2}\right\rVert}_{H^{1}(\mathcal{C})}$
$\displaystyle\lesssim$
$\displaystyle\;\left|\int_{\mathcal{C}}\left(|\sigma+\rho_{0}|^{p-2}-\sigma^{p-2}\right)\rho_{2}^{2}\right|+\left(\sum_{j=1}^{\nu}|\beta_{j}|+\sum_{j=1}^{\nu}\sum_{k=1}^{n}|\beta_{jk}|\right)\int_{\mathcal{C}}\left||\sigma+\rho_{0}|^{p-2}-\sigma^{p-2}\right|\cdot\sigma|\rho_{2}|$
$\displaystyle+\left({\left\lVert\rho_{1}\right\rVert}_{H^{1}(\mathcal{C})}^{\min\\{2,p-1\\}}+{\left\lVert
H_{1}(v)\right\rVert}_{H^{-1}(\mathcal{C})}\right){\left\lVert\rho_{2}\right\rVert}_{H^{1}(\mathcal{C})}$
$\displaystyle\lesssim$
$\displaystyle\;o_{\delta}(1){\left\lVert\rho_{2}\right\rVert}_{H^{1}(\mathcal{C})}^{2}+F_{3}(Q)\left(\sum_{j=1}^{\nu}|\beta_{j}|+\sum_{j=1}^{\nu}\sum_{k=1}^{n}|\beta_{jk}|\right){\left\lVert\rho_{2}\right\rVert}_{H^{1}(\mathcal{C})}$
(4.21)
$\displaystyle+\left({\left\lVert\rho_{1}\right\rVert}_{H^{1}(\mathcal{C})}^{\min\\{2,p-1\\}}+{\left\lVert
H_{1}(v)\right\rVert}_{H^{-1}(\mathcal{C})}\right){\left\lVert\rho_{2}\right\rVert}_{H^{1}(\mathcal{C})}$

It follows from Lemma 2.4 and a standard localization argument (see for
example [19, Proposition 3.10]) that

${\left\lVert\rho_{2}\right\rVert}_{H^{1}(\mathcal{C})}^{2}\geq(\gamma_{3}-o_{\delta}(1))\int_{\mathcal{C}}\sigma^{p-2}\rho_{2}^{2}.$
(4.22)

Combining (4) and (4.22) gives

$\displaystyle{\left\lVert\rho_{2}\right\rVert}_{H^{1}(\mathcal{C})}\lesssim
o_{\delta}(1)\sum_{j=1}^{\nu}|\beta_{j}|+{\left\lVert
H_{1}(v)\right\rVert}_{H^{-1}(\mathcal{C})}+\sum_{j=1}^{\nu}\sum_{k=1}^{n}|\beta_{jk}|^{\min\\{2,p-1\\}}+F_{3}(Q)\sum_{j=1}^{\nu}\sum_{k=1}^{n}|\beta_{jk}|.$
(4.23)

From (4.20) and (4.23) we get

$\displaystyle\sum_{i=1}^{\nu}|\beta_{i}|+{\left\lVert\rho_{2}\right\rVert}_{H^{1}(\mathcal{C})}\lesssim
Q^{2}+{\left\lVert
H_{1}(v)\right\rVert}_{H^{-1}(\mathcal{C})}+\sum_{j=1}^{\nu}\sum_{k=1}^{n}|\beta_{jk}|^{\min\\{2,p-1\\}}+F_{3}(Q)\sum_{j=1}^{\nu}\sum_{k=1}^{n}|\beta_{jk}|$
(4.24)

Substituting (4.14), (4.17) and (4.24) into (4.2) yields

$\displaystyle{\left\lVert
H_{1}(v)\right\rVert}_{H^{-1}(\mathcal{C})}+\sum_{j=1}^{\nu}\sum_{k=1}^{n}|\beta_{jk}|^{2}+\max_{1\leq
k\leq\nu}\left|\int_{\mathcal{C}}\sigma^{p-2}\rho\;\partial_{s}V_{t_{k}}\right|\gtrsim
Q.$ (4.25)

Note that

$\displaystyle\left|\int_{\mathcal{C}}\sigma^{p-2}\rho\;\partial_{s}V_{t_{k}}\right|\lesssim$
$\displaystyle\;\left|\int_{\mathcal{C}}\sigma^{p-2}\rho_{0}\;\partial_{s}V_{t_{k}}\right|+\left|\int_{\mathcal{C}}\sigma^{p-2}\rho_{1}\;\partial_{s}V_{t_{k}}\right|$
$\displaystyle\lesssim$
$\displaystyle\;\left|\int_{\mathcal{C}}\left(\sigma^{p-2}-V_{t_{k}}^{p-2}\right)\rho_{0}\;\partial_{s}V_{t_{k}}\right|+\sum_{i=1}^{\nu}|\beta_{i}|\left|\int_{\mathcal{C}}\sigma^{p-2}V_{t_{i}}\;\partial_{s}V_{t_{k}}\right|$
$\displaystyle+\sum_{i=1}^{\nu}|\hat{\beta}_{i}|\left|\int_{\mathcal{C}}\sigma^{p-2}\partial_{s}V_{t_{i}}\;\partial_{s}V_{t_{k}}\right|+\left|\int_{\mathcal{C}}\sigma^{p-2}\rho_{2}\;\partial_{s}V_{t_{k}}\right|$
$\displaystyle\lesssim$
$\displaystyle\;o_{\delta}(1)Q+|\beta_{k}|\left|\int_{\mathcal{C}}\left(\sigma^{p-2}-V_{t_{k}}^{p-2}\right)V_{t_{k}}\;\partial_{s}V_{t_{k}}\right|$
$\displaystyle+\left|\int_{\mathcal{C}}\left(\sigma^{p-2}-V_{t_{k}}^{p-2}\right)\rho_{2}\;\partial_{s}V_{t_{k}}\right|$
$\displaystyle\lesssim$
$\displaystyle\;o_{\delta}(1)Q+F_{3}(Q){\left\lVert\rho_{2}\right\rVert}_{H^{1}(\mathcal{C})}.$
(4.26)

From (4.24), (4.25) and (4) we derive

$\displaystyle{\left\lVert
H_{1}(v)\right\rVert}_{H^{-1}(\mathcal{C})}+\sum_{j=1}^{\nu}\sum_{k=1}^{n}|\beta_{jk}|^{2}+F_{3}(Q)\sum_{j=1}^{\nu}\sum_{k=1}^{n}|\beta_{jk}|^{\min\\{2,p-1\\}}\gtrsim
Q.$ (4.27)

Now we define

$\displaystyle\psi_{1}:=\Pi_{Y}(v)=\sum_{i=1}^{\nu}\sum_{j=1}^{n}\beta_{ij}V_{t_{i}}^{\frac{p}{2}}\theta_{j}$

and

$\displaystyle\psi_{2}:=\Pi_{Y}^{\perp}(v)-\sigma=\rho_{0}+\sum_{i=1}^{\nu}\beta_{i}V_{t_{i}}+\sum_{i=1}^{\nu}\hat{\beta}_{i}\partial_{s}V_{t_{i}}+\rho_{2}.$

Based on the above arguments, we have the following two estimates:

$\displaystyle{\left\lVert\psi_{2}\right\rVert}_{H^{1}(\mathcal{C})}\lesssim
F_{1}(Q)+{\left\lVert
H_{1}(v)\right\rVert}_{H^{-1}(\mathcal{C})}+F_{3}(Q){\left\lVert\psi_{1}\right\rVert}_{H^{1}(\mathcal{C})}+{\left\lVert\psi_{1}\right\rVert}^{\min\\{2,p-1\\}}_{H^{1}(\mathcal{C})}$
(4.28)

and

$\displaystyle Q\lesssim{\left\lVert
H_{1}(v)\right\rVert}_{H^{-1}(\mathcal{C})}+F_{3}(Q){\left\lVert\psi_{1}\right\rVert}^{\min\\{2,p-1\\}}_{H^{1}(\mathcal{C})}+{\left\lVert\psi_{1}\right\rVert}^{2}_{H^{1}(\mathcal{C})}.$
(4.29)

If

$\displaystyle
F_{3}(Q){\left\lVert\psi_{1}\right\rVert}^{\min\\{2,p-1\\}}_{H^{1}(\mathcal{C})}+{\left\lVert\psi_{1}\right\rVert}^{2}_{H^{1}(\mathcal{C})}\leq{\left\lVert
H_{1}(v)\right\rVert}_{H^{-1}(\mathcal{C})},$

from (4.29) we get

$\displaystyle Q\lesssim{\left\lVert
H_{1}(v)\right\rVert}_{H^{-1}(\mathcal{C})},\quad{\left\lVert\psi_{1}\right\rVert}_{H^{1}(\mathcal{C})}\leq{\left\lVert
H_{1}(v)\right\rVert}_{H^{-1}(\mathcal{C})}^{\frac{1}{2}}.$ (4.30)

Combining (4.28) and (4.30) yields

$\displaystyle{\left\lVert\psi_{2}\right\rVert}_{H^{1}(\mathcal{C})}\lesssim
F_{1}\left({\left\lVert H_{1}(v)\right\rVert}_{H^{-1}(\mathcal{C})}\right),$
(4.31)

which indicates (2.6). If

$\displaystyle
F_{3}(Q){\left\lVert\psi_{1}\right\rVert}^{\min\\{2,p-1\\}}_{H^{1}(\mathcal{C})}+{\left\lVert\psi_{1}\right\rVert}^{2}_{H^{1}(\mathcal{C})}\geq{\left\lVert
H_{1}(v)\right\rVert}_{H^{-1}(\mathcal{C})},$

from (4.29) we obtain

$\displaystyle
Q\lesssim{\left\lVert\psi_{1}\right\rVert}_{H^{1}(\mathcal{C})}^{2},\quad{\left\lVert
H_{1}(v)\right\rVert}_{H^{-1}(\mathcal{C})}\lesssim{\left\lVert\psi_{1}\right\rVert}_{H^{1}(\mathcal{C})}^{2}.$
(4.32)

Combining (4.28) and (4.32) gives

$\displaystyle{\left\lVert\psi_{2}\right\rVert}_{H^{1}(\mathcal{C})}\lesssim
F_{1}\left({\left\lVert\psi_{1}\right\rVert}^{2}_{H^{1}(\mathcal{C})}\right),$
(4.33)

which contradicts our assumptions. The proof is complete. ∎

###### Proof of Theorem 1.2.

Thanks to the global compactness principle, there exists a number
$\varepsilon=\varepsilon(a,n,\nu)$ such that, when

$\displaystyle{\left\lVert
H(u)\right\rVert}_{D_{a}^{-1,2}(\mathbb{R}^{n})}\leq\varepsilon,$ (4.34)

we have

$\displaystyle\inf_{\begin{subarray}{c}s_{i}\in\mathbb{R}_{+}\\\ 1\leq
i\leq\nu\end{subarray}}{\left\lVert
u-\sum_{i=1}^{\nu}U_{s_{i}}\right\rVert}_{D_{a}^{1,2}(\mathbb{R}^{n})}\leq\delta$

and the infimum can be attained by $U_{\lambda_{i}},1\leq i\leq\nu$ with

$\displaystyle\min\left\\{\frac{\lambda_{i}}{\lambda_{j}},\frac{\lambda_{j}}{\lambda_{i}}\right\\}\leq\delta,\quad\forall\;i\neq
j.$

Thus when (4.34) holds, (1.13) comes from (2.6). If (4.34) does not hold,
(1.13) follows from the fact that the left side of it has an obvious upper
bound. ∎

Declarations

Conflict of interest On behalf of all authors, the corresponding author states
that there is no Conflict of interest.
Data Availability Statements All data generated or analyzed during this study
are included in this article.

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