#  Stability of Caffarelli-Kohn-Nirenberg inequality

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

In this paper, we consider the Caffarelli-Kohn-Nirenberg (CKN) inequality:

$\displaystyle\bigg{(}\int_{\mathbb{R}^{N}}|x|^{-b(p+1)}|u|^{p+1}dx\bigg{)}^{\frac{2}{p+1}}\leq
C_{a,b,N}\int_{\mathbb{R}^{N}}|x|^{-2a}|\nabla u|^{2}dx$

where $N\geq 3$, $a<\frac{N-2}{2}$, $a\leq b\leq a+1$ and
$p=\frac{N+2(1+a-b)}{N-2(1+a-b)}$. It is well-known that up to dilations
$\tau^{\frac{N-2}{2}-a}u(\tau x)$ and scalar multiplications $Cu(x)$, the CKN
inequality has a unique extremal function $W(x)$ which is positive and
radially symmetric in the parameter region $b_{FS}(a)\leq b<a+1$ with $a<0$
and $a\leq b<a+1$ with $a\geq 0$ and $a+b>0$, where $b_{FS}(a)$ is the Felli-
Schneider curve. We prove that in the above parameter region the following
stabilities hold:

  1. $(1)$

stability of CKN inequality in the functional inequality setting

$dist_{D^{1,2}_{a}}^{2}(u,\mathcal{Z})\lesssim\|u\|^{2}_{D^{1,2}_{a}(\mathbb{R}^{N})}-C_{a,b,N}^{-1}\|u\|^{2}_{L^{p+1}(|x|^{-b(p+1)},\mathbb{R}^{N})}$

where $\mathcal{Z}=\\{cW_{\tau}\mid
c\in\mathbb{R}\backslash\\{0\\},\tau>0\\}$;

  2. $(2)$

stability of CKN inequality in the critical point setting (in the class of
nonnegative functions)

$\displaystyle
dist_{D_{a}^{1,2}}(u,\mathcal{Z}_{0}^{\nu})\lesssim\left\\{\begin{aligned}
&\Gamma(u),\quad p>2\text{ or }\nu=1,\\\
&\Gamma(u)|\log\Gamma(u)|^{\frac{1}{2}},\quad p=2\text{ and }\nu\geq 2,\\\
&\Gamma(u)^{\frac{p}{2}},\quad 1<p<2\text{ and }\nu\geq
2,\end{aligned}\right.$

where $\Gamma(u)=\|div(|x|^{-a}\nabla
u)+|x|^{-b(p+1)}|u|^{p-1}u\|_{(D^{1,2}_{a})^{{}^{\prime}}}$ and

$\mathcal{Z}_{0}^{\nu}=\\{(W_{\tau_{1}},W_{\tau_{2}},\cdots,W_{\tau_{\nu}})\mid\tau_{i}>0\\}.$

Our results generalize the recent works in [7, 15, 11] on the sharp stability
of profile decompositions for the the special case $a=b=0$ (the Sobolev
inequality) to the above parameter region of the CKN inequality. This
parameter region is optimal for such stabilities in the sense that in the
region $a<b<b_{FS}(a)$ with $a<0$, the nonnegative solution of the Euler-
Lagrange equation of CKN inequality is no longer unique.

Keywords: Caffarelli-Kohn-Nirenberg inequality; Sharp stability; Profile
decomposition.

AMS Subject Classification 2010: 35B09; 35B33; 35B40; 35J20.

##  1\. Introduction

In this paper, we consider the following Caffarelli-Kohn-Nirenberg (CKN for
short) inequality:

$\displaystyle\bigg{(}\int_{\mathbb{R}^{N}}|x|^{-b(p+1)}|u|^{p+1}dx\bigg{)}^{\frac{2}{p+1}}\leq
C_{a,b,N}\int_{\mathbb{R}^{N}}|x|^{-2a}|\nabla u|^{2}dx,$ (1.1)

where

$\displaystyle N\geq 3,\quad-\infty<a<\frac{N-2}{2},\quad a\leq b\leq
a+1\quad\text{and}\quad p=\frac{N+2(1+a-b)}{N-2(1+a-b)},$ (1.2)

$u\in D^{1,2}_{a}(\mathbb{R}^{N})$ and $D^{1,2}_{a}(\mathbb{R}^{N})$ is the
Hilbert space given by

$\displaystyle D^{1,2}_{a}(\mathbb{R}^{N})=\\{u\in
D^{1,2}(\mathbb{R}^{N})\mid\int_{\mathbb{R}^{N}}|x|^{-2a}|\nabla
u|^{2}dx<+\infty\\}$ (1.3)

with the inner product

$\displaystyle\langle
u,v\rangle_{D^{1,2}_{a}(\mathbb{R}^{N})}=\int_{\mathbb{R}^{N}}|x|^{-2a}\nabla
u\nabla vdx$

and $D^{1,2}(\mathbb{R}^{N})=\dot{W}^{1,2}(\mathbb{R}^{N})$ being the usual
homogeneous Sobolev space (cf. [15, Definition 2.1]).

(1.1) is established in the celebrated paper [4] by Caffarelli, Kohn and
Nirenberg, as it is named, for a much more general version. Moreover, as
pointed out in [5], a fundamental task in understanding the CKN inequality
(1.1) is to study the best constants, existence (and nonexistence) of extremal
functions, as well as their qualitative properties for parameters $a$ and $b$
in the full region (1.2), since (1.1) contains the classical Sobolev
inequality ($a=b=0$) and the classical Hardy inequality ($a=0,b=1$) as special
cases, which have played important roles in many applications by virtue of the
complete knowledge about the best constants, extremal functions, and their
qualitative properties.

Under the condition (1.2), it is well-known (cf. [1, 6, 5, 17, 23]) that (1.1)
has extremal functions if and only if either for $a<b<a+1$ with $a<0$ or for
$a\leq b<a+1$ with $a\geq 0$. Moreover, let

$\displaystyle
b_{FS}(a)=\frac{N(a_{c}-a)}{2\sqrt{(a_{c}-a)^{2}+(N-1)}}+a-a_{c}>a$ (1.4)

be the Felli-Schneider curve found in [16], then it is also well-known (cf.
[1, 6, 12, 13, 14, 16, 17, 23]) that up to dilations $\tau^{a_{c}-a}u(\tau x)$
and scalar multiplications $Cu(x)$ (also up to translations $u(x+y)$ in the
special case $a=b=0$), (1.1) has a unique extremal function

$\displaystyle
W(x)=(2(p+1)(a_{c}-a)^{2})^{\frac{1}{(p-1)}}\bigg{(}1+|x|^{(a_{c}-a)(p-1)}\bigg{)}^{-\frac{2}{p-1}}$
(1.5)

either for $b_{FS}(a)\leq b<a+1$ with $a<0$ or for $a\leq b<a+1$ with $a\geq
0$ while, extremal functions of (1.1) must be non-radial for $a<b<b_{FS}(a)$
with $a<0$. Here, for the sake of simplicity, we denote $a_{c}=\frac{N-2}{2}$,
as that in [12, 13, 14]. On the other hand, it has been proved in [18] that
extremal functions of (1.1) must have $\mathcal{O}(N-1)$ symmetry for
$a<b<b_{FS}(a)$ with $a<0$, that is, extremal functions of (1.1) must depend
on the radius $r$ and the angle $\theta_{N}$ between the positive $x_{N}$-axis
and $\overrightarrow{Ox}$ for $a<b<b_{FS}(a)$ with $a<0$ up to rotations. To
our best knowledge, whether the extremal function of (1.1) is unique or not
for $a<b<b_{FS}(a)$ with $a<0$ is still unknown.

Once the extremal functions of (1.1) are well understood, it is natural to
study quantitative stability for the CKN inequality (1.1) by asking whether
the deviation of a given function from attaining equality in (1.1) controls
its distance to the family of extremal functions. These studies were initialed
by Brezís and Lieb in [3] by raising an open question for the classical
Sobolev inequality ($a=b=0$) which was settled by Bianchi and Egnell in [2]
while, quantitative stability for the Hardy-Sobolev inequality ($a=0$,
$0<b<1$) was studied in [20]. Since the extremal function of (1.1) is unique
up to dilations $\tau^{a_{c}-a}u(\tau x)$ and scalar multiplications $Cu(x)$
either for $b_{FS}(a)\leq b<a+1$ with $a<0$ or for $a\leq b<a+1$ with $a\geq
0$ and $a+b>0$, the smooth manifold

$\displaystyle\mathcal{Z}=\\{cW_{\tau}(x)\mid
c\in\mathbb{R}\backslash\\{0\\}\text{ and }\tau>0\\}$

is all extremal functions of (1.1) in the above parameter region. Thus, it is
natural to extend the quantitative stability for the Sobolev inequality
($a=b=0$) and the Hardy-Sobolev inequality ($a=0$, $0<b<1$) to the CKN
inequality (1.1) in the above parameter region. Our first result in this
paper, which devoted to this aspect, can be stated as follows.

######  Theorem 1.1.

Let $b_{FS}(a)$ be the Felli-Schneider curve given by (1.4) and assume that
either

  1. $(1)$

$b_{FS}(a)\leq b<a+1$ with $a<0$ or

  2. $(2)$

$a\leq b<a+1$ with $a\geq 0$ and $a+b>0$.

Then

$\displaystyle
dist_{D^{1,2}_{a}}^{2}(u,\mathcal{Z})\lesssim\|u\|^{2}_{D^{1,2}_{a}(\mathbb{R}^{N})}-C_{a,b,N}^{-1}\|u\|^{2}_{L^{p+1}(|x|^{-b(p+1)},\mathbb{R}^{N})}$

for all $u\in D^{1,2}_{a}(\mathbb{R}^{N})$, where
$L^{p+1}(|x|^{-b(p+1)},\mathbb{R}^{N})$ is the usual weighted Lebesgue space
and its usual norm is given by

$\displaystyle\|u\|_{L^{p+1}(|x|^{-b(p+1)},\mathbb{R}^{N})}=\bigg{(}\int_{\mathbb{R}^{N}}|x|^{-b(p+1)}|u|^{p+1}dx\bigg{)}^{\frac{1}{p+1}}.$

######  Remark 1.1.

The conditions $b_{FS}(a)\leq b<a+1$ with $a<0$ and $a\leq b<a+1$ with $a\geq
0$ and $a+b>0$ in Theorem 1.1 is sharp in the sense that, for $a<b<b_{FS}(a)$
with $a<0$, the extremal function of (1.1) is no longer $cW_{\tau}$.

On the other hand, it is well-known that the Euler-Lagrange equation of the
Sobolev inequality ($a=b=0$) is given by

$\displaystyle-\Delta u=|u|^{\frac{4}{N-2}}u,\quad\text{in }\mathbb{R}^{N}$
(1.6)

and the Aubin-Talanti bubbles, given by

$\displaystyle
U[z,\lambda]=[N(N-2)]^{\frac{N-2}{4}}\bigg{(}\frac{\lambda}{\lambda^{2}+|x-z|^{2}}\bigg{)}^{\frac{N-2}{2}},$

are the only poaitive solutions of (1.6), where $z\in\mathbb{R}^{N}$ and
$\lambda>0$. Thus, the smooth manifold (except $c=0$)

$\displaystyle\mathcal{M}=\\{cU[z,\lambda]\mid
c\in\mathbb{R},z\in\mathbb{R}^{N},\lambda>0\\}$

is all nonnagetive solutions of (1.6). Moreover, Struwe proved in [21] the
following well-known stability of profile decompositions to (1.6) for
nonnegative functions.

######  Theorem 1.2.

Let $N\geq 3$ and $\nu\geq 1$ be positive integers. Let $\\{u_{n}\\}\subset
D^{1,2}(\mathbb{R}^{N})$ be a nonnegative sequence with

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

where $S$ is the best Sobolev constant. Assume that $\|\Delta
u_{n}+|u_{n}|^{\frac{4}{N-2}}u_{n}\|_{H^{-1}}\to 0$ as $n\to\infty$, then
there exist a sequence $(z_{1}^{(n)},z_{2}^{(n)},\cdots,z_{\nu}^{(n)})$ of
$\nu$-tuples of points in $\mathbb{R}^{N}$ and a sequence of
$(\lambda_{1}^{(n)},\lambda_{2}^{(n)},\cdots,\lambda_{\nu}^{(n)})$ of
$\nu$-tuples of positive real numbers such that

$\displaystyle\|\nabla u_{n}-\sum_{i=1}^{\nu}\nabla
U[z_{i}^{(n)},\lambda_{i}^{(n)}]\|_{L^{2}(\mathbb{R}^{N})}\to 0\quad\text{as
}n\to\infty.$

In the recent papers [7, 15], Figalli et al. initialed a study on the
quantitative version of Theorem 1.2 by proving

  1. $(1)$

(Ciraolo-Figalli-Maggi [7]) Let $N\geq 3$ and $u\in D^{1,2}(\mathbb{R}^{N})$
be positive such that $\|\nabla
u\|_{L^{2}(\mathbb{R}^{N})}^{2}\leq\frac{3}{2}S^{\frac{N}{2}}$ and $\|\Delta
u+|u|^{\frac{4}{N-2}}u\|_{H^{-1}}\leq\delta$ for some $\delta>0$ sufficiently
small, then

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

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

  2. $(2)$

(Figalli-Glaudo [15]) Let $u\in D^{1,2}(\mathbb{R}^{N})$ be nonnegative such
that

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

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

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

where

$\displaystyle\mathcal{M}_{0}^{\nu}=\\{\sum_{i=1}^{\nu}U[z_{i},\lambda_{i}]\mid
z_{i}\in\mathbb{R}^{N},\lambda_{i}>0\\}.$

######  Remark 1.2.

The stability obtained in [15] is more general than the conclusion $(2)$
stated here in the sense that, $u$ could be sign-changing if $u$ is close to
the sum of $U[z_{i},\lambda_{i}]$ in $D^{1,2}(\mathbb{R}^{N})$ where
$U[z_{i},\lambda_{i}]$ are weakly interacting (the definition of weakly
interaction can be found in [15, Definition 3.1]). We choose to state the
conclusion $(2)$ since it is more close to Struwe’s theorem on the stability
of profile decompositions to (1.6) for nonnegative functions.

As pointed out in [15], it is rather surprisingly that the conclusion $(2)$ is
false for $N\geq 6$. Figalli and Glaudo constructed a counterexample of the
conclusion $(2)$ for $N\geq 6$ with two bubbles and conjectured in [15] that
the quantitative version of Theorem 1.2 for $N\geq 6$ and $\nu>1$ will be

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

with $0<\gamma(N)<1$ under the same assumptions of the conclusion $(2)$. In
the very recent work [11], the first author of the current paper, together
with Deng and Sun, proved that the quantitative version of Theorem 1.2 for
$N\geq 6$ and $\nu>1$ is actually

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

under the same assumptions of the conclusion $(2)$, where the orders of the
right hand sides in above estimates are optimal. We would like to refer the
recent works [7, 15] once more for more motivations, discussions and
applications of the study on the quantitative version of Theorem 1.2.

Since the Sobolev inequality ($a=b=0$) is a special case of the CKN inequality
(1.1) and the Euler-Lagrange equation of (1.1) is given by

$\displaystyle-div(|x|^{-a}\nabla u)=|x|^{-b(p+1)}|u|^{p-1}u,\quad\text{in
}\mathbb{R}^{N},$ (1.7)

it is natural to ask whether the stability of profile decompositions to (1.7)
for nonnegative functions which is similar to that of (1.6) holds or not. Our
second main result of this paper, which devoted to this natural question, can
be stated as follows.

######  Theorem 1.3.

Let $N\geq 3$ and $\nu\geq 1$ be positive integers. Let $b_{FS}(a)$ be the
Felli-Schneider curve given by (1.4) and assume that either $b_{FS}(a)\leq
b<a+1$ with $a<0$ or $a\leq b<a+1$ with $a\geq 0$ and $a+b>0$. Then for any
nonnegative $u\in D^{1,2}_{a}(\mathbb{R}^{N})$ such that

$\displaystyle(\nu-\frac{1}{2})(C_{a,b,N}^{-1})^{\frac{p+1}{p-1}}<\|u\|_{D^{1,2}_{a}(\mathbb{R}^{N})}^{2}<(\nu+\frac{1}{2})(C_{a,b,N}^{-1})^{\frac{p+1}{p-1}}$
(1.8)

and $\Gamma(u)\leq\delta$ with some $\delta>0$ sufficiently small, we have

$\displaystyle
dist_{D_{a}^{1,2}}(u,\mathcal{Z}_{0}^{\nu})\lesssim\left\\{\begin{aligned}
&\Gamma(u),\quad p>2\text{ or }\nu=1,\\\
&\Gamma(u)|\log\Gamma(u)|^{\frac{1}{2}},\quad p=2\text{ and }\nu\geq 2,\\\
&\Gamma(u)^{\frac{p}{2}},\quad 1<p<2\text{ and }\nu\geq
2.\end{aligned}\right.$

where $\Gamma(u)=\|div(|x|^{-a}\nabla
u)+|x|^{-b(p+1)}|u|^{p-1}u\|_{(D^{1,2}_{a})^{{}^{\prime}}}$ and

$\mathcal{Z}_{0}^{\nu}=\\{\sum_{j=1}^{\nu}W_{\tau_{j}}\mid\tau_{j}>0\\}.$

Moreover, this stability is sharp in the sense that, there exists nonnegative
$u_{*}\in D^{1,2}_{a}(\mathbb{R}^{N})$, satisfying (1.8) and
$\Gamma(u_{*})\leq\delta$ for some $\delta>0$ sufficiently small, such that

$\displaystyle
dist_{D_{a}^{1,2}}(u_{*},\mathcal{Z}_{0}^{\nu})\gtrsim\left\\{\begin{aligned}
&\Gamma(u_{*}),\quad p>2\text{ or }\nu=1,\\\
&\Gamma(u_{*})|\log\Gamma(u_{*})|^{\frac{1}{2}},\quad p=2\text{ and }\nu\geq
2,\\\ &\Gamma(u_{*})^{\frac{p}{2}},\quad 1<p<2\text{ and }\nu\geq
2.\end{aligned}\right.$

######  Remark 1.3.

  1. $(1)$

The conditions $b_{FS}(a)\leq b<a+1$ with $a<0$ and $a\leq b<a+1$ with $a\geq
0$ and $a+b>0$ in Theorem 1.3 is optimal in the sense that, for
$a<b<b_{FS}(a)$ with $a<0$, the nonnegative solutions of (1.7) is no longer
unique.

  2. $(2)$

$(2)$ of Theorem 1.3 can be generalized to more general class of functions as
that in [15, Theorem 3.3] by introducing the concept of weakly interaction for
the nonnegative solutions of (1.7) as that in [15, Definition 3.1].

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

##  2\. Preliminaries

Let $D^{1,2}_{a}(\mathbb{R}^{N})$ be the Hilbert space given by (1.3) with the
norm $\|\cdot\|_{D^{1,2}_{a}(\mathbb{R}^{N})}$. By [5, Proposition 2.2],
$D^{1,2}_{a}(\mathbb{R}^{N})$ is isomorphic to the Hilbert space
$H^{1}(\mathcal{C})$ by the transformation

$\displaystyle u(x)=|x|^{-(a_{c}-a)}v(-\ln|x|,\frac{x}{|x|}),$ (2.1)

where $\mathcal{C}=\mathbb{R}\times\mathbb{S}^{N-1}$ is the standard cylinder,
the inner product in $H^{1}(\mathcal{C})$ is given by

$\displaystyle\langle w,v\rangle_{H^{1}(\mathcal{C})}=\int_{\mathcal{C}}\nabla
w\nabla v+(a_{c}-a)^{2}uvd\mu$

with $d\mu$ being the volume element on $\mathcal{C}$, $u\in
D^{1,2}_{a}(\mathbb{R}^{N})$ and $w,v\in H^{1}(\mathcal{C})$.

The CKN inequality (1.1) can be rewritten as the following minimizing problem:

$\displaystyle C_{a,b,N}^{-1}=\inf_{u\in
D^{1,2}_{a}(\mathbb{R}^{N})\backslash\\{0\\}}\frac{\|u\|^{2}_{D^{1,2}_{a}(\mathbb{R}^{N})}}{\|u\|^{2}_{L^{p+1}(|x|^{-b(p+1)},\mathbb{R}^{N})}},$
(2.2)

where $L^{p+1}(|x|^{-b(p+1)},\mathbb{R}^{N})$ is the usual weighted Lebesgue
space and its usual norm is given by
$\|u\|_{L^{p+1}(|x|^{-b(p+1)},\mathbb{R}^{N})}=\bigg{(}\int_{\mathbb{R}^{N}}|x|^{-b(p+1)}|u|^{p+1}dx\bigg{)}^{\frac{1}{p+1}}$.
The Euler-Lagrange equation of the minimizing problem (2.2) is given by (1.7).
By (2.1), (2.2) is equivalent to the following minimizing problem:

$\displaystyle C_{a,b,N}^{-1}=\inf_{v\in
H^{1}(\mathcal{C})\backslash\\{0\\}}\frac{\|v\|^{2}_{H^{1}(\mathcal{C})}}{\|u\|^{2}_{L^{p+1}(\mathcal{C})}},$
(2.3)

where $\|\cdot\|_{L^{p+1}(\mathcal{C})}$ is the usual norm in the Lebesgue
space $L^{p+1}(\mathcal{C})$. Let $t=-\ln|x|$ and $\theta=\frac{x}{|x|}$ for
$x\in\mathbb{R}^{N}\backslash\\{0\\}$, then by [5, Proposition 2.2], (1.7) is
equivalent to the following equation of $v$:

$\displaystyle-\Delta_{\theta}v-\partial_{t}^{2}v+(a_{c}-a)^{2}v=|v|^{p-1}v,\quad\text{in
}\mathcal{C}$ (2.4)

where $\Delta_{\theta}$ is the Laplace-Beltrami operator on
$\mathbb{S}^{N-1}$.

Clearly, minimizers of (2.2) are ground states of (1.7). It is also well-known
(cf. [6, 5, 16]) that up to dilations $u_{\tau}(x)=\tau^{a_{c}-a}u(\tau x)$
and scalar multiplications $Cu(x)$ (also up to translations $u(x+y)$ for the
spacial case $a=b=0$), the radial function $W(x)$ given by (1.5) is also the
unique minimizer of the following minimizing problem

$\displaystyle C_{a,b,N,rad}^{-1}=\inf_{u\in
D^{1,2}_{a,rad}(\mathbb{R}^{N})\backslash\\{0\\}}\frac{\|u\|^{2}_{D^{1,2}_{a}(\mathbb{R}^{N})}}{\|u\|^{2}_{L^{p+1}(|x|^{-b(p+1)},\mathbb{R}^{N})}}$

under the condition (1.2), where

$\displaystyle D^{1,2}_{a,rad}(\mathbb{R}^{N})=\\{u\in
D^{1,2}_{a}(\mathbb{R}^{N})\mid u\text{ is radially symmetric}\\}.$

Thus, $W(x)$ is always a solution of (1.7) under the condition (1.2). It has
been proved in [6, 14] that $W(x)$ is the unique nonnegative solution of (1.7)
in $D^{1,2}_{a}(\mathbb{R}^{N})$ either for $b_{FS}(a)\leq b<a+1$ with $a<0$
or for $a\leq b<a+1$ with $a\geq 0$. Moreover, it has also been proved in [16]
that $W(x)$ is nondegenerate in $D^{1,2}_{a}(\mathbb{R}^{N})$ under the
condition (1.2). That is, up to scalar multiplications $CV(x)$,

$\displaystyle V(x):=\nabla W(x)\cdot
x-(a_{c}-a)W(x)=\frac{\partial}{\partial\lambda}(\lambda^{-(a_{c}-a)}W(\lambda
x))|_{\lambda=1}$ (2.5)

is the only nonzero solution in $D^{1,2}_{a}(\mathbb{R}^{N})$ to the
linearization of (1.7) around $W$ which is given by

$\displaystyle-div(|x|^{-a}\nabla u)=p|x|^{-b(p+1)}W^{p-1}u,\quad\text{in
}\mathbb{R}^{N}.$ (2.6)

By the transformation (2.1), the linear equation (2.6) can be rewritten as
follows:

$\displaystyle-\Delta_{\theta}v-\partial_{t}^{2}v+(a_{c}-a)^{2}v=p\Psi^{p-1}v,\quad\text{in
}\mathcal{C},$ (2.7)

where $\Delta_{\theta}$ is the Laplace-Beltrami operator on
$\mathbb{S}^{N-1}$, $t=-\ln|x|$ and $\theta=\frac{x}{|x|}$ for
$x\in\mathbb{R}^{N}\backslash\\{0\\}$, and

$\displaystyle\Psi(t)=\bigg{(}\frac{(p+1)(a_{c}-a)^{2}}{2}\bigg{)}^{\frac{1}{p-1}}\bigg{(}cosh(\frac{(a_{c}-a)(p-1)}{2}t)\bigg{)}^{-\frac{2}{p-1}}.$
(2.8)

It follows from the transformation (2.1) that

$\displaystyle\Psi_{s}^{\prime}(t)=\Psi^{\prime}(t-\log
s)=\frac{\partial}{\partial t}\Psi(t-\log s)=-s\frac{\partial}{\partial
s}\Psi(t-\log s)$

is the only nonzero solution of (2.8) in $H^{1}(\mathcal{C})$ .

##  3\. Profile decompositions of nonnegative functions

It is well-known that all minimizers of (2.2) are positive in
$\mathbb{R}^{N}\backslash\\{0\\}$. Indeed, let

$\displaystyle\mathcal{E}(u)=\frac{1}{2}\|u\|^{2}_{D^{1,2}_{a}(\mathbb{R}^{N})}-\frac{1}{p+1}\|u\|^{p+1}_{L^{p+1}(|x|^{-b(p+1)},\mathbb{R}^{N})},$

then by (1.1), $\mathcal{E}(u)$ is of class $C^{2}$ in
$D^{1,2}_{a}(\mathbb{R}^{N})$. Since it is well-known that extremal functions
of (2.2) are ground states of (1.7), extremal functions of (2.2) are also
minimizers of the minimizing problem

$\displaystyle c=\inf_{u\in\mathcal{N}}\mathcal{E}(u),$

where

$\displaystyle\mathcal{N}=\\{u\in
D^{1,2}_{a}(\mathbb{R}^{N})\backslash\\{0\\}\mid\mathcal{E}^{\prime}(u)u=0\\}$

is the usual Nehari manifold. Since $p>1$ for $a\leq b<a+1$, it is standard to
use the fibering maps to show the double-energy property of $\mathcal{E}(u)$,
that is, $c_{sg}\geq 2c$, where
$c_{sg}=\inf_{u\in\mathcal{N}_{sg}}\mathcal{E}(u)$ with

$\displaystyle\mathcal{N}_{sg}=\\{u\in
D^{1,2}_{a}(\mathbb{R}^{N})\backslash\\{0\\}\mid\mathcal{E}^{\prime}(u^{\pm})u^{\pm}=0\\}$

and $u^{\pm}=\max\\{\pm u,0\\}$. Thus, by the the double-energy property of
$\mathcal{E}(u)$, all minimizers of $\mathcal{E}(u)$ in $\mathcal{N}$ at the
energy level $c$ are nonnegative which implies that all extremal functions of
(2.2) are nonnegative. It follows from the maximum principle that all extremal
functions of (2.2) are positive in $\mathbb{R}^{N}\backslash\\{0\\}$.

As that of the Sobolev and Hardy-Sobolev inequalities, we have the following
relatively compactness of minimizing sequences of (2.2).

######  Proposition 3.1.

Suppose that $\\{u_{n}\\}\subset D^{1,2}_{a}(\mathbb{R}^{N})$ be a minimizing
sequence of (2.2) either for $a<b<a+1$ with $a<0$ or for $a\leq b<a+1$ with
$a\geq 0$ and $a+b>0$. Then there exists $\\{\tau_{n}\\}\subset(0,+\infty)$
such that $(u_{n})_{\tau_{n}}\to u_{0}$ strongly in
$D^{1,2}_{a}(\mathbb{R}^{N})$ as $n\to\infty$ up to a subsequence, where
$u_{0}$ is a minimizer of (2.2). Moreover, we have $u_{0}=CW_{\tau_{0}}$ with
some $\tau_{0}>0$ and $C\in\mathbb{R}\backslash\\{0\\}$ either for
$b_{FS}(a)\leq b<a+1$ with $a<0$ or for $a\leq b<a+1$ with $a\geq 0$ and
$a+b>0$, where $b_{FS}(a)$ is the Felli-Schneider curve given by (1.4).

###### Proof.

Since the case $a\geq 0$ is considered in [24, Theorem 4], we shall only give
the proof for $a<0$. Moreover, the proof is rather standard nowadays (cf.
[22]), so we only sketch it here. Without loss of generality, we may assume
that

$\displaystyle\|u_{n}\|^{2}_{L^{p+1}(|x|^{-b(p+1)},\mathbb{R}^{N})}=1.$

Then $\\{u_{n}\\}$ is bounded in $D^{1,2}_{a}(\mathbb{R}^{N})$ and thus,
$u_{n}\rightharpoonup\widehat{u}_{0}$ weakly in $D^{1,2}_{a}(\mathbb{R}^{N})$
as $n\to\infty$ up to a subsequence. Moreover, by the double-energy property
of $\mathcal{E}(u)$, we may also assume that $\widehat{u}_{0}\geq 0$ without
loss of generality. By (2.1), the corresponding $v_{n}\rightharpoonup v_{0}$
weakly in $H^{1}(\mathcal{C})$ as $n\to\infty$ up to a subsequence with
$v_{0}\geq 0$. Since $a<b<a+1$ for $a<0$, we have $1<p<\frac{N+2}{N-2}$ by
(1.2). Thus, by [5, Lemma 4.1], there exists $\\{\tau_{n}\\}\subset\mathbb{R}$
such that

$\displaystyle\overline{v}_{n}=v_{n}(t-\tau_{n},\theta)\rightharpoonup\overline{v}_{0}\not=0\quad\text{weakly
in $H^{1}(\mathcal{C})$ as $n\to\infty$}.$

It follows from the Brezís-Lieb lemma and the concavity of the function
$t^{\frac{2}{p+1}}$ for $0<t<1$ with $p>1$ that

$\displaystyle 1+o_{n}(1)$ $\displaystyle=$ $\displaystyle
C_{a,b,N}(\|\overline{v}_{n}-\overline{v}_{0}\|_{H^{1}(\mathcal{C})}^{2}+\|\overline{v}_{0}\|_{H^{1}(\mathcal{C})}^{2})$
$\displaystyle\geq$
$\displaystyle(1-\|\overline{v}_{0}\|_{L^{p+1}(\mathcal{C})}^{p+1}+o_{n}(1))^{\frac{2}{p+1}}+(\|\overline{v}_{0}\|_{L^{p+1}(\mathcal{C})}^{p+1})^{\frac{2}{p+1}}$
$\displaystyle\geq$ $\displaystyle 1+o_{n}(1),$

which implies that $\overline{v}_{n}\to\overline{v}_{0}$ strongly in
$L^{p+1}(\mathcal{C})$ as $n\to\infty$. Correspondingly, by (2.1), we have
$(u_{n})_{\tau_{n}}\to u_{0}$ strongly in
$L^{p+1}(|x|^{-b(p+1)},\mathbb{R}^{N})$ as $n\to\infty$. It is then easy to
show that $u_{0}$ is a minimizer of (2.2). In the cases $b_{FS}(a)\leq b<a+1$
with $a<0$ or $a\leq b<a+1$ with $a\geq 0$ and $a+b>0$, $W$ is the unique
minimizer of (2.2) up to dilations $u_{\tau}(x)=\tau^{a_{c}-a}u(\tau x)$ and
scalar multiplications $Cu(x)$. Thus, we must have $u_{0}=CW_{\tau_{0}}$ with
some $\tau_{0}>0$ and $C\in\mathbb{R}\backslash\\{0\\}$. ∎

As the well-known results of profile decompositions to the Sobolev inequality
due to Struwe (cf. [21, 22]), we have the following profile decompositions of
(1.7) for nonnegative functions which, to our best knowledge, is new.

######  Proposition 3.2.

Let $\\{w_{n}\\}$ be a nonnegative $(PS)$ sequence of $\mathcal{E}(u)$ with

$\displaystyle(\nu-\frac{1}{2})(C_{a,b,N}^{-1})^{\frac{p+1}{p-1}}<\|w_{n}\|_{D^{1,2}_{a}(\mathbb{R}^{N})}^{2}<(\nu+\frac{1}{2})(C_{a,b,N}^{-1})^{\frac{p+1}{p-1}}$

with some $\nu\in\mathbb{N}$ either for $b_{FS}(a)\leq b<a+1$ with $a<0$ or
for $a\leq b<a+1$ with $a\geq 0$ and $a+b>0$ where $b_{FS}(a)$ is the Felli-
Schneider curve given by (1.4). Then there exists
$\\{\tau_{i,n}\\}\subset\mathbb{R}_{+}$, satisfying

$\displaystyle\min_{i\not=j}\bigg{\\{}\max\bigg{\\{}\frac{\tau_{i,n}}{\tau_{j,n}},\frac{\tau_{j,n}}{\tau_{i,n}}\bigg{\\}}\bigg{\\}}\to+\infty$

as $n\to\infty$ for $\nu\geq 2$, such that

  1. $(1)$

$w_{n}=\sum_{i=1}^{\nu}(W)_{\tau_{i,n}}+o_{n}(1)$ in
$D^{1,2}_{a}(\mathbb{R}^{N})$.

  2. $(2)$

$\|w_{n}\|_{D^{1,2}_{a}(\mathbb{R}^{N})}^{2}=\nu\|W\|_{D^{1,2}_{a}(\mathbb{R}^{N})}^{2}+o_{n}(1)$.

###### Proof.

Since $p=\frac{N+2}{N-2}$ for $a=b$ and $p<\frac{N+2}{N-2}$ for $a<b$, We
shall divide the proof into two parts which is devoted to the case
$p<\frac{N+2}{N-2}$ and $p=\frac{N+2}{N-2}$, respectively.

The case $p<\frac{N+2}{N-2}(a<b)$.

In this case, we use the transformation (2.1) to $w_{n}$. Then the related
$\widetilde{w}_{n}(t,\theta)$ satisfy

$\displaystyle(\nu-\frac{1}{2})(C_{a,b,N}^{-1})^{\frac{p+1}{p-1}}<\|\widetilde{w}_{n}\|_{H^{1}(\mathcal{C})}^{2}<(\nu+\frac{1}{2})(C_{a,b,N}^{-1})^{\frac{p+1}{p-1}}$

and $\mathcal{J}^{\prime}(\widetilde{w}_{n})\to 0$ in $H^{-1}(\mathcal{C})$ as
$n\to\infty$, where $H^{-1}(\mathcal{C})$ is the dual space of
$H^{1}(\mathcal{C})$ and

$\displaystyle\mathcal{J}(u)=\frac{1}{2}\|u\|^{2}_{H^{1}(\mathcal{C})}-\frac{1}{p+1}\|u\|^{p+1}_{L^{p+1}(\mathcal{C})}.$

Since $p<\frac{N+2}{N-2}$ and $\Psi(t)$ is the unique nonnegative solution of
(2.8) in $H^{1}(\mathcal{C})$ either for $b_{FS}(a)\leq b<a+1$ with $a<0$ or
for $a<b<a+1$ with $a\geq 0$ and $a+b>0$, the conclusion then follows from
(2.1) and adapting [5, Lemma 4.1] in a standard way.

The case $p=\frac{N+2}{N-2}(a=b)$.

In this case, we have $a>0$ by the assumptions. Moreover, [5, Lemma 4.1] is
invalid to drive the conclusion and thus, we shall mainly follows Struwe’s
idea in proving [22, Theorem 3.1]. However, according to the singular
potential $|x|^{-2a}$, the argument is more involved. Let $U_{\varepsilon}$ be
the standard Aubin-Talanti bubbles, that is,

$\displaystyle
U_{\varepsilon}=[N(N-2)]^{\frac{N-2}{4}}\bigg{(}\frac{\varepsilon}{\varepsilon^{2}+|x|^{2}}\bigg{)}^{\frac{N-2}{2}}.$

By [24, Lemma 1],

$\displaystyle C_{a,a,N}^{-1}<S\quad\text{for }a>0.$ (3.1)

Thus, there exists $R_{\varepsilon}>0$ such that

$\displaystyle\int_{B_{R_{\varepsilon}}(0)}|\nabla
U_{\varepsilon}|^{2}dx>\frac{1}{L_{R_{\varepsilon}}}(C_{a,b,N}^{-1})^{\frac{N}{2}},$
(3.2)

where $L_{R_{\varepsilon}}$ is the number such that the ball
$B_{2R_{\varepsilon}}(0)$ is covered by $L_{R_{\varepsilon}}$ balls with
radius $R_{\varepsilon}$. Here, we have used the fact that $2\leq
L_{R_{\varepsilon}}\leq 2^{N}$. Let

$\displaystyle
Q_{n}(r)=\sup_{y\in\mathbb{R}^{N}}\int_{B_{r}(y)}|x|^{-2a}|\nabla
w_{n}|^{2}dx$

be the well-known concentration function of $w_{n}$. Since

$\displaystyle(\nu-\frac{1}{2})(C_{a,a,N}^{-1})^{\frac{N}{2}}<\|w_{n}\|_{D^{1,2}_{a}(\mathbb{R}^{N})}^{2}<(\nu+\frac{1}{2})(C_{a,a,N}^{-1})^{\frac{N}{2}}$

for some $\nu\in\mathbb{N}$, we can choose $r_{n}>0$ and
$y_{n}\in\mathbb{R}^{N}$ such that

$\displaystyle Q_{n}(r_{n})=\int_{B_{r_{n}}(y_{n})}|x|^{-2a}|\nabla
w_{n}|^{2}dx=\frac{1}{2L_{R_{\varepsilon}}}(C_{a,a,N}^{-1})^{\frac{N}{2}}.$

Let

$\displaystyle
v_{n}=(r_{n}R_{\varepsilon}^{-1})^{-(a_{c}-a)}w_{n}(r_{n}R_{\varepsilon}^{-1}x),$

then

$\displaystyle\sup_{y\in\mathbb{R}^{N}}\int_{B_{R_{\varepsilon}}(y)}|x|^{-2a}|\nabla
v_{n}|^{2}dx=\int_{B_{R_{\varepsilon}}(\frac{R_{\varepsilon}y_{n}}{r_{n}})}|x|^{-2a}|\nabla
v_{n}|^{2}dx=\frac{1}{2L_{R_{\varepsilon}}}(C_{a,a,N}^{-1})^{\frac{N}{2}}.$
(3.3)

Since $\|\cdot\|_{D^{1,2}_{a}}$ is invariant under the dilation
$u_{\tau}(x)=\tau^{a_{c}-a}u(\tau x)$, $\\{v_{n}\\}$ is bounded in
$D^{1,2}(\mathbb{R}^{N})$ and thus, $v_{n}\rightharpoonup v_{0}$ weakly in
$D^{1,2}(\mathbb{R}^{N})$ as $n\to\infty$ up to a subsequence. Clearly,
$v_{0}\geq 0$. We define $\varpi_{n}=(v_{n}-v_{0})\varphi$, where $\varphi$ is
a smooth cut-off function such that $\varphi=1$ in $B_{R_{\varepsilon}}(z)$
and $\varphi=0$ in $B^{c}_{\frac{3}{2}R_{\varepsilon}}(z)$ for any
$z\in\mathbb{R}^{N}$. Then by [24, Lemma 2],

$\displaystyle\|\varpi_{n}\|_{D^{1,2}_{a}(\mathbb{R}^{N})}^{2}\lesssim\|v_{n}-v_{0}\|_{D^{1,2}_{a}(\mathbb{R}^{N})}^{2}+\int_{B_{2R_{\varepsilon}}(z)\backslash
B_{R_{\varepsilon}}(z)}|x|^{-2a}|v_{n}-v_{0}|^{2}dx\lesssim 1$

and

$\displaystyle\int_{B_{2R_{\varepsilon}}(z)\backslash
B_{R_{\varepsilon}}(z)}|x|^{-2a}|v_{n}-v_{0}|^{2}dx\to 0\quad\text{as
}n\to\infty.$

Note that by the fact that $\mathcal{E}^{\prime}(w_{n})\to 0$ in
$D^{-1,2}_{a}(\mathbb{R}^{N})$ as $n\to\infty$ and the invariance of
$\mathcal{E}(w_{n})$ under the the dilation $u_{\tau}(x)=\tau^{a_{c}-a}u(\tau
x)$, $\mathcal{E}^{\prime}(v_{n})\to 0$ in $D^{-1,2}_{a}(\mathbb{R}^{N})$ as
$n\to\infty$. It follows that $\mathcal{E}^{\prime}(v_{0})=0$, which, together
with the Brezís-Lieb lemma, implies

$\displaystyle o_{n}(1)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}(|x|^{-2a}\nabla(v_{n}-v_{0})\nabla(\varpi_{n}\varphi)-|x|^{-\frac{2Na}{N-2}}(v_{n}^{\frac{N+2}{N-2}}-v_{0}^{\frac{N+2}{N-2}})\varpi_{n}\varphi)dx$
(3.4) $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}(|x|^{-2a}|\nabla\varpi_{n}|^{2}-|x|^{-\frac{2Na}{N-2}}|v_{n}-v_{0}|^{\frac{4}{N-2}}|\varpi_{n}|^{2})dx+o_{n}(1)$
$\displaystyle\geq$
$\displaystyle\|\varpi_{n}\|_{D^{1,2}_{a}(\mathbb{R}^{N})}^{2}-\|\varpi_{n,*}\|_{L^{\frac{2N}{N-2}}(|x|^{-\frac{2Na}{N-2}},\mathbb{R}^{N})}^{\frac{4}{N-2}}\|\varpi_{n}\|_{L^{\frac{2N}{N-2}}(|x|^{-\frac{2Na}{N-2}},\mathbb{R}^{N})}^{2}$
$\displaystyle+o_{n}(1).$

Here, $\varpi_{n,*}=(v_{n}-v_{0})\varphi_{*}$, where $\varphi_{*}$ is a smooth
cut-off function such that $\varphi_{*}=1$ in
$B_{\frac{3}{2}R_{\varepsilon}}(z)$ and $\varphi_{*}=0$ in
$B^{c}_{2R_{\varepsilon}}(z)$. By the Brezís-Lieb lemma once more,

$\displaystyle\|\varpi_{n,*}\|_{L^{\frac{2N}{N-2}}(|x|^{-\frac{2Na}{N-2}},\mathbb{R}^{N})}^{\frac{2N}{N-2}}$
$\displaystyle\leq$
$\displaystyle\int_{B_{2R_{\varepsilon}}(z)}|x|^{-\frac{2Na}{N-2}}|v_{n}-v_{0}|^{\frac{2N}{N-2}}dx$
$\displaystyle\leq$
$\displaystyle\int_{B_{2R_{\varepsilon}}(z)}|x|^{-\frac{2Na}{N-2}}|v_{n}|^{\frac{2N}{N-2}}dx+o_{n}(1).$

It follows from the CKN inequality (1.1), (3.3) and (3.4) that $\varpi_{n}\to
0$ strongly in $D^{1,2}_{a}(\mathbb{R}^{N})$ as $n\to\infty$, which implies
that $v_{n}\to v_{0}$ strongly in $D^{1,2}_{a}(B_{R_{0}}(z))$ as $n\to\infty$
for any $z\in\mathbb{R}^{N}$. Thus, by a standard covering argument, $v_{n}\to
v_{0}$ strongly $D^{1,2}_{a,loc}(\mathbb{R}^{N})$ as $n\to\infty$. We claim
that $v_{0}\not=0$. Assume the contrary that $v_{0}=0$, then by (3.3),
$|\frac{R_{\varepsilon}y_{n}}{r_{n}}|\to+\infty$ as $n\to\infty$. It follows
that

$\displaystyle\int_{B_{\frac{1}{2}|\frac{R_{\varepsilon}y_{n}}{r_{n}}|}(\frac{R_{\varepsilon}y_{n}}{r_{n}})}|\nabla(\frac{v_{n}}{|\frac{R_{\varepsilon}y_{n}}{r_{n}}|^{a}})|^{2}\sim\int_{B_{\frac{1}{2}|\frac{R_{\varepsilon}y_{n}}{r_{n}}|}(\frac{R_{\varepsilon}y_{n}}{r_{n}})}|x|^{-2a}|\nabla
v_{n}|^{2}\lesssim\|v_{n}\|_{D^{1,2}_{a}(\mathbb{R}^{N})}^{2},$

which implies that $\\{\overline{v}_{n}\\}$ is bounded in
$D^{1,2}_{loc}(\mathbb{R}^{N})$ and thus,
$\overline{v}_{n}\rightharpoonup\overline{v}_{0}$ weakly in
$D^{1,2}_{loc}(\mathbb{R}^{N})$ as $n\to\infty$, where
$\overline{v}_{n}=\frac{v_{n}(x+\frac{R_{\varepsilon}y_{n}}{r_{n}})}{|\frac{R_{\varepsilon}y_{n}}{r_{n}}|^{a}}$.
Now, by $\mathcal{E}^{\prime}(v_{n})\to 0$ in $D^{-1,2}_{a}(\mathbb{R}^{N})$
as $n\to\infty$, we know that

$\displaystyle-\Delta\overline{v}_{n}=\overline{v}_{n}^{\frac{N+2}{N-2}}+o_{n}(1)\quad\text{in
}D^{1,2}_{loc}(\mathbb{R}^{N}).$ (3.5)

By (3.1), (3.3) and $|\frac{R_{\varepsilon}y_{n}}{r_{n}}|\to+\infty$ as
$n\to\infty$,

$\displaystyle\int_{B_{R_{\varepsilon}}(0)}|\nabla\overline{v}_{n}|^{2}dx$
$\displaystyle=$
$\displaystyle\sup_{y\in\mathbb{R}^{N}}\int_{B_{R_{\varepsilon}}(y)}|\nabla\overline{v}_{n}|^{2}dx$
$\displaystyle=$
$\displaystyle\frac{1}{2L_{R_{\varepsilon}}}(C_{a,b,N}^{-1})^{\frac{N}{2}}+o_{n}(1)$
$\displaystyle<$
$\displaystyle\frac{1}{2L_{R_{\varepsilon}}}S^{\frac{N}{2}}+o_{n}(1)$

Thus, by applying similar arguments as that used for (3.4) to (3.5), we can
show that $\overline{v}_{n}\to\overline{v}_{0}$ strongly in
$D^{1,2}(B_{R_{\varepsilon}}(0))$ as $n\to\infty$. By (3),
$\overline{v}_{0}\not=0$ and thus, $\overline{v}_{0}=U_{\varepsilon}$ for some
$\varepsilon>0$ by (3.5). It is impossible since (3.2) and (3) hold at the
same time now. Therefore, we must have $v_{0}\not=0$. Since $v_{0}\geq 0$, by
[6, Theorem B and Proposition 4.4] and [14, Theorem 1.2], we have $v_{0}=W$
either for $b_{FS}(a)\leq b<a+1$ with $a<0$ or for $a\leq b<a+1$ with $a\geq
0$ and $a+b>0$. Thus, $w_{n}\rightharpoonup(W)_{r_{n}R_{\varepsilon}^{-1}}$
weakly in $D^{1,2}_{a}(\mathbb{R}^{N})$ as $n\to\infty$. By running the above
argument to $w_{n}-(W)_{r_{n}R_{\varepsilon}^{-1}}$, we will arrive at that
$w_{n}\rightharpoonup(W)_{r_{n,1}r_{n}R_{\varepsilon}^{-1}R_{\varepsilon_{1}}^{-1}}+(W)_{r_{n,1}R_{\varepsilon_{1}}^{-1}}$
weakly in $D^{1,2}_{a}(\mathbb{R}^{N})$ as $n\to\infty$ for some $r_{n,1}>0$
and $\varepsilon_{1}>0$. The conclusion then follows from iterating the above
arguments for $\nu$ times and using the fact that $W(|x|)$ is the unique
nonnegative solution of (1.7) in $D^{1,2}_{a}(\mathbb{R}^{N})$ either for
$b_{FS}(a)\leq b<a+1$ with $a<0$ or for $a\leq b<a+1$ with $a\geq 0$ and
$a+b>0$. ∎

##  4\. Stability of CKN inequality in the functional inequality setting

It is well-known that the minimizing problem (2.2) and the equation (1.7) are
invariant under the dilation $u_{\tau}(x)=\tau^{\frac{N-2-2a}{2}}u(\tau x)$.
Thus, the smooth manifold

$\displaystyle\mathcal{Z}=\\{cW_{\tau}(x)\mid
c\in\mathbb{R}\backslash\\{0\\}\text{ and }\tau>0\\}$

is all extremal functions of the minimizing problem (2.2). Let

$\displaystyle d^{2}(u)=\inf_{c\in\mathbb{R},\ \
\tau>0}\|u-cW_{\tau}\|^{2}_{D^{1,2}_{a}(\mathbb{R}^{N})},$

where $u\in D^{1,2}_{a}(\mathbb{R}^{N})$. Then we have the following stability
for the CKN inequality (1.1).

######  Proposition 4.1.

Let
$e(u):=\|u\|^{2}_{D^{1,2}_{a}(\mathbb{R}^{N})}-C_{a,b,N}^{-1}\|u\|^{2}_{L^{p+1}(|x|^{-b(p+1)},\mathbb{R}^{N})}$.
Then $e(u)\gtrsim d^{2}(u)$ for all $u\in D^{1,2}_{a}(\mathbb{R}^{N})$ in the
following two cases:

  1. $(1)$

$b_{FS}(a)\leq b<a+1$ with $a<0$,

  2. $(2)$

$a\leq b<a+1$ with $a\geq 0$ and $a+b>0$.

###### Proof.

The proof mainly follows the arguments in [2] for the stability of the Sobolev
inequality. It is easy to see that $d^{2}(u)$ can be attained by some
$c_{0}\not=0$ and $\tau_{0}>0$. Indeed,

$\displaystyle\|u-cW_{\tau}\|^{2}_{D^{1,2}_{a}(\mathbb{R}^{N})}=\|u\|^{2}_{D^{1,2}_{a}(\mathbb{R}^{N})}+c^{2}\|W_{\tau}\|^{2}_{D^{1,2}_{a}(\mathbb{R}^{N})}-c\langle
u,W_{\tau}\rangle_{D^{1,2}_{a}(\mathbb{R}^{N})}.$

Thus, by taking $(c,\tau)\in(\mathbb{R},\mathbb{R}^{+})$ such that $c\langle
u,W_{\tau}\rangle_{D^{1,2}_{a}(\mathbb{R}^{N})}>0$ with $|c|>0$ sufficiently
small, we have $d^{2}(u)<\|u\|^{2}_{D^{1,2}_{a}(\mathbb{R}^{N})}$. By the
invariance of the norm $\|\cdot\|_{D^{1,2}_{a}(\mathbb{R}^{N})}$ under the
dilation $u_{\tau}(x)=\tau^{a_{c}-a}u(\tau x)$,

$\displaystyle\|u-cW_{\tau}\|^{2}_{D^{1,2}_{a}(\mathbb{R}^{N})}$
$\displaystyle=$
$\displaystyle\|u\|^{2}_{D^{1,2}_{a}(\mathbb{R}^{N})}+c^{2}\|W_{\tau}\|^{2}_{D^{1,2}_{a}(\mathbb{R}^{N})}-c\langle
u,W_{\tau}\rangle_{D^{1,2}_{a}(\mathbb{R}^{N})}$ $\displaystyle\geq$
$\displaystyle\|u\|^{2}_{D^{1,2}_{a}(\mathbb{R}^{N})}+c^{2}\|W\|^{2}_{D^{1,2}_{a}(\mathbb{R}^{N})}-|c|\|u\|_{D^{1,2}_{a}(\mathbb{R}^{N})}\|W\|_{D^{1,2}_{a}(\mathbb{R}^{N})}.$

Thus, the minimizing sequence of $d^{2}(u)$, say $\\{(c_{n},\tau_{n})\\}$,
must satisfy $|c_{n}|\sim 1$. On the other hand,

$\displaystyle|\int_{|\tau x|\leq\rho}|x|^{-2a}\nabla u\nabla W_{\tau}|$
$\displaystyle\leq$ $\displaystyle\int_{|y|\leq\rho}|y|^{-2a}|\nabla
u_{\frac{1}{\tau}}(y)\nabla W(y)|$ $\displaystyle\leq$
$\displaystyle\|u\|_{D^{1,2}_{a}(\mathbb{R}^{N})}\bigg{(}\int_{|y|\leq\rho}|y|^{-2a}|\nabla
W|^{2}\bigg{)}^{\frac{1}{2}}$ $\displaystyle=$ $\displaystyle o_{\rho}(1)$

as $\rho\to 0$ which is uniformly for $\tau>0$ and

$\displaystyle|\int_{|\tau x|\geq\rho}|x|^{-2a}\nabla u\nabla W_{\tau}|$
$\displaystyle\leq$
$\displaystyle\|W\|_{D^{1,2}_{a}(\mathbb{R}^{N})}\bigg{(}\int_{|x|\geq\frac{\rho}{\tau}}|x|^{-2a}|\nabla
u|^{2}\bigg{)}^{\frac{1}{2}}$ $\displaystyle=$ $\displaystyle o_{\tau}(1)$

as $\tau\to 0$ for any fixed $\rho>0$. By taking $\tau\to 0$ first and
$\rho\to 0$ next, we have $|\int_{\mathbb{R}^{N}}|x|^{-2a}\nabla u\nabla
W_{\tau}|\to 0$ as $\tau\to 0$. Note that $1+|\tau x|\sim 1$ for $|\tau
x|\lesssim 1$, by (1.5),

$\displaystyle|\int_{|\tau x|\leq R}|x|^{-2a}\nabla u\nabla
W_{\tau}|\lesssim\tau^{a_{c}-a}\|u\|_{D^{1,2}_{a}(\mathbb{R}^{N})}\bigg{(}\int_{0}^{\frac{R}{\tau}}r^{-2a+N-1}\bigg{)}^{\frac{1}{2}}=o_{\tau}(1)$

as $\tau\to+\infty$ for any fixed $R>0$ and

$\displaystyle|\int_{|\tau x|\geq R}|x|^{-2a}\nabla u\nabla W_{\tau}|$
$\displaystyle\leq$ $\displaystyle\int_{|y|\geq R}|y|^{-2a}|\nabla
u_{\frac{1}{\tau}}(y)\nabla W(y)|$ $\displaystyle\leq$
$\displaystyle\|u\|_{D^{1,2}_{a}(\mathbb{R}^{N})}\bigg{(}\int_{|y|\geq
R}|y|^{-2a}|\nabla W|^{2}\bigg{)}^{\frac{1}{2}}$ $\displaystyle=$
$\displaystyle o_{R}(1)$

as $R\to+\infty$ which is uniformly for $\tau>0$. Thus, by taking
$\tau\to+\infty$ first and $R\to+\infty$ next, we also have
$|\int_{\mathbb{R}^{N}}|x|^{-2a}\nabla u\nabla W_{\tau}|\to 0$ as
$\tau\to+\infty$. It follows from (4) and
$d^{2}(u)<\|u\|^{2}_{D^{1,2}_{a}(\mathbb{R}^{N})}$ that the minimizing
sequence $\\{(c_{n},\tau_{n})\\}$ must satisfy $|\tau_{n}|\sim 1$. Thus,
$d^{2}(u)$ can be attained by some $c_{0}\not=0$ and $\tau_{0}>0$, which
implies

$\displaystyle\langle
u,c_{0}W_{\tau_{0}}\rangle_{D^{1,2}_{a}(\mathbb{R}^{N})}=\|c_{0}W_{\tau_{0}}\|_{D^{1,2}_{a}(\mathbb{R}^{N})}^{2}\quad\text{and}\quad\langle
u,\partial_{\tau}W_{\tau}|_{\tau=\tau_{0}}\rangle_{D^{1,2}_{a}(\mathbb{R}^{N})}=0.$

Note that

$\displaystyle\mathcal{T}_{W_{\tau_{0}}}\mathcal{Z}=\text{span}\\{\partial_{\tau}W_{\tau}|_{\tau=\tau_{0}}\\},$

and $\partial_{\tau}W_{\tau}|_{\tau=\tau_{0}}=V_{\tau_{0}}$, where $V(x)$ is
given by (2.5). Thus, by the nondegneracy of $W_{\tau_{0}}$ in
$D^{1,2}_{a}(\mathbb{R}^{N})$,

$\displaystyle u=c_{0}W_{\tau_{0}}+\phi_{\tau_{0}}$ (4.2)

in $D^{1,2}_{a}(\mathbb{R}^{N})$, where

$\displaystyle\langle\phi_{\tau_{0}},W_{\tau_{0}}\rangle_{D^{1,2}_{a}(\mathbb{R}^{N})}=\langle\phi_{\tau_{0}},V_{\tau_{0}}\rangle_{D^{1,2}_{a}(\mathbb{R}^{N})}=0.$
(4.3)

It follows that
$d^{2}(u)=\|\phi_{\tau_{0}}\|_{D^{1,2}_{a}(\mathbb{R}^{N})}^{2}$. Since
$W_{\tau_{0}}$ is the ground state, the Morse index of $W_{\tau_{0}}$ is equal
to $1$. It follows from the nondegneracy of $W_{\tau_{0}}$ in
$D^{1,2}_{a}(\mathbb{R}^{N})$ that

$\displaystyle\|\phi_{\tau_{0}}\|_{D^{1,2}_{a}(\mathbb{R}^{N})}^{2}>p\int_{\mathbb{R}^{N}}|x|^{-b(p+1)}W_{\tau_{0}}^{p-1}\phi_{\tau_{0}}^{2}.$
(4.4)

Let us first consider the case that $d(u)>0$ is sufficiently small, then by
the elementary inequality

$\displaystyle\bigg{|}|\alpha+\beta|^{q}-|\alpha|^{q}-q|\alpha|^{q-2}\alpha\beta-\frac{q(q-1)}{2}|\alpha|^{q-2}\beta^{2}\bigg{|}\lesssim|\beta|^{q}+|\alpha|^{q-3}|\beta|^{3}\chi_{q\geq
3}$

for $q>2$, where $\chi_{q\geq 3}=1$ for $q\geq 3$ and $\chi_{q\geq 3}=0$ for
$2<q<3$, and the CKN inequality (1.1),

$\displaystyle\|W_{\tau_{0}}+\frac{\phi_{\tau_{0}}}{c_{0}}\|^{p+1}_{L^{p+1}(|x|^{-b(p+1)},\mathbb{R}^{N})}$
$\displaystyle=$
$\displaystyle\|W_{\tau_{0}}\|^{p+1}_{L^{p+1}(|x|^{-b(p+1)},\mathbb{R}^{N})}+o(d^{2}(u))$
$\displaystyle+(p+1)\int_{\mathbb{R}^{N}}|x|^{-b(p+1)}W_{\tau_{0}}^{p}\frac{\phi_{\tau_{0}}}{c_{0}}$
$\displaystyle+\frac{p(p+1)}{2}\int_{\mathbb{R}^{N}}|x|^{-b(p+1)}W_{\tau_{0}}^{p-1}(\frac{\phi_{\tau_{0}}}{c_{0}})^{2},$

which, together with
$\langle\phi_{\tau_{0}},W_{\tau_{0}}\rangle_{D^{1,2}_{a}(\mathbb{R}^{N})}=0$
and the fact that $W_{\tau_{0}}$ is a solution of (1.7), implies that

$\displaystyle\|W_{\tau_{0}}+\frac{\phi_{\tau_{0}}}{c_{0}}\|^{p+1}_{L^{p+1}(|x|^{-b(p+1)},\mathbb{R}^{N})}$
$\displaystyle=$
$\displaystyle\|W_{\tau_{0}}\|^{p+1}_{L^{p+1}(|x|^{-b(p+1)},\mathbb{R}^{N})}+o(d^{2}(u))$
$\displaystyle+\frac{p(p+1)}{2}\int_{\mathbb{R}^{N}}|x|^{-b(p+1)}W_{\tau_{0}}^{p-1}(\frac{\phi_{\tau_{0}}}{c_{0}})^{2}.$

On the other hand, by the fact that $W_{\tau_{0}}$ is a solution of (1.7) and
it is also a minimizer of (2.2), we have

$\displaystyle
C_{a,b,N}^{-1}=\frac{\|W_{\tau_{0}}\|^{2}_{D^{1,2}_{a}(\mathbb{R}^{N})}}{\|W_{\tau_{0}}\|^{2}_{L^{p+1}(|x|^{-b(p+1)},\mathbb{R}^{N})}}=\|W_{\tau_{0}}\|^{p-1}_{L^{p+1}(|x|^{-b(p+1)},\mathbb{R}^{N})}.$

It follows from (4.2), (4.3) and (4.4) that for $d(u)>0$ is sufficiently
small,

$\displaystyle e(u)$ $\displaystyle:=$
$\displaystyle\|u\|^{2}_{D^{1,2}_{a}(\mathbb{R}^{N})}-C_{a,b,N}^{-1}\|u\|^{2}_{L^{p+1}(|x|^{-b(p+1)},\mathbb{R}^{N})}$
$\displaystyle=$ $\displaystyle
c_{0}^{2}\bigg{(}\|\frac{\phi_{\tau_{0}}}{c_{0}}\|^{2}_{D^{1,2}_{a}(\mathbb{R}^{N})}-p\int_{\mathbb{R}^{N}}|x|^{-b(p+1)}W_{\tau_{0}}^{p-1}(\frac{\phi_{\tau_{0}}}{c_{0}})^{2}+o(d^{2}(u))\bigg{)}$
$\displaystyle\gtrsim$ $\displaystyle d^{2}(u).$

It remains to consider the case $d(u)\gtrsim 1$. Assume that $e(u)\gtrsim
d^{2}(u)$ does not hold for all $u\in D^{1,2}_{a}(\mathbb{R}^{N})$. Then there
exists $\\{u_{n}\\}\subset D^{1,2}_{a}(\mathbb{R}^{N})\backslash\\{0\\}$ such
that $e(u_{n})=o(d^{2}(u_{n}))$. Thus, $e(u_{n})\to 0$ as $n\to\infty$ in this
case. It follows that $\\{u_{n}\\}$ is a minimizing sequence of (2.2). By
Proposition 3.1, we have $d(u_{n})\to 0$ as $n\to\infty$, which is a
contradiction. ∎

We close this section by the proof of Theorem 1.1.

Proof of Theorem 1.1: It follows immediately from Proposition 4.1. $\Box$

##  5\. Stability of profile decompositions to nonnegative functions

###  5.1. The one-bubble case

In this section, we will consider the one-bubble case and prove the following
result.

######  Proposition 5.1.

Let $v\in H^{1}(\mathcal{C})$ be nonnegative such that

$\displaystyle\frac{1}{2}(C_{a,b,N}^{-1})^{\frac{p+1}{p-1}}<\|v\|_{H^{1}(\mathcal{C})}^{2}<\frac{3}{2}(C_{a,b,N}^{-1})^{\frac{p+1}{p-1}}\quad\text{and}\quad\|f\|_{H^{-1}(\mathcal{C})}\leq\delta$

for $\delta>0$ sufficiently small, where
$f=-\Delta_{\theta}v-\partial_{t}^{2}v+(a_{c}-a)^{2}v-v^{p}$. Then either for
$b_{FS}(a)\leq b<a+1$ with $a<0$ or for $a\leq b<a+1$ with $a\geq 0$ and
$a+b>0$, we have

$\displaystyle d_{0}(v)\lesssim\|f\|_{H^{-1}(\mathcal{C})}$ (5.1)

where
$d_{0}^{2}(v)=\inf_{s\in\mathbb{R}}\|v-\Psi_{s}\|^{2}_{H^{1}(\mathcal{C})}$.

###### Proof.

We shall mainly adapt the ideas in [7] to prove this proposition. As that in
the proof of Proposition 4.1,

$\displaystyle\widetilde{d}_{0}(v)=\inf_{c\in\mathbb{R},\ \
s\in\mathbb{R}}\|v-c\Psi_{s}\|_{H^{1}(\mathcal{C})}$

is attained by some $c_{0}\not=0$ and $s_{0}\in\mathbb{R}$, which implies that
$v=c_{0}\Psi_{s_{0}}+\psi_{0}$ and
$\widetilde{d}_{0}(v)=\|\psi_{0}\|_{H^{1}(\mathcal{C})}$, where

$\displaystyle\langle\Psi_{s_{0}},\psi_{0}\rangle_{H^{1}(\mathcal{C})}=0\quad\text{and}\quad\langle\Psi_{s_{0}}^{\prime},\psi_{0}\rangle_{H^{1}(\mathcal{C})}=0.$
(5.2)

By Proposition 3.2, we also have that $\|\psi_{0}\|_{H^{1}(\mathcal{C})}\to 0$
and $c_{0}=1+\alpha_{0}$ with $\alpha_{0}\to 0$ as $\delta\to 0$. Now, by the
orthogonal condition (5.2),

$\displaystyle\|\psi_{0}\|_{H^{1}(\mathcal{C})}^{2}=\langle\psi_{0},v\rangle_{H^{1}(\mathcal{C})}=\int_{\mathcal{C}}v^{p}\psi_{0}+\int_{\mathcal{C}}f\psi_{0}.$

By the Taylor expansion and some elementary inequalities,

$\displaystyle\int_{\mathcal{C}}v^{p}\psi_{0}=c_{0}^{p}\int_{\mathcal{C}}\Psi_{s_{0}}^{p}\psi_{0}+pc_{0}^{p-1}\int_{\mathcal{C}}\Psi_{s_{0}}^{p-1}\psi_{0}^{2}+O(\|\psi_{0}\|_{H^{1}(\mathcal{C})}^{\sigma_{p}+1}),$

where $\sigma_{p}=2$ for $p\geq 2$ and $\sigma_{p}=p$ for $1<p<2$. Since
$\Psi$ is a solution of (2.4),

$\displaystyle\int_{\mathcal{C}}v^{p}\psi_{0}=pc_{0}^{p-1}\int_{\mathcal{C}}\Psi_{s_{0}}^{p-1}\psi_{0}^{2}+O(\|\psi_{0}\|_{H^{1}(\mathcal{C})}^{\sigma_{p}+1}).$

Note that $\Psi$ is the ground state of (2.4), thus, the Morse index of $\Psi$
is equal to $1$. It follows from the orthogonal condition (5.2) and the
nondegeneracy of $\Psi$ in $H^{1}(\mathcal{C})$ that

$\displaystyle
p\int_{\mathcal{C}}\Psi_{s_{0}}^{p-1}\psi_{0}^{2}<\|\psi_{0}\|_{H^{1}(\mathcal{C})}^{2}.$

Thus, by $\|\psi_{0}\|_{H^{1}(\mathcal{C})}\to 0$ and $c_{0}=1+\alpha_{0}$
with $\alpha_{0}\to 0$ as $\delta\to 0$,

$\displaystyle\|\psi_{0}\|_{H^{1}(\mathcal{C})}\lesssim\|f\|_{H^{-1}(\mathcal{C})}$
(5.3)

for $\delta>0$ sufficiently small. On the other hand, we have

$\displaystyle\|v\|_{H^{1}(\mathcal{C})}^{2}=\|v\|_{L^{p+1}(\mathcal{C})}^{p+1}+\int_{\mathcal{C}}fv.$

Since $c_{0}=1+\alpha_{0}$ with $\alpha_{0}\to 0$ as $\delta\to 0$, by the
orthogonal condition (5.2),

$\displaystyle\|v\|_{H^{1}(\mathcal{C})}^{2}=(1+2\alpha_{0}+O(\alpha_{0}^{2}))\|\Psi\|_{H^{1}(\mathcal{C})}^{2}+\|\psi_{0}\|_{H^{1}(\mathcal{C})}^{2}.$

By the Taylor expansion, the orthogonal condition (5.2) and some elementary
inequalities,

$\displaystyle\|v\|_{L^{p+1}(\mathcal{C})}^{p+1}=(1+(p+1)\alpha_{0}+O(\alpha_{0}^{2}))\|\Psi\|_{L^{p+1}(\mathcal{C})}^{p+1}+O(\|\psi_{0}\|_{H^{1}(\mathcal{C})}^{2}),$

which, together with (5.3) and the fact that $\Psi$ is a solution of (2.4),
implies that

$\displaystyle(p-1)|\alpha_{0}|\lesssim\|f\|_{H^{-1}(\mathcal{C})}^{2}+\|f\|_{H^{-1}(\mathcal{C})}\sim\|f\|_{H^{-1}(\mathcal{C})}$

for $\delta>0$ sufficiently small. Now, (5.1) then follows from rewriting
$v=\Psi_{s_{0}}+\alpha_{0}\Psi_{s_{0}}+\psi_{0}$. ∎

###  5.2. The multi-bubble case

Let us first compute the interaction of two bubbles, which plays an important
role in the stability for the multi-bubble case.

######  Lemma 5.1.

Let $W_{\tau_{1}}$ and $W_{\tau_{2}}$ be two bubbles such that
$\tau_{1}\not=\tau_{2}$. Then

$\displaystyle\langle
W_{\tau_{1}},W_{\tau_{2}}\rangle_{D^{1,2}_{a}(\mathbb{R}^{N})}\sim\bigg{(}\frac{\min\\{\tau_{1},\tau_{2}\\}}{\max\\{\tau_{1},\tau_{2}\\}}\bigg{)}^{a_{c}-a}.$

Moreover, by the transformation (2.1), we also have

$\displaystyle\langle\Psi_{s_{1}},\Psi_{s_{2}}\rangle_{H^{1}(\mathcal{C})}\sim
e^{-(a_{c}-a)|s_{1}-s_{2}|},$ (5.4)

where $\Psi(t)$ is given by (2.8), $\Psi_{s}(t)=\Psi(t-s)$ and
$s_{i}=\ln\tau_{i}$.

###### Proof.

Without loss of generality, we may assume that $\tau_{1}=1$ and
$\tau_{2}:=\tau<1$ by the invariance of
$\langle\cdot,\cdot\rangle_{D^{1,2}_{a}(\mathbb{R}^{N})}$ under the dilation
$u_{\tau}=\tau^{a_{c}-a}u(\tau x)$. Since $W$ is a solution of (1.7),

$\displaystyle\langle W,W_{\tau}\rangle_{D^{1,2}_{a}(\mathbb{R}^{N})}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}|x|^{-b(p+1)}W^{p}W_{\tau}$
$\displaystyle=$ $\displaystyle\int_{|x|\leq
1}|x|^{-b(p+1)}W^{p}W_{\tau}+\int_{1<|x|\leq\frac{1}{\tau}}|x|^{-b(p+1)}W^{p}W_{\tau}$
$\displaystyle+\int_{\frac{1}{\tau}<|x|}|x|^{-b(p+1)}W^{p}W_{\tau}.$

Since $\tau<1$, by (1.5), $W(x)\sim 1$ and $W_{\tau}(x)\sim\tau^{a_{c}-a}$ in
the region $\\{x\in\mathbb{R}^{N}\mid|x|\leq 1\\}$. It follows that

$\displaystyle\int_{|x|\leq
1}|x|^{-b(p+1)}W^{p}W_{\tau}\sim\tau^{a_{c}-a}\int_{0}^{1}r^{N-1-b(p+1)}\sim\tau^{a_{c}-a}.$

Here, we have used the fact that $N-b(p+1)=(p+1)(a_{c}-a)>0$. In the region
$\\{x\in\mathbb{R}^{N}\mid 1<|x|\leq\frac{1}{\tau}\\}$,
$W_{\tau}(x)\sim\tau^{a_{c}-a}$ and $W(x)\sim|x|^{-2(a_{c}-a)}$ by (1.5).
Thus,

$\displaystyle\int_{1<|x|\leq\frac{1}{\tau}}|x|^{-b(p+1)}W^{p}W_{\tau}\sim\tau^{a_{c}-a}\int_{1}^{\frac{1}{\tau}}r^{N-1-b(p+1)-2p(a_{c}-a)}\sim\tau^{a_{c}-a},$

where we have used the fact that $N-b(p+1)-2p(a_{c}-a)=(1-p)(a_{c}-a)<0$ and
$\tau<1$. In the region $\\{x\in\mathbb{R}^{N}\mid\frac{1}{\tau}<|x|\\}$,
$W_{\tau}(x)\sim\tau^{-(a_{c}-a)}|x|^{-2(a_{c}-a)}$ and
$W(x)\sim|x|^{-2(a_{c}-a)}$ by (1.5). Therefore,

$\displaystyle\int_{\frac{1}{\tau}<|x|}|x|^{-b(p+1)}W^{p}W_{\tau}\sim\tau^{-(a_{c}-a)}\int_{\frac{1}{\tau}}^{+\infty}r^{N-1-b(p+1)-2(p+1)(a_{c}-a)}\sim\tau^{p(a_{c}-a)},$

where we have used the fact that $N-b(p+1)-2(p+1)(a_{c}-a)=-(p+1)(a_{c}-a)<0$.
Thus, by $\tau<1$ and $p>1$,

$\displaystyle\langle
W,W_{\tau}\rangle_{D^{1,2}_{a}(\mathbb{R}^{N})}\sim\tau^{a_{c}-a}.$ (5.5)

By (2.1), we have $\langle\Psi,\Psi_{s}\rangle_{H^{1}(\mathcal{C})}=\langle
W,W_{\tau}\rangle_{D^{1,2}_{a}(\mathbb{R}^{N})}$, where $\Psi(t)$ is given by
(2.8), $\Psi_{s}(t)=\Psi(t-s)$ and $s=\ln\tau$. Then (5.4) follows immediately
from (5.5). ∎

Let $v\in H^{1}(\mathcal{C})$ be nonnegative such that

$\displaystyle(\nu-\frac{1}{2})(C_{a,b,N}^{-1})^{\frac{p+1}{p-1}}<\|v\|_{H^{1}(\mathcal{C})}^{2}<(\nu+\frac{1}{2})(C_{a,b,N}^{-1})^{\frac{p+1}{p-1}}$

for some positive integer $\nu\geq 2$ and denote

$\displaystyle f:=-\Delta_{\theta}v-\partial_{t}^{2}v+(a_{c}-a)^{2}v-v^{p}.$
(5.6)

Then $f\in H^{-1}(\mathcal{C})$. As that in [7, 11, 15], we consider the
following minimizing problem:

$\displaystyle
d_{*}^{2}(v)=\min_{s_{j}\in\mathbb{R}}\|v-\sum_{j=1}^{\nu}\Psi_{s_{j}}\|^{2}_{H^{1}(\mathcal{C})}.$

By similar arguments as that used in the proof of Proposition 4.1, we can show
that $d_{*}^{2}(v)$ is attained at some $\\{s_{j}\\}\in\mathbb{R}^{\nu}$ and
thus, we can write $v=\sum_{j=1}^{\nu}\Psi_{s_{j}}+\rho$, where $\rho$
satisfies the following orthogonal conditions:

$\displaystyle\langle\Psi_{s_{j}}^{\prime},\rho\rangle_{H^{1}(\mathcal{C})}=0\quad\text{for
all }j=1,2,\cdots,\nu.$ (5.7)

Clearly, $d_{*}^{2}(v)=\|\rho\|_{H^{1}(\mathcal{C})}^{2}$. Moreover, by
Proposition 3.2, we know that $d_{*}(v)\to 0$ as
$\|f\|_{H^{-1}(\mathcal{C})}\to 0$ either for $b_{FS}(a)\leq b<a+1$ with $a<0$
or for $a\leq b<a+1$ with $a\geq 0$ and $a+b>0$. Thus, if
$\|f\|_{H^{-1}(\mathcal{C})}\leq\delta$ for $\delta>0$ sufficiently small, we
have $\|\rho\|_{H^{1}(\mathcal{C})}\leq\delta^{\prime}$ either for
$b_{FS}(a)\leq b<a+1$ with $a<0$ or for $a\leq b<a+1$ with $a\geq 0$ and
$a+b>0$, where $\delta^{\prime}\to 0$ as $\delta\to 0$.

Since $\Psi_{s_{j}}$ are solutions of (2.4), by (5.7), we can rewrite (5.6) as
follows:

$\displaystyle\left\\{\begin{aligned}
&-\Delta_{\theta}\rho-\partial_{t}^{2}\rho+(a_{c}-a)^{2}\rho=(\sum_{j=1}^{\nu}\Psi_{s_{j}}+\rho)^{p}-\sum_{j=1}^{\nu}\Psi_{s_{j}}^{p}+f,\quad\text{in
}\mathcal{C},\\\
&\langle\Psi_{s_{j}}^{\prime},\rho\rangle_{H^{1}(\mathcal{C})}=0\quad\text{for
all }j=1,2,\cdots,\nu.\end{aligned}\right.$ (5.8)

In what follows, for the sake of simplicity, we denote

$\displaystyle R=\min_{i\not=j}|s_{i}-s_{j}|\quad\text{and}\quad
Q=e^{-(a_{c}-a)\min_{i\not=j}|s_{i}-s_{j}|}$

as that in [11]. Moreover, we also assume that $s_{1}<s_{2}<\cdots<s_{\nu}$
without loss of generality. For the sake of simplicity, we also denote
$s_{0}=-\infty$ and $s_{\nu+1}=+\infty$.

######  Lemma 5.2.

Let $b_{FS}(a)\leq b<a+1$ for $a<0$ and $a\leq b<a+1$ for $a\geq 0$ and
$a+b>0$. Then

$\displaystyle\|f\|_{H^{-1}(\mathcal{C})}\gtrsim
Q+O(Q^{\frac{1}{2}}\|\rho\|_{H^{1}(\mathcal{C})}+\|\rho\|_{H^{1}(\mathcal{C})}^{p+1})$
(5.9)

for $\|f\|_{H^{-1}(\mathcal{C})}\leq\delta$ with $\delta>0$ sufficiently
small.

###### Proof.

Suppose that $R=s_{j_{0}+1}-s_{j_{0}}$ for some
$j_{0}\in\\{1,2,\cdots,\nu-1\\}$. Multiplying (5.8) with
$-\Psi_{s_{j_{0}}}^{\prime}$ and integrating by parts, we have

$\displaystyle\|f\|_{H^{-1}(\mathcal{C})}\gtrsim\langle(\sum_{j=1}^{\nu}\Psi_{s_{j}}+\rho)^{p}-\sum_{j=1}^{\nu}\Psi_{s_{j}}^{p},-\Psi_{s_{j_{0}}}^{\prime}\rangle_{L^{2}(\mathcal{C})}.$

In the region $\\{3|\rho|<\sum_{j=1}^{\nu}\Psi_{s_{j}}\\}$, by the Taylor
expansion,

$\displaystyle(\sum_{j=1}^{\nu}\Psi_{s_{j}}+\rho)^{p}-(\sum_{j=1}^{\nu}\Psi_{s_{j}})^{p}-p(\sum_{j=1}^{\nu}\Psi_{s_{j}})^{p-1}\rho$
$\displaystyle=$ $\displaystyle
p(p-1)(\sum_{j=1}^{\nu}\Psi_{s_{j}}+\xi\rho)^{p-2}\rho^{2}$ (5.10)
$\displaystyle\sim$
$\displaystyle(\sum_{j=1}^{\nu}\Psi_{s_{j}})^{p-2}\rho^{2}>0,$

where $\xi\in(0,1)$. In the region $\\{\sum_{j=1}^{\nu}\Psi_{s_{j}}\leq
3|\rho|\\}$, since $\Psi_{s_{j}}$ are all positive, we also have
$\Psi_{s_{j_{0}}}\leq 3|\rho|$ which, together with
$-\Psi_{s_{j_{0}}}^{\prime}\sim\Psi_{s_{j_{0}}}$, implies that

$\displaystyle\bigg{(}(\sum_{j=1}^{\nu}\Psi_{s_{j}}+\rho)^{p}-(\sum_{j=1}^{\nu}\Psi_{s_{j}})^{p}-p(\sum_{j=1}^{\nu}\Psi_{s_{j}})^{p-1}\rho\bigg{)}\Psi_{s_{j_{0}}}^{\prime}\lesssim\rho^{p+1}.$
(5.11)

By Proposition 3.2 and Lemma 5.1, $R\to+\infty$ as $\delta\to 0$. Thus,

$\displaystyle e^{-(a_{c}-a)|t-s_{i}|}\gtrsim
e^{-(a_{c}-a)|t-s_{i-1}|}+e^{-(a_{c}-a)|t-s_{i+1}|}\sim\sum_{j\not=i}e^{-(a_{c}-a)|t-s_{j}|}$

in $(s_{i}-\frac{R}{2}+O(1),s_{i}+\frac{R}{2}+O(1))$ for all
$i=1,2,\cdots,\nu$, which, together with (2.8), implies that
$\\{\Psi_{s_{i}}\gtrsim\sum_{j\not=i}\Psi_{s_{j}}\\}$ in the region
$(s_{i}-\frac{R}{2}+O(1),s_{i}+\frac{R}{2}+O(1))$ for all $i=1,2,\cdots,\nu$.
It follows from the Taylor expansion that

$\displaystyle(\sum_{j=1}^{\nu}\Psi_{s_{j}})^{p}-\sum_{j=1}^{\nu}\Psi_{s_{j}}^{p}=p(\Psi_{s_{i}}+\xi_{i}\sum_{j\not=i}\Psi_{s_{j}})^{p-1}\sum_{j\not=i}\Psi_{s_{j}}\sim\Psi_{s_{i}}^{p-1}\sum_{j\not=i}\Psi_{s_{j}}$
(5.12)

in the region $(s_{i}-\frac{R}{2}+O(1),s_{i}+\frac{R}{2}+O(1))$ for all
$i=1,2,\cdots,\nu$, where $\xi_{i}\in(0,1)$. In the region
$\mathbb{R}\backslash(\cup_{i=1}^{\nu}(s_{i}-\frac{R}{2}+O(1),s_{i}+\frac{R}{2}+O(1)))$,
by (2.8),

$\displaystyle|(\sum_{j=1}^{\nu}\Psi_{s_{j}})^{p}-\sum_{j=1}^{\nu}\Psi_{s_{j}}^{p}|\lesssim\sum_{j=1}^{\nu}\Psi_{s_{j}}^{p}\sim\sum_{j=1}^{\nu}e^{-p(a_{c}-a)|t-s_{j}|}.$
(5.13)

Thus, by $-\Psi_{s_{j_{0}}}^{\prime}\sim\Psi_{s_{j_{0}}}$ once more, (5.13),
$p>1$ and the orthogonal conditions in (5.8),

$\displaystyle\|f\|_{H^{-1}(\mathcal{C})}$ $\displaystyle\gtrsim$
$\displaystyle-\int_{\mathcal{C}}((\sum_{j=1}^{\nu}\Psi_{s_{j}})^{p}-\sum_{j=1}^{\nu}\Psi_{s_{j}}^{p}+p((\sum_{j=1}^{\nu}\Psi_{s_{j}})^{p-1}-\Psi_{s_{j_{0}}}^{p-1})\rho)\Psi_{s_{j_{0}}}^{\prime})$
$\displaystyle+O(\|\rho\|_{H^{1}(\mathcal{C})}^{p+1})$ $\displaystyle\gtrsim$
$\displaystyle\sum_{i=1}^{\nu}\int_{(s_{i}-\frac{R}{2}+O(1),s_{i}+\frac{R}{2}+O(1))}\Psi_{s_{i}}^{p-1}\Psi_{s_{j_{0}}}\sum_{j\not=i}^{\nu}\Psi_{s_{j}}+o(Q)$
$\displaystyle-\int_{(s_{j_{0}}-\frac{R}{2}+O(1),s_{j_{0}}+\frac{R}{2}+O(1))}\Psi_{s_{j_{0}}}^{p-1}\sum_{j\not=j_{0}}\Psi_{s_{j}}|\rho|$
$\displaystyle-\int_{(s_{j_{0}}-\frac{R}{2}+O(1),s_{j_{0}}+\frac{R}{2}+O(1))^{c}}\Psi_{s_{j_{0}}}\sum_{j=1}^{\nu}\Psi_{s_{j}}^{p-1}|\rho|+O(\|\rho\|_{H^{1}(\mathcal{C})}^{p+1})$
$\displaystyle\sim$
$\displaystyle\sum_{i=1}^{\nu}\int_{(s_{i}-\frac{R}{2}+O(1),s_{i}+\frac{R}{2}+O(1))}\Psi_{s_{i}}^{p-1}\Psi_{s_{j_{0}}}\sum_{j\not=i}^{\nu}\Psi_{s_{j}}+o(Q)$
$\displaystyle+O(Q^{\frac{1}{2}}\|\rho\|_{H^{1}(\mathcal{C})}+\|\rho\|_{H^{1}(\mathcal{C})}^{p+1}).$

By (2.8), $p>1$ and [25, Lemma 4.1],

$\displaystyle\sum_{i=1}^{\nu}\int_{(s_{i}-\frac{R}{2}+O(1),s_{i}+\frac{R}{2}+O(1))}\Psi_{s_{i}}^{p-1}\Psi_{s_{j_{0}}}\sum_{j\not=i}^{\nu}\Psi_{s_{j}}$
$\displaystyle\sim$
$\displaystyle\int_{(s_{j_{0}}-\frac{R}{2}+O(1),s_{j_{0}}+\frac{R}{2}+O(1))}\Psi_{s_{j_{0}}}^{p}(\Psi_{s_{j_{0}+1}}+\Psi_{s_{j_{0}-1}})$
$\displaystyle+\int_{(s_{j_{0}+1}-\frac{R}{2}+O(1),s_{j_{0}+1}+\frac{R}{2}+O(1))}\Psi_{s_{j_{0}}}^{2}\Psi_{s_{j_{0}+1}}^{p-1}$
$\displaystyle+\int_{(s_{j_{0}-1}-\frac{R}{2}+O(1),s_{j_{0}-1}+\frac{R}{2}+O(1))}\Psi_{s_{j_{0}}}^{2}\Psi_{s_{j_{0}-1}}^{p-1}+o(Q)$
$\displaystyle\sim$
$\displaystyle\int_{0}^{\frac{R}{2}}e^{-(a_{c}-a)pr}e^{-(a_{c}-a)(R-r)}+\int_{0}^{\frac{R}{2}}e^{-2(a_{c}-a)(R-r)}e^{-(a_{c}-a)(p-1)r}+o(Q)$
$\displaystyle\sim$ $\displaystyle Q.$

It follows that (5.9) holds for $\delta>0$ sufficiently small. ∎

As that in [11], we want to drive the precise behavior of first approximation
of $\rho$ by considering the following equation:

$\displaystyle\left\\{\begin{aligned}
&-\Delta_{\theta}\phi-\partial_{t}^{2}\phi+(a_{c}-a)^{2}\phi\\\
&=|\sum_{j=1}^{\nu}\Psi_{s_{j}}+\phi|^{p-1}(\sum_{j=1}^{\nu}\Psi_{s_{j}}+\phi)\\\
&-\sum_{j=1}^{\nu}\Psi_{s_{j}}^{p}+\sum_{j=1}^{\nu}c_{j}\Psi_{s_{j}}^{p-1}\Psi_{s_{j}}^{\prime},\quad\text{in
}\mathcal{C},\\\
&\langle\Psi_{s_{j}}^{\prime},\phi\rangle_{H^{1}(\mathcal{C})}=0\quad\text{for
all }j=1,2,\cdots,\nu,\end{aligned}\right.$ (5.14)

where $c_{j}$ and $\phi$ are all unknowns. By (2.8) and some elementary
inequalities, we can rewrite

$\displaystyle|\sum_{j=1}^{\nu}\Psi_{s_{j}}+\phi|^{p-1}(\sum_{j=1}^{\nu}\Psi_{s_{j}}+\phi)-\sum_{j=1}^{\nu}\Psi_{s_{j}}^{p}$
$\displaystyle=$ $\displaystyle
p(\sum_{j=1}^{\nu}\Psi_{s_{j}})^{p-1}\phi+(\sum_{j=1}^{\nu}\Psi_{s_{j}})^{p}$
$\displaystyle-\sum_{j=1}^{\nu}\Psi_{s_{j}}^{p}+O(|\phi|^{\sigma_{p}}),$

where $\sigma_{p}=2$ for $p\geq 2$ and $\sigma_{p}=p$ for $1<p<2$. Thus,
(5.14) can be rewritten as follows:

$\displaystyle\left\\{\begin{aligned}
&\mathcal{L}(\phi)=E+N(\phi)+\sum_{j=1}^{\nu}c_{j}\Psi_{s_{j}}^{p-1}\Psi_{s_{j}}^{\prime},\quad\text{in
}\mathcal{C},\\\
&\langle\Psi_{s_{j}}^{\prime},\phi\rangle_{H^{1}(\mathcal{C})}=0\quad\text{for
all }j=1,2,\cdots,\nu,\end{aligned}\right.$ (5.15)

where the linear operator $\mathcal{L}(\phi)$ is given by

$\displaystyle\mathcal{L}(\phi):=-\Delta_{\theta}\phi-\partial_{t}^{2}\phi+(a_{c}-a)^{2}\phi-p(\sum_{j=1}^{\nu}\Psi_{s_{j}})^{p-1}\phi,$
(5.16)

$E=(\sum_{j=1}^{\nu}\Psi_{s_{j}})^{p}-\sum_{j=1}^{\nu}\Psi_{s_{j}}^{p}$ is the
error and $N(\phi)=O(|\phi|^{\sigma_{p}})$ is the nonlinear part.

######  Lemma 5.3.

For $\delta>0$ sufficiently small, we have

$\displaystyle\|E\|_{\natural}:=\sum_{i=1}^{\nu}\sup_{t\in(\frac{s_{i}+s_{i-1}}{2},\frac{s_{i+1}+s_{i}}{2})}\frac{|E|}{Qe^{-(a_{c}-a)(p-2)|t-s_{i}|}}\lesssim
1$ (5.17)

for $1<p<3$ and

$\displaystyle\|E\|_{\sharp}:=\sum_{i=1}^{\nu}\sup_{t\in(\frac{s_{i}+s_{i-1}}{2},\frac{s_{i+1}+s_{i}}{2})}\frac{|E|}{Qe^{-(1-\varsigma)(a_{c}-a)|t-s_{i}|}}\lesssim
1$ (5.18)

for $p\geq 3$ with $\varsigma>0$ sufficiently small.

###### Proof.

By (2.8) and similar arguments as that used for (5.12),

$\displaystyle E$ $\displaystyle\sim$
$\displaystyle\Psi_{s_{i}}^{p-1}\Psi_{s_{i+1}}$ (5.19) $\displaystyle\sim$
$\displaystyle e^{-(p-1)(a_{c}-a)|t-s_{i}|}e^{-(a_{c}-a)|t-s_{i+1}|}$
$\displaystyle\sim$ $\displaystyle
e^{-(a_{c}-a)|s_{i}-s_{i+1}|}e^{-(a_{c}-a)(p-2)|t-s_{i}|}$

in the region $(s_{i},\frac{s_{i+1}+s_{i}}{2})$ for all $i=1,2,\cdots,\nu-1$
and

$\displaystyle E$ $\displaystyle\sim$
$\displaystyle\Psi_{s_{i}}^{p-1}\Psi_{s_{i-1}}$ (5.20) $\displaystyle\sim$
$\displaystyle e^{-(p-1)(a_{c}-a)|t-s_{i}|}e^{-(a_{c}-a)|t-s_{i-1}|}$
$\displaystyle\sim$ $\displaystyle
e^{-(a_{c}-a)|s_{i}-s_{i-1}|}e^{-(a_{c}-a)(p-2)|t-s_{i}|}$

in the region $(\frac{s_{i-1}+s_{i}}{2},s_{i})$ for all $i=2,3,\cdots,\nu$. In
the region $(-\infty,s_{1})$, since $s_{1}<s_{2}<\cdots<s_{\nu-1}<s_{\nu}$,

$\displaystyle E$ $\displaystyle\sim$
$\displaystyle\Psi_{s_{1}}^{p-1}\Psi_{s_{2}}$ (5.21) $\displaystyle\sim$
$\displaystyle e^{-(p-1)(a_{c}-a)|t-s_{1}|}e^{-(a_{c}-a)|t-s_{2}|}$
$\displaystyle\sim$ $\displaystyle
e^{-(a_{c}-a)|s_{1}-s_{2}|}e^{-p(a_{c}-a)|t-s_{1}|}.$

In the region $(s_{\nu},+\infty)$, since
$s_{1}<s_{2}<\cdots<s_{\nu-1}<s_{\nu}$,

$\displaystyle E$ $\displaystyle\sim$
$\displaystyle\Psi_{s_{\nu}}^{p-1}\Psi_{s_{\nu-1}}$ (5.22) $\displaystyle\sim$
$\displaystyle e^{-(p-1)(a_{c}-a)|t-s_{\nu}|}e^{-(a_{c}-a)|t-s_{\nu-1}|}$
$\displaystyle\sim$ $\displaystyle
e^{-(a_{c}-a)|s_{\nu}-s_{\nu-1}|}e^{-p(a_{c}-a)|t-s_{i}|}.$

(5.17) and (5.18) then follow immediately from (5.19), (5.20) and (5.21),
(5.22). ∎

To solve (5.14), we shall use the fix point argument, which leads us to
establish a good linear theory by considering the following linear equation:

$\displaystyle\left\\{\begin{aligned} &\mathcal{L}(\phi)=g,\quad\text{in
}\mathcal{C},\\\
&\langle\Psi_{s_{j}}^{\prime},\phi\rangle_{H^{1}(\mathcal{C})}=0\quad\text{for
all }j=1,2,\cdots,\nu,\end{aligned}\right.$

where $g$ satisfies
$\langle\Psi_{s_{j}}^{\prime},g\rangle_{L^{2}(\mathcal{C})}=0$ for all
$j=1,2,\cdots,\nu$ and $\mathcal{L}(\phi)$ is given by (5.16). Based on Lemma
5.3, we shall introduce the following spaces:

$\displaystyle X=\\{\phi\in
H^{1}(\mathcal{C})\mid\|\phi\|_{\natural}<+\infty\\}\quad,Y=\\{\phi\in
L^{2}(\mathcal{C})\mid\|\phi\|_{\natural}<+\infty\\},$

and

$\displaystyle\widehat{X}=\\{\phi\in
H^{1}(\mathcal{C})\mid\|\phi\|_{\sharp}<+\infty\\}\quad,\widehat{Y}=\\{\phi\in
L^{2}(\mathcal{C})\mid\|\phi\|_{\sharp}<+\infty\\}.$

Clearly, $X$, $Y$ and $\widehat{X}$, $\widehat{Y}$ are all Banach spaces. Let

$\displaystyle X^{\perp}$ $\displaystyle=$ $\displaystyle\\{\phi\in
X\mid\langle\Psi_{s_{j}}^{\prime},\phi\rangle_{H^{1}(\mathcal{C})}=0\quad\text{for
all }j=1,2,\cdots,\nu\\},$ $\displaystyle Y^{\perp}$ $\displaystyle=$
$\displaystyle\\{\phi\in
Y\mid\langle\Psi_{s_{j}}^{\prime},\phi\rangle_{L^{2}(\mathcal{C})}=0\quad\text{for
all }j=1,2,\cdots,\nu\\}$

and

$\displaystyle\widehat{X}^{\perp}$ $\displaystyle=$
$\displaystyle\\{\phi\in\widehat{X}\mid\langle\Psi_{s_{j}}^{\prime},\phi\rangle_{H^{1}(\mathcal{C})}=0\quad\text{for
all }j=1,2,\cdots,\nu\\},$ $\displaystyle\widehat{Y}^{\perp}$ $\displaystyle=$
$\displaystyle\\{\phi\in\widehat{Y}\mid\langle\Psi_{s_{j}}^{\prime},\phi\rangle_{L^{2}(\mathcal{C})}=0\quad\text{for
all }j=1,2,\cdots,\nu\\},$

then we have the following.

######  Lemma 5.4.

Let $b_{FS}(a)\leq b<a+1$ for $a<0$ and $a\leq b<a+1$ for $a\geq 0$ and
$a+b>0$.

  1. $(1)$

If $p\geq 3$, then for $\delta>0$ sufficiently small, there exists a unique
$\phi\in\widehat{X}^{\perp}$ such that $\mathcal{L}(\phi)=g$ and
$\|\phi\|_{\sharp}\lesssim\|g\|_{\sharp}$ for every $g\in\widehat{Y}^{\perp}$.

  2. $(2)$

If $1<p<3$, then for $\delta>0$ sufficiently small, there exists a unique
$\phi\in X^{\perp}$ such that $\mathcal{L}(\phi)=g$ and
$\|\phi\|_{\natural}\lesssim\|g\|_{\natural}$ for every $g\in Y^{\perp}$.

Here $\mathcal{L}(\phi)$ is given by (5.16).

###### Proof.

Since the proof is rather standard nowadays (cf. [8, 9, 10, 26, 27]), we only
sketch it here. We start by proving the a-priori estimates
$\|\phi\|_{\sharp}\lesssim\|g\|_{\sharp}$ for $p\geq 3$ and
$\|\phi\|_{\natural}\lesssim\|g\|_{\natural}$ for $1<p<3$. Assuming the
contrary, that is, there exist $\\{g_{n}\\}$ and $\\{\delta_{n}\\}$ such that
$\|g_{n}\|_{\sharp}\to 0$ and $\delta_{n}\to 0$ as $n\to\infty$ and
$\|\phi_{n}\|_{\sharp}=1$ for $p\geq 3$ while, $\|g_{n}\|_{\natural}\to 0$ and
$\delta_{n}\to 0$ as $n\to\infty$ and $\|\phi_{n}\|_{\natural}=1$ for $1<p<3$.
Since $\delta_{n}\to 0$ as $n\to\infty$, by proposition 3.2,

$\displaystyle R_{n}=\min_{i\not=j}|s_{i,n}-s_{j,n}|\to+\infty\quad\text{as
}n\to\infty.$

By the definition of the norms $\|\cdot\|_{\natural}$ and $\|\cdot\|_{\sharp}$
given by (5.17) and (5.18),

$\displaystyle|\mathcal{L}(\phi_{n})|\lesssim\left\\{\begin{aligned}
&\|g_{n}\|_{\natural}\sum_{i=1}^{\nu}Q_{n}\varphi_{i,n}(t)\chi_{i,n}(t),\quad
1<p<3,\\\
&\|g_{n}\|_{\sharp}\sum_{i=1}^{\nu}Q_{n}\varphi_{i,n}(t)\chi_{i,n}(t),\quad
p\geq 3,\end{aligned}\right.$

where

$\displaystyle\varphi_{i,n}(t)=\left\\{\begin{aligned}
&e^{-(1-\varsigma)(a_{c}-a)|t-s_{i,n}|},\quad p\geq 3,\\\
&e^{-(p-2)(a_{c}-a)|t-s_{i,n}|},\quad 1<p<3,\end{aligned}\right.$ (5.23)

and $\chi_{i,n}$ is a cut-off function such that

$\displaystyle\chi_{i,n}(t)=\left\\{\begin{aligned} &1,\quad
t\in(\frac{s_{i,n}+s_{i-1,n}}{2},\frac{s_{i+1,n}+s_{i,n}}{2}),\\\ &0,\quad
t\in(\frac{s_{i,n}+s_{i-1,n}}{2},\frac{s_{i+1,n}+s_{i,n}}{2})^{c}.\end{aligned}\right.$
(5.24)

By (2.8), it is easy to see that

$\displaystyle\mathcal{L}(\varphi_{i,n})\gtrsim\left\\{\begin{aligned}
&e^{-(1-\varsigma)(a_{c}-a)|t-s_{i,n}|},\quad p\geq 3,\\\
&e^{-(p-2)(a_{c}-a)|t-s_{i,n}|},\quad 1<p<3\end{aligned}\right.$

in
$(\frac{s_{i,n}+s_{i-1,n}}{2},\frac{s_{i+1,n}+s_{i,n}}{2})\backslash(s_{i}-T,s_{i}+T)$
for a sufficiently large $T>0$. Thus, by the maximum principle,

$\displaystyle|\phi_{n}|\lesssim\left\\{\begin{aligned}
&\|g_{n}\|_{\natural}Q_{n}\varphi_{i,n}(t),\quad 1<p<3,\\\
&\|g_{n}\|_{\sharp}Q_{n}\varphi_{i,n}(t),\quad p\geq 3\end{aligned}\right.$
(5.25)

in
$(\frac{s_{i,n}+s_{i-1,n}}{2},\frac{s_{i+1,n}+s_{i,n}}{2})\backslash(s_{i,n}-T,s_{i,n}+T)$
for all $i=1,2,\cdots,\nu$. On the other hand, by $\|\phi_{n}\|_{\sharp}=1$
and $\|g_{n}\|_{\sharp}=o_{n}(1)$ for $p\geq 3$ while
$\|\phi_{n}\|_{\natural}=1$ and $\|g_{n}\|_{\natural}=o_{n}(1)$ for $1<p<3$,
it is standard to use the Moser iteration and the Sobolev embedding theorem to
show that $Q_{n}^{-1}\phi_{n}(\cdot+s_{i,n})\to\widehat{\phi}$ uniformly in
every compact set of $\mathcal{C}$ as $n\to\infty$ for all $i=1,2,\cdots,\nu$,
where $\widehat{\phi}$ is a solution of (2.7). We recall that by the
nondegeneracy of $\Psi$ in $H^{1}(\mathcal{C})$, $\Psi^{\prime}$ is the only
nonzero solution of (2.7). Thus, we must have that
$\widehat{\phi}=C\Psi^{\prime}$, which together with the orthogonal condition
in $X^{\perp}$ for $1<p<3$ and the orthogonal condition in
$\widehat{X}^{\perp}$ for $p\geq 3$, implies that $\widehat{\phi}=0$. Since
$\varphi_{i,n}(t)\sim 1$ in $[s_{i,n}-T,s_{i,n}+T]$ for fixed $T>0$,
$\frac{|\phi_{n}|}{Q_{n}\varphi_{i,n}(t)}=o_{n}(1)$ in $[s_{i,n}-T,s_{i,n}+T]$
for fixed $T>0$. Thus, by (5.25), $\|\phi_{n}\|_{\sharp}=o_{n}(1)$ for $p\geq
3$ and $\|\phi_{n}\|_{\natural}=o_{n}(1)$ for $1<p<3$. It contradicts
$\|\phi_{n}\|_{\sharp}=1$ for $p\geq 3$ and $\|\phi_{n}\|_{\natural}=1$ for
$1<p<3$. The a-priori estimates $\|\phi\|_{\sharp}\lesssim\|g\|_{\sharp}$ for
$p\geq 3$ and $\|\phi\|_{\natural}\lesssim\|g\|_{\natural}$ for $1<p<3$ imply
that $\mathcal{L}:X^{\perp}\to Y^{\perp}$ for $1<p<3$ and
$\mathcal{L}:\widehat{X}^{\perp}\to\widehat{Y}^{\perp}$ for $p\geq 3$ are
injective for $\delta>0$ sufficiently small. Since
$(\sum_{j=1}^{\nu}\Psi_{s_{j}})^{p-1}\to 0$ as $|t|\to+\infty$ by (2.8), it is
standard to use the the Fredholm alternative to show that for $\delta>0$
sufficiently small, $\mathcal{L}(\phi)=g$ is unique solvable in $X^{\perp}$
for every $g\in Y^{\perp}$ in the case of $1<p<3$ and $\mathcal{L}(\phi)=g$ is
unique solvable in $\widehat{X}^{\perp}$ for every $g\in\widehat{Y}^{\perp}$
in the case of $p\geq 3$. ∎

Let us go back to (5.15), then we have the following.

######  Lemma 5.5.

Let $b_{FS}(a)\leq b<a+1$ for $a<0$ and $a\leq b<a+1$ for $a\geq 0$ and
$a+b>0$. Then (5.15) has a unique solution $(\psi,c_{1},c_{2},\cdots,c_{\nu})$
for $\delta>0$ sufficiently small. Moreover,

$\displaystyle\|\phi\|_{H^{1}(\mathcal{C})}\lesssim\left\\{\begin{aligned}
&Q,\quad p>2,\\\ &Q|\log(Q)|^{\frac{1}{2}},\quad p=2,\\\
&Q^{\frac{p}{2}},\quad 1<p<2.\end{aligned}\right.$ (5.26)

and $\sum_{j=1}^{\nu}|c_{l}|\lesssim Q$.

###### Proof.

Since $R\to+\infty$ as $\delta\to 0$ and $p>1$, by [19, Lemma 6] and (2.8),

$\displaystyle\langle\Psi_{s_{j}}^{p-1}\Psi_{s_{j}}^{\prime},\Psi_{s_{i}}^{\prime}\rangle_{L^{2}(\mathcal{C})}\sim
Q$ (5.27)

for $i\not=j$ for $\delta>0$ sufficiently small. Thus, $\\{c_{j}\\}$ in (5.15)
can be chosen to be the unique solution of the following equation:

$\displaystyle(\langle\Psi_{s_{j}}^{p-1}\Psi_{s_{j}}^{\prime},\Psi_{s_{i}}^{\prime}\rangle_{L^{2}(\mathcal{C})})_{i,j=1,2,\cdots,\nu}\bullet(c_{j})_{j=1,2,\cdots,\nu}=-(\langle
E+N(\phi),\Psi_{s_{i}}^{\prime}\rangle_{L^{2}(\mathcal{C})})_{i=1,2,\cdots,\nu}.$

By Lemmas 5.3 and 5.4, and adapting the fix point arguments in a standard way
(cf. [8, 9, 10, 26, 27]), (5.15) is unique solvable in the set
$\widehat{B}=\\{\phi\in\widehat{X}^{\perp}\mid\|\phi\|_{\sharp}\leq C\\}$ in
the case of $p\geq 3$ and in the set $B=\\{\phi\in
X^{\perp}\mid\|\phi\|_{\natural}\leq C\\}$ in the case of $1<p<3$ for a
sufficiently large $C>0$. Note that $N(\phi)=O(\phi^{2})$ for $p\geq 2$, by
$-\Psi^{\prime}\sim\Psi$ and [19, Lemma 6],

$\displaystyle|\langle
N(\phi),\Psi_{s_{j}}^{\prime}\rangle|\lesssim\int_{\mathcal{C}}\sum_{i=1}^{\nu}(Q\varphi_{i}(t)\chi_{i}(t))^{2}\Psi_{s_{j}}\lesssim
Q^{2},$

where $\varphi_{i}$ is given by (5.23) and $\chi_{i}$ is a cut-off function
given by (5.24). For $1<p<2$, $N(\phi)=O(|\phi|^{p})$. Thus, by
$-\Psi^{\prime}\sim\Psi$ and (2.8),

$\displaystyle|\langle
N(\phi),\Psi_{s_{j}}^{\prime}\rangle|\lesssim\int_{\mathcal{C}}\sum_{i=1}^{\nu}(Q\varphi_{i}(t)\chi_{i}(t))^{p}\Psi_{s_{j}}\sim
Q^{p}\int_{0}^{\frac{R}{2}}e^{(2-p)p(a_{c}-a)r-r}=O(Q^{\frac{p^{2}+1}{2}}).$

On the other hand, by (2.8), (5.19), (5.20), (5.21), (5.22) and
$-\Psi_{s_{j}}^{\prime}\sim\Psi_{s_{j}}$,

$\displaystyle|\langle
E,\Psi_{s_{j}}^{\prime}\rangle_{L^{2}(\mathcal{C})}|\sim
Q\sum_{i=1}^{\nu}\int_{(\frac{s_{i-1}+s_{i}}{2},\frac{s_{i}+s_{i+1}}{2})}\Psi_{s_{i}}^{p-2}\Psi_{s_{j}}\sim
Q\int_{(\frac{s_{j-1}+s_{j}}{2},\frac{s_{j}+s_{j+1}}{2})}\Psi_{s_{j}}^{p-1}\sim
Q.$

It follows from $p>1$ that $\sum_{j=1}^{\nu}|c_{l}|\lesssim Q$ and

$\displaystyle\|\phi\|_{H^{1}(\mathcal{C})}^{2}$ $\displaystyle=$
$\displaystyle\langle
E+N(\phi)+\sum_{j=1}^{\nu}c_{j}\Psi_{s_{j}}^{p-1}\Psi_{s_{j}}^{\prime},\phi\rangle_{L^{2}(\mathcal{C})}$
(5.28) $\displaystyle\leq$ $\displaystyle\langle
E,\phi\rangle_{L^{2}(\mathcal{C})}+O(Q\|\phi\|_{H^{1}(\mathcal{C})}+\|\phi\|_{H^{1}(\mathcal{C})}^{\sigma_{p}+1}),$

where $\sigma_{p}=2$ for $p\geq 2$ and $\sigma_{p}=p$ for $1<p<2$. Since
$\|\phi\|_{\sharp}\leq C$ for $p\geq 3$, by (5.18),

$\displaystyle\langle
E,\phi\rangle_{L^{2}(\mathcal{C})}\lesssim\sum_{i=1}^{\nu}Q^{2}\int_{(\frac{s_{i-1}+s_{i}}{2},\frac{s_{i}+s_{i+1}}{2})}\Psi_{s_{i}}^{(2-2\varsigma)}\sim
Q^{2}$

for $p\geq 3$. For $1<p<3$, $\|\phi\|_{\natural}\leq C$. Thus, by (5.17),

$\displaystyle\langle
E,\phi\rangle_{L^{2}(\mathcal{C})}\lesssim\sum_{i=1}^{\nu}Q^{2}\int_{(\frac{s_{i-1}+s_{i}}{2},\frac{s_{i}+s_{i+1}}{2})}\Psi_{s_{i}}^{2(p-2)}\sim\left\\{\begin{aligned}
&Q^{2},\quad p>2,\\\ &Q^{2}\log(Q),\quad p=2,\\\ &Q^{p},\quad
1<p<2.\end{aligned}\right.$

(5.26) then follows from (5.28). ∎

Let $\varphi=\rho-\phi$, then by (5.8) and (5.14),

$\displaystyle\left\\{\begin{aligned}
&-\Delta_{\theta}\varphi-\partial_{t}^{2}\varphi+(a_{c}-a)^{2}\varphi\\\
&=(\sum_{j=1}^{\nu}\Psi_{s_{j}}+\phi+\varphi)^{p}-|\sum_{j=1}^{\nu}\Psi_{s_{j}}+\phi|^{p-1}(\sum_{j=1}^{\nu}\Psi_{s_{j}}+\phi)\\\
&-\sum_{j=1}^{\nu}c_{j}\Psi_{s_{j}}^{p-1}\Psi_{s_{j}}^{\prime}+f,\quad\text{in
}\mathcal{C},\\\
&\langle\Psi_{s_{j}}^{\prime},\varphi\rangle_{H^{1}(\mathcal{C})}=0\quad\text{for
all }j=1,2,\cdots,\nu.\end{aligned}\right.$ (5.29)

Let $M_{0}=\text{span}\\{\Psi_{s_{j}}\\}$ and
$M=\text{span}\\{\Psi_{s_{j}}^{\prime}\\}$. Then by the orthogonal conditions
satisfied by $\varphi$, we can decompose
$\varphi=\sum_{j=1}^{\nu}\beta_{j}\Psi_{s_{j}}+\Psi^{\perp}$, where
$\Psi^{\perp}\in(M_{0}\oplus M)^{\perp}$ in $H^{1}(\mathcal{C})$.

######  Lemma 5.6.

Let $b_{FS}(a)\leq b<a+1$ for $a<0$ and $a\leq b<a+1$ for $a\geq 0$ and
$a+b>0$. Then for $\delta>0$ sufficiently small, we have

$\displaystyle|\beta_{j}|\lesssim\|f\|_{H^{-1}(\mathcal{C})}+Q^{2}$ (5.30)

and

$\displaystyle\|\Psi^{\perp}\|_{H^{1}(\mathcal{C})}\lesssim\|f\|_{H^{-1}(\mathcal{C})}+Q^{2}.$
(5.31)

###### Proof.

Since $\Psi$ is the minimizer of (2.3), the Morse index of $\Psi$ is equal to
$1$. It follows from the nondegeneracy of $\Psi$ that

$\displaystyle\int_{\mathcal{C}}|\nabla_{\theta}v|^{2}+|\partial_{t}v|^{2}+(a_{c}-a)^{2}v^{2}>(p+2\varepsilon)\int_{\mathcal{C}}\Psi^{p-1}v^{2}$
(5.32)

for all $v\in\text{span}\\{\Psi,\Psi^{\prime}\\}^{\perp}$ with some
$\varepsilon>0$ sufficiently small. Since $R\to+\infty$ as $\delta\to 0$ by
Proposition 3.2 and $p>1$, for $\delta>0$ sufficiently small, it is standard
to use (5.32) and the exponential decay of $\Psi$ at infinity given by (2.8)
to show that

$\displaystyle\int_{\mathcal{C}}|\nabla_{\theta}v|^{2}+|\partial_{t}v|^{2}+(a_{c}-a)^{2}v^{2}>(p+\varepsilon)\int_{\mathcal{C}}(\sum_{j=1}^{\nu}\Psi_{s_{j}})^{p-1}v^{2}$
(5.33)

for all $v\in(M_{0}\oplus M)^{\perp}$. By (2.8), $\|\phi\|_{\sharp}\leq C$ for
$p\geq 3$ and $\|\phi\|_{\natural}\leq C$ for $1<p<3$,

$\displaystyle(\sum_{j=1}^{\nu}\Psi_{s_{j}}+\phi+\varphi)^{p}-|\sum_{j=1}^{\nu}\Psi_{s_{j}}+\phi|^{p-1}(\sum_{j=1}^{\nu}\Psi_{s_{j}}+\phi)$
$\displaystyle=$ $\displaystyle
p|\sum_{j=1}^{\nu}\Psi_{s_{j}}+\phi|^{p-1}\varphi+O(|\varphi|^{\sigma_{p}})$
$\displaystyle=$ $\displaystyle
p(\sum_{j=1}^{\nu}\Psi_{s_{j}})^{p-1}\varphi+q(t,\theta)\varphi+O(|\varphi|^{\sigma_{p}}),$

where $\|q\|_{L^{\infty}(\mathcal{C})}\leq\widehat{\delta}$ with
$\widehat{\delta}\to 0$ as $\delta\to 0$, $\sigma_{p}=2$ for $p\geq 2$ and
$\sigma_{p}=p$ for $1<p<2$. Now, multiplying (5.29) with $\Psi_{s_{j}}$ for
all $j$ and $\Psi^{\perp}$, respectively, and integrating by parts,

$\displaystyle(1-p)\beta_{j}\|\Psi\|_{H^{1}(\mathcal{C})}^{2}$
$\displaystyle=$
$\displaystyle\int_{\mathcal{C}}(p(\sum_{l=1}^{\nu}\Psi_{s_{l}})^{p-1}+q(t,\theta))(\sum_{i\not=j}\beta_{i}\Psi_{s_{i}}+\Psi^{\perp})\Psi_{s_{j}}$
$\displaystyle+\beta_{j}\int_{\mathcal{C}}(p(\sum_{l=1}^{\nu}\Psi_{s_{l}})^{p-1}-p\Psi_{s_{j}}^{p-1}+q(t,\theta))\Psi_{s_{j}}^{2}$
$\displaystyle+\langle
f,\Psi_{s_{j}}\rangle_{L^{2}(\mathcal{C})}+\sum_{i\not=j}c_{i}\langle\Psi_{s_{i}}^{p-1}\Psi_{s_{i}}^{\prime},\Psi_{s_{j}}\rangle_{L^{2}(\mathcal{C})}$
$\displaystyle+O(\sum_{i=1}^{\nu}\beta_{i}^{\sigma_{p}+1}+\|\Psi^{\perp}\|_{H^{1}(\mathcal{C})}^{\sigma_{p}+1})$

and

$\displaystyle\|\Psi^{\perp}\|_{H^{1}(\mathcal{C})}^{2}$ $\displaystyle=$
$\displaystyle\int_{\mathcal{C}}(p(\sum_{l=1}^{\nu}\Psi_{s_{l}})^{p-1}+q(t,\theta))(\sum_{i=1}^{\nu}\beta_{i}\Psi_{s_{i}}+\Psi^{\perp})\Psi^{\perp}$
$\displaystyle+\langle
f,\Psi^{\perp}\rangle_{L^{2}(\mathcal{C})}+O(\sum_{i=1}^{\nu}\beta_{i}^{\sigma_{p}+1}+\|\Psi^{\perp}\|_{H^{1}(\mathcal{C})}^{\sigma_{p}+1}).$

By (5.19), (5.20), (5.21), (5.22) and [19, Lemma 6],

$\displaystyle|\beta_{j}|\lesssim\widetilde{\delta}(\sum_{i\not=j}|\beta_{i}|+\|\Psi^{\perp}\|_{H^{1}(\mathcal{C})})+\|f\|_{H^{-1}(\mathcal{C})}+\sum_{i\not=j}|c_{i}\langle\Psi_{s_{i}}^{p-1}\Psi_{s_{i}}^{\prime},\Psi_{s_{j}}\rangle_{L^{2}(\mathcal{C})}|$

for all $j=1,2,\cdots,\nu$ and by (5.33),

$\displaystyle\|\Psi^{\perp}\|_{H^{1}(\mathcal{C})}\lesssim\widetilde{\delta}\sum_{i=1}^{\nu}|\beta_{i}|+\|f\|_{H^{-1}(\mathcal{C})},$

where $\widetilde{\delta}\to 0$ as $\delta\to 0$. It follows from Lemma 5.5
and (5.27) that (5.30) and (5.31) hold for $\delta>0$ sufficiently small. ∎

We are now in the position to prove the following stability.

######  Proposition 5.2.

Let $v\in H^{1}(\mathcal{C})$ be nonnegative such that

$\displaystyle(\nu-\frac{1}{2})(C_{a,b,N}^{-1})^{\frac{p+1}{p-1}}<\|v\|_{H^{1}(\mathcal{C})}^{2}<(\nu+\frac{1}{2})(C_{a,b,N}^{-1})^{\frac{p+1}{p-1}}$

with $\nu\geq 2$ and $\|f\|_{H^{-1}(\mathcal{C})}\leq\delta$ for $\delta>0$
sufficiently small, where $f$ is given by (5.6). Then either for
$b_{FS}(a)\leq b<a+1$ with $a<0$ or for $a\leq b<a+1$ with $a\geq 0$ and
$a+b>0$, we have

$\displaystyle d_{*}(v)\lesssim\left\\{\begin{aligned}
&\|f\|_{H^{-1}(\mathcal{C})},\quad p>2,\\\
&\|f\|_{H^{-1}(\mathcal{C})}|\log(\|f\|_{H^{-1}(\mathcal{C})})|^{\frac{1}{2}},\quad
p=2,\\\ &\|f\|_{H^{-1}(\mathcal{C})}^{\frac{p}{2}},\quad
1<p<2.\end{aligned}\right.$

###### Proof.

We recall that $d_{*}^{2}(v)=\|\rho\|_{H^{1}(\mathcal{C})}^{2}$ and
$\rho=\phi+\varphi$. The conclusion then follows immediately from Lemmas 5.2,
5.5 and 5.6. ∎

##  6\. Optimality of the stability to profile decompositions

In this section, we will construct examples, as that in [7, 11], to show that
the orders in Propositions 5.1 and 5.2 are sharp. Let us begin with the
examples for $\nu=1$.

######  Proposition 6.1.

Let $b_{FS}(a)\leq b<a+1$ for $a<0$ and $a\leq b<a+1$ for $a\geq 0$ and
$a+b>0$. Then the stability stated in Proposition 5.1 is sharp in the sense
that, there exists nonnegative $v_{*}\in H^{1}(\mathcal{C})$, with

$\displaystyle\frac{1}{2}(C_{a,b,N}^{-1})^{\frac{p+1}{p-1}}<\|v_{*}\|_{H^{1}(\mathcal{C})}^{2}<\frac{3}{2}(C_{a,b,N}^{-1})^{\frac{p+1}{p-1}}\quad\text{and}\quad\|f_{*}\|_{H^{-1}(\mathcal{C})}\leq\delta$

for $\delta>0$ sufficiently small, such that
$d_{0}(v_{*})\gtrsim\|f_{*}\|_{H^{-1}(\mathcal{C})}$.

###### Proof.

Let $v_{\varepsilon}=\Psi+\varepsilon\varphi$ where $\varphi\in
C^{\infty}_{0}(\mathcal{C})$ is positive and even such that
$\langle\Psi^{\prime},\varphi\rangle_{H^{1}(\mathcal{C})}=0.$ Then as that in
the proof of Proposition 5.1, we have

$\displaystyle f_{\varepsilon}$ $\displaystyle=$
$\displaystyle-\Delta_{\theta}v-\partial_{t}^{2}v+(a_{c}-a)^{2}v-v^{p}$
$\displaystyle=$
$\displaystyle\varepsilon(-\Delta_{\theta}\varphi-\partial_{t}^{2}\varphi+(a_{c}-a)^{2}\varphi-p\Psi^{p-1}\varphi)+O((\varepsilon\varphi)^{\sigma_{p}}).$

It follows that $\|f_{\varepsilon}\|_{H^{-1}(\mathcal{C})}\lesssim\varepsilon$
for $\varepsilon>0$ sufficiently small. As that in the proof of Proposition
5.1, it is easy to see that $d_{0}(v_{\varepsilon})\lesssim\varepsilon$ is
attained by some $s_{\varepsilon}\in\mathbb{R}$. Thus, we can rewrite
$v_{\varepsilon}=\Psi_{s_{\varepsilon}}+\widetilde{\varphi}_{\varepsilon}$,
where
$d_{0}(v_{\varepsilon})=\|\widetilde{\varphi}_{\varepsilon}\|_{H^{1}(\mathcal{C})}\lesssim\varepsilon$
and

$\displaystyle\langle\Psi_{s_{\varepsilon}},\widetilde{\varphi}_{\varepsilon}\rangle_{H^{1}(\mathcal{C})}=\langle\Psi_{s_{\varepsilon}}^{\prime},\widetilde{\varphi}_{\varepsilon}\rangle_{H^{1}(\mathcal{C})}=0.$

Note that

$\displaystyle\|\Psi_{s_{\varepsilon}}-\Psi\|_{H^{1}(\mathcal{C})}=\|\widetilde{\varphi}_{\varepsilon}-\varepsilon\varphi\|_{H^{1}(\mathcal{C})}\lesssim\varepsilon,$

we have $s_{\varepsilon}=o_{\varepsilon}(1)$. Clearly,
$\widetilde{\varphi}_{\varepsilon}$ satisfies

$\displaystyle-\Delta_{\theta}\widetilde{\varphi}_{\varepsilon}-\partial_{t}^{2}\widetilde{\varphi}_{\varepsilon}+(a_{c}-a)^{2}\widetilde{\varphi}_{\varepsilon}=(\Psi_{s_{\varepsilon}}+\widetilde{\varphi}_{\varepsilon})^{p}-\Psi_{s_{\varepsilon}}^{p}+f_{\varepsilon}.$

Let $\widetilde{f}_{\varepsilon}$ be the projection of $f_{\varepsilon}$ in
$H^{1}(\mathcal{C})$, then by the Taylor expansion and some elementary
inequalities,

$\displaystyle|\langle\widetilde{\varphi}_{\varepsilon},\widetilde{f}_{\varepsilon}\rangle_{H^{1}(\mathcal{C})}|\gtrsim-|\langle\widetilde{\varphi}_{\varepsilon},\widetilde{f}_{\varepsilon}\rangle_{H^{1}(\mathcal{C})}|-|\langle\widetilde{\varphi}_{\varepsilon}^{\sigma_{p}},\widetilde{f}_{\varepsilon}\rangle_{L^{2}(\mathcal{C})}|+\|f_{\varepsilon}\|_{H^{-1}}^{2}.$

It follows that
$d_{0}(v_{\varepsilon})=\|\widetilde{\varphi}_{\varepsilon}\|_{H^{1}(\mathcal{C})}\gtrsim\|f_{\varepsilon}\|_{H^{-1}}$.
∎

We next construct examples for $\nu\geq 2$.

######  Proposition 6.2.

Let $b_{FS}(a)\leq b<a+1$ for $a<0$ and $a\leq b<a+1$ for $a\geq 0$ and
$a+b>0$. Then the stability stated in Proposition 5.2 is sharp in the sense
that, there exists nonnegative $v_{*}\in H^{1}(\mathcal{C})$ such that

$\displaystyle d_{*}(v_{*})\gtrsim\left\\{\begin{aligned}
&\|f_{*}\|_{H^{-1}(\mathcal{C})},\quad p>2,\\\
&\|f_{*}\|_{H^{-1}(\mathcal{C})}|\log(\|f_{*}\|_{H^{-1}(\mathcal{C})})|^{\frac{1}{2}},\quad
p=2,\\\ &\|f_{*}\|_{H^{-1}(\mathcal{C})}^{\frac{p}{2}},\quad
1<p<2.\end{aligned}\right.$

###### Proof.

Let us consider the following equation:

$\displaystyle\left\\{\begin{aligned}
&-\Delta_{\theta}\widetilde{\phi}_{R}-\partial_{t}^{2}\widetilde{\phi}_{R}+(a_{c}-a)^{2}\widetilde{\phi}_{R}\\\
&=|\sum_{j=1}^{2}\Psi_{s_{j,R}}+\widetilde{\phi}_{R}|^{p-1}(\sum_{j=1}^{2}\Psi_{s_{j,R}}+\widetilde{\phi}_{R})\\\
&-\sum_{j=1}^{2}\Psi_{s_{j,R}}^{p}+\sum_{j=1}^{2}c_{j,R}\Psi_{s_{j,R}}^{p-1}\Psi_{s_{j,R}}^{\prime},\quad\text{in
}\mathcal{C},\\\
&\langle\Psi_{s_{j,R}}^{\prime},\widetilde{\phi}_{R}\rangle_{H^{1}(\mathcal{C})}=0\quad\text{for
all }j=1,2,\end{aligned}\right.$ (6.1)

where $s_{1,R}=-\frac{R}{2}$ and $s_{2,R}=\frac{R}{2}$. By Lemma 5.5, (6.1) is
solvable for $R>0$ sufficiently large with $|c_{1,R}|+|c_{2,R}|\lesssim
Q=e^{-(a_{c}-a)R}$ either for $b_{FS}(a)\leq b<a+1$ with $a<0$ or for $a\leq
b<a+1$ with $a\geq 0$ and $a+b>0$. Let
$v_{R}=\sum_{j=1}^{2}\Psi_{s_{j,R}}+\widetilde{\phi}_{R}$, then

$\displaystyle f_{R}$ $\displaystyle:=$
$\displaystyle-\Delta_{\theta}v_{R}-\partial_{t}^{2}v_{R}+(a_{c}-a)^{2}v_{R}-|v_{R}|^{p-1}v_{R}$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{2}c_{j,R}\Psi_{s_{j,R}}^{p-1}\Psi_{s_{j,R}}^{\prime}.$

which, together with Lemma 5.2 and Proposition 5.2, implies that

$\displaystyle\|f_{R}\|_{H^{-1}(\mathcal{C})}\sim\sum_{j=1}^{2}|c_{j,R}|\sim
Q$ (6.3)

for $R>0$ sufficiently large. Note that as that in the proof of Proposition
4.1, we can show that
$d_{*}^{2}(v_{R})\leq\|\widetilde{\phi}_{R}\|_{H^{1}(\mathcal{C})}^{2}$ is
attained at $\sum_{j=1}^{2}\Psi_{s^{\prime}_{j,R}}$ for some
$s^{\prime}_{1,R}$ and $s^{\prime}_{2,R}$. Thus, we can rewrite
$v_{R}=\sum_{j=1}^{2}\Psi_{s^{\prime}_{j,R}}+\widetilde{\varphi}_{R}$, where
$\widetilde{\varphi}_{R}\in(\text{span}\\{\Psi_{s^{\prime}_{j,R}}^{\prime}\\})^{\perp}$
in $H^{1}(\mathcal{C})$. It follows that
$d_{*}^{2}(v_{R})=\|\widetilde{\varphi}_{R}\|_{H^{1}(\mathcal{C})}^{2}\leq\|\widetilde{\phi}_{R}\|_{H^{1}(\mathcal{C})}^{2}$.
Since

$\displaystyle\|\sum_{j=1}^{2}\Psi_{s^{\prime}_{j,R}}-\sum_{j=1}^{2}\Psi_{s_{j,R}}\|_{H^{1}(\mathcal{C})}\lesssim\|\widetilde{\phi}_{R}\|_{H^{1}(\mathcal{C})}\to
0\quad\text{as }R\to+\infty,$

we have $s^{\prime}_{j,R}=s_{j,R}+o_{R}(1)$. Clearly, by (6.1),
$\widetilde{\varphi}_{R}$ satisfies the following equation:

$\displaystyle\left\\{\begin{aligned}
&-\Delta_{\theta}\widetilde{\varphi}_{R}-\partial_{t}^{2}\widetilde{\varphi}_{R}+(a_{c}-a)^{2}\widetilde{\varphi}_{R}\\\
&=|\sum_{j=1}^{2}\Psi_{s^{\prime}_{j,R}}+\widetilde{\varphi}_{R}|^{p-1}(\sum_{j=1}^{2}\Psi_{s^{\prime}_{j,R}}+\widetilde{\varphi}_{R})\\\
&-\sum_{j=1}^{2}\Psi_{s^{\prime}_{j,R}}^{p}+\sum_{j=1}^{2}c_{j,R}\Psi_{s_{j,R}}^{p-1}\Psi_{s_{j,R}}^{\prime},\quad\text{in
}\mathcal{C},\\\
&\langle\Psi_{s^{\prime}_{j,R}}^{\prime},\widetilde{\varphi}_{R}\rangle_{H^{1}(\mathcal{C})}=0\quad\text{for
all }j=1,2.\end{aligned}\right.$ (6.4)

Let $\varrho_{R}:[0,1]\to[0,1]$ be a smooth cut-off function such that

$\displaystyle\varrho_{R}(t)=\left\\{\begin{aligned} &1,\quad
s^{\prime}_{1,R}+\frac{R}{2}-3\leq t\leq s^{\prime}_{1,R}+\frac{R}{2}-2,\\\
&0,\quad t\leq s^{\prime}_{1,R}+\frac{R}{2}-4\text{ or }t\geq
s^{\prime}_{1,R}+\frac{R}{2}-1.\end{aligned}\right.$ (6.5)

Then $\|\varrho_{R}\|_{H^{1}(\mathcal{C})}\lesssim 1$. Similar to that of
(5.10), (5.11) and (5.12),

$\displaystyle|\sum_{j=1}^{2}\Psi_{s^{\prime}_{j,R}}+\widetilde{\varphi}_{R}|^{p-1}(\sum_{j=1}^{2}\Psi_{s^{\prime}_{j,R}}+\widetilde{\varphi}_{R})-\sum_{j=1}^{2}\Psi_{s^{\prime}_{j,R}}^{p}\gtrsim\Psi_{s^{\prime}_{1,R}}^{p-1}\Psi_{s^{\prime}_{2,R}}+p\Psi_{s^{\prime}_{1,R}}^{p-1}\widetilde{\varphi}_{R}+O(|\widetilde{\varphi}_{R}|^{\sigma_{p}})$

in the region
$[s^{\prime}_{1,R}+\frac{R}{2}-4,s^{\prime}_{1,R}+\frac{R}{2}-1]$. Thus, by
multiplying (6.4) with $\varrho_{R}$ and integrating by parts,

$\displaystyle\|\widetilde{\varphi}_{R}\|_{H^{1}(\mathcal{C})}$
$\displaystyle\gtrsim$
$\displaystyle\int_{\mathcal{C}}(\Psi_{s^{\prime}_{1,R}}^{p-1}\Psi_{s^{\prime}_{2,R}}+p\Psi_{s^{\prime}_{1,R}}^{p-1}\widetilde{\varphi}_{R}+O(|\widetilde{\varphi}_{R}|^{\sigma_{p}}))\varrho_{R}+\int_{\mathcal{C}}\sum_{j=1}^{2}c_{j,R}\Psi_{s_{j,R}}^{p-1}\Psi_{s_{j,R}}^{\prime}\varrho_{R}$
$\displaystyle\gtrsim$
$\displaystyle\int_{\mathcal{C}}\Psi_{s^{\prime}_{1,R}}^{p-1}\Psi_{s^{\prime}_{2,R}}\varrho_{R}-\sum_{j=1}^{2}|c_{j,R}|\int_{\mathcal{C}}\Psi_{s_{j,R}}^{p-1}\Psi_{s_{j,R}}^{\prime}\varrho_{R}-\|\widetilde{\varphi}_{R}\|_{H^{1}(\mathcal{C})}.$

By (2.8) and (6.5),

$\displaystyle\sum_{j=1}^{2}|c_{j,R}|\int_{\mathcal{C}}\Psi_{s_{j,R}}^{p-1}\Psi_{s_{j,R}}^{\prime}\varrho_{R}=o(\sum_{j=1}^{2}|c_{j,R}|)\quad\text{as
}R\to+\infty$

and

$\displaystyle\int_{\mathcal{C}}\Psi_{s^{\prime}_{1,R}}^{p-1}\Psi_{s^{\prime}_{2,R}}\varrho_{R}\gtrsim\int_{\frac{R}{2}-3}^{\frac{R}{2}-2}e^{-(p-1)(a_{c}-a)r}e^{-(a_{c}-a)(R-r)}\sim\left\\{\begin{aligned}
&Q,\quad p\geq 2,\\\ &Q^{\frac{p}{2}},\quad 1<p<2\end{aligned}\right.$

for $R>0$ sufficiently large. It follows from (6.3) that

$\displaystyle d_{*}(v_{R})\gtrsim\left\\{\begin{aligned}
&\|f_{R}\|_{H^{-1}(\mathcal{C})},\quad p>2,\\\
&\|f_{R}\|_{H^{-1}(\mathcal{C})}^{\frac{p}{2}},\quad
1<p<2\end{aligned}\right.$

for $R>0$ sufficiently large. For $p=2$, we shall modify the test function
$\varrho_{R}$ by

$\displaystyle\widetilde{\varrho}_{R}(t)=\left\\{\begin{aligned} &1,\quad
s^{\prime}_{1,R}+\frac{R}{4}\leq t\leq s^{\prime}_{1,R}+\frac{R}{2}-2,\\\
&0,\quad t\leq s^{\prime}_{1,R}+\frac{R}{4}-1\text{ or }t\geq
s^{\prime}_{1,R}+\frac{R}{2}-1.\end{aligned}\right.$ (6.6)

Then $\|\widetilde{\varrho}_{R}\|_{H^{1}(\mathcal{C})}\lesssim\sqrt{R}$ for
$R>0$ sufficiently large. Thus, by multiplying (6.4) with
$\widetilde{\varrho}_{R}$ and integrating by parts,

$\displaystyle\sqrt{R}\|\widetilde{\varphi}_{R}\|_{H^{1}(\mathcal{C})}$
$\displaystyle\gtrsim$
$\displaystyle\int_{\mathcal{C}}(\Psi_{s^{\prime}_{1,R}}\Psi_{s^{\prime}_{2,R}}+p\Psi_{s^{\prime}_{1,R}}\widetilde{\varphi}_{R}+O(\widetilde{\varphi}_{R}^{2}))\widetilde{\varrho}_{R}+\int_{\mathcal{C}}\sum_{j=1}^{2}c_{j,R}\Psi_{s_{j,R}}\Psi_{s_{j,R}}^{\prime}\widetilde{\varrho}_{R}$
$\displaystyle\gtrsim$
$\displaystyle\int_{\mathcal{C}}\Psi_{s^{\prime}_{1,R}}\Psi_{s^{\prime}_{2,R}}\widetilde{\varrho}_{R}-\sum_{j=1}^{2}|c_{j,R}|\int_{\mathcal{C}}\Psi_{s_{j,R}}\Psi_{s_{j,R}}^{\prime}\widetilde{\varrho}_{R}-\|\widetilde{\varphi}_{R}\|_{H^{1}(\mathcal{C})}\sqrt{R}.$

By (2.8) and (6.6),

$\displaystyle\sum_{j=1}^{2}|c_{j,R}|\int_{\mathcal{C}}\Psi_{s_{j,R}}\Psi_{s_{j,R}}^{\prime}\widetilde{\varrho}_{R}\sim\sum_{j=1}^{2}|c_{j,R}|\int_{\frac{R}{4}}^{\frac{R}{2}}e^{-2(a_{c}-a)r}\sim\sum_{j=1}^{2}|c_{j,R}|e^{-(a_{c}-a)\frac{R}{2}}$

as $R\to+\infty$. On the other hand,

$\displaystyle\int_{\mathcal{C}}\Psi_{s^{\prime}_{1,R}}\Psi_{s^{\prime}_{2,R}}\widetilde{\varrho}_{R}\gtrsim\int_{\frac{R}{4}}^{\frac{R}{2}-1}e^{-(a_{c}-a)r}e^{-(a_{c}-a)(R-r)}\sim
Re^{-(a_{c}-a)R}$

for $R>0$ sufficiently large. It follows from (6.3) that

$\displaystyle
d_{*}(v_{R})\gtrsim\|f_{R}\|_{H^{-1}(\mathcal{C})}|\log(\|f_{R}\|_{H^{-1}(\mathcal{C})})|^{\frac{1}{2}}$

for $p=2$ and $R>0$ sufficiently large. Now, we take $v_{*}=v_{R}^{+}$ then
$v_{*}=v_{R}+v_{R}^{-}$, where $v_{R}^{\pm}=\max\\{\pm v_{R},0\\}$. Clearly,
we have $0\leq v_{R}^{-}\leq|\widetilde{\phi}_{R}|$ since
$\sum_{j=1}^{2}\Psi_{s_{j,R}}$ is positive. It follows from (6) that

$\displaystyle\|v_{R}^{-}\|_{H^{1}(\mathcal{C})}^{2}=\langle
v_{R}^{-},V_{R}\rangle_{H^{1}(\mathcal{C})}=\int_{\mathcal{C}}|v_{R}|^{p-1}v_{R}v_{R}^{-}+\int_{\mathcal{C}}f_{R}v_{R}^{-}\lesssim\|v_{R}^{-}\|_{H^{1}(\mathcal{C})}^{p+1}+\int_{\mathcal{C}}|f_{R}||v_{R}^{-}|.$

For $1<p\leq 2$, by Lemma 5.5 and (6.3),

$\displaystyle\|v_{R}^{-}\|_{H^{1}(\mathcal{C})}^{2}\lesssim\int_{\mathcal{C}}|f_{R}||\widetilde{\phi}_{R}|\lesssim\left\\{\begin{aligned}
&Q^{2}|\log(Q)|^{\frac{1}{2}},\quad p=2,\\\ &Q^{1+\frac{p}{2}},\quad
1<p<2.\end{aligned}\right.$

For $p>2$, recall that by Lemma 5.5,
$\|\widetilde{\phi}_{R}\|_{\sharp}\lesssim 1$. Thus, by (2.8) and (5.18),
$v_{R}^{-}=0$ for $|t-s_{j,R}|\leq\frac{R}{2}$. It follows that

$\displaystyle\|v_{R}^{-}\|_{H^{1}(\mathcal{C})}^{2}\lesssim\int_{\mathcal{C}}|f_{R}||v_{R}^{-}|\lesssim\int_{(\cup_{j=1}^{2}\\{|t-s_{j,R}|\leq
R\\})^{c}}|f_{R}||\widetilde{\phi}_{R}|=o(Q^{2}).$

The conclusion then follows from $d_{*}(v_{*})\geq
d_{*}(v_{R})-\|v_{R}^{-}\|_{H^{1}(\mathcal{C})}$. ∎

We close this section by the proof of Theorem 1.3.

Proof of Theorem 1.3: It follows immediately from (2.1) and Propositions 5.1,
5.2, 6.1 and 6.2. $\Box$

##  7\. Acknowledgements

The research of J. Wei is partially supported by NSERC of Canada and the
research of Y. Wu is supported by NSFC (No. 11971339).

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