Here we prove Theorem
5.1. We discuss both of the cases when
\(n=1\) or
M is spherical. For convenience, we denote the CR conformal sub-Laplacian of a contact form
\(\theta \) by
\(L_\theta \), i.e.
$$\begin{aligned} L_\theta =-\left( 2+\frac{2}{n}\right) \Delta _{\theta }+R_{\theta }. \end{aligned}$$
Let
\(\{u_\nu \}\) be a sequence of positive functions satisfying (
5.1) and (
5.2). Note that
\(u_\nu \) is uniformly bounded in
\(S_1^2(M)\). To see this, it follows from integration by parts and Hölder’s inequality that
$$\begin{aligned}&\int _M\left( \left( 2+\frac{2}{n}\right) |\nabla _{\theta _0}u_\nu |^2_{\theta _0}+R_{\theta _0}u_\nu ^2\right) dV_{\theta _0}\\&\quad \le \left( \int _M|L_{\theta _0}u_\nu -r_\infty u_\nu ^{1+\frac{2}{n}}|^{\frac{2n+2}{n+2}} dV_{\theta _0} \right) ^{\frac{n+2}{2n+2}}\left( \int _Mu_\nu ^{2+\frac{2}{n}}\right) ^{\frac{n}{2n+2}} +r_\infty \int _Mu_\nu ^{2+\frac{2}{n}} dV_{\theta _0}, \end{aligned}$$
which is uniformly bounded by (
5.1) and (
5.2). Thus, by passing to a subsequence if necessary, we assume that
\(u_\nu \) converges to
\(u_\infty \) weakly in
\(S_1^2(M)\). Since the Folland–Stein embedding
\(S_1^2(M)\hookrightarrow L^s(M)\) is compact for
\(1<s<2+\displaystyle \frac{2}{n}\) (see Proposition 5.6 in [
28]),
\(u_\nu \) converges to
\(u_\infty \) in
\(L^s(M)\) for
\(1<s<2+\displaystyle \frac{2}{n}\).
If
\(u_\nu \) converges to
\(u_\infty \) strongly in
\(S_1^2(M)\), then Theorem
5.1 follows. Therefore, we assume that
\(u_\nu \) converges to
\(u_\infty \) weakly in
\(S_1^2(M)\), but not strongly in
\(S_1^2(M)\). On the other hand, it follows from (
5.2), (
5.3) and (
B.4) that
$$\begin{aligned}&\int _M\Bigg |L_{\theta _0}(u_\nu -u_\infty )-r_\infty |u_\nu -u_\infty |^{\frac{2}{n}}(u_\nu -u_\infty )\Bigg |^{\frac{2n+2}{n+2}}dV_{\theta _0}\\&\quad \le C\int _M\Bigg |L_{\theta _0}u_\nu -r_\infty u_\nu ^{1+\frac{2}{n}}\Bigg |^{\frac{2n+2}{n+2}}dV_{\theta _0}\\&\qquad +\,C\, r_\infty \int _M\Bigg |u_\nu ^{1+\frac{2}{n}}-u_\infty ^{1+\frac{2}{n}}-|u_\nu -u_\infty |^{\frac{2}{n}}(u_\nu -u_\infty )\Bigg |^{\frac{2n+2}{n+2}}dV_{\theta _0} \rightarrow 0 \end{aligned}$$
as
\(\nu \rightarrow \infty \). That is to say, if we let
\(v_\nu :=u_\nu -u_\infty \), then
\(\{v_\nu \}\) is a sequence of functions such that
\(v_\nu \) converges to 0 weakly in
\(S_1^2(M)\), but not strongly in
\(S_1^2(M)\), and satisfies
$$\begin{aligned} \int _M\Bigg |L_{\theta _0}v_\nu -r_\infty |v_\nu |^{\frac{2}{n}}v_\nu \Bigg |^{\frac{2n+2}{n+2}}dV_{\theta _0} \rightarrow 0 \quad \text{ as } \nu \rightarrow \infty . \end{aligned}$$
(B.9)
We are going to extract bubbles from
\(v_\nu \). To do this, we mainly follow the proof of Proposition 8 in [
19]. The argument is almost the same except with some small modifications. However, there are some parts in the argument of [
19] which are not very precise. So we are going to provide all the details of the proof.
As before, we denote
$$\begin{aligned} B_r(x)=\{y\in M:d(x,y)<r\}, \end{aligned}$$
where
d is the Carnot–Carathéodory distance on
M with respect to the contact form
\(\theta _0\).
Before we give the proof of Lemma B.4, we remark that our constant in Lemma B.4 is different from that of [19]. But one can see from the following arguments that a uniform constant will be sufficient for our purpose.
It follows from Lemma
B.4 that for any
\(x\in M\) satisfying (
B.10) and for any given
\(a_0<1\), and
\(\nu \) sufficiently large, there exists a
\(\rho _\nu (x)\) such that
$$\begin{aligned} \int _{B_{\rho _\nu (x)}(x)}\left( \left( 2+\frac{2}{n}\right) |\nabla _{\theta _0}v_\nu |_{\theta _0}^2+R_{\theta _0}v_\nu ^2\right) dV_{\theta _0} = a_0 r_\infty ^{-n} Y(M,\theta _0)^{n+1}. \end{aligned}$$
Which means for and
\(\rho >\rho _\nu (x)\), we have
$$\begin{aligned} \int _{B_{\rho _\nu (x)}(x)}\left( \left( 2+\frac{2}{n}\right) |\nabla _{\theta _0}v_\nu |_{\theta _0}^2+R_{\theta _0}v_\nu ^2\right) dV_{\theta _0} >a_0 r_\infty ^{-n} Y(M,\theta _0)^{n+1}. \end{aligned}$$
Then for every
\(\nu \) sufficiently large, we define:
$$\begin{aligned} \rho _{1,\nu }=\inf \rho _\nu (x), \end{aligned}$$
(B.32)
where the infimum is taken among all
\(x\in M\) satisfying (
B.10), which is a closed set. Thus the infimum is attained. That is, there exists
\(x^*_{1,\nu }\in M\) such that
$$\begin{aligned} \rho _{1,\nu }=\rho _\nu (x^*_{1,\nu }). \end{aligned}$$
(B.33)
For any
\(\rho _0>0\), by the proof of Lemma
B.4, we have
$$\begin{aligned} \int _{B_{{\widetilde{\rho }}}(x)}\left( \left( 2+\frac{2}{n}\right) |\nabla _{\theta _0}v_\nu |_{\theta _0}^2+R_{\theta _0}v_\nu ^2\right) dV_{\theta _0} \ge a_0 r_\infty ^{-n} Y(M,\theta _0)^{n+1} \end{aligned}$$
for all
\(\nu \) sufficiently large. Since
\(\rho _0\) is arbitrary, and
\(\rho _\nu (x)\le \rho _0\), then we have the following lemma:
As explained in Appendix A, we can find
\(\rho >0\) which is independent of
x (since
M is compact) with
\({\widehat{\theta }}_x=\varphi _x^{\frac{2}{n}}\theta _0\) in a neighborhood
\(B_{3\delta }(x)\) of
x such that (
A.1) and (
A.4) are satisfied when
\(n=1\) and (
A.9) is satisfied when
M is spherical. We define a sequence of functions
\(\{ {\widetilde{v}}_\nu \}\) in
\(B_{2\delta }(x)\) as follows:
$$\begin{aligned} {\widetilde{v}}_\nu =\varphi _x^{-1}v_\nu . \end{aligned}$$
Since
\({\widehat{\theta }}_x=\varphi _x^{\frac{2}{n}}\theta _0\), by the CR transformation law, we have
$$\begin{aligned} L_{\theta _0}v_\nu =\varphi _x^{1+\frac{2}{n}}L_{{\widehat{\theta }}_x}{\widetilde{v}}_\nu , \end{aligned}$$
(B.34)
which holds in
\(B_{2\delta }(x)\subset M\). It follows from (
A.8) that (
B.34) holds in the CR normal coordinates
\(\{(z,t):(t^2+|z|^4)^{\frac{1}{4}}<{\widehat{\rho }}\}\subset {\mathbb {H}}^n\), where
\({\widehat{\rho }}>0\) is independent of
x. By (
B.9) and (
B.34), we have
$$\begin{aligned} \int _{\{(t^2+|z|^4)^{\frac{1}{4}}<{\widehat{\rho }}\}}\Bigg |L_{{\widehat{\theta }}_x}{\widetilde{v}}_\nu -r_\infty |{\widetilde{v}}_\nu |^{\frac{2}{n}}{\widetilde{v}}_\nu \Bigg |^{\frac{2n+2}{n+2}}dV_{{\widehat{\theta }}_x} \rightarrow 0 \quad \text{ as } \nu \rightarrow \infty . \end{aligned}$$
(B.35)
By the properties of
\(v_\nu \), we know that
\({\widetilde{v}}_\nu \) is bounded in
\(L^{2+\frac{2}{n}}\) and
\({\widetilde{v}}_\nu \rightarrow 0\) in
\(L^s\) for all
\(s<2+\displaystyle \frac{2}{n}\) as
\(\nu \rightarrow \infty \).
When
\(n=1\), it follows from (
A.1), (
A.3), (
A.4) and (
B.35) that
$$\begin{aligned}&\int _{\{(t^2+|z|^4)^{\frac{1}{4}}<{\widehat{\rho }}\}}\left| L_{\theta _{{\mathbb {H}}^n}}{\widetilde{v}}_\nu -r_\infty |{\widetilde{v}}_\nu |^{\frac{2}{n}}{\widetilde{v}}_\nu \right| ^{\frac{2n+2}{n+2}}dV_{\theta _{{\mathbb {H}}^n}}\nonumber \\&\quad \le C\int _{\{(t^2+|z|^4)^{\frac{1}{4}}<{\widehat{\rho }}\}}\left| L_{\theta _{{\mathbb {H}}^n}} {\widetilde{v}}_\nu -L_{{\widehat{\theta }}_x}{\widetilde{v}}_\nu \right| ^{\frac{2n+2}{n+2}}dV_{\theta _{{\mathbb {H}}^n}}\nonumber \\&\qquad +\,C\int _{\{(t^2+|z|^4)^{\frac{1}{4}}<{\widehat{\rho }}\}}\left| L_{{\widehat{\theta }}_x}{\widetilde{v}}_\nu -r_\infty {\widetilde{v}}_\nu ^{1+\frac{2}{n}}\right| ^{\frac{2n+2}{n+2}}\Bigg |dV_{\theta _{{\mathbb {H}}^n}}-dV_{{\widehat{\theta }}_x}\Bigg |\nonumber \\&\qquad +\,C\int _{\{(t^2+|z|^4)^{\frac{1}{4}}<{\widehat{\rho }}\}}\left| L_{{\widehat{\theta }}_x}{\widetilde{v}}_\nu -r_\infty |{\widetilde{v}}_\nu |^{\frac{2}{n}}{\widetilde{v}}_\nu \right| ^{\frac{2n+2}{n+2}}dV_{{\widehat{\theta }}_x} \rightarrow 0 \end{aligned}$$
(B.36)
as
\(\nu \rightarrow \infty \). When
M is spherical, then it follows from (
A.9) and (
B.35) that
$$\begin{aligned} \int _{\{(t^2+|z|^4)^{\frac{1}{4}}<{\widehat{\rho }}\}}\Bigg |L_{\theta _{{\mathbb {H}}^n}}{\widetilde{v}}_\nu -r_\infty |{\widetilde{v}}_\nu |^{\frac{2}{n}}{\widetilde{v}}_\nu \Bigg |^{\frac{2n+2}{n+2}}dV_{\theta _{{\mathbb {H}}^n}} \rightarrow 0 \quad \text{ as } \nu \rightarrow \infty , \end{aligned}$$
(B.37)
since
\(R_{\theta _{{\mathbb {H}}^n}}=0\).
Let
\({\widehat{\chi }}\) be a cut off function such that
\({\widehat{\chi }}(s)=1\) if
\(0\le s\le \displaystyle \frac{{\widehat{\rho }}}{2}\) and 0 if
\(s\ge {\widehat{\rho }}\). Let
\(\{{\widetilde{V}}_\nu \}\) be a sequence of functions in
\({\mathbb {H}}^n\) defined by
$$\begin{aligned} {\widetilde{V}}_\nu (z,t)= & {} \left\{ \begin{array}{ll} (\rho _{1,\nu })^n\,{\widehat{\chi }}\Bigg (\rho _{1,\nu }(t^2+|z|^4)^{\frac{1}{4}}\Bigg )\,{\widetilde{v}}_\nu \Bigg (\rho _{1,\nu }z,(\rho _{1,\nu })^2t\Bigg ), &{} \hbox { for } (t^2+|z|^4)^{\frac{1}{4}}<\displaystyle \frac{{\widehat{\rho }}}{\rho _{1,\nu }}; \\ 0, &{} \hbox { otherwise,} \end{array} \right. \nonumber \\ \end{aligned}$$
(B.38)
where
\(\rho _{1,\nu }\) is defined as in (
B.32). Then we have the following:
It follows from Proposition
B.6(ii) that, by passing to subsequence if necessary,
\({\widetilde{V}}_\nu \) converges to
\({\widetilde{V}}\) weakly in
\(S_1^2(B)\) as
\(\nu \rightarrow \infty \) on each ball
B of
\({\mathbb {H}}^n\). Since the Folland–Stein embedding
\(S_1^2(B)\hookrightarrow L^s(B)\) is compact for
\(1<s<2+\displaystyle \frac{2}{n}\) on each ball
B of
\({\mathbb {H}}^n\),
\({\widetilde{V}}_\nu \) converges to
\({\widetilde{V}}\) in
\(L^s(B)\) for
\(1<s<2+\displaystyle \frac{2}{n}\). On the other hand, it follows from Proposition
B.6(i) that
\({\widetilde{V}}\) satisfies
$$\begin{aligned} \left( 2+\frac{2}{n}\right) \Delta _{\theta _{{\mathbb {H}}^n}}{\widetilde{V}}=r_\infty |{\widetilde{V}}|^{\frac{2}{n}}{\widetilde{V}}\quad \text{ in } {\mathbb {H}}^n. \end{aligned}$$
(B.39)
Since
\({\widetilde{V}}\) satisfies (
B.39), it follows from the result of Jerison and Lee in [
26] that there exists
\((z_0,t_0)\in {\mathbb {H}}^n\) and
\(\gamma _1>0\) such that
$$\begin{aligned} {\widetilde{V}}(z,t)=W\circ T_{(z_0,t_0)}(z,t), \end{aligned}$$
where
$$\begin{aligned} W(z,t)=\left( \frac{n(2n+2)}{r_\infty }\right) ^{\frac{n}{2}}\left( \frac{\gamma _1^2}{\gamma _1^4t^2+(\gamma _1^2|z|^2+1)^2}\right) ^{\frac{n}{2}} \end{aligned}$$
and
$$\begin{aligned} T_{(z_0,t_0)}(z,t)=(z+z_0,t+t_0+2Im(z\cdot z_0)) \text{ for } (z,t)\in {\mathbb {H}}^n \end{aligned}$$
is the translation in
\({\mathbb {H}}^n\). By the optimality of
\((x^*_{1,\nu }, \rho _{1,\nu })\), we can conclude that
\((z_0,t_0)=(0,0)\); for if
\((z_0,t_0)\ne (0,0)\), we can find
\(({\widetilde{x}}^*_{1,\nu }, {\widetilde{\rho }}_{1,\nu })\) with
\({\widetilde{\rho }}_{1,\nu }<\rho _{1,\nu }\) such that
$$\begin{aligned} \int _{B_{{\widetilde{\rho }}_{1,\nu }}({\widetilde{x}}^*_{1,\nu })}\left( \left( 2+\frac{2}{n}\right) |\nabla _{\theta _0}v_\nu |_{\theta _0}^2+R_{\theta _0}v_\nu ^2\right) dV_{\theta _0} \ge a_0 r_\infty ^{-n} Y(M,\theta _0)^{n+1}. \end{aligned}$$
Therefore, we have
$$\begin{aligned} {\widetilde{V}}(z,t)=\left( \frac{n(2n+2)}{r_\infty }\right) ^{\frac{n}{2}}\left( \frac{\gamma _1^2}{\gamma _1^4t^2+(\gamma _1^2|z|^2+1)^2}\right) ^{\frac{n}{2}}. \end{aligned}$$
(B.43)
We remark that it was claimed in [
19] that
\(\gamma _1=1\) (see the last line in P.146 in [
19]), which does not seem to be true. In fact, we will show that
\(\gamma _1\ge C_1\). Here
\(C_1\) is a positive constant depending only on
\(a_0\),
\(r_{\infty }\), and
M. We need the following:
For any
\((x,\lambda )\in M\times (0,\infty )\), we can find a unique solution
\({\widehat{\omega }}(x,\lambda )\) of the following equation:
$$\begin{aligned} L_{\theta _0}{\widehat{\omega }}(x,\lambda )=r_\infty \omega '(x,\lambda )^{1+\frac{2}{n}} \quad \text{ in } M, \end{aligned}$$
(B.58)
where
\(\omega '(x,\lambda )\) is defined as
$$\begin{aligned} \omega '(x,\lambda )(y)= \left\{ \begin{array}{ll} \chi _\delta (\rho _x(y))\varphi _x(y)\omega (x,\lambda )(y), &{} \hbox { for } y\in B_{2\delta }(x); \\ 0, &{} \hbox { otherwise.} \end{array} \right. \end{aligned}$$
Here
\(\chi _\delta \) is the cut-off function defined in (
A.11),
\(\varphi _x\) is the conformal factor such that
\({\widehat{\theta }}_x=\varphi _x^{\frac{2}{n}}\theta _0\) in a neighborhood
\(B_{3\delta }(x)\) of
x. Moreover,
\(\omega (x,\lambda )(y)\) is given by
$$\begin{aligned} \omega (x,\lambda )(y)=\left( \frac{n(2n+2)}{r_\infty }\right) ^{\frac{n}{2}}\left( \frac{\lambda ^2}{\lambda ^4t^2+(\lambda ^2|z|^2+1)^2}\right) ^{\frac{n}{2}}, \end{aligned}$$
(B.59)
where (
z,
t) is CR normal coordinates of
y centered at
x. It follows from the definition of
\({\widehat{\omega }}(x,\lambda )\) that
$$\begin{aligned} {\widehat{\omega }}(x,\lambda )(y)=\omega '(x,\lambda )(y)=0 \quad \hbox { for }y\in M-B_{2\delta }(x). \end{aligned}$$
(B.60)
When
\(n=1\), there holds (see Proposition 1 in [
18])
$$\begin{aligned} |{\widehat{\omega }}(x,\lambda )(y)-\omega '(x,\lambda )(y)|\le C\lambda ^{-1}(1+|\log (\lambda ^{-2}+\rho _x(y)^2)|) \quad \text{ for } y\in B_{2\delta }(x)\nonumber \\ \end{aligned}$$
(B.61)
and (see (3.6) in [
18])
$$\begin{aligned} \Bigg |L_{\theta _0}\Bigg ({\widehat{\omega }}(x,\lambda )(y)-\omega '(x,\lambda )(y)\Bigg )\Bigg |\le \inf \Bigg \{1,\frac{C}{\rho _x(y)^2+\lambda ^{-2}}\Bigg \} \quad \text{ for } y\in B_{2\delta }(x).\nonumber \\ \end{aligned}$$
(B.62)
It follows from (
B.60)–(
B.62) that
$$\begin{aligned} \Vert {\widehat{\omega }}(x,\lambda )-\omega '(x,\lambda )\Vert _{S_1^2(M)}\rightarrow 0 \quad \text{ as } \lambda \rightarrow \infty . \end{aligned}$$
(B.63)
Similarly, when
M is spherical, it follows from Lemmas 3 and 4 in [
19] that
$$\begin{aligned} |{\widehat{\omega }}(x,\lambda )(y)-\omega '(x,\lambda )(y)|\le \frac{C}{\lambda ^n} \quad \text{ for } y\in B_{2\delta }(x) \end{aligned}$$
(B.64)
and
$$\begin{aligned} \Vert {\widehat{\omega }}(x,\lambda )-\omega '(x,\lambda )\Vert _{S_1^2(M)}=O\left( \frac{1}{\lambda }\right) . \end{aligned}$$
(B.65)
We have the following:
We can now extract from
\(\{v_\nu \}\) the first bubble and consider the following new sequence of functions:
$$\begin{aligned} v_\nu ^1(x)=v_\nu (x)-{\widehat{\omega }}\left( x^*_{1,\nu },\frac{\gamma _1}{\rho _{1,\nu }}\right) (x). \end{aligned}$$
Iterating the above procedure, either
\(v_\nu ^1\) converges strongly to 0 in
\(S_1^2(M)\) as
\(\nu \rightarrow \infty \), or we can find a new sequence
\((x^*_{2,\nu },\rho _{2,\nu })\) and extract another bubble by defining
$$\begin{aligned} v_\nu ^2(x)=v_\nu ^1(x)-{\widehat{\omega }}\left( x^*_{2,\nu },\frac{\gamma _2}{\rho _{2,\nu }}\right) (x) \end{aligned}$$
and show that
$$\begin{aligned} \int _M\Bigg |L_{\theta _0}v_\nu ^2-r_\infty |v_\nu ^2|^{\frac{2}{n}}v_\nu ^2\Bigg |^{\frac{2n+2}{n+2}}dV_{\theta _0} \rightarrow 0 \quad \text{ as } \nu \rightarrow \infty . \end{aligned}$$
On the other hand, it can be shown that (see Lemmas 15 and 16 in [
19])
$$\begin{aligned} \rho _{2,\nu }\ge \frac{1}{2}\rho _{1,\nu } \quad \text{ and } \quad \frac{\rho _{2,\nu }}{\rho _{1,\nu }}+\frac{d(x^*_{1,\nu },x^*_{2,\nu })^2}{\rho _{1,\nu }\,\rho _{2,\nu }} \rightarrow \infty \end{aligned}$$
(B.74)
as
\(\nu \rightarrow \infty \). Here
d is Carnot–Carathéodory distance on
M with respect to the contact form
\(\theta _0\). This argument can be iterated as long as the new sequence
\(\{v_\nu ^l\}\) does not coverage strongly to 0 in
\(S_1^2(M)\). And we claim that the iteration must terminate in finite steps. To see this, note that
$$\begin{aligned}&\int _M \left( \left( 2+\frac{2}{n}\right) |\nabla _{\theta _0}v_\nu ^{l}|^2_{\theta _0} +R_{\theta _0}(v_\nu ^{l})^2\right) dV_{\theta _0}\\&\quad =\int _M \left( \left( 2+\frac{2}{n}\right) \Bigg |\nabla _{\theta _0}\Bigg (v_\nu ^{l-1}-{\widehat{\omega }}\Bigg (x^*_{l,\nu },\frac{\gamma _{l}}{\rho _{l,\nu }}\Bigg )\Bigg )\Bigg |^2_{\theta _0}\right. \\&\qquad \left. +\,R_{\theta _0}\Bigg (v_\nu ^{l-1}-{\widehat{\omega }}\Bigg (x^*_{l,\nu },\frac{\gamma _{l}}{\rho _{l,\nu }}\Bigg )\Bigg )^2\right) dV_{\theta _0}\\&\quad =\int _M \left( \left( 2+\frac{2}{n}\right) |\nabla _{\theta _0}v_\nu ^{l-1}|^2_{\theta _0} +R_{\theta _0}(v_\nu ^{l-1})^2\right) dV_{\theta _0}\\&\qquad +\,r_\infty \int _M{\widehat{\omega }}\left( x^*_{l,\nu },\frac{\gamma _{l}}{\rho _{l,\nu }}\right) \omega '\left( x^*_{l,\nu },\frac{\gamma _{l}}{\rho _{l,\nu }}\right) ^{1+\frac{2}{n}}dV_{\theta _0} +o(1), \end{aligned}$$
where the first equality follows from the fact that
\(v_\nu ^{l-1}\) converges to 0 weakly in
\(S_1^2(M)\) as
\(\nu \rightarrow \infty \), and the last equality follows from (
B.58). We compute
$$\begin{aligned}&\int _M{\widehat{\omega }}\left( x^*_{l,\nu },\frac{\gamma _{l}}{\rho _{l,\nu }}\right) \omega '\left( x^*_{l,\nu },\frac{\gamma _{l}}{\rho _{l,\nu }}\right) ^{1+\frac{2}{n}}dV_{\theta _0}\\&\quad \ge \int _{B_{\delta }(x^*_{l,\nu })}\Bigg (\varphi _{x^*_{l,\nu }}(y)\omega \Bigg (x^*_{l,\nu },\frac{\gamma _{l}}{\rho _{l,\nu }}\Bigg )(y)\Bigg )^{2+\frac{2}{n}}dV_{\theta _0}+o(1)\\&\quad \ge \int _{\left\{ (|z|^4+t^2)^{\frac{1}{4}}\le \frac{\delta }{C_0}\right\} }\omega \left( x^*_{l,\nu },\frac{\gamma _{l}}{\rho _{l,\nu }}\right) (z,t)^{2+\frac{2}{n}}dV_{\theta _{{\mathbb {H}}^n}}+o(1)\\&\quad =\left( \frac{n(2n+2)}{r_\infty }\right) ^{n+1} \int _{\left\{ (|{\tilde{z}}|^4+{\tilde{t}}^2)^{\frac{1}{4}}\le \frac{\delta }{C_0\rho _{l,\nu }}\right\} } \left( \frac{\gamma _{l}^2}{\gamma _{l}^4{\tilde{t}}^2+(\gamma _{l}^2|{\tilde{z}}|^2+1)^2}\right) ^{n+1}dV_{\theta _{{\mathbb {H}}^n}}+o(1)\\&\quad =\left( \frac{n(2n+2)}{r_\infty }\right) ^{n+1} \int _{\left\{ (|{\widehat{z}}|^4+{\widehat{t}}^2)^{\frac{1}{4}}\le \frac{\gamma _{l}\delta }{C_0\rho _{l,\nu }}\right\} } \left( \frac{1}{{\widehat{t}}^2+(|{\widehat{z}}|^2+1)^2}\right) ^{n+1}dV_{\theta _{{\mathbb {H}}^n}}+o(1)\\&\quad \ge \left( \frac{n(2n+2)}{r_\infty }\right) ^{n+1} \int _{\left\{ (|{\widehat{z}}|^4+{\widehat{t}}^2)^{\frac{1}{4}}\le \frac{C_1\delta }{C_0}\right\} } \left( \frac{1}{{\widehat{t}}^2+(|{\widehat{z}}|^2+1)^2}\right) ^{n+1}dV_{\theta _{{\mathbb {H}}^n}}+o(1), \end{aligned}$$
where the first inequality follows from (
B.60), (
B.61), (
B.64) and the definition of
\(\omega '(x^*_{l,\nu },\frac{\gamma _{l}}{\rho _{l,\nu }})\), the second inequality follows from (
A.1) and (
A.8), the first equality follows from the change of variables
\(({\tilde{z}},{\tilde{t}})=(\frac{z}{\rho _{l,\nu }},\frac{t}{(\rho _{l,\nu })^2})\), the second equality follows from the change of variables
\(({\widehat{z}},{\widehat{t}})=(\gamma _l{\tilde{z}},\gamma _{l}^2{\tilde{t}})\), and the last inequality follows from Proposition
B.10 and Lemma
B.5. Hence, if we let
$$\begin{aligned} C_2=r_\infty \left( \frac{n(2n+2)}{r_\infty }\right) ^{n+1} \int _{\{(|{\widehat{z}}|^4+{\widehat{t}}^2)^{\frac{1}{4}}\le \frac{C_1\delta }{C_0}\}} \left( \frac{1}{{\widehat{t}}^2+(|{\widehat{z}}|^2+1)^2}\right) ^{n+1}dV_{\theta _{{\mathbb {H}}^n}}, \end{aligned}$$
then it follows from the above computation that
$$\begin{aligned}&\int _M \left( \left( 2+\frac{2}{n}\right) |\nabla _{\theta _0}v_\nu ^{l}|^2_{\theta _0} +R_{\theta _0}(v_\nu ^{l})^2\right) dV_{\theta _0}\\&\quad \ge \int _M \left( \left( 2+\frac{2}{n}\right) |\nabla _{\theta _0}v_\nu ^{l-1}|^2_{\theta _0} +R_{\theta _0}(v_\nu ^{l-1})^2\right) dV_{\theta _0}+C_2+o(1). \end{aligned}$$
That is to say, the quantity
\(\int _M \left( \left( 2+\frac{2}{n}\right) |\nabla _{\theta _0}v_\nu ^{l-1}|^2_{\theta _0} +R_{\theta _0}(v_\nu ^{l-1})^2\right) dV_{\theta _0}\) at the
l-th step decreases by at least
\(C_2\) after extraction of a bubble. Therefore, the iteration must stop after finite steps.
Therefore, there exists an integer
m and a sequence of
m-tuples
\((x^*_{k,\nu },\varepsilon ^*_{k,\nu })_{1\le k\le m}\) where
\(\varepsilon ^*_{k,\nu }=\displaystyle \frac{\rho _{k,\nu }}{\gamma _k}\) such that
$$\begin{aligned} \varepsilon ^*_{k,\nu }\rightarrow 0 \quad \text{ as } \nu \rightarrow \infty \text{ for } \text{ all } 1\le k\le m, \end{aligned}$$
by Lemma
B.5 and Proposition
B.10. Also, we have
$$\begin{aligned} \Bigg \Vert v_\nu -\sum _{k=1}^m{\widehat{\omega }}\left( x^*_{k,\nu },\frac{1}{\varepsilon ^*_{k,\nu }}\right) \Bigg \Vert _{S^2_1(M)}\rightarrow 0 \quad \text{ as } \quad \nu \rightarrow \infty . \end{aligned}$$
(B.75)
Now (
5.4) follows from (
B.74) and Proposition
B.10. On the other hand, we have
$$\begin{aligned}&\Bigg \Vert u_\nu -u_\infty -\sum _{k=1}^m{\overline{u}}_{\Bigg (x^*_{k,\nu },\varepsilon ^*_{k,\nu }\Bigg )}\Bigg \Vert _{S^2_1(M)}\\&\quad =\left\| v_\nu -\sum _{k=1}^m{\widehat{\omega }}\left( x^*_{k,\nu },\frac{1}{\varepsilon ^*_{k,\nu }}\right) -\sum _{k=1}^m\left( \omega '\left( x^*_{k,\nu },\frac{1}{\varepsilon ^*_{k,\nu }}\right) -{\widehat{\omega }}\left( x^*_{k,\nu },\frac{1}{\varepsilon ^*_{k,\nu }}\right) \right) \right. \\&\qquad \left. -\sum _{k=1}^m \varphi _{x^*_{k,\nu }}(y) \left( \frac{n(2n+2)}{r_\infty }\right) ^{\frac{n}{2}} (\varepsilon ^*_{k,\nu })^n \Bigg (1-\chi _\delta (\rho _{x^*_{k,\nu }}(y))\Bigg )G_{x^*_{k,\nu }}(y)\right\| _{S^2_1(M)}=o(1) \end{aligned}$$
where the first equality follows from (
A.10) and (
A.11), and the last equality follows from (
B.59), (
B.63), (
B.65), (
B.75), Lemma
B.5 and the fact that the Green’s function
\(G_{x^*_{k,\nu }}(y)\) is bounded in
\(S_1^2(K)\) for any compact set
\(K\subset M-\{x^*_{k,\nu }\}\) (see (
A.6) and (
A.7)). This proves (
5.5) and this completes the proof of Theorem
5.1.