$\displaystyle-\frac{h}{2(p-1)}\int_{\mathbb{R}^{n}}w^{2}(y\cdot y_{0})\rho
dy+\frac{h}{p+1}\int_{\mathbb{R}^{n}}|w|^{p+1}(y\cdot y_{0})\rho dy$
$\displaystyle+h\int_{\mathbb{R}^{n}}(\nabla w\cdot y)(\nabla w\cdot
y_{0})\rho dy-\frac{h}{2}\int_{\mathbb{R}^{n}}|\nabla w|^{2}(y\cdot y_{0})\rho
dy$ $\displaystyle=$ $\displaystyle-\frac{1}{2}\int_{\mathbb{R}^{n}}(\nabla
w\cdot y_{0})(\nabla w\cdot y_{0})\rho
dy-\frac{h^{2}}{4}\int_{\mathbb{R}^{n}}(\nabla w\cdot y)(\nabla w\cdot y)\rho
dy$ $\displaystyle+h\int_{\mathbb{R}^{n}}(\nabla w\cdot y)(\nabla w\cdot
y_{0})\rho dy+\frac{h^{2}}{2}\int_{\mathbb{R}^{n}}|\nabla
w|^{2}[\frac{n}{2}-\frac{|y|^{2}}{4}]\rho dy$
$\displaystyle+\frac{1}{2}\int_{\mathbb{R}^{n}}|\nabla
w|^{2}[\frac{|y_{0}|^{2}}{2}+\frac{|y|^{2}}{4}]\rho
dy+\frac{1}{2(p-1)}\int_{\mathbb{R}^{n}}w^{2}[\frac{|y_{0}|^{2}}{2}+\frac{|y|^{2}}{4}]\rho
dy$
$\displaystyle-\frac{1}{p+1}\int_{\mathbb{R}^{n}}|w|^{p+1}[\frac{|y_{0}|^{2}}{2}+\frac{|y|^{2}}{4}]\rho
dy,$

where we have applied (3.7).

Then

$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{n}}|\nabla
w|^{2}\rho\\{[\frac{nh}{2}+\frac{y\cdot
y_{0}}{2}-\frac{h|y|^{2}}{4}]^{2}+\frac{nh^{2}}{2}-\frac{|y_{0}|^{2}}{2}-\frac{h^{2}|y|^{2}}{2}\\}dy$
$\displaystyle+\frac{1}{2(p-1)}\int_{\mathbb{R}^{n}}w^{2}\rho\\{[\frac{nh}{2}+\frac{y\cdot
y_{0}}{2}-\frac{h|y|^{2}}{4}]^{2}+\frac{nh^{2}}{2}-\frac{|y_{0}|^{2}}{2}-\frac{h^{2}|y|^{2}}{2}\\}dy$
$\displaystyle-\frac{1}{p+1}\int_{\mathbb{R}^{n}}|w|^{p+1}\rho\\{[\frac{nh}{2}+\frac{y\cdot
y_{0}}{2}-\frac{h|y|^{2}}{4}]^{2}+\frac{nh^{2}}{2}-\frac{|y_{0}|^{2}}{2}-\frac{h^{2}|y|^{2}}{2}\\}dy$
$\displaystyle=$ $\displaystyle h^{2}\int_{\mathbb{R}^{n}}|\nabla
w|^{2}[\frac{n}{2}-\frac{|y|^{2}}{4}]\rho
dy-\frac{h^{2}}{p+1}\int_{\mathbb{R}^{n}}|w|^{p+1}[\frac{n}{2}-\frac{|y|^{2}}{4}]\rho
dy$
$\displaystyle+\frac{h^{2}}{2(p-1)}\int_{\mathbb{R}^{n}}w^{2}[\frac{n}{2}-\frac{|y|^{2}}{4}]\rho
dy-\frac{h^{2}}{4}\int_{\mathbb{R}^{n}}(\nabla w\cdot y)(\nabla w\cdot y)\rho
dy$ $\displaystyle-\frac{1}{2}\int_{\mathbb{R}^{n}}(\nabla w\cdot
y_{0})(\nabla w\cdot y_{0})\rho dy+h\int_{\mathbb{R}^{n}}(\nabla w\cdot
y)(\nabla w\cdot y_{0})\rho dy.$

Plugging this into (Proof of Theorem 4.3.), we have

$\displaystyle\frac{\partial^{2}}{\partial
s^{2}}(F_{x(s),t(s)}(w+s\phi))|_{s=0}$
$\displaystyle=\int_{\mathbb{R}^{n}}|\nabla\phi|^{2}\rho
dy-p\int_{\mathbb{R}^{n}}|w|^{p-1}\phi^{2}\rho
dy+\frac{1}{p-1}\int_{\mathbb{R}^{n}}\phi^{2}\rho dy$
$\displaystyle+h\int_{\mathbb{R}^{n}}(\frac{2}{p-1}w+\nabla w\cdot y)\phi\rho
dy-\int_{\mathbb{R}^{n}}(\nabla w\cdot y_{0})\phi\rho dy$
$\displaystyle-\frac{(p+1)h}{p-1}\frac{1}{2}\int_{\mathbb{R}^{n}}|\nabla
w|^{2}\rho[\frac{nh}{2}-\frac{h|y|^{2}}{4}]dy$
$\displaystyle+\frac{(p+1)h}{p-1}\frac{1}{p+1}\int_{\mathbb{R}^{n}}|w|^{p+1}\rho[\frac{nh}{2}-\frac{h|y|^{2}}{4}]dy$
$\displaystyle-\frac{(p+1)h}{p-1}\frac{1}{2(p-1)}\int_{\mathbb{R}^{n}}w^{2}\rho[\frac{nh}{2}-\frac{h|y|^{2}}{4}]dy$
(12.12)
$\displaystyle+\frac{h}{p-1}\int_{\mathbb{R}^{n}}w^{2}\rho[\frac{nh}{2}+\frac{y\cdot
y_{0}}{2}-\frac{h|y|^{2}}{4}]dy-\frac{h^{2}}{(p-1)^{2}}\int_{\mathbb{R}^{n}}w^{2}\rho
dy$ $\displaystyle+\frac{h^{2}}{2}\int_{\mathbb{R}^{n}}|\nabla
w|^{2}[\frac{n}{2}-\frac{|y|^{2}}{4}]\rho
dy-\frac{h^{2}}{4}\int_{\mathbb{R}^{n}}(\nabla w\cdot y)(\nabla w\cdot y)\rho
dy$ $\displaystyle-\frac{1}{2}\int_{\mathbb{R}^{n}}(\nabla w\cdot
y_{0})(\nabla w\cdot y_{0})\rho dy+h\int_{\mathbb{R}^{n}}(\nabla w\cdot
y)(\nabla w\cdot y_{0})\rho dy.$

Multiplying both sides of (3.2) by $(\frac{n}{2}-\frac{|y|^{2}}{4})w\rho$ and
integrating by parts, we get

$\displaystyle 0=$ $\displaystyle\int_{\mathbb{R}^{n}}[\frac{1}{\rho}{\rm
div}(\rho\nabla
w)-\frac{1}{p-1}w+|w|^{p-1}w](\frac{n}{2}-\frac{|y|^{2}}{4})w\rho dy$
$\displaystyle=$ $\displaystyle-\int_{\mathbb{R}^{n}}\rho\nabla
w\nabla[(\frac{n}{2}-\frac{|y|^{2}}{4})w]-\frac{1}{p-1}\int_{\mathbb{R}^{n}}w^{2}(\frac{n}{2}-\frac{|y|^{2}}{4})\rho
dy$
$\displaystyle+\int_{\mathbb{R}^{n}}|w|^{p+1}(\frac{n}{2}-\frac{|y|^{2}}{4})\rho
dy$ $\displaystyle=$ $\displaystyle-\int_{\mathbb{R}^{n}}|\nabla
w|^{2}(\frac{n}{2}-\frac{|y|^{2}}{4})\rho
dy+\frac{1}{2}\int_{\mathbb{R}^{n}}w(\nabla w\cdot y)\rho dy$
$\displaystyle-\frac{1}{p-1}\int_{\mathbb{R}^{n}}w^{2}(\frac{n}{2}-\frac{|y|^{2}}{4})\rho
dy+\int_{\mathbb{R}^{n}}|w|^{p+1}(\frac{n}{2}-\frac{|y|^{2}}{4})\rho dy$
$\displaystyle=$ $\displaystyle-\int_{\mathbb{R}^{n}}|\nabla
w|^{2}(\frac{n}{2}-\frac{|y|^{2}}{4})\rho+\frac{1}{4}\int_{\mathbb{R}^{n}}(\nabla
w^{2}\cdot y)\rho dy$
$\displaystyle+\frac{1}{p-1}\int_{\mathbb{R}^{n}}w^{2}(\frac{n}{2}-\frac{|y|^{2}}{4})\rho
dy+\int_{\mathbb{R}^{n}}|w|^{p+1}(\frac{n}{2}-\frac{|y|^{2}}{4})\rho dy$
$\displaystyle=$ $\displaystyle-\int_{\mathbb{R}^{n}}|\nabla
w|^{2}(\frac{n}{2}-\frac{|y|^{2}}{4})\rho
dy-\frac{1}{2}\int_{\mathbb{R}^{n}}w^{2}(\frac{n}{2}-\frac{|y|^{2}}{4})\rho
dy$
$\displaystyle-\frac{1}{p-1}\int_{\mathbb{R}^{n}}w^{2}(\frac{n}{2}-\frac{|y|^{2}}{4})\rho
dy+\int_{\mathbb{R}^{n}}|w|^{p+1}(\frac{n}{2}-\frac{|y|^{2}}{4})\rho dy.$

Therefore

$\displaystyle 0=$ $\displaystyle-\frac{1}{p+1}\int_{\mathbb{R}^{n}}|\nabla
w|^{2}(\frac{n}{2}-\frac{|y|^{2}}{4})\rho
dy-\frac{1}{2(p-1)}\int_{\mathbb{R}^{n}}w^{2}(\frac{n}{2}-\frac{|y|^{2}}{4})\rho
dy$ (12.13)
$\displaystyle+\frac{1}{p+1}\int_{\mathbb{R}^{n}}|w|^{p+1}(\frac{n}{2}-\frac{|y|^{2}}{4})\rho
dy.$

By (3.2) and (12.13), we have

$\displaystyle 0=\frac{1}{p-1}\int_{\mathbb{R}^{n}}|\nabla w|^{2}\rho
dy-\frac{1}{p+1}\int_{\mathbb{R}^{n}}|\nabla
w|^{2}(\frac{n}{2}-\frac{|y|^{2}}{4})\rho dy.$ (12.14)

Substituting (12.14) and (3.2) into (Proof of Theorem 4.3.) gives

$\displaystyle\frac{\partial^{2}}{\partial
s^{2}}(F_{x(s),t(s)}(w+s\phi))|_{s=0}$
$\displaystyle=\int_{\mathbb{R}^{n}}|\nabla\phi|^{2}\rho
dy-p\int_{\mathbb{R}^{n}}|w|^{p-1}\phi^{2}\rho
dy+\frac{1}{p-1}\int_{\mathbb{R}^{n}}\phi^{2}\rho dy$
$\displaystyle+h\int_{\mathbb{R}^{n}}(\frac{2}{p-1}w+\nabla w\cdot y)\phi\rho
dy-\int_{\mathbb{R}^{n}}(\nabla w\cdot y_{0})\phi\rho dy$ (12.15)
$\displaystyle+\frac{h}{p-1}\int_{\mathbb{R}^{n}}w^{2}\rho[\frac{nh}{2}+\frac{y\cdot
y_{0}}{2}-\frac{h|y|^{2}}{4}]dy-\frac{h^{2}}{(p-1)^{2}}\int_{\mathbb{R}^{n}}w^{2}\rho
dy$ $\displaystyle-\frac{h^{2}}{4}\int_{\mathbb{R}^{n}}(\nabla w\cdot
y)(\nabla w\cdot y)\rho dy-\frac{1}{2}\int_{\mathbb{R}^{n}}(\nabla w\cdot
y_{0})(\nabla w\cdot y_{0})\rho dy$
$\displaystyle+h\int_{\mathbb{R}^{n}}(\nabla w\cdot y)(\nabla w\cdot
y_{0})\rho dy.$

Let $L$ be the linear operator defined by

$L\psi=\Delta\psi-\frac{y}{2}\cdot\nabla\psi-\frac{1}{p-1}\psi+p|w|^{p-1}\psi.$

We know from Lemma 5.4 that $2/(p-1)w+y\cdot\nabla w$ is an eigenfunction of
$L$ associated to the eigenvalue $-1$ and $w_{i},i=1,2,\cdots,n$ are
eigenfunctions of $L$ associated to the eigenvalue $-1/2$. Since $L$ is self-
adjoint, for any $y_{0}\in\mathbb{R}^{n}$,

$\int_{\mathbb{R}^{n}}\left(\frac{2}{p-1}w+y\cdot\nabla w\right)(\nabla w\cdot
y_{0})\rho dy=0.$ (12.16)

By (12.16), we have

$\displaystyle\int_{\mathbb{R}^{n}}(\nabla w\cdot y)(\nabla w\cdot y_{0})\rho
dy$ $\displaystyle=$ $\displaystyle-\frac{2}{p-1}\int_{\mathbb{R}^{n}}w(\nabla
w\cdot y_{0})\rho dy$ (12.17) $\displaystyle=$
$\displaystyle-\frac{1}{p-1}\int_{\mathbb{R}^{n}}(\nabla w^{2}\cdot y_{0})\rho
dy$ $\displaystyle=$
$\displaystyle-\frac{1}{2(p-1)}\int_{\mathbb{R}^{n}}w^{2}\rho(y\cdot
y_{0})dy.$

Finally, we have

$\displaystyle\frac{h^{2}}{p-1}\int_{\mathbb{R}^{n}}w^{2}\rho[\frac{n}{2}-\frac{|y|^{2}}{4}]dy$
$\displaystyle=$
$\displaystyle-\frac{h^{2}}{p-1}\int_{\mathbb{R}^{n}}w^{2}\Delta\rho$
$\displaystyle=$ $\displaystyle\frac{h^{2}}{p-1}\int_{\mathbb{R}^{n}}\nabla
w^{2}\cdot\nabla\rho dy$ (12.18) $\displaystyle=$
$\displaystyle-\frac{h^{2}}{p-1}\int_{\mathbb{R}^{n}}w(\nabla w\cdot y)\rho
dy.$

Substituting (Proof of Theorem 4.3.) and (Proof of Theorem 4.3.) into (Proof
of Theorem 4.3.) gives

$\displaystyle\frac{\partial^{2}}{\partial
s^{2}}(F_{x(s),t(s)}(w+s\phi))|_{s=0}$
$\displaystyle=\int_{\mathbb{R}^{n}}|\nabla\phi|^{2}\rho
dy-p\int_{\mathbb{R}^{n}}|w|^{p-1}\phi^{2}\rho
dy+\frac{1}{p-1}\int_{\mathbb{R}^{n}}\phi^{2}\rho dy$
$\displaystyle+h\int_{\mathbb{R}^{n}}(\frac{2}{p-1}w+\nabla w\cdot y)\phi\rho
dy-\int_{\mathbb{R}^{n}}(\nabla w\cdot y_{0})\phi\rho dy$ (12.19)
$\displaystyle-\frac{h^{2}}{(p-1)^{2}}\int_{\mathbb{R}^{n}}w^{2}\rho
dy-\frac{h^{2}}{p-1}\int_{\mathbb{R}^{n}}w(\nabla w\cdot y)\rho dy$
$\displaystyle-\frac{h^{2}}{4}\int_{\mathbb{R}^{n}}(\nabla w\cdot y)(\nabla
w\cdot y)\rho dy-\frac{1}{2}\int_{\mathbb{R}^{n}}(\nabla w\cdot y_{0})(\nabla
w\cdot y_{0})\rho dy.$

Since

$\displaystyle-\frac{h^{2}}{(p-1)^{2}}\int_{\mathbb{R}^{n}}w^{2}\rho
dy-\frac{h^{2}}{p-1}\int_{\mathbb{R}^{n}}w(\nabla w\cdot y)\rho
dy-\frac{h^{2}}{4}\int_{\mathbb{R}^{n}}(\nabla w\cdot y)(\nabla w\cdot y)\rho
dy$ $\displaystyle=$
$\displaystyle-h^{2}\int_{\mathbb{R}^{n}}(\frac{1}{p-1}w+\frac{y}{2}\cdot\nabla
w)^{2}\rho dy,$

we get (4.3) with the help of (Proof of Theorem 4.3.). ∎

## Appendix B

In this appendix, our main objective is to prove (11.4).

###### Proof of (11.4).

By the definition of $\rho$, we have

$\displaystyle\int_{\mathbb{R}^{n}}|y|^{-\frac{2(p+1)}{p-1}}\rho dy$
$\displaystyle=$
$\displaystyle(4\pi)^{-\frac{n}{2}}\int_{\mathbb{R}^{n}}|y|^{-\frac{2(p+1)}{p-1}}e^{-\frac{|y|^{2}}{4}}dy$
(12.20) $\displaystyle=$
$\displaystyle(4\pi)^{-\frac{n}{2}}\omega_{n-1}\int_{0}^{+\infty}r^{n-1-\frac{2(p+1)}{p-1}}e^{-\frac{r^{2}}{4}}dr,$

where $\omega_{n-1}$ is the area of the unit sphere $\mathbb{S}^{n-1}$ in
$\mathbb{R}^{n}$. Recall that we have assumed $p>(n+2)/(n-2)$, so the above
integral is well defined. Let $r=2\sqrt{s}$. Then $dr=s^{-1/2}ds$ and

$\displaystyle E(w)=$ $\displaystyle
c(n,p)\int_{0}^{+\infty}r^{n-1-\frac{2(p+1)}{p-1}}e^{-\frac{r^{2}}{4}}dr$
$\displaystyle=$ $\displaystyle
c(n,p)\int_{0}^{+\infty}s^{-\frac{1}{2}}(2\sqrt{s})^{n-1-\frac{2(p+1)}{p-1}}e^{-s}ds$
$\displaystyle=$ $\displaystyle
2^{n-3-\frac{4}{p-1}}c(n,p)\int_{0}^{\infty}s^{\frac{n-4-\frac{4}{p-1}}{2}}e^{-s}ds$
$\displaystyle=$ $\displaystyle
2^{n-3-\frac{4}{p-1}}c(n,p)\Gamma(\frac{n-2-\frac{4}{p-1}}{2}),$

where

$c(n,p)=\left(\frac{1}{2}-\frac{1}{p+1}\right)\left[\frac{2}{p-1}\left(n-2-\frac{2}{p-1}\right)\right]^{\frac{p+1}{p-1}}(4\pi)^{-\frac{n}{2}}\omega_{n-1}$

and

$\Gamma(\tau)=\int_{0}^{+\infty}s^{\tau-1}e^{-s}ds$

is the $\Gamma$-function. Recall that the area of the unit sphere is

$\omega_{n-1}=\frac{2\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2})}.$

Then

$\displaystyle E(w)=$ $\displaystyle
2^{n-3-\frac{4}{p-1}}c(n,p)\Gamma\left(\frac{n-2-\frac{4}{p-1}}{2}\right)$
(12.21) $\displaystyle=$ $\displaystyle
2^{-2-\frac{4}{p-1}}\left(\frac{1}{2}-\frac{1}{p+1}\right)\left[\frac{2}{p-1}\left(n-2-\frac{2}{p-1}\right)\right]^{\frac{p+1}{p-1}}\frac{\Gamma\left(\frac{n-2-\frac{4}{p-1}}{2}\right)}{\Gamma\left(\frac{n}{2}\right)}.$

In order that Lemma 11.2 holds, we need only to show

$2^{-2-\frac{4}{p-1}}\left[\frac{2}{p-1}\left(n-2-\frac{2}{p-1}\right)\right]^{\frac{p+1}{p-1}}\frac{\Gamma\left(\frac{n-2-\frac{4}{p-1}}{2}\right)}{\Gamma\left(\frac{n}{2}\right)}>(\frac{1}{p-1})^{\frac{p+1}{p-1}}.$

This is equivalent to

$\left(\frac{n-2}{2}-\frac{1}{p-1}\right)^{\frac{p+1}{p-1}}\frac{\Gamma\left(\frac{n-2-\frac{4}{p-1}}{2}\right)}{\Gamma\left(\frac{n}{2}\right)}>1.$
(12.22)

To obtain (12.22), we set $\alpha=2/(p-1)$. Using $p>(n+2)/(n-2)$ again, we
have $\alpha\in(0,(n-2)/2)$. Let

$x=\frac{n}{2},\quad
f(x)=\frac{\Gamma(x-1-\alpha)}{\Gamma(x)}\left(x-1-\frac{\alpha}{2}\right)^{1+\alpha}$

and

$\phi(x)=\log
f(x)=\log\Gamma(x-1-\alpha)-\log\Gamma(x)+(1+\alpha)\log\left(x-1-\frac{\alpha}{2}\right).$

Then

$\phi^{\prime}(x)=\frac{\Gamma^{\prime}(x-1-\alpha)}{\Gamma(x-1-\alpha)}-\frac{\Gamma^{\prime}(x)}{\Gamma(x)}+\frac{1+\alpha}{x-1-\frac{\alpha}{2}}$
(12.23)

and

$\displaystyle\phi^{\prime\prime}(x)=$
$\displaystyle\left(\frac{\Gamma^{\prime}(x-1-\alpha)}{\Gamma(x-1-\alpha)}\right)^{\prime}-\left(\frac{\Gamma^{\prime}(x)}{\Gamma(x)}\right)^{\prime}-\frac{1+\alpha}{\left(x-1-\frac{\alpha}{2}\right)^{2}}$
$\displaystyle=$
$\displaystyle\sum^{\infty}_{l=0}(x-1-\alpha+l)^{-2}-\sum^{\infty}_{l=0}(x+l)^{-2}-\frac{1+\alpha}{\left(x-1-\frac{\alpha}{2}\right)^{2}}$

Notice that for any $\tau>1$,

$\left(\tau-1-\frac{\alpha}{2}\right)\left(\tau+\frac{\alpha}{2}\right)\leq\tau(\tau-1),$

so

$\displaystyle\frac{1}{1+\alpha}\left[\frac{1}{\left(\tau-1-\frac{\alpha}{2}\right)^{2}}-\frac{1}{\left(\tau+\frac{\alpha}{2}\right)^{2}}\right]-\left[\frac{1}{(\tau-1)^{2}}-\frac{1}{\tau^{2}}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{1+\alpha}\frac{(1+\alpha)(2\tau-1)}{\left(\tau-1-\frac{\alpha}{2}\right)^{2}\left(\tau+\frac{\alpha}{2}\right)^{2}}-\frac{2\tau-1}{\tau^{2}(\tau-1)^{2}}$
$\displaystyle=$
$\displaystyle(2\tau-1)\left[\frac{1}{\left(\tau-1-\frac{\alpha}{2})^{2}(\tau+\frac{\alpha}{2}\right)^{2}}-\frac{1}{\tau^{2}(\tau-1)^{2}}\right]$
$\displaystyle\geq$ $\displaystyle 0.$

Take $\tau=x+l-\alpha/2$ in the above inequality, which gives

$\phi^{\prime\prime}(x)\geq(1+\alpha)\sum_{l=0}^{\infty}\left[\frac{1}{(x-1+l-\alpha/2)^{2}}-\frac{1}{(x+l-\alpha/2)^{2}}\right]-\frac{1+\alpha}{(x-1-\frac{\alpha}{2})^{2}}=0.$

Thus $\phi$ is convex. By the mean value theorem, we have

$(1+\alpha)\left[\frac{1}{x-1-\frac{\alpha}{2}}-\sum_{l=0}^{\infty}(x+l)^{-2}\right]\leq\phi^{\prime}(x)\leq(1+\alpha)\left[\frac{1}{x-1-\frac{\alpha}{2}}-\sum_{l=0}^{\infty}(x-1-\alpha+l)^{-2}\right].$

Hence $\lim_{x\rightarrow\infty}\phi^{\prime}(x)=0$; and for any
$\alpha\in(0,(n-2)/2)$, $\phi$ is decreasing in $[3/2,+\infty)$. Moreover, we
see from (12.23) that $\phi^{\prime}(x)=O(|x|^{-2})$ as $x\rightarrow\infty$.
Thus there exists a constant $c_{0}$ such that
$\lim_{x\rightarrow\infty}\phi(x)=c_{0}$. By the Stirling’s formula (see
[23]), we have for $m=1,2,\cdots$,

$\log\Gamma(x)=\left(x-\frac{1}{2}\right)\log
x-x+\frac{1}{2}\log(2\pi)+\sum_{l=1}^{m}\frac{B_{2l}}{2l(2l-1)}x^{-2l+1}+O(x^{-2m-1}),$

where $B_{2l}$ are the Bernoulli numbers. Therefore, as $x\rightarrow\infty$,

$\displaystyle\phi(x)=$
$\displaystyle\log\Gamma(x-1-\alpha)-\log\Gamma(x)+(1+\alpha)\log\left(x-1-\frac{\alpha}{2}\right)$
$\displaystyle=$
$\displaystyle(x-1-\alpha-\frac{1}{2})\log(x-1-\alpha)-(x-1-\alpha)$
$\displaystyle-(x-\frac{1}{2})\log
x+x+(1+\alpha)\log\left(x-1-\frac{\alpha}{2}\right)+O(|x|^{-1})$
$\displaystyle=$
$\displaystyle(x-\frac{1}{2})\log\left(1-\frac{1+\alpha}{x}\right)+(1+\alpha)\left[\log\left(1+\frac{\alpha}{2(x-1-\alpha)}\right)+1\right]+O(|x|^{-1}).$

Because

$\lim_{x\to\infty}\left(x-\frac{1}{2}\right)\log\left(1-\frac{1+\alpha}{x}\right)=-(1+\alpha),$

we get

$\phi(x)=O(|x|^{-1}),\quad\text{as}~{}x\to+\infty.$

This then implies that for any $\alpha\in(0,(n-2)/2)$, if $x>1$, then
$\phi(x)>0$. ∎

## References

  * [1] Jacob Bernstein and Lu Wang.  A sharp lower bound for the entropy of closed hypersurfaces up to dimension six.  Invent. Math., 206(3):601–627, 2016.
  * [2] Marie-Françoise Bidaut-Véron and Laurent Véron.  Nonlinear elliptic equations on compact riemannian manifolds and asymptotics of Emden equations.  Invent. Math., 106(3):489–539, 1991.
  * [3] KeKyeongsu Choi and Jiuzhou Huang.  Linearly stable self-similar solutions of semilinear heat equations.  arXiv:2403.16641, pages 1–15, 2024.
  * [4] Tobias H. Colding and William P. Minicozzi, II.  Generic mean curvature flow I: generic singularities.  Ann. of Math. (2), 175(2):755–833, 2012.
  * [5] Tobias Holck Colding, Tom Ilmanen, William P. Minicozzi, II, and Brian White.  The round sphere minimizes entropy among closed self-shrinkers.  J. Differential Geom., 95(1):53–69, 2013.
  * [6] Tobias Holck Colding and William P. Minicozzi, II.  The singular set of mean curvature flow with generic singularities.  Invent. Math., 204(2):443–471, 2016.
  * [7] Charles Collot, Frank Merle;, and Pierre Raphaël.  Strongly anisotropic type ii blow up at an isolated point.  J. Amer. Math. Soc., 33(2):527–607, 2020.
  * [8] del Pino Manuel, Lai Chen-Chih, Musso Monica, Wei Juncheng, and Zhou Yifu.  New type ii Finite time blow-up for the energy supercritical heat equation.  arXiv:2006.00716, 2020.
  * [9] Jean Dolbeault, Maria J. Esteban, and Michael Loss.  Nonlinear flows and rigidity results on compact manifolds.  J. Funct. Anal., 267(5):1338–1363, 2014.
  * [10] Shi-Zhong Du.  Energy non-collapsing and refined blowup for a semilinear heat equation.  J. Differential Equations, 266(9):5942–5970, 2019.
  * [11] Hiroshi Fujita.  On the blowing up of solutions of the cauchy problem for $u_{t}=\Delta u+u^{1+\alpha}$.  J. Fac. Sci. Univ. Tokyo Sect. I, 13:109–124, 1966.
  * [12] Basilis Gidas and Joel Spruck.  Global and local behavior of positive solutions of nonlinear elliptic equations.  Comm. Pure Appl. Math., 34(4):525–598, 1981.
  * [13] Yoshikazu Giga and Robert V Kohn.  Asymptotically self-similar blow-up of semilinear heat equations.  Comm. Pure Appl. Math., 38(3):297–319, 1985.
  * [14] Yoshikazu Giga and Robert V. Kohn.  Characterizing blowup using similarity variables.  Indiana Univ. Math. J., 36(1):1–40, 1987.
  * [15] Yoshikazu Giga and Robert V. Kohn.  Nondegeneracy of blowup for semilinear heat equations.  Comm. Pure Appl. Math., 42(6):845–884, 1989.
  * [16] David Gilbarg and Neil S. Trudinger.  Elliptic partial differential equations of second order. Reprint of the 1998 edition.  Springer-Verlag, Berlin, 2001.
  * [17] Hiroshi Matano and Frank Merle.  Threshold and generic type I behaviors for a supercritical nonlinear heat equation.  J. Funct. Anal., 261(3):716–748, 2011.
  * [18] Frank Merle and Hatem Zaag.  Optimal estimates for blowup rate and behavior for nonlinear heat equations.  Comm. Pure Appl. Math., 51(2):139–196, 1998.
  * [19] J. J. L. Velázquez.  Estimates on the $(n-1)$-dimensional Hausdorff measure of the blow-up set for a semilinear heat equation.  Indiana Univ. Math. J., 42(2):445–476, 1993.
  * [20] J. J. L Velázquez.  Curvature blow-up in perturbations of minimal cones evolving by mean curvature flow.  Ann. Scuola Norm. Sup. Pisa Cl. Sci., 21(4):595–628, 1994.
  * [21] Kelei Wang and Juncheng Wei.  Analysis of blow-up locus and existence of weak solutions for nonlinear supercritical problems.  Int. Math. Res. Not. IMRN, 2015(10):2634–2670, 2015.
  * [22] Kelei Wang and Juncheng Wei.  Refined blowup analysis and nonexistence of Type II blowups for an energy critical nonlinear heat equation.  arXiv:2101.07186, 2021.
  * [23] Magnus Wilhelm, Oberhettinger Fritz, and Soni Raj Pal.  Formulas and Theorems for the Special Functions of Mathematical Physics.  Springer-Verlag, New-York, 1966.