Blow-up phenomena for the constant scalar curvature and constant boundary mean curvature equation (after Chen and Wu)

TL;DR

This paper investigates the blow-up phenomena in solutions to the constant scalar curvature and boundary mean curvature equation, extending known counterexamples to higher dimensions and analyzing solution compactness.

Contribution

It extends the existence of counterexamples for solution non-compactness to dimensions not less than 35, improving previous results for dimensions 62 and above.

Findings

Counterexamples exist for dimensions ≥35.

Solution compactness fails in high dimensions.

Extends previous results by Chen and Wu.

Abstract

In this paper, the compactness of the solutions to the constant scalar curvature and constant boundary mean curvature equation is considered. Chen and Wu constructed a smooth counterexample showing that the compactness of the set of ``lower energy" solutions to the above equation fails when the dimension of the manifold is not less than 62. We prove that a smooth counterexample still exists when the dimension of the manifold is not less than 35.

Equations

$$\frac{4(n-1)}{n-2}{\Delta }_{g}u-R_{g}u+c{u}^{\frac{n+2}{n-2}}=0.$$

$$\begin{array}{rl}-\frac{4(n-1)}{n-2}{\Delta }_{g}u+R_{g}u& =R_{\tilde{g}}{u}^{\frac{n+2}{n-2}}\text{mbox}inM,\\ \frac{2}{n-2}\frac{\partial u}{\partial {\nu }_g}+H_{g}u& =H_{\tilde{g}}{u}^{\frac{n}{n-2}}\text{mbox}on\partial M.\end{array}$$

$$\begin{array}{rl}-\frac{4(n-1)}{n-2}{\Delta }_{g}u+R_{g}u& =c{u}^{\frac{n+2}{n-2}}\text{mbox}inM,\\ \frac{2}{n-2}\frac{\partial u}{\partial {\nu }_g}+H_{g}u& =0\text{mbox}on\partial M,\end{array}$$

$$\begin{array}{rl}-\frac{4(n-1)}{n-2}{\Delta }_{g}u+R_{g}u& =0\text{mbox}inM,\\ \frac{2}{n-2}\frac{\partial u}{\partial {\nu }_g}+H_{g}u& =c{u}^{\frac{n}{n-2}}\text{mbox}on\partial M.\end{array}$$

$$\begin{array}{rl}-{\Delta }_{g}u+\frac{n-2}{4(n-1)}{R}_{g}u& =c_1{u}^{\frac{n+2}{n-2}}\text{mbox}inM,\\ \frac{\partial u}{\partial {\nu }_g}+\frac{n-2}{2}{H}_{g}u& =c{u}^{\frac{n}{n-2}}\text{mbox}on\partial M\end{array}$$

$$\begin{array}{rl}{I}_{(M,g)}[u]& ={\int }_M\left(\frac{4(n-1)}{n-2}|{\nabla }_{g}u{|}^2+R_g{u}^2\right)d{\mu }_g+2(n-1){\int }_{\partial M}{H}_g{u}^{2}d{\sigma }_g\\ & -\frac{4(n-1)}{n}{c}_1{\int }_M{u}_{+}^{\frac{2n}{n-2}}d{\mu }_g-4c{\int }_{\partial M}{u}_{+}^{\frac{2(n-1)}{n-2}}d{\sigma }_g\end{array}$$

$$W(x)={\left(\frac{n(n-2)}{c_1}\right)}^{\frac{n-2}{4}}(1+|x-T_c{\mathbf{e}}_n{|}^2{)}^{\frac{2-n}{2}}\text{mbox}with{T}_c=-\frac{c}{n-2}\sqrt{\frac{n(n-2)}{c_1}},$$

$$\begin{array}{rl}{S}_c& =I_{({\mathbb{R}}_{+}^n,|dx{|}^2)}[W]\\ & ={\int }_{{\mathbb{R}}_{+}^n}\frac{4(n-1)}{n-2}|\nabla W{|}^{2}dx-\frac{4(n-1)}{n}{c}_1{\int }_{{\mathbb{R}}_{+}^n}{W}^{\frac{2n}{n-2}}dx-4c{\int }_{\partial {\mathbb{R}}_{+}^n}{W}^{\frac{2(n-1)}{n-2}}d\sigma \\ & =\frac{4}{n-2}{\int }_{{\mathbb{R}}_{+}^n}|\nabla W{|}^{2}dx+\frac{4c_1}{n-2}{\int }_{{\mathbb{R}}_{+}^n}{W}^{\frac{2n}{n-2}}dx>0.\end{array}$$

$$u_{(\xi ,ϵ)}(x)={\left(\frac{ϵ}{{ϵ}^2+(x_n-T_cϵ{)}^2+|x^{\prime }-\xi {|}^2}\right)}^{\frac{n-2}{2}},$$

$$\begin{array}{rl}{u}_{(\xi ,ϵ,i)}& ={\left(\frac{ϵ}{{ϵ}^2+(x_n-T_cϵ{)}^2+|x^{\prime }-\xi {|}^2}\right)}^{\frac{n+2}{2}}\frac{2ϵ(x_i-{\xi }_i)}{{ϵ}^2+(x_n-T_cϵ{)}^2+|x^{\prime }-\xi {|}^2},\\ {\hat{u}}_{(\xi ,ϵ,i)}& ={\left(\frac{ϵ}{{ϵ}^2+(x_n-T_cϵ{)}^2+|x^{\prime }-\xi {|}^2}\right)}^{\frac{n}{2}}\frac{2ϵ(x_i-{\xi }_i)}{{ϵ}^2+(x_n-T_cϵ{)}^2+|x^{\prime }-\xi {|}^2}\end{array}$$

$$\begin{array}{rl}{u}_{(\xi ,ϵ,n)}& ={\left(\frac{ϵ}{{ϵ}^2+(x_n-T_cϵ{)}^2+|x^{\prime }-\xi {|}^2}\right)}^{\frac{n+2}{2}}\frac{(1+T_c^2){ϵ}^2-x_n^2-|x^{\prime }-\xi {|}^2}{{ϵ}^2+(x_n-T_cϵ{)}^2+|x^{\prime }-\xi {|}^2},\\ {\hat{u}}_{(\xi ,ϵ,n)}& ={\left(\frac{ϵ}{{ϵ}^2+(x_n-T_cϵ{)}^2+|x^{\prime }-\xi {|}^2}\right)}^{\frac{n}{2}}\frac{(1+T_c^2){ϵ}^2-x_n^2-|x^{\prime }-\xi {|}^2}{{ϵ}^2+(x_n-T_cϵ{)}^2+|x^{\prime }-\xi {|}^2}.\end{array}$$

$$\mathcal{E}=\left\{w\in L^{\frac{2n}{n-2}}({\mathbb{R}}_{+}^n)\cap L^{\frac{2(n-1)}{n-2}}(\partial {\mathbb{R}}_{+}^n)\cap H_{loc}^1({\mathbb{R}}_{+}^n):{\int }_{{\mathbb{R}}_{+}^n}|\nabla w{|}^2<\infty \right\}$$

$${\mathcal{E}}_{(\xi ,ϵ)}=\left\{w\in \mathcal{E}:2n{\int }_{{\mathbb{R}}_{+}^n}w{u}_{(\xi ,ϵ,a)}-T_c{\int }_{\partial {\mathbb{R}}_{+}^n}w{\hat{u}}_{(\xi ,ϵ,a)}=0\text{mbox}forall1\le a\le n\right\}.$$

$$\sum _{i,j,k,l=1}^{n-1}({\overline{W}}_{ikjl}+{\overline{W}}_{iljk}{)}^2>0.$$

$$H_{ij}(x)=H_{ij}(x^{\prime })={\overline{W}}_{ikjl}{x}_k{x}_l\text{mbox}and{H}_{na}(x)=0,$$

$$\begin{array}{rl}\mathcal{F}(\xi ,ϵ)& =\frac{1}{2}{\int }_{{\mathbb{R}}_{+}^n}\sum _{i,j,l=1}^{n-1}{\overline{H}}_{il}{\overline{H}}_{jl}{\partial }_i{u}_{(\xi ,ϵ)}{\partial }_j{u}_{(\xi ,ϵ)}-\frac{n-2}{16(n-1)}{\int }_{{\mathbb{R}}_{+}^n}\sum _{i,j,l=1}^{n-1}\sum _{i,j,l=1}^{n-1}({\partial }_l{\overline{H}}_{ij}{)}^2{u}_{(\xi ,ϵ)}^2\\ & +{\int }_{{\mathbb{R}}_{+}^n}\sum _{i,j=1}^{n-1}{\overline{H}}_{ij}{\partial }_i{\partial }_j{u}_{(\xi ,ϵ)}{z}_{(\xi ,ϵ)}\end{array}$$

$$\begin{array}{rl}& {\int }_{{\mathbb{R}}_{+}^n}\left(⟨\nabla z_{(\xi ,ϵ)},\nabla \phi ⟩-n(n+2)u_{(\xi ,ϵ)}^{\frac{4}{n-2}}{z}_{(\xi ,ϵ)}\phi \right)+n{T}_c{\int }_{\partial {\mathbb{R}}_{+}^n}{u}_{(\xi ,ϵ)}^{\frac{2}{n-2}}{w}_{(\xi ,ϵ)}\phi \\ & =-{\int }_{{\mathbb{R}}_{+}^n}\mu {\lambda }^{2d}f({\lambda }^{-2}|x^{\prime }{|}^2)H_{ij}(x){\partial }_i{\partial }_j{u}_{(\xi ,ϵ)}\phi \end{array}$$

$$\sum _{q=0}^{2d}{\alpha }_q{s}^q:=(n+1)f(s{)}^2+4sf(s)f^{\prime }(s)+2s^2{f}^{\prime }(s{)}^2.$$

$$c_q={\int }_0^{\infty }(1+(t-T_c{)}^2{)}^{\frac{5+2q-n}{2}}dt,q\in \mathbb{N}\text{mbox}and0\le q\le 2d.$$

$$I(s)=\sum _{q=0}^{2d}\left(c_q{\alpha }_q{s}^{q+2}\prod _{j=0}^q\frac{n-1+2j}{n-5-2j}\right).$$

$$2f(s)f^{\prime }(s)+s{f}^{\prime }(s):=\sum _{q=0}^{2d-1}{\beta }_q{s}^q.$$

$$J(s)=\sum _{q=0}^{2q-1}\left(c_q{\beta }_q{s}^{q+2}\prod _{j=0}^q\frac{n+3+2j}{n-5-2j}\right).$$

$$\left\{\begin{array}[]{ll}h_{ij}(x)=\mu\lambda^{2}f(\lambda^{-2}|x^{\prime}|^{2})H_{ij}(x),&\hbox{in $B_{\rho}^{+}$;}\\ h_{ab}(x)=0,&\hbox{in $\overline{\mathbb{R}^{n}_{+}}\setminus B_{\rho}^{+}$;}\\ h_{na}(x)=0,&\hbox{in $\overline{\mathbb{R}^{n}_{+}}$,}\end{array}\right.$$

$$\left\{\begin{array}[]{ll}\displaystyle\Delta_{g}v-\frac{n-1}{4(n-2)}R_{g}v+n(n-2)v^{\frac{n+2}{n-2}}=0,&\hbox{in $\mathbb{R}^{n}_{+}$;}\\ \displaystyle\frac{\partial v}{\partial x_{n}}=(n-2)T_{c}v^{\frac{n}{n-2}},&\hbox{on $\partial\mathbb{R}^{n}_{+}$.}\end{array}\right.$$

$$\underset{B_{\lambda }^{+}(0)}{\sup }v\ge C{\lambda }^{\frac{2-n}{2}}$$

$$\begin{array}{rl}& 2(n-2){\int }_{{\mathbb{R}}_{+}^n}{v}^{\frac{2n}{n-2}}-\frac{n-2}{n-1}{T}_c{\int }_{\partial {\mathbb{R}}_{+}^n}{v}^{\frac{2(n-1)}{n-2}}\\ & <2(n-2){\int }_{{\mathbb{R}}_{+}^n}{u}_{(0,1)}^{\frac{2n}{n-2}}-\frac{n-2}{n-1}{T}_c{\int }_{\partial {\mathbb{R}}_{+}^n}{u}_{(0,1)}^{\frac{2(n-1)}{n-2}}.\end{array}$$

$$\begin{array}{rl}{\int }_{{\mathbb{R}}_{+}^n}& \left(⟨\nabla v_{(\xi ,ϵ)},\nabla \phi {⟩}_g,c_n{R}_g{v}_{(\xi ,ϵ)}\phi -n(n-2)|v_{(\xi ,ϵ)}{|}^{\frac{4}{n-2}}{v}_{(\xi ,ϵ)}\phi \right)\\ +& {\int }_{\partial {\mathbb{R}}_{+}^n}\left(d_n{h}_g{v}_{(\xi ,ϵ)}+(n-2)T_c|v_{(\xi ,ϵ)}^{\frac{2}{n-2}}{v}_{(\xi ,ϵ)}\right)\phi =0\end{array}$$

$$\begin{array}{rl}\Vert & v_{(\xi ,ϵ)}-u_{(\xi ,ϵ)}{\Vert }_{\mathcal{E}}\\ & \le C\Vert {\Delta }_g{u}_{(\xi ,ϵ)}-c_n{R}_g{u}_{(\xi ,ϵ)}+n(n-2)u_{(\xi ,ϵ)}^{\frac{n+2}{n-2}}{\Vert }_{L^{\frac{2n}{n+2}}({\mathbb{R}}_{+}^n)}+C\Vert h_g{u}_{(\xi ,ϵ)}{\Vert }_{L^{\frac{2(n-1)}{n}}(\partial {\mathbb{R}}_{+}^n)}.\end{array}$$

$$\begin{array}{rl}& {\mathcal{F}}_g(\xi ,ϵ)\\ :=& {\int }_{{\mathbb{R}}_{+}^n}\left(|\nabla v_{(\xi ,ϵ)}{|}_g^2+c_n{R}_g{v}_{(\xi ,ϵ)}^2-(n-2{)}^2|v_{(\xi ,ϵ)}^{\frac{2n}{n-2}}\right)+d_n{\int }_{\partial {\mathbb{R}}_{+}^n}{h}_g{v}_{(\xi ,ϵ)}^2\\ & +\frac{(n-2{)}^2}{n-1}{T}_c{\int }_{\partial {\mathbb{R}}_{+}^n}|v_{(\xi ,ϵ)}{|}^{\frac{2(n-1)}{n-2}}-2(n-2){\int }_{{\mathbb{R}}_{+}^n}{u}_{(\xi ,ϵ)}^{\frac{2n}{n-2}}+\frac{n-2}{n-1}{T}_c{\int }_{\partial {\mathbb{R}}_{+}^n}{u}_{(\xi ,ϵ)}^{\frac{2(n-1)}{n-2}}.\end{array}$$

$$\begin{split}\left\{\begin{array}[]{ll}-\Delta_{g}v_{(\xi,\epsilon)}+c_{n}R_{g}v_{(\xi,\epsilon)}=n(n-2)v_{(\xi,\epsilon)}^{\frac{n+2}{n-2}},&\hbox{in $\mathbb{R}^{n}_{+}$,}\\ \frac{\partial v_{(\xi,\epsilon)}}{\partial x_{n}}-d_{n}h_{g}v_{(\xi,\epsilon)}=(n-2)T_{c}v_{(\xi,\epsilon)}^{\frac{n}{n-2}},&\hbox{on $\partial\mathbb{R}^{n}_{+}$.}\end{array}\right.\end{split}$$

Full text

1. Introduction

Given an $n$-dimensional closed (i.e. compact without boundary) Riemannian manifold $(M,g)$ with $n\geq 3$, the Yamabe problem is to find a metric $\tilde{g}$ conformal to $g$ such that the scalar curvature $R_{\tilde{g}}$ of $\tilde{g}$ is constant. If we write $\tilde{g}=u^{\frac{4}{n-2}}g$ with $0<u\in C^{\infty}(M)$, then $\tilde{g}$ has constant scalar curvature $c$ if and only if

[TABLE]

The Yamabe problem is solved by Aubin [6], Trudinger [34], and Schoen [31]. In other words, there exists at least one solution to (1.1).

Solutions to (1.1) are usually not unique. In [29], Pollack has used gluing techniques to prove the following: given any conformal class with positive Yamabe constant and any positive integer $N$, there exists a new conformal class which is close to the original one in the $C^{0}$-norm and contains at least $N$ metrics of constant scalar curvature.

It is an interesting question whether the set of all solutions to (1.1) is compact in the $C^{2}$-topology. A conjecture due to Schoen states that this should be true unless $(M,g)$ is conformally equivalent to the standard sphere (see [30, 32, 33]). This conjecture has been verified in low dimensions [16, 26, 27] and in the locally conformally flat case [32, 33]. In particular, Khuri, Marques and Schoen [24] proved that the compactness conjecture is true up to dimension $24$, assuming that the positive mass theorem holds.

It turns out that the conjecture is false in higher dimension. Brendle [7] constructed Riemannian manifolds $(M,g)$ such that the set of constant scalar curvature metrics in the conformal class of $g$ is non-compact, when the dimension $n\geq 52$. Modifying the arguments in [7], Brendle and Marques [9] were able to construct such Riemannian manifold when the dimension $25\leq n\leq 52$.

Now suppose that $(M,g)$ is an $n$-dimensional compact Riemannian manifold with boundary $\partial M$. The Yamabe problem can also be formulated on Riemannian manifolds with boundary. And there are two types:

(I) Find $\tilde{g}$ conformal to $g$ such that the scalar curvature of $\tilde{g}$ is constant in $M$ and the mean curvature of $\tilde{g}$ is zero on $\partial M$.

(II) Find $\tilde{g}$ conformal to $g$ such that the scalar curvature of $\tilde{g}$ is zero in $M$ and the mean curvature of $\tilde{g}$ is constant on $\partial M$.

If we write $\tilde{g}=u^{\frac{4}{n-2}}g$ with $0<u\in C^{\infty}(M)$, then

[TABLE]

Here $\displaystyle\frac{\partial}{\partial\nu_{g}}$ is the outward normal derivative with respect to $g$, and $H_{g}$ (and $H_{\tilde{g}}$ respectively) is the mean curvature of $g$ (and $\tilde{g}$ respectively). Therefore, the Yamabe problem with boundary (I) is equivalent to solving (1.2) with $R_{\tilde{g}}\equiv c$ and $H_{\tilde{g}}\equiv 0$, i.e.

[TABLE]

and the Yamabe problem with boundary (II) is equivalent to solving (1.2) with $R_{\tilde{g}}\equiv 0$ and $H_{\tilde{g}}\equiv c$, i.e.

[TABLE]

The Yamabe problem with boundary (I) and (II) have been studied intensively. See [3, 8, 14, 17, 18, 28] and the references therein.

Inspired by the compactness conjecture in the closed case, one can ask if the set of all solutions to (1.3) (and to (1.4) respectively) is compact in $C^{2}$-topology. This was studied in [1, 3, 15, 19, 20, 21, 25]

More generally, one can try to find a metric $\tilde{g}$ conformal to $g$ such that the scalar curvature of $\tilde{g}$ is equal to $a$ in $M$ and the mean curvature of $\tilde{g}$ is equal to $b$ on $\partial M$, for some constants $a$ and $b$. This includes the Yamabe problem with boundary (I) and (II) as special cases. In view of (1.2), it is equivalent to solving the following:

[TABLE]

where $c_{1}=\displaystyle\frac{n-2}{4(n-1)}a$ and $c=\displaystyle\frac{n-2}{2}b$. This problem of finding conformal metric of constant scalar curvature and constant boundary mean curvature has been studied in [10, 11, 12, 22, 23].

Similar to the Yamabe problem on closed manifolds and the Yamabe problem with boundary (I) and (II), it would be interesting to study the compactness and non-compactness of (1.5). Almaraz and Wang [5] obtained a compactness result of (1.5) when the dimension $n=3$. Chen and Wu [13] were able to construct Riemannian manifolds $(M,g)$ with boundary $\partial M$ such that the set of all solutions to (1.5) is non-compact, when the dimension $n\geq 62$. In order to state their result, we need some notations. We define the functional

[TABLE]

for any $u\in H^{1}(M,g)$, where $u_{+}=\max\{u,0\}$. Let $W$ be a single bubble in $(\mathbb{R}^{n}_{+},|dx|^{2})$, i.e.

[TABLE]

and $S_{c}$ the energy of $W$, i.e.

[TABLE]

The following theorem was proved in [13, Theorem 1.1].

Theorem 1.1**.**

*Let $c\in\mathbb{R}_{+}$ and $n\geq 62$, there exists a metric $g$ on $\mathbb{S}^{n}_{+}$ such that the Yamabe constant with boundary $Y(\mathbb{S}^{n}_{+},\partial\mathbb{S}^{n}_{+},g)>0$, and a sequence of positive smooth functions $\{u_{\nu}\}_{\nu\in\mathbb{N}}$ with the following properties:

(i) $g$ is not locally conformally flat,

(ii) $\partial\mathbb{S}^{n}_{+}$ is umbilic with respect to $g$,

(iii) $u_{\nu}$ is a positive smooth solution to (1.5) with $c_{1}=n(n-2)$ and $M=\mathbb{S}^{n}_{+}$,

(iv) $I_{(\mathbb{S}^{n}_{+},g)}[u_{\nu}]<S_{c}$, and

(v) $\sup_{\partial\mathbb{S}^{n}_{+}}u_{\nu}\to\infty$ as $\nu\to\infty$.*

In this paper, by following and modifying the arguments of Chen and Wu in [13], we can lower the dimension further to $n\geq 35$. The following is our main theorem.

Theorem 1.2**.**

Let $n\geq 35$. There exists a constant $\overline{c}\in\mathbb{R}^{+}$ depending only on $n$, such that for all $c\in\mathbb{R}^{+}$ with $c<\overline{c}$, there exist a metric $g$ on $\mathbb{S}^{n}_{+}$ whose Yamabe constant with boundary $Y(\mathbb{S}^{n}_{+},\partial\mathbb{S}^{n}_{+},g)>0$, and a sequence of positive smooth functions $\{u_{\nu}\}_{\nu\in\mathbb{N}}$ with properties (i)-(v) in Theorem 1.1.

We would like to emphasize the main difference between Theorem 1.1 and Theorem 1.2. Chen and Wu were able to prove that Theorem 1.1 holds for all $c\in\mathbb{R}^{+}$, while we are only able to prove that Theorem 1.2 holds for all $c\in\mathbb{R}^{+}$ less than a particular value $\overline{c}$.

In Section 2, we will summarize the argument of Chen and Wu used in the proof of Theorem 1.1. In Section 3, we will then point out the exact place we need to modify the argument of Chen and Wu in order to prove Theorem 1.2.

2. Summary of the proof of Theorem 1.1

In this section, we provide a concise exposition of the principal arguments employed by Chen and Wu in the proof of Theorem 1.1. This overview serves to clarify the framework within which the subsequent modifications will be introduced.

From now on, we let $c_{1}=n(n-2)$ and $T_{c}=-c/(n-2)$, where $c$ is given in (1.5). Given a pair $(\xi,\epsilon)\in\mathbb{R}^{n-1}\times(0,\infty)$, we define

[TABLE]

where $x=(x^{\prime},x_{n})\in\mathbb{R}^{n}_{+}$. Define

[TABLE]

for $1\leq i\leq n-1$, and

[TABLE]

Define

[TABLE]

and

[TABLE]

We define a norm on $\mathcal{E}$ by $\|w\|_{\mathcal{E}}=\displaystyle\left(\int_{\mathbb{R}^{n}_{+}}|\nabla w|^{2}\right)^{\frac{1}{2}}$ Clearly, $u_{(\xi,\epsilon)}\in\mathcal{E}_{(\xi,\epsilon)}$.

We introduce a multi-linear form $\overline{W}:\mathbb{R}^{n-1}\times\mathbb{R}^{n-1}\times\mathbb{R}^{n-1}\times\mathbb{R}^{n-1}\to\mathbb{R}$ satisfying the same algebraic properties of the Weyl tensor on $\partial\mathbb{R}^{n}_{+}$. Moreover, we assume

[TABLE]

If $x=(x^{\prime},x_{n})\in\mathbb{R}^{n}_{+}$, we identify $x^{\prime}$ with $(x^{\prime},0)\in\partial\mathbb{R}^{n}_{+}$ and define

[TABLE]

as well as $\overline{H}_{ab}(x)=f(|x^{\prime}|^{2})H_{ab}(x)$, where $f(s)$ is a polynomial of degree $d$ for $0\leq d<(n-6)/4$ and to be determined.

We then define

[TABLE]

for $(\xi,\epsilon)\in\mathbb{R}^{n}_{+}\times(0,\infty)$, where $z_{(\xi,\epsilon)}(x)=\mu^{-1}w_{(\xi,\epsilon)}(x)$ satisfying

[TABLE]

for all test function $\varphi\in\mathcal{E}_{(\xi,\epsilon)}$.

If $f(s)=\displaystyle\sum_{i=0}^{d}a_{i}s^{i}$, $a_{i}\in\mathbb{R}$ for $1\leq i\leq d$, then $\alpha_{q}$ are constants defined as

[TABLE]

Also, we define

[TABLE]

Then we have

[TABLE]

Let

[TABLE]

Then

[TABLE]

In order to show that $\mathcal{F}(\xi,\epsilon)$ has a strict local minimum at $(0,1)$, Chen and Wu showed that it suffices to show that $I(1)>0$, $I^{\prime}(1)=0$, $I^{\prime\prime}(1)<0$ and $J(1)<0$. By choosing $d=1$, i.e. $f$ is a polynomial of degree $1$, Chen and Wu was able to show this when $n\geq 62$.

Proposition 2.1** (Proposition 5.9 in [13]).**

Let $n\geq 62$. There exists a polynomial $f(s)=-s+a_{0}$ such that $I(1)>0$, $I^{\prime}(1)=0$, $I^{\prime\prime}(1)<0$ and $J(1)<0$. This implies that $\mathcal{F}(\xi,\epsilon)$ has a strict local minimum at the point $(0,1)$.

Using Proposition 2.1, Chen and Wu proved the following:

Proposition 2.2** (Proposition 6.1 in [13]).**

For $T_{c}\in\mathbb{R}_{-}$ and $n\geq n_{0}$, let $g=\exp(h)$ be a smooth Riemannian metric in $\overline{\mathbb{R}^{n}_{+}}$, where $h$ is a symmetric trace-free two tensor in $\overline{\mathbb{R}^{n}_{+}}$ satisfying

[TABLE]

where $0<\mu\leq 1$, $0<\lambda\leq\rho\leq 1$, $1\leq i\leq n-1$, $1\leq a,b\leq n$ and $H_{ab}$ is defined as in (2.1). Assume that $|h(x)|+|\partial h(x)|+|\partial^{2}h(x)|\leq\alpha$ for all $x\in\overline{\mathbb{R}^{n}_{+}}$. If $\alpha$ and $\mu^{-2}\lambda^{n-10}\rho^{2-n}$ are sufficiently small, then there exists a positive smooth solution of

[TABLE]

Moreover, there exists $C=C(n,T_{c})>0$ such that

[TABLE]

and

[TABLE]

Sketch of the Proof.

For the sake of completeness, we provide the proof here. However, this is only a sketch, and for the full proof, the reader is referred to [13].

Step 1. If $h$ satisfies the conditions of Proposition 2.2 and $g=\exp(h)$, then we can establish the following: Given any pair $(\xi,\epsilon)\in\mathbb{R}^{n-1}\times(0,\infty)$, there exists a unique function $v_{(\xi,\epsilon)}\in\mathcal{E}$ such that $v_{(\xi,\epsilon)}-u_{(\xi,\epsilon)}\in\mathcal{E}_{(\xi,\epsilon)}$, and

[TABLE]

for all $\varphi\in\mathcal{E}_{(\xi,\epsilon)}$. Moreover, there exists a positive constant $C$, depending only on $T_{c}$ and $n$ such that

[TABLE]

Step 2. Given a pair $(\xi,\epsilon)\in\mathbb{R}^{n-1}\times(0,\infty)$, we define the following energy functional

[TABLE]

Then the functional $\mathcal{F}_{g}$ is continuously differentiable. Moreover, if $(\xi,\epsilon)$ is a critical point of $\mathcal{F}_{g}$, then the function $v_{(\xi,\epsilon)}$ is a positive smooth solution of

[TABLE]

Step 3. Applying Step 1 to each pair $(\xi,\epsilon)\in\mathbb{R}^{n-1}\times(0,\infty)$, we choose $v_{(\xi,\epsilon)}$ to be the unique element of $\mathcal{E}$ such that $v_{(\xi,\epsilon)}-u_{(\xi,\epsilon)}\in\mathcal{E}_{(\xi,\epsilon)}$ and

[TABLE]

for all $\varphi\in\mathcal{E}_{(\xi,\epsilon)}$. Let $\Omega=\left\{(\xi,\epsilon)\in\mathbb{R}^{n-1}\times(0,\infty):|\xi|<1,\frac{1}{2}<\epsilon<1\right\}$. We define the function $w_{(\xi,\epsilon)}$ as the unique element of $\mathcal{E}_{(\xi,\epsilon)}$ satisfying

[TABLE]

for all $\varphi\in\mathcal{E}_{(\xi,\epsilon)}$. Then we can establish the following: For any $(\xi,\epsilon)\in\lambda\Omega$, there holds

[TABLE]

Step 4. It follows from Proposition 2.1 that $(0,1)$ is a strict local minimum point of $\mathcal{F}(\xi,\epsilon)$. Hence, we can find an open set $\Omega^{\prime}\subset\Omega$ such that $(0,1)\in\Omega^{\prime}$ and $\mathcal{F}(0,1)<\inf_{(\xi,\epsilon)\in\partial\Omega^{\prime}}\mathcal{F}(\xi,\epsilon)<0$. By Step 3 with $d=1$, we have

[TABLE]

for all $(\xi,\epsilon)\in\Omega$, equivalently,

[TABLE]

for all $(\xi,\epsilon)\in\Omega$. If $\mu^{-2}\lambda^{n-10}\rho^{2-n}$ is sufficiently small, then we have

[TABLE]

Consequently, there exists $(\overline{\xi},\overline{\epsilon})\in\Omega^{\prime}$ such that

[TABLE]

It follows from Step 2 that that the function $v=v_{(\lambda\overline{\xi},\lambda\overline{\epsilon})}$ obtained in Step 1 is a positive smooth solution to (2.7). By definition of $\mathcal{F}_{g}$ we have

[TABLE]

whence

[TABLE]

Using (2.8), we estimate

[TABLE]

Then

[TABLE]

Hence, if $\alpha$ is sufficiently small, then we obtain

[TABLE]

This completes the proof. ∎

We remark that in the original statement of Proposition 6.1 in [13], it is assumed that $f(s)=-s+a_{0}$ and $n_{0}=62$. However, one can see from the proof of Proposition 6.1 in [13] that Proposition 2.2 is still true.

With Proposition 2.2, one can follow the proof of Theorem 6.2 in [13] to prove the following:

Theorem 2.3** (Theorem 6.2 in [13]).**

*For $T_{c}\in\mathbb{R}_{-}$ and $n\geq n_{0}$, there exists a smooth Riemannian metric $g$ in $\overline{\mathbb{R}^{n}_{+}}$ with the following properties:

(i) $g_{ab}(x)=\delta_{ab}$ for $x\in\overline{\mathbb{R}^{n}_{+}}\setminus B_{1/2}^{+}(0)$,

(ii) $g$ is not locally conformally flat,

(iii) $\partial\mathbb{R}^{n}_{+}$ is totally geodesic with respect to $g$,

(iv) there exists a sequence of positive smooth functions $\{v_{\nu}\}_{\nu\in\mathbb{N}}$ satisfying*

[TABLE]

for all $\nu$. Moreover, there hold

[TABLE]

for all $\nu$, i.e. $I_{(\mathbb{R}^{n}_{+},|dx|^{2})}[v_{\nu}]<S_{c}$, and $\sup_{B_{1}(0)^{+}}v_{\nu}\to\infty$ as $\nu\to\infty$.

Now Theorem 1.1 is a direct consequence of Theorem 2.3 by taking $n_{0}=62$.

3. Proof of Theorem 1.2

In order to lower the dimension further, we are going to choose $f(s)$ to be a polynomial of degree $6$. In addition, we will establish the corresponding version of Proposition 2.1. More precisely, we prove in this section the following:

Proposition 3.1**.**

Let $n\geq 35$. There exists a polynomial $f(s)=\displaystyle\sum_{i=0}^{6}a_{i}s^{i}$ such that $I(1)>0$, $I^{\prime}(1)=0$, $I^{\prime\prime}(1)<0$ and $J(1)<0$, provided that $T_{c}=-\displaystyle\frac{c}{n-2}$ is sufficiently close to [math]. This implies that $\mathcal{F}(\xi,\epsilon)$ has a strict local minimum at the point $(0,1)$.

With Proposition 3.1, Theorem 1.2 now follows from Proposition 2.2 and Theorem 2.3 with $n_{0}=35$. Thus it boils down to proving Proposition 3.1.

Now we choose $d=6$ such that $f(s)=a_{0}+a_{1}s+a_{2}s^{2}+a_{3}s^{3}+a_{4}s^{4}+a_{5}s^{5}+a_{6}s^{6}$, where $a_{1},a_{3},a_{5}$ are negative constants and $a_{2},a_{4},a_{6}$ are positive constants to be chosen later. It follows from (2.2) that

[TABLE]

Differentiating (2.4) with respect to $s$ yields

[TABLE]

Combining this with (3.1), one can see that $I^{\prime}(1)$ can be considered as a quadratic polynomial in $a_{0}$. More precisely, $p_{n}(a_{0}):=I^{\prime}(1)=A_{n}a_{0}^{2}+B_{n}a_{0}+C_{n}$ where

[TABLE]

Note that it follows from (2.3) that $c_{q}$ is a continuous function of $T_{c}$, i.e. $c_{q}=c_{q}(T_{c})$. Hence, $A_{n}=A_{n}(T_{c})$, $B_{n}=B_{n}(T_{c})$, and $C_{n}=C_{n}(T_{c})$ can also be regarded as continuous functions of $T_{c}$. Let $discrim(p_{n})(T_{c}):=B_{n}(T_{c})^{2}-4A_{n}(T_{c})C_{n}(T_{c})$, which is the discriminant of $I^{\prime}(1)$, the quadratic polynomial in $a_{0}$. We will show that $discrim(p_{n})(0)>0$. By continuity, this implies that $discrim(p_{n})(T_{c})>0$ when $T_{c}=-\displaystyle\frac{c}{n-2}$ is sufficiently close to [math]. When $T_{c}=0$, the explicit value of $c_{q}(0)$ can be computed by using (2.3):

[TABLE]

where $\Gamma(s)$ denotes the gamma function.

Now we choose $a_{1}=-10$, $a_{2}=10^{-4}$, $a_{3}=-10^{-3}$, $a_{4}=1.84\times 10^{-1}$, $a_{5}=-2.65\times 10^{-2}$, and $a_{6}=7.37\times 10^{-4}$. By (3.3) and (3.4), a direct computation with the help of Mathematica shows that $discrim(p_{n})(0)>0$ for $35\leq n\leq 62$ (c.f. [35]). This tells us that there exists $a_{0}\in\mathbb{R}$ such that $I^{\prime}(1)=0$, when $T_{c}=-\displaystyle\frac{c}{n-2}$ is sufficiently close to [math].

Differentiating (3.2) with respect to $s$ yields

[TABLE]

Combining this with (3.1), we have the following:

[TABLE]

Note that $I^{\prime\prime}(1)=I^{\prime\prime}(1)(T_{c})$ can also be regarded as a continuous function of $T_{c}$. We will show that $I^{\prime\prime}(1)(0)<0$, which implies that $I^{\prime\prime}(1)(T_{c})<0$ when $T_{c}=-\displaystyle\frac{c}{n-2}$ is close to [math]. Let $a_{0}=a_{0}(n)$ be the largest root of the quadratic polynomial $p_{n}$ when $T_{c}=0$, i.e.

[TABLE]

We remark that, unlike other $a_{i}$’s that have a fixed value, $a_{0}$ depends on $n$. For example, we have

[TABLE]

when $n=35$, while

[TABLE]

when $n=62$. Then direct computations with the help of Mathematica show that $I^{\prime\prime}(1)(0)<0$ for $35\leq n\leq 62$ (c.f. [35]), which by continuity implies that $I^{\prime\prime}(1)(T_{c})<0$ for $35\leq n\leq 62$ when $T_{c}$ is sufficiently close to [math]. Likewise, with the help of Mathematica, we can show that $I(1)>0$ when $T_{c}=0$ (c.f. [35]), which implies that $I(1)>0$ when $T_{c}$ is sufficiently close to [math].

It remains to show that $J(1)<0$ when $T_{c}$ is sufficiently close to [math]. It follows from (2.5) that

[TABLE]

Combining this with (2.6), we have

[TABLE]

Viewing $J(1)$ as a continuous function of $T_{c}$, with our choice of $a_{i}$, we can show that $J(1)(0)<0$ for $35\leq n\leq 62$ with the help of Mathematica (c.f. [35]). This implies that, by continuity, $J(1)<0$ when $T_{c}$ is sufficiently close to [math]. This finishes the proof of Proposition 3.1.

Acknowledgement

The first author was supported by the National Science and Technology Council (NSTC), Taiwan, with grant Number: 112-2115-M-032-006-MY2.