Blow-up phenomena for the constant scalar curvature and constant boundary mean curvature equation

TL;DR

This paper constructs examples demonstrating non-uniqueness and lack of compactness in solutions to the constant scalar curvature and boundary mean curvature equation, especially in high dimensions.

Contribution

It provides explicit counterexamples showing non-uniqueness and non-compactness of solutions in certain high-dimensional warped product manifolds.

Findings

Non-uniqueness of solutions demonstrated

Counterexamples show failure of compactness in dimensions ≥62

Warped product manifolds used to illustrate phenomena

Abstract

We first present a warped product manifold with boundary to show the non-uniqueness of the positive constant scalar curvature and positive constant boundary mean curvature equation. Next, we construct a smooth counterexample to show 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 $62$.

Equations

$$\{\begin{array}{ll}-{\Delta }_{g}u+c_n{R}_{g}u=c_1{u}^{\frac{n+2}{n-2}},& \mathrm{in}M,\\ \frac{\partial u}{\partial {\nu }_g}+d_n{h}_{g}u=c{u}^{\frac{n}{n-2}},& \mathrm{on}\partial M,\end{array}$$

$$I_{(M,g)}[u]=$$

$$-\frac{4(n-1)}{n}{c}_1{\int }_M{u}_{+}^{\frac{2n}{n-2}}\mathrm{d}{\mu }_g-4c{\int }_{\partial M}{u}_{+}^{\frac{2(n-1)}{n-2}}\mathrm{d}{\sigma }_g$$

$$L_g(u\phi )=u^{\frac{n+2}{n-2}}{L}_{\tilde{g}}(\phi )\mathrm{and}{B}_g(u\phi )=u^{\frac{n}{n-2}}{B}_{\tilde{g}}(\phi ).$$

$$Y(M,\partial M,[g])=\underset{\tilde{g}\in [g]}{\inf }\frac{{\int }_M{R}_{\tilde{g}}\mathrm{d}{\mu }_{\tilde{g}}+2(n-1){\int }_{\partial M}{h}_{\tilde{g}}\mathrm{d}{\sigma }_{\tilde{g}}}{({\int }_M\mathrm{d}{\mu }_{\tilde{g}}{)}^{\frac{n-2}{n}}}.$$

$${\pi }_{ij}=k^{\frac{1}{2}}{\pi }_{ij}^{g_1},{\pi }_{iI}=0,{\pi }_{IJ}=01\le i,j\le n_1-1,1\le I,J\le n_2,$$

$$\{\begin{array}{ll}-{\Delta }_{g}u+c_n{R}_{g}u=c_n{R}_g{u}^{\frac{n+2}{n-2}},& \mathrm{in}M,\\ \frac{\partial u}{\partial {\nu }_g}+d_n{h}_{g}u=d_n{h}_g{u}^{\frac{n}{n-2}},& \mathrm{on}\partial M,\end{array}$$

$$I_{(M,g)}[u_0]<S_c(k),$$

$$W(x)={\left(\frac{n(n-1)}{R_g}\right)}^{\frac{n-2}{4}}{\left(\frac{2}{1+|x-T_c{\text{e}}_n{|}^2}\right)}^{\frac{n-2}{2}}\mathrm{with}{T}_c=-\frac{h_g}{2},$$

$$S_c(k)=$$

$$=$$

$$\to$$

$$=$$

$$V_1=\mathrm{Vol}(M^{n_1},g_1),{\hat{V}}_1=\mathrm{Vol}(\partial M^{n_1},g_1),V_2=\mathrm{Vol}(M^{n_2},g_2).$$

$$I_{(M,g)}[1]=$$

$$=$$

$$>$$

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

$$\{\begin{array}{ll}-\Delta u_{(\xi ,ϵ)}=n(n-2)u_{(\xi ,ϵ)}^{\frac{n+2}{n-2}},& \mathrm{in}{\mathbb{R}}_{+}^n,\\ \frac{\partial u_{(\xi ,ϵ)}}{\partial x_n}=(n-2)T_c{u}_{(\xi ,ϵ)}^{\frac{n}{n-2}},& \mathrm{on}\partial {\mathbb{R}}_{+}^n.\end{array}$$

$${\partial }_a{u}_{(\xi ,ϵ)}{\partial }_b{u}_{(\xi ,ϵ)}-c_n{\partial }_a{\partial }_b{u}_{(\xi ,ϵ)}^2=\frac{1}{n}\left(|\nabla u_{(\xi ,ϵ)}{|}^2-c_n\Delta u_{(\xi ,ϵ)}^2\right){\delta }_{ab}$$

$$u_{(\xi ,ϵ,i)}(x)=$$

$${\hat{u}}_{(\xi ,ϵ,i)}(x)=$$

$$u_{(\xi ,ϵ,n)}(x)=$$

$${\hat{u}}_{(\xi ,ϵ,n)}(x)=$$

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

$${\mathcal{E}}_{(\xi ,ϵ)}=\left\{w\in \mathcal{E};2n{\int }_{{\mathbb{R}}_{+}^n}w{u}_{(\xi ,ϵ,i)}-T_c{\int }_{\partial {\mathbb{R}}_{+}^n}w{\hat{u}}_{(\xi ,ϵ,i)}=0\mathrm{for}\mathrm{all}1\le i\le n\right\}.$$

$${\left({\int }_{{\mathbb{R}}_{+}^n}{w}^{\frac{2n}{n-2}}\right)}^{\frac{n-2}{n}}+{\left({\int }_{\partial {\mathbb{R}}_{+}^n}{w}^{\frac{2(n-1)}{n-2}}\right)}^{\frac{n-2}{n-1}}\le K{\int }_{{\mathbb{R}}_{+}^n}|\nabla w{|}^2$$

$$\{\begin{array}{l}{\eta }_i=\frac{2x_i}{1+(T_c-x_n{)}^2+|x^{\prime }{|}^2},\mathrm{for}1\le i\le n-1,\\ {\eta }_n=\frac{2(x_n-T_c)}{1+(T_c-x_n{)}^2+|x^{\prime }{|}^2},\\ {\eta }_{n+1}=\frac{|x^{\prime }{|}^2+(x_n-T_c{)}^2-1}{1+(T_c-x_n{)}^2+|x^{\prime }{|}^2}.\end{array}$$

$$\{\begin{array}{l}{\zeta }_i={\eta }_i,\mathrm{for}1\le i\le n-1,\\ {\zeta }_n=-\frac{T_c{\eta }_n+{\eta }_{n+1}}{\sqrt{1+T_c^2}},\\ {\zeta }_{n+1}=\frac{{\eta }_n-T_c{\eta }_{n+1}}{\sqrt{1+T_c^2}}.\end{array}$$

$$\frac{1}{2}\left[{\int }_{{\mathbb{R}}_{+}^n}\left(|\nabla w{|}^2-n(n+2)u_{(\xi ,ϵ)}^{\frac{4}{n-2}}{w}^2\right)+n{T}_c{\int }_{\partial {\mathbb{R}}_{+}^n}{u}_{(\xi ,ϵ)}^{\frac{2}{n-2}}{w}^2\right]$$

$$+\frac{(1+(n-2)\mu )2^{n-1}}{n|\Sigma |-T_c|\partial \Sigma |}{\left[2n{\int }_{{\mathbb{R}}_{+}^n}{u}_{(\xi ,ϵ)}^{\frac{n+2}{n-2}}w-T_c{\int }_{\partial {\mathbb{R}}_{+}^n}{u}_{(\xi ,ϵ)}^{\frac{n}{n-2}}w\right]}^2\ge \mu \Vert w{\Vert }_{\mathcal{E}}^2$$

$$\varphi (x)=2^{\frac{n-2}{2}}\Phi \circ {\pi }^{-1}(x)u_{(\xi ,ϵ)}(x),x\in {\mathbb{R}}_{+}^n.$$

$$Q_2({\Phi }_0):=$$

$$+\frac{1}{2(n|\Sigma |-T_c|\partial \Sigma |)}{\left(n{\int }_{\Sigma }{\Phi }_0-T_c{\int }_{\partial \Sigma }{\Phi }_0\right)}^2$$

$$\ge$$

Full text

$\mathbf{2010}$** MSC:** Primary 53C21, 35J20; Secondary 35B33, 34B18.

Keywords: Manifold with boundary, conformal metrics, scalar curvature, boundary mean curvature.

1 Introduction

Let $(M^{n},g)$ be a smooth compact Riemannian manifold with boundary $\partial M$ and $n\geq 3$. In 1999, Zheng-Chao Han and Yan Yan Li [13] proposed a question of finding a conformal metric with positive constant scalar curvature and any constant boundary mean curvature in the positive Yamabe constant conformal class. In analytical terms, it corresponds to the existence of a positive solution to

[TABLE]

for $c_{1}\in\mathbb{R}_{+},c\in\mathbb{R}$, where $c_{n}=(n-2)/[4(n-1)]$ and $d_{n}=(n-2)/2$, $R_{g}$ is the scalar curvature and $h_{g}$ is the mean curvature of $\partial M$ and $\nu_{g}$ is the outward unit normal on $\partial M$. This existence problem has been studied by Z. C. Han and Y. Y. Li [12, 13], and recently by the first named author and his collaborators [6, 5]. The closely related works are referred to J. Escobar [10, 9] etc. X. Chen, Y. Ruan an L. Sun [5] introduced a “free” functional

[TABLE]

for any $u\in H^{1}(M,g)$, where $u_{+}=\max\{u,0\}$. The authors applied the Mountain Pass Lemma to show the existence of PDE (1.1) for all $c\in\mathbb{R}$, in addition that the energy of solutions below a threshold $S_{c}$, except for the case that $n\geq 8$, $\partial M$ is umbilic, the Weyl tensor of $M$ vanishes on $\partial M$ and has an interior non-zero point. Here the geometric meaning of $S_{c}$ is the energy $I_{({\mathbb{R}}^{n}_{+},|\mathrm{d}x|^{2})}$ of a single bubble $u_{(0,1)}(x)$; see Section 3 or [5]. We will present an example in Section 2 to show the non-uniqueness of PDE (1.1) when $c_{1},c\in\mathbb{R}_{+}$. Indeed, Han-Li [13] established the compactness of the full set of positive solutions to PDE (1.1) for all $c\leq\bar{c}$ with any given positive constant $\bar{c}$ (see [12, Conjecture 2] and [13, Theorem 0.1]), provided that $M$ is locally conformally flat with umbilic boundary, and is not conformally equivalent to the standard hemisphere $\mathbb{S}^{n}_{+}$.

Denote by $L_{g}=-\Delta_{g}+c_{n}R_{g}$ the conformal Laplacian and $B_{g}=\frac{\partial}{\partial\nu_{g}}+d_{n}h_{g}$ the boundary conformally covariant operator, respectively. Both $L_{g}$ and $B_{g}$ have the following conformally covariant properties: Let $\tilde{g}=u^{4/(n-2)}g$, then for any $\varphi\in C^{\infty}(\overline{M})$, there hold

[TABLE]

The Yamabe constant is defined by

[TABLE]

For the closed manifolds, the question of compactness of the full set of solutions to the Yamabe equation was initiated by R. Schoen in $1988$. A necessary condition is that the manifold is not conformally equivalent to the standard sphere $\mathbb{S}^{n}$. It has been extensively studied by R. Schoen [21, 22], Y.Y. Li and M. Zhu [17], O. Druet [8], F. Marques [20], Y.Y. Li and L. Zhang [18, 19], etc. Eventually, the compactness for dimensions $3\leq n\leq 24$ with assuming positive mass theorem was established by M. Khuri, F. Marques and R. Schoen [15]. For the non-compactness part, S. Brendle [3] discovered the first smooth counterexamples in dimensions $n\geq 52$. S. Brendle and F. Marques [4] extended the above counterexample to the remaining dimensions $25\leq n\leq 51$.

For the manifolds with boundary, the blow-up phenomena in dimensions $n\geq 25$ were discovered by Almaraz [1] corresponding to $c_{1}=0,c>0$ in (1.1). Such blow-up phenomena in large dimensions also appear in the $Q$-curvature equation (see J. Wei and C. Zhao [23] ) and in the fractional Yamabe problem (see S. Kim, M. Musso and J. Wei [14]).

It is natural to expect that the blow-up phenomena of PDE (1.1) occur in large dimensions. Now we confirm it in dimensions $n\geq 62$ for the set of solutions whose energy of $I_{(M,g)}$ is below $S_{c}$.

Theorem 1.1**.**

For $n\geq 62$, there exists a smooth Riemannian metric $g$ on $\mathbb{S}^{n}_{+}$, such that $Y(\mathbb{S}^{n}_{+},\partial\mathbb{S}^{n}_{+},[g])>0$ and a sequence of positive smooth functions $\{v_{\nu};\nu\in\mathbb{N}\}$ with the following properties:

1. (i)

$g$* is not conformally flat;* 2. (ii)

$\partial\mathbb{S}^{n}_{+}$* is umbilic with respect to $g$;* 3. (iii)

for all $\nu$, $v_{\nu}$ is a positive solution to PDE (1.1) with $c_{1},c>0$ and $M=\mathbb{S}^{n}_{+}$; 4. (iv)

$I_{(\mathbb{S}^{n}_{+},g)}[v_{\nu}]<S_{c}$; 5. (v)

$\sup_{\partial\mathbb{S}^{n}_{+}}v_{\nu}\to\infty$* as $\nu\to\infty$.*

Since the compactness results of PDE (1.1) are not abundant yet, the critical dimension of the non-compactness is not a main issue in this paper and thus left to future study.

The paper is arranged as follows. We show the multiplicity of PDE (1.1) on a warped product manifold with boundary, which is presented in Section 2. In Section 3, we describe how the problem can be reduced to finding critical points of a certain function $\mathcal{F}_{g}(\xi,\epsilon)$, where $\xi$ is a vector in $\mathbb{R}^{n-1}$ and $\epsilon$ is a positive real number. In Section 4, we show that the function $\mathcal{F}_{g}(\xi,\epsilon)$ can be approximated by an auxiliary function $F(\xi,\epsilon)$. In Section 5, we prove that the function $\mathcal{F}(\xi,\epsilon)$ has a strict local minimum point $(0,1)$. Finally, in Section 6, we use a perturbation argument to find critical points of $\mathcal{F}_{g}(\xi,\epsilon)$ and then show the non-compactness.

Acknowledgement: Part of this work was carried out while both authors were visiting the Department of Mathematics at Rutgers University, to which they are grateful for providing the very stimulating research environment and supports.

2 Non-uniqueness: an example

The purpose of this section is to construct a warped product manifold with boundary, which demonstrates the multiplicity of solutions of (1.1). This is somewhat inspired by the one for the Yamabe problem in [2, p.178].

Proposition 2.1**.**

If $n\geq 5$ and $c_{1},c_{2}>0$, then PDE (1.1) admits at least two positive smooth solutions.

Proof.

Suppose $(M^{n_{1}},g_{1})$ is an $n_{1}$-dimensional ($n_{1}\geq 3$) smooth compact manifold with boundary such that $R_{g_{1}}$ and $h_{g_{1}}$ are two positive constants, which is guaranteed by [5, Theorem 1.1] for almost all smooth manifolds with boundary. Let $(M^{n_{2}},g_{2})$ be an $n_{2}$-dimensional ($n_{2}\geq 2$) smooth closed manifold with positive constant scalar curvature $R_{g_{2}}$. Consider a warped product manifold $(M^{n},g)=(M^{n_{1}}\times M^{n_{2}},kg_{1}+g_{2})$, where $n=n_{1}+n_{2}\geq 5$ and $k$ is a positive constant. Obviously, $\partial M=\partial M^{n_{1}}\times M^{n_{2}}$ and the second fundamental form $\pi$ on $\partial M$ satisfies:

[TABLE]

which means $h_{g}=k^{-\frac{1}{2}}h_{g_{1}}$ and $R_{g}=k^{-1}R_{g_{1}}+R_{g_{2}}$.

Next we claim that the following PDE

[TABLE]

has at least two positive smooth solutions.

To that end, first notice that $1$ is a solution of (2.1). On the other hand, it follows from [5, Theorem 1.1] that there exists a positive smooth mountain critical point $u_{0}$ of $I_{(M,g)}$, which is the one in (1) with $c_{1}=c_{n}R_{g}$ and $c=d_{n}h_{g}$, such that

[TABLE]

where we use $S_{c}(k)$ instead of $S_{c}$ to emphasize $k$. If we replace $(M,g)$ by $({\mathbb{R}}^{n}_{+},|\mathrm{d}x|^{2})$ in (2.1) with its positive solution denoted by

[TABLE]

which is also called a standard bubble. Furthermore, as $k\to\infty$, by the dominated convergence theorem we have

[TABLE]

For simplicity, we let

[TABLE]

Indeed, if $k$ is large enough, then $u_{0}$ is distinct from $1$. This follows from

[TABLE]

if $k$ is sufficiently large. ∎

3 Lyapunov-Schmidt reduction

From now on, let $c_{1}=n(n-2)$ and $T_{c}=-c/(n-2)$ be a negative real number for brevity. Given a pair $(\xi,\epsilon)\in\mathbb{R}^{n-1}\times(0,\infty)$ we define

[TABLE]

where $x=(x^{\prime},x_{n})\in{\mathbb{R}}^{n}_{+}$. Then $u_{(\xi,\epsilon)}$ satisfies

[TABLE]

This implies that the metric $u_{(\xi,\epsilon)}^{4/(n-2)}|\mathrm{d}x|^{2}$ is Einstein, then there holds

[TABLE]

for $1\leq a,b\leq n$. Define

[TABLE]

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

[TABLE]

Obviously, $\|{u_{(\xi,\epsilon,a)}}\|_{L^{\frac{2n}{n+2}}({\mathbb{R}}^{n}_{+})}$ and $\|{u_{(\xi,\epsilon,a)}}\|_{L^{\frac{2(n-1)}{n}}(\partial{\mathbb{R}}^{n}_{+})}$ are constant in $\xi$ and $\epsilon$ for $1\leq a\leq n$.

Define

[TABLE]

and

[TABLE]

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

It follows from [10, Theorem 3.3] that there exists an optimal constant $K=K(n)>0$ such that

[TABLE]

for all $w\in\mathcal{E}$.

Let $\pi:\eta\in\mathbb{S}^{n}\mapsto x\in{\mathbb{R}}^{n}_{+}$ be the stereographic projection (see [6, Fig.1 on p.9]), which is given by

[TABLE]

Let $\Sigma=\pi^{-1}({\mathbb{R}}^{n}_{+})$ be a spherical cap equipped with the standard round metric $g_{\Sigma}=g_{\mathbb{S}^{n}}$. If we choose the center of $\Sigma$ as the north pole, the coordinate system $\xi$ is changed to another coordinate system $\zeta$ by

[TABLE]

Proposition 3.1**.**

There exist a positive constant $\mu$ depending only on $n$ and $T_{c}$, such that

[TABLE]

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

Proof.

Given a function $\Phi$ defined on $\Sigma$, we define

[TABLE]

It follows from [12, Proposition 3.4] that there exists a positive constant $\mu$ depending on $n$ and $T_{c}$, such that

[TABLE]

for all $\Phi_{0}\in\mathrm{span}\{1,\zeta_{1},\zeta_{2},\cdots,\zeta_{n}\}^{\bot}$, which denotes the orthogonal complement of $\mathrm{span}\{1,\zeta_{1},\zeta_{2},\cdots,\zeta_{n}\}$ in $H^{1}(\Sigma)$.

For any $\Phi\in\mathrm{span}\{\zeta_{1},\zeta_{2},\cdots,\zeta_{n}\}^{\bot}$, we set $\Phi=\Phi_{0}+\bar{\Phi}$, where $\bar{\Phi}=\frac{1}{|\Sigma|}\int_{\Sigma}\Phi$. Then by (3) we have

[TABLE]

By (1.3) we have

[TABLE]

and

[TABLE]

Therefore, we combine these facts together to obtain the desired estimate. ∎

Proposition 3.2**.**

Consider a Riemannian metric in $\overline{\mathbb{R}_{+}^{n}}$ of the form $g(x)=\exp(h(x))$, where $h(x)$ is a trace-free symmetric two-tensor in $\overline{\mathbb{R}_{+}^{n}}$ satisfying $|h(x)|+|\partial h(x)|+|{\partial}^{2}h(x)|\leq\alpha\leq 1$, and $h_{na}(x)=0$ for all $x\in\overline{{\mathbb{R}}^{n}_{+}}$ and $h(x)=0$ for all $x\in\overline{{\mathbb{R}}^{n}_{+}}\setminus B_{1}^{+}(0)$. Then there exists a constant $C$, depending only on $n$ and $T_{c}$, such that

[TABLE]

Proof.

It is not hard to verify that $h_{g}=O(|h|)$, which together with [1, Proposition 2.3] yields the desired estimate. ∎

Proposition 3.3**.**

Consider a Riemannian metric in $\overline{\mathbb{R}_{+}^{n}}$ of the form $g(x)=\exp(h(x))$, where $h(x)$ is a trace-free symmetric two-tensor in $\overline{\mathbb{R}_{+}^{n}}$ satisfying $|h(x)|+|\partial h(x)|+|{\partial}^{2}h(x)|\leq\alpha\leq 1$ and $h_{na}(x)=0$ for all $x\in\overline{{\mathbb{R}}^{n}_{+}}$ and $h(x)=0$ for all $x\in\overline{{\mathbb{R}}^{n}_{+}}\setminus B_{1}^{+}(0)$. Here $\alpha$ depends only on $n$ and $T_{c}$. Then, given any pair $(\xi,\epsilon)\in\mathbb{R}^{n-1}\times(0,\infty)$ and any functions $f\in L^{\frac{2n}{n+2}}({\mathbb{R}}^{n}_{+}),\hat{f}\in L^{\frac{2(n-1)}{n}}(\partial{\mathbb{R}}^{n}_{+})$, there exists a unique function $w:=G(\xi,\epsilon)(f,\hat{f})\in\mathcal{E}_{(\xi,\epsilon)}$, such that

[TABLE]

for all $\varphi\in\mathcal{E}_{(\xi,\epsilon)}$. Furthermore, there holds

[TABLE]

Proof.

By Propositions 3.1 and 3.2, and Hölder’s inequality, we can follow nearly the same lines in [3, Corollary 3] that there exist two positive constants $\alpha$ and $C$, depending only on $n$ and $T_{c}$, such that $|h(x)|+|\partial h(x)|+|{\partial}^{2}h(x)|\leq\alpha$ and

[TABLE]

for all $w\in\mathcal{E}_{(\xi,\epsilon)}$, where

[TABLE]

Suppose that $w\in\mathcal{E}_{(\xi,\epsilon)}$ satisfies (3.3), then

[TABLE]

Since $u_{(\xi,\epsilon)}\in\mathcal{E}_{(\xi,\epsilon)}$, we have

[TABLE]

Then by (3.3), (3) and (3), we have

[TABLE]

Hence it follows from Young’s inequality that

[TABLE]

This implies the uniqueness of the solutions to (3.3).

For the existence part, thanks to the coercive estimate (3), it suffices to minimize the following functional

[TABLE]

over all $w\in\mathcal{E}_{(\xi,\epsilon)}$. ∎

Proposition 3.4**.**

Consider a Riemannian metric in $\overline{\mathbb{R}_{+}^{n}}$ of the form $g(x)=\exp(h(x))$, where $h(x)$ is a trace-free symmetric two-tensor in $\overline{\mathbb{R}_{+}^{n}}$ satisfying $|h(x)|+|\partial h(x)|+|{\partial}^{2}h(x)|\leq\alpha\leq 1$ and $h_{na}(x)=0$ for all $x\in\overline{{\mathbb{R}}^{n}_{+}}$ and $h(x)=0$ for all $x\in\overline{{\mathbb{R}}^{n}_{+}}\setminus B_{1}^{+}(0)$. Here $\alpha$ depends only on $n$ and $T_{c}$. Then, 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]

In particular, $v_{(\xi,\epsilon)}\not\equiv 0$ if $\alpha$ is sufficiently small.

Proof.

Let $G_{(\xi,\epsilon)}:L^{\frac{2n}{n+2}}({\mathbb{R}}^{n}_{+})\times L^{\frac{2(n-1)}{n}}(\partial{\mathbb{R}}^{n}_{+})\to\mathcal{E}_{(\xi,\epsilon)}$ be the solution operator constructed in Proposition 3.3, and we define a nonlinear operator $\Phi_{(\xi,\epsilon)}$ on $\mathcal{E}_{(\xi,\epsilon)}$ by

[TABLE]

In particular, it follows from Propositions 3.2 and 3.3 that $\|\Phi_{(\xi,\epsilon)}(0)\|_{\mathcal{E}}\leq C\alpha$.

Using the pointwise estimates

[TABLE]

and

[TABLE]

and Proposition 3.3, we obtain

[TABLE]

for $w,\widetilde{w}\in\mathcal{E}_{(\xi,\epsilon)}$. Hence, if $\alpha$ is sufficiently small, then the contraction mapping principle implies that $\Phi_{(\xi,\epsilon)}$ has a unique fixed point $w_{(\xi,\epsilon)}$ within $\mathcal{E}_{(\xi,\epsilon)}$. Hence $v_{(\xi,\epsilon)}=u_{(\xi,\epsilon)}+w_{(\xi,\epsilon)}$ is the desired solution, and not identically zero, which follows from (3.4) and Proposition 3.2. ∎

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

[TABLE]

Proposition 3.5**.**

The function $\mathcal{F}_{g}$ 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]

Proof.

By definition of $v_{(\xi,\epsilon)}$, we can find real numbers $b_{a}=b_{a}(\xi,\epsilon),1\leq a\leq n$, such that

[TABLE]

for all test function $\varphi\in\mathcal{E}$. This implies

[TABLE]

and

[TABLE]

for $1\leq j\leq n-1$. On the other hand, we have

[TABLE]

since $v_{(\xi,\epsilon)}-u_{(\xi,\epsilon)}\in\mathcal{E}_{(\xi,\epsilon)}$. Differentiating the above equation with respect to $\epsilon$ and $\xi_{k}$, we obtain

[TABLE]

where $1\leq a\leq n$, $\bar{c}_{n}$ is a nonzero constant independent of $\xi$ and $\epsilon$, and

[TABLE]

where each $\bar{c}_{i},1\leq i\leq n-1$ is a nonzero constant independent of $\xi$ and $\epsilon$.

Therefore, putting these facts together, we conclude that

[TABLE]

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

[TABLE]

Hence, if $(\xi,\epsilon)$ is a critical point of $\mathcal{F}_{g}$, then

[TABLE]

By (3.3) and (3.4) we have

[TABLE]

Thus, if $\alpha$ is sufficiently small, we obtain

[TABLE]

Consequently, we have

[TABLE]

for all $\varphi\in\mathcal{E}$.

Finally, we follow the same lines in [3, Proposition 6] that $v_{(\xi,\epsilon)}\geq 0$ in $\overline{{\mathbb{R}}^{n}_{+}}$. Together with $v_{(\xi,\epsilon)}\not\equiv 0$ by Proposition 3.4, the strong maximum principle and the Hopf boundary point lemma give $v_{(\xi,\epsilon)}>0$ in $\overline{{\mathbb{R}}^{n}_{+}}$. By the regularity theory of P. Cherrier [7], we show that $v_{(\xi,\epsilon)}$ is smooth. ∎

4 An estimate for the energy of a bubble

We first 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 $x=(x^{\prime},x_{n})\in{\mathbb{R}}^{n}_{+}$, then 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 is to be determined later. Then $H$ is symmetric, trace-free, independent of the variable $x_{n}$, and satisfies

[TABLE]

We define a Riemannian metric $g=\exp(h)$ in $\overline{{\mathbb{R}}^{n}_{+}}$, where $h$ is a trace-free symmetric two-tensor in $\overline{{\mathbb{R}}^{n}_{+}}$ and $h_{na}=\partial_{n}h_{ab}(x)=0$ for all $x\in\overline{{\mathbb{R}}^{n}_{+}}$, and satisfies

[TABLE]

Here $0<\mu\leq 1,0<\lambda\leq\rho\leq 1$. This gives $h_{ab}(x)=O(\mu(\lambda+|x|)^{2d+2})$. In addition, we require that $|h|+|\partial h|+|\partial^{2}h|\leq\alpha$, where $\alpha$ is the constant given in Proposition 3.4. The boundary $\partial{\mathbb{R}}^{n}_{+}$ is totally geodesic with respect to $g$, since the second fundamental form vanishes on $\partial{\mathbb{R}}^{n}_{+}$ , explicitly

[TABLE]

Applying Proposition 3.4 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 $\Omega=\left\{(\xi,\epsilon)\in\mathbb{R}^{n-1}\times(0,\infty);|\xi|<1,\frac{1}{2}<\epsilon<1\right\}$. Similar to [3, Proposition 7 and Corollary 8] and [4, Proposition 5 and Corollary 6], for any pair $(\xi,\epsilon)\in\lambda\Omega$ we obtain

[TABLE]

[TABLE]

and together with Proposition 3.4,

[TABLE]

By Proposition 3.3 with $h=0$, we define the function $w_{(\xi,\epsilon)}$ as the unique element of $\mathcal{E}_{(\xi,\epsilon)}$ satisfying

[TABLE]

for all $\varphi\in\mathcal{E}_{(\xi,\epsilon)}$. In particular, $w_{(0,\epsilon)}=0$, since $x^{i}H_{ij}(x)=0$ for any $x\in\overline{{\mathbb{R}}^{n}_{+}}$.

Proposition 4.1**.**

The function $w_{(\xi,\epsilon)}$ is smooth and satisfies that given any $(\xi,\epsilon)\in\lambda\Omega$, $|\partial^{k}w_{(\xi,\epsilon)}(x)|\leq C\lambda^{\frac{n-2}{2}}\mu(\lambda+|x|)^{2d+4-k-n}$ for all $x\in{\mathbb{R}}^{n}_{+},k=0,1,2$.

Proof.

By definition of $\mathcal{E}_{(\xi,\epsilon)}$, there exist real numbers $\bar{b}_{a}(\xi,\epsilon),1\leq a\leq n$ such that

[TABLE]

for all $\phi\in\mathcal{E}$. Hence it follows from standard elliptic estimates that $w_{(\xi,\epsilon)}$ is smooth. Since

[TABLE]

then by (3.3) and (3.7) we have

[TABLE]

Choosing $\phi=u_{(\xi,\epsilon,a)}$ in $\eqref{linearize expansion}$, we obtain

[TABLE]

Hence, we have

[TABLE]

for all $x\in{\mathbb{R}}^{n}_{+}$, and

[TABLE]

for all $x\in\partial{\mathbb{R}}^{n}_{+}$. We let $r=(\lambda+|x_{0}|)/2$ for any fixed $x_{0}\in\overline{{\mathbb{R}}^{n}_{+}}$. Then $\lambda+|x|\geq r$ for any $x\in B_{r}^{+}(x_{0})$. Based on the above facts, we obtain

[TABLE]

By [w, Theorems 8.25 and 8.26]e have

[TABLE]

since $d<(n-6)/4$. Then we obtain

[TABLE]

By Green’s representation formula, we have

[TABLE]

for any $x\in{\mathbb{R}}^{n}_{+}$, where $x^{\ast}=(x_{1},x_{2},\cdots,x_{n-1},-x_{n})$. From this we obtain

[TABLE]

for all $0<\beta<n-2$. Since

[TABLE]

and

[TABLE]

we conclude that

[TABLE]

for all $0<\beta\leq n-4-2d$. Iterating this inequality, we obtain

[TABLE]

Differentiating the equation (4) twice and repeating the argument above, we obtain the estimates of the first and second derivatives of $w_{(\xi,\epsilon)}$. ∎

Proposition 4.2**.**

There holds

[TABLE]

for all $(\xi,\epsilon)\in\lambda\Omega$.

Proof.

Consider the functions

[TABLE]

and

[TABLE]

By definition of $w_{(\xi,\epsilon)}$, we have

[TABLE]

for all $\varphi\in\mathcal{E}_{(\xi,\epsilon)}$. Since $w_{(\xi,\epsilon)}\in\mathcal{E}_{(\xi,\epsilon)}$, we obtain

[TABLE]

By definitions of $v_{(\xi,\epsilon)}$ and $u_{(\xi,\epsilon)}$ we have

[TABLE]

where

[TABLE]

and

[TABLE]

Thus, we obtain

[TABLE]

It follows from (3.4) that

[TABLE]

Following the same lines in [4, Corollary 8] and [3, Proposition 7], together with Proposition 4.1 and (4) we obtain

[TABLE]

and by (4.2), (4),

[TABLE]

Therefore, putting these facts together, we obtain the desired estimate. ∎

Proposition 4.3**.**

There holds

[TABLE]

for $(\xi,\epsilon)\in\lambda\Omega$.

Proof.

By definition of $v_{(\xi,\epsilon)}$, we have

[TABLE]

By (3.1), (4) and (4), an integration by parts gives

[TABLE]

On the other hand, by Proposition 4.2 we have

[TABLE]

Putting these facts together, we obtain the desired estimate. ∎

Proposition 4.4**.**

There hold

[TABLE]

and

[TABLE]

for $(\xi,\epsilon)\in\lambda\Omega$.

Proof.

We only need to prove the second assertion, since the first one is similar to [3, Proposition 12] together with (4). Observe that

[TABLE]

This together with (4) implies

[TABLE]

This proves the assertion. ∎

Proposition 4.5**.**

There holds

[TABLE]

for $(\xi,\epsilon)\in\lambda\Omega$.

Proof.

By equation (3.1) of $u_{(\xi,\epsilon)}$, we have

[TABLE]

Then the LHS of (4.5) becomes

[TABLE]

Notice that

[TABLE]

for $x\in B_{\rho}^{+}(0)$. This implies

[TABLE]

Since $\partial_{a}h_{ab}(x)=0$ in $B_{\rho}^{+}(0)$, it follows from [4, Proposition 4] that

[TABLE]

for $x\in B_{\rho}^{+}(0)$. This implies

[TABLE]

Since $h_{ab}(x)$ is trace-free, by (3.2) we obtain

[TABLE]

then

[TABLE]

Again by $\partial_{a}h_{ab}(x)=0$ in $B_{\rho}^{+}(0)$, we obtain

[TABLE]

Then the desired estimate follows from all the above facts. ∎

Consequently, collecting Propositions 4.3- 4.5 together, we arrive at the following key estimate.

Corollary 4.6**.**

Let $\mathcal{F}_{g}(\xi,\epsilon)$ be the function defined in (3), then for any $(\xi,\epsilon)\in\lambda\Omega$, there holds

[TABLE]

where $w_{(\xi,\epsilon)}$ satisfies (4).

5 Finding a critical point of an auxiliary function

We define

[TABLE]

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

[TABLE]

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

Next we show that the function $\mathcal{F}{(\xi,\epsilon)}$ has a strict local minimum. Throughout this section we use indices $1\leq i,j,k,l,m,p,q,r\leq n-1$ .

Since $\overline{H}_{ab}(-x)=\overline{H}_{ab}(x)$ for any $x\in{\mathbb{R}}^{n}_{+}$, the function $\mathcal{F}(\xi,\epsilon)$ satisfies $\mathcal{F}(\xi,\epsilon)=\mathcal{F}(-\xi,\epsilon)$ for all $(\xi,\epsilon)\in\mathbb{R}^{n-1}\times(0,\infty)$. This implies

[TABLE]

for all $\epsilon>0$ .

Proposition 5.1**.**

There hold

[TABLE]

and

[TABLE]

Proof.

The proof is similar to [3, Proposition 16]. ∎

Proposition 5.2**.**

There holds

[TABLE]

Proof.

Since

[TABLE]

and by Euler’s formula we obtain

[TABLE]

Hence, the assertion follows from Proposition 5.1. ∎

Corollary 5.3**.**

There holds

[TABLE]

Proposition 5.4**.**

There holds

[TABLE]

Proof.

Since $\sum_{i,j=1}^{n-1}\bar{H}_{ij}(x)\partial_{i}\partial_{j}u_{(0,\epsilon)}(x)=0$, then $z_{(0,\epsilon)}=0$, and by symmetry we have

[TABLE]

Then we have

[TABLE]

Hence, the result follows from Corollary 5.3. ∎

By Proposition 5.4, we rewrite

[TABLE]

where $\alpha_{q}$ are constants defined by

[TABLE]

Then we obtain

[TABLE]

where

[TABLE]

For clarity, we rewrite

[TABLE]

where

[TABLE]

Proposition 5.5**.**

There holds

[TABLE]

Proof.

As in [3, Proposition 21], similarly we obtain

[TABLE]

This together with Proposition 5.2 and Corollary 5.3 gives the desired assertion. ∎

For brevity, we let

[TABLE]

By definition (5.4) of $c_{q}$, a direct computation yields

[TABLE]

where

[TABLE]

In order to show that $\mathcal{F}(\xi,\epsilon)$ has a strict local minimum at $(0,1)$, By (5.2),(5.5) and Proposition 5.5, our strategy is to find some polynomials $f(s):=\sum_{i=0}^{d}a_{i}s^{i},a_{i}\in\mathbb{R}$ for $1\leq i\leq d$, such that $I(1)>0,I^{\prime}(1)=0,I^{\prime\prime}(1)<0$ and $J(1)<0$.

Before proceeding to find such polynomials $f$, we first need the following elementary result.

Lemma 5.6**.**

Let $T_{c}<0,n\geq 25$ and $c_{q}$ be defined in (5.4), there holds

[TABLE]

for $0\leq q\leq 2d$. In particular,

[TABLE]

Proof.

Let $a=-T_{c}>0$ and define

[TABLE]

An integration by parts gives

[TABLE]

Notice that

[TABLE]

From this, we iterate (5.9) to obtain

[TABLE]

Then we have

[TABLE]

Since

[TABLE]

then

[TABLE]

In particular, it follows from (5.4) and the above inequality that for $0\leq q\leq 2d$,

[TABLE]

[TABLE]

And the remained estimates follow from the above estimate. ∎

Now we choose $d=1$ and let $f(s)=a_{0}+a_{1}s$. Then by (5.3) we obtain

[TABLE]

Differentiating (5.6) with respect to $s$, we obtain

[TABLE]

We set $a_{1}=-1$ and define

[TABLE]

then

[TABLE]

Notice that the discriminant $d(p_{n})$ of $p_{n}$ is given by

[TABLE]

By Lemma 5.6, we have

[TABLE]

Define

[TABLE]

then

[TABLE]

Notice that $q^{\prime}(x)>0$ for $x\geq 62$ and $q(62)>0,q(61)<0$, then $d(p_{n})>0$ for $n\geq 62$.

Hence, we can choose

[TABLE]

such that $I^{\prime}(1)=0$. From this and Lemma 5.6, we obtain

[TABLE]

for $n\geq 62$.

Lemma 5.7**.**

There holds $I^{\prime\prime}(1)<0$ for $n\geq 62$.

Proof.

By definition (5.6) of $I$, we have

[TABLE]

Notice that

[TABLE]

By definition (5.10) of $a_{0}$ and $I^{\prime}(1)=0$, we have

[TABLE]

for $n\geq 62$. This implies the desired result. ∎

Lemma 5.8**.**

There holds $J(1)<0$ for $n\geq 62$.

Proof.

By definition (5) of $J$, letting $a_{1}=-1$ and $a_{0}$ be chosen as in (5.10) we obtain $\beta_{0}=-2a_{0}$, $\beta_{1}=3$ by virtue of (5.7), and

[TABLE]

Thus we have

[TABLE]

By Lemma 5.6 we have

[TABLE]

We consider

[TABLE]

Then a direct computation shows that $\mathcal{P}^{\prime\prime}(n)=6(\alpha-1)n+18-32\alpha>0$ for $n\geq 62$ and $\mathcal{P}^{\prime}(62)>0,\mathcal{P}(62)>0$. This implies that $\mathcal{P}(n)>0$  for $n\geq 62$. Observe that

[TABLE]

where the last inequality follows from $n\geq 62$ and the choice of $\alpha$. This yields $J(1)<0$ for $n\geq 62$. ∎

Combing the above facts with Lemmas 5.7-5.8 we arrive at

Proposition 5.9**.**

Let $n\geq 62$, then there exists a polynomial $f(s)=-s+a_{0}$ with

[TABLE]

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 $(0,1)$.

6 Proof of Theorem 1.1

Proposition 6.1**.**

For $n\geq 62$, let $g=\exp(h)$ is a smooth Riemannian metric on $\overline{{\mathbb{R}}^{n}_{+}}$, where $h$ is a symmetric trace-free two tensor on $\overline{{\mathbb{R}}^{n}_{+}}$ satisfying

[TABLE]

where $f(s)=a_{0}-s$ with the constant $a_{0}$ given in Proposition 5.9, $0<\mu\leq 1,0<\lambda\leq\rho\leq 1$, $1\leq i\leq n-1$, $1\leq a,b\leq n$ and $H_{ab}$ is defined in (4.1). 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]

Proof.

It follows from Proposition 5.9 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 $\mathcal{F}(0,1)<\displaystyle{\inf_{(\xi,\epsilon)\in\partial\Omega^{\prime}}}\mathcal{F}(\xi,\epsilon)<0$. By Corollary 4.6 with $d=1$, we have

[TABLE]

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

[TABLE]

for all $(\xi,\epsilon)\in\Omega$. If $\mu^{-2}\lambda^{n-4d-6}\rho^{2-n}$ is sufficiently small, then we have

[TABLE]

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

[TABLE]

It follows from Proposition 3.5 that the function $v=v_{(\lambda\bar{\xi},\lambda\bar{\epsilon})}$ obtained in Proposition 3.4 is a positive smooth solution to (6.1). By definition of $\mathcal{F}_{g}$ we have

[TABLE]

whence

[TABLE]

By (3.4) and Proposition 3.2 we estimate

[TABLE]

Then,

[TABLE]

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

[TABLE]

This completes the proof. ∎

Theorem 6.2**.**

Let $n\geq 62$, then there exists a smooth Riemannian metric $g$ on $\overline{\mathbb{R}}^{n}_{+}$ with the following properties:

1. (a)

$g_{ab}(x)=\delta_{ab}$* for $\overline{\mathbb{R}}^{n}_{+}\setminus B_{1/2}^{+}(0)$;* 2. (b)

$g$* is not conformally flat;* 3. (c)

$\partial{\mathbb{R}}^{n}_{+}$* is totally geodesic with respect to the induced metric of $g$;* 4. (d)

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. $I_{({\mathbb{R}}^{n}_{+},g)}[v_{\nu}]<S_{c}$, and $\displaystyle{\sup_{B_{1}^{+}(0)}}v_{\nu}\to\infty$ as $\nu\to\infty$.

Proof.

Let $\chi$ be a smooth cut-off function in $\mathbb{R}$ such that $0\leq\chi(t)\leq 1$ for $t\in\mathbb{R}$, $\chi(t)=1$ for $t\leq 1$ and $\chi(t)=0$ for $t\geq 2$. We define a trace-free symmetric two-tensor in $\overline{{\mathbb{R}}^{n}_{+}}$ by

[TABLE]

where $x_{N}=(\frac{1}{N},0,\cdots,0)\in\partial{\mathbb{R}}^{n}_{+}$. Observe that $h$ is smooth and satisfies $h_{an}(x)=0$ and $\partial_{n}h_{ab}(x)=0$ on $\partial{\mathbb{R}}^{n}_{+}$. We choose $\alpha>0$ to be the constant in Proposition 6.1 and $N_{0}$ sufficiently large, then $h_{ab}(x)=0$ for $|x|\geq\frac{1}{2}$ and $|h(x)|+|\partial h(x)|+|\partial^{2}h(x)|\leq\alpha$ for all $x\in\overline{\mathbb{R}}^{n}_{+}$. Thus, the desired assertion follows from Proposition 6.1 with $\lambda=2^{-N/2},\rho=(2N)^{-2},\mu=2^{-N}$. ∎