# On Fully Nonlinear Loewner-Nirenberg Problem of Ricci curvature

Zhenan Sui  Institute for Advanced Study in Mathematics of HIT, Harbin
Institute of Technology, Harbin, China<EMAIL_ADDRESS>
###### Abstract.

We prove the existence of a smooth complete conformal metric with constant kth
elementary symmetric function of negative Ricci curvature under certain
condition on general domain in Euclidean space. We then formulate this problem
for more general equations.

###### 2010 Mathematics Subject Classification:

Primary 53C21; Secondary 35J60

##  1\. Introduction

In this paper, we discuss the fully nonlinear version of Loewner-Nirenberg
problem: let $\Omega\subsetneq\mathbb{R}^{n}$ be a domain in Euclidean space
with $n\geq 3$ and $g$ the Euclidean metric. Assume that $\partial\Omega$
consists of a finite number of disjoint, non-self-intersecting, smooth,
compact surfaces. We want to find a smooth complete metric
$g_{u}=u^{\frac{4}{n-2}}g$ with $u>0$, which satisfies

(1.1)
$\sigma_{k}\Big{(}-g_{u}^{-1}\text{Ric}_{g_{u}}\Big{)}:=\sigma_{k}\Big{(}\lambda\big{(}-g_{u}^{-1}\text{Ric}_{g_{u}}\big{)}\Big{)}=1\quad\text{
in }\Omega,$

where $\text{Ric}_{g}$ is the Ricci tensor of $g$,
$\lambda(A)=(\lambda_{1},\ldots,\lambda_{n})$ are the eigenvalues of the
matrix $A$, and

$\sigma_{k}(\lambda)=\sum\limits_{1\leq i_{1}<\cdots<i_{k}\leq
n}\lambda_{i_{1}}\cdots\lambda_{i_{k}}$

is the $k$th elementary symmetric function defined on Garding’s cone

$\Gamma_{k}=\\{\lambda\in\mathbb{R}^{n}:\sigma_{j}(\lambda)>0,\,j=1,\ldots,k\\}.$

Under conformal deformation of metric $g_{u}=u^{\frac{4}{n-2}}g$, the Ricci
tensor $\text{Ric}_{g_{u}}$ and $\text{Ric}_{g}$ are related by the formula

$\frac{n-2}{2}\mbox{Ric}_{g_{u}}=\frac{n-2}{2}\mbox{Ric}_{g}-(n-2)\frac{\nabla^{2}u}{u}-\bigg{(}\frac{\Delta
u}{u}+\frac{|\nabla u|^{2}}{u^{2}}\bigg{)}g+\frac{n}{u^{2}}du\otimes du,$

where $\nabla{u}$, $\nabla^{2}u$ and $\Delta u$ are the gradient, Hessian and
Laplace-Beltrami operator of $u$ with respect to the background metric $g$
respectively. We note that $\mbox{Ric}_{g}\equiv 0$ if $g$ is Euclidean.

Consequently, the geometric problem (1.1) is equivalent to the second order
partial differential equation

(1.2)
$\sigma_{k}^{\frac{1}{k}}\big{(}W[u]\big{)}=\frac{n-2}{2}u^{\frac{n+2}{n-2}}\quad\text{
in }\Omega$

along with the boundary condition

(1.3) $u=\infty\quad\text{ on }\partial\Omega,$

where

(1.4) $W[u]=g^{-1}\bigg{(}(n-2)\nabla^{2}u-\frac{n}{u}du\otimes
du+\Big{(}\Delta u+\frac{|\nabla
u|^{2}}{u}\Big{)}g-\frac{n-2}{2}u\text{Ric}_{g}\bigg{)}.$

We shall call a $C^{2}$ function $u$ $k$-admissible (or admissible if there is
no ambiguity) in $\Omega$ if

$\lambda\big{(}W[u]\big{)}(x)\in\Gamma_{k}$

for any $x\in\Omega$. We note that equation (1.2) is elliptic if $u$ is
admissible. For convenience, we call equation (1.2) the conformal $k$-Ricci
curvature equation. Our goal in this paper is to seek smooth positive
admissible solutions to (1.2).

When $k=1$, (1.2)-(1.3) reduces to

(1.5) $\left\\{\begin{aligned} \frac{4(n-1)}{n-2}\Delta
u-S_{g}u=&u^{\frac{n+2}{n-2}}\quad\text{ in }\,\,\Omega,\\\
u=&\infty\quad\text{ on }\,\,\partial\Omega,\end{aligned}\right.$

where $S_{g}$ is the scalar curvature with respect to $g$. In this case, the
above problem is known as Loewner-Nirenberg problem. If $\Gamma$ is a smooth
compact submanifold of $\mathbb{R}^{n}$ of codimension $m$ such that
$\partial\Omega\setminus\Gamma$ is also compact, Loewner and Nirenberg [13]
proved the existence of a smooth conformally flat metric with constant
negative scalar curvature in $\Omega$ satisfying

$u(x)\rightarrow\infty\quad\text{ as }x\rightarrow\Gamma$

if $m<\frac{n}{2}+1$. They also gave the nonexistence result if
$m>\frac{n}{2}+1$ and conjectured the nonexistence for the borderline case
$m=\frac{n}{2}+1$, which was later proved by Aviles and Véron [2, 15].

When $k>1$, equation (1.2) becomes fully nonlinear. Following [5], we call the
above problem fully nonlinear Loewner-Nirenberg problem. If $\text{Ric}_{g}$
in (1.1) is replaced by the Schouten tensor $A_{g}$:

$A_{g}=\frac{1}{n-2}\Big{(}Ric_{g}-\frac{S_{g}}{2(n-1)}g\Big{)},$

González, Li and Luc [5] proved the existence of a Lipschitz continuous
complete conformally flat metric $g_{u}=u^{\frac{4}{n-2}}g$ satisfying

$\sigma_{k}\Big{(}-g_{u}^{-1}A_{g_{u}}\Big{)}=1$

if the vector

$u_{m}=\big{(}\underbrace{1,\ldots,1}_{n-m+1},\underbrace{-1,\ldots,-1}_{m-1}\big{)}\in\Gamma_{k}.$

They also gave the nonexistence result if $u_{m}\notin\overline{\Gamma}_{k}$.
The borderline case

(1.6) $u_{m}\in\partial\Gamma_{k}$

remains open.

Motivated by the above literatures, we investigate fully nonlinear Loewner-
Nirenberg problem of Ricci curvature. Different from the Schouten tensor case,
we are able to obtain smooth solutions, thanks to the interior second order
estimates of Guan [6] and Evans-Krylov theory [3, 11].

Our first result is an existence result when $\partial\Omega$ is composed of
hypersurfaces.

######  Theorem 1.1.

Let $\Omega\subsetneq\mathbb{R}^{n}$ be a domain whose boundary
$\partial\Omega$ consists of finitely many disjoint non-self-intersecting
smooth compact hypersurfaces. Then there exists a smooth complete metric
$g_{u}=u^{\frac{4}{n-2}}g$ satisfying (1.1), or equivalently, there exists a
smooth positive admissible solution $u$ to (1.2)-(1.3). Moreover, the
conformal factor $u$ has the following growth rate

(1.7)
${\rho}^{\frac{n}{2}-1}(x)u(x)\rightarrow\big{(}(n-1)(C_{n}^{k})^{\frac{1}{k}}\big{)}^{\frac{n-2}{4}}\quad\text{
as }x\rightarrow\partial\Omega,$

where $\rho(x)$ is the distance of $x$ to $\partial\Omega$. If $\Omega$ is
bounded or $k=1$, the solution $u$ is unique.

We remark that on smooth compact Riemannian manifold with boundary, the growth
rate (1.7) was discovered by Gursky, Streets and Warren [10]. In Theorem 2.8,
we shall give a simplified proof for general smooth manifolds with compact
boundaries.

Our main result on the solvability of fully nonlinear Loewner-Nirenberg
problem of Ricci curvature is as follows.

######  Theorem 1.2.

Let $\Omega\subsetneq\mathbb{R}^{n}$ be a domain. Suppose that the boundary
$\partial\Omega$ consists of finitely many disjoint non-self-intersecting
smooth compact surfaces, and the maximal solution $u_{\Omega}$ of (1.2)
satisfies $u_{\Omega}\not\equiv 0$. If each component of $\partial\Omega$ has
codimension $m$ satisfying

(1.8)
$v_{m}=\big{(}\underbrace{n-m,\ldots,n-m}_{n-m+1},\underbrace{2-m,\ldots,2-m}_{m-1}\big{)}\in\Gamma_{k},$

then there exists a smooth complete metric $g_{u}=u^{\frac{4}{n-2}}g$
satisfying (1.1), or equivalently, there exists a smooth positive admissible
solution to (1.2)-(1.3). If some component has codimension $m$ satisfying
$v_{m}\notin\overline{\Gamma}_{k}$, then there does not exist a complete
conformal metric satisfying (1.1), or equivalently, equation (1.2)-(1.3) is
not solvable.

The borderline case $v_{m}\in\partial\Gamma_{k}$ is still open. The difficulty
lies in the lack of a suitable upper barrier. We observe that when $k=1$, the
borderline case (1.6) agrees with (1.8), which was solved in [2, 15] by some
integral estimates. We wish to reinvestigate the borderline case in the
future.

Motivated by [9], the second goal in this paper is to investigate the
existence of complete conformal metrics subject to the following more general
equation

(1.9)
$\sigma_{k}\Big{(}-g_{u}^{-1}\text{Ric}_{g_{u}}\Big{)}+\alpha(x)\sigma_{k-1}\Big{(}-g_{u}^{-1}\text{Ric}_{g_{u}}\Big{)}=\alpha_{0}(x),$

where $\alpha_{0}(x)>0$ and $\alpha(x)$ are real valued smooth functions on
$\overline{\Omega}$. Under conformal deformation of metric
$g_{u}=u^{\frac{4}{n-2}}g$, (1.9) is equivalent to

(1.10)
$\displaystyle\sigma_{k}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}+\alpha(x)\sigma_{k-1}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}=\alpha_{0}(x).$

As proved in [9], equation (1.10) is elliptic when $u$ is $(k-1)$-admissible.
In what follows, we may assume that $k\geq 2$ in (1.10).

Our first result associated to (1.10) shows that similar to (1.2), there
exists smooth positive $(k-1)$-admissible solutions to Dirichlet problems.

######  Theorem 1.3.

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$ whose boundary is
composed of finitely many disjoint non-self-intersecting smooth compact
hypersurfaces. For any smooth positive function $\varphi$ defined on
$\partial\Omega$, there exists a smooth positive $(k-1)$-admissible solution
$u$ to equation (1.10) which satisfies the boundary condition

(1.11) $u=\varphi\quad\text{ on }\partial\Omega.$

The proof of Theorem 1.3 relies on the establishment of $C^{2}$ a priori
estimates (see Theorem 4.2, 4.4, 4.5) and the existence of a subsolution. We
notice that the method for $C^{2}$ estimates is similar to Guan [6] but may be
more complicated due to some extra terms. In addition, the estimates in
section 4 depend explicitly on $\inf\alpha_{0}$. Also, we give a new proof of
the second order boundary estimate. For deriving preliminary estimates and
conducting continuity process, we construct a smooth subsolution to the
Dirichlet problem (1.10)–(1.11), synthesizing the ideas of Guan [6] and Guan
[8].

Our next results associated to (1.10) are on the formulation of fully
nonlinear Loewner-Nirenberg type problem. When the boundary $\partial\Omega$
is composed of hypersurfaces, we have the following result.

######  Theorem 1.4.

Let $\Omega$ be a domain in $\mathbb{R}^{n}$ with nonempty smooth compact
boundary which is composed of closed hypersurfaces. Then there exists a smooth
complete metric $g_{u}=u^{\frac{4}{n-2}}g$ satisfying (1.9), or equivalently,
there exists a smooth positive $(k-1)$-admissible solution $u$ to equation
(1.10)–(1.3), provided that

$\inf\limits_{\Omega}\alpha_{0}(x)>0,\quad\inf\limits_{\Omega}\alpha(x)>-\infty,\quad\sup\limits_{\Omega}\alpha_{0}(x)<\infty,\quad\sup\limits_{\Omega}\alpha(x)<\infty.$

The proof of Theorem 1.4 relies further on the interior second order estimates
for equation (1.10), which are derived in Theorem 4.1, 4.3. Also, we need to
find a global upper barrier and a global lower barrier to conduct the diagonal
process. The latter is given by Proposition 5.4 and the former is provided by
Loewner and Nirenberg [13].

When $\Omega$ is a general domain in $\mathbb{R}^{n}$ with smooth compact
boundary, in order to define the maximal solution (see Definition 5.6), we
have to assume that $\alpha(x)\leq 0$ in $\Omega$ so that we can apply the
maximum principle Theorem 5.3. We obtain the following result.

######  Theorem 1.5.

Let $\Omega\subsetneq\mathbb{R}^{n}$ be a domain with finitely many disjoint
non-self-intersecting smooth compact surfaces as the boundary. Suppose that

$\alpha(x)\leq 0\text{ in
}\Omega,\quad\inf\limits_{\Omega}\alpha_{0}(x)>0,\quad\inf\limits_{\Omega}\alpha(x)>-\infty,\quad\sup\limits_{\Omega}\alpha_{0}(x)<\infty.$

In addition, suppose that the maximal solution $u_{\Omega}$ of (1.10)
satisfies

$u_{\Omega}\not\equiv 0\quad\text{ in }\Omega.$

If each component of $\partial\Omega$ has codimension $m$ satisfying
$v_{m}\in\Gamma_{k}$, then there exists a smooth complete metric
$g_{u}=u^{\frac{4}{n-2}}g$ satisfying (1.9), or equivalently, there exists a
smooth positive $k$-admissible solution to (1.10)–(1.3). If some component has
codimension $m$ satisfying $v_{m}\notin\overline{\Gamma}_{k}$, then there does
not exist a complete conformal metric satisfying (1.9), or equivalently,
equation (1.10)–(1.3) is not solvable.

This paper is organized as follows. We prove Theorem 1.1 in section 2 and
Theorem 1.2 in section 3. Section 4 is devoted to $C^{2}$ a priori estimates
and interior $C^{2}$ estimates for equation (1.10). Then we apply continuity
method and degree theory to prove Theorem 1.3. In section 5, we discuss the
fully nonlinear Loewner-Nirenberg problem associated to equation (1.10) and
give the proof of Theorem 1.4 and Theorem 1.5.

##  2\. Solutions on Euclidean domains with smooth compact boundary consisting
of closed hypersurfaces

In this paper, we shall mainly discuss admissible solutions to equation
(1.2)-(1.3) in Euclidean domains which have smooth compact boundaries. Within
this section, we focus on the case when these boundaries are composed of
closed hypersurfaces.

###  2.1. Preliminaries

######  Definition 2.1.

A function $0<\underline{u}\in C^{2}(\Omega)$ is a subsolution of (1.2) in
$\Omega$ if

$\lambda\big{(}W[\underline{u}]\big{)}\in\Gamma_{k}\text{ and
}\sigma_{k}^{\frac{1}{k}}\big{(}W[\underline{u}]\big{)}\geq\frac{n-2}{2}\underline{u}^{\frac{n+2}{n-2}}\quad\text{
in }\Omega.$

A function $0<\overline{u}\in C^{2}(\Omega)$ is a supersolution of (1.2) in
$\Omega$ if

$\text{either }\lambda\big{(}W[\overline{u}]\big{)}\notin\Gamma_{k}\text{ or
}\sigma_{k}^{\frac{1}{k}}\big{(}W[\overline{u}]\big{)}\leq\frac{n-2}{2}\overline{u}^{\frac{n+2}{n-2}}\quad\text{
in }\Omega.$

In view of the form of equation (1.2), we first give a general property on
supersolutions and subsolutions.

######  Proposition 2.2.

If $u$ is a positive subsolution of (1.2), so does $cu$ for any constant
$0<c<1$. If $u$ is a positive supersolution of (1.2), so does $cu$ for any
constant $c>1$; if in addition $\lambda\big{(}W[u]\big{)}\notin\Gamma_{k}$
everywhere in $\Omega$, then $cu$ is a supersolution of (1.2) for any $c>0$.

###### Proof.

The conclusion follows from the fact that $W[cu]=cW[u]$ and

$\sigma_{k}^{\frac{1}{k}}(W[cu])=\sigma_{k}^{\frac{1}{k}}(cW[u])=c\sigma_{k}^{\frac{1}{k}}(W[u])\quad\text{if
}\lambda\big{(}W[u]\big{)}\in\Gamma_{k}.$

∎

We will also need the following maximum principle. The proof is similar to
[10].

######  Theorem 2.3.

Let $M$ be a smooth compact manifold with boundary. Suppose that $u$ and $v$
are $C^{2}$ positive subsolution and supersolution of (1.2) respectively on
$M$. If $u\leq v$ on $\partial M$, then $u\leq v$ on $M$.

###### Proof.

Suppose that $u>v$ somewhere in the interior of $M$. Let $C$ be the maximum of
$\frac{u}{v}$ on $M$, which is attained at $x_{0}$ in the interior of $M$.
Since $C>1$, by Theorem 2.2 we know that $w=\frac{u}{C}$ is a strict
subsolution, that is,

$\sigma_{k}^{\frac{1}{k}}(W[w])>\frac{n-2}{2}w^{\frac{n+2}{n-2}}.$

On the other hand, since $w(x_{0})=v(x_{0})$ while $w\leq v$ near $x_{0}$,
thus at $x_{0}$ we have

$\nabla w(x_{0})=\nabla
v(x_{0}),\quad\nabla^{2}w(x_{0})\leq\nabla^{2}v(x_{0})$

and consequently $W[w](x_{0})\leq W[v](x_{0})$. It follows that

$\sigma_{k}^{\frac{1}{k}}(W[v])(x_{0})\geq\sigma_{k}^{\frac{1}{k}}(W[w])(x_{0})>\frac{n-2}{2}w^{\frac{n+2}{n-2}}(x_{0})=\frac{n-2}{2}v^{\frac{n+2}{n-2}}(x_{0}),$

contradicting with the fact that $v$ is a supersolution. ∎

Next, we provide some special solutions on balls and exterior of balls in
$\mathbb{R}^{n}$ of equation (1.2), which will be used frequently as
supersolutions or subsolutions. In Euclidean coordinates, $W[u]$ can be
expressed as

(2.1) $W_{ij}[u]=(n-2)u_{ij}-n\frac{u_{i}u_{j}}{u}+\Big{(}\Delta
u+\frac{|\nabla u|^{2}}{u}\Big{)}\delta_{ij}.$

Let

$B_{R}(x_{0})=\big{\\{}x\in\mathbb{R}^{n}\big{|}|x-x_{0}|<R\big{\\}}.$

We can verify the following proposition.

######  Proposition 2.4.

For any fixed $s>0$,

$u(x)=\Big{(}4(n-1)(C_{n}^{k})^{\frac{1}{k}}s^{2}\Big{)}^{\frac{n-2}{4}}\Big{(}s^{2}-{|x-x_{0}|}^{2}\Big{)}^{1-\frac{n}{2}}\quad\text{
in }B_{s}(x_{0}),$
$v(x)=\Big{(}4(n-1)(C_{n}^{k})^{\frac{1}{k}}s^{2}\Big{)}^{\frac{n-2}{4}}\Big{(}{|x-x_{0}|}^{2}-s^{2}\Big{)}^{1-\frac{n}{2}}\quad\text{
in }\mathbb{R}^{n}\setminus\overline{B_{s}(x_{0})}$

are admissible solutions of (1.2) which approach to $\infty$ on $\partial
B_{s}(x_{0})$.

###### Proof.

We first prove that $u(x)$ is an admissible solution of (1.2). For
convenience, denote

$c:=\Big{(}4(n-1)(C_{n}^{k})^{\frac{1}{k}}s^{2}\Big{)}^{\frac{n-2}{4}},\quad
x=(x^{1},\ldots,x^{n}),\quad x_{0}=(x_{0}^{1},\ldots,x_{0}^{n}).$

Direct calculation shows that

$W_{ij}[u]=2(n-1)(n-2)c\Big{(}s^{2}-|x-x_{0}|^{2}\Big{)}^{-\frac{n}{2}-1}s^{2}\delta_{ij}.$

Hence

$\sigma_{k}^{\frac{1}{k}}\big{(}W[u]\big{)}=2(n-1)(n-2)c\Big{(}s^{2}-|x-x_{0}|^{2}\Big{)}^{-\frac{n}{2}-1}s^{2}\big{(}C_{n}^{k}\big{)}^{\frac{1}{k}},$

which agrees with the right hand side

$\frac{n-2}{2}u^{\frac{n+2}{n-2}}=\frac{n-2}{2}c^{\frac{n+2}{n-2}}\Big{(}s^{2}-|x-x_{0}|^{2}\Big{)}^{-\frac{n}{2}-1}.$

$v(x)$ can be verified similarly. ∎

The following result is a direct consequence of Proposition 2.4 and Theorem
2.3.

######  Corollary 2.5.

For any positive admissible solution $u$ of (1.2) in $B_{R}(x_{0})$, we have

(2.2)
$u(x)\leq\Big{(}4(n-1)(C_{n}^{k})^{\frac{1}{k}}R^{2}\Big{)}^{\frac{n-2}{4}}\Big{(}R^{2}-{|x-x_{0}|}^{2}\Big{)}^{1-\frac{n}{2}}\quad\text{
in }B_{R}(x_{0}).$

In particular, we have

(2.3)
$u(x_{0})\leq\Big{(}4(n-1)(C_{n}^{k})^{\frac{1}{k}}\Big{)}^{\frac{n-2}{4}}R^{1-\frac{n}{2}}.$

As a consequence of (2.3), we immediately obtain

######  Corollary 2.6.

There does not exist a positive admissible solution to (1.2) on
$\mathbb{R}^{n}$.

###### Proof.

If $u$ is a positive admissible solution to (1.2) on $\mathbb{R}^{n}$, then
$u$ is obviously a positive admissible solution on $B_{R}(0)$ for any $R>0$.
By (2.3) we have

$u(0)\leq\Big{(}4(n-1)(C_{n}^{k})^{\frac{1}{k}}\Big{)}^{\frac{n-2}{4}}R^{1-\frac{n}{2}}.$

Letting $R\rightarrow\infty$ yields

$u(0)\leq 0,$

which contradicts with the fact that $u(0)>0$. ∎

We also find the decay property at $\infty$ of any positive admissible
solution of (1.2).

######  Corollary 2.7.

Let $\Omega$ be an unbounded domain in $\mathbb{R}^{n}$ with smooth compact
boundary. If $u$ is a smooth positive admissible solution of (1.2), then

(2.4) $u(x)\rightarrow 0\quad\text{ as }\,\,|x|\rightarrow\infty.$

###### Proof.

Applying (2.3) in a ball centered at $x$ of radius $|x|/2$ for $|x|$ large, we
have

$u(x)\leq\Big{(}4(n-1)\big{(}C_{n}^{k}\big{)}^{\frac{1}{k}}\Big{)}^{\frac{n-2}{4}}\Big{(}{\frac{|x|}{2}}\Big{)}^{1-\frac{n}{2}},$

which implies (2.4). ∎

###  2.2. Growth rate near codimension $1$ boundary

In this section, we assume that the boundary of $\Omega$ is composed of smooth
compact hypersurfaces. For positive admissible solution of (1.2) which
approaches $\infty$ at $\partial\Omega$, we characterize the growth rate.

######  Theorem 2.8.

Let $(M,g)$ be a smooth manifold with boundary. Assume that $\partial M$ is
compact. If $u$ is a smooth positive admissible solution of (1.2) on $M$ which
approaches to $\infty$ at $\partial M$, then

(2.5)
${\rho}^{\frac{n}{2}-1}(x)u(x)\rightarrow\big{(}(n-1)(C_{n}^{k})^{\frac{1}{k}}\big{)}^{\frac{n-2}{4}}\quad\text{
as }x\rightarrow\partial M,$

where $\rho(x)$ is the distance from $x$ to $\partial M$.

###### Proof.

We first construct a positive admissible subsolution in a neighborhood of
$\partial M$. Consider

$\underline{u}=c_{0}(\rho+\epsilon)^{1-\frac{n}{2}}e^{w},$

where $\epsilon$ is a small positive constant,

$c_{0}=\big{(}(n-1)(C_{n}^{k})^{\frac{1}{k}}\big{)}^{\frac{n-2}{4}},$

and

$w=w(\rho)=\frac{1}{\rho+\delta}-\frac{1}{\delta}$

with $\delta$ a small positive constant to be chosen later.

For any point $x\in M\setminus\partial M$ at which $\rho$ is smooth, we choose
a local orthonormal frame $e_{1},\ldots,e_{n}$ around $x$ such that
$e_{1}=\frac{\partial}{\partial\rho}$ (recall that $\nabla\rho$ =
$\frac{\partial}{\partial\rho}$ and $|\nabla\rho|=1$ at $x$). Elementary
calculation shows that at $x$,

$\nabla\underline{u}=c_{0}e^{w}(\rho+\epsilon)^{-\frac{n}{2}}\Big{(}1-\frac{n}{2}+(\rho+\epsilon)w^{\prime}\Big{)}\nabla\rho,$

$\displaystyle\nabla^{2}\underline{u}=$ $\displaystyle
c_{0}e^{w}(\rho+\epsilon)^{-\frac{n}{2}-1}\Bigg{(}\bigg{(}\Big{(}1-\frac{n}{2}\Big{)}(\rho+\epsilon)+(\rho+\epsilon)^{2}w^{\prime}\bigg{)}\nabla^{2}\rho$
$\displaystyle+\bigg{(}\frac{n}{2}\Big{(}\frac{n}{2}-1\Big{)}+(\rho+\epsilon)^{2}w^{\prime\prime}-(n-2)(\rho+\epsilon)w^{\prime}+(\rho+\epsilon)^{2}w^{\prime
2}\bigg{)}\nabla\rho\otimes\nabla\rho\Bigg{)},$

and consequently,

$\displaystyle W[\underline{u}]=$ $\displaystyle
c_{0}e^{w}(\rho+\epsilon)^{-\frac{n}{2}-1}\Bigg{(}\Big{(}(n-2)(\rho+\epsilon)^{2}w^{\prime}-\frac{1}{2}(n-2)^{2}(\rho+\epsilon)\Big{)}\nabla^{2}\rho$
$\displaystyle+\Big{(}(n-2)(\rho+\epsilon)^{2}w^{\prime\prime}+2(n-2)(\rho+\epsilon)w^{\prime}-2(\rho+\epsilon)^{2}w^{\prime
2}\Big{)}\nabla\rho\otimes\nabla\rho$
$\displaystyle+\bigg{(}\Big{(}(\rho+\epsilon)^{2}w^{\prime}-\frac{1}{2}(n-2)(\rho+\epsilon)\Big{)}\Delta\rho$
$\displaystyle+(\rho+\epsilon)^{2}w^{\prime\prime}-2(n-2)(\rho+\epsilon)w^{\prime}+2(\rho+\epsilon)^{2}w^{\prime
2}+\frac{1}{2}(n-1)(n-2)\bigg{)}I$
$\displaystyle-\frac{n-2}{2}(\rho+\epsilon)^{2}\text{Ric}_{g}\Bigg{)}.$

For $\rho_{0}>0$, denote

$M_{\rho_{0}}=\big{\\{}x\in M\big{|}\rho(x)<\rho_{0}\big{\\}}.$

We may choose $\rho_{0}$ sufficiently small such that on $M_{\rho_{0}}$,
$u>c_{0}$ and

(2.6) $\Delta\rho I+(n-2)\nabla^{2}\rho\leq\lambda I,\quad\text{Ric}_{g}\leq
C_{0}I,$

where $I$ is the unit matrix and $\lambda$, $C_{0}$ are positive constants
depending only on $g=I$. Denoting

$\displaystyle\Psi=$
$\displaystyle(n-2)(\rho+\epsilon)^{2}w^{\prime\prime}\nabla\rho\otimes\nabla\rho+(n-2)(\rho+\epsilon)^{2}w^{\prime}\nabla^{2}\rho+(\rho+\epsilon)^{2}w^{\prime\prime}I$
$\displaystyle+2(\rho+\epsilon)^{2}w^{\prime
2}(I-\nabla\rho\otimes\nabla\rho)+(\rho+\epsilon)^{2}w^{\prime}\Delta\rho
I-\frac{n-2}{2}(\rho+\epsilon)^{2}\text{Ric}_{g},$ $\displaystyle\Phi=$
$\displaystyle-\frac{1}{2}(n-2)(\rho+\epsilon)\Delta\rho
I-\frac{1}{2}(n-2)^{2}(\rho+\epsilon)\nabla^{2}\rho$
$\displaystyle+2(n-2)(\rho+\epsilon)w^{\prime}(\nabla\rho\otimes\nabla\rho-I)+\frac{1}{2}(n-1)(n-2)I,$

then

$W[\underline{u}]=c_{0}e^{w}(\rho+\epsilon)^{-\frac{n}{2}-1}(\Psi+\Phi).$

Note that

$w^{\prime}=\frac{-1}{(\rho+\delta)^{2}}<0,\quad
w^{\prime\prime}=\frac{2}{(\rho+\delta)^{3}}>0.$

In view of (2.6) we obtain

$\Psi\geq(\rho+\epsilon)^{2}\Big{(}w^{\prime\prime}+\lambda
w^{\prime}-\frac{n-2}{2}C_{0}\Big{)}I.$

We note that

$w^{\prime\prime}+\lambda
w^{\prime}-\frac{n-2}{2}C_{0}=\frac{2}{(\rho+\delta)^{3}}-\frac{\lambda}{(\rho+\delta)^{2}}-\frac{n-2}{2}C_{0}\geq
0$

for $\rho_{0}+\delta$ sufficiently small depending only on $n$, $\lambda$ and
$C_{0}$. Hence we conclude that $\Psi\geq 0$.

Also, by (2.6) and for $\rho_{0}+\delta\leq\sqrt{\frac{2}{\lambda}}$, we have

$\displaystyle\Phi\geq$ $\displaystyle-\frac{n-2}{2}(\rho+\epsilon)\lambda
I+\frac{2(n-2)(\rho+\epsilon)}{(\rho+\delta)^{2}}(I-\nabla\rho\otimes\nabla\rho)+\frac{1}{2}(n-1)(n-2)I$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}(n-1)(n-2)\left(\begin{array}[]{cccc}1-\frac{\lambda(\rho+\epsilon)}{n-1}&&&\\\
&1+\frac{\lambda(\rho+\epsilon)}{n-1}&&\\\ &&\ddots&\\\
&&&1+\frac{\lambda(\rho+\epsilon)}{n-1}\\\ \end{array}\right).$

Letting $\epsilon<\rho_{0}$ and $\rho_{0}<\frac{n-1}{2\lambda}$, we have
$\Phi>0$ and

$\displaystyle\sigma_{k}^{\frac{1}{k}}(W[\underline{u}])\geq$ $\displaystyle
c_{0}e^{w}(\rho+\epsilon)^{-\frac{n}{2}-1}\sigma_{k}^{\frac{1}{k}}(\Phi)$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}(n-1)(n-2)c_{0}e^{w}(\rho+\epsilon)^{-\frac{n}{2}-1}(C_{n}^{k})^{\frac{1}{k}}\cdot$
$\displaystyle\Big{(}1+\frac{\lambda(\rho+\epsilon)}{n-1}\Big{)}^{\frac{k-1}{k}}\bigg{(}1+\Big{(}1-\frac{2k}{n}\Big{)}\frac{\lambda(\rho+\epsilon)}{n-1}\bigg{)}^{\frac{1}{k}}.$

Elementary calculation shows that

$\Big{(}1+\frac{\lambda(\rho+\epsilon)}{n-1}\Big{)}^{k-1}\bigg{(}1+\Big{(}1-\frac{2k}{n}\Big{)}\frac{\lambda(\rho+\epsilon)}{n-1}\bigg{)}\geq
1$

when $k\leq\frac{n}{2}$, or when $k>\frac{n}{2}$ and
$\rho_{0}\leq\frac{(n-1)(n-2)k}{2(2k-n)(k-1)\lambda}$. Therefore,

$\sigma_{k}^{\frac{1}{k}}(W[\underline{u}])\geq\frac{1}{2}(n-1)(n-2)c_{0}(C_{n}^{k})^{\frac{1}{k}}e^{w}(\rho+\epsilon)^{-\frac{n}{2}-1}.$

On the other hand,

$\frac{n-2}{2}\underline{u}^{\frac{n+2}{n-2}}=\frac{n-2}{2}{c_{0}}^{\frac{n+2}{n-2}}(\rho+\epsilon)^{-\frac{n}{2}-1}e^{\frac{n+2}{n-2}w}.$

Note that $w\leq 0$. Hence $\underline{u}$ is a subsolution of (1.2).

Observe that

$\underline{u}=c_{0}\epsilon^{1-\frac{n}{2}}\leq u=\infty\quad\text{ on
}\partial M,$

and on $\rho=\rho_{0}$,

$\underline{u}\leq
c_{0}{\rho_{0}}^{1-\frac{n}{2}}e^{(\rho_{0}+\delta)^{-1}-{\delta}^{-1}}\leq
c_{0}{\rho_{0}}^{1-\frac{n}{2}}e^{-\delta^{-1}/2}\leq c_{0}\leq u$

for $\delta$ sufficiently small depending in addition on $\rho_{0}$. By the
maximum principle,

$\underline{u}\leq u\quad\text{ on }M_{\rho_{0}}.$

Letting $\epsilon\rightarrow 0^{+}$ and then $x\rightarrow\partial M$ we
obtain

$\liminf\limits_{x\rightarrow\partial M}\rho^{\frac{n}{2}-1}u\geq c_{0}.$

We next prove

(2.7) $\limsup\limits_{x\rightarrow\partial M}\rho^{\frac{n}{2}-1}u\leq
c_{0}.$

For this, we construct a supersolution in
$M_{\rho_{1}}\setminus\overline{M_{\epsilon}}$ for any $0<\epsilon<\rho_{1}$:

$\overline{u}=c_{0}(\rho-\epsilon)^{1-\frac{n}{2}}e^{v},$

where

$v=v(\rho)=\frac{1}{2}\Big{(}\ln(\delta+\rho)-\ln\delta\Big{)}$

with $\rho_{1}$, $\delta$ small positive constants to be chosen later.

From the above calculation, it is easy to obtain

$\displaystyle\sigma_{1}\big{(}W[\overline{u}]\big{)}=$ $\displaystyle
c_{0}e^{v}(\rho-\epsilon)^{-\frac{n}{2}-1}\bigg{(}\Big{(}2(n-1)(\rho-\epsilon)^{2}v^{\prime}-(n-2)(n-1)(\rho-\epsilon)\Big{)}\Delta\rho$
$\displaystyle+2(n-1)(\rho-\epsilon)^{2}v^{\prime\prime}-2(n-2)(n-1)(\rho-\epsilon)v^{\prime}+2(n-1)(\rho-\epsilon)^{2}v^{\prime
2}$
$\displaystyle+\frac{n(n-1)(n-2)}{2}-\frac{n-2}{2}(\rho-\epsilon)^{2}S_{g}\bigg{)},$

where $S_{g}$ is the scalar curvature on $M$. We may choose $\rho_{1}$
sufficiently small such that on $M_{\rho_{1}}$,

$|\Delta\rho|\leq C_{1},\quad S_{g}\geq-C_{1},$

where $C_{1}$ is a positive constant depending only on $g$. Note that

$v^{\prime}=\frac{1}{2(\delta+\rho)},\quad
v^{\prime\prime}=-\frac{1}{2(\delta+\rho)^{2}}.$

It follows that

$\displaystyle
2(n-1)v^{\prime}\Delta\rho+2(n-1)v^{\prime\prime}+2(n-1)v^{\prime
2}-\frac{n-2}{2}S_{g}$ $\displaystyle\leq$
$\displaystyle\frac{(n-1)C_{1}}{\delta+\rho}-\frac{n-1}{2(\delta+\rho)^{2}}+\frac{n-2}{2}C_{1}\leq
0,$

and

$\displaystyle-(n-2)(n-1)\Delta\rho-2(n-2)(n-1)v^{\prime}\leq(n-2)(n-1)\Big{(}C_{1}-\frac{1}{\delta+\rho}\Big{)}\leq
0$

for $\rho_{1}+\delta$ sufficiently small depending only on $n$ and $C_{1}$.
Consequently,

$\sigma_{1}\big{(}W[\overline{u}]\big{)}\leq
c_{0}e^{v}(\rho-\epsilon)^{-\frac{n}{2}-1}\frac{n(n-1)(n-2)}{2}.$

Then applying Newton-Maclaurin inequality we arrive at

$\sigma_{k}^{\frac{1}{k}}\big{(}W[\overline{u}]\big{)}\leq\frac{(C_{n}^{k})^{\frac{1}{k}}}{n}\sigma_{1}\big{(}W[\overline{u}]\big{)}\leq\frac{n-2}{2}{\overline{u}}^{\frac{n+2}{n-2}}\quad\text{
in }M_{\rho_{1}}\setminus\overline{M_{\epsilon}}.$

Also, we note that on $\rho=\epsilon$,

$\overline{u}=\infty>u,$

and on $\rho=\rho_{1}$,

$\overline{u}\geq
c_{0}{\rho_{1}}^{1-\frac{n}{2}}\sqrt{\frac{\delta+\rho_{1}}{\delta}}\geq u$

if $\delta$ is chosen sufficiently small depending in addition on $\rho_{1}$
and $u$. By the maximum principle, we have

$\overline{u}\geq u\quad\text{ on }M_{\rho_{1}}\setminus M_{\epsilon}.$

Letting $\epsilon\rightarrow 0^{+}$ and then $x\rightarrow\partial M$ we
obtain (2.7). ∎

######  Remark 2.9.

In [10], a global subsolution

$\underline{u}=c_{0}(\rho+\epsilon)^{1-\frac{n}{2}}e^{A\big{(}(\rho+\delta)^{-p}-{\delta}^{-p}\big{)}}$

is constructed on compact manifold with boundary which yields the lower bound
of the growth rate. We simplify the proof by defining the subsolution only in
a neighborhood of the boundary $\partial M$. For the upper bound, we also
adopt a new supersolution near $\partial M$. For the special case when $M$ is
a domain in $\mathbb{R}^{n}$, we may use González-Li-Nguyen’s method (see
Lemma 3.4 in [5]) to give an easier proof.

###  2.3. Existence of smooth solutions

We now cite an important result of Loewner and Nirenberg [13] when the
boundary of the domain is composed of hypersurfaces for the scalar case
($k=1$), which will be used as an upper barrier later.

######  Lemma 2.10.

Let $\Omega$ be a domain in $\mathbb{R}^{n}$ having smooth compact
hypersurfaces as boundary. There exists a unique positive solution $u\in
C^{\infty}(\Omega)$ of (1.5). In addition, if $\rho(x)$ denotes the distance
of $x$ to the boundary of $\Omega$, then

${\rho}^{\frac{n}{2}-1}(x)u(x)\rightarrow\big{(}n(n-1)\big{)}^{\frac{n-2}{4}}\quad\text{
as }x\rightarrow\partial\Omega.$

If $\Omega$ is unbounded, the solution $u(x)$ has the property that

$|x|^{n-2}u(x)\rightarrow c\quad\text{ as }|x|\rightarrow\infty,$

where $c$ is a positive constant depending only on $n$ and $\Omega$.

We can now state the main theorem in this section.

######  Theorem 2.11.

Let $\Omega$ be a domain in $\mathbb{R}^{n}$ with nonempty smooth compact
boundary which are composed of closed hypersurfaces. Then there exists a
positive admissible solution $u\in C^{\infty}(\Omega)$ to equation (1.2) which
tends to $\infty$ on $\partial\Omega$. If $\Omega$ is bounded or $k=1$, the
solution is unique.

###### Proof.

If $\Omega$ is a bounded domain, by Corollary 1.3 in [6], there exists a
positive admissible solution $u\in C^{\infty}(\Omega)$ to equation (1.2) with
infinite boundary value.

When $\Omega$ is unbounded, we may assume without loss of generality that
$0\notin\overline{\Omega}$. Let $B_{s}(0)$ be a fixed ball such that
$\overline{\Omega}\subset\mathbb{R}^{n}\setminus\overline{B_{s}(0)}$. By
Proposition 2.4, there exists a smooth solution

$\underline{u}=C_{1}(|x|^{2}-s^{2})^{1-\frac{n}{2}},\quad
C_{1}=\big{(}4(n-1)(C_{n}^{k})^{\frac{1}{k}}s^{2}\big{)}^{\frac{n-2}{4}}$

to equation (1.2) in $\mathbb{R}^{n}\setminus\overline{B_{s}(0)}$.

For any $R>\max\limits_{\partial\Omega}\underline{u}$ large enough such that
$\partial\Omega\subset B_{R}(0)$, by Corollary 5.4 in [6], there exists a
smooth positive admissible solution $u_{R}$ to the Dirichlet problem

(2.8) $\left\\{\begin{aligned}
&\sigma_{k}^{\frac{1}{k}}(W[u])=\frac{n-2}{2}{u}^{\frac{n+2}{n-2}}\quad\text{
in }B_{R}(0)\cap\Omega,\\\ &u=R\quad\text{ on }\partial\Omega,\\\
&u=\underline{u}\quad\text{ on }\partial B_{R}(0).\end{aligned}\right.$

By Lemma 2.10, we are able to find a positive solution $\bar{u}\in
C^{\infty}(\Omega)$ of (1.5) which tends to $\infty$ at $\partial\Omega$ with
the growth rate

$\rho^{\frac{n}{2}-1}(x)\overline{u}(x)\rightarrow\big{(}n(n-1)\big{)}^{\frac{n-2}{4}}\quad\text{
as }x\rightarrow\partial\Omega,$

and decays to $0$ at $\infty$ with the decay rate

$|x|^{n-2}\overline{u}(x)\rightarrow c\quad\text{ as }|x|\rightarrow\infty,$

where $c$ is the positive constant in Lemma 2.10.

Comparing the decay rate as $|x|\rightarrow\infty$, we have on $\partial
B_{R}(0)$ with $R$ sufficiently large,

$u_{R}(x)=\underline{u}(x)\leq\frac{2C_{1}}{c}\overline{u}(x).$

Meanwhile, we have

$u_{R}(x)=R<\infty=\frac{2C_{1}}{c}\overline{u}(x)\quad\text{ on
}\partial\Omega.$

By the maximum principle, we obtain

$u_{R}(x)\leq\max\Big{\\{}\frac{2C_{1}}{c},1\Big{\\}}\overline{u}(x)\quad\text{
on }B_{R}(0)\cap\Omega.$

In proving the above inequality, we have applied the Newton-Maclaurin
inequality as well as Proposition 2.2. On the other hand, by the maximum
principle we have

$u_{R}(x)\geq\underline{u}(x)\quad\text{ on }B_{R}(0)\cap\Omega.$

Now we can apply the interior estimates of Guan (see Theorems 2.1 and 3.1 in
[6]), followed by the Evans-Krylov interior estimates [3, 11] and a standard
diagonal process to obtain a smooth positive admissible solution $u$ of (1.2)
on $\Omega$ which tends to $\infty$ at $\partial\Omega$.

When $\Omega$ is bounded, we shall prove the uniqueness. Suppose $v$ is
another positive admissible solution satisfying (1.2)-(1.3). By Theorem 2.8,
$u$ and $v$ have the same growth rate as $x\rightarrow\partial\Omega$. Hence
for any $\epsilon>0$, on $\\{\rho=c\\}$ where $c>0$ is a sufficiently small
positive constant, we have

$u\leq(1+\epsilon)v.$

By Proposition 2.2 and the maximum principle,

$u\leq(1+\epsilon)v\quad\text{ in }\,\,\\{x\in\Omega|\rho(x)>c\\}.$

Letting $c\rightarrow 0$, we have

$u\leq(1+\epsilon)v\quad\text{ in }\,\,\Omega.$

Then let $\epsilon\rightarrow 0$. It follows that

$u\leq v\quad\text{ in }\,\,\Omega.$

Similarly, we can show that $v\leq u$ in $\Omega$. Hence we proved the
uniqueness. ∎

##  3\. Maximal solution on general domain in $\mathbb{R}^{n}$

For any domain $\Omega\subset\mathbb{R}^{n}$ with smooth compact boundary
$\partial\Omega$, we hope to associate a smooth positive admissible solution
$u_{\Omega}$ of (1.2) which is maximal, in the sense that it is greater than
or equal to any smooth positive admissible solution of (1.2) in $\Omega$. We
will then investigate when $u_{\Omega}$ tends to $\infty$ on $\partial\Omega$.

###  3.1. Construction of the maximal solution

Let $\Omega_{(1)}\Subset\Omega_{(2)}\Subset\ldots$ be an increasing sequence
of bounded subdomains of $\Omega$ with smooth compact boundaries
$\partial\Omega_{(j)}$ which are closed hypersurfaces such that
$\Omega=\cup\Omega_{(j)}$. By Theorem 2.11, we can find a unique smooth
positive admissible solution $u_{(j)}$ of (1.2) in $\Omega_{(j)}$ which tends
to $\infty$ on the boundary $\partial\Omega_{(j)}$. By the maximum principle,
we see that $\\{u_{(j)}\\}$ is a monotone decreasing sequence of positive
functions. It follows that $u_{(j)}$ converges to a nonnegative function
$u_{\Omega}$ in $\Omega$.

We have the following dichotomy, which is similar to [5].

######  Lemma 3.1.

We have either $u_{\Omega}>0$ in $\Omega$ or $u_{\Omega}\equiv 0$ in $\Omega$.

###### Proof.

For the sake of completeness, we provide the proof. Suppose that
$u_{\Omega}\not\equiv 0$. Then we must have $u_{\Omega}>c$ on some
$B(x_{0},r_{0})\subset\Omega$ for some constant $c>0$. We may choose
$0<r_{1}<r_{0}$ such that the solution

$v=\big{(}4(n-1)(C_{n}^{k})^{\frac{1}{k}}r_{1}^{2}\big{)}^{\frac{n-2}{4}}\big{(}{|x-x_{0}|}^{2}-r_{1}^{2}\big{)}^{1-\frac{n}{2}}\quad\text{
in }\mathbb{R}^{n}\setminus\overline{B_{r_{1}}(x_{0})}$

(see Proposition 2.4) satisfies

$\big{(}4(n-1)(C_{n}^{k})^{\frac{1}{k}}r_{1}^{2}\big{)}^{\frac{n-2}{4}}\big{(}r_{0}^{2}-r_{1}^{2}\big{)}^{1-\frac{n}{2}}=c.$

Let $\Omega_{(j)}$ and $u_{(j)}$ be as above. By the maximum principle, we
know that $u_{(j)}\geq v$ in $\Omega_{(j)}\setminus B(x_{0},r_{0})$ for any
$j$. Hence $u_{\Omega}\geq v>0$ in $\Omega\setminus B(x_{0},r_{0})$. We thus
have $u_{\Omega}>0$ in $\Omega$. ∎

######  Remark 3.2.

Several remarks are as follows.

  1. (1)

In view of Lemma 3.1 and the interior regularity of Guan [6] and Evans-Krylov
[3, 11], we know that $u_{\Omega}$ is smooth.

  2. (2)

If there exists a positive admissible subsolution $v$ of (1.2) in $\Omega$
(for example, when $\mathbb{R}^{n}\setminus\overline{\Omega}\neq\emptyset$),
then $u_{\Omega}>0$ in $\Omega$.

  3. (3)

When $u_{\Omega}>0$ in $\Omega$, $u_{\Omega}$ is the so-called maximal
solution of equation (1.2) in $\Omega$. In fact, if $w$ is any smooth positive
admissible solution of (1.2) in $\Omega$, then $u_{(j)}\geq w$ on
$\Omega_{(j)}$ by the maximum principle. Hence $u_{\Omega}\geq w$ in $\Omega$.

  4. (4)

When $\Omega$ is bounded and $\partial\Omega$ is composed of closed
hypersurfaces, $u_{\Omega}$ coincides with the solution given by Theorem 2.11.

As in [13], we call a compact subset $\Gamma\subset\partial\Omega$ regular, if

(3.1) $u_{\Omega}(x)\rightarrow\infty\quad\text{ as }x\rightarrow\Gamma.$

The rest of this section discusses the validity of (3.1).

###  3.2. Regularity of a portion on the boundary

We consider a portion $\Gamma\subset\partial\Omega$ which is a smooth compact
non-self-intersecting surface of codimension $m$. We also assume that
$\partial\Omega\setminus\Gamma$ is smooth compact. As in [13, 5], we first
give a necessary and sufficient condition for (3.1).

######  Theorem 3.3.

Let $\Omega$ be a domain in $\mathbb{R}^{n}$ and $\Gamma$ be a compact subset
of $\partial\Omega$ such that $\partial\Omega\setminus\Gamma$ is also compact.
Suppose that $u_{\Omega}\not\equiv 0$. Then $u_{\Omega}(x)\rightarrow\infty$
as $x\rightarrow\Gamma$ if and only if there exists an open neighborhood $U$
of $\Gamma$ and a $C^{2}$ positive admissible subsolution $\phi(x)$ of (1.2)
defined in $\Omega\cap U$ which tends to $\infty$ as $x\rightarrow\Gamma$.

###### Proof.

Necessity is obvious. For sufficiency, without loss of generality we may
assume that $\overline{U}$ is compact,
$\overline{U}\cap(\partial\Omega\setminus\Gamma)=\emptyset$ and $0<\phi\in
C^{2}(\overline{U}\cap\Omega)$. For $j$ sufficiently large, we have

$\partial(U\cap\Omega_{(j)})=\partial U\cup(U\cap\partial\Omega_{(j)}).$

Since $u_{\Omega}$ is positive in $\Omega$ and $\partial U\subset\Omega$ is
compact, we have $\inf\limits_{\partial U}u_{\Omega}:=m>0$. Denote
$M:=\sup\limits_{\partial U}\phi(x)>0$ and $A:=\max\\{\frac{M}{m},1\\}$. We
note that $u_{(j)}\geq u_{\Omega}\geq m\geq\frac{\phi}{A}$ on $\partial U$,
and $u_{(j)}=\infty>\frac{\phi}{A}$ on $U\cap\partial\Omega_{(j)}$. By
Proposition 2.2, $\frac{\phi}{A}$ is again a positive admissible subsolution
of (1.2). By the maximum principle, we arrive at $u_{(j)}\geq\frac{\phi}{A}$
in $U\cap\Omega_{(j)}$. Consequently, we obtain $u_{\Omega}\geq\frac{\phi}{A}$
in $U\cap\Omega$. Hence we proved the sufficiency. ∎

Now we aim at constructing $\phi(x)$ in a neighborhood of $\Gamma$ as in
Theorem 3.3. Let $\rho(x)$ denote the distance of $x$ to $\Gamma$. Define

$\Gamma_{\rho_{0}}=\\{x\in\Omega|\rho(x)<\rho_{0}\\}.$

For $\rho_{0}$ sufficiently small, $\rho$ is a smooth function in
$\Gamma_{\rho_{0}}\setminus\Gamma$. Define on
$\Gamma_{\rho_{0}}\setminus\Gamma$

$\phi(x)=c\rho^{1-\frac{n}{2}}(x),$

where $c>0$ is a constant to be determined later. Recall that
$|\nabla\rho|=1$. Direct calculation shows that

$\phi_{i}=c\Big{(}1-\frac{n}{2}\Big{)}{\rho}^{-\frac{n}{2}}\rho_{i},$
$\phi_{ij}=c\Big{(}\frac{n}{2}-1\Big{)}\frac{n}{2}{\rho}^{-\frac{n}{2}-1}\rho_{i}\rho_{j}+c\Big{(}1-\frac{n}{2}\Big{)}{\rho}^{-\frac{n}{2}}\rho_{ij},$
$|\nabla\phi|^{2}=c^{2}\Big{(}1-\frac{n}{2}\Big{)}^{2}{\rho}^{-n},$
$\Delta\phi=c\Big{(}\frac{n}{2}-1\Big{)}\frac{n}{2}{\rho}^{-\frac{n}{2}-1}+c\Big{(}1-\frac{n}{2}\Big{)}{\rho}^{-\frac{n}{2}}\Delta\rho.$

Thus, we have

$W_{ij}[\phi]=\frac{1}{2}(n-2)c{\rho}^{-\frac{n}{2}-1}\Big{(}(n-1-\rho\Delta\rho)\delta_{ij}-(n-2)\rho\rho_{ij}\Big{)}.$

As shown in [5], as $\rho\rightarrow 0$,

$\lambda(\rho\nabla^{2}\rho)\rightarrow\big{(}\underbrace{0,\ldots,0}_{n-m+1},\underbrace{1,\ldots,1}_{m-1}\big{)}.$

Consequently, as $\rho\rightarrow 0$,

$\lambda\big{(}{\rho}^{\frac{n}{2}+1}W_{ij}[\phi]\big{)}\rightarrow\frac{1}{2}(n-2)c\big{(}\underbrace{n-m,\ldots,n-m}_{n-m+1},\underbrace{2-m,\ldots,2-m}_{m-1}\big{)}.$

If we assume that

$v_{m}=\big{(}\underbrace{n-m,\ldots,n-m}_{n-m+1},\underbrace{2-m,\ldots,2-m}_{m-1}\big{)}\in\Gamma_{k},$

then we have as $\rho\rightarrow 0$,

${\rho}^{\frac{n}{2}+1}\sigma_{k}^{\frac{1}{k}}\big{(}W_{ij}[\phi]\big{)}\rightarrow\frac{1}{2}(n-2)c\sigma_{k}^{\frac{1}{k}}(v_{m}).$

On the other hand,

$\frac{n-2}{2}\phi^{\frac{n+2}{n-2}}=\frac{n-2}{2}c^{\frac{n+2}{n-2}}\rho^{-\frac{n}{2}-1}.$

If we choose

$0<c<\Big{(}\sigma_{k}^{\frac{1}{k}}(v_{m})\Big{)}^{\frac{n-2}{4}},$

then $\phi$ is the one as in Theorem 3.3, which implies that
$u_{\Omega}\rightarrow\infty$ as $x\rightarrow\Gamma$ if $u_{\Omega}\not\equiv
0$.

###  3.3. The case when $v_{m}\notin\overline{\Gamma}_{k}$

We adopt the same function as [5]

$\psi=\psi_{c,d}=(c\rho^{-\alpha}+d)^{\beta},$

where $\alpha$, $\beta$, $c$ and $d$ are positive constants. Recall that
$\rho(x)$ is the distance of $x$ to $\Gamma$. By direct calculation, we have

$\displaystyle\psi_{i}=$ $\displaystyle-\alpha\beta
c(c\rho^{-\alpha}+d)^{\beta-1}\rho^{-\alpha-1}\rho_{i},$
$\displaystyle\psi_{ij}=$
$\displaystyle\alpha^{2}\beta(\beta-1)c^{2}(c\rho^{-\alpha}+d)^{\beta-2}\rho^{-2\alpha-2}\rho_{i}\rho_{j}$
$\displaystyle+\alpha(\alpha+1)\beta
c(c\rho^{-\alpha}+d)^{\beta-1}\rho^{-\alpha-2}\rho_{i}\rho_{j}-\alpha\beta
c(c\rho^{-\alpha}+d)^{\beta-1}\rho^{-\alpha-1}\rho_{ij},$
$\displaystyle|\nabla\psi|^{2}=$
$\displaystyle\alpha^{2}\beta^{2}c^{2}(c\rho^{-\alpha}+d)^{2\beta-2}\rho^{-2\alpha-2},$
$\displaystyle\Delta\psi=$
$\displaystyle\alpha^{2}\beta(\beta-1)c^{2}(c\rho^{-\alpha}+d)^{\beta-2}\rho^{-2\alpha-2}+\alpha(\alpha+1)\beta
c(c\rho^{-\alpha}+d)^{\beta-1}\rho^{-\alpha-2}$ $\displaystyle-\alpha\beta
c(c\rho^{-\alpha}+d)^{\beta-1}\rho^{-\alpha-1}\Delta\rho.$

It follows that

$\displaystyle\psi^{-\frac{n+2}{n-2}}W_{ij}[\psi]=$ $\displaystyle\alpha\beta
c\rho^{-\alpha-2}(c\rho^{-\alpha}+d)^{-\frac{4\beta}{n-2}-1}\bigg{(}-(n-2)\rho\rho_{ij}$
$\displaystyle+\Big{(}\alpha(-n-2\beta+2)\frac{c\rho^{-\alpha}}{c\rho^{-\alpha}+d}+(n-2)(\alpha+1)\Big{)}\rho_{i}\rho_{j}$
$\displaystyle+\Big{(}\alpha(2\beta-1)\frac{c\rho^{-\alpha}}{c\rho^{-\alpha}+d}+\alpha+1-\rho\Delta\rho\Big{)}\delta_{ij}\bigg{)}.$

Choosing an appropriate coordinate system as in [5] such that as
$\rho\rightarrow 0$,

$\rho\nabla^{2}\rho=\left(\begin{array}[]{ccc}\mathcal{O}(\rho)_{(n-m)\times(n-m)}&&\\\
&0&\\\ &&I_{(m-1)\times(m-1)}\\\ \end{array}\right)$

and

$\nabla\rho\otimes\nabla\rho=\left(\begin{array}[]{ccc}0_{(n-m)\times(n-m)}&&\\\
&1&\\\ &&0_{(m-1)\times(m-1)}\\\ \end{array}\right),$

we thus obtain

(3.2) $\frac{1}{\alpha\beta
c}\rho^{\alpha+2}(c\rho^{-\alpha}+d)^{\frac{4\beta}{n-2}+1}\psi^{-\frac{n+2}{n-2}}W_{ij}[\psi]=A_{m}+B_{\alpha\beta}(\zeta)+\mathcal{O}(\rho),$

where $\zeta=\frac{c\rho^{-\alpha}}{c\rho^{-\alpha}+d}$,

$A_{m}=\left(\begin{array}[]{ccc}(n-m)I&&\\\ &n-m&\\\ &&(2-m)I\\\
\end{array}\right),$

and

$\tiny{B_{\alpha\beta}(\zeta)=\left(\begin{array}[]{ccc}\big{(}2\alpha\beta\zeta+\alpha(1-\zeta)+2-n\big{)}I&&\\\
&(n-1)\alpha(1-\zeta)&\\\
&&\big{(}2\alpha\beta\zeta+\alpha(1-\zeta)+2-n\big{)}I\\\
\end{array}\right).}$

Since $v_{m}\notin\overline{\Gamma}_{k}$, we may choose $\alpha$ sufficiently
small, and then an appropriate $\beta$ such that $\alpha\beta$ is slightly
larger than $\frac{n}{2}-1$ in order to make

$\lambda(A_{m}+B_{\alpha\beta}(\zeta))\notin\overline{\Gamma}_{k}$

for all $0<\zeta<1$.

Then we can choose a sufficiently small $0<\rho_{0}<1$ such that on
$0<\rho<\rho_{0}$ and for all $0<\zeta<1$, the righthand side of (3.2)
satisfies

$\lambda\big{(}A_{m}+B_{\alpha\beta}(\zeta)+\mathcal{O}(\rho)\big{)}\notin\overline{\Gamma}_{k}.$

It follows that $\lambda\big{(}W_{ij}[\psi]\big{)}\notin\overline{\Gamma}_{k}$
on $\\{0<\rho<\rho_{0}\\}$ for all $c>0$ and $d>0$, which means that $\psi$ is
a supersolution of (1.2).

Now we compare $u_{\Omega}$ and $\psi$. Choosing $d$ sufficiently large
depending on $\rho_{0}$ such that on $\rho=\rho_{0}$ we have $\psi\geq
d^{\beta}\geq u_{\Omega}$. By comparing $u_{\Omega}$ with the first function
in Proposition 2.4 in the ball $B_{\rho(x)}(x)$ we can deduce that

$u_{\Omega}(x)\leq C(n,k)\rho^{1-\frac{n}{2}}\leq
c^{\beta}\rho^{-\alpha\beta}<\psi$

in $0<\rho\leq\delta$, where $\delta$ is a sufficiently small positive
constant depending on $c$. By the maximum principle we have
$u_{\Omega}\leq\psi$ in $\delta<\rho<\rho_{0}$. Letting $\delta\rightarrow 0$,
we arrive at $u_{\Omega}\leq\psi$ in $0<\rho<\rho_{0}$. Next letting
$c\rightarrow 0$ we deduce that $u_{\Omega}\leq d^{\beta}$ in
$0<\rho<\rho_{0}$.

##  4\. More general equations

By change of variable $u=e^{\frac{n-2}{2}v}$, equation (1.10) is equivalent to

(4.1)
$\sigma_{k}\bigg{(}(n-2)e^{-2v}\mathcal{W}[v]\bigg{)}+\alpha(x)\sigma_{k-1}\bigg{(}(n-2)e^{-2v}\mathcal{W}[v]\bigg{)}=\alpha_{0}(x),$

where

$\mathcal{W}[v]=g^{-1}\bigg{(}\nabla^{2}v-dv\otimes dv+\Big{(}\frac{\Delta
v}{n-2}+|\nabla v|^{2}\Big{)}g-\frac{\text{Ric}_{g}}{n-2}\bigg{)}.$

Now we derive a priori estimates for $(k-1)$-admissible solutions to equation
(4.1) on a smooth compact manifold with boundary, that is, on
$\overline{M}:=M\cup\partial M$. We write (4.1) in the following form

(4.2)
$F\big{(}\mathcal{W}[v]\big{)}:=\frac{\sigma_{k}\big{(}\mathcal{W}[v]\big{)}}{\sigma_{k-1}\big{(}\mathcal{W}[v]\big{)}}-\frac{\alpha_{0}(x)e^{2kv}}{(n-2)^{k}}\frac{1}{\sigma_{k-1}\big{(}\mathcal{W}[v]\big{)}}=-\frac{\alpha(x)}{n-2}e^{2v}.$

As shown in [9], for $(k-1)$-admissible $v$, namely
$\lambda\big{(}\mathcal{W}[v]\big{)}\in\Gamma_{k-1}$, equation (4.2) is
elliptic and the operator $F\big{(}\mathcal{W}[v]\big{)}$ is concave with
respect to $\\{\mathcal{W}_{ij}\\}$.

###  4.1. Gradient estimates

Let $v\in C^{3}(M)\cap C^{1}(\overline{M})$ be a $(k-1)$-admissible solution
of (4.1). We adopt the same test function as [6]: $\Phi=\zeta(x)we^{\eta(v)}$,
where $w=\frac{1}{2}|\nabla v|^{2}$, $\zeta$ and $\eta$ are functions to be
chosen later. Assume that $\Phi$ attains its maximum at an interior point
$x_{0}\in M$. Choose a local orthonormal frame $e_{1},\ldots,e_{n}$ about
$x_{0}$. Then, at $x_{0}$, we have

(4.3)
$\frac{\nabla_{i}\zeta}{\zeta}+\frac{\nabla_{i}w}{w}+\eta^{\prime}\nabla_{i}v=0,$
(4.4)
$F^{ij}\Big{(}\frac{\nabla_{ij}\zeta}{\zeta}-\frac{\nabla_{i}\zeta\nabla_{j}\zeta}{\zeta^{2}}+\frac{\nabla_{ij}w}{w}-\frac{\nabla_{i}w\nabla_{j}w}{w^{2}}+\eta^{\prime}\nabla_{ij}v+\eta^{\prime\prime}\nabla_{i}v\nabla_{j}v\Big{)}\leq
0,$

where $F^{ij}=\frac{\partial
F}{\partial\mathcal{W}_{ij}}\big{(}\mathcal{W}[v]\big{)}$.

By direct calculation,

$\nabla_{i}w=\nabla_{im}v\nabla_{m}v,$
$\nabla_{ij}w=\nabla_{ijm}v\nabla_{m}v+\nabla_{im}v\nabla_{jm}v.$

By (4.3), we have

(4.5) $F^{ij}\frac{\nabla_{i}w\nabla_{j}w}{w^{2}}\leq
3F^{ij}\frac{\nabla_{i}\zeta\nabla_{j}\zeta}{\zeta^{2}}+\frac{3\eta^{\prime
2}}{2}F^{ij}\nabla_{i}v\nabla_{j}v.$

Hence (4.4) becomes

(4.6) $\displaystyle
F^{ij}\Big{(}\frac{\nabla_{ij}\zeta}{\zeta}-\frac{5\nabla_{i}\zeta\nabla_{j}\zeta}{2\zeta^{2}}\Big{)}+\frac{1}{w}\Big{(}\delta_{lm}-\frac{\nabla_{l}v\nabla_{m}v}{2w}\Big{)}F^{ij}\nabla_{il}v\nabla_{jm}v$
$\displaystyle+\frac{F^{ij}\nabla_{ijm}v\nabla_{m}v}{w}+\eta^{\prime}F^{ij}\nabla_{ij}v+\Big{(}\eta^{\prime\prime}-\frac{3}{4}\eta^{\prime
2}\Big{)}F^{ij}\nabla_{i}v\nabla_{j}v\leq 0.$

Choose a smooth function $\zeta$ such that

$0\leq\zeta\leq 1,\quad|\nabla\zeta|\leq
b_{0}\sqrt{\zeta},\quad|\nabla^{2}\zeta|\leq b_{0}.$

Then (4.6) reduces to

(4.7)
$\frac{F^{ij}\nabla_{ijm}v\nabla_{m}v}{w}+\eta^{\prime}F^{ij}\nabla_{ij}v+\Big{(}\eta^{\prime\prime}-\frac{3}{4}\eta^{\prime
2}\Big{)}F^{ij}\nabla_{i}v\nabla_{j}v\leq\frac{C}{\zeta}\sum F^{ii}.$

Similarly, we have

(4.8) $\displaystyle\frac{\Delta\nabla_{m}v\nabla_{m}v}{w}+\eta^{\prime}\Delta
v+\Big{(}\eta^{\prime\prime}-\frac{3}{4}\eta^{\prime 2}\Big{)}|\nabla
v|^{2}\leq\frac{C}{\zeta}.$

Differentiating (4.2) yields

(4.9)
$F^{ij}\nabla_{l}\mathcal{W}_{ij}-\frac{\nabla_{l}\alpha_{0}e^{2kv}+\alpha_{0}e^{2kv}2k\nabla_{l}v}{(n-2)^{k}}\frac{1}{\sigma_{k-1}\big{(}\mathcal{W}[v]\big{)}}=-\frac{\nabla_{l}\alpha
e^{2v}+\alpha e^{2v}2\nabla_{l}v}{n-2},$

where

$\nabla_{l}\mathcal{W}_{ij}[v]=\nabla_{lij}v-\nabla_{li}v\nabla_{j}v-\nabla_{i}v\nabla_{lj}v+\Big{(}\frac{\nabla_{l}\Delta
v}{n-2}+2\nabla_{l}w\Big{)}\delta_{ij}-\nabla_{l}\mathcal{R}_{ij}.$

Since

$\nabla_{ijl}v=\nabla_{lij}v+R_{lij}^{m}\nabla_{m}v,$

we have

$\nabla_{l}\Delta v=\Delta\nabla_{l}v-\sum\limits_{m}R_{lmm}^{s}\nabla_{s}v.$

Assume that $|\nabla v|\geq 1$. Combining (4.7), (4.8) and (4.9) we obtain

(4.10) $\displaystyle\eta^{\prime}F^{ij}\nabla_{ij}v+\eta^{\prime}\frac{\Delta
v}{n-2}\sum F^{ii}$
$\displaystyle+\Big{(}\eta^{\prime\prime}-\frac{3}{4}\eta^{\prime
2}\Big{)}F^{ij}\nabla_{i}v\nabla_{j}v+\Big{(}\eta^{\prime\prime}-\frac{3}{4}\eta^{\prime
2}\Big{)}\frac{|\nabla v|^{2}}{n-2}\sum F^{ii}$
$\displaystyle+\frac{e^{2kv}\nabla_{l}\alpha_{0}\nabla_{l}v}{(n-2)^{k}\sigma_{k-1}w}+\frac{4k\alpha_{0}e^{2kv}}{(n-2)^{k}\sigma_{k-1}}-\frac{e^{2v}\nabla_{l}\alpha\nabla_{l}v}{(n-2)w}-\frac{4\alpha
e^{2v}}{n-2}$ $\displaystyle\leq$ $\displaystyle\frac{C}{\zeta}\sum
F^{ii}-\frac{F^{ij}\nabla_{li}v\nabla_{j}v\nabla_{l}v}{w}-\frac{F^{ij}\nabla_{i}v\nabla_{lj}v\nabla_{l}v}{w}+\frac{2\nabla_{l}w\nabla_{l}v}{w}\sum
F^{ii}.$

By (4.3),

(4.11)
$\displaystyle-\frac{F^{ij}\nabla_{li}v\nabla_{j}v\nabla_{l}v}{w}-\frac{F^{ij}\nabla_{i}v\nabla_{lj}v\nabla_{l}v}{w}+\frac{2\nabla_{l}w\nabla_{l}v}{w}\sum
F^{ii}$ $\displaystyle=$ $\displaystyle
F^{ij}\Big{(}\frac{\nabla_{i}\zeta}{\zeta}+\eta^{\prime}\nabla_{i}v\Big{)}\nabla_{j}v+F^{ij}\nabla_{i}v\Big{(}\frac{\nabla_{j}\zeta}{\zeta}+\eta^{\prime}\nabla_{j}v\Big{)}$
$\displaystyle-2\nabla_{l}v\Big{(}\frac{\nabla_{l}\zeta}{\zeta}+\eta^{\prime}\nabla_{l}v\Big{)}\sum
F^{ii}$ $\displaystyle\leq$ $\displaystyle\frac{C}{\sqrt{\zeta}}|\nabla v|\sum
F^{ii}+2\eta^{\prime}F^{ij}\nabla_{i}v\nabla_{j}v-2\eta^{\prime}|\nabla
v|^{2}\sum F^{ii}.$

Also, we have

$F^{ij}\mathcal{W}_{ij}=\frac{\sigma_{k}}{\sigma_{k-1}}+(k-1)\frac{\alpha_{0}e^{2kv}}{(n-2)^{k}\sigma_{k-1}},$

and

$F^{ij}\mathcal{W}_{ij}=F^{ij}\nabla_{ij}v-F^{ij}\nabla_{i}v\nabla_{j}v+\frac{\Delta
v}{n-2}\sum F^{ii}+|\nabla v|^{2}\sum F^{ii}-F^{ij}\mathcal{R}_{ij}.$

Therefore,

(4.12) $\displaystyle\eta^{\prime}F^{ij}\nabla_{ij}v+\eta^{\prime}\frac{\Delta
v}{n-2}\sum F^{ii}$ $\displaystyle\geq$
$\displaystyle\eta^{\prime}\frac{\sigma_{k}}{\sigma_{k-1}}+(k-1)\frac{\alpha_{0}e^{2kv}}{(n-2)^{k}\sigma_{k-1}}\eta^{\prime}$
$\displaystyle+\eta^{\prime}F^{ij}\nabla_{i}v\nabla_{j}v-\eta^{\prime}|\nabla
v|^{2}\sum F^{ii}-C|\eta^{\prime}|\sum F^{ii}.$

Taking (4.11) and (4.12) into (4.10), and in view of (4.2), we obtain

(4.13)
$\displaystyle\frac{e^{2kv}}{(n-2)^{k}\sigma_{k-1}}\Big{(}k\eta^{\prime}\alpha_{0}+4k\alpha_{0}+\frac{\nabla_{l}\alpha_{0}\nabla_{l}v}{w}\Big{)}$
$\displaystyle+\Big{(}\eta^{\prime\prime}-\frac{3}{4}\eta^{\prime
2}-\eta^{\prime}\Big{)}F^{ij}\nabla_{i}v\nabla_{j}v+\Big{(}\eta^{\prime\prime}-\frac{3}{4}\eta^{\prime
2}+(n-2)\eta^{\prime}\Big{)}\frac{|\nabla v|^{2}}{n-2}\sum F^{ii}$
$\displaystyle\leq$ $\displaystyle C\Big{(}\frac{1}{\zeta}+\frac{|\nabla
v|}{\sqrt{\zeta}}+|\eta^{\prime}|\Big{)}\sum
F^{ii}+\Big{(}\eta^{\prime}\alpha+4\alpha+\frac{\nabla_{l}\alpha\nabla_{l}v}{w}\Big{)}\frac{e^{2v}}{n-2}.$

Choose

$\eta(v)=\Big{(}\frac{3}{2}+v-\inf_{\\{\zeta>0\\}}v\Big{)}^{-N},$

where $N\geq 1$ is sufficiently large such that

$\displaystyle\eta^{\prime\prime}-\frac{3}{4}\eta^{\prime 2}-\eta^{\prime}$
$\displaystyle\geq$ $\displaystyle\eta^{\prime\prime}-\frac{3}{4}\eta^{\prime
2}+(n-2)\eta^{\prime}$ $\displaystyle\geq$
$\displaystyle\frac{N^{2}}{2}\Big{(}\frac{3}{2}+v-\inf_{\\{\zeta>0\\}}v\Big{)}^{-N-2}.$

By Newton-Maclaurin inequality

$\sigma_{k}\sigma_{k-2}\leq\frac{(n-k+1)(k-1)}{(n-k+2)k}\sigma_{k-1}^{2},$

we know that

$\displaystyle\sum F^{ii}=$
$\displaystyle(n-k+1)-\frac{(n-k+2)\sigma_{k}\sigma_{k-2}}{\sigma_{k-1}^{2}}+\frac{(n-k+2)\alpha_{0}e^{2kv}\sigma_{k-2}}{(n-2)^{k}\sigma_{k-1}^{2}}$
$\displaystyle\geq$ $\displaystyle\frac{n-k+1}{k}.$

Therefore,

$\bigg{(}\eta^{\prime}\alpha+4\alpha+\frac{\nabla_{l}\alpha\nabla_{l}v}{w}\bigg{)}\frac{e^{2v}}{n-2}\leq
C\bigg{(}|\eta^{\prime}|+1+\frac{1}{|\nabla v|}\bigg{)}\sum F^{ii}.$

Also, we know that

$k\eta^{\prime}\alpha_{0}+4k\alpha_{0}+\frac{\nabla_{l}\alpha_{0}\nabla_{l}v}{w}\geq\alpha_{0}\bigg{(}-kN\Big{(}\frac{2}{3}\Big{)}^{N+1}+4k-\frac{C}{\alpha_{0}|\nabla
v|}\bigg{)}>0$

for $N$ sufficiently large and $|\nabla v|\geq\frac{C}{2k\inf\alpha_{0}}$.
Hence, (4.13) reduces to

$|\nabla v|^{2}\leq C\Big{(}\frac{1}{\zeta}+\frac{|\nabla
v|}{\sqrt{\zeta}}\Big{)}.$

Consequently,

(4.14) $\sqrt{\zeta}|\nabla v|\leq C.$

Taking $\zeta$ to be a standard cutoff function in a geodesic ball $B_{r}$ of
radius $r>0$ satisfying

$|\nabla\zeta|\leq\frac{C}{r},$

by (4.14) we obtain

######  Theorem 4.1.

If $v\in C^{3}(B_{r})$ is a $(k-1)$-admissible solution of (4.1) in a geodesic
ball $B_{r}\subset M$ of radius $r>0$, then

$\sup\limits_{B_{\frac{r}{2}}}|\nabla v|\leq C,$

where $C$ depends on $r^{-1}$, $n$, $k$, $\|v\|_{C^{0}(B_{r})}$,
$\|\alpha\|_{C^{1}(B_{r})}$, $\|\alpha_{0}\|_{C^{1}{(B_{r})}}$ and
$\inf\limits_{B_{r}}\alpha_{0}$.

Let $\zeta\equiv 1$. By (4.14) we have

######  Theorem 4.2.

Let $v\in C^{3}(M)\cap C^{1}(\overline{M})$ be a $(k-1)$-admissible solution
of (4.1) in $\overline{M}$. Then

$\max\limits_{\overline{M}}|\nabla v|\leq C,$

where $C$ depends on $n$, $k$, $\|v\|_{C^{0}(\overline{M})}$,
$\max\limits_{\partial M}|\nabla v|$, $\|\alpha\|_{C^{1}(M)}$,
$\|\alpha_{0}\|_{C^{1}{(M)}}$ and $\inf\limits_{M}\alpha_{0}$.

###  4.2. Interior and global estimates for second derivatives

Let $v\in C^{4}(M)\cap C^{2}(\overline{M})$ be a $(k-1)$-admissible solution
of (4.1). Consider the test function $\Psi(x)=\zeta
e^{\eta(w)}\nabla_{\xi\xi}v$, where $\xi$ is a unit tangent vector to
$\overline{M}$ at $x$, $w=\frac{1}{2}|\nabla v|^{2}$, $\zeta$ and $\eta$ are
functions to be chosen later, with $\zeta$ smooth and satisfying

$0\leq\zeta\leq 1,\quad|\nabla\zeta|\leq b_{0},\quad|\nabla^{2}\zeta|\leq
b_{0}.$

Assume that $\Psi$ is attained at some interior point $x_{0}\in M$ and for
some unit vector $\xi\in T_{x_{0}}\overline{M}$. Choose a smooth local
orthonormal frame $e_{1},\ldots,e_{n}$ about $x_{0}$ such that
$e_{1}(x_{0})=\xi$ and $\big{\\{}\mathcal{W}_{ij}[v](x_{0})\big{\\}}$ is
diagonal. Assume that at $x_{0}$, $\nabla_{11}v\geq 1$. Since
$\tilde{\Psi}:=\zeta e^{\eta(w)}\nabla_{11}v$ locally attains a maximum at
$x_{0}$, thus at $x_{0}$,

(4.15)
$\frac{\nabla_{i11}v}{\nabla_{11}v}+\nabla_{i}\eta+\frac{\nabla_{i}\zeta}{\zeta}=0,$

and

(4.16)
$F^{ii}\bigg{(}\frac{\nabla_{ii11}v}{\nabla_{11}v}-\Big{(}\frac{\nabla_{i11}v}{\nabla_{11}v}\Big{)}^{2}+\nabla_{ii}\eta+\frac{\nabla_{ii}\zeta}{\zeta}-\Big{(}\frac{\nabla_{i}\zeta}{\zeta}\Big{)}^{2}\bigg{)}\leq
0.$

By (4.15) and Cauchy-Schwartz inequality,

$\bigg{(}\frac{\nabla_{i11}v}{\nabla_{11}v}\bigg{)}^{2}\leq(1+\epsilon)(\nabla_{i}\eta)^{2}+\Big{(}1+\frac{1}{\epsilon}\Big{)}\Big{(}\frac{\nabla_{i}\zeta}{\zeta}\Big{)}^{2},\quad\epsilon>0.$

Therefore, (4.16) reduces to

(4.17)
$F^{ii}\bigg{(}\frac{\nabla_{ii11}v}{\nabla_{11}v}+\nabla_{ii}\eta-(1+\epsilon)\big{(}\nabla_{i}\eta\big{)}^{2}\bigg{)}\leq\frac{C}{\zeta^{2}}\sum
F^{ii}.$

Similarly,

(4.18)
$\frac{\Delta\nabla_{11}v}{\nabla_{11}v}+\Delta\eta-(1+\epsilon)|\nabla\eta|^{2}\leq\frac{C}{\zeta^{2}}.$

Besides, we notice that

(4.19) $\displaystyle F^{ii}\nabla_{11}\bigg{(}\nabla_{ii}v+\frac{\Delta
v}{n-2}\bigg{)}=F^{ii}\nabla_{11}\Big{(}\mathcal{W}_{ii}+\big{(}\nabla_{i}v\big{)}^{2}-|\nabla
v|^{2}+\mathcal{R}_{ii}\Big{)}$ $\displaystyle\geq$ $\displaystyle
F^{ii}\nabla_{11}\mathcal{W}_{ii}+2F^{ii}\nabla_{i}v\nabla_{11i}v+2F^{ii}\big{(}\nabla_{1i}v\big{)}^{2}$
$\displaystyle-2\sum\limits_{k}\nabla_{11k}v\nabla_{k}v\sum\limits_{i}F^{ii}-2\sum\limits_{k}\big{(}\nabla_{1k}v\big{)}^{2}\sum\limits_{i}F^{ii}-C\sum
F^{ii}.$

By (4.15) and the formula for interchanging order of covariant derivatives,

$2F^{ii}\nabla_{i}v\nabla_{11i}v-2\sum\limits_{k}\nabla_{11k}v\nabla_{k}v\sum\limits_{i}F^{ii}\geq-C\bigg{(}|\nabla\eta|+\frac{1}{\zeta}\bigg{)}\nabla_{11}v\sum
F^{ii}.$

Applying the formula for interchanging order of covariant derivatives, (4.19)
reduces to

(4.20) $\displaystyle F^{ii}\nabla_{ii11}v+\frac{\Delta\nabla_{11}v}{n-2}\sum
F^{ii}$ $\displaystyle\geq$ $\displaystyle
F^{ii}\nabla_{11}\mathcal{W}_{ii}+2F^{ii}\big{(}\nabla_{1i}v\big{)}^{2}-2\sum\limits_{k}\big{(}\nabla_{1k}v\big{)}^{2}\sum\limits_{i}F^{ii}$
$\displaystyle-C\bigg{(}|\nabla\eta|\nabla_{11}v+\frac{\nabla_{11}v}{\zeta}+\sum\limits_{j,l}|\nabla_{jl}v|\bigg{)}\sum
F^{ii}.$

Differentiating equation (4.2) twice and by the concavity of
$\frac{\sigma_{k}}{\sigma_{k-1}}$,

(4.21) $\displaystyle
F^{ii}\nabla_{11}\mathcal{W}_{ii}+\frac{\alpha_{0}e^{2kv}}{(n-2)^{k}\sigma_{k-1}}\bigg{(}\frac{\sigma_{k-1}^{ij,rs}}{\sigma_{k-1}}-\frac{2\sigma_{k-1}^{ij}\sigma_{k-1}^{rs}}{\sigma_{k-1}^{2}}\bigg{)}\nabla_{1}\mathcal{W}_{ij}\nabla_{1}\mathcal{W}_{rs}$
$\displaystyle+\frac{\big{(}2\nabla_{1}\alpha_{0}e^{2kv}+4k\alpha_{0}e^{2kv}\nabla_{1}v\big{)}\sigma_{k-1}^{ii}\nabla_{1}\mathcal{W}_{ii}}{(n-2)^{k}\sigma_{k-1}^{2}}$
$\displaystyle-\frac{\nabla_{1}\alpha_{0}e^{2kv}4k\nabla_{1}v+\alpha_{0}e^{2kv}4k^{2}\big{(}\nabla_{1}v\big{)}^{2}+\alpha_{0}e^{2kv}2k\nabla_{11}v+\nabla_{11}\alpha_{0}e^{2kv}}{(n-2)^{k}\sigma_{k-1}}$
$\displaystyle\geq$ $\displaystyle-\frac{\nabla_{11}\alpha
e^{2v}+4\nabla_{1}\alpha e^{2v}\nabla_{1}v+4\alpha
e^{2v}\big{(}\nabla_{1}v\big{)}^{2}+2\alpha e^{2v}\nabla_{11}v}{n-2}.$

By the concavity of $\sigma_{k-1}^{\frac{1}{k-1}}$,

$\bigg{(}\frac{\sigma_{k-1}^{ij,rs}}{\sigma_{k-1}}+\Big{(}\frac{1}{k-1}-1\Big{)}\frac{\sigma_{k-1}^{ij}\sigma_{k-1}^{rs}}{\sigma_{k-1}^{2}}\bigg{)}\nabla_{1}\mathcal{W}_{ij}\nabla_{1}\mathcal{W}_{rs}\leq
0.$

Consequently,

$\displaystyle\frac{\alpha_{0}e^{2kv}}{(n-2)^{k}\sigma_{k-1}}\bigg{(}\frac{\sigma_{k-1}^{ij,rs}}{\sigma_{k-1}}-\frac{2\sigma_{k-1}^{ij}\sigma_{k-1}^{rs}}{\sigma_{k-1}^{2}}\bigg{)}\nabla_{1}\mathcal{W}_{ij}\nabla_{1}\mathcal{W}_{rs}$
$\displaystyle+\frac{\big{(}2\nabla_{1}\alpha_{0}e^{2kv}+4k\alpha_{0}e^{2kv}\nabla_{1}v\big{)}\sigma_{k-1}^{ii}\nabla_{1}\mathcal{W}_{ii}}{(n-2)^{k}\sigma_{k-1}^{2}}$
$\displaystyle\leq$
$\displaystyle-\frac{\alpha_{0}e^{2kv}}{(n-2)^{k}\sigma_{k-1}}\frac{k}{k-1}\frac{\sigma_{k-1}^{ij}\sigma_{k-1}^{rs}}{\sigma_{k-1}^{2}}\nabla_{1}\mathcal{W}_{ij}\nabla_{1}\mathcal{W}_{rs}$
$\displaystyle+\frac{\big{(}2\nabla_{1}\alpha_{0}e^{2kv}+4k\alpha_{0}e^{2kv}\nabla_{1}v\big{)}\sigma_{k-1}^{ii}\nabla_{1}\mathcal{W}_{ii}}{(n-2)^{k}\sigma_{k-1}^{2}}$
$\displaystyle\leq$
$\displaystyle\frac{(k-1)e^{2kv}\big{(}\nabla_{1}\alpha_{0}+2k\alpha_{0}\nabla_{1}v\big{)}^{2}}{k(n-2)^{k}\alpha_{0}\sigma_{k-1}}.$

By (4.2) and Newton-Maclaurin inequality,

$\displaystyle\frac{\alpha_{0}}{\sigma_{k-1}}=\frac{(n-2)^{k}}{e^{2kv}}\bigg{(}\frac{\sigma_{k}}{\sigma_{k-1}}+\frac{\alpha
e^{2v}}{n-2}\bigg{)}\leq C\Big{(}\sigma_{k-1}^{\frac{1}{k-1}}+1\Big{)}\leq
C(\sigma_{1}+1)\leq C\nabla_{11}v.$

Also,

$|\nabla_{ij}v|\leq C\nabla_{11}v.$

Therefore, (4.21) reduces to

$F^{ii}\nabla_{11}\mathcal{W}_{ii}\geq-C\nabla_{11}^{2}v.$

Combining this inequality with (4.20), (4.17) and (4.18) yields,

(4.22) $\displaystyle
F^{ii}\bigg{(}\nabla_{ii}\eta-(1+\epsilon)\big{(}\nabla_{i}\eta\big{)}^{2}\bigg{)}+\frac{1}{n-2}\bigg{(}\Delta\eta-(1+\epsilon)|\nabla\eta|^{2}\bigg{)}\sum
F^{ii}$ $\displaystyle\leq$ $\displaystyle
C\bigg{(}|\nabla\eta|+\nabla_{11}v+\frac{1}{\zeta^{2}}\bigg{)}\sum F^{ii}.$

Let

$\eta(w)=\bigg{(}1-\frac{3w}{4M}\bigg{)}^{-1/2},$

where

$w=\frac{|\nabla v|^{2}}{2},\quad M=\sup\limits_{\\{\zeta>0\\}}w.$

Choosing $\epsilon=\frac{1}{2}$, we have

$\eta^{\prime\prime}-(1+\epsilon)\eta^{\prime
2}=\frac{9}{64M^{2}}\bigg{(}1-\frac{3w}{4M}\bigg{)}^{-\frac{5}{2}}\Bigg{(}3-(1+\epsilon)\bigg{(}1-\frac{3w}{4M}\bigg{)}^{-\frac{1}{2}}\Bigg{)}\geq
0.$

Therefore,

$\displaystyle
F^{ii}\bigg{(}\nabla_{ii}\eta-(1+\epsilon)\big{(}\nabla_{i}\eta\big{)}^{2}\bigg{)}+\frac{1}{n-2}\bigg{(}\Delta\eta-(1+\epsilon)|\nabla\eta|^{2}\bigg{)}\sum
F^{ii}$ $\displaystyle\geq$
$\displaystyle\eta^{\prime}F^{ii}\big{(}\nabla_{iil}v\nabla_{l}v+\big{(}\nabla_{il}v\big{)}^{2}\big{)}+\frac{1}{n-2}\eta^{\prime}\bigg{(}\Delta\nabla_{l}v\nabla_{l}v+\sum\limits_{jl}\big{(}\nabla_{jl}v\big{)}^{2}\bigg{)}\sum
F^{ii}.$

By (4.9) and interchanging order of covariant derivatives,

$F^{ii}\nabla_{iil}v\nabla_{l}v+\frac{1}{n-2}\Delta\nabla_{l}v\nabla_{l}v\sum
F^{ii}\geq-C\nabla_{11}v\sum F^{ii}.$

We note that

$\frac{3}{8M}\leq\eta^{\prime}\leq\frac{3}{M},\quad|\nabla\eta|\leq
C\nabla_{11}v.$

Hence (4.22) reduces to

$\big{(}\nabla_{11}v\big{)}^{2}\leq
C\bigg{(}\nabla_{11}v+\frac{1}{\zeta^{2}}\bigg{)}.$

Consequently,

(4.23) $\zeta\nabla_{11}v\leq C.$

Finally, choosing $\zeta$ to be an appropriate cutoff function with support in
$B_{r}$, we obtain the second order interior estimate for equation (4.1).

######  Theorem 4.3.

Let $v\in C^{4}(B_{r})$ be a $(k-1)$-admissible solution of (4.1) in a
geodesic ball $B_{r}\subset M$ of radius $r>0$. Then

$\sup\limits_{B_{\frac{r}{2}}}|\nabla^{2}v|\leq C,$

where $C$ depends on $r^{-1}$, $n$, $k$, $\|v\|_{C^{1}(B_{r})}$,
$\|\alpha\|_{C^{2}(B_{r})}$, $\|\alpha_{0}\|_{C^{2}{(B_{r})}}$ and
$\inf\limits_{B_{r}}\alpha_{0}$.

If we choose $\zeta\equiv 1$ in (4.23), we obtain the global second order
estimate.

######  Theorem 4.4.

Let $v\in C^{4}(M)\cap C^{2}(\overline{M})$ be a $(k-1)$-admissible solution
of (4.1). Then

$\max\limits_{\overline{M}}|\nabla^{2}v|\leq C,$

where $C$ depends on $n$, $k$, $\|v\|_{C^{1}(\overline{M})}$,
$\max\limits_{\partial M}|\nabla^{2}v|$, $\|\alpha\|_{C^{2}(M)}$,
$\|\alpha_{0}\|_{C^{2}{(M)}}$ and $\inf\limits_{M}\alpha_{0}$.

###  4.3. Boundary estimates for second derivatives

######  Theorem 4.5.

Let $v\in C^{3}(\overline{M})$ be a $(k-1)$-admissible solution of (4.1)
satisfying

$v=\varphi\quad\text{ on }\partial M,$

where $\varphi\in C^{\infty}(\overline{M})$. Then we have the estimate

$|\nabla^{2}v|\leq C\quad\text{ on }\partial M,$

where $C>0$ is a constant depending on $n$, $k$,
$\|v\|_{C^{1}(\overline{M})}$, $\|\varphi\|_{C^{3}(\overline{M})}$,
$\|\alpha\|_{C^{1}(\overline{M})}$, $\|\alpha_{0}\|_{C^{1}(\overline{M})}$ and
$\min\limits_{\overline{M}}\alpha_{0}$.

###### Proof.

Let $x_{0}$ be an arbitrary point on $\partial M$. Choose a smooth local
orthonormal frame $e_{1},\ldots,e_{n}$ around $x_{0}$ such that $e_{n}$ is the
interior unit normal vector field to $\partial M$ along $\partial M$. We
obtain the pure tangential second derivative bound immediately

(4.24) $\big{|}\nabla_{st}v(x_{0})\big{|}\leq C,\quad\forall\,s,t<n.$

For the tangential-normal second derivative estimate

(4.25) $\big{|}\nabla_{sn}v(x_{0})\big{|}\leq C,\quad\forall\,s<n,$

we use the function

$\Theta=\beta\bigg{(}\rho-\frac{N}{2}\rho^{2}\bigg{)},$

where $\beta$ and $N$ are positive constants, and $\rho(x)$ is the distance
from $x$ to $\partial M$. Set

$M_{\delta}=\\{x\in M\,|\,\rho(x)<\delta\\}.$

Choose $\delta>0$ sufficiently small such that $\rho(x)$ is smooth in
$M_{\delta}$, on which, we have

$|\nabla\rho|=1,\quad|\nabla^{2}\rho|\leq C.$

We consider the linearized operator $\mathcal{L}$ which is locally defined by

$\mathcal{L}\eta=F^{ij}\big{(}\nabla_{ij}\eta-2\nabla_{i}v\nabla_{j}\eta\big{)}+\bigg{(}\frac{\Delta\eta}{n-2}+2\nabla_{m}v\nabla_{m}\eta\bigg{)}\sum
F^{ii}.$

It can be seen that

$\displaystyle\mathcal{L}\Theta\leq\beta\bigg{(}C(1+N\delta)-\frac{N}{n-2}\bigg{)}\sum
F^{ii}.$

Choosing

$N\geq(n-2)(2C+1),\quad\text{ and }\quad\delta\leq\frac{1}{N},$

we have

(4.26) $\mathcal{L}\Theta\leq-\beta\sum F^{ii}\quad\text{ in }M_{\delta}$

and

(4.27) $\Theta\geq\frac{\beta}{2}\rho\quad\text{ in }M_{\delta}.$

We note that

$\nabla_{ij}(\nabla_{k}v)=\nabla_{kij}v+\Gamma_{ik}^{l}\nabla_{jl}v+\Gamma_{jk}^{l}\nabla_{il}v+\nabla_{k}\Gamma_{ij}^{l}\nabla_{l}v.$

Therefore, for $t<n$,

$\displaystyle\mathcal{L}\big{(}\nabla_{t}(v-\varphi)\big{)}$ $\displaystyle=$
$\displaystyle
F^{ij}\Big{(}\nabla_{tij}(v-\varphi)+\Gamma_{it}^{l}\nabla_{jl}(v-\varphi)+\Gamma_{jt}^{l}\nabla_{il}(v-\varphi)+\nabla_{t}\Gamma_{ij}^{l}\nabla_{l}(v-\varphi)\Big{)}$
$\displaystyle-2F^{ij}\nabla_{i}v\Big{(}\nabla_{jt}(v-\varphi)+\Gamma_{jt}^{l}\nabla_{l}(v-\varphi)\Big{)}$
$\displaystyle+\Bigg{(}\frac{\nabla_{t}\Delta(v-\varphi)+2\sum_{j}\Gamma_{jt}^{l}\nabla_{jl}(v-\varphi)+\sum_{j}\nabla_{t}\Gamma_{jj}^{l}\nabla_{l}(v-\varphi)}{n-2}$
$\displaystyle+2\nabla_{m}v\Big{(}\nabla_{mt}(v-\varphi)+\Gamma_{mt}^{l}\nabla_{l}(v-\varphi)\Big{)}\Bigg{)}\sum
F^{ii}.$

By (4.9), we have

$\displaystyle
F^{ij}\Big{(}\nabla_{tij}v-2\nabla_{i}v\nabla_{tj}v\Big{)}+\Bigg{(}\frac{\nabla_{t}\Delta
v}{n-2}+\nabla_{t}|\nabla v|^{2}\Bigg{)}\sum F^{ii}$ $\displaystyle\leq$
$\displaystyle C\sum F^{ii}+\frac{C\alpha_{0}}{\sigma_{k-1}}\leq C\sum
F^{ii}+C\big{(}\sigma_{1}+1\big{)}\leq C\sum F^{ii}+C\sigma_{1}.$

Consequently, for $t<n$,

(4.28) $\displaystyle\mathcal{L}\big{(}\nabla_{t}(v-\varphi)\big{)}$
$\displaystyle\leq$ $\displaystyle C\sum
F^{ii}+C\sigma_{1}+C\sqrt{\sum_{jl}(\nabla_{jl}v)^{2}}\sum F^{ii}$
$\displaystyle\leq$ $\displaystyle
C\Bigg{(}1+\sqrt{\sum_{jl}(\nabla_{jl}v)^{2}}\Bigg{)}\sum F^{ii}.$

Similarly, we may verify that

(4.29) $\displaystyle\mathcal{L}\big{(}-\nabla_{t}(v-\varphi)\big{)}\leq
C\Bigg{(}1+\sqrt{\sum_{jl}(\nabla_{jl}v)^{2}}\Bigg{)}\sum F^{ii}.$

We also have

(4.30)
$\displaystyle\mathcal{L}\Bigg{(}\sum\limits_{s<n}\big{|}\nabla_{s}(v-\varphi)\big{|}^{2}\Bigg{)}=\sum\limits_{s<n}2\nabla_{s}(v-\varphi)\mathcal{L}\big{(}\nabla_{s}(v-\varphi)\big{)}$
$\displaystyle+\sum\limits_{s<n}2F^{ij}\nabla_{i}\big{(}\nabla_{s}(v-\varphi)\big{)}\nabla_{j}\big{(}\nabla_{s}(v-\varphi)\big{)}+\sum\limits_{s<n}\frac{2\Big{|}\nabla\big{(}\nabla_{s}(v-\varphi)\big{)}\Big{|}^{2}}{n-2}\sum
F^{ii}$ $\displaystyle\geq$
$\displaystyle-C\Bigg{(}1+\sqrt{\sum_{jl}(\nabla_{jl}v)^{2}}\Bigg{)}\sum
F^{ii}+0+\frac{\sum\limits_{l}\sum\limits_{s<n}(\nabla_{ls}v)^{2}}{n-2}\sum
F^{ii}-C\sum F^{ii}.$

We observe that if $\nabla_{nn}v\leq 0$, by the fact that

$0\leq\sigma_{1}\big{(}\mathcal{W}[v]\big{)}\leq\frac{2(n-1)}{n-2}\Delta v+C,$

we have

$0\geq\nabla_{nn}v\geq-\sum\limits_{s<n}\nabla_{ss}v-C.$

If $\nabla_{nn}v\geq 0$, by the concavity of $F$ with respect to
$\\{\mathcal{W}_{ij}\\}$, that is,

$\sum\limits_{ij}F^{ij}\big{(}\delta_{ij}-\mathcal{W}_{ij}\big{)}\geq
F\big{(}I\big{)}-F\big{(}\mathcal{W}[v]\big{)}\geq-C,$

we have

$\displaystyle\sum F^{ii}+C\geq\sum\limits_{ij}F^{ij}\mathcal{W}_{ij}$
$\displaystyle\geq$
$\displaystyle\sum\limits_{ij}F^{ij}\bigg{(}\nabla_{ij}v+\frac{\Delta
v}{n-2}\delta_{ij}\bigg{)}-C\sum F^{ii}$ $\displaystyle=$
$\displaystyle\sum\limits_{i}\sum\limits_{s<n}F^{is}\nabla_{is}v+\sum\limits_{s<n}F^{sn}\nabla_{sn}v+F^{nn}\nabla_{nn}v+\frac{\Delta
v}{n-2}\sum F^{ii}-C\sum F^{ii}$ $\displaystyle\geq$
$\displaystyle-C\sqrt{\sum_{l}\sum_{s<n}(\nabla_{ls}v)^{2}}\sum
F^{ii}+\frac{\Delta v}{n-2}\sum F^{ii}-C\sum F^{ii},$

which implies that

$0\leq\nabla_{nn}v\leq C+C\sqrt{\sum_{l}\sum_{s<n}(\nabla_{ls}v)^{2}}.$

By the above observation, (4.26), (4.28), (4.29) and (4.30), we have

$\displaystyle\mathcal{L}\Bigg{(}\Theta-\sum\limits_{s<n}\big{|}\nabla_{s}(v-\varphi)\big{|}^{2}\pm\nabla_{t}(v-\varphi)\Bigg{)}$
$\displaystyle\leq$ $\displaystyle-\beta\sum
F^{ii}+C\sqrt{\sum\limits_{l}\sum\limits_{s<n}(\nabla_{ls}v)^{2}}\sum
F^{ii}-\frac{\sum\limits_{l}\sum\limits_{s<n}(\nabla_{ls}v)^{2}}{n-2}\sum
F^{ii}+C\sum F^{ii}.$

If

$\sqrt{\sum\limits_{l}\sum\limits_{s<n}(\nabla_{ls}v)^{2}}\leq C(n-2),$

then we obtain

$\mathcal{L}\Bigg{(}\Theta-\sum\limits_{s<n}\big{|}\nabla_{s}(v-\varphi)\big{|}^{2}\pm\nabla_{t}(v-\varphi)\Bigg{)}\leq-\beta\sum
F^{ii}+C^{2}(n-2)\sum F^{ii}+C\sum F^{ii}.$

If

$\sqrt{\sum\limits_{l}\sum\limits_{s<n}(\nabla_{ls}v)^{2}}\geq C(n-2),$

then we obtain

$\mathcal{L}\Bigg{(}\Theta-\sum\limits_{s<n}\big{|}\nabla_{s}(v-\varphi)\big{|}^{2}\pm\nabla_{t}(v-\varphi)\Bigg{)}\leq-\beta\sum
F^{ii}+C\sum F^{ii}.$

Thus, choosing $\beta$ sufficiently large, we arrive at

$\mathcal{L}\Bigg{(}\Theta-\sum\limits_{s<n}\big{|}\nabla_{s}(v-\varphi)\big{|}^{2}\pm\nabla_{t}(v-\varphi)\Bigg{)}\leq
0.$

Also,

$\Theta-\sum\limits_{s<n}\big{|}\nabla_{s}(v-\varphi)\big{|}^{2}\pm\nabla_{t}(v-\varphi)=0\quad\text{
on }\partial M,$

and by (4.27), we may choose $\beta$ further large depending on $\delta$ such
that

$\Theta-\sum\limits_{s<n}\big{|}\nabla_{s}(v-\varphi)\big{|}^{2}\pm\nabla_{t}(v-\varphi)\geq\frac{\beta}{2}\delta-C\geq
0\quad\text{ on }\\{x\in M\,|\,\rho(x)=\delta\\}.$

By the maximum principle,

$\Theta-\sum\limits_{s<n}\big{|}\nabla_{s}(v-\varphi)\big{|}^{2}\pm\nabla_{t}(v-\varphi)\geq
0\quad\text{ in }M_{\delta}.$

Hence we obtain (4.25).

Next, we derive the double normal second derivative estimate

(4.31) $\nabla_{nn}v(x_{0})\leq C.$

By (4.1), that is,

$\sigma_{k}\big{(}\mathcal{W}[v]\big{)}+\frac{\alpha
e^{2v}}{n-2}\sigma_{k-1}\big{(}\mathcal{W}[v]\big{)}=\frac{\alpha_{0}e^{2kv}}{(n-2)^{k}},$

if $\nabla_{nn}v(x_{0})$ is sufficiently large, in view of (4.24) and (4.25),
we have at $x_{0}$,

$\sigma_{k}\bigg{(}\frac{\Delta
v}{n-2}\delta_{ij}-C\delta_{ij}\bigg{)}-C\sigma_{k-1}\bigg{(}\frac{\Delta
v}{n-2}\delta_{ij}+\nabla_{nn}v\delta_{ij}+C\delta_{ij}\bigg{)}\leq C,$

which further implies that

$\bigg{(}\frac{1}{n-2}\nabla_{nn}v-C\bigg{)}^{k}\sigma_{k}(I)-C\bigg{(}\frac{n-1}{n-2}\nabla_{nn}v+C\bigg{)}^{k-1}\sigma_{k-1}(I)\leq
C.$

Hence we obtain (4.31).

∎

######  Remark 4.6.

When deriving the estimates, we can also directly obtain

$\frac{1}{\sigma_{k-1}\big{(}\mathcal{W}[v]\big{)}}\leq C$

by (4.1). In fact, we have

$\displaystyle\inf\alpha_{0}\leq$
$\displaystyle\alpha_{0}(x)=\sigma_{k}\bigg{(}(n-2)e^{-2v}\mathcal{W}[v]\bigg{)}+\alpha(x)\sigma_{k-1}\bigg{(}(n-2)e^{-2v}\mathcal{W}[v]\bigg{)}$
$\displaystyle\leq$
$\displaystyle\sigma_{k-1}^{\frac{k}{k-1}}\bigg{(}(n-2)e^{-2v}\mathcal{W}[v]\bigg{)}+\sup\alpha\sigma_{k-1}\bigg{(}(n-2)e^{-2v}\mathcal{W}[v]\bigg{)}.$

Next, we shall use continuity method and degree theory to prove the existence
of a smooth $(k-1)$-admissible solution to the Dirichlet problem

(4.32) $\left\\{\begin{aligned}
F\big{(}\mathcal{W}[v]\big{)}=&-\frac{\alpha(x)}{n-2}e^{2v}\quad&\text{ in
}\Omega,\\\ v=&\varphi\quad&\text{ on }\partial\Omega,\end{aligned}\right.$

where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with smooth compact
boundary which are composed of closed hypersurfaces.

###  4.4. Existence of subsolutions

In this subsection, we construct a subsolution synthesizing the ideas of Guan
[6] and Guan [8]. We note that there exist sufficiently large $R>r>0$ such
that $\Omega\subset B_{r}(0)$, and

$w(x)=-\ln\big{(}R^{2}-|x|^{2}\big{)}-C$

satisfies

$\mathcal{W}[w]\geq\nabla^{2}w+\frac{\Delta
w}{n-2}g\geq\frac{4R^{2}}{(n-2)\big{(}R^{2}-|x|^{2}\big{)}^{2}}g.$

It follows that

$(n-2)e^{-2w}\mathcal{W}[w]\geq e^{2C}4R^{2}g>e^{2C}2R^{2}g.$

Choosing $C>0$ sufficiently large such that

(4.33)
$\displaystyle\frac{\sigma_{k}}{\sigma_{k-1}}\bigg{(}(n-2)e^{-2w}\mathcal{W}[w]\bigg{)}-\frac{\alpha_{0}(x)}{\sigma_{k-1}\bigg{(}(n-2)e^{-2w}\mathcal{W}[w]\bigg{)}}$
$\displaystyle\geq$
$\displaystyle\frac{\sigma_{k}}{\sigma_{k-1}}\bigg{(}e^{2C}2R^{2}g\bigg{)}-\frac{\alpha_{0}(x)}{\sigma_{k-1}\bigg{(}e^{2C}2R^{2}g\bigg{)}}>-\alpha(x)\quad\text{
on }\overline{\Omega}$

and

$w\leq-\ln\big{(}R^{2}-r^{2}\big{)}-C<\varphi\quad\text{ on }\partial\Omega.$

Also, we consider

$\eta=2\ln\delta-\ln(\rho+\delta^{2})+\varphi.$

For $\delta>0$ sufficiently small,

$\mathcal{W}[\eta]\geq\nabla^{2}\eta+\frac{\Delta\eta}{n-2}g\geq\frac{1}{2(n-2)(\rho+\delta^{2})^{2}}g\quad\text{
on }\\{0\leq\rho\leq\delta\\}.$

Consequently,

$(n-2)e^{-2\eta}\mathcal{W}[\eta]\geq\frac{e^{-2\varphi}}{2\delta^{4}}g>\frac{e^{-2\varphi}}{4\delta^{4}}g\quad\text{
on }\\{0\leq\rho\leq\delta\\}.$

Choosing $\delta>0$ further small such that

(4.34)
$\displaystyle\frac{\sigma_{k}}{\sigma_{k-1}}\bigg{(}(n-2)e^{-2\eta}\mathcal{W}[\eta]\bigg{)}-\frac{\alpha_{0}(x)}{\sigma_{k-1}\bigg{(}(n-2)e^{-2\eta}\mathcal{W}[\eta]\bigg{)}}$
$\displaystyle\geq$
$\displaystyle\frac{\sigma_{k}}{\sigma_{k-1}}\bigg{(}\frac{e^{-2\varphi}}{4\delta^{4}}g\bigg{)}-\frac{\alpha_{0}(x)}{\sigma_{k-1}\bigg{(}\frac{e^{-2\varphi}}{4\delta^{4}}g\bigg{)}}>-\alpha(x)\quad\text{
on }\\{0\leq\rho\leq\delta\\},$

and

$\eta<2\ln\delta-\ln\delta+\varphi<w\quad\text{ on }\rho=\delta.$

Now we need a lemma from Guan [8].

######  Lemma 4.7.

For all $\epsilon>0$, there is an even function $h(t)\in
C^{\infty}(\mathbb{R})$ such that

  1. (1)

$h(t)\geq|t|$ for all $t\in\mathbb{R}$, and $h(t)=|t|$ for all
$|t|\geq\epsilon$;

  2. (2)

$\big{|}h^{\prime}(t)\big{|}\leq 1$ and $h^{\prime\prime}(t)\geq 0$ for all
$t\in\mathbb{R}$, and $h^{\prime}(t)\geq 0$ for all $t\geq 0$.

Define

$\underline{v}=\left\\{\begin{aligned}
&\frac{1}{2}(\eta+w)+\frac{1}{2}h(\eta-w)\quad\text{ on
}\\{0\leq\rho\leq\delta\\}\\\ &w\quad\text{ on
}\\{\rho\geq\delta\\}\end{aligned}\right..$

Direct calculation shows that on $\\{0\leq\rho\leq\delta\\}$,

$\nabla_{i}\underline{v}=\frac{1}{2}\Big{(}\nabla_{i}\eta+\nabla_{i}w\Big{)}+\frac{1}{2}h^{\prime}(\eta-w)\Big{(}\nabla_{i}\eta-\nabla_{i}w\Big{)},$
$\displaystyle\nabla_{ij}\underline{v}=$
$\displaystyle\frac{1}{2}\Big{(}\nabla_{ij}\eta+\nabla_{ij}w\Big{)}+\frac{1}{2}h^{\prime}\Big{(}\nabla_{ij}\eta-\nabla_{ij}w\Big{)}$
$\displaystyle+\frac{1}{2}h^{\prime\prime}\Big{(}\nabla_{i}\eta-\nabla_{i}w\Big{)}\Big{(}\nabla_{j}\eta-\nabla_{j}w\Big{)}.$

It follows that within $\\{0\leq\rho\leq\delta\\}$,

$\displaystyle\mathcal{W}[\underline{v}]\geq$
$\displaystyle\nabla^{2}\underline{v}+\frac{\Delta\underline{v}}{n-2}g$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\Big{(}\nabla^{2}\eta+\nabla^{2}w\Big{)}+\frac{1}{2}h^{\prime}\Big{(}\nabla^{2}\eta-\nabla^{2}w\Big{)}$
$\displaystyle+\frac{1}{2(n-2)}\Big{(}\Delta\eta+\Delta
w\Big{)}g+\frac{1}{2(n-2)}h^{\prime}\Big{(}\Delta\eta-\Delta w\Big{)}g$
$\displaystyle=$
$\displaystyle\frac{1+h^{\prime}}{2}\bigg{(}\nabla^{2}\eta+\frac{\Delta\eta}{n-2}g\bigg{)}+\frac{1-h^{\prime}}{2}\bigg{(}\nabla^{2}w+\frac{\Delta
w}{n-2}g\bigg{)}$ $\displaystyle\geq$
$\displaystyle\frac{1+h^{\prime}}{2}\frac{1}{2(n-2)(\rho+\delta^{2})^{2}}g+\frac{1-h^{\prime}}{2}\frac{4R^{2}}{(n-2)\big{(}R^{2}-|x|^{2}\big{)}^{2}}g.$

Since on $\\{0\leq\rho\leq\delta\\}$,

$\underline{v}\leq\sup\\{\eta,w\\}+\frac{\epsilon}{2},$

we have

$\displaystyle(n-2)e^{-2\underline{v}}\mathcal{W}[\underline{v}]$
$\displaystyle\geq$
$\displaystyle(n-2)e^{-2\underline{v}}\frac{1+h^{\prime}}{2}\frac{1}{2(n-2)(\rho+\delta^{2})^{2}}g+(n-2)e^{-2\underline{v}}\frac{1-h^{\prime}}{2}\frac{4R^{2}}{(n-2)\big{(}R^{2}-|x|^{2}\big{)}^{2}}g$
$\displaystyle\geq$ $\displaystyle\left\\{\begin{aligned}
(n-2)e^{-2\eta-\epsilon}\frac{1}{2}\frac{1}{2(n-2)(\rho+\delta^{2})^{2}}g\quad\text{
on }\\{0\leq\rho\leq\delta\\}\cap\\{\eta\geq w\\}\\\
(n-2)e^{-2w-\epsilon}\frac{1}{2}\frac{4R^{2}}{(n-2)\big{(}R^{2}-|x|^{2}\big{)}^{2}}g\quad\text{
on }\\{0\leq\rho\leq\delta\\}\cap\\{\eta\leq w\\}\end{aligned}\right.$
$\displaystyle\geq$ $\displaystyle\left\\{\begin{aligned}
\frac{1}{4\delta^{4}}e^{-2\varphi-\epsilon}g\quad\text{ on
}\\{0\leq\rho\leq\delta\\}\cap\\{\eta\geq w\\}\\\
e^{2C-\epsilon}2R^{2}g\quad\text{ on }\\{0\leq\rho\leq\delta\\}\cap\\{\eta\leq
w\\}\end{aligned}\right..$

In view of (4.33) and (4.34), we may choose $\epsilon>0$ sufficiently small
such that

$\frac{\sigma_{k}}{\sigma_{k-1}}\bigg{(}(n-2)e^{-2\underline{v}}\mathcal{W}[\underline{v}]\bigg{)}-\frac{\alpha_{0}(x)}{\sigma_{k-1}\bigg{(}(n-2)e^{-2\underline{v}}\mathcal{W}[\underline{v}]\bigg{)}}>-\alpha(x)\quad\text{
on }\overline{\Omega}.$

###  4.5. Preliminary estimates

For a $(k-1)$-admissible function $v\in C^{2}(\overline{\Omega})$ with

$v\geq\underline{v}\text{ in }\overline{\Omega}\quad\text{ and
}v=\varphi\text{ on }\partial\Omega,$

we change $v$ back into $u$ by $v=\frac{2}{n-2}\ln u$ to see that

$\sigma_{1}\big{(}W[u]\big{)}=2(n-1)\Delta u\geq 0.$

Let $h$ be the solution to

$\left\\{\begin{aligned} \Delta h=&0\quad&\text{ in }\Omega,\\\
h=&e^{\frac{n-2}{2}\varphi}\quad&\text{ on
}\partial\Omega.\end{aligned}\right.$

By the maximum principle, $u\leq h$ in $\Omega$ and hence

$v\leq\overline{v}:=\frac{2}{n-2}\ln h\quad\text{ in }\Omega.$

Then we have

$\nabla_{\nu}\underline{v}\leq\nabla_{\nu}v\leq\nabla_{\nu}\overline{v}\quad\text{
on }\partial\Omega,$

where $\nu$ is the interior unit normal to $\partial\Omega$.

###  4.6. Existence of solutions

Denote

$\displaystyle G[v]:=$ $\displaystyle G(\nabla^{2}v,\nabla
v,v)=F\big{(}\mathcal{W}[v]\big{)},$ $\displaystyle G^{ij}[v]:=$
$\displaystyle G^{ij}(\nabla^{2}v,\nabla v,v)=\frac{\partial
G}{\partial\nabla_{ij}v},$ $\displaystyle G^{i}[v]:=$ $\displaystyle
G^{i}(\nabla^{2}v,\nabla v,v)=\frac{\partial G}{\partial\nabla_{i}v},$
$\displaystyle G_{v}[v]:=$ $\displaystyle G_{v}(\nabla^{2}v,\nabla
v,v)=\frac{\partial G}{\partial v}.$

Let $C_{0}$ be a positive constant such that

(4.35)
$G[\underline{v}]=G(\nabla^{2}\underline{v},\nabla\underline{v},\underline{v})>-C_{0}\quad\text{
in }\overline{\Omega}.$

For $t\in[0,1]$, we consider the following two equations (similar construction
of the equations can be found in Su [14]).

(4.36) $\left\\{\begin{aligned} G[v]=&(1-t)G[\underline{v}]-tC_{0}\quad&\text{
in }\Omega,\\\ v=&\varphi\quad&\text{ on }\partial\Omega.\end{aligned}\right.$
(4.37) $\left\\{\begin{aligned}
G[v]=&-(1-t)C_{0}-t\frac{\alpha(x)}{n-2}e^{2v}\quad&\text{ in }\Omega,\\\
v=&\varphi\quad&\text{ on }\partial\Omega.\end{aligned}\right.$

######  Remark 4.8.

For $x\in\overline{\Omega}$ and a $C^{2}$ function $v$ which is
$(k-1)$-admissible near $x$, we have

$G_{v}(x)=-\frac{\alpha_{0}(x)e^{2kv}2k}{(n-2)^{k}}\frac{1}{\sigma_{k-1}\big{(}\mathcal{W}[v]\big{)}}<0$

if $\alpha_{0}(x)>0$.

######  Lemma 4.9.

For $t\in[0,1]$, let $\underline{V}$ and $v$ be any $(k-1)$-admissible
subsolution and solution of (4.36). Then $v\geq\underline{V}$ in $\Omega$. In
particular, (4.36) has at most one $(k-1)$-admissible solution.

###### Proof.

Suppose that $\underline{V}-v$ achieves a positive maximum at
$x_{0}\in\Omega$, at which we have

$\underline{V}(x_{0})>v(x_{0}),\quad\nabla\underline{V}(x_{0})=\nabla
v(x_{0}),\quad\nabla^{2}\underline{V}(x_{0})\leq\nabla^{2}v(x_{0}).$

It follows that $\mathcal{W}[\underline{V}](x_{0})\leq\mathcal{W}[v](x_{0})$,
and therefore

$\displaystyle F\big{(}\mathcal{W}[\underline{V}]\big{)}(x_{0})=$
$\displaystyle\frac{\sigma_{k}}{\sigma_{k-1}}\big{(}\mathcal{W}[\underline{V}]\big{)}(x_{0})-\frac{\alpha_{0}(x_{0})e^{2k\underline{V}(x_{0})}}{(n-2)^{k}}\frac{1}{\sigma_{k-1}\big{(}\mathcal{W}[\underline{V}]\big{)}(x_{0})}$
$\displaystyle\leq$
$\displaystyle\frac{\sigma_{k}}{\sigma_{k-1}}\big{(}\mathcal{W}[v]\big{)}(x_{0})-\frac{\alpha_{0}(x_{0})e^{2k\underline{V}(x_{0})}}{(n-2)^{k}}\frac{1}{\sigma_{k-1}\big{(}\mathcal{W}[v]\big{)}(x_{0})}$
$\displaystyle<$
$\displaystyle\frac{\sigma_{k}}{\sigma_{k-1}}\big{(}\mathcal{W}[v]\big{)}(x_{0})-\frac{\alpha_{0}(x_{0})e^{2kv(x_{0})}}{(n-2)^{k}}\frac{1}{\sigma_{k-1}\big{(}\mathcal{W}[v]\big{)}(x_{0})}$
$\displaystyle=$ $\displaystyle F\big{(}\mathcal{W}[v]\big{)}(x_{0}).$

But

$F\big{(}\mathcal{W}[\underline{V}]\big{)}(x_{0})\geq(1-t)G[\underline{v}](x_{0})-tC_{0}=F\big{(}\mathcal{W}[v]\big{)}(x_{0}),$

which is a contradiction. ∎

######  Theorem 4.10.

For $t\in[0,1]$, (4.36) has a unique $(k-1)$-admissible solution
$v\geq\underline{v}$.

###### Proof.

By Lemma 4.9, we immediately obtain uniqueness. For existence, we use standard
continuity method. By assumption (4.35), $\underline{v}$ is a subsolution of
(4.36). We note that the $C^{2}$ estimate for $(k-1)$-admissible solution
$v\geq\underline{v}$ of (4.36) implies uniform ellipticity of this equation
and hence gives $C^{2,\alpha}$ estimate by Evans-Krylov theory [3, 11]

(4.38) $\|v\|_{C^{2,\alpha}(\overline{\Omega})}\leq C,$

where $C$ is independent of $t$. Denote

$C_{0}^{2,\alpha}(\overline{\Omega})=\\{w\in
C^{2,\alpha}(\overline{\Omega})\,|\,w=0\text{ on }\partial\Omega\\},$

and consider the open subset of $C_{0}^{2,\alpha}(\overline{\Omega})$

$\mathcal{U}=\\{w\in
C_{0}^{2,\alpha}(\overline{\Omega})\,|\,\underline{v}+w\text{ is
}(k-1)\text{-admissible in }\overline{\Omega}\\}.$

Define $\mathcal{L}:\mathcal{U}\times[0,1]\rightarrow
C^{\alpha}(\overline{\Omega})$,

$\mathcal{L}(w,t)=G[\underline{v}+w]-(1-t)G[\underline{v}]+tC_{0},$

and set

$\mathcal{S}=\\{t\in[0,1]\,|\,\mathcal{L}(w,t)=0\text{ has a solution }w\text{
in }\mathcal{U}\\}.$

$\mathcal{S}\neq\emptyset$ since $\mathcal{L}(0,0)=0$.

$\mathcal{S}$ is open in $[0,1]$. In fact, for any $t_{0}\in\mathcal{S}$,
there exists $w_{0}\in\mathcal{U}$ such that $\mathcal{L}(w_{0},t_{0})=0$.
Note that the Fréchet derivative of $\mathcal{L}$ with respect to $w$ at
$(w_{0},t_{0})$ is a linear elliptic operator from
$C^{2,\alpha}_{0}(\overline{\Omega})$ to $C^{\alpha}(\overline{\Omega})$,

$\mathcal{L}_{w}\big{|}_{(w_{0},t_{0})}(h)=G^{ij}[\underline{v}+w_{0}]\nabla_{ij}h+G^{i}[\underline{v}+w_{0}]\nabla_{i}h+G_{v}[\underline{v}+w_{0}]h.$

Remark 4.8 implies $\mathcal{L}_{w}\big{|}_{(w_{0},t_{0})}$ is invertible.
Thus a neighborhood of $t_{0}$ is also contained in $\mathcal{S}$ by implicit
function theorem.

$\mathcal{S}$ is closed in $[0,1]$. In fact, let $t_{i}$ be a sequence in
$\mathcal{S}$ converging to $t_{0}\in[0,1]$ and $w_{i}\in\mathcal{U}$ be the
unique solution to $\mathcal{L}(w_{i},t_{i})=0$. Lemma 4.9 implies $w_{i}\geq
0$. By (4.38), $v_{i}:=\underline{v}+w_{i}$ is a bounded sequence in
$C^{2,\alpha}(\overline{\Omega})$, which possesses a subsequence converging to
a $(k-1)$-admissible solution $v_{0}$ of (4.36). Since
$w_{0}=v_{0}-\underline{v}\in\mathcal{U}$ and $\mathcal{L}(w_{0},t_{0})=0$, we
know that $t_{0}\in\mathcal{S}$. ∎

Next we may assume that $\underline{v}$ is not a solution of (4.32).

######  Lemma 4.11.

If $v\geq\underline{v}$ is a $(k-1)$-admissible solution of (4.37), then
$v>\underline{v}$ in $\Omega$ and $\nabla_{\nu}(v-\underline{v})>0$ on
$\partial\Omega$.

###### Proof.

We note that

$G[\underline{v}]-G[v]\geq-t\frac{\alpha(x)}{n-2}(e^{2\underline{v}}-e^{2v}),$

and

$\displaystyle
G[\underline{v}]-G[v]=F\big{(}\mathcal{W}[\underline{v}]\big{)}-F\big{(}\mathcal{W}[v]\big{)}$
$\displaystyle=$
$\displaystyle\int_{0}^{1}\frac{d}{ds}\Bigg{(}\frac{\sigma_{k}}{\sigma_{k-1}}\Big{(}(1-s)\mathcal{W}[v]+s\mathcal{W}[\underline{v}]\Big{)}-\frac{\alpha_{0}(x)e^{2k((1-s)v+s\underline{v})}}{(n-2)^{k}\sigma_{k-1}\Big{(}(1-s)\mathcal{W}[v]+s\mathcal{W}[\underline{v}]\Big{)}}\Bigg{)}ds$
$\displaystyle=$
$\displaystyle\underbrace{\int_{0}^{1}\Bigg{(}\Big{(}\frac{\sigma_{k}}{\sigma_{k-1}}\Big{)}^{ij}+\frac{\alpha_{0}(x)e^{2k((1-s)v+s\underline{v})}\sigma_{k-1}^{ij}}{(n-2)^{k}\sigma_{k-1}^{2}}\Bigg{)}\Big{(}(1-s)\mathcal{W}[v]+s\mathcal{W}[\underline{v}]\Big{)}ds}_{\text{denoted
by }a_{ij}}$
$\displaystyle\cdot\Big{(}\mathcal{W}_{ij}[\underline{v}]-\mathcal{W}_{ij}[v]\Big{)}-\underbrace{\int_{0}^{1}\frac{\alpha_{0}(x)e^{2k((1-s)v+s\underline{v})}2k}{(n-2)^{k}\sigma_{k-1}\Big{(}(1-s)\mathcal{W}[v]+s\mathcal{W}[\underline{v}]\Big{)}}ds}_{\text{denoted
by }c}\cdot(\underline{v}-v)$ $\displaystyle=$ $\displaystyle
a_{ij}\bigg{(}\nabla_{ij}(\underline{v}-v)+\frac{\Delta(\underline{v}-v)}{n-2}\delta_{ij}\bigg{)}-a_{ij}\nabla_{i}\underline{v}\nabla_{j}(\underline{v}-v)-a_{ij}\nabla_{j}v\nabla_{i}(\underline{v}-v)$
$\displaystyle+\nabla(\underline{v}+v)\cdot\nabla(\underline{v}-v)\sum
a_{ii}-c(\underline{v}-v).$

Applying the Maximum Principle and Lemma H (see p. 212 of [4]) we proved the
lemma. ∎

######  Theorem 4.12.

For any $t\in[0,1]$, there is a $(k-1)$-admissible solution
$v\geq\underline{v}$ to Dirichlet problem (4.37).

###### Proof.

We obtain by classical Schauder theory the $C^{4,\alpha}$ estimate

(4.39) $\|v\|_{C^{4,\alpha}(\overline{\Omega})}<C_{4}.$

Also, we have

(4.40)
$\text{dist}\big{(}\lambda(\mathcal{W}[v]),\partial\Gamma_{k-1}\big{)}>c_{2}>0\quad\text{
in }\overline{\Omega},$

where $C_{4}$ and $c_{2}$ are independent of $t$. Denote

$C_{0}^{4,\alpha}(\overline{\Omega})=\big{\\{}w\in
C^{4,\alpha}(\overline{\Omega})\,|\,w=0\text{ on }\partial\Omega\big{\\}}$

and the open bounded subset of $C_{0}^{4,\alpha}(\overline{\Omega})$

$\mathcal{O}=\Bigg{\\{}w\in
C_{0}^{4,\alpha}(\overline{\Omega})\left|\footnotesize\begin{aligned}
&w>0\text{ in }\Omega,\nabla_{\nu}w>0\text{ on
}\partial\Omega,\|w{\|}_{C^{4,\alpha}(\overline{\Omega})}<C_{4}+\|\underline{v}\|_{C^{4,\alpha}(\overline{\Omega})},\\\
&\underline{v}+w\text{ is }(k-1)\text{-admissible in
}\overline{\Omega},\text{dist}\Big{(}\lambda\big{(}\mathcal{W}[\underline{v}+w]\big{)},\partial\Gamma_{k-1}\Big{)}>c_{2}\text{
in }\overline{\Omega}\end{aligned}\right.\Bigg{\\}}.$

Define a map $\mathcal{M}_{t}(w):\mathcal{O}\times[0,1]\rightarrow
C^{2,\alpha}(\overline{\Omega})$,

$\mathcal{M}_{t}(w)=G[\underline{v}+w]+(1-t)C_{0}+t\frac{\alpha(x)}{n-2}e^{2(\underline{v}+w)}.$

By Theorem 4.10 and Lemma 4.9, there is a unique $(k-1)$-admissible solution
$v^{0}$ of (4.36) at $t=1$, which is also the unique $(k-1)$-admissible
solution of (4.37) at $t=0$. By Lemma 4.9, $w^{0}=v^{0}-\underline{v}\geq 0$
in $\Omega$. Consequently, $w^{0}>0$ in $\Omega$ and $\nabla_{\nu}w^{0}>0$ on
$\partial\Omega$ by Lemma 4.11. We also note that $\underline{v}+w^{0}$
satisfies (4.39) and (4.40). Thus, $w^{0}\in\mathcal{O}$. By Lemma 4.11,
(4.39) and (4.40), $\mathcal{M}_{t}(w)=0$ has no solution on
$\partial\mathcal{O}$ for any $t\in[0,1]$. Since $\mathcal{M}_{t}$ is
uniformly elliptic on $\mathcal{O}$ independent of $t$, the degree of
$\mathcal{M}_{t}$ on $\mathcal{O}$ at $0$ can be defined independent of $t$.
This degree is nonzero at $t=0$. In fact, since $\mathcal{M}_{0}(w)=0$ has a
unique solution $w^{0}\in\mathcal{O}$. The Fréchet derivative of
$\mathcal{M}_{0}$ with respect to $w$ at $w^{0}$ is a linear elliptic operator
from $C^{4,\alpha}_{0}(\overline{\Omega})$ to
$C^{2,\alpha}(\overline{\Omega})$,

$\mathcal{M}_{0,w}|_{w^{0}}(h)=G^{ij}[v^{0}]\nabla_{ij}h+G^{i}[v^{0}]\nabla_{i}h+G_{v}[v^{0}]h.$

By Remark 4.8, $G_{v}[v^{0}]<0$ in $\overline{\Omega}$. Thus,
$\mathcal{M}_{0,w}|_{w^{0}}$ is invertible. The degree theory in [12] implies
that the degree at $t=0$ is nonzero, which implies that (4.37) has at least
one $(k-1)$-admissible solution $v\geq\underline{v}$ for any $t\in[0,1]$. ∎

##  5\. Fully nonlinear Loewner-Nirenberg problem for general equations

In this section, we discuss fully nonlinear Loewner-Nirenberg problem related
to general equation (1.10).

###  5.1. The case of smooth compact boundary consisting of closed
hypersurfaces

We first give the definition of supersolutions and subsolutions to equation
(1.10).

######  Definition 5.1.

A function $0<\underline{u}\in C^{2}(\Omega)$ is a subsolution of (1.10) in
$\Omega$ if

$\displaystyle\lambda\big{(}W[\underline{u}]\big{)}\in\Gamma_{k-1}\text{ and
}$
$\displaystyle\sigma_{k}\bigg{(}\frac{2}{n-2}\underline{u}^{-\frac{n+2}{n-2}}W[\underline{u}]\bigg{)}$
$\displaystyle+\alpha(x)\sigma_{k-1}\bigg{(}\frac{2}{n-2}\underline{u}^{-\frac{n+2}{n-2}}W[\underline{u}]\bigg{)}\geq\alpha_{0}(x)\quad\text{
in }\Omega.$

A function $0<\overline{u}\in C^{2}(\Omega)$ is a supersolution of (1.10) in
$\Omega$ if

$\displaystyle\text{either
}\lambda\big{(}W[\overline{u}]\big{)}\notin\Gamma_{k-1}\text{ or }$
$\displaystyle\sigma_{k}\bigg{(}\frac{2}{n-2}\overline{u}^{-\frac{n+2}{n-2}}W[\overline{u}]\bigg{)}$
$\displaystyle+\alpha(x)\sigma_{k-1}\bigg{(}\frac{2}{n-2}\overline{u}^{-\frac{n+2}{n-2}}W[\overline{u}]\bigg{)}\leq\alpha_{0}(x)\quad\text{
in }\Omega.$

######  Proposition 5.2.

If $\underline{u}$ is a subsolution of (1.10) satisfying

(5.1)
$\lambda\big{(}W[\underline{u}]\big{)}\in\overline{\Gamma}_{k}\quad\text{ in
}\Omega,$

so does $c\underline{u}$ for any constant $0<c<1$. If $\overline{u}$ is a
supersolution of (1.10) satisfying

$\lambda\big{(}W[\overline{u}]\big{)}\notin\Gamma_{k-1}\quad\text{everywhere
in }\Omega,$

then $c\overline{u}$ is a supersolution of (1.10) for any $c>0$.

###### Proof.

When $\lambda\big{(}W[\underline{u}]\big{)}\in\overline{\Gamma}_{k}$,

$\displaystyle\frac{\sigma_{k}}{\sigma_{k-1}}\bigg{(}\frac{2}{n-2}c^{-\frac{4}{n-2}}\underline{u}^{-\frac{n+2}{n-2}}W[\underline{u}]\bigg{)}-\frac{\alpha_{0}(x)}{\sigma_{k-1}\bigg{(}\frac{2}{n-2}c^{-\frac{4}{n-2}}\underline{u}^{-\frac{n+2}{n-2}}W[\underline{u}]\bigg{)}}$
$\displaystyle\geq$
$\displaystyle\frac{\sigma_{k}}{\sigma_{k-1}}\bigg{(}\frac{2}{n-2}\underline{u}^{-\frac{n+2}{n-2}}W[\underline{u}]\bigg{)}-\frac{\alpha_{0}(x)}{\sigma_{k-1}\bigg{(}\frac{2}{n-2}\underline{u}^{-\frac{n+2}{n-2}}W[\underline{u}]\bigg{)}}\geq-\alpha(x).$

Hence we proved the first statement. The second statement is straightforward
by definition. ∎

Then we observe the following maximum principle of (1.10).

######  Theorem 5.3.

Let $M$ be a smooth compact manifold with boundary. Suppose that
$\underline{u}$ is a subsolution of

$\displaystyle\sigma_{k}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}+\alpha(x)\sigma_{k-1}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}=\alpha_{0}(x)$

satisfying

$\lambda\big{(}W[\underline{u}]\big{)}\in\Gamma_{k}\quad\text{ in }\Omega,$

and $\overline{u}$ is a supersolution of

$\displaystyle\sigma_{k}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}+\beta(x)\sigma_{k-1}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}=\beta_{0}(x),$

with $\alpha(x)\leq\beta(x)$ and $\alpha_{0}(x)\geq\beta_{0}(x)$ on $M$. If
$\underline{u}\leq\overline{u}$ on $\partial M$, then
$\underline{u}\leq\overline{u}$ on $M$.

###### Proof.

Suppose that $\underline{u}>\overline{u}$ somewhere in the interior of $M$.
Let $C$ be the maximum of $\frac{\underline{u}}{\overline{u}}$ on $M$, which
is attained at $x_{0}$ in the interior of $M$. Since $C>1$, by Proposition
5.2, we know that $\underline{u}_{1}:=\frac{\underline{u}}{C}$ satisfies

$\frac{\sigma_{k}}{\sigma_{k-1}}\bigg{(}\frac{2}{n-2}\underline{u}_{1}^{-\frac{n+2}{n-2}}W[\underline{u}_{1}]\bigg{)}-\frac{\alpha_{0}(x)}{\sigma_{k-1}\bigg{(}\frac{2}{n-2}\underline{u}_{1}^{-\frac{n+2}{n-2}}W[\underline{u}_{1}]\bigg{)}}>-\alpha(x).$

On the other hand, since $\underline{u}_{1}(x_{0})=\overline{u}(x_{0})$ while
$\underline{u}_{1}\leq\overline{u}$ near $x_{0}$, thus at $x_{0}$

$\nabla\underline{u}_{1}(x_{0})=\nabla\overline{u}(x_{0}),\quad\nabla^{2}\underline{u}_{1}(x_{0})\leq\nabla^{2}\overline{u}(x_{0}).$

It follows that $W[\underline{u}_{1}](x_{0})\leq W[\overline{u}](x_{0})$,
which further implies

$\displaystyle-\alpha(x_{0})<\frac{\sigma_{k}}{\sigma_{k-1}}\bigg{(}\frac{2}{n-2}\underline{u}_{1}^{-\frac{n+2}{n-2}}W[\underline{u}_{1}]\bigg{)}(x_{0})-\frac{\alpha_{0}(x_{0})}{\sigma_{k-1}\bigg{(}\frac{2}{n-2}\underline{u}_{1}^{-\frac{n+2}{n-2}}W[\underline{u}_{1}]\bigg{)}(x_{0})}$
$\displaystyle\leq$
$\displaystyle\frac{\sigma_{k}}{\sigma_{k-1}}\bigg{(}\frac{2}{n-2}\overline{u}^{-\frac{n+2}{n-2}}W[\overline{u}]\bigg{)}(x_{0})-\frac{\beta_{0}(x_{0})}{\sigma_{k-1}\bigg{(}\frac{2}{n-2}\overline{u}^{-\frac{n+2}{n-2}}W[\overline{u}]\bigg{)}(x_{0})}\leq-\beta(x_{0}),$

which is a contradiction. ∎

Similar to Proposition 2.4, we observe the following fact.

######  Proposition 5.4.

For any real number $\alpha$, any positive constant $\alpha_{0}$, any fixed
$s>0$, there exists a positive constant $c$ such that

$u(x)=c\Big{(}s^{2}-{|x-x_{0}|}^{2}\Big{)}^{1-\frac{n}{2}}\quad\text{ in
}B_{s}(x_{0}),$

and

$v(x)=c\Big{(}{|x-x_{0}|}^{2}-s^{2}\Big{)}^{1-\frac{n}{2}}\quad\text{ in
}\mathbb{R}^{n}\setminus\overline{B_{s}(x_{0})},$

are admissible solutions of

$\displaystyle\sigma_{k}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}+\alpha\sigma_{k-1}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}=\alpha_{0},$

which approach to $\infty$ on $\partial B_{s}(x_{0})$.

###### Proof.

Consider

$u(x)=c\Big{(}s^{2}-{|x-x_{0}|}^{2}\Big{)}^{1-\frac{n}{2}}.$

We have

$W_{ij}[u]=2(n-1)(n-2)c\Big{(}s^{2}-|x-x_{0}|^{2}\Big{)}^{-\frac{n}{2}-1}s^{2}\delta_{ij},$

and therefore

$\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W_{ij}[u]=4(n-1)c^{-\frac{4}{n-2}}s^{2}\delta_{ij}.$

Hence

$\displaystyle\sigma_{k}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}+\alpha\sigma_{k-1}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}$
$\displaystyle=$
$\displaystyle\bigg{(}4(n-1)c^{-\frac{4}{n-2}}s^{2}\bigg{)}^{k}\sigma_{k}\big{(}I\big{)}+\alpha\bigg{(}4(n-1)c^{-\frac{4}{n-2}}s^{2}\bigg{)}^{k-1}\sigma_{k-1}\big{(}I\big{)}.$

Since $k\geq 2$, by intermediate value theorem, there exists a positive
constant $c$ such that the above quantity equals $\alpha_{0}$. ∎

######  Remark 5.5.

The relation between $c$ and $s$ in $u(x)$ and $v(x)$ in Proposition 5.4 can
be obtained once $\alpha$ and $\alpha_{0}$ are fixed. In fact, let
$\tilde{\mu}$ be a positive root of

$\mu^{k}\sigma_{k}\big{(}I\big{)}+\alpha\mu^{k-1}\sigma_{k-1}\big{(}I\big{)}=\alpha_{0}.$

Then choose

$4(n-1)c^{-\frac{4}{n-2}}s^{2}=\tilde{\mu}$

to obtain

$c=\bigg{(}\frac{4(n-1)}{\tilde{\mu}}\bigg{)}^{\frac{n-2}{4}}s^{\frac{n-2}{2}}.$

Proof of Theorem 1.4 If $\Omega$ is a bounded domain, for any positive integer
$m$, we consider the Dirichlet problem

(5.2) $\left\\{\begin{aligned}
F\big{(}\mathcal{W}[v]\big{)}=&-\frac{\alpha(x)}{n-2}e^{2v}\quad&\text{ in
}\Omega,\\\ v=&m\quad&\text{ on }\partial\Omega.\end{aligned}\right.$

By Theorem 4.12, there exists a smooth $(k-1)$-admissible solution $v_{m}$ to
(5.2). Let $\mu_{0}$ be the smallest positive root of the equation

$\mu^{k}+\overline{\alpha}\mu^{k-1}=\underline{\alpha}_{0},$

where

$\overline{\alpha}=\max\Big{\\{}\sup\limits_{\Omega}\alpha(x),0.01\Big{\\}}>0,\quad\underline{\alpha}_{0}=\inf\limits_{\Omega}\alpha_{0}(x)>0.$

By Loewner and Nirenberg [13], there exists a positive solution $\overline{u}$
to

(5.3) $\left\\{\begin{aligned}
\sigma_{1}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}=&\mu_{0}\quad\text{
in }\Omega,\\\ u=&\infty\quad\text{ on }\partial\Omega.\end{aligned}\right.$

Since $u_{m}=e^{\frac{n-2}{2}v_{m}}$ satisfies

$\displaystyle\underline{\alpha}_{0}\leq\alpha_{0}(x)=$
$\displaystyle\sigma_{k}\bigg{(}\frac{2}{n-2}u_{m}^{-\frac{n+2}{n-2}}W[u_{m}]\bigg{)}+\alpha(x)\sigma_{k-1}\bigg{(}\frac{2}{n-2}u_{m}^{-\frac{n+2}{n-2}}W[u_{m}]\bigg{)}$
$\displaystyle\leq$
$\displaystyle\sigma_{1}^{k}\bigg{(}\frac{2}{n-2}u_{m}^{-\frac{n+2}{n-2}}W[u_{m}]\bigg{)}+\overline{\alpha}\sigma_{1}^{k-1}\bigg{(}\frac{2}{n-2}u_{m}^{-\frac{n+2}{n-2}}W[u_{m}]\bigg{)},$

we have

$\sigma_{1}\bigg{(}\frac{2}{n-2}u_{m}^{-\frac{n+2}{n-2}}W[u_{m}]\bigg{)}\geq\mu_{0}.$

By the maximum principle Theorem 2.3, we have

$u_{m}\leq\overline{u}\quad\text{ in }\Omega\quad\text{ for any }m.$

Denote

$\underline{\alpha}=\inf\limits_{\Omega}\alpha(x),\quad\overline{\alpha}_{0}=\sup\limits_{\Omega}\alpha_{0}(x).$

Let $t>0$ be sufficiently large such that $\overline{\Omega}\subset B_{t}(0)$.
Consider the equation

(5.4)
$\sigma_{k}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}+\underline{\alpha}\sigma_{k-1}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}=\overline{\alpha}_{0}.$

For any fixed $s>t>0$, by Proposition 5.4 and Remark 5.5, there exists a
smooth solution

$\underline{u}=\bigg{(}\frac{4(n-1)}{\tilde{\mu}}\bigg{)}^{\frac{n-2}{4}}s^{\frac{n-2}{2}}\Big{(}s^{2}-|x|^{2}\Big{)}^{1-\frac{n}{2}}$

to equation (5.4) in $B_{s}(0)$, which is a subsolution of (1.10). We may
choose $s$ sufficiently large so that

$\displaystyle\underline{u}\leq\bigg{(}\frac{4(n-1)}{\tilde{\mu}}\bigg{)}^{\frac{n-2}{4}}s^{\frac{n-2}{2}}\Big{(}s^{2}-t^{2}\Big{)}^{1-\frac{n}{2}}\leq
e^{\frac{n-2}{2}}\leq e^{\frac{(n-2)m}{2}}=u_{m}\quad\text{ on
}\partial\Omega.$

By Theorem 5.3 we know that

$u_{m}\geq\underline{u}\quad\text{ in }\Omega\quad\text{ for any }m.$

By interior $C^{2}$ estimate established in last section and Evans-Krylov
theory [3, 11], we obtain a smooth positive $(k-1)$-admissible solution $u$ to
equation (1.10) which tends to $\infty$ on $\partial\Omega$.

When $\Omega$ is unbounded, we may assume without loss of generality that
$0\notin\overline{\Omega}$. Let $B_{s}(0)$ be a fixed ball such that
$\overline{\Omega}\subset\mathbb{R}^{n}\setminus\overline{B_{s}(0)}$. By
Proposition 5.4, there exists a smooth solution

$\underline{u}=c_{0}\Big{(}|x|^{2}-s^{2}\Big{)}^{1-\frac{n}{2}}\quad\text{ for
some positive constant }c_{0}$

to equation (5.4) in $\mathbb{R}^{n}\setminus\overline{B_{s}(0)}$, which is a
subsolution of (1.10).

For any $R>\max\limits_{\partial\Omega}\underline{u}$ large enough such that
$\partial\Omega\subset B_{R}(0)$, by Theorem 4.12, there exists a smooth
positive $(k-1)$-admissible solution $u_{R}$ to the Dirichlet problem

$\left\\{\begin{aligned}
&\sigma_{k}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}+\alpha\sigma_{k-1}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}=\alpha_{0}\quad\text{
in }B_{R}(0)\cap\Omega,\\\ &u=R\quad\text{ on }\partial\Omega,\\\
&u=\underline{u}\quad\text{ on }\partial B_{R}(0).\end{aligned}\right.$

By Theorem 5.3, we know that

$u_{R}\geq\underline{u}\quad\text{ in }B_{R}(0)\cap\Omega.$

By Lemma 2.10, we are able to find a positive solution $\overline{u}\in
C^{\infty}(\Omega)$ of (5.3) which decays to $0$ at $\infty$ with the decay
rate

$|x|^{n-2}\overline{u}(x)\rightarrow c\quad\text{ as }|x|\rightarrow\infty$

for some positive constant $c$. Comparing the decay rate as
$|x|\rightarrow\infty$, we have on $\partial B_{R}(0)$ with $R$ sufficiently
large,

$u_{R}(x)=\underline{u}(x)\leq\frac{2c_{0}}{c}\overline{u}(x).$

Meanwhile, we have

$u_{R}(x)=R<\infty=\frac{2c_{0}}{c}\overline{u}(x)\quad\text{ on
}\partial\Omega.$

By Proposition 2.2 and Theorem 2.3, we obtain

$u_{R}(x)\leq\max\Big{\\{}\frac{2c_{0}}{c},1\Big{\\}}\overline{u}(x)\quad\text{
in }B_{R}(0)\cap\Omega.$

Applying the interior estimates in section 4, Evans-Krylov interior estimates
[3, 11] and a standard diagonal process, we obtain a smooth positive
$(k-1)$-admissible solution $u$ of (1.10) in $\Omega$ which tends to $\infty$
on $\partial\Omega$. ∎

###  5.2. Maximal solution on general domain in $\mathbb{R}^{n}$ when
$\alpha\leq 0$

We first observe the fact that when $\alpha(x)\leq 0$, any $(k-1)$-admissible
solution to equation (1.10) must be $k$-admissible. Then we are able to apply
the maximum principle Theorem 5.3. As in section 3, we can define the maximal
solution of (1.10).

Assume throughout this subsection that $\Omega\subset\mathbb{R}^{n}$ is a
domain with smooth compact boundary $\partial\Omega$. Moreover, we assume that

$\alpha(x)\leq 0\text{ in
}\Omega,\quad\underline{\alpha}_{0}=\inf\limits_{\Omega}\alpha_{0}(x)>0,\quad\underline{\alpha}=\inf\limits_{\Omega}\alpha(x)>-\infty,\quad\overline{\alpha}_{0}=\sup\limits_{\Omega}\alpha_{0}(x)<\infty.$

######  Definition 5.6.

A smooth positive $k$-admissible solution $u_{\Omega}$ of (1.10) is said to be
maximal in $\Omega$, if it is greater than or equal to any smooth positive
$k$-admissible solution of (1.10) in $\Omega$.

Let $\Omega_{(1)}\Subset\Omega_{(2)}\Subset\ldots$ be an increasing sequence
of bounded subdomains of $\Omega$ with smooth compact boundaries
$\partial\Omega_{(j)}$ which are closed hypersurfaces such that
$\Omega=\cup\Omega_{(j)}$. By Theorem 1.4, we can find a smooth positive
$k$-admissible solution $u_{(j)}$ of (1.10) in $\Omega_{(j)}$ which tends to
$\infty$ on $\partial\Omega_{(j)}$. By Theorem 5.3, we see that
$\\{u_{(j)}\\}$ is a monotone decreasing sequence of positive functions. It
follows that $u_{(j)}$ converges to a nonnegative function $u_{\Omega}$ in
$\Omega$.

######  Lemma 5.7.

Either $u_{\Omega}>0$ in $\Omega$ or $u_{\Omega}\equiv 0$ in $\Omega$.

###### Proof.

Suppose that $u_{\Omega}\not\equiv 0$. Then we must have $u_{\Omega}>c$ on
some $B(x_{0},r_{0})\subset\Omega$ for some constant $c>0$. By Proposition 5.4
and Remark 5.5, we may choose $0<r_{1}<r_{0}$ such that

$v=\bigg{(}\frac{4(n-1)}{\tilde{\mu}}\bigg{)}^{\frac{n-2}{4}}r_{1}^{\frac{n-2}{2}}\Big{(}{|x-x_{0}|}^{2}-r_{1}^{2}\Big{)}^{1-\frac{n}{2}}\quad\text{
in }\mathbb{R}^{n}\setminus\overline{B_{r_{1}}(x_{0})}$

is a solution of

$\sigma_{k}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}+\underline{\alpha}\sigma_{k-1}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}=\overline{\alpha}_{0}$

satisfying

$\bigg{(}\frac{4(n-1)}{\tilde{\mu}}\bigg{)}^{\frac{n-2}{4}}r_{1}^{\frac{n-2}{2}}\Big{(}r_{0}^{2}-r_{1}^{2}\Big{)}^{1-\frac{n}{2}}=c.$

By Theorem 5.3, we know that $u_{(j)}\geq v$ in $\Omega_{(j)}\setminus
B(x_{0},r_{0})$ for any $j$. Hence $u_{\Omega}\geq v>0$ in $\Omega\setminus
B(x_{0},r_{0})$. We thus have $u_{\Omega}>0$ in $\Omega$. ∎

######  Remark 5.8.

  1. (1)

In view of Lemma 5.7, the interior regularity in section 4, and Evans-Krylov
theory [3, 11], we know that $u_{\Omega}$ is smooth.

  2. (2)

If there exists a positive $k$-admissible subsolution of (1.10) in $\Omega$
(for example, when $\mathbb{R}^{n}\setminus\overline{\Omega}\neq\emptyset$),
then $u_{\Omega}>0$ in $\Omega$.

  3. (3)

When $u_{\Omega}>0$ in $\Omega$, $u_{\Omega}$ is the maximal solution of
equation (1.10) in $\Omega$. In fact, if $w$ is any smooth positive
$k$-admissible solution of (1.10) in $\Omega$, then $u_{(j)}\geq w$ in
$\Omega_{(j)}$ by Theorem 5.3. Hence $u_{\Omega}\geq w$ in $\Omega$.

######  Definition 5.9.

We call a compact subset $\Gamma\subset\partial\Omega$ regular, if

$u_{\Omega}(x)\rightarrow\infty\quad\text{ as }x\rightarrow\Gamma.$

Now we consider a portion $\Gamma\subset\partial\Omega$ which is a smooth
compact non-self-intersecting surface of codimension $m$. Meanwhile, we assume
that $\partial\Omega\setminus\Gamma$ is smooth compact.

######  Theorem 5.10.

Let $\Omega$ be a domain in $\mathbb{R}^{n}$ and $\Gamma$ be a compact subset
of $\partial\Omega$ such that $\partial\Omega\setminus\Gamma$ is also compact.
Suppose that $u_{\Omega}\not\equiv 0$. Then $\Gamma$ is regular if there
exists an open neighborhood $U$ of $\Gamma$ and a $C^{2}$ positive
$k$-admissible subsolution $\phi(x)$ of (1.10) defined in $\Omega\cap U$ which
tends to $\infty$ as $x\rightarrow\Gamma$.

###### Proof.

Without loss of generality we may assume that $\overline{U}$ is compact,

$\overline{U}\cap(\partial\Omega\setminus\Gamma)=\emptyset\text{ and
}0<\phi\in C^{2}(\overline{U}\cap\Omega).$

For $j$ sufficiently large, we have

$\partial(U\cap\Omega_{(j)})=\partial U\cup(U\cap\partial\Omega_{(j)}).$

Since $u_{\Omega}$ is positive in $\Omega$ and $\partial U\subset\Omega$ is
compact, we have $\inf\limits_{\partial U}u_{\Omega}:=m>0$. Denote
$M:=\sup\limits_{\partial U}\phi(x)>0$ and $A:=\max\\{\frac{M}{m},1\\}$. We
note that $u_{(j)}\geq u_{\Omega}\geq m\geq\frac{\phi}{A}$ on $\partial U$,
and $u_{(j)}=\infty>\frac{\phi}{A}$ on $U\cap\partial\Omega_{(j)}$. By
Proposition 5.2, $\frac{\phi}{A}$ is again a positive $k$-admissible
subsolution of (1.10). By Theorem 5.3, we arrive at
$u_{(j)}\geq\frac{\phi}{A}$ in $U\cap\Omega_{(j)}$. Consequently, we obtain
$u_{\Omega}\geq\frac{\phi}{A}$ in $U\cap\Omega$. ∎

In what follows we assume that $u_{\Omega}\not\equiv 0$. Similar to section 3,
let $\rho(x)$ be the distance of $x$ to $\Gamma$, and

$\Gamma_{\rho_{0}}=\big{\\{}x\in\Omega\,\big{|}\,\rho(x)<\rho_{0}\big{\\}}.$

For $\rho_{0}$ sufficiently small, define on
$\Gamma_{\rho_{0}}\setminus\Gamma$

$\phi(x)=c\rho^{1-\frac{n}{2}}(x),$

where $c>0$ is a constant to be determined later.

Denote

$v_{m}=\big{(}\underbrace{n-m,\ldots,n-m}_{n-m+1},\underbrace{2-m,\ldots,2-m}_{m-1}\big{)}.$

As $\rho\rightarrow 0$,

$\lambda\bigg{(}\frac{2}{n-2}\phi^{-\frac{n+2}{n-2}}W[\phi]\bigg{)}\rightarrow
c^{-\frac{4}{n-2}}v_{m}.$

If we assume that $v_{m}\in\Gamma_{k}$, then there exists $c>0$ sufficiently
small such that

$\displaystyle\sigma_{k}\Big{(}c^{-\frac{4}{n-2}}v_{m}\Big{)}+\underline{\alpha}\sigma_{k-1}\Big{(}c^{-\frac{4}{n-2}}v_{m}\Big{)}>\overline{\alpha}_{0}.$

As $\rho_{0}>0$ sufficiently small, within $\\{0<\rho<\rho_{0}\\}$,

$\sigma_{k}\bigg{(}\frac{2}{n-2}{\phi}^{-\frac{n+2}{n-2}}W[\phi]\bigg{)}+\underline{\alpha}\sigma_{k-1}\bigg{(}\frac{2}{n-2}{\phi}^{-\frac{n+2}{n-2}}W[\phi]\bigg{)}>\overline{\alpha}_{0}.$

Therefore, $\Gamma$ is regular by Theorem 5.10.

If $v_{m}\in\mathbb{R}^{n}\setminus\overline{\Gamma}_{k}$, as in section 3, we
consider

$\psi=\psi_{c,d}=(c\rho^{-a}+d)^{b},$

where $a$, $b$, $c$ and $d$ are positive constants. Direct calculation shows
that

(5.5)
$\frac{1}{abc}\rho^{a+2}(c\rho^{-a}+d)^{\frac{4b}{n-2}+1}\psi^{-\frac{n+2}{n-2}}W[\psi]=A_{m}+B_{ab}(\zeta)+\mathcal{O}(\rho),$

where $\zeta$, $A_{m}$ and $B_{ab}(\zeta)$ are as in (3.2). Choose $a$
sufficiently small, and then choose an appropriate $b$ such that $ab$ is
slightly larger than $\frac{n}{2}-1$ such that

$\lambda\big{(}A_{m}+B_{ab}(\zeta)\big{)}\notin\overline{\Gamma}_{k}\quad\text{for
all }0<\zeta<1.$

Then choose $0<\rho_{0}<1$ further small such that within $0<\rho<\rho_{0}$
and for all $0<\zeta<1$,

$\lambda\big{(}A_{m}+B_{ab}(\zeta)+\mathcal{O}(\rho)\big{)}\notin\overline{\Gamma}_{k},$

which implies that

$\lambda\big{(}W[\psi]\big{)}\notin\overline{\Gamma}_{k}\text{ in
}\\{0<\rho<\rho_{0}\\}\text{ for all }c>0\text{ and }d>0.$

Therefore, we have either $\lambda\big{(}W[\psi]\big{)}\notin\Gamma_{k-1}$,
or, if $\lambda\big{(}W[\psi]\big{)}\in\Gamma_{k-1}$, then

$\sigma_{k}\bigg{(}\frac{2}{n-2}\psi^{-\frac{n+2}{n-2}}W[\psi]\bigg{)}+\alpha\sigma_{k-1}\bigg{(}\frac{2}{n-2}\psi^{-\frac{n+2}{n-2}}W[\psi]\bigg{)}\leq
0<\alpha_{0}.$

That is, $\psi$ is a supersolution of (1.10). Now, choosing $d$ sufficiently
large depending on $\rho_{0}$ such that on $\rho=\rho_{0}$ we have $\psi\geq
d^{b}\geq u_{\Omega}$. By Proposition 5.4, Remark 5.5 and Theorem 5.3, the
solution of the equation

$\sigma_{k}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}=\underline{\alpha}_{0}$

in the ball $B_{\rho(x)}(x)$ can serve as an upper bound for $u_{\Omega}$.
Thus, we can deduce that

$u_{\Omega}\leq\bigg{(}\frac{4(n-1)}{\tilde{\mu}}\bigg{)}^{\frac{n-2}{4}}\rho^{1-\frac{n}{2}}\leq
c^{b}\rho^{-ab}<\psi$

within $0<\rho\leq\delta$, where $\delta>0$ is a sufficiently small constant
depending on $c$. By Theorem 5.3 we have $u_{\Omega}\leq\psi$ in
$\delta<\rho<\rho_{0}$. Letting $\delta\rightarrow 0$, we arrive at
$u_{\Omega}\leq\psi$ in $0<\rho<\rho_{0}$. Next letting $c\rightarrow 0$ we
deduce that $u_{\Omega}\leq d^{b}$ in $0<\rho<\rho_{0}$. Hence $\Gamma$ is not
regular according to Definition 5.9.

## References

  * [1]
  * [2] P. Aviles, A study of the singularities of solutions of a class of nonlinear elliptic partial equations, Comm. Partial Differential Equations 7 (1982), 609–643.
  * [3] L. C. Evans, Classical solutions of fully nonlinear, convex, second order elliptic equations, Communications on Pure and Applied Mathematics 35 (1982), 333-363.
  * [4] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209–243.
  * [5] M. González, Y. Li and L. Nguyen, Existence and uniqueness to a fully nonlinear version of the Loewner-Nirenberg problem, Commun. Math. Stat. 6 (2018), 269–288.
  * [6] B. Guan, Complete conformal metrics of negative Ricci curvature on compact manifolds with boundary, Int. Math. Res. Not. IMRN 2008, Art. ID rnn pp 105, 25.
  * [7] B. Guan, Addendum to: Complete conformal metrics of negative Ricci curvature on compact manifolds with boundary, Int. Math. Res. Not. IMRN 22 (2009), 4354–4355.
  * [8] P. Guan, The extremal function associated to intrinsic norms, Ann. of Math. 156 (2002), 197–211.
  * [9] P. Guan and X. Zhang, A class of curvature type equations, Pure Appl. Math. Q. 17 (2021), 865–907.
  * [10] M. Gursky, J. Streets and M. Warren, Existence of complete conformal metrics of negative Ricci curvature on manifolds with boundary, Calc. Var. Partial Differential Equations 41 (2011), 21–43.
  * [11] N. V. Krylov, Boundedly nonhomogeneous elliptic and parabolic equations in a domain, Izvestiya Rossiiskoi Akademii Nauk, Seriya Matematicheskaya 47 (1983), 75–108.
  * [12] Y. Li, Degree theory for second order nonlinear elliptic operators and its applications, Comm. Partial Differential Equations 14 (1989), 1541–1578.
  * [13] C. Loewner and L. Nirenberg, Partial differential equations invariant under conformal or projective transformations, 245–272. Contributions to Analysis. [A collection of papers dedicated to Lipman Bers]. New York: Academic Press, 1974.
  * [14] C. Su, Starshaped locally convex hypersurfaces with prescribed curvature and boundary, J. Geom. Anal. 26 (2016), 1730–1753.
  * [15] L. Véron, Singularités éliminables d’équations elliptiques non linéaires, J. Differential Equations 41 (1981), 87–95.