# Fully nonlinear elliptic equations on Hermitian manifolds for symmetric
functions
of partial Laplacians

Mathew George  Department of Mathematics, Ohio State University, Columbus, OH
43210, USA<EMAIL_ADDRESS>,  Bo Guan  Department of
Mathematics, Ohio State University, Columbus, OH 43210, USA<EMAIL_ADDRESS>state.edu and  Chunhui Qiu  School of Mathematical Sciences, Xiamen
University, Xiamen, 361005, China<EMAIL_ADDRESS>
###### Abstract.

We consider a class of fully nonlinear second order elliptic equations on
Hermitian manifolds closely related to the general notion of
𝔾-plurisubharmonicity of Harvey-Lawson and an equation treated by Székelyhidi-
Tosatti-Weinkove in the proof of Gauduchon conjecture. Under fairly general
assumptions we derive interior estimates and establish the existence of smooth
solutions for the Dirichlet problem as well as for equations on closed
manifolds.

Mathematical Subject Classification (MSC2020): 35J15, 35J60, 58J05.

The third author was partially supported by NSFC (Grant No. 11971401).

##  1\. Introduction

In a remarkable paper where the Gauduchon conjecture which extends Calabi-Yau
Theorem to non-Kähler metrics was proved, Székelyhidi-Tosatti-Weinkove [23]
solved the Monge-Ampère type equation

(1.1) $(\Delta u\omega-\sqrt{-1}\partial\bar{\partial}u+\chi(\partial
u,\bar{\partial}u))^{n}=\psi\omega^{n}$

on a compact Hermitian manifold $(M^{n},\omega)$, where for a function $u\in
C^{2}(M)$, $\chi(\partial u,\bar{\partial}u)$ is a real $(1,1)$ form which
depends on $\partial u$, $\bar{\partial}u$ linearly and satisfies some
additional assumptions (which for convenience we shall refer as STW
conditions). This equation also has close connections to the form-type Monge-
Ampère equations studied by Fu-Wang-Wu [4, 5], and the notion of $n-1$
plurisubharmonic functions in the sense of Harvey-Lawson [14, 15, 16].

In the same paper [23], the authors also treated more general equations of the
form

(1.2) $f(\lambda(\Delta u\omega-\sqrt{-1}\partial\bar{\partial}u+\chi(\partial
u,\bar{\partial}u))=\psi$

under the same (STW) conditions on $\chi$, where $\lambda(U)$ denotes the
eigenvalues of a real $(1,1)$ form $U$ with respect to $\omega$, and $f$ is a
symmetric function of $n$ variables satisfying the structure conditions of
Caffarelli-Nirenberg-Spruck [1], i.e. $f$ is assumed to be defined in a
symmetric open and convex cone $\Gamma\subset\hbox{\bbbld R}^{n}$ with vertex
at the origin,

(1.3) $\Gamma_{n}=\\{\lambda\in\hbox{\bbbld
R}^{n}:\lambda_{i}>0\\}\subset\Gamma,$

and satisfies the conditions

(1.4) $f_{i}\equiv\frac{\partial f}{\partial\lambda_{i}}>0\;\;\mbox{in
$\Gamma$},\;\;1\leq i\leq n,$ (1.5) $\mbox{$f$ is a concave function in
$\Gamma$},$ (1.6) $\sup_{\partial\Gamma}f<\inf_{M}\psi$

and, in addition

(1.7)
$\lim_{t\rightarrow\infty}f(t\lambda)=\sup_{\Gamma}f,\;\;\forall\;\lambda\in\Gamma.$

There are other important fully nonlinear equations of second order in complex
geometry with explicit dependence on the solution and its gradient, such as
the Fu-Yau equation [6, 7] arising from the Strominger system in superstring
theory and its higher dimension extensions studied by Phong-Picard-Zhang [18,
19, 20] and Chu-Huang-Zhu [2]. In general, the presence of these terms and the
gradients especially, cause substantial difficulties in solving the equations,
even in the case of linear dependence for the Monge-Ampère type equation
(1.1).

It is clear that Székelyhidi-Tosatti-Weinkove [23] primarily focused on
equation (1.1) in order to prove the Gauduchon conjecture. It seems worthwhile
to take a closer look at equation (1.2) from different angles, and from a more
PDE point of view. This is one of our main purposes in the current paper. We
shall assume in place of (1.4) that $f$ satisfies the weaker condition

(1.8) $f_{i}\equiv\frac{\partial f}{\partial\lambda_{i}}\geq 0\;\;\mbox{in
$\Gamma$},\;\;1\leq i\leq n.$

In order to describe our results, we recall some notions from [9] and [11].
For a fixed real number $\sigma\in(\sup_{\partial\Gamma}f,\sup_{\Gamma}f)$
define

$\Gamma^{\sigma}=\\{\lambda\in\Gamma:f(\lambda)>\sigma\\}.$

######  Lemma 1.1.

Under conditions (1.8) and (1.5), the level hypersurface of $f$

$\partial\Gamma^{\sigma}=\\{\lambda\in\Gamma:f(\lambda)=\sigma\\},$

which is the boundary of $\Gamma^{\sigma}$, is smooth and convex.

###### Proof.

If $f$ satisfies both (1.4) and (1.5), then $\partial\Gamma^{\sigma}$ is
clearly smooth and convex. This is still true under the weaker condition (1.8)
in place of (1.4). For if $Df(\lambda_{0})=0$ for some $\lambda_{0}\in\Gamma$
then by (1.8) and (1.5) we see that $Df=0$ in the cone
$\lambda_{0}+\Gamma_{n}$ and therefore $f(\lambda_{0})=\sup_{\Gamma}f>\sigma$,
showing that $Df\neq 0$ on $\partial\Gamma^{\sigma}$. So
$\partial\Gamma^{\sigma}$ is smooth. ∎

For $\lambda\in\partial\Gamma^{\sigma}$ let

$\nu_{\lambda}=\frac{Df(\lambda)}{|Df(\lambda)|}$

denote the unit normal vector to $\partial\Gamma^{\sigma}$ at $\lambda$.

######  Definition 1.2 ([9]).

For $\mu\in\hbox{\bbbld R}^{n}$ let

$S^{\sigma}_{\mu}=\\{\lambda\in\partial\Gamma^{\sigma}:\nu_{\lambda}\cdot(\mu-\lambda)\leq
0\\}.$

The tangent cone at infinity to $\Gamma^{\sigma}$ is defined as

$\displaystyle\mathcal{C}^{+}_{\sigma}$ $\displaystyle=\\{\mu\in\hbox{\bbbld
R}^{n}:S^{\sigma}_{\mu}\;\mbox{is compact}\\}.$

Clearly $\mathcal{C}^{+}_{\sigma}$ is a symmetric convex cone. As in [9] one
can show that $\mathcal{C}^{+}_{\sigma}$ is open.

######  Definition 1.3 ([11]).

The rank of ${\mathcal{C}}_{\sigma}^{+}$ is defined to be

$\min\\{r(\nu):\mbox{$\nu$ is the unit normal vector of a supporting plane to
${\mathcal{C}}_{\sigma}^{+}$}\\}$

where $r(\nu)$ denotes the number of non-zero components of $\nu$.

Our first result concerns a priori interior estimates for second order
derivatives. Let $u\in C^{4}(M)$ be an admissible solution of the equation

(1.9) $f(\lambda(\Delta
u\omega-\sqrt{-1}\partial\bar{\partial}u+\chi[u]))=\psi[u]$

where $\chi[u]=\chi(z,u,\partial u,\bar{\partial}u)$ and
$\psi[u]=\psi(z,u,\partial u,\bar{\partial}u)$ are $C^{2}$ in $(z,u,\partial
u,\bar{\partial}u)$. We assume

(1.10) $\lim_{t\rightarrow+\infty}f(t{\bf{1}})-\sup_{M}\psi[u]\geq c_{0}>0$

where ${\bf{1}}=(1,\ldots,1)\in\Gamma$. So $c_{0}$ may depend on
$|u|_{C^{1}(M)}$. This is clearly a necessary and fairly mild condition for
the following result to hold.

######  Theorem 1.4.

In addition to (1.5), (1.6), (1.8) and (1.10) assume that $f$ satisfies

(1.11) $\sum f_{i}\lambda_{i}\geq-C_{0}\sum f_{i}\;\;\mbox{in $\Gamma$}$

for some constant $C_{0}\geq 0$, and

(1.12) $\mbox{rank of}\;\;\mathcal{C}_{\sigma}^{+}\geq
2,\;\;\forall\;\inf_{M}\psi[u]\leq\sigma\leq\sup_{M}\psi[u].$

Let $B_{r}$ be a geodesic ball of radius $r$ in $M$. Then

(1.13) $\sup_{B_{\frac{r}{2}}}|\partial\bar{\partial}u|\leq\frac{C}{r^{2}}$

where $C$ depends on $c_{0}$, $C_{0}$, $|u|_{C^{1}(\bar{B}_{r})}$ and other
known data.

Geometrically, condition (1.11) means that at any point
$\lambda_{0}\in\Gamma$, the distance from the origin to the tangent plane at
$\lambda_{0}$ of the level hypersuface $\partial\Gamma^{f(\lambda_{0})}$ has a
uniform bound $C_{0}$. It is satisfied in most applications and is weaker than
assumption (1.7) which implies

(1.14) $\sum f_{i}\lambda_{i}\geq 0\;\;\mbox{in $\Gamma$}.$

By an inequality of Lin-Trudinger [17], the rank of $\mathcal{C}_{\sigma}^{+}$
is $n-k+1$ for $f=\sigma_{k}^{\frac{1}{k}}$ and $\sigma>0$, where

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

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

$\Gamma_{k}=\\{\lambda\in\hbox{\bbbld R}^{n}:\sigma_{j}(\lambda)>0,\;\mbox{for
$1\leq j\leq k$}\\}.$

So Theorem 1.4 applies to $f=\sigma_{k}^{\frac{1}{k}}$ for $k\leq n-1$ but
excludes equation (1.1) which corresponds to $f=\sigma_{n}^{\frac{1}{n}}$.

Theorem 1.4 heavily relies on a crucial lemma in Section 2. From the proof we
shall see that there exists $\beta_{0}>1$ such that Theorem 1.4 extends to the
equation

(1.15) $f(\lambda(\Delta
u\omega-\beta\sqrt{-1}\partial\bar{\partial}u+\chi[u]))=\psi[u],$

for all $\beta<\beta_{0}$. It would be interesting to decide the supremum
value, $\sup\beta_{0}$, of such $\beta_{0}$ and study the equation in the
limiting case. For equation (1.1), $\sup{\beta}_{0}=1$.

The second purpose of the current paper is to extend Theorems 1.4 and 1.5 to a
more general setting.

Let $K\leq n$ be a fixed positive integer. Set

$\mathfrak{I}_{K}=\\{(i_{1},\ldots,i_{K}):1\leq i_{1}<\cdots<i_{K}\leq
n,\;i_{j}\in\hbox{\bbbld N}\\}$

and for $I=(i_{1},\ldots,i_{K})\in\mathfrak{I}_{K}$ define

(1.16) $\Lambda_{I}(\lambda)=\sum_{i\in
I}\lambda_{i}=\lambda_{i_{1}}+\cdots+\lambda_{i_{K}},\;\;\lambda=(\lambda_{1},\ldots,\lambda_{n})\in\hbox{\bbbld
R}^{n}.$

For convenience we fix an order for the elements in $\mathfrak{I}_{K}$:

$I_{1},\ldots,I_{N},\;\mbox{where}\;N=\frac{n!}{K!(n-K)!}$

and write

$\Lambda(\lambda)=(\Lambda_{I_{1}}(\lambda),\ldots,\Lambda_{I_{N}}(\lambda))$

or simply,

$\Lambda(\lambda)=(\Lambda_{1}(\lambda),\ldots,\Lambda_{N}(\lambda)).$

We consider the equation

(1.17) $f(\Lambda(\sqrt{-1}\partial\bar{\partial}u+X[u]))=\psi[u],$

where

$\Lambda(\sqrt{-1}\partial\bar{\partial}u+X[u])=\Lambda(\lambda(\sqrt{-1}\partial\bar{\partial}u+X[u]))$

and

$\lambda(\sqrt{-1}\partial\bar{\partial}u+X[u])=(\lambda_{1},\ldots,\lambda_{n})$

denotes the eigenvalues of $\sqrt{-1}\partial\bar{\partial}u+X[u]$ with
respect to $\omega$, $X[u]=X(z,u,\partial u,\bar{\partial}u)$ is a real
$(1,1)$ form depending on $u$, $\partial u$ and $\bar{\partial}u$, and $f$ is
a symmetric concave function defined in an open symmetric convex cone
$\Gamma\subset\hbox{\bbbld R}^{N}$ with vertex at the origin,
$\Gamma_{N}\subset\Gamma$, and satisfies

(1.18) $f_{i}\equiv f_{\Lambda_{i}}\equiv\frac{\partial
f}{\partial\Lambda_{i}}\geq 0\;\;\mbox{in $\Gamma$},\;\;1\leq i\leq N.$

We may similarly define admissible functions by requiring
$\Lambda(\sqrt{-1}\partial\bar{\partial}u+X[u])\in\Gamma$. Clearly, equations
(1.2) and (1.17) coincide when $K=n-1$.

Equation (1.17) is closely related to the general notion of
𝔾-plurisubharmonicity of Harvey-Lawson introduced and studied in a long series
of papers (see e.g. [14, 15, 16]). As in [11] we denote

$\rho_{k}(\lambda):=\prod_{1\leq i_{1}<\cdots<i_{k}\leq
n}(\lambda_{i_{1}}+\cdots+\lambda_{i_{k}}),\;1\leq k\leq n$

defined in the cone

$\mathcal{P}_{k}:=\\{\lambda\in\hbox{\bbbld
R}^{n}:\lambda_{i_{1}}+\cdots+\lambda_{i_{k}}>0,\;\forall\;1\leq
i_{1}<\cdots<i_{k}\leq n\\}.$

A function $u$ is $k$-plurisubharmonic in the sense of Harvey-Lawson [15] if

$\lambda(\sqrt{-1}\partial\bar{\partial}u+X)\in\mathcal{P}_{k}.$

For $f(\lambda)=\log\rho_{k}(\lambda)$ which is concave in $\mathcal{P}_{k}$,
the rank of $\mathcal{C}_{\sigma}^{+}$ is $k$. Clearly,

$\rho_{1}(\lambda)=\sigma_{n}(\lambda),\;\;\rho_{n}(\lambda)=\sigma_{1}(\lambda),$

and equation (1.1) is equivalent to

$\rho_{n-1}(\lambda(\sqrt{-1}\partial\bar{\partial}u+X))=\psi.$

Note that $\rho_{K}(\lambda)=\sigma_{N}(\Lambda(\lambda))$. So the equation

$\rho_{K}(\lambda(-\partial\bar{\partial}u+\chi))=\psi$

for $1\leq K\leq n$ is included in the examples of equation (1.17) given by

$f(\Lambda)=\sigma_{k}^{1/k}(\Lambda),\;1\leq k\leq N.$

Theorem 1.4 holds for equation (1.17) with the modification of replacing the
rank condition (1.12) by

(1.19) $\mbox{rank
of}\;\;\mathcal{C}_{\sigma}^{+}\geq\frac{N(n-K)}{n}+1,\;\;\forall\;\sup_{\partial\Gamma}f<\sigma<\sup_{\Gamma}f.$

Indeed it extends to the equation

(1.20)
$f(\Lambda(\sqrt{-1}\partial\bar{\partial}u+X[u])-\beta\Lambda^{\prime}(\sqrt{-1}\partial\bar{\partial}u+X[u]))=\psi[u],$

for some $\beta>0$, where
$\Lambda^{\prime}=(\Lambda_{I^{\prime}_{1}},\ldots,\Lambda_{I^{\prime}_{N}})$
and $I^{\prime}_{j}$ denotes the complement of $I_{j}$ in $\\{1,\ldots,n\\}$;
we state a more complete result for equation (1.20) to include global and
boundary estimates on second derivatives.

######  Theorem 1.5.

Under conditions (1.5), (1.18), (1.11) and (1.19), there exists constant
$\beta_{0}>1$ such that for any $\beta<\beta_{0}$ and admissible solution
$u\in C^{4}(M)$ of equation (1.20) satisfying (1.10), the interior estimate
(1.13) holds for any geodesic ball $B_{r}$ in $M$.

Suppose in addition that $(M^{n},\omega)$ is a Hermitian manifold with smooth
boundary $\partial M$ and compact closure $\bar{M}=M\cup\partial M$, and $u\in
C^{4}(\bar{M})$. Then

(1.21) $\max_{{\bar{M}}}|\partial\bar{\partial}u|\leq C\Big{(}1+\max_{\partial
M}|\partial\bar{\partial}u|\Big{)}$

and

(1.22) $\max_{\partial M}|\nabla^{2}u|\leq C.$

One may also derive interior gradient estimates under suitable growth
conditions of $X$ and $\psi$ on $u$ and $\nabla u$; see Theorem 4.1 for the
precise statement.

We now turn our attentions to existence of admissible solutions for equation
(1.17). As we shall see from the proofs the results extend to equation (1.20)
for $\beta<\beta_{0}$ (as in Theorem 1.5) without difficulty. For simplicity
we only state our results here for $X$ and $\psi$ both independent of $u$ and
$\nabla u$; more general results are stated in Section 5. We first consider
the Dirichlet problem.

######  Theorem 1.6.

Let $(M^{n},\omega)$ be a Hermitian manifold with smooth boundary $\partial M$
and compact closure $\bar{M}=M\cup\partial M$, $X$ a smooth real $(1,1)$ form
on $\bar{M}$, $\psi\in C^{\infty}(\bar{M})$ and $\varphi\in C^{0}(\partial
M)$. Suppose (1.5), (1.6), (1.18), (1.11) and (1.19) hold, and that there
exists an admissible viscosity subsolution $\underline{u}\in C^{0}(\bar{M})$
satisfying

(1.23) $\left\\{\begin{aligned}
f(\Lambda(\sqrt{-1}\partial\bar{\partial}\underline{u}+X))\geq\,&\psi\;\mbox{in
$\bar{M}$},\\\ \underline{u}=\varphi\;\mbox{on $\partial
M$}.\end{aligned}\right.$

Equation (1.17) admits a unique admissible solution $u\in C^{\infty}(M)\cap
C^{0}(\bar{M})$ with $u=\varphi$ on $\partial M$ and, for any subdomain
$\Omega\subset\overline{\Omega}\subset M$,

(1.24) $|u|_{C^{2,\alpha}(\Omega)}\leq C,\;0<\alpha\leq 1$

where $C$ depends on the distance from $\Omega$ to $\partial M$.

If in addition $\varphi\in C^{\infty}(\partial M)$ then $u\in
C^{\infty}(\bar{M})$ and satisfies the global estimate

(1.25) $|u|_{C^{2,\alpha}(\bar{M})}\leq C,\;0<\alpha\leq 1.$

More precisely, for a domain $\Omega$ in $M$ we have

(1.26) $|\nabla
u|^{2}+|\nabla^{2}u|\leq\frac{C}{d^{2}}\Big{\\{}1+\sup_{M}u-u\Big{\\}}\;\;\mbox{in
$\Omega$}$

and

(1.27)
$|\nabla^{2}u|_{C^{\alpha}(\Omega)}\leq\frac{C}{d^{2+\alpha}}\Big{\\{}1+\sup_{M}u-\inf_{\Omega}u\Big{\\}},\;0<\alpha\leq
1$

where $d$ is the distance from $\Omega$ to $\partial M$, and $C$ is
independent of $d$.

An interesting question would be when there exist admissible subsolutions.
Under suitable conditions on $f$, one should be able to follow the
construction of Caffarelli-Nirenberg-Spruck [1] with appropriate
modifications, for instance, on manifolds with $K$-plurisubharmonic exhausting
functions.

Equation (1.17) belongs to the more general class of equations of the form

(1.28) $f(\lambda(\sqrt{-1}\partial\bar{\partial}u+X[u]))=\psi[u].$

Consequently, when $X$ and $\psi$ are both independent of $u$ and its gradient
$\nabla u$, results of Szekelyhidi [22] apply to equation (1.17) on compact
Hermitian manifold (without boundary). Under assumption (1.19), we obtain the
following existence result for equation (1.17) without the key
$\mathcal{C}$-subsolution condition in [22] (which, in terms of the tangent
cone at infinity, is equivalent to the condition introduced in [9] for a type
I cone $\Gamma$ that there exists a function $\underline{u}\in C^{2}(M)$ with
$\lambda(\sqrt{-1}\partial\bar{\partial}\underline{u}+X)\in\mathcal{C}^{+}_{\psi}$
on $M$ [10]; recall from [1] that $\Gamma$ is of type I if each positive
$\lambda_{i}$ axis is contained in $\partial\Gamma$).

######  Theorem 1.7.

Let $(M^{n},\omega)$ be a compact Hermitian manifold, $X$ a smooth positive
$(1,1)$ form and $\psi\in C^{\infty}(M)$. Suppose (1.5), (1.6), (1.18), (1.11)
and (1.19) hold; without loss of generality (by adding a constant if
necessary) we may assume in place of (1.6) that

(1.29) $\sup_{\partial\Gamma}f\leq 0<\psi.$

Assume in addition that

(1.30) $\sup_{\Gamma}f=\lim_{t\rightarrow+\infty}f(t{\bf{1}})>\sup_{M}\psi.$

Then there exists a unique constant $b$ such that the equation

(1.31) $f(\Lambda(\sqrt{-1}\partial\bar{\partial}u+X))=e^{b}\psi$

has a unique admissible solution $u\in C^{\infty}(M)$ with

$\sup_{M}u=0.$

Without the assumption $X>0$ equation (5.1) may become degenerate. Note that
when $X>0$ constants are admissible functions. On the other hand, if $X\leq 0$
there are no admissible functions on compact manifolds (without boundary).

Comparing to equation (1.17), the general equation (1.28) is much more
difficult, and most of the results in this paper no longer hold without
additional assumptions even in the case $X$ and $\psi$ are independent of $u$
and $\nabla u$. We also note that, among others, equation (1.28) will only be
degenerate elliptic if condition (1.4) is replaced by (1.8). For
$X[u]=X(\partial u,\bar{\partial}u)$ and $\psi[u]=\psi(\partial
u,\bar{\partial}u)$, Guan-Nie [11] derived the second order estimates for
equation (1.28) on closed Hermitian manifolds under rather strong conditions;
we refer the reader to [11] for more references.

For $f=(\sigma_{n}/\sigma_{k})^{\frac{1}{n-k}}$, the Dirichlet problem for
equation (1.2) was recently studied by Feng-Ge-Zheng [3]. In [12], Guan-Qiu-
Yuan treated equations of the form

(1.32) $F(\gamma\Delta
u\omega-\beta\sqrt{-1}\partial\bar{\partial}u+\chi[u])=\psi[u]$

for $\gamma>0$, $\beta<\gamma$. One should be able to extend most of results
described in the current paper to the limiting case $\beta=\gamma$ for
equation (1.32). It also seems interesting to study equation (1.17) with
$\Lambda_{I}$ defined in (1.16) replaced by other symmetric functions of $K$
variables.

The rest of this paper is organized as follows. In Section 2 we prove some key
properties of the tangent cone at infinity related to its rank. The second
derivative and gradient estimates are derived in Sections 3 and 4,
respectively. We also establish the Harnack inequality using the gradient
estimates for positive solutions when $X$ and $\psi$ are independent of $u$
and $\nabla u$. Section 5 concerns the existence of admissible solutions,
where we prove Theorems 1.6, 1.7 and their extensions.

The second author wishes to thank Duong Phong for stimulating communications,
especially for his comments on results described in [12], which motivated us
to carry out the current work.

##  2\. Rank of tangent cone at infinity

In this section $f$ is assumed to satisfy (1.5), (1.8) and (1.11). We shall
prove the following key lemmas.

######  Lemma 2.1.

Let $P=\\{\mu\in\hbox{\bbbld R}^{n}:\nu\cdot\mu=c\\}$ be a hyperplane, where
$\nu$ is a unit vector. Suppose that there exists a sequence
$\\{\lambda_{k}\\}$ in $\partial\Gamma^{\sigma}$ with

(2.1)
$\lim_{k\rightarrow+\infty}\nu_{\lambda_{k}}=\nu,\;\;\lim_{k\rightarrow+\infty}\nu_{\lambda_{k}}\cdot\lambda_{k}=c,\;\;\lim_{k\rightarrow+\infty}|\lambda_{k}|=+\infty.$

Then $P$ is a supporting hyperplane to $\mathcal{C}_{\sigma}^{+}$ at a non-
vertex point.

###### Proof.

Assume $\mu_{0}\in\hbox{\bbbld R}^{n}$ with $\nu\cdot\mu_{0}\leq c-\epsilon$
for some $\epsilon>0$. Then

$\lim_{k\rightarrow+\infty}\nu_{\lambda_{k}}\cdot(\mu_{0}-\lambda_{k})=\nu\cdot\mu_{0}-c\leq-\epsilon.$

This means that $S_{\mu_{0}}^{\sigma}$ is noncompact and therefore
$\mu_{0}\notin\mathcal{C}_{\sigma}^{+}$. As $\epsilon>0$ is arbitrary, we see
that $\mathcal{C}_{\sigma}^{+}$ is contained in the half space
$P^{+}=\\{\mu\in\hbox{\bbbld R}^{n}:\nu\cdot\mu>c\\}$.

By symmetry the vertex of $\mathcal{C}_{\sigma}^{+}$ is at a point ${\bf
a}=(a,\ldots,a)$ for some real number $a$. It is easy to see that ${\bf a}\in
P$, i.e.

$a=\frac{c}{\sum\nu^{i}}$

where $\nu=(\nu^{1},\ldots,\nu^{n})$. Indeed, clearly $a\geq
c(\sum\nu^{i})^{-1}$ for otherwise ${\bf a}\notin\overline{P^{+}}$. Let
$R_{k}$ be the ray from ${\bf a}$ passing $\lambda_{k}$ so $R_{k}$ lies in
$\mathcal{C}_{\sigma}^{+}\subset P^{+}$. On the other hand, $R_{k}$ would not
completely lie in $\overline{P^{+}}$ for $k$ large if $a>c(\sum\nu^{i})^{-1}$,
a contradiction.

Let $\mu_{k}$ be the point of intersection of $R_{k}$ and the unit sphere
centered at ${\bf a}$. Passing to a subsequence we may assume
$\mu_{k}\rightarrow\mu\in\overline{\mathcal{C}_{\sigma}^{+}}$ as
$k\rightarrow+\infty$. Clearly,

$0\geq\nu_{k}\cdot(\mu_{k}-\lambda_{k})\geq\nu_{k}\cdot({\bf a}-\lambda_{k}).$

Taking limit as $k\rightarrow+\infty$ we obtain $\nu\cdot\mu=c$. This show
that $\mu\in P\cap\partial\mathcal{C}_{\sigma}^{+}$. ∎

######  Lemma 2.2.

Let $r\geq 1$ be the rank of $\mathcal{C}_{\sigma}^{+}$. There exists
$c_{1}>0$ such that at any point $\lambda\in\partial\Gamma^{\sigma}$ where
without loss of generality we assume $f_{1}\leq\cdots\leq f_{n}$,

$\sum_{i\leq n-r+1}f_{i}\geq c_{1}\sum f_{i}.$

###### Proof.

We prove by contradiction. Suppose that for each $k=1,2,\ldots$ there exists
$\lambda_{k}\in\partial\Gamma^{\sigma}$ with $f_{1}(\lambda_{k})\leq\cdots\leq
f_{n}(\lambda_{k})$ and

(2.2) $\sum_{i\leq n-r+1}f_{i}(\lambda_{k})\leq\frac{1}{k}\sum
f_{i}(\lambda_{k}).$

Note that there is $t>0$ such that $t{\bf{1}}\in\partial\Gamma^{\sigma}$, that
is, $f(t{\bf{1}})=\sigma$. By the concavity of $f$,

$\sum f_{i}(t-\lambda_{i})\geq f(t{\bf{1}})-f(\lambda)=0.$

Therefore

$\sum f_{i}\lambda_{i}\leq t\sum f_{i}.$

By (1.11), we can pass to a subsequence and assume that

$\lim_{k\rightarrow+\infty}\nu_{\lambda_{k}}=\nu,\;\;\lim_{k\rightarrow+\infty}\nu_{\lambda_{k}}\cdot\lambda_{k}=c,\;\;\lim_{k\rightarrow+\infty}|\lambda_{k}|=+\infty.$

According to Lemma 2.1, the hyperplane $P=\\{\mu\in\hbox{\bbbld
R}^{n}:\nu\cdot\mu=c\\}$ is a supporting hyperplane to
$\mathcal{C}_{\sigma}^{+}$. However, $\nu$ has at most $r-1$ nonzero
components, which is a contradiction. ∎

######  Lemma 2.3.

Let $P=\\{\mu\in\hbox{\bbbld R}^{n}:\nu\cdot\mu=c\\}$, where $\nu$ is a unit
vector, be a tangent hyperplane to $\partial\mathcal{C}_{\sigma}^{+}$. Then
there exists a sequence $\\{\lambda_{k}\\}$ in $\partial\Gamma^{\sigma}$ with

(2.3)
$\lim_{k\rightarrow+\infty}\nu_{\lambda_{k}}=\nu,\;\;\lim_{k\rightarrow+\infty}\nu_{\lambda_{k}}\cdot\lambda_{k}=c,\;\;\lim_{k\rightarrow+\infty}|\lambda_{k}|=+\infty.$

###### Proof.

We may assume $\partial\Gamma^{\sigma}\cap\partial\mathcal{C}_{\sigma}^{+}\cap
P=\emptyset$ for otherwise it contains a ray (see [9]) and (2.3) holds for
some
$\lambda_{k}\in\partial\Gamma^{\sigma}\cap\partial\mathcal{C}_{\sigma}^{+}\cap
P$.

Suppose that $P$ is the tangent hyperplane to
$\partial\mathcal{C}_{\sigma}^{+}$ at $\mu\in
P\cap\partial\mathcal{C}_{\sigma}^{+}$. We have
$\mu\notin\partial\Gamma^{\sigma}$ and
$\mu_{k}=\mu+\frac{\nu}{k}\in\mathcal{C}_{\sigma}^{+}$ for all $k\geq 1$ as
$\nu\in\overline{\Gamma_{n}}$ and $\mu+\Gamma_{n}\in\mathcal{C}_{\sigma}^{+}$.
Let

$S_{\mu_{k}}^{\sigma,0}=\\{\lambda\in\partial\Gamma^{\sigma}:\nu_{\lambda}\cdot(\mu_{k}-\lambda)=0\\}$

and choose $\lambda_{k}\in S_{\mu_{k}}^{\sigma,0}$ satisfying

$\nu\cdot(\lambda_{k}-\mu)=\min_{\lambda\in
S_{\mu_{k}}^{\sigma,0}}\nu\cdot(\lambda-\mu)\geq 0;$

such $\lambda_{k}$ exists as $S_{\mu_{k}}^{\sigma,0}$ is compact. As
$\partial\Gamma^{\sigma}\cap\partial\mathcal{C}_{\sigma}^{+}\cap P=\emptyset$,
we have

$\lim_{k\rightarrow+\infty}|\lambda_{k}|=+\infty.$

So the last limit in (2.3) holds. Next,

$\nu_{\lambda_{k}}\cdot(\lambda_{k}-\mu)=\nu_{\lambda_{k}}\cdot(\lambda_{k}-\mu_{k})+\nu_{\lambda_{k}}\cdot(\mu_{k}-\mu)=\frac{\nu_{k}\cdot\nu}{k}.$

Therefore

(2.4) $\lim_{k\rightarrow+\infty}\nu_{\lambda_{k}}\cdot(\lambda_{k}-\mu)=0.$

Passing to a subsequence we may assume

$\lim_{k\rightarrow+\infty}\nu_{\lambda_{k}}=\nu^{\prime}.$

By (2.4) we have

$\lim_{k\rightarrow+\infty}\nu_{\lambda_{k}}\cdot\lambda_{k}=\nu^{\prime}\cdot\mu\equiv
c^{\prime}.$

By Lemma 2.1 we see that $P^{\prime}=\\{\eta\in\hbox{\bbbld
R}^{n}:\nu^{\prime}\cdot\eta=c^{\prime}\\}$ is a supporting plane to
$\mathcal{C}_{\sigma}^{+}$ at $\mu$. By uniqueness of the tangent plane,
$P^{\prime}=P$ and therefore $\nu^{\prime}=\nu$. ∎

##  3\. The second order estimates

In this section we establish the second order estimates in Theorems 1.4 and
1.5 for admissible solutions. We first fix some notation. In local coordinates
$z=(z_{1},\ldots,z_{n})$, we shall write

(3.1) $\omega=\sqrt{-1}g_{i\bar{j}}dz_{i}\wedge dz_{\bar{j}}$

and $\\{g^{i\bar{j}}\\}=\\{g_{i\bar{j}}\\}^{-1}$. The Christoffel symbols
$\Gamma_{ij}^{k}$, torsion and curvature tensors are given respectively by

$\nabla_{\frac{\partial}{\partial z_{i}}}\frac{\partial}{\partial
z_{j}}=\Gamma_{ij}^{k}\frac{\partial}{\partial
z_{k}},\;\;T^{k}_{ij}=\Gamma_{ij}^{k}-\Gamma_{ji}^{k}$

and

$R_{i\bar{j}k\bar{l}}=-g_{m\bar{l}}\frac{\partial\Gamma_{ik}^{m}}{\partial\bar{z}_{j}}=-\frac{\partial
g_{k\bar{l}}}{\partial z_{i}\partial\bar{z}_{j}}+g^{p\bar{q}}\frac{\partial
g_{k\bar{q}}}{\partial z_{i}}\frac{\partial
g_{p\bar{l}}}{\partial\bar{z}_{j}}$

where $\nabla$ is the Chern connection on $(M^{n},\omega)$.

Let $u\in C^{4}(M)$ be an admissible solution of equation (1.17). We denote

$\mathfrak{g}[u]=\sqrt{-1}\partial\bar{\partial}u+X[u]$

and define $G$ by

$G(\mathfrak{g}[u])=f(\Lambda(\mathfrak{g}[u])).$

Consequently, equation (1.17) takes the form

(3.2) $G(\mathfrak{g}[u])=\psi[u].$

When $K=n-1$, we have $N=n$ and

$\Lambda(\mathfrak{g}[u])=\lambda((\mbox{tr}\mathfrak{g}[u])\omega-\mathfrak{g}[u]).$

We see that equation (1.9) is covered by equation (3.2) for

$X[u]=\frac{\mbox{tr}\chi[u]}{n-1}\omega-\chi[u].$

In local coordinates, we write

$\mathfrak{g}[u]=\sqrt{-1}\mathfrak{g}_{i\bar{j}}dz_{i}\wedge
dz_{\bar{j}}=\sqrt{-1}(\partial_{i}\bar{\partial}_{j}u+X_{i\bar{j}})dz_{i}\wedge
dz_{\bar{j}}$

and differentiate equation (3.2) twice to obtain

(3.3) $G^{i\bar{j}}\nabla_{p}\mathfrak{g}_{i\bar{j}}=\nabla_{p}\psi[u]$

and

(3.4)
$G^{i\bar{j}}\nabla_{\bar{q}}\nabla_{p}\mathfrak{g}_{i\bar{j}}=\nabla_{\bar{q}}\nabla_{p}\psi[u]-G^{i\bar{j},k\bar{l}}\nabla_{p}\mathfrak{g}_{i\bar{j}}\nabla_{\bar{q}}\mathfrak{g}_{k\bar{l}},$

where

$G^{i\bar{j}}=\frac{\partial
G}{\partial\mathfrak{g}_{i\bar{j}}}(\mathfrak{g}[u]),\;\;G^{i\bar{j},k\bar{l}}=\frac{\partial^{2}G}{\partial\mathfrak{g}_{i\bar{j}}\partial\mathfrak{g}_{k\bar{l}}}(\mathfrak{g}[u]).$

To estimate $|\partial\bar{\partial}u|$, as in [11] we follow ideas of
Tosatti-Weinkove [24] and consider the quantity which is given in local
coordinates by

(3.5) $A:=\sup_{z\in M}\max_{\xi\in
T^{1,0}_{z}M}\;e^{(1+\gamma)\phi}\mathfrak{g}_{p\bar{q}}\xi_{p}\bar{\xi_{q}}(g^{k\bar{l}}\mathfrak{g}_{i\bar{l}}\mathfrak{g}_{k\bar{j}}\xi_{i}\bar{\xi_{j}})^{\frac{\gamma}{2}}/|\xi|^{2+\gamma}$

where $\phi$ is a function depending on $u$ and $|\nabla u|$, and $\gamma>0$
is a small constant to be determined.

Assume that $A$ is achieved at an interior point $z_{0}\in M$ for some $\xi\in
T^{1,0}_{z_{0}}M$. We choose local coordinates around $z_{0}$ such that
$g_{i\bar{j}}=\delta_{ij}$ and $T_{ij}^{k}=2\Gamma_{ij}^{k}$ using the lemma
of Streets and Tian [21], and that $\mathfrak{g}_{i\bar{j}}$ is diagonal at
$z_{0}$ with

$\mathfrak{g}_{1\bar{1}}\geq\mathfrak{g}_{2\bar{2}}\geq\cdots\geq\mathfrak{g}_{n\bar{n}}.$

As noted in [24] the maximum $A$ is achieved for $\xi=\partial_{1}$ at $z_{0}$
when $\gamma>0$ is sufficiently small; see also [11]. We assume
$\mathfrak{g}_{1\bar{1}}\geq 1$; otherwise we are done.

Let
$W=g_{1\bar{1}}^{-1}g^{k\bar{l}}\mathfrak{g}_{1\bar{l}}\mathfrak{g}_{k\bar{1}}$.
We see that the function
$e^{(1+\gamma)\phi}g_{1\bar{1}}^{-1}\mathfrak{g}_{1\bar{1}}W^{\frac{\gamma}{2}}$
which is locally well defined attains a maximum
$A=(e^{\phi}\mathfrak{g}_{1\bar{1}})^{1+\gamma}$ at $z_{0}$ where
$W=\mathfrak{g}_{1\bar{1}}^{2}$,

(3.6)
$\displaystyle\frac{\partial_{i}(g_{1\bar{1}}^{-1}\mathfrak{g}_{1\bar{1}})}{\mathfrak{g}_{1\bar{1}}}+\frac{\gamma\partial_{i}W}{2W}+(1+\gamma)\partial_{i}\phi=$
$\displaystyle 0,$
$\displaystyle\frac{\bar{\partial}_{i}(g_{1\bar{1}}^{-1}\mathfrak{g}_{1\bar{1}})}{\mathfrak{g}_{1\bar{1}}}+\frac{\gamma\bar{\partial}_{i}W}{2W}+(1+\gamma)\bar{\partial}_{i}\phi=$
$\displaystyle 0$

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

(3.7) $\displaystyle 0\geq$
$\displaystyle\frac{1}{\mathfrak{g}_{1\bar{1}}}G^{i\bar{i}}\bar{\partial}_{i}\partial_{i}(g_{1\bar{1}}^{-1}\mathfrak{g}_{1\bar{1}})-\frac{1}{\mathfrak{g}_{1\bar{1}}^{2}}G^{i\bar{i}}\partial_{i}(g_{1\bar{1}}^{-1}\mathfrak{g}_{1\bar{1}})\bar{\partial}_{i}(g_{1\bar{1}}^{-1}\mathfrak{g}_{1\bar{1}})$
$\displaystyle+\frac{\gamma}{2W}G^{i\bar{i}}\bar{\partial}_{i}\partial_{i}W-\frac{\gamma}{2W^{2}}G^{i\bar{i}}\partial_{i}W\bar{\partial}_{i}W+(1+\gamma)G^{i\bar{i}}\bar{\partial}_{i}\partial_{i}\phi.$

In what follows we shall make use of calculations in [11]; for the reader’s
convenience we give a brief outline. First,

(3.8)
$\displaystyle\partial_{i}(g_{1\bar{1}}^{-1}\mathfrak{g}_{1\bar{1}})=\nabla_{i}\mathfrak{g}_{1\bar{1}},\;\;\partial_{i}W=2\mathfrak{g}_{1\bar{1}}\nabla_{i}\mathfrak{g}_{1\bar{1}},$
(3.9)
$\displaystyle\bar{\partial}_{j}\partial_{i}(g_{1\bar{1}}^{-1}\mathfrak{g}_{1\bar{1}})=$
$\displaystyle\nabla_{\bar{j}}\nabla_{i}\mathfrak{g}_{1\bar{1}}+(\overline{\Gamma_{j1}^{m}}\nabla_{i}\mathfrak{g}_{1\bar{m}}-\overline{\Gamma_{j1}^{1}}\nabla_{i}\mathfrak{g}_{1\bar{1}})$
$\displaystyle+(\Gamma_{i1}^{m}\nabla_{\bar{j}}\mathfrak{g}_{m\bar{1}}-\Gamma_{i1}^{1}\nabla_{\bar{j}}\mathfrak{g}_{1\bar{1}})+(\Gamma_{i1}^{1}\overline{\Gamma_{j1}^{1}}-\Gamma_{i1}^{m}\overline{\Gamma_{j1}^{m}})\mathfrak{g}_{1\bar{1}}.$

and

(3.10) $\displaystyle\bar{\partial}_{j}\partial_{i}W=$ $\displaystyle
2\mathfrak{g}_{1\bar{1}}\nabla_{\bar{j}}\nabla_{i}\mathfrak{g}_{1\bar{1}}+2\nabla_{i}\mathfrak{g}_{1\bar{1}}\nabla_{\bar{j}}\mathfrak{g}_{1\bar{1}}+\sum_{l>1}\nabla_{i}\mathfrak{g}_{l\bar{1}}\nabla_{\bar{j}}\mathfrak{g}_{1\bar{l}}$
$\displaystyle+\sum_{l>1}(\nabla_{i}\mathfrak{g}_{1\bar{l}}+{\Gamma_{i1}^{l}}\mathfrak{g}_{l\bar{l}})(\nabla_{\bar{j}}\mathfrak{g}_{l\bar{1}}+\overline{\Gamma_{j1}^{l}}\mathfrak{g}_{l\bar{l}})$
$\displaystyle+\mathfrak{g}_{1\bar{1}}\sum_{l>1}(\overline{\Gamma_{j1}^{l}}\nabla_{i}\mathfrak{g}_{1\bar{l}}+\Gamma_{i1}^{l}\nabla_{\bar{j}}\mathfrak{g}_{l\bar{1}})$
$\displaystyle-\mathfrak{g}_{1\bar{1}}\sum_{l>1}\Gamma_{i1}^{m}\overline{\Gamma_{j1}^{m}}(\mathfrak{g}_{1\bar{1}}+\mathfrak{g}_{l\bar{l})}.$

It follows that

(3.11) $\displaystyle
G^{i\bar{i}}\partial_{i}W\bar{\partial}_{i}W=4\mathfrak{g}_{1\bar{1}}^{2}G^{i\bar{i}}\nabla_{i}\mathfrak{g}_{1\bar{1}}\nabla_{\bar{i}}\mathfrak{g}_{1\bar{1}},$
(3.12) $\displaystyle
G^{i\bar{i}}\partial_{i}(g_{1\bar{1}}^{-1}\mathfrak{g}_{1\bar{1}})\bar{\partial}_{i}(g_{1\bar{1}}^{-1}\mathfrak{g}_{1\bar{1}})=G^{i\bar{i}}\nabla_{i}\mathfrak{g}_{1\bar{1}}\nabla_{\bar{i}}\mathfrak{g}_{1\bar{1}}$

and, by Cauchy-Schwarz inequality,

(3.13) $\displaystyle G^{i\bar{i}}\bar{\partial}_{i}\partial_{i}W\geq$
$\displaystyle
2\mathfrak{g}_{1\bar{1}}G^{i\bar{i}}\nabla_{\bar{i}}\nabla_{i}\mathfrak{g}_{1\bar{1}}+2G^{i\bar{i}}\nabla_{i}\mathfrak{g}_{1\bar{1}}\nabla_{\bar{i}}\mathfrak{g}_{1\bar{1}}$
$\displaystyle+$
$\displaystyle\sum_{l>1}G^{i\bar{i}}\nabla_{i}\mathfrak{g}_{1\bar{l}}\nabla_{\bar{i}}\mathfrak{g}_{l\bar{1}}+\frac{1}{2}\sum_{l>1}G^{i\bar{i}}\nabla_{i}\mathfrak{g}_{1\bar{l}}\nabla_{\bar{i}}\mathfrak{g}_{l\bar{1}}-C\mathfrak{g}_{1\bar{1}}^{2}\sum
G^{i\bar{i}},$ (3.14) $\displaystyle
G^{i\bar{i}}\bar{\partial}_{i}\partial_{i}(g_{1\bar{1}}^{-1}\mathfrak{g}_{1\bar{1}})\geq$
$\displaystyle
G^{i\bar{i}}\nabla_{\bar{i}}\nabla_{i}\mathfrak{g}_{1\bar{1}}-\frac{\gamma}{8\mathfrak{g}_{1\bar{1}}}\sum_{l>1}G^{i\bar{i}}\nabla_{i}\mathfrak{g}_{1\bar{l}}\nabla_{\bar{i}}\mathfrak{g}_{l\bar{1}}-C\mathfrak{g}_{1\bar{1}}\sum
G^{i\bar{i}}.$

We derive from (3.6), (3.7) and (3.11)-(3.14) that

(3.15)
$\displaystyle\nabla_{i}\mathfrak{g}_{1\bar{1}}+\mathfrak{g}_{1\bar{1}}\partial_{i}\phi=0,\;\nabla_{\bar{i}}\mathfrak{g}_{1\bar{1}}+\mathfrak{g}_{1\bar{1}}\bar{\partial}_{i}\phi=0$

and

(3.16) $\displaystyle 0\geq$
$\displaystyle\frac{1}{\mathfrak{g}_{1\bar{1}}}G^{i\bar{i}}\nabla_{\bar{i}}\nabla_{i}\mathfrak{g}_{1\bar{1}}-\frac{1}{\mathfrak{g}_{1\bar{1}}^{2}}G^{i\bar{i}}\nabla_{i}\mathfrak{g}_{1\bar{1}}\nabla_{\bar{i}}\mathfrak{g}_{1\bar{1}}+G^{i\bar{i}}\bar{\partial}_{i}\partial_{i}\phi$
$\displaystyle+\frac{\gamma}{\mathfrak{g}_{1\bar{1}}^{2}}\sum_{l>1}G^{i\bar{i}}\nabla_{i}\mathfrak{g}_{1\bar{l}}\nabla_{\bar{i}}\mathfrak{g}_{l\bar{1}}+\frac{\gamma}{16\mathfrak{g}_{1\bar{1}}^{2}}\sum_{l>1}G^{i\bar{i}}\nabla_{i}\mathfrak{g}_{1\bar{l}}\nabla_{\bar{i}}\mathfrak{g}_{l\bar{1}}-C\sum
G^{i\bar{i}}.$

As in [11] and [12] we have

(3.17)
$\displaystyle\nabla_{\bar{i}}\nabla_{i}\mathfrak{g}_{1\bar{1}}-\nabla_{\bar{1}}\nabla_{1}\mathfrak{g}_{i\bar{i}}=$
$\displaystyle
R_{i\bar{i}1\bar{1}}\mathfrak{g}_{1\bar{1}}-R_{1\bar{1}i\bar{i}}\mathfrak{g}_{i\bar{i}}-T_{i1}^{l}\nabla_{\bar{i}}\mathfrak{g}_{l\bar{1}}-\overline{T_{i1}^{l}}\nabla_{i}\mathfrak{g}_{1\bar{l}}$
$\displaystyle-
T_{i1}^{l}\overline{T_{i1}^{l}}\mathfrak{g}_{l\bar{l}}+H_{i\bar{i}}$

where

$\displaystyle H_{i\bar{i}}=$
$\displaystyle\nabla_{\bar{i}}\nabla_{i}X_{1\bar{1}}-\nabla_{\bar{1}}\nabla_{1}X_{i\bar{i}}-2\mathfrak{Re}\\{T_{i1}^{l}\nabla_{\bar{i}}X_{l\bar{1}}\\}+R_{i\bar{i}1\bar{l}}X_{l\bar{1}}-R_{1\bar{1}i\bar{l}}X_{l\bar{i}}-T_{i1}^{j}\overline{T_{i1}^{l}}X_{j\bar{l}}.$

It follows from Schwarz inequality that

(3.18) $\displaystyle
G^{i\bar{i}}\nabla_{\bar{i}}\nabla_{i}\mathfrak{g}_{1\bar{1}}\geq$
$\displaystyle
G^{i\bar{i}}\nabla_{\bar{1}}\nabla_{1}\mathfrak{g}_{i\bar{i}}-\frac{\gamma}{32\mathfrak{g}_{1\bar{1}}}G^{i\bar{i}}\nabla_{i}\mathfrak{g}_{1\bar{l}}\nabla_{\bar{i}}\mathfrak{g}_{l\bar{1}}$
$\displaystyle-C\mathfrak{g}_{1\bar{1}}\sum
G^{i\bar{i}}+G^{i\bar{i}}H_{i\bar{i}}.$

From (3.4), (3.15), (3.16) and (3.18) we derive

(3.19)
$\displaystyle\mathfrak{g}_{1\bar{1}}G^{i\bar{i}}\bar{\partial}_{i}\partial_{i}\phi\leq$
$\displaystyle-\nabla_{\bar{1}}\nabla_{1}\psi+\frac{1+\gamma}{\mathfrak{g}_{1\bar{1}}}G^{i\bar{i}}\nabla_{i}\mathfrak{g}_{1\bar{1}}\nabla_{\bar{i}}\mathfrak{g}_{1\bar{1}}-G^{i\bar{i}}H_{i\bar{i}}$
$\displaystyle-\frac{\gamma}{32\mathfrak{g}_{1\bar{1}}}\sum_{l>1}G^{i\bar{i}}(\nabla_{i}\mathfrak{g}_{l\bar{1}}\nabla_{\bar{i}}\mathfrak{g}_{1\bar{l}}+\nabla_{i}\mathfrak{g}_{1\bar{l}}\nabla_{\bar{i}}\mathfrak{g}_{l\bar{1}})+C\mathfrak{g}_{1\bar{1}}\sum
G^{i\bar{i}}$ $\displaystyle\leq$
$\displaystyle-\nabla_{\bar{1}}\nabla_{1}\psi+(1+\gamma)\mathfrak{g}_{1\bar{1}}G^{i\bar{i}}\nabla_{i}\phi\nabla_{\bar{i}}\phi-G^{i\bar{i}}H_{i\bar{i}}+C\mathfrak{g}_{1\bar{1}}\sum
G^{i\bar{i}}.$

By straightforward calculations (see e.g. [11], [12]),

(3.20) $\displaystyle G^{i\bar{i}}H_{i\bar{i}}\geq$ $\displaystyle
2G^{i\bar{i}}\mathfrak{Re}\\{X_{1\bar{1},\zeta_{\alpha}}\nabla_{\alpha}\nabla_{\bar{i}}\nabla_{i}u\\}-2G^{i\bar{i}}\mathfrak{Re}\\{X_{i\bar{i},\zeta_{\alpha}}\nabla_{\alpha}\nabla_{\bar{1}}\nabla_{1}u\\}$
$\displaystyle-C\mathfrak{g}_{1\bar{1}}^{2}\sum
G^{i\bar{i}}-C\sum_{i,k}G^{i\bar{i}}|\nabla_{i}\nabla_{k}u|^{2}-C\sum_{k}|\nabla_{1}\nabla_{k}u|^{2}\sum
G^{i\bar{i}}$ $\displaystyle\geq$ $\displaystyle
2\mathfrak{Re}\\{X_{1\bar{1},\zeta_{\alpha}}\nabla_{\alpha}\psi\\}+2\mathfrak{g}_{1\bar{1}}G^{i\bar{i}}\mathfrak{Re}\\{X_{i\bar{i},\zeta_{\alpha}}\nabla_{\alpha}\phi\\}-C|A|^{2}\sum
G^{i\bar{i}}.$

Henceforth for convenience we denote

$|A_{i}|^{2}=\mathfrak{g}_{i\bar{i}}^{2}+\sum_{k}|\nabla_{i}\nabla_{k}u|^{2},\;\;|A|^{2}=\sum|A_{i}|^{2}.$

Next,

(3.21)
$\nabla_{\alpha}\psi=\psi_{\alpha}+\psi_{u}\nabla_{\alpha}u+\psi_{\zeta_{\beta}}\partial_{\alpha}\partial_{\beta}u+\psi_{\bar{\zeta}_{\beta}}\partial_{\alpha}\bar{\partial}_{\beta}u$

and

(3.22) $\displaystyle\nabla_{\bar{1}}\nabla_{1}\psi\geq$
$\displaystyle\psi_{\zeta_{\alpha}}\nabla_{\alpha}\nabla_{\bar{1}}\nabla_{1}u+\psi_{\bar{\zeta}_{\alpha}}\nabla_{\bar{\alpha}}\nabla_{\bar{1}}\nabla_{1}u-C|A_{1}|^{2}$
$\displaystyle\geq$
$\displaystyle-\mathfrak{g}_{1\bar{1}}\psi_{\zeta_{\alpha}}\nabla_{\alpha}\phi-\mathfrak{g}_{1\bar{1}}\psi_{\bar{\zeta}_{\alpha}}\nabla_{\bar{\alpha}}\phi-C|A|^{2}.$

Finally, from (3.19)-(3.22) we derive

(3.23)
$\displaystyle\mathfrak{g}_{1\bar{1}}G^{i\bar{i}}(\bar{\partial}_{i}\partial_{i}\phi-(1+\gamma)\bar{\partial}_{i}\phi\partial_{i}\phi)\leq$
$\displaystyle C(\mathfrak{g}_{1\bar{1}}|\nabla\phi|+|A|^{2})\Big{(}1+\sum
G^{i\bar{i}}\Big{)}.$

Let $\phi=\log(\eta/h)$ where $h=(1-\gamma|\nabla u|^{2})$, $\gamma$ as
before, and $\eta$ is a smooth function to be chosen; we require $\gamma$
small to satisfy $2\gamma|\nabla u|^{2}\leq 1$. By straightforward
calculations,

(3.24) $\partial_{i}|\nabla
u|^{2}=\nabla_{k}u\nabla_{i}\nabla_{\bar{k}}u+\nabla_{\bar{k}}u\nabla_{i}\nabla_{k}u$

and

(3.25) $\displaystyle\bar{\partial}_{i}\partial_{i}|\nabla u|^{2}=$
$\displaystyle\nabla_{i}\nabla_{\bar{k}}u\nabla_{k}\nabla_{\bar{i}}u+\nabla_{i}\nabla_{k}u\nabla_{\bar{i}}\nabla_{\bar{k}}u$
$\displaystyle+\nabla_{\bar{k}}u\nabla_{\bar{i}}\nabla_{i}\nabla_{k}u+\nabla_{k}u\nabla_{\bar{i}}\nabla_{i}\nabla_{\bar{k}}u$
$\displaystyle=$
$\displaystyle\nabla_{i}\nabla_{\bar{k}}u\nabla_{k}\nabla_{\bar{i}}u+\nabla_{i}\nabla_{k}u\nabla_{\bar{i}}\nabla_{\bar{k}}u$
$\displaystyle+\nabla_{\bar{k}}u\nabla_{k}\nabla_{\bar{i}}\nabla_{i}u+\nabla_{k}u\nabla_{\bar{k}}\nabla_{\bar{i}}\nabla_{i}u$
$\displaystyle+R_{i\bar{i}k\bar{l}}\nabla_{l}u\nabla_{\bar{k}}u-T^{l}_{ik}\nabla_{l\bar{i}}u\nabla_{\bar{k}}u-\overline{T^{l}_{ik}}\nabla_{i\bar{l}}u\nabla_{k}u$
$\displaystyle\geq$
$\displaystyle(1-\gamma)|A_{i}|^{2}+\nabla_{\bar{k}}u\nabla_{k}\nabla_{\bar{i}}\nabla_{i}u+\nabla_{k}u\nabla_{\bar{k}}\nabla_{\bar{i}}\nabla_{i}u-C|\nabla
u|^{2}$

It follows that

$G^{i\bar{i}}\bar{\partial}_{i}h\partial_{i}h\leq
C\gamma^{2}G^{i\bar{i}}|A_{i}|^{2}$

and

(3.26) $\displaystyle-G^{i\bar{i}}\bar{\partial}_{i}\partial_{i}h\geq$
$\displaystyle\gamma(1-\gamma)\sum G^{i\bar{i}}|A_{i}|^{2}-C|A|\sum
G^{i\bar{i}}-C|A|.$

Clearly $|\nabla\phi|\leq|\nabla h|/h+|\nabla\eta|/\eta$ and

(3.27) $\displaystyle G^{i\bar{i}}\bar{\partial}_{i}\phi\partial_{i}\phi\leq$
$\displaystyle\frac{1+\gamma}{h^{2}}G^{i\bar{i}}\bar{\partial}_{i}h\partial_{i}h+\frac{1+\gamma}{\gamma\eta^{2}}G^{i\bar{i}}\bar{\partial}_{i}\eta\partial_{i}\eta.$

We see that

$\displaystyle
G^{i\bar{i}}(\bar{\partial}_{i}\partial_{i}\phi-(1+\gamma)\bar{\partial}_{i}\phi\partial_{i}\phi)\geq$
$\displaystyle-G^{i\bar{i}}\Big{(}\frac{\bar{\partial}_{i}\partial_{i}h}{h}+\frac{3\gamma\bar{\partial}_{i}h\partial_{i}h}{h^{2}}\Big{)}-\frac{C}{\gamma}G^{i\bar{i}}\bar{\partial}_{i}\eta\partial_{i}\eta+G^{i\bar{i}}\bar{\partial}_{i}\partial_{i}\eta$
$\displaystyle\geq$ $\displaystyle\gamma(1-\gamma-C\gamma^{2})\sum
G^{i\bar{i}}|A_{i}|^{2}-\frac{C}{\gamma\eta^{2}}G^{i\bar{i}}\bar{\partial}_{i}\eta\partial_{i}\eta$
$\displaystyle+\frac{1}{\eta}G^{i\bar{i}}\bar{\partial}_{i}\partial_{i}\eta-C|A|\sum
G^{i\bar{i}}-C|A|.$

We can now fix $\gamma$ sufficiently small, further requiring that
$2\gamma+C\gamma^{2}\leq 1$, to derive

(3.28) $\displaystyle\gamma^{2}\mathfrak{g}_{1\bar{1}}\sum
G^{i\bar{i}}|A_{i}|^{2}+\mathfrak{g}_{1\bar{1}}G^{i\bar{i}}\Big{(}\frac{\bar{\partial}_{i}\partial_{i}\eta}{\eta}-\frac{C\bar{\partial}_{i}\eta\partial_{i}\eta}{\eta^{2}}\Big{)}\leq
C|A|^{2}\Big{(}1+\sum G^{i\bar{i}}\Big{)}.$

From Lemma 2.2 we have the following key inequality for each $i\geq 1$,

(3.29)
$G^{i\bar{i}}=\sum_{l=1}^{N}f_{\Lambda_{l}}\frac{\partial\Lambda_{l}}{\partial\lambda_{i}}=\sum_{i\in
I}f_{\Lambda_{I}}\geq c_{1}\sum_{l=1}^{N}f_{\Lambda_{l}},$

where the sum $\sum_{i\in I}f_{\Lambda_{I}}$ is taken over all
$I\in\mathfrak{I}_{K}$ with $i\in I$, since
$\lambda(\mathfrak{g})=(\mathfrak{g}_{1\bar{1}},\ldots,\mathfrak{g}_{n\bar{n}})$.
Note that when $\mathfrak{g}_{i\bar{j}}$ is diagonal, so is $G^{i\bar{j}}$.

Consequently, by (3.28) we obtain

(3.30)
$\displaystyle\frac{c_{1}\gamma^{2}}{2}\mathfrak{g}_{1\bar{1}}|A|^{2}\sum
f_{\Lambda_{l}}+\mathfrak{g}_{1\bar{1}}G^{i\bar{i}}\Big{(}\frac{\bar{\partial}_{i}\partial_{i}\eta}{\eta}-\frac{C\bar{\partial}_{i}\eta\partial_{i}\eta}{\eta^{2}}\Big{)}\leq
C|A|^{2}$

provided that $\mathfrak{g}_{1\bar{1}}$ is large enough.

By the concavity of $f$ and Schwarz inequality, we derive

$\displaystyle\sqrt{\mathfrak{g}_{1\bar{1}}}\sum f_{\Lambda_{l}}=$
$\displaystyle\sqrt{\mathfrak{g}_{1\bar{1}}}\sum f_{\Lambda_{l}}-\sum
f_{\Lambda_{l}}\Lambda_{l}(\mathfrak{g})+\sum
f_{\Lambda_{l}}\Lambda_{l}(\mathfrak{g})$ $\displaystyle\geq$ $\displaystyle
f(\sqrt{\mathfrak{g}_{1\bar{1}}}{\bf{1}})-f(\Lambda(\mathfrak{g}))-\epsilon\sum
f_{\Lambda_{l}}\Lambda_{l}^{2}-\frac{1}{4\epsilon}\sum f_{\Lambda_{l}}$
$\displaystyle\geq$ $\displaystyle\frac{c_{0}}{2}-\epsilon\sum
f_{\Lambda_{l}}\Lambda_{l}^{2}-\frac{1}{4\epsilon}\sum f_{\Lambda_{l}}$

by assumption (1.10), provided that $\mathfrak{g}_{1\bar{1}}$ is sufficiently
large. So from (3.30) we obtain

(3.31) $\displaystyle\mathfrak{g}_{1\bar{1}}|A|^{2}\sum
f_{\Lambda_{l}}+\sqrt{\mathfrak{g}_{1\bar{1}}}|A|^{2}+C\mathfrak{g}_{1\bar{1}}G^{i\bar{i}}\Big{(}\frac{\bar{\partial}_{i}\partial_{i}\eta}{\eta}-\frac{C\bar{\partial}_{i}\eta\partial_{i}\eta}{\eta^{2}}\Big{)}\leq
C|A|^{2}.$

To derive the interior estimate, following [13] we take $\eta$ to be a smooth
function with compact support in $B_{r}\subset M$ satisfying

(3.32) $0\leq\eta\leq 1,~{}~{}\eta|_{B_{\frac{r}{2}}}\equiv
1,~{}~{}|\nabla\eta|\leq\frac{C\sqrt{\eta}}{r},~{}~{}|\nabla^{2}\eta|\leq\frac{C}{r^{2}},$

so that

(3.33) $\displaystyle
G^{i\bar{i}}\bar{\partial}_{i}\partial_{i}\eta-\frac{C}{\eta}G^{i\bar{i}}\bar{\partial}_{i}\eta\partial_{i}\eta\leq\frac{C}{r^{2}}\sum
f_{\Lambda_{i}}.$

We derive a bound $\eta\mathfrak{g}_{1\bar{1}}\leq C/r^{2}$ at $z_{0}$ which
gives (1.13).

Suppose that $(M^{n},\omega)$ is a Hermitian manifold with smooth boundary
$\partial M$. Taking $\eta=1$ we obtain the global estimate (1.21). The
boundary estimate (1.22) may be derived as in [12] with some minor
modifications; we shall omit the details here.

######  Remark 3.1.

When $X$ and $\psi$ are independent of $u$ and $\nabla u$, we obtain

(3.34) $\sup_{B_{\frac{r}{2}}}|\bar{\partial}\partial
u|\leq\frac{C}{r^{2}}\Big{\\{}1+\sup_{B_{r}}|\partial u|^{2}\Big{\\}}.$

##  4\. Gradient estimates

In this section we derive gradient estimates. As usual it requires suitable
growth conditions of $X$ and $\psi$ on $u$ and its gradient. Concerning $X$,
we assume the following sub-linear growth assumption for $D_{\zeta}X$ when
$|\zeta|$ is sufficiently large,

(4.1)
$|D_{\zeta}X(z,u,\zeta,\bar{\zeta})|\leq\varrho_{0}|\zeta|,\;\;D_{u}X\leq(\varrho_{0}|\zeta|^{2}+\varrho_{1})\omega$

where $\varrho_{1}=\varrho_{1}(z,u)$ and
$\varrho_{0}=\varrho_{0}(z,u,|\zeta|)\rightarrow 0^{+}$ as
$|\zeta|\rightarrow\infty$; clearly we may assume $t\varrho_{0}(z,u,t)$ to be
increasing in $t>0$. It follows that

$|X|\leq C\varrho_{0}|\zeta|^{2}+\varrho_{1}(z,u)$

for some function $\varrho_{1}$; we shall only need (for convenience we drop
the constant $C$)

(4.2) $X\leq(\varrho_{0}|\zeta|^{2}+\varrho_{1})\omega$

which will also be crucial to the existence, more precisely to the upper
bound, of admissible solutions of the Dirichlet problem in Theorem 5.2 in
Section 5. We impose a similar condition on $\psi$ but, as in [12], associate
it with the growth of $f$:

(4.3)
$|D_{\zeta}\psi(z,u,\zeta,\bar{\zeta})|\leq\varrho_{0}f(|\zeta|^{2}{\bf{1}})/|\zeta|,\;\;-D_{u}\psi\leq\varrho_{0}f(|\zeta|^{2}{\bf{1}})+\varrho_{1}(z,u)$

which in particular implies

(4.4) $\psi\leq\varrho_{0}f(|\zeta|^{2}{\bf{1}})+\varrho_{1}(z,u).$

######  Theorem 4.1.

Let $u\in C^{3}(\bar{B}_{r})$ be an admissible solution of equation (1.17).
Under conditions (1.5), (1.6), (1.18), (1.11), (1.19), (4.1) and (4.3), $u$
satisfies the following interior gradient estimate

(4.5) $|\partial
u|^{2}\leq\frac{C}{r^{2}}\Big{\\{}1+\sup_{\bar{B}_{r}}u-u\Big{\\}}\;\;\mbox{in
${B}_{\frac{r}{2}}$}$

for some uniform constant $C>0$.

######  Remark 4.2.

We may replace (4.1) and (4.3) by the weaker assumptions (4.2), (4.4) and the
linear-like growth conditions

(4.6) $|D_{\zeta}X(z,u,\zeta,\bar{\zeta})|\leq\varrho(z,u)|\zeta|,$ (4.7)
$|D_{\zeta}\psi(z,u,\zeta,\bar{\zeta})|\leq\varrho(z,u)f(|\zeta|^{2}{\bf{1}})/|\zeta|,$

where $\varrho>0$. In this case we can derive the bound

(4.8) $|\partial
u|^{2}\leq\frac{C}{r^{2}}\Big{\\{}1+u-\inf_{\bar{B}_{r}}u\Big{\\}}^{N}\;\;\mbox{in
${B}_{\frac{r}{2}}$}$

for sufficiently large $N>0$; see e.g. [12].

######  Remark 4.3.

When $\omega$ is Kähler, (4.6) and (4.7) can be replaced by the assumptions

(4.9)
$D^{2}_{p_{i}p_{j}}X(z,u,p)\xi_{i}\xi_{j}\leq\varrho(z,u)|\xi|^{2},\;\;\mbox{for
$\xi\in T_{z}M$},$

that is, $X$ is semi-concave in $\nabla u$, the real gradient of $u$, and

(4.10)
$D^{2}_{p_{i}p_{j}}\psi(z,u,p)\xi_{i}\xi_{j}\geq-\varrho(z,u)f(|p|^{2}{\bf{1}})|\xi|^{2}/|p|^{2},\;\;\xi\in
T_{z}M.$

###### Proof of Theorem 4.1.

Consider

$\max_{\bar{M}}e^{\phi}|\nabla u|^{2}$

where $\phi$ is a function to be chosen. Suppose it is attained at an interior
point $z_{0}\in M$. We choose local coordinates $(z_{1},\ldots,z_{n})$ such
that $g_{i\bar{j}}=\delta_{ij}$, $T_{ij}^{k}=2\Gamma_{ij}^{k}$ and
$\mathfrak{g}_{i\bar{j}}$ are diagonal at $z_{0}$. The function $\log|\nabla
u|^{2}+\phi$ achieves a maximum at $z_{0}$ where we assume $|\nabla u|\geq 1$.
Therefore at $z_{0}$,

(4.11) $\partial_{i}|\nabla u|^{2}+|\nabla
u|^{2}\partial_{i}\phi=0,\;\;\bar{\partial}_{i}|\nabla u|^{2}+|\nabla
u|^{2}\bar{\partial}_{i}\phi=0$

and

(4.12) $G^{i\bar{i}}\frac{\bar{\partial}_{i}\partial_{i}|\nabla
u|^{2}}{|\nabla u|^{2}}-G^{i\bar{i}}\frac{\bar{\partial}_{i}|\nabla
u|^{2}\partial_{i}|\nabla u|^{2}}{|\nabla
u|^{4}}+G^{i\bar{i}}\bar{\partial}_{i}\partial_{i}\phi\leq 0.$

By (3.24), (3.25) and Schwarz inequality,

(4.13) $\bar{\partial}_{i}|\nabla u|^{2}\partial_{i}|\nabla u|^{2}\leq
2|\nabla u|^{2}|Q_{i}|^{2}$

and

(4.14) $\displaystyle\bar{\partial}_{i}\partial_{i}|\nabla u|^{2}\geq$
$\displaystyle(1-\gamma)|Q_{i}|^{2}+\nabla_{\bar{k}}u\nabla_{k}\nabla_{\bar{i}}\nabla_{i}u+\nabla_{k}u\nabla_{\bar{k}}\nabla_{\bar{i}}\nabla_{i}u-C|\nabla
u|^{2}.$

where $0<\gamma<1/6$ and

$|Q_{i}|^{2}=\nabla_{i}\nabla_{\bar{k}}u\nabla_{k}\nabla_{\bar{i}}u+\nabla_{i}\nabla_{k}u\nabla_{\bar{i}}\nabla_{\bar{k}}u=\sum_{k}(|\nabla_{i}\nabla_{\bar{k}}u|^{2}+|\nabla_{i}\nabla_{k}u|^{2}).$

It follows from (3.3) that

(4.15)
$G^{i\bar{i}}\nabla_{k}\nabla_{\bar{i}}\nabla_{i}u=G^{i\bar{i}}(\nabla_{k}\mathfrak{g}_{i\bar{i}}-\nabla_{k}{X}_{i\bar{i}})=\nabla_{k}\psi-G^{i\bar{i}}\nabla_{k}{X}_{i\bar{i}}.$

So

(4.16) $\displaystyle G^{i\bar{i}}\bar{\partial}_{i}\partial_{i}|\nabla
u|^{2}\geq$ $\displaystyle G^{i\bar{i}}(1-\gamma)|Q_{i}|^{2}-C|\nabla
u|^{2}\sum G^{i\bar{i}}+R$

where

$R=2\mathfrak{Re}\\{(\nabla_{k}\psi-G^{i\bar{i}}\nabla_{k}X_{i\bar{i}})\nabla_{\bar{k}}u\\}.$

We derive

(4.17) $\displaystyle
G^{i\bar{i}}\bar{\partial}_{i}\partial_{i}\phi-\frac{1+\gamma}{2}G^{i\bar{i}}\bar{\partial}_{i}\phi\partial_{i}\phi\leq$
$\displaystyle-\frac{R}{|\nabla u|^{2}}+C\sum G^{i\bar{i}}.$

Let $\phi=\log(\eta/h)$ where $h=1+\sup_{M}u-u$. So

$G^{i\bar{i}}\bar{\partial}_{i}\partial_{i}\phi=\frac{1}{h}G^{i\bar{i}}\bar{\partial}_{i}\partial_{i}u+\frac{1}{h^{2}}G^{i\bar{i}}\bar{\partial}_{i}u\partial_{i}u+\frac{1}{\eta}G^{i\bar{i}}\bar{\partial}_{i}\partial_{i}\eta-\frac{1}{\eta^{2}}G^{i\bar{i}}\bar{\partial}_{i}\eta\partial_{i}\eta.$

As before, $|\nabla\phi|\leq|\nabla u|/h+|\nabla\eta|/\eta$ and

$\displaystyle G^{i\bar{i}}\bar{\partial}_{i}\phi\partial_{i}\phi\leq$
$\displaystyle\frac{1+\gamma}{h^{2}}G^{i\bar{i}}\bar{\partial}_{i}u\partial_{i}u+\frac{1+\gamma}{\gamma\eta^{2}}G^{i\bar{i}}\bar{\partial}_{i}\eta\partial_{i}\eta.$

From (3.29) it follows that

(4.18)
$\displaystyle\frac{1}{h^{2}}G^{i\bar{i}}\bar{\partial}_{i}u\partial_{i}u-\frac{1+\gamma}{2}G^{i\bar{i}}\bar{\partial}_{i}\phi\partial_{i}\phi$
$\displaystyle\geq\frac{1-3\gamma}{2h^{2}}G^{i\bar{i}}\bar{\partial}_{i}u\partial_{i}u-\frac{C}{\eta^{2}}G^{i\bar{i}}\bar{\partial}_{i}\eta\partial_{i}\eta$
$\displaystyle\geq\frac{c_{1}|\nabla u|^{2}}{4h^{2}}\sum
G^{i\bar{i}}-\frac{C}{\eta^{2}}G^{i\bar{i}}\bar{\partial}_{i}\eta\partial_{i}\eta.$

By the concavity of $f$ and assumption (1.11),

$\displaystyle|\nabla u|^{2}\sum G^{i\bar{i}}\geq$ $\displaystyle f(|\nabla
u|^{2}{\bf{1}})-f(\Lambda)+G^{i\bar{i}}\mathfrak{g}_{i\bar{i}}\geq f(|\nabla
u|^{2}{\bf{1}})-\psi-C\sum G^{i\bar{i}},$

Similarly,

$\begin{aligned}
G^{i\bar{i}}\bar{\partial}_{i}\partial_{i}u=\,&G^{i\bar{i}}\mathfrak{g}_{i\bar{i}}-G^{i\bar{i}}X_{i\bar{i}}\geq-G^{i\bar{i}}X_{i\bar{i}}-C\sum
G^{i\bar{i}}\end{aligned}.$

Combining the above inequalities, and taking $\eta$ as in the previous
section, we derive

(4.19) $\displaystyle\frac{c_{1}|\nabla u|^{2}}{8h^{2}}\sum
G^{i\bar{i}}+\frac{c_{1}}{8h^{2}}f(|\nabla u|^{2}{\bf{1}})\leq$
$\displaystyle-\frac{1}{h}G^{i\bar{i}}\bar{\partial}_{i}\partial_{i}u-\frac{1}{\eta}G^{i\bar{i}}\bar{\partial}_{i}\partial_{i}\eta+\frac{C}{\eta^{2}}G^{i\bar{i}}\bar{\partial}_{i}\eta\partial_{i}\eta$
$\displaystyle+\frac{c_{1}\psi}{8h^{2}}+C\sum G^{i\bar{i}}-\frac{R}{|\nabla
u|^{2}}$ $\displaystyle\leq$
$\displaystyle\frac{1}{h}G^{i\bar{i}}X_{i\bar{i}}+\frac{c_{1}\psi}{8h^{2}}-\frac{R}{|\nabla
u|^{2}}+\frac{C}{\eta}\sum G^{i\bar{i}}.$

Consequently, we obtain a bound $\eta|\nabla u|^{2}\leq C/r^{2}$ if $\psi$ and
$X$ are independent of $u$ and $\partial u$. In the general case, by (3.21)
and (4.11),

(4.20) $\displaystyle\mathfrak{Re}\\{\nabla_{k}\psi\nabla_{\bar{k}}u\\}=$
$\displaystyle\psi_{u}|\nabla
u|^{2}+\mathfrak{Re}\\{\psi_{k}\nabla_{\bar{k}}u+\psi_{\zeta_{\alpha}}\partial_{\alpha}|\nabla
u|^{2}+\psi_{\zeta_{\alpha}}\Gamma_{\alpha
k}^{l}\nabla_{l}u\nabla_{\bar{k}}u\\}$ $\displaystyle=$ $\displaystyle|\nabla
u|^{2}(\psi_{u}-\mathfrak{Re}\\{\psi_{\zeta_{\alpha}}\partial_{\alpha}\phi\\})+\mathfrak{Re}\\{\psi_{k}\nabla_{\bar{k}}u+\psi_{\zeta_{\alpha}}\Gamma_{\alpha
k}^{l}\nabla_{l}u\nabla_{\bar{k}}u\\}$ $\displaystyle=$ $\displaystyle|\nabla
u|^{2}A-\frac{|\nabla
u|^{2}}{\eta}\mathfrak{Re}\\{\psi_{\zeta_{\alpha}}\partial_{\alpha}\eta\\}$
$\displaystyle\geq$ $\displaystyle|\nabla u|^{2}A-\frac{C|\nabla
u|^{2}|D_{\zeta}\psi|}{\sqrt{\eta}}$

and similarly,

(4.21) $\displaystyle
G^{i\bar{i}}\mathfrak{Re}\\{\nabla_{\bar{k}}u\nabla_{k}X_{i\bar{i}}\\}=$
$\displaystyle|\nabla u|^{2}B-\frac{|\nabla
u|^{2}}{\eta}G^{i\bar{i}}\mathfrak{Re}\\{X_{i\bar{i},\zeta_{\alpha}}\partial_{\alpha}\eta\\}$
$\displaystyle\leq|\nabla u|^{2}B+\frac{C|\nabla
u|^{2}|D_{\zeta}X|}{\sqrt{\eta}}\sum G^{i\bar{i}}$

where

$A=\psi_{u}-\frac{1}{h}\mathfrak{Re}\\{\psi_{\zeta_{\alpha}}\partial_{\alpha}u\\}+\frac{1}{|\nabla
u|^{2}}\mathfrak{Re}\\{\psi_{k}\nabla_{\bar{k}}u+\psi_{\zeta_{\alpha}}\Gamma_{\alpha
k}^{l}\nabla_{l}u\nabla_{\bar{k}}u\\},$
$B=G^{i\bar{i}}X_{i\bar{i},u}-\frac{1}{h}G^{i\bar{i}}\mathfrak{Re}\\{X_{i\bar{i},\zeta_{\alpha}}\partial_{\alpha}u\\}+\frac{1}{|\nabla
u|^{2}}G^{i\bar{i}}\mathfrak{Re}\\{(X_{i\bar{i}k}+X_{i\bar{i},\zeta_{\alpha}}\Gamma_{\alpha
k}^{l}\nabla_{l}u)\nabla_{\bar{k}}u\\}.$

By assumptions (4.1) and (4.3),

(4.22)
$\displaystyle\frac{1}{h}G^{i\bar{i}}X_{i\bar{i}}+\frac{c_{1}\psi}{8h^{2}}-\frac{R}{|\nabla
u|^{2}}\leq$ $\displaystyle CH\sum G^{i\bar{i}}+CE$
$\displaystyle+\frac{C\varrho_{0}|\nabla u|}{\sqrt{\eta}}\sum
G^{i\bar{i}}+\frac{C\varrho_{0}f(|\nabla u|^{2}{\bf{1}})}{|\nabla
u|\sqrt{\eta}}$

where

$E=|\nabla_{z}\psi||\nabla
u|^{-1}+(\psi_{u})^{-}+\psi^{+}+|D_{\zeta}\psi||\nabla
u|\leq\varrho_{0}f(|\nabla u|^{2}{\bf{1}})$

by (4.3) and (4.4), and

$H=|\nabla_{z}X||\nabla
u|^{-1}+\mbox{tr}X^{+}+\mbox{tr}(D_{u}X)^{+}+|D_{\zeta}X||\nabla
u|\leq\varrho_{0}|\nabla u|^{2}$

by (4.1) and (4.2). Plugging these back to (4.22), we obtain a bound
$\eta|\nabla u|^{2}\leq C/r^{2}$ at $z_{0}$ from which (4.5) follows. ∎

######  Remark 4.4.

Assume both $X$ and $\psi$ are independent of $u$. Taking $\eta=1$ we obtain

$|\nabla u|^{2}\leq C\Big{(}1+\sup_{M}u-u\Big{)}\;\;\mbox{on M}$

for any admissible solution $u\in C^{3}(M)$ of equation (1.17). It follows
that

$\Big{|}\Big{(}1+\sup_{M}u-u(x)\Big{)}^{\frac{1}{2}}-\Big{(}1+\sup_{M}u-u(y)\Big{)}^{\frac{1}{2}}\Big{|}\leq
C\mbox{dist}(x,y),\;\;\forall x,y\in M$

where $\mbox{dist}(x,y)$ denotes the distance between $x$ and $y$ in
$(M,\omega)$. In particular,

(4.23) $\sup_{M}u-\inf_{M}u\leq Cd^{2}$

where $d$ is the diameter of $M$.

######  Remark 4.5.

Assume $u>0$ on $B_{r}$ and take $h=u^{2}$ in the previous proof. We can
derive a bound $\eta|\nabla u|^{2}\leq C/r^{2}$ at $z_{0}$ if $\psi$ and $X$
are independent of $u$ and $\partial u$. This gives

$\frac{|\nabla u|^{2}}{u^{2}}\leq C\;\;\mbox{in $B_{\frac{r}{2}}$}.$

Consequently, we obtain the Harnack inequality

$\sup_{B_{\frac{r}{2}}}u\leq C\inf_{B_{\frac{r}{2}}}u.$

######  Remark 4.6.

For a Hermitian manifold $(M^{n},\omega)$ with boundary $\partial M$, taking
$\eta=1$ we derive the global gradient estimate

$\max_{\bar{M}}|\partial u|\leq C\max_{\partial M}|\partial u|+C.$

##  5\. Existence

In this section we outline proofs of Theorems 1.6, 1.7. With the aid of the a
priori estimates in the specific forms derived in the previous sections we can
extend these theorems to cover more general cases where $X$ and $\psi$ are
allowed to depend on $u$ and $\nabla u$.

The following result extends Theorem 1.7.

######  Theorem 5.1.

Let $(M^{n},\omega)$ be a compact Hermitian manifold. Assume that (1.5),
(1.18), (1.11), (1.19), (1.29), (1.30), (4.1) and (4.3) hold. In addition,
assume that $X=X(z,\nabla u)$ and $\psi=\psi(z,\nabla u)$ are both independent
of $u$, and $X(z,0)>0$ on $M$. Then there exists a unique constant $b$ such
that the equation

(5.1) $f(\Lambda(\sqrt{-1}\partial\bar{\partial}u+X[u]))=e^{b}\psi[u]$

has a unique admissible solution $u\in C^{\infty}(M)$ up to a constant.

###### Proof.

To begin with we assume $X[u]=X(z,u,\nabla u)$ and $\psi[u]=\psi(z,u,\nabla
u)$ satisfying

(5.2) $-D_{u}X,\;D_{u}\psi\geq 0.$

First we choose $A>0$ sufficiently large such that $H\equiv
f(\Lambda(A\omega))>0$. For $t\in[0,1]$ consider the equation

(5.3)
$f(\Lambda(\sqrt{-1}\partial\bar{\partial}u+X[u]+tA\omega))=e^{tu}\psi^{t}[u]$

where $\psi^{t}[u]=tH+(1-t)\psi[u]$. Note that for $t>0$, by (5.2)

$D_{u}(e^{tu}\psi^{t}[u])\geq Ht^{2}e^{tu}>0.$

So the solution of equation (5.3) is unique and its linearized operator is
nonsingular for $t>0$. When $t=1$ the solution is $u^{1}=0$.

Suppose $u\in C^{\infty}(M)$ is the admissible solution of equation (5.3) for
fixed $t>0$, and assume $\sup_{M}u\geq 0$. At a point where $u$ attains its
maximum value, by (5.2) we have

$\displaystyle e^{tu}\psi^{t}(z,0,0)\leq$ $\displaystyle
e^{tu}\psi^{t}(z,u,0)$ $\displaystyle\leq$ $\displaystyle
f(\Lambda(X(z,u,0)+tA\omega)\leq f(\Lambda(X(z,0,0)+tA\omega).$

Therefore,

(5.4) $t\sup_{M}u\leq\sup_{z\in
M}\log\frac{f(\Lambda(X(z,0,0)+A\omega)}{\psi^{t}(z,0,0)}\leq C.$

Similarly, suppose $\inf_{M}u\leq 0$. Then

(5.5) $t\inf_{M}u\geq\inf_{z\in
M}\log\frac{f(\Lambda(X(z,0,0))}{\psi^{t}(z,0,0)}\geq-C.$

Consequently, we may apply the continuity method to obtain a unique admissible
solution $u^{t}\in C^{\infty}(M)$ of equation (5.3) for all $t\in(0,1]$ with
the bound

(5.6) $-C\leq t\inf_{M}u^{t}\leq t\sup_{M}u^{t}\leq C,\;\;0<t\leq 1$

where $C$ is independent of $t$.

By (5.6) we can find a sequence $t_{k}\rightarrow 0$ such that both of the
following limits exist

$\lim_{k\rightarrow\infty}t_{k}\inf_{M}u^{t_{k}}=a,\;\;\lim_{k\rightarrow\infty}t_{k}\sup_{M}u^{t_{k}}=b.$

Suppose now that $X$ and $\psi$ are both independent of $u$. By (4.23) we see
that $a=b$ and therefore

$\lim_{k\rightarrow\infty}t_{k}u^{t_{k}}=b\;\;\mbox{on $M$}.$

Moreover, by (4.23) and the estimates we have established there exists a
subsequence of

$\Big{\\{}u^{t_{k}}-\sup_{M}u^{t_{k}}\Big{\\}}$

converging in $C^{2,\alpha}(M)$ to a smooth admissible solution $u$ of
equation (5.1). Clearly,

$\inf_{M}u\geq-Cd^{2},\;\;\sup_{M}u=0$

where $d$ is the diameter of $M$. ∎

We next consider the Dirichlet problem

(5.7) $\left\\{\begin{aligned}
f(\Lambda(\sqrt{-1}\partial\bar{\partial}u+X[u]))=\,&\psi[u]\;\mbox{in
$\bar{M}$},\\\ u=\varphi\;\mbox{on $\partial M$}.\end{aligned}\right.$

Here $(M^{n},\omega)$ is assumed as in Theorem 1.6 to be a Hermitian manifold
with smooth boundary $\partial M$ and compact closure $\bar{M}=M\cup\partial
M$, and $\varphi\in C^{0}(\partial M)$.

######  Theorem 5.2.

In addition to (1.5), (1.6), (1.18), (1.11), (1.19), (4.1), (4.3) and (5.2),
assume that there exists an admissible subsolution $\underline{u}\in
C^{0}(\bar{M})$ satisfying

(5.8) $\left\\{\begin{aligned}
f(\Lambda(\sqrt{-1}\partial\bar{\partial}\underline{u}+X[\underline{u}]))\geq\,&\psi[\underline{u}]\;\mbox{in
$\bar{M}$},\\\ \underline{u}=\varphi\;\mbox{on $\partial
M$}\end{aligned}\right.$

in the viscosity sense. The Dirichlet problem (5.7) admits a unique admissible
solution $u\in C^{\infty}(M)\cap C^{0}(\bar{M})$. Moreover, $u\in
C^{\infty}(\bar{M})$ if $\varphi\in C^{\infty}(\partial M)$.

###### Proof.

We assume $\underline{u}\in C^{\infty}(\bar{M})$ and $\varphi\in
C^{\infty}(\partial M)$; the general case follows from approximation. Let
$u\in C^{4}(M)\cap C^{0}(\bar{M})$ be an admissible solution of problem (5.7).
From the comparison principle and (5.2) we see that $u\geq\underline{u}$ on
$M$ and therefore

(5.9) $\frac{\partial
u}{\partial\nu}\leq\frac{\partial\underline{u}}{\partial\nu}\;\;\mbox{on
$\partial M$}$

where $\nu$ denotes the exterior unit normal to $\partial M$. By assumptions
(4.1) and (5.2),

$\mbox{tr}X[u]\leq\mbox{tr}X(z,\inf_{M}\underline{u},\partial
u,\bar{\partial}u)\leq\varrho_{1}(z,\inf_{M}\underline{u})+\varrho_{0}(z,\inf_{M}\underline{u},|\partial
u|)|\partial u|^{2}.$

Since $\Delta u+\mbox{tr}X[u]\geq 0$, we have

(5.10) $\Delta u+\mbox{tr}X(z,\inf_{M}\underline{u},\partial
u,\bar{\partial}u)\geq 0.$

It follows from the maximum principle that $u\leq h$ on $\bar{M}$ where $h\in
C^{\infty}(\bar{M})$ is the unique solution of

(5.11) $\Delta h+\mbox{tr}X(z,\inf_{M}\underline{u},\partial
h,\bar{\partial}h)=0\;\;\mbox{in $M$},\;\;h=\varphi\;\;\mbox{on $\partial
M$}.$

By the assumption $\varrho_{0}=\varrho_{0}(z,u,t)\rightarrow 0^{+}$ as
$t\rightarrow\infty$ in (4.1) we see that problem (5.11) is solvable.
Combining with the gradient estimate in Section 4 (see Remark 4.6), we have
established the global $C^{1}$ bound

(5.12) $\sup_{\bar{M}}|u|+\sup_{\bar{M}}|\nabla u|\leq C.$

By Theorem 1.5 and Evans-Krylov Theorem we obtain the $C^{2,\alpha}$ estimate

(5.13) $|u|_{C^{2,\alpha}(\bar{M})}\leq C.$

The existence and smoothness of a unique admissible solution now follows from
the standard continuity method and the classical Schauder theory for linear
uniformly elliptic equations. ∎

######  Remark 5.3.

The assumption $\varrho_{0}=\varrho_{0}(z,u,t)\rightarrow 0^{+}$ as
$t\rightarrow\infty$ in (4.1) is critical to the existence of solution of
problem (5.11).

Consider the problem in a domain $\Omega\subset\hbox{\bbbld R}^{n}$

(5.14) $\Delta h+|\nabla h|^{2}+b=0\;\;\mbox{in $\Omega$},\;\;h=0\;\;\mbox{on
$\partial\Omega$}.$

Clearly $h$ satisfies $\Delta e^{h}+be^{h}=0$. Therefore, problem (5.14) has
no solution if $b$ is the first eigenvalue of $\Omega$. Indeed, let $\phi_{1}$
be the first eigenfunction

(5.15) $\Delta\phi_{1}+b\phi_{1}=0\;\;\mbox{in
$\Omega$},\;\;\phi_{1}=0\;\;\mbox{on $\partial\Omega$},$

normalized so that

$\sup_{\Omega}\phi_{1}=1.$

Note that $e^{h}>1\geq\phi_{1}$ in $\Omega$ since $\Delta e^{h}<0$ in
$\Omega$. We can find $t>1$ such that

$0<\inf_{\Omega}(e^{h}-t\phi_{1})<1.$

Consequently, in the interior point $x_{0}$ where $e^{h}-t\phi_{1}$ attains
the minimal value,

$\Delta(e^{h}-t\phi_{1})+b(e^{h}-t\phi_{1})\geq b(e^{h}-t\phi_{1})>0$

which is a contradiction.

When $u$ is a radial function, in spherical coordinates equation (5.14)
reduces to the simple Riccati equation for $y=u^{\prime}$

$y^{\prime}+\frac{n-1}{r}y+y^{2}+b=0.$

In the special case $n=1$ and $b=1$, the solution is $u(x)=-\log\cos x$ which
is only defined for $-\frac{\pi}{2}<x<\frac{\pi}{2}$.

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