1 Introduction

Let \((M^{n}, g)\) be a compact Riemannian manifold of dimension \(n \geq3\) with smooth boundary ∂M, \(\bar{M}:= M \cup \partial M\). In conformal geometry, it is interesting to find a complete metric \(\tilde{g} \in[g]\), the conformal class of g, with which the manifold has prescribed curvature. In general, such conformal deformation can be interpreted by certain partial differential equations. See [8, 13, 22, 25, 26] for more details.

In [8], Guan studied the existence of a complete conformal metric g̃ of negative Ricci curvature on M satisfying

$$ f \bigl( - \lambda\bigl(\tilde{g}^{-1} \mathrm{Ric}_{\tilde{g}}\bigr)\bigr) = \psi\quad \mbox{in } M, $$
(1.1)

where \(\mathrm{Ric}_{\tilde{g}}\) is the Ricci tensor of g̃, and \(\lambda(\tilde{g}^{-1} \mathrm{Ric}_{\tilde{g}}) = (\lambda_{1}, \ldots , \lambda_{n})\) are the eigenvalues of \(\tilde{g}^{-1} \mathrm {Ric}_{\tilde{g}}\). The transformation formula for the Ricci tensor under conformal deformation \(\tilde{g} = e^{2 u} g\) is given by

$$\frac{1}{n - 2} \mathrm{Ric}_{\tilde{g}} = \frac{1}{n - 2} \mathrm{Ric}_{g} - \nabla^{2} u - \biggl(\frac {\Delta u}{n - 2} + \vert \nabla u \vert ^{2} \biggr) g + du \otimes du, $$

where ∇u, \(\nabla^{2} u\), and Δu denote the gradient, Hessian, and Laplacian of u with respect to the metric g, respectively. When f is homogenous of degree one, it is easy to verify that equation (1.1) is equivalent to the following form:

$$ f \biggl(\lambda \biggl(g^{-1} \biggl[ \nabla^{2} u + \frac{\Delta u}{n - 2} g + \vert \nabla u \vert ^{2} g - du \otimes du - \frac{\mathrm{Ric}_{g}}{n - 2} \biggr] \biggr) \biggr) = \frac{\psi(x)}{n - 2} e^{2u}. $$
(1.2)

In this paper, we study the obstacle problem of equation (1.2). More generally, let

$$T [u] := \nabla^{2} u + s \, du \otimes du + \biggl( \gamma\Delta u - \frac{t}{2} \vert \nabla u \vert ^{2} \biggr) g + \chi, $$

where χ is a smooth \((0,2)\) tensor, \(\gamma> 0\) is a constant, and \(s, t \in\mathbb{R}\). We consider the following equation:

$$ \max \bigl\{ u -h, - \bigl(f \bigl(\lambda\bigl(g^{-1} T [u]\bigr)\bigr) - \psi[u]\bigr) \bigr\} = 0 \quad\mbox{in } M $$
(1.3)

with the Dirichlet boundary condition

$$ u = \varphi\quad\mbox{on } \partial M, $$
(1.4)

where \(h \in C^{3} (\bar{M})\), \(\varphi\in C^{4} (\partial M)\), \(h > \varphi\) on ∂M, \(\psi[u] = \psi(x, u)\) is a positive function in \(C^{3} (\bar{M} \times\mathbb{R})\).

Equations as (1.1) and (1.3) are the Hessian equations, which were well studied by many authors such as [2, 7, 9–12, 23, 24]. Generally, \(f \in C^{2} (\Gamma) \cap C^{0} (\bar{\Gamma})\) is a symmetric function of \(\lambda\in\mathbb{R}^{n}\), defined in an open, convex, and symmetric cone \(\Gamma\subsetneqq\mathbb{R}^{n}\), with vertex at the origin, which contains the positive cone: \(\Gamma_{n}^{+} : =\{\lambda\in\mathbb{R}^{n}: \mbox{each component } \lambda_{i}>0\}\) and satisfies the following fundamental structure conditions:

$$ f_{i} \equiv\frac{\partial f}{\partial\lambda_{i}} > 0\quad \mbox{in } \Gamma, 1 \leq i \leq n, $$
(1.5)
$$ \mbox{$f$ is a concave function}, $$
(1.6)

and

$$ f > 0 \quad\mbox{in } \Gamma, \qquad f = 0 \quad\mbox{on } \partial\Gamma. $$
(1.7)

Here, for convenience, we also assume that

$$ \mbox{$f$ is homogeneous of degree one}. $$
(1.8)

We observe that by the concavity and homogeneity of f,

$$ \sum f_{i} (\lambda) = f (\lambda) + \sum f_{i} (\lambda) (1 - \lambda _{i}) \geq f (1, \ldots, 1) > 0 \quad\mbox{in } \Gamma. $$
(1.9)

Important classes of f are the elementary symmetric functions and their quotients, i.e.,

$$f (\lambda) = (\sigma_{k})^{\frac{1}{k}} (\lambda) : = \biggl(\sum _{1 \leq i_{1} < \cdots< i_{k} \leq n} \lambda_{i_{1}} \cdots \lambda_{i_{k}} \biggr)^{\frac {1}{k}}, \quad 1 \leq k \leq n, $$

and

$$f (\lambda) = \biggl(\frac{\sigma_{k}}{\sigma_{l}} \biggr)^{\frac {1}{k-l}}, \quad0 \leq l < k \leq n. $$

Let F be defined by \(F (r) = f (\lambda(r))\) for \(r = \{r_{ij}\} \in\mathcal{S}^{n \times n}\) with \(\lambda(r) \in\Gamma\), where \(\mathcal{S}^{n \times n}\) is the set of \(n \times n\) symmetric matrices. It is shown in [2] that (1.5) implies F is an elliptic operator and (1.6) ensures that F is concave.

A function \(u \in C^{2} (M)\) is called admissible at \(x \in M\) if \(\lambda(g^{-1} T [u]) (x) \in\Gamma\), and we call it admissible in M when it is admissible at each x in M. In this paper, we prove the existence of an admissible viscosity solution of (1.3) and (1.4) in \(C^{1, 1} (\bar{M})\) (see [1, 3] for the definition of viscosity solutions).

Many authors have studied various obstacle problems. In [6], Gerhardt considered a hypersurface bounded from below by an obstacle with prescribed mean curvature in \(\mathbb{R}^{n}\). Lee [17] considered the obstacle problem for the Monge–Ampère equation (i.e., \(f = (\sigma_{n})^{\frac{1}{n}}\)) for the case that \(T [u] = D^{2} u\), \(\psi\equiv1\), and \(\varphi\equiv0\), and proved the \(C^{1,1}\) regularity of the viscosity solution in a strictly convex domain in \(\mathbb{R}^{n}\). Xiong and Bao [27] extended the work of Lee to a nonconvex domain in \(\mathbb{R}^{n}\) with general ψ and φ under additional assumptions. Bao, Dong, and Jiao treated a class of obstacle problems in [1] assuming that \(T[u] = \nabla^{2} u + A(x, u, \nabla u)\), under a certain technical assumption. Because of the term \(\gamma\Delta u\) (\(\gamma> 0\)), here we only need a minimal amount of assumptions. For other works, see [4, 14, 15, 18–21].

Our main result is the following theorem.

Theorem 1.1

Assume that (1.5)–(1.8) and either the following condition

$$ \lim_{z \rightarrow+ \infty} \psi(x, z) = +\infty, \quad\forall x \in\bar{M}, $$
(1.10)

or

$$ \frac{2s - n t}{1 + n\gamma} < 2 \lambda_{1} $$
(1.11)

hold, where \(\lambda_{1}\) is the first eigenvalue of the problem

$$ \textstyle\begin{cases} \Delta u + \lambda(\operatorname{tr} \chi)^{+} u = 0 & \textit{on } \bar{M},\\ u = 0 & \textit{on } \partial M \end{cases} $$
(1.12)

(\(\lambda_{1} = + \infty\) if \(\operatorname{tr} \chi\leq0\)). Then there exists a viscosity solution \(u \in C^{1,1} (\bar{M})\) to (1.3) and (1.4), if there exists a subsolution \(\underline{u} \in C^{0} (\bar{M}) \cap C^{1} (\bar{M} _{\delta})\) for some \(\delta> 0\) such that

$$ \textstyle\begin{cases} f (\lambda(g^{-1} T [\underline{u}])) \geq\psi[\underline{u}], & \textit{in } M,\\ \underline{u} = \varphi, & \textit{on } \partial M, \\ \underline{u} \leq h, & \textit{in } M, \end{cases} $$
(1.13)

where \(M_{\delta}= \{x \in M: \operatorname{dist} (x, \partial M) \leq \delta\}\). Moreover, we have that \(u \in C^{3, \alpha} (E)\) for any \(\alpha\in (0, 1)\), and \(f(\lambda(g^{-1} T [u])) = \psi[u]\) in E, where \(E:= \{x \in M: u (x) < h (x)\}\).

Remark 1.2

(1.10), as well as (1.11), is used in Lemma 3.2 to derive an upper bound for u. Assumption (1.13) is just applied to derive a lower bound for u on M and \(\nabla_{\nu} u\) on ∂M, where ν is the interior unit normal to ∂M.

Remark 1.3

We can construct some subsolutions of (1.2) satisfying (1.13) as in [15] following ideas from [2] and [7] since

$$\vert \nabla u \vert ^{2} g - du \otimes du $$

is positive definite and that we can obtain a priori upper bound of any admissible function (Lemma 3.2) under additional conditions that there exists a sufficiently large number \(R > 0\) such that at each point \(x \in\partial M\),

$$ (\kappa_{1}, \ldots, \kappa_{n-1}, R) \in \Gamma, $$
(1.14)

where \(\kappa_{1}, \ldots, \kappa_{n-1}\) are the principal curvatures of ∂M with respect to the interior normal, and that for every \(C > 0\) and every compact set K in Γ there is a number \(R = R (C, K)\) such that

$$ f (R \lambda) \geq C \quad\mbox{for all } \lambda\in K. $$
(1.15)

We use a penalization technique to prove the existence of viscosity solutions to (1.3) and (1.4). We shall consider the following singular perturbation problem:

$$ \textstyle\begin{cases} f (\lambda( g^{-1} T [u] )) = \psi[u] + \beta_{\varepsilon}(u-h) & \mbox{in } M,\\ u = \varphi& \mbox{on } \partial M, \end{cases} $$
(1.16)

where the penalty function \(\beta_{\varepsilon}\in C^{2} (\mathbb{R})\) satisfies

$$ \begin{aligned} & \beta_{\varepsilon}, \beta'_{\varepsilon}, \beta''_{\varepsilon}\geq 0 \quad\mbox{on } \mathbb{R}, \beta_{\varepsilon}(z) = 0, \mbox{ whenever } z \leq0; \\ & \beta_{\varepsilon}(z) \rightarrow\infty\quad\mbox{as } \varepsilon \rightarrow0^{+}, \mbox{ whenever } z > 0. \end{aligned} $$
(1.17)

An example given in [27] is

$$ \beta_{\varepsilon}(z) = \textstyle\begin{cases} 0, & z \leq0,\\ z^{3} / \varepsilon, & z > 0, \end{cases} $$
(1.18)

for \(\varepsilon\in(0, 1)\). Observe that \(\underline{u}\) is also a subsolution to (1.16).

Let

$$\mathscr{U} = \bigl\{ u_{\varepsilon} | u_{\varepsilon} \in C^{4}( \bar{M}) \mbox{ is an admissible solution of } (1.16) \mbox{ with } u_{\varepsilon} \geq\underline{u} \mbox{ on } \bar{M} \bigr\} . $$

We aim to derive the uniform bound

$$ \vert u_{\varepsilon} \vert _{C^{2} (\bar{M})} \leq C $$
(1.19)

for \(u_{\varepsilon}\in\mathscr{U}\), where C is independent of ε. After establishing (1.19), the equation (1.16) becomes uniformly elliptic by (1.7). By Evans–Krylov [5], [16] theorem, we can derive the \(C^{2, \alpha}\) estimates (which may depend on ε) of \(u_{\varepsilon}\). Higher estimates can be derived by Schauder theory. Following the proof as in [8] or [1], we can prove there exists an admissible solution \(u_{\varepsilon}\) to (1.16). Then we can conclude by (1.19) that there exists a viscosity solution \(u \in C^{1, 1} (\bar{M})\) to (1.3) and (1.4), see [1, 27].

Thus, our main work is focused on the a priori estimates for admissible solutions up to their second order derivatives. In Sect. 2, we achieve the estimates for second order derivatives. Finally, we end this paper with gradient and \(C^{0}\) estimates in Sect. 3.

2 Estimates for second order derivatives

In this section, we prove a priori estimates of second order derivatives for admissible solutions. From now on, we drop the subscript ε when there is no possible confusion.

Theorem 2.1

Assume that f satisfies (1.5)–(1.8) and \(u \in C^{4} (\bar{M})\) is an admissible solution to (1.16). Then

$$ \sup_{M} \bigl\vert \nabla^{2} u \bigr\vert \leq C \Bigl(1 + \sup_{\partial M} \bigl\vert \nabla^{2} u \bigr\vert \Bigr), $$
(2.1)

where C depends on \(|u|_{C^{1}(\bar{M})}\) and other known data.

Proof

Set

$$W(x) = \max_{ \xi\in T_{x} M, \vert \xi \vert = 1} \bigl( \nabla_{\xi\xi} u + s \vert \nabla_{\xi} u \vert ^{2} \bigr) e^{\phi}, \quad x \in\bar{M}, $$

where ϕ is a function to be determined. Assume that W is achieved at an interior point \(x_{0} \in M\) and a unit direction \(\xi\in T_{x_{0}} M\). Choose a smooth orthonormal local frame \(e_{1}, \ldots, e_{n}\) about \(x_{0}\) such that \(\xi= e_{1}\), \(\nabla_{i} e_{j} (x_{0}) = 0\) and that \(T_{ij} (x_{0})\) is diagonal. We write \(G = \nabla_{11} u + s |\nabla _{1} u|^{2} \). Assume \(G (x_{0}) > 0\) (otherwise we are done).

At the point \(x_{0}\), where the function \(\log G + \phi\) (defined near \(x_{0}\)) attains its maximum, we have

$$ \frac{\nabla_{i} G}{G} + \nabla_{i} \phi= 0, \quad i = 1, \ldots, n, $$
(2.2)

and

$$ \frac{\nabla_{ii} G}{G} - \biggl(\frac{\nabla_{i} G}{G} \biggr)^{2} + \nabla_{ii} \phi\leq0. $$
(2.3)

By (2.3) we have

$$ F^{ii} \bigl(\nabla_{ii} G + G \nabla_{ii} \phi- G \vert \nabla_{i} \phi \vert ^{2} \bigr) \leq0 $$
(2.4)

and

$$ \Delta G + G \Delta\phi- G \vert \nabla\phi \vert ^{2} \leq0. $$
(2.5)

Since \(\gamma> 0\), we obtain

$$ F^{ii} \bigl(\nabla_{ii} G + \gamma\Delta G + G \nabla_{ii} \phi+ \gamma G \Delta\phi- G \vert \nabla_{i} \phi \vert ^{2} - \gamma G \vert \nabla\phi \vert ^{2} \bigr) \leq0. $$
(2.6)

By calculation, we get

$$ \nabla_{i} G = \nabla_{i11} u + 2 s \nabla_{1} u \nabla_{i1} u, $$
(2.7)

and

$$ \nabla_{ii} G = \nabla_{ii11} u + 2 s \bigl( \vert \nabla_{i1} u \vert ^{2} + \nabla_{1} u \nabla_{ii1} u\bigr). $$
(2.8)

Recall the formula for interchanging order of covariant derivatives

$$ \nabla_{ijk} v - \nabla_{kij} v = R^{l}_{kij} \nabla_{l} v, $$
(2.9)

and

$$ \begin{aligned}[b] \nabla_{ijkl} v - \nabla_{klij} v = {}&R^{m}_{ljk} \nabla_{im} v + \nabla_{i} R^{m}_{ljk} \nabla_{m} v + R^{m}_{lik} \nabla_{jm} v \\ &{}+ R^{m}_{jik} \nabla_{lm} v + R^{m}_{jil} \nabla_{km} v + \nabla_{k} R^{m}_{jil} \nabla_{m} v. \end{aligned} $$
(2.10)

It follows from (2.10)

$$ \nabla_{ii} G \geq\nabla_{11ii} u + 2 s \bigl( \vert \nabla_{i1} u \vert ^{2} + \nabla _{1} u \nabla_{1ii} u\bigr) - C ( 1 + G), $$
(2.11)

and

$$ \begin{aligned}[b] \nabla_{ii} G + \gamma \Delta G \geq{}& \nabla_{11ii} u + 2 s \bigl( \vert \nabla_{i1} u \vert ^{2} + \nabla_{1} u \nabla_{1ii} u\bigr) + \gamma\nabla _{11} (\Delta u) \\ & {} + 2 s \gamma \bigl( \vert \nabla_{i1} u \vert ^{2} + \nabla_{1} u \nabla_{1} (\Delta u) \bigr) - C (1 + G). \end{aligned} $$
(2.12)

Differentiating equation (1.16) once at \(x_{0}\), we obtain for \(1 \leq k \leq n\),

$$ \nabla_{k} F = F^{ii} \nabla_{k} T_{ii} = \psi_{x_{k}} + \psi_{z} \nabla_{k} u + \nabla_{k} \beta_{\varepsilon}(u - h). $$
(2.13)

It is easy to see that

$$ \begin{aligned}[b] F^{ii} \nabla_{1} (\nabla_{ii} u + \gamma\Delta u) &= F^{ii} \nabla_{1} \biggl(T_{ii} [u] - s \vert \nabla_{i} u \vert ^{2} + \frac{t}{2} \vert \nabla u \vert ^{2} - \chi_{ii} \biggr) \\ &\geq\nabla_{1} F - 2s F^{ii} \nabla_{i} u \nabla_{1i} u + t \nabla_{k} u \nabla_{1k} u\sum _{i} F^{ii} - \sum _{i} F^{ii} \end{aligned} $$
(2.14)

and that

$$ \begin{aligned}[b] F^{ii} \nabla_{11} (\nabla_{ii} u + \gamma\Delta u) = {}& F^{ii} \nabla_{11} \biggl(T_{ii} [u] - s \vert \nabla_{i} u \vert ^{2} + \frac {t}{2} \vert \nabla u \vert ^{2} - \chi_{ii} \biggr) \\ \geq{}& F^{ii} \nabla_{11} T_{ii} [u] - 2s F^{ii} \bigl( \nabla_{i} u \nabla_{11i} u + \vert \nabla_{1i} u \vert ^{2} \bigr) \\ &{} + t \sum_{k} \bigl(\nabla_{k} u \nabla_{11k} u + \vert \nabla_{1k} u \vert ^{2} \bigr) \sum_{i} F^{ii} - C \sum _{i} F^{ii}. \end{aligned} $$
(2.15)

With (2.9) we see

$$ \begin{aligned}[b] 2s \nabla_{i} u \nabla_{11i} u & \leq2s \nabla_{i} u (\nabla_{i} G - 2s \nabla_{1} u \nabla_{i1} u) + C \\ & \leq- 4s^{2} \nabla_{i} u \nabla_{1} u \nabla_{i1} u + C \bigl( 1 + G \vert \nabla\phi \vert \bigr), \end{aligned} $$
(2.16)

and similarly

$$ t \nabla_{k} u \nabla_{11k} u \geq- 2st \nabla_{k} u \nabla_{1} u \nabla _{k1} u - C \bigl( 1 + G \vert \nabla\phi \vert \bigr). $$
(2.17)

With (2.12), (2.14)–(2.17), and the concavity of F, we derive

$$ \begin{aligned} F^{ii} ( \nabla_{ii} G + \gamma\Delta G) \geq{}& \nabla_{11} F + 2s \nabla_{1} u \nabla_{1} F + (2s \gamma+ t) \sum _{k} \vert \nabla_{1k} u \vert ^{2} \sum F^{ii} \\ & {} - C \biggl(G + G \vert \nabla\phi \vert + \sum _{j,k} \vert \nabla_{jk} u \vert \biggr) \\ \geq{} & \nabla_{11} F + 2s \nabla_{1} u \nabla_{1} F - C \bigl(G^{2} + G \vert \nabla\phi \vert \bigr). \end{aligned} $$
(2.18)

By (1.9) and \(\beta_{\varepsilon}'' > 0\) it follows from (2.6) and (2.18) that

$$ \begin{aligned}[b] &F^{ii} \bigl( \nabla_{ii} \phi- \vert \nabla_{i} \phi \vert ^{2} \bigr) + \gamma \bigl( \Delta\phi- \vert \nabla\phi \vert ^{2} \bigr) \sum F^{ii} \\ &\quad\leq C \bigl(G + \vert \nabla\phi \vert \bigr) \sum F^{ii} + \biggl(\frac {C}{G} - 1 \biggr)\beta'_{\varepsilon} (u - h). \end{aligned} $$
(2.19)

Let

$$\phi:= \eta(w) = \biggl(1 - \frac{w}{2 a} \biggr)^{-1/2} , \quad w = \frac{ \vert \nabla u \vert ^{2}}{2}, $$

where \(a > \sup_{M} w\) is a constant to be determined. We have

$$1 \leq\eta< \sqrt{2} , \quad\eta' = \frac{\eta^{3}}{4 a}, \qquad \eta'' = \frac{3 \eta^{\prime2}}{\eta} $$

and

$$ \nabla_{ii} \phi- \vert \nabla_{i} \phi \vert ^{2} = \eta' \nabla_{ii} w + \bigl( \eta'' - \eta^{\prime2} \bigr) \vert \nabla_{i} w \vert ^{2} \geq\eta' \nabla_{ii} w. $$
(2.20)

Next, by (2.14)

$$ \begin{aligned}[b] &F^{ii} ( \nabla_{ii} w + \gamma\Delta w ) \\ &\quad= F^{ii} \biggl(\sum_{l} \vert \nabla_{il} u \vert ^{2} + \gamma\sum _{k,l} \vert \nabla_{kl} u \vert ^{2} \biggr) + F^{ii} \nabla_{l} u \bigl( \nabla_{iil} u + \gamma\Delta(\nabla_{l} u) \bigr) \\ &\quad\geq F^{ii} \nabla_{l} u \biggl(\nabla_{lii} u + \gamma\sum_{k}\nabla_{lkk} u \biggr) + \bigl(\gamma G^{2} - C G \bigr) \sum F^{ii} \\ &\quad\geq- C \beta'_{\varepsilon} (u - h) + \bigl(\gamma G^{2} - C G \bigr) \sum F^{ii}. \end{aligned} $$
(2.21)

Combining (2.19), (2.20), (2.21), and \(|\nabla \phi| \leq C \eta' G\), we have

$$ \eta' \bigl(\gamma G^{2} - C G \bigr) \sum F^{ii} \leq C \bigl(G + \eta' G \bigr) \sum F^{ii} + \biggl(\frac{C}{G} - 1 + C \eta' \biggr)\beta '_{\varepsilon} (u - h). $$
(2.22)

We could assume that \(G \geq2 C\). When \(a > 2 C\), the coefficient of \(\beta'_{\varepsilon}(u - h)\) is negative. Then we can derive \(G \leq\frac{4 a C}{\gamma}\). □

To derive the boundary estimates for \(\nabla^{2} u\), we note that \(\operatorname{tr} (s du \otimes du - \frac{t}{2} |\nabla u|^{2} g + \chi ) \leq C\) on M̄, where C is independent of ε, though it may depend on \(|u|_{C^{1}(\bar{M})}\). As in [1, 4], let H be the solution to

$$\textstyle\begin{cases} (1 + n \gamma )\Delta H + C = 0 & \mbox{in } M, \\ H = \varphi& \mbox{on } \partial M. \end{cases} $$

Then we have \(u \leq H\) in M by the maximum principle and \(\beta_{\varepsilon}(u - h) \equiv0\) in \(M_{\delta}= \{x \in M: \operatorname{dist} (x, \partial M) \leq\delta\}\), where δ is sufficiently small. Thus,

$$ \textstyle\begin{cases} f (\lambda( g^{-1} T[u] )) = \psi[u] & \mbox{in } M_{\delta},\\ u = \varphi& \mbox{on } \partial M. \end{cases} $$
(2.23)

By the same arguments of Sect. 4 in [8], we obtain that

$$ \sup_{\partial M} \bigl\vert \nabla^{2} u \bigr\vert \leq C, $$
(2.24)

where C depends on \(|u|_{C^{1}(\bar{M})}\) and other known data.

Combining (2.1) and (2.24), we therefore get the full estimates for second order derivatives.

3 Gradient estimates, maximum principle, and existence

For the gradient estimates, we have the following theorem.

Theorem 3.1

Assume that (1.5)–(1.8) hold. Let \(u \in C^{3} (\bar{M})\) be an admissible solution to (1.16). Then

$$ \sup_{M} \vert \nabla u \vert \leq C \Bigl(1 + \sup_{\partial M} \vert \nabla u \vert \Bigr), $$
(3.1)

where C depends on \(|u|_{C^{0} (\bar{M})}\) and other known data.

Proof

Suppose that \(we^{\phi}\), where \(w = \frac{|\nabla u|^{2}}{2}\) and \(\phi = \phi(u)\) to be determined satisfying that \(\phi' (u) > 0\), achieves a maximum at an interior point \(x_{0} \in M\). As before, we choose a smooth orthonormal local frame \(e_{1}, \ldots, e_{n}\) about \(x_{0}\) such that \(\nabla_{e_{i}} e_{j} = 0\) at \(x_{0}\) and \(\{T_{ij} (x_{0})\}\) is diagonal. Differentiating \(we^{\phi}\) at \(x_{0}\) twice, we have

$$ \nabla_{i} w + w \nabla_{i} \phi= 0 $$
(3.2)

and

$$ \nabla_{ii} w - w (\nabla_{i} \phi)^{2} + w \nabla_{ii} \phi\leq0. $$
(3.3)

Differentiating w, we see

$$\nabla_{i} w = \sum_{k} \nabla_{k} u \nabla_{ik} u, \qquad \nabla_{ii} w = \sum_{k} (\nabla_{ik} u)^{2} + \sum_{k} \nabla_{k} u \nabla_{iik} u . $$

Using (3.2) it follows from (3.3) that

$$ F^{ii} \biggl(\delta_{kl} - \frac{\nabla_{k} u \nabla_{l} u}{2 w} \biggr) \nabla_{ik} u \nabla_{il} u + F^{ii} \nabla_{k} u \nabla_{iik} u - w F^{ii} \biggl( \frac{(\nabla_{i} \phi)^{2}}{2} - \nabla_{ii} \phi \biggr) \leq0 $$
(3.4)

and

$$ \sum_{i,k,l} \biggl( \delta_{kl} - \frac{\nabla_{k} u \nabla_{l} u}{2 w} \biggr) \nabla_{ik} u \nabla_{il} u + \nabla_{k} u \Delta(\nabla_{k} u) - \frac{w}{2} \vert \nabla\phi \vert ^{2} + w \Delta\phi \leq0. $$
(3.5)

Note that the first term in (3.4) and (3.5) is nonnegative. Multiply \(\gamma\sum F^{ii}\) to (3.5) and add what we got to (3.4). Thus, by (2.9) we obtain

$$ \begin{aligned}[b] & F^{ii} \nabla_{k} u (\nabla_{kii} u + \gamma\nabla_{k} \Delta u ) - \frac{w}{2} F^{ii} \bigl( \vert \nabla_{i} \phi \vert ^{2} + \gamma \vert \nabla\phi \vert ^{2} \bigr) \\ &\quad{}+ w F^{ii} ( \nabla_{ii} \phi+ \gamma\Delta\phi ) \leq C \vert \nabla u \vert ^{2} \sum F^{ii} . \end{aligned} $$
(3.6)

Now we compute the first term in (3.6). Firstly, we have

$$\nabla_{i} \phi= \phi' \nabla_{i} u, \qquad \nabla_{ii} \phi= \phi' \nabla_{ii} u + \phi'' (\nabla_{i} u)^{2}. $$

Using (3.2), we easily get that

$$ \begin{aligned}[b] & F^{ii} \nabla_{k} u (\nabla_{kii} u + \gamma\nabla_{k} \Delta u ) \\ & \quad= F^{ii} \nabla_{k} u \nabla_{k} \biggl( T_{ii} - s \vert \nabla_{i} u \vert ^{2} + \frac{t}{2} \vert \nabla u \vert ^{2} - \chi_{ii} \biggr) \\ & \quad= \nabla_{k} u \nabla_{k} (\psi+ \beta_{\varepsilon}) + w \phi ' F^{ii} \bigl( 2 s \vert \nabla_{i} u \vert ^{2} - t \vert \nabla u \vert ^{2} \bigr) - F^{ii} \nabla_{k} u \nabla_{k} \chi_{ii}. \end{aligned} $$
(3.7)

By the homogeneity of F, we also get

$$ \begin{aligned}[b] & F^{ii} ( \nabla_{ii} \phi+ \gamma\Delta\phi ) \\ & \quad= \phi'' F^{ii} \bigl( \vert \nabla_{i} u \vert ^{2} + \gamma \vert \nabla u \vert ^{2} \bigr) + \phi' F^{ii} \biggl( T_{ii} - s \vert \nabla_{i} u \vert ^{2} + \frac {t}{2} \vert \nabla u \vert ^{2} - \chi_{ii} \biggr) \\ & \quad= \phi'' F^{ii} \bigl( \vert \nabla_{i} u \vert ^{2} + \gamma \vert \nabla u \vert ^{2} \bigr) + \phi' \biggl(F - s F^{ii} \vert \nabla_{i} u \vert ^{2} + \frac{t}{2} F^{ii} \vert \nabla u \vert ^{2} - F^{ii} \chi_{ii} \biggr). \end{aligned} $$
(3.8)

According to (3.7) and (3.8), it follows from (3.6)

$$ \begin{aligned}[b] & \gamma \vert \nabla u \vert ^{2} \biggl( \phi'' - \frac{1}{2} \bigl(\phi'\bigr)^{2} - \frac {t}{2 \gamma} \phi' \biggr) \sum F^{ii} + \biggl( \phi'' - \frac{1}{2} \bigl(\phi' \bigr)^{2} + s \phi' \biggr) F^{ii} ( \nabla_{i} u)^{2} \\ & \quad\leq- \phi' \bigl(\psi+ \beta_{\varepsilon} - F^{ii} \chi_{ii} \bigr) + C \sum F^{ii} - \frac{\nabla_{k} u \nabla_{k} (\psi+ \beta_{\varepsilon})}{w} \\ & \quad\leq- \biggl( \phi' \beta_{\varepsilon} + \frac{\nabla_{k} u \nabla_{k} (u - h) \beta'_{\varepsilon}}{w} \biggr) + C \sum F^{ii} + C . \end{aligned} $$
(3.9)

Let

$$\phi(u) = v^{- a}, \qquad v = 1 - u + \sup_{M} u. $$

We have

$$\phi' (u) = a v^{- a - 1}, \qquad\phi'' (u) = \frac{(a + 1 )\phi '}{v} , $$

and

$$\phi'' - \frac{1}{2} \bigl( \phi'\bigr)^{2} = \phi' \biggl( \frac{a + 1}{v} - \frac{a v^{- a}}{2v} \biggr) \geq\frac{\phi' a}{2v} > 0 $$

since \(v^{- a} \leq1\). When \(|\nabla u (x_{0})|\) is sufficiently large, we see \(\nabla_{k} u \nabla_{k} (u - h) > 0\). Hence we have that the first term on the right-hand side of (3.9) is negative as \(\beta_{\varepsilon}, \beta'_{\varepsilon} > 0\). From (3.9) and (1.9) when a is sufficiently large, we then obtain that

$$ \frac{\phi' a \gamma \vert \nabla u \vert ^{2} }{4 v} \leq C, $$
(3.10)

from which we conclude that (3.1) holds. □

In order to prove (1.19), it remains to bound \(\sup_{M} |u| + \sup_{\partial M} |\nabla u|\). We quote two lemmas in [8], the ingredients of whose proofs are the maximum principle.

Lemma 3.2

If either (1.10) or (1.11) holds, then any admissible solution u of (1.16) admits the a priori bound

$$ \sup_{M} u \leq c_{0}. $$
(3.11)

Lemma 3.3

If u is admissible such that \(\operatorname{tr} T[u] \geq0\) and \(|u|_{C^{0}(M)} \leq\mu\), then

$$ \sup_{\partial M} \nabla_{\nu}u \leq c_{1} (\mu), $$
(3.12)

where ν is the interior unit normal to ∂M.

Now with the above two lemmas and the fact \(\nabla_{\nu}u \geq\nabla _{\nu}\underline{u}\) on ∂M when \(u \in\mathscr{U}\), we then have the following.

Theorem 3.4

Suppose that (1.5)–(1.8), and either (1.10) or (1.11) hold. Then, for \(u \in\mathscr{U}\), (1.19) holds.

Therefore, the uniform estimates (1.19) ensure that there exist a subsequence \(\{u_{\varepsilon_{k}}\}\) of \(\{u_{\varepsilon}\}\) and a function \(u \in C^{1,1} (\bar{M})\) such that \(u_{\varepsilon_{k}} \rightarrow u\) in M as \(\varepsilon_{k} \rightarrow0\). It is easy to verify that u satisfies (1.3) and (1.4) and \(u \in C^{3, \alpha} (E)\) for any \(\alpha\in(0, 1)\). Consequently, Theorem 1.1 is established.