Asymptotic Dirichlet problems in warped products

We study the asymptotic Dirichlet problem for Killing graphs with prescribed mean curvature H in warped product manifolds M×ϱR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M\times _\varrho \mathbb {R}$$\end{document}. In the first part of the paper, we prove the existence of Killing graphs with prescribed boundary on geodesic balls under suitable assumptions on H and the mean curvature of the Killing cylinders over geodesic spheres. In the process we obtain a uniform interior gradient estimate improving previous results by Dajczer and de Lira. In the second part we solve the asymptotic Dirichlet problem in a large class of manifolds whose sectional curvatures are allowed to go to 0 or to -∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\infty $$\end{document} provided that H satisfies certain bounds with respect to the sectional curvatures of M and the norm of the Killing vector field. Finally we obtain non-existence results if the prescribed mean curvature function H grows too fast.


Introduction
Let N be a Riemannian manifold of the form N = M × R, where M is a complete ndimensional Riemannian manifold and ∈ C ∞ (M) is a smooth (warping) function. This means that the Riemannian metricḡ in N is of the form g = ( • π 1 ) 2 π * 2 dt 2 + π * 1 g, (1.1) where g denotes the Riemannian metric in M whereas t is the natural coordinate in R and π 1 : M × R → M and π 2 : M × R → R are the standard projections. It follows that the coordinate vector field X = ∂ t is a Killing field and that = |X | on M. Since the norm of X is preserved along its flow lines, we may extend to a smooth function = |X | ∈ C ∞ (N ). From now on, we suppose that > 0 on M.
In this paper we study Killing graphs with prescribed mean curvature. Such graphs were introduced by Dajczer and Ripoll in [9], where the Dirichlet problem for a graph of constant mean curvature H with C 2,α boundary values was solved in a bounded domain contained in a normal geodesic disk D ⊂ M of radius r 0 under hypothesis involving r 0 , data on , and the curvature of the ambient 3-dimensional space N . A bit later in [10] the Dirichlet problem for prescribed mean curvature H ∈ C α with C 2,α boundary values was solved in bounded domains ⊂ M with C 2,α boundary again under hypothesis involving data on and the Ricci curvature of the ambient space N . Recall that given a domain ⊂ M, the Killing graph of a C 2 function u : → R is the hypersurface given by In other words, where : × R → N is the flow generated by X . In [11] the Dirichlet problem was solved with merely continuous boundary data. Furthermore, the authors proved the existence and uniqueness of so-called radial graphs in the hyperbolic space H n+1 with prescribed mean curvature and asymptotic boundary data at infinity thus solving the asymptotic Dirichlet problem in H n × cosh r R. One of our goals in the current paper is to solve the asymptotic Dirichlet problem with prescribed mean curvature in a large class of negatively curved manifolds.
On the other hand, it is an interesting question under which conditions on a Riemannian manifold M every entire constant mean curvature graph over M is a slice, i.e. a graph of a constant function. The first such result is the celebrated theorem due to Bombieri, De Giorgi, and Miranda [3] that an entire minimal positive graph over R n is a totally geodesic slice. Their result was extended by Rosenberg, Schulze, and Spruck [18] to a complete Riemannian manifold M with nonnegative Ricci curvature and the sectional curvature bounded from below by a negative constant. Ding, Jost, and Xin considered in [12] complete, noncompact Riemannian manifolds with nonnegative Ricci curvature, Euclidean volume growth, and quadratic decay of the curvature tensor. They proved that an entire minimal graph over such a manifold M must be a slice if its height function has at most linear growth on one side unless M is isometric to Euclidean space. In the recent paper [5] Casteras, Heinonen, and Holopainen showed that a minimal positive graph over a complete Riemannian manifold with asymptotically nonnegative sectional curvature and only one end is a slice if its height function has at most linear growth. Entire Killing graphs in M × R with constant mean curvature were studied in [7,8]. In particular, it was shown in [7] that a bounded entire Killing graph of constant mean curvature must be a slice if Ric M ≥ 0, K M ≥ −K 0 for some K 0 ≥ 0, and if ≥ 0 > 0, with || || C 2 (M) < ∞.
Our current paper is inspired by the above mentioned research [7,8,10,11] on Killing graphs with prescribed mean curvature as well as by the recent paper [4]. In the latter, the asymptotic Dirichlet problem for f -minimal graphs in Cartan-Hadamard manifolds M has been studied. Recall that f -minimal hypersurfaces are natural generalizations of selfshrinkers which play a crucial role in the study of mean curvature flow. Moreover, they are minimal hypersurfaces of weighted manifolds M f = M, g, e − f d vol M , where (M, g) is a complete Riemannian manifold with the Riemannian volume element d vol M .
Returning to the Killing graph u of a function u, we note that the induced metric in u has components g i j + 2 (x)u i u j , (1.2) where g i j are local components of the metric g. The induced volume element in u (or equivalently, on the domain ⊂ M) is given by d = −2 + |∇u| 2 dM.
We consider the constrained area functional Note that this is the divergence-form operator that fits well with the weighted measure dM in the sense that a suitable version of the divergence theorem is still valid in this context. Reasoning another way around, since is oriented by the normal vector field where∇ is the Riemannian connection in N , we can interpret as a weighted mean curvature of the submanifold u in the Riemannian product M × R in the sense that the Euler-Lagrange PDE may be rewritten as div ∇u More generally, if f is an arbitrary density in M we consider a weighted area functional of the form For the time being, we restrict ourselves to the case where f = 0. Intrinsically, given a hypersurface ⊂ N and denoting u = t| , the parametric counterpart of (1.3) is where is the Laplace-Beltrami operator in . Indeed if ∇ denotes the intrinsic covariant derivative in , we have where T denotes tangential projection onto T . Hence we obtain from where the formula (1.5) above follows.
In particular, minimal graphs in N = M × R have height function that satisfies the weighted harmonic equation This may be considered as a PDE in if we replace the metric g by the induced metric with components given by (1.2). Denoting we can write (1.3) in non-divergence form as (1.7)

Main results
The existence of Killing graphs with prescribed mean curvature H over bounded domains ⊂ M with continuous boundary data on ∂ was established in [11,Theorem 2] under suitable conditions on the Ricci curvature on , the mean curvature function H , and on the mean curvature of the Killing cylinder over ∂ ; see also [10].
In this paper we mainly focus on the setting where M is a Cartan-Hadamard manifold with sectional curvatures controlled from above and below by some radial functions. We prove quantitative a priori height and gradient estimates for solutions of (1.3) on geodesic balls = B(o, k) ⊂ M under natural conditions on the prescribed mean curvature function in terms of sectional curvatures K M and the warping function . These estimates allow us to use the continuity method (the Leray-Schauder method) and hence are enough to guarantee the existence of solutions to the following Dirichlet problem where ϕ ∈ C(∂ ). We formulate the (local) existence result in geodesic balls on Cartan-Hadamard manifolds.
and therefore can be estimated from below in terms of a suitable model manifold M −a 2 (r ) × + R, where M −a 2 (r ) is a rotationally symmetric Cartan-Hadamard manifold with radial sectional curvatures equal to −a 2 (r ) and + : M → (0, ∞) is a positive rotationally symmetric To formulate the next corollary and for later purposes we denote by for all x ∈¯ , then there exists a unique solution u ∈ C 2,α ( ) ∩ C(¯ ) to (2.1).
As mentioned above the proofs of Theorem 2.1 and Corollary 2.2 for boundary data ϕ ∈ C 2,α (∂ ) follow from the well-known continuity method once the a priori height and gradient estimates are at our disposal. The case of a continuous boundary values ϕ ∈ C(∂ ) can be treated as in [11]; see also [4].
Our main object in this paper is the asymptotic Dirichlet problem for Killing graphs with prescribed mean curvature and behaviour at infinity. To solve the problem, we extend the given boundary value function ϕ ∈ C(∂ ∞ M) to a continuous function ϕ ∈ C(M); see Sect. 5 for the notation. Then we apply Corollary 2.2 for an exhaustion k = B(o, k), k ∈ N, of M to obtain a sequence of solutions u k with boundary values u k |∂ k = ϕ. Under a suitable bound on |H | in terms of a comparison manifold M −a 2 (r ) × + R we obtain a global height estimate and, consequently together with Schauder estimates, the sequence is uniformly bounded in the C 2,α -norm. Hence there exists a subsequence that converges in the C 2,α -norm to a global solution u to the equation div ∇u Finally, under suitable curvature upper and lower bounds as well as conditions on |H | we are able to construct (local) barriers at infinity and prove that the solution u extends continuously to ∂ ∞ M and attains the given boundary values ϕ there.
The following two solvability theorems will be proven in Sect. 6.

Remark 2.5
The following example illustrates the need of our assumptions about the warping function in Theorems 2.3 and 2.4. Let N be the (n + 1)-dimensional hyperbolic space H n+1 and consider the Killing vector field X in H n+1 corresponding to a one-parameter family of parabolic isometries of H n+1 preserving a given ideal point, say p 0 ∈ ∂ ∞ H n+1 . This configuration cannot be directly compared with a rotationally invariant model (that is, invariant by a one-parameter family of elliptic isometries) as we have assumed for instance in conditions 4.13 and 4.14. This borderline case of a one-parameter family of parabolic isometries and the corresponding Killing field in H n+1 were studied by Ripoll and Telichevsky in [17] using different techniques relying on a variant of the Perron method.

A priori height and gradient estimates
Throughout this section we denote by k = B(o, k) the geodesic ball centered at a given point o ∈ M with radius k ∈ N, and by d(·) = dist(·, ∂ k ) the distance function to the boundary of k .

Height estimate
Fix k ∈ N and suppose that u k ∈ C 2 ( k ) is a solution of the Dirichlet problem (2.1). We aim to show that the function where h will be determined later, is an upper barrier for the solution u k . It suffices to show (see [19, p. 795] or [10, pp. 239-240]) that v k is a barrier in an open neighbourhood of ∂ k in which the points can be joined to ∂ k by unique geodesics. In this neighbourhood the distance function d has the same regularity as ∂ k and therefore the derivatives of d in the following computations are well-defined. Since X is Killing field, we have where κ is the principal curvature of the Killing cylinder C k−d over the geodesic sphere where ∂ d denotes the derivative to the direction ∇d. However, where H k−d is the mean curvature of the cylinder C k−d , and we have Hence it follows that Suppose that the principal curvature of the Killing cylinder C k−d satisfies where 0 is a smooth positive increasing function on [0, ∞). We note already at this point that, in the case of Cartan-Hadamard manifolds, ∇d = −∇r and this agrees with the assumption (4.13). Then define the function h as for some constant C > 0 to be fixed later. Now, since h > 0, we have Assuming that in¯ k and choosing the constant C as is a lower barrier for u k and together these barriers give the following height estimate.

Lemma 3.1 Assume that
in¯ k and that u k is a solution to the Dirichlet problem (2.1). Then there exists a constant

Boundary gradient estimate
For given ε > 0 we define an annulus In order to obtain a boundary gradient estimate, we aim to show that a function of the form is an upper barrier in the set U k (ε) for a fixed ε ∈ (0, 1/2) chosen so that d is smooth in U k (ε).
Using the expression (3.5) we obtain that and combining with the previous reasoning, this results to We note that and, on the other hand, Moreover, the matrix (σ i j ) has eigenvalues 1/( 2 W 3 ) and 1/W which can be estimated as When ≥ 1, this is trivial, and when < 1 we can choose the constant K in the definition (3.7) of g such that this holds. Therefore we are able to estimate where Then, for K large, we have We also have which implies that Hence we obtain However, Combining these with the fact that W ≥ K 2 , we obtain the estimate Finally observe that Taking (3.6), (3.8), (3.9) and (3.10) into account, we can choose . This suffices for the following boundary gradient estimate.

Interior gradient estimate
In this subsection we prove a quantitative interior gradient estimate that is interesting on its own. The proof is based on the technique due to Korevaar and Simon [14], and further developed by Wang [20]. We will perform the computations in a coordinate free way.
where the functions η, γ and ψ will be specified later. Suppose that χ attains its maximum at x 0 ∈ B( p, R), and without loss of generality, that and therefore Moreover, the matrix is non-positive at x 0 . Applying the Ricci identities for the Hessian of u we have and this yields On the other hand, denoting and differentiating both sides in (1.7) we have (3.14) Contracting (3.14) with u k , we get . Using the previous identity, (3.13) and noticing that Notice that (3.13) yields to Plugging this into the previous estimate, we get Suppose that |∇u|(x 0 ) > 1. Otherwise we are done. Hence, following [20], we set (3.15) where t = |∇u| 2 . Then we have By modifying the argument in [18, Proof of Theorem 4.1, Case 2] we may assume that the maximum point x 0 is not in the cut-locus C( p) of p. Then we choose η as As in [20], we set where M > 0 is a constant to be fixed later. Then γ = 0 and hence (3.20) Let L = L( p, R) ≥ 0 be chosen in such a way that in B( p, R). Then we obtain We consider first the case Then we have On the other hand, when which implies that Hence at x 0 We have proven the following quantitative gradient estimate. Here we denote by R B the Riemannian curvature tensor in a set B.
where η, γ , and ψ are as in the previous proof. If χ attains its maximum in an interior point x 0 ∈ B( p, R) ∩ , the proof of Lemma 3.3 applies and we have a desired upper bound. Otherwise, χ attains its maximum at x 0 ∈ ∂ , but then |∇u(x 0 )| ≤ max ∂ |∇u| and again we are done.
We remark that a global gradient estimate for bounded Killing graphs follows immediately from (3.23), (3.20), and (3.21) in the case of bounded warping functions under some assumptions on the curvature.

Global barriers
In this section we present two methods to obtain global (upper and lower) barriers for solutions to (2.1).
In the case when H is constant along flow lines of X , that is, when H is a function in M, there is a conservation law (a flux formula) corresponding to the invariance of A H with respect to the flow generated by X . This flux formula for graphs is stated as where = ∂ and ν is the outward unit normal vector field along ⊂ M.
Suppose for a while that M is a model manifold with respect to a fixed pole o ∈ M and that = |X | is a radial function. In terms of polar coordinates (r , ϑ) ∈ R + × S n−1 centered at o the metric in M is of the form where dϑ 2 stands for the canonical metric in S n−1 . Suppose that H and u are also radial functions. Applying (4.1) to = B(o, r ), the geodesic ball centered at o with radius r , we obtain This is a first integral of (1.3) in this rotationally invariant setting. Indeed, taking derivatives on both sides of (4.2) with respect to r we get On the other hand in this particular setting (1.3) becomes It is convenient to write (4.2) in a "quadrature" form as follows For instance, in the case when H is constant we have to impose a condition such as in order to guarantee the existence of radial solutions u = u(r ) to (1.3) for model manifolds.
Note that the right-hand side in (4.4) is a sort of weighted isoperimetric ratio in M with respect to the density (r (x)) = |X (x)|. By l'Hospital's rule we see that (4.4) is equivalent to the requirement This discussion motivates us to define in the general case a function of the form for some nonnegative functions + (r (x)), ξ + (r (x)) and H (r (x)) to be chosen later.
Plugging u + (x) = u + (r (x)) into the differential operator Moreover, suppose that Hence we obtain In order to prove that u + is indeed an upper barrier we next check that Note that u + ≤ 0. We observe that ∂ ∂r if and only if But now integrating (4.8) we get and furthermore assuming 3 we see that (4.10) holds. Therefore we are left to show that The conditions (4.4) and (4.5) in our mind, we choose H as with some ε ∈ (0, 1). Note that then and we see that with this choice the denominator in the definition of u + stays bounded from 0. Moreover, we have and therefore u + is well defined, positive and decreasing function if Now we can compute −2 , and for example, taking ε = 1 − √ 2/2 we have −2 .
For the prescribed mean curvature we obtain the bound All together, we have obtained the following.

Lemma 4.1 Let M be a complete Riemannian manifold with a pole o and consider the warped product manifold M × R, where satisfies
for some positive and increasing C 1 -function + : [0, ∞) → (0, ∞) such that (4.14) Furthermore, assume that the radial sectional curvatures of M are bounded from above by and that the prescribed mean curvature function satisfies for some ε ∈ (0, 1). Then the function u + defined by (4.6) and (4.11) satisfies Q[u + ] ≤ 0 and u + ≥ ||ϕ|| C 0 in M with u + (r ) → ||ϕ|| C 0 as r → ∞.
with f a as in (2.3).
In a rotationally symmetric case if = + (r ) (and (4.12) holds), we see that the bound for the mean curvature is

Example: hyperbolic space
We consider the warped model of H n+1 given by H n × cosh r R, where r is a radial coordinate in H n defined with respect to a fixed reference point o ∈ H n . Then the hyperbolic metric is expressed as where dϑ 2 stands for the standard metric in S n−1 ⊂ T o H n . The flow of the Killing field X = ∂ t is given by the hyperbolic translations generated by a geodesic γ orthogonal to H n through o. Since (r ) = cosh r and ξ(r ) = sinh r in this case, we obtain Therefore a natural bound to the mean curvature function according (4.4) is that is, below the mean curvature of horospheres. We also have for |H | < 1 Therefore we have u 2 (r ) ≤ 1.

Global barrier V
In this subsection we construct a global barrier using an idea of Mastrolia, Monticelli, and Punzo [15]; see also [4]. Recall that + : [0, ∞) → (0, ∞) is an increasing smooth function satisfying + (0) = (o) and for all x ∈ M. Then we have an estimate We define where D is the constant given by (4.23). Denoting V (r ) = V (r (x)), we observe that .
Since V (r ) < 0, the limit exists. Furthermore, D ≤ 0 (see [15, (4.5)]) and finite by (4.21) and therefore V is well defined. Next we write and aim to prove that Q[V ] ≤ 0. First we estimate the weighted Laplacian of V by using (4.20) and thus the first term of (4.24) can be estimated as Then, for the last term of (4.24) we have . .
Finally, if the prescribed mean curvature function satisfies Hence we have proved the following uniform height estimate. Next we discuss possible choices of the functions + and a 0 and their influence on the bound of |H |. Notice that the right hand side of (4.25) can be written as −2 f a (r ) Hence if we can choose the comparison manifold M −a 2 (r ) × + R and a 0 such that V (r ) → −∞ and −2 asymptotically as r → ∞.

Barrier at infinity
In this section we assume that M is a Cartan-Hadamard manifold of dimension n ≥ 2, Throughout this section, we assume that the sectional curvatures of M are bounded from below and above by x) is the distance to a fixed point o ∈ M and P x is any 2-dimensional subspace of T x M. The functions a, b : [0, ∞) → [0, ∞) are assumed to be smooth such that a(t) = 0 and b(t) is constant for t ∈ [0, T 0 ] for some T 0 > 0, and that assumptions (A1)-(A7) hold. These curvature bounds are needed to control the first two derivatives of "barrier" functions that we will construct in the next subsection. We assume that function b in (5.1) is monotonic and that there exist positive constants T 1 ≥ T 0 , C 1 , C 2 , C 3 , and Q ∈ (0, 1) such that for all t ≥ T 1 and for all t ≥ 0. In addition, we assume that and that there exists a constant C 4 > 0 such that see (2.3) for the definition of f a . We recall from [13] the following two examples of functions a and b.

Construction of a barrier
Following [13], we construct a barrier function for each boundary point Thus P(ϕ) is an integral average of ϕ with respect to χ similar to that in [1, p. 436] except that here the function b is taken into account explicitly. If ϕ ∈ C(M), we extend P(ϕ) : M → R to a functionM → R by setting P(ϕ)(x) = ϕ(x) whenever x ∈ M(∞). Then the extended function P(ϕ) is C ∞ -smooth in M and continuous inM; see [13,Lemma 3.13]. In particular, applying P to the functionh yields an appropriate smooth extension Furthermore, we denote by Hess x u the norm of the Hessian of a smooth function u at x, that is The following lemma gives the desired estimates for derivatives of h. We refer to [13] for the proofs of these estimates; see also [6]. ,

5)
for all x ∈ 3 \ B(o, R 1 ). In addition, Let A > 0 be a fixed constant, and R 3 > 0 and δ > 0 constants that will be determined later, and h the function defined in (5.4). We will show that a function is a supersolution In the proof we shall use the following estimates obtained in [13]: Lemma 5.4 [13,Lemma 3.17] There exist constants R 2 = R 2 (C) and c 2 = c 2 (C) with the following property. If δ ∈ (0, 1), then Let us denote where C 1 and C 4 are constants defined in (A1) and (A7), respectively.

Lemma 5.5 Assume that the prescribed mean curvature function H satisfies
for some positive constants C 0 > 1 and δ < min{δ 1 , φ − 1}, and that the warping function satisfies Proof In the proof we will denote by c those positive constants whose actual value is irrelevant and may vary even within a line. Furthermore, the estimates will be done in 3 \B(o, R 3 ), with R 3 large enough. Note that and hence we only need to find R 3 = R 3 (C, C 0 , δ) ≥ R 2 so that

Solving the asymptotic Dirichlet problem
In this section we solve the asymptotic Dirichlet problem (5.1) on a Cartan-Hadamard manifold M with given boundary data ϕ ∈ C(∂ ∞ M). If the ambient manifold N = M × R is a Cartan-Hadamard manifold, too, we will interpret the graph S = {(x, u(x)) : x ∈ M} of the solution u as a Killing graph with prescribed mean curvature H and continuous boundary values at infinity. We recall from [2, 7.7] that N is a Cartan-Hadamard manifold if and only if the warping function is convex. In that case we may consider ∂ ∞ M as a subset of ∂ ∞ N in the sense that a representative γ of a boundary point x 0 ∈ ∂ ∞ M is also a representative of a pointx 0 ∈ ∂ ∞ N since M is a totally geodesic submanifold of N . Given ϕ ∈ C(∂ ∞ M) we define its Killing graph on ∂ ∞ N as follows. For x ∈ ∂ ∞ M, take the (totally geodesic) leaf where is the flow generated by X . Let γ x be any geodesic on M representing x. Theñ γ x : t → (γ x (t), ϕ(x)) is a geodesic on M ϕ(x) and also on N since (·, ϕ(x)) is an isometry. Henceγ x defines a point in ∂ ∞ N which we, by abusing the notation, denote by (x, ϕ(x)). Using this notation, we call the set the Killing graph of ϕ. Note that, in general, ∂ ∞ N has no canonical smooth structure. Proof Suppose first that x ∈ ∂ ∞ S and let (x i , u(x i )) be a sequence in S converging to x in the cone topology ofN . SinceM is compact, there exist x 0 ∈ ∂ ∞ M and a subsequence (x i j , u(x i j )) such that x i j → x 0 ∈ ∂ ∞ M in the cone topology ofM. Hence u(x i j ) → ϕ(x 0 ), and consequently (x i j , u(x i j )) → (x 0 , ϕ(x 0 )) in the product topology ofM × R. On the other hand, (x i j , ϕ(x 0 )) → (x 0 , ϕ(x 0 )) in the cone topology of M ϕ(x 0 ) . We need to verify that (x i j , u(x i j )) → (x 0 , ϕ(x 0 )) in the cone topology ofN which then implies that x = (x 0 , ϕ(x 0 )) ∈ . Towards this end, let V be an arbitrary cone neighborhood inN of (x 0 , ϕ(x 0 )) and let σ be a geodesic ray emanating from (o, ϕ(x 0 )) representing (x 0 , ϕ(x 0 )). It is a geodesic ray both in N and in M ϕ(x 0 ) . Let T (σ 0 , 2α, r ) ⊂ V be a truncated cone inN and T := T M (σ 0 , α, 2r ) a truncated cone inM ϕ(x 0 ) . Then (T , (ϕ(x 0 )−δ, ϕ(x 0 )+δ)) ⊂ V for sufficiently small δ > 0. It follows that (x i j , u(x i j )) ∈ V for all i j large enough, and therefore x = (x 0 , ϕ(x 0 )) ∈ . Conversely, if (x 0 , ϕ(x 0 )) ∈ , let x i ∈ M be a sequence such that x i → x 0 in the cone topology ofM. Then (x i , u(x i )) ∈ S and (x i , u(x i )) → (x 0 , ϕ(x 0 )) in the product topology ofM × R. We need to show that (x i , u(x i )) → (x 0 , ϕ(x 0 )) ∈ in the cone topology ofN . To prove this, fix o = (x, ϕ(x 0 )) ∈ M ϕ(x 0 ) and let σ be a geodesic ray in N (and in M ϕ(x 0 ) ) representing (x 0 , ϕ(x 0 )). Let V = T (σ 0 , 2α, r ) be an arbitrary truncated cone neighborhood inN of (x 0 , ϕ(x 0 )). Furthermore, let δ > 0 be so small that ∈ U for all sufficiently large i. Hence (x i , u(x i )) → (x 0 , ϕ(x 0 )) ∈ in the cone topology ofN .
We formulate our global existence results in the following two theorems depending on the assumption on the prescribed mean curvature function H . By Lemma 4.1, we see that the sequence (u k ) is uniformly bounded. Applying the gradient estimates in compact domains and then the diagonal argument, we obtain a subsequence converging locally uniformly with respect to C 2 -norm to a solution u. Next we show that u extends continuously to the boundary ∂ ∞ M with u|∂ ∞ M = ϕ. Let x 0 ∈ ∂ ∞ M and ε > 0 be fixed. By the continuity of the function ϕ we find a constant L ∈ (8/π, ∞) so that |ϕ(y) − ϕ(x 0 )| < ε/2 whenever y ∈ C(v 0 , 4/L) ∩ ∂ ∞ M, where v 0 =γ o,x 0 0 is the initial direction of the geodesic ray representing x 0 . Taking (4.16) into account, we can choose R 3 in Lemma 5.5 so big that u + (r ) ≤ ||ϕ|| ∞ + ε/2 when r ≥ R 3 .
Again, by the continuity of the function ϕ inM, we can choose k 0 such that ∂ B k ∩ U = ∅ and |ϕ(x) − ϕ(x 0 )| < ε/2 (5.3) for every x ∈ ∂ B k ∩ U when k ≥ k 0 . We denote V k = B k ∩ U for k ≥ k 0 and note that We prove (5.2) by showing that w − ≤ u k ≤ w + (5.4) holds in V k for every k ≥ k 0 . Let k ≥ k 0 and x ∈ ∂ B k ∩Ū . Since u k |∂ B k = ϕ|∂ B k , (5.3) implies on ∂U ∩B k . Similarly we have u k ≥ w − on ∂U ∩B k and therefore w − ≤ u k ≤ w + on ∂ V k . By Lemma 5.5 ψ is a supersolution in U and hence the comparison principle yields u k ≤ w + in U . On the other hand, −ψ is a subsolution in U , so u k ≥ w − in U , and (5.4) follows. This is true for every k ≥ k 0 so we have (5.2). Since lim x→x 0 ψ(x) = 0, we have lim sup The point x 0 ∈ ∂ ∞ M and constant ε > 0 were arbitrary so this shows that u extends continuously to C(M) and u|∂ ∞ M = ϕ. Finally, the uniqueness follows from the comparison principle.

Non-existence result
In the following, we state a non-existence result for the prescribed weighted mean curvature graph equation by adapting the approach of Pigola, Rigoli and Setti in [16]. We denote by Then, if q ≡ 0, there are no solutions to (5.3).

Proof
The proof is very similar to that in [16], the only differences being our use of the divergence operator with respect to the weighted volume form dM and a suitable form of the Mikljukov-Hwang-Collin-Krust inequality which in our setting reads as follows Together these result in the extra factors of 0 in (5.1), (5.2), and on the right hand side of (5.3). Taking into account these differences the proof in [16] applies almost verbatim.
As direct corollaries of the previous theorem, we have Proof By choosing p(s) = (s 2 log s) −1 in (i), we see that (5.1) holds, and therefore the claim follows. Similarly, choosing p(s) = (s log s) −1 in (ii) or p(s) = (log s) −1 in (iii), the condition (5.2) holds and the claim follows.