Let M be Hadamard manifold with sectional curvature K M ≤ −k 2 , k > 0. Denote by ∂ ∞ M the asymptotic boundary of M . We say that M satisfies the strict convexity condition (SC condition) if, given x ∈ ∂ ∞ M and a relatively open subset W ⊂ ∂ ∞ M containing x, there exists a C 2 open subset Ω ⊂ M such that x ∈ Int (∂ ∞ Ω) ⊂ W and M \ Ω is convex. We prove that the SC condition implies that M is regular at infinity relative to the operator Q [u] := div a(|∇u|) |∇u| ∇u , subject to some conditions. It follows that under the SC condition, the Dirichlet problem for the minimal hypersurface and the p-Laplacian (p > 1) equations are solvable for any prescribed continuous asymptotic boundary data. It is also proved that if M is rotationally symmetric or if inf B R+1 K M ≥ −e 2kR /R 2+2ǫ , R ≥ R * , for some R * and ǫ > 0, where B R+1 is the geodesic ball with radius R + 1 centered at a fixed point of M, then M satisfies the SC condition.

This result can be seen as a limit case of the existence theorem proved by B. Guan and J. Spruck in [6] on CMC |H| < 1 radial graphs with prescribed C 0 asymptotic boundary data. arXiv:1503.08083v2 [math.DG] 1 Apr 2015

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Given an unbounded domain Ω of a Hadamard manifold M , it makes sense to consider the problem of finding minimal graphs with prescribed continuous data on its cone-topology-boundary, i.e., on its ordinary boundary together with its asymptotic boundary. In this article it is proved that under the hypothesis that the sectional curvature of M is ≤ −1 this Dirichlet problem is solvable if Ω satisfies certain convexity condition at infinity and if ∂Ω is mean convex. We also prove that mean convexity of ∂Ω is a necessary condition, extending to unbounded domains some results that are valid on bounded ones.