In this talk I will discuss the Dirichlet problem for the Laplace and minimal hypersurface PDEs with prescribed asymptotic boun- day in a Hadamard manifold M in an uni ed way by considering the family of Dirichlet problems $$\left\{ \begin{array}{ll} Q_t(u):=\mbox{div}\frac{\nabla u}{\sqrt{1+|t\nabla u|^2}}=0 &\mbox{in } M, u^{C^{\infty}(M)\cap C^0(\overline{M}),\\ u_{|\partial_\infty M}=\psi,& t\in [0,1] \end{array} \right.$$ I shall explain some results which I recently obtained in collaboration with Miriam Telichevesky. In the harmonic case a partial extension and improvements of well known results of E. Hsu and Choi. In the minimal case a partial extension of a theorem of J. A. Gálvez and H. Rosenberg by allowing the sectional curvature of M degenerate to 0 at in finity.