The Dirichlet problem for the minimal surface equation -with possible   infinite boundary data- over domains in a Riemannian surface
  In this paper, we study existence and uniqueness of solutions to Jenkins-Serrin type problems on domains in a Riemannian surface. In the case of unbounded domains, the study is focused on the hyperbolic plane. 
L. Mazet, M. M. Rodriguez, H. Rosenberg