The geometric problem of prescribing positive curvature to noncompact surfaces of finite topological type is studied with the help of the conformal Gauss curvature equation Deltau + Ke2u -- k = 0. New asymptotic conditions on the admissible set of prescribed curvature functions are elucidated. In particular, a compact surface punctured at two points admits a positive curvature function with arbitrary polynomial growth about one of the removed points. The main theorem generalizes previous results of McOwen and Aviles for the Euclidean plane. The relation with the theory of Hulin and Troyanov is discussed. The method used to prove the existence of solutions is a refined version of the variational method of Berger-Moser based on the pseudomonotonicity of the potential operator or Gateaux derivative of the functional. The technique reveals the relation with the monotonicity methods normally used to deal with the negative curvature case. The geometric instrument that makes the reduction of the equation possible is a lemma of Kalka and Yang
